Newform orbit 700.2.r.b

• Label: 700.2.r.b
• Level: $700$
• Weight: $2$
• Character orbit: 700.r
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \zeta_{12} q^{3} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} - 2 \zeta_{12}^{2} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(-\zeta_{12}^{2}\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

149.1

−0.866025	+	0.500000i
0.866025	−	0.500000i
−0.866025	−	0.500000i
0.866025	+	0.500000i

0

−0.866025

+

0.500000i

0

0

0

1.73205

−

2.00000i

0

−1.00000

+

1.73205i

0

149.2

0

0.866025

−

0.500000i

0

0

0

−1.73205

+

2.00000i

0

−1.00000

+

1.73205i

0

249.1

0

−0.866025

−

0.500000i

0

0

0

1.73205

+

2.00000i

0

−1.00000

−

1.73205i

0

249.2

0

0.866025

+

0.500000i

0

0

0

−1.73205

−

2.00000i

0

−1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.r.b		4
5.b	even	2	1	inner	700.2.r.b		4
5.c	odd	4	1		28.2.e.a	✓	2
5.c	odd	4	1		700.2.i.c		2
7.c	even	3	1	inner	700.2.r.b		4
7.c	even	3	1		4900.2.e.i		2
7.d	odd	6	1		4900.2.e.h		2
15.e	even	4	1		252.2.k.c		2
20.e	even	4	1		112.2.i.b		2
35.f	even	4	1		196.2.e.a		2
35.i	odd	6	1		4900.2.e.h		2
35.j	even	6	1	inner	700.2.r.b		4
35.j	even	6	1		4900.2.e.i		2
35.k	even	12	1		196.2.a.a		1
35.k	even	12	1		196.2.e.a		2
35.k	even	12	1		4900.2.a.n		1
35.l	odd	12	1		28.2.e.a	✓	2
35.l	odd	12	1		196.2.a.b		1
35.l	odd	12	1		700.2.i.c		2
35.l	odd	12	1		4900.2.a.g		1
40.i	odd	4	1		448.2.i.e		2
40.k	even	4	1		448.2.i.c		2
45.k	odd	12	1		2268.2.i.a		2
45.k	odd	12	1		2268.2.l.h		2
45.l	even	12	1		2268.2.i.h		2
45.l	even	12	1		2268.2.l.a		2
60.l	odd	4	1		1008.2.s.p		2
105.k	odd	4	1		1764.2.k.b		2
105.w	odd	12	1		1764.2.a.j		1
105.w	odd	12	1		1764.2.k.b		2
105.x	even	12	1		252.2.k.c		2
105.x	even	12	1		1764.2.a.a		1
140.j	odd	4	1		784.2.i.d		2
140.w	even	12	1		112.2.i.b		2
140.w	even	12	1		784.2.a.d		1
140.x	odd	12	1		784.2.a.g		1
140.x	odd	12	1		784.2.i.d		2
280.bp	odd	12	1		3136.2.a.k		1
280.br	even	12	1		448.2.i.c		2
280.br	even	12	1		3136.2.a.s		1
280.bt	odd	12	1		448.2.i.e		2
280.bt	odd	12	1		3136.2.a.h		1
280.bv	even	12	1		3136.2.a.v		1
315.bt	odd	12	1		2268.2.l.h		2
315.bv	even	12	1		2268.2.l.a		2
315.bx	even	12	1		2268.2.i.h		2
315.ch	odd	12	1		2268.2.i.a		2
420.bp	odd	12	1		1008.2.s.p		2
420.bp	odd	12	1		7056.2.a.f		1
420.br	even	12	1		7056.2.a.bw		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.e.a	✓	2	5.c	odd	4	1
28.2.e.a	✓	2	35.l	odd	12	1
112.2.i.b		2	20.e	even	4	1
112.2.i.b		2	140.w	even	12	1
196.2.a.a		1	35.k	even	12	1
196.2.a.b		1	35.l	odd	12	1
196.2.e.a		2	35.f	even	4	1
196.2.e.a		2	35.k	even	12	1
252.2.k.c		2	15.e	even	4	1
252.2.k.c		2	105.x	even	12	1
448.2.i.c		2	40.k	even	4	1
448.2.i.c		2	280.br	even	12	1
448.2.i.e		2	40.i	odd	4	1
448.2.i.e		2	280.bt	odd	12	1
700.2.i.c		2	5.c	odd	4	1
700.2.i.c		2	35.l	odd	12	1
700.2.r.b		4	1.a	even	1	1	trivial
700.2.r.b		4	5.b	even	2	1	inner
700.2.r.b		4	7.c	even	3	1	inner
700.2.r.b		4	35.j	even	6	1	inner
784.2.a.d		1	140.w	even	12	1
784.2.a.g		1	140.x	odd	12	1
784.2.i.d		2	140.j	odd	4	1
784.2.i.d		2	140.x	odd	12	1
1008.2.s.p		2	60.l	odd	4	1
1008.2.s.p		2	420.bp	odd	12	1
1764.2.a.a		1	105.x	even	12	1
1764.2.a.j		1	105.w	odd	12	1
1764.2.k.b		2	105.k	odd	4	1
1764.2.k.b		2	105.w	odd	12	1
2268.2.i.a		2	45.k	odd	12	1
2268.2.i.a		2	315.ch	odd	12	1
2268.2.i.h		2	45.l	even	12	1
2268.2.i.h		2	315.bx	even	12	1
2268.2.l.a		2	45.l	even	12	1
2268.2.l.a		2	315.bv	even	12	1
2268.2.l.h		2	45.k	odd	12	1
2268.2.l.h		2	315.bt	odd	12	1
3136.2.a.h		1	280.bt	odd	12	1
3136.2.a.k		1	280.bp	odd	12	1
3136.2.a.s		1	280.br	even	12	1
3136.2.a.v		1	280.bv	even	12	1
4900.2.a.g		1	35.l	odd	12	1
4900.2.a.n		1	35.k	even	12	1
4900.2.e.h		2	7.d	odd	6	1
4900.2.e.h		2	35.i	odd	6	1
4900.2.e.i		2	7.c	even	3	1
4900.2.e.i		2	35.j	even	6	1
7056.2.a.f		1	420.bp	odd	12	1
7056.2.a.bw		1	420.br	even	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{4} - T_{3}^{2} + 1 \)

\( T_{11}^{2} - 3T_{11} + 9 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} - T^{2} + 1 \)
$5$	\( T^{4} \)
$7$	\( T^{4} + 2T^{2} + 49 \)
$11$	\( (T^{2} - 3 T + 9)^{2} \)