Newform orbit 1274.2.h.a

• Label: 1274.2.h.a
• Level: $1274$
• Weight: $2$
• Character orbit: 1274.h
• Analytic conductor: $10.173$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1729412175\)
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 26)
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.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} + (\zeta_{6} - 1) q^{5} + q^{8} - 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - q^{5} + 2 q^{8} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(885\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(-1 + \zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

263.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−0.500000

−

0.866025i

0

0

1.00000

−3.00000

1.00000

373.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−0.500000

+

0.866025i

0

0

1.00000

−3.00000

1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1274.2.h.a		2
7.b	odd	2	1		1274.2.h.b		2
7.c	even	3	1		1274.2.e.m		2
7.c	even	3	1		1274.2.g.a		2
7.d	odd	6	1		26.2.c.a	✓	2
7.d	odd	6	1		1274.2.e.n		2
13.c	even	3	1		1274.2.e.m		2
21.g	even	6	1		234.2.h.c		2
28.f	even	6	1		208.2.i.b		2
35.i	odd	6	1		650.2.e.c		2
35.k	even	12	2		650.2.o.c		4
56.j	odd	6	1		832.2.i.e		2
56.m	even	6	1		832.2.i.f		2
84.j	odd	6	1		1872.2.t.k		2
91.g	even	3	1	inner	1274.2.h.a		2
91.h	even	3	1		1274.2.g.a		2
91.l	odd	6	1		338.2.c.e		2
91.m	odd	6	1		338.2.a.e		1
91.m	odd	6	1		1274.2.h.b		2
91.n	odd	6	1		1274.2.e.n		2
91.p	odd	6	1		338.2.a.c		1
91.s	odd	6	1		338.2.c.e		2
91.v	odd	6	1		26.2.c.a	✓	2
91.w	even	12	2		338.2.b.b		2
91.ba	even	12	2		338.2.e.b		4
91.bb	even	12	2		338.2.e.b		4
273.r	even	6	1		234.2.h.c		2
273.y	even	6	1		3042.2.a.k		1
273.bf	even	6	1		3042.2.a.e		1
273.ch	odd	12	2		3042.2.b.e		2
364.ba	even	6	1		208.2.i.b		2
364.bp	even	6	1		2704.2.a.i		1
364.br	even	6	1		2704.2.a.h		1
364.cg	odd	12	2		2704.2.f.g		2
455.y	odd	6	1		650.2.e.c		2
455.bk	odd	6	1		8450.2.a.s		1
455.bw	odd	6	1		8450.2.a.f		1
455.cs	even	12	2		650.2.o.c		4
728.bq	odd	6	1		832.2.i.e		2
728.cl	even	6	1		832.2.i.f		2
1092.bt	odd	6	1		1872.2.t.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.c.a	✓	2	7.d	odd	6	1
26.2.c.a	✓	2	91.v	odd	6	1
208.2.i.b		2	28.f	even	6	1
208.2.i.b		2	364.ba	even	6	1
234.2.h.c		2	21.g	even	6	1
234.2.h.c		2	273.r	even	6	1
338.2.a.c		1	91.p	odd	6	1
338.2.a.e		1	91.m	odd	6	1
338.2.b.b		2	91.w	even	12	2
338.2.c.e		2	91.l	odd	6	1
338.2.c.e		2	91.s	odd	6	1
338.2.e.b		4	91.ba	even	12	2
338.2.e.b		4	91.bb	even	12	2
650.2.e.c		2	35.i	odd	6	1
650.2.e.c		2	455.y	odd	6	1
650.2.o.c		4	35.k	even	12	2
650.2.o.c		4	455.cs	even	12	2
832.2.i.e		2	56.j	odd	6	1
832.2.i.e		2	728.bq	odd	6	1
832.2.i.f		2	56.m	even	6	1
832.2.i.f		2	728.cl	even	6	1
1274.2.e.m		2	7.c	even	3	1
1274.2.e.m		2	13.c	even	3	1
1274.2.e.n		2	7.d	odd	6	1
1274.2.e.n		2	91.n	odd	6	1
1274.2.g.a		2	7.c	even	3	1
1274.2.g.a		2	91.h	even	3	1
1274.2.h.a		2	1.a	even	1	1	trivial
1274.2.h.a		2	91.g	even	3	1	inner
1274.2.h.b		2	7.b	odd	2	1
1274.2.h.b		2	91.m	odd	6	1
1872.2.t.k		2	84.j	odd	6	1
1872.2.t.k		2	1092.bt	odd	6	1
2704.2.a.h		1	364.br	even	6	1
2704.2.a.i		1	364.bp	even	6	1
2704.2.f.g		2	364.cg	odd	12	2
3042.2.a.e		1	273.bf	even	6	1
3042.2.a.k		1	273.y	even	6	1
3042.2.b.e		2	273.ch	odd	12	2
8450.2.a.f		1	455.bw	odd	6	1
8450.2.a.s		1	455.bk	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\):

\( T_{3} \)

\( T_{5}^{2} + T_{5} + 1 \)

\( T_{11} - 4 \)

Hecke characteristic polynomials

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