Properties

Label 4016.2.a.g
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 0
Dimension 7
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{5} q^{5} \) \( + ( 1 + \beta_{6} ) q^{7} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{5} q^{5} \) \( + ( 1 + \beta_{6} ) q^{7} \) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{9} \) \( + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{11} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{15} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} \) \( + ( 2 + \beta_{2} + \beta_{4} ) q^{19} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{23} \) \( + ( -1 - \beta_{3} ) q^{25} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{27} \) \( + ( - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{29} \) \( + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{31} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} \) \( + ( 2 - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{35} \) \( + ( - \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{37} \) \( + ( 2 + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{39} \) \( + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{41} \) \( + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{43} \) \( + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{45} \) \( + ( 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{47} \) \( + ( - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{49} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{51} \) \( + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} \) \( + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{55} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{57} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{59} \) \( + ( 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{61} \) \( + ( -1 + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{63} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{65} \) \( + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{67} \) \( + ( 2 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{69} \) \( + ( 3 - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{71} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{73} \) \( + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{75} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{77} \) \( + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{79} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{6} ) q^{81} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{83} \) \( + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{85} \) \( + ( 1 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{89} \) \( + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{91} \) \( + ( 3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{93} \) \( + ( 1 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{95} \) \( + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{97} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(7q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 15q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut -\mathstrut 9q^{25} \) \(\mathstrut +\mathstrut 9q^{27} \) \(\mathstrut +\mathstrut 21q^{31} \) \(\mathstrut +\mathstrut 7q^{35} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut 23q^{39} \) \(\mathstrut -\mathstrut 10q^{41} \) \(\mathstrut +\mathstrut 23q^{43} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut +\mathstrut 10q^{47} \) \(\mathstrut -\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 20q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut +\mathstrut 23q^{55} \) \(\mathstrut -\mathstrut 6q^{57} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 3q^{61} \) \(\mathstrut +\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 28q^{67} \) \(\mathstrut +\mathstrut 18q^{69} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 7q^{73} \) \(\mathstrut -\mathstrut 11q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 30q^{79} \) \(\mathstrut -\mathstrut 5q^{81} \) \(\mathstrut -\mathstrut 13q^{83} \) \(\mathstrut +\mathstrut q^{85} \) \(\mathstrut +\mathstrut 7q^{87} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut +\mathstrut 36q^{93} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut 10q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(6\) \(x^{5}\mathstrut +\mathstrut \) \(18\) \(x^{4}\mathstrut +\mathstrut \) \(8\) \(x^{3}\mathstrut -\mathstrut \) \(17\) \(x^{2}\mathstrut -\mathstrut \) \(9\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 13 \nu^{3} + 15 \nu^{2} - 17 \nu - 5 \)\()/3\)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 6 \nu^{4} - 18 \nu^{3} - 7 \nu^{2} + 17 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 18 \nu^{3} + 8 \nu^{2} - 18 \nu - 7 \)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{6} - 16 \nu^{5} - 25 \nu^{4} + 92 \nu^{3} + 9 \nu^{2} - 73 \nu - 16 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{6} - 23 \nu^{5} - 35 \nu^{4} + 136 \nu^{3} + 15 \nu^{2} - 125 \nu - 29 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{5}\)\(=\)\(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(38\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(41\) \(\beta_{4}\mathstrut +\mathstrut \) \(76\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{2}\mathstrut +\mathstrut \) \(64\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)

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
−2.18229
−0.844838
−0.358013
−0.164919
1.40474
2.29157
2.85375
0 −2.18229 0 −1.66714 0 1.81734 0 1.76240 0
1.2 0 −0.844838 0 −2.47034 0 0.972158 0 −2.28625 0
1.3 0 −0.358013 0 2.25358 0 4.66949 0 −2.87183 0
1.4 0 −0.164919 0 −1.38175 0 −1.87004 0 −2.97280 0
1.5 0 1.40474 0 2.59588 0 −1.10443 0 −1.02671 0
1.6 0 2.29157 0 −1.78468 0 −0.671050 0 2.25130 0
1.7 0 2.85375 0 0.454452 0 2.18653 0 5.14389 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(T_{3}^{7} \) \(\mathstrut -\mathstrut 3 T_{3}^{6} \) \(\mathstrut -\mathstrut 6 T_{3}^{5} \) \(\mathstrut +\mathstrut 18 T_{3}^{4} \) \(\mathstrut +\mathstrut 8 T_{3}^{3} \) \(\mathstrut -\mathstrut 17 T_{3}^{2} \) \(\mathstrut -\mathstrut 9 T_{3} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).