# Properties

 Label 819.2.a.f.1.1 Level $819$ Weight $2$ Character 819.1 Self dual yes Analytic conductor $6.540$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.a (trivial)

## 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: yes Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 819.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +6.00000 q^{10} +6.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} +5.00000 q^{19} +6.00000 q^{20} +12.0000 q^{22} -3.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} -2.00000 q^{28} +5.00000 q^{29} -3.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} -3.00000 q^{35} -4.00000 q^{37} +10.0000 q^{38} +6.00000 q^{41} -1.00000 q^{43} +12.0000 q^{44} -6.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -2.00000 q^{52} +9.00000 q^{53} +18.0000 q^{55} +10.0000 q^{58} -8.00000 q^{59} -10.0000 q^{61} -6.00000 q^{62} -8.00000 q^{64} -3.00000 q^{65} -6.00000 q^{67} -8.00000 q^{68} -6.00000 q^{70} +8.00000 q^{71} -13.0000 q^{73} -8.00000 q^{74} +10.0000 q^{76} -6.00000 q^{77} +3.00000 q^{79} -12.0000 q^{80} +12.0000 q^{82} -15.0000 q^{83} -12.0000 q^{85} -2.00000 q^{86} -3.00000 q^{89} +1.00000 q^{91} -6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} +7.00000 q^{97} +2.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 6.00000 1.89737
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ 6.00000 1.34164
$$21$$ 0 0
$$22$$ 12.0000 2.55841
$$23$$ −3.00000 −0.625543 −0.312772 0.949828i $$-0.601257\pi$$
−0.312772 + 0.949828i $$0.601257\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 10.0000 1.62221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 12.0000 1.80907
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 8.00000 1.13137
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 0 0
$$55$$ 18.0000 2.42712
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ −6.00000 −0.733017 −0.366508 0.930415i $$-0.619447\pi$$
−0.366508 + 0.930415i $$0.619447\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 0 0
$$70$$ −6.00000 −0.717137
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −13.0000 −1.52153 −0.760767 0.649025i $$-0.775177\pi$$
−0.760767 + 0.649025i $$0.775177\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 10.0000 1.14708
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ −12.0000 −1.34164
$$81$$ 0 0
$$82$$ 12.0000 1.32518
$$83$$ −15.0000 −1.64646 −0.823232 0.567705i $$-0.807831\pi$$
−0.823232 + 0.567705i $$0.807831\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.00000 −0.317999 −0.159000 0.987279i $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ −6.00000 −0.625543
$$93$$ 0 0
$$94$$ −14.0000 −1.44399
$$95$$ 15.0000 1.53897
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 0 0
$$100$$ 8.00000 0.800000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 36.0000 3.43247
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ 3.00000 0.282216 0.141108 0.989994i $$-0.454933\pi$$
0.141108 + 0.989994i $$0.454933\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ −16.0000 −1.47292
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ −20.0000 −1.81071
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ −6.00000 −0.526235
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ −5.00000 −0.433555
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −4.00000 −0.341743 −0.170872 0.985293i $$-0.554658\pi$$
−0.170872 + 0.985293i $$0.554658\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ −6.00000 −0.507093
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ 15.0000 1.24568
$$146$$ −26.0000 −2.15178
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ −9.00000 −0.722897
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ 6.00000 0.477334
$$159$$ 0 0
$$160$$ −24.0000 −1.89737
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ −30.0000 −2.32845
$$167$$ −5.00000 −0.386912 −0.193456 0.981109i $$-0.561970\pi$$
−0.193456 + 0.981109i $$0.561970\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −24.0000 −1.84072
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 8.00000 0.608229 0.304114 0.952636i $$-0.401639\pi$$
0.304114 + 0.952636i $$0.401639\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ −24.0000 −1.80907
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ −23.0000 −1.71910 −0.859550 0.511051i $$-0.829256\pi$$
−0.859550 + 0.511051i $$0.829256\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ −24.0000 −1.75505
$$188$$ −14.0000 −1.02105
$$189$$ 0 0
$$190$$ 30.0000 2.17643
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 28.0000 1.97007
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ 30.0000 2.07514
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ 18.0000 1.23625
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ −3.00000 −0.204598
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ 36.0000 2.42712
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 15.0000 1.00447 0.502237 0.864730i $$-0.332510\pi$$
0.502237 + 0.864730i $$0.332510\pi$$
$$224$$ 8.00000 0.534522
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ −18.0000 −1.18688
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −15.0000 −0.982683 −0.491341 0.870967i $$-0.663493\pi$$
−0.491341 + 0.870967i $$0.663493\pi$$
$$234$$ 0 0
$$235$$ −21.0000 −1.36989
$$236$$ −16.0000 −1.04151
$$237$$ 0 0
$$238$$ 8.00000 0.518563
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ 50.0000 3.21412
$$243$$ 0 0
$$244$$ −20.0000 −1.28037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −6.00000 −0.379473
$$251$$ 26.0000 1.64111 0.820553 0.571571i $$-0.193666\pi$$
0.820553 + 0.571571i $$0.193666\pi$$
$$252$$ 0 0
$$253$$ −18.0000 −1.13165
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ −6.00000 −0.372104
$$261$$ 0 0
$$262$$ −16.0000 −0.988483
$$263$$ 15.0000 0.924940 0.462470 0.886635i $$-0.346963\pi$$
0.462470 + 0.886635i $$0.346963\pi$$
$$264$$ 0 0
$$265$$ 27.0000 1.65860
$$266$$ −10.0000 −0.613139
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 16.0000 0.970143
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ 24.0000 1.44725
$$276$$ 0 0
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ −36.0000 −2.15914
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 30.0000 1.76166
$$291$$ 0 0
$$292$$ −26.0000 −1.52153
$$293$$ 19.0000 1.10999 0.554996 0.831853i $$-0.312720\pi$$
0.554996 + 0.831853i $$0.312720\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 36.0000 2.08542
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −20.0000 −1.14708
$$305$$ −30.0000 −1.71780
$$306$$ 0 0
$$307$$ −33.0000 −1.88341 −0.941705 0.336440i $$-0.890777\pi$$
−0.941705 + 0.336440i $$0.890777\pi$$
$$308$$ −12.0000 −0.683763
$$309$$ 0 0
$$310$$ −18.0000 −1.02233
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −16.0000 −0.902932
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ −24.0000 −1.34164
$$321$$ 0 0
$$322$$ 6.00000 0.334367
$$323$$ −20.0000 −1.11283
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 7.00000 0.385922
$$330$$ 0 0
$$331$$ 22.0000 1.20923 0.604615 0.796518i $$-0.293327\pi$$
0.604615 + 0.796518i $$0.293327\pi$$
$$332$$ −30.0000 −1.64646
$$333$$ 0 0
$$334$$ −10.0000 −0.547176
$$335$$ −18.0000 −0.983445
$$336$$ 0 0
$$337$$ 17.0000 0.926049 0.463025 0.886345i $$-0.346764\pi$$
0.463025 + 0.886345i $$0.346764\pi$$
$$338$$ 2.00000 0.108786
$$339$$ 0 0
$$340$$ −24.0000 −1.30158
$$341$$ −18.0000 −0.974755
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 16.0000 0.860165
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ 0 0
$$349$$ 11.0000 0.588817 0.294408 0.955680i $$-0.404877\pi$$
0.294408 + 0.955680i $$0.404877\pi$$
$$350$$ −8.00000 −0.427618
$$351$$ 0 0
$$352$$ −48.0000 −2.55841
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ 0 0
$$355$$ 24.0000 1.27379
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −46.0000 −2.43118
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 28.0000 1.47165
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ −39.0000 −2.04135
$$366$$ 0 0
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 0 0
$$370$$ −24.0000 −1.24770
$$371$$ −9.00000 −0.467257
$$372$$ 0 0
$$373$$ 30.0000 1.55334 0.776671 0.629907i $$-0.216907\pi$$
0.776671 + 0.629907i $$0.216907\pi$$
$$374$$ −48.0000 −2.48202
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 30.0000 1.53897
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ 36.0000 1.83951 0.919757 0.392488i $$-0.128386\pi$$
0.919757 + 0.392488i $$0.128386\pi$$
$$384$$ 0 0
$$385$$ −18.0000 −0.917365
$$386$$ 44.0000 2.23954
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4.00000 −0.201517
$$395$$ 9.00000 0.452839
$$396$$ 0 0
$$397$$ −13.0000 −0.652451 −0.326226 0.945292i $$-0.605777\pi$$
−0.326226 + 0.945292i $$0.605777\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ −16.0000 −0.800000
$$401$$ 32.0000 1.59800 0.799002 0.601329i $$-0.205362\pi$$
0.799002 + 0.601329i $$0.205362\pi$$
$$402$$ 0 0
$$403$$ 3.00000 0.149441
$$404$$ 28.0000 1.39305
$$405$$ 0 0
$$406$$ −10.0000 −0.496292
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ −13.0000 −0.642809 −0.321404 0.946942i $$-0.604155\pi$$
−0.321404 + 0.946942i $$0.604155\pi$$
$$410$$ 36.0000 1.77791
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ −45.0000 −2.20896
$$416$$ 8.00000 0.392232
$$417$$ 0 0
$$418$$ 60.0000 2.93470
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 0 0
$$421$$ −12.0000 −0.584844 −0.292422 0.956289i $$-0.594461\pi$$
−0.292422 + 0.956289i $$0.594461\pi$$
$$422$$ −10.0000 −0.486792
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −16.0000 −0.776114
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ 8.00000 0.386695
$$429$$ 0 0
$$430$$ −6.00000 −0.289346
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ 0 0
$$433$$ 12.0000 0.576683 0.288342 0.957528i $$-0.406896\pi$$
0.288342 + 0.957528i $$0.406896\pi$$
$$434$$ 6.00000 0.288009
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ −15.0000 −0.717547
$$438$$ 0 0
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 8.00000 0.380521
$$443$$ −19.0000 −0.902717 −0.451359 0.892343i $$-0.649060\pi$$
−0.451359 + 0.892343i $$0.649060\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ 30.0000 1.42054
$$447$$ 0 0
$$448$$ 8.00000 0.377964
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −40.0000 −1.87729
$$455$$ 3.00000 0.140642
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 28.0000 1.30835
$$459$$ 0 0
$$460$$ −18.0000 −0.839254
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ −20.0000 −0.928477
$$465$$ 0 0
$$466$$ −30.0000 −1.38972
$$467$$ 22.0000 1.01804 0.509019 0.860755i $$-0.330008\pi$$
0.509019 + 0.860755i $$0.330008\pi$$
$$468$$ 0 0
$$469$$ 6.00000 0.277054
$$470$$ −42.0000 −1.93732
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ 20.0000 0.917663
$$476$$ 8.00000 0.366679
$$477$$ 0 0
$$478$$ 8.00000 0.365911
$$479$$ 11.0000 0.502603 0.251301 0.967909i $$-0.419141\pi$$
0.251301 + 0.967909i $$0.419141\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −34.0000 −1.54866
$$483$$ 0 0
$$484$$ 50.0000 2.27273
$$485$$ 21.0000 0.953561
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ −10.0000 −0.449921
$$495$$ 0 0
$$496$$ 12.0000 0.538816
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ −6.00000 −0.268328
$$501$$ 0 0
$$502$$ 52.0000 2.32087
$$503$$ −2.00000 −0.0891756 −0.0445878 0.999005i $$-0.514197\pi$$
−0.0445878 + 0.999005i $$0.514197\pi$$
$$504$$ 0 0
$$505$$ 42.0000 1.86898
$$506$$ −36.0000 −1.60040
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 19.0000 0.842160 0.421080 0.907023i $$-0.361651\pi$$
0.421080 + 0.907023i $$0.361651\pi$$
$$510$$ 0 0
$$511$$ 13.0000 0.575086
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ 4.00000 0.176432
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ −42.0000 −1.84716
$$518$$ 8.00000 0.351500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.0000 −1.75243 −0.876216 0.481919i $$-0.839940\pi$$
−0.876216 + 0.481919i $$0.839940\pi$$
$$522$$ 0 0
$$523$$ 10.0000 0.437269 0.218635 0.975807i $$-0.429840\pi$$
0.218635 + 0.975807i $$0.429840\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 54.0000 2.34561
$$531$$ 0 0
$$532$$ −10.0000 −0.433555
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −40.0000 −1.71973 −0.859867 0.510518i $$-0.829454\pi$$
−0.859867 + 0.510518i $$0.829454\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 32.0000 1.37199
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −7.00000 −0.299298 −0.149649 0.988739i $$-0.547814\pi$$
−0.149649 + 0.988739i $$0.547814\pi$$
$$548$$ −8.00000 −0.341743
$$549$$ 0 0
$$550$$ 48.0000 2.04673
$$551$$ 25.0000 1.06504
$$552$$ 0 0
$$553$$ −3.00000 −0.127573
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −36.0000 −1.52674
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 0 0
$$559$$ 1.00000 0.0422955
$$560$$ 12.0000 0.507093
$$561$$ 0 0
$$562$$ 60.0000 2.53095
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 9.00000 0.378633
$$566$$ 32.0000 1.34506
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7.00000 −0.293455 −0.146728 0.989177i $$-0.546874\pi$$
−0.146728 + 0.989177i $$0.546874\pi$$
$$570$$ 0 0
$$571$$ −17.0000 −0.711428 −0.355714 0.934595i $$-0.615762\pi$$
−0.355714 + 0.934595i $$0.615762\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ 0 0
$$574$$ −12.0000 −0.500870
$$575$$ −12.0000 −0.500435
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −2.00000 −0.0831890
$$579$$ 0 0
$$580$$ 30.0000 1.24568
$$581$$ 15.0000 0.622305
$$582$$ 0 0
$$583$$ 54.0000 2.23645
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 38.0000 1.56977
$$587$$ −39.0000 −1.60970 −0.804851 0.593477i $$-0.797755\pi$$
−0.804851 + 0.593477i $$0.797755\pi$$
$$588$$ 0 0
$$589$$ −15.0000 −0.618064
$$590$$ −48.0000 −1.97613
$$591$$ 0 0
$$592$$ 16.0000 0.657596
$$593$$ 27.0000 1.10876 0.554379 0.832265i $$-0.312956\pi$$
0.554379 + 0.832265i $$0.312956\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ 36.0000 1.47462
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ −11.0000 −0.449448 −0.224724 0.974422i $$-0.572148\pi$$
−0.224724 + 0.974422i $$0.572148\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 2.00000 0.0815139
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 75.0000 3.04918
$$606$$ 0 0
$$607$$ 2.00000 0.0811775 0.0405887 0.999176i $$-0.487077\pi$$
0.0405887 + 0.999176i $$0.487077\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ 0 0
$$610$$ −60.0000 −2.42933
$$611$$ 7.00000 0.283190
$$612$$ 0 0
$$613$$ 8.00000 0.323117 0.161558 0.986863i $$-0.448348\pi$$
0.161558 + 0.986863i $$0.448348\pi$$
$$614$$ −66.0000 −2.66354
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −18.0000 −0.722897
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 3.00000 0.120192
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 44.0000 1.75859
$$627$$ 0 0
$$628$$ −16.0000 −0.638470
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 22.0000 0.875806 0.437903 0.899022i $$-0.355721\pi$$
0.437903 + 0.899022i $$0.355721\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 48.0000 1.90632
$$635$$ −12.0000 −0.476205
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 60.0000 2.37542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ −40.0000 −1.57378
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ −8.00000 −0.313786
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ −24.0000 −0.937043
$$657$$ 0 0
$$658$$ 14.0000 0.545777
$$659$$ −17.0000 −0.662226 −0.331113 0.943591i $$-0.607424\pi$$
−0.331113 + 0.943591i $$0.607424\pi$$
$$660$$ 0 0
$$661$$ −33.0000 −1.28355 −0.641776 0.766892i $$-0.721802\pi$$
−0.641776 + 0.766892i $$0.721802\pi$$
$$662$$ 44.0000 1.71011
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −15.0000 −0.581675
$$666$$ 0 0
$$667$$ −15.0000 −0.580802
$$668$$ −10.0000 −0.386912
$$669$$ 0 0
$$670$$ −36.0000 −1.39080
$$671$$ −60.0000 −2.31627
$$672$$ 0 0
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −36.0000 −1.37851
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ −2.00000 −0.0763604
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ 11.0000 0.418460 0.209230 0.977866i $$-0.432904\pi$$
0.209230 + 0.977866i $$0.432904\pi$$
$$692$$ 16.0000 0.608229
$$693$$ 0 0
$$694$$ 64.0000 2.42941
$$695$$ −54.0000 −2.04834
$$696$$ 0 0
$$697$$ −24.0000 −0.909065
$$698$$ 22.0000 0.832712
$$699$$ 0 0
$$700$$ −8.00000 −0.302372
$$701$$ 27.0000 1.01978 0.509888 0.860241i $$-0.329687\pi$$
0.509888 + 0.860241i $$0.329687\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ −48.0000 −1.80907
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ −14.0000 −0.526524
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 48.0000 1.80141
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 9.00000 0.337053
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ −46.0000 −1.71910
$$717$$ 0 0
$$718$$ −40.0000 −1.49279
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 12.0000 0.446594
$$723$$ 0 0
$$724$$ 28.0000 1.04061
$$725$$ 20.0000 0.742781
$$726$$ 0 0
$$727$$ 46.0000 1.70605 0.853023 0.521874i $$-0.174767\pi$$
0.853023 + 0.521874i $$0.174767\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −78.0000 −2.88691
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ 51.0000 1.88373 0.941864 0.335994i $$-0.109072\pi$$
0.941864 + 0.335994i $$0.109072\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 24.0000 0.884652
$$737$$ −36.0000 −1.32608
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ −24.0000 −0.882258
$$741$$ 0 0
$$742$$ −18.0000 −0.660801
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 0 0
$$745$$ 54.0000 1.97841
$$746$$ 60.0000 2.19676
$$747$$ 0 0
$$748$$ −48.0000 −1.75505
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ −17.0000 −0.620339 −0.310169 0.950681i $$-0.600386\pi$$
−0.310169 + 0.950681i $$0.600386\pi$$
$$752$$ 28.0000 1.02105
$$753$$ 0 0
$$754$$ −10.0000 −0.364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.0000 −0.545184 −0.272592 0.962130i $$-0.587881\pi$$
−0.272592 + 0.962130i $$0.587881\pi$$
$$758$$ −12.0000 −0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 72.0000 2.60147
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ −36.0000 −1.29735
$$771$$ 0 0
$$772$$ 44.0000 1.58359
$$773$$ −54.0000 −1.94225 −0.971123 0.238581i $$-0.923318\pi$$
−0.971123 + 0.238581i $$0.923318\pi$$
$$774$$ 0 0
$$775$$ −12.0000 −0.431053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −60.0000 −2.15110
$$779$$ 30.0000 1.07486
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ −4.00000 −0.142857
$$785$$ −24.0000 −0.856597
$$786$$ 0 0
$$787$$ 37.0000 1.31891 0.659454 0.751745i $$-0.270788\pi$$
0.659454 + 0.751745i $$0.270788\pi$$
$$788$$ −4.00000 −0.142494
$$789$$ 0 0
$$790$$ 18.0000 0.640411
$$791$$ −3.00000 −0.106668
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ −26.0000 −0.922705
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ −32.0000 −1.13137
$$801$$ 0 0
$$802$$ 64.0000 2.25992
$$803$$ −78.0000 −2.75256
$$804$$ 0 0
$$805$$ 9.00000 0.317208
$$806$$ 6.00000 0.211341
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 31.0000 1.08990 0.544951 0.838468i $$-0.316548\pi$$
0.544951 + 0.838468i $$0.316548\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ −10.0000 −0.350931
$$813$$ 0 0
$$814$$ −48.0000 −1.68240
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 36.0000 1.25717
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 16.0000 0.556711
$$827$$ −4.00000 −0.139094 −0.0695468 0.997579i $$-0.522155\pi$$
−0.0695468 + 0.997579i $$0.522155\pi$$
$$828$$ 0 0
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ −90.0000 −3.12395
$$831$$ 0 0
$$832$$ 8.00000 0.277350
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ −15.0000 −0.519096
$$836$$ 60.0000 2.07514
$$837$$ 0 0
$$838$$ 20.0000 0.690889
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −24.0000 −0.827095
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ −25.0000 −0.859010
$$848$$ −36.0000 −1.23625
$$849$$ 0 0
$$850$$ −32.0000 −1.09759
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 45.0000 1.54077 0.770385 0.637579i $$-0.220064\pi$$
0.770385 + 0.637579i $$0.220064\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ −6.00000 −0.204598
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ 24.0000 0.815553
$$867$$ 0 0
$$868$$ 6.00000 0.203653
$$869$$ 18.0000 0.610608
$$870$$ 0 0
$$871$$ 6.00000 0.203302
$$872$$ 0 0
$$873$$ 0 0
$$874$$ −30.0000 −1.01477
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ −44.0000 −1.48493
$$879$$ 0 0
$$880$$ −72.0000 −2.42712
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ −38.0000 −1.27663
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ −18.0000 −0.603361
$$891$$ 0 0
$$892$$ 30.0000 1.00447
$$893$$ −35.0000 −1.17123
$$894$$ 0 0
$$895$$ −69.0000 −2.30642
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −72.0000 −2.40267
$$899$$ −15.0000 −0.500278
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 72.0000 2.39734
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 42.0000 1.39613
$$906$$ 0 0
$$907$$ 7.00000 0.232431 0.116216 0.993224i $$-0.462924\pi$$
0.116216 + 0.993224i $$0.462924\pi$$
$$908$$ −40.0000 −1.32745
$$909$$ 0 0
$$910$$ 6.00000 0.198898
$$911$$ 15.0000 0.496972 0.248486 0.968635i $$-0.420067\pi$$
0.248486 + 0.968635i $$0.420067\pi$$
$$912$$ 0 0
$$913$$ −90.0000 −2.97857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ 8.00000 0.264183
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 44.0000 1.44906
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ −28.0000 −0.920137
$$927$$ 0 0
$$928$$ −40.0000 −1.31306
$$929$$ −5.00000 −0.164045 −0.0820223 0.996630i $$-0.526138\pi$$
−0.0820223 + 0.996630i $$0.526138\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ −30.0000 −0.982683
$$933$$ 0 0
$$934$$ 44.0000 1.43972
$$935$$ −72.0000 −2.35465
$$936$$ 0 0
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 0 0
$$940$$ −42.0000 −1.36989
$$941$$ −55.0000 −1.79295 −0.896474 0.443096i $$-0.853880\pi$$
−0.896474 + 0.443096i $$0.853880\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ 32.0000 1.04151
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ 0 0
$$949$$ 13.0000 0.421998
$$950$$ 40.0000 1.29777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 39.0000 1.26333 0.631667 0.775240i $$-0.282371\pi$$
0.631667 + 0.775240i $$0.282371\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ 22.0000 0.710788
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 8.00000 0.257930
$$963$$ 0 0
$$964$$ −34.0000 −1.09507
$$965$$ 66.0000 2.12462
$$966$$ 0 0
$$967$$ 22.0000 0.707472 0.353736 0.935345i $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 42.0000 1.34854
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 0 0
$$973$$ 18.0000 0.577054
$$974$$ −52.0000 −1.66619
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ −10.0000 −0.319928 −0.159964 0.987123i $$-0.551138\pi$$
−0.159964 + 0.987123i $$0.551138\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 6.00000 0.191663
$$981$$ 0 0
$$982$$ 24.0000 0.765871
$$983$$ −17.0000 −0.542216 −0.271108 0.962549i $$-0.587390\pi$$
−0.271108 + 0.962549i $$0.587390\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ −40.0000 −1.27386
$$987$$ 0 0
$$988$$ −10.0000 −0.318142
$$989$$ 3.00000 0.0953945
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 24.0000 0.762001
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ 12.0000 0.380426
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 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 819.2.a.f.1.1 1
3.2 odd 2 91.2.a.a.1.1 1
7.6 odd 2 5733.2.a.l.1.1 1
12.11 even 2 1456.2.a.g.1.1 1
15.14 odd 2 2275.2.a.h.1.1 1
21.2 odd 6 637.2.e.e.508.1 2
21.5 even 6 637.2.e.d.508.1 2
21.11 odd 6 637.2.e.e.79.1 2
21.17 even 6 637.2.e.d.79.1 2
21.20 even 2 637.2.a.a.1.1 1
24.5 odd 2 5824.2.a.s.1.1 1
24.11 even 2 5824.2.a.t.1.1 1
39.5 even 4 1183.2.c.b.337.2 2
39.8 even 4 1183.2.c.b.337.1 2
39.38 odd 2 1183.2.a.b.1.1 1
273.272 even 2 8281.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.a.1.1 1 3.2 odd 2
637.2.a.a.1.1 1 21.20 even 2
637.2.e.d.79.1 2 21.17 even 6
637.2.e.d.508.1 2 21.5 even 6
637.2.e.e.79.1 2 21.11 odd 6
637.2.e.e.508.1 2 21.2 odd 6
819.2.a.f.1.1 1 1.1 even 1 trivial
1183.2.a.b.1.1 1 39.38 odd 2
1183.2.c.b.337.1 2 39.8 even 4
1183.2.c.b.337.2 2 39.5 even 4
1456.2.a.g.1.1 1 12.11 even 2
2275.2.a.h.1.1 1 15.14 odd 2
5733.2.a.l.1.1 1 7.6 odd 2
5824.2.a.s.1.1 1 24.5 odd 2
5824.2.a.t.1.1 1 24.11 even 2
8281.2.a.l.1.1 1 273.272 even 2