Properties

 Label 115.2.b.b Level $115$ Weight $2$ Character orbit 115.b Analytic conductor $0.918$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.527896576.2 Defining polynomial: $$x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{5} + ( -2 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{6} + ( \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{8} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{2} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{3} + ( -1 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{4} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{5} + ( -2 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{6} + ( \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{8} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{9} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{3} + 3 \beta_{5} - 3 \beta_{7} ) q^{12} + ( \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{13} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{14} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{15} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{16} + ( \beta_{1} + \beta_{3} - 4 \beta_{5} + 2 \beta_{7} ) q^{17} + 2 \beta_{5} q^{18} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{19} + ( -1 - \beta_{2} - \beta_{3} - 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} + ( -2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{21} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} -\beta_{5} q^{23} + ( 3 + 2 \beta_{2} ) q^{24} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{25} + ( 2 + 5 \beta_{2} - \beta_{4} + \beta_{6} ) q^{26} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{28} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{29} + ( -\beta_{2} + \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{30} + ( 1 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{31} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{33} + ( -4 - 4 \beta_{2} + \beta_{4} - \beta_{6} ) q^{34} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} + ( -4 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{36} + ( -\beta_{4} - \beta_{6} - 4 \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{38} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{40} + ( -5 + 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{41} + ( -\beta_{1} - \beta_{3} - 3 \beta_{4} + 8 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{42} + ( 5 \beta_{1} + 5 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{43} + ( -3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{44} + ( 4 - 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{45} -\beta_{2} q^{46} + ( -3 \beta_{1} - 3 \beta_{3} + \beta_{5} - 6 \beta_{7} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{48} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} ) q^{49} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{50} + ( 4 - 5 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{51} + ( \beta_{1} + \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} ) q^{52} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -8 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} ) q^{54} + ( -3 + 5 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{55} + ( 4 + 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{56} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{58} + 2 \beta_{2} q^{59} + ( -1 - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{60} + ( -6 + 5 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + 10 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{62} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{5} ) q^{63} + ( 7 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{64} + ( -5 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{65} + ( -2 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{66} + ( \beta_{4} + \beta_{6} ) q^{67} + ( 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{68} + ( 1 - \beta_{1} + \beta_{3} ) q^{69} + ( -5 - \beta_{1} - 3 \beta_{3} + 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{70} + ( -5 - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{71} + ( 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{72} + ( -5 \beta_{1} - 5 \beta_{3} - 4 \beta_{4} + 9 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 10 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 3 \beta_{6} ) q^{74} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{75} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{76} + ( 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{77} + ( 4 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} ) q^{78} + ( -6 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{79} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{80} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 4 \beta_{6} ) q^{81} + ( 2 \beta_{1} + 2 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{82} + ( -7 \beta_{1} - 7 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{83} + ( -2 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{84} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{85} + ( 6 - \beta_{1} + 6 \beta_{2} + \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{86} + ( \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{7} ) q^{87} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 4 \beta_{7} ) q^{88} + ( 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{6} ) q^{89} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{90} + ( 8 + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{91} + ( -\beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{92} + ( 3 \beta_{1} + 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{93} + ( 12 + \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{94} + ( -1 + 3 \beta_{1} + \beta_{3} + 3 \beta_{5} ) q^{95} + ( 9 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{96} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} ) q^{97} + ( 4 \beta_{1} + 4 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{98} + ( -8 + 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{4} - 6q^{5} - 12q^{6} - 8q^{9} + O(q^{10})$$ $$8q - 4q^{4} - 6q^{5} - 12q^{6} - 8q^{9} + 6q^{10} + 4q^{11} + 8q^{14} + 6q^{15} + 4q^{16} + 8q^{19} - 8q^{20} - 4q^{21} + 24q^{24} - 16q^{25} + 12q^{26} - 8q^{29} - 2q^{30} - 28q^{34} + 28q^{35} - 16q^{36} + 16q^{39} - 10q^{40} - 16q^{41} - 12q^{44} + 24q^{45} + 28q^{50} + 20q^{51} - 44q^{54} - 16q^{55} + 28q^{56} - 16q^{60} - 16q^{61} + 40q^{64} - 14q^{65} - 16q^{66} + 4q^{69} - 28q^{70} - 48q^{71} + 72q^{74} - 36q^{76} - 48q^{79} - 2q^{80} + 16q^{81} - 4q^{84} + 12q^{85} + 28q^{86} + 16q^{89} - 4q^{90} + 52q^{91} + 84q^{94} - 4q^{95} + 60q^{96} - 72q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-27 \nu^{7} + 31 \nu^{6} + 7 \nu^{5} - 221 \nu^{4} - 187 \nu^{3} + 128 \nu^{2} + 66 \nu - 882$$$$)/467$$ $$\beta_{3}$$ $$=$$ $$($$$$52 \nu^{7} - 77 \nu^{6} + 73 \nu^{5} + 97 \nu^{4} + 585 \nu^{3} - 333 \nu^{2} + 288 \nu + 142$$$$)/467$$ $$\beta_{4}$$ $$=$$ $$($$$$138 \nu^{7} - 366 \nu^{6} + 535 \nu^{5} - 12 \nu^{4} + 852 \nu^{3} - 1692 \nu^{2} + 2309 \nu - 162$$$$)/467$$ $$\beta_{5}$$ $$=$$ $$($$$$-142 \nu^{7} + 336 \nu^{6} - 361 \nu^{5} - 211 \nu^{4} - 897 \nu^{3} + 2005 \nu^{2} - 1469 \nu - 280$$$$)/467$$ $$\beta_{6}$$ $$=$$ $$($$$$273 \nu^{7} - 521 \nu^{6} + 500 \nu^{5} + 626 \nu^{4} + 1787 \nu^{3} - 2332 \nu^{2} + 1979 \nu + 979$$$$)/467$$ $$\beta_{7}$$ $$=$$ $$($$$$-284 \nu^{7} + 672 \nu^{6} - 722 \nu^{5} - 422 \nu^{4} - 1794 \nu^{3} + 3543 \nu^{2} - 2938 \nu - 560$$$$)/467$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2 \beta_{5}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{4} - 5 \beta_{2} - 7$$ $$\nu^{5}$$ $$=$$ $$6 \beta_{7} - 6 \beta_{5} + 5 \beta_{4} - 6 \beta_{2} - 5 \beta_{1} - 6$$ $$\nu^{6}$$ $$=$$ $$22 \beta_{7} + 6 \beta_{6} - 27 \beta_{5} + 6 \beta_{4} - \beta_{3} - \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$29 \beta_{7} + 22 \beta_{6} - 30 \beta_{5} - 15 \beta_{3} + 29 \beta_{2} + 30$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 1.47984 + 1.47984i −0.199724 + 0.199724i 0.790245 − 0.790245i −1.07037 − 1.07037i −1.07037 + 1.07037i 0.790245 + 0.790245i −0.199724 − 0.199724i 1.47984 − 1.47984i
2.37988i 1.95969i −3.66382 0.479844 + 2.18398i −4.66382 2.28394i 3.95969i −0.840379 5.19760 1.14197i
24.2 1.92022i 1.39945i −1.68725 −1.19972 1.88697i −2.68725 4.60747i 0.600553i 1.04155 −3.62340 + 2.30373i
24.3 0.751024i 0.580491i 1.43596 −0.209755 + 2.22621i 0.435963 0.315061i 2.58049i 2.66303 1.67194 + 0.157531i
24.4 0.291367i 3.14073i 1.91511 −2.07037 0.844739i 0.915105 1.20647i 1.14073i −6.86420 −0.246129 + 0.603236i
24.5 0.291367i 3.14073i 1.91511 −2.07037 + 0.844739i 0.915105 1.20647i 1.14073i −6.86420 −0.246129 0.603236i
24.6 0.751024i 0.580491i 1.43596 −0.209755 2.22621i 0.435963 0.315061i 2.58049i 2.66303 1.67194 0.157531i
24.7 1.92022i 1.39945i −1.68725 −1.19972 + 1.88697i −2.68725 4.60747i 0.600553i 1.04155 −3.62340 2.30373i
24.8 2.37988i 1.95969i −3.66382 0.479844 2.18398i −4.66382 2.28394i 3.95969i −0.840379 5.19760 + 1.14197i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.8 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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.b.b 8
3.b odd 2 1 1035.2.b.e 8
4.b odd 2 1 1840.2.e.d 8
5.b even 2 1 inner 115.2.b.b 8
5.c odd 4 1 575.2.a.i 4
5.c odd 4 1 575.2.a.j 4
15.d odd 2 1 1035.2.b.e 8
15.e even 4 1 5175.2.a.bv 4
15.e even 4 1 5175.2.a.bw 4
20.d odd 2 1 1840.2.e.d 8
20.e even 4 1 9200.2.a.ck 4
20.e even 4 1 9200.2.a.cq 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 1.a even 1 1 trivial
115.2.b.b 8 5.b even 2 1 inner
575.2.a.i 4 5.c odd 4 1
575.2.a.j 4 5.c odd 4 1
1035.2.b.e 8 3.b odd 2 1
1035.2.b.e 8 15.d odd 2 1
1840.2.e.d 8 4.b odd 2 1
1840.2.e.d 8 20.d odd 2 1
5175.2.a.bv 4 15.e even 4 1
5175.2.a.bw 4 15.e even 4 1
9200.2.a.ck 4 20.e even 4 1
9200.2.a.cq 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 10 T_{2}^{6} + 27 T_{2}^{4} + 14 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(115, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 14 T^{2} + 27 T^{4} + 10 T^{6} + T^{8}$$
$3$ $$25 + 96 T^{2} + 70 T^{4} + 16 T^{6} + T^{8}$$
$5$ $$625 + 750 T + 650 T^{2} + 410 T^{3} + 206 T^{4} + 82 T^{5} + 26 T^{6} + 6 T^{7} + T^{8}$$
$7$ $$16 + 176 T^{2} + 152 T^{4} + 28 T^{6} + T^{8}$$
$11$ $$( -28 + 44 T - 16 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$3721 + 5872 T^{2} + 1174 T^{4} + 64 T^{6} + T^{8}$$
$17$ $$400 + 1776 T^{2} + 1612 T^{4} + 80 T^{6} + T^{8}$$
$19$ $$( -20 + 28 T - 6 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$23$ $$( 1 + T^{2} )^{4}$$
$29$ $$( 5 - 4 T - 22 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( -167 - 256 T - 74 T^{2} + T^{4} )^{2}$$
$37$ $$226576 + 75856 T^{2} + 5752 T^{4} + 148 T^{6} + T^{8}$$
$41$ $$( 2485 - 368 T - 94 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$43$ $$3857296 + 653600 T^{2} + 21340 T^{4} + 252 T^{6} + T^{8}$$
$47$ $$20367169 + 1273144 T^{2} + 28734 T^{4} + 280 T^{6} + T^{8}$$
$53$ $$4804864 + 468480 T^{2} + 16096 T^{4} + 224 T^{6} + T^{8}$$
$59$ $$( 16 - 16 T - 20 T^{2} + T^{4} )^{2}$$
$61$ $$( -2756 - 1148 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$16 + 208 T^{2} + 120 T^{4} + 20 T^{6} + T^{8}$$
$71$ $$( -7435 - 1568 T + 66 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$73$ $$69538921 + 3070400 T^{2} + 50550 T^{4} + 368 T^{6} + T^{8}$$
$79$ $$( 28 + 412 T + 178 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$83$ $$223442704 + 9024496 T^{2} + 114508 T^{4} + 576 T^{6} + T^{8}$$
$89$ $$( 2380 + 372 T - 86 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$97$ $$21864976 + 3638896 T^{2} + 69592 T^{4} + 460 T^{6} + T^{8}$$