Properties

Label 4032.2.b.m
Level 4032
Weight 2
Character orbit 4032.b
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM discriminant -7
Inner twists 8

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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 \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{3} q^{7} -\beta_{1} q^{11} + 2 \beta_{1} q^{23} + 5 q^{25} -\beta_{2} q^{29} + 6 q^{37} + 2 \beta_{3} q^{43} -7 q^{49} + \beta_{2} q^{53} + 6 \beta_{3} q^{67} -4 \beta_{1} q^{71} + \beta_{2} q^{77} + 6 \beta_{3} q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 20q^{25} + 24q^{37} - 28q^{49} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} - 2 \nu \)
\(\beta_{2}\)\(=\)\( -2 \nu^{3} + 10 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 5 \beta_{1}\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
−1.32288 + 0.500000i
1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 0.500000i
0 0 0 0 0 2.64575i 0 0 0
3583.2 0 0 0 0 0 2.64575i 0 0 0
3583.3 0 0 0 0 0 2.64575i 0 0 0
3583.4 0 0 0 0 0 2.64575i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.b.m 4
3.b odd 2 1 inner 4032.2.b.m 4
4.b odd 2 1 inner 4032.2.b.m 4
7.b odd 2 1 CM 4032.2.b.m 4
8.b even 2 1 252.2.b.c 4
8.d odd 2 1 252.2.b.c 4
12.b even 2 1 inner 4032.2.b.m 4
21.c even 2 1 inner 4032.2.b.m 4
24.f even 2 1 252.2.b.c 4
24.h odd 2 1 252.2.b.c 4
28.d even 2 1 inner 4032.2.b.m 4
56.e even 2 1 252.2.b.c 4
56.h odd 2 1 252.2.b.c 4
84.h odd 2 1 inner 4032.2.b.m 4
168.e odd 2 1 252.2.b.c 4
168.i even 2 1 252.2.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.b.c 4 8.b even 2 1
252.2.b.c 4 8.d odd 2 1
252.2.b.c 4 24.f even 2 1
252.2.b.c 4 24.h odd 2 1
252.2.b.c 4 56.e even 2 1
252.2.b.c 4 56.h odd 2 1
252.2.b.c 4 168.e odd 2 1
252.2.b.c 4 168.i even 2 1
4032.2.b.m 4 1.a even 1 1 trivial
4032.2.b.m 4 3.b odd 2 1 inner
4032.2.b.m 4 4.b odd 2 1 inner
4032.2.b.m 4 7.b odd 2 1 CM
4032.2.b.m 4 12.b even 2 1 inner
4032.2.b.m 4 21.c even 2 1 inner
4032.2.b.m 4 28.d even 2 1 inner
4032.2.b.m 4 84.h odd 2 1 inner

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 16 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - 5 T^{2} )^{4} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 54 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2}( 1 + 12 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 6 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2}( 1 + 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 + 114 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 - 89 T^{2} )^{4} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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