# Properties

 Label 4600.2.a.d.1.1 Level $4600$ Weight $2$ Character 4600.1 Self dual yes Analytic conductor $36.731$ 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] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +5.00000 q^{13} +4.00000 q^{17} +1.00000 q^{19} -6.00000 q^{21} +1.00000 q^{23} +4.00000 q^{27} +9.00000 q^{29} -2.00000 q^{31} -10.0000 q^{33} +2.00000 q^{37} -10.0000 q^{39} +3.00000 q^{41} +7.00000 q^{43} +12.0000 q^{47} +2.00000 q^{49} -8.00000 q^{51} -12.0000 q^{53} -2.00000 q^{57} -6.00000 q^{59} -10.0000 q^{61} +3.00000 q^{63} +8.00000 q^{67} -2.00000 q^{69} +2.00000 q^{71} -1.00000 q^{73} +15.0000 q^{77} -11.0000 q^{79} -11.0000 q^{81} -9.00000 q^{83} -18.0000 q^{87} -14.0000 q^{89} +15.0000 q^{91} +4.00000 q^{93} +16.0000 q^{97} +5.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −1.30931
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 0 0
$$33$$ −10.0000 −1.74078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −10.0000 −1.60128
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 3.00000 0.377964
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −2.00000 −0.240772
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0 0
$$73$$ −1.00000 −0.117041 −0.0585206 0.998286i $$-0.518638\pi$$
−0.0585206 + 0.998286i $$0.518638\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 15.0000 1.70941
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −18.0000 −1.92980
$$88$$ 0 0
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 15.0000 1.57243
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 9.00000 0.886796 0.443398 0.896325i $$-0.353773\pi$$
0.443398 + 0.896325i $$0.353773\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.0000 −1.54678 −0.773389 0.633932i $$-0.781440\pi$$
−0.773389 + 0.633932i $$0.781440\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ −8.00000 −0.752577 −0.376288 0.926503i $$-0.622800\pi$$
−0.376288 + 0.926503i $$0.622800\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.00000 0.462250
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.0000 0.887357 0.443678 0.896186i $$-0.353673\pi$$
0.443678 + 0.896186i $$0.353673\pi$$
$$128$$ 0 0
$$129$$ −14.0000 −1.23263
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 3.00000 0.260133
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −24.0000 −2.02116
$$142$$ 0 0
$$143$$ 25.0000 2.09061
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −4.00000 −0.329914
$$148$$ 0 0
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ 4.00000 0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ 0 0
$$159$$ 24.0000 1.90332
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.0000 1.08335 0.541676 0.840587i $$-0.317790\pi$$
0.541676 + 0.840587i $$0.317790\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −5.00000 −0.380143 −0.190071 0.981770i $$-0.560872\pi$$
−0.190071 + 0.981770i $$0.560872\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.0000 1.46254
$$188$$ 0 0
$$189$$ 12.0000 0.872872
$$190$$ 0 0
$$191$$ −1.00000 −0.0723575 −0.0361787 0.999345i $$-0.511519\pi$$
−0.0361787 + 0.999345i $$0.511519\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.0000 1.63868 0.819341 0.573306i $$-0.194340\pi$$
0.819341 + 0.573306i $$0.194340\pi$$
$$198$$ 0 0
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 0 0
$$203$$ 27.0000 1.89503
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ −4.00000 −0.274075
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 20.0000 1.34535
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −30.0000 −1.98246 −0.991228 0.132164i $$-0.957808\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ −30.0000 −1.97386
$$232$$ 0 0
$$233$$ 25.0000 1.63780 0.818902 0.573933i $$-0.194583\pi$$
0.818902 + 0.573933i $$0.194583\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 22.0000 1.42905
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −4.00000 −0.257663 −0.128831 0.991667i $$-0.541123\pi$$
−0.128831 + 0.991667i $$0.541123\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5.00000 0.318142
$$248$$ 0 0
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ 0 0
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 28.0000 1.71357
$$268$$ 0 0
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 0 0
$$273$$ −30.0000 −1.81568
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0000 −0.781094 −0.390547 0.920583i $$-0.627714\pi$$
−0.390547 + 0.920583i $$0.627714\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.00000 0.531253
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −32.0000 −1.87587
$$292$$ 0 0
$$293$$ −28.0000 −1.63578 −0.817889 0.575376i $$-0.804856\pi$$
−0.817889 + 0.575376i $$0.804856\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 20.0000 1.16052
$$298$$ 0 0
$$299$$ 5.00000 0.289157
$$300$$ 0 0
$$301$$ 21.0000 1.21042
$$302$$ 0 0
$$303$$ 28.0000 1.60856
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −18.0000 −1.02398
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −17.0000 −0.954815 −0.477408 0.878682i $$-0.658423\pi$$
−0.477408 + 0.878682i $$0.658423\pi$$
$$318$$ 0 0
$$319$$ 45.0000 2.51952
$$320$$ 0 0
$$321$$ 32.0000 1.78607
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 20.0000 1.10600
$$328$$ 0 0
$$329$$ 36.0000 1.98474
$$330$$ 0 0
$$331$$ 26.0000 1.42909 0.714545 0.699590i $$-0.246634\pi$$
0.714545 + 0.699590i $$0.246634\pi$$
$$332$$ 0 0
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −32.0000 −1.74315 −0.871576 0.490261i $$-0.836901\pi$$
−0.871576 + 0.490261i $$0.836901\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ −7.00000 −0.374701 −0.187351 0.982293i $$-0.559990\pi$$
−0.187351 + 0.982293i $$0.559990\pi$$
$$350$$ 0 0
$$351$$ 20.0000 1.06752
$$352$$ 0 0
$$353$$ −21.0000 −1.11772 −0.558859 0.829263i $$-0.688761\pi$$
−0.558859 + 0.829263i $$0.688761\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −24.0000 −1.27021
$$358$$ 0 0
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ −28.0000 −1.46962
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 15.0000 0.782994 0.391497 0.920179i $$-0.371957\pi$$
0.391497 + 0.920179i $$0.371957\pi$$
$$368$$ 0 0
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ −36.0000 −1.86903
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 45.0000 2.31762
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ −20.0000 −1.02463
$$382$$ 0 0
$$383$$ 23.0000 1.17525 0.587623 0.809135i $$-0.300064\pi$$
0.587623 + 0.809135i $$0.300064\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 7.00000 0.355830
$$388$$ 0 0
$$389$$ −4.00000 −0.202808 −0.101404 0.994845i $$-0.532333\pi$$
−0.101404 + 0.994845i $$0.532333\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ −10.0000 −0.498135
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10.0000 0.495682
$$408$$ 0 0
$$409$$ −35.0000 −1.73064 −0.865319 0.501221i $$-0.832884\pi$$
−0.865319 + 0.501221i $$0.832884\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 0 0
$$413$$ −18.0000 −0.885722
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −24.0000 −1.17529
$$418$$ 0 0
$$419$$ 3.00000 0.146560 0.0732798 0.997311i $$-0.476653\pi$$
0.0732798 + 0.997311i $$0.476653\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −30.0000 −1.45180
$$428$$ 0 0
$$429$$ −50.0000 −2.41402
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1.00000 0.0478365
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −14.0000 −0.665160 −0.332580 0.943075i $$-0.607919\pi$$
−0.332580 + 0.943075i $$0.607919\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −28.0000 −1.32435
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 15.0000 0.706322
$$452$$ 0 0
$$453$$ 36.0000 1.69143
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 28.0000 1.30978 0.654892 0.755722i $$-0.272714\pi$$
0.654892 + 0.755722i $$0.272714\pi$$
$$458$$ 0 0
$$459$$ 16.0000 0.746816
$$460$$ 0 0
$$461$$ 41.0000 1.90956 0.954780 0.297313i $$-0.0960904\pi$$
0.954780 + 0.297313i $$0.0960904\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ 16.0000 0.737241
$$472$$ 0 0
$$473$$ 35.0000 1.60930
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ −17.0000 −0.776750 −0.388375 0.921501i $$-0.626963\pi$$
−0.388375 + 0.921501i $$0.626963\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ −2.00000 −0.0902587 −0.0451294 0.998981i $$-0.514370\pi$$
−0.0451294 + 0.998981i $$0.514370\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 0 0
$$501$$ −28.0000 −1.25095
$$502$$ 0 0
$$503$$ 5.00000 0.222939 0.111469 0.993768i $$-0.464444\pi$$
0.111469 + 0.993768i $$0.464444\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −24.0000 −1.06588
$$508$$ 0 0
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ −3.00000 −0.132712
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 60.0000 2.63880
$$518$$ 0 0
$$519$$ 10.0000 0.438951
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ −7.00000 −0.306089 −0.153044 0.988219i $$-0.548908\pi$$
−0.153044 + 0.988219i $$0.548908\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 15.0000 0.649722
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ −23.0000 −0.988847 −0.494424 0.869221i $$-0.664621\pi$$
−0.494424 + 0.869221i $$0.664621\pi$$
$$542$$ 0 0
$$543$$ 40.0000 1.71656
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ 0 0
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 9.00000 0.383413
$$552$$ 0 0
$$553$$ −33.0000 −1.40330
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ 35.0000 1.48034
$$560$$ 0 0
$$561$$ −40.0000 −1.68880
$$562$$ 0 0
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −33.0000 −1.38587
$$568$$ 0 0
$$569$$ −4.00000 −0.167689 −0.0838444 0.996479i $$-0.526720\pi$$
−0.0838444 + 0.996479i $$0.526720\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ 2.00000 0.0835512
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.0000 0.957503 0.478751 0.877951i $$-0.341090\pi$$
0.478751 + 0.877951i $$0.341090\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −27.0000 −1.12015
$$582$$ 0 0
$$583$$ −60.0000 −2.48495
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −30.0000 −1.23823 −0.619116 0.785299i $$-0.712509\pi$$
−0.619116 + 0.785299i $$0.712509\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ −46.0000 −1.89219
$$592$$ 0 0
$$593$$ 17.0000 0.698106 0.349053 0.937103i $$-0.386503\pi$$
0.349053 + 0.937103i $$0.386503\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −10.0000 −0.409273
$$598$$ 0 0
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 0 0
$$609$$ −54.0000 −2.18819
$$610$$ 0 0
$$611$$ 60.0000 2.42734
$$612$$ 0 0
$$613$$ −18.0000 −0.727013 −0.363507 0.931592i $$-0.618421\pi$$
−0.363507 + 0.931592i $$0.618421\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 32.0000 1.28827 0.644136 0.764911i $$-0.277217\pi$$
0.644136 + 0.764911i $$0.277217\pi$$
$$618$$ 0 0
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −42.0000 −1.68269
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −10.0000 −0.399362
$$628$$ 0 0
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −25.0000 −0.995234 −0.497617 0.867397i $$-0.665792\pi$$
−0.497617 + 0.867397i $$0.665792\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 10.0000 0.396214
$$638$$ 0 0
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ −29.0000 −1.14365 −0.571824 0.820376i $$-0.693764\pi$$
−0.571824 + 0.820376i $$0.693764\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ 29.0000 1.13486 0.567429 0.823422i $$-0.307938\pi$$
0.567429 + 0.823422i $$0.307938\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −1.00000 −0.0390137
$$658$$ 0 0
$$659$$ −45.0000 −1.75295 −0.876476 0.481446i $$-0.840112\pi$$
−0.876476 + 0.481446i $$0.840112\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 0 0
$$663$$ −40.0000 −1.55347
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.00000 0.348481
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −50.0000 −1.93023
$$672$$ 0 0
$$673$$ −15.0000 −0.578208 −0.289104 0.957298i $$-0.593357\pi$$
−0.289104 + 0.957298i $$0.593357\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 48.0000 1.84207
$$680$$ 0 0
$$681$$ 40.0000 1.53280
$$682$$ 0 0
$$683$$ 34.0000 1.30097 0.650487 0.759517i $$-0.274565\pi$$
0.650487 + 0.759517i $$0.274565\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 60.0000 2.28914
$$688$$ 0 0
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 0 0
$$693$$ 15.0000 0.569803
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ −50.0000 −1.89117
$$700$$ 0 0
$$701$$ 8.00000 0.302156 0.151078 0.988522i $$-0.451726\pi$$
0.151078 + 0.988522i $$0.451726\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −42.0000 −1.57957
$$708$$ 0 0
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ 0 0
$$713$$ −2.00000 −0.0749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ −34.0000 −1.26799 −0.633993 0.773339i $$-0.718585\pi$$
−0.633993 + 0.773339i $$0.718585\pi$$
$$720$$ 0 0
$$721$$ 27.0000 1.00553
$$722$$ 0 0
$$723$$ 8.00000 0.297523
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −16.0000 −0.593407 −0.296704 0.954970i $$-0.595887\pi$$
−0.296704 + 0.954970i $$0.595887\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 28.0000 1.03562
$$732$$ 0 0
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 40.0000 1.47342
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ −10.0000 −0.367359
$$742$$ 0 0
$$743$$ −21.0000 −0.770415 −0.385208 0.922830i $$-0.625870\pi$$
−0.385208 + 0.922830i $$0.625870\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9.00000 −0.329293
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 19.0000 0.693320 0.346660 0.937991i $$-0.387316\pi$$
0.346660 + 0.937991i $$0.387316\pi$$
$$752$$ 0 0
$$753$$ −40.0000 −1.45768
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 0 0
$$759$$ −10.0000 −0.362977
$$760$$ 0 0
$$761$$ 1.00000 0.0362500 0.0181250 0.999836i $$-0.494230\pi$$
0.0181250 + 0.999836i $$0.494230\pi$$
$$762$$ 0 0
$$763$$ −30.0000 −1.08607
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −30.0000 −1.08324
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −4.00000 −0.144056
$$772$$ 0 0
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −12.0000 −0.430498
$$778$$ 0 0
$$779$$ 3.00000 0.107486
$$780$$ 0 0
$$781$$ 10.0000 0.357828
$$782$$ 0 0
$$783$$ 36.0000 1.28654
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25.0000 0.891154 0.445577 0.895244i $$-0.352999\pi$$
0.445577 + 0.895244i $$0.352999\pi$$
$$788$$ 0 0
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ −50.0000 −1.77555
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ −14.0000 −0.494666
$$802$$ 0 0
$$803$$ −5.00000 −0.176446
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ 0 0
$$809$$ −23.0000 −0.808637 −0.404318 0.914618i $$-0.632491\pi$$
−0.404318 + 0.914618i $$0.632491\pi$$
$$810$$ 0 0
$$811$$ 40.0000 1.40459 0.702295 0.711886i $$-0.252159\pi$$
0.702295 + 0.711886i $$0.252159\pi$$
$$812$$ 0 0
$$813$$ −24.0000 −0.841717
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.00000 0.244899
$$818$$ 0 0
$$819$$ 15.0000 0.524142
$$820$$ 0 0
$$821$$ 51.0000 1.77991 0.889956 0.456046i $$-0.150735\pi$$
0.889956 + 0.456046i $$0.150735\pi$$
$$822$$ 0 0
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.00000 −0.243414 −0.121707 0.992566i $$-0.538837\pi$$
−0.121707 + 0.992566i $$0.538837\pi$$
$$828$$ 0 0
$$829$$ −31.0000 −1.07667 −0.538337 0.842729i $$-0.680947\pi$$
−0.538337 + 0.842729i $$0.680947\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ 0 0
$$833$$ 8.00000 0.277184
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.00000 −0.276520
$$838$$ 0 0
$$839$$ 15.0000 0.517858 0.258929 0.965896i $$-0.416631\pi$$
0.258929 + 0.965896i $$0.416631\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ 16.0000 0.551069
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 42.0000 1.44314
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ −19.0000 −0.650548 −0.325274 0.945620i $$-0.605456\pi$$
−0.325274 + 0.945620i $$0.605456\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 0 0
$$861$$ −18.0000 −0.613438
$$862$$ 0 0
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 2.00000 0.0679236
$$868$$ 0 0
$$869$$ −55.0000 −1.86575
$$870$$ 0 0
$$871$$ 40.0000 1.35535
$$872$$ 0 0
$$873$$ 16.0000 0.541518
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ 56.0000 1.88883
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ 0 0
$$891$$ −55.0000 −1.84257
$$892$$ 0 0
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −10.0000 −0.333890
$$898$$ 0 0
$$899$$ −18.0000 −0.600334
$$900$$ 0 0
$$901$$ −48.0000 −1.59911
$$902$$ 0 0
$$903$$ −42.0000 −1.39767
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −15.0000 −0.498067 −0.249033 0.968495i $$-0.580113\pi$$
−0.249033 + 0.968495i $$0.580113\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 5.00000 0.165657 0.0828287 0.996564i $$-0.473605\pi$$
0.0828287 + 0.996564i $$0.473605\pi$$
$$912$$ 0 0
$$913$$ −45.0000 −1.48928
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.0000 0.594412
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 32.0000 1.05444
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 9.00000 0.295599
$$928$$ 0 0
$$929$$ −13.0000 −0.426516 −0.213258 0.976996i $$-0.568408\pi$$
−0.213258 + 0.976996i $$0.568408\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 0 0
$$933$$ −36.0000 −1.17859
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 44.0000 1.43742 0.718709 0.695311i $$-0.244734\pi$$
0.718709 + 0.695311i $$0.244734\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 0 0
$$943$$ 3.00000 0.0976934
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ −5.00000 −0.162307
$$950$$ 0 0
$$951$$ 34.0000 1.10253
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −90.0000 −2.90929
$$958$$ 0 0
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ −16.0000 −0.515593
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −12.0000 −0.385894 −0.192947 0.981209i $$-0.561805\pi$$
−0.192947 + 0.981209i $$0.561805\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −41.0000 −1.31575 −0.657876 0.753126i $$-0.728545\pi$$
−0.657876 + 0.753126i $$0.728545\pi$$
$$972$$ 0 0
$$973$$ 36.0000 1.15411
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 20.0000 0.639857 0.319928 0.947442i $$-0.396341\pi$$
0.319928 + 0.947442i $$0.396341\pi$$
$$978$$ 0 0
$$979$$ −70.0000 −2.23721
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 0 0
$$983$$ 1.00000 0.0318950 0.0159475 0.999873i $$-0.494924\pi$$
0.0159475 + 0.999873i $$0.494924\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −72.0000 −2.29179
$$988$$ 0 0
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ −52.0000 −1.65017
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −5.00000 −0.158352 −0.0791758 0.996861i $$-0.525229\pi$$
−0.0791758 + 0.996861i $$0.525229\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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 4600.2.a.d.1.1 1
4.3 odd 2 9200.2.a.bc.1.1 1
5.2 odd 4 4600.2.e.d.4049.2 2
5.3 odd 4 4600.2.e.d.4049.1 2
5.4 even 2 4600.2.a.m.1.1 yes 1
20.19 odd 2 9200.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.d.1.1 1 1.1 even 1 trivial
4600.2.a.m.1.1 yes 1 5.4 even 2
4600.2.e.d.4049.1 2 5.3 odd 4
4600.2.e.d.4049.2 2 5.2 odd 4
9200.2.a.j.1.1 1 20.19 odd 2
9200.2.a.bc.1.1 1 4.3 odd 2