Newform orbit 338.4.c.g

• Label: 338.4.c.g
• Level: $338$
• Weight: $4$
• Character orbit: 338.c
• Analytic conductor: $19.943$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.9426455819\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 2) q^{2} + ( - \zeta_{6} + 1) q^{3} - 4 \zeta_{6} q^{4} - 17 q^{5} - 2 \zeta_{6} q^{6} - 35 \zeta_{6} q^{7} - 8 q^{8} + 26 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} - 4 q^{4} - 34 q^{5} - 2 q^{6} - 35 q^{7} - 16 q^{8} + 26 q^{9}+O(q^{10}) \)

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

1.00000

−

1.73205i

0.500000

−

0.866025i

−2.00000

−

3.46410i

−17.0000

−1.00000

−

1.73205i

−17.5000

−

30.3109i

−8.00000

13.0000

+

22.5167i

−17.0000

+

29.4449i

315.1

1.00000

+

1.73205i

0.500000

+

0.866025i

−2.00000

+

3.46410i

−17.0000

−1.00000

+

1.73205i

−17.5000

+

30.3109i

−8.00000

13.0000

−

22.5167i

−17.0000

−

29.4449i

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	338.4.c.g		2
13.b	even	2	1		338.4.c.c		2
13.c	even	3	1		338.4.a.b		1
13.c	even	3	1	inner	338.4.c.g		2
13.d	odd	4	2		338.4.e.c		4
13.e	even	6	1		26.4.a.b	✓	1
13.e	even	6	1		338.4.c.c		2
13.f	odd	12	2		338.4.b.b		2
13.f	odd	12	2		338.4.e.c		4
39.h	odd	6	1		234.4.a.a		1
52.i	odd	6	1		208.4.a.e		1
65.l	even	6	1		650.4.a.c		1
65.r	odd	12	2		650.4.b.d		2
91.t	odd	6	1		1274.4.a.f		1
104.p	odd	6	1		832.4.a.g		1
104.s	even	6	1		832.4.a.j		1
156.r	even	6	1		1872.4.a.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.4.a.b	✓	1	13.e	even	6	1
208.4.a.e		1	52.i	odd	6	1
234.4.a.a		1	39.h	odd	6	1
338.4.a.b		1	13.c	even	3	1
338.4.b.b		2	13.f	odd	12	2
338.4.c.c		2	13.b	even	2	1
338.4.c.c		2	13.e	even	6	1
338.4.c.g		2	1.a	even	1	1	trivial
338.4.c.g		2	13.c	even	3	1	inner
338.4.e.c		4	13.d	odd	4	2
338.4.e.c		4	13.f	odd	12	2
650.4.a.c		1	65.l	even	6	1
650.4.b.d		2	65.r	odd	12	2
832.4.a.g		1	104.p	odd	6	1
832.4.a.j		1	104.s	even	6	1
1274.4.a.f		1	91.t	odd	6	1
1872.4.a.b		1	156.r	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)

\( T_{5} + 17 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 2T + 4 \)
$3$	\( T^{2} - T + 1 \)
$5$	\( (T + 17)^{2} \)
$7$	\( T^{2} + 35T + 1225 \)
$11$	\( T^{2} - 2T + 4 \)