Newform orbit 585.2.cm.a

• Label: 585.2.cm.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.cm
• Analytic conductor: $4.671$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.cm (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.67124851824\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 24 q^{6} + 4 q^{7} - 36 q^{8}+O(q^{10}) \)

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	−1.97601	+	1.97601i	−1.72888	+	0.104699i		−	5.80920i	−0.965926	−	0.258819i	3.20940	−	3.62317i	3.76955	+	1.01005i	7.52701	+	7.52701i	2.97808	−	0.362024i	2.42010	−	1.39725i
11.2	−1.95804	+	1.95804i	1.69798	−	0.341849i		−	5.66785i	0.965926	+	0.258819i	−2.65536	+	3.99407i	−1.12826	−	0.302317i	7.18181	+	7.18181i	2.76628	−	1.16091i	−2.39810	+	1.38454i
11.3	−1.80633	+	1.80633i	0.0891243	−	1.72976i		−	4.52566i	−0.965926	−	0.258819i	2.96352	+	3.28550i	−3.55730	−	0.953176i	4.56218	+	4.56218i	−2.98411	−	0.308327i	2.21229	−	1.27727i
11.4	−1.63542	+	1.63542i	1.32331	+	1.11752i		−	3.34923i	−0.965926	−	0.258819i	−3.99179	+	0.336552i	2.92001	+	0.782415i	2.20656	+	2.20656i	0.502295	+	2.95765i	2.00298	−	1.15642i
11.5	−1.61918	+	1.61918i	−1.59381	+	0.678060i		−	3.24348i	0.965926	+	0.258819i	1.48276	−	3.67856i	−0.733430	−	0.196522i	2.01341	+	2.01341i	2.08047	−	2.16140i	−1.98308	+	1.14493i
11.6	−1.59769	+	1.59769i	−0.917692	+	1.46896i		−	3.10520i	0.965926	+	0.258819i	−0.880751	−	3.81312i	2.65105	+	0.710347i	1.76576	+	1.76576i	−1.31568	−	2.69611i	−1.95676	+	1.12973i
11.7	−1.53708	+	1.53708i	0.382966	−	1.68918i		−	2.72522i	0.965926	+	0.258819i	2.00776	+	3.18505i	0.186034	+	0.0498475i	1.11472	+	1.11472i	−2.70667	−	1.29380i	−1.88253	+	1.08688i
11.8	−1.52334	+	1.52334i	1.49953	+	0.866843i		−	2.64114i	0.965926	+	0.258819i	−3.60479	+	0.963794i	1.24140	+	0.332631i	0.976667	+	0.976667i	1.49717	+	2.59971i	−1.86570	+	1.07716i
11.9	−1.46697	+	1.46697i	−0.594527	−	1.62682i		−	2.30401i	−0.965926	−	0.258819i	3.25865	+	1.51434i	2.58328	+	0.692188i	0.445979	+	0.445979i	−2.29307	+	1.93438i	1.79667	−	1.03731i
11.10	−1.42201	+	1.42201i	−0.734312	+	1.56869i		−	2.04423i	−0.965926	−	0.258819i	−1.18649	−	3.27489i	−1.26945	−	0.340148i	0.0628911	+	0.0628911i	−1.92157	−	2.30381i	1.74160	−	1.00551i
11.11	−1.34437	+	1.34437i	−1.72320	+	0.174851i		−	1.61467i	−0.965926	−	0.258819i	2.08156	−	2.55169i	−3.00692	−	0.805701i	−0.518028	−	0.518028i	2.93885	−	0.602608i	1.64651	−	0.950614i
11.12	−1.31977	+	1.31977i	0.842789	+	1.51318i		−	1.48358i	0.965926	+	0.258819i	−3.10933	−	0.884758i	−4.84339	−	1.29778i	−0.681550	−	0.681550i	−1.57941	+	2.55058i	−1.61638	+	0.933218i
11.13	−1.24886	+	1.24886i	1.17421	−	1.27328i		−	1.11930i	0.965926	+	0.258819i	0.123720	+	3.05657i	−3.10288	−	0.831413i	−1.09987	−	1.09987i	−0.242464	−	2.99019i	−1.52953	+	0.883077i
11.14	−1.21445	+	1.21445i	1.70129	−	0.325000i		−	0.949755i	−0.965926	−	0.258819i	−1.67142	+	2.46081i	−3.76044	−	1.00761i	−1.27547	−	1.27547i	2.78875	−	1.10583i	1.48739	−	0.858742i
11.15	−1.08997	+	1.08997i	−1.49138	−	0.880790i		−	0.376070i	0.965926	+	0.258819i	2.58559	−	0.665522i	4.54048	+	1.21662i	−1.77004	−	1.77004i	1.44842	+	2.62718i	−1.33494	+	0.770725i
11.16	−1.04871	+	1.04871i	1.68095	−	0.417629i		−	0.199578i	−0.965926	−	0.258819i	−1.32485	+	2.20080i	−0.536552	−	0.143769i	−1.88812	−	1.88812i	2.65117	−	1.40403i	1.28440	−	0.741549i
11.17	−0.841957	+	0.841957i	0.384397	+	1.68886i			0.582217i	−0.965926	−	0.258819i	−1.74559	−	1.09830i	3.93897	+	1.05544i	−2.17412	−	2.17412i	−2.70448	+	1.29838i	1.03118	−	0.595353i
11.18	−0.814519	+	0.814519i	0.169661	+	1.72372i			0.673118i	0.965926	+	0.258819i	−1.54220	−	1.26581i	2.16964	+	0.581354i	−2.17731	−	2.17731i	−2.94243	+	0.584896i	−0.997578	+	0.575952i
11.19	−0.765254	+	0.765254i	−1.61613	+	0.623007i			0.828771i	−0.965926	−	0.258819i	0.759988	−	1.71351i	3.59901	+	0.964350i	−2.16473	−	2.16473i	2.22372	−	2.01372i	0.937241	−	0.541117i
11.20	−0.757603	+	0.757603i	−0.500254	−	1.65824i			0.852077i	0.965926	+	0.258819i	1.63528	+	0.877290i	−0.842470	−	0.225739i	−2.16074	−	2.16074i	−2.49949	+	1.65908i	−0.927870	+	0.535706i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	585.2.cm.a	✓	224
9.d	odd	6	1		585.2.dd.a	yes	224
13.f	odd	12	1		585.2.dd.a	yes	224
117.x	even	12	1	inner	585.2.cm.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.cm.a	✓	224	1.a	even	1	1	trivial
585.2.cm.a	✓	224	117.x	even	12	1	inner
585.2.dd.a	yes	224	9.d	odd	6	1
585.2.dd.a	yes	224	13.f	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(585, [\chi])\).