# Properties

 Label 2520.2.t.g.1009.5 Level $2520$ Weight $2$ Character 2520.1009 Analytic conductor $20.122$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2520.t (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.1223013094$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.5161984.1 Defining polynomial: $$x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1009.5 Root $$-1.75233 + 1.75233i$$ of defining polynomial Character $$\chi$$ $$=$$ 2520.1009 Dual form 2520.2.t.g.1009.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.75233 - 1.38900i) q^{5} +1.00000i q^{7} +O(q^{10})$$ $$q+(1.75233 - 1.38900i) q^{5} +1.00000i q^{7} -5.14134 q^{11} +4.64600i q^{13} -3.86799i q^{17} +0.778008 q^{19} +5.00933i q^{23} +(1.14134 - 4.86799i) q^{25} +9.42401 q^{29} +4.72666 q^{31} +(1.38900 + 1.75233i) q^{35} +6.00000i q^{37} +1.00933 q^{41} +7.00933i q^{43} +11.4240i q^{47} -1.00000 q^{49} +7.55602i q^{53} +(-9.00933 + 7.14134i) q^{55} +12.5140 q^{59} +11.5047 q^{61} +(6.45331 + 8.14134i) q^{65} -11.7360i q^{67} -2.72666 q^{71} -5.00933i q^{73} -5.14134i q^{77} -5.68802 q^{79} -4.67531i q^{83} +(-5.37266 - 6.77801i) q^{85} -2.82936 q^{89} -4.64600 q^{91} +(1.36333 - 1.08066i) q^{95} -1.58532i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + O(q^{10})$$ $$6q - 14q^{11} - 8q^{19} - 10q^{25} + 6q^{29} + 20q^{31} + 2q^{35} - 36q^{41} - 6q^{49} - 12q^{55} + 12q^{59} + 48q^{61} + 22q^{65} - 8q^{71} - 34q^{79} + 14q^{85} + 10q^{91} + 4q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$$.

 $$n$$ $$281$$ $$631$$ $$1081$$ $$1261$$ $$2017$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.75233 1.38900i 0.783667 0.621181i
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.14134 −1.55017 −0.775086 0.631856i $$-0.782293\pi$$
−0.775086 + 0.631856i $$0.782293\pi$$
$$12$$ 0 0
$$13$$ 4.64600i 1.28857i 0.764786 + 0.644284i $$0.222845\pi$$
−0.764786 + 0.644284i $$0.777155\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.86799i 0.938126i −0.883165 0.469063i $$-0.844592\pi$$
0.883165 0.469063i $$-0.155408\pi$$
$$18$$ 0 0
$$19$$ 0.778008 0.178487 0.0892436 0.996010i $$-0.471555\pi$$
0.0892436 + 0.996010i $$0.471555\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.00933i 1.04452i 0.852787 + 0.522259i $$0.174910\pi$$
−0.852787 + 0.522259i $$0.825090\pi$$
$$24$$ 0 0
$$25$$ 1.14134 4.86799i 0.228267 0.973599i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 9.42401 1.74999 0.874997 0.484128i $$-0.160863\pi$$
0.874997 + 0.484128i $$0.160863\pi$$
$$30$$ 0 0
$$31$$ 4.72666 0.848933 0.424466 0.905444i $$-0.360462\pi$$
0.424466 + 0.905444i $$0.360462\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.38900 + 1.75233i 0.234785 + 0.296198i
$$36$$ 0 0
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.00933 0.157631 0.0788153 0.996889i $$-0.474886\pi$$
0.0788153 + 0.996889i $$0.474886\pi$$
$$42$$ 0 0
$$43$$ 7.00933i 1.06891i 0.845196 + 0.534456i $$0.179484\pi$$
−0.845196 + 0.534456i $$0.820516\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.4240i 1.66636i 0.553000 + 0.833181i $$0.313483\pi$$
−0.553000 + 0.833181i $$0.686517\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 7.55602i 1.03790i 0.854805 + 0.518949i $$0.173677\pi$$
−0.854805 + 0.518949i $$0.826323\pi$$
$$54$$ 0 0
$$55$$ −9.00933 + 7.14134i −1.21482 + 0.962938i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 12.5140 1.62918 0.814592 0.580035i $$-0.196961\pi$$
0.814592 + 0.580035i $$0.196961\pi$$
$$60$$ 0 0
$$61$$ 11.5047 1.47302 0.736511 0.676426i $$-0.236472\pi$$
0.736511 + 0.676426i $$0.236472\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.45331 + 8.14134i 0.800435 + 1.00981i
$$66$$ 0 0
$$67$$ 11.7360i 1.43378i −0.697187 0.716889i $$-0.745565\pi$$
0.697187 0.716889i $$-0.254435\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.72666 −0.323595 −0.161797 0.986824i $$-0.551729\pi$$
−0.161797 + 0.986824i $$0.551729\pi$$
$$72$$ 0 0
$$73$$ 5.00933i 0.586298i −0.956067 0.293149i $$-0.905297\pi$$
0.956067 0.293149i $$-0.0947031\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.14134i 0.585910i
$$78$$ 0 0
$$79$$ −5.68802 −0.639953 −0.319976 0.947426i $$-0.603675\pi$$
−0.319976 + 0.947426i $$0.603675\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 4.67531i 0.513181i −0.966520 0.256591i $$-0.917401\pi$$
0.966520 0.256591i $$-0.0825992\pi$$
$$84$$ 0 0
$$85$$ −5.37266 6.77801i −0.582746 0.735178i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −2.82936 −0.299911 −0.149956 0.988693i $$-0.547913\pi$$
−0.149956 + 0.988693i $$0.547913\pi$$
$$90$$ 0 0
$$91$$ −4.64600 −0.487033
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.36333 1.08066i 0.139875 0.110873i
$$96$$ 0 0
$$97$$ 1.58532i 0.160965i −0.996756 0.0804824i $$-0.974354\pi$$
0.996756 0.0804824i $$-0.0256461\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.06068 0.901571 0.450786 0.892632i $$-0.351144\pi$$
0.450786 + 0.892632i $$0.351144\pi$$
$$102$$ 0 0
$$103$$ 5.14134i 0.506591i −0.967389 0.253295i $$-0.918486\pi$$
0.967389 0.253295i $$-0.0815145\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 3.97070 0.380324 0.190162 0.981753i $$-0.439099\pi$$
0.190162 + 0.981753i $$0.439099\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.54669i 0.427716i 0.976865 + 0.213858i $$0.0686030\pi$$
−0.976865 + 0.213858i $$0.931397\pi$$
$$114$$ 0 0
$$115$$ 6.95798 + 8.77801i 0.648835 + 0.818553i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.86799 0.354578
$$120$$ 0 0
$$121$$ 15.4333 1.40303
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −4.76166 10.1157i −0.425896 0.904772i
$$126$$ 0 0
$$127$$ 16.1214i 1.43054i 0.698849 + 0.715270i $$0.253696\pi$$
−0.698849 + 0.715270i $$0.746304\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.0513514 −0.00448659 −0.00224330 0.999997i $$-0.500714\pi$$
−0.00224330 + 0.999997i $$0.500714\pi$$
$$132$$ 0 0
$$133$$ 0.778008i 0.0674618i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 14.0187i 1.19769i 0.800863 + 0.598847i $$0.204374\pi$$
−0.800863 + 0.598847i $$0.795626\pi$$
$$138$$ 0 0
$$139$$ −12.6167 −1.07013 −0.535067 0.844810i $$-0.679714\pi$$
−0.535067 + 0.844810i $$0.679714\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 23.8867i 1.99750i
$$144$$ 0 0
$$145$$ 16.5140 13.0900i 1.37141 1.08706i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −11.4533 −0.938292 −0.469146 0.883121i $$-0.655438\pi$$
−0.469146 + 0.883121i $$0.655438\pi$$
$$150$$ 0 0
$$151$$ 5.86799 0.477530 0.238765 0.971077i $$-0.423257\pi$$
0.238765 + 0.971077i $$0.423257\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.28267 6.56534i 0.665280 0.527341i
$$156$$ 0 0
$$157$$ 5.78734i 0.461880i −0.972968 0.230940i $$-0.925820\pi$$
0.972968 0.230940i $$-0.0741801\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.00933 −0.394790
$$162$$ 0 0
$$163$$ 3.27334i 0.256388i −0.991749 0.128194i $$-0.959082\pi$$
0.991749 0.128194i $$-0.0409180\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 24.8773i 1.92506i 0.271166 + 0.962532i $$0.412591\pi$$
−0.271166 + 0.962532i $$0.587409\pi$$
$$168$$ 0 0
$$169$$ −8.58532 −0.660409
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.62734i 0.503868i 0.967744 + 0.251934i $$0.0810665\pi$$
−0.967744 + 0.251934i $$0.918933\pi$$
$$174$$ 0 0
$$175$$ 4.86799 + 1.14134i 0.367986 + 0.0862769i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −10.0187 −0.748830 −0.374415 0.927261i $$-0.622156\pi$$
−0.374415 + 0.927261i $$0.622156\pi$$
$$180$$ 0 0
$$181$$ 1.78734 0.132852 0.0664258 0.997791i $$-0.478840\pi$$
0.0664258 + 0.997791i $$0.478840\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 8.33402 + 10.5140i 0.612730 + 0.773004i
$$186$$ 0 0
$$187$$ 19.8867i 1.45426i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.59465 0.621887 0.310943 0.950428i $$-0.399355\pi$$
0.310943 + 0.950428i $$0.399355\pi$$
$$192$$ 0 0
$$193$$ 5.17064i 0.372191i 0.982532 + 0.186095i $$0.0595834\pi$$
−0.982532 + 0.186095i $$0.940417\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.9160i 1.70394i −0.523590 0.851971i $$-0.675407\pi$$
0.523590 0.851971i $$-0.324593\pi$$
$$198$$ 0 0
$$199$$ 15.3107 1.08534 0.542672 0.839945i $$-0.317413\pi$$
0.542672 + 0.839945i $$0.317413\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 9.42401i 0.661436i
$$204$$ 0 0
$$205$$ 1.76868 1.40196i 0.123530 0.0979172i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ −1.03863 −0.0715025 −0.0357512 0.999361i $$-0.511382\pi$$
−0.0357512 + 0.999361i $$0.511382\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 9.73599 + 12.2827i 0.663989 + 0.837671i
$$216$$ 0 0
$$217$$ 4.72666i 0.320866i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 17.9707 1.20884
$$222$$ 0 0
$$223$$ 9.86799i 0.660810i 0.943839 + 0.330405i $$0.107185\pi$$
−0.943839 + 0.330405i $$0.892815\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 21.6553i 1.43731i 0.695364 + 0.718657i $$0.255243\pi$$
−0.695364 + 0.718657i $$0.744757\pi$$
$$228$$ 0 0
$$229$$ −20.2313 −1.33692 −0.668462 0.743747i $$-0.733047\pi$$
−0.668462 + 0.743747i $$0.733047\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5.45331i 0.357258i −0.983916 0.178629i $$-0.942834\pi$$
0.983916 0.178629i $$-0.0571663\pi$$
$$234$$ 0 0
$$235$$ 15.8680 + 20.0187i 1.03511 + 1.30587i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −11.5853 −0.749392 −0.374696 0.927148i $$-0.622253\pi$$
−0.374696 + 0.927148i $$0.622253\pi$$
$$240$$ 0 0
$$241$$ −5.00933 −0.322679 −0.161340 0.986899i $$-0.551581\pi$$
−0.161340 + 0.986899i $$0.551581\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1.75233 + 1.38900i −0.111952 + 0.0887402i
$$246$$ 0 0
$$247$$ 3.61462i 0.229993i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 23.4206 1.47830 0.739148 0.673543i $$-0.235228\pi$$
0.739148 + 0.673543i $$0.235228\pi$$
$$252$$ 0 0
$$253$$ 25.7546i 1.61918i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 13.0093i 0.811500i −0.913984 0.405750i $$-0.867010\pi$$
0.913984 0.405750i $$-0.132990\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6.01866i 0.371126i 0.982632 + 0.185563i $$0.0594109\pi$$
−0.982632 + 0.185563i $$0.940589\pi$$
$$264$$ 0 0
$$265$$ 10.4953 + 13.2406i 0.644723 + 0.813367i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 3.24065 0.197586 0.0987929 0.995108i $$-0.468502\pi$$
0.0987929 + 0.995108i $$0.468502\pi$$
$$270$$ 0 0
$$271$$ −29.1307 −1.76956 −0.884782 0.466006i $$-0.845693\pi$$
−0.884782 + 0.466006i $$0.845693\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −5.86799 + 25.0280i −0.353853 + 1.50924i
$$276$$ 0 0
$$277$$ 4.44398i 0.267013i −0.991048 0.133507i $$-0.957376\pi$$
0.991048 0.133507i $$-0.0426237\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 24.6974 1.47332 0.736660 0.676263i $$-0.236402\pi$$
0.736660 + 0.676263i $$0.236402\pi$$
$$282$$ 0 0
$$283$$ 7.73937i 0.460058i −0.973184 0.230029i $$-0.926118\pi$$
0.973184 0.230029i $$-0.0738821\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.00933i 0.0595788i
$$288$$ 0 0
$$289$$ 2.03863 0.119920
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 25.1086i 1.46686i 0.679764 + 0.733431i $$0.262082\pi$$
−0.679764 + 0.733431i $$0.737918\pi$$
$$294$$ 0 0
$$295$$ 21.9287 17.3820i 1.27674 1.01202i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −23.2733 −1.34593
$$300$$ 0 0
$$301$$ −7.00933 −0.404011
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 20.1600 15.9800i 1.15436 0.915014i
$$306$$ 0 0
$$307$$ 33.9193i 1.93588i −0.251183 0.967940i $$-0.580820\pi$$
0.251183 0.967940i $$-0.419180\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −0.565344 −0.0320577 −0.0160289 0.999872i $$-0.505102\pi$$
−0.0160289 + 0.999872i $$0.505102\pi$$
$$312$$ 0 0
$$313$$ 27.3400i 1.54535i 0.634804 + 0.772673i $$0.281081\pi$$
−0.634804 + 0.772673i $$0.718919\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 22.7267i 1.27646i −0.769847 0.638228i $$-0.779668\pi$$
0.769847 0.638228i $$-0.220332\pi$$
$$318$$ 0 0
$$319$$ −48.4520 −2.71279
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.00933i 0.167444i
$$324$$ 0 0
$$325$$ 22.6167 + 5.30265i 1.25455 + 0.294138i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −11.4240 −0.629826
$$330$$ 0 0
$$331$$ 0.462642 0.0254291 0.0127145 0.999919i $$-0.495953\pi$$
0.0127145 + 0.999919i $$0.495953\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −16.3013 20.5653i −0.890637 1.12360i
$$336$$ 0 0
$$337$$ 10.5653i 0.575531i −0.957701 0.287765i $$-0.907088\pi$$
0.957701 0.287765i $$-0.0929124\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −24.3013 −1.31599
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 31.8760i 1.71119i −0.517643 0.855597i $$-0.673190\pi$$
0.517643 0.855597i $$-0.326810\pi$$
$$348$$ 0 0
$$349$$ 9.62602 0.515269 0.257635 0.966242i $$-0.417057\pi$$
0.257635 + 0.966242i $$0.417057\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 26.2534i 1.39733i −0.715451 0.698663i $$-0.753779\pi$$
0.715451 0.698663i $$-0.246221\pi$$
$$354$$ 0 0
$$355$$ −4.77801 + 3.78734i −0.253590 + 0.201011i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −4.56534 −0.240950 −0.120475 0.992716i $$-0.538442\pi$$
−0.120475 + 0.992716i $$0.538442\pi$$
$$360$$ 0 0
$$361$$ −18.3947 −0.968142
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.95798 8.77801i −0.364197 0.459462i
$$366$$ 0 0
$$367$$ 3.79073i 0.197874i 0.995094 + 0.0989371i $$0.0315443\pi$$
−0.995094 + 0.0989371i $$0.968456\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7.55602 −0.392289
$$372$$ 0 0
$$373$$ 20.7453i 1.07415i −0.843534 0.537076i $$-0.819529\pi$$
0.843534 0.537076i $$-0.180471\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 43.7839i 2.25499i
$$378$$ 0 0
$$379$$ −3.00933 −0.154579 −0.0772894 0.997009i $$-0.524627\pi$$
−0.0772894 + 0.997009i $$0.524627\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 7.00933i 0.358160i 0.983835 + 0.179080i $$0.0573121\pi$$
−0.983835 + 0.179080i $$0.942688\pi$$
$$384$$ 0 0
$$385$$ −7.14134 9.00933i −0.363956 0.459158i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −19.7653 −1.00214 −0.501070 0.865407i $$-0.667060\pi$$
−0.501070 + 0.865407i $$0.667060\pi$$
$$390$$ 0 0
$$391$$ 19.3760 0.979889
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −9.96731 + 7.90069i −0.501510 + 0.397527i
$$396$$ 0 0
$$397$$ 9.37266i 0.470400i −0.971947 0.235200i $$-0.924425\pi$$
0.971947 0.235200i $$-0.0755745\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 16.1507 0.806526 0.403263 0.915084i $$-0.367876\pi$$
0.403263 + 0.915084i $$0.367876\pi$$
$$402$$ 0 0
$$403$$ 21.9600i 1.09391i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 30.8480i 1.52908i
$$408$$ 0 0
$$409$$ −15.2920 −0.756141 −0.378070 0.925777i $$-0.623412\pi$$
−0.378070 + 0.925777i $$0.623412\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 12.5140i 0.615773i
$$414$$ 0 0
$$415$$ −6.49402 8.19269i −0.318779 0.402163i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −33.1820 −1.62105 −0.810524 0.585705i $$-0.800818\pi$$
−0.810524 + 0.585705i $$0.800818\pi$$
$$420$$ 0 0
$$421$$ 27.8094 1.35535 0.677673 0.735363i $$-0.262988\pi$$
0.677673 + 0.735363i $$0.262988\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −18.8294 4.41468i −0.913358 0.214143i
$$426$$ 0 0
$$427$$ 11.5047i 0.556750i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.5454 1.03780 0.518902 0.854834i $$-0.326341\pi$$
0.518902 + 0.854834i $$0.326341\pi$$
$$432$$ 0 0
$$433$$ 36.0187i 1.73095i 0.500955 + 0.865473i $$0.332982\pi$$
−0.500955 + 0.865473i $$0.667018\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.89730i 0.186433i
$$438$$ 0 0
$$439$$ 21.1893 1.01131 0.505655 0.862736i $$-0.331251\pi$$
0.505655 + 0.862736i $$0.331251\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.1120i 0.527949i −0.964530 0.263974i $$-0.914967\pi$$
0.964530 0.263974i $$-0.0850334\pi$$
$$444$$ 0 0
$$445$$ −4.95798 + 3.92999i −0.235031 + 0.186299i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0.961367 0.0453697 0.0226848 0.999743i $$-0.492779\pi$$
0.0226848 + 0.999743i $$0.492779\pi$$
$$450$$ 0 0
$$451$$ −5.18930 −0.244355
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −8.14134 + 6.45331i −0.381672 + 0.302536i
$$456$$ 0 0
$$457$$ 28.2241i 1.32027i 0.751149 + 0.660133i $$0.229500\pi$$
−0.751149 + 0.660133i $$0.770500\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −26.0887 −1.21507 −0.607535 0.794293i $$-0.707842\pi$$
−0.607535 + 0.794293i $$0.707842\pi$$
$$462$$ 0 0
$$463$$ 2.90663i 0.135082i 0.997716 + 0.0675412i $$0.0215154\pi$$
−0.997716 + 0.0675412i $$0.978485\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 12.2020i 0.564642i −0.959320 0.282321i $$-0.908896\pi$$
0.959320 0.282321i $$-0.0911043\pi$$
$$468$$ 0 0
$$469$$ 11.7360 0.541917
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 36.0373i 1.65700i
$$474$$ 0 0
$$475$$ 0.887968 3.78734i 0.0407428 0.173775i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −17.6519 −0.806538 −0.403269 0.915082i $$-0.632126\pi$$
−0.403269 + 0.915082i $$0.632126\pi$$
$$480$$ 0 0
$$481$$ −27.8760 −1.27104
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2.20202 2.77801i −0.0999884 0.126143i
$$486$$ 0 0
$$487$$ 1.57467i 0.0713553i −0.999363 0.0356776i $$-0.988641\pi$$
0.999363 0.0356776i $$-0.0113590\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 3.26270 0.147243 0.0736217 0.997286i $$-0.476544\pi$$
0.0736217 + 0.997286i $$0.476544\pi$$
$$492$$ 0 0
$$493$$ 36.4520i 1.64172i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.72666i 0.122307i
$$498$$ 0 0
$$499$$ −42.5293 −1.90387 −0.951936 0.306298i $$-0.900910\pi$$
−0.951936 + 0.306298i $$0.900910\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 18.6974i 0.833674i −0.908981 0.416837i $$-0.863139\pi$$
0.908981 0.416837i $$-0.136861\pi$$
$$504$$ 0 0
$$505$$ 15.8773 12.5853i 0.706532 0.560039i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 20.2313 0.896738 0.448369 0.893849i $$-0.352005\pi$$
0.448369 + 0.893849i $$0.352005\pi$$
$$510$$ 0 0
$$511$$ 5.00933 0.221600
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −7.14134 9.00933i −0.314685 0.396998i
$$516$$ 0 0
$$517$$ 58.7347i 2.58315i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −21.8387 −0.956770 −0.478385 0.878150i $$-0.658778\pi$$
−0.478385 + 0.878150i $$0.658778\pi$$
$$522$$ 0 0
$$523$$ 20.4554i 0.894451i 0.894421 + 0.447226i $$0.147588\pi$$
−0.894421 + 0.447226i $$0.852412\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 18.2827i 0.796406i
$$528$$ 0 0
$$529$$ −2.09337 −0.0910163
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.68934i 0.203118i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 5.14134 0.221453
$$540$$ 0 0
$$541$$ 27.3400 1.17544 0.587718 0.809066i $$-0.300026\pi$$
0.587718 + 0.809066i $$0.300026\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.95798 5.51531i 0.298047 0.236250i
$$546$$ 0 0
$$547$$ 22.6867i 0.970013i −0.874510 0.485007i $$-0.838817\pi$$
0.874510 0.485007i $$-0.161183\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7.33195 0.312352
$$552$$ 0 0
$$553$$ 5.68802i 0.241879i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.7453i 0.540036i 0.962855 + 0.270018i $$0.0870297\pi$$
−0.962855 + 0.270018i $$0.912970\pi$$
$$558$$ 0 0
$$559$$ −32.5653 −1.37737
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 5.20333i 0.219294i −0.993971 0.109647i $$-0.965028\pi$$
0.993971 0.109647i $$-0.0349721\pi$$
$$564$$ 0 0
$$565$$ 6.31537 + 7.96731i 0.265689 + 0.335187i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7.37605 −0.309220 −0.154610 0.987976i $$-0.549412\pi$$
−0.154610 + 0.987976i $$0.549412\pi$$
$$570$$ 0 0
$$571$$ 15.8973 0.665281 0.332641 0.943054i $$-0.392060\pi$$
0.332641 + 0.943054i $$0.392060\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 24.3854 + 5.71733i 1.01694 + 0.238429i
$$576$$ 0 0
$$577$$ 19.9707i 0.831391i −0.909504 0.415695i $$-0.863538\pi$$
0.909504 0.415695i $$-0.136462\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4.67531 0.193964
$$582$$ 0 0
$$583$$ 38.8480i 1.60892i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 13.8060i 0.569834i 0.958552 + 0.284917i $$0.0919661\pi$$
−0.958552 + 0.284917i $$0.908034\pi$$
$$588$$ 0 0
$$589$$ 3.67738 0.151524
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 20.0666i 0.824037i 0.911175 + 0.412019i $$0.135176\pi$$
−0.911175 + 0.412019i $$0.864824\pi$$
$$594$$ 0 0
$$595$$ 6.77801 5.37266i 0.277871 0.220257i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 46.0267 1.88060 0.940299 0.340349i $$-0.110545\pi$$
0.940299 + 0.340349i $$0.110545\pi$$
$$600$$ 0 0
$$601$$ −1.37605 −0.0561301 −0.0280651 0.999606i $$-0.508935\pi$$
−0.0280651 + 0.999606i $$0.508935\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 27.0443 21.4370i 1.09951 0.871537i
$$606$$ 0 0
$$607$$ 18.7933i 0.762796i 0.924411 + 0.381398i $$0.124557\pi$$
−0.924411 + 0.381398i $$0.875443\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −53.0759 −2.14722
$$612$$ 0 0
$$613$$ 24.8480i 1.00360i −0.864983 0.501801i $$-0.832671\pi$$
0.864983 0.501801i $$-0.167329\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 43.1680i 1.73788i −0.494919 0.868939i $$-0.664802\pi$$
0.494919 0.868939i $$-0.335198\pi$$
$$618$$ 0 0
$$619$$ −31.9486 −1.28412 −0.642062 0.766652i $$-0.721921\pi$$
−0.642062 + 0.766652i $$0.721921\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 2.82936i 0.113356i
$$624$$ 0 0
$$625$$ −22.3947 11.1120i −0.895788 0.444481i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 23.2080 0.925362
$$630$$ 0 0
$$631$$ −8.59465 −0.342148 −0.171074 0.985258i $$-0.554724\pi$$
−0.171074 + 0.985258i $$0.554724\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 22.3926 + 28.2500i 0.888625 + 1.12107i
$$636$$ 0 0
$$637$$ 4.64600i 0.184081i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.2266 −1.31237 −0.656186 0.754599i $$-0.727831\pi$$
−0.656186 + 0.754599i $$0.727831\pi$$
$$642$$ 0 0
$$643$$ 17.4940i 0.689897i −0.938622 0.344948i $$-0.887896\pi$$
0.938622 0.344948i $$-0.112104\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.91595i 0.389836i −0.980820 0.194918i $$-0.937556\pi$$
0.980820 0.194918i $$-0.0624441\pi$$
$$648$$ 0 0
$$649$$ −64.3386 −2.52551
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.6240i 0.572280i 0.958188 + 0.286140i $$0.0923722\pi$$
−0.958188 + 0.286140i $$0.907628\pi$$
$$654$$ 0 0
$$655$$ −0.0899847 + 0.0713273i −0.00351599 + 0.00278699i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 20.2534 0.788959 0.394480 0.918905i $$-0.370925\pi$$
0.394480 + 0.918905i $$0.370925\pi$$
$$660$$ 0 0
$$661$$ −12.4299 −0.483469 −0.241734 0.970342i $$-0.577716\pi$$
−0.241734 + 0.970342i $$0.577716\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.08066 + 1.36333i 0.0419060 + 0.0528676i
$$666$$ 0 0
$$667$$ 47.2080i 1.82790i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −59.1493 −2.28344
$$672$$ 0 0
$$673$$ 47.6774i 1.83783i 0.394458 + 0.918914i $$0.370932\pi$$
−0.394458 + 0.918914i $$0.629068\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.00339i 0.230729i −0.993323 0.115364i $$-0.963196\pi$$
0.993323 0.115364i $$-0.0368036\pi$$
$$678$$ 0 0
$$679$$ 1.58532 0.0608390
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 28.5653i 1.09302i −0.837452 0.546511i $$-0.815956\pi$$
0.837452 0.546511i $$-0.184044\pi$$
$$684$$ 0 0
$$685$$ 19.4720 + 24.5653i 0.743986 + 0.938594i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −35.1053 −1.33740
$$690$$ 0 0
$$691$$ 47.8247 1.81934 0.909668 0.415337i $$-0.136336\pi$$
0.909668 + 0.415337i $$0.136336\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −22.1086 + 17.5246i −0.838629 + 0.664747i
$$696$$ 0 0
$$697$$ 3.90408i 0.147877i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 4.80005 0.181296 0.0906478 0.995883i $$-0.471106\pi$$
0.0906478 + 0.995883i $$0.471106\pi$$
$$702$$ 0 0
$$703$$ 4.66805i 0.176059i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 9.06068i 0.340762i
$$708$$ 0 0
$$709$$ −11.7653 −0.441855 −0.220927 0.975290i $$-0.570908\pi$$
−0.220927 + 0.975290i $$0.570908\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 23.6774i 0.886725i
$$714$$ 0 0
$$715$$ −33.1787 41.8573i −1.24081 1.56538i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −40.7640 −1.52024 −0.760120 0.649783i $$-0.774860\pi$$
−0.760120 + 0.649783i $$0.774860\pi$$
$$720$$ 0 0
$$721$$ 5.14134 0.191473
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 10.7560 45.8760i 0.399466 1.70379i
$$726$$ 0 0
$$727$$ 22.1214i 0.820436i 0.911988 + 0.410218i $$0.134547\pi$$
−0.911988 + 0.410218i $$0.865453\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 27.1120 1.00277
$$732$$ 0 0
$$733$$ 13.0500i 0.482014i −0.970523 0.241007i $$-0.922522\pi$$
0.970523 0.241007i $$-0.0774777\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 60.3386i 2.22260i
$$738$$ 0 0
$$739$$ 34.5734 1.27180 0.635901 0.771771i $$-0.280629\pi$$
0.635901 + 0.771771i $$0.280629\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 3.55602i 0.130458i −0.997870 0.0652288i $$-0.979222\pi$$
0.997870 0.0652288i $$-0.0207777\pi$$
$$744$$ 0 0
$$745$$ −20.0700 + 15.9087i −0.735308 + 0.582850i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −23.6226 −0.862002 −0.431001 0.902351i $$-0.641839\pi$$
−0.431001 + 0.902351i $$0.641839\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 10.2827 8.15066i 0.374225 0.296633i
$$756$$ 0 0
$$757$$ 15.1893i 0.552064i 0.961148 + 0.276032i $$0.0890196\pi$$
−0.961148 + 0.276032i $$0.910980\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.4626 −1.24927 −0.624635 0.780917i $$-0.714752\pi$$
−0.624635 + 0.780917i $$0.714752\pi$$
$$762$$ 0 0
$$763$$ 3.97070i 0.143749i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 58.1400i 2.09931i
$$768$$ 0 0
$$769$$ −40.3854 −1.45633 −0.728167 0.685400i $$-0.759627\pi$$
−0.728167 + 0.685400i $$0.759627\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 10.7674i 0.387275i −0.981073 0.193638i $$-0.937971\pi$$
0.981073 0.193638i $$-0.0620286\pi$$
$$774$$ 0 0
$$775$$ 5.39470 23.0093i 0.193783 0.826519i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0.785266 0.0281351
$$780$$ 0 0
$$781$$ 14.0187 0.501627
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −8.03863 10.1413i −0.286911 0.361960i
$$786$$ 0 0
$$787$$ 9.19269i 0.327684i −0.986487 0.163842i $$-0.947611\pi$$
0.986487 0.163842i $$-0.0523887\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.54669 −0.161662
$$792$$ 0 0
$$793$$ 53.4507i 1.89809i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 21.1086i 0.747706i −0.927488 0.373853i $$-0.878036\pi$$
0.927488 0.373853i $$-0.121964\pi$$
$$798$$ 0 0
$$799$$ 44.1880 1.56326
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 25.7546i 0.908862i
$$804$$ 0 0
$$805$$ −8.77801 + 6.95798i −0.309384 + 0.245236i
$$806$$ 0 0
$$807$$ 0 0