Properties

Label 156.2.a.b
Level $156$
Weight $2$
Character orbit 156.a
Self dual yes
Analytic conductor $1.246$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.24566627153\)
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

\(f(q)\) \(=\) \( q + q^{3} + 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{7} + q^{9} + q^{13} - 6q^{17} + 2q^{19} + 2q^{21} - 5q^{25} + q^{27} - 6q^{29} + 2q^{31} + 2q^{37} + q^{39} - 12q^{41} - 4q^{43} - 3q^{49} - 6q^{51} + 6q^{53} + 2q^{57} + 12q^{59} + 2q^{61} + 2q^{63} - 10q^{67} + 12q^{71} + 14q^{73} - 5q^{75} + 8q^{79} + q^{81} + 12q^{83} - 6q^{87} + 2q^{91} + 2q^{93} - 10q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)

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 156.2.a.b 1
3.b odd 2 1 468.2.a.c 1
4.b odd 2 1 624.2.a.b 1
5.b even 2 1 3900.2.a.a 1
5.c odd 4 2 3900.2.h.e 2
7.b odd 2 1 7644.2.a.a 1
8.b even 2 1 2496.2.a.h 1
8.d odd 2 1 2496.2.a.v 1
9.c even 3 2 4212.2.i.f 2
9.d odd 6 2 4212.2.i.g 2
12.b even 2 1 1872.2.a.i 1
13.b even 2 1 2028.2.a.e 1
13.c even 3 2 2028.2.i.b 2
13.d odd 4 2 2028.2.b.d 2
13.e even 6 2 2028.2.i.c 2
13.f odd 12 4 2028.2.q.d 4
24.f even 2 1 7488.2.a.bb 1
24.h odd 2 1 7488.2.a.bf 1
39.d odd 2 1 6084.2.a.h 1
39.f even 4 2 6084.2.b.a 2
52.b odd 2 1 8112.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.b 1 1.a even 1 1 trivial
468.2.a.c 1 3.b odd 2 1
624.2.a.b 1 4.b odd 2 1
1872.2.a.i 1 12.b even 2 1
2028.2.a.e 1 13.b even 2 1
2028.2.b.d 2 13.d odd 4 2
2028.2.i.b 2 13.c even 3 2
2028.2.i.c 2 13.e even 6 2
2028.2.q.d 4 13.f odd 12 4
2496.2.a.h 1 8.b even 2 1
2496.2.a.v 1 8.d odd 2 1
3900.2.a.a 1 5.b even 2 1
3900.2.h.e 2 5.c odd 4 2
4212.2.i.f 2 9.c even 3 2
4212.2.i.g 2 9.d odd 6 2
6084.2.a.h 1 39.d odd 2 1
6084.2.b.a 2 39.f even 4 2
7488.2.a.bb 1 24.f even 2 1
7488.2.a.bf 1 24.h odd 2 1
7644.2.a.a 1 7.b odd 2 1
8112.2.a.i 1 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(156))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -1 + T \)
$17$ \( 6 + T \)
$19$ \( -2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( -2 + T \)
$37$ \( -2 + T \)
$41$ \( 12 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( -6 + T \)
$59$ \( -12 + T \)
$61$ \( -2 + T \)
$67$ \( 10 + T \)
$71$ \( -12 + T \)
$73$ \( -14 + T \)
$79$ \( -8 + T \)
$83$ \( -12 + T \)
$89$ \( T \)
$97$ \( 10 + T \)
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