Newform orbit 1440.2.x.e

• Label: 1440.2.x.e
• Level: $1440$
• Weight: $2$
• Character orbit: 1440.x
• Analytic conductor: $11.498$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.4984578911\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (2 i - 1) q^{5} + ( - i + 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(641\)	\(901\)	\(991\)
\(\chi(n)\)	\(i\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

		1.00000i
	−	1.00000i

0

0

0

−1.00000

+

2.00000i

0

1.00000

−

1.00000i

0

0

0

703.1

0

0

0

−1.00000

−

2.00000i

0

1.00000

+

1.00000i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1440.2.x.e		2
3.b	odd	2	1		160.2.n.e	yes	2
4.b	odd	2	1		1440.2.x.b		2
5.c	odd	4	1		1440.2.x.b		2
12.b	even	2	1		160.2.n.b	✓	2
15.d	odd	2	1		800.2.n.c		2
15.e	even	4	1		160.2.n.b	✓	2
15.e	even	4	1		800.2.n.h		2
20.e	even	4	1	inner	1440.2.x.e		2
24.f	even	2	1		320.2.n.f		2
24.h	odd	2	1		320.2.n.c		2
48.i	odd	4	1		1280.2.o.d		2
48.i	odd	4	1		1280.2.o.n		2
48.k	even	4	1		1280.2.o.e		2
48.k	even	4	1		1280.2.o.k		2
60.h	even	2	1		800.2.n.h		2
60.l	odd	4	1		160.2.n.e	yes	2
60.l	odd	4	1		800.2.n.c		2
120.i	odd	2	1		1600.2.n.j		2
120.m	even	2	1		1600.2.n.e		2
120.q	odd	4	1		320.2.n.c		2
120.q	odd	4	1		1600.2.n.j		2
120.w	even	4	1		320.2.n.f		2
120.w	even	4	1		1600.2.n.e		2
240.z	odd	4	1		1280.2.o.n		2
240.bb	even	4	1		1280.2.o.e		2
240.bd	odd	4	1		1280.2.o.d		2
240.bf	even	4	1		1280.2.o.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.n.b	✓	2	12.b	even	2	1
160.2.n.b	✓	2	15.e	even	4	1
160.2.n.e	yes	2	3.b	odd	2	1
160.2.n.e	yes	2	60.l	odd	4	1
320.2.n.c		2	24.h	odd	2	1
320.2.n.c		2	120.q	odd	4	1
320.2.n.f		2	24.f	even	2	1
320.2.n.f		2	120.w	even	4	1
800.2.n.c		2	15.d	odd	2	1
800.2.n.c		2	60.l	odd	4	1
800.2.n.h		2	15.e	even	4	1
800.2.n.h		2	60.h	even	2	1
1280.2.o.d		2	48.i	odd	4	1
1280.2.o.d		2	240.bd	odd	4	1
1280.2.o.e		2	48.k	even	4	1
1280.2.o.e		2	240.bb	even	4	1
1280.2.o.k		2	48.k	even	4	1
1280.2.o.k		2	240.bf	even	4	1
1280.2.o.n		2	48.i	odd	4	1
1280.2.o.n		2	240.z	odd	4	1
1440.2.x.b		2	4.b	odd	2	1
1440.2.x.b		2	5.c	odd	4	1
1440.2.x.e		2	1.a	even	1	1	trivial
1440.2.x.e		2	20.e	even	4	1	inner
1600.2.n.e		2	120.m	even	2	1
1600.2.n.e		2	120.w	even	4	1
1600.2.n.j		2	120.i	odd	2	1
1600.2.n.j		2	120.q	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

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

\( T_{11}^{2} + 36 \)

\( T_{17}^{2} + 2T_{17} + 2 \)

\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 2T + 5 \)
$7$	\( T^{2} - 2T + 2 \)
$11$	\( T^{2} + 36 \)