Newform orbit 385.2.bu.a

• Label: 385.2.bu.a
• Level: $385$
• Weight: $2$
• Character orbit: 385.bu
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $704$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.bu (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(704\)
Relative dimension:	\(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 704 q - 6 q^{2} - 18 q^{3} - 30 q^{5} - 10 q^{7} - 8 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.56734	+	0.985509i	0.638433	−	0.414603i	4.13371	−	3.72201i	2.19449	+	0.429203i	−1.23048	+	1.69361i	−1.10595	−	2.40351i	−4.44762	+	8.72895i	−0.984509	+	2.21124i	−6.05698	+	1.06078i
3.2	−2.53871	+	0.974519i	2.00766	−	1.30379i	4.00907	−	3.60978i	−2.21413	−	0.312475i	−3.82630	+	5.26645i	−1.91643	+	1.82408i	−4.19096	+	8.22522i	1.11063	−	2.49451i	5.92554	−	1.36443i
3.3	−2.34383	+	0.899710i	−2.46734	+	1.60231i	3.19775	−	2.87927i	−0.883059	−	2.05431i	4.34140	−	5.97543i	−1.90881	−	1.83206i	−2.62491	+	5.15167i	2.30016	−	5.16625i	3.91802	+	4.02046i
3.4	−2.26308	+	0.868713i	−1.40298	+	0.911106i	2.88056	−	2.59367i	−2.20498	+	0.371567i	2.38356	−	3.28069i	2.54410	+	0.726338i	−2.06475	+	4.05229i	−0.0819700	+	0.184108i	4.66725	−	2.75638i
3.5	−2.22696	+	0.854851i	0.560315	−	0.363873i	2.74230	−	2.46918i	0.162798	+	2.23013i	−0.936743	+	1.28932i	2.14209	+	1.55288i	−1.83032	+	3.59221i	−1.03866	+	2.33287i	−2.26898	−	4.82726i
3.6	−2.12155	+	0.814386i	−1.95784	+	1.27144i	2.35145	−	2.11726i	0.483951	+	2.18307i	3.11822	−	4.29186i	−2.26181	+	1.37266i	−1.20108	+	2.35726i	0.996386	−	2.23792i	−2.80459	−	4.23737i
3.7	−2.09118	+	0.802728i	0.286091	−	0.185790i	2.24236	−	2.01903i	0.164779	−	2.22999i	−0.449129	+	0.618173i	1.39842	−	2.24598i	−1.03461	+	2.03053i	−1.17288	+	2.63433i	1.44549	+	4.79557i
3.8	−1.91771	+	0.736142i	2.14393	−	1.39228i	1.64943	−	1.48516i	0.929098	−	2.03391i	−3.08652	+	4.24823i	1.72482	+	2.00624i	−0.204729	+	0.401803i	1.43776	−	3.22926i	−0.284501	+	4.58440i
3.9	−1.70354	+	0.653929i	0.387441	−	0.251607i	0.988145	−	0.889730i	−2.21985	+	0.268845i	−0.495489	+	0.681982i	−2.33291	−	1.24800i	0.555303	−	1.08984i	−1.13341	+	2.54567i	3.60580	−	1.90961i
3.10	−1.62535	+	0.623912i	2.69986	−	1.75331i	0.766195	−	0.689885i	2.18730	+	0.464466i	−3.29430	+	4.53422i	−1.99433	−	1.73857i	0.765873	−	1.50311i	2.99495	−	6.72678i	−3.84490	+	0.609764i
3.11	−1.48586	+	0.570367i	−1.99706	+	1.29690i	0.396160	−	0.356704i	1.08446	+	1.95549i	2.22763	−	3.06606i	0.454341	−	2.60645i	1.05993	−	2.08023i	1.08606	−	2.43934i	−2.72670	−	2.28704i
3.12	−1.33588	+	0.512796i	−0.836856	+	0.543460i	0.0353231	−	0.0318051i	−1.84280	−	1.26652i	0.839254	−	1.15513i	0.199364	+	2.63823i	1.26837	−	2.48932i	−0.815232	+	1.83104i	3.11123	+	0.746943i
3.13	−1.31895	+	0.506299i	−0.825752	+	0.536250i	−0.00298924	+	0.00269153i	2.11719	−	0.719392i	0.817626	−	1.12537i	−2.50450	+	0.852923i	1.28537	−	2.52267i	−0.825907	+	1.85502i	−2.42824	+	2.02077i
3.14	−1.28592	+	0.493617i	0.655666	−	0.425795i	−0.0763673	+	0.0687615i	−0.835203	+	2.07423i	−0.632952	+	0.871184i	0.285557	−	2.63030i	1.31492	−	2.58067i	−0.971613	+	2.18228i	0.0501245	−	3.07956i
3.15	−1.01764	+	0.390634i	0.704903	−	0.457769i	−0.603301	+	0.543214i	2.09076	+	0.792913i	−0.538515	+	0.741202i	2.38921	+	1.13653i	1.39147	−	2.73092i	−0.932874	+	2.09527i	−2.43738	+	0.00982560i
3.16	−0.984404	+	0.377877i	−2.81619	+	1.82886i	−0.660030	+	0.594294i	0.567978	−	2.16273i	2.08119	−	2.86451i	1.29548	+	2.30689i	1.38258	−	2.71346i	3.36602	−	7.56021i	0.258127	+	2.34362i
3.17	−0.977612	+	0.375270i	2.17839	−	1.41467i	−0.671392	+	0.604524i	−1.30680	+	1.81447i	−1.59874	+	2.20048i	−1.14215	+	2.38652i	1.38031	−	2.70900i	1.52391	−	3.42276i	0.596624	−	2.26425i
3.18	−0.591629	+	0.227105i	2.27192	−	1.47540i	−1.18784	+	1.06954i	−2.13730	−	0.657215i	−1.00906	+	1.38886i	2.64389	−	0.0991708i	1.03527	−	2.03183i	1.76460	−	3.96336i	1.41375	−	0.0965653i
3.19	−0.481580	+	0.184861i	−0.422474	+	0.274358i	−1.28854	+	1.16021i	−0.903391	−	2.04545i	0.152737	−	0.210224i	2.13334	−	1.56488i	0.874433	−	1.71617i	−1.11700	+	2.50882i	0.813180	+	0.818047i
3.20	−0.361507	+	0.138769i	−1.15726	+	0.751536i	−1.37486	+	1.23793i	−1.22428	+	1.87113i	0.314069	−	0.432278i	−1.02244	+	2.44021i	0.676828	−	1.32835i	−0.445756	+	1.00119i	0.182928	−	0.846320i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.d	odd	6	1	inner
11.c	even	5	1	inner
35.k	even	12	1	inner
55.k	odd	20	1	inner
77.p	odd	30	1	inner
385.bu	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	385.2.bu.a	✓	704
5.c	odd	4	1	inner	385.2.bu.a	✓	704
7.d	odd	6	1	inner	385.2.bu.a	✓	704
11.c	even	5	1	inner	385.2.bu.a	✓	704
35.k	even	12	1	inner	385.2.bu.a	✓	704
55.k	odd	20	1	inner	385.2.bu.a	✓	704
77.p	odd	30	1	inner	385.2.bu.a	✓	704
385.bu	even	60	1	inner	385.2.bu.a	✓	704

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
385.2.bu.a	✓	704	1.a	even	1	1	trivial
385.2.bu.a	✓	704	5.c	odd	4	1	inner
385.2.bu.a	✓	704	7.d	odd	6	1	inner
385.2.bu.a	✓	704	11.c	even	5	1	inner
385.2.bu.a	✓	704	35.k	even	12	1	inner
385.2.bu.a	✓	704	55.k	odd	20	1	inner
385.2.bu.a	✓	704	77.p	odd	30	1	inner
385.2.bu.a	✓	704	385.bu	even	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(385, [\chi])\).