Properties

Label 57.2.a.b
Level 57
Weight 2
Character orbit 57.a
Self dual Yes
Analytic conductor 0.455
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + q^{9} - 2q^{10} - 3q^{11} + 2q^{12} - 6q^{13} - 6q^{14} + q^{15} - 4q^{16} + 3q^{17} - 2q^{18} - q^{19} + 2q^{20} + 3q^{21} + 6q^{22} + 4q^{23} - 4q^{25} + 12q^{26} + q^{27} + 6q^{28} - 10q^{29} - 2q^{30} + 2q^{31} + 8q^{32} - 3q^{33} - 6q^{34} + 3q^{35} + 2q^{36} + 8q^{37} + 2q^{38} - 6q^{39} - 8q^{41} - 6q^{42} - q^{43} - 6q^{44} + q^{45} - 8q^{46} + 3q^{47} - 4q^{48} + 2q^{49} + 8q^{50} + 3q^{51} - 12q^{52} - 6q^{53} - 2q^{54} - 3q^{55} - q^{57} + 20q^{58} + 2q^{60} + 7q^{61} - 4q^{62} + 3q^{63} - 8q^{64} - 6q^{65} + 6q^{66} + 8q^{67} + 6q^{68} + 4q^{69} - 6q^{70} + 12q^{71} - 11q^{73} - 16q^{74} - 4q^{75} - 2q^{76} - 9q^{77} + 12q^{78} - 4q^{80} + q^{81} + 16q^{82} + 4q^{83} + 6q^{84} + 3q^{85} + 2q^{86} - 10q^{87} + 10q^{89} - 2q^{90} - 18q^{91} + 8q^{92} + 2q^{93} - 6q^{94} - q^{95} + 8q^{96} - 2q^{97} - 4q^{98} - 3q^{99} + O(q^{100}) \)

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 1.00000 −2.00000 3.00000 0 1.00000 −2.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\):

\( T_{2} + 2 \)
\( T_{5} - 1 \)