# Properties

 Label 2850.2.d.p.799.2 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: $$x^{2} + 1$$ 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 799.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.p.799.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +4.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -6.00000 q^{46} +6.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +1.00000i q^{57} -6.00000i q^{58} +12.0000 q^{59} +14.0000 q^{61} +2.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +8.00000i q^{67} -6.00000i q^{68} +6.00000 q^{69} +1.00000i q^{72} -14.0000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +12.0000i q^{83} +4.00000 q^{84} -4.00000 q^{86} +6.00000i q^{87} +6.00000 q^{89} +16.0000 q^{91} -6.00000i q^{92} -2.00000i q^{93} -6.00000 q^{94} +1.00000 q^{96} -10.0000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} - 2 q^{19} - 8 q^{21} - 2 q^{24} - 8 q^{26} - 12 q^{29} + 4 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 8 q^{56} + 24 q^{59} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 8 q^{74} + 2 q^{76} + 20 q^{79} + 2 q^{81} + 8 q^{84} - 8 q^{86} + 12 q^{89} + 32 q^{91} - 12 q^{94} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 - 2 * q^19 - 8 * q^21 - 2 * q^24 - 8 * q^26 - 12 * q^29 + 4 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 + 12 * q^41 - 12 * q^46 - 18 * q^49 + 12 * q^51 - 2 * q^54 - 8 * q^56 + 24 * q^59 + 28 * q^61 - 2 * q^64 + 12 * q^69 + 8 * q^74 + 2 * q^76 + 20 * q^79 + 2 * q^81 + 8 * q^84 - 8 * q^86 + 12 * q^89 + 32 * q^91 - 12 * q^94 + 2 * q^96

## Character values

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

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

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 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
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000i 0.192450i
$$28$$ 4.00000i 0.755929i
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 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
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ − 4.00000i − 0.554700i
$$53$$ − 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
$$56$$ −4.00000 −0.534522
$$57$$ 1.00000i 0.132453i
$$58$$ − 6.00000i − 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 2.00000i 0.254000i
$$63$$ 4.00000i 0.503953i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 4.00000i 0.452911i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000i 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 16.0000 1.67726
$$92$$ − 6.00000i − 0.625543i
$$93$$ − 2.00000i − 0.207390i
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ 10.0000i 0.985329i 0.870219 + 0.492665i $$0.163977\pi$$
−0.870219 + 0.492665i $$0.836023\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ − 4.00000i − 0.377964i
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 4.00000i − 0.369800i
$$118$$ 12.0000i 1.10469i
$$119$$ 24.0000 2.20008
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 14.0000i 1.26750i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 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 $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 9.00000i 0.742307i
$$148$$ 4.00000i 0.328798i
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 1.00000i 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 4.00000i 0.308607i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 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
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 16.0000i 1.18600i
$$183$$ − 14.0000i − 1.03491i
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ 0 0
$$188$$ − 6.00000i − 0.437595i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −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
$$202$$ 0 0
$$203$$ 24.0000i 1.68447i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ − 6.00000i − 0.417029i
$$208$$ 4.00000i 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 8.00000i − 0.543075i
$$218$$ 16.0000i 1.08366i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ − 4.00000i − 0.268462i
$$223$$ 22.0000i 1.47323i 0.676313 + 0.736614i $$0.263577\pi$$
−0.676313 + 0.736614i $$0.736423\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 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
$$236$$ −12.0000 −0.781133
$$237$$ − 10.0000i − 0.649570i
$$238$$ 24.0000i 1.55569i
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ − 4.00000i − 0.254514i
$$248$$ − 2.00000i − 0.127000i
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 4.00000i − 0.251976i
$$253$$ 0 0
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ 18.0000i 1.10993i 0.831875 + 0.554964i $$0.187268\pi$$
−0.831875 + 0.554964i $$0.812732\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ − 6.00000i − 0.367194i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ − 16.0000i − 0.968364i
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ − 20.0000i − 1.19952i
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.0000i − 1.41668i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000i 0.819288i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ − 12.0000i − 0.695141i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ − 10.0000i − 0.575435i
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ − 16.0000i − 0.913168i −0.889680 0.456584i $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.568880
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 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
$$322$$ 24.0000i 1.33747i
$$323$$ − 6.00000i − 0.333849i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ − 16.0000i − 0.884802i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 4.00000i 0.219199i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000i 0.0540738i
$$343$$ 8.00000i 0.431959i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ − 6.00000i − 0.321634i
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ − 24.0000i − 1.27021i
$$358$$ − 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
$$362$$ − 16.0000i − 0.840941i
$$363$$ 11.0000i 0.577350i
$$364$$ −16.0000 −0.838628
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ − 4.00000i − 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 2.00000i 0.103695i
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ − 24.0000i − 1.23606i
$$378$$ 4.00000i 0.205738i
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 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
$$386$$ −10.0000 −0.508987
$$387$$ − 4.00000i − 0.203331i
$$388$$ 10.0000i 0.507673i
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 9.00000i 0.454569i
$$393$$ − 12.0000i − 0.605320i
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 26.0000i 1.30490i 0.757831 + 0.652451i $$0.226259\pi$$
−0.757831 + 0.652451i $$0.773741\pi$$
$$398$$ − 20.0000i − 1.00251i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ − 10.0000i − 0.492665i
$$413$$ − 48.0000i − 2.36193i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ − 6.00000i − 0.291730i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 56.0000i − 2.71003i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 10.0000i 0.480569i 0.970702 + 0.240285i $$0.0772408\pi$$
−0.970702 + 0.240285i $$0.922759\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ − 6.00000i − 0.287019i
$$438$$ − 14.0000i − 0.668946i
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ − 24.0000i − 1.14156i
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −22.0000 −1.04173
$$447$$ 12.0000i 0.567581i
$$448$$ 4.00000i 0.188982i
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 18.0000i − 0.846649i
$$453$$ 10.0000i 0.469841i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 2.00000i 0.0935561i 0.998905 + 0.0467780i $$0.0148953\pi$$
−0.998905 + 0.0467780i $$0.985105\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ − 36.0000i − 1.66588i −0.553362 0.832941i $$-0.686655\pi$$
0.553362 0.832941i $$-0.313345\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ − 12.0000i − 0.552345i
$$473$$ 0 0
$$474$$ 10.0000 0.459315
$$475$$ 0 0
$$476$$ −24.0000 −1.10004
$$477$$ 6.00000i 0.274721i
$$478$$ 18.0000i 0.823301i
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 14.0000i 0.637683i
$$483$$ − 24.0000i − 1.09204i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ − 14.0000i − 0.633750i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ − 36.0000i − 1.62136i
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 12.0000i 0.537733i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 12.0000i 0.535586i
$$503$$ − 6.00000i − 0.267527i −0.991013 0.133763i $$-0.957294\pi$$
0.991013 0.133763i $$-0.0427062\pi$$
$$504$$ 4.00000 0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.00000i 0.133235i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 1.00000i 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ − 16.0000i − 0.703000i
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ − 8.00000i − 0.349816i −0.984585 0.174908i $$-0.944037\pi$$
0.984585 0.174908i $$-0.0559627\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −18.0000 −0.784837
$$527$$ 12.0000i 0.522728i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ − 4.00000i − 0.173422i
$$533$$ 24.0000i 1.03956i
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 12.0000i 0.517838i
$$538$$ 6.00000i 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ 16.0000i 0.686626i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 16.0000 0.684737
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ −14.0000 −0.597505
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ − 6.00000i − 0.255377i
$$553$$ − 40.0000i − 1.70097i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 12.0000i 0.508456i 0.967144 + 0.254228i $$0.0818214\pi$$
−0.967144 + 0.254228i $$0.918179\pi$$
$$558$$ − 2.00000i − 0.0846668i
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 18.0000i − 0.759284i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ − 4.00000i − 0.167984i
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ − 18.0000i − 0.751961i
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ − 10.0000i − 0.414513i
$$583$$ 0 0
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ − 4.00000i − 0.164399i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ 20.0000i 0.818546i
$$598$$ − 24.0000i − 0.981433i
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 34.0000i − 1.38002i −0.723801 0.690009i $$-0.757607\pi$$
0.723801 0.690009i $$-0.242393\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000i 0.242536i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 10.0000i 0.402259i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 30.0000i 1.20289i
$$623$$ − 24.0000i − 0.961540i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 4.00000i 0.158986i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 36.0000i − 1.42637i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ − 4.00000i − 0.156652i
$$653$$ − 36.0000i − 1.40879i −0.709809 0.704394i $$-0.751219\pi$$
0.709809 0.704394i $$-0.248781\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 14.0000i 0.546192i
$$658$$ 24.0000i 0.935617i
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 8.00000i 0.310929i
$$663$$ 24.0000i 0.932083i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ − 36.0000i − 1.39393i
$$668$$ 12.0000i 0.464294i
$$669$$ 22.0000 0.850569
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 4.00000i − 0.154303i
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 18.0000i 0.691286i
$$679$$ −40.0000 −1.53506
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ − 10.0000i − 0.381524i
$$688$$ 4.00000i 0.152499i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 36.0000i 1.36360i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 4.00000i 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ − 6.00000i − 0.224860i
$$713$$ 12.0000i 0.449404i
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ − 18.0000i − 0.672222i
$$718$$ 6.00000i 0.223918i
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 40.0000 1.48968
$$722$$ 1.00000i 0.0372161i
$$723$$ − 14.0000i − 0.520666i
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ − 16.0000i − 0.592999i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 14.0000i 0.517455i
$$733$$ − 50.0000i − 1.84679i −0.383849 0.923396i $$-0.625402\pi$$
0.383849 0.923396i $$-0.374598\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 0 0
$$738$$ − 6.00000i − 0.220863i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ − 24.0000i − 0.881068i
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 6.00000i 0.218797i
$$753$$ − 12.0000i − 0.437304i
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 46.0000i − 1.67190i −0.548807 0.835949i $$-0.684918\pi$$
0.548807 0.835949i $$-0.315082\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ − 64.0000i − 2.31696i
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 48.0000i 1.73318i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 10.0000i − 0.359908i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 16.0000i 0.573997i
$$778$$ − 24.0000i − 0.860442i
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 36.0000i − 1.28736i
$$783$$ − 6.00000i − 0.214423i
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 20.0000i 0.712923i 0.934310 + 0.356462i $$0.116017\pi$$
−0.934310 + 0.356462i $$0.883983\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ 18.0000 0.640817
$$790$$ 0 0
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ 56.0000i 1.98862i
$$794$$ −26.0000 −0.922705
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 18.0000i 0.637593i 0.947823 + 0.318796i $$0.103279\pi$$
−0.947823 + 0.318796i $$0.896721\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 30.0000i 1.05934i
$$803$$ 0 0
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ − 6.00000i − 0.211210i
$$808$$ 0 0
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ − 24.0000i − 0.842235i
$$813$$ − 20.0000i − 0.701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ − 4.00000i − 0.139942i
$$818$$ − 14.0000i − 0.489499i
$$819$$ −16.0000 −0.559085
$$820$$ 0 0
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ − 18.0000i − 0.627822i
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ 10.0000 0.348367
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ 16.0000 0.555703 0.277851 0.960624i $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ − 4.00000i − 0.138675i
$$833$$ − 54.0000i − 1.87099i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.00000i 0.0691301i
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 28.0000i − 0.964944i
$$843$$ 18.0000i 0.619953i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 44.0000i 1.51186i
$$848$$ − 6.00000i − 0.206041i
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ − 38.0000i − 1.30110i −0.759465 0.650548i $$-0.774539\pi$$
0.759465 0.650548i $$-0.225461\pi$$
$$854$$ 56.0000 1.91628
$$855$$ 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
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 12.0000i − 0.408722i
$$863$$ − 12.0000i − 0.408485i −0.978920 0.204242i $$-0.934527\pi$$
0.978920 0.204242i $$-0.0654731\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −10.0000 −0.339814
$$867$$ 19.0000i 0.645274i
$$868$$ 8.00000i 0.271538i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ − 16.0000i − 0.541828i
$$873$$ 10.0000i 0.338449i
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 44.0000i 1.48577i 0.669417 + 0.742887i $$0.266544\pi$$
−0.669417 + 0.742887i $$0.733456\pi$$
$$878$$ − 26.0000i − 0.877457i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ 4.00000i 0.134231i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 22.0000i − 0.736614i
$$893$$ − 6.00000i − 0.200782i
$$894$$ −12.0000 −0.401340
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 24.0000i 0.801337i
$$898$$ 30.0000i 1.00111i
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −10.0000 −0.332228
$$907$$ 8.00000i 0.265636i 0.991140 + 0.132818i $$0.0424025\pi$$
−0.991140 + 0.132818i $$0.957597\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 0 0
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ − 48.0000i − 1.58510i
$$918$$ − 6.00000i − 0.198030i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ 12.0000i 0.395199i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ − 10.0000i − 0.328443i
$$928$$ − 6.00000i − 0.196960i
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 30.0000i − 0.982156i
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ − 58.0000i − 1.89478i −0.320085 0.947389i $$-0.603712\pi$$
0.320085 0.947389i $$-0.396288\pi$$
$$938$$ 32.0000i 1.04484i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ 36.0000i 1.17232i
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ 56.0000 1.81784
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ − 24.0000i − 0.777844i
$$953$$ − 54.0000i − 1.74923i −0.484817 0.874616i $$-0.661114\pi$$
0.484817 0.874616i $$-0.338886\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −18.0000 −0.582162
$$957$$ 0 0
$$958$$ 6.00000i 0.193851i
$$959$$ −72.0000 −2.32500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 16.0000i 0.515861i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 24.0000 0.772187
$$967$$ − 28.0000i − 0.900419i −0.892923 0.450210i $$-0.851349\pi$$
0.892923 0.450210i $$-0.148651\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −6.00000 −0.192748
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 80.0000i 2.56468i
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ − 18.0000i − 0.575871i −0.957650 0.287936i $$-0.907031\pi$$
0.957650 0.287936i $$-0.0929689\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 36.0000i 1.14881i
$$983$$ 48.0000i 1.53096i 0.643458 + 0.765481i $$0.277499\pi$$
−0.643458 + 0.765481i $$0.722501\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ − 24.0000i − 0.763928i
$$988$$ 4.00000i 0.127257i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 26.0000 0.825917 0.412959 0.910750i $$-0.364495\pi$$
0.412959 + 0.910750i $$0.364495\pi$$
$$992$$ 2.00000i 0.0635001i
$$993$$ − 8.00000i − 0.253872i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 26.0000i 0.823428i 0.911313 + 0.411714i $$0.135070\pi$$
−0.911313 + 0.411714i $$0.864930\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 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.p.799.2 2
5.2 odd 4 2850.2.a.g.1.1 1
5.3 odd 4 114.2.a.c.1.1 1
5.4 even 2 inner 2850.2.d.p.799.1 2
15.2 even 4 8550.2.a.bj.1.1 1
15.8 even 4 342.2.a.c.1.1 1
20.3 even 4 912.2.a.c.1.1 1
35.13 even 4 5586.2.a.u.1.1 1
40.3 even 4 3648.2.a.bc.1.1 1
40.13 odd 4 3648.2.a.i.1.1 1
60.23 odd 4 2736.2.a.o.1.1 1
95.18 even 4 2166.2.a.a.1.1 1
285.113 odd 4 6498.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.c.1.1 1 5.3 odd 4
342.2.a.c.1.1 1 15.8 even 4
912.2.a.c.1.1 1 20.3 even 4
2166.2.a.a.1.1 1 95.18 even 4
2736.2.a.o.1.1 1 60.23 odd 4
2850.2.a.g.1.1 1 5.2 odd 4
2850.2.d.p.799.1 2 5.4 even 2 inner
2850.2.d.p.799.2 2 1.1 even 1 trivial
3648.2.a.i.1.1 1 40.13 odd 4
3648.2.a.bc.1.1 1 40.3 even 4
5586.2.a.u.1.1 1 35.13 even 4
6498.2.a.t.1.1 1 285.113 odd 4
8550.2.a.bj.1.1 1 15.2 even 4