Newform orbit 720.2.ce.a

• Label: 720.2.ce.a
• Level: $720$
• Weight: $2$
• Character orbit: 720.ce
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $560$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.ce (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(560\)
Relative dimension:	\(140\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 4 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
229.1	−1.41421		0.000315556i	−1.67590	+	0.437429i	2.00000		0.000892526i	−2.08909	−	0.797299i	2.36995	−	0.619147i	0.413786	+	0.716698i	−2.82843	+	0.00189333i	2.61731	−	1.46618i	2.95468	+	1.12689i
229.2	−1.41389	−	0.0303798i	−0.463626	−	1.66885i	1.99815	+	0.0859071i	1.89019	+	1.19465i	0.604816	+	2.37365i	0.557227	+	0.965146i	−2.82255	−	0.182166i	−2.57010	+	1.54744i	−2.63622	−	1.74653i
229.3	−1.41382	−	0.0333574i	1.29435	−	1.15094i	1.99777	+	0.0943228i	−0.578106	+	2.16004i	−1.86836	+	1.58405i	1.06549	+	1.84549i	−2.82135	−	0.199996i	0.350662	−	2.97944i	0.889392	−	3.03463i
229.4	−1.41369	+	0.0384668i	0.801465	+	1.53547i	1.99704	−	0.108760i	−0.792486	−	2.09092i	−1.19209	−	2.13984i	2.49869	+	4.32785i	−2.81901	+	0.230573i	−1.71531	+	2.46124i	1.20076	+	2.92544i
229.5	−1.41302	+	0.0580371i	0.239220	−	1.71545i	1.99326	−	0.164015i	−2.10476	+	0.754973i	−0.238463	+	2.43785i	−1.85818	−	3.21847i	−2.80701	+	0.347440i	−2.88555	−	0.820739i	2.93026	−	1.18895i
229.6	−1.40758	−	0.136781i	1.57600	−	0.718480i	1.96258	+	0.385062i	−0.106626	−	2.23352i	−2.31663	+	0.795753i	−0.805016	−	1.39433i	−2.70983	−	0.810452i	1.96757	−	2.26465i	−0.155419	+	3.15846i
229.7	−1.39691	+	0.220541i	−0.314334	+	1.70329i	1.90272	−	0.616153i	1.86069	−	1.24009i	0.0634517	−	2.44867i	−0.630230	−	1.09159i	−2.52205	+	1.28034i	−2.80239	−	1.07080i	−2.32573	+	2.14265i
229.8	−1.38957	+	0.262858i	1.68223	+	0.412450i	1.86181	−	0.730521i	0.976053	+	2.01180i	−2.44599	−	0.130941i	1.36034	+	2.35618i	−2.39509	+	1.50450i	2.65977	+	1.38767i	−1.88511	−	2.53897i
229.9	−1.38161	−	0.301937i	1.60380	+	0.654073i	1.81767	+	0.834316i	2.17137	−	0.534008i	−2.01834	−	1.38792i	−0.632866	−	1.09616i	−2.25939	−	1.70152i	2.14438	+	2.09801i	−3.16121	+	0.0821728i
229.10	−1.38114	+	0.304064i	−1.54992	+	0.773134i	1.81509	−	0.839909i	0.926564	+	2.03506i	1.90558	−	1.53908i	−1.81438	−	3.14260i	−2.25151	+	1.71194i	1.80453	−	2.39660i	−1.89850	−	2.52897i
229.11	−1.36409	+	0.373178i	1.55039	+	0.772195i	1.72148	−	1.01810i	−2.02588	+	0.946481i	−2.40304	−	0.474771i	−1.84315	−	3.19243i	−1.96832	+	2.03119i	1.80743	+	2.39441i	2.41027	−	2.04710i
229.12	−1.36051	−	0.386018i	−1.17060	+	1.27659i	1.70198	+	1.05036i	−1.23018	+	1.86726i	2.08540	−	1.28495i	1.16693	+	2.02118i	−1.91010	−	2.08603i	−0.259389	−	2.98877i	2.39447	−	2.06555i
229.13	−1.34206	+	0.445957i	−1.39788	−	1.02271i	1.60224	−	1.19700i	−0.848191	−	2.06895i	2.33212	+	0.749151i	−0.631472	−	1.09374i	−1.61650	+	2.32098i	0.908109	+	2.85925i	2.06099	+	2.39840i
229.14	−1.34126	−	0.448367i	0.522669	−	1.65131i	1.59793	+	1.20275i	0.505734	−	2.17813i	−1.44142	+	1.98048i	1.69266	+	2.93177i	−1.60397	−	2.32965i	−2.45363	−	1.72618i	−1.65492	+	2.69467i
229.15	−1.33281	−	0.472872i	0.421301	+	1.68003i	1.55278	+	1.26050i	1.18135	+	1.89853i	0.232923	−	2.43839i	−1.70751	−	2.95750i	−1.47352	−	2.41428i	−2.64501	+	1.41560i	−0.676756	−	3.08901i
229.16	−1.32705	−	0.488803i	−1.70864	−	0.283799i	1.52214	+	1.29733i	2.17604	−	0.514633i	2.12874	+	1.21181i	1.63908	+	2.83897i	−1.38583	−	2.46566i	2.83892	+	0.969823i	−3.13928	−	0.380709i
229.17	−1.32585	+	0.492049i	−0.567192	−	1.63655i	1.51577	−	1.30477i	1.43456	−	1.71524i	1.55728	+	1.89074i	0.108847	+	0.188529i	−1.36768	+	2.47577i	−2.35659	+	1.85648i	−1.05803	+	2.98003i
229.18	−1.32506	−	0.494170i	0.0981206	+	1.72927i	1.51159	+	1.30961i	−2.08371	−	0.811276i	0.724537	−	2.33988i	−1.68304	−	2.91511i	−1.35578	−	2.48231i	−2.98074	+	0.339354i	2.36014	+	2.10470i
229.19	−1.30458	−	0.545969i	−1.41349	−	1.00103i	1.40384	+	1.42452i	−0.587288	+	2.15757i	1.29747	+	2.07764i	−0.884485	−	1.53197i	−1.05367	−	2.62484i	0.995888	+	2.82988i	1.94413	−	2.49407i
229.20	−1.27635	−	0.609033i	1.38067	+	1.04582i	1.25816	+	1.55468i	−1.90644	+	1.16854i	−1.12528	−	2.17572i	1.47110	+	2.54803i	−0.659004	−	2.75058i	0.812502	+	2.88788i	3.14497	−	0.330391i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
16.e	even	4	1	inner
45.j	even	6	1	inner
80.q	even	4	1	inner
144.x	even	12	1	inner
720.ce	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.ce.a	✓	560
5.b	even	2	1	inner	720.2.ce.a	✓	560
9.c	even	3	1	inner	720.2.ce.a	✓	560
16.e	even	4	1	inner	720.2.ce.a	✓	560
45.j	even	6	1	inner	720.2.ce.a	✓	560
80.q	even	4	1	inner	720.2.ce.a	✓	560
144.x	even	12	1	inner	720.2.ce.a	✓	560
720.ce	even	12	1	inner	720.2.ce.a	✓	560

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.2.ce.a	✓	560	1.a	even	1	1	trivial
720.2.ce.a	✓	560	5.b	even	2	1	inner
720.2.ce.a	✓	560	9.c	even	3	1	inner
720.2.ce.a	✓	560	16.e	even	4	1	inner
720.2.ce.a	✓	560	45.j	even	6	1	inner
720.2.ce.a	✓	560	80.q	even	4	1	inner
720.2.ce.a	✓	560	144.x	even	12	1	inner
720.2.ce.a	✓	560	720.ce	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).