Newform orbit 819.2.cf.a

• Label: 819.2.cf.a
• Level: $819$
• Weight: $2$
• Character orbit: 819.cf
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.cf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 40 q^{4} + 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
647.1	−1.38197	+	2.39365i		0		−2.81970	−	4.88386i	−0.158230	+	0.0913544i		0		1.37309	−	2.26155i	10.0591				0			−	0.504997i
647.2	−1.34285	+	2.32588i		0		−2.60649	−	4.51458i	−3.82736	+	2.20973i		0		−2.63625	−	0.223979i	8.62911				0			−	11.8693i
647.3	−1.24039	+	2.14842i		0		−2.07714	−	3.59771i	0.324415	−	0.187301i		0		−1.96749	+	1.76890i	5.34429				0				0.929307i
647.4	−1.23010	+	2.13060i		0		−2.02630	−	3.50966i	2.79545	−	1.61395i		0		0.0205372	−	2.64567i	5.04982				0				7.94130i
647.5	−1.16719	+	2.02163i		0		−1.72465	−	2.98718i	−0.181641	+	0.104870i		0		2.52998	+	0.774094i	3.38321				0			−	0.489613i
647.6	−1.08364	+	1.87692i		0		−1.34856	−	2.33577i	3.03807	−	1.75403i		0		1.98010	+	1.75477i	1.51085				0				7.60296i
647.7	−1.00178	+	1.73513i		0		−1.00712	−	1.74438i	2.16163	−	1.24802i		0		−1.81619	+	1.92391i	0.0285282				0				5.00094i
647.8	−0.922471	+	1.59777i		0		−0.701906	−	1.21574i	−1.93273	+	1.11586i		0		−2.64503	−	0.0617161i	−1.09993				0			−	4.11741i
647.9	−0.895613	+	1.55125i		0		−0.604244	−	1.04658i	−3.03909	+	1.75462i		0		2.02998	−	1.69682i	−1.41778				0			−	6.28583i
647.10	−0.879233	+	1.52288i		0		−0.546102	−	0.945876i	−1.50845	+	0.870906i		0		−0.283342	−	2.63054i	−1.59633				0			−	3.06292i
647.11	−0.762619	+	1.32089i		0		−0.163176	−	0.282629i	2.81914	−	1.62763i		0		0.113312	−	2.64332i	−2.55271				0				4.96505i
647.12	−0.635731	+	1.10112i		0		0.191692	+	0.332020i	−1.83217	+	1.05780i		0		0.819503	+	2.51563i	−3.03038				0			−	2.68992i
647.13	−0.427250	+	0.740019i		0		0.634915	+	1.09970i	1.37336	−	0.792911i		0		−1.63907	+	2.07688i	−2.79407				0				1.35508i
647.14	−0.420660	+	0.728605i		0		0.646090	+	1.11906i	−0.753428	+	0.434992i		0		1.45824	+	2.20761i	−2.76978				0			−	0.731935i
647.15	−0.366778	+	0.635277i		0		0.730948	+	1.26604i	1.13351	−	0.654435i		0		−1.75031	−	1.98404i	−2.53949				0				0.960128i
647.16	−0.354910	+	0.614722i		0		0.748078	+	1.29571i	3.36240	−	1.94128i		0		0.746093	+	2.53837i	−2.48164				0				2.75592i
647.17	−0.293468	+	0.508301i		0		0.827753	+	1.43371i	−2.53491	+	1.46353i		0		2.19659	−	1.47478i	−2.14555				0			−	1.71800i
647.18	−0.252565	+	0.437456i		0		0.872422	+	1.51108i	−0.706530	+	0.407915i		0		2.30708	−	1.29515i	−1.89163				0			−	0.412100i
647.19	−0.115146	+	0.199439i		0		0.973483	+	1.68612i	1.39654	−	0.806292i		0		−2.33681	−	1.24070i	−0.908955				0				0.371366i
647.20	0.115146	−	0.199439i		0		0.973483	+	1.68612i	−1.39654	+	0.806292i		0		−2.33681	−	1.24070i	0.908955				0				0.371366i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
91.p	odd	6	1	inner
273.y	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	819.2.cf.a	✓	76
3.b	odd	2	1	inner	819.2.cf.a	✓	76
7.d	odd	6	1		819.2.ei.a	yes	76
13.e	even	6	1		819.2.ei.a	yes	76
21.g	even	6	1		819.2.ei.a	yes	76
39.h	odd	6	1		819.2.ei.a	yes	76
91.p	odd	6	1	inner	819.2.cf.a	✓	76
273.y	even	6	1	inner	819.2.cf.a	✓	76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
819.2.cf.a	✓	76	1.a	even	1	1	trivial
819.2.cf.a	✓	76	3.b	odd	2	1	inner
819.2.cf.a	✓	76	91.p	odd	6	1	inner
819.2.cf.a	✓	76	273.y	even	6	1	inner
819.2.ei.a	yes	76	7.d	odd	6	1
819.2.ei.a	yes	76	13.e	even	6	1
819.2.ei.a	yes	76	21.g	even	6	1
819.2.ei.a	yes	76	39.h	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).