Newform orbit 432.2.bj.a

• Label: 432.2.bj.a
• Level: $432$
• Weight: $2$
• Character orbit: 432.bj
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.bj (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41347	−	0.0457693i	1.41081	+	1.00480i	1.99581	+	0.129387i	−0.451047	+	0.210327i	−1.94815	−	1.48483i	0.650509	−	3.68922i	−2.81510	−	0.274232i	0.980759	+	2.83516i	0.647170	−	0.276647i
11.2	−1.41262	+	0.0671088i	−1.64576	−	0.539886i	1.99099	−	0.189599i	−2.14457	+	1.00003i	2.36106	+	0.652210i	−0.446923	+	2.53462i	−2.79979	+	0.401444i	2.41705	+	1.77705i	2.96236	−	1.55658i
11.3	−1.41099	−	0.0954482i	1.71137	−	0.266838i	1.98178	+	0.269353i	−2.91453	+	1.35907i	−2.44020	+	0.213158i	−0.433865	+	2.46057i	−2.77056	−	0.569211i	2.85760	−	0.913318i	4.24209	−	1.63944i
11.4	−1.40904	−	0.120812i	−0.646554	+	1.60685i	1.97081	+	0.340458i	1.63314	−	0.761545i	1.10515	−	2.18601i	−0.638985	+	3.62387i	−2.73582	−	0.717817i	−2.16394	−	2.07783i	−2.39317	+	0.875748i
11.5	−1.40536	−	0.158024i	−1.72785	+	0.120515i	1.95006	+	0.444159i	3.64812	−	1.70115i	2.44729	+	0.103674i	0.736424	−	4.17647i	−2.67034	−	0.932357i	2.97095	−	0.416466i	−5.39573	+	1.81423i
11.6	−1.36790	−	0.358938i	0.748873	−	1.56179i	1.74233	+	0.981986i	1.61065	−	0.751058i	−1.58497	+	1.86758i	−0.149029	+	0.845186i	−2.03087	−	1.96865i	−1.87838	−	2.33917i	−2.47280	+	0.449253i
11.7	−1.36652	+	0.364177i	−0.589257	−	1.62873i	1.73475	−	0.995311i	−2.14350	+	0.999531i	1.39838	+	2.01110i	0.824762	−	4.67746i	−2.00810	+	1.99187i	−2.30555	+	1.91949i	2.56513	−	2.14649i
11.8	−1.34888	+	0.424869i	1.40837	+	1.00821i	1.63897	−	1.14620i	1.83680	−	0.856513i	−2.32808	−	0.761591i	−0.211132	+	1.19739i	−1.72380	+	2.24243i	0.967010	+	2.83988i	−2.11372	+	1.93573i
11.9	−1.33243	+	0.473947i	1.23188	−	1.21757i	1.55075	−	1.26300i	2.78882	−	1.30045i	−1.06433	+	2.20617i	0.243644	−	1.38177i	−1.46767	+	2.41784i	0.0350450	−	2.99980i	−3.09957	+	3.05451i
11.10	−1.32255	−	0.500851i	−1.23524	+	1.21416i	1.49830	+	1.32480i	−2.38847	+	1.11376i	2.24179	−	0.987116i	0.330679	−	1.87537i	−1.31805	−	2.50255i	0.0516452	−	2.99956i	3.71671	−	0.276743i
11.11	−1.25692	−	0.648191i	−0.892926	−	1.48414i	1.15970	+	1.62945i	0.756449	−	0.352738i	0.160328	+	2.44424i	−0.222648	+	1.26270i	−0.401452	−	2.79979i	−1.40537	+	2.65046i	−1.17944	−	0.0469600i
11.12	−1.25580	+	0.650350i	−0.416170	+	1.68131i	1.15409	−	1.63343i	0.706106	−	0.329263i	−0.570812	−	2.38205i	0.241012	−	1.36684i	−0.387011	+	2.80182i	−2.65360	−	1.39942i	−0.672595	+	0.872706i
11.13	−1.17422	+	0.788161i	0.237306	−	1.71572i	0.757603	−	1.85096i	−0.814785	+	0.379940i	1.07361	+	2.20167i	−0.728449	+	4.13124i	0.569257	+	2.77055i	−2.88737	−	0.814300i	0.657285	−	1.08832i
11.14	−1.12355	−	0.858857i	0.837200	+	1.51628i	0.524731	+	1.92994i	−1.42689	+	0.665372i	0.361629	−	2.42265i	−0.255884	+	1.45119i	1.06798	−	2.61905i	−1.59819	+	2.53885i	2.17465	+	0.477919i
11.15	−1.11141	+	0.874510i	−1.49998	−	0.866058i	0.470465	−	1.94388i	2.31389	−	1.07899i	2.42447	−	0.349202i	−0.254246	+	1.44190i	1.17706	+	2.57187i	1.49989	+	2.59814i	−1.62810	+	3.22272i
11.16	−1.09183	−	0.898834i	0.592090	−	1.62771i	0.384194	+	1.96275i	−2.62704	+	1.22501i	−2.10950	+	1.24499i	0.576268	−	3.26818i	1.34471	−	2.48832i	−2.29886	−	1.92750i	3.96937	+	1.02377i
11.17	−1.09036	+	0.900614i	−1.63814	+	0.562587i	0.377790	−	1.96399i	−1.73541	+	0.809237i	1.27949	−	2.08875i	0.446922	−	2.53462i	1.35687	+	2.48171i	2.36699	−	1.84319i	1.16342	−	2.44530i
11.18	−1.03330	+	0.965550i	1.61462	−	0.626890i	0.135428	−	1.99541i	−2.00571	+	0.935279i	−1.06310	+	2.20677i	0.295687	−	1.67692i	1.78673	+	2.19262i	2.21402	−	2.02438i	1.16945	−	2.90304i
11.19	−0.956562	−	1.04163i	−1.71601	+	0.235149i	−0.169980	+	1.99276i	1.98476	−	0.925510i	1.88641	+	1.56252i	−0.864532	+	4.90300i	2.23832	−	1.72915i	2.88941	−	0.807036i	−2.86258	−	1.18208i
11.20	−0.948532	−	1.04895i	1.73143	−	0.0462528i	−0.200575	+	1.98992i	1.65781	−	0.773051i	−1.69084	−	1.77231i	0.761547	−	4.31895i	2.27757	−	1.67711i	2.99572	−	0.160167i	−2.38338	−	1.00569i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner
27.f	odd	18	1	inner
432.bj	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	432.2.bj.a	✓	840
16.f	odd	4	1	inner	432.2.bj.a	✓	840
27.f	odd	18	1	inner	432.2.bj.a	✓	840
432.bj	even	36	1	inner	432.2.bj.a	✓	840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
432.2.bj.a	✓	840	1.a	even	1	1	trivial
432.2.bj.a	✓	840	16.f	odd	4	1	inner
432.2.bj.a	✓	840	27.f	odd	18	1	inner
432.2.bj.a	✓	840	432.bj	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(432, [\chi])\).