Newform orbit 208.2.n.a

• Label: 208.2.n.a
• Level: $208$
• Weight: $2$
• Character orbit: 208.n
• Analytic conductor: $1.661$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{4} - 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.41282	−	0.0627417i	−1.56058	+	1.56058i	1.99213	+	0.177286i	3.08656	+	3.08656i	2.30273	−	2.10690i		−	2.75937i	−2.80340	−	0.375462i		−	1.87081i	−4.16711	−	4.55442i
53.2	−1.40847	+	0.127312i	−2.17257	+	2.17257i	1.96758	−	0.358630i	−2.25200	−	2.25200i	2.78341	−	3.33660i			2.10748i	−2.72563	+	0.755616i		−	6.44015i	3.45858	+	2.88517i
53.3	−1.40279	−	0.179405i	1.23013	−	1.23013i	1.93563	+	0.503335i	−0.845708	−	0.845708i	−1.94630	+	1.50492i		−	3.63419i	−2.62497	−	1.05333i		−	0.0264265i	1.03462	+	1.33807i
53.4	−1.33938	−	0.453937i	0.474808	−	0.474808i	1.58788	+	1.21599i	0.481789	+	0.481789i	−0.851482	+	0.420416i			4.68072i	−1.57480	−	2.34947i			2.54911i	−0.426597	−	0.864001i
53.5	−1.09059	+	0.900345i	−0.326558	+	0.326558i	0.378759	−	1.96381i	−0.499927	−	0.499927i	0.0621250	−	0.650155i			0.517149i	1.35504	+	2.48272i			2.78672i	0.995320	+	0.0951070i
53.6	−1.07039	+	0.924262i	2.16798	−	2.16798i	0.291481	−	1.97865i	−2.73093	−	2.73093i	−0.316809	+	4.32437i			2.40663i	1.51679	+	2.38733i		−	6.40026i	5.44726	+	0.399074i
53.7	−0.888975	−	1.09987i	2.33011	−	2.33011i	−0.419447	+	1.95552i	1.49813	+	1.49813i	−4.63423	−	0.491418i			0.00620215i	2.52371	−	1.27707i		−	7.85879i	0.315955	−	2.97956i
53.8	−0.638200	+	1.26202i	1.33118	−	1.33118i	−1.18540	−	1.61085i	1.78745	+	1.78745i	0.830420	+	2.52954i		−	4.71876i	2.78945	−	0.467961i		−	0.544086i	−3.39655	+	1.11505i
53.9	−0.625246	+	1.26849i	−1.56862	+	1.56862i	−1.21814	−	1.58624i	2.41649	+	2.41649i	−1.00901	−	2.97055i			3.76757i	2.77376	−	0.553404i		−	1.92113i	−4.57619	+	1.55439i
53.10	−0.395977	−	1.35765i	−0.140536	+	0.140536i	−1.68640	+	1.07519i	1.99922	+	1.99922i	0.246447	+	0.135149i			0.679168i	2.12751	+	1.86379i			2.96050i	1.92259	−	3.50588i
53.11	−0.0872990	+	1.41152i	−0.316792	+	0.316792i	−1.98476	−	0.246448i	−2.69572	−	2.69572i	−0.419501	−	0.474812i		−	1.94658i	0.521133	−	2.78000i			2.79929i	4.04038	−	3.56972i
53.12	−0.0687403	+	1.41254i	0.559046	−	0.559046i	−1.99055	−	0.194197i	0.101861	+	0.101861i	0.751247	+	0.828105i			3.07699i	0.411142	−	2.79839i			2.37494i	−0.150885	+	0.136881i
53.13	0.0884103	−	1.41145i	−1.08262	+	1.08262i	−1.98437	−	0.249573i	−2.20245	−	2.20245i	1.43234	+	1.62377i			4.01772i	−0.527698	+	2.77877i			0.655887i	−3.30336	+	2.91392i
53.14	0.389205	−	1.35960i	1.57408	−	1.57408i	−1.69704	−	1.05833i	−0.836237	−	0.836237i	−1.52748	−	2.75277i			0.226329i	−2.09940	+	1.89539i		−	1.95547i	−1.46242	+	0.811482i
53.15	0.435971	−	1.34534i	−2.05439	+	2.05439i	−1.61986	−	1.17306i	0.347700	+	0.347700i	1.86819	+	3.65951i		−	5.08331i	−2.28437	+	1.66784i		−	5.44107i	0.619360	−	0.316186i
53.16	0.579160	+	1.29018i	−1.99491	+	1.99491i	−1.32915	+	1.49445i	−0.414832	−	0.414832i	−3.72918	−	1.41843i			0.607342i	−2.69790	−	0.849322i		−	4.95936i	0.294955	−	0.775463i
53.17	0.620633	+	1.27075i	2.03774	−	2.03774i	−1.22963	+	1.57734i	1.14590	+	1.14590i	3.85416	+	1.32478i			2.61154i	−2.76756	−	0.583605i		−	5.30481i	−0.744971	+	2.16733i
53.18	0.899312	+	1.09144i	−0.152350	+	0.152350i	−0.382476	+	1.96309i	1.59716	+	1.59716i	−0.303291	−	0.0292705i		−	3.10196i	−2.48655	+	1.34798i			2.95358i	−0.306857	+	3.17955i
53.19	0.986819	−	1.01301i	0.718176	−	0.718176i	−0.0523779	−	1.99931i	2.02033	+	2.02033i	−0.0188099	−	1.43623i		−	0.407753i	−2.07701	−	1.91990i			1.96845i	4.04031	−	0.0529148i
53.20	1.14676	−	0.827610i	−0.390961	+	0.390961i	0.630122	−	1.89814i	−2.58479	−	2.58479i	−0.124775	+	0.771902i		−	2.97780i	−0.848323	−	2.69821i			2.69430i	−5.10333	−	0.824937i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.2.n.a	✓	48
4.b	odd	2	1		832.2.n.a		48
8.b	even	2	1		1664.2.n.b		48
8.d	odd	2	1		1664.2.n.a		48
16.e	even	4	1	inner	208.2.n.a	✓	48
16.e	even	4	1		1664.2.n.b		48
16.f	odd	4	1		832.2.n.a		48
16.f	odd	4	1		1664.2.n.a		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.2.n.a	✓	48	1.a	even	1	1	trivial
208.2.n.a	✓	48	16.e	even	4	1	inner
832.2.n.a		48	4.b	odd	2	1
832.2.n.a		48	16.f	odd	4	1
1664.2.n.a		48	8.d	odd	2	1
1664.2.n.a		48	16.f	odd	4	1
1664.2.n.b		48	8.b	even	2	1
1664.2.n.b		48	16.e	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(208, [\chi])\).