Newform orbit 2268.2.j.a

• Label: 2268.2.j.a
• Level: $2268$
• Weight: $2$
• Character orbit: 2268.j
• Analytic conductor: $18.110$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.j (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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
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.

\(f(q)\)	\(=\)	\( q - 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + q^{7}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

757.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

0

−2.00000

+

3.46410i

0

0.500000

+

0.866025i

0

0

0

1513.1

0

0

0

−2.00000

−

3.46410i

0

0.500000

−

0.866025i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	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.j.a		2
3.b	odd	2	1		2268.2.j.n		2
9.c	even	3	1		84.2.a.a	✓	1
9.c	even	3	1	inner	2268.2.j.a		2
9.d	odd	6	1		252.2.a.a		1
9.d	odd	6	1		2268.2.j.n		2
36.f	odd	6	1		336.2.a.f		1
36.h	even	6	1		1008.2.a.a		1
45.h	odd	6	1		6300.2.a.w		1
45.j	even	6	1		2100.2.a.r		1
45.k	odd	12	2		2100.2.k.i		2
45.l	even	12	2		6300.2.k.g		2
63.g	even	3	1		588.2.i.e		2
63.h	even	3	1		588.2.i.e		2
63.i	even	6	1		1764.2.k.a		2
63.j	odd	6	1		1764.2.k.k		2
63.k	odd	6	1		588.2.i.d		2
63.l	odd	6	1		588.2.a.d		1
63.n	odd	6	1		1764.2.k.k		2
63.o	even	6	1		1764.2.a.k		1
63.s	even	6	1		1764.2.k.a		2
63.t	odd	6	1		588.2.i.d		2
72.j	odd	6	1		4032.2.a.bm		1
72.l	even	6	1		4032.2.a.bn		1
72.n	even	6	1		1344.2.a.k		1
72.p	odd	6	1		1344.2.a.a		1
144.v	odd	12	2		5376.2.c.p		2
144.x	even	12	2		5376.2.c.q		2
180.p	odd	6	1		8400.2.a.e		1
252.n	even	6	1		2352.2.q.z		2
252.s	odd	6	1		7056.2.a.cd		1
252.u	odd	6	1		2352.2.q.b		2
252.bi	even	6	1		2352.2.a.a		1
252.bj	even	6	1		2352.2.q.z		2
252.bl	odd	6	1		2352.2.q.b		2
504.be	even	6	1		9408.2.a.df		1
504.bn	odd	6	1		9408.2.a.bn		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.a.a	✓	1	9.c	even	3	1
252.2.a.a		1	9.d	odd	6	1
336.2.a.f		1	36.f	odd	6	1
588.2.a.d		1	63.l	odd	6	1
588.2.i.d		2	63.k	odd	6	1
588.2.i.d		2	63.t	odd	6	1
588.2.i.e		2	63.g	even	3	1
588.2.i.e		2	63.h	even	3	1
1008.2.a.a		1	36.h	even	6	1
1344.2.a.a		1	72.p	odd	6	1
1344.2.a.k		1	72.n	even	6	1
1764.2.a.k		1	63.o	even	6	1
1764.2.k.a		2	63.i	even	6	1
1764.2.k.a		2	63.s	even	6	1
1764.2.k.k		2	63.j	odd	6	1
1764.2.k.k		2	63.n	odd	6	1
2100.2.a.r		1	45.j	even	6	1
2100.2.k.i		2	45.k	odd	12	2
2268.2.j.a		2	1.a	even	1	1	trivial
2268.2.j.a		2	9.c	even	3	1	inner
2268.2.j.n		2	3.b	odd	2	1
2268.2.j.n		2	9.d	odd	6	1
2352.2.a.a		1	252.bi	even	6	1
2352.2.q.b		2	252.u	odd	6	1
2352.2.q.b		2	252.bl	odd	6	1
2352.2.q.z		2	252.n	even	6	1
2352.2.q.z		2	252.bj	even	6	1
4032.2.a.bm		1	72.j	odd	6	1
4032.2.a.bn		1	72.l	even	6	1
5376.2.c.p		2	144.v	odd	12	2
5376.2.c.q		2	144.x	even	12	2
6300.2.a.w		1	45.h	odd	6	1
6300.2.k.g		2	45.l	even	12	2
7056.2.a.cd		1	252.s	odd	6	1
8400.2.a.e		1	180.p	odd	6	1
9408.2.a.bn		1	504.bn	odd	6	1
9408.2.a.df		1	504.be	even	6	1

Hecke kernels

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

\( T_{5}^{2} + 4T_{5} + 16 \)

\( T_{11}^{2} + 2T_{11} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4T + 16 \)
$7$	\( T^{2} - T + 1 \)
$11$	\( T^{2} + 2T + 4 \)