Properties

Label 8036.2.a.p
Level 8036
Weight 2
Character orbit 8036.a
Self dual Yes
Analytic conductor 64.168
Analytic rank 0
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{2} q^{5} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{2} q^{5} \) \( + ( 1 + \beta_{2} + \beta_{3} ) q^{9} \) \( + \beta_{6} q^{11} \) \( + ( 1 + \beta_{7} ) q^{13} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{15} \) \( + ( 1 - \beta_{4} + \beta_{9} ) q^{17} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{19} \) \( + ( -\beta_{4} + \beta_{5} ) q^{23} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} \) \( + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{27} \) \( + ( 1 + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} ) q^{29} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{31} \) \( + ( \beta_{3} + \beta_{6} - \beta_{9} ) q^{33} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{37} \) \( + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{39} \) \(+ q^{41}\) \( + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{43} \) \( + ( 5 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{45} \) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{47} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{51} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} \) \( + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} \) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{59} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{8} ) q^{61} \) \( + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{65} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{67} \) \( + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{69} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} \) \( + ( 4 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} \) \( + ( -9 + 6 \beta_{1} - 6 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{75} \) \( + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{79} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{81} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{83} \) \( + ( \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{85} \) \( + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{87} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{89} \) \( + ( 2 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{6} + 3 \beta_{7} - 3 \beta_{9} ) q^{93} \) \( + ( 1 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{95} \) \( + ( 6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} ) q^{97} \) \( + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 12q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 16q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 44q^{45} \) \(\mathstrut -\mathstrut 14q^{47} \) \(\mathstrut +\mathstrut 14q^{51} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut +\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 14q^{61} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut +\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 44q^{73} \) \(\mathstrut -\mathstrut 50q^{75} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut -\mathstrut 14q^{81} \) \(\mathstrut +\mathstrut 20q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut 20q^{87} \) \(\mathstrut +\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 4q^{95} \) \(\mathstrut +\mathstrut 46q^{97} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(18\) \(x^{8}\mathstrut +\mathstrut \) \(32\) \(x^{7}\mathstrut +\mathstrut \) \(110\) \(x^{6}\mathstrut -\mathstrut \) \(154\) \(x^{5}\mathstrut -\mathstrut \) \(282\) \(x^{4}\mathstrut +\mathstrut \) \(256\) \(x^{3}\mathstrut +\mathstrut \) \(253\) \(x^{2}\mathstrut -\mathstrut \) \(126\) \(x\mathstrut -\mathstrut \) \(49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{9} - 3 \nu^{8} + 36 \nu^{7} + 41 \nu^{6} - 213 \nu^{5} - 154 \nu^{4} + 445 \nu^{3} + 132 \nu^{2} - 275 \nu - 28 \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{9} + 3 \nu^{8} - 36 \nu^{7} - 41 \nu^{6} + 213 \nu^{5} + 154 \nu^{4} - 445 \nu^{3} - 111 \nu^{2} + 275 \nu - 56 \)\()/21\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{9} - \nu^{8} + 47 \nu^{7} + 16 \nu^{6} - 218 \nu^{5} - 84 \nu^{4} + 293 \nu^{3} + 72 \nu^{2} - 73 \nu - 7 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{9} + 9 \nu^{8} - 213 \nu^{7} - 151 \nu^{6} + 1080 \nu^{5} + 812 \nu^{4} - 1811 \nu^{3} - 1173 \nu^{2} + 895 \nu + 371 \)\()/21\)
\(\beta_{6}\)\(=\)\((\)\( -22 \nu^{9} - 12 \nu^{8} + 354 \nu^{7} + 199 \nu^{6} - 1734 \nu^{5} - 1043 \nu^{4} + 2711 \nu^{3} + 1221 \nu^{2} - 1240 \nu - 266 \)\()/21\)
\(\beta_{7}\)\(=\)\((\)\( -29 \nu^{9} - 12 \nu^{8} + 459 \nu^{7} + 206 \nu^{6} - 2196 \nu^{5} - 1162 \nu^{4} + 3334 \nu^{3} + 1452 \nu^{2} - 1415 \nu - 343 \)\()/21\)
\(\beta_{8}\)\(=\)\((\)\( 4 \nu^{9} + 3 \nu^{8} - 63 \nu^{7} - 49 \nu^{6} + 297 \nu^{5} + 251 \nu^{4} - 425 \nu^{3} - 309 \nu^{2} + 175 \nu + 74 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 12 \nu^{9} + 11 \nu^{8} - 195 \nu^{7} - 176 \nu^{6} + 970 \nu^{5} + 868 \nu^{4} - 1529 \nu^{3} - 1072 \nu^{2} + 691 \nu + 252 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(11\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(53\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(25\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(77\) \(\beta_{3}\mathstrut +\mathstrut \) \(61\) \(\beta_{2}\mathstrut -\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(194\)
\(\nu^{7}\)\(=\)\(87\) \(\beta_{9}\mathstrut -\mathstrut \) \(86\) \(\beta_{8}\mathstrut -\mathstrut \) \(101\) \(\beta_{7}\mathstrut +\mathstrut \) \(88\) \(\beta_{6}\mathstrut -\mathstrut \) \(102\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\) \(\beta_{4}\mathstrut -\mathstrut \) \(108\) \(\beta_{3}\mathstrut -\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(417\) \(\beta_{1}\mathstrut -\mathstrut \) \(50\)
\(\nu^{8}\)\(=\)\(-\)\(149\) \(\beta_{9}\mathstrut +\mathstrut \) \(151\) \(\beta_{8}\mathstrut +\mathstrut \) \(150\) \(\beta_{7}\mathstrut -\mathstrut \) \(33\) \(\beta_{6}\mathstrut +\mathstrut \) \(249\) \(\beta_{5}\mathstrut -\mathstrut \) \(129\) \(\beta_{4}\mathstrut +\mathstrut \) \(655\) \(\beta_{3}\mathstrut +\mathstrut \) \(474\) \(\beta_{2}\mathstrut -\mathstrut \) \(269\) \(\beta_{1}\mathstrut +\mathstrut \) \(1522\)
\(\nu^{9}\)\(=\)\(737\) \(\beta_{9}\mathstrut -\mathstrut \) \(722\) \(\beta_{8}\mathstrut -\mathstrut \) \(884\) \(\beta_{7}\mathstrut +\mathstrut \) \(750\) \(\beta_{6}\mathstrut -\mathstrut \) \(902\) \(\beta_{5}\mathstrut +\mathstrut \) \(277\) \(\beta_{4}\mathstrut -\mathstrut \) \(1026\) \(\beta_{3}\mathstrut -\mathstrut \) \(420\) \(\beta_{2}\mathstrut +\mathstrut \) \(3352\) \(\beta_{1}\mathstrut -\mathstrut \) \(745\)

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
−2.96176
−1.91097
−1.73395
−1.05099
−0.284914
0.729469
1.20067
2.58799
2.70664
2.71781
0 −2.96176 0 4.34462 0 0 0 5.77201 0
1.2 0 −1.91097 0 1.95825 0 0 0 0.651806 0
1.3 0 −1.73395 0 −3.41343 0 0 0 0.00658743 0
1.4 0 −1.05099 0 −1.02904 0 0 0 −1.89542 0
1.5 0 −0.284914 0 2.38934 0 0 0 −2.91882 0
1.6 0 0.729469 0 −3.02160 0 0 0 −2.46788 0
1.7 0 1.20067 0 −0.960855 0 0 0 −1.55839 0
1.8 0 2.58799 0 3.78481 0 0 0 3.69770 0
1.9 0 2.70664 0 0.469317 0 0 0 4.32591 0
1.10 0 2.71781 0 −0.521406 0 0 0 4.38650 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
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)