# Properties

 Label 864.3.e.e.161.4 Level $864$ Weight $3$ Character 864.161 Analytic conductor $23.542$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(161,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("864.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.e (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: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2441150464.4 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196$$ x^8 - 14*x^6 + 77*x^4 - 188*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.4 Root $$2.27249 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 864.161 Dual form 864.3.e.e.161.6

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.88259i q^{5} +10.9726 q^{7} +O(q^{10})$$ $$q-1.88259i q^{5} +10.9726 q^{7} -0.171573i q^{11} +6.48528 q^{13} +27.2699i q^{17} +5.32478 q^{19} +3.51472i q^{23} +21.4558 q^{25} -34.8003i q^{29} -26.9469 q^{31} -20.6569i q^{35} +46.4264 q^{37} -23.5047i q^{41} -55.1858 q^{43} +57.5980i q^{47} +71.3970 q^{49} +51.7436i q^{53} -0.323002 q^{55} -82.2843i q^{59} +79.8823 q^{61} -12.2091i q^{65} +44.5362 q^{67} +41.2304i q^{71} +66.3381 q^{73} -1.88259i q^{77} +115.050 q^{79} +36.3381i q^{83} +51.3381 q^{85} -158.027i q^{89} +71.1601 q^{91} -10.0244i q^{95} +62.8823 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100})$$ 8 * q - 16 * q^13 - 32 * q^25 + 32 * q^37 + 96 * q^49 + 96 * q^61 - 216 * q^73 - 336 * q^85 - 40 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$5$$ − 1.88259i − 0.376519i −0.982119 0.188259i $$-0.939715\pi$$
0.982119 0.188259i $$-0.0602845\pi$$
$$6$$ 0 0
$$7$$ 10.9726 1.56751 0.783754 0.621071i $$-0.213302\pi$$
0.783754 + 0.621071i $$0.213302\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 0.171573i − 0.0155975i −0.999970 0.00779877i $$-0.997518\pi$$
0.999970 0.00779877i $$-0.00248245\pi$$
$$12$$ 0 0
$$13$$ 6.48528 0.498868 0.249434 0.968392i $$-0.419755\pi$$
0.249434 + 0.968392i $$0.419755\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 27.2699i 1.60411i 0.597249 + 0.802056i $$0.296260\pi$$
−0.597249 + 0.802056i $$0.703740\pi$$
$$18$$ 0 0
$$19$$ 5.32478 0.280251 0.140126 0.990134i $$-0.455249\pi$$
0.140126 + 0.990134i $$0.455249\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.51472i 0.152814i 0.997077 + 0.0764069i $$0.0243448\pi$$
−0.997077 + 0.0764069i $$0.975655\pi$$
$$24$$ 0 0
$$25$$ 21.4558 0.858234
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 34.8003i − 1.20001i −0.799997 0.600004i $$-0.795165\pi$$
0.799997 0.600004i $$-0.204835\pi$$
$$30$$ 0 0
$$31$$ −26.9469 −0.869254 −0.434627 0.900610i $$-0.643120\pi$$
−0.434627 + 0.900610i $$0.643120\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 20.6569i − 0.590196i
$$36$$ 0 0
$$37$$ 46.4264 1.25477 0.627384 0.778710i $$-0.284126\pi$$
0.627384 + 0.778710i $$0.284126\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 23.5047i − 0.573285i −0.958038 0.286643i $$-0.907461\pi$$
0.958038 0.286643i $$-0.0925393\pi$$
$$42$$ 0 0
$$43$$ −55.1858 −1.28339 −0.641695 0.766960i $$-0.721769\pi$$
−0.641695 + 0.766960i $$0.721769\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 57.5980i 1.22549i 0.790281 + 0.612744i $$0.209935\pi$$
−0.790281 + 0.612744i $$0.790065\pi$$
$$48$$ 0 0
$$49$$ 71.3970 1.45708
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 51.7436i 0.976294i 0.872761 + 0.488147i $$0.162327\pi$$
−0.872761 + 0.488147i $$0.837673\pi$$
$$54$$ 0 0
$$55$$ −0.323002 −0.00587276
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 82.2843i − 1.39465i −0.716756 0.697324i $$-0.754374\pi$$
0.716756 0.697324i $$-0.245626\pi$$
$$60$$ 0 0
$$61$$ 79.8823 1.30955 0.654773 0.755826i $$-0.272764\pi$$
0.654773 + 0.755826i $$0.272764\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 12.2091i − 0.187833i
$$66$$ 0 0
$$67$$ 44.5362 0.664720 0.332360 0.943153i $$-0.392155\pi$$
0.332360 + 0.943153i $$0.392155\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 41.2304i 0.580711i 0.956919 + 0.290355i $$0.0937735\pi$$
−0.956919 + 0.290355i $$0.906227\pi$$
$$72$$ 0 0
$$73$$ 66.3381 0.908741 0.454371 0.890813i $$-0.349864\pi$$
0.454371 + 0.890813i $$0.349864\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.88259i − 0.0244493i
$$78$$ 0 0
$$79$$ 115.050 1.45633 0.728167 0.685400i $$-0.240373\pi$$
0.728167 + 0.685400i $$0.240373\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 36.3381i 0.437808i 0.975746 + 0.218904i $$0.0702482\pi$$
−0.975746 + 0.218904i $$0.929752\pi$$
$$84$$ 0 0
$$85$$ 51.3381 0.603978
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 158.027i − 1.77558i −0.460245 0.887792i $$-0.652238\pi$$
0.460245 0.887792i $$-0.347762\pi$$
$$90$$ 0 0
$$91$$ 71.1601 0.781979
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 10.0244i − 0.105520i
$$96$$ 0 0
$$97$$ 62.8823 0.648271 0.324135 0.946011i $$-0.394927\pi$$
0.324135 + 0.946011i $$0.394927\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 130.702i 1.29408i 0.762458 + 0.647038i $$0.223992\pi$$
−0.762458 + 0.647038i $$0.776008\pi$$
$$102$$ 0 0
$$103$$ −72.4521 −0.703419 −0.351709 0.936109i $$-0.614399\pi$$
−0.351709 + 0.936109i $$0.614399\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 168.255i − 1.57248i −0.617924 0.786238i $$-0.712026\pi$$
0.617924 0.786238i $$-0.287974\pi$$
$$108$$ 0 0
$$109$$ 103.397 0.948596 0.474298 0.880364i $$-0.342702\pi$$
0.474298 + 0.880364i $$0.342702\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 140.115i 1.23995i 0.784621 + 0.619976i $$0.212858\pi$$
−0.784621 + 0.619976i $$0.787142\pi$$
$$114$$ 0 0
$$115$$ 6.61678 0.0575373
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 299.220i 2.51446i
$$120$$ 0 0
$$121$$ 120.971 0.999757
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 87.4574i − 0.699660i
$$126$$ 0 0
$$127$$ −120.052 −0.945292 −0.472646 0.881252i $$-0.656701\pi$$
−0.472646 + 0.881252i $$0.656701\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 198.421i − 1.51467i −0.653028 0.757333i $$-0.726502\pi$$
0.653028 0.757333i $$-0.273498\pi$$
$$132$$ 0 0
$$133$$ 58.4264 0.439296
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 120.375i 0.878650i 0.898328 + 0.439325i $$0.144782\pi$$
−0.898328 + 0.439325i $$0.855218\pi$$
$$138$$ 0 0
$$139$$ −11.9416 −0.0859105 −0.0429553 0.999077i $$-0.513677\pi$$
−0.0429553 + 0.999077i $$0.513677\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 1.11270i − 0.00778111i
$$144$$ 0 0
$$145$$ −65.5147 −0.451826
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 55.5088i − 0.372542i −0.982498 0.186271i $$-0.940360\pi$$
0.982498 0.186271i $$-0.0596403\pi$$
$$150$$ 0 0
$$151$$ −203.154 −1.34539 −0.672695 0.739920i $$-0.734863\pi$$
−0.672695 + 0.739920i $$0.734863\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 50.7300i 0.327290i
$$156$$ 0 0
$$157$$ −164.971 −1.05077 −0.525384 0.850865i $$-0.676078\pi$$
−0.525384 + 0.850865i $$0.676078\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 38.5654i 0.239537i
$$162$$ 0 0
$$163$$ −178.948 −1.09784 −0.548919 0.835875i $$-0.684961\pi$$
−0.548919 + 0.835875i $$0.684961\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 43.8924i 0.262828i 0.991328 + 0.131414i $$0.0419518\pi$$
−0.991328 + 0.131414i $$0.958048\pi$$
$$168$$ 0 0
$$169$$ −126.941 −0.751131
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 191.858i − 1.10901i −0.832181 0.554504i $$-0.812908\pi$$
0.832181 0.554504i $$-0.187092\pi$$
$$174$$ 0 0
$$175$$ 235.425 1.34529
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 144.765i − 0.808740i −0.914595 0.404370i $$-0.867491\pi$$
0.914595 0.404370i $$-0.132509\pi$$
$$180$$ 0 0
$$181$$ −0.735065 −0.00406113 −0.00203057 0.999998i $$-0.500646\pi$$
−0.00203057 + 0.999998i $$0.500646\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 87.4020i − 0.472443i
$$186$$ 0 0
$$187$$ 4.67877 0.0250202
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 290.132i 1.51902i 0.650498 + 0.759508i $$0.274560\pi$$
−0.650498 + 0.759508i $$0.725440\pi$$
$$192$$ 0 0
$$193$$ 63.6030 0.329549 0.164775 0.986331i $$-0.447310\pi$$
0.164775 + 0.986331i $$0.447310\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 88.3710i − 0.448584i −0.974522 0.224292i $$-0.927993\pi$$
0.974522 0.224292i $$-0.0720069\pi$$
$$198$$ 0 0
$$199$$ 318.204 1.59902 0.799508 0.600656i $$-0.205094\pi$$
0.799508 + 0.600656i $$0.205094\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 381.848i − 1.88102i
$$204$$ 0 0
$$205$$ −44.2498 −0.215853
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 0.913587i − 0.00437123i
$$210$$ 0 0
$$211$$ 46.6310 0.221000 0.110500 0.993876i $$-0.464755\pi$$
0.110500 + 0.993876i $$0.464755\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 103.892i 0.483220i
$$216$$ 0 0
$$217$$ −295.676 −1.36256
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 176.853i 0.800239i
$$222$$ 0 0
$$223$$ −47.2770 −0.212004 −0.106002 0.994366i $$-0.533805\pi$$
−0.106002 + 0.994366i $$0.533805\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 342.500i 1.50881i 0.656410 + 0.754404i $$0.272074\pi$$
−0.656410 + 0.754404i $$0.727926\pi$$
$$228$$ 0 0
$$229$$ −381.647 −1.66658 −0.833290 0.552836i $$-0.813545\pi$$
−0.833290 + 0.552836i $$0.813545\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 383.716i 1.64685i 0.567424 + 0.823426i $$0.307940\pi$$
−0.567424 + 0.823426i $$0.692060\pi$$
$$234$$ 0 0
$$235$$ 108.434 0.461419
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 315.161i − 1.31867i −0.751850 0.659334i $$-0.770838\pi$$
0.751850 0.659334i $$-0.229162\pi$$
$$240$$ 0 0
$$241$$ −256.558 −1.06456 −0.532279 0.846569i $$-0.678664\pi$$
−0.532279 + 0.846569i $$0.678664\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 134.411i − 0.548618i
$$246$$ 0 0
$$247$$ 34.5327 0.139808
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 246.000i − 0.980080i −0.871700 0.490040i $$-0.836982\pi$$
0.871700 0.490040i $$-0.163018\pi$$
$$252$$ 0 0
$$253$$ 0.603030 0.00238352
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 443.046i − 1.72391i −0.506982 0.861957i $$-0.669239\pi$$
0.506982 0.861957i $$-0.330761\pi$$
$$258$$ 0 0
$$259$$ 509.416 1.96686
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 479.897i 1.82470i 0.409409 + 0.912351i $$0.365735\pi$$
−0.409409 + 0.912351i $$0.634265\pi$$
$$264$$ 0 0
$$265$$ 97.4121 0.367593
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 418.406i 1.55541i 0.628628 + 0.777706i $$0.283617\pi$$
−0.628628 + 0.777706i $$0.716383\pi$$
$$270$$ 0 0
$$271$$ −278.347 −1.02711 −0.513555 0.858057i $$-0.671672\pi$$
−0.513555 + 0.858057i $$0.671672\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 3.68124i − 0.0133863i
$$276$$ 0 0
$$277$$ −192.617 −0.695369 −0.347685 0.937612i $$-0.613032\pi$$
−0.347685 + 0.937612i $$0.613032\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 435.515i − 1.54988i −0.632037 0.774939i $$-0.717781\pi$$
0.632037 0.774939i $$-0.282219\pi$$
$$282$$ 0 0
$$283$$ −518.774 −1.83312 −0.916562 0.399894i $$-0.869047\pi$$
−0.916562 + 0.399894i $$0.869047\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 257.907i − 0.898629i
$$288$$ 0 0
$$289$$ −454.647 −1.57317
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 117.634i − 0.401482i −0.979644 0.200741i $$-0.935665\pi$$
0.979644 0.200741i $$-0.0643350\pi$$
$$294$$ 0 0
$$295$$ −154.908 −0.525111
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 22.7939i 0.0762339i
$$300$$ 0 0
$$301$$ −605.529 −2.01172
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 150.386i − 0.493068i
$$306$$ 0 0
$$307$$ −371.618 −1.21048 −0.605241 0.796042i $$-0.706923\pi$$
−0.605241 + 0.796042i $$0.706923\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 27.8680i 0.0896076i 0.998996 + 0.0448038i $$0.0142663\pi$$
−0.998996 + 0.0448038i $$0.985734\pi$$
$$312$$ 0 0
$$313$$ −279.368 −0.892548 −0.446274 0.894896i $$-0.647249\pi$$
−0.446274 + 0.894896i $$0.647249\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 546.367i 1.72355i 0.507287 + 0.861777i $$0.330648\pi$$
−0.507287 + 0.861777i $$0.669352\pi$$
$$318$$ 0 0
$$319$$ −5.97078 −0.0187172
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 145.206i 0.449554i
$$324$$ 0 0
$$325$$ 139.147 0.428145
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 631.997i 1.92096i
$$330$$ 0 0
$$331$$ −305.294 −0.922337 −0.461168 0.887313i $$-0.652570\pi$$
−0.461168 + 0.887313i $$0.652570\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 83.8436i − 0.250279i
$$336$$ 0 0
$$337$$ 159.206 0.472422 0.236211 0.971702i $$-0.424094\pi$$
0.236211 + 0.971702i $$0.424094\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.62335i 0.0135582i
$$342$$ 0 0
$$343$$ 245.752 0.716478
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 296.387i − 0.854141i −0.904218 0.427070i $$-0.859546\pi$$
0.904218 0.427070i $$-0.140454\pi$$
$$348$$ 0 0
$$349$$ 108.839 0.311858 0.155929 0.987768i $$-0.450163\pi$$
0.155929 + 0.987768i $$0.450163\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 12.3200i − 0.0349008i −0.999848 0.0174504i $$-0.994445\pi$$
0.999848 0.0174504i $$-0.00555492\pi$$
$$354$$ 0 0
$$355$$ 77.6201 0.218648
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 92.2254i 0.256895i 0.991716 + 0.128448i $$0.0409994\pi$$
−0.991716 + 0.128448i $$0.959001\pi$$
$$360$$ 0 0
$$361$$ −332.647 −0.921459
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 124.888i − 0.342158i
$$366$$ 0 0
$$367$$ 13.0673 0.0356058 0.0178029 0.999842i $$-0.494333\pi$$
0.0178029 + 0.999842i $$0.494333\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 567.759i 1.53035i
$$372$$ 0 0
$$373$$ −233.632 −0.626361 −0.313180 0.949694i $$-0.601394\pi$$
−0.313180 + 0.949694i $$0.601394\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 225.689i − 0.598646i
$$378$$ 0 0
$$379$$ 683.528 1.80351 0.901753 0.432253i $$-0.142281\pi$$
0.901753 + 0.432253i $$0.142281\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 279.858i − 0.730699i −0.930870 0.365350i $$-0.880949\pi$$
0.930870 0.365350i $$-0.119051\pi$$
$$384$$ 0 0
$$385$$ −3.54416 −0.00920560
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 260.434i − 0.669497i −0.942308 0.334748i $$-0.891349\pi$$
0.942308 0.334748i $$-0.108651\pi$$
$$390$$ 0 0
$$391$$ −95.8460 −0.245130
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 216.593i − 0.548337i
$$396$$ 0 0
$$397$$ 167.809 0.422693 0.211346 0.977411i $$-0.432215\pi$$
0.211346 + 0.977411i $$0.432215\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 60.2430i − 0.150232i −0.997175 0.0751159i $$-0.976067\pi$$
0.997175 0.0751159i $$-0.0239327\pi$$
$$402$$ 0 0
$$403$$ −174.758 −0.433643
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 7.96551i − 0.0195713i
$$408$$ 0 0
$$409$$ 532.765 1.30260 0.651301 0.758819i $$-0.274223\pi$$
0.651301 + 0.758819i $$0.274223\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 902.869i − 2.18612i
$$414$$ 0 0
$$415$$ 68.4098 0.164843
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 483.255i 1.15335i 0.816973 + 0.576676i $$0.195651\pi$$
−0.816973 + 0.576676i $$0.804349\pi$$
$$420$$ 0 0
$$421$$ −29.2061 −0.0693731 −0.0346865 0.999398i $$-0.511043\pi$$
−0.0346865 + 0.999398i $$0.511043\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 585.098i 1.37670i
$$426$$ 0 0
$$427$$ 876.512 2.05272
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 634.971i − 1.47325i −0.676302 0.736625i $$-0.736418\pi$$
0.676302 0.736625i $$-0.263582\pi$$
$$432$$ 0 0
$$433$$ −28.5004 −0.0658209 −0.0329104 0.999458i $$-0.510478\pi$$
−0.0329104 + 0.999458i $$0.510478\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 18.7151i 0.0428263i
$$438$$ 0 0
$$439$$ 287.547 0.655006 0.327503 0.944850i $$-0.393793\pi$$
0.327503 + 0.944850i $$0.393793\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 0.176624i 0 0.000398699i −1.00000 0.000199349i $$-0.999937\pi$$
1.00000 0.000199349i $$-6.34549e-5\pi$$
$$444$$ 0 0
$$445$$ −297.500 −0.668540
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 36.5166i − 0.0813287i −0.999173 0.0406644i $$-0.987053\pi$$
0.999173 0.0406644i $$-0.0129474\pi$$
$$450$$ 0 0
$$451$$ −4.03277 −0.00894184
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 133.966i − 0.294430i
$$456$$ 0 0
$$457$$ −243.235 −0.532244 −0.266122 0.963939i $$-0.585742\pi$$
−0.266122 + 0.963939i $$0.585742\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 155.231i 0.336726i 0.985725 + 0.168363i $$0.0538481\pi$$
−0.985725 + 0.168363i $$0.946152\pi$$
$$462$$ 0 0
$$463$$ −444.550 −0.960151 −0.480076 0.877227i $$-0.659391\pi$$
−0.480076 + 0.877227i $$0.659391\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 853.357i 1.82732i 0.406482 + 0.913659i $$0.366755\pi$$
−0.406482 + 0.913659i $$0.633245\pi$$
$$468$$ 0 0
$$469$$ 488.676 1.04195
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.46838i 0.0200177i
$$474$$ 0 0
$$475$$ 114.248 0.240521
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 445.882i − 0.930861i −0.885085 0.465430i $$-0.845900\pi$$
0.885085 0.465430i $$-0.154100\pi$$
$$480$$ 0 0
$$481$$ 301.088 0.625963
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 118.382i − 0.244086i
$$486$$ 0 0
$$487$$ −438.413 −0.900232 −0.450116 0.892970i $$-0.648617\pi$$
−0.450116 + 0.892970i $$0.648617\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 186.754i 0.380355i 0.981750 + 0.190178i $$0.0609064\pi$$
−0.981750 + 0.190178i $$0.939094\pi$$
$$492$$ 0 0
$$493$$ 948.999 1.92495
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 452.403i 0.910268i
$$498$$ 0 0
$$499$$ −432.775 −0.867284 −0.433642 0.901085i $$-0.642772\pi$$
−0.433642 + 0.901085i $$0.642772\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 45.6913i 0.0908377i 0.998968 + 0.0454188i $$0.0144622\pi$$
−0.998968 + 0.0454188i $$0.985538\pi$$
$$504$$ 0 0
$$505$$ 246.058 0.487244
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 267.051i 0.524658i 0.964978 + 0.262329i $$0.0844906\pi$$
−0.964978 + 0.262329i $$0.915509\pi$$
$$510$$ 0 0
$$511$$ 727.898 1.42446
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 136.398i 0.264850i
$$516$$ 0 0
$$517$$ 9.88225 0.0191146
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 144.101i − 0.276586i −0.990391 0.138293i $$-0.955838\pi$$
0.990391 0.138293i $$-0.0441616\pi$$
$$522$$ 0 0
$$523$$ −631.083 −1.20666 −0.603330 0.797491i $$-0.706160\pi$$
−0.603330 + 0.797491i $$0.706160\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 734.839i − 1.39438i
$$528$$ 0 0
$$529$$ 516.647 0.976648
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 152.435i − 0.285994i
$$534$$ 0 0
$$535$$ −316.755 −0.592066
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 12.2498i − 0.0227269i
$$540$$ 0 0
$$541$$ 235.765 0.435794 0.217897 0.975972i $$-0.430080\pi$$
0.217897 + 0.975972i $$0.430080\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 194.654i − 0.357164i
$$546$$ 0 0
$$547$$ −813.418 −1.48705 −0.743526 0.668707i $$-0.766848\pi$$
−0.743526 + 0.668707i $$0.766848\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 185.304i − 0.336304i
$$552$$ 0 0
$$553$$ 1262.40 2.28281
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 175.773i 0.315571i 0.987473 + 0.157786i $$0.0504355\pi$$
−0.987473 + 0.157786i $$0.949565\pi$$
$$558$$ 0 0
$$559$$ −357.895 −0.640242
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 572.927i 1.01763i 0.860875 + 0.508816i $$0.169917\pi$$
−0.860875 + 0.508816i $$0.830083\pi$$
$$564$$ 0 0
$$565$$ 263.779 0.466865
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 814.664i − 1.43175i −0.698230 0.715873i $$-0.746029\pi$$
0.698230 0.715873i $$-0.253971\pi$$
$$570$$ 0 0
$$571$$ −128.284 −0.224665 −0.112333 0.993671i $$-0.535832\pi$$
−0.112333 + 0.993671i $$0.535832\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 75.4113i 0.131150i
$$576$$ 0 0
$$577$$ 30.4996 0.0528589 0.0264294 0.999651i $$-0.491586\pi$$
0.0264294 + 0.999651i $$0.491586\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 398.722i 0.686268i
$$582$$ 0 0
$$583$$ 8.87780 0.0152278
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 193.794i − 0.330143i −0.986282 0.165071i $$-0.947215\pi$$
0.986282 0.165071i $$-0.0527855\pi$$
$$588$$ 0 0
$$589$$ −143.486 −0.243610
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 684.488i 1.15428i 0.816645 + 0.577140i $$0.195831\pi$$
−0.816645 + 0.577140i $$0.804169\pi$$
$$594$$ 0 0
$$595$$ 563.310 0.946740
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 814.244i 1.35934i 0.733519 + 0.679669i $$0.237877\pi$$
−0.733519 + 0.679669i $$0.762123\pi$$
$$600$$ 0 0
$$601$$ −531.368 −0.884139 −0.442069 0.896981i $$-0.645756\pi$$
−0.442069 + 0.896981i $$0.645756\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 227.738i − 0.376427i
$$606$$ 0 0
$$607$$ 464.880 0.765865 0.382933 0.923776i $$-0.374914\pi$$
0.382933 + 0.923776i $$0.374914\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 373.539i 0.611357i
$$612$$ 0 0
$$613$$ 1120.59 1.82804 0.914019 0.405672i $$-0.132962\pi$$
0.914019 + 0.405672i $$0.132962\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 561.372i − 0.909841i −0.890532 0.454921i $$-0.849668\pi$$
0.890532 0.454921i $$-0.150332\pi$$
$$618$$ 0 0
$$619$$ −471.340 −0.761454 −0.380727 0.924687i $$-0.624326\pi$$
−0.380727 + 0.924687i $$0.624326\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 1733.96i − 2.78324i
$$624$$ 0 0
$$625$$ 371.749 0.594799
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1266.04i 2.01279i
$$630$$ 0 0
$$631$$ 431.160 0.683296 0.341648 0.939828i $$-0.389015\pi$$
0.341648 + 0.939828i $$0.389015\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 226.009i 0.355920i
$$636$$ 0 0
$$637$$ 463.029 0.726891
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 399.580i − 0.623370i −0.950186 0.311685i $$-0.899107\pi$$
0.950186 0.311685i $$-0.100893\pi$$
$$642$$ 0 0
$$643$$ 1013.66 1.57646 0.788231 0.615380i $$-0.210997\pi$$
0.788231 + 0.615380i $$0.210997\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 28.4710i − 0.0440046i −0.999758 0.0220023i $$-0.992996\pi$$
0.999758 0.0220023i $$-0.00700412\pi$$
$$648$$ 0 0
$$649$$ −14.1177 −0.0217531
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 274.692i − 0.420662i −0.977630 0.210331i $$-0.932546\pi$$
0.977630 0.210331i $$-0.0674542\pi$$
$$654$$ 0 0
$$655$$ −373.547 −0.570300
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 87.7065i 0.133090i 0.997783 + 0.0665451i $$0.0211976\pi$$
−0.997783 + 0.0665451i $$0.978802\pi$$
$$660$$ 0 0
$$661$$ −212.368 −0.321282 −0.160641 0.987013i $$-0.551356\pi$$
−0.160641 + 0.987013i $$0.551356\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 109.993i − 0.165403i
$$666$$ 0 0
$$667$$ 122.313 0.183378
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ − 13.7056i − 0.0204257i
$$672$$ 0 0
$$673$$ 1099.17 1.63325 0.816623 0.577171i $$-0.195843\pi$$
0.816623 + 0.577171i $$0.195843\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 715.745i 1.05723i 0.848862 + 0.528615i $$0.177288\pi$$
−0.848862 + 0.528615i $$0.822712\pi$$
$$678$$ 0 0
$$679$$ 689.979 1.01617
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 588.823i 0.862112i 0.902325 + 0.431056i $$0.141859\pi$$
−0.902325 + 0.431056i $$0.858141\pi$$
$$684$$ 0 0
$$685$$ 226.617 0.330828
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 335.572i 0.487042i
$$690$$ 0 0
$$691$$ 466.975 0.675796 0.337898 0.941183i $$-0.390284\pi$$
0.337898 + 0.941183i $$0.390284\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 22.4811i 0.0323469i
$$696$$ 0 0
$$697$$ 640.971 0.919613
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 374.525i 0.534273i 0.963659 + 0.267136i $$0.0860774\pi$$
−0.963659 + 0.267136i $$0.913923\pi$$
$$702$$ 0 0
$$703$$ 247.210 0.351650
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1434.13i 2.02847i
$$708$$ 0 0
$$709$$ −259.647 −0.366215 −0.183108 0.983093i $$-0.558616\pi$$
−0.183108 + 0.983093i $$0.558616\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 94.7107i − 0.132834i
$$714$$ 0 0
$$715$$ −2.09476 −0.00292973
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 1124.90i − 1.56454i −0.622942 0.782268i $$-0.714063\pi$$
0.622942 0.782268i $$-0.285937\pi$$
$$720$$ 0 0
$$721$$ −794.985 −1.10261
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 746.669i − 1.02989i
$$726$$ 0 0
$$727$$ 185.241 0.254803 0.127401 0.991851i $$-0.459336\pi$$
0.127401 + 0.991851i $$0.459336\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 1504.91i − 2.05870i
$$732$$ 0 0
$$733$$ 62.0732 0.0846837 0.0423419 0.999103i $$-0.486518\pi$$
0.0423419 + 0.999103i $$0.486518\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 7.64121i − 0.0103680i
$$738$$ 0 0
$$739$$ −156.357 −0.211579 −0.105789 0.994389i $$-0.533737\pi$$
−0.105789 + 0.994389i $$0.533737\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 371.220i 0.499624i 0.968294 + 0.249812i $$0.0803688\pi$$
−0.968294 + 0.249812i $$0.919631\pi$$
$$744$$ 0 0
$$745$$ −104.500 −0.140269
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 1846.19i − 2.46487i
$$750$$ 0 0
$$751$$ −380.653 −0.506861 −0.253431 0.967354i $$-0.581559\pi$$
−0.253431 + 0.967354i $$0.581559\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 382.456i 0.506564i
$$756$$ 0 0
$$757$$ 351.088 0.463789 0.231895 0.972741i $$-0.425508\pi$$
0.231895 + 0.972741i $$0.425508\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 300.772i − 0.395232i −0.980280 0.197616i $$-0.936680\pi$$
0.980280 0.197616i $$-0.0633199\pi$$
$$762$$ 0 0
$$763$$ 1134.53 1.48693
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 533.637i − 0.695745i
$$768$$ 0 0
$$769$$ −378.955 −0.492790 −0.246395 0.969170i $$-0.579246\pi$$
−0.246395 + 0.969170i $$0.579246\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 249.305i − 0.322516i −0.986912 0.161258i $$-0.948445\pi$$
0.986912 0.161258i $$-0.0515552\pi$$
$$774$$ 0 0
$$775$$ −578.168 −0.746023
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 125.157i − 0.160664i
$$780$$ 0 0
$$781$$ 7.07403 0.00905765
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 310.572i 0.395634i
$$786$$ 0 0
$$787$$ −1126.62 −1.43154 −0.715769 0.698337i $$-0.753924\pi$$
−0.715769 + 0.698337i $$0.753924\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1537.42i 1.94364i
$$792$$ 0 0
$$793$$ 518.059 0.653290
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 451.324i − 0.566278i −0.959079 0.283139i $$-0.908624\pi$$
0.959079 0.283139i $$-0.0913758\pi$$
$$798$$ 0 0
$$799$$ −1570.69 −1.96582
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 11.3818i − 0.0141741i
$$804$$ 0 0
$$805$$ 72.6030 0.0901901
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 684.710i − 0.846365i −0.906044 0.423183i $$-0.860913\pi$$
0.906044 0.423183i $$-0.139087\pi$$
$$810$$ 0 0
$$811$$ −1324.62 −1.63331 −0.816656 0.577125i $$-0.804174\pi$$
−0.816656 + 0.577125i $$0.804174\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 336.886i 0.413357i
$$816$$ 0 0
$$817$$ −293.852 −0.359672
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 837.836i 1.02051i 0.860024 + 0.510253i $$0.170448\pi$$
−0.860024 + 0.510253i $$0.829552\pi$$
$$822$$ 0 0
$$823$$ 217.033 0.263710 0.131855 0.991269i $$-0.457907\pi$$
0.131855 + 0.991269i $$0.457907\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 458.215i − 0.554069i −0.960860 0.277035i $$-0.910648\pi$$
0.960860 0.277035i $$-0.0893517\pi$$
$$828$$ 0 0
$$829$$ −335.706 −0.404953 −0.202476 0.979287i $$-0.564899\pi$$
−0.202476 + 0.979287i $$0.564899\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1946.99i 2.33732i
$$834$$ 0 0
$$835$$ 82.6314 0.0989598
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 1202.59i − 1.43336i −0.697401 0.716681i $$-0.745660\pi$$
0.697401 0.716681i $$-0.254340\pi$$
$$840$$ 0 0
$$841$$ −370.058 −0.440021
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 238.978i 0.282815i
$$846$$ 0 0
$$847$$ 1327.36 1.56713
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 163.176i 0.191746i
$$852$$ 0 0
$$853$$ −730.471 −0.856355 −0.428178 0.903695i $$-0.640844\pi$$
−0.428178 + 0.903695i $$0.640844\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1244.03i 1.45161i 0.687899 + 0.725807i $$0.258533\pi$$
−0.687899 + 0.725807i $$0.741467\pi$$
$$858$$ 0 0
$$859$$ −1584.24 −1.84428 −0.922141 0.386855i $$-0.873561\pi$$
−0.922141 + 0.386855i $$0.873561\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 746.392i 0.864881i 0.901663 + 0.432440i $$0.142347\pi$$
−0.901663 + 0.432440i $$0.857653\pi$$
$$864$$ 0 0
$$865$$ −361.191 −0.417562
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 19.7395i − 0.0227152i
$$870$$ 0 0
$$871$$ 288.830 0.331607
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 959.632i − 1.09672i
$$876$$ 0 0
$$877$$ −1351.97 −1.54158 −0.770792 0.637087i $$-0.780139\pi$$
−0.770792 + 0.637087i $$0.780139\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 649.799i − 0.737569i −0.929515 0.368785i $$-0.879774\pi$$
0.929515 0.368785i $$-0.120226\pi$$
$$882$$ 0 0
$$883$$ 829.881 0.939843 0.469922 0.882708i $$-0.344282\pi$$
0.469922 + 0.882708i $$0.344282\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 296.662i 0.334455i 0.985918 + 0.167228i $$0.0534815\pi$$
−0.985918 + 0.167228i $$0.946519\pi$$
$$888$$ 0 0
$$889$$ −1317.28 −1.48175
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 306.696i 0.343445i
$$894$$ 0 0
$$895$$ −272.533 −0.304506
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 937.759i 1.04311i
$$900$$ 0 0
$$901$$ −1411.04 −1.56608
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1.38383i 0.00152909i
$$906$$ 0 0
$$907$$ 964.450 1.06334 0.531670 0.846952i $$-0.321565\pi$$
0.531670 + 0.846952i $$0.321565\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 399.941i − 0.439013i −0.975611 0.219507i $$-0.929555\pi$$
0.975611 0.219507i $$-0.0704448\pi$$
$$912$$ 0 0
$$913$$ 6.23463 0.00682873
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 2177.19i − 2.37425i
$$918$$ 0 0
$$919$$ −1363.99 −1.48421 −0.742107 0.670281i $$-0.766174\pi$$
−0.742107 + 0.670281i $$0.766174\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 267.391i 0.289698i
$$924$$ 0 0
$$925$$ 996.118 1.07688
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 78.7364i 0.0847539i 0.999102 + 0.0423770i $$0.0134930\pi$$
−0.999102 + 0.0423770i $$0.986507\pi$$
$$930$$ 0 0
$$931$$ 380.173 0.408349
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 8.80822i − 0.00942056i
$$936$$ 0 0
$$937$$ −861.705 −0.919642 −0.459821 0.888012i $$-0.652086\pi$$
−0.459821 + 0.888012i $$0.652086\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 641.991i 0.682243i 0.940019 + 0.341122i $$0.110807\pi$$
−0.940019 + 0.341122i $$0.889193\pi$$
$$942$$ 0 0
$$943$$ 82.6124 0.0876060
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1259.50i − 1.32999i −0.746849 0.664994i $$-0.768434\pi$$
0.746849 0.664994i $$-0.231566\pi$$
$$948$$ 0 0
$$949$$ 430.221 0.453342
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 867.155i 0.909921i 0.890512 + 0.454961i $$0.150347\pi$$
−0.890512 + 0.454961i $$0.849653\pi$$
$$954$$ 0 0
$$955$$ 546.200 0.571938
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1320.82i 1.37729i
$$960$$ 0 0
$$961$$ −234.865 −0.244397
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 119.739i − 0.124081i
$$966$$ 0 0
$$967$$ −157.972 −0.163363 −0.0816813 0.996659i $$-0.526029\pi$$
−0.0816813 + 0.996659i $$0.526029\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 826.466i − 0.851149i −0.904923 0.425575i $$-0.860072\pi$$
0.904923 0.425575i $$-0.139928\pi$$
$$972$$ 0 0
$$973$$ −131.029 −0.134665
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 1777.33i − 1.81917i −0.415514 0.909587i $$-0.636398\pi$$
0.415514 0.909587i $$-0.363602\pi$$
$$978$$ 0 0
$$979$$ −27.1131 −0.0276947
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 587.324i 0.597481i 0.954334 + 0.298740i $$0.0965665\pi$$
−0.954334 + 0.298740i $$0.903434\pi$$
$$984$$ 0 0
$$985$$ −166.367 −0.168900
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 193.962i − 0.196120i
$$990$$ 0 0
$$991$$ 1132.76 1.14304 0.571522 0.820587i $$-0.306353\pi$$
0.571522 + 0.820587i $$0.306353\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 599.049i − 0.602059i
$$996$$ 0 0
$$997$$ −1296.22 −1.30012 −0.650060 0.759883i $$-0.725256\pi$$
−0.650060 + 0.759883i $$0.725256\pi$$
$$998$$ 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 864.3.e.e.161.4 yes 8
3.2 odd 2 inner 864.3.e.e.161.6 yes 8
4.3 odd 2 inner 864.3.e.e.161.3 8
8.3 odd 2 1728.3.e.v.1025.5 8
8.5 even 2 1728.3.e.v.1025.6 8
12.11 even 2 inner 864.3.e.e.161.5 yes 8
24.5 odd 2 1728.3.e.v.1025.4 8
24.11 even 2 1728.3.e.v.1025.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.e.161.3 8 4.3 odd 2 inner
864.3.e.e.161.4 yes 8 1.1 even 1 trivial
864.3.e.e.161.5 yes 8 12.11 even 2 inner
864.3.e.e.161.6 yes 8 3.2 odd 2 inner
1728.3.e.v.1025.3 8 24.11 even 2
1728.3.e.v.1025.4 8 24.5 odd 2
1728.3.e.v.1025.5 8 8.3 odd 2
1728.3.e.v.1025.6 8 8.5 even 2