Newform orbit 720.2.cf.a

• Label: 720.2.cf.a
• Level: $720$
• Weight: $2$
• Character orbit: 720.cf
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(96\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q - 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41388	−	0.0306609i	−0.425155	−	1.67906i	1.99812	+	0.0867018i	−0.965926	−	0.258819i	0.549637	+	2.38703i	−0.547298	+	0.947947i	−2.82245	−	0.183850i	−2.63849	+	1.42772i	1.35777	+	0.395556i
11.2	−1.41368	−	0.0389030i	1.34720	−	1.08860i	1.99697	+	0.109993i	−0.965926	−	0.258819i	−1.94686	+	1.48652i	2.30038	−	3.98438i	−2.81880	−	0.233182i	0.629893	−	2.93313i	1.35544	+	0.403464i
11.3	−1.39931	+	0.204795i	−1.63594	+	0.568941i	1.91612	−	0.573142i	0.965926	+	0.258819i	2.17267	−	1.13115i	−0.548548	+	0.950114i	−2.56386	+	1.19441i	2.35261	−	1.86151i	−1.40463	−	0.164351i
11.4	−1.39749	−	0.216861i	−0.285126	+	1.70842i	1.90594	+	0.606122i	0.965926	+	0.258819i	0.768950	−	2.32566i	0.981834	−	1.70059i	−2.53209	−	1.26037i	−2.83741	−	0.974230i	−1.29374	−	0.571168i
11.5	−1.38912	−	0.265243i	−1.12623	−	1.31590i	1.85929	+	0.736907i	0.965926	+	0.258819i	1.21543	+	2.12667i	−0.827119	+	1.43261i	−2.38732	−	1.51681i	−0.463207	+	2.96402i	−1.27313	−	0.615735i
11.6	−1.37924	+	0.312549i	1.52098	+	0.828632i	1.80463	−	0.862163i	−0.965926	−	0.258819i	−2.35679	−	0.667505i	−0.303417	+	0.525534i	−2.21955	+	1.75317i	1.62674	+	2.52066i	1.41314	+	0.0550751i
11.7	−1.37170	−	0.344144i	−0.677155	+	1.59420i	1.76313	+	0.944125i	−0.965926	−	0.258819i	1.47749	−	1.95372i	−2.36298	+	4.09280i	−2.09357	−	1.90183i	−2.08292	−	2.15903i	1.23589	+	0.687440i
11.8	−1.37101	−	0.346869i	−1.64895	−	0.530062i	1.75936	+	0.951126i	−0.965926	−	0.258819i	2.07687	+	1.29869i	0.779023	−	1.34931i	−2.08220	−	1.91428i	2.43807	+	1.74809i	1.23452	+	0.689895i
11.9	−1.35922	+	0.390551i	1.43839	−	0.964897i	1.69494	−	1.06169i	−0.965926	−	0.258819i	−1.57825	+	1.87327i	−1.27997	+	2.21698i	−1.88915	+	2.10502i	1.13795	−	2.77580i	1.41398	−	0.0254518i
11.10	−1.35814	−	0.394290i	1.72697	−	0.132597i	1.68907	+	1.07100i	0.965926	+	0.258819i	−2.39774	−	0.500841i	0.977504	−	1.69309i	−1.87170	−	2.12055i	2.96484	−	0.457982i	−1.20981	−	0.732367i
11.11	−1.35472	+	0.405858i	−0.711754	−	1.57905i	1.67056	−	1.09965i	0.965926	+	0.258819i	1.60510	+	1.85031i	1.79564	−	3.11014i	−1.81684	+	2.16774i	−1.98681	+	2.24779i	−1.41361	+	0.0414005i
11.12	−1.35406	+	0.408062i	1.72254	+	0.181237i	1.66697	−	1.10508i	0.965926	+	0.258819i	−2.40639	+	0.457497i	−1.88096	+	3.25792i	−1.80624	+	2.17658i	2.93431	+	0.624378i	−1.41354	+	0.0437002i
11.13	−1.35213	−	0.414413i	0.970230	−	1.43480i	1.65652	+	1.12068i	0.965926	+	0.258819i	−1.90648	+	1.53796i	−2.03154	+	3.51873i	−1.77541	−	2.20180i	−1.11731	−	2.78417i	−1.19880	−	0.750250i
11.14	−1.33386	+	0.469914i	0.806234	−	1.53297i	1.55836	−	1.25360i	0.965926	+	0.258819i	−0.355040	+	2.42362i	0.915344	−	1.58542i	−1.48955	+	2.40442i	−1.69997	−	2.47186i	−1.41003	+	0.108674i
11.15	−1.29240	−	0.574192i	1.44158	+	0.960133i	1.34061	+	1.48417i	−0.965926	−	0.258819i	−1.31180	−	2.06862i	−0.232701	+	0.403050i	−0.880402	−	2.68792i	1.15629	+	2.76821i	1.09975	+	0.889125i
11.16	−1.26298	+	0.636308i	0.418989	+	1.68061i	1.19023	−	1.60728i	−0.965926	−	0.258819i	−1.59856	−	1.85597i	−0.905428	+	1.56825i	−0.480500	+	2.78731i	−2.64890	+	1.40831i	1.38463	−	0.287743i
11.17	−1.24501	+	0.670782i	−0.782024	+	1.54546i	1.10010	−	1.67026i	−0.965926	−	0.258819i	−0.0630363	−	2.44868i	1.60665	−	2.78281i	−0.249259	+	2.81742i	−1.77688	−	2.41717i	1.37620	−	0.325693i
11.18	−1.18889	+	0.765853i	0.969209	+	1.43549i	0.826939	−	1.82104i	0.965926	+	0.258819i	−2.25166	−	0.964374i	2.31488	−	4.00948i	0.411503	+	2.79833i	−1.12127	+	2.78258i	−1.34660	+	0.432049i
11.19	−1.12949	+	0.851030i	−1.61143	−	0.635048i	0.551495	−	1.92246i	−0.965926	−	0.258819i	2.36054	−	0.654098i	1.14469	−	1.98265i	1.01316	+	2.64074i	2.19343	+	2.04667i	1.31127	−	0.529699i
11.20	−1.12540	−	0.856433i	0.847953	−	1.51029i	0.533043	+	1.92766i	−0.965926	−	0.258819i	−2.24775	+	0.973463i	−1.39447	+	2.41529i	1.05102	−	2.62590i	−1.56195	−	2.56131i	0.865390	+	1.11853i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
16.f	odd	4	1	inner
144.u	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.cf.a	✓	384
9.d	odd	6	1	inner	720.2.cf.a	✓	384
16.f	odd	4	1	inner	720.2.cf.a	✓	384
144.u	even	12	1	inner	720.2.cf.a	✓	384

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.2.cf.a	✓	384	1.a	even	1	1	trivial
720.2.cf.a	✓	384	9.d	odd	6	1	inner
720.2.cf.a	✓	384	16.f	odd	4	1	inner
720.2.cf.a	✓	384	144.u	even	12	1	inner

Hecke kernels

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