Newform orbit 1620.1.l.a

• Label: 1620.1.l.a
• Level: $1620$
• Weight: $1$
• Character orbit: 1620.l
• Analytic conductor: $0.808$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $S_{4}$
• CM/RM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1620.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.808485320465\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.0.40500.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	q + z * q^5 + (z - 1) * q^7 - q^11 + (z - 1) * q^17 - z * q^19 + (-z - 1) * q^23 - q^25 + z * q^29 + q^31 + (-z - 1) * q^35 - q^41 - z * q^49 + (z + 1) * q^53 - z * q^55 + z * q^59 + (z - 1) * q^67 + q^71 + (z + 1) * q^73 + (-z + 1) * q^77 + (-z - 1) * q^85 - z * q^89 + q^95
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 2 q^{11} - 2 q^{17} - 2 q^{23} - 2 q^{25} + 2 q^{31} - 2 q^{35} - 2 q^{41} + 2 q^{53} - 2 q^{67} + 2 q^{71} + 2 q^{73} + 2 q^{77} - 2 q^{85} + 2 q^{95}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\)	\(811\)	\(1297\)	\(1541\)
\(\chi(n)\)	\(1\)	\(-i\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

973.1

		1.00000i
	−	1.00000i

0

0

0

1.00000i

0

−1.00000

+

1.00000i

0

0

0

1297.1

0

0

0

−

1.00000i

0

−1.00000

−

1.00000i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1620.1.l.a	✓	2
3.b	odd	2	1		1620.1.l.b	yes	2
5.c	odd	4	1	inner	1620.1.l.a	✓	2
9.c	even	3	2		1620.1.v.b		4
9.d	odd	6	2		1620.1.v.a		4
15.e	even	4	1		1620.1.l.b	yes	2
45.k	odd	12	2		1620.1.v.b		4
45.l	even	12	2		1620.1.v.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1620.1.l.a	✓	2	1.a	even	1	1	trivial
1620.1.l.a	✓	2	5.c	odd	4	1	inner
1620.1.l.b	yes	2	3.b	odd	2	1
1620.1.l.b	yes	2	15.e	even	4	1
1620.1.v.a		4	9.d	odd	6	2
1620.1.v.a		4	45.l	even	12	2
1620.1.v.b		4	9.c	even	3	2
1620.1.v.b		4	45.k	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

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