Newform orbit 861.2.cl.a

• Label: 861.2.cl.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.cl
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $3456$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.cl (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(3456\)
Relative dimension:	\(108\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3456 q - 16 q^{3} - 40 q^{4} - 64 q^{6} - 64 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−2.14683	+	1.73847i	0.280422	+	1.70920i	1.17079	−	5.50812i	−0.113959	−	2.17447i	−3.57341	−	3.18186i	−2.62774	+	0.308152i	4.55397	+	8.93767i	−2.84273	+	0.958594i	4.02491	+	4.47011i
11.2	−2.13226	+	1.72667i	1.42361	+	0.986582i	1.14932	−	5.40713i	0.144919	+	2.76522i	−4.73900	+	0.354448i	1.06439	+	2.42220i	4.39445	+	8.62459i	1.05331	+	2.80901i	−5.08363	−	5.64594i
11.3	−2.12204	+	1.71840i	−1.72955	+	0.0929614i	1.13435	−	5.33672i	−0.0498201	−	0.950624i	3.51044	−	3.16933i	2.18565	−	1.49095i	4.28415	+	8.40812i	2.98272	−	0.321564i	1.73927	+	1.93165i
11.4	−2.02688	+	1.64134i	0.0118789	−	1.73201i	0.998445	−	4.69731i	−0.0210000	−	0.400703i	2.81874	+	3.53008i	0.816823	+	2.51651i	3.31804	+	6.51201i	−2.99972	−	0.0411488i	0.700254	+	0.777711i
11.5	−2.00868	+	1.62660i	1.72136	+	0.192107i	0.973158	−	4.57835i	−0.101767	−	1.94183i	−3.77016	+	2.41409i	−0.807550	−	2.51950i	3.14552	+	6.17344i	2.92619	+	0.661371i	3.36300	+	3.73499i
11.6	−2.00652	+	1.62485i	−1.21968	−	1.22979i	0.970168	−	4.56428i	−0.194613	−	3.71344i	4.44553	+	0.485817i	−2.57407	+	0.611670i	3.12528	+	6.13370i	−0.0247829	+	2.99990i	6.42426	+	7.13487i
11.7	−1.99920	+	1.61892i	−1.65097	+	0.523746i	0.960081	−	4.51683i	0.172703	+	3.29537i	2.45272	−	3.71986i	−2.23338	+	1.41845i	3.05722	+	6.00014i	2.45138	−	1.72937i	−5.68022	−	6.30853i
11.8	−1.94956	+	1.57872i	1.59370	−	0.678329i	0.892594	−	4.19932i	−0.147248	−	2.80966i	−2.03611	+	3.83844i	2.57619	+	0.602679i	2.61163	+	5.12560i	2.07974	−	2.16210i	4.72274	+	5.24513i
11.9	−1.92752	+	1.56088i	−0.522952	+	1.65122i	0.863184	−	4.06096i	−0.0790320	−	1.50802i	−1.56935	−	3.99903i	1.95821	+	1.77916i	2.42283	+	4.75507i	−2.45304	−	1.72702i	2.50617	+	2.78338i
11.10	−1.89915	+	1.53791i	1.28592	−	1.16035i	0.825813	−	3.88515i	0.0841046	+	1.60481i	−0.657643	+	4.18131i	−2.60080	+	0.485607i	2.18776	+	4.29371i	0.307164	−	2.98423i	−2.62778	−	2.91844i
11.11	−1.88035	+	1.52268i	−0.575511	+	1.63364i	0.801345	−	3.77003i	0.139139	+	2.65494i	−1.40535	−	3.94813i	0.781138	−	2.52781i	2.03682	+	3.99748i	−2.33757	−	1.88036i	−4.30424	−	4.78034i
11.12	−1.83059	+	1.48238i	−0.881337	−	1.49106i	0.737778	−	3.47097i	0.121671	+	2.32161i	3.82368	+	1.42303i	−2.05332	−	1.66850i	1.65596	+	3.25001i	−1.44649	+	2.62824i	−3.66425	−	4.06956i
11.13	−1.76155	+	1.42648i	1.25942	+	1.18906i	0.652406	−	3.06933i	0.112534	+	2.14727i	−3.91470	−	0.298066i	−0.255185	−	2.63342i	1.17096	+	2.29815i	0.172262	+	2.99505i	−3.26127	−	3.62201i
11.14	−1.67452	+	1.35600i	0.395157	−	1.68637i	0.549462	−	2.58502i	−0.182763	−	3.48733i	1.62502	+	3.35970i	0.132198	−	2.64245i	0.628765	+	1.23402i	−2.68770	−	1.33276i	5.03487	+	5.59179i
11.15	−1.63902	+	1.32726i	−1.63279	−	0.577911i	0.508969	−	2.39451i	0.00793193	+	0.151350i	3.44323	−	1.21993i	0.676578	+	2.55778i	0.428957	+	0.841875i	2.33204	+	1.88722i	−0.213881	−	0.237539i
11.16	−1.62258	+	1.31394i	1.63334	−	0.576358i	0.490507	−	2.30765i	0.128063	+	2.44359i	−1.89293	+	3.08130i	2.53425	−	0.759995i	0.340483	+	0.668236i	2.33562	−	1.88278i	−3.41853	−	3.79666i
11.17	−1.62071	+	1.31243i	−1.38039	+	1.04620i	0.488419	−	2.29783i	−0.172480	−	3.29112i	0.864149	−	3.50724i	−1.02471	−	2.43926i	0.330583	+	0.648806i	0.810935	−	2.88832i	4.59889	+	5.10758i
11.18	−1.54003	+	1.24709i	1.49098	+	0.881456i	0.400627	−	1.88480i	−0.176456	−	3.36699i	−3.39541	+	0.501924i	0.405753	+	2.61445i	−0.0657569	−	0.129055i	1.44607	+	2.62848i	4.47068	+	4.96519i
11.19	−1.47569	+	1.19499i	−1.09367	−	1.34309i	0.333838	−	1.57058i	−0.0389693	−	0.743578i	3.21890	+	0.675057i	2.17358	−	1.50850i	−0.339936	−	0.667162i	−0.607770	+	2.93779i	0.946077	+	1.05072i
11.20	−1.46892	+	1.18951i	1.08431	+	1.35066i	0.326977	−	1.53831i	0.0633342	+	1.20849i	−3.19939	−	0.694212i	2.61937	+	0.372707i	−0.366694	−	0.719678i	−0.648544	+	2.92906i	−1.53054	−	1.69984i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner
41.h	odd	40	1	inner
123.o	even	40	1	inner
287.bf	odd	120	1	inner
861.cl	even	120	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.cl.a	✓	3456
3.b	odd	2	1	inner	861.2.cl.a	✓	3456
7.c	even	3	1	inner	861.2.cl.a	✓	3456
21.h	odd	6	1	inner	861.2.cl.a	✓	3456
41.h	odd	40	1	inner	861.2.cl.a	✓	3456
123.o	even	40	1	inner	861.2.cl.a	✓	3456
287.bf	odd	120	1	inner	861.2.cl.a	✓	3456
861.cl	even	120	1	inner	861.2.cl.a	✓	3456

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.cl.a	✓	3456	1.a	even	1	1	trivial
861.2.cl.a	✓	3456	3.b	odd	2	1	inner
861.2.cl.a	✓	3456	7.c	even	3	1	inner
861.2.cl.a	✓	3456	21.h	odd	6	1	inner
861.2.cl.a	✓	3456	41.h	odd	40	1	inner
861.2.cl.a	✓	3456	123.o	even	40	1	inner
861.2.cl.a	✓	3456	287.bf	odd	120	1	inner
861.2.cl.a	✓	3456	861.cl	even	120	1	inner

Hecke kernels

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