Properties

Label 38.2.a.a
Level 38
Weight 2
Character orbit 38.a
Self dual Yes
Analytic conductor 0.303
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 38 = 2 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 38.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.303431527681\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2q^{9} - 6q^{11} + q^{12} + 5q^{13} + q^{14} + q^{16} + 3q^{17} + 2q^{18} + q^{19} - q^{21} + 6q^{22} + 3q^{23} - q^{24} - 5q^{25} - 5q^{26} - 5q^{27} - q^{28} + 9q^{29} - 4q^{31} - q^{32} - 6q^{33} - 3q^{34} - 2q^{36} + 2q^{37} - q^{38} + 5q^{39} + q^{42} + 8q^{43} - 6q^{44} - 3q^{46} + q^{48} - 6q^{49} + 5q^{50} + 3q^{51} + 5q^{52} - 3q^{53} + 5q^{54} + q^{56} + q^{57} - 9q^{58} + 9q^{59} - 10q^{61} + 4q^{62} + 2q^{63} + q^{64} + 6q^{66} + 5q^{67} + 3q^{68} + 3q^{69} - 6q^{71} + 2q^{72} - 7q^{73} - 2q^{74} - 5q^{75} + q^{76} + 6q^{77} - 5q^{78} - 10q^{79} + q^{81} - 6q^{83} - q^{84} - 8q^{86} + 9q^{87} + 6q^{88} - 12q^{89} - 5q^{91} + 3q^{92} - 4q^{93} - q^{96} - 10q^{97} + 6q^{98} + 12q^{99} + O(q^{100}) \)

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
−1.00000 1.00000 1.00000 0 −1.00000 −1.00000 −1.00000 −2.00000 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
\(2\) \(1\)
\(19\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\).