Properties

Label 4004.2.a.h
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{4} q^{5} \) \(+ q^{7}\) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{4} q^{5} \) \(+ q^{7}\) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} \) \(+ q^{11}\) \(- q^{13}\) \( + ( -2 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{15} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{8} ) q^{17} \) \( + ( 1 - \beta_{2} + \beta_{4} ) q^{19} \) \( + \beta_{1} q^{21} \) \( + ( 1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{23} \) \( + ( 1 + \beta_{3} + \beta_{6} ) q^{25} \) \( + ( 3 + 2 \beta_{1} + 2 \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{29} \) \( + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{31} \) \( + \beta_{1} q^{33} \) \( -\beta_{4} q^{35} \) \( + ( -\beta_{3} - \beta_{4} - \beta_{8} ) q^{37} \) \( -\beta_{1} q^{39} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{43} \) \( + ( -1 - \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{45} \) \( + ( 1 + \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{47} \) \(+ q^{49}\) \( + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{51} \) \( + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{53} \) \( -\beta_{4} q^{55} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{57} \) \( + ( 3 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{59} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{63} \) \( + \beta_{4} q^{65} \) \( + ( 1 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{67} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{69} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{71} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 3 \beta_{7} - \beta_{8} ) q^{73} \) \( + ( 4 + \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{75} \) \(+ q^{77}\) \( + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{79} \) \( + ( 8 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{8} ) q^{81} \) \( + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{83} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{85} \) \( + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{87} \) \( + ( 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{89} \) \(- q^{91}\) \( + ( 1 - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{93} \) \( + ( -8 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{95} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{97} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut 9q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 9q^{15} \) \(\mathstrut +\mathstrut 5q^{17} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 27q^{27} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut +\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 3q^{39} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 10q^{47} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 15q^{51} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut +\mathstrut 23q^{59} \) \(\mathstrut +\mathstrut 34q^{61} \) \(\mathstrut +\mathstrut 20q^{63} \) \(\mathstrut +\mathstrut 10q^{67} \) \(\mathstrut -\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 4q^{71} \) \(\mathstrut +\mathstrut 9q^{73} \) \(\mathstrut +\mathstrut 30q^{75} \) \(\mathstrut +\mathstrut 9q^{77} \) \(\mathstrut -\mathstrut 34q^{79} \) \(\mathstrut +\mathstrut 69q^{81} \) \(\mathstrut +\mathstrut 15q^{83} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut 39q^{87} \) \(\mathstrut -\mathstrut 9q^{91} \) \(\mathstrut +\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 64q^{95} \) \(\mathstrut +\mathstrut 15q^{97} \) \(\mathstrut +\mathstrut 20q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(19\) \(x^{7}\mathstrut +\mathstrut \) \(51\) \(x^{6}\mathstrut +\mathstrut \) \(116\) \(x^{5}\mathstrut -\mathstrut \) \(247\) \(x^{4}\mathstrut -\mathstrut \) \(249\) \(x^{3}\mathstrut +\mathstrut \) \(288\) \(x^{2}\mathstrut +\mathstrut \) \(189\) \(x\mathstrut -\mathstrut \) \(14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{8} - 1526 \nu^{7} + 1071 \nu^{6} + 27349 \nu^{5} - 11387 \nu^{4} - 132924 \nu^{3} + 26935 \nu^{2} + 141008 \nu + 36543 \)\()/15311\)
\(\beta_{3}\)\(=\)\((\)\( 38 \nu^{8} - 296 \nu^{7} - 916 \nu^{6} + 7131 \nu^{5} + 7323 \nu^{4} - 46071 \nu^{3} - 19277 \nu^{2} + 55726 \nu - 10711 \)\()/15311\)
\(\beta_{4}\)\(=\)\((\)\( 49 \nu^{8} + 1230 \nu^{7} - 1987 \nu^{6} - 20218 \nu^{5} + 18710 \nu^{4} + 86853 \nu^{3} - 61523 \nu^{2} - 69971 \nu + 29301 \)\()/15311\)
\(\beta_{5}\)\(=\)\((\)\( 95 \nu^{8} - 740 \nu^{7} - 2290 \nu^{6} + 10172 \nu^{5} + 25963 \nu^{4} - 30967 \nu^{3} - 117092 \nu^{2} + 16827 \nu + 103366 \)\()/15311\)
\(\beta_{6}\)\(=\)\((\)\( 626 \nu^{8} - 847 \nu^{7} - 9449 \nu^{6} + 9491 \nu^{5} + 32800 \nu^{4} - 14361 \nu^{3} + 7997 \nu^{2} - 64309 \nu - 41874 \)\()/15311\)
\(\beta_{7}\)\(=\)\((\)\( -702 \nu^{8} + 1439 \nu^{7} + 11281 \nu^{6} - 23753 \nu^{5} - 47446 \nu^{4} + 121814 \nu^{3} + 30557 \nu^{2} - 169631 \nu + 17363 \)\()/15311\)
\(\beta_{8}\)\(=\)\((\)\( -979 \nu^{8} + 1985 \nu^{7} + 18764 \nu^{6} - 31010 \nu^{5} - 110094 \nu^{4} + 127655 \nu^{3} + 192431 \nu^{2} - 81863 \nu - 70160 \)\()/15311\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(44\)
\(\nu^{5}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(31\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(74\) \(\beta_{1}\mathstrut +\mathstrut \) \(49\)
\(\nu^{6}\)\(=\)\(19\) \(\beta_{8}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut +\mathstrut \) \(29\) \(\beta_{5}\mathstrut -\mathstrut \) \(81\) \(\beta_{4}\mathstrut +\mathstrut \) \(148\) \(\beta_{3}\mathstrut -\mathstrut \) \(95\) \(\beta_{2}\mathstrut +\mathstrut \) \(124\) \(\beta_{1}\mathstrut +\mathstrut \) \(434\)
\(\nu^{7}\)\(=\)\(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(110\) \(\beta_{7}\mathstrut +\mathstrut \) \(133\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(394\) \(\beta_{3}\mathstrut -\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(734\) \(\beta_{1}\mathstrut +\mathstrut \) \(673\)
\(\nu^{8}\)\(=\)\(249\) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(310\) \(\beta_{6}\mathstrut +\mathstrut \) \(337\) \(\beta_{5}\mathstrut -\mathstrut \) \(741\) \(\beta_{4}\mathstrut +\mathstrut \) \(1649\) \(\beta_{3}\mathstrut -\mathstrut \) \(924\) \(\beta_{2}\mathstrut +\mathstrut \) \(1440\) \(\beta_{1}\mathstrut +\mathstrut \) \(4485\)

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
−2.98175
−2.56319
−1.21819
−0.648104
0.0675528
1.32285
2.29815
3.32951
3.39317
0 −2.98175 0 2.28847 0 1.00000 0 5.89080 0
1.2 0 −2.56319 0 −1.31026 0 1.00000 0 3.56996 0
1.3 0 −1.21819 0 3.23173 0 1.00000 0 −1.51602 0
1.4 0 −0.648104 0 −1.99671 0 1.00000 0 −2.57996 0
1.5 0 0.0675528 0 −1.58844 0 1.00000 0 −2.99544 0
1.6 0 1.32285 0 −0.265081 0 1.00000 0 −1.25008 0
1.7 0 2.29815 0 0.952431 0 1.00000 0 2.28151 0
1.8 0 3.32951 0 2.67963 0 1.00000 0 8.08561 0
1.9 0 3.39317 0 −3.99177 0 1.00000 0 8.51361 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).