# Properties

 Label 4400.2.b.m.4049.1 Level $4400$ Weight $2$ Character 4400.4049 Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 4049.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.4049 Dual form 4400.2.b.m.4049.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{7} +3.00000 q^{9} +1.00000 q^{11} -8.00000 q^{19} +8.00000i q^{23} -10.0000 q^{29} -8.00000 q^{31} +10.0000i q^{37} -2.00000 q^{41} +6.00000i q^{43} -8.00000i q^{47} +3.00000 q^{49} +14.0000i q^{53} -4.00000 q^{59} +10.0000 q^{61} -6.00000i q^{63} +4.00000i q^{67} -8.00000i q^{73} -2.00000i q^{77} -4.00000 q^{79} +9.00000 q^{81} -10.0000i q^{83} -6.00000 q^{89} +10.0000i q^{97} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} + 2 q^{11} - 16 q^{19} - 20 q^{29} - 16 q^{31} - 4 q^{41} + 6 q^{49} - 8 q^{59} + 20 q^{61} - 8 q^{79} + 18 q^{81} - 12 q^{89} + 6 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 + 2 * q^11 - 16 * q^19 - 20 * q^29 - 16 * q^31 - 4 * q^41 + 6 * q^49 - 8 * q^59 + 20 * q^61 - 8 * q^79 + 18 * q^81 - 12 * q^89 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 14.0000i 1.92305i 0.274721 + 0.961524i $$0.411414\pi$$
−0.274721 + 0.961524i $$0.588586\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ − 6.00000i − 0.755929i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.00000i − 0.227921i
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ − 10.0000i − 1.09764i −0.835940 0.548821i $$-0.815077\pi$$
0.835940 0.548821i $$-0.184923\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.0000i 0.887357i 0.896186 + 0.443678i $$0.146327\pi$$
−0.896186 + 0.443678i $$0.853673\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ − 8.00000i − 0.626608i −0.949653 0.313304i $$-0.898564\pi$$
0.949653 0.313304i $$-0.101436\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.0000i 0.773823i 0.922117 + 0.386912i $$0.126458\pi$$
−0.922117 + 0.386912i $$0.873542\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −24.0000 −1.83533
$$172$$ 0 0
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ − 8.00000i − 0.575853i −0.957653 0.287926i $$-0.907034\pi$$
0.957653 0.287926i $$-0.0929658\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.0000i 1.40372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 24.0000i 1.66812i
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 18.0000i − 1.19470i −0.801980 0.597351i $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 4.00000i − 0.262049i −0.991379 0.131024i $$-0.958173\pi$$
0.991379 0.131024i $$-0.0418266\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 28.0000 1.81117 0.905585 0.424165i $$-0.139432\pi$$
0.905585 + 0.424165i $$0.139432\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 8.00000i 0.502956i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ −30.0000 −1.85695
$$262$$ 0 0
$$263$$ 10.0000i 0.616626i 0.951285 + 0.308313i $$0.0997645\pi$$
−0.951285 + 0.308313i $$0.900236\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ 0 0
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ − 10.0000i − 0.594438i −0.954809 0.297219i $$-0.903941\pi$$
0.954809 0.297219i $$-0.0960592\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.00000i 0.236113i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 26.0000i 1.48390i 0.670456 + 0.741949i $$0.266098\pi$$
−0.670456 + 0.741949i $$0.733902\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ − 6.00000i − 0.339140i −0.985518 0.169570i $$-0.945762\pi$$
0.985518 0.169570i $$-0.0542379\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 0 0
$$319$$ −10.0000 −0.559893
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 30.0000i 1.64399i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 32.0000i − 1.74315i −0.490261 0.871576i $$-0.663099\pi$$
0.490261 0.871576i $$-0.336901\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.0000i 0.536828i 0.963304 + 0.268414i $$0.0864995\pi$$
−0.963304 + 0.268414i $$0.913500\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −12.0000 −0.633336 −0.316668 0.948536i $$-0.602564\pi$$
−0.316668 + 0.948536i $$0.602564\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 24.0000i − 1.25279i −0.779506 0.626395i $$-0.784530\pi$$
0.779506 0.626395i $$-0.215470\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 28.0000 1.45369
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 4.00000i − 0.204390i −0.994764 0.102195i $$-0.967413\pi$$
0.994764 0.102195i $$-0.0325866\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 18.0000i 0.914991i
$$388$$ 0 0
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10.0000i 0.495682i
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.00000i 0.393654i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ − 24.0000i − 1.16692i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 64.0000i − 3.06154i
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 12.0000i − 0.561336i −0.959805 0.280668i $$-0.909444\pi$$
0.959805 0.280668i $$-0.0905560\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6.00000i 0.275880i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 42.0000i 1.92305i
$$478$$ 0 0
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 24.0000i − 1.08754i −0.839233 0.543772i $$-0.816996\pi$$
0.839233 0.543772i $$-0.183004\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 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
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 8.00000i − 0.351840i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 30.0000i 1.28271i 0.767245 + 0.641354i $$0.221627\pi$$
−0.767245 + 0.641354i $$0.778373\pi$$
$$548$$ 0 0
$$549$$ 30.0000 1.28037
$$550$$ 0 0
$$551$$ 80.0000 3.40811
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.0000i 0.508456i 0.967144 + 0.254228i $$0.0818214\pi$$
−0.967144 + 0.254228i $$0.918179\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 42.0000i 1.77009i 0.465506 + 0.885044i $$0.345872\pi$$
−0.465506 + 0.885044i $$0.654128\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 18.0000i − 0.755929i
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −20.0000 −0.829740
$$582$$ 0 0
$$583$$ 14.0000i 0.579821i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ 64.0000 2.63707
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8.00000i 0.328521i 0.986417 + 0.164260i $$0.0525237\pi$$
−0.986417 + 0.164260i $$0.947476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 42.0000i 1.70473i 0.522949 + 0.852364i $$0.324832\pi$$
−0.522949 + 0.852364i $$0.675168\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 4.00000i − 0.161558i −0.996732 0.0807792i $$-0.974259\pi$$
0.996732 0.0807792i $$-0.0257409\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 0 0
$$619$$ 36.0000 1.44696 0.723481 0.690344i $$-0.242541\pi$$
0.723481 + 0.690344i $$0.242541\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0000i 0.480770i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 0 0
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 24.0000i − 0.936329i
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 80.0000i − 3.09761i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ 8.00000i 0.308377i 0.988041 + 0.154189i $$0.0492764\pi$$
−0.988041 + 0.154189i $$0.950724\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 24.0000i 0.922395i 0.887298 + 0.461197i $$0.152580\pi$$
−0.887298 + 0.461197i $$0.847420\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ 0 0
$$693$$ − 6.00000i − 0.227921i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −46.0000 −1.73740 −0.868698 0.495342i $$-0.835043\pi$$
−0.868698 + 0.495342i $$0.835043\pi$$
$$702$$ 0 0
$$703$$ − 80.0000i − 3.01726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 28.0000i − 1.05305i
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 0 0
$$713$$ − 64.0000i − 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000i 1.03846i 0.854634 + 0.519231i $$0.173782\pi$$
−0.854634 + 0.519231i $$0.826218\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 4.00000i 0.147743i 0.997268 + 0.0738717i $$0.0235355\pi$$
−0.997268 + 0.0738717i $$0.976464\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4.00000i 0.147342i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 6.00000i − 0.220119i −0.993925 0.110059i $$-0.964896\pi$$
0.993925 0.110059i $$-0.0351041\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 30.0000i − 1.09764i
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.0000i 0.508839i 0.967094 + 0.254419i $$0.0818843\pi$$
−0.967094 + 0.254419i $$0.918116\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ − 4.00000i − 0.144810i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18.0000i 0.641631i 0.947142 + 0.320815i $$0.103957\pi$$
−0.947142 + 0.320815i $$0.896043\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −20.0000 −0.711118
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.0000i 1.34603i 0.739629 + 0.673015i $$0.235001\pi$$
−0.739629 + 0.673015i $$0.764999\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 0 0
$$803$$ − 8.00000i − 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 48.0000i − 1.67931i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26.0000 0.907406 0.453703 0.891153i $$-0.350103\pi$$
0.453703 + 0.891153i $$0.350103\pi$$
$$822$$ 0 0
$$823$$ 28.0000i 0.976019i 0.872838 + 0.488009i $$0.162277\pi$$
−0.872838 + 0.488009i $$0.837723\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 22.0000i − 0.765015i −0.923952 0.382507i $$-0.875061\pi$$
0.923952 0.382507i $$-0.124939\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −80.0000 −2.74236
$$852$$ 0 0
$$853$$ 4.00000i 0.136957i 0.997653 + 0.0684787i $$0.0218145\pi$$
−0.997653 + 0.0684787i $$0.978185\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 48.0000i − 1.63965i −0.572615 0.819824i $$-0.694071\pi$$
0.572615 0.819824i $$-0.305929\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 52.0000i 1.77010i 0.465495 + 0.885050i $$0.345876\pi$$
−0.465495 + 0.885050i $$0.654124\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 30.0000i 1.01535i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 20.0000i − 0.675352i −0.941262 0.337676i $$-0.890359\pi$$
0.941262 0.337676i $$-0.109641\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −22.0000 −0.741199 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$882$$ 0 0
$$883$$ − 56.0000i − 1.88455i −0.334840 0.942275i $$-0.608682\pi$$
0.334840 0.942275i $$-0.391318\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 26.0000i 0.872995i 0.899706 + 0.436497i $$0.143781\pi$$
−0.899706 + 0.436497i $$0.856219\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 9.00000 0.301511
$$892$$ 0 0
$$893$$ 64.0000i 2.14168i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 80.0000 2.66815
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 0 0
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ − 10.0000i − 0.330952i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 32.0000i 1.05673i
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 12.0000i 0.394132i
$$928$$ 0 0
$$929$$ −2.00000 −0.0656179 −0.0328089 0.999462i $$-0.510445\pi$$
−0.0328089 + 0.999462i $$0.510445\pi$$
$$930$$ 0 0
$$931$$ −24.0000 −0.786568
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 20.0000i 0.653372i 0.945133 + 0.326686i $$0.105932\pi$$
−0.945133 + 0.326686i $$0.894068\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ 0 0
$$943$$ − 16.0000i − 0.521032i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 52.0000i − 1.68445i −0.539130 0.842223i $$-0.681247\pi$$
0.539130 0.842223i $$-0.318753\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 6.00000i 0.193347i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 14.0000i − 0.450210i −0.974335 0.225105i $$-0.927728\pi$$
0.974335 0.225105i $$-0.0722725\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ 8.00000i 0.256468i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.00000i 0.0639857i 0.999488 + 0.0319928i $$0.0101854\pi$$
−0.999488 + 0.0319928i $$0.989815\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ 0 0
$$983$$ 44.0000i 1.40338i 0.712481 + 0.701691i $$0.247571\pi$$
−0.712481 + 0.701691i $$0.752429\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 4.00000i 0.126681i 0.997992 + 0.0633406i $$0.0201755\pi$$
−0.997992 + 0.0633406i $$0.979825\pi$$
$$998$$ 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 4400.2.b.m.4049.1 2
4.3 odd 2 2200.2.b.d.1849.2 2
5.2 odd 4 880.2.a.f.1.1 1
5.3 odd 4 4400.2.a.o.1.1 1
5.4 even 2 inner 4400.2.b.m.4049.2 2
15.2 even 4 7920.2.a.bg.1.1 1
20.3 even 4 2200.2.a.f.1.1 1
20.7 even 4 440.2.a.a.1.1 1
20.19 odd 2 2200.2.b.d.1849.1 2
40.27 even 4 3520.2.a.t.1.1 1
40.37 odd 4 3520.2.a.u.1.1 1
55.32 even 4 9680.2.a.n.1.1 1
60.47 odd 4 3960.2.a.p.1.1 1
220.87 odd 4 4840.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.a.1.1 1 20.7 even 4
880.2.a.f.1.1 1 5.2 odd 4
2200.2.a.f.1.1 1 20.3 even 4
2200.2.b.d.1849.1 2 20.19 odd 2
2200.2.b.d.1849.2 2 4.3 odd 2
3520.2.a.t.1.1 1 40.27 even 4
3520.2.a.u.1.1 1 40.37 odd 4
3960.2.a.p.1.1 1 60.47 odd 4
4400.2.a.o.1.1 1 5.3 odd 4
4400.2.b.m.4049.1 2 1.1 even 1 trivial
4400.2.b.m.4049.2 2 5.4 even 2 inner
4840.2.a.c.1.1 1 220.87 odd 4
7920.2.a.bg.1.1 1 15.2 even 4
9680.2.a.n.1.1 1 55.32 even 4