# Properties

 Label 3744.2.m.h.1585.8 Level $3744$ Weight $2$ Character 3744.1585 Analytic conductor $29.896$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3744.m (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.8959905168$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625$$ x^16 + 2*x^14 - 16*x^12 - 72*x^10 + 26*x^8 + 360*x^6 + 725*x^4 + 1000*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{20}$$ Twist minimal: no (minimal twist has level 936) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1585.8 Root $$-1.90184 - 0.0324487i$$ of defining polynomial Character $$\chi$$ $$=$$ 3744.1585 Dual form 3744.2.m.h.1585.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.66251 q^{5} +3.57266i q^{7} +O(q^{10})$$ $$q-1.66251 q^{5} +3.57266i q^{7} +1.02749 q^{11} +(1.87826 - 3.07768i) q^{13} +5.05251 q^{17} -1.16083 q^{19} +8.17513 q^{23} -2.23607 q^{25} +4.29792i q^{29} -7.98872i q^{31} -5.93958i q^{35} -9.83470 q^{37} -1.62054i q^{41} +2.35114i q^{43} +7.73877i q^{47} -5.76393 q^{49} -11.2521i q^{53} -1.70820 q^{55} +9.73249 q^{59} -11.4127i q^{61} +(-3.12262 + 5.11667i) q^{65} +7.23901 q^{67} +10.5672i q^{71} +4.41606i q^{73} +3.67086i q^{77} +6.94427 q^{79} +2.29753 q^{83} -8.39984 q^{85} +9.02546i q^{89} +(10.9955 + 6.71040i) q^{91} +1.92989 q^{95} +11.5614i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q - 128 q^{49} + 80 q^{55} - 32 q^{79}+O(q^{100})$$ 16 * q - 128 * q^49 + 80 * q^55 - 32 * q^79

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$2017$$ $$2081$$ $$2341$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.66251 −0.743496 −0.371748 0.928334i $$-0.621241\pi$$
−0.371748 + 0.928334i $$0.621241\pi$$
$$6$$ 0 0
$$7$$ 3.57266i 1.35034i 0.737662 + 0.675170i $$0.235930\pi$$
−0.737662 + 0.675170i $$0.764070\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.02749 0.309799 0.154899 0.987930i $$-0.450495\pi$$
0.154899 + 0.987930i $$0.450495\pi$$
$$12$$ 0 0
$$13$$ 1.87826 3.07768i 0.520936 0.853596i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.05251 1.22541 0.612707 0.790310i $$-0.290081\pi$$
0.612707 + 0.790310i $$0.290081\pi$$
$$18$$ 0 0
$$19$$ −1.16083 −0.266312 −0.133156 0.991095i $$-0.542511\pi$$
−0.133156 + 0.991095i $$0.542511\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.17513 1.70463 0.852317 0.523026i $$-0.175197\pi$$
0.852317 + 0.523026i $$0.175197\pi$$
$$24$$ 0 0
$$25$$ −2.23607 −0.447214
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 4.29792i 0.798104i 0.916928 + 0.399052i $$0.130661\pi$$
−0.916928 + 0.399052i $$0.869339\pi$$
$$30$$ 0 0
$$31$$ 7.98872i 1.43482i −0.696653 0.717408i $$-0.745328\pi$$
0.696653 0.717408i $$-0.254672\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 5.93958i 1.00397i
$$36$$ 0 0
$$37$$ −9.83470 −1.61682 −0.808408 0.588623i $$-0.799670\pi$$
−0.808408 + 0.588623i $$0.799670\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.62054i 0.253087i −0.991961 0.126543i $$-0.959612\pi$$
0.991961 0.126543i $$-0.0403883\pi$$
$$42$$ 0 0
$$43$$ 2.35114i 0.358546i 0.983799 + 0.179273i $$0.0573745\pi$$
−0.983799 + 0.179273i $$0.942626\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.73877i 1.12882i 0.825496 + 0.564408i $$0.190895\pi$$
−0.825496 + 0.564408i $$0.809105\pi$$
$$48$$ 0 0
$$49$$ −5.76393 −0.823419
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 11.2521i 1.54560i −0.634652 0.772798i $$-0.718857\pi$$
0.634652 0.772798i $$-0.281143\pi$$
$$54$$ 0 0
$$55$$ −1.70820 −0.230334
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 9.73249 1.26706 0.633531 0.773717i $$-0.281605\pi$$
0.633531 + 0.773717i $$0.281605\pi$$
$$60$$ 0 0
$$61$$ 11.4127i 1.46124i −0.682782 0.730622i $$-0.739230\pi$$
0.682782 0.730622i $$-0.260770\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.12262 + 5.11667i −0.387314 + 0.634645i
$$66$$ 0 0
$$67$$ 7.23901 0.884386 0.442193 0.896920i $$-0.354201\pi$$
0.442193 + 0.896920i $$0.354201\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.5672i 1.25410i 0.778981 + 0.627048i $$0.215737\pi$$
−0.778981 + 0.627048i $$0.784263\pi$$
$$72$$ 0 0
$$73$$ 4.41606i 0.516860i 0.966030 + 0.258430i $$0.0832052\pi$$
−0.966030 + 0.258430i $$0.916795\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.67086i 0.418334i
$$78$$ 0 0
$$79$$ 6.94427 0.781292 0.390646 0.920541i $$-0.372252\pi$$
0.390646 + 0.920541i $$0.372252\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 2.29753 0.252187 0.126093 0.992018i $$-0.459756\pi$$
0.126093 + 0.992018i $$0.459756\pi$$
$$84$$ 0 0
$$85$$ −8.39984 −0.911090
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 9.02546i 0.956697i 0.878170 + 0.478349i $$0.158764\pi$$
−0.878170 + 0.478349i $$0.841236\pi$$
$$90$$ 0 0
$$91$$ 10.9955 + 6.71040i 1.15264 + 0.703441i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.92989 0.198002
$$96$$ 0 0
$$97$$ 11.5614i 1.17388i 0.809630 + 0.586940i $$0.199668\pi$$
−0.809630 + 0.586940i $$0.800332\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 7.96879i 0.792924i 0.918051 + 0.396462i $$0.129762\pi$$
−0.918051 + 0.396462i $$0.870238\pi$$
$$102$$ 0 0
$$103$$ 11.7082 1.15364 0.576822 0.816870i $$-0.304293\pi$$
0.576822 + 0.816870i $$0.304293\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.65626i 0.256791i 0.991723 + 0.128395i $$0.0409826\pi$$
−0.991723 + 0.128395i $$0.959017\pi$$
$$108$$ 0 0
$$109$$ 12.1564 1.16437 0.582184 0.813057i $$-0.302198\pi$$
0.582184 + 0.813057i $$0.302198\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 5.05251 0.475300 0.237650 0.971351i $$-0.423623\pi$$
0.237650 + 0.971351i $$0.423623\pi$$
$$114$$ 0 0
$$115$$ −13.5912 −1.26739
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 18.0509i 1.65473i
$$120$$ 0 0
$$121$$ −9.94427 −0.904025
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 12.0300 1.07600
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.5646i 1.44726i −0.690189 0.723629i $$-0.742473\pi$$
0.690189 0.723629i $$-0.257527\pi$$
$$132$$ 0 0
$$133$$ 4.14725i 0.359612i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 13.6020i 1.16209i 0.813870 + 0.581047i $$0.197357\pi$$
−0.813870 + 0.581047i $$0.802643\pi$$
$$138$$ 0 0
$$139$$ 19.3642i 1.64245i 0.570607 + 0.821223i $$0.306708\pi$$
−0.570607 + 0.821223i $$0.693292\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.92989 3.16228i 0.161385 0.264443i
$$144$$ 0 0
$$145$$ 7.14533i 0.593387i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −7.04250 −0.576944 −0.288472 0.957488i $$-0.593147\pi$$
−0.288472 + 0.957488i $$0.593147\pi$$
$$150$$ 0 0
$$151$$ 3.57266i 0.290739i −0.989377 0.145370i $$-0.953563\pi$$
0.989377 0.145370i $$-0.0464371\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 13.2813i 1.06678i
$$156$$ 0 0
$$157$$ 0.555029i 0.0442961i −0.999755 0.0221481i $$-0.992949\pi$$
0.999755 0.0221481i $$-0.00705053\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 29.2070i 2.30183i
$$162$$ 0 0
$$163$$ 7.23901 0.567003 0.283501 0.958972i $$-0.408504\pi$$
0.283501 + 0.958972i $$0.408504\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.32611i 0.566911i 0.958985 + 0.283456i $$0.0914809\pi$$
−0.958985 + 0.283456i $$0.908519\pi$$
$$168$$ 0 0
$$169$$ −5.94427 11.5614i −0.457252 0.889337i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 11.2521i 0.855482i −0.903901 0.427741i $$-0.859310\pi$$
0.903901 0.427741i $$-0.140690\pi$$
$$174$$ 0 0
$$175$$ 7.98872i 0.603891i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 21.8772i 1.63518i 0.575804 + 0.817588i $$0.304689\pi$$
−0.575804 + 0.817588i $$0.695311\pi$$
$$180$$ 0 0
$$181$$ 12.8658i 0.956305i 0.878277 + 0.478152i $$0.158693\pi$$
−0.878277 + 0.478152i $$0.841307\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 16.3503 1.20210
$$186$$ 0 0
$$187$$ 5.19139 0.379632
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.92989 0.139642 0.0698209 0.997560i $$-0.477757\pi$$
0.0698209 + 0.997560i $$0.477757\pi$$
$$192$$ 0 0
$$193$$ 4.41606i 0.317875i 0.987289 + 0.158937i $$0.0508068\pi$$
−0.987289 + 0.158937i $$0.949193\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.1825 1.65168 0.825841 0.563903i $$-0.190701\pi$$
0.825841 + 0.563903i $$0.190701\pi$$
$$198$$ 0 0
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −15.3550 −1.07771
$$204$$ 0 0
$$205$$ 2.69417i 0.188169i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.19274 −0.0825033
$$210$$ 0 0
$$211$$ 10.5146i 0.723856i 0.932206 + 0.361928i $$0.117881\pi$$
−0.932206 + 0.361928i $$0.882119\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 3.90879i 0.266577i
$$216$$ 0 0
$$217$$ 28.5410 1.93749
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.48993 15.5500i 0.638362 1.04601i
$$222$$ 0 0
$$223$$ 7.98872i 0.534964i 0.963563 + 0.267482i $$0.0861916\pi$$
−0.963563 + 0.267482i $$0.913808\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −4.35250 −0.288886 −0.144443 0.989513i $$-0.546139\pi$$
−0.144443 + 0.989513i $$0.546139\pi$$
$$228$$ 0 0
$$229$$ −1.43486 −0.0948185 −0.0474092 0.998876i $$-0.515096\pi$$
−0.0474092 + 0.998876i $$0.515096\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 16.3503 1.07114 0.535571 0.844490i $$-0.320096\pi$$
0.535571 + 0.844490i $$0.320096\pi$$
$$234$$ 0 0
$$235$$ 12.8658i 0.839270i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 7.07107i 0.457389i −0.973498 0.228695i $$-0.926554\pi$$
0.973498 0.228695i $$-0.0734457\pi$$
$$240$$ 0 0
$$241$$ 15.9774i 1.02920i −0.857431 0.514599i $$-0.827941\pi$$
0.857431 0.514599i $$-0.172059\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 9.58258 0.612209
$$246$$ 0 0
$$247$$ −2.18034 + 3.57266i −0.138732 + 0.227323i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 22.5042i 1.42045i −0.703973 0.710227i $$-0.748592\pi$$
0.703973 0.710227i $$-0.251408\pi$$
$$252$$ 0 0
$$253$$ 8.39984 0.528093
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.5955 1.40947 0.704735 0.709471i $$-0.251066\pi$$
0.704735 + 0.709471i $$0.251066\pi$$
$$258$$ 0 0
$$259$$ 35.1361i 2.18325i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.3503 1.00820 0.504100 0.863645i $$-0.331824\pi$$
0.504100 + 0.863645i $$0.331824\pi$$
$$264$$ 0 0
$$265$$ 18.7067i 1.14914i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 7.58124i 0.462237i −0.972926 0.231118i $$-0.925762\pi$$
0.972926 0.231118i $$-0.0742384\pi$$
$$270$$ 0 0
$$271$$ 7.98872i 0.485280i 0.970116 + 0.242640i $$0.0780134\pi$$
−0.970116 + 0.242640i $$0.921987\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −2.29753 −0.138546
$$276$$ 0 0
$$277$$ 18.1231i 1.08891i 0.838790 + 0.544455i $$0.183263\pi$$
−0.838790 + 0.544455i $$0.816737\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6.19704i 0.369684i −0.982768 0.184842i $$-0.940823\pi$$
0.982768 0.184842i $$-0.0591774\pi$$
$$282$$ 0 0
$$283$$ 2.90617i 0.172754i 0.996263 + 0.0863769i $$0.0275289\pi$$
−0.996263 + 0.0863769i $$0.972471\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 5.78966 0.341753
$$288$$ 0 0
$$289$$ 8.52786 0.501639
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −0.877578 −0.0512687 −0.0256343 0.999671i $$-0.508161\pi$$
−0.0256343 + 0.999671i $$0.508161\pi$$
$$294$$ 0 0
$$295$$ −16.1803 −0.942056
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 15.3550 25.1605i 0.888005 1.45507i
$$300$$ 0 0
$$301$$ −8.39984 −0.484159
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 18.9737i 1.08643i
$$306$$ 0 0
$$307$$ −14.7521 −0.841944 −0.420972 0.907074i $$-0.638311\pi$$
−0.420972 + 0.907074i $$0.638311\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.5254 1.39071 0.695354 0.718667i $$-0.255248\pi$$
0.695354 + 0.718667i $$0.255248\pi$$
$$312$$ 0 0
$$313$$ 6.47214 0.365827 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −31.1025 −1.74689 −0.873446 0.486921i $$-0.838120\pi$$
−0.873446 + 0.486921i $$0.838120\pi$$
$$318$$ 0 0
$$319$$ 4.41606i 0.247252i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5.86510 −0.326343
$$324$$ 0 0
$$325$$ −4.19992 + 6.88191i −0.232970 + 0.381740i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −27.6480 −1.52428
$$330$$ 0 0
$$331$$ −10.9955 −0.604369 −0.302185 0.953249i $$-0.597716\pi$$
−0.302185 + 0.953249i $$0.597716\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −12.0349 −0.657537
$$336$$ 0 0
$$337$$ 5.52786 0.301122 0.150561 0.988601i $$-0.451892\pi$$
0.150561 + 0.988601i $$0.451892\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.20830i 0.444504i
$$342$$ 0 0
$$343$$ 4.41606i 0.238445i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 20.4750i 1.09916i 0.835442 + 0.549578i $$0.185212\pi$$
−0.835442 + 0.549578i $$0.814788\pi$$
$$348$$ 0 0
$$349$$ 13.5912 0.727522 0.363761 0.931492i $$-0.381493\pi$$
0.363761 + 0.931492i $$0.381493\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 20.5942i 1.09612i −0.836439 0.548060i $$-0.815367\pi$$
0.836439 0.548060i $$-0.184633\pi$$
$$354$$ 0 0
$$355$$ 17.5680i 0.932415i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.1142i 1.69493i 0.530855 + 0.847463i $$0.321871\pi$$
−0.530855 + 0.847463i $$0.678129\pi$$
$$360$$ 0 0
$$361$$ −17.6525 −0.929078
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 7.34173i 0.384284i
$$366$$ 0 0
$$367$$ −31.1246 −1.62469 −0.812346 0.583176i $$-0.801810\pi$$
−0.812346 + 0.583176i $$0.801810\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 40.2000 2.08708
$$372$$ 0 0
$$373$$ 26.0746i 1.35009i −0.737777 0.675045i $$-0.764124\pi$$
0.737777 0.675045i $$-0.235876\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 13.2276 + 8.07262i 0.681258 + 0.415761i
$$378$$ 0 0
$$379$$ −19.3954 −0.996273 −0.498137 0.867099i $$-0.665982\pi$$
−0.498137 + 0.867099i $$0.665982\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.57494i 0.182671i −0.995820 0.0913354i $$-0.970886\pi$$
0.995820 0.0913354i $$-0.0291135\pi$$
$$384$$ 0 0
$$385$$ 6.10284i 0.311030i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 35.3980i 1.79475i 0.441270 + 0.897374i $$0.354528\pi$$
−0.441270 + 0.897374i $$0.645472\pi$$
$$390$$ 0 0
$$391$$ 41.3050 2.08888
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −11.5449 −0.580887
$$396$$ 0 0
$$397$$ 20.5562 1.03169 0.515843 0.856683i $$-0.327479\pi$$
0.515843 + 0.856683i $$0.327479\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.4149i 1.51885i −0.650596 0.759424i $$-0.725481\pi$$
0.650596 0.759424i $$-0.274519\pi$$
$$402$$ 0 0
$$403$$ −24.5868 15.0049i −1.22475 0.747447i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.1050 −0.500887
$$408$$ 0 0
$$409$$ 34.6842i 1.71502i −0.514466 0.857511i $$-0.672010\pi$$
0.514466 0.857511i $$-0.327990\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 34.7709i 1.71097i
$$414$$ 0 0
$$415$$ −3.81966 −0.187500
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 7.96879i 0.389301i 0.980873 + 0.194650i $$0.0623572\pi$$
−0.980873 + 0.194650i $$0.937643\pi$$
$$420$$ 0 0
$$421$$ −8.39984 −0.409383 −0.204692 0.978827i $$-0.565619\pi$$
−0.204692 + 0.978827i $$0.565619\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −11.2978 −0.548022
$$426$$ 0 0
$$427$$ 40.7737 1.97318
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.82688i 0.0879975i −0.999032 0.0439987i $$-0.985990\pi$$
0.999032 0.0439987i $$-0.0140098\pi$$
$$432$$ 0 0
$$433$$ −8.18034 −0.393122 −0.196561 0.980492i $$-0.562977\pi$$
−0.196561 + 0.980492i $$0.562977\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −9.48993 −0.453965
$$438$$ 0 0
$$439$$ −1.81966 −0.0868476 −0.0434238 0.999057i $$-0.513827\pi$$
−0.0434238 + 0.999057i $$0.513827\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.2521i 0.534604i −0.963613 0.267302i $$-0.913868\pi$$
0.963613 0.267302i $$-0.0861321\pi$$
$$444$$ 0 0
$$445$$ 15.0049i 0.711301i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 4.86163i 0.229435i 0.993398 + 0.114717i $$0.0365962\pi$$
−0.993398 + 0.114717i $$0.963404\pi$$
$$450$$ 0 0
$$451$$ 1.66509i 0.0784059i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −18.2802 11.1561i −0.856987 0.523005i
$$456$$ 0 0
$$457$$ 17.0199i 0.796159i 0.917351 + 0.398079i $$0.130323\pi$$
−0.917351 + 0.398079i $$0.869677\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −21.9124 −1.02056 −0.510282 0.860007i $$-0.670459\pi$$
−0.510282 + 0.860007i $$0.670459\pi$$
$$462$$ 0 0
$$463$$ 3.57266i 0.166036i 0.996548 + 0.0830179i $$0.0264559\pi$$
−0.996548 + 0.0830179i $$0.973544\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.59584i 0.397768i 0.980023 + 0.198884i $$0.0637317\pi$$
−0.980023 + 0.198884i $$0.936268\pi$$
$$468$$ 0 0
$$469$$ 25.8626i 1.19422i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2.41577i 0.111077i
$$474$$ 0 0
$$475$$ 2.59569 0.119099
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 29.2858i 1.33810i 0.743216 + 0.669052i $$0.233300\pi$$
−0.743216 + 0.669052i $$0.766700\pi$$
$$480$$ 0 0
$$481$$ −18.4721 + 30.2681i −0.842257 + 1.38011i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 19.2209i 0.872776i
$$486$$ 0 0
$$487$$ 0.843392i 0.0382177i 0.999817 + 0.0191089i $$0.00608291\pi$$
−0.999817 + 0.0191089i $$0.993917\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2.02920i 0.0915767i 0.998951 + 0.0457883i $$0.0145800\pi$$
−0.998951 + 0.0457883i $$0.985420\pi$$
$$492$$ 0 0
$$493$$ 21.7153i 0.978008i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −37.7530 −1.69346
$$498$$ 0 0
$$499$$ −33.5347 −1.50122 −0.750609 0.660747i $$-0.770240\pi$$
−0.750609 + 0.660747i $$0.770240\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.24525 0.278462 0.139231 0.990260i $$-0.455537\pi$$
0.139231 + 0.990260i $$0.455537\pi$$
$$504$$ 0 0
$$505$$ 13.2482i 0.589536i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9.58258 0.424740 0.212370 0.977189i $$-0.431882\pi$$
0.212370 + 0.977189i $$0.431882\pi$$
$$510$$ 0 0
$$511$$ −15.7771 −0.697937
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −19.4650 −0.857729
$$516$$ 0 0
$$517$$ 7.95148i 0.349706i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1.19274 −0.0522547 −0.0261274 0.999659i $$-0.508318\pi$$
−0.0261274 + 0.999659i $$0.508318\pi$$
$$522$$ 0 0
$$523$$ 24.0664i 1.05235i 0.850376 + 0.526176i $$0.176375\pi$$
−0.850376 + 0.526176i $$0.823625\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 40.3631i 1.75824i
$$528$$ 0 0
$$529$$ 43.8328 1.90577
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −4.98752 3.04381i −0.216034 0.131842i
$$534$$ 0 0
$$535$$ 4.41606i 0.190923i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −5.92236 −0.255094
$$540$$ 0 0
$$541$$ −3.20845 −0.137942 −0.0689711 0.997619i $$-0.521972\pi$$
−0.0689711 + 0.997619i $$0.521972\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −20.2100 −0.865703
$$546$$ 0 0
$$547$$ 42.1895i 1.80389i −0.431847 0.901947i $$-0.642138\pi$$
0.431847 0.901947i $$-0.357862\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.98915i 0.212545i
$$552$$ 0 0
$$553$$ 24.8096i 1.05501i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24.4525 1.03609 0.518043 0.855355i $$-0.326661\pi$$
0.518043 + 0.855355i $$0.326661\pi$$
$$558$$ 0 0
$$559$$ 7.23607 + 4.41606i 0.306053 + 0.186779i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0.627058i 0.0264274i 0.999913 + 0.0132137i $$0.00420617\pi$$
−0.999913 + 0.0132137i $$0.995794\pi$$
$$564$$ 0 0
$$565$$ −8.39984 −0.353384
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.4553 −1.10906 −0.554532 0.832163i $$-0.687103\pi$$
−0.554532 + 0.832163i $$0.687103\pi$$
$$570$$ 0 0
$$571$$ 24.6215i 1.03038i 0.857077 + 0.515188i $$0.172278\pi$$
−0.857077 + 0.515188i $$0.827722\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −18.2802 −0.762335
$$576$$ 0 0
$$577$$ 44.5588i 1.85501i −0.373816 0.927503i $$-0.621951\pi$$
0.373816 0.927503i $$-0.378049\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 8.20830i 0.340538i
$$582$$ 0 0
$$583$$ 11.5614i 0.478824i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 10.5174 0.434100 0.217050 0.976160i $$-0.430356\pi$$
0.217050 + 0.976160i $$0.430356\pi$$
$$588$$ 0 0
$$589$$ 9.27354i 0.382110i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 26.6637i 1.09495i −0.836823 0.547474i $$-0.815589\pi$$
0.836823 0.547474i $$-0.184411\pi$$
$$594$$ 0 0
$$595$$ 30.0098i 1.23028i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −46.6653 −1.90669 −0.953347 0.301877i $$-0.902387\pi$$
−0.953347 + 0.301877i $$0.902387\pi$$
$$600$$ 0 0
$$601$$ 19.4164 0.792012 0.396006 0.918248i $$-0.370396\pi$$
0.396006 + 0.918248i $$0.370396\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 16.5324 0.672139
$$606$$ 0 0
$$607$$ −15.7082 −0.637576 −0.318788 0.947826i $$-0.603276\pi$$
−0.318788 + 0.947826i $$0.603276\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 23.8175 + 14.5354i 0.963552 + 0.588040i
$$612$$ 0 0
$$613$$ 6.96497 0.281313 0.140656 0.990058i $$-0.455079\pi$$
0.140656 + 0.990058i $$0.455079\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.36861i 0.135615i −0.997698 0.0678075i $$-0.978400\pi$$
0.997698 0.0678075i $$-0.0216004\pi$$
$$618$$ 0 0
$$619$$ 29.7781 1.19688 0.598442 0.801166i $$-0.295787\pi$$
0.598442 + 0.801166i $$0.295787\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −32.2450 −1.29187
$$624$$ 0 0
$$625$$ −8.81966 −0.352786
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −49.6899 −1.98127
$$630$$ 0 0
$$631$$ 22.2794i 0.886928i −0.896292 0.443464i $$-0.853749\pi$$
0.896292 0.443464i $$-0.146251\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −9.97505 −0.395848
$$636$$ 0 0
$$637$$ −10.8262 + 17.7396i −0.428948 + 0.702867i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 43.9983 1.73783 0.868914 0.494963i $$-0.164818\pi$$
0.868914 + 0.494963i $$0.164818\pi$$
$$642$$ 0 0
$$643$$ 33.8734 1.33584 0.667918 0.744235i $$-0.267186\pi$$
0.667918 + 0.744235i $$0.267186\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.5603 1.43733 0.718667 0.695354i $$-0.244753\pi$$
0.718667 + 0.695354i $$0.244753\pi$$
$$648$$ 0 0
$$649$$ 10.0000 0.392534
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 33.3688i 1.30582i 0.757435 + 0.652911i $$0.226452\pi$$
−0.757435 + 0.652911i $$0.773548\pi$$
$$654$$ 0 0
$$655$$ 27.5388i 1.07603i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 36.4126i 1.41843i −0.704991 0.709216i $$-0.749049\pi$$
0.704991 0.709216i $$-0.250951\pi$$
$$660$$ 0 0
$$661$$ 46.3038 1.80101 0.900504 0.434847i $$-0.143198\pi$$
0.900504 + 0.434847i $$0.143198\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 6.89484i 0.267370i
$$666$$ 0 0
$$667$$ 35.1361i 1.36047i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 11.7264i 0.452692i
$$672$$ 0 0
$$673$$ −12.6525 −0.487717 −0.243859 0.969811i $$-0.578413\pi$$
−0.243859 + 0.969811i $$0.578413\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 17.8187i 0.684830i −0.939549 0.342415i $$-0.888755\pi$$
0.939549 0.342415i $$-0.111245\pi$$
$$678$$ 0 0
$$679$$ −41.3050 −1.58514
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 10.5174 0.402438 0.201219 0.979546i $$-0.435510\pi$$
0.201219 + 0.979546i $$0.435510\pi$$
$$684$$ 0 0
$$685$$ 22.6134i 0.864012i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −34.6304 21.1344i −1.31931 0.805156i
$$690$$ 0 0
$$691$$ −4.91735 −0.187065 −0.0935324 0.995616i $$-0.529816\pi$$
−0.0935324 + 0.995616i $$0.529816\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 32.1931i 1.22115i
$$696$$ 0 0
$$697$$ 8.18782i 0.310136i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.8021i 1.01230i −0.862445 0.506151i $$-0.831068\pi$$
0.862445 0.506151i $$-0.168932\pi$$
$$702$$ 0 0
$$703$$ 11.4164 0.430578
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −28.4698 −1.07072
$$708$$ 0 0
$$709$$ 8.39984 0.315463 0.157731 0.987482i $$-0.449582\pi$$
0.157731 + 0.987482i $$0.449582\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 65.3089i 2.44584i
$$714$$ 0 0
$$715$$ −3.20845 + 5.25731i −0.119989 + 0.196612i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 12.0349 0.448826 0.224413 0.974494i $$-0.427953\pi$$
0.224413 + 0.974494i $$0.427953\pi$$
$$720$$ 0 0
$$721$$ 41.8295i 1.55781i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 9.61045i 0.356923i
$$726$$ 0 0
$$727$$ 22.3607 0.829312 0.414656 0.909978i $$-0.363902\pi$$
0.414656 + 0.909978i $$0.363902\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 11.8792i 0.439367i
$$732$$ 0 0
$$733$$ −46.8519 −1.73051 −0.865256 0.501330i $$-0.832844\pi$$
−0.865256 + 0.501330i $$0.832844\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7.43798 0.273982
$$738$$ 0 0
$$739$$ −19.9434 −0.733631 −0.366816 0.930294i $$-0.619552\pi$$
−0.366816 + 0.930294i $$0.619552\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 47.5918i 1.74597i −0.487744 0.872987i $$-0.662180\pi$$
0.487744 0.872987i $$-0.337820\pi$$
$$744$$ 0 0
$$745$$ 11.7082 0.428955
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −9.48993 −0.346755
$$750$$ 0 0
$$751$$ 54.5410 1.99023 0.995115 0.0987222i $$-0.0314755\pi$$
0.995115 + 0.0987222i $$0.0314755\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 5.93958i 0.216164i
$$756$$ 0 0
$$757$$ 8.50651i 0.309174i −0.987979 0.154587i $$-0.950595\pi$$
0.987979 0.154587i $$-0.0494047\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 16.4304i 0.595601i −0.954628 0.297800i $$-0.903747\pi$$
0.954628 0.297800i $$-0.0962530\pi$$
$$762$$ 0 0
$$763$$ 43.4306i 1.57229i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.2802 29.9535i 0.660058 1.08156i
$$768$$ 0 0
$$769$$ 24.1653i 0.871422i −0.900087 0.435711i $$-0.856497\pi$$
0.900087 0.435711i $$-0.143503\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −30.3176 −1.09045 −0.545224 0.838290i $$-0.683555\pi$$
−0.545224 + 0.838290i $$0.683555\pi$$
$$774$$ 0 0
$$775$$ 17.8633i 0.641669i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.88118i 0.0674001i
$$780$$ 0 0
$$781$$ 10.8576i 0.388517i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0.922740i 0.0329340i
$$786$$ 0 0
$$787$$ −40.4996 −1.44366 −0.721828 0.692072i $$-0.756698\pi$$
−0.721828 + 0.692072i $$0.756698\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 18.0509i 0.641817i
$$792$$ 0 0
$$793$$ −35.1246 21.4360i −1.24731 0.761214i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 21.4896i 0.761201i 0.924740 + 0.380601i $$0.124283\pi$$
−0.924740 + 0.380601i $$0.875717\pi$$
$$798$$ 0 0
$$799$$ 39.1002i 1.38327i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 4.53744i 0.160123i
$$804$$ 0 0
$$805$$ 48.5569i 1.71141i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0