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 $2$

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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:

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

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