# Properties

 Label 531.8.a.d.1.11 Level $531$ Weight $8$ Character 531.1 Self dual yes Analytic conductor $165.876$ Analytic rank $0$ Dimension $17$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$531 = 3^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 531.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$165.876448532$$ Analytic rank: $$0$$ Dimension: $$17$$ Coefficient field: $$\mathbb{Q}[x]/(x^{17} - \cdots)$$ Defining polynomial: $$x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: multiple of $$2^{10}\cdot 3^{5}$$ Twist minimal: no (minimal twist has level 177) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.11 Root $$-4.85375$$ of defining polynomial Character $$\chi$$ $$=$$ 531.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +O(q^{10})$$ $$q+6.85375 q^{2} -81.0262 q^{4} +190.727 q^{5} -799.288 q^{7} -1432.61 q^{8} +1307.20 q^{10} -972.348 q^{11} -1278.77 q^{13} -5478.12 q^{14} +552.586 q^{16} -36467.5 q^{17} +1900.72 q^{19} -15453.9 q^{20} -6664.23 q^{22} -80181.7 q^{23} -41748.1 q^{25} -8764.35 q^{26} +64763.2 q^{28} +201337. q^{29} +106900. q^{31} +187162. q^{32} -249939. q^{34} -152446. q^{35} -246438. q^{37} +13027.1 q^{38} -273238. q^{40} +348348. q^{41} +303791. q^{43} +78785.6 q^{44} -549545. q^{46} -1.12223e6 q^{47} -184682. q^{49} -286131. q^{50} +103614. q^{52} -534787. q^{53} -185453. q^{55} +1.14507e6 q^{56} +1.37991e6 q^{58} +205379. q^{59} -2.14348e6 q^{61} +732666. q^{62} +1.21203e6 q^{64} -243896. q^{65} -158432. q^{67} +2.95482e6 q^{68} -1.04483e6 q^{70} +3.94945e6 q^{71} +3.25842e6 q^{73} -1.68903e6 q^{74} -154008. q^{76} +777186. q^{77} -5.28930e6 q^{79} +105393. q^{80} +2.38749e6 q^{82} +3.63337e6 q^{83} -6.95534e6 q^{85} +2.08211e6 q^{86} +1.39300e6 q^{88} +1.34439e6 q^{89} +1.02210e6 q^{91} +6.49682e6 q^{92} -7.69146e6 q^{94} +362519. q^{95} +2.79048e6 q^{97} -1.26576e6 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10})$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 6.85375 0.605791 0.302896 0.953024i $$-0.402047\pi$$
0.302896 + 0.953024i $$0.402047\pi$$
$$3$$ 0 0
$$4$$ −81.0262 −0.633017
$$5$$ 190.727 0.682367 0.341183 0.939997i $$-0.389172\pi$$
0.341183 + 0.939997i $$0.389172\pi$$
$$6$$ 0 0
$$7$$ −799.288 −0.880765 −0.440383 0.897810i $$-0.645157\pi$$
−0.440383 + 0.897810i $$0.645157\pi$$
$$8$$ −1432.61 −0.989267
$$9$$ 0 0
$$10$$ 1307.20 0.413372
$$11$$ −972.348 −0.220266 −0.110133 0.993917i $$-0.535128\pi$$
−0.110133 + 0.993917i $$0.535128\pi$$
$$12$$ 0 0
$$13$$ −1278.77 −0.161432 −0.0807161 0.996737i $$-0.525721\pi$$
−0.0807161 + 0.996737i $$0.525721\pi$$
$$14$$ −5478.12 −0.533560
$$15$$ 0 0
$$16$$ 552.586 0.0337272
$$17$$ −36467.5 −1.80026 −0.900128 0.435625i $$-0.856527\pi$$
−0.900128 + 0.435625i $$0.856527\pi$$
$$18$$ 0 0
$$19$$ 1900.72 0.0635742 0.0317871 0.999495i $$-0.489880\pi$$
0.0317871 + 0.999495i $$0.489880\pi$$
$$20$$ −15453.9 −0.431950
$$21$$ 0 0
$$22$$ −6664.23 −0.133435
$$23$$ −80181.7 −1.37413 −0.687065 0.726596i $$-0.741101\pi$$
−0.687065 + 0.726596i $$0.741101\pi$$
$$24$$ 0 0
$$25$$ −41748.1 −0.534376
$$26$$ −8764.35 −0.0977942
$$27$$ 0 0
$$28$$ 64763.2 0.557539
$$29$$ 201337. 1.53296 0.766479 0.642270i $$-0.222007\pi$$
0.766479 + 0.642270i $$0.222007\pi$$
$$30$$ 0 0
$$31$$ 106900. 0.644484 0.322242 0.946657i $$-0.395564\pi$$
0.322242 + 0.946657i $$0.395564\pi$$
$$32$$ 187162. 1.00970
$$33$$ 0 0
$$34$$ −249939. −1.09058
$$35$$ −152446. −0.601005
$$36$$ 0 0
$$37$$ −246438. −0.799838 −0.399919 0.916550i $$-0.630962\pi$$
−0.399919 + 0.916550i $$0.630962\pi$$
$$38$$ 13027.1 0.0385127
$$39$$ 0 0
$$40$$ −273238. −0.675043
$$41$$ 348348. 0.789350 0.394675 0.918821i $$-0.370857\pi$$
0.394675 + 0.918821i $$0.370857\pi$$
$$42$$ 0 0
$$43$$ 303791. 0.582687 0.291343 0.956619i $$-0.405898\pi$$
0.291343 + 0.956619i $$0.405898\pi$$
$$44$$ 78785.6 0.139432
$$45$$ 0 0
$$46$$ −549545. −0.832436
$$47$$ −1.12223e6 −1.57666 −0.788330 0.615252i $$-0.789054\pi$$
−0.788330 + 0.615252i $$0.789054\pi$$
$$48$$ 0 0
$$49$$ −184682. −0.224253
$$50$$ −286131. −0.323720
$$51$$ 0 0
$$52$$ 103614. 0.102189
$$53$$ −534787. −0.493418 −0.246709 0.969090i $$-0.579349\pi$$
−0.246709 + 0.969090i $$0.579349\pi$$
$$54$$ 0 0
$$55$$ −185453. −0.150302
$$56$$ 1.14507e6 0.871312
$$57$$ 0 0
$$58$$ 1.37991e6 0.928652
$$59$$ 205379. 0.130189
$$60$$ 0 0
$$61$$ −2.14348e6 −1.20911 −0.604555 0.796564i $$-0.706649\pi$$
−0.604555 + 0.796564i $$0.706649\pi$$
$$62$$ 732666. 0.390423
$$63$$ 0 0
$$64$$ 1.21203e6 0.577940
$$65$$ −243896. −0.110156
$$66$$ 0 0
$$67$$ −158432. −0.0643548 −0.0321774 0.999482i $$-0.510244\pi$$
−0.0321774 + 0.999482i $$0.510244\pi$$
$$68$$ 2.95482e6 1.13959
$$69$$ 0 0
$$70$$ −1.04483e6 −0.364084
$$71$$ 3.94945e6 1.30958 0.654790 0.755811i $$-0.272757\pi$$
0.654790 + 0.755811i $$0.272757\pi$$
$$72$$ 0 0
$$73$$ 3.25842e6 0.980342 0.490171 0.871626i $$-0.336934\pi$$
0.490171 + 0.871626i $$0.336934\pi$$
$$74$$ −1.68903e6 −0.484535
$$75$$ 0 0
$$76$$ −154008. −0.0402435
$$77$$ 777186. 0.194003
$$78$$ 0 0
$$79$$ −5.28930e6 −1.20699 −0.603494 0.797367i $$-0.706225\pi$$
−0.603494 + 0.797367i $$0.706225\pi$$
$$80$$ 105393. 0.0230143
$$81$$ 0 0
$$82$$ 2.38749e6 0.478181
$$83$$ 3.63337e6 0.697487 0.348744 0.937218i $$-0.386608\pi$$
0.348744 + 0.937218i $$0.386608\pi$$
$$84$$ 0 0
$$85$$ −6.95534e6 −1.22844
$$86$$ 2.08211e6 0.352987
$$87$$ 0 0
$$88$$ 1.39300e6 0.217902
$$89$$ 1.34439e6 0.202143 0.101072 0.994879i $$-0.467773\pi$$
0.101072 + 0.994879i $$0.467773\pi$$
$$90$$ 0 0
$$91$$ 1.02210e6 0.142184
$$92$$ 6.49682e6 0.869848
$$93$$ 0 0
$$94$$ −7.69146e6 −0.955128
$$95$$ 362519. 0.0433809
$$96$$ 0 0
$$97$$ 2.79048e6 0.310441 0.155220 0.987880i $$-0.450391\pi$$
0.155220 + 0.987880i $$0.450391\pi$$
$$98$$ −1.26576e6 −0.135850
$$99$$ 0 0
$$100$$ 3.38269e6 0.338269
$$101$$ 1.43415e7 1.38506 0.692532 0.721387i $$-0.256495\pi$$
0.692532 + 0.721387i $$0.256495\pi$$
$$102$$ 0 0
$$103$$ 1.62771e6 0.146773 0.0733867 0.997304i $$-0.476619\pi$$
0.0733867 + 0.997304i $$0.476619\pi$$
$$104$$ 1.83198e6 0.159700
$$105$$ 0 0
$$106$$ −3.66529e6 −0.298908
$$107$$ 7.86386e6 0.620572 0.310286 0.950643i $$-0.399575\pi$$
0.310286 + 0.950643i $$0.399575\pi$$
$$108$$ 0 0
$$109$$ 2.63958e6 0.195228 0.0976141 0.995224i $$-0.468879\pi$$
0.0976141 + 0.995224i $$0.468879\pi$$
$$110$$ −1.27105e6 −0.0910518
$$111$$ 0 0
$$112$$ −441675. −0.0297057
$$113$$ 2.91448e7 1.90015 0.950074 0.312026i $$-0.101008\pi$$
0.950074 + 0.312026i $$0.101008\pi$$
$$114$$ 0 0
$$115$$ −1.52928e7 −0.937661
$$116$$ −1.63135e7 −0.970388
$$117$$ 0 0
$$118$$ 1.40762e6 0.0788673
$$119$$ 2.91480e7 1.58560
$$120$$ 0 0
$$121$$ −1.85417e7 −0.951483
$$122$$ −1.46909e7 −0.732468
$$123$$ 0 0
$$124$$ −8.66170e6 −0.407969
$$125$$ −2.28631e7 −1.04701
$$126$$ 0 0
$$127$$ −3.71615e7 −1.60983 −0.804915 0.593390i $$-0.797789\pi$$
−0.804915 + 0.593390i $$0.797789\pi$$
$$128$$ −1.56498e7 −0.659588
$$129$$ 0 0
$$130$$ −1.67160e6 −0.0667315
$$131$$ −1.50104e7 −0.583369 −0.291684 0.956515i $$-0.594216\pi$$
−0.291684 + 0.956515i $$0.594216\pi$$
$$132$$ 0 0
$$133$$ −1.51922e6 −0.0559939
$$134$$ −1.08585e6 −0.0389856
$$135$$ 0 0
$$136$$ 5.22437e7 1.78094
$$137$$ −1.75306e7 −0.582470 −0.291235 0.956652i $$-0.594066\pi$$
−0.291235 + 0.956652i $$0.594066\pi$$
$$138$$ 0 0
$$139$$ −1.16871e7 −0.369110 −0.184555 0.982822i $$-0.559084\pi$$
−0.184555 + 0.982822i $$0.559084\pi$$
$$140$$ 1.23521e7 0.380446
$$141$$ 0 0
$$142$$ 2.70685e7 0.793332
$$143$$ 1.24341e6 0.0355580
$$144$$ 0 0
$$145$$ 3.84004e7 1.04604
$$146$$ 2.23324e7 0.593883
$$147$$ 0 0
$$148$$ 1.99679e7 0.506311
$$149$$ 6.76391e7 1.67512 0.837559 0.546346i $$-0.183982\pi$$
0.837559 + 0.546346i $$0.183982\pi$$
$$150$$ 0 0
$$151$$ 4.31830e7 1.02069 0.510345 0.859970i $$-0.329518\pi$$
0.510345 + 0.859970i $$0.329518\pi$$
$$152$$ −2.72300e6 −0.0628919
$$153$$ 0 0
$$154$$ 5.32664e6 0.117525
$$155$$ 2.03888e7 0.439775
$$156$$ 0 0
$$157$$ 4.62920e7 0.954679 0.477340 0.878719i $$-0.341601\pi$$
0.477340 + 0.878719i $$0.341601\pi$$
$$158$$ −3.62515e7 −0.731183
$$159$$ 0 0
$$160$$ 3.56968e7 0.688985
$$161$$ 6.40883e7 1.21029
$$162$$ 0 0
$$163$$ 7.22501e7 1.30672 0.653359 0.757048i $$-0.273359\pi$$
0.653359 + 0.757048i $$0.273359\pi$$
$$164$$ −2.82253e7 −0.499672
$$165$$ 0 0
$$166$$ 2.49022e7 0.422532
$$167$$ −3.24667e7 −0.539425 −0.269713 0.962941i $$-0.586929\pi$$
−0.269713 + 0.962941i $$0.586929\pi$$
$$168$$ 0 0
$$169$$ −6.11133e7 −0.973940
$$170$$ −4.76702e7 −0.744175
$$171$$ 0 0
$$172$$ −2.46150e7 −0.368851
$$173$$ −1.31381e7 −0.192918 −0.0964589 0.995337i $$-0.530752\pi$$
−0.0964589 + 0.995337i $$0.530752\pi$$
$$174$$ 0 0
$$175$$ 3.33687e7 0.470659
$$176$$ −537306. −0.00742895
$$177$$ 0 0
$$178$$ 9.21409e6 0.122457
$$179$$ 1.99708e7 0.260261 0.130131 0.991497i $$-0.458460\pi$$
0.130131 + 0.991497i $$0.458460\pi$$
$$180$$ 0 0
$$181$$ 9.38499e7 1.17641 0.588205 0.808712i $$-0.299835\pi$$
0.588205 + 0.808712i $$0.299835\pi$$
$$182$$ 7.00524e6 0.0861337
$$183$$ 0 0
$$184$$ 1.14869e8 1.35938
$$185$$ −4.70025e7 −0.545783
$$186$$ 0 0
$$187$$ 3.54591e7 0.396535
$$188$$ 9.09298e7 0.998053
$$189$$ 0 0
$$190$$ 2.48462e6 0.0262798
$$191$$ 7.50591e7 0.779447 0.389724 0.920932i $$-0.372571\pi$$
0.389724 + 0.920932i $$0.372571\pi$$
$$192$$ 0 0
$$193$$ −7.13691e7 −0.714594 −0.357297 0.933991i $$-0.616302\pi$$
−0.357297 + 0.933991i $$0.616302\pi$$
$$194$$ 1.91253e7 0.188062
$$195$$ 0 0
$$196$$ 1.49641e7 0.141956
$$197$$ 1.40656e8 1.31077 0.655384 0.755296i $$-0.272507\pi$$
0.655384 + 0.755296i $$0.272507\pi$$
$$198$$ 0 0
$$199$$ 2.79140e7 0.251094 0.125547 0.992088i $$-0.459931\pi$$
0.125547 + 0.992088i $$0.459931\pi$$
$$200$$ 5.98088e7 0.528640
$$201$$ 0 0
$$202$$ 9.82931e7 0.839060
$$203$$ −1.60926e8 −1.35018
$$204$$ 0 0
$$205$$ 6.64394e7 0.538626
$$206$$ 1.11559e7 0.0889140
$$207$$ 0 0
$$208$$ −706629. −0.00544465
$$209$$ −1.84816e6 −0.0140032
$$210$$ 0 0
$$211$$ 1.66531e8 1.22041 0.610207 0.792242i $$-0.291086\pi$$
0.610207 + 0.792242i $$0.291086\pi$$
$$212$$ 4.33317e7 0.312342
$$213$$ 0 0
$$214$$ 5.38969e7 0.375937
$$215$$ 5.79412e7 0.397606
$$216$$ 0 0
$$217$$ −8.54439e7 −0.567639
$$218$$ 1.80910e7 0.118268
$$219$$ 0 0
$$220$$ 1.50266e7 0.0951438
$$221$$ 4.66334e7 0.290619
$$222$$ 0 0
$$223$$ 8.94297e6 0.0540026 0.0270013 0.999635i $$-0.491404\pi$$
0.0270013 + 0.999635i $$0.491404\pi$$
$$224$$ −1.49596e8 −0.889308
$$225$$ 0 0
$$226$$ 1.99751e8 1.15109
$$227$$ 6.88172e7 0.390487 0.195243 0.980755i $$-0.437450\pi$$
0.195243 + 0.980755i $$0.437450\pi$$
$$228$$ 0 0
$$229$$ 1.97507e8 1.08682 0.543411 0.839467i $$-0.317132\pi$$
0.543411 + 0.839467i $$0.317132\pi$$
$$230$$ −1.04813e8 −0.568027
$$231$$ 0 0
$$232$$ −2.88438e8 −1.51651
$$233$$ −2.46365e8 −1.27595 −0.637973 0.770058i $$-0.720227\pi$$
−0.637973 + 0.770058i $$0.720227\pi$$
$$234$$ 0 0
$$235$$ −2.14039e8 −1.07586
$$236$$ −1.66411e7 −0.0824118
$$237$$ 0 0
$$238$$ 1.99773e8 0.960545
$$239$$ 3.78259e8 1.79224 0.896120 0.443812i $$-0.146374\pi$$
0.896120 + 0.443812i $$0.146374\pi$$
$$240$$ 0 0
$$241$$ −2.71983e8 −1.25165 −0.625824 0.779964i $$-0.715237\pi$$
−0.625824 + 0.779964i $$0.715237\pi$$
$$242$$ −1.27080e8 −0.576400
$$243$$ 0 0
$$244$$ 1.73678e8 0.765387
$$245$$ −3.52239e7 −0.153023
$$246$$ 0 0
$$247$$ −2.43058e6 −0.0102629
$$248$$ −1.53146e8 −0.637567
$$249$$ 0 0
$$250$$ −1.56698e8 −0.634268
$$251$$ −1.67816e8 −0.669847 −0.334923 0.942245i $$-0.608710\pi$$
−0.334923 + 0.942245i $$0.608710\pi$$
$$252$$ 0 0
$$253$$ 7.79645e7 0.302674
$$254$$ −2.54695e8 −0.975221
$$255$$ 0 0
$$256$$ −2.62399e8 −0.977513
$$257$$ 3.77370e8 1.38676 0.693381 0.720571i $$-0.256120\pi$$
0.693381 + 0.720571i $$0.256120\pi$$
$$258$$ 0 0
$$259$$ 1.96975e8 0.704470
$$260$$ 1.97619e7 0.0697305
$$261$$ 0 0
$$262$$ −1.02878e8 −0.353400
$$263$$ 2.37395e8 0.804686 0.402343 0.915489i $$-0.368196\pi$$
0.402343 + 0.915489i $$0.368196\pi$$
$$264$$ 0 0
$$265$$ −1.01998e8 −0.336692
$$266$$ −1.04124e7 −0.0339206
$$267$$ 0 0
$$268$$ 1.28371e7 0.0407377
$$269$$ −2.56851e8 −0.804542 −0.402271 0.915521i $$-0.631779\pi$$
−0.402271 + 0.915521i $$0.631779\pi$$
$$270$$ 0 0
$$271$$ −2.99436e8 −0.913927 −0.456964 0.889485i $$-0.651063\pi$$
−0.456964 + 0.889485i $$0.651063\pi$$
$$272$$ −2.01514e7 −0.0607176
$$273$$ 0 0
$$274$$ −1.20150e8 −0.352855
$$275$$ 4.05937e7 0.117705
$$276$$ 0 0
$$277$$ −1.07342e8 −0.303453 −0.151727 0.988422i $$-0.548483\pi$$
−0.151727 + 0.988422i $$0.548483\pi$$
$$278$$ −8.01006e7 −0.223604
$$279$$ 0 0
$$280$$ 2.18396e8 0.594555
$$281$$ −1.26999e8 −0.341451 −0.170726 0.985319i $$-0.554611\pi$$
−0.170726 + 0.985319i $$0.554611\pi$$
$$282$$ 0 0
$$283$$ 6.79420e8 1.78191 0.890955 0.454091i $$-0.150036\pi$$
0.890955 + 0.454091i $$0.150036\pi$$
$$284$$ −3.20009e8 −0.828986
$$285$$ 0 0
$$286$$ 8.52200e6 0.0215407
$$287$$ −2.78430e8 −0.695232
$$288$$ 0 0
$$289$$ 9.19537e8 2.24092
$$290$$ 2.63187e8 0.633682
$$291$$ 0 0
$$292$$ −2.64018e8 −0.620573
$$293$$ −3.54246e8 −0.822752 −0.411376 0.911466i $$-0.634952\pi$$
−0.411376 + 0.911466i $$0.634952\pi$$
$$294$$ 0 0
$$295$$ 3.91714e7 0.0888366
$$296$$ 3.53051e8 0.791254
$$297$$ 0 0
$$298$$ 4.63581e8 1.01477
$$299$$ 1.02534e8 0.221829
$$300$$ 0 0
$$301$$ −2.42816e8 −0.513210
$$302$$ 2.95966e8 0.618325
$$303$$ 0 0
$$304$$ 1.05031e6 0.00214418
$$305$$ −4.08821e8 −0.825056
$$306$$ 0 0
$$307$$ −4.47178e8 −0.882056 −0.441028 0.897493i $$-0.645386\pi$$
−0.441028 + 0.897493i $$0.645386\pi$$
$$308$$ −6.29724e7 −0.122807
$$309$$ 0 0
$$310$$ 1.39739e8 0.266412
$$311$$ 1.03177e9 1.94501 0.972506 0.232879i $$-0.0748145\pi$$
0.972506 + 0.232879i $$0.0748145\pi$$
$$312$$ 0 0
$$313$$ −1.47667e8 −0.272194 −0.136097 0.990695i $$-0.543456\pi$$
−0.136097 + 0.990695i $$0.543456\pi$$
$$314$$ 3.17274e8 0.578336
$$315$$ 0 0
$$316$$ 4.28571e8 0.764044
$$317$$ 8.08129e8 1.42486 0.712431 0.701742i $$-0.247594\pi$$
0.712431 + 0.701742i $$0.247594\pi$$
$$318$$ 0 0
$$319$$ −1.95769e8 −0.337658
$$320$$ 2.31167e8 0.394367
$$321$$ 0 0
$$322$$ 4.39245e8 0.733181
$$323$$ −6.93145e7 −0.114450
$$324$$ 0 0
$$325$$ 5.33861e7 0.0862654
$$326$$ 4.95184e8 0.791599
$$327$$ 0 0
$$328$$ −4.99047e8 −0.780878
$$329$$ 8.96983e8 1.38867
$$330$$ 0 0
$$331$$ −1.71934e8 −0.260594 −0.130297 0.991475i $$-0.541593\pi$$
−0.130297 + 0.991475i $$0.541593\pi$$
$$332$$ −2.94398e8 −0.441521
$$333$$ 0 0
$$334$$ −2.22519e8 −0.326779
$$335$$ −3.02173e7 −0.0439136
$$336$$ 0 0
$$337$$ 2.64986e8 0.377154 0.188577 0.982058i $$-0.439613\pi$$
0.188577 + 0.982058i $$0.439613\pi$$
$$338$$ −4.18855e8 −0.590004
$$339$$ 0 0
$$340$$ 5.63565e8 0.777620
$$341$$ −1.03944e8 −0.141958
$$342$$ 0 0
$$343$$ 8.05862e8 1.07828
$$344$$ −4.35215e8 −0.576433
$$345$$ 0 0
$$346$$ −9.00455e7 −0.116868
$$347$$ −5.14421e8 −0.660945 −0.330473 0.943816i $$-0.607208\pi$$
−0.330473 + 0.943816i $$0.607208\pi$$
$$348$$ 0 0
$$349$$ −6.09218e7 −0.0767157 −0.0383578 0.999264i $$-0.512213\pi$$
−0.0383578 + 0.999264i $$0.512213\pi$$
$$350$$ 2.28701e8 0.285121
$$351$$ 0 0
$$352$$ −1.81986e8 −0.222402
$$353$$ 1.90611e8 0.230641 0.115320 0.993328i $$-0.463211\pi$$
0.115320 + 0.993328i $$0.463211\pi$$
$$354$$ 0 0
$$355$$ 7.53268e8 0.893614
$$356$$ −1.08931e8 −0.127960
$$357$$ 0 0
$$358$$ 1.36875e8 0.157664
$$359$$ 1.39955e9 1.59646 0.798229 0.602354i $$-0.205770\pi$$
0.798229 + 0.602354i $$0.205770\pi$$
$$360$$ 0 0
$$361$$ −8.90259e8 −0.995958
$$362$$ 6.43224e8 0.712659
$$363$$ 0 0
$$364$$ −8.28171e7 −0.0900047
$$365$$ 6.21471e8 0.668953
$$366$$ 0 0
$$367$$ −1.50266e9 −1.58683 −0.793413 0.608684i $$-0.791698\pi$$
−0.793413 + 0.608684i $$0.791698\pi$$
$$368$$ −4.43073e7 −0.0463455
$$369$$ 0 0
$$370$$ −3.22143e8 −0.330631
$$371$$ 4.27449e8 0.434585
$$372$$ 0 0
$$373$$ 8.89561e7 0.0887554 0.0443777 0.999015i $$-0.485869\pi$$
0.0443777 + 0.999015i $$0.485869\pi$$
$$374$$ 2.43027e8 0.240218
$$375$$ 0 0
$$376$$ 1.60772e9 1.55974
$$377$$ −2.57463e8 −0.247469
$$378$$ 0 0
$$379$$ −7.27075e8 −0.686028 −0.343014 0.939330i $$-0.611448\pi$$
−0.343014 + 0.939330i $$0.611448\pi$$
$$380$$ −2.93736e7 −0.0274608
$$381$$ 0 0
$$382$$ 5.14436e8 0.472182
$$383$$ −2.06745e9 −1.88036 −0.940178 0.340683i $$-0.889342\pi$$
−0.940178 + 0.340683i $$0.889342\pi$$
$$384$$ 0 0
$$385$$ 1.48231e8 0.132381
$$386$$ −4.89145e8 −0.432895
$$387$$ 0 0
$$388$$ −2.26102e8 −0.196514
$$389$$ 8.24531e7 0.0710205 0.0355102 0.999369i $$-0.488694\pi$$
0.0355102 + 0.999369i $$0.488694\pi$$
$$390$$ 0 0
$$391$$ 2.92402e9 2.47379
$$392$$ 2.64578e8 0.221846
$$393$$ 0 0
$$394$$ 9.64019e8 0.794051
$$395$$ −1.00881e9 −0.823609
$$396$$ 0 0
$$397$$ 2.60792e8 0.209183 0.104592 0.994515i $$-0.466646\pi$$
0.104592 + 0.994515i $$0.466646\pi$$
$$398$$ 1.91316e8 0.152111
$$399$$ 0 0
$$400$$ −2.30694e7 −0.0180230
$$401$$ 5.57165e8 0.431498 0.215749 0.976449i $$-0.430781\pi$$
0.215749 + 0.976449i $$0.430781\pi$$
$$402$$ 0 0
$$403$$ −1.36700e8 −0.104040
$$404$$ −1.16204e9 −0.876769
$$405$$ 0 0
$$406$$ −1.10295e9 −0.817925
$$407$$ 2.39624e8 0.176177
$$408$$ 0 0
$$409$$ −1.59001e9 −1.14913 −0.574564 0.818460i $$-0.694828\pi$$
−0.574564 + 0.818460i $$0.694828\pi$$
$$410$$ 4.55359e8 0.326295
$$411$$ 0 0
$$412$$ −1.31887e8 −0.0929100
$$413$$ −1.64157e8 −0.114666
$$414$$ 0 0
$$415$$ 6.92983e8 0.475942
$$416$$ −2.39336e8 −0.162998
$$417$$ 0 0
$$418$$ −1.26668e7 −0.00848303
$$419$$ −5.88021e8 −0.390520 −0.195260 0.980751i $$-0.562555\pi$$
−0.195260 + 0.980751i $$0.562555\pi$$
$$420$$ 0 0
$$421$$ −7.50924e8 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$422$$ 1.14136e9 0.739316
$$423$$ 0 0
$$424$$ 7.66142e8 0.488123
$$425$$ 1.52245e9 0.962013
$$426$$ 0 0
$$427$$ 1.71326e9 1.06494
$$428$$ −6.37178e8 −0.392833
$$429$$ 0 0
$$430$$ 3.97114e8 0.240866
$$431$$ 2.36435e9 1.42247 0.711233 0.702957i $$-0.248137\pi$$
0.711233 + 0.702957i $$0.248137\pi$$
$$432$$ 0 0
$$433$$ 6.97398e7 0.0412832 0.0206416 0.999787i $$-0.493429\pi$$
0.0206416 + 0.999787i $$0.493429\pi$$
$$434$$ −5.85611e8 −0.343871
$$435$$ 0 0
$$436$$ −2.13875e8 −0.123583
$$437$$ −1.52403e8 −0.0873592
$$438$$ 0 0
$$439$$ −2.54774e9 −1.43724 −0.718620 0.695403i $$-0.755226\pi$$
−0.718620 + 0.695403i $$0.755226\pi$$
$$440$$ 2.65683e8 0.148689
$$441$$ 0 0
$$442$$ 3.19614e8 0.176055
$$443$$ −1.15431e9 −0.630825 −0.315412 0.948955i $$-0.602143\pi$$
−0.315412 + 0.948955i $$0.602143\pi$$
$$444$$ 0 0
$$445$$ 2.56411e8 0.137936
$$446$$ 6.12929e7 0.0327143
$$447$$ 0 0
$$448$$ −9.68759e8 −0.509029
$$449$$ −5.55994e8 −0.289873 −0.144937 0.989441i $$-0.546298\pi$$
−0.144937 + 0.989441i $$0.546298\pi$$
$$450$$ 0 0
$$451$$ −3.38715e8 −0.173867
$$452$$ −2.36150e9 −1.20283
$$453$$ 0 0
$$454$$ 4.71655e8 0.236554
$$455$$ 1.94943e8 0.0970215
$$456$$ 0 0
$$457$$ 3.93559e8 0.192887 0.0964435 0.995338i $$-0.469253\pi$$
0.0964435 + 0.995338i $$0.469253\pi$$
$$458$$ 1.35366e9 0.658388
$$459$$ 0 0
$$460$$ 1.23912e9 0.593555
$$461$$ 2.99382e9 1.42322 0.711611 0.702574i $$-0.247966\pi$$
0.711611 + 0.702574i $$0.247966\pi$$
$$462$$ 0 0
$$463$$ −1.18369e9 −0.554248 −0.277124 0.960834i $$-0.589381\pi$$
−0.277124 + 0.960834i $$0.589381\pi$$
$$464$$ 1.11256e8 0.0517023
$$465$$ 0 0
$$466$$ −1.68852e9 −0.772957
$$467$$ −3.15650e9 −1.43416 −0.717079 0.696992i $$-0.754521\pi$$
−0.717079 + 0.696992i $$0.754521\pi$$
$$468$$ 0 0
$$469$$ 1.26633e8 0.0566815
$$470$$ −1.46697e9 −0.651747
$$471$$ 0 0
$$472$$ −2.94228e8 −0.128792
$$473$$ −2.95390e8 −0.128346
$$474$$ 0 0
$$475$$ −7.93515e7 −0.0339725
$$476$$ −2.36175e9 −1.00371
$$477$$ 0 0
$$478$$ 2.59249e9 1.08572
$$479$$ −2.27050e9 −0.943945 −0.471972 0.881613i $$-0.656458\pi$$
−0.471972 + 0.881613i $$0.656458\pi$$
$$480$$ 0 0
$$481$$ 3.15137e8 0.129120
$$482$$ −1.86410e9 −0.758238
$$483$$ 0 0
$$484$$ 1.50236e9 0.602305
$$485$$ 5.32222e8 0.211834
$$486$$ 0 0
$$487$$ 2.04995e8 0.0804252 0.0402126 0.999191i $$-0.487196\pi$$
0.0402126 + 0.999191i $$0.487196\pi$$
$$488$$ 3.07078e9 1.19613
$$489$$ 0 0
$$490$$ −2.41416e8 −0.0926999
$$491$$ −1.08333e9 −0.413023 −0.206512 0.978444i $$-0.566211\pi$$
−0.206512 + 0.978444i $$0.566211\pi$$
$$492$$ 0 0
$$493$$ −7.34224e9 −2.75972
$$494$$ −1.66586e7 −0.00621718
$$495$$ 0 0
$$496$$ 5.90715e7 0.0217366
$$497$$ −3.15675e9 −1.15343
$$498$$ 0 0
$$499$$ 1.23774e9 0.445942 0.222971 0.974825i $$-0.428425\pi$$
0.222971 + 0.974825i $$0.428425\pi$$
$$500$$ 1.85251e9 0.662773
$$501$$ 0 0
$$502$$ −1.15017e9 −0.405787
$$503$$ −3.42069e9 −1.19847 −0.599234 0.800574i $$-0.704528\pi$$
−0.599234 + 0.800574i $$0.704528\pi$$
$$504$$ 0 0
$$505$$ 2.73532e9 0.945122
$$506$$ 5.34349e8 0.183357
$$507$$ 0 0
$$508$$ 3.01105e9 1.01905
$$509$$ 3.62806e9 1.21944 0.609722 0.792616i $$-0.291281\pi$$
0.609722 + 0.792616i $$0.291281\pi$$
$$510$$ 0 0
$$511$$ −2.60442e9 −0.863451
$$512$$ 2.04753e8 0.0674195
$$513$$ 0 0
$$514$$ 2.58640e9 0.840088
$$515$$ 3.10449e8 0.100153
$$516$$ 0 0
$$517$$ 1.09120e9 0.347285
$$518$$ 1.35002e9 0.426762
$$519$$ 0 0
$$520$$ 3.49408e8 0.108974
$$521$$ 1.36162e9 0.421817 0.210909 0.977506i $$-0.432358\pi$$
0.210909 + 0.977506i $$0.432358\pi$$
$$522$$ 0 0
$$523$$ 1.07415e9 0.328329 0.164165 0.986433i $$-0.447507\pi$$
0.164165 + 0.986433i $$0.447507\pi$$
$$524$$ 1.21624e9 0.369282
$$525$$ 0 0
$$526$$ 1.62705e9 0.487472
$$527$$ −3.89838e9 −1.16024
$$528$$ 0 0
$$529$$ 3.02428e9 0.888234
$$530$$ −6.99072e8 −0.203965
$$531$$ 0 0
$$532$$ 1.23097e8 0.0354451
$$533$$ −4.45456e8 −0.127426
$$534$$ 0 0
$$535$$ 1.49985e9 0.423458
$$536$$ 2.26972e8 0.0636642
$$537$$ 0 0
$$538$$ −1.76039e9 −0.487385
$$539$$ 1.79575e8 0.0493953
$$540$$ 0 0
$$541$$ −6.13348e9 −1.66539 −0.832696 0.553730i $$-0.813204\pi$$
−0.832696 + 0.553730i $$0.813204\pi$$
$$542$$ −2.05226e9 −0.553649
$$543$$ 0 0
$$544$$ −6.82531e9 −1.81772
$$545$$ 5.03440e8 0.133217
$$546$$ 0 0
$$547$$ −6.39180e9 −1.66981 −0.834906 0.550392i $$-0.814478\pi$$
−0.834906 + 0.550392i $$0.814478\pi$$
$$548$$ 1.42043e9 0.368713
$$549$$ 0 0
$$550$$ 2.78219e8 0.0713045
$$551$$ 3.82685e8 0.0974565
$$552$$ 0 0
$$553$$ 4.22767e9 1.06307
$$554$$ −7.35697e8 −0.183829
$$555$$ 0 0
$$556$$ 9.46963e8 0.233653
$$557$$ −5.20568e9 −1.27639 −0.638197 0.769873i $$-0.720319\pi$$
−0.638197 + 0.769873i $$0.720319\pi$$
$$558$$ 0 0
$$559$$ −3.88478e8 −0.0940643
$$560$$ −8.42396e7 −0.0202702
$$561$$ 0 0
$$562$$ −8.70420e8 −0.206848
$$563$$ −3.32429e9 −0.785089 −0.392545 0.919733i $$-0.628405\pi$$
−0.392545 + 0.919733i $$0.628405\pi$$
$$564$$ 0 0
$$565$$ 5.55872e9 1.29660
$$566$$ 4.65657e9 1.07947
$$567$$ 0 0
$$568$$ −5.65803e9 −1.29553
$$569$$ −1.61858e9 −0.368333 −0.184166 0.982895i $$-0.558959\pi$$
−0.184166 + 0.982895i $$0.558959\pi$$
$$570$$ 0 0
$$571$$ −6.26853e8 −0.140909 −0.0704546 0.997515i $$-0.522445\pi$$
−0.0704546 + 0.997515i $$0.522445\pi$$
$$572$$ −1.00749e8 −0.0225088
$$573$$ 0 0
$$574$$ −1.90829e9 −0.421165
$$575$$ 3.34743e9 0.734302
$$576$$ 0 0
$$577$$ 2.33845e9 0.506772 0.253386 0.967365i $$-0.418456\pi$$
0.253386 + 0.967365i $$0.418456\pi$$
$$578$$ 6.30228e9 1.35753
$$579$$ 0 0
$$580$$ −3.11144e9 −0.662161
$$581$$ −2.90411e9 −0.614323
$$582$$ 0 0
$$583$$ 5.19999e8 0.108683
$$584$$ −4.66806e9 −0.969820
$$585$$ 0 0
$$586$$ −2.42791e9 −0.498416
$$587$$ 5.68734e9 1.16058 0.580291 0.814409i $$-0.302939\pi$$
0.580291 + 0.814409i $$0.302939\pi$$
$$588$$ 0 0
$$589$$ 2.03187e8 0.0409725
$$590$$ 2.68471e8 0.0538164
$$591$$ 0 0
$$592$$ −1.36178e8 −0.0269763
$$593$$ 6.92748e9 1.36422 0.682109 0.731250i $$-0.261063\pi$$
0.682109 + 0.731250i $$0.261063\pi$$
$$594$$ 0 0
$$595$$ 5.55932e9 1.08196
$$596$$ −5.48053e9 −1.06038
$$597$$ 0 0
$$598$$ 7.02741e8 0.134382
$$599$$ −1.04577e7 −0.00198812 −0.000994062 1.00000i $$-0.500316\pi$$
−0.000994062 1.00000i $$0.500316\pi$$
$$600$$ 0 0
$$601$$ 5.50968e9 1.03530 0.517649 0.855593i $$-0.326807\pi$$
0.517649 + 0.855593i $$0.326807\pi$$
$$602$$ −1.66420e9 −0.310898
$$603$$ 0 0
$$604$$ −3.49896e9 −0.646114
$$605$$ −3.53641e9 −0.649260
$$606$$ 0 0
$$607$$ −1.08515e10 −1.96939 −0.984693 0.174298i $$-0.944234\pi$$
−0.984693 + 0.174298i $$0.944234\pi$$
$$608$$ 3.55742e8 0.0641908
$$609$$ 0 0
$$610$$ −2.80195e9 −0.499812
$$611$$ 1.43507e9 0.254524
$$612$$ 0 0
$$613$$ −5.00263e9 −0.877177 −0.438588 0.898688i $$-0.644521\pi$$
−0.438588 + 0.898688i $$0.644521\pi$$
$$614$$ −3.06485e9 −0.534342
$$615$$ 0 0
$$616$$ −1.11341e9 −0.191920
$$617$$ −1.47593e9 −0.252970 −0.126485 0.991969i $$-0.540370\pi$$
−0.126485 + 0.991969i $$0.540370\pi$$
$$618$$ 0 0
$$619$$ −9.36559e9 −1.58715 −0.793575 0.608472i $$-0.791783\pi$$
−0.793575 + 0.608472i $$0.791783\pi$$
$$620$$ −1.65202e9 −0.278385
$$621$$ 0 0
$$622$$ 7.07150e9 1.17827
$$623$$ −1.07455e9 −0.178041
$$624$$ 0 0
$$625$$ −1.09904e9 −0.180067
$$626$$ −1.01207e9 −0.164893
$$627$$ 0 0
$$628$$ −3.75087e9 −0.604328
$$629$$ 8.98698e9 1.43991
$$630$$ 0 0
$$631$$ −1.19966e10 −1.90088 −0.950438 0.310914i $$-0.899365\pi$$
−0.950438 + 0.310914i $$0.899365\pi$$
$$632$$ 7.57751e9 1.19403
$$633$$ 0 0
$$634$$ 5.53871e9 0.863170
$$635$$ −7.08771e9 −1.09849
$$636$$ 0 0
$$637$$ 2.36165e8 0.0362016
$$638$$ −1.34175e9 −0.204551
$$639$$ 0 0
$$640$$ −2.98484e9 −0.450081
$$641$$ 1.25626e10 1.88398 0.941989 0.335644i $$-0.108954\pi$$
0.941989 + 0.335644i $$0.108954\pi$$
$$642$$ 0 0
$$643$$ −4.11211e9 −0.609995 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$644$$ −5.19283e9 −0.766131
$$645$$ 0 0
$$646$$ −4.75064e8 −0.0693327
$$647$$ 8.36038e9 1.21356 0.606780 0.794870i $$-0.292461\pi$$
0.606780 + 0.794870i $$0.292461\pi$$
$$648$$ 0 0
$$649$$ −1.99700e8 −0.0286762
$$650$$ 3.65895e8 0.0522588
$$651$$ 0 0
$$652$$ −5.85415e9 −0.827175
$$653$$ −5.92578e8 −0.0832817 −0.0416409 0.999133i $$-0.513259\pi$$
−0.0416409 + 0.999133i $$0.513259\pi$$
$$654$$ 0 0
$$655$$ −2.86290e9 −0.398072
$$656$$ 1.92492e8 0.0266225
$$657$$ 0 0
$$658$$ 6.14769e9 0.841243
$$659$$ −3.61916e9 −0.492617 −0.246308 0.969192i $$-0.579218\pi$$
−0.246308 + 0.969192i $$0.579218\pi$$
$$660$$ 0 0
$$661$$ 1.49686e9 0.201593 0.100797 0.994907i $$-0.467861\pi$$
0.100797 + 0.994907i $$0.467861\pi$$
$$662$$ −1.17839e9 −0.157866
$$663$$ 0 0
$$664$$ −5.20521e9 −0.690002
$$665$$ −2.89757e8 −0.0382084
$$666$$ 0 0
$$667$$ −1.61435e10 −2.10648
$$668$$ 2.63066e9 0.341465
$$669$$ 0 0
$$670$$ −2.07102e8 −0.0266025
$$671$$ 2.08421e9 0.266326
$$672$$ 0 0
$$673$$ 1.55009e10 1.96022 0.980109 0.198461i $$-0.0635943\pi$$
0.980109 + 0.198461i $$0.0635943\pi$$
$$674$$ 1.81615e9 0.228476
$$675$$ 0 0
$$676$$ 4.95177e9 0.616520
$$677$$ 7.97429e9 0.987715 0.493857 0.869543i $$-0.335586\pi$$
0.493857 + 0.869543i $$0.335586\pi$$
$$678$$ 0 0
$$679$$ −2.23040e9 −0.273425
$$680$$ 9.96431e9 1.21525
$$681$$ 0 0
$$682$$ −7.12407e8 −0.0859969
$$683$$ −7.37364e9 −0.885542 −0.442771 0.896635i $$-0.646005\pi$$
−0.442771 + 0.896635i $$0.646005\pi$$
$$684$$ 0 0
$$685$$ −3.34356e9 −0.397458
$$686$$ 5.52317e9 0.653212
$$687$$ 0 0
$$688$$ 1.67871e8 0.0196524
$$689$$ 6.83868e8 0.0796535
$$690$$ 0 0
$$691$$ 5.70889e9 0.658232 0.329116 0.944290i $$-0.393249\pi$$
0.329116 + 0.944290i $$0.393249\pi$$
$$692$$ 1.06453e9 0.122120
$$693$$ 0 0
$$694$$ −3.52571e9 −0.400395
$$695$$ −2.22905e9 −0.251868
$$696$$ 0 0
$$697$$ −1.27034e10 −1.42103
$$698$$ −4.17543e8 −0.0464737
$$699$$ 0 0
$$700$$ −2.70374e9 −0.297935
$$701$$ −1.23928e10 −1.35880 −0.679402 0.733767i $$-0.737761\pi$$
−0.679402 + 0.733767i $$0.737761\pi$$
$$702$$ 0 0
$$703$$ −4.68410e8 −0.0508491
$$704$$ −1.17851e9 −0.127300
$$705$$ 0 0
$$706$$ 1.30640e9 0.139720
$$707$$ −1.14630e10 −1.21992
$$708$$ 0 0
$$709$$ 1.39979e10 1.47504 0.737518 0.675328i $$-0.235998\pi$$
0.737518 + 0.675328i $$0.235998\pi$$
$$710$$ 5.16271e9 0.541344
$$711$$ 0 0
$$712$$ −1.92599e9 −0.199974
$$713$$ −8.57143e9 −0.885605
$$714$$ 0 0
$$715$$ 2.37152e8 0.0242636
$$716$$ −1.61815e9 −0.164750
$$717$$ 0 0
$$718$$ 9.59215e9 0.967120
$$719$$ −7.66012e9 −0.768572 −0.384286 0.923214i $$-0.625552\pi$$
−0.384286 + 0.923214i $$0.625552\pi$$
$$720$$ 0 0
$$721$$ −1.30101e9 −0.129273
$$722$$ −6.10161e9 −0.603343
$$723$$ 0 0
$$724$$ −7.60430e9 −0.744688
$$725$$ −8.40543e9 −0.819175
$$726$$ 0 0
$$727$$ 1.02618e10 0.990493 0.495247 0.868752i $$-0.335078\pi$$
0.495247 + 0.868752i $$0.335078\pi$$
$$728$$ −1.46428e9 −0.140658
$$729$$ 0 0
$$730$$ 4.25940e9 0.405246
$$731$$ −1.10785e10 −1.04899
$$732$$ 0 0
$$733$$ −1.67714e10 −1.57292 −0.786459 0.617643i $$-0.788088\pi$$
−0.786459 + 0.617643i $$0.788088\pi$$
$$734$$ −1.02988e10 −0.961285
$$735$$ 0 0
$$736$$ −1.50069e10 −1.38746
$$737$$ 1.54051e8 0.0141752
$$738$$ 0 0
$$739$$ 5.49527e9 0.500880 0.250440 0.968132i $$-0.419425\pi$$
0.250440 + 0.968132i $$0.419425\pi$$
$$740$$ 3.80843e9 0.345490
$$741$$ 0 0
$$742$$ 2.92962e9 0.263268
$$743$$ 1.77986e9 0.159193 0.0795966 0.996827i $$-0.474637\pi$$
0.0795966 + 0.996827i $$0.474637\pi$$
$$744$$ 0 0
$$745$$ 1.29006e10 1.14305
$$746$$ 6.09683e8 0.0537673
$$747$$ 0 0
$$748$$ −2.87311e9 −0.251013
$$749$$ −6.28548e9 −0.546578
$$750$$ 0 0
$$751$$ −7.12062e9 −0.613448 −0.306724 0.951798i $$-0.599233\pi$$
−0.306724 + 0.951798i $$0.599233\pi$$
$$752$$ −6.20127e8 −0.0531763
$$753$$ 0 0
$$754$$ −1.76459e9 −0.149914
$$755$$ 8.23619e9 0.696485
$$756$$ 0 0
$$757$$ 6.43143e9 0.538855 0.269428 0.963021i $$-0.413166\pi$$
0.269428 + 0.963021i $$0.413166\pi$$
$$758$$ −4.98319e9 −0.415590
$$759$$ 0 0
$$760$$ −5.19350e8 −0.0429153
$$761$$ −1.46912e10 −1.20840 −0.604202 0.796831i $$-0.706508\pi$$
−0.604202 + 0.796831i $$0.706508\pi$$
$$762$$ 0 0
$$763$$ −2.10979e9 −0.171950
$$764$$ −6.08175e9 −0.493403
$$765$$ 0 0
$$766$$ −1.41698e10 −1.13910
$$767$$ −2.62632e8 −0.0210167
$$768$$ 0 0
$$769$$ 1.24049e10 0.983677 0.491838 0.870687i $$-0.336325\pi$$
0.491838 + 0.870687i $$0.336325\pi$$
$$770$$ 1.01594e9 0.0801952
$$771$$ 0 0
$$772$$ 5.78276e9 0.452350
$$773$$ 2.65186e9 0.206501 0.103250 0.994655i $$-0.467076\pi$$
0.103250 + 0.994655i $$0.467076\pi$$
$$774$$ 0 0
$$775$$ −4.46287e9 −0.344396
$$776$$ −3.99768e9 −0.307109
$$777$$ 0 0
$$778$$ 5.65113e8 0.0430236
$$779$$ 6.62112e8 0.0501823
$$780$$ 0 0
$$781$$ −3.84024e9 −0.288456
$$782$$ 2.00405e10 1.49860
$$783$$ 0 0
$$784$$ −1.02053e8 −0.00756342
$$785$$ 8.82916e9 0.651441
$$786$$ 0 0
$$787$$ −4.05957e9 −0.296871 −0.148436 0.988922i $$-0.547424\pi$$
−0.148436 + 0.988922i $$0.547424\pi$$
$$788$$ −1.13968e10 −0.829738
$$789$$ 0 0
$$790$$ −6.91415e9 −0.498935
$$791$$ −2.32951e10 −1.67358
$$792$$ 0 0
$$793$$ 2.74102e9 0.195189
$$794$$ 1.78740e9 0.126721
$$795$$ 0 0
$$796$$ −2.26177e9 −0.158947
$$797$$ −2.36833e10 −1.65706 −0.828530 0.559944i $$-0.810823\pi$$
−0.828530 + 0.559944i $$0.810823\pi$$
$$798$$ 0 0
$$799$$ 4.09248e10 2.83839
$$800$$ −7.81364e9 −0.539558
$$801$$ 0 0
$$802$$ 3.81867e9 0.261398
$$803$$ −3.16832e9 −0.215936
$$804$$ 0 0
$$805$$ 1.22234e10 0.825859
$$806$$ −9.36910e8 −0.0630268
$$807$$ 0 0
$$808$$ −2.05458e10 −1.37020
$$809$$ 9.66394e9 0.641704 0.320852 0.947129i $$-0.396031\pi$$
0.320852 + 0.947129i $$0.396031\pi$$
$$810$$ 0 0
$$811$$ 1.27359e10 0.838413 0.419207 0.907891i $$-0.362308\pi$$
0.419207 + 0.907891i $$0.362308\pi$$
$$812$$ 1.30392e10 0.854684
$$813$$ 0 0
$$814$$ 1.64232e9 0.106727
$$815$$ 1.37801e10 0.891661
$$816$$ 0 0
$$817$$ 5.77422e8 0.0370438
$$818$$ −1.08975e10 −0.696132
$$819$$ 0 0
$$820$$ −5.38333e9 −0.340959
$$821$$ −1.18429e10 −0.746893 −0.373446 0.927652i $$-0.621824\pi$$
−0.373446 + 0.927652i $$0.621824\pi$$
$$822$$ 0 0
$$823$$ 1.12076e10 0.700829 0.350415 0.936595i $$-0.386041\pi$$
0.350415 + 0.936595i $$0.386041\pi$$
$$824$$ −2.33188e9 −0.145198
$$825$$ 0 0
$$826$$ −1.12509e9 −0.0694636
$$827$$ 1.69838e10 1.04416 0.522078 0.852898i $$-0.325157\pi$$
0.522078 + 0.852898i $$0.325157\pi$$
$$828$$ 0 0
$$829$$ −4.59293e9 −0.279994 −0.139997 0.990152i $$-0.544709\pi$$
−0.139997 + 0.990152i $$0.544709\pi$$
$$830$$ 4.74953e9 0.288322
$$831$$ 0 0
$$832$$ −1.54990e9 −0.0932980
$$833$$ 6.73488e9 0.403713
$$834$$ 0 0
$$835$$ −6.19230e9 −0.368086
$$836$$ 1.49749e8 0.00886428
$$837$$ 0 0
$$838$$ −4.03015e9 −0.236574
$$839$$ 2.30945e10 1.35002 0.675012 0.737806i $$-0.264138\pi$$
0.675012 + 0.737806i $$0.264138\pi$$
$$840$$ 0 0
$$841$$ 2.32866e10 1.34996
$$842$$ −5.14664e9 −0.297120
$$843$$ 0 0
$$844$$ −1.34934e10 −0.772542
$$845$$ −1.16560e10 −0.664584
$$846$$ 0 0
$$847$$ 1.48202e10 0.838033
$$848$$ −2.95516e8 −0.0166416
$$849$$ 0 0
$$850$$ 1.04345e10 0.582779
$$851$$ 1.97598e10 1.09908
$$852$$ 0 0
$$853$$ 2.22496e10 1.22744 0.613721 0.789523i $$-0.289672\pi$$
0.613721 + 0.789523i $$0.289672\pi$$
$$854$$ 1.17422e10 0.645132
$$855$$ 0 0
$$856$$ −1.12659e10 −0.613912
$$857$$ −1.07105e10 −0.581270 −0.290635 0.956834i $$-0.593867\pi$$
−0.290635 + 0.956834i $$0.593867\pi$$
$$858$$ 0 0
$$859$$ 3.19391e10 1.71928 0.859639 0.510902i $$-0.170688\pi$$
0.859639 + 0.510902i $$0.170688\pi$$
$$860$$ −4.69475e9 −0.251691
$$861$$ 0 0
$$862$$ 1.62047e10 0.861717
$$863$$ −2.90663e10 −1.53940 −0.769700 0.638406i $$-0.779594\pi$$
−0.769700 + 0.638406i $$0.779594\pi$$
$$864$$ 0 0
$$865$$ −2.50580e9 −0.131641
$$866$$ 4.77979e8 0.0250090
$$867$$ 0 0
$$868$$ 6.92319e9 0.359325
$$869$$ 5.14304e9 0.265858
$$870$$ 0 0
$$871$$ 2.02598e8 0.0103889
$$872$$ −3.78150e9 −0.193133
$$873$$ 0 0
$$874$$ −1.04453e9 −0.0529215
$$875$$ 1.82742e10 0.922167
$$876$$ 0 0
$$877$$ −1.87443e9 −0.0938364 −0.0469182 0.998899i $$-0.514940\pi$$
−0.0469182 + 0.998899i $$0.514940\pi$$
$$878$$ −1.74616e10 −0.870668
$$879$$ 0 0
$$880$$ −1.02479e8 −0.00506927
$$881$$ 3.84330e10 1.89360 0.946802 0.321818i $$-0.104294\pi$$
0.946802 + 0.321818i $$0.104294\pi$$
$$882$$ 0 0
$$883$$ 1.17045e10 0.572125 0.286063 0.958211i $$-0.407653\pi$$
0.286063 + 0.958211i $$0.407653\pi$$
$$884$$ −3.77853e9 −0.183967
$$885$$ 0 0
$$886$$ −7.91134e9 −0.382148
$$887$$ −3.09354e9 −0.148841 −0.0744205 0.997227i $$-0.523711\pi$$
−0.0744205 + 0.997227i $$0.523711\pi$$
$$888$$ 0 0
$$889$$ 2.97027e10 1.41788
$$890$$ 1.75738e9 0.0835604
$$891$$ 0 0
$$892$$ −7.24615e8 −0.0341846
$$893$$ −2.13304e9 −0.100235
$$894$$ 0 0
$$895$$ 3.80897e9 0.177594
$$896$$ 1.25087e10 0.580942
$$897$$ 0 0
$$898$$ −3.81064e9 −0.175603
$$899$$ 2.15229e10 0.987967
$$900$$ 0 0
$$901$$ 1.95023e10 0.888279
$$902$$ −2.32147e9 −0.105327
$$903$$ 0 0
$$904$$ −4.17533e10 −1.87975
$$905$$ 1.78997e10 0.802744
$$906$$ 0 0
$$907$$ −5.95597e9 −0.265050 −0.132525 0.991180i $$-0.542308\pi$$
−0.132525 + 0.991180i $$0.542308\pi$$
$$908$$ −5.57599e9 −0.247185
$$909$$ 0 0
$$910$$ 1.33609e9 0.0587748
$$911$$ 4.08258e10 1.78904 0.894521 0.447026i $$-0.147517\pi$$
0.894521 + 0.447026i $$0.147517\pi$$
$$912$$ 0 0
$$913$$ −3.53290e9 −0.153633
$$914$$ 2.69735e9 0.116849
$$915$$ 0 0
$$916$$ −1.60032e10 −0.687977
$$917$$ 1.19976e10 0.513811
$$918$$ 0 0
$$919$$ −1.12519e10 −0.478214 −0.239107 0.970993i $$-0.576855\pi$$
−0.239107 + 0.970993i $$0.576855\pi$$
$$920$$ 2.19087e10 0.927598
$$921$$ 0 0
$$922$$ 2.05189e10 0.862176
$$923$$ −5.05043e9 −0.211408
$$924$$ 0 0
$$925$$ 1.02883e10 0.427414
$$926$$ −8.11271e9 −0.335759
$$927$$ 0 0
$$928$$ 3.76825e10 1.54783
$$929$$ −1.44429e10 −0.591018 −0.295509 0.955340i $$-0.595489\pi$$
−0.295509 + 0.955340i $$0.595489\pi$$
$$930$$ 0 0
$$931$$ −3.51029e8 −0.0142567
$$932$$ 1.99620e10 0.807696
$$933$$ 0 0
$$934$$ −2.16339e10 −0.868801
$$935$$ 6.76301e9 0.270582
$$936$$ 0 0
$$937$$ 2.43718e10 0.967829 0.483915 0.875115i $$-0.339215\pi$$
0.483915 + 0.875115i $$0.339215\pi$$
$$938$$ 8.67909e8 0.0343372
$$939$$ 0 0
$$940$$ 1.73428e10 0.681038
$$941$$ 1.04613e10 0.409280 0.204640 0.978837i $$-0.434398\pi$$
0.204640 + 0.978837i $$0.434398\pi$$
$$942$$ 0 0
$$943$$ −2.79311e10 −1.08467
$$944$$ 1.13490e8 0.00439090
$$945$$ 0 0
$$946$$ −2.02453e9 −0.0777509
$$947$$ 6.80097e9 0.260223 0.130112 0.991499i $$-0.458466\pi$$
0.130112 + 0.991499i $$0.458466\pi$$
$$948$$ 0 0
$$949$$ −4.16677e9 −0.158259
$$950$$ −5.43855e8 −0.0205802
$$951$$ 0 0
$$952$$ −4.17578e10 −1.56859
$$953$$ 9.17306e8 0.0343312 0.0171656 0.999853i $$-0.494536\pi$$
0.0171656 + 0.999853i $$0.494536\pi$$
$$954$$ 0 0
$$955$$ 1.43158e10 0.531869
$$956$$ −3.06488e10 −1.13452
$$957$$ 0 0
$$958$$ −1.55614e10 −0.571833
$$959$$ 1.40120e10 0.513019
$$960$$ 0 0
$$961$$ −1.60850e10 −0.584640
$$962$$ 2.15987e9 0.0782195
$$963$$ 0 0
$$964$$ 2.20377e10 0.792314
$$965$$ −1.36120e10 −0.487615
$$966$$ 0 0
$$967$$ −3.69970e10 −1.31575 −0.657875 0.753127i $$-0.728544\pi$$
−0.657875 + 0.753127i $$0.728544\pi$$
$$968$$ 2.65631e10 0.941271
$$969$$ 0 0
$$970$$ 3.64771e9 0.128327
$$971$$ 1.31210e10 0.459937 0.229968 0.973198i $$-0.426138\pi$$
0.229968 + 0.973198i $$0.426138\pi$$
$$972$$ 0 0
$$973$$ 9.34138e9 0.325099
$$974$$ 1.40498e9 0.0487209
$$975$$ 0 0
$$976$$ −1.18446e9 −0.0407798
$$977$$ 3.90962e10 1.34123 0.670615 0.741806i $$-0.266030\pi$$
0.670615 + 0.741806i $$0.266030\pi$$
$$978$$ 0 0
$$979$$ −1.30721e9 −0.0445253
$$980$$ 2.85406e9 0.0968660
$$981$$ 0 0
$$982$$ −7.42485e9 −0.250206
$$983$$ 3.75059e10 1.25940 0.629699 0.776840i $$-0.283178\pi$$
0.629699 + 0.776840i $$0.283178\pi$$
$$984$$ 0 0
$$985$$ 2.68269e10 0.894424
$$986$$ −5.03219e10 −1.67181
$$987$$ 0 0
$$988$$ 1.96941e8 0.00649660
$$989$$ −2.43585e10 −0.800688
$$990$$ 0 0
$$991$$ 5.39247e10 1.76007 0.880036 0.474907i $$-0.157518\pi$$
0.880036 + 0.474907i $$0.157518\pi$$
$$992$$ 2.00076e10 0.650735
$$993$$ 0 0
$$994$$ −2.16355e10 −0.698739
$$995$$ 5.32397e9 0.171338
$$996$$ 0 0
$$997$$ 2.01581e10 0.644192 0.322096 0.946707i $$-0.395613\pi$$
0.322096 + 0.946707i $$0.395613\pi$$
$$998$$ 8.48317e9 0.270148
$$999$$ 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 531.8.a.d.1.11 17
3.2 odd 2 177.8.a.b.1.7 17

By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.7 17 3.2 odd 2
531.8.a.d.1.11 17 1.1 even 1 trivial