Newform orbit 342.2.g.b

• Label: 342.2.g.b
• Level: $342$
• Weight: $2$
• Character orbit: 342.g
• Analytic conductor: $2.731$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.73088374913\)
Analytic rank:	\(1\)
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 38)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

163.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

0

0

−4.00000

1.00000

0

0

235.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

0

0

−4.00000

1.00000

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	342.2.g.b		2
3.b	odd	2	1		38.2.c.a	✓	2
4.b	odd	2	1		2736.2.s.m		2
12.b	even	2	1		304.2.i.c		2
15.d	odd	2	1		950.2.e.d		2
15.e	even	4	2		950.2.j.e		4
19.c	even	3	1	inner	342.2.g.b		2
19.c	even	3	1		6498.2.a.s		1
19.d	odd	6	1		6498.2.a.e		1
24.f	even	2	1		1216.2.i.d		2
24.h	odd	2	1		1216.2.i.h		2
57.d	even	2	1		722.2.c.b		2
57.f	even	6	1		722.2.a.d		1
57.f	even	6	1		722.2.c.b		2
57.h	odd	6	1		38.2.c.a	✓	2
57.h	odd	6	1		722.2.a.c		1
57.j	even	18	6		722.2.e.i		6
57.l	odd	18	6		722.2.e.j		6
76.g	odd	6	1		2736.2.s.m		2
228.m	even	6	1		304.2.i.c		2
228.m	even	6	1		5776.2.a.g		1
228.n	odd	6	1		5776.2.a.n		1
285.n	odd	6	1		950.2.e.d		2
285.v	even	12	2		950.2.j.e		4
456.u	even	6	1		1216.2.i.d		2
456.x	odd	6	1		1216.2.i.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.2.c.a	✓	2	3.b	odd	2	1
38.2.c.a	✓	2	57.h	odd	6	1
304.2.i.c		2	12.b	even	2	1
304.2.i.c		2	228.m	even	6	1
342.2.g.b		2	1.a	even	1	1	trivial
342.2.g.b		2	19.c	even	3	1	inner
722.2.a.c		1	57.h	odd	6	1
722.2.a.d		1	57.f	even	6	1
722.2.c.b		2	57.d	even	2	1
722.2.c.b		2	57.f	even	6	1
722.2.e.i		6	57.j	even	18	6
722.2.e.j		6	57.l	odd	18	6
950.2.e.d		2	15.d	odd	2	1
950.2.e.d		2	285.n	odd	6	1
950.2.j.e		4	15.e	even	4	2
950.2.j.e		4	285.v	even	12	2
1216.2.i.d		2	24.f	even	2	1
1216.2.i.d		2	456.u	even	6	1
1216.2.i.h		2	24.h	odd	2	1
1216.2.i.h		2	456.x	odd	6	1
2736.2.s.m		2	4.b	odd	2	1
2736.2.s.m		2	76.g	odd	6	1
5776.2.a.g		1	228.m	even	6	1
5776.2.a.n		1	228.n	odd	6	1
6498.2.a.e		1	19.d	odd	6	1
6498.2.a.s		1	19.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\):

\( T_{5} \)

\( T_{7} + 4 \)

Hecke characteristic polynomials

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