Properties

Label 4020.2.q.i
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.q (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} - q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{1} ) q^{13} - q^{15} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( -\beta_{1} + \beta_{3} ) q^{21} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + q^{25} + q^{27} + ( -\beta_{1} - \beta_{3} ) q^{29} + ( -\beta_{1} - 2 \beta_{3} ) q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( \beta_{1} - \beta_{3} ) q^{35} + ( -2 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 1 - \beta_{1} ) q^{39} + ( 7 \beta_{1} + \beta_{3} ) q^{41} + ( 1 + \beta_{2} ) q^{43} - q^{45} + ( 5 \beta_{1} - \beta_{3} ) q^{47} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -4 - 2 \beta_{2} ) q^{53} + ( -\beta_{1} - \beta_{3} ) q^{55} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{57} + ( 2 - 2 \beta_{2} ) q^{59} + ( -11 + 11 \beta_{1} ) q^{61} + ( -\beta_{1} + \beta_{3} ) q^{63} + ( -1 + \beta_{1} ) q^{65} + ( 6 - 6 \beta_{1} - \beta_{2} - \beta_{3} ) q^{67} + ( -4 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{71} + ( 13 - 13 \beta_{1} ) q^{73} + q^{75} + ( -8 + 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( 11 \beta_{1} - 2 \beta_{3} ) q^{79} + q^{81} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 4 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{3} ) q^{87} + ( -2 - 4 \beta_{2} ) q^{89} + \beta_{2} q^{91} + ( -\beta_{1} - 2 \beta_{3} ) q^{93} + ( 4 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{95} + ( 1 - \beta_{1} ) q^{97} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} - q^{7} + 4q^{9} + 3q^{11} + 2q^{13} - 4q^{15} - 9q^{17} - 7q^{19} - q^{21} - 9q^{23} + 4q^{25} + 4q^{27} - 3q^{29} - 4q^{31} + 3q^{33} + q^{35} - q^{37} + 2q^{39} + 15q^{41} + 2q^{43} - 4q^{45} + 9q^{47} - 3q^{49} - 9q^{51} - 12q^{53} - 3q^{55} - 7q^{57} + 12q^{59} - 22q^{61} - q^{63} - 2q^{65} + 13q^{67} - 9q^{69} - 9q^{71} + 26q^{73} + 4q^{75} - 15q^{77} + 20q^{79} + 4q^{81} + 3q^{83} + 9q^{85} - 3q^{87} - 2q^{91} - 4q^{93} + 7q^{95} + 2q^{97} + 3q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 1.00000 0 −1.00000 0 −1.68614 + 2.92048i 0 1.00000 0
841.2 0 1.00000 0 −1.00000 0 1.18614 2.05446i 0 1.00000 0
3781.1 0 1.00000 0 −1.00000 0 −1.68614 2.92048i 0 1.00000 0
3781.2 0 1.00000 0 −1.00000 0 1.18614 + 2.05446i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
67.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\( T_{7}^{4} + T_{7}^{3} + 9 T_{7}^{2} - 8 T_{7} + 64 \)
\( T_{11}^{4} - 3 T_{11}^{3} + 15 T_{11}^{2} + 18 T_{11} + 36 \)
\( T_{17}^{4} + 9 T_{17}^{3} + 69 T_{17}^{2} + 108 T_{17} + 144 \)