Newform orbit 156.2.i.a

• Label: 156.2.i.a
• Level: $156$
• Weight: $2$
• Character orbit: 156.i
• Analytic conductor: $1.246$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.24566627153\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{3} + 2 q^{5} - \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 4 q^{5} - q^{7} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(53\)	\(79\)	\(145\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

61.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0.500000

−

0.866025i

0

2.00000

0

−0.500000

−

0.866025i

0

−0.500000

−

0.866025i

0

133.1

0

0.500000

+

0.866025i

0

2.00000

0

−0.500000

+

0.866025i

0

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	156.2.i.a	✓	2
3.b	odd	2	1		468.2.l.b		2
4.b	odd	2	1		624.2.q.d		2
5.b	even	2	1		3900.2.q.e		2
5.c	odd	4	2		3900.2.by.c		4
12.b	even	2	1		1872.2.t.e		2
13.b	even	2	1		2028.2.i.f		2
13.c	even	3	1	inner	156.2.i.a	✓	2
13.c	even	3	1		2028.2.a.b		1
13.d	odd	4	2		2028.2.q.g		4
13.e	even	6	1		2028.2.a.a		1
13.e	even	6	1		2028.2.i.f		2
13.f	odd	12	2		2028.2.b.c		2
13.f	odd	12	2		2028.2.q.g		4
39.h	odd	6	1		6084.2.a.l		1
39.i	odd	6	1		468.2.l.b		2
39.i	odd	6	1		6084.2.a.e		1
39.k	even	12	2		6084.2.b.b		2
52.i	odd	6	1		8112.2.a.u		1
52.j	odd	6	1		624.2.q.d		2
52.j	odd	6	1		8112.2.a.bd		1
65.n	even	6	1		3900.2.q.e		2
65.q	odd	12	2		3900.2.by.c		4
156.p	even	6	1		1872.2.t.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.i.a	✓	2	1.a	even	1	1	trivial
156.2.i.a	✓	2	13.c	even	3	1	inner
468.2.l.b		2	3.b	odd	2	1
468.2.l.b		2	39.i	odd	6	1
624.2.q.d		2	4.b	odd	2	1
624.2.q.d		2	52.j	odd	6	1
1872.2.t.e		2	12.b	even	2	1
1872.2.t.e		2	156.p	even	6	1
2028.2.a.a		1	13.e	even	6	1
2028.2.a.b		1	13.c	even	3	1
2028.2.b.c		2	13.f	odd	12	2
2028.2.i.f		2	13.b	even	2	1
2028.2.i.f		2	13.e	even	6	1
2028.2.q.g		4	13.d	odd	4	2
2028.2.q.g		4	13.f	odd	12	2
3900.2.q.e		2	5.b	even	2	1
3900.2.q.e		2	65.n	even	6	1
3900.2.by.c		4	5.c	odd	4	2
3900.2.by.c		4	65.q	odd	12	2
6084.2.a.e		1	39.i	odd	6	1
6084.2.a.l		1	39.h	odd	6	1
6084.2.b.b		2	39.k	even	12	2
8112.2.a.u		1	52.i	odd	6	1
8112.2.a.bd		1	52.j	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(156, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( (T - 2)^{2} \)
$7$	\( T^{2} + T + 1 \)
$11$	\( T^{2} + 2T + 4 \)