Properties

Label 57.2.a.a
Level 57
Weight 2
Character orbit 57.a
Self dual Yes
Analytic conductor 0.455
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.455147291521\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + q^{9} + 6q^{10} + q^{11} - 2q^{12} + 2q^{13} + 10q^{14} + 3q^{15} - 4q^{16} - q^{17} - 2q^{18} - q^{19} - 6q^{20} + 5q^{21} - 2q^{22} - 4q^{23} + 4q^{25} - 4q^{26} - q^{27} - 10q^{28} - 2q^{29} - 6q^{30} - 6q^{31} + 8q^{32} - q^{33} + 2q^{34} + 15q^{35} + 2q^{36} + 2q^{38} - 2q^{39} - 10q^{42} - q^{43} + 2q^{44} - 3q^{45} + 8q^{46} - 9q^{47} + 4q^{48} + 18q^{49} - 8q^{50} + q^{51} + 4q^{52} + 10q^{53} + 2q^{54} - 3q^{55} + q^{57} + 4q^{58} - 8q^{59} + 6q^{60} - q^{61} + 12q^{62} - 5q^{63} - 8q^{64} - 6q^{65} + 2q^{66} + 8q^{67} - 2q^{68} + 4q^{69} - 30q^{70} - 12q^{71} - 11q^{73} - 4q^{75} - 2q^{76} - 5q^{77} + 4q^{78} + 16q^{79} + 12q^{80} + q^{81} + 12q^{83} + 10q^{84} + 3q^{85} + 2q^{86} + 2q^{87} - 6q^{89} + 6q^{90} - 10q^{91} - 8q^{92} + 6q^{93} + 18q^{94} + 3q^{95} - 8q^{96} - 10q^{97} - 36q^{98} + q^{99} + O(q^{100}) \)

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
−2.00000 −1.00000 2.00000 −3.00000 2.00000 −5.00000 0 1.00000 6.00000
\(n\): 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
\(3\) \(1\)
\(19\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\):

\( T_{2} + 2 \)
\( T_{5} + 3 \)