# Properties

 Label 2850.2.d.p.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: $$x^{2} + 1$$ 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.p.799.2

## $q$-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 $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 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 $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 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 $$-0.784877\pi$$
0.780189 0.625543i $$-0.215123\pi$$
$$24$$ −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 $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 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 $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 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 $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −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 $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 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 $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 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 $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 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 $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 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 $$-0.836023\pi$$
0.870219 0.492665i $$-0.163977\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 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 $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ −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 $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 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 $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ − 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 $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ − 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 $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 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 $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −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 $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 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 $$-0.736423\pi$$
0.676313 0.736614i $$-0.263577\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
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 $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 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 $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 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 $$-0.812732\pi$$
0.831875 0.554964i $$-0.187268\pi$$
$$264$$ 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 $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 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 $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 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 $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ −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 $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 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 $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 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 $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ 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 $$-0.777193\pi$$
0.764862 0.644194i $$-0.222807\pi$$
$$348$$ 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 $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 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 $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ − 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 $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 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 $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ −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 $$-0.773741\pi$$
0.757831 0.652451i $$-0.226259\pi$$
$$398$$ 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 $$-0.922759\pi$$
0.970702 0.240285i $$-0.0772408\pi$$
$$434$$ 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 $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 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 $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ − 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 $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ − 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 $$-0.669856\pi$$
0.508652 0.860972i $$-0.330144\pi$$
$$488$$ 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 $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 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 $$0.0559627\pi$$
−0.984585 + 0.174908i $$0.944037\pi$$
$$524$$ −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 $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ − 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 $$-0.918179\pi$$
0.967144 0.254228i $$-0.0818214\pi$$
$$558$$ 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 $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ −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 $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ 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 $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 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 $$0.242393\pi$$
−0.723801 + 0.690009i $$0.757607\pi$$
$$608$$ 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 $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 16.0000 0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ − 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 $$-0.974868\pi$$
0.996885 0.0788723i $$-0.0251319\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 6.00000i 0.235884i 0.993020 + 0.117942i $$0.0376297\pi$$
−0.993020 + 0.117942i $$0.962370\pi$$
$$648$$ 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 $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 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 $$-0.938264\pi$$
0.981251 0.192736i $$-0.0617360\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ − 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 $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ −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 $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 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 $$0.374598\pi$$
−0.383849 + 0.923396i $$0.625402\pi$$
$$734$$ 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 $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ −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 $$0.315082\pi$$
−0.548807 + 0.835949i $$0.684918\pi$$
$$758$$ 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 $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 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 $$-0.883983\pi$$
0.934310 0.356462i $$-0.116017\pi$$
$$788$$ 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 $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ − 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 $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 10.0000 0.348367
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 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 $$0.225461\pi$$
−0.759465 + 0.650548i $$0.774539\pi$$
$$854$$ 56.0000 1.91628
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 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 $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ −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 $$-0.733456\pi$$
0.669417 0.742887i $$-0.266544\pi$$
$$878$$ 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 $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ − 12.0000i − 0.402921i −0.979497 0.201460i $$-0.935431\pi$$
0.979497 0.201460i $$-0.0645687\pi$$
$$888$$ − 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 $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 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 $$0.396288\pi$$
−0.320085 + 0.947389i $$0.603712\pi$$
$$938$$ − 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 $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ − 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 $$0.338886\pi$$
−0.484817 + 0.874616i $$0.661114\pi$$
$$954$$ −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 $$0.148651\pi$$
−0.892923 + 0.450210i $$0.851349\pi$$
$$968$$ − 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 $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ − 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 $$-0.722501\pi$$
0.643458 0.765481i $$-0.277499\pi$$
$$984$$ −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 $$-0.864930\pi$$
0.911313 0.411714i $$-0.135070\pi$$
$$998$$ − 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.799.1 2
5.2 odd 4 114.2.a.c.1.1 1
5.3 odd 4 2850.2.a.g.1.1 1
5.4 even 2 inner 2850.2.d.p.799.2 2
15.2 even 4 342.2.a.c.1.1 1
15.8 even 4 8550.2.a.bj.1.1 1
20.7 even 4 912.2.a.c.1.1 1
35.27 even 4 5586.2.a.u.1.1 1
40.27 even 4 3648.2.a.bc.1.1 1
40.37 odd 4 3648.2.a.i.1.1 1
60.47 odd 4 2736.2.a.o.1.1 1
95.37 even 4 2166.2.a.a.1.1 1
285.227 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 5.2 odd 4
342.2.a.c.1.1 1 15.2 even 4
912.2.a.c.1.1 1 20.7 even 4
2166.2.a.a.1.1 1 95.37 even 4
2736.2.a.o.1.1 1 60.47 odd 4
2850.2.a.g.1.1 1 5.3 odd 4
2850.2.d.p.799.1 2 1.1 even 1 trivial
2850.2.d.p.799.2 2 5.4 even 2 inner
3648.2.a.i.1.1 1 40.37 odd 4
3648.2.a.bc.1.1 1 40.27 even 4
5586.2.a.u.1.1 1 35.27 even 4
6498.2.a.t.1.1 1 285.227 odd 4
8550.2.a.bj.1.1 1 15.8 even 4