Newform orbit 693.2.bx.a

• Label: 693.2.bx.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.bx
• 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.bx (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(736\)
Relative dimension:	\(92\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q + 3 q^{2} - 3 q^{3} + 91 q^{4} - 6 q^{5} - 8 q^{6} - 3 q^{7} - 16 q^{8} + 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.53739	+	1.12972i	−1.20327	−	1.24585i	3.82382	−	4.24678i	0.972382	−	0.706477i	4.46063	+	1.80185i	2.12867	−	1.57123i	−3.18825	+	9.81243i	−0.104277	+	2.99819i	−1.66919	+	2.89112i
4.2	−2.46378	+	1.09695i	−1.71484	−	0.243568i	3.52866	−	3.91898i	−1.74638	+	1.26882i	4.49217	−	1.28099i	−1.41098	+	2.23811i	−2.72814	+	8.39635i	2.88135	+	0.835361i	2.91087	−	5.04178i
4.3	−2.45170	+	1.09157i	1.66996	−	0.459590i	3.48107	−	3.86612i	0.192489	−	0.139851i	−3.59258	+	2.94966i	1.44401	+	2.21694i	−2.65578	+	8.17365i	2.57755	−	1.53500i	−0.319268	+	0.552988i
4.4	−2.40658	+	1.07148i	−0.227296	+	1.71707i	3.30529	−	3.67089i	−0.509835	+	0.370417i	−1.29280	−	4.37581i	0.315005	−	2.62693i	−2.39305	+	7.36504i	−2.89667	−	0.780566i	0.830063	−	1.43771i
4.5	−2.36051	+	1.05097i	1.73033	−	0.0770926i	3.12920	−	3.47533i	−1.43487	+	1.04249i	−4.00344	+	2.00050i	−1.07205	−	2.41883i	−2.13711	+	6.57736i	2.98811	−	0.266792i	2.29140	−	3.96881i
4.6	−2.32378	+	1.03461i	−1.25659	+	1.19204i	2.99125	−	3.32212i	3.25094	−	2.36195i	1.68674	−	4.07013i	1.31412	+	2.29632i	−1.94181	+	5.97627i	0.158062	−	2.99583i	−5.11076	+	8.85209i
4.7	−2.29931	+	1.02372i	0.618154	+	1.61799i	2.90057	−	3.22141i	−3.55799	+	2.58503i	−3.07770	−	3.08744i	0.999696	+	2.44961i	−1.81596	+	5.58896i	−2.23577	+	2.00033i	5.53458	−	9.58617i
4.8	−2.27845	+	1.01443i	0.797951	−	1.53729i	2.82402	−	3.13639i	2.62001	−	1.90355i	−0.258613	+	4.31212i	−2.10365	+	1.60457i	−1.71131	+	5.26688i	−1.72655	−	2.45337i	−4.03855	+	6.99497i
4.9	−2.09397	+	0.932294i	1.45574	+	0.938517i	2.17726	−	2.41809i	0.510986	−	0.371253i	−3.92325	−	0.608043i	−2.61968	+	0.370501i	−0.888122	+	2.73336i	1.23837	+	2.73248i	−0.723870	+	1.25378i
4.10	−2.08664	+	0.929030i	0.579181	−	1.63234i	2.15269	−	2.39081i	−1.88802	+	1.37173i	0.307957	+	3.94419i	2.64136	−	0.152297i	−0.859096	+	2.64403i	−2.32910	−	1.89085i	2.66523	−	4.61632i
4.11	−2.07619	+	0.924379i	−1.73176	+	0.0319770i	2.11782	−	2.35208i	2.20578	−	1.60259i	3.56589	−	1.66719i	−2.03415	−	1.69182i	−0.818200	+	2.51816i	2.99795	−	0.110753i	−3.09821	+	5.36626i
4.12	−2.04102	+	0.908720i	−0.754935	−	1.55887i	2.00172	−	2.22314i	0.915203	−	0.664934i	2.95741	+	2.49566i	−2.54245	−	0.732084i	−0.684548	+	2.10682i	−1.86015	+	2.35369i	−1.26371	+	2.18880i
4.13	−2.00100	+	0.890904i	0.776950	+	1.54801i	1.87204	−	2.07911i	1.60942	−	1.16931i	−2.93381	−	2.40539i	2.42005	−	1.06927i	−0.539949	+	1.66179i	−1.79270	+	2.40546i	−2.17871	+	3.77364i
4.14	−1.95311	+	0.869580i	−1.64408	+	0.544965i	1.72020	−	1.91048i	−2.89466	+	2.10309i	2.73718	−	2.49404i	0.0603888	−	2.64506i	−0.377105	+	1.16061i	2.40603	−	1.79194i	3.82477	−	6.62470i
4.15	−1.92861	+	0.858672i	−0.660573	−	1.60114i	1.64396	−	1.82580i	−0.715413	+	0.519778i	2.64884	+	2.52076i	0.285440	+	2.63031i	−0.298039	+	0.917269i	−2.12729	+	2.11534i	0.933433	−	1.61675i
4.16	−1.90615	+	0.848672i	−1.50360	+	0.859767i	1.57490	−	1.74910i	−0.664340	+	0.482671i	2.13642	−	2.91490i	2.54119	+	0.736434i	−0.228024	+	0.701787i	1.52160	−	2.58548i	0.856701	−	1.48385i
4.17	−1.86903	+	0.832146i	1.72521	−	0.153780i	1.46255	−	1.62432i	3.51643	−	2.55483i	−3.09650	+	1.72305i	1.60924	−	2.10008i	−0.117431	+	0.361416i	2.95270	−	0.530607i	−4.44631	+	7.70124i
4.18	−1.68993	+	0.752405i	1.07887	−	1.35500i	0.951485	−	1.05673i	−1.73809	+	1.26280i	−0.803711	+	3.10161i	−2.60264	+	0.475650i	0.330422	−	1.01693i	−0.672060	−	2.92375i	1.98712	−	3.44179i
4.19	−1.63297	+	0.727046i	1.43418	+	0.971152i	0.799742	−	0.888203i	0.921715	−	0.669665i	−3.04804	−	0.543152i	0.167771	+	2.64043i	0.444551	−	1.36819i	1.11373	+	2.78561i	−1.01826	+	1.76367i
4.20	−1.51959	+	0.676563i	−0.785427	+	1.54373i	0.513143	−	0.569903i	1.59172	−	1.15645i	0.149092	−	2.87722i	−1.09470	−	2.40866i	0.633844	−	1.95077i	−1.76621	−	2.42497i	−1.63635	+	2.83423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
63.g	even	3	1	inner
693.bx	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.bx.a	yes	736
7.c	even	3	1		693.2.bw.a	✓	736
9.c	even	3	1		693.2.bw.a	✓	736
11.c	even	5	1	inner	693.2.bx.a	yes	736
63.g	even	3	1	inner	693.2.bx.a	yes	736
77.m	even	15	1		693.2.bw.a	✓	736
99.m	even	15	1		693.2.bw.a	✓	736
693.bx	even	15	1	inner	693.2.bx.a	yes	736

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.bw.a	✓	736	7.c	even	3	1
693.2.bw.a	✓	736	9.c	even	3	1
693.2.bw.a	✓	736	77.m	even	15	1
693.2.bw.a	✓	736	99.m	even	15	1
693.2.bx.a	yes	736	1.a	even	1	1	trivial
693.2.bx.a	yes	736	11.c	even	5	1	inner
693.2.bx.a	yes	736	63.g	even	3	1	inner
693.2.bx.a	yes	736	693.bx	even	15	1	inner

Hecke kernels

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