Properties

Label 8036.2.a.n
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 1
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{5} q^{5} + ( 1 + \beta_{2} ) q^{9} + ( -1 - \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{4} + \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{4} ) q^{15} + ( -\beta_{1} + \beta_{4} - \beta_{6} ) q^{17} + ( -1 - \beta_{2} - \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{27} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{31} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( -4 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{39} - q^{41} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{45} + ( -3 - \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{51} + ( -\beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{57} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{61} + ( -3 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{65} + ( -\beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( 2 + 3 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + ( -4 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{81} + ( 5 - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{85} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{89} + ( -4 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{9} - 8q^{11} + 7q^{13} - q^{15} + q^{17} - 4q^{19} - 3q^{23} - 4q^{25} + 12q^{27} - 4q^{29} - 4q^{31} - 23q^{33} - 31q^{37} + 5q^{39} - 8q^{41} - 8q^{43} - q^{45} - 24q^{47} - 23q^{51} - q^{53} - 2q^{55} - 15q^{57} - 4q^{59} + 4q^{61} - 24q^{65} + 21q^{69} + 8q^{71} - 11q^{73} + 15q^{75} + 14q^{79} - 28q^{81} + 42q^{83} - 20q^{85} - 25q^{87} + 11q^{89} - 27q^{93} - 15q^{95} + 16q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 14 x^{6} - 4 x^{5} + 60 x^{4} + 31 x^{3} - 75 x^{2} - 60 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 13 \nu^{4} - 34 \nu^{3} + 10 \nu^{2} + 46 \nu + 19 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 4 \nu^{6} - 17 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} - 29 \nu^{2} - 26 \nu - 8 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 16 \nu^{5} - 11 \nu^{4} + 73 \nu^{3} + 35 \nu^{2} - 94 \nu - 40 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - 4 \nu^{6} - 20 \nu^{5} + 29 \nu^{4} + 65 \nu^{3} - 41 \nu^{2} - 80 \nu - 17 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + \nu^{6} + 26 \nu^{5} - 5 \nu^{4} - 101 \nu^{3} - 7 \nu^{2} + 113 \nu + 44 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + 8 \beta_{2} + \beta_{1} + 23\)
\(\nu^{5}\)\(=\)\(10 \beta_{7} + 10 \beta_{6} + 10 \beta_{5} + \beta_{4} + 12 \beta_{3} + 13 \beta_{2} + 28 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(15 \beta_{7} + 23 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 36 \beta_{3} + 62 \beta_{2} + 15 \beta_{1} + 149\)
\(\nu^{7}\)\(=\)\(83 \beta_{7} + 86 \beta_{6} + 85 \beta_{5} + 14 \beta_{4} + 116 \beta_{3} + 126 \beta_{2} + 173 \beta_{1} + 210\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.28881
−2.24904
−1.33338
−0.503623
−0.308117
1.71291
2.10300
2.86706
0 −2.28881 0 −0.350035 0 0 0 2.23863 0
1.2 0 −2.24904 0 1.47097 0 0 0 2.05817 0
1.3 0 −1.33338 0 −1.76687 0 0 0 −1.22209 0
1.4 0 −0.503623 0 −2.23729 0 0 0 −2.74636 0
1.5 0 −0.308117 0 3.30124 0 0 0 −2.90506 0
1.6 0 1.71291 0 −2.15120 0 0 0 −0.0659453 0
1.7 0 2.10300 0 2.93524 0 0 0 1.42260 0
1.8 0 2.86706 0 −1.20206 0 0 0 5.22005 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{8} - 14 T_{3}^{6} - 4 T_{3}^{5} + 60 T_{3}^{4} + 31 T_{3}^{3} - 75 T_{3}^{2} - 60 T_{3} - 11 \)
\( T_{5}^{8} - 18 T_{5}^{6} - 12 T_{5}^{5} + 98 T_{5}^{4} + 117 T_{5}^{3} - 115 T_{5}^{2} - 196 T_{5} - 51 \)
\(T_{11}^{8} + \cdots\)