Properties

Label 4016.2.a.h
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{1} q^{3} \) \( + ( -1 - \beta_{5} ) q^{5} \) \( + ( - \beta_{6} + \beta_{7} ) q^{7} \) \( + \beta_{2} q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + ( -1 - \beta_{5} ) q^{5} \) \( + ( - \beta_{6} + \beta_{7} ) q^{7} \) \( + \beta_{2} q^{9} \) \( + ( - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} \) \( + ( - \beta_{4} + \beta_{5} ) q^{13} \) \( + ( - \beta_{1} - \beta_{7} ) q^{15} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} ) q^{17} \) \( + ( - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{19} \) \( + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{21} \) \( + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} ) q^{23} \) \( + ( -1 - \beta_{2} + \beta_{5} + \beta_{8} ) q^{25} \) \( + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} ) q^{27} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{29} \) \( + ( -1 - \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{31} \) \( + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{33} \) \( + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{35} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{37} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{39} \) \( + ( -1 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{41} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{43} \) \( + ( -1 - \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{45} \) \( + ( \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{7} ) q^{47} \) \( + ( -2 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{49} \) \( + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{51} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{53} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{55} \) \( + ( -5 - \beta_{1} + 3 \beta_{3} - \beta_{5} - 3 \beta_{7} - \beta_{8} ) q^{57} \) \( + ( 2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{59} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{63} \) \( + ( -3 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{65} \) \( + ( 1 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{67} \) \( + ( -2 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{69} \) \( + ( 3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} + 3 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{71} \) \( + ( -5 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} - \beta_{8} ) q^{73} \) \( + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{7} - \beta_{8} ) q^{75} \) \( + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} ) q^{77} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} \) \( + ( -4 + \beta_{1} - 3 \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{81} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{83} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{8} ) q^{85} \) \( + ( -2 + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{87} \) \( + ( -5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{89} \) \( + ( -2 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{91} \) \( + ( -7 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{7} - 2 \beta_{8} ) q^{93} \) \( + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{95} \) \( + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{97} \) \( + ( - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(9q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut -\mathstrut 12q^{25} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 9q^{29} \) \(\mathstrut -\mathstrut 3q^{31} \) \(\mathstrut -\mathstrut 14q^{33} \) \(\mathstrut +\mathstrut 8q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut -\mathstrut 23q^{41} \) \(\mathstrut -\mathstrut 10q^{45} \) \(\mathstrut +\mathstrut 11q^{47} \) \(\mathstrut -\mathstrut 21q^{49} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 21q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut -\mathstrut 21q^{57} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 11q^{61} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 29q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 14q^{69} \) \(\mathstrut +\mathstrut 19q^{71} \) \(\mathstrut -\mathstrut 31q^{73} \) \(\mathstrut -\mathstrut 16q^{75} \) \(\mathstrut -\mathstrut 26q^{77} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 27q^{81} \) \(\mathstrut +\mathstrut 22q^{83} \) \(\mathstrut +\mathstrut 4q^{85} \) \(\mathstrut +\mathstrut 6q^{87} \) \(\mathstrut -\mathstrut 36q^{89} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 32q^{93} \) \(\mathstrut +\mathstrut 3q^{95} \) \(\mathstrut -\mathstrut 38q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(11\) \(x^{7}\mathstrut +\mathstrut \) \(7\) \(x^{6}\mathstrut +\mathstrut \) \(40\) \(x^{5}\mathstrut -\mathstrut \) \(11\) \(x^{4}\mathstrut -\mathstrut \) \(53\) \(x^{3}\mathstrut -\mathstrut \) \(2\) \(x^{2}\mathstrut +\mathstrut \) \(13\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - 4 \nu^{7} - 3 \nu^{6} + 28 \nu^{5} - 16 \nu^{4} - 47 \nu^{3} + 36 \nu^{2} + 14 \nu + 3 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{8} + 4 \nu^{7} + 7 \nu^{6} - 32 \nu^{5} - 16 \nu^{4} + 67 \nu^{3} + 24 \nu^{2} - 30 \nu - 11 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{8} - 8 \nu^{7} - 21 \nu^{6} + 60 \nu^{5} + 32 \nu^{4} - 117 \nu^{3} + 50 \nu - 7 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{8} + 4 \nu^{7} + 29 \nu^{6} - 28 \nu^{5} - 88 \nu^{4} + 49 \nu^{3} + 92 \nu^{2} - 14 \nu - 13 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{8} + 12 \nu^{7} + 39 \nu^{6} - 88 \nu^{5} - 84 \nu^{4} + 159 \nu^{3} + 56 \nu^{2} - 46 \nu - 3 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{8} - 8 \nu^{7} - 21 \nu^{6} + 58 \nu^{5} + 34 \nu^{4} - 101 \nu^{3} - 10 \nu^{2} + 22 \nu - 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{6}\)\(=\)\(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(21\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)
\(\nu^{7}\)\(=\)\(55\) \(\beta_{8}\mathstrut +\mathstrut \) \(77\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(51\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(101\) \(\beta_{1}\mathstrut +\mathstrut \) \(132\)
\(\nu^{8}\)\(=\)\(92\) \(\beta_{8}\mathstrut +\mathstrut \) \(170\) \(\beta_{7}\mathstrut -\mathstrut \) \(67\) \(\beta_{6}\mathstrut +\mathstrut \) \(26\) \(\beta_{5}\mathstrut -\mathstrut \) \(52\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\) \(\beta_{3}\mathstrut +\mathstrut \) \(194\) \(\beta_{2}\mathstrut +\mathstrut \) \(95\) \(\beta_{1}\mathstrut +\mathstrut \) \(462\)

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
−2.17295
−1.49215
−1.42772
−0.600027
−0.0778757
0.540607
1.71581
1.92428
2.59002
0 −2.17295 0 −1.48785 0 1.36809 0 1.72170 0
1.2 0 −1.49215 0 0.0991275 0 1.31645 0 −0.773492 0
1.3 0 −1.42772 0 −3.45271 0 −2.96003 0 −0.961621 0
1.4 0 −0.600027 0 2.23756 0 −0.532333 0 −2.63997 0
1.5 0 −0.0778757 0 1.70938 0 3.05552 0 −2.99394 0
1.6 0 0.540607 0 −2.60958 0 −0.631745 0 −2.70774 0
1.7 0 1.71581 0 −0.169057 0 0.581118 0 −0.0559961 0
1.8 0 1.92428 0 0.603023 0 −4.00229 0 0.702869 0
1.9 0 2.59002 0 −1.92989 0 1.80522 0 3.70818 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
\(2\) \(1\)
\(251\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).