# Properties

 Label 2760.3.g.a.2161.14 Level $2760$ Weight $3$ Character 2760.2161 Analytic conductor $75.205$ Analytic rank $0$ Dimension $96$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,3,Mod(2161,2760)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2760.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2760.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$75.2045529634$$ Analytic rank: $$0$$ Dimension: $$96$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2161.14 Character $$\chi$$ $$=$$ 2760.2161 Dual form 2760.3.g.a.2161.35

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} -2.23607i q^{5} +3.56454i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{3} -2.23607i q^{5} +3.56454i q^{7} +3.00000 q^{9} -4.23478i q^{11} -15.5305 q^{13} +3.87298i q^{15} +9.96060i q^{17} +9.40924i q^{19} -6.17397i q^{21} +(18.1725 - 14.0982i) q^{23} -5.00000 q^{25} -5.19615 q^{27} +29.5135 q^{29} -57.3969 q^{31} +7.33486i q^{33} +7.97056 q^{35} +17.4527i q^{37} +26.8997 q^{39} -19.0465 q^{41} +2.74875i q^{43} -6.70820i q^{45} +25.1833 q^{47} +36.2940 q^{49} -17.2523i q^{51} -36.1205i q^{53} -9.46927 q^{55} -16.2973i q^{57} +13.6254 q^{59} +28.2845i q^{61} +10.6936i q^{63} +34.7274i q^{65} +19.6093i q^{67} +(-31.4757 + 24.4188i) q^{69} -42.1382 q^{71} +49.2899 q^{73} +8.66025 q^{75} +15.0951 q^{77} -124.063i q^{79} +9.00000 q^{81} +55.4945i q^{83} +22.2726 q^{85} -51.1189 q^{87} -47.5283i q^{89} -55.3593i q^{91} +99.4144 q^{93} +21.0397 q^{95} -91.3507i q^{97} -12.7044i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$96 q + 288 q^{9}+O(q^{10})$$ 96 * q + 288 * q^9 $$96 q + 288 q^{9} - 16 q^{23} - 480 q^{25} - 80 q^{31} + 80 q^{35} + 48 q^{39} + 112 q^{41} + 32 q^{47} - 688 q^{49} - 80 q^{55} - 496 q^{59} - 96 q^{69} - 416 q^{71} - 320 q^{73} + 864 q^{81} + 192 q^{93}+O(q^{100})$$ 96 * q + 288 * q^9 - 16 * q^23 - 480 * q^25 - 80 * q^31 + 80 * q^35 + 48 * q^39 + 112 * q^41 + 32 * q^47 - 688 * q^49 - 80 * q^55 - 496 * q^59 - 96 * q^69 - 416 * q^71 - 320 * q^73 + 864 * q^81 + 192 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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
$$3$$ −1.73205 −0.577350
$$4$$ 0 0
$$5$$ 2.23607i 0.447214i
$$6$$ 0 0
$$7$$ 3.56454i 0.509220i 0.967044 + 0.254610i $$0.0819472\pi$$
−0.967044 + 0.254610i $$0.918053\pi$$
$$8$$ 0 0
$$9$$ 3.00000 0.333333
$$10$$ 0 0
$$11$$ 4.23478i 0.384980i −0.981299 0.192490i $$-0.938344\pi$$
0.981299 0.192490i $$-0.0616564\pi$$
$$12$$ 0 0
$$13$$ −15.5305 −1.19466 −0.597329 0.801996i $$-0.703771\pi$$
−0.597329 + 0.801996i $$0.703771\pi$$
$$14$$ 0 0
$$15$$ 3.87298i 0.258199i
$$16$$ 0 0
$$17$$ 9.96060i 0.585917i 0.956125 + 0.292959i $$0.0946399\pi$$
−0.956125 + 0.292959i $$0.905360\pi$$
$$18$$ 0 0
$$19$$ 9.40924i 0.495223i 0.968859 + 0.247612i $$0.0796457\pi$$
−0.968859 + 0.247612i $$0.920354\pi$$
$$20$$ 0 0
$$21$$ 6.17397i 0.293999i
$$22$$ 0 0
$$23$$ 18.1725 14.0982i 0.790110 0.612965i
$$24$$ 0 0
$$25$$ −5.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.19615 −0.192450
$$28$$ 0 0
$$29$$ 29.5135 1.01771 0.508853 0.860853i $$-0.330070\pi$$
0.508853 + 0.860853i $$0.330070\pi$$
$$30$$ 0 0
$$31$$ −57.3969 −1.85151 −0.925757 0.378119i $$-0.876571\pi$$
−0.925757 + 0.378119i $$0.876571\pi$$
$$32$$ 0 0
$$33$$ 7.33486i 0.222269i
$$34$$ 0 0
$$35$$ 7.97056 0.227730
$$36$$ 0 0
$$37$$ 17.4527i 0.471695i 0.971790 + 0.235847i $$0.0757866\pi$$
−0.971790 + 0.235847i $$0.924213\pi$$
$$38$$ 0 0
$$39$$ 26.8997 0.689736
$$40$$ 0 0
$$41$$ −19.0465 −0.464548 −0.232274 0.972650i $$-0.574617\pi$$
−0.232274 + 0.972650i $$0.574617\pi$$
$$42$$ 0 0
$$43$$ 2.74875i 0.0639244i 0.999489 + 0.0319622i $$0.0101756\pi$$
−0.999489 + 0.0319622i $$0.989824\pi$$
$$44$$ 0 0
$$45$$ 6.70820i 0.149071i
$$46$$ 0 0
$$47$$ 25.1833 0.535816 0.267908 0.963445i $$-0.413668\pi$$
0.267908 + 0.963445i $$0.413668\pi$$
$$48$$ 0 0
$$49$$ 36.2940 0.740695
$$50$$ 0 0
$$51$$ 17.2523i 0.338280i
$$52$$ 0 0
$$53$$ 36.1205i 0.681520i −0.940150 0.340760i $$-0.889316\pi$$
0.940150 0.340760i $$-0.110684\pi$$
$$54$$ 0 0
$$55$$ −9.46927 −0.172168
$$56$$ 0 0
$$57$$ 16.2973i 0.285917i
$$58$$ 0 0
$$59$$ 13.6254 0.230939 0.115469 0.993311i $$-0.463163\pi$$
0.115469 + 0.993311i $$0.463163\pi$$
$$60$$ 0 0
$$61$$ 28.2845i 0.463680i 0.972754 + 0.231840i $$0.0744746\pi$$
−0.972754 + 0.231840i $$0.925525\pi$$
$$62$$ 0 0
$$63$$ 10.6936i 0.169740i
$$64$$ 0 0
$$65$$ 34.7274i 0.534267i
$$66$$ 0 0
$$67$$ 19.6093i 0.292676i 0.989235 + 0.146338i $$0.0467488\pi$$
−0.989235 + 0.146338i $$0.953251\pi$$
$$68$$ 0 0
$$69$$ −31.4757 + 24.4188i −0.456170 + 0.353896i
$$70$$ 0 0
$$71$$ −42.1382 −0.593496 −0.296748 0.954956i $$-0.595902\pi$$
−0.296748 + 0.954956i $$0.595902\pi$$
$$72$$ 0 0
$$73$$ 49.2899 0.675204 0.337602 0.941289i $$-0.390384\pi$$
0.337602 + 0.941289i $$0.390384\pi$$
$$74$$ 0 0
$$75$$ 8.66025 0.115470
$$76$$ 0 0
$$77$$ 15.0951 0.196040
$$78$$ 0 0
$$79$$ 124.063i 1.57041i −0.619234 0.785207i $$-0.712557\pi$$
0.619234 0.785207i $$-0.287443\pi$$
$$80$$ 0 0
$$81$$ 9.00000 0.111111
$$82$$ 0 0
$$83$$ 55.4945i 0.668608i 0.942465 + 0.334304i $$0.108501\pi$$
−0.942465 + 0.334304i $$0.891499\pi$$
$$84$$ 0 0
$$85$$ 22.2726 0.262030
$$86$$ 0 0
$$87$$ −51.1189 −0.587573
$$88$$ 0 0
$$89$$ 47.5283i 0.534026i −0.963693 0.267013i $$-0.913963\pi$$
0.963693 0.267013i $$-0.0860366\pi$$
$$90$$ 0 0
$$91$$ 55.3593i 0.608344i
$$92$$ 0 0
$$93$$ 99.4144 1.06897
$$94$$ 0 0
$$95$$ 21.0397 0.221471
$$96$$ 0 0
$$97$$ 91.3507i 0.941759i −0.882197 0.470880i $$-0.843937\pi$$
0.882197 0.470880i $$-0.156063\pi$$
$$98$$ 0 0
$$99$$ 12.7044i 0.128327i
$$100$$ 0 0
$$101$$ 98.5205 0.975450 0.487725 0.872997i $$-0.337827\pi$$
0.487725 + 0.872997i $$0.337827\pi$$
$$102$$ 0 0
$$103$$ 65.7326i 0.638181i 0.947724 + 0.319090i $$0.103377\pi$$
−0.947724 + 0.319090i $$0.896623\pi$$
$$104$$ 0 0
$$105$$ −13.8054 −0.131480
$$106$$ 0 0
$$107$$ 202.746i 1.89482i 0.320015 + 0.947412i $$0.396312\pi$$
−0.320015 + 0.947412i $$0.603688\pi$$
$$108$$ 0 0
$$109$$ 88.4157i 0.811153i −0.914061 0.405576i $$-0.867071\pi$$
0.914061 0.405576i $$-0.132929\pi$$
$$110$$ 0 0
$$111$$ 30.2290i 0.272333i
$$112$$ 0 0
$$113$$ 2.51013i 0.0222136i 0.999938 + 0.0111068i $$0.00353547\pi$$
−0.999938 + 0.0111068i $$0.996465\pi$$
$$114$$ 0 0
$$115$$ −31.5245 40.6350i −0.274126 0.353348i
$$116$$ 0 0
$$117$$ −46.5916 −0.398219
$$118$$ 0 0
$$119$$ −35.5050 −0.298361
$$120$$ 0 0
$$121$$ 103.067 0.851790
$$122$$ 0 0
$$123$$ 32.9894 0.268207
$$124$$ 0 0
$$125$$ 11.1803i 0.0894427i
$$126$$ 0 0
$$127$$ −26.8952 −0.211773 −0.105886 0.994378i $$-0.533768\pi$$
−0.105886 + 0.994378i $$0.533768\pi$$
$$128$$ 0 0
$$129$$ 4.76097i 0.0369068i
$$130$$ 0 0
$$131$$ −138.064 −1.05393 −0.526964 0.849888i $$-0.676670\pi$$
−0.526964 + 0.849888i $$0.676670\pi$$
$$132$$ 0 0
$$133$$ −33.5397 −0.252178
$$134$$ 0 0
$$135$$ 11.6190i 0.0860663i
$$136$$ 0 0
$$137$$ 220.431i 1.60899i −0.593962 0.804493i $$-0.702437\pi$$
0.593962 0.804493i $$-0.297563\pi$$
$$138$$ 0 0
$$139$$ −142.183 −1.02290 −0.511450 0.859313i $$-0.670892\pi$$
−0.511450 + 0.859313i $$0.670892\pi$$
$$140$$ 0 0
$$141$$ −43.6188 −0.309353
$$142$$ 0 0
$$143$$ 65.7685i 0.459920i
$$144$$ 0 0
$$145$$ 65.9942i 0.455132i
$$146$$ 0 0
$$147$$ −62.8631 −0.427640
$$148$$ 0 0
$$149$$ 80.4962i 0.540243i −0.962826 0.270121i $$-0.912936\pi$$
0.962826 0.270121i $$-0.0870638\pi$$
$$150$$ 0 0
$$151$$ −30.4233 −0.201479 −0.100739 0.994913i $$-0.532121\pi$$
−0.100739 + 0.994913i $$0.532121\pi$$
$$152$$ 0 0
$$153$$ 29.8818i 0.195306i
$$154$$ 0 0
$$155$$ 128.343i 0.828022i
$$156$$ 0 0
$$157$$ 6.66993i 0.0424836i −0.999774 0.0212418i $$-0.993238\pi$$
0.999774 0.0212418i $$-0.00676199\pi$$
$$158$$ 0 0
$$159$$ 62.5626i 0.393476i
$$160$$ 0 0
$$161$$ 50.2536 + 64.7767i 0.312134 + 0.402340i
$$162$$ 0 0
$$163$$ 201.010 1.23319 0.616594 0.787281i $$-0.288512\pi$$
0.616594 + 0.787281i $$0.288512\pi$$
$$164$$ 0 0
$$165$$ 16.4013 0.0994015
$$166$$ 0 0
$$167$$ 13.2877 0.0795671 0.0397836 0.999208i $$-0.487333\pi$$
0.0397836 + 0.999208i $$0.487333\pi$$
$$168$$ 0 0
$$169$$ 72.1979 0.427206
$$170$$ 0 0
$$171$$ 28.2277i 0.165074i
$$172$$ 0 0
$$173$$ 187.705 1.08500 0.542499 0.840057i $$-0.317478\pi$$
0.542499 + 0.840057i $$0.317478\pi$$
$$174$$ 0 0
$$175$$ 17.8227i 0.101844i
$$176$$ 0 0
$$177$$ −23.5999 −0.133333
$$178$$ 0 0
$$179$$ 54.1869 0.302720 0.151360 0.988479i $$-0.451635\pi$$
0.151360 + 0.988479i $$0.451635\pi$$
$$180$$ 0 0
$$181$$ 21.9178i 0.121093i 0.998165 + 0.0605463i $$0.0192843\pi$$
−0.998165 + 0.0605463i $$0.980716\pi$$
$$182$$ 0 0
$$183$$ 48.9902i 0.267706i
$$184$$ 0 0
$$185$$ 39.0255 0.210948
$$186$$ 0 0
$$187$$ 42.1810 0.225567
$$188$$ 0 0
$$189$$ 18.5219i 0.0979995i
$$190$$ 0 0
$$191$$ 344.941i 1.80597i −0.429668 0.902987i $$-0.641370\pi$$
0.429668 0.902987i $$-0.358630\pi$$
$$192$$ 0 0
$$193$$ 206.578 1.07035 0.535175 0.844741i $$-0.320246\pi$$
0.535175 + 0.844741i $$0.320246\pi$$
$$194$$ 0 0
$$195$$ 60.1496i 0.308459i
$$196$$ 0 0
$$197$$ 154.646 0.785003 0.392502 0.919751i $$-0.371610\pi$$
0.392502 + 0.919751i $$0.371610\pi$$
$$198$$ 0 0
$$199$$ 128.107i 0.643754i −0.946781 0.321877i $$-0.895686\pi$$
0.946781 0.321877i $$-0.104314\pi$$
$$200$$ 0 0
$$201$$ 33.9643i 0.168977i
$$202$$ 0 0
$$203$$ 105.202i 0.518237i
$$204$$ 0 0
$$205$$ 42.5892i 0.207752i
$$206$$ 0 0
$$207$$ 54.5176 42.2946i 0.263370 0.204322i
$$208$$ 0 0
$$209$$ 39.8461 0.190651
$$210$$ 0 0
$$211$$ 40.8218 0.193468 0.0967341 0.995310i $$-0.469160\pi$$
0.0967341 + 0.995310i $$0.469160\pi$$
$$212$$ 0 0
$$213$$ 72.9855 0.342655
$$214$$ 0 0
$$215$$ 6.14639 0.0285879
$$216$$ 0 0
$$217$$ 204.594i 0.942829i
$$218$$ 0 0
$$219$$ −85.3726 −0.389829
$$220$$ 0 0
$$221$$ 154.694i 0.699971i
$$222$$ 0 0
$$223$$ 193.085 0.865851 0.432926 0.901430i $$-0.357481\pi$$
0.432926 + 0.901430i $$0.357481\pi$$
$$224$$ 0 0
$$225$$ −15.0000 −0.0666667
$$226$$ 0 0
$$227$$ 87.9176i 0.387302i −0.981070 0.193651i $$-0.937967\pi$$
0.981070 0.193651i $$-0.0620330\pi$$
$$228$$ 0 0
$$229$$ 387.053i 1.69019i −0.534618 0.845094i $$-0.679545\pi$$
0.534618 0.845094i $$-0.320455\pi$$
$$230$$ 0 0
$$231$$ −26.1454 −0.113184
$$232$$ 0 0
$$233$$ −135.592 −0.581940 −0.290970 0.956732i $$-0.593978\pi$$
−0.290970 + 0.956732i $$0.593978\pi$$
$$234$$ 0 0
$$235$$ 56.3116i 0.239624i
$$236$$ 0 0
$$237$$ 214.883i 0.906679i
$$238$$ 0 0
$$239$$ 461.517 1.93103 0.965516 0.260344i $$-0.0838359\pi$$
0.965516 + 0.260344i $$0.0838359\pi$$
$$240$$ 0 0
$$241$$ 368.653i 1.52968i −0.644221 0.764840i $$-0.722818\pi$$
0.644221 0.764840i $$-0.277182\pi$$
$$242$$ 0 0
$$243$$ −15.5885 −0.0641500
$$244$$ 0 0
$$245$$ 81.1559i 0.331249i
$$246$$ 0 0
$$247$$ 146.131i 0.591622i
$$248$$ 0 0
$$249$$ 96.1192i 0.386021i
$$250$$ 0 0
$$251$$ 77.7032i 0.309575i 0.987948 + 0.154787i $$0.0494692\pi$$
−0.987948 + 0.154787i $$0.950531\pi$$
$$252$$ 0 0
$$253$$ −59.7029 76.9567i −0.235980 0.304177i
$$254$$ 0 0
$$255$$ −38.5772 −0.151283
$$256$$ 0 0
$$257$$ −157.196 −0.611659 −0.305829 0.952086i $$-0.598934\pi$$
−0.305829 + 0.952086i $$0.598934\pi$$
$$258$$ 0 0
$$259$$ −62.2109 −0.240197
$$260$$ 0 0
$$261$$ 88.5405 0.339235
$$262$$ 0 0
$$263$$ 452.069i 1.71889i −0.511225 0.859447i $$-0.670808\pi$$
0.511225 0.859447i $$-0.329192\pi$$
$$264$$ 0 0
$$265$$ −80.7680 −0.304785
$$266$$ 0 0
$$267$$ 82.3214i 0.308320i
$$268$$ 0 0
$$269$$ −18.6332 −0.0692685 −0.0346343 0.999400i $$-0.511027\pi$$
−0.0346343 + 0.999400i $$0.511027\pi$$
$$270$$ 0 0
$$271$$ 238.072 0.878493 0.439247 0.898367i $$-0.355245\pi$$
0.439247 + 0.898367i $$0.355245\pi$$
$$272$$ 0 0
$$273$$ 95.8851i 0.351228i
$$274$$ 0 0
$$275$$ 21.1739i 0.0769961i
$$276$$ 0 0
$$277$$ 223.227 0.805872 0.402936 0.915228i $$-0.367990\pi$$
0.402936 + 0.915228i $$0.367990\pi$$
$$278$$ 0 0
$$279$$ −172.191 −0.617171
$$280$$ 0 0
$$281$$ 436.729i 1.55420i −0.629380 0.777098i $$-0.716691\pi$$
0.629380 0.777098i $$-0.283309\pi$$
$$282$$ 0 0
$$283$$ 23.3890i 0.0826466i 0.999146 + 0.0413233i $$0.0131574\pi$$
−0.999146 + 0.0413233i $$0.986843\pi$$
$$284$$ 0 0
$$285$$ −36.4418 −0.127866
$$286$$ 0 0
$$287$$ 67.8919i 0.236557i
$$288$$ 0 0
$$289$$ 189.787 0.656701
$$290$$ 0 0
$$291$$ 158.224i 0.543725i
$$292$$ 0 0
$$293$$ 156.918i 0.535558i 0.963480 + 0.267779i $$0.0862897\pi$$
−0.963480 + 0.267779i $$0.913710\pi$$
$$294$$ 0 0
$$295$$ 30.4673i 0.103279i
$$296$$ 0 0
$$297$$ 22.0046i 0.0740895i
$$298$$ 0 0
$$299$$ −282.229 + 218.953i −0.943911 + 0.732284i
$$300$$ 0 0
$$301$$ −9.79803 −0.0325516
$$302$$ 0 0
$$303$$ −170.642 −0.563176
$$304$$ 0 0
$$305$$ 63.2460 0.207364
$$306$$ 0 0
$$307$$ 271.068 0.882957 0.441479 0.897272i $$-0.354454\pi$$
0.441479 + 0.897272i $$0.354454\pi$$
$$308$$ 0 0
$$309$$ 113.852i 0.368454i
$$310$$ 0 0
$$311$$ −111.747 −0.359315 −0.179657 0.983729i $$-0.557499\pi$$
−0.179657 + 0.983729i $$0.557499\pi$$
$$312$$ 0 0
$$313$$ 599.676i 1.91590i −0.286939 0.957949i $$-0.592638\pi$$
0.286939 0.957949i $$-0.407362\pi$$
$$314$$ 0 0
$$315$$ 23.9117 0.0759101
$$316$$ 0 0
$$317$$ 54.9678 0.173400 0.0867000 0.996234i $$-0.472368\pi$$
0.0867000 + 0.996234i $$0.472368\pi$$
$$318$$ 0 0
$$319$$ 124.983i 0.391797i
$$320$$ 0 0
$$321$$ 351.167i 1.09398i
$$322$$ 0 0
$$323$$ −93.7217 −0.290160
$$324$$ 0 0
$$325$$ 77.6527 0.238931
$$326$$ 0 0
$$327$$ 153.140i 0.468319i
$$328$$ 0 0
$$329$$ 89.7671i 0.272848i
$$330$$ 0 0
$$331$$ −30.2790 −0.0914774 −0.0457387 0.998953i $$-0.514564\pi$$
−0.0457387 + 0.998953i $$0.514564\pi$$
$$332$$ 0 0
$$333$$ 52.3581i 0.157232i
$$334$$ 0 0
$$335$$ 43.8478 0.130889
$$336$$ 0 0
$$337$$ 527.421i 1.56505i −0.622621 0.782523i $$-0.713932\pi$$
0.622621 0.782523i $$-0.286068\pi$$
$$338$$ 0 0
$$339$$ 4.34768i 0.0128250i
$$340$$ 0 0
$$341$$ 243.064i 0.712797i
$$342$$ 0 0
$$343$$ 304.034i 0.886397i
$$344$$ 0 0
$$345$$ 54.6021 + 70.3819i 0.158267 + 0.204005i
$$346$$ 0 0
$$347$$ −94.2652 −0.271658 −0.135829 0.990732i $$-0.543370\pi$$
−0.135829 + 0.990732i $$0.543370\pi$$
$$348$$ 0 0
$$349$$ 70.5283 0.202087 0.101043 0.994882i $$-0.467782\pi$$
0.101043 + 0.994882i $$0.467782\pi$$
$$350$$ 0 0
$$351$$ 80.6991 0.229912
$$352$$ 0 0
$$353$$ 53.2357 0.150809 0.0754047 0.997153i $$-0.475975\pi$$
0.0754047 + 0.997153i $$0.475975\pi$$
$$354$$ 0 0
$$355$$ 94.2238i 0.265419i
$$356$$ 0 0
$$357$$ 61.4964 0.172259
$$358$$ 0 0
$$359$$ 406.022i 1.13098i 0.824755 + 0.565491i $$0.191313\pi$$
−0.824755 + 0.565491i $$0.808687\pi$$
$$360$$ 0 0
$$361$$ 272.466 0.754754
$$362$$ 0 0
$$363$$ −178.517 −0.491781
$$364$$ 0 0
$$365$$ 110.216i 0.301961i
$$366$$ 0 0
$$367$$ 59.1029i 0.161043i 0.996753 + 0.0805216i $$0.0256586\pi$$
−0.996753 + 0.0805216i $$0.974341\pi$$
$$368$$ 0 0
$$369$$ −57.1394 −0.154849
$$370$$ 0 0
$$371$$ 128.753 0.347044
$$372$$ 0 0
$$373$$ 165.580i 0.443915i −0.975056 0.221957i $$-0.928755\pi$$
0.975056 0.221957i $$-0.0712446\pi$$
$$374$$ 0 0
$$375$$ 19.3649i 0.0516398i
$$376$$ 0 0
$$377$$ −458.361 −1.21581
$$378$$ 0 0
$$379$$ 104.043i 0.274521i 0.990535 + 0.137260i $$0.0438297\pi$$
−0.990535 + 0.137260i $$0.956170\pi$$
$$380$$ 0 0
$$381$$ 46.5838 0.122267
$$382$$ 0 0
$$383$$ 197.855i 0.516593i −0.966066 0.258296i $$-0.916839\pi$$
0.966066 0.258296i $$-0.0831611\pi$$
$$384$$ 0 0
$$385$$ 33.7536i 0.0876717i
$$386$$ 0 0
$$387$$ 8.24625i 0.0213081i
$$388$$ 0 0
$$389$$ 290.651i 0.747174i −0.927595 0.373587i $$-0.878128\pi$$
0.927595 0.373587i $$-0.121872\pi$$
$$390$$ 0 0
$$391$$ 140.427 + 181.009i 0.359147 + 0.462939i
$$392$$ 0 0
$$393$$ 239.135 0.608485
$$394$$ 0 0
$$395$$ −277.413 −0.702310
$$396$$ 0 0
$$397$$ 590.952 1.48854 0.744272 0.667877i $$-0.232797\pi$$
0.744272 + 0.667877i $$0.232797\pi$$
$$398$$ 0 0
$$399$$ 58.0924 0.145595
$$400$$ 0 0
$$401$$ 541.538i 1.35047i 0.737603 + 0.675234i $$0.235957\pi$$
−0.737603 + 0.675234i $$0.764043\pi$$
$$402$$ 0 0
$$403$$ 891.406 2.21193
$$404$$ 0 0
$$405$$ 20.1246i 0.0496904i
$$406$$ 0 0
$$407$$ 73.9085 0.181593
$$408$$ 0 0
$$409$$ 17.9115 0.0437934 0.0218967 0.999760i $$-0.493030\pi$$
0.0218967 + 0.999760i $$0.493030\pi$$
$$410$$ 0 0
$$411$$ 381.798i 0.928948i
$$412$$ 0 0
$$413$$ 48.5683i 0.117599i
$$414$$ 0 0
$$415$$ 124.089 0.299011
$$416$$ 0 0
$$417$$ 246.268 0.590572
$$418$$ 0 0
$$419$$ 744.342i 1.77647i −0.459387 0.888236i $$-0.651931\pi$$
0.459387 0.888236i $$-0.348069\pi$$
$$420$$ 0 0
$$421$$ 172.505i 0.409751i −0.978788 0.204876i $$-0.934321\pi$$
0.978788 0.204876i $$-0.0656790\pi$$
$$422$$ 0 0
$$423$$ 75.5500 0.178605
$$424$$ 0 0
$$425$$ 49.8030i 0.117183i
$$426$$ 0 0
$$427$$ −100.821 −0.236115
$$428$$ 0 0
$$429$$ 113.914i 0.265535i
$$430$$ 0 0
$$431$$ 371.378i 0.861666i 0.902432 + 0.430833i $$0.141780\pi$$
−0.902432 + 0.430833i $$0.858220\pi$$
$$432$$ 0 0
$$433$$ 329.060i 0.759955i 0.924996 + 0.379977i $$0.124068\pi$$
−0.924996 + 0.379977i $$0.875932\pi$$
$$434$$ 0 0
$$435$$ 114.305i 0.262771i
$$436$$ 0 0
$$437$$ 132.653 + 170.990i 0.303555 + 0.391281i
$$438$$ 0 0
$$439$$ −666.702 −1.51868 −0.759341 0.650693i $$-0.774479\pi$$
−0.759341 + 0.650693i $$0.774479\pi$$
$$440$$ 0 0
$$441$$ 108.882 0.246898
$$442$$ 0 0
$$443$$ 532.068 1.20106 0.600528 0.799604i $$-0.294957\pi$$
0.600528 + 0.799604i $$0.294957\pi$$
$$444$$ 0 0
$$445$$ −106.277 −0.238824
$$446$$ 0 0
$$447$$ 139.423i 0.311909i
$$448$$ 0 0
$$449$$ 298.428 0.664650 0.332325 0.943165i $$-0.392167\pi$$
0.332325 + 0.943165i $$0.392167\pi$$
$$450$$ 0 0
$$451$$ 80.6576i 0.178842i
$$452$$ 0 0
$$453$$ 52.6947 0.116324
$$454$$ 0 0
$$455$$ −123.787 −0.272060
$$456$$ 0 0
$$457$$ 641.898i 1.40459i 0.711886 + 0.702295i $$0.247841\pi$$
−0.711886 + 0.702295i $$0.752159\pi$$
$$458$$ 0 0
$$459$$ 51.7568i 0.112760i
$$460$$ 0 0
$$461$$ −199.215 −0.432137 −0.216069 0.976378i $$-0.569323\pi$$
−0.216069 + 0.976378i $$0.569323\pi$$
$$462$$ 0 0
$$463$$ 89.7770 0.193903 0.0969514 0.995289i $$-0.469091\pi$$
0.0969514 + 0.995289i $$0.469091\pi$$
$$464$$ 0 0
$$465$$ 222.297i 0.478059i
$$466$$ 0 0
$$467$$ 393.352i 0.842295i −0.906992 0.421148i $$-0.861627\pi$$
0.906992 0.421148i $$-0.138373\pi$$
$$468$$ 0 0
$$469$$ −69.8983 −0.149037
$$470$$ 0 0
$$471$$ 11.5527i 0.0245279i
$$472$$ 0 0
$$473$$ 11.6404 0.0246096
$$474$$ 0 0
$$475$$ 47.0462i 0.0990447i
$$476$$ 0 0
$$477$$ 108.362i 0.227173i
$$478$$ 0 0
$$479$$ 166.947i 0.348533i 0.984699 + 0.174266i $$0.0557554\pi$$
−0.984699 + 0.174266i $$0.944245\pi$$
$$480$$ 0 0
$$481$$ 271.050i 0.563514i
$$482$$ 0 0
$$483$$ −87.0419 112.197i −0.180211 0.232291i
$$484$$ 0 0
$$485$$ −204.266 −0.421168
$$486$$ 0 0
$$487$$ −16.4883 −0.0338568 −0.0169284 0.999857i $$-0.505389\pi$$
−0.0169284 + 0.999857i $$0.505389\pi$$
$$488$$ 0 0
$$489$$ −348.159 −0.711981
$$490$$ 0 0
$$491$$ 278.396 0.566997 0.283499 0.958973i $$-0.408505\pi$$
0.283499 + 0.958973i $$0.408505\pi$$
$$492$$ 0 0
$$493$$ 293.972i 0.596292i
$$494$$ 0 0
$$495$$ −28.4078 −0.0573895
$$496$$ 0 0
$$497$$ 150.203i 0.302220i
$$498$$ 0 0
$$499$$ −627.776 −1.25807 −0.629034 0.777378i $$-0.716549\pi$$
−0.629034 + 0.777378i $$0.716549\pi$$
$$500$$ 0 0
$$501$$ −23.0150 −0.0459381
$$502$$ 0 0
$$503$$ 328.082i 0.652250i 0.945327 + 0.326125i $$0.105743\pi$$
−0.945327 + 0.326125i $$0.894257\pi$$
$$504$$ 0 0
$$505$$ 220.298i 0.436235i
$$506$$ 0 0
$$507$$ −125.050 −0.246648
$$508$$ 0 0
$$509$$ −77.9340 −0.153112 −0.0765560 0.997065i $$-0.524392\pi$$
−0.0765560 + 0.997065i $$0.524392\pi$$
$$510$$ 0 0
$$511$$ 175.696i 0.343828i
$$512$$ 0 0
$$513$$ 48.8919i 0.0953058i
$$514$$ 0 0
$$515$$ 146.983 0.285403
$$516$$ 0 0
$$517$$ 106.646i 0.206279i
$$518$$ 0 0
$$519$$ −325.114 −0.626423
$$520$$ 0 0
$$521$$ 202.724i 0.389105i 0.980892 + 0.194552i $$0.0623254\pi$$
−0.980892 + 0.194552i $$0.937675\pi$$
$$522$$ 0 0
$$523$$ 253.141i 0.484018i 0.970274 + 0.242009i $$0.0778063\pi$$
−0.970274 + 0.242009i $$0.922194\pi$$
$$524$$ 0 0
$$525$$ 30.8698i 0.0587997i
$$526$$ 0 0
$$527$$ 571.708i 1.08483i
$$528$$ 0 0
$$529$$ 131.481 512.400i 0.248547 0.968620i
$$530$$ 0 0
$$531$$ 40.8762 0.0769796
$$532$$ 0 0
$$533$$ 295.802 0.554975
$$534$$ 0 0
$$535$$ 453.354 0.847391
$$536$$ 0 0
$$537$$ −93.8544 −0.174775
$$538$$ 0 0
$$539$$ 153.697i 0.285153i
$$540$$ 0 0
$$541$$ 254.574 0.470561 0.235281 0.971928i $$-0.424399\pi$$
0.235281 + 0.971928i $$0.424399\pi$$
$$542$$ 0 0
$$543$$ 37.9627i 0.0699128i
$$544$$ 0 0
$$545$$ −197.703 −0.362759
$$546$$ 0 0
$$547$$ −499.840 −0.913784 −0.456892 0.889522i $$-0.651037\pi$$
−0.456892 + 0.889522i $$0.651037\pi$$
$$548$$ 0 0
$$549$$ 84.8534i 0.154560i
$$550$$ 0 0
$$551$$ 277.700i 0.503992i
$$552$$ 0 0
$$553$$ 442.227 0.799687
$$554$$ 0 0
$$555$$ −67.5941 −0.121791
$$556$$ 0 0
$$557$$ 530.592i 0.952589i −0.879286 0.476295i $$-0.841980\pi$$
0.879286 0.476295i $$-0.158020\pi$$
$$558$$ 0 0
$$559$$ 42.6896i 0.0763678i
$$560$$ 0 0
$$561$$ −73.0596 −0.130231
$$562$$ 0 0
$$563$$ 907.847i 1.61252i 0.591564 + 0.806258i $$0.298511\pi$$
−0.591564 + 0.806258i $$0.701489\pi$$
$$564$$ 0 0
$$565$$ 5.61283 0.00993421
$$566$$ 0 0
$$567$$ 32.0809i 0.0565800i
$$568$$ 0 0
$$569$$ 132.208i 0.232352i 0.993229 + 0.116176i $$0.0370637\pi$$
−0.993229 + 0.116176i $$0.962936\pi$$
$$570$$ 0 0
$$571$$ 314.412i 0.550635i −0.961353 0.275317i $$-0.911217\pi$$
0.961353 0.275317i $$-0.0887829\pi$$
$$572$$ 0 0
$$573$$ 597.455i 1.04268i
$$574$$ 0 0
$$575$$ −90.8626 + 70.4910i −0.158022 + 0.122593i
$$576$$ 0 0
$$577$$ 928.302 1.60884 0.804421 0.594060i $$-0.202476\pi$$
0.804421 + 0.594060i $$0.202476\pi$$
$$578$$ 0 0
$$579$$ −357.803 −0.617967
$$580$$ 0 0
$$581$$ −197.812 −0.340469
$$582$$ 0 0
$$583$$ −152.963 −0.262372
$$584$$ 0 0
$$585$$ 104.182i 0.178089i
$$586$$ 0 0
$$587$$ −696.989 −1.18737 −0.593687 0.804696i $$-0.702328\pi$$
−0.593687 + 0.804696i $$0.702328\pi$$
$$588$$ 0 0
$$589$$ 540.062i 0.916913i
$$590$$ 0 0
$$591$$ −267.854 −0.453222
$$592$$ 0 0
$$593$$ 26.7574 0.0451221 0.0225610 0.999745i $$-0.492818\pi$$
0.0225610 + 0.999745i $$0.492818\pi$$
$$594$$ 0 0
$$595$$ 79.3915i 0.133431i
$$596$$ 0 0
$$597$$ 221.888i 0.371672i
$$598$$ 0 0
$$599$$ −589.134 −0.983529 −0.491765 0.870728i $$-0.663648\pi$$
−0.491765 + 0.870728i $$0.663648\pi$$
$$600$$ 0 0
$$601$$ 161.884 0.269358 0.134679 0.990889i $$-0.457000\pi$$
0.134679 + 0.990889i $$0.457000\pi$$
$$602$$ 0 0
$$603$$ 58.8280i 0.0975588i
$$604$$ 0 0
$$605$$ 230.464i 0.380932i
$$606$$ 0 0
$$607$$ −440.997 −0.726519 −0.363259 0.931688i $$-0.618336\pi$$
−0.363259 + 0.931688i $$0.618336\pi$$
$$608$$ 0 0
$$609$$ 182.215i 0.299204i
$$610$$ 0 0
$$611$$ −391.111 −0.640116
$$612$$ 0 0
$$613$$ 900.764i 1.46944i −0.678373 0.734718i $$-0.737315\pi$$
0.678373 0.734718i $$-0.262685\pi$$
$$614$$ 0 0
$$615$$ 73.7666i 0.119946i
$$616$$ 0 0
$$617$$ 252.328i 0.408959i 0.978871 + 0.204480i $$0.0655502\pi$$
−0.978871 + 0.204480i $$0.934450\pi$$
$$618$$ 0 0
$$619$$ 65.5081i 0.105829i 0.998599 + 0.0529144i $$0.0168511\pi$$
−0.998599 + 0.0529144i $$0.983149\pi$$
$$620$$ 0 0
$$621$$ −94.4272 + 73.2564i −0.152057 + 0.117965i
$$622$$ 0 0
$$623$$ 169.417 0.271937
$$624$$ 0 0
$$625$$ 25.0000 0.0400000
$$626$$ 0 0
$$627$$ −69.0155 −0.110073
$$628$$ 0 0
$$629$$ −173.839 −0.276374
$$630$$ 0 0
$$631$$ 619.416i 0.981642i −0.871261 0.490821i $$-0.836697\pi$$
0.871261 0.490821i $$-0.163303\pi$$
$$632$$ 0 0
$$633$$ −70.7054 −0.111699
$$634$$ 0 0
$$635$$ 60.1394i 0.0947078i
$$636$$ 0 0
$$637$$ −563.666 −0.884876
$$638$$ 0 0
$$639$$ −126.415 −0.197832
$$640$$ 0 0
$$641$$ 204.217i 0.318592i −0.987231 0.159296i $$-0.949078\pi$$
0.987231 0.159296i $$-0.0509224\pi$$
$$642$$ 0 0
$$643$$ 955.967i 1.48673i 0.668886 + 0.743365i $$0.266771\pi$$
−0.668886 + 0.743365i $$0.733229\pi$$
$$644$$ 0 0
$$645$$ −10.6459 −0.0165052
$$646$$ 0 0
$$647$$ −845.973 −1.30753 −0.653766 0.756697i $$-0.726812\pi$$
−0.653766 + 0.756697i $$0.726812\pi$$
$$648$$ 0 0
$$649$$ 57.7006i 0.0889069i
$$650$$ 0 0
$$651$$ 354.367i 0.544342i
$$652$$ 0 0
$$653$$ 337.119 0.516261 0.258131 0.966110i $$-0.416894\pi$$
0.258131 + 0.966110i $$0.416894\pi$$
$$654$$ 0 0
$$655$$ 308.722i 0.471331i
$$656$$ 0 0
$$657$$ 147.870 0.225068
$$658$$ 0 0
$$659$$ 56.3995i 0.0855835i −0.999084 0.0427917i $$-0.986375\pi$$
0.999084 0.0427917i $$-0.0136252\pi$$
$$660$$ 0 0
$$661$$ 563.767i 0.852900i −0.904511 0.426450i $$-0.859764\pi$$
0.904511 0.426450i $$-0.140236\pi$$
$$662$$ 0 0
$$663$$ 267.937i 0.404128i
$$664$$ 0 0
$$665$$ 74.9969i 0.112777i
$$666$$ 0 0
$$667$$ 536.335 416.087i 0.804100 0.623819i
$$668$$ 0 0
$$669$$ −334.433 −0.499899
$$670$$ 0 0
$$671$$ 119.779 0.178508
$$672$$ 0 0
$$673$$ 576.120 0.856048 0.428024 0.903767i $$-0.359210\pi$$
0.428024 + 0.903767i $$0.359210\pi$$
$$674$$ 0 0
$$675$$ 25.9808 0.0384900
$$676$$ 0 0
$$677$$ 1003.34i 1.48204i 0.671482 + 0.741020i $$0.265658\pi$$
−0.671482 + 0.741020i $$0.734342\pi$$
$$678$$ 0 0
$$679$$ 325.623 0.479563
$$680$$ 0 0
$$681$$ 152.278i 0.223609i
$$682$$ 0 0
$$683$$ 603.291 0.883296 0.441648 0.897188i $$-0.354394\pi$$
0.441648 + 0.897188i $$0.354394\pi$$
$$684$$ 0 0
$$685$$ −492.899 −0.719560
$$686$$ 0 0
$$687$$ 670.395i 0.975830i
$$688$$ 0 0
$$689$$ 560.972i 0.814183i
$$690$$ 0 0
$$691$$ −710.505 −1.02823 −0.514113 0.857722i $$-0.671879\pi$$
−0.514113 + 0.857722i $$0.671879\pi$$
$$692$$ 0 0
$$693$$ 45.2852 0.0653466
$$694$$ 0 0
$$695$$ 317.931i 0.457455i
$$696$$ 0 0
$$697$$ 189.714i 0.272187i
$$698$$ 0 0
$$699$$ 234.852 0.335983
$$700$$ 0 0
$$701$$ 1303.64i 1.85968i 0.367961 + 0.929841i $$0.380056\pi$$
−0.367961 + 0.929841i $$0.619944\pi$$
$$702$$ 0 0
$$703$$ −164.217 −0.233594
$$704$$ 0 0
$$705$$ 97.5346i 0.138347i
$$706$$ 0 0
$$707$$ 351.180i 0.496719i
$$708$$ 0 0
$$709$$ 322.823i 0.455322i −0.973740 0.227661i $$-0.926892\pi$$
0.973740 0.227661i $$-0.0731077\pi$$
$$710$$ 0 0
$$711$$ 372.188i 0.523471i
$$712$$ 0 0
$$713$$ −1043.05 + 809.194i −1.46290 + 1.13491i
$$714$$ 0 0
$$715$$ 147.063 0.205682
$$716$$ 0 0
$$717$$ −799.370 −1.11488
$$718$$ 0 0
$$719$$ −972.662 −1.35280 −0.676399 0.736535i $$-0.736461\pi$$
−0.676399 + 0.736535i $$0.736461\pi$$
$$720$$ 0 0
$$721$$ −234.307 −0.324975
$$722$$ 0 0
$$723$$ 638.525i 0.883161i
$$724$$ 0 0
$$725$$ −147.567 −0.203541
$$726$$ 0 0
$$727$$ 1114.68i 1.53326i −0.642092 0.766628i $$-0.721933\pi$$
0.642092 0.766628i $$-0.278067\pi$$
$$728$$ 0 0
$$729$$ 27.0000 0.0370370
$$730$$ 0 0
$$731$$ −27.3792 −0.0374544
$$732$$ 0 0
$$733$$ 689.439i 0.940572i 0.882514 + 0.470286i $$0.155849\pi$$
−0.882514 + 0.470286i $$0.844151\pi$$
$$734$$ 0 0
$$735$$ 140.566i 0.191247i
$$736$$ 0 0
$$737$$ 83.0413 0.112675
$$738$$ 0 0
$$739$$ −1116.55 −1.51090 −0.755448 0.655209i $$-0.772581\pi$$
−0.755448 + 0.655209i $$0.772581\pi$$
$$740$$ 0 0
$$741$$ 253.106i 0.341573i
$$742$$ 0 0
$$743$$ 551.192i 0.741847i 0.928663 + 0.370924i $$0.120959\pi$$
−0.928663 + 0.370924i $$0.879041\pi$$
$$744$$ 0 0
$$745$$ −179.995 −0.241604
$$746$$ 0 0
$$747$$ 166.483i 0.222869i
$$748$$ 0 0
$$749$$ −722.698 −0.964883
$$750$$ 0 0
$$751$$ 405.850i 0.540413i −0.962802 0.270207i $$-0.912908\pi$$
0.962802 0.270207i $$-0.0870920\pi$$
$$752$$ 0 0
$$753$$ 134.586i 0.178733i
$$754$$ 0 0
$$755$$ 68.0285i 0.0901040i
$$756$$ 0 0
$$757$$ 793.111i 1.04770i 0.851810 + 0.523852i $$0.175505\pi$$
−0.851810 + 0.523852i $$0.824495\pi$$
$$758$$ 0 0
$$759$$ 103.408 + 133.293i 0.136243 + 0.175617i
$$760$$ 0 0
$$761$$ −563.287 −0.740193 −0.370097 0.928993i $$-0.620675\pi$$
−0.370097 + 0.928993i $$0.620675\pi$$
$$762$$ 0 0
$$763$$ 315.161 0.413055
$$764$$ 0 0
$$765$$ 66.8177 0.0873434
$$766$$ 0 0
$$767$$ −211.610 −0.275893
$$768$$ 0 0
$$769$$ 342.516i 0.445405i 0.974886 + 0.222702i $$0.0714878\pi$$
−0.974886 + 0.222702i $$0.928512\pi$$
$$770$$ 0 0
$$771$$ 272.272 0.353141
$$772$$ 0 0
$$773$$ 927.041i 1.19928i 0.800271 + 0.599638i $$0.204689\pi$$
−0.800271 + 0.599638i $$0.795311\pi$$
$$774$$ 0 0
$$775$$ 286.985 0.370303
$$776$$ 0 0
$$777$$ 107.752 0.138678
$$778$$ 0 0
$$779$$ 179.213i 0.230055i
$$780$$ 0 0
$$781$$ 178.446i 0.228484i
$$782$$ 0 0
$$783$$ −153.357 −0.195858
$$784$$ 0 0
$$785$$ −14.9144 −0.0189992
$$786$$ 0 0
$$787$$ 866.621i 1.10117i −0.834779 0.550585i $$-0.814405\pi$$
0.834779 0.550585i $$-0.185595\pi$$
$$788$$ 0 0
$$789$$ 783.006i 0.992404i
$$790$$ 0 0
$$791$$ −8.94748 −0.0113116
$$792$$ 0 0
$$793$$ 439.273i 0.553939i
$$794$$ 0 0
$$795$$ 139.894 0.175968
$$796$$ 0 0
$$797$$ 425.944i 0.534434i −0.963636 0.267217i $$-0.913896\pi$$
0.963636 0.267217i $$-0.0861041\pi$$
$$798$$ 0 0
$$799$$