Embedded newform 729.2.g.a.622.3

• Label: 729.2.g.a.622.3
• Level: $729$
• Weight: $2$
• Character: 729.622
• Analytic conductor: $5.821$
• Analytic rank: $0$
• Dimension: $144$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.g (of order \(27\), degree \(18\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.82109430735\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	no (minimal twist has level 81)
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			622.3
Character	\(\chi\)	\(=\)	729.622
Dual form			729.2.g.a.109.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31250 + 1.39117i) q^{2} +(-0.0964037 - 1.65519i) q^{4} +(2.46793 + 0.288460i) q^{5} +(-4.06877 + 2.04341i) q^{7} +(-0.501084 - 0.420459i) q^{8} +(-3.64045 + 3.05470i) q^{10} +(-0.125874 + 0.291810i) q^{11} +(-3.31064 + 0.784636i) q^{13} +(2.49752 - 8.34231i) q^{14} +(4.53616 - 0.530202i) q^{16} +(-0.885565 - 5.02229i) q^{17} +(-0.216164 + 1.22593i) q^{19} +(0.239538 - 4.11270i) q^{20} +(-0.240746 - 0.558112i) q^{22} +(1.70487 + 0.856218i) q^{23} +(1.14226 + 0.270720i) q^{25} +(3.25365 - 5.63548i) q^{26} +(3.77448 + 6.53759i) q^{28} +(-0.993943 - 3.32000i) q^{29} +(-5.67009 + 3.72928i) q^{31} +(-4.43488 + 5.95708i) q^{32} +(8.14914 + 5.35978i) q^{34} +(-10.6309 + 3.86933i) q^{35} +(-8.02231 - 2.91988i) q^{37} +(-1.42176 - 1.90975i) q^{38} +(-1.11536 - 1.18221i) q^{40} +(-4.35826 - 4.61948i) q^{41} +(-3.63719 - 4.88559i) q^{43} +(0.495135 + 0.180214i) q^{44} +(-3.42878 + 1.24797i) q^{46} +(3.40689 + 2.24075i) q^{47} +(8.19925 - 11.0135i) q^{49} +(-1.87583 + 1.23375i) q^{50} +(1.61788 + 5.40409i) q^{52} +(3.88136 + 6.72271i) q^{53} +(-0.394825 + 0.683857i) q^{55} +(2.89797 + 0.686831i) q^{56} +(5.92322 + 2.97475i) q^{58} +(-1.57205 - 3.64441i) q^{59} +(-0.142985 + 2.45496i) q^{61} +(2.25394 - 12.7827i) q^{62} +(-0.880398 - 4.99299i) q^{64} +(-8.39677 + 0.981441i) q^{65} +(0.988137 - 3.30061i) q^{67} +(-8.22746 + 1.94994i) q^{68} +(8.57015 - 19.8678i) q^{70} +(2.16099 - 1.81328i) q^{71} +(-3.24023 - 2.71887i) q^{73} +(14.5913 - 7.32803i) q^{74} +(2.04998 + 0.239609i) q^{76} +(-0.0841338 - 1.44452i) q^{77} +(4.97152 - 5.26950i) q^{79} +11.3479 q^{80} +12.1467 q^{82} +(-8.29076 + 8.78769i) q^{83} +(-0.736786 - 12.6501i) q^{85} +(11.5705 + 1.35239i) q^{86} +(0.185768 - 0.0932961i) q^{88} +(-2.86182 - 2.40135i) q^{89} +(11.8669 - 9.95751i) q^{91} +(1.25285 - 2.90442i) q^{92} +(-7.58878 + 1.79857i) q^{94} +(-0.887111 + 2.96316i) q^{95} +(-14.5265 + 1.69791i) q^{97} +(4.56012 + 25.8617i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	144 * q - 9 * q^2 + 9 * q^4 - 9 * q^5 + 9 * q^7 + 18 * q^8 - 18 * q^10 - 9 * q^11 + 9 * q^13 - 9 * q^14 + 9 * q^16 + 18 * q^17 - 18 * q^19 - 45 * q^20 + 9 * q^22 + 45 * q^23 + 9 * q^25 - 45 * q^26 - 9 * q^28 - 36 * q^29 + 9 * q^31 + 99 * q^32 + 9 * q^34 - 9 * q^35 - 18 * q^37 - 18 * q^38 + 9 * q^40 - 27 * q^41 + 9 * q^43 - 54 * q^44 - 18 * q^46 + 99 * q^47 + 9 * q^49 - 126 * q^50 - 27 * q^52 - 45 * q^53 - 9 * q^55 + 225 * q^56 + 9 * q^58 - 72 * q^59 + 9 * q^61 - 81 * q^62 - 18 * q^64 + 81 * q^65 - 45 * q^67 - 117 * q^68 - 99 * q^70 + 90 * q^71 - 18 * q^73 - 81 * q^74 - 153 * q^76 - 81 * q^77 - 99 * q^79 + 288 * q^80 - 36 * q^82 - 45 * q^83 - 99 * q^85 - 81 * q^86 - 153 * q^88 + 81 * q^89 - 18 * q^91 - 207 * q^92 - 99 * q^94 + 171 * q^95 - 45 * q^97 - 81 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{20}{27}\right)\)

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31250	+	1.39117i	−0.928076	+	0.983703i	−0.999906	−	0.0137139i	\(-0.995635\pi\)
							0.0718301	+	0.997417i	\(0.477116\pi\)
\(3\)		0			0
							\(5\)	2.46793	+	0.288460i	1.10369	+	0.129003i	0.648372	−	0.761324i	\(-0.275450\pi\)
0.455321	+	0.890327i	\(0.349524\pi\)
\(7\)	−4.06877	+	2.04341i	−1.53785	+	0.772338i	−0.997605	−	0.0691692i	\(-0.977965\pi\)
							−0.540246	+	0.841507i	\(0.681669\pi\)
\(11\)	−0.125874	+	0.291810i	−0.0379526	+	0.0879840i	−0.936138	−	0.351633i	\(-0.885627\pi\)
							0.898185	+	0.439617i	\(0.144886\pi\)
\(13\)	−3.31064	+	0.784636i	−0.918206	+	0.217619i	−0.662436	−	0.749119i	\(-0.730477\pi\)
							−0.255770	+	0.966738i	\(0.582329\pi\)
\(17\)	−0.885565	−	5.02229i	−0.214781	−	1.21808i	−0.881286	−	0.472584i	\(-0.843321\pi\)
							0.666505	−	0.745501i	\(-0.267790\pi\)
\(19\)	−0.216164	+	1.22593i	−0.0495915	+	0.281248i	−0.999512	−	0.0312449i	\(-0.990053\pi\)
							0.949920	+	0.312493i	\(0.101164\pi\)
\(23\)	1.70487	+	0.856218i	0.355490	+	0.178534i	0.617577	−	0.786510i	\(-0.288114\pi\)
							−0.262087	+	0.965044i	\(0.584411\pi\)
\(29\)	−0.993943	−	3.32000i	−0.184571	−	0.616509i	−0.999326	−	0.0367041i	\(-0.988314\pi\)
							0.814756	−	0.579805i	\(-0.196871\pi\)
\(31\)	−5.67009	+	3.72928i	−1.01838	+	0.669798i	−0.944853	−	0.327496i	\(-0.893795\pi\)
							−0.0735253	+	0.997293i	\(0.523425\pi\)
\(37\)	−8.02231	−	2.91988i	−1.31886	−	0.480026i	−0.415767	−	0.909471i	\(-0.636487\pi\)
							−0.903093	+	0.429445i	\(0.858709\pi\)
\(41\)	−4.35826	−	4.61948i	−0.680646	−	0.721442i	0.291304	−	0.956630i	\(-0.405911\pi\)
							−0.971950	+	0.235188i	\(0.924429\pi\)
\(43\)	−3.63719	−	4.88559i	−0.554666	−	0.745046i	0.433039	−	0.901375i	\(-0.357441\pi\)
							−0.987705	+	0.156329i	\(0.950034\pi\)
\(47\)	3.40689	+	2.24075i	0.496946	+	0.326846i	0.773119	−	0.634262i	\(-0.218696\pi\)
							−0.276173	+	0.961108i	\(0.589066\pi\)
\(53\)	3.88136	+	6.72271i	0.533146	+	0.923435i	0.999251	+	0.0387058i	\(0.0123235\pi\)
							−0.466105	+	0.884729i	\(0.654343\pi\)
\(59\)	−1.57205	−	3.64441i	−0.204663	−	0.474462i	0.784710	−	0.619863i	\(-0.212812\pi\)
							−0.989373	+	0.145401i	\(0.953553\pi\)
\(61\)	−0.142985	+	2.45496i	−0.0183074	+	0.314325i	0.976706	+	0.214582i	\(0.0688390\pi\)
							−0.995013	+	0.0997428i	\(0.968198\pi\)
\(67\)	0.988137	−	3.30061i	0.120720	−	0.403233i	−0.876173	−	0.481996i	\(-0.839912\pi\)
							0.996893	+	0.0787630i	\(0.0250970\pi\)
\(71\)	2.16099	−	1.81328i	0.256462	−	0.215197i	−0.505487	−	0.862834i	\(-0.668687\pi\)
							0.761949	+	0.647637i	\(0.224243\pi\)
\(73\)	−3.24023	−	2.71887i	−0.379240	−	0.318220i	0.433164	−	0.901315i	\(-0.357397\pi\)
							−0.812404	+	0.583095i	\(0.801842\pi\)
\(79\)	4.97152	−	5.26950i	0.559339	−	0.592865i	−0.384559	−	0.923100i	\(-0.625647\pi\)
							0.943899	+	0.330235i	\(0.107128\pi\)
\(83\)	−8.29076	+	8.78769i	−0.910029	+	0.964575i	−0.999506	−	0.0314144i	\(-0.989999\pi\)
							0.0894773	+	0.995989i	\(0.471480\pi\)
\(89\)	−2.86182	−	2.40135i	−0.303352	−	0.254543i	0.478386	−	0.878150i	\(-0.341222\pi\)
							−0.781738	+	0.623607i	\(0.785666\pi\)
\(97\)	−14.5265	+	1.69791i	−1.47495	+	0.172396i	−0.815238	−	0.579126i	\(-0.803394\pi\)
							−0.659707	+	0.751523i	\(0.729320\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.g.a.622.3		144
3.2	odd	2		729.2.g.d.622.6		144
9.2	odd	6		81.2.g.a.16.3	✓	144
9.4	even	3		729.2.g.b.379.3		144
9.5	odd	6		729.2.g.c.379.6		144
9.7	even	3		243.2.g.a.208.6		144
81.5	odd	54		729.2.g.c.352.6		144
81.22	even	27	inner	729.2.g.a.109.3		144
81.32	odd	54		81.2.g.a.76.3	yes	144
81.34	even	27		6561.2.a.d.1.58		72
81.47	odd	54		6561.2.a.c.1.15		72
81.49	even	27		243.2.g.a.118.6		144
81.59	odd	54		729.2.g.d.109.6		144
81.76	even	27		729.2.g.b.352.3		144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.16.3	✓	144	9.2	odd	6
81.2.g.a.76.3	yes	144	81.32	odd	54
243.2.g.a.118.6		144	81.49	even	27
243.2.g.a.208.6		144	9.7	even	3
729.2.g.a.109.3		144	81.22	even	27	inner
729.2.g.a.622.3		144	1.1	even	1	trivial
729.2.g.b.352.3		144	81.76	even	27
729.2.g.b.379.3		144	9.4	even	3
729.2.g.c.352.6		144	81.5	odd	54
729.2.g.c.379.6		144	9.5	odd	6
729.2.g.d.109.6		144	81.59	odd	54
729.2.g.d.622.6		144	3.2	odd	2
6561.2.a.c.1.15		72	81.47	odd	54
6561.2.a.d.1.58		72	81.34	even	27