# Properties

 Label 1323.4.a.bo.1.2 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9$$ x^8 - 54*x^6 - 6*x^5 + 555*x^4 + 642*x^3 - 218*x^2 - 54*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.07765$$ 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)

 $$f(q)$$ $$=$$ $$q-3.49236 q^{2} +4.19658 q^{4} +7.87531 q^{5} +13.2829 q^{8} +O(q^{10})$$ $$q-3.49236 q^{2} +4.19658 q^{4} +7.87531 q^{5} +13.2829 q^{8} -27.5034 q^{10} -54.0035 q^{11} -48.9137 q^{13} -79.9614 q^{16} -96.9775 q^{17} -142.327 q^{19} +33.0494 q^{20} +188.600 q^{22} -103.787 q^{23} -62.9795 q^{25} +170.824 q^{26} -24.5329 q^{29} +187.379 q^{31} +172.991 q^{32} +338.680 q^{34} +146.555 q^{37} +497.058 q^{38} +104.607 q^{40} -314.180 q^{41} +173.165 q^{43} -226.630 q^{44} +362.462 q^{46} +259.235 q^{47} +219.947 q^{50} -205.270 q^{52} -620.858 q^{53} -425.295 q^{55} +85.6778 q^{58} +443.110 q^{59} +113.296 q^{61} -654.396 q^{62} +35.5454 q^{64} -385.210 q^{65} +628.975 q^{67} -406.974 q^{68} -41.3042 q^{71} +447.111 q^{73} -511.822 q^{74} -597.287 q^{76} +434.706 q^{79} -629.720 q^{80} +1097.23 q^{82} +329.158 q^{83} -763.728 q^{85} -604.756 q^{86} -717.324 q^{88} +24.8709 q^{89} -435.551 q^{92} -905.340 q^{94} -1120.87 q^{95} -499.239 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 48 q^{4}+O(q^{10})$$ 8 * q + 48 * q^4 $$8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100})$$ 8 * q + 48 * q^4 + 44 * q^10 - 84 * q^13 + 156 * q^16 + 12 * q^19 + 224 * q^22 + 408 * q^25 + 800 * q^31 - 948 * q^34 + 692 * q^37 + 96 * q^40 + 1456 * q^43 + 1524 * q^46 + 1972 * q^52 - 1280 * q^55 + 2372 * q^58 + 216 * q^61 + 4964 * q^64 + 684 * q^67 - 4564 * q^73 - 380 * q^76 + 556 * q^79 + 3340 * q^82 + 1296 * q^85 + 6696 * q^88 - 492 * q^94 + 584 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.49236 −1.23474 −0.617368 0.786675i $$-0.711801\pi$$
−0.617368 + 0.786675i $$0.711801\pi$$
$$3$$ 0 0
$$4$$ 4.19658 0.524573
$$5$$ 7.87531 0.704389 0.352195 0.935927i $$-0.385436\pi$$
0.352195 + 0.935927i $$0.385436\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 13.2829 0.587027
$$9$$ 0 0
$$10$$ −27.5034 −0.869735
$$11$$ −54.0035 −1.48024 −0.740122 0.672473i $$-0.765232\pi$$
−0.740122 + 0.672473i $$0.765232\pi$$
$$12$$ 0 0
$$13$$ −48.9137 −1.04355 −0.521777 0.853082i $$-0.674731\pi$$
−0.521777 + 0.853082i $$0.674731\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −79.9614 −1.24940
$$17$$ −96.9775 −1.38356 −0.691780 0.722109i $$-0.743173\pi$$
−0.691780 + 0.722109i $$0.743173\pi$$
$$18$$ 0 0
$$19$$ −142.327 −1.71853 −0.859266 0.511530i $$-0.829079\pi$$
−0.859266 + 0.511530i $$0.829079\pi$$
$$20$$ 33.0494 0.369503
$$21$$ 0 0
$$22$$ 188.600 1.82771
$$23$$ −103.787 −0.940918 −0.470459 0.882422i $$-0.655912\pi$$
−0.470459 + 0.882422i $$0.655912\pi$$
$$24$$ 0 0
$$25$$ −62.9795 −0.503836
$$26$$ 170.824 1.28851
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −24.5329 −0.157091 −0.0785456 0.996911i $$-0.525028\pi$$
−0.0785456 + 0.996911i $$0.525028\pi$$
$$30$$ 0 0
$$31$$ 187.379 1.08562 0.542812 0.839855i $$-0.317360\pi$$
0.542812 + 0.839855i $$0.317360\pi$$
$$32$$ 172.991 0.955647
$$33$$ 0 0
$$34$$ 338.680 1.70833
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 146.555 0.651174 0.325587 0.945512i $$-0.394438\pi$$
0.325587 + 0.945512i $$0.394438\pi$$
$$38$$ 497.058 2.12193
$$39$$ 0 0
$$40$$ 104.607 0.413496
$$41$$ −314.180 −1.19675 −0.598374 0.801217i $$-0.704186\pi$$
−0.598374 + 0.801217i $$0.704186\pi$$
$$42$$ 0 0
$$43$$ 173.165 0.614127 0.307064 0.951689i $$-0.400654\pi$$
0.307064 + 0.951689i $$0.400654\pi$$
$$44$$ −226.630 −0.776495
$$45$$ 0 0
$$46$$ 362.462 1.16179
$$47$$ 259.235 0.804537 0.402269 0.915522i $$-0.368222\pi$$
0.402269 + 0.915522i $$0.368222\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 219.947 0.622104
$$51$$ 0 0
$$52$$ −205.270 −0.547420
$$53$$ −620.858 −1.60908 −0.804541 0.593896i $$-0.797589\pi$$
−0.804541 + 0.593896i $$0.797589\pi$$
$$54$$ 0 0
$$55$$ −425.295 −1.04267
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 85.6778 0.193966
$$59$$ 443.110 0.977764 0.488882 0.872350i $$-0.337405\pi$$
0.488882 + 0.872350i $$0.337405\pi$$
$$60$$ 0 0
$$61$$ 113.296 0.237803 0.118902 0.992906i $$-0.462063\pi$$
0.118902 + 0.992906i $$0.462063\pi$$
$$62$$ −654.396 −1.34046
$$63$$ 0 0
$$64$$ 35.5454 0.0694245
$$65$$ −385.210 −0.735069
$$66$$ 0 0
$$67$$ 628.975 1.14689 0.573444 0.819245i $$-0.305607\pi$$
0.573444 + 0.819245i $$0.305607\pi$$
$$68$$ −406.974 −0.725777
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −41.3042 −0.0690410 −0.0345205 0.999404i $$-0.510990\pi$$
−0.0345205 + 0.999404i $$0.510990\pi$$
$$72$$ 0 0
$$73$$ 447.111 0.716854 0.358427 0.933558i $$-0.383313\pi$$
0.358427 + 0.933558i $$0.383313\pi$$
$$74$$ −511.822 −0.804028
$$75$$ 0 0
$$76$$ −597.287 −0.901494
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 434.706 0.619092 0.309546 0.950884i $$-0.399823\pi$$
0.309546 + 0.950884i $$0.399823\pi$$
$$80$$ −629.720 −0.880061
$$81$$ 0 0
$$82$$ 1097.23 1.47767
$$83$$ 329.158 0.435299 0.217649 0.976027i $$-0.430161\pi$$
0.217649 + 0.976027i $$0.430161\pi$$
$$84$$ 0 0
$$85$$ −763.728 −0.974564
$$86$$ −604.756 −0.758285
$$87$$ 0 0
$$88$$ −717.324 −0.868943
$$89$$ 24.8709 0.0296214 0.0148107 0.999890i $$-0.495285\pi$$
0.0148107 + 0.999890i $$0.495285\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −435.551 −0.493580
$$93$$ 0 0
$$94$$ −905.340 −0.993391
$$95$$ −1120.87 −1.21051
$$96$$ 0 0
$$97$$ −499.239 −0.522578 −0.261289 0.965261i $$-0.584148\pi$$
−0.261289 + 0.965261i $$0.584148\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −264.299 −0.264299
$$101$$ 258.076 0.254253 0.127127 0.991887i $$-0.459425\pi$$
0.127127 + 0.991887i $$0.459425\pi$$
$$102$$ 0 0
$$103$$ 960.317 0.918669 0.459334 0.888263i $$-0.348088\pi$$
0.459334 + 0.888263i $$0.348088\pi$$
$$104$$ −649.716 −0.612595
$$105$$ 0 0
$$106$$ 2168.26 1.98679
$$107$$ −2054.26 −1.85601 −0.928004 0.372570i $$-0.878477\pi$$
−0.928004 + 0.372570i $$0.878477\pi$$
$$108$$ 0 0
$$109$$ 1784.10 1.56776 0.783881 0.620911i $$-0.213237\pi$$
0.783881 + 0.620911i $$0.213237\pi$$
$$110$$ 1485.28 1.28742
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1296.36 −1.07921 −0.539607 0.841917i $$-0.681427\pi$$
−0.539607 + 0.841917i $$0.681427\pi$$
$$114$$ 0 0
$$115$$ −817.356 −0.662773
$$116$$ −102.954 −0.0824058
$$117$$ 0 0
$$118$$ −1547.50 −1.20728
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1585.38 1.19112
$$122$$ −395.669 −0.293624
$$123$$ 0 0
$$124$$ 786.352 0.569488
$$125$$ −1480.40 −1.05929
$$126$$ 0 0
$$127$$ −312.058 −0.218037 −0.109018 0.994040i $$-0.534771\pi$$
−0.109018 + 0.994040i $$0.534771\pi$$
$$128$$ −1508.06 −1.04137
$$129$$ 0 0
$$130$$ 1345.29 0.907616
$$131$$ 83.2704 0.0555372 0.0277686 0.999614i $$-0.491160\pi$$
0.0277686 + 0.999614i $$0.491160\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −2196.61 −1.41610
$$135$$ 0 0
$$136$$ −1288.14 −0.812187
$$137$$ 2909.58 1.81447 0.907233 0.420628i $$-0.138190\pi$$
0.907233 + 0.420628i $$0.138190\pi$$
$$138$$ 0 0
$$139$$ 1414.91 0.863392 0.431696 0.902019i $$-0.357915\pi$$
0.431696 + 0.902019i $$0.357915\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 144.249 0.0852474
$$143$$ 2641.51 1.54472
$$144$$ 0 0
$$145$$ −193.204 −0.110653
$$146$$ −1561.47 −0.885125
$$147$$ 0 0
$$148$$ 615.029 0.341588
$$149$$ 2186.63 1.20225 0.601125 0.799155i $$-0.294719\pi$$
0.601125 + 0.799155i $$0.294719\pi$$
$$150$$ 0 0
$$151$$ 1639.26 0.883450 0.441725 0.897151i $$-0.354367\pi$$
0.441725 + 0.897151i $$0.354367\pi$$
$$152$$ −1890.52 −1.00882
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1475.67 0.764701
$$156$$ 0 0
$$157$$ 1258.98 0.639986 0.319993 0.947420i $$-0.396320\pi$$
0.319993 + 0.947420i $$0.396320\pi$$
$$158$$ −1518.15 −0.764415
$$159$$ 0 0
$$160$$ 1362.35 0.673147
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3159.01 −1.51799 −0.758995 0.651096i $$-0.774309\pi$$
−0.758995 + 0.651096i $$0.774309\pi$$
$$164$$ −1318.48 −0.627782
$$165$$ 0 0
$$166$$ −1149.54 −0.537479
$$167$$ −1848.90 −0.856718 −0.428359 0.903609i $$-0.640908\pi$$
−0.428359 + 0.903609i $$0.640908\pi$$
$$168$$ 0 0
$$169$$ 195.548 0.0890068
$$170$$ 2667.21 1.20333
$$171$$ 0 0
$$172$$ 726.703 0.322154
$$173$$ −623.157 −0.273860 −0.136930 0.990581i $$-0.543723\pi$$
−0.136930 + 0.990581i $$0.543723\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4318.20 1.84941
$$177$$ 0 0
$$178$$ −86.8580 −0.0365746
$$179$$ −2340.66 −0.977370 −0.488685 0.872460i $$-0.662523\pi$$
−0.488685 + 0.872460i $$0.662523\pi$$
$$180$$ 0 0
$$181$$ −461.896 −0.189682 −0.0948411 0.995492i $$-0.530234\pi$$
−0.0948411 + 0.995492i $$0.530234\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −1378.60 −0.552345
$$185$$ 1154.16 0.458680
$$186$$ 0 0
$$187$$ 5237.13 2.04800
$$188$$ 1087.90 0.422038
$$189$$ 0 0
$$190$$ 3914.48 1.49467
$$191$$ −2665.79 −1.00990 −0.504948 0.863150i $$-0.668488\pi$$
−0.504948 + 0.863150i $$0.668488\pi$$
$$192$$ 0 0
$$193$$ 130.569 0.0486972 0.0243486 0.999704i $$-0.492249\pi$$
0.0243486 + 0.999704i $$0.492249\pi$$
$$194$$ 1743.52 0.645245
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3729.51 1.34882 0.674408 0.738359i $$-0.264399\pi$$
0.674408 + 0.738359i $$0.264399\pi$$
$$198$$ 0 0
$$199$$ −3772.75 −1.34394 −0.671968 0.740580i $$-0.734551\pi$$
−0.671968 + 0.740580i $$0.734551\pi$$
$$200$$ −836.551 −0.295765
$$201$$ 0 0
$$202$$ −901.296 −0.313935
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2474.27 −0.842977
$$206$$ −3353.77 −1.13431
$$207$$ 0 0
$$208$$ 3911.20 1.30381
$$209$$ 7686.17 2.54384
$$210$$ 0 0
$$211$$ 2398.28 0.782484 0.391242 0.920288i $$-0.372046\pi$$
0.391242 + 0.920288i $$0.372046\pi$$
$$212$$ −2605.48 −0.844081
$$213$$ 0 0
$$214$$ 7174.22 2.29168
$$215$$ 1363.73 0.432585
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6230.73 −1.93577
$$219$$ 0 0
$$220$$ −1784.78 −0.546955
$$221$$ 4743.53 1.44382
$$222$$ 0 0
$$223$$ −3338.23 −1.00244 −0.501220 0.865320i $$-0.667115\pi$$
−0.501220 + 0.865320i $$0.667115\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 4527.35 1.33254
$$227$$ 1269.04 0.371054 0.185527 0.982639i $$-0.440601\pi$$
0.185527 + 0.982639i $$0.440601\pi$$
$$228$$ 0 0
$$229$$ −2051.48 −0.591990 −0.295995 0.955190i $$-0.595651\pi$$
−0.295995 + 0.955190i $$0.595651\pi$$
$$230$$ 2854.50 0.818349
$$231$$ 0 0
$$232$$ −325.868 −0.0922169
$$233$$ 4950.44 1.39191 0.695953 0.718088i $$-0.254982\pi$$
0.695953 + 0.718088i $$0.254982\pi$$
$$234$$ 0 0
$$235$$ 2041.55 0.566707
$$236$$ 1859.55 0.512908
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 262.671 0.0710910 0.0355455 0.999368i $$-0.488683\pi$$
0.0355455 + 0.999368i $$0.488683\pi$$
$$240$$ 0 0
$$241$$ 2902.37 0.775761 0.387880 0.921710i $$-0.373207\pi$$
0.387880 + 0.921710i $$0.373207\pi$$
$$242$$ −5536.73 −1.47072
$$243$$ 0 0
$$244$$ 475.454 0.124745
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 6961.75 1.79338
$$248$$ 2488.94 0.637290
$$249$$ 0 0
$$250$$ 5170.08 1.30794
$$251$$ −6265.68 −1.57564 −0.787821 0.615904i $$-0.788791\pi$$
−0.787821 + 0.615904i $$0.788791\pi$$
$$252$$ 0 0
$$253$$ 5604.87 1.39279
$$254$$ 1089.82 0.269218
$$255$$ 0 0
$$256$$ 4982.33 1.21639
$$257$$ −3420.96 −0.830325 −0.415163 0.909747i $$-0.636275\pi$$
−0.415163 + 0.909747i $$0.636275\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −1616.57 −0.385597
$$261$$ 0 0
$$262$$ −290.810 −0.0685738
$$263$$ 8222.52 1.92784 0.963921 0.266188i $$-0.0857642\pi$$
0.963921 + 0.266188i $$0.0857642\pi$$
$$264$$ 0 0
$$265$$ −4889.45 −1.13342
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2639.54 0.601626
$$269$$ −5493.33 −1.24511 −0.622555 0.782576i $$-0.713905\pi$$
−0.622555 + 0.782576i $$0.713905\pi$$
$$270$$ 0 0
$$271$$ −1096.48 −0.245781 −0.122891 0.992420i $$-0.539216\pi$$
−0.122891 + 0.992420i $$0.539216\pi$$
$$272$$ 7754.45 1.72861
$$273$$ 0 0
$$274$$ −10161.3 −2.24039
$$275$$ 3401.12 0.745800
$$276$$ 0 0
$$277$$ −1555.47 −0.337398 −0.168699 0.985668i $$-0.553957\pi$$
−0.168699 + 0.985668i $$0.553957\pi$$
$$278$$ −4941.39 −1.06606
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 995.950 0.211435 0.105718 0.994396i $$-0.466286\pi$$
0.105718 + 0.994396i $$0.466286\pi$$
$$282$$ 0 0
$$283$$ −8635.47 −1.81387 −0.906936 0.421269i $$-0.861585\pi$$
−0.906936 + 0.421269i $$0.861585\pi$$
$$284$$ −173.337 −0.0362170
$$285$$ 0 0
$$286$$ −9225.11 −1.90732
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 4491.64 0.914236
$$290$$ 674.739 0.136628
$$291$$ 0 0
$$292$$ 1876.34 0.376042
$$293$$ 2180.01 0.434667 0.217333 0.976097i $$-0.430264\pi$$
0.217333 + 0.976097i $$0.430264\pi$$
$$294$$ 0 0
$$295$$ 3489.63 0.688726
$$296$$ 1946.67 0.382257
$$297$$ 0 0
$$298$$ −7636.49 −1.48446
$$299$$ 5076.61 0.981900
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −5724.88 −1.09083
$$303$$ 0 0
$$304$$ 11380.7 2.14713
$$305$$ 892.238 0.167506
$$306$$ 0 0
$$307$$ 5281.18 0.981800 0.490900 0.871216i $$-0.336668\pi$$
0.490900 + 0.871216i $$0.336668\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −5153.57 −0.944204
$$311$$ −4297.47 −0.783561 −0.391780 0.920059i $$-0.628141\pi$$
−0.391780 + 0.920059i $$0.628141\pi$$
$$312$$ 0 0
$$313$$ −2360.96 −0.426356 −0.213178 0.977013i $$-0.568381\pi$$
−0.213178 + 0.977013i $$0.568381\pi$$
$$314$$ −4396.82 −0.790213
$$315$$ 0 0
$$316$$ 1824.28 0.324759
$$317$$ 9092.66 1.61102 0.805512 0.592580i $$-0.201891\pi$$
0.805512 + 0.592580i $$0.201891\pi$$
$$318$$ 0 0
$$319$$ 1324.86 0.232533
$$320$$ 279.931 0.0489019
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 13802.5 2.37769
$$324$$ 0 0
$$325$$ 3080.56 0.525780
$$326$$ 11032.4 1.87432
$$327$$ 0 0
$$328$$ −4173.23 −0.702524
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4379.01 0.727167 0.363584 0.931562i $$-0.381553\pi$$
0.363584 + 0.931562i $$0.381553\pi$$
$$332$$ 1381.34 0.228346
$$333$$ 0 0
$$334$$ 6457.02 1.05782
$$335$$ 4953.37 0.807856
$$336$$ 0 0
$$337$$ 3314.59 0.535778 0.267889 0.963450i $$-0.413674\pi$$
0.267889 + 0.963450i $$0.413674\pi$$
$$338$$ −682.924 −0.109900
$$339$$ 0 0
$$340$$ −3205.05 −0.511230
$$341$$ −10119.1 −1.60699
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2300.14 0.360510
$$345$$ 0 0
$$346$$ 2176.29 0.338144
$$347$$ −8553.87 −1.32333 −0.661666 0.749799i $$-0.730150\pi$$
−0.661666 + 0.749799i $$0.730150\pi$$
$$348$$ 0 0
$$349$$ 9832.96 1.50816 0.754078 0.656785i $$-0.228084\pi$$
0.754078 + 0.656785i $$0.228084\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −9342.10 −1.41459
$$353$$ 6171.41 0.930513 0.465257 0.885176i $$-0.345962\pi$$
0.465257 + 0.885176i $$0.345962\pi$$
$$354$$ 0 0
$$355$$ −325.284 −0.0486317
$$356$$ 104.373 0.0155386
$$357$$ 0 0
$$358$$ 8174.43 1.20679
$$359$$ 9547.02 1.40354 0.701772 0.712401i $$-0.252392\pi$$
0.701772 + 0.712401i $$0.252392\pi$$
$$360$$ 0 0
$$361$$ 13398.0 1.95335
$$362$$ 1613.11 0.234207
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3521.14 0.504944
$$366$$ 0 0
$$367$$ −4028.68 −0.573012 −0.286506 0.958078i $$-0.592494\pi$$
−0.286506 + 0.958078i $$0.592494\pi$$
$$368$$ 8298.96 1.17558
$$369$$ 0 0
$$370$$ −4030.76 −0.566349
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 20.0509 0.00278336 0.00139168 0.999999i $$-0.499557\pi$$
0.00139168 + 0.999999i $$0.499557\pi$$
$$374$$ −18289.9 −2.52874
$$375$$ 0 0
$$376$$ 3443.39 0.472285
$$377$$ 1199.99 0.163933
$$378$$ 0 0
$$379$$ −3787.79 −0.513366 −0.256683 0.966496i $$-0.582630\pi$$
−0.256683 + 0.966496i $$0.582630\pi$$
$$380$$ −4703.82 −0.635003
$$381$$ 0 0
$$382$$ 9309.92 1.24695
$$383$$ −8001.72 −1.06754 −0.533771 0.845629i $$-0.679226\pi$$
−0.533771 + 0.845629i $$0.679226\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −455.994 −0.0601282
$$387$$ 0 0
$$388$$ −2095.10 −0.274130
$$389$$ −3484.64 −0.454186 −0.227093 0.973873i $$-0.572922\pi$$
−0.227093 + 0.973873i $$0.572922\pi$$
$$390$$ 0 0
$$391$$ 10065.0 1.30182
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −13024.8 −1.66543
$$395$$ 3423.45 0.436082
$$396$$ 0 0
$$397$$ −7037.76 −0.889710 −0.444855 0.895603i $$-0.646745\pi$$
−0.444855 + 0.895603i $$0.646745\pi$$
$$398$$ 13175.8 1.65941
$$399$$ 0 0
$$400$$ 5035.93 0.629491
$$401$$ 8389.73 1.04480 0.522398 0.852702i $$-0.325038\pi$$
0.522398 + 0.852702i $$0.325038\pi$$
$$402$$ 0 0
$$403$$ −9165.41 −1.13291
$$404$$ 1083.04 0.133374
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7914.47 −0.963896
$$408$$ 0 0
$$409$$ −13545.5 −1.63761 −0.818804 0.574074i $$-0.805362\pi$$
−0.818804 + 0.574074i $$0.805362\pi$$
$$410$$ 8641.03 1.04085
$$411$$ 0 0
$$412$$ 4030.05 0.481908
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 2592.22 0.306620
$$416$$ −8461.61 −0.997270
$$417$$ 0 0
$$418$$ −26842.9 −3.14098
$$419$$ −1450.22 −0.169089 −0.0845443 0.996420i $$-0.526943\pi$$
−0.0845443 + 0.996420i $$0.526943\pi$$
$$420$$ 0 0
$$421$$ −923.118 −0.106865 −0.0534324 0.998571i $$-0.517016\pi$$
−0.0534324 + 0.998571i $$0.517016\pi$$
$$422$$ −8375.64 −0.966161
$$423$$ 0 0
$$424$$ −8246.80 −0.944575
$$425$$ 6107.60 0.697087
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −8620.87 −0.973611
$$429$$ 0 0
$$430$$ −4762.64 −0.534128
$$431$$ 1238.21 0.138382 0.0691908 0.997603i $$-0.477958\pi$$
0.0691908 + 0.997603i $$0.477958\pi$$
$$432$$ 0 0
$$433$$ 10786.5 1.19715 0.598575 0.801067i $$-0.295734\pi$$
0.598575 + 0.801067i $$0.295734\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7487.13 0.822405
$$437$$ 14771.7 1.61700
$$438$$ 0 0
$$439$$ 3353.49 0.364586 0.182293 0.983244i $$-0.441648\pi$$
0.182293 + 0.983244i $$0.441648\pi$$
$$440$$ −5649.15 −0.612074
$$441$$ 0 0
$$442$$ −16566.1 −1.78274
$$443$$ −8090.36 −0.867685 −0.433843 0.900989i $$-0.642843\pi$$
−0.433843 + 0.900989i $$0.642843\pi$$
$$444$$ 0 0
$$445$$ 195.866 0.0208650
$$446$$ 11658.3 1.23775
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4211.19 −0.442625 −0.221312 0.975203i $$-0.571034\pi$$
−0.221312 + 0.975203i $$0.571034\pi$$
$$450$$ 0 0
$$451$$ 16966.8 1.77148
$$452$$ −5440.27 −0.566126
$$453$$ 0 0
$$454$$ −4431.95 −0.458154
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9777.62 −1.00083 −0.500413 0.865787i $$-0.666819\pi$$
−0.500413 + 0.865787i $$0.666819\pi$$
$$458$$ 7164.51 0.730951
$$459$$ 0 0
$$460$$ −3430.10 −0.347672
$$461$$ −12360.4 −1.24876 −0.624381 0.781120i $$-0.714649\pi$$
−0.624381 + 0.781120i $$0.714649\pi$$
$$462$$ 0 0
$$463$$ −16107.0 −1.61675 −0.808377 0.588666i $$-0.799653\pi$$
−0.808377 + 0.588666i $$0.799653\pi$$
$$464$$ 1961.68 0.196269
$$465$$ 0 0
$$466$$ −17288.7 −1.71864
$$467$$ −13485.4 −1.33625 −0.668125 0.744049i $$-0.732903\pi$$
−0.668125 + 0.744049i $$0.732903\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −7129.84 −0.699734
$$471$$ 0 0
$$472$$ 5885.80 0.573974
$$473$$ −9351.55 −0.909058
$$474$$ 0 0
$$475$$ 8963.69 0.865858
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −917.340 −0.0877786
$$479$$ −390.965 −0.0372936 −0.0186468 0.999826i $$-0.505936\pi$$
−0.0186468 + 0.999826i $$0.505936\pi$$
$$480$$ 0 0
$$481$$ −7168.53 −0.679536
$$482$$ −10136.1 −0.957860
$$483$$ 0 0
$$484$$ 6653.18 0.624830
$$485$$ −3931.66 −0.368098
$$486$$ 0 0
$$487$$ −15791.5 −1.46937 −0.734683 0.678410i $$-0.762669\pi$$
−0.734683 + 0.678410i $$0.762669\pi$$
$$488$$ 1504.89 0.139597
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 191.606 0.0176111 0.00880557 0.999961i $$-0.497197\pi$$
0.00880557 + 0.999961i $$0.497197\pi$$
$$492$$ 0 0
$$493$$ 2379.14 0.217345
$$494$$ −24312.9 −2.21435
$$495$$ 0 0
$$496$$ −14983.1 −1.35637
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 5621.00 0.504270 0.252135 0.967692i $$-0.418867\pi$$
0.252135 + 0.967692i $$0.418867\pi$$
$$500$$ −6212.61 −0.555672
$$501$$ 0 0
$$502$$ 21882.0 1.94550
$$503$$ −16936.3 −1.50129 −0.750647 0.660704i $$-0.770258\pi$$
−0.750647 + 0.660704i $$0.770258\pi$$
$$504$$ 0 0
$$505$$ 2032.43 0.179093
$$506$$ −19574.2 −1.71973
$$507$$ 0 0
$$508$$ −1309.58 −0.114376
$$509$$ −4225.77 −0.367984 −0.183992 0.982928i $$-0.558902\pi$$
−0.183992 + 0.982928i $$0.558902\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −5335.61 −0.460552
$$513$$ 0 0
$$514$$ 11947.2 1.02523
$$515$$ 7562.80 0.647100
$$516$$ 0 0
$$517$$ −13999.6 −1.19091
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −5116.71 −0.431505
$$521$$ 4084.11 0.343432 0.171716 0.985146i $$-0.445069\pi$$
0.171716 + 0.985146i $$0.445069\pi$$
$$522$$ 0 0
$$523$$ 16971.4 1.41894 0.709471 0.704734i $$-0.248934\pi$$
0.709471 + 0.704734i $$0.248934\pi$$
$$524$$ 349.451 0.0291333
$$525$$ 0 0
$$526$$ −28716.0 −2.38038
$$527$$ −18171.6 −1.50202
$$528$$ 0 0
$$529$$ −1395.22 −0.114673
$$530$$ 17075.7 1.39947
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 15367.7 1.24887
$$534$$ 0 0
$$535$$ −16177.9 −1.30735
$$536$$ 8354.62 0.673255
$$537$$ 0 0
$$538$$ 19184.7 1.53738
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 630.987 0.0501446 0.0250723 0.999686i $$-0.492018\pi$$
0.0250723 + 0.999686i $$0.492018\pi$$
$$542$$ 3829.32 0.303475
$$543$$ 0 0
$$544$$ −16776.2 −1.32219
$$545$$ 14050.4 1.10431
$$546$$ 0 0
$$547$$ 14377.3 1.12382 0.561910 0.827198i $$-0.310067\pi$$
0.561910 + 0.827198i $$0.310067\pi$$
$$548$$ 12210.3 0.951819
$$549$$ 0 0
$$550$$ −11877.9 −0.920866
$$551$$ 3491.70 0.269966
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 5432.27 0.416598
$$555$$ 0 0
$$556$$ 5937.80 0.452912
$$557$$ −12749.6 −0.969867 −0.484934 0.874551i $$-0.661156\pi$$
−0.484934 + 0.874551i $$0.661156\pi$$
$$558$$ 0 0
$$559$$ −8470.16 −0.640876
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −3478.21 −0.261067
$$563$$ 14080.0 1.05400 0.527001 0.849865i $$-0.323316\pi$$
0.527001 + 0.849865i $$0.323316\pi$$
$$564$$ 0 0
$$565$$ −10209.2 −0.760187
$$566$$ 30158.2 2.23965
$$567$$ 0 0
$$568$$ −548.640 −0.0405289
$$569$$ −977.908 −0.0720493 −0.0360246 0.999351i $$-0.511469\pi$$
−0.0360246 + 0.999351i $$0.511469\pi$$
$$570$$ 0 0
$$571$$ −4877.53 −0.357475 −0.178737 0.983897i $$-0.557201\pi$$
−0.178737 + 0.983897i $$0.557201\pi$$
$$572$$ 11085.3 0.810315
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6536.46 0.474068
$$576$$ 0 0
$$577$$ 8873.67 0.640235 0.320117 0.947378i $$-0.396278\pi$$
0.320117 + 0.947378i $$0.396278\pi$$
$$578$$ −15686.4 −1.12884
$$579$$ 0 0
$$580$$ −810.797 −0.0580457
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 33528.5 2.38183
$$584$$ 5938.93 0.420813
$$585$$ 0 0
$$586$$ −7613.37 −0.536699
$$587$$ −17470.2 −1.22840 −0.614201 0.789150i $$-0.710521\pi$$
−0.614201 + 0.789150i $$0.710521\pi$$
$$588$$ 0 0
$$589$$ −26669.2 −1.86568
$$590$$ −12187.1 −0.850395
$$591$$ 0 0
$$592$$ −11718.7 −0.813575
$$593$$ 17570.5 1.21675 0.608376 0.793649i $$-0.291821\pi$$
0.608376 + 0.793649i $$0.291821\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 9176.35 0.630668
$$597$$ 0 0
$$598$$ −17729.4 −1.21239
$$599$$ −2470.61 −0.168525 −0.0842623 0.996444i $$-0.526853\pi$$
−0.0842623 + 0.996444i $$0.526853\pi$$
$$600$$ 0 0
$$601$$ −7760.05 −0.526688 −0.263344 0.964702i $$-0.584825\pi$$
−0.263344 + 0.964702i $$0.584825\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 6879.28 0.463433
$$605$$ 12485.4 0.839013
$$606$$ 0 0
$$607$$ −23384.1 −1.56365 −0.781823 0.623500i $$-0.785710\pi$$
−0.781823 + 0.623500i $$0.785710\pi$$
$$608$$ −24621.3 −1.64231
$$609$$ 0 0
$$610$$ −3116.01 −0.206826
$$611$$ −12680.1 −0.839579
$$612$$ 0 0
$$613$$ 19483.4 1.28373 0.641865 0.766817i $$-0.278161\pi$$
0.641865 + 0.766817i $$0.278161\pi$$
$$614$$ −18443.8 −1.21226
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11177.6 0.729327 0.364664 0.931139i $$-0.381184\pi$$
0.364664 + 0.931139i $$0.381184\pi$$
$$618$$ 0 0
$$619$$ 6253.28 0.406043 0.203021 0.979174i $$-0.434924\pi$$
0.203021 + 0.979174i $$0.434924\pi$$
$$620$$ 6192.77 0.401141
$$621$$ 0 0
$$622$$ 15008.3 0.967491
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −3786.15 −0.242313
$$626$$ 8245.33 0.526437
$$627$$ 0 0
$$628$$ 5283.42 0.335719
$$629$$ −14212.5 −0.900938
$$630$$ 0 0
$$631$$ −19733.2 −1.24496 −0.622478 0.782637i $$-0.713874\pi$$
−0.622478 + 0.782637i $$0.713874\pi$$
$$632$$ 5774.16 0.363424
$$633$$ 0 0
$$634$$ −31754.8 −1.98919
$$635$$ −2457.55 −0.153583
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −4626.90 −0.287117
$$639$$ 0 0
$$640$$ −11876.5 −0.733528
$$641$$ 143.768 0.00885882 0.00442941 0.999990i $$-0.498590\pi$$
0.00442941 + 0.999990i $$0.498590\pi$$
$$642$$ 0 0
$$643$$ 29565.4 1.81329 0.906645 0.421894i $$-0.138634\pi$$
0.906645 + 0.421894i $$0.138634\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −48203.4 −2.93582
$$647$$ −23780.7 −1.44500 −0.722501 0.691370i $$-0.757008\pi$$
−0.722501 + 0.691370i $$0.757008\pi$$
$$648$$ 0 0
$$649$$ −23929.5 −1.44733
$$650$$ −10758.4 −0.649200
$$651$$ 0 0
$$652$$ −13257.0 −0.796296
$$653$$ −1741.59 −0.104370 −0.0521851 0.998637i $$-0.516619\pi$$
−0.0521851 + 0.998637i $$0.516619\pi$$
$$654$$ 0 0
$$655$$ 655.780 0.0391198
$$656$$ 25122.3 1.49521
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 8493.45 0.502061 0.251030 0.967979i $$-0.419231\pi$$
0.251030 + 0.967979i $$0.419231\pi$$
$$660$$ 0 0
$$661$$ 25014.7 1.47195 0.735976 0.677008i $$-0.236724\pi$$
0.735976 + 0.677008i $$0.236724\pi$$
$$662$$ −15293.1 −0.897860
$$663$$ 0 0
$$664$$ 4372.18 0.255532
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2546.20 0.147810
$$668$$ −7759.05 −0.449411
$$669$$ 0 0
$$670$$ −17299.0 −0.997489
$$671$$ −6118.36 −0.352007
$$672$$ 0 0
$$673$$ 29696.5 1.70091 0.850456 0.526046i $$-0.176326\pi$$
0.850456 + 0.526046i $$0.176326\pi$$
$$674$$ −11575.7 −0.661544
$$675$$ 0 0
$$676$$ 820.632 0.0466905
$$677$$ 3713.27 0.210801 0.105401 0.994430i $$-0.466388\pi$$
0.105401 + 0.994430i $$0.466388\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −10144.5 −0.572096
$$681$$ 0 0
$$682$$ 35339.7 1.98420
$$683$$ 24254.7 1.35883 0.679414 0.733755i $$-0.262234\pi$$
0.679414 + 0.733755i $$0.262234\pi$$
$$684$$ 0 0
$$685$$ 22913.8 1.27809
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −13846.5 −0.767289
$$689$$ 30368.4 1.67917
$$690$$ 0 0
$$691$$ 14298.7 0.787191 0.393595 0.919284i $$-0.371231\pi$$
0.393595 + 0.919284i $$0.371231\pi$$
$$692$$ −2615.13 −0.143659
$$693$$ 0 0
$$694$$ 29873.2 1.63396
$$695$$ 11142.9 0.608164
$$696$$ 0 0
$$697$$ 30468.4 1.65577
$$698$$ −34340.2 −1.86217
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −28913.3 −1.55783 −0.778916 0.627128i $$-0.784230\pi$$
−0.778916 + 0.627128i $$0.784230\pi$$
$$702$$ 0 0
$$703$$ −20858.7 −1.11906
$$704$$ −1919.57 −0.102765
$$705$$ 0 0
$$706$$ −21552.8 −1.14894
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10575.2 0.560169 0.280085 0.959975i $$-0.409637\pi$$
0.280085 + 0.959975i $$0.409637\pi$$
$$710$$ 1136.01 0.0600473
$$711$$ 0 0
$$712$$ 330.357 0.0173886
$$713$$ −19447.6 −1.02148
$$714$$ 0 0
$$715$$ 20802.7 1.08808
$$716$$ −9822.77 −0.512701
$$717$$ 0 0
$$718$$ −33341.6 −1.73301
$$719$$ −5347.27 −0.277357 −0.138678 0.990337i $$-0.544285\pi$$
−0.138678 + 0.990337i $$0.544285\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −46790.7 −2.41187
$$723$$ 0 0
$$724$$ −1938.39 −0.0995021
$$725$$ 1545.07 0.0791482
$$726$$ 0 0
$$727$$ −19629.1 −1.00138 −0.500689 0.865627i $$-0.666920\pi$$
−0.500689 + 0.865627i $$0.666920\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −12297.1 −0.623473
$$731$$ −16793.2 −0.849682
$$732$$ 0 0
$$733$$ 2404.87 0.121181 0.0605906 0.998163i $$-0.480702\pi$$
0.0605906 + 0.998163i $$0.480702\pi$$
$$734$$ 14069.6 0.707519
$$735$$ 0 0
$$736$$ −17954.2 −0.899186
$$737$$ −33966.9 −1.69767
$$738$$ 0 0
$$739$$ −19309.0 −0.961153 −0.480577 0.876953i $$-0.659573\pi$$
−0.480577 + 0.876953i $$0.659573\pi$$
$$740$$ 4843.54 0.240611
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −801.314 −0.0395658 −0.0197829 0.999804i $$-0.506297\pi$$
−0.0197829 + 0.999804i $$0.506297\pi$$
$$744$$ 0 0
$$745$$ 17220.4 0.846852
$$746$$ −70.0249 −0.00343672
$$747$$ 0 0
$$748$$ 21978.0 1.07433
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 14872.7 0.722654 0.361327 0.932439i $$-0.382324\pi$$
0.361327 + 0.932439i $$0.382324\pi$$
$$752$$ −20728.7 −1.00519
$$753$$ 0 0
$$754$$ −4190.81 −0.202414
$$755$$ 12909.7 0.622292
$$756$$ 0 0
$$757$$ 9127.52 0.438237 0.219118 0.975698i $$-0.429682\pi$$
0.219118 + 0.975698i $$0.429682\pi$$
$$758$$ 13228.3 0.633871
$$759$$ 0 0
$$760$$ −14888.4 −0.710605
$$761$$ 30994.4 1.47641 0.738203 0.674578i $$-0.235675\pi$$
0.738203 + 0.674578i $$0.235675\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −11187.2 −0.529764
$$765$$ 0 0
$$766$$ 27944.9 1.31813
$$767$$ −21674.2 −1.02035
$$768$$ 0 0
$$769$$ 12979.5 0.608653 0.304326 0.952568i $$-0.401569\pi$$
0.304326 + 0.952568i $$0.401569\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 547.944 0.0255452
$$773$$ −28808.1 −1.34043 −0.670217 0.742165i $$-0.733799\pi$$
−0.670217 + 0.742165i $$0.733799\pi$$
$$774$$ 0 0
$$775$$ −11801.1 −0.546976
$$776$$ −6631.35 −0.306767
$$777$$ 0 0
$$778$$ 12169.6 0.560799
$$779$$ 44716.4 2.05665
$$780$$ 0 0
$$781$$ 2230.57 0.102197
$$782$$ −35150.7 −1.60740
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9914.88 0.450799
$$786$$ 0 0
$$787$$ −3658.15 −0.165691 −0.0828456 0.996562i $$-0.526401\pi$$
−0.0828456 + 0.996562i $$0.526401\pi$$
$$788$$ 15651.2 0.707552
$$789$$ 0 0
$$790$$ −11955.9 −0.538446
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −5541.70 −0.248161
$$794$$ 24578.4 1.09856
$$795$$ 0 0
$$796$$ −15832.7 −0.704992
$$797$$ 20171.4 0.896497 0.448249 0.893909i $$-0.352048\pi$$
0.448249 + 0.893909i $$0.352048\pi$$
$$798$$ 0 0
$$799$$ −25139.9 −1.11312
$$800$$ −10894.9 −0.481489
$$801$$ 0 0
$$802$$ −29300.0 −1.29005
$$803$$ −24145.6 −1.06112
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 32008.9 1.39884
$$807$$ 0 0
$$808$$ 3428.01 0.149253
$$809$$ 30508.2 1.32585 0.662924 0.748686i $$-0.269315\pi$$
0.662924 + 0.748686i $$0.269315\pi$$
$$810$$ 0 0
$$811$$ 2550.91 0.110449 0.0552247 0.998474i $$-0.482412\pi$$
0.0552247 + 0.998474i $$0.482412\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 27640.2 1.19016
$$815$$ −24878.1 −1.06926
$$816$$ 0 0
$$817$$ −24646.1 −1.05540
$$818$$ 47305.7 2.02201
$$819$$ 0 0
$$820$$ −10383.5 −0.442202
$$821$$ 32657.1 1.38824 0.694118 0.719861i $$-0.255794\pi$$
0.694118 + 0.719861i $$0.255794\pi$$
$$822$$ 0 0
$$823$$ 30390.4 1.28717 0.643587 0.765373i $$-0.277446\pi$$
0.643587 + 0.765373i $$0.277446\pi$$
$$824$$ 12755.8 0.539284
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 13100.8 0.550859 0.275430 0.961321i $$-0.411180\pi$$
0.275430 + 0.961321i $$0.411180\pi$$
$$828$$ 0 0
$$829$$ −24522.6 −1.02739 −0.513694 0.857974i $$-0.671723\pi$$
−0.513694 + 0.857974i $$0.671723\pi$$
$$830$$ −9052.97 −0.378594
$$831$$ 0 0
$$832$$ −1738.65 −0.0724483
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −14560.6 −0.603463
$$836$$ 32255.6 1.33443
$$837$$ 0 0
$$838$$ 5064.71 0.208780
$$839$$ 38095.2 1.56757 0.783785 0.621032i $$-0.213286\pi$$
0.783785 + 0.621032i $$0.213286\pi$$
$$840$$ 0 0
$$841$$ −23787.1 −0.975322
$$842$$ 3223.86 0.131950
$$843$$ 0 0
$$844$$ 10064.6 0.410470
$$845$$ 1540.00 0.0626954
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 49644.6 2.01038
$$849$$ 0 0
$$850$$ −21329.9 −0.860718
$$851$$ −15210.5 −0.612702
$$852$$ 0 0
$$853$$ 27500.3 1.10386 0.551929 0.833891i $$-0.313892\pi$$
0.551929 + 0.833891i $$0.313892\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −27286.6 −1.08953
$$857$$ −13903.0 −0.554163 −0.277082 0.960846i $$-0.589367\pi$$
−0.277082 + 0.960846i $$0.589367\pi$$
$$858$$ 0 0
$$859$$ −41524.3 −1.64935 −0.824674 0.565608i $$-0.808641\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$860$$ 5723.01 0.226922
$$861$$ 0 0
$$862$$ −4324.28 −0.170865
$$863$$ 1744.09 0.0687942 0.0343971 0.999408i $$-0.489049\pi$$
0.0343971 + 0.999408i $$0.489049\pi$$
$$864$$ 0 0
$$865$$ −4907.55 −0.192904
$$866$$ −37670.3 −1.47816
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −23475.7 −0.916407
$$870$$ 0 0
$$871$$ −30765.5 −1.19684
$$872$$ 23698.1 0.920319
$$873$$ 0 0
$$874$$ −51588.2 −1.99656
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 46717.4 1.79879 0.899393 0.437140i $$-0.144009\pi$$
0.899393 + 0.437140i $$0.144009\pi$$
$$878$$ −11711.6 −0.450168
$$879$$ 0 0
$$880$$ 34007.1 1.30270
$$881$$ 17214.0 0.658291 0.329145 0.944279i $$-0.393239\pi$$
0.329145 + 0.944279i $$0.393239\pi$$
$$882$$ 0 0
$$883$$ 20487.2 0.780804 0.390402 0.920645i $$-0.372336\pi$$
0.390402 + 0.920645i $$0.372336\pi$$
$$884$$ 19906.6 0.757388
$$885$$ 0 0
$$886$$ 28254.5 1.07136
$$887$$ 18307.5 0.693016 0.346508 0.938047i $$-0.387367\pi$$
0.346508 + 0.938047i $$0.387367\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −684.034 −0.0257628
$$891$$ 0 0
$$892$$ −14009.1 −0.525853
$$893$$ −36896.1 −1.38262
$$894$$ 0 0
$$895$$ −18433.4 −0.688449
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 14707.0 0.546524
$$899$$ −4596.96 −0.170542
$$900$$ 0 0
$$901$$ 60209.3 2.22626
$$902$$ −59254.3 −2.18731
$$903$$ 0 0
$$904$$ −17219.4 −0.633528
$$905$$ −3637.58 −0.133610
$$906$$ 0 0
$$907$$ 38336.3 1.40346 0.701730 0.712443i $$-0.252411\pi$$
0.701730 + 0.712443i $$0.252411\pi$$
$$908$$ 5325.63 0.194645
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30148.4 −1.09644 −0.548222 0.836333i $$-0.684695\pi$$
−0.548222 + 0.836333i $$0.684695\pi$$
$$912$$ 0 0
$$913$$ −17775.7 −0.644348
$$914$$ 34147.0 1.23576
$$915$$ 0 0
$$916$$ −8609.21 −0.310542
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −13799.4 −0.495323 −0.247661 0.968847i $$-0.579662\pi$$
−0.247661 + 0.968847i $$0.579662\pi$$
$$920$$ −10856.9 −0.389066
$$921$$ 0 0
$$922$$ 43166.8 1.54189
$$923$$ 2020.34 0.0720480
$$924$$ 0 0
$$925$$ −9229.94 −0.328085
$$926$$ 56251.5 1.99626
$$927$$ 0 0
$$928$$ −4243.96 −0.150124
$$929$$ −42953.7 −1.51697 −0.758485 0.651691i $$-0.774060\pi$$
−0.758485 + 0.651691i $$0.774060\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 20774.9 0.730155
$$933$$ 0 0
$$934$$ 47095.8 1.64991
$$935$$ 41244.0 1.44259
$$936$$ 0 0
$$937$$ −6369.72 −0.222081 −0.111040 0.993816i $$-0.535418\pi$$
−0.111040 + 0.993816i $$0.535418\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 8567.54 0.297279
$$941$$ 6002.48 0.207944 0.103972 0.994580i $$-0.466845\pi$$
0.103972 + 0.994580i $$0.466845\pi$$
$$942$$ 0 0
$$943$$ 32607.9 1.12604
$$944$$ −35431.7 −1.22161
$$945$$ 0 0
$$946$$ 32659.0 1.12245
$$947$$ 12113.1 0.415652 0.207826 0.978166i $$-0.433361\pi$$
0.207826 + 0.978166i $$0.433361\pi$$
$$948$$ 0 0
$$949$$ −21869.8 −0.748077
$$950$$ −31304.4 −1.06911
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 10130.6 0.344346 0.172173 0.985067i $$-0.444921\pi$$
0.172173 + 0.985067i $$0.444921\pi$$
$$954$$ 0 0
$$955$$ −20994.0 −0.711360
$$956$$ 1102.32 0.0372924
$$957$$ 0 0
$$958$$ 1365.39 0.0460478
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5320.00 0.178578
$$962$$ 25035.1 0.839047
$$963$$ 0 0
$$964$$ 12180.0 0.406943
$$965$$ 1028.27 0.0343018
$$966$$ 0 0
$$967$$ −17651.7 −0.587013 −0.293506 0.955957i $$-0.594822\pi$$
−0.293506 + 0.955957i $$0.594822\pi$$
$$968$$ 21058.5 0.699221
$$969$$ 0 0
$$970$$ 13730.8 0.454504
$$971$$ 32208.7 1.06450 0.532248 0.846588i $$-0.321347\pi$$
0.532248 + 0.846588i $$0.321347\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 55149.6 1.81428
$$975$$ 0 0
$$976$$ −9059.27 −0.297111
$$977$$ 24098.8 0.789138 0.394569 0.918866i $$-0.370894\pi$$
0.394569 + 0.918866i $$0.370894\pi$$
$$978$$ 0 0
$$979$$ −1343.11 −0.0438469
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −669.158 −0.0217451
$$983$$ 15156.9 0.491790 0.245895 0.969296i $$-0.420918\pi$$
0.245895 + 0.969296i $$0.420918\pi$$
$$984$$ 0 0
$$985$$ 29371.1 0.950092
$$986$$ −8308.82 −0.268364
$$987$$ 0 0
$$988$$ 29215.5 0.940759
$$989$$ −17972.4 −0.577844
$$990$$ 0 0
$$991$$ 18357.4 0.588437 0.294219 0.955738i $$-0.404941\pi$$
0.294219 + 0.955738i $$0.404941\pi$$
$$992$$ 32414.9 1.03747
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −29711.6 −0.946654
$$996$$ 0 0
$$997$$ 18552.6 0.589335 0.294668 0.955600i $$-0.404791\pi$$
0.294668 + 0.955600i $$0.404791\pi$$
$$998$$ −19630.6 −0.622640
$$999$$ 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.bo.1.2 8
3.2 odd 2 inner 1323.4.a.bo.1.7 8
7.2 even 3 189.4.e.h.109.7 yes 16
7.4 even 3 189.4.e.h.163.7 yes 16
7.6 odd 2 1323.4.a.bn.1.2 8
21.2 odd 6 189.4.e.h.109.2 16
21.11 odd 6 189.4.e.h.163.2 yes 16
21.20 even 2 1323.4.a.bn.1.7 8

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.h.109.2 16 21.2 odd 6
189.4.e.h.109.7 yes 16 7.2 even 3
189.4.e.h.163.2 yes 16 21.11 odd 6
189.4.e.h.163.7 yes 16 7.4 even 3
1323.4.a.bn.1.2 8 7.6 odd 2
1323.4.a.bn.1.7 8 21.20 even 2
1323.4.a.bo.1.2 8 1.1 even 1 trivial
1323.4.a.bo.1.7 8 3.2 odd 2 inner