Newform orbit 3136.2.a.a

• Label: 3136.2.a.a
• Level: $3136$
• Weight: $2$
• Character orbit: 3136.a
• Self dual: yes
• Analytic conductor: $25.041$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

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 56)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 3 q^{3} - q^{5} + 6 q^{9}+O(q^{10}) \)

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

−3.00000

0

−1.00000

0

0

0

6.00000

0

Atkin-Lehner signs

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

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	3136.2.a.a		1
4.b	odd	2	1		3136.2.a.bb		1
7.b	odd	2	1		3136.2.a.bc		1
7.d	odd	6	2		448.2.i.a		2
8.b	even	2	1		784.2.a.j		1
8.d	odd	2	1		392.2.a.a		1
24.f	even	2	1		3528.2.a.k		1
24.h	odd	2	1		7056.2.a.s		1
28.d	even	2	1		3136.2.a.b		1
28.f	even	6	2		448.2.i.f		2
40.e	odd	2	1		9800.2.a.bp		1
56.e	even	2	1		392.2.a.f		1
56.h	odd	2	1		784.2.a.a		1
56.j	odd	6	2		112.2.i.c		2
56.k	odd	6	2		392.2.i.f		2
56.m	even	6	2		56.2.i.a	✓	2
56.p	even	6	2		784.2.i.a		2
168.e	odd	2	1		3528.2.a.r		1
168.i	even	2	1		7056.2.a.bi		1
168.v	even	6	2		3528.2.s.o		2
168.ba	even	6	2		1008.2.s.e		2
168.be	odd	6	2		504.2.s.e		2
280.n	even	2	1		9800.2.a.b		1
280.ba	even	6	2		1400.2.q.g		2
280.bp	odd	12	4		1400.2.bh.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.i.a	✓	2	56.m	even	6	2
112.2.i.c		2	56.j	odd	6	2
392.2.a.a		1	8.d	odd	2	1
392.2.a.f		1	56.e	even	2	1
392.2.i.f		2	56.k	odd	6	2
448.2.i.a		2	7.d	odd	6	2
448.2.i.f		2	28.f	even	6	2
504.2.s.e		2	168.be	odd	6	2
784.2.a.a		1	56.h	odd	2	1
784.2.a.j		1	8.b	even	2	1
784.2.i.a		2	56.p	even	6	2
1008.2.s.e		2	168.ba	even	6	2
1400.2.q.g		2	280.ba	even	6	2
1400.2.bh.f		4	280.bp	odd	12	4
3136.2.a.a		1	1.a	even	1	1	trivial
3136.2.a.b		1	28.d	even	2	1
3136.2.a.bb		1	4.b	odd	2	1
3136.2.a.bc		1	7.b	odd	2	1
3528.2.a.k		1	24.f	even	2	1
3528.2.a.r		1	168.e	odd	2	1
3528.2.s.o		2	168.v	even	6	2
7056.2.a.s		1	24.h	odd	2	1
7056.2.a.bi		1	168.i	even	2	1
9800.2.a.b		1	280.n	even	2	1
9800.2.a.bp		1	40.e	odd	2	1

Hecke kernels

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

\( T_{3} + 3 \)

\( T_{5} + 1 \)

\( T_{11} + 1 \)

Hecke characteristic polynomials

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