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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \zeta_{16}^{4} \)
\(\beta_{2}\)\(=\)\( -\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \)
\(\beta_{3}\)\(=\)\( 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 2 \zeta_{16} \)
\(\beta_{4}\)\(=\)\( -\zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{3} + \zeta_{16}^{2} + \zeta_{16} \)
\(\beta_{5}\)\(=\)\( \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16} \)
\(\beta_{6}\)\(=\)\( \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} - \zeta_{16} \)
\(\beta_{7}\)\(=\)\( -2 \zeta_{16}^{6} + 2 \zeta_{16}^{2} \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{16}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\)\()/8\)
\(\zeta_{16}^{2}\)\(=\)\((\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/4\)
\(\zeta_{16}^{3}\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\)\()/8\)
\(\zeta_{16}^{4}\)\(=\)\(\beta_{1}\)\(/2\)
\(\zeta_{16}^{5}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\)\()/8\)
\(\zeta_{16}^{6}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/4\)
\(\zeta_{16}^{7}\)\(=\)\((\)\(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\)\()/8\)

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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

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