Newform orbit 338.2.c.e

• Label: 338.2.c.e
• Level: $338$
• Weight: $2$
• Character orbit: 338.c
• Analytic conductor: $2.699$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.69894358832\)
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.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + q^{5} + 4 \zeta_{6} q^{7} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} + 4 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\)	\(171\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

191.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

1.00000

0

2.00000

+

3.46410i

−1.00000

1.50000

+

2.59808i

0.500000

−

0.866025i

315.1

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

1.00000

0

2.00000

−

3.46410i

−1.00000

1.50000

−

2.59808i

0.500000

+

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	338.2.c.e		2
13.b	even	2	1		26.2.c.a	✓	2
13.c	even	3	1		338.2.a.c		1
13.c	even	3	1	inner	338.2.c.e		2
13.d	odd	4	2		338.2.e.b		4
13.e	even	6	1		26.2.c.a	✓	2
13.e	even	6	1		338.2.a.e		1
13.f	odd	12	2		338.2.b.b		2
13.f	odd	12	2		338.2.e.b		4
39.d	odd	2	1		234.2.h.c		2
39.h	odd	6	1		234.2.h.c		2
39.h	odd	6	1		3042.2.a.e		1
39.i	odd	6	1		3042.2.a.k		1
39.k	even	12	2		3042.2.b.e		2
52.b	odd	2	1		208.2.i.b		2
52.i	odd	6	1		208.2.i.b		2
52.i	odd	6	1		2704.2.a.h		1
52.j	odd	6	1		2704.2.a.i		1
52.l	even	12	2		2704.2.f.g		2
65.d	even	2	1		650.2.e.c		2
65.h	odd	4	2		650.2.o.c		4
65.l	even	6	1		650.2.e.c		2
65.l	even	6	1		8450.2.a.f		1
65.n	even	6	1		8450.2.a.s		1
65.r	odd	12	2		650.2.o.c		4
91.b	odd	2	1		1274.2.g.a		2
91.k	even	6	1		1274.2.h.b		2
91.l	odd	6	1		1274.2.h.a		2
91.p	odd	6	1		1274.2.e.m		2
91.r	even	6	1		1274.2.e.n		2
91.r	even	6	1		1274.2.h.b		2
91.s	odd	6	1		1274.2.e.m		2
91.s	odd	6	1		1274.2.h.a		2
91.t	odd	6	1		1274.2.g.a		2
91.u	even	6	1		1274.2.e.n		2
104.e	even	2	1		832.2.i.e		2
104.h	odd	2	1		832.2.i.f		2
104.p	odd	6	1		832.2.i.f		2
104.s	even	6	1		832.2.i.e		2
156.h	even	2	1		1872.2.t.k		2
156.r	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	13.b	even	2	1
26.2.c.a	✓	2	13.e	even	6	1
208.2.i.b		2	52.b	odd	2	1
208.2.i.b		2	52.i	odd	6	1
234.2.h.c		2	39.d	odd	2	1
234.2.h.c		2	39.h	odd	6	1
338.2.a.c		1	13.c	even	3	1
338.2.a.e		1	13.e	even	6	1
338.2.b.b		2	13.f	odd	12	2
338.2.c.e		2	1.a	even	1	1	trivial
338.2.c.e		2	13.c	even	3	1	inner
338.2.e.b		4	13.d	odd	4	2
338.2.e.b		4	13.f	odd	12	2
650.2.e.c		2	65.d	even	2	1
650.2.e.c		2	65.l	even	6	1
650.2.o.c		4	65.h	odd	4	2
650.2.o.c		4	65.r	odd	12	2
832.2.i.e		2	104.e	even	2	1
832.2.i.e		2	104.s	even	6	1
832.2.i.f		2	104.h	odd	2	1
832.2.i.f		2	104.p	odd	6	1
1274.2.e.m		2	91.p	odd	6	1
1274.2.e.m		2	91.s	odd	6	1
1274.2.e.n		2	91.r	even	6	1
1274.2.e.n		2	91.u	even	6	1
1274.2.g.a		2	91.b	odd	2	1
1274.2.g.a		2	91.t	odd	6	1
1274.2.h.a		2	91.l	odd	6	1
1274.2.h.a		2	91.s	odd	6	1
1274.2.h.b		2	91.k	even	6	1
1274.2.h.b		2	91.r	even	6	1
1872.2.t.k		2	156.h	even	2	1
1872.2.t.k		2	156.r	even	6	1
2704.2.a.h		1	52.i	odd	6	1
2704.2.a.i		1	52.j	odd	6	1
2704.2.f.g		2	52.l	even	12	2
3042.2.a.e		1	39.h	odd	6	1
3042.2.a.k		1	39.i	odd	6	1
3042.2.b.e		2	39.k	even	12	2
8450.2.a.f		1	65.l	even	6	1
8450.2.a.s		1	65.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\):

\( T_{3} \)

\( T_{5} - 1 \)

Hecke characteristic polynomials

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