Properties

Label 8036.2.a.m
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.

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

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

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

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
2.86706
2.10300
1.71291
−0.308117
−0.503623
−1.33338
−2.24904
−2.28881
0 −2.86706 0 1.20206 0 0 0 5.22005 0
1.2 0 −2.10300 0 −2.93524 0 0 0 1.42260 0
1.3 0 −1.71291 0 2.15120 0 0 0 −0.0659453 0
1.4 0 0.308117 0 −3.30124 0 0 0 −2.90506 0
1.5 0 0.503623 0 2.23729 0 0 0 −2.74636 0
1.6 0 1.33338 0 1.76687 0 0 0 −1.22209 0
1.7 0 2.24904 0 −1.47097 0 0 0 2.05817 0
1.8 0 2.28881 0 0.350035 0 0 0 2.23863 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

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