Newform orbit 869.2.v.a

• Label: 869.2.v.a
• Level: $869$
• Weight: $2$
• Character orbit: 869.v
• Analytic conductor: $6.939$
• Analytic rank: $0$
• Dimension: $3744$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	869.v (of order \(65\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.93899993565\)
Analytic rank:	\(0\)
Dimension:	\(3744\)
Relative dimension:	\(78\) over \(\Q(\zeta_{65})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{65}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3744 q - 37 q^{2} - 33 q^{3} + 39 q^{4} - 33 q^{5} - 45 q^{6} - 45 q^{7} - 111 q^{8} + 53 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−1.99078	−	1.94325i	0.0689341	−	0.123968i	0.138686	+	5.73775i	0.244501	+	0.196368i	−0.378133	+	0.112837i	−1.89124	−	2.03358i	7.08461	−	7.61779i	1.57230	+	2.53131i	−0.105158	−	0.866053i
38.2	−1.97784	−	1.93061i	1.23275	−	2.21691i	0.136261	+	5.63744i	−2.62922	−	2.11162i	−6.71815	+	2.00473i	−0.283651	−	0.304999i	6.84968	−	7.36518i	−1.81210	−	2.91737i	1.12345	+	9.25243i
38.3	−1.88854	−	1.84344i	0.406439	−	0.730920i	0.119970	+	4.96342i	2.22000	+	1.78296i	−2.11499	+	0.631123i	3.50700	+	3.77093i	5.32867	−	5.72970i	1.21387	+	1.95425i	−0.905763	−	7.45963i
38.4	−1.84225	−	1.79826i	−1.49746	+	2.69295i	0.111821	+	4.62627i	−1.44092	−	1.15725i	7.60132	−	2.26827i	−1.94588	−	2.09233i	4.60680	−	4.95350i	−3.42669	−	5.51676i	0.573487	+	4.72309i
38.5	−1.83615	−	1.79231i	−0.912387	+	1.64079i	0.110763	+	4.58250i	−1.92275	−	1.54423i	4.61609	−	1.37746i	2.53946	+	2.73057i	4.51505	−	4.85485i	−0.276821	−	0.445665i	0.762725	+	6.28161i
38.6	−1.81940	−	1.77596i	−0.539107	+	0.969502i	0.107873	+	4.46296i	1.56088	+	1.25360i	2.70265	−	0.806485i	−0.231046	−	0.248434i	4.26681	−	4.58793i	0.933622	+	1.50307i	−0.613528	−	5.05285i
38.7	−1.75961	−	1.71759i	−1.50402	+	2.70476i	0.0977749	+	4.04517i	1.86748	+	1.49985i	7.29217	−	2.17602i	0.898097	+	0.965687i	3.42675	−	3.68465i	−3.47072	−	5.58765i	−0.709920	−	5.84672i
38.8	−1.75170	−	1.70988i	1.12198	−	2.01771i	0.0964652	+	3.99098i	3.19901	+	2.56924i	−5.41540	+	1.61598i	−1.94219	−	2.08836i	3.32100	−	3.57094i	−1.22938	−	1.97923i	−1.21063	−	9.97046i
38.9	−1.71801	−	1.67698i	1.52567	−	2.74369i	0.0909480	+	3.76272i	0.730938	+	0.587044i	−7.22223	+	2.15515i	2.13426	+	2.29488i	2.88381	−	3.10085i	−3.61723	−	5.82352i	−0.271295	−	2.23432i
38.10	−1.64057	−	1.60139i	0.465637	−	0.837378i	0.0786751	+	3.25497i	−3.17389	−	2.54906i	−2.10488	+	0.628108i	0.777111	+	0.835596i	1.96084	−	2.10841i	1.09854	+	1.76857i	1.12492	+	9.26455i
38.11	−1.61883	−	1.58017i	0.491342	−	0.883605i	0.0753314	+	3.11663i	−0.937973	−	0.753321i	−2.19164	+	0.653998i	−0.0394044	−	0.0423700i	1.72167	−	1.85124i	1.04358	+	1.68010i	0.328040	+	2.70165i
38.12	−1.57234	−	1.53479i	−0.572023	+	1.02870i	0.0683308	+	2.82700i	−1.93185	−	1.55154i	2.47825	−	0.739522i	−3.15368	−	3.39103i	1.23871	−	1.33194i	0.851913	+	1.37153i	0.656230	+	5.40454i
38.13	−1.57194	−	1.53440i	0.245401	−	0.441318i	0.0682714	+	2.82454i	−0.0961014	−	0.0771826i	−1.06291	+	0.317179i	−0.328826	−	0.353573i	1.23472	−	1.32764i	1.44838	+	2.33180i	0.0326363	+	0.268784i
38.14	−1.38043	−	1.34747i	0.692437	−	1.24524i	0.0415918	+	1.72075i	1.81527	+	1.45791i	−2.63378	+	0.785935i	0.456635	+	0.491002i	−0.366201	+	0.393761i	0.511759	+	0.823900i	−0.541367	−	4.45856i
38.15	−1.35274	−	1.32044i	−0.959727	+	1.72592i	0.0380211	+	1.57302i	1.73592	+	1.39418i	3.57725	−	1.06747i	2.36146	+	2.53918i	−0.549092	+	0.590416i	−0.474819	−	0.764429i	−0.507320	−	4.17816i
38.16	−1.34710	−	1.31494i	−0.510799	+	0.918596i	0.0373026	+	1.54329i	2.85717	+	2.29470i	1.89599	−	0.565775i	−3.08513	−	3.31731i	−0.584921	+	0.628942i	1.00002	+	1.60997i	−0.831522	−	6.84820i
38.17	−1.33922	−	1.30724i	−0.973944	+	1.75149i	0.0363028	+	1.50193i	−1.34103	−	1.07703i	3.59393	−	1.07245i	0.903809	+	0.971829i	−0.634231	+	0.681963i	−0.536234	−	0.863303i	0.387993	+	3.19541i
38.18	−1.17989	−	1.15171i	1.19368	−	2.14665i	0.0173637	+	0.718373i	−1.96129	−	1.57519i	−3.88074	+	1.15803i	−2.78540	−	2.99503i	−1.43886	+	1.54715i	−1.60032	−	2.57642i	0.499942	+	4.11739i
38.19	−1.14137	−	1.11412i	0.823346	−	1.48066i	0.0131446	+	0.543819i	−1.16420	−	0.935012i	−2.58937	+	0.772683i	3.23348	+	3.47683i	−1.58155	+	1.70057i	0.0684530	+	0.110205i	0.287072	+	2.36425i
38.20	−1.13017	−	1.10319i	−1.05905	+	1.90453i	0.0119443	+	0.494164i	−3.14912	−	2.52918i	3.29796	−	0.984129i	0.268195	+	0.288379i	−1.61945	+	1.74133i	−0.922752	−	1.48557i	0.768901	+	6.33247i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
79.e	even	13	1	inner
869.v	even	65	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	869.2.v.a	✓	3744
11.c	even	5	1	inner	869.2.v.a	✓	3744
79.e	even	13	1	inner	869.2.v.a	✓	3744
869.v	even	65	1	inner	869.2.v.a	✓	3744

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
869.2.v.a	✓	3744	1.a	even	1	1	trivial
869.2.v.a	✓	3744	11.c	even	5	1	inner
869.2.v.a	✓	3744	79.e	even	13	1	inner
869.2.v.a	✓	3744	869.v	even	65	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(869, [\chi])\).