# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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