# Properties

 Label 180.3.f.f.19.4 Level $180$ Weight $3$ Character 180.19 Analytic conductor $4.905$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.90464475849$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{15})$$ Defining polynomial: $$x^{4} - 7 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 19.4 Root $$1.93649 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 180.19 Dual form 180.3.f.f.19.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.93649 + 0.500000i) q^{2} +(3.50000 + 1.93649i) q^{4} +5.00000i q^{5} +(5.80948 + 5.50000i) q^{8} +O(q^{10})$$ $$q+(1.93649 + 0.500000i) q^{2} +(3.50000 + 1.93649i) q^{4} +5.00000i q^{5} +(5.80948 + 5.50000i) q^{8} +(-2.50000 + 9.68246i) q^{10} +(8.50000 + 13.5554i) q^{16} +14.0000i q^{17} -30.9839i q^{19} +(-9.68246 + 17.5000i) q^{20} +30.9839 q^{23} -25.0000 q^{25} -61.9677i q^{31} +(9.68246 + 30.5000i) q^{32} +(-7.00000 + 27.1109i) q^{34} +(15.4919 - 60.0000i) q^{38} +(-27.5000 + 29.0474i) q^{40} +(60.0000 + 15.4919i) q^{46} -92.9516 q^{47} -49.0000 q^{49} +(-48.4123 - 12.5000i) q^{50} -86.0000i q^{53} +118.000 q^{61} +(30.9839 - 120.000i) q^{62} +(3.50000 + 63.9042i) q^{64} +(-27.1109 + 49.0000i) q^{68} +(60.0000 - 108.444i) q^{76} +123.935i q^{79} +(-67.7772 + 42.5000i) q^{80} +61.9677 q^{83} -70.0000 q^{85} +(108.444 + 60.0000i) q^{92} +(-180.000 - 46.4758i) q^{94} +154.919 q^{95} +(-94.8881 - 24.5000i) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 14q^{4} + O(q^{10})$$ $$4q + 14q^{4} - 10q^{10} + 34q^{16} - 100q^{25} - 28q^{34} - 110q^{40} + 240q^{46} - 196q^{49} + 472q^{61} + 14q^{64} + 240q^{76} - 280q^{85} - 720q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$91$$ $$101$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.93649 + 0.500000i 0.968246 + 0.250000i
$$3$$ 0 0
$$4$$ 3.50000 + 1.93649i 0.875000 + 0.484123i
$$5$$ 5.00000i 1.00000i
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 5.80948 + 5.50000i 0.726184 + 0.687500i
$$9$$ 0 0
$$10$$ −2.50000 + 9.68246i −0.250000 + 0.968246i
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 8.50000 + 13.5554i 0.531250 + 0.847215i
$$17$$ 14.0000i 0.823529i 0.911290 + 0.411765i $$0.135087\pi$$
−0.911290 + 0.411765i $$0.864913\pi$$
$$18$$ 0 0
$$19$$ 30.9839i 1.63073i −0.578947 0.815365i $$-0.696536\pi$$
0.578947 0.815365i $$-0.303464\pi$$
$$20$$ −9.68246 + 17.5000i −0.484123 + 0.875000i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 30.9839 1.34712 0.673562 0.739130i $$-0.264763\pi$$
0.673562 + 0.739130i $$0.264763\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 61.9677i 1.99896i −0.0322581 0.999480i $$-0.510270\pi$$
0.0322581 0.999480i $$-0.489730\pi$$
$$32$$ 9.68246 + 30.5000i 0.302577 + 0.953125i
$$33$$ 0 0
$$34$$ −7.00000 + 27.1109i −0.205882 + 0.797379i
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 15.4919 60.0000i 0.407682 1.57895i
$$39$$ 0 0
$$40$$ −27.5000 + 29.0474i −0.687500 + 0.726184i
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 60.0000 + 15.4919i 1.30435 + 0.336781i
$$47$$ −92.9516 −1.97769 −0.988847 0.148936i $$-0.952415\pi$$
−0.988847 + 0.148936i $$0.952415\pi$$
$$48$$ 0 0
$$49$$ −49.0000 −1.00000
$$50$$ −48.4123 12.5000i −0.968246 0.250000i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 86.0000i 1.62264i −0.584601 0.811321i $$-0.698749\pi$$
0.584601 0.811321i $$-0.301251\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 118.000 1.93443 0.967213 0.253966i $$-0.0817352\pi$$
0.967213 + 0.253966i $$0.0817352\pi$$
$$62$$ 30.9839 120.000i 0.499740 1.93548i
$$63$$ 0 0
$$64$$ 3.50000 + 63.9042i 0.0546875 + 0.998504i
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ −27.1109 + 49.0000i −0.398689 + 0.720588i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 60.0000 108.444i 0.789474 1.42689i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 123.935i 1.56880i 0.620253 + 0.784402i $$0.287030\pi$$
−0.620253 + 0.784402i $$0.712970\pi$$
$$80$$ −67.7772 + 42.5000i −0.847215 + 0.531250i
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 61.9677 0.746599 0.373300 0.927711i $$-0.378226\pi$$
0.373300 + 0.927711i $$0.378226\pi$$
$$84$$ 0 0
$$85$$ −70.0000 −0.823529
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 108.444 + 60.0000i 1.17873 + 0.652174i
$$93$$ 0 0
$$94$$ −180.000 46.4758i −1.91489 0.494423i
$$95$$ 154.919 1.63073
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ −94.8881 24.5000i −0.968246 0.250000i
$$99$$ 0 0
$$100$$ −87.5000 48.4123i −0.875000 0.484123i
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 43.0000 166.538i 0.405660 1.57112i
$$107$$ −185.903 −1.73741 −0.868707 0.495327i $$-0.835048\pi$$
−0.868707 + 0.495327i $$0.835048\pi$$
$$108$$ 0 0
$$109$$ −22.0000 −0.201835 −0.100917 0.994895i $$-0.532178\pi$$
−0.100917 + 0.994895i $$0.532178\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 206.000i 1.82301i 0.411290 + 0.911504i $$0.365078\pi$$
−0.411290 + 0.911504i $$0.634922\pi$$
$$114$$ 0 0
$$115$$ 154.919i 1.34712i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 121.000 1.00000
$$122$$ 228.506 + 59.0000i 1.87300 + 0.483607i
$$123$$ 0 0
$$124$$ 120.000 216.887i 0.967742 1.74909i
$$125$$ 125.000i 1.00000i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −25.1744 + 125.500i −0.196675 + 0.980469i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ −77.0000 + 81.3327i −0.566176 + 0.598034i
$$137$$ 226.000i 1.64964i 0.565399 + 0.824818i $$0.308722\pi$$
−0.565399 + 0.824818i $$0.691278\pi$$
$$138$$ 0 0
$$139$$ 92.9516i 0.668717i 0.942446 + 0.334358i $$0.108520\pi$$
−0.942446 + 0.334358i $$0.891480\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 185.903i 1.23115i 0.788079 + 0.615574i $$0.211076\pi$$
−0.788079 + 0.615574i $$0.788924\pi$$
$$152$$ 170.411 180.000i 1.12113 1.18421i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 309.839 1.99896
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ −61.9677 + 240.000i −0.392201 + 1.51899i
$$159$$ 0 0
$$160$$ −152.500 + 48.4123i −0.953125 + 0.302577i
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 120.000 + 30.9839i 0.722892 + 0.186650i
$$167$$ −216.887 −1.29872 −0.649362 0.760479i $$-0.724964\pi$$
−0.649362 + 0.760479i $$0.724964\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ −135.554 35.0000i −0.797379 0.205882i
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 154.000i 0.890173i −0.895487 0.445087i $$-0.853173\pi$$
0.895487 0.445087i $$-0.146827\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −122.000 −0.674033 −0.337017 0.941499i $$-0.609418\pi$$
−0.337017 + 0.941499i $$0.609418\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 180.000 + 170.411i 0.978261 + 0.926148i
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −325.331 180.000i −1.73048 0.957447i
$$189$$ 0 0
$$190$$ 300.000 + 77.4597i 1.57895 + 0.407682i
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −171.500 94.8881i −0.875000 0.484123i
$$197$$ 374.000i 1.89848i −0.314557 0.949239i $$-0.601856\pi$$
0.314557 0.949239i $$-0.398144\pi$$
$$198$$ 0 0
$$199$$ 371.806i 1.86837i −0.356784 0.934187i $$-0.616127\pi$$
0.356784 0.934187i $$-0.383873\pi$$
$$200$$ −145.237 137.500i −0.726184 0.687500i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 216.887i 1.02790i −0.857820 0.513950i $$-0.828182\pi$$
0.857820 0.513950i $$-0.171818\pi$$
$$212$$ 166.538 301.000i 0.785558 1.41981i
$$213$$ 0 0
$$214$$ −360.000 92.9516i −1.68224 0.434353i
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −42.6028 11.0000i −0.195426 0.0504587i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −103.000 + 398.917i −0.455752 + 1.76512i
$$227$$ −433.774 −1.91090 −0.955450 0.295154i $$-0.904629\pi$$
−0.955450 + 0.295154i $$0.904629\pi$$
$$228$$ 0 0
$$229$$ −218.000 −0.951965 −0.475983 0.879455i $$-0.657907\pi$$
−0.475983 + 0.879455i $$0.657907\pi$$
$$230$$ −77.4597 + 300.000i −0.336781 + 1.30435i
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 34.0000i 0.145923i 0.997335 + 0.0729614i $$0.0232450\pi$$
−0.997335 + 0.0729614i $$0.976755\pi$$
$$234$$ 0 0
$$235$$ 464.758i 1.97769i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −478.000 −1.98340 −0.991701 0.128564i $$-0.958963\pi$$
−0.991701 + 0.128564i $$0.958963\pi$$
$$242$$ 234.315 + 60.5000i 0.968246 + 0.250000i
$$243$$ 0 0
$$244$$ 413.000 + 228.506i 1.69262 + 0.936500i
$$245$$ 245.000i 1.00000i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 340.823 360.000i 1.37428 1.45161i
$$249$$ 0 0
$$250$$ 62.5000 242.061i 0.250000 0.968246i
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −111.500 + 230.443i −0.435547 + 0.900166i
$$257$$ 466.000i 1.81323i 0.421959 + 0.906615i $$0.361343\pi$$
−0.421959 + 0.906615i $$0.638657\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 278.855 1.06028 0.530142 0.847909i $$-0.322139\pi$$
0.530142 + 0.847909i $$0.322139\pi$$
$$264$$ 0 0
$$265$$ 430.000 1.62264
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 247.871i 0.914653i −0.889299 0.457326i $$-0.848807\pi$$
0.889299 0.457326i $$-0.151193\pi$$
$$272$$ −189.776 + 119.000i −0.697707 + 0.437500i
$$273$$ 0 0
$$274$$ −113.000 + 437.647i −0.412409 + 1.59725i
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ −46.4758 + 180.000i −0.167179 + 0.647482i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 93.0000 0.321799
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 394.000i 1.34471i −0.740229 0.672355i $$-0.765283\pi$$
0.740229 0.672355i $$-0.234717\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −92.9516 + 360.000i −0.307787 + 1.19205i
$$303$$ 0 0
$$304$$ 420.000 263.363i 1.38158 0.866325i
$$305$$ 590.000i 1.93443i
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 600.000 + 154.919i 1.93548 + 0.499740i
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −240.000 + 433.774i −0.759494 + 1.37270i
$$317$$ 134.000i 0.422713i 0.977409 + 0.211356i $$0.0677881\pi$$
−0.977409 + 0.211356i $$0.932212\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −319.521 + 17.5000i −0.998504 + 0.0546875i
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 433.774 1.34295
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 650.661i 1.96574i 0.184290 + 0.982872i $$0.441001\pi$$
−0.184290 + 0.982872i $$0.558999\pi$$
$$332$$ 216.887 + 120.000i 0.653274 + 0.361446i
$$333$$ 0 0
$$334$$ −420.000 108.444i −1.25749 0.324681i
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 327.267 + 84.5000i 0.968246 + 0.250000i
$$339$$ 0 0
$$340$$ −245.000 135.554i −0.720588 0.398689i
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 77.0000 298.220i 0.222543 0.861907i
$$347$$ 371.806 1.07149 0.535744 0.844380i $$-0.320031\pi$$
0.535744 + 0.844380i $$0.320031\pi$$
$$348$$ 0 0
$$349$$ 458.000 1.31232 0.656160 0.754621i $$-0.272179\pi$$
0.656160 + 0.754621i $$0.272179\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 274.000i 0.776204i 0.921616 + 0.388102i $$0.126869\pi$$
−0.921616 + 0.388102i $$0.873131\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ −599.000 −1.65928
$$362$$ −236.252 61.0000i −0.652630 0.168508i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 263.363 + 420.000i 0.715660 + 1.14130i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −540.000 511.234i −1.43617 1.35966i
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 154.919i 0.408758i −0.978892 0.204379i $$-0.934482\pi$$
0.978892 0.204379i $$-0.0655175\pi$$
$$380$$ 542.218 + 300.000i 1.42689 + 0.789474i
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −340.823 −0.889876 −0.444938 0.895561i $$-0.646774\pi$$
−0.444938 + 0.895561i $$0.646774\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 433.774i 1.10940i
$$392$$ −284.664 269.500i −0.726184 0.687500i
$$393$$ 0 0
$$394$$ 187.000 724.248i 0.474619 1.83819i
$$395$$ −619.677 −1.56880
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 185.903 720.000i 0.467093 1.80905i
$$399$$ 0 0
$$400$$ −212.500 338.886i −0.531250 0.847215i
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −142.000 −0.347188 −0.173594 0.984817i $$-0.555538\pi$$
−0.173594 + 0.984817i $$0.555538\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 309.839i 0.746599i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 602.000 1.42993 0.714964 0.699161i $$-0.246443\pi$$
0.714964 + 0.699161i $$0.246443\pi$$
$$422$$ 108.444 420.000i 0.256975 0.995261i
$$423$$ 0 0
$$424$$ 473.000 499.615i 1.11557 1.17834i
$$425$$ 350.000i 0.823529i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −650.661 360.000i −1.52024 0.841121i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −77.0000 42.6028i −0.176606 0.0977129i
$$437$$ 960.000i 2.19680i
$$438$$ 0 0
$$439$$ 619.677i 1.41157i 0.708428 + 0.705783i $$0.249405\pi$$
−0.708428 + 0.705783i $$0.750595\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −681.645 −1.53870 −0.769351 0.638826i $$-0.779420\pi$$
−0.769351 + 0.638826i $$0.779420\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −398.917 + 721.000i −0.882560 + 1.59513i
$$453$$ 0 0
$$454$$ −840.000 216.887i −1.85022 0.477725i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ −422.155 109.000i −0.921736 0.237991i
$$459$$ 0 0
$$460$$ −300.000 + 542.218i −0.652174 + 1.17873i
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −17.0000 + 65.8407i −0.0364807 + 0.141289i
$$467$$ 867.548 1.85771 0.928853 0.370450i $$-0.120796\pi$$
0.928853 + 0.370450i $$0.120796\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 232.379 900.000i 0.494423 1.91489i
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 774.597i 1.63073i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −925.643 239.000i −1.92042 0.495851i
$$483$$ 0 0
$$484$$ 423.500 + 234.315i 0.875000 + 0.484123i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 685.518 + 649.000i 1.40475 + 1.32992i
$$489$$ 0 0
$$490$$ 122.500 474.440i 0.250000 0.968246i
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 840.000 526.726i 1.69355 1.06195i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 340.823i 0.683011i 0.939880 + 0.341506i $$0.110937\pi$$
−0.939880 + 0.341506i $$0.889063\pi$$
$$500$$ 242.061 437.500i 0.484123 0.875000i
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −154.919 −0.307991 −0.153995 0.988072i $$-0.549214\pi$$
−0.153995 + 0.988072i $$0.549214\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −331.140 + 390.500i −0.646758 + 0.762695i
$$513$$ 0 0
$$514$$ −233.000 + 902.405i −0.453307 + 1.75565i
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 540.000 + 139.427i 1.02662 + 0.265071i
$$527$$ 867.548 1.64620
$$528$$ 0 0
$$529$$ 431.000 0.814745
$$530$$ 832.691 + 215.000i 1.57112 + 0.405660i
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 929.516i 1.73741i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1078.00 1.99261 0.996303 0.0859072i $$-0.0273789\pi$$
0.996303 + 0.0859072i $$0.0273789\pi$$
$$542$$ 123.935 480.000i 0.228663 0.885609i
$$543$$ 0 0
$$544$$ −427.000 + 135.554i −0.784926 + 0.249181i
$$545$$ 110.000i 0.201835i
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ −437.647 + 791.000i −0.798626 + 1.44343i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −180.000 + 325.331i −0.323741 + 0.585127i
$$557$$ 614.000i 1.10233i 0.834395 + 0.551167i $$0.185817\pi$$
−0.834395 + 0.551167i $$0.814183\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1115.42 −1.98121 −0.990603 0.136767i $$-0.956329\pi$$
−0.990603 + 0.136767i $$0.956329\pi$$
$$564$$ 0 0
$$565$$ −1030.00 −1.82301
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 1084.44i 1.89919i −0.313485 0.949593i $$-0.601497\pi$$
0.313485 0.949593i $$-0.398503\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −774.597 −1.34712
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 180.094 + 46.5000i 0.311581 + 0.0804498i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 197.000 762.978i 0.336177 1.30201i
$$587$$ 805.581 1.37237 0.686184 0.727428i $$-0.259284\pi$$
0.686184 + 0.727428i $$0.259284\pi$$
$$588$$ 0 0
$$589$$ −1920.00 −3.25976
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1166.00i 1.96627i 0.182873 + 0.983137i $$0.441460\pi$$
−0.182873 + 0.983137i $$0.558540\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ −242.000 −0.402662 −0.201331 0.979523i $$-0.564527\pi$$
−0.201331 + 0.979523i $$0.564527\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −360.000 + 650.661i −0.596026 + 1.07725i
$$605$$ 605.000i 1.00000i
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 945.008 300.000i 1.55429 0.493421i
$$609$$ 0 0
$$610$$ −295.000 + 1142.53i −0.483607 + 1.87300i
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1186.00i 1.92220i −0.276193 0.961102i $$-0.589073\pi$$
0.276193 0.961102i $$-0.410927\pi$$
$$618$$ 0 0
$$619$$ 1022.47i 1.65181i −0.563813 0.825903i $$-0.690666\pi$$
0.563813 0.825903i $$-0.309334\pi$$
$$620$$ 1084.44 + 600.000i 1.74909 + 0.967742i
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 1239.35i 1.96411i −0.188590 0.982056i $$-0.560392\pi$$
0.188590 0.982056i $$-0.439608\pi$$
$$632$$ −681.645 + 720.000i −1.07855 + 1.13924i
$$633$$ 0 0
$$634$$ −67.0000 + 259.490i −0.105678 + 0.409290i
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −627.500 125.872i −0.980469 0.196675i
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 840.000 + 216.887i 1.30031 + 0.335738i
$$647$$ 1084.44 1.67610 0.838049 0.545595i $$-0.183696\pi$$
0.838049 + 0.545595i $$0.183696\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1114.00i 1.70597i 0.521933 + 0.852986i $$0.325211\pi$$
−0.521933 + 0.852986i $$0.674789\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 838.000 1.26778 0.633888 0.773425i $$-0.281458\pi$$
0.633888 + 0.773425i $$0.281458\pi$$
$$662$$ −325.331 + 1260.00i −0.491436 + 1.90332i
$$663$$ 0 0
$$664$$ 360.000 + 340.823i 0.542169 + 0.513287i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −759.105 420.000i −1.13638 0.628743i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 591.500 + 327.267i 0.875000 + 0.484123i
$$677$$ 374.000i 0.552437i 0.961095 + 0.276219i $$0.0890814\pi$$
−0.961095 + 0.276219i $$0.910919\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −406.663 385.000i −0.598034 0.566176i
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1363.29 1.99603 0.998016 0.0629575i $$-0.0200533\pi$$
0.998016 + 0.0629575i $$0.0200533\pi$$
$$684$$ 0 0
$$685$$ −1130.00 −1.64964
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 402.790i 0.582909i −0.956585 0.291455i $$-0.905861\pi$$
0.956585 0.291455i $$-0.0941392\pi$$
$$692$$ 298.220 539.000i 0.430953 0.778902i
$$693$$ 0 0
$$694$$ 720.000 + 185.903i 1.03746 + 0.267872i
$$695$$ −464.758 −0.668717
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 886.913 + 229.000i 1.27065 + 0.328080i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −137.000 + 530.599i −0.194051 + 0.751556i
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 742.000 1.04654 0.523272 0.852166i $$-0.324711\pi$$
0.523272 + 0.852166i $$0.324711\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1920.00i 2.69285i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −1159.96 299.500i −1.60659 0.414820i
$$723$$ 0 0
$$724$$ −427.000 236.252i −0.589779 0.326315i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 300.000 + 945.008i 0.407609 + 1.28398i
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 216.887i 0.293487i 0.989175 + 0.146744i $$0.0468792\pi$$
−0.989175 + 0.146744i $$0.953121\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −1394.27 −1.87655 −0.938273 0.345895i $$-0.887575\pi$$
−0.938273 + 0.345895i $$0.887575\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 433.774i 0.577595i 0.957390 + 0.288798i $$0.0932555\pi$$
−0.957390 + 0.288798i $$0.906745\pi$$
$$752$$ −790.089 1260.00i −1.05065 1.67553i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −929.516 −1.23115
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 77.4597 300.000i 0.102190 0.395778i
$$759$$ 0 0
$$760$$ 900.000 + 852.056i 1.18421 + 1.12113i
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −660.000 170.411i −0.861619 0.222469i
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 578.000 0.751625 0.375813 0.926696i $$-0.377364\pi$$
0.375813 + 0.926696i $$0.377364\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1526.00i 1.97413i −0.160330 0.987063i $$-0.551256\pi$$
0.160330 0.987063i $$-0.448744\pi$$
$$774$$ 0 0
$$775$$ 1549.19i 1.99896i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −216.887 + 840.000i −0.277349 + 1.07417i
$$783$$ 0 0
$$784$$ −416.500 664.217i −0.531250 0.847215i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 724.248 1309.00i 0.919096 1.66117i
$$789$$ 0 0
$$790$$ −1200.00 309.839i −1.51899 0.392201i
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 720.000 1301.32i 0.904523 1.63483i
$$797$$ 826.000i 1.03639i 0.855264 + 0.518193i $$0.173395\pi$$
−0.855264 + 0.518193i $$0.826605\pi$$
$$798$$ 0 0
$$799$$ 1301.32i 1.62869i
$$800$$ −242.061 762.500i −0.302577 0.953125i
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$