Properties

Label 4008.2.a.h
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{1} q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \( + ( \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} \) \( + ( -1 - \beta_{2} - \beta_{5} ) q^{13} \) \( - \beta_{1} q^{15} \) \( + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{17} \) \( + ( -2 \beta_{1} - \beta_{4} + \beta_{7} ) q^{19} \) \( - \beta_{3} q^{21} \) \( + ( 1 - \beta_{3} ) q^{23} \) \( + ( - \beta_{6} + \beta_{7} ) q^{25} \) \(- q^{27}\) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} \) \( + ( -3 - \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} \) \( + ( - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{33} \) \( + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{35} \) \( + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} \) \( + ( 1 + \beta_{2} + \beta_{5} ) q^{39} \) \( + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} \) \( + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( -5 - 2 \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{47} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{49} \) \( + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} ) q^{51} \) \( + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{53} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} ) q^{55} \) \( + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{57} \) \( + ( -2 + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{59} \) \( + ( -1 - \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{61} \) \( + \beta_{3} q^{63} \) \( + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{65} \) \( + ( 3 - \beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{67} \) \( + ( -1 + \beta_{3} ) q^{69} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{71} \) \( + ( -5 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} \) \( + ( \beta_{6} - \beta_{7} ) q^{75} \) \( + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{77} \) \( + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{79} \) \(+ q^{81}\) \( + ( -4 + \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{83} \) \( + ( 2 - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{85} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{87} \) \( + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{89} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{91} \) \( + ( 3 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} \) \( + ( -8 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{95} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{97} \) \( + ( \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(8q \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 8q^{3} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 8q^{13} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 17q^{29} \) \(\mathstrut -\mathstrut 23q^{31} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 15q^{35} \) \(\mathstrut +\mathstrut 8q^{37} \) \(\mathstrut +\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 8q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut -\mathstrut 34q^{47} \) \(\mathstrut +\mathstrut 5q^{49} \) \(\mathstrut +\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 16q^{59} \) \(\mathstrut -\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut +\mathstrut 21q^{67} \) \(\mathstrut -\mathstrut 9q^{69} \) \(\mathstrut -\mathstrut 29q^{71} \) \(\mathstrut -\mathstrut 38q^{73} \) \(\mathstrut -\mathstrut 6q^{75} \) \(\mathstrut +\mathstrut 20q^{77} \) \(\mathstrut -\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut -\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 23q^{85} \) \(\mathstrut -\mathstrut 17q^{87} \) \(\mathstrut +\mathstrut 11q^{89} \) \(\mathstrut -\mathstrut 5q^{91} \) \(\mathstrut +\mathstrut 23q^{93} \) \(\mathstrut -\mathstrut 67q^{95} \) \(\mathstrut +\mathstrut 8q^{97} \) \(\mathstrut -\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(23\) \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut +\mathstrut \) \(163\) \(x^{4}\mathstrut +\mathstrut \) \(13\) \(x^{3}\mathstrut -\mathstrut \) \(418\) \(x^{2}\mathstrut +\mathstrut \) \(4\) \(x\mathstrut +\mathstrut \) \(269\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 185 \nu^{7} - 1337 \nu^{6} - 3411 \nu^{5} + 24599 \nu^{4} + 18970 \nu^{3} - 110475 \nu^{2} - 18219 \nu + 87357 \)\()/8453\)
\(\beta_{3}\)\(=\)\((\)\( -211 \nu^{7} - 714 \nu^{6} + 3799 \nu^{5} + 13889 \nu^{4} - 10670 \nu^{3} - 61564 \nu^{2} - 16368 \nu + 64354 \)\()/8453\)
\(\beta_{4}\)\(=\)\((\)\( -253 \nu^{7} - 776 \nu^{6} + 5076 \nu^{5} + 14290 \nu^{4} - 24572 \nu^{3} - 48820 \nu^{2} + 35699 \nu + 9019 \)\()/8453\)
\(\beta_{5}\)\(=\)\((\)\( -312 \nu^{7} + 747 \nu^{6} + 4656 \nu^{5} - 11512 \nu^{4} - 10289 \nu^{3} + 48782 \nu^{2} - 26206 \nu - 39128 \)\()/8453\)
\(\beta_{6}\)\(=\)\((\)\( 545 \nu^{7} + 402 \nu^{6} - 10734 \nu^{5} - 10235 \nu^{4} + 53600 \nu^{3} + 47164 \nu^{2} - 66009 \nu - 38506 \)\()/8453\)
\(\beta_{7}\)\(=\)\((\)\( 545 \nu^{7} + 402 \nu^{6} - 10734 \nu^{5} - 10235 \nu^{4} + 53600 \nu^{3} + 55617 \nu^{2} - 66009 \nu - 80771 \)\()/8453\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(41\)
\(\nu^{5}\)\(=\)\(54\) \(\beta_{7}\mathstrut -\mathstrut \) \(59\) \(\beta_{6}\mathstrut -\mathstrut \) \(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(36\) \(\beta_{4}\mathstrut +\mathstrut \) \(47\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\)
\(\nu^{6}\)\(=\)\(191\) \(\beta_{7}\mathstrut -\mathstrut \) \(247\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(102\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(29\) \(\beta_{2}\mathstrut +\mathstrut \) \(61\) \(\beta_{1}\mathstrut +\mathstrut \) \(422\)
\(\nu^{7}\)\(=\)\(804\) \(\beta_{7}\mathstrut -\mathstrut \) \(902\) \(\beta_{6}\mathstrut -\mathstrut \) \(332\) \(\beta_{5}\mathstrut -\mathstrut \) \(531\) \(\beta_{4}\mathstrut +\mathstrut \) \(624\) \(\beta_{3}\mathstrut -\mathstrut \) \(240\) \(\beta_{2}\mathstrut +\mathstrut \) \(1139\) \(\beta_{1}\mathstrut +\mathstrut \) \(333\)

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
−3.24252
−2.57145
−2.12535
−0.938106
1.02139
1.94540
2.18690
3.72373
0 −1.00000 0 −3.24252 0 −3.21024 0 1.00000 0
1.2 0 −1.00000 0 −2.57145 0 1.34479 0 1.00000 0
1.3 0 −1.00000 0 −2.12535 0 2.08900 0 1.00000 0
1.4 0 −1.00000 0 −0.938106 0 4.96672 0 1.00000 0
1.5 0 −1.00000 0 1.02139 0 −1.14465 0 1.00000 0
1.6 0 −1.00000 0 1.94540 0 −4.16489 0 1.00000 0
1.7 0 −1.00000 0 2.18690 0 0.195760 0 1.00000 0
1.8 0 −1.00000 0 3.72373 0 −1.07650 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{8} \) \(\mathstrut -\mathstrut 23 T_{5}^{6} \) \(\mathstrut -\mathstrut 3 T_{5}^{5} \) \(\mathstrut +\mathstrut 163 T_{5}^{4} \) \(\mathstrut +\mathstrut 13 T_{5}^{3} \) \(\mathstrut -\mathstrut 418 T_{5}^{2} \) \(\mathstrut +\mathstrut 4 T_{5} \) \(\mathstrut +\mathstrut 269 \)
\(T_{7}^{8} + \cdots\)