Properties

Label 4032.2.c.q
Level 4032
Weight 2
Character orbit 4032.c
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.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.897122304.10
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{5} q^{5} \) \(- q^{7}\) \(+O(q^{10})\) \( q\) \( -\beta_{5} q^{5} \) \(- q^{7}\) \( + \beta_{5} q^{11} \) \( + ( -\beta_{4} - \beta_{6} ) q^{13} \) \( + \beta_{1} q^{17} \) \( + \beta_{4} q^{19} \) \( -\beta_{2} q^{23} \) \( + ( -5 - \beta_{7} ) q^{25} \) \( -\beta_{3} q^{29} \) \( + 2 q^{31} \) \( + \beta_{5} q^{35} \) \( + ( \beta_{4} + 4 \beta_{6} ) q^{37} \) \( -\beta_{1} q^{41} \) \( + ( 2 \beta_{4} + \beta_{6} ) q^{43} \) \( + ( -\beta_{1} - \beta_{2} ) q^{47} \) \(+ q^{49}\) \( + ( 10 + \beta_{7} ) q^{55} \) \( + \beta_{3} q^{59} \) \( + ( \beta_{4} + \beta_{6} ) q^{61} \) \( + ( \beta_{1} - 3 \beta_{2} ) q^{65} \) \( -5 \beta_{6} q^{67} \) \( + ( 2 \beta_{1} - \beta_{2} ) q^{71} \) \( + ( 6 - \beta_{7} ) q^{73} \) \( -\beta_{5} q^{77} \) \( + ( 4 - \beta_{7} ) q^{79} \) \( + ( -\beta_{3} - 2 \beta_{5} ) q^{83} \) \( + ( -\beta_{4} + 4 \beta_{6} ) q^{85} \) \( + ( -\beta_{1} + 2 \beta_{2} ) q^{89} \) \( + ( \beta_{4} + \beta_{6} ) q^{91} \) \( + 2 \beta_{2} q^{95} \) \( + ( 2 - \beta_{7} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 40q^{25} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut +\mathstrut 80q^{55} \) \(\mathstrut +\mathstrut 48q^{73} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(8\) \(x^{6}\mathstrut +\mathstrut \) \(51\) \(x^{4}\mathstrut -\mathstrut \) \(104\) \(x^{2}\mathstrut +\mathstrut \) \(169\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{7} + 17 \nu^{5} + 85 \nu^{3} - 403 \nu \)\()/663\)
\(\beta_{2}\)\(=\)\((\)\( -14 \nu^{7} + 255 \nu^{5} - 1377 \nu^{3} + 4771 \nu \)\()/663\)
\(\beta_{3}\)\(=\)\((\)\( 32 \nu^{7} - 204 \nu^{5} + 1632 \nu^{3} - 676 \nu \)\()/663\)
\(\beta_{4}\)\(=\)\((\)\( 32 \nu^{6} - 204 \nu^{4} + 1632 \nu^{2} - 2002 \)\()/663\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{7} - 255 \nu^{5} + 1377 \nu^{3} - 845 \nu \)\()/663\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{6} - 16 \nu^{4} + 76 \nu^{2} - 104 \)\()/39\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{6} + 400 \)\()/51\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(24\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(38\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(32\) \(\beta_{5}\mathstrut +\mathstrut \) \(27\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(81\) \(\beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(51\) \(\beta_{7}\mathstrut -\mathstrut \) \(400\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(204\) \(\beta_{5}\mathstrut -\mathstrut \) \(151\) \(\beta_{3}\mathstrut +\mathstrut \) \(53\) \(\beta_{2}\mathstrut +\mathstrut \) \(453\) \(\beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2017.1
1.30421 + 0.752986i
−1.30421 + 0.752986i
−2.07341 1.19709i
2.07341 1.19709i
2.07341 + 1.19709i
−2.07341 + 1.19709i
−1.30421 0.752986i
1.30421 0.752986i
0 0 0 4.11439i 0 −1.00000 0 0 0
2017.2 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.3 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.4 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.5 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.6 0 0 0 1.75265i 0 −1.00000 0 0 0
2017.7 0 0 0 4.11439i 0 −1.00000 0 0 0
2017.8 0 0 0 4.11439i 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2017.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
8.b Even 1 yes
24.h Odd 1 yes

Hecke kernels

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

\(T_{5}^{4} \) \(\mathstrut +\mathstrut 20 T_{5}^{2} \) \(\mathstrut +\mathstrut 52 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 20 T_{11}^{2} \) \(\mathstrut +\mathstrut 52 \)
\(T_{13}^{4} \) \(\mathstrut +\mathstrut 32 T_{13}^{2} \) \(\mathstrut +\mathstrut 64 \)
\(T_{17}^{4} \) \(\mathstrut -\mathstrut 44 T_{17}^{2} \) \(\mathstrut +\mathstrut 52 \)
\(T_{23}^{4} \) \(\mathstrut -\mathstrut 60 T_{23}^{2} \) \(\mathstrut +\mathstrut 468 \)
\(T_{31} \) \(\mathstrut -\mathstrut 2 \)