Newform orbit 630.2.u.a

• Label: 630.2.u.a
• Level: $630$
• Weight: $2$
• Character orbit: 630.u
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.03057532734\)
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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
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.

\(f(q)\)	\(=\)	\( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{5} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 4 q^{5}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(281\)	\(451\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 + \zeta_{12}^{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

109.1

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

−0.866025

+

0.500000i

0

0.500000

−

0.866025i

−0.133975

−

2.23205i

0

−1.73205

−

2.00000i

1.00000i

0

1.23205

+

1.86603i

109.2

0.866025

−

0.500000i

0

0.500000

−

0.866025i

−1.86603

−

1.23205i

0

1.73205

+

2.00000i

−

1.00000i

0

−2.23205

−

0.133975i

289.1

−0.866025

−

0.500000i

0

0.500000

+

0.866025i

−0.133975

+

2.23205i

0

−1.73205

+

2.00000i

−

1.00000i

0

1.23205

−

1.86603i

289.2

0.866025

+

0.500000i

0

0.500000

+

0.866025i

−1.86603

+

1.23205i

0

1.73205

−

2.00000i

1.00000i

0

−2.23205

+

0.133975i

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	630.2.u.a		4
3.b	odd	2	1		70.2.i.b	✓	4
5.b	even	2	1	inner	630.2.u.a		4
7.c	even	3	1	inner	630.2.u.a		4
12.b	even	2	1		560.2.bw.d		4
15.d	odd	2	1		70.2.i.b	✓	4
15.e	even	4	1		350.2.e.c		2
15.e	even	4	1		350.2.e.j		2
21.c	even	2	1		490.2.i.a		4
21.g	even	6	1		490.2.c.d		2
21.g	even	6	1		490.2.i.a		4
21.h	odd	6	1		70.2.i.b	✓	4
21.h	odd	6	1		490.2.c.a		2
35.j	even	6	1	inner	630.2.u.a		4
60.h	even	2	1		560.2.bw.d		4
84.n	even	6	1		560.2.bw.d		4
105.g	even	2	1		490.2.i.a		4
105.o	odd	6	1		70.2.i.b	✓	4
105.o	odd	6	1		490.2.c.a		2
105.p	even	6	1		490.2.c.d		2
105.p	even	6	1		490.2.i.a		4
105.w	odd	12	1		2450.2.a.j		1
105.w	odd	12	1		2450.2.a.bb		1
105.x	even	12	1		350.2.e.c		2
105.x	even	12	1		350.2.e.j		2
105.x	even	12	1		2450.2.a.k		1
105.x	even	12	1		2450.2.a.ba		1
420.ba	even	6	1		560.2.bw.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.i.b	✓	4	3.b	odd	2	1
70.2.i.b	✓	4	15.d	odd	2	1
70.2.i.b	✓	4	21.h	odd	6	1
70.2.i.b	✓	4	105.o	odd	6	1
350.2.e.c		2	15.e	even	4	1
350.2.e.c		2	105.x	even	12	1
350.2.e.j		2	15.e	even	4	1
350.2.e.j		2	105.x	even	12	1
490.2.c.a		2	21.h	odd	6	1
490.2.c.a		2	105.o	odd	6	1
490.2.c.d		2	21.g	even	6	1
490.2.c.d		2	105.p	even	6	1
490.2.i.a		4	21.c	even	2	1
490.2.i.a		4	21.g	even	6	1
490.2.i.a		4	105.g	even	2	1
490.2.i.a		4	105.p	even	6	1
560.2.bw.d		4	12.b	even	2	1
560.2.bw.d		4	60.h	even	2	1
560.2.bw.d		4	84.n	even	6	1
560.2.bw.d		4	420.ba	even	6	1
630.2.u.a		4	1.a	even	1	1	trivial
630.2.u.a		4	5.b	even	2	1	inner
630.2.u.a		4	7.c	even	3	1	inner
630.2.u.a		4	35.j	even	6	1	inner
2450.2.a.j		1	105.w	odd	12	1
2450.2.a.k		1	105.x	even	12	1
2450.2.a.ba		1	105.x	even	12	1
2450.2.a.bb		1	105.w	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

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

\( T_{37}^{4} - T_{37}^{2} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - T^{2} + 1 \)
$3$	\( T^{4} \)
$5$	T^4 + 4*T^3 + 11*T^2 + 20*T + 25
$7$	\( T^{4} + 2T^{2} + 49 \)
$11$	\( (T^{2} - 3 T + 9)^{2} \)