# Properties

 Label 387.4.a.h Level 387 Weight 4 Character orbit 387.a Self dual yes Analytic conductor 22.834 Analytic rank 1 Dimension 6 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 387.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.8337391722$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 4 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( -7 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{7} + ( -10 + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 4 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} + ( -7 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{5} ) q^{7} + ( -10 + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} ) q^{8} + ( 9 - 5 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + ( 3 - 6 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} ) q^{11} + ( 7 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} ) q^{13} + ( 30 + \beta_{1} - 5 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{14} + ( -10 - 14 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{16} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -11 - 10 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{19} + ( -23 + 6 \beta_{1} + 12 \beta_{2} - 19 \beta_{3} + 7 \beta_{4} + 11 \beta_{5} ) q^{20} + ( -83 - 5 \beta_{1} + 19 \beta_{2} - 14 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} ) q^{22} + ( -24 - 6 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} + 10 \beta_{4} + 7 \beta_{5} ) q^{23} + ( 18 - 20 \beta_{1} - 9 \beta_{2} + 13 \beta_{3} + 12 \beta_{4} + 11 \beta_{5} ) q^{25} + ( -3 + 3 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - \beta_{4} + 10 \beta_{5} ) q^{26} + ( -70 + 24 \beta_{1} + 16 \beta_{2} - 26 \beta_{3} - 6 \beta_{5} ) q^{28} + ( -81 - 24 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 18 \beta_{4} + 8 \beta_{5} ) q^{29} + ( 42 - 18 \beta_{1} - 6 \beta_{2} + \beta_{3} - 21 \beta_{4} - \beta_{5} ) q^{31} + ( -86 - 16 \beta_{1} + 23 \beta_{2} - 12 \beta_{3} - 17 \beta_{4} + 14 \beta_{5} ) q^{32} + ( -14 - 17 \beta_{1} - 9 \beta_{3} - 17 \beta_{4} + 15 \beta_{5} ) q^{34} + ( -32 - 18 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} - 28 \beta_{4} - 30 \beta_{5} ) q^{35} + ( 47 - 18 \beta_{1} + 31 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 9 \beta_{5} ) q^{37} + ( -95 - 2 \beta_{1} - 2 \beta_{2} - 11 \beta_{3} + 19 \beta_{4} - 11 \beta_{5} ) q^{38} + ( 107 - 22 \beta_{1} - 26 \beta_{2} + 55 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} ) q^{40} + ( -78 + 28 \beta_{1} - 27 \beta_{2} + 17 \beta_{3} - 16 \beta_{4} - 3 \beta_{5} ) q^{41} -43 q^{43} + ( 74 - 41 \beta_{1} - 43 \beta_{2} + 69 \beta_{3} - 12 \beta_{4} - 22 \beta_{5} ) q^{44} + ( -6 - 71 \beta_{1} + 3 \beta_{2} - \beta_{3} - 26 \beta_{4} + 3 \beta_{5} ) q^{46} + ( -79 + 44 \beta_{1} + 9 \beta_{2} - 5 \beta_{3} + 40 \beta_{4} - 23 \beta_{5} ) q^{47} + ( 43 - 30 \beta_{1} - 32 \beta_{2} - 6 \beta_{3} + 46 \beta_{4} - 8 \beta_{5} ) q^{49} + ( -228 - 9 \beta_{1} + 45 \beta_{2} - 61 \beta_{3} + 8 \beta_{4} - 9 \beta_{5} ) q^{50} + ( 6 + 3 \beta_{1} - 9 \beta_{2} + 33 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{52} + ( -67 - 20 \beta_{1} + 37 \beta_{2} + 30 \beta_{3} - 7 \beta_{4} - 26 \beta_{5} ) q^{53} + ( -290 + 78 \beta_{1} + 27 \beta_{2} - 40 \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{55} + ( 144 - 92 \beta_{1} - 38 \beta_{2} + 84 \beta_{3} - 6 \beta_{4} + 32 \beta_{5} ) q^{56} + ( -189 - 47 \beta_{1} - 12 \beta_{2} - 58 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} ) q^{58} + ( -62 - 42 \beta_{1} + 8 \beta_{2} - 30 \beta_{3} + 4 \beta_{4} + 10 \beta_{5} ) q^{59} + ( -208 + 46 \beta_{1} + 25 \beta_{2} - 4 \beta_{3} - 46 \beta_{4} + 13 \beta_{5} ) q^{61} + ( -272 + 75 \beta_{1} - 6 \beta_{2} - 59 \beta_{3} + 3 \beta_{4} + 39 \beta_{5} ) q^{62} + ( 62 + 68 \beta_{1} - 45 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} - 38 \beta_{5} ) q^{64} + ( 12 - 18 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} + 28 \beta_{4} + 53 \beta_{5} ) q^{65} + ( -113 + 14 \beta_{1} - 69 \beta_{2} + 78 \beta_{3} - \beta_{4} - 56 \beta_{5} ) q^{67} + ( -95 + 3 \beta_{1} - 18 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{68} + ( -324 + 70 \beta_{1} + 50 \beta_{2} - 102 \beta_{3} + 56 \beta_{4} + 66 \beta_{5} ) q^{70} + ( 62 + 56 \beta_{1} + 38 \beta_{2} + 28 \beta_{3} - 116 \beta_{4} + 46 \beta_{5} ) q^{71} + ( 148 + 146 \beta_{1} - 51 \beta_{2} + 92 \beta_{3} + 26 \beta_{4} + 9 \beta_{5} ) q^{73} + ( -229 + 100 \beta_{1} - 53 \beta_{2} + 87 \beta_{3} + 114 \beta_{4} - 87 \beta_{5} ) q^{74} + ( 167 - 68 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} - 7 \beta_{4} - 15 \beta_{5} ) q^{76} + ( -434 + 14 \beta_{1} + 55 \beta_{2} - 24 \beta_{3} + 50 \beta_{4} + 75 \beta_{5} ) q^{77} + ( -297 + 96 \beta_{1} + 14 \beta_{2} - 53 \beta_{3} + 95 \beta_{4} - 35 \beta_{5} ) q^{79} + ( -409 + 198 \beta_{1} + 66 \beta_{2} - 85 \beta_{3} + 25 \beta_{4} - 11 \beta_{5} ) q^{80} + ( 300 - 63 \beta_{1} + 75 \beta_{2} - 103 \beta_{3} - 72 \beta_{4} + 83 \beta_{5} ) q^{82} + ( 79 + 136 \beta_{1} - 37 \beta_{2} - 102 \beta_{3} + 5 \beta_{4} - 64 \beta_{5} ) q^{83} + ( 14 + 74 \beta_{1} + 8 \beta_{2} + 26 \beta_{3} + 7 \beta_{4} - 27 \beta_{5} ) q^{85} + ( 43 - 43 \beta_{1} ) q^{86} + ( -120 + 212 \beta_{1} + 82 \beta_{2} - 208 \beta_{3} + 32 \beta_{4} + 46 \beta_{5} ) q^{88} + ( -576 + 2 \beta_{1} - 89 \beta_{2} + 62 \beta_{3} + 34 \beta_{4} - 17 \beta_{5} ) q^{89} + ( -594 + 66 \beta_{1} + 51 \beta_{2} + 108 \beta_{3} - 114 \beta_{4} + 63 \beta_{5} ) q^{91} + ( -595 + 95 \beta_{1} + 3 \beta_{2} - 46 \beta_{3} - 5 \beta_{4} - 41 \beta_{5} ) q^{92} + ( 553 - 128 \beta_{1} + 35 \beta_{2} + 121 \beta_{3} - 4 \beta_{4} - 39 \beta_{5} ) q^{94} + ( 13 + 30 \beta_{1} + 33 \beta_{2} - 37 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{95} + ( 4 + 102 \beta_{1} - 3 \beta_{2} - 29 \beta_{3} + 22 \beta_{4} + 133 \beta_{5} ) q^{97} + ( -371 - 117 \beta_{1} + 106 \beta_{2} - 68 \beta_{3} - 70 \beta_{4} + 64 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{2} + 22q^{4} - 43q^{5} + 8q^{7} - 54q^{8} + O(q^{10})$$ $$6q - 6q^{2} + 22q^{4} - 43q^{5} + 8q^{7} - 54q^{8} + 57q^{10} + 28q^{11} + 56q^{13} + 184q^{14} - 54q^{16} - 19q^{17} - 75q^{19} - 135q^{20} - 504q^{22} - 131q^{23} + 105q^{25} - 44q^{26} - 404q^{28} - 515q^{29} + 237q^{31} - 558q^{32} - 107q^{34} - 198q^{35} + 269q^{37} - 527q^{38} + 613q^{40} - 471q^{41} - 258q^{43} + 428q^{44} - 67q^{46} - 415q^{47} + 350q^{49} - 1335q^{50} - 8q^{52} - 450q^{53} - 1732q^{55} + 780q^{56} - 1055q^{58} - 356q^{59} - 1328q^{61} - 1603q^{62} + 466q^{64} + 62q^{65} - 632q^{67} - 571q^{68} - 1902q^{70} + 144q^{71} + 864q^{73} - 1207q^{74} + 1005q^{76} - 2660q^{77} - 1613q^{79} - 2399q^{80} + 1673q^{82} + 682q^{83} + 84q^{85} + 258q^{86} - 608q^{88} - 3378q^{89} - 3900q^{91} - 3491q^{92} + 3197q^{94} + 79q^{95} - 55q^{97} - 2398q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu^{4} - 16 \nu^{3} - 36 \nu^{2} - 25 \nu - 34$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 36 \nu^{2} + 103 \nu + 30$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 44 \nu^{2} + 111 \nu + 118$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 36 \nu^{3} + 24 \nu^{2} + 331 \nu + 182$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{1} + 11$$ $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 16 \beta_{1} + 8$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{5} - 15 \beta_{4} + 20 \beta_{3} - 3 \beta_{2} + 24 \beta_{1} + 179$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5} - 6 \beta_{4} - 20 \beta_{3} + 30 \beta_{2} + 269 \beta_{1} + 200$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.15251 −3.17112 −0.847740 −0.299707 4.15653 4.31455
−5.15251 0 18.5484 −17.2665 0 −23.3206 −54.3507 0 88.9661
1.2 −4.17112 0 9.39827 7.54340 0 4.58222 −5.83236 0 −31.4645
1.3 −1.84774 0 −4.58586 −2.98245 0 −26.0720 23.2554 0 5.51080
1.4 −1.29971 0 −6.31076 −20.4116 0 29.9522 18.5998 0 26.5291
1.5 3.15653 0 1.96369 −1.36370 0 13.0131 −19.0538 0 −4.30455
1.6 3.31455 0 2.98627 −8.51910 0 9.84502 −16.6183 0 −28.2370
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.4.a.h 6
3.b odd 2 1 43.4.a.b 6
12.b even 2 1 688.4.a.i 6
15.d odd 2 1 1075.4.a.b 6
21.c even 2 1 2107.4.a.c 6
129.d even 2 1 1849.4.a.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.a.b 6 3.b odd 2 1
387.4.a.h 6 1.a even 1 1 trivial
688.4.a.i 6 12.b even 2 1
1075.4.a.b 6 15.d odd 2 1
1849.4.a.c 6 129.d even 2 1
2107.4.a.c 6 21.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$43$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 6 T_{2}^{5} - 17 T_{2}^{4} - 124 T_{2}^{3} + 26 T_{2}^{2} + 608 T_{2} + 540$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(387))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 31 T^{2} + 116 T^{3} + 442 T^{4} + 1472 T^{5} + 4668 T^{6} + 11776 T^{7} + 28288 T^{8} + 59392 T^{9} + 126976 T^{10} + 196608 T^{11} + 262144 T^{12}$$
$3$ 1
$5$ $$1 + 43 T + 1247 T^{2} + 26367 T^{3} + 452519 T^{4} + 6421366 T^{5} + 77975134 T^{6} + 802670750 T^{7} + 7070609375 T^{8} + 51498046875 T^{9} + 304443359375 T^{10} + 1312255859375 T^{11} + 3814697265625 T^{12}$$
$7$ $$1 - 8 T + 886 T^{2} - 4080 T^{3} + 442155 T^{4} - 3221432 T^{5} + 186242508 T^{6} - 1104951176 T^{7} + 52019093595 T^{8} - 164642716560 T^{9} + 12263380460086 T^{10} - 37980492079544 T^{11} + 1628413597910449 T^{12}$$
$11$ $$1 - 28 T + 3144 T^{2} - 41080 T^{3} + 3977536 T^{4} - 3139972 T^{5} + 4111332998 T^{6} - 4179302732 T^{7} + 7046447653696 T^{8} - 96864491146280 T^{9} + 9867218816410824 T^{10} - 116962948743638228 T^{11} + 5559917313492231481 T^{12}$$
$13$ $$1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 4725231126824 T^{7} + 180879261248920 T^{8} - 5629759045135732 T^{9} + 204463995034893256 T^{10} - 2866410008789082392 T^{11} +$$$$11\!\cdots\!29$$$$T^{12}$$
$17$ $$1 + 19 T + 23143 T^{2} + 543393 T^{3} + 241535186 T^{4} + 5650313095 T^{5} + 1493450611759 T^{6} + 27759988235735 T^{7} + 5830072218002834 T^{8} + 64439821973334321 T^{9} + 13483626436208358823 T^{10} + 54386037978686500067 T^{11} +$$$$14\!\cdots\!09$$$$T^{12}$$
$19$ $$1 + 75 T + 38259 T^{2} + 2365781 T^{3} + 629552155 T^{4} + 31151517862 T^{5} + 5681041321490 T^{6} + 213668261015458 T^{7} + 29617835767423555 T^{8} + 763408424339300399 T^{9} + 84679215488552253699 T^{10} +$$$$11\!\cdots\!25$$$$T^{11} +$$$$10\!\cdots\!41$$$$T^{12}$$
$23$ $$1 + 131 T + 52195 T^{2} + 3677795 T^{3} + 974730114 T^{4} + 33210519163 T^{5} + 11858751245947 T^{6} + 404072386656221 T^{7} + 144295038961061346 T^{8} + 6624270252565314085 T^{9} +$$$$11\!\cdots\!95$$$$T^{10} +$$$$34\!\cdots\!17$$$$T^{11} +$$$$32\!\cdots\!69$$$$T^{12}$$
$29$ $$1 + 515 T + 204583 T^{2} + 58068807 T^{3} + 13900558631 T^{4} + 2733986934494 T^{5} + 464005745217070 T^{6} + 66679207345374166 T^{7} + 8268376448646633551 T^{8} +$$$$84\!\cdots\!83$$$$T^{9} +$$$$72\!\cdots\!03$$$$T^{10} +$$$$44\!\cdots\!35$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$31$ $$1 - 237 T + 125373 T^{2} - 21016589 T^{3} + 7321362670 T^{4} - 1031063540341 T^{5} + 275109610824401 T^{6} - 30716413930298731 T^{7} + 6497736319560988270 T^{8} -$$$$55\!\cdots\!19$$$$T^{9} +$$$$98\!\cdots\!53$$$$T^{10} -$$$$55\!\cdots\!87$$$$T^{11} +$$$$69\!\cdots\!41$$$$T^{12}$$
$37$ $$1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 65968300092541106 T^{7} + 20981383282418309715 T^{8} -$$$$39\!\cdots\!61$$$$T^{9} +$$$$83\!\cdots\!91$$$$T^{10} -$$$$89\!\cdots\!17$$$$T^{11} +$$$$16\!\cdots\!29$$$$T^{12}$$
$41$ $$1 + 471 T + 349763 T^{2} + 112600045 T^{3} + 50459129866 T^{4} + 12922113800443 T^{5} + 4372223136871043 T^{6} + 890605005240332003 T^{7} +$$$$23\!\cdots\!06$$$$T^{8} +$$$$36\!\cdots\!45$$$$T^{9} +$$$$78\!\cdots\!03$$$$T^{10} +$$$$73\!\cdots\!71$$$$T^{11} +$$$$10\!\cdots\!21$$$$T^{12}$$
$43$ $$( 1 + 43 T )^{6}$$
$47$ $$1 + 415 T + 421631 T^{2} + 116866317 T^{3} + 77411983523 T^{4} + 16052264514750 T^{5} + 9154052369892234 T^{6} + 1666594258714889250 T^{7} +$$$$83\!\cdots\!67$$$$T^{8} +$$$$13\!\cdots\!39$$$$T^{9} +$$$$48\!\cdots\!71$$$$T^{10} +$$$$50\!\cdots\!45$$$$T^{11} +$$$$12\!\cdots\!89$$$$T^{12}$$
$53$ $$1 + 450 T + 321704 T^{2} + 149982378 T^{3} + 77929548632 T^{4} + 28654270442506 T^{5} + 12746558079363422 T^{6} + 4265961820668965762 T^{7} +$$$$17\!\cdots\!28$$$$T^{8} +$$$$49\!\cdots\!74$$$$T^{9} +$$$$15\!\cdots\!64$$$$T^{10} +$$$$32\!\cdots\!50$$$$T^{11} +$$$$10\!\cdots\!89$$$$T^{12}$$
$59$ $$1 + 356 T + 1153950 T^{2} + 347607228 T^{3} + 570632518215 T^{4} + 139273869185096 T^{5} + 154351054402988548 T^{6} + 28603927979365831384 T^{7} +$$$$24\!\cdots\!15$$$$T^{8} +$$$$30\!\cdots\!92$$$$T^{9} +$$$$20\!\cdots\!50$$$$T^{10} +$$$$13\!\cdots\!44$$$$T^{11} +$$$$75\!\cdots\!21$$$$T^{12}$$
$61$ $$1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} +$$$$14\!\cdots\!08$$$$T^{7} +$$$$58\!\cdots\!39$$$$T^{8} +$$$$17\!\cdots\!84$$$$T^{9} +$$$$47\!\cdots\!74$$$$T^{10} +$$$$80\!\cdots\!28$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$67$ $$1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 18881302583762735248 T^{7} +$$$$16\!\cdots\!92$$$$T^{8} +$$$$68\!\cdots\!04$$$$T^{9} +$$$$75\!\cdots\!08$$$$T^{10} +$$$$15\!\cdots\!76$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12}$$
$71$ $$1 - 144 T + 863230 T^{2} - 72744496 T^{3} + 475313592223 T^{4} - 62333254020128 T^{5} + 209643801201434276 T^{6} - 22309757279598032608 T^{7} +$$$$60\!\cdots\!83$$$$T^{8} -$$$$33\!\cdots\!76$$$$T^{9} +$$$$14\!\cdots\!30$$$$T^{10} -$$$$84\!\cdots\!44$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$73$ $$1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 81531488761123023064 T^{7} +$$$$69\!\cdots\!63$$$$T^{8} -$$$$25\!\cdots\!44$$$$T^{9} +$$$$24\!\cdots\!18$$$$T^{10} -$$$$76\!\cdots\!48$$$$T^{11} +$$$$34\!\cdots\!69$$$$T^{12}$$
$79$ $$1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} +$$$$13\!\cdots\!02$$$$T^{7} +$$$$79\!\cdots\!19$$$$T^{8} +$$$$37\!\cdots\!69$$$$T^{9} +$$$$16\!\cdots\!21$$$$T^{10} +$$$$46\!\cdots\!87$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$83$ $$1 - 682 T + 1120004 T^{2} - 1783287782 T^{3} + 1291656570448 T^{4} - 1149121488958882 T^{5} + 1227368803599345682 T^{6} -$$$$65\!\cdots\!34$$$$T^{7} +$$$$42\!\cdots\!12$$$$T^{8} -$$$$33\!\cdots\!46$$$$T^{9} +$$$$11\!\cdots\!44$$$$T^{10} -$$$$41\!\cdots\!74$$$$T^{11} +$$$$34\!\cdots\!09$$$$T^{12}$$
$89$ $$1 + 3378 T + 7851850 T^{2} + 13055260850 T^{3} + 17370332054203 T^{4} + 19017874978895668 T^{5} + 17369747195795800052 T^{6} +$$$$13\!\cdots\!92$$$$T^{7} +$$$$86\!\cdots\!83$$$$T^{8} +$$$$45\!\cdots\!50$$$$T^{9} +$$$$19\!\cdots\!50$$$$T^{10} +$$$$58\!\cdots\!22$$$$T^{11} +$$$$12\!\cdots\!81$$$$T^{12}$$
$97$ $$1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} -$$$$15\!\cdots\!33$$$$T^{7} +$$$$27\!\cdots\!30$$$$T^{8} -$$$$71\!\cdots\!87$$$$T^{9} +$$$$17\!\cdots\!11$$$$T^{10} +$$$$34\!\cdots\!15$$$$T^{11} +$$$$57\!\cdots\!89$$$$T^{12}$$