Properties

Label 4032.2.b.a
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 1
Dimension 2
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -3 + 2 \zeta_{6} ) q^{7} + ( -4 + 8 \zeta_{6} ) q^{13} -8 q^{19} + 5 q^{25} + 4 q^{31} -10 q^{37} + ( 6 - 12 \zeta_{6} ) q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -4 + 8 \zeta_{6} ) q^{61} + ( 2 - 4 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{73} + ( 10 - 20 \zeta_{6} ) q^{79} + ( -4 - 16 \zeta_{6} ) q^{91} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{7} - 16q^{19} + 10q^{25} + 8q^{31} - 20q^{37} + 2q^{49} - 24q^{91} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −2.00000 1.73205i 0 0 0
3583.2 0 0 0 0 0 −2.00000 + 1.73205i 0 0 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
28.d Even 1 yes
84.h Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5} \)
\( T_{11} \)
\( T_{19} + 8 \)