Newform orbit 588.6.f.c

• Label: 588.6.f.c
• Level: $588$
• Weight: $6$
• Character orbit: 588.f
• Analytic conductor: $94.306$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(94.3056860500\)
Analytic rank:	\(0\)
Dimension:	\(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 440 q^{9} - 1224 q^{15} + 19896 q^{25} + 26224 q^{37} - 17384 q^{39} - 21584 q^{43} - 52776 q^{51} + 34880 q^{57} + 89776 q^{67} - 288240 q^{79} + 72504 q^{81} - 50640 q^{85} + 263376 q^{93} - 87056 q^{99}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
293.1		0		−15.4583	−	2.01056i		0		−55.4468				0			0			0		234.915	+	62.1595i		0
293.2		0		−15.4583	+	2.01056i		0		−55.4468				0			0			0		234.915	−	62.1595i		0
293.3		0		−14.9371	−	4.45900i		0		55.8613				0			0			0		203.235	+	133.209i		0
293.4		0		−14.9371	+	4.45900i		0		55.8613				0			0			0		203.235	−	133.209i		0
293.5		0		−14.6607	−	5.29761i		0		−76.9520				0			0			0		186.871	+	155.333i		0
293.6		0		−14.6607	+	5.29761i		0		−76.9520				0			0			0		186.871	−	155.333i		0
293.7		0		−14.3469	−	6.09642i		0		105.900				0			0			0		168.667	+	174.930i		0
293.8		0		−14.3469	+	6.09642i		0		105.900				0			0			0		168.667	−	174.930i		0
293.9		0		−12.2329	−	9.66213i		0		−9.49643				0			0			0		56.2864	+	236.391i		0
293.10		0		−12.2329	+	9.66213i		0		−9.49643				0			0			0		56.2864	−	236.391i		0
293.11		0		−10.9639	−	11.0812i		0		30.7995				0			0			0		−2.58436	+	242.986i		0
293.12		0		−10.9639	+	11.0812i		0		30.7995				0			0			0		−2.58436	−	242.986i		0
293.13		0		−7.24147	−	13.8044i		0		44.4937				0			0			0		−138.122	+	199.928i		0
293.14		0		−7.24147	+	13.8044i		0		44.4937				0			0			0		−138.122	−	199.928i		0
293.15		0		−5.99954	−	14.3877i		0		−29.1880				0			0			0		−171.011	+	172.639i		0
293.16		0		−5.99954	+	14.3877i		0		−29.1880				0			0			0		−171.011	−	172.639i		0
293.17		0		−5.18992	−	14.6991i		0		−64.3163				0			0			0		−189.129	+	152.575i		0
293.18		0		−5.18992	+	14.6991i		0		−64.3163				0			0			0		−189.129	−	152.575i		0
293.19		0		−1.39153	−	15.5262i		0		−69.8984				0			0			0		−239.127	+	43.2103i		0
293.20		0		−1.39153	+	15.5262i		0		−69.8984				0			0			0		−239.127	−	43.2103i		0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	588.6.f.c	✓	40
3.b	odd	2	1	inner	588.6.f.c	✓	40
7.b	odd	2	1	inner	588.6.f.c	✓	40
21.c	even	2	1	inner	588.6.f.c	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.6.f.c	✓	40	1.a	even	1	1	trivial
588.6.f.c	✓	40	3.b	odd	2	1	inner
588.6.f.c	✓	40	7.b	odd	2	1	inner
588.6.f.c	✓	40	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^20 - 36224*T5^18 + 542808650*T5^16 - 4443328193216*T5^14 + 21939099407211468*T5^12 - 67718461239150675200*T5^10 + 130432890502619115701304*T5^8 - 151142571134404466562686720*T5^6 + 96734860607747674712082167296*T5^4 - 28186131076502910563122562981888*T5^2 + 1857776694286958271972806850043904