Properties

Label 528.2.a.f
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 264)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 4q^{5} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + 4q^{5} + 2q^{7} + q^{9} + q^{11} - 4q^{15} - 6q^{17} - 4q^{19} - 2q^{21} + 6q^{23} + 11q^{25} - q^{27} + 6q^{29} - q^{33} + 8q^{35} + 6q^{37} - 10q^{41} + 8q^{43} + 4q^{45} - 6q^{47} - 3q^{49} + 6q^{51} - 12q^{53} + 4q^{55} + 4q^{57} + 8q^{59} + 4q^{61} + 2q^{63} + 12q^{67} - 6q^{69} - 10q^{71} + 2q^{73} - 11q^{75} + 2q^{77} - 2q^{79} + q^{81} - 12q^{83} - 24q^{85} - 6q^{87} - 6q^{89} - 16q^{95} + 14q^{97} + q^{99} + O(q^{100}) \)

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 4.00000 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.f 1
3.b odd 2 1 1584.2.a.a 1
4.b odd 2 1 264.2.a.d 1
8.b even 2 1 2112.2.a.p 1
8.d odd 2 1 2112.2.a.a 1
11.b odd 2 1 5808.2.a.p 1
12.b even 2 1 792.2.a.a 1
20.d odd 2 1 6600.2.a.k 1
20.e even 4 2 6600.2.d.c 2
24.f even 2 1 6336.2.a.ck 1
24.h odd 2 1 6336.2.a.cn 1
44.c even 2 1 2904.2.a.o 1
132.d odd 2 1 8712.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.d 1 4.b odd 2 1
528.2.a.f 1 1.a even 1 1 trivial
792.2.a.a 1 12.b even 2 1
1584.2.a.a 1 3.b odd 2 1
2112.2.a.a 1 8.d odd 2 1
2112.2.a.p 1 8.b even 2 1
2904.2.a.o 1 44.c even 2 1
5808.2.a.p 1 11.b odd 2 1
6336.2.a.ck 1 24.f even 2 1
6336.2.a.cn 1 24.h odd 2 1
6600.2.a.k 1 20.d odd 2 1
6600.2.d.c 2 20.e even 4 2
8712.2.a.a 1 132.d odd 2 1

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} - 4 \)
\( T_{7} - 2 \)
\( T_{13} \)