Newform orbit 756.2.bs.a

• Label: 756.2.bs.a
• Level: $756$
• Weight: $2$
• Character orbit: 756.bs
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.bs (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(140\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 3 q^{2} - 3 q^{4} - 6 q^{5} - 12 q^{6} - 18 q^{8} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41413	+	0.0149120i	0.154673	+	1.72513i	1.99956	−	0.0421753i	3.70389	−	0.653095i	−0.244453	−	2.43726i	−2.38269	−	1.15013i	−2.82701	+	0.0894589i	−2.95215	+	0.533661i	−5.22806	+	0.978797i
11.2	−1.41382	+	0.0335363i	−1.23240	−	1.21704i	1.99775	−	0.0948283i	−1.44666	+	0.255085i	1.78320	+	1.67934i	1.11827	−	2.39781i	−2.82127	+	0.201067i	0.0376115	+	2.99976i	2.03676	−	0.409160i
11.3	−1.41197	+	0.0796147i	−1.59653	+	0.671641i	1.98732	−	0.224827i	0.781464	−	0.137793i	2.20078	−	1.07544i	2.60906	+	0.439114i	−2.78814	+	0.475670i	2.09780	−	2.14459i	−1.09243	+	0.256776i
11.4	−1.41190	+	0.0808315i	−1.10309	+	1.33536i	1.98693	−	0.228252i	−1.49610	+	0.263803i	1.44951	−	1.97457i	−1.99141	+	1.74193i	−2.78690	+	0.482877i	−0.566395	−	2.94605i	2.09102	−	0.493396i
11.5	−1.41111	+	0.0937031i	1.68184	−	0.414019i	1.98244	−	0.264450i	−2.78746	+	0.491505i	−2.33446	+	0.741819i	2.41910	+	1.07144i	−2.77265	+	0.558927i	2.65718	−	1.39263i	3.88735	−	0.954760i
11.6	−1.40534	+	0.158152i	0.564160	+	1.63760i	1.94998	−	0.444515i	−3.91223	+	0.689832i	−1.05183	−	2.21216i	−0.574615	−	2.58260i	−2.67008	+	0.933089i	−2.36345	+	1.84773i	5.38892	−	1.58818i
11.7	−1.40267	−	0.180347i	0.0171663	+	1.73197i	1.93495	+	0.505934i	1.75070	−	0.308695i	0.288276	−	2.43247i	2.14695	+	1.54616i	−2.62285	−	1.05862i	−2.99941	+	0.0594627i	−2.51132	+	0.117263i
11.8	−1.39274	−	0.245534i	−1.54124	−	0.790313i	1.87943	+	0.683928i	1.51908	−	0.267855i	1.95248	+	1.47912i	−1.79492	+	1.94378i	−2.44962	−	1.41399i	1.75081	+	2.43612i	−2.18145		6.52995e-5i
11.9	−1.39272	−	0.245617i	1.54124	+	0.790313i	1.87934	+	0.684153i	1.51908	−	0.267855i	−1.95240	−	1.47924i	1.79492	−	1.94378i	−2.44936	−	1.41443i	1.75081	+	2.43612i	−2.18145		6.52995e-5i
11.10	−1.38987	+	0.261245i	0.749350	−	1.56156i	1.86350	−	0.726195i	−2.82395	+	0.497938i	−0.633553	+	2.36614i	−2.63326	−	0.256777i	−2.40032	+	1.49615i	−1.87695	−	2.34031i	3.79485	−	1.42981i
11.11	−1.37976	−	0.310270i	−0.0171663	−	1.73197i	1.80747	+	0.856194i	1.75070	−	0.308695i	−0.513691	+	2.39502i	−2.14695	−	1.54616i	−2.22821	−	1.74214i	−2.99941	+	0.0594627i	−2.51132	−	0.117263i
11.12	−1.35727	+	0.397255i	−1.72583	+	0.146654i	1.68438	−	1.07837i	3.07035	−	0.541386i	2.28416	−	0.884644i	−1.38755	−	2.25271i	−1.85777	+	2.13276i	2.95699	−	0.506199i	−3.95223	+	1.95452i
11.13	−1.35660	+	0.399529i	0.345844	−	1.69717i	1.68075	−	1.08401i	0.871331	−	0.153639i	0.208897	+	2.44057i	0.218427	+	2.63672i	−1.84702	+	2.14208i	−2.76078	−	1.17391i	−1.12067	+	0.556550i
11.14	−1.35563	+	0.402825i	1.69073	−	0.376077i	1.67546	−	1.09216i	3.16927	−	0.558828i	−2.14051	+	1.19089i	−1.27230	+	2.31975i	−1.83136	+	2.15549i	2.71713	−	1.27169i	−4.07125	+	2.03423i
11.15	−1.32375	−	0.497675i	−0.154673	−	1.72513i	1.50464	+	1.31760i	3.70389	−	0.653095i	−0.653807	+	2.36062i	2.38269	+	1.15013i	−1.33603	−	2.49299i	−2.95215	+	0.533661i	−5.22806	−	0.978797i
11.16	−1.31708	−	0.515067i	1.23240	+	1.21704i	1.46941	+	1.35677i	−1.44666	+	0.255085i	−0.996311	−	2.23771i	−1.11827	+	2.39781i	−1.23651	−	2.54383i	0.0376115	+	2.99976i	2.03676	+	0.409160i
11.17	−1.29959	−	0.557736i	1.59653	−	0.671641i	1.37786	+	1.44965i	0.781464	−	0.137793i	−2.44943	−	0.0175830i	−2.60906	−	0.439114i	−0.982129	−	2.65244i	2.09780	−	2.14459i	−1.09243	−	0.256776i
11.18	−1.29911	−	0.558856i	1.10309	−	1.33536i	1.37536	+	1.45203i	−1.49610	+	0.263803i	−2.17931	+	1.11831i	1.99141	−	1.74193i	−0.975268	−	2.65497i	−0.566395	−	2.94605i	2.09102	+	0.493396i
11.19	−1.29808	+	0.561242i	−1.72841	−	0.112253i	1.37001	−	1.45707i	−3.58626	+	0.632355i	2.30661	−	0.824343i	1.09900	+	2.40670i	−0.960616	+	2.66030i	2.97480	+	0.388039i	4.30035	−	2.83361i
11.20	−1.29496	+	0.568405i	1.56875	+	0.734184i	1.35383	−	1.47212i	−1.56831	+	0.276535i	−2.44878	−	0.0590536i	2.42629	+	1.05504i	−0.916395	+	2.67586i	1.92195	+	2.30350i	1.87371	−	1.24953i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
189.bf	odd	18	1	inner
756.bs	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.bs.a	✓	840
4.b	odd	2	1	inner	756.2.bs.a	✓	840
7.c	even	3	1		756.2.ci.a	yes	840
27.f	odd	18	1		756.2.ci.a	yes	840
28.g	odd	6	1		756.2.ci.a	yes	840
108.l	even	18	1		756.2.ci.a	yes	840
189.bf	odd	18	1	inner	756.2.bs.a	✓	840
756.bs	even	18	1	inner	756.2.bs.a	✓	840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.bs.a	✓	840	1.a	even	1	1	trivial
756.2.bs.a	✓	840	4.b	odd	2	1	inner
756.2.bs.a	✓	840	189.bf	odd	18	1	inner
756.2.bs.a	✓	840	756.bs	even	18	1	inner
756.2.ci.a	yes	840	7.c	even	3	1
756.2.ci.a	yes	840	27.f	odd	18	1
756.2.ci.a	yes	840	28.g	odd	6	1
756.2.ci.a	yes	840	108.l	even	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(756, [\chi])\).