Properties

Label 4008.2.a.g
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{6}\mathstrut -\mathstrut \) \(7\) \(x^{5}\mathstrut +\mathstrut \) \(11\) \(x^{4}\mathstrut +\mathstrut \) \(12\) \(x^{3}\mathstrut -\mathstrut \) \(14\) \(x^{2}\mathstrut -\mathstrut \) \(6\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 2 \nu^{2} - 8 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + \nu^{5} + 7 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} - \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} + 3 \nu - 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 3 \nu^{3} - 15 \nu^{2} + 6 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{5}\)\(=\)\((\)\(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/2\)
\(\nu^{6}\)\(=\)\(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(21\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(50\) \(\beta_{2}\mathstrut +\mathstrut \) \(49\) \(\beta_{1}\mathstrut +\mathstrut \) \(34\)

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
2.80982
1.47270
−0.674271
−2.05123
1.31154
−1.20126
0.332704
0 −1.00000 0 −3.20729 0 3.68779 0 1.00000 0
1.2 0 −1.00000 0 −3.01562 0 −1.84677 0 1.00000 0
1.3 0 −1.00000 0 −1.68509 0 −2.23045 0 1.00000 0
1.4 0 −1.00000 0 −0.597616 0 2.60994 0 1.00000 0
1.5 0 −1.00000 0 0.0621653 0 0.782308 0 1.00000 0
1.6 0 −1.00000 0 1.82640 0 2.26944 0 1.00000 0
1.7 0 −1.00000 0 3.61705 0 2.72774 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{7} \) \(\mathstrut +\mathstrut 3 T_{5}^{6} \) \(\mathstrut -\mathstrut 15 T_{5}^{5} \) \(\mathstrut -\mathstrut 50 T_{5}^{4} \) \(\mathstrut +\mathstrut 23 T_{5}^{3} \) \(\mathstrut +\mathstrut 133 T_{5}^{2} \) \(\mathstrut +\mathstrut 56 T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7}^{7} \) \(\mathstrut -\mathstrut 8 T_{7}^{6} \) \(\mathstrut +\mathstrut 11 T_{7}^{5} \) \(\mathstrut +\mathstrut 55 T_{7}^{4} \) \(\mathstrut -\mathstrut 147 T_{7}^{3} \) \(\mathstrut -\mathstrut 37 T_{7}^{2} \) \(\mathstrut +\mathstrut 336 T_{7} \) \(\mathstrut -\mathstrut 192 \)