Properties

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

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.0.2312.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{5} -\beta_{3} q^{7} + \beta_{2} q^{11} + 2 \beta_{2} q^{13} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 3 + \beta_{1} - \beta_{3} ) q^{19} + \beta_{2} q^{23} + ( -2 - \beta_{1} + \beta_{3} ) q^{25} -2 q^{29} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{35} + ( 3 - \beta_{1} + \beta_{3} ) q^{37} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{41} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( 4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{49} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -1 + \beta_{1} - \beta_{3} ) q^{55} -4 q^{59} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( 3 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 2 - 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -1 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{77} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -5 - 3 \beta_{1} + 3 \beta_{3} ) q^{85} + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{89} + ( -2 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -6 + 6 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{7} + O(q^{10}) \) \( 4q - 2q^{7} + 12q^{19} - 8q^{25} - 8q^{29} + 12q^{35} + 12q^{37} - 8q^{47} + 8q^{49} + 16q^{53} - 4q^{55} - 16q^{59} - 8q^{65} - 8q^{77} + 8q^{83} - 20q^{85} - 16q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x + 4\):

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

Character Values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
1.28078 0.599676i
−0.780776 1.17915i
−0.780776 + 1.17915i
1.28078 + 0.599676i
0 0 0 3.33513i 0 1.56155 + 2.13578i 0 0 0
3583.2 0 0 0 1.69614i 0 −2.56155 0.662153i 0 0 0
3583.3 0 0 0 1.69614i 0 −2.56155 + 0.662153i 0 0 0
3583.4 0 0 0 3.33513i 0 1.56155 2.13578i 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
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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 14 T_{5}^{2} + 32 \)
\( T_{11}^{4} + 10 T_{11}^{2} + 8 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)