Newform orbit 400.5.b.a

• Label: 400.5.b.a
• Level: $400$
• Weight: $5$
• Character orbit: 400.b
• Self dual: yes
• Analytic conductor: $41.348$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	400.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.3479852335\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 81 q^{9} - 240 q^{13} + 480 q^{17} - 82 q^{29} - 1680 q^{37} + 3038 q^{41} + 2401 q^{49} + 5040 q^{53} + 6958 q^{61} - 10560 q^{73} + 6561 q^{81} + 9758 q^{89} + 18720 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

351.1

0

0

0

0

0

0

0

0

81.0000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.5.b.a		1
4.b	odd	2	1	CM	400.5.b.a		1
5.b	even	2	1		400.5.b.b		1
5.c	odd	4	2		80.5.h.b	✓	2
15.e	even	4	2		720.5.j.a		2
20.d	odd	2	1		400.5.b.b		1
20.e	even	4	2		80.5.h.b	✓	2
40.i	odd	4	2		320.5.h.c		2
40.k	even	4	2		320.5.h.c		2
60.l	odd	4	2		720.5.j.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.5.h.b	✓	2	5.c	odd	4	2
80.5.h.b	✓	2	20.e	even	4	2
320.5.h.c		2	40.i	odd	4	2
320.5.h.c		2	40.k	even	4	2
400.5.b.a		1	1.a	even	1	1	trivial
400.5.b.a		1	4.b	odd	2	1	CM
400.5.b.b		1	5.b	even	2	1
400.5.b.b		1	20.d	odd	2	1
720.5.j.a		2	15.e	even	4	2
720.5.j.a		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(400, [\chi])\):

\( T_{3} \)

\( T_{13} + 240 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T \)