# Properties

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

# Related objects

## 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: $$x^{2} + 1$$ 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.j.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} -1.00000i q^{12} +2.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -2.00000 q^{21} +6.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -4.00000 q^{41} +2.00000i q^{42} +6.00000i q^{43} +6.00000 q^{44} -4.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} -8.00000i q^{58} +4.00000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -8.00000i q^{67} -2.00000i q^{68} +4.00000 q^{69} -1.00000i q^{72} -6.00000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} -4.00000i q^{83} +2.00000 q^{84} +6.00000 q^{86} +8.00000i q^{87} -6.00000i q^{88} +4.00000 q^{89} +4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} +12.0000i q^{97} -3.00000i q^{98} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 12q^{11} + 4q^{14} + 2q^{16} + 2q^{19} - 4q^{21} - 2q^{24} + 16q^{29} - 16q^{31} + 4q^{34} + 2q^{36} - 8q^{41} + 12q^{44} - 8q^{46} + 6q^{49} - 4q^{51} - 2q^{54} - 4q^{56} + 8q^{59} + 4q^{61} - 2q^{64} - 12q^{66} + 8q^{69} - 8q^{74} - 2q^{76} - 16q^{79} + 2q^{81} + 4q^{84} + 12q^{86} + 8q^{89} - 24q^{94} + 2q^{96} + 12q^{99} + O(q^{100})$$

## 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
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 6.00000i 1.27920i
$$23$$ − 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
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 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
$$33$$ − 6.00000i − 1.04447i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 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
$$52$$ 0 0
$$53$$ − 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
$$58$$ − 8.00000i − 1.05045i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 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
$$66$$ −6.00000 −0.738549
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ − 12.0000i − 1.36753i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 4.00000i 0.441726i
$$83$$ − 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
$$87$$ 8.00000i 0.857690i
$$88$$ − 6.00000i − 0.639602i
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 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 $$0.208512\pi$$
−0.793011 + 0.609208i $$0.791488\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 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
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 2.00000i 0.188982i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −8.00000 −0.742781
$$117$$ 0 0
$$118$$ − 4.00000i − 0.368230i
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 4.00000i − 0.360668i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ 22.0000 1.92215 0.961074 0.276289i $$-0.0891049\pi$$
0.961074 + 0.276289i $$0.0891049\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ 2.00000i 0.173422i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 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
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ 3.00000i 0.247436i
$$148$$ 4.00000i 0.328798i
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 2.00000i − 0.161690i
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 8.00000i 0.636446i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 6.00000i − 0.469956i −0.972001 0.234978i $$-0.924498\pi$$
0.972001 0.234978i $$-0.0755019\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 6.00000i − 0.457496i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 8.00000 0.606478
$$175$$ 0 0
$$176$$ −6.00000 −0.452267
$$177$$ 4.00000i 0.300658i
$$178$$ − 4.00000i − 0.299813i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ − 12.0000i − 0.877527i
$$188$$ 12.0000i 0.875190i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −22.0000 −1.59186 −0.795932 0.605386i $$-0.793019\pi$$
−0.795932 + 0.605386i $$0.793019\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 8.00000i 0.575853i 0.957653 + 0.287926i $$0.0929658\pi$$
−0.957653 + 0.287926i $$0.907034\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 14.0000i 0.997459i 0.866758 + 0.498729i $$0.166200\pi$$
−0.866758 + 0.498729i $$0.833800\pi$$
$$198$$ − 6.00000i − 0.426401i
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 10.0000i 0.703598i
$$203$$ 16.0000i 1.12298i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 16.0000i − 1.08615i
$$218$$ − 14.0000i − 0.948200i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 4.00000i − 0.268462i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 28.0000i − 1.85843i −0.369546 0.929213i $$-0.620487\pi$$
0.369546 0.929213i $$-0.379513\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 12.0000 0.789542
$$232$$ 8.00000i 0.525226i
$$233$$ − 14.0000i − 0.917170i −0.888650 0.458585i $$-0.848356\pi$$
0.888650 0.458585i $$-0.151644\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ − 8.00000i − 0.519656i
$$238$$ 4.00000i 0.259281i
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ − 25.0000i − 1.60706i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −4.00000 −0.255031
$$247$$ 0 0
$$248$$ − 8.00000i − 0.508001i
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 24.0000i 1.50887i
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 6.00000i 0.373544i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −8.00000 −0.495188
$$262$$ − 22.0000i − 1.35916i
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 4.00000i 0.244796i
$$268$$ 8.00000i 0.488678i
$$269$$ −8.00000 −0.487769 −0.243884 0.969804i $$-0.578422\pi$$
−0.243884 + 0.969804i $$0.578422\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ − 12.0000i − 0.714590i
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 6.00000i 0.351123i
$$293$$ − 10.0000i − 0.584206i −0.956387 0.292103i $$-0.905645\pi$$
0.956387 0.292103i $$-0.0943550\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 6.00000i 0.348155i
$$298$$ 22.0000i 1.27443i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ − 10.0000i − 0.574485i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 8.00000i − 0.456584i −0.973593 0.228292i $$-0.926686\pi$$
0.973593 0.228292i $$-0.0733141\pi$$
$$308$$ 12.0000i 0.683763i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ −48.0000 −2.68748
$$320$$ 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
$$326$$ −6.00000 −0.332309
$$327$$ 14.0000i 0.774202i
$$328$$ − 4.00000i − 0.220863i
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 4.00000i 0.219529i
$$333$$ 4.00000i 0.219199i
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ − 20.0000i − 1.08947i −0.838608 0.544735i $$-0.816630\pi$$
0.838608 0.544735i $$-0.183370\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 48.0000 2.59935
$$342$$ 1.00000i 0.0540738i
$$343$$ 20.0000i 1.07990i
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 36.0000i 1.93258i 0.257454 + 0.966291i $$0.417117\pi$$
−0.257454 + 0.966291i $$0.582883\pi$$
$$348$$ − 8.00000i − 0.428845i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6.00000i 0.319801i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −4.00000 −0.212000
$$357$$ − 4.00000i − 0.211702i
$$358$$ 12.0000i 0.634220i
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 2.00000i − 0.105118i
$$363$$ 25.0000i 1.31216i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 10.0000i − 0.521996i −0.965339 0.260998i $$-0.915948\pi$$
0.965339 0.260998i $$-0.0840516\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 4.00000 0.208232
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 8.00000i 0.414781i
$$373$$ 24.0000i 1.24267i 0.783544 + 0.621336i $$0.213410\pi$$
−0.783544 + 0.621336i $$0.786590\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 0 0
$$378$$ − 2.00000i − 0.102869i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 22.0000i 1.12562i
$$383$$ − 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
$$386$$ 8.00000 0.407189
$$387$$ − 6.00000i − 0.304997i
$$388$$ − 12.0000i − 0.609208i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 3.00000i 0.151523i
$$393$$ 22.0000i 1.10975i
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ 22.0000i 1.10415i 0.833795 + 0.552074i $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ 0 0
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ 16.0000 0.794067
$$407$$ 24.0000i 1.18964i
$$408$$ − 2.00000i − 0.0990148i
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 6.00000i 0.293470i
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 28.0000i 1.36302i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −14.0000 −0.670478
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 6.00000i − 0.286691i
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ − 22.0000i − 1.04056i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 4.00000 0.188772 0.0943858 0.995536i $$-0.469911\pi$$
0.0943858 + 0.995536i $$0.469911\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 14.0000i 0.658505i
$$453$$ 0 0
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ − 14.0000i − 0.654892i −0.944870 0.327446i $$-0.893812\pi$$
0.944870 0.327446i $$-0.106188\pi$$
$$458$$ 18.0000i 0.841085i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ − 12.0000i − 0.558291i
$$463$$ 26.0000i 1.20832i 0.796862 + 0.604161i $$0.206492\pi$$
−0.796862 + 0.604161i $$0.793508\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ − 8.00000i − 0.370196i −0.982720 0.185098i $$-0.940740\pi$$
0.982720 0.185098i $$-0.0592602\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 4.00000i 0.184115i
$$473$$ − 36.0000i − 1.65528i
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ 6.00000i 0.274721i
$$478$$ − 18.0000i − 0.823301i
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 22.0000i 1.00207i
$$483$$ 8.00000i 0.364013i
$$484$$ −25.0000 −1.13636
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 6.00000 0.271329
$$490$$ 0 0
$$491$$ −34.0000 −1.53440 −0.767199 0.641409i $$-0.778350\pi$$
−0.767199 + 0.641409i $$0.778350\pi$$
$$492$$ 4.00000i 0.180334i
$$493$$ 16.0000i 0.720604i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ − 4.00000i − 0.179244i
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 14.0000i 0.624851i
$$503$$ − 40.0000i − 1.78351i −0.452517 0.891756i $$-0.649474\pi$$
0.452517 0.891756i $$-0.350526\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ 13.0000i 0.577350i
$$508$$ 12.0000i 0.532414i
$$509$$ −36.0000 −1.59567 −0.797836 0.602875i $$-0.794022\pi$$
−0.797836 + 0.602875i $$0.794022\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 6.00000 0.264135
$$517$$ 72.0000i 3.16656i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ 24.0000 1.05146 0.525730 0.850652i $$-0.323792\pi$$
0.525730 + 0.850652i $$0.323792\pi$$
$$522$$ 8.00000i 0.350150i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −22.0000 −0.961074
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ − 16.0000i − 0.696971i
$$528$$ − 6.00000i − 0.261116i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 0 0
$$534$$ 4.00000 0.173097
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ − 12.0000i − 0.517838i
$$538$$ 8.00000i 0.344904i
$$539$$ −18.0000 −0.775315
$$540$$ 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
$$543$$ 2.00000i 0.0858282i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 40.0000i 1.71028i 0.518400 + 0.855138i $$0.326528\pi$$
−0.518400 + 0.855138i $$0.673472\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 4.00000i 0.170251i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 22.0000i − 0.932170i −0.884740 0.466085i $$-0.845664\pi$$
0.884740 0.466085i $$-0.154336\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ − 16.0000i − 0.674919i
$$563$$ − 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
$$566$$ −22.0000 −0.924729
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ − 22.0000i − 0.919063i
$$574$$ −8.00000 −0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 38.0000i − 1.58196i −0.611842 0.790980i $$-0.709571\pi$$
0.611842 0.790980i $$-0.290429\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ 8.00000 0.331896
$$582$$ 12.0000i 0.497416i
$$583$$ 36.0000i 1.49097i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 24.0000i 0.990586i 0.868726 + 0.495293i $$0.164939\pi$$
−0.868726 + 0.495293i $$0.835061\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −14.0000 −0.575883
$$592$$ − 4.00000i − 0.164399i
$$593$$ − 14.0000i − 0.574911i −0.957794 0.287456i $$-0.907191\pi$$
0.957794 0.287456i $$-0.0928094\pi$$
$$594$$ 6.00000 0.246183
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ − 20.0000i − 0.818546i
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 12.0000i 0.489083i
$$603$$ 8.00000i 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ −16.0000 −0.648353
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 2.00000i 0.0808452i
$$613$$ 10.0000i 0.403896i 0.979396 + 0.201948i $$0.0647272\pi$$
−0.979396 + 0.201948i $$0.935273\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ − 38.0000i − 1.52982i −0.644136 0.764911i $$-0.722783\pi$$
0.644136 0.764911i $$-0.277217\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 6.00000i 0.240578i
$$623$$ 8.00000i 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ − 6.00000i − 0.239617i
$$628$$ − 6.00000i − 0.239426i
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ − 8.00000i − 0.318223i
$$633$$ − 28.0000i − 1.11290i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 48.0000i 1.90034i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 26.0000i 1.02534i 0.858586 + 0.512670i $$0.171344\pi$$
−0.858586 + 0.512670i $$0.828656\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 6.00000i 0.234978i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ −4.00000 −0.156174
$$657$$ 6.00000i 0.234082i
$$658$$ − 24.0000i − 0.935617i
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 26.0000 1.01128 0.505641 0.862744i $$-0.331256\pi$$
0.505641 + 0.862744i $$0.331256\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ − 32.0000i − 1.23904i
$$668$$ 16.0000i 0.619059i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −12.0000 −0.463255
$$672$$ 2.00000i 0.0771517i
$$673$$ 28.0000i 1.07932i 0.841883 + 0.539660i $$0.181447\pi$$
−0.841883 + 0.539660i $$0.818553\pi$$
$$674$$ −20.0000 −0.770371
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ − 14.0000i − 0.537667i
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ 28.0000 1.07296
$$682$$ − 48.0000i − 1.83801i
$$683$$ 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
$$687$$ − 18.0000i − 0.686743i
$$688$$ 6.00000i 0.228748i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 12.0000i 0.455842i
$$694$$ 36.0000 1.36654
$$695$$ 0 0
$$696$$ −8.00000 −0.303239
$$697$$ − 8.00000i − 0.303022i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ − 20.0000i − 0.752177i
$$708$$ − 4.00000i − 0.150329i
$$709$$ −50.0000 −1.87779 −0.938895 0.344204i $$-0.888149\pi$$
−0.938895 + 0.344204i $$0.888149\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 4.00000i 0.149906i
$$713$$ 32.0000i 1.19841i
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 18.0000i 0.672222i
$$718$$ − 6.00000i − 0.223918i
$$719$$ −14.0000 −0.522112 −0.261056 0.965324i $$-0.584071\pi$$
−0.261056 + 0.965324i $$0.584071\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 22.0000i − 0.818189i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 25.0000 0.927837
$$727$$ 14.0000i 0.519231i 0.965712 + 0.259616i $$0.0835959\pi$$
−0.965712 + 0.259616i $$0.916404\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 18.0000i − 0.664845i −0.943131 0.332423i $$-0.892134\pi$$
0.943131 0.332423i $$-0.107866\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ 48.0000i 1.76810i
$$738$$ − 4.00000i − 0.147242i
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 12.0000i − 0.440534i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 24.0000 0.878702
$$747$$ 4.00000i 0.146352i
$$748$$ 12.0000i 0.438763i
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ − 14.0000i − 0.510188i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ − 38.0000i − 1.38113i −0.723269 0.690567i $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ −24.0000 −0.871145
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ 28.0000i 1.01367i
$$764$$ 22.0000 0.795932
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ − 8.00000i − 0.287926i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ −12.0000 −0.430775
$$777$$ 8.00000i 0.286998i
$$778$$ − 30.0000i − 1.07555i
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 8.00000i − 0.286079i
$$783$$ − 8.00000i − 0.285897i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ 22.0000 0.784714
$$787$$ 44.0000i 1.56843i 0.620489 + 0.784215i $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ 28.0000 0.995565
$$792$$ 6.00000i 0.213201i
$$793$$ 0 0
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ −4.00000 −0.141333
$$802$$ 0 0
$$803$$ 36.0000i 1.27041i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 8.00000i − 0.281613i
$$808$$ − 10.0000i − 0.351799i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ − 16.0000i − 0.561490i
$$813$$ 20.0000i 0.701431i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 6.00000i 0.209913i
$$818$$ 14.0000i 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −50.0000 −1.74501 −0.872506 0.488603i $$-0.837507\pi$$
−0.872506 + 0.488603i $$0.837507\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ − 14.0000i − 0.488009i −0.969774 0.244005i $$-0.921539\pi$$
0.969774 0.244005i $$-0.0784612\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 0 0
$$833$$ 6.00000i 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ 8.00000i 0.276520i
$$838$$ − 14.0000i − 0.483622i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 2.00000i 0.0689246i
$$843$$ 16.0000i 0.551069i
$$844$$ 28.0000 0.963800
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 50.0000i 1.71802i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ −16.0000 −0.548473
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ − 36.0000i − 1.22616i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 16.0000i 0.543075i
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 14.0000i 0.474100i
$$873$$ − 12.0000i − 0.406138i
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 10.0000 0.337292
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ − 14.0000i − 0.471138i −0.971858 0.235569i $$-0.924305\pi$$
0.971858 0.235569i $$-0.0756953\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ − 8.00000i − 0.268614i −0.990940 0.134307i $$-0.957119\pi$$
0.990940 0.134307i $$-0.0428808\pi$$
$$888$$ 4.00000i 0.134231i
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ − 16.0000i − 0.535720i
$$893$$ − 12.0000i − 0.401565i
$$894$$ −22.0000 −0.735790
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ − 4.00000i − 0.133482i
$$899$$ −64.0000 −2.13452
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 24.0000i − 0.799113i
$$903$$ − 12.0000i − 0.399335i
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 28.0000i 0.929213i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 24.0000i 0.794284i
$$914$$ −14.0000 −0.463079
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ 44.0000i 1.45301i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 8.00000 0.263609
$$922$$ 34.0000i 1.11973i
$$923$$ 0 0
$$924$$ −12.0000 −0.394771
$$925$$ 0 0
$$926$$ 26.0000 0.854413
$$927$$ 0 0
$$928$$ − 8.00000i − 0.262613i
$$929$$ 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
$$932$$ 14.0000i 0.458585i
$$933$$ − 6.00000i − 0.196431i
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 34.0000i − 1.11073i −0.831606 0.555366i $$-0.812578\pi$$
0.831606 0.555366i $$-0.187422\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ −40.0000 −1.30396 −0.651981 0.758235i $$-0.726062\pi$$
−0.651981 + 0.758235i $$0.726062\pi$$
$$942$$ 6.00000i 0.195491i
$$943$$ 16.0000i 0.521032i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −36.0000 −1.17046
$$947$$ 44.0000i 1.42981i 0.699223 + 0.714904i $$0.253530\pi$$
−0.699223 + 0.714904i $$0.746470\pi$$
$$948$$ 8.00000i 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ − 4.00000i − 0.129641i
$$953$$ 34.0000i 1.10137i 0.834714 + 0.550684i $$0.185633\pi$$
−0.834714 + 0.550684i $$0.814367\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −18.0000 −0.582162
$$957$$ − 48.0000i − 1.55162i
$$958$$ 18.0000i 0.581554i
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ − 18.0000i − 0.578841i −0.957202 0.289420i $$-0.906537\pi$$
0.957202 0.289420i $$-0.0934626\pi$$
$$968$$ 25.0000i 0.803530i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 6.00000i − 0.191859i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 34.0000i 1.08498i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 4.00000 0.127515
$$985$$ 0 0
$$986$$ 16.0000 0.509544
$$987$$ 24.0000i 0.763928i
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −4.00000 −0.126745
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ 24.0000i 0.759707i
$$999$$ −4.00000 −0.126554
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.j.799.1 2
5.2 odd 4 2850.2.a.y.1.1 1
5.3 odd 4 570.2.a.b.1.1 1
5.4 even 2 inner 2850.2.d.j.799.2 2
15.2 even 4 8550.2.a.g.1.1 1
15.8 even 4 1710.2.a.s.1.1 1
20.3 even 4 4560.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.b.1.1 1 5.3 odd 4
1710.2.a.s.1.1 1 15.8 even 4
2850.2.a.y.1.1 1 5.2 odd 4
2850.2.d.j.799.1 2 1.1 even 1 trivial
2850.2.d.j.799.2 2 5.4 even 2 inner
4560.2.a.r.1.1 1 20.3 even 4
8550.2.a.g.1.1 1 15.2 even 4