Properties

Label 4020.2.g.a
Level 4020
Weight 2
Character orbit 4020.g
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.g (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + i q^{3} \) \( + ( 1 - 2 i ) q^{5} \) \(- q^{9}\) \(+O(q^{10})\) \( q\) \( + i q^{3} \) \( + ( 1 - 2 i ) q^{5} \) \(- q^{9}\) \( + 2 i q^{13} \) \( + ( 2 + i ) q^{15} \) \( -2 i q^{17} \) \( -8 i q^{23} \) \( + ( -3 - 4 i ) q^{25} \) \( - i q^{27} \) \( -6 q^{29} \) \( -2 q^{31} \) \( -2 q^{39} \) \( -6 q^{41} \) \( + 4 i q^{43} \) \( + ( -1 + 2 i ) q^{45} \) \( + 7 q^{49} \) \( + 2 q^{51} \) \( -4 i q^{53} \) \( -2 q^{59} \) \( -10 q^{61} \) \( + ( 4 + 2 i ) q^{65} \) \( - i q^{67} \) \( + 8 q^{69} \) \( -10 q^{71} \) \( -8 i q^{73} \) \( + ( 4 - 3 i ) q^{75} \) \( + 14 q^{79} \) \(+ q^{81}\) \( -12 i q^{83} \) \( + ( -4 - 2 i ) q^{85} \) \( -6 i q^{87} \) \( + 2 q^{89} \) \( -2 i q^{93} \) \( + 14 i q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(2q \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut -\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 16q^{69} \) \(\mathstrut -\mathstrut 20q^{71} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut +\mathstrut 28q^{79} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1609.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 0 0 −1.00000 0
1609.2 0 1.00000i 0 1.00000 2.00000i 0 0 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
5.b Even 1 no

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7} \) acting on \(S_{2}^{\mathrm{new}}(4020, \chi)\).