Properties

Label 156.2.a.a
Level $156$
Weight $2$
Character orbit 156.a
Self dual yes
Analytic conductor $1.246$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{3} - 4q^{5} - 2q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - 4q^{5} - 2q^{7} + q^{9} - 4q^{11} + q^{13} + 4q^{15} + 2q^{17} - 2q^{19} + 2q^{21} + 11q^{25} - q^{27} - 6q^{29} - 10q^{31} + 4q^{33} + 8q^{35} + 10q^{37} - q^{39} + 8q^{41} + 4q^{43} - 4q^{45} - 4q^{47} - 3q^{49} - 2q^{51} - 10q^{53} + 16q^{55} + 2q^{57} - 8q^{59} - 14q^{61} - 2q^{63} - 4q^{65} + 2q^{67} + 16q^{71} - 10q^{73} - 11q^{75} + 8q^{77} - 16q^{79} + q^{81} - 8q^{85} + 6q^{87} - 4q^{89} - 2q^{91} + 10q^{93} + 8q^{95} - 2q^{97} - 4q^{99} + O(q^{100}) \)

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
0 −1.00000 0 −4.00000 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.a 1
3.b odd 2 1 468.2.a.d 1
4.b odd 2 1 624.2.a.e 1
5.b even 2 1 3900.2.a.m 1
5.c odd 4 2 3900.2.h.b 2
7.b odd 2 1 7644.2.a.k 1
8.b even 2 1 2496.2.a.bc 1
8.d odd 2 1 2496.2.a.o 1
9.c even 3 2 4212.2.i.l 2
9.d odd 6 2 4212.2.i.b 2
12.b even 2 1 1872.2.a.s 1
13.b even 2 1 2028.2.a.c 1
13.c even 3 2 2028.2.i.e 2
13.d odd 4 2 2028.2.b.a 2
13.e even 6 2 2028.2.i.g 2
13.f odd 12 4 2028.2.q.h 4
24.f even 2 1 7488.2.a.d 1
24.h odd 2 1 7488.2.a.c 1
39.d odd 2 1 6084.2.a.b 1
39.f even 4 2 6084.2.b.j 2
52.b odd 2 1 8112.2.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 1.a even 1 1 trivial
468.2.a.d 1 3.b odd 2 1
624.2.a.e 1 4.b odd 2 1
1872.2.a.s 1 12.b even 2 1
2028.2.a.c 1 13.b even 2 1
2028.2.b.a 2 13.d odd 4 2
2028.2.i.e 2 13.c even 3 2
2028.2.i.g 2 13.e even 6 2
2028.2.q.h 4 13.f odd 12 4
2496.2.a.o 1 8.d odd 2 1
2496.2.a.bc 1 8.b even 2 1
3900.2.a.m 1 5.b even 2 1
3900.2.h.b 2 5.c odd 4 2
4212.2.i.b 2 9.d odd 6 2
4212.2.i.l 2 9.c even 3 2
6084.2.a.b 1 39.d odd 2 1
6084.2.b.j 2 39.f even 4 2
7488.2.a.c 1 24.h odd 2 1
7488.2.a.d 1 24.f even 2 1
7644.2.a.k 1 7.b odd 2 1
8112.2.a.bi 1 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(156))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( 4 + T \)
$7$ \( 2 + T \)
$11$ \( 4 + T \)
$13$ \( -1 + T \)
$17$ \( -2 + T \)
$19$ \( 2 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( -10 + T \)
$41$ \( -8 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( 10 + T \)
$59$ \( 8 + T \)
$61$ \( 14 + T \)
$67$ \( -2 + T \)
$71$ \( -16 + T \)
$73$ \( 10 + T \)
$79$ \( 16 + T \)
$83$ \( T \)
$89$ \( 4 + T \)
$97$ \( 2 + T \)
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