Newform orbit 555.2.cc.a

• Label: 555.2.cc.a
• Level: $555$
• Weight: $2$
• Character orbit: 555.cc
• Analytic conductor: $4.432$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 555 = 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	555.cc (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.43169731218\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(72\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 864 q - 18 q^{3} - 48 q^{6} - 24 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.16749	−	2.50369i	−1.61656	−	0.621882i	−3.61985	+	4.31397i	1.58461	+	1.57766i	0.330317	+	4.77340i	−0.439316	−	0.627408i	9.69020	+	2.59648i	2.22653	+	2.01062i	2.09995	−	5.80927i
53.2	−1.16534	−	2.49909i	0.673302	−	1.59583i	−3.60185	+	4.29251i	−1.69798	+	1.45495i	−4.77274	+	0.177047i	2.04917	+	2.92652i	9.59780	+	2.57172i	−2.09333	−	2.14895i	5.61477	+	2.54789i
53.3	−1.13132	−	2.42611i	−1.30276	+	1.14141i	−3.32058	+	3.95731i	−0.441655	−	2.19202i	4.24303	+	1.86934i	0.715683	+	1.02210i	8.18608	+	2.19345i	0.394354	−	2.97397i	−4.81843	+	3.55137i
53.4	−1.10675	−	2.37344i	1.71580	−	0.236719i	−3.12272	+	3.72152i	−1.52163	−	1.63849i	−2.46080	−	3.81035i	−1.54122	−	2.20109i	7.22973	+	1.93720i	2.88793	−	0.812326i	−2.20478	+	5.42489i
53.5	−1.06683	−	2.28783i	1.20397	+	1.24518i	−2.81046	+	3.34938i	1.80201	−	1.32391i	1.56432	−	4.08287i	1.17611	+	1.67967i	5.78445	+	1.54994i	−0.100933	+	2.99830i	−4.95134	−	2.71031i
53.6	−0.988252	−	2.11931i	−1.10763	+	1.33160i	−2.22927	+	2.65674i	−1.87675	+	1.21566i	3.91669	+	1.03146i	−2.55841	−	3.65379i	3.31610	+	0.888547i	−0.546305	−	2.94984i	4.43106	+	2.77604i
53.7	−0.985998	−	2.11448i	1.45178	−	0.944637i	−2.21326	+	2.63766i	2.18739	+	0.464047i	−3.42887	−	2.13834i	−1.16748	−	1.66734i	3.25239	+	0.871475i	1.21532	−	2.74281i	−1.17554	−	5.08273i
53.8	−0.965213	−	2.06991i	−0.219008	+	1.71815i	−2.06730	+	2.46371i	0.389364	+	2.20191i	3.76780	−	1.20505i	2.13886	+	3.05461i	2.68290	+	0.718880i	−2.90407	−	0.752576i	4.18192	−	2.93126i
53.9	−0.940837	−	2.01763i	−1.47382	−	0.909870i	−1.90008	+	2.26443i	−2.10130	+	0.764539i	−0.449161	+	3.82966i	−1.13036	−	1.61432i	2.05575	+	0.550837i	1.34427	+	2.68196i	3.51954	+	3.52035i
53.10	−0.936649	−	2.00865i	−1.35255	−	1.08195i	−1.87179	+	2.23071i	−1.24170	−	1.85962i	−0.906393	+	3.73020i	1.93449	+	2.76274i	1.95236	+	0.523133i	0.658775	+	2.92678i	−2.57228	+	4.23595i
53.11	−0.907611	−	1.94638i	0.996813	+	1.41646i	−1.67905	+	2.00102i	−2.23607	−	0.00211852i	1.85225	−	3.22577i	−0.186940	−	0.266978i	1.26983	+	0.340251i	−1.01273	+	2.82390i	2.02535	+	4.35415i
53.12	−0.900912	−	1.93201i	1.71207	+	0.262335i	−1.63545	+	1.94906i	−0.520545	+	2.17463i	−1.03559	−	3.54408i	0.242575	+	0.346433i	1.12080	+	0.300317i	2.86236	+	0.898271i	4.67038	−	0.953456i
53.13	−0.831909	−	1.78404i	−0.207279	−	1.71960i	−1.20513	+	1.43622i	0.471453	+	2.18580i	−2.89540	+	1.80035i	−1.63411	−	2.33376i	−0.237951	−	0.0637587i	−2.91407	+	0.712877i	3.50734	−	2.65948i
53.14	−0.808170	−	1.73313i	1.06686	−	1.36448i	−1.06501	+	1.26923i	0.801198	−	2.08760i	−3.22702	−	0.746279i	2.29342	+	3.27535i	−0.633824	−	0.169833i	−0.723603	−	2.91143i	−4.26558	+	0.298560i
53.15	−0.799109	−	1.71370i	−1.64819	+	0.532407i	−1.01260	+	1.20677i	2.23577	−	0.0368002i	2.22947	+	2.39905i	0.898316	+	1.28293i	−0.775631	−	0.207830i	2.43309	−	1.75502i	−1.84969	−	3.80201i
53.16	−0.769311	−	1.64979i	−1.71208	−	0.262258i	−0.844403	+	1.00632i	1.11334	−	1.93920i	0.884451	+	3.02634i	−2.34399	−	3.34756i	−1.20681	−	0.323363i	2.86244	+	0.898015i	−4.05578	−	0.344928i
53.17	−0.718080	−	1.53993i	0.277771	+	1.70963i	−0.570163	+	0.679494i	−0.149554	−	2.23106i	2.43325	−	1.65540i	−1.65121	−	2.35817i	−1.82666	−	0.489452i	−2.84569	+	0.949772i	−3.32828	+	1.83238i
53.18	−0.716262	−	1.53603i	−0.656449	−	1.60283i	−0.560778	+	0.668309i	1.82552	+	1.29131i	−1.99181	+	2.15637i	2.41139	+	3.44382i	−1.84594	−	0.494617i	−2.13815	+	2.10436i	0.675935	−	3.72897i
53.19	−0.618669	−	1.32674i	1.05504	−	1.37364i	−0.0919102	+	0.109534i	−0.785359	−	2.09361i	−2.47519	−	0.549927i	−1.68195	−	2.40208i	−2.62584	−	0.703592i	−0.773797	−	2.89849i	−2.29180	+	2.33722i
53.20	−0.613853	−	1.31641i	0.683226	−	1.59160i	−0.0705488	+	0.0840768i	−2.03880	+	0.918302i	−2.51461	+	0.0776030i	0.131992	+	0.188504i	−2.65203	−	0.710609i	−2.06640	−	2.17485i	2.46039	+	2.12020i

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.f	even	9	1	inner
111.p	odd	18	1	inner
185.bd	odd	36	1	inner
555.cc	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	555.2.cc.a	✓	864
3.b	odd	2	1	inner	555.2.cc.a	✓	864
5.c	odd	4	1	inner	555.2.cc.a	✓	864
15.e	even	4	1	inner	555.2.cc.a	✓	864
37.f	even	9	1	inner	555.2.cc.a	✓	864
111.p	odd	18	1	inner	555.2.cc.a	✓	864
185.bd	odd	36	1	inner	555.2.cc.a	✓	864
555.cc	even	36	1	inner	555.2.cc.a	✓	864

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
555.2.cc.a	✓	864	1.a	even	1	1	trivial
555.2.cc.a	✓	864	3.b	odd	2	1	inner
555.2.cc.a	✓	864	5.c	odd	4	1	inner
555.2.cc.a	✓	864	15.e	even	4	1	inner
555.2.cc.a	✓	864	37.f	even	9	1	inner
555.2.cc.a	✓	864	111.p	odd	18	1	inner
555.2.cc.a	✓	864	185.bd	odd	36	1	inner
555.2.cc.a	✓	864	555.cc	even	36	1	inner

Hecke kernels

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