Newform orbit 693.2.p.a

• Label: 693.2.p.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.p
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 6 q^{3} - 180 q^{4} - 6 q^{5} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
241.1		−	2.77664i	−0.321710	+	1.70191i	−5.70973			0.339756	+	0.196158i	4.72559	+	0.893271i	1.49184	+	2.18505i			10.3006i	−2.79301	−	1.09504i	0.544661	−	0.943380i
241.2		−	2.71760i	1.54698	−	0.779003i	−5.38536			−1.18511	−	0.684225i	−2.11702	−	4.20408i	−2.63166	+	0.272734i			9.20005i	1.78631	−	2.41021i	−1.85945	+	3.22066i
241.3		−	2.70840i	0.911806	−	1.47262i	−5.33543			3.32943	+	1.92225i	−3.98844	−	2.46954i	1.35128	−	2.27465i			9.03367i	−1.33722	−	2.68549i	5.20621	−	9.01742i
241.4		−	2.66202i	−1.72938	−	0.0961242i	−5.08634			−1.42552	−	0.823025i	−0.255884	+	4.60364i	2.64113	−	0.156265i			8.21589i	2.98152	+	0.332471i	−2.19091	+	3.79476i
241.5		−	2.52678i	−1.03594	−	1.38810i	−4.38461			2.85112	+	1.64609i	−3.50743	+	2.61759i	−0.162671	+	2.64075i			6.02539i	−0.853668	+	2.87598i	4.15931	−	7.20414i
241.6		−	2.52360i	−0.177604	−	1.72292i	−4.36856			−1.73874	−	1.00386i	−4.34796	+	0.448202i	−0.438353	+	2.60919i			5.97730i	−2.93691	+	0.611996i	−2.53335	+	4.38789i
241.7		−	2.51008i	1.04986	+	1.37760i	−4.30049			2.50979	+	1.44903i	3.45789	−	2.63523i	−2.45158	+	0.994867i			5.77442i	−0.795582	+	2.89259i	3.63718	−	6.29977i
241.8		−	2.44978i	−1.45044	+	0.946692i	−4.00141			−3.18089	−	1.83649i	2.31918	+	3.55325i	−2.41016	−	1.09139i			4.90301i	1.20755	−	2.74624i	−4.49898	+	7.79247i
241.9		−	2.40059i	−0.802734	−	1.53480i	−3.76282			−1.27349	−	0.735249i	−3.68443	+	1.92703i	−1.68575	−	2.03917i			4.23180i	−1.71124	+	2.46408i	−1.76503	+	3.05712i
241.10		−	2.38659i	−1.67959	+	0.423056i	−3.69582			2.27722	+	1.31475i	1.00966	+	4.00850i	−2.64399	−	0.0964694i			4.04724i	2.64205	−	1.42112i	3.13778	−	5.43480i
241.11		−	2.30621i	1.69832	+	0.340145i	−3.31861			0.535504	+	0.309174i	0.784447	−	3.91669i	1.75900	−	1.97634i			3.04098i	2.76860	+	1.15535i	0.713019	−	1.23499i
241.12		−	2.21720i	−0.233637	+	1.71622i	−2.91599			−0.744825	−	0.430025i	3.80521	+	0.518020i	0.850851	−	2.50521i			2.03095i	−2.89083	−	0.801944i	−0.953453	+	1.65143i
241.13		−	2.17389i	0.909348	+	1.47414i	−2.72582			−3.37456	−	1.94830i	3.20462	−	1.97683i	1.51809	+	2.16689i			1.57785i	−1.34617	+	2.68101i	−4.23540	+	7.33593i
241.14		−	2.16160i	1.24216	−	1.20708i	−2.67251			−1.18312	−	0.683074i	−2.60923	−	2.68504i	2.27650	+	1.34816i			1.45369i	0.0859008	−	2.99877i	−1.47653	+	2.55743i
241.15		−	2.12939i	−1.54922	−	0.774543i	−2.53429			0.578025	+	0.333723i	−1.64930	+	3.29889i	1.56581	−	2.13266i			1.13771i	1.80017	+	2.39987i	0.710625	−	1.23084i
241.16		−	1.96785i	1.61782	+	0.618603i	−1.87243			−1.93048	−	1.11457i	1.21732	−	3.18362i	−2.25844	+	1.37820i		−	0.251034i	2.23466	+	2.00157i	−2.19330	+	3.79890i
241.17		−	1.89872i	1.27872	−	1.16828i	−1.60516			−3.68429	−	2.12713i	−2.21824	−	2.42793i	0.124769	−	2.64281i		−	0.749700i	0.270238	−	2.98780i	−4.03883	+	6.99546i
241.18		−	1.87847i	−1.68413	+	0.404593i	−1.52865			0.949548	+	0.548222i	0.760016	+	3.16359i	−0.156565	+	2.64111i		−	0.885415i	2.67261	−	1.36278i	1.02982	−	1.78370i
241.19		−	1.85540i	1.06220	+	1.36811i	−1.44250			3.19862	+	1.84672i	2.53839	−	1.97080i	2.64321	−	0.116009i		−	1.03439i	−0.743457	+	2.90642i	3.42640	−	5.93470i
241.20		−	1.84408i	−0.821003	+	1.52511i	−1.40065			−0.838235	−	0.483955i	2.81243	+	1.51400i	−1.60484	+	2.10344i		−	1.10526i	−1.65191	−	2.50424i	−0.892455	+	1.54578i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
63.t	odd	6	1	inner
693.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.p.a	✓	184
7.d	odd	6	1		693.2.ba.a	yes	184
9.c	even	3	1		693.2.ba.a	yes	184
11.b	odd	2	1	inner	693.2.p.a	✓	184
63.t	odd	6	1	inner	693.2.p.a	✓	184
77.i	even	6	1		693.2.ba.a	yes	184
99.h	odd	6	1		693.2.ba.a	yes	184
693.p	even	6	1	inner	693.2.p.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.p.a	✓	184	1.a	even	1	1	trivial
693.2.p.a	✓	184	11.b	odd	2	1	inner
693.2.p.a	✓	184	63.t	odd	6	1	inner
693.2.p.a	✓	184	693.p	even	6	1	inner
693.2.ba.a	yes	184	7.d	odd	6	1
693.2.ba.a	yes	184	9.c	even	3	1
693.2.ba.a	yes	184	77.i	even	6	1
693.2.ba.a	yes	184	99.h	odd	6	1

Hecke kernels

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