Newform orbit 704.1.h.a

• Label: 704.1.h.a
• Level: $704$
• Weight: $1$
• Character orbit: 704.h
• Self dual: yes
• Analytic conductor: $0.351$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -11
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.351341768894\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 44)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.44.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.247808.1
Stark unit:	Root of $x^{6} - 54x^{5} + 431x^{4} - 2164x^{3} + 431x^{2} - 54x + 1$

$q$-expansion

\(f(q)\)

\(=\)

q - q^3 + q^5 + q^11 - q^15 + q^23 + q^27 + q^31 - q^33 + q^37 - 2 * q^47 + q^49 - 2 * q^53 + q^55 - q^59 - q^67 - q^69 + q^71 - q^81 - q^89 - q^93 - q^97

Character values

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

\(n\)	\(133\)	\(321\)	\(639\)
\(\chi(n)\)	\(0\)	\(1\)	\(0\)

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

−1.00000

0

1.00000

0

0

0

0

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	704.1.h.a		1
4.b	odd	2	1		704.1.h.b		1
8.b	even	2	1		176.1.h.a		1
8.d	odd	2	1		44.1.d.a	✓	1
11.b	odd	2	1	CM	704.1.h.a		1
16.e	even	4	2		2816.1.b.a		2
16.f	odd	4	2		2816.1.b.b		2
24.f	even	2	1		396.1.f.a		1
24.h	odd	2	1		1584.1.j.a		1
40.e	odd	2	1		1100.1.f.a		1
40.k	even	4	2		1100.1.e.a		2
44.c	even	2	1		704.1.h.b		1
56.e	even	2	1		2156.1.h.a		1
56.k	odd	6	2		2156.1.k.b		2
56.m	even	6	2		2156.1.k.a		2
72.l	even	6	2		3564.1.m.a		2
72.p	odd	6	2		3564.1.m.b		2
88.b	odd	2	1		176.1.h.a		1
88.g	even	2	1		44.1.d.a	✓	1
88.k	even	10	4		484.1.f.a		4
88.l	odd	10	4		484.1.f.a		4
88.o	even	10	4		1936.1.n.a		4
88.p	odd	10	4		1936.1.n.a		4
176.i	even	4	2		2816.1.b.b		2
176.l	odd	4	2		2816.1.b.a		2
264.m	even	2	1		1584.1.j.a		1
264.p	odd	2	1		396.1.f.a		1
440.c	even	2	1		1100.1.f.a		1
440.w	odd	4	2		1100.1.e.a		2
616.g	odd	2	1		2156.1.h.a		1
616.y	even	6	2		2156.1.k.b		2
616.z	odd	6	2		2156.1.k.a		2
792.s	odd	6	2		3564.1.m.a		2
792.z	even	6	2		3564.1.m.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
44.1.d.a	✓	1	8.d	odd	2	1
44.1.d.a	✓	1	88.g	even	2	1
176.1.h.a		1	8.b	even	2	1
176.1.h.a		1	88.b	odd	2	1
396.1.f.a		1	24.f	even	2	1
396.1.f.a		1	264.p	odd	2	1
484.1.f.a		4	88.k	even	10	4
484.1.f.a		4	88.l	odd	10	4
704.1.h.a		1	1.a	even	1	1	trivial
704.1.h.a		1	11.b	odd	2	1	CM
704.1.h.b		1	4.b	odd	2	1
704.1.h.b		1	44.c	even	2	1
1100.1.e.a		2	40.k	even	4	2
1100.1.e.a		2	440.w	odd	4	2
1100.1.f.a		1	40.e	odd	2	1
1100.1.f.a		1	440.c	even	2	1
1584.1.j.a		1	24.h	odd	2	1
1584.1.j.a		1	264.m	even	2	1
1936.1.n.a		4	88.o	even	10	4
1936.1.n.a		4	88.p	odd	10	4
2156.1.h.a		1	56.e	even	2	1
2156.1.h.a		1	616.g	odd	2	1
2156.1.k.a		2	56.m	even	6	2
2156.1.k.a		2	616.z	odd	6	2
2156.1.k.b		2	56.k	odd	6	2
2156.1.k.b		2	616.y	even	6	2
2816.1.b.a		2	16.e	even	4	2
2816.1.b.a		2	176.l	odd	4	2
2816.1.b.b		2	16.f	odd	4	2
2816.1.b.b		2	176.i	even	4	2
3564.1.m.a		2	72.l	even	6	2
3564.1.m.a		2	792.s	odd	6	2
3564.1.m.b		2	72.p	odd	6	2
3564.1.m.b		2	792.z	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(704, [\chi])\).

Hecke characteristic polynomials

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