Properties

Label 8016.2.a.bb
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + ( 1 - \beta_{1} ) q^{5} \) \( + \beta_{6} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( 1 - \beta_{1} ) q^{5} \) \( + \beta_{6} q^{7} \) \(+ q^{9}\) \( + ( -1 + \beta_{5} ) q^{11} \) \( + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} \) \( + ( -1 + \beta_{1} ) q^{15} \) \( + ( 1 - \beta_{2} + \beta_{3} + \beta_{8} ) q^{17} \) \( + ( \beta_{2} + \beta_{7} ) q^{19} \) \( -\beta_{6} q^{21} \) \( + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{23} \) \( + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} \) \(- q^{27}\) \( + ( \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{29} \) \( + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{31} \) \( + ( 1 - \beta_{5} ) q^{33} \) \( + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{35} \) \( + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{37} \) \( + ( -1 + \beta_{4} - \beta_{6} - \beta_{7} ) q^{39} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} \) \( + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{43} \) \( + ( 1 - \beta_{1} ) q^{45} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{47} \) \( + ( 2 - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{49} \) \( + ( -1 + \beta_{2} - \beta_{3} - \beta_{8} ) q^{51} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{53} \) \( + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{55} \) \( + ( -\beta_{2} - \beta_{7} ) q^{57} \) \( + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{59} \) \( + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} \) \( + \beta_{6} q^{63} \) \( + ( 2 - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{65} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{67} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} ) q^{69} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{71} \) \( + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} \) \( + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{75} \) \( + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{77} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{79} \) \(+ q^{81}\) \( + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{83} \) \( + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} \) \( + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{87} \) \( + ( 4 + \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{89} \) \( + ( 4 - \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{91} \) \( + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{93} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{95} \) \( + ( 4 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{97} \) \( + ( -1 + \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(9q \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 7q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 9q^{15} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 19q^{23} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut -\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 13q^{29} \) \(\mathstrut -\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 7q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 15q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 18q^{41} \) \(\mathstrut +\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 9q^{45} \) \(\mathstrut -\mathstrut 25q^{47} \) \(\mathstrut +\mathstrut 19q^{49} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut 3q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 3q^{59} \) \(\mathstrut +\mathstrut 14q^{61} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 14q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 19q^{69} \) \(\mathstrut -\mathstrut 17q^{71} \) \(\mathstrut -\mathstrut 20q^{73} \) \(\mathstrut -\mathstrut 22q^{75} \) \(\mathstrut +\mathstrut 14q^{77} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut +\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut -\mathstrut 13q^{87} \) \(\mathstrut +\mathstrut 36q^{89} \) \(\mathstrut +\mathstrut 41q^{91} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 31q^{97} \) \(\mathstrut -\mathstrut 7q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(29\) \(x^{7}\mathstrut -\mathstrut \) \(7\) \(x^{6}\mathstrut +\mathstrut \) \(266\) \(x^{5}\mathstrut +\mathstrut \) \(69\) \(x^{4}\mathstrut -\mathstrut \) \(901\) \(x^{3}\mathstrut -\mathstrut \) \(199\) \(x^{2}\mathstrut +\mathstrut \) \(875\) \(x\mathstrut +\mathstrut \) \(391\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149 \)\()/51607\)
\(\beta_{3}\)\(=\)\((\)\( 1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465 \)\()/51607\)
\(\beta_{4}\)\(=\)\((\)\( 2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049 \)\()/51607\)
\(\beta_{5}\)\(=\)\((\)\( 2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431 \)\()/51607\)
\(\beta_{6}\)\(=\)\((\)\( 3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617 \)\()/51607\)
\(\beta_{7}\)\(=\)\((\)\( 3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141 \)\()/51607\)
\(\beta_{8}\)\(=\)\((\)\( -3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783 \)\()/51607\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(79\)
\(\nu^{5}\)\(=\)\(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(38\) \(\beta_{7}\mathstrut -\mathstrut \) \(23\) \(\beta_{6}\mathstrut +\mathstrut \) \(36\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(91\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(91\)
\(\nu^{6}\)\(=\)\(115\) \(\beta_{8}\mathstrut +\mathstrut \) \(102\) \(\beta_{7}\mathstrut -\mathstrut \) \(21\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\) \(\beta_{5}\mathstrut +\mathstrut \) \(207\) \(\beta_{4}\mathstrut +\mathstrut \) \(46\) \(\beta_{3}\mathstrut +\mathstrut \) \(486\) \(\beta_{2}\mathstrut +\mathstrut \) \(163\) \(\beta_{1}\mathstrut +\mathstrut \) \(1059\)
\(\nu^{7}\)\(=\)\(431\) \(\beta_{8}\mathstrut +\mathstrut \) \(648\) \(\beta_{7}\mathstrut -\mathstrut \) \(397\) \(\beta_{6}\mathstrut +\mathstrut \) \(555\) \(\beta_{5}\mathstrut +\mathstrut \) \(464\) \(\beta_{4}\mathstrut -\mathstrut \) \(97\) \(\beta_{3}\mathstrut +\mathstrut \) \(1703\) \(\beta_{2}\mathstrut +\mathstrut \) \(1714\) \(\beta_{1}\mathstrut +\mathstrut \) \(2087\)
\(\nu^{8}\)\(=\)\(2125\) \(\beta_{8}\mathstrut +\mathstrut \) \(2069\) \(\beta_{7}\mathstrut -\mathstrut \) \(679\) \(\beta_{6}\mathstrut +\mathstrut \) \(965\) \(\beta_{5}\mathstrut +\mathstrut \) \(3251\) \(\beta_{4}\mathstrut +\mathstrut \) \(758\) \(\beta_{3}\mathstrut +\mathstrut \) \(7981\) \(\beta_{2}\mathstrut +\mathstrut \) \(3699\) \(\beta_{1}\mathstrut +\mathstrut \) \(15652\)

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
4.16840
2.69402
1.79204
1.61974
−0.529979
−0.907808
−2.42964
−3.19525
−3.21153
0 −1.00000 0 −3.16840 0 0.230890 0 1.00000 0
1.2 0 −1.00000 0 −1.69402 0 4.12928 0 1.00000 0
1.3 0 −1.00000 0 −0.792043 0 −3.80237 0 1.00000 0
1.4 0 −1.00000 0 −0.619742 0 −1.05844 0 1.00000 0
1.5 0 −1.00000 0 1.52998 0 1.05249 0 1.00000 0
1.6 0 −1.00000 0 1.90781 0 −2.81337 0 1.00000 0
1.7 0 −1.00000 0 3.42964 0 −3.44225 0 1.00000 0
1.8 0 −1.00000 0 4.19525 0 −1.43344 0 1.00000 0
1.9 0 −1.00000 0 4.21153 0 5.13720 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)
\(T_{13}^{9} - \cdots\)