Properties

Label 4032.2.b.e
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 2
CM disc. -7
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
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 \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{7} \) \(+O(q^{10})\) \( q\) \( -\beta q^{7} \) \( + 2 \beta q^{11} \) \( + 2 \beta q^{23} \) \( + 5 q^{25} \) \( -2 q^{29} \) \( -6 q^{37} \) \( -2 \beta q^{43} \) \( -7 q^{49} \) \( -10 q^{53} \) \( + 6 \beta q^{67} \) \( + 2 \beta q^{71} \) \( + 14 q^{77} \) \( + 6 \beta q^{79} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 20q^{53} \) \(\mathstrut +\mathstrut 28q^{77} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 2.64575i 0 0 0
3583.2 0 0 0 0 0 2.64575i 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
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
4.b Odd 1 yes
28.d Even 1 yes

Hecke kernels

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

\(T_{5} \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 28 \)
\(T_{19} \)