Newform orbit 912.1.b.a

• Label: 912.1.b.a
• Level: $912$
• Weight: $1$
• Character orbit: 912.b
• Self dual: yes
• Analytic conductor: $0.455$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -228, 76
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$912 = 2^{4} \cdot 3 \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	912.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$0.455147291521$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-3}, \sqrt{19})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.2736.2
Stark unit:	Root of $x^{4} - 52x^{3} - 6x^{2} - 52x + 1$

$q$-expansion

$f(q)$

$=$

$q - q^{3} + q^{9} + q^{19} + q^{25} - q^{27} + 2 q^{31} + q^{49} - q^{57} - 2 q^{61} + 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{79} + q^{81} - 2 q^{93}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

911.1

0

0

−1.00000

0

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
76.d	even	2	1	RM by $\Q(\sqrt{19})$
228.b	odd	2	1	CM by $\Q(\sqrt{-57})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	912.1.b.a	✓	1
3.b	odd	2	1	CM	912.1.b.a	✓	1
4.b	odd	2	1		912.1.b.b	yes	1
8.b	even	2	1		3648.1.b.b		1
8.d	odd	2	1		3648.1.b.a		1
12.b	even	2	1		912.1.b.b	yes	1
19.b	odd	2	1		912.1.b.b	yes	1
24.f	even	2	1		3648.1.b.a		1
24.h	odd	2	1		3648.1.b.b		1
57.d	even	2	1		912.1.b.b	yes	1
76.d	even	2	1	RM	912.1.b.a	✓	1
152.b	even	2	1		3648.1.b.b		1
152.g	odd	2	1		3648.1.b.a		1
228.b	odd	2	1	CM	912.1.b.a	✓	1
456.l	odd	2	1		3648.1.b.b		1
456.p	even	2	1		3648.1.b.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
912.1.b.a	✓	1	1.a	even	1	1	trivial
912.1.b.a	✓	1	3.b	odd	2	1	CM
912.1.b.a	✓	1	76.d	even	2	1	RM
912.1.b.a	✓	1	228.b	odd	2	1	CM
912.1.b.b	yes	1	4.b	odd	2	1
912.1.b.b	yes	1	12.b	even	2	1
912.1.b.b	yes	1	19.b	odd	2	1
912.1.b.b	yes	1	57.d	even	2	1
3648.1.b.a		1	8.d	odd	2	1
3648.1.b.a		1	24.f	even	2	1
3648.1.b.a		1	152.g	odd	2	1
3648.1.b.a		1	456.p	even	2	1
3648.1.b.b		1	8.b	even	2	1
3648.1.b.b		1	24.h	odd	2	1
3648.1.b.b		1	152.b	even	2	1
3648.1.b.b		1	456.l	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{31} - 2$ acting on $S_{1}^{\mathrm{new}}(912, [\chi])$.

Hecke characteristic polynomials

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