# Properties

 Label 5850.2.e.be.5149.2 Level $5850$ Weight $2$ Character 5850.5149 Analytic conductor $46.712$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.7124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1950) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 5149.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5850.5149 Dual form 5850.2.e.be.5149.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +5.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} -7.00000 q^{29} -9.00000 q^{31} +1.00000i q^{32} -5.00000 q^{34} +8.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} -5.00000 q^{44} -9.00000i q^{47} +6.00000 q^{49} +1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} -7.00000i q^{58} +1.00000 q^{59} -7.00000 q^{61} -9.00000i q^{62} -1.00000 q^{64} +15.0000i q^{67} -5.00000i q^{68} +8.00000 q^{71} +4.00000i q^{73} -8.00000 q^{74} +5.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} +9.00000i q^{83} -8.00000 q^{86} -5.00000i q^{88} +16.0000 q^{89} +1.00000 q^{91} +9.00000 q^{94} -2.00000i q^{97} +6.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 10q^{11} - 2q^{14} + 2q^{16} + 2q^{26} - 14q^{29} - 18q^{31} - 10q^{34} + 4q^{41} - 10q^{44} + 12q^{49} + 2q^{56} + 2q^{59} - 14q^{61} - 2q^{64} + 16q^{71} - 16q^{74} + 8q^{79} - 16q^{86} + 32q^{89} + 2q^{91} + 18q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$$.

 $$n$$ $$2251$$ $$3251$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i 0.981981 + 0.188982i $$0.0605189\pi$$
−0.981981 + 0.188982i $$0.939481\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.00000i 1.21268i 0.795206 + 0.606339i $$0.207363\pi$$
−0.795206 + 0.606339i $$0.792637\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 5.00000i 1.06600i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ −7.00000 −1.29987 −0.649934 0.759991i $$-0.725203\pi$$
−0.649934 + 0.759991i $$0.725203\pi$$
$$30$$ 0 0
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −5.00000 −0.857493
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ − 11.0000i − 1.51097i −0.655168 0.755483i $$-0.727402\pi$$
0.655168 0.755483i $$-0.272598\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ − 7.00000i − 0.919145i
$$59$$ 1.00000 0.130189 0.0650945 0.997879i $$-0.479265\pi$$
0.0650945 + 0.997879i $$0.479265\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ − 9.00000i − 1.14300i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 15.0000i 1.83254i 0.400559 + 0.916271i $$0.368816\pi$$
−0.400559 + 0.916271i $$0.631184\pi$$
$$68$$ − 5.00000i − 0.606339i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.00000i 0.569803i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.00000i 0.220863i
$$83$$ 9.00000i 0.987878i 0.869496 + 0.493939i $$0.164443\pi$$
−0.869496 + 0.493939i $$0.835557\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ − 5.00000i − 0.533002i
$$89$$ 16.0000 1.69600 0.847998 0.529999i $$-0.177808\pi$$
0.847998 + 0.529999i $$0.177808\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 9.00000 0.928279
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 6.00000i 0.606092i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.00000 0.696526 0.348263 0.937397i $$-0.386772\pi$$
0.348263 + 0.937397i $$0.386772\pi$$
$$102$$ 0 0
$$103$$ 6.00000i 0.591198i 0.955312 + 0.295599i $$0.0955191\pi$$
−0.955312 + 0.295599i $$0.904481\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.00000 0.649934
$$117$$ 0 0
$$118$$ 1.00000i 0.0920575i
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ − 7.00000i − 0.633750i
$$123$$ 0 0
$$124$$ 9.00000 0.808224
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −15.0000 −1.29580
$$135$$ 0 0
$$136$$ 5.00000 0.428746
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000i 0.671345i
$$143$$ − 5.00000i − 0.418121i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ − 8.00000i − 0.657596i
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −21.0000 −1.70896 −0.854478 0.519488i $$-0.826123\pi$$
−0.854478 + 0.519488i $$0.826123\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −5.00000 −0.402911
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.00000i 0.399043i 0.979893 + 0.199522i $$0.0639388\pi$$
−0.979893 + 0.199522i $$0.936061\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −9.00000 −0.698535
$$167$$ − 4.00000i − 0.309529i −0.987951 0.154765i $$-0.950538\pi$$
0.987951 0.154765i $$-0.0494619\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ 11.0000i 0.836315i 0.908375 + 0.418157i $$0.137324\pi$$
−0.908375 + 0.418157i $$0.862676\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 5.00000 0.376889
$$177$$ 0 0
$$178$$ 16.0000i 1.19925i
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 0 0
$$181$$ −11.0000 −0.817624 −0.408812 0.912619i $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 1.00000i 0.0741249i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 25.0000i 1.82818i
$$188$$ 9.00000i 0.656392i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ − 10.0000i − 0.719816i −0.932988 0.359908i $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 7.00000i 0.492518i
$$203$$ − 7.00000i − 0.491304i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −6.00000 −0.418040
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 11.0000i 0.755483i
$$213$$ 0 0
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 9.00000i − 0.610960i
$$218$$ − 6.00000i − 0.406371i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 29.0000i 1.92480i 0.271640 + 0.962399i $$0.412434\pi$$
−0.271640 + 0.962399i $$0.587566\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 7.00000i 0.459573i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1.00000 −0.0650945
$$237$$ 0 0
$$238$$ − 5.00000i − 0.324102i
$$239$$ 1.00000 0.0646846 0.0323423 0.999477i $$-0.489703\pi$$
0.0323423 + 0.999477i $$0.489703\pi$$
$$240$$ 0 0
$$241$$ −22.0000 −1.41714 −0.708572 0.705638i $$-0.750660\pi$$
−0.708572 + 0.705638i $$0.750660\pi$$
$$242$$ 14.0000i 0.899954i
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 9.00000i 0.571501i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 27.0000i 1.68421i 0.539311 + 0.842107i $$0.318685\pi$$
−0.539311 + 0.842107i $$0.681315\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.00000i 0.123560i
$$263$$ − 26.0000i − 1.60323i −0.597841 0.801614i $$-0.703975\pi$$
0.597841 0.801614i $$-0.296025\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 15.0000i − 0.916271i
$$269$$ 9.00000 0.548740 0.274370 0.961624i $$-0.411531\pi$$
0.274370 + 0.961624i $$0.411531\pi$$
$$270$$ 0 0
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ 5.00000i 0.303170i
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ 2.00000i 0.119952i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ − 22.0000i − 1.30776i −0.756596 0.653882i $$-0.773139\pi$$
0.756596 0.653882i $$-0.226861\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 5.00000 0.295656
$$287$$ 2.00000i 0.118056i
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 4.00000i − 0.234082i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 0 0
$$298$$ − 22.0000i − 1.27443i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ − 21.0000i − 1.20841i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ − 5.00000i − 0.284901i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 21.0000i 1.18699i 0.804838 + 0.593495i $$0.202252\pi$$
−0.804838 + 0.593495i $$0.797748\pi$$
$$314$$ −5.00000 −0.282166
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ −35.0000 −1.95962
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ − 2.00000i − 0.110432i
$$329$$ 9.00000 0.496186
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 9.00000i − 0.493939i
$$333$$ 0 0
$$334$$ 4.00000 0.218870
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 13.0000i 0.708155i 0.935216 + 0.354078i $$0.115205\pi$$
−0.935216 + 0.354078i $$0.884795\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −45.0000 −2.43689
$$342$$ 0 0
$$343$$ 13.0000i 0.701934i
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −11.0000 −0.591364
$$347$$ − 6.00000i − 0.322097i −0.986947 0.161048i $$-0.948512\pi$$
0.986947 0.161048i $$-0.0514875\pi$$
$$348$$ 0 0
$$349$$ −32.0000 −1.71292 −0.856460 0.516213i $$-0.827341\pi$$
−0.856460 + 0.516213i $$0.827341\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.00000i 0.266501i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −16.0000 −0.847998
$$357$$ 0 0
$$358$$ 2.00000i 0.105703i
$$359$$ 27.0000 1.42501 0.712503 0.701669i $$-0.247562\pi$$
0.712503 + 0.701669i $$0.247562\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ − 11.0000i − 0.578147i
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 36.0000i 1.87918i 0.342296 + 0.939592i $$0.388796\pi$$
−0.342296 + 0.939592i $$0.611204\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11.0000 0.571092
$$372$$ 0 0
$$373$$ − 17.0000i − 0.880227i −0.897942 0.440113i $$-0.854938\pi$$
0.897942 0.440113i $$-0.145062\pi$$
$$374$$ −25.0000 −1.29272
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 7.00000i 0.360518i
$$378$$ 0 0
$$379$$ 15.0000 0.770498 0.385249 0.922813i $$-0.374116\pi$$
0.385249 + 0.922813i $$0.374116\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 18.0000i − 0.920960i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ 2.00000i 0.101535i
$$389$$ 2.00000 0.101404 0.0507020 0.998714i $$-0.483854\pi$$
0.0507020 + 0.998714i $$0.483854\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 6.00000i − 0.303046i
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 34.0000i 1.70641i 0.521575 + 0.853206i $$0.325345\pi$$
−0.521575 + 0.853206i $$0.674655\pi$$
$$398$$ − 18.0000i − 0.902258i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 9.00000i 0.448322i
$$404$$ −7.00000 −0.348263
$$405$$ 0 0
$$406$$ 7.00000 0.347404
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 6.00000i − 0.295599i
$$413$$ 1.00000i 0.0492068i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.00000 −0.0977064 −0.0488532 0.998806i $$-0.515557\pi$$
−0.0488532 + 0.998806i $$0.515557\pi$$
$$420$$ 0 0
$$421$$ 28.0000 1.36464 0.682318 0.731055i $$-0.260972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ 16.0000i 0.778868i
$$423$$ 0 0
$$424$$ −11.0000 −0.534207
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 7.00000i − 0.338754i
$$428$$ − 6.00000i − 0.290021i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 9.00000 0.432014
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 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
$$441$$ 0 0
$$442$$ 5.00000i 0.237826i
$$443$$ 32.0000i 1.52037i 0.649709 + 0.760183i $$0.274891\pi$$
−0.649709 + 0.760183i $$0.725109\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ 10.0000 0.470882
$$452$$ 2.00000i 0.0940721i
$$453$$ 0 0
$$454$$ −29.0000 −1.36104
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.00000i − 0.0935561i −0.998905 0.0467780i $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ 2.00000i 0.0934539i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ − 11.0000i − 0.511213i −0.966781 0.255607i $$-0.917725\pi$$
0.966781 0.255607i $$-0.0822752\pi$$
$$464$$ −7.00000 −0.324967
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 32.0000i − 1.48078i −0.672176 0.740392i $$-0.734640\pi$$
0.672176 0.740392i $$-0.265360\pi$$
$$468$$ 0 0
$$469$$ −15.0000 −0.692636
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 1.00000i − 0.0460287i
$$473$$ 40.0000i 1.83920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 5.00000 0.229175
$$477$$ 0 0
$$478$$ 1.00000i 0.0457389i
$$479$$ 29.0000 1.32504 0.662522 0.749043i $$-0.269486\pi$$
0.662522 + 0.749043i $$0.269486\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ − 22.0000i − 1.00207i
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 1.00000i 0.0453143i 0.999743 + 0.0226572i $$0.00721262\pi$$
−0.999743 + 0.0226572i $$0.992787\pi$$
$$488$$ 7.00000i 0.316875i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ − 35.0000i − 1.57632i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −9.00000 −0.404112
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ −11.0000 −0.492428 −0.246214 0.969216i $$-0.579187\pi$$
−0.246214 + 0.969216i $$0.579187\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 6.00000i 0.267793i
$$503$$ 2.00000i 0.0891756i 0.999005 + 0.0445878i $$0.0141974\pi$$
−0.999005 + 0.0445878i $$0.985803\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ − 8.00000i − 0.354943i
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −27.0000 −1.19092
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 45.0000i − 1.97910i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ − 18.0000i − 0.787085i −0.919306 0.393543i $$-0.871249\pi$$
0.919306 0.393543i $$-0.128751\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ 0 0
$$526$$ 26.0000 1.13365
$$527$$ − 45.0000i − 1.96023i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 2.00000i − 0.0866296i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 15.0000 0.647901
$$537$$ 0 0
$$538$$ 9.00000i 0.388018i
$$539$$ 30.0000 1.29219
$$540$$ 0 0
$$541$$ 28.0000 1.20381 0.601907 0.798566i $$-0.294408\pi$$
0.601907 + 0.798566i $$0.294408\pi$$
$$542$$ 7.00000i 0.300676i
$$543$$ 0 0
$$544$$ −5.00000 −0.214373
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 2.00000i 0.0855138i 0.999086 + 0.0427569i $$0.0136141\pi$$
−0.999086 + 0.0427569i $$0.986386\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 4.00000i 0.170097i
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −2.00000 −0.0848189
$$557$$ 2.00000i 0.0847427i 0.999102 + 0.0423714i $$0.0134913\pi$$
−0.999102 + 0.0423714i $$0.986509\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 30.0000i − 1.26547i
$$563$$ 24.0000i 1.01148i 0.862686 + 0.505740i $$0.168780\pi$$
−0.862686 + 0.505740i $$0.831220\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 22.0000 0.924729
$$567$$ 0 0
$$568$$ − 8.00000i − 0.335673i
$$569$$ −9.00000 −0.377300 −0.188650 0.982044i $$-0.560411\pi$$
−0.188650 + 0.982044i $$0.560411\pi$$
$$570$$ 0 0
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 5.00000i 0.209061i
$$573$$ 0 0
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −9.00000 −0.373383
$$582$$ 0 0
$$583$$ − 55.0000i − 2.27787i
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ − 15.0000i − 0.619116i −0.950881 0.309558i $$-0.899819\pi$$
0.950881 0.309558i $$-0.100181\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8.00000i 0.328798i
$$593$$ − 22.0000i − 0.903432i −0.892162 0.451716i $$-0.850812\pi$$
0.892162 0.451716i $$-0.149188\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6.00000 0.245153 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$600$$ 0 0
$$601$$ −35.0000 −1.42768 −0.713840 0.700309i $$-0.753046\pi$$
−0.713840 + 0.700309i $$0.753046\pi$$
$$602$$ − 8.00000i − 0.326056i
$$603$$ 0 0
$$604$$ 21.0000 0.854478
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2.00000i − 0.0811775i −0.999176 0.0405887i $$-0.987077\pi$$
0.999176 0.0405887i $$-0.0129233\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −9.00000 −0.364101
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 5.00000 0.201456
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 0 0
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000i 0.721734i
$$623$$ 16.0000i 0.641026i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −21.0000 −0.839329
$$627$$ 0 0
$$628$$ − 5.00000i − 0.199522i
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 6.00000i − 0.237729i
$$638$$ − 35.0000i − 1.38566i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ − 28.0000i − 1.10421i −0.833774 0.552106i $$-0.813824\pi$$
0.833774 0.552106i $$-0.186176\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 48.0000i − 1.88707i −0.331266 0.943537i $$-0.607476\pi$$
0.331266 0.943537i $$-0.392524\pi$$
$$648$$ 0 0
$$649$$ 5.00000 0.196267
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 27.0000i 1.05659i 0.849060 + 0.528296i $$0.177169\pi$$
−0.849060 + 0.528296i $$0.822831\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 9.00000i 0.350857i
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 4.00000i 0.154765i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −35.0000 −1.35116
$$672$$ 0 0
$$673$$ 11.0000i 0.424019i 0.977268 + 0.212009i $$0.0680008\pi$$
−0.977268 + 0.212009i $$0.931999\pi$$
$$674$$ −13.0000 −0.500741
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ 2.00000 0.0767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 45.0000i − 1.72314i
$$683$$ − 19.0000i − 0.727015i −0.931591 0.363507i $$-0.881579\pi$$
0.931591 0.363507i $$-0.118421\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ −11.0000 −0.419067
$$690$$ 0 0
$$691$$ 7.00000 0.266293 0.133146 0.991096i $$-0.457492\pi$$
0.133146 + 0.991096i $$0.457492\pi$$
$$692$$ − 11.0000i − 0.418157i
$$693$$ 0 0
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10.0000i 0.378777i
$$698$$ − 32.0000i − 1.21122i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 25.0000 0.944237 0.472118 0.881535i $$-0.343489\pi$$
0.472118 + 0.881535i $$0.343489\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −5.00000 −0.188445
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 7.00000i 0.263262i
$$708$$ 0 0
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 16.0000i − 0.599625i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ 0 0
$$718$$ 27.0000i 1.00763i
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ −6.00000 −0.223452
$$722$$ − 19.0000i − 0.707107i
$$723$$ 0 0
$$724$$ 11.0000 0.408812
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.0000i 1.40934i 0.709534 + 0.704671i $$0.248905\pi$$
−0.709534 + 0.704671i $$0.751095\pi$$
$$728$$ − 1.00000i − 0.0370625i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ 0 0
$$733$$ 20.0000i 0.738717i 0.929287 + 0.369358i $$0.120423\pi$$
−0.929287 + 0.369358i $$0.879577\pi$$
$$734$$ −36.0000 −1.32878
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 75.0000i 2.76266i
$$738$$ 0 0
$$739$$ −3.00000 −0.110357 −0.0551784 0.998477i $$-0.517573\pi$$
−0.0551784 + 0.998477i $$0.517573\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.0000i 0.403823i
$$743$$ − 39.0000i − 1.43077i −0.698730 0.715386i $$-0.746251\pi$$
0.698730 0.715386i $$-0.253749\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 17.0000 0.622414
$$747$$ 0 0
$$748$$ − 25.0000i − 0.914091i
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ − 9.00000i − 0.328196i
$$753$$ 0 0
$$754$$ −7.00000 −0.254925
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 33.0000i − 1.19941i −0.800223 0.599703i $$-0.795286\pi$$
0.800223 0.599703i $$-0.204714\pi$$
$$758$$ 15.0000i 0.544825i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 8.00000 0.290000 0.145000 0.989432i $$-0.453682\pi$$
0.145000 + 0.989432i $$0.453682\pi$$
$$762$$ 0 0
$$763$$ − 6.00000i − 0.217215i
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1.00000i − 0.0361079i
$$768$$ 0 0
$$769$$ 8.00000 0.288487 0.144244 0.989542i $$-0.453925\pi$$
0.144244 + 0.989542i $$0.453925\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000i 0.359908i
$$773$$ − 14.0000i − 0.503545i −0.967786 0.251773i $$-0.918987\pi$$
0.967786 0.251773i $$-0.0810135\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 2.00000i 0.0717035i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 40.0000 1.43131
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 6.00000 0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 5.00000i 0.178231i 0.996021 + 0.0891154i $$0.0284040\pi$$
−0.996021 + 0.0891154i $$0.971596\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.00000 0.0711118
$$792$$ 0 0
$$793$$ 7.00000i 0.248577i
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ − 13.0000i − 0.460484i −0.973133 0.230242i $$-0.926048\pi$$
0.973133 0.230242i $$-0.0739517\pi$$
$$798$$ 0 0
$$799$$ 45.0000 1.59199
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 20.0000i 0.705785i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −9.00000 −0.317011
$$807$$ 0 0
$$808$$ − 7.00000i − 0.246259i
$$809$$ −34.0000 −1.19538 −0.597688 0.801729i $$-0.703914\pi$$
−0.597688 + 0.801729i $$0.703914\pi$$
$$810$$ 0 0
$$811$$ −29.0000 −1.01833 −0.509164 0.860670i $$-0.670045\pi$$
−0.509164 + 0.860670i $$0.670045\pi$$
$$812$$ 7.00000i 0.245652i
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 14.0000i 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16.0000 0.558404 0.279202 0.960232i $$-0.409930\pi$$
0.279202 + 0.960232i $$0.409930\pi$$
$$822$$ 0 0
$$823$$ − 28.0000i − 0.976019i −0.872838 0.488009i $$-0.837723\pi$$
0.872838 0.488009i $$-0.162277\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ −1.00000 −0.0347945
$$827$$ − 21.0000i − 0.730242i −0.930960 0.365121i $$-0.881028\pi$$
0.930960 0.365121i $$-0.118972\pi$$
$$828$$ 0 0
$$829$$ 37.0000 1.28506 0.642532 0.766259i $$-0.277884\pi$$
0.642532 + 0.766259i $$0.277884\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000i 0.0346688i
$$833$$ 30.0000i 1.03944i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 2.00000i − 0.0690889i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ 28.0000i 0.964944i
$$843$$ 0 0
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.0000i 0.481046i
$$848$$ − 11.0000i − 0.377742i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 8.00000i − 0.273915i −0.990577 0.136957i $$-0.956268\pi$$
0.990577 0.136957i $$-0.0437323\pi$$
$$854$$ 7.00000 0.239535
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ − 26.0000i − 0.888143i −0.895991 0.444072i $$-0.853534\pi$$
0.895991 0.444072i $$-0.146466\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 12.0000i − 0.408722i
$$863$$ − 27.0000i − 0.919091i −0.888154 0.459545i $$-0.848012\pi$$
0.888154 0.459545i $$-0.151988\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ 9.00000i 0.305480i
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ 15.0000 0.508256
$$872$$ 6.00000i 0.203186i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 40.0000i − 1.35070i −0.737496 0.675352i $$-0.763992\pi$$
0.737496 0.675352i $$-0.236008\pi$$
$$878$$ − 4.00000i − 0.134993i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 0 0
$$883$$ 32.0000i 1.07689i 0.842662 + 0.538443i $$0.180987\pi$$
−0.842662 + 0.538443i $$0.819013\pi$$
$$884$$ −5.00000 −0.168168
$$885$$ 0 0
$$886$$ −32.0000 −1.07506
$$887$$ 4.00000i 0.134307i 0.997743 + 0.0671534i $$0.0213917\pi$$
−0.997743 + 0.0671534i $$0.978608\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 36.0000i 1.20134i
$$899$$ 63.0000 2.10117
$$900$$ 0 0
$$901$$ 55.0000 1.83232
$$902$$ 10.0000i 0.332964i
$$903$$ 0 0
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 40.0000i − 1.32818i −0.747653 0.664089i $$-0.768820\pi$$
0.747653 0.664089i $$-0.231180\pi$$
$$908$$ − 29.0000i − 0.962399i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ 45.0000i 1.48928i
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ 2.00000i 0.0660458i
$$918$$ 0 0
$$919$$ 50.0000 1.64935 0.824674 0.565608i $$-0.191359\pi$$
0.824674 + 0.565608i $$0.191359\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 6.00000i 0.197599i
$$923$$ − 8.00000i − 0.263323i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 11.0000 0.361482
$$927$$ 0 0
$$928$$ − 7.00000i − 0.229786i
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ 32.0000 1.04707
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 51.0000i − 1.66610i −0.553200 0.833049i $$-0.686593\pi$$
0.553200 0.833049i $$-0.313407\pi$$
$$938$$ − 15.0000i − 0.489767i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −8.00000 −0.260793 −0.130396 0.991462i $$-0.541625\pi$$
−0.130396 + 0.991462i $$0.541625\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 1.00000 0.0325472
$$945$$ 0 0
$$946$$ −40.0000 −1.30051
$$947$$ − 41.0000i − 1.33232i −0.745808 0.666160i $$-0.767937\pi$$
0.745808 0.666160i $$-0.232063\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 5.00000i 0.162051i
$$953$$ − 51.0000i − 1.65205i −0.563632 0.826026i $$-0.690596\pi$$
0.563632 0.826026i $$-0.309404\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1.00000 −0.0323423
$$957$$ 0 0
$$958$$ 29.0000i 0.936947i
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 50.0000 1.61290
$$962$$ 8.00000i 0.257930i
$$963$$ 0 0
$$964$$ 22.0000 0.708572
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 17.0000i 0.546683i 0.961917 + 0.273342i $$0.0881289\pi$$
−0.961917 + 0.273342i $$0.911871\pi$$
$$968$$ − 14.0000i − 0.449977i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 34.0000 1.09111 0.545556 0.838074i $$-0.316319\pi$$
0.545556 + 0.838074i $$0.316319\pi$$
$$972$$ 0 0
$$973$$ 2.00000i 0.0641171i
$$974$$ −1.00000 −0.0320421
$$975$$ 0 0
$$976$$ −7.00000 −0.224065
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ 0 0
$$979$$ 80.0000 2.55681
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 30.0000i − 0.957338i
$$983$$ 31.0000i 0.988746i 0.869250 + 0.494373i $$0.164602\pi$$
−0.869250 + 0.494373i $$0.835398\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 35.0000 1.11463
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 36.0000 1.14358 0.571789 0.820401i $$-0.306250\pi$$
0.571789 + 0.820401i $$0.306250\pi$$
$$992$$ − 9.00000i − 0.285750i
$$993$$ 0 0
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 11.0000i 0.348373i 0.984713 + 0.174187i $$0.0557296\pi$$
−0.984713 + 0.174187i $$0.944270\pi$$
$$998$$ − 11.0000i − 0.348199i
$$999$$ 0 0
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 5850.2.e.be.5149.2 2
3.2 odd 2 1950.2.e.a.1249.1 2
5.2 odd 4 5850.2.a.k.1.1 1
5.3 odd 4 5850.2.a.bu.1.1 1
5.4 even 2 inner 5850.2.e.be.5149.1 2
15.2 even 4 1950.2.a.p.1.1 yes 1
15.8 even 4 1950.2.a.l.1.1 1
15.14 odd 2 1950.2.e.a.1249.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.l.1.1 1 15.8 even 4
1950.2.a.p.1.1 yes 1 15.2 even 4
1950.2.e.a.1249.1 2 3.2 odd 2
1950.2.e.a.1249.2 2 15.14 odd 2
5850.2.a.k.1.1 1 5.2 odd 4
5850.2.a.bu.1.1 1 5.3 odd 4
5850.2.e.be.5149.1 2 5.4 even 2 inner
5850.2.e.be.5149.2 2 1.1 even 1 trivial