Newform orbit 2156.2.i.c

• Label: 2156.2.i.c
• Level: $2156$
• Weight: $2$
• Character orbit: 2156.i
• Analytic conductor: $17.216$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.2157466758\)
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:	no (minimal twist has level 44)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\)	\(981\)	\(1079\)	\(1277\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

177.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0.500000

+

0.866025i

0

−1.50000

+

2.59808i

0

0

0

1.00000

−

1.73205i

0

1145.1

0

0.500000

−

0.866025i

0

−1.50000

−

2.59808i

0

0

0

1.00000

+

1.73205i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2156.2.i.c		2
7.b	odd	2	1		2156.2.i.b		2
7.c	even	3	1		2156.2.a.a		1
7.c	even	3	1	inner	2156.2.i.c		2
7.d	odd	6	1		44.2.a.a	✓	1
7.d	odd	6	1		2156.2.i.b		2
21.g	even	6	1		396.2.a.c		1
28.f	even	6	1		176.2.a.a		1
28.g	odd	6	1		8624.2.a.w		1
35.i	odd	6	1		1100.2.a.b		1
35.k	even	12	2		1100.2.b.c		2
56.j	odd	6	1		704.2.a.f		1
56.m	even	6	1		704.2.a.i		1
63.i	even	6	1		3564.2.i.a		2
63.k	odd	6	1		3564.2.i.j		2
63.s	even	6	1		3564.2.i.a		2
63.t	odd	6	1		3564.2.i.j		2
77.i	even	6	1		484.2.a.a		1
77.n	even	30	4		484.2.e.b		4
77.p	odd	30	4		484.2.e.a		4
84.j	odd	6	1		1584.2.a.p		1
91.s	odd	6	1		7436.2.a.d		1
105.p	even	6	1		9900.2.a.h		1
105.w	odd	12	2		9900.2.c.g		2
112.v	even	12	2		2816.2.c.k		2
112.x	odd	12	2		2816.2.c.e		2
140.s	even	6	1		4400.2.a.v		1
140.x	odd	12	2		4400.2.b.k		2
168.ba	even	6	1		6336.2.a.j		1
168.be	odd	6	1		6336.2.a.i		1
231.k	odd	6	1		4356.2.a.j		1
308.m	odd	6	1		1936.2.a.c		1
616.s	even	6	1		7744.2.a.m		1
616.z	odd	6	1		7744.2.a.bc		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.2.a.a	✓	1	7.d	odd	6	1
176.2.a.a		1	28.f	even	6	1
396.2.a.c		1	21.g	even	6	1
484.2.a.a		1	77.i	even	6	1
484.2.e.a		4	77.p	odd	30	4
484.2.e.b		4	77.n	even	30	4
704.2.a.f		1	56.j	odd	6	1
704.2.a.i		1	56.m	even	6	1
1100.2.a.b		1	35.i	odd	6	1
1100.2.b.c		2	35.k	even	12	2
1584.2.a.p		1	84.j	odd	6	1
1936.2.a.c		1	308.m	odd	6	1
2156.2.a.a		1	7.c	even	3	1
2156.2.i.b		2	7.b	odd	2	1
2156.2.i.b		2	7.d	odd	6	1
2156.2.i.c		2	1.a	even	1	1	trivial
2156.2.i.c		2	7.c	even	3	1	inner
2816.2.c.e		2	112.x	odd	12	2
2816.2.c.k		2	112.v	even	12	2
3564.2.i.a		2	63.i	even	6	1
3564.2.i.a		2	63.s	even	6	1
3564.2.i.j		2	63.k	odd	6	1
3564.2.i.j		2	63.t	odd	6	1
4356.2.a.j		1	231.k	odd	6	1
4400.2.a.v		1	140.s	even	6	1
4400.2.b.k		2	140.x	odd	12	2
6336.2.a.i		1	168.be	odd	6	1
6336.2.a.j		1	168.ba	even	6	1
7436.2.a.d		1	91.s	odd	6	1
7744.2.a.m		1	616.s	even	6	1
7744.2.a.bc		1	616.z	odd	6	1
8624.2.a.w		1	28.g	odd	6	1
9900.2.a.h		1	105.p	even	6	1
9900.2.c.g		2	105.w	odd	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2156, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)

\( T_{5}^{2} + 3T_{5} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - T + 1 \)