Properties

Label 4020.2.a.g
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: 6.6.195727752.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(15\) \(x^{4}\mathstrut -\mathstrut \) \(25\) \(x^{3}\mathstrut -\mathstrut \) \(3\) \(x^{2}\mathstrut +\mathstrut \) \(9\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 14 \nu^{3} - 11 \nu^{2} + 8 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{5} + \nu^{4} + 30 \nu^{3} + 34 \nu^{2} - 16 \nu - 12 \)
\(\beta_{4}\)\(=\)\( -3 \nu^{5} + 2 \nu^{4} + 43 \nu^{3} + 48 \nu^{2} - 17 \nu - 16 \)
\(\beta_{5}\)\(=\)\( 4 \nu^{5} - 2 \nu^{4} - 59 \nu^{3} - 70 \nu^{2} + 22 \nu + 21 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{4}\)\(=\)\(18\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(68\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{5}\)\(=\)\(71\) \(\beta_{5}\mathstrut +\mathstrut \) \(55\) \(\beta_{4}\mathstrut +\mathstrut \) \(86\) \(\beta_{3}\mathstrut +\mathstrut \) \(54\) \(\beta_{2}\mathstrut +\mathstrut \) \(317\) \(\beta_{1}\mathstrut +\mathstrut \) \(313\)

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
−1.75666
−0.457832
0.600664
−0.587102
−2.33297
4.53390
0 1.00000 0 −1.00000 0 −3.63834 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −2.59172 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 1.05225 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 1.68378 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 3.15496 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 3.33907 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 \(T_{7}^{6} \) \(\mathstrut -\mathstrut 3 T_{7}^{5} \) \(\mathstrut -\mathstrut 18 T_{7}^{4} \) \(\mathstrut +\mathstrut 60 T_{7}^{3} \) \(\mathstrut +\mathstrut 51 T_{7}^{2} \) \(\mathstrut -\mathstrut 264 T_{7} \) \(\mathstrut +\mathstrut 176 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).