Embedded newform 243.2.i.a.4.15

• Label: 243.2.i.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.337931 + 0.0131133i) q^{2} +(0.210129 + 1.71926i) q^{3} +(-1.87996 - 0.146122i) q^{4} +(-0.812985 - 1.00794i) q^{5} +(0.0484641 + 0.583746i) q^{6} +(-2.68541 + 1.22067i) q^{7} +(-1.30518 - 0.152554i) q^{8} +(-2.91169 + 0.722532i) q^{9} +(-0.261516 - 0.351276i) q^{10} +(0.115263 + 5.94292i) q^{11} +(-0.143813 - 3.26284i) q^{12} +(-0.569671 - 0.182658i) q^{13} +(-0.923490 + 0.377287i) q^{14} +(1.56208 - 1.60953i) q^{15} +(3.28691 + 0.514063i) q^{16} +(-0.953954 - 3.18643i) q^{17} +(-0.993427 + 0.205984i) q^{18} +(0.963453 - 0.228343i) q^{19} +(1.38110 + 2.01369i) q^{20} +(-2.66292 - 4.36041i) q^{21} +(-0.0389802 + 2.00981i) q^{22} +(-5.10672 + 3.65006i) q^{23} +(-0.0119772 - 2.27600i) q^{24} +(0.703515 - 3.24779i) q^{25} +(-0.190115 - 0.0691960i) q^{26} +(-1.85405 - 4.85412i) q^{27} +(5.22683 - 1.90241i) q^{28} +(-0.116746 - 0.854732i) q^{29} +(0.548982 - 0.523426i) q^{30} +(6.40844 + 6.28535i) q^{31} +(3.68287 + 0.723293i) q^{32} +(-10.1932 + 1.44695i) q^{33} +(-0.280587 - 1.08930i) q^{34} +(3.41356 + 1.71435i) q^{35} +(5.57944 - 0.932869i) q^{36} +(7.86448 + 5.17255i) q^{37} +(0.328575 - 0.0645301i) q^{38} +(0.194331 - 1.01779i) q^{39} +(0.907325 + 1.43957i) q^{40} +(-2.43195 + 4.61695i) q^{41} +(-0.842706 - 1.50844i) q^{42} +(2.37654 - 6.94571i) q^{43} +(0.651702 - 11.1893i) q^{44} +(3.09543 + 2.34741i) q^{45} +(-1.77359 + 1.16651i) q^{46} +(-7.16675 + 7.02910i) q^{47} +(-0.193132 + 5.75906i) q^{48} +(1.11874 - 1.28193i) q^{49} +(0.280329 - 1.08830i) q^{50} +(5.27784 - 2.30965i) q^{51} +(1.04427 + 0.426631i) q^{52} +(5.98136 + 5.01896i) q^{53} +(-0.562888 - 1.66467i) q^{54} +(5.89641 - 4.94768i) q^{55} +(3.69116 - 1.18352i) q^{56} +(0.595029 + 1.60844i) q^{57} +(-0.0282438 - 0.290372i) q^{58} +(-3.67320 + 2.02678i) q^{59} +(-3.17184 + 2.79759i) q^{60} +(-13.8754 + 1.07848i) q^{61} +(2.08319 + 2.20805i) q^{62} +(6.93711 - 5.49450i) q^{63} +(-5.23930 - 1.24174i) q^{64} +(0.279025 + 0.722694i) q^{65} +(-3.46357 + 0.355302i) q^{66} +(0.663106 - 4.85479i) q^{67} +(1.32779 + 6.12975i) q^{68} +(-7.34847 - 8.01278i) q^{69} +(1.13107 + 0.624097i) q^{70} +(-3.22866 + 7.48487i) q^{71} +(3.91050 - 0.498845i) q^{72} +(7.31400 - 9.82441i) q^{73} +(2.58983 + 1.85110i) q^{74} +(5.73161 + 0.527069i) q^{75} +(-1.84462 + 0.288493i) q^{76} +(-7.56385 - 15.8185i) q^{77} +(0.0790172 - 0.341396i) q^{78} +(-2.89701 + 4.59641i) q^{79} +(-2.15406 - 3.73094i) q^{80} +(7.95590 - 4.20758i) q^{81} +(-0.882376 + 1.52832i) q^{82} +(-0.712665 - 1.35296i) q^{83} +(4.36904 + 8.58651i) q^{84} +(-2.43619 + 3.55205i) q^{85} +(0.894189 - 2.31601i) q^{86} +(1.44497 - 0.380321i) q^{87} +(0.756175 - 7.77416i) q^{88} +(-4.02409 - 9.32889i) q^{89} +(1.01526 + 0.833855i) q^{90} +(1.75276 - 0.204869i) q^{91} +(10.1338 - 6.11577i) q^{92} +(-9.45953 + 12.3385i) q^{93} +(-2.51404 + 2.28137i) q^{94} +(-1.01343 - 0.785467i) q^{95} +(-0.469648 + 6.48379i) q^{96} +(2.85688 - 3.54198i) q^{97} +(0.394866 - 0.418534i) q^{98} +(-4.62956 - 17.2207i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.337931	+	0.0131133i	0.238954	+	0.00927249i	0.157974	−	0.987443i	\(-0.449504\pi\)
							0.0809796	+	0.996716i	\(0.474195\pi\)
\(3\)	0.210129	+	1.71926i	0.121318	+	0.992614i
							\(5\)	−0.812985	−	1.00794i	−0.363578	−	0.450766i	0.563231	−	0.826300i	\(-0.309558\pi\)
−0.926809	+	0.375534i	\(0.877459\pi\)
\(7\)	−2.68541	+	1.22067i	−1.01499	+	0.461369i	−0.851041	−	0.525100i	\(-0.824028\pi\)
							−0.163948	+	0.986469i	\(0.552423\pi\)
\(11\)	0.115263	+	5.94292i	0.0347530	+	1.79186i	0.461152	+	0.887321i	\(0.347436\pi\)
							−0.426399	+	0.904535i	\(0.640218\pi\)
\(13\)	−0.569671	−	0.182658i	−0.157998	−	0.0506601i	0.225274	−	0.974295i	\(-0.427672\pi\)
							−0.383273	+	0.923635i	\(0.625203\pi\)
\(17\)	−0.953954	−	3.18643i	−0.231368	−	0.772823i	−0.992285	−	0.123974i	\(-0.960436\pi\)
							0.760918	−	0.648849i	\(-0.224749\pi\)
\(19\)	0.963453	−	0.228343i	0.221031	−	0.0523854i	−0.118608	−	0.992941i	\(-0.537843\pi\)
							0.339639	+	0.940556i	\(0.389695\pi\)
\(23\)	−5.10672	+	3.65006i	−1.06482	+	0.761091i	−0.972092	−	0.234600i	\(-0.924622\pi\)
							−0.0927328	+	0.995691i	\(0.529560\pi\)
\(29\)	−0.116746	−	0.854732i	−0.0216792	−	0.158720i	0.976777	−	0.214256i	\(-0.0687328\pi\)
							−0.998457	+	0.0555366i	\(0.982313\pi\)
\(31\)	6.40844	+	6.28535i	1.15099	+	1.12888i	0.989874	+	0.141946i	\(0.0453360\pi\)
							0.161115	+	0.986936i	\(0.448491\pi\)
\(37\)	7.86448	+	5.17255i	1.29291	+	0.850362i	0.994567	−	0.104095i	\(-0.0331948\pi\)
							0.298346	+	0.954458i	\(0.403565\pi\)
\(41\)	−2.43195	+	4.61695i	−0.379807	+	0.721046i	−0.997880	−	0.0650747i	\(-0.979271\pi\)
							0.618073	+	0.786120i	\(0.287913\pi\)
\(43\)	2.37654	−	6.94571i	0.362419	−	1.05921i	−0.602828	−	0.797871i	\(-0.705959\pi\)
							0.965247	−	0.261339i	\(-0.0841641\pi\)
\(47\)	−7.16675	+	7.02910i	−1.04538	+	1.02530i	−0.0457252	+	0.998954i	\(0.514560\pi\)
							−0.999653	+	0.0263455i	\(0.991613\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.15	✓	1404
3.2	odd	2		729.2.i.a.64.12		1404
243.61	even	81	inner	243.2.i.a.61.15	yes	1404
243.182	odd	162		729.2.i.a.262.12		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.15	✓	1404	1.1	even	1	trivial
243.2.i.a.61.15	yes	1404	243.61	even	81	inner
729.2.i.a.64.12		1404	3.2	odd	2
729.2.i.a.262.12		1404	243.182	odd	162