# Properties

 Label 585.2.bu.c Level $585$ Weight $2$ Character orbit 585.bu Analytic conductor $4.671$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.bu (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{2} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{5} + \beta_{7} ) q^{2} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{7} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{8} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{11} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{13} + ( -2 - \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{14} + ( -3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{16} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{17} + ( 2 - 4 \beta_{2} - \beta_{6} ) q^{19} + ( -3 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{22} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{23} - q^{25} + ( -3 + \beta_{1} + 6 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 5 \beta_{7} ) q^{26} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( 5 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 2 + 2 \beta_{2} - 4 \beta_{6} + 2 \beta_{7} ) q^{31} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{32} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{34} + ( \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{35} + ( -1 + 3 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{37} + ( 5 - \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + \beta_{7} ) q^{38} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{40} + ( -1 - \beta_{6} + 2 \beta_{7} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{43} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{44} + ( -7 - \beta_{1} + 3 \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{46} + ( 6 - 4 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{6} ) q^{47} + ( -2 + 6 \beta_{1} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{49} + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{50} + ( -7 + 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{52} + ( 4 - 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{55} + ( -2 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 12 \beta_{7} ) q^{56} + ( 13 - 5 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} - 4 \beta_{6} ) q^{58} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} ) q^{61} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{62} + ( 1 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{7} ) q^{64} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{65} + ( 1 - \beta_{1} + \beta_{4} - 7 \beta_{7} ) q^{67} + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{68} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{70} + ( -6 - 2 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} ) q^{71} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{74} + ( 4 + \beta_{1} + 5 \beta_{3} - \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 7 \beta_{7} ) q^{76} + ( 5 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - 5 \beta_{7} ) q^{77} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{79} + ( 1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{80} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{82} + ( 6 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 2 + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{85} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{86} + ( -3 - 3 \beta_{1} - 10 \beta_{2} - 3 \beta_{4} + 6 \beta_{6} - 5 \beta_{7} ) q^{88} + ( -1 - 2 \beta_{1} + 8 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} - 3 \beta_{6} + 8 \beta_{7} ) q^{89} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} ) q^{91} + ( -8 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 2 \beta_{7} ) q^{92} + ( 12 - 4 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} - 6 \beta_{7} ) q^{94} + ( \beta_{2} + 4 \beta_{6} + 2 \beta_{7} ) q^{95} + ( -6 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{97} + ( -8 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} - 6q^{7} + O(q^{10})$$ $$8q + 2q^{4} - 6q^{7} - 2q^{10} - 4q^{14} - 2q^{16} + 2q^{17} + 12q^{19} - 12q^{20} - 12q^{22} + 10q^{23} - 8q^{25} - 10q^{26} - 18q^{28} + 8q^{29} - 6q^{32} - 10q^{35} + 6q^{37} + 16q^{38} - 12q^{40} - 12q^{41} - 2q^{43} - 42q^{46} + 12q^{49} - 6q^{52} + 24q^{53} - 12q^{56} + 36q^{58} + 12q^{59} - 28q^{61} - 4q^{62} - 8q^{64} + 8q^{65} + 6q^{67} + 14q^{68} - 10q^{74} + 54q^{76} + 36q^{77} - 16q^{79} + 4q^{82} + 18q^{85} - 18q^{88} - 24q^{89} + 28q^{91} - 44q^{92} + 32q^{94} + 16q^{95} - 30q^{97} - 72q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 8 \nu - 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 10 \nu^{2} - 16 \nu + 8$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} - 3 \nu^{2} + 18 \nu - 16$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{7} - 5 \nu^{6} + 2 \nu^{5} + 7 \nu^{4} - 8 \nu^{3} - 9 \nu^{2} + 28 \nu - 20$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 7 \nu^{6} - 3 \nu^{5} - 11 \nu^{4} + 15 \nu^{3} + 11 \nu^{2} - 40 \nu + 32$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{7} - 20 \nu^{6} + 11 \nu^{5} + 30 \nu^{4} - 45 \nu^{3} - 28 \nu^{2} + 116 \nu - 88$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 1$$ $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 3 \beta_{6} - 5 \beta_{5} + \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 4 \beta_{1} - 1$$ $$\nu^{7}$$ $$=$$ $$-6 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + \beta_{1} + 6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 0.665665 − 1.24775i 1.40994 − 0.109843i −1.27597 + 0.609843i 1.20036 + 0.747754i 0.665665 + 1.24775i 1.40994 + 0.109843i −1.27597 − 0.609843i 1.20036 − 0.747754i
−1.29515 0.747754i 0 0.118272 + 0.204852i 1.00000i 0 −4.18016 + 2.41342i 2.63726i 0 0.747754 1.29515i
316.2 −1.05628 0.609843i 0 −0.256182 0.443720i 1.00000i 0 3.11786 1.80010i 3.06430i 0 −0.609843 + 1.05628i
316.3 0.190254 + 0.109843i 0 −0.975869 1.69025i 1.00000i 0 −0.287734 + 0.166123i 0.868145i 0 0.109843 0.190254i
316.4 2.16117 + 1.24775i 0 2.11378 + 3.66117i 1.00000i 0 −1.64996 + 0.952606i 5.55889i 0 −1.24775 + 2.16117i
361.1 −1.29515 + 0.747754i 0 0.118272 0.204852i 1.00000i 0 −4.18016 2.41342i 2.63726i 0 0.747754 + 1.29515i
361.2 −1.05628 + 0.609843i 0 −0.256182 + 0.443720i 1.00000i 0 3.11786 + 1.80010i 3.06430i 0 −0.609843 1.05628i
361.3 0.190254 0.109843i 0 −0.975869 + 1.69025i 1.00000i 0 −0.287734 0.166123i 0.868145i 0 0.109843 + 0.190254i
361.4 2.16117 1.24775i 0 2.11378 3.66117i 1.00000i 0 −1.64996 0.952606i 5.55889i 0 −1.24775 2.16117i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bu.c 8
3.b odd 2 1 65.2.m.a 8
12.b even 2 1 1040.2.da.b 8
13.e even 6 1 inner 585.2.bu.c 8
13.f odd 12 1 7605.2.a.cf 4
13.f odd 12 1 7605.2.a.cj 4
15.d odd 2 1 325.2.n.d 8
15.e even 4 1 325.2.m.b 8
15.e even 4 1 325.2.m.c 8
39.d odd 2 1 845.2.m.g 8
39.f even 4 1 845.2.e.m 8
39.f even 4 1 845.2.e.n 8
39.h odd 6 1 65.2.m.a 8
39.h odd 6 1 845.2.c.g 8
39.i odd 6 1 845.2.c.g 8
39.i odd 6 1 845.2.m.g 8
39.k even 12 1 845.2.a.l 4
39.k even 12 1 845.2.a.m 4
39.k even 12 1 845.2.e.m 8
39.k even 12 1 845.2.e.n 8
156.r even 6 1 1040.2.da.b 8
195.y odd 6 1 325.2.n.d 8
195.bf even 12 1 325.2.m.b 8
195.bf even 12 1 325.2.m.c 8
195.bh even 12 1 4225.2.a.bi 4
195.bh even 12 1 4225.2.a.bl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 3.b odd 2 1
65.2.m.a 8 39.h odd 6 1
325.2.m.b 8 15.e even 4 1
325.2.m.b 8 195.bf even 12 1
325.2.m.c 8 15.e even 4 1
325.2.m.c 8 195.bf even 12 1
325.2.n.d 8 15.d odd 2 1
325.2.n.d 8 195.y odd 6 1
585.2.bu.c 8 1.a even 1 1 trivial
585.2.bu.c 8 13.e even 6 1 inner
845.2.a.l 4 39.k even 12 1
845.2.a.m 4 39.k even 12 1
845.2.c.g 8 39.h odd 6 1
845.2.c.g 8 39.i odd 6 1
845.2.e.m 8 39.f even 4 1
845.2.e.m 8 39.k even 12 1
845.2.e.n 8 39.f even 4 1
845.2.e.n 8 39.k even 12 1
845.2.m.g 8 39.d odd 2 1
845.2.m.g 8 39.i odd 6 1
1040.2.da.b 8 12.b even 2 1
1040.2.da.b 8 156.r even 6 1
4225.2.a.bi 4 195.bh even 12 1
4225.2.a.bl 4 195.bh even 12 1
7605.2.a.cf 4 13.f odd 12 1
7605.2.a.cj 4 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 5 T_{2}^{6} + 24 T_{2}^{4} + 30 T_{2}^{3} + 7 T_{2}^{2} - 6 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T + 7 T^{2} + 30 T^{3} + 24 T^{4} - 5 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$121 + 726 T + 1606 T^{2} + 924 T^{3} + 75 T^{4} - 84 T^{5} - 2 T^{6} + 6 T^{7} + T^{8}$$
$11$ $$1089 - 990 T^{2} + 867 T^{4} - 30 T^{6} + T^{8}$$
$13$ $$28561 + 2704 T^{2} + 1248 T^{3} + 30 T^{4} + 96 T^{5} + 16 T^{6} + T^{8}$$
$17$ $$169 + 130 T + 334 T^{2} - 128 T^{3} + 331 T^{4} + 16 T^{5} + 22 T^{6} - 2 T^{7} + T^{8}$$
$19$ $$( 169 + 78 T - T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$89401 - 43654 T + 23110 T^{2} - 5104 T^{3} + 1795 T^{4} - 352 T^{5} + 94 T^{6} - 10 T^{7} + T^{8}$$
$29$ $$1 + 40 T + 1618 T^{2} - 704 T^{3} + 643 T^{4} + 64 T^{5} + 82 T^{6} - 8 T^{7} + T^{8}$$
$31$ $$( 64 + 32 T^{2} + T^{4} )^{2}$$
$37$ $$1 + 66 T + 1402 T^{2} - 3300 T^{3} + 2367 T^{4} + 300 T^{5} - 38 T^{6} - 6 T^{7} + T^{8}$$
$41$ $$( 1 - 6 T + 11 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$43$ $$169 - 130 T + 334 T^{2} + 128 T^{3} + 331 T^{4} - 16 T^{5} + 22 T^{6} + 2 T^{7} + T^{8}$$
$47$ $$1763584 + 350464 T^{2} + 14304 T^{4} + 208 T^{6} + T^{8}$$
$53$ $$( -48 + 36 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$59$ $$9 - 108 T + 486 T^{2} - 648 T^{3} + 183 T^{4} + 216 T^{5} + 30 T^{6} - 12 T^{7} + T^{8}$$
$61$ $$1590121 + 1215604 T + 603958 T^{2} + 178096 T^{3} + 38311 T^{4} + 5296 T^{5} + 526 T^{6} + 28 T^{7} + T^{8}$$
$67$ $$7667361 + 847314 T - 284454 T^{2} - 34884 T^{3} + 9615 T^{4} + 684 T^{5} - 102 T^{6} - 6 T^{7} + T^{8}$$
$71$ $$109767529 - 2263032 T - 2268434 T^{2} + 47088 T^{3} + 37047 T^{4} - 218 T^{6} + T^{8}$$
$73$ $$2930944 + 404608 T^{2} + 16944 T^{4} + 232 T^{6} + T^{8}$$
$79$ $$( 4432 - 640 T - 132 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$83$ $$36864 + 73728 T^{2} + 7104 T^{4} + 192 T^{6} + T^{8}$$
$89$ $$78375609 - 34420464 T + 3179718 T^{2} + 816480 T^{3} + 4143 T^{4} - 5040 T^{5} - 18 T^{6} + 24 T^{7} + T^{8}$$
$97$ $$196249 + 71766 T - 16946 T^{2} - 9396 T^{3} + 2187 T^{4} + 1740 T^{5} + 358 T^{6} + 30 T^{7} + T^{8}$$