# Properties

 Label 2850.2.d.b.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 114) 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.b.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} +6.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} +2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -1.00000i q^{57} -2.00000i q^{58} -12.0000 q^{59} +14.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +12.0000i q^{67} -6.00000i q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -6.00000i q^{73} -10.0000 q^{74} -1.00000 q^{76} -2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} +4.00000 q^{86} -2.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} +4.00000i q^{92} -4.00000i q^{93} +4.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} + 2q^{19} + 2q^{24} + 4q^{26} + 4q^{29} + 8q^{31} + 12q^{34} + 2q^{36} + 4q^{39} + 20q^{41} + 8q^{44} - 8q^{46} + 14q^{49} + 12q^{51} + 2q^{54} - 24q^{59} + 28q^{61} - 2q^{64} + 8q^{66} - 8q^{69} + 16q^{71} - 20q^{74} - 2q^{76} + 8q^{79} + 2q^{81} + 8q^{86} + 12q^{89} + 8q^{94} - 2q^{96} + 8q^{99} + O(q^{100})$$

## 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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ − 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
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 2.00000i − 0.262613i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ − 2.00000i − 0.226455i
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 2.00000i − 0.214423i
$$88$$ − 4.00000i − 0.426401i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000i 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 7.00000i − 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ − 2.00000i − 0.184900i
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 14.0000i − 1.26750i
$$123$$ − 10.0000i − 0.901670i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.0000i 1.06483i 0.846484 + 0.532414i $$0.178715\pi$$
−0.846484 + 0.532414i $$0.821285\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ 14.0000i 1.19610i 0.801459 + 0.598050i $$0.204058\pi$$
−0.801459 + 0.598050i $$0.795942\pi$$
$$138$$ 4.00000i 0.340503i
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ − 8.00000i − 0.671345i
$$143$$ − 8.00000i − 0.668994i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ − 7.00000i − 0.577350i
$$148$$ 10.0000i 0.821995i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ − 22.0000i − 1.75579i −0.478852 0.877896i $$-0.658947\pi$$
0.478852 0.877896i $$-0.341053\pi$$
$$158$$ − 4.00000i − 0.318223i
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 4.00000i − 0.304997i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000i 0.901975i
$$178$$ − 6.00000i − 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ − 14.0000i − 1.03491i
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ − 24.0000i − 1.75505i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ − 2.00000i − 0.140720i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ 4.00000i 0.278019i
$$208$$ 2.00000i 0.138675i
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ − 8.00000i − 0.548151i
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ − 6.00000i − 0.406371i
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ 10.0000i 0.671156i
$$223$$ − 28.0000i − 1.87502i −0.347960 0.937509i $$-0.613126\pi$$
0.347960 0.937509i $$-0.386874\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ − 28.0000i − 1.85843i −0.369546 0.929213i $$-0.620487\pi$$
0.369546 0.929213i $$-0.379513\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ − 4.00000i − 0.259828i
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 2.00000i 0.127257i
$$248$$ 4.00000i 0.254000i
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 12.0000 0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 6.00000i − 0.367194i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ 14.0000 0.845771
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 12.0000i 0.713326i 0.934233 + 0.356663i $$0.116086\pi$$
−0.934233 + 0.356663i $$0.883914\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 6.00000i 0.351123i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ 10.0000 0.581238
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 6.00000i − 0.347571i
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 20.0000i − 1.15087i
$$303$$ − 2.00000i − 0.114897i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ −12.0000 −0.682656
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 22.0000i − 1.24351i −0.783210 0.621757i $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 6.00000i 0.333849i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ − 6.00000i − 0.331801i
$$328$$ 10.0000i 0.552158i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 1.00000i 0.0540738i
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 20.0000i 1.07366i 0.843692 + 0.536828i $$0.180378\pi$$
−0.843692 + 0.536828i $$0.819622\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 4.00000i 0.213201i
$$353$$ 18.0000i 0.958043i 0.877803 + 0.479022i $$0.159008\pi$$
−0.877803 + 0.479022i $$0.840992\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000i 0.735824i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 4.00000i 0.207390i
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 4.00000i 0.206010i
$$378$$ 0 0
$$379$$ 36.0000 1.84920 0.924598 0.380945i $$-0.124401\pi$$
0.924598 + 0.380945i $$0.124401\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ − 4.00000i − 0.204658i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ − 4.00000i − 0.203331i
$$388$$ 10.0000i 0.507673i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 7.00000i 0.353553i
$$393$$ − 12.0000i − 0.605320i
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 10.0000i 0.501886i 0.968002 + 0.250943i $$0.0807406\pi$$
−0.968002 + 0.250943i $$0.919259\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ − 12.0000i − 0.598506i
$$403$$ 8.00000i 0.398508i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000i 1.98273i
$$408$$ 6.00000i 0.297044i
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ 12.0000i 0.591198i
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 12.0000i 0.587643i
$$418$$ 4.00000i 0.195646i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ − 4.00000i − 0.194487i
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ − 4.00000i − 0.193347i
$$429$$ −8.00000 −0.386244
$$430$$ 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
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −6.00000 −0.287348
$$437$$ − 4.00000i − 0.191346i
$$438$$ 6.00000i 0.286691i
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 12.0000i 0.570782i
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −28.0000 −1.32584
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ − 2.00000i − 0.0940721i
$$453$$ − 20.0000i − 0.939682i
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ − 12.0000i − 0.552345i
$$473$$ − 16.0000i − 0.735681i
$$474$$ −4.00000 −0.183726
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 12.0000i 0.548867i
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ − 10.0000i − 0.455488i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 4.00000i − 0.181257i −0.995885 0.0906287i $$-0.971112\pi$$
0.995885 0.0906287i $$-0.0288876\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 10.0000i 0.450835i
$$493$$ 12.0000i 0.540453i
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 28.0000i 1.24970i
$$503$$ 36.0000i 1.60516i 0.596544 + 0.802580i $$0.296540\pi$$
−0.596544 + 0.802580i $$0.703460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 0.711287
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 12.0000i − 0.532414i
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ − 16.0000i − 0.703679i
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 24.0000i 1.04546i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 12.0000i 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ 14.0000i 0.600798i
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ − 14.0000i − 0.598050i
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ − 4.00000i − 0.170251i
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ − 34.0000i − 1.44063i −0.693649 0.720313i $$-0.743998\pi$$
0.693649 0.720313i $$-0.256002\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ − 10.0000i − 0.421825i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 8.00000i 0.335673i
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ − 4.00000i − 0.167102i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 10.0000i 0.414513i
$$583$$ 40.0000i 1.65663i
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ − 10.0000i − 0.410997i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 8.00000i − 0.327144i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ − 12.0000i − 0.488678i
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 4.00000i 0.162355i 0.996700 + 0.0811775i $$0.0258681\pi$$
−0.996700 + 0.0811775i $$0.974132\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 6.00000i 0.242536i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 12.0000i 0.482711i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 4.00000i − 0.160385i
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 4.00000i 0.159745i
$$628$$ 22.0000i 0.877896i
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 4.00000i 0.159111i
$$633$$ − 12.0000i − 0.476957i
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 14.0000i 0.554700i
$$638$$ 8.00000i 0.316723i
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 4.00000i 0.157256i 0.996904 + 0.0786281i $$0.0250540\pi$$
−0.996904 + 0.0786281i $$0.974946\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 20.0000i − 0.783260i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 4.00000i 0.155464i
$$663$$ 12.0000i 0.466041i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 10.0000 0.387492
$$667$$ − 8.00000i − 0.309761i
$$668$$ 0 0
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 26.0000i 0.999261i 0.866239 + 0.499631i $$0.166531\pi$$
−0.866239 + 0.499631i $$0.833469\pi$$
$$678$$ − 2.00000i − 0.0768095i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ 16.0000i 0.612672i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 10.0000i − 0.381524i
$$688$$ 4.00000i 0.152499i
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 60.0000i 2.27266i
$$698$$ − 26.0000i − 0.984115i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 10.0000i − 0.377157i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ − 12.0000i − 0.450988i
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 6.00000i 0.224860i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 12.0000i 0.448148i
$$718$$ 12.0000i 0.447836i
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 1.00000i − 0.0372161i
$$723$$ − 10.0000i − 0.371904i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 14.0000i 0.517455i
$$733$$ 6.00000i 0.221615i 0.993842 + 0.110808i $$0.0353437\pi$$
−0.993842 + 0.110808i $$0.964656\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ − 48.0000i − 1.76810i
$$738$$ 10.0000i 0.368105i
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ 32.0000i 1.17397i 0.809599 + 0.586983i $$0.199684\pi$$
−0.809599 + 0.586983i $$0.800316\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ − 12.0000i − 0.439057i
$$748$$ 24.0000i 0.877527i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 28.0000i 1.02038i
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ − 36.0000i − 1.30758i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ − 12.0000i − 0.434714i
$$763$$ 0 0
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ − 24.0000i − 0.866590i
$$768$$ − 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
$$771$$ −2.00000 −0.0720282
$$772$$ 6.00000i 0.215945i
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 18.0000i 0.645331i
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ − 24.0000i − 0.858238i
$$783$$ 2.00000i 0.0714742i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ − 22.0000i − 0.783718i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 4.00000i 0.142134i
$$793$$ 28.0000i 0.994309i
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 14.0000i 0.494357i
$$803$$ 24.0000i 0.846942i
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 6.00000i 0.211210i
$$808$$ 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
$$813$$ 8.00000i 0.280572i
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 4.00000i 0.139942i
$$818$$ − 14.0000i − 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ − 14.0000i − 0.488306i
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 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
$$829$$ −42.0000 −1.45872 −0.729360 0.684130i $$-0.760182\pi$$
−0.729360 + 0.684130i $$0.760182\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 42.0000i 1.45521i
$$834$$ 12.0000 0.415526
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 4.00000i 0.138260i
$$838$$ − 12.0000i − 0.414533i
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 26.0000i − 0.896019i
$$843$$ − 10.0000i − 0.344418i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 0 0
$$848$$ − 10.0000i − 0.343401i
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ 8.00000i 0.274075i
$$853$$ 6.00000i 0.205436i 0.994711 + 0.102718i $$0.0327539\pi$$
−0.994711 + 0.102718i $$0.967246\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 42.0000i − 1.43469i −0.696717 0.717346i $$-0.745357\pi$$
0.696717 0.717346i $$-0.254643\pi$$
$$858$$ 8.00000i 0.273115i
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.0000i 0.817443i
$$863$$ − 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
$$866$$ 26.0000 0.883516
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 6.00000i 0.203186i
$$873$$ 10.0000i 0.338449i
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ − 34.0000i − 1.14810i −0.818821 0.574049i $$-0.805372\pi$$
0.818821 0.574049i $$-0.194628\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 32.0000i 1.07445i 0.843437 + 0.537227i $$0.180528\pi$$
−0.843437 + 0.537227i $$0.819472\pi$$
$$888$$ − 10.0000i − 0.335578i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 28.0000i 0.937509i
$$893$$ 4.00000i 0.133855i
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 8.00000i − 0.267112i
$$898$$ 34.0000i 1.13459i
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 60.0000 1.99889
$$902$$ 40.0000i 1.33185i
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ −20.0000 −0.664455
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ 28.0000i 0.929213i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 48.0000i − 1.58857i
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 2.00000i − 0.0658665i
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 12.0000i 0.394132i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 22.0000 0.721797 0.360898 0.932605i $$-0.382470\pi$$
0.360898 + 0.932605i $$0.382470\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ 6.00000i 0.196537i
$$933$$ − 4.00000i − 0.130954i
$$934$$ 20.0000 0.654420
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ − 26.0000i − 0.849383i −0.905338 0.424691i $$-0.860383\pi$$
0.905338 0.424691i $$-0.139617\pi$$
$$938$$ 0 0
$$939$$ −22.0000 −0.717943
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 22.0000i 0.716799i
$$943$$ − 40.0000i − 1.30258i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 4.00000i 0.129914i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 8.00000i 0.258603i
$$958$$ 4.00000i 0.129234i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 20.0000i − 0.644826i
$$963$$ − 4.00000i − 0.128898i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ −4.00000 −0.128168
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 46.0000i 1.47167i 0.677161 + 0.735835i $$0.263210\pi$$
−0.677161 + 0.735835i $$0.736790\pi$$
$$978$$ − 20.0000i − 0.639529i
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ − 20.0000i − 0.638226i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ − 2.00000i − 0.0636285i
$$989$$ 16.0000 0.508770
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ 4.00000i 0.126936i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 2.00000i 0.0633406i 0.999498 + 0.0316703i $$0.0100827\pi$$
−0.999498 + 0.0316703i $$0.989917\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ 10.0000 0.316386
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.b.799.1 2
5.2 odd 4 114.2.a.b.1.1 1
5.3 odd 4 2850.2.a.j.1.1 1
5.4 even 2 inner 2850.2.d.b.799.2 2
15.2 even 4 342.2.a.b.1.1 1
15.8 even 4 8550.2.a.ba.1.1 1
20.7 even 4 912.2.a.k.1.1 1
35.27 even 4 5586.2.a.y.1.1 1
40.27 even 4 3648.2.a.c.1.1 1
40.37 odd 4 3648.2.a.x.1.1 1
60.47 odd 4 2736.2.a.d.1.1 1
95.37 even 4 2166.2.a.d.1.1 1
285.227 odd 4 6498.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.b.1.1 1 5.2 odd 4
342.2.a.b.1.1 1 15.2 even 4
912.2.a.k.1.1 1 20.7 even 4
2166.2.a.d.1.1 1 95.37 even 4
2736.2.a.d.1.1 1 60.47 odd 4
2850.2.a.j.1.1 1 5.3 odd 4
2850.2.d.b.799.1 2 1.1 even 1 trivial
2850.2.d.b.799.2 2 5.4 even 2 inner
3648.2.a.c.1.1 1 40.27 even 4
3648.2.a.x.1.1 1 40.37 odd 4
5586.2.a.y.1.1 1 35.27 even 4
6498.2.a.p.1.1 1 285.227 odd 4
8550.2.a.ba.1.1 1 15.8 even 4