Newform orbit 728.1.l.a

• Label: 728.1.l.a
• Level: $728$
• Weight: $1$
• Character orbit: 728.l
• Self dual: yes
• Analytic conductor: $0.363$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -728
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$728 = 2^{3} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	728.l (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$0.363319329197$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.728.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.6889792.2
Stark unit:	Root of $x^{6} - 15x^{5} + 68x^{4} - 121x^{3} + 68x^{2} - 15x + 1$

$q$-expansion

$f(q)$

$=$

q - q^2 - q^3 + q^4 + q^6 - q^7 - q^8 + q^11 - q^12 + q^13 + q^14 + q^16 + q^21 - q^22 - q^23 + q^24 + q^25 - q^26 + q^27 - q^28 + q^31 - q^32 - q^33 + q^37 - q^39 + q^41 - q^42 + q^44 + q^46 + q^47 - q^48 + q^49 - q^50 + q^52 - q^54 + q^56 - q^61 - q^62 + q^64 + q^66 + q^67 + q^69 + q^73 - q^74 - q^75 - q^77 + q^78 - q^79 - q^81 - q^82 + q^84 - q^88 - 2 * q^89 - q^91 - q^92 - q^93 - q^94 + q^96 + q^97 - q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$.

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

181.1

0

−1.00000

−1.00000

1.00000

0

1.00000

−1.00000

−1.00000

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
728.l	odd	2	1	CM by $\Q(\sqrt{-182})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	728.1.l.a	✓	1
4.b	odd	2	1		2912.1.l.d		1
7.b	odd	2	1		728.1.l.b	yes	1
8.b	even	2	1		728.1.l.d	yes	1
8.d	odd	2	1		2912.1.l.b		1
13.b	even	2	1		728.1.l.c	yes	1
28.d	even	2	1		2912.1.l.a		1
52.b	odd	2	1		2912.1.l.c		1
56.e	even	2	1		2912.1.l.c		1
56.h	odd	2	1		728.1.l.c	yes	1
91.b	odd	2	1		728.1.l.d	yes	1
104.e	even	2	1		728.1.l.b	yes	1
104.h	odd	2	1		2912.1.l.a		1
364.h	even	2	1		2912.1.l.b		1
728.b	even	2	1		2912.1.l.d		1
728.l	odd	2	1	CM	728.1.l.a	✓	1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
728.1.l.a	✓	1	1.a	even	1	1	trivial
728.1.l.a	✓	1	728.l	odd	2	1	CM
728.1.l.b	yes	1	7.b	odd	2	1
728.1.l.b	yes	1	104.e	even	2	1
728.1.l.c	yes	1	13.b	even	2	1
728.1.l.c	yes	1	56.h	odd	2	1
728.1.l.d	yes	1	8.b	even	2	1
728.1.l.d	yes	1	91.b	odd	2	1
2912.1.l.a		1	28.d	even	2	1
2912.1.l.a		1	104.h	odd	2	1
2912.1.l.b		1	8.d	odd	2	1
2912.1.l.b		1	364.h	even	2	1
2912.1.l.c		1	52.b	odd	2	1
2912.1.l.c		1	56.e	even	2	1
2912.1.l.d		1	4.b	odd	2	1
2912.1.l.d		1	728.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(728, [\chi])$:

$T_{3} + 1$

$T_{11} - 1$

Hecke characteristic polynomials

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