Newform orbit 861.2.r.a

• Label: 861.2.r.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.r
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(108\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 100 q^{4} + 2 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
122.1	−2.41684	+	1.39536i	−1.71237	+	0.260366i	2.89408	−	5.01269i	−1.81414	−	3.14219i	3.77522	−	3.01864i	−1.60965	+	2.09977i			10.5717i	2.86442	−	0.891685i	8.76899	+	5.06278i
122.2	−2.41684	+	1.39536i	1.71237	−	0.260366i	2.89408	−	5.01269i	−1.81414	−	3.14219i	−3.77522	+	3.01864i	1.60965	−	2.09977i			10.5717i	2.86442	−	0.891685i	8.76899	+	5.06278i
122.3	−2.26904	+	1.31003i	−0.950726	−	1.44780i	2.43236	−	4.21298i	0.289899	+	0.502120i	4.05390	+	2.03963i	−1.52606	−	2.16128i			7.50576i	−1.19224	+	2.75292i	−1.31559	−	0.759554i
122.4	−2.26904	+	1.31003i	0.950726	+	1.44780i	2.43236	−	4.21298i	0.289899	+	0.502120i	−4.05390	−	2.03963i	1.52606	+	2.16128i			7.50576i	−1.19224	+	2.75292i	−1.31559	−	0.759554i
122.5	−2.24087	+	1.29376i	−0.968785	+	1.43578i	2.34765	−	4.06626i	0.458388	+	0.793951i	0.313360	−	4.47076i	1.86005	−	1.88155i			6.97419i	−1.12291	−	2.78192i	−2.05437	−	1.18609i
122.6	−2.24087	+	1.29376i	0.968785	−	1.43578i	2.34765	−	4.06626i	0.458388	+	0.793951i	−0.313360	+	4.47076i	−1.86005	+	1.88155i			6.97419i	−1.12291	−	2.78192i	−2.05437	−	1.18609i
122.7	−2.15729	+	1.24551i	−1.72475	−	0.158902i	2.10259	−	3.64180i	0.529472	+	0.917072i	3.91869	−	1.80539i	2.59501	+	0.515664i			5.49316i	2.94950	+	0.548131i	−2.28445	−	1.31893i
122.8	−2.15729	+	1.24551i	1.72475	+	0.158902i	2.10259	−	3.64180i	0.529472	+	0.917072i	−3.91869	+	1.80539i	−2.59501	−	0.515664i			5.49316i	2.94950	+	0.548131i	−2.28445	−	1.31893i
122.9	−2.13925	+	1.23510i	−1.22577	−	1.22372i	2.05094	−	3.55233i	2.12878	+	3.68715i	4.13364	+	1.10391i	0.134524	+	2.64233i			5.19206i	0.00500695	+	3.00000i	−9.10800	−	5.25850i
122.10	−2.13925	+	1.23510i	1.22577	+	1.22372i	2.05094	−	3.55233i	2.12878	+	3.68715i	−4.13364	−	1.10391i	−0.134524	−	2.64233i			5.19206i	0.00500695	+	3.00000i	−9.10800	−	5.25850i
122.11	−2.05288	+	1.18523i	−0.203740	+	1.72003i	1.80954	−	3.13422i	−1.19120	−	2.06322i	−1.62038	−	3.77249i	−2.11074	−	1.59523i			3.83799i	−2.91698	−	0.700875i	4.89078	+	2.82369i
122.12	−2.05288	+	1.18523i	0.203740	−	1.72003i	1.80954	−	3.13422i	−1.19120	−	2.06322i	1.62038	+	3.77249i	2.11074	+	1.59523i			3.83799i	−2.91698	−	0.700875i	4.89078	+	2.82369i
122.13	−1.91182	+	1.10379i	−1.56998	−	0.731544i	1.43670	−	2.48843i	−1.06119	−	1.83803i	3.80899	−	0.334350i	0.763084	−	2.53332i			1.92809i	1.92969	+	2.29702i	4.05759	+	2.34265i
122.14	−1.91182	+	1.10379i	1.56998	+	0.731544i	1.43670	−	2.48843i	−1.06119	−	1.83803i	−3.80899	+	0.334350i	−0.763084	+	2.53332i			1.92809i	1.92969	+	2.29702i	4.05759	+	2.34265i
122.15	−1.90275	+	1.09855i	−0.950044	+	1.44825i	1.41364	−	2.44849i	1.45802	+	2.52537i	0.216720	−	3.79932i	−2.61579	+	0.397023i			1.81762i	−1.19483	−	2.75179i	−5.54851	−	3.20344i
122.16	−1.90275	+	1.09855i	0.950044	−	1.44825i	1.41364	−	2.44849i	1.45802	+	2.52537i	−0.216720	+	3.79932i	2.61579	−	0.397023i			1.81762i	−1.19483	−	2.75179i	−5.54851	−	3.20344i
122.17	−1.79465	+	1.03614i	−0.935954	−	1.45739i	1.14717	−	1.98696i	−0.983506	−	1.70348i	3.18977	+	1.64572i	−2.21456	+	1.44766i			0.609959i	−1.24798	+	2.72810i	3.53009	+	2.03810i
122.18	−1.79465	+	1.03614i	0.935954	+	1.45739i	1.14717	−	1.98696i	−0.983506	−	1.70348i	−3.18977	−	1.64572i	2.21456	−	1.44766i			0.609959i	−1.24798	+	2.72810i	3.53009	+	2.03810i
122.19	−1.75852	+	1.01528i	−1.54661	+	0.779739i	1.06160	−	1.83875i	0.385863	+	0.668335i	1.92810	−	2.94144i	−0.418379	+	2.61246i			0.250172i	1.78401	−	2.41191i	−1.35710	−	0.783521i
122.20	−1.75852	+	1.01528i	1.54661	−	0.779739i	1.06160	−	1.83875i	0.385863	+	0.668335i	−1.92810	+	2.94144i	0.418379	−	2.61246i			0.250172i	1.78401	−	2.41191i	−1.35710	−	0.783521i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner
41.b	even	2	1	inner
123.b	odd	2	1	inner
287.i	odd	6	1	inner
861.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.r.a	✓	216
3.b	odd	2	1	inner	861.2.r.a	✓	216
7.d	odd	6	1	inner	861.2.r.a	✓	216
21.g	even	6	1	inner	861.2.r.a	✓	216
41.b	even	2	1	inner	861.2.r.a	✓	216
123.b	odd	2	1	inner	861.2.r.a	✓	216
287.i	odd	6	1	inner	861.2.r.a	✓	216
861.r	even	6	1	inner	861.2.r.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.r.a	✓	216	1.a	even	1	1	trivial
861.2.r.a	✓	216	3.b	odd	2	1	inner
861.2.r.a	✓	216	7.d	odd	6	1	inner
861.2.r.a	✓	216	21.g	even	6	1	inner
861.2.r.a	✓	216	41.b	even	2	1	inner
861.2.r.a	✓	216	123.b	odd	2	1	inner
861.2.r.a	✓	216	287.i	odd	6	1	inner
861.2.r.a	✓	216	861.r	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).