Newform orbit 165.3.u.a

• Label: 165.3.u.a
• Level: $165$
• Weight: $3$
• Character orbit: 165.u
• Analytic conductor: $4.496$
• Analytic rank: $0$
• Dimension: $352$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	165.u (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.49592436194\)
Analytic rank:	\(0\)
Dimension:	\(352\)
Relative dimension:	\(44\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q - 4 q^{3} - 20 q^{6} - 20 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−3.41566	+	1.74037i	−2.85296	+	0.927699i	6.28673	−	8.65294i	2.96147	−	4.02861i	8.13021	−	8.13390i	−5.04392	+	0.798878i	−4.01529	+	25.3516i	7.27875	−	5.29338i	−3.10413	+	18.9144i
2.2	−3.35793	+	1.71095i	2.99991	+	0.0230476i	5.99720	−	8.25443i	3.92617	+	3.09599i	−10.1129	+	5.05531i	−3.81061	+	0.603541i	−3.65702	+	23.0895i	8.99894	+	0.138282i	−18.4809	−	3.67865i
2.3	−3.16844	+	1.61440i	0.753650	−	2.90379i	5.08156	−	6.99417i	−0.0329201	−	4.99989i	2.29999	+	10.4172i	7.87318	−	1.24699i	−2.58409	+	16.3153i	−7.86402	−	4.37689i	8.17612	+	15.7887i
2.4	−3.12189	+	1.59068i	−0.753644	+	2.90379i	4.86479	−	6.69581i	−2.80484	+	4.13919i	−2.26622	−	10.2641i	−0.274295	+	0.0434441i	−2.34399	+	14.7994i	−7.86404	−	4.37686i	2.17226	−	17.3837i
2.5	−2.89885	+	1.47704i	2.34370	+	1.87272i	3.87053	−	5.32733i	−4.07210	−	2.90137i	−9.56010	−	1.96698i	6.99164	−	1.10737i	−1.31561	+	8.30645i	1.98587	+	8.77817i	16.0898	+	2.39599i
2.6	−2.70586	+	1.37870i	−2.11065	−	2.13194i	3.06971	−	4.22509i	−4.99099	−	0.300069i	8.65044	+	2.85876i	−12.8458	+	2.03458i	−0.580774	+	3.66686i	−0.0903072	+	8.99955i	13.9186	−	6.06915i
2.7	−2.61603	+	1.33293i	2.08120	−	2.16070i	2.71577	−	3.73793i	−3.71190	+	3.34989i	−2.56442	+	8.42655i	−0.717689	+	0.113671i	−0.284921	+	1.79892i	−0.337214	−	8.99368i	5.24527	−	13.7111i
2.8	−2.59272	+	1.32106i	−0.984643	−	2.83381i	2.62586	−	3.61418i	4.66289	+	1.80485i	6.29652	+	6.04650i	−0.180347	+	0.0285641i	−0.212751	+	1.34326i	−7.06095	+	5.58058i	−14.4739	+	1.48048i
2.9	−2.45055	+	1.24862i	−2.99014	−	0.242973i	2.09502	−	2.88355i	1.88570	+	4.63078i	7.63089	−	3.13813i	5.23794	−	0.829608i	0.187475	−	1.18367i	8.88193	+	1.45305i	−10.4031	−	8.99346i
2.10	−2.31412	+	1.17910i	2.09492	+	2.14739i	1.61373	−	2.22112i	2.53431	−	4.31014i	−7.37991	−	2.49920i	−10.9433	+	1.73326i	0.509713	−	3.21820i	−0.222604	+	8.99725i	−0.782609	+	12.9624i
2.11	−2.10253	+	1.07129i	−1.04272	+	2.81296i	0.921830	−	1.26879i	4.04150	−	2.94385i	−0.821151	−	7.03139i	8.53769	−	1.35224i	0.897640	−	5.66748i	−6.82547	−	5.86626i	−5.34365	+	10.5192i
2.12	−1.75611	+	0.894784i	−2.06562	+	2.17560i	−0.0678477	+	0.0933843i	−4.41189	−	2.35270i	1.68077	−	5.66888i	−3.42896	+	0.543093i	1.26888	−	8.01137i	−0.466455	−	8.98790i	9.85294	+	0.183912i
2.13	−1.54850	+	0.788998i	2.72683	−	1.25075i	−0.575819	+	0.792547i	4.99166	−	0.288596i	−3.23565	+	4.08825i	0.645086	−	0.102172i	1.35382	−	8.54766i	5.87123	−	6.82119i	−7.50187	+	4.38530i
2.14	−1.40349	+	0.715114i	0.567889	+	2.94576i	−0.892746	+	1.22876i	1.46972	+	4.77911i	−2.90358	−	3.72824i	−10.8778	+	1.72287i	1.35990	−	8.58609i	−8.35500	+	3.34573i	−5.48035	−	5.65642i
2.15	−1.36249	+	0.694225i	−2.34721	−	1.86832i	−0.976703	+	1.34432i	−0.225156	−	4.99493i	4.49509	+	0.916076i	4.44182	−	0.703515i	1.35435	−	8.55101i	2.01879	+	8.77066i	3.77438	+	6.64924i
2.16	−1.26083	+	0.642427i	2.47776	+	1.69137i	−1.17415	+	1.61608i	1.51839	+	4.76387i	−4.21062	−	0.540754i	13.0676	−	2.06971i	1.32766	−	8.38250i	3.27856	+	8.38159i	−4.97488	−	5.03100i
2.17	−1.17271	+	0.597524i	2.75827	−	1.17980i	−1.33294	+	1.83463i	−2.88973	−	4.08038i	−2.52969	+	3.03169i	−0.540506	+	0.0856077i	1.29048	−	8.14776i	6.21615	−	6.50841i	5.82692	+	3.05840i
2.18	−0.884720	+	0.450787i	−0.430470	−	2.96896i	−1.77162	+	2.43843i	−4.30427	+	2.54427i	1.71921	+	2.43264i	9.39583	−	1.48815i	1.08950	−	6.87883i	−8.62939	+	2.55609i	2.66115	−	4.19127i
2.19	−0.527698	+	0.268876i	−2.91682	+	0.701549i	−2.14497	+	2.95230i	−3.13577	+	3.89447i	1.35057	−	1.15447i	1.22951	−	0.194736i	0.708689	−	4.47448i	8.01566	−	4.09258i	0.607614	−	2.89824i
2.20	−0.510389	+	0.260056i	0.754487	−	2.90358i	−2.15827	+	2.97061i	1.85470	+	4.64328i	0.370010	+	1.67816i	−9.47897	+	1.50132i	0.687471	−	4.34052i	−7.86150	−	4.38142i	−2.15413	−	1.88755i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
11.d	odd	10	1	inner
15.e	even	4	1	inner
33.f	even	10	1	inner
55.l	even	20	1	inner
165.u	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	165.3.u.a	✓	352
3.b	odd	2	1	inner	165.3.u.a	✓	352
5.c	odd	4	1	inner	165.3.u.a	✓	352
11.d	odd	10	1	inner	165.3.u.a	✓	352
15.e	even	4	1	inner	165.3.u.a	✓	352
33.f	even	10	1	inner	165.3.u.a	✓	352
55.l	even	20	1	inner	165.3.u.a	✓	352
165.u	odd	20	1	inner	165.3.u.a	✓	352

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.3.u.a	✓	352	1.a	even	1	1	trivial
165.3.u.a	✓	352	3.b	odd	2	1	inner
165.3.u.a	✓	352	5.c	odd	4	1	inner
165.3.u.a	✓	352	11.d	odd	10	1	inner
165.3.u.a	✓	352	15.e	even	4	1	inner
165.3.u.a	✓	352	33.f	even	10	1	inner
165.3.u.a	✓	352	55.l	even	20	1	inner
165.3.u.a	✓	352	165.u	odd	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).