Properties

Label 138.2.a.b
Level $138$
Weight $2$
Character orbit 138.a
Self dual yes
Analytic conductor $1.102$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{12} + 2 q^{13} - 2 q^{14} + q^{16} - q^{18} + 2 q^{19} + 2 q^{21} - q^{23} - q^{24} - 5 q^{25} - 2 q^{26} + q^{27} + 2 q^{28} - 6 q^{29} - 4 q^{31} - q^{32} + q^{36} - 10 q^{37} - 2 q^{38} + 2 q^{39} - 6 q^{41} - 2 q^{42} + 2 q^{43} + q^{46} + q^{48} - 3 q^{49} + 5 q^{50} + 2 q^{52} + 12 q^{53} - q^{54} - 2 q^{56} + 2 q^{57} + 6 q^{58} + 12 q^{59} - 10 q^{61} + 4 q^{62} + 2 q^{63} + q^{64} + 14 q^{67} - q^{69} - q^{72} + 2 q^{73} + 10 q^{74} - 5 q^{75} + 2 q^{76} - 2 q^{78} - 10 q^{79} + q^{81} + 6 q^{82} + 2 q^{84} - 2 q^{86} - 6 q^{87} + 12 q^{89} + 4 q^{91} - q^{92} - 4 q^{93} - q^{96} - 10 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 2.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.b 1
3.b odd 2 1 414.2.a.c 1
4.b odd 2 1 1104.2.a.b 1
5.b even 2 1 3450.2.a.o 1
5.c odd 4 2 3450.2.d.g 2
7.b odd 2 1 6762.2.a.g 1
8.b even 2 1 4416.2.a.i 1
8.d odd 2 1 4416.2.a.t 1
12.b even 2 1 3312.2.a.h 1
23.b odd 2 1 3174.2.a.d 1
69.c even 2 1 9522.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.b 1 1.a even 1 1 trivial
414.2.a.c 1 3.b odd 2 1
1104.2.a.b 1 4.b odd 2 1
3174.2.a.d 1 23.b odd 2 1
3312.2.a.h 1 12.b even 2 1
3450.2.a.o 1 5.b even 2 1
3450.2.d.g 2 5.c odd 4 2
4416.2.a.i 1 8.b even 2 1
4416.2.a.t 1 8.d odd 2 1
6762.2.a.g 1 7.b odd 2 1
9522.2.a.k 1 69.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 14 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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