Newform orbit 520.2.cs.a

• Label: 520.2.cs.a
• Level: $520$
• Weight: $2$
• Character orbit: 520.cs
• Analytic conductor: $4.152$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.cs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.15222090511\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 6 q^{2} - 4 q^{3} - 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.41357	+	0.0427482i	1.07727	−	0.288654i	1.99635	−	0.120855i	2.21831	−	0.281212i	−1.51045	+	0.454083i	−1.80137	−	0.482675i	−2.81680	+	0.256177i	−1.52089	+	0.878084i	−3.12372	+	0.492341i
43.2	−1.41280	+	0.0632856i	2.98062	−	0.798656i	1.99199	−	0.178819i	0.0738413	−	2.23485i	−4.16047	+	1.31697i	1.55096	+	0.415580i	−2.80296	+	0.378699i	5.64819	−	3.26099i	0.0371109	+	3.16206i
43.3	−1.41033	−	0.104701i	−1.89726	+	0.508369i	1.97808	+	0.295327i	−1.26250	+	1.84556i	2.72899	−	0.518323i	0.346608	+	0.0928734i	−2.75882	−	0.623617i	0.743070	−	0.429012i	1.97377	−	2.47067i
43.4	−1.40759	−	0.136690i	−3.13823	+	0.840887i	1.96263	+	0.384809i	1.61237	−	1.54927i	4.53229	−	0.754660i	2.01184	+	0.539072i	−2.70999	−	0.809927i	6.54334	−	3.77780i	−2.48134	+	1.96035i
43.5	−1.39651	+	0.223040i	−1.59499	+	0.427377i	1.90051	−	0.622957i	−1.58043	−	1.58185i	2.13211	−	0.952585i	−4.27539	−	1.14559i	−2.51514	+	1.29386i	−0.236729	+	0.136676i	2.55991	+	1.85657i
43.6	−1.38296	+	0.295672i	0.361127	−	0.0967636i	1.82516	−	0.817804i	0.996316	+	2.00184i	−0.470814	+	0.240595i	−0.863735	−	0.231437i	−2.28232	+	1.67064i	−2.47703	+	1.43011i	−1.96975	−	2.47388i
43.7	−1.36847	−	0.356792i	2.23757	−	0.599554i	1.74540	+	0.976515i	−2.23509	−	0.0661214i	−3.27595	+	0.0221246i	−3.07029	−	0.822683i	−2.04011	−	1.95907i	2.04916	−	1.18308i	3.03505	+	0.887946i
43.8	−1.34551	+	0.435421i	0.361127	−	0.0967636i	1.62082	−	1.17173i	−0.996316	−	2.00184i	−0.443768	+	0.287439i	0.863735	+	0.231437i	−1.67064	+	2.28232i	−2.47703	+	1.43011i	2.21220	+	2.25969i
43.9	−1.34290	−	0.443421i	−0.898963	+	0.240876i	1.60675	+	1.19094i	−2.12690	+	0.690146i	1.31403	+	0.0751467i	2.50362	+	0.670843i	−1.62962	−	2.31178i	−1.84796	+	1.06692i	3.16224	+	0.0163157i
43.10	−1.32094	+	0.505099i	−1.59499	+	0.427377i	1.48975	−	1.33441i	1.58043	+	1.58185i	1.89102	−	1.37017i	4.27539	+	1.14559i	−1.29386	+	2.51514i	−0.236729	+	0.136676i	−2.88664	−	1.29125i
43.11	−1.27391	−	0.614123i	−2.05802	+	0.551446i	1.24570	+	1.56468i	1.73823	+	1.40662i	2.96040	+	0.561387i	−4.09638	−	1.09762i	−0.626013	−	2.75828i	1.33330	−	0.769779i	−1.35051	−	2.85939i
43.12	−1.25516	+	0.651592i	2.98062	−	0.798656i	1.15086	−	1.63570i	−0.0738413	+	2.23485i	−3.22077	+	2.94459i	−1.55096	−	0.415580i	−0.378699	+	2.80296i	5.64819	−	3.26099i	−1.36353	−	2.85321i
43.13	−1.25497	−	0.651960i	0.185227	−	0.0496314i	1.14990	+	1.63638i	−0.921281	−	2.03746i	−0.264812	−	0.0584748i	4.16087	+	1.11490i	−0.376228	−	2.80329i	−2.56623	+	1.48161i	−0.172164	+	3.15759i
43.14	−1.24661	−	0.667812i	1.23251	−	0.330249i	1.10805	+	1.66500i	−0.413825	+	2.19744i	−1.75699	−	0.411392i	−1.69252	−	0.453510i	−0.269402	−	2.81557i	−1.18807	+	0.685932i	1.98335	−	2.46299i
43.15	−1.24556	+	0.669763i	1.07727	−	0.288654i	1.10284	−	1.66846i	−2.21831	+	0.281212i	−1.14847	+	1.08105i	1.80137	+	0.482675i	−0.256177	+	2.81680i	−1.52089	+	0.878084i	2.57470	−	1.83601i
43.16	−1.17932	−	0.780512i	2.20750	−	0.591498i	0.781602	+	1.84095i	2.21421	−	0.311866i	−3.06503	−	1.02541i	2.30184	+	0.616775i	0.515124	−	2.78112i	1.92512	−	1.11147i	−2.85469	−	1.36043i
43.17	−1.16903	+	0.795840i	−1.89726	+	0.508369i	0.733277	−	1.86073i	1.26250	−	1.84556i	1.81338	−	2.10421i	−0.346608	−	0.0928734i	0.623617	+	2.75882i	0.743070	−	0.429012i	−0.00712500	+	3.16227i
43.18	−1.15067	+	0.822173i	−3.13823	+	0.840887i	0.648062	−	1.89209i	−1.61237	+	1.54927i	2.91970	−	3.54775i	−2.01184	−	0.539072i	0.809927	+	2.70999i	6.54334	−	3.77780i	0.581532	−	3.10835i
43.19	−1.07600	−	0.917733i	0.862140	−	0.231010i	0.315533	+	1.97495i	0.553447	−	2.16649i	−1.13966	−	0.542649i	−3.68844	−	0.988314i	1.47297	−	2.41462i	−1.90816	+	1.10167i	−2.58377	+	1.82322i
43.20	−1.00759	−	0.992353i	−3.14910	+	0.843799i	0.0304711	+	1.99977i	−1.89924	−	1.18020i	4.01035	+	2.27482i	0.412808	+	0.110612i	1.95377	−	2.04518i	6.60677	−	3.81442i	0.742480	+	3.07388i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
8.d	odd	2	1	inner
13.e	even	6	1	inner
40.k	even	4	1	inner
65.r	odd	12	1	inner
104.p	odd	6	1	inner
520.cs	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	520.2.cs.a	✓	320
5.c	odd	4	1	inner	520.2.cs.a	✓	320
8.d	odd	2	1	inner	520.2.cs.a	✓	320
13.e	even	6	1	inner	520.2.cs.a	✓	320
40.k	even	4	1	inner	520.2.cs.a	✓	320
65.r	odd	12	1	inner	520.2.cs.a	✓	320
104.p	odd	6	1	inner	520.2.cs.a	✓	320
520.cs	even	12	1	inner	520.2.cs.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.cs.a	✓	320	1.a	even	1	1	trivial
520.2.cs.a	✓	320	5.c	odd	4	1	inner
520.2.cs.a	✓	320	8.d	odd	2	1	inner
520.2.cs.a	✓	320	13.e	even	6	1	inner
520.2.cs.a	✓	320	40.k	even	4	1	inner
520.2.cs.a	✓	320	65.r	odd	12	1	inner
520.2.cs.a	✓	320	104.p	odd	6	1	inner
520.2.cs.a	✓	320	520.cs	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(520, [\chi])\).