Properties

Label 4004.2.a.g
Level 4004
Weight 2
Character orbit 4004.a
Self dual Yes
Analytic conductor 31.972
Analytic rank 1
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.246302029.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + ( -1 - \beta_{4} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + ( -1 - \beta_{4} ) q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{9} \) \(- q^{11}\) \(- q^{13}\) \( + ( \beta_{1} + \beta_{2} ) q^{15} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} \) \( + ( -2 + \beta_{1} + \beta_{5} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( 1 + \beta_{2} + \beta_{4} ) q^{23} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{25} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{27} \) \( + ( -3 + \beta_{1} + \beta_{2} ) q^{29} \) \( + ( 1 - 2 \beta_{2} + 2 \beta_{4} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + ( -1 - \beta_{4} ) q^{35} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( 3 - \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{41} \) \( + ( -3 + \beta_{3} - \beta_{4} ) q^{43} \) \( + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{45} \) \( + ( -2 + 3 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{47} \) \(+ q^{49}\) \( + ( -1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{51} \) \( + ( 1 + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} \) \( + ( 1 + \beta_{4} ) q^{55} \) \( + ( -5 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} \) \( + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{59} \) \( + ( -4 + \beta_{2} + 2 \beta_{5} ) q^{61} \) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{63} \) \( + ( 1 + \beta_{4} ) q^{65} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{67} \) \( + ( -2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{69} \) \( + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{71} \) \( + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{73} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{75} \) \(- q^{77}\) \( + ( -2 + 4 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{79} \) \( + ( 3 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{81} \) \( + ( -4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{83} \) \( + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{85} \) \( + ( -6 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{87} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{89} \) \(- q^{91}\) \( + ( 4 - 3 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} ) q^{93} \) \( + ( 5 - \beta_{1} - 2 \beta_{2} + 4 \beta_{4} ) q^{95} \) \( + ( -4 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{97} \) \( + ( -1 - \beta_{3} - \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut -\mathstrut 3q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 14q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 20q^{45} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 6q^{49} \) \(\mathstrut -\mathstrut 5q^{51} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 3q^{55} \) \(\mathstrut -\mathstrut 22q^{57} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 26q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut +\mathstrut 9q^{67} \) \(\mathstrut -\mathstrut 11q^{69} \) \(\mathstrut +\mathstrut 3q^{71} \) \(\mathstrut -\mathstrut 7q^{73} \) \(\mathstrut -\mathstrut 6q^{75} \) \(\mathstrut -\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 6q^{81} \) \(\mathstrut -\mathstrut 15q^{83} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut -\mathstrut 23q^{87} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 12q^{95} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut -\mathstrut \) \(9\) \(x^{4}\mathstrut +\mathstrut \) \(14\) \(x^{3}\mathstrut +\mathstrut \) \(15\) \(x^{2}\mathstrut -\mathstrut \) \(13\) \(x\mathstrut -\mathstrut \) \(10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 9 \nu^{2} + 5 \nu + 10 \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 2 \nu^{4} + 8 \nu^{3} - 12 \nu^{2} - 6 \nu + 4 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 13 \nu^{2} + 6 \nu - 8 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{5} - 6 \nu^{4} - 13 \nu^{3} + 42 \nu^{2} - 3 \nu - 29 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(34\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)

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
3.12467
1.73486
1.23083
−0.714665
−0.820857
−2.55484
0 −3.12467 0 −1.82266 0 1.00000 0 6.76356 0
1.2 0 −1.73486 0 1.63787 0 1.00000 0 0.00974910 0
1.3 0 −1.23083 0 −3.39686 0 1.00000 0 −1.48506 0
1.4 0 0.714665 0 2.43635 0 1.00000 0 −2.48925 0
1.5 0 0.820857 0 0.0215726 0 1.00000 0 −2.32619 0
1.6 0 2.55484 0 −1.87627 0 1.00000 0 3.52720 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{6} \) \(\mathstrut +\mathstrut 2 T_{3}^{5} \) \(\mathstrut -\mathstrut 9 T_{3}^{4} \) \(\mathstrut -\mathstrut 14 T_{3}^{3} \) \(\mathstrut +\mathstrut 15 T_{3}^{2} \) \(\mathstrut +\mathstrut 13 T_{3} \) \(\mathstrut -\mathstrut 10 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).