Newform orbit 1274.2.e.l

• Label: 1274.2.e.l
• 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:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 182)
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.

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

Character values

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

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

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

1.50000

+

2.59808i

1.00000

−1.50000

−

2.59808i

−1.50000

−

2.59808i

0

−1.00000

−3.00000

+

5.19615i

1.50000

+

2.59808i

471.1

−1.00000

1.50000

−

2.59808i

1.00000

−1.50000

+

2.59808i

−1.50000

+

2.59808i

0

−1.00000

−3.00000

−

5.19615i

1.50000

−

2.59808i

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.l		2
7.b	odd	2	1		1274.2.e.a		2
7.c	even	3	1		1274.2.g.k		2
7.c	even	3	1		1274.2.h.c		2
7.d	odd	6	1		182.2.g.a	✓	2
7.d	odd	6	1		1274.2.h.n		2
13.c	even	3	1		1274.2.h.c		2
21.g	even	6	1		1638.2.r.n		2
28.f	even	6	1		1456.2.s.g		2
91.g	even	3	1		1274.2.g.k		2
91.h	even	3	1	inner	1274.2.e.l		2
91.l	odd	6	1		2366.2.a.p		1
91.m	odd	6	1		182.2.g.a	✓	2
91.n	odd	6	1		1274.2.h.n		2
91.v	odd	6	1		1274.2.e.a		2
91.v	odd	6	1		2366.2.a.g		1
91.ba	even	12	2		2366.2.d.h		2
273.bf	even	6	1		1638.2.r.n		2
364.br	even	6	1		1456.2.s.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
182.2.g.a	✓	2	7.d	odd	6	1
182.2.g.a	✓	2	91.m	odd	6	1
1274.2.e.a		2	7.b	odd	2	1
1274.2.e.a		2	91.v	odd	6	1
1274.2.e.l		2	1.a	even	1	1	trivial
1274.2.e.l		2	91.h	even	3	1	inner
1274.2.g.k		2	7.c	even	3	1
1274.2.g.k		2	91.g	even	3	1
1274.2.h.c		2	7.c	even	3	1
1274.2.h.c		2	13.c	even	3	1
1274.2.h.n		2	7.d	odd	6	1
1274.2.h.n		2	91.n	odd	6	1
1456.2.s.g		2	28.f	even	6	1
1456.2.s.g		2	364.br	even	6	1
1638.2.r.n		2	21.g	even	6	1
1638.2.r.n		2	273.bf	even	6	1
2366.2.a.g		1	91.v	odd	6	1
2366.2.a.p		1	91.l	odd	6	1
2366.2.d.h		2	91.ba	even	12	2

Hecke kernels

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

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

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

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} - 3T + 9 \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 4T + 16 \)