# Properties

 Label 588.2.o.b Level $588$ Weight $2$ Character orbit 588.o Analytic conductor $4.695$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.69520363885$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.562828176.1 Defining polynomial: $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} + 2 \nu^{3} + 8 \nu^{2} + 4 \nu - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 4 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} - 4 \nu + 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} + 2 \nu^{3} - 4 \nu + 8$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 4 \nu^{6} - 3 \nu^{5} - 8 \nu^{4} + 10 \nu^{3} - 12 \nu + 40$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 2 \nu^{6} + \nu^{5} - 6 \nu^{4} + 10 \nu^{3} + 4 \nu^{2} - 12 \nu + 32$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} - 2 \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{6} - 2 \beta_{5} - \beta_{4} + 6 \beta_{3} + \beta_{2} - 4$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} + 8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$493$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.0777157 + 1.41208i −1.33790 − 0.458297i 1.40376 + 0.171630i 0.856419 − 1.12541i 0.0777157 − 1.41208i −1.33790 + 0.458297i 1.40376 − 0.171630i 0.856419 + 1.12541i
−1.18404 0.773342i −0.500000 0.866025i 0.803884 + 1.83133i −0.380152 0.219481i −0.0777157 + 1.41208i 0 0.464416 2.79004i −0.500000 + 0.866025i 0.280380 + 0.553861i
19.2 −0.272050 + 1.38780i −0.500000 0.866025i −1.85198 0.755103i −2.12403 1.22631i 1.33790 0.458297i 0 1.55176 2.36475i −0.500000 + 0.866025i 2.27971 2.61411i
19.3 0.553244 1.30151i −0.500000 0.866025i −1.38784 1.44010i −0.834598 0.481855i −1.40376 + 0.171630i 0 −2.64212 + 1.00956i −0.500000 + 0.866025i −1.08887 + 0.819652i
19.4 1.40284 0.178976i −0.500000 0.866025i 1.93594 0.502151i 3.33878 + 1.92764i −0.856419 1.12541i 0 2.62594 1.05092i −0.500000 + 0.866025i 5.02878 + 2.10662i
31.1 −1.18404 + 0.773342i −0.500000 + 0.866025i 0.803884 1.83133i −0.380152 + 0.219481i −0.0777157 1.41208i 0 0.464416 + 2.79004i −0.500000 0.866025i 0.280380 0.553861i
31.2 −0.272050 1.38780i −0.500000 + 0.866025i −1.85198 + 0.755103i −2.12403 + 1.22631i 1.33790 + 0.458297i 0 1.55176 + 2.36475i −0.500000 0.866025i 2.27971 + 2.61411i
31.3 0.553244 + 1.30151i −0.500000 + 0.866025i −1.38784 + 1.44010i −0.834598 + 0.481855i −1.40376 0.171630i 0 −2.64212 1.00956i −0.500000 0.866025i −1.08887 0.819652i
31.4 1.40284 + 0.178976i −0.500000 + 0.866025i 1.93594 + 0.502151i 3.33878 1.92764i −0.856419 + 1.12541i 0 2.62594 + 1.05092i −0.500000 0.866025i 5.02878 2.10662i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.o.b 8
4.b odd 2 1 588.2.o.d 8
7.b odd 2 1 84.2.o.b yes 8
7.c even 3 1 84.2.o.a 8
7.c even 3 1 588.2.b.b 8
7.d odd 6 1 588.2.b.a 8
7.d odd 6 1 588.2.o.d 8
21.c even 2 1 252.2.bf.f 8
21.g even 6 1 1764.2.b.j 8
21.h odd 6 1 252.2.bf.g 8
21.h odd 6 1 1764.2.b.i 8
28.d even 2 1 84.2.o.a 8
28.f even 6 1 588.2.b.b 8
28.f even 6 1 inner 588.2.o.b 8
28.g odd 6 1 84.2.o.b yes 8
28.g odd 6 1 588.2.b.a 8
56.e even 2 1 1344.2.bl.j 8
56.h odd 2 1 1344.2.bl.i 8
56.k odd 6 1 1344.2.bl.i 8
56.p even 6 1 1344.2.bl.j 8
84.h odd 2 1 252.2.bf.g 8
84.j odd 6 1 1764.2.b.i 8
84.n even 6 1 252.2.bf.f 8
84.n even 6 1 1764.2.b.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 7.c even 3 1
84.2.o.a 8 28.d even 2 1
84.2.o.b yes 8 7.b odd 2 1
84.2.o.b yes 8 28.g odd 6 1
252.2.bf.f 8 21.c even 2 1
252.2.bf.f 8 84.n even 6 1
252.2.bf.g 8 21.h odd 6 1
252.2.bf.g 8 84.h odd 2 1
588.2.b.a 8 7.d odd 6 1
588.2.b.a 8 28.g odd 6 1
588.2.b.b 8 7.c even 3 1
588.2.b.b 8 28.f even 6 1
588.2.o.b 8 1.a even 1 1 trivial
588.2.o.b 8 28.f even 6 1 inner
588.2.o.d 8 4.b odd 2 1
588.2.o.d 8 7.d odd 6 1
1344.2.bl.i 8 56.h odd 2 1
1344.2.bl.i 8 56.k odd 6 1
1344.2.bl.j 8 56.e even 2 1
1344.2.bl.j 8 56.p even 6 1
1764.2.b.i 8 21.h odd 6 1
1764.2.b.i 8 84.j odd 6 1
1764.2.b.j 8 21.g even 6 1
1764.2.b.j 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(588, [\chi])$$:

 $$T_{5}^{8} - 11 T_{5}^{6} + 125 T_{5}^{4} + 264 T_{5}^{3} + 236 T_{5}^{2} + 96 T_{5} + 16$$ $$T_{11}^{8} + \cdots$$ $$T_{19}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 8 T + 4 T^{2} - 4 T^{3} - 2 T^{5} + T^{6} - T^{7} + T^{8}$$
$3$ $$( 1 + T + T^{2} )^{4}$$
$5$ $$16 + 96 T + 236 T^{2} + 264 T^{3} + 125 T^{4} - 11 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$400 - 240 T - 212 T^{2} + 156 T^{3} + 125 T^{4} - 78 T^{5} - T^{6} + 6 T^{7} + T^{8}$$
$13$ $$256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8}$$
$17$ $$1024 - 1536 T - 128 T^{2} + 1344 T^{3} + 752 T^{4} - 28 T^{6} + T^{8}$$
$19$ $$16 - 240 T + 3628 T^{2} + 372 T^{3} + 405 T^{4} + 78 T^{5} + 43 T^{6} + 6 T^{7} + T^{8}$$
$23$ $$16384 - 12288 T - 2048 T^{2} + 3840 T^{3} + 1472 T^{4} - 40 T^{6} + T^{8}$$
$29$ $$( -512 - 352 T - 45 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$31$ $$4173849 + 551610 T + 244512 T^{2} + 1836 T^{3} + 6633 T^{4} - 36 T^{5} + 120 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$355216 - 214560 T + 103972 T^{2} - 22632 T^{3} + 4605 T^{4} - 462 T^{5} + 79 T^{6} - 6 T^{7} + T^{8}$$
$41$ $$350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8}$$
$43$ $$1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8}$$
$47$ $$4096 + 1024 T + 2048 T^{2} + 64 T^{3} + 784 T^{4} + 80 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$
$53$ $$64 - 928 T + 12968 T^{2} - 7012 T^{3} + 3265 T^{4} - 476 T^{5} + 77 T^{6} + 4 T^{7} + T^{8}$$
$59$ $$1420864 - 548320 T + 243784 T^{2} - 20956 T^{3} + 5977 T^{4} + 542 T^{5} + 223 T^{6} + 14 T^{7} + T^{8}$$
$61$ $$1048576 + 1572864 T + 901120 T^{2} + 172032 T^{3} + 7424 T^{4} - 1344 T^{5} - 64 T^{6} + 12 T^{7} + T^{8}$$
$67$ $$4129024 - 146304 T - 337616 T^{2} + 12024 T^{3} + 30929 T^{4} + 7014 T^{5} + 755 T^{6} + 42 T^{7} + T^{8}$$
$71$ $$200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8}$$
$73$ $$952576 - 1124352 T + 494096 T^{2} - 61056 T^{3} - 3127 T^{4} + 954 T^{5} + 55 T^{6} - 18 T^{7} + T^{8}$$
$79$ $$241081 + 67758 T - 89888 T^{2} - 27048 T^{3} + 37649 T^{4} + 1176 T^{5} - 184 T^{6} - 6 T^{7} + T^{8}$$
$83$ $$( 196 - 304 T - 103 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$89$ $$4096 + 18432 T + 33536 T^{2} + 26496 T^{3} + 8528 T^{4} - 92 T^{6} + T^{8}$$
$97$ $$246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8}$$