Newform orbit 1134.2.h.a

• Label: 1134.2.h.a
• Level: $1134$
• Weight: $2$
• Character orbit: 1134.h
• Analytic conductor: $9.055$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05503558921\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 q^{5} + ( - 2 \zeta_{6} - 1) q^{7} + q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 6 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(-1 + \zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

109.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−3.00000

0

−2.00000

−

1.73205i

1.00000

0

1.50000

−

2.59808i

541.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−3.00000

0

−2.00000

+

1.73205i

1.00000

0

1.50000

+

2.59808i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1134.2.h.a		2
3.b	odd	2	1		1134.2.h.p		2
7.c	even	3	1		1134.2.e.p		2
9.c	even	3	1		126.2.g.b		2
9.c	even	3	1		1134.2.e.p		2
9.d	odd	6	1		42.2.e.b	✓	2
9.d	odd	6	1		1134.2.e.a		2
21.h	odd	6	1		1134.2.e.a		2
36.f	odd	6	1		1008.2.s.n		2
36.h	even	6	1		336.2.q.d		2
45.h	odd	6	1		1050.2.i.e		2
45.l	even	12	2		1050.2.o.b		4
63.g	even	3	1		882.2.a.g		1
63.g	even	3	1	inner	1134.2.h.a		2
63.h	even	3	1		126.2.g.b		2
63.i	even	6	1		294.2.e.f		2
63.j	odd	6	1		42.2.e.b	✓	2
63.k	odd	6	1		882.2.a.k		1
63.l	odd	6	1		882.2.g.b		2
63.n	odd	6	1		294.2.a.d		1
63.n	odd	6	1		1134.2.h.p		2
63.o	even	6	1		294.2.e.f		2
63.s	even	6	1		294.2.a.a		1
63.t	odd	6	1		882.2.g.b		2
72.j	odd	6	1		1344.2.q.v		2
72.l	even	6	1		1344.2.q.j		2
252.n	even	6	1		7056.2.a.bz		1
252.o	even	6	1		2352.2.a.m		1
252.r	odd	6	1		2352.2.q.m		2
252.s	odd	6	1		2352.2.q.m		2
252.u	odd	6	1		1008.2.s.n		2
252.bb	even	6	1		336.2.q.d		2
252.bl	odd	6	1		7056.2.a.g		1
252.bn	odd	6	1		2352.2.a.n		1
315.u	even	6	1		7350.2.a.cw		1
315.v	odd	6	1		7350.2.a.ce		1
315.br	odd	6	1		1050.2.i.e		2
315.bv	even	12	2		1050.2.o.b		4
504.u	odd	6	1		9408.2.a.bm		1
504.y	even	6	1		9408.2.a.db		1
504.bi	odd	6	1		1344.2.q.v		2
504.bt	even	6	1		1344.2.q.j		2
504.cy	even	6	1		9408.2.a.bu		1
504.db	odd	6	1		9408.2.a.d		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.2.e.b	✓	2	9.d	odd	6	1
42.2.e.b	✓	2	63.j	odd	6	1
126.2.g.b		2	9.c	even	3	1
126.2.g.b		2	63.h	even	3	1
294.2.a.a		1	63.s	even	6	1
294.2.a.d		1	63.n	odd	6	1
294.2.e.f		2	63.i	even	6	1
294.2.e.f		2	63.o	even	6	1
336.2.q.d		2	36.h	even	6	1
336.2.q.d		2	252.bb	even	6	1
882.2.a.g		1	63.g	even	3	1
882.2.a.k		1	63.k	odd	6	1
882.2.g.b		2	63.l	odd	6	1
882.2.g.b		2	63.t	odd	6	1
1008.2.s.n		2	36.f	odd	6	1
1008.2.s.n		2	252.u	odd	6	1
1050.2.i.e		2	45.h	odd	6	1
1050.2.i.e		2	315.br	odd	6	1
1050.2.o.b		4	45.l	even	12	2
1050.2.o.b		4	315.bv	even	12	2
1134.2.e.a		2	9.d	odd	6	1
1134.2.e.a		2	21.h	odd	6	1
1134.2.e.p		2	7.c	even	3	1
1134.2.e.p		2	9.c	even	3	1
1134.2.h.a		2	1.a	even	1	1	trivial
1134.2.h.a		2	63.g	even	3	1	inner
1134.2.h.p		2	3.b	odd	2	1
1134.2.h.p		2	63.n	odd	6	1
1344.2.q.j		2	72.l	even	6	1
1344.2.q.j		2	504.bt	even	6	1
1344.2.q.v		2	72.j	odd	6	1
1344.2.q.v		2	504.bi	odd	6	1
2352.2.a.m		1	252.o	even	6	1
2352.2.a.n		1	252.bn	odd	6	1
2352.2.q.m		2	252.r	odd	6	1
2352.2.q.m		2	252.s	odd	6	1
7056.2.a.g		1	252.bl	odd	6	1
7056.2.a.bz		1	252.n	even	6	1
7350.2.a.ce		1	315.v	odd	6	1
7350.2.a.cw		1	315.u	even	6	1
9408.2.a.d		1	504.db	odd	6	1
9408.2.a.bm		1	504.u	odd	6	1
9408.2.a.bu		1	504.cy	even	6	1
9408.2.a.db		1	504.y	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5} + 3 \)

\( T_{11} + 3 \)

\( T_{17} \)

\( T_{23} \)

Hecke characteristic polynomials

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