Newform orbit 468.2.cc.a

• Label: 468.2.cc.a
• Level: $468$
• Weight: $2$
• Character orbit: 468.cc
• Analytic conductor: $3.737$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.cc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.73699881460\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 4 q^{5} + 16 q^{6} - 14 q^{8} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
115.1	−1.41386	−	0.0314817i	1.53773	−	0.797113i	1.99802	+	0.0890216i	−0.811368	+	3.02807i	−2.19923	+	1.07860i	0.000497755	−	0.00185765i	−2.82212	−	0.188765i	1.72922	−	2.45149i	1.24249	−	4.25573i
115.2	−1.41215	−	0.0764560i	0.754680	−	1.55899i	1.98831	+	0.215934i	0.0534430	−	0.199452i	−1.18491	+	2.14382i	−0.369700	+	1.37974i	−2.79127	−	0.456948i	−1.86092	−	2.35308i	−0.0907186	+	0.277569i
115.3	−1.41187	+	0.0813270i	−0.964885	−	1.43840i	1.98677	−	0.229647i	0.528383	−	1.97195i	1.47928	+	1.95237i	1.08687	−	4.05627i	−2.78639	+	0.485810i	−1.13799	+	2.77578i	−0.585637	+	2.82712i
115.4	−1.39423	−	0.236908i	−1.38695	+	1.03749i	1.88775	+	0.660609i	0.149518	−	0.558008i	2.17951	−	1.11792i	−0.185558	+	0.692511i	−2.47545	−	1.36826i	0.847239	−	2.87788i	−0.340659	+	0.742568i
115.5	−1.39181	+	0.250728i	−0.444216	+	1.67412i	1.87427	−	0.697932i	−0.853671	+	3.18594i	0.198516	−	2.44143i	1.01605	−	3.79193i	−2.43364	+	1.44132i	−2.60534	−	1.48734i	0.389342	−	4.64827i
115.6	−1.38917	−	0.264983i	−1.14894	−	1.29612i	1.85957	+	0.736212i	0.854512	−	3.18908i	1.25262	+	2.10498i	−0.948665	+	3.54046i	−2.38816	−	1.51548i	−0.359865	+	2.97834i	−2.03211	+	4.20373i
115.7	−1.37919	+	0.312773i	−1.60341	−	0.655028i	1.80435	−	0.862750i	−0.981823	+	3.66421i	2.41629	+	0.401905i	−1.20364	+	4.49205i	−2.21870	+	1.75425i	2.14188	+	2.10056i	0.208055	−	5.36074i
115.8	−1.37328	+	0.337770i	1.39007	+	1.03329i	1.77182	−	0.927710i	0.686092	−	2.56053i	−2.25798	−	0.949481i	1.00005	−	3.73222i	−2.11986	+	1.87248i	0.864607	+	2.87271i	−0.0773283	+	3.74808i
115.9	−1.37187	−	0.343481i	1.31550	+	1.12670i	1.76404	+	0.942420i	−0.274018	+	1.02265i	−1.41769	−	1.99754i	−0.0510972	+	0.190697i	−2.09633	−	1.89879i	0.461073	+	2.96436i	0.727177	−	1.30882i
115.10	−1.34980	+	0.421942i	−0.0914946	+	1.72963i	1.64393	−	1.13908i	0.782778	−	2.92137i	−0.606306	−	2.37327i	−0.419278	+	1.56477i	−1.73835	+	2.23117i	−2.98326	−	0.316504i	0.176053	+	4.27355i
115.11	−1.23645	+	0.686437i	−1.48542	+	0.890801i	1.05761	−	1.69749i	−0.0247001	+	0.0921819i	1.22517	−	2.12108i	−0.223238	+	0.833136i	−0.142463	+	2.82484i	1.41295	−	2.64643i	−0.0327367	−	0.130933i
115.12	−1.23263	−	0.693271i	1.72931	−	0.0974217i	1.03875	+	1.70909i	1.08036	−	4.03196i	−2.19914	−	1.07880i	−0.618047	+	2.30658i	−0.0955285	−	2.82681i	2.98102	−	0.336944i	−4.12692	+	4.22093i
115.13	−1.21204	+	0.728668i	−0.162541	−	1.72441i	0.938086	−	1.76635i	−0.651818	+	2.43262i	1.45353	+	1.97161i	0.521026	−	1.94450i	0.150086	+	2.82444i	−2.94716	+	0.560574i	−0.982540	−	3.42339i
115.14	−1.21159	−	0.729411i	−1.01384	−	1.40432i	0.935918	+	1.76750i	−0.783682	+	2.92474i	0.204035	+	2.44098i	0.484520	−	1.80825i	0.155283	−	2.82416i	−0.944248	+	2.84752i	3.08285	−	2.97197i
115.15	−1.20458	−	0.740939i	0.998880	−	1.41500i	0.902019	+	1.78504i	0.471183	−	1.75848i	−2.25166	+	0.964376i	1.26471	−	4.71995i	0.236051	−	2.81856i	−1.00448	−	2.82684i	−1.87050	+	1.76911i
115.16	−1.20232	+	0.744590i	1.68572	+	0.397923i	0.891171	−	1.79048i	0.0495489	−	0.184919i	−2.32307	+	0.776740i	−1.29538	+	4.83441i	0.261697	+	2.81629i	2.68331	+	1.34157i	0.0781151	+	0.259226i
115.17	−1.17382	−	0.788769i	−1.73195	−	0.0187955i	0.755688	+	1.85174i	−0.0602769	+	0.224957i	2.01816	+	1.38817i	0.0867686	−	0.323825i	0.573556	−	2.76966i	2.99929	+	0.0651056i	0.248193	−	0.216513i
115.18	−1.10245	+	0.885777i	1.68495	−	0.401183i	0.430798	−	1.95305i	−0.273463	+	1.02058i	−1.50221	+	1.93477i	0.758954	−	2.83245i	1.25503	+	2.53474i	2.67810	−	1.35195i	−0.602525	−	1.36737i
115.19	−1.09282	+	0.897637i	−1.72587	−	0.146238i	0.388495	−	1.96191i	0.747664	−	2.79032i	2.01732	−	1.38939i	0.403684	−	1.50657i	1.33653	+	2.49273i	2.95723	+	0.504775i	1.68764	+	3.72044i
115.20	−1.02045	−	0.979120i	0.243831	−	1.71480i	0.0826497	+	1.99829i	−0.175064	+	0.653347i	−1.92781	+	1.51114i	−1.04572	+	3.90270i	1.87223	−	2.12009i	−2.88109	−	0.836243i	0.818349	−	0.495301i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
117.w	odd	12	1	inner
468.cc	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	468.2.cc.a	✓	320
4.b	odd	2	1	inner	468.2.cc.a	✓	320
9.c	even	3	1		468.2.cf.a	yes	320
13.f	odd	12	1		468.2.cf.a	yes	320
36.f	odd	6	1		468.2.cf.a	yes	320
52.l	even	12	1		468.2.cf.a	yes	320
117.w	odd	12	1	inner	468.2.cc.a	✓	320
468.cc	even	12	1	inner	468.2.cc.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.2.cc.a	✓	320	1.a	even	1	1	trivial
468.2.cc.a	✓	320	4.b	odd	2	1	inner
468.2.cc.a	✓	320	117.w	odd	12	1	inner
468.2.cc.a	✓	320	468.cc	even	12	1	inner
468.2.cf.a	yes	320	9.c	even	3	1
468.2.cf.a	yes	320	13.f	odd	12	1
468.2.cf.a	yes	320	36.f	odd	6	1
468.2.cf.a	yes	320	52.l	even	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).