Newform orbit 693.2.ce.a

• Label: 693.2.ce.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.ce
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $736$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.ce (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(736\)
Relative dimension:	\(92\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 9 q^{2} - 9 q^{3} - 85 q^{4} - 18 q^{5} + 12 q^{6} - 3 q^{7} + 3 q^{9}+O(q^{10}) \)

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	−1.13367	+	2.54627i	1.22240	+	1.22709i	−3.86000	−	4.28697i	−0.771567	−	0.560576i	−4.51029	+	1.72144i	1.41545	+	2.23528i	9.99010	−	3.24598i	−0.0114824	+	2.99998i	2.30208	−	1.32911i
47.2	−1.10695	+	2.48625i	−1.24302	+	1.20620i	−3.61784	−	4.01801i	2.39965	+	1.74345i	−1.62295	−	4.42565i	0.0922054	−	2.64414i	8.81788	−	2.86510i	0.0901761	−	2.99864i	−6.99094	+	4.03622i
47.3	−1.08752	+	2.44261i	0.773505	−	1.54974i	−3.44538	−	3.82648i	1.46082	+	1.06134i	2.94420	+	3.57474i	1.06128	−	2.42357i	8.00770	−	2.60186i	−1.80338	−	2.39746i	−4.18111	+	2.41397i
47.4	−1.08287	+	2.43218i	−0.0800370	−	1.73020i	−3.40460	−	3.78119i	−1.76825	−	1.28471i	4.29482	+	1.67893i	−0.976176	+	2.45908i	7.81920	−	2.54061i	−2.98719	+	0.276960i	5.03942	−	2.90951i
47.5	−1.06804	+	2.39885i	−1.51269	−	0.843665i	−3.27552	−	3.63783i	−1.84635	−	1.34145i	3.63944	−	2.72765i	2.38282	−	1.14986i	7.23030	−	2.34927i	1.57646	+	2.55241i	5.18992	−	2.99640i
47.6	−1.06346	+	2.38858i	−1.02741	+	1.39442i	−3.23610	−	3.59405i	−2.63685	−	1.91578i	−2.23808	−	3.93698i	−2.64483	−	0.0699196i	7.05285	−	2.29161i	−0.888838	−	2.86530i	7.38021	−	4.26096i
47.7	−1.05030	+	2.35901i	1.71966	+	0.206767i	−3.12355	−	3.46905i	−0.650624	−	0.472706i	−2.29393	+	3.83954i	−2.19806	−	1.47259i	6.55245	−	2.12902i	2.91450	+	0.711138i	1.79847	−	1.03835i
47.8	−1.00648	+	2.26059i	−1.73204	+	0.00690604i	−2.75902	−	3.06420i	1.91344	+	1.39020i	1.72765	−	3.92238i	−0.0265246	+	2.64562i	4.99699	−	1.62362i	2.99990	−	0.0239230i	−5.06851	+	2.92631i
47.9	−0.916118	+	2.05763i	1.27497	+	1.17237i	−2.05633	−	2.28378i	3.51611	+	2.55461i	−3.58033	+	1.54940i	−2.54671	+	0.717121i	2.29878	−	0.746918i	0.251108	+	2.98947i	−8.47762	+	4.89455i
47.10	−0.901969	+	2.02586i	−0.365336	−	1.69308i	−1.95228	−	2.16823i	3.22480	+	2.34296i	3.75946	+	0.786990i	2.30234	+	1.30354i	1.93534	−	0.628830i	−2.73306	+	1.23709i	−7.65516	+	4.41971i
47.11	−0.900181	+	2.02184i	0.650422	+	1.60529i	−1.93924	−	2.15375i	−1.38666	−	1.00747i	−3.83113	−	0.130002i	0.464064	−	2.60474i	1.89049	−	0.614257i	−2.15390	+	2.08823i	3.28519	−	1.89671i
47.12	−0.899019	+	2.01923i	1.63625	−	0.568049i	−1.93079	−	2.14436i	0.975409	+	0.708676i	−0.324001	+	3.81466i	0.548462	+	2.58828i	1.86150	−	0.604839i	2.35464	−	1.85894i	−2.30789	+	1.33246i
47.13	−0.898665	+	2.01843i	−0.796142	−	1.53823i	−1.92822	−	2.14150i	0.701857	+	0.509929i	3.82028	−	0.224606i	−2.48052	−	0.920348i	1.85267	−	0.601969i	−1.73232	+	2.44930i	−1.65999	+	0.958396i
47.14	−0.891575	+	2.00251i	0.831250	−	1.51955i	−1.87688	−	2.08449i	−2.65706	−	1.93047i	2.30179	+	3.01938i	−0.788919	−	2.52539i	1.67811	−	0.545252i	−1.61805	−	2.52625i	6.23474	−	3.59963i
47.15	−0.862246	+	1.93664i	−0.0654541	+	1.73081i	−1.66883	−	1.85343i	1.08719	+	0.789893i	−3.29552	−	1.61915i	−2.40609	+	1.10034i	0.996041	−	0.323633i	−2.99143	−	0.226578i	−2.46716	+	1.42442i
47.16	−0.855224	+	1.92087i	1.71988	−	0.204970i	−1.62005	−	1.79925i	1.07493	+	0.780984i	−1.07716	+	3.47895i	2.44426	−	1.01271i	0.842157	−	0.273633i	2.91597	−	0.705047i	−2.41947	+	1.39688i
47.17	−0.854218	+	1.91860i	−1.70191	+	0.321739i	−1.61309	−	1.79152i	0.167832	+	0.121937i	0.836508	−	3.54012i	−0.290695	+	2.62973i	0.820384	−	0.266559i	2.79297	−	1.09514i	−0.377313	+	0.217842i
47.18	−0.814797	+	1.83006i	−0.865297	+	1.50042i	−1.34698	−	1.49597i	−3.15717	−	2.29381i	−2.04082	−	2.80609i	1.95329	+	1.78456i	0.0248130	−	0.00806223i	−1.50252	−	2.59662i	6.77027	−	3.90882i
47.19	−0.767741	+	1.72438i	1.59234	−	0.681504i	−1.04578	−	1.16146i	−2.69228	−	1.95606i	−0.0473377	+	3.26901i	−1.84582	+	1.89551i	−0.784676	+	0.254957i	2.07110	−	2.17037i	5.43995	−	3.14076i
47.20	−0.763341	+	1.71449i	−1.55715	−	0.758467i	−1.01853	−	1.13119i	2.00621	+	1.45760i	2.48902	−	2.09076i	−0.212305	−	2.63722i	−0.852875	+	0.277116i	1.84945	+	2.36210i	−4.03046	+	2.32699i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
63.s	even	6	1	inner
693.ce	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.ce.a	✓	736
7.d	odd	6	1		693.2.db.a	yes	736
9.d	odd	6	1		693.2.db.a	yes	736
11.c	even	5	1	inner	693.2.ce.a	✓	736
63.s	even	6	1	inner	693.2.ce.a	✓	736
77.p	odd	30	1		693.2.db.a	yes	736
99.n	odd	30	1		693.2.db.a	yes	736
693.ce	even	30	1	inner	693.2.ce.a	✓	736

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.ce.a	✓	736	1.a	even	1	1	trivial
693.2.ce.a	✓	736	11.c	even	5	1	inner
693.2.ce.a	✓	736	63.s	even	6	1	inner
693.2.ce.a	✓	736	693.ce	even	30	1	inner
693.2.db.a	yes	736	7.d	odd	6	1
693.2.db.a	yes	736	9.d	odd	6	1
693.2.db.a	yes	736	77.p	odd	30	1
693.2.db.a	yes	736	99.n	odd	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).