# Properties

 Label 1600.4.a.bx.1.1 Level $1600$ Weight $4$ Character 1600.1 Self dual yes Analytic conductor $94.403$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$94.4030560092$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000 q^{3} -4.00000 q^{7} +37.0000 q^{9} +O(q^{10})$$ $$q+8.00000 q^{3} -4.00000 q^{7} +37.0000 q^{9} +12.0000 q^{11} -58.0000 q^{13} -66.0000 q^{17} -100.000 q^{19} -32.0000 q^{21} +132.000 q^{23} +80.0000 q^{27} +90.0000 q^{29} -152.000 q^{31} +96.0000 q^{33} -34.0000 q^{37} -464.000 q^{39} -438.000 q^{41} -32.0000 q^{43} -204.000 q^{47} -327.000 q^{49} -528.000 q^{51} +222.000 q^{53} -800.000 q^{57} +420.000 q^{59} -902.000 q^{61} -148.000 q^{63} +1024.00 q^{67} +1056.00 q^{69} -432.000 q^{71} -362.000 q^{73} -48.0000 q^{77} +160.000 q^{79} -359.000 q^{81} -72.0000 q^{83} +720.000 q^{87} +810.000 q^{89} +232.000 q^{91} -1216.00 q^{93} -1106.00 q^{97} +444.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 8.00000 1.53960 0.769800 0.638285i $$-0.220356\pi$$
0.769800 + 0.638285i $$0.220356\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.00000 −0.215980 −0.107990 0.994152i $$-0.534441\pi$$
−0.107990 + 0.994152i $$0.534441\pi$$
$$8$$ 0 0
$$9$$ 37.0000 1.37037
$$10$$ 0 0
$$11$$ 12.0000 0.328921 0.164461 0.986384i $$-0.447412\pi$$
0.164461 + 0.986384i $$0.447412\pi$$
$$12$$ 0 0
$$13$$ −58.0000 −1.23741 −0.618704 0.785624i $$-0.712342\pi$$
−0.618704 + 0.785624i $$0.712342\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −66.0000 −0.941609 −0.470804 0.882238i $$-0.656036\pi$$
−0.470804 + 0.882238i $$0.656036\pi$$
$$18$$ 0 0
$$19$$ −100.000 −1.20745 −0.603726 0.797192i $$-0.706318\pi$$
−0.603726 + 0.797192i $$0.706318\pi$$
$$20$$ 0 0
$$21$$ −32.0000 −0.332522
$$22$$ 0 0
$$23$$ 132.000 1.19669 0.598346 0.801238i $$-0.295825\pi$$
0.598346 + 0.801238i $$0.295825\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 80.0000 0.570222
$$28$$ 0 0
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ 0 0
$$33$$ 96.0000 0.506408
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −34.0000 −0.151069 −0.0755347 0.997143i $$-0.524066\pi$$
−0.0755347 + 0.997143i $$0.524066\pi$$
$$38$$ 0 0
$$39$$ −464.000 −1.90511
$$40$$ 0 0
$$41$$ −438.000 −1.66839 −0.834196 0.551467i $$-0.814068\pi$$
−0.834196 + 0.551467i $$0.814068\pi$$
$$42$$ 0 0
$$43$$ −32.0000 −0.113487 −0.0567437 0.998389i $$-0.518072\pi$$
−0.0567437 + 0.998389i $$0.518072\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −204.000 −0.633116 −0.316558 0.948573i $$-0.602527\pi$$
−0.316558 + 0.948573i $$0.602527\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ −528.000 −1.44970
$$52$$ 0 0
$$53$$ 222.000 0.575359 0.287680 0.957727i $$-0.407116\pi$$
0.287680 + 0.957727i $$0.407116\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −800.000 −1.85899
$$58$$ 0 0
$$59$$ 420.000 0.926769 0.463384 0.886157i $$-0.346635\pi$$
0.463384 + 0.886157i $$0.346635\pi$$
$$60$$ 0 0
$$61$$ −902.000 −1.89327 −0.946633 0.322312i $$-0.895540\pi$$
−0.946633 + 0.322312i $$0.895540\pi$$
$$62$$ 0 0
$$63$$ −148.000 −0.295972
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1024.00 1.86719 0.933593 0.358334i $$-0.116655\pi$$
0.933593 + 0.358334i $$0.116655\pi$$
$$68$$ 0 0
$$69$$ 1056.00 1.84243
$$70$$ 0 0
$$71$$ −432.000 −0.722098 −0.361049 0.932547i $$-0.617581\pi$$
−0.361049 + 0.932547i $$0.617581\pi$$
$$72$$ 0 0
$$73$$ −362.000 −0.580396 −0.290198 0.956967i $$-0.593721\pi$$
−0.290198 + 0.956967i $$0.593721\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −48.0000 −0.0710404
$$78$$ 0 0
$$79$$ 160.000 0.227866 0.113933 0.993488i $$-0.463655\pi$$
0.113933 + 0.993488i $$0.463655\pi$$
$$80$$ 0 0
$$81$$ −359.000 −0.492455
$$82$$ 0 0
$$83$$ −72.0000 −0.0952172 −0.0476086 0.998866i $$-0.515160\pi$$
−0.0476086 + 0.998866i $$0.515160\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 720.000 0.887266
$$88$$ 0 0
$$89$$ 810.000 0.964717 0.482359 0.875974i $$-0.339780\pi$$
0.482359 + 0.875974i $$0.339780\pi$$
$$90$$ 0 0
$$91$$ 232.000 0.267255
$$92$$ 0 0
$$93$$ −1216.00 −1.35584
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1106.00 −1.15770 −0.578852 0.815433i $$-0.696499\pi$$
−0.578852 + 0.815433i $$0.696499\pi$$
$$98$$ 0 0
$$99$$ 444.000 0.450744
$$100$$ 0 0
$$101$$ 258.000 0.254178 0.127089 0.991891i $$-0.459437\pi$$
0.127089 + 0.991891i $$0.459437\pi$$
$$102$$ 0 0
$$103$$ −988.000 −0.945151 −0.472575 0.881290i $$-0.656676\pi$$
−0.472575 + 0.881290i $$0.656676\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 24.0000 0.0216838 0.0108419 0.999941i $$-0.496549\pi$$
0.0108419 + 0.999941i $$0.496549\pi$$
$$108$$ 0 0
$$109$$ −950.000 −0.834803 −0.417401 0.908722i $$-0.637059\pi$$
−0.417401 + 0.908722i $$0.637059\pi$$
$$110$$ 0 0
$$111$$ −272.000 −0.232586
$$112$$ 0 0
$$113$$ 1038.00 0.864131 0.432066 0.901842i $$-0.357785\pi$$
0.432066 + 0.901842i $$0.357785\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −2146.00 −1.69571
$$118$$ 0 0
$$119$$ 264.000 0.203368
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ 0 0
$$123$$ −3504.00 −2.56866
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −124.000 −0.0866395 −0.0433198 0.999061i $$-0.513793\pi$$
−0.0433198 + 0.999061i $$0.513793\pi$$
$$128$$ 0 0
$$129$$ −256.000 −0.174725
$$130$$ 0 0
$$131$$ 132.000 0.0880374 0.0440187 0.999031i $$-0.485984\pi$$
0.0440187 + 0.999031i $$0.485984\pi$$
$$132$$ 0 0
$$133$$ 400.000 0.260785
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1254.00 0.782018 0.391009 0.920387i $$-0.372126\pi$$
0.391009 + 0.920387i $$0.372126\pi$$
$$138$$ 0 0
$$139$$ −2860.00 −1.74519 −0.872597 0.488440i $$-0.837566\pi$$
−0.872597 + 0.488440i $$0.837566\pi$$
$$140$$ 0 0
$$141$$ −1632.00 −0.974746
$$142$$ 0 0
$$143$$ −696.000 −0.407010
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −2616.00 −1.46778
$$148$$ 0 0
$$149$$ −750.000 −0.412365 −0.206183 0.978514i $$-0.566104\pi$$
−0.206183 + 0.978514i $$0.566104\pi$$
$$150$$ 0 0
$$151$$ 448.000 0.241442 0.120721 0.992686i $$-0.461479\pi$$
0.120721 + 0.992686i $$0.461479\pi$$
$$152$$ 0 0
$$153$$ −2442.00 −1.29035
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2246.00 1.14172 0.570861 0.821047i $$-0.306610\pi$$
0.570861 + 0.821047i $$0.306610\pi$$
$$158$$ 0 0
$$159$$ 1776.00 0.885824
$$160$$ 0 0
$$161$$ −528.000 −0.258461
$$162$$ 0 0
$$163$$ 568.000 0.272940 0.136470 0.990644i $$-0.456424\pi$$
0.136470 + 0.990644i $$0.456424\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1524.00 −0.706172 −0.353086 0.935591i $$-0.614868\pi$$
−0.353086 + 0.935591i $$0.614868\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ −3700.00 −1.65466
$$172$$ 0 0
$$173$$ 3702.00 1.62692 0.813462 0.581618i $$-0.197580\pi$$
0.813462 + 0.581618i $$0.197580\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3360.00 1.42685
$$178$$ 0 0
$$179$$ 3180.00 1.32785 0.663923 0.747801i $$-0.268890\pi$$
0.663923 + 0.747801i $$0.268890\pi$$
$$180$$ 0 0
$$181$$ 2098.00 0.861564 0.430782 0.902456i $$-0.358238\pi$$
0.430782 + 0.902456i $$0.358238\pi$$
$$182$$ 0 0
$$183$$ −7216.00 −2.91487
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −792.000 −0.309715
$$188$$ 0 0
$$189$$ −320.000 −0.123156
$$190$$ 0 0
$$191$$ −4392.00 −1.66384 −0.831921 0.554894i $$-0.812759\pi$$
−0.831921 + 0.554894i $$0.812759\pi$$
$$192$$ 0 0
$$193$$ 2158.00 0.804851 0.402425 0.915453i $$-0.368167\pi$$
0.402425 + 0.915453i $$0.368167\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1074.00 −0.388423 −0.194212 0.980960i $$-0.562215\pi$$
−0.194212 + 0.980960i $$0.562215\pi$$
$$198$$ 0 0
$$199$$ −2840.00 −1.01167 −0.505835 0.862630i $$-0.668815\pi$$
−0.505835 + 0.862630i $$0.668815\pi$$
$$200$$ 0 0
$$201$$ 8192.00 2.87472
$$202$$ 0 0
$$203$$ −360.000 −0.124468
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 4884.00 1.63991
$$208$$ 0 0
$$209$$ −1200.00 −0.397157
$$210$$ 0 0
$$211$$ −2668.00 −0.870487 −0.435243 0.900313i $$-0.643338\pi$$
−0.435243 + 0.900313i $$0.643338\pi$$
$$212$$ 0 0
$$213$$ −3456.00 −1.11174
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 608.000 0.190202
$$218$$ 0 0
$$219$$ −2896.00 −0.893578
$$220$$ 0 0
$$221$$ 3828.00 1.16515
$$222$$ 0 0
$$223$$ 1772.00 0.532116 0.266058 0.963957i $$-0.414279\pi$$
0.266058 + 0.963957i $$0.414279\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2784.00 0.814011 0.407006 0.913426i $$-0.366573\pi$$
0.407006 + 0.913426i $$0.366573\pi$$
$$228$$ 0 0
$$229$$ −350.000 −0.100998 −0.0504992 0.998724i $$-0.516081\pi$$
−0.0504992 + 0.998724i $$0.516081\pi$$
$$230$$ 0 0
$$231$$ −384.000 −0.109374
$$232$$ 0 0
$$233$$ −1962.00 −0.551652 −0.275826 0.961208i $$-0.588951\pi$$
−0.275826 + 0.961208i $$0.588951\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1280.00 0.350823
$$238$$ 0 0
$$239$$ 4320.00 1.16919 0.584597 0.811324i $$-0.301252\pi$$
0.584597 + 0.811324i $$0.301252\pi$$
$$240$$ 0 0
$$241$$ −478.000 −0.127762 −0.0638811 0.997958i $$-0.520348\pi$$
−0.0638811 + 0.997958i $$0.520348\pi$$
$$242$$ 0 0
$$243$$ −5032.00 −1.32841
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5800.00 1.49411
$$248$$ 0 0
$$249$$ −576.000 −0.146596
$$250$$ 0 0
$$251$$ 2652.00 0.666903 0.333452 0.942767i $$-0.391787\pi$$
0.333452 + 0.942767i $$0.391787\pi$$
$$252$$ 0 0
$$253$$ 1584.00 0.393617
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2334.00 0.566502 0.283251 0.959046i $$-0.408587\pi$$
0.283251 + 0.959046i $$0.408587\pi$$
$$258$$ 0 0
$$259$$ 136.000 0.0326279
$$260$$ 0 0
$$261$$ 3330.00 0.789739
$$262$$ 0 0
$$263$$ −3948.00 −0.925643 −0.462822 0.886451i $$-0.653163\pi$$
−0.462822 + 0.886451i $$0.653163\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6480.00 1.48528
$$268$$ 0 0
$$269$$ −1590.00 −0.360387 −0.180193 0.983631i $$-0.557672\pi$$
−0.180193 + 0.983631i $$0.557672\pi$$
$$270$$ 0 0
$$271$$ −4952.00 −1.11001 −0.555005 0.831847i $$-0.687284\pi$$
−0.555005 + 0.831847i $$0.687284\pi$$
$$272$$ 0 0
$$273$$ 1856.00 0.411466
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1646.00 0.357034 0.178517 0.983937i $$-0.442870\pi$$
0.178517 + 0.983937i $$0.442870\pi$$
$$278$$ 0 0
$$279$$ −5624.00 −1.20681
$$280$$ 0 0
$$281$$ −1158.00 −0.245838 −0.122919 0.992417i $$-0.539226\pi$$
−0.122919 + 0.992417i $$0.539226\pi$$
$$282$$ 0 0
$$283$$ −6992.00 −1.46866 −0.734331 0.678792i $$-0.762504\pi$$
−0.734331 + 0.678792i $$0.762504\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1752.00 0.360339
$$288$$ 0 0
$$289$$ −557.000 −0.113373
$$290$$ 0 0
$$291$$ −8848.00 −1.78240
$$292$$ 0 0
$$293$$ −258.000 −0.0514421 −0.0257210 0.999669i $$-0.508188\pi$$
−0.0257210 + 0.999669i $$0.508188\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 960.000 0.187558
$$298$$ 0 0
$$299$$ −7656.00 −1.48080
$$300$$ 0 0
$$301$$ 128.000 0.0245110
$$302$$ 0 0
$$303$$ 2064.00 0.391332
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8944.00 1.66274 0.831370 0.555720i $$-0.187557\pi$$
0.831370 + 0.555720i $$0.187557\pi$$
$$308$$ 0 0
$$309$$ −7904.00 −1.45515
$$310$$ 0 0
$$311$$ −1392.00 −0.253804 −0.126902 0.991915i $$-0.540503\pi$$
−0.126902 + 0.991915i $$0.540503\pi$$
$$312$$ 0 0
$$313$$ 5878.00 1.06148 0.530742 0.847534i $$-0.321913\pi$$
0.530742 + 0.847534i $$0.321913\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10326.0 1.82955 0.914773 0.403969i $$-0.132370\pi$$
0.914773 + 0.403969i $$0.132370\pi$$
$$318$$ 0 0
$$319$$ 1080.00 0.189556
$$320$$ 0 0
$$321$$ 192.000 0.0333844
$$322$$ 0 0
$$323$$ 6600.00 1.13695
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7600.00 −1.28526
$$328$$ 0 0
$$329$$ 816.000 0.136740
$$330$$ 0 0
$$331$$ −4228.00 −0.702090 −0.351045 0.936359i $$-0.614174\pi$$
−0.351045 + 0.936359i $$0.614174\pi$$
$$332$$ 0 0
$$333$$ −1258.00 −0.207021
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1106.00 −0.178776 −0.0893882 0.995997i $$-0.528491\pi$$
−0.0893882 + 0.995997i $$0.528491\pi$$
$$338$$ 0 0
$$339$$ 8304.00 1.33042
$$340$$ 0 0
$$341$$ −1824.00 −0.289663
$$342$$ 0 0
$$343$$ 2680.00 0.421885
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9336.00 −1.44433 −0.722165 0.691720i $$-0.756853\pi$$
−0.722165 + 0.691720i $$0.756853\pi$$
$$348$$ 0 0
$$349$$ 11770.0 1.80525 0.902627 0.430424i $$-0.141636\pi$$
0.902627 + 0.430424i $$0.141636\pi$$
$$350$$ 0 0
$$351$$ −4640.00 −0.705598
$$352$$ 0 0
$$353$$ −8322.00 −1.25477 −0.627387 0.778707i $$-0.715876\pi$$
−0.627387 + 0.778707i $$0.715876\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2112.00 0.313106
$$358$$ 0 0
$$359$$ −10680.0 −1.57011 −0.785054 0.619427i $$-0.787365\pi$$
−0.785054 + 0.619427i $$0.787365\pi$$
$$360$$ 0 0
$$361$$ 3141.00 0.457938
$$362$$ 0 0
$$363$$ −9496.00 −1.37303
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −5884.00 −0.836900 −0.418450 0.908240i $$-0.637426\pi$$
−0.418450 + 0.908240i $$0.637426\pi$$
$$368$$ 0 0
$$369$$ −16206.0 −2.28632
$$370$$ 0 0
$$371$$ −888.000 −0.124266
$$372$$ 0 0
$$373$$ −2098.00 −0.291234 −0.145617 0.989341i $$-0.546517\pi$$
−0.145617 + 0.989341i $$0.546517\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5220.00 −0.713113
$$378$$ 0 0
$$379$$ 3860.00 0.523153 0.261576 0.965183i $$-0.415758\pi$$
0.261576 + 0.965183i $$0.415758\pi$$
$$380$$ 0 0
$$381$$ −992.000 −0.133390
$$382$$ 0 0
$$383$$ −9588.00 −1.27917 −0.639587 0.768718i $$-0.720895\pi$$
−0.639587 + 0.768718i $$0.720895\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1184.00 −0.155520
$$388$$ 0 0
$$389$$ 13410.0 1.74785 0.873925 0.486060i $$-0.161566\pi$$
0.873925 + 0.486060i $$0.161566\pi$$
$$390$$ 0 0
$$391$$ −8712.00 −1.12682
$$392$$ 0 0
$$393$$ 1056.00 0.135542
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13114.0 −1.65787 −0.828933 0.559348i $$-0.811052\pi$$
−0.828933 + 0.559348i $$0.811052\pi$$
$$398$$ 0 0
$$399$$ 3200.00 0.401505
$$400$$ 0 0
$$401$$ −5838.00 −0.727022 −0.363511 0.931590i $$-0.618422\pi$$
−0.363511 + 0.931590i $$0.618422\pi$$
$$402$$ 0 0
$$403$$ 8816.00 1.08972
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −408.000 −0.0496899
$$408$$ 0 0
$$409$$ 9530.00 1.15215 0.576074 0.817398i $$-0.304584\pi$$
0.576074 + 0.817398i $$0.304584\pi$$
$$410$$ 0 0
$$411$$ 10032.0 1.20400
$$412$$ 0 0
$$413$$ −1680.00 −0.200163
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −22880.0 −2.68690
$$418$$ 0 0
$$419$$ 7260.00 0.846478 0.423239 0.906018i $$-0.360893\pi$$
0.423239 + 0.906018i $$0.360893\pi$$
$$420$$ 0 0
$$421$$ −12062.0 −1.39636 −0.698178 0.715924i $$-0.746006\pi$$
−0.698178 + 0.715924i $$0.746006\pi$$
$$422$$ 0 0
$$423$$ −7548.00 −0.867604
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3608.00 0.408907
$$428$$ 0 0
$$429$$ −5568.00 −0.626633
$$430$$ 0 0
$$431$$ 13608.0 1.52082 0.760411 0.649442i $$-0.224998\pi$$
0.760411 + 0.649442i $$0.224998\pi$$
$$432$$ 0 0
$$433$$ 3838.00 0.425964 0.212982 0.977056i $$-0.431682\pi$$
0.212982 + 0.977056i $$0.431682\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −13200.0 −1.44495
$$438$$ 0 0
$$439$$ −7400.00 −0.804516 −0.402258 0.915526i $$-0.631775\pi$$
−0.402258 + 0.915526i $$0.631775\pi$$
$$440$$ 0 0
$$441$$ −12099.0 −1.30645
$$442$$ 0 0
$$443$$ −8352.00 −0.895746 −0.447873 0.894097i $$-0.647818\pi$$
−0.447873 + 0.894097i $$0.647818\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −6000.00 −0.634878
$$448$$ 0 0
$$449$$ 10770.0 1.13200 0.566000 0.824405i $$-0.308490\pi$$
0.566000 + 0.824405i $$0.308490\pi$$
$$450$$ 0 0
$$451$$ −5256.00 −0.548770
$$452$$ 0 0
$$453$$ 3584.00 0.371724
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6694.00 0.685191 0.342595 0.939483i $$-0.388694\pi$$
0.342595 + 0.939483i $$0.388694\pi$$
$$458$$ 0 0
$$459$$ −5280.00 −0.536927
$$460$$ 0 0
$$461$$ 3018.00 0.304907 0.152454 0.988311i $$-0.451283\pi$$
0.152454 + 0.988311i $$0.451283\pi$$
$$462$$ 0 0
$$463$$ 14492.0 1.45464 0.727322 0.686296i $$-0.240765\pi$$
0.727322 + 0.686296i $$0.240765\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −7776.00 −0.770515 −0.385257 0.922809i $$-0.625887\pi$$
−0.385257 + 0.922809i $$0.625887\pi$$
$$468$$ 0 0
$$469$$ −4096.00 −0.403274
$$470$$ 0 0
$$471$$ 17968.0 1.75780
$$472$$ 0 0
$$473$$ −384.000 −0.0373284
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 8214.00 0.788455
$$478$$ 0 0
$$479$$ 13680.0 1.30492 0.652458 0.757825i $$-0.273738\pi$$
0.652458 + 0.757825i $$0.273738\pi$$
$$480$$ 0 0
$$481$$ 1972.00 0.186934
$$482$$ 0 0
$$483$$ −4224.00 −0.397927
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7916.00 0.736567 0.368284 0.929714i $$-0.379946\pi$$
0.368284 + 0.929714i $$0.379946\pi$$
$$488$$ 0 0
$$489$$ 4544.00 0.420218
$$490$$ 0 0
$$491$$ 13932.0 1.28053 0.640267 0.768152i $$-0.278824\pi$$
0.640267 + 0.768152i $$0.278824\pi$$
$$492$$ 0 0
$$493$$ −5940.00 −0.542645
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1728.00 0.155959
$$498$$ 0 0
$$499$$ −8260.00 −0.741019 −0.370509 0.928829i $$-0.620817\pi$$
−0.370509 + 0.928829i $$0.620817\pi$$
$$500$$ 0 0
$$501$$ −12192.0 −1.08722
$$502$$ 0 0
$$503$$ −11148.0 −0.988200 −0.494100 0.869405i $$-0.664502\pi$$
−0.494100 + 0.869405i $$0.664502\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9336.00 0.817803
$$508$$ 0 0
$$509$$ 9690.00 0.843815 0.421907 0.906639i $$-0.361361\pi$$
0.421907 + 0.906639i $$0.361361\pi$$
$$510$$ 0 0
$$511$$ 1448.00 0.125354
$$512$$ 0 0
$$513$$ −8000.00 −0.688516
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2448.00 −0.208245
$$518$$ 0 0
$$519$$ 29616.0 2.50481
$$520$$ 0 0
$$521$$ −16038.0 −1.34863 −0.674316 0.738443i $$-0.735562\pi$$
−0.674316 + 0.738443i $$0.735562\pi$$
$$522$$ 0 0
$$523$$ −992.000 −0.0829391 −0.0414695 0.999140i $$-0.513204\pi$$
−0.0414695 + 0.999140i $$0.513204\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10032.0 0.829223
$$528$$ 0 0
$$529$$ 5257.00 0.432070
$$530$$ 0 0
$$531$$ 15540.0 1.27002
$$532$$ 0 0
$$533$$ 25404.0 2.06448
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 25440.0 2.04435
$$538$$ 0 0
$$539$$ −3924.00 −0.313578
$$540$$ 0 0
$$541$$ −7142.00 −0.567576 −0.283788 0.958887i $$-0.591591\pi$$
−0.283788 + 0.958887i $$0.591591\pi$$
$$542$$ 0 0
$$543$$ 16784.0 1.32646
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7616.00 −0.595314 −0.297657 0.954673i $$-0.596205\pi$$
−0.297657 + 0.954673i $$0.596205\pi$$
$$548$$ 0 0
$$549$$ −33374.0 −2.59448
$$550$$ 0 0
$$551$$ −9000.00 −0.695849
$$552$$ 0 0
$$553$$ −640.000 −0.0492144
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −10314.0 −0.784593 −0.392296 0.919839i $$-0.628319\pi$$
−0.392296 + 0.919839i $$0.628319\pi$$
$$558$$ 0 0
$$559$$ 1856.00 0.140430
$$560$$ 0 0
$$561$$ −6336.00 −0.476838
$$562$$ 0 0
$$563$$ 7128.00 0.533587 0.266793 0.963754i $$-0.414036\pi$$
0.266793 + 0.963754i $$0.414036\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1436.00 0.106360
$$568$$ 0 0
$$569$$ 2010.00 0.148091 0.0740453 0.997255i $$-0.476409\pi$$
0.0740453 + 0.997255i $$0.476409\pi$$
$$570$$ 0 0
$$571$$ −23188.0 −1.69945 −0.849726 0.527224i $$-0.823233\pi$$
−0.849726 + 0.527224i $$0.823233\pi$$
$$572$$ 0 0
$$573$$ −35136.0 −2.56165
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −22466.0 −1.62092 −0.810461 0.585793i $$-0.800783\pi$$
−0.810461 + 0.585793i $$0.800783\pi$$
$$578$$ 0 0
$$579$$ 17264.0 1.23915
$$580$$ 0 0
$$581$$ 288.000 0.0205650
$$582$$ 0 0
$$583$$ 2664.00 0.189248
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −22776.0 −1.60148 −0.800738 0.599015i $$-0.795559\pi$$
−0.800738 + 0.599015i $$0.795559\pi$$
$$588$$ 0 0
$$589$$ 15200.0 1.06334
$$590$$ 0 0
$$591$$ −8592.00 −0.598016
$$592$$ 0 0
$$593$$ 21198.0 1.46796 0.733978 0.679174i $$-0.237662\pi$$
0.733978 + 0.679174i $$0.237662\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −22720.0 −1.55757
$$598$$ 0 0
$$599$$ −15960.0 −1.08866 −0.544330 0.838871i $$-0.683216\pi$$
−0.544330 + 0.838871i $$0.683216\pi$$
$$600$$ 0 0
$$601$$ 5882.00 0.399221 0.199610 0.979875i $$-0.436032\pi$$
0.199610 + 0.979875i $$0.436032\pi$$
$$602$$ 0 0
$$603$$ 37888.0 2.55874
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8516.00 0.569446 0.284723 0.958610i $$-0.408098\pi$$
0.284723 + 0.958610i $$0.408098\pi$$
$$608$$ 0 0
$$609$$ −2880.00 −0.191631
$$610$$ 0 0
$$611$$ 11832.0 0.783423
$$612$$ 0 0
$$613$$ 8462.00 0.557548 0.278774 0.960357i $$-0.410072\pi$$
0.278774 + 0.960357i $$0.410072\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11094.0 0.723870 0.361935 0.932203i $$-0.382116\pi$$
0.361935 + 0.932203i $$0.382116\pi$$
$$618$$ 0 0
$$619$$ 2180.00 0.141553 0.0707767 0.997492i $$-0.477452\pi$$
0.0707767 + 0.997492i $$0.477452\pi$$
$$620$$ 0 0
$$621$$ 10560.0 0.682380
$$622$$ 0 0
$$623$$ −3240.00 −0.208359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −9600.00 −0.611463
$$628$$ 0 0
$$629$$ 2244.00 0.142248
$$630$$ 0 0
$$631$$ 26848.0 1.69382 0.846911 0.531734i $$-0.178459\pi$$
0.846911 + 0.531734i $$0.178459\pi$$
$$632$$ 0 0
$$633$$ −21344.0 −1.34020
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18966.0 1.17969
$$638$$ 0 0
$$639$$ −15984.0 −0.989542
$$640$$ 0 0
$$641$$ 26322.0 1.62193 0.810965 0.585095i $$-0.198943\pi$$
0.810965 + 0.585095i $$0.198943\pi$$
$$642$$ 0 0
$$643$$ 10168.0 0.623619 0.311809 0.950145i $$-0.399065\pi$$
0.311809 + 0.950145i $$0.399065\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23604.0 −1.43426 −0.717132 0.696937i $$-0.754546\pi$$
−0.717132 + 0.696937i $$0.754546\pi$$
$$648$$ 0 0
$$649$$ 5040.00 0.304834
$$650$$ 0 0
$$651$$ 4864.00 0.292834
$$652$$ 0 0
$$653$$ 16422.0 0.984139 0.492069 0.870556i $$-0.336241\pi$$
0.492069 + 0.870556i $$0.336241\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −13394.0 −0.795357
$$658$$ 0 0
$$659$$ −26100.0 −1.54281 −0.771405 0.636345i $$-0.780446\pi$$
−0.771405 + 0.636345i $$0.780446\pi$$
$$660$$ 0 0
$$661$$ 3058.00 0.179943 0.0899716 0.995944i $$-0.471322\pi$$
0.0899716 + 0.995944i $$0.471322\pi$$
$$662$$ 0 0
$$663$$ 30624.0 1.79387
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 11880.0 0.689648
$$668$$ 0 0
$$669$$ 14176.0 0.819246
$$670$$ 0 0
$$671$$ −10824.0 −0.622736
$$672$$ 0 0
$$673$$ −10802.0 −0.618702 −0.309351 0.950948i $$-0.600112\pi$$
−0.309351 + 0.950948i $$0.600112\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −10674.0 −0.605960 −0.302980 0.952997i $$-0.597982\pi$$
−0.302980 + 0.952997i $$0.597982\pi$$
$$678$$ 0 0
$$679$$ 4424.00 0.250041
$$680$$ 0 0
$$681$$ 22272.0 1.25325
$$682$$ 0 0
$$683$$ 28608.0 1.60272 0.801358 0.598185i $$-0.204111\pi$$
0.801358 + 0.598185i $$0.204111\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −2800.00 −0.155497
$$688$$ 0 0
$$689$$ −12876.0 −0.711954
$$690$$ 0 0
$$691$$ −2428.00 −0.133669 −0.0668346 0.997764i $$-0.521290\pi$$
−0.0668346 + 0.997764i $$0.521290\pi$$
$$692$$ 0 0
$$693$$ −1776.00 −0.0973516
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 28908.0 1.57097
$$698$$ 0 0
$$699$$ −15696.0 −0.849324
$$700$$ 0 0
$$701$$ 6618.00 0.356574 0.178287 0.983979i $$-0.442944\pi$$
0.178287 + 0.983979i $$0.442944\pi$$
$$702$$ 0 0
$$703$$ 3400.00 0.182409
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1032.00 −0.0548972
$$708$$ 0 0
$$709$$ −20510.0 −1.08642 −0.543208 0.839598i $$-0.682791\pi$$
−0.543208 + 0.839598i $$0.682791\pi$$
$$710$$ 0 0
$$711$$ 5920.00 0.312261
$$712$$ 0 0
$$713$$ −20064.0 −1.05386
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 34560.0 1.80009
$$718$$ 0 0
$$719$$ −31680.0 −1.64321 −0.821603 0.570061i $$-0.806920\pi$$
−0.821603 + 0.570061i $$0.806920\pi$$
$$720$$ 0 0
$$721$$ 3952.00 0.204133
$$722$$ 0 0
$$723$$ −3824.00 −0.196703
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 13196.0 0.673195 0.336597 0.941649i $$-0.390724\pi$$
0.336597 + 0.941649i $$0.390724\pi$$
$$728$$ 0 0
$$729$$ −30563.0 −1.55276
$$730$$ 0 0
$$731$$ 2112.00 0.106861
$$732$$ 0 0
$$733$$ 8102.00 0.408259 0.204130 0.978944i $$-0.434564\pi$$
0.204130 + 0.978944i $$0.434564\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 12288.0 0.614158
$$738$$ 0 0
$$739$$ −12580.0 −0.626201 −0.313101 0.949720i $$-0.601368\pi$$
−0.313101 + 0.949720i $$0.601368\pi$$
$$740$$ 0 0
$$741$$ 46400.0 2.30033
$$742$$ 0 0
$$743$$ 29892.0 1.47595 0.737975 0.674828i $$-0.235782\pi$$
0.737975 + 0.674828i $$0.235782\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2664.00 −0.130483
$$748$$ 0 0
$$749$$ −96.0000 −0.00468326
$$750$$ 0 0
$$751$$ 40408.0 1.96339 0.981697 0.190450i $$-0.0609946\pi$$
0.981697 + 0.190450i $$0.0609946\pi$$
$$752$$ 0 0
$$753$$ 21216.0 1.02676
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 32366.0 1.55398 0.776990 0.629513i $$-0.216746\pi$$
0.776990 + 0.629513i $$0.216746\pi$$
$$758$$ 0 0
$$759$$ 12672.0 0.606014
$$760$$ 0 0
$$761$$ −17238.0 −0.821126 −0.410563 0.911832i $$-0.634668\pi$$
−0.410563 + 0.911832i $$0.634668\pi$$
$$762$$ 0 0
$$763$$ 3800.00 0.180300
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24360.0 −1.14679
$$768$$ 0 0
$$769$$ 10850.0 0.508792 0.254396 0.967100i $$-0.418123\pi$$
0.254396 + 0.967100i $$0.418123\pi$$
$$770$$ 0 0
$$771$$ 18672.0 0.872186
$$772$$ 0 0
$$773$$ 9102.00 0.423514 0.211757 0.977322i $$-0.432081\pi$$
0.211757 + 0.977322i $$0.432081\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1088.00 0.0502340
$$778$$ 0 0
$$779$$ 43800.0 2.01450
$$780$$ 0 0
$$781$$ −5184.00 −0.237514
$$782$$ 0 0
$$783$$ 7200.00 0.328617
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25504.0 1.15517 0.577585 0.816330i $$-0.303995\pi$$
0.577585 + 0.816330i $$0.303995\pi$$
$$788$$ 0 0
$$789$$ −31584.0 −1.42512
$$790$$ 0 0
$$791$$ −4152.00 −0.186635
$$792$$ 0 0
$$793$$ 52316.0 2.34274
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14166.0 0.629593 0.314796 0.949159i $$-0.398064\pi$$
0.314796 + 0.949159i $$0.398064\pi$$
$$798$$ 0 0
$$799$$ 13464.0 0.596148
$$800$$ 0 0
$$801$$ 29970.0 1.32202
$$802$$ 0 0
$$803$$ −4344.00 −0.190905
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12720.0 −0.554852
$$808$$ 0 0
$$809$$ 33210.0 1.44327 0.721633 0.692276i $$-0.243392\pi$$
0.721633 + 0.692276i $$0.243392\pi$$
$$810$$ 0 0
$$811$$ 39212.0 1.69780 0.848902 0.528550i $$-0.177264\pi$$
0.848902 + 0.528550i $$0.177264\pi$$
$$812$$ 0 0
$$813$$ −39616.0 −1.70897
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3200.00 0.137030
$$818$$ 0 0
$$819$$ 8584.00 0.366238
$$820$$ 0 0
$$821$$ −6222.00 −0.264494 −0.132247 0.991217i $$-0.542219\pi$$
−0.132247 + 0.991217i $$0.542219\pi$$
$$822$$ 0 0
$$823$$ 31172.0 1.32028 0.660138 0.751144i $$-0.270498\pi$$
0.660138 + 0.751144i $$0.270498\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 264.000 0.0111006 0.00555029 0.999985i $$-0.498233\pi$$
0.00555029 + 0.999985i $$0.498233\pi$$
$$828$$ 0 0
$$829$$ 29050.0 1.21707 0.608533 0.793528i $$-0.291758\pi$$
0.608533 + 0.793528i $$0.291758\pi$$
$$830$$ 0 0
$$831$$ 13168.0 0.549691
$$832$$ 0 0
$$833$$ 21582.0 0.897685
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −12160.0 −0.502164
$$838$$ 0 0
$$839$$ 21720.0 0.893752 0.446876 0.894596i $$-0.352537\pi$$
0.446876 + 0.894596i $$0.352537\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ 0 0
$$843$$ −9264.00 −0.378492
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4748.00 0.192613
$$848$$ 0 0
$$849$$ −55936.0 −2.26115
$$850$$ 0 0
$$851$$ −4488.00 −0.180783
$$852$$ 0 0
$$853$$ −6658.00 −0.267252 −0.133626 0.991032i $$-0.542662\pi$$
−0.133626 + 0.991032i $$0.542662\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 13974.0 0.556993 0.278496 0.960437i $$-0.410164\pi$$
0.278496 + 0.960437i $$0.410164\pi$$
$$858$$ 0 0
$$859$$ 23780.0 0.944544 0.472272 0.881453i $$-0.343434\pi$$
0.472272 + 0.881453i $$0.343434\pi$$
$$860$$ 0 0
$$861$$ 14016.0 0.554778
$$862$$ 0 0
$$863$$ −12228.0 −0.482324 −0.241162 0.970485i $$-0.577529\pi$$
−0.241162 + 0.970485i $$0.577529\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −4456.00 −0.174549
$$868$$ 0 0
$$869$$ 1920.00 0.0749500
$$870$$ 0 0
$$871$$ −59392.0 −2.31047
$$872$$ 0 0
$$873$$ −40922.0 −1.58648
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11606.0 0.446872 0.223436 0.974719i $$-0.428273\pi$$
0.223436 + 0.974719i $$0.428273\pi$$
$$878$$ 0 0
$$879$$ −2064.00 −0.0792002
$$880$$ 0 0
$$881$$ −32958.0 −1.26037 −0.630183 0.776446i $$-0.717020\pi$$
−0.630183 + 0.776446i $$0.717020\pi$$
$$882$$ 0 0
$$883$$ −8072.00 −0.307638 −0.153819 0.988099i $$-0.549157\pi$$
−0.153819 + 0.988099i $$0.549157\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 15756.0 0.596431 0.298216 0.954498i $$-0.403609\pi$$
0.298216 + 0.954498i $$0.403609\pi$$
$$888$$ 0 0
$$889$$ 496.000 0.0187124
$$890$$ 0 0
$$891$$ −4308.00 −0.161979
$$892$$ 0 0
$$893$$ 20400.0 0.764457
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −61248.0 −2.27983
$$898$$ 0 0
$$899$$ −13680.0 −0.507512
$$900$$ 0 0
$$901$$ −14652.0 −0.541763
$$902$$ 0 0
$$903$$ 1024.00 0.0377371
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −18776.0 −0.687372 −0.343686 0.939085i $$-0.611676\pi$$
−0.343686 + 0.939085i $$0.611676\pi$$
$$908$$ 0 0
$$909$$ 9546.00 0.348318
$$910$$ 0 0
$$911$$ 20568.0 0.748022 0.374011 0.927424i $$-0.377982\pi$$
0.374011 + 0.927424i $$0.377982\pi$$
$$912$$ 0 0
$$913$$ −864.000 −0.0313190
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −528.000 −0.0190143
$$918$$ 0 0
$$919$$ 6280.00 0.225417 0.112708 0.993628i $$-0.464047\pi$$
0.112708 + 0.993628i $$0.464047\pi$$
$$920$$ 0 0
$$921$$ 71552.0 2.55996
$$922$$ 0 0
$$923$$ 25056.0 0.893530
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −36556.0 −1.29521
$$928$$ 0 0
$$929$$ −20430.0 −0.721514 −0.360757 0.932660i $$-0.617482\pi$$
−0.360757 + 0.932660i $$0.617482\pi$$
$$930$$ 0 0
$$931$$ 32700.0 1.15113
$$932$$ 0 0
$$933$$ −11136.0 −0.390757
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −8906.00 −0.310508 −0.155254 0.987875i $$-0.549620\pi$$
−0.155254 + 0.987875i $$0.549620\pi$$
$$938$$ 0 0
$$939$$ 47024.0 1.63426
$$940$$ 0 0
$$941$$ 17418.0 0.603412 0.301706 0.953401i $$-0.402444\pi$$
0.301706 + 0.953401i $$0.402444\pi$$
$$942$$ 0 0
$$943$$ −57816.0 −1.99655
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2544.00 0.0872956 0.0436478 0.999047i $$-0.486102\pi$$
0.0436478 + 0.999047i $$0.486102\pi$$
$$948$$ 0 0
$$949$$ 20996.0 0.718187
$$950$$ 0 0
$$951$$ 82608.0 2.81677
$$952$$ 0 0
$$953$$ −15402.0 −0.523525 −0.261763 0.965132i $$-0.584304\pi$$
−0.261763 + 0.965132i $$0.584304\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8640.00 0.291841
$$958$$ 0 0
$$959$$ −5016.00 −0.168900
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ 0 0
$$963$$ 888.000 0.0297148
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −49444.0 −1.64427 −0.822136 0.569291i $$-0.807218\pi$$
−0.822136 + 0.569291i $$0.807218\pi$$
$$968$$ 0 0
$$969$$ 52800.0 1.75044
$$970$$ 0 0
$$971$$ −25188.0 −0.832463 −0.416231 0.909259i $$-0.636649\pi$$
−0.416231 + 0.909259i $$0.636649\pi$$
$$972$$ 0 0
$$973$$ 11440.0 0.376927
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2946.00 −0.0964697 −0.0482348 0.998836i $$-0.515360\pi$$
−0.0482348 + 0.998836i $$0.515360\pi$$
$$978$$ 0 0
$$979$$ 9720.00 0.317316
$$980$$ 0 0
$$981$$ −35150.0 −1.14399
$$982$$ 0 0
$$983$$ 15012.0 0.487089 0.243544 0.969890i $$-0.421690\pi$$
0.243544 + 0.969890i $$0.421690\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6528.00 0.210525
$$988$$ 0 0
$$989$$ −4224.00 −0.135809
$$990$$ 0 0
$$991$$ 5128.00 0.164376 0.0821878 0.996617i $$-0.473809\pi$$
0.0821878 + 0.996617i $$0.473809\pi$$
$$992$$ 0 0
$$993$$ −33824.0 −1.08094
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −49714.0 −1.57920 −0.789598 0.613625i $$-0.789711\pi$$
−0.789598 + 0.613625i $$0.789711\pi$$
$$998$$ 0 0
$$999$$ −2720.00 −0.0861431
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 1600.4.a.bx.1.1 1
4.3 odd 2 1600.4.a.d.1.1 1
5.4 even 2 320.4.a.b.1.1 1
8.3 odd 2 50.4.a.c.1.1 1
8.5 even 2 400.4.a.b.1.1 1
20.19 odd 2 320.4.a.m.1.1 1
24.11 even 2 450.4.a.q.1.1 1
40.3 even 4 50.4.b.a.49.2 2
40.13 odd 4 400.4.c.c.49.1 2
40.19 odd 2 10.4.a.a.1.1 1
40.27 even 4 50.4.b.a.49.1 2
40.29 even 2 80.4.a.f.1.1 1
40.37 odd 4 400.4.c.c.49.2 2
56.27 even 2 2450.4.a.b.1.1 1
80.19 odd 4 1280.4.d.j.641.2 2
80.29 even 4 1280.4.d.g.641.1 2
80.59 odd 4 1280.4.d.j.641.1 2
80.69 even 4 1280.4.d.g.641.2 2
120.29 odd 2 720.4.a.j.1.1 1
120.59 even 2 90.4.a.a.1.1 1
120.83 odd 4 450.4.c.d.199.1 2
120.107 odd 4 450.4.c.d.199.2 2
280.19 even 6 490.4.e.a.361.1 2
280.59 even 6 490.4.e.a.471.1 2
280.139 even 2 490.4.a.o.1.1 1
280.179 odd 6 490.4.e.i.471.1 2
280.219 odd 6 490.4.e.i.361.1 2
360.59 even 6 810.4.e.w.541.1 2
360.139 odd 6 810.4.e.c.541.1 2
360.259 odd 6 810.4.e.c.271.1 2
360.299 even 6 810.4.e.w.271.1 2
440.219 even 2 1210.4.a.b.1.1 1
520.259 odd 2 1690.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 40.19 odd 2
50.4.a.c.1.1 1 8.3 odd 2
50.4.b.a.49.1 2 40.27 even 4
50.4.b.a.49.2 2 40.3 even 4
80.4.a.f.1.1 1 40.29 even 2
90.4.a.a.1.1 1 120.59 even 2
320.4.a.b.1.1 1 5.4 even 2
320.4.a.m.1.1 1 20.19 odd 2
400.4.a.b.1.1 1 8.5 even 2
400.4.c.c.49.1 2 40.13 odd 4
400.4.c.c.49.2 2 40.37 odd 4
450.4.a.q.1.1 1 24.11 even 2
450.4.c.d.199.1 2 120.83 odd 4
450.4.c.d.199.2 2 120.107 odd 4
490.4.a.o.1.1 1 280.139 even 2
490.4.e.a.361.1 2 280.19 even 6
490.4.e.a.471.1 2 280.59 even 6
490.4.e.i.361.1 2 280.219 odd 6
490.4.e.i.471.1 2 280.179 odd 6
720.4.a.j.1.1 1 120.29 odd 2
810.4.e.c.271.1 2 360.259 odd 6
810.4.e.c.541.1 2 360.139 odd 6
810.4.e.w.271.1 2 360.299 even 6
810.4.e.w.541.1 2 360.59 even 6
1210.4.a.b.1.1 1 440.219 even 2
1280.4.d.g.641.1 2 80.29 even 4
1280.4.d.g.641.2 2 80.69 even 4
1280.4.d.j.641.1 2 80.59 odd 4
1280.4.d.j.641.2 2 80.19 odd 4
1600.4.a.d.1.1 1 4.3 odd 2
1600.4.a.bx.1.1 1 1.1 even 1 trivial
1690.4.a.a.1.1 1 520.259 odd 2
2450.4.a.b.1.1 1 56.27 even 2