Newform orbit 605.2.r.a

• Label: 605.2.r.a
• Level: $605$
• Weight: $2$
• Character orbit: 605.r
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.r (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83094932229\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1280 q - 22 q^{2} - 40 q^{3} - 14 q^{5} - 44 q^{6} - 22 q^{7} - 22 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−1.65403	−	2.20953i	−1.20427	+	1.20427i	−1.58273	+	5.39029i	0.661016	+	2.13613i	4.65276	+	0.668966i	−0.992826	+	4.56395i	9.35584	−	3.48955i			0.0994872i	3.62650	−	4.99377i
32.2	−1.63965	−	2.19032i	1.62268	−	1.62268i	−1.54558	+	5.26376i	0.347156	−	2.20896i	−6.21481	−	0.893555i	0.0527075	−	0.242292i	8.93645	−	3.33312i		−	2.26615i	−5.40754	+	2.86154i
32.3	−1.60536	−	2.14451i	−1.49754	+	1.49754i	−1.45828	+	4.96643i	1.95274	−	1.08941i	5.61559	+	0.807400i	0.692711	−	3.18434i	7.97176	−	2.97331i		−	1.48527i	−5.47109	−	2.43878i
32.4	−1.55810	−	2.08138i	−0.762046	+	0.762046i	−1.34100	+	4.56704i	−2.23099	+	0.150681i	2.77346	+	0.398763i	0.945323	−	4.34558i	6.72311	−	2.50759i			1.83857i	3.78973	+	4.40876i
32.5	−1.55047	−	2.07118i	1.30510	−	1.30510i	−1.32238	+	4.50362i	−2.13842	+	0.653577i	−4.72661	−	0.679584i	−0.374253	+	1.72041i	6.52992	−	2.43554i		−	0.406559i	4.66923	+	3.41571i
32.6	−1.36166	−	1.81897i	0.545918	−	0.545918i	−0.891058	+	3.03466i	−0.445983	−	2.19114i	−1.73637	−	0.249652i	−0.413052	+	1.89877i	2.47545	−	0.923297i			2.40395i	−3.37834	+	3.79483i
32.7	−1.34436	−	1.79586i	−0.787015	+	0.787015i	−0.854335	+	2.90960i	−1.17893	−	1.90004i	2.47140	+	0.355334i	−0.306060	+	1.40694i	2.17003	−	0.809380i			1.76121i	−1.82729	+	4.67153i
32.8	−1.34419	−	1.79563i	−0.436724	+	0.436724i	−0.853977	+	2.90838i	−0.335531	+	2.21075i	1.37124	+	0.197154i	0.125054	−	0.574863i	2.16709	−	0.808283i			2.61854i	4.42072	−	2.36919i
32.9	−1.31487	−	1.75646i	0.251483	−	0.251483i	−0.792800	+	2.70003i	2.23442	+	0.0859194i	−0.772387	−	0.111053i	0.518840	−	2.38507i	1.67342	−	0.624152i			2.87351i	−2.78705	−	4.03763i
32.10	−1.29735	−	1.73306i	2.35487	−	2.35487i	−0.756913	+	2.57781i	0.140101	+	2.23167i	−7.13625	−	1.02604i	−0.147963	+	0.680177i	1.39275	−	0.519471i		−	8.09087i	3.68587	−	3.13808i
32.11	−1.29475	−	1.72958i	−2.08349	+	2.08349i	−0.751609	+	2.55974i	−2.21633	−	0.296468i	6.30115	+	0.905969i	−0.317966	+	1.46166i	1.35184	−	0.504211i		−	5.68187i	2.35682	+	4.21716i
32.12	−1.20651	−	1.61170i	1.27231	−	1.27231i	−0.578464	+	1.97007i	2.15875	+	0.582930i	−3.58564	−	0.515537i	−1.05425	+	4.84632i	0.100425	−	0.0374565i		−	0.237553i	−1.66503	−	4.18257i
32.13	−1.18351	−	1.58098i	2.15121	−	2.15121i	−0.535343	+	1.82321i	1.95281	−	1.08928i	−5.94699	−	0.855048i	0.546359	−	2.51157i	−0.184700	+	0.0688894i		−	6.25540i	−4.03329	−	1.79819i
32.14	−1.17119	−	1.56452i	−2.21909	+	2.21909i	−0.512590	+	1.74572i	1.83519	+	1.27754i	6.07079	+	0.872848i	0.0862130	−	0.396315i	−0.330664	+	0.123331i		−	6.84872i	−0.150612	−	4.36743i
32.15	−1.05562	−	1.41014i	1.43490	−	1.43490i	−0.310697	+	1.05814i	−2.23306	−	0.115999i	−3.53811	−	0.508704i	0.943957	−	4.33930i	−1.48074	+	0.552288i		−	1.11789i	2.19367	+	3.27137i
32.16	−1.02585	−	1.37037i	−1.69135	+	1.69135i	−0.262091	+	0.892598i	1.54512	−	1.61636i	4.05284	+	0.582710i	−0.361306	+	1.66090i	−1.71570	+	0.639922i		−	2.72132i	−3.80007	−	0.459245i
32.17	−0.989614	−	1.32197i	−0.0346243	+	0.0346243i	−0.204802	+	0.697492i	0.766483	+	2.10060i	0.0800369	+	0.0115076i	0.0122172	−	0.0561614i	−1.96972	+	0.734666i			2.99760i	2.01840	−	3.09205i
32.18	−0.832083	−	1.11153i	1.00692	−	1.00692i	0.0203215	−	0.0692087i	−1.62222	+	1.53896i	−1.95707	−	0.281384i	−0.335672	+	1.54306i	−2.69570	+	1.00545i			0.972218i	3.06043	+	0.522603i
32.19	−0.826501	−	1.10408i	−1.13965	+	1.13965i	0.0275847	−	0.0939450i	−1.95581	+	1.08388i	2.20018	+	0.316338i	−1.02748	+	4.72324i	−2.71093	+	1.01113i			0.402407i	2.81317	+	1.26353i
32.20	−0.823163	−	1.09962i	1.35528	−	1.35528i	0.0319051	−	0.108659i	−1.65144	−	1.50756i	−2.60591	−	0.374674i	0.0674781	−	0.310192i	−2.71972	+	1.01440i		−	0.673588i	−0.298338	+	3.05692i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
121.f	odd	22	1	inner
605.r	even	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	605.2.r.a	✓	1280
5.c	odd	4	1	inner	605.2.r.a	✓	1280
121.f	odd	22	1	inner	605.2.r.a	✓	1280
605.r	even	44	1	inner	605.2.r.a	✓	1280

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
605.2.r.a	✓	1280	1.a	even	1	1	trivial
605.2.r.a	✓	1280	5.c	odd	4	1	inner
605.2.r.a	✓	1280	121.f	odd	22	1	inner
605.2.r.a	✓	1280	605.r	even	44	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(605, [\chi])\).