Newform orbit 525.2.bs.a

• Label: 525.2.bs.a
• Level: $525$
• Weight: $2$
• Character orbit: 525.bs
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $1216$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(1216\)
Relative dimension:	\(76\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1216 q - 8 q^{3} - 20 q^{4} - 24 q^{6} - 28 q^{7} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−2.14544	−	1.73734i	1.64793	+	0.533222i	1.16873	+	5.49845i	2.07100	+	0.843193i	−2.60915	−	4.00702i	0.291962	+	2.62959i	4.53862	−	8.90754i	2.43135	+	1.75743i	−2.97829	−	5.40705i
2.2	−2.10462	−	1.70428i	0.495015	−	1.65981i	1.10900	+	5.21743i	0.865799	−	2.06165i	−3.87060	+	2.64961i	−2.02686	−	1.70054i	4.09903	−	8.04480i	−2.50992	−	1.64326i	−5.33581	+	2.86341i
2.3	−2.02843	−	1.64259i	−1.68808	−	0.387799i	1.00060	+	4.70747i	2.22126	−	0.256881i	2.78716	+	3.55945i	2.15806	−	1.53061i	3.33287	−	6.54113i	2.69922	+	1.30927i	−4.92763	−	3.12756i
2.4	−2.02020	−	1.63593i	−1.04782	−	1.37916i	0.989132	+	4.65350i	−1.01425	+	1.99281i	−0.139395	+	4.50033i	−2.56962	+	0.630106i	3.25423	−	6.38679i	−0.804150	+	2.89022i	5.30908	−	2.36665i
2.5	−1.93861	−	1.56985i	1.72611	−	0.143382i	0.877938	+	4.13037i	−1.96574	+	1.06578i	−3.57133	−	2.43177i	0.878403	−	2.49568i	2.51712	−	4.94013i	2.95888	−	0.494984i	5.48391	+	1.01980i
2.6	−1.90594	−	1.54340i	0.518584	+	1.65260i	0.834698	+	3.92695i	0.362947	−	2.20642i	1.56222	−	3.95012i	2.39774	−	1.11841i	2.24315	−	4.40243i	−2.46214	+	1.71402i	−4.09713	+	3.64512i
2.7	−1.87622	−	1.51933i	−1.55929	+	0.754067i	0.796003	+	3.74490i	−2.20949	−	0.343734i	4.07124	+	0.954284i	−0.488296	−	2.60030i	2.00418	−	3.93343i	1.86277	−	2.35162i	3.62324	+	4.00187i
2.8	−1.82030	−	1.47405i	1.15573	−	1.29008i	0.724840	+	3.41011i	−1.83998	−	1.27061i	−4.00540	+	0.644724i	0.479622	+	2.60192i	1.58048	−	3.10187i	−0.328590	−	2.98195i	1.47637	+	5.02511i
2.9	−1.81817	−	1.47233i	−1.51942	+	0.831477i	0.722180	+	3.39759i	−0.209118	+	2.22627i	3.98678	+	0.725319i	0.560858	+	2.58562i	1.56505	−	3.07158i	1.61729	−	2.52673i	3.65801	−	3.73985i
2.10	−1.76827	−	1.43192i	−0.780419	+	1.54627i	0.660569	+	3.10773i	2.17999	−	0.497639i	3.59412	−	1.61672i	−2.64461	−	0.0776153i	1.21599	−	2.38651i	−1.78189	−	2.41347i	−4.56739	−	2.24160i
2.11	−1.63394	−	1.32314i	0.407314	+	1.68348i	0.503241	+	2.36756i	−1.47395	+	1.68151i	1.56195	−	3.28963i	2.44053	+	1.02168i	0.401327	−	0.787649i	−2.66819	+	1.37141i	4.63321	−	0.797245i
2.12	−1.61952	−	1.31146i	0.852586	+	1.50768i	0.487094	+	2.29160i	0.657350	+	2.13726i	0.596485	−	3.55986i	−2.28662	−	1.33093i	0.324313	−	0.636500i	−1.54619	+	2.57085i	1.73835	−	4.32344i
2.13	−1.49559	−	1.21111i	−0.875641	−	1.49441i	0.354192	+	1.66634i	2.19499	−	0.426619i	−0.500284	+	3.29552i	0.855774	+	2.50353i	−0.258988	+	0.508293i	−1.46651	+	2.61713i	−3.79949	−	2.02032i
2.14	−1.49464	−	1.21034i	1.52200	−	0.826756i	0.353218	+	1.66176i	2.03858	+	0.918808i	−3.27550	−	0.606427i	−0.413910	−	2.61317i	−0.262916	+	0.516001i	1.63295	−	2.51664i	−1.93488	−	3.84066i
2.15	−1.47969	−	1.19823i	1.49107	+	0.881317i	0.337908	+	1.58973i	−1.81017	−	1.31275i	−1.15030	−	3.09072i	−1.79797	+	1.94095i	−0.323938	+	0.635763i	1.44656	+	2.62820i	1.10552	+	4.11146i
2.16	−1.40992	−	1.14173i	−1.65433	−	0.513008i	0.268505	+	1.26322i	−0.187678	−	2.22818i	1.74677	+	2.61211i	−2.52135	+	0.801735i	−0.583603	+	1.14539i	2.47365	+	1.69737i	−2.27937	+	3.35584i
2.17	−1.40053	−	1.13413i	0.668427	−	1.59788i	0.259417	+	1.22046i	1.53018	+	1.63051i	−2.74835	+	1.47979i	−1.57002	+	2.12957i	−0.615476	+	1.20794i	−2.10641	−	2.13612i	−0.293854	−	4.01899i
2.18	−1.29573	−	1.04926i	1.71222	−	0.261325i	0.162144	+	0.762826i	1.09567	−	1.94923i	−2.49278	−	1.45796i	2.61928	+	0.373301i	−0.923561	+	1.81259i	2.86342	−	0.894893i	−3.46495	+	1.37604i
2.19	−1.27995	−	1.03648i	−0.162421	−	1.72442i	0.148148	+	0.696980i	0.135304	−	2.23197i	−1.57943	+	2.37551i	1.74728	−	1.98671i	−0.962644	+	1.88930i	−2.94724	+	0.560165i	−2.48657	+	2.71656i
2.20	−1.20229	−	0.973599i	−1.55126	−	0.770447i	0.0817938	+	0.384809i	0.300868	+	2.21573i	1.11497	+	2.43661i	0.513705	−	2.59540i	−1.12839	+	2.21460i	1.81282	+	2.39033i	1.79550	−	2.95689i

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
25.f	odd	20	1	inner
75.l	even	20	1	inner
175.w	odd	60	1	inner
525.bs	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.bs.a	✓	1216
3.b	odd	2	1	inner	525.2.bs.a	✓	1216
7.c	even	3	1	inner	525.2.bs.a	✓	1216
21.h	odd	6	1	inner	525.2.bs.a	✓	1216
25.f	odd	20	1	inner	525.2.bs.a	✓	1216
75.l	even	20	1	inner	525.2.bs.a	✓	1216
175.w	odd	60	1	inner	525.2.bs.a	✓	1216
525.bs	even	60	1	inner	525.2.bs.a	✓	1216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.2.bs.a	✓	1216	1.a	even	1	1	trivial
525.2.bs.a	✓	1216	3.b	odd	2	1	inner
525.2.bs.a	✓	1216	7.c	even	3	1	inner
525.2.bs.a	✓	1216	21.h	odd	6	1	inner
525.2.bs.a	✓	1216	25.f	odd	20	1	inner
525.2.bs.a	✓	1216	75.l	even	20	1	inner
525.2.bs.a	✓	1216	175.w	odd	60	1	inner
525.2.bs.a	✓	1216	525.bs	even	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).