# Properties

 Label 688.4.a.i Level 688 Weight 4 Character orbit 688.a Self dual yes Analytic conductor 40.593 Analytic rank 0 Dimension 6 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 688.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.5933140839$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 7 + \beta_{1} - \beta_{4} ) q^{5} + ( -2 - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( 12 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} + ( 7 + \beta_{1} - \beta_{4} ) q^{5} + ( -2 - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( 12 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{9} + ( 3 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{11} + ( 7 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{13} + ( 22 + 8 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{15} + ( 4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{17} + ( 11 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{19} + ( 4 + 16 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{21} + ( -24 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 11 \beta_{4} - 6 \beta_{5} ) q^{23} + ( 18 + 13 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} - 7 \beta_{5} ) q^{25} + ( -30 + 8 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{27} + ( 81 - 5 \beta_{1} + 13 \beta_{2} - \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{29} + ( -42 - \beta_{1} + 11 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - 14 \beta_{5} ) q^{31} + ( 104 + 32 \beta_{1} - 16 \beta_{2} + 21 \beta_{3} + 3 \beta_{4} + 14 \beta_{5} ) q^{33} + ( -32 + 16 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 25 \beta_{4} + 33 \beta_{5} ) q^{35} + ( 47 - 3 \beta_{1} + 7 \beta_{2} - 16 \beta_{3} + 17 \beta_{4} - 14 \beta_{5} ) q^{37} + ( -26 + 46 \beta_{1} - 10 \beta_{2} + 23 \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{39} + ( 78 - 17 \beta_{1} + 3 \beta_{2} - 17 \beta_{3} + 4 \beta_{4} - 23 \beta_{5} ) q^{41} + 43 q^{43} + ( 56 + 8 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 28 \beta_{4} + 24 \beta_{5} ) q^{45} + ( -79 - 5 \beta_{1} + 29 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} - 13 \beta_{5} ) q^{47} + ( 43 - 6 \beta_{1} - 2 \beta_{2} - 13 \beta_{3} - 35 \beta_{4} - 3 \beta_{5} ) q^{49} + ( 203 + 3 \beta_{1} + 25 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 8 \beta_{5} ) q^{51} + ( 67 - 30 \beta_{1} - 9 \beta_{2} + 19 \beta_{3} - 35 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 290 + 40 \beta_{1} - 25 \beta_{2} - 14 \beta_{3} - 17 \beta_{4} + 10 \beta_{5} ) q^{55} + ( -156 + 18 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} - 26 \beta_{5} ) q^{57} + ( -62 - 30 \beta_{1} - 10 \beta_{2} - 11 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} ) q^{59} + ( -208 - 4 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} + 29 \beta_{4} + 4 \beta_{5} ) q^{61} + ( 340 - 66 \beta_{1} + 9 \beta_{2} - 52 \beta_{3} - 29 \beta_{4} + 2 \beta_{5} ) q^{63} + ( -12 + 6 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} + 45 \beta_{4} + 36 \beta_{5} ) q^{65} + ( 113 - 78 \beta_{1} - 7 \beta_{3} + 6 \beta_{4} - 63 \beta_{5} ) q^{67} + ( -186 - 48 \beta_{1} - 22 \beta_{2} + \beta_{3} + 8 \beta_{4} ) q^{69} + ( 62 + 28 \beta_{1} - 17 \beta_{2} + 45 \beta_{3} + 54 \beta_{4} + 16 \beta_{5} ) q^{71} + ( 148 + 92 \beta_{1} + 28 \beta_{2} + 45 \beta_{3} - 43 \beta_{4} + 8 \beta_{5} ) q^{73} + ( 402 - 8 \beta_{1} - 17 \beta_{2} + 22 \beta_{4} + 76 \beta_{5} ) q^{75} + ( 434 + 24 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} + 35 \beta_{4} + 90 \beta_{5} ) q^{77} + ( 297 + 53 \beta_{1} - 60 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} + 37 \beta_{5} ) q^{79} + ( -61 - 140 \beta_{1} + 13 \beta_{2} - 66 \beta_{3} - 22 \beta_{4} - 39 \beta_{5} ) q^{81} + ( 79 - 102 \beta_{1} + 42 \beta_{2} + 26 \beta_{3} + 11 \beta_{4} + 48 \beta_{5} ) q^{83} + ( 14 + 26 \beta_{1} + 29 \beta_{2} + 8 \beta_{3} + 14 \beta_{4} + 6 \beta_{5} ) q^{85} + ( -33 + 95 \beta_{1} + 44 \beta_{2} + 46 \beta_{3} + 31 \beta_{4} - 45 \beta_{5} ) q^{87} + ( 576 - 62 \beta_{1} + 9 \beta_{2} - 10 \beta_{3} + 53 \beta_{4} - 36 \beta_{5} ) q^{89} + ( 594 - 108 \beta_{1} + 15 \beta_{2} - 48 \beta_{3} - 51 \beta_{4} ) q^{91} + ( 313 - 71 \beta_{1} + 71 \beta_{2} + 47 \beta_{3} + 43 \beta_{4} - 16 \beta_{5} ) q^{93} + ( 13 - 37 \beta_{1} + 15 \beta_{2} + 17 \beta_{4} - 16 \beta_{5} ) q^{95} + ( 4 - 29 \beta_{1} - 3 \beta_{2} + 54 \beta_{3} - 79 \beta_{4} - 76 \beta_{5} ) q^{97} + ( 213 - 64 \beta_{1} + 18 \beta_{2} - 114 \beta_{3} + 37 \beta_{4} + 40 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 7q^{3} + 43q^{5} - 8q^{7} + 81q^{9} + O(q^{10})$$ $$6q - 7q^{3} + 43q^{5} - 8q^{7} + 81q^{9} + 28q^{11} + 56q^{13} + 124q^{15} + 19q^{17} + 75q^{19} - 18q^{21} - 131q^{23} + 105q^{25} - 238q^{27} + 515q^{29} - 237q^{31} + 540q^{33} - 198q^{35} + 269q^{37} - 290q^{39} + 471q^{41} + 258q^{43} + 334q^{45} - 415q^{47} + 350q^{49} + 1241q^{51} + 450q^{53} + 1732q^{55} - 1000q^{57} - 356q^{59} - 1328q^{61} + 2290q^{63} - 62q^{65} + 632q^{67} - 1130q^{69} + 144q^{71} + 864q^{73} + 2494q^{75} + 2660q^{77} + 1613q^{79} - 102q^{81} + 682q^{83} + 84q^{85} - 449q^{87} + 3378q^{89} + 3900q^{91} + 1879q^{93} + 79q^{95} - 55q^{97} + 1612q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 36 \nu^{2} + 103 \nu + 30$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 3 \nu^{3} - 17 \nu^{2} - 51 \nu - 16$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} - 3 \nu^{3} + 17 \nu^{2} + 59 \nu + 16$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{3} + \nu^{2} - 17 \nu - 19$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 26 \nu^{3} + 6 \nu^{2} - 153 \nu - 74$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} + 22$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 17 \beta_{2} - 4 \beta_{1} + 32$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$10 \beta_{5} + 7 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 20 \beta_{1} + 179$$ $$\nu^{5}$$ $$=$$ $$($$$$-56 \beta_{5} + 64 \beta_{4} + 249 \beta_{3} + 289 \beta_{2} - 80 \beta_{1} + 800$$$$)/4$$

## 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
 −0.847740 4.15653 −3.17112 −0.299707 −4.15251 4.31455
0 −9.49653 0 2.98245 0 26.0720 0 63.1842 0
1.2 0 −7.20925 0 1.36370 0 −13.0131 0 24.9733 0
1.3 0 −2.46717 0 −7.54340 0 −4.58222 0 −20.9131 0
1.4 0 −1.43046 0 20.4116 0 −29.9522 0 −24.9538 0
1.5 0 6.49933 0 17.2665 0 23.3206 0 15.2413 0
1.6 0 7.10409 0 8.51910 0 −9.84502 0 23.4681 0
 $$n$$: 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 688.4.a.i 6
4.b odd 2 1 43.4.a.b 6
12.b even 2 1 387.4.a.h 6
20.d odd 2 1 1075.4.a.b 6
28.d even 2 1 2107.4.a.c 6
172.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 4.b odd 2 1
387.4.a.h 6 12.b even 2 1
688.4.a.i 6 1.a even 1 1 trivial
1075.4.a.b 6 20.d odd 2 1
1849.4.a.c 6 172.d even 2 1
2107.4.a.c 6 28.d even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 7 T_{3}^{5} - 97 T_{3}^{4} - 588 T_{3}^{3} + 2140 T_{3}^{2} + 11756 T_{3} + 11156$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(688))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 7 T + 65 T^{2} + 357 T^{3} + 2599 T^{4} + 15158 T^{5} + 96098 T^{6} + 409266 T^{7} + 1894671 T^{8} + 7026831 T^{9} + 34543665 T^{10} + 100442349 T^{11} + 387420489 T^{12}$$
$5$ $$1 - 43 T + 1247 T^{2} - 26367 T^{3} + 452519 T^{4} - 6421366 T^{5} + 77975134 T^{6} - 802670750 T^{7} + 7070609375 T^{8} - 51498046875 T^{9} + 304443359375 T^{10} - 1312255859375 T^{11} + 3814697265625 T^{12}$$
$7$ $$1 + 8 T + 886 T^{2} + 4080 T^{3} + 442155 T^{4} + 3221432 T^{5} + 186242508 T^{6} + 1104951176 T^{7} + 52019093595 T^{8} + 164642716560 T^{9} + 12263380460086 T^{10} + 37980492079544 T^{11} + 1628413597910449 T^{12}$$
$11$ $$1 - 28 T + 3144 T^{2} - 41080 T^{3} + 3977536 T^{4} - 3139972 T^{5} + 4111332998 T^{6} - 4179302732 T^{7} + 7046447653696 T^{8} - 96864491146280 T^{9} + 9867218816410824 T^{10} - 116962948743638228 T^{11} + 5559917313492231481 T^{12}$$
$13$ $$1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 4725231126824 T^{7} + 180879261248920 T^{8} - 5629759045135732 T^{9} + 204463995034893256 T^{10} - 2866410008789082392 T^{11} +$$$$11\!\cdots\!29$$$$T^{12}$$
$17$ $$1 - 19 T + 23143 T^{2} - 543393 T^{3} + 241535186 T^{4} - 5650313095 T^{5} + 1493450611759 T^{6} - 27759988235735 T^{7} + 5830072218002834 T^{8} - 64439821973334321 T^{9} + 13483626436208358823 T^{10} - 54386037978686500067 T^{11} +$$$$14\!\cdots\!09$$$$T^{12}$$
$19$ $$1 - 75 T + 38259 T^{2} - 2365781 T^{3} + 629552155 T^{4} - 31151517862 T^{5} + 5681041321490 T^{6} - 213668261015458 T^{7} + 29617835767423555 T^{8} - 763408424339300399 T^{9} + 84679215488552253699 T^{10} -$$$$11\!\cdots\!25$$$$T^{11} +$$$$10\!\cdots\!41$$$$T^{12}$$
$23$ $$1 + 131 T + 52195 T^{2} + 3677795 T^{3} + 974730114 T^{4} + 33210519163 T^{5} + 11858751245947 T^{6} + 404072386656221 T^{7} + 144295038961061346 T^{8} + 6624270252565314085 T^{9} +$$$$11\!\cdots\!95$$$$T^{10} +$$$$34\!\cdots\!17$$$$T^{11} +$$$$32\!\cdots\!69$$$$T^{12}$$
$29$ $$1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 66679207345374166 T^{7} + 8268376448646633551 T^{8} -$$$$84\!\cdots\!83$$$$T^{9} +$$$$72\!\cdots\!03$$$$T^{10} -$$$$44\!\cdots\!35$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$31$ $$1 + 237 T + 125373 T^{2} + 21016589 T^{3} + 7321362670 T^{4} + 1031063540341 T^{5} + 275109610824401 T^{6} + 30716413930298731 T^{7} + 6497736319560988270 T^{8} +$$$$55\!\cdots\!19$$$$T^{9} +$$$$98\!\cdots\!53$$$$T^{10} +$$$$55\!\cdots\!87$$$$T^{11} +$$$$69\!\cdots\!41$$$$T^{12}$$
$37$ $$1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 65968300092541106 T^{7} + 20981383282418309715 T^{8} -$$$$39\!\cdots\!61$$$$T^{9} +$$$$83\!\cdots\!91$$$$T^{10} -$$$$89\!\cdots\!17$$$$T^{11} +$$$$16\!\cdots\!29$$$$T^{12}$$
$41$ $$1 - 471 T + 349763 T^{2} - 112600045 T^{3} + 50459129866 T^{4} - 12922113800443 T^{5} + 4372223136871043 T^{6} - 890605005240332003 T^{7} +$$$$23\!\cdots\!06$$$$T^{8} -$$$$36\!\cdots\!45$$$$T^{9} +$$$$78\!\cdots\!03$$$$T^{10} -$$$$73\!\cdots\!71$$$$T^{11} +$$$$10\!\cdots\!21$$$$T^{12}$$
$43$ $$( 1 - 43 T )^{6}$$
$47$ $$1 + 415 T + 421631 T^{2} + 116866317 T^{3} + 77411983523 T^{4} + 16052264514750 T^{5} + 9154052369892234 T^{6} + 1666594258714889250 T^{7} +$$$$83\!\cdots\!67$$$$T^{8} +$$$$13\!\cdots\!39$$$$T^{9} +$$$$48\!\cdots\!71$$$$T^{10} +$$$$50\!\cdots\!45$$$$T^{11} +$$$$12\!\cdots\!89$$$$T^{12}$$
$53$ $$1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 4265961820668965762 T^{7} +$$$$17\!\cdots\!28$$$$T^{8} -$$$$49\!\cdots\!74$$$$T^{9} +$$$$15\!\cdots\!64$$$$T^{10} -$$$$32\!\cdots\!50$$$$T^{11} +$$$$10\!\cdots\!89$$$$T^{12}$$
$59$ $$1 + 356 T + 1153950 T^{2} + 347607228 T^{3} + 570632518215 T^{4} + 139273869185096 T^{5} + 154351054402988548 T^{6} + 28603927979365831384 T^{7} +$$$$24\!\cdots\!15$$$$T^{8} +$$$$30\!\cdots\!92$$$$T^{9} +$$$$20\!\cdots\!50$$$$T^{10} +$$$$13\!\cdots\!44$$$$T^{11} +$$$$75\!\cdots\!21$$$$T^{12}$$
$61$ $$1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} +$$$$14\!\cdots\!08$$$$T^{7} +$$$$58\!\cdots\!39$$$$T^{8} +$$$$17\!\cdots\!84$$$$T^{9} +$$$$47\!\cdots\!74$$$$T^{10} +$$$$80\!\cdots\!28$$$$T^{11} +$$$$13\!\cdots\!81$$$$T^{12}$$
$67$ $$1 - 632 T + 927628 T^{2} - 253029932 T^{3} + 179674044568 T^{4} + 62778009874096 T^{5} - 5212390617577006 T^{6} + 18881302583762735248 T^{7} +$$$$16\!\cdots\!92$$$$T^{8} -$$$$68\!\cdots\!04$$$$T^{9} +$$$$75\!\cdots\!08$$$$T^{10} -$$$$15\!\cdots\!76$$$$T^{11} +$$$$74\!\cdots\!09$$$$T^{12}$$
$71$ $$1 - 144 T + 863230 T^{2} - 72744496 T^{3} + 475313592223 T^{4} - 62333254020128 T^{5} + 209643801201434276 T^{6} - 22309757279598032608 T^{7} +$$$$60\!\cdots\!83$$$$T^{8} -$$$$33\!\cdots\!76$$$$T^{9} +$$$$14\!\cdots\!30$$$$T^{10} -$$$$84\!\cdots\!44$$$$T^{11} +$$$$21\!\cdots\!61$$$$T^{12}$$
$73$ $$1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 81531488761123023064 T^{7} +$$$$69\!\cdots\!63$$$$T^{8} -$$$$25\!\cdots\!44$$$$T^{9} +$$$$24\!\cdots\!18$$$$T^{10} -$$$$76\!\cdots\!48$$$$T^{11} +$$$$34\!\cdots\!69$$$$T^{12}$$
$79$ $$1 - 1613 T + 2731081 T^{2} - 3130876751 T^{3} + 3268965374139 T^{4} - 2799662350208018 T^{5} + 2111366737833161318 T^{6} -$$$$13\!\cdots\!02$$$$T^{7} +$$$$79\!\cdots\!19$$$$T^{8} -$$$$37\!\cdots\!69$$$$T^{9} +$$$$16\!\cdots\!21$$$$T^{10} -$$$$46\!\cdots\!87$$$$T^{11} +$$$$14\!\cdots\!61$$$$T^{12}$$
$83$ $$1 - 682 T + 1120004 T^{2} - 1783287782 T^{3} + 1291656570448 T^{4} - 1149121488958882 T^{5} + 1227368803599345682 T^{6} -$$$$65\!\cdots\!34$$$$T^{7} +$$$$42\!\cdots\!12$$$$T^{8} -$$$$33\!\cdots\!46$$$$T^{9} +$$$$11\!\cdots\!44$$$$T^{10} -$$$$41\!\cdots\!74$$$$T^{11} +$$$$34\!\cdots\!09$$$$T^{12}$$
$89$ $$1 - 3378 T + 7851850 T^{2} - 13055260850 T^{3} + 17370332054203 T^{4} - 19017874978895668 T^{5} + 17369747195795800052 T^{6} -$$$$13\!\cdots\!92$$$$T^{7} +$$$$86\!\cdots\!83$$$$T^{8} -$$$$45\!\cdots\!50$$$$T^{9} +$$$$19\!\cdots\!50$$$$T^{10} -$$$$58\!\cdots\!22$$$$T^{11} +$$$$12\!\cdots\!81$$$$T^{12}$$
$97$ $$1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} -$$$$15\!\cdots\!33$$$$T^{7} +$$$$27\!\cdots\!30$$$$T^{8} -$$$$71\!\cdots\!87$$$$T^{9} +$$$$17\!\cdots\!11$$$$T^{10} +$$$$34\!\cdots\!15$$$$T^{11} +$$$$57\!\cdots\!89$$$$T^{12}$$