Properties

 Label 2850.2.d.d.799.1 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 799.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.d.799.2

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} +2.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} -2.00000i q^{42} -8.00000i q^{43} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +6.00000 q^{59} +2.00000 q^{61} -2.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} +1.00000 q^{76} +2.00000i q^{78} -2.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -2.00000 q^{84} -8.00000 q^{86} -6.00000i q^{87} +12.0000 q^{89} +4.00000 q^{91} -2.00000i q^{93} -1.00000 q^{96} -10.0000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} + 2 q^{24} - 4 q^{26} + 12 q^{29} + 4 q^{31} + 2 q^{36} - 4 q^{39} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} + 4 q^{74} + 2 q^{76} - 4 q^{79} + 2 q^{81} - 4 q^{84} - 16 q^{86} + 24 q^{89} + 8 q^{91} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 + 2 * q^24 - 4 * q^26 + 12 * q^29 + 4 * q^31 + 2 * q^36 - 4 * q^39 + 6 * q^49 + 2 * q^54 - 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 + 4 * q^74 + 2 * q^76 - 4 * q^79 + 2 * q^81 - 4 * q^84 - 16 * q^86 + 24 * q^89 + 8 * q^91 - 2 * q^96

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 1.00000i 0.132453i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ − 2.00000i − 0.254000i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 2.00000i 0.226455i
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ − 6.00000i − 0.643268i
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ − 2.00000i − 0.207390i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 2.00000i 0.188982i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 2.00000i 0.184900i
$$118$$ − 6.00000i − 0.552345i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 2.00000i − 0.181071i
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ − 2.00000i − 0.173422i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 12.0000i − 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ 2.00000i 0.159111i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 8.00000i 0.609994i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 6.00000i − 0.450988i
$$178$$ − 12.0000i − 0.899438i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ − 2.00000i − 0.147844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −2.00000 −0.146647
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 6.00000i 0.422159i
$$203$$ 12.0000i 0.842235i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 4.00000i 0.271538i
$$218$$ − 16.0000i − 1.08366i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 2.00000i − 0.134231i
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ − 24.0000i − 1.57229i −0.618041 0.786146i $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −6.00000 −0.390567
$$237$$ 2.00000i 0.129914i
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 2.00000i 0.127000i
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ − 12.0000i − 0.734388i
$$268$$ 4.00000i 0.244339i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 0 0
$$273$$ − 4.00000i − 0.242091i
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 28.0000i − 1.68236i −0.540758 0.841178i $$-0.681862\pi$$
0.540758 0.841178i $$-0.318138\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 24.0000 1.43172 0.715860 0.698244i $$-0.246035\pi$$
0.715860 + 0.698244i $$0.246035\pi$$
$$282$$ 0 0
$$283$$ − 32.0000i − 1.90220i −0.308879 0.951101i $$-0.599954\pi$$
0.308879 0.951101i $$-0.400046\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000i 0.819288i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ − 6.00000i − 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 10.0000i 0.575435i
$$303$$ 6.00000i 0.344691i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ − 16.0000i − 0.884802i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 1.00000i − 0.0540738i
$$343$$ 20.0000i 1.07990i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 18.0000i 0.966291i 0.875540 + 0.483145i $$0.160506\pi$$
−0.875540 + 0.483145i $$0.839494\pi$$
$$348$$ 6.00000i 0.321634i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ − 12.0000i − 0.638696i −0.947638 0.319348i $$-0.896536\pi$$
0.947638 0.319348i $$-0.103464\pi$$
$$354$$ −6.00000 −0.318896
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ 0 0
$$358$$ 6.00000i 0.317110i
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 16.0000i 0.840941i
$$363$$ 11.0000i 0.577350i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 10.0000i − 0.521996i −0.965339 0.260998i $$-0.915948\pi$$
0.965339 0.260998i $$-0.0840516\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 2.00000i 0.103695i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 2.00000i 0.102869i
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 8.00000i 0.406663i
$$388$$ 10.0000i 0.507673i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 3.00000i 0.151523i
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8.00000i 0.401508i 0.979642 + 0.200754i $$0.0643393\pi$$
−0.979642 + 0.200754i $$0.935661\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ − 4.00000i − 0.199254i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 8.00000i 0.394132i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ − 4.00000i − 0.195881i
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 2.00000i − 0.0961139i −0.998845 0.0480569i $$-0.984697\pi$$
0.998845 0.0480569i $$-0.0153029\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ 0 0
$$438$$ 14.0000i 0.668946i
$$439$$ 22.0000 1.05000 0.525001 0.851101i $$-0.324065\pi$$
0.525001 + 0.851101i $$0.324065\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 30.0000i 1.42534i 0.701498 + 0.712672i $$0.252515\pi$$
−0.701498 + 0.712672i $$0.747485\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 10.0000i 0.469841i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −24.0000 −1.11178
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 6.00000i 0.276172i
$$473$$ 0 0
$$474$$ 2.00000 0.0918630
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 6.00000i 0.274721i
$$478$$ 24.0000i 1.09773i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 22.0000i 1.00207i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 6.00000i 0.268866i
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 16.0000i 0.709885i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ 28.0000 1.23865
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 2.00000i 0.0867110i
$$533$$ 0 0
$$534$$ −12.0000 −0.519291
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 6.00000i 0.258919i
$$538$$ 18.0000i 0.776035i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ 16.0000i 0.686626i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ − 4.00000i − 0.170097i
$$554$$ −28.0000 −1.18961
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 6.00000i 0.254228i 0.991888 + 0.127114i $$0.0405714\pi$$
−0.991888 + 0.127114i $$0.959429\pi$$
$$558$$ 2.00000i 0.0846668i
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 24.0000i − 1.01238i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −32.0000 −1.34506
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 10.0000i 0.414513i
$$583$$ 0 0
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 2.00000i 0.0821995i
$$593$$ − 36.0000i − 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 4.00000i − 0.163709i
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ 4.00000i 0.162893i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16.0000i 0.646234i 0.946359 + 0.323117i $$0.104731\pi$$
−0.946359 + 0.323117i $$0.895269\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000i 0.483102i 0.970388 + 0.241551i $$0.0776561\pi$$
−0.970388 + 0.241551i $$0.922344\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000i 0.961540i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 4.00000i 0.159617i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ − 2.00000i − 0.0795557i
$$633$$ − 8.00000i − 0.317971i
$$634$$ −30.0000 −1.19145
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 6.00000i − 0.237729i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ − 4.00000i − 0.156652i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ −16.0000 −0.625650
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000i 0.546192i
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 14.0000i − 0.539660i −0.962908 0.269830i $$-0.913032\pi$$
0.962908 0.269830i $$-0.0869676\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 30.0000i − 1.15299i −0.817099 0.576497i $$-0.804419\pi$$
0.817099 0.576497i $$-0.195581\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ − 10.0000i − 0.381524i
$$688$$ − 8.00000i − 0.304997i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ − 18.0000i − 0.684257i
$$693$$ 0 0
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 0 0
$$698$$ − 10.0000i − 0.378506i
$$699$$ −24.0000 −0.907763
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ − 2.00000i − 0.0754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ − 12.0000i − 0.451306i
$$708$$ 6.00000i 0.225494i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ 12.0000i 0.449719i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.00000 0.224231
$$717$$ 24.0000i 0.896296i
$$718$$ 24.0000i 0.895672i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 22.0000i 0.818189i
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ − 34.0000i − 1.26099i −0.776193 0.630495i $$-0.782852\pi$$
0.776193 0.630495i $$-0.217148\pi$$
$$728$$ 4.00000i 0.148250i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 2.00000i 0.0739221i
$$733$$ − 32.0000i − 1.18195i −0.806691 0.590973i $$-0.798744\pi$$
0.806691 0.590973i $$-0.201256\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ − 12.0000i − 0.440534i
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −22.0000 −0.802791 −0.401396 0.915905i $$-0.631475\pi$$
−0.401396 + 0.915905i $$0.631475\pi$$
$$752$$ 0 0
$$753$$ − 12.0000i − 0.437304i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 32.0000i 1.16306i 0.813525 + 0.581530i $$0.197546\pi$$
−0.813525 + 0.581530i $$0.802454\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 32.0000i 1.15848i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 12.0000i − 0.433295i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 34.0000 1.22607 0.613036 0.790055i $$-0.289948\pi$$
0.613036 + 0.790055i $$0.289948\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 2.00000i 0.0719816i
$$773$$ 18.0000i 0.647415i 0.946157 + 0.323708i $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 4.00000i 0.143499i
$$778$$ − 30.0000i − 1.07555i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 44.0000i 1.56843i 0.620489 + 0.784215i $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ − 4.00000i − 0.142044i
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −12.0000 −0.423999
$$802$$ − 12.0000i − 0.423735i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 18.0000i 0.633630i
$$808$$ − 6.00000i − 0.211079i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ − 12.0000i − 0.421117i
$$813$$ − 20.0000i − 0.701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000i 0.279885i
$$818$$ − 22.0000i − 0.769212i
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 42.0000 1.46581 0.732905 0.680331i $$-0.238164\pi$$
0.732905 + 0.680331i $$0.238164\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ 22.0000i 0.766872i 0.923567 + 0.383436i $$0.125259\pi$$
−0.923567 + 0.383436i $$0.874741\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ 0 0
$$829$$ −8.00000 −0.277851 −0.138926 0.990303i $$-0.544365\pi$$
−0.138926 + 0.990303i $$0.544365\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.00000i 0.0691301i
$$838$$ 12.0000i 0.414533i
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 8.00000i − 0.275698i
$$843$$ − 24.0000i − 0.826604i
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 22.0000i − 0.755929i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −32.0000 −1.09824
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 44.0000i − 1.50653i −0.657716 0.753266i $$-0.728477\pi$$
0.657716 0.753266i $$-0.271523\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.0000i 0.817443i
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ − 17.0000i − 0.577350i
$$868$$ − 4.00000i − 0.135769i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 16.0000i 0.541828i
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 38.0000i 1.28317i 0.767052 + 0.641584i $$0.221723\pi$$
−0.767052 + 0.641584i $$0.778277\pi$$
$$878$$ − 22.0000i − 0.742464i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 30.0000 1.00787
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 4.00000i − 0.133930i
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 24.0000i 0.800890i
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ 44.0000i 1.46100i 0.682915 + 0.730498i $$0.260712\pi$$
−0.682915 + 0.730498i $$0.739288\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 60.0000 1.98789 0.993944 0.109885i $$-0.0350482\pi$$
0.993944 + 0.109885i $$0.0350482\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 0 0
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ − 30.0000i − 0.987997i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −26.0000 −0.854413
$$927$$ 8.00000i 0.262754i
$$928$$ − 6.00000i − 0.196960i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 24.0000i 0.786146i
$$933$$ 0 0
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 4.00000i 0.130327i
$$943$$ 0 0
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 18.0000i − 0.584921i −0.956278 0.292461i $$-0.905526\pi$$
0.956278 0.292461i $$-0.0944741\pi$$
$$948$$ − 2.00000i − 0.0649570i
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ − 30.0000i − 0.971795i −0.874016 0.485898i $$-0.838493\pi$$
0.874016 0.485898i $$-0.161507\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 4.00000i − 0.128965i
$$963$$ 0 0
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 26.0000i 0.836104i 0.908423 + 0.418052i $$0.137287\pi$$
−0.908423 + 0.418052i $$0.862713\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −42.0000 −1.34784 −0.673922 0.738802i $$-0.735392\pi$$
−0.673922 + 0.738802i $$0.735392\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 8.00000i 0.256468i
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ − 4.00000i − 0.127906i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 2.00000i − 0.0636285i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −58.0000 −1.84243 −0.921215 0.389053i $$-0.872802\pi$$
−0.921215 + 0.389053i $$0.872802\pi$$
$$992$$ − 2.00000i − 0.0635001i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ 44.0000i 1.39349i 0.717317 + 0.696747i $$0.245370\pi$$
−0.717317 + 0.696747i $$0.754630\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ −2.00000 −0.0632772
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 2850.2.d.d.799.1 2
5.2 odd 4 2850.2.a.q.1.1 1
5.3 odd 4 570.2.a.f.1.1 1
5.4 even 2 inner 2850.2.d.d.799.2 2
15.2 even 4 8550.2.a.e.1.1 1
15.8 even 4 1710.2.a.o.1.1 1
20.3 even 4 4560.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.f.1.1 1 5.3 odd 4
1710.2.a.o.1.1 1 15.8 even 4
2850.2.a.q.1.1 1 5.2 odd 4
2850.2.d.d.799.1 2 1.1 even 1 trivial
2850.2.d.d.799.2 2 5.4 even 2 inner
4560.2.a.m.1.1 1 20.3 even 4
8550.2.a.e.1.1 1 15.2 even 4