Newform orbit 243.2.i.a

• Label: 243.2.i.a
• Level: $243$
• Weight: $2$
• Character orbit: 243.i
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.i (of order \(81\), degree \(54\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(1404\)
Relative dimension:	\(26\) over \(\Q(\zeta_{81})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{81}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1404 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 54 q^{5} - 54 q^{6} - 54 q^{7} - 54 q^{8} - 54 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.78387	−	0.108027i	−1.71620	+	0.233793i	5.74428	+	0.446480i	2.42193	+	3.00272i	4.80293	−	0.465452i	0.433636	−	0.197112i	−10.4088	−	1.21662i	2.89068	−	0.802469i	−6.41795	−	8.62081i
4.2	−2.42984	−	0.0942887i	1.64930	−	0.528965i	3.90123	+	0.303227i	−0.723370	−	0.896838i	−4.05741	+	1.12979i	1.78613	−	0.811895i	−4.62031	−	0.540037i	2.44039	−	1.74485i	1.67311	+	2.24737i
4.3	−2.34586	−	0.0910299i	−0.823564	−	1.52373i	3.50077	+	0.272102i	−1.67953	−	2.08230i	1.79326	+	3.64941i	−3.42025	+	1.55470i	−3.52404	−	0.411902i	−1.64349	+	2.50977i	3.75040	+	5.03765i
4.4	−2.16400	−	0.0839729i	−1.61426	+	0.627826i	2.68184	+	0.208449i	−2.41283	−	2.99144i	3.54597	−	1.22306i	2.23840	−	1.01748i	−1.48402	−	0.173458i	2.21167	−	2.02695i	4.97016	+	6.67608i
4.5	−2.03967	−	0.0791483i	−0.137599	−	1.72658i	2.15999	+	0.167888i	1.00485	+	1.24581i	0.144001	+	3.53253i	2.22760	−	1.01257i	−0.337568	−	0.0394561i	−2.96213	+	0.475151i	−1.95095	−	2.62057i
4.6	−1.94351	−	0.0754171i	1.58908	+	0.689068i	1.77756	+	0.138163i	2.24632	+	2.78500i	−3.03643	−	1.45905i	−3.31801	+	1.50822i	0.419355	+	0.0490156i	2.05037	+	2.18997i	−4.15571	−	5.58208i
4.7	−1.55794	−	0.0604553i	−0.167548	+	1.72393i	0.429550	+	0.0333873i	1.31451	+	1.62974i	0.365251	−	2.67566i	3.10528	−	1.41152i	2.42995	+	0.284021i	−2.94386	−	0.577680i	−1.94941	−	2.61851i
4.8	−1.53351	−	0.0595072i	−1.14640	+	1.29838i	0.354129	+	0.0275251i	0.161696	+	0.200471i	1.83527	−	1.92285i	−4.41863	+	2.00851i	2.50715	+	0.293044i	−0.371555	−	2.97690i	−0.236033	−	0.317047i
4.9	−1.04754	−	0.0406495i	0.956457	+	1.44402i	−0.898290	−	0.0698206i	−2.13661	−	2.64898i	−0.943232	−	1.55155i	1.55684	−	0.707671i	3.02065	+	0.353063i	−1.17038	+	2.76228i	2.13051	+	2.86177i
4.10	−0.829136	−	0.0321742i	0.755643	−	1.55853i	−1.30755	−	0.101631i	−1.06671	−	1.32252i	−0.676676	+	1.26792i	−2.20706	+	1.00323i	2.72917	+	0.318994i	−1.85801	−	2.35538i	0.841900	+	1.13087i
4.11	−0.815384	−	0.0316406i	1.60488	−	0.651417i	−1.33014	−	0.103386i	0.303466	+	0.376239i	−1.32921	+	0.480376i	0.274968	−	0.124988i	2.70226	+	0.315849i	2.15131	−	2.09090i	−0.235537	−	0.316381i
4.12	−0.479029	−	0.0185885i	−1.73158	−	0.0404413i	−1.76486	−	0.137176i	−0.169194	−	0.209768i	0.828725	+	0.0515600i	0.242417	−	0.110192i	1.79517	+	0.209825i	2.99673	+	0.140055i	0.0771496	+	0.103630i
4.13	0.0783119	+	0.00303886i	−0.825726	−	1.52256i	−1.98786	−	0.154509i	2.71771	+	3.36944i	−0.0600373	−	0.121743i	−3.85378	+	1.75176i	−0.310885	−	0.0363373i	−1.63635	+	2.51443i	0.202590	+	0.272126i
4.14	0.242273	+	0.00940130i	1.46469	+	0.924490i	−1.93538	−	0.150429i	1.25182	+	1.55201i	0.346164	+	0.237749i	1.52000	−	0.690924i	−0.949108	−	0.110935i	1.29064	+	2.70818i	0.288691	+	0.387780i
4.15	0.337931	+	0.0131133i	0.210129	+	1.71926i	−1.87996	−	0.146122i	−0.812985	−	1.00794i	0.0484641	+	0.583746i	−2.68541	+	1.22067i	−1.30518	−	0.152554i	−2.91169	+	0.722532i	−0.261516	−	0.351276i
4.16	0.498870	+	0.0193584i	0.114249	−	1.72828i	−1.74549	−	0.135670i	0.793310	+	0.983550i	0.0904521	−	0.859974i	4.69382	−	2.13360i	−1.85988	−	0.217389i	−2.97389	−	0.394908i	0.376718	+	0.506021i
4.17	0.905616	+	0.0351420i	−1.29585	+	1.14925i	−1.17508	−	0.0913345i	−1.68295	−	2.08654i	−1.21393	+	0.995240i	0.467604	−	0.212552i	−2.86130	−	0.334438i	0.358453	−	2.97851i	−1.45079	−	1.94874i
4.18	1.08187	+	0.0419816i	1.65295	−	0.517453i	−0.825297	−	0.0641472i	−2.72770	−	3.38181i	1.81001	−	0.490425i	3.01742	−	1.37158i	−3.04091	−	0.355431i	2.46448	−	1.71065i	−2.80905	−	3.77321i
4.19	1.14737	+	0.0445232i	−1.35679	−	1.07662i	−0.679512	−	0.0528158i	−1.27759	−	1.58397i	−1.50880	−	1.29569i	−1.57880	+	0.717652i	−3.05824	−	0.357457i	0.681758	+	2.92151i	−1.39535	−	1.87428i
4.20	1.55634	+	0.0603929i	−0.263751	+	1.71185i	0.424549	+	0.0329986i	1.93917	+	2.40419i	−0.513869	+	2.64829i	−0.871256	+	0.396034i	−2.43520	−	0.284635i	−2.86087	−	0.903006i	2.87280	+	3.85885i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
243.i	even	81	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.2.i.a	✓	1404
3.b	odd	2	1		729.2.i.a		1404
243.i	even	81	1	inner	243.2.i.a	✓	1404
243.j	odd	162	1		729.2.i.a		1404

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.2.i.a	✓	1404	1.a	even	1	1	trivial
243.2.i.a	✓	1404	243.i	even	81	1	inner
729.2.i.a		1404	3.b	odd	2	1
729.2.i.a		1404	243.j	odd	162	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(243, [\chi])\).