Properties

Label 231.2.i.e
Level $231$
Weight $2$
Character orbit 231.i
Analytic conductor $1.845$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{4} + ( -\beta_{1} - \beta_{4} + \beta_{6} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{7} ) q^{8} -\beta_{4} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{4} ) q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{4} + ( -\beta_{1} - \beta_{4} + \beta_{6} ) q^{5} -\beta_{3} q^{6} + ( -\beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( 2 - 2 \beta_{2} - \beta_{3} - \beta_{7} ) q^{8} -\beta_{4} q^{9} + ( -2 + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{10} + ( 1 - \beta_{4} ) q^{11} + ( -\beta_{1} - \beta_{6} ) q^{12} -\beta_{7} q^{13} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{14} + ( 1 + \beta_{2} - \beta_{3} ) q^{15} + ( -4 \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{16} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{17} + ( \beta_{1} + \beta_{3} ) q^{18} + ( -\beta_{1} + \beta_{5} + 2 \beta_{6} ) q^{19} + ( -1 - \beta_{2} - \beta_{3} ) q^{20} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{21} + \beta_{3} q^{22} + ( 2 \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{23} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{24} + ( -1 - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{25} + ( \beta_{1} + \beta_{4} - \beta_{6} ) q^{26} + q^{27} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{28} + ( 2 + \beta_{3} + \beta_{7} ) q^{29} + ( -2 \beta_{4} + \beta_{5} ) q^{30} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{31} + ( -2 + 5 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} ) q^{32} + \beta_{4} q^{33} + ( -2 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{7} ) q^{34} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{35} + ( -\beta_{2} - \beta_{3} ) q^{36} + ( \beta_{1} + \beta_{6} ) q^{37} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{38} + ( \beta_{5} + \beta_{7} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{40} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{41} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} ) q^{42} + ( 4 + 2 \beta_{2} - 3 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{6} ) q^{44} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{45} + ( 2 + \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{6} ) q^{46} + ( -\beta_{1} - 2 \beta_{4} - \beta_{6} ) q^{47} + ( 1 - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{7} ) q^{48} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{49} + ( 2 + 4 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{50} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{51} + ( 2 - 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{52} + ( 3 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{53} -\beta_{1} q^{54} + ( -1 - \beta_{2} + \beta_{3} ) q^{55} + ( 4 - \beta_{1} - \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{56} + ( 2 \beta_{2} - \beta_{3} + \beta_{7} ) q^{57} + ( -2 \beta_{1} + \beta_{4} + 2 \beta_{6} ) q^{58} + ( -5 - 2 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{60} + ( -2 \beta_{1} - \beta_{5} ) q^{61} + ( -4 + 6 \beta_{3} + \beta_{7} ) q^{62} + ( -\beta_{1} + \beta_{7} ) q^{63} + ( 8 - 5 \beta_{2} - 7 \beta_{3} ) q^{64} + ( 2 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} ) q^{65} + ( -\beta_{1} - \beta_{3} ) q^{66} + ( -9 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{67} + ( 7 \beta_{1} + 8 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{68} + ( 1 + 2 \beta_{3} + 2 \beta_{7} ) q^{69} + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{70} + ( 4 - \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{71} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{72} + ( 2 + 6 \beta_{2} - 2 \beta_{4} - \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{73} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{74} + ( -\beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{75} + ( -2 + 2 \beta_{2} + 3 \beta_{3} ) q^{76} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{77} + ( -1 - \beta_{2} + \beta_{3} ) q^{78} + ( 4 \beta_{1} + 3 \beta_{5} - 2 \beta_{6} ) q^{79} + ( -3 - \beta_{1} + 5 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{80} + ( -1 + \beta_{4} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{82} + ( -2 - 2 \beta_{3} - 4 \beta_{7} ) q^{83} + ( 2 - 3 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{84} + ( 6 + 4 \beta_{2} + \beta_{7} ) q^{85} + ( -3 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{86} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{87} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{88} + ( 2 \beta_{1} - 6 \beta_{6} ) q^{89} + ( 2 + \beta_{7} ) q^{90} + ( 5 + \beta_{2} + \beta_{3} - 6 \beta_{4} + 2 \beta_{6} ) q^{91} + ( 3 \beta_{2} + 5 \beta_{3} ) q^{92} + ( \beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{93} + ( -2 + 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{94} + ( -8 - 4 \beta_{2} + 8 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{95} + ( -5 \beta_{1} - 2 \beta_{4} - 4 \beta_{6} ) q^{96} + ( -10 + \beta_{2} - \beta_{3} - 2 \beta_{7} ) q^{97} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - 4q^{3} - 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 24q^{8} - 4q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - 4q^{3} - 4q^{4} - 4q^{5} + 4q^{6} + 2q^{7} + 24q^{8} - 4q^{9} - 10q^{10} + 4q^{11} - 4q^{12} - 4q^{13} - 4q^{14} + 8q^{15} - 12q^{16} - 2q^{17} - 2q^{18} - 4q^{21} - 4q^{22} - 4q^{23} - 12q^{24} - 4q^{25} + 4q^{26} + 8q^{27} - 22q^{28} + 16q^{29} - 10q^{30} + 12q^{31} - 26q^{32} + 4q^{33} - 32q^{34} - 2q^{35} + 8q^{36} + 4q^{37} - 8q^{38} + 2q^{39} + 6q^{40} + 4q^{41} + 20q^{42} + 36q^{43} + 4q^{44} - 4q^{45} + 14q^{46} - 12q^{47} + 24q^{48} - 4q^{49} + 4q^{50} - 2q^{51} + 6q^{52} + 12q^{53} - 2q^{54} - 8q^{55} + 48q^{56} + 4q^{58} - 12q^{59} - 2q^{61} - 52q^{62} + 2q^{63} + 112q^{64} + 4q^{65} + 2q^{66} - 28q^{67} + 48q^{68} + 8q^{69} - 32q^{70} + 24q^{71} - 12q^{72} - 6q^{73} + 16q^{74} - 4q^{75} - 36q^{76} + 4q^{77} - 8q^{78} - 2q^{79} - 16q^{80} - 4q^{81} + 12q^{82} - 24q^{83} - 4q^{84} + 36q^{85} - 36q^{86} - 8q^{87} + 12q^{88} - 8q^{89} + 20q^{90} + 12q^{91} - 32q^{92} + 12q^{93} - 20q^{94} - 34q^{95} - 26q^{96} - 88q^{97} + 16q^{98} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 68 \nu^{7} - 215 \nu^{6} + 357 \nu^{5} + 646 \nu^{4} - 1444 \nu^{3} + 1156 \nu^{2} + 561 \nu + 5468 \)\()/4243\)
\(\beta_{3}\)\(=\)\((\)\( 84 \nu^{7} - 16 \nu^{6} + 441 \nu^{5} + 798 \nu^{4} + 3208 \nu^{3} + 1428 \nu^{2} + 693 \nu + 2262 \)\()/4243\)
\(\beta_{4}\)\(=\)\((\)\( -754 \nu^{7} + 1760 \nu^{6} - 6080 \nu^{5} + 1323 \nu^{4} - 13440 \nu^{3} + 12640 \nu^{2} - 16828 \nu + 5760 \)\()/12729\)
\(\beta_{5}\)\(=\)\((\)\( -815 \nu^{7} + 3388 \nu^{6} - 11704 \nu^{5} + 15594 \nu^{4} - 25872 \nu^{3} + 24332 \nu^{2} - 56579 \nu + 11088 \)\()/12729\)
\(\beta_{6}\)\(=\)\((\)\( 1052 \nu^{7} - 2827 \nu^{6} + 9766 \nu^{5} - 6978 \nu^{4} + 21588 \nu^{3} - 20303 \nu^{2} + 17165 \nu - 9252 \)\()/12729\)
\(\beta_{7}\)\(=\)\((\)\( -556 \nu^{7} + 510 \nu^{6} - 2919 \nu^{5} - 5282 \nu^{4} - 8909 \nu^{3} - 9452 \nu^{2} - 4587 \nu - 13760 \)\()/4243\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 5 \beta_{3} + 2 \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{6} - 2 \beta_{5} - 9 \beta_{4} - 10 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-8 \beta_{7} - 20 \beta_{6} - 8 \beta_{5} - 18 \beta_{4} - 33 \beta_{3} - 20 \beta_{2} - 33 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(-20 \beta_{7} - 83 \beta_{3} - 61 \beta_{2} + 58\)
\(\nu^{7}\)\(=\)\(164 \beta_{6} + 61 \beta_{5} + 146 \beta_{4} + 243 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.39083 + 2.40898i
0.643668 + 1.11487i
−0.276205 0.478401i
−0.758290 1.31340i
1.39083 2.40898i
0.643668 1.11487i
−0.276205 + 0.478401i
−0.758290 + 1.31340i
−1.39083 2.40898i −0.500000 + 0.866025i −2.86880 + 4.96890i −0.412855 0.715087i 2.78165 2.63323 + 0.257073i 10.3967 −0.500000 0.866025i −1.14842 + 1.98912i
67.2 −0.643668 1.11487i −0.500000 + 0.866025i 0.171383 0.296844i −1.95872 3.39260i 1.28734 −0.234193 + 2.63537i −3.01593 −0.500000 0.866025i −2.52153 + 4.36742i
67.3 0.276205 + 0.478401i −0.500000 + 0.866025i 0.847422 1.46778i −0.795012 1.37700i −0.552409 0.886763 2.49272i 2.04107 −0.500000 0.866025i 0.439172 0.760669i
67.4 0.758290 + 1.31340i −0.500000 + 0.866025i −0.150007 + 0.259820i 1.16659 + 2.02059i −1.51658 −2.28580 + 1.33233i 2.57816 −0.500000 0.866025i −1.76922 + 3.06438i
100.1 −1.39083 + 2.40898i −0.500000 0.866025i −2.86880 4.96890i −0.412855 + 0.715087i 2.78165 2.63323 0.257073i 10.3967 −0.500000 + 0.866025i −1.14842 1.98912i
100.2 −0.643668 + 1.11487i −0.500000 0.866025i 0.171383 + 0.296844i −1.95872 + 3.39260i 1.28734 −0.234193 2.63537i −3.01593 −0.500000 + 0.866025i −2.52153 4.36742i
100.3 0.276205 0.478401i −0.500000 0.866025i 0.847422 + 1.46778i −0.795012 + 1.37700i −0.552409 0.886763 + 2.49272i 2.04107 −0.500000 + 0.866025i 0.439172 + 0.760669i
100.4 0.758290 1.31340i −0.500000 0.866025i −0.150007 0.259820i 1.16659 2.02059i −1.51658 −2.28580 1.33233i 2.57816 −0.500000 + 0.866025i −1.76922 3.06438i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.i.e 8
3.b odd 2 1 693.2.i.i 8
7.c even 3 1 inner 231.2.i.e 8
7.c even 3 1 1617.2.a.z 4
7.d odd 6 1 1617.2.a.x 4
21.g even 6 1 4851.2.a.bu 4
21.h odd 6 1 693.2.i.i 8
21.h odd 6 1 4851.2.a.bt 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 1.a even 1 1 trivial
231.2.i.e 8 7.c even 3 1 inner
693.2.i.i 8 3.b odd 2 1
693.2.i.i 8 21.h odd 6 1
1617.2.a.x 4 7.d odd 6 1
1617.2.a.z 4 7.c even 3 1
4851.2.a.bt 4 21.h odd 6 1
4851.2.a.bu 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2 T_{2}^{7} + 8 T_{2}^{6} + 21 T_{2}^{4} + 4 T_{2}^{3} + 28 T_{2}^{2} - 12 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 12 T + 28 T^{2} + 4 T^{3} + 21 T^{4} + 8 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 144 + 240 T + 352 T^{2} + 176 T^{3} + 108 T^{4} + 24 T^{5} + 20 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( 2401 - 686 T + 196 T^{2} - 70 T^{3} - 22 T^{4} - 10 T^{5} + 4 T^{6} - 2 T^{7} + T^{8} \)
$11$ \( ( 1 - T + T^{2} )^{4} \)
$13$ \( ( -4 - 16 T - 8 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$17$ \( 225 - 1860 T + 15976 T^{2} + 4900 T^{3} + 1833 T^{4} + 168 T^{5} + 44 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( 7921 - 10858 T + 11324 T^{2} - 4880 T^{3} + 1689 T^{4} - 244 T^{5} + 40 T^{6} + T^{8} \)
$23$ \( 216225 - 35340 T + 25306 T^{2} - 528 T^{3} + 1603 T^{4} - 16 T^{5} + 58 T^{6} + 4 T^{7} + T^{8} \)
$29$ \( ( 3 + 14 T + 12 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( 1948816 - 698000 T + 238832 T^{2} - 37504 T^{3} + 7460 T^{4} - 904 T^{5} + 152 T^{6} - 12 T^{7} + T^{8} \)
$37$ \( 1 + 10 T^{2} + 8 T^{3} + 99 T^{4} + 40 T^{5} + 26 T^{6} - 4 T^{7} + T^{8} \)
$41$ \( ( 60 - 64 T - 44 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$43$ \( ( -1385 + 308 T + 60 T^{2} - 18 T^{3} + T^{4} )^{2} \)
$47$ \( 81 + 360 T + 1258 T^{2} + 1304 T^{3} + 955 T^{4} + 376 T^{5} + 106 T^{6} + 12 T^{7} + T^{8} \)
$53$ \( 39087504 + 6026928 T + 1704544 T^{2} + 30512 T^{3} + 20692 T^{4} - 440 T^{5} + 268 T^{6} - 12 T^{7} + T^{8} \)
$59$ \( 62001 - 52788 T + 49426 T^{2} - 2160 T^{3} + 2619 T^{4} + 208 T^{5} + 162 T^{6} + 12 T^{7} + T^{8} \)
$61$ \( 400 - 640 T + 1504 T^{2} + 688 T^{3} + 620 T^{4} + 16 T^{5} + 28 T^{6} + 2 T^{7} + T^{8} \)
$67$ \( 2131600 + 192720 T + 297744 T^{2} + 56416 T^{3} + 42020 T^{4} + 5640 T^{5} + 592 T^{6} + 28 T^{7} + T^{8} \)
$71$ \( ( -699 + 296 T + 2 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$73$ \( 294328336 - 12215072 T + 5173376 T^{2} - 12208 T^{3} + 61100 T^{4} - 208 T^{5} + 308 T^{6} + 6 T^{7} + T^{8} \)
$79$ \( 150544 + 167616 T + 244048 T^{2} - 65488 T^{3} + 20652 T^{4} - 1160 T^{5} + 152 T^{6} + 2 T^{7} + T^{8} \)
$83$ \( ( 2592 - 944 T - 96 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$89$ \( 145154304 - 14650368 T + 4273792 T^{2} + 89344 T^{3} + 51504 T^{4} + 576 T^{5} + 296 T^{6} + 8 T^{7} + T^{8} \)
$97$ \( ( 8501 + 4280 T + 682 T^{2} + 44 T^{3} + T^{4} )^{2} \)
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