# Properties

 Label 27.10.c.a Level 27 Weight 10 Character orbit 27.c Analytic conductor 13.906 Analytic rank 0 Dimension 16 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 27.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$13.9059675764$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}\cdot 3^{40}\cdot 17^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -2 + \beta_{8} + 2 \beta_{9} ) q^{2}$$ $$+ ( 2 \beta_{1} - 2 \beta_{8} - 224 \beta_{9} - \beta_{10} ) q^{4}$$ $$+ ( \beta_{4} - 56 \beta_{9} - \beta_{10} - \beta_{11} ) q^{5}$$ $$+ ( -51 + 67 \beta_{8} + 51 \beta_{9} + \beta_{11} + \beta_{12} ) q^{7}$$ $$+ ( 930 - 210 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -2 + \beta_{8} + 2 \beta_{9} ) q^{2}$$ $$+ ( 2 \beta_{1} - 2 \beta_{8} - 224 \beta_{9} - \beta_{10} ) q^{4}$$ $$+ ( \beta_{4} - 56 \beta_{9} - \beta_{10} - \beta_{11} ) q^{5}$$ $$+ ( -51 + 67 \beta_{8} + 51 \beta_{9} + \beta_{11} + \beta_{12} ) q^{7}$$ $$+ ( 930 - 210 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{8}$$ $$+ ( 99 - 357 \beta_{1} - 12 \beta_{2} - 3 \beta_{3} - 21 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{10}$$ $$+ ( -12361 + 46 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} - \beta_{7} - 122 \beta_{8} + 12361 \beta_{9} + 46 \beta_{10} - 6 \beta_{11} + 5 \beta_{12} - \beta_{13} + 4 \beta_{14} + 2 \beta_{15} ) q^{11}$$ $$+ ( -195 \beta_{1} + 25 \beta_{4} - \beta_{5} + 195 \beta_{8} + 4094 \beta_{9} - 25 \beta_{10} - 25 \beta_{11} + \beta_{12} - 8 \beta_{13} - 6 \beta_{14} + 5 \beta_{15} ) q^{13}$$ $$+ ( 17 \beta_{1} - 12 \beta_{4} - 7 \beta_{5} - 17 \beta_{8} - 49287 \beta_{9} - 202 \beta_{10} + 12 \beta_{11} + 7 \beta_{12} - \beta_{13} + 2 \beta_{14} + 11 \beta_{15} ) q^{14}$$ $$+ ( -41142 + 149 \beta_{2} - 3 \beta_{3} + 4 \beta_{6} - 20 \beta_{7} + 1522 \beta_{8} + 41142 \beta_{9} + 149 \beta_{10} + 69 \beta_{11} - 7 \beta_{12} - 20 \beta_{13} + 3 \beta_{14} - 4 \beta_{15} ) q^{16}$$ $$+ ( 51805 + 293 \beta_{1} - 277 \beta_{2} + 6 \beta_{3} + 29 \beta_{4} + 15 \beta_{5} + 7 \beta_{6} - 28 \beta_{7} ) q^{17}$$ $$+ ( -9814 - 6752 \beta_{1} + 88 \beta_{2} + 36 \beta_{3} + 5 \beta_{4} - 19 \beta_{5} + 63 \beta_{6} + 3 \beta_{7} ) q^{19}$$ $$+ ( -231816 + 794 \beta_{2} + 51 \beta_{3} - 62 \beta_{6} - 94 \beta_{7} + 2356 \beta_{8} + 231816 \beta_{9} + 794 \beta_{10} + 263 \beta_{11} - 75 \beta_{12} - 94 \beta_{13} - 51 \beta_{14} + 62 \beta_{15} ) q^{20}$$ $$+ ( 37155 \beta_{1} - 465 \beta_{4} + 11 \beta_{5} - 37155 \beta_{8} + 61098 \beta_{9} + 672 \beta_{10} + 465 \beta_{11} - 11 \beta_{12} - 94 \beta_{13} + 87 \beta_{14} + 100 \beta_{15} ) q^{22}$$ $$+ ( -156 \beta_{1} - 281 \beta_{4} + 94 \beta_{5} + 156 \beta_{8} - 133227 \beta_{9} + 180 \beta_{10} + 281 \beta_{11} - 94 \beta_{12} - 40 \beta_{13} - 32 \beta_{14} + 125 \beta_{15} ) q^{23}$$ $$+ ( -293349 + 143 \beta_{2} + 48 \beta_{3} + 38 \beta_{6} - 124 \beta_{7} + 53128 \beta_{8} + 293349 \beta_{9} + 143 \beta_{10} - 327 \beta_{11} - 20 \beta_{12} - 124 \beta_{13} - 48 \beta_{14} - 38 \beta_{15} ) q^{25}$$ $$+ ( -151485 - 835 \beta_{1} + 1756 \beta_{2} + 45 \beta_{3} + 163 \beta_{4} - 90 \beta_{5} + 21 \beta_{6} - 147 \beta_{7} ) q^{26}$$ $$+ ( 91444 - 114468 \beta_{1} + 1040 \beta_{2} - 141 \beta_{3} + 575 \beta_{4} + 9 \beta_{5} + 110 \beta_{6} + 38 \beta_{7} ) q^{28}$$ $$+ ( 163126 - 4409 \beta_{2} - 230 \beta_{3} - 55 \beta_{6} + 4 \beta_{7} - 9279 \beta_{8} - 163126 \beta_{9} - 4409 \beta_{10} - 361 \beta_{11} + 463 \beta_{12} + 4 \beta_{13} + 230 \beta_{14} + 55 \beta_{15} ) q^{29}$$ $$+ ( 77344 \beta_{1} + 211 \beta_{4} - 20 \beta_{5} - 77344 \beta_{8} - 304065 \beta_{9} - 972 \beta_{10} - 211 \beta_{11} + 20 \beta_{12} - 66 \beta_{13} - 504 \beta_{14} - 99 \beta_{15} ) q^{31}$$ $$+ ( -7546 \beta_{1} + 1043 \beta_{4} - 465 \beta_{5} + 7546 \beta_{8} - 719834 \beta_{9} + 2701 \beta_{10} - 1043 \beta_{11} + 465 \beta_{12} + 140 \beta_{13} + 219 \beta_{14} - 100 \beta_{15} ) q^{32}$$ $$+ ( 98165 - 1448 \beta_{2} - 321 \beta_{3} - 151 \beta_{6} + 293 \beta_{7} + 192650 \beta_{8} - 98165 \beta_{9} - 1448 \beta_{10} - 291 \beta_{11} + 316 \beta_{12} + 293 \beta_{13} + 321 \beta_{14} + 151 \beta_{15} ) q^{34}$$ $$+ ( 1947641 - 34121 \beta_{1} + 5074 \beta_{2} - 572 \beta_{3} - 1375 \beta_{4} + 247 \beta_{5} - 38 \beta_{6} + 638 \beta_{7} ) q^{35}$$ $$+ ( 1042796 - 172511 \beta_{1} - 6230 \beta_{2} + 18 \beta_{3} - 4798 \beta_{4} + 599 \beta_{5} - 807 \beta_{6} + 12 \beta_{7} ) q^{37}$$ $$+ ( -4930771 + 2962 \beta_{2} + 253 \beta_{3} + 617 \beta_{6} + 781 \beta_{7} - 40124 \beta_{8} + 4930771 \beta_{9} + 2962 \beta_{10} - 1309 \beta_{11} - 1430 \beta_{12} + 781 \beta_{13} - 253 \beta_{14} - 617 \beta_{15} ) q^{38}$$ $$+ ( 406724 \beta_{1} + 4758 \beta_{4} - 282 \beta_{5} - 406724 \beta_{8} - 2142844 \beta_{9} - 5188 \beta_{10} - 4758 \beta_{11} + 282 \beta_{12} + 1536 \beta_{13} + 1350 \beta_{14} - 696 \beta_{15} ) q^{40}$$ $$+ ( -35267 \beta_{1} - 310 \beta_{4} + 739 \beta_{5} + 35267 \beta_{8} - 6839647 \beta_{9} + 3554 \beta_{10} + 310 \beta_{11} - 739 \beta_{12} - 232 \beta_{13} - 818 \beta_{14} - 1507 \beta_{15} ) q^{41}$$ $$+ ( 1877041 + 866 \beta_{2} + 1104 \beta_{3} + 121 \beta_{6} + 1447 \beta_{7} + 248519 \beta_{8} - 1877041 \beta_{9} + 866 \beta_{10} + 7606 \beta_{11} - 822 \beta_{12} + 1447 \beta_{13} - 1104 \beta_{14} - 121 \beta_{15} ) q^{43}$$ $$+ ( 20738136 + 253686 \beta_{1} - 34479 \beta_{2} + 2224 \beta_{3} + 4912 \beta_{4} - 152 \beta_{5} - 624 \beta_{6} + 264 \beta_{7} ) q^{44}$$ $$+ ( 167035 - 78089 \beta_{1} + 9782 \beta_{2} + 1326 \beta_{3} + 16164 \beta_{4} - 3173 \beta_{5} - 1303 \beta_{6} - 1381 \beta_{7} ) q^{46}$$ $$+ ( -19571853 + 4408 \beta_{2} + 1224 \beta_{3} + 1986 \beta_{6} + 1308 \beta_{7} + 283031 \beta_{8} + 19571853 \beta_{9} + 4408 \beta_{10} - 533 \beta_{11} + 1845 \beta_{12} + 1308 \beta_{13} - 1224 \beta_{14} - 1986 \beta_{15} ) q^{47}$$ $$+ ( -407791 \beta_{1} + 889 \beta_{4} + 2043 \beta_{5} + 407791 \beta_{8} + 1954605 \beta_{9} + 6577 \beta_{10} - 889 \beta_{11} - 2043 \beta_{12} + 668 \beta_{13} - 930 \beta_{14} - 1853 \beta_{15} ) q^{49}$$ $$+ ( 429040 \beta_{1} - 2265 \beta_{4} + 1772 \beta_{5} - 429040 \beta_{8} - 39484621 \beta_{9} - 49984 \beta_{10} + 2265 \beta_{11} - 1772 \beta_{12} + 1465 \beta_{13} + 1709 \beta_{14} - 77 \beta_{15} ) q^{50}$$ $$+ ( -2368328 - 6028 \beta_{2} - 1737 \beta_{3} - 1254 \beta_{6} - 1926 \beta_{7} - 887128 \beta_{8} + 2368328 \beta_{9} - 6028 \beta_{10} - 24773 \beta_{11} - 1295 \beta_{12} - 1926 \beta_{13} + 1737 \beta_{14} + 1254 \beta_{15} ) q^{52}$$ $$+ ( 32898504 - 429823 \beta_{1} + 10954 \beta_{2} - 2966 \beta_{3} - 16786 \beta_{4} - 653 \beta_{5} + 1709 \beta_{6} + 1552 \beta_{7} ) q^{53}$$ $$+ ( -668975 + 1497898 \beta_{1} - 12496 \beta_{2} - 2832 \beta_{3} - 12975 \beta_{4} + 5830 \beta_{5} + 2081 \beta_{6} + 3542 \beta_{7} ) q^{55}$$ $$+ ( -58732792 + 53530 \beta_{2} - 3544 \beta_{3} - 5856 \beta_{6} - 3144 \beta_{7} - 273824 \beta_{8} + 58732792 \beta_{9} + 53530 \beta_{10} + 9192 \beta_{11} + 1424 \beta_{12} - 3144 \beta_{13} + 3544 \beta_{14} + 5856 \beta_{15} ) q^{56}$$ $$+ ( -2235223 \beta_{1} - 42441 \beta_{4} - 5694 \beta_{5} + 2235223 \beta_{8} + 7182659 \beta_{9} + 27164 \beta_{10} + 42441 \beta_{11} + 5694 \beta_{12} - 6141 \beta_{13} - 3375 \beta_{14} + 7629 \beta_{15} ) q^{58}$$ $$+ ( -872655 \beta_{1} + 12604 \beta_{4} - 7974 \beta_{5} + 872655 \beta_{8} - 38389595 \beta_{9} + 76010 \beta_{10} - 12604 \beta_{11} + 7974 \beta_{12} - 4209 \beta_{13} - 1440 \beta_{14} + 5133 \beta_{15} ) q^{59}$$ $$+ ( 8943420 - 21509 \beta_{2} - 522 \beta_{3} + 4983 \beta_{6} - 468 \beta_{7} - 2416069 \beta_{8} - 8943420 \beta_{9} - 21509 \beta_{10} + 8799 \beta_{11} + 9597 \beta_{12} - 468 \beta_{13} + 522 \beta_{14} - 4983 \beta_{15} ) q^{61}$$ $$+ ( 57272655 - 685559 \beta_{1} + 66230 \beta_{2} - 4040 \beta_{3} + 30510 \beta_{4} + 181 \beta_{5} - 3143 \beta_{6} - 10729 \beta_{7} ) q^{62}$$ $$+ ( -25410538 + 1622470 \beta_{1} + 65555 \beta_{2} - 2109 \beta_{3} - 19653 \beta_{4} + 3383 \beta_{5} + 9724 \beta_{6} - 2828 \beta_{7} ) q^{64}$$ $$+ ( -60235322 - 179423 \beta_{2} - 1970 \beta_{3} - 77 \beta_{6} - 7324 \beta_{7} - 1178955 \beta_{8} + 60235322 \beta_{9} - 179423 \beta_{10} - 28537 \beta_{11} - 8225 \beta_{12} - 7324 \beta_{13} + 1970 \beta_{14} + 77 \beta_{15} ) q^{65}$$ $$+ ( -2200274 \beta_{1} + 58152 \beta_{4} + 4327 \beta_{5} + 2200274 \beta_{8} + 2108783 \beta_{9} - 78350 \beta_{10} - 58152 \beta_{11} - 4327 \beta_{12} - 5837 \beta_{13} + 6564 \beta_{14} - 1240 \beta_{15} ) q^{67}$$ $$+ ( -586134 \beta_{1} - 24771 \beta_{4} + 3775 \beta_{5} + 586134 \beta_{8} - 114354928 \beta_{9} - 19121 \beta_{10} + 24771 \beta_{11} - 3775 \beta_{12} + 4914 \beta_{13} - 2543 \beta_{14} + 9630 \beta_{15} ) q^{68}$$ $$+ ( -28486201 + 143338 \beta_{2} + 6498 \beta_{3} - 7143 \beta_{6} - 17709 \beta_{7} - 785239 \beta_{8} + 28486201 \beta_{9} + 143338 \beta_{10} + 79320 \beta_{11} - 9729 \beta_{12} - 17709 \beta_{13} - 6498 \beta_{14} + 7143 \beta_{15} ) q^{70}$$ $$+ ( 77152576 + 3082137 \beta_{1} + 181852 \beta_{2} + 16448 \beta_{3} + 11046 \beta_{4} + 5909 \beta_{5} + 10901 \beta_{6} + 3520 \beta_{7} ) q^{71}$$ $$+ ( 37000133 + 2781501 \beta_{1} - 160593 \beta_{2} + 10998 \beta_{3} - 9639 \beta_{4} - 30273 \beta_{5} + 327 \beta_{6} + 6372 \beta_{7} ) q^{73}$$ $$+ ( -128151670 + 80536 \beta_{2} + 16222 \beta_{3} - 13450 \beta_{6} - 5690 \beta_{7} + 2908894 \beta_{8} + 128151670 \beta_{9} + 80536 \beta_{10} + 91010 \beta_{11} + 12028 \beta_{12} - 5690 \beta_{13} - 16222 \beta_{14} + 13450 \beta_{15} ) q^{74}$$ $$+ ( 2836386 \beta_{1} + 15595 \beta_{4} + 16477 \beta_{5} - 2836386 \beta_{8} + 14600804 \beta_{9} + 102083 \beta_{10} - 15595 \beta_{11} - 16477 \beta_{12} + 642 \beta_{13} + 351 \beta_{14} + 11838 \beta_{15} ) q^{76}$$ $$+ ( 3579288 \beta_{1} - 75311 \beta_{4} + 21956 \beta_{5} - 3579288 \beta_{8} - 90306574 \beta_{9} + 244159 \beta_{10} + 75311 \beta_{11} - 21956 \beta_{12} - 10548 \beta_{13} + 11912 \beta_{14} - 9720 \beta_{15} ) q^{77}$$ $$+ ( 4243949 + 59584 \beta_{2} - 8592 \beta_{3} + 13571 \beta_{6} + 29354 \beta_{7} - 3682188 \beta_{8} - 4243949 \beta_{9} + 59584 \beta_{10} - 71675 \beta_{11} - 28060 \beta_{12} + 29354 \beta_{13} + 8592 \beta_{14} - 13571 \beta_{15} ) q^{79}$$ $$+ ( 183192248 - 2972360 \beta_{1} - 395264 \beta_{2} - 10048 \beta_{3} - 118896 \beta_{4} - 9544 \beta_{5} - 18616 \beta_{6} - 5240 \beta_{7} ) q^{80}$$ $$+ ( -11925292 - 5471785 \beta_{1} + 58816 \beta_{2} + 2502 \beta_{3} + 45042 \beta_{4} + 36048 \beta_{5} - 17454 \beta_{6} - 20358 \beta_{7} ) q^{82}$$ $$+ ( -146806607 + 200486 \beta_{2} - 2788 \beta_{3} + 42316 \beta_{6} + 27386 \beta_{7} - 1219435 \beta_{8} + 146806607 \beta_{9} + 200486 \beta_{10} + 7957 \beta_{11} - 12823 \beta_{12} + 27386 \beta_{13} + 2788 \beta_{14} - 42316 \beta_{15} ) q^{83}$$ $$+ ( -1250297 \beta_{1} - 60222 \beta_{4} - 49255 \beta_{5} + 1250297 \beta_{8} + 717820 \beta_{9} + 114466 \beta_{10} + 60222 \beta_{11} + 49255 \beta_{12} + 37580 \beta_{13} + 2238 \beta_{14} - 47405 \beta_{15} ) q^{85}$$ $$+ ( -2991027 \beta_{1} + 248023 \beta_{4} - 17867 \beta_{5} + 2991027 \beta_{8} - 177664332 \beta_{9} - 670332 \beta_{10} - 248023 \beta_{11} + 17867 \beta_{12} + 23752 \beta_{13} - 23141 \beta_{14} - 28334 \beta_{15} ) q^{86}$$ $$+ ( 111239218 - 679735 \beta_{2} + 3447 \beta_{3} - 29616 \beta_{6} + 2208 \beta_{7} + 19393222 \beta_{8} - 111239218 \beta_{9} - 679735 \beta_{10} - 24105 \beta_{11} + 63111 \beta_{12} + 2208 \beta_{13} - 3447 \beta_{14} + 29616 \beta_{15} ) q^{88}$$ $$+ ( 207454586 - 1730401 \beta_{1} - 299520 \beta_{2} - 10078 \beta_{3} + 155228 \beta_{4} - 15991 \beta_{5} + 15471 \beta_{6} + 58896 \beta_{7} ) q^{89}$$ $$+ ( -45489921 - 3719136 \beta_{1} + 490610 \beta_{2} - 22080 \beta_{3} + 168149 \beta_{4} + 17308 \beta_{5} - 41423 \beta_{6} + 26326 \beta_{7} ) q^{91}$$ $$+ ( 9668348 - 52852 \beta_{2} - 44119 \beta_{3} - 5846 \beta_{6} + 38162 \beta_{7} - 1492548 \beta_{8} - 9668348 \beta_{9} - 52852 \beta_{10} - 453491 \beta_{11} + 11915 \beta_{12} + 38162 \beta_{13} + 44119 \beta_{14} + 5846 \beta_{15} ) q^{92}$$ $$+ ( 21842023 \beta_{1} - 53088 \beta_{4} + 39955 \beta_{5} - 21842023 \beta_{8} - 246721889 \beta_{9} - 814862 \beta_{10} + 53088 \beta_{11} - 39955 \beta_{12} + 13633 \beta_{13} - 33918 \beta_{14} + 7517 \beta_{15} ) q^{94}$$ $$+ ( -1974074 \beta_{1} - 35184 \beta_{4} - 48070 \beta_{5} + 1974074 \beta_{8} + 49192268 \beta_{9} - 60040 \beta_{10} + 35184 \beta_{11} + 48070 \beta_{12} - 19268 \beta_{13} + 17936 \beta_{14} - 29846 \beta_{15} ) q^{95}$$ $$+ ( -34072601 + 22850 \beta_{2} - 11406 \beta_{3} + 23539 \beta_{6} + 55048 \beta_{7} + 4570295 \beta_{8} + 34072601 \beta_{9} + 22850 \beta_{10} - 359990 \beta_{11} + 15681 \beta_{12} + 55048 \beta_{13} + 11406 \beta_{14} - 23539 \beta_{15} ) q^{97}$$ $$+ ( -302145943 + 4751942 \beta_{1} + 329232 \beta_{2} - 16669 \beta_{3} + 53993 \beta_{4} + 56060 \beta_{5} - 35179 \beta_{6} - 18719 \beta_{7} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q$$ $$\mathstrut -\mathstrut 15q^{2}$$ $$\mathstrut -\mathstrut 1793q^{4}$$ $$\mathstrut -\mathstrut 453q^{5}$$ $$\mathstrut -\mathstrut 343q^{7}$$ $$\mathstrut +\mathstrut 14478q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$16q$$ $$\mathstrut -\mathstrut 15q^{2}$$ $$\mathstrut -\mathstrut 1793q^{4}$$ $$\mathstrut -\mathstrut 453q^{5}$$ $$\mathstrut -\mathstrut 343q^{7}$$ $$\mathstrut +\mathstrut 14478q^{8}$$ $$\mathstrut +\mathstrut 1020q^{10}$$ $$\mathstrut -\mathstrut 99150q^{11}$$ $$\mathstrut +\mathstrut 32435q^{13}$$ $$\mathstrut -\mathstrut 394824q^{14}$$ $$\mathstrut -\mathstrut 328193q^{16}$$ $$\mathstrut +\mathstrut 831078q^{17}$$ $$\mathstrut -\mathstrut 170554q^{19}$$ $$\mathstrut -\mathstrut 1855164q^{20}$$ $$\mathstrut +\mathstrut 529359q^{22}$$ $$\mathstrut -\mathstrut 1064559q^{23}$$ $$\mathstrut -\mathstrut 2293229q^{25}$$ $$\mathstrut -\mathstrut 2436312q^{26}$$ $$\mathstrut +\mathstrut 1225724q^{28}$$ $$\mathstrut +\mathstrut 1309053q^{29}$$ $$\mathstrut -\mathstrut 2359819q^{31}$$ $$\mathstrut -\mathstrut 5760063q^{32}$$ $$\mathstrut +\mathstrut 981801q^{34}$$ $$\mathstrut +\mathstrut 31066554q^{35}$$ $$\mathstrut +\mathstrut 16391516q^{37}$$ $$\mathstrut -\mathstrut 39490203q^{38}$$ $$\mathstrut -\mathstrut 16760496q^{40}$$ $$\mathstrut -\mathstrut 54747318q^{41}$$ $$\mathstrut +\mathstrut 15249608q^{43}$$ $$\mathstrut +\mathstrut 332509926q^{44}$$ $$\mathstrut +\mathstrut 2390520q^{46}$$ $$\mathstrut -\mathstrut 156295545q^{47}$$ $$\mathstrut +\mathstrut 15239583q^{49}$$ $$\mathstrut -\mathstrut 315590163q^{50}$$ $$\mathstrut -\mathstrut 19773358q^{52}$$ $$\mathstrut +\mathstrut 525516228q^{53}$$ $$\mathstrut -\mathstrut 7579770q^{55}$$ $$\mathstrut -\mathstrut 470339790q^{56}$$ $$\mathstrut +\mathstrut 55408560q^{58}$$ $$\mathstrut -\mathstrut 307774074q^{59}$$ $$\mathstrut +\mathstrut 69192125q^{61}$$ $$\mathstrut +\mathstrut 914436924q^{62}$$ $$\mathstrut -\mathstrut 403588478q^{64}$$ $$\mathstrut -\mathstrut 482470359q^{65}$$ $$\mathstrut +\mathstrut 14328044q^{67}$$ $$\mathstrut -\mathstrut 915409575q^{68}$$ $$\mathstrut -\mathstrut 229271934q^{70}$$ $$\mathstrut +\mathstrut 1239601392q^{71}$$ $$\mathstrut +\mathstrut 598613198q^{73}$$ $$\mathstrut -\mathstrut 1022736000q^{74}$$ $$\mathstrut +\mathstrut 119954093q^{76}$$ $$\mathstrut -\mathstrut 717995541q^{77}$$ $$\mathstrut +\mathstrut 30257531q^{79}$$ $$\mathstrut +\mathstrut 2927826528q^{80}$$ $$\mathstrut -\mathstrut 202376022q^{82}$$ $$\mathstrut -\mathstrut 1176168291q^{83}$$ $$\mathstrut +\mathstrut 4818366q^{85}$$ $$\mathstrut -\mathstrut 1426944009q^{86}$$ $$\mathstrut +\mathstrut 911312427q^{88}$$ $$\mathstrut +\mathstrut 3317041296q^{89}$$ $$\mathstrut -\mathstrut 739230122q^{91}$$ $$\mathstrut +\mathstrut 76813998q^{92}$$ $$\mathstrut -\mathstrut 1954316784q^{94}$$ $$\mathstrut +\mathstrut 391400652q^{95}$$ $$\mathstrut -\mathstrut 267311278q^{97}$$ $$\mathstrut -\mathstrut 4827300318q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut -\mathstrut$$ $$8$$ $$x^{15}\mathstrut +\mathstrut$$ $$1984$$ $$x^{14}\mathstrut -\mathstrut$$ $$13748$$ $$x^{13}\mathstrut +\mathstrut$$ $$1552498$$ $$x^{12}\mathstrut -\mathstrut$$ $$9136628$$ $$x^{11}\mathstrut +\mathstrut$$ $$609566956$$ $$x^{10}\mathstrut -\mathstrut$$ $$2964409064$$ $$x^{9}\mathstrut +\mathstrut$$ $$126210674407$$ $$x^{8}\mathstrut -\mathstrut$$ $$487156186164$$ $$x^{7}\mathstrut +\mathstrut$$ $$13162328064828$$ $$x^{6}\mathstrut -\mathstrut$$ $$37794288146040$$ $$x^{5}\mathstrut +\mathstrut$$ $$578928267028062$$ $$x^{4}\mathstrut -\mathstrut$$ $$1095428523832956$$ $$x^{3}\mathstrut +\mathstrut$$ $$5807598664427172$$ $$x^{2}\mathstrut -\mathstrut$$ $$5266103139591300$$ $$x\mathstrut +\mathstrut$$ $$1336504689748125$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$5356486831063$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$37495407817441$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$10768266373819476$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$64122157941290123$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$8310631469581187951$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$40966264540663762638$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2967296540749851082915$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11624092387537542263167$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$446086543743461225215899$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1297746662449888706706558$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$12093535794335783679581031$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$22037640345037600160817339$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1405524052550259266715171108$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1416328506724533399940110039$$ $$\nu\mathstrut +\mathstrut$$ $$1381004908806684163861395225$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$5356486831063$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$37495407817441$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$10768266373819476$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$64122157941290123$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$8310631469581187951$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$40966264540663762638$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2967296540749851082915$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11624092387537542263167$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$446086543743461225215899$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1297746662449888706706558$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$12093535794335783679581031$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$22037640345037600160817339$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$10869389437978694340188911908$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$10880193892152968473413850839$$ $$\nu\mathstrut +\mathstrut$$ $$2313718780748487653782612064025$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$3190978127243561323583$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$22336846890704929265081$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$8034793659454601645533716$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$47918382947148445792756243$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$7613188257612656606439500791$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$37627221805898650746578056158$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$3433021618005421630100984433515$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13506849330778968857169651171047$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$755556202687840422931447079182659$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$2219552143593417965105720831438478$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$73267616574320149625797901748579471$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$142851662575494040836135985274431299$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2418308188384789405385919011757748572$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$2347250042561060879643728140159184001$$ $$\nu\mathstrut +\mathstrut$$ $$22878652635112254000161226174555843375$$$$)/$$$$37\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$170744966576564209457$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$1195214766035949466199$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$331342322691927499875564$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$1972516144193097656192797$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$252943029675077190426543289$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$1246662236338873080413226882$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$96456395276061641511616706885$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$378367256530739010740841682313$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$19187660465639707264540628322861$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$56243910334051285861834688612562$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1863830909549477931316684349261409$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3634360914506626043761272576970221$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$69613029521953215370734358268785188$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$67805160360904996047226727227553679$$ $$\nu\mathstrut -\mathstrut$$ $$364543962387371302108247838654631425$$$$)/$$$$15\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$13842211326319491175153$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$96895479284236438226071$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$23388131431185117020247756$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$139069147356415628424547613$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$14234945547803167286564078681$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$69902236563838300807373094978$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$3558946115011529805080589508165$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13817896842411570660016313392777$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$213787630780965051289452660952269$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$593292317184217918143522382855698$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$41208308833910348289470104499105439$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$83389456358503798877550175176755091$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$4352056692332435826519811709462858748$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$4310265364228028114067762593391364209$$ $$\nu\mathstrut -\mathstrut$$ $$20096039530342015492593793091239338975$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$45240689715040240238087$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$316684828005281681666609$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$86315655478842843602442324$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$513777030108988399752988027$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$64005415700831974403336165599$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$315325003383228270899024825262$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$23372239808966679816363734761235$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$91602647824061290010949364868783$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$4422629561233404548398280196955851$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$12948598143368656611448480578722142$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$425132950264679872327566516962135319$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$828791145548445449677178304707754411$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$18949720133548220927513242764023823708$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$18537467487526984502325653896985696889$$ $$\nu\mathstrut -\mathstrut$$ $$183190395189927280496285555295598351575$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$34242099001410319729021$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$239694693009872238103147$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$65250506334015183384928092$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$388387006994962761214227641$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$48260064231162871651345662917$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$237745819648543934900606019546$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$17462196448561967221364572423105$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$68426573340193235463076567579189$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$3193007628343364739514692382513833$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$9340524148807767164483037364417386$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$269914513390685287692917146838053677$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$524340844174883711922280974371565513$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$7215721027983503613859047629621664564$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6955096026492851785300454456103062787$$ $$\nu\mathstrut +\mathstrut$$ $$30227210200152465769576633682327757075$$$$)/$$$$55\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$963182093599559192897$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$6219671132668440846766$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$1899809881580565193484573$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$10266138112371853394043841$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$1476165301545136250039046596$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$6492116909861801572498815841$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$574639850081881823738428361027$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$2003978922330222716974351811788$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$117826386375574617650129475694124$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$324498714302536863306620773508253$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$12188548381532347005066996889008591$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$27798008711725172814887799708592500$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$537315675233907184908147851931934479$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1045805110644539559188578543458690267$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$3825677001573191545260597486690017994$$ $$\nu\mathstrut +\mathstrut$$ $$1909585594937825490821755815793200375$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$1926364187199118385794$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$14447731403993387893455$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$3813678487131725930368607$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$24569786240062318831011878$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$2976372899770566430271445375$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$16100265372735276438727863853$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$1164639805275507285445853175052$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$5120531501904433853949357622791$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$240011170640318916729918942363155$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$816255446334091963006747570285185$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24863680672218979932695488349476500$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$60130430874528986589724302272887851$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1082894258397303009061902996575855877$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$1564615694784806544011339311775505666$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$10668739496637667697066532009995579169$$ $$\nu\mathstrut -\mathstrut$$ $$4484180365292282197649888246231336625$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!69$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$10\!\cdots\!79$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$27\!\cdots\!20$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$17\!\cdots\!81$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$21\!\cdots\!29$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$11\!\cdots\!46$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$85\!\cdots\!81$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$37\!\cdots\!41$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$17\!\cdots\!81$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$59\!\cdots\!62$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$18\!\cdots\!69$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$44\!\cdots\!29$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$79\!\cdots\!44$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$11\!\cdots\!07$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$78\!\cdots\!77$$ $$\nu\mathstrut +\mathstrut$$ $$32\!\cdots\!00$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$16\!\cdots\!35$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$12\!\cdots\!56$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$31\!\cdots\!57$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$20\!\cdots\!47$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$24\!\cdots\!26$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$13\!\cdots\!17$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$97\!\cdots\!01$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$43\!\cdots\!10$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$20\!\cdots\!34$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$70\!\cdots\!13$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$20\!\cdots\!01$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$52\!\cdots\!82$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$89\!\cdots\!83$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$13\!\cdots\!69$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$84\!\cdots\!32$$ $$\nu\mathstrut -\mathstrut$$ $$44\!\cdots\!75$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$25\!\cdots\!04$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$18\!\cdots\!17$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$50\!\cdots\!51$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$31\!\cdots\!92$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$39\!\cdots\!77$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$20\!\cdots\!77$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15\!\cdots\!14$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$65\!\cdots\!61$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$31\!\cdots\!53$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$10\!\cdots\!61$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$32\!\cdots\!82$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$80\!\cdots\!45$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14\!\cdots\!73$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$21\!\cdots\!64$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$13\!\cdots\!63$$ $$\nu\mathstrut -\mathstrut$$ $$68\!\cdots\!25$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!88$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$27\!\cdots\!01$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$69\!\cdots\!31$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$47\!\cdots\!08$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$54\!\cdots\!61$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$31\!\cdots\!93$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$21\!\cdots\!90$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10\!\cdots\!57$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$44\!\cdots\!69$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$16\!\cdots\!13$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$46\!\cdots\!46$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!29$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$20\!\cdots\!57$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$32\!\cdots\!76$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$21\!\cdots\!35$$ $$\nu\mathstrut +\mathstrut$$ $$91\!\cdots\!25$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$42\!\cdots\!51$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$31\!\cdots\!18$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$84\!\cdots\!59$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$53\!\cdots\!63$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$65\!\cdots\!08$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$35\!\cdots\!03$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$25\!\cdots\!61$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$11\!\cdots\!84$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$52\!\cdots\!12$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$17\!\cdots\!19$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$54\!\cdots\!53$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$12\!\cdots\!20$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$23\!\cdots\!97$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$32\!\cdots\!21$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$22\!\cdots\!62$$ $$\nu\mathstrut -\mathstrut$$ $$94\!\cdots\!25$$$$)/$$$$41\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!21$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$81\!\cdots\!94$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$21\!\cdots\!01$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$13\!\cdots\!05$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$17\!\cdots\!04$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$91\!\cdots\!45$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$67\!\cdots\!47$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$29\!\cdots\!44$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13\!\cdots\!96$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$46\!\cdots\!17$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$14\!\cdots\!19$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$34\!\cdots\!12$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$62\!\cdots\!35$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$87\!\cdots\!35$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$60\!\cdots\!54$$ $$\nu\mathstrut +\mathstrut$$ $$25\!\cdots\!75$$$$)/$$$$41\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{9}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$732$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$2$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$650$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2459$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1225$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3619$$$$)/9$$ $$\nu^{4}$$ $$=$$ $$($$$$4$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1297$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$4924$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$20$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$61$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1673$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5254$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$897597$$$$)/9$$ $$\nu^{5}$$ $$=$$ $$($$$$260$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$4609$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$20$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3301$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$5177$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18299$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3061305$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3434573$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$160$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$160$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1553$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3061$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2372$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$21742$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1696324$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$8283966$$$$)/27$$ $$\nu^{6}$$ $$=$$ $$($$$$260$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$4619$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$20$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3311$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$5187$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$18314$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3064546$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3446886$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$15844$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$372$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$13558$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$54082$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$6371$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$884856$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$4557954$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$416029317$$$$)/9$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$217192$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$2838597$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$60904$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1616397$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$3534069$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$16044840$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$3664900239$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1700973318$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$197444$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$113972$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$682881$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2363781$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1510680$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$17453646$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$820650909$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$6240494943$$$$)/27$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$872408$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$11419082$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$243896$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$6511970$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$14208922$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$64435798$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$14702513673$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$6852184156$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$29613428$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2123452$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$27578497$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$105743203$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$12855947$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1393099177$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$10484928722$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$611798904726$$$$)/27$$ $$\nu^{9}$$ $$=$$ $$($$$$139152644$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1628842083$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$62926964$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$803003703$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2107275867$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$11535326505$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3199116483192$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$871249976205$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$166275256$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$62016088$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$291298233$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1565034885$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$898514310$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$12238230366$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$403681120635$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$4412210621253$$$$)/27$$ $$\nu^{10}$$ $$=$$ $$($$$$702311740$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$8229950589$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$316464460$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$4063927881$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$10643055237$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$58160285694$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$16105915643415$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$4407713720688$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$16986252316$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2103348596$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17056178094$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$61045770666$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$7901098473$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$730801936887$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$7312986626220$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$308392716149673$$$$)/27$$ $$\nu^{11}$$ $$=$$ $$($$$$-$$$$81064649544$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$907087393079$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$52356195816$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$413813942399$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1160396803111$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$7614270716503$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2363188262583039$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$455273967622513$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$121710376068$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$30826022100$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$123262213954$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$949912916918$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$520403963161$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$8045904210740$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$199243622889908$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2947876547781504$$$$)/27$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$494127738296$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$5533242534858$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$317622309560$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$2527694527314$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$7079689216026$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$46326451841340$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$14356537441375596$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2780241945439524$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$9366423352040$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1424824399864$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$10133907926244$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$33180779267148$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4705092327594$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$383666900310720$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$4796162524664940$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$157825132383954063$$$$)/27$$ $$\nu^{13}$$ $$=$$ $$($$$$14959139996928$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$166029476861446$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$13187437396064$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$73452664879366$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$202228149215110$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1598371224283296$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$530417264916344769$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$80318650692450198$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$27646773282568$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$4885634882120$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$17144529841390$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$182281593468854$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$99493848361760$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1699070417689220$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$32656276570103239$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$630392297080274195$$$$)/9$$ $$\nu^{14}$$ $$=$$ $$($$$$321659706889952$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$3570815283072724$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$281764033821056$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1580978729125636$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$4354522560733268$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$34270358790402812$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$11357041833973044279$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1729006229062488206$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$5051435949109352$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$842469541014040$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5895687763494122$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$17444745779202638$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$2762070405598318$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$201517958377799309$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3021976542102457018$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$81469375415446412352$$$$)/27$$ $$\nu^{15}$$ $$=$$ $$($$$$-$$$$8008982644022936$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$90546390925340368$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$9352862502209144$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$40029781508985688$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$101850517331209824$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$976934360147824111$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$337822352899425184334$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$42874127580991539567$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$18027108894576960$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2254850726372960$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$6907940213042681$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$101406335587943713$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$56931292398587519$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1050912554298351990$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$15911644964328716013$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$393198532068822377367$$$$)/9$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 − 22.2094i 0.5 − 19.2639i 0.5 − 9.36376i 0.5 − 3.55897i 0.5 − 0.103648i 0.5 + 13.2694i 0.5 + 15.6774i 0.5 + 23.8209i 0.5 + 22.2094i 0.5 + 19.2639i 0.5 + 9.36376i 0.5 + 3.55897i 0.5 + 0.103648i 0.5 − 13.2694i 0.5 − 15.6774i 0.5 − 23.8209i
−19.9839 34.6131i 0 −542.712 + 940.005i −1103.58 + 1911.45i 0 −2172.76 3763.33i 22918.5 0 88215.0
10.2 −17.4330 30.1949i 0 −351.819 + 609.369i 1003.26 1737.70i 0 −2433.94 4215.71i 6681.68 0 −69959.4
10.3 −8.85925 15.3447i 0 99.0272 171.520i −369.939 + 640.753i 0 3013.67 + 5219.83i −12581.1 0 13109.5
10.4 −3.83216 6.63749i 0 226.629 392.533i 1105.48 1914.74i 0 3116.58 + 5398.07i −7398.05 0 −16945.4
10.5 −0.839762 1.45451i 0 254.590 440.962i −769.303 + 1332.47i 0 −2828.47 4899.05i −1715.10 0 2584.13
10.6 10.7416 + 18.6050i 0 25.2357 43.7094i −22.4201 + 38.8327i 0 −4672.78 8093.49i 12083.7 0 −963.310
10.7 12.8270 + 22.2170i 0 −73.0637 + 126.550i 353.226 611.805i 0 2284.21 + 3956.36i 9386.09 0 18123.3
10.8 19.8795 + 34.4322i 0 −534.387 + 925.585i −423.223 + 733.045i 0 3521.99 + 6100.27i −22136.7 0 −33653.8
19.1 −19.9839 + 34.6131i 0 −542.712 940.005i −1103.58 1911.45i 0 −2172.76 + 3763.33i 22918.5 0 88215.0
19.2 −17.4330 + 30.1949i 0 −351.819 609.369i 1003.26 + 1737.70i 0 −2433.94 + 4215.71i 6681.68 0 −69959.4
19.3 −8.85925 + 15.3447i 0 99.0272 + 171.520i −369.939 640.753i 0 3013.67 5219.83i −12581.1 0 13109.5
19.4 −3.83216 + 6.63749i 0 226.629 + 392.533i 1105.48 + 1914.74i 0 3116.58 5398.07i −7398.05 0 −16945.4
19.5 −0.839762 + 1.45451i 0 254.590 + 440.962i −769.303 1332.47i 0 −2828.47 + 4899.05i −1715.10 0 2584.13
19.6 10.7416 18.6050i 0 25.2357 + 43.7094i −22.4201 38.8327i 0 −4672.78 + 8093.49i 12083.7 0 −963.310
19.7 12.8270 22.2170i 0 −73.0637 126.550i 353.226 + 611.805i 0 2284.21 3956.36i 9386.09 0 18123.3
19.8 19.8795 34.4322i 0 −534.387 925.585i −423.223 733.045i 0 3521.99 6100.27i −22136.7 0 −33653.8
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.8 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
9.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{10}^{\mathrm{new}}(27, [\chi])$$.