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

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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
0.500000 22.2094i
0.500000 19.2639i
0.500000 9.36376i
0.500000 3.55897i
0.500000 0.103648i
0.500000 + 13.2694i
0.500000 + 15.6774i
0.500000 + 23.8209i
0.500000 + 22.2094i
0.500000 + 19.2639i
0.500000 + 9.36376i
0.500000 + 3.55897i
0.500000 + 0.103648i
0.500000 13.2694i
0.500000 15.6774i
0.500000 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:

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 \(S_{10}^{\mathrm{new}}(27, [\chi])\).