# Properties

 Label 230.2.e.a Level $230$ Weight $2$ Character orbit 230.e Analytic conductor $1.837$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.110166016.2 Defining polynomial: $$x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 10 x^{6} + 19 x^{4} + 10 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} + 10 \nu^{4} + 9 \nu^{3} + 18 \nu^{2} + 9 \nu + 5$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - \nu^{5} + 10 \nu^{4} - 9 \nu^{3} + 18 \nu^{2} - 9 \nu + 5$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + 9 \nu^{4} + 11 \nu^{2} + 4$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + 10 \nu^{5} + 9 \nu^{4} + 18 \nu^{3} + 9 \nu^{2} + 5 \nu - 4$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 18 \nu^{3} - 9 \nu^{2} + 5 \nu$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} + 19 \nu^{5} + 29 \nu^{3} + 9 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + \beta_{2} - 4 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 20$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{7} + 18 \beta_{6} + 18 \beta_{5} + 7 \beta_{3} - 7 \beta_{2} + 27 \beta_{1} + 18$$ $$\nu^{6}$$ $$=$$ $$52 \beta_{6} - 52 \beta_{5} + 72 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} - 151$$ $$\nu^{7}$$ $$=$$ $$72 \beta_{7} - 142 \beta_{6} - 142 \beta_{5} - 52 \beta_{3} + 52 \beta_{2} - 203 \beta_{1} - 142$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-\beta_{7}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 − 0.360409i 2.77462i 0.814115i − 1.22833i 0.360409i − 2.77462i − 0.814115i 1.22833i
−0.707107 0.707107i −1.96195 + 1.96195i 1.00000i 1.77462 1.36041i 2.77462 0.105561 0.105561i 0.707107 0.707107i 4.69853i −2.21680 0.292893i
137.2 −0.707107 0.707107i 0.254848 0.254848i 1.00000i −1.36041 + 1.77462i −0.360409 −0.812668 + 0.812668i 0.707107 0.707107i 2.87011i 2.21680 0.292893i
137.3 0.707107 + 0.707107i −0.868559 + 0.868559i 1.00000i −2.22833 0.185885i −1.22833 −1.38978 + 1.38978i −0.707107 + 0.707107i 1.49121i −1.44423 1.70711i
137.4 0.707107 + 0.707107i 0.575666 0.575666i 1.00000i −0.185885 2.22833i 0.814115 2.09689 2.09689i −0.707107 + 0.707107i 2.33722i 1.44423 1.70711i
183.1 −0.707107 + 0.707107i −1.96195 1.96195i 1.00000i 1.77462 + 1.36041i 2.77462 0.105561 + 0.105561i 0.707107 + 0.707107i 4.69853i −2.21680 + 0.292893i
183.2 −0.707107 + 0.707107i 0.254848 + 0.254848i 1.00000i −1.36041 1.77462i −0.360409 −0.812668 0.812668i 0.707107 + 0.707107i 2.87011i 2.21680 + 0.292893i
183.3 0.707107 0.707107i −0.868559 0.868559i 1.00000i −2.22833 + 0.185885i −1.22833 −1.38978 1.38978i −0.707107 0.707107i 1.49121i −1.44423 + 1.70711i
183.4 0.707107 0.707107i 0.575666 + 0.575666i 1.00000i −0.185885 + 2.22833i 0.814115 2.09689 + 2.09689i −0.707107 0.707107i 2.33722i 1.44423 + 1.70711i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 183.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.e.a 8
5.b even 2 1 1150.2.e.c 8
5.c odd 4 1 230.2.e.b yes 8
5.c odd 4 1 1150.2.e.b 8
23.b odd 2 1 230.2.e.b yes 8
115.c odd 2 1 1150.2.e.b 8
115.e even 4 1 inner 230.2.e.a 8
115.e even 4 1 1150.2.e.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.e.a 8 1.a even 1 1 trivial
230.2.e.a 8 115.e even 4 1 inner
230.2.e.b yes 8 5.c odd 4 1
230.2.e.b yes 8 23.b odd 2 1
1150.2.e.b 8 5.c odd 4 1
1150.2.e.b 8 115.c odd 2 1
1150.2.e.c 8 5.b even 2 1
1150.2.e.c 8 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(230, [\chi])$$:

 $$T_{3}^{8} + 4 T_{3}^{7} + 8 T_{3}^{6} - T_{3}^{4} + 8 T_{3}^{2} - 4 T_{3} + 1$$ $$T_{7}^{8} + 8 T_{7}^{5} + 47 T_{7}^{4} + 56 T_{7}^{3} + 32 T_{7}^{2} - 8 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$1 - 4 T + 8 T^{2} - T^{4} + 8 T^{6} + 4 T^{7} + T^{8}$$
$5$ $$625 + 500 T + 200 T^{2} + 60 T^{3} + 14 T^{4} + 12 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$7$ $$1 - 8 T + 32 T^{2} + 56 T^{3} + 47 T^{4} + 8 T^{5} + T^{8}$$
$11$ $$289 + 886 T^{2} + 471 T^{4} + 50 T^{6} + T^{8}$$
$13$ $$5041 - 4260 T + 1800 T^{2} + 1860 T^{3} + 819 T^{4} + 60 T^{5} + T^{8}$$
$17$ $$30625 + 59500 T + 57800 T^{2} + 30140 T^{3} + 9851 T^{4} + 2084 T^{5} + 288 T^{6} + 24 T^{7} + T^{8}$$
$19$ $$( -49 + 34 T + 3 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$23$ $$279841 - 194672 T + 67712 T^{2} - 18768 T^{3} + 4418 T^{4} - 816 T^{5} + 128 T^{6} - 16 T^{7} + T^{8}$$
$29$ $$16 + 384 T^{2} + 760 T^{4} + 64 T^{6} + T^{8}$$
$31$ $$( -49 - 86 T - 35 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$37$ $$795664 + 21408 T + 288 T^{2} - 5536 T^{3} + 4940 T^{4} - 352 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$41$ $$( -119 + 82 T - 5 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$43$ $$3083536 + 1475040 T + 352800 T^{2} + 6560 T^{3} - 2356 T^{4} - 160 T^{5} + 200 T^{6} - 20 T^{7} + T^{8}$$
$47$ $$4624 + 1088 T + 128 T^{2} + 576 T^{3} + 1160 T^{4} + 528 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$53$ $$614656 - 627200 T + 320000 T^{2} - 86400 T^{3} + 13232 T^{4} - 800 T^{5} + T^{8}$$
$59$ $$226576 + 53008 T^{2} + 3868 T^{4} + 108 T^{6} + T^{8}$$
$61$ $$13315201 + 1217618 T^{2} + 35179 T^{4} + 338 T^{6} + T^{8}$$
$67$ $$9265936 - 6404576 T + 2213408 T^{2} - 403520 T^{3} + 40684 T^{4} - 1360 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$71$ $$( -1193 - 594 T + 67 T^{2} + 22 T^{3} + T^{4} )^{2}$$
$73$ $$1882384 - 2403744 T + 1534752 T^{2} - 48928 T^{3} + 2780 T^{4} - 1584 T^{5} + 392 T^{6} - 28 T^{7} + T^{8}$$
$79$ $$( -5348 - 2080 T - 210 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$83$ $$10000 - 236000 T + 2784800 T^{2} - 111360 T^{3} + 2316 T^{4} - 1072 T^{5} + 392 T^{6} - 28 T^{7} + T^{8}$$
$89$ $$( -2500 + 2000 T - 50 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$97$ $$246584209 + 28328212 T + 1627208 T^{2} - 420988 T^{3} + 60403 T^{4} + 620 T^{5} + 32 T^{6} - 8 T^{7} + T^{8}$$