Properties

Label 735.2.s.k
Level 735
Weight 2
Character orbit 735.s
Analytic conductor 5.869
Analytic rank 0
Dimension 8
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} + 10 \nu + 2 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 7 \nu^{4} + 10 \nu^{2} - 2 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} + 2 \nu^{3} + 22 \nu^{2} + 10 \nu + 8 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} - 2 \nu^{3} + 22 \nu^{2} - 10 \nu + 8 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} - 10 \nu^{5} - 9 \nu^{4} - 29 \nu^{3} - 20 \nu^{2} - 20 \nu - 6 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 29 \nu^{3} - 20 \nu^{2} + 20 \nu - 6 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-6 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(7 \beta_{5} - 7 \beta_{4} + 4 \beta_{2} + 25 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(32 \beta_{7} + 32 \beta_{6} + 25 \beta_{5} + 25 \beta_{4} + 18 \beta_{3} + 9 \beta_{1} - 4\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - 2 \beta_{6} - 41 \beta_{5} + 41 \beta_{4} - 40 \beta_{2} - 125 \beta_{1} + 20\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
2.33086i
1.07834i
0.385731i
2.06288i
2.33086i
1.07834i
0.385731i
2.06288i
−2.01859 + 1.16543i 1.21646 + 1.23297i 1.71646 2.97300i −0.500000 0.866025i −3.89248 1.07116i 0 3.33995i −0.0404447 + 2.99973i 2.01859 + 1.16543i
521.2 −0.933868 + 0.539169i −0.918594 1.46840i −0.418594 + 0.725026i −0.500000 0.866025i 1.64956 + 0.876010i 0 3.05945i −1.31237 + 2.69772i 0.933868 + 0.539169i
521.3 −0.334053 + 0.192865i −1.42561 + 0.983691i −0.925606 + 1.60320i −0.500000 0.866025i 0.286507 0.603555i 0 1.48553i 1.06470 2.80471i 0.334053 + 0.192865i
521.4 1.78651 1.03144i 0.627739 1.61429i 1.12774 1.95330i −0.500000 0.866025i −0.543588 3.53142i 0 0.527019i −2.21189 2.02671i −1.78651 1.03144i
656.1 −2.01859 1.16543i 1.21646 1.23297i 1.71646 + 2.97300i −0.500000 + 0.866025i −3.89248 + 1.07116i 0 3.33995i −0.0404447 2.99973i 2.01859 1.16543i
656.2 −0.933868 0.539169i −0.918594 + 1.46840i −0.418594 0.725026i −0.500000 + 0.866025i 1.64956 0.876010i 0 3.05945i −1.31237 2.69772i 0.933868 0.539169i
656.3 −0.334053 0.192865i −1.42561 0.983691i −0.925606 1.60320i −0.500000 + 0.866025i 0.286507 + 0.603555i 0 1.48553i 1.06470 + 2.80471i 0.334053 0.192865i
656.4 1.78651 + 1.03144i 0.627739 + 1.61429i 1.12774 + 1.95330i −0.500000 + 0.866025i −0.543588 + 3.53142i 0 0.527019i −2.21189 + 2.02671i −1.78651 + 1.03144i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 656.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.s.k 8
3.b odd 2 1 735.2.s.l 8
7.b odd 2 1 105.2.s.c 8
7.c even 3 1 105.2.s.d yes 8
7.c even 3 1 735.2.b.c 8
7.d odd 6 1 735.2.b.d 8
7.d odd 6 1 735.2.s.l 8
21.c even 2 1 105.2.s.d yes 8
21.g even 6 1 735.2.b.c 8
21.g even 6 1 inner 735.2.s.k 8
21.h odd 6 1 105.2.s.c 8
21.h odd 6 1 735.2.b.d 8
35.c odd 2 1 525.2.t.g 8
35.f even 4 2 525.2.q.f 16
35.j even 6 1 525.2.t.f 8
35.l odd 12 2 525.2.q.e 16
105.g even 2 1 525.2.t.f 8
105.k odd 4 2 525.2.q.e 16
105.o odd 6 1 525.2.t.g 8
105.x even 12 2 525.2.q.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c 8 7.b odd 2 1
105.2.s.c 8 21.h odd 6 1
105.2.s.d yes 8 7.c even 3 1
105.2.s.d yes 8 21.c even 2 1
525.2.q.e 16 35.l odd 12 2
525.2.q.e 16 105.k odd 4 2
525.2.q.f 16 35.f even 4 2
525.2.q.f 16 105.x even 12 2
525.2.t.f 8 35.j even 6 1
525.2.t.f 8 105.g even 2 1
525.2.t.g 8 35.c odd 2 1
525.2.t.g 8 105.o odd 6 1
735.2.b.c 8 7.c even 3 1
735.2.b.c 8 21.g even 6 1
735.2.b.d 8 7.d odd 6 1
735.2.b.d 8 21.h odd 6 1
735.2.s.k 8 1.a even 1 1 trivial
735.2.s.k 8 21.g even 6 1 inner
735.2.s.l 8 3.b odd 2 1
735.2.s.l 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\(T_{2}^{8} + \cdots\)
\( T_{13}^{8} + 21 T_{13}^{6} + 123 T_{13}^{4} + 135 T_{13}^{2} + 36 \)
\(T_{17}^{8} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \)
$3$ \( 1 + T + 3 T^{2} + 4 T^{3} + 16 T^{4} + 12 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T + T^{2} )^{4} \)
$7$ 1
$11$ \( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \)
$17$ \( 1 - 12 T + 34 T^{2} - 12 T^{3} + 1078 T^{4} - 6882 T^{5} + 8740 T^{6} - 70272 T^{7} + 637627 T^{8} - 1194624 T^{9} + 2525860 T^{10} - 33811266 T^{11} + 90035638 T^{12} - 17038284 T^{13} + 820677346 T^{14} - 4924064076 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 9 T + 100 T^{2} + 657 T^{3} + 4723 T^{4} + 26244 T^{5} + 148996 T^{6} + 704196 T^{7} + 3331528 T^{8} + 13379724 T^{9} + 53787556 T^{10} + 180007596 T^{11} + 615506083 T^{12} + 1626797043 T^{13} + 4704588100 T^{14} + 8044845651 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 27 T + 391 T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 136108917 T^{9} + 612549202 T^{10} - 2504881125 T^{11} + 8857247491 T^{12} - 25719626628 T^{13} + 57882032599 T^{14} - 91930287069 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \)
$31$ \( 1 - 21 T + 262 T^{2} - 2415 T^{3} + 17293 T^{4} - 101304 T^{5} + 505090 T^{6} - 2328618 T^{7} + 11769748 T^{8} - 72187158 T^{9} + 485391490 T^{10} - 3017947464 T^{11} + 15970448653 T^{12} - 69139399665 T^{13} + 232525964422 T^{14} - 577764896331 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 7 T - 24 T^{2} + 493 T^{3} - 973 T^{4} - 16188 T^{5} + 118336 T^{6} + 258098 T^{7} - 5756772 T^{8} + 9549626 T^{9} + 162001984 T^{10} - 819970764 T^{11} - 1823558653 T^{12} + 34186570801 T^{13} - 61577433816 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \)
$41$ \( ( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 6 T - 116 T^{2} + 252 T^{3} + 10126 T^{4} - 1986 T^{5} - 595736 T^{6} + 157218 T^{7} + 25623007 T^{8} + 7389246 T^{9} - 1315980824 T^{10} - 206192478 T^{11} + 49411649806 T^{12} + 57794941764 T^{13} - 1250388978164 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 87932088 T^{9} + 2107199440 T^{10} - 26815725240 T^{11} + 229218473050 T^{12} - 1485430391136 T^{13} + 7535882783860 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 15 T + 223 T^{2} + 2220 T^{3} + 19711 T^{4} + 141723 T^{5} + 816310 T^{6} + 5175267 T^{7} + 31433836 T^{8} + 315691287 T^{9} + 3037489510 T^{10} + 32168428263 T^{11} + 272915371951 T^{12} + 1875003788220 T^{13} + 11489043482503 T^{14} + 47141142540315 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 4 T - 234 T^{2} + 412 T^{3} + 35255 T^{4} - 28434 T^{5} - 3551522 T^{6} + 717722 T^{7} + 271900824 T^{8} + 48087374 T^{9} - 15942782258 T^{10} - 8551895142 T^{11} + 710427770855 T^{12} + 556251544084 T^{13} - 21167261427546 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16} \)
$83$ \( ( 1 + 15 T + 380 T^{2} + 3759 T^{3} + 49260 T^{4} + 311997 T^{5} + 2617820 T^{6} + 8576805 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \)
show more
show less