Properties

Label 4020.2.a.f
Level 4020
Weight 2
Character orbit 4020.a
Self dual Yes
Analytic conductor 32.100
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut -\mathstrut \) \(28\) \(x^{4}\mathstrut -\mathstrut \) \(12\) \(x^{3}\mathstrut +\mathstrut \) \(209\) \(x^{2}\mathstrut +\mathstrut \) \(360\) \(x\mathstrut +\mathstrut \) \(144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{5} - 17 \nu^{4} - 104 \nu^{3} + 180 \nu^{2} + 685 \nu + 348 \)\()/24\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{5} + 19 \nu^{4} + 184 \nu^{3} - 204 \nu^{2} - 1415 \nu - 924 \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{5} - 19 \nu^{4} - 160 \nu^{3} + 204 \nu^{2} + 1055 \nu + 492 \)\()/24\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{5} + 19 \nu^{4} + 160 \nu^{3} - 180 \nu^{2} - 1103 \nu - 708 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{3}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(18\)
\(\nu^{4}\)\(=\)\(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(45\) \(\beta_{1}\mathstrut +\mathstrut \) \(143\)
\(\nu^{5}\)\(=\)\(-\)\(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(46\) \(\beta_{4}\mathstrut +\mathstrut \) \(62\) \(\beta_{3}\mathstrut -\mathstrut \) \(19\) \(\beta_{2}\mathstrut +\mathstrut \) \(256\) \(\beta_{1}\mathstrut +\mathstrut \) \(467\)

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
−3.20623
−2.21014
−1.71821
−0.620094
4.08027
4.67440
0 −1.00000 0 1.00000 0 −3.20623 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.21014 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −1.71821 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 −0.620094 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 4.08027 0 1.00000 0
1.6 0 −1.00000 0 1.00000 0 4.67440 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(67\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{6} \) \(\mathstrut -\mathstrut T_{7}^{5} \) \(\mathstrut -\mathstrut 28 T_{7}^{4} \) \(\mathstrut -\mathstrut 12 T_{7}^{3} \) \(\mathstrut +\mathstrut 209 T_{7}^{2} \) \(\mathstrut +\mathstrut 360 T_{7} \) \(\mathstrut +\mathstrut 144 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).