Properties

Label 4032.2.a.bb
Level 4032
Weight 2
Character orbit 4032.a
Self dual yes
Analytic conductor 32.196
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} - q^{7} + O(q^{10}) \) \( q + 2q^{5} - q^{7} - 4q^{11} - 2q^{13} + 6q^{17} - 8q^{19} - q^{25} + 6q^{29} + 8q^{31} - 2q^{35} + 2q^{37} - 2q^{41} + 4q^{43} + 8q^{47} + q^{49} + 6q^{53} - 8q^{55} + 6q^{61} - 4q^{65} + 4q^{67} + 8q^{71} + 10q^{73} + 4q^{77} + 16q^{79} + 8q^{83} + 12q^{85} + 6q^{89} + 2q^{91} - 16q^{95} - 6q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 2.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.a.bb 1
3.b odd 2 1 448.2.a.d 1
4.b odd 2 1 4032.2.a.bk 1
8.b even 2 1 504.2.a.c 1
8.d odd 2 1 1008.2.a.d 1
12.b even 2 1 448.2.a.e 1
21.c even 2 1 3136.2.a.q 1
24.f even 2 1 112.2.a.b 1
24.h odd 2 1 56.2.a.a 1
48.i odd 4 2 1792.2.b.i 2
48.k even 4 2 1792.2.b.d 2
56.e even 2 1 7056.2.a.bo 1
56.h odd 2 1 3528.2.a.x 1
56.j odd 6 2 3528.2.s.e 2
56.p even 6 2 3528.2.s.t 2
84.h odd 2 1 3136.2.a.p 1
120.i odd 2 1 1400.2.a.g 1
120.m even 2 1 2800.2.a.p 1
120.q odd 4 2 2800.2.g.p 2
120.w even 4 2 1400.2.g.g 2
168.e odd 2 1 784.2.a.e 1
168.i even 2 1 392.2.a.d 1
168.s odd 6 2 392.2.i.c 2
168.v even 6 2 784.2.i.e 2
168.ba even 6 2 392.2.i.d 2
168.be odd 6 2 784.2.i.g 2
264.m even 2 1 6776.2.a.g 1
312.b odd 2 1 9464.2.a.c 1
840.u even 2 1 9800.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 24.h odd 2 1
112.2.a.b 1 24.f even 2 1
392.2.a.d 1 168.i even 2 1
392.2.i.c 2 168.s odd 6 2
392.2.i.d 2 168.ba even 6 2
448.2.a.d 1 3.b odd 2 1
448.2.a.e 1 12.b even 2 1
504.2.a.c 1 8.b even 2 1
784.2.a.e 1 168.e odd 2 1
784.2.i.e 2 168.v even 6 2
784.2.i.g 2 168.be odd 6 2
1008.2.a.d 1 8.d odd 2 1
1400.2.a.g 1 120.i odd 2 1
1400.2.g.g 2 120.w even 4 2
1792.2.b.d 2 48.k even 4 2
1792.2.b.i 2 48.i odd 4 2
2800.2.a.p 1 120.m even 2 1
2800.2.g.p 2 120.q odd 4 2
3136.2.a.p 1 84.h odd 2 1
3136.2.a.q 1 21.c even 2 1
3528.2.a.x 1 56.h odd 2 1
3528.2.s.e 2 56.j odd 6 2
3528.2.s.t 2 56.p even 6 2
4032.2.a.bb 1 1.a even 1 1 trivial
4032.2.a.bk 1 4.b odd 2 1
6776.2.a.g 1 264.m even 2 1
7056.2.a.bo 1 56.e even 2 1
9464.2.a.c 1 312.b odd 2 1
9800.2.a.u 1 840.u even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5} - 2 \)
\( T_{11} + 4 \)
\( T_{13} + 2 \)
\( T_{17} - 6 \)
\( T_{19} + 8 \)