Newform orbit 3240.1.p.f

• Label: 3240.1.p.f
• Level: $3240$
• Weight: $1$
• Character orbit: 3240.p
• Self dual: yes
• Analytic conductor: $1.617$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -40
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3240.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.61697064093\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.251942400.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} - q^{5} - \beta q^{7} + q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\)	\(1297\)	\(1621\)	\(2431\)	\(3161\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1459.1

1.73205
−1.73205

1.00000

0

1.00000

−1.00000

0

−1.73205

1.00000

0

−1.00000

1459.2

1.00000

0

1.00000

−1.00000

0

1.73205

1.00000

0

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by \(\Q(\sqrt{-10}) \)
15.d	odd	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3240.1.p.f	yes	2
3.b	odd	2	1		3240.1.p.e	✓	2
5.b	even	2	1		3240.1.p.e	✓	2
8.d	odd	2	1		3240.1.p.e	✓	2
9.c	even	3	2		3240.1.z.k		4
9.d	odd	6	2		3240.1.z.l		4
15.d	odd	2	1	inner	3240.1.p.f	yes	2
24.f	even	2	1	inner	3240.1.p.f	yes	2
40.e	odd	2	1	CM	3240.1.p.f	yes	2
45.h	odd	6	2		3240.1.z.k		4
45.j	even	6	2		3240.1.z.l		4
72.l	even	6	2		3240.1.z.k		4
72.p	odd	6	2		3240.1.z.l		4
120.m	even	2	1		3240.1.p.e	✓	2
360.z	odd	6	2		3240.1.z.k		4
360.bd	even	6	2		3240.1.z.l		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3240.1.p.e	✓	2	3.b	odd	2	1
3240.1.p.e	✓	2	5.b	even	2	1
3240.1.p.e	✓	2	8.d	odd	2	1
3240.1.p.e	✓	2	120.m	even	2	1
3240.1.p.f	yes	2	1.a	even	1	1	trivial
3240.1.p.f	yes	2	15.d	odd	2	1	inner
3240.1.p.f	yes	2	24.f	even	2	1	inner
3240.1.p.f	yes	2	40.e	odd	2	1	CM
3240.1.z.k		4	9.c	even	3	2
3240.1.z.k		4	45.h	odd	6	2
3240.1.z.k		4	72.l	even	6	2
3240.1.z.k		4	360.z	odd	6	2
3240.1.z.l		4	9.d	odd	6	2
3240.1.z.l		4	45.j	even	6	2
3240.1.z.l		4	72.p	odd	6	2
3240.1.z.l		4	360.bd	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{2} - 3 \)

\( T_{11} \)

\( T_{23} - 1 \)

Hecke characteristic polynomials

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