# Properties

 Label 9576.2.a.br.1.1 Level $9576$ Weight $2$ Character 9576.1 Self dual yes Analytic conductor $76.465$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 9576.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.732051 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q-0.732051 q^{5} -1.00000 q^{7} -3.46410 q^{11} +4.00000 q^{13} +3.26795 q^{17} +1.00000 q^{19} -0.535898 q^{23} -4.46410 q^{25} -2.73205 q^{29} -3.46410 q^{31} +0.732051 q^{35} -7.46410 q^{37} +11.4641 q^{41} -8.92820 q^{43} +5.66025 q^{47} +1.00000 q^{49} +8.19615 q^{53} +2.53590 q^{55} +8.00000 q^{59} +4.53590 q^{61} -2.92820 q^{65} -4.00000 q^{67} -6.19615 q^{71} +0.535898 q^{73} +3.46410 q^{77} -5.46410 q^{79} -0.196152 q^{83} -2.39230 q^{85} -10.3923 q^{89} -4.00000 q^{91} -0.732051 q^{95} +6.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 $$2 q + 2 q^{5} - 2 q^{7} + 8 q^{13} + 10 q^{17} + 2 q^{19} - 8 q^{23} - 2 q^{25} - 2 q^{29} - 2 q^{35} - 8 q^{37} + 16 q^{41} - 4 q^{43} - 6 q^{47} + 2 q^{49} + 6 q^{53} + 12 q^{55} + 16 q^{59} + 16 q^{61} + 8 q^{65} - 8 q^{67} - 2 q^{71} + 8 q^{73} - 4 q^{79} + 10 q^{83} + 16 q^{85} - 8 q^{91} + 2 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^13 + 10 * q^17 + 2 * q^19 - 8 * q^23 - 2 * q^25 - 2 * q^29 - 2 * q^35 - 8 * q^37 + 16 * q^41 - 4 * q^43 - 6 * q^47 + 2 * q^49 + 6 * q^53 + 12 * q^55 + 16 * q^59 + 16 * q^61 + 8 * q^65 - 8 * q^67 - 2 * q^71 + 8 * q^73 - 4 * q^79 + 10 * q^83 + 16 * q^85 - 8 * q^91 + 2 * q^95 + 12 * q^97

## 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
$$5$$ −0.732051 −0.327383 −0.163692 0.986512i $$-0.552340\pi$$
−0.163692 + 0.986512i $$0.552340\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.26795 0.792594 0.396297 0.918122i $$-0.370295\pi$$
0.396297 + 0.918122i $$0.370295\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.535898 −0.111743 −0.0558713 0.998438i $$-0.517794\pi$$
−0.0558713 + 0.998438i $$0.517794\pi$$
$$24$$ 0 0
$$25$$ −4.46410 −0.892820
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.73205 −0.507329 −0.253665 0.967292i $$-0.581636\pi$$
−0.253665 + 0.967292i $$0.581636\pi$$
$$30$$ 0 0
$$31$$ −3.46410 −0.622171 −0.311086 0.950382i $$-0.600693\pi$$
−0.311086 + 0.950382i $$0.600693\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.732051 0.123739
$$36$$ 0 0
$$37$$ −7.46410 −1.22709 −0.613545 0.789659i $$-0.710257\pi$$
−0.613545 + 0.789659i $$0.710257\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.4641 1.79039 0.895196 0.445673i $$-0.147036\pi$$
0.895196 + 0.445673i $$0.147036\pi$$
$$42$$ 0 0
$$43$$ −8.92820 −1.36154 −0.680769 0.732498i $$-0.738354\pi$$
−0.680769 + 0.732498i $$0.738354\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.66025 0.825633 0.412816 0.910814i $$-0.364545\pi$$
0.412816 + 0.910814i $$0.364545\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.19615 1.12583 0.562914 0.826515i $$-0.309680\pi$$
0.562914 + 0.826515i $$0.309680\pi$$
$$54$$ 0 0
$$55$$ 2.53590 0.341940
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 4.53590 0.580762 0.290381 0.956911i $$-0.406218\pi$$
0.290381 + 0.956911i $$0.406218\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2.92820 −0.363199
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.19615 −0.735348 −0.367674 0.929955i $$-0.619846\pi$$
−0.367674 + 0.929955i $$0.619846\pi$$
$$72$$ 0 0
$$73$$ 0.535898 0.0627222 0.0313611 0.999508i $$-0.490016\pi$$
0.0313611 + 0.999508i $$0.490016\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.46410 0.394771
$$78$$ 0 0
$$79$$ −5.46410 −0.614759 −0.307380 0.951587i $$-0.599452\pi$$
−0.307380 + 0.951587i $$0.599452\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −0.196152 −0.0215305 −0.0107653 0.999942i $$-0.503427\pi$$
−0.0107653 + 0.999942i $$0.503427\pi$$
$$84$$ 0 0
$$85$$ −2.39230 −0.259482
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.3923 −1.10158 −0.550791 0.834643i $$-0.685674\pi$$
−0.550791 + 0.834643i $$0.685674\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −0.732051 −0.0751068
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 14.5885 1.45161 0.725803 0.687903i $$-0.241468\pi$$
0.725803 + 0.687903i $$0.241468\pi$$
$$102$$ 0 0
$$103$$ 6.92820 0.682656 0.341328 0.939944i $$-0.389123\pi$$
0.341328 + 0.939944i $$0.389123\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.19615 0.212310 0.106155 0.994350i $$-0.466146\pi$$
0.106155 + 0.994350i $$0.466146\pi$$
$$108$$ 0 0
$$109$$ 15.8564 1.51877 0.759384 0.650643i $$-0.225500\pi$$
0.759384 + 0.650643i $$0.225500\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.73205 −0.257010 −0.128505 0.991709i $$-0.541018\pi$$
−0.128505 + 0.991709i $$0.541018\pi$$
$$114$$ 0 0
$$115$$ 0.392305 0.0365826
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −3.26795 −0.299572
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ 2.92820 0.259836 0.129918 0.991525i $$-0.458529\pi$$
0.129918 + 0.991525i $$0.458529\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.19615 0.716101 0.358051 0.933702i $$-0.383442\pi$$
0.358051 + 0.933702i $$0.383442\pi$$
$$132$$ 0 0
$$133$$ −1.00000 −0.0867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −10.9282 −0.933659 −0.466830 0.884347i $$-0.654604\pi$$
−0.466830 + 0.884347i $$0.654604\pi$$
$$138$$ 0 0
$$139$$ 2.53590 0.215092 0.107546 0.994200i $$-0.465701\pi$$
0.107546 + 0.994200i $$0.465701\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −13.8564 −1.15873
$$144$$ 0 0
$$145$$ 2.00000 0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −9.46410 −0.775329 −0.387665 0.921800i $$-0.626718\pi$$
−0.387665 + 0.921800i $$0.626718\pi$$
$$150$$ 0 0
$$151$$ −21.4641 −1.74672 −0.873362 0.487072i $$-0.838065\pi$$
−0.873362 + 0.487072i $$0.838065\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2.53590 0.203688
$$156$$ 0 0
$$157$$ 14.3923 1.14863 0.574315 0.818634i $$-0.305268\pi$$
0.574315 + 0.818634i $$0.305268\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0.535898 0.0422347
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4.39230 −0.339887 −0.169943 0.985454i $$-0.554358\pi$$
−0.169943 + 0.985454i $$0.554358\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −10.3923 −0.790112 −0.395056 0.918657i $$-0.629275\pi$$
−0.395056 + 0.918657i $$0.629275\pi$$
$$174$$ 0 0
$$175$$ 4.46410 0.337454
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 25.5167 1.90720 0.953602 0.301069i $$-0.0973434\pi$$
0.953602 + 0.301069i $$0.0973434\pi$$
$$180$$ 0 0
$$181$$ −23.8564 −1.77323 −0.886616 0.462506i $$-0.846951\pi$$
−0.886616 + 0.462506i $$0.846951\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 5.46410 0.401729
$$186$$ 0 0
$$187$$ −11.3205 −0.827838
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.3923 −1.33082 −0.665410 0.746478i $$-0.731743\pi$$
−0.665410 + 0.746478i $$0.731743\pi$$
$$192$$ 0 0
$$193$$ 17.3205 1.24676 0.623379 0.781920i $$-0.285760\pi$$
0.623379 + 0.781920i $$0.285760\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.92820 0.493614 0.246807 0.969065i $$-0.420619\pi$$
0.246807 + 0.969065i $$0.420619\pi$$
$$198$$ 0 0
$$199$$ −16.7846 −1.18983 −0.594915 0.803789i $$-0.702814\pi$$
−0.594915 + 0.803789i $$0.702814\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2.73205 0.191752
$$204$$ 0 0
$$205$$ −8.39230 −0.586144
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −3.46410 −0.239617
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.53590 0.445745
$$216$$ 0 0
$$217$$ 3.46410 0.235159
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 13.0718 0.879304
$$222$$ 0 0
$$223$$ 13.3205 0.892007 0.446004 0.895031i $$-0.352847\pi$$
0.446004 + 0.895031i $$0.352847\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 13.4641 0.893644 0.446822 0.894623i $$-0.352556\pi$$
0.446822 + 0.894623i $$0.352556\pi$$
$$228$$ 0 0
$$229$$ 0.928203 0.0613374 0.0306687 0.999530i $$-0.490236\pi$$
0.0306687 + 0.999530i $$0.490236\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.5359 0.952278 0.476139 0.879370i $$-0.342036\pi$$
0.476139 + 0.879370i $$0.342036\pi$$
$$234$$ 0 0
$$235$$ −4.14359 −0.270298
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 19.8564 1.28440 0.642202 0.766535i $$-0.278021\pi$$
0.642202 + 0.766535i $$0.278021\pi$$
$$240$$ 0 0
$$241$$ 15.0718 0.970860 0.485430 0.874276i $$-0.338663\pi$$
0.485430 + 0.874276i $$0.338663\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.732051 −0.0467690
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.7321 0.677401 0.338701 0.940894i $$-0.390013\pi$$
0.338701 + 0.940894i $$0.390013\pi$$
$$252$$ 0 0
$$253$$ 1.85641 0.116711
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −0.143594 −0.00895712 −0.00447856 0.999990i $$-0.501426\pi$$
−0.00447856 + 0.999990i $$0.501426\pi$$
$$258$$ 0 0
$$259$$ 7.46410 0.463797
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.9282 1.04384 0.521919 0.852995i $$-0.325216\pi$$
0.521919 + 0.852995i $$0.325216\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 12.9282 0.788246 0.394123 0.919058i $$-0.371048\pi$$
0.394123 + 0.919058i $$0.371048\pi$$
$$270$$ 0 0
$$271$$ 23.3205 1.41662 0.708310 0.705902i $$-0.249458\pi$$
0.708310 + 0.705902i $$0.249458\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 15.4641 0.932520
$$276$$ 0 0
$$277$$ 22.2487 1.33680 0.668398 0.743804i $$-0.266980\pi$$
0.668398 + 0.743804i $$0.266980\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.7321 1.11746 0.558730 0.829349i $$-0.311289\pi$$
0.558730 + 0.829349i $$0.311289\pi$$
$$282$$ 0 0
$$283$$ 21.8564 1.29923 0.649614 0.760264i $$-0.274930\pi$$
0.649614 + 0.760264i $$0.274930\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −11.4641 −0.676705
$$288$$ 0 0
$$289$$ −6.32051 −0.371795
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 20.5359 1.19972 0.599860 0.800105i $$-0.295223\pi$$
0.599860 + 0.800105i $$0.295223\pi$$
$$294$$ 0 0
$$295$$ −5.85641 −0.340973
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.14359 −0.123967
$$300$$ 0 0
$$301$$ 8.92820 0.514613
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.32051 −0.190132
$$306$$ 0 0
$$307$$ −21.3205 −1.21683 −0.608413 0.793621i $$-0.708193\pi$$
−0.608413 + 0.793621i $$0.708193\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 25.2679 1.43281 0.716407 0.697683i $$-0.245785\pi$$
0.716407 + 0.697683i $$0.245785\pi$$
$$312$$ 0 0
$$313$$ 21.3205 1.20511 0.602553 0.798079i $$-0.294150\pi$$
0.602553 + 0.798079i $$0.294150\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.7321 −0.827434 −0.413717 0.910405i $$-0.635770\pi$$
−0.413717 + 0.910405i $$0.635770\pi$$
$$318$$ 0 0
$$319$$ 9.46410 0.529888
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.26795 0.181834
$$324$$ 0 0
$$325$$ −17.8564 −0.990495
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −5.66025 −0.312060
$$330$$ 0 0
$$331$$ −16.3923 −0.901003 −0.450501 0.892776i $$-0.648755\pi$$
−0.450501 + 0.892776i $$0.648755\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 2.92820 0.159985
$$336$$ 0 0
$$337$$ 16.9282 0.922138 0.461069 0.887364i $$-0.347466\pi$$
0.461069 + 0.887364i $$0.347466\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 12.0000 0.649836
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8.14359 0.437171 0.218586 0.975818i $$-0.429856\pi$$
0.218586 + 0.975818i $$0.429856\pi$$
$$348$$ 0 0
$$349$$ −3.85641 −0.206429 −0.103214 0.994659i $$-0.532913\pi$$
−0.103214 + 0.994659i $$0.532913\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.1962 −0.968483 −0.484242 0.874934i $$-0.660904\pi$$
−0.484242 + 0.874934i $$0.660904\pi$$
$$354$$ 0 0
$$355$$ 4.53590 0.240740
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11.4641 0.605052 0.302526 0.953141i $$-0.402170\pi$$
0.302526 + 0.953141i $$0.402170\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −0.392305 −0.0205342
$$366$$ 0 0
$$367$$ 1.85641 0.0969036 0.0484518 0.998826i $$-0.484571\pi$$
0.0484518 + 0.998826i $$0.484571\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8.19615 −0.425523
$$372$$ 0 0
$$373$$ −29.7128 −1.53847 −0.769236 0.638965i $$-0.779363\pi$$
−0.769236 + 0.638965i $$0.779363\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −10.9282 −0.562831
$$378$$ 0 0
$$379$$ 18.2487 0.937373 0.468687 0.883364i $$-0.344727\pi$$
0.468687 + 0.883364i $$0.344727\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 14.5359 0.742750 0.371375 0.928483i $$-0.378886\pi$$
0.371375 + 0.928483i $$0.378886\pi$$
$$384$$ 0 0
$$385$$ −2.53590 −0.129241
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 9.46410 0.479849 0.239924 0.970792i $$-0.422877\pi$$
0.239924 + 0.970792i $$0.422877\pi$$
$$390$$ 0 0
$$391$$ −1.75129 −0.0885665
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.00000 0.201262
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.0526 1.10125 0.550626 0.834752i $$-0.314389\pi$$
0.550626 + 0.834752i $$0.314389\pi$$
$$402$$ 0 0
$$403$$ −13.8564 −0.690237
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 25.8564 1.28165
$$408$$ 0 0
$$409$$ −19.7128 −0.974736 −0.487368 0.873197i $$-0.662043\pi$$
−0.487368 + 0.873197i $$0.662043\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 0.143594 0.00704873
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0.196152 0.00958267 0.00479134 0.999989i $$-0.498475\pi$$
0.00479134 + 0.999989i $$0.498475\pi$$
$$420$$ 0 0
$$421$$ 9.32051 0.454254 0.227127 0.973865i $$-0.427067\pi$$
0.227127 + 0.973865i $$0.427067\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −14.5885 −0.707644
$$426$$ 0 0
$$427$$ −4.53590 −0.219508
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0526 0.773225 0.386612 0.922242i $$-0.373645\pi$$
0.386612 + 0.922242i $$0.373645\pi$$
$$432$$ 0 0
$$433$$ 8.92820 0.429062 0.214531 0.976717i $$-0.431178\pi$$
0.214531 + 0.976717i $$0.431178\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −0.535898 −0.0256355
$$438$$ 0 0
$$439$$ 4.53590 0.216487 0.108243 0.994124i $$-0.465477\pi$$
0.108243 + 0.994124i $$0.465477\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7.85641 0.373269 0.186635 0.982429i $$-0.440242\pi$$
0.186635 + 0.982429i $$0.440242\pi$$
$$444$$ 0 0
$$445$$ 7.60770 0.360639
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3.41154 0.161001 0.0805003 0.996755i $$-0.474348\pi$$
0.0805003 + 0.996755i $$0.474348\pi$$
$$450$$ 0 0
$$451$$ −39.7128 −1.87000
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.92820 0.137276
$$456$$ 0 0
$$457$$ 12.3923 0.579688 0.289844 0.957074i $$-0.406397\pi$$
0.289844 + 0.957074i $$0.406397\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 41.5167 1.93362 0.966812 0.255490i $$-0.0822366\pi$$
0.966812 + 0.255490i $$0.0822366\pi$$
$$462$$ 0 0
$$463$$ −13.8564 −0.643962 −0.321981 0.946746i $$-0.604349\pi$$
−0.321981 + 0.946746i $$0.604349\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6.73205 −0.311522 −0.155761 0.987795i $$-0.549783\pi$$
−0.155761 + 0.987795i $$0.549783\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 30.9282 1.42208
$$474$$ 0 0
$$475$$ −4.46410 −0.204827
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 29.2679 1.33729 0.668643 0.743583i $$-0.266875\pi$$
0.668643 + 0.743583i $$0.266875\pi$$
$$480$$ 0 0
$$481$$ −29.8564 −1.36133
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.39230 −0.199444
$$486$$ 0 0
$$487$$ −6.14359 −0.278393 −0.139196 0.990265i $$-0.544452\pi$$
−0.139196 + 0.990265i $$0.544452\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7.85641 −0.354555 −0.177277 0.984161i $$-0.556729\pi$$
−0.177277 + 0.984161i $$0.556729\pi$$
$$492$$ 0 0
$$493$$ −8.92820 −0.402106
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.19615 0.277935
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −5.66025 −0.252378 −0.126189 0.992006i $$-0.540275\pi$$
−0.126189 + 0.992006i $$0.540275\pi$$
$$504$$ 0 0
$$505$$ −10.6795 −0.475231
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −14.7846 −0.655316 −0.327658 0.944796i $$-0.606259\pi$$
−0.327658 + 0.944796i $$0.606259\pi$$
$$510$$ 0 0
$$511$$ −0.535898 −0.0237067
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −5.07180 −0.223490
$$516$$ 0 0
$$517$$ −19.6077 −0.862345
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −17.7128 −0.776012 −0.388006 0.921657i $$-0.626836\pi$$
−0.388006 + 0.921657i $$0.626836\pi$$
$$522$$ 0 0
$$523$$ −0.535898 −0.0234332 −0.0117166 0.999931i $$-0.503730\pi$$
−0.0117166 + 0.999931i $$0.503730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.3205 −0.493129
$$528$$ 0 0
$$529$$ −22.7128 −0.987514
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 45.8564 1.98626
$$534$$ 0 0
$$535$$ −1.60770 −0.0695067
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −3.46410 −0.149209
$$540$$ 0 0
$$541$$ 31.8564 1.36961 0.684807 0.728725i $$-0.259887\pi$$
0.684807 + 0.728725i $$0.259887\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −11.6077 −0.497219
$$546$$ 0 0
$$547$$ 40.1051 1.71477 0.857386 0.514675i $$-0.172087\pi$$
0.857386 + 0.514675i $$0.172087\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.73205 −0.116389
$$552$$ 0 0
$$553$$ 5.46410 0.232357
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −13.4641 −0.570492 −0.285246 0.958454i $$-0.592075\pi$$
−0.285246 + 0.958454i $$0.592075\pi$$
$$558$$ 0 0
$$559$$ −35.7128 −1.51049
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 14.5359 0.612615 0.306308 0.951933i $$-0.400906\pi$$
0.306308 + 0.951933i $$0.400906\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −22.0526 −0.924491 −0.462246 0.886752i $$-0.652956\pi$$
−0.462246 + 0.886752i $$0.652956\pi$$
$$570$$ 0 0
$$571$$ −17.0718 −0.714432 −0.357216 0.934022i $$-0.616274\pi$$
−0.357216 + 0.934022i $$0.616274\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2.39230 0.0997660
$$576$$ 0 0
$$577$$ 45.3205 1.88672 0.943359 0.331775i $$-0.107647\pi$$
0.943359 + 0.331775i $$0.107647\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.196152 0.00813777
$$582$$ 0 0
$$583$$ −28.3923 −1.17589
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −22.0526 −0.910207 −0.455103 0.890439i $$-0.650398\pi$$
−0.455103 + 0.890439i $$0.650398\pi$$
$$588$$ 0 0
$$589$$ −3.46410 −0.142736
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 30.9808 1.27223 0.636114 0.771595i $$-0.280541\pi$$
0.636114 + 0.771595i $$0.280541\pi$$
$$594$$ 0 0
$$595$$ 2.39230 0.0980749
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −20.0526 −0.819325 −0.409663 0.912237i $$-0.634354\pi$$
−0.409663 + 0.912237i $$0.634354\pi$$
$$600$$ 0 0
$$601$$ 40.6410 1.65778 0.828891 0.559410i $$-0.188972\pi$$
0.828891 + 0.559410i $$0.188972\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −0.732051 −0.0297621
$$606$$ 0 0
$$607$$ −24.7846 −1.00598 −0.502988 0.864293i $$-0.667766\pi$$
−0.502988 + 0.864293i $$0.667766\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 22.6410 0.915957
$$612$$ 0 0
$$613$$ 7.60770 0.307272 0.153636 0.988128i $$-0.450902\pi$$
0.153636 + 0.988128i $$0.450902\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 35.7128 1.43774 0.718872 0.695143i $$-0.244659\pi$$
0.718872 + 0.695143i $$0.244659\pi$$
$$618$$ 0 0
$$619$$ −13.4641 −0.541168 −0.270584 0.962696i $$-0.587217\pi$$
−0.270584 + 0.962696i $$0.587217\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ 17.2487 0.689948
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −24.3923 −0.972585
$$630$$ 0 0
$$631$$ −20.9282 −0.833139 −0.416569 0.909104i $$-0.636768\pi$$
−0.416569 + 0.909104i $$0.636768\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2.14359 −0.0850659
$$636$$ 0 0
$$637$$ 4.00000 0.158486
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −7.12436 −0.281395 −0.140698 0.990053i $$-0.544935\pi$$
−0.140698 + 0.990053i $$0.544935\pi$$
$$642$$ 0 0
$$643$$ 41.5692 1.63933 0.819665 0.572843i $$-0.194160\pi$$
0.819665 + 0.572843i $$0.194160\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −19.1244 −0.751856 −0.375928 0.926649i $$-0.622676\pi$$
−0.375928 + 0.926649i $$0.622676\pi$$
$$648$$ 0 0
$$649$$ −27.7128 −1.08782
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −13.0718 −0.511539 −0.255769 0.966738i $$-0.582329\pi$$
−0.255769 + 0.966738i $$0.582329\pi$$
$$654$$ 0 0
$$655$$ −6.00000 −0.234439
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −16.7321 −0.651788 −0.325894 0.945406i $$-0.605665\pi$$
−0.325894 + 0.945406i $$0.605665\pi$$
$$660$$ 0 0
$$661$$ 26.6410 1.03622 0.518108 0.855315i $$-0.326637\pi$$
0.518108 + 0.855315i $$0.326637\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.732051 0.0283877
$$666$$ 0 0
$$667$$ 1.46410 0.0566902
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −15.7128 −0.606586
$$672$$ 0 0
$$673$$ −32.2487 −1.24310 −0.621548 0.783376i $$-0.713496\pi$$
−0.621548 + 0.783376i $$0.713496\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.92820 0.189406 0.0947031 0.995506i $$-0.469810\pi$$
0.0947031 + 0.995506i $$0.469810\pi$$
$$678$$ 0 0
$$679$$ −6.00000 −0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −15.2679 −0.584212 −0.292106 0.956386i $$-0.594356\pi$$
−0.292106 + 0.956386i $$0.594356\pi$$
$$684$$ 0 0
$$685$$ 8.00000 0.305664
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 32.7846 1.24899
$$690$$ 0 0
$$691$$ −4.67949 −0.178016 −0.0890081 0.996031i $$-0.528370\pi$$
−0.0890081 + 0.996031i $$0.528370\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.85641 −0.0704175
$$696$$ 0 0
$$697$$ 37.4641 1.41905
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ −7.46410 −0.281514
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −14.5885 −0.548655
$$708$$ 0 0
$$709$$ 14.2487 0.535122 0.267561 0.963541i $$-0.413782\pi$$
0.267561 + 0.963541i $$0.413782\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1.85641 0.0695230
$$714$$ 0 0
$$715$$ 10.1436 0.379349
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −7.41154 −0.276404 −0.138202 0.990404i $$-0.544132\pi$$
−0.138202 + 0.990404i $$0.544132\pi$$
$$720$$ 0 0
$$721$$ −6.92820 −0.258020
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 12.1962 0.452954
$$726$$ 0 0
$$727$$ −8.39230 −0.311253 −0.155627 0.987816i $$-0.549740\pi$$
−0.155627 + 0.987816i $$0.549740\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −29.1769 −1.07915
$$732$$ 0 0
$$733$$ 7.46410 0.275693 0.137846 0.990454i $$-0.455982\pi$$
0.137846 + 0.990454i $$0.455982\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 13.8564 0.510407
$$738$$ 0 0
$$739$$ 27.0718 0.995852 0.497926 0.867219i $$-0.334095\pi$$
0.497926 + 0.867219i $$0.334095\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −34.3013 −1.25839 −0.629196 0.777247i $$-0.716616\pi$$
−0.629196 + 0.777247i $$0.716616\pi$$
$$744$$ 0 0
$$745$$ 6.92820 0.253830
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −2.19615 −0.0802457
$$750$$ 0 0
$$751$$ −16.3923 −0.598164 −0.299082 0.954227i $$-0.596680\pi$$
−0.299082 + 0.954227i $$0.596680\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 15.7128 0.571848
$$756$$ 0 0
$$757$$ −5.71281 −0.207636 −0.103818 0.994596i $$-0.533106\pi$$
−0.103818 + 0.994596i $$0.533106\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7.26795 0.263463 0.131731 0.991285i $$-0.457946\pi$$
0.131731 + 0.991285i $$0.457946\pi$$
$$762$$ 0 0
$$763$$ −15.8564 −0.574040
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 32.0000 1.15545
$$768$$ 0 0
$$769$$ 39.4641 1.42311 0.711556 0.702629i $$-0.247991\pi$$
0.711556 + 0.702629i $$0.247991\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −9.60770 −0.345565 −0.172782 0.984960i $$-0.555276\pi$$
−0.172782 + 0.984960i $$0.555276\pi$$
$$774$$ 0 0
$$775$$ 15.4641 0.555487
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 11.4641 0.410744
$$780$$ 0 0
$$781$$ 21.4641 0.768046
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −10.5359 −0.376042
$$786$$ 0 0
$$787$$ 16.5359 0.589441 0.294721 0.955583i $$-0.404773\pi$$
0.294721 + 0.955583i $$0.404773\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 2.73205 0.0971405
$$792$$ 0 0
$$793$$ 18.1436 0.644298
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 36.2487 1.28400 0.641998 0.766707i $$-0.278106\pi$$
0.641998 + 0.766707i $$0.278106\pi$$
$$798$$ 0 0
$$799$$ 18.4974 0.654392
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1.85641 −0.0655112
$$804$$ 0 0
$$805$$ −0.392305 −0.0138269
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −38.9282 −1.36864 −0.684321 0.729181i $$-0.739901\pi$$
−0.684321 + 0.729181i $$0.739901\pi$$
$$810$$ 0 0
$$811$$ −27.7128 −0.973128 −0.486564 0.873645i $$-0.661750\pi$$
−0.486564 + 0.873645i $$0.661750\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 7.32051 0.256426
$$816$$ 0 0
$$817$$ −8.92820 −0.312358
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −19.3205 −0.674290 −0.337145 0.941453i $$-0.609461\pi$$
−0.337145 + 0.941453i $$0.609461\pi$$
$$822$$ 0 0
$$823$$ −10.0000 −0.348578 −0.174289 0.984695i $$-0.555763\pi$$
−0.174289 + 0.984695i $$0.555763\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 45.5167 1.58277 0.791385 0.611318i $$-0.209361\pi$$
0.791385 + 0.611318i $$0.209361\pi$$
$$828$$ 0 0
$$829$$ 39.0718 1.35702 0.678510 0.734591i $$-0.262626\pi$$
0.678510 + 0.734591i $$0.262626\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 3.26795 0.113228
$$834$$ 0 0
$$835$$ 3.21539 0.111273
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −32.7846 −1.13185 −0.565925 0.824457i $$-0.691481\pi$$
−0.565925 + 0.824457i $$0.691481\pi$$
$$840$$ 0 0
$$841$$ −21.5359 −0.742617
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −2.19615 −0.0755499
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ −19.0718 −0.653006 −0.326503 0.945196i $$-0.605870\pi$$
−0.326503 + 0.945196i $$0.605870\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 14.3923 0.491632 0.245816 0.969317i $$-0.420944\pi$$
0.245816 + 0.969317i $$0.420944\pi$$
$$858$$ 0 0
$$859$$ −40.7846 −1.39155 −0.695776 0.718258i $$-0.744940\pi$$
−0.695776 + 0.718258i $$0.744940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −16.0526 −0.546435 −0.273218 0.961952i $$-0.588088\pi$$
−0.273218 + 0.961952i $$0.588088\pi$$
$$864$$ 0 0
$$865$$ 7.60770 0.258669
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 18.9282 0.642095
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −6.92820 −0.234216
$$876$$ 0 0
$$877$$ 13.2154 0.446252 0.223126 0.974790i $$-0.428374\pi$$
0.223126 + 0.974790i $$0.428374\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 17.5167 0.590151 0.295076 0.955474i $$-0.404655\pi$$
0.295076 + 0.955474i $$0.404655\pi$$
$$882$$ 0 0
$$883$$ −39.7128 −1.33644 −0.668221 0.743963i $$-0.732944\pi$$
−0.668221 + 0.743963i $$0.732944\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.1051 −1.21229 −0.606146 0.795354i $$-0.707285\pi$$
−0.606146 + 0.795354i $$0.707285\pi$$
$$888$$ 0 0
$$889$$ −2.92820 −0.0982088
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 5.66025 0.189413
$$894$$ 0 0
$$895$$ −18.6795 −0.624387
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9.46410 0.315645
$$900$$ 0 0
$$901$$ 26.7846 0.892325
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 17.4641 0.580526
$$906$$ 0 0
$$907$$ 38.6410 1.28305 0.641527 0.767101i $$-0.278301\pi$$
0.641527 + 0.767101i $$0.278301\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0526 1.59205 0.796026 0.605262i $$-0.206932\pi$$
0.796026 + 0.605262i $$0.206932\pi$$
$$912$$ 0 0
$$913$$ 0.679492 0.0224879
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −8.19615 −0.270661
$$918$$ 0 0
$$919$$ 13.8564 0.457081 0.228540 0.973534i $$-0.426605\pi$$
0.228540 + 0.973534i $$0.426605\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −24.7846 −0.815795
$$924$$ 0 0
$$925$$ 33.3205 1.09557
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −21.8038 −0.715361 −0.357681 0.933844i $$-0.616432\pi$$
−0.357681 + 0.933844i $$0.616432\pi$$
$$930$$ 0 0
$$931$$ 1.00000 0.0327737
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 8.28719 0.271020
$$936$$ 0 0
$$937$$ 39.8564 1.30205 0.651026 0.759055i $$-0.274339\pi$$
0.651026 + 0.759055i $$0.274339\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 40.6410 1.32486 0.662430 0.749124i $$-0.269525\pi$$
0.662430 + 0.749124i $$0.269525\pi$$
$$942$$ 0 0
$$943$$ −6.14359 −0.200063
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 57.0333 1.85333 0.926667 0.375883i $$-0.122661\pi$$
0.926667 + 0.375883i $$0.122661\pi$$
$$948$$ 0 0
$$949$$ 2.14359 0.0695840
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 3.41154 0.110511 0.0552554 0.998472i $$-0.482403\pi$$
0.0552554 + 0.998472i $$0.482403\pi$$
$$954$$ 0 0
$$955$$ 13.4641 0.435688
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.9282 0.352890
$$960$$ 0 0
$$961$$ −19.0000 −0.612903
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −12.6795 −0.408167
$$966$$ 0 0
$$967$$ −38.7846 −1.24723 −0.623614 0.781732i $$-0.714336\pi$$
−0.623614 + 0.781732i $$0.714336\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −41.0718 −1.31806 −0.659028 0.752118i $$-0.729032\pi$$
−0.659028 + 0.752118i $$0.729032\pi$$
$$972$$ 0 0
$$973$$ −2.53590 −0.0812972
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.12436 0.0999570 0.0499785 0.998750i $$-0.484085\pi$$
0.0499785 + 0.998750i $$0.484085\pi$$
$$978$$ 0 0
$$979$$ 36.0000 1.15056
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −12.7846 −0.407766 −0.203883 0.978995i $$-0.565356\pi$$
−0.203883 + 0.978995i $$0.565356\pi$$
$$984$$ 0 0
$$985$$ −5.07180 −0.161601
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 4.78461 0.152142
$$990$$ 0 0
$$991$$ −52.7846 −1.67676 −0.838379 0.545087i $$-0.816496\pi$$
−0.838379 + 0.545087i $$0.816496\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 12.2872 0.389530
$$996$$ 0 0
$$997$$ −46.1051 −1.46016 −0.730082 0.683360i $$-0.760518\pi$$
−0.730082 + 0.683360i $$0.760518\pi$$
$$998$$ 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 9576.2.a.br.1.1 yes 2
3.2 odd 2 9576.2.a.bf.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9576.2.a.bf.1.2 2 3.2 odd 2
9576.2.a.br.1.1 yes 2 1.1 even 1 trivial