Newform orbit 3072.1.i.d

• Label: 3072.1.i.d
• Level: $3072$
• Weight: $1$
• Character orbit: 3072.i
• Analytic conductor: $1.533$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{4}$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$3072 = 2^{10} \cdot 3$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3072.i (of order $4$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.53312771881$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1536)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.9216.1

$q$-expansion

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	q + q^3 + q^9 + (-z + 1) * q^11 - 2*z * q^17 + (z - 1) * q^19 + z * q^25 + q^27 + (-z + 1) * q^33 + (-z - 1) * q^43 + q^49 - 2*z * q^51 + (z - 1) * q^57 + (z - 1) * q^59 + (z - 1) * q^67 + z * q^75 + q^81 + (z + 1) * q^83 + 2 * q^89 + (-z + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{3} + 2 q^{9} + 2 q^{11} - 2 q^{19} + 2 q^{27} + 2 q^{33} - 2 q^{43} + 2 q^{49} - 2 q^{57} - 2 q^{59} - 2 q^{67} + 2 q^{81} + 2 q^{83} + 4 q^{89} + 2 q^{99}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$-1$	$1$	$i$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

257.1

	−	1.00000i
		1.00000i

0

1.00000

0

0

0

0

0

1.00000

0

1793.1

0

1.00000

0

0

0

0

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2})$
48.i	odd	4	1	inner
48.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3072.1.i.d		2
3.b	odd	2	1		3072.1.i.b		2
4.b	odd	2	1		3072.1.i.a		2
8.b	even	2	1		3072.1.i.a		2
8.d	odd	2	1	CM	3072.1.i.d		2
12.b	even	2	1		3072.1.i.c		2
16.e	even	4	1		3072.1.i.b		2
16.e	even	4	1		3072.1.i.c		2
16.f	odd	4	1		3072.1.i.b		2
16.f	odd	4	1		3072.1.i.c		2
24.f	even	2	1		3072.1.i.b		2
24.h	odd	2	1		3072.1.i.c		2
32.g	even	8	2		1536.1.e.b	✓	4
32.g	even	8	2		1536.1.h.c		4
32.h	odd	8	2		1536.1.e.b	✓	4
32.h	odd	8	2		1536.1.h.c		4
48.i	odd	4	1		3072.1.i.a		2
48.i	odd	4	1	inner	3072.1.i.d		2
48.k	even	4	1		3072.1.i.a		2
48.k	even	4	1	inner	3072.1.i.d		2
96.o	even	8	2		1536.1.e.b	✓	4
96.o	even	8	2		1536.1.h.c		4
96.p	odd	8	2		1536.1.e.b	✓	4
96.p	odd	8	2		1536.1.h.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1536.1.e.b	✓	4	32.g	even	8	2
1536.1.e.b	✓	4	32.h	odd	8	2
1536.1.e.b	✓	4	96.o	even	8	2
1536.1.e.b	✓	4	96.p	odd	8	2
1536.1.h.c		4	32.g	even	8	2
1536.1.h.c		4	32.h	odd	8	2
1536.1.h.c		4	96.o	even	8	2
1536.1.h.c		4	96.p	odd	8	2
3072.1.i.a		2	4.b	odd	2	1
3072.1.i.a		2	8.b	even	2	1
3072.1.i.a		2	48.i	odd	4	1
3072.1.i.a		2	48.k	even	4	1
3072.1.i.b		2	3.b	odd	2	1
3072.1.i.b		2	16.e	even	4	1
3072.1.i.b		2	16.f	odd	4	1
3072.1.i.b		2	24.f	even	2	1
3072.1.i.c		2	12.b	even	2	1
3072.1.i.c		2	16.e	even	4	1
3072.1.i.c		2	16.f	odd	4	1
3072.1.i.c		2	24.h	odd	2	1
3072.1.i.d		2	1.a	even	1	1	trivial
3072.1.i.d		2	8.d	odd	2	1	CM
3072.1.i.d		2	48.i	odd	4	1	inner
3072.1.i.d		2	48.k	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3072, [\chi])$:

$T_{5}$

$T_{11}^{2} - 2T_{11} + 2$

$T_{19}^{2} + 2T_{19} + 2$

Hecke characteristic polynomials

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