Newform orbit 144.4.a.d

• Label: 144.4.a.d
• Level: $144$
• Weight: $4$
• Character orbit: 144.a
• Self dual: yes
• Analytic conductor: $8.496$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$144 = 2^{4} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	144.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$8.49627504083$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

$f(q)$

$=$

$q - 20 q^{7} - 70 q^{13} - 56 q^{19} - 125 q^{25} - 308 q^{31} + 110 q^{37} + 520 q^{43} + 57 q^{49} + 182 q^{61} + 880 q^{67} + 1190 q^{73} - 884 q^{79} + 1400 q^{91} - 1330 q^{97}+O(q^{100})$

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

−20.0000

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	144.4.a.d		1
3.b	odd	2	1	CM	144.4.a.d		1
4.b	odd	2	1		9.4.a.a	✓	1
8.b	even	2	1		576.4.a.l		1
8.d	odd	2	1		576.4.a.m		1
12.b	even	2	1		9.4.a.a	✓	1
20.d	odd	2	1		225.4.a.d		1
20.e	even	4	2		225.4.b.g		2
24.f	even	2	1		576.4.a.m		1
24.h	odd	2	1		576.4.a.l		1
28.d	even	2	1		441.4.a.f		1
28.f	even	6	2		441.4.e.j		2
28.g	odd	6	2		441.4.e.i		2
36.f	odd	6	2		81.4.c.b		2
36.h	even	6	2		81.4.c.b		2
44.c	even	2	1		1089.4.a.g		1
52.b	odd	2	1		1521.4.a.g		1
60.h	even	2	1		225.4.a.d		1
60.l	odd	4	2		225.4.b.g		2
84.h	odd	2	1		441.4.a.f		1
84.j	odd	6	2		441.4.e.j		2
84.n	even	6	2		441.4.e.i		2
132.d	odd	2	1		1089.4.a.g		1
156.h	even	2	1		1521.4.a.g		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.4.a.a	✓	1	4.b	odd	2	1
9.4.a.a	✓	1	12.b	even	2	1
81.4.c.b		2	36.f	odd	6	2
81.4.c.b		2	36.h	even	6	2
144.4.a.d		1	1.a	even	1	1	trivial
144.4.a.d		1	3.b	odd	2	1	CM
225.4.a.d		1	20.d	odd	2	1
225.4.a.d		1	60.h	even	2	1
225.4.b.g		2	20.e	even	4	2
225.4.b.g		2	60.l	odd	4	2
441.4.a.f		1	28.d	even	2	1
441.4.a.f		1	84.h	odd	2	1
441.4.e.i		2	28.g	odd	6	2
441.4.e.i		2	84.n	even	6	2
441.4.e.j		2	28.f	even	6	2
441.4.e.j		2	84.j	odd	6	2
576.4.a.l		1	8.b	even	2	1
576.4.a.l		1	24.h	odd	2	1
576.4.a.m		1	8.d	odd	2	1
576.4.a.m		1	24.f	even	2	1
1089.4.a.g		1	44.c	even	2	1
1089.4.a.g		1	132.d	odd	2	1
1521.4.a.g		1	52.b	odd	2	1
1521.4.a.g		1	156.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}$ acting on $S_{4}^{\mathrm{new}}(\Gamma_0(144))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T$
$5$	$T$
$7$	$T + 20$
$11$	$T$