# Properties

 Label 2850.2.d.q.799.2 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## 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 799.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.q.799.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} +2.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} +1.00000i q^{38} -4.00000 q^{39} -8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} -2.00000 q^{44} +4.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} +4.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} -1.00000i q^{57} +2.00000 q^{61} -8.00000i q^{62} +2.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -8.00000 q^{71} +1.00000i q^{72} -14.0000i q^{73} -8.00000 q^{74} -1.00000 q^{76} -4.00000i q^{77} -4.00000i q^{78} +1.00000 q^{81} -8.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} -6.00000 q^{86} -2.00000i q^{88} -8.00000 q^{91} +4.00000i q^{92} +8.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} -12.0000i q^{97} +3.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 16 q^{31} + 4 q^{34} + 2 q^{36} - 8 q^{39} - 16 q^{41} - 4 q^{44} + 8 q^{46} + 6 q^{49} - 4 q^{51} - 2 q^{54} - 4 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 8 q^{69} - 16 q^{71} - 16 q^{74} - 2 q^{76} + 2 q^{81} + 4 q^{84} - 12 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^21 - 2 * q^24 + 8 * q^26 - 16 * q^31 + 4 * q^34 + 2 * q^36 - 8 * q^39 - 16 * q^41 - 4 * q^44 + 8 * q^46 + 6 * q^49 - 4 * q^51 - 2 * q^54 - 4 * q^56 + 4 * q^61 - 2 * q^64 + 4 * q^66 - 8 * q^69 - 16 * q^71 - 16 * q^74 - 2 * q^76 + 2 * q^81 + 4 * q^84 - 12 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96 - 4 * q^99

## 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 $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 2.00000i 0.426401i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ 1.00000i 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 4.00000i 0.554700i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ − 1.00000i − 0.132453i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ − 4.00000i − 0.455842i
$$78$$ − 4.00000i − 0.452911i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 8.00000i − 0.883452i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ 0 0
$$88$$ − 2.00000i − 0.213201i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ 4.00000i 0.417029i
$$93$$ 8.00000i 0.829561i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 12.0000i − 1.21842i −0.793011 0.609208i $$-0.791488\pi$$
0.793011 0.609208i $$-0.208512\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ − 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
$$116$$ 0 0
$$117$$ 4.00000i 0.369800i
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000i 0.181071i
$$123$$ 8.00000i 0.721336i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 6.00000 0.528271
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ − 2.00000i − 0.173422i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ − 22.0000i − 1.87959i −0.341743 0.939793i $$-0.611017\pi$$
0.341743 0.939793i $$-0.388983\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ − 8.00000i − 0.671345i
$$143$$ − 8.00000i − 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 8.00000i − 0.657596i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 2.00000i 0.161690i
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ 1.00000i 0.0785674i
$$163$$ − 14.0000i − 1.09656i −0.836293 0.548282i $$-0.815282\pi$$
0.836293 0.548282i $$-0.184718\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 6.00000i − 0.457496i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ − 2.00000i − 0.147844i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ − 4.00000i − 0.292509i
$$188$$ 12.0000i 0.875190i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 24.0000i − 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ − 2.00000i − 0.142134i
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 2.00000i 0.140720i
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ 4.00000i 0.278019i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 8.00000i 0.548151i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 16.0000i 1.08615i
$$218$$ − 10.0000i − 0.677285i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 8.00000i 0.536925i
$$223$$ − 4.00000i − 0.267860i −0.990991 0.133930i $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ − 4.00000i − 0.259281i
$$239$$ −10.0000 −0.646846 −0.323423 0.946254i $$-0.604834\pi$$
−0.323423 + 0.946254i $$0.604834\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −8.00000 −0.510061
$$247$$ − 4.00000i − 0.254514i
$$248$$ 8.00000i 0.508001i
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ − 8.00000i − 0.502956i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 6.00000i 0.373544i
$$259$$ 16.0000 0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 18.0000i − 1.11204i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ − 8.00000i − 0.488678i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 8.00000i 0.484182i
$$274$$ 22.0000 1.32907
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 18.0000i 1.08152i 0.841178 + 0.540758i $$0.181862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 16.0000i 0.944450i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 14.0000i 0.819288i
$$293$$ − 14.0000i − 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 2.00000i 0.116052i
$$298$$ − 10.0000i − 0.579284i
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 8.00000i − 0.460348i
$$303$$ − 2.00000i − 0.114897i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 32.0000i − 1.82634i −0.407583 0.913168i $$-0.633628\pi$$
0.407583 0.913168i $$-0.366372\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ 6.00000i 0.339140i 0.985518 + 0.169570i $$0.0542379\pi$$
−0.985518 + 0.169570i $$0.945762\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ − 8.00000i − 0.445823i
$$323$$ − 2.00000i − 0.111283i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 14.0000 0.775388
$$327$$ 10.0000i 0.553001i
$$328$$ 8.00000i 0.441726i
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ − 8.00000i − 0.438397i
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ 28.0000i 1.52526i 0.646837 + 0.762629i $$0.276092\pi$$
−0.646837 + 0.762629i $$0.723908\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 20.0000i − 1.07990i
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 2.00000i 0.106600i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.00000i 0.211702i
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 2.00000i 0.105118i
$$363$$ 7.00000i 0.367405i
$$364$$ 8.00000 0.419314
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 18.0000i 0.939592i 0.882775 + 0.469796i $$0.155673\pi$$
−0.882775 + 0.469796i $$0.844327\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ − 8.00000i − 0.414781i
$$373$$ 36.0000i 1.86401i 0.362446 + 0.932005i $$0.381942\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ 2.00000i 0.102869i
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ − 18.0000i − 0.920960i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 24.0000 1.22157
$$387$$ − 6.00000i − 0.304997i
$$388$$ 12.0000i 0.609208i
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ − 3.00000i − 0.151523i
$$393$$ 18.0000i 0.907980i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ 32.0000i 1.59403i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 16.0000i 0.793091i
$$408$$ 2.00000i 0.0990148i
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ −22.0000 −1.08518
$$412$$ 4.00000i 0.197066i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 4.00000 0.196116
$$417$$ 0 0
$$418$$ 2.00000i 0.0978232i
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 12.0000i 0.584151i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ − 4.00000i − 0.193574i
$$428$$ 12.0000i 0.580042i
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 14.0000i − 0.668946i
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ − 8.00000i − 0.380521i
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ 10.0000i 0.472984i
$$448$$ 2.00000i 0.0944911i
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ − 6.00000i − 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ − 4.00000i − 0.186097i
$$463$$ 6.00000i 0.278844i 0.990233 + 0.139422i $$0.0445244\pi$$
−0.990233 + 0.139422i $$0.955476\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 8.00000i 0.370196i 0.982720 + 0.185098i $$0.0592602\pi$$
−0.982720 + 0.185098i $$0.940740\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 12.0000i 0.551761i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ − 6.00000i − 0.274721i
$$478$$ − 10.0000i − 0.457389i
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 32.0000 1.45907
$$482$$ 2.00000i 0.0910975i
$$483$$ 8.00000i 0.364013i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 28.0000i 1.26880i 0.773004 + 0.634401i $$0.218753\pi$$
−0.773004 + 0.634401i $$0.781247\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ −14.0000 −0.633102
$$490$$ 0 0
$$491$$ 22.0000 0.992846 0.496423 0.868081i $$-0.334646\pi$$
0.496423 + 0.868081i $$0.334646\pi$$
$$492$$ − 8.00000i − 0.360668i
$$493$$ 0 0
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 16.0000i 0.717698i
$$498$$ − 4.00000i − 0.179244i
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 2.00000i 0.0892644i
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 3.00000i 0.133235i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ −28.0000 −1.23865
$$512$$ 1.00000i 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −6.00000 −0.264135
$$517$$ − 24.0000i − 1.05552i
$$518$$ 16.0000i 0.703000i
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 16.0000i 0.696971i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000i 0.0867110i
$$533$$ 32.0000i 1.38607i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 12.0000i 0.515444i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ 22.0000i 0.939793i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ −18.0000 −0.764747
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 42.0000i − 1.77960i −0.456354 0.889799i $$-0.650845\pi$$
0.456354 0.889799i $$-0.349155\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 12.0000i 0.506189i
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 8.00000i 0.335673i
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 18.0000i 0.751961i
$$574$$ −16.0000 −0.667827
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 22.0000i − 0.915872i −0.888985 0.457936i $$-0.848589\pi$$
0.888985 0.457936i $$-0.151411\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ −8.00000 −0.331896
$$582$$ − 12.0000i − 0.497416i
$$583$$ 12.0000i 0.496989i
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 8.00000i 0.328798i
$$593$$ 46.0000i 1.88899i 0.328521 + 0.944497i $$0.393450\pi$$
−0.328521 + 0.944497i $$0.606550\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 20.0000i − 0.818546i
$$598$$ − 16.0000i − 0.654289i
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 12.0000i 0.489083i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ − 12.0000i − 0.487065i −0.969893 0.243532i $$-0.921694\pi$$
0.969893 0.243532i $$-0.0783062\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 6.00000i 0.242338i 0.992632 + 0.121169i $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ 32.0000 1.29141
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 40.0000 1.60774 0.803868 0.594808i $$-0.202772\pi$$
0.803868 + 0.594808i $$0.202772\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 18.0000i − 0.721734i
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −6.00000 −0.239808
$$627$$ − 2.00000i − 0.0798723i
$$628$$ − 18.0000i − 0.718278i
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ − 12.0000i − 0.476957i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ − 12.0000i − 0.475457i
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 32.0000 1.26392 0.631962 0.774999i $$-0.282250\pi$$
0.631962 + 0.774999i $$0.282250\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ − 32.0000i − 1.25805i −0.777385 0.629025i $$-0.783454\pi$$
0.777385 0.629025i $$-0.216546\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 14.0000i 0.548282i
$$653$$ 46.0000i 1.80012i 0.435767 + 0.900060i $$0.356477\pi$$
−0.435767 + 0.900060i $$0.643523\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ 14.0000i 0.546192i
$$658$$ − 24.0000i − 0.935617i
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 42.0000 1.63361 0.816805 0.576913i $$-0.195743\pi$$
0.816805 + 0.576913i $$0.195743\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 8.00000i 0.310694i
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ − 8.00000i − 0.309529i
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 4.00000i − 0.154189i −0.997024 0.0770943i $$-0.975436\pi$$
0.997024 0.0770943i $$-0.0245643\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ − 16.0000i − 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 10.0000i 0.381524i
$$688$$ 6.00000i 0.228748i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 4.00000i 0.151947i
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000i 0.606043i
$$698$$ 10.0000i 0.378506i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 4.00000i 0.150970i
$$703$$ 8.00000i 0.301726i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ − 4.00000i − 0.150435i
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32.0000i 1.19841i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 10.0000i 0.373457i
$$718$$ 10.0000i 0.373197i
$$719$$ −50.0000 −1.86469 −0.932343 0.361576i $$-0.882239\pi$$
−0.932343 + 0.361576i $$0.882239\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 1.00000i 0.0372161i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 18.0000i 0.667583i 0.942647 + 0.333792i $$0.108328\pi$$
−0.942647 + 0.333792i $$0.891672\pi$$
$$728$$ 8.00000i 0.296500i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 2.00000i 0.0739221i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ −18.0000 −0.664392
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 16.0000i 0.589368i
$$738$$ 8.00000i 0.294484i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 12.0000i 0.440534i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −36.0000 −1.31805
$$747$$ 4.00000i 0.146352i
$$748$$ 4.00000i 0.146254i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ − 2.00000i − 0.0728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ −8.00000 −0.290382
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 20.0000i 0.724049i
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −50.0000 −1.80305 −0.901523 0.432731i $$-0.857550\pi$$
−0.901523 + 0.432731i $$0.857550\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 24.0000i 0.863779i
$$773$$ − 34.0000i − 1.22290i −0.791285 0.611448i $$-0.790588\pi$$
0.791285 0.611448i $$-0.209412\pi$$
$$774$$ 6.00000 0.215666
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ − 16.0000i − 0.573997i
$$778$$ − 30.0000i − 1.07555i
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ − 8.00000i − 0.286079i
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 2.00000i 0.0710669i
$$793$$ − 8.00000i − 0.284088i
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ − 2.00000i − 0.0708436i −0.999372 0.0354218i $$-0.988723\pi$$
0.999372 0.0354218i $$-0.0112775\pi$$
$$798$$ − 2.00000i − 0.0707992i
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 12.0000i 0.423735i
$$803$$ − 28.0000i − 0.988099i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −32.0000 −1.12715
$$807$$ 0 0
$$808$$ − 2.00000i − 0.0703598i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ − 12.0000i − 0.420858i
$$814$$ −16.0000 −0.560800
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 6.00000i 0.209913i
$$818$$ − 30.0000i − 1.04893i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ − 22.0000i − 0.767338i
$$823$$ − 34.0000i − 1.18517i −0.805510 0.592583i $$-0.798108\pi$$
0.805510 0.592583i $$-0.201892\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ 18.0000 0.624413
$$832$$ 4.00000i 0.138675i
$$833$$ − 6.00000i − 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ − 8.00000i − 0.276520i
$$838$$ 30.0000i 1.03633i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 22.0000i 0.758170i
$$843$$ − 12.0000i − 0.413302i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 14.0000i 0.481046i
$$848$$ 6.00000i 0.206041i
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ 12.0000i 0.408722i
$$863$$ 16.0000i 0.544646i 0.962206 + 0.272323i $$0.0877920\pi$$
−0.962206 + 0.272323i $$0.912208\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ − 13.0000i − 0.441503i
$$868$$ − 16.0000i − 0.543075i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 32.0000 1.08428
$$872$$ 10.0000i 0.338643i
$$873$$ 12.0000i 0.406138i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ − 12.0000i − 0.405211i −0.979260 0.202606i $$-0.935059\pi$$
0.979260 0.202606i $$-0.0649409\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 26.0000i 0.874970i 0.899226 + 0.437485i $$0.144131\pi$$
−0.899226 + 0.437485i $$0.855869\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 48.0000i 1.61168i 0.592132 + 0.805841i $$0.298286\pi$$
−0.592132 + 0.805841i $$0.701714\pi$$
$$888$$ − 8.00000i − 0.268462i
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 4.00000i 0.133930i
$$893$$ − 12.0000i − 0.401565i
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 16.0000i 0.534224i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 16.0000i − 0.532742i
$$903$$ − 12.0000i − 0.399335i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 8.00000i − 0.264761i
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 36.0000i 1.18882i
$$918$$ 2.00000i 0.0660098i
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ 2.00000i 0.0658665i
$$923$$ 32.0000i 1.05329i
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −6.00000 −0.197172
$$927$$ 4.00000i 0.131377i
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ − 6.00000i − 0.196537i
$$933$$ 18.0000i 0.589294i
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ 18.0000i 0.586472i
$$943$$ 32.0000i 1.04206i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ −56.0000 −1.81784
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 4.00000i 0.129641i
$$953$$ − 34.0000i − 1.10137i −0.834714 0.550684i $$-0.814367\pi$$
0.834714 0.550684i $$-0.185633\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 10.0000 0.323423
$$957$$ 0 0
$$958$$ 10.0000i 0.323085i
$$959$$ −44.0000 −1.42083
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 32.0000i 1.03172i
$$963$$ 12.0000i 0.386695i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 58.0000i 1.86515i 0.360971 + 0.932577i $$0.382445\pi$$
−0.360971 + 0.932577i $$0.617555\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ −28.0000 −0.897178
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 38.0000i 1.21573i 0.794041 + 0.607864i $$0.207973\pi$$
−0.794041 + 0.607864i $$0.792027\pi$$
$$978$$ − 14.0000i − 0.447671i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 10.0000 0.319275
$$982$$ 22.0000i 0.702048i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24.0000i 0.763928i
$$988$$ 4.00000i 0.127257i
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ − 12.0000i − 0.380808i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 62.0000i − 1.96356i −0.190022 0.981780i $$-0.560856\pi$$
0.190022 0.981780i $$-0.439144\pi$$
$$998$$ 40.0000i 1.26618i
$$999$$ −8.00000 −0.253109
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.q.799.2 2
5.2 odd 4 2850.2.a.e.1.1 1
5.3 odd 4 570.2.a.l.1.1 1
5.4 even 2 inner 2850.2.d.q.799.1 2
15.2 even 4 8550.2.a.bf.1.1 1
15.8 even 4 1710.2.a.b.1.1 1
20.3 even 4 4560.2.a.o.1.1 1

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