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.

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

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

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

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 no

Hecke kernels

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

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