Properties

Label 4004.2.a.k
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{4} q^{5} \) \(- q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{4} q^{5} \) \(- q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(+ q^{11}\) \(+ q^{13}\) \( + ( \beta_{1} - \beta_{6} - \beta_{8} ) q^{15} \) \( + \beta_{8} q^{17} \) \( + ( 1 + \beta_{9} ) q^{19} \) \( -\beta_{1} q^{21} \) \( -\beta_{5} q^{23} \) \( + ( 2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{25} \) \( + ( -1 + 2 \beta_{1} + \beta_{7} + \beta_{8} ) q^{27} \) \( + ( 1 - \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{29} \) \( + ( \beta_{3} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + \beta_{4} q^{35} \) \( + ( 2 - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{41} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{43} \) \( + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{45} \) \( -\beta_{5} q^{47} \) \(+ q^{49}\) \( + ( 2 + \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{51} \) \( + ( -1 + \beta_{1} - \beta_{4} - \beta_{7} ) q^{53} \) \( -\beta_{4} q^{55} \) \( + ( 2 + \beta_{1} - \beta_{3} + \beta_{6} ) q^{57} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{9} ) q^{59} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{61} \) \( + ( -2 - \beta_{2} ) q^{63} \) \( -\beta_{4} q^{65} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} ) q^{67} \) \( + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{69} \) \( + ( 3 - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{71} \) \( + ( 2 - \beta_{3} + \beta_{4} + \beta_{6} ) q^{73} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{75} \) \(- q^{77}\) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{79} \) \( + ( 5 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{81} \) \( + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} ) q^{83} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{85} \) \( + ( -6 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{87} \) \( + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{89} \) \(- q^{91}\) \( + ( 4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{93} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{95} \) \( + ( 6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{9} ) q^{97} \) \( + ( 2 + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 17q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 17q^{9} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 10q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 6q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut +\mathstrut 8q^{45} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut +\mathstrut 11q^{51} \) \(\mathstrut -\mathstrut 5q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 11q^{59} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut 17q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 2q^{67} \) \(\mathstrut +\mathstrut 10q^{69} \) \(\mathstrut +\mathstrut 28q^{71} \) \(\mathstrut +\mathstrut 11q^{73} \) \(\mathstrut -\mathstrut 6q^{75} \) \(\mathstrut -\mathstrut 10q^{77} \) \(\mathstrut -\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 46q^{81} \) \(\mathstrut +\mathstrut 7q^{83} \) \(\mathstrut +\mathstrut 33q^{85} \) \(\mathstrut -\mathstrut 47q^{87} \) \(\mathstrut +\mathstrut 30q^{89} \) \(\mathstrut -\mathstrut 10q^{91} \) \(\mathstrut +\mathstrut 41q^{93} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut 55q^{97} \) \(\mathstrut +\mathstrut 17q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(x^{9}\mathstrut -\mathstrut \) \(23\) \(x^{8}\mathstrut +\mathstrut \) \(23\) \(x^{7}\mathstrut +\mathstrut \) \(170\) \(x^{6}\mathstrut -\mathstrut \) \(165\) \(x^{5}\mathstrut -\mathstrut \) \(411\) \(x^{4}\mathstrut +\mathstrut \) \(360\) \(x^{3}\mathstrut +\mathstrut \) \(111\) \(x^{2}\mathstrut -\mathstrut \) \(48\) \(x\mathstrut -\mathstrut \) \(12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\( -194 \nu^{9} + 241 \nu^{8} + 4395 \nu^{7} - 5633 \nu^{6} - 31377 \nu^{5} + 41297 \nu^{4} + 67636 \nu^{3} - 92755 \nu^{2} + 6654 \nu + 9704 \)\()/557\)
\(\beta_{4}\)\(=\)\((\)\( -905 \nu^{9} + 1219 \nu^{8} + 20557 \nu^{7} - 27785 \nu^{6} - 147644 \nu^{5} + 198405 \nu^{4} + 324953 \nu^{3} - 433194 \nu^{2} + 9553 \nu + 43902 \)\()/2228\)
\(\beta_{5}\)\(=\)\((\)\( -945 \nu^{9} + 1039 \nu^{8} + 21601 \nu^{7} - 24077 \nu^{6} - 157444 \nu^{5} + 173385 \nu^{4} + 363085 \nu^{3} - 374890 \nu^{2} - 39607 \nu + 31662 \)\()/2228\)
\(\beta_{6}\)\(=\)\((\)\( -721 \nu^{9} + 933 \nu^{8} + 16423 \nu^{7} - 21225 \nu^{6} - 118160 \nu^{5} + 150853 \nu^{4} + 259609 \nu^{3} - 325306 \nu^{2} + 10661 \nu + 28910 \)\()/1114\)
\(\beta_{7}\)\(=\)\((\)\( -439 \nu^{9} + 531 \nu^{8} + 9954 \nu^{7} - 12164 \nu^{6} - 71350 \nu^{5} + 87177 \nu^{4} + 157210 \nu^{3} - 190155 \nu^{2} + 202 \nu + 17727 \)\()/557\)
\(\beta_{8}\)\(=\)\((\)\( 439 \nu^{9} - 531 \nu^{8} - 9954 \nu^{7} + 12164 \nu^{6} + 71350 \nu^{5} - 87177 \nu^{4} - 156653 \nu^{3} + 190155 \nu^{2} - 4658 \nu - 17170 \)\()/557\)
\(\beta_{9}\)\(=\)\((\)\( 1955 \nu^{9} - 2621 \nu^{8} - 44063 \nu^{7} + 60231 \nu^{6} + 312432 \nu^{5} - 433387 \nu^{4} - 666987 \nu^{3} + 952474 \nu^{2} - 62587 \nu - 99134 \)\()/2228\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(41\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(68\) \(\beta_{1}\mathstrut -\mathstrut \) \(14\)
\(\nu^{6}\)\(=\)\(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(29\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(81\) \(\beta_{2}\mathstrut -\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(362\)
\(\nu^{7}\)\(=\)\(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(148\) \(\beta_{8}\mathstrut +\mathstrut \) \(144\) \(\beta_{7}\mathstrut +\mathstrut \) \(38\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(596\) \(\beta_{1}\mathstrut -\mathstrut \) \(161\)
\(\nu^{8}\)\(=\)\(175\) \(\beta_{9}\mathstrut -\mathstrut \) \(58\) \(\beta_{8}\mathstrut +\mathstrut \) \(154\) \(\beta_{7}\mathstrut -\mathstrut \) \(157\) \(\beta_{6}\mathstrut -\mathstrut \) \(198\) \(\beta_{5}\mathstrut +\mathstrut \) \(349\) \(\beta_{4}\mathstrut +\mathstrut \) \(87\) \(\beta_{3}\mathstrut +\mathstrut \) \(744\) \(\beta_{2}\mathstrut -\mathstrut \) \(178\) \(\beta_{1}\mathstrut +\mathstrut \) \(3313\)
\(\nu^{9}\)\(=\)\(108\) \(\beta_{9}\mathstrut +\mathstrut \) \(1614\) \(\beta_{8}\mathstrut +\mathstrut \) \(1506\) \(\beta_{7}\mathstrut +\mathstrut \) \(536\) \(\beta_{6}\mathstrut -\mathstrut \) \(195\) \(\beta_{5}\mathstrut -\mathstrut \) \(517\) \(\beta_{4}\mathstrut +\mathstrut \) \(358\) \(\beta_{3}\mathstrut -\mathstrut \) \(236\) \(\beta_{2}\mathstrut +\mathstrut \) \(5329\) \(\beta_{1}\mathstrut -\mathstrut \) \(1740\)

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
−3.18458
−2.90534
−2.02595
−0.290239
−0.257439
0.418919
1.02188
2.24381
2.85872
3.12022
0 −3.18458 0 −2.71592 0 −1.00000 0 7.14153 0
1.2 0 −2.90534 0 1.80243 0 −1.00000 0 5.44100 0
1.3 0 −2.02595 0 3.89239 0 −1.00000 0 1.10449 0
1.4 0 −0.290239 0 0.725041 0 −1.00000 0 −2.91576 0
1.5 0 −0.257439 0 −3.68825 0 −1.00000 0 −2.93373 0
1.6 0 0.418919 0 0.0566209 0 −1.00000 0 −2.82451 0
1.7 0 1.02188 0 3.35739 0 −1.00000 0 −1.95576 0
1.8 0 2.24381 0 −2.78728 0 −1.00000 0 2.03468 0
1.9 0 2.85872 0 3.60422 0 −1.00000 0 5.17231 0
1.10 0 3.12022 0 −0.246641 0 −1.00000 0 6.73575 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).