Newform orbit 1350.2.e.a

• Label: 1350.2.e.a
• Level: $1350$
• Weight: $2$
• Character orbit: 1350.e
• Analytic conductor: $10.780$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7798042729\)
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 90)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\)	\(1001\)	\(1027\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

451.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

0

0

−0.500000

+

0.866025i

1.00000

0

0

901.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

0

0

−0.500000

−

0.866025i

1.00000

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1350.2.e.a		2
3.b	odd	2	1		450.2.e.g		2
5.b	even	2	1		1350.2.e.i		2
5.c	odd	4	2		270.2.i.a		4
9.c	even	3	1	inner	1350.2.e.a		2
9.c	even	3	1		4050.2.a.be		1
9.d	odd	6	1		450.2.e.g		2
9.d	odd	6	1		4050.2.a.j		1
15.d	odd	2	1		450.2.e.b		2
15.e	even	4	2		90.2.i.a	✓	4
20.e	even	4	2		2160.2.by.b		4
45.h	odd	6	1		450.2.e.b		2
45.h	odd	6	1		4050.2.a.x		1
45.j	even	6	1		1350.2.e.i		2
45.j	even	6	1		4050.2.a.g		1
45.k	odd	12	2		270.2.i.a		4
45.k	odd	12	2		810.2.c.c		2
45.l	even	12	2		90.2.i.a	✓	4
45.l	even	12	2		810.2.c.b		2
60.l	odd	4	2		720.2.by.a		4
180.v	odd	12	2		720.2.by.a		4
180.x	even	12	2		2160.2.by.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.2.i.a	✓	4	15.e	even	4	2
90.2.i.a	✓	4	45.l	even	12	2
270.2.i.a		4	5.c	odd	4	2
270.2.i.a		4	45.k	odd	12	2
450.2.e.b		2	15.d	odd	2	1
450.2.e.b		2	45.h	odd	6	1
450.2.e.g		2	3.b	odd	2	1
450.2.e.g		2	9.d	odd	6	1
720.2.by.a		4	60.l	odd	4	2
720.2.by.a		4	180.v	odd	12	2
810.2.c.b		2	45.l	even	12	2
810.2.c.c		2	45.k	odd	12	2
1350.2.e.a		2	1.a	even	1	1	trivial
1350.2.e.a		2	9.c	even	3	1	inner
1350.2.e.i		2	5.b	even	2	1
1350.2.e.i		2	45.j	even	6	1
2160.2.by.b		4	20.e	even	4	2
2160.2.by.b		4	180.x	even	12	2
4050.2.a.g		1	45.j	even	6	1
4050.2.a.j		1	9.d	odd	6	1
4050.2.a.x		1	45.h	odd	6	1
4050.2.a.be		1	9.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 \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \)

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

\( T_{17} + 2 \)

Hecke characteristic polynomials

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