Properties

Label 4032.2.p.f
Level 4032
Weight 2
Character orbit 4032.p
Analytic conductor 32.196
Analytic rank 0
Dimension 8
CM no
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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 448)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{11} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{19} -3 q^{25} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{31} + ( \zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{35} + ( -4 + 8 \zeta_{24}^{4} ) q^{37} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{41} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( 5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( -8 + 16 \zeta_{24}^{4} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{59} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{61} -6 q^{65} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{67} -6 \zeta_{24}^{6} q^{71} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 2 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{77} + 14 \zeta_{24}^{6} q^{79} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{83} + ( 4 - 8 \zeta_{24}^{4} ) q^{85} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{89} + ( -3 \zeta_{24} - 12 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{91} + 6 \zeta_{24}^{6} q^{95} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{25} + 40q^{49} - 48q^{65} + O(q^{100}) \)

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
0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.965926 0.258819i
0 0 0 −1.41421 0 −2.44949 1.00000i 0 0 0
1567.2 0 0 0 −1.41421 0 −2.44949 + 1.00000i 0 0 0
1567.3 0 0 0 −1.41421 0 2.44949 1.00000i 0 0 0
1567.4 0 0 0 −1.41421 0 2.44949 + 1.00000i 0 0 0
1567.5 0 0 0 1.41421 0 −2.44949 1.00000i 0 0 0
1567.6 0 0 0 1.41421 0 −2.44949 + 1.00000i 0 0 0
1567.7 0 0 0 1.41421 0 2.44949 1.00000i 0 0 0
1567.8 0 0 0 1.41421 0 2.44949 + 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.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.p.f 8
3.b odd 2 1 448.2.e.b 8
4.b odd 2 1 inner 4032.2.p.f 8
7.b odd 2 1 inner 4032.2.p.f 8
8.b even 2 1 inner 4032.2.p.f 8
8.d odd 2 1 inner 4032.2.p.f 8
12.b even 2 1 448.2.e.b 8
21.c even 2 1 448.2.e.b 8
24.f even 2 1 448.2.e.b 8
24.h odd 2 1 448.2.e.b 8
28.d even 2 1 inner 4032.2.p.f 8
48.i odd 4 2 1792.2.f.j 8
48.k even 4 2 1792.2.f.j 8
56.e even 2 1 inner 4032.2.p.f 8
56.h odd 2 1 inner 4032.2.p.f 8
84.h odd 2 1 448.2.e.b 8
168.e odd 2 1 448.2.e.b 8
168.i even 2 1 448.2.e.b 8
336.v odd 4 2 1792.2.f.j 8
336.y even 4 2 1792.2.f.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.b 8 3.b odd 2 1
448.2.e.b 8 12.b even 2 1
448.2.e.b 8 21.c even 2 1
448.2.e.b 8 24.f even 2 1
448.2.e.b 8 24.h odd 2 1
448.2.e.b 8 84.h odd 2 1
448.2.e.b 8 168.e odd 2 1
448.2.e.b 8 168.i even 2 1
1792.2.f.j 8 48.i odd 4 2
1792.2.f.j 8 48.k even 4 2
1792.2.f.j 8 336.v odd 4 2
1792.2.f.j 8 336.y even 4 2
4032.2.p.f 8 1.a even 1 1 trivial
4032.2.p.f 8 4.b odd 2 1 inner
4032.2.p.f 8 7.b odd 2 1 inner
4032.2.p.f 8 8.b even 2 1 inner
4032.2.p.f 8 8.d odd 2 1 inner
4032.2.p.f 8 28.d even 2 1 inner
4032.2.p.f 8 56.e even 2 1 inner
4032.2.p.f 8 56.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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{2} - 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 8 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 8 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 10 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 20 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{8} \)
$29$ \( ( 1 - 29 T^{2} )^{8} \)
$31$ \( ( 1 + 38 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )^{4}( 1 + 10 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 14 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 74 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 + 86 T^{2} + 2809 T^{4} )^{4} \)
$59$ \( ( 1 - 68 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + 104 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 26 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 106 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 122 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 38 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 68 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 38 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \)
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