Properties

Label 4020.2.q.a
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \(+ q^{5}\) \( -3 \zeta_{6} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \(+ q^{5}\) \( -3 \zeta_{6} q^{7} \) \(+ q^{9}\) \( -5 \zeta_{6} q^{11} \) \( + ( -1 + \zeta_{6} ) q^{13} \) \(- q^{15}\) \( + ( 7 - 7 \zeta_{6} ) q^{17} \) \( + ( 5 - 5 \zeta_{6} ) q^{19} \) \( + 3 \zeta_{6} q^{21} \) \( + ( -1 + \zeta_{6} ) q^{23} \) \(+ q^{25}\) \(- q^{27}\) \( -3 \zeta_{6} q^{29} \) \( + 3 \zeta_{6} q^{31} \) \( + 5 \zeta_{6} q^{33} \) \( -3 \zeta_{6} q^{35} \) \( + ( -1 + \zeta_{6} ) q^{37} \) \( + ( 1 - \zeta_{6} ) q^{39} \) \( + 5 \zeta_{6} q^{41} \) \( -4 q^{43} \) \(+ q^{45}\) \( -3 \zeta_{6} q^{47} \) \( + ( -2 + 2 \zeta_{6} ) q^{49} \) \( + ( -7 + 7 \zeta_{6} ) q^{51} \) \( + 6 q^{53} \) \( -5 \zeta_{6} q^{55} \) \( + ( -5 + 5 \zeta_{6} ) q^{57} \) \( + 4 q^{59} \) \( + ( -7 + 7 \zeta_{6} ) q^{61} \) \( -3 \zeta_{6} q^{63} \) \( + ( -1 + \zeta_{6} ) q^{65} \) \( + ( -7 - 2 \zeta_{6} ) q^{67} \) \( + ( 1 - \zeta_{6} ) q^{69} \) \( -\zeta_{6} q^{71} \) \( + ( -5 + 5 \zeta_{6} ) q^{73} \) \(- q^{75}\) \( + ( -15 + 15 \zeta_{6} ) q^{77} \) \( -9 \zeta_{6} q^{79} \) \(+ q^{81}\) \( + ( -9 + 9 \zeta_{6} ) q^{83} \) \( + ( 7 - 7 \zeta_{6} ) q^{85} \) \( + 3 \zeta_{6} q^{87} \) \( -10 q^{89} \) \( + 3 q^{91} \) \( -3 \zeta_{6} q^{93} \) \( + ( 5 - 5 \zeta_{6} ) q^{95} \) \( + ( 7 - 7 \zeta_{6} ) q^{97} \) \( -5 \zeta_{6} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 3q^{7} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 5q^{19} \) \(\mathstrut +\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut -\mathstrut 3q^{29} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 5q^{33} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut q^{39} \) \(\mathstrut +\mathstrut 5q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 3q^{47} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 7q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 5q^{55} \) \(\mathstrut -\mathstrut 5q^{57} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 7q^{61} \) \(\mathstrut -\mathstrut 3q^{63} \) \(\mathstrut -\mathstrut q^{65} \) \(\mathstrut -\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut -\mathstrut 5q^{73} \) \(\mathstrut -\mathstrut 2q^{75} \) \(\mathstrut -\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut 9q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 9q^{83} \) \(\mathstrut +\mathstrut 7q^{85} \) \(\mathstrut +\mathstrut 3q^{87} \) \(\mathstrut -\mathstrut 20q^{89} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut -\mathstrut 3q^{93} \) \(\mathstrut +\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut -\mathstrut 5q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0 1.00000 0 −1.50000 + 2.59808i 0 1.00000 0
3781.1 0 −1.00000 0 1.00000 0 −1.50000 2.59808i 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 \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\(T_{7}^{2} \) \(\mathstrut +\mathstrut 3 T_{7} \) \(\mathstrut +\mathstrut 9 \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 5 T_{11} \) \(\mathstrut +\mathstrut 25 \)
\(T_{17}^{2} \) \(\mathstrut -\mathstrut 7 T_{17} \) \(\mathstrut +\mathstrut 49 \)