# Properties

 Label 2800.2.g.i.449.1 Level $2800$ Weight $2$ Character 2800.449 Analytic conductor $22.358$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2800 = 2^{4} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2800.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.3581125660$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2800.449 Dual form 2800.2.g.i.449.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} -7.00000 q^{19} +1.00000 q^{21} -5.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +3.00000i q^{33} -8.00000i q^{37} +2.00000 q^{39} -9.00000 q^{41} -8.00000i q^{43} -6.00000i q^{47} -1.00000 q^{49} -3.00000 q^{51} -12.0000i q^{53} +7.00000i q^{57} +12.0000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -7.00000i q^{67} -6.00000 q^{71} +5.00000i q^{73} -3.00000i q^{77} +14.0000 q^{79} +1.00000 q^{81} +9.00000i q^{83} -6.00000i q^{87} +15.0000 q^{89} -2.00000 q^{91} -4.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9} + O(q^{10})$$ $$2 q + 4 q^{9} - 6 q^{11} - 14 q^{19} + 2 q^{21} + 12 q^{29} + 8 q^{31} + 4 q^{39} - 18 q^{41} - 2 q^{49} - 6 q^{51} + 24 q^{59} - 20 q^{61} - 12 q^{71} + 28 q^{79} + 2 q^{81} + 30 q^{89} - 4 q^{91} - 12 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$801$$ $$2101$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.00000i − 0.577350i −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.00000i − 0.962250i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 3.00000i 0.522233i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 6.00000i − 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 7.00000i 0.927173i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 7.00000i − 0.855186i −0.903971 0.427593i $$-0.859362\pi$$
0.903971 0.427593i $$-0.140638\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 5.00000i 0.585206i 0.956234 + 0.292603i $$0.0945214\pi$$
−0.956234 + 0.292603i $$0.905479\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 3.00000i − 0.341882i
$$78$$ 0 0
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 6.00000i − 0.643268i
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ − 20.0000i − 1.97066i −0.170664 0.985329i $$-0.554591\pi$$
0.170664 0.985329i $$-0.445409\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ − 9.00000i − 0.846649i −0.905978 0.423324i $$-0.860863\pi$$
0.905978 0.423324i $$-0.139137\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 4.00000i 0.369800i
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 9.00000i 0.811503i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ − 7.00000i − 0.606977i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 21.0000i − 1.79415i −0.441877 0.897076i $$-0.645687\pi$$
0.441877 0.897076i $$-0.354313\pi$$
$$138$$ 0 0
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ − 6.00000i − 0.501745i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000i 0.0824786i
$$148$$ 0 0
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 20.0000i − 1.59617i −0.602542 0.798087i $$-0.705846\pi$$
0.602542 0.798087i $$-0.294154\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 5.00000i − 0.391630i −0.980641 0.195815i $$-0.937265\pi$$
0.980641 0.195815i $$-0.0627352\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −14.0000 −1.07061
$$172$$ 0 0
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ 0 0
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 10.0000i 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000i 0.658145i
$$188$$ 0 0
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 0 0
$$193$$ 5.00000i 0.359908i 0.983675 + 0.179954i $$0.0575949\pi$$
−0.983675 + 0.179954i $$0.942405\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 21.0000 1.45260
$$210$$ 0 0
$$211$$ −17.0000 −1.17033 −0.585164 0.810915i $$-0.698970\pi$$
−0.585164 + 0.810915i $$0.698970\pi$$
$$212$$ 0 0
$$213$$ 6.00000i 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000i 0.271538i
$$218$$ 0 0
$$219$$ 5.00000 0.337869
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ − 14.0000i − 0.937509i −0.883328 0.468755i $$-0.844703\pi$$
0.883328 0.468755i $$-0.155297\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 14.0000i − 0.909398i
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 0 0
$$243$$ − 16.0000i − 1.02640i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 14.0000i − 0.890799i
$$248$$ 0 0
$$249$$ 9.00000 0.570352
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 30.0000i 1.87135i 0.352865 + 0.935674i $$0.385208\pi$$
−0.352865 + 0.935674i $$0.614792\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ − 6.00000i − 0.369976i −0.982741 0.184988i $$-0.940775\pi$$
0.982741 0.184988i $$-0.0592246\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 15.0000i − 0.917985i
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 0 0
$$273$$ 2.00000i 0.121046i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 1.00000i 0.0594438i 0.999558 + 0.0297219i $$0.00946217\pi$$
−0.999558 + 0.0297219i $$0.990538\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 9.00000i − 0.531253i
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 15.0000i 0.870388i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.00000i − 0.399511i −0.979846 0.199756i $$-0.935985\pi$$
0.979846 0.199756i $$-0.0640148\pi$$
$$308$$ 0 0
$$309$$ −20.0000 −1.13776
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ 0 0
$$323$$ 21.0000i 1.16847i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 14.0000i 0.774202i
$$328$$ 0 0
$$329$$ 6.00000 0.330791
$$330$$ 0 0
$$331$$ 25.0000 1.37412 0.687062 0.726599i $$-0.258900\pi$$
0.687062 + 0.726599i $$0.258900\pi$$
$$332$$ 0 0
$$333$$ − 16.0000i − 0.876795i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 13.0000i 0.708155i 0.935216 + 0.354078i $$0.115205\pi$$
−0.935216 + 0.354078i $$0.884795\pi$$
$$338$$ 0 0
$$339$$ −9.00000 −0.488813
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 21.0000i − 1.12734i −0.826000 0.563670i $$-0.809389\pi$$
0.826000 0.563670i $$-0.190611\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 0 0
$$351$$ 10.0000 0.533761
$$352$$ 0 0
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 3.00000i − 0.158777i
$$358$$ 0 0
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 2.00000i 0.104973i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ 17.0000 0.873231 0.436616 0.899648i $$-0.356177\pi$$
0.436616 + 0.899648i $$0.356177\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 0 0
$$383$$ 30.0000i 1.53293i 0.642287 + 0.766464i $$0.277986\pi$$
−0.642287 + 0.766464i $$0.722014\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 16.0000i − 0.813326i
$$388$$ 0 0
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ −7.00000 −0.350438
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000i 1.18964i
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ −21.0000 −1.03585
$$412$$ 0 0
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 7.00000i 0.342791i
$$418$$ 0 0
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ 0 0
$$423$$ − 12.0000i − 0.583460i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 10.0000i − 0.483934i
$$428$$ 0 0
$$429$$ −6.00000 −0.289683
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 11.0000i 0.528626i 0.964437 + 0.264313i $$0.0851452\pi$$
−0.964437 + 0.264313i $$0.914855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ 0 0
$$443$$ − 21.0000i − 0.997740i −0.866677 0.498870i $$-0.833748\pi$$
0.866677 0.498870i $$-0.166252\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.0000i 0.567581i
$$448$$ 0 0
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 27.0000 1.27138
$$452$$ 0 0
$$453$$ 8.00000i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 17.0000i − 0.795226i −0.917553 0.397613i $$-0.869839\pi$$
0.917553 0.397613i $$-0.130161\pi$$
$$458$$ 0 0
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 0 0
$$469$$ 7.00000 0.323230
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ 24.0000i 1.10352i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 24.0000i − 1.09888i
$$478$$ 0 0
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 34.0000i − 1.54069i −0.637629 0.770344i $$-0.720085\pi$$
0.637629 0.770344i $$-0.279915\pi$$
$$488$$ 0 0
$$489$$ −5.00000 −0.226108
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ − 18.0000i − 0.810679i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 9.00000i − 0.399704i
$$508$$ 0 0
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ −5.00000 −0.221187
$$512$$ 0 0
$$513$$ 35.0000i 1.54529i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.0000i 0.791639i
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ 0 0
$$523$$ 7.00000i 0.306089i 0.988219 + 0.153044i $$0.0489077\pi$$
−0.988219 + 0.153044i $$0.951092\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 12.0000i − 0.522728i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ − 18.0000i − 0.779667i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 3.00000i − 0.129460i
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 0 0
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 35.0000i 1.49649i 0.663421 + 0.748246i $$0.269104\pi$$
−0.663421 + 0.748246i $$0.730896\pi$$
$$548$$ 0 0
$$549$$ −20.0000 −0.853579
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ 0 0
$$553$$ 14.0000i 0.595341i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 36.0000i − 1.52537i −0.646771 0.762684i $$-0.723881\pi$$
0.646771 0.762684i $$-0.276119\pi$$
$$558$$ 0 0
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 9.00000 0.379980
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000i 0.0419961i
$$568$$ 0 0
$$569$$ 27.0000 1.13190 0.565949 0.824440i $$-0.308510\pi$$
0.565949 + 0.824440i $$0.308510\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ − 6.00000i − 0.250654i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 7.00000i 0.291414i 0.989328 + 0.145707i $$0.0465456\pi$$
−0.989328 + 0.145707i $$0.953454\pi$$
$$578$$ 0 0
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ −9.00000 −0.373383
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 39.0000i 1.60970i 0.593477 + 0.804851i $$0.297755\pi$$
−0.593477 + 0.804851i $$0.702245\pi$$
$$588$$ 0 0
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 0 0
$$593$$ − 27.0000i − 1.10876i −0.832265 0.554379i $$-0.812956\pi$$
0.832265 0.554379i $$-0.187044\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 14.0000i − 0.572982i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ 0 0
$$603$$ − 14.0000i − 0.570124i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 44.0000i 1.78590i 0.450151 + 0.892952i $$0.351370\pi$$
−0.450151 + 0.892952i $$0.648630\pi$$
$$608$$ 0 0
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 15.0000i 0.600962i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 21.0000i − 0.838659i
$$628$$ 0 0
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 17.0000i 0.675689i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 0 0
$$653$$ 36.0000i 1.40879i 0.709809 + 0.704394i $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 9.00000 0.350590 0.175295 0.984516i $$-0.443912\pi$$
0.175295 + 0.984516i $$0.443912\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ − 6.00000i − 0.233021i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −14.0000 −0.541271
$$670$$ 0 0
$$671$$ 30.0000 1.15814
$$672$$ 0 0
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ − 3.00000i − 0.114792i −0.998351 0.0573959i $$-0.981720\pi$$
0.998351 0.0573959i $$-0.0182797\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 26.0000i 0.991962i
$$688$$ 0 0
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 19.0000 0.722794 0.361397 0.932412i $$-0.382300\pi$$
0.361397 + 0.932412i $$0.382300\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 27.0000i 1.02270i
$$698$$ 0 0
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 56.0000i 2.11208i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 0 0
$$711$$ 28.0000 1.05008
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 12.0000i 0.448148i
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 20.0000 0.744839
$$722$$ 0 0
$$723$$ 25.0000i 0.929760i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 34.0000i − 1.26099i −0.776193 0.630495i $$-0.782852\pi$$
0.776193 0.630495i $$-0.217148\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ − 40.0000i − 1.47743i −0.674016 0.738717i $$-0.735432\pi$$
0.674016 0.738717i $$-0.264568\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0000i 0.773545i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −14.0000 −0.514303
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 18.0000i 0.658586i
$$748$$ 0 0
$$749$$ −3.00000 −0.109618
$$750$$ 0 0
$$751$$ 46.0000 1.67856 0.839282 0.543696i $$-0.182976\pi$$
0.839282 + 0.543696i $$0.182976\pi$$
$$752$$ 0 0
$$753$$ 15.0000i 0.546630i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 34.0000i 1.23575i 0.786276 + 0.617876i $$0.212006\pi$$
−0.786276 + 0.617876i $$0.787994\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ 0 0
$$763$$ − 14.0000i − 0.506834i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000i 0.866590i
$$768$$ 0 0
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ 0 0
$$773$$ − 12.0000i − 0.431610i −0.976436 0.215805i $$-0.930762\pi$$
0.976436 0.215805i $$-0.0692376\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 8.00000i − 0.286998i
$$778$$ 0 0
$$779$$ 63.0000 2.25721
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ − 30.0000i − 1.07211i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 0 0
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ − 15.0000i − 0.529339i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 6.00000i − 0.211210i
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ 0 0
$$813$$ 2.00000i 0.0701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 56.0000i 1.95919i
$$818$$ 0 0
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ 0 0
$$823$$ − 26.0000i − 0.906303i −0.891434 0.453152i $$-0.850300\pi$$
0.891434 0.453152i $$-0.149700\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 9.00000i 0.312961i 0.987681 + 0.156480i $$0.0500148\pi$$
−0.987681 + 0.156480i $$0.949985\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 3.00000i 0.103944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 20.0000i − 0.691301i
$$838$$ 0 0
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 18.0000i 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ 1.00000 0.0343199
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 15.0000i − 0.512390i −0.966625 0.256195i $$-0.917531\pi$$
0.966625 0.256195i $$-0.0824690\pi$$
$$858$$ 0 0
$$859$$ −31.0000 −1.05771 −0.528853 0.848713i $$-0.677378\pi$$
−0.528853 + 0.848713i $$0.677378\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 0 0
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 8.00000i − 0.271694i
$$868$$ 0 0
$$869$$ −42.0000 −1.42475
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 0 0
$$873$$ 20.0000i 0.676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ 0 0
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 0 0
$$883$$ − 47.0000i − 1.58168i −0.612026 0.790838i $$-0.709645\pi$$
0.612026 0.790838i $$-0.290355\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.00000i 0.201460i 0.994914 + 0.100730i $$0.0321179\pi$$
−0.994914 + 0.100730i $$0.967882\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ 0 0
$$893$$ 42.0000i 1.40548i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ − 8.00000i − 0.266223i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ − 27.0000i − 0.893570i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −34.0000 −1.12156 −0.560778 0.827966i $$-0.689498\pi$$
−0.560778 + 0.827966i $$0.689498\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ 0 0
$$923$$ − 12.0000i − 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 40.0000i − 1.31377i
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ 0 0
$$933$$ − 18.0000i − 0.589294i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 29.0000i − 0.947389i −0.880689 0.473694i $$-0.842920\pi$$
0.880689 0.473694i $$-0.157080\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ 57.0000i 1.84641i 0.384307 + 0.923206i $$0.374441\pi$$
−0.384307 + 0.923206i $$0.625559\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 18.0000i 0.581857i
$$958$$ 0 0
$$959$$ 21.0000 0.678125
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 6.00000i 0.193347i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 34.0000i − 1.09337i −0.837340 0.546683i $$-0.815890\pi$$
0.837340 0.546683i $$-0.184110\pi$$
$$968$$ 0 0
$$969$$ 21.0000 0.674617
$$970$$ 0 0
$$971$$ −9.00000 −0.288824 −0.144412 0.989518i $$-0.546129\pi$$
−0.144412 + 0.989518i $$0.546129\pi$$
$$972$$ 0 0
$$973$$ − 7.00000i − 0.224410i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.00000i 0.0959785i 0.998848 + 0.0479893i $$0.0152813\pi$$
−0.998848 + 0.0479893i $$0.984719\pi$$
$$978$$ 0 0
$$979$$ −45.0000 −1.43821
$$980$$ 0 0
$$981$$ −28.0000 −0.893971
$$982$$ 0 0
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 6.00000i − 0.190982i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −20.0000 −0.635321 −0.317660 0.948205i $$-0.602897\pi$$
−0.317660 + 0.948205i $$0.602897\pi$$
$$992$$ 0 0
$$993$$ − 25.0000i − 0.793351i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 62.0000i − 1.96356i −0.190022 0.981780i $$-0.560856\pi$$
0.190022 0.981780i $$-0.439144\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 2800.2.g.i.449.1 2
4.3 odd 2 350.2.c.c.99.1 2
5.2 odd 4 2800.2.a.h.1.1 1
5.3 odd 4 2800.2.a.x.1.1 1
5.4 even 2 inner 2800.2.g.i.449.2 2
12.11 even 2 3150.2.g.f.2899.2 2
20.3 even 4 350.2.a.a.1.1 1
20.7 even 4 350.2.a.e.1.1 yes 1
20.19 odd 2 350.2.c.c.99.2 2
28.27 even 2 2450.2.c.h.99.1 2
60.23 odd 4 3150.2.a.x.1.1 1
60.47 odd 4 3150.2.a.m.1.1 1
60.59 even 2 3150.2.g.f.2899.1 2
140.27 odd 4 2450.2.a.x.1.1 1
140.83 odd 4 2450.2.a.m.1.1 1
140.139 even 2 2450.2.c.h.99.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 20.3 even 4
350.2.a.e.1.1 yes 1 20.7 even 4
350.2.c.c.99.1 2 4.3 odd 2
350.2.c.c.99.2 2 20.19 odd 2
2450.2.a.m.1.1 1 140.83 odd 4
2450.2.a.x.1.1 1 140.27 odd 4
2450.2.c.h.99.1 2 28.27 even 2
2450.2.c.h.99.2 2 140.139 even 2
2800.2.a.h.1.1 1 5.2 odd 4
2800.2.a.x.1.1 1 5.3 odd 4
2800.2.g.i.449.1 2 1.1 even 1 trivial
2800.2.g.i.449.2 2 5.4 even 2 inner
3150.2.a.m.1.1 1 60.47 odd 4
3150.2.a.x.1.1 1 60.23 odd 4
3150.2.g.f.2899.1 2 60.59 even 2
3150.2.g.f.2899.2 2 12.11 even 2