Newform orbit 48.5.e.b

• Label: 48.5.e.b
• Level: $48$
• Weight: $5$
• Character orbit: 48.e
• Analytic conductor: $4.962$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$48 = 2^{4} \cdot 3$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.96175822802$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
Defining polynomial:	$x^{2} + 2$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 6\sqrt{-2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 3) * q^3 + 2*b * q^5 - 26 * q^7 + (6*b - 63) * q^9 + 14*b * q^11 + 50 * q^13 + (6*b - 144) * q^15 + 24*b * q^17 + 358 * q^19 + (-26*b - 78) * q^21 - 44*b * q^23 + 337 * q^25 + (-45*b - 621) * q^27 - 170*b * q^29 + 742 * q^31 + (42*b - 1008) * q^33 - 52*b * q^35 + 1874 * q^37 + (50*b + 150) * q^39 + 284*b * q^41 + 262 * q^43 + (-126*b - 864) * q^45 + 200*b * q^47 - 1725 * q^49 + (72*b - 1728) * q^51 - 54*b * q^53 - 2016 * q^55 + (358*b + 1074) * q^57 + 214*b * q^59 - 1486 * q^61 + (-156*b + 1638) * q^63 + 100*b * q^65 + 4486 * q^67 + (-132*b + 3168) * q^69 - 420*b * q^71 + 290 * q^73 + (337*b + 1011) * q^75 - 364*b * q^77 - 9818 * q^79 + (-756*b + 1377) * q^81 - 838*b * q^83 - 3456 * q^85 + (-510*b + 12240) * q^87 - 924*b * q^89 - 1300 * q^91 + (742*b + 2226) * q^93 + 716*b * q^95 - 478 * q^97 + (-882*b - 6048) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 6 * q^3 - 52 * q^7 - 126 * q^9 + 100 * q^13 - 288 * q^15 + 716 * q^19 - 156 * q^21 + 674 * q^25 - 1242 * q^27 + 1484 * q^31 - 2016 * q^33 + 3748 * q^37 + 300 * q^39 + 524 * q^43 - 1728 * q^45 - 3450 * q^49 - 3456 * q^51 - 4032 * q^55 + 2148 * q^57 - 2972 * q^61 + 3276 * q^63 + 8972 * q^67 + 6336 * q^69 + 580 * q^73 + 2022 * q^75 - 19636 * q^79 + 2754 * q^81 - 6912 * q^85 + 24480 * q^87 - 2600 * q^91 + 4452 * q^93 - 956 * q^97 - 12096 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

17.1

	−	1.41421i
		1.41421i

0

3.00000

−

8.48528i

0

−

16.9706i

0

−26.0000

0

−63.0000

−

50.9117i

0

17.2

0

3.00000

+

8.48528i

0

16.9706i

0

−26.0000

0

−63.0000

+

50.9117i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	48.5.e.b		2
3.b	odd	2	1	inner	48.5.e.b		2
4.b	odd	2	1		6.5.b.a	✓	2
8.b	even	2	1		192.5.e.c		2
8.d	odd	2	1		192.5.e.d		2
12.b	even	2	1		6.5.b.a	✓	2
20.d	odd	2	1		150.5.d.a		2
20.e	even	4	2		150.5.b.a		4
24.f	even	2	1		192.5.e.d		2
24.h	odd	2	1		192.5.e.c		2
28.d	even	2	1		294.5.b.a		2
36.f	odd	6	2		162.5.d.a		4
36.h	even	6	2		162.5.d.a		4
60.h	even	2	1		150.5.d.a		2
60.l	odd	4	2		150.5.b.a		4
84.h	odd	2	1		294.5.b.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.5.b.a	✓	2	4.b	odd	2	1
6.5.b.a	✓	2	12.b	even	2	1
48.5.e.b		2	1.a	even	1	1	trivial
48.5.e.b		2	3.b	odd	2	1	inner
150.5.b.a		4	20.e	even	4	2
150.5.b.a		4	60.l	odd	4	2
150.5.d.a		2	20.d	odd	2	1
150.5.d.a		2	60.h	even	2	1
162.5.d.a		4	36.f	odd	6	2
162.5.d.a		4	36.h	even	6	2
192.5.e.c		2	8.b	even	2	1
192.5.e.c		2	24.h	odd	2	1
192.5.e.d		2	8.d	odd	2	1
192.5.e.d		2	24.f	even	2	1
294.5.b.a		2	28.d	even	2	1
294.5.b.a		2	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 288$ acting on $S_{5}^{\mathrm{new}}(48, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 6T + 81$
$5$	$T^{2} + 288$
$7$	$(T + 26)^{2}$
$11$	$T^{2} + 14112$