# Properties

 Label 1728.2.c.d.1727.4 Level $1728$ Weight $2$ Character 1728.1727 Analytic conductor $13.798$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1727.4 Root $$1.22474 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1727 Dual form 1728.2.c.d.1727.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.82843i q^{5} +1.73205i q^{7} +O(q^{10})$$ $$q+2.82843i q^{5} +1.73205i q^{7} +4.89898 q^{11} +1.00000 q^{13} -2.82843i q^{17} +5.19615i q^{19} +4.89898 q^{23} -3.00000 q^{25} -5.65685i q^{29} +3.46410i q^{31} -4.89898 q^{35} +1.00000 q^{37} +5.65685i q^{41} +3.46410i q^{43} -4.89898 q^{47} +4.00000 q^{49} -5.65685i q^{53} +13.8564i q^{55} -4.89898 q^{59} -11.0000 q^{61} +2.82843i q^{65} +12.1244i q^{67} -1.00000 q^{73} +8.48528i q^{77} +1.73205i q^{79} -9.79796 q^{83} +8.00000 q^{85} -2.82843i q^{89} +1.73205i q^{91} -14.6969 q^{95} -13.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{13} - 12q^{25} + 4q^{37} + 16q^{49} - 44q^{61} - 4q^{73} + 32q^{85} - 52q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$5$$ 2.82843i 1.26491i 0.774597 + 0.632456i $$0.217953\pi$$
−0.774597 + 0.632456i $$0.782047\pi$$
$$6$$ 0 0
$$7$$ 1.73205i 0.654654i 0.944911 + 0.327327i $$0.106148\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.89898 1.47710 0.738549 0.674200i $$-0.235511\pi$$
0.738549 + 0.674200i $$0.235511\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.82843i − 0.685994i −0.939336 0.342997i $$-0.888558\pi$$
0.939336 0.342997i $$-0.111442\pi$$
$$18$$ 0 0
$$19$$ 5.19615i 1.19208i 0.802955 + 0.596040i $$0.203260\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.89898 1.02151 0.510754 0.859727i $$-0.329366\pi$$
0.510754 + 0.859727i $$0.329366\pi$$
$$24$$ 0 0
$$25$$ −3.00000 −0.600000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 5.65685i − 1.05045i −0.850963 0.525226i $$-0.823981\pi$$
0.850963 0.525226i $$-0.176019\pi$$
$$30$$ 0 0
$$31$$ 3.46410i 0.622171i 0.950382 + 0.311086i $$0.100693\pi$$
−0.950382 + 0.311086i $$0.899307\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.89898 −0.828079
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.65685i 0.883452i 0.897150 + 0.441726i $$0.145634\pi$$
−0.897150 + 0.441726i $$0.854366\pi$$
$$42$$ 0 0
$$43$$ 3.46410i 0.528271i 0.964486 + 0.264135i $$0.0850865\pi$$
−0.964486 + 0.264135i $$0.914913\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.89898 −0.714590 −0.357295 0.933992i $$-0.616301\pi$$
−0.357295 + 0.933992i $$0.616301\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 5.65685i − 0.777029i −0.921443 0.388514i $$-0.872988\pi$$
0.921443 0.388514i $$-0.127012\pi$$
$$54$$ 0 0
$$55$$ 13.8564i 1.86840i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ 0 0
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.82843i 0.350823i
$$66$$ 0 0
$$67$$ 12.1244i 1.48123i 0.671932 + 0.740613i $$0.265465\pi$$
−0.671932 + 0.740613i $$0.734535\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.48528i 0.966988i
$$78$$ 0 0
$$79$$ 1.73205i 0.194871i 0.995242 + 0.0974355i $$0.0310640\pi$$
−0.995242 + 0.0974355i $$0.968936\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.79796 −1.07547 −0.537733 0.843115i $$-0.680719\pi$$
−0.537733 + 0.843115i $$0.680719\pi$$
$$84$$ 0 0
$$85$$ 8.00000 0.867722
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 2.82843i − 0.299813i −0.988700 0.149906i $$-0.952103\pi$$
0.988700 0.149906i $$-0.0478972\pi$$
$$90$$ 0 0
$$91$$ 1.73205i 0.181568i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −14.6969 −1.50787
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.3137i 1.12576i 0.826540 + 0.562878i $$0.190306\pi$$
−0.826540 + 0.562878i $$0.809694\pi$$
$$102$$ 0 0
$$103$$ 8.66025i 0.853320i 0.904412 + 0.426660i $$0.140310\pi$$
−0.904412 + 0.426660i $$0.859690\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 14.6969 1.42081 0.710403 0.703795i $$-0.248513\pi$$
0.710403 + 0.703795i $$0.248513\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 2.82843i − 0.266076i −0.991111 0.133038i $$-0.957527\pi$$
0.991111 0.133038i $$-0.0424732\pi$$
$$114$$ 0 0
$$115$$ 13.8564i 1.29212i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.89898 0.449089
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 5.65685i 0.505964i
$$126$$ 0 0
$$127$$ − 10.3923i − 0.922168i −0.887357 0.461084i $$-0.847461\pi$$
0.887357 0.461084i $$-0.152539\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.79796 0.856052 0.428026 0.903767i $$-0.359209\pi$$
0.428026 + 0.903767i $$0.359209\pi$$
$$132$$ 0 0
$$133$$ −9.00000 −0.780399
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 19.7990i − 1.69154i −0.533546 0.845771i $$-0.679141\pi$$
0.533546 0.845771i $$-0.320859\pi$$
$$138$$ 0 0
$$139$$ − 8.66025i − 0.734553i −0.930112 0.367277i $$-0.880290\pi$$
0.930112 0.367277i $$-0.119710\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.89898 0.409673
$$144$$ 0 0
$$145$$ 16.0000 1.32873
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 22.6274i − 1.85371i −0.375419 0.926855i $$-0.622501\pi$$
0.375419 0.926855i $$-0.377499\pi$$
$$150$$ 0 0
$$151$$ 22.5167i 1.83238i 0.400744 + 0.916190i $$0.368752\pi$$
−0.400744 + 0.916190i $$0.631248\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −9.79796 −0.786991
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.48528i 0.668734i
$$162$$ 0 0
$$163$$ 5.19615i 0.406994i 0.979076 + 0.203497i $$0.0652307\pi$$
−0.979076 + 0.203497i $$0.934769\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 4.89898 0.379094 0.189547 0.981872i $$-0.439298\pi$$
0.189547 + 0.981872i $$0.439298\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 5.65685i − 0.430083i −0.976605 0.215041i $$-0.931011\pi$$
0.976605 0.215041i $$-0.0689886\pi$$
$$174$$ 0 0
$$175$$ − 5.19615i − 0.392792i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −11.0000 −0.817624 −0.408812 0.912619i $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2.82843i 0.207950i
$$186$$ 0 0
$$187$$ − 13.8564i − 1.01328i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.4949 1.77239 0.886194 0.463314i $$-0.153340\pi$$
0.886194 + 0.463314i $$0.153340\pi$$
$$192$$ 0 0
$$193$$ −1.00000 −0.0719816 −0.0359908 0.999352i $$-0.511459\pi$$
−0.0359908 + 0.999352i $$0.511459\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.82843i 0.201517i 0.994911 + 0.100759i $$0.0321270\pi$$
−0.994911 + 0.100759i $$0.967873\pi$$
$$198$$ 0 0
$$199$$ − 5.19615i − 0.368345i −0.982894 0.184173i $$-0.941039\pi$$
0.982894 0.184173i $$-0.0589606\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 9.79796 0.687682
$$204$$ 0 0
$$205$$ −16.0000 −1.11749
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 25.4558i 1.76082i
$$210$$ 0 0
$$211$$ 12.1244i 0.834675i 0.908752 + 0.417338i $$0.137037\pi$$
−0.908752 + 0.417338i $$0.862963\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −9.79796 −0.668215
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 2.82843i − 0.190261i
$$222$$ 0 0
$$223$$ − 24.2487i − 1.62381i −0.583787 0.811907i $$-0.698430\pi$$
0.583787 0.811907i $$-0.301570\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 19.5959 1.30063 0.650313 0.759666i $$-0.274638\pi$$
0.650313 + 0.759666i $$0.274638\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.6274i 1.48237i 0.671300 + 0.741186i $$0.265736\pi$$
−0.671300 + 0.741186i $$0.734264\pi$$
$$234$$ 0 0
$$235$$ − 13.8564i − 0.903892i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 19.5959 1.26755 0.633777 0.773516i $$-0.281504\pi$$
0.633777 + 0.773516i $$0.281504\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 11.3137i 0.722806i
$$246$$ 0 0
$$247$$ 5.19615i 0.330623i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −29.3939 −1.85533 −0.927663 0.373420i $$-0.878185\pi$$
−0.927663 + 0.373420i $$0.878185\pi$$
$$252$$ 0 0
$$253$$ 24.0000 1.50887
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 28.2843i − 1.76432i −0.470946 0.882162i $$-0.656087\pi$$
0.470946 0.882162i $$-0.343913\pi$$
$$258$$ 0 0
$$259$$ 1.73205i 0.107624i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −19.5959 −1.20834 −0.604168 0.796857i $$-0.706494\pi$$
−0.604168 + 0.796857i $$0.706494\pi$$
$$264$$ 0 0
$$265$$ 16.0000 0.982872
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2.82843i 0.172452i 0.996276 + 0.0862261i $$0.0274808\pi$$
−0.996276 + 0.0862261i $$0.972519\pi$$
$$270$$ 0 0
$$271$$ − 5.19615i − 0.315644i −0.987468 0.157822i $$-0.949553\pi$$
0.987468 0.157822i $$-0.0504472\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −14.6969 −0.886259
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.65685i 0.337460i 0.985662 + 0.168730i $$0.0539665\pi$$
−0.985662 + 0.168730i $$0.946033\pi$$
$$282$$ 0 0
$$283$$ − 24.2487i − 1.44144i −0.693228 0.720718i $$-0.743812\pi$$
0.693228 0.720718i $$-0.256188\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.79796 −0.578355
$$288$$ 0 0
$$289$$ 9.00000 0.529412
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.82843i 0.165238i 0.996581 + 0.0826192i $$0.0263285\pi$$
−0.996581 + 0.0826192i $$0.973671\pi$$
$$294$$ 0 0
$$295$$ − 13.8564i − 0.806751i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.89898 0.283315
$$300$$ 0 0
$$301$$ −6.00000 −0.345834
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 31.1127i − 1.78151i
$$306$$ 0 0
$$307$$ − 10.3923i − 0.593120i −0.955014 0.296560i $$-0.904160\pi$$
0.955014 0.296560i $$-0.0958395\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.4949 −1.38898 −0.694489 0.719503i $$-0.744370\pi$$
−0.694489 + 0.719503i $$0.744370\pi$$
$$312$$ 0 0
$$313$$ 11.0000 0.621757 0.310878 0.950450i $$-0.399377\pi$$
0.310878 + 0.950450i $$0.399377\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 5.65685i − 0.317721i −0.987301 0.158860i $$-0.949218\pi$$
0.987301 0.158860i $$-0.0507819\pi$$
$$318$$ 0 0
$$319$$ − 27.7128i − 1.55162i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 14.6969 0.817760
$$324$$ 0 0
$$325$$ −3.00000 −0.166410
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 8.48528i − 0.467809i
$$330$$ 0 0
$$331$$ − 22.5167i − 1.23763i −0.785538 0.618814i $$-0.787614\pi$$
0.785538 0.618814i $$-0.212386\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −34.2929 −1.87362
$$336$$ 0 0
$$337$$ 11.0000 0.599208 0.299604 0.954064i $$-0.403145\pi$$
0.299604 + 0.954064i $$0.403145\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.9706i 0.919007i
$$342$$ 0 0
$$343$$ 19.0526i 1.02874i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −19.5959 −1.05196 −0.525982 0.850496i $$-0.676302\pi$$
−0.525982 + 0.850496i $$0.676302\pi$$
$$348$$ 0 0
$$349$$ −23.0000 −1.23116 −0.615581 0.788074i $$-0.711079\pi$$
−0.615581 + 0.788074i $$0.711079\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 11.3137i − 0.602168i −0.953598 0.301084i $$-0.902652\pi$$
0.953598 0.301084i $$-0.0973484\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 14.6969 0.775675 0.387837 0.921728i $$-0.373222\pi$$
0.387837 + 0.921728i $$0.373222\pi$$
$$360$$ 0 0
$$361$$ −8.00000 −0.421053
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 2.82843i − 0.148047i
$$366$$ 0 0
$$367$$ − 19.0526i − 0.994535i −0.867597 0.497268i $$-0.834337\pi$$
0.867597 0.497268i $$-0.165663\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9.79796 0.508685
$$372$$ 0 0
$$373$$ 25.0000 1.29445 0.647225 0.762299i $$-0.275929\pi$$
0.647225 + 0.762299i $$0.275929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 5.65685i − 0.291343i
$$378$$ 0 0
$$379$$ − 15.5885i − 0.800725i −0.916357 0.400363i $$-0.868884\pi$$
0.916357 0.400363i $$-0.131116\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 19.5959 1.00130 0.500652 0.865648i $$-0.333094\pi$$
0.500652 + 0.865648i $$0.333094\pi$$
$$384$$ 0 0
$$385$$ −24.0000 −1.22315
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 14.1421i − 0.717035i −0.933523 0.358517i $$-0.883282\pi$$
0.933523 0.358517i $$-0.116718\pi$$
$$390$$ 0 0
$$391$$ − 13.8564i − 0.700749i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.89898 −0.246494
$$396$$ 0 0
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.6274i 1.12996i 0.825105 + 0.564980i $$0.191116\pi$$
−0.825105 + 0.564980i $$0.808884\pi$$
$$402$$ 0 0
$$403$$ 3.46410i 0.172559i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.89898 0.242833
$$408$$ 0 0
$$409$$ −1.00000 −0.0494468 −0.0247234 0.999694i $$-0.507871\pi$$
−0.0247234 + 0.999694i $$0.507871\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 8.48528i − 0.417533i
$$414$$ 0 0
$$415$$ − 27.7128i − 1.36037i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.89898 −0.239331 −0.119665 0.992814i $$-0.538182\pi$$
−0.119665 + 0.992814i $$0.538182\pi$$
$$420$$ 0 0
$$421$$ 25.0000 1.21843 0.609213 0.793007i $$-0.291486\pi$$
0.609213 + 0.793007i $$0.291486\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8.48528i 0.411597i
$$426$$ 0 0
$$427$$ − 19.0526i − 0.922018i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 14.6969 0.707927 0.353963 0.935259i $$-0.384834\pi$$
0.353963 + 0.935259i $$0.384834\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 25.4558i 1.21772i
$$438$$ 0 0
$$439$$ 17.3205i 0.826663i 0.910581 + 0.413331i $$0.135635\pi$$
−0.910581 + 0.413331i $$0.864365\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9.79796 −0.465515 −0.232758 0.972535i $$-0.574775\pi$$
−0.232758 + 0.972535i $$0.574775\pi$$
$$444$$ 0 0
$$445$$ 8.00000 0.379236
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 31.1127i 1.46830i 0.678988 + 0.734150i $$0.262419\pi$$
−0.678988 + 0.734150i $$0.737581\pi$$
$$450$$ 0 0
$$451$$ 27.7128i 1.30495i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.89898 −0.229668
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 14.1421i − 0.658665i −0.944214 0.329332i $$-0.893176\pi$$
0.944214 0.329332i $$-0.106824\pi$$
$$462$$ 0 0
$$463$$ − 12.1244i − 0.563467i −0.959493 0.281733i $$-0.909091\pi$$
0.959493 0.281733i $$-0.0909093\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.6969 0.680093 0.340047 0.940409i $$-0.389557\pi$$
0.340047 + 0.940409i $$0.389557\pi$$
$$468$$ 0 0
$$469$$ −21.0000 −0.969690
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 16.9706i 0.780307i
$$474$$ 0 0
$$475$$ − 15.5885i − 0.715247i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9.79796 0.447680 0.223840 0.974626i $$-0.428141\pi$$
0.223840 + 0.974626i $$0.428141\pi$$
$$480$$ 0 0
$$481$$ 1.00000 0.0455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 36.7696i − 1.66962i
$$486$$ 0 0
$$487$$ − 25.9808i − 1.17730i −0.808388 0.588650i $$-0.799659\pi$$
0.808388 0.588650i $$-0.200341\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.4949 1.10544 0.552720 0.833367i $$-0.313590\pi$$
0.552720 + 0.833367i $$0.313590\pi$$
$$492$$ 0 0
$$493$$ −16.0000 −0.720604
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 24.2487i − 1.08552i −0.839887 0.542761i $$-0.817379\pi$$
0.839887 0.542761i $$-0.182621\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14.6969 0.655304 0.327652 0.944798i $$-0.393743\pi$$
0.327652 + 0.944798i $$0.393743\pi$$
$$504$$ 0 0
$$505$$ −32.0000 −1.42398
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 19.7990i 0.877575i 0.898591 + 0.438787i $$0.144592\pi$$
−0.898591 + 0.438787i $$0.855408\pi$$
$$510$$ 0 0
$$511$$ − 1.73205i − 0.0766214i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −24.4949 −1.07937
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 19.7990i − 0.867409i −0.901055 0.433705i $$-0.857206\pi$$
0.901055 0.433705i $$-0.142794\pi$$
$$522$$ 0 0
$$523$$ − 15.5885i − 0.681636i −0.940129 0.340818i $$-0.889296\pi$$
0.940129 0.340818i $$-0.110704\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.79796 0.426806
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 5.65685i 0.245026i
$$534$$ 0 0
$$535$$ 41.5692i 1.79719i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 19.5959 0.844056
$$540$$ 0 0
$$541$$ 1.00000 0.0429934 0.0214967 0.999769i $$-0.493157\pi$$
0.0214967 + 0.999769i $$0.493157\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 28.2843i 1.21157i
$$546$$ 0 0
$$547$$ − 22.5167i − 0.962743i −0.876517 0.481371i $$-0.840139\pi$$
0.876517 0.481371i $$-0.159861\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 29.3939 1.25222
$$552$$ 0 0
$$553$$ −3.00000 −0.127573
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 2.82843i 0.119844i 0.998203 + 0.0599222i $$0.0190852\pi$$
−0.998203 + 0.0599222i $$0.980915\pi$$
$$558$$ 0 0
$$559$$ 3.46410i 0.146516i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 9.79796 0.412935 0.206467 0.978453i $$-0.433803\pi$$
0.206467 + 0.978453i $$0.433803\pi$$
$$564$$ 0 0
$$565$$ 8.00000 0.336563
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 19.7990i − 0.830017i −0.909818 0.415008i $$-0.863779\pi$$
0.909818 0.415008i $$-0.136221\pi$$
$$570$$ 0 0
$$571$$ 32.9090i 1.37720i 0.725143 + 0.688599i $$0.241774\pi$$
−0.725143 + 0.688599i $$0.758226\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −14.6969 −0.612905
$$576$$ 0 0
$$577$$ 35.0000 1.45707 0.728535 0.685009i $$-0.240202\pi$$
0.728535 + 0.685009i $$0.240202\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 16.9706i − 0.704058i
$$582$$ 0 0
$$583$$ − 27.7128i − 1.14775i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.89898 0.202203 0.101101 0.994876i $$-0.467763\pi$$
0.101101 + 0.994876i $$0.467763\pi$$
$$588$$ 0 0
$$589$$ −18.0000 −0.741677
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 28.2843i − 1.16150i −0.814083 0.580748i $$-0.802760\pi$$
0.814083 0.580748i $$-0.197240\pi$$
$$594$$ 0 0
$$595$$ 13.8564i 0.568057i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −9.79796 −0.400334 −0.200167 0.979762i $$-0.564148\pi$$
−0.200167 + 0.979762i $$0.564148\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 36.7696i 1.49489i
$$606$$ 0 0
$$607$$ − 12.1244i − 0.492112i −0.969256 0.246056i $$-0.920865\pi$$
0.969256 0.246056i $$-0.0791348\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.89898 −0.198191
$$612$$ 0 0
$$613$$ −11.0000 −0.444286 −0.222143 0.975014i $$-0.571305\pi$$
−0.222143 + 0.975014i $$0.571305\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.82843i − 0.113868i −0.998378 0.0569341i $$-0.981868\pi$$
0.998378 0.0569341i $$-0.0181325\pi$$
$$618$$ 0 0
$$619$$ 19.0526i 0.765787i 0.923792 + 0.382893i $$0.125072\pi$$
−0.923792 + 0.382893i $$0.874928\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4.89898 0.196273
$$624$$ 0 0
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 2.82843i − 0.112777i
$$630$$ 0 0
$$631$$ 36.3731i 1.44799i 0.689806 + 0.723994i $$0.257696\pi$$
−0.689806 + 0.723994i $$0.742304\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 29.3939 1.16646
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 45.2548i − 1.78746i −0.448607 0.893729i $$-0.648080\pi$$
0.448607 0.893729i $$-0.351920\pi$$
$$642$$ 0 0
$$643$$ 17.3205i 0.683054i 0.939872 + 0.341527i $$0.110944\pi$$
−0.939872 + 0.341527i $$0.889056\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −29.3939 −1.15559 −0.577796 0.816181i $$-0.696087\pi$$
−0.577796 + 0.816181i $$0.696087\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 5.65685i − 0.221370i −0.993856 0.110685i $$-0.964696\pi$$
0.993856 0.110685i $$-0.0353044\pi$$
$$654$$ 0 0
$$655$$ 27.7128i 1.08283i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9.79796 −0.381674 −0.190837 0.981622i $$-0.561120\pi$$
−0.190837 + 0.981622i $$0.561120\pi$$
$$660$$ 0 0
$$661$$ 37.0000 1.43913 0.719567 0.694423i $$-0.244340\pi$$
0.719567 + 0.694423i $$0.244340\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 25.4558i − 0.987135i
$$666$$ 0 0
$$667$$ − 27.7128i − 1.07304i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −53.8888 −2.08035
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 14.1421i − 0.543526i −0.962364 0.271763i $$-0.912393\pi$$
0.962364 0.271763i $$-0.0876068\pi$$
$$678$$ 0 0
$$679$$ − 22.5167i − 0.864110i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 56.0000 2.13965
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 5.65685i − 0.215509i
$$690$$ 0 0
$$691$$ 45.0333i 1.71315i 0.516024 + 0.856574i $$0.327412\pi$$
−0.516024 + 0.856574i $$0.672588\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 24.4949 0.929144
$$696$$ 0 0
$$697$$ 16.0000 0.606043
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.82843i 0.106828i 0.998572 + 0.0534141i $$0.0170103\pi$$
−0.998572 + 0.0534141i $$0.982990\pi$$
$$702$$ 0 0
$$703$$ 5.19615i 0.195977i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −19.5959 −0.736980
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 16.9706i 0.635553i
$$714$$ 0 0
$$715$$ 13.8564i 0.518200i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −29.3939 −1.09621 −0.548103 0.836411i $$-0.684650\pi$$
−0.548103 + 0.836411i $$0.684650\pi$$
$$720$$ 0 0
$$721$$ −15.0000 −0.558629
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 16.9706i 0.630271i
$$726$$ 0 0
$$727$$ 17.3205i 0.642382i 0.947014 + 0.321191i $$0.104083\pi$$
−0.947014 + 0.321191i $$0.895917\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 9.79796 0.362391
$$732$$ 0 0
$$733$$ −38.0000 −1.40356 −0.701781 0.712393i $$-0.747612\pi$$
−0.701781 + 0.712393i $$0.747612\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 59.3970i 2.18792i
$$738$$ 0 0
$$739$$ − 10.3923i − 0.382287i −0.981562 0.191144i $$-0.938780\pi$$
0.981562 0.191144i $$-0.0612196\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 34.2929 1.25808 0.629041 0.777372i $$-0.283448\pi$$
0.629041 + 0.777372i $$0.283448\pi$$
$$744$$ 0 0
$$745$$ 64.0000 2.34478
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 25.4558i 0.930136i
$$750$$ 0 0
$$751$$ 29.4449i 1.07446i 0.843436 + 0.537229i $$0.180529\pi$$
−0.843436 + 0.537229i $$0.819471\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −63.6867 −2.31780
$$756$$ 0 0
$$757$$ −47.0000 −1.70824 −0.854122 0.520073i $$-0.825905\pi$$
−0.854122 + 0.520073i $$0.825905\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 2.82843i − 0.102530i −0.998685 0.0512652i $$-0.983675\pi$$
0.998685 0.0512652i $$-0.0163254\pi$$
$$762$$ 0 0
$$763$$ 17.3205i 0.627044i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −4.89898 −0.176892
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 28.2843i 1.01731i 0.860969 + 0.508657i $$0.169858\pi$$
−0.860969 + 0.508657i $$0.830142\pi$$
$$774$$ 0 0
$$775$$ − 10.3923i − 0.373303i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −29.3939 −1.05314
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 28.2843i 1.00951i
$$786$$ 0 0
$$787$$ − 8.66025i − 0.308705i −0.988016 0.154352i $$-0.950671\pi$$
0.988016 0.154352i $$-0.0493291\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 4.89898 0.174188
$$792$$ 0 0
$$793$$ −11.0000 −0.390621
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 22.6274i − 0.801504i −0.916187 0.400752i $$-0.868749\pi$$
0.916187 0.400752i $$-0.131251\pi$$
$$798$$ 0 0
$$799$$ 13.8564i 0.490204i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −4.89898 −0.172881
$$804$$ 0 0
$$805$$ −24.0000 −0.845889
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 22.6274i 0.795538i 0.917486 + 0.397769i $$0.130215\pi$$
−0.917486 + 0.397769i $$0.869785\pi$$
$$810$$ 0 0
$$811$$ 31.1769i 1.09477i 0.836881 + 0.547385i $$0.184377\pi$$
−0.836881 + 0.547385i $$0.815623\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −14.6969 −0.514811
$$816$$ 0 0
$$817$$ −18.0000 −0.629740
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 31.1127i − 1.08584i −0.839784 0.542920i $$-0.817319\pi$$
0.839784 0.542920i $$-0.182681\pi$$
$$822$$ 0 0
$$823$$ 50.2295i 1.75089i 0.483318 + 0.875445i $$0.339431\pi$$
−0.483318 + 0.875445i $$0.660569\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14.6969 0.511063 0.255531 0.966801i $$-0.417750\pi$$
0.255531 + 0.966801i $$0.417750\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 11.3137i − 0.391997i
$$834$$ 0 0
$$835$$ 13.8564i 0.479521i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −48.9898 −1.69132 −0.845658 0.533726i $$-0.820792\pi$$
−0.845658 + 0.533726i $$0.820792\pi$$
$$840$$ 0 0
$$841$$ −3.00000 −0.103448
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 33.9411i − 1.16761i
$$846$$ 0 0
$$847$$ 22.5167i 0.773682i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4.89898 0.167935
$$852$$ 0 0
$$853$$ 49.0000 1.67773 0.838864 0.544341i $$-0.183220\pi$$
0.838864 + 0.544341i $$0.183220\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 56.5685i 1.93234i 0.257897 + 0.966172i $$0.416970\pi$$
−0.257897 + 0.966172i $$0.583030\pi$$
$$858$$ 0 0
$$859$$ 12.1244i 0.413678i 0.978375 + 0.206839i $$0.0663176\pi$$
−0.978375 + 0.206839i $$0.933682\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 14.6969 0.500290 0.250145 0.968208i $$-0.419522\pi$$
0.250145 + 0.968208i $$0.419522\pi$$
$$864$$ 0 0
$$865$$ 16.0000 0.544016
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 8.48528i 0.287843i
$$870$$ 0 0
$$871$$ 12.1244i 0.410818i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −9.79796 −0.331231
$$876$$ 0 0
$$877$$ 1.00000 0.0337676 0.0168838 0.999857i $$-0.494625\pi$$
0.0168838 + 0.999857i $$0.494625\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 19.7990i − 0.667045i −0.942742 0.333522i $$-0.891763\pi$$
0.942742 0.333522i $$-0.108237\pi$$
$$882$$ 0 0
$$883$$ 5.19615i 0.174864i 0.996170 + 0.0874322i $$0.0278661\pi$$
−0.996170 + 0.0874322i $$0.972134\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 48.9898 1.64492 0.822458 0.568826i $$-0.192602\pi$$
0.822458 + 0.568826i $$0.192602\pi$$
$$888$$ 0 0
$$889$$ 18.0000 0.603701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 25.4558i − 0.851847i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 19.5959 0.653560
$$900$$ 0 0
$$901$$ −16.0000 −0.533037
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 31.1127i − 1.03422i
$$906$$ 0 0
$$907$$ − 1.73205i − 0.0575118i −0.999586 0.0287559i $$-0.990845\pi$$
0.999586 0.0287559i $$-0.00915455\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9.79796 0.324621 0.162310 0.986740i $$-0.448105\pi$$
0.162310 + 0.986740i $$0.448105\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.9706i 0.560417i
$$918$$ 0 0
$$919$$ − 10.3923i − 0.342811i −0.985201 0.171405i $$-0.945169\pi$$
0.985201 0.171405i $$-0.0548307\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 45.2548i − 1.48476i −0.669977 0.742381i $$-0.733696\pi$$
0.669977 0.742381i $$-0.266304\pi$$
$$930$$ 0 0
$$931$$ 20.7846i 0.681188i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 39.1918 1.28171
$$936$$ 0 0
$$937$$ −37.0000 −1.20874 −0.604369 0.796705i $$-0.706575\pi$$
−0.604369 + 0.796705i $$0.706575\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 48.0833i − 1.56747i −0.621096 0.783735i $$-0.713312\pi$$
0.621096 0.783735i $$-0.286688\pi$$
$$942$$ 0 0
$$943$$ 27.7128i 0.902453i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 34.2929 1.11437 0.557184 0.830389i $$-0.311882\pi$$
0.557184 + 0.830389i $$0.311882\pi$$
$$948$$ 0 0
$$949$$ −1.00000 −0.0324614
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 2.82843i − 0.0916217i −0.998950 0.0458109i $$-0.985413\pi$$
0.998950 0.0458109i $$-0.0145872\pi$$
$$954$$ 0 0
$$955$$ 69.2820i 2.24191i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 34.2929 1.10737
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 2.82843i − 0.0910503i
$$966$$ 0 0
$$967$$ − 53.6936i − 1.72667i −0.504632 0.863334i $$-0.668372\pi$$
0.504632 0.863334i $$-0.331628\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.6969 −0.471647 −0.235824 0.971796i $$-0.575779\pi$$
−0.235824 + 0.971796i $$0.575779\pi$$
$$972$$ 0 0
$$973$$ 15.0000 0.480878
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 5.65685i 0.180979i 0.995897 + 0.0904894i $$0.0288431\pi$$
−0.995897 + 0.0904894i $$0.971157\pi$$
$$978$$ 0 0
$$979$$ − 13.8564i − 0.442853i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −34.2929 −1.09377 −0.546886 0.837207i $$-0.684187\pi$$
−0.546886 + 0.837207i $$0.684187\pi$$
$$984$$ 0 0
$$985$$ −8.00000 −0.254901
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.9706i 0.539633i
$$990$$ 0 0
$$991$$ 36.3731i 1.15543i 0.816239 + 0.577714i $$0.196055\pi$$
−0.816239 + 0.577714i $$0.803945\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 14.6969 0.465924
$$996$$ 0 0
$$997$$ 34.0000 1.07679 0.538395 0.842692i $$-0.319031\pi$$
0.538395 + 0.842692i $$0.319031\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.c.d.1727.4 4
3.2 odd 2 inner 1728.2.c.d.1727.2 4
4.3 odd 2 inner 1728.2.c.d.1727.3 4
8.3 odd 2 108.2.b.b.107.2 yes 4
8.5 even 2 108.2.b.b.107.4 yes 4
12.11 even 2 inner 1728.2.c.d.1727.1 4
24.5 odd 2 108.2.b.b.107.1 4
24.11 even 2 108.2.b.b.107.3 yes 4
72.5 odd 6 324.2.h.b.107.2 4
72.11 even 6 324.2.h.b.215.1 4
72.13 even 6 324.2.h.b.107.1 4
72.29 odd 6 324.2.h.a.215.2 4
72.43 odd 6 324.2.h.b.215.2 4
72.59 even 6 324.2.h.a.107.2 4
72.61 even 6 324.2.h.a.215.1 4
72.67 odd 6 324.2.h.a.107.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.b.b.107.1 4 24.5 odd 2
108.2.b.b.107.2 yes 4 8.3 odd 2
108.2.b.b.107.3 yes 4 24.11 even 2
108.2.b.b.107.4 yes 4 8.5 even 2
324.2.h.a.107.1 4 72.67 odd 6
324.2.h.a.107.2 4 72.59 even 6
324.2.h.a.215.1 4 72.61 even 6
324.2.h.a.215.2 4 72.29 odd 6
324.2.h.b.107.1 4 72.13 even 6
324.2.h.b.107.2 4 72.5 odd 6
324.2.h.b.215.1 4 72.11 even 6
324.2.h.b.215.2 4 72.43 odd 6
1728.2.c.d.1727.1 4 12.11 even 2 inner
1728.2.c.d.1727.2 4 3.2 odd 2 inner
1728.2.c.d.1727.3 4 4.3 odd 2 inner
1728.2.c.d.1727.4 4 1.1 even 1 trivial