Newform orbit 578.2.c.d

• Label: 578.2.c.d
• Level: $578$
• Weight: $2$
• Character orbit: 578.c
• Analytic conductor: $4.615$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.c (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.61535323683\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - i q^{2} + (2 i + 2) q^{3} - q^{4} + (2 i + 2) q^{5} + ( - 2 i + 2) q^{6} + i q^{8} + 5 i q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{4} + 4 q^{5} + 4 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/578\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(i\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

251.1

		1.00000i
	−	1.00000i

−

1.00000i

2.00000

+

2.00000i

−1.00000

2.00000

+

2.00000i

2.00000

−

2.00000i

0

1.00000i

5.00000i

2.00000

−

2.00000i

327.1

1.00000i

2.00000

−

2.00000i

−1.00000

2.00000

−

2.00000i

2.00000

+

2.00000i

0

−

1.00000i

−

5.00000i

2.00000

+

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	578.2.c.d		2
17.b	even	2	1		578.2.c.a		2
17.c	even	4	1		578.2.c.a		2
17.c	even	4	1	inner	578.2.c.d		2
17.d	even	8	2		34.2.b.a	✓	2
17.d	even	8	2		578.2.a.d		2
17.e	odd	16	8		578.2.d.g		8
51.g	odd	8	2		306.2.b.d		2
51.g	odd	8	2		5202.2.a.u		2
68.g	odd	8	2		272.2.b.a		2
68.g	odd	8	2		4624.2.a.s		2
85.k	odd	8	2		850.2.d.i		4
85.m	even	8	2		850.2.b.f		2
85.n	odd	8	2		850.2.d.i		4
119.l	odd	8	2		1666.2.b.c		2
136.o	even	8	2		1088.2.b.b		2
136.p	odd	8	2		1088.2.b.a		2
204.p	even	8	2		2448.2.c.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
34.2.b.a	✓	2	17.d	even	8	2
272.2.b.a		2	68.g	odd	8	2
306.2.b.d		2	51.g	odd	8	2
578.2.a.d		2	17.d	even	8	2
578.2.c.a		2	17.b	even	2	1
578.2.c.a		2	17.c	even	4	1
578.2.c.d		2	1.a	even	1	1	trivial
578.2.c.d		2	17.c	even	4	1	inner
578.2.d.g		8	17.e	odd	16	8
850.2.b.f		2	85.m	even	8	2
850.2.d.i		4	85.k	odd	8	2
850.2.d.i		4	85.n	odd	8	2
1088.2.b.a		2	136.p	odd	8	2
1088.2.b.b		2	136.o	even	8	2
1666.2.b.c		2	119.l	odd	8	2
2448.2.c.n		2	204.p	even	8	2
4624.2.a.s		2	68.g	odd	8	2
5202.2.a.u		2	51.g	odd	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 4T_{3} + 8 \) acting on \(S_{2}^{\mathrm{new}}(578, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 1 \)
$3$	\( T^{2} - 4T + 8 \)
$5$	\( T^{2} - 4T + 8 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T + 8 \)