Properties

Label 4004.2.a.j
Level 4004
Weight 2
Character orbit 4004.a
Self dual Yes
Analytic conductor 31.972
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{3} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(- q^{11}\) \(- q^{13}\) \( + ( \beta_{1} - \beta_{6} + \beta_{8} ) q^{15} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{17} \) \( + ( 1 + \beta_{5} + \beta_{8} ) q^{19} \) \( + \beta_{1} q^{21} \) \( + ( -\beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} \) \( + ( 3 - \beta_{1} + \beta_{3} + \beta_{7} ) q^{25} \) \( + ( -1 - 2 \beta_{1} - \beta_{7} - \beta_{8} ) q^{27} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} ) q^{29} \) \( + ( -2 - \beta_{2} + \beta_{4} - \beta_{8} ) q^{31} \) \( + \beta_{1} q^{33} \) \( -\beta_{3} q^{35} \) \( + ( 1 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{41} \) \( + ( 3 + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{43} \) \( + ( -2 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{9} ) q^{45} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( 2 - 3 \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{51} \) \( + ( 2 - \beta_{4} - \beta_{5} + \beta_{8} ) q^{53} \) \( -\beta_{3} q^{55} \) \( + ( 1 - \beta_{2} + 3 \beta_{3} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{57} \) \( + ( -3 - \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{59} \) \( + ( 1 - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} \) \( + ( -1 - \beta_{2} ) q^{63} \) \( -\beta_{3} q^{65} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{67} \) \( + ( 6 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{69} \) \( + ( -3 + \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{71} \) \( + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{73} \) \( + ( 4 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{75} \) \(+ q^{77}\) \( + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{79} \) \( + ( 5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{81} \) \( + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{83} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{85} \) \( + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{87} \) \( + ( 4 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{89} \) \(+ q^{91}\) \( + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{8} - \beta_{9} ) q^{93} \) \( + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{95} \) \( + ( -1 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} ) q^{97} \) \( + ( -1 - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut -\mathstrut 10q^{11} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 13q^{29} \) \(\mathstrut -\mathstrut 17q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 21q^{43} \) \(\mathstrut -\mathstrut 15q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut +\mathstrut 19q^{51} \) \(\mathstrut +\mathstrut 22q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 10q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 19q^{67} \) \(\mathstrut +\mathstrut 51q^{69} \) \(\mathstrut -\mathstrut 25q^{71} \) \(\mathstrut +\mathstrut 22q^{73} \) \(\mathstrut +\mathstrut 28q^{75} \) \(\mathstrut +\mathstrut 10q^{77} \) \(\mathstrut +\mathstrut 41q^{79} \) \(\mathstrut +\mathstrut 54q^{81} \) \(\mathstrut +\mathstrut 8q^{83} \) \(\mathstrut +\mathstrut 9q^{85} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut +\mathstrut 36q^{89} \) \(\mathstrut +\mathstrut 10q^{91} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut +\mathstrut 13q^{95} \) \(\mathstrut -\mathstrut 5q^{97} \) \(\mathstrut -\mathstrut 10q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(18\) \(x^{8}\mathstrut +\mathstrut \) \(32\) \(x^{7}\mathstrut +\mathstrut \) \(93\) \(x^{6}\mathstrut -\mathstrut \) \(119\) \(x^{5}\mathstrut -\mathstrut \) \(174\) \(x^{4}\mathstrut +\mathstrut \) \(107\) \(x^{3}\mathstrut +\mathstrut \) \(51\) \(x^{2}\mathstrut -\mathstrut \) \(17\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 8 \nu^{9} + 6 \nu^{8} - 215 \nu^{7} - 54 \nu^{6} + 1783 \nu^{5} - 130 \nu^{4} - 4362 \nu^{3} + 148 \nu^{2} + 1715 \nu - 26 \)\()/125\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{9} + 34 \nu^{8} + 240 \nu^{7} - 631 \nu^{6} - 1263 \nu^{5} + 3330 \nu^{4} + 2132 \nu^{3} - 5628 \nu^{2} - 515 \nu + 1061 \)\()/125\)
\(\beta_{5}\)\(=\)\((\)\( -31 \nu^{9} + 83 \nu^{8} + 505 \nu^{7} - 1322 \nu^{6} - 2056 \nu^{5} + 4935 \nu^{4} + 2509 \nu^{3} - 4836 \nu^{2} + 845 \nu + 432 \)\()/125\)
\(\beta_{6}\)\(=\)\((\)\( -31 \nu^{9} + 33 \nu^{8} + 630 \nu^{7} - 572 \nu^{6} - 3906 \nu^{5} + 2460 \nu^{4} + 7959 \nu^{3} - 2286 \nu^{2} - 2055 \nu + 357 \)\()/125\)
\(\beta_{7}\)\(=\)\((\)\( 53 \nu^{9} - 104 \nu^{8} - 940 \nu^{7} + 1611 \nu^{6} + 4728 \nu^{5} - 5430 \nu^{4} - 8542 \nu^{3} + 3593 \nu^{2} + 1290 \nu - 466 \)\()/125\)
\(\beta_{8}\)\(=\)\((\)\( -53 \nu^{9} + 104 \nu^{8} + 940 \nu^{7} - 1611 \nu^{6} - 4728 \nu^{5} + 5430 \nu^{4} + 8667 \nu^{3} - 3593 \nu^{2} - 2290 \nu + 341 \)\()/125\)
\(\beta_{9}\)\(=\)\((\)\( -59 \nu^{9} + 112 \nu^{8} + 1070 \nu^{7} - 1758 \nu^{6} - 5634 \nu^{5} + 6115 \nu^{4} + 10951 \nu^{3} - 4029 \nu^{2} - 3570 \nu - 27 \)\()/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{5}\)\(=\)\(-\)\(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(75\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{6}\)\(=\)\(-\)\(31\) \(\beta_{9}\mathstrut +\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(117\) \(\beta_{2}\mathstrut -\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(300\)
\(\nu^{7}\)\(=\)\(-\)\(34\) \(\beta_{9}\mathstrut +\mathstrut \) \(166\) \(\beta_{8}\mathstrut +\mathstrut \) \(145\) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(32\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(43\) \(\beta_{3}\mathstrut -\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(754\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{8}\)\(=\)\(-\)\(377\) \(\beta_{9}\mathstrut +\mathstrut \) \(372\) \(\beta_{8}\mathstrut -\mathstrut \) \(185\) \(\beta_{7}\mathstrut -\mathstrut \) \(228\) \(\beta_{6}\mathstrut -\mathstrut \) \(137\) \(\beta_{5}\mathstrut +\mathstrut \) \(145\) \(\beta_{4}\mathstrut -\mathstrut \) \(269\) \(\beta_{3}\mathstrut +\mathstrut \) \(1236\) \(\beta_{2}\mathstrut -\mathstrut \) \(577\) \(\beta_{1}\mathstrut +\mathstrut \) \(2996\)
\(\nu^{9}\)\(=\)\(-\)\(427\) \(\beta_{9}\mathstrut +\mathstrut \) \(1849\) \(\beta_{8}\mathstrut +\mathstrut \) \(1566\) \(\beta_{7}\mathstrut -\mathstrut \) \(175\) \(\beta_{6}\mathstrut +\mathstrut \) \(413\) \(\beta_{5}\mathstrut -\mathstrut \) \(185\) \(\beta_{4}\mathstrut -\mathstrut \) \(637\) \(\beta_{3}\mathstrut -\mathstrut \) \(426\) \(\beta_{2}\mathstrut +\mathstrut \) \(7826\) \(\beta_{1}\mathstrut -\mathstrut \) \(492\)

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
3.21473
2.78432
2.16679
0.672934
0.357256
−0.0962745
−0.497051
−1.59443
−1.69862
−3.30965
0 −3.21473 0 −2.32236 0 −1.00000 0 7.33449 0
1.2 0 −2.78432 0 2.77311 0 −1.00000 0 4.75245 0
1.3 0 −2.16679 0 −1.26627 0 −1.00000 0 1.69497 0
1.4 0 −0.672934 0 0.538228 0 −1.00000 0 −2.54716 0
1.5 0 −0.357256 0 3.31742 0 −1.00000 0 −2.87237 0
1.6 0 0.0962745 0 −1.48698 0 −1.00000 0 −2.99073 0
1.7 0 0.497051 0 −2.93955 0 −1.00000 0 −2.75294 0
1.8 0 1.59443 0 4.37026 0 −1.00000 0 −0.457782 0
1.9 0 1.69862 0 −3.84237 0 −1.00000 0 −0.114675 0
1.10 0 3.30965 0 −1.14150 0 −1.00000 0 7.95375 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(T_{3}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).