Newform orbit 1368.2.s.g

• Label: 1368.2.s.g
• Level: $1368$
• Weight: $2$
• Character orbit: 1368.s
• Analytic conductor: $10.924$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.9235349965\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 152)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\)	\(343\)	\(685\)	\(1009\)	\(1217\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

505.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0

0

1.50000

−

2.59808i

0

0

0

0

0

577.1

0

0

0

1.50000

+

2.59808i

0

0

0

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	1368.2.s.g		2
3.b	odd	2	1		152.2.i.a	✓	2
4.b	odd	2	1		2736.2.s.q		2
12.b	even	2	1		304.2.i.b		2
19.c	even	3	1	inner	1368.2.s.g		2
24.f	even	2	1		1216.2.i.e		2
24.h	odd	2	1		1216.2.i.i		2
57.f	even	6	1		2888.2.a.c		1
57.h	odd	6	1		152.2.i.a	✓	2
57.h	odd	6	1		2888.2.a.e		1
76.g	odd	6	1		2736.2.s.q		2
228.m	even	6	1		304.2.i.b		2
228.m	even	6	1		5776.2.a.h		1
228.n	odd	6	1		5776.2.a.o		1
456.u	even	6	1		1216.2.i.e		2
456.x	odd	6	1		1216.2.i.i		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.i.a	✓	2	3.b	odd	2	1
152.2.i.a	✓	2	57.h	odd	6	1
304.2.i.b		2	12.b	even	2	1
304.2.i.b		2	228.m	even	6	1
1216.2.i.e		2	24.f	even	2	1
1216.2.i.e		2	456.u	even	6	1
1216.2.i.i		2	24.h	odd	2	1
1216.2.i.i		2	456.x	odd	6	1
1368.2.s.g		2	1.a	even	1	1	trivial
1368.2.s.g		2	19.c	even	3	1	inner
2736.2.s.q		2	4.b	odd	2	1
2736.2.s.q		2	76.g	odd	6	1
2888.2.a.c		1	57.f	even	6	1
2888.2.a.e		1	57.h	odd	6	1
5776.2.a.h		1	228.m	even	6	1
5776.2.a.o		1	228.n	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

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

\( T_{7} \)

Hecke characteristic polynomials

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