Properties

Label 4032.2.b.p
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM No
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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{5} + ( \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{7} + 2 \zeta_{16}^{4} q^{11} + ( -3 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{13} + ( -4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{17} + ( -3 \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{19} + ( 4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{23} + ( 1 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{25} + ( -2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{31} + ( \zeta_{16} + 2 \zeta_{16}^{2} + \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{35} + ( 2 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{37} + ( 6 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{41} + ( -4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{43} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{47} + ( 1 + 2 \zeta_{16} - 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{49} -2 q^{53} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( \zeta_{16} + 5 \zeta_{16}^{3} - 5 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{61} + ( 4 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{65} + ( 4 \zeta_{16}^{2} - 10 \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{67} + ( -2 \zeta_{16}^{2} + 6 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{71} + ( 6 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{73} + ( -2 - 2 \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( -2 \zeta_{16}^{2} + 10 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{79} + ( 3 \zeta_{16} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{83} -8 q^{85} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{89} + ( 5 \zeta_{16} - 6 \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} - 6 \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{91} + ( -2 \zeta_{16}^{2} + 8 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{95} + ( 4 \zeta_{16} - 8 \zeta_{16}^{3} - 8 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} - 16q^{29} + 16q^{37} + 8q^{49} - 16q^{53} + 32q^{65} - 16q^{77} - 64q^{85} + O(q^{100}) \)

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
−0.382683 0.923880i
0.382683 0.923880i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.923880 0.382683i
0.923880 0.382683i
−0.382683 + 0.923880i
0.382683 + 0.923880i
0 0 0 2.61313i 0 −2.61313 + 0.414214i 0 0 0
3583.2 0 0 0 2.61313i 0 2.61313 0.414214i 0 0 0
3583.3 0 0 0 1.08239i 0 −1.08239 2.41421i 0 0 0
3583.4 0 0 0 1.08239i 0 1.08239 + 2.41421i 0 0 0
3583.5 0 0 0 1.08239i 0 −1.08239 + 2.41421i 0 0 0
3583.6 0 0 0 1.08239i 0 1.08239 2.41421i 0 0 0
3583.7 0 0 0 2.61313i 0 −2.61313 0.414214i 0 0 0
3583.8 0 0 0 2.61313i 0 2.61313 + 0.414214i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3583.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
7.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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 8 T_{5}^{2} + 8 \)
\( T_{11}^{2} + 4 \)
\( T_{19}^{4} - 40 T_{19}^{2} + 392 \)