Properties

Label 4016.2.a.f
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.60853001.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{4} q^{3} \) \( + ( -1 + \beta_{1} - \beta_{3} ) q^{5} \) \( + ( -1 + \beta_{5} ) q^{7} \) \( + ( 3 - \beta_{1} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{4} q^{3} \) \( + ( -1 + \beta_{1} - \beta_{3} ) q^{5} \) \( + ( -1 + \beta_{5} ) q^{7} \) \( + ( 3 - \beta_{1} + \beta_{5} ) q^{9} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} \) \( + ( -1 + \beta_{1} + \beta_{3} ) q^{13} \) \( + ( -1 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{15} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{17} \) \( + ( - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} \) \( + ( -1 - \beta_{1} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{21} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} \) \( + ( 1 - 2 \beta_{1} + \beta_{4} ) q^{25} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{27} \) \( + ( 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{29} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{31} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{33} \) \( + ( - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{35} \) \( + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{37} \) \( + ( 1 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{39} \) \( + ( 1 - 3 \beta_{2} + \beta_{3} - \beta_{5} ) q^{41} \) \( + ( -1 + 3 \beta_{1} - 2 \beta_{4} ) q^{43} \) \( + ( -7 + 4 \beta_{1} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{45} \) \( + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{49} \) \( + ( 5 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{51} \) \( + ( 1 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{53} \) \( + ( 6 - 2 \beta_{1} ) q^{55} \) \( + ( -4 + 2 \beta_{5} ) q^{57} \) \( + ( -1 + \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} ) q^{59} \) \( + ( -11 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{61} \) \( + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{63} \) \( + ( 2 - 2 \beta_{1} + 4 \beta_{3} - \beta_{4} ) q^{65} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{67} \) \( + ( -6 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{69} \) \( + ( -2 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} ) q^{71} \) \( + ( 1 - \beta_{1} + 4 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{73} \) \( + ( 6 - \beta_{1} - 4 \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{75} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{77} \) \( + ( -5 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{79} \) \( + ( -2 \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{81} \) \( + ( -4 + 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} \) \( + ( -6 + 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{85} \) \( + ( -2 + 8 \beta_{1} - 6 \beta_{3} + 4 \beta_{5} ) q^{87} \) \( + ( 5 - \beta_{1} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{89} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{91} \) \( + ( 2 + 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{93} \) \( + ( -2 - 2 \beta_{2} + 4 \beta_{3} ) q^{95} \) \( + ( 2 - 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{97} \) \( + ( -5 + 5 \beta_{1} - \beta_{2} - 5 \beta_{3} + 4 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(6q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut 5q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut -\mathstrut 14q^{21} \) \(\mathstrut -\mathstrut 18q^{23} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 16q^{27} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut -\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 16q^{33} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 13q^{37} \) \(\mathstrut +\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 5q^{43} \) \(\mathstrut -\mathstrut 23q^{45} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut +\mathstrut 16q^{51} \) \(\mathstrut -\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 30q^{55} \) \(\mathstrut -\mathstrut 24q^{57} \) \(\mathstrut +\mathstrut 5q^{59} \) \(\mathstrut -\mathstrut 61q^{61} \) \(\mathstrut +\mathstrut 27q^{63} \) \(\mathstrut -\mathstrut q^{65} \) \(\mathstrut +\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 21q^{69} \) \(\mathstrut -\mathstrut 22q^{71} \) \(\mathstrut +\mathstrut 6q^{73} \) \(\mathstrut +\mathstrut 30q^{75} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 28q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 14q^{83} \) \(\mathstrut -\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 24q^{87} \) \(\mathstrut +\mathstrut 18q^{89} \) \(\mathstrut +\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 27q^{93} \) \(\mathstrut -\mathstrut 20q^{95} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(3\) \(x^{5}\mathstrut -\mathstrut \) \(8\) \(x^{4}\mathstrut +\mathstrut \) \(15\) \(x^{3}\mathstrut +\mathstrut \) \(20\) \(x^{2}\mathstrut -\mathstrut \) \(12\) \(x\mathstrut -\mathstrut \) \(5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 4 \nu^{3} - 15 \nu^{2} - 9 \nu + 1 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} - 4 \nu^{3} + 19 \nu^{2} + 5 \nu - 17 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 17 \nu^{2} - 27 \nu - 3 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 7 \nu^{3} + 12 \nu^{2} + 15 \nu - 4 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{4}\)\(=\)\(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(19\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{5}\)\(=\)\(24\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(73\) \(\beta_{2}\mathstrut +\mathstrut \) \(76\) \(\beta_{1}\mathstrut +\mathstrut \) \(97\)

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
0.730357
−1.76567
2.15902
−0.303212
3.75626
−1.57676
0 −3.20334 0 1.15583 0 3.99175 0 7.26139 0
1.2 0 −2.70836 0 −2.61208 0 −1.43044 0 4.33524 0
1.3 0 −1.28113 0 0.633107 0 −3.19968 0 −1.35870 0
1.4 0 1.63228 0 2.87032 0 −4.63886 0 −0.335646 0
1.5 0 1.68934 0 0.420498 0 −0.389856 0 −0.146119 0
1.6 0 2.87121 0 −3.46767 0 −0.332919 0 5.24384 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(T_{3}^{6} \) \(\mathstrut +\mathstrut T_{3}^{5} \) \(\mathstrut -\mathstrut 16 T_{3}^{4} \) \(\mathstrut -\mathstrut 9 T_{3}^{3} \) \(\mathstrut +\mathstrut 74 T_{3}^{2} \) \(\mathstrut +\mathstrut 8 T_{3} \) \(\mathstrut -\mathstrut 88 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).