Properties

Label 490.2.i.f
Level 490
Weight 2
Character orbit 490.i
Analytic conductor 3.913
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 490.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + \zeta_{24}^{6} q^{8} + 3 \zeta_{24}^{4} q^{9} +O(q^{10})\) \( q + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{2} + ( -2 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{3} + ( 1 - \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + \zeta_{24}^{6} q^{8} + 3 \zeta_{24}^{4} q^{9} + ( 1 + 2 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{10} + ( -4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( -\zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{12} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{13} + ( 3 + 2 \zeta_{24}^{3} + 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{15} -\zeta_{24}^{4} q^{16} + 2 \zeta_{24}^{2} q^{17} -3 \zeta_{24}^{2} q^{18} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{19} + ( 1 - \zeta_{24}^{3} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{20} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{22} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{23} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{24} + ( 4 \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{25} + ( -\zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{26} + ( -2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{29} + ( -2 \zeta_{24} - 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{30} + ( 4 - 4 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{2} q^{32} + ( 12 \zeta_{24}^{2} - 12 \zeta_{24}^{6} ) q^{33} -2 q^{34} + 3 q^{36} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{37} + ( 2 \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{38} + ( -6 + 4 \zeta_{24} - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{39} + ( \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{40} + ( 6 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{41} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{44} + ( -3 - 3 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{45} + ( -2 - 4 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{46} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{48} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{50} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{51} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{52} + ( 4 \zeta_{24} - 6 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{53} + ( 6 - 4 \zeta_{24} - 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{55} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{57} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{58} + ( 4 + 2 \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{59} + ( 3 + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{60} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{61} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{62} - q^{64} + ( 2 \zeta_{24} - 5 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{65} + ( -12 + 12 \zeta_{24}^{4} ) q^{66} + 8 \zeta_{24}^{2} q^{67} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{68} + ( -12 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{69} + ( -6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{71} + ( -3 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{72} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( -2 + 2 \zeta_{24}^{4} ) q^{74} + ( \zeta_{24} - 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{75} + ( -4 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{76} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{78} + ( 2 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{79} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{80} + ( 9 - 9 \zeta_{24}^{4} ) q^{81} + ( -2 \zeta_{24} - 6 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{82} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{83} + ( -2 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{85} + ( -2 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{86} + ( 4 \zeta_{24} - 12 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{87} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{88} + 10 \zeta_{24}^{4} q^{89} + ( 3 + 3 \zeta_{24} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{90} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{92} + ( -4 \zeta_{24} + 12 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 12 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{93} + ( 4 - 4 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 1 + 4 \zeta_{24} + 7 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{95} + ( \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{96} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{97} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 4q^{5} + 12q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 4q^{5} + 12q^{9} + 4q^{10} + 24q^{15} - 4q^{16} - 16q^{19} + 8q^{20} - 8q^{26} - 16q^{29} - 12q^{30} + 16q^{31} - 16q^{34} + 24q^{36} - 24q^{39} - 4q^{40} + 48q^{41} - 12q^{45} - 8q^{46} - 8q^{50} + 48q^{55} + 16q^{59} + 12q^{60} + 24q^{61} - 8q^{64} + 4q^{65} - 48q^{66} - 96q^{69} - 48q^{71} - 8q^{74} - 32q^{76} + 8q^{79} + 4q^{80} + 36q^{81} - 16q^{85} + 16q^{86} + 40q^{89} + 24q^{90} + 16q^{94} + 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.866025 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i −2.03906 0.917738i 2.44949 0 1.00000i 1.50000 2.59808i 1.30701 + 1.81431i
79.2 −0.866025 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.30701 1.81431i −2.44949 0 1.00000i 1.50000 2.59808i −2.03906 + 0.917738i
79.3 0.866025 + 0.500000i −2.12132 + 1.22474i 0.500000 + 0.866025i 0.917738 2.03906i −2.44949 0 1.00000i 1.50000 2.59808i 1.81431 1.30701i
79.4 0.866025 + 0.500000i 2.12132 1.22474i 0.500000 + 0.866025i 1.81431 + 1.30701i 2.44949 0 1.00000i 1.50000 2.59808i 0.917738 + 2.03906i
459.1 −0.866025 + 0.500000i −2.12132 1.22474i 0.500000 0.866025i −2.03906 + 0.917738i 2.44949 0 1.00000i 1.50000 + 2.59808i 1.30701 1.81431i
459.2 −0.866025 + 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.30701 + 1.81431i −2.44949 0 1.00000i 1.50000 + 2.59808i −2.03906 0.917738i
459.3 0.866025 0.500000i −2.12132 1.22474i 0.500000 0.866025i 0.917738 + 2.03906i −2.44949 0 1.00000i 1.50000 + 2.59808i 1.81431 + 1.30701i
459.4 0.866025 0.500000i 2.12132 + 1.22474i 0.500000 0.866025i 1.81431 1.30701i 2.44949 0 1.00000i 1.50000 + 2.59808i 0.917738 2.03906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.i.f 8
5.b even 2 1 inner 490.2.i.f 8
7.b odd 2 1 490.2.i.c 8
7.c even 3 1 490.2.c.e 4
7.c even 3 1 inner 490.2.i.f 8
7.d odd 6 1 70.2.c.a 4
7.d odd 6 1 490.2.i.c 8
21.g even 6 1 630.2.g.g 4
28.f even 6 1 560.2.g.e 4
35.c odd 2 1 490.2.i.c 8
35.i odd 6 1 70.2.c.a 4
35.i odd 6 1 490.2.i.c 8
35.j even 6 1 490.2.c.e 4
35.j even 6 1 inner 490.2.i.f 8
35.k even 12 1 350.2.a.g 2
35.k even 12 1 350.2.a.h 2
35.l odd 12 1 2450.2.a.bl 2
35.l odd 12 1 2450.2.a.bq 2
56.j odd 6 1 2240.2.g.j 4
56.m even 6 1 2240.2.g.i 4
84.j odd 6 1 5040.2.t.t 4
105.p even 6 1 630.2.g.g 4
105.w odd 12 1 3150.2.a.bs 2
105.w odd 12 1 3150.2.a.bt 2
140.s even 6 1 560.2.g.e 4
140.x odd 12 1 2800.2.a.bl 2
140.x odd 12 1 2800.2.a.bm 2
280.ba even 6 1 2240.2.g.i 4
280.bk odd 6 1 2240.2.g.j 4
420.be odd 6 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 7.d odd 6 1
70.2.c.a 4 35.i odd 6 1
350.2.a.g 2 35.k even 12 1
350.2.a.h 2 35.k even 12 1
490.2.c.e 4 7.c even 3 1
490.2.c.e 4 35.j even 6 1
490.2.i.c 8 7.b odd 2 1
490.2.i.c 8 7.d odd 6 1
490.2.i.c 8 35.c odd 2 1
490.2.i.c 8 35.i odd 6 1
490.2.i.f 8 1.a even 1 1 trivial
490.2.i.f 8 5.b even 2 1 inner
490.2.i.f 8 7.c even 3 1 inner
490.2.i.f 8 35.j even 6 1 inner
560.2.g.e 4 28.f even 6 1
560.2.g.e 4 140.s even 6 1
630.2.g.g 4 21.g even 6 1
630.2.g.g 4 105.p even 6 1
2240.2.g.i 4 56.m even 6 1
2240.2.g.i 4 280.ba even 6 1
2240.2.g.j 4 56.j odd 6 1
2240.2.g.j 4 280.bk odd 6 1
2450.2.a.bl 2 35.l odd 12 1
2450.2.a.bq 2 35.l odd 12 1
2800.2.a.bl 2 140.x odd 12 1
2800.2.a.bm 2 140.x odd 12 1
3150.2.a.bs 2 105.w odd 12 1
3150.2.a.bt 2 105.w odd 12 1
5040.2.t.t 4 84.j odd 6 1
5040.2.t.t 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{4} - 6 T_{3}^{2} + 36 \)
\( T_{11}^{4} + 24 T_{11}^{2} + 576 \)
\( T_{19}^{4} + 8 T_{19}^{3} + 54 T_{19}^{2} + 80 T_{19} + 100 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - 9 T^{4} + 81 T^{8} )^{2} \)
$5$ \( 1 - 4 T + 8 T^{2} + 8 T^{3} - 41 T^{4} + 40 T^{5} + 200 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 32 T^{2} + 498 T^{4} - 5408 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 8 T + 16 T^{2} + 80 T^{3} + 727 T^{4} + 1520 T^{5} + 5776 T^{6} + 54872 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 + 36 T^{2} + 298 T^{4} - 2160 T^{6} + 30579 T^{8} - 1142640 T^{10} + 83392618 T^{12} + 5329292004 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 8 T + 10 T^{2} + 64 T^{3} - 29 T^{4} + 1984 T^{5} + 9610 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 - 12 T + 107 T^{2} - 444 T^{3} + 1369 T^{4} )^{2}( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 12 T + 94 T^{2} - 492 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 92 T^{2} + 4278 T^{4} - 170108 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 108 T^{2} + 5866 T^{4} + 149040 T^{6} + 2971347 T^{8} + 329229360 T^{10} + 28624208746 T^{12} + 1164155255532 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 92 T^{2} + 4186 T^{4} - 123280 T^{6} - 13364573 T^{8} - 346293520 T^{10} + 33029553466 T^{12} + 2039121223868 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 8 T - 64 T^{2} - 80 T^{3} + 9127 T^{4} - 4720 T^{5} - 222784 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 12 T - 8 T^{2} - 360 T^{3} + 10599 T^{4} - 21960 T^{5} - 29768 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( 1 + 236 T^{2} + 31498 T^{4} + 3195440 T^{6} + 260340979 T^{8} + 17028499760 T^{10} + 894487795018 T^{12} + 35714877404204 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 4 T - 122 T^{2} + 80 T^{3} + 11539 T^{4} + 6320 T^{5} - 761402 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 160 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 124 T^{2} + 8838 T^{4} - 1166716 T^{6} + 88529281 T^{8} )^{2} \)
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