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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).