Properties

Label 330.2.m.f
Level $330$
Weight $2$
Character orbit 330.m
Analytic conductor $2.635$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.2769390625.1
Defining polynomial: \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + 21 x^{4} + 75 x^{3} - 198 x^{2} - 87 x + 841\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + 21 x^{4} + 75 x^{3} - 198 x^{2} - 87 x + 841\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3622 \nu^{7} - 168690 \nu^{6} + 210024 \nu^{5} - 119321 \nu^{4} + 621001 \nu^{3} - 2170527 \nu^{2} - 10517880 \nu + 20083080 \)\()/11583209\)
\(\beta_{3}\)\(=\)\((\)\( 4915 \nu^{7} - 90523 \nu^{6} + 209410 \nu^{5} + 136080 \nu^{4} + 66153 \nu^{3} - 2218842 \nu^{2} - 4202349 \nu + 23823587 \)\()/11583209\)
\(\beta_{4}\)\(=\)\((\)\( 908 \nu^{7} - 3211 \nu^{6} + 11167 \nu^{5} + 3050 \nu^{4} + 96671 \nu^{3} - 39405 \nu^{2} - 93476 \nu + 949895 \)\()/399421\)
\(\beta_{5}\)\(=\)\((\)\( -31462 \nu^{7} + 137254 \nu^{6} - 28223 \nu^{5} + 415469 \nu^{4} - 1154253 \nu^{3} + 298519 \nu^{2} + 11658276 \nu - 7703821 \)\()/11583209\)
\(\beta_{6}\)\(=\)\((\)\( -3648 \nu^{7} + 3143 \nu^{6} - 19751 \nu^{5} + 17019 \nu^{4} - 91661 \nu^{3} - 187200 \nu^{2} + 360035 \nu - 912398 \)\()/399421\)
\(\beta_{7}\)\(=\)\((\)\( 5692 \nu^{7} - 7492 \nu^{6} + 4739 \nu^{5} - 18791 \nu^{4} + 84213 \nu^{3} + 337956 \nu^{2} - 703386 \nu + 105038 \)\()/399421\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + 5 \beta_{5} - 6 \beta_{3} + 6 \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 5 \beta_{4} - 6 \beta_{3} + \beta_{2} - \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(9 \beta_{7} + 11 \beta_{6} + 21 \beta_{5} + 11 \beta_{4} + 10 \beta_{3} - 9 \beta_{1} - 10\)
\(\nu^{5}\)\(=\)\(-8 \beta_{7} - 20 \beta_{6} + 21 \beta_{5} - 8 \beta_{4} + 21 \beta_{2} - 47\)
\(\nu^{6}\)\(=\)\(-29 \beta_{7} - 29 \beta_{6} - 60 \beta_{5} + 60 \beta_{3} - 128 \beta_{2} - 59 \beta_{1}\)
\(\nu^{7}\)\(=\)\(9 \beta_{7} - 324 \beta_{5} - 31 \beta_{4} + 557 \beta_{3} - 557 \beta_{2} + 31 \beta_{1} - 324\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.77873 1.10367i
−2.08774 + 0.152618i
1.88778 1.74766i
−1.07877 + 2.33544i
1.77873 + 1.10367i
−2.08774 0.152618i
1.88778 + 1.74766i
−1.07877 2.33544i
−0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −0.309017 0.951057i −0.309017 0.951057i −1.27873 0.929049i 0.809017 0.587785i 0.309017 0.951057i 1.00000
31.2 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i −0.309017 0.951057i −0.309017 0.951057i 2.58774 + 1.88011i 0.809017 0.587785i 0.309017 0.951057i 1.00000
91.1 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i 0.809017 + 0.587785i −1.38778 + 4.27116i −0.309017 0.951057i −0.809017 + 0.587785i 1.00000
91.2 0.809017 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i 0.809017 + 0.587785i 1.57877 4.85894i −0.309017 0.951057i −0.809017 + 0.587785i 1.00000
181.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 + 0.951057i −1.27873 + 0.929049i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000
181.2 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 + 0.951057i 2.58774 1.88011i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000
301.1 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i 0.809017 0.587785i −1.38778 4.27116i −0.309017 + 0.951057i −0.809017 0.587785i 1.00000
301.2 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i 0.809017 0.587785i 1.57877 + 4.85894i −0.309017 + 0.951057i −0.809017 0.587785i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.m.f 8
3.b odd 2 1 990.2.n.i 8
11.c even 5 1 inner 330.2.m.f 8
11.c even 5 1 3630.2.a.bq 4
11.d odd 10 1 3630.2.a.bs 4
33.h odd 10 1 990.2.n.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.f 8 1.a even 1 1 trivial
330.2.m.f 8 11.c even 5 1 inner
990.2.n.i 8 3.b odd 2 1
990.2.n.i 8 33.h odd 10 1
3630.2.a.bq 4 11.c even 5 1
3630.2.a.bs 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$7$ \( 13456 + 7192 T + 808 T^{2} - 896 T^{3} + 505 T^{4} - 76 T^{5} + 38 T^{6} - 3 T^{7} + T^{8} \)
$11$ \( 14641 - 13310 T + 3509 T^{2} + 330 T^{3} - 349 T^{4} + 30 T^{5} + 29 T^{6} - 10 T^{7} + T^{8} \)
$13$ \( 14641 + 7381 T - 898 T^{2} - 2097 T^{3} + 1505 T^{4} - 123 T^{5} + 62 T^{6} - T^{7} + T^{8} \)
$17$ \( 256 + 192 T + 608 T^{2} + 1104 T^{3} + 1105 T^{4} - 276 T^{5} + 38 T^{6} - 3 T^{7} + T^{8} \)
$19$ \( 16 + 112 T + 2488 T^{2} - 2226 T^{3} + 575 T^{4} + 309 T^{5} + 88 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( ( 55 + 130 T - 65 T^{2} + T^{4} )^{2} \)
$29$ \( 524176 - 143352 T + 122528 T^{2} - 62604 T^{3} + 18505 T^{4} - 3204 T^{5} + 378 T^{6} - 27 T^{7} + T^{8} \)
$31$ \( 26896 - 43296 T + 32112 T^{2} - 9178 T^{3} + 2955 T^{4} + 263 T^{5} + 42 T^{6} + 6 T^{7} + T^{8} \)
$37$ \( 3031081 - 111424 T + 161147 T^{2} - 26832 T^{3} + 4250 T^{4} + 192 T^{5} + 32 T^{6} + 4 T^{7} + T^{8} \)
$41$ \( 3254416 + 4387328 T + 2420328 T^{2} + 313946 T^{3} + 45705 T^{4} + 4536 T^{5} + 408 T^{6} + 23 T^{7} + T^{8} \)
$43$ \( ( -4 - 54 T - 39 T^{2} + T^{3} + T^{4} )^{2} \)
$47$ \( 555025 - 182525 T + 84900 T^{2} - 7925 T^{3} + 585 T^{4} + 45 T^{5} + 30 T^{6} - 5 T^{7} + T^{8} \)
$53$ \( 1936 + 4488 T + 4668 T^{2} + 1826 T^{3} + 1275 T^{4} + 311 T^{5} + 138 T^{6} + 18 T^{7} + T^{8} \)
$59$ \( ( 3025 + 825 T + 190 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$61$ \( 1638400 - 614400 T + 204800 T^{2} - 32000 T^{3} + 6160 T^{4} - 280 T^{5} + 160 T^{6} + 20 T^{7} + T^{8} \)
$67$ \( ( 76 - 74 T - 89 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$71$ \( 1638400 - 819200 T + 153600 T^{2} + 16000 T^{3} + 9360 T^{4} + 960 T^{5} + 120 T^{6} + 10 T^{7} + T^{8} \)
$73$ \( ( 5776 - 1672 T + 244 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$79$ \( 82373776 - 13486936 T + 1077372 T^{2} + 37522 T^{3} + 24005 T^{4} + 1678 T^{5} + 222 T^{6} + 11 T^{7} + T^{8} \)
$83$ \( ( 4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$89$ \( ( -6724 - 1178 T + 79 T^{2} + 23 T^{3} + T^{4} )^{2} \)
$97$ \( 1290496 + 518016 T + 389312 T^{2} + 31168 T^{3} + 2800 T^{4} + 272 T^{5} + 112 T^{6} - 16 T^{7} + T^{8} \)
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