Newform orbit 2240.2.k.e

• Label: 2240.2.k.e
• Level: $2240$
• Weight: $2$
• Character orbit: 2240.k
• Analytic conductor: $17.886$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.8864900528\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.342102016.5
Defining polynomial:	\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} - \beta_{2}) q^{3} + \beta_1 q^{5} + ( - \beta_{5} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_1 + 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} - 3\nu^{5} - 5\nu^{4} - 2\nu^{3} + 2\nu^{2} - 8 ) / 16 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} + \nu^{6} - \nu^{5} + 3\nu^{4} + 6\nu^{3} + 10\nu^{2} - 8\nu + 8 ) / 16 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} + 3\nu^{4} - 6\nu^{3} + 10\nu^{2} + 8\nu + 8 ) / 16 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} - \nu^{6} - 3\nu^{5} + 5\nu^{4} - 2\nu^{3} - 2\nu^{2} + 8 ) / 16 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 4\nu^{6} + 3\nu^{5} - 4\nu^{4} + 10\nu^{3} + 24\nu - 16 ) / 16 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} + 2\nu^{6} + \nu^{5} + 2\nu^{4} + 4\nu^{3} + 12\nu + 8 ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\)	\(897\)	\(1471\)	\(1541\)	\(1921\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1791.1

1.17915	−	0.780776i
1.17915	+	0.780776i
−0.599676	+	1.28078i
−0.599676	−	1.28078i
0.599676	+	1.28078i
0.599676	−	1.28078i
−1.17915	−	0.780776i
−1.17915	+	0.780776i

0

−3.02045

0

−

1.00000i

0

−2.17238

+

1.51022i

0

6.12311

0

1791.2

0

−3.02045

0

1.00000i

0

−2.17238

−

1.51022i

0

6.12311

0

1791.3

0

−0.936426

0

−

1.00000i

0

−2.60399

+

0.468213i

0

−2.12311

0

1791.4

0

−0.936426

0

1.00000i

0

−2.60399

−

0.468213i

0

−2.12311

0

1791.5

0

0.936426

0

−

1.00000i

0

2.60399

−

0.468213i

0

−2.12311

0

1791.6

0

0.936426

0

1.00000i

0

2.60399

+

0.468213i

0

−2.12311

0

1791.7

0

3.02045

0

−

1.00000i

0

2.17238

−

1.51022i

0

6.12311

0

1791.8

0

3.02045

0

1.00000i

0

2.17238

+

1.51022i

0

6.12311

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2240.2.k.e		8
4.b	odd	2	1	inner	2240.2.k.e		8
7.b	odd	2	1	inner	2240.2.k.e		8
8.b	even	2	1		140.2.g.c	✓	8
8.d	odd	2	1		140.2.g.c	✓	8
24.f	even	2	1		1260.2.c.c		8
24.h	odd	2	1		1260.2.c.c		8
28.d	even	2	1	inner	2240.2.k.e		8
40.e	odd	2	1		700.2.g.j		8
40.f	even	2	1		700.2.g.j		8
40.i	odd	4	1		700.2.c.i		8
40.i	odd	4	1		700.2.c.j		8
40.k	even	4	1		700.2.c.i		8
40.k	even	4	1		700.2.c.j		8
56.e	even	2	1		140.2.g.c	✓	8
56.h	odd	2	1		140.2.g.c	✓	8
56.j	odd	6	2		980.2.o.e		16
56.k	odd	6	2		980.2.o.e		16
56.m	even	6	2		980.2.o.e		16
56.p	even	6	2		980.2.o.e		16
168.e	odd	2	1		1260.2.c.c		8
168.i	even	2	1		1260.2.c.c		8
280.c	odd	2	1		700.2.g.j		8
280.n	even	2	1		700.2.g.j		8
280.s	even	4	1		700.2.c.i		8
280.s	even	4	1		700.2.c.j		8
280.y	odd	4	1		700.2.c.i		8
280.y	odd	4	1		700.2.c.j		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.g.c	✓	8	8.b	even	2	1
140.2.g.c	✓	8	8.d	odd	2	1
140.2.g.c	✓	8	56.e	even	2	1
140.2.g.c	✓	8	56.h	odd	2	1
700.2.c.i		8	40.i	odd	4	1
700.2.c.i		8	40.k	even	4	1
700.2.c.i		8	280.s	even	4	1
700.2.c.i		8	280.y	odd	4	1
700.2.c.j		8	40.i	odd	4	1
700.2.c.j		8	40.k	even	4	1
700.2.c.j		8	280.s	even	4	1
700.2.c.j		8	280.y	odd	4	1
700.2.g.j		8	40.e	odd	2	1
700.2.g.j		8	40.f	even	2	1
700.2.g.j		8	280.c	odd	2	1
700.2.g.j		8	280.n	even	2	1
980.2.o.e		16	56.j	odd	6	2
980.2.o.e		16	56.k	odd	6	2
980.2.o.e		16	56.m	even	6	2
980.2.o.e		16	56.p	even	6	2
1260.2.c.c		8	24.f	even	2	1
1260.2.c.c		8	24.h	odd	2	1
1260.2.c.c		8	168.e	odd	2	1
1260.2.c.c		8	168.i	even	2	1
2240.2.k.e		8	1.a	even	1	1	trivial
2240.2.k.e		8	4.b	odd	2	1	inner
2240.2.k.e		8	7.b	odd	2	1	inner
2240.2.k.e		8	28.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{4} - 10T_{3}^{2} + 8 \)

\( T_{19}^{4} - 28T_{19}^{2} + 128 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( (T^{4} - 10 T^{2} + 8)^{2} \)
$5$	\( (T^{2} + 1)^{4} \)
$7$	T^8 - 18*T^6 + 162*T^4 - 882*T^2 + 2401
$11$	\( (T^{4} + 28 T^{2} + 128)^{2} \)