Embedded newform 243.2.i.a.4.6

• Label: 243.2.i.a.4.6
• Level: $243$
• Weight: $2$
• Character: 243.4
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $1404$
• 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:	$\mathrm{SU}(2)[C_{81}]$

Embedding invariants

Embedding label			4.6
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94351 - 0.0754171i) q^{2} +(1.58908 + 0.689068i) q^{3} +(1.77756 + 0.138163i) q^{4} +(2.24632 + 2.78500i) q^{5} +(-3.03643 - 1.45905i) q^{6} +(-3.31801 + 1.50822i) q^{7} +(0.419355 + 0.0490156i) q^{8} +(2.05037 + 2.18997i) q^{9} +(-4.15571 - 5.58208i) q^{10} +(0.0227097 + 1.17090i) q^{11} +(2.72949 + 1.44441i) q^{12} +(-5.58550 - 1.79092i) q^{13} +(6.56233 - 2.68101i) q^{14} +(1.65053 + 5.97346i) q^{15} +(-4.33434 - 0.677879i) q^{16} +(-0.837086 - 2.79606i) q^{17} +(-3.81976 - 4.41087i) q^{18} +(6.10696 - 1.44738i) q^{19} +(3.60818 + 5.26086i) q^{20} +(-6.31186 + 0.110355i) q^{21} +(0.0441697 - 2.27738i) q^{22} +(-0.447514 + 0.319863i) q^{23} +(0.632615 + 0.366854i) q^{24} +(-1.65174 + 7.62530i) q^{25} +(10.7204 + 3.90191i) q^{26} +(1.74917 + 4.89289i) q^{27} +(-6.10635 + 2.22253i) q^{28} +(1.21646 + 8.90609i) q^{29} +(-2.75733 - 11.7340i) q^{30} +(1.14389 + 1.12192i) q^{31} +(7.54413 + 1.48162i) q^{32} +(-0.770744 + 1.87631i) q^{33} +(1.41602 + 5.49731i) q^{34} +(-11.6537 - 5.85270i) q^{35} +(3.34209 + 4.17609i) q^{36} +(3.68159 + 2.42142i) q^{37} +(-11.9781 + 2.35242i) q^{38} +(-7.64176 - 6.69470i) q^{39} +(0.805496 + 1.27801i) q^{40} +(0.239638 - 0.454941i) q^{41} +(12.2755 + 0.261546i) q^{42} +(3.67014 - 10.7264i) q^{43} +(-0.121408 + 2.08449i) q^{44} +(-1.49328 + 10.6296i) q^{45} +(0.893871 - 0.587908i) q^{46} +(5.57420 - 5.46714i) q^{47} +(-6.42053 - 4.06386i) q^{48} +(4.13181 - 4.73453i) q^{49} +(3.78526 - 14.6953i) q^{50} +(0.596477 - 5.01999i) q^{51} +(-9.68112 - 3.95517i) q^{52} +(5.29113 + 4.43979i) q^{53} +(-3.03053 - 9.64131i) q^{54} +(-3.20995 + 2.69347i) q^{55} +(-1.46535 + 0.469846i) q^{56} +(10.7018 + 1.90811i) q^{57} +(-1.69254 - 17.4008i) q^{58} +(3.01295 - 1.66247i) q^{59} +(2.10862 + 10.8462i) q^{60} +(0.761049 - 0.0591534i) q^{61} +(-2.13855 - 2.26673i) q^{62} +(-10.1061 - 4.17394i) q^{63} +(-6.01280 - 1.42506i) q^{64} +(-7.55910 - 19.5786i) q^{65} +(1.63946 - 3.58851i) q^{66} +(1.71436 - 12.5513i) q^{67} +(-1.10166 - 5.08583i) q^{68} +(-0.931544 + 0.199923i) q^{69} +(22.2077 + 12.2537i) q^{70} +(0.319770 - 0.741310i) q^{71} +(0.752491 + 1.01888i) q^{72} +(-5.40871 + 7.26516i) q^{73} +(-6.97259 - 4.98371i) q^{74} +(-7.87911 + 10.9791i) q^{75} +(11.0555 - 1.72904i) q^{76} +(-1.84133 - 3.85082i) q^{77} +(14.3469 + 13.5875i) q^{78} +(-0.860602 + 1.36544i) q^{79} +(-7.84842 - 13.5939i) q^{80} +(-0.591952 + 8.98051i) q^{81} +(-0.500049 + 0.866110i) q^{82} +(0.582126 + 1.10514i) q^{83} +(-11.2350 - 0.675903i) q^{84} +(5.90666 - 8.61213i) q^{85} +(-7.94191 + 20.5701i) q^{86} +(-4.20384 + 14.9907i) q^{87} +(-0.0478691 + 0.492137i) q^{88} +(1.38007 + 3.19935i) q^{89} +(3.70386 - 20.5462i) q^{90} +(21.2338 - 2.48188i) q^{91} +(-0.839676 + 0.506747i) q^{92} +(1.04466 + 2.57104i) q^{93} +(-11.2458 + 10.2051i) q^{94} +(17.7491 + 13.7566i) q^{95} +(10.9673 + 7.55283i) q^{96} +(1.43287 - 1.77647i) q^{97} +(-8.38728 + 8.89000i) q^{98} +(-2.51768 + 2.45052i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	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 - 54 * q^10 - 54 * q^11 - 54 * q^12 - 54 * q^13 - 54 * q^14 - 54 * q^15 - 54 * q^16 - 54 * q^17 - 54 * q^18 - 54 * q^19 - 54 * q^20 - 54 * q^21 - 54 * q^22 - 54 * q^23 - 54 * q^24 - 54 * q^25 - 54 * q^26 - 54 * q^27 - 54 * q^28 - 54 * q^29 - 54 * q^30 - 54 * q^31 - 54 * q^32 - 54 * q^33 - 54 * q^34 - 54 * q^35 - 54 * q^36 - 54 * q^37 - 54 * q^38 - 54 * q^39 - 54 * q^40 - 54 * q^41 - 54 * q^42 - 54 * q^43 - 54 * q^44 - 54 * q^45 - 54 * q^46 - 54 * q^47 - 54 * q^48 - 54 * q^49 - 54 * q^50 - 54 * q^51 - 54 * q^52 - 54 * q^53 - 54 * q^54 - 54 * q^55 - 54 * q^56 - 54 * q^57 - 54 * q^58 - 54 * q^59 - 54 * q^60 - 54 * q^61 - 54 * q^62 - 54 * q^63 - 54 * q^64 + 54 * q^66 - 54 * q^67 + 135 * q^68 + 54 * q^69 - 54 * q^70 + 54 * q^71 + 270 * q^72 - 54 * q^73 + 162 * q^74 + 81 * q^75 - 54 * q^76 + 162 * q^77 + 162 * q^78 - 54 * q^79 + 351 * q^80 + 54 * q^81 - 27 * q^82 + 54 * q^83 + 324 * q^84 - 54 * q^85 + 162 * q^86 + 162 * q^87 - 54 * q^88 + 81 * q^89 + 162 * q^90 - 54 * q^91 + 270 * q^92 + 54 * q^93 - 54 * q^94 + 54 * q^95 + 135 * q^96 - 54 * q^97 + 54 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{81}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94351	−	0.0754171i	−1.37427	−	0.0533279i	−0.658982	−	0.752158i	\(-0.729013\pi\)
							−0.715287	+	0.698830i	\(0.753704\pi\)
\(3\)	1.58908	+	0.689068i	0.917458	+	0.397833i
							\(5\)	2.24632	+	2.78500i	1.00458	+	1.24549i	0.969241	+	0.246113i	\(0.0791534\pi\)
0.0353423	+	0.999375i	\(0.488748\pi\)
\(7\)	−3.31801	+	1.50822i	−1.25409	+	0.570054i	−0.927043	−	0.374954i	\(-0.877658\pi\)
							−0.327047	+	0.945008i	\(0.606053\pi\)
\(11\)	0.0227097	+	1.17090i	0.00684722	+	0.353041i	0.987530	+	0.157434i	\(0.0503220\pi\)
							−0.980682	+	0.195607i	\(0.937332\pi\)
\(13\)	−5.58550	−	1.79092i	−1.54914	−	0.496711i	−0.596964	−	0.802268i	\(-0.703627\pi\)
							−0.952174	+	0.305557i	\(0.901157\pi\)
\(17\)	−0.837086	−	2.79606i	−0.203023	−	0.678145i	−0.997488	−	0.0708348i	\(-0.977434\pi\)
							0.794465	−	0.607310i	\(-0.207751\pi\)
\(19\)	6.10696	−	1.44738i	1.40103	−	0.332051i	0.540456	−	0.841372i	\(-0.318252\pi\)
							0.860576	+	0.509321i	\(0.170103\pi\)
\(23\)	−0.447514	+	0.319863i	−0.0933130	+	0.0666961i	−0.627192	−	0.778865i	\(-0.715796\pi\)
							0.533879	+	0.845561i	\(0.320734\pi\)
\(29\)	1.21646	+	8.90609i	0.225892	+	1.65382i	0.658753	+	0.752359i	\(0.271084\pi\)
							−0.432861	+	0.901461i	\(0.642496\pi\)
\(31\)	1.14389	+	1.12192i	0.205449	+	0.201503i	0.795655	−	0.605749i	\(-0.207127\pi\)
							−0.590207	+	0.807252i	\(0.700954\pi\)
\(37\)	3.68159	+	2.42142i	0.605250	+	0.398079i	0.814820	−	0.579715i	\(-0.196836\pi\)
							−0.209570	+	0.977794i	\(0.567206\pi\)
\(41\)	0.239638	−	0.454941i	0.0374251	−	0.0710498i	−0.865336	−	0.501193i	\(-0.832895\pi\)
							0.902761	+	0.430143i	\(0.141537\pi\)
\(43\)	3.67014	−	10.7264i	0.559691	−	1.63576i	−0.195832	−	0.980637i	\(-0.562741\pi\)
							0.755523	−	0.655122i	\(-0.227383\pi\)
\(47\)	5.57420	−	5.46714i	0.813081	−	0.797464i	−0.169061	−	0.985606i	\(-0.554074\pi\)
							0.982142	+	0.188142i	\(0.0602464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.6	✓	1404
3.2	odd	2		729.2.i.a.64.21		1404
243.61	even	81	inner	243.2.i.a.61.6	yes	1404
243.182	odd	162		729.2.i.a.262.21		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.6	✓	1404	1.1	even	1	trivial
243.2.i.a.61.6	yes	1404	243.61	even	81	inner
729.2.i.a.64.21		1404	3.2	odd	2
729.2.i.a.262.21		1404	243.182	odd	162