Properties

Label 38.2.a.b
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} - 4q^{10} + 2q^{11} - q^{12} - q^{13} + 3q^{14} + 4q^{15} + q^{16} + 3q^{17} - 2q^{18} - q^{19} - 4q^{20} - 3q^{21} + 2q^{22} - q^{23} - q^{24} + 11q^{25} - q^{26} + 5q^{27} + 3q^{28} - 5q^{29} + 4q^{30} - 8q^{31} + q^{32} - 2q^{33} + 3q^{34} - 12q^{35} - 2q^{36} - 2q^{37} - q^{38} + q^{39} - 4q^{40} - 8q^{41} - 3q^{42} + 4q^{43} + 2q^{44} + 8q^{45} - q^{46} + 8q^{47} - q^{48} + 2q^{49} + 11q^{50} - 3q^{51} - q^{52} - q^{53} + 5q^{54} - 8q^{55} + 3q^{56} + q^{57} - 5q^{58} + 15q^{59} + 4q^{60} + 2q^{61} - 8q^{62} - 6q^{63} + q^{64} + 4q^{65} - 2q^{66} + 3q^{67} + 3q^{68} + q^{69} - 12q^{70} + 2q^{71} - 2q^{72} + 9q^{73} - 2q^{74} - 11q^{75} - q^{76} + 6q^{77} + q^{78} - 10q^{79} - 4q^{80} + q^{81} - 8q^{82} - 6q^{83} - 3q^{84} - 12q^{85} + 4q^{86} + 5q^{87} + 2q^{88} + 8q^{90} - 3q^{91} - q^{92} + 8q^{93} + 8q^{94} + 4q^{95} - q^{96} - 2q^{97} + 2q^{98} - 4q^{99} + O(q^{100}) \)

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 −4.00000 −1.00000 3.00000 1.00000 −2.00000 −4.00000
\(n\): 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
\(2\) \(-1\)
\(19\) \(1\)

Hecke kernels

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