Newform orbit 208.2.bf.a

• Label: 208.2.bf.a
• Level: $208$
• Weight: $2$
• Character orbit: 208.bf
• Analytic conductor: $1.661$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.66088836204\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 2 q^{6} - 8 q^{7} + 2 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.39460	+	0.234728i	−0.751939	+	2.80627i	1.88981	−	0.654703i		−	3.13443i	0.389940	−	4.09012i	−1.02597	−	3.82896i	−2.48184	+	1.35664i	−4.71168	−	2.72029i	0.735740	+	4.37128i
11.2	−1.38619	+	0.280119i	0.349108	−	1.30289i	1.84307	−	0.776598i			0.268801i	−0.118967	+	1.90385i	0.897134	+	3.34815i	−2.33731	+	1.59279i	1.02243	+	0.590303i	−0.0752962	−	0.372610i
11.3	−1.34204	−	0.446019i	−0.176107	+	0.657240i	1.60213	+	1.19715i		−	0.0882347i	0.529484	−	0.803494i	0.0817171	+	0.304972i	−1.61617	−	2.32120i	2.19713	+	1.26851i	−0.0393544	+	0.118414i
11.4	−1.27287	+	0.616273i	0.642342	−	2.39725i	1.24042	−	1.56888i		−	1.50782i	0.659742	+	3.44726i	−0.689815	−	2.57443i	−0.612036	+	2.76141i	−2.73615	−	1.57971i	0.929230	+	1.91927i
11.5	−1.20240	−	0.744472i	0.653472	−	2.43879i	0.891524	+	1.79030i			1.71756i	−2.60134	+	2.44591i	−1.08088	−	4.03389i	0.260863	−	2.81637i	−2.92260	−	1.68736i	1.27868	−	2.06519i
11.6	−1.17392	+	0.788613i	−0.397626	+	1.48396i	0.756179	−	1.85154i			2.09477i	−0.703489	−	2.05562i	0.0583636	+	0.217816i	0.572453	+	2.76989i	0.554044	+	0.319878i	−1.65196	−	2.45909i
11.7	−0.796291	+	1.16873i	−0.257060	+	0.959359i	−0.731841	−	1.86129i		−	4.16815i	−0.916534	−	1.06436i	1.22167	+	4.55935i	2.75810	+	0.626809i	1.74379	+	1.00677i	4.87143	+	3.31906i
11.8	−0.744187	−	1.20257i	0.295974	−	1.10459i	−0.892370	+	1.78988i			2.78152i	−1.54861	+	0.466091i	1.02035	+	3.80799i	2.81656	−	0.258866i	1.46556	+	0.846141i	3.34499	−	2.06997i
11.9	−0.625215	−	1.26851i	0.568060	−	2.12003i	−1.21821	+	1.58618i		−	4.07874i	−3.04443	+	0.604887i	0.302237	+	1.12797i	2.77372	+	0.553608i	−1.57376	−	0.908610i	−5.17390	+	2.55009i
11.10	−0.592777	−	1.28398i	−0.329252	+	1.22878i	−1.29723	+	1.52223i		−	0.827228i	1.77291	−	0.305640i	−0.595448	−	2.22224i	2.72349	+	0.763282i	1.19657	+	0.690842i	−1.06215	+	0.490361i
11.11	−0.585119	+	1.28749i	0.598790	−	2.23472i	−1.31527	−	1.50667i			4.10099i	2.52681	+	2.07851i	0.302935	+	1.13057i	2.70942	−	0.811819i	−2.03733	−	1.17625i	−5.28000	−	2.39957i
11.12	−0.332154	+	1.37465i	0.0600954	−	0.224279i	−1.77935	−	0.913194i		−	1.67010i	0.288345	+	0.157106i	−1.15044	−	4.29352i	1.84634	−	2.14267i	2.55139	+	1.47304i	2.29581	+	0.554730i
11.13	−0.105362	+	1.41028i	−0.582115	+	2.17248i	−1.97780	−	0.297180i			2.01220i	−3.00248	−	1.04984i	−0.00731967	−	0.0273174i	0.627493	−	2.75794i	−1.78275	−	1.02927i	−2.83777	−	0.212009i
11.14	0.0647741	−	1.41273i	−0.855339	+	3.19217i	−1.99161	−	0.183017i		−	1.40751i	4.45427	+	1.41513i	0.610442	+	2.27820i	−0.387557	+	2.80175i	−6.86026	−	3.96077i	−1.98844	−	0.0911704i
11.15	0.244870	−	1.39285i	−0.225052	+	0.839904i	−1.88008	−	0.682136i			3.61097i	1.11475	+	0.519131i	0.197106	+	0.735610i	−1.41049	+	2.45164i	1.94329	+	1.12196i	5.02954	+	0.884217i
11.16	0.315709	+	1.37852i	0.737906	−	2.75390i	−1.80066	+	0.870424i		−	2.35573i	4.02928	+	0.147790i	0.105146	+	0.392411i	−1.76838	−	2.20745i	−4.44140	−	2.56424i	3.24743	−	0.743724i
11.17	0.395756	−	1.35771i	0.0539276	−	0.201261i	−1.68675	−	1.07464i		−	1.90833i	−0.251911	−	0.152868i	−0.560762	−	2.09279i	−2.12660	+	1.86483i	2.56048	+	1.47829i	−2.59096	−	0.755232i
11.18	0.830378	+	1.14476i	−0.675637	+	2.52151i	−0.620945	+	1.90116i		−	1.60478i	−3.44756	+	1.32037i	0.522704	+	1.95076i	−2.69199	+	0.867852i	−3.30346	−	1.90725i	1.83708	−	1.33257i
11.19	0.900498	+	1.09046i	0.0854467	−	0.318891i	−0.378206	+	1.96391i			3.12859i	0.424683	−	0.193985i	−0.641339	−	2.39351i	−2.48214	+	1.35608i	2.50369	+	1.44550i	−3.41160	+	2.81729i
11.20	0.924475	−	1.07021i	0.519709	−	1.93958i	−0.290693	−	1.97876i		−	0.884276i	−1.59530	−	2.34929i	0.991011	+	3.69851i	−2.38643	−	1.51821i	−0.893795	−	0.516033i	−0.946360	−	0.817491i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
208.bf	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	208.2.bf.a	✓	104
4.b	odd	2	1		832.2.bn.a		104
13.f	odd	12	1		208.2.bk.a	yes	104
16.e	even	4	1		832.2.bs.a		104
16.f	odd	4	1		208.2.bk.a	yes	104
52.l	even	12	1		832.2.bs.a		104
208.be	odd	12	1		832.2.bn.a		104
208.bf	even	12	1	inner	208.2.bf.a	✓	104

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.2.bf.a	✓	104	1.a	even	1	1	trivial
208.2.bf.a	✓	104	208.bf	even	12	1	inner
208.2.bk.a	yes	104	13.f	odd	12	1
208.2.bk.a	yes	104	16.f	odd	4	1
832.2.bn.a		104	4.b	odd	2	1
832.2.bn.a		104	208.be	odd	12	1
832.2.bs.a		104	16.e	even	4	1
832.2.bs.a		104	52.l	even	12	1

Hecke kernels

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