Newform orbit 4900.2.e.h

• Label: 4900.2.e.h
• Level: $4900$
• Weight: $2$
• Character orbit: 4900.e
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(39.1266969904\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + i q^{3} + 2 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(1177\)	\(2451\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2549.1

	−	1.00000i
		1.00000i

0

−

1.00000i

0

0

0

0

0

2.00000

0

2549.2

0

1.00000i

0

0

0

0

0

2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4900.2.e.h		2
5.b	even	2	1	inner	4900.2.e.h		2
5.c	odd	4	1		196.2.a.a		1
5.c	odd	4	1		4900.2.a.n		1
7.b	odd	2	1		4900.2.e.i		2
7.d	odd	6	2		700.2.r.b		4
15.e	even	4	1		1764.2.a.j		1
20.e	even	4	1		784.2.a.g		1
35.c	odd	2	1		4900.2.e.i		2
35.f	even	4	1		196.2.a.b		1
35.f	even	4	1		4900.2.a.g		1
35.i	odd	6	2		700.2.r.b		4
35.k	even	12	2		28.2.e.a	✓	2
35.k	even	12	2		700.2.i.c		2
35.l	odd	12	2		196.2.e.a		2
40.i	odd	4	1		3136.2.a.v		1
40.k	even	4	1		3136.2.a.k		1
60.l	odd	4	1		7056.2.a.bw		1
105.k	odd	4	1		1764.2.a.a		1
105.w	odd	12	2		252.2.k.c		2
105.x	even	12	2		1764.2.k.b		2
140.j	odd	4	1		784.2.a.d		1
140.w	even	12	2		784.2.i.d		2
140.x	odd	12	2		112.2.i.b		2
280.s	even	4	1		3136.2.a.h		1
280.y	odd	4	1		3136.2.a.s		1
280.bp	odd	12	2		448.2.i.c		2
280.bv	even	12	2		448.2.i.e		2
315.bs	even	12	2		2268.2.i.a		2
315.bu	odd	12	2		2268.2.i.h		2
315.bw	odd	12	2		2268.2.l.a		2
315.cg	even	12	2		2268.2.l.h		2
420.w	even	4	1		7056.2.a.f		1
420.br	even	12	2		1008.2.s.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.e.a	✓	2	35.k	even	12	2
112.2.i.b		2	140.x	odd	12	2
196.2.a.a		1	5.c	odd	4	1
196.2.a.b		1	35.f	even	4	1
196.2.e.a		2	35.l	odd	12	2
252.2.k.c		2	105.w	odd	12	2
448.2.i.c		2	280.bp	odd	12	2
448.2.i.e		2	280.bv	even	12	2
700.2.i.c		2	35.k	even	12	2
700.2.r.b		4	7.d	odd	6	2
700.2.r.b		4	35.i	odd	6	2
784.2.a.d		1	140.j	odd	4	1
784.2.a.g		1	20.e	even	4	1
784.2.i.d		2	140.w	even	12	2
1008.2.s.p		2	420.br	even	12	2
1764.2.a.a		1	105.k	odd	4	1
1764.2.a.j		1	15.e	even	4	1
1764.2.k.b		2	105.x	even	12	2
2268.2.i.a		2	315.bs	even	12	2
2268.2.i.h		2	315.bu	odd	12	2
2268.2.l.a		2	315.bw	odd	12	2
2268.2.l.h		2	315.cg	even	12	2
3136.2.a.h		1	280.s	even	4	1
3136.2.a.k		1	40.k	even	4	1
3136.2.a.s		1	280.y	odd	4	1
3136.2.a.v		1	40.i	odd	4	1
4900.2.a.g		1	35.f	even	4	1
4900.2.a.n		1	5.c	odd	4	1
4900.2.e.h		2	1.a	even	1	1	trivial
4900.2.e.h		2	5.b	even	2	1	inner
4900.2.e.i		2	7.b	odd	2	1
4900.2.e.i		2	35.c	odd	2	1
7056.2.a.f		1	420.w	even	4	1
7056.2.a.bw		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4900, [\chi])\):

\( T_{3}^{2} + 1 \)

\( T_{11} + 3 \)

\( T_{19} + 1 \)

\( T_{31} - 7 \)

Hecke characteristic polynomials

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