Properties

Label 8016.2.a.r
Level 8016
Weight 2
Character orbit 8016.a
Self dual Yes
Analytic conductor 64.008
Analytic rank 1
Dimension 5
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.11256624.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + \beta_{2} q^{5} \) \( + ( -2 + \beta_{4} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + \beta_{2} q^{5} \) \( + ( -2 + \beta_{4} ) q^{7} \) \(+ q^{9}\) \( -\beta_{3} q^{11} \) \( + ( 1 + \beta_{1} ) q^{13} \) \( -\beta_{2} q^{15} \) \( + ( 1 - \beta_{1} ) q^{17} \) \( + ( -2 + \beta_{3} ) q^{19} \) \( + ( 2 - \beta_{4} ) q^{21} \) \( + \beta_{3} q^{23} \) \( + ( 3 - 2 \beta_{1} + \beta_{4} ) q^{25} \) \(- q^{27}\) \( + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{29} \) \( + ( -2 - \beta_{3} - \beta_{4} ) q^{31} \) \( + \beta_{3} q^{33} \) \( + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{35} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} \) \( + ( -1 - \beta_{1} ) q^{39} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} \) \( + ( -4 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{43} \) \( + \beta_{2} q^{45} \) \( + ( 2 - 2 \beta_{1} - \beta_{4} ) q^{47} \) \( + ( 3 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{49} \) \( + ( -1 + \beta_{1} ) q^{51} \) \( + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} \) \( + ( 2 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{55} \) \( + ( 2 - \beta_{3} ) q^{57} \) \( + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} \) \( + ( 4 - 2 \beta_{1} - 2 \beta_{4} ) q^{61} \) \( + ( -2 + \beta_{4} ) q^{63} \) \( + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{65} \) \( + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{67} \) \( -\beta_{3} q^{69} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{71} \) \( + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{73} \) \( + ( -3 + 2 \beta_{1} - \beta_{4} ) q^{75} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{77} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} \) \(+ q^{81}\) \( + ( -1 + \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{83} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{85} \) \( + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{87} \) \( + ( -2 + 2 \beta_{1} + 3 \beta_{4} ) q^{89} \) \( + ( -2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{91} \) \( + ( 2 + \beta_{3} + \beta_{4} ) q^{93} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{95} \) \( + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{97} \) \( -\beta_{3} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 12q^{19} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut +\mathstrut 14q^{25} \) \(\mathstrut -\mathstrut 5q^{27} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut -\mathstrut 18q^{43} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut +\mathstrut 7q^{47} \) \(\mathstrut +\mathstrut 16q^{49} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 3q^{53} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 12q^{57} \) \(\mathstrut -\mathstrut 13q^{59} \) \(\mathstrut +\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 9q^{63} \) \(\mathstrut -\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut 9q^{67} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut -\mathstrut 14q^{75} \) \(\mathstrut +\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut 6q^{79} \) \(\mathstrut +\mathstrut 5q^{81} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut +\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut 6q^{87} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 12q^{91} \) \(\mathstrut +\mathstrut 9q^{93} \) \(\mathstrut -\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(x^{4}\mathstrut -\mathstrut \) \(16\) \(x^{3}\mathstrut +\mathstrut \) \(20\) \(x^{2}\mathstrut +\mathstrut \) \(31\) \(x\mathstrut -\mathstrut \) \(11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 7 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 11 \nu - 3 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 14 \nu^{2} + 8 \nu + 17 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(28\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(81\)

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
0.308735
−1.14410
2.50259
3.20970
−3.87693
0 −1.00000 0 −3.45234 0 2.53613 0 1.00000 0
1.2 0 −1.00000 0 −2.84552 0 −4.19124 0 1.00000 0
1.3 0 −1.00000 0 −0.368511 0 −4.85901 0 1.00000 0
1.4 0 −1.00000 0 1.65109 0 −0.854515 0 1.00000 0
1.5 0 −1.00000 0 4.01528 0 −1.63136 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{5} \) \(\mathstrut +\mathstrut T_{5}^{4} \) \(\mathstrut -\mathstrut 19 T_{5}^{3} \) \(\mathstrut -\mathstrut 21 T_{5}^{2} \) \(\mathstrut +\mathstrut 60 T_{5} \) \(\mathstrut +\mathstrut 24 \)
\(T_{7}^{5} \) \(\mathstrut +\mathstrut 9 T_{7}^{4} \) \(\mathstrut +\mathstrut 15 T_{7}^{3} \) \(\mathstrut -\mathstrut 49 T_{7}^{2} \) \(\mathstrut -\mathstrut 132 T_{7} \) \(\mathstrut -\mathstrut 72 \)
\(T_{11}^{5} \) \(\mathstrut -\mathstrut 2 T_{11}^{4} \) \(\mathstrut -\mathstrut 28 T_{11}^{3} \) \(\mathstrut +\mathstrut 60 T_{11}^{2} \) \(\mathstrut +\mathstrut 144 T_{11} \) \(\mathstrut -\mathstrut 288 \)
\(T_{13}^{5} \) \(\mathstrut -\mathstrut 6 T_{13}^{4} \) \(\mathstrut -\mathstrut 2 T_{13}^{3} \) \(\mathstrut +\mathstrut 52 T_{13}^{2} \) \(\mathstrut -\mathstrut 48 T_{13} \) \(\mathstrut -\mathstrut 8 \)