Newform orbit 704.3.h.b

• Label: 704.3.h.b
• Level: $704$
• Weight: $3$
• Character orbit: 704.h
• Self dual: yes
• Analytic conductor: $19.183$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -11
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	704.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(19.1826106110\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 5 q^{3} + q^{5} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/704\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(321\)	\(639\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0

0

5.00000

0

1.00000

0

0

0

16.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by \(\Q(\sqrt{-11}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.3.h.b		1
4.b	odd	2	1		704.3.h.a		1
8.b	even	2	1		11.3.b.a	✓	1
8.d	odd	2	1		176.3.h.a		1
11.b	odd	2	1	CM	704.3.h.b		1
24.f	even	2	1		1584.3.j.a		1
24.h	odd	2	1		99.3.c.a		1
40.f	even	2	1		275.3.c.a		1
40.i	odd	4	2		275.3.d.a		2
44.c	even	2	1		704.3.h.a		1
56.h	odd	2	1		539.3.c.a		1
88.b	odd	2	1		11.3.b.a	✓	1
88.g	even	2	1		176.3.h.a		1
88.o	even	10	4		121.3.d.b		4
88.p	odd	10	4		121.3.d.b		4
264.m	even	2	1		99.3.c.a		1
264.p	odd	2	1		1584.3.j.a		1
440.o	odd	2	1		275.3.c.a		1
440.t	even	4	2		275.3.d.a		2
616.o	even	2	1		539.3.c.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.3.b.a	✓	1	8.b	even	2	1
11.3.b.a	✓	1	88.b	odd	2	1
99.3.c.a		1	24.h	odd	2	1
99.3.c.a		1	264.m	even	2	1
121.3.d.b		4	88.o	even	10	4
121.3.d.b		4	88.p	odd	10	4
176.3.h.a		1	8.d	odd	2	1
176.3.h.a		1	88.g	even	2	1
275.3.c.a		1	40.f	even	2	1
275.3.c.a		1	440.o	odd	2	1
275.3.d.a		2	40.i	odd	4	2
275.3.d.a		2	440.t	even	4	2
539.3.c.a		1	56.h	odd	2	1
539.3.c.a		1	616.o	even	2	1
704.3.h.a		1	4.b	odd	2	1
704.3.h.a		1	44.c	even	2	1
704.3.h.b		1	1.a	even	1	1	trivial
704.3.h.b		1	11.b	odd	2	1	CM
1584.3.j.a		1	24.f	even	2	1
1584.3.j.a		1	264.p	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{3}^{\mathrm{new}}(704, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T - 5 \)
$5$	\( T - 1 \)
$7$	\( T \)
$11$	\( T - 11 \)