Properties

Label 2646.2.d.f
Level $2646$
Weight $2$
Character orbit 2646.d
Analytic conductor $21.128$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{48}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{48}^{12} q^{2} - q^{4} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{5} -\zeta_{48}^{12} q^{8} +O(q^{10})\) \( q + \zeta_{48}^{12} q^{2} - q^{4} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{5} -\zeta_{48}^{12} q^{8} + ( 1 - \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{8} - \zeta_{48}^{13} ) q^{10} + ( 2 \zeta_{48} - \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 2 \zeta_{48}^{11} - \zeta_{48}^{12} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{11} + ( \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - \zeta_{48}^{6} + \zeta_{48}^{9} + \zeta_{48}^{10} - 2 \zeta_{48}^{13} + 2 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{13} + q^{16} + ( \zeta_{48}^{2} + \zeta_{48}^{6} + \zeta_{48}^{10} - 2 \zeta_{48}^{14} ) q^{17} + ( -1 - 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{8} - 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 4 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{19} + ( 2 \zeta_{48}^{4} + \zeta_{48}^{9} - \zeta_{48}^{12} - \zeta_{48}^{15} ) q^{20} + ( 1 - 2 \zeta_{48} + \zeta_{48}^{3} + \zeta_{48}^{5} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{22} + ( -\zeta_{48}^{2} + \zeta_{48}^{3} - \zeta_{48}^{5} + \zeta_{48}^{6} + \zeta_{48}^{10} - 2 \zeta_{48}^{11} - 3 \zeta_{48}^{12} - \zeta_{48}^{13} ) q^{23} + ( -\zeta_{48}^{2} + 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - \zeta_{48}^{6} + \zeta_{48}^{10} - 4 \zeta_{48}^{11} + 2 \zeta_{48}^{13} ) q^{25} + ( -\zeta_{48}^{2} - \zeta_{48}^{3} - \zeta_{48}^{5} - \zeta_{48}^{6} + 2 \zeta_{48}^{9} - \zeta_{48}^{10} + \zeta_{48}^{13} + 2 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{26} + ( -4 \zeta_{48} - \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + \zeta_{48}^{6} + 4 \zeta_{48}^{7} + 2 \zeta_{48}^{9} + \zeta_{48}^{10} + 2 \zeta_{48}^{11} + 2 \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{29} + ( 1 - \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + \zeta_{48}^{6} - 2 \zeta_{48}^{8} - 2 \zeta_{48}^{9} - \zeta_{48}^{10} - \zeta_{48}^{13} - 2 \zeta_{48}^{14} - 2 \zeta_{48}^{15} ) q^{31} + \zeta_{48}^{12} q^{32} + ( \zeta_{48}^{2} - \zeta_{48}^{6} + \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{34} + ( 1 + 4 \zeta_{48} - 3 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} - 3 \zeta_{48}^{6} + 4 \zeta_{48}^{7} - 2 \zeta_{48}^{9} + 3 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{37} + ( 2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{4} + 2 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{10} + \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 4 \zeta_{48}^{14} ) q^{38} + ( -1 + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{8} + \zeta_{48}^{13} ) q^{40} + ( \zeta_{48}^{2} - \zeta_{48}^{3} + 2 \zeta_{48}^{4} - \zeta_{48}^{5} + \zeta_{48}^{6} - 3 \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{14} + 3 \zeta_{48}^{15} ) q^{41} + ( -2 + 2 \zeta_{48} + 2 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{43} + ( -2 \zeta_{48} + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{12} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{44} + ( 3 + 2 \zeta_{48} - \zeta_{48}^{2} - \zeta_{48}^{6} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{15} ) q^{46} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 4 \zeta_{48}^{4} + 2 \zeta_{48}^{5} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{12} - 2 \zeta_{48}^{13} + 4 \zeta_{48}^{14} ) q^{47} + ( -4 \zeta_{48} + \zeta_{48}^{2} - \zeta_{48}^{6} + 4 \zeta_{48}^{7} + 2 \zeta_{48}^{9} - \zeta_{48}^{10} - 2 \zeta_{48}^{15} ) q^{50} + ( -\zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + \zeta_{48}^{6} - \zeta_{48}^{9} - \zeta_{48}^{10} + 2 \zeta_{48}^{13} - 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{52} + ( 2 \zeta_{48} - 3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} - 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 6 \zeta_{48}^{11} + 2 \zeta_{48}^{12} + 3 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{53} + ( -3 - 2 \zeta_{48}^{2} + 4 \zeta_{48}^{3} - 4 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 6 \zeta_{48}^{8} - 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 4 \zeta_{48}^{13} - 4 \zeta_{48}^{14} - 3 \zeta_{48}^{15} ) q^{55} + ( -2 - 2 \zeta_{48} - \zeta_{48}^{2} - 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} - \zeta_{48}^{6} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} + \zeta_{48}^{10} + 4 \zeta_{48}^{11} - 2 \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{58} + ( 2 \zeta_{48}^{2} - 5 \zeta_{48}^{3} - 4 \zeta_{48}^{4} - 5 \zeta_{48}^{5} + 2 \zeta_{48}^{6} - \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 2 \zeta_{48}^{12} + 5 \zeta_{48}^{13} - 4 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{59} + ( -2 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 2 \zeta_{48}^{13} - 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{61} + ( \zeta_{48}^{2} + 2 \zeta_{48}^{3} + 2 \zeta_{48}^{4} + 2 \zeta_{48}^{5} + \zeta_{48}^{6} + \zeta_{48}^{9} + \zeta_{48}^{10} - \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{62} - q^{64} + ( -6 \zeta_{48} + 6 \zeta_{48}^{2} + 2 \zeta_{48}^{3} - 2 \zeta_{48}^{5} - 6 \zeta_{48}^{6} + 6 \zeta_{48}^{7} + 3 \zeta_{48}^{9} - 6 \zeta_{48}^{10} - 4 \zeta_{48}^{11} + 4 \zeta_{48}^{12} - 2 \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{65} + ( -2 + 4 \zeta_{48} + \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{7} - 2 \zeta_{48}^{9} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 2 \zeta_{48}^{15} ) q^{67} + ( -\zeta_{48}^{2} - \zeta_{48}^{6} - \zeta_{48}^{10} + 2 \zeta_{48}^{14} ) q^{68} + ( -6 \zeta_{48} + 2 \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - 2 \zeta_{48}^{6} + 6 \zeta_{48}^{7} + 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + 4 \zeta_{48}^{11} - \zeta_{48}^{12} + 2 \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{71} + ( -2 - \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{8} + 3 \zeta_{48}^{9} - \zeta_{48}^{13} + 3 \zeta_{48}^{15} ) q^{73} + ( -2 \zeta_{48} + 3 \zeta_{48}^{2} - 2 \zeta_{48}^{3} + 2 \zeta_{48}^{5} - 3 \zeta_{48}^{6} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} - 3 \zeta_{48}^{10} + 4 \zeta_{48}^{11} + \zeta_{48}^{12} + 2 \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{74} + ( 1 + 2 \zeta_{48}^{2} - 2 \zeta_{48}^{6} - 2 \zeta_{48}^{8} + 2 \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{76} + ( 6 \zeta_{48} + 4 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} + 4 \zeta_{48}^{6} + 6 \zeta_{48}^{7} - 3 \zeta_{48}^{9} - 4 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} - 3 \zeta_{48}^{15} ) q^{79} + ( -2 \zeta_{48}^{4} - \zeta_{48}^{9} + \zeta_{48}^{12} + \zeta_{48}^{15} ) q^{80} + ( -1 + \zeta_{48}^{2} - 3 \zeta_{48}^{3} + 3 \zeta_{48}^{5} - \zeta_{48}^{6} + 2 \zeta_{48}^{8} - \zeta_{48}^{9} + \zeta_{48}^{10} - 3 \zeta_{48}^{13} + 2 \zeta_{48}^{14} - \zeta_{48}^{15} ) q^{82} + ( 4 \zeta_{48}^{2} - \zeta_{48}^{3} - \zeta_{48}^{5} + 4 \zeta_{48}^{6} + 4 \zeta_{48}^{9} + 4 \zeta_{48}^{10} + \zeta_{48}^{13} - 8 \zeta_{48}^{14} - 4 \zeta_{48}^{15} ) q^{83} + ( -2 \zeta_{48} - 3 \zeta_{48}^{2} + \zeta_{48}^{3} + \zeta_{48}^{5} - 3 \zeta_{48}^{6} - 2 \zeta_{48}^{7} + \zeta_{48}^{9} + 3 \zeta_{48}^{10} - 2 \zeta_{48}^{11} + \zeta_{48}^{13} + \zeta_{48}^{15} ) q^{85} + ( -2 \zeta_{48} - 2 \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 2 \zeta_{48}^{7} + \zeta_{48}^{9} + 2 \zeta_{48}^{10} + 2 \zeta_{48}^{11} - 2 \zeta_{48}^{12} + \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{86} + ( -1 + 2 \zeta_{48} - \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{7} - \zeta_{48}^{9} + 2 \zeta_{48}^{11} - \zeta_{48}^{13} - \zeta_{48}^{15} ) q^{88} + ( \zeta_{48}^{2} - \zeta_{48}^{3} - 10 \zeta_{48}^{4} - \zeta_{48}^{5} + \zeta_{48}^{6} - \zeta_{48}^{9} + \zeta_{48}^{10} + 5 \zeta_{48}^{12} + \zeta_{48}^{13} - 2 \zeta_{48}^{14} + \zeta_{48}^{15} ) q^{89} + ( \zeta_{48}^{2} - \zeta_{48}^{3} + \zeta_{48}^{5} - \zeta_{48}^{6} - \zeta_{48}^{10} + 2 \zeta_{48}^{11} + 3 \zeta_{48}^{12} + \zeta_{48}^{13} ) q^{92} + ( 2 - 2 \zeta_{48}^{2} + 2 \zeta_{48}^{6} - 4 \zeta_{48}^{8} + 2 \zeta_{48}^{9} - 2 \zeta_{48}^{10} - 4 \zeta_{48}^{14} + 2 \zeta_{48}^{15} ) q^{94} + ( 6 \zeta_{48} - 8 \zeta_{48}^{2} - 4 \zeta_{48}^{3} + 4 \zeta_{48}^{5} + 8 \zeta_{48}^{6} - 6 \zeta_{48}^{7} - 3 \zeta_{48}^{9} + 8 \zeta_{48}^{10} + 8 \zeta_{48}^{11} - 3 \zeta_{48}^{12} + 4 \zeta_{48}^{13} + 3 \zeta_{48}^{15} ) q^{95} + ( -4 - 2 \zeta_{48}^{2} + \zeta_{48}^{3} - \zeta_{48}^{5} + 2 \zeta_{48}^{6} + 8 \zeta_{48}^{8} + 3 \zeta_{48}^{9} - 2 \zeta_{48}^{10} + \zeta_{48}^{13} - 4 \zeta_{48}^{14} + 3 \zeta_{48}^{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} + O(q^{10}) \) \( 16q - 16q^{4} + 16q^{16} + 16q^{22} + 16q^{37} - 32q^{43} + 48q^{46} - 32q^{58} - 16q^{64} - 32q^{67} - 16q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2645.1
0.130526 + 0.991445i
0.991445 0.130526i
−0.991445 + 0.130526i
0.793353 0.608761i
−0.130526 0.991445i
−0.608761 0.793353i
0.608761 + 0.793353i
−0.793353 + 0.608761i
0.130526 0.991445i
0.991445 + 0.130526i
−0.991445 0.130526i
0.793353 + 0.608761i
−0.130526 + 0.991445i
−0.608761 + 0.793353i
0.608761 0.793353i
−0.793353 0.608761i
1.00000i 0 −1.00000 −3.57981 0 0 1.00000i 0 3.57981i
2645.2 1.00000i 0 −1.00000 −2.49742 0 0 1.00000i 0 2.49742i
2645.3 1.00000i 0 −1.00000 −0.966684 0 0 1.00000i 0 0.966684i
2645.4 1.00000i 0 −1.00000 −0.115708 0 0 1.00000i 0 0.115708i
2645.5 1.00000i 0 −1.00000 0.115708 0 0 1.00000i 0 0.115708i
2645.6 1.00000i 0 −1.00000 0.966684 0 0 1.00000i 0 0.966684i
2645.7 1.00000i 0 −1.00000 2.49742 0 0 1.00000i 0 2.49742i
2645.8 1.00000i 0 −1.00000 3.57981 0 0 1.00000i 0 3.57981i
2645.9 1.00000i 0 −1.00000 −3.57981 0 0 1.00000i 0 3.57981i
2645.10 1.00000i 0 −1.00000 −2.49742 0 0 1.00000i 0 2.49742i
2645.11 1.00000i 0 −1.00000 −0.966684 0 0 1.00000i 0 0.966684i
2645.12 1.00000i 0 −1.00000 −0.115708 0 0 1.00000i 0 0.115708i
2645.13 1.00000i 0 −1.00000 0.115708 0 0 1.00000i 0 0.115708i
2645.14 1.00000i 0 −1.00000 0.966684 0 0 1.00000i 0 0.966684i
2645.15 1.00000i 0 −1.00000 2.49742 0 0 1.00000i 0 2.49742i
2645.16 1.00000i 0 −1.00000 3.57981 0 0 1.00000i 0 3.57981i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2645.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.f 16
3.b odd 2 1 inner 2646.2.d.f 16
7.b odd 2 1 inner 2646.2.d.f 16
21.c even 2 1 inner 2646.2.d.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2646.2.d.f 16 1.a even 1 1 trivial
2646.2.d.f 16 3.b odd 2 1 inner
2646.2.d.f 16 7.b odd 2 1 inner
2646.2.d.f 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 20 T_{5}^{6} + 98 T_{5}^{4} - 76 T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( T^{16} \)
$5$ \( ( 1 - 76 T^{2} + 98 T^{4} - 20 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 2401 + 3700 T^{2} + 774 T^{4} + 52 T^{6} + T^{8} )^{2} \)
$13$ \( ( 6724 + 4160 T^{2} + 860 T^{4} + 64 T^{6} + T^{8} )^{2} \)
$17$ \( ( -6 + T^{2} )^{8} \)
$19$ \( ( 22801 + 83308 T^{2} + 6134 T^{4} + 140 T^{6} + T^{8} )^{2} \)
$23$ \( ( 18769 + 10556 T^{2} + 1458 T^{4} + 68 T^{6} + T^{8} )^{2} \)
$29$ \( ( 329476 + 85920 T^{2} + 5948 T^{4} + 144 T^{6} + T^{8} )^{2} \)
$31$ \( ( 1 + 20 T^{2} + 98 T^{4} + 76 T^{6} + T^{8} )^{2} \)
$37$ \( ( -1337 + 692 T - 90 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$41$ \( ( 49 - 34516 T^{2} + 3734 T^{4} - 116 T^{6} + T^{8} )^{2} \)
$43$ \( ( -392 - 224 T - 16 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$47$ \( ( 256 - 27904 T^{2} + 4640 T^{4} - 176 T^{6} + T^{8} )^{2} \)
$53$ \( ( 9388096 + 788992 T^{2} + 22512 T^{4} + 256 T^{6} + T^{8} )^{2} \)
$59$ \( ( 41886784 - 2309888 T^{2} + 44144 T^{4} - 352 T^{6} + T^{8} )^{2} \)
$61$ \( ( 4096 + 38912 T^{2} + 6272 T^{4} + 160 T^{6} + T^{8} )^{2} \)
$67$ \( ( -206 - 208 T - 36 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$71$ \( ( 17497489 + 1388772 T^{2} + 35954 T^{4} + 348 T^{6} + T^{8} )^{2} \)
$73$ \( ( 3136 + 8704 T^{2} + 4208 T^{4} + 128 T^{6} + T^{8} )^{2} \)
$79$ \( ( -2744 - 1344 T - 184 T^{2} + T^{4} )^{4} \)
$83$ \( ( 7761796 - 3073808 T^{2} + 73172 T^{4} - 520 T^{6} + T^{8} )^{2} \)
$89$ \( ( 15062161 - 1608308 T^{2} + 38582 T^{4} - 340 T^{6} + T^{8} )^{2} \)
$97$ \( ( 24840256 + 2070400 T^{2} + 45392 T^{4} + 368 T^{6} + T^{8} )^{2} \)
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