# Properties

 Label 5.7.c.a Level 5 Weight 7 Character orbit 5.c Analytic conductor 1.150 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 5.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.15027041810$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{201})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{2} + ( 7 + 8 \beta_{1} + \beta_{2} ) q^{3} + ( -49 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{4} + ( -25 + 60 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{5} + ( -128 - 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{6} + ( 143 - 143 \beta_{1} + 11 \beta_{3} ) q^{7} + ( 460 + 470 \beta_{1} + 10 \beta_{2} ) q^{8} + ( -516 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{2} + ( 7 + 8 \beta_{1} + \beta_{2} ) q^{3} + ( -49 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{4} + ( -25 + 60 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{5} + ( -128 - 10 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{6} + ( 143 - 143 \beta_{1} + 11 \beta_{3} ) q^{7} + ( 460 + 470 \beta_{1} + 10 \beta_{2} ) q^{8} + ( -516 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{9} + ( -1050 + 285 \beta_{1} - 65 \beta_{2} + 20 \beta_{3} ) q^{10} + ( -338 + 75 \beta_{1} + 75 \beta_{2} - 75 \beta_{3} ) q^{11} + ( 808 - 808 \beta_{1} - 124 \beta_{3} ) q^{12} + ( 557 + 423 \beta_{1} - 134 \beta_{2} ) q^{13} + ( -418 \beta_{1} + 110 \beta_{2} + 110 \beta_{3} ) q^{14} + ( -25 + 1270 \beta_{1} + 95 \beta_{2} + 90 \beta_{3} ) q^{15} + ( -24 - 170 \beta_{1} - 170 \beta_{2} + 170 \beta_{3} ) q^{16} + ( -667 + 667 \beta_{1} + 306 \beta_{3} ) q^{17} + ( -438 + 3 \beta_{1} + 441 \beta_{2} ) q^{18} + ( 1110 \beta_{1} - 570 \beta_{2} - 570 \beta_{3} ) q^{19} + ( 4700 - 6295 \beta_{1} + 105 \beta_{2} - 665 \beta_{3} ) q^{20} + ( 902 + 55 \beta_{1} + 55 \beta_{2} - 55 \beta_{3} ) q^{21} + ( -6824 + 6824 \beta_{1} + 112 \beta_{3} ) q^{22} + ( -9593 - 9912 \beta_{1} - 319 \beta_{2} ) q^{23} + ( 7980 \beta_{1} + 540 \beta_{2} + 540 \beta_{3} ) q^{24} + ( 8125 + 5175 \beta_{1} + 175 \beta_{2} + 1225 \beta_{3} ) q^{25} + ( 11172 - 155 \beta_{1} - 155 \beta_{2} + 155 \beta_{3} ) q^{26} + ( 7320 - 7320 \beta_{1} - 1020 \beta_{3} ) q^{27} + ( -792 - 1628 \beta_{1} - 836 \beta_{2} ) q^{28} + ( 10440 \beta_{1} + 1120 \beta_{2} + 1120 \beta_{3} ) q^{29} + ( -11800 - 13130 \beta_{1} - 1730 \beta_{2} - 10 \beta_{3} ) q^{30} + ( -10018 + 1725 \beta_{1} + 1725 \beta_{2} - 1725 \beta_{3} ) q^{31} + ( -12392 + 12392 \beta_{1} - 404 \beta_{3} ) q^{32} + ( 5134 + 5996 \beta_{1} + 862 \beta_{2} ) q^{33} + ( -34853 \beta_{1} - 1585 \beta_{2} - 1585 \beta_{3} ) q^{34} + ( -7425 + 16335 \beta_{1} + 110 \beta_{2} - 2255 \beta_{3} ) q^{35} + ( -8364 - 1845 \beta_{1} - 1845 \beta_{2} + 1845 \beta_{3} ) q^{36} + ( 38113 - 38113 \beta_{1} + 2796 \beta_{3} ) q^{37} + ( 53640 + 55380 \beta_{1} + 1740 \beta_{2} ) q^{38} + ( -6117 \beta_{1} - 515 \beta_{2} - 515 \beta_{3} ) q^{39} + ( -29500 + 28300 \beta_{1} + 6800 \beta_{2} + 2850 \beta_{3} ) q^{40} + ( -48458 - 5025 \beta_{1} - 5025 \beta_{2} + 5025 \beta_{3} ) q^{41} + ( -7304 + 7304 \beta_{1} + 1232 \beta_{3} ) q^{42} + ( -17193 - 18832 \beta_{1} - 1639 \beta_{2} ) q^{43} + ( -62488 \beta_{1} - 2360 \beta_{2} - 2360 \beta_{3} ) q^{44} + ( 19050 + 32145 \beta_{1} - 3630 \beta_{2} - 4635 \beta_{3} ) q^{45} + ( 70272 + 10550 \beta_{1} + 10550 \beta_{2} - 10550 \beta_{3} ) q^{46} + ( -2297 + 2297 \beta_{1} - 5009 \beta_{3} ) q^{47} + ( -17168 - 19912 \beta_{1} - 2744 \beta_{2} ) q^{48} + ( 67676 \beta_{1} + 3025 \beta_{2} + 3025 \beta_{3} ) q^{49} + ( -43750 - 125450 \beta_{1} - 9200 \beta_{2} + 4975 \beta_{3} ) q^{50} + ( -39938 - 3115 \beta_{1} - 3115 \beta_{2} + 3115 \beta_{3} ) q^{51} + ( -42492 + 42492 \beta_{1} + 1666 \beta_{3} ) q^{52} + ( -11143 - 3667 \beta_{1} + 7476 \beta_{2} ) q^{53} + ( 141660 \beta_{1} + 10380 \beta_{2} + 10380 \beta_{3} ) q^{54} + ( 120950 + 20595 \beta_{1} - 5 \beta_{2} + 6565 \beta_{3} ) q^{55} + ( 120560 - 3740 \beta_{1} - 3740 \beta_{2} + 3740 \beta_{3} ) q^{56} + ( 45240 - 45240 \beta_{1} - 7440 \beta_{3} ) q^{57} + ( -130640 - 146680 \beta_{1} - 16040 \beta_{2} ) q^{58} + ( 20430 \beta_{1} - 1810 \beta_{2} - 1810 \beta_{3} ) q^{59} + ( 144200 + 9460 \beta_{1} + 10860 \beta_{2} - 10880 \beta_{3} ) q^{60} + ( -31138 + 3375 \beta_{1} + 3375 \beta_{2} - 3375 \beta_{3} ) q^{61} + ( -152464 + 152464 \beta_{1} + 332 \beta_{3} ) q^{62} + ( -92433 - 82632 \beta_{1} + 9801 \beta_{2} ) q^{63} + ( -32764 \beta_{1} - 22060 \beta_{2} - 22060 \beta_{3} ) q^{64} + ( -108775 - 110380 \beta_{1} + 9695 \beta_{2} - 4585 \beta_{3} ) q^{65} + ( -106736 - 7720 \beta_{1} - 7720 \beta_{2} + 7720 \beta_{3} ) q^{66} + ( -23217 + 23217 \beta_{1} + 27031 \beta_{3} ) q^{67} + ( 182348 + 205542 \beta_{1} + 23194 \beta_{2} ) q^{68} + ( -178347 \beta_{1} - 12145 \beta_{2} - 12145 \beta_{3} ) q^{69} + ( -28600 + 168410 \beta_{1} - 9790 \beta_{2} - 330 \beta_{3} ) q^{70} + ( 317422 + 10125 \beta_{1} + 10125 \beta_{2} - 10125 \beta_{3} ) q^{71} + ( 229260 - 229260 \beta_{1} + 8790 \beta_{3} ) q^{72} + ( 149857 + 126913 \beta_{1} - 22944 \beta_{2} ) q^{73} + ( -97423 \beta_{1} + 29725 \beta_{2} + 29725 \beta_{3} ) q^{74} + ( -100625 + 109100 \beta_{1} - 275 \beta_{2} + 16200 \beta_{3} ) q^{75} + ( -496080 - 22380 \beta_{1} - 22380 \beta_{2} + 22380 \beta_{3} ) q^{76} + ( -130834 + 130834 \beta_{1} - 23518 \beta_{3} ) q^{77} + ( 62704 + 71396 \beta_{1} + 8692 \beta_{2} ) q^{78} + ( 149040 \beta_{1} + 23120 \beta_{2} + 23120 \beta_{3} ) q^{79} + ( -254400 - 94090 \beta_{1} - 7890 \beta_{2} - 10930 \beta_{3} ) q^{80} + ( -182619 + 26415 \beta_{1} + 26415 \beta_{2} - 26415 \beta_{3} ) q^{81} + ( 599416 - 599416 \beta_{1} - 78608 \beta_{3} ) q^{82} + ( 164607 + 103828 \beta_{1} - 60779 \beta_{2} ) q^{83} + ( -102168 \beta_{1} - 7480 \beta_{2} - 7480 \beta_{3} ) q^{84} + ( -322675 + 82810 \beta_{1} - 20165 \beta_{2} + 6945 \beta_{3} ) q^{85} + ( 232672 + 22110 \beta_{1} + 22110 \beta_{2} - 22110 \beta_{3} ) q^{86} + ( -177240 + 177240 \beta_{1} + 27240 \beta_{3} ) q^{87} + ( -80480 - 13360 \beta_{1} + 67120 \beta_{2} ) q^{88} + ( -541680 \beta_{1} - 20640 \beta_{2} - 20640 \beta_{3} ) q^{89} + ( 253350 + 419070 \beta_{1} - 10980 \beta_{2} + 22065 \beta_{3} ) q^{90} + ( 306702 - 23815 \beta_{1} - 23815 \beta_{2} + 23815 \beta_{3} ) q^{91} + ( -581592 + 581592 \beta_{1} + 113156 \beta_{3} ) q^{92} + ( 102374 + 119956 \beta_{1} + 17582 \beta_{2} ) q^{93} + ( 504442 \beta_{1} + 12730 \beta_{2} + 12730 \beta_{3} ) q^{94} + ( 201000 - 851550 \beta_{1} + 45450 \beta_{2} - 8850 \beta_{3} ) q^{95} + ( -133088 - 9160 \beta_{1} - 9160 \beta_{2} + 9160 \beta_{3} ) q^{96} + ( -11647 + 11647 \beta_{1} + 6416 \beta_{3} ) q^{97} + ( -431802 - 514603 \beta_{1} - 82801 \beta_{2} ) q^{98} + ( 361833 \beta_{1} - 42645 \beta_{2} - 42645 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 10q^{2} + 30q^{3} - 70q^{5} - 552q^{6} + 550q^{7} + 1860q^{8} + O(q^{10})$$ $$4q - 10q^{2} + 30q^{3} - 70q^{5} - 552q^{6} + 550q^{7} + 1860q^{8} - 4370q^{10} - 1052q^{11} + 3480q^{12} + 1960q^{13} - 90q^{15} - 776q^{16} - 3280q^{17} - 870q^{18} + 20340q^{20} + 3828q^{21} - 27520q^{22} - 39010q^{23} + 30400q^{25} + 44068q^{26} + 31320q^{27} - 4840q^{28} - 50640q^{30} - 33172q^{31} - 48760q^{32} + 22260q^{33} - 24970q^{35} - 40836q^{36} + 146860q^{37} + 218040q^{38} - 110100q^{40} - 213932q^{41} - 31680q^{42} - 72050q^{43} + 78210q^{45} + 323288q^{46} + 830q^{47} - 74160q^{48} - 203350q^{50} - 172212q^{51} - 173300q^{52} - 29620q^{53} + 470660q^{55} + 467280q^{56} + 195840q^{57} - 554640q^{58} + 620280q^{60} - 111052q^{61} - 610520q^{62} - 350130q^{63} - 406540q^{65} - 457824q^{66} - 146930q^{67} + 775780q^{68} - 133320q^{70} + 1310188q^{71} + 899460q^{72} + 553540q^{73} - 435450q^{75} - 2073840q^{76} - 476300q^{77} + 268200q^{78} - 1011520q^{80} - 624816q^{81} + 2554880q^{82} + 536870q^{83} - 1344920q^{85} + 1019128q^{86} - 763440q^{87} - 187680q^{88} + 947310q^{90} + 1131548q^{91} - 2552680q^{92} + 444660q^{93} + 912600q^{95} - 568992q^{96} - 59420q^{97} - 1892810q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 101 x^{2} + 2500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 51 \nu$$$$)/50$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu + 51$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 50 \nu^{2} + 101 \nu - 2550$$$$)/50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + \beta_{1} - 102$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-51 \beta_{3} - 51 \beta_{2} + 151 \beta_{1}$$$$)/2$$

## 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
 − 6.58872i 7.58872i 6.58872i − 7.58872i
−9.58872 9.58872i 14.5887 14.5887i 119.887i 88.8309 87.9436i −279.774 59.5240 + 59.5240i 535.887 535.887i 303.338i −1695.04 8.50745i
2.2 4.58872 + 4.58872i 0.411277 0.411277i 21.8872i −123.831 17.0564i 3.77447 215.476 + 215.476i 394.113 394.113i 728.662i −489.959 646.493i
3.1 −9.58872 + 9.58872i 14.5887 + 14.5887i 119.887i 88.8309 + 87.9436i −279.774 59.5240 59.5240i 535.887 + 535.887i 303.338i −1695.04 + 8.50745i
3.2 4.58872 4.58872i 0.411277 + 0.411277i 21.8872i −123.831 + 17.0564i 3.77447 215.476 215.476i 394.113 + 394.113i 728.662i −489.959 + 646.493i
 $$n$$: e.g. 2-40 or 990-1000 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 5.7.c.a 4
3.b odd 2 1 45.7.g.a 4
4.b odd 2 1 80.7.p.b 4
5.b even 2 1 25.7.c.c 4
5.c odd 4 1 inner 5.7.c.a 4
5.c odd 4 1 25.7.c.c 4
15.d odd 2 1 225.7.g.c 4
15.e even 4 1 45.7.g.a 4
15.e even 4 1 225.7.g.c 4
20.e even 4 1 80.7.p.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.7.c.a 4 1.a even 1 1 trivial
5.7.c.a 4 5.c odd 4 1 inner
25.7.c.c 4 5.b even 2 1
25.7.c.c 4 5.c odd 4 1
45.7.g.a 4 3.b odd 2 1
45.7.g.a 4 15.e even 4 1
80.7.p.b 4 4.b odd 2 1
80.7.p.b 4 20.e even 4 1
225.7.g.c 4 15.d odd 2 1
225.7.g.c 4 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(5, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 10 T + 50 T^{2} - 240 T^{3} - 6592 T^{4} - 15360 T^{5} + 204800 T^{6} + 2621440 T^{7} + 16777216 T^{8}$$
$3$ $$1 - 30 T + 450 T^{2} - 22230 T^{3} + 1098018 T^{4} - 16205670 T^{5} + 239148450 T^{6} - 11622614670 T^{7} + 282429536481 T^{8}$$
$5$ $$1 + 70 T - 12750 T^{2} + 1093750 T^{3} + 244140625 T^{4}$$
$7$ $$1 - 550 T + 151250 T^{2} - 78815550 T^{3} + 40412328098 T^{4} - 9272570641950 T^{5} + 2093494689151250 T^{6} - 895627478850746950 T^{7} +$$$$19\!\cdots\!01$$$$T^{8}$$
$11$ $$( 1 + 526 T + 2481666 T^{2} + 931841086 T^{3} + 3138428376721 T^{4} )^{2}$$
$13$ $$1 - 1960 T + 1920800 T^{2} - 6864764760 T^{3} + 22780068732638 T^{4} - 33134908326450840 T^{5} + 44750961903261504800 T^{6} -$$$$22\!\cdots\!40$$$$T^{7} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$1 + 3280 T + 5379200 T^{2} + 52715999280 T^{3} + 451561024170878 T^{4} + 1272436070024950320 T^{5} +$$$$31\!\cdots\!00$$$$T^{6} +$$$$46\!\cdots\!20$$$$T^{7} +$$$$33\!\cdots\!21$$$$T^{8}$$
$19$ $$1 - 55109524 T^{2} + 4864046077848966 T^{4} -$$$$12\!\cdots\!64$$$$T^{6} +$$$$48\!\cdots\!21$$$$T^{8}$$
$23$ $$1 + 39010 T + 760890050 T^{2} + 12796505733210 T^{3} + 182810834786595458 T^{4} +$$$$18\!\cdots\!90$$$$T^{5} +$$$$16\!\cdots\!50$$$$T^{6} +$$$$12\!\cdots\!90$$$$T^{7} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$1 - 1657037284 T^{2} + 1284148562793102246 T^{4} -$$$$58\!\cdots\!44$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8}$$
$31$ $$( 1 + 16586 T + 1245680586 T^{2} + 14720136053066 T^{3} + 787662783788549761 T^{4} )^{2}$$
$37$ $$1 - 146860 T + 10783929800 T^{2} - 657351006913860 T^{3} + 36420548331850749038 T^{4} -$$$$16\!\cdots\!40$$$$T^{5} +$$$$70\!\cdots\!00$$$$T^{6} -$$$$24\!\cdots\!40$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$( 1 + 106966 T + 7285264146 T^{2} + 508099650242806 T^{3} + 22563490300366186081 T^{4} )^{2}$$
$43$ $$1 + 72050 T + 2595601250 T^{2} + 482755757677050 T^{3} + 89644137077771169698 T^{4} +$$$$30\!\cdots\!50$$$$T^{5} +$$$$10\!\cdots\!50$$$$T^{6} +$$$$18\!\cdots\!50$$$$T^{7} +$$$$15\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 830 T + 344450 T^{2} - 6853931089830 T^{3} +$$$$13\!\cdots\!18$$$$T^{4} -$$$$73\!\cdots\!70$$$$T^{5} +$$$$40\!\cdots\!50$$$$T^{6} -$$$$10\!\cdots\!70$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 + 29620 T + 438672200 T^{2} + 493381118739420 T^{3} +$$$$52\!\cdots\!18$$$$T^{4} +$$$$10\!\cdots\!80$$$$T^{5} +$$$$21\!\cdots\!00$$$$T^{6} +$$$$32\!\cdots\!80$$$$T^{7} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$1 - 166570372564 T^{2} +$$$$10\!\cdots\!86$$$$T^{4} -$$$$29\!\cdots\!84$$$$T^{6} +$$$$31\!\cdots\!61$$$$T^{8}$$
$61$ $$( 1 + 55526 T + 101522017266 T^{2} + 2860720306768886 T^{3} +$$$$26\!\cdots\!21$$$$T^{4} )^{2}$$
$67$ $$1 + 146930 T + 10794212450 T^{2} + 2898062262927930 T^{3} -$$$$42\!\cdots\!22$$$$T^{4} +$$$$26\!\cdots\!70$$$$T^{5} +$$$$88\!\cdots\!50$$$$T^{6} +$$$$10\!\cdots\!70$$$$T^{7} +$$$$66\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 - 655094 T + 342881964426 T^{2} - 83917727394943574 T^{3} +$$$$16\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 - 553540 T + 153203265800 T^{2} - 75685034633171340 T^{3} +$$$$37\!\cdots\!58$$$$T^{4} -$$$$11\!\cdots\!60$$$$T^{5} +$$$$35\!\cdots\!00$$$$T^{6} -$$$$19\!\cdots\!60$$$$T^{7} +$$$$52\!\cdots\!41$$$$T^{8}$$
$79$ $$1 - 713041150084 T^{2} +$$$$23\!\cdots\!46$$$$T^{4} -$$$$42\!\cdots\!44$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$1 - 536870 T + 144114698450 T^{2} + 4448869644102930 T^{3} -$$$$11\!\cdots\!22$$$$T^{4} +$$$$14\!\cdots\!70$$$$T^{5} +$$$$15\!\cdots\!50$$$$T^{6} -$$$$18\!\cdots\!30$$$$T^{7} +$$$$11\!\cdots\!21$$$$T^{8}$$
$89$ $$1 - 1229834859844 T^{2} +$$$$77\!\cdots\!26$$$$T^{4} -$$$$30\!\cdots\!24$$$$T^{6} +$$$$61\!\cdots\!41$$$$T^{8}$$
$97$ $$1 + 59420 T + 1765368200 T^{2} + 49275595300926420 T^{3} +$$$$13\!\cdots\!18$$$$T^{4} +$$$$41\!\cdots\!80$$$$T^{5} +$$$$12\!\cdots\!00$$$$T^{6} +$$$$34\!\cdots\!80$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8}$$