Newform orbit 208.3.o.a

• Label: 208.3.o.a
• Level: $208$
• Weight: $3$
• Character orbit: 208.o
• Analytic conductor: $5.668$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	208.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.66758949869\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 4 q^{3} - 4 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
51.1	−1.99724	−	0.105118i	−1.51644	+	1.51644i	3.97790	+	0.419890i	−1.00552	−	1.00552i	3.18809	−	2.86928i		−	2.82265i	−7.90067	−	1.25677i			4.40083i	1.90256	+	2.11396i
51.2	−1.99159	+	0.183173i	1.20204	−	1.20204i	3.93290	−	0.729612i	5.48867	+	5.48867i	−2.17379	+	2.61415i			4.93207i	−7.69909	+	2.17349i			6.11021i	−11.9366	−	9.92583i
51.3	−1.97408	+	0.320925i	−0.390830	+	0.390830i	3.79401	−	1.26707i	−5.69669	−	5.69669i	0.646104	−	0.896958i			8.61858i	−7.08307	+	3.71889i			8.69450i	13.0740	+	9.41753i
51.4	−1.96385	−	0.378564i	3.21597	−	3.21597i	3.71338	+	1.48688i	0.128865	+	0.128865i	−7.53311	+	5.09821i		−	4.32596i	−6.72962	−	4.32576i		−	11.6849i	−0.204287	−	0.301854i
51.5	−1.91539	−	0.575562i	−3.94823	+	3.94823i	3.33746	+	2.20485i	−3.27230	−	3.27230i	9.83486	−	5.28996i		−	1.88275i	−5.12351	−	6.14407i		−	22.1770i	4.38433	+	8.15115i
51.6	−1.80322	+	0.865093i	−3.09353	+	3.09353i	2.50323	−	3.11991i	1.17271	+	1.17271i	2.90214	−	8.25453i			9.10856i	−1.81487	+	7.79142i		−	10.1399i	−3.12917	−	1.10016i
51.7	−1.79541	+	0.881196i	0.226931	−	0.226931i	2.44699	−	3.16421i	−3.37333	−	3.37333i	−0.207463	+	0.607403i		−	12.1930i	−1.60505	+	7.83733i			8.89701i	9.02908	+	3.08395i
51.8	−1.79436	+	0.883325i	−2.86493	+	2.86493i	2.43947	−	3.17001i	5.01219	+	5.01219i	2.61006	−	7.67139i		−	11.7945i	−1.57715	+	7.84300i		−	7.41566i	−13.4211	−	4.56629i
51.9	−1.75620	−	0.956946i	0.0623252	−	0.0623252i	2.16851	+	3.36118i	3.14131	+	3.14131i	−0.169098	+	0.0498140i		−	9.71316i	−0.591878	−	7.97808i			8.99223i	−2.51072	−	8.52284i
51.10	−1.74445	+	0.978202i	3.11641	−	3.11641i	2.08624	−	3.41286i	−2.58097	−	2.58097i	−2.38796	+	8.48491i		−	0.0373756i	−0.300892	+	7.99434i		−	10.4240i	7.02709	+	1.97767i
51.11	−1.72465	−	1.01272i	1.55752	−	1.55752i	1.94881	+	3.49315i	−2.66443	−	2.66443i	−4.26350	+	1.10885i			10.9809i	0.176561	−	7.99805i			4.14826i	1.89689	+	7.29351i
51.12	−1.64357	−	1.13959i	−2.80100	+	2.80100i	1.40267	+	3.74600i	6.19239	+	6.19239i	7.79564	−	1.41166i			10.2384i	1.96351	−	7.75530i		−	6.69120i	−3.12086	−	17.2344i
51.13	−1.54265	+	1.27289i	1.85513	−	1.85513i	0.759509	−	3.92723i	2.88559	+	2.88559i	−0.500435	+	5.22319i			1.32860i	3.82727	+	7.02510i			2.11695i	−8.12447	−	0.778407i
51.14	−1.49085	−	1.33318i	−1.34315	+	1.34315i	0.445278	+	3.97514i	−1.34010	−	1.34010i	3.79310	−	0.211781i			0.780718i	4.63572	−	6.51998i			5.39188i	0.211300	+	3.78449i
51.15	−1.14357	−	1.64081i	3.98032	−	3.98032i	−1.38449	+	3.75276i	6.01613	+	6.01613i	−11.0827	−	1.97915i			4.78302i	7.74081	−	2.01986i		−	22.6858i	2.99143	−	16.7512i
51.16	−1.13966	+	1.64353i	−1.44409	+	1.44409i	−1.40235	−	3.74612i	1.32511	+	1.32511i	−0.727628	−	4.01917i			7.99609i	7.75505	+	1.96450i			4.82920i	−3.68803	+	0.667678i
51.17	−1.11098	−	1.66304i	2.79401	−	2.79401i	−1.53143	+	3.69523i	−4.42244	−	4.42244i	−7.75066	−	1.54246i		−	4.49012i	7.84672	−	1.55850i		−	6.61297i	−2.44145	+	12.2680i
51.18	−0.977226	+	1.74500i	−1.78017	+	1.78017i	−2.09006	−	3.41052i	−3.55706	−	3.55706i	−1.36677	−	4.84603i		−	3.21216i	7.99382	−	0.314304i			2.66197i	9.68313	−	2.73102i
51.19	−0.965476	−	1.75153i	−2.49968	+	2.49968i	−2.13571	+	3.38212i	−5.43804	−	5.43804i	6.79165	+	1.96488i		−	4.33985i	7.98586	+	0.475407i		−	3.49682i	−4.27459	+	14.7752i
51.20	−0.879637	+	1.79617i	2.31366	−	2.31366i	−2.45248	−	3.15996i	−6.07553	−	6.07553i	2.12055	+	6.19091i			5.73060i	7.83313	−	1.62546i		−	1.70602i	16.2570	−	5.56844i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner
16.f	odd	4	1	inner
208.o	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.3.o.a	✓	108
13.b	even	2	1	inner	208.3.o.a	✓	108
16.f	odd	4	1	inner	208.3.o.a	✓	108
208.o	odd	4	1	inner	208.3.o.a	✓	108

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.3.o.a	✓	108	1.a	even	1	1	trivial
208.3.o.a	✓	108	13.b	even	2	1	inner
208.3.o.a	✓	108	16.f	odd	4	1	inner
208.3.o.a	✓	108	208.o	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(208, [\chi])\).