Properties

Label 4008.2.a.j
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\)  \(=\)  \( q\) \(- q^{3}\) \( + ( -1 - \beta_{1} ) q^{5} \) \( + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( -1 - \beta_{1} ) q^{5} \) \( + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} \) \(+ q^{9}\) \( + ( - \beta_{4} - \beta_{7} ) q^{11} \) \( + ( -1 + \beta_{1} + \beta_{8} ) q^{13} \) \( + ( 1 + \beta_{1} ) q^{15} \) \( + ( \beta_{3} - \beta_{8} ) q^{17} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{19} \) \( + ( - \beta_{2} - \beta_{4} - \beta_{7} ) q^{21} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} \) \( + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{25} \) \(- q^{27}\) \( + ( -1 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{29} \) \( + ( 2 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{31} \) \( + ( \beta_{4} + \beta_{7} ) q^{33} \) \( + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{35} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{37} \) \( + ( 1 - \beta_{1} - \beta_{8} ) q^{39} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{41} \) \( + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{43} \) \( + ( -1 - \beta_{1} ) q^{45} \) \( + ( 1 - \beta_{2} - \beta_{4} ) q^{47} \) \( + ( 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{49} \) \( + ( - \beta_{3} + \beta_{8} ) q^{51} \) \( + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{53} \) \( + ( 3 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{55} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{57} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{59} \) \( + ( 1 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} \) \( + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{63} \) \( + ( -1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{65} \) \( + ( -2 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{67} \) \( + ( - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{69} \) \( + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{71} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} \) \( + ( - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{75} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} \) \( + ( 3 - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{79} \) \(+ q^{81}\) \( + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{83} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{85} \) \( + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} \) \( + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{89} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{91} \) \( + ( -2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{93} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{95} \) \( + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} \) \( + ( - \beta_{4} - \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(10q \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut q^{7} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut 10q^{15} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 10q^{27} \) \(\mathstrut -\mathstrut 13q^{29} \) \(\mathstrut +\mathstrut 23q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 10q^{45} \) \(\mathstrut +\mathstrut 10q^{47} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut +\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 26q^{53} \) \(\mathstrut +\mathstrut 11q^{55} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 10q^{59} \) \(\mathstrut -\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut q^{63} \) \(\mathstrut -\mathstrut 22q^{65} \) \(\mathstrut -\mathstrut 5q^{67} \) \(\mathstrut -\mathstrut 7q^{69} \) \(\mathstrut +\mathstrut 25q^{71} \) \(\mathstrut -\mathstrut 8q^{73} \) \(\mathstrut -\mathstrut 12q^{75} \) \(\mathstrut -\mathstrut 46q^{77} \) \(\mathstrut +\mathstrut 26q^{79} \) \(\mathstrut +\mathstrut 10q^{81} \) \(\mathstrut -\mathstrut 14q^{83} \) \(\mathstrut +\mathstrut 9q^{85} \) \(\mathstrut +\mathstrut 13q^{87} \) \(\mathstrut -\mathstrut 31q^{89} \) \(\mathstrut -\mathstrut 3q^{91} \) \(\mathstrut -\mathstrut 23q^{93} \) \(\mathstrut -\mathstrut 5q^{95} \) \(\mathstrut -\mathstrut 32q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(26\) \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut +\mathstrut \) \(220\) \(x^{6}\mathstrut +\mathstrut \) \(42\) \(x^{5}\mathstrut -\mathstrut \) \(675\) \(x^{4}\mathstrut -\mathstrut \) \(67\) \(x^{3}\mathstrut +\mathstrut \) \(628\) \(x^{2}\mathstrut -\mathstrut \) \(48\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 31274 \nu^{9} + 188403 \nu^{8} - 799085 \nu^{7} - 4445172 \nu^{6} + 5651462 \nu^{5} + 31352276 \nu^{4} - 4209233 \nu^{3} - 62769314 \nu^{2} - 26831938 \nu + 17959625 \)\()/6386707\)
\(\beta_{3}\)\(=\)\((\)\( 45076 \nu^{9} - 26608 \nu^{8} - 1096194 \nu^{7} + 535650 \nu^{6} + 9006576 \nu^{5} - 3054363 \nu^{4} - 31921253 \nu^{3} + 4461694 \nu^{2} + 45128857 \nu + 3445799 \)\()/6386707\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(140074\) \(\nu^{9}\mathstrut -\mathstrut \) \(197290\) \(\nu^{8}\mathstrut +\mathstrut \) \(4018521\) \(\nu^{7}\mathstrut +\mathstrut \) \(4959639\) \(\nu^{6}\mathstrut -\mathstrut \) \(37660725\) \(\nu^{5}\mathstrut -\mathstrut \) \(36535738\) \(\nu^{4}\mathstrut +\mathstrut \) \(125932814\) \(\nu^{3}\mathstrut +\mathstrut \) \(66704449\) \(\nu^{2}\mathstrut -\mathstrut \) \(128029606\) \(\nu\mathstrut -\mathstrut \) \(3074005\)\()/6386707\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(252127\) \(\nu^{9}\mathstrut -\mathstrut \) \(49534\) \(\nu^{8}\mathstrut +\mathstrut \) \(6474592\) \(\nu^{7}\mathstrut +\mathstrut \) \(2278935\) \(\nu^{6}\mathstrut -\mathstrut \) \(53041455\) \(\nu^{5}\mathstrut -\mathstrut \) \(25138542\) \(\nu^{4}\mathstrut +\mathstrut \) \(150476284\) \(\nu^{3}\mathstrut +\mathstrut \) \(61255546\) \(\nu^{2}\mathstrut -\mathstrut \) \(122806757\) \(\nu\mathstrut -\mathstrut \) \(20051371\)\()/6386707\)
\(\beta_{6}\)\(=\)\((\)\( -400908 \nu^{9} + 156741 \nu^{8} + 10079448 \nu^{7} - 2045962 \nu^{6} - 81378952 \nu^{5} + 2450805 \nu^{4} + 233226318 \nu^{3} - 5489909 \nu^{2} - 195374082 \nu + 15210925 \)\()/6386707\)
\(\beta_{7}\)\(=\)\((\)\( -446372 \nu^{9} + 13553 \nu^{8} + 11479221 \nu^{7} + 1022842 \nu^{6} - 95483893 \nu^{5} - 17429855 \nu^{4} + 282874842 \nu^{3} + 36533942 \nu^{2} - 233347950 \nu + 3313559 \)\()/6386707\)
\(\beta_{8}\)\(=\)\((\)\( -581351 \nu^{9} + 4817 \nu^{8} + 14786968 \nu^{7} + 1678767 \nu^{6} - 120466276 \nu^{5} - 24247147 \nu^{4} + 344299563 \nu^{3} + 37019495 \nu^{2} - 291434843 \nu + 24062029 \)\()/6386707\)
\(\beta_{9}\)\(=\)\((\)\(676598\) \(\nu^{9}\mathstrut +\mathstrut \) \(130521\) \(\nu^{8}\mathstrut -\mathstrut \) \(17690130\) \(\nu^{7}\mathstrut -\mathstrut \) \(4911181\) \(\nu^{6}\mathstrut +\mathstrut \) \(151157770\) \(\nu^{5}\mathstrut +\mathstrut \) \(47856867\) \(\nu^{4}\mathstrut -\mathstrut \) \(472650162\) \(\nu^{3}\mathstrut -\mathstrut \) \(87928296\) \(\nu^{2}\mathstrut +\mathstrut \) \(458021977\) \(\nu\mathstrut -\mathstrut \) \(18894784\)\()/6386707\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(24\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{5}\)\(=\)\(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(93\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(132\) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(81\) \(\beta_{7}\mathstrut +\mathstrut \) \(59\) \(\beta_{6}\mathstrut -\mathstrut \) \(40\) \(\beta_{5}\mathstrut +\mathstrut \) \(167\) \(\beta_{4}\mathstrut -\mathstrut \) \(268\) \(\beta_{3}\mathstrut -\mathstrut \) \(20\) \(\beta_{2}\mathstrut -\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(341\)
\(\nu^{7}\)\(=\)\(81\) \(\beta_{9}\mathstrut +\mathstrut \) \(246\) \(\beta_{8}\mathstrut -\mathstrut \) \(103\) \(\beta_{7}\mathstrut -\mathstrut \) \(253\) \(\beta_{6}\mathstrut +\mathstrut \) \(212\) \(\beta_{5}\mathstrut +\mathstrut \) \(51\) \(\beta_{4}\mathstrut -\mathstrut \) \(139\) \(\beta_{3}\mathstrut +\mathstrut \) \(245\) \(\beta_{2}\mathstrut +\mathstrut \) \(1024\) \(\beta_{1}\mathstrut +\mathstrut \) \(45\)
\(\nu^{8}\)\(=\)\(1477\) \(\beta_{9}\mathstrut -\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(784\) \(\beta_{7}\mathstrut +\mathstrut \) \(876\) \(\beta_{6}\mathstrut -\mathstrut \) \(447\) \(\beta_{5}\mathstrut +\mathstrut \) \(1957\) \(\beta_{4}\mathstrut -\mathstrut \) \(2992\) \(\beta_{3}\mathstrut -\mathstrut \) \(265\) \(\beta_{2}\mathstrut -\mathstrut \) \(319\) \(\beta_{1}\mathstrut +\mathstrut \) \(3816\)
\(\nu^{9}\)\(=\)\(1188\) \(\beta_{9}\mathstrut +\mathstrut \) \(3084\) \(\beta_{8}\mathstrut -\mathstrut \) \(1004\) \(\beta_{7}\mathstrut -\mathstrut \) \(3245\) \(\beta_{6}\mathstrut +\mathstrut \) \(2675\) \(\beta_{5}\mathstrut +\mathstrut \) \(861\) \(\beta_{4}\mathstrut -\mathstrut \) \(1359\) \(\beta_{3}\mathstrut +\mathstrut \) \(3283\) \(\beta_{2}\mathstrut +\mathstrut \) \(11672\) \(\beta_{1}\mathstrut +\mathstrut \) \(459\)

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
3.44741
3.20371
1.76045
1.15141
0.108417
−0.0299246
−1.39234
−2.25873
−2.53098
−3.45943
0 −1.00000 0 −4.44741 0 2.42708 0 1.00000 0
1.2 0 −1.00000 0 −4.20371 0 −3.93441 0 1.00000 0
1.3 0 −1.00000 0 −2.76045 0 3.28542 0 1.00000 0
1.4 0 −1.00000 0 −2.15141 0 −2.22742 0 1.00000 0
1.5 0 −1.00000 0 −1.10842 0 −3.58606 0 1.00000 0
1.6 0 −1.00000 0 −0.970075 0 4.67245 0 1.00000 0
1.7 0 −1.00000 0 0.392343 0 0.0476950 0 1.00000 0
1.8 0 −1.00000 0 1.25873 0 −0.924269 0 1.00000 0
1.9 0 −1.00000 0 1.53098 0 1.95354 0 1.00000 0
1.10 0 −1.00000 0 2.45943 0 −0.714036 0 1.00000 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
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)