Embedded newform 243.2.i.a.4.10

• Label: 243.2.i.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	243.4
Dual form			243.2.i.a.61.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.829136 - 0.0321742i) q^{2} +(0.755643 - 1.55853i) q^{3} +(-1.30755 - 0.101631i) q^{4} +(-1.06671 - 1.32252i) q^{5} +(-0.676676 + 1.26792i) q^{6} +(-2.20706 + 1.00323i) q^{7} +(2.72917 + 0.318994i) q^{8} +(-1.85801 - 2.35538i) q^{9} +(0.841900 + 1.13087i) q^{10} +(0.0547589 + 2.82335i) q^{11} +(-1.14644 + 1.96106i) q^{12} +(-5.52920 - 1.77287i) q^{13} +(1.86223 - 0.760804i) q^{14} +(-2.86723 + 0.663149i) q^{15} +(0.338906 + 0.0530039i) q^{16} +(0.381447 + 1.27412i) q^{17} +(1.46476 + 2.01271i) q^{18} +(-3.22382 + 0.764059i) q^{19} +(1.26038 + 1.83768i) q^{20} +(-0.104188 + 4.19784i) q^{21} +(0.0454367 - 2.34271i) q^{22} +(1.54117 - 1.10156i) q^{23} +(2.55944 - 4.01244i) q^{24} +(0.447345 - 2.06517i) q^{25} +(4.52742 + 1.64785i) q^{26} +(-5.07491 + 1.11592i) q^{27} +(2.98781 - 1.08747i) q^{28} +(-0.282681 - 2.06959i) q^{29} +(2.39866 - 0.457590i) q^{30} +(-3.04514 - 2.98665i) q^{31} +(-5.67178 - 1.11390i) q^{32} +(4.44165 + 2.04811i) q^{33} +(-0.275278 - 1.06869i) q^{34} +(3.68109 + 1.84871i) q^{35} +(2.19006 + 3.26862i) q^{36} +(-3.01579 - 1.98352i) q^{37} +(2.69757 - 0.529785i) q^{38} +(-6.94116 + 7.27775i) q^{39} +(-2.48937 - 3.94965i) q^{40} +(1.91949 - 3.64407i) q^{41} +(0.221448 - 3.47723i) q^{42} +(1.79685 - 5.25149i) q^{43} +(0.215341 - 3.69725i) q^{44} +(-1.13307 + 4.96976i) q^{45} +(-1.31328 + 0.863761i) q^{46} +(3.87549 - 3.80105i) q^{47} +(0.338700 - 0.488141i) q^{48} +(-0.738023 + 0.845680i) q^{49} +(-0.437355 + 1.69792i) q^{50} +(2.27399 + 0.368287i) q^{51} +(7.04955 + 2.88006i) q^{52} +(-2.96543 - 2.48829i) q^{53} +(4.24369 - 0.761970i) q^{54} +(3.67552 - 3.08413i) q^{55} +(-6.34345 + 2.03395i) q^{56} +(-1.24525 + 5.60176i) q^{57} +(0.167793 + 1.72507i) q^{58} +(13.1182 - 7.23834i) q^{59} +(3.81646 - 0.575704i) q^{60} +(3.50428 - 0.272375i) q^{61} +(2.42874 + 2.57431i) q^{62} +(6.46371 + 3.33445i) q^{63} +(3.99928 + 0.947846i) q^{64} +(3.55343 + 9.20361i) q^{65} +(-3.61683 - 1.84106i) q^{66} +(-1.81968 + 13.3224i) q^{67} +(-0.369273 - 1.70475i) q^{68} +(-0.552239 - 3.23435i) q^{69} +(-2.99264 - 1.65127i) q^{70} +(-0.433406 + 1.00475i) q^{71} +(-4.31946 - 7.02092i) q^{72} +(-3.80885 + 5.11617i) q^{73} +(2.43669 + 1.74164i) q^{74} +(-2.88059 - 2.25773i) q^{75} +(4.29297 - 0.671408i) q^{76} +(-2.95333 - 6.17636i) q^{77} +(5.98932 - 5.81091i) q^{78} +(8.07966 - 12.8192i) q^{79} +(-0.291417 - 0.504749i) q^{80} +(-2.09563 + 8.75262i) q^{81} +(-1.70877 + 2.95967i) q^{82} +(-3.14312 - 5.96706i) q^{83} +(0.562862 - 5.47832i) q^{84} +(1.27816 - 1.86360i) q^{85} +(-1.65880 + 4.29639i) q^{86} +(-3.43912 - 1.12331i) q^{87} +(-0.751187 + 7.72288i) q^{88} +(-0.186829 - 0.433118i) q^{89} +(1.09937 - 4.08415i) q^{90} +(13.9818 - 1.63424i) q^{91} +(-2.12712 + 1.28373i) q^{92} +(-6.95581 + 2.48908i) q^{93} +(-3.33560 + 3.02690i) q^{94} +(4.44937 + 3.44852i) q^{95} +(-6.02189 + 7.99790i) q^{96} +(1.41603 - 1.75560i) q^{97} +(0.639130 - 0.677439i) q^{98} +(6.54833 - 5.37479i) 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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.829136	−	0.0321742i	−0.586288	−	0.0227506i	−0.256075	−	0.966657i	\(-0.582429\pi\)
							−0.330213	+	0.943906i	\(0.607121\pi\)
\(3\)	0.755643	−	1.55853i	0.436271	−	0.899815i
							\(5\)	−1.06671	−	1.32252i	−0.477049	−	0.591448i	0.481475	−	0.876460i	\(-0.340101\pi\)
−0.958524	+	0.285012i	\(0.908002\pi\)
\(7\)	−2.20706	+	1.00323i	−0.834189	+	0.379185i	−0.784918	−	0.619600i	\(-0.787295\pi\)
							−0.0492712	+	0.998785i	\(0.515690\pi\)
\(11\)	0.0547589	+	2.82335i	0.0165104	+	0.851273i	0.912942	+	0.408090i	\(0.133805\pi\)
							−0.896431	+	0.443183i	\(0.853849\pi\)
\(13\)	−5.52920	−	1.77287i	−1.53352	−	0.491705i	−0.585708	−	0.810522i	\(-0.699183\pi\)
							−0.947816	+	0.318817i	\(0.896714\pi\)
\(17\)	0.381447	+	1.27412i	0.0925146	+	0.309020i	0.991705	−	0.128538i	\(-0.0410285\pi\)
							−0.899190	+	0.437558i	\(0.855843\pi\)
\(19\)	−3.22382	+	0.764059i	−0.739594	+	0.175287i	−0.583110	−	0.812393i	\(-0.698164\pi\)
							−0.156484	+	0.987680i	\(0.550016\pi\)
\(23\)	1.54117	−	1.10156i	0.321357	−	0.229692i	−0.409360	−	0.912373i	\(-0.634248\pi\)
							0.730717	+	0.682681i	\(0.239186\pi\)
\(29\)	−0.282681	−	2.06959i	−0.0524925	−	0.384313i	−0.998235	−	0.0593949i	\(-0.981083\pi\)
							0.945742	−	0.324919i	\(-0.105337\pi\)
\(31\)	−3.04514	−	2.98665i	−0.546923	−	0.536418i	0.373353	−	0.927689i	\(-0.378208\pi\)
							−0.920275	+	0.391272i	\(0.872035\pi\)
\(37\)	−3.01579	−	1.98352i	−0.495794	−	0.326089i	0.276867	−	0.960908i	\(-0.410704\pi\)
							−0.772660	+	0.634820i	\(0.781074\pi\)
\(41\)	1.91949	−	3.64407i	0.299774	−	0.569108i	−0.687867	−	0.725837i	\(-0.741453\pi\)
							0.987642	+	0.156729i	\(0.0500949\pi\)
\(43\)	1.79685	−	5.25149i	0.274017	−	0.800845i	−0.720230	−	0.693735i	\(-0.755964\pi\)
							0.994247	−	0.107110i	\(-0.0341597\pi\)
\(47\)	3.87549	−	3.80105i	0.565298	−	0.554441i	−0.360354	−	0.932816i	\(-0.617344\pi\)
							0.925652	+	0.378375i	\(0.123517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.i.a.4.10	✓	1404
3.2	odd	2		729.2.i.a.64.17		1404
243.61	even	81	inner	243.2.i.a.61.10	yes	1404
243.182	odd	162		729.2.i.a.262.17		1404

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.i.a.4.10	✓	1404	1.1	even	1	trivial
243.2.i.a.61.10	yes	1404	243.61	even	81	inner
729.2.i.a.64.17		1404	3.2	odd	2
729.2.i.a.262.17		1404	243.182	odd	162