Properties

Label 4020.2.a.h
Level 4020
Weight 2
Character orbit 4020.a
Self dual Yes
Analytic conductor 32.100
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(28\) \(x^{5}\mathstrut +\mathstrut \) \(90\) \(x^{4}\mathstrut +\mathstrut \) \(143\) \(x^{3}\mathstrut -\mathstrut \) \(418\) \(x^{2}\mathstrut -\mathstrut \) \(256\) \(x\mathstrut +\mathstrut \) \(160\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 24 \nu^{4} - 10 \nu^{3} + 99 \nu^{2} + 86 \nu + 60 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + \nu^{5} + 26 \nu^{4} - 34 \nu^{3} - 115 \nu^{2} + 64 \nu + 60 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 28 \nu^{4} - 14 \nu^{3} + 187 \nu^{2} + 78 \nu - 180 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 27 \nu^{4} - 10 \nu^{3} - 141 \nu^{2} - 7 \nu + 30 \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{6} + 3 \nu^{5} - 188 \nu^{4} - 18 \nu^{3} + 1017 \nu^{2} + 610 \nu - 360 \)\()/40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(25\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(23\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(144\)
\(\nu^{5}\)\(=\)\(-\)\(46\) \(\beta_{6}\mathstrut -\mathstrut \) \(83\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut +\mathstrut \) \(47\) \(\beta_{3}\mathstrut +\mathstrut \) \(28\) \(\beta_{2}\mathstrut +\mathstrut \) \(318\) \(\beta_{1}\mathstrut -\mathstrut \) \(264\)
\(\nu^{6}\)\(=\)\(74\) \(\beta_{6}\mathstrut +\mathstrut \) \(554\) \(\beta_{5}\mathstrut +\mathstrut \) \(281\) \(\beta_{4}\mathstrut -\mathstrut \) \(490\) \(\beta_{3}\mathstrut +\mathstrut \) \(98\) \(\beta_{2}\mathstrut -\mathstrut \) \(690\) \(\beta_{1}\mathstrut +\mathstrut \) \(2709\)

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
−4.78907
−2.11010
−0.941196
0.403734
3.15736
3.41543
3.86385
0 −1.00000 0 −1.00000 0 −4.78907 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.11010 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −0.941196 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 0.403734 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 3.15736 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 3.41543 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 3.86385 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(67\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{7} \) \(\mathstrut -\mathstrut 3 T_{7}^{6} \) \(\mathstrut -\mathstrut 28 T_{7}^{5} \) \(\mathstrut +\mathstrut 90 T_{7}^{4} \) \(\mathstrut +\mathstrut 143 T_{7}^{3} \) \(\mathstrut -\mathstrut 418 T_{7}^{2} \) \(\mathstrut -\mathstrut 256 T_{7} \) \(\mathstrut +\mathstrut 160 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).