Properties

Label 350.3.i.a
Level $350$
Weight $3$
Character orbit 350.i
Analytic conductor $9.537$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( 6 \zeta_{24} + 6 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{2} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + 2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( 6 \zeta_{24} + 6 \zeta_{24}^{7} ) q^{9} + ( 9 - 3 \zeta_{24}^{3} - 9 \zeta_{24}^{4} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{11} + ( -2 \zeta_{24} - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{12} + ( -2 \zeta_{24} - 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{13} + ( 8 + \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{14} -4 \zeta_{24}^{4} q^{16} + ( 4 \zeta_{24} - 5 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 10 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{17} -12 \zeta_{24}^{2} q^{18} + ( -2 - \zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{19} + ( 19 - 2 \zeta_{24} - 10 \zeta_{24}^{3} - 11 \zeta_{24}^{4} + 10 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{21} + ( -9 \zeta_{24} + 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{22} + ( -9 \zeta_{24} - 15 \zeta_{24}^{2} + 15 \zeta_{24}^{6} + 9 \zeta_{24}^{7} ) q^{23} + ( -4 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{24} + ( 8 + 6 \zeta_{24} + 12 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{26} + ( -3 \zeta_{24} - 6 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{27} + ( -8 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{28} + ( -12 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{29} + ( -7 - 30 \zeta_{24} - 15 \zeta_{24}^{3} - 7 \zeta_{24}^{4} + 15 \zeta_{24}^{5} - 15 \zeta_{24}^{7} ) q^{31} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{32} + ( -12 \zeta_{24} - 15 \zeta_{24}^{2} + 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} - 15 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{33} + ( -4 - 5 \zeta_{24} + 5 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 5 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{34} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{36} + ( 24 \zeta_{24} - 31 \zeta_{24}^{2} + 31 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{37} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{38} + ( -6 + 12 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{39} + ( 2 - 10 \zeta_{24} + 10 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 10 \zeta_{24}^{5} - 20 \zeta_{24}^{7} ) q^{41} + ( -19 \zeta_{24} + 4 \zeta_{24}^{2} + 11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} - 20 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{42} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{43} + ( 6 \zeta_{24} - 18 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{44} + ( 18 + 15 \zeta_{24}^{3} - 18 \zeta_{24}^{4} - 15 \zeta_{24}^{5} - 15 \zeta_{24}^{7} ) q^{46} + ( \zeta_{24} - 29 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 29 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{47} + ( 4 \zeta_{24} - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{48} + ( -15 - 22 \zeta_{24} - 26 \zeta_{24}^{3} + 40 \zeta_{24}^{4} + 26 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( -21 \zeta_{24} + 27 \zeta_{24}^{4} - 21 \zeta_{24}^{7} ) q^{51} + ( -8 \zeta_{24} - 12 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 24 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{52} + ( -39 \zeta_{24}^{2} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{53} + ( 12 + 3 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{54} + ( 20 + 6 \zeta_{24} + 2 \zeta_{24}^{3} - 16 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{56} -3 \zeta_{24}^{6} q^{57} + ( 12 \zeta_{24} - 12 \zeta_{24}^{2} + 12 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{58} + ( 13 - 50 \zeta_{24} - 25 \zeta_{24}^{3} + 13 \zeta_{24}^{4} + 25 \zeta_{24}^{5} - 25 \zeta_{24}^{7} ) q^{59} + ( -14 - 32 \zeta_{24} - 64 \zeta_{24}^{3} + 7 \zeta_{24}^{4} + 64 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{61} + ( 7 \zeta_{24} + 60 \zeta_{24}^{2} + 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 30 \zeta_{24}^{6} - 14 \zeta_{24}^{7} ) q^{62} + ( 6 \zeta_{24} + 12 \zeta_{24}^{2} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 60 \zeta_{24}^{6} - 18 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( -24 + 30 \zeta_{24} + 15 \zeta_{24}^{3} - 24 \zeta_{24}^{4} - 15 \zeta_{24}^{5} + 15 \zeta_{24}^{7} ) q^{66} + ( 29 \zeta_{24}^{2} - 45 \zeta_{24}^{3} - 45 \zeta_{24}^{5} + 45 \zeta_{24}^{7} ) q^{67} + ( 4 \zeta_{24} + 10 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 10 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{68} + ( 3 - 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{69} + ( -6 + 30 \zeta_{24} + 30 \zeta_{24}^{3} - 30 \zeta_{24}^{5} ) q^{71} + ( -24 \zeta_{24}^{2} + 24 \zeta_{24}^{6} ) q^{72} + ( 32 \zeta_{24} - 53 \zeta_{24}^{2} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 106 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{73} + ( -48 + 31 \zeta_{24}^{3} + 48 \zeta_{24}^{4} - 31 \zeta_{24}^{5} - 31 \zeta_{24}^{7} ) q^{74} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{76} + ( -42 \zeta_{24} + 21 \zeta_{24}^{2} - 42 \zeta_{24}^{6} + 42 \zeta_{24}^{7} ) q^{77} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 24 \zeta_{24}^{6} ) q^{78} + ( 15 \zeta_{24} - 55 \zeta_{24}^{4} + 15 \zeta_{24}^{7} ) q^{79} + ( 9 - 54 \zeta_{24}^{3} - 9 \zeta_{24}^{4} + 54 \zeta_{24}^{5} + 54 \zeta_{24}^{7} ) q^{81} + ( -2 \zeta_{24} + 20 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 20 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{82} + ( 4 \zeta_{24} + 136 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 68 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{83} + ( 16 + 16 \zeta_{24} - 4 \zeta_{24}^{3} - 38 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{84} + ( 2 \zeta_{24} + 12 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{86} + ( 36 \zeta_{24} - 24 \zeta_{24}^{2} - 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 48 \zeta_{24}^{6} - 18 \zeta_{24}^{7} ) q^{87} + ( -12 \zeta_{24}^{2} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{88} + ( 126 - 24 \zeta_{24} - 48 \zeta_{24}^{3} - 63 \zeta_{24}^{4} + 48 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{89} + ( -18 + 52 \zeta_{24} + 8 \zeta_{24}^{3} + 48 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 44 \zeta_{24}^{7} ) q^{91} + ( -18 \zeta_{24} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} + 30 \zeta_{24}^{6} ) q^{92} + ( -24 \zeta_{24} + 69 \zeta_{24}^{2} - 69 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{93} + ( 2 + 58 \zeta_{24} + 29 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 29 \zeta_{24}^{5} + 29 \zeta_{24}^{7} ) q^{94} + ( -16 + 4 \zeta_{24} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{96} + ( 26 \zeta_{24} + 44 \zeta_{24}^{2} + 26 \zeta_{24}^{3} + 26 \zeta_{24}^{5} - 22 \zeta_{24}^{6} - 52 \zeta_{24}^{7} ) q^{97} + ( 15 \zeta_{24} + 44 \zeta_{24}^{2} - 40 \zeta_{24}^{3} - 40 \zeta_{24}^{5} - 52 \zeta_{24}^{6} + 25 \zeta_{24}^{7} ) q^{98} + ( -36 + 54 \zeta_{24} + 54 \zeta_{24}^{3} - 54 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + O(q^{10}) \) \( 8 q + 8 q^{4} + 36 q^{11} + 72 q^{14} - 16 q^{16} - 12 q^{19} + 108 q^{21} - 48 q^{24} + 48 q^{26} - 96 q^{29} - 84 q^{31} - 24 q^{39} - 72 q^{44} + 72 q^{46} + 40 q^{49} + 108 q^{51} + 72 q^{54} + 96 q^{56} + 156 q^{59} - 84 q^{61} - 64 q^{64} - 288 q^{66} - 48 q^{71} - 192 q^{74} - 220 q^{79} + 36 q^{81} - 24 q^{84} + 48 q^{86} + 756 q^{89} + 48 q^{91} + 24 q^{94} - 96 q^{96} - 288 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
−1.22474 + 0.707107i −2.09077 + 3.62132i 1.00000 1.73205i 0 5.91359i −6.63103 2.24264i 2.82843i −4.24264 7.34847i 0
199.2 −1.22474 + 0.707107i −0.358719 + 0.621320i 1.00000 1.73205i 0 1.01461i −3.16693 6.24264i 2.82843i 4.24264 + 7.34847i 0
199.3 1.22474 0.707107i 0.358719 0.621320i 1.00000 1.73205i 0 1.01461i 3.16693 + 6.24264i 2.82843i 4.24264 + 7.34847i 0
199.4 1.22474 0.707107i 2.09077 3.62132i 1.00000 1.73205i 0 5.91359i 6.63103 + 2.24264i 2.82843i −4.24264 7.34847i 0
299.1 −1.22474 0.707107i −2.09077 3.62132i 1.00000 + 1.73205i 0 5.91359i −6.63103 + 2.24264i 2.82843i −4.24264 + 7.34847i 0
299.2 −1.22474 0.707107i −0.358719 0.621320i 1.00000 + 1.73205i 0 1.01461i −3.16693 + 6.24264i 2.82843i 4.24264 7.34847i 0
299.3 1.22474 + 0.707107i 0.358719 + 0.621320i 1.00000 + 1.73205i 0 1.01461i 3.16693 6.24264i 2.82843i 4.24264 7.34847i 0
299.4 1.22474 + 0.707107i 2.09077 + 3.62132i 1.00000 + 1.73205i 0 5.91359i 6.63103 2.24264i 2.82843i −4.24264 + 7.34847i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.i.a 8
5.b even 2 1 inner 350.3.i.a 8
5.c odd 4 1 14.3.d.a 4
5.c odd 4 1 350.3.k.a 4
7.d odd 6 1 inner 350.3.i.a 8
15.e even 4 1 126.3.n.c 4
20.e even 4 1 112.3.s.b 4
35.f even 4 1 98.3.d.a 4
35.i odd 6 1 inner 350.3.i.a 8
35.k even 12 1 14.3.d.a 4
35.k even 12 1 98.3.b.b 4
35.k even 12 1 350.3.k.a 4
35.l odd 12 1 98.3.b.b 4
35.l odd 12 1 98.3.d.a 4
40.i odd 4 1 448.3.s.d 4
40.k even 4 1 448.3.s.c 4
60.l odd 4 1 1008.3.cg.l 4
105.k odd 4 1 882.3.n.b 4
105.w odd 12 1 126.3.n.c 4
105.w odd 12 1 882.3.c.f 4
105.x even 12 1 882.3.c.f 4
105.x even 12 1 882.3.n.b 4
140.j odd 4 1 784.3.s.c 4
140.w even 12 1 784.3.c.e 4
140.w even 12 1 784.3.s.c 4
140.x odd 12 1 112.3.s.b 4
140.x odd 12 1 784.3.c.e 4
280.bp odd 12 1 448.3.s.c 4
280.bv even 12 1 448.3.s.d 4
420.br even 12 1 1008.3.cg.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 5.c odd 4 1
14.3.d.a 4 35.k even 12 1
98.3.b.b 4 35.k even 12 1
98.3.b.b 4 35.l odd 12 1
98.3.d.a 4 35.f even 4 1
98.3.d.a 4 35.l odd 12 1
112.3.s.b 4 20.e even 4 1
112.3.s.b 4 140.x odd 12 1
126.3.n.c 4 15.e even 4 1
126.3.n.c 4 105.w odd 12 1
350.3.i.a 8 1.a even 1 1 trivial
350.3.i.a 8 5.b even 2 1 inner
350.3.i.a 8 7.d odd 6 1 inner
350.3.i.a 8 35.i odd 6 1 inner
350.3.k.a 4 5.c odd 4 1
350.3.k.a 4 35.k even 12 1
448.3.s.c 4 40.k even 4 1
448.3.s.c 4 280.bp odd 12 1
448.3.s.d 4 40.i odd 4 1
448.3.s.d 4 280.bv even 12 1
784.3.c.e 4 140.w even 12 1
784.3.c.e 4 140.x odd 12 1
784.3.s.c 4 140.j odd 4 1
784.3.s.c 4 140.w even 12 1
882.3.c.f 4 105.w odd 12 1
882.3.c.f 4 105.x even 12 1
882.3.n.b 4 105.k odd 4 1
882.3.n.b 4 105.x even 12 1
1008.3.cg.l 4 60.l odd 4 1
1008.3.cg.l 4 420.br even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 18 T_{3}^{6} + 315 T_{3}^{4} + 162 T_{3}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( 81 + 162 T^{2} + 315 T^{4} + 18 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 5764801 - 48020 T^{2} + 294 T^{4} - 20 T^{6} + T^{8} \)
$11$ \( ( 3969 - 1134 T + 261 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$13$ \( ( 7056 - 264 T^{2} + T^{4} )^{2} \)
$17$ \( 6765201 + 514998 T^{2} + 36603 T^{4} + 198 T^{6} + T^{8} \)
$19$ \( ( 9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$23$ \( 15752961 - 3072006 T^{2} + 595107 T^{4} - 774 T^{6} + T^{8} \)
$29$ \( ( 72 + 24 T + T^{2} )^{4} \)
$31$ \( ( 1447209 - 50526 T - 615 T^{2} + 42 T^{3} + T^{4} )^{2} \)
$37$ \( 1330863361 - 154168706 T^{2} + 17822595 T^{4} - 4226 T^{6} + T^{8} \)
$41$ \( ( 345744 + 1224 T^{2} + T^{4} )^{2} \)
$43$ \( ( 4624 + 152 T^{2} + T^{4} )^{2} \)
$47$ \( 40135886713521 + 32043891762 T^{2} + 19248075 T^{4} + 5058 T^{6} + T^{8} \)
$53$ \( 2311278643521 - 5500405602 T^{2} + 11569635 T^{4} - 3618 T^{6} + T^{8} \)
$59$ \( ( 10517049 + 252954 T - 1215 T^{2} - 78 T^{3} + T^{4} )^{2} \)
$61$ \( ( 35964009 - 251874 T - 5409 T^{2} + 42 T^{3} + T^{4} )^{2} \)
$67$ \( 106042233977761 - 100731915542 T^{2} + 85389843 T^{4} - 9782 T^{6} + T^{8} \)
$71$ \( ( -1764 + 12 T + T^{2} )^{4} \)
$73$ \( 2254908894346161 + 946203664806 T^{2} + 349559595 T^{4} + 19926 T^{6} + T^{8} \)
$79$ \( ( 6630625 + 283250 T + 9525 T^{2} + 110 T^{3} + T^{4} )^{2} \)
$83$ \( ( 189778176 - 27936 T^{2} + T^{4} )^{2} \)
$89$ \( ( 71419401 - 3194478 T + 56079 T^{2} - 378 T^{3} + T^{4} )^{2} \)
$97$ \( ( 6780816 - 11016 T^{2} + T^{4} )^{2} \)
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