# Properties

 Label 1323.4.a.t.1.2 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ 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] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.24264 q^{2} +10.0000 q^{4} +16.9706 q^{5} +8.48528 q^{8} +O(q^{10})$$ $$q+4.24264 q^{2} +10.0000 q^{4} +16.9706 q^{5} +8.48528 q^{8} +72.0000 q^{10} +16.9706 q^{11} -29.0000 q^{13} -44.0000 q^{16} +50.9117 q^{17} -29.0000 q^{19} +169.706 q^{20} +72.0000 q^{22} +84.8528 q^{23} +163.000 q^{25} -123.037 q^{26} +271.529 q^{29} +268.000 q^{31} -254.558 q^{32} +216.000 q^{34} +83.0000 q^{37} -123.037 q^{38} +144.000 q^{40} +271.529 q^{41} -232.000 q^{43} +169.706 q^{44} +360.000 q^{46} +390.323 q^{47} +691.550 q^{50} -290.000 q^{52} +305.470 q^{53} +288.000 q^{55} +1152.00 q^{58} -288.500 q^{59} -767.000 q^{61} +1137.03 q^{62} -728.000 q^{64} -492.146 q^{65} -511.000 q^{67} +509.117 q^{68} +712.764 q^{71} -137.000 q^{73} +352.139 q^{74} -290.000 q^{76} -475.000 q^{79} -746.705 q^{80} +1152.00 q^{82} -576.999 q^{83} +864.000 q^{85} -984.293 q^{86} +144.000 q^{88} +254.558 q^{89} +848.528 q^{92} +1656.00 q^{94} -492.146 q^{95} -821.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{4}+O(q^{10})$$ 2 * q + 20 * q^4 $$2 q + 20 q^{4} + 144 q^{10} - 58 q^{13} - 88 q^{16} - 58 q^{19} + 144 q^{22} + 326 q^{25} + 536 q^{31} + 432 q^{34} + 166 q^{37} + 288 q^{40} - 464 q^{43} + 720 q^{46} - 580 q^{52} + 576 q^{55} + 2304 q^{58} - 1534 q^{61} - 1456 q^{64} - 1022 q^{67} - 274 q^{73} - 580 q^{76} - 950 q^{79} + 2304 q^{82} + 1728 q^{85} + 288 q^{88} + 3312 q^{94} - 1642 q^{97}+O(q^{100})$$ 2 * q + 20 * q^4 + 144 * q^10 - 58 * q^13 - 88 * q^16 - 58 * q^19 + 144 * q^22 + 326 * q^25 + 536 * q^31 + 432 * q^34 + 166 * q^37 + 288 * q^40 - 464 * q^43 + 720 * q^46 - 580 * q^52 + 576 * q^55 + 2304 * q^58 - 1534 * q^61 - 1456 * q^64 - 1022 * q^67 - 274 * q^73 - 580 * q^76 - 950 * q^79 + 2304 * q^82 + 1728 * q^85 + 288 * q^88 + 3312 * q^94 - 1642 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.24264 1.50000 0.750000 0.661438i $$-0.230053\pi$$
0.750000 + 0.661438i $$0.230053\pi$$
$$3$$ 0 0
$$4$$ 10.0000 1.25000
$$5$$ 16.9706 1.51789 0.758947 0.651153i $$-0.225714\pi$$
0.758947 + 0.651153i $$0.225714\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 8.48528 0.375000
$$9$$ 0 0
$$10$$ 72.0000 2.27684
$$11$$ 16.9706 0.465165 0.232583 0.972577i $$-0.425282\pi$$
0.232583 + 0.972577i $$0.425282\pi$$
$$12$$ 0 0
$$13$$ −29.0000 −0.618704 −0.309352 0.950948i $$-0.600112\pi$$
−0.309352 + 0.950948i $$0.600112\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −44.0000 −0.687500
$$17$$ 50.9117 0.726347 0.363173 0.931722i $$-0.381693\pi$$
0.363173 + 0.931722i $$0.381693\pi$$
$$18$$ 0 0
$$19$$ −29.0000 −0.350161 −0.175080 0.984554i $$-0.556019\pi$$
−0.175080 + 0.984554i $$0.556019\pi$$
$$20$$ 169.706 1.89737
$$21$$ 0 0
$$22$$ 72.0000 0.697748
$$23$$ 84.8528 0.769262 0.384631 0.923070i $$-0.374329\pi$$
0.384631 + 0.923070i $$0.374329\pi$$
$$24$$ 0 0
$$25$$ 163.000 1.30400
$$26$$ −123.037 −0.928056
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 271.529 1.73868 0.869339 0.494216i $$-0.164545\pi$$
0.869339 + 0.494216i $$0.164545\pi$$
$$30$$ 0 0
$$31$$ 268.000 1.55272 0.776358 0.630292i $$-0.217065\pi$$
0.776358 + 0.630292i $$0.217065\pi$$
$$32$$ −254.558 −1.40625
$$33$$ 0 0
$$34$$ 216.000 1.08952
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 83.0000 0.368787 0.184393 0.982853i $$-0.440968\pi$$
0.184393 + 0.982853i $$0.440968\pi$$
$$38$$ −123.037 −0.525241
$$39$$ 0 0
$$40$$ 144.000 0.569210
$$41$$ 271.529 1.03429 0.517143 0.855899i $$-0.326996\pi$$
0.517143 + 0.855899i $$0.326996\pi$$
$$42$$ 0 0
$$43$$ −232.000 −0.822783 −0.411391 0.911459i $$-0.634957\pi$$
−0.411391 + 0.911459i $$0.634957\pi$$
$$44$$ 169.706 0.581456
$$45$$ 0 0
$$46$$ 360.000 1.15389
$$47$$ 390.323 1.21137 0.605686 0.795704i $$-0.292899\pi$$
0.605686 + 0.795704i $$0.292899\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 691.550 1.95600
$$51$$ 0 0
$$52$$ −290.000 −0.773380
$$53$$ 305.470 0.791690 0.395845 0.918317i $$-0.370452\pi$$
0.395845 + 0.918317i $$0.370452\pi$$
$$54$$ 0 0
$$55$$ 288.000 0.706071
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1152.00 2.60802
$$59$$ −288.500 −0.636601 −0.318300 0.947990i $$-0.603112\pi$$
−0.318300 + 0.947990i $$0.603112\pi$$
$$60$$ 0 0
$$61$$ −767.000 −1.60991 −0.804953 0.593338i $$-0.797810\pi$$
−0.804953 + 0.593338i $$0.797810\pi$$
$$62$$ 1137.03 2.32908
$$63$$ 0 0
$$64$$ −728.000 −1.42188
$$65$$ −492.146 −0.939127
$$66$$ 0 0
$$67$$ −511.000 −0.931770 −0.465885 0.884845i $$-0.654264\pi$$
−0.465885 + 0.884845i $$0.654264\pi$$
$$68$$ 509.117 0.907934
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 712.764 1.19140 0.595701 0.803207i $$-0.296874\pi$$
0.595701 + 0.803207i $$0.296874\pi$$
$$72$$ 0 0
$$73$$ −137.000 −0.219653 −0.109826 0.993951i $$-0.535029\pi$$
−0.109826 + 0.993951i $$0.535029\pi$$
$$74$$ 352.139 0.553180
$$75$$ 0 0
$$76$$ −290.000 −0.437701
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −475.000 −0.676477 −0.338238 0.941060i $$-0.609831\pi$$
−0.338238 + 0.941060i $$0.609831\pi$$
$$80$$ −746.705 −1.04355
$$81$$ 0 0
$$82$$ 1152.00 1.55143
$$83$$ −576.999 −0.763059 −0.381529 0.924357i $$-0.624603\pi$$
−0.381529 + 0.924357i $$0.624603\pi$$
$$84$$ 0 0
$$85$$ 864.000 1.10252
$$86$$ −984.293 −1.23417
$$87$$ 0 0
$$88$$ 144.000 0.174437
$$89$$ 254.558 0.303181 0.151591 0.988443i $$-0.451560\pi$$
0.151591 + 0.988443i $$0.451560\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 848.528 0.961578
$$93$$ 0 0
$$94$$ 1656.00 1.81706
$$95$$ −492.146 −0.531507
$$96$$ 0 0
$$97$$ −821.000 −0.859381 −0.429690 0.902976i $$-0.641377\pi$$
−0.429690 + 0.902976i $$0.641377\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1630.00 1.63000
$$101$$ 543.058 0.535013 0.267506 0.963556i $$-0.413800\pi$$
0.267506 + 0.963556i $$0.413800\pi$$
$$102$$ 0 0
$$103$$ −839.000 −0.802613 −0.401306 0.915944i $$-0.631444\pi$$
−0.401306 + 0.915944i $$0.631444\pi$$
$$104$$ −246.073 −0.232014
$$105$$ 0 0
$$106$$ 1296.00 1.18753
$$107$$ −763.675 −0.689975 −0.344987 0.938607i $$-0.612117\pi$$
−0.344987 + 0.938607i $$0.612117\pi$$
$$108$$ 0 0
$$109$$ 218.000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 1221.88 1.05911
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1510.38 1.25739 0.628693 0.777654i $$-0.283590\pi$$
0.628693 + 0.777654i $$0.283590\pi$$
$$114$$ 0 0
$$115$$ 1440.00 1.16766
$$116$$ 2715.29 2.17335
$$117$$ 0 0
$$118$$ −1224.00 −0.954901
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1043.00 −0.783621
$$122$$ −3254.11 −2.41486
$$123$$ 0 0
$$124$$ 2680.00 1.94090
$$125$$ 644.881 0.461440
$$126$$ 0 0
$$127$$ 1244.00 0.869190 0.434595 0.900626i $$-0.356891\pi$$
0.434595 + 0.900626i $$0.356891\pi$$
$$128$$ −1052.17 −0.726562
$$129$$ 0 0
$$130$$ −2088.00 −1.40869
$$131$$ 2511.64 1.67514 0.837570 0.546330i $$-0.183976\pi$$
0.837570 + 0.546330i $$0.183976\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2167.99 −1.39765
$$135$$ 0 0
$$136$$ 432.000 0.272380
$$137$$ 1340.67 0.836070 0.418035 0.908431i $$-0.362719\pi$$
0.418035 + 0.908431i $$0.362719\pi$$
$$138$$ 0 0
$$139$$ −947.000 −0.577867 −0.288933 0.957349i $$-0.593301\pi$$
−0.288933 + 0.957349i $$0.593301\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3024.00 1.78710
$$143$$ −492.146 −0.287800
$$144$$ 0 0
$$145$$ 4608.00 2.63913
$$146$$ −581.242 −0.329479
$$147$$ 0 0
$$148$$ 830.000 0.460984
$$149$$ −576.999 −0.317246 −0.158623 0.987339i $$-0.550705\pi$$
−0.158623 + 0.987339i $$0.550705\pi$$
$$150$$ 0 0
$$151$$ −2311.00 −1.24547 −0.622737 0.782431i $$-0.713979\pi$$
−0.622737 + 0.782431i $$0.713979\pi$$
$$152$$ −246.073 −0.131310
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4548.11 2.35686
$$156$$ 0 0
$$157$$ −1622.00 −0.824520 −0.412260 0.911066i $$-0.635261\pi$$
−0.412260 + 0.911066i $$0.635261\pi$$
$$158$$ −2015.25 −1.01472
$$159$$ 0 0
$$160$$ −4320.00 −2.13454
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 2243.00 1.07782 0.538912 0.842362i $$-0.318836\pi$$
0.538912 + 0.842362i $$0.318836\pi$$
$$164$$ 2715.29 1.29286
$$165$$ 0 0
$$166$$ −2448.00 −1.14459
$$167$$ −2732.26 −1.26604 −0.633020 0.774135i $$-0.718185\pi$$
−0.633020 + 0.774135i $$0.718185\pi$$
$$168$$ 0 0
$$169$$ −1356.00 −0.617205
$$170$$ 3665.64 1.65378
$$171$$ 0 0
$$172$$ −2320.00 −1.02848
$$173$$ 1357.65 0.596646 0.298323 0.954465i $$-0.403573\pi$$
0.298323 + 0.954465i $$0.403573\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −746.705 −0.319801
$$177$$ 0 0
$$178$$ 1080.00 0.454772
$$179$$ −2341.94 −0.977903 −0.488952 0.872311i $$-0.662621\pi$$
−0.488952 + 0.872311i $$0.662621\pi$$
$$180$$ 0 0
$$181$$ 1591.00 0.653360 0.326680 0.945135i $$-0.394070\pi$$
0.326680 + 0.945135i $$0.394070\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 720.000 0.288473
$$185$$ 1408.56 0.559779
$$186$$ 0 0
$$187$$ 864.000 0.337871
$$188$$ 3903.23 1.51421
$$189$$ 0 0
$$190$$ −2088.00 −0.797260
$$191$$ −4768.73 −1.80656 −0.903280 0.429051i $$-0.858848\pi$$
−0.903280 + 0.429051i $$0.858848\pi$$
$$192$$ 0 0
$$193$$ −3481.00 −1.29828 −0.649140 0.760669i $$-0.724871\pi$$
−0.649140 + 0.760669i $$0.724871\pi$$
$$194$$ −3483.21 −1.28907
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2392.85 −0.865398 −0.432699 0.901538i $$-0.642439\pi$$
−0.432699 + 0.901538i $$0.642439\pi$$
$$198$$ 0 0
$$199$$ −2351.00 −0.837477 −0.418739 0.908107i $$-0.637528\pi$$
−0.418739 + 0.908107i $$0.637528\pi$$
$$200$$ 1383.10 0.489000
$$201$$ 0 0
$$202$$ 2304.00 0.802519
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4608.00 1.56994
$$206$$ −3559.58 −1.20392
$$207$$ 0 0
$$208$$ 1276.00 0.425359
$$209$$ −492.146 −0.162883
$$210$$ 0 0
$$211$$ 1703.00 0.555637 0.277818 0.960634i $$-0.410389\pi$$
0.277818 + 0.960634i $$0.410389\pi$$
$$212$$ 3054.70 0.989612
$$213$$ 0 0
$$214$$ −3240.00 −1.03496
$$215$$ −3937.17 −1.24890
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 924.896 0.287348
$$219$$ 0 0
$$220$$ 2880.00 0.882589
$$221$$ −1476.44 −0.449394
$$222$$ 0 0
$$223$$ −1388.00 −0.416804 −0.208402 0.978043i $$-0.566826\pi$$
−0.208402 + 0.978043i $$0.566826\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 6408.00 1.88608
$$227$$ 4717.82 1.37944 0.689719 0.724077i $$-0.257734\pi$$
0.689719 + 0.724077i $$0.257734\pi$$
$$228$$ 0 0
$$229$$ −434.000 −0.125238 −0.0626191 0.998038i $$-0.519945\pi$$
−0.0626191 + 0.998038i $$0.519945\pi$$
$$230$$ 6109.40 1.75149
$$231$$ 0 0
$$232$$ 2304.00 0.652004
$$233$$ 3461.99 0.973403 0.486701 0.873568i $$-0.338200\pi$$
0.486701 + 0.873568i $$0.338200\pi$$
$$234$$ 0 0
$$235$$ 6624.00 1.83873
$$236$$ −2885.00 −0.795751
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2817.11 −0.762443 −0.381222 0.924484i $$-0.624497\pi$$
−0.381222 + 0.924484i $$0.624497\pi$$
$$240$$ 0 0
$$241$$ 2095.00 0.559962 0.279981 0.960006i $$-0.409672\pi$$
0.279981 + 0.960006i $$0.409672\pi$$
$$242$$ −4425.07 −1.17543
$$243$$ 0 0
$$244$$ −7670.00 −2.01238
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 841.000 0.216646
$$248$$ 2274.06 0.582269
$$249$$ 0 0
$$250$$ 2736.00 0.692159
$$251$$ −203.647 −0.0512114 −0.0256057 0.999672i $$-0.508151\pi$$
−0.0256057 + 0.999672i $$0.508151\pi$$
$$252$$ 0 0
$$253$$ 1440.00 0.357834
$$254$$ 5277.85 1.30379
$$255$$ 0 0
$$256$$ 1360.00 0.332031
$$257$$ 1798.88 0.436619 0.218309 0.975880i $$-0.429946\pi$$
0.218309 + 0.975880i $$0.429946\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −4921.46 −1.17391
$$261$$ 0 0
$$262$$ 10656.0 2.51271
$$263$$ 373.352 0.0875357 0.0437679 0.999042i $$-0.486064\pi$$
0.0437679 + 0.999042i $$0.486064\pi$$
$$264$$ 0 0
$$265$$ 5184.00 1.20170
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −5110.00 −1.16471
$$269$$ −7484.02 −1.69631 −0.848157 0.529744i $$-0.822288\pi$$
−0.848157 + 0.529744i $$0.822288\pi$$
$$270$$ 0 0
$$271$$ 3319.00 0.743966 0.371983 0.928239i $$-0.378678\pi$$
0.371983 + 0.928239i $$0.378678\pi$$
$$272$$ −2240.11 −0.499364
$$273$$ 0 0
$$274$$ 5688.00 1.25410
$$275$$ 2766.20 0.606575
$$276$$ 0 0
$$277$$ 8354.00 1.81207 0.906035 0.423203i $$-0.139094\pi$$
0.906035 + 0.423203i $$0.139094\pi$$
$$278$$ −4017.78 −0.866800
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2579.53 −0.547621 −0.273811 0.961784i $$-0.588284\pi$$
−0.273811 + 0.961784i $$0.588284\pi$$
$$282$$ 0 0
$$283$$ 6208.00 1.30398 0.651992 0.758226i $$-0.273934\pi$$
0.651992 + 0.758226i $$0.273934\pi$$
$$284$$ 7127.64 1.48925
$$285$$ 0 0
$$286$$ −2088.00 −0.431699
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −2321.00 −0.472420
$$290$$ 19550.1 3.95869
$$291$$ 0 0
$$292$$ −1370.00 −0.274566
$$293$$ −6194.26 −1.23506 −0.617529 0.786548i $$-0.711866\pi$$
−0.617529 + 0.786548i $$0.711866\pi$$
$$294$$ 0 0
$$295$$ −4896.00 −0.966292
$$296$$ 704.278 0.138295
$$297$$ 0 0
$$298$$ −2448.00 −0.475869
$$299$$ −2460.73 −0.475946
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −9804.74 −1.86821
$$303$$ 0 0
$$304$$ 1276.00 0.240736
$$305$$ −13016.4 −2.44367
$$306$$ 0 0
$$307$$ 2320.00 0.431301 0.215650 0.976471i $$-0.430813\pi$$
0.215650 + 0.976471i $$0.430813\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 19296.0 3.53529
$$311$$ −797.616 −0.145430 −0.0727149 0.997353i $$-0.523166\pi$$
−0.0727149 + 0.997353i $$0.523166\pi$$
$$312$$ 0 0
$$313$$ −1307.00 −0.236026 −0.118013 0.993012i $$-0.537652\pi$$
−0.118013 + 0.993012i $$0.537652\pi$$
$$314$$ −6881.56 −1.23678
$$315$$ 0 0
$$316$$ −4750.00 −0.845596
$$317$$ −2070.41 −0.366832 −0.183416 0.983035i $$-0.558716\pi$$
−0.183416 + 0.983035i $$0.558716\pi$$
$$318$$ 0 0
$$319$$ 4608.00 0.808773
$$320$$ −12354.6 −2.15825
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1476.44 −0.254338
$$324$$ 0 0
$$325$$ −4727.00 −0.806790
$$326$$ 9516.24 1.61674
$$327$$ 0 0
$$328$$ 2304.00 0.387857
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −5173.00 −0.859014 −0.429507 0.903063i $$-0.641313\pi$$
−0.429507 + 0.903063i $$0.641313\pi$$
$$332$$ −5769.99 −0.953824
$$333$$ 0 0
$$334$$ −11592.0 −1.89906
$$335$$ −8671.96 −1.41433
$$336$$ 0 0
$$337$$ 2621.00 0.423665 0.211832 0.977306i $$-0.432057\pi$$
0.211832 + 0.977306i $$0.432057\pi$$
$$338$$ −5753.02 −0.925808
$$339$$ 0 0
$$340$$ 8640.00 1.37815
$$341$$ 4548.11 0.722270
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −1968.59 −0.308544
$$345$$ 0 0
$$346$$ 5760.00 0.894970
$$347$$ −7704.64 −1.19195 −0.595975 0.803003i $$-0.703234\pi$$
−0.595975 + 0.803003i $$0.703234\pi$$
$$348$$ 0 0
$$349$$ −1955.00 −0.299853 −0.149927 0.988697i $$-0.547904\pi$$
−0.149927 + 0.988697i $$0.547904\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4320.00 −0.654139
$$353$$ −1187.94 −0.179115 −0.0895576 0.995982i $$-0.528545\pi$$
−0.0895576 + 0.995982i $$0.528545\pi$$
$$354$$ 0 0
$$355$$ 12096.0 1.80842
$$356$$ 2545.58 0.378977
$$357$$ 0 0
$$358$$ −9936.00 −1.46685
$$359$$ −560.029 −0.0823320 −0.0411660 0.999152i $$-0.513107\pi$$
−0.0411660 + 0.999152i $$0.513107\pi$$
$$360$$ 0 0
$$361$$ −6018.00 −0.877387
$$362$$ 6750.04 0.980039
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2324.97 −0.333409
$$366$$ 0 0
$$367$$ −7895.00 −1.12293 −0.561465 0.827500i $$-0.689762\pi$$
−0.561465 + 0.827500i $$0.689762\pi$$
$$368$$ −3733.52 −0.528868
$$369$$ 0 0
$$370$$ 5976.00 0.839669
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 9803.00 1.36080 0.680402 0.732839i $$-0.261805\pi$$
0.680402 + 0.732839i $$0.261805\pi$$
$$374$$ 3665.64 0.506807
$$375$$ 0 0
$$376$$ 3312.00 0.454264
$$377$$ −7874.34 −1.07573
$$378$$ 0 0
$$379$$ 10505.0 1.42376 0.711881 0.702300i $$-0.247844\pi$$
0.711881 + 0.702300i $$0.247844\pi$$
$$380$$ −4921.46 −0.664384
$$381$$ 0 0
$$382$$ −20232.0 −2.70984
$$383$$ 1086.12 0.144903 0.0724516 0.997372i $$-0.476918\pi$$
0.0724516 + 0.997372i $$0.476918\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14768.6 −1.94742
$$387$$ 0 0
$$388$$ −8210.00 −1.07423
$$389$$ 2155.26 0.280915 0.140458 0.990087i $$-0.455143\pi$$
0.140458 + 0.990087i $$0.455143\pi$$
$$390$$ 0 0
$$391$$ 4320.00 0.558751
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −10152.0 −1.29810
$$395$$ −8061.02 −1.02682
$$396$$ 0 0
$$397$$ −12422.0 −1.57038 −0.785192 0.619253i $$-0.787436\pi$$
−0.785192 + 0.619253i $$0.787436\pi$$
$$398$$ −9974.45 −1.25622
$$399$$ 0 0
$$400$$ −7172.00 −0.896500
$$401$$ 15511.1 1.93164 0.965819 0.259216i $$-0.0834642\pi$$
0.965819 + 0.259216i $$0.0834642\pi$$
$$402$$ 0 0
$$403$$ −7772.00 −0.960672
$$404$$ 5430.58 0.668766
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1408.56 0.171547
$$408$$ 0 0
$$409$$ −7265.00 −0.878316 −0.439158 0.898410i $$-0.644723\pi$$
−0.439158 + 0.898410i $$0.644723\pi$$
$$410$$ 19550.1 2.35490
$$411$$ 0 0
$$412$$ −8390.00 −1.00327
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −9792.00 −1.15824
$$416$$ 7382.19 0.870053
$$417$$ 0 0
$$418$$ −2088.00 −0.244324
$$419$$ 3173.50 0.370013 0.185006 0.982737i $$-0.440769\pi$$
0.185006 + 0.982737i $$0.440769\pi$$
$$420$$ 0 0
$$421$$ 3413.00 0.395106 0.197553 0.980292i $$-0.436701\pi$$
0.197553 + 0.980292i $$0.436701\pi$$
$$422$$ 7225.22 0.833455
$$423$$ 0 0
$$424$$ 2592.00 0.296884
$$425$$ 8298.61 0.947156
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −7636.75 −0.862468
$$429$$ 0 0
$$430$$ −16704.0 −1.87335
$$431$$ −12677.0 −1.41678 −0.708388 0.705824i $$-0.750577\pi$$
−0.708388 + 0.705824i $$0.750577\pi$$
$$432$$ 0 0
$$433$$ −8642.00 −0.959141 −0.479570 0.877503i $$-0.659208\pi$$
−0.479570 + 0.877503i $$0.659208\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2180.00 0.239457
$$437$$ −2460.73 −0.269366
$$438$$ 0 0
$$439$$ −524.000 −0.0569685 −0.0284842 0.999594i $$-0.509068\pi$$
−0.0284842 + 0.999594i $$0.509068\pi$$
$$440$$ 2443.76 0.264777
$$441$$ 0 0
$$442$$ −6264.00 −0.674091
$$443$$ −18158.5 −1.94749 −0.973743 0.227649i $$-0.926896\pi$$
−0.973743 + 0.227649i $$0.926896\pi$$
$$444$$ 0 0
$$445$$ 4320.00 0.460197
$$446$$ −5888.79 −0.625206
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 3309.26 0.347825 0.173913 0.984761i $$-0.444359\pi$$
0.173913 + 0.984761i $$0.444359\pi$$
$$450$$ 0 0
$$451$$ 4608.00 0.481114
$$452$$ 15103.8 1.57173
$$453$$ 0 0
$$454$$ 20016.0 2.06916
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9466.00 −0.968930 −0.484465 0.874811i $$-0.660986\pi$$
−0.484465 + 0.874811i $$0.660986\pi$$
$$458$$ −1841.31 −0.187857
$$459$$ 0 0
$$460$$ 14400.0 1.45957
$$461$$ 3241.38 0.327475 0.163738 0.986504i $$-0.447645\pi$$
0.163738 + 0.986504i $$0.447645\pi$$
$$462$$ 0 0
$$463$$ 11315.0 1.13575 0.567875 0.823115i $$-0.307766\pi$$
0.567875 + 0.823115i $$0.307766\pi$$
$$464$$ −11947.3 −1.19534
$$465$$ 0 0
$$466$$ 14688.0 1.46010
$$467$$ −17462.7 −1.73036 −0.865180 0.501462i $$-0.832796\pi$$
−0.865180 + 0.501462i $$0.832796\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 28103.3 2.75810
$$471$$ 0 0
$$472$$ −2448.00 −0.238725
$$473$$ −3937.17 −0.382730
$$474$$ 0 0
$$475$$ −4727.00 −0.456610
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −11952.0 −1.14366
$$479$$ −8926.52 −0.851488 −0.425744 0.904844i $$-0.639988\pi$$
−0.425744 + 0.904844i $$0.639988\pi$$
$$480$$ 0 0
$$481$$ −2407.00 −0.228170
$$482$$ 8888.33 0.839943
$$483$$ 0 0
$$484$$ −10430.0 −0.979527
$$485$$ −13932.8 −1.30445
$$486$$ 0 0
$$487$$ −18493.0 −1.72073 −0.860367 0.509674i $$-0.829766\pi$$
−0.860367 + 0.509674i $$0.829766\pi$$
$$488$$ −6508.21 −0.603715
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12812.8 1.17766 0.588831 0.808256i $$-0.299588\pi$$
0.588831 + 0.808256i $$0.299588\pi$$
$$492$$ 0 0
$$493$$ 13824.0 1.26288
$$494$$ 3568.06 0.324969
$$495$$ 0 0
$$496$$ −11792.0 −1.06749
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −12256.0 −1.09951 −0.549753 0.835327i $$-0.685278\pi$$
−0.549753 + 0.835327i $$0.685278\pi$$
$$500$$ 6448.81 0.576799
$$501$$ 0 0
$$502$$ −864.000 −0.0768171
$$503$$ 7382.19 0.654385 0.327193 0.944958i $$-0.393897\pi$$
0.327193 + 0.944958i $$0.393897\pi$$
$$504$$ 0 0
$$505$$ 9216.00 0.812092
$$506$$ 6109.40 0.536751
$$507$$ 0 0
$$508$$ 12440.0 1.08649
$$509$$ 20992.6 1.82806 0.914028 0.405652i $$-0.132956\pi$$
0.914028 + 0.405652i $$0.132956\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 14187.4 1.22461
$$513$$ 0 0
$$514$$ 7632.00 0.654928
$$515$$ −14238.3 −1.21828
$$516$$ 0 0
$$517$$ 6624.00 0.563488
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −4176.00 −0.352173
$$521$$ −12269.7 −1.03176 −0.515879 0.856661i $$-0.672535\pi$$
−0.515879 + 0.856661i $$0.672535\pi$$
$$522$$ 0 0
$$523$$ −6833.00 −0.571293 −0.285646 0.958335i $$-0.592208\pi$$
−0.285646 + 0.958335i $$0.592208\pi$$
$$524$$ 25116.4 2.09392
$$525$$ 0 0
$$526$$ 1584.00 0.131304
$$527$$ 13644.3 1.12781
$$528$$ 0 0
$$529$$ −4967.00 −0.408235
$$530$$ 21993.8 1.80255
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7874.34 −0.639917
$$534$$ 0 0
$$535$$ −12960.0 −1.04731
$$536$$ −4335.98 −0.349414
$$537$$ 0 0
$$538$$ −31752.0 −2.54447
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10555.0 −0.838808 −0.419404 0.907800i $$-0.637761\pi$$
−0.419404 + 0.907800i $$0.637761\pi$$
$$542$$ 14081.3 1.11595
$$543$$ 0 0
$$544$$ −12960.0 −1.02143
$$545$$ 3699.58 0.290776
$$546$$ 0 0
$$547$$ 17291.0 1.35157 0.675786 0.737098i $$-0.263804\pi$$
0.675786 + 0.737098i $$0.263804\pi$$
$$548$$ 13406.7 1.04509
$$549$$ 0 0
$$550$$ 11736.0 0.909863
$$551$$ −7874.34 −0.608817
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 35443.0 2.71810
$$555$$ 0 0
$$556$$ −9470.00 −0.722334
$$557$$ 10335.1 0.786196 0.393098 0.919497i $$-0.371403\pi$$
0.393098 + 0.919497i $$0.371403\pi$$
$$558$$ 0 0
$$559$$ 6728.00 0.509059
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10944.0 −0.821432
$$563$$ −16427.5 −1.22973 −0.614864 0.788633i $$-0.710789\pi$$
−0.614864 + 0.788633i $$0.710789\pi$$
$$564$$ 0 0
$$565$$ 25632.0 1.90858
$$566$$ 26338.3 1.95598
$$567$$ 0 0
$$568$$ 6048.00 0.446775
$$569$$ 15697.8 1.15656 0.578282 0.815837i $$-0.303723\pi$$
0.578282 + 0.815837i $$0.303723\pi$$
$$570$$ 0 0
$$571$$ −4075.00 −0.298658 −0.149329 0.988788i $$-0.547711\pi$$
−0.149329 + 0.988788i $$0.547711\pi$$
$$572$$ −4921.46 −0.359749
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 13831.0 1.00312
$$576$$ 0 0
$$577$$ −6995.00 −0.504689 −0.252345 0.967637i $$-0.581202\pi$$
−0.252345 + 0.967637i $$0.581202\pi$$
$$578$$ −9847.17 −0.708630
$$579$$ 0 0
$$580$$ 46080.0 3.29891
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 5184.00 0.368266
$$584$$ −1162.48 −0.0823697
$$585$$ 0 0
$$586$$ −26280.0 −1.85259
$$587$$ 5583.32 0.392586 0.196293 0.980545i $$-0.437110\pi$$
0.196293 + 0.980545i $$0.437110\pi$$
$$588$$ 0 0
$$589$$ −7772.00 −0.543701
$$590$$ −20772.0 −1.44944
$$591$$ 0 0
$$592$$ −3652.00 −0.253541
$$593$$ 14968.0 1.03653 0.518266 0.855219i $$-0.326578\pi$$
0.518266 + 0.855219i $$0.326578\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5769.99 −0.396557
$$597$$ 0 0
$$598$$ −10440.0 −0.713919
$$599$$ −18192.4 −1.24094 −0.620470 0.784230i $$-0.713058\pi$$
−0.620470 + 0.784230i $$0.713058\pi$$
$$600$$ 0 0
$$601$$ 6550.00 0.444559 0.222280 0.974983i $$-0.428650\pi$$
0.222280 + 0.974983i $$0.428650\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −23110.0 −1.55684
$$605$$ −17700.3 −1.18945
$$606$$ 0 0
$$607$$ −12827.0 −0.857713 −0.428857 0.903373i $$-0.641083\pi$$
−0.428857 + 0.903373i $$0.641083\pi$$
$$608$$ 7382.19 0.492414
$$609$$ 0 0
$$610$$ −55224.0 −3.66550
$$611$$ −11319.4 −0.749480
$$612$$ 0 0
$$613$$ 18767.0 1.23653 0.618264 0.785970i $$-0.287836\pi$$
0.618264 + 0.785970i $$0.287836\pi$$
$$614$$ 9842.93 0.646951
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −7551.90 −0.492752 −0.246376 0.969174i $$-0.579240\pi$$
−0.246376 + 0.969174i $$0.579240\pi$$
$$618$$ 0 0
$$619$$ −24581.0 −1.59611 −0.798056 0.602583i $$-0.794138\pi$$
−0.798056 + 0.602583i $$0.794138\pi$$
$$620$$ 45481.1 2.94607
$$621$$ 0 0
$$622$$ −3384.00 −0.218145
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9431.00 −0.603584
$$626$$ −5545.13 −0.354038
$$627$$ 0 0
$$628$$ −16220.0 −1.03065
$$629$$ 4225.67 0.267867
$$630$$ 0 0
$$631$$ −18223.0 −1.14968 −0.574838 0.818267i $$-0.694935\pi$$
−0.574838 + 0.818267i $$0.694935\pi$$
$$632$$ −4030.51 −0.253679
$$633$$ 0 0
$$634$$ −8784.00 −0.550248
$$635$$ 21111.4 1.31934
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 19550.1 1.21316
$$639$$ 0 0
$$640$$ −17856.0 −1.10284
$$641$$ 1595.23 0.0982963 0.0491481 0.998792i $$-0.484349\pi$$
0.0491481 + 0.998792i $$0.484349\pi$$
$$642$$ 0 0
$$643$$ 26296.0 1.61277 0.806386 0.591389i $$-0.201420\pi$$
0.806386 + 0.591389i $$0.201420\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6264.00 −0.381507
$$647$$ 25659.5 1.55916 0.779582 0.626301i $$-0.215432\pi$$
0.779582 + 0.626301i $$0.215432\pi$$
$$648$$ 0 0
$$649$$ −4896.00 −0.296125
$$650$$ −20055.0 −1.21019
$$651$$ 0 0
$$652$$ 22430.0 1.34728
$$653$$ 441.235 0.0264423 0.0132212 0.999913i $$-0.495791\pi$$
0.0132212 + 0.999913i $$0.495791\pi$$
$$654$$ 0 0
$$655$$ 42624.0 2.54268
$$656$$ −11947.3 −0.711071
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 27051.1 1.59903 0.799515 0.600647i $$-0.205090\pi$$
0.799515 + 0.600647i $$0.205090\pi$$
$$660$$ 0 0
$$661$$ −623.000 −0.0366594 −0.0183297 0.999832i $$-0.505835\pi$$
−0.0183297 + 0.999832i $$0.505835\pi$$
$$662$$ −21947.2 −1.28852
$$663$$ 0 0
$$664$$ −4896.00 −0.286147
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 23040.0 1.33750
$$668$$ −27322.6 −1.58255
$$669$$ 0 0
$$670$$ −36792.0 −2.12149
$$671$$ −13016.4 −0.748872
$$672$$ 0 0
$$673$$ 27875.0 1.59659 0.798293 0.602269i $$-0.205737\pi$$
0.798293 + 0.602269i $$0.205737\pi$$
$$674$$ 11120.0 0.635497
$$675$$ 0 0
$$676$$ −13560.0 −0.771507
$$677$$ 6601.55 0.374768 0.187384 0.982287i $$-0.439999\pi$$
0.187384 + 0.982287i $$0.439999\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 7331.28 0.413444
$$681$$ 0 0
$$682$$ 19296.0 1.08340
$$683$$ −15680.8 −0.878491 −0.439245 0.898367i $$-0.644754\pi$$
−0.439245 + 0.898367i $$0.644754\pi$$
$$684$$ 0 0
$$685$$ 22752.0 1.26906
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 10208.0 0.565663
$$689$$ −8858.63 −0.489822
$$690$$ 0 0
$$691$$ 10600.0 0.583564 0.291782 0.956485i $$-0.405752\pi$$
0.291782 + 0.956485i $$0.405752\pi$$
$$692$$ 13576.5 0.745808
$$693$$ 0 0
$$694$$ −32688.0 −1.78792
$$695$$ −16071.1 −0.877140
$$696$$ 0 0
$$697$$ 13824.0 0.751250
$$698$$ −8294.36 −0.449780
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 13593.4 0.732406 0.366203 0.930535i $$-0.380658\pi$$
0.366203 + 0.930535i $$0.380658\pi$$
$$702$$ 0 0
$$703$$ −2407.00 −0.129135
$$704$$ −12354.6 −0.661407
$$705$$ 0 0
$$706$$ −5040.00 −0.268673
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −33523.0 −1.77572 −0.887858 0.460117i $$-0.847807\pi$$
−0.887858 + 0.460117i $$0.847807\pi$$
$$710$$ 51319.0 2.71263
$$711$$ 0 0
$$712$$ 2160.00 0.113693
$$713$$ 22740.6 1.19445
$$714$$ 0 0
$$715$$ −8352.00 −0.436849
$$716$$ −23419.4 −1.22238
$$717$$ 0 0
$$718$$ −2376.00 −0.123498
$$719$$ −31870.7 −1.65310 −0.826549 0.562865i $$-0.809699\pi$$
−0.826549 + 0.562865i $$0.809699\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −25532.2 −1.31608
$$723$$ 0 0
$$724$$ 15910.0 0.816700
$$725$$ 44259.2 2.26724
$$726$$ 0 0
$$727$$ 13084.0 0.667481 0.333741 0.942665i $$-0.391689\pi$$
0.333741 + 0.942665i $$0.391689\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −9864.00 −0.500114
$$731$$ −11811.5 −0.597626
$$732$$ 0 0
$$733$$ 19222.0 0.968596 0.484298 0.874903i $$-0.339075\pi$$
0.484298 + 0.874903i $$0.339075\pi$$
$$734$$ −33495.6 −1.68440
$$735$$ 0 0
$$736$$ −21600.0 −1.08178
$$737$$ −8671.96 −0.433427
$$738$$ 0 0
$$739$$ 6320.00 0.314594 0.157297 0.987551i $$-0.449722\pi$$
0.157297 + 0.987551i $$0.449722\pi$$
$$740$$ 14085.6 0.699724
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4157.79 0.205295 0.102648 0.994718i $$-0.467269\pi$$
0.102648 + 0.994718i $$0.467269\pi$$
$$744$$ 0 0
$$745$$ −9792.00 −0.481545
$$746$$ 41590.6 2.04121
$$747$$ 0 0
$$748$$ 8640.00 0.422339
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 20333.0 0.987965 0.493982 0.869472i $$-0.335541\pi$$
0.493982 + 0.869472i $$0.335541\pi$$
$$752$$ −17174.2 −0.832818
$$753$$ 0 0
$$754$$ −33408.0 −1.61359
$$755$$ −39219.0 −1.89050
$$756$$ 0 0
$$757$$ −14011.0 −0.672706 −0.336353 0.941736i $$-0.609194\pi$$
−0.336353 + 0.941736i $$0.609194\pi$$
$$758$$ 44568.9 2.13564
$$759$$ 0 0
$$760$$ −4176.00 −0.199315
$$761$$ 25981.9 1.23764 0.618820 0.785533i $$-0.287611\pi$$
0.618820 + 0.785533i $$0.287611\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −47687.3 −2.25820
$$765$$ 0 0
$$766$$ 4608.00 0.217355
$$767$$ 8366.49 0.393867
$$768$$ 0 0
$$769$$ 6289.00 0.294912 0.147456 0.989069i $$-0.452892\pi$$
0.147456 + 0.989069i $$0.452892\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −34810.0 −1.62285
$$773$$ 7229.46 0.336385 0.168192 0.985754i $$-0.446207\pi$$
0.168192 + 0.985754i $$0.446207\pi$$
$$774$$ 0 0
$$775$$ 43684.0 2.02474
$$776$$ −6966.42 −0.322268
$$777$$ 0 0
$$778$$ 9144.00 0.421373
$$779$$ −7874.34 −0.362166
$$780$$ 0 0
$$781$$ 12096.0 0.554198
$$782$$ 18328.2 0.838127
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −27526.3 −1.25153
$$786$$ 0 0
$$787$$ 25675.0 1.16292 0.581458 0.813576i $$-0.302482\pi$$
0.581458 + 0.813576i $$0.302482\pi$$
$$788$$ −23928.5 −1.08175
$$789$$ 0 0
$$790$$ −34200.0 −1.54023
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 22243.0 0.996056
$$794$$ −52702.1 −2.35558
$$795$$ 0 0
$$796$$ −23510.0 −1.04685
$$797$$ 3326.23 0.147831 0.0739154 0.997265i $$-0.476451\pi$$
0.0739154 + 0.997265i $$0.476451\pi$$
$$798$$ 0 0
$$799$$ 19872.0 0.879876
$$800$$ −41493.0 −1.83375
$$801$$ 0 0
$$802$$ 65808.0 2.89746
$$803$$ −2324.97 −0.102175
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −32973.8 −1.44101
$$807$$ 0 0
$$808$$ 4608.00 0.200630
$$809$$ 10080.5 0.438087 0.219043 0.975715i $$-0.429706\pi$$
0.219043 + 0.975715i $$0.429706\pi$$
$$810$$ 0 0
$$811$$ −14312.0 −0.619682 −0.309841 0.950788i $$-0.600276\pi$$
−0.309841 + 0.950788i $$0.600276\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 5976.00 0.257320
$$815$$ 38065.0 1.63602
$$816$$ 0 0
$$817$$ 6728.00 0.288106
$$818$$ −30822.8 −1.31747
$$819$$ 0 0
$$820$$ 46080.0 1.96242
$$821$$ 2766.20 0.117590 0.0587948 0.998270i $$-0.481274\pi$$
0.0587948 + 0.998270i $$0.481274\pi$$
$$822$$ 0 0
$$823$$ −33343.0 −1.41223 −0.706114 0.708098i $$-0.749553\pi$$
−0.706114 + 0.708098i $$0.749553\pi$$
$$824$$ −7119.15 −0.300980
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −18379.1 −0.772799 −0.386399 0.922332i $$-0.626281\pi$$
−0.386399 + 0.922332i $$0.626281\pi$$
$$828$$ 0 0
$$829$$ −3593.00 −0.150531 −0.0752654 0.997164i $$-0.523980\pi$$
−0.0752654 + 0.997164i $$0.523980\pi$$
$$830$$ −41543.9 −1.73736
$$831$$ 0 0
$$832$$ 21112.0 0.879720
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −46368.0 −1.92171
$$836$$ −4921.46 −0.203603
$$837$$ 0 0
$$838$$ 13464.0 0.555019
$$839$$ 17140.3 0.705301 0.352651 0.935755i $$-0.385280\pi$$
0.352651 + 0.935755i $$0.385280\pi$$
$$840$$ 0 0
$$841$$ 49339.0 2.02300
$$842$$ 14480.1 0.592658
$$843$$ 0 0
$$844$$ 17030.0 0.694546
$$845$$ −23012.1 −0.936852
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −13440.7 −0.544287
$$849$$ 0 0
$$850$$ 35208.0 1.42073
$$851$$ 7042.78 0.283694
$$852$$ 0 0
$$853$$ 4741.00 0.190303 0.0951517 0.995463i $$-0.469666\pi$$
0.0951517 + 0.995463i $$0.469666\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −6480.00 −0.258740
$$857$$ −11981.2 −0.477562 −0.238781 0.971073i $$-0.576748\pi$$
−0.238781 + 0.971073i $$0.576748\pi$$
$$858$$ 0 0
$$859$$ −6887.00 −0.273552 −0.136776 0.990602i $$-0.543674\pi$$
−0.136776 + 0.990602i $$0.543674\pi$$
$$860$$ −39371.7 −1.56112
$$861$$ 0 0
$$862$$ −53784.0 −2.12516
$$863$$ −8400.43 −0.331349 −0.165674 0.986181i $$-0.552980\pi$$
−0.165674 + 0.986181i $$0.552980\pi$$
$$864$$ 0 0
$$865$$ 23040.0 0.905646
$$866$$ −36664.9 −1.43871
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8061.02 −0.314674
$$870$$ 0 0
$$871$$ 14819.0 0.576490
$$872$$ 1849.79 0.0718370
$$873$$ 0 0
$$874$$ −10440.0 −0.404048
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 13475.0 0.518835 0.259418 0.965765i $$-0.416469\pi$$
0.259418 + 0.965765i $$0.416469\pi$$
$$878$$ −2223.14 −0.0854527
$$879$$ 0 0
$$880$$ −12672.0 −0.485424
$$881$$ 5243.90 0.200535 0.100268 0.994961i $$-0.468030\pi$$
0.100268 + 0.994961i $$0.468030\pi$$
$$882$$ 0 0
$$883$$ −7909.00 −0.301426 −0.150713 0.988578i $$-0.548157\pi$$
−0.150713 + 0.988578i $$0.548157\pi$$
$$884$$ −14764.4 −0.561742
$$885$$ 0 0
$$886$$ −77040.0 −2.92123
$$887$$ −35672.1 −1.35034 −0.675171 0.737662i $$-0.735930\pi$$
−0.675171 + 0.737662i $$0.735930\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 18328.2 0.690295
$$891$$ 0 0
$$892$$ −13880.0 −0.521005
$$893$$ −11319.4 −0.424175
$$894$$ 0 0
$$895$$ −39744.0 −1.48435
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 14040.0 0.521738
$$899$$ 72769.8 2.69967
$$900$$ 0 0
$$901$$ 15552.0 0.575041
$$902$$ 19550.1 0.721670
$$903$$ 0 0
$$904$$ 12816.0 0.471520
$$905$$ 27000.2 0.991730
$$906$$ 0 0
$$907$$ −16999.0 −0.622318 −0.311159 0.950358i $$-0.600717\pi$$
−0.311159 + 0.950358i $$0.600717\pi$$
$$908$$ 47178.2 1.72430
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −39032.3 −1.41954 −0.709768 0.704435i $$-0.751200\pi$$
−0.709768 + 0.704435i $$0.751200\pi$$
$$912$$ 0 0
$$913$$ −9792.00 −0.354948
$$914$$ −40160.8 −1.45339
$$915$$ 0 0
$$916$$ −4340.00 −0.156548
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −28348.0 −1.01753 −0.508767 0.860904i $$-0.669899\pi$$
−0.508767 + 0.860904i $$0.669899\pi$$
$$920$$ 12218.8 0.437872
$$921$$ 0 0
$$922$$ 13752.0 0.491213
$$923$$ −20670.1 −0.737125
$$924$$ 0 0
$$925$$ 13529.0 0.480898
$$926$$ 48005.5 1.70363
$$927$$ 0 0
$$928$$ −69120.0 −2.44502
$$929$$ −33160.5 −1.17111 −0.585554 0.810633i $$-0.699123\pi$$
−0.585554 + 0.810633i $$0.699123\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 34619.9 1.21675
$$933$$ 0 0
$$934$$ −74088.0 −2.59554
$$935$$ 14662.6 0.512853
$$936$$ 0 0
$$937$$ 133.000 0.00463706 0.00231853 0.999997i $$-0.499262\pi$$
0.00231853 + 0.999997i $$0.499262\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 66240.0 2.29842
$$941$$ 48790.4 1.69024 0.845122 0.534573i $$-0.179527\pi$$
0.845122 + 0.534573i $$0.179527\pi$$
$$942$$ 0 0
$$943$$ 23040.0 0.795637
$$944$$ 12694.0 0.437663
$$945$$ 0 0
$$946$$ −16704.0 −0.574095
$$947$$ 45328.4 1.55541 0.777705 0.628629i $$-0.216383\pi$$
0.777705 + 0.628629i $$0.216383\pi$$
$$948$$ 0 0
$$949$$ 3973.00 0.135900
$$950$$ −20055.0 −0.684915
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 43122.2 1.46576 0.732878 0.680360i $$-0.238177\pi$$
0.732878 + 0.680360i $$0.238177\pi$$
$$954$$ 0 0
$$955$$ −80928.0 −2.74217
$$956$$ −28171.1 −0.953054
$$957$$ 0 0
$$958$$ −37872.0 −1.27723
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 42033.0 1.41093
$$962$$ −10212.0 −0.342255
$$963$$ 0 0
$$964$$ 20950.0 0.699952
$$965$$ −59074.5 −1.97065
$$966$$ 0 0
$$967$$ 22061.0 0.733644 0.366822 0.930291i $$-0.380446\pi$$
0.366822 + 0.930291i $$0.380446\pi$$
$$968$$ −8850.15 −0.293858
$$969$$ 0 0
$$970$$ −59112.0 −1.95667
$$971$$ −33449.0 −1.10549 −0.552744 0.833351i $$-0.686419\pi$$
−0.552744 + 0.833351i $$0.686419\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −78459.2 −2.58110
$$975$$ 0 0
$$976$$ 33748.0 1.10681
$$977$$ 20602.3 0.674642 0.337321 0.941390i $$-0.390479\pi$$
0.337321 + 0.941390i $$0.390479\pi$$
$$978$$ 0 0
$$979$$ 4320.00 0.141029
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 54360.0 1.76649
$$983$$ −11930.3 −0.387098 −0.193549 0.981091i $$-0.562000\pi$$
−0.193549 + 0.981091i $$0.562000\pi$$
$$984$$ 0 0
$$985$$ −40608.0 −1.31358
$$986$$ 58650.3 1.89433
$$987$$ 0 0
$$988$$ 8410.00 0.270807
$$989$$ −19685.9 −0.632936
$$990$$ 0 0
$$991$$ −35017.0 −1.12245 −0.561227 0.827662i $$-0.689670\pi$$
−0.561227 + 0.827662i $$0.689670\pi$$
$$992$$ −68221.7 −2.18351
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −39897.8 −1.27120
$$996$$ 0 0
$$997$$ −13646.0 −0.433474 −0.216737 0.976230i $$-0.569541\pi$$
−0.216737 + 0.976230i $$0.569541\pi$$
$$998$$ −51997.8 −1.64926
$$999$$ 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 1323.4.a.t.1.2 2
3.2 odd 2 inner 1323.4.a.t.1.1 2
7.6 odd 2 27.4.a.c.1.2 yes 2
21.20 even 2 27.4.a.c.1.1 2
28.27 even 2 432.4.a.q.1.1 2
35.13 even 4 675.4.b.i.649.1 4
35.27 even 4 675.4.b.i.649.4 4
35.34 odd 2 675.4.a.n.1.1 2
56.13 odd 2 1728.4.a.bp.1.2 2
56.27 even 2 1728.4.a.bk.1.2 2
63.13 odd 6 81.4.c.e.55.1 4
63.20 even 6 81.4.c.e.28.2 4
63.34 odd 6 81.4.c.e.28.1 4
63.41 even 6 81.4.c.e.55.2 4
84.83 odd 2 432.4.a.q.1.2 2
105.62 odd 4 675.4.b.i.649.2 4
105.83 odd 4 675.4.b.i.649.3 4
105.104 even 2 675.4.a.n.1.2 2
168.83 odd 2 1728.4.a.bk.1.1 2
168.125 even 2 1728.4.a.bp.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
27.4.a.c.1.1 2 21.20 even 2
27.4.a.c.1.2 yes 2 7.6 odd 2
81.4.c.e.28.1 4 63.34 odd 6
81.4.c.e.28.2 4 63.20 even 6
81.4.c.e.55.1 4 63.13 odd 6
81.4.c.e.55.2 4 63.41 even 6
432.4.a.q.1.1 2 28.27 even 2
432.4.a.q.1.2 2 84.83 odd 2
675.4.a.n.1.1 2 35.34 odd 2
675.4.a.n.1.2 2 105.104 even 2
675.4.b.i.649.1 4 35.13 even 4
675.4.b.i.649.2 4 105.62 odd 4
675.4.b.i.649.3 4 105.83 odd 4
675.4.b.i.649.4 4 35.27 even 4
1323.4.a.t.1.1 2 3.2 odd 2 inner
1323.4.a.t.1.2 2 1.1 even 1 trivial
1728.4.a.bk.1.1 2 168.83 odd 2
1728.4.a.bk.1.2 2 56.27 even 2
1728.4.a.bp.1.1 2 168.125 even 2
1728.4.a.bp.1.2 2 56.13 odd 2