Properties

Label 4032.2.b.k
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM disc. -84
Inner twists 8

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 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.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{5} \) \( -\beta_{3} q^{7} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{5} \) \( -\beta_{3} q^{7} \) \( -\beta_{1} q^{11} \) \( -\beta_{2} q^{17} \) \( + 2 \beta_{3} q^{19} \) \( -5 \beta_{1} q^{23} \) \( -9 q^{25} \) \( + 4 \beta_{3} q^{31} \) \( -7 \beta_{1} q^{35} \) \( -8 q^{37} \) \( -\beta_{2} q^{41} \) \( + 7 q^{49} \) \( + 2 \beta_{3} q^{55} \) \( -11 \beta_{1} q^{71} \) \( + \beta_{2} q^{77} \) \( + 14 q^{85} \) \( + 5 \beta_{2} q^{89} \) \( + 14 \beta_{1} q^{95} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 36q^{25} \) \(\mathstrut -\mathstrut 32q^{37} \) \(\mathstrut +\mathstrut 28q^{49} \) \(\mathstrut +\mathstrut 56q^{85} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(8\) \(x^{2}\mathstrut +\mathstrut \) \(9\):

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

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
1.16372i
2.57794i
1.16372i
2.57794i
0 0 0 3.74166i 0 −2.64575 0 0 0
3583.2 0 0 0 3.74166i 0 2.64575 0 0 0
3583.3 0 0 0 3.74166i 0 −2.64575 0 0 0
3583.4 0 0 0 3.74166i 0 2.64575 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
84.h Odd 1 CM by \(\Q(\sqrt{-21}) \) yes
3.b Odd 1 yes
4.b Odd 1 yes
7.b Odd 1 yes
12.b Even 1 yes
21.c Even 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 14 \)
\(T_{11}^{2} \) \(\mathstrut +\mathstrut 2 \)
\(T_{19}^{2} \) \(\mathstrut -\mathstrut 28 \)