Newform orbit 2352.1.z.c

• Label: 2352.1.z.c
• Level: $2352$
• Weight: $1$
• Character orbit: 2352.z
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{2}$
• CM/RM discs: -3, -84, 28
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2352.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 336)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\)
Artin image:	$C_3\times D_4$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{9} - 2 \zeta_{6} q^{19} - \zeta_{6}^{2} q^{25} - q^{27} - 2 \zeta_{6}^{2} q^{31} + 2 \zeta_{6} q^{37} - 2 q^{57} - \zeta_{6} q^{75} + \zeta_{6}^{2} q^{81} - 2 \zeta_{6} q^{93} +O(q^{100}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{9} - 2 q^{19} + q^{25} - 2 q^{27} + 2 q^{31} + 2 q^{37} - 4 q^{57} - q^{75} - q^{81} - 2 q^{93}+O(q^{100}) \)

Character values

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

\(n\)	\(785\)	\(1471\)	\(1765\)	\(2257\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

815.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0.500000

−

0.866025i

0

0

0

0

0

−0.500000

−

0.866025i

0

1391.1

0

0.500000

+

0.866025i

0

0

0

0

0

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
28.d	even	2	1	RM by \(\Q(\sqrt{7}) \)
84.h	odd	2	1	CM by \(\Q(\sqrt{-21}) \)
7.c	even	3	1	inner
21.h	odd	6	1	inner
28.f	even	6	1	inner
84.j	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2352.1.z.c		2
3.b	odd	2	1	CM	2352.1.z.c		2
4.b	odd	2	1		2352.1.z.b		2
7.b	odd	2	1		2352.1.z.b		2
7.c	even	3	1		336.1.o.a	✓	1
7.c	even	3	1	inner	2352.1.z.c		2
7.d	odd	6	1		336.1.o.b	yes	1
7.d	odd	6	1		2352.1.z.b		2
12.b	even	2	1		2352.1.z.b		2
21.c	even	2	1		2352.1.z.b		2
21.g	even	6	1		336.1.o.b	yes	1
21.g	even	6	1		2352.1.z.b		2
21.h	odd	6	1		336.1.o.a	✓	1
21.h	odd	6	1	inner	2352.1.z.c		2
28.d	even	2	1	RM	2352.1.z.c		2
28.f	even	6	1		336.1.o.a	✓	1
28.f	even	6	1	inner	2352.1.z.c		2
28.g	odd	6	1		336.1.o.b	yes	1
28.g	odd	6	1		2352.1.z.b		2
56.j	odd	6	1		1344.1.o.a		1
56.k	odd	6	1		1344.1.o.a		1
56.m	even	6	1		1344.1.o.b		1
56.p	even	6	1		1344.1.o.b		1
84.h	odd	2	1	CM	2352.1.z.c		2
84.j	odd	6	1		336.1.o.a	✓	1
84.j	odd	6	1	inner	2352.1.z.c		2
84.n	even	6	1		336.1.o.b	yes	1
84.n	even	6	1		2352.1.z.b		2
168.s	odd	6	1		1344.1.o.b		1
168.v	even	6	1		1344.1.o.a		1
168.ba	even	6	1		1344.1.o.a		1
168.be	odd	6	1		1344.1.o.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.1.o.a	✓	1	7.c	even	3	1
336.1.o.a	✓	1	21.h	odd	6	1
336.1.o.a	✓	1	28.f	even	6	1
336.1.o.a	✓	1	84.j	odd	6	1
336.1.o.b	yes	1	7.d	odd	6	1
336.1.o.b	yes	1	21.g	even	6	1
336.1.o.b	yes	1	28.g	odd	6	1
336.1.o.b	yes	1	84.n	even	6	1
1344.1.o.a		1	56.j	odd	6	1
1344.1.o.a		1	56.k	odd	6	1
1344.1.o.a		1	168.v	even	6	1
1344.1.o.a		1	168.ba	even	6	1
1344.1.o.b		1	56.m	even	6	1
1344.1.o.b		1	56.p	even	6	1
1344.1.o.b		1	168.s	odd	6	1
1344.1.o.b		1	168.be	odd	6	1
2352.1.z.b		2	4.b	odd	2	1
2352.1.z.b		2	7.b	odd	2	1
2352.1.z.b		2	7.d	odd	6	1
2352.1.z.b		2	12.b	even	2	1
2352.1.z.b		2	21.c	even	2	1
2352.1.z.b		2	21.g	even	6	1
2352.1.z.b		2	28.g	odd	6	1
2352.1.z.b		2	84.n	even	6	1
2352.1.z.c		2	1.a	even	1	1	trivial
2352.1.z.c		2	3.b	odd	2	1	CM
2352.1.z.c		2	7.c	even	3	1	inner
2352.1.z.c		2	21.h	odd	6	1	inner
2352.1.z.c		2	28.d	even	2	1	RM
2352.1.z.c		2	28.f	even	6	1	inner
2352.1.z.c		2	84.h	odd	2	1	CM
2352.1.z.c		2	84.j	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\):

\( T_{13} \)

\( T_{19}^{2} + 2T_{19} + 4 \)

Hecke characteristic polynomials

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