Newform orbit 288.3.e.b

• Label: 288.3.e.b
• Level: $288$
• Weight: $3$
• Character orbit: 288.e
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$288 = 2^{5} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	288.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
Defining polynomial:	$x^{2} + 2$
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{-2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 7*b * q^5 - 24 * q^13 - 7*b * q^17 - 73 * q^25 + 41*b * q^29 + 70 * q^37 + 31*b * q^41 - 49 * q^49 + 17*b * q^53 - 22 * q^61 - 168*b * q^65 + 96 * q^73 + 98 * q^85 - 41*b * q^89 + 144 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 48 q^{13} - 146 q^{25} + 140 q^{37} - 98 q^{49} - 44 q^{61} + 192 q^{73} + 196 q^{85} + 288 q^{97}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

161.1

	−	1.41421i
		1.41421i

0

0

0

−

9.89949i

0

0

0

0

0

161.2

0

0

0

9.89949i

0

0

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.e.b	✓	2
3.b	odd	2	1	inner	288.3.e.b	✓	2
4.b	odd	2	1	CM	288.3.e.b	✓	2
8.b	even	2	1		576.3.e.e		2
8.d	odd	2	1		576.3.e.e		2
12.b	even	2	1	inner	288.3.e.b	✓	2
16.e	even	4	2		2304.3.h.e		4
16.f	odd	4	2		2304.3.h.e		4
24.f	even	2	1		576.3.e.e		2
24.h	odd	2	1		576.3.e.e		2
48.i	odd	4	2		2304.3.h.e		4
48.k	even	4	2		2304.3.h.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.3.e.b	✓	2	1.a	even	1	1	trivial
288.3.e.b	✓	2	3.b	odd	2	1	inner
288.3.e.b	✓	2	4.b	odd	2	1	CM
288.3.e.b	✓	2	12.b	even	2	1	inner
576.3.e.e		2	8.b	even	2	1
576.3.e.e		2	8.d	odd	2	1
576.3.e.e		2	24.f	even	2	1
576.3.e.e		2	24.h	odd	2	1
2304.3.h.e		4	16.e	even	4	2
2304.3.h.e		4	16.f	odd	4	2
2304.3.h.e		4	48.i	odd	4	2
2304.3.h.e		4	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(288, [\chi])$:

$T_{5}^{2} + 98$

$T_{7}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 98$
$7$	$T^{2}$
$11$	$T^{2}$