Properties

Label 8036.2.a.l
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.470117.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} - \beta_{4} ) q^{13} + ( -2 - \beta_{3} + \beta_{4} ) q^{15} + ( -2 \beta_{2} - \beta_{3} ) q^{17} + ( 2 \beta_{3} + \beta_{4} ) q^{19} + ( -2 - \beta_{1} + 2 \beta_{4} ) q^{23} + ( -1 + \beta_{3} - \beta_{4} ) q^{25} + ( 2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{27} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( 1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 - \beta_{1} - \beta_{2} ) q^{33} + ( -4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} - q^{41} + ( -6 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{43} + ( \beta_{3} + \beta_{4} ) q^{45} + ( 2 + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{51} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{55} + ( -3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{57} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( -1 + \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{61} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{65} + ( -5 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{67} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{69} + ( -5 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( -3 - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -3 + 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{85} + ( 7 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{87} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{89} + ( 4 + 3 \beta_{3} - 3 \beta_{4} ) q^{93} + ( 2 + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{95} + ( -2 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( -7 + \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 2q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( 5q + 2q^{3} + q^{5} + q^{9} - 2q^{11} + q^{13} - 9q^{15} + 3q^{17} + 4q^{19} - 8q^{23} - 6q^{25} + 8q^{27} - 9q^{29} + 11q^{31} - 5q^{33} - 11q^{37} - 17q^{39} - 5q^{41} - 27q^{43} + 3q^{45} + 3q^{47} - 3q^{51} - 19q^{53} + 13q^{55} - 11q^{57} + 15q^{59} + 7q^{65} - 21q^{67} - 14q^{69} - 16q^{71} + 10q^{73} - 12q^{75} - 14q^{79} - 7q^{81} + 2q^{83} - 21q^{85} + 36q^{87} - 6q^{89} + 17q^{93} + 9q^{95} - 20q^{97} - 17q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 6 x^{3} + 8 x^{2} + 7 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 8 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 9 \nu - 5 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + 3 \nu^{3} + 5 \nu^{2} - 13 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{2} + 5 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 5 \beta_{3} + 8 \beta_{2} + 7 \beta_{1} + 17\)

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
−2.05768
−0.475832
−0.189142
1.94177
2.78088
0 −2.05768 0 2.77770 0 0 0 1.23404 0
1.2 0 −0.475832 0 −1.19617 0 0 0 −2.77358 0
1.3 0 −0.189142 0 1.51194 0 0 0 −2.96423 0
1.4 0 1.94177 0 −2.68629 0 0 0 0.770473 0
1.5 0 2.78088 0 0.592821 0 0 0 4.73330 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{5} - 2 T_{3}^{4} - 6 T_{3}^{3} + 8 T_{3}^{2} + 7 T_{3} + 1 \)
\( T_{5}^{5} - T_{5}^{4} - 9 T_{5}^{3} + 8 T_{5}^{2} + 12 T_{5} - 8 \)
\( T_{11}^{5} + 2 T_{11}^{4} - 29 T_{11}^{3} - 98 T_{11}^{2} - 88 T_{11} - 24 \)