Newform orbit 207.3.p.a

• Label: 207.3.p.a
• Level: $207$
• Weight: $3$
• Character orbit: 207.p
• Analytic conductor: $5.640$
• Analytic rank: $0$
• Dimension: $920$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	207.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.64034147226\)
Analytic rank:	\(0\)
Dimension:	\(920\)
Relative dimension:	\(46\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 920 q - 9 q^{2} - 16 q^{3} + 79 q^{4} - 11 q^{5} - 13 q^{6} - 11 q^{7} - 70 q^{8} + 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.27268	−	3.67716i	2.99759	−	0.120256i	−8.75762	+	6.88707i	2.12854	+	1.51572i	−4.25717	−	10.8696i	8.14249	+	1.97535i	23.3766	+	15.0233i	8.97108	−	0.720957i	2.86462	−	9.75601i
7.2	−1.22574	−	3.54155i	−1.10465	+	2.78922i	−7.89594	+	6.20944i	4.52852	+	3.22475i	11.2322	+	0.493293i	−4.86503	−	1.18024i	19.0585	+	12.2481i	−6.55951	−	6.16221i	5.86981	−	19.9907i
7.3	−1.15544	−	3.33843i	−2.83228	−	0.989034i	−6.66588	+	5.24211i	1.71939	+	1.22437i	−0.0292864	+	10.5982i	9.73668	+	2.36209i	13.3148	+	8.55689i	7.04362	+	5.60244i	2.10083	−	7.15478i
7.4	−1.10845	−	3.20264i	−0.0772054	+	2.99901i	−5.88406	+	4.62728i	−7.11239	−	5.06471i	9.69033	−	3.07697i	6.69124	+	1.62328i	9.93753	+	6.38646i	−8.98808	−	0.463079i	−8.33676	+	28.3924i
7.5	−1.09022	−	3.15000i	−2.99991	−	0.0231242i	−5.58968	+	4.39578i	−5.03024	−	3.58202i	3.19774	+	9.47492i	−9.32741	−	2.26280i	8.72400	+	5.60658i	8.99893	+	0.138741i	−5.79926	+	19.7505i
7.6	−1.07231	−	3.09825i	−1.03595	−	2.81546i	−5.30508	+	4.17196i	1.66303	+	1.18424i	−7.61212	+	6.22870i	−5.47715	−	1.32874i	7.58207	+	4.87270i	−6.85360	+	5.83336i	1.88578	−	6.42238i
7.7	−1.05439	−	3.04647i	2.58572	+	1.52121i	−5.02501	+	3.95171i	−2.46457	−	1.75501i	1.90794	−	9.48125i	−8.89932	−	2.15895i	6.48905	+	4.17026i	4.37186	+	7.86682i	−2.74796	+	9.35869i
7.8	−0.972781	−	2.81067i	2.06860	−	2.17276i	−3.80933	+	2.99569i	−4.42049	−	3.14782i	−8.11920	−	3.70053i	0.779816	+	0.189181i	2.11714	+	1.36060i	−0.441770	−	8.98915i	−4.54730	+	15.4867i
7.9	−0.903763	−	2.61125i	0.786458	−	2.89508i	−2.85764	+	2.24727i	6.70533	+	4.77485i	−8.27056	+	0.562825i	5.78622	+	1.40372i	−0.847469	−	0.544635i	−7.76297	−	4.55372i	6.40830	−	21.8246i
7.10	−0.829996	−	2.39812i	2.06495	+	2.17623i	−1.91786	+	1.50822i	4.91069	+	3.49688i	3.50495	−	6.75826i	1.67111	+	0.405408i	−3.33065	−	2.14048i	−0.471956	+	8.98762i	4.31009	−	14.6788i
7.11	−0.744077	−	2.14987i	−1.89905	+	2.32242i	−0.924073	+	0.726699i	3.30593	+	2.35415i	6.40593	+	2.35464i	−3.91580	−	0.949962i	−5.40549	−	3.47390i	−1.78725	−	8.82076i	2.60123	−	8.85899i
7.12	−0.715440	−	2.06713i	−2.62803	+	1.44689i	−0.616954	+	0.485178i	−0.803977	−	0.572510i	4.87109	+	4.39731i	8.28005	+	2.00872i	−5.91643	−	3.80226i	4.81304	−	7.60491i	−0.608253	+	2.07152i
7.13	−0.609575	−	1.76125i	2.92012	−	0.687660i	0.413788	−	0.325406i	2.10535	+	1.49921i	−2.99118	−	4.72389i	−5.01490	−	1.21660i	−7.09692	−	4.56092i	8.05425	−	4.01610i	1.35712	−	4.62193i
7.14	−0.586173	−	1.69364i	−1.31237	−	2.69772i	0.619406	−	0.487106i	−4.74071	−	3.37584i	−3.79968	+	3.80401i	3.38933	+	0.822242i	−7.21886	−	4.63928i	−5.55536	+	7.08081i	−2.93857	+	10.0079i
7.15	−0.532010	−	1.53714i	−2.86155	−	0.900848i	1.06445	−	0.837089i	6.19695	+	4.41283i	0.137642	+	4.87787i	−11.4735	−	2.78343i	−7.32656	−	4.70849i	7.37695	+	5.15565i	3.48630	−	11.8732i
7.16	−0.473892	−	1.36922i	0.970202	+	2.83879i	1.49402	−	1.17491i	−3.88296	−	2.76504i	3.42715	−	2.67370i	−9.94573	−	2.41281i	−7.19232	−	4.62222i	−7.11742	+	5.50839i	−1.94585	+	6.62695i
7.17	−0.404726	−	1.16938i	2.70540	+	1.29646i	1.94057	−	1.52608i	−3.91559	−	2.78828i	0.421104	−	3.68835i	12.1228	+	2.94097i	−6.73396	−	4.32765i	5.63840	+	7.01488i	−1.67581	+	5.70729i
7.18	−0.288402	−	0.833282i	2.50659	−	1.64833i	2.53303	−	1.99200i	3.73171	+	2.65734i	−2.09643	−	1.61332i	4.57304	+	1.10941i	−5.35762	−	3.44313i	3.56602	−	8.26338i	1.13808	−	3.87595i
7.19	−0.273874	−	0.791306i	−0.519571	+	2.95467i	2.59305	−	2.03920i	−2.94145	−	2.09460i	2.48034	−	0.398065i	−2.46692	−	0.598468i	−5.14153	−	3.30426i	−8.46009	−	3.07032i	−0.851882	+	2.90124i
7.20	−0.245059	−	0.708051i	−2.38873	−	1.81493i	2.70293	−	2.12561i	4.94001	+	3.51777i	−0.699684	+	2.13611i	8.42154	+	2.04304i	−4.68868	−	3.01323i	2.41205	+	8.67076i	1.28016	−	4.35984i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
23.d	odd	22	1	inner
207.p	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	207.3.p.a	✓	920
9.c	even	3	1	inner	207.3.p.a	✓	920
23.d	odd	22	1	inner	207.3.p.a	✓	920
207.p	odd	66	1	inner	207.3.p.a	✓	920

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
207.3.p.a	✓	920	1.a	even	1	1	trivial
207.3.p.a	✓	920	9.c	even	3	1	inner
207.3.p.a	✓	920	23.d	odd	22	1	inner
207.3.p.a	✓	920	207.p	odd	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(207, [\chi])\).