# Properties

 Label 147.4.g.a.68.1 Level $147$ Weight $4$ Character 147.68 Analytic conductor $8.673$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## Embedding invariants

 Embedding label 68.1 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 147.68 Dual form 147.4.g.a.80.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$$ $$q+(-4.50000 + 2.59808i) q^{3} +(-4.00000 - 6.92820i) q^{4} +(13.5000 - 23.3827i) q^{9} +(36.0000 + 20.7846i) q^{12} +62.3538i q^{13} +(-32.0000 + 55.4256i) q^{16} +(135.000 + 77.9423i) q^{19} +(62.5000 + 108.253i) q^{25} +140.296i q^{27} +(-135.000 + 77.9423i) q^{31} -216.000 q^{36} +(55.0000 - 95.2628i) q^{37} +(-162.000 - 280.592i) q^{39} +520.000 q^{43} -332.554i q^{48} +(432.000 - 249.415i) q^{52} -810.000 q^{57} +(-810.000 - 467.654i) q^{61} +512.000 q^{64} +(440.000 + 762.102i) q^{67} +(-324.000 + 187.061i) q^{73} +(-562.500 - 324.760i) q^{75} -1247.08i q^{76} +(-442.000 + 765.566i) q^{79} +(-364.500 - 631.333i) q^{81} +(405.000 - 701.481i) q^{93} +1371.78i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + O(q^{10})$$ $$2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64} + 880 q^{67} - 648 q^{73} - 1125 q^{75} - 884 q^{79} - 729 q^{81} + 810 q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$3$$ −4.50000 + 2.59808i −0.866025 + 0.500000i
$$4$$ −4.00000 6.92820i −0.500000 0.866025i
$$5$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 13.5000 23.3827i 0.500000 0.866025i
$$10$$ 0 0
$$11$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$12$$ 36.0000 + 20.7846i 0.866025 + 0.500000i
$$13$$ 62.3538i 1.33030i 0.746712 + 0.665148i $$0.231631\pi$$
−0.746712 + 0.665148i $$0.768369\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −32.0000 + 55.4256i −0.500000 + 0.866025i
$$17$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$18$$ 0 0
$$19$$ 135.000 + 77.9423i 1.63006 + 0.941115i 0.984073 + 0.177766i $$0.0568871\pi$$
0.645986 + 0.763349i $$0.276446\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$24$$ 0 0
$$25$$ 62.5000 + 108.253i 0.500000 + 0.866025i
$$26$$ 0 0
$$27$$ 140.296i 1.00000i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −135.000 + 77.9423i −0.782152 + 0.451576i −0.837192 0.546908i $$-0.815805\pi$$
0.0550403 + 0.998484i $$0.482471\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −216.000 −1.00000
$$37$$ 55.0000 95.2628i 0.244377 0.423273i −0.717579 0.696477i $$-0.754750\pi$$
0.961956 + 0.273204i $$0.0880833\pi$$
$$38$$ 0 0
$$39$$ −162.000 280.592i −0.665148 1.15207i
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 520.000 1.84417 0.922084 0.386989i $$-0.126485\pi$$
0.922084 + 0.386989i $$0.126485\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$48$$ 332.554i 1.00000i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 432.000 249.415i 1.15207 0.665148i
$$53$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −810.000 −1.88223
$$58$$ 0 0
$$59$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$60$$ 0 0
$$61$$ −810.000 467.654i −1.70016 0.981589i −0.945584 0.325379i $$-0.894508\pi$$
−0.754578 0.656210i $$-0.772158\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 512.000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 440.000 + 762.102i 0.802307 + 1.38964i 0.918094 + 0.396362i $$0.129728\pi$$
−0.115787 + 0.993274i $$0.536939\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −324.000 + 187.061i −0.519470 + 0.299916i −0.736718 0.676200i $$-0.763625\pi$$
0.217248 + 0.976117i $$0.430292\pi$$
$$74$$ 0 0
$$75$$ −562.500 324.760i −0.866025 0.500000i
$$76$$ 1247.08i 1.88223i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −442.000 + 765.566i −0.629480 + 1.09029i 0.358177 + 0.933654i $$0.383399\pi$$
−0.987656 + 0.156637i $$0.949935\pi$$
$$80$$ 0 0
$$81$$ −364.500 631.333i −0.500000 0.866025i
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 405.000 701.481i 0.451576 0.782152i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1371.78i 1.43591i 0.696088 + 0.717957i $$0.254922\pi$$
−0.696088 + 0.717957i $$0.745078\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 500.000 866.025i 0.500000 0.866025i
$$101$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$102$$ 0 0
$$103$$ 891.000 + 514.419i 0.852358 + 0.492109i 0.861446 0.507850i $$-0.169560\pi$$
−0.00908799 + 0.999959i $$0.502893\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$108$$ 972.000 561.184i 0.866025 0.500000i
$$109$$ −323.000 559.452i −0.283833 0.491613i 0.688493 0.725243i $$-0.258273\pi$$
−0.972325 + 0.233630i $$0.924939\pi$$
$$110$$ 0 0
$$111$$ 571.577i 0.488754i
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1458.00 + 841.777i 1.15207 + 0.665148i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −665.500 + 1152.68i −0.500000 + 0.866025i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 1080.00 + 623.538i 0.782152 + 0.451576i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 380.000 0.265508 0.132754 0.991149i $$-0.457618\pi$$
0.132754 + 0.991149i $$0.457618\pi$$
$$128$$ 0 0
$$129$$ −2340.00 + 1351.00i −1.59710 + 0.922084i
$$130$$ 0 0
$$131$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$138$$ 0 0
$$139$$ 2026.50i 1.23659i 0.785948 + 0.618293i $$0.212175\pi$$
−0.785948 + 0.618293i $$0.787825\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 864.000 + 1496.49i 0.500000 + 0.866025i
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ −880.000 −0.488754
$$149$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$150$$ 0 0
$$151$$ 874.000 + 1513.81i 0.471027 + 0.815843i 0.999451 0.0331378i $$-0.0105500\pi$$
−0.528424 + 0.848981i $$0.677217\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −1296.00 + 2244.74i −0.665148 + 1.15207i
$$157$$ −702.000 + 405.300i −0.356852 + 0.206028i −0.667699 0.744432i $$-0.732721\pi$$
0.310847 + 0.950460i $$0.399387\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1700.00 2944.49i 0.816897 1.41491i −0.0910600 0.995845i $$-0.529026\pi$$
0.907957 0.419062i $$-0.137641\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −1691.00 −0.769686
$$170$$ 0 0
$$171$$ 3645.00 2104.44i 1.63006 0.941115i
$$172$$ −2080.00 3602.67i −0.922084 1.59710i
$$173$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$180$$ 0 0
$$181$$ 3429.46i 1.40834i −0.710031 0.704171i $$-0.751319\pi$$
0.710031 0.704171i $$-0.248681\pi$$
$$182$$ 0 0
$$183$$ 4860.00 1.96318
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$192$$ −2304.00 + 1330.22i −0.866025 + 0.500000i
$$193$$ −575.000 995.929i −0.214453 0.371443i 0.738650 0.674089i $$-0.235464\pi$$
−0.953103 + 0.302646i $$0.902130\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 1755.00 1013.25i 0.625169 0.360942i −0.153710 0.988116i $$-0.549122\pi$$
0.778879 + 0.627175i $$0.215789\pi$$
$$200$$ 0 0
$$201$$ −3960.00 2286.31i −1.38964 0.802307i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −3456.00 1995.32i −1.15207 0.665148i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6032.00 −1.96806 −0.984028 0.178011i $$-0.943034\pi$$
−0.984028 + 0.178011i $$0.943034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 972.000 1683.55i 0.299916 0.519470i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5830.08i 1.75072i −0.483469 0.875362i $$-0.660623\pi$$
0.483469 0.875362i $$-0.339377\pi$$
$$224$$ 0 0
$$225$$ 3375.00 1.00000
$$226$$ 0 0
$$227$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$228$$ 3240.00 + 5611.84i 0.941115 + 1.63006i
$$229$$ −4590.00 2650.04i −1.32452 0.764714i −0.340076 0.940398i $$-0.610453\pi$$
−0.984447 + 0.175684i $$0.943786\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 4593.40i 1.25896i
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −1080.00 + 623.538i −0.288668 + 0.166662i −0.637341 0.770582i $$-0.719966\pi$$
0.348673 + 0.937244i $$0.386632\pi$$
$$242$$ 0 0
$$243$$ 3280.50 + 1894.00i 0.866025 + 0.500000i
$$244$$ 7482.46i 1.96318i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4860.00 + 8417.77i −1.25196 + 2.16846i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −2048.00 3547.24i −0.500000 0.866025i
$$257$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 3520.00 6096.82i 0.802307 1.38964i
$$269$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$270$$ 0 0
$$271$$ 7695.00 + 4442.71i 1.72486 + 0.995850i 0.907935 + 0.419111i $$0.137658\pi$$
0.816928 + 0.576739i $$0.195675\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2015.00 + 3490.08i 0.437074 + 0.757035i 0.997462 0.0711951i $$-0.0226813\pi$$
−0.560388 + 0.828230i $$0.689348\pi$$
$$278$$ 0 0
$$279$$ 4208.88i 0.903151i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 6669.00 3850.35i 1.40082 0.808761i 0.406340 0.913722i $$-0.366805\pi$$
0.994476 + 0.104961i $$0.0334717\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2456.50 4254.78i 0.500000 0.866025i
$$290$$ 0 0
$$291$$ −3564.00 6173.03i −0.717957 1.24354i
$$292$$ 2592.00 + 1496.49i 0.519470 + 0.299916i
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 5196.15i 1.00000i
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −8640.00 + 4988.31i −1.63006 + 0.941115i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1590.02i 0.295594i 0.989018 + 0.147797i $$0.0472182\pi$$
−0.989018 + 0.147797i $$0.952782\pi$$
$$308$$ 0 0
$$309$$ −5346.00 −0.984218
$$310$$ 0 0
$$311$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$312$$ 0 0
$$313$$ 4104.00 + 2369.45i 0.741124 + 0.427888i 0.822478 0.568797i $$-0.192591\pi$$
−0.0813539 + 0.996685i $$0.525924\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 7072.00 1.25896
$$317$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −2916.00 + 5050.66i −0.500000 + 0.866025i
$$325$$ −6750.00 + 3897.11i −1.15207 + 0.665148i
$$326$$ 0 0
$$327$$ 2907.00 + 1678.36i 0.491613 + 0.283833i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 496.000 859.097i 0.0823644 0.142659i −0.821901 0.569631i $$-0.807086\pi$$
0.904265 + 0.426971i $$0.140420\pi$$
$$332$$ 0 0
$$333$$ −1485.00 2572.10i −0.244377 0.423273i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4930.00 0.796897 0.398448 0.917191i $$-0.369549\pi$$
0.398448 + 0.917191i $$0.369549\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$348$$ 0 0
$$349$$ 5300.08i 0.812913i 0.913670 + 0.406456i $$0.133236\pi$$
−0.913670 + 0.406456i $$0.866764\pi$$
$$350$$ 0 0
$$351$$ −8748.00 −1.33030
$$352$$ 0 0
$$353$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$360$$ 0 0
$$361$$ 8720.50 + 15104.3i 1.27140 + 2.20212i
$$362$$ 0 0
$$363$$ 6916.08i 1.00000i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11583.0 6687.45i 1.64749 0.951177i 0.669420 0.742884i $$-0.266543\pi$$
0.978066 0.208293i $$-0.0667908\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −6480.00 −0.903151
$$373$$ −6175.00 + 10695.4i −0.857183 + 1.48469i 0.0174213 + 0.999848i $$0.494454\pi$$
−0.874605 + 0.484837i $$0.838879\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8584.00 1.16340 0.581702 0.813402i $$-0.302387\pi$$
0.581702 + 0.813402i $$0.302387\pi$$
$$380$$ 0 0
$$381$$ −1710.00 + 987.269i −0.229937 + 0.132754i
$$382$$ 0 0
$$383$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7020.00 12159.0i 0.922084 1.59710i
$$388$$ 9504.00 5487.14i 1.24354 0.717957i
$$389$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13662.0 7887.76i −1.72714 0.997167i −0.901182 0.433441i $$-0.857299\pi$$
−0.825962 0.563726i $$-0.809368\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −8000.00 −1.00000
$$401$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$402$$ 0 0
$$403$$ −4860.00 8417.77i −0.600729 1.04049i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −12420.0 + 7170.69i −1.50154 + 0.866914i −0.501541 + 0.865134i $$0.667233\pi$$
−0.999998 + 0.00177990i $$0.999433\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8230.71i 0.984218i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −5265.00 9119.25i −0.618293 1.07091i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 17138.0 1.98398 0.991989 0.126322i $$-0.0403172\pi$$
0.991989 + 0.126322i $$0.0403172\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$432$$ −7776.00 4489.48i −0.866025 0.500000i
$$433$$ 17833.2i 1.97923i −0.143727 0.989617i $$-0.545909\pi$$
0.143727 0.989617i $$-0.454091\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2584.00 + 4475.62i −0.283833 + 0.491613i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −9315.00 5378.02i −1.01271 0.584690i −0.100728 0.994914i $$-0.532117\pi$$
−0.911985 + 0.410224i $$0.865450\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$444$$ 3960.00 2286.31i 0.423273 0.244377i
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −7866.00 4541.44i −0.815843 0.471027i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6355.00 11007.2i 0.650491 1.12668i −0.332513 0.943099i $$-0.607897\pi$$
0.983004 0.183585i $$-0.0587702\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −19780.0 −1.98543 −0.992716 0.120482i $$-0.961556\pi$$
−0.992716 + 0.120482i $$0.961556\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$468$$ 13468.4i 1.33030i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2106.00 3647.70i 0.206028 0.356852i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 19485.6i 1.88223i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$480$$ 0 0
$$481$$ 5940.00 + 3429.46i 0.563078 + 0.325093i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 10648.0 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10450.0 + 18099.9i 0.972351 + 1.68416i 0.688415 + 0.725317i $$0.258307\pi$$
0.283936 + 0.958843i $$0.408360\pi$$
$$488$$ 0 0
$$489$$ 17666.9i 1.63379i
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 9976.61i 0.903151i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −7568.00 + 13108.2i −0.678938 + 1.17596i 0.296363 + 0.955075i $$0.404226\pi$$
−0.975301 + 0.220880i $$0.929107\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 7609.50 4393.35i 0.666568 0.384843i
$$508$$ −1520.00 2632.72i −0.132754 0.229937i
$$509$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −10935.0 + 18940.0i −0.941115 + 1.63006i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 18720.0 + 10808.0i 1.59710 + 0.922084i
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$522$$ 0 0
$$523$$ 17901.0 + 10335.1i 1.49667 + 0.864100i 0.999993 0.00383755i $$-0.00122153\pi$$
0.496673 + 0.867938i $$0.334555\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −6083.50 10536.9i −0.500000 0.866025i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 11339.0 19639.7i 0.901112 1.56077i 0.0750596 0.997179i $$-0.476085\pi$$
0.826053 0.563593i $$-0.190581\pi$$
$$542$$ 0 0
$$543$$ 8910.00 + 15432.6i 0.704171 + 1.21966i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1640.00 0.128193 0.0640963 0.997944i $$-0.479584\pi$$
0.0640963 + 0.997944i $$0.479584\pi$$
$$548$$ 0 0
$$549$$ −21870.0 + 12626.7i −1.70016 + 0.981589i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 14040.0 8106.00i 1.07091 0.618293i
$$557$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$558$$ 0 0
$$559$$ 32424.0i 2.45329i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$570$$ 0 0
$$571$$ −11656.0 20188.8i −0.854270 1.47964i −0.877320 0.479905i $$-0.840671\pi$$
0.0230498 0.999734i $$-0.492662\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 6912.00 11971.9i 0.500000 0.866025i
$$577$$ −18468.0 + 10662.5i −1.33247 + 0.769300i −0.985677 0.168644i $$-0.946061\pi$$
−0.346789 + 0.937943i $$0.612728\pi$$
$$578$$ 0 0
$$579$$ 5175.00 + 2987.79i 0.371443 + 0.214453i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ −24300.0 −1.69994
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3520.00 + 6096.82i 0.244377 + 0.423273i
$$593$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5265.00 + 9119.25i −0.360942 + 0.625169i
$$598$$ 0 0
$$599$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$600$$ 0 0
$$601$$ 3117.69i 0.211603i 0.994387 + 0.105801i $$0.0337408\pi$$
−0.994387 + 0.105801i $$0.966259\pi$$
$$602$$ 0 0
$$603$$ 23760.0 1.60461
$$604$$ 6992.00 12110.5i 0.471027 0.815843i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8073.00 + 4660.95i 0.539824 + 0.311667i 0.745007 0.667056i $$-0.232446\pi$$
−0.205184 + 0.978723i $$0.565779\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −8695.00 15060.2i −0.572900 0.992292i −0.996266 0.0863334i $$-0.972485\pi$$
0.423366 0.905959i $$-0.360848\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ −13365.0 + 7716.29i −0.867827 + 0.501040i −0.866625 0.498959i $$-0.833716\pi$$
−0.00120126 + 0.999999i $$0.500382\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 20736.0 1.33030
$$625$$ −7812.50 + 13531.6i −0.500000 + 0.866025i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 5616.00 + 3242.40i 0.356852 + 0.206028i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1892.00 0.119365 0.0596825 0.998217i $$-0.480991\pi$$
0.0596825 + 0.998217i $$0.480991\pi$$
$$632$$ 0 0
$$633$$ 27144.0 15671.6i 1.70439 0.984028i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$642$$ 0 0
$$643$$ 29836.3i 1.82991i −0.403561 0.914953i $$-0.632228\pi$$
0.403561 0.914953i $$-0.367772\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −27200.0 −1.63379
$$653$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10101.3i 0.599833i
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 23490.0 13562.0i 1.38223 0.798032i 0.389808 0.920896i $$-0.372541\pi$$
0.992423 + 0.122864i $$0.0392080\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 15147.0 + 26235.4i 0.875362 + 1.51617i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −24050.0 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$674$$ 0 0
$$675$$ −15187.5 + 8768.51i −0.866025 + 0.500000i
$$676$$ 6764.00 + 11715.6i 0.384843 + 0.666568i
$$677$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$684$$ −29160.0 16835.5i −1.63006 0.941115i
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 27540.0 1.52943
$$688$$ −16640.0 + 28821.3i −0.922084 + 1.59710i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −28215.0 16289.9i −1.55333 0.896814i −0.997868 0.0652705i $$-0.979209\pi$$
−0.555460 0.831543i $$-0.687458\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 14850.0 8573.65i 0.796698 0.459974i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 18073.0 31303.4i 0.957328 1.65814i 0.228381 0.973572i $$-0.426657\pi$$
0.728948 0.684569i $$-0.240010\pi$$
$$710$$ 0 0
$$711$$ 11934.0 + 20670.3i 0.629480 + 1.09029i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 3240.00 5611.84i 0.166662 0.288668i
$$724$$ −23760.0 + 13717.8i −1.21966 + 0.704171i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 37692.9i 1.92290i −0.274971 0.961452i $$-0.588668\pi$$
0.274971 0.961452i $$-0.411332\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ −19440.0 33671.1i −0.981589 1.70016i
$$733$$ −31806.0 18363.2i −1.60270 0.925321i −0.990944 0.134277i $$-0.957129\pi$$
−0.611759 0.791044i $$-0.709538\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −15688.0 27172.4i −0.780910 1.35258i −0.931412 0.363966i $$-0.881422\pi$$
0.150502 0.988610i $$-0.451911\pi$$
$$740$$ 0 0
$$741$$ 50506.6i 2.50392i
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −11726.0 + 20310.0i −0.569757 + 0.986849i 0.426832 + 0.904331i $$0.359630\pi$$
−0.996590 + 0.0825179i $$0.973704\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 41470.0 1.99109 0.995543 0.0943039i $$-0.0300625\pi$$
0.995543 + 0.0943039i $$0.0300625\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 18432.0 + 10641.7i 0.866025 + 0.500000i
$$769$$ 42400.6i 1.98830i 0.107995 + 0.994151i $$0.465557\pi$$
−0.107995 + 0.994151i $$0.534443\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4600.00 + 7967.43i −0.214453 + 0.371443i
$$773$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$774$$ 0 0
$$775$$ −16875.0 9742.79i −0.782152 0.451576i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7047.00 4068.59i 0.319185 0.184281i −0.331844 0.943334i $$-0.607671\pi$$
0.651029 + 0.759053i $$0.274338\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 29160.0 50506.6i 1.30580 2.26172i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −14040.0 8106.00i −0.625169 0.360942i
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0