Newform orbit 3900.2.a.a

• Label: 3900.2.a.a
• Level: $3900$
• Weight: $2$
• Character orbit: 3900.a
• Self dual: yes
• Analytic conductor: $31.142$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.1416567883\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \)

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

0

0

−2.00000

0

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)
\(5\)	\(1\)
\(13\)	\(1\)

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	3900.2.a.a		1
5.b	even	2	1		156.2.a.b	✓	1
5.c	odd	4	2		3900.2.h.e		2
15.d	odd	2	1		468.2.a.c		1
20.d	odd	2	1		624.2.a.b		1
35.c	odd	2	1		7644.2.a.a		1
40.e	odd	2	1		2496.2.a.v		1
40.f	even	2	1		2496.2.a.h		1
45.h	odd	6	2		4212.2.i.g		2
45.j	even	6	2		4212.2.i.f		2
60.h	even	2	1		1872.2.a.i		1
65.d	even	2	1		2028.2.a.e		1
65.g	odd	4	2		2028.2.b.d		2
65.l	even	6	2		2028.2.i.c		2
65.n	even	6	2		2028.2.i.b		2
65.s	odd	12	4		2028.2.q.d		4
120.i	odd	2	1		7488.2.a.bf		1
120.m	even	2	1		7488.2.a.bb		1
195.e	odd	2	1		6084.2.a.h		1
195.n	even	4	2		6084.2.b.a		2
260.g	odd	2	1		8112.2.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
156.2.a.b	✓	1	5.b	even	2	1
468.2.a.c		1	15.d	odd	2	1
624.2.a.b		1	20.d	odd	2	1
1872.2.a.i		1	60.h	even	2	1
2028.2.a.e		1	65.d	even	2	1
2028.2.b.d		2	65.g	odd	4	2
2028.2.i.b		2	65.n	even	6	2
2028.2.i.c		2	65.l	even	6	2
2028.2.q.d		4	65.s	odd	12	4
2496.2.a.h		1	40.f	even	2	1
2496.2.a.v		1	40.e	odd	2	1
3900.2.a.a		1	1.a	even	1	1	trivial
3900.2.h.e		2	5.c	odd	4	2
4212.2.i.f		2	45.j	even	6	2
4212.2.i.g		2	45.h	odd	6	2
6084.2.a.h		1	195.e	odd	2	1
6084.2.b.a		2	195.n	even	4	2
7488.2.a.bb		1	120.m	even	2	1
7488.2.a.bf		1	120.i	odd	2	1
7644.2.a.a		1	35.c	odd	2	1
8112.2.a.i		1	260.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3900))\):

\( T_{7} + 2 \)

\( T_{11} \)

\( T_{17} - 6 \)

Hecke characteristic polynomials

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