# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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