Newform orbit 560.2.db.a

• Label: 560.2.db.a
• Level: $560$
• Weight: $2$
• Character orbit: 560.db
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.db (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 368 q - 2 q^{2} - 2 q^{5} - 16 q^{6} - 8 q^{7} + 4 q^{8} + 168 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.41419	−	0.00860625i	2.02991	+	1.17197i	1.99985	+	0.0243417i	0.392134	+	2.20142i	−2.86058	−	1.67485i	1.14116	+	2.38700i	−2.82796	−	0.0516349i	1.24701	+	2.15989i	−0.535606	−	3.11659i
67.2	−1.41418	+	0.00983984i	−0.227319	−	0.131243i	1.99981	−	0.0278306i	−2.23470	−	0.0782722i	0.322761	+	0.183364i	−0.196814	+	2.63842i	−2.82781	+	0.0590352i	−1.46555	−	2.53841i	3.16103	+	0.0887019i
67.3	−1.41328	−	0.0513799i	1.96509	+	1.13455i	1.99472	+	0.145228i	−0.311326	−	2.21429i	−2.71893	−	1.70440i	2.45023	−	0.998181i	−2.81164	−	0.307737i	1.07439	+	1.86089i	0.326221	+	3.14541i
67.4	−1.41314	−	0.0549767i	−0.890684	−	0.514237i	1.99396	+	0.155380i	2.14658	−	0.626243i	1.23039	+	0.775658i	1.93786	−	1.80131i	−2.80920	−	0.329195i	−0.971121	−	1.68203i	−3.06786	+	0.766960i
67.5	−1.41215	+	0.0763356i	−2.64925	−	1.52954i	1.98835	−	0.215595i	−0.802745	+	2.08701i	3.85790	+	1.95772i	0.247859	−	2.63412i	−2.79139	+	0.456234i	3.17900	+	5.50620i	0.974286	−	3.00845i
67.6	−1.41095	−	0.0960155i	1.64151	+	0.947726i	1.98156	+	0.270946i	2.23494	+	0.0710535i	−2.22509	−	1.49480i	−2.64574	+	0.00710298i	−2.76987	−	0.572552i	0.296368	+	0.513324i	−3.14657	−	0.314842i
67.7	−1.36767	+	0.359825i	−2.55251	−	1.47369i	1.74105	−	0.984244i	−1.06060	−	1.96853i	4.02127	+	1.09707i	2.05824	+	1.66242i	−2.02703	+	1.97260i	2.84355	+	4.92518i	2.15888	+	2.31068i
67.8	−1.36703	−	0.362272i	−1.05512	−	0.609177i	1.73752	+	0.990469i	−2.01981	−	0.959361i	1.22170	+	1.21500i	−1.87159	−	1.87007i	−2.01641	−	1.98345i	−0.757808	−	1.31256i	2.41358	+	2.04319i
67.9	−1.34452	−	0.438480i	0.606999	+	0.350451i	1.61547	+	1.17909i	0.596526	+	2.15503i	−0.662456	−	0.737345i	−1.47069	−	2.19933i	−1.65503	−	2.29366i	−1.25437	−	2.17263i	0.142896	−	3.15905i
67.10	−1.33721	+	0.460280i	−1.68596	−	0.973392i	1.57628	−	1.23099i	1.88239	+	1.20690i	2.70253	+	0.525618i	−1.16389	+	2.37600i	−1.54123	+	2.37162i	0.394983	+	0.684130i	−3.07267	−	0.747464i
67.11	−1.31349	+	0.524168i	0.567224	+	0.327487i	1.45050	−	1.37698i	−1.75608	+	1.38426i	−0.916699	−	0.132829i	2.42974	−	1.04707i	−1.18344	+	2.56894i	−1.28550	−	2.22656i	1.58101	−	2.73869i
67.12	−1.31124	+	0.529772i	1.11240	+	0.642246i	1.43868	−	1.38931i	−0.158075	−	2.23047i	−1.79887	−	0.252817i	−2.63964	+	0.179738i	−1.15043	+	2.58389i	−0.675039	−	1.16920i	1.38892	+	2.84093i
67.13	−1.30713	−	0.539814i	−2.20671	−	1.27404i	1.41720	+	1.41122i	0.886736	−	2.05273i	2.19672	+	2.85656i	−2.52890	+	0.777586i	−1.09067	−	2.60968i	1.74637	+	3.02480i	−2.26718	+	2.20452i
67.14	−1.24380	−	0.673031i	−0.966445	−	0.557977i	1.09406	+	1.67423i	0.981672	+	2.00906i	0.826525	+	1.34446i	2.42995	+	1.04659i	−0.233982	−	2.81873i	−0.877322	−	1.51957i	0.131158	−	3.15956i
67.15	−1.22708	−	0.703040i	2.64992	+	1.52993i	1.01147	+	1.72538i	−1.94089	−	1.11038i	−2.17607	−	3.74035i	−1.89277	+	1.84863i	−0.0281454	−	2.82829i	3.18138	+	5.51031i	1.60099	+	2.72706i
67.16	−1.21719	+	0.720028i	−0.387277	−	0.223595i	0.963119	−	1.75283i	−0.941111	+	2.02838i	0.632386	−	0.00669261i	−2.53424	−	0.760028i	0.0897823	+	2.82700i	−1.40001	−	2.42489i	−0.314974	−	3.14655i
67.17	−1.18785	−	0.767467i	1.38259	+	0.798236i	0.821989	+	1.82328i	−2.08827	+	0.799452i	−1.02969	−	2.00928i	2.06071	−	1.65936i	0.422902	−	2.79663i	−0.225637	−	0.390815i	3.09411	+	0.653049i
67.18	−1.17700	+	0.784006i	2.75664	+	1.59154i	0.770668	−	1.84555i	2.23485	+	0.0738742i	−4.49235	+	0.287969i	1.06433	−	2.42223i	0.539849	+	2.77643i	3.56603	+	6.17654i	−2.68834	+	1.66518i
67.19	−1.10932	−	0.877154i	−2.44963	−	1.41430i	0.461201	+	1.94610i	−1.45280	+	1.69982i	1.47688	+	3.71762i	−0.560159	+	2.58577i	1.19541	−	2.56340i	2.50047	+	4.33093i	3.10262	−	0.611325i
67.20	−1.08959	+	0.901549i	−1.13160	−	0.653332i	0.374420	−	1.96464i	−1.28743	−	1.82826i	1.82200	−	0.308332i	1.31202	−	2.29752i	1.36325	+	2.47821i	−0.646314	−	1.11945i	3.05104	+	0.831373i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
80.j	even	4	1	inner
560.db	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	560.2.db.a	yes	368
5.c	odd	4	1		560.2.cf.a	✓	368
7.c	even	3	1	inner	560.2.db.a	yes	368
16.f	odd	4	1		560.2.cf.a	✓	368
35.l	odd	12	1		560.2.cf.a	✓	368
80.j	even	4	1	inner	560.2.db.a	yes	368
112.u	odd	12	1		560.2.cf.a	✓	368
560.db	even	12	1	inner	560.2.db.a	yes	368

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
560.2.cf.a	✓	368	5.c	odd	4	1
560.2.cf.a	✓	368	16.f	odd	4	1
560.2.cf.a	✓	368	35.l	odd	12	1
560.2.cf.a	✓	368	112.u	odd	12	1
560.2.db.a	yes	368	1.a	even	1	1	trivial
560.2.db.a	yes	368	7.c	even	3	1	inner
560.2.db.a	yes	368	80.j	even	4	1	inner
560.2.db.a	yes	368	560.db	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(560, [\chi])\).