Newform orbit 555.2.bi.a

• Label: 555.2.bi.a
• Level: $555$
• Weight: $2$
• Character orbit: 555.bi
• Analytic conductor: $4.432$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 555 = 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	555.bi (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.43169731218\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 16 q^{6} - 4 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−0.730116	−	2.72483i	1.50999	−	0.848488i	−5.15958	+	2.97888i	1.73787	+	1.40706i	−3.41445	−	3.49497i	0.175375	−	0.654510i	7.89461	+	7.89461i	1.56013	−	2.56242i	2.56515	−	5.76271i
47.2	−0.702268	−	2.62090i	1.38524	+	1.03977i	−4.64388	+	2.68115i	−0.537633	−	2.17047i	1.75231	−	4.36077i	−0.703013	+	2.62368i	6.45101	+	6.45101i	0.837774	+	2.88065i	−5.31103	+	2.93334i
47.3	−0.658434	−	2.45731i	−0.842875	−	1.51313i	−3.87278	+	2.23595i	1.16787	−	1.90685i	−3.16325	+	3.06750i	1.02555	−	3.82742i	4.44664	+	4.44664i	−1.57912	+	2.55076i	−5.45469	−	1.61428i
47.4	−0.650943	−	2.42935i	−0.714457	−	1.57783i	−3.74597	+	2.16274i	−2.20042	−	0.397657i	−3.36804	+	2.76274i	−0.786417	+	2.93495i	4.13564	+	4.13564i	−1.97910	+	2.25458i	0.466301	+	5.60446i
47.5	−0.640755	−	2.39133i	−1.57649	−	0.717406i	−3.57584	+	2.06451i	−0.456366	+	2.18900i	−0.705409	+	4.22959i	0.495613	−	1.84965i	3.72701	+	3.72701i	1.97066	+	2.26197i	5.52704	−	0.311292i
47.6	−0.638472	−	2.38281i	−0.357768	+	1.69470i	−3.53808	+	2.04271i	2.19081	+	0.447615i	4.26657	−	0.229525i	0.482533	−	1.80084i	3.63769	+	3.63769i	−2.74400	−	1.21262i	−0.332189	−	5.50607i
47.7	−0.629578	−	2.34962i	0.870003	+	1.49770i	−3.39227	+	1.95853i	−1.82936	+	1.28586i	2.97128	−	2.98709i	0.920063	−	3.43372i	3.29742	+	3.29742i	−1.48619	+	2.60600i	4.17300	+	3.48875i
47.8	−0.627819	−	2.34305i	−1.27383	+	1.17361i	−3.36369	+	1.94203i	−0.905250	−	2.04463i	3.54956	+	2.24785i	0.577617	−	2.15570i	3.23160	+	3.23160i	0.245299	−	2.98995i	−4.22235	+	3.40471i
47.9	−0.583531	−	2.17777i	1.41634	−	0.996979i	−2.67011	+	1.54159i	−1.71736	−	1.43202i	−2.99767	−	2.50270i	0.265798	−	0.991970i	1.72685	+	1.72685i	1.01207	−	2.82413i	−2.11647	+	4.57564i
47.10	−0.578972	−	2.16075i	−0.231467	−	1.71651i	−2.60160	+	1.50203i	2.08250	+	0.814371i	−3.57495	+	1.49396i	−1.33384	+	4.97795i	1.58821	+	1.58821i	−2.89285	+	0.794631i	0.553946	−	4.97126i
47.11	−0.538717	−	2.01052i	1.73189	+	0.0235861i	−2.01992	+	1.16620i	−1.23915	+	1.86132i	−0.885578	−	3.49470i	−0.789178	+	2.94525i	0.489232	+	0.489232i	2.99889	+	0.0816971i	4.40977	+	1.48861i
47.12	−0.507879	−	1.89543i	1.35688	+	1.07651i	−1.60266	+	0.925299i	2.11640	−	0.721709i	1.35132	−	3.11861i	0.0592084	−	0.220969i	−0.207304	−	0.207304i	0.682256	+	2.92139i	−2.44282	−	3.64494i
47.13	−0.506687	−	1.89098i	−1.68229	+	0.412189i	−1.58703	+	0.916272i	1.28315	+	1.83126i	1.63184	+	2.97233i	−0.202179	+	0.754544i	−0.231809	−	0.231809i	2.66020	−	1.38684i	2.81272	−	3.35430i
47.14	−0.503825	−	1.88030i	1.07495	−	1.35811i	−1.54964	+	0.894683i	1.41979	−	1.72748i	−3.09525	−	1.33698i	0.0498104	−	0.185895i	−0.289933	−	0.289933i	−0.688949	−	2.91982i	−3.96351	−	1.79929i
47.15	−0.483150	−	1.80314i	−1.68112	−	0.416918i	−1.28583	+	0.742372i	−1.47075	−	1.68431i	0.0604730	+	3.23274i	−0.657877	+	2.45523i	−0.680132	−	0.680132i	2.65236	+	1.40178i	−2.32645	+	3.46574i
47.16	−0.470395	−	1.75554i	−0.342400	+	1.69787i	−1.12860	+	0.651596i	0.701344	−	2.12323i	3.14174	−	0.197573i	−1.04052	+	3.88326i	−0.895500	−	0.895500i	−2.76552	−	1.16270i	−4.05733	−	0.232478i
47.17	−0.457681	−	1.70809i	−1.20821	+	1.24106i	−0.976041	+	0.563518i	−2.00778	+	0.984280i	2.67281	+	1.49573i	0.0854025	−	0.318727i	−1.09156	−	1.09156i	−0.0804428	−	2.99892i	2.60016	+	2.97898i
47.18	−0.427747	−	1.59638i	0.651239	−	1.60496i	−0.633395	+	0.365691i	−0.687583	+	2.12773i	−2.84068	−	0.353105i	−0.216635	+	0.808491i	−1.48254	−	1.48254i	−2.15178	−	2.09042i	3.69076	+	0.187511i
47.19	−0.401669	−	1.49905i	1.72836	−	0.113060i	−0.353763	+	0.204245i	1.14612	+	1.92000i	−0.863710	−	2.54548i	1.31722	−	4.91592i	−1.74649	−	1.74649i	2.97444	−	0.390814i	2.41782	−	2.48930i
47.20	−0.359567	−	1.34192i	−0.149542	−	1.72558i	0.0605850	−	0.0349788i	−2.09743	−	0.775107i	−2.26183	+	0.821137i	1.02636	−	3.83043i	−2.03343	−	2.03343i	−2.95527	+	0.516095i	−0.285968	+	3.09329i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
37.c	even	3	1	inner
111.i	odd	6	1	inner
185.s	odd	12	1	inner
555.bi	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	555.2.bi.a	✓	288
3.b	odd	2	1	inner	555.2.bi.a	✓	288
5.c	odd	4	1	inner	555.2.bi.a	✓	288
15.e	even	4	1	inner	555.2.bi.a	✓	288
37.c	even	3	1	inner	555.2.bi.a	✓	288
111.i	odd	6	1	inner	555.2.bi.a	✓	288
185.s	odd	12	1	inner	555.2.bi.a	✓	288
555.bi	even	12	1	inner	555.2.bi.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
555.2.bi.a	✓	288	1.a	even	1	1	trivial
555.2.bi.a	✓	288	3.b	odd	2	1	inner
555.2.bi.a	✓	288	5.c	odd	4	1	inner
555.2.bi.a	✓	288	15.e	even	4	1	inner
555.2.bi.a	✓	288	37.c	even	3	1	inner
555.2.bi.a	✓	288	111.i	odd	6	1	inner
555.2.bi.a	✓	288	185.s	odd	12	1	inner
555.2.bi.a	✓	288	555.bi	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).