Newform orbit 588.2.e.f

• Label: 588.2.e.f
• Level: $588$
• Weight: $2$
• Character orbit: 588.e
• Analytic conductor: $4.695$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$588 = 2^{2} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	588.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.69520363885$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ $24 q - 24 q^{16} - 12 q^{18} - 24 q^{25} - 48 q^{30} + 12 q^{36} + 72 q^{46} - 24 q^{57} + 72 q^{58} + 72 q^{60} - 48 q^{64} + 108 q^{72} - 24 q^{78} - 24 q^{81} - 48 q^{85} - 48 q^{88} + 48 q^{93}+O(q^{100})$

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
491.1	−1.29871	−	0.559784i	−1.60392	−	0.653796i	1.37328	+	1.45399i			3.97283i	1.71704	+	1.74694i		0		−0.969574	−	2.65705i	2.14510	+	2.09727i	2.22392	−	5.15954i
491.2	−1.29871	−	0.559784i	1.60392	+	0.653796i	1.37328	+	1.45399i		−	3.97283i	−1.71704	−	1.74694i		0		−0.969574	−	2.65705i	2.14510	+	2.09727i	−2.22392	+	5.15954i
491.3	−1.29871	+	0.559784i	−1.60392	+	0.653796i	1.37328	−	1.45399i		−	3.97283i	1.71704	−	1.74694i		0		−0.969574	+	2.65705i	2.14510	−	2.09727i	2.22392	+	5.15954i
491.4	−1.29871	+	0.559784i	1.60392	−	0.653796i	1.37328	−	1.45399i			3.97283i	−1.71704	+	1.74694i		0		−0.969574	+	2.65705i	2.14510	−	2.09727i	−2.22392	−	5.15954i
491.5	−1.05533	−	0.941421i	−0.406768	−	1.68361i	0.227452	+	1.98702i			1.25365i	−1.15571	+	2.15971i		0		1.63059	−	2.31110i	−2.66908	+	1.36968i	1.18022	−	1.32302i
491.6	−1.05533	−	0.941421i	0.406768	+	1.68361i	0.227452	+	1.98702i		−	1.25365i	1.15571	−	2.15971i		0		1.63059	−	2.31110i	−2.66908	+	1.36968i	−1.18022	+	1.32302i
491.7	−1.05533	+	0.941421i	−0.406768	+	1.68361i	0.227452	−	1.98702i		−	1.25365i	−1.15571	−	2.15971i		0		1.63059	+	2.31110i	−2.66908	−	1.36968i	1.18022	+	1.32302i
491.8	−1.05533	+	0.941421i	0.406768	−	1.68361i	0.227452	−	1.98702i			1.25365i	1.15571	+	2.15971i		0		1.63059	+	2.31110i	−2.66908	−	1.36968i	−1.18022	−	1.32302i
491.9	−0.446802	−	1.34178i	−1.32740	+	1.11266i	−1.60074	+	1.19902i		−	0.803124i	2.08603	+	1.28394i		0		2.32403	+	1.61211i	0.523976	−	2.95389i	−1.07761	+	0.358837i
491.10	−0.446802	−	1.34178i	1.32740	−	1.11266i	−1.60074	+	1.19902i			0.803124i	−2.08603	−	1.28394i		0		2.32403	+	1.61211i	0.523976	−	2.95389i	1.07761	−	0.358837i
491.11	−0.446802	+	1.34178i	−1.32740	−	1.11266i	−1.60074	−	1.19902i			0.803124i	2.08603	−	1.28394i		0		2.32403	−	1.61211i	0.523976	+	2.95389i	−1.07761	−	0.358837i
491.12	−0.446802	+	1.34178i	1.32740	+	1.11266i	−1.60074	−	1.19902i		−	0.803124i	−2.08603	+	1.28394i		0		2.32403	−	1.61211i	0.523976	+	2.95389i	1.07761	+	0.358837i
491.13	0.446802	−	1.34178i	−1.32740	+	1.11266i	−1.60074	−	1.19902i			0.803124i	0.899858	+	2.27821i		0		−2.32403	+	1.61211i	0.523976	−	2.95389i	1.07761	+	0.358837i
491.14	0.446802	−	1.34178i	1.32740	−	1.11266i	−1.60074	−	1.19902i		−	0.803124i	−0.899858	−	2.27821i		0		−2.32403	+	1.61211i	0.523976	−	2.95389i	−1.07761	−	0.358837i
491.15	0.446802	+	1.34178i	−1.32740	−	1.11266i	−1.60074	+	1.19902i		−	0.803124i	0.899858	−	2.27821i		0		−2.32403	−	1.61211i	0.523976	+	2.95389i	1.07761	−	0.358837i
491.16	0.446802	+	1.34178i	1.32740	+	1.11266i	−1.60074	+	1.19902i			0.803124i	−0.899858	+	2.27821i		0		−2.32403	−	1.61211i	0.523976	+	2.95389i	−1.07761	+	0.358837i
491.17	1.05533	−	0.941421i	−0.406768	−	1.68361i	0.227452	−	1.98702i		−	1.25365i	−2.01426	−	1.39383i		0		−1.63059	−	2.31110i	−2.66908	+	1.36968i	−1.18022	−	1.32302i
491.18	1.05533	−	0.941421i	0.406768	+	1.68361i	0.227452	−	1.98702i			1.25365i	2.01426	+	1.39383i		0		−1.63059	−	2.31110i	−2.66908	+	1.36968i	1.18022	+	1.32302i
491.19	1.05533	+	0.941421i	−0.406768	+	1.68361i	0.227452	+	1.98702i			1.25365i	−2.01426	+	1.39383i		0		−1.63059	+	2.31110i	−2.66908	−	1.36968i	−1.18022	+	1.32302i
491.20	1.05533	+	0.941421i	0.406768	−	1.68361i	0.227452	+	1.98702i		−	1.25365i	2.01426	−	1.39383i		0		−1.63059	+	2.31110i	−2.66908	−	1.36968i	1.18022	−	1.32302i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.b	odd	2	1	inner
12.b	even	2	1	inner
21.c	even	2	1	inner
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.2.e.f	✓	24
3.b	odd	2	1	inner	588.2.e.f	✓	24
4.b	odd	2	1	inner	588.2.e.f	✓	24
7.b	odd	2	1	inner	588.2.e.f	✓	24
7.c	even	3	1		588.2.n.d		24
7.c	even	3	1		588.2.n.h		24
7.d	odd	6	1		588.2.n.d		24
7.d	odd	6	1		588.2.n.h		24
12.b	even	2	1	inner	588.2.e.f	✓	24
21.c	even	2	1	inner	588.2.e.f	✓	24
21.g	even	6	1		588.2.n.d		24
21.g	even	6	1		588.2.n.h		24
21.h	odd	6	1		588.2.n.d		24
21.h	odd	6	1		588.2.n.h		24
28.d	even	2	1	inner	588.2.e.f	✓	24
28.f	even	6	1		588.2.n.d		24
28.f	even	6	1		588.2.n.h		24
28.g	odd	6	1		588.2.n.d		24
28.g	odd	6	1		588.2.n.h		24
84.h	odd	2	1	inner	588.2.e.f	✓	24
84.j	odd	6	1		588.2.n.d		24
84.j	odd	6	1		588.2.n.h		24
84.n	even	6	1		588.2.n.d		24
84.n	even	6	1		588.2.n.h		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.2.e.f	✓	24	1.a	even	1	1	trivial
588.2.e.f	✓	24	3.b	odd	2	1	inner
588.2.e.f	✓	24	4.b	odd	2	1	inner
588.2.e.f	✓	24	7.b	odd	2	1	inner
588.2.e.f	✓	24	12.b	even	2	1	inner
588.2.e.f	✓	24	21.c	even	2	1	inner
588.2.e.f	✓	24	28.d	even	2	1	inner
588.2.e.f	✓	24	84.h	odd	2	1	inner
588.2.n.d		24	7.c	even	3	1
588.2.n.d		24	7.d	odd	6	1
588.2.n.d		24	21.g	even	6	1
588.2.n.d		24	21.h	odd	6	1
588.2.n.d		24	28.f	even	6	1
588.2.n.d		24	28.g	odd	6	1
588.2.n.d		24	84.j	odd	6	1
588.2.n.d		24	84.n	even	6	1
588.2.n.h		24	7.c	even	3	1
588.2.n.h		24	7.d	odd	6	1
588.2.n.h		24	21.g	even	6	1
588.2.n.h		24	21.h	odd	6	1
588.2.n.h		24	28.f	even	6	1
588.2.n.h		24	28.g	odd	6	1
588.2.n.h		24	84.j	odd	6	1
588.2.n.h		24	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(588, [\chi])$:

$T_{5}^{6} + 18T_{5}^{4} + 36T_{5}^{2} + 16$

$T_{11}^{6} - 42T_{11}^{4} + 480T_{11}^{2} - 1536$

$T_{13}^{6} - 48T_{13}^{4} + 576T_{13}^{2} - 392$