# Properties

 Label 540.3.b.a.269.1 Level $540$ Weight $3$ Character 540.269 Analytic conductor $14.714$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 540.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7139342755$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-11}, \sqrt{-19})$$ Defining polynomial: $$x^{4} + 15x^{2} + 4$$ x^4 + 15*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 269.1 Root $$-0.521137i$$ of defining polynomial Character $$\chi$$ $$=$$ 540.269 Dual form 540.3.b.a.269.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-4.36421 - 2.44002i) q^{5} +2.79549i q^{7} +O(q^{10})$$ $$q+(-4.36421 - 2.44002i) q^{5} +2.79549i q^{7} +18.1465i q^{11} -23.0266i q^{13} +5.72842 q^{17} +23.1852 q^{19} +0.271584 q^{23} +(13.0926 + 21.2975i) q^{25} -39.7995i q^{29} +47.3705 q^{31} +(6.82104 - 12.2001i) q^{35} -34.8712i q^{37} -13.2665i q^{41} -46.7158i q^{43} -40.9137 q^{47} +41.1852 q^{49} +91.3705 q^{53} +(44.2779 - 79.1953i) q^{55} +78.8398i q^{59} +31.1852 q^{61} +(-56.1852 + 100.493i) q^{65} -6.91631i q^{67} -81.5870i q^{71} +106.084i q^{73} -50.7284 q^{77} +63.5557 q^{79} -0.284161 q^{83} +(-25.0000 - 13.9774i) q^{85} -28.5210i q^{89} +64.3705 q^{91} +(-101.185 - 56.5724i) q^{95} -92.9138i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{5}+O(q^{10})$$ 4 * q - 3 * q^5 $$4 q - 3 q^{5} - 6 q^{17} + 6 q^{19} + 30 q^{23} + 9 q^{25} + 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} + 192 q^{53} + 47 q^{55} + 38 q^{61} - 138 q^{65} - 174 q^{77} - 6 q^{79} + 288 q^{83} - 100 q^{85} + 84 q^{91} - 318 q^{95}+O(q^{100})$$ 4 * q - 3 * q^5 - 6 * q^17 + 6 * q^19 + 30 * q^23 + 9 * q^25 + 16 * q^31 - 45 * q^35 - 48 * q^47 + 78 * q^49 + 192 * q^53 + 47 * q^55 + 38 * q^61 - 138 * q^65 - 174 * q^77 - 6 * q^79 + 288 * q^83 - 100 * q^85 + 84 * q^91 - 318 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$461$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −4.36421 2.44002i −0.872842 0.488004i
$$6$$ 0 0
$$7$$ 2.79549i 0.399355i 0.979862 + 0.199678i $$0.0639895\pi$$
−0.979862 + 0.199678i $$0.936010\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 18.1465i 1.64969i 0.565363 + 0.824843i $$0.308736\pi$$
−0.565363 + 0.824843i $$0.691264\pi$$
$$12$$ 0 0
$$13$$ 23.0266i 1.77127i −0.464378 0.885637i $$-0.653722\pi$$
0.464378 0.885637i $$-0.346278\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.72842 0.336966 0.168483 0.985705i $$-0.446113\pi$$
0.168483 + 0.985705i $$0.446113\pi$$
$$18$$ 0 0
$$19$$ 23.1852 1.22028 0.610138 0.792295i $$-0.291114\pi$$
0.610138 + 0.792295i $$0.291114\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.271584 0.0118080 0.00590400 0.999983i $$-0.498121\pi$$
0.00590400 + 0.999983i $$0.498121\pi$$
$$24$$ 0 0
$$25$$ 13.0926 + 21.2975i 0.523705 + 0.851900i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 39.7995i 1.37240i −0.727415 0.686198i $$-0.759278\pi$$
0.727415 0.686198i $$-0.240722\pi$$
$$30$$ 0 0
$$31$$ 47.3705 1.52808 0.764040 0.645169i $$-0.223213\pi$$
0.764040 + 0.645169i $$0.223213\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 6.82104 12.2001i 0.194887 0.348574i
$$36$$ 0 0
$$37$$ 34.8712i 0.942465i −0.882009 0.471232i $$-0.843809\pi$$
0.882009 0.471232i $$-0.156191\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 13.2665i 0.323573i −0.986826 0.161787i $$-0.948274\pi$$
0.986826 0.161787i $$-0.0517256\pi$$
$$42$$ 0 0
$$43$$ 46.7158i 1.08641i −0.839599 0.543207i $$-0.817210\pi$$
0.839599 0.543207i $$-0.182790\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −40.9137 −0.870504 −0.435252 0.900309i $$-0.643341\pi$$
−0.435252 + 0.900309i $$0.643341\pi$$
$$48$$ 0 0
$$49$$ 41.1852 0.840515
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 91.3705 1.72397 0.861986 0.506932i $$-0.169221\pi$$
0.861986 + 0.506932i $$0.169221\pi$$
$$54$$ 0 0
$$55$$ 44.2779 79.1953i 0.805052 1.43991i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 78.8398i 1.33627i 0.744041 + 0.668134i $$0.232907\pi$$
−0.744041 + 0.668134i $$0.767093\pi$$
$$60$$ 0 0
$$61$$ 31.1852 0.511234 0.255617 0.966778i $$-0.417721\pi$$
0.255617 + 0.966778i $$0.417721\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −56.1852 + 100.493i −0.864388 + 1.54604i
$$66$$ 0 0
$$67$$ 6.91631i 0.103229i −0.998667 0.0516143i $$-0.983563\pi$$
0.998667 0.0516143i $$-0.0164366\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 81.5870i 1.14911i −0.818465 0.574556i $$-0.805175\pi$$
0.818465 0.574556i $$-0.194825\pi$$
$$72$$ 0 0
$$73$$ 106.084i 1.45320i 0.687060 + 0.726601i $$0.258901\pi$$
−0.687060 + 0.726601i $$0.741099\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −50.7284 −0.658811
$$78$$ 0 0
$$79$$ 63.5557 0.804503 0.402252 0.915529i $$-0.368228\pi$$
0.402252 + 0.915529i $$0.368228\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −0.284161 −0.00342363 −0.00171182 0.999999i $$-0.500545\pi$$
−0.00171182 + 0.999999i $$0.500545\pi$$
$$84$$ 0 0
$$85$$ −25.0000 13.9774i −0.294118 0.164440i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 28.5210i 0.320461i −0.987080 0.160230i $$-0.948776\pi$$
0.987080 0.160230i $$-0.0512237\pi$$
$$90$$ 0 0
$$91$$ 64.3705 0.707368
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −101.185 56.5724i −1.06511 0.595499i
$$96$$ 0 0
$$97$$ 92.9138i 0.957874i −0.877849 0.478937i $$-0.841022\pi$$
0.877849 0.478937i $$-0.158978\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 46.6675i 0.462055i 0.972947 + 0.231027i $$0.0742087\pi$$
−0.972947 + 0.231027i $$0.925791\pi$$
$$102$$ 0 0
$$103$$ 11.1820i 0.108563i 0.998526 + 0.0542813i $$0.0172868\pi$$
−0.998526 + 0.0542813i $$0.982713\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 147.297 1.37661 0.688303 0.725424i $$-0.258356\pi$$
0.688303 + 0.725424i $$0.258356\pi$$
$$108$$ 0 0
$$109$$ −121.556 −1.11519 −0.557595 0.830113i $$-0.688276\pi$$
−0.557595 + 0.830113i $$0.688276\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 97.6295 0.863978 0.431989 0.901879i $$-0.357812\pi$$
0.431989 + 0.901879i $$0.357812\pi$$
$$114$$ 0 0
$$115$$ −1.18525 0.662669i −0.0103065 0.00576234i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 16.0137i 0.134569i
$$120$$ 0 0
$$121$$ −208.297 −1.72146
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −5.17267 124.893i −0.0413814 0.999143i
$$126$$ 0 0
$$127$$ 88.6481i 0.698017i 0.937120 + 0.349008i $$0.113482\pi$$
−0.937120 + 0.349008i $$0.886518\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 89.9735i 0.686820i 0.939186 + 0.343410i $$0.111582\pi$$
−0.939186 + 0.343410i $$0.888418\pi$$
$$132$$ 0 0
$$133$$ 64.8141i 0.487324i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −229.926 −1.67829 −0.839147 0.543905i $$-0.816945\pi$$
−0.839147 + 0.543905i $$0.816945\pi$$
$$138$$ 0 0
$$139$$ 57.6295 0.414601 0.207300 0.978277i $$-0.433532\pi$$
0.207300 + 0.978277i $$0.433532\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 417.852 2.92205
$$144$$ 0 0
$$145$$ −97.1115 + 173.693i −0.669734 + 1.19788i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 11.1337i 0.0747227i −0.999302 0.0373613i $$-0.988105\pi$$
0.999302 0.0373613i $$-0.0118953\pi$$
$$150$$ 0 0
$$151$$ −22.1852 −0.146922 −0.0734611 0.997298i $$-0.523404\pi$$
−0.0734611 + 0.997298i $$0.523404\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −206.735 115.585i −1.33377 0.745709i
$$156$$ 0 0
$$157$$ 225.337i 1.43527i 0.696419 + 0.717635i $$0.254775\pi$$
−0.696419 + 0.717635i $$0.745225\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0.759209i 0.00471559i
$$162$$ 0 0
$$163$$ 302.948i 1.85858i −0.369352 0.929290i $$-0.620420\pi$$
0.369352 0.929290i $$-0.379580\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −238.667 −1.42915 −0.714573 0.699561i $$-0.753379\pi$$
−0.714573 + 0.699561i $$0.753379\pi$$
$$168$$ 0 0
$$169$$ −361.223 −2.13741
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 108.802 0.628914 0.314457 0.949272i $$-0.398178\pi$$
0.314457 + 0.949272i $$0.398178\pi$$
$$174$$ 0 0
$$175$$ −59.5369 + 36.6003i −0.340211 + 0.209144i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 168.054i 0.938849i −0.882973 0.469425i $$-0.844461\pi$$
0.882973 0.469425i $$-0.155539\pi$$
$$180$$ 0 0
$$181$$ −154.297 −0.852468 −0.426234 0.904613i $$-0.640160\pi$$
−0.426234 + 0.904613i $$0.640160\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −85.0863 + 152.185i −0.459926 + 0.822622i
$$186$$ 0 0
$$187$$ 103.951i 0.555887i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 154.932i 0.811164i 0.914059 + 0.405582i $$0.132931\pi$$
−0.914059 + 0.405582i $$0.867069\pi$$
$$192$$ 0 0
$$193$$ 64.0066i 0.331640i 0.986156 + 0.165820i $$0.0530271\pi$$
−0.986156 + 0.165820i $$0.946973\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 9.56832 0.0485702 0.0242851 0.999705i $$-0.492269\pi$$
0.0242851 + 0.999705i $$0.492269\pi$$
$$198$$ 0 0
$$199$$ −117.815 −0.592034 −0.296017 0.955183i $$-0.595658\pi$$
−0.296017 + 0.955183i $$0.595658\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 111.259 0.548074
$$204$$ 0 0
$$205$$ −32.3705 + 57.8978i −0.157905 + 0.282428i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 420.732i 2.01307i
$$210$$ 0 0
$$211$$ −48.8148 −0.231350 −0.115675 0.993287i $$-0.536903\pi$$
−0.115675 + 0.993287i $$0.536903\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −113.987 + 203.878i −0.530174 + 0.948268i
$$216$$ 0 0
$$217$$ 132.424i 0.610247i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 131.906i 0.596859i
$$222$$ 0 0
$$223$$ 319.721i 1.43373i 0.697213 + 0.716864i $$0.254423\pi$$
−0.697213 + 0.716864i $$0.745577\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.2716 −0.0804916 −0.0402458 0.999190i $$-0.512814\pi$$
−0.0402458 + 0.999190i $$0.512814\pi$$
$$228$$ 0 0
$$229$$ 134.074 0.585475 0.292737 0.956193i $$-0.405434\pi$$
0.292737 + 0.956193i $$0.405434\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 284.173 1.21963 0.609813 0.792546i $$-0.291245\pi$$
0.609813 + 0.792546i $$0.291245\pi$$
$$234$$ 0 0
$$235$$ 178.556 + 99.8301i 0.759812 + 0.424809i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 206.770i 0.865145i −0.901599 0.432572i $$-0.857606\pi$$
0.901599 0.432572i $$-0.142394\pi$$
$$240$$ 0 0
$$241$$ −162.667 −0.674968 −0.337484 0.941331i $$-0.609576\pi$$
−0.337484 + 0.941331i $$0.609576\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −179.741 100.493i −0.733637 0.410174i
$$246$$ 0 0
$$247$$ 533.877i 2.16144i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 347.387i 1.38401i −0.721893 0.692005i $$-0.756728\pi$$
0.721893 0.692005i $$-0.243272\pi$$
$$252$$ 0 0
$$253$$ 4.92831i 0.0194795i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 21.0377 0.0818589 0.0409294 0.999162i $$-0.486968\pi$$
0.0409294 + 0.999162i $$0.486968\pi$$
$$258$$ 0 0
$$259$$ 97.4820 0.376378
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 215.457 0.819227 0.409614 0.912259i $$-0.365663\pi$$
0.409614 + 0.912259i $$0.365663\pi$$
$$264$$ 0 0
$$265$$ −398.760 222.946i −1.50475 0.841304i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 180.237i 0.670024i −0.942214 0.335012i $$-0.891260\pi$$
0.942214 0.335012i $$-0.108740\pi$$
$$270$$ 0 0
$$271$$ 231.741 0.855133 0.427566 0.903984i $$-0.359371\pi$$
0.427566 + 0.903984i $$0.359371\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −386.476 + 237.586i −1.40537 + 0.863948i
$$276$$ 0 0
$$277$$ 105.566i 0.381104i 0.981677 + 0.190552i $$0.0610278\pi$$
−0.981677 + 0.190552i $$0.938972\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 504.597i 1.79572i −0.440284 0.897859i $$-0.645122\pi$$
0.440284 0.897859i $$-0.354878\pi$$
$$282$$ 0 0
$$283$$ 151.619i 0.535756i 0.963453 + 0.267878i $$0.0863224\pi$$
−0.963453 + 0.267878i $$0.913678\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 37.0863 0.129221
$$288$$ 0 0
$$289$$ −256.185 −0.886454
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 549.038 1.87385 0.936924 0.349532i $$-0.113659\pi$$
0.936924 + 0.349532i $$0.113659\pi$$
$$294$$ 0 0
$$295$$ 192.370 344.073i 0.652103 1.16635i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.25364i 0.0209152i
$$300$$ 0 0
$$301$$ 130.593 0.433865
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −136.099 76.0926i −0.446226 0.249484i
$$306$$ 0 0
$$307$$ 146.401i 0.476876i −0.971158 0.238438i $$-0.923365\pi$$
0.971158 0.238438i $$-0.0766355\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 487.824i 1.56856i 0.620404 + 0.784282i $$0.286969\pi$$
−0.620404 + 0.784282i $$0.713031\pi$$
$$312$$ 0 0
$$313$$ 109.687i 0.350437i −0.984530 0.175218i $$-0.943937\pi$$
0.984530 0.175218i $$-0.0560631\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −363.568 −1.14690 −0.573452 0.819239i $$-0.694396\pi$$
−0.573452 + 0.819239i $$0.694396\pi$$
$$318$$ 0 0
$$319$$ 722.223 2.26402
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 132.815 0.411191
$$324$$ 0 0
$$325$$ 490.408 301.478i 1.50895 0.927625i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 114.374i 0.347640i
$$330$$ 0 0
$$331$$ 300.223 0.907018 0.453509 0.891252i $$-0.350172\pi$$
0.453509 + 0.891252i $$0.350172\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −16.8759 + 30.1842i −0.0503759 + 0.0901022i
$$336$$ 0 0
$$337$$ 373.353i 1.10787i 0.832559 + 0.553937i $$0.186875\pi$$
−0.832559 + 0.553937i $$0.813125\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 859.610i 2.52085i
$$342$$ 0 0
$$343$$ 252.112i 0.735020i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 17.4443 0.0502716 0.0251358 0.999684i $$-0.491998\pi$$
0.0251358 + 0.999684i $$0.491998\pi$$
$$348$$ 0 0
$$349$$ 234.149 0.670915 0.335457 0.942055i $$-0.391109\pi$$
0.335457 + 0.942055i $$0.391109\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 381.284 1.08013 0.540063 0.841625i $$-0.318401\pi$$
0.540063 + 0.841625i $$0.318401\pi$$
$$354$$ 0 0
$$355$$ −199.074 + 356.063i −0.560771 + 1.00299i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 222.024i 0.618451i 0.950989 + 0.309226i $$0.100070\pi$$
−0.950989 + 0.309226i $$0.899930\pi$$
$$360$$ 0 0
$$361$$ 176.556 0.489074
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 258.846 462.971i 0.709168 1.26841i
$$366$$ 0 0
$$367$$ 449.205i 1.22399i −0.790861 0.611995i $$-0.790367\pi$$
0.790861 0.611995i $$-0.209633\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 255.425i 0.688477i
$$372$$ 0 0
$$373$$ 322.082i 0.863492i −0.901995 0.431746i $$-0.857898\pi$$
0.901995 0.431746i $$-0.142102\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −916.446 −2.43089
$$378$$ 0 0
$$379$$ −216.074 −0.570115 −0.285058 0.958510i $$-0.592013\pi$$
−0.285058 + 0.958510i $$0.592013\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −373.977 −0.976440 −0.488220 0.872721i $$-0.662354\pi$$
−0.488220 + 0.872721i $$0.662354\pi$$
$$384$$ 0 0
$$385$$ 221.389 + 123.778i 0.575037 + 0.321502i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 223.108i 0.573543i −0.957999 0.286771i $$-0.907418\pi$$
0.957999 0.286771i $$-0.0925820\pi$$
$$390$$ 0 0
$$391$$ 1.55575 0.00397889
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −277.370 155.077i −0.702204 0.392600i
$$396$$ 0 0
$$397$$ 610.535i 1.53787i −0.639325 0.768936i $$-0.720786\pi$$
0.639325 0.768936i $$-0.279214\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 411.441i 1.02604i −0.858377 0.513019i $$-0.828527\pi$$
0.858377 0.513019i $$-0.171473\pi$$
$$402$$ 0 0
$$403$$ 1090.78i 2.70665i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 632.791 1.55477
$$408$$ 0 0
$$409$$ 590.630 1.44408 0.722041 0.691850i $$-0.243204\pi$$
0.722041 + 0.691850i $$0.243204\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −220.396 −0.533646
$$414$$ 0 0
$$415$$ 1.24014 + 0.693359i 0.00298829 + 0.00167074i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 239.556i 0.571733i 0.958269 + 0.285867i $$0.0922814\pi$$
−0.958269 + 0.285867i $$0.907719\pi$$
$$420$$ 0 0
$$421$$ 593.408 1.40952 0.704760 0.709445i $$-0.251055\pi$$
0.704760 + 0.709445i $$0.251055\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 75.0000 + 122.001i 0.176471 + 0.287061i
$$426$$ 0 0
$$427$$ 87.1780i 0.204164i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 436.566i 1.01291i −0.862265 0.506457i $$-0.830955\pi$$
0.862265 0.506457i $$-0.169045\pi$$
$$432$$ 0 0
$$433$$ 231.736i 0.535187i 0.963532 + 0.267593i $$0.0862284\pi$$
−0.963532 + 0.267593i $$0.913772\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.29674 0.0144090
$$438$$ 0 0
$$439$$ −419.223 −0.954950 −0.477475 0.878645i $$-0.658448\pi$$
−0.477475 + 0.878645i $$0.658448\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 446.520 1.00795 0.503973 0.863720i $$-0.331871\pi$$
0.503973 + 0.863720i $$0.331871\pi$$
$$444$$ 0 0
$$445$$ −69.5918 + 124.472i −0.156386 + 0.279711i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 495.776i 1.10418i 0.833785 + 0.552089i $$0.186169\pi$$
−0.833785 + 0.552089i $$0.813831\pi$$
$$450$$ 0 0
$$451$$ 240.741 0.533794
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −280.926 157.065i −0.617420 0.345198i
$$456$$ 0 0
$$457$$ 683.363i 1.49532i 0.664080 + 0.747662i $$0.268824\pi$$
−0.664080 + 0.747662i $$0.731176\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 400.128i 0.867956i 0.900923 + 0.433978i $$0.142890\pi$$
−0.900923 + 0.433978i $$0.857110\pi$$
$$462$$ 0 0
$$463$$ 316.263i 0.683074i −0.939868 0.341537i $$-0.889053\pi$$
0.939868 0.341537i $$-0.110947\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 419.741 0.898803 0.449401 0.893330i $$-0.351637\pi$$
0.449401 + 0.893330i $$0.351637\pi$$
$$468$$ 0 0
$$469$$ 19.3345 0.0412249
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 847.730 1.79224
$$474$$ 0 0
$$475$$ 303.556 + 493.788i 0.639065 + 1.03955i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 844.970i 1.76403i −0.471222 0.882015i $$-0.656187\pi$$
0.471222 0.882015i $$-0.343813\pi$$
$$480$$ 0 0
$$481$$ −802.964 −1.66936
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −226.711 + 405.495i −0.467446 + 0.836072i
$$486$$ 0 0
$$487$$ 546.384i 1.12194i 0.827837 + 0.560969i $$0.189571\pi$$
−0.827837 + 0.560969i $$0.810429\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 647.456i 1.31865i −0.751859 0.659324i $$-0.770843\pi$$
0.751859 0.659324i $$-0.229157\pi$$
$$492$$ 0 0
$$493$$ 227.988i 0.462450i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 228.075 0.458904
$$498$$ 0 0
$$499$$ 582.815 1.16797 0.583983 0.811766i $$-0.301494\pi$$
0.583983 + 0.811766i $$0.301494\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −752.520 −1.49606 −0.748032 0.663663i $$-0.769001\pi$$
−0.748032 + 0.663663i $$0.769001\pi$$
$$504$$ 0 0
$$505$$ 113.870 203.667i 0.225484 0.403301i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 291.538i 0.572767i 0.958115 + 0.286383i $$0.0924531\pi$$
−0.958115 + 0.286383i $$0.907547\pi$$
$$510$$ 0 0
$$511$$ −296.556 −0.580344
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 27.2842 48.8004i 0.0529790 0.0947580i
$$516$$ 0 0
$$517$$ 742.441i 1.43606i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 137.690i 0.264280i −0.991231 0.132140i $$-0.957815\pi$$
0.991231 0.132140i $$-0.0421848\pi$$
$$522$$ 0 0
$$523$$ 330.987i 0.632862i 0.948616 + 0.316431i $$0.102485\pi$$
−0.948616 + 0.316431i $$0.897515\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 271.358 0.514911
$$528$$ 0 0
$$529$$ −528.926 −0.999861
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −305.482 −0.573137
$$534$$ 0 0
$$535$$ −642.834 359.407i −1.20156 0.671788i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 747.370i 1.38659i
$$540$$ 0 0
$$541$$ 556.593 1.02882 0.514412 0.857543i $$-0.328010\pi$$
0.514412 + 0.857543i $$0.328010\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 530.495 + 296.598i 0.973385 + 0.544217i
$$546$$ 0 0
$$547$$ 465.170i 0.850402i 0.905099 + 0.425201i $$0.139797\pi$$
−0.905099 + 0.425201i $$0.860203\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 922.761i 1.67470i
$$552$$ 0 0
$$553$$ 177.669i 0.321283i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −1059.37 −1.90192 −0.950962 0.309306i $$-0.899903\pi$$
−0.950962 + 0.309306i $$0.899903\pi$$
$$558$$ 0 0
$$559$$ −1075.70 −1.92434
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −386.160 −0.685897 −0.342949 0.939354i $$-0.611426\pi$$
−0.342949 + 0.939354i $$0.611426\pi$$
$$564$$ 0 0
$$565$$ −426.075 238.218i −0.754116 0.421624i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 686.062i 1.20573i −0.797842 0.602866i $$-0.794025\pi$$
0.797842 0.602866i $$-0.205975\pi$$
$$570$$ 0 0
$$571$$ −821.631 −1.43893 −0.719467 0.694527i $$-0.755614\pi$$
−0.719467 + 0.694527i $$0.755614\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.55575 + 5.78406i 0.00618390 + 0.0100592i
$$576$$ 0 0
$$577$$ 183.840i 0.318613i −0.987229 0.159306i $$-0.949074\pi$$
0.987229 0.159306i $$-0.0509258\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.794370i 0.00136725i
$$582$$ 0 0
$$583$$ 1658.06i 2.84401i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 320.407 0.545837 0.272919 0.962037i $$-0.412011\pi$$
0.272919 + 0.962037i $$0.412011\pi$$
$$588$$ 0 0
$$589$$ 1098.30 1.86468
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −1061.34 −1.78977 −0.894887 0.446292i $$-0.852744\pi$$
−0.894887 + 0.446292i $$0.852744\pi$$
$$594$$ 0 0
$$595$$ 39.0738 69.8872i 0.0656702 0.117457i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 977.056i 1.63115i 0.578655 + 0.815573i $$0.303578\pi$$
−0.578655 + 0.815573i $$0.696422\pi$$
$$600$$ 0 0
$$601$$ −261.816 −0.435635 −0.217817 0.975990i $$-0.569894\pi$$
−0.217817 + 0.975990i $$0.569894\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 909.050 + 508.248i 1.50256 + 0.840079i
$$606$$ 0 0
$$607$$ 668.806i 1.10182i 0.834564 + 0.550911i $$0.185720\pi$$
−0.834564 + 0.550911i $$0.814280\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 942.101i 1.54190i
$$612$$ 0 0
$$613$$ 220.119i 0.359086i −0.983750 0.179543i $$-0.942538\pi$$
0.983750 0.179543i $$-0.0574618\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 321.063 0.520361 0.260181 0.965560i $$-0.416218\pi$$
0.260181 + 0.965560i $$0.416218\pi$$
$$618$$ 0 0
$$619$$ −891.187 −1.43972 −0.719860 0.694119i $$-0.755794\pi$$
−0.719860 + 0.694119i $$0.755794\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 79.7301 0.127978
$$624$$ 0 0
$$625$$ −282.166 + 557.680i −0.451466 + 0.892288i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 199.757i 0.317578i
$$630$$ 0 0
$$631$$ −583.669 −0.924990 −0.462495 0.886622i $$-0.653046\pi$$
−0.462495 + 0.886622i $$0.653046\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 216.303 386.879i 0.340635 0.609258i
$$636$$ 0 0
$$637$$ 948.355i 1.48878i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 696.292i 1.08626i 0.839649 + 0.543129i $$0.182761\pi$$
−0.839649 + 0.543129i $$0.817239\pi$$
$$642$$ 0 0
$$643$$ 801.664i 1.24676i 0.781920 + 0.623378i $$0.214240\pi$$
−0.781920 + 0.623378i $$0.785760\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 216.221 0.334191 0.167095 0.985941i $$-0.446561\pi$$
0.167095 + 0.985941i $$0.446561\pi$$
$$648$$ 0 0
$$649$$ −1430.67 −2.20442
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −438.802 −0.671979 −0.335989 0.941866i $$-0.609071\pi$$
−0.335989 + 0.941866i $$0.609071\pi$$
$$654$$ 0 0
$$655$$ 219.537 392.663i 0.335171 0.599485i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 366.112i 0.555557i 0.960645 + 0.277779i $$0.0895982\pi$$
−0.960645 + 0.277779i $$0.910402\pi$$
$$660$$ 0 0
$$661$$ 158.075 0.239146 0.119573 0.992825i $$-0.461847\pi$$
0.119573 + 0.992825i $$0.461847\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 158.148 282.862i 0.237816 0.425357i
$$666$$ 0 0
$$667$$ 10.8089i 0.0162052i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 565.904i 0.843374i
$$672$$ 0 0
$$673$$ 648.575i 0.963708i 0.876252 + 0.481854i $$0.160036\pi$$
−0.876252 + 0.481854i $$0.839964\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −202.345 −0.298885 −0.149443 0.988770i $$-0.547748\pi$$
−0.149443 + 0.988770i $$0.547748\pi$$
$$678$$ 0 0
$$679$$ 259.739 0.382532
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 173.829 0.254508 0.127254 0.991870i $$-0.459384\pi$$
0.127254 + 0.991870i $$0.459384\pi$$
$$684$$ 0 0
$$685$$ 1003.45 + 561.024i 1.46488 + 0.819013i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2103.95i 3.05363i
$$690$$ 0 0
$$691$$ −846.743 −1.22539 −0.612694 0.790320i $$-0.709914\pi$$
−0.612694 + 0.790320i $$0.709914\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −251.507 140.617i −0.361881 0.202327i
$$696$$ 0 0
$$697$$ 75.9960i 0.109033i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 72.2614i 0.103083i −0.998671 0.0515416i $$-0.983587\pi$$
0.998671 0.0515416i $$-0.0164135\pi$$
$$702$$ 0 0
$$703$$ 808.497i 1.15007i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −130.459 −0.184524
$$708$$ 0 0
$$709$$ 287.557 0.405582 0.202791 0.979222i $$-0.434999\pi$$
0.202791 + 0.979222i $$0.434999\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 12.8651 0.0180436
$$714$$ 0 0
$$715$$ −1823.60 1019.57i −2.55048 1.42597i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1015.27i 1.41205i −0.708185 0.706027i $$-0.750486\pi$$
0.708185 0.706027i $$-0.249514\pi$$
$$720$$ 0 0
$$721$$ −31.2590 −0.0433551
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 847.630 521.080i 1.16914 0.718731i
$$726$$ 0 0
$$727$$ 311.418i 0.428361i 0.976794 + 0.214180i $$0.0687080\pi$$
−0.976794 + 0.214180i $$0.931292\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 267.608i 0.366084i
$$732$$ 0 0
$$733$$ 321.047i 0.437990i 0.975726 + 0.218995i $$0.0702778\pi$$
−0.975726 + 0.218995i $$0.929722\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 125.507 0.170295
$$738$$ 0 0
$$739$$ −657.185 −0.889290 −0.444645 0.895707i $$-0.646670\pi$$
−0.444645 + 0.895707i $$0.646670\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1024.96 −1.37949 −0.689747 0.724051i $$-0.742278\pi$$
−0.689747 + 0.724051i $$0.742278\pi$$
$$744$$ 0 0
$$745$$ −27.1664 + 48.5897i −0.0364649 + 0.0652211i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 411.766i 0.549755i
$$750$$ 0 0
$$751$$ −63.1475 −0.0840846 −0.0420423 0.999116i $$-0.513386\pi$$
−0.0420423 + 0.999116i $$0.513386\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 96.8210 + 54.1324i 0.128240 + 0.0716986i
$$756$$ 0 0
$$757$$ 1452.21i 1.91837i 0.282781 + 0.959185i $$0.408743\pi$$
−0.282781 + 0.959185i $$0.591257\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 487.824i 0.641030i 0.947243 + 0.320515i $$0.103856\pi$$
−0.947243 + 0.320515i $$0.896144\pi$$
$$762$$ 0 0
$$763$$ 339.808i 0.445357i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1815.41 2.36690
$$768$$ 0 0
$$769$$ 677.148 0.880556 0.440278 0.897862i $$-0.354880\pi$$
0.440278 + 0.897862i $$0.354880\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −949.926 −1.22888 −0.614441 0.788963i $$-0.710619\pi$$
−0.614441 + 0.788963i $$0.710619\pi$$
$$774$$ 0 0
$$775$$ 620.204 + 1008.87i 0.800263 + 1.30177i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 307.587i 0.394849i
$$780$$ 0 0
$$781$$ 1480.52 1.89567
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 549.827 983.419i 0.700417 1.25276i
$$786$$ 0 0
$$787$$ 1144.33i 1.45404i 0.686617 + 0.727020i $$0.259095\pi$$
−0.686617 + 0.727020i $$0.740905\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 272.922i 0.345034i
$$792$$ 0 0
$$793$$ 718.089i 0.905535i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 651.101 0.816939 0.408470 0.912772i $$-0.366063\pi$$
0.408470 + 0.912772i $$0.366063\pi$$
$$798$$ 0 0
$$799$$ −234.370 −0.293330
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1925.05 −2.39733
$$804$$ 0 0
$$805$$ 1.85248 3.31335i 0.00230122 0.00411596i
$$806$$