Newform orbit 40.5.e.a

• Label: 40.5.e.a
• Level: $40$
• Weight: $5$
• Character orbit: 40.e
• Self dual: yes
• Analytic conductor: $4.135$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -40
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	40.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.13479852335\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 4 q^{2} + 16 q^{4} - 25 q^{5} + 62 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

0

−4.00000

0

16.0000

−25.0000

0

62.0000

−64.0000

81.0000

100.000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	40.5.e.a	✓	1
4.b	odd	2	1		160.5.e.a		1
5.b	even	2	1		40.5.e.b	yes	1
5.c	odd	4	2		200.5.g.c		2
8.b	even	2	1		160.5.e.b		1
8.d	odd	2	1		40.5.e.b	yes	1
20.d	odd	2	1		160.5.e.b		1
20.e	even	4	2		800.5.g.c		2
40.e	odd	2	1	CM	40.5.e.a	✓	1
40.f	even	2	1		160.5.e.a		1
40.i	odd	4	2		800.5.g.c		2
40.k	even	4	2		200.5.g.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.5.e.a	✓	1	1.a	even	1	1	trivial
40.5.e.a	✓	1	40.e	odd	2	1	CM
40.5.e.b	yes	1	5.b	even	2	1
40.5.e.b	yes	1	8.d	odd	2	1
160.5.e.a		1	4.b	odd	2	1
160.5.e.a		1	40.f	even	2	1
160.5.e.b		1	8.b	even	2	1
160.5.e.b		1	20.d	odd	2	1
200.5.g.c		2	5.c	odd	4	2
200.5.g.c		2	40.k	even	4	2
800.5.g.c		2	20.e	even	4	2
800.5.g.c		2	40.i	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(40, [\chi])\):

\( T_{3} \)

\( T_{7} - 62 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 4 \)
$3$	\( T \)
$5$	\( T + 25 \)
$7$	\( T - 62 \)
$11$	\( T - 82 \)