Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
1.69513 1.23158i
−0.386111 + 0.280526i
−0.227943 + 0.701538i
0.418926 1.28932i
−0.227943 0.701538i
0.418926 + 1.28932i
1.69513 + 1.23158i
−0.386111 0.280526i
−0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −0.518200 + 1.59485i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 −0.809017 + 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −0.0268854 + 0.0827449i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i 1.00000 −1.00000 1.15245 + 0.837304i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 0.309017 0.951057i −0.309017 0.951057i −0.809017 0.587785i 1.00000 −1.00000 3.89263 + 2.82816i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 1.15245 0.837304i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 0.309017 + 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 3.89263 2.82816i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −0.518200 1.59485i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 −0.809017 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −0.0268854 0.0827449i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.a 8
31.d even 5 1 inner 930.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.a 8 1.a even 1 1 trivial
930.2.n.a 8 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$5$ \( ( -1 + T )^{8} \)
$7$ \( 1 + 6 T + 125 T^{2} - 141 T^{3} + 94 T^{4} - 51 T^{5} + 35 T^{6} - 9 T^{7} + T^{8} \)
$11$ \( 410881 - 5769 T + 29090 T^{2} - 2131 T^{3} + 949 T^{4} + 19 T^{5} + 10 T^{6} + T^{7} + T^{8} \)
$13$ \( 121 - 44 T + 177 T^{2} - 247 T^{3} + 180 T^{4} + 137 T^{5} + 37 T^{6} - T^{7} + T^{8} \)
$17$ \( 121 - 737 T + 1875 T^{2} - 1417 T^{3} + 894 T^{4} - 313 T^{5} + 85 T^{6} - 13 T^{7} + T^{8} \)
$19$ \( 7921 + 5518 T + 5705 T^{2} + 1158 T^{3} + 179 T^{4} - 18 T^{5} + 5 T^{6} + 2 T^{7} + T^{8} \)
$23$ \( 1681 + 3034 T + 2919 T^{2} + 1538 T^{3} + 659 T^{4} + 86 T^{5} + 21 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( 323761 - 41537 T + 57786 T^{2} + 5229 T^{3} + 1869 T^{4} - 53 T^{5} - 6 T^{6} + T^{7} + T^{8} \)
$31$ \( 923521 - 89373 T - 1922 T^{2} - 5301 T^{3} + 1275 T^{4} - 171 T^{5} - 2 T^{6} - 3 T^{7} + T^{8} \)
$37$ \( ( -139 + 124 T - 23 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$41$ \( 346921 - 305102 T + 172461 T^{2} - 63301 T^{3} + 16754 T^{4} - 3103 T^{5} + 399 T^{6} - 29 T^{7} + T^{8} \)
$43$ \( 225625 - 59375 T + 157875 T^{2} + 48125 T^{3} + 9350 T^{4} + 1625 T^{5} + 285 T^{6} + 25 T^{7} + T^{8} \)
$47$ \( 346921 - 325717 T + 228195 T^{2} - 86697 T^{3} + 18894 T^{4} - 2053 T^{5} + 165 T^{6} - 13 T^{7} + T^{8} \)
$53$ \( 361 + 589 T + 1335 T^{2} + 2841 T^{3} + 4284 T^{4} + 1111 T^{5} + 195 T^{6} + 19 T^{7} + T^{8} \)
$59$ \( 6974881 + 4708903 T + 1276170 T^{2} - 5107 T^{3} + 21789 T^{4} - 2933 T^{5} + 370 T^{6} - 23 T^{7} + T^{8} \)
$61$ \( ( -9 + 3 T + T^{2} )^{4} \)
$67$ \( ( -8209 - 2613 T - 172 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$71$ \( 244328161 + 3173093 T + 3175935 T^{2} + 249893 T^{3} + 24354 T^{4} - 1303 T^{5} + 125 T^{6} - 3 T^{7} + T^{8} \)
$73$ \( 144264121 - 29715214 T + 2732397 T^{2} + 37878 T^{3} + 70455 T^{4} + 4262 T^{5} + 297 T^{6} + 14 T^{7} + T^{8} \)
$79$ \( 17147881 + 8000412 T + 3779911 T^{2} + 713796 T^{3} + 81509 T^{4} + 6768 T^{5} + 589 T^{6} + 34 T^{7} + T^{8} \)
$83$ \( 1907161 - 2411226 T + 1356334 T^{2} - 380532 T^{3} + 53684 T^{4} - 2964 T^{5} + 361 T^{6} - 8 T^{7} + T^{8} \)
$89$ \( 25391521 - 4006005 T + 971198 T^{2} - 185855 T^{3} + 27649 T^{4} - 2665 T^{5} + 322 T^{6} - 25 T^{7} + T^{8} \)
$97$ \( 214300321 - 9412877 T + 192133 T^{2} + 54571 T^{3} + 53580 T^{4} + 6641 T^{5} + 603 T^{6} + 28 T^{7} + T^{8} \)
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