Properties

Label 528.2.a.h
Level 528
Weight 2
Character orbit 528.a
Self dual yes
Analytic conductor 4.216
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{5} - 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{5} - 2q^{7} + q^{9} + q^{11} + 6q^{13} + 2q^{15} - 4q^{17} + 2q^{19} - 2q^{21} + 8q^{23} - q^{25} + q^{27} + q^{33} - 4q^{35} - 6q^{37} + 6q^{39} - 10q^{43} + 2q^{45} - 3q^{49} - 4q^{51} + 14q^{53} + 2q^{55} + 2q^{57} + 12q^{59} - 14q^{61} - 2q^{63} + 12q^{65} - 4q^{67} + 8q^{69} + 6q^{73} - q^{75} - 2q^{77} - 2q^{79} + q^{81} - 16q^{83} - 8q^{85} - 14q^{89} - 12q^{91} + 4q^{95} - 2q^{97} + q^{99} + O(q^{100}) \)

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 1.00000 0 2.00000 0 −2.00000 0 1.00000 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 528.2.a.h 1
3.b odd 2 1 1584.2.a.c 1
4.b odd 2 1 132.2.a.a 1
8.b even 2 1 2112.2.a.b 1
8.d odd 2 1 2112.2.a.t 1
11.b odd 2 1 5808.2.a.bf 1
12.b even 2 1 396.2.a.b 1
20.d odd 2 1 3300.2.a.k 1
20.e even 4 2 3300.2.c.c 2
24.f even 2 1 6336.2.a.cf 1
24.h odd 2 1 6336.2.a.bz 1
28.d even 2 1 6468.2.a.l 1
36.f odd 6 2 3564.2.i.b 2
36.h even 6 2 3564.2.i.g 2
44.c even 2 1 1452.2.a.c 1
44.g even 10 4 1452.2.i.l 4
44.h odd 10 4 1452.2.i.k 4
60.h even 2 1 9900.2.a.g 1
60.l odd 4 2 9900.2.c.k 2
132.d odd 2 1 4356.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.a 1 4.b odd 2 1
396.2.a.b 1 12.b even 2 1
528.2.a.h 1 1.a even 1 1 trivial
1452.2.a.c 1 44.c even 2 1
1452.2.i.k 4 44.h odd 10 4
1452.2.i.l 4 44.g even 10 4
1584.2.a.c 1 3.b odd 2 1
2112.2.a.b 1 8.b even 2 1
2112.2.a.t 1 8.d odd 2 1
3300.2.a.k 1 20.d odd 2 1
3300.2.c.c 2 20.e even 4 2
3564.2.i.b 2 36.f odd 6 2
3564.2.i.g 2 36.h even 6 2
4356.2.a.c 1 132.d odd 2 1
5808.2.a.bf 1 11.b odd 2 1
6336.2.a.bz 1 24.h odd 2 1
6336.2.a.cf 1 24.f even 2 1
6468.2.a.l 1 28.d even 2 1
9900.2.a.g 1 60.h even 2 1
9900.2.c.k 2 60.l odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} - 2 \)
\( T_{7} + 2 \)
\( T_{13} - 6 \)