# Properties

 Label 630.3.v.c Level 630 Weight 3 Character orbit 630.v Analytic conductor 17.166 Analytic rank 0 Dimension 16 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 630.v (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.1662566547$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 7$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{6} ) q^{2} + 2 \beta_{3} q^{4} -\beta_{4} q^{5} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{10} ) q^{7} + 2 \beta_{1} q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{6} ) q^{2} + 2 \beta_{3} q^{4} -\beta_{4} q^{5} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{10} ) q^{7} + 2 \beta_{1} q^{8} -\beta_{7} q^{10} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{8} + \beta_{13} + \beta_{15} ) q^{11} + ( 2 + 4 \beta_{3} - \beta_{5} - \beta_{8} - \beta_{11} - \beta_{13} - \beta_{15} ) q^{13} + ( -\beta_{1} + \beta_{3} + \beta_{13} ) q^{14} + ( -4 - 4 \beta_{3} ) q^{16} + ( -1 + \beta_{1} + \beta_{5} + 5 \beta_{7} - \beta_{11} - \beta_{12} ) q^{17} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{19} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{20} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{22} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{23} -5 \beta_{3} q^{25} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{26} + ( 4 + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{28} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{29} + ( 9 + \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - 7 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{31} -4 \beta_{6} q^{32} + ( -1 - 2 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{34} + ( -1 - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{12} + \beta_{15} ) q^{35} + ( 6 + 8 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} - 9 \beta_{6} + 12 \beta_{7} - 7 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{37} + ( 6 + 7 \beta_{1} + 2 \beta_{3} - 5 \beta_{6} + 5 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{14} - \beta_{15} ) q^{38} + 2 \beta_{8} q^{40} + ( 3 + 6 \beta_{1} + 7 \beta_{3} + 3 \beta_{4} + \beta_{5} - 10 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{11} + \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{41} + ( -4 - 6 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{43} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{15} ) q^{44} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 4 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} - 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{46} + ( -2 - 2 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} - 8 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{15} ) q^{47} + ( -6 + 13 \beta_{1} - \beta_{2} - 6 \beta_{3} + 10 \beta_{4} + 19 \beta_{5} + 4 \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{49} -5 \beta_{1} q^{50} + ( -4 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{52} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 14 \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{53} + ( 2 + \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{55} + ( -2 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} ) q^{56} + ( -9 + \beta_{1} - 7 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + 11 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + \beta_{14} + \beta_{15} ) q^{58} + ( -9 - 20 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 14 \beta_{5} + 16 \beta_{6} + 14 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} + 3 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{59} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 15 \beta_{8} + 3 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} - 3 \beta_{12} + 4 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{61} + ( -2 - 5 \beta_{1} - 4 \beta_{3} + 12 \beta_{4} + 14 \beta_{5} + 10 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{15} ) q^{62} + 8 q^{64} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{65} + ( -3 + 2 \beta_{1} + 17 \beta_{3} - 6 \beta_{4} - \beta_{5} + 28 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} + 5 \beta_{10} + \beta_{11} + \beta_{12} + 4 \beta_{13} + \beta_{14} + \beta_{15} ) q^{67} + ( 2 - 2 \beta_{1} + 4 \beta_{4} - 10 \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} ) q^{68} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{70} + ( 6 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - \beta_{13} - 2 \beta_{14} ) q^{71} + ( -28 - 29 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} + \beta_{4} - 4 \beta_{5} + 18 \beta_{6} - 7 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + 3 \beta_{13} - 3 \beta_{15} ) q^{73} + ( -1 - 3 \beta_{2} - 12 \beta_{3} - 23 \beta_{4} - 14 \beta_{5} + 7 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{74} + ( -8 - 2 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} - 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{15} ) q^{76} + ( 6 + 30 \beta_{1} + 2 \beta_{2} + 22 \beta_{3} - 12 \beta_{4} - 17 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} + 19 \beta_{8} + \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{77} + ( 35 + \beta_{1} + \beta_{2} + 34 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} - \beta_{6} + 22 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{14} ) q^{79} -4 \beta_{5} q^{80} + ( 7 + \beta_{1} - 4 \beta_{2} - 13 \beta_{3} - 2 \beta_{5} + 7 \beta_{6} + \beta_{7} + 8 \beta_{8} + 4 \beta_{9} + \beta_{10} + 2 \beta_{12} + 6 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{82} + ( -11 + \beta_{1} + \beta_{2} - 25 \beta_{3} - 5 \beta_{4} + \beta_{5} - 5 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 8 \beta_{11} - \beta_{12} + 3 \beta_{13} + 4 \beta_{14} + 3 \beta_{15} ) q^{83} + ( 5 + 27 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + 4 \beta_{11} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{85} + ( 11 + 2 \beta_{1} + 13 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + \beta_{14} + 5 \beta_{15} ) q^{86} + ( -2 - 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{88} + ( 16 - 13 \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + \beta_{7} - 31 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} - 7 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{89} + ( -7 + 14 \beta_{1} - 2 \beta_{2} + 24 \beta_{3} + 11 \beta_{4} + 17 \beta_{5} - 34 \beta_{6} + 5 \beta_{7} - 16 \beta_{8} + 3 \beta_{9} - 4 \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{91} + ( -6 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{92} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 24 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{94} + ( -1 - 2 \beta_{2} + 25 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{95} + ( -14 + 29 \beta_{1} - \beta_{2} - 26 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 57 \beta_{6} - 17 \beta_{7} + 20 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} + 3 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{97} + ( -35 + 2 \beta_{1} + \beta_{2} - 23 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} + 16 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + 4q^{7} + O(q^{10})$$ $$16q - 16q^{4} + 4q^{7} + 4q^{11} - 8q^{14} - 32q^{16} - 12q^{17} - 72q^{19} - 48q^{22} + 12q^{23} + 40q^{25} + 32q^{28} - 72q^{29} + 120q^{31} + 20q^{35} + 44q^{37} + 72q^{38} - 56q^{43} + 8q^{44} + 8q^{46} + 24q^{47} - 40q^{49} - 72q^{52} - 32q^{53} - 16q^{56} - 88q^{58} - 132q^{59} + 96q^{61} + 128q^{64} - 20q^{65} - 164q^{67} + 24q^{68} + 136q^{71} - 348q^{73} + 112q^{74} - 96q^{77} + 280q^{79} + 264q^{82} + 120q^{85} + 88q^{86} + 48q^{88} + 300q^{89} - 272q^{91} - 48q^{92} - 200q^{95} - 384q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + 595763862 x^{4} - 630430976 x^{3} + 1087013404 x^{2} + 294123256 x + 101626561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-57754008957283349 \nu^{15} - 676562680999836294 \nu^{14} - 8239103585403158159 \nu^{13} - 54078590776251682475 \nu^{12} - 455975368108340251668 \nu^{11} - 2793391292439344623383 \nu^{10} - 16545566601055515429559 \nu^{9} - 76261143494901216401580 \nu^{8} - 306572240997688184788012 \nu^{7} - 1467974756910134628134465 \nu^{6} - 3497231496721962844392765 \nu^{5} - 10591214719874231223793348 \nu^{4} + 4685579583089306764764583 \nu^{3} - 35874681437029057265441973 \nu^{2} - 9955887688266404429946106 \nu + 1202728311133779179554909109$$$$)/$$$$85\!\cdots\!24$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$84\!\cdots\!59$$$$\nu^{15} +$$$$18\!\cdots\!59$$$$\nu^{14} -$$$$10\!\cdots\!76$$$$\nu^{13} +$$$$17\!\cdots\!32$$$$\nu^{12} -$$$$93\!\cdots\!61$$$$\nu^{11} +$$$$11\!\cdots\!31$$$$\nu^{10} -$$$$46\!\cdots\!43$$$$\nu^{9} +$$$$37\!\cdots\!13$$$$\nu^{8} -$$$$14\!\cdots\!75$$$$\nu^{7} +$$$$92\!\cdots\!21$$$$\nu^{6} -$$$$29\!\cdots\!01$$$$\nu^{5} +$$$$12\!\cdots\!27$$$$\nu^{4} -$$$$35\!\cdots\!88$$$$\nu^{3} +$$$$11\!\cdots\!10$$$$\nu^{2} -$$$$23\!\cdots\!43$$$$\nu +$$$$10\!\cdots\!97$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{3}$$ $$=$$ $$($$$$17\!\cdots\!52$$$$\nu^{15} -$$$$54\!\cdots\!68$$$$\nu^{14} +$$$$15\!\cdots\!20$$$$\nu^{13} -$$$$24\!\cdots\!26$$$$\nu^{12} +$$$$10\!\cdots\!84$$$$\nu^{11} -$$$$16\!\cdots\!08$$$$\nu^{10} +$$$$34\!\cdots\!20$$$$\nu^{9} -$$$$61\!\cdots\!59$$$$\nu^{8} +$$$$84\!\cdots\!08$$$$\nu^{7} -$$$$14\!\cdots\!20$$$$\nu^{6} +$$$$11\!\cdots\!24$$$$\nu^{5} -$$$$18\!\cdots\!30$$$$\nu^{4} +$$$$10\!\cdots\!24$$$$\nu^{3} -$$$$13\!\cdots\!92$$$$\nu^{2} +$$$$18\!\cdots\!72$$$$\nu -$$$$13\!\cdots\!85$$$$)/$$$$62\!\cdots\!01$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!96$$$$\nu^{15} +$$$$38\!\cdots\!85$$$$\nu^{14} -$$$$13\!\cdots\!72$$$$\nu^{13} +$$$$20\!\cdots\!38$$$$\nu^{12} -$$$$88\!\cdots\!08$$$$\nu^{11} +$$$$14\!\cdots\!26$$$$\nu^{10} -$$$$29\!\cdots\!68$$$$\nu^{9} +$$$$53\!\cdots\!48$$$$\nu^{8} -$$$$73\!\cdots\!36$$$$\nu^{7} +$$$$12\!\cdots\!58$$$$\nu^{6} -$$$$99\!\cdots\!36$$$$\nu^{5} +$$$$16\!\cdots\!18$$$$\nu^{4} -$$$$93\!\cdots\!40$$$$\nu^{3} +$$$$12\!\cdots\!43$$$$\nu^{2} -$$$$16\!\cdots\!76$$$$\nu +$$$$68\!\cdots\!16$$$$)/$$$$43\!\cdots\!02$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!20$$$$\nu^{15} +$$$$67\!\cdots\!15$$$$\nu^{14} -$$$$14\!\cdots\!96$$$$\nu^{13} +$$$$23\!\cdots\!76$$$$\nu^{12} -$$$$92\!\cdots\!60$$$$\nu^{11} +$$$$15\!\cdots\!89$$$$\nu^{10} -$$$$31\!\cdots\!16$$$$\nu^{9} +$$$$59\!\cdots\!84$$$$\nu^{8} -$$$$77\!\cdots\!16$$$$\nu^{7} +$$$$13\!\cdots\!95$$$$\nu^{6} -$$$$10\!\cdots\!04$$$$\nu^{5} +$$$$17\!\cdots\!02$$$$\nu^{4} -$$$$94\!\cdots\!28$$$$\nu^{3} +$$$$12\!\cdots\!24$$$$\nu^{2} -$$$$16\!\cdots\!36$$$$\nu -$$$$10\!\cdots\!40$$$$)/$$$$43\!\cdots\!02$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!02$$$$\nu^{15} +$$$$21\!\cdots\!67$$$$\nu^{14} -$$$$12\!\cdots\!27$$$$\nu^{13} +$$$$17\!\cdots\!98$$$$\nu^{12} -$$$$77\!\cdots\!07$$$$\nu^{11} +$$$$11\!\cdots\!47$$$$\nu^{10} -$$$$26\!\cdots\!76$$$$\nu^{9} +$$$$45\!\cdots\!39$$$$\nu^{8} -$$$$63\!\cdots\!65$$$$\nu^{7} +$$$$10\!\cdots\!73$$$$\nu^{6} -$$$$83\!\cdots\!68$$$$\nu^{5} +$$$$14\!\cdots\!09$$$$\nu^{4} -$$$$76\!\cdots\!85$$$$\nu^{3} +$$$$10\!\cdots\!02$$$$\nu^{2} -$$$$13\!\cdots\!99$$$$\nu +$$$$98\!\cdots\!05$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!08$$$$\nu^{15} +$$$$23\!\cdots\!47$$$$\nu^{14} -$$$$34\!\cdots\!47$$$$\nu^{13} +$$$$63\!\cdots\!32$$$$\nu^{12} -$$$$22\!\cdots\!79$$$$\nu^{11} +$$$$42\!\cdots\!09$$$$\nu^{10} -$$$$74\!\cdots\!82$$$$\nu^{9} +$$$$15\!\cdots\!65$$$$\nu^{8} -$$$$18\!\cdots\!33$$$$\nu^{7} +$$$$35\!\cdots\!35$$$$\nu^{6} -$$$$24\!\cdots\!50$$$$\nu^{5} +$$$$45\!\cdots\!11$$$$\nu^{4} -$$$$22\!\cdots\!31$$$$\nu^{3} +$$$$30\!\cdots\!16$$$$\nu^{2} -$$$$38\!\cdots\!01$$$$\nu -$$$$23\!\cdots\!35$$$$)/$$$$73\!\cdots\!88$$ $$\beta_{8}$$ $$=$$ $$($$$$38\!\cdots\!37$$$$\nu^{15} -$$$$24\!\cdots\!57$$$$\nu^{14} +$$$$35\!\cdots\!20$$$$\nu^{13} -$$$$47\!\cdots\!39$$$$\nu^{12} +$$$$22\!\cdots\!83$$$$\nu^{11} -$$$$32\!\cdots\!10$$$$\nu^{10} +$$$$75\!\cdots\!95$$$$\nu^{9} -$$$$12\!\cdots\!37$$$$\nu^{8} +$$$$18\!\cdots\!93$$$$\nu^{7} -$$$$28\!\cdots\!18$$$$\nu^{6} +$$$$24\!\cdots\!61$$$$\nu^{5} -$$$$41\!\cdots\!79$$$$\nu^{4} +$$$$22\!\cdots\!80$$$$\nu^{3} -$$$$29\!\cdots\!03$$$$\nu^{2} +$$$$39\!\cdots\!71$$$$\nu -$$$$16\!\cdots\!40$$$$)/$$$$73\!\cdots\!88$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$22\!\cdots\!59$$$$\nu^{15} +$$$$37\!\cdots\!28$$$$\nu^{14} -$$$$21\!\cdots\!43$$$$\nu^{13} +$$$$56\!\cdots\!51$$$$\nu^{12} -$$$$14\!\cdots\!38$$$$\nu^{11} +$$$$38\!\cdots\!49$$$$\nu^{10} -$$$$51\!\cdots\!07$$$$\nu^{9} +$$$$13\!\cdots\!14$$$$\nu^{8} -$$$$13\!\cdots\!66$$$$\nu^{7} +$$$$32\!\cdots\!47$$$$\nu^{6} -$$$$19\!\cdots\!69$$$$\nu^{5} +$$$$43\!\cdots\!94$$$$\nu^{4} -$$$$20\!\cdots\!81$$$$\nu^{3} +$$$$37\!\cdots\!81$$$$\nu^{2} -$$$$69\!\cdots\!98$$$$\nu +$$$$21\!\cdots\!05$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$25\!\cdots\!09$$$$\nu^{15} +$$$$22\!\cdots\!54$$$$\nu^{14} -$$$$23\!\cdots\!02$$$$\nu^{13} +$$$$50\!\cdots\!26$$$$\nu^{12} -$$$$15\!\cdots\!37$$$$\nu^{11} +$$$$34\!\cdots\!02$$$$\nu^{10} -$$$$53\!\cdots\!69$$$$\nu^{9} +$$$$12\!\cdots\!16$$$$\nu^{8} -$$$$13\!\cdots\!55$$$$\nu^{7} +$$$$30\!\cdots\!18$$$$\nu^{6} -$$$$18\!\cdots\!19$$$$\nu^{5} +$$$$41\!\cdots\!96$$$$\nu^{4} -$$$$18\!\cdots\!74$$$$\nu^{3} +$$$$32\!\cdots\!08$$$$\nu^{2} -$$$$55\!\cdots\!99$$$$\nu +$$$$13\!\cdots\!50$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!04$$$$\nu^{15} -$$$$80\!\cdots\!61$$$$\nu^{14} -$$$$10\!\cdots\!02$$$$\nu^{13} +$$$$53\!\cdots\!56$$$$\nu^{12} -$$$$64\!\cdots\!50$$$$\nu^{11} +$$$$40\!\cdots\!77$$$$\nu^{10} -$$$$21\!\cdots\!84$$$$\nu^{9} +$$$$18\!\cdots\!11$$$$\nu^{8} -$$$$51\!\cdots\!94$$$$\nu^{7} +$$$$37\!\cdots\!87$$$$\nu^{6} -$$$$67\!\cdots\!44$$$$\nu^{5} +$$$$53\!\cdots\!65$$$$\nu^{4} -$$$$60\!\cdots\!66$$$$\nu^{3} +$$$$23\!\cdots\!24$$$$\nu^{2} -$$$$76\!\cdots\!18$$$$\nu -$$$$58\!\cdots\!05$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{12}$$ $$=$$ $$($$$$44\!\cdots\!37$$$$\nu^{15} +$$$$47\!\cdots\!06$$$$\nu^{14} +$$$$41\!\cdots\!13$$$$\nu^{13} -$$$$54\!\cdots\!13$$$$\nu^{12} +$$$$25\!\cdots\!84$$$$\nu^{11} -$$$$70\!\cdots\!69$$$$\nu^{10} +$$$$85\!\cdots\!95$$$$\nu^{9} -$$$$43\!\cdots\!24$$$$\nu^{8} +$$$$20\!\cdots\!56$$$$\nu^{7} -$$$$99\!\cdots\!11$$$$\nu^{6} +$$$$27\!\cdots\!61$$$$\nu^{5} -$$$$16\!\cdots\!64$$$$\nu^{4} +$$$$24\!\cdots\!53$$$$\nu^{3} -$$$$10\!\cdots\!51$$$$\nu^{2} +$$$$33\!\cdots\!52$$$$\nu +$$$$25\!\cdots\!67$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!37$$$$\nu^{15} -$$$$19\!\cdots\!29$$$$\nu^{14} -$$$$12\!\cdots\!46$$$$\nu^{13} -$$$$12\!\cdots\!35$$$$\nu^{12} -$$$$78\!\cdots\!85$$$$\nu^{11} +$$$$40\!\cdots\!96$$$$\nu^{10} -$$$$26\!\cdots\!89$$$$\nu^{9} +$$$$87\!\cdots\!99$$$$\nu^{8} -$$$$62\!\cdots\!15$$$$\nu^{7} +$$$$18\!\cdots\!68$$$$\nu^{6} -$$$$80\!\cdots\!59$$$$\nu^{5} +$$$$36\!\cdots\!45$$$$\nu^{4} -$$$$69\!\cdots\!82$$$$\nu^{3} +$$$$84\!\cdots\!51$$$$\nu^{2} -$$$$80\!\cdots\!23$$$$\nu -$$$$85\!\cdots\!04$$$$)/$$$$95\!\cdots\!44$$ $$\beta_{14}$$ $$=$$ $$($$$$59\!\cdots\!59$$$$\nu^{15} -$$$$28\!\cdots\!45$$$$\nu^{14} +$$$$55\!\cdots\!96$$$$\nu^{13} -$$$$64\!\cdots\!31$$$$\nu^{12} +$$$$35\!\cdots\!19$$$$\nu^{11} -$$$$44\!\cdots\!96$$$$\nu^{10} +$$$$12\!\cdots\!89$$$$\nu^{9} -$$$$17\!\cdots\!59$$$$\nu^{8} +$$$$30\!\cdots\!97$$$$\nu^{7} -$$$$39\!\cdots\!28$$$$\nu^{6} +$$$$41\!\cdots\!51$$$$\nu^{5} -$$$$52\!\cdots\!65$$$$\nu^{4} +$$$$38\!\cdots\!94$$$$\nu^{3} -$$$$38\!\cdots\!01$$$$\nu^{2} +$$$$83\!\cdots\!49$$$$\nu +$$$$12\!\cdots\!86$$$$)/$$$$31\!\cdots\!48$$ $$\beta_{15}$$ $$=$$ $$($$$$18\!\cdots\!59$$$$\nu^{15} +$$$$13\!\cdots\!69$$$$\nu^{14} +$$$$17\!\cdots\!63$$$$\nu^{13} -$$$$81\!\cdots\!10$$$$\nu^{12} +$$$$10\!\cdots\!74$$$$\nu^{11} -$$$$63\!\cdots\!71$$$$\nu^{10} +$$$$35\!\cdots\!95$$$$\nu^{9} -$$$$28\!\cdots\!33$$$$\nu^{8} +$$$$85\!\cdots\!70$$$$\nu^{7} -$$$$62\!\cdots\!85$$$$\nu^{6} +$$$$11\!\cdots\!37$$$$\nu^{5} -$$$$89\!\cdots\!95$$$$\nu^{4} +$$$$98\!\cdots\!29$$$$\nu^{3} -$$$$44\!\cdots\!90$$$$\nu^{2} +$$$$13\!\cdots\!30$$$$\nu +$$$$11\!\cdots\!41$$$$)/$$$$56\!\cdots\!32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 7 \beta_{11} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - \beta_{2} - \beta_{1} - 2$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{15} - 2 \beta_{14} + \beta_{13} + 3 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} - 9 \beta_{9} - 3 \beta_{8} + \beta_{7} - 109 \beta_{6} - 80 \beta_{5} - 47 \beta_{4} - 331 \beta_{3} + \beta_{2} + 106 \beta_{1} - 327$$$$)/14$$ $$\nu^{3}$$ $$=$$ $$($$$$-8 \beta_{15} + 3 \beta_{14} + 21 \beta_{13} + 14 \beta_{12} - 31 \beta_{11} - 33 \beta_{10} + 13 \beta_{9} - 10 \beta_{8} + 5 \beta_{7} + 49 \beta_{6} - 12 \beta_{5} + 24 \beta_{4} + 36 \beta_{3} + 7 \beta_{2} - 91 \beta_{1} + 66$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-142 \beta_{15} + 50 \beta_{14} - 340 \beta_{13} - 124 \beta_{12} + 56 \beta_{11} - 182 \beta_{10} - 34 \beta_{9} + 446 \beta_{8} - 410 \beta_{7} + 2228 \beta_{6} + 1090 \beta_{5} + 2232 \beta_{4} + 5783 \beta_{3} + 108 \beta_{2} - 74 \beta_{1} + 216$$$$)/7$$ $$\nu^{5}$$ $$=$$ $$($$$$1117 \beta_{15} - 5413 \beta_{14} - 1560 \beta_{13} + 577 \beta_{12} - 406 \beta_{11} + 8127 \beta_{10} - 7107 \beta_{9} - 7045 \beta_{8} + 8100 \beta_{7} - 34857 \beta_{6} - 6863 \beta_{5} - 8342 \beta_{4} - 26959 \beta_{3} - 1560 \beta_{2} + 34280 \beta_{1} - 27942$$$$)/14$$ $$\nu^{6}$$ $$=$$ $$($$$$373 \beta_{15} + 811 \beta_{14} + 5121 \beta_{13} + 1811 \beta_{12} - 3871 \beta_{11} - 1876 \beta_{10} + 5494 \beta_{9} - 2336 \beta_{8} - 4770 \beta_{7} + 8181 \beta_{6} + 10768 \beta_{5} - 14013 \beta_{4} + 2687 \beta_{3} - 2061 \beta_{2} - 36767 \beta_{1} + 65215$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-28571 \beta_{15} + 240228 \beta_{14} - 307833 \beta_{13} - 259745 \beta_{12} + 352709 \beta_{11} + 7448 \beta_{10} + 46482 \beta_{9} + 577363 \beta_{8} - 419357 \beta_{7} + 1109420 \beta_{6} + 502001 \beta_{5} + 624328 \beta_{4} + 1298817 \beta_{3} + 75053 \beta_{2} - 19517 \beta_{1} + 48088$$$$)/14$$ $$\nu^{8}$$ $$=$$ $$($$$$315240 \beta_{15} - 585096 \beta_{14} + 42060 \beta_{13} + 106188 \beta_{12} + 254436 \beta_{11} + 755412 \beta_{10} - 1006524 \beta_{9} - 1679676 \beta_{8} + 2668068 \beta_{7} - 6863628 \beta_{6} - 4565496 \beta_{5} - 2871168 \beta_{4} - 10909879 \beta_{3} + 42060 \beta_{2} + 6757440 \beta_{1} - 10761631$$$$)/7$$ $$\nu^{9}$$ $$=$$ $$($$$$-211658 \beta_{15} + 112891 \beta_{14} + 2584593 \beta_{13} + 1733618 \beta_{12} - 2283949 \beta_{11} - 2269825 \beta_{10} + 2372935 \beta_{9} - 1234934 \beta_{8} - 1436811 \beta_{7} + 4755651 \beta_{6} + 1540690 \beta_{5} - 1855458 \beta_{4} + 2382716 \beta_{3} - 413535 \beta_{2} - 13124849 \beta_{1} + 13687566$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-35876621 \beta_{15} + 61134047 \beta_{14} - 113664034 \beta_{13} - 69460730 \beta_{12} + 64929816 \beta_{11} - 18918823 \beta_{10} - 2265457 \beta_{9} + 256629859 \beta_{8} - 186279675 \beta_{7} + 527802720 \beta_{6} + 286533740 \beta_{5} + 518729804 \beta_{4} + 973723816 \beta_{3} + 33611164 \beta_{2} - 8326683 \beta_{1} + 44203304$$$$)/14$$ $$\nu^{11}$$ $$=$$ $$($$$$243360247 \beta_{15} - 660025487 \beta_{14} - 50569868 \beta_{13} + 36476625 \beta_{12} + 22383382 \beta_{11} + 783548605 \beta_{10} - 939862359 \beta_{9} - 1317124749 \beta_{8} + 1937747386 \beta_{7} - 4955270443 \beta_{6} - 2514880763 \beta_{5} - 1789133120 \beta_{4} - 5659756251 \beta_{3} - 50569868 \beta_{2} + 4918793818 \beta_{1} - 5673849494$$$$)/14$$ $$\nu^{12}$$ $$=$$ $$2844970 \beta_{15} + 25153442 \beta_{14} + 437037738 \beta_{13} + 258149834 \beta_{12} - 330765190 \beta_{11} - 277613336 \beta_{10} + 439882708 \beta_{9} - 281146500 \beta_{8} - 415417460 \beta_{7} + 742649486 \beta_{6} + 639763490 \beta_{5} - 799187892 \beta_{4} + 302766778 \beta_{3} - 131425990 \beta_{2} - 2641680030 \beta_{1} + 3560760219$$ $$\nu^{13}$$ $$=$$ $$($$$$-11420160217 \beta_{15} + 32491949352 \beta_{14} - 47557113003 \beta_{13} - 34314451069 \beta_{12} + 38031705061 \beta_{11} - 1786376438 \beta_{10} + 1858626996 \beta_{9} + 98554916267 \beta_{8} - 72126701137 \beta_{7} + 177814191664 \beta_{6} + 98397713389 \beta_{5} + 158727596438 \beta_{4} + 277548890873 \beta_{3} + 13278787213 \beta_{2} - 1822501717 \beta_{1} + 13242661934$$$$)/14$$ $$\nu^{14}$$ $$=$$ $$($$$$96789315536 \beta_{15} - 221473349274 \beta_{14} - 697022381 \beta_{13} + 16084665733 \beta_{12} + 31472309085 \beta_{11} + 254339703121 \beta_{10} - 334347330543 \beta_{9} - 547413084613 \beta_{8} + 857268154219 \beta_{7} - 2003444141563 \beta_{6} - 1256838728428 \beta_{5} - 812026206409 \beta_{4} - 2699608177345 \beta_{3} - 697022381 \beta_{2} + 1987359475830 \beta_{1} - 2684220533993$$$$)/14$$ $$\nu^{15}$$ $$=$$ $$($$$$-10562294418 \beta_{15} + 13560958727 \beta_{14} + 374237620401 \beta_{13} + 244437298584 \beta_{12} - 295682469183 \beta_{11} - 279122846147 \beta_{10} + 363675325983 \beta_{9} - 207584657090 \beta_{8} - 289138472617 \beta_{7} + 656359130857 \beta_{6} + 391530627116 \beta_{5} - 486645401370 \beta_{4} + 292683804874 \beta_{3} - 92116109945 \beta_{2} - 2010794977127 \beta_{1} + 2399013198412$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.848921 + 1.47037i −0.141814 − 0.245629i −2.10711 − 3.64962i 2.81422 + 4.87437i 1.92573 + 3.33546i −2.63284 − 4.56021i −3.67087 − 6.35814i 2.96377 + 5.13339i 0.848921 − 1.47037i −0.141814 + 0.245629i −2.10711 + 3.64962i 2.81422 − 4.87437i 1.92573 − 3.33546i −2.63284 + 4.56021i −3.67087 + 6.35814i 2.96377 − 5.13339i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 −6.38854 2.86123i 2.82843 0 2.73861 1.58114i
271.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 4.24494 + 5.56601i 2.82843 0 2.73861 1.58114i
271.3 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 −5.26304 + 4.61524i 2.82843 0 −2.73861 + 1.58114i
271.4 −0.707107 + 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 6.99242 + 0.325616i 2.82843 0 −2.73861 + 1.58114i
271.5 0.707107 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 −2.67372 6.46925i −2.82843 0 −2.73861 + 1.58114i
271.6 0.707107 1.22474i 0 −1.00000 1.73205i −1.93649 1.11803i 0 1.94434 + 6.72455i −2.82843 0 −2.73861 + 1.58114i
271.7 0.707107 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 −2.59373 + 6.50174i −2.82843 0 2.73861 1.58114i
271.8 0.707107 1.22474i 0 −1.00000 1.73205i 1.93649 + 1.11803i 0 5.73733 4.01037i −2.82843 0 2.73861 1.58114i
451.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −6.38854 + 2.86123i 2.82843 0 2.73861 + 1.58114i
451.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 4.24494 5.56601i 2.82843 0 2.73861 + 1.58114i
451.3 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 −5.26304 4.61524i 2.82843 0 −2.73861 1.58114i
451.4 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 6.99242 0.325616i 2.82843 0 −2.73861 1.58114i
451.5 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −2.67372 + 6.46925i −2.82843 0 −2.73861 1.58114i
451.6 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 1.94434 6.72455i −2.82843 0 −2.73861 1.58114i
451.7 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 −2.59373 6.50174i −2.82843 0 2.73861 + 1.58114i
451.8 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.93649 1.11803i 0 5.73733 + 4.01037i −2.82843 0 2.73861 + 1.58114i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.3.v.c 16
3.b odd 2 1 210.3.o.b 16
7.d odd 6 1 inner 630.3.v.c 16
15.d odd 2 1 1050.3.p.i 16
15.e even 4 2 1050.3.q.e 32
21.g even 6 1 210.3.o.b 16
21.g even 6 1 1470.3.f.d 16
21.h odd 6 1 1470.3.f.d 16
105.p even 6 1 1050.3.p.i 16
105.w odd 12 2 1050.3.q.e 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 3.b odd 2 1
210.3.o.b 16 21.g even 6 1
630.3.v.c 16 1.a even 1 1 trivial
630.3.v.c 16 7.d odd 6 1 inner
1050.3.p.i 16 15.d odd 2 1
1050.3.p.i 16 105.p even 6 1
1050.3.q.e 32 15.e even 4 2
1050.3.q.e 32 105.w odd 12 2
1470.3.f.d 16 21.g even 6 1
1470.3.f.d 16 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{16} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(630, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} )^{4}$$
$3$ 1
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{4}$$
$7$ $$1 - 4 T + 28 T^{2} - 352 T^{3} + 2898 T^{4} - 16836 T^{5} + 70272 T^{6} - 1158948 T^{7} + 7794283 T^{8} - 56788452 T^{9} + 168723072 T^{10} - 1980738564 T^{11} + 16706393298 T^{12} - 99431287648 T^{13} + 387556041628 T^{14} - 2712892291396 T^{15} + 33232930569601 T^{16}$$
$11$ $$1 - 4 T - 364 T^{2} + 2552 T^{3} + 54108 T^{4} - 655724 T^{5} - 3422504 T^{6} + 97911028 T^{7} - 146055574 T^{8} - 11147160032 T^{9} + 52178728220 T^{10} + 1271915177612 T^{11} - 12776124769680 T^{12} - 140823145942564 T^{13} + 3084355376442156 T^{14} + 7698059096283072 T^{15} - 472018303966569005 T^{16} + 931465150650251712 T^{17} + 45158047066489605996 T^{18} -$$$$24\!\cdots\!04$$$$T^{19} -$$$$27\!\cdots\!80$$$$T^{20} +$$$$32\!\cdots\!12$$$$T^{21} +$$$$16\!\cdots\!20$$$$T^{22} -$$$$42\!\cdots\!12$$$$T^{23} -$$$$67\!\cdots\!14$$$$T^{24} +$$$$54\!\cdots\!68$$$$T^{25} -$$$$23\!\cdots\!04$$$$T^{26} -$$$$53\!\cdots\!04$$$$T^{27} +$$$$53\!\cdots\!28$$$$T^{28} +$$$$30\!\cdots\!72$$$$T^{29} -$$$$52\!\cdots\!84$$$$T^{30} -$$$$69\!\cdots\!04$$$$T^{31} +$$$$21\!\cdots\!21$$$$T^{32}$$
$13$ $$1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} -$$$$21\!\cdots\!12$$$$T^{18} +$$$$12\!\cdots\!16$$$$T^{20} -$$$$31\!\cdots\!76$$$$T^{22} +$$$$51\!\cdots\!70$$$$T^{24} -$$$$58\!\cdots\!64$$$$T^{26} +$$$$44\!\cdots\!28$$$$T^{28} -$$$$20\!\cdots\!56$$$$T^{30} +$$$$44\!\cdots\!81$$$$T^{32}$$
$17$ $$1 + 12 T + 420 T^{2} + 4464 T^{3} + 92044 T^{4} + 785292 T^{5} + 23989560 T^{6} - 227912772 T^{7} - 3236850998 T^{8} - 177940653384 T^{9} - 2004860299572 T^{10} - 43692613137348 T^{11} - 466147839356624 T^{12} - 8985915182672820 T^{13} - 29797959923559588 T^{14} + 769960022203743096 T^{15} + 34516652247342574675 T^{16} +$$$$22\!\cdots\!44$$$$T^{17} -$$$$24\!\cdots\!48$$$$T^{18} -$$$$21\!\cdots\!80$$$$T^{19} -$$$$32\!\cdots\!84$$$$T^{20} -$$$$88\!\cdots\!52$$$$T^{21} -$$$$11\!\cdots\!92$$$$T^{22} -$$$$29\!\cdots\!36$$$$T^{23} -$$$$15\!\cdots\!38$$$$T^{24} -$$$$32\!\cdots\!48$$$$T^{25} +$$$$97\!\cdots\!60$$$$T^{26} +$$$$92\!\cdots\!88$$$$T^{27} +$$$$31\!\cdots\!24$$$$T^{28} +$$$$43\!\cdots\!16$$$$T^{29} +$$$$11\!\cdots\!20$$$$T^{30} +$$$$98\!\cdots\!88$$$$T^{31} +$$$$23\!\cdots\!61$$$$T^{32}$$
$19$ $$1 + 72 T + 3812 T^{2} + 150048 T^{3} + 4883342 T^{4} + 136887912 T^{5} + 3384543112 T^{6} + 75691203624 T^{7} + 1567163213049 T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 4327693265152902096 T^{13} + 88720456587214840044 T^{14} +$$$$17\!\cdots\!80$$$$T^{15} +$$$$34\!\cdots\!48$$$$T^{16} +$$$$64\!\cdots\!80$$$$T^{17} +$$$$11\!\cdots\!24$$$$T^{18} +$$$$20\!\cdots\!76$$$$T^{19} +$$$$36\!\cdots\!54$$$$T^{20} +$$$$67\!\cdots\!64$$$$T^{21} +$$$$12\!\cdots\!60$$$$T^{22} +$$$$24\!\cdots\!64$$$$T^{23} +$$$$45\!\cdots\!69$$$$T^{24} +$$$$78\!\cdots\!84$$$$T^{25} +$$$$12\!\cdots\!12$$$$T^{26} +$$$$18\!\cdots\!32$$$$T^{27} +$$$$23\!\cdots\!82$$$$T^{28} +$$$$26\!\cdots\!88$$$$T^{29} +$$$$24\!\cdots\!92$$$$T^{30} +$$$$16\!\cdots\!72$$$$T^{31} +$$$$83\!\cdots\!61$$$$T^{32}$$
$23$ $$1 - 12 T - 1804 T^{2} + 41256 T^{3} + 1078172 T^{4} - 48109188 T^{5} - 226519400 T^{6} + 29579538300 T^{7} + 68149425194 T^{8} - 16885870838304 T^{9} + 1241401844668 T^{10} + 10296857200481412 T^{11} - 100210062266615184 T^{12} - 3750005340988049388 T^{13} + 80231845600155444876 T^{14} +$$$$51\!\cdots\!40$$$$T^{15} -$$$$38\!\cdots\!85$$$$T^{16} +$$$$27\!\cdots\!60$$$$T^{17} +$$$$22\!\cdots\!16$$$$T^{18} -$$$$55\!\cdots\!32$$$$T^{19} -$$$$78\!\cdots\!04$$$$T^{20} +$$$$42\!\cdots\!88$$$$T^{21} +$$$$27\!\cdots\!28$$$$T^{22} -$$$$19\!\cdots\!36$$$$T^{23} +$$$$41\!\cdots\!34$$$$T^{24} +$$$$95\!\cdots\!00$$$$T^{25} -$$$$38\!\cdots\!00$$$$T^{26} -$$$$43\!\cdots\!52$$$$T^{27} +$$$$51\!\cdots\!52$$$$T^{28} +$$$$10\!\cdots\!84$$$$T^{29} -$$$$24\!\cdots\!24$$$$T^{30} -$$$$85\!\cdots\!88$$$$T^{31} +$$$$37\!\cdots\!21$$$$T^{32}$$
$29$ $$( 1 + 36 T + 1444 T^{2} - 19116 T^{3} - 1274860 T^{4} - 68800452 T^{5} + 84102156 T^{6} + 30035473452 T^{7} + 2426711187974 T^{8} + 25259833173132 T^{9} + 59483856997836 T^{10} - 40924113344941092 T^{11} - 637744142027460460 T^{12} - 8042239471766642316 T^{13} +$$$$51\!\cdots\!04$$$$T^{14} +$$$$10\!\cdots\!16$$$$T^{15} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} +$$$$21\!\cdots\!82$$$$T^{12} -$$$$69\!\cdots\!28$$$$T^{13} +$$$$23\!\cdots\!88$$$$T^{14} -$$$$74\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!16$$$$T^{16} -$$$$71\!\cdots\!40$$$$T^{17} +$$$$21\!\cdots\!48$$$$T^{18} -$$$$61\!\cdots\!68$$$$T^{19} +$$$$17\!\cdots\!62$$$$T^{20} -$$$$52\!\cdots\!60$$$$T^{21} +$$$$15\!\cdots\!16$$$$T^{22} -$$$$44\!\cdots\!56$$$$T^{23} +$$$$12\!\cdots\!25$$$$T^{24} -$$$$31\!\cdots\!72$$$$T^{25} +$$$$76\!\cdots\!76$$$$T^{26} -$$$$16\!\cdots\!64$$$$T^{27} +$$$$30\!\cdots\!74$$$$T^{28} -$$$$48\!\cdots\!60$$$$T^{29} +$$$$66\!\cdots\!08$$$$T^{30} -$$$$66\!\cdots\!20$$$$T^{31} +$$$$52\!\cdots\!61$$$$T^{32}$$
$37$ $$1 - 44 T - 1956 T^{2} - 11728 T^{3} + 3652806 T^{4} + 242885524 T^{5} - 1534355576 T^{6} - 317275895964 T^{7} - 9605253544351 T^{8} + 75954637404076 T^{9} + 6072970847630504 T^{10} + 248299820626414612 T^{11} + 4168440621293518230 T^{12} +$$$$29\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!28$$$$T^{14} -$$$$69\!\cdots\!04$$$$T^{15} -$$$$27\!\cdots\!16$$$$T^{16} -$$$$95\!\cdots\!76$$$$T^{17} +$$$$20\!\cdots\!08$$$$T^{18} +$$$$75\!\cdots\!40$$$$T^{19} +$$$$14\!\cdots\!30$$$$T^{20} +$$$$11\!\cdots\!88$$$$T^{21} +$$$$39\!\cdots\!24$$$$T^{22} +$$$$68\!\cdots\!64$$$$T^{23} -$$$$11\!\cdots\!91$$$$T^{24} -$$$$53\!\cdots\!56$$$$T^{25} -$$$$35\!\cdots\!76$$$$T^{26} +$$$$76\!\cdots\!56$$$$T^{27} +$$$$15\!\cdots\!66$$$$T^{28} -$$$$69\!\cdots\!52$$$$T^{29} -$$$$15\!\cdots\!76$$$$T^{30} -$$$$48\!\cdots\!56$$$$T^{31} +$$$$15\!\cdots\!81$$$$T^{32}$$
$41$ $$1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} +$$$$50\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!32$$$$T^{14} +$$$$18\!\cdots\!14$$$$T^{16} -$$$$28\!\cdots\!52$$$$T^{18} +$$$$40\!\cdots\!76$$$$T^{20} -$$$$49\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!40$$$$T^{24} -$$$$47\!\cdots\!64$$$$T^{26} +$$$$33\!\cdots\!04$$$$T^{28} -$$$$16\!\cdots\!36$$$$T^{30} +$$$$40\!\cdots\!81$$$$T^{32}$$
$43$ $$( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 1611056221164 T^{7} + 122468758030675 T^{8} + 2978842952932236 T^{9} + 204841330302006704 T^{10} + 4087494160581305452 T^{11} +$$$$25\!\cdots\!10$$$$T^{12} +$$$$41\!\cdots\!12$$$$T^{13} +$$$$22\!\cdots\!32$$$$T^{14} +$$$$20\!\cdots\!72$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$1 - 24 T + 10040 T^{2} - 236352 T^{3} + 48191588 T^{4} - 529473000 T^{5} + 142018937488 T^{6} + 1533668366952 T^{7} + 289429105887690 T^{8} + 11114511768111792 T^{9} + 616929826339364168 T^{10} + 28449623988679606104 T^{11} +$$$$19\!\cdots\!48$$$$T^{12} +$$$$39\!\cdots\!52$$$$T^{13} +$$$$60\!\cdots\!84$$$$T^{14} +$$$$28\!\cdots\!36$$$$T^{15} +$$$$14\!\cdots\!27$$$$T^{16} +$$$$62\!\cdots\!24$$$$T^{17} +$$$$29\!\cdots\!04$$$$T^{18} +$$$$42\!\cdots\!08$$$$T^{19} +$$$$46\!\cdots\!28$$$$T^{20} +$$$$14\!\cdots\!96$$$$T^{21} +$$$$71\!\cdots\!88$$$$T^{22} +$$$$28\!\cdots\!48$$$$T^{23} +$$$$16\!\cdots\!90$$$$T^{24} +$$$$19\!\cdots\!28$$$$T^{25} +$$$$39\!\cdots\!88$$$$T^{26} -$$$$32\!\cdots\!00$$$$T^{27} +$$$$65\!\cdots\!28$$$$T^{28} -$$$$70\!\cdots\!08$$$$T^{29} +$$$$66\!\cdots\!40$$$$T^{30} -$$$$34\!\cdots\!76$$$$T^{31} +$$$$32\!\cdots\!41$$$$T^{32}$$
$53$ $$1 + 32 T - 12072 T^{2} - 345280 T^{3} + 71529764 T^{4} + 1574568864 T^{5} - 290206976304 T^{6} - 2770742930720 T^{7} + 1006589576083018 T^{8} - 3889510628997504 T^{9} - 3360408915215254232 T^{10} + 25194864185334307040 T^{11} +$$$$10\!\cdots\!32$$$$T^{12} -$$$$34\!\cdots\!88$$$$T^{13} -$$$$28\!\cdots\!00$$$$T^{14} +$$$$63\!\cdots\!04$$$$T^{15} +$$$$75\!\cdots\!55$$$$T^{16} +$$$$17\!\cdots\!36$$$$T^{17} -$$$$22\!\cdots\!00$$$$T^{18} -$$$$75\!\cdots\!52$$$$T^{19} +$$$$64\!\cdots\!52$$$$T^{20} +$$$$44\!\cdots\!60$$$$T^{21} -$$$$16\!\cdots\!12$$$$T^{22} -$$$$53\!\cdots\!76$$$$T^{23} +$$$$39\!\cdots\!78$$$$T^{24} -$$$$30\!\cdots\!80$$$$T^{25} -$$$$88\!\cdots\!04$$$$T^{26} +$$$$13\!\cdots\!76$$$$T^{27} +$$$$17\!\cdots\!84$$$$T^{28} -$$$$23\!\cdots\!20$$$$T^{29} -$$$$22\!\cdots\!92$$$$T^{30} +$$$$17\!\cdots\!68$$$$T^{31} +$$$$15\!\cdots\!41$$$$T^{32}$$
$59$ $$1 + 132 T + 9660 T^{2} + 508464 T^{3} + 17070268 T^{4} + 1368116580 T^{5} + 85370423112 T^{6} + 5735724829716 T^{7} + 274189008275050 T^{8} + 2776853544451464 T^{9} + 245103035882958612 T^{10} - 4667418467579246988 T^{11} +$$$$23\!\cdots\!68$$$$T^{12} +$$$$15\!\cdots\!44$$$$T^{13} +$$$$45\!\cdots\!16$$$$T^{14} +$$$$18\!\cdots\!68$$$$T^{15} -$$$$23\!\cdots\!33$$$$T^{16} +$$$$62\!\cdots\!08$$$$T^{17} +$$$$55\!\cdots\!76$$$$T^{18} +$$$$65\!\cdots\!04$$$$T^{19} +$$$$34\!\cdots\!28$$$$T^{20} -$$$$23\!\cdots\!88$$$$T^{21} +$$$$43\!\cdots\!72$$$$T^{22} +$$$$17\!\cdots\!04$$$$T^{23} +$$$$59\!\cdots\!50$$$$T^{24} +$$$$43\!\cdots\!36$$$$T^{25} +$$$$22\!\cdots\!12$$$$T^{26} +$$$$12\!\cdots\!80$$$$T^{27} +$$$$54\!\cdots\!48$$$$T^{28} +$$$$56\!\cdots\!24$$$$T^{29} +$$$$37\!\cdots\!60$$$$T^{30} +$$$$17\!\cdots\!32$$$$T^{31} +$$$$46\!\cdots\!81$$$$T^{32}$$
$61$ $$1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} -$$$$25\!\cdots\!96$$$$T^{11} +$$$$16\!\cdots\!12$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{13} +$$$$72\!\cdots\!12$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$28\!\cdots\!43$$$$T^{16} -$$$$16\!\cdots\!04$$$$T^{17} +$$$$10\!\cdots\!92$$$$T^{18} -$$$$56\!\cdots\!92$$$$T^{19} +$$$$31\!\cdots\!72$$$$T^{20} -$$$$18\!\cdots\!96$$$$T^{21} +$$$$89\!\cdots\!72$$$$T^{22} -$$$$51\!\cdots\!40$$$$T^{23} +$$$$23\!\cdots\!86$$$$T^{24} -$$$$12\!\cdots\!00$$$$T^{25} +$$$$55\!\cdots\!72$$$$T^{26} -$$$$24\!\cdots\!60$$$$T^{27} +$$$$11\!\cdots\!80$$$$T^{28} -$$$$36\!\cdots\!12$$$$T^{29} +$$$$17\!\cdots\!44$$$$T^{30} -$$$$34\!\cdots\!96$$$$T^{31} +$$$$13\!\cdots\!21$$$$T^{32}$$
$67$ $$1 + 164 T - 6340 T^{2} - 2597968 T^{3} - 31915482 T^{4} + 22697073892 T^{5} + 967634972296 T^{6} - 123392891205548 T^{7} - 9455087268413407 T^{8} + 333882071018564764 T^{9} + 51639925158822741896 T^{10} +$$$$26\!\cdots\!32$$$$T^{11} -$$$$16\!\cdots\!86$$$$T^{12} -$$$$60\!\cdots\!36$$$$T^{13} +$$$$18\!\cdots\!60$$$$T^{14} +$$$$15\!\cdots\!28$$$$T^{15} +$$$$54\!\cdots\!88$$$$T^{16} +$$$$70\!\cdots\!92$$$$T^{17} +$$$$36\!\cdots\!60$$$$T^{18} -$$$$55\!\cdots\!84$$$$T^{19} -$$$$67\!\cdots\!26$$$$T^{20} +$$$$47\!\cdots\!68$$$$T^{21} +$$$$42\!\cdots\!56$$$$T^{22} +$$$$12\!\cdots\!56$$$$T^{23} -$$$$15\!\cdots\!67$$$$T^{24} -$$$$91\!\cdots\!32$$$$T^{25} +$$$$32\!\cdots\!96$$$$T^{26} +$$$$33\!\cdots\!88$$$$T^{27} -$$$$21\!\cdots\!22$$$$T^{28} -$$$$78\!\cdots\!92$$$$T^{29} -$$$$85\!\cdots\!40$$$$T^{30} +$$$$99\!\cdots\!36$$$$T^{31} +$$$$27\!\cdots\!61$$$$T^{32}$$
$71$ $$( 1 - 68 T + 26092 T^{2} - 2195028 T^{3} + 351820772 T^{4} - 28773766364 T^{5} + 3188424927972 T^{6} - 217589962697708 T^{7} + 19655929349959526 T^{8} - 1096871001959146028 T^{9} + 81023237162072440932 T^{10} -$$$$36\!\cdots\!44$$$$T^{11} +$$$$22\!\cdots\!92$$$$T^{12} -$$$$71\!\cdots\!28$$$$T^{13} +$$$$42\!\cdots\!72$$$$T^{14} -$$$$56\!\cdots\!08$$$$T^{15} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$1 + 348 T + 82708 T^{2} + 14734320 T^{3} + 2214323678 T^{4} + 290251205388 T^{5} + 34312832193656 T^{6} + 3733330194438684 T^{7} + 379285162573239545 T^{8} + 36467064302261366820 T^{9} +$$$$33\!\cdots\!56$$$$T^{10} +$$$$29\!\cdots\!16$$$$T^{11} +$$$$24\!\cdots\!70$$$$T^{12} +$$$$20\!\cdots\!16$$$$T^{13} +$$$$16\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!12$$$$T^{15} +$$$$91\!\cdots\!12$$$$T^{16} +$$$$65\!\cdots\!48$$$$T^{17} +$$$$45\!\cdots\!52$$$$T^{18} +$$$$30\!\cdots\!24$$$$T^{19} +$$$$20\!\cdots\!70$$$$T^{20} +$$$$12\!\cdots\!84$$$$T^{21} +$$$$76\!\cdots\!76$$$$T^{22} +$$$$44\!\cdots\!80$$$$T^{23} +$$$$24\!\cdots\!45$$$$T^{24} +$$$$12\!\cdots\!96$$$$T^{25} +$$$$63\!\cdots\!56$$$$T^{26} +$$$$28\!\cdots\!52$$$$T^{27} +$$$$11\!\cdots\!98$$$$T^{28} +$$$$41\!\cdots\!80$$$$T^{29} +$$$$12\!\cdots\!48$$$$T^{30} +$$$$27\!\cdots\!52$$$$T^{31} +$$$$42\!\cdots\!21$$$$T^{32}$$
$79$ $$1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 1123575678148193512 T^{9} - 61265347082641448728 T^{10} +$$$$14\!\cdots\!40$$$$T^{11} +$$$$44\!\cdots\!62$$$$T^{12} -$$$$73\!\cdots\!60$$$$T^{13} +$$$$53\!\cdots\!92$$$$T^{14} -$$$$10\!\cdots\!56$$$$T^{15} +$$$$35\!\cdots\!48$$$$T^{16} -$$$$66\!\cdots\!96$$$$T^{17} +$$$$20\!\cdots\!52$$$$T^{18} -$$$$17\!\cdots\!60$$$$T^{19} +$$$$67\!\cdots\!82$$$$T^{20} +$$$$13\!\cdots\!40$$$$T^{21} -$$$$36\!\cdots\!48$$$$T^{22} -$$$$41\!\cdots\!72$$$$T^{23} +$$$$23\!\cdots\!13$$$$T^{24} +$$$$23\!\cdots\!80$$$$T^{25} +$$$$49\!\cdots\!24$$$$T^{26} -$$$$17\!\cdots\!96$$$$T^{27} +$$$$12\!\cdots\!78$$$$T^{28} +$$$$35\!\cdots\!24$$$$T^{29} +$$$$19\!\cdots\!52$$$$T^{30} -$$$$23\!\cdots\!80$$$$T^{31} +$$$$52\!\cdots\!41$$$$T^{32}$$
$83$ $$1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} -$$$$83\!\cdots\!72$$$$T^{10} +$$$$84\!\cdots\!60$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!06$$$$T^{16} -$$$$34\!\cdots\!84$$$$T^{18} +$$$$19\!\cdots\!60$$$$T^{20} -$$$$88\!\cdots\!92$$$$T^{22} +$$$$34\!\cdots\!16$$$$T^{24} -$$$$10\!\cdots\!60$$$$T^{26} +$$$$24\!\cdots\!68$$$$T^{28} -$$$$35\!\cdots\!44$$$$T^{30} +$$$$25\!\cdots\!61$$$$T^{32}$$
$89$ $$1 - 300 T + 54668 T^{2} - 7400400 T^{3} + 757620572 T^{4} - 53632718700 T^{5} + 870152589352 T^{6} + 480156239482500 T^{7} - 98736006237054870 T^{8} + 12632355432189921000 T^{9} -$$$$12\!\cdots\!92$$$$T^{10} +$$$$89\!\cdots\!00$$$$T^{11} -$$$$37\!\cdots\!44$$$$T^{12} -$$$$17\!\cdots\!40$$$$T^{13} +$$$$60\!\cdots\!12$$$$T^{14} -$$$$83\!\cdots\!20$$$$T^{15} +$$$$83\!\cdots\!23$$$$T^{16} -$$$$66\!\cdots\!20$$$$T^{17} +$$$$37\!\cdots\!92$$$$T^{18} -$$$$84\!\cdots\!40$$$$T^{19} -$$$$14\!\cdots\!64$$$$T^{20} +$$$$27\!\cdots\!00$$$$T^{21} -$$$$30\!\cdots\!32$$$$T^{22} +$$$$24\!\cdots\!00$$$$T^{23} -$$$$15\!\cdots\!70$$$$T^{24} +$$$$58\!\cdots\!00$$$$T^{25} +$$$$84\!\cdots\!52$$$$T^{26} -$$$$41\!\cdots\!00$$$$T^{27} +$$$$46\!\cdots\!52$$$$T^{28} -$$$$35\!\cdots\!00$$$$T^{29} +$$$$20\!\cdots\!08$$$$T^{30} -$$$$90\!\cdots\!00$$$$T^{31} +$$$$24\!\cdots\!21$$$$T^{32}$$
$97$ $$1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} -$$$$21\!\cdots\!72$$$$T^{10} +$$$$29\!\cdots\!68$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$34\!\cdots\!38$$$$T^{16} -$$$$30\!\cdots\!04$$$$T^{18} +$$$$23\!\cdots\!48$$$$T^{20} -$$$$14\!\cdots\!52$$$$T^{22} +$$$$81\!\cdots\!08$$$$T^{24} -$$$$36\!\cdots\!36$$$$T^{26} +$$$$12\!\cdots\!92$$$$T^{28} -$$$$30\!\cdots\!28$$$$T^{30} +$$$$37\!\cdots\!41$$$$T^{32}$$