Newform orbit 2304.3.g.i

• Label: 2304.3.g.i
• Level: $2304$
• Weight: $3$
• Character orbit: 2304.g
• Analytic conductor: $62.779$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -24
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$2304 = 2^{8} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	2304.g (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6})$
Defining polynomial:	$x^{2} + 6$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 4\sqrt{-6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 2 * q^5 - b * q^7 + 2*b * q^11 - 21 * q^25 + 50 * q^29 - 5*b * q^31 + 2*b * q^35 - 47 * q^49 - 94 * q^53 - 4*b * q^55 - 12*b * q^59 - 50 * q^73 + 192 * q^77 + 15*b * q^79 - 10*b * q^83 - 190 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{5} - 42 q^{25} + 100 q^{29} - 94 q^{49} - 188 q^{53} - 100 q^{73} + 384 q^{77} - 380 q^{97}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1279.1

		2.44949i
	−	2.44949i

0

0

0

−2.00000

0

−

9.79796i

0

0

0

1279.2

0

0

0

−2.00000

0

9.79796i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
4.b	odd	2	1	inner
24.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2304.3.g.i		2
3.b	odd	2	1		2304.3.g.l		2
4.b	odd	2	1	inner	2304.3.g.i		2
8.b	even	2	1		2304.3.g.l		2
8.d	odd	2	1		2304.3.g.l		2
12.b	even	2	1		2304.3.g.l		2
16.e	even	4	2		1152.3.b.f	✓	4
16.f	odd	4	2		1152.3.b.f	✓	4
24.f	even	2	1	inner	2304.3.g.i		2
24.h	odd	2	1	CM	2304.3.g.i		2
48.i	odd	4	2		1152.3.b.f	✓	4
48.k	even	4	2		1152.3.b.f	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.3.b.f	✓	4	16.e	even	4	2
1152.3.b.f	✓	4	16.f	odd	4	2
1152.3.b.f	✓	4	48.i	odd	4	2
1152.3.b.f	✓	4	48.k	even	4	2
2304.3.g.i		2	1.a	even	1	1	trivial
2304.3.g.i		2	4.b	odd	2	1	inner
2304.3.g.i		2	24.f	even	2	1	inner
2304.3.g.i		2	24.h	odd	2	1	CM
2304.3.g.l		2	3.b	odd	2	1
2304.3.g.l		2	8.b	even	2	1
2304.3.g.l		2	8.d	odd	2	1
2304.3.g.l		2	12.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 $S_{3}^{\mathrm{new}}(2304, [\chi])$:

$T_{5} + 2$

$T_{7}^{2} + 96$

$T_{11}^{2} + 384$

$T_{13}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$(T + 2)^{2}$
$7$	$T^{2} + 96$
$11$	$T^{2} + 384$