Newform orbit 812.1.g.c

• Label: 812.1.g.c
• Level: $812$
• Weight: $1$
• Character orbit: 812.g
• Self dual: yes
• Analytic conductor: $0.405$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -203
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$812 = 2^{2} \cdot 7 \cdot 29$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	812.g (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$0.405240790258$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
Defining polynomial:	$x^{2} - 3$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.4615408.1

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - b * q^3 - q^7 + 2 * q^9 + b * q^19 + b * q^21 + q^23 + q^25 - b * q^27 - q^29 + b * q^41 + b * q^47 + q^49 - q^53 - 3 * q^57 - 2 * q^63 + q^67 - b * q^69 - q^71 - b * q^73 - b * q^75 + q^81 + b * q^87 - b * q^89 + b * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{7} + 4 q^{9} + 2 q^{23} + 2 q^{25} - 2 q^{29} + 2 q^{49} - 2 q^{53} - 6 q^{57} - 4 q^{63} + 2 q^{67} - 2 q^{71} + 2 q^{81}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/812\mathbb{Z}\right)^\times$.

$n$	$407$	$465$	$785$
$\chi(n)$	$1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

405.1

1.73205
−1.73205

0

−1.73205

0

0

0

−1.00000

0

2.00000

0

405.2

0

1.73205

0

0

0

−1.00000

0

2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
203.c	odd	2	1	CM by $\Q(\sqrt{-203})$
7.b	odd	2	1	inner
29.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	812.1.g.c	✓	2
4.b	odd	2	1		3248.1.k.d		2
7.b	odd	2	1	inner	812.1.g.c	✓	2
28.d	even	2	1		3248.1.k.d		2
29.b	even	2	1	inner	812.1.g.c	✓	2
116.d	odd	2	1		3248.1.k.d		2
203.c	odd	2	1	CM	812.1.g.c	✓	2
812.c	even	2	1		3248.1.k.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
812.1.g.c	✓	2	1.a	even	1	1	trivial
812.1.g.c	✓	2	7.b	odd	2	1	inner
812.1.g.c	✓	2	29.b	even	2	1	inner
812.1.g.c	✓	2	203.c	odd	2	1	CM
3248.1.k.d		2	4.b	odd	2	1
3248.1.k.d		2	28.d	even	2	1
3248.1.k.d		2	116.d	odd	2	1
3248.1.k.d		2	812.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} - 3$ acting on $S_{1}^{\mathrm{new}}(812, [\chi])$.

Hecke characteristic polynomials

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