Newform orbit 432.3.e.b

• Label: 432.3.e.b
• Level: $432$
• Weight: $3$
• Character orbit: 432.e
• Self dual: yes
• Analytic conductor: $11.771$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.7711474204\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 13 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

161.1

0

0

0

0

0

0

13.0000

0

0

0

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	432.3.e.b		1
3.b	odd	2	1	CM	432.3.e.b		1
4.b	odd	2	1		27.3.b.a	✓	1
8.b	even	2	1		1728.3.e.d		1
8.d	odd	2	1		1728.3.e.a		1
9.c	even	3	2		1296.3.q.a		2
9.d	odd	6	2		1296.3.q.a		2
12.b	even	2	1		27.3.b.a	✓	1
20.d	odd	2	1		675.3.c.c		1
20.e	even	4	2		675.3.d.c		2
24.f	even	2	1		1728.3.e.a		1
24.h	odd	2	1		1728.3.e.d		1
36.f	odd	6	2		81.3.d.a		2
36.h	even	6	2		81.3.d.a		2
60.h	even	2	1		675.3.c.c		1
60.l	odd	4	2		675.3.d.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
27.3.b.a	✓	1	4.b	odd	2	1
27.3.b.a	✓	1	12.b	even	2	1
81.3.d.a		2	36.f	odd	6	2
81.3.d.a		2	36.h	even	6	2
432.3.e.b		1	1.a	even	1	1	trivial
432.3.e.b		1	3.b	odd	2	1	CM
675.3.c.c		1	20.d	odd	2	1
675.3.c.c		1	60.h	even	2	1
675.3.d.c		2	20.e	even	4	2
675.3.d.c		2	60.l	odd	4	2
1296.3.q.a		2	9.c	even	3	2
1296.3.q.a		2	9.d	odd	6	2
1728.3.e.a		1	8.d	odd	2	1
1728.3.e.a		1	24.f	even	2	1
1728.3.e.d		1	8.b	even	2	1
1728.3.e.d		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(432, [\chi])\):

\( T_{5} \)

\( T_{7} - 13 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T - 13 \)
$11$	\( T \)