Properties

Label 528.2.a.g
Level 528
Weight 2
Character orbit 528.a
Self dual Yes
Analytic conductor 4.216
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{5} - 4q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{5} - 4q^{7} + q^{9} - q^{11} - 2q^{13} - 2q^{15} - 2q^{17} - 4q^{21} - 8q^{23} - q^{25} + q^{27} - 6q^{29} + 8q^{31} - q^{33} + 8q^{35} + 6q^{37} - 2q^{39} - 2q^{41} - 2q^{45} - 8q^{47} + 9q^{49} - 2q^{51} + 6q^{53} + 2q^{55} + 4q^{59} + 6q^{61} - 4q^{63} + 4q^{65} + 4q^{67} - 8q^{69} - 14q^{73} - q^{75} + 4q^{77} + 4q^{79} + q^{81} - 12q^{83} + 4q^{85} - 6q^{87} - 6q^{89} + 8q^{91} + 8q^{93} + 2q^{97} - q^{99} + O(q^{100}) \)

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 −4.00000 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.

Atkin-Lehner signs

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

Hecke kernels

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

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