# Properties

 Label 4400.2.a.bu.1.2 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1100) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.79129 q^{3} +0.208712 q^{7} +4.79129 q^{9} +O(q^{10})$$ $$q+2.79129 q^{3} +0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.791288 q^{17} -6.58258 q^{19} +0.582576 q^{21} +3.79129 q^{23} +5.00000 q^{27} +6.79129 q^{29} +8.58258 q^{31} +2.79129 q^{33} -2.58258 q^{37} -2.79129 q^{39} -1.41742 q^{41} +10.0000 q^{43} +1.41742 q^{47} -6.95644 q^{49} +2.20871 q^{51} +11.3739 q^{53} -18.3739 q^{57} +10.5826 q^{59} +4.20871 q^{61} +1.00000 q^{63} +4.00000 q^{67} +10.5826 q^{69} +10.7477 q^{71} -7.79129 q^{73} +0.208712 q^{77} +15.5390 q^{79} -0.417424 q^{81} +9.95644 q^{83} +18.9564 q^{87} -0.791288 q^{89} -0.208712 q^{91} +23.9564 q^{93} -6.20871 q^{97} +4.79129 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 5 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 + 5 * q^7 + 5 * q^9 $$2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49} + 9 q^{51} + 9 q^{53} - 23 q^{57} + 12 q^{59} + 13 q^{61} + 2 q^{63} + 8 q^{67} + 12 q^{69} - 6 q^{71} - 11 q^{73} + 5 q^{77} - q^{79} - 10 q^{81} - 3 q^{83} + 15 q^{87} + 3 q^{89} - 5 q^{91} + 25 q^{93} - 17 q^{97} + 5 q^{99}+O(q^{100})$$ 2 * q + q^3 + 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 4 * q^19 - 8 * q^21 + 3 * q^23 + 10 * q^27 + 9 * q^29 + 8 * q^31 + q^33 + 4 * q^37 - q^39 - 12 * q^41 + 20 * q^43 + 12 * q^47 + 9 * q^49 + 9 * q^51 + 9 * q^53 - 23 * q^57 + 12 * q^59 + 13 * q^61 + 2 * q^63 + 8 * q^67 + 12 * q^69 - 6 * q^71 - 11 * q^73 + 5 * q^77 - q^79 - 10 * q^81 - 3 * q^83 + 15 * q^87 + 3 * q^89 - 5 * q^91 + 25 * q^93 - 17 * q^97 + 5 * q^99

## 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
$$3$$ 2.79129 1.61155 0.805775 0.592221i $$-0.201749\pi$$
0.805775 + 0.592221i $$0.201749\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.208712 0.0788858 0.0394429 0.999222i $$-0.487442\pi$$
0.0394429 + 0.999222i $$0.487442\pi$$
$$8$$ 0 0
$$9$$ 4.79129 1.59710
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0.791288 0.191915 0.0959577 0.995385i $$-0.469409\pi$$
0.0959577 + 0.995385i $$0.469409\pi$$
$$18$$ 0 0
$$19$$ −6.58258 −1.51015 −0.755073 0.655640i $$-0.772399\pi$$
−0.755073 + 0.655640i $$0.772399\pi$$
$$20$$ 0 0
$$21$$ 0.582576 0.127128
$$22$$ 0 0
$$23$$ 3.79129 0.790538 0.395269 0.918565i $$-0.370651\pi$$
0.395269 + 0.918565i $$0.370651\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 6.79129 1.26111 0.630555 0.776144i $$-0.282827\pi$$
0.630555 + 0.776144i $$0.282827\pi$$
$$30$$ 0 0
$$31$$ 8.58258 1.54148 0.770738 0.637152i $$-0.219888\pi$$
0.770738 + 0.637152i $$0.219888\pi$$
$$32$$ 0 0
$$33$$ 2.79129 0.485901
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.58258 −0.424573 −0.212286 0.977207i $$-0.568091\pi$$
−0.212286 + 0.977207i $$0.568091\pi$$
$$38$$ 0 0
$$39$$ −2.79129 −0.446964
$$40$$ 0 0
$$41$$ −1.41742 −0.221364 −0.110682 0.993856i $$-0.535304\pi$$
−0.110682 + 0.993856i $$0.535304\pi$$
$$42$$ 0 0
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.41742 0.206753 0.103376 0.994642i $$-0.467035\pi$$
0.103376 + 0.994642i $$0.467035\pi$$
$$48$$ 0 0
$$49$$ −6.95644 −0.993777
$$50$$ 0 0
$$51$$ 2.20871 0.309282
$$52$$ 0 0
$$53$$ 11.3739 1.56232 0.781160 0.624331i $$-0.214628\pi$$
0.781160 + 0.624331i $$0.214628\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −18.3739 −2.43368
$$58$$ 0 0
$$59$$ 10.5826 1.37773 0.688867 0.724888i $$-0.258108\pi$$
0.688867 + 0.724888i $$0.258108\pi$$
$$60$$ 0 0
$$61$$ 4.20871 0.538870 0.269435 0.963019i $$-0.413163\pi$$
0.269435 + 0.963019i $$0.413163\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 10.5826 1.27399
$$70$$ 0 0
$$71$$ 10.7477 1.27552 0.637760 0.770235i $$-0.279861\pi$$
0.637760 + 0.770235i $$0.279861\pi$$
$$72$$ 0 0
$$73$$ −7.79129 −0.911901 −0.455951 0.890005i $$-0.650701\pi$$
−0.455951 + 0.890005i $$0.650701\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0.208712 0.0237850
$$78$$ 0 0
$$79$$ 15.5390 1.74828 0.874138 0.485678i $$-0.161427\pi$$
0.874138 + 0.485678i $$0.161427\pi$$
$$80$$ 0 0
$$81$$ −0.417424 −0.0463805
$$82$$ 0 0
$$83$$ 9.95644 1.09286 0.546431 0.837504i $$-0.315986\pi$$
0.546431 + 0.837504i $$0.315986\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 18.9564 2.03234
$$88$$ 0 0
$$89$$ −0.791288 −0.0838763 −0.0419382 0.999120i $$-0.513353\pi$$
−0.0419382 + 0.999120i $$0.513353\pi$$
$$90$$ 0 0
$$91$$ −0.208712 −0.0218790
$$92$$ 0 0
$$93$$ 23.9564 2.48417
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.20871 −0.630399 −0.315200 0.949025i $$-0.602071\pi$$
−0.315200 + 0.949025i $$0.602071\pi$$
$$98$$ 0 0
$$99$$ 4.79129 0.481543
$$100$$ 0 0
$$101$$ −17.3739 −1.72876 −0.864382 0.502836i $$-0.832290\pi$$
−0.864382 + 0.502836i $$0.832290\pi$$
$$102$$ 0 0
$$103$$ −5.95644 −0.586905 −0.293453 0.955974i $$-0.594804\pi$$
−0.293453 + 0.955974i $$0.594804\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −19.7477 −1.90908 −0.954542 0.298075i $$-0.903655\pi$$
−0.954542 + 0.298075i $$0.903655\pi$$
$$108$$ 0 0
$$109$$ −4.79129 −0.458922 −0.229461 0.973318i $$-0.573696\pi$$
−0.229461 + 0.973318i $$0.573696\pi$$
$$110$$ 0 0
$$111$$ −7.20871 −0.684221
$$112$$ 0 0
$$113$$ 1.41742 0.133340 0.0666700 0.997775i $$-0.478763\pi$$
0.0666700 + 0.997775i $$0.478763\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −4.79129 −0.442955
$$118$$ 0 0
$$119$$ 0.165151 0.0151394
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −3.95644 −0.356740
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.3739 −1.45295 −0.726473 0.687195i $$-0.758842\pi$$
−0.726473 + 0.687195i $$0.758842\pi$$
$$128$$ 0 0
$$129$$ 27.9129 2.45759
$$130$$ 0 0
$$131$$ 0.626136 0.0547058 0.0273529 0.999626i $$-0.491292\pi$$
0.0273529 + 0.999626i $$0.491292\pi$$
$$132$$ 0 0
$$133$$ −1.37386 −0.119129
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −11.5390 −0.985845 −0.492922 0.870073i $$-0.664071\pi$$
−0.492922 + 0.870073i $$0.664071\pi$$
$$138$$ 0 0
$$139$$ −9.74773 −0.826791 −0.413396 0.910551i $$-0.635657\pi$$
−0.413396 + 0.910551i $$0.635657\pi$$
$$140$$ 0 0
$$141$$ 3.95644 0.333192
$$142$$ 0 0
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −19.4174 −1.60152
$$148$$ 0 0
$$149$$ 3.16515 0.259299 0.129650 0.991560i $$-0.458615\pi$$
0.129650 + 0.991560i $$0.458615\pi$$
$$150$$ 0 0
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 0 0
$$153$$ 3.79129 0.306507
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 17.1652 1.36993 0.684964 0.728577i $$-0.259818\pi$$
0.684964 + 0.728577i $$0.259818\pi$$
$$158$$ 0 0
$$159$$ 31.7477 2.51776
$$160$$ 0 0
$$161$$ 0.791288 0.0623622
$$162$$ 0 0
$$163$$ 21.3739 1.67413 0.837065 0.547103i $$-0.184270\pi$$
0.837065 + 0.547103i $$0.184270\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.5826 1.28320 0.641599 0.767040i $$-0.278271\pi$$
0.641599 + 0.767040i $$0.278271\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −31.5390 −2.41185
$$172$$ 0 0
$$173$$ −13.7477 −1.04522 −0.522610 0.852572i $$-0.675042\pi$$
−0.522610 + 0.852572i $$0.675042\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 29.5390 2.22029
$$178$$ 0 0
$$179$$ −22.1216 −1.65345 −0.826723 0.562610i $$-0.809797\pi$$
−0.826723 + 0.562610i $$0.809797\pi$$
$$180$$ 0 0
$$181$$ −3.37386 −0.250777 −0.125389 0.992108i $$-0.540018\pi$$
−0.125389 + 0.992108i $$0.540018\pi$$
$$182$$ 0 0
$$183$$ 11.7477 0.868417
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.791288 0.0578647
$$188$$ 0 0
$$189$$ 1.04356 0.0759079
$$190$$ 0 0
$$191$$ −17.2087 −1.24518 −0.622589 0.782549i $$-0.713919\pi$$
−0.622589 + 0.782549i $$0.713919\pi$$
$$192$$ 0 0
$$193$$ −7.16515 −0.515759 −0.257879 0.966177i $$-0.583024\pi$$
−0.257879 + 0.966177i $$0.583024\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.5390 1.89083 0.945413 0.325874i $$-0.105658\pi$$
0.945413 + 0.325874i $$0.105658\pi$$
$$198$$ 0 0
$$199$$ 1.62614 0.115274 0.0576369 0.998338i $$-0.481643\pi$$
0.0576369 + 0.998338i $$0.481643\pi$$
$$200$$ 0 0
$$201$$ 11.1652 0.787529
$$202$$ 0 0
$$203$$ 1.41742 0.0994837
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 18.1652 1.26257
$$208$$ 0 0
$$209$$ −6.58258 −0.455326
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 0 0
$$213$$ 30.0000 2.05557
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1.79129 0.121601
$$218$$ 0 0
$$219$$ −21.7477 −1.46958
$$220$$ 0 0
$$221$$ −0.791288 −0.0532278
$$222$$ 0 0
$$223$$ 20.5826 1.37831 0.689156 0.724613i $$-0.257982\pi$$
0.689156 + 0.724613i $$0.257982\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −11.5390 −0.765871 −0.382936 0.923775i $$-0.625087\pi$$
−0.382936 + 0.923775i $$0.625087\pi$$
$$228$$ 0 0
$$229$$ −7.62614 −0.503949 −0.251975 0.967734i $$-0.581080\pi$$
−0.251975 + 0.967734i $$0.581080\pi$$
$$230$$ 0 0
$$231$$ 0.582576 0.0383307
$$232$$ 0 0
$$233$$ 15.7913 1.03452 0.517261 0.855828i $$-0.326952\pi$$
0.517261 + 0.855828i $$0.326952\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 43.3739 2.81744
$$238$$ 0 0
$$239$$ 2.20871 0.142870 0.0714349 0.997445i $$-0.477242\pi$$
0.0714349 + 0.997445i $$0.477242\pi$$
$$240$$ 0 0
$$241$$ −23.1216 −1.48939 −0.744696 0.667404i $$-0.767406\pi$$
−0.744696 + 0.667404i $$0.767406\pi$$
$$242$$ 0 0
$$243$$ −16.1652 −1.03699
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6.58258 0.418839
$$248$$ 0 0
$$249$$ 27.7913 1.76120
$$250$$ 0 0
$$251$$ 11.5390 0.728336 0.364168 0.931333i $$-0.381353\pi$$
0.364168 + 0.931333i $$0.381353\pi$$
$$252$$ 0 0
$$253$$ 3.79129 0.238356
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −0.539015 −0.0334928
$$260$$ 0 0
$$261$$ 32.5390 2.01411
$$262$$ 0 0
$$263$$ −24.1652 −1.49009 −0.745044 0.667016i $$-0.767571\pi$$
−0.745044 + 0.667016i $$0.767571\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.20871 −0.135171
$$268$$ 0 0
$$269$$ −20.7042 −1.26236 −0.631178 0.775638i $$-0.717428\pi$$
−0.631178 + 0.775638i $$0.717428\pi$$
$$270$$ 0 0
$$271$$ −15.7477 −0.956606 −0.478303 0.878195i $$-0.658748\pi$$
−0.478303 + 0.878195i $$0.658748\pi$$
$$272$$ 0 0
$$273$$ −0.582576 −0.0352591
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.58258 −0.515677 −0.257838 0.966188i $$-0.583010\pi$$
−0.257838 + 0.966188i $$0.583010\pi$$
$$278$$ 0 0
$$279$$ 41.1216 2.46189
$$280$$ 0 0
$$281$$ −13.7477 −0.820121 −0.410060 0.912058i $$-0.634492\pi$$
−0.410060 + 0.912058i $$0.634492\pi$$
$$282$$ 0 0
$$283$$ 30.7042 1.82517 0.912587 0.408883i $$-0.134082\pi$$
0.912587 + 0.408883i $$0.134082\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.295834 −0.0174625
$$288$$ 0 0
$$289$$ −16.3739 −0.963168
$$290$$ 0 0
$$291$$ −17.3303 −1.01592
$$292$$ 0 0
$$293$$ −2.83485 −0.165614 −0.0828068 0.996566i $$-0.526388\pi$$
−0.0828068 + 0.996566i $$0.526388\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000 0.290129
$$298$$ 0 0
$$299$$ −3.79129 −0.219256
$$300$$ 0 0
$$301$$ 2.08712 0.120300
$$302$$ 0 0
$$303$$ −48.4955 −2.78599
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 27.2087 1.55288 0.776442 0.630189i $$-0.217023\pi$$
0.776442 + 0.630189i $$0.217023\pi$$
$$308$$ 0 0
$$309$$ −16.6261 −0.945828
$$310$$ 0 0
$$311$$ 5.83485 0.330864 0.165432 0.986221i $$-0.447098\pi$$
0.165432 + 0.986221i $$0.447098\pi$$
$$312$$ 0 0
$$313$$ 6.74773 0.381404 0.190702 0.981648i $$-0.438924\pi$$
0.190702 + 0.981648i $$0.438924\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.37386 −0.470323 −0.235162 0.971956i $$-0.575562\pi$$
−0.235162 + 0.971956i $$0.575562\pi$$
$$318$$ 0 0
$$319$$ 6.79129 0.380239
$$320$$ 0 0
$$321$$ −55.1216 −3.07659
$$322$$ 0 0
$$323$$ −5.20871 −0.289820
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −13.3739 −0.739576
$$328$$ 0 0
$$329$$ 0.295834 0.0163098
$$330$$ 0 0
$$331$$ −17.3303 −0.952560 −0.476280 0.879294i $$-0.658015\pi$$
−0.476280 + 0.879294i $$0.658015\pi$$
$$332$$ 0 0
$$333$$ −12.3739 −0.678084
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 0 0
$$339$$ 3.95644 0.214884
$$340$$ 0 0
$$341$$ 8.58258 0.464773
$$342$$ 0 0
$$343$$ −2.91288 −0.157281
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.6261 −0.677807 −0.338903 0.940821i $$-0.610056\pi$$
−0.338903 + 0.940821i $$0.610056\pi$$
$$348$$ 0 0
$$349$$ −14.5826 −0.780587 −0.390294 0.920690i $$-0.627627\pi$$
−0.390294 + 0.920690i $$0.627627\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ 18.1652 0.966833 0.483417 0.875390i $$-0.339396\pi$$
0.483417 + 0.875390i $$0.339396\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0.460985 0.0243979
$$358$$ 0 0
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 24.3303 1.28054
$$362$$ 0 0
$$363$$ 2.79129 0.146505
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.62614 −0.450281 −0.225140 0.974326i $$-0.572284\pi$$
−0.225140 + 0.974326i $$0.572284\pi$$
$$368$$ 0 0
$$369$$ −6.79129 −0.353540
$$370$$ 0 0
$$371$$ 2.37386 0.123245
$$372$$ 0 0
$$373$$ −19.3303 −1.00089 −0.500443 0.865770i $$-0.666829\pi$$
−0.500443 + 0.865770i $$0.666829\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.79129 −0.349769
$$378$$ 0 0
$$379$$ 31.0000 1.59236 0.796182 0.605058i $$-0.206850\pi$$
0.796182 + 0.605058i $$0.206850\pi$$
$$380$$ 0 0
$$381$$ −45.7042 −2.34150
$$382$$ 0 0
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 47.9129 2.43555
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 3.00000 0.151717
$$392$$ 0 0
$$393$$ 1.74773 0.0881612
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7.20871 0.361795 0.180897 0.983502i $$-0.442100\pi$$
0.180897 + 0.983502i $$0.442100\pi$$
$$398$$ 0 0
$$399$$ −3.83485 −0.191983
$$400$$ 0 0
$$401$$ −14.8348 −0.740817 −0.370408 0.928869i $$-0.620782\pi$$
−0.370408 + 0.928869i $$0.620782\pi$$
$$402$$ 0 0
$$403$$ −8.58258 −0.427529
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.58258 −0.128014
$$408$$ 0 0
$$409$$ 17.0000 0.840596 0.420298 0.907386i $$-0.361926\pi$$
0.420298 + 0.907386i $$0.361926\pi$$
$$410$$ 0 0
$$411$$ −32.2087 −1.58874
$$412$$ 0 0
$$413$$ 2.20871 0.108684
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −27.2087 −1.33242
$$418$$ 0 0
$$419$$ −27.0000 −1.31904 −0.659518 0.751689i $$-0.729240\pi$$
−0.659518 + 0.751689i $$0.729240\pi$$
$$420$$ 0 0
$$421$$ 25.3739 1.23665 0.618323 0.785924i $$-0.287812\pi$$
0.618323 + 0.785924i $$0.287812\pi$$
$$422$$ 0 0
$$423$$ 6.79129 0.330204
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.878409 0.0425092
$$428$$ 0 0
$$429$$ −2.79129 −0.134765
$$430$$ 0 0
$$431$$ −10.9129 −0.525655 −0.262827 0.964843i $$-0.584655\pi$$
−0.262827 + 0.964843i $$0.584655\pi$$
$$432$$ 0 0
$$433$$ −13.3303 −0.640613 −0.320307 0.947314i $$-0.603786\pi$$
−0.320307 + 0.947314i $$0.603786\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −24.9564 −1.19383
$$438$$ 0 0
$$439$$ −11.9564 −0.570650 −0.285325 0.958431i $$-0.592101\pi$$
−0.285325 + 0.958431i $$0.592101\pi$$
$$440$$ 0 0
$$441$$ −33.3303 −1.58716
$$442$$ 0 0
$$443$$ −22.4174 −1.06508 −0.532542 0.846404i $$-0.678763\pi$$
−0.532542 + 0.846404i $$0.678763\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 8.83485 0.417874
$$448$$ 0 0
$$449$$ 26.2087 1.23687 0.618433 0.785838i $$-0.287768\pi$$
0.618433 + 0.785838i $$0.287768\pi$$
$$450$$ 0 0
$$451$$ −1.41742 −0.0667439
$$452$$ 0 0
$$453$$ −13.9564 −0.655731
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.3739 0.625603 0.312801 0.949819i $$-0.398733\pi$$
0.312801 + 0.949819i $$0.398733\pi$$
$$458$$ 0 0
$$459$$ 3.95644 0.184671
$$460$$ 0 0
$$461$$ −24.4955 −1.14087 −0.570434 0.821344i $$-0.693225\pi$$
−0.570434 + 0.821344i $$0.693225\pi$$
$$462$$ 0 0
$$463$$ 2.25227 0.104672 0.0523360 0.998630i $$-0.483333\pi$$
0.0523360 + 0.998630i $$0.483333\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10.5826 −0.489703 −0.244852 0.969561i $$-0.578739\pi$$
−0.244852 + 0.969561i $$0.578739\pi$$
$$468$$ 0 0
$$469$$ 0.834849 0.0385497
$$470$$ 0 0
$$471$$ 47.9129 2.20771
$$472$$ 0 0
$$473$$ 10.0000 0.459800
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 54.4955 2.49518
$$478$$ 0 0
$$479$$ −0.165151 −0.00754596 −0.00377298 0.999993i $$-0.501201\pi$$
−0.00377298 + 0.999993i $$0.501201\pi$$
$$480$$ 0 0
$$481$$ 2.58258 0.117755
$$482$$ 0 0
$$483$$ 2.20871 0.100500
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 29.7477 1.34800 0.673999 0.738732i $$-0.264575\pi$$
0.673999 + 0.738732i $$0.264575\pi$$
$$488$$ 0 0
$$489$$ 59.6606 2.69795
$$490$$ 0 0
$$491$$ 7.41742 0.334744 0.167372 0.985894i $$-0.446472\pi$$
0.167372 + 0.985894i $$0.446472\pi$$
$$492$$ 0 0
$$493$$ 5.37386 0.242027
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.24318 0.100620
$$498$$ 0 0
$$499$$ −14.9564 −0.669542 −0.334771 0.942299i $$-0.608659\pi$$
−0.334771 + 0.942299i $$0.608659\pi$$
$$500$$ 0 0
$$501$$ 46.2867 2.06794
$$502$$ 0 0
$$503$$ −16.4174 −0.732017 −0.366008 0.930612i $$-0.619276\pi$$
−0.366008 + 0.930612i $$0.619276\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −33.4955 −1.48759
$$508$$ 0 0
$$509$$ 10.1216 0.448632 0.224316 0.974517i $$-0.427985\pi$$
0.224316 + 0.974517i $$0.427985\pi$$
$$510$$ 0 0
$$511$$ −1.62614 −0.0719360
$$512$$ 0 0
$$513$$ −32.9129 −1.45314
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.41742 0.0623382
$$518$$ 0 0
$$519$$ −38.3739 −1.68443
$$520$$ 0 0
$$521$$ 1.58258 0.0693339 0.0346670 0.999399i $$-0.488963\pi$$
0.0346670 + 0.999399i $$0.488963\pi$$
$$522$$ 0 0
$$523$$ 25.1652 1.10040 0.550198 0.835034i $$-0.314552\pi$$
0.550198 + 0.835034i $$0.314552\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.79129 0.295833
$$528$$ 0 0
$$529$$ −8.62614 −0.375049
$$530$$ 0 0
$$531$$ 50.7042 2.20037
$$532$$ 0 0
$$533$$ 1.41742 0.0613955
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −61.7477 −2.66461
$$538$$ 0 0
$$539$$ −6.95644 −0.299635
$$540$$ 0 0
$$541$$ −10.6261 −0.456853 −0.228427 0.973561i $$-0.573358\pi$$
−0.228427 + 0.973561i $$0.573358\pi$$
$$542$$ 0 0
$$543$$ −9.41742 −0.404140
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7.37386 −0.315284 −0.157642 0.987496i $$-0.550389\pi$$
−0.157642 + 0.987496i $$0.550389\pi$$
$$548$$ 0 0
$$549$$ 20.1652 0.860628
$$550$$ 0 0
$$551$$ −44.7042 −1.90446
$$552$$ 0 0
$$553$$ 3.24318 0.137914
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −45.6606 −1.93470 −0.967351 0.253441i $$-0.918438\pi$$
−0.967351 + 0.253441i $$0.918438\pi$$
$$558$$ 0 0
$$559$$ −10.0000 −0.422955
$$560$$ 0 0
$$561$$ 2.20871 0.0932519
$$562$$ 0 0
$$563$$ 40.1216 1.69092 0.845462 0.534036i $$-0.179325\pi$$
0.845462 + 0.534036i $$0.179325\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −0.0871215 −0.00365876
$$568$$ 0 0
$$569$$ 18.9564 0.794695 0.397348 0.917668i $$-0.369931\pi$$
0.397348 + 0.917668i $$0.369931\pi$$
$$570$$ 0 0
$$571$$ −27.2867 −1.14191 −0.570957 0.820980i $$-0.693428\pi$$
−0.570957 + 0.820980i $$0.693428\pi$$
$$572$$ 0 0
$$573$$ −48.0345 −2.00667
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 11.9564 0.497753 0.248877 0.968535i $$-0.419939\pi$$
0.248877 + 0.968535i $$0.419939\pi$$
$$578$$ 0 0
$$579$$ −20.0000 −0.831172
$$580$$ 0 0
$$581$$ 2.07803 0.0862112
$$582$$ 0 0
$$583$$ 11.3739 0.471057
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −15.9564 −0.658593 −0.329296 0.944227i $$-0.606812\pi$$
−0.329296 + 0.944227i $$0.606812\pi$$
$$588$$ 0 0
$$589$$ −56.4955 −2.32785
$$590$$ 0 0
$$591$$ 74.0780 3.04716
$$592$$ 0 0
$$593$$ −22.5826 −0.927355 −0.463678 0.886004i $$-0.653470\pi$$
−0.463678 + 0.886004i $$0.653470\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.53901 0.185770
$$598$$ 0 0
$$599$$ −44.7042 −1.82656 −0.913281 0.407329i $$-0.866460\pi$$
−0.913281 + 0.407329i $$0.866460\pi$$
$$600$$ 0 0
$$601$$ 28.5390 1.16413 0.582065 0.813142i $$-0.302245\pi$$
0.582065 + 0.813142i $$0.302245\pi$$
$$602$$ 0 0
$$603$$ 19.1652 0.780465
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.0000 1.38002 0.690009 0.723801i $$-0.257607\pi$$
0.690009 + 0.723801i $$0.257607\pi$$
$$608$$ 0 0
$$609$$ 3.95644 0.160323
$$610$$ 0 0
$$611$$ −1.41742 −0.0573428
$$612$$ 0 0
$$613$$ 33.4519 1.35111 0.675555 0.737310i $$-0.263904\pi$$
0.675555 + 0.737310i $$0.263904\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −28.7477 −1.15734 −0.578670 0.815562i $$-0.696428\pi$$
−0.578670 + 0.815562i $$0.696428\pi$$
$$618$$ 0 0
$$619$$ −6.25227 −0.251300 −0.125650 0.992075i $$-0.540102\pi$$
−0.125650 + 0.992075i $$0.540102\pi$$
$$620$$ 0 0
$$621$$ 18.9564 0.760696
$$622$$ 0 0
$$623$$ −0.165151 −0.00661665
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −18.3739 −0.733781
$$628$$ 0 0
$$629$$ −2.04356 −0.0814821
$$630$$ 0 0
$$631$$ −18.1216 −0.721409 −0.360705 0.932680i $$-0.617464\pi$$
−0.360705 + 0.932680i $$0.617464\pi$$
$$632$$ 0 0
$$633$$ −13.9564 −0.554719
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 6.95644 0.275624
$$638$$ 0 0
$$639$$ 51.4955 2.03713
$$640$$ 0 0
$$641$$ −23.8348 −0.941420 −0.470710 0.882288i $$-0.656002\pi$$
−0.470710 + 0.882288i $$0.656002\pi$$
$$642$$ 0 0
$$643$$ 43.4955 1.71529 0.857647 0.514239i $$-0.171926\pi$$
0.857647 + 0.514239i $$0.171926\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21.3303 −0.838581 −0.419290 0.907852i $$-0.637721\pi$$
−0.419290 + 0.907852i $$0.637721\pi$$
$$648$$ 0 0
$$649$$ 10.5826 0.415402
$$650$$ 0 0
$$651$$ 5.00000 0.195965
$$652$$ 0 0
$$653$$ 26.7042 1.04501 0.522507 0.852635i $$-0.324997\pi$$
0.522507 + 0.852635i $$0.324997\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −37.3303 −1.45639
$$658$$ 0 0
$$659$$ −0.460985 −0.0179574 −0.00897871 0.999960i $$-0.502858\pi$$
−0.00897871 + 0.999960i $$0.502858\pi$$
$$660$$ 0 0
$$661$$ 30.5826 1.18952 0.594762 0.803902i $$-0.297246\pi$$
0.594762 + 0.803902i $$0.297246\pi$$
$$662$$ 0 0
$$663$$ −2.20871 −0.0857793
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 25.7477 0.996956
$$668$$ 0 0
$$669$$ 57.4519 2.22122
$$670$$ 0 0
$$671$$ 4.20871 0.162476
$$672$$ 0 0
$$673$$ −38.9129 −1.49998 −0.749991 0.661448i $$-0.769942\pi$$
−0.749991 + 0.661448i $$0.769942\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −1.58258 −0.0608233 −0.0304117 0.999537i $$-0.509682\pi$$
−0.0304117 + 0.999537i $$0.509682\pi$$
$$678$$ 0 0
$$679$$ −1.29583 −0.0497295
$$680$$ 0 0
$$681$$ −32.2087 −1.23424
$$682$$ 0 0
$$683$$ −34.9129 −1.33590 −0.667952 0.744204i $$-0.732829\pi$$
−0.667952 + 0.744204i $$0.732829\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −21.2867 −0.812140
$$688$$ 0 0
$$689$$ −11.3739 −0.433310
$$690$$ 0 0
$$691$$ 32.1216 1.22196 0.610981 0.791645i $$-0.290775\pi$$
0.610981 + 0.791645i $$0.290775\pi$$
$$692$$ 0 0
$$693$$ 1.00000 0.0379869
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.12159 −0.0424833
$$698$$ 0 0
$$699$$ 44.0780 1.66718
$$700$$ 0 0
$$701$$ 20.8348 0.786921 0.393461 0.919341i $$-0.371278\pi$$
0.393461 + 0.919341i $$0.371278\pi$$
$$702$$ 0 0
$$703$$ 17.0000 0.641167
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3.62614 −0.136375
$$708$$ 0 0
$$709$$ 29.4955 1.10773 0.553863 0.832608i $$-0.313153\pi$$
0.553863 + 0.832608i $$0.313153\pi$$
$$710$$ 0 0
$$711$$ 74.4519 2.79216
$$712$$ 0 0
$$713$$ 32.5390 1.21860
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.16515 0.230242
$$718$$ 0 0
$$719$$ −6.49545 −0.242240 −0.121120 0.992638i $$-0.538649\pi$$
−0.121120 + 0.992638i $$0.538649\pi$$
$$720$$ 0 0
$$721$$ −1.24318 −0.0462985
$$722$$ 0 0
$$723$$ −64.5390 −2.40023
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 23.1216 0.857532 0.428766 0.903416i $$-0.358948\pi$$
0.428766 + 0.903416i $$0.358948\pi$$
$$728$$ 0 0
$$729$$ −43.8693 −1.62479
$$730$$ 0 0
$$731$$ 7.91288 0.292668
$$732$$ 0 0
$$733$$ 31.2432 1.15399 0.576997 0.816747i $$-0.304225\pi$$
0.576997 + 0.816747i $$0.304225\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000 0.147342
$$738$$ 0 0
$$739$$ −42.1216 −1.54947 −0.774734 0.632287i $$-0.782116\pi$$
−0.774734 + 0.632287i $$0.782116\pi$$
$$740$$ 0 0
$$741$$ 18.3739 0.674981
$$742$$ 0 0
$$743$$ −39.7913 −1.45980 −0.729900 0.683554i $$-0.760434\pi$$
−0.729900 + 0.683554i $$0.760434\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 47.7042 1.74540
$$748$$ 0 0
$$749$$ −4.12159 −0.150600
$$750$$ 0 0
$$751$$ −13.2087 −0.481993 −0.240996 0.970526i $$-0.577474\pi$$
−0.240996 + 0.970526i $$0.577474\pi$$
$$752$$ 0 0
$$753$$ 32.2087 1.17375
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −7.33030 −0.266424 −0.133212 0.991088i $$-0.542529\pi$$
−0.133212 + 0.991088i $$0.542529\pi$$
$$758$$ 0 0
$$759$$ 10.5826 0.384123
$$760$$ 0 0
$$761$$ 30.3303 1.09947 0.549736 0.835338i $$-0.314728\pi$$
0.549736 + 0.835338i $$0.314728\pi$$
$$762$$ 0 0
$$763$$ −1.00000 −0.0362024
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −10.5826 −0.382115
$$768$$ 0 0
$$769$$ −41.7477 −1.50546 −0.752731 0.658328i $$-0.771264\pi$$
−0.752731 + 0.658328i $$0.771264\pi$$
$$770$$ 0 0
$$771$$ 50.2432 1.80946
$$772$$ 0 0
$$773$$ 26.2087 0.942662 0.471331 0.881956i $$-0.343774\pi$$
0.471331 + 0.881956i $$0.343774\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1.50455 −0.0539753
$$778$$ 0 0
$$779$$ 9.33030 0.334293
$$780$$ 0 0
$$781$$ 10.7477 0.384584
$$782$$ 0 0
$$783$$ 33.9564 1.21350
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17.7477 0.632638 0.316319 0.948653i $$-0.397553\pi$$
0.316319 + 0.948653i $$0.397553\pi$$
$$788$$ 0 0
$$789$$ −67.4519 −2.40135
$$790$$ 0 0
$$791$$ 0.295834 0.0105186
$$792$$ 0 0
$$793$$ −4.20871 −0.149456
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 34.2867 1.21450 0.607249 0.794511i $$-0.292273\pi$$
0.607249 + 0.794511i $$0.292273\pi$$
$$798$$ 0 0
$$799$$ 1.12159 0.0396790
$$800$$ 0 0
$$801$$ −3.79129 −0.133959
$$802$$ 0 0
$$803$$ −7.79129 −0.274949
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −57.7913 −2.03435
$$808$$ 0 0
$$809$$ −45.4955 −1.59953 −0.799767 0.600310i $$-0.795044\pi$$
−0.799767 + 0.600310i $$0.795044\pi$$
$$810$$ 0 0
$$811$$ 55.6606 1.95451 0.977254 0.212072i $$-0.0680210\pi$$
0.977254 + 0.212072i $$0.0680210\pi$$
$$812$$ 0 0
$$813$$ −43.9564 −1.54162
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −65.8258 −2.30295
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ 48.6606 1.69827 0.849133 0.528178i $$-0.177125\pi$$
0.849133 + 0.528178i $$0.177125\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10.4174 0.362249 0.181125 0.983460i $$-0.442026\pi$$
0.181125 + 0.983460i $$0.442026\pi$$
$$828$$ 0 0
$$829$$ −37.9564 −1.31828 −0.659141 0.752020i $$-0.729080\pi$$
−0.659141 + 0.752020i $$0.729080\pi$$
$$830$$ 0 0
$$831$$ −23.9564 −0.831040
$$832$$ 0 0
$$833$$ −5.50455 −0.190721
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 42.9129 1.48329
$$838$$ 0 0
$$839$$ 23.7042 0.818359 0.409179 0.912454i $$-0.365815\pi$$
0.409179 + 0.912454i $$0.365815\pi$$
$$840$$ 0 0
$$841$$ 17.1216 0.590400
$$842$$ 0 0
$$843$$ −38.3739 −1.32167
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.208712 0.00717143
$$848$$ 0 0
$$849$$ 85.7042 2.94136
$$850$$ 0 0
$$851$$ −9.79129 −0.335641
$$852$$ 0 0
$$853$$ 3.12159 0.106881 0.0534406 0.998571i $$-0.482981\pi$$
0.0534406 + 0.998571i $$0.482981\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −30.4955 −1.04170 −0.520852 0.853647i $$-0.674386\pi$$
−0.520852 + 0.853647i $$0.674386\pi$$
$$858$$ 0 0
$$859$$ 2.08712 0.0712117 0.0356058 0.999366i $$-0.488664\pi$$
0.0356058 + 0.999366i $$0.488664\pi$$
$$860$$ 0 0
$$861$$ −0.825757 −0.0281417
$$862$$ 0 0
$$863$$ 27.3303 0.930334 0.465167 0.885223i $$-0.345994\pi$$
0.465167 + 0.885223i $$0.345994\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −45.7042 −1.55219
$$868$$ 0 0
$$869$$ 15.5390 0.527125
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ −29.7477 −1.00681
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −16.6606 −0.562589 −0.281294 0.959622i $$-0.590764\pi$$
−0.281294 + 0.959622i $$0.590764\pi$$
$$878$$ 0 0
$$879$$ −7.91288 −0.266895
$$880$$ 0 0
$$881$$ 1.87841 0.0632852 0.0316426 0.999499i $$-0.489926\pi$$
0.0316426 + 0.999499i $$0.489926\pi$$
$$882$$ 0 0
$$883$$ −26.1652 −0.880527 −0.440264 0.897869i $$-0.645115\pi$$
−0.440264 + 0.897869i $$0.645115\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 39.3303 1.32058 0.660291 0.751010i $$-0.270433\pi$$
0.660291 + 0.751010i $$0.270433\pi$$
$$888$$ 0 0
$$889$$ −3.41742 −0.114617
$$890$$ 0 0
$$891$$ −0.417424 −0.0139842
$$892$$ 0 0
$$893$$ −9.33030 −0.312227
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −10.5826 −0.353342
$$898$$ 0 0
$$899$$ 58.2867 1.94397
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 5.82576 0.193869
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −32.6606 −1.08448 −0.542239 0.840224i $$-0.682423\pi$$
−0.542239 + 0.840224i $$0.682423\pi$$
$$908$$ 0 0
$$909$$ −83.2432 −2.76100
$$910$$ 0 0
$$911$$ 12.3303 0.408521 0.204261 0.978917i $$-0.434521\pi$$
0.204261 + 0.978917i $$0.434521\pi$$
$$912$$ 0 0
$$913$$ 9.95644 0.329510
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0.130682 0.00431551
$$918$$ 0 0
$$919$$ −28.8348 −0.951174 −0.475587 0.879669i $$-0.657764\pi$$
−0.475587 + 0.879669i $$0.657764\pi$$
$$920$$ 0 0
$$921$$ 75.9473 2.50255
$$922$$ 0 0
$$923$$ −10.7477 −0.353766
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −28.5390 −0.937344
$$928$$ 0 0
$$929$$ 13.7477 0.451048 0.225524 0.974238i $$-0.427591\pi$$
0.225524 + 0.974238i $$0.427591\pi$$
$$930$$ 0 0
$$931$$ 45.7913 1.50075
$$932$$ 0 0
$$933$$ 16.2867 0.533204
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 35.1652 1.14880 0.574398 0.818576i $$-0.305236\pi$$
0.574398 + 0.818576i $$0.305236\pi$$
$$938$$ 0 0
$$939$$ 18.8348 0.614652
$$940$$ 0 0
$$941$$ −50.0780 −1.63250 −0.816249 0.577701i $$-0.803950\pi$$
−0.816249 + 0.577701i $$0.803950\pi$$
$$942$$ 0 0
$$943$$ −5.37386 −0.174997
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.91288 0.159647 0.0798235 0.996809i $$-0.474564\pi$$
0.0798235 + 0.996809i $$0.474564\pi$$
$$948$$ 0 0
$$949$$ 7.79129 0.252916
$$950$$ 0 0
$$951$$ −23.3739 −0.757949
$$952$$ 0 0
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 18.9564 0.612775
$$958$$ 0 0
$$959$$ −2.40833 −0.0777691
$$960$$ 0 0
$$961$$ 42.6606 1.37615
$$962$$ 0 0
$$963$$ −94.6170 −3.04899
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −43.5390 −1.40012 −0.700060 0.714084i $$-0.746844\pi$$
−0.700060 + 0.714084i $$0.746844\pi$$
$$968$$ 0 0
$$969$$ −14.5390 −0.467060
$$970$$ 0 0
$$971$$ −18.9564 −0.608341 −0.304171 0.952618i $$-0.598379\pi$$
−0.304171 + 0.952618i $$0.598379\pi$$
$$972$$ 0 0
$$973$$ −2.03447 −0.0652221
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −25.7477 −0.823743 −0.411871 0.911242i $$-0.635125\pi$$
−0.411871 + 0.911242i $$0.635125\pi$$
$$978$$ 0 0
$$979$$ −0.791288 −0.0252897
$$980$$ 0 0
$$981$$ −22.9564 −0.732943
$$982$$ 0 0
$$983$$ 48.6606 1.55203 0.776016 0.630713i $$-0.217237\pi$$
0.776016 + 0.630713i $$0.217237\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0.825757 0.0262841
$$988$$ 0 0
$$989$$ 37.9129 1.20556
$$990$$ 0 0
$$991$$ −55.8693 −1.77475 −0.887374 0.461051i $$-0.847473\pi$$
−0.887374 + 0.461051i $$0.847473\pi$$
$$992$$ 0 0
$$993$$ −48.3739 −1.53510
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 40.3739 1.27865 0.639327 0.768935i $$-0.279213\pi$$
0.639327 + 0.768935i $$0.279213\pi$$
$$998$$ 0 0
$$999$$ −12.9129 −0.408545
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 4400.2.a.bu.1.2 2
4.3 odd 2 1100.2.a.g.1.1 2
5.2 odd 4 4400.2.b.s.4049.1 4
5.3 odd 4 4400.2.b.s.4049.4 4
5.4 even 2 4400.2.a.bi.1.1 2
12.11 even 2 9900.2.a.bh.1.2 2
20.3 even 4 1100.2.b.d.749.1 4
20.7 even 4 1100.2.b.d.749.4 4
20.19 odd 2 1100.2.a.h.1.2 yes 2
60.23 odd 4 9900.2.c.x.5149.3 4
60.47 odd 4 9900.2.c.x.5149.2 4
60.59 even 2 9900.2.a.bz.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.1 2 4.3 odd 2
1100.2.a.h.1.2 yes 2 20.19 odd 2
1100.2.b.d.749.1 4 20.3 even 4
1100.2.b.d.749.4 4 20.7 even 4
4400.2.a.bi.1.1 2 5.4 even 2
4400.2.a.bu.1.2 2 1.1 even 1 trivial
4400.2.b.s.4049.1 4 5.2 odd 4
4400.2.b.s.4049.4 4 5.3 odd 4
9900.2.a.bh.1.2 2 12.11 even 2
9900.2.a.bz.1.1 2 60.59 even 2
9900.2.c.x.5149.2 4 60.47 odd 4
9900.2.c.x.5149.3 4 60.23 odd 4