Properties

Label 6032.2.a.bc
Level 6032
Weight 2
Character orbit 6032.a
Self dual Yes
Analytic conductor 48.166
Analytic rank 0
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6032.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( + \beta_{5} q^{5} \) \( + \beta_{3} q^{7} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 - \beta_{1} ) q^{3} \) \( + \beta_{5} q^{5} \) \( + \beta_{3} q^{7} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{9} \) \( + ( 1 + \beta_{8} ) q^{11} \) \(+ q^{13}\) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{15} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{17} \) \( + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{19} \) \( + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{21} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{10} ) q^{23} \) \( + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{25} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} ) q^{27} \) \(- q^{29}\) \( + ( 1 + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{31} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{33} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{35} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{37} \) \( + ( 1 - \beta_{1} ) q^{39} \) \( + ( -\beta_{1} + \beta_{3} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{41} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{9} + \beta_{10} ) q^{43} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{45} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{47} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{49} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{51} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{53} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{55} \) \( + ( 3 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{8} + \beta_{10} ) q^{57} \) \( + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{10} ) q^{59} \) \( + ( -\beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{61} \) \( + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{63} \) \( + \beta_{5} q^{65} \) \( + ( 3 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{67} \) \( + ( -\beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{69} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{71} \) \( + ( -\beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{73} \) \( + ( 4 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} ) q^{75} \) \( + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{77} \) \( + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{79} \) \( + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{81} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{83} \) \( + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{85} \) \( + ( -1 + \beta_{1} ) q^{87} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{89} \) \( + \beta_{3} q^{91} \) \( + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{93} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{95} \) \( + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{10} ) q^{97} \) \( + ( 3 - 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut 13q^{11} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut +\mathstrut 24q^{27} \) \(\mathstrut -\mathstrut 11q^{29} \) \(\mathstrut +\mathstrut 11q^{31} \) \(\mathstrut +\mathstrut 17q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 9q^{41} \) \(\mathstrut +\mathstrut 30q^{43} \) \(\mathstrut -\mathstrut 16q^{45} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 13q^{51} \) \(\mathstrut -\mathstrut 9q^{53} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 9q^{59} \) \(\mathstrut -\mathstrut 5q^{61} \) \(\mathstrut +\mathstrut 6q^{63} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 25q^{67} \) \(\mathstrut +\mathstrut 26q^{71} \) \(\mathstrut +\mathstrut 10q^{73} \) \(\mathstrut +\mathstrut 41q^{75} \) \(\mathstrut -\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 14q^{79} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut +\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 19q^{85} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut -\mathstrut 11q^{89} \) \(\mathstrut +\mathstrut 3q^{91} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut +\mathstrut 31q^{95} \) \(\mathstrut +\mathstrut 12q^{97} \) \(\mathstrut +\mathstrut 35q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(5\) \(x^{10}\mathstrut -\mathstrut \) \(9\) \(x^{9}\mathstrut +\mathstrut \) \(65\) \(x^{8}\mathstrut +\mathstrut \) \(19\) \(x^{7}\mathstrut -\mathstrut \) \(298\) \(x^{6}\mathstrut +\mathstrut \) \(17\) \(x^{5}\mathstrut +\mathstrut \) \(541\) \(x^{4}\mathstrut -\mathstrut \) \(60\) \(x^{3}\mathstrut -\mathstrut \) \(287\) \(x^{2}\mathstrut +\mathstrut \) \(16\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{10} - 186 \nu^{9} + 493 \nu^{8} - 202 \nu^{7} - 4619 \nu^{6} + 11193 \nu^{5} + 8936 \nu^{4} - 31635 \nu^{3} - 1873 \nu^{2} + 10278 \nu + 636 \)\()/2308\)
\(\beta_{4}\)\(=\)\((\)\( -44 \nu^{10} + 239 \nu^{9} - 376 \nu^{8} - 547 \nu^{7} + 6416 \nu^{6} - 8175 \nu^{5} - 23773 \nu^{4} + 26550 \nu^{3} + 25769 \nu^{2} - 10787 \nu - 780 \)\()/2308\)
\(\beta_{5}\)\(=\)\((\)\( -53 \nu^{10} + 301 \nu^{9} + 229 \nu^{8} - 3557 \nu^{7} + 1801 \nu^{6} + 15790 \nu^{5} - 13673 \nu^{4} - 29837 \nu^{3} + 25624 \nu^{2} + 15791 \nu - 5608 \)\()/2308\)
\(\beta_{6}\)\(=\)\((\)\( 92 \nu^{10} - 185 \nu^{9} - 1312 \nu^{8} + 829 \nu^{7} + 8196 \nu^{6} + 4609 \nu^{5} - 22785 \nu^{4} - 25090 \nu^{3} + 19241 \nu^{2} + 23289 \nu + 2680 \)\()/2308\)
\(\beta_{7}\)\(=\)\((\)\( -59 \nu^{10} + 150 \nu^{9} + 1017 \nu^{8} - 2294 \nu^{7} - 5507 \nu^{6} + 11187 \nu^{5} + 10178 \nu^{4} - 21653 \nu^{3} - 3515 \nu^{2} + 15238 \nu + 790 \)\()/1154\)
\(\beta_{8}\)\(=\)\((\)\( 125 \nu^{10} - 797 \nu^{9} - 453 \nu^{8} + 9173 \nu^{7} - 1809 \nu^{6} - 39026 \nu^{5} + 2113 \nu^{4} + 67801 \nu^{3} + 19388 \nu^{2} - 29927 \nu - 11160 \)\()/2308\)
\(\beta_{9}\)\(=\)\((\)\( 154 \nu^{10} - 1125 \nu^{9} + 162 \nu^{8} + 12589 \nu^{7} - 12070 \nu^{6} - 48417 \nu^{5} + 44835 \nu^{4} + 71520 \nu^{3} - 40281 \nu^{2} - 32351 \nu + 3884 \)\()/2308\)
\(\beta_{10}\)\(=\)\((\)\( 373 \nu^{10} - 1672 \nu^{9} - 3789 \nu^{8} + 21016 \nu^{7} + 11755 \nu^{6} - 90147 \nu^{5} - 8406 \nu^{4} + 143815 \nu^{3} - 11557 \nu^{2} - 52268 \nu + 7504 \)\()/2308\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{5}\)\(=\)\(-\)\(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(62\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{6}\)\(=\)\(-\)\(61\) \(\beta_{10}\mathstrut +\mathstrut \) \(59\) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(40\) \(\beta_{7}\mathstrut +\mathstrut \) \(81\) \(\beta_{6}\mathstrut -\mathstrut \) \(72\) \(\beta_{5}\mathstrut +\mathstrut \) \(52\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(72\) \(\beta_{2}\mathstrut +\mathstrut \) \(171\) \(\beta_{1}\mathstrut +\mathstrut \) \(128\)
\(\nu^{7}\)\(=\)\(-\)\(253\) \(\beta_{10}\mathstrut +\mathstrut \) \(246\) \(\beta_{9}\mathstrut +\mathstrut \) \(29\) \(\beta_{8}\mathstrut -\mathstrut \) \(173\) \(\beta_{7}\mathstrut +\mathstrut \) \(329\) \(\beta_{6}\mathstrut -\mathstrut \) \(274\) \(\beta_{5}\mathstrut +\mathstrut \) \(202\) \(\beta_{4}\mathstrut -\mathstrut \) \(128\) \(\beta_{3}\mathstrut +\mathstrut \) \(196\) \(\beta_{2}\mathstrut +\mathstrut \) \(644\) \(\beta_{1}\mathstrut +\mathstrut \) \(311\)
\(\nu^{8}\)\(=\)\(-\)\(924\) \(\beta_{10}\mathstrut +\mathstrut \) \(880\) \(\beta_{9}\mathstrut +\mathstrut \) \(128\) \(\beta_{8}\mathstrut -\mathstrut \) \(597\) \(\beta_{7}\mathstrut +\mathstrut \) \(1227\) \(\beta_{6}\mathstrut -\mathstrut \) \(1040\) \(\beta_{5}\mathstrut +\mathstrut \) \(717\) \(\beta_{4}\mathstrut -\mathstrut \) \(510\) \(\beta_{3}\mathstrut +\mathstrut \) \(736\) \(\beta_{2}\mathstrut +\mathstrut \) \(2045\) \(\beta_{1}\mathstrut +\mathstrut \) \(1190\)
\(\nu^{9}\)\(=\)\(-\)\(3493\) \(\beta_{10}\mathstrut +\mathstrut \) \(3330\) \(\beta_{9}\mathstrut +\mathstrut \) \(510\) \(\beta_{8}\mathstrut -\mathstrut \) \(2286\) \(\beta_{7}\mathstrut +\mathstrut \) \(4612\) \(\beta_{6}\mathstrut -\mathstrut \) \(3802\) \(\beta_{5}\mathstrut +\mathstrut \) \(2616\) \(\beta_{4}\mathstrut -\mathstrut \) \(2005\) \(\beta_{3}\mathstrut +\mathstrut \) \(2307\) \(\beta_{2}\mathstrut +\mathstrut \) \(7308\) \(\beta_{1}\mathstrut +\mathstrut \) \(3548\)
\(\nu^{10}\)\(=\)\(-\)\(12651\) \(\beta_{10}\mathstrut +\mathstrut \) \(11937\) \(\beta_{9}\mathstrut +\mathstrut \) \(2005\) \(\beta_{8}\mathstrut -\mathstrut \) \(8105\) \(\beta_{7}\mathstrut +\mathstrut \) \(16828\) \(\beta_{6}\mathstrut -\mathstrut \) \(13955\) \(\beta_{5}\mathstrut +\mathstrut \) \(9297\) \(\beta_{4}\mathstrut -\mathstrut \) \(7515\) \(\beta_{3}\mathstrut +\mathstrut \) \(8198\) \(\beta_{2}\mathstrut +\mathstrut \) \(24630\) \(\beta_{1}\mathstrut +\mathstrut \) \(12646\)

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
3.53326
2.67951
2.59661
1.86331
0.933335
0.278926
−0.224040
−0.820253
−1.73315
−1.98153
−2.12599
0 −2.53326 0 −3.09469 0 −0.871610 0 3.41739 0
1.2 0 −1.67951 0 1.05669 0 1.62834 0 −0.179232 0
1.3 0 −1.59661 0 −0.374330 0 3.78268 0 −0.450834 0
1.4 0 −0.863310 0 −1.89826 0 −2.78049 0 −2.25470 0
1.5 0 0.0666645 0 3.14719 0 −2.33769 0 −2.99556 0
1.6 0 0.721074 0 0.0376400 0 1.18778 0 −2.48005 0
1.7 0 1.22404 0 −3.27866 0 −0.601962 0 −1.50173 0
1.8 0 1.82025 0 1.95827 0 3.06441 0 0.313322 0
1.9 0 2.73315 0 3.52761 0 2.06502 0 4.47010 0
1.10 0 2.98153 0 0.681937 0 −4.55922 0 5.88949 0
1.11 0 3.12599 0 −3.76339 0 2.42274 0 6.77179 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

\(T_{3}^{11} - \cdots\)
\(T_{5}^{11} + \cdots\)