# Properties

 Label 930.2.j.f Level $930$ Weight $2$ Character orbit 930.j Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.1698758656.6 Defining polynomial: $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 97 x^{4} + 176 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 18 \nu^{5} + 28 \nu^{4} + 89 \nu^{3} + 74 \nu^{2} + 104 \nu - 16$$$$)/64$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 18 \nu^{5} - 28 \nu^{4} + 89 \nu^{3} - 74 \nu^{2} + 104 \nu + 16$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} - 10 \nu^{5} - 20 \nu^{4} + 15 \nu^{3} - 2 \nu^{2} + 120 \nu + 80$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} - 10 \nu^{5} - 20 \nu^{4} + 15 \nu^{3} - 2 \nu^{2} + 184 \nu + 80$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 10 \nu^{5} - 20 \nu^{4} - 15 \nu^{3} - 2 \nu^{2} - 120 \nu + 80$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 14 \nu^{4} + 45 \nu^{2} + 32$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + 46 \nu^{5} + 179 \nu^{3} + 168 \nu$$$$)/64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} - \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2 \beta_{2} - 2 \beta_{1} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{7} - 2 \beta_{5} - 5 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - 22 \beta_{2} + 22 \beta_{1} + 37$$ $$\nu^{5}$$ $$=$$ $$-52 \beta_{7} + 22 \beta_{5} + 37 \beta_{4} - 59 \beta_{3} + 56 \beta_{2} + 56 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$89 \beta_{6} - 56 \beta_{5} - 56 \beta_{3} + 218 \beta_{2} - 218 \beta_{1} - 325$$ $$\nu^{7}$$ $$=$$ $$580 \beta_{7} - 218 \beta_{5} - 325 \beta_{4} + 543 \beta_{3} - 620 \beta_{2} - 620 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$\beta_{7}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
497.1
 0.692297i − 1.69230i − 3.16053i 2.16053i − 0.692297i 1.69230i 3.16053i − 2.16053i
−0.707107 + 0.707107i 1.70711 + 0.292893i 1.00000i −1.19663 + 1.88893i −1.41421 + 1.00000i 3.59604 + 3.59604i 0.707107 + 0.707107i 2.82843 + 1.00000i −0.489528 2.18183i
497.2 −0.707107 + 0.707107i 1.70711 + 0.292893i 1.00000i 0.489528 2.18183i −1.41421 + 1.00000i −0.474719 0.474719i 0.707107 + 0.707107i 2.82843 + 1.00000i 1.19663 + 1.88893i
497.3 0.707107 0.707107i 0.292893 + 1.70711i 1.00000i −1.52773 1.63280i 1.41421 + 1.00000i −1.33991 1.33991i −0.707107 0.707107i −2.82843 + 1.00000i −2.23483 0.0743018i
497.4 0.707107 0.707107i 0.292893 + 1.70711i 1.00000i 2.23483 0.0743018i 1.41421 + 1.00000i 0.218591 + 0.218591i −0.707107 0.707107i −2.82843 + 1.00000i 1.52773 1.63280i
683.1 −0.707107 0.707107i 1.70711 0.292893i 1.00000i −1.19663 1.88893i −1.41421 1.00000i 3.59604 3.59604i 0.707107 0.707107i 2.82843 1.00000i −0.489528 + 2.18183i
683.2 −0.707107 0.707107i 1.70711 0.292893i 1.00000i 0.489528 + 2.18183i −1.41421 1.00000i −0.474719 + 0.474719i 0.707107 0.707107i 2.82843 1.00000i 1.19663 1.88893i
683.3 0.707107 + 0.707107i 0.292893 1.70711i 1.00000i −1.52773 + 1.63280i 1.41421 1.00000i −1.33991 + 1.33991i −0.707107 + 0.707107i −2.82843 1.00000i −2.23483 + 0.0743018i
683.4 0.707107 + 0.707107i 0.292893 1.70711i 1.00000i 2.23483 + 0.0743018i 1.41421 1.00000i 0.218591 0.218591i −0.707107 + 0.707107i −2.82843 1.00000i 1.52773 + 1.63280i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.f yes 8
3.b odd 2 1 930.2.j.c 8
5.c odd 4 1 930.2.j.c 8
15.e even 4 1 inner 930.2.j.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.c 8 3.b odd 2 1
930.2.j.c 8 5.c odd 4 1
930.2.j.f yes 8 1.a even 1 1 trivial
930.2.j.f yes 8 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{8} - \cdots$$ $$T_{11}^{8} + 22 T_{11}^{6} + 97 T_{11}^{4} + 144 T_{11}^{2} + 64$$ $$T_{17}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 9 - 12 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$5$ $$625 + 50 T^{2} - 80 T^{3} + 2 T^{4} - 16 T^{5} + 2 T^{6} + T^{8}$$
$7$ $$4 - 8 T + 8 T^{2} + 52 T^{3} + 117 T^{4} + 48 T^{5} + 8 T^{6} - 4 T^{7} + T^{8}$$
$11$ $$64 + 144 T^{2} + 97 T^{4} + 22 T^{6} + T^{8}$$
$13$ $$3136 + 6272 T + 6272 T^{2} + 2688 T^{3} + 596 T^{4} + 24 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$17$ $$3717184 - 339328 T + 15488 T^{2} + 7424 T^{3} + 3540 T^{4} - 168 T^{5} + 8 T^{6} + 4 T^{7} + T^{8}$$
$19$ $$107584 + 53776 T^{2} + 4417 T^{4} + 118 T^{6} + T^{8}$$
$23$ $$37636 - 42680 T + 24200 T^{2} - 3868 T^{3} + 437 T^{4} - 136 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$29$ $$( 248 - 8 T - 62 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$31$ $$( -1 + T )^{8}$$
$37$ $$( 8 + 4 T + T^{2} )^{4}$$
$41$ $$1048576 + 208896 T^{2} + 11140 T^{4} + 196 T^{6} + T^{8}$$
$43$ $$85264 - 56064 T + 18432 T^{2} + 4768 T^{3} + 785 T^{4} - 104 T^{5} + 32 T^{6} + 8 T^{7} + T^{8}$$
$47$ $$16384 - 16384 T + 8192 T^{2} + 16128 T^{3} + 9860 T^{4} + 2616 T^{5} + 392 T^{6} + 28 T^{7} + T^{8}$$
$53$ $$15376 + 75392 T + 184832 T^{2} - 68432 T^{3} + 12977 T^{4} + 772 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$59$ $$( -1024 - 512 T - 30 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$61$ $$( -7792 + 720 T + 186 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$67$ $$2408704 - 1489920 T + 460800 T^{2} + 59648 T^{3} + 4640 T^{4} - 448 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$71$ $$583696 + 369784 T^{2} + 22561 T^{4} + 286 T^{6} + T^{8}$$
$73$ $$38416 + 120736 T + 189728 T^{2} + 114296 T^{3} + 38417 T^{4} + 6476 T^{5} + 648 T^{6} + 36 T^{7} + T^{8}$$
$79$ $$602176 + 346288 T^{2} + 20289 T^{4} + 278 T^{6} + T^{8}$$
$83$ $$18800896 - 15193344 T + 6139008 T^{2} - 815904 T^{3} + 56196 T^{4} - 888 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$89$ $$( 248 + 288 T + 113 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$97$ $$4129024 + 2178304 T + 574592 T^{2} - 102112 T^{3} + 9860 T^{4} + 344 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$