Properties

Label 231.2.j.f
Level 231
Weight 2
Character orbit 231.j
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.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

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 \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.418926 1.28932i
−0.227943 + 0.701538i
0.418926 + 1.28932i
−0.227943 0.701538i
−0.386111 0.280526i
1.69513 + 1.23158i
−0.386111 + 0.280526i
1.69513 1.23158i
0.0501062 0.154211i 0.809017 0.587785i 1.59676 + 1.16012i −1.35567 4.17234i −0.0501062 0.154211i 0.809017 + 0.587785i 0.521270 0.378725i 0.309017 0.951057i −0.711349
64.2 0.449894 1.38463i 0.809017 0.587785i −0.0967635 0.0703028i 0.737640 + 2.27022i −0.449894 1.38463i 0.809017 + 0.587785i 2.21480 1.60914i 0.309017 0.951057i 3.47528
148.1 0.0501062 + 0.154211i 0.809017 + 0.587785i 1.59676 1.16012i −1.35567 + 4.17234i −0.0501062 + 0.154211i 0.809017 0.587785i 0.521270 + 0.378725i 0.309017 + 0.951057i −0.711349
148.2 0.449894 + 1.38463i 0.809017 + 0.587785i −0.0967635 + 0.0703028i 0.737640 2.27022i −0.449894 + 1.38463i 0.809017 0.587785i 2.21480 + 1.60914i 0.309017 + 0.951057i 3.47528
169.1 −1.43376 1.04169i −0.309017 + 0.951057i 0.352519 + 1.08494i −0.477260 + 0.346750i 1.43376 1.04169i −0.309017 0.951057i −0.470553 + 1.44821i −0.809017 0.587785i 1.04548
169.2 1.93376 + 1.40496i −0.309017 + 0.951057i 1.14748 + 3.53158i 2.09529 1.52232i −1.93376 + 1.40496i −0.309017 0.951057i −1.26552 + 3.89486i −0.809017 0.587785i 6.19059
190.1 −1.43376 + 1.04169i −0.309017 0.951057i 0.352519 1.08494i −0.477260 0.346750i 1.43376 + 1.04169i −0.309017 + 0.951057i −0.470553 1.44821i −0.809017 + 0.587785i 1.04548
190.2 1.93376 1.40496i −0.309017 0.951057i 1.14748 3.53158i 2.09529 + 1.52232i −1.93376 1.40496i −0.309017 + 0.951057i −1.26552 3.89486i −0.809017 + 0.587785i 6.19059
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 190.2
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 231.2.j.f 8
3.b odd 2 1 693.2.m.f 8
11.c even 5 1 inner 231.2.j.f 8
11.c even 5 1 2541.2.a.bn 4
11.d odd 10 1 2541.2.a.bm 4
33.f even 10 1 7623.2.a.cl 4
33.h odd 10 1 693.2.m.f 8
33.h odd 10 1 7623.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 1.a even 1 1 trivial
231.2.j.f 8 11.c even 5 1 inner
693.2.m.f 8 3.b odd 2 1
693.2.m.f 8 33.h odd 10 1
2541.2.a.bm 4 11.d odd 10 1
2541.2.a.bn 4 11.c even 5 1
7623.2.a.ci 4 33.h odd 10 1
7623.2.a.cl 4 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - 3 T^{2} + 6 T^{3} + 3 T^{4} - 6 T^{5} + 9 T^{6} - 2 T^{7} - 27 T^{8} - 4 T^{9} + 36 T^{10} - 48 T^{11} + 48 T^{12} + 192 T^{13} - 192 T^{14} - 256 T^{15} + 256 T^{16} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$5$ \( 1 - 2 T + 10 T^{2} - 42 T^{3} + 99 T^{4} - 338 T^{5} + 900 T^{6} - 2008 T^{7} + 5281 T^{8} - 10040 T^{9} + 22500 T^{10} - 42250 T^{11} + 61875 T^{12} - 131250 T^{13} + 156250 T^{14} - 156250 T^{15} + 390625 T^{16} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$11$ \( ( 1 + 11 T + 51 T^{2} + 121 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 + 8 T - 6 T^{2} - 254 T^{3} - 569 T^{4} + 3664 T^{5} + 18260 T^{6} - 21760 T^{7} - 308143 T^{8} - 282880 T^{9} + 3085940 T^{10} + 8049808 T^{11} - 16251209 T^{12} - 94308422 T^{13} - 28960854 T^{14} + 501988136 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 4 T - 10 T^{2} - 76 T^{3} - 101 T^{4} + 3812 T^{5} + 15336 T^{6} - 19792 T^{7} - 179499 T^{8} - 336464 T^{9} + 4432104 T^{10} + 18728356 T^{11} - 8435621 T^{12} - 107909132 T^{13} - 241375690 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 10 T^{2} + 70 T^{3} + 319 T^{4} - 280 T^{5} + 300 T^{6} + 20400 T^{7} + 134681 T^{8} + 387600 T^{9} + 108300 T^{10} - 1920520 T^{11} + 41572399 T^{12} + 173326930 T^{13} - 470458810 T^{14} + 16983563041 T^{16} \)
$23$ \( ( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2869 T^{4} + 11500 T^{5} + 43907 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( 1 - 106 T^{2} + 4095 T^{4} - 50924 T^{6} - 177451 T^{8} - 42827084 T^{10} + 2896315695 T^{12} - 63051272026 T^{14} + 500246412961 T^{16} \)
$31$ \( 1 - 24 T + 238 T^{2} - 1194 T^{3} + 1727 T^{4} + 14880 T^{5} - 38008 T^{6} - 810792 T^{7} + 7640665 T^{8} - 25134552 T^{9} - 36525688 T^{10} + 443290080 T^{11} + 1594920767 T^{12} - 34183206294 T^{13} + 211225876078 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 6 T + 51 T^{2} - 228 T^{3} + 3191 T^{4} - 8808 T^{5} + 60945 T^{6} - 124230 T^{7} + 3928552 T^{8} - 4596510 T^{9} + 83433705 T^{10} - 446151624 T^{11} + 5980447751 T^{12} - 15810422196 T^{13} + 130852046859 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 20 T + 170 T^{2} - 1070 T^{3} + 8519 T^{4} - 70060 T^{5} + 465660 T^{6} - 2505360 T^{7} + 13832161 T^{8} - 102719760 T^{9} + 782774460 T^{10} - 4828605260 T^{11} + 24072657959 T^{12} - 123966135070 T^{13} + 807517720970 T^{14} - 3895085477620 T^{15} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 16856 T^{5} + 127581 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 + 22 T + 222 T^{2} + 1554 T^{3} + 9963 T^{4} + 18726 T^{5} - 507516 T^{6} - 6126008 T^{7} - 43788087 T^{8} - 287922376 T^{9} - 1121102844 T^{10} + 1944189498 T^{11} + 48616261803 T^{12} + 356402140878 T^{13} + 2392985803038 T^{14} + 11145708650186 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 20 T + 117 T^{2} + 500 T^{3} + 10595 T^{4} + 96940 T^{5} + 362607 T^{6} + 3666700 T^{7} + 43463984 T^{8} + 194335100 T^{9} + 1018563063 T^{10} + 14432136380 T^{11} + 83599646195 T^{12} + 209097746500 T^{13} + 2593230252093 T^{14} + 23494222796740 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 18 T + 22 T^{2} + 1598 T^{3} - 11653 T^{4} + 17790 T^{5} + 84228 T^{6} - 3028056 T^{7} + 40795865 T^{8} - 178655304 T^{9} + 293197668 T^{10} + 3653692410 T^{11} - 141203607733 T^{12} + 1142449029802 T^{13} + 927971740102 T^{14} - 44795726726742 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 2 T - 74 T^{2} - 552 T^{3} + 1931 T^{4} + 48386 T^{5} + 199304 T^{6} - 1537696 T^{7} - 12160643 T^{8} - 93799456 T^{9} + 741610184 T^{10} + 10982702666 T^{11} + 26736318971 T^{12} - 466217158152 T^{13} - 3812507702714 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \)
$67$ \( ( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( 1 - 14 T - 77 T^{2} + 2716 T^{3} - 11793 T^{4} - 191640 T^{5} + 2135337 T^{6} + 4898798 T^{7} - 179455320 T^{8} + 347814658 T^{9} + 10764233817 T^{10} - 68590064040 T^{11} - 299679954033 T^{12} + 4900286917316 T^{13} - 9863721861917 T^{14} - 127331682217474 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 2 T - 146 T^{2} + 446 T^{3} + 13211 T^{4} - 19066 T^{5} - 955660 T^{6} + 409840 T^{7} + 76882177 T^{8} + 29918320 T^{9} - 5092712140 T^{10} - 7416998122 T^{11} + 375169161851 T^{12} + 924589930478 T^{13} - 22094797038194 T^{14} - 22094797038194 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 20 T + 25 T^{2} + 2320 T^{3} - 19901 T^{4} + 85160 T^{5} - 408605 T^{6} - 11176060 T^{7} + 215428736 T^{8} - 882908740 T^{9} - 2550103805 T^{10} + 41987201240 T^{11} - 775145561981 T^{12} + 7138770845680 T^{13} + 6077186388025 T^{14} - 384078179723180 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 8 T - 10 T^{2} - 98 T^{3} + 5199 T^{4} + 59504 T^{5} + 80364 T^{6} - 1623536 T^{7} + 2309081 T^{8} - 134753488 T^{9} + 553627596 T^{10} + 34023613648 T^{11} + 246735810879 T^{12} - 386025983014 T^{13} - 3269403733690 T^{14} + 217088407917016 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 16 T + 308 T^{2} + 3096 T^{3} + 38678 T^{4} + 275544 T^{5} + 2439668 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 4 T + 26 T^{2} - 1532 T^{3} + 8091 T^{4} + 62828 T^{5} + 1010040 T^{6} - 5954240 T^{7} - 108387563 T^{8} - 577561280 T^{9} + 9503466360 T^{10} + 57341419244 T^{11} + 716290412571 T^{12} - 13155805273724 T^{13} + 21657272128154 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} \)
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