# Properties

 Label 8550.2.a.ct.1.1 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.788.1 Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.35386$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.35386 q^{7} +1.00000 q^{8} +4.89449 q^{11} -1.89449 q^{13} -2.35386 q^{14} +1.00000 q^{16} -2.35386 q^{17} -1.00000 q^{19} +4.89449 q^{22} -6.16707 q^{23} -1.89449 q^{26} -2.35386 q^{28} +4.27258 q^{29} +7.70771 q^{31} +1.00000 q^{32} -2.35386 q^{34} +7.78899 q^{37} -1.00000 q^{38} +3.54064 q^{41} -1.73194 q^{43} +4.89449 q^{44} -6.16707 q^{46} -5.32962 q^{47} -1.45936 q^{49} -1.89449 q^{52} -1.97577 q^{53} -2.35386 q^{56} +4.27258 q^{58} +6.81322 q^{59} +3.97577 q^{61} +7.70771 q^{62} +1.00000 q^{64} +5.19131 q^{67} -2.35386 q^{68} -4.14284 q^{71} +4.37809 q^{73} +7.78899 q^{74} -1.00000 q^{76} -11.5209 q^{77} +9.49670 q^{79} +3.54064 q^{82} -4.43513 q^{83} -1.73194 q^{86} +4.89449 q^{88} +3.62191 q^{89} +4.45936 q^{91} -6.16707 q^{92} -5.32962 q^{94} +7.06157 q^{97} -1.45936 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} + 2 q^{7} + 3 q^{8} + 5 q^{11} + 4 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} + 5 q^{22} - q^{23} + 4 q^{26} + 2 q^{28} + 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{37} - 3 q^{38} + 10 q^{41} + 2 q^{43} + 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} + 4 q^{52} + 5 q^{53} + 2 q^{56} + 5 q^{58} + 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{68} + 16 q^{71} + 15 q^{73} + 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} + 10 q^{82} - 3 q^{83} + 2 q^{86} + 5 q^{88} + 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} - 6 q^{97} - 5 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 2 * q^7 + 3 * q^8 + 5 * q^11 + 4 * q^13 + 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 + 5 * q^22 - q^23 + 4 * q^26 + 2 * q^28 + 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^37 - 3 * q^38 + 10 * q^41 + 2 * q^43 + 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 + 4 * q^52 + 5 * q^53 + 2 * q^56 + 5 * q^58 + 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^68 + 16 * q^71 + 15 * q^73 + 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 + 10 * q^82 - 3 * q^83 + 2 * q^86 + 5 * q^88 + 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 - 6 * q^97 - 5 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.35386 −0.889674 −0.444837 0.895612i $$-0.646738\pi$$
−0.444837 + 0.895612i $$0.646738\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.89449 1.47575 0.737873 0.674940i $$-0.235831\pi$$
0.737873 + 0.674940i $$0.235831\pi$$
$$12$$ 0 0
$$13$$ −1.89449 −0.525438 −0.262719 0.964872i $$-0.584619\pi$$
−0.262719 + 0.964872i $$0.584619\pi$$
$$14$$ −2.35386 −0.629094
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.35386 −0.570894 −0.285447 0.958395i $$-0.592142\pi$$
−0.285447 + 0.958395i $$0.592142\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.89449 1.04351
$$23$$ −6.16707 −1.28592 −0.642962 0.765898i $$-0.722295\pi$$
−0.642962 + 0.765898i $$0.722295\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.89449 −0.371541
$$27$$ 0 0
$$28$$ −2.35386 −0.444837
$$29$$ 4.27258 0.793398 0.396699 0.917949i $$-0.370156\pi$$
0.396699 + 0.917949i $$0.370156\pi$$
$$30$$ 0 0
$$31$$ 7.70771 1.38435 0.692173 0.721732i $$-0.256654\pi$$
0.692173 + 0.721732i $$0.256654\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −2.35386 −0.403683
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.78899 1.28050 0.640251 0.768166i $$-0.278830\pi$$
0.640251 + 0.768166i $$0.278830\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.54064 0.552955 0.276477 0.961020i $$-0.410833\pi$$
0.276477 + 0.961020i $$0.410833\pi$$
$$42$$ 0 0
$$43$$ −1.73194 −0.264119 −0.132060 0.991242i $$-0.542159\pi$$
−0.132060 + 0.991242i $$0.542159\pi$$
$$44$$ 4.89449 0.737873
$$45$$ 0 0
$$46$$ −6.16707 −0.909286
$$47$$ −5.32962 −0.777405 −0.388703 0.921363i $$-0.627077\pi$$
−0.388703 + 0.921363i $$0.627077\pi$$
$$48$$ 0 0
$$49$$ −1.45936 −0.208480
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.89449 −0.262719
$$53$$ −1.97577 −0.271392 −0.135696 0.990750i $$-0.543327\pi$$
−0.135696 + 0.990750i $$0.543327\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.35386 −0.314547
$$57$$ 0 0
$$58$$ 4.27258 0.561017
$$59$$ 6.81322 0.887006 0.443503 0.896273i $$-0.353736\pi$$
0.443503 + 0.896273i $$0.353736\pi$$
$$60$$ 0 0
$$61$$ 3.97577 0.509045 0.254522 0.967067i $$-0.418082\pi$$
0.254522 + 0.967067i $$0.418082\pi$$
$$62$$ 7.70771 0.978880
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.19131 0.634219 0.317110 0.948389i $$-0.397288\pi$$
0.317110 + 0.948389i $$0.397288\pi$$
$$68$$ −2.35386 −0.285447
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.14284 −0.491665 −0.245832 0.969312i $$-0.579061\pi$$
−0.245832 + 0.969312i $$0.579061\pi$$
$$72$$ 0 0
$$73$$ 4.37809 0.512417 0.256208 0.966622i $$-0.417527\pi$$
0.256208 + 0.966622i $$0.417527\pi$$
$$74$$ 7.78899 0.905451
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −11.5209 −1.31293
$$78$$ 0 0
$$79$$ 9.49670 1.06846 0.534231 0.845339i $$-0.320601\pi$$
0.534231 + 0.845339i $$0.320601\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.54064 0.390998
$$83$$ −4.43513 −0.486819 −0.243409 0.969924i $$-0.578266\pi$$
−0.243409 + 0.969924i $$0.578266\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.73194 −0.186760
$$87$$ 0 0
$$88$$ 4.89449 0.521755
$$89$$ 3.62191 0.383922 0.191961 0.981403i $$-0.438515\pi$$
0.191961 + 0.981403i $$0.438515\pi$$
$$90$$ 0 0
$$91$$ 4.45936 0.467468
$$92$$ −6.16707 −0.642962
$$93$$ 0 0
$$94$$ −5.32962 −0.549709
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.06157 0.716994 0.358497 0.933531i $$-0.383289\pi$$
0.358497 + 0.933531i $$0.383289\pi$$
$$98$$ −1.45936 −0.147418
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.10551 0.607521 0.303760 0.952748i $$-0.401758\pi$$
0.303760 + 0.952748i $$0.401758\pi$$
$$102$$ 0 0
$$103$$ 11.2483 1.10833 0.554166 0.832406i $$-0.313037\pi$$
0.554166 + 0.832406i $$0.313037\pi$$
$$104$$ −1.89449 −0.185770
$$105$$ 0 0
$$106$$ −1.97577 −0.191903
$$107$$ 15.3099 1.48007 0.740033 0.672571i $$-0.234810\pi$$
0.740033 + 0.672571i $$0.234810\pi$$
$$108$$ 0 0
$$109$$ −14.7077 −1.40874 −0.704372 0.709831i $$-0.748771\pi$$
−0.704372 + 0.709831i $$0.748771\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.35386 −0.222418
$$113$$ −3.29681 −0.310138 −0.155069 0.987904i $$-0.549560\pi$$
−0.155069 + 0.987904i $$0.549560\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.27258 0.396699
$$117$$ 0 0
$$118$$ 6.81322 0.627208
$$119$$ 5.54064 0.507909
$$120$$ 0 0
$$121$$ 12.9561 1.17782
$$122$$ 3.97577 0.359949
$$123$$ 0 0
$$124$$ 7.70771 0.692173
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.62644 −0.588001 −0.294001 0.955805i $$-0.594987\pi$$
−0.294001 + 0.955805i $$0.594987\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.9803 1.48358 0.741788 0.670635i $$-0.233978\pi$$
0.741788 + 0.670635i $$0.233978\pi$$
$$132$$ 0 0
$$133$$ 2.35386 0.204105
$$134$$ 5.19131 0.448461
$$135$$ 0 0
$$136$$ −2.35386 −0.201841
$$137$$ 12.8703 1.09958 0.549790 0.835303i $$-0.314708\pi$$
0.549790 + 0.835303i $$0.314708\pi$$
$$138$$ 0 0
$$139$$ 9.30992 0.789657 0.394828 0.918755i $$-0.370804\pi$$
0.394828 + 0.918755i $$0.370804\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.14284 −0.347660
$$143$$ −9.27258 −0.775412
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.37809 0.362333
$$147$$ 0 0
$$148$$ 7.78899 0.640251
$$149$$ −16.1231 −1.32086 −0.660429 0.750888i $$-0.729626\pi$$
−0.660429 + 0.750888i $$0.729626\pi$$
$$150$$ 0 0
$$151$$ −15.6638 −1.27470 −0.637350 0.770575i $$-0.719969\pi$$
−0.637350 + 0.770575i $$0.719969\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −11.5209 −0.928383
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.2483 −1.13714 −0.568571 0.822634i $$-0.692504\pi$$
−0.568571 + 0.822634i $$0.692504\pi$$
$$158$$ 9.49670 0.755517
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 14.5164 1.14405
$$162$$ 0 0
$$163$$ 6.00000 0.469956 0.234978 0.972001i $$-0.424498\pi$$
0.234978 + 0.972001i $$0.424498\pi$$
$$164$$ 3.54064 0.276477
$$165$$ 0 0
$$166$$ −4.43513 −0.344233
$$167$$ 0.756178 0.0585148 0.0292574 0.999572i $$-0.490686\pi$$
0.0292574 + 0.999572i $$0.490686\pi$$
$$168$$ 0 0
$$169$$ −9.41090 −0.723915
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1.73194 −0.132060
$$173$$ 13.6880 1.04068 0.520340 0.853959i $$-0.325805\pi$$
0.520340 + 0.853959i $$0.325805\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.89449 0.368936
$$177$$ 0 0
$$178$$ 3.62191 0.271474
$$179$$ 23.2044 1.73438 0.867189 0.497978i $$-0.165924\pi$$
0.867189 + 0.497978i $$0.165924\pi$$
$$180$$ 0 0
$$181$$ 6.65067 0.494340 0.247170 0.968972i $$-0.420499\pi$$
0.247170 + 0.968972i $$0.420499\pi$$
$$182$$ 4.45936 0.330550
$$183$$ 0 0
$$184$$ −6.16707 −0.454643
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −11.5209 −0.842494
$$188$$ −5.32962 −0.388703
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 14.7890 1.07009 0.535047 0.844822i $$-0.320294\pi$$
0.535047 + 0.844822i $$0.320294\pi$$
$$192$$ 0 0
$$193$$ 11.5977 0.834819 0.417410 0.908718i $$-0.362938\pi$$
0.417410 + 0.908718i $$0.362938\pi$$
$$194$$ 7.06157 0.506991
$$195$$ 0 0
$$196$$ −1.45936 −0.104240
$$197$$ 12.6022 0.897870 0.448935 0.893564i $$-0.351803\pi$$
0.448935 + 0.893564i $$0.351803\pi$$
$$198$$ 0 0
$$199$$ −23.7405 −1.68292 −0.841460 0.540319i $$-0.818304\pi$$
−0.841460 + 0.540319i $$0.818304\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.10551 0.429582
$$203$$ −10.0570 −0.705866
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 11.2483 0.783710
$$207$$ 0 0
$$208$$ −1.89449 −0.131359
$$209$$ −4.89449 −0.338559
$$210$$ 0 0
$$211$$ 19.9848 1.37581 0.687906 0.725800i $$-0.258530\pi$$
0.687906 + 0.725800i $$0.258530\pi$$
$$212$$ −1.97577 −0.135696
$$213$$ 0 0
$$214$$ 15.3099 1.04656
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −18.1428 −1.23162
$$218$$ −14.7077 −0.996132
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.45936 0.299969
$$222$$ 0 0
$$223$$ 18.7122 1.25306 0.626532 0.779396i $$-0.284474\pi$$
0.626532 + 0.779396i $$0.284474\pi$$
$$224$$ −2.35386 −0.157274
$$225$$ 0 0
$$226$$ −3.29681 −0.219301
$$227$$ 13.0813 0.868235 0.434117 0.900856i $$-0.357060\pi$$
0.434117 + 0.900856i $$0.357060\pi$$
$$228$$ 0 0
$$229$$ −9.89902 −0.654146 −0.327073 0.944999i $$-0.606062\pi$$
−0.327073 + 0.944999i $$0.606062\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 4.27258 0.280509
$$233$$ 12.6395 0.828044 0.414022 0.910267i $$-0.364124\pi$$
0.414022 + 0.910267i $$0.364124\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.81322 0.443503
$$237$$ 0 0
$$238$$ 5.54064 0.359146
$$239$$ −20.1605 −1.30407 −0.652036 0.758188i $$-0.726085\pi$$
−0.652036 + 0.758188i $$0.726085\pi$$
$$240$$ 0 0
$$241$$ −11.7693 −0.758126 −0.379063 0.925371i $$-0.623754\pi$$
−0.379063 + 0.925371i $$0.623754\pi$$
$$242$$ 12.9561 0.832847
$$243$$ 0 0
$$244$$ 3.97577 0.254522
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.89449 0.120544
$$248$$ 7.70771 0.489440
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −18.3715 −1.15960 −0.579799 0.814760i $$-0.696869\pi$$
−0.579799 + 0.814760i $$0.696869\pi$$
$$252$$ 0 0
$$253$$ −30.1847 −1.89770
$$254$$ −6.62644 −0.415780
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 3.83293 0.239091 0.119546 0.992829i $$-0.461856\pi$$
0.119546 + 0.992829i $$0.461856\pi$$
$$258$$ 0 0
$$259$$ −18.3341 −1.13923
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 16.9803 1.04905
$$263$$ −21.2483 −1.31023 −0.655115 0.755529i $$-0.727380\pi$$
−0.655115 + 0.755529i $$0.727380\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.35386 0.144324
$$267$$ 0 0
$$268$$ 5.19131 0.317110
$$269$$ 5.78899 0.352961 0.176480 0.984304i $$-0.443529\pi$$
0.176480 + 0.984304i $$0.443529\pi$$
$$270$$ 0 0
$$271$$ 4.01971 0.244180 0.122090 0.992519i $$-0.461040\pi$$
0.122090 + 0.992519i $$0.461040\pi$$
$$272$$ −2.35386 −0.142723
$$273$$ 0 0
$$274$$ 12.8703 0.777521
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 12.0989 0.726953 0.363476 0.931603i $$-0.381590\pi$$
0.363476 + 0.931603i $$0.381590\pi$$
$$278$$ 9.30992 0.558372
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 9.21101 0.549483 0.274742 0.961518i $$-0.411408\pi$$
0.274742 + 0.961518i $$0.411408\pi$$
$$282$$ 0 0
$$283$$ −8.49670 −0.505076 −0.252538 0.967587i $$-0.581265\pi$$
−0.252538 + 0.967587i $$0.581265\pi$$
$$284$$ −4.14284 −0.245832
$$285$$ 0 0
$$286$$ −9.27258 −0.548299
$$287$$ −8.33415 −0.491949
$$288$$ 0 0
$$289$$ −11.4594 −0.674080
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4.37809 0.256208
$$293$$ 12.0989 0.706825 0.353413 0.935468i $$-0.385021\pi$$
0.353413 + 0.935468i $$0.385021\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7.78899 0.452726
$$297$$ 0 0
$$298$$ −16.1231 −0.933988
$$299$$ 11.6835 0.675673
$$300$$ 0 0
$$301$$ 4.07675 0.234980
$$302$$ −15.6638 −0.901349
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.9318 −1.19464 −0.597321 0.802002i $$-0.703768\pi$$
−0.597321 + 0.802002i $$0.703768\pi$$
$$308$$ −11.5209 −0.656466
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 14.9561 0.848080 0.424040 0.905643i $$-0.360612\pi$$
0.424040 + 0.905643i $$0.360612\pi$$
$$312$$ 0 0
$$313$$ −32.2902 −1.82515 −0.912575 0.408909i $$-0.865909\pi$$
−0.912575 + 0.408909i $$0.865909\pi$$
$$314$$ −14.2483 −0.804081
$$315$$ 0 0
$$316$$ 9.49670 0.534231
$$317$$ 24.2726 1.36328 0.681642 0.731686i $$-0.261266\pi$$
0.681642 + 0.731686i $$0.261266\pi$$
$$318$$ 0 0
$$319$$ 20.9121 1.17085
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 14.5164 0.808968
$$323$$ 2.35386 0.130972
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 6.00000 0.332309
$$327$$ 0 0
$$328$$ 3.54064 0.195499
$$329$$ 12.5452 0.691637
$$330$$ 0 0
$$331$$ 6.80869 0.374240 0.187120 0.982337i $$-0.440085\pi$$
0.187120 + 0.982337i $$0.440085\pi$$
$$332$$ −4.43513 −0.243409
$$333$$ 0 0
$$334$$ 0.756178 0.0413762
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.1231 0.878283 0.439142 0.898418i $$-0.355283\pi$$
0.439142 + 0.898418i $$0.355283\pi$$
$$338$$ −9.41090 −0.511885
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 37.7253 2.04294
$$342$$ 0 0
$$343$$ 19.9121 1.07515
$$344$$ −1.73194 −0.0933802
$$345$$ 0 0
$$346$$ 13.6880 0.735872
$$347$$ 23.3670 1.25440 0.627202 0.778857i $$-0.284200\pi$$
0.627202 + 0.778857i $$0.284200\pi$$
$$348$$ 0 0
$$349$$ 16.5977 0.888453 0.444227 0.895914i $$-0.353478\pi$$
0.444227 + 0.895914i $$0.353478\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.89449 0.260877
$$353$$ −5.32105 −0.283211 −0.141605 0.989923i $$-0.545226\pi$$
−0.141605 + 0.989923i $$0.545226\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3.62191 0.191961
$$357$$ 0 0
$$358$$ 23.2044 1.22639
$$359$$ 6.67038 0.352049 0.176025 0.984386i $$-0.443676\pi$$
0.176025 + 0.984386i $$0.443676\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 6.65067 0.349551
$$363$$ 0 0
$$364$$ 4.45936 0.233734
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.14284 0.425053 0.212526 0.977155i $$-0.431831\pi$$
0.212526 + 0.977155i $$0.431831\pi$$
$$368$$ −6.16707 −0.321481
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.65067 0.241451
$$372$$ 0 0
$$373$$ 3.94296 0.204159 0.102079 0.994776i $$-0.467450\pi$$
0.102079 + 0.994776i $$0.467450\pi$$
$$374$$ −11.5209 −0.595733
$$375$$ 0 0
$$376$$ −5.32962 −0.274854
$$377$$ −8.09438 −0.416882
$$378$$ 0 0
$$379$$ −33.2044 −1.70560 −0.852798 0.522241i $$-0.825096\pi$$
−0.852798 + 0.522241i $$0.825096\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 14.7890 0.756670
$$383$$ −19.3867 −0.990612 −0.495306 0.868719i $$-0.664944\pi$$
−0.495306 + 0.868719i $$0.664944\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 11.5977 0.590306
$$387$$ 0 0
$$388$$ 7.06157 0.358497
$$389$$ −6.87026 −0.348336 −0.174168 0.984716i $$-0.555724\pi$$
−0.174168 + 0.984716i $$0.555724\pi$$
$$390$$ 0 0
$$391$$ 14.5164 0.734126
$$392$$ −1.45936 −0.0737090
$$393$$ 0 0
$$394$$ 12.6022 0.634890
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 21.8879 1.09852 0.549261 0.835651i $$-0.314909\pi$$
0.549261 + 0.835651i $$0.314909\pi$$
$$398$$ −23.7405 −1.19000
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7.49670 0.374367 0.187184 0.982325i $$-0.440064\pi$$
0.187184 + 0.982325i $$0.440064\pi$$
$$402$$ 0 0
$$403$$ −14.6022 −0.727388
$$404$$ 6.10551 0.303760
$$405$$ 0 0
$$406$$ −10.0570 −0.499123
$$407$$ 38.1231 1.88969
$$408$$ 0 0
$$409$$ 4.21101 0.208221 0.104111 0.994566i $$-0.466800\pi$$
0.104111 + 0.994566i $$0.466800\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 11.2483 0.554166
$$413$$ −16.0373 −0.789146
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.89449 −0.0928852
$$417$$ 0 0
$$418$$ −4.89449 −0.239397
$$419$$ 10.8703 0.531047 0.265523 0.964104i $$-0.414455\pi$$
0.265523 + 0.964104i $$0.414455\pi$$
$$420$$ 0 0
$$421$$ 33.7496 1.64485 0.822427 0.568871i $$-0.192620\pi$$
0.822427 + 0.568871i $$0.192620\pi$$
$$422$$ 19.9848 0.972846
$$423$$ 0 0
$$424$$ −1.97577 −0.0959517
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −9.35838 −0.452884
$$428$$ 15.3099 0.740033
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −15.7011 −0.756296 −0.378148 0.925745i $$-0.623439\pi$$
−0.378148 + 0.925745i $$0.623439\pi$$
$$432$$ 0 0
$$433$$ 15.1847 0.729730 0.364865 0.931060i $$-0.381115\pi$$
0.364865 + 0.931060i $$0.381115\pi$$
$$434$$ −18.1428 −0.870884
$$435$$ 0 0
$$436$$ −14.7077 −0.704372
$$437$$ 6.16707 0.295011
$$438$$ 0 0
$$439$$ −25.4482 −1.21458 −0.607289 0.794481i $$-0.707743\pi$$
−0.607289 + 0.794481i $$0.707743\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.45936 0.212110
$$443$$ −24.4351 −1.16095 −0.580474 0.814279i $$-0.697133\pi$$
−0.580474 + 0.814279i $$0.697133\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 18.7122 0.886050
$$447$$ 0 0
$$448$$ −2.35386 −0.111209
$$449$$ 26.5012 1.25067 0.625335 0.780356i $$-0.284962\pi$$
0.625335 + 0.780356i $$0.284962\pi$$
$$450$$ 0 0
$$451$$ 17.3296 0.816020
$$452$$ −3.29681 −0.155069
$$453$$ 0 0
$$454$$ 13.0813 0.613935
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −36.3230 −1.69912 −0.849560 0.527493i $$-0.823132\pi$$
−0.849560 + 0.527493i $$0.823132\pi$$
$$458$$ −9.89902 −0.462551
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 23.0813 1.07500 0.537501 0.843263i $$-0.319368\pi$$
0.537501 + 0.843263i $$0.319368\pi$$
$$462$$ 0 0
$$463$$ 31.0419 1.44264 0.721319 0.692603i $$-0.243536\pi$$
0.721319 + 0.692603i $$0.243536\pi$$
$$464$$ 4.27258 0.198350
$$465$$ 0 0
$$466$$ 12.6395 0.585515
$$467$$ −16.6925 −0.772438 −0.386219 0.922407i $$-0.626219\pi$$
−0.386219 + 0.922407i $$0.626219\pi$$
$$468$$ 0 0
$$469$$ −12.2196 −0.564248
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.81322 0.313604
$$473$$ −8.47699 −0.389772
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 5.54064 0.253955
$$477$$ 0 0
$$478$$ −20.1605 −0.922118
$$479$$ −12.5891 −0.575211 −0.287605 0.957749i $$-0.592859\pi$$
−0.287605 + 0.957749i $$0.592859\pi$$
$$480$$ 0 0
$$481$$ −14.7562 −0.672824
$$482$$ −11.7693 −0.536076
$$483$$ 0 0
$$484$$ 12.9561 0.588912
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 20.7077 0.938356 0.469178 0.883104i $$-0.344550\pi$$
0.469178 + 0.883104i $$0.344550\pi$$
$$488$$ 3.97577 0.179975
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0747 0.905957 0.452979 0.891521i $$-0.350361\pi$$
0.452979 + 0.891521i $$0.350361\pi$$
$$492$$ 0 0
$$493$$ −10.0570 −0.452946
$$494$$ 1.89449 0.0852373
$$495$$ 0 0
$$496$$ 7.70771 0.346086
$$497$$ 9.75165 0.437421
$$498$$ 0 0
$$499$$ 41.5956 1.86207 0.931037 0.364924i $$-0.118905\pi$$
0.931037 + 0.364924i $$0.118905\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −18.3715 −0.819959
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −30.1847 −1.34187
$$507$$ 0 0
$$508$$ −6.62644 −0.294001
$$509$$ 25.8879 1.14746 0.573730 0.819044i $$-0.305496\pi$$
0.573730 + 0.819044i $$0.305496\pi$$
$$510$$ 0 0
$$511$$ −10.3054 −0.455884
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 3.83293 0.169063
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −26.0858 −1.14725
$$518$$ −18.3341 −0.805556
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −24.5386 −1.07505 −0.537527 0.843247i $$-0.680641\pi$$
−0.537527 + 0.843247i $$0.680641\pi$$
$$522$$ 0 0
$$523$$ −4.21101 −0.184135 −0.0920674 0.995753i $$-0.529348\pi$$
−0.0920674 + 0.995753i $$0.529348\pi$$
$$524$$ 16.9803 0.741788
$$525$$ 0 0
$$526$$ −21.2483 −0.926472
$$527$$ −18.1428 −0.790315
$$528$$ 0 0
$$529$$ 15.0328 0.653600
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.35386 0.102053
$$533$$ −6.70771 −0.290543
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 5.19131 0.224230
$$537$$ 0 0
$$538$$ 5.78899 0.249581
$$539$$ −7.14284 −0.307664
$$540$$ 0 0
$$541$$ −12.7208 −0.546910 −0.273455 0.961885i $$-0.588167\pi$$
−0.273455 + 0.961885i $$0.588167\pi$$
$$542$$ 4.01971 0.172661
$$543$$ 0 0
$$544$$ −2.35386 −0.100921
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −5.97577 −0.255505 −0.127753 0.991806i $$-0.540776\pi$$
−0.127753 + 0.991806i $$0.540776\pi$$
$$548$$ 12.8703 0.549790
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.27258 −0.182018
$$552$$ 0 0
$$553$$ −22.3539 −0.950583
$$554$$ 12.0989 0.514033
$$555$$ 0 0
$$556$$ 9.30992 0.394828
$$557$$ 29.6441 1.25606 0.628030 0.778189i $$-0.283862\pi$$
0.628030 + 0.778189i $$0.283862\pi$$
$$558$$ 0 0
$$559$$ 3.28116 0.138778
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 9.21101 0.388543
$$563$$ 24.3427 1.02592 0.512962 0.858411i $$-0.328548\pi$$
0.512962 + 0.858411i $$0.328548\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.49670 −0.357143
$$567$$ 0 0
$$568$$ −4.14284 −0.173830
$$569$$ −23.6264 −0.990472 −0.495236 0.868759i $$-0.664918\pi$$
−0.495236 + 0.868759i $$0.664918\pi$$
$$570$$ 0 0
$$571$$ 8.81322 0.368822 0.184411 0.982849i $$-0.440962\pi$$
0.184411 + 0.982849i $$0.440962\pi$$
$$572$$ −9.27258 −0.387706
$$573$$ 0 0
$$574$$ −8.33415 −0.347861
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −36.9121 −1.53667 −0.768336 0.640047i $$-0.778915\pi$$
−0.768336 + 0.640047i $$0.778915\pi$$
$$578$$ −11.4594 −0.476647
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 10.4397 0.433110
$$582$$ 0 0
$$583$$ −9.67038 −0.400506
$$584$$ 4.37809 0.181167
$$585$$ 0 0
$$586$$ 12.0989 0.499801
$$587$$ −15.9737 −0.659305 −0.329652 0.944102i $$-0.606932\pi$$
−0.329652 + 0.944102i $$0.606932\pi$$
$$588$$ 0 0
$$589$$ −7.70771 −0.317591
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 7.78899 0.320125
$$593$$ 4.22007 0.173297 0.0866487 0.996239i $$-0.472384\pi$$
0.0866487 + 0.996239i $$0.472384\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −16.1231 −0.660429
$$597$$ 0 0
$$598$$ 11.6835 0.477773
$$599$$ −21.6552 −0.884807 −0.442404 0.896816i $$-0.645874\pi$$
−0.442404 + 0.896816i $$0.645874\pi$$
$$600$$ 0 0
$$601$$ −30.0944 −1.22758 −0.613788 0.789471i $$-0.710355\pi$$
−0.613788 + 0.789471i $$0.710355\pi$$
$$602$$ 4.07675 0.166156
$$603$$ 0 0
$$604$$ −15.6638 −0.637350
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −14.9979 −0.608747 −0.304373 0.952553i $$-0.598447\pi$$
−0.304373 + 0.952553i $$0.598447\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.0969 0.408478
$$612$$ 0 0
$$613$$ −16.7562 −0.676776 −0.338388 0.941007i $$-0.609882\pi$$
−0.338388 + 0.941007i $$0.609882\pi$$
$$614$$ −20.9318 −0.844740
$$615$$ 0 0
$$616$$ −11.5209 −0.464192
$$617$$ 18.1141 0.729245 0.364623 0.931155i $$-0.381198\pi$$
0.364623 + 0.931155i $$0.381198\pi$$
$$618$$ 0 0
$$619$$ −36.0176 −1.44767 −0.723835 0.689973i $$-0.757622\pi$$
−0.723835 + 0.689973i $$0.757622\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 14.9561 0.599683
$$623$$ −8.52546 −0.341565
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −32.2902 −1.29058
$$627$$ 0 0
$$628$$ −14.2483 −0.568571
$$629$$ −18.3341 −0.731030
$$630$$ 0 0
$$631$$ 17.6749 0.703627 0.351813 0.936070i $$-0.385565\pi$$
0.351813 + 0.936070i $$0.385565\pi$$
$$632$$ 9.49670 0.377758
$$633$$ 0 0
$$634$$ 24.2726 0.963987
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.76475 0.109543
$$638$$ 20.9121 0.827919
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.08580 −0.0823842 −0.0411921 0.999151i $$-0.513116\pi$$
−0.0411921 + 0.999151i $$0.513116\pi$$
$$642$$ 0 0
$$643$$ 37.5295 1.48002 0.740010 0.672596i $$-0.234821\pi$$
0.740010 + 0.672596i $$0.234821\pi$$
$$644$$ 14.5164 0.572026
$$645$$ 0 0
$$646$$ 2.35386 0.0926112
$$647$$ −9.12521 −0.358749 −0.179375 0.983781i $$-0.557407\pi$$
−0.179375 + 0.983781i $$0.557407\pi$$
$$648$$ 0 0
$$649$$ 33.3473 1.30899
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 6.00000 0.234978
$$653$$ −13.2529 −0.518625 −0.259313 0.965793i $$-0.583496\pi$$
−0.259313 + 0.965793i $$0.583496\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 3.54064 0.138239
$$657$$ 0 0
$$658$$ 12.5452 0.489061
$$659$$ 24.9364 0.971382 0.485691 0.874130i $$-0.338568\pi$$
0.485691 + 0.874130i $$0.338568\pi$$
$$660$$ 0 0
$$661$$ −29.9606 −1.16533 −0.582666 0.812712i $$-0.697990\pi$$
−0.582666 + 0.812712i $$0.697990\pi$$
$$662$$ 6.80869 0.264627
$$663$$ 0 0
$$664$$ −4.43513 −0.172116
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −26.3493 −1.02025
$$668$$ 0.756178 0.0292574
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 19.4594 0.751220
$$672$$ 0 0
$$673$$ 2.85055 0.109881 0.0549404 0.998490i $$-0.482503\pi$$
0.0549404 + 0.998490i $$0.482503\pi$$
$$674$$ 16.1231 0.621040
$$675$$ 0 0
$$676$$ −9.41090 −0.361958
$$677$$ 42.5562 1.63557 0.817784 0.575526i $$-0.195203\pi$$
0.817784 + 0.575526i $$0.195203\pi$$
$$678$$ 0 0
$$679$$ −16.6219 −0.637890
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 37.7253 1.44458
$$683$$ 12.2286 0.467916 0.233958 0.972247i $$-0.424832\pi$$
0.233958 + 0.972247i $$0.424832\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 19.9121 0.760248
$$687$$ 0 0
$$688$$ −1.73194 −0.0660298
$$689$$ 3.74308 0.142600
$$690$$ 0 0
$$691$$ 21.7581 0.827719 0.413859 0.910341i $$-0.364180\pi$$
0.413859 + 0.910341i $$0.364180\pi$$
$$692$$ 13.6880 0.520340
$$693$$ 0 0
$$694$$ 23.3670 0.886998
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −8.33415 −0.315678
$$698$$ 16.5977 0.628231
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 28.6592 1.08244 0.541222 0.840880i $$-0.317962\pi$$
0.541222 + 0.840880i $$0.317962\pi$$
$$702$$ 0 0
$$703$$ −7.78899 −0.293767
$$704$$ 4.89449 0.184468
$$705$$ 0 0
$$706$$ −5.32105 −0.200260
$$707$$ −14.3715 −0.540495
$$708$$ 0 0
$$709$$ 6.69461 0.251421 0.125711 0.992067i $$-0.459879\pi$$
0.125711 + 0.992067i $$0.459879\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 3.62191 0.135737
$$713$$ −47.5340 −1.78016
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23.2044 0.867189
$$717$$ 0 0
$$718$$ 6.67038 0.248936
$$719$$ 24.7980 0.924811 0.462405 0.886669i $$-0.346986\pi$$
0.462405 + 0.886669i $$0.346986\pi$$
$$720$$ 0 0
$$721$$ −26.4770 −0.986055
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 6.65067 0.247170
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −33.0328 −1.22512 −0.612560 0.790424i $$-0.709860\pi$$
−0.612560 + 0.790424i $$0.709860\pi$$
$$728$$ 4.45936 0.165275
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 4.07675 0.150784
$$732$$ 0 0
$$733$$ −42.5098 −1.57014 −0.785068 0.619410i $$-0.787372\pi$$
−0.785068 + 0.619410i $$0.787372\pi$$
$$734$$ 8.14284 0.300558
$$735$$ 0 0
$$736$$ −6.16707 −0.227321
$$737$$ 25.4088 0.935946
$$738$$ 0 0
$$739$$ 1.20441 0.0443049 0.0221524 0.999755i $$-0.492948\pi$$
0.0221524 + 0.999755i $$0.492948\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.65067 0.170731
$$743$$ −21.5870 −0.791951 −0.395976 0.918261i $$-0.629594\pi$$
−0.395976 + 0.918261i $$0.629594\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 3.94296 0.144362
$$747$$ 0 0
$$748$$ −11.5209 −0.421247
$$749$$ −36.0373 −1.31678
$$750$$ 0 0
$$751$$ −42.2089 −1.54023 −0.770113 0.637908i $$-0.779800\pi$$
−0.770113 + 0.637908i $$0.779800\pi$$
$$752$$ −5.32962 −0.194351
$$753$$ 0 0
$$754$$ −8.09438 −0.294780
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −32.3867 −1.17711 −0.588557 0.808456i $$-0.700304\pi$$
−0.588557 + 0.808456i $$0.700304\pi$$
$$758$$ −33.2044 −1.20604
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −43.1362 −1.56369 −0.781844 0.623475i $$-0.785721\pi$$
−0.781844 + 0.623475i $$0.785721\pi$$
$$762$$ 0 0
$$763$$ 34.6198 1.25332
$$764$$ 14.7890 0.535047
$$765$$ 0 0
$$766$$ −19.3867 −0.700469
$$767$$ −12.9076 −0.466066
$$768$$ 0 0
$$769$$ −33.7541 −1.21720 −0.608602 0.793476i $$-0.708269\pi$$
−0.608602 + 0.793476i $$0.708269\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 11.5977 0.417410
$$773$$ −44.9055 −1.61514 −0.807570 0.589772i $$-0.799217\pi$$
−0.807570 + 0.589772i $$0.799217\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 7.06157 0.253495
$$777$$ 0 0
$$778$$ −6.87026 −0.246311
$$779$$ −3.54064 −0.126856
$$780$$ 0 0
$$781$$ −20.2771 −0.725572
$$782$$ 14.5164 0.519106
$$783$$ 0 0
$$784$$ −1.45936 −0.0521201
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 23.0550 0.821821 0.410910 0.911676i $$-0.365211\pi$$
0.410910 + 0.911676i $$0.365211\pi$$
$$788$$ 12.6022 0.448935
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7.76023 0.275922
$$792$$ 0 0
$$793$$ −7.53206 −0.267471
$$794$$ 21.8879 0.776772
$$795$$ 0 0
$$796$$ −23.7405 −0.841460
$$797$$ −35.7384 −1.26592 −0.632960 0.774184i $$-0.718160\pi$$
−0.632960 + 0.774184i $$0.718160\pi$$
$$798$$ 0 0
$$799$$ 12.5452 0.443816
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 7.49670 0.264718
$$803$$ 21.4285 0.756196
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −14.6022 −0.514341
$$807$$ 0 0
$$808$$ 6.10551 0.214791
$$809$$ 21.0525 0.740167 0.370084 0.928998i $$-0.379329\pi$$
0.370084 + 0.928998i $$0.379329\pi$$
$$810$$ 0 0
$$811$$ 17.3912 0.610687 0.305344 0.952242i $$-0.401229\pi$$
0.305344 + 0.952242i $$0.401229\pi$$
$$812$$ −10.0570 −0.352933
$$813$$ 0 0
$$814$$ 38.1231 1.33622
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.73194 0.0605931
$$818$$ 4.21101 0.147235
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.43966 0.224746 0.112373 0.993666i $$-0.464155\pi$$
0.112373 + 0.993666i $$0.464155\pi$$
$$822$$ 0 0
$$823$$ −22.8308 −0.795833 −0.397917 0.917422i $$-0.630267\pi$$
−0.397917 + 0.917422i $$0.630267\pi$$
$$824$$ 11.2483 0.391855
$$825$$ 0 0
$$826$$ −16.0373 −0.558010
$$827$$ 29.2705 1.01784 0.508918 0.860815i $$-0.330046\pi$$
0.508918 + 0.860815i $$0.330046\pi$$
$$828$$ 0 0
$$829$$ −22.5053 −0.781640 −0.390820 0.920467i $$-0.627809\pi$$
−0.390820 + 0.920467i $$0.627809\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.89449 −0.0656797
$$833$$ 3.43513 0.119020
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −4.89449 −0.169280
$$837$$ 0 0
$$838$$ 10.8703 0.375507
$$839$$ −25.2241 −0.870833 −0.435417 0.900229i $$-0.643399\pi$$
−0.435417 + 0.900229i $$0.643399\pi$$
$$840$$ 0 0
$$841$$ −10.7450 −0.370519
$$842$$ 33.7496 1.16309
$$843$$ 0 0
$$844$$ 19.9848 0.687906
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −30.4967 −1.04788
$$848$$ −1.97577 −0.0678481
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −48.0353 −1.64663
$$852$$ 0 0
$$853$$ −19.4548 −0.666121 −0.333060 0.942905i $$-0.608081\pi$$
−0.333060 + 0.942905i $$0.608081\pi$$
$$854$$ −9.35838 −0.320237
$$855$$ 0 0
$$856$$ 15.3099 0.523282
$$857$$ −50.2351 −1.71600 −0.858000 0.513650i $$-0.828293\pi$$
−0.858000 + 0.513650i $$0.828293\pi$$
$$858$$ 0 0
$$859$$ −5.82840 −0.198862 −0.0994312 0.995044i $$-0.531702\pi$$
−0.0994312 + 0.995044i $$0.531702\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −15.7011 −0.534782
$$863$$ −8.37356 −0.285039 −0.142520 0.989792i $$-0.545520\pi$$
−0.142520 + 0.989792i $$0.545520\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 15.1847 0.515997
$$867$$ 0 0
$$868$$ −18.1428 −0.615808
$$869$$ 46.4815 1.57678
$$870$$ 0 0
$$871$$ −9.83490 −0.333243
$$872$$ −14.7077 −0.498066
$$873$$ 0 0
$$874$$ 6.16707 0.208604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −23.9031 −0.807149 −0.403575 0.914947i $$-0.632232\pi$$
−0.403575 + 0.914947i $$0.632232\pi$$
$$878$$ −25.4482 −0.858836
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25.0419 0.843682 0.421841 0.906670i $$-0.361384\pi$$
0.421841 + 0.906670i $$0.361384\pi$$
$$882$$ 0 0
$$883$$ −9.51188 −0.320100 −0.160050 0.987109i $$-0.551166\pi$$
−0.160050 + 0.987109i $$0.551166\pi$$
$$884$$ 4.45936 0.149985
$$885$$ 0 0
$$886$$ −24.4351 −0.820914
$$887$$ −40.7824 −1.36934 −0.684669 0.728854i $$-0.740053\pi$$
−0.684669 + 0.728854i $$0.740053\pi$$
$$888$$ 0 0
$$889$$ 15.5977 0.523129
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 18.7122 0.626532
$$893$$ 5.32962 0.178349
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.35386 −0.0786368
$$897$$ 0 0
$$898$$ 26.5012 0.884357
$$899$$ 32.9318 1.09834
$$900$$ 0 0
$$901$$ 4.65067 0.154936
$$902$$ 17.3296 0.577013
$$903$$ 0 0
$$904$$ −3.29681 −0.109650
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −9.05299 −0.300600 −0.150300 0.988640i $$-0.548024\pi$$
−0.150300 + 0.988640i $$0.548024\pi$$
$$908$$ 13.0813 0.434117
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 28.2947 0.937446 0.468723 0.883345i $$-0.344714\pi$$
0.468723 + 0.883345i $$0.344714\pi$$
$$912$$ 0 0
$$913$$ −21.7077 −0.718420
$$914$$ −36.3230 −1.20146
$$915$$ 0 0
$$916$$ −9.89902 −0.327073
$$917$$ −39.9692 −1.31990
$$918$$ 0 0
$$919$$ 42.3801 1.39799 0.698995 0.715127i $$-0.253631\pi$$
0.698995 + 0.715127i $$0.253631\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 23.0813 0.760141
$$923$$ 7.84858 0.258339
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31.0419 1.02010
$$927$$ 0 0
$$928$$ 4.27258 0.140254
$$929$$ −38.7430 −1.27112 −0.635558 0.772053i $$-0.719230\pi$$
−0.635558 + 0.772053i $$0.719230\pi$$
$$930$$ 0 0
$$931$$ 1.45936 0.0478287
$$932$$ 12.6395 0.414022
$$933$$ 0 0
$$934$$ −16.6925 −0.546196
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.20441 0.104683 0.0523417 0.998629i $$-0.483331\pi$$
0.0523417 + 0.998629i $$0.483331\pi$$
$$938$$ −12.2196 −0.398984
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −3.14492 −0.102521 −0.0512607 0.998685i $$-0.516324\pi$$
−0.0512607 + 0.998685i $$0.516324\pi$$
$$942$$ 0 0
$$943$$ −21.8354 −0.711058
$$944$$ 6.81322 0.221751
$$945$$ 0 0
$$946$$ −8.47699 −0.275611
$$947$$ −29.0509 −0.944028 −0.472014 0.881591i $$-0.656473\pi$$
−0.472014 + 0.881591i $$0.656473\pi$$
$$948$$ 0 0
$$949$$ −8.29426 −0.269243
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 5.54064 0.179573
$$953$$ −11.5452 −0.373985 −0.186992 0.982361i $$-0.559874\pi$$
−0.186992 + 0.982361i $$0.559874\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −20.1605 −0.652036
$$957$$ 0 0
$$958$$ −12.5891 −0.406735
$$959$$ −30.2947 −0.978268
$$960$$ 0 0
$$961$$ 28.4088 0.916413
$$962$$ −14.7562 −0.475758
$$963$$ 0 0
$$964$$ −11.7693 −0.379063
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −17.2044 −0.553256 −0.276628 0.960977i $$-0.589217\pi$$
−0.276628 + 0.960977i $$0.589217\pi$$
$$968$$ 12.9561 0.416424
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −38.2771 −1.22837 −0.614185 0.789162i $$-0.710515\pi$$
−0.614185 + 0.789162i $$0.710515\pi$$
$$972$$ 0 0
$$973$$ −21.9142 −0.702537
$$974$$ 20.7077 0.663518
$$975$$ 0 0
$$976$$ 3.97577 0.127261
$$977$$ 16.5340 0.528971 0.264485 0.964390i $$-0.414798\pi$$
0.264485 + 0.964390i $$0.414798\pi$$
$$978$$ 0 0
$$979$$ 17.7274 0.566571
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 20.0747 0.640608
$$983$$ 41.7783 1.33252 0.666261 0.745719i $$-0.267894\pi$$
0.666261 + 0.745719i $$0.267894\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −10.0570 −0.320281
$$987$$ 0 0
$$988$$ 1.89449 0.0602718
$$989$$ 10.6810 0.339637
$$990$$ 0 0
$$991$$ 27.7845 0.882602 0.441301 0.897359i $$-0.354517\pi$$
0.441301 + 0.897359i $$0.354517\pi$$
$$992$$ 7.70771 0.244720
$$993$$ 0 0
$$994$$ 9.75165 0.309304
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16.9297 0.536170 0.268085 0.963395i $$-0.413609\pi$$
0.268085 + 0.963395i $$0.413609\pi$$
$$998$$ 41.5956 1.31669
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.ct.1.1 yes 3
3.2 odd 2 8550.2.a.ch.1.1 yes 3
5.4 even 2 8550.2.a.cc.1.3 3
15.14 odd 2 8550.2.a.cm.1.3 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
8550.2.a.cc.1.3 3 5.4 even 2
8550.2.a.ch.1.1 yes 3 3.2 odd 2
8550.2.a.cm.1.3 yes 3 15.14 odd 2
8550.2.a.ct.1.1 yes 3 1.1 even 1 trivial