Newform orbit 2312.1.e.a

• Label: 2312.1.e.a
• Level: $2312$
• Weight: $1$
• Character orbit: 2312.e
• Analytic conductor: $1.154$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2312.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.15383830921\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 136)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.314432.1
Artin image:	$\SD_{16}$
Artin field:	Galois closure of 8.2.210093400576.2

$q$-expansion

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1155.1

		1.41421i
	−	1.41421i

−1.00000

−

1.41421i

1.00000

0

1.41421i

0

−1.00000

−1.00000

0

1155.2

−1.00000

1.41421i

1.00000

0

−

1.41421i

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	inner
136.e	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.1.e.a		2
8.d	odd	2	1	CM	2312.1.e.a		2
17.b	even	2	1	inner	2312.1.e.a		2
17.c	even	4	2		2312.1.f.b		2
17.d	even	8	2		136.1.j.a	✓	2
17.d	even	8	2		2312.1.j.b		2
17.e	odd	16	8		2312.1.p.e		8
51.g	odd	8	2		1224.1.s.a		2
68.g	odd	8	2		544.1.n.a		2
85.k	odd	8	2		3400.1.bc.b		2
85.m	even	8	2		3400.1.y.a		2
85.n	odd	8	2		3400.1.bc.a		2
136.e	odd	2	1	inner	2312.1.e.a		2
136.j	odd	4	2		2312.1.f.b		2
136.o	even	8	2		544.1.n.a		2
136.p	odd	8	2		136.1.j.a	✓	2
136.p	odd	8	2		2312.1.j.b		2
136.s	even	16	8		2312.1.p.e		8
408.bd	even	8	2		1224.1.s.a		2
680.bq	odd	8	2		3400.1.y.a		2
680.bw	even	8	2		3400.1.bc.b		2
680.bz	even	8	2		3400.1.bc.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.1.j.a	✓	2	17.d	even	8	2
136.1.j.a	✓	2	136.p	odd	8	2
544.1.n.a		2	68.g	odd	8	2
544.1.n.a		2	136.o	even	8	2
1224.1.s.a		2	51.g	odd	8	2
1224.1.s.a		2	408.bd	even	8	2
2312.1.e.a		2	1.a	even	1	1	trivial
2312.1.e.a		2	8.d	odd	2	1	CM
2312.1.e.a		2	17.b	even	2	1	inner
2312.1.e.a		2	136.e	odd	2	1	inner
2312.1.f.b		2	17.c	even	4	2
2312.1.f.b		2	136.j	odd	4	2
2312.1.j.b		2	17.d	even	8	2
2312.1.j.b		2	136.p	odd	8	2
2312.1.p.e		8	17.e	odd	16	8
2312.1.p.e		8	136.s	even	16	8
3400.1.y.a		2	85.m	even	8	2
3400.1.y.a		2	680.bq	odd	8	2
3400.1.bc.a		2	85.n	odd	8	2
3400.1.bc.a		2	680.bz	even	8	2
3400.1.bc.b		2	85.k	odd	8	2
3400.1.bc.b		2	680.bw	even	8	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2312, [\chi])\).

Hecke characteristic polynomials

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