Properties

Label 4032.2.c.n
Level 4032
Weight 2
Character orbit 4032.c
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{5} \) \(+ q^{7}\) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{5} \) \(+ q^{7}\) \( + 2 \beta_{1} q^{11} \) \( + ( -2 \beta_{1} + \beta_{3} ) q^{13} \) \( -2 q^{17} \) \( -\beta_{3} q^{19} \) \( -\beta_{2} q^{23} \) \( + ( 1 + \beta_{2} ) q^{25} \) \( + ( -\beta_{1} - \beta_{3} ) q^{29} \) \( + 4 q^{31} \) \( -\beta_{1} q^{35} \) \( + ( -\beta_{1} - \beta_{3} ) q^{37} \) \( + 2 q^{41} \) \( -2 \beta_{1} q^{43} \) \( + ( 4 + 2 \beta_{2} ) q^{47} \) \(+ q^{49}\) \( + ( -3 \beta_{1} + 3 \beta_{3} ) q^{53} \) \( + ( 8 - 2 \beta_{2} ) q^{55} \) \( + ( -4 \beta_{1} - \beta_{3} ) q^{59} \) \( + ( 5 \beta_{1} + 2 \beta_{3} ) q^{61} \) \( + ( -8 + 3 \beta_{2} ) q^{65} \) \( + ( -2 \beta_{1} + 2 \beta_{3} ) q^{67} \) \( + ( 4 + 2 \beta_{2} ) q^{71} \) \( + ( 6 - 2 \beta_{2} ) q^{73} \) \( + 2 \beta_{1} q^{77} \) \( + ( -4 + 2 \beta_{2} ) q^{79} \) \( + ( -3 \beta_{1} - 2 \beta_{3} ) q^{83} \) \( + 2 \beta_{1} q^{85} \) \( + ( 2 + 4 \beta_{2} ) q^{89} \) \( + ( -2 \beta_{1} + \beta_{3} ) q^{91} \) \( -\beta_{2} q^{95} \) \( + ( 2 - 2 \beta_{2} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 32q^{55} \) \(\mathstrut -\mathstrut 32q^{65} \) \(\mathstrut +\mathstrut 16q^{71} \) \(\mathstrut +\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring:

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1 \)
\(\beta_{2}\)\(=\)\( -2 \zeta_{12}^{3} + 4 \zeta_{12} \)
\(\beta_{3}\)\(=\)\( 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\zeta_{12}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/8\)
\(\zeta_{12}^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/8\)
\(\zeta_{12}^{3}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2017.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 2.73205i 0 1.00000 0 0 0
2017.2 0 0 0 0.732051i 0 1.00000 0 0 0
2017.3 0 0 0 0.732051i 0 1.00000 0 0 0
2017.4 0 0 0 2.73205i 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
8.b Even 1 yes

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 8 T_{5}^{2} \) \(\mathstrut +\mathstrut 4 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 32 T_{11}^{2} \) \(\mathstrut +\mathstrut 64 \)
\(T_{13}^{4} \) \(\mathstrut +\mathstrut 56 T_{13}^{2} \) \(\mathstrut +\mathstrut 484 \)
\(T_{17} \) \(\mathstrut +\mathstrut 2 \)
\(T_{23}^{2} \) \(\mathstrut -\mathstrut 12 \)
\(T_{31} \) \(\mathstrut -\mathstrut 4 \)