# Properties

 Label 3528.2.s.ba.3313.1 Level $3528$ Weight $2$ Character 3528.3313 Analytic conductor $28.171$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3528.s (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.1712218331$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## Embedding invariants

 Embedding label 3313.1 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 3528.3313 Dual form 3528.2.s.ba.361.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(2.00000 - 3.46410i) q^{5} +O(q^{10})$$ $$q+(2.00000 - 3.46410i) q^{5} +(1.00000 + 1.73205i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(4.00000 - 6.92820i) q^{23} +(-5.50000 - 9.52628i) q^{25} -2.00000 q^{29} +(2.00000 + 3.46410i) q^{31} +(3.00000 - 5.19615i) q^{37} -2.00000 q^{41} +8.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(-5.00000 - 8.66025i) q^{53} +(-3.00000 - 5.19615i) q^{59} +(2.00000 - 3.46410i) q^{61} +(6.00000 + 10.3923i) q^{67} +(-7.00000 - 12.1244i) q^{73} +(4.00000 - 6.92820i) q^{79} +6.00000 q^{83} +8.00000 q^{85} +(-5.00000 + 8.66025i) q^{89} +(4.00000 + 6.92820i) q^{95} +2.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 2q^{17} - 2q^{19} + 8q^{23} - 11q^{25} - 4q^{29} + 4q^{31} + 6q^{37} - 4q^{41} + 16q^{43} + 4q^{47} - 10q^{53} - 6q^{59} + 4q^{61} + 12q^{67} - 14q^{73} + 8q^{79} + 12q^{83} + 16q^{85} - 10q^{89} + 8q^{95} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$1765$$ $$2647$$ $$\chi(n)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.00000 3.46410i 0.894427 1.54919i 0.0599153 0.998203i $$-0.480917\pi$$
0.834512 0.550990i $$-0.185750\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 0.275029i $$-0.0886875\pi$$
−0.718900 + 0.695113i $$0.755354\pi$$
$$18$$ 0 0
$$19$$ −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 0.287984i $$-0.907015\pi$$
0.728219 + 0.685344i $$0.240348\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 6.92820i 0.834058 1.44463i −0.0607377 0.998154i $$-0.519345\pi$$
0.894795 0.446476i $$-0.147321\pi$$
$$24$$ 0 0
$$25$$ −5.50000 9.52628i −1.10000 1.90526i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i $$-0.0497126\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 5.19615i 0.493197 0.854242i −0.506772 0.862080i $$-0.669162\pi$$
0.999969 + 0.00783774i $$0.00249486\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000 3.46410i 0.291730 0.505291i −0.682489 0.730896i $$-0.739102\pi$$
0.974219 + 0.225605i $$0.0724358\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −5.00000 8.66025i −0.686803 1.18958i −0.972867 0.231367i $$-0.925680\pi$$
0.286064 0.958211i $$-0.407653\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i $$-0.294388\pi$$
−0.992524 + 0.122047i $$0.961054\pi$$
$$60$$ 0 0
$$61$$ 2.00000 3.46410i 0.256074 0.443533i −0.709113 0.705095i $$-0.750904\pi$$
0.965187 + 0.261562i $$0.0842377\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.00000 + 10.3923i 0.733017 + 1.26962i 0.955588 + 0.294706i $$0.0952216\pi$$
−0.222571 + 0.974916i $$0.571445\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −7.00000 12.1244i −0.819288 1.41905i −0.906208 0.422833i $$-0.861036\pi$$
0.0869195 0.996215i $$-0.472298\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i $$-0.684745\pi$$
0.998388 + 0.0567635i $$0.0180781\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −5.00000 + 8.66025i −0.529999 + 0.917985i 0.469389 + 0.882992i $$0.344474\pi$$
−0.999388 + 0.0349934i $$0.988859\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.00000 + 6.92820i 0.410391 + 0.710819i
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i $$-0.963017\pi$$
0.396236 0.918149i $$-0.370316\pi$$
$$102$$ 0 0
$$103$$ −6.00000 + 10.3923i −0.591198 + 1.02398i 0.402874 + 0.915255i $$0.368011\pi$$
−0.994071 + 0.108729i $$0.965322\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −6.00000 + 10.3923i −0.580042 + 1.00466i 0.415432 + 0.909624i $$0.363630\pi$$
−0.995474 + 0.0950377i $$0.969703\pi$$
$$108$$ 0 0
$$109$$ −5.00000 8.66025i −0.478913 0.829502i 0.520794 0.853682i $$-0.325636\pi$$
−0.999708 + 0.0241802i $$0.992302\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −16.0000 27.7128i −1.49201 2.58423i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.50000 9.52628i 0.500000 0.866025i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −7.00000 + 12.1244i −0.611593 + 1.05931i 0.379379 + 0.925241i $$0.376138\pi$$
−0.990972 + 0.134069i $$0.957196\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1.00000 + 1.73205i 0.0854358 + 0.147979i 0.905577 0.424182i $$-0.139438\pi$$
−0.820141 + 0.572161i $$0.806105\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.00000 + 6.92820i −0.332182 + 0.575356i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.00000 + 1.73205i −0.0819232 + 0.141895i −0.904076 0.427372i $$-0.859440\pi$$
0.822153 + 0.569267i $$0.192773\pi$$
$$150$$ 0 0
$$151$$ −8.00000 13.8564i −0.651031 1.12762i −0.982873 0.184284i $$-0.941004\pi$$
0.331842 0.943335i $$-0.392330\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.0000 1.28515
$$156$$ 0 0
$$157$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −8.00000 + 13.8564i −0.626608 + 1.08532i 0.361619 + 0.932326i $$0.382224\pi$$
−0.988227 + 0.152992i $$0.951109\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −4.00000 + 6.92820i −0.304114 + 0.526742i −0.977064 0.212947i $$-0.931694\pi$$
0.672949 + 0.739689i $$0.265027\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2.00000 3.46410i −0.149487 0.258919i 0.781551 0.623841i $$-0.214429\pi$$
−0.931038 + 0.364922i $$0.881096\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −12.0000 20.7846i −0.882258 1.52811i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −4.00000 + 6.92820i −0.289430 + 0.501307i −0.973674 0.227946i $$-0.926799\pi$$
0.684244 + 0.729253i $$0.260132\pi$$
$$192$$ 0 0
$$193$$ 9.00000 + 15.5885i 0.647834 + 1.12208i 0.983639 + 0.180150i $$0.0576584\pi$$
−0.335805 + 0.941932i $$0.609008\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ −2.00000 3.46410i −0.141776 0.245564i 0.786389 0.617731i $$-0.211948\pi$$
−0.928166 + 0.372168i $$0.878615\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −4.00000 + 6.92820i −0.279372 + 0.483887i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 16.0000 27.7128i 1.09119 1.89000i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −7.00000 12.1244i −0.464606 0.804722i 0.534577 0.845120i $$-0.320471\pi$$
−0.999184 + 0.0403978i $$0.987137\pi$$
$$228$$ 0 0
$$229$$ −8.00000 + 13.8564i −0.528655 + 0.915657i 0.470787 + 0.882247i $$0.343970\pi$$
−0.999442 + 0.0334101i $$0.989363\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 13.0000 22.5167i 0.851658 1.47512i −0.0280525 0.999606i $$-0.508931\pi$$
0.879711 0.475509i $$-0.157736\pi$$
$$234$$ 0 0
$$235$$ −8.00000 13.8564i −0.521862 0.903892i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −1.00000 1.73205i −0.0644157 0.111571i 0.832019 0.554747i $$-0.187185\pi$$
−0.896435 + 0.443176i $$0.853852\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.0000 −0.883672 −0.441836 0.897096i $$-0.645673\pi$$
−0.441836 + 0.897096i $$0.645673\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.00000 + 15.5885i −0.561405 + 0.972381i 0.435970 + 0.899961i $$0.356405\pi$$
−0.997374 + 0.0724199i $$0.976928\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$264$$ 0 0
$$265$$ −40.0000 −2.45718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 12.0000 + 20.7846i 0.731653 + 1.26726i 0.956176 + 0.292791i $$0.0945841\pi$$
−0.224523 + 0.974469i $$0.572083\pi$$
$$270$$ 0 0
$$271$$ 16.0000 27.7128i 0.971931 1.68343i 0.282218 0.959350i $$-0.408930\pi$$
0.689713 0.724083i $$-0.257737\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i $$-0.936830\pi$$
0.319447 0.947604i $$-0.396503\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −5.00000 8.66025i −0.297219 0.514799i 0.678280 0.734804i $$-0.262726\pi$$
−0.975499 + 0.220005i $$0.929393\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 6.50000 11.2583i 0.382353 0.662255i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −8.00000 13.8564i −0.458079 0.793416i
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.0000 + 20.7846i 0.680458 + 1.17859i 0.974841 + 0.222900i $$0.0715523\pi$$
−0.294384 + 0.955687i $$0.595114\pi$$
$$312$$ 0 0
$$313$$ 7.00000 12.1244i 0.395663 0.685309i −0.597522 0.801852i $$-0.703848\pi$$
0.993186 + 0.116543i $$0.0371814\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.00000 5.19615i 0.168497 0.291845i −0.769395 0.638774i $$-0.779442\pi$$
0.937892 + 0.346929i $$0.112775\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 + 6.92820i −0.219860 + 0.380808i −0.954765 0.297361i $$-0.903893\pi$$
0.734905 + 0.678170i $$0.237227\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 48.0000 2.62252
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i $$-0.943860\pi$$
0.340293 0.940319i $$-0.389474\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 15.0000 + 25.9808i 0.798369 + 1.38282i 0.920677 + 0.390324i $$0.127637\pi$$
−0.122308 + 0.992492i $$0.539030\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$360$$ 0 0
$$361$$ 7.50000 + 12.9904i 0.394737 + 0.683704i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −56.0000 −2.93117
$$366$$ 0 0
$$367$$ 4.00000 + 6.92820i 0.208798 + 0.361649i 0.951336 0.308155i $$-0.0997115\pi$$
−0.742538 + 0.669804i $$0.766378\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 17.0000 29.4449i 0.880227 1.52460i 0.0291379 0.999575i $$-0.490724\pi$$
0.851089 0.525022i $$-0.175943\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −6.00000 + 10.3923i −0.306586 + 0.531022i −0.977613 0.210411i $$-0.932520\pi$$
0.671027 + 0.741433i $$0.265853\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 5.00000 + 8.66025i 0.253510 + 0.439092i 0.964490 0.264120i $$-0.0850816\pi$$
−0.710980 + 0.703213i $$0.751748\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −16.0000 27.7128i −0.805047 1.39438i
$$396$$ 0 0
$$397$$ −4.00000 + 6.92820i −0.200754 + 0.347717i −0.948772 0.315963i $$-0.897673\pi$$
0.748017 + 0.663679i $$0.231006\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15.0000 25.9808i 0.749064 1.29742i −0.199207 0.979957i $$-0.563837\pi$$
0.948272 0.317460i $$-0.102830\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 3.00000 + 5.19615i 0.148340 + 0.256933i 0.930614 0.366002i $$-0.119274\pi$$
−0.782274 + 0.622935i $$0.785940\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 20.7846i 0.589057 1.02028i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 11.0000 19.0526i 0.533578 0.924185i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 8.00000 + 13.8564i 0.382692 + 0.662842i
$$438$$ 0 0
$$439$$ 12.0000 20.7846i 0.572729 0.991995i −0.423556 0.905870i $$-0.639218\pi$$
0.996284 0.0861252i $$-0.0274485\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −2.00000 + 3.46410i −0.0950229 + 0.164584i −0.909618 0.415445i $$-0.863626\pi$$
0.814595 + 0.580030i $$0.196959\pi$$
$$444$$ 0 0
$$445$$ 20.0000 + 34.6410i 0.948091 + 1.64214i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 + 19.0526i −0.514558 + 0.891241i 0.485299 + 0.874348i $$0.338711\pi$$
−0.999857 + 0.0168929i $$0.994623\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3.00000 + 5.19615i −0.138823 + 0.240449i −0.927052 0.374934i $$-0.877665\pi$$
0.788228 + 0.615383i $$0.210999\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 22.0000 1.00943
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −2.00000 3.46410i −0.0913823 0.158279i 0.816711 0.577047i $$-0.195795\pi$$
−0.908093 + 0.418769i $$0.862462\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.00000 6.92820i 0.181631 0.314594i
$$486$$ 0 0
$$487$$ 4.00000 + 6.92820i 0.181257 + 0.313947i 0.942309 0.334744i $$-0.108650\pi$$
−0.761052 + 0.648691i $$0.775317\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ −2.00000 3.46410i −0.0900755 0.156015i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.00000 + 3.46410i −0.0895323 + 0.155074i −0.907314 0.420455i $$-0.861871\pi$$
0.817781 + 0.575529i $$0.195204\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ −48.0000 −2.13597
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 12.0000 20.7846i 0.531891 0.921262i −0.467416 0.884037i $$-0.654815\pi$$
0.999307 0.0372243i $$-0.0118516\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 24.0000 + 41.5692i 1.05757 + 1.83176i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 9.00000 + 15.5885i 0.394297 + 0.682943i 0.993011 0.118020i $$-0.0376547\pi$$
−0.598714 + 0.800963i $$0.704321\pi$$
$$522$$ 0 0
$$523$$ −17.0000 + 29.4449i −0.743358 + 1.28753i 0.207600 + 0.978214i $$0.433435\pi$$
−0.950958 + 0.309320i $$0.899899\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.00000 + 6.92820i −0.174243 + 0.301797i
$$528$$ 0 0
$$529$$ −20.5000 35.5070i −0.891304 1.54378i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 24.0000 + 41.5692i 1.03761 + 1.79719i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −11.0000 + 19.0526i −0.472927 + 0.819133i −0.999520 0.0309841i $$-0.990136\pi$$
0.526593 + 0.850118i $$0.323469\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −40.0000 −1.71341
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.00000 3.46410i 0.0852029 0.147576i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −13.0000 22.5167i −0.550828 0.954062i −0.998215 0.0597213i $$-0.980979\pi$$
0.447387 0.894340i $$-0.352355\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −17.0000 29.4449i −0.716465 1.24095i −0.962392 0.271665i $$-0.912426\pi$$
0.245927 0.969288i $$-0.420908\pi$$
$$564$$ 0 0
$$565$$ −12.0000 + 20.7846i −0.504844 + 0.874415i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −21.0000 + 36.3731i −0.880366 + 1.52484i −0.0294311 + 0.999567i $$0.509370\pi$$
−0.850935 + 0.525271i $$0.823964\pi$$
$$570$$ 0 0
$$571$$ 8.00000 + 13.8564i 0.334790 + 0.579873i 0.983444 0.181210i $$-0.0580014\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −88.0000 −3.66985
$$576$$ 0 0
$$577$$ 9.00000 + 15.5885i 0.374675 + 0.648956i 0.990278 0.139100i $$-0.0444210\pi$$
−0.615603 + 0.788056i $$0.711088\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 10.0000 0.412744 0.206372 0.978474i $$-0.433834\pi$$
0.206372 + 0.978474i $$0.433834\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −21.0000 + 36.3731i −0.862367 + 1.49366i 0.00727173 + 0.999974i $$0.497685\pi$$
−0.869638 + 0.493689i $$0.835648\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 20.0000 + 34.6410i 0.817178 + 1.41539i 0.907754 + 0.419504i $$0.137796\pi$$
−0.0905757 + 0.995890i $$0.528871\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −22.0000 38.1051i −0.894427 1.54919i
$$606$$ 0 0
$$607$$ −16.0000 + 27.7128i −0.649420 + 1.12483i 0.333842 + 0.942629i $$0.391655\pi$$
−0.983262 + 0.182199i $$0.941678\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −5.00000 8.66025i −0.201948 0.349784i 0.747208 0.664590i $$-0.231394\pi$$
−0.949156 + 0.314806i $$0.898061\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 3.00000 + 5.19615i 0.120580 + 0.208851i 0.919997 0.391926i $$-0.128191\pi$$
−0.799416 + 0.600777i $$0.794858\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −20.5000 + 35.5070i −0.820000 + 1.42028i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16.0000 27.7128i 0.634941 1.09975i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −1.00000 1.73205i −0.0394976 0.0684119i 0.845601 0.533816i $$-0.179242\pi$$
−0.885098 + 0.465404i $$0.845909\pi$$
$$642$$ 0 0
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 18.0000 + 31.1769i 0.707653 + 1.22569i 0.965726 + 0.259565i $$0.0835793\pi$$
−0.258073 + 0.966126i $$0.583087\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.0000 + 19.0526i −0.430463 + 0.745584i −0.996913 0.0785119i $$-0.974983\pi$$
0.566450 + 0.824096i $$0.308316\pi$$
$$654$$ 0 0
$$655$$ 28.0000 + 48.4974i 1.09405 + 1.89495i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 40.0000 1.55818 0.779089 0.626913i $$-0.215682\pi$$
0.779089 + 0.626913i $$0.215682\pi$$
$$660$$ 0 0
$$661$$ 10.0000 + 17.3205i 0.388955 + 0.673690i 0.992309 0.123784i $$-0.0395028\pi$$
−0.603354 + 0.797473i $$0.706170\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.00000 + 13.8564i −0.309761 + 0.536522i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 12.0000 20.7846i 0.461197 0.798817i −0.537823 0.843057i $$-0.680753\pi$$
0.999021 + 0.0442400i $$0.0140866\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 6.00000 + 10.3923i 0.229584 + 0.397650i 0.957685 0.287819i $$-0.0929302\pi$$
−0.728101 + 0.685470i $$0.759597\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 7.00000 12.1244i 0.266293 0.461232i −0.701609 0.712562i $$-0.747535\pi$$
0.967901 + 0.251330i $$0.0808679\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −36.0000 + 62.3538i −1.36556 + 2.36522i
$$696$$ 0 0
$$697$$ −2.00000 3.46410i −0.0757554 0.131212i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ 0 0
$$703$$ 6.00000 + 10.3923i 0.226294 + 0.391953i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −5.00000 + 8.66025i −0.187779 + 0.325243i −0.944509 0.328484i $$-0.893462\pi$$
0.756730 + 0.653727i $$0.226796\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −18.0000 + 31.1769i −0.671287 + 1.16270i 0.306253 + 0.951950i $$0.400925\pi$$
−0.977539 + 0.210752i $$0.932409\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 11.0000 + 19.0526i 0.408530 + 0.707594i
$$726$$ 0 0
$$727$$ −20.0000 −0.741759 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 + 13.8564i 0.295891 + 0.512498i
$$732$$ 0 0
$$733$$ 26.0000 45.0333i 0.960332 1.66334i 0.238667 0.971102i $$-0.423290\pi$$
0.721665 0.692242i $$-0.243377\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −8.00000 13.8564i −0.294285 0.509716i 0.680534 0.732717i $$-0.261748\pi$$
−0.974818 + 0.223001i $$0.928415\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ 4.00000 + 6.92820i 0.146549 + 0.253830i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 + 27.7128i −0.583848 + 1.01125i 0.411170 + 0.911559i $$0.365120\pi$$
−0.995018 + 0.0996961i $$0.968213\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −64.0000 −2.32920
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i $$-0.557920\pi$$
0.942207 0.335032i $$-0.108747\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −2.00000 3.46410i −0.0719350 0.124595i 0.827814 0.561002i $$-0.189584\pi$$
−0.899749 + 0.436407i $$0.856251\pi$$
$$774$$ 0 0
$$775$$ 22.0000 38.1051i 0.790263 1.36878i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.00000 3.46410i 0.0716574 0.124114i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 11.0000 + 19.0526i 0.392108 + 0.679150i 0.992727 0.120384i $$-0.0384127\pi$$
−0.600620 + 0.799535i $$0.705079\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −24.0000 −0.850124 −0.425062 0.905164i $$-0.639748\pi$$
−0.425062 + 0.905164i $$0.639748\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$