Newform orbit 624.2.c.a

• Label: 624.2.c.a
• Level: $624$
• Weight: $2$
• Character orbit: 624.c
• Analytic conductor: $4.983$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.98266508613\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\)	\(79\)	\(145\)	\(209\)	\(469\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

337.1

	−	1.00000i
		1.00000i

0

−1.00000

0

−

2.00000i

0

−

2.00000i

0

1.00000

0

337.2

0

−1.00000

0

2.00000i

0

2.00000i

0

1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	624.2.c.a		2
3.b	odd	2	1		1872.2.c.b		2
4.b	odd	2	1		78.2.b.a	✓	2
8.b	even	2	1		2496.2.c.m		2
8.d	odd	2	1		2496.2.c.f		2
12.b	even	2	1		234.2.b.a		2
13.b	even	2	1	inner	624.2.c.a		2
13.d	odd	4	1		8112.2.a.g		1
13.d	odd	4	1		8112.2.a.j		1
20.d	odd	2	1		1950.2.b.c		2
20.e	even	4	1		1950.2.f.d		2
20.e	even	4	1		1950.2.f.g		2
28.d	even	2	1		3822.2.c.d		2
39.d	odd	2	1		1872.2.c.b		2
52.b	odd	2	1		78.2.b.a	✓	2
52.f	even	4	1		1014.2.a.b		1
52.f	even	4	1		1014.2.a.g		1
52.i	odd	6	2		1014.2.i.c		4
52.j	odd	6	2		1014.2.i.c		4
52.l	even	12	2		1014.2.e.b		2
52.l	even	12	2		1014.2.e.e		2
104.e	even	2	1		2496.2.c.m		2
104.h	odd	2	1		2496.2.c.f		2
156.h	even	2	1		234.2.b.a		2
156.l	odd	4	1		3042.2.a.c		1
156.l	odd	4	1		3042.2.a.n		1
260.g	odd	2	1		1950.2.b.c		2
260.p	even	4	1		1950.2.f.d		2
260.p	even	4	1		1950.2.f.g		2
364.h	even	2	1		3822.2.c.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.2.b.a	✓	2	4.b	odd	2	1
78.2.b.a	✓	2	52.b	odd	2	1
234.2.b.a		2	12.b	even	2	1
234.2.b.a		2	156.h	even	2	1
624.2.c.a		2	1.a	even	1	1	trivial
624.2.c.a		2	13.b	even	2	1	inner
1014.2.a.b		1	52.f	even	4	1
1014.2.a.g		1	52.f	even	4	1
1014.2.e.b		2	52.l	even	12	2
1014.2.e.e		2	52.l	even	12	2
1014.2.i.c		4	52.i	odd	6	2
1014.2.i.c		4	52.j	odd	6	2
1872.2.c.b		2	3.b	odd	2	1
1872.2.c.b		2	39.d	odd	2	1
1950.2.b.c		2	20.d	odd	2	1
1950.2.b.c		2	260.g	odd	2	1
1950.2.f.d		2	20.e	even	4	1
1950.2.f.d		2	260.p	even	4	1
1950.2.f.g		2	20.e	even	4	1
1950.2.f.g		2	260.p	even	4	1
2496.2.c.f		2	8.d	odd	2	1
2496.2.c.f		2	104.h	odd	2	1
2496.2.c.m		2	8.b	even	2	1
2496.2.c.m		2	104.e	even	2	1
3042.2.a.c		1	156.l	odd	4	1
3042.2.a.n		1	156.l	odd	4	1
3822.2.c.d		2	28.d	even	2	1
3822.2.c.d		2	364.h	even	2	1
8112.2.a.g		1	13.d	odd	4	1
8112.2.a.j		1	13.d	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{2} + 4 \)

\( T_{7}^{2} + 4 \)

\( T_{11} \)

Hecke characteristic polynomials

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