Properties

Label 4008.2.a.i
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 1
Dimension 9
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( + ( -2 + \beta_{5} + \beta_{7} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + ( -1 + \beta_{1} ) q^{5} \) \( + ( -2 + \beta_{5} + \beta_{7} ) q^{7} \) \(+ q^{9}\) \( + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{11} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{13} \) \( + ( -1 + \beta_{1} ) q^{15} \) \( + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{17} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{19} \) \( + ( -2 + \beta_{5} + \beta_{7} ) q^{21} \) \( + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{23} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{25} \) \(+ q^{27}\) \( + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{29} \) \( + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} \) \( + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{33} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{35} \) \( + ( 1 - \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{37} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} ) q^{39} \) \( + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{41} \) \( + ( -4 - 2 \beta_{3} + \beta_{7} ) q^{43} \) \( + ( -1 + \beta_{1} ) q^{45} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{49} \) \( + ( -1 - \beta_{2} - \beta_{6} - \beta_{7} ) q^{51} \) \( + ( -3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{53} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{55} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{57} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{8} ) q^{59} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{61} \) \( + ( -2 + \beta_{5} + \beta_{7} ) q^{63} \) \( + ( 2 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{65} \) \( + ( 1 - 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{67} \) \( + ( -\beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{69} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{71} \) \( + ( 2 + \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{73} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{75} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{7} + 2 \beta_{8} ) q^{77} \) \( + ( -4 + 2 \beta_{2} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{79} \) \(+ q^{81}\) \( + ( -2 - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{83} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{85} \) \( + ( -5 + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{87} \) \( + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{89} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} \) \( + ( -3 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{93} \) \( + ( -5 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{95} \) \( + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{97} \) \( + ( 2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 9q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 11q^{21} \) \(\mathstrut -\mathstrut 7q^{23} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut -\mathstrut 9q^{29} \) \(\mathstrut -\mathstrut 25q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 5q^{35} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 24q^{43} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 9q^{51} \) \(\mathstrut -\mathstrut 26q^{53} \) \(\mathstrut -\mathstrut 29q^{55} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut -\mathstrut 11q^{63} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 25q^{67} \) \(\mathstrut -\mathstrut 7q^{69} \) \(\mathstrut -\mathstrut 15q^{71} \) \(\mathstrut -\mathstrut 10q^{73} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut -\mathstrut 20q^{77} \) \(\mathstrut -\mathstrut 34q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 13q^{85} \) \(\mathstrut -\mathstrut 9q^{87} \) \(\mathstrut +\mathstrut 13q^{89} \) \(\mathstrut -\mathstrut 21q^{91} \) \(\mathstrut -\mathstrut 25q^{93} \) \(\mathstrut -\mathstrut 7q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(16\) \(x^{7}\mathstrut +\mathstrut \) \(45\) \(x^{6}\mathstrut +\mathstrut \) \(67\) \(x^{5}\mathstrut -\mathstrut \) \(166\) \(x^{4}\mathstrut -\mathstrut \) \(83\) \(x^{3}\mathstrut +\mathstrut \) \(152\) \(x^{2}\mathstrut +\mathstrut \) \(51\) \(x\mathstrut -\mathstrut \) \(10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -253 \nu^{8} + 10903 \nu^{7} - 17171 \nu^{6} - 175482 \nu^{5} + 247955 \nu^{4} + 716568 \nu^{3} - 563660 \nu^{2} - 635601 \nu + 68945 \)\()/102665\)
\(\beta_{3}\)\(=\)\((\)\( -428 \nu^{8} + 9923 \nu^{7} - 2266 \nu^{6} - 178372 \nu^{5} + 67645 \nu^{4} + 878658 \nu^{3} - 38080 \nu^{2} - 988001 \nu - 301330 \)\()/102665\)
\(\beta_{4}\)\(=\)\((\)\( -3031 \nu^{8} + 7666 \nu^{7} + 48718 \nu^{6} - 99334 \nu^{5} - 199070 \nu^{4} + 207921 \nu^{3} + 208150 \nu^{2} + 200063 \nu - 130065 \)\()/102665\)
\(\beta_{5}\)\(=\)\((\)\( 3811 \nu^{8} - 23831 \nu^{7} - 35953 \nu^{6} + 387944 \nu^{5} - 79645 \nu^{4} - 1646101 \nu^{3} + 520415 \nu^{2} + 1605297 \nu + 255125 \)\()/102665\)
\(\beta_{6}\)\(=\)\((\)\( 5684 \nu^{8} - 5129 \nu^{7} - 118627 \nu^{6} + 65121 \nu^{5} + 751965 \nu^{4} - 184819 \nu^{3} - 1469865 \nu^{2} + 296533 \nu + 445920 \)\()/102665\)
\(\beta_{7}\)\(=\)\((\)\( -6589 \nu^{8} + 20594 \nu^{7} + 101842 \nu^{6} - 311796 \nu^{5} - 367380 \nu^{4} + 1137454 \nu^{3} + 148730 \nu^{2} - 769633 \nu + 59190 \)\()/102665\)
\(\beta_{8}\)\(=\)\((\)\( 16128 \nu^{8} - 24668 \nu^{7} - 293609 \nu^{6} + 290982 \nu^{5} + 1500975 \nu^{4} - 446023 \nu^{3} - 2000020 \nu^{2} - 396149 \nu + 286160 \)\()/102665\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(12\) \(\beta_{7}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(17\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(54\)
\(\nu^{5}\)\(=\)\(-\)\(30\) \(\beta_{8}\mathstrut -\mathstrut \) \(45\) \(\beta_{7}\mathstrut +\mathstrut \) \(33\) \(\beta_{6}\mathstrut -\mathstrut \) \(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(36\) \(\beta_{4}\mathstrut +\mathstrut \) \(34\) \(\beta_{3}\mathstrut -\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut +\mathstrut \) \(109\)
\(\nu^{6}\)\(=\)\(-\)\(56\) \(\beta_{8}\mathstrut -\mathstrut \) \(145\) \(\beta_{7}\mathstrut +\mathstrut \) \(106\) \(\beta_{6}\mathstrut -\mathstrut \) \(132\) \(\beta_{5}\mathstrut +\mathstrut \) \(54\) \(\beta_{4}\mathstrut +\mathstrut \) \(105\) \(\beta_{3}\mathstrut -\mathstrut \) \(225\) \(\beta_{2}\mathstrut -\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(635\)
\(\nu^{7}\)\(=\)\(-\)\(391\) \(\beta_{8}\mathstrut -\mathstrut \) \(576\) \(\beta_{7}\mathstrut +\mathstrut \) \(466\) \(\beta_{6}\mathstrut -\mathstrut \) \(324\) \(\beta_{5}\mathstrut -\mathstrut \) \(395\) \(\beta_{4}\mathstrut +\mathstrut \) \(506\) \(\beta_{3}\mathstrut -\mathstrut \) \(459\) \(\beta_{2}\mathstrut +\mathstrut \) \(691\) \(\beta_{1}\mathstrut +\mathstrut \) \(1496\)
\(\nu^{8}\)\(=\)\(-\)\(846\) \(\beta_{8}\mathstrut -\mathstrut \) \(1799\) \(\beta_{7}\mathstrut +\mathstrut \) \(1544\) \(\beta_{6}\mathstrut -\mathstrut \) \(1474\) \(\beta_{5}\mathstrut +\mathstrut \) \(418\) \(\beta_{4}\mathstrut +\mathstrut \) \(1662\) \(\beta_{3}\mathstrut -\mathstrut \) \(2818\) \(\beta_{2}\mathstrut -\mathstrut \) \(429\) \(\beta_{1}\mathstrut +\mathstrut \) \(7652\)

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
−3.17025
−1.99382
−0.985991
−0.482381
0.141931
1.41388
1.58856
2.94484
3.54323
0 1.00000 0 −4.17025 0 −1.78655 0 1.00000 0
1.2 0 1.00000 0 −2.99382 0 −4.65862 0 1.00000 0
1.3 0 1.00000 0 −1.98599 0 −0.0909950 0 1.00000 0
1.4 0 1.00000 0 −1.48238 0 −1.03908 0 1.00000 0
1.5 0 1.00000 0 −0.858069 0 2.33225 0 1.00000 0
1.6 0 1.00000 0 0.413878 0 2.72537 0 1.00000 0
1.7 0 1.00000 0 0.588564 0 −1.23518 0 1.00000 0
1.8 0 1.00000 0 1.94484 0 −3.19705 0 1.00000 0
1.9 0 1.00000 0 2.54323 0 −4.05015 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(1\)

Hecke kernels

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

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)