Properties

Label 4032.2.v.a
Level 4032
Weight 2
Character orbit 4032.v
Analytic conductor 32.196
Analytic rank 0
Dimension 4
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.v (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{2} + \beta_{3} ) q^{5} \) \(- q^{7}\) \(+O(q^{10})\) \( q\) \( + ( \beta_{2} + \beta_{3} ) q^{5} \) \(- q^{7}\) \( + ( -\beta_{2} + \beta_{3} ) q^{11} \) \( + ( 2 + 2 \beta_{1} ) q^{13} \) \( -5 \beta_{2} q^{23} \) \( -\beta_{1} q^{25} \) \( + ( \beta_{2} - \beta_{3} ) q^{29} \) \( -8 \beta_{1} q^{31} \) \( + ( -\beta_{2} - \beta_{3} ) q^{35} \) \( + ( 5 - 5 \beta_{1} ) q^{37} \) \( + 2 \beta_{3} q^{41} \) \( + ( 3 + 3 \beta_{1} ) q^{43} \) \( + 4 \beta_{3} q^{47} \) \(+ q^{49}\) \( + 4 q^{55} \) \( + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} \) \( + ( -2 - 2 \beta_{1} ) q^{61} \) \( + 4 \beta_{2} q^{65} \) \( + ( 5 - 5 \beta_{1} ) q^{67} \) \( + 7 \beta_{2} q^{71} \) \( + 14 \beta_{1} q^{73} \) \( + ( \beta_{2} - \beta_{3} ) q^{77} \) \( -16 \beta_{1} q^{79} \) \( + ( -2 \beta_{2} - 2 \beta_{3} ) q^{83} \) \( + 8 \beta_{3} q^{89} \) \( + ( -2 - 2 \beta_{1} ) q^{91} \) \( -6 q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 8q^{13} \) \(\mathstrut +\mathstrut 20q^{37} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 16q^{55} \) \(\mathstrut -\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 20q^{67} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut -\mathstrut 24q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \zeta_{8}^{2} \)
\(\beta_{2}\)\(=\)\( \zeta_{8}^{3} + \zeta_{8} \)
\(\beta_{3}\)\(=\)\( -\zeta_{8}^{3} + \zeta_{8} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{8}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)
\(\zeta_{8}^{2}\)\(=\)\(\beta_{1}\)
\(\zeta_{8}^{3}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\)\()/2\)

Character Values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 −1.41421 1.41421i 0 −1.00000 0 0 0
1583.2 0 0 0 1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.1 0 0 0 −1.41421 + 1.41421i 0 −1.00000 0 0 0
3599.2 0 0 0 1.41421 1.41421i 0 −1.00000 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 yes
16.f Odd 1 yes
48.k Even 1 yes

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 16 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 16 \)