Newform orbit 671.2.bk.a

• Label: 671.2.bk.a
• Level: $671$
• Weight: $2$
• Character orbit: 671.bk
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.bk (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 3 q^{2} - 2 q^{3} - 235 q^{4} - 16 q^{5} - 14 q^{6} - 11 q^{7} - 114 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
15.1	−1.41308	+	2.44752i	0.672776	+	2.07059i	−2.99357	−	5.18501i	3.19177	+	0.678431i	−6.01849	−	1.27927i	−1.51348	−	1.68089i	11.2682			−1.40767	+	1.02273i	−6.17068	+	6.85323i
15.2	−1.35616	+	2.34895i	−0.793092	−	2.44088i	−2.67836	−	4.63906i	−0.868718	−	0.184652i	6.80907	+	1.44731i	1.00757	+	1.11902i	9.10454			−2.90187	+	2.10833i	1.61186	−	1.79015i
15.3	−1.28076	+	2.21833i	0.471724	+	1.45182i	−2.28067	−	3.95024i	−1.53953	−	0.327238i	−3.82478	−	0.812981i	2.90716	+	3.22873i	6.56091			0.541804	−	0.393643i	2.69769	−	2.99608i
15.4	−1.24641	+	2.15884i	−0.162972	−	0.501575i	−2.10707	−	3.64956i	2.45529	+	0.521888i	1.28595	+	0.273337i	1.25152	+	1.38996i	5.51947			2.20203	−	1.59987i	−4.18697	+	4.65010i
15.5	−1.24068	+	2.14893i	0.842245	+	2.59216i	−2.07859	−	3.60022i	−1.88044	−	0.399699i	−6.61533	−	1.40613i	−0.735327	−	0.816663i	5.35274			−3.58289	+	2.60312i	3.19195	−	3.54502i
15.6	−1.20450	+	2.08626i	−0.162391	−	0.499788i	−1.90164	−	3.29374i	1.48670	+	0.316008i	1.23828	+	0.263206i	−1.95894	−	2.17562i	4.34412			2.20363	−	1.60103i	−2.45001	+	2.72101i
15.7	−1.16856	+	2.02400i	−0.679390	−	2.09095i	−1.73104	−	2.99826i	0.419327	+	0.0891308i	5.02597	+	1.06830i	−0.125385	−	0.139254i	3.41706			−1.48344	+	1.07778i	−0.670408	+	0.744563i
15.8	−1.14169	+	1.97746i	0.284866	+	0.876728i	−1.60690	−	2.78323i	−3.44424	−	0.732095i	−2.05892	−	0.437637i	−0.754530	−	0.837991i	2.77154			1.73955	−	1.26386i	5.37993	−	5.97501i
15.9	−1.02617	+	1.77737i	−0.480248	−	1.47805i	−1.10603	−	1.91571i	−3.70450	−	0.787417i	3.11986	+	0.663147i	3.05524	+	3.39319i	0.435230			0.473052	−	0.343692i	5.20097	−	5.77626i
15.10	−0.989366	+	1.71363i	0.274596	+	0.845119i	−0.957690	−	1.65877i	−1.20480	−	0.256088i	−1.71990	−	0.365576i	−0.574724	−	0.638296i	−0.167440			1.78823	−	1.29922i	1.63083	−	1.81122i
15.11	−0.985185	+	1.70639i	0.929389	+	2.86037i	−0.941178	−	1.63017i	2.41704	+	0.513758i	−5.79652	−	1.23209i	2.57514	+	2.85999i	−0.231800			−4.89088	+	3.55343i	−3.25791	+	3.61827i
15.12	−0.984817	+	1.70575i	−0.870426	−	2.67890i	−0.939728	−	1.62766i	−3.83447	−	0.815043i	5.42674	+	1.15349i	−2.57452	−	2.85929i	−0.237428			−3.99179	+	2.90021i	5.16651	−	5.73800i
15.13	−0.867714	+	1.50292i	0.408148	+	1.25615i	−0.505856	−	0.876168i	3.64869	+	0.775553i	−2.24206	−	0.476564i	0.133329	+	0.148076i	−1.71510			1.01572	−	0.737965i	−4.33162	+	4.81075i
15.14	−0.856822	+	1.48406i	0.0686466	+	0.211273i	−0.468290	−	0.811101i	1.60031	+	0.340156i	−0.372359	−	0.0791474i	2.98071	+	3.31041i	−1.82233			2.38713	−	1.73435i	−1.87599	+	2.08350i
15.15	−0.814697	+	1.41110i	−0.498474	−	1.53414i	−0.327463	−	0.567182i	3.54855	+	0.754268i	2.57093	+	0.546468i	−2.54573	−	2.82732i	−2.19166			0.321929	−	0.233895i	−3.95534	+	4.39285i
15.16	−0.777227	+	1.34620i	−0.382830	−	1.17823i	−0.208165	−	0.360552i	−1.18337	−	0.251533i	1.88367	+	0.400387i	−0.455747	−	0.506159i	−2.46174			1.18539	−	0.861235i	1.25836	−	1.39755i
15.17	−0.776663	+	1.34522i	−0.962618	−	2.96263i	−0.206410	−	0.357512i	2.04419	+	0.434506i	4.73302	+	1.00603i	1.95596	+	2.17231i	−2.46541			−5.42351	+	3.94041i	−2.17215	+	2.41242i
15.18	−0.692326	+	1.19914i	0.978135	+	3.01039i	0.0413684	+	0.0716523i	−0.932625	−	0.198235i	−4.28708	−	0.911247i	−3.17317	−	3.52416i	−2.88387			−5.67865	+	4.12578i	0.883394	−	0.981108i
15.19	−0.653945	+	1.13267i	0.317351	+	0.976707i	0.144711	+	0.250647i	0.294142	+	0.0625219i	−1.31381	−	0.279260i	−3.23963	−	3.59798i	−2.99431			1.57381	−	1.14344i	−0.263170	+	0.292279i
15.20	−0.534691	+	0.926113i	0.0934351	+	0.287564i	0.428210	+	0.741682i	−1.29501	−	0.275262i	−0.316275	−	0.0672264i	0.486997	+	0.540865i	−3.05461			2.35309	−	1.70962i	0.947353	−	1.05214i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
671.bk	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	671.2.bk.a	✓	480
11.c	even	5	1		671.2.bo.a	yes	480
61.i	even	15	1		671.2.bo.a	yes	480
671.bk	even	15	1	inner	671.2.bk.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
671.2.bk.a	✓	480	1.a	even	1	1	trivial
671.2.bk.a	✓	480	671.bk	even	15	1	inner
671.2.bo.a	yes	480	11.c	even	5	1
671.2.bo.a	yes	480	61.i	even	15	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).