Newform orbit 2268.2.i.h

• Label: 2268.2.i.h
• Level: $2268$
• Weight: $2$
• Character orbit: 2268.i
• Analytic conductor: $18.110$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.1100711784\)
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:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - 3 \zeta_{6} + 3) q^{5} + ( - 3 \zeta_{6} + 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1135\)	\(1541\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

865.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

1.50000

−

2.59808i

0

−0.500000

−

2.59808i

0

0

0

2053.1

0

0

0

1.50000

+

2.59808i

0

−0.500000

+

2.59808i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2268.2.i.h		2
3.b	odd	2	1		2268.2.i.a		2
7.c	even	3	1		2268.2.l.a		2
9.c	even	3	1		252.2.k.c		2
9.c	even	3	1		2268.2.l.a		2
9.d	odd	6	1		28.2.e.a	✓	2
9.d	odd	6	1		2268.2.l.h		2
21.h	odd	6	1		2268.2.l.h		2
36.f	odd	6	1		1008.2.s.p		2
36.h	even	6	1		112.2.i.b		2
45.h	odd	6	1		700.2.i.c		2
45.l	even	12	2		700.2.r.b		4
63.g	even	3	1		252.2.k.c		2
63.h	even	3	1		1764.2.a.a		1
63.h	even	3	1	inner	2268.2.i.h		2
63.i	even	6	1		196.2.a.a		1
63.j	odd	6	1		196.2.a.b		1
63.j	odd	6	1		2268.2.i.a		2
63.k	odd	6	1		1764.2.k.b		2
63.l	odd	6	1		1764.2.k.b		2
63.n	odd	6	1		28.2.e.a	✓	2
63.o	even	6	1		196.2.e.a		2
63.s	even	6	1		196.2.e.a		2
63.t	odd	6	1		1764.2.a.j		1
72.j	odd	6	1		448.2.i.e		2
72.l	even	6	1		448.2.i.c		2
252.o	even	6	1		112.2.i.b		2
252.r	odd	6	1		784.2.a.g		1
252.s	odd	6	1		784.2.i.d		2
252.u	odd	6	1		7056.2.a.f		1
252.bb	even	6	1		784.2.a.d		1
252.bj	even	6	1		7056.2.a.bw		1
252.bl	odd	6	1		1008.2.s.p		2
252.bn	odd	6	1		784.2.i.d		2
315.v	odd	6	1		700.2.i.c		2
315.bq	even	6	1		4900.2.a.n		1
315.br	odd	6	1		4900.2.a.g		1
315.bu	odd	12	2		4900.2.e.h		2
315.bv	even	12	2		4900.2.e.i		2
315.bx	even	12	2		700.2.r.b		4
504.bi	odd	6	1		3136.2.a.h		1
504.bt	even	6	1		3136.2.a.s		1
504.ca	even	6	1		3136.2.a.v		1
504.cm	odd	6	1		3136.2.a.k		1
504.cy	even	6	1		448.2.i.c		2
504.db	odd	6	1		448.2.i.e		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.2.e.a	✓	2	9.d	odd	6	1
28.2.e.a	✓	2	63.n	odd	6	1
112.2.i.b		2	36.h	even	6	1
112.2.i.b		2	252.o	even	6	1
196.2.a.a		1	63.i	even	6	1
196.2.a.b		1	63.j	odd	6	1
196.2.e.a		2	63.o	even	6	1
196.2.e.a		2	63.s	even	6	1
252.2.k.c		2	9.c	even	3	1
252.2.k.c		2	63.g	even	3	1
448.2.i.c		2	72.l	even	6	1
448.2.i.c		2	504.cy	even	6	1
448.2.i.e		2	72.j	odd	6	1
448.2.i.e		2	504.db	odd	6	1
700.2.i.c		2	45.h	odd	6	1
700.2.i.c		2	315.v	odd	6	1
700.2.r.b		4	45.l	even	12	2
700.2.r.b		4	315.bx	even	12	2
784.2.a.d		1	252.bb	even	6	1
784.2.a.g		1	252.r	odd	6	1
784.2.i.d		2	252.s	odd	6	1
784.2.i.d		2	252.bn	odd	6	1
1008.2.s.p		2	36.f	odd	6	1
1008.2.s.p		2	252.bl	odd	6	1
1764.2.a.a		1	63.h	even	3	1
1764.2.a.j		1	63.t	odd	6	1
1764.2.k.b		2	63.k	odd	6	1
1764.2.k.b		2	63.l	odd	6	1
2268.2.i.a		2	3.b	odd	2	1
2268.2.i.a		2	63.j	odd	6	1
2268.2.i.h		2	1.a	even	1	1	trivial
2268.2.i.h		2	63.h	even	3	1	inner
2268.2.l.a		2	7.c	even	3	1
2268.2.l.a		2	9.c	even	3	1
2268.2.l.h		2	9.d	odd	6	1
2268.2.l.h		2	21.h	odd	6	1
3136.2.a.h		1	504.bi	odd	6	1
3136.2.a.k		1	504.cm	odd	6	1
3136.2.a.s		1	504.bt	even	6	1
3136.2.a.v		1	504.ca	even	6	1
4900.2.a.g		1	315.br	odd	6	1
4900.2.a.n		1	315.bq	even	6	1
4900.2.e.h		2	315.bu	odd	12	2
4900.2.e.i		2	315.bv	even	12	2
7056.2.a.f		1	252.u	odd	6	1
7056.2.a.bw		1	252.bj	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 9 \)

\( T_{13}^{2} + 2T_{13} + 4 \)

\( T_{19}^{2} - T_{19} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 3T + 9 \)
$7$	\( T^{2} + T + 7 \)
$11$	\( T^{2} + 3T + 9 \)