Properties

Label 138.2.a.a
Level 138
Weight 2
Character orbit 138.a
Self dual yes
Analytic conductor 1.102
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.10193554789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −2.00000 1.00000 −2.00000 −1.00000 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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.2.a.a 1
3.b odd 2 1 414.2.a.d 1
4.b odd 2 1 1104.2.a.e 1
5.b even 2 1 3450.2.a.y 1
5.c odd 4 2 3450.2.d.j 2
7.b odd 2 1 6762.2.a.q 1
8.b even 2 1 4416.2.a.z 1
8.d odd 2 1 4416.2.a.m 1
12.b even 2 1 3312.2.a.n 1
23.b odd 2 1 3174.2.a.b 1
69.c even 2 1 9522.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.a 1 1.a even 1 1 trivial
414.2.a.d 1 3.b odd 2 1
1104.2.a.e 1 4.b odd 2 1
3174.2.a.b 1 23.b odd 2 1
3312.2.a.n 1 12.b even 2 1
3450.2.a.y 1 5.b even 2 1
3450.2.d.j 2 5.c odd 4 2
4416.2.a.m 1 8.d odd 2 1
4416.2.a.z 1 8.b even 2 1
6762.2.a.q 1 7.b odd 2 1
9522.2.a.i 1 69.c even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(23\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(138))\).