Newform orbit 859.2.f.a

• Label: 859.2.f.a
• Level: $859$
• Weight: $2$
• Character orbit: 859.f
• Analytic conductor: $6.859$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	859.f (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.85914953363\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
100.1	−2.48316	+	1.30326i	−0.107649	+	0.155957i	3.33145	−	4.82644i	0.315645	+	0.457291i	0.0640574	−	0.527560i	−0.162393	−	1.33743i	−1.30634	+	10.7587i	1.05108	+	2.77147i	−1.37977	−	0.724157i
100.2	−2.33911	+	1.22766i	1.03151	−	1.49440i	2.82816	−	4.09729i	−2.42491	−	3.51309i	−0.578201	+	4.76191i	0.415688	+	3.42350i	−0.948446	+	7.81115i	−0.105405	−	0.277931i	9.98501	+	5.24054i
100.3	−2.32231	+	1.21884i	−1.57145	+	2.27663i	2.77143	−	4.01511i	1.33663	+	1.93644i	0.874530	−	7.20240i	−0.169976	−	1.39987i	−0.910065	+	7.49506i	−1.64980	−	4.35017i	−5.46428	−	2.86788i
100.4	−2.20441	+	1.15696i	1.29739	−	1.87959i	2.38474	−	3.45489i	−0.179952	−	0.260706i	−0.685356	+	5.64442i	0.00469214	+	0.0386433i	−0.659588	+	5.43220i	−0.785826	−	2.07205i	0.698316	+	0.366504i
100.5	−2.15874	+	1.13300i	−0.194366	+	0.281587i	2.24036	−	3.24573i	0.305534	+	0.442643i	0.100548	−	0.828091i	0.433769	+	3.57241i	−0.571237	+	4.70456i	1.02230	+	2.69559i	−1.16108	−	0.609383i
100.6	−2.15192	+	1.12941i	−0.589027	+	0.853353i	2.21904	−	3.21484i	−0.851299	−	1.23332i	0.303749	−	2.50160i	−0.487978	−	4.01886i	−0.558435	+	4.59913i	0.682556	+	1.79975i	3.22485	+	1.69253i
100.7	−2.07491	+	1.08900i	0.871263	−	1.26224i	1.98321	−	2.87317i	1.93783	+	2.80743i	−0.433214	+	3.56784i	−0.369958	−	3.04688i	−0.421190	+	3.46881i	0.229659	+	0.605561i	−7.07810	−	3.71488i
100.8	−2.05140	+	1.07666i	−1.67977	+	2.43357i	1.91291	−	2.77132i	−1.51519	−	2.19513i	0.825761	−	6.80075i	−0.0387319	−	0.318985i	−0.381856	+	3.14487i	−2.03682	−	5.37064i	5.47165	+	2.87174i
100.9	−2.04140	+	1.07141i	−1.39826	+	2.02573i	1.88326	−	2.72837i	−0.257120	−	0.372503i	0.684023	−	5.63344i	0.629477	+	5.18421i	−0.365492	+	3.01010i	−1.08464	−	2.85997i	0.923988	+	0.484946i
100.10	−1.96210	+	1.02979i	−0.0513722	+	0.0744255i	1.65324	−	2.39513i	−1.35574	−	1.96412i	0.0241548	−	0.198933i	0.0102623	+	0.0845172i	−0.243141	+	2.00245i	1.06091	+	2.79740i	4.68272	+	2.45768i
100.11	−1.87296	+	0.983006i	−1.03794	+	1.50372i	1.40555	−	2.03629i	2.19485	+	3.17979i	0.465860	−	3.83670i	0.0206728	+	0.170256i	−0.120925	+	0.995909i	−0.120026	−	0.316484i	−7.23662	−	3.79807i
100.12	−1.74565	+	0.916187i	−0.487732	+	0.706602i	1.07176	−	1.55271i	−2.01300	−	2.91634i	0.204029	−	1.68033i	−0.327841	−	2.70001i	0.0269275	−	0.221768i	0.802411	+	2.11578i	6.18590	+	3.24661i
100.13	−1.74103	+	0.913764i	0.895484	−	1.29733i	1.06010	−	1.53582i	1.65604	+	2.39919i	−0.373610	+	3.07696i	0.380352	+	3.13248i	0.0317215	−	0.261250i	0.182634	+	0.481567i	−5.07552	−	2.66384i
100.14	−1.70030	+	0.892384i	1.83435	−	2.65752i	0.958526	−	1.38866i	−0.866259	−	1.25499i	−0.747414	+	6.15551i	−0.526271	−	4.33423i	0.0723652	−	0.595981i	−2.63374	−	6.94460i	2.59283	+	1.36082i
100.15	−1.63364	+	0.857399i	1.08831	−	1.57669i	0.797506	−	1.15539i	−0.352636	−	0.510881i	−0.426053	+	3.50886i	−0.0548376	−	0.451628i	0.132564	−	1.09177i	−0.237723	−	0.626824i	1.01411	+	0.532244i
100.16	−1.45273	+	0.762453i	−1.72610	+	2.50069i	0.392967	−	0.569310i	−0.441076	−	0.639009i	0.600905	−	4.94890i	−0.419640	−	3.45605i	0.258716	−	2.13072i	−2.21020	−	5.82783i	1.12798	+	0.592009i
100.17	−1.42855	+	0.749761i	1.40492	−	2.03538i	0.342484	−	0.496173i	1.21140	+	1.75502i	−0.480952	+	3.96099i	0.541445	+	4.45920i	0.271692	−	2.23758i	−1.10515	−	2.91404i	−3.04640	−	1.59887i
100.18	−1.42185	+	0.746244i	1.80935	−	2.62130i	0.328646	−	0.476126i	−1.98586	−	2.87701i	−0.616497	+	5.07731i	0.446566	+	3.67780i	0.275132	−	2.26592i	−2.53364	−	6.68065i	4.97054	+	2.60874i
100.19	−1.40677	+	0.738332i	−0.456049	+	0.660700i	0.297748	−	0.431363i	2.02811	+	2.93823i	0.153741	−	1.26617i	−0.0376523	−	0.310095i	0.282631	−	2.32768i	0.835270	+	2.20243i	−5.02248	−	2.63600i
100.20	−1.37011	+	0.719088i	−0.554289	+	0.803027i	0.223979	−	0.324490i	0.478083	+	0.692622i	0.181989	−	1.49882i	0.117665	+	0.969059i	0.299485	−	2.46648i	0.726199	+	1.91483i	−1.15308	−	0.605184i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
859.f	even	13	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	859.2.f.a	✓	840
859.f	even	13	1	inner	859.2.f.a	✓	840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
859.2.f.a	✓	840	1.a	even	1	1	trivial
859.2.f.a	✓	840	859.f	even	13	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(859, [\chi])\).