Newform orbit 1274.2.e.n

• Label: 1274.2.e.n
• Level: $1274$
• Weight: $2$
• Character orbit: 1274.e
• 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.e (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:	\(\Z[a_1, \ldots, a_{5}]\)
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.

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

165.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

0

1.00000

0.500000

+

0.866025i

0

0

1.00000

1.50000

−

2.59808i

0.500000

+

0.866025i

471.1

1.00000

0

1.00000

0.500000

−

0.866025i

0

0

1.00000

1.50000

+

2.59808i

0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.h	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.e.n		2
7.b	odd	2	1		1274.2.e.m		2
7.c	even	3	1		26.2.c.a	✓	2
7.c	even	3	1		1274.2.h.b		2
7.d	odd	6	1		1274.2.g.a		2
7.d	odd	6	1		1274.2.h.a		2
13.c	even	3	1		1274.2.h.b		2
21.h	odd	6	1		234.2.h.c		2
28.g	odd	6	1		208.2.i.b		2
35.j	even	6	1		650.2.e.c		2
35.l	odd	12	2		650.2.o.c		4
56.k	odd	6	1		832.2.i.f		2
56.p	even	6	1		832.2.i.e		2
84.n	even	6	1		1872.2.t.k		2
91.g	even	3	1		26.2.c.a	✓	2
91.h	even	3	1		338.2.a.e		1
91.h	even	3	1	inner	1274.2.e.n		2
91.k	even	6	1		338.2.a.c		1
91.m	odd	6	1		1274.2.g.a		2
91.n	odd	6	1		1274.2.h.a		2
91.r	even	6	1		338.2.c.e		2
91.u	even	6	1		338.2.c.e		2
91.v	odd	6	1		1274.2.e.m		2
91.x	odd	12	2		338.2.b.b		2
91.z	odd	12	2		338.2.e.b		4
91.bd	odd	12	2		338.2.e.b		4
273.s	odd	6	1		3042.2.a.e		1
273.bm	odd	6	1		234.2.h.c		2
273.bp	odd	6	1		3042.2.a.k		1
273.bv	even	12	2		3042.2.b.e		2
364.q	odd	6	1		208.2.i.b		2
364.bi	odd	6	1		2704.2.a.h		1
364.bk	odd	6	1		2704.2.a.i		1
364.ca	even	12	2		2704.2.f.g		2
455.ba	even	6	1		8450.2.a.f		1
455.bm	even	6	1		650.2.e.c		2
455.bz	even	6	1		8450.2.a.s		1
455.cx	odd	12	2		650.2.o.c		4
728.bg	even	6	1		832.2.i.e		2
728.di	odd	6	1		832.2.i.f		2
1092.dc	even	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.c	even	3	1
26.2.c.a	✓	2	91.g	even	3	1
208.2.i.b		2	28.g	odd	6	1
208.2.i.b		2	364.q	odd	6	1
234.2.h.c		2	21.h	odd	6	1
234.2.h.c		2	273.bm	odd	6	1
338.2.a.c		1	91.k	even	6	1
338.2.a.e		1	91.h	even	3	1
338.2.b.b		2	91.x	odd	12	2
338.2.c.e		2	91.r	even	6	1
338.2.c.e		2	91.u	even	6	1
338.2.e.b		4	91.z	odd	12	2
338.2.e.b		4	91.bd	odd	12	2
650.2.e.c		2	35.j	even	6	1
650.2.e.c		2	455.bm	even	6	1
650.2.o.c		4	35.l	odd	12	2
650.2.o.c		4	455.cx	odd	12	2
832.2.i.e		2	56.p	even	6	1
832.2.i.e		2	728.bg	even	6	1
832.2.i.f		2	56.k	odd	6	1
832.2.i.f		2	728.di	odd	6	1
1274.2.e.m		2	7.b	odd	2	1
1274.2.e.m		2	91.v	odd	6	1
1274.2.e.n		2	1.a	even	1	1	trivial
1274.2.e.n		2	91.h	even	3	1	inner
1274.2.g.a		2	7.d	odd	6	1
1274.2.g.a		2	91.m	odd	6	1
1274.2.h.a		2	7.d	odd	6	1
1274.2.h.a		2	91.n	odd	6	1
1274.2.h.b		2	7.c	even	3	1
1274.2.h.b		2	13.c	even	3	1
1872.2.t.k		2	84.n	even	6	1
1872.2.t.k		2	1092.dc	even	6	1
2704.2.a.h		1	364.bi	odd	6	1
2704.2.a.i		1	364.bk	odd	6	1
2704.2.f.g		2	364.ca	even	12	2
3042.2.a.e		1	273.s	odd	6	1
3042.2.a.k		1	273.bp	odd	6	1
3042.2.b.e		2	273.bv	even	12	2
8450.2.a.f		1	455.ba	even	6	1
8450.2.a.s		1	455.bz	even	6	1

Hecke kernels

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

\( T_{3} \)

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

\( T_{11}^{2} + 4T_{11} + 16 \)

Hecke characteristic polynomials

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