# Properties

 Label 315.3.w.a Level 315 Weight 3 Character orbit 315.w Analytic conductor 8.583 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 315.w (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.58312832735$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.523596960000.16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{4} -\beta_{6} q^{5} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{6} ) q^{7} + ( 5 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{4} -\beta_{6} q^{5} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{6} ) q^{7} + ( 5 + 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{10} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{11} + ( 2 - 4 \beta_{1} + 2 \beta_{3} + 5 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} ) q^{13} + ( 5 - \beta_{1} + \beta_{2} + 6 \beta_{3} + 11 \beta_{4} - \beta_{6} ) q^{14} + ( -5 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{16} + ( 3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{17} + ( 2 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{19} + ( -5 + 2 \beta_{1} - \beta_{3} - 9 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} + ( -5 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 5 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{22} + ( -3 \beta_{1} - 4 \beta_{2} + 12 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{23} + ( 5 + 5 \beta_{4} ) q^{25} + ( -20 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 10 \beta_{4} + \beta_{5} + 14 \beta_{6} - \beta_{7} ) q^{26} + ( -13 - 3 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{28} + ( 5 - 10 \beta_{2} - 5 \beta_{4} - \beta_{5} - 9 \beta_{6} + 5 \beta_{7} ) q^{29} + ( -8 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 12 \beta_{4} - 7 \beta_{5} ) q^{31} + ( -5 - \beta_{1} + 3 \beta_{2} + \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + 10 \beta_{6} - 6 \beta_{7} ) q^{32} + ( -20 + 4 \beta_{1} - 2 \beta_{3} - 38 \beta_{4} + 14 \beta_{5} + 14 \beta_{6} + 2 \beta_{7} ) q^{34} + ( 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + 7 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{35} + ( 8 \beta_{1} - 4 \beta_{2} + 22 \beta_{4} - 10 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{37} + ( -10 - 7 \beta_{1} - 8 \beta_{2} - 7 \beta_{3} + 2 \beta_{4} - 22 \beta_{5} ) q^{38} + ( 10 - 3 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{40} + ( -4 + 6 \beta_{1} - 3 \beta_{3} - 5 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} ) q^{41} + ( 49 + 10 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} ) q^{43} + ( 7 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{44} + ( -15 + 17 \beta_{1} - 2 \beta_{2} - 17 \beta_{3} - 13 \beta_{4} + 9 \beta_{5} + 22 \beta_{6} + 4 \beta_{7} ) q^{46} + ( 14 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 7 \beta_{4} + \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{47} + ( 11 + 9 \beta_{1} + 2 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} - 14 \beta_{5} - 7 \beta_{6} - 2 \beta_{7} ) q^{49} -5 \beta_{3} q^{50} + ( -13 + 15 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 10 \beta_{4} + 12 \beta_{5} ) q^{52} + ( 13 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} + 2 \beta_{5} + 14 \beta_{6} + 10 \beta_{7} ) q^{53} + ( -6 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{55} + ( -35 + 14 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 12 \beta_{4} + 21 \beta_{5} + 13 \beta_{6} + 6 \beta_{7} ) q^{56} + ( -32 \beta_{1} - 4 \beta_{2} - 8 \beta_{4} + 16 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} ) q^{58} + ( -3 + 14 \beta_{1} - \beta_{2} + 14 \beta_{3} + 2 \beta_{4} - 14 \beta_{5} ) q^{59} + ( -60 - 4 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} - 30 \beta_{4} + 10 \beta_{5} - 4 \beta_{6} - 10 \beta_{7} ) q^{61} + ( -5 + 40 \beta_{1} - 20 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 7 \beta_{7} ) q^{62} + ( -9 - 12 \beta_{2} - \beta_{3} - 6 \beta_{4} - 15 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{64} + ( -14 \beta_{1} + 2 \beta_{2} - 21 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{65} + ( 47 + 5 \beta_{1} - \beta_{2} - 5 \beta_{3} + 48 \beta_{4} - 11 \beta_{5} - 20 \beta_{6} + 2 \beta_{7} ) q^{67} + ( 44 - 20 \beta_{1} + 12 \beta_{2} + 40 \beta_{3} + 22 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} ) q^{68} + ( -7 \beta_{1} + 3 \beta_{2} + 13 \beta_{3} + 3 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} ) q^{70} + ( 1 + 10 \beta_{2} - 28 \beta_{3} + 5 \beta_{4} + 9 \beta_{5} + \beta_{6} - 5 \beta_{7} ) q^{71} + ( 31 - 11 \beta_{1} - 7 \beta_{2} - 11 \beta_{3} - 38 \beta_{4} - 23 \beta_{5} ) q^{73} + ( 40 + 31 \beta_{1} - 7 \beta_{2} - 31 \beta_{3} + 47 \beta_{4} - 7 \beta_{5} + 14 \beta_{7} ) q^{74} + ( -31 + 46 \beta_{1} - 23 \beta_{3} - 56 \beta_{4} + 21 \beta_{5} + 21 \beta_{6} + 6 \beta_{7} ) q^{76} + ( -2 + 13 \beta_{1} + 11 \beta_{2} - 9 \beta_{3} + 21 \beta_{4} + 7 \beta_{5} - 13 \beta_{6} - 5 \beta_{7} ) q^{77} + ( -5 \beta_{1} + 6 \beta_{2} + 52 \beta_{4} + 12 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} ) q^{79} + ( 5 - 9 \beta_{1} + \beta_{2} - 9 \beta_{3} - 4 \beta_{4} ) q^{80} + ( 30 + 5 \beta_{1} - 9 \beta_{2} - 10 \beta_{3} + 15 \beta_{4} - 9 \beta_{5} - 15 \beta_{6} + 9 \beta_{7} ) q^{82} + ( -9 - 24 \beta_{1} + 12 \beta_{3} - 13 \beta_{4} - 37 \beta_{5} - 37 \beta_{6} + 5 \beta_{7} ) q^{83} + ( -5 - 6 \beta_{2} - 14 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{85} + ( -48 \beta_{1} - 3 \beta_{2} + 19 \beta_{4} - 33 \beta_{5} - 15 \beta_{6} - 3 \beta_{7} ) q^{86} + ( 15 - 17 \beta_{1} - 5 \beta_{2} + 17 \beta_{3} + 20 \beta_{4} + 10 \beta_{6} + 10 \beta_{7} ) q^{88} + ( -46 - 10 \beta_{1} + 11 \beta_{2} + 20 \beta_{3} - 23 \beta_{4} + 11 \beta_{5} - 11 \beta_{6} - 11 \beta_{7} ) q^{89} + ( -33 + 15 \beta_{1} + 6 \beta_{2} + 15 \beta_{3} - 4 \beta_{4} - 35 \beta_{5} - 48 \beta_{6} + 14 \beta_{7} ) q^{91} + ( 5 - 6 \beta_{2} + 53 \beta_{3} - 3 \beta_{4} + 23 \beta_{5} - 29 \beta_{6} + 3 \beta_{7} ) q^{92} + ( 5 - \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + 5 \beta_{5} ) q^{94} + ( 20 - 14 \beta_{1} - 2 \beta_{2} + 14 \beta_{3} + 22 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( 6 - 28 \beta_{1} + 14 \beta_{3} + 22 \beta_{4} + 16 \beta_{5} + 16 \beta_{6} + 10 \beta_{7} ) q^{97} + ( 45 + 19 \beta_{1} - 9 \beta_{2} - 10 \beta_{3} - 38 \beta_{4} + 14 \beta_{5} + 16 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 6q^{4} - 16q^{7} + 32q^{8} + O(q^{10})$$ $$8q - 2q^{2} - 6q^{4} - 16q^{7} + 32q^{8} - 20q^{11} + 16q^{14} - 2q^{16} + 18q^{17} - 16q^{22} - 62q^{23} + 20q^{25} - 120q^{26} - 120q^{28} + 100q^{29} - 126q^{31} - 36q^{32} - 80q^{37} - 114q^{38} + 90q^{40} + 352q^{43} + 18q^{44} - 82q^{46} + 72q^{47} + 38q^{49} - 20q^{50} - 48q^{52} + 76q^{53} - 196q^{56} - 40q^{58} + 54q^{59} - 396q^{61} - 4q^{64} + 60q^{65} + 184q^{67} + 312q^{68} - 164q^{71} + 348q^{73} + 140q^{74} - 152q^{77} - 206q^{79} + 204q^{82} - 60q^{85} - 178q^{86} + 124q^{88} - 282q^{89} - 114q^{91} + 288q^{92} + 30q^{94} + 120q^{95} + 592q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 13 x^{6} - 2 x^{5} + 91 x^{4} - 50 x^{3} + 190 x^{2} + 100 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 20 \nu^{6} - 51 \nu^{5} + 304 \nu^{4} - 193 \nu^{3} + 1752 \nu^{2} - 2510 \nu + 2630$$$$)/630$$ $$\beta_{3}$$ $$=$$ $$($$$$-87 \nu^{7} + 24 \nu^{6} - 841 \nu^{5} - 1276 \nu^{4} - 10117 \nu^{3} - 4640 \nu^{2} - 2900 \nu - 13700$$$$)/21630$$ $$\beta_{4}$$ $$=$$ $$($$$$137 \nu^{7} - 361 \nu^{6} + 1805 \nu^{5} - 1115 \nu^{4} + 11191 \nu^{3} - 16967 \nu^{2} + 21390 \nu - 10830$$$$)/21630$$ $$\beta_{5}$$ $$=$$ $$($$$$472 \nu^{7} - 1249 \nu^{6} + 6966 \nu^{5} - 8699 \nu^{4} + 48092 \nu^{3} - 74565 \nu^{2} + 78220 \nu - 131200$$$$)/64890$$ $$\beta_{6}$$ $$=$$ $$($$$$661 \nu^{7} - 2047 \nu^{6} + 8793 \nu^{5} - 5927 \nu^{4} + 44711 \nu^{3} - 64485 \nu^{2} + 84520 \nu + 126050$$$$)/64890$$ $$\beta_{7}$$ $$=$$ $$($$$$-977 \nu^{7} + 1985 \nu^{6} - 15693 \nu^{5} + 4657 \nu^{4} - 114889 \nu^{3} + 25521 \nu^{2} - 381050 \nu - 55810$$$$)/64890$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{6} + \beta_{5} - 5 \beta_{4} - \beta_{3} + \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} - 9 \beta_{3} - 2 \beta_{2} - 5$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{7} - 11 \beta_{6} - 24 \beta_{5} + 41 \beta_{4} - 2 \beta_{2} - 17 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-26 \beta_{7} - 42 \beta_{6} - 8 \beta_{5} + 72 \beta_{4} + 95 \beta_{3} + 13 \beta_{2} - 95 \beta_{1} + 85$$ $$\nu^{6}$$ $$=$$ $$-34 \beta_{7} - 121 \beta_{6} + 189 \beta_{5} + 34 \beta_{4} + 243 \beta_{3} + 68 \beta_{2} + 475$$ $$\nu^{7}$$ $$=$$ $$155 \beta_{7} + 311 \beta_{6} + 777 \beta_{5} - 905 \beta_{4} + 155 \beta_{2} + 1081 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 1.76021 + 3.04878i 0.836732 + 1.44926i −0.336732 − 0.583237i −1.26021 − 2.18275i 1.76021 − 3.04878i 0.836732 − 1.44926i −0.336732 + 0.583237i −1.26021 + 2.18275i
−1.76021 3.04878i 0 −4.19671 + 7.26891i 1.93649 1.11803i 0 0.244004 + 6.99575i 15.4667 0 −6.81728 3.93596i
136.2 −0.836732 1.44926i 0 0.599760 1.03881i −1.93649 + 1.11803i 0 4.76104 + 5.13152i −8.70121 0 3.24065 + 1.87099i
136.3 0.336732 + 0.583237i 0 1.77322 3.07131i −1.93649 + 1.11803i 0 −6.82455 1.55742i 5.08226 0 −1.30416 0.752955i
136.4 1.26021 + 2.18275i 0 −1.17628 + 2.03737i 1.93649 1.11803i 0 −6.18050 + 3.28656i 4.15226 0 4.88079 + 2.81792i
271.1 −1.76021 + 3.04878i 0 −4.19671 7.26891i 1.93649 + 1.11803i 0 0.244004 6.99575i 15.4667 0 −6.81728 + 3.93596i
271.2 −0.836732 + 1.44926i 0 0.599760 + 1.03881i −1.93649 1.11803i 0 4.76104 5.13152i −8.70121 0 3.24065 1.87099i
271.3 0.336732 0.583237i 0 1.77322 + 3.07131i −1.93649 1.11803i 0 −6.82455 + 1.55742i 5.08226 0 −1.30416 + 0.752955i
271.4 1.26021 2.18275i 0 −1.17628 2.03737i 1.93649 + 1.11803i 0 −6.18050 3.28656i 4.15226 0 4.88079 2.81792i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.4 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 315.3.w.a 8
3.b odd 2 1 105.3.n.a 8
7.d odd 6 1 inner 315.3.w.a 8
15.d odd 2 1 525.3.o.l 8
15.e even 4 2 525.3.s.h 16
21.g even 6 1 105.3.n.a 8
21.g even 6 1 735.3.h.a 8
21.h odd 6 1 735.3.h.a 8
105.p even 6 1 525.3.o.l 8
105.w odd 12 2 525.3.s.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.a 8 3.b odd 2 1
105.3.n.a 8 21.g even 6 1
315.3.w.a 8 1.a even 1 1 trivial
315.3.w.a 8 7.d odd 6 1 inner
525.3.o.l 8 15.d odd 2 1
525.3.o.l 8 105.p even 6 1
525.3.s.h 16 15.e even 4 2
525.3.s.h 16 105.w odd 12 2
735.3.h.a 8 21.g even 6 1
735.3.h.a 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(315, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T - 3 T^{2} - 14 T^{3} - 13 T^{4} + 18 T^{5} + 70 T^{6} + 44 T^{7} - 156 T^{8} + 176 T^{9} + 1120 T^{10} + 1152 T^{11} - 3328 T^{12} - 14336 T^{13} - 12288 T^{14} + 32768 T^{15} + 65536 T^{16}$$
$3$ 1
$5$ $$( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 + 16 T + 109 T^{2} + 784 T^{3} + 6664 T^{4} + 38416 T^{5} + 261709 T^{6} + 1882384 T^{7} + 5764801 T^{8}$$
$11$ $$1 + 20 T - 147 T^{2} - 2960 T^{3} + 57131 T^{4} + 519480 T^{5} - 8882912 T^{6} - 8003440 T^{7} + 1602642534 T^{8} - 968416240 T^{9} - 130054714592 T^{10} + 920290508280 T^{11} + 12246537230411 T^{12} - 76774776818960 T^{13} - 461348971377987 T^{14} + 7594996671664820 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 251281048928 T^{10} + 32487291694546 T^{12} - 4380040003026428 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 - 18 T + 766 T^{2} - 11844 T^{3} + 262438 T^{4} - 1906254 T^{5} + 22625848 T^{6} + 382636314 T^{7} - 5169040877 T^{8} + 110581894746 T^{9} + 1889733450808 T^{10} - 46012337456526 T^{11} + 1830703831301158 T^{12} - 23877431756917956 T^{13} + 446288633717996926 T^{14} - 3030800878069216722 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 + 598 T^{2} + 183481 T^{4} - 682560 T^{5} - 48973562 T^{6} - 501474240 T^{7} - 32269961996 T^{8} - 181032200640 T^{9} - 6382283573402 T^{10} - 32111636535360 T^{11} + 3116161130325721 T^{12} + 1323562321601564278 T^{14} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$1 + 62 T + 1497 T^{2} - 6014 T^{3} - 1196893 T^{4} - 31086552 T^{5} - 143771420 T^{6} + 13896561704 T^{7} + 513552019554 T^{8} + 7351281141416 T^{9} - 40233137944220 T^{10} - 4601925361264728 T^{11} - 93729870105931933 T^{12} - 249139038438885086 T^{13} + 32806192774734420537 T^{14} +$$$$71\!\cdots\!58$$$$T^{15} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 - 50 T + 1234 T^{2} + 15850 T^{3} - 1164374 T^{4} + 13329850 T^{5} + 872784754 T^{6} - 29741166050 T^{7} + 500246412961 T^{8} )^{2}$$
$31$ $$1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1194991513915968 T^{9} + 35241648542229442 T^{10} + 957696435880658100 T^{11} + 23204023931078714797 T^{12} +$$$$47\!\cdots\!66$$$$T^{13} +$$$$77\!\cdots\!63$$$$T^{14} +$$$$95\!\cdots\!46$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 80 T + 1194 T^{2} + 28960 T^{3} + 3461705 T^{4} - 28416960 T^{5} - 6540374054 T^{6} - 198858748720 T^{7} - 6603314864556 T^{8} - 272237626997680 T^{9} - 12257713977418694 T^{10} - 72910144735496640 T^{11} + 12159167688035595305 T^{12} +$$$$13\!\cdots\!40$$$$T^{13} +$$$$78\!\cdots\!14$$$$T^{14} +$$$$72\!\cdots\!20$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} +$$$$39\!\cdots\!45$$$$T^{12} -$$$$22\!\cdots\!86$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 2107763852 T^{5} + 58177736617 T^{6} - 1112559896624 T^{7} + 11688200277601 T^{8} )^{2}$$
$47$ $$1 - 72 T + 10951 T^{2} - 664056 T^{3} + 65519473 T^{4} - 3342900456 T^{5} + 246192812578 T^{6} - 10750018584384 T^{7} + 651041931981118 T^{8} - 23746791052904256 T^{9} + 1201342389873427618 T^{10} - 36033843838636290024 T^{11} +$$$$15\!\cdots\!53$$$$T^{12} -$$$$34\!\cdots\!44$$$$T^{13} +$$$$12\!\cdots\!91$$$$T^{14} -$$$$18\!\cdots\!68$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 - 76 T - 3069 T^{2} + 443764 T^{3} + 2229785 T^{4} - 1396117872 T^{5} + 34651196266 T^{6} + 1991894657480 T^{7} - 169369280357850 T^{8} + 5595232092861320 T^{9} + 273414605764143946 T^{10} - 30944060693658997488 T^{11} +$$$$13\!\cdots\!85$$$$T^{12} +$$$$77\!\cdots\!36$$$$T^{13} -$$$$15\!\cdots\!29$$$$T^{14} -$$$$10\!\cdots\!44$$$$T^{15} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 - 54 T + 7198 T^{2} - 336204 T^{3} + 19932742 T^{4} + 202333950 T^{5} - 35478676088 T^{6} + 7641841019598 T^{7} - 385856896323245 T^{8} + 26601248589220638 T^{9} - 429907925960363768 T^{10} + 8534553984691411950 T^{11} +$$$$29\!\cdots\!82$$$$T^{12} -$$$$17\!\cdots\!04$$$$T^{13} +$$$$12\!\cdots\!38$$$$T^{14} -$$$$33\!\cdots\!94$$$$T^{15} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 3072245831066337948 T^{9} +$$$$15\!\cdots\!72$$$$T^{10} +$$$$70\!\cdots\!88$$$$T^{11} +$$$$27\!\cdots\!18$$$$T^{12} +$$$$87\!\cdots\!32$$$$T^{13} +$$$$22\!\cdots\!44$$$$T^{14} +$$$$39\!\cdots\!36$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 - 184 T + 6231 T^{2} + 140176 T^{3} + 74370665 T^{4} - 7237038408 T^{5} + 24063966106 T^{6} - 15184872524680 T^{7} + 3087140085953070 T^{8} - 68164892763288520 T^{9} + 484915892741904826 T^{10} -$$$$65\!\cdots\!52$$$$T^{11} +$$$$30\!\cdots\!65$$$$T^{12} +$$$$25\!\cdots\!24$$$$T^{13} +$$$$50\!\cdots\!91$$$$T^{14} -$$$$67\!\cdots\!36$$$$T^{15} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$( 1 + 82 T + 12166 T^{2} + 846262 T^{3} + 94594474 T^{4} + 4266006742 T^{5} + 309158511046 T^{6} + 10504223281522 T^{7} + 645753531245761 T^{8} )^{2}$$
$73$ $$1 - 348 T + 68263 T^{2} - 9707460 T^{3} + 1072498525 T^{4} - 96253557984 T^{5} + 7434307414846 T^{6} - 526544361727584 T^{7} + 37365046682274814 T^{8} - 2805954903646295136 T^{9} +$$$$21\!\cdots\!86$$$$T^{10} -$$$$14\!\cdots\!76$$$$T^{11} +$$$$86\!\cdots\!25$$$$T^{12} -$$$$41\!\cdots\!40$$$$T^{13} +$$$$15\!\cdots\!23$$$$T^{14} -$$$$42\!\cdots\!32$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 + 206 T + 5583 T^{2} - 659438 T^{3} + 124066817 T^{4} + 19494076044 T^{5} + 428008398310 T^{6} + 33090623674568 T^{7} + 7605703397631354 T^{8} + 206518582352978888 T^{9} + 16670961782854763110 T^{10} +$$$$47\!\cdots\!24$$$$T^{11} +$$$$18\!\cdots\!37$$$$T^{12} -$$$$62\!\cdots\!38$$$$T^{13} +$$$$32\!\cdots\!03$$$$T^{14} +$$$$75\!\cdots\!86$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} -$$$$10\!\cdots\!56$$$$T^{10} +$$$$50\!\cdots\!80$$$$T^{12} -$$$$22\!\cdots\!92$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16}$$
$89$ $$1 + 282 T + 59686 T^{2} + 9356196 T^{3} + 1240796086 T^{4} + 138656838366 T^{5} + 14271061565800 T^{6} + 1337157406377822 T^{7} + 121622616146107507 T^{8} + 10591623815918728062 T^{9} +$$$$89\!\cdots\!00$$$$T^{10} +$$$$68\!\cdots\!26$$$$T^{11} +$$$$48\!\cdots\!66$$$$T^{12} +$$$$29\!\cdots\!96$$$$T^{13} +$$$$14\!\cdots\!06$$$$T^{14} +$$$$55\!\cdots\!62$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} -$$$$10\!\cdots\!68$$$$T^{10} +$$$$72\!\cdots\!08$$$$T^{12} -$$$$30\!\cdots\!16$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16}$$