Properties

Label 4004.2.a.i
Level 4004
Weight 2
Character orbit 4004.a
Self dual Yes
Analytic conductor 31.972
Analytic rank 0
Dimension 9
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{7} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{7} q^{5} \) \(+ q^{7}\) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{9} \) \(- q^{11}\) \(+ q^{13}\) \( + ( -1 + 2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{15} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{17} \) \( + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} \) \( + \beta_{1} q^{21} \) \( + ( -1 - \beta_{4} - \beta_{6} ) q^{23} \) \( + ( 5 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{8} ) q^{25} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{27} \) \( + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{29} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} ) q^{31} \) \( -\beta_{1} q^{33} \) \( -\beta_{7} q^{35} \) \( + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{43} \) \( + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{45} \) \( + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( 3 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{51} \) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{53} \) \( + \beta_{7} q^{55} \) \( + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{57} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{59} \) \( + ( 5 - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{61} \) \( + ( 1 + \beta_{3} + \beta_{4} ) q^{63} \) \( -\beta_{7} q^{65} \) \( + ( 2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{67} \) \( + ( 3 - 4 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{69} \) \( + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{71} \) \( + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{73} \) \( + ( -4 + 8 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{75} \) \(- q^{77}\) \( + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} \) \( + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{81} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{83} \) \( + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} \) \( + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{87} \) \( + ( -\beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{89} \) \(+ q^{91}\) \( + ( -1 + \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{93} \) \( + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{95} \) \( + ( 3 - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} \) \( + ( -1 - \beta_{3} - \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 13q^{9} \) \(\mathstrut -\mathstrut 9q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 5q^{17} \) \(\mathstrut +\mathstrut 17q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 35q^{25} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 3q^{29} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 7q^{37} \) \(\mathstrut +\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 21q^{43} \) \(\mathstrut +\mathstrut 43q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 25q^{51} \) \(\mathstrut -\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 30q^{61} \) \(\mathstrut +\mathstrut 13q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 15q^{67} \) \(\mathstrut +\mathstrut 9q^{69} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut +\mathstrut 18q^{73} \) \(\mathstrut -\mathstrut 32q^{75} \) \(\mathstrut -\mathstrut 9q^{77} \) \(\mathstrut +\mathstrut 13q^{79} \) \(\mathstrut +\mathstrut 13q^{81} \) \(\mathstrut +\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 45q^{85} \) \(\mathstrut +\mathstrut 21q^{87} \) \(\mathstrut +\mathstrut 9q^{91} \) \(\mathstrut -\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut 11q^{95} \) \(\mathstrut +\mathstrut 29q^{97} \) \(\mathstrut -\mathstrut 13q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(12\) \(x^{7}\mathstrut +\mathstrut \) \(60\) \(x^{6}\mathstrut +\mathstrut \) \(15\) \(x^{5}\mathstrut -\mathstrut \) \(233\) \(x^{4}\mathstrut +\mathstrut \) \(74\) \(x^{3}\mathstrut +\mathstrut \) \(271\) \(x^{2}\mathstrut -\mathstrut \) \(67\) \(x\mathstrut -\mathstrut \) \(87\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -10 \nu^{8} - 189 \nu^{7} + 627 \nu^{6} + 2658 \nu^{5} - 6837 \nu^{4} - 8299 \nu^{3} + 17667 \nu^{2} + 5318 \nu - 5712 \)\()/681\)
\(\beta_{3}\)\(=\)\((\)\( -46 \nu^{8} + 84 \nu^{7} + 705 \nu^{6} - 1257 \nu^{5} - 2712 \nu^{4} + 5000 \nu^{3} + 2817 \nu^{2} - 5365 \nu - 1623 \)\()/681\)
\(\beta_{4}\)\(=\)\((\)\( 46 \nu^{8} - 84 \nu^{7} - 705 \nu^{6} + 1257 \nu^{5} + 2712 \nu^{4} - 5000 \nu^{3} - 2136 \nu^{2} + 5365 \nu - 1101 \)\()/681\)
\(\beta_{5}\)\(=\)\((\)\( -77 \nu^{8} + 111 \nu^{7} + 1491 \nu^{6} - 1734 \nu^{5} - 9129 \nu^{4} + 7126 \nu^{3} + 20334 \nu^{2} - 6449 \nu - 11703 \)\()/681\)
\(\beta_{6}\)\(=\)\((\)\( 176 \nu^{8} - 351 \nu^{7} - 2727 \nu^{6} + 4839 \nu^{5} + 11235 \nu^{4} - 14926 \nu^{3} - 15249 \nu^{2} + 6374 \nu + 6417 \)\()/681\)
\(\beta_{7}\)\(=\)\((\)\( 298 \nu^{8} - 633 \nu^{7} - 4656 \nu^{6} + 8913 \nu^{5} + 19464 \nu^{4} - 29312 \nu^{3} - 26214 \nu^{2} + 18175 \nu + 9774 \)\()/681\)
\(\beta_{8}\)\(=\)\((\)\( -143 \nu^{8} + 271 \nu^{7} + 2315 \nu^{6} - 3804 \nu^{5} - 10533 \nu^{4} + 12553 \nu^{3} + 16717 \nu^{2} - 8215 \nu - 7044 \)\()/227\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(27\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(74\) \(\beta_{1}\mathstrut -\mathstrut \) \(18\)
\(\nu^{6}\)\(=\)\(-\)\(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(46\) \(\beta_{7}\mathstrut +\mathstrut \) \(35\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(116\) \(\beta_{4}\mathstrut +\mathstrut \) \(95\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(209\)
\(\nu^{7}\)\(=\)\(139\) \(\beta_{8}\mathstrut -\mathstrut \) \(43\) \(\beta_{7}\mathstrut +\mathstrut \) \(311\) \(\beta_{6}\mathstrut -\mathstrut \) \(171\) \(\beta_{5}\mathstrut +\mathstrut \) \(211\) \(\beta_{4}\mathstrut +\mathstrut \) \(109\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(727\) \(\beta_{1}\mathstrut -\mathstrut \) \(212\)
\(\nu^{8}\)\(=\)\(-\)\(133\) \(\beta_{8}\mathstrut -\mathstrut \) \(552\) \(\beta_{7}\mathstrut +\mathstrut \) \(466\) \(\beta_{6}\mathstrut -\mathstrut \) \(182\) \(\beta_{5}\mathstrut +\mathstrut \) \(1220\) \(\beta_{4}\mathstrut +\mathstrut \) \(920\) \(\beta_{3}\mathstrut +\mathstrut \) \(171\) \(\beta_{2}\mathstrut +\mathstrut \) \(135\) \(\beta_{1}\mathstrut +\mathstrut \) \(1876\)

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
−3.11788
−1.90247
−0.980714
−0.554873
0.848270
1.89826
2.10249
2.42810
3.27882
0 −3.11788 0 4.40203 0 1.00000 0 6.72118 0
1.2 0 −1.90247 0 −3.63968 0 1.00000 0 0.619386 0
1.3 0 −0.980714 0 −1.42373 0 1.00000 0 −2.03820 0
1.4 0 −0.554873 0 3.11431 0 1.00000 0 −2.69212 0
1.5 0 0.848270 0 −0.844151 0 1.00000 0 −2.28044 0
1.6 0 1.89826 0 3.07273 0 1.00000 0 0.603374 0
1.7 0 2.10249 0 −4.18458 0 1.00000 0 1.42045 0
1.8 0 2.42810 0 0.790216 0 1.00000 0 2.89567 0
1.9 0 3.27882 0 2.71285 0 1.00000 0 7.75069 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).