Newform orbit 2352.1.d.a

• Label: 2352.1.d.a
• Level: $2352$
• Weight: $1$
• Character orbit: 2352.d
• Self dual: yes
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2352.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.588.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.5531904.1

$q$-expansion

\(f(q)\)

\(=\)

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(1471\)	\(1765\)	\(2257\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

785.1

0

0

−1.00000

0

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2352.1.d.a		1
3.b	odd	2	1	CM	2352.1.d.a		1
4.b	odd	2	1		588.1.c.b		1
7.b	odd	2	1		2352.1.d.b		1
7.c	even	3	2		336.1.bn.a		2
7.d	odd	6	2		2352.1.bn.a		2
12.b	even	2	1		588.1.c.b		1
21.c	even	2	1		2352.1.d.b		1
21.g	even	6	2		2352.1.bn.a		2
21.h	odd	6	2		336.1.bn.a		2
28.d	even	2	1		588.1.c.a		1
28.f	even	6	2		588.1.p.a		2
28.g	odd	6	2		84.1.p.a	✓	2
56.k	odd	6	2		1344.1.bn.b		2
56.p	even	6	2		1344.1.bn.a		2
84.h	odd	2	1		588.1.c.a		1
84.j	odd	6	2		588.1.p.a		2
84.n	even	6	2		84.1.p.a	✓	2
140.p	odd	6	2		2100.1.bn.c		2
140.w	even	12	4		2100.1.bh.a		4
168.s	odd	6	2		1344.1.bn.a		2
168.v	even	6	2		1344.1.bn.b		2
252.o	even	6	2		2268.1.m.a		2
252.u	odd	6	2		2268.1.bh.b		2
252.bb	even	6	2		2268.1.bh.b		2
252.bl	odd	6	2		2268.1.m.a		2
420.ba	even	6	2		2100.1.bn.c		2
420.bp	odd	12	4		2100.1.bh.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.1.p.a	✓	2	28.g	odd	6	2
84.1.p.a	✓	2	84.n	even	6	2
336.1.bn.a		2	7.c	even	3	2
336.1.bn.a		2	21.h	odd	6	2
588.1.c.a		1	28.d	even	2	1
588.1.c.a		1	84.h	odd	2	1
588.1.c.b		1	4.b	odd	2	1
588.1.c.b		1	12.b	even	2	1
588.1.p.a		2	28.f	even	6	2
588.1.p.a		2	84.j	odd	6	2
1344.1.bn.a		2	56.p	even	6	2
1344.1.bn.a		2	168.s	odd	6	2
1344.1.bn.b		2	56.k	odd	6	2
1344.1.bn.b		2	168.v	even	6	2
2100.1.bh.a		4	140.w	even	12	4
2100.1.bh.a		4	420.bp	odd	12	4
2100.1.bn.c		2	140.p	odd	6	2
2100.1.bn.c		2	420.ba	even	6	2
2268.1.m.a		2	252.o	even	6	2
2268.1.m.a		2	252.bl	odd	6	2
2268.1.bh.b		2	252.u	odd	6	2
2268.1.bh.b		2	252.bb	even	6	2
2352.1.d.a		1	1.a	even	1	1	trivial
2352.1.d.a		1	3.b	odd	2	1	CM
2352.1.d.b		1	7.b	odd	2	1
2352.1.d.b		1	21.c	even	2	1
2352.1.bn.a		2	7.d	odd	6	2
2352.1.bn.a		2	21.g	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

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