# Properties

 Label 7.17.d.a Level 7 Weight 17 Character orbit 7.d Analytic conductor 11.363 Analytic rank 0 Dimension 20 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$17$$ Character orbit: $$[\chi]$$ = 7.d (of order $$6$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$11.36271807$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{38}\cdot 3^{16}\cdot 7^{22}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 9 - \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{2}$$ $$+ ( 438 + \beta_{1} + 219 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{3}$$ $$+ ( 10 \beta_{1} + 28595 \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{4}$$ $$+ ( 8103 - 201 \beta_{1} - 8102 \beta_{2} + 202 \beta_{3} - 3 \beta_{6} - \beta_{9} - \beta_{10} ) q^{5}$$ $$+ ( -71449 - 1886 \beta_{1} - 142897 \beta_{2} - 943 \beta_{3} - 52 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} - \beta_{10} + \beta_{12} ) q^{6}$$ $$+ ( -432974 - 194 \beta_{1} - 680825 \beta_{2} - 878 \beta_{3} + 115 \beta_{4} + 21 \beta_{5} - 27 \beta_{6} - 13 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{7}$$ $$+ ( 1200801 - \beta_{1} - \beta_{2} + 34631 \beta_{3} - 300 \beta_{4} - 42 \beta_{5} + 601 \beta_{6} + 42 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{16} ) q^{8}$$ $$+ ( 14605406 - 21134 \beta_{1} + 14605422 \beta_{2} - 21149 \beta_{3} - 1331 \beta_{4} + 57 \beta_{5} + 664 \beta_{6} + \beta_{8} + 14 \beta_{9} - 16 \beta_{10} + 3 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} - \beta_{18} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 9 - \beta_{1} + 9 \beta_{2} - \beta_{3} ) q^{2}$$ $$+ ( 438 + \beta_{1} + 219 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{3}$$ $$+ ( 10 \beta_{1} + 28595 \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{4}$$ $$+ ( 8103 - 201 \beta_{1} - 8102 \beta_{2} + 202 \beta_{3} - 3 \beta_{6} - \beta_{9} - \beta_{10} ) q^{5}$$ $$+ ( -71449 - 1886 \beta_{1} - 142897 \beta_{2} - 943 \beta_{3} - 52 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} - \beta_{10} + \beta_{12} ) q^{6}$$ $$+ ( -432974 - 194 \beta_{1} - 680825 \beta_{2} - 878 \beta_{3} + 115 \beta_{4} + 21 \beta_{5} - 27 \beta_{6} - 13 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{7}$$ $$+ ( 1200801 - \beta_{1} - \beta_{2} + 34631 \beta_{3} - 300 \beta_{4} - 42 \beta_{5} + 601 \beta_{6} + 42 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} - \beta_{16} ) q^{8}$$ $$+ ( 14605406 - 21134 \beta_{1} + 14605422 \beta_{2} - 21149 \beta_{3} - 1331 \beta_{4} + 57 \beta_{5} + 664 \beta_{6} + \beta_{8} + 14 \beta_{9} - 16 \beta_{10} + 3 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} - \beta_{18} ) q^{9}$$ $$+ ( -37848956 - 22179 \beta_{1} - 18924484 \beta_{2} - 44325 \beta_{3} + 3639 \beta_{4} + 297 \beta_{5} - 3633 \beta_{6} - 150 \beta_{7} - 7 \beta_{8} - 27 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + 6 \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{10}$$ $$+ ( -8 + 71636 \beta_{1} - 48689418 \beta_{2} + 23 \beta_{3} + 99 \beta_{4} + 4 \beta_{5} + 106 \beta_{6} + 173 \beta_{7} + 12 \beta_{8} - 20 \beta_{9} - 50 \beta_{10} + 9 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} + 2 \beta_{16} + 2 \beta_{17} + 3 \beta_{19} ) q^{11}$$ $$+ ( -74750909 + 279299 \beta_{1} + 74750832 \beta_{2} - 279332 \beta_{3} - 11 \beta_{4} + 769 \beta_{5} - 21303 \beta_{6} - 1538 \beta_{7} + 18 \beta_{8} + 42 \beta_{9} + 5 \beta_{10} + 9 \beta_{11} + 19 \beta_{12} - 4 \beta_{13} - 15 \beta_{14} + 45 \beta_{15} + 18 \beta_{16} + 6 \beta_{17} + 2 \beta_{18} + 6 \beta_{19} ) q^{12}$$ $$+ ( 63890330 - 960858 \beta_{1} + 127780847 \beta_{2} - 480401 \beta_{3} + 21739 \beta_{4} + 595 \beta_{5} - 26 \beta_{6} + 559 \beta_{7} + 49 \beta_{8} - 13 \beta_{9} - 181 \beta_{10} + 18 \beta_{11} - 206 \beta_{12} - 3 \beta_{13} - 17 \beta_{14} - 35 \beta_{15} - 18 \beta_{16} + 10 \beta_{17} - 3 \beta_{18} + 12 \beta_{19} ) q^{13}$$ $$+ ( 67557631 + 2061809 \beta_{1} + 75435369 \beta_{2} + 649317 \beta_{3} - 29890 \beta_{4} + 3070 \beta_{5} + 32909 \beta_{6} - 1833 \beta_{7} + 11 \beta_{8} + 951 \beta_{9} + 1448 \beta_{10} - 35 \beta_{11} - 405 \beta_{12} + 7 \beta_{13} + 436 \beta_{14} - 7 \beta_{15} + 7 \beta_{16} + 20 \beta_{17} - 7 \beta_{18} + 23 \beta_{19} ) q^{14}$$ $$+ ( -104497627 - 309 \beta_{1} - 1493 \beta_{2} + 2400839 \beta_{3} + 47137 \beta_{4} - 5364 \beta_{5} - 93839 \beta_{6} + 5337 \beta_{7} - 316 \beta_{8} + 2364 \beta_{9} + 1424 \beta_{10} - 48 \beta_{11} + 740 \beta_{12} - 2 \beta_{13} - 1383 \beta_{14} + 347 \beta_{15} - 48 \beta_{16} + 33 \beta_{17} + 2 \beta_{18} + 39 \beta_{19} ) q^{15}$$ $$+ ( -1353271500 - 4742130 \beta_{1} - 1353268458 \beta_{2} - 4745192 \beta_{3} + 66226 \beta_{4} + 43246 \beta_{5} - 33218 \beta_{6} - 124 \beta_{7} - 152 \beta_{8} + 2876 \beta_{9} - 3066 \beta_{10} + 24 \beta_{11} + 982 \beta_{12} + 814 \beta_{14} - 198 \beta_{15} + 58 \beta_{17} + 30 \beta_{18} + 66 \beta_{19} ) q^{16}$$ $$+ ( 1098829900 + 511180 \beta_{1} + 549414516 \beta_{2} + 1030952 \beta_{3} + 3730 \beta_{4} - 1876 \beta_{5} - 3432 \beta_{6} + 851 \beta_{7} - 580 \beta_{8} - 8311 \beta_{9} + 293 \beta_{10} + 24 \beta_{11} + 155 \beta_{12} + 23 \beta_{13} + 1031 \beta_{14} + 187 \beta_{15} + 12 \beta_{16} + 58 \beta_{17} - 46 \beta_{18} + 74 \beta_{19} ) q^{17}$$ $$+ ( -258 - 11066710 \beta_{1} + 2161157803 \beta_{2} + 14509 \beta_{3} - 328129 \beta_{4} + 57 \beta_{5} - 327793 \beta_{6} + 4572 \beta_{7} + 655 \beta_{8} - 14165 \beta_{9} - 28944 \beta_{10} + 2897 \beta_{12} - 113 \beta_{13} - 1487 \beta_{14} + 106 \beta_{15} + 15 \beta_{16} + 81 \beta_{17} + 69 \beta_{19} ) q^{18}$$ $$+ ( -991065170 - 3188216 \beta_{1} + 991053617 \beta_{2} + 3176116 \beta_{3} + 94 \beta_{4} + 59457 \beta_{5} + 354475 \beta_{6} - 118800 \beta_{7} - 273 \beta_{8} + 11498 \beta_{9} + 11885 \beta_{10} - 146 \beta_{11} + 1558 \beta_{12} + 124 \beta_{13} - 1511 \beta_{14} + 391 \beta_{15} - 292 \beta_{16} + 23 \beta_{17} - 62 \beta_{18} + 61 \beta_{19} ) q^{19}$$ $$+ ( 1306825601 + 37391782 \beta_{1} + 2613651160 \beta_{2} + 18695646 \beta_{3} - 154171 \beta_{4} - 106795 \beta_{5} - 136 \beta_{6} - 106783 \beta_{7} + 150 \beta_{8} - 212 \beta_{9} + 448 \beta_{10} - 334 \beta_{11} - 9808 \beta_{12} + 86 \beta_{13} + 104 \beta_{14} - 376 \beta_{15} + 334 \beta_{16} + 10 \beta_{17} + 86 \beta_{18} - 4 \beta_{19} ) q^{20}$$ $$+ ( -2094000048 - 2533771 \beta_{1} - 7212418653 \beta_{2} - 20132654 \beta_{3} - 158010 \beta_{4} - 1933 \beta_{5} + 1414383 \beta_{6} + 258434 \beta_{7} - 299 \beta_{8} + 10321 \beta_{9} + 68962 \beta_{10} + 714 \beta_{11} - 5753 \beta_{12} - 210 \beta_{13} + 7366 \beta_{14} + 772 \beta_{15} - 168 \beta_{16} - 162 \beta_{17} + 203 \beta_{18} - 180 \beta_{19} ) q^{21}$$ $$+ ( 7181570553 + 4230 \beta_{1} - 35842 \beta_{2} - 60335240 \beta_{3} + 536250 \beta_{4} - 246856 \beta_{5} - 1078053 \beta_{6} + 247123 \beta_{7} + 4185 \beta_{8} + 79280 \beta_{9} + 35812 \beta_{10} + 1341 \beta_{11} + 2145 \beta_{12} + 57 \beta_{13} - 5289 \beta_{14} - 3905 \beta_{15} + 1341 \beta_{16} - 352 \beta_{17} - 57 \beta_{18} - 437 \beta_{19} ) q^{22}$$ $$+ ( -19913254111 - 20323389 \beta_{1} - 19913228719 \beta_{2} - 20348684 \beta_{3} + 4878361 \beta_{4} + 690342 \beta_{5} - 2437638 \beta_{6} + 1813 \beta_{7} + 2001 \beta_{8} + 27846 \beta_{9} - 25119 \beta_{10} - 572 \beta_{11} - 953 \beta_{12} + 1491 \beta_{14} + 2818 \beta_{15} - 861 \beta_{17} - 412 \beta_{18} - 952 \beta_{19} ) q^{23}$$ $$+ ( 40932655031 + 87064854 \beta_{1} + 20466335126 \beta_{2} + 174324703 \beta_{3} - 1465949 \beta_{4} - 1614868 \beta_{5} + 1460295 \beta_{6} + 808928 \beta_{7} + 9681 \beta_{8} - 200635 \beta_{9} - 4915 \beta_{10} - 1278 \beta_{11} - 2480 \beta_{12} - 194 \beta_{13} - 3247 \beta_{14} - 4401 \beta_{15} - 639 \beta_{16} - 996 \beta_{17} + 388 \beta_{18} - 1290 \beta_{19} ) q^{24}$$ $$+ ( 6342 + 19074182 \beta_{1} - 21266751894 \beta_{2} + 62402 \beta_{3} - 2846954 \beta_{4} - 1454 \beta_{5} - 2854066 \beta_{6} - 496282 \beta_{7} - 13536 \beta_{8} - 68900 \beta_{9} - 124816 \beta_{10} - 19444 \beta_{12} + 1416 \beta_{13} + 10214 \beta_{14} - 1816 \beta_{15} - 1116 \beta_{16} - 1742 \beta_{17} - 1598 \beta_{19} ) q^{25}$$ $$+ ( -45311744356 + 2113982 \beta_{1} + 45311735606 \beta_{2} - 2122560 \beta_{3} + 929 \beta_{4} + 1478593 \beta_{5} + 13335756 \beta_{6} - 2959547 \beta_{7} + 370 \beta_{8} + 15917 \beta_{9} + 16997 \beta_{10} + 555 \beta_{11} - 12665 \beta_{12} - 1754 \beta_{13} + 11224 \beta_{14} - 13501 \beta_{15} + 1110 \beta_{16} - 1351 \beta_{17} + 877 \beta_{18} - 2138 \beta_{19} ) q^{26}$$ $$+ ( 29386417800 - 437768034 \beta_{1} + 58772927295 \beta_{2} - 218885193 \beta_{3} - 14893227 \beta_{4} - 536355 \beta_{5} + 6675 \beta_{6} - 529623 \beta_{7} - 11763 \beta_{8} + 5592 \beta_{9} - 100071 \beta_{10} + 2058 \beta_{11} + 105849 \beta_{12} - 1092 \beta_{13} + 1221 \beta_{14} + 12129 \beta_{15} - 2058 \beta_{16} - 2175 \beta_{17} - 1092 \beta_{18} - 2244 \beta_{19} ) q^{27}$$ $$+ ( 88858832399 + 139620168 \beta_{1} - 75517970383 \beta_{2} + 156229287 \beta_{3} - 33384546 \beta_{4} - 1335570 \beta_{5} + 29392868 \beta_{6} + 4196543 \beta_{7} + 4873 \beta_{8} - 79599 \beta_{9} - 123291 \beta_{10} - 5516 \beta_{11} + 89280 \beta_{12} + 2842 \beta_{13} - 106179 \beta_{14} - 12445 \beta_{15} + 1841 \beta_{16} - 898 \beta_{17} - 2618 \beta_{18} - 1200 \beta_{19} ) q^{28}$$ $$+ ( 21973277834 - 13100 \beta_{1} + 35331 \beta_{2} + 297840033 \beta_{3} + 3036925 \beta_{4} - 944425 \beta_{5} - 6062570 \beta_{6} + 945231 \beta_{7} - 10175 \beta_{8} - 81409 \beta_{9} - 22595 \beta_{10} - 13126 \beta_{11} - 91758 \beta_{12} - 715 \beta_{13} + 181787 \beta_{14} + 1417 \beta_{15} - 13126 \beta_{16} - 338 \beta_{17} + 715 \beta_{18} + 130 \beta_{19} ) q^{29}$$ $$+ ( -229703618525 - 173227537 \beta_{1} - 229703836016 \beta_{2} - 173008430 \beta_{3} + 113795543 \beta_{4} + 5734521 \beta_{5} - 56902858 \beta_{6} - 8529 \beta_{7} - 6796 \beta_{8} - 226984 \beta_{9} + 217842 \beta_{10} + 5889 \beta_{11} - 86816 \beta_{12} - 96025 \beta_{14} - 9025 \beta_{15} + 4323 \beta_{17} + 3409 \beta_{18} + 4206 \beta_{19} ) q^{30}$$ $$+ ( 69137988487 - 30410641 \beta_{1} + 34568951510 \beta_{2} - 61909392 \beta_{3} + 28231558 \beta_{4} - 3704933 \beta_{5} - 28196783 \beta_{6} + 1845526 \beta_{7} - 48068 \beta_{8} + 1127600 \beta_{9} + 24400 \beta_{10} + 16568 \beta_{11} + 11997 \beta_{12} + 326 \beta_{13} - 70107 \beta_{14} + 35896 \beta_{15} + 8284 \beta_{16} + 4627 \beta_{17} - 652 \beta_{18} + 7044 \beta_{19} ) q^{31}$$ $$+ ( -53442 + 2356631644 \beta_{1} + 352910340452 \beta_{2} - 789982 \beta_{3} - 74906698 \beta_{4} + 7864 \beta_{5} - 74854002 \beta_{6} - 2177836 \beta_{7} + 97258 \beta_{8} + 835486 \beta_{9} + 1569666 \beta_{10} - 85244 \beta_{12} - 10112 \beta_{13} + 41106 \beta_{14} + 13526 \beta_{15} + 15690 \beta_{16} + 12584 \beta_{17} + 10224 \beta_{19} ) q^{32}$$ $$+ ( 13832925217 - 373390302 \beta_{1} - 13832664303 \beta_{2} + 373687644 \beta_{3} - 19472 \beta_{4} + 2819902 \beta_{5} + 160997660 \beta_{6} - 5617862 \beta_{7} + 22068 \beta_{8} - 320706 \beta_{9} - 359428 \beta_{10} + 4932 \beta_{11} - 38768 \beta_{12} + 14804 \beta_{13} + 48330 \beta_{14} + 87648 \beta_{15} + 9864 \beta_{16} + 9474 \beta_{17} - 7402 \beta_{18} + 16788 \beta_{19} ) q^{33}$$ $$+ ( -43484334257 - 1434499288 \beta_{1} - 86969121219 \beta_{2} - 717224416 \beta_{3} - 54439068 \beta_{4} + 915498 \beta_{5} - 49601 \beta_{6} + 849561 \beta_{7} + 104569 \beta_{8} - 35868 \beta_{9} + 515297 \beta_{10} + 1011 \beta_{11} - 124056 \beta_{12} + 7857 \beta_{13} - 14511 \beta_{14} - 84691 \beta_{15} - 1011 \beta_{16} + 21590 \beta_{17} + 7857 \beta_{18} + 21979 \beta_{19} ) q^{34}$$ $$+ ( -303710616981 - 1818977540 \beta_{1} + 118222769682 \beta_{2} - 3112852826 \beta_{3} - 199294972 \beta_{4} + 4307184 \beta_{5} + 162919366 \beta_{6} - 3894749 \beta_{7} - 35794 \beta_{8} - 255200 \beta_{9} - 2695502 \beta_{10} + 13132 \beta_{11} - 223017 \beta_{12} - 22554 \beta_{13} + 207584 \beta_{14} + 82154 \beta_{15} - 12586 \beta_{16} + 15012 \beta_{17} + 19292 \beta_{18} + 19445 \beta_{19} ) q^{35}$$ $$+ ( -124671073044 - 88002 \beta_{1} + 1488358 \beta_{2} + 403619948 \beta_{3} + 145536098 \beta_{4} + 4581372 \beta_{5} - 290890996 \beta_{6} - 4606296 \beta_{7} - 124446 \beta_{8} - 3268064 \beta_{9} - 1610608 \beta_{10} + 60474 \beta_{11} + 407644 \beta_{12} + 5022 \beta_{13} - 741992 \beta_{14} + 182472 \beta_{15} + 60474 \beta_{16} + 22758 \beta_{17} - 5022 \beta_{18} + 20592 \beta_{19} ) q^{36}$$ $$+ ( 531234354891 - 2355755613 \beta_{1} + 531234063022 \beta_{2} - 2355497135 \beta_{3} + 338753643 \beta_{4} - 11097767 \beta_{5} - 169397925 \beta_{6} - 418 \beta_{7} - 41867 \beta_{8} - 303003 \beta_{9} + 267695 \beta_{10} - 32970 \beta_{11} + 492457 \beta_{12} + 460966 \beta_{14} - 43366 \beta_{15} - 3820 \beta_{17} - 18777 \beta_{18} + 4238 \beta_{19} ) q^{37}$$ $$+ ( -603315627752 + 4933787425 \beta_{1} - 301657764638 \beta_{2} + 9863198286 \beta_{3} + 11547031 \beta_{4} + 27203985 \beta_{5} - 11618227 \beta_{6} - 13614714 \beta_{7} - 20207 \beta_{8} + 4237826 \beta_{9} + 5234 \beta_{10} - 112750 \beta_{11} + 11013 \beta_{12} + 7053 \beta_{13} + 544734 \beta_{14} - 164588 \beta_{15} - 56375 \beta_{16} + 8481 \beta_{17} - 14106 \beta_{18} - 4521 \beta_{19} ) q^{38}$$ $$+ ( 233904 + 7279769309 \beta_{1} - 1323459476061 \beta_{2} - 742993 \beta_{3} - 161486208 \beta_{4} + 9150 \beta_{5} - 161668929 \beta_{6} + 22712841 \beta_{7} - 292334 \beta_{8} + 605658 \beta_{9} + 1606924 \beta_{10} + 1170541 \beta_{12} + 42940 \beta_{13} - 600354 \beta_{14} - 67310 \beta_{15} - 115140 \beta_{16} - 39588 \beta_{17} - 15219 \beta_{19} ) q^{39}$$ $$+ ( 502024717958 - 7758616833 \beta_{1} - 502022518281 \beta_{2} + 7760519922 \beta_{3} + 143923 \beta_{4} - 33189594 \beta_{5} + 644176318 \beta_{6} + 66256884 \beta_{7} - 240446 \beta_{8} - 2063002 \beta_{9} - 1787673 \beta_{10} - 62207 \beta_{11} + 740177 \beta_{12} - 81176 \beta_{13} - 746809 \beta_{14} - 39869 \beta_{15} - 124414 \beta_{16} - 6812 \beta_{17} + 40588 \beta_{18} - 47580 \beta_{19} ) q^{40}$$ $$+ ( 45726038626 - 8377888034 \beta_{1} + 91450597227 \beta_{2} - 4189085697 \beta_{3} - 136363097 \beta_{4} + 442971 \beta_{5} + 88286 \beta_{6} + 690273 \beta_{7} - 371855 \beta_{8} + 30107 \beta_{9} + 1265415 \beta_{10} - 84568 \beta_{11} - 2042202 \beta_{12} - 32901 \beta_{13} + 40887 \beta_{14} + 135685 \beta_{15} + 84568 \beta_{16} - 91080 \beta_{17} - 32901 \beta_{18} - 82434 \beta_{19} ) q^{41}$$ $$+ ( 1760466520719 + 663577540 \beta_{1} + 1904554656072 \beta_{2} - 23575689935 \beta_{3} - 595684221 \beta_{4} + 11889769 \beta_{5} + 539281726 \beta_{6} - 84643895 \beta_{7} + 116998 \beta_{8} + 2708111 \beta_{9} + 7793809 \beta_{10} + 85911 \beta_{11} - 951911 \beta_{12} + 112868 \beta_{13} + 1849592 \beta_{14} - 202037 \beta_{15} + 64386 \beta_{16} - 50769 \beta_{17} - 84665 \beta_{18} - 77130 \beta_{19} ) q^{42}$$ $$+ ( -29507481536 + 813546 \beta_{1} - 827536 \beta_{2} + 7928797534 \beta_{3} + 53308542 \beta_{4} - 1116322 \beta_{5} - 108024300 \beta_{6} + 1280680 \beta_{7} + 1061622 \beta_{8} + 3712748 \beta_{9} + 1274998 \beta_{10} - 81912 \beta_{11} + 100638 \beta_{12} - 19956 \beta_{13} - 650544 \beta_{14} - 1097948 \beta_{15} - 81912 \beta_{16} - 143104 \beta_{17} + 19956 \beta_{18} - 121850 \beta_{19} ) q^{43}$$ $$+ ( 2513975681599 - 15867586753 \beta_{1} + 2513978988604 \beta_{2} - 15870585038 \beta_{3} + 246842933 \beta_{4} - 139020423 \beta_{5} - 123291243 \beta_{6} + 117692 \beta_{7} + 493944 \beta_{8} + 3335916 \beta_{9} - 3104409 \beta_{10} + 98387 \beta_{11} - 258793 \beta_{12} + 90435 \beta_{14} + 299039 \beta_{15} - 25080 \beta_{17} + 71392 \beta_{18} - 92612 \beta_{19} ) q^{44}$$ $$+ ( -5658862560688 + 26800413830 \beta_{1} - 2829430902783 \beta_{2} + 53631426278 \beta_{3} + 650704354 \beta_{4} + 69756168 \beta_{5} - 650780630 \beta_{6} - 34634265 \beta_{7} + 1029402 \beta_{8} - 30194059 \beta_{9} - 436949 \beta_{10} + 460788 \beta_{11} - 330031 \beta_{12} - 67609 \beta_{13} - 1484461 \beta_{14} + 609489 \beta_{15} + 230394 \beta_{16} - 162546 \beta_{17} + 135218 \beta_{18} - 99876 \beta_{19} ) q^{45}$$ $$+ ( -626796 + 74267768283 \beta_{1} + 1576383842741 \beta_{2} + 10624361 \beta_{3} - 196714416 \beta_{4} - 256316 \beta_{5} - 196418246 \beta_{6} - 47557316 \beta_{7} + 42276 \beta_{8} - 10566965 \beta_{9} - 21964948 \beta_{10} - 4050598 \beta_{12} - 92214 \beta_{13} + 2199431 \beta_{14} + 348626 \beta_{15} + 514788 \beta_{16} + 37336 \beta_{17} - 109490 \beta_{19} ) q^{46}$$ $$+ ( 3565983502924 - 2965995800 \beta_{1} - 3565999270657 \beta_{2} + 2951299501 \beta_{3} - 602669 \beta_{4} - 76101109 \beta_{5} + 218165046 \beta_{6} + 152671049 \beta_{7} + 1270478 \beta_{8} + 16404540 \beta_{9} + 15443058 \beta_{10} + 278172 \beta_{11} - 3033457 \beta_{12} + 288668 \beta_{13} + 2812040 \beta_{14} - 2007161 \beta_{15} + 556344 \beta_{16} - 209474 \beta_{17} - 144334 \beta_{18} - 53197 \beta_{19} ) q^{47}$$ $$+ ( -3072795297874 - 126665445382 \beta_{1} - 6145582910350 \beta_{2} - 63332345048 \beta_{3} + 479217264 \beta_{4} + 73976434 \beta_{5} + 577980 \beta_{6} + 73816846 \beta_{7} + 345894 \beta_{8} + 673104 \beta_{9} - 7418488 \beta_{10} + 511842 \beta_{11} + 7202452 \beta_{12} + 61266 \beta_{13} + 111264 \beta_{14} + 1044696 \beta_{15} - 511842 \beta_{16} + 156390 \beta_{17} + 61266 \beta_{18} + 53196 \beta_{19} ) q^{48}$$ $$+ ( 45423153153 + 13195023562 \beta_{1} + 5409891784115 \beta_{2} - 44440517597 \beta_{3} + 1103322003 \beta_{4} - 68834899 \beta_{5} - 1400280518 \beta_{6} + 23724211 \beta_{7} - 31871 \beta_{8} - 3881591 \beta_{9} + 19035471 \beta_{10} - 811832 \beta_{11} + 3150070 \beta_{12} - 343049 \beta_{13} - 8457239 \beta_{14} - 457191 \beta_{15} - 284200 \beta_{16} - 116382 \beta_{17} + 186151 \beta_{18} - 15372 \beta_{19} ) q^{49}$$ $$+ ( 1797486895198 - 2283104 \beta_{1} - 31238468 \beta_{2} + 17034829314 \beta_{3} - 764749228 \beta_{4} + 91146016 \beta_{5} + 1533499836 \beta_{6} - 91710288 \beta_{7} - 3416816 \beta_{8} + 57240004 \beta_{9} + 31111724 \beta_{10} - 488448 \beta_{11} - 2903720 \beta_{12} + 30628 \beta_{13} + 6992072 \beta_{14} + 1994552 \beta_{15} - 488448 \beta_{16} + 384668 \beta_{17} - 30628 \beta_{18} + 205064 \beta_{19} ) q^{50}$$ $$+ ( 418287036918 - 30122595900 \beta_{1} + 418284612295 \beta_{2} - 30121973992 \beta_{3} - 2605537034 \beta_{4} + 54722811 \beta_{5} + 1303226563 \beta_{6} - 110844 \beta_{7} - 2227821 \beta_{8} - 797150 \beta_{9} + 1481837 \beta_{10} - 83250 \beta_{11} - 3783950 \beta_{12} - 4606175 \beta_{14} + 370425 \beta_{15} - 101709 \beta_{17} - 184566 \beta_{18} + 212553 \beta_{19} ) q^{51}$$ $$+ ( 8475954109334 + 107169450934 \beta_{1} + 4237976275878 \beta_{2} + 214324904760 \beta_{3} - 2037849032 \beta_{4} + 33207728 \beta_{5} + 2037701754 \beta_{6} - 17554138 \beta_{7} - 3238750 \beta_{8} + 11725580 \beta_{9} + 1014708 \beta_{10} - 1143260 \beta_{11} + 1257136 \beta_{12} + 290230 \beta_{13} + 3505468 \beta_{14} - 2576296 \beta_{15} - 571630 \beta_{16} + 633516 \beta_{17} - 580460 \beta_{18} + 333390 \beta_{19} ) q^{52}$$ $$+ ( 1555678 + 120910478179 \beta_{1} - 2882296916936 \beta_{2} + 24484125 \beta_{3} + 2704738129 \beta_{4} + 1026336 \beta_{5} + 2704027699 \beta_{6} + 167708595 \beta_{7} + 1602536 \beta_{8} - 24040944 \beta_{9} - 46434305 \beta_{10} + 6822605 \beta_{12} - 39603 \beta_{13} - 4233733 \beta_{14} - 1913621 \beta_{15} - 1424170 \beta_{16} - 43836 \beta_{17} + 491250 \beta_{19} ) q^{53}$$ $$+ ( -21165703720275 - 70906061829 \beta_{1} + 21165721471980 \beta_{2} + 70922432778 \beta_{3} + 1542351 \beta_{4} + 332093439 \beta_{5} - 8861579088 \beta_{6} - 665544897 \beta_{7} - 3916818 \beta_{8} - 22750824 \beta_{9} - 21219252 \beta_{10} - 469431 \beta_{11} + 4930296 \beta_{12} - 570198 \beta_{13} - 3645507 \beta_{14} + 10071693 \beta_{15} - 938862 \beta_{16} + 1117215 \beta_{17} + 285099 \beta_{18} + 664542 \beta_{19} ) q^{54}$$ $$+ ( -2823306813856 - 153936813093 \beta_{1} - 5646585684530 \beta_{2} - 76968943079 \beta_{3} + 563456455 \beta_{4} - 123369185 \beta_{5} - 3243592 \beta_{6} - 124959206 \beta_{7} + 478485 \beta_{8} - 3363004 \beta_{9} - 28165235 \beta_{10} - 1364248 \beta_{11} + 7139299 \beta_{12} + 120032 \beta_{13} - 939362 \beta_{14} - 5547822 \beta_{15} + 1364248 \beta_{16} + 620 \beta_{17} + 120032 \beta_{18} + 530007 \beta_{19} ) q^{55}$$ $$+ ( 5430138684445 - 136870148171 \beta_{1} - 12061478801227 \beta_{2} - 340892270707 \beta_{3} + 8638001650 \beta_{4} + 168246712 \beta_{5} - 7507433525 \beta_{6} + 487369648 \beta_{7} - 828729 \beta_{8} - 21863445 \beta_{9} - 55259476 \beta_{10} + 2844877 \beta_{11} + 16959037 \beta_{12} + 436870 \beta_{13} - 1007600 \beta_{14} + 4305646 \beta_{15} + 1109087 \beta_{16} + 1367054 \beta_{17} + 116914 \beta_{18} + 1001848 \beta_{19} ) q^{56}$$ $$+ ( 18322298687355 + 567504 \beta_{1} + 60944836 \beta_{2} + 6288397988 \beta_{3} - 1994039036 \beta_{4} - 70382892 \beta_{5} + 3985162432 \beta_{6} + 71645232 \beta_{7} + 4430604 \beta_{8} - 120866468 \beta_{9} - 65096284 \beta_{10} + 2536020 \beta_{11} - 8166536 \beta_{12} + 100464 \beta_{13} + 15347740 \beta_{14} + 3845412 \beta_{15} + 2536020 \beta_{16} - 361932 \beta_{17} - 100464 \beta_{18} + 538476 \beta_{19} ) q^{57}$$ $$+ ( -27990198255682 - 87531813776 \beta_{1} - 27990179337542 \beta_{2} - 87543632064 \beta_{3} - 11753903141 \beta_{4} + 1198916587 \beta_{5} + 5871362830 \beta_{6} - 1902215 \beta_{7} + 6186716 \beta_{8} + 6927879 \beta_{9} - 15950903 \beta_{10} - 377169 \beta_{11} + 1260185 \beta_{12} + 170750 \beta_{14} - 8033945 \beta_{15} + 1445647 \beta_{17} + 280269 \beta_{18} + 456568 \beta_{19} ) q^{58}$$ $$+ ( -6533753476512 + 180320707801 \beta_{1} - 3266882071947 \beta_{2} + 360546682066 \beta_{3} - 854362335 \beta_{4} - 1452188286 \beta_{5} + 859776165 \beta_{6} + 727529652 \beta_{7} - 818532 \beta_{8} + 106304968 \beta_{9} + 2891124 \beta_{10} + 1600248 \beta_{11} - 1317918 \beta_{12} - 563244 \beta_{13} - 23805138 \beta_{14} + 9850548 \beta_{15} + 800124 \beta_{16} - 957006 \beta_{17} + 1126488 \beta_{18} + 202332 \beta_{19} ) q^{59}$$ $$+ ( -5520159 + 583936222380 \beta_{1} + 17474018516679 \beta_{2} - 197798757 \beta_{3} + 5215559442 \beta_{4} - 1139652 \beta_{5} + 5221222386 \beta_{6} - 794203821 \beta_{7} + 455547 \beta_{8} + 199235613 \beta_{9} + 390378483 \beta_{10} - 24470202 \beta_{12} + 822828 \beta_{13} + 14350107 \beta_{14} + 7776057 \beta_{15} + 2124747 \beta_{16} + 1131804 \beta_{17} - 3924 \beta_{19} ) q^{60}$$ $$+ ( -30127458236501 - 99269908033 \beta_{1} + 30127467002492 \beta_{2} + 99275542065 \beta_{3} - 2329537 \beta_{4} + 361119691 \beta_{5} - 11080586249 \beta_{6} - 719207744 \beta_{7} + 6934639 \beta_{8} + 4255399 \beta_{9} + 4595083 \beta_{10} - 736642 \beta_{11} - 4874087 \beta_{12} - 69082 \beta_{13} + 1587976 \beta_{14} - 18422028 \beta_{15} - 1473284 \beta_{16} - 2241024 \beta_{17} + 34541 \beta_{18} - 1230478 \beta_{19} ) q^{61}$$ $$+ ( 2641519794463 - 256733063852 \beta_{1} + 5282884170882 \beta_{2} - 128365323782 \beta_{3} + 5443594522 \beta_{4} - 529981910 \beta_{5} + 3764611 \beta_{6} - 526408013 \beta_{7} + 6304551 \beta_{8} + 5203166 \beta_{9} + 160553560 \beta_{10} + 730477 \beta_{11} - 64695371 \beta_{12} - 1056539 \beta_{13} + 1708075 \beta_{14} + 5821147 \beta_{15} - 730477 \beta_{16} + 382016 \beta_{17} - 1056539 \beta_{18} - 1191299 \beta_{19} ) q^{62}$$ $$+ ( 70544321995481 - 45207803339 \beta_{1} + 22342267794446 \beta_{2} - 324672391135 \beta_{3} + 8516160656 \beta_{4} - 302348403 \beta_{5} + 1804444754 \beta_{6} + 862374462 \beta_{7} + 933559 \beta_{8} + 134010610 \beta_{9} - 135736071 \beta_{10} - 3837372 \beta_{11} - 96187594 \beta_{12} + 1044246 \beta_{13} + 63012637 \beta_{14} - 9657953 \beta_{15} - 3356640 \beta_{16} - 4007991 \beta_{17} - 2044630 \beta_{18} - 2541495 \beta_{19} ) q^{63}$$ $$+ ( 125195233009376 + 9121584 \beta_{1} + 133679376 \beta_{2} + 427387015364 \beta_{3} - 879677216 \beta_{4} - 1726622804 \beta_{5} + 1753513468 \beta_{6} + 1723803596 \beta_{7} - 1085688 \beta_{8} - 251017044 \beta_{9} - 123015768 \beta_{10} - 3128808 \beta_{11} + 70441092 \beta_{12} - 608148 \beta_{13} - 141028920 \beta_{14} - 19330848 \beta_{15} - 3128808 \beta_{16} + 153804 \beta_{17} + 608148 \beta_{18} - 2511600 \beta_{19} ) q^{64}$$ $$+ ( -50044102409288 - 127408639380 \beta_{1} - 50044301996418 \beta_{2} - 127227359173 \beta_{3} - 7873087371 \beta_{4} + 534546027 \beta_{5} + 3954891180 \beta_{6} + 7486462 \beta_{7} - 13056627 \beta_{8} - 163309736 \beta_{9} + 192878632 \beta_{10} + 732768 \beta_{11} + 47087403 \beta_{12} + 54991460 \beta_{14} + 27332730 \beta_{15} - 4861314 \beta_{17} + 28613 \beta_{18} - 2625148 \beta_{19} ) q^{65}$$ $$+ ( -63418576377510 + 56617218599 \beta_{1} - 31709264302235 \beta_{2} + 112960916470 \beta_{3} - 10798703532 \beta_{4} + 642313324 \beta_{5} + 10780414300 \beta_{6} - 321405368 \beta_{7} + 20802636 \beta_{8} + 244170240 \beta_{9} - 15330568 \beta_{10} - 1157160 \beta_{11} - 2965220 \beta_{12} - 458372 \beta_{13} + 108528144 \beta_{14} - 21579504 \beta_{15} - 578580 \beta_{16} + 165804 \beta_{17} + 916744 \beta_{18} - 2672652 \beta_{19} ) q^{66}$$ $$+ ( 15675594 - 88893389843 \beta_{1} + 3315768064197 \beta_{2} + 219389410 \beta_{3} - 1123182985 \beta_{4} - 3160090 \beta_{5} - 1145051027 \beta_{6} + 1282781074 \beta_{7} - 21361020 \beta_{8} - 232098748 \beta_{9} - 434787266 \beta_{10} + 130992652 \beta_{12} - 1897422 \beta_{13} - 67982984 \beta_{14} - 18427046 \beta_{15} - 43536 \beta_{16} - 5210686 \beta_{17} - 4185388 \beta_{19} ) q^{67}$$ $$+ ( -32158232441747 + 38141344970 \beta_{1} + 32158536859003 \beta_{2} - 37824559738 \beta_{3} + 2127988 \beta_{4} + 193448475 \beta_{5} + 2654961767 \beta_{6} - 391209570 \beta_{7} - 6588212 \beta_{8} - 315528316 \beta_{9} - 318664288 \beta_{10} + 4270778 \beta_{11} + 30551232 \beta_{12} + 4172292 \beta_{13} - 26781456 \beta_{14} - 3493508 \beta_{15} + 8541556 \beta_{16} + 246090 \beta_{17} - 2086146 \beta_{18} - 1191450 \beta_{19} ) q^{68}$$ $$+ ( -153342540569927 - 401204062236 \beta_{1} - 306685024292533 \beta_{2} - 200607283253 \beta_{3} + 3075520509 \beta_{4} - 68969852 \beta_{5} + 12630400 \beta_{6} - 63849698 \beta_{7} - 43489956 \beta_{8} + 5521142 \beta_{9} - 84820035 \beta_{10} + 5420964 \beta_{11} + 30052194 \beta_{12} + 2497274 \beta_{13} + 1298726 \beta_{14} + 23962074 \beta_{15} - 5420964 \beta_{16} - 4611984 \beta_{17} + 2497274 \beta_{18} - 1706718 \beta_{19} ) q^{69}$$ $$+ ( 120393558532016 + 433297533247 \beta_{1} + 297551097178798 \beta_{2} + 590223561176 \beta_{3} - 29838914846 \beta_{4} - 742196820 \beta_{5} + 6556052725 \beta_{6} - 2228813551 \beta_{7} + 8041875 \beta_{8} - 428466945 \beta_{9} + 185149347 \beta_{10} - 5475673 \beta_{11} + 138264764 \beta_{12} - 5995213 \beta_{13} - 62073352 \beta_{14} - 2466847 \beta_{15} + 5866063 \beta_{16} + 3465300 \beta_{17} + 5192537 \beta_{18} - 1324083 \beta_{19} ) q^{70}$$ $$+ ( 155521579470208 - 6418460 \beta_{1} - 170721402 \beta_{2} - 186995563418 \beta_{3} - 2173316722 \beta_{4} + 3340617280 \beta_{5} + 4335794034 \beta_{6} - 3333211668 \beta_{7} + 12572692 \beta_{8} + 351437502 \beta_{9} + 177008514 \beta_{10} - 8799432 \beta_{11} - 116746346 \beta_{12} + 779038 \beta_{13} + 224419472 \beta_{14} + 2636396 \beta_{15} - 8799432 \beta_{16} - 3284086 \beta_{17} - 779038 \beta_{18} + 837440 \beta_{19} ) q^{71}$$ $$+ ( 93652908215032 - 44645684140 \beta_{1} + 93653423374100 \beta_{2} - 45136399680 \beta_{3} + 37714888644 \beta_{4} - 4786221660 \beta_{5} - 18876624948 \beta_{6} - 6200112 \beta_{7} + 21273248 \beta_{8} + 477002688 \beta_{9} - 506069572 \beta_{10} + 3366768 \beta_{11} - 101449732 \beta_{12} - 107854716 \beta_{14} - 26037052 \beta_{15} + 4614972 \beta_{17} - 1649564 \beta_{18} + 1585140 \beta_{19} ) q^{72}$$ $$+ ( -76525881091060 - 156748447838 \beta_{1} - 38262951271293 \beta_{2} - 312416240462 \beta_{3} + 26610922842 \beta_{4} + 1975760406 \beta_{5} - 26608638034 \beta_{6} - 985782012 \beta_{7} - 9801760 \beta_{8} - 1071703926 \beta_{9} + 4732294 \beta_{10} + 501984 \beta_{11} + 5211666 \beta_{12} + 5353866 \beta_{13} - 225063024 \beta_{14} + 2634910 \beta_{15} + 250992 \beta_{16} - 1398794 \beta_{17} - 10707732 \beta_{18} + 1256594 \beta_{19} ) q^{73}$$ $$+ ( -13964500 - 902259534039 \beta_{1} + 215938313958973 \beta_{2} + 35889410 \beta_{3} - 23858648000 \beta_{4} + 9087472 \beta_{5} - 23830987990 \beta_{6} - 957027400 \beta_{7} + 41536616 \beta_{8} - 13299854 \beta_{9} - 62188222 \beta_{10} - 330747328 \beta_{12} - 154850 \beta_{13} + 163847926 \beta_{14} + 15869338 \beta_{15} - 7504664 \beta_{16} + 7156388 \beta_{17} + 8121930 \beta_{19} ) q^{74}$$ $$+ ( -164596095491434 + 462245967612 \beta_{1} + 164594707976004 \beta_{2} - 463629710336 \beta_{3} - 4471286 \beta_{4} - 5098985022 \beta_{5} + 91262549112 \beta_{6} + 10198259772 \beta_{7} + 9724062 \beta_{8} + 1374972710 \beta_{9} + 1362823222 \beta_{10} - 2001036 \beta_{11} - 120906034 \beta_{12} - 12365188 \beta_{13} + 120027968 \beta_{14} + 50707548 \beta_{15} - 4002072 \beta_{16} + 5401104 \beta_{17} + 6182594 \beta_{18} + 5497680 \beta_{19} ) q^{75}$$ $$+ ( -402028133641814 + 2007534025219 \beta_{1} - 804055954081775 \beta_{2} + 1003775299703 \beta_{3} - 32145470341 \beta_{4} + 5285467909 \beta_{5} - 31597229 \beta_{6} + 5263264969 \beta_{7} + 80382521 \beta_{8} - 20773481 \beta_{9} - 258673552 \beta_{10} - 12474289 \beta_{11} + 217379381 \beta_{12} - 114256 \beta_{13} - 4206724 \beta_{14} - 58987734 \beta_{15} + 12474289 \beta_{16} + 10709492 \beta_{17} - 114256 \beta_{18} + 7400980 \beta_{19} ) q^{76}$$ $$+ ( 272617362733058 - 46078821419 \beta_{1} + 412666219957912 \beta_{2} + 1040684586988 \beta_{3} - 33896138540 \beta_{4} + 4945733391 \beta_{5} + 24963432701 \beta_{6} - 5305290605 \beta_{7} - 26843867 \beta_{8} + 952605152 \beta_{9} + 986382711 \beta_{10} + 28743470 \beta_{11} + 7967514 \beta_{12} + 10533873 \beta_{13} - 212167861 \beta_{14} + 46986817 \beta_{15} + 2506378 \beta_{16} + 6733746 \beta_{17} - 474439 \beta_{18} + 15480476 \beta_{19} ) q^{77}$$ $$+ ( 686889671216446 - 43054782 \beta_{1} - 1201005005 \beta_{2} - 2401381710721 \beta_{3} + 46477508458 \beta_{4} + 824634876 \beta_{5} - 92871659711 \beta_{6} - 834430827 \beta_{7} - 54618677 \beta_{8} + 2263430630 \beta_{9} + 1139388563 \beta_{10} + 29991351 \beta_{11} - 32884188 \beta_{12} + 2886563 \beta_{13} + 99161411 \beta_{14} + 91615837 \beta_{15} + 29991351 \beta_{16} + 10168824 \beta_{17} - 2886563 \beta_{18} + 10541697 \beta_{19} ) q^{78}$$ $$+ ( 13023404387369 + 1883970912139 \beta_{1} + 13022919273089 \beta_{2} + 1884473146406 \beta_{3} + 138074806307 \beta_{4} - 9011278830 \beta_{5} - 69072365584 \beta_{6} - 19901725 \beta_{7} - 788045 \beta_{8} - 553868698 \beta_{9} + 491095359 \beta_{10} - 15627168 \beta_{11} - 41449115 \beta_{12} - 62521283 \beta_{14} - 60779646 \beta_{15} + 10947709 \beta_{17} + 5789880 \beta_{18} + 8954016 \beta_{19} ) q^{79}$$ $$+ ( -1251907573578700 - 2202644466218 \beta_{1} - 625953907544306 \beta_{2} - 4404689149502 \beta_{3} + 67808622122 \beta_{4} + 16102952192 \beta_{5} - 67702114696 \beta_{6} - 8066259070 \beta_{7} - 125650692 \beta_{8} - 466386558 \beta_{9} + 84559474 \beta_{10} + 8821592 \beta_{11} + 14978216 \beta_{12} - 11134786 \beta_{13} + 176353226 \beta_{14} + 108744746 \beta_{15} + 4410796 \beta_{16} + 9855316 \beta_{17} + 22269572 \beta_{18} + 16257686 \beta_{19} ) q^{80}$$ $$+ ( -53441766 - 4432319877798 \beta_{1} + 207260709396546 \beta_{2} + 1283589024 \beta_{3} - 68277817002 \beta_{4} + 3913794 \beta_{5} - 68232883890 \beta_{6} + 4320023013 \beta_{7} + 49002066 \beta_{8} - 1263672663 \beta_{9} - 2603956557 \beta_{10} + 295471587 \beta_{12} + 8090541 \beta_{13} - 138810867 \beta_{14} + 26267409 \beta_{15} + 22181760 \beta_{16} + 10420002 \beta_{17} + 7166898 \beta_{19} ) q^{81}$$ $$+ ( -397162936908432 - 40595773750 \beta_{1} + 397165005960714 \beta_{2} + 42579721452 \beta_{3} + 17140409 \beta_{4} - 400135835 \beta_{5} + 141585875572 \beta_{6} + 810332281 \beta_{7} - 38522870 \beta_{8} - 2046957611 \beta_{9} - 1983010751 \beta_{10} - 20825833 \beta_{11} + 152883931 \beta_{12} + 12750374 \beta_{13} - 151995416 \beta_{14} + 4981155 \beta_{15} - 41651666 \beta_{16} - 2133135 \beta_{17} - 6375187 \beta_{18} + 1220402 \beta_{19} ) q^{82}$$ $$+ ( -436244325303274 + 4245365239808 \beta_{1} - 872487240360086 \beta_{2} + 2122701205744 \beta_{3} - 122193709660 \beta_{4} - 4549497752 \beta_{5} - 40395666 \beta_{6} - 4562004416 \beta_{7} + 61422532 \beta_{8} - 23535206 \beta_{9} - 1400597130 \beta_{10} - 718128 \beta_{11} + 37999522 \beta_{12} - 13287482 \beta_{13} - 18052280 \beta_{14} - 62739052 \beta_{15} + 718128 \beta_{16} + 3572978 \beta_{17} - 13287482 \beta_{18} + 4168888 \beta_{19} ) q^{83}$$ $$+ ( 1814880395991103 - 157780056084 \beta_{1} + 1920644744200917 \beta_{2} + 6107184218842 \beta_{3} - 176947225896 \beta_{4} - 9136018087 \beta_{5} + 184968624715 \beta_{6} + 174777260 \beta_{7} + 3585414 \beta_{8} - 1614592504 \beta_{9} - 689074372 \beta_{10} - 35229432 \beta_{11} + 172492516 \beta_{12} + 939918 \beta_{13} + 209492780 \beta_{14} - 53225928 \beta_{15} - 45534426 \beta_{16} - 16130772 \beta_{17} - 25976076 \beta_{18} - 15189762 \beta_{19} ) q^{84}$$ $$+ ( 1396367275371015 + 54773286 \beta_{1} + 1602451460 \beta_{2} - 298775834346 \beta_{3} + 16922900356 \beta_{4} + 13575631011 \beta_{5} - 33862881812 \beta_{6} - 13598531223 \beta_{7} - 10049751 \beta_{8} - 3128611057 \beta_{9} - 1562774894 \beta_{10} - 559878 \beta_{11} + 53020560 \beta_{12} - 11985837 \beta_{13} - 103234983 \beta_{14} - 87394673 \beta_{15} - 559878 \beta_{16} + 4930658 \beta_{17} + 11985837 \beta_{18} - 13038896 \beta_{19} ) q^{85}$$ $$+ ( -749413711543598 - 30585272362 \beta_{1} - 749412586483684 \beta_{2} - 31861732844 \beta_{3} + 48528775408 \beta_{4} + 10381528428 \beta_{5} - 24154373616 \beta_{6} + 35893404 \beta_{7} - 137809056 \beta_{8} + 1372102658 \beta_{9} - 1191965590 \beta_{10} + 13061908 \beta_{11} + 222318214 \beta_{12} + 231197050 \beta_{14} + 157835176 \beta_{15} - 29097648 \beta_{17} - 11508432 \beta_{18} - 6795756 \beta_{19} ) q^{86}$$ $$+ ( -279117493151478 - 3307710038059 \beta_{1} - 139558538537351 \beta_{2} - 6617285991971 \beta_{3} - 61493013420 \beta_{4} - 24840839864 \beta_{5} + 61319836177 \beta_{6} + 12422777581 \beta_{7} + 172099320 \beta_{8} + 1598803290 \beta_{9} - 116575384 \beta_{10} - 66933504 \beta_{11} - 28256411 \beta_{12} - 988886 \beta_{13} - 115875594 \beta_{14} - 226635570 \beta_{15} - 33466752 \beta_{16} - 1571766 \beta_{17} + 1977772 \beta_{18} - 25695759 \beta_{19} ) q^{87}$$ $$+ ( 172953393 - 7523816998032 \beta_{1} + 1038383607499928 \beta_{2} - 3379184795 \beta_{3} + 1273181783 \beta_{4} - 36183172 \beta_{5} + 1108525165 \beta_{6} - 7568774500 \beta_{7} - 263326857 \beta_{8} + 3268654455 \beta_{9} + 6823006413 \beta_{10} + 100608262 \beta_{12} - 9088026 \beta_{13} - 58168799 \beta_{14} - 52470813 \beta_{15} - 51121257 \beta_{16} - 40610712 \beta_{17} - 38396942 \beta_{19} ) q^{88}$$ $$+ ( -1134025558181419 + 3377615813966 \beta_{1} + 1134023018283937 \beta_{2} - 3380045928826 \beta_{3} - 19353886 \beta_{4} + 1591844294 \beta_{5} - 3784717150 \beta_{6} - 3186060796 \beta_{7} + 35995362 \beta_{8} + 2525565910 \beta_{9} + 2460838162 \beta_{10} + 32599748 \beta_{11} + 175301750 \beta_{12} + 21747308 \beta_{13} - 171798604 \beta_{14} - 106841680 \beta_{15} + 65199496 \beta_{16} - 8161244 \beta_{17} - 10873654 \beta_{18} - 8951980 \beta_{19} ) q^{89}$$ $$+ ( -2560783779672964 + 9614970974474 \beta_{1} - 5121572456828835 \beta_{2} + 4807402052417 \beta_{3} + 308109286756 \beta_{4} + 1405760430 \beta_{5} + 165289405 \beta_{6} + 1506532755 \beta_{7} - 412119645 \beta_{8} + 88070420 \beta_{9} + 4687754587 \beta_{10} + 21159105 \beta_{11} - 1083867040 \beta_{12} + 28995395 \beta_{13} + 47953355 \beta_{14} + 282625455 \beta_{15} - 21159105 \beta_{16} - 48223590 \beta_{17} + 28995395 \beta_{18} - 33590775 \beta_{19} ) q^{90}$$ $$+ ( 1180800466131755 + 4025636951854 \beta_{1} + 1671038024623961 \beta_{2} + 4740048926648 \beta_{3} + 110036530422 \beta_{4} + 16619548463 \beta_{5} - 185919512135 \beta_{6} + 11377027179 \beta_{7} + 133782313 \beta_{8} + 2932931624 \beta_{9} - 3176850593 \beta_{10} - 4938514 \beta_{11} - 992817595 \beta_{12} - 36170134 \beta_{13} + 795807425 \beta_{14} - 76631261 \beta_{15} + 118122046 \beta_{16} + 11933635 \beta_{17} + 49222460 \beta_{18} - 35920780 \beta_{19} ) q^{91}$$ $$+ ( 5651926300618496 + 114947465 \beta_{1} + 2700167381 \beta_{2} + 5002862743561 \beta_{3} - 37068321647 \beta_{4} - 65249462287 \beta_{5} + 73715862819 \beta_{6} + 65360987847 \beta_{7} + 359779001 \beta_{8} - 4849697431 \beta_{9} - 2541474626 \beta_{10} - 104654535 \beta_{11} + 510677093 \beta_{12} + 6963254 \beta_{13} - 1208698382 \beta_{14} - 215834324 \beta_{15} - 104654535 \beta_{16} - 64769150 \beta_{17} - 6963254 \beta_{18} - 18012740 \beta_{19} ) q^{92}$$ $$+ ( -1559115656046703 + 3236446790977 \beta_{1} - 1559119937954710 \beta_{2} + 3241013808995 \beta_{3} - 66286604535 \beta_{4} + 29406388995 \beta_{5} + 33085770057 \beta_{6} + 24885474 \beta_{7} + 357085879 \beta_{8} - 4488585569 \beta_{9} + 4423179189 \beta_{10} + 67217082 \beta_{11} + 161672459 \beta_{12} + 278171074 \beta_{14} - 18315986 \beta_{15} + 11102460 \beta_{17} + 8534813 \beta_{18} - 35987934 \beta_{19} ) q^{93}$$ $$+ ( -482390633361852 - 8845972791493 \beta_{1} - 241195063018352 \beta_{2} - 17698790641198 \beta_{3} - 203628303027 \beta_{4} - 26828761545 \beta_{5} + 203431481881 \beta_{6} + 13518660666 \beta_{7} + 452669903 \beta_{8} + 6770493942 \beta_{9} - 267145184 \beta_{10} + 180466126 \beta_{11} - 70700185 \beta_{12} + 44124349 \beta_{13} + 531416642 \beta_{14} + 12542750 \beta_{15} + 90233063 \beta_{16} - 69519929 \beta_{17} - 88248698 \beta_{18} - 45304605 \beta_{19} ) q^{94}$$ $$+ ( -155026473 - 4961605158384 \beta_{1} + 1955821662348947 \beta_{2} - 389975501 \beta_{3} + 179836238070 \beta_{4} + 18148611 \beta_{5} + 179932349568 \beta_{6} + 1598472843 \beta_{7} + 244696014 \beta_{8} + 475098218 \beta_{9} + 707434240 \beta_{10} - 325026954 \beta_{12} - 17611854 \beta_{13} + 157627059 \beta_{14} - 40087176 \beta_{15} + 84902604 \beta_{16} + 23374623 \beta_{17} + 20761617 \beta_{19} ) q^{95}$$ $$+ ( -4577033634215012 + 4144521341854 \beta_{1} + 4577041548534750 \beta_{2} - 4136328244700 \beta_{3} - 94216666 \beta_{4} + 47944968024 \beta_{5} - 715072900528 \beta_{6} - 95911858752 \beta_{7} + 266915484 \beta_{8} - 7738038692 \beta_{9} - 7978744378 \beta_{10} + 69797982 \beta_{11} - 880226486 \beta_{12} - 92682776 \beta_{13} + 809051902 \beta_{14} - 366372426 \beta_{15} + 139595964 \beta_{16} - 32140764 \beta_{17} + 46341388 \beta_{18} - 39448332 \beta_{19} ) q^{96}$$ $$+ ( -2110577608826772 + 3252190123964 \beta_{1} - 4221156451251417 \beta_{2} + 1626171824031 \beta_{3} - 235726523661 \beta_{4} - 41560252973 \beta_{5} + 129550426 \beta_{6} - 41529316577 \beta_{7} + 356540577 \beta_{8} + 188657889 \beta_{9} + 1540704005 \beta_{10} + 45621304 \beta_{11} - 192656940 \beta_{12} - 5475623 \beta_{13} + 68780481 \beta_{14} + 190320371 \beta_{15} - 45621304 \beta_{16} + 53631840 \beta_{17} - 5475623 \beta_{18} - 10312132 \beta_{19} ) q^{97}$$ $$+ ( 5338480411994149 - 2573121062771 \beta_{1} + 4128712437574501 \beta_{2} + 6167336921893 \beta_{3} + 645172732113 \beta_{4} - 49624091167 \beta_{5} - 130909317854 \beta_{6} + 48276896283 \beta_{7} - 263480924 \beta_{8} - 5704912787 \beta_{9} - 838015297 \beta_{10} + 14098525 \beta_{11} - 463455937 \beta_{12} + 56335496 \beta_{13} - 652671614 \beta_{14} + 145270461 \beta_{15} - 103493488 \beta_{16} - 44154467 \beta_{17} + 22080919 \beta_{18} + 73686256 \beta_{19} ) q^{98}$$ $$+ ( 7133365373253121 - 172944468 \beta_{1} - 949845605 \beta_{2} - 2148219803638 \beta_{3} - 274208356220 \beta_{4} + 62279383257 \beta_{5} + 548947734817 \beta_{6} - 62414483187 \beta_{7} - 541562219 \beta_{8} + 1188938606 \beta_{9} + 865721639 \beta_{10} + 43467798 \beta_{11} + 303752793 \beta_{12} + 51196010 \beta_{13} - 333331597 \beta_{14} + 131546431 \beta_{15} + 43467798 \beta_{16} + 74325993 \beta_{17} - 51196010 \beta_{18} + 13552056 \beta_{19} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q$$ $$\mathstrut +\mathstrut 92q^{2}$$ $$\mathstrut +\mathstrut 6558q^{3}$$ $$\mathstrut -\mathstrut 285924q^{4}$$ $$\mathstrut +\mathstrut 241890q^{5}$$ $$\mathstrut -\mathstrut 1847944q^{7}$$ $$\mathstrut +\mathstrut 23873872q^{8}$$ $$\mathstrut +\mathstrut 146092512q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$20q$$ $$\mathstrut +\mathstrut 92q^{2}$$ $$\mathstrut +\mathstrut 6558q^{3}$$ $$\mathstrut -\mathstrut 285924q^{4}$$ $$\mathstrut +\mathstrut 241890q^{5}$$ $$\mathstrut -\mathstrut 1847944q^{7}$$ $$\mathstrut +\mathstrut 23873872q^{8}$$ $$\mathstrut +\mathstrut 146092512q^{9}$$ $$\mathstrut -\mathstrut 567579804q^{10}$$ $$\mathstrut +\mathstrut 487037030q^{11}$$ $$\mathstrut -\mathstrut 2240722092q^{12}$$ $$\mathstrut +\mathstrut 598121216q^{14}$$ $$\mathstrut -\mathstrut 2098975212q^{15}$$ $$\mathstrut -\mathstrut 13522996616q^{16}$$ $$\mathstrut +\mathstrut 16479299850q^{17}$$ $$\mathstrut -\mathstrut 21631627512q^{18}$$ $$\mathstrut -\mathstrut 29753076894q^{19}$$ $$\mathstrut +\mathstrut 30310552398q^{21}$$ $$\mathstrut +\mathstrut 143879888720q^{22}$$ $$\mathstrut -\mathstrut 199076938822q^{23}$$ $$\mathstrut +\mathstrut 613456300512q^{24}$$ $$\mathstrut +\mathstrut 212723266388q^{25}$$ $$\mathstrut -\mathstrut 1359419612544q^{26}$$ $$\mathstrut +\mathstrut 2531834927748q^{28}$$ $$\mathstrut +\mathstrut 438309600272q^{29}$$ $$\mathstrut -\mathstrut 2296351012392q^{30}$$ $$\mathstrut +\mathstrut 1037434908306q^{31}$$ $$\mathstrut -\mathstrut 3523947158064q^{32}$$ $$\mathstrut +\mathstrut 411779151054q^{33}$$ $$\mathstrut -\mathstrut 7248579242478q^{35}$$ $$\mathstrut -\mathstrut 2493315404256q^{36}$$ $$\mathstrut +\mathstrut 5318067734218q^{37}$$ $$\mathstrut -\mathstrut 9079217417208q^{38}$$ $$\mathstrut +\mathstrut 13250117821332q^{39}$$ $$\mathstrut +\mathstrut 15010293809208q^{40}$$ $$\mathstrut +\mathstrut 16256122433712q^{42}$$ $$\mathstrut -\mathstrut 621187489400q^{43}$$ $$\mathstrut +\mathstrut 25172272315980q^{44}$$ $$\mathstrut -\mathstrut 85040191344096q^{45}$$ $$\mathstrut -\mathstrut 15614039192704q^{46}$$ $$\mathstrut +\mathstrut 106960600327866q^{47}$$ $$\mathstrut -\mathstrut 52978127838580q^{49}$$ $$\mathstrut +\mathstrut 35872835226128q^{50}$$ $$\mathstrut +\mathstrut 4235281588962q^{51}$$ $$\mathstrut +\mathstrut 126484190926632q^{52}$$ $$\mathstrut +\mathstrut 29048763888218q^{53}$$ $$\mathstrut -\mathstrut 635343594055560q^{54}$$ $$\mathstrut +\mathstrut 230352840277168q^{56}$$ $$\mathstrut +\mathstrut 366396034764636q^{57}$$ $$\mathstrut -\mathstrut 279762037805080q^{58}$$ $$\mathstrut -\mathstrut 99092116656282q^{59}$$ $$\mathstrut -\mathstrut 173605217618196q^{60}$$ $$\mathstrut -\mathstrut 904353032308434q^{61}$$ $$\mathstrut +\mathstrut 1188663557133192q^{63}$$ $$\mathstrut +\mathstrut 2502182430870944q^{64}$$ $$\mathstrut -\mathstrut 500211967540404q^{65}$$ $$\mathstrut -\mathstrut 951679982792988q^{66}$$ $$\mathstrut -\mathstrut 33326890567694q^{67}$$ $$\mathstrut -\mathstrut 964537664673492q^{68}$$ $$\mathstrut -\mathstrut 569178593140080q^{70}$$ $$\mathstrut +\mathstrut 3111156268483352q^{71}$$ $$\mathstrut +\mathstrut 936737429904432q^{72}$$ $$\mathstrut -\mathstrut 1146801010325370q^{73}$$ $$\mathstrut -\mathstrut 2161044304782140q^{74}$$ $$\mathstrut -\mathstrut 4935640048894356q^{75}$$ $$\mathstrut +\mathstrut 1321278185203718q^{77}$$ $$\mathstrut +\mathstrut 13747974994685232q^{78}$$ $$\mathstrut +\mathstrut 126874036566250q^{79}$$ $$\mathstrut -\mathstrut 18764997660746184q^{80}$$ $$\mathstrut -\mathstrut 2081051947014102q^{81}$$ $$\mathstrut -\mathstrut 11916006908925168q^{82}$$ $$\mathstrut +\mathstrut 17065299612720828q^{84}$$ $$\mathstrut +\mathstrut 27928718307758508q^{85}$$ $$\mathstrut -\mathstrut 7493915124549152q^{86}$$ $$\mathstrut -\mathstrut 4167264363403248q^{87}$$ $$\mathstrut -\mathstrut 10398917942882000q^{88}$$ $$\mathstrut -\mathstrut 34000447101257730q^{89}$$ $$\mathstrut +\mathstrut 6895892767523928q^{91}$$ $$\mathstrut +\mathstrut 113018032389709752q^{92}$$ $$\mathstrut -\mathstrut 15597881218045122q^{93}$$ $$\mathstrut -\mathstrut 7183919979926376q^{94}$$ $$\mathstrut -\mathstrut 19569222088313322q^{95}$$ $$\mathstrut -\mathstrut 137281944028491360q^{96}$$ $$\mathstrut +\mathstrut 65453414234371436q^{98}$$ $$\mathstrut +\mathstrut 142672614742345392q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20}\mathstrut -\mathstrut$$ $$2$$ $$x^{19}\mathstrut +\mathstrut$$ $$470221$$ $$x^{18}\mathstrut -\mathstrut$$ $$17792382$$ $$x^{17}\mathstrut +\mathstrut$$ $$149610140535$$ $$x^{16}\mathstrut -\mathstrut$$ $$5885709493452$$ $$x^{15}\mathstrut +\mathstrut$$ $$25413927970368382$$ $$x^{14}\mathstrut -\mathstrut$$ $$921982218074544860$$ $$x^{13}\mathstrut +\mathstrut$$ $$3092605513604061172669$$ $$x^{12}\mathstrut -\mathstrut$$ $$78217760507911051825254$$ $$x^{11}\mathstrut +\mathstrut$$ $$217775891572550635786539387$$ $$x^{10}\mathstrut -\mathstrut$$ $$772894989756975230685011586$$ $$x^{9}\mathstrut +\mathstrut$$ $$10848813785145958185022965942709$$ $$x^{8}\mathstrut +\mathstrut$$ $$50328102663961537370536178622316$$ $$x^{7}\mathstrut +\mathstrut$$ $$292280160701254761307491650568975646$$ $$x^{6}\mathstrut +\mathstrut$$ $$6884264619587692201297746565224842172$$ $$x^{5}\mathstrut +\mathstrut$$ $$5465648386210730671112398026225318665199$$ $$x^{4}\mathstrut +\mathstrut$$ $$69498817213798305204024309903431721187110$$ $$x^{3}\mathstrut +\mathstrut$$ $$44558299163904360256389313185385225104835245$$ $$x^{2}\mathstrut -\mathstrut$$ $$34923314543744821707168835219945993579648950$$ $$x\mathstrut +\mathstrut$$ $$261490399505792227817024003348104130194641159225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$81\!\cdots\!00$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$47\!\cdots\!70$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$33\!\cdots\!44$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$22\!\cdots\!01$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$18\!\cdots\!76$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$69\!\cdots\!40$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$41\!\cdots\!00$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$11\!\cdots\!08$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$56\!\cdots\!04$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$13\!\cdots\!78$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$39\!\cdots\!12$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$92\!\cdots\!80$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$29\!\cdots\!88$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$44\!\cdots\!80$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$28\!\cdots\!84$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$10\!\cdots\!48$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$30\!\cdots\!20$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$22\!\cdots\!82$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$18\!\cdots\!60$$$$)/$$$$12\!\cdots\!35$$ $$\beta_{3}$$ $$=$$ $$($$$$12\!\cdots\!30$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$11\!\cdots\!24$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$58\!\cdots\!79$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$16\!\cdots\!04$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$18\!\cdots\!60$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$53\!\cdots\!00$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$30\!\cdots\!32$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$81\!\cdots\!16$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$36\!\cdots\!62$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$57\!\cdots\!48$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$24\!\cdots\!20$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$15\!\cdots\!48$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11\!\cdots\!20$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$13\!\cdots\!36$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$28\!\cdots\!92$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$90\!\cdots\!80$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$58\!\cdots\!78$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$40\!\cdots\!00$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$15\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$55\!\cdots\!00$$$$)/$$$$32\!\cdots\!65$$ $$\beta_{4}$$ $$=$$ $$($$$$21\!\cdots\!78$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$89\!\cdots\!13$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$88\!\cdots\!57$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$39\!\cdots\!45$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$24\!\cdots\!50$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$12\!\cdots\!54$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$29\!\cdots\!68$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$22\!\cdots\!84$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$25\!\cdots\!14$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$28\!\cdots\!65$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$81\!\cdots\!61$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$21\!\cdots\!55$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$26\!\cdots\!28$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10\!\cdots\!04$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$46\!\cdots\!80$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$27\!\cdots\!66$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$49\!\cdots\!20$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$33\!\cdots\!35$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$13\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$21\!\cdots\!75$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{5}$$ $$=$$ $$($$$$20\!\cdots\!61$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$13\!\cdots\!38$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$10\!\cdots\!20$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$67\!\cdots\!51$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$35\!\cdots\!00$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$21\!\cdots\!62$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$66\!\cdots\!24$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$35\!\cdots\!16$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$86\!\cdots\!81$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$42\!\cdots\!62$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$67\!\cdots\!02$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$27\!\cdots\!23$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$30\!\cdots\!64$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$12\!\cdots\!68$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$84\!\cdots\!58$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$29\!\cdots\!28$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$67\!\cdots\!97$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$46\!\cdots\!60$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$92\!\cdots\!10$$ $$\nu\mathstrut -\mathstrut$$ $$17\!\cdots\!75$$$$)/$$$$18\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$95\!\cdots\!79$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$36\!\cdots\!36$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$43\!\cdots\!66$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$31\!\cdots\!55$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$13\!\cdots\!80$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$97\!\cdots\!78$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$22\!\cdots\!44$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$14\!\cdots\!48$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$26\!\cdots\!47$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$12\!\cdots\!60$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$17\!\cdots\!88$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$19\!\cdots\!55$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$78\!\cdots\!96$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$40\!\cdots\!32$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$15\!\cdots\!30$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!88$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$19\!\cdots\!95$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$12\!\cdots\!30$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$37\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$15\!\cdots\!25$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$38\!\cdots\!71$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$77\!\cdots\!31$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$17\!\cdots\!09$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$30\!\cdots\!10$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$53\!\cdots\!90$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$97\!\cdots\!08$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$82\!\cdots\!56$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$17\!\cdots\!88$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$93\!\cdots\!03$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$22\!\cdots\!15$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$54\!\cdots\!67$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$18\!\cdots\!00$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$25\!\cdots\!24$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$93\!\cdots\!48$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$41\!\cdots\!10$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$27\!\cdots\!42$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$89\!\cdots\!25$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$43\!\cdots\!55$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$19\!\cdots\!85$$ $$\nu\mathstrut -\mathstrut$$ $$35\!\cdots\!00$$$$)/$$$$18\!\cdots\!20$$ $$\beta_{8}$$ $$=$$ $$($$$$60\!\cdots\!39$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$62\!\cdots\!54$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$68\!\cdots\!40$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$29\!\cdots\!59$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$45\!\cdots\!44$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$96\!\cdots\!42$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$15\!\cdots\!96$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$16\!\cdots\!04$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$26\!\cdots\!89$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$20\!\cdots\!42$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$29\!\cdots\!30$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$14\!\cdots\!35$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$16\!\cdots\!96$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$68\!\cdots\!92$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$68\!\cdots\!54$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$17\!\cdots\!40$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$89\!\cdots\!85$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$22\!\cdots\!08$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!50$$ $$\nu\mathstrut +\mathstrut$$ $$16\!\cdots\!75$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$25\!\cdots\!27$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$80\!\cdots\!47$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$12\!\cdots\!83$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$41\!\cdots\!12$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$42\!\cdots\!26$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$13\!\cdots\!44$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$78\!\cdots\!00$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$21\!\cdots\!56$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$98\!\cdots\!23$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$24\!\cdots\!55$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$74\!\cdots\!21$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$14\!\cdots\!58$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$34\!\cdots\!56$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$62\!\cdots\!40$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$91\!\cdots\!94$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$10\!\cdots\!54$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$68\!\cdots\!57$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$12\!\cdots\!67$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$61\!\cdots\!45$$ $$\nu\mathstrut -\mathstrut$$ $$39\!\cdots\!90$$$$)/$$$$18\!\cdots\!20$$ $$\beta_{10}$$ $$=$$ $$($$$$91\!\cdots\!86$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$16\!\cdots\!45$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$39\!\cdots\!33$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$63\!\cdots\!61$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$11\!\cdots\!70$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$20\!\cdots\!18$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$17\!\cdots\!28$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$37\!\cdots\!12$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$18\!\cdots\!90$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$48\!\cdots\!17$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$84\!\cdots\!57$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$39\!\cdots\!83$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$31\!\cdots\!64$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$18\!\cdots\!24$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$13\!\cdots\!48$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$52\!\cdots\!02$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$46\!\cdots\!48$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$57\!\cdots\!55$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$57\!\cdots\!95$$ $$\nu\mathstrut +\mathstrut$$ $$37\!\cdots\!95$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$56\!\cdots\!08$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$65\!\cdots\!31$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$11\!\cdots\!87$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$30\!\cdots\!89$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$32\!\cdots\!98$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$10\!\cdots\!34$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$11\!\cdots\!88$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$18\!\cdots\!28$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$25\!\cdots\!00$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$23\!\cdots\!35$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$36\!\cdots\!13$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$17\!\cdots\!11$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$23\!\cdots\!24$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$82\!\cdots\!56$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$10\!\cdots\!68$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$22\!\cdots\!82$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$20\!\cdots\!34$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$26\!\cdots\!89$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$27\!\cdots\!85$$ $$\nu\mathstrut -\mathstrut$$ $$21\!\cdots\!35$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{12}$$ $$=$$ $$($$$$57\!\cdots\!42$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$18\!\cdots\!57$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$23\!\cdots\!89$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$79\!\cdots\!81$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$68\!\cdots\!54$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$25\!\cdots\!06$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$89\!\cdots\!92$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$45\!\cdots\!64$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$85\!\cdots\!82$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$57\!\cdots\!13$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$22\!\cdots\!57$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$43\!\cdots\!75$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$37\!\cdots\!00$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$20\!\cdots\!56$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$78\!\cdots\!56$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$55\!\cdots\!02$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$69\!\cdots\!80$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$68\!\cdots\!27$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$22\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$43\!\cdots\!55$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!17$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$70\!\cdots\!82$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$38\!\cdots\!60$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$32\!\cdots\!15$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$14\!\cdots\!84$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$10\!\cdots\!54$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$32\!\cdots\!80$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$16\!\cdots\!20$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$44\!\cdots\!85$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$19\!\cdots\!14$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$40\!\cdots\!22$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$11\!\cdots\!23$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$20\!\cdots\!00$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$45\!\cdots\!40$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$81\!\cdots\!82$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$67\!\cdots\!68$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$12\!\cdots\!77$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$59\!\cdots\!28$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$22\!\cdots\!10$$ $$\nu\mathstrut +\mathstrut$$ $$65\!\cdots\!95$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$59\!\cdots\!31$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$10\!\cdots\!59$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$28\!\cdots\!03$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$58\!\cdots\!36$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$92\!\cdots\!66$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$18\!\cdots\!12$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$16\!\cdots\!36$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$30\!\cdots\!20$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$19\!\cdots\!47$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$34\!\cdots\!83$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!69$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$19\!\cdots\!10$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$67\!\cdots\!72$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$82\!\cdots\!72$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$17\!\cdots\!26$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$13\!\cdots\!86$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$19\!\cdots\!05$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$17\!\cdots\!07$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$13\!\cdots\!25$$ $$\nu\mathstrut +\mathstrut$$ $$40\!\cdots\!90$$$$)/$$$$23\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$58\!\cdots\!77$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$25\!\cdots\!89$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$26\!\cdots\!73$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$20\!\cdots\!68$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$84\!\cdots\!94$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$63\!\cdots\!24$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$13\!\cdots\!84$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$91\!\cdots\!92$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$16\!\cdots\!49$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$82\!\cdots\!85$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$10\!\cdots\!67$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$15\!\cdots\!42$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$44\!\cdots\!00$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$33\!\cdots\!08$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$79\!\cdots\!86$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$65\!\cdots\!42$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$84\!\cdots\!83$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$54\!\cdots\!67$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$16\!\cdots\!05$$ $$\nu\mathstrut +\mathstrut$$ $$10\!\cdots\!30$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{16}$$ $$=$$ $$($$$$39\!\cdots\!75$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$21\!\cdots\!79$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$18\!\cdots\!83$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$16\!\cdots\!84$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$57\!\cdots\!58$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$50\!\cdots\!60$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$95\!\cdots\!48$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$76\!\cdots\!68$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$11\!\cdots\!95$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$74\!\cdots\!91$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$74\!\cdots\!61$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$24\!\cdots\!58$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$33\!\cdots\!04$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$69\!\cdots\!36$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$68\!\cdots\!06$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$23\!\cdots\!46$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$81\!\cdots\!57$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$96\!\cdots\!29$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$96\!\cdots\!75$$ $$\nu\mathstrut +\mathstrut$$ $$32\!\cdots\!50$$$$)/$$$$18\!\cdots\!20$$ $$\beta_{17}$$ $$=$$ $$($$$$16\!\cdots\!76$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$41\!\cdots\!79$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$73\!\cdots\!31$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$16\!\cdots\!61$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$21\!\cdots\!86$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$53\!\cdots\!98$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$31\!\cdots\!60$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$94\!\cdots\!28$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$34\!\cdots\!36$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$12\!\cdots\!11$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$16\!\cdots\!79$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$93\!\cdots\!51$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$67\!\cdots\!72$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$45\!\cdots\!00$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$19\!\cdots\!00$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!54$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$10\!\cdots\!26$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$14\!\cdots\!53$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$42\!\cdots\!85$$ $$\nu\mathstrut +\mathstrut$$ $$71\!\cdots\!95$$$$)/$$$$54\!\cdots\!60$$ $$\beta_{18}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!52$$ $$\nu^{19}\mathstrut +\mathstrut$$ $$22\!\cdots\!61$$ $$\nu^{18}\mathstrut -\mathstrut$$ $$15\!\cdots\!11$$ $$\nu^{17}\mathstrut +\mathstrut$$ $$60\!\cdots\!11$$ $$\nu^{16}\mathstrut -\mathstrut$$ $$47\!\cdots\!34$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$19\!\cdots\!94$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$80\!\cdots\!48$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$26\!\cdots\!84$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$97\!\cdots\!08$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$19\!\cdots\!61$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$66\!\cdots\!27$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$60\!\cdots\!19$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$32\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$63\!\cdots\!44$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$80\!\cdots\!80$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$42\!\cdots\!62$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$13\!\cdots\!34$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$38\!\cdots\!77$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$82\!\cdots\!75$$ $$\nu\mathstrut -\mathstrut$$ $$26\!\cdots\!15$$$$)/$$$$90\!\cdots\!60$$ $$\beta_{19}$$ $$=$$ $$($$$$10\!\cdots\!29$$ $$\nu^{19}\mathstrut -\mathstrut$$ $$21\!\cdots\!50$$ $$\nu^{18}\mathstrut +\mathstrut$$ $$48\!\cdots\!08$$ $$\nu^{17}\mathstrut -\mathstrut$$ $$11\!\cdots\!35$$ $$\nu^{16}\mathstrut +\mathstrut$$ $$16\!\cdots\!04$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$36\!\cdots\!18$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$28\!\cdots\!52$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$59\!\cdots\!24$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$35\!\cdots\!81$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$67\!\cdots\!14$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$26\!\cdots\!06$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$39\!\cdots\!63$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$12\!\cdots\!08$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$16\!\cdots\!84$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$35\!\cdots\!62$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$26\!\cdots\!64$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$46\!\cdots\!97$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$27\!\cdots\!28$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$41\!\cdots\!90$$ $$\nu\mathstrut -\mathstrut$$ $$15\!\cdots\!55$$$$)/$$$$27\!\cdots\!80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-$$$$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$94050$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$94050$$ $$\nu^{3}$$ $$=$$ $$-$$$$\beta_{16}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$69$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$655$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$69$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$327$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$166216$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2561232$$ $$\nu^{4}$$ $$=$$ $$-$$$$58$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$60$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$322$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$1018$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$30$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1868$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$6226$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$3182$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$228$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$241728$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$240786$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$124$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$240884$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$3194$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$15611695021$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$12681254$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$84$$ $$\nu^{5}$$ $$=$$ $$5330$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$11462$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$7614$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$353662$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$343722$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$348250$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$279724$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$609942$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$976548$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$107400$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2284$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$164045100$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$24014760$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$328488826$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$33347217281$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1175557986832$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$33346549431$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1175558557942$$ $$\nu^{6}$$ $$=$$ $$-$$$$5822296$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$9804750$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$21680706$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$26039904$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$78063824$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$764668228$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$9804750$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$354715430$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$26039904$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1214579876$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$2331993974$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$121742584$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$55553213005$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$117870497120$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$55515673889$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$59009755671$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4157688645514$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1258086668$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$46432988$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3125881494314572$$ $$\nu^{7}$$ $$=$$ $$2664712038$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$5821239808$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$68823969001$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$42645819003$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$111445124815$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4403496238$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$217824162558$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$513912323259$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$311033613673$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$93619233705$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$7443388954577$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$52354302007474$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$491815732$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$52380743494476$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$292322925781$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$386691566415949625$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$7260676725597081$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$86287688425$$ $$\nu^{8}$$ $$=$$ $$5753793941412$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$2559168325212$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$3525587873364$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$1401784311396$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$107589624644868$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$122291838143892$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$8879612336952$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$331173005265420$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$331999427022072$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$23910404742960$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$9279381814776$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$15463200288215488$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$12888876483140104$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$30928488784707524$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1230299811210798640$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$679570155184569203817$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1229962920006877228$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$679570479672956265093$$ $$\nu^{9}$$ $$=$$ $$-$$$$1848262918151904$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$1308953624892972$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$1160551491220932$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$16533054034022440$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$17255608461329136$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$64455704998231976$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1308953624892972$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$31141502074731076$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$16533054034022440$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$83740601860291336$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$205408381681471700$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$7869884302216984$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2194887815619767808$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$28790065968146214964$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$2194414975555477848$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$14394524913111673332$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1653197563569005162341$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$105818444648769488$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4970636460821056$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$114618599492234493232836$$ $$\nu^{10}$$ $$=$$ $$-$$$$754194184393796692$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$692508456794619248$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$2741351669243205792$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$6678861484594127932$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$32833742582720959708$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$645019606882400412$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$61972396848759072008$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$182062299398706892684$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$94801860707800107188$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$493774223354773536$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3028946934413999407665$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4152782153391573683085$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$2200896825582212632$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4156998889859940576017$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$93526018371722264972$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$154559389390099258686011914$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$347510760008298743352632$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4818877039627063536$$ $$\nu^{11}$$ $$=$$ $$358869220345396032308$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$355349397896945665436$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$157311026231262479244$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$5974432348048056410389$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$8749876434331494797613$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$9471134697470737229101$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$3960998174521194171745$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$23417960044586815398579$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$29908572639211530320520$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$4327480344278951278608$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$201558194114133553064$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3725482819197009498875823$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$624917324236930117797525$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$7458485091659348579329819$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$386707571870615887901269577$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$32403148486521930657757451464$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$386682092709407442902191662$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$32403170355941235514597315387$$ $$\nu^{12}$$ $$=$$ $$-$$$$196534731509028437124568$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$162824597230676089715898$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$357888649867318796276262$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$799556281982758179844404$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$2092991099170222382988704$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$15935814470341294774474852$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$162824597230676089715898$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$7512486558985007237680982$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$799556281982758179844404$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$24044851210283194066736372$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$47395017201788731518262622$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1739374401366825741498100$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$721122046901008501477556068$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2235815361737771506782870206$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$720602804332782892322128112$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1119153582465448046962728954$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$95602944577501971816964170598$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$25434011686793037557954480$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$898894675297021598330960$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$36132329500542398633208365132443$$ $$\nu^{13}$$ $$=$$ $$33681865156779064240587162$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$136080594451456508654811376$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$951857625486880956583870900$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$157861089227745853685236098$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$2603663519345086792882526686$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$92557028411842580896333234$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$4992687879217983833418713964$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$13349771772260272887484042254$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$7605334996433867738446557322$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1339062873251240503820060340$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$173274509862335857227174373964$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$939411684408742092753241959442$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$68716864137898380173637052$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$940058405515842596232275429160$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$7206934913962333076000014174$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$8912405834922208560738691939528016$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$92061291682852732551316739369067$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1360099408841250482548305028$$ $$\nu^{14}$$ $$=$$ $$88\!\cdots\!90$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$41\!\cdots\!06$$ $$\beta_{18}\mathstrut +\mathstrut$$ $$34\!\cdots\!90$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$18\!\cdots\!22$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$18\!\cdots\!38$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$21\!\cdots\!02$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$22\!\cdots\!60$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$56\!\cdots\!86$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$58\!\cdots\!08$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$50\!\cdots\!96$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$12\!\cdots\!80$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$29\!\cdots\!25$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$17\!\cdots\!17$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$59\!\cdots\!88$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$25\!\cdots\!72$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$85\!\cdots\!86$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$25\!\cdots\!70$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$85\!\cdots\!72$$ $$\nu^{15}$$ $$=$$ $$-$$$$34\!\cdots\!56$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$23\!\cdots\!62$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$28\!\cdots\!22$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$22\!\cdots\!25$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$33\!\cdots\!32$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$13\!\cdots\!22$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$23\!\cdots\!62$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$63\!\cdots\!59$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$22\!\cdots\!25$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18\!\cdots\!34$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$41\!\cdots\!17$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$22\!\cdots\!77$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$47\!\cdots\!29$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$46\!\cdots\!61$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$47\!\cdots\!41$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$23\!\cdots\!57$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$22\!\cdots\!06$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$22\!\cdots\!27$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$66\!\cdots\!99$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$24\!\cdots\!26$$ $$\nu^{16}$$ $$=$$ $$-$$$$76\!\cdots\!96$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$14\!\cdots\!76$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$61\!\cdots\!24$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$86\!\cdots\!04$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$52\!\cdots\!44$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$10\!\cdots\!56$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$99\!\cdots\!28$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$28\!\cdots\!36$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$15\!\cdots\!36$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$36\!\cdots\!60$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$41\!\cdots\!16$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$78\!\cdots\!40$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$29\!\cdots\!68$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$79\!\cdots\!16$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$14\!\cdots\!24$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$20\!\cdots\!73$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$68\!\cdots\!12$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$10\!\cdots\!52$$ $$\nu^{17}$$ $$=$$ $$74\!\cdots\!92$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$59\!\cdots\!04$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$13\!\cdots\!20$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$83\!\cdots\!36$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$16\!\cdots\!24$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$18\!\cdots\!96$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$55\!\cdots\!16$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$46\!\cdots\!12$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$55\!\cdots\!60$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$75\!\cdots\!80$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$61\!\cdots\!72$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$57\!\cdots\!48$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$12\!\cdots\!72$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$11\!\cdots\!80$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$53\!\cdots\!45$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$64\!\cdots\!48$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$53\!\cdots\!73$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$64\!\cdots\!24$$ $$\nu^{18}$$ $$=$$ $$-$$$$36\!\cdots\!84$$ $$\beta_{19}\mathstrut -\mathstrut$$ $$26\!\cdots\!64$$ $$\beta_{18}\mathstrut -\mathstrut$$ $$53\!\cdots\!84$$ $$\beta_{17}\mathstrut -\mathstrut$$ $$16\!\cdots\!00$$ $$\beta_{16}\mathstrut -\mathstrut$$ $$37\!\cdots\!20$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$24\!\cdots\!20$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$26\!\cdots\!64$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$11\!\cdots\!16$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$16\!\cdots\!00$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$36\!\cdots\!56$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$74\!\cdots\!36$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$21\!\cdots\!08$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$10\!\cdots\!49$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$41\!\cdots\!58$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$10\!\cdots\!65$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$20\!\cdots\!41$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$18\!\cdots\!32$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$39\!\cdots\!08$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$12\!\cdots\!88$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$50\!\cdots\!82$$ $$\nu^{19}$$ $$=$$ $$25\!\cdots\!44$$ $$\beta_{19}\mathstrut +\mathstrut$$ $$22\!\cdots\!40$$ $$\beta_{17}\mathstrut +\mathstrut$$ $$13\!\cdots\!85$$ $$\beta_{16}\mathstrut +\mathstrut$$ $$11\!\cdots\!95$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$48\!\cdots\!21$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$15\!\cdots\!76$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$92\!\cdots\!30$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$25\!\cdots\!33$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$14\!\cdots\!67$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$18\!\cdots\!01$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$33\!\cdots\!81$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$14\!\cdots\!08$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17\!\cdots\!52$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$14\!\cdots\!64$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$13\!\cdots\!67$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$16\!\cdots\!41$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$13\!\cdots\!79$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$20\!\cdots\!05$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −251.134 + 434.977i −187.565 + 324.873i −120.823 + 209.272i −64.4555 + 111.640i −58.5903 + 101.481i 45.0238 − 77.9836i 90.3350 − 156.465i 137.362 − 237.918i 180.712 − 313.002i 230.136 − 398.607i −251.134 − 434.977i −187.565 − 324.873i −120.823 − 209.272i −64.4555 − 111.640i −58.5903 − 101.481i 45.0238 + 77.9836i 90.3350 + 156.465i 137.362 + 237.918i 180.712 + 313.002i 230.136 + 398.607i
−246.634 427.182i 6439.04 + 3717.58i −88888.5 + 153959.i 360858. 208341.i 3.66753e6i −3.56165e6 4.53294e6i 5.53649e7 6.11747e6 + 1.05958e7i −1.77999e8 1.02768e8i
3.2 −183.065 317.078i −5337.54 3081.63i −34257.8 + 59336.2i −149102. + 86084.1i 2.25656e6i 303703. + 5.75680e6i 1.09092e6 −2.53044e6 4.38285e6i 5.45908e7 + 3.15180e7i
3.3 −116.323 201.477i 2610.90 + 1507.40i 5705.88 9882.88i −301331. + 173973.i 701383.i 2.93221e6 4.96337e6i −1.79016e7 −1.69788e7 2.94082e7i 7.01035e7 + 4.04742e7i
3.4 −59.9555 103.846i −7943.79 4586.35i 25578.7 44303.6i 625327. 361033.i 1.09991e6i −1.45978e6 5.57691e6i −1.39928e7 2.05458e7 + 3.55864e7i −7.49836e7 4.32918e7i
3.5 −54.0903 93.6871i 8486.99 + 4899.97i 26916.5 46620.7i 214019. 123564.i 1.06016e6i 516705. + 5.74160e6i −1.29134e7 2.64960e7 + 4.58924e7i −2.31527e7 1.33672e7i
3.6 49.5238 + 85.7778i 289.874 + 167.359i 27862.8 48259.7i −159884. + 92309.1i 33153.0i −5.76147e6 + 195847.i 1.20107e7 −2.14673e7 3.71825e7i −1.58361e7 9.14300e6i
3.7 94.8350 + 164.259i −9421.11 5439.28i 14780.6 25600.8i −477118. + 275464.i 2.06334e6i 5.57858e6 + 1.45338e6i 1.80371e7 3.76482e7 + 6.52085e7i −9.04950e7 5.22473e7i
3.8 141.862 + 245.712i 1577.48 + 910.759i −7481.54 + 12958.4i 402298. 232267.i 516808.i 5.58849e6 + 1.41483e6i 1.43487e7 −1.98644e7 3.44061e7i 1.14141e8 + 6.58996e7i
3.9 185.212 + 320.796i 10622.1 + 6132.66i −35838.6 + 62074.3i −453136. + 261618.i 4.54336e6i 694120. 5.72286e6i −2.27485e6 5.36956e7 + 9.30036e7i −1.67852e8 9.69094e7i
3.10 234.636 + 406.401i −4044.92 2335.34i −77340.0 + 133957.i 59013.4 34071.4i 2.19181e6i −5.75487e6 + 338210.i −4.18327e7 −1.06158e7 1.83871e7i 2.76933e7 + 1.59887e7i
5.1 −246.634 + 427.182i 6439.04 3717.58i −88888.5 153959.i 360858. + 208341.i 3.66753e6i −3.56165e6 + 4.53294e6i 5.53649e7 6.11747e6 1.05958e7i −1.77999e8 + 1.02768e8i
5.2 −183.065 + 317.078i −5337.54 + 3081.63i −34257.8 59336.2i −149102. 86084.1i 2.25656e6i 303703. 5.75680e6i 1.09092e6 −2.53044e6 + 4.38285e6i 5.45908e7 3.15180e7i
5.3 −116.323 + 201.477i 2610.90 1507.40i 5705.88 + 9882.88i −301331. 173973.i 701383.i 2.93221e6 + 4.96337e6i −1.79016e7 −1.69788e7 + 2.94082e7i 7.01035e7 4.04742e7i
5.4 −59.9555 + 103.846i −7943.79 + 4586.35i 25578.7 + 44303.6i 625327. + 361033.i 1.09991e6i −1.45978e6 + 5.57691e6i −1.39928e7 2.05458e7 3.55864e7i −7.49836e7 + 4.32918e7i
5.5 −54.0903 + 93.6871i 8486.99 4899.97i 26916.5 + 46620.7i 214019. + 123564.i 1.06016e6i 516705. 5.74160e6i −1.29134e7 2.64960e7 4.58924e7i −2.31527e7 + 1.33672e7i
5.6 49.5238 85.7778i 289.874 167.359i 27862.8 + 48259.7i −159884. 92309.1i 33153.0i −5.76147e6 195847.i 1.20107e7 −2.14673e7 + 3.71825e7i −1.58361e7 + 9.14300e6i
5.7 94.8350 164.259i −9421.11 + 5439.28i 14780.6 + 25600.8i −477118. 275464.i 2.06334e6i 5.57858e6 1.45338e6i 1.80371e7 3.76482e7 6.52085e7i −9.04950e7 + 5.22473e7i
5.8 141.862 245.712i 1577.48 910.759i −7481.54 12958.4i 402298. + 232267.i 516808.i 5.58849e6 1.41483e6i 1.43487e7 −1.98644e7 + 3.44061e7i 1.14141e8 6.58996e7i
5.9 185.212 320.796i 10622.1 6132.66i −35838.6 62074.3i −453136. 261618.i 4.54336e6i 694120. + 5.72286e6i −2.27485e6 5.36956e7 9.30036e7i −1.67852e8 + 9.69094e7i
5.10 234.636 406.401i −4044.92 + 2335.34i −77340.0 133957.i 59013.4 + 34071.4i 2.19181e6i −5.75487e6 338210.i −4.18327e7 −1.06158e7 + 1.83871e7i 2.76933e7 1.59887e7i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.d Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{17}^{\mathrm{new}}(7, [\chi])$$.