Properties

Label 8036.2.a.k
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.287349.1
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{3} \) \( + ( 1 - \beta_{2} + \beta_{3} ) q^{5} \) \( + ( \beta_{2} + \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{3} \) \( + ( 1 - \beta_{2} + \beta_{3} ) q^{5} \) \( + ( \beta_{2} + \beta_{4} ) q^{9} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{11} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} \) \( + ( 2 - \beta_{1} + \beta_{4} ) q^{15} \) \( + ( \beta_{3} + \beta_{4} ) q^{17} \) \( + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{19} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{23} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{25} \) \( + ( -2 - \beta_{1} + \beta_{2} ) q^{27} \) \( + ( 2 - 3 \beta_{1} - \beta_{4} ) q^{29} \) \( + ( -3 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} \) \( + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{33} \) \( + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{37} \) \( + ( 4 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{39} \) \(+ q^{41}\) \( + ( 6 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{43} \) \( + ( -2 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{45} \) \( + ( -2 - 3 \beta_{4} ) q^{47} \) \( + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{51} \) \( + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{55} \) \( + ( 3 + 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{57} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{59} \) \( + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{61} \) \( + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{65} \) \( + ( 3 - 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{67} \) \( + ( -3 + 8 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{69} \) \( + ( 3 + 7 \beta_{1} + \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{71} \) \( + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{73} \) \( + ( 3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} \) \( + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{79} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{81} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{83} \) \( + ( 3 - \beta_{2} - \beta_{3} ) q^{85} \) \( + ( -1 + 7 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{87} \) \( + ( 5 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} ) q^{89} \) \( + ( -2 + 9 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{93} \) \( + ( 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{95} \) \( + ( 6 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{97} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 9q^{15} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 11q^{29} \) \(\mathstrut -\mathstrut 13q^{31} \) \(\mathstrut -\mathstrut 11q^{33} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut +\mathstrut 23q^{39} \) \(\mathstrut +\mathstrut 5q^{41} \) \(\mathstrut +\mathstrut 29q^{43} \) \(\mathstrut -\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut 7q^{47} \) \(\mathstrut -\mathstrut 3q^{51} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut 19q^{55} \) \(\mathstrut +\mathstrut 9q^{57} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut +\mathstrut 8q^{61} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 3q^{67} \) \(\mathstrut -\mathstrut 10q^{69} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut +\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 18q^{75} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 15q^{81} \) \(\mathstrut +\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 13q^{85} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 5q^{93} \) \(\mathstrut -\mathstrut 7q^{95} \) \(\mathstrut +\mathstrut 24q^{97} \) \(\mathstrut +\mathstrut 27q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)

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
2.13797
−2.05679
−1.36870
1.14203
0.145487
0 −2.57090 0 −0.350332 0 0 0 3.60954 0
1.2 0 −2.23037 0 −1.70418 0 0 0 1.97454 0
1.3 0 0.126667 0 4.03743 0 0 0 −2.98396 0
1.4 0 0.695770 0 −1.38288 0 0 0 −2.51590 0
1.5 0 1.97883 0 2.39996 0 0 0 0.915782 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{5} \) \(\mathstrut +\mathstrut 2 T_{3}^{4} \) \(\mathstrut -\mathstrut 6 T_{3}^{3} \) \(\mathstrut -\mathstrut 8 T_{3}^{2} \) \(\mathstrut +\mathstrut 9 T_{3} \) \(\mathstrut -\mathstrut 1 \)
\(T_{5}^{5} \) \(\mathstrut -\mathstrut 3 T_{5}^{4} \) \(\mathstrut -\mathstrut 9 T_{5}^{3} \) \(\mathstrut +\mathstrut 12 T_{5}^{2} \) \(\mathstrut +\mathstrut 28 T_{5} \) \(\mathstrut +\mathstrut 8 \)
\(T_{11}^{5} \) \(\mathstrut -\mathstrut 6 T_{11}^{4} \) \(\mathstrut -\mathstrut 7 T_{11}^{3} \) \(\mathstrut +\mathstrut 68 T_{11}^{2} \) \(\mathstrut -\mathstrut 84 T_{11} \) \(\mathstrut +\mathstrut 24 \)