Newform orbit 2352.2.b.a

• Label: 2352.2.b.a
• Level: $2352$
• Weight: $2$
• Character orbit: 2352.b
• Analytic conductor: $18.781$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.7808145554\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{3} - 2 \beta q^{5} + q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1567.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.00000

0

−

3.46410i

0

0

0

1.00000

0

1567.2

0

−1.00000

0

3.46410i

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2352.2.b.a		2
3.b	odd	2	1		7056.2.b.a		2
4.b	odd	2	1		2352.2.b.h		2
7.b	odd	2	1		2352.2.b.h		2
7.c	even	3	1		336.2.bl.e	yes	2
7.c	even	3	1		2352.2.bl.l		2
7.d	odd	6	1		336.2.bl.a	✓	2
7.d	odd	6	1		2352.2.bl.f		2
12.b	even	2	1		7056.2.b.l		2
21.c	even	2	1		7056.2.b.l		2
21.g	even	6	1		1008.2.cs.n		2
21.h	odd	6	1		1008.2.cs.m		2
28.d	even	2	1	inner	2352.2.b.a		2
28.f	even	6	1		336.2.bl.e	yes	2
28.f	even	6	1		2352.2.bl.l		2
28.g	odd	6	1		336.2.bl.a	✓	2
28.g	odd	6	1		2352.2.bl.f		2
56.j	odd	6	1		1344.2.bl.h		2
56.k	odd	6	1		1344.2.bl.h		2
56.m	even	6	1		1344.2.bl.d		2
56.p	even	6	1		1344.2.bl.d		2
84.h	odd	2	1		7056.2.b.a		2
84.j	odd	6	1		1008.2.cs.m		2
84.n	even	6	1		1008.2.cs.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.2.bl.a	✓	2	7.d	odd	6	1
336.2.bl.a	✓	2	28.g	odd	6	1
336.2.bl.e	yes	2	7.c	even	3	1
336.2.bl.e	yes	2	28.f	even	6	1
1008.2.cs.m		2	21.h	odd	6	1
1008.2.cs.m		2	84.j	odd	6	1
1008.2.cs.n		2	21.g	even	6	1
1008.2.cs.n		2	84.n	even	6	1
1344.2.bl.d		2	56.m	even	6	1
1344.2.bl.d		2	56.p	even	6	1
1344.2.bl.h		2	56.j	odd	6	1
1344.2.bl.h		2	56.k	odd	6	1
2352.2.b.a		2	1.a	even	1	1	trivial
2352.2.b.a		2	28.d	even	2	1	inner
2352.2.b.h		2	4.b	odd	2	1
2352.2.b.h		2	7.b	odd	2	1
2352.2.bl.f		2	7.d	odd	6	1
2352.2.bl.f		2	28.g	odd	6	1
2352.2.bl.l		2	7.c	even	3	1
2352.2.bl.l		2	28.f	even	6	1
7056.2.b.a		2	3.b	odd	2	1
7056.2.b.a		2	84.h	odd	2	1
7056.2.b.l		2	12.b	even	2	1
7056.2.b.l		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{2} + 12 \)

\( T_{11}^{2} + 12 \)

\( T_{13}^{2} + 27 \)

\( T_{19} + 7 \)

\( T_{31} + 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( (T + 1)^{2} \)
$5$	\( T^{2} + 12 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 12 \)