Newform orbit 288.6.a.l

• Label: 288.6.a.l
• Level: $288$
• Weight: $6$
• Character orbit: 288.a
• Self dual: yes
• Analytic conductor: $46.191$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$288 = 2^{5} \cdot 3^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	288.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$46.1905401061$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
Defining polynomial:	$x^{2} - 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{4}\cdot 3$
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 48\sqrt{3}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 46 * q^5 + 2*b * q^7 + b * q^11 - 42 * q^13 - 962 * q^17 + 25*b * q^19 - 38*b * q^23 - 1009 * q^25 + 2554 * q^29 - 24*b * q^31 - 92*b * q^35 + 11950 * q^37 + 5078 * q^41 + 151*b * q^43 + 148*b * q^47 + 10841 * q^49 + 19714 * q^53 - 46*b * q^55 - 107*b * q^59 + 29318 * q^61 + 1932 * q^65 - 203*b * q^67 + 974*b * q^71 + 37914 * q^73 + 13824 * q^77 - 1068*b * q^79 - 473*b * q^83 + 44252 * q^85 - 13930 * q^89 - 84*b * q^91 - 1150*b * q^95 + 163602 * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 92 * q^5 - 84 * q^13 - 1924 * q^17 - 2018 * q^25 + 5108 * q^29 + 23900 * q^37 + 10156 * q^41 + 21682 * q^49 + 39428 * q^53 + 58636 * q^61 + 3864 * q^65 + 75828 * q^73 + 27648 * q^77 + 88504 * q^85 - 27860 * q^89 + 327204 * q^97

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−1.73205
1.73205

0

0

0

−46.0000

0

−166.277

0

0

0

1.2

0

0

0

−46.0000

0

166.277

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.6.a.l		2
3.b	odd	2	1		32.6.a.d	✓	2
4.b	odd	2	1	inner	288.6.a.l		2
8.b	even	2	1		576.6.a.bp		2
8.d	odd	2	1		576.6.a.bp		2
12.b	even	2	1		32.6.a.d	✓	2
15.d	odd	2	1		800.6.a.k		2
15.e	even	4	2		800.6.c.d		4
24.f	even	2	1		64.6.a.h		2
24.h	odd	2	1		64.6.a.h		2
48.i	odd	4	2		256.6.b.l		4
48.k	even	4	2		256.6.b.l		4
60.h	even	2	1		800.6.a.k		2
60.l	odd	4	2		800.6.c.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.6.a.d	✓	2	3.b	odd	2	1
32.6.a.d	✓	2	12.b	even	2	1
64.6.a.h		2	24.f	even	2	1
64.6.a.h		2	24.h	odd	2	1
256.6.b.l		4	48.i	odd	4	2
256.6.b.l		4	48.k	even	4	2
288.6.a.l		2	1.a	even	1	1	trivial
288.6.a.l		2	4.b	odd	2	1	inner
576.6.a.bp		2	8.b	even	2	1
576.6.a.bp		2	8.d	odd	2	1
800.6.a.k		2	15.d	odd	2	1
800.6.a.k		2	60.h	even	2	1
800.6.c.d		4	15.e	even	4	2
800.6.c.d		4	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(288))$:

$T_{5} + 46$

$T_{7}^{2} - 27648$

$T_{11}^{2} - 6912$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$(T + 46)^{2}$
$7$	$T^{2} - 27648$
$11$	$T^{2} - 6912$