# Properties

 Label 3900.2.h.b.1249.1 Level $3900$ Weight $2$ Character 3900.1249 Analytic conductor $31.142$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3900.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.1416567883$$ 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 156) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3900.1249 Dual form 3900.2.h.b.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{13} -2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{27} +6.00000 q^{29} -10.0000 q^{31} +4.00000i q^{33} -10.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -10.0000i q^{53} -2.00000i q^{57} +8.00000 q^{59} -14.0000 q^{61} -2.00000i q^{63} -2.00000i q^{67} +16.0000 q^{71} -10.0000i q^{73} -8.00000i q^{77} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{87} +4.00000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +2.00000i q^{97} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 8q^{11} + 4q^{19} + 4q^{21} + 12q^{29} - 20q^{31} + 2q^{39} + 16q^{41} + 6q^{49} - 4q^{51} + 16q^{59} - 28q^{61} + 32q^{71} + 32q^{79} + 2q^{81} + 8q^{89} - 4q^{91} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1951$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 0 0
$$33$$ 4.00000i 0.696311i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 8.00000 1.24939 0.624695 0.780869i $$-0.285223\pi$$
0.624695 + 0.780869i $$0.285223\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 2.00000i − 0.244339i −0.992509 0.122169i $$-0.961015\pi$$
0.992509 0.122169i $$-0.0389851\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 0 0
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 8.00000i − 0.911685i
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 6.00000i − 0.643268i
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 0 0
$$93$$ 10.0000i 1.03695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 0 0
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 8.00000i − 0.683486i −0.939793 0.341743i $$-0.888983\pi$$
0.939793 0.341743i $$-0.111017\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 3.00000i − 0.247436i
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2.00000i 0.156652i 0.996928 + 0.0783260i $$0.0249575\pi$$
−0.996928 + 0.0783260i $$0.975042\pi$$
$$164$$ 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
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 8.00000i − 0.601317i
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 14.0000i 1.03491i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ 0 0
$$203$$ 12.0000i 0.842235i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ − 16.0000i − 1.09630i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 20.0000i − 1.35769i
$$218$$ 0 0
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 0 0
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 30.0000i − 1.87135i −0.352865 0.935674i $$-0.614792\pi$$
0.352865 0.935674i $$-0.385208\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ − 16.0000i − 0.986602i −0.869859 0.493301i $$-0.835790\pi$$
0.869859 0.493301i $$-0.164210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 4.00000i − 0.244796i
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −10.0000 −0.607457 −0.303728 0.952759i $$-0.598232\pi$$
−0.303728 + 0.952759i $$0.598232\pi$$
$$272$$ 0 0
$$273$$ 2.00000i 0.121046i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 0 0
$$279$$ 10.0000 0.598684
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 16.0000i 0.944450i
$$288$$ 0 0
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ 8.00000i 0.467365i 0.972313 + 0.233682i $$0.0750776\pi$$
−0.972313 + 0.233682i $$0.924922\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 4.00000i − 0.232104i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ − 10.0000i − 0.574485i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 22.0000i − 1.25561i −0.778372 0.627803i $$-0.783954\pi$$
0.778372 0.627803i $$-0.216046\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 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
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ − 4.00000i − 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2.00000i − 0.110600i
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 30.0000i 1.63420i 0.576493 + 0.817102i $$0.304421\pi$$
−0.576493 + 0.817102i $$0.695579\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 40.0000 2.16612
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ − 16.0000i − 0.851594i −0.904819 0.425797i $$-0.859994\pi$$
0.904819 0.425797i $$-0.140006\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 4.00000i − 0.211702i
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 4.00000i − 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 0 0
$$369$$ −8.00000 −0.416463
$$370$$ 0 0
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000i 0.309016i
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 4.00000i − 0.203331i
$$388$$ 0 0
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 4.00000i − 0.201773i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 18.0000i − 0.903394i −0.892171 0.451697i $$-0.850819\pi$$
0.892171 0.451697i $$-0.149181\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ − 10.0000i − 0.498135i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ −8.00000 −0.394611
$$412$$ 0 0
$$413$$ 16.0000i 0.787309i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ − 4.00000i − 0.194487i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 28.0000i − 1.35501i
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ − 18.0000i − 0.865025i −0.901628 0.432512i $$-0.857627\pi$$
0.901628 0.432512i $$-0.142373\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −12.0000 −0.566315 −0.283158 0.959073i $$-0.591382\pi$$
−0.283158 + 0.959073i $$0.591382\pi$$
$$450$$ 0 0
$$451$$ −32.0000 −1.50682
$$452$$ 0 0
$$453$$ − 18.0000i − 0.845714i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ 0 0
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 22.0000i 1.02243i 0.859454 + 0.511213i $$0.170804\pi$$
−0.859454 + 0.511213i $$0.829196\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.0000i 0.457869i
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000i 1.17817i 0.808070 + 0.589086i $$0.200512\pi$$
−0.808070 + 0.589086i $$0.799488\pi$$
$$488$$ 0 0
$$489$$ 2.00000 0.0904431
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000i 1.43540i
$$498$$ 0 0
$$499$$ 6.00000 0.268597 0.134298 0.990941i $$-0.457122\pi$$
0.134298 + 0.990941i $$0.457122\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000i 0.0444116i
$$508$$ 0 0
$$509$$ −12.0000 −0.531891 −0.265945 0.963988i $$-0.585684\pi$$
−0.265945 + 0.963988i $$0.585684\pi$$
$$510$$ 0 0
$$511$$ 20.0000 0.884748
$$512$$ 0 0
$$513$$ 2.00000i 0.0883022i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 16.0000i − 0.703679i
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −2.00000 −0.0876216 −0.0438108 0.999040i $$-0.513950\pi$$
−0.0438108 + 0.999040i $$0.513950\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.0000i 0.871214i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ 8.00000i 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.0000i 0.517838i
$$538$$ 0 0
$$539$$ −12.0000 −0.516877
$$540$$ 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
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 32.0000i 1.36078i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 24.0000i − 1.01691i −0.861088 0.508456i $$-0.830216\pi$$
0.861088 0.508456i $$-0.169784\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ 28.0000i 1.18006i 0.807382 + 0.590030i $$0.200884\pi$$
−0.807382 + 0.590030i $$0.799116\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −8.00000 −0.334790 −0.167395 0.985890i $$-0.553535\pi$$
−0.167395 + 0.985890i $$0.553535\pi$$
$$572$$ 0 0
$$573$$ 8.00000i 0.334205i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000i 1.65663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ 0 0
$$589$$ −20.0000 −0.824086
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 0 0
$$593$$ − 36.0000i − 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.00000i 0.163709i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 2.00000i 0.0814463i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4.00000i 0.162355i 0.996700 + 0.0811775i $$0.0258681\pi$$
−0.996700 + 0.0811775i $$0.974132\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ −4.00000 −0.161823
$$612$$ 0 0
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 48.0000i 1.93241i 0.257780 + 0.966204i $$0.417009\pi$$
−0.257780 + 0.966204i $$0.582991\pi$$
$$618$$ 0 0
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 8.00000i 0.320513i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 8.00000i 0.319489i
$$628$$ 0 0
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −30.0000 −1.19428 −0.597141 0.802137i $$-0.703697\pi$$
−0.597141 + 0.802137i $$0.703697\pi$$
$$632$$ 0 0
$$633$$ 12.0000i 0.476957i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.00000i 0.118864i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ 26.0000i 1.02534i 0.858586 + 0.512670i $$0.171344\pi$$
−0.858586 + 0.512670i $$0.828656\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 0 0
$$649$$ −32.0000 −1.25611
$$650$$ 0 0
$$651$$ −20.0000 −0.783862
$$652$$ 0 0
$$653$$ − 46.0000i − 1.80012i −0.435767 0.900060i $$-0.643523\pi$$
0.435767 0.900060i $$-0.356477\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 10.0000 0.388955 0.194477 0.980907i $$-0.437699\pi$$
0.194477 + 0.980907i $$0.437699\pi$$
$$662$$ 0 0
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 56.0000 2.16186
$$672$$ 0 0
$$673$$ 22.0000i 0.848038i 0.905653 + 0.424019i $$0.139381\pi$$
−0.905653 + 0.424019i $$0.860619\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 0 0
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 20.0000i 0.765279i 0.923898 + 0.382639i $$0.124985\pi$$
−0.923898 + 0.382639i $$0.875015\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 2.00000i − 0.0763048i
$$688$$ 0 0
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 0 0
$$693$$ 8.00000i 0.303895i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 16.0000i − 0.606043i
$$698$$ 0 0
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ − 20.0000i − 0.754314i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 20.0000i 0.752177i
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 24.0000i − 0.896296i
$$718$$ 0 0
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 2.00000i 0.0743808i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 44.0000i − 1.63187i −0.578144 0.815935i $$-0.696223\pi$$
0.578144 0.815935i $$-0.303777\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.00000i 0.294684i
$$738$$ 0 0
$$739$$ −22.0000 −0.809283 −0.404642 0.914475i $$-0.632604\pi$$
−0.404642 + 0.914475i $$0.632604\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 0 0
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 20.0000i 0.728841i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 18.0000i − 0.654221i −0.944986 0.327111i $$-0.893925\pi$$
0.944986 0.327111i $$-0.106075\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 40.0000 1.45000 0.724999 0.688749i $$-0.241840\pi$$
0.724999 + 0.688749i $$0.241840\pi$$
$$762$$ 0 0
$$763$$ 4.00000i 0.144810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.00000i 0.288863i
$$768$$ 0 0
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ 0 0
$$773$$ − 48.0000i − 1.72644i −0.504828 0.863220i $$-0.668444\pi$$
0.504828 0.863220i $$-0.331556\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 20.0000i − 0.717496i
$$778$$ 0 0
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 50.0000i 1.78231i 0.453701 + 0.891154i $$0.350103\pi$$
−0.453701 + 0.891154i $$0.649897\pi$$
$$788$$ 0 0
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ − 14.0000i − 0.497155i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 26.0000i 0.920967i 0.887668 + 0.460484i $$0.152324\pi$$
−0.887668 + 0.460484i $$0.847676\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ −4.00000 −0.141333
$$802$$ 0 0
$$803$$ 40.0000i 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 14.0000i − 0.492823i
$$808$$ 0 0
$$809$$ −38.0000 −1.33601 −0.668004 0.744157i $$-0.732851\pi$$
−0.668004 + 0.744157i $$0.732851\pi$$
$$810$$ 0 0
$$811$$ 6.00000 0.210688 0.105344 0.994436i $$-0.466406\pi$$
0.105344 + 0.994436i $$0.466406\pi$$
$$812$$ 0 0
$$813$$ 10.0000i 0.350715i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000i 0.279885i
$$818$$ 0 0
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ 0 0
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.0000i 0.695468i 0.937593 + 0.347734i $$0.113049\pi$$
−0.937593 + 0.347734i $$0.886951\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 0 0
$$833$$ − 6.00000i − 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 10.0000i − 0.345651i
$$838$$ 0 0
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 8.00000i 0.275535i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000i 0.343604i
$$848$$ 0 0
$$849$$ −16.0000 −0.549119
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 0 0
$$861$$ 16.0000 0.545279
$$862$$ 0 0
$$863$$ 4.00000i 0.136162i 0.997680 + 0.0680808i $$0.0216876\pi$$
−0.997680 + 0.0680808i $$0.978312\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ 0 0
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 10.0000i − 0.337676i −0.985644 0.168838i $$-0.945999\pi$$
0.985644 0.168838i $$-0.0540015\pi$$
$$878$$ 0 0
$$879$$ 8.00000 0.269833
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ − 32.0000i − 1.07689i −0.842662 0.538443i $$-0.819013\pi$$
0.842662 0.538443i $$-0.180987\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 0 0
$$889$$ 24.0000 0.804934
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 8.00000i 0.267710i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −60.0000 −2.00111
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 56.0000i 1.85945i 0.368255 + 0.929725i $$0.379955\pi$$
−0.368255 + 0.929725i $$0.620045\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 8.00000i 0.264183i
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 0 0
$$921$$ −22.0000 −0.724925
$$922$$ 0 0
$$923$$ 16.0000i 0.526646i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000i 0.262754i
$$928$$ 0 0
$$929$$ −16.0000 −0.524943 −0.262471 0.964940i $$-0.584538\pi$$
−0.262471 + 0.964940i $$0.584538\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34.0000i 1.11073i 0.831606 + 0.555366i $$0.187422\pi$$
−0.831606 + 0.555366i $$0.812578\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 8.00000 0.260793 0.130396 0.991462i $$-0.458375\pi$$
0.130396 + 0.991462i $$0.458375\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 0 0
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ − 42.0000i − 1.36051i −0.732974 0.680257i $$-0.761868\pi$$
0.732974 0.680257i $$-0.238132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 24.0000i 0.775810i
$$958$$ 0 0
$$959$$ 16.0000 0.516667
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 22.0000i − 0.707472i −0.935345 0.353736i $$-0.884911\pi$$
0.935345 0.353736i $$-0.115089\pi$$
$$968$$ 0 0
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 12.0000i 0.383914i 0.981403 + 0.191957i $$0.0614834\pi$$
−0.981403 + 0.191957i $$0.938517\pi$$
$$978$$ 0 0
$$979$$ −16.0000 −0.511362
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 0 0
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 8.00000i 0.254643i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −36.0000 −1.14358 −0.571789 0.820401i $$-0.693750\pi$$
−0.571789 + 0.820401i $$0.693750\pi$$
$$992$$ 0 0
$$993$$ − 2.00000i − 0.0634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 22.0000i − 0.696747i −0.937356 0.348373i $$-0.886734\pi$$
0.937356 0.348373i $$-0.113266\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 3900.2.h.b.1249.1 2
5.2 odd 4 156.2.a.a.1.1 1
5.3 odd 4 3900.2.a.m.1.1 1
5.4 even 2 inner 3900.2.h.b.1249.2 2
15.2 even 4 468.2.a.d.1.1 1
20.7 even 4 624.2.a.e.1.1 1
35.27 even 4 7644.2.a.k.1.1 1
40.27 even 4 2496.2.a.o.1.1 1
40.37 odd 4 2496.2.a.bc.1.1 1
45.2 even 12 4212.2.i.b.1405.1 2
45.7 odd 12 4212.2.i.l.1405.1 2
45.22 odd 12 4212.2.i.l.2809.1 2
45.32 even 12 4212.2.i.b.2809.1 2
60.47 odd 4 1872.2.a.s.1.1 1
65.2 even 12 2028.2.q.h.1837.2 4
65.7 even 12 2028.2.q.h.361.1 4
65.12 odd 4 2028.2.a.c.1.1 1
65.17 odd 12 2028.2.i.g.2005.1 2
65.22 odd 12 2028.2.i.e.2005.1 2
65.32 even 12 2028.2.q.h.361.2 4
65.37 even 12 2028.2.q.h.1837.1 4
65.42 odd 12 2028.2.i.e.529.1 2
65.47 even 4 2028.2.b.a.337.1 2
65.57 even 4 2028.2.b.a.337.2 2
65.62 odd 12 2028.2.i.g.529.1 2
120.77 even 4 7488.2.a.c.1.1 1
120.107 odd 4 7488.2.a.d.1.1 1
195.47 odd 4 6084.2.b.j.4393.2 2
195.77 even 4 6084.2.a.b.1.1 1
195.122 odd 4 6084.2.b.j.4393.1 2
260.207 even 4 8112.2.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.a.1.1 1 5.2 odd 4
468.2.a.d.1.1 1 15.2 even 4
624.2.a.e.1.1 1 20.7 even 4
1872.2.a.s.1.1 1 60.47 odd 4
2028.2.a.c.1.1 1 65.12 odd 4
2028.2.b.a.337.1 2 65.47 even 4
2028.2.b.a.337.2 2 65.57 even 4
2028.2.i.e.529.1 2 65.42 odd 12
2028.2.i.e.2005.1 2 65.22 odd 12
2028.2.i.g.529.1 2 65.62 odd 12
2028.2.i.g.2005.1 2 65.17 odd 12
2028.2.q.h.361.1 4 65.7 even 12
2028.2.q.h.361.2 4 65.32 even 12
2028.2.q.h.1837.1 4 65.37 even 12
2028.2.q.h.1837.2 4 65.2 even 12
2496.2.a.o.1.1 1 40.27 even 4
2496.2.a.bc.1.1 1 40.37 odd 4
3900.2.a.m.1.1 1 5.3 odd 4
3900.2.h.b.1249.1 2 1.1 even 1 trivial
3900.2.h.b.1249.2 2 5.4 even 2 inner
4212.2.i.b.1405.1 2 45.2 even 12
4212.2.i.b.2809.1 2 45.32 even 12
4212.2.i.l.1405.1 2 45.7 odd 12
4212.2.i.l.2809.1 2 45.22 odd 12
6084.2.a.b.1.1 1 195.77 even 4
6084.2.b.j.4393.1 2 195.122 odd 4
6084.2.b.j.4393.2 2 195.47 odd 4
7488.2.a.c.1.1 1 120.77 even 4
7488.2.a.d.1.1 1 120.107 odd 4
7644.2.a.k.1.1 1 35.27 even 4
8112.2.a.bi.1.1 1 260.207 even 4