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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 8 T^{3} - 11 T^{4} + 4 T^{5} + 20 T^{6} + 8 T^{7} - 23 T^{8} + 16 T^{9} + 80 T^{10} + 32 T^{11} - 176 T^{12} - 256 T^{13} + 256 T^{15} + 256 T^{16} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 1 + 4 T - 16 T^{3} - 2 T^{4} + 56 T^{5} + 32 T^{6} - 380 T^{7} - 1421 T^{8} - 1900 T^{9} + 800 T^{10} + 7000 T^{11} - 1250 T^{12} - 50000 T^{13} + 312500 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 2 T + 4 T^{2} - 10 T^{3} - 22 T^{4} - 70 T^{5} + 196 T^{6} - 686 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 - T + T^{2} )^{4} \)
$13$ \( ( 1 + 2 T + 44 T^{2} + 62 T^{3} + 802 T^{4} + 806 T^{5} + 7436 T^{6} + 4394 T^{7} + 28561 T^{8} )^{2} \)
$17$ \( 1 + 2 T - 24 T^{2} + 100 T^{3} + 439 T^{4} - 2546 T^{5} + 6524 T^{6} + 38396 T^{7} - 127700 T^{8} + 652732 T^{9} + 1885436 T^{10} - 12508498 T^{11} + 36665719 T^{12} + 141985700 T^{13} - 579301656 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 36 T^{2} - 244 T^{3} + 739 T^{4} + 6710 T^{5} + 20824 T^{6} - 98942 T^{7} - 570724 T^{8} - 1879898 T^{9} + 7517464 T^{10} + 46023890 T^{11} + 96307219 T^{12} - 604168156 T^{13} - 1693651716 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 + 4 T - 34 T^{2} - 200 T^{3} - 7 T^{4} + 944 T^{5} - 10850 T^{6} + 35868 T^{7} + 780852 T^{8} + 824964 T^{9} - 5739650 T^{10} + 11485648 T^{11} - 1958887 T^{12} - 1287268600 T^{13} - 5033220226 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} \)
$29$ \( ( 1 - 8 T + 128 T^{2} - 682 T^{3} + 5745 T^{4} - 19778 T^{5} + 107648 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( 1 - 12 T + 28 T^{2} - 160 T^{3} + 2190 T^{4} + 2424 T^{5} - 76128 T^{6} + 348684 T^{7} - 2030189 T^{8} + 10809204 T^{9} - 73159008 T^{10} + 72213384 T^{11} + 2022510990 T^{12} - 4580664160 T^{13} + 24850103068 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 4 T - 122 T^{2} + 336 T^{3} + 9793 T^{4} - 17900 T^{5} - 522282 T^{6} + 256928 T^{7} + 22257092 T^{8} + 9506336 T^{9} - 715004058 T^{10} - 906688700 T^{11} + 18353658673 T^{12} + 23299569552 T^{13} - 313018621898 T^{14} - 379727508532 T^{15} + 3512479453921 T^{16} \)
$41$ \( ( 1 - 2 T + 120 T^{2} - 310 T^{3} + 6538 T^{4} - 12710 T^{5} + 201720 T^{6} - 137842 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 18 T + 232 T^{2} - 2014 T^{3} + 14869 T^{4} - 86602 T^{5} + 428968 T^{6} - 1431126 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 12 T - 82 T^{2} - 752 T^{3} + 13457 T^{4} + 68796 T^{5} - 783266 T^{6} - 399704 T^{7} + 54619956 T^{8} - 18786088 T^{9} - 1730234594 T^{10} + 7142607108 T^{11} + 65665867217 T^{12} - 172467445264 T^{13} - 883895656978 T^{14} + 6079477445556 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 12 T + 56 T^{2} + 832 T^{3} - 13546 T^{4} + 105984 T^{5} - 91520 T^{6} - 5351324 T^{7} + 65244435 T^{8} - 283620172 T^{9} - 257079680 T^{10} + 15778579968 T^{11} - 106884455626 T^{12} + 347938650176 T^{13} + 1241204223224 T^{14} - 14096533678044 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 12 T - 74 T^{2} - 1208 T^{3} + 5569 T^{4} + 73360 T^{5} - 324634 T^{6} - 1264412 T^{7} + 27329972 T^{8} - 74600308 T^{9} - 1130050954 T^{10} + 15066603440 T^{11} + 67481583409 T^{12} - 863628553192 T^{13} - 3121359489434 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 2 T - 216 T^{2} - 228 T^{3} + 28314 T^{4} + 16182 T^{5} - 2563912 T^{6} - 387746 T^{7} + 178127415 T^{8} - 23652506 T^{9} - 9540316552 T^{10} + 3673006542 T^{11} + 392031142074 T^{12} - 192567956628 T^{13} - 11128400861976 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 28 T + 324 T^{2} + 1888 T^{3} + 6510 T^{4} + 29080 T^{5} + 472480 T^{6} + 8872436 T^{7} + 99414595 T^{8} + 594453212 T^{9} + 2120962720 T^{10} + 8746188040 T^{11} + 131183797710 T^{12} + 2549036202016 T^{13} + 29308515822756 T^{14} + 169699924949044 T^{15} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 12 T + 286 T^{2} - 2260 T^{3} + 29831 T^{4} - 160460 T^{5} + 1441726 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( 1 + 6 T + 16 T^{2} - 1084 T^{3} - 12630 T^{4} - 57030 T^{5} + 123528 T^{6} + 6885378 T^{7} + 59722615 T^{8} + 502632594 T^{9} + 658280712 T^{10} - 22185639510 T^{11} - 358669783830 T^{12} - 2247209606812 T^{13} + 2421347620624 T^{14} + 66284391114582 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 2 T - 164 T^{2} - 1476 T^{3} + 11962 T^{4} + 165982 T^{5} + 347064 T^{6} - 7874110 T^{7} - 67195297 T^{8} - 622054690 T^{9} + 2166026424 T^{10} + 81835599298 T^{11} + 465920868922 T^{12} - 4541735244924 T^{13} - 39866342705444 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} \)
$83$ \( ( 1 + 12 T + 236 T^{2} + 2044 T^{3} + 27990 T^{4} + 169652 T^{5} + 1625804 T^{6} + 6861444 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( 1 + 8 T - 60 T^{2} - 848 T^{3} - 10262 T^{4} - 96488 T^{5} - 110704 T^{6} + 10006904 T^{7} + 164456179 T^{8} + 890614456 T^{9} - 876886384 T^{10} - 68021048872 T^{11} - 643860877142 T^{12} - 4735282412752 T^{13} - 29818877457660 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 44 T + 1070 T^{2} + 17084 T^{3} + 197263 T^{4} + 1657148 T^{5} + 10067630 T^{6} + 40157612 T^{7} + 88529281 T^{8} )^{2} \)
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