Properties

Label 4020.2.a.e
Level 4020
Weight 2
Character orbit 4020.a
Self dual yes
Analytic conductor 32.100
Analytic rank 1
Dimension 5
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1257629.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 8 \nu^{2} + 11 \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 9 \nu^{2} - 6 \nu + 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 8 \beta_{2} + 21\)

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
1.14431
−2.49431
−1.67454
1.48010
2.54445
0 −1.00000 0 1.00000 0 −4.78538 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.20813 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −0.300647 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 0.397192 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 2.89696 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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.a.e 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.a.e 5 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{5} + 3 T_{7}^{4} - 12 T_{7}^{3} - 16 T_{7}^{2} + 3 T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).