Newform orbit 400.6.a.m

• Label: 400.6.a.m
• Level: $400$
• Weight: $6$
• Character orbit: 400.a
• Self dual: yes
• Analytic conductor: $64.154$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q + 22 * q^3 + 218 * q^7 + 241 * q^9 + 480 * q^11 + 622 * q^13 - 186 * q^17 + 1204 * q^19 + 4796 * q^21 - 3186 * q^23 - 44 * q^27 + 5526 * q^29 - 9356 * q^31 + 10560 * q^33 - 5618 * q^37 + 13684 * q^39 - 14394 * q^41 - 370 * q^43 + 16146 * q^47 + 30717 * q^49 - 4092 * q^51 + 4374 * q^53 + 26488 * q^57 + 11748 * q^59 + 13202 * q^61 + 52538 * q^63 - 11542 * q^67 - 70092 * q^69 + 29532 * q^71 - 33698 * q^73 + 104640 * q^77 - 31208 * q^79 - 59531 * q^81 - 38466 * q^83 + 121572 * q^87 + 119514 * q^89 + 135596 * q^91 - 205832 * q^93 - 94658 * q^97 + 115680 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

0

0

22.0000

0

0

0

218.000

0

241.000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.6.a.m		1
4.b	odd	2	1		100.6.a.a		1
5.b	even	2	1		80.6.a.b		1
5.c	odd	4	2		400.6.c.c		2
12.b	even	2	1		900.6.a.b		1
15.d	odd	2	1		720.6.a.l		1
20.d	odd	2	1		20.6.a.a	✓	1
20.e	even	4	2		100.6.c.a		2
40.e	odd	2	1		320.6.a.c		1
40.f	even	2	1		320.6.a.n		1
60.h	even	2	1		180.6.a.e		1
60.l	odd	4	2		900.6.d.h		2
140.c	even	2	1		980.6.a.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.6.a.a	✓	1	20.d	odd	2	1
80.6.a.b		1	5.b	even	2	1
100.6.a.a		1	4.b	odd	2	1
100.6.c.a		2	20.e	even	4	2
180.6.a.e		1	60.h	even	2	1
320.6.a.c		1	40.e	odd	2	1
320.6.a.n		1	40.f	even	2	1
400.6.a.m		1	1.a	even	1	1	trivial
400.6.c.c		2	5.c	odd	4	2
720.6.a.l		1	15.d	odd	2	1
900.6.a.b		1	12.b	even	2	1
900.6.d.h		2	60.l	odd	4	2
980.6.a.b		1	140.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 22$ acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 22$
$5$	$T$
$7$	$T - 218$
$11$	$T - 480$