Newform orbit 3136.2.a.m

• Label: 3136.2.a.m
• Level: $3136$
• Weight: $2$
• Character orbit: 3136.a
• Self dual: yes
• Analytic conductor: $25.041$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.0410860739\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	\(+1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 2 q^{5} - 3 q^{9} + 6 q^{13} - 2 q^{17} - q^{25} + 10 q^{29} + 2 q^{37} - 10 q^{41} + 6 q^{45} - 14 q^{53} - 10 q^{61} - 12 q^{65} + 6 q^{73} + 9 q^{81} + 4 q^{85} - 10 q^{89} - 18 q^{97}+O(q^{100}) \)

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

0

0

−2.00000

0

0

0

−3.00000

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3136.2.a.m		1
4.b	odd	2	1	CM	3136.2.a.m		1
7.b	odd	2	1		64.2.a.a		1
8.b	even	2	1		1568.2.a.e		1
8.d	odd	2	1		1568.2.a.e		1
21.c	even	2	1		576.2.a.c		1
28.d	even	2	1		64.2.a.a		1
35.c	odd	2	1		1600.2.a.n		1
35.f	even	4	2		1600.2.c.l		2
56.e	even	2	1		32.2.a.a	✓	1
56.h	odd	2	1		32.2.a.a	✓	1
56.j	odd	6	2		1568.2.i.g		2
56.k	odd	6	2		1568.2.i.f		2
56.m	even	6	2		1568.2.i.g		2
56.p	even	6	2		1568.2.i.f		2
77.b	even	2	1		7744.2.a.v		1
84.h	odd	2	1		576.2.a.c		1
112.j	even	4	2		256.2.b.b		2
112.l	odd	4	2		256.2.b.b		2
140.c	even	2	1		1600.2.a.n		1
140.j	odd	4	2		1600.2.c.l		2
168.e	odd	2	1		288.2.a.d		1
168.i	even	2	1		288.2.a.d		1
224.v	odd	8	4		1024.2.e.j		4
224.x	even	8	4		1024.2.e.j		4
280.c	odd	2	1		800.2.a.d		1
280.n	even	2	1		800.2.a.d		1
280.s	even	4	2		800.2.c.e		2
280.y	odd	4	2		800.2.c.e		2
308.g	odd	2	1		7744.2.a.v		1
336.v	odd	4	2		2304.2.d.j		2
336.y	even	4	2		2304.2.d.j		2
504.be	even	6	2		2592.2.i.t		2
504.bn	odd	6	2		2592.2.i.t		2
504.cc	even	6	2		2592.2.i.e		2
504.co	odd	6	2		2592.2.i.e		2
616.g	odd	2	1		3872.2.a.f		1
616.o	even	2	1		3872.2.a.f		1
728.b	even	2	1		5408.2.a.g		1
728.l	odd	2	1		5408.2.a.g		1
840.b	odd	2	1		7200.2.a.v		1
840.u	even	2	1		7200.2.a.v		1
840.bm	even	4	2		7200.2.f.m		2
840.bp	odd	4	2		7200.2.f.m		2
952.e	odd	2	1		9248.2.a.f		1
952.k	even	2	1		9248.2.a.f		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.2.a.a	✓	1	56.e	even	2	1
32.2.a.a	✓	1	56.h	odd	2	1
64.2.a.a		1	7.b	odd	2	1
64.2.a.a		1	28.d	even	2	1
256.2.b.b		2	112.j	even	4	2
256.2.b.b		2	112.l	odd	4	2
288.2.a.d		1	168.e	odd	2	1
288.2.a.d		1	168.i	even	2	1
576.2.a.c		1	21.c	even	2	1
576.2.a.c		1	84.h	odd	2	1
800.2.a.d		1	280.c	odd	2	1
800.2.a.d		1	280.n	even	2	1
800.2.c.e		2	280.s	even	4	2
800.2.c.e		2	280.y	odd	4	2
1024.2.e.j		4	224.v	odd	8	4
1024.2.e.j		4	224.x	even	8	4
1568.2.a.e		1	8.b	even	2	1
1568.2.a.e		1	8.d	odd	2	1
1568.2.i.f		2	56.k	odd	6	2
1568.2.i.f		2	56.p	even	6	2
1568.2.i.g		2	56.j	odd	6	2
1568.2.i.g		2	56.m	even	6	2
1600.2.a.n		1	35.c	odd	2	1
1600.2.a.n		1	140.c	even	2	1
1600.2.c.l		2	35.f	even	4	2
1600.2.c.l		2	140.j	odd	4	2
2304.2.d.j		2	336.v	odd	4	2
2304.2.d.j		2	336.y	even	4	2
2592.2.i.e		2	504.cc	even	6	2
2592.2.i.e		2	504.co	odd	6	2
2592.2.i.t		2	504.be	even	6	2
2592.2.i.t		2	504.bn	odd	6	2
3136.2.a.m		1	1.a	even	1	1	trivial
3136.2.a.m		1	4.b	odd	2	1	CM
3872.2.a.f		1	616.g	odd	2	1
3872.2.a.f		1	616.o	even	2	1
5408.2.a.g		1	728.b	even	2	1
5408.2.a.g		1	728.l	odd	2	1
7200.2.a.v		1	840.b	odd	2	1
7200.2.a.v		1	840.u	even	2	1
7200.2.f.m		2	840.bm	even	4	2
7200.2.f.m		2	840.bp	odd	4	2
7744.2.a.v		1	77.b	even	2	1
7744.2.a.v		1	308.g	odd	2	1
9248.2.a.f		1	952.e	odd	2	1
9248.2.a.f		1	952.k	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3} \)

\( T_{5} + 2 \)

\( T_{11} \)

Hecke characteristic polynomials

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