Properties

Label 5.9.c.a
Level 5
Weight 9
Character orbit 5.c
Analytic conductor 2.037
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.03689305031\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( -12 - 12 \beta_{1} - \beta_{2} - \beta_{5} ) q^{3} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( 48 + 39 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 20 \beta_{5} ) q^{5} + ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6} + ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7} + ( -1326 - 1326 \beta_{1} + 2 \beta_{2} - 130 \beta_{5} ) q^{8} + ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( -12 - 12 \beta_{1} - \beta_{2} - \beta_{5} ) q^{3} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( 48 + 39 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} - 4 \beta_{4} - 20 \beta_{5} ) q^{5} + ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6} + ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7} + ( -1326 - 1326 \beta_{1} + 2 \beta_{2} - 130 \beta_{5} ) q^{8} + ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9} + ( 5308 - 4006 \beta_{1} - 37 \beta_{2} - 190 \beta_{3} + 41 \beta_{4} - 295 \beta_{5} ) q^{10} + ( 3792 - 85 \beta_{2} + 25 \beta_{3} - 85 \beta_{4} + 25 \beta_{5} ) q^{11} + ( -7596 + 7596 \beta_{1} - 76 \beta_{3} + 92 \beta_{4} ) q^{12} + ( -20015 - 20015 \beta_{1} + 80 \beta_{2} + 554 \beta_{5} ) q^{13} + ( 31626 \beta_{1} + 77 \beta_{2} + 1260 \beta_{3} - 77 \beta_{4} - 1260 \beta_{5} ) q^{14} + ( 39246 - 22122 \beta_{1} + 181 \beta_{2} + 1470 \beta_{3} - 108 \beta_{4} + 1585 \beta_{5} ) q^{15} + ( 37132 + 374 \beta_{2} - 670 \beta_{3} + 374 \beta_{4} - 670 \beta_{5} ) q^{16} + ( -43617 + 43617 \beta_{1} - 2366 \beta_{3} - 466 \beta_{4} ) q^{17} + ( -75822 - 75822 \beta_{1} - 606 \beta_{2} - 171 \beta_{5} ) q^{18} + ( 54264 \beta_{1} - 684 \beta_{2} + 930 \beta_{3} + 684 \beta_{4} - 930 \beta_{5} ) q^{19} + ( 68634 - 38088 \beta_{1} - 451 \beta_{2} - 2745 \beta_{3} - 307 \beta_{4} + 2215 \beta_{5} ) q^{20} + ( 40068 - 511 \beta_{2} - 2695 \beta_{3} - 511 \beta_{4} - 2695 \beta_{5} ) q^{21} + ( -6310 + 6310 \beta_{1} + 2068 \beta_{3} + 970 \beta_{4} ) q^{22} + ( 4722 + 4722 \beta_{1} + 1581 \beta_{2} + 1859 \beta_{5} ) q^{23} + ( 49272 \beta_{1} + 2268 \beta_{2} - 6060 \beta_{3} - 2268 \beta_{4} + 6060 \beta_{5} ) q^{24} + ( -51065 - 12420 \beta_{1} + 785 \beta_{2} + 4825 \beta_{3} + 2245 \beta_{4} - 18775 \beta_{5} ) q^{25} + ( -147684 - 1034 \beta_{2} + 20145 \beta_{3} - 1034 \beta_{4} + 20145 \beta_{5} ) q^{26} + ( 58572 - 58572 \beta_{1} + 20460 \beta_{3} + 156 \beta_{4} ) q^{27} + ( 223748 + 223748 \beta_{1} - 196 \beta_{2} - 18844 \beta_{5} ) q^{28} + ( -451044 \beta_{1} - 2386 \beta_{2} - 15980 \beta_{3} + 2386 \beta_{4} + 15980 \beta_{5} ) q^{29} + ( -422334 + 390588 \beta_{1} - 1849 \beta_{2} - 26380 \beta_{3} - 3493 \beta_{4} + 13160 \beta_{5} ) q^{30} + ( -99628 + 3045 \beta_{2} - 34425 \beta_{3} + 3045 \beta_{4} - 34425 \beta_{5} ) q^{31} + ( 516180 - 516180 \beta_{1} - 35236 \beta_{3} - 3660 \beta_{4} ) q^{32} + ( 697956 + 697956 \beta_{1} - 6862 \beta_{2} + 33818 \beta_{5} ) q^{33} + ( -631220 \beta_{1} - 5162 \beta_{2} + 47165 \beta_{3} + 5162 \beta_{4} - 47165 \beta_{5} ) q^{34} + ( -854112 + 233184 \beta_{1} + 2968 \beta_{2} + 47285 \beta_{3} - 3549 \beta_{4} + 28630 \beta_{5} ) q^{35} + ( -674778 + 3039 \beta_{2} + 22005 \beta_{3} + 3039 \beta_{4} + 22005 \beta_{5} ) q^{36} + ( -72653 + 72653 \beta_{1} - 7536 \beta_{3} + 1506 \beta_{4} ) q^{37} + ( 250116 + 250116 \beta_{1} + 10068 \beta_{2} - 18420 \beta_{5} ) q^{38} + ( -375300 \beta_{1} + 17417 \beta_{2} + 20785 \beta_{3} - 17417 \beta_{4} - 20785 \beta_{5} ) q^{39} + ( 438150 + 627450 \beta_{1} + 6400 \beta_{2} + 29250 \beta_{3} + 14550 \beta_{4} + 19000 \beta_{5} ) q^{40} + ( 422352 - 12505 \beta_{2} - 23675 \beta_{3} - 12505 \beta_{4} - 23675 \beta_{5} ) q^{41} + ( 718914 - 718914 \beta_{1} - 33292 \beta_{3} + 11522 \beta_{4} ) q^{42} + ( 105376 + 105376 \beta_{1} + 5323 \beta_{2} + 85519 \beta_{5} ) q^{43} + ( 1524720 \beta_{1} - 13872 \beta_{2} - 21760 \beta_{3} + 13872 \beta_{4} + 21760 \beta_{5} ) q^{44} + ( 75771 - 2545872 \beta_{1} - 33219 \beta_{2} - 81405 \beta_{3} - 6258 \beta_{4} - 80040 \beta_{5} ) q^{45} + ( -500818 - 11345 \beta_{2} - 47600 \beta_{3} - 11345 \beta_{4} - 47600 \beta_{5} ) q^{46} + ( -2583906 + 2583906 \beta_{1} + 90779 \beta_{3} - 4263 \beta_{4} ) q^{47} + ( -3565608 - 3565608 \beta_{1} - 15784 \beta_{2} - 157816 \beta_{5} ) q^{48} + ( 1195649 \beta_{1} - 8575 \beta_{2} - 219275 \beta_{3} + 8575 \beta_{4} + 219275 \beta_{5} ) q^{49} + ( 4991010 + 1292430 \beta_{1} + 32360 \beta_{2} - 118675 \beta_{3} - 4230 \beta_{4} - 57400 \beta_{5} ) q^{50} + ( 5798544 + 52723 \beta_{2} + 246835 \beta_{3} + 52723 \beta_{4} + 246835 \beta_{5} ) q^{51} + ( -230594 + 230594 \beta_{1} + 275554 \beta_{3} - 48362 \beta_{4} ) q^{52} + ( -2162457 - 2162457 \beta_{1} - 4186 \beta_{2} - 271376 \beta_{5} ) q^{53} + ( 5442984 \beta_{1} + 21396 \beta_{2} + 38580 \beta_{3} - 21396 \beta_{4} - 38580 \beta_{5} ) q^{54} + ( 710936 - 5132052 \beta_{1} + 41621 \beta_{2} + 160645 \beta_{3} - 25703 \beta_{4} - 53515 \beta_{5} ) q^{55} + ( -3082968 + 308 \beta_{2} + 11060 \beta_{3} + 308 \beta_{4} + 11060 \beta_{5} ) q^{56} + ( -5636916 + 5636916 \beta_{1} - 338880 \beta_{3} + 63132 \beta_{4} ) q^{57} + ( -4241136 - 4241136 \beta_{1} - 3328 \beta_{2} + 768320 \beta_{5} ) q^{58} + ( -1372608 \beta_{1} - 30152 \beta_{2} + 508490 \beta_{3} + 30152 \beta_{4} - 508490 \beta_{5} ) q^{59} + ( 2170068 + 3015924 \beta_{1} - 77052 \beta_{2} + 126760 \beta_{3} - 1064 \beta_{4} + 129180 \beta_{5} ) q^{60} + ( 4266032 - 95625 \beta_{2} - 466875 \beta_{3} - 95625 \beta_{4} - 466875 \beta_{5} ) q^{61} + ( 9144870 - 9144870 \beta_{1} - 445592 \beta_{3} + 32310 \beta_{4} ) q^{62} + ( 7604394 + 7604394 \beta_{1} + 27237 \beta_{2} - 367101 \beta_{5} ) q^{63} + ( 118376 \beta_{1} + 38548 \beta_{2} - 400060 \beta_{3} - 38548 \beta_{4} + 400060 \beta_{5} ) q^{64} + ( -5227299 - 2571057 \beta_{1} - 12939 \beta_{2} - 362305 \beta_{3} + 137927 \beta_{4} + 697385 \beta_{5} ) q^{65} + ( -8968140 + 7354 \beta_{2} - 302420 \beta_{3} + 7354 \beta_{4} - 302420 \beta_{5} ) q^{66} + ( -5539832 + 5539832 \beta_{1} + 101959 \beta_{3} - 103661 \beta_{4} ) q^{67} + ( 1400586 + 1400586 \beta_{1} + 36978 \beta_{2} - 105434 \beta_{5} ) q^{68} + ( -14569164 \beta_{1} + 7561 \beta_{2} + 447455 \beta_{3} - 7561 \beta_{4} - 447455 \beta_{5} ) q^{69} + ( -7627452 + 12563614 \beta_{1} - 20447 \beta_{2} + 1252860 \beta_{3} - 72429 \beta_{4} - 541520 \beta_{5} ) q^{70} + ( -2912988 + 78125 \beta_{2} + 949375 \beta_{3} + 78125 \beta_{4} + 949375 \beta_{5} ) q^{71} + ( 13544946 - 13544946 \beta_{1} + 738030 \beta_{3} + 74658 \beta_{4} ) q^{72} + ( 18480997 + 18480997 \beta_{1} + 12756 \beta_{2} + 499584 \beta_{5} ) q^{73} + ( -1998552 \beta_{1} + 1500 \beta_{2} - 14725 \beta_{3} - 1500 \beta_{4} + 14725 \beta_{5} ) q^{74} + ( -20653380 - 1097340 \beta_{1} + 197195 \beta_{2} - 1296600 \beta_{3} - 154260 \beta_{4} - 96925 \beta_{5} ) q^{75} + ( -9032136 + 133116 \beta_{2} - 436380 \beta_{3} + 133116 \beta_{4} - 436380 \beta_{5} ) q^{76} + ( 4353552 - 4353552 \beta_{1} - 41342 \beta_{3} - 110754 \beta_{4} ) q^{77} + ( 5459142 + 5459142 \beta_{1} - 167434 \beta_{2} - 982072 \beta_{5} ) q^{78} + ( -22879824 \beta_{1} - 122456 \beta_{2} - 1313680 \beta_{3} + 122456 \beta_{4} + 1313680 \beta_{5} ) q^{79} + ( 4670928 + 25409004 \beta_{1} - 19442 \beta_{2} - 1455290 \beta_{3} - 58494 \beta_{4} - 1112470 \beta_{5} ) q^{80} + ( 10214919 - 179613 \beta_{2} + 956565 \beta_{3} - 179613 \beta_{4} + 956565 \beta_{5} ) q^{81} + ( 6347570 - 6347570 \beta_{1} + 166228 \beta_{3} + 197410 \beta_{4} ) q^{82} + ( -2441256 - 2441256 \beta_{1} - 212813 \beta_{2} - 273261 \beta_{5} ) q^{83} + ( 1447824 \beta_{1} - 94976 \beta_{2} - 575680 \beta_{3} + 94976 \beta_{4} + 575680 \beta_{5} ) q^{84} + ( -3531827 - 25557811 \beta_{1} - 286147 \beta_{2} + 1708485 \beta_{3} + 200271 \beta_{4} - 1055395 \beta_{5} ) q^{85} + ( -22769346 - 117457 \beta_{2} + 146560 \beta_{3} - 117457 \beta_{4} + 146560 \beta_{5} ) q^{86} + ( -22156764 + 22156764 \beta_{1} - 284520 \beta_{3} - 150972 \beta_{4} ) q^{87} + ( -7348032 - 7348032 \beta_{1} + 371264 \beta_{2} + 137840 \beta_{5} ) q^{88} + ( 38523288 \beta_{1} + 442572 \beta_{2} + 666960 \beta_{3} - 442572 \beta_{4} - 666960 \beta_{5} ) q^{89} + ( 21423516 - 21678762 \beta_{1} + 160401 \beta_{2} + 419745 \beta_{3} + 398307 \beta_{4} + 3442410 \beta_{5} ) q^{90} + ( 29991584 + 224203 \beta_{2} - 2877665 \beta_{3} + 224203 \beta_{4} - 2877665 \beta_{5} ) q^{91} + ( 11498148 - 11498148 \beta_{1} + 297684 \beta_{3} - 173396 \beta_{4} ) q^{92} + ( -16384884 - 16384884 \beta_{1} + 679018 \beta_{2} + 1434658 \beta_{5} ) q^{93} + ( 24130162 \beta_{1} + 65201 \beta_{2} + 3178480 \beta_{3} - 65201 \beta_{4} - 3178480 \beta_{5} ) q^{94} + ( 38898900 + 17396700 \beta_{1} - 213600 \beta_{2} + 1454250 \beta_{3} - 410700 \beta_{4} + 1055250 \beta_{5} ) q^{95} + ( 29428560 - 328088 \beta_{2} + 1746040 \beta_{3} - 328088 \beta_{4} + 1746040 \beta_{5} ) q^{96} + ( -31017911 + 31017911 \beta_{1} + 586904 \beta_{3} + 861272 \beta_{4} ) q^{97} + ( -58292850 - 58292850 \beta_{1} - 335650 \beta_{2} + 1563051 \beta_{5} ) q^{98} + ( 9802896 \beta_{1} - 486789 \beta_{2} + 615855 \beta_{3} + 486789 \beta_{4} - 615855 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + O(q^{10}) \) \( 6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + 30870q^{10} + 23192q^{11} - 45912q^{12} - 119142q^{13} + 241440q^{15} + 218616q^{16} - 265502q^{17} - 454062q^{18} + 412260q^{20} + 231672q^{21} - 35664q^{22} + 28888q^{23} - 340350q^{25} - 801388q^{26} + 392040q^{27} + 1305192q^{28} - 2549760q^{30} - 747648q^{31} + 3033928q^{32} + 4269096q^{33} - 4971680q^{35} - 3972804q^{36} - 454002q^{37} + 1443720q^{38} + 2683500q^{40} + 2489432q^{41} + 4223856q^{42} + 792648q^{43} + 210690q^{45} - 3149928q^{46} - 15313352q^{47} - 21677712q^{48} + 29537650q^{50} + 35567712q^{51} - 735732q^{52} - 13509122q^{53} + 4448040q^{55} - 18454800q^{56} - 34625520q^{57} - 23903520q^{58} + 13688520q^{60} + 24111192q^{61} + 53913416q^{62} + 44837688q^{63} - 30943610q^{65} - 55047936q^{66} - 32827752q^{67} + 8118692q^{68} - 44156280q^{70} - 13992928q^{71} + 82596420q^{72} + 111859638q^{73} - 126793200q^{75} - 56470800q^{76} + 26260136q^{77} + 31125576q^{78} + 23045920q^{80} + 65834226q^{81} + 38023056q^{82} - 14768432q^{83} - 19713030q^{85} - 135560008q^{86} - 133207680q^{87} - 44555040q^{88} + 135147990q^{90} + 167542032q^{91} + 69931048q^{92} - 96798024q^{93} + 239661000q^{95} + 184867872q^{96} - 186656202q^{97} - 345959698q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 39 \nu^{5} - 22 \nu^{4} - 10 \nu^{3} + 790 \nu^{2} + 41631 \nu - 62928 \)\()/66000\)
\(\beta_{2}\)\(=\)\((\)\( -273 \nu^{5} + 154 \nu^{4} + 70 \nu^{3} - 5530 \nu^{2} + 1028583 \nu - 21504 \)\()/66000\)
\(\beta_{3}\)\(=\)\((\)\( 761 \nu^{5} - 2178 \nu^{4} - 4990 \nu^{3} - 45790 \nu^{2} + 838569 \nu - 2347872 \)\()/66000\)
\(\beta_{4}\)\(=\)\((\)\( -77 \nu^{5} + 146 \nu^{4} - 3570 \nu^{3} + 2030 \nu^{2} - 83733 \nu + 244704 \)\()/6000\)
\(\beta_{5}\)\(=\)\((\)\( -921 \nu^{5} - 1342 \nu^{4} - 610 \nu^{3} + 48190 \nu^{2} - 823209 \nu + 187392 \)\()/66000\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 7 \beta_{1} + 7\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(10 \beta_{5} + \beta_{4} - 10 \beta_{3} - \beta_{2} + 446 \beta_{1}\)\()/20\)
\(\nu^{3}\)\(=\)\((\)\(-31 \beta_{4} - 20 \beta_{3} - 283 \beta_{1} + 283\)\()/20\)
\(\nu^{4}\)\(=\)\((\)\(-35 \beta_{5} + 5 \beta_{4} - 35 \beta_{3} + 5 \beta_{2} - 1348\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-400 \beta_{5} - 1019 \beta_{2} + 17267 \beta_{1} + 17267\)\()/20\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
3.70505 3.70505i
−4.23471 + 4.23471i
1.52966 1.52966i
3.70505 + 3.70505i
−4.23471 4.23471i
1.52966 + 1.52966i
−11.8649 11.8649i −90.9660 + 90.9660i 25.5528i −434.373 449.383i 2158.61 508.219 + 508.219i −2734.24 + 2734.24i 9988.61i −178.084 + 10485.7i
2.2 −4.39608 4.39608i 75.2981 75.2981i 217.349i −14.1685 + 624.839i −662.032 730.992 + 730.992i −2080.88 + 2080.88i 4778.60i 2809.13 2684.56i
2.3 15.2610 + 15.2610i −20.3321 + 20.3321i 209.796i 558.542 280.457i −620.576 −2415.21 2415.21i 705.116 705.116i 5734.21i 12804.0 + 4243.86i
3.1 −11.8649 + 11.8649i −90.9660 90.9660i 25.5528i −434.373 + 449.383i 2158.61 508.219 508.219i −2734.24 2734.24i 9988.61i −178.084 10485.7i
3.2 −4.39608 + 4.39608i 75.2981 + 75.2981i 217.349i −14.1685 624.839i −662.032 730.992 730.992i −2080.88 2080.88i 4778.60i 2809.13 + 2684.56i
3.3 15.2610 15.2610i −20.3321 20.3321i 209.796i 558.542 + 280.457i −620.576 −2415.21 + 2415.21i 705.116 + 705.116i 5734.21i 12804.0 4243.86i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.9.c.a 6
3.b odd 2 1 45.9.g.a 6
4.b odd 2 1 80.9.p.c 6
5.b even 2 1 25.9.c.b 6
5.c odd 4 1 inner 5.9.c.a 6
5.c odd 4 1 25.9.c.b 6
15.e even 4 1 45.9.g.a 6
20.e even 4 1 80.9.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.9.c.a 6 1.a even 1 1 trivial
5.9.c.a 6 5.c odd 4 1 inner
25.9.c.b 6 5.b even 2 1
25.9.c.b 6 5.c odd 4 1
45.9.g.a 6 3.b odd 2 1
45.9.g.a 6 15.e even 4 1
80.9.p.c 6 4.b odd 2 1
80.9.p.c 6 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 2912 T^{3} - 51136 T^{4} - 1135744 T^{5} + 2070656 T^{6} - 290750464 T^{7} - 3351248896 T^{8} + 48855252992 T^{9} + 8589934592 T^{10} + 2199023255552 T^{11} + 281474976710656 T^{12} \)
$3$ \( 1 + 72 T + 2592 T^{2} + 88992 T^{3} - 43043121 T^{4} - 3659977224 T^{5} - 147990802464 T^{6} - 24013110566664 T^{7} - 1852865220656241 T^{8} + 25133969310517152 T^{9} + 4803028329503971872 T^{10} + \)\(87\!\cdots\!72\)\( T^{11} + \)\(79\!\cdots\!61\)\( T^{12} \)
$5$ \( 1 - 220 T + 194375 T^{2} - 199375000 T^{3} + 75927734375 T^{4} - 33569335937500 T^{5} + 59604644775390625 T^{6} \)
$7$ \( 1 + 2352 T + 2765952 T^{2} - 2361533048 T^{3} + 6289666320399 T^{4} + 128493496173721896 T^{5} + \)\(28\!\cdots\!96\)\( T^{6} + \)\(74\!\cdots\!96\)\( T^{7} + \)\(20\!\cdots\!99\)\( T^{8} - \)\(45\!\cdots\!48\)\( T^{9} + \)\(30\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} \)
$11$ \( ( 1 - 11596 T + 493098215 T^{2} - 5106545516920 T^{3} + 105699981590497415 T^{4} - \)\(53\!\cdots\!56\)\( T^{5} + \)\(98\!\cdots\!41\)\( T^{6} )^{2} \)
$13$ \( 1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(38\!\cdots\!76\)\( T^{7} + \)\(91\!\cdots\!19\)\( T^{8} + \)\(18\!\cdots\!02\)\( T^{9} + \)\(31\!\cdots\!42\)\( T^{10} + \)\(43\!\cdots\!42\)\( T^{11} + \)\(29\!\cdots\!21\)\( T^{12} \)
$17$ \( 1 + 265502 T + 35245656002 T^{2} + 3505767378301982 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} + \)\(40\!\cdots\!76\)\( T^{5} + \)\(37\!\cdots\!76\)\( T^{6} + \)\(28\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!39\)\( T^{8} + \)\(11\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!22\)\( T^{10} + \)\(43\!\cdots\!02\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$19$ \( 1 - 66391003446 T^{2} + \)\(21\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(60\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!06\)\( T^{10} + \)\(23\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 - 28888 T + 417258272 T^{2} - 2207314762520128 T^{3} + \)\(86\!\cdots\!39\)\( T^{4} - \)\(83\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(65\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!79\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(15\!\cdots\!12\)\( T^{10} - \)\(85\!\cdots\!88\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} \)
$29$ \( 1 - 1726260912966 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(34\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \)
$31$ \( ( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(27\!\cdots\!44\)\( T^{5} + \)\(62\!\cdots\!21\)\( T^{6} )^{2} \)
$37$ \( 1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!56\)\( T^{7} + \)\(45\!\cdots\!19\)\( T^{8} + \)\(68\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!62\)\( T^{10} + \)\(24\!\cdots\!02\)\( T^{11} + \)\(18\!\cdots\!21\)\( T^{12} \)
$41$ \( ( 1 - 1244716 T + 19779856453415 T^{2} - 17348876621060070520 T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(79\!\cdots\!56\)\( T^{5} + \)\(50\!\cdots\!61\)\( T^{6} )^{2} \)
$43$ \( 1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!96\)\( T^{6} - \)\(22\!\cdots\!04\)\( T^{7} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(58\!\cdots\!52\)\( T^{10} - \)\(17\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 + 15313352 T + 117249374737952 T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(63\!\cdots\!79\)\( T^{4} + \)\(35\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} + \)\(85\!\cdots\!76\)\( T^{7} + \)\(36\!\cdots\!59\)\( T^{8} + \)\(11\!\cdots\!32\)\( T^{9} + \)\(37\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} \)
$53$ \( 1 + 13509122 T + 91248188605442 T^{2} + \)\(98\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} + \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} + \)\(51\!\cdots\!36\)\( T^{7} + \)\(46\!\cdots\!59\)\( T^{8} + \)\(23\!\cdots\!02\)\( T^{9} + \)\(13\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!22\)\( T^{11} + \)\(58\!\cdots\!61\)\( T^{12} \)
$59$ \( 1 - 413223229068726 T^{2} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} \)
$61$ \( ( 1 - 12055596 T + 200152007609415 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!56\)\( T^{5} + \)\(70\!\cdots\!41\)\( T^{6} )^{2} \)
$67$ \( 1 + 32827752 T + 538830650686752 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(90\!\cdots\!39\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} + \)\(14\!\cdots\!72\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{11} + \)\(44\!\cdots\!41\)\( T^{12} \)
$71$ \( ( 1 + 6996464 T + 1071469029384215 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} + \)\(29\!\cdots\!44\)\( T^{5} + \)\(26\!\cdots\!81\)\( T^{6} )^{2} \)
$73$ \( 1 - 111859638 T + 6256289306745522 T^{2} - \)\(29\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(33\!\cdots\!84\)\( T^{7} + \)\(79\!\cdots\!79\)\( T^{8} - \)\(15\!\cdots\!98\)\( T^{9} + \)\(26\!\cdots\!62\)\( T^{10} - \)\(38\!\cdots\!38\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \)
$79$ \( 1 - 4185433961698566 T^{2} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(22\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} \)
$83$ \( 1 + 14768432 T + 109053291869312 T^{2} + \)\(28\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(51\!\cdots\!39\)\( T^{8} + \)\(32\!\cdots\!52\)\( T^{9} + \)\(28\!\cdots\!32\)\( T^{10} + \)\(85\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$89$ \( 1 - 8165894455313286 T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(37\!\cdots\!81\)\( T^{12} \)
$97$ \( 1 + 186656202 T + 17420268872532402 T^{2} + \)\(93\!\cdots\!22\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(87\!\cdots\!76\)\( T^{7} + \)\(41\!\cdots\!59\)\( T^{8} + \)\(45\!\cdots\!82\)\( T^{9} + \)\(65\!\cdots\!82\)\( T^{10} + \)\(55\!\cdots\!02\)\( T^{11} + \)\(23\!\cdots\!61\)\( T^{12} \)
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