Newform orbit 432.1.e.a

• Label: 432.1.e.a
• Level: $432$
• Weight: $1$
• Character orbit: 432.e
• Self dual: yes
• Analytic conductor: $0.216$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.215596085457\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 108)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of \(\Q(\sqrt[3]{2})\)
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.186624.1
Stark unit:	Root of $x^{6} - 42x^{5} + 375x^{4} - 1100x^{3} + 375x^{2} - 42x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{7} - q^{13} + q^{19} + q^{25} - 2 q^{31} - q^{37} - 2 q^{43} - q^{61} + q^{67} - q^{73} + q^{79} - q^{91} - q^{97}+O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(12z)^{3}\eta(36z)^{3}}{\eta(6z)\eta(18z)\eta(24z)\eta(72z)}=q\prod_{n=1}^\infty(1 - q^{6n})^{-1}(1 - q^{12n})^{3}(1 - q^{18n})^{-1}(1 - q^{24n})^{-1}(1 - q^{36n})^{3}(1 - q^{72n})^{-1}\)

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

1.00000

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	432.1.e.a		1
3.b	odd	2	1	CM	432.1.e.a		1
4.b	odd	2	1		108.1.c.a	✓	1
8.b	even	2	1		1728.1.e.b		1
8.d	odd	2	1		1728.1.e.a		1
9.c	even	3	2		1296.1.q.a		2
9.d	odd	6	2		1296.1.q.a		2
12.b	even	2	1		108.1.c.a	✓	1
20.d	odd	2	1		2700.1.g.b		1
20.e	even	4	2		2700.1.b.b		2
24.f	even	2	1		1728.1.e.a		1
24.h	odd	2	1		1728.1.e.b		1
36.f	odd	6	2		324.1.g.a		2
36.h	even	6	2		324.1.g.a		2
60.h	even	2	1		2700.1.g.b		1
60.l	odd	4	2		2700.1.b.b		2
108.j	odd	18	6		2916.1.k.c		6
108.l	even	18	6		2916.1.k.c		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.1.c.a	✓	1	4.b	odd	2	1
108.1.c.a	✓	1	12.b	even	2	1
324.1.g.a		2	36.f	odd	6	2
324.1.g.a		2	36.h	even	6	2
432.1.e.a		1	1.a	even	1	1	trivial
432.1.e.a		1	3.b	odd	2	1	CM
1296.1.q.a		2	9.c	even	3	2
1296.1.q.a		2	9.d	odd	6	2
1728.1.e.a		1	8.d	odd	2	1
1728.1.e.a		1	24.f	even	2	1
1728.1.e.b		1	8.b	even	2	1
1728.1.e.b		1	24.h	odd	2	1
2700.1.b.b		2	20.e	even	4	2
2700.1.b.b		2	60.l	odd	4	2
2700.1.g.b		1	20.d	odd	2	1
2700.1.g.b		1	60.h	even	2	1
2916.1.k.c		6	108.j	odd	18	6
2916.1.k.c		6	108.l	even	18	6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(432, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T - 1 \)
$11$	\( T \)