Properties

Label 4032.2.p.c
Level 4032
Weight 2
Character orbit 4032.p
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM discriminant -3
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.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + 2 q^{13} + 8 \zeta_{12}^{3} q^{19} -5 q^{25} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{31} + ( -4 + 8 \zeta_{12}^{2} ) q^{37} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{43} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} -14 q^{61} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} + ( 8 - 16 \zeta_{12}^{2} ) q^{73} + 4 \zeta_{12}^{3} q^{79} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} + ( 8 - 16 \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{13} - 20q^{25} - 4q^{49} - 56q^{61} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 0 0 −1.73205 2.00000i 0 0 0
1567.2 0 0 0 0 0 −1.73205 + 2.00000i 0 0 0
1567.3 0 0 0 0 0 1.73205 2.00000i 0 0 0
1567.4 0 0 0 0 0 1.73205 + 2.00000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
168.e odd 2 1 inner
168.i even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{5} \)
\( T_{11} \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 + 26 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 - 22 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 - 59 T^{2} )^{4} \)
$61$ \( ( 1 + 14 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 122 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 - 89 T^{2} )^{4} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
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