Newform orbit 9800.2.a.c

• Label: 9800.2.a.c
• Level: $9800$
• Weight: $2$
• Character orbit: 9800.a
• Self dual: yes
• Analytic conductor: $78.253$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(78.2533939809\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

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

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

0

0

0

0

6.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(7\)	\(-1\)

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	9800.2.a.c		1
5.b	even	2	1		9800.2.a.bq		1
7.b	odd	2	1		200.2.a.e	yes	1
21.c	even	2	1		1800.2.a.h		1
28.d	even	2	1		400.2.a.a		1
35.c	odd	2	1		200.2.a.a	✓	1
35.f	even	4	2		200.2.c.a		2
56.e	even	2	1		1600.2.a.y		1
56.h	odd	2	1		1600.2.a.a		1
84.h	odd	2	1		3600.2.a.bf		1
105.g	even	2	1		1800.2.a.r		1
105.k	odd	4	2		1800.2.f.f		2
140.c	even	2	1		400.2.a.h		1
140.j	odd	4	2		400.2.c.a		2
280.c	odd	2	1		1600.2.a.x		1
280.n	even	2	1		1600.2.a.b		1
280.s	even	4	2		1600.2.c.a		2
280.y	odd	4	2		1600.2.c.b		2
420.o	odd	2	1		3600.2.a.m		1
420.w	even	4	2		3600.2.f.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.a.a	✓	1	35.c	odd	2	1
200.2.a.e	yes	1	7.b	odd	2	1
200.2.c.a		2	35.f	even	4	2
400.2.a.a		1	28.d	even	2	1
400.2.a.h		1	140.c	even	2	1
400.2.c.a		2	140.j	odd	4	2
1600.2.a.a		1	56.h	odd	2	1
1600.2.a.b		1	280.n	even	2	1
1600.2.a.x		1	280.c	odd	2	1
1600.2.a.y		1	56.e	even	2	1
1600.2.c.a		2	280.s	even	4	2
1600.2.c.b		2	280.y	odd	4	2
1800.2.a.h		1	21.c	even	2	1
1800.2.a.r		1	105.g	even	2	1
1800.2.f.f		2	105.k	odd	4	2
3600.2.a.m		1	420.o	odd	2	1
3600.2.a.bf		1	84.h	odd	2	1
3600.2.f.n		2	420.w	even	4	2
9800.2.a.c		1	1.a	even	1	1	trivial
9800.2.a.bq		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9800))\):

\( T_{3} + 3 \)

\( T_{11} - 1 \)

\( T_{13} - 4 \)

\( T_{19} + 1 \)

\( T_{23} - 2 \)

Hecke characteristic polynomials

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