Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - \beta q^{5} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta q^{5} + q^{9} - \beta q^{11} + ( - \beta - 1) q^{13} + \beta q^{15} + 6 q^{17} + 2 \beta q^{19} - 7 q^{25} - q^{27} - 6 q^{29} + 2 \beta q^{31} + \beta q^{33} + (\beta + 1) q^{39} + \beta q^{41} + 8 q^{43} - \beta q^{45} - \beta q^{47} + 7 q^{49} - 6 q^{51} + 6 q^{53} - 12 q^{55} - 2 \beta q^{57} + \beta q^{59} + 10 q^{61} + (\beta - 12) q^{65} - 4 \beta q^{67} - 3 \beta q^{71} + 2 \beta q^{73} + 7 q^{75} - 8 q^{79} + q^{81} + \beta q^{83} - 6 \beta q^{85} + 6 q^{87} + 5 \beta q^{89} - 2 \beta q^{93} + 24 q^{95} - 2 \beta q^{97} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 2 q^{13} + 12 q^{17} - 14 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{39} + 16 q^{43} + 14 q^{49} - 12 q^{51} + 12 q^{53} - 24 q^{55} + 20 q^{61} - 24 q^{65} + 14 q^{75} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.00000 0 3.46410i 0 0 0 1.00000 0
25.2 0 −1.00000 0 3.46410i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.b.a 2
3.b odd 2 1 468.2.b.a 2
4.b odd 2 1 624.2.c.f 2
5.b even 2 1 3900.2.c.c 2
5.c odd 4 2 3900.2.j.h 4
7.b odd 2 1 7644.2.e.g 2
8.b even 2 1 2496.2.c.l 2
8.d odd 2 1 2496.2.c.e 2
12.b even 2 1 1872.2.c.c 2
13.b even 2 1 inner 156.2.b.a 2
13.c even 3 1 2028.2.q.b 2
13.c even 3 1 2028.2.q.c 2
13.d odd 4 2 2028.2.a.g 2
13.e even 6 1 2028.2.q.b 2
13.e even 6 1 2028.2.q.c 2
13.f odd 12 4 2028.2.i.i 4
39.d odd 2 1 468.2.b.a 2
39.f even 4 2 6084.2.a.v 2
52.b odd 2 1 624.2.c.f 2
52.f even 4 2 8112.2.a.bs 2
65.d even 2 1 3900.2.c.c 2
65.h odd 4 2 3900.2.j.h 4
91.b odd 2 1 7644.2.e.g 2
104.e even 2 1 2496.2.c.l 2
104.h odd 2 1 2496.2.c.e 2
156.h even 2 1 1872.2.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.a 2 1.a even 1 1 trivial
156.2.b.a 2 13.b even 2 1 inner
468.2.b.a 2 3.b odd 2 1
468.2.b.a 2 39.d odd 2 1
624.2.c.f 2 4.b odd 2 1
624.2.c.f 2 52.b odd 2 1
1872.2.c.c 2 12.b even 2 1
1872.2.c.c 2 156.h even 2 1
2028.2.a.g 2 13.d odd 4 2
2028.2.i.i 4 13.f odd 12 4
2028.2.q.b 2 13.c even 3 1
2028.2.q.b 2 13.e even 6 1
2028.2.q.c 2 13.c even 3 1
2028.2.q.c 2 13.e even 6 1
2496.2.c.e 2 8.d odd 2 1
2496.2.c.e 2 104.h odd 2 1
2496.2.c.l 2 8.b even 2 1
2496.2.c.l 2 104.e even 2 1
3900.2.c.c 2 5.b even 2 1
3900.2.c.c 2 65.d even 2 1
3900.2.j.h 4 5.c odd 4 2
3900.2.j.h 4 65.h odd 4 2
6084.2.a.v 2 39.f even 4 2
7644.2.e.g 2 7.b odd 2 1
7644.2.e.g 2 91.b odd 2 1
8112.2.a.bs 2 52.f even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 48 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 192 \) Copy content Toggle raw display
$71$ \( T^{2} + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 300 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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