Newform orbit 208.2.bk.a

• Label: 208.2.bk.a
• Level: $208$
• Weight: $2$
• Character orbit: 208.bk
• 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.bk (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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 4 q^{2} - 2 q^{3} - 6 q^{4} - 8 q^{5} + 6 q^{6} - 8 q^{7} - 10 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
115.1	−1.40465	+	0.164214i	−0.887781	−	0.237880i	1.94607	−	0.461325i	−1.45725			1.28608	+	0.188352i	0.176423	+	0.658420i	−2.65778	+	0.967571i	−1.86651	−	1.07763i	2.04693	−	0.239301i
115.2	−1.34367	−	0.441065i	−3.12349	−	0.836936i	1.61092	+	1.18530i	1.80598			3.82781	+	2.50223i	0.370074	+	1.38114i	−1.64177	−	2.30317i	6.45765	+	3.72832i	−2.42665	−	0.796554i
115.3	−1.32508	+	0.494117i	−0.318891	−	0.0854467i	1.51170	−	1.30949i	3.12859			0.464779	−	0.0443455i	−0.641339	−	2.39351i	−1.35608	+	2.48214i	−2.50369	−	1.44550i	−4.14564	+	1.54589i
115.4	−1.30505	−	0.544827i	2.67645	+	0.717153i	1.40633	+	1.42206i	1.73779			−3.10219	−	2.39413i	−1.11993	−	4.17965i	−1.06056	−	2.62206i	4.05101	+	2.33885i	−2.26790	−	0.946792i
115.5	−1.29151	+	0.576201i	2.52151	+	0.675637i	1.33598	−	1.48834i	−1.60478			−3.64585	+	0.580308i	0.522704	+	1.95076i	−0.867852	+	2.69199i	3.30346	+	1.90725i	2.07258	−	0.924673i
115.6	−1.25680	−	0.648425i	0.562891	+	0.150826i	1.15909	+	1.62988i	−3.28554			−0.609641	−	0.554551i	−0.106277	−	0.396633i	−0.399888	−	2.80002i	−2.30398	−	1.33020i	4.12926	+	2.13043i
115.7	−0.962674	+	1.03598i	−2.75390	−	0.737906i	−0.146519	−	1.99463i	−2.35573			3.41557	−	2.14263i	0.105146	+	0.392411i	2.20745	+	1.76838i	4.44140	+	2.56424i	2.26780	−	2.44049i
115.8	−0.726014	−	1.21363i	1.79959	+	0.482199i	−0.945808	+	1.76223i	0.860828			−0.721315	−	2.53412i	0.822827	+	3.07083i	2.82537	−	0.131538i	0.407934	+	0.235521i	−0.624973	−	1.04473i
115.9	−0.621394	−	1.27038i	−1.21919	−	0.326681i	−1.22774	+	1.57881i	2.25812			0.342588	+	1.75184i	−0.969905	−	3.61974i	2.76861	+	0.578632i	−1.21837	−	0.703426i	−1.40318	−	2.86867i
115.10	−0.613895	+	1.27402i	2.17248	+	0.582115i	−1.24626	−	1.56423i	2.01220			−2.07531	+	2.41043i	−0.00731967	−	0.0273174i	2.75794	−	0.627493i	1.78275	+	1.02927i	−1.23528	+	2.56359i
115.11	−0.399673	+	1.35656i	−0.224279	−	0.0600954i	−1.68052	−	1.08436i	−1.67010			0.171161	−	0.280230i	−1.15044	−	4.29352i	2.14267	−	1.84634i	−2.55139	−	1.47304i	0.667493	−	2.26559i
115.12	−0.265514	−	1.38907i	−1.93958	−	0.519709i	−1.85900	+	0.737633i	−0.884276			−0.206924	+	2.83219i	0.991011	+	3.69851i	1.51821	+	2.38643i	0.893795	+	0.516033i	0.234788	+	1.22832i
115.13	−0.137018	+	1.40756i	−2.23472	−	0.598790i	−1.96245	−	0.385722i	4.10099			1.14903	−	3.06345i	0.302935	+	1.13057i	0.811819	−	2.70942i	2.03733	+	1.17625i	−0.561910	+	5.77240i
115.14	0.105245	+	1.41029i	0.959359	+	0.257060i	−1.97785	+	0.296853i	−4.16815			−0.261561	+	1.38003i	1.22167	+	4.55935i	−0.626809	−	2.75810i	−1.74379	−	1.00677i	−0.438679	−	5.87831i
115.15	0.336120	−	1.37369i	−0.201261	−	0.0539276i	−1.77405	−	0.923450i	−1.90833			−0.141728	+	0.258343i	−0.560762	−	2.09279i	−1.86483	+	2.12660i	−2.56048	−	1.47829i	−0.641428	+	2.62145i
115.16	0.484363	−	1.32868i	0.839904	+	0.225052i	−1.53079	−	1.28713i	3.61097			0.705840	−	1.00696i	0.197106	+	0.735610i	−2.45164	+	1.41049i	−1.94329	−	1.12196i	1.74902	−	4.79782i
115.17	0.622338	+	1.26992i	1.48396	+	0.397626i	−1.22539	+	1.58064i	2.09477			0.418573	+	2.13197i	0.0583636	+	0.217816i	−2.76989	−	0.572453i	−0.554044	−	0.319878i	1.30366	+	2.66019i
115.18	0.650269	−	1.25585i	3.19217	+	0.855339i	−1.15430	−	1.63328i	−1.40751			3.14994	−	3.45267i	0.610442	+	2.27820i	−2.80175	+	0.387557i	6.86026	+	3.96077i	−0.915262	+	1.76762i
115.19	0.794205	+	1.17014i	−2.39725	−	0.642342i	−0.738478	+	1.85867i	−1.50782			−1.15228	−	3.31529i	−0.689815	−	2.57443i	−2.76141	+	0.612036i	2.73615	+	1.57971i	−1.19752	−	1.76437i
115.20	1.06042	+	0.935687i	−1.30289	−	0.349108i	0.248979	+	1.98444i	0.268801			−1.05495	−	1.58930i	0.897134	+	3.34815i	−1.59279	+	2.33731i	−1.02243	−	0.590303i	0.285042	+	0.251513i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
208.bk	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.bk.a	yes	104
4.b	odd	2	1		832.2.bs.a		104
13.f	odd	12	1		208.2.bf.a	✓	104
16.e	even	4	1		832.2.bn.a		104
16.f	odd	4	1		208.2.bf.a	✓	104
52.l	even	12	1		832.2.bn.a		104
208.bk	even	12	1	inner	208.2.bk.a	yes	104
208.bl	odd	12	1		832.2.bs.a		104

    

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

Hecke kernels

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