Newform orbit 2312.1.f.a

• Label: 2312.1.f.a
• Level: $2312$
• Weight: $1$
• Character orbit: 2312.f
• Self dual: yes
• Analytic conductor: $1.154$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -8, -136, 17
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2312.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.15383830921\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 136)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-2}, \sqrt{17})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.2.39304.1

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{2} + q^{4} + q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\)	\(1157\)	\(1735\)	\(1737\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

579.1

0

1.00000

0

1.00000

0

0

0

1.00000

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
17.b	even	2	1	RM by \(\Q(\sqrt{17}) \)
136.e	odd	2	1	CM by \(\Q(\sqrt{-34}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.1.f.a		1
8.d	odd	2	1	CM	2312.1.f.a		1
17.b	even	2	1	RM	2312.1.f.a		1
17.c	even	4	2		136.1.e.a	✓	1
17.d	even	8	4		2312.1.j.a		2
17.e	odd	16	8		2312.1.p.c		4
51.f	odd	4	2		1224.1.n.a		1
68.f	odd	4	2		544.1.e.a		1
85.f	odd	4	2		3400.1.k.a		2
85.i	odd	4	2		3400.1.k.a		2
85.j	even	4	2		3400.1.g.a		1
136.e	odd	2	1	CM	2312.1.f.a		1
136.i	even	4	2		544.1.e.a		1
136.j	odd	4	2		136.1.e.a	✓	1
136.p	odd	8	4		2312.1.j.a		2
136.s	even	16	8		2312.1.p.c		4
408.q	even	4	2		1224.1.n.a		1
680.t	even	4	2		3400.1.k.a		2
680.bc	odd	4	2		3400.1.g.a		1
680.bl	even	4	2		3400.1.k.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.1.e.a	✓	1	17.c	even	4	2
136.1.e.a	✓	1	136.j	odd	4	2
544.1.e.a		1	68.f	odd	4	2
544.1.e.a		1	136.i	even	4	2
1224.1.n.a		1	51.f	odd	4	2
1224.1.n.a		1	408.q	even	4	2
2312.1.f.a		1	1.a	even	1	1	trivial
2312.1.f.a		1	8.d	odd	2	1	CM
2312.1.f.a		1	17.b	even	2	1	RM
2312.1.f.a		1	136.e	odd	2	1	CM
2312.1.j.a		2	17.d	even	8	4
2312.1.j.a		2	136.p	odd	8	4
2312.1.p.c		4	17.e	odd	16	8
2312.1.p.c		4	136.s	even	16	8
3400.1.g.a		1	85.j	even	4	2
3400.1.g.a		1	680.bc	odd	4	2
3400.1.k.a		2	85.f	odd	4	2
3400.1.k.a		2	85.i	odd	4	2
3400.1.k.a		2	680.t	even	4	2
3400.1.k.a		2	680.bl	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(2312, [\chi])\).

Hecke characteristic polynomials

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