Newform orbit 2156.2.a.a

• Label: 2156.2.a.a
• Level: $2156$
• Weight: $2$
• Character orbit: 2156.a
• Self dual: yes
• Analytic conductor: $17.216$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(17.2157466758\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} + 3 q^{5} - 2 q^{9}+O(q^{10}) \)

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

3.00000

0

0

0

−2.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(7\)	\(-1\)
\(11\)	\(1\)

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	2156.2.a.a		1
4.b	odd	2	1		8624.2.a.w		1
7.b	odd	2	1		44.2.a.a	✓	1
7.c	even	3	2		2156.2.i.c		2
7.d	odd	6	2		2156.2.i.b		2
21.c	even	2	1		396.2.a.c		1
28.d	even	2	1		176.2.a.a		1
35.c	odd	2	1		1100.2.a.b		1
35.f	even	4	2		1100.2.b.c		2
56.e	even	2	1		704.2.a.i		1
56.h	odd	2	1		704.2.a.f		1
63.l	odd	6	2		3564.2.i.j		2
63.o	even	6	2		3564.2.i.a		2
77.b	even	2	1		484.2.a.a		1
77.j	odd	10	4		484.2.e.a		4
77.l	even	10	4		484.2.e.b		4
84.h	odd	2	1		1584.2.a.p		1
91.b	odd	2	1		7436.2.a.d		1
105.g	even	2	1		9900.2.a.h		1
105.k	odd	4	2		9900.2.c.g		2
112.j	even	4	2		2816.2.c.k		2
112.l	odd	4	2		2816.2.c.e		2
140.c	even	2	1		4400.2.a.v		1
140.j	odd	4	2		4400.2.b.k		2
168.e	odd	2	1		6336.2.a.i		1
168.i	even	2	1		6336.2.a.j		1
231.h	odd	2	1		4356.2.a.j		1
308.g	odd	2	1		1936.2.a.c		1
616.g	odd	2	1		7744.2.a.bc		1
616.o	even	2	1		7744.2.a.m		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.2.a.a	✓	1	7.b	odd	2	1
176.2.a.a		1	28.d	even	2	1
396.2.a.c		1	21.c	even	2	1
484.2.a.a		1	77.b	even	2	1
484.2.e.a		4	77.j	odd	10	4
484.2.e.b		4	77.l	even	10	4
704.2.a.f		1	56.h	odd	2	1
704.2.a.i		1	56.e	even	2	1
1100.2.a.b		1	35.c	odd	2	1
1100.2.b.c		2	35.f	even	4	2
1584.2.a.p		1	84.h	odd	2	1
1936.2.a.c		1	308.g	odd	2	1
2156.2.a.a		1	1.a	even	1	1	trivial
2156.2.i.b		2	7.d	odd	6	2
2156.2.i.c		2	7.c	even	3	2
2816.2.c.e		2	112.l	odd	4	2
2816.2.c.k		2	112.j	even	4	2
3564.2.i.a		2	63.o	even	6	2
3564.2.i.j		2	63.l	odd	6	2
4356.2.a.j		1	231.h	odd	2	1
4400.2.a.v		1	140.c	even	2	1
4400.2.b.k		2	140.j	odd	4	2
6336.2.a.i		1	168.e	odd	2	1
6336.2.a.j		1	168.i	even	2	1
7436.2.a.d		1	91.b	odd	2	1
7744.2.a.m		1	616.o	even	2	1
7744.2.a.bc		1	616.g	odd	2	1
8624.2.a.w		1	4.b	odd	2	1
9900.2.a.h		1	105.g	even	2	1
9900.2.c.g		2	105.k	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2156))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 1 \)
$5$	\( T - 3 \)
$7$	\( T \)
$11$	\( T + 1 \)