Newform orbit 36.8.a.b

• Label: 36.8.a.b
• Level: $36$
• Weight: $8$
• Character orbit: 36.a
• Self dual: yes
• Analytic conductor: $11.246$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$36 = 2^{2} \cdot 3^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	36.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$11.2458609174$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

$f(q)$

$=$

q - 508 * q^7 - 14614 * q^13 - 57448 * q^19 - 78125 * q^25 + 178916 * q^31 + 279710 * q^37 + 1035224 * q^43 - 565479 * q^49 - 3535546 * q^61 - 385072 * q^67 - 6274810 * q^73 + 8763044 * q^79 + 7423912 * q^91 + 12245198 * q^97

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

0

0

0

0

−508.000

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	36.8.a.b	✓	1
3.b	odd	2	1	CM	36.8.a.b	✓	1
4.b	odd	2	1		144.8.a.e		1
8.b	even	2	1		576.8.a.q		1
8.d	odd	2	1		576.8.a.r		1
9.c	even	3	2		324.8.e.c		2
9.d	odd	6	2		324.8.e.c		2
12.b	even	2	1		144.8.a.e		1
24.f	even	2	1		576.8.a.r		1
24.h	odd	2	1		576.8.a.q		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.8.a.b	✓	1	1.a	even	1	1	trivial
36.8.a.b	✓	1	3.b	odd	2	1	CM
144.8.a.e		1	4.b	odd	2	1
144.8.a.e		1	12.b	even	2	1
324.8.e.c		2	9.c	even	3	2
324.8.e.c		2	9.d	odd	6	2
576.8.a.q		1	8.b	even	2	1
576.8.a.q		1	24.h	odd	2	1
576.8.a.r		1	8.d	odd	2	1
576.8.a.r		1	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ acting on $S_{8}^{\mathrm{new}}(\Gamma_0(36))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T$
$5$	$T$
$7$	$T + 508$
$11$	$T$