Properties

Label 136.2.a.b
Level $136$
Weight $2$
Character orbit 136.a
Self dual yes
Analytic conductor $1.086$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.08596546749\)
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

\(f(q)\) \(=\) \( q + 2 q^{3} + q^{9} + O(q^{10}) \) \( q + 2 q^{3} + q^{9} + 2 q^{11} - 6 q^{13} - q^{17} + 4 q^{19} + 4 q^{23} - 5 q^{25} - 4 q^{27} - 8 q^{31} + 4 q^{33} - 4 q^{37} - 12 q^{39} + 6 q^{41} + 8 q^{43} - 8 q^{47} - 7 q^{49} - 2 q^{51} + 10 q^{53} + 8 q^{57} + 12 q^{61} + 8 q^{67} + 8 q^{69} + 12 q^{71} + 2 q^{73} - 10 q^{75} - 4 q^{79} - 11 q^{81} + 16 q^{83} + 10 q^{89} - 16 q^{93} - 18 q^{97} + 2 q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

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 136.2.a.b 1
3.b odd 2 1 1224.2.a.d 1
4.b odd 2 1 272.2.a.a 1
5.b even 2 1 3400.2.a.b 1
5.c odd 4 2 3400.2.e.c 2
7.b odd 2 1 6664.2.a.b 1
8.b even 2 1 1088.2.a.c 1
8.d odd 2 1 1088.2.a.m 1
12.b even 2 1 2448.2.a.j 1
17.b even 2 1 2312.2.a.a 1
17.c even 4 2 2312.2.b.b 2
20.d odd 2 1 6800.2.a.w 1
24.f even 2 1 9792.2.a.bd 1
24.h odd 2 1 9792.2.a.be 1
68.d odd 2 1 4624.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.b 1 1.a even 1 1 trivial
272.2.a.a 1 4.b odd 2 1
1088.2.a.c 1 8.b even 2 1
1088.2.a.m 1 8.d odd 2 1
1224.2.a.d 1 3.b odd 2 1
2312.2.a.a 1 17.b even 2 1
2312.2.b.b 2 17.c even 4 2
2448.2.a.j 1 12.b even 2 1
3400.2.a.b 1 5.b even 2 1
3400.2.e.c 2 5.c odd 4 2
4624.2.a.f 1 68.d odd 2 1
6664.2.a.b 1 7.b odd 2 1
6800.2.a.w 1 20.d odd 2 1
9792.2.a.bd 1 24.f even 2 1
9792.2.a.be 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(136))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( 6 + T \)
$17$ \( 1 + T \)
$19$ \( -4 + T \)
$23$ \( -4 + T \)
$29$ \( T \)
$31$ \( 8 + T \)
$37$ \( 4 + T \)
$41$ \( -6 + T \)
$43$ \( -8 + T \)
$47$ \( 8 + T \)
$53$ \( -10 + T \)
$59$ \( T \)
$61$ \( -12 + T \)
$67$ \( -8 + T \)
$71$ \( -12 + T \)
$73$ \( -2 + T \)
$79$ \( 4 + T \)
$83$ \( -16 + T \)
$89$ \( -10 + T \)
$97$ \( 18 + T \)
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