# Properties

 Label 531.8.a.d.1.10 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 1.10 Root $$-4.01497$$ of defining polynomial Character $$\chi$$ $$=$$ 531.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.01497 q^{2} -91.8201 q^{4} +385.807 q^{5} +847.649 q^{7} -1322.21 q^{8} +O(q^{10})$$ $$q+6.01497 q^{2} -91.8201 q^{4} +385.807 q^{5} +847.649 q^{7} -1322.21 q^{8} +2320.62 q^{10} +5526.47 q^{11} +2987.05 q^{13} +5098.58 q^{14} +3799.90 q^{16} +6651.02 q^{17} +47421.5 q^{19} -35424.9 q^{20} +33241.6 q^{22} -16902.5 q^{23} +70722.2 q^{25} +17967.0 q^{26} -77831.2 q^{28} -2638.69 q^{29} +138735. q^{31} +192099. q^{32} +40005.7 q^{34} +327029. q^{35} -91219.9 q^{37} +285239. q^{38} -510119. q^{40} -549406. q^{41} -240843. q^{43} -507441. q^{44} -101668. q^{46} +1.17880e6 q^{47} -105035. q^{49} +425392. q^{50} -274271. q^{52} +100058. q^{53} +2.13215e6 q^{55} -1.12077e6 q^{56} -15871.6 q^{58} +205379. q^{59} +71464.3 q^{61} +834486. q^{62} +669085. q^{64} +1.15243e6 q^{65} -912586. q^{67} -610697. q^{68} +1.96707e6 q^{70} +274897. q^{71} -5.34005e6 q^{73} -548685. q^{74} -4.35425e6 q^{76} +4.68451e6 q^{77} +1.12488e6 q^{79} +1.46603e6 q^{80} -3.30466e6 q^{82} +4.95003e6 q^{83} +2.56601e6 q^{85} -1.44866e6 q^{86} -7.30717e6 q^{88} +8.64528e6 q^{89} +2.53197e6 q^{91} +1.55199e6 q^{92} +7.09048e6 q^{94} +1.82956e7 q^{95} -2.69412e6 q^{97} -631782. q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10})$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 6.01497 0.531654 0.265827 0.964021i $$-0.414355\pi$$
0.265827 + 0.964021i $$0.414355\pi$$
$$3$$ 0 0
$$4$$ −91.8201 −0.717345
$$5$$ 385.807 1.38031 0.690153 0.723664i $$-0.257543\pi$$
0.690153 + 0.723664i $$0.257543\pi$$
$$6$$ 0 0
$$7$$ 847.649 0.934055 0.467028 0.884243i $$-0.345325\pi$$
0.467028 + 0.884243i $$0.345325\pi$$
$$8$$ −1322.21 −0.913032
$$9$$ 0 0
$$10$$ 2320.62 0.733844
$$11$$ 5526.47 1.25191 0.625956 0.779859i $$-0.284709\pi$$
0.625956 + 0.779859i $$0.284709\pi$$
$$12$$ 0 0
$$13$$ 2987.05 0.377087 0.188543 0.982065i $$-0.439623\pi$$
0.188543 + 0.982065i $$0.439623\pi$$
$$14$$ 5098.58 0.496594
$$15$$ 0 0
$$16$$ 3799.90 0.231928
$$17$$ 6651.02 0.328335 0.164167 0.986432i $$-0.447506\pi$$
0.164167 + 0.986432i $$0.447506\pi$$
$$18$$ 0 0
$$19$$ 47421.5 1.58613 0.793064 0.609139i $$-0.208485\pi$$
0.793064 + 0.609139i $$0.208485\pi$$
$$20$$ −35424.9 −0.990155
$$21$$ 0 0
$$22$$ 33241.6 0.665583
$$23$$ −16902.5 −0.289670 −0.144835 0.989456i $$-0.546265\pi$$
−0.144835 + 0.989456i $$0.546265\pi$$
$$24$$ 0 0
$$25$$ 70722.2 0.905244
$$26$$ 17967.0 0.200479
$$27$$ 0 0
$$28$$ −77831.2 −0.670040
$$29$$ −2638.69 −0.0200907 −0.0100454 0.999950i $$-0.503198\pi$$
−0.0100454 + 0.999950i $$0.503198\pi$$
$$30$$ 0 0
$$31$$ 138735. 0.836410 0.418205 0.908353i $$-0.362659\pi$$
0.418205 + 0.908353i $$0.362659\pi$$
$$32$$ 192099. 1.03634
$$33$$ 0 0
$$34$$ 40005.7 0.174560
$$35$$ 327029. 1.28928
$$36$$ 0 0
$$37$$ −91219.9 −0.296063 −0.148031 0.988983i $$-0.547294\pi$$
−0.148031 + 0.988983i $$0.547294\pi$$
$$38$$ 285239. 0.843270
$$39$$ 0 0
$$40$$ −510119. −1.26026
$$41$$ −549406. −1.24494 −0.622472 0.782642i $$-0.713871\pi$$
−0.622472 + 0.782642i $$0.713871\pi$$
$$42$$ 0 0
$$43$$ −240843. −0.461949 −0.230974 0.972960i $$-0.574191\pi$$
−0.230974 + 0.972960i $$0.574191\pi$$
$$44$$ −507441. −0.898052
$$45$$ 0 0
$$46$$ −101668. −0.154004
$$47$$ 1.17880e6 1.65615 0.828074 0.560618i $$-0.189437\pi$$
0.828074 + 0.560618i $$0.189437\pi$$
$$48$$ 0 0
$$49$$ −105035. −0.127540
$$50$$ 425392. 0.481276
$$51$$ 0 0
$$52$$ −274271. −0.270501
$$53$$ 100058. 0.0923184 0.0461592 0.998934i $$-0.485302\pi$$
0.0461592 + 0.998934i $$0.485302\pi$$
$$54$$ 0 0
$$55$$ 2.13215e6 1.72802
$$56$$ −1.12077e6 −0.852823
$$57$$ 0 0
$$58$$ −15871.6 −0.0106813
$$59$$ 205379. 0.130189
$$60$$ 0 0
$$61$$ 71464.3 0.0403120 0.0201560 0.999797i $$-0.493584\pi$$
0.0201560 + 0.999797i $$0.493584\pi$$
$$62$$ 834486. 0.444680
$$63$$ 0 0
$$64$$ 669085. 0.319045
$$65$$ 1.15243e6 0.520495
$$66$$ 0 0
$$67$$ −912586. −0.370691 −0.185345 0.982673i $$-0.559340\pi$$
−0.185345 + 0.982673i $$0.559340\pi$$
$$68$$ −610697. −0.235529
$$69$$ 0 0
$$70$$ 1.96707e6 0.685451
$$71$$ 274897. 0.0911520 0.0455760 0.998961i $$-0.485488\pi$$
0.0455760 + 0.998961i $$0.485488\pi$$
$$72$$ 0 0
$$73$$ −5.34005e6 −1.60663 −0.803315 0.595555i $$-0.796932\pi$$
−0.803315 + 0.595555i $$0.796932\pi$$
$$74$$ −548685. −0.157403
$$75$$ 0 0
$$76$$ −4.35425e6 −1.13780
$$77$$ 4.68451e6 1.16935
$$78$$ 0 0
$$79$$ 1.12488e6 0.256692 0.128346 0.991729i $$-0.459033\pi$$
0.128346 + 0.991729i $$0.459033\pi$$
$$80$$ 1.46603e6 0.320131
$$81$$ 0 0
$$82$$ −3.30466e6 −0.661878
$$83$$ 4.95003e6 0.950243 0.475122 0.879920i $$-0.342404\pi$$
0.475122 + 0.879920i $$0.342404\pi$$
$$84$$ 0 0
$$85$$ 2.56601e6 0.453202
$$86$$ −1.44866e6 −0.245597
$$87$$ 0 0
$$88$$ −7.30717e6 −1.14304
$$89$$ 8.64528e6 1.29991 0.649956 0.759972i $$-0.274787\pi$$
0.649956 + 0.759972i $$0.274787\pi$$
$$90$$ 0 0
$$91$$ 2.53197e6 0.352220
$$92$$ 1.55199e6 0.207793
$$93$$ 0 0
$$94$$ 7.09048e6 0.880497
$$95$$ 1.82956e7 2.18934
$$96$$ 0 0
$$97$$ −2.69412e6 −0.299720 −0.149860 0.988707i $$-0.547882\pi$$
−0.149860 + 0.988707i $$0.547882\pi$$
$$98$$ −631782. −0.0678073
$$99$$ 0 0
$$100$$ −6.49372e6 −0.649372
$$101$$ −1.51552e7 −1.46365 −0.731826 0.681492i $$-0.761332\pi$$
−0.731826 + 0.681492i $$0.761332\pi$$
$$102$$ 0 0
$$103$$ −1.40028e7 −1.26266 −0.631328 0.775516i $$-0.717490\pi$$
−0.631328 + 0.775516i $$0.717490\pi$$
$$104$$ −3.94952e6 −0.344292
$$105$$ 0 0
$$106$$ 601849. 0.0490814
$$107$$ 8.87562e6 0.700415 0.350208 0.936672i $$-0.386111\pi$$
0.350208 + 0.936672i $$0.386111\pi$$
$$108$$ 0 0
$$109$$ −2.49654e7 −1.84648 −0.923242 0.384219i $$-0.874471\pi$$
−0.923242 + 0.384219i $$0.874471\pi$$
$$110$$ 1.28248e7 0.918708
$$111$$ 0 0
$$112$$ 3.22098e6 0.216633
$$113$$ −1.44492e7 −0.942038 −0.471019 0.882123i $$-0.656114\pi$$
−0.471019 + 0.882123i $$0.656114\pi$$
$$114$$ 0 0
$$115$$ −6.52110e6 −0.399833
$$116$$ 242285. 0.0144120
$$117$$ 0 0
$$118$$ 1.23535e6 0.0692154
$$119$$ 5.63773e6 0.306683
$$120$$ 0 0
$$121$$ 1.10547e7 0.567283
$$122$$ 429856. 0.0214320
$$123$$ 0 0
$$124$$ −1.27386e7 −0.599994
$$125$$ −2.85605e6 −0.130792
$$126$$ 0 0
$$127$$ 1.71357e7 0.742315 0.371157 0.928570i $$-0.378961\pi$$
0.371157 + 0.928570i $$0.378961\pi$$
$$128$$ −2.05642e7 −0.866716
$$129$$ 0 0
$$130$$ 6.93181e6 0.276723
$$131$$ 1.82887e7 0.710777 0.355388 0.934719i $$-0.384349\pi$$
0.355388 + 0.934719i $$0.384349\pi$$
$$132$$ 0 0
$$133$$ 4.01968e7 1.48153
$$134$$ −5.48918e6 −0.197079
$$135$$ 0 0
$$136$$ −8.79406e6 −0.299780
$$137$$ 4.79157e7 1.59205 0.796023 0.605266i $$-0.206933\pi$$
0.796023 + 0.605266i $$0.206933\pi$$
$$138$$ 0 0
$$139$$ 2.76336e7 0.872740 0.436370 0.899767i $$-0.356264\pi$$
0.436370 + 0.899767i $$0.356264\pi$$
$$140$$ −3.00278e7 −0.924860
$$141$$ 0 0
$$142$$ 1.65350e6 0.0484613
$$143$$ 1.65079e7 0.472079
$$144$$ 0 0
$$145$$ −1.01803e6 −0.0277313
$$146$$ −3.21203e7 −0.854170
$$147$$ 0 0
$$148$$ 8.37582e6 0.212379
$$149$$ −6.17248e6 −0.152865 −0.0764325 0.997075i $$-0.524353\pi$$
−0.0764325 + 0.997075i $$0.524353\pi$$
$$150$$ 0 0
$$151$$ −2.68093e7 −0.633673 −0.316837 0.948480i $$-0.602621\pi$$
−0.316837 + 0.948480i $$0.602621\pi$$
$$152$$ −6.27013e7 −1.44819
$$153$$ 0 0
$$154$$ 2.81772e7 0.621692
$$155$$ 5.35249e7 1.15450
$$156$$ 0 0
$$157$$ 1.87147e7 0.385952 0.192976 0.981203i $$-0.438186\pi$$
0.192976 + 0.981203i $$0.438186\pi$$
$$158$$ 6.76614e6 0.136471
$$159$$ 0 0
$$160$$ 7.41134e7 1.43046
$$161$$ −1.43274e7 −0.270568
$$162$$ 0 0
$$163$$ 9.00787e7 1.62917 0.814583 0.580047i $$-0.196966\pi$$
0.814583 + 0.580047i $$0.196966\pi$$
$$164$$ 5.04465e7 0.893053
$$165$$ 0 0
$$166$$ 2.97743e7 0.505200
$$167$$ 6.92715e7 1.15093 0.575463 0.817828i $$-0.304822\pi$$
0.575463 + 0.817828i $$0.304822\pi$$
$$168$$ 0 0
$$169$$ −5.38260e7 −0.857806
$$170$$ 1.54345e7 0.240947
$$171$$ 0 0
$$172$$ 2.21142e7 0.331376
$$173$$ 1.01646e8 1.49255 0.746277 0.665635i $$-0.231839\pi$$
0.746277 + 0.665635i $$0.231839\pi$$
$$174$$ 0 0
$$175$$ 5.99476e7 0.845548
$$176$$ 2.10001e7 0.290353
$$177$$ 0 0
$$178$$ 5.20011e7 0.691103
$$179$$ −1.59338e7 −0.207650 −0.103825 0.994596i $$-0.533108\pi$$
−0.103825 + 0.994596i $$0.533108\pi$$
$$180$$ 0 0
$$181$$ 8.29983e7 1.04038 0.520192 0.854049i $$-0.325860\pi$$
0.520192 + 0.854049i $$0.325860\pi$$
$$182$$ 1.52297e7 0.187259
$$183$$ 0 0
$$184$$ 2.23487e7 0.264478
$$185$$ −3.51933e7 −0.408657
$$186$$ 0 0
$$187$$ 3.67567e7 0.411046
$$188$$ −1.08238e8 −1.18803
$$189$$ 0 0
$$190$$ 1.10047e8 1.16397
$$191$$ 2.94448e7 0.305768 0.152884 0.988244i $$-0.451144\pi$$
0.152884 + 0.988244i $$0.451144\pi$$
$$192$$ 0 0
$$193$$ 8.61854e7 0.862945 0.431473 0.902126i $$-0.357994\pi$$
0.431473 + 0.902126i $$0.357994\pi$$
$$194$$ −1.62051e7 −0.159347
$$195$$ 0 0
$$196$$ 9.64432e6 0.0914904
$$197$$ 1.74170e8 1.62308 0.811541 0.584296i $$-0.198629\pi$$
0.811541 + 0.584296i $$0.198629\pi$$
$$198$$ 0 0
$$199$$ 9.07182e7 0.816035 0.408017 0.912974i $$-0.366220\pi$$
0.408017 + 0.912974i $$0.366220\pi$$
$$200$$ −9.35098e7 −0.826517
$$201$$ 0 0
$$202$$ −9.11583e7 −0.778155
$$203$$ −2.23668e6 −0.0187658
$$204$$ 0 0
$$205$$ −2.11965e8 −1.71840
$$206$$ −8.42266e7 −0.671296
$$207$$ 0 0
$$208$$ 1.13505e7 0.0874568
$$209$$ 2.62074e8 1.98569
$$210$$ 0 0
$$211$$ −1.22589e8 −0.898387 −0.449194 0.893434i $$-0.648289\pi$$
−0.449194 + 0.893434i $$0.648289\pi$$
$$212$$ −9.18737e6 −0.0662241
$$213$$ 0 0
$$214$$ 5.33866e7 0.372378
$$215$$ −9.29188e7 −0.637631
$$216$$ 0 0
$$217$$ 1.17598e8 0.781254
$$218$$ −1.50166e8 −0.981690
$$219$$ 0 0
$$220$$ −1.95775e8 −1.23959
$$221$$ 1.98669e7 0.123811
$$222$$ 0 0
$$223$$ −5.48773e7 −0.331380 −0.165690 0.986178i $$-0.552985\pi$$
−0.165690 + 0.986178i $$0.552985\pi$$
$$224$$ 1.62833e8 0.967997
$$225$$ 0 0
$$226$$ −8.69114e7 −0.500838
$$227$$ −1.64956e8 −0.936003 −0.468001 0.883728i $$-0.655026\pi$$
−0.468001 + 0.883728i $$0.655026\pi$$
$$228$$ 0 0
$$229$$ 1.18120e8 0.649980 0.324990 0.945717i $$-0.394639\pi$$
0.324990 + 0.945717i $$0.394639\pi$$
$$230$$ −3.92243e7 −0.212573
$$231$$ 0 0
$$232$$ 3.48891e6 0.0183435
$$233$$ 1.12335e8 0.581795 0.290898 0.956754i $$-0.406046\pi$$
0.290898 + 0.956754i $$0.406046\pi$$
$$234$$ 0 0
$$235$$ 4.54791e8 2.28599
$$236$$ −1.88579e7 −0.0933903
$$237$$ 0 0
$$238$$ 3.39108e7 0.163049
$$239$$ −2.96755e8 −1.40607 −0.703033 0.711158i $$-0.748171\pi$$
−0.703033 + 0.711158i $$0.748171\pi$$
$$240$$ 0 0
$$241$$ 1.37243e7 0.0631583 0.0315791 0.999501i $$-0.489946\pi$$
0.0315791 + 0.999501i $$0.489946\pi$$
$$242$$ 6.64939e7 0.301598
$$243$$ 0 0
$$244$$ −6.56186e6 −0.0289176
$$245$$ −4.05232e7 −0.176045
$$246$$ 0 0
$$247$$ 1.41651e8 0.598107
$$248$$ −1.83437e8 −0.763670
$$249$$ 0 0
$$250$$ −1.71791e7 −0.0695359
$$251$$ 2.78781e7 0.111277 0.0556385 0.998451i $$-0.482281\pi$$
0.0556385 + 0.998451i $$0.482281\pi$$
$$252$$ 0 0
$$253$$ −9.34112e7 −0.362641
$$254$$ 1.03071e8 0.394654
$$255$$ 0 0
$$256$$ −2.09336e8 −0.779837
$$257$$ −1.86791e8 −0.686419 −0.343209 0.939259i $$-0.611514\pi$$
−0.343209 + 0.939259i $$0.611514\pi$$
$$258$$ 0 0
$$259$$ −7.73224e7 −0.276539
$$260$$ −1.05816e8 −0.373374
$$261$$ 0 0
$$262$$ 1.10006e8 0.377887
$$263$$ −2.51415e8 −0.852210 −0.426105 0.904674i $$-0.640115\pi$$
−0.426105 + 0.904674i $$0.640115\pi$$
$$264$$ 0 0
$$265$$ 3.86033e7 0.127428
$$266$$ 2.41783e8 0.787661
$$267$$ 0 0
$$268$$ 8.37937e7 0.265913
$$269$$ 4.66642e8 1.46168 0.730838 0.682551i $$-0.239130\pi$$
0.730838 + 0.682551i $$0.239130\pi$$
$$270$$ 0 0
$$271$$ 4.71085e8 1.43783 0.718914 0.695099i $$-0.244640\pi$$
0.718914 + 0.695099i $$0.244640\pi$$
$$272$$ 2.52732e7 0.0761500
$$273$$ 0 0
$$274$$ 2.88212e8 0.846417
$$275$$ 3.90844e8 1.13329
$$276$$ 0 0
$$277$$ −4.94424e8 −1.39772 −0.698860 0.715259i $$-0.746309\pi$$
−0.698860 + 0.715259i $$0.746309\pi$$
$$278$$ 1.66215e8 0.463995
$$279$$ 0 0
$$280$$ −4.32402e8 −1.17716
$$281$$ 1.88387e8 0.506499 0.253249 0.967401i $$-0.418501\pi$$
0.253249 + 0.967401i $$0.418501\pi$$
$$282$$ 0 0
$$283$$ 3.45526e8 0.906208 0.453104 0.891458i $$-0.350317\pi$$
0.453104 + 0.891458i $$0.350317\pi$$
$$284$$ −2.52411e7 −0.0653874
$$285$$ 0 0
$$286$$ 9.92943e7 0.250982
$$287$$ −4.65703e8 −1.16285
$$288$$ 0 0
$$289$$ −3.66103e8 −0.892196
$$290$$ −6.12340e6 −0.0147435
$$291$$ 0 0
$$292$$ 4.90324e8 1.15251
$$293$$ −4.35989e8 −1.01260 −0.506301 0.862357i $$-0.668988\pi$$
−0.506301 + 0.862357i $$0.668988\pi$$
$$294$$ 0 0
$$295$$ 7.92367e7 0.179701
$$296$$ 1.20612e8 0.270315
$$297$$ 0 0
$$298$$ −3.71273e7 −0.0812712
$$299$$ −5.04886e7 −0.109231
$$300$$ 0 0
$$301$$ −2.04150e8 −0.431486
$$302$$ −1.61257e8 −0.336895
$$303$$ 0 0
$$304$$ 1.80197e8 0.367867
$$305$$ 2.75714e7 0.0556429
$$306$$ 0 0
$$307$$ 9.90146e8 1.95306 0.976529 0.215387i $$-0.0691013\pi$$
0.976529 + 0.215387i $$0.0691013\pi$$
$$308$$ −4.30132e8 −0.838830
$$309$$ 0 0
$$310$$ 3.21951e8 0.613795
$$311$$ 2.78364e8 0.524750 0.262375 0.964966i $$-0.415494\pi$$
0.262375 + 0.964966i $$0.415494\pi$$
$$312$$ 0 0
$$313$$ −7.58142e8 −1.39748 −0.698740 0.715376i $$-0.746256\pi$$
−0.698740 + 0.715376i $$0.746256\pi$$
$$314$$ 1.12568e8 0.205193
$$315$$ 0 0
$$316$$ −1.03287e8 −0.184137
$$317$$ 3.93211e8 0.693296 0.346648 0.937995i $$-0.387320\pi$$
0.346648 + 0.937995i $$0.387320\pi$$
$$318$$ 0 0
$$319$$ −1.45826e7 −0.0251518
$$320$$ 2.58138e8 0.440379
$$321$$ 0 0
$$322$$ −8.61788e7 −0.143848
$$323$$ 3.15402e8 0.520781
$$324$$ 0 0
$$325$$ 2.11251e8 0.341355
$$326$$ 5.41821e8 0.866152
$$327$$ 0 0
$$328$$ 7.26431e8 1.13667
$$329$$ 9.99212e8 1.54693
$$330$$ 0 0
$$331$$ −9.50914e7 −0.144126 −0.0720632 0.997400i $$-0.522958\pi$$
−0.0720632 + 0.997400i $$0.522958\pi$$
$$332$$ −4.54512e8 −0.681652
$$333$$ 0 0
$$334$$ 4.16666e8 0.611894
$$335$$ −3.52082e8 −0.511667
$$336$$ 0 0
$$337$$ −6.91531e8 −0.984254 −0.492127 0.870524i $$-0.663780\pi$$
−0.492127 + 0.870524i $$0.663780\pi$$
$$338$$ −3.23762e8 −0.456055
$$339$$ 0 0
$$340$$ −2.35611e8 −0.325102
$$341$$ 7.66714e8 1.04711
$$342$$ 0 0
$$343$$ −7.87108e8 −1.05319
$$344$$ 3.18445e8 0.421774
$$345$$ 0 0
$$346$$ 6.11400e8 0.793522
$$347$$ 5.15844e8 0.662774 0.331387 0.943495i $$-0.392484\pi$$
0.331387 + 0.943495i $$0.392484\pi$$
$$348$$ 0 0
$$349$$ −8.89896e8 −1.12060 −0.560300 0.828290i $$-0.689314\pi$$
−0.560300 + 0.828290i $$0.689314\pi$$
$$350$$ 3.60583e8 0.449539
$$351$$ 0 0
$$352$$ 1.06163e9 1.29740
$$353$$ 3.85325e8 0.466246 0.233123 0.972447i $$-0.425105\pi$$
0.233123 + 0.972447i $$0.425105\pi$$
$$354$$ 0 0
$$355$$ 1.06057e8 0.125818
$$356$$ −7.93810e8 −0.932485
$$357$$ 0 0
$$358$$ −9.58411e7 −0.110398
$$359$$ 1.51447e9 1.72755 0.863777 0.503875i $$-0.168093\pi$$
0.863777 + 0.503875i $$0.168093\pi$$
$$360$$ 0 0
$$361$$ 1.35493e9 1.51580
$$362$$ 4.99232e8 0.553124
$$363$$ 0 0
$$364$$ −2.32486e8 −0.252663
$$365$$ −2.06023e9 −2.21764
$$366$$ 0 0
$$367$$ −8.13769e8 −0.859350 −0.429675 0.902984i $$-0.641372\pi$$
−0.429675 + 0.902984i $$0.641372\pi$$
$$368$$ −6.42279e7 −0.0671825
$$369$$ 0 0
$$370$$ −2.11687e8 −0.217264
$$371$$ 8.48144e7 0.0862305
$$372$$ 0 0
$$373$$ −1.20212e9 −1.19940 −0.599702 0.800223i $$-0.704714\pi$$
−0.599702 + 0.800223i $$0.704714\pi$$
$$374$$ 2.21090e8 0.218534
$$375$$ 0 0
$$376$$ −1.55863e9 −1.51212
$$377$$ −7.88190e6 −0.00757594
$$378$$ 0 0
$$379$$ −4.34931e8 −0.410377 −0.205189 0.978722i $$-0.565781\pi$$
−0.205189 + 0.978722i $$0.565781\pi$$
$$380$$ −1.67990e9 −1.57051
$$381$$ 0 0
$$382$$ 1.77110e8 0.162563
$$383$$ −3.58726e7 −0.0326263 −0.0163131 0.999867i $$-0.505193\pi$$
−0.0163131 + 0.999867i $$0.505193\pi$$
$$384$$ 0 0
$$385$$ 1.80732e9 1.61407
$$386$$ 5.18403e8 0.458788
$$387$$ 0 0
$$388$$ 2.47374e8 0.215003
$$389$$ 1.13638e9 0.978815 0.489407 0.872055i $$-0.337213\pi$$
0.489407 + 0.872055i $$0.337213\pi$$
$$390$$ 0 0
$$391$$ −1.12419e8 −0.0951087
$$392$$ 1.38878e8 0.116448
$$393$$ 0 0
$$394$$ 1.04762e9 0.862917
$$395$$ 4.33988e8 0.354314
$$396$$ 0 0
$$397$$ −5.01657e8 −0.402383 −0.201192 0.979552i $$-0.564481\pi$$
−0.201192 + 0.979552i $$0.564481\pi$$
$$398$$ 5.45667e8 0.433848
$$399$$ 0 0
$$400$$ 2.68738e8 0.209951
$$401$$ −6.17044e8 −0.477871 −0.238936 0.971035i $$-0.576798\pi$$
−0.238936 + 0.971035i $$0.576798\pi$$
$$402$$ 0 0
$$403$$ 4.14408e8 0.315399
$$404$$ 1.39155e9 1.04994
$$405$$ 0 0
$$406$$ −1.34536e7 −0.00997692
$$407$$ −5.04124e8 −0.370644
$$408$$ 0 0
$$409$$ 1.67124e9 1.20783 0.603915 0.797048i $$-0.293606\pi$$
0.603915 + 0.797048i $$0.293606\pi$$
$$410$$ −1.27496e9 −0.913595
$$411$$ 0 0
$$412$$ 1.28574e9 0.905760
$$413$$ 1.74089e8 0.121604
$$414$$ 0 0
$$415$$ 1.90976e9 1.31163
$$416$$ 5.73811e8 0.390789
$$417$$ 0 0
$$418$$ 1.57637e9 1.05570
$$419$$ 6.96670e7 0.0462677 0.0231339 0.999732i $$-0.492636\pi$$
0.0231339 + 0.999732i $$0.492636\pi$$
$$420$$ 0 0
$$421$$ −2.02237e9 −1.32091 −0.660457 0.750864i $$-0.729637\pi$$
−0.660457 + 0.750864i $$0.729637\pi$$
$$422$$ −7.37371e8 −0.477631
$$423$$ 0 0
$$424$$ −1.32298e8 −0.0842896
$$425$$ 4.70375e8 0.297223
$$426$$ 0 0
$$427$$ 6.05766e7 0.0376537
$$428$$ −8.14960e8 −0.502439
$$429$$ 0 0
$$430$$ −5.58904e8 −0.338999
$$431$$ 2.20133e9 1.32439 0.662193 0.749333i $$-0.269626\pi$$
0.662193 + 0.749333i $$0.269626\pi$$
$$432$$ 0 0
$$433$$ −1.11885e9 −0.662316 −0.331158 0.943575i $$-0.607439\pi$$
−0.331158 + 0.943575i $$0.607439\pi$$
$$434$$ 7.07351e8 0.415356
$$435$$ 0 0
$$436$$ 2.29232e9 1.32457
$$437$$ −8.01542e8 −0.459453
$$438$$ 0 0
$$439$$ 3.23226e8 0.182340 0.0911698 0.995835i $$-0.470939\pi$$
0.0911698 + 0.995835i $$0.470939\pi$$
$$440$$ −2.81916e9 −1.57774
$$441$$ 0 0
$$442$$ 1.19499e8 0.0658244
$$443$$ 2.28214e9 1.24718 0.623591 0.781751i $$-0.285673\pi$$
0.623591 + 0.781751i $$0.285673\pi$$
$$444$$ 0 0
$$445$$ 3.33541e9 1.79428
$$446$$ −3.30086e8 −0.176179
$$447$$ 0 0
$$448$$ 5.67149e8 0.298005
$$449$$ −2.89839e9 −1.51111 −0.755553 0.655087i $$-0.772632\pi$$
−0.755553 + 0.655087i $$0.772632\pi$$
$$450$$ 0 0
$$451$$ −3.03627e9 −1.55856
$$452$$ 1.32672e9 0.675766
$$453$$ 0 0
$$454$$ −9.92205e8 −0.497629
$$455$$ 9.76852e8 0.486171
$$456$$ 0 0
$$457$$ −1.61157e9 −0.789844 −0.394922 0.918715i $$-0.629228\pi$$
−0.394922 + 0.918715i $$0.629228\pi$$
$$458$$ 7.10490e8 0.345564
$$459$$ 0 0
$$460$$ 5.98768e8 0.286818
$$461$$ −4.98049e8 −0.236766 −0.118383 0.992968i $$-0.537771\pi$$
−0.118383 + 0.992968i $$0.537771\pi$$
$$462$$ 0 0
$$463$$ −1.13218e9 −0.530131 −0.265065 0.964230i $$-0.585394\pi$$
−0.265065 + 0.964230i $$0.585394\pi$$
$$464$$ −1.00268e7 −0.00465959
$$465$$ 0 0
$$466$$ 6.75693e8 0.309313
$$467$$ −3.04377e9 −1.38294 −0.691470 0.722405i $$-0.743037\pi$$
−0.691470 + 0.722405i $$0.743037\pi$$
$$468$$ 0 0
$$469$$ −7.73552e8 −0.346246
$$470$$ 2.73556e9 1.21536
$$471$$ 0 0
$$472$$ −2.71555e8 −0.118867
$$473$$ −1.33101e9 −0.578319
$$474$$ 0 0
$$475$$ 3.35376e9 1.43583
$$476$$ −5.17657e8 −0.219997
$$477$$ 0 0
$$478$$ −1.78497e9 −0.747539
$$479$$ −1.85843e9 −0.772630 −0.386315 0.922367i $$-0.626252\pi$$
−0.386315 + 0.922367i $$0.626252\pi$$
$$480$$ 0 0
$$481$$ −2.72479e8 −0.111641
$$482$$ 8.25512e7 0.0335783
$$483$$ 0 0
$$484$$ −1.01505e9 −0.406937
$$485$$ −1.03941e9 −0.413705
$$486$$ 0 0
$$487$$ −3.95871e8 −0.155311 −0.0776555 0.996980i $$-0.524743\pi$$
−0.0776555 + 0.996980i $$0.524743\pi$$
$$488$$ −9.44909e7 −0.0368062
$$489$$ 0 0
$$490$$ −2.43746e8 −0.0935948
$$491$$ −8.60610e8 −0.328111 −0.164056 0.986451i $$-0.552458\pi$$
−0.164056 + 0.986451i $$0.552458\pi$$
$$492$$ 0 0
$$493$$ −1.75500e7 −0.00659648
$$494$$ 8.52025e8 0.317986
$$495$$ 0 0
$$496$$ 5.27179e8 0.193987
$$497$$ 2.33016e8 0.0851410
$$498$$ 0 0
$$499$$ 2.92320e9 1.05319 0.526595 0.850116i $$-0.323469\pi$$
0.526595 + 0.850116i $$0.323469\pi$$
$$500$$ 2.62243e8 0.0938228
$$501$$ 0 0
$$502$$ 1.67686e8 0.0591609
$$503$$ −2.74156e9 −0.960529 −0.480264 0.877124i $$-0.659459\pi$$
−0.480264 + 0.877124i $$0.659459\pi$$
$$504$$ 0 0
$$505$$ −5.84700e9 −2.02029
$$506$$ −5.61866e8 −0.192799
$$507$$ 0 0
$$508$$ −1.57340e9 −0.532495
$$509$$ −3.66404e9 −1.23154 −0.615769 0.787927i $$-0.711154\pi$$
−0.615769 + 0.787927i $$0.711154\pi$$
$$510$$ 0 0
$$511$$ −4.52649e9 −1.50068
$$512$$ 1.37307e9 0.452113
$$513$$ 0 0
$$514$$ −1.12354e9 −0.364937
$$515$$ −5.40239e9 −1.74285
$$516$$ 0 0
$$517$$ 6.51463e9 2.07335
$$518$$ −4.65092e8 −0.147023
$$519$$ 0 0
$$520$$ −1.52375e9 −0.475229
$$521$$ 4.43075e9 1.37260 0.686302 0.727317i $$-0.259233\pi$$
0.686302 + 0.727317i $$0.259233\pi$$
$$522$$ 0 0
$$523$$ −5.16773e9 −1.57959 −0.789794 0.613372i $$-0.789813\pi$$
−0.789794 + 0.613372i $$0.789813\pi$$
$$524$$ −1.67927e9 −0.509872
$$525$$ 0 0
$$526$$ −1.51226e9 −0.453080
$$527$$ 9.22728e8 0.274623
$$528$$ 0 0
$$529$$ −3.11913e9 −0.916091
$$530$$ 2.32198e8 0.0677473
$$531$$ 0 0
$$532$$ −3.69087e9 −1.06277
$$533$$ −1.64110e9 −0.469451
$$534$$ 0 0
$$535$$ 3.42428e9 0.966787
$$536$$ 1.20663e9 0.338453
$$537$$ 0 0
$$538$$ 2.80684e9 0.777105
$$539$$ −5.80473e8 −0.159669
$$540$$ 0 0
$$541$$ 5.16094e9 1.40132 0.700662 0.713493i $$-0.252888\pi$$
0.700662 + 0.713493i $$0.252888\pi$$
$$542$$ 2.83356e9 0.764426
$$543$$ 0 0
$$544$$ 1.27766e9 0.340266
$$545$$ −9.63183e9 −2.54871
$$546$$ 0 0
$$547$$ −1.77725e9 −0.464295 −0.232147 0.972681i $$-0.574575\pi$$
−0.232147 + 0.972681i $$0.574575\pi$$
$$548$$ −4.39962e9 −1.14205
$$549$$ 0 0
$$550$$ 2.35092e9 0.602515
$$551$$ −1.25131e8 −0.0318664
$$552$$ 0 0
$$553$$ 9.53505e8 0.239765
$$554$$ −2.97394e9 −0.743102
$$555$$ 0 0
$$556$$ −2.53732e9 −0.626055
$$557$$ 2.65766e9 0.651637 0.325819 0.945432i $$-0.394360\pi$$
0.325819 + 0.945432i $$0.394360\pi$$
$$558$$ 0 0
$$559$$ −7.19409e8 −0.174195
$$560$$ 1.24268e9 0.299020
$$561$$ 0 0
$$562$$ 1.13314e9 0.269282
$$563$$ 2.85664e9 0.674646 0.337323 0.941389i $$-0.390478\pi$$
0.337323 + 0.941389i $$0.390478\pi$$
$$564$$ 0 0
$$565$$ −5.57459e9 −1.30030
$$566$$ 2.07833e9 0.481789
$$567$$ 0 0
$$568$$ −3.63472e8 −0.0832247
$$569$$ 1.18310e9 0.269233 0.134617 0.990898i $$-0.457020\pi$$
0.134617 + 0.990898i $$0.457020\pi$$
$$570$$ 0 0
$$571$$ 7.04152e8 0.158285 0.0791426 0.996863i $$-0.474782\pi$$
0.0791426 + 0.996863i $$0.474782\pi$$
$$572$$ −1.51575e9 −0.338643
$$573$$ 0 0
$$574$$ −2.80119e9 −0.618231
$$575$$ −1.19538e9 −0.262222
$$576$$ 0 0
$$577$$ −5.26118e9 −1.14017 −0.570083 0.821587i $$-0.693089\pi$$
−0.570083 + 0.821587i $$0.693089\pi$$
$$578$$ −2.20210e9 −0.474339
$$579$$ 0 0
$$580$$ 9.34752e7 0.0198929
$$581$$ 4.19589e9 0.887580
$$582$$ 0 0
$$583$$ 5.52970e8 0.115574
$$584$$ 7.06068e9 1.46690
$$585$$ 0 0
$$586$$ −2.62246e9 −0.538354
$$587$$ −7.97467e9 −1.62734 −0.813672 0.581324i $$-0.802535\pi$$
−0.813672 + 0.581324i $$0.802535\pi$$
$$588$$ 0 0
$$589$$ 6.57902e9 1.32665
$$590$$ 4.76607e8 0.0955384
$$591$$ 0 0
$$592$$ −3.46627e8 −0.0686651
$$593$$ 5.84550e9 1.15115 0.575573 0.817750i $$-0.304779\pi$$
0.575573 + 0.817750i $$0.304779\pi$$
$$594$$ 0 0
$$595$$ 2.17508e9 0.423316
$$596$$ 5.66758e8 0.109657
$$597$$ 0 0
$$598$$ −3.03688e8 −0.0580729
$$599$$ 2.93990e9 0.558906 0.279453 0.960159i $$-0.409847\pi$$
0.279453 + 0.960159i $$0.409847\pi$$
$$600$$ 0 0
$$601$$ −6.68201e9 −1.25559 −0.627793 0.778380i $$-0.716042\pi$$
−0.627793 + 0.778380i $$0.716042\pi$$
$$602$$ −1.22796e9 −0.229401
$$603$$ 0 0
$$604$$ 2.46163e9 0.454562
$$605$$ 4.26500e9 0.783024
$$606$$ 0 0
$$607$$ 4.48841e8 0.0814577 0.0407289 0.999170i $$-0.487032\pi$$
0.0407289 + 0.999170i $$0.487032\pi$$
$$608$$ 9.10965e9 1.64376
$$609$$ 0 0
$$610$$ 1.65841e8 0.0295827
$$611$$ 3.52115e9 0.624511
$$612$$ 0 0
$$613$$ 7.78075e9 1.36430 0.682150 0.731212i $$-0.261045\pi$$
0.682150 + 0.731212i $$0.261045\pi$$
$$614$$ 5.95570e9 1.03835
$$615$$ 0 0
$$616$$ −6.19391e9 −1.06766
$$617$$ 6.97925e9 1.19622 0.598109 0.801414i $$-0.295919\pi$$
0.598109 + 0.801414i $$0.295919\pi$$
$$618$$ 0 0
$$619$$ −3.49030e9 −0.591488 −0.295744 0.955267i $$-0.595568\pi$$
−0.295744 + 0.955267i $$0.595568\pi$$
$$620$$ −4.91466e9 −0.828176
$$621$$ 0 0
$$622$$ 1.67435e9 0.278985
$$623$$ 7.32816e9 1.21419
$$624$$ 0 0
$$625$$ −6.62706e9 −1.08578
$$626$$ −4.56021e9 −0.742975
$$627$$ 0 0
$$628$$ −1.71838e9 −0.276861
$$629$$ −6.06705e8 −0.0972077
$$630$$ 0 0
$$631$$ −7.42461e9 −1.17644 −0.588222 0.808700i $$-0.700172\pi$$
−0.588222 + 0.808700i $$0.700172\pi$$
$$632$$ −1.48733e9 −0.234368
$$633$$ 0 0
$$634$$ 2.36516e9 0.368593
$$635$$ 6.61107e9 1.02462
$$636$$ 0 0
$$637$$ −3.13745e8 −0.0480938
$$638$$ −8.77142e7 −0.0133720
$$639$$ 0 0
$$640$$ −7.93382e9 −1.19633
$$641$$ 2.52065e9 0.378015 0.189007 0.981976i $$-0.439473\pi$$
0.189007 + 0.981976i $$0.439473\pi$$
$$642$$ 0 0
$$643$$ −7.35494e9 −1.09104 −0.545520 0.838098i $$-0.683668\pi$$
−0.545520 + 0.838098i $$0.683668\pi$$
$$644$$ 1.31554e9 0.194090
$$645$$ 0 0
$$646$$ 1.89713e9 0.276875
$$647$$ 9.35379e8 0.135776 0.0678879 0.997693i $$-0.478374\pi$$
0.0678879 + 0.997693i $$0.478374\pi$$
$$648$$ 0 0
$$649$$ 1.13502e9 0.162985
$$650$$ 1.27067e9 0.181483
$$651$$ 0 0
$$652$$ −8.27103e9 −1.16867
$$653$$ −1.22407e10 −1.72032 −0.860161 0.510022i $$-0.829637\pi$$
−0.860161 + 0.510022i $$0.829637\pi$$
$$654$$ 0 0
$$655$$ 7.05591e9 0.981089
$$656$$ −2.08769e9 −0.288737
$$657$$ 0 0
$$658$$ 6.01023e9 0.822433
$$659$$ −5.21215e9 −0.709444 −0.354722 0.934972i $$-0.615425\pi$$
−0.354722 + 0.934972i $$0.615425\pi$$
$$660$$ 0 0
$$661$$ −1.11350e10 −1.49964 −0.749820 0.661642i $$-0.769860\pi$$
−0.749820 + 0.661642i $$0.769860\pi$$
$$662$$ −5.71972e8 −0.0766253
$$663$$ 0 0
$$664$$ −6.54499e9 −0.867603
$$665$$ 1.55082e10 2.04497
$$666$$ 0 0
$$667$$ 4.46004e7 0.00581967
$$668$$ −6.36052e9 −0.825610
$$669$$ 0 0
$$670$$ −2.11776e9 −0.272029
$$671$$ 3.94945e8 0.0504671
$$672$$ 0 0
$$673$$ 9.15611e9 1.15787 0.578933 0.815375i $$-0.303469\pi$$
0.578933 + 0.815375i $$0.303469\pi$$
$$674$$ −4.15954e9 −0.523282
$$675$$ 0 0
$$676$$ 4.94231e9 0.615342
$$677$$ −1.14733e10 −1.42111 −0.710557 0.703639i $$-0.751557\pi$$
−0.710557 + 0.703639i $$0.751557\pi$$
$$678$$ 0 0
$$679$$ −2.28367e9 −0.279955
$$680$$ −3.39281e9 −0.413788
$$681$$ 0 0
$$682$$ 4.61176e9 0.556701
$$683$$ 5.63555e9 0.676806 0.338403 0.941001i $$-0.390113\pi$$
0.338403 + 0.941001i $$0.390113\pi$$
$$684$$ 0 0
$$685$$ 1.84862e10 2.19751
$$686$$ −4.73443e9 −0.559930
$$687$$ 0 0
$$688$$ −9.15179e8 −0.107139
$$689$$ 2.98880e8 0.0348120
$$690$$ 0 0
$$691$$ −6.22407e9 −0.717632 −0.358816 0.933408i $$-0.616819\pi$$
−0.358816 + 0.933408i $$0.616819\pi$$
$$692$$ −9.33317e9 −1.07068
$$693$$ 0 0
$$694$$ 3.10279e9 0.352366
$$695$$ 1.06612e10 1.20465
$$696$$ 0 0
$$697$$ −3.65411e9 −0.408758
$$698$$ −5.35270e9 −0.595771
$$699$$ 0 0
$$700$$ −5.50439e9 −0.606550
$$701$$ −8.57820e9 −0.940553 −0.470276 0.882519i $$-0.655846\pi$$
−0.470276 + 0.882519i $$0.655846\pi$$
$$702$$ 0 0
$$703$$ −4.32579e9 −0.469593
$$704$$ 3.69768e9 0.399416
$$705$$ 0 0
$$706$$ 2.31772e9 0.247881
$$707$$ −1.28463e10 −1.36713
$$708$$ 0 0
$$709$$ 2.72444e9 0.287088 0.143544 0.989644i $$-0.454150\pi$$
0.143544 + 0.989644i $$0.454150\pi$$
$$710$$ 6.37932e8 0.0668914
$$711$$ 0 0
$$712$$ −1.14309e10 −1.18686
$$713$$ −2.34496e9 −0.242283
$$714$$ 0 0
$$715$$ 6.36885e9 0.651613
$$716$$ 1.46304e9 0.148957
$$717$$ 0 0
$$718$$ 9.10952e9 0.918460
$$719$$ 2.39574e9 0.240375 0.120188 0.992751i $$-0.461650\pi$$
0.120188 + 0.992751i $$0.461650\pi$$
$$720$$ 0 0
$$721$$ −1.18695e10 −1.17939
$$722$$ 8.14988e9 0.805881
$$723$$ 0 0
$$724$$ −7.62091e9 −0.746314
$$725$$ −1.86614e8 −0.0181870
$$726$$ 0 0
$$727$$ 1.12427e9 0.108517 0.0542586 0.998527i $$-0.482720\pi$$
0.0542586 + 0.998527i $$0.482720\pi$$
$$728$$ −3.34780e9 −0.321588
$$729$$ 0 0
$$730$$ −1.23922e10 −1.17902
$$731$$ −1.60185e9 −0.151674
$$732$$ 0 0
$$733$$ 1.04255e10 0.977759 0.488880 0.872351i $$-0.337406\pi$$
0.488880 + 0.872351i $$0.337406\pi$$
$$734$$ −4.89480e9 −0.456876
$$735$$ 0 0
$$736$$ −3.24696e9 −0.300196
$$737$$ −5.04338e9 −0.464072
$$738$$ 0 0
$$739$$ −1.61439e8 −0.0147147 −0.00735737 0.999973i $$-0.502342\pi$$
−0.00735737 + 0.999973i $$0.502342\pi$$
$$740$$ 3.23145e9 0.293148
$$741$$ 0 0
$$742$$ 5.10156e8 0.0458447
$$743$$ −1.59100e10 −1.42302 −0.711508 0.702678i $$-0.751987\pi$$
−0.711508 + 0.702678i $$0.751987\pi$$
$$744$$ 0 0
$$745$$ −2.38139e9 −0.211000
$$746$$ −7.23070e9 −0.637668
$$747$$ 0 0
$$748$$ −3.37500e9 −0.294862
$$749$$ 7.52341e9 0.654226
$$750$$ 0 0
$$751$$ 9.63052e9 0.829679 0.414839 0.909895i $$-0.363838\pi$$
0.414839 + 0.909895i $$0.363838\pi$$
$$752$$ 4.47934e9 0.384107
$$753$$ 0 0
$$754$$ −4.74094e7 −0.00402777
$$755$$ −1.03432e10 −0.874663
$$756$$ 0 0
$$757$$ −1.23984e10 −1.03880 −0.519398 0.854532i $$-0.673844\pi$$
−0.519398 + 0.854532i $$0.673844\pi$$
$$758$$ −2.61610e9 −0.218178
$$759$$ 0 0
$$760$$ −2.41906e10 −1.99894
$$761$$ 3.84482e9 0.316249 0.158125 0.987419i $$-0.449455\pi$$
0.158125 + 0.987419i $$0.449455\pi$$
$$762$$ 0 0
$$763$$ −2.11619e10 −1.72472
$$764$$ −2.70363e9 −0.219341
$$765$$ 0 0
$$766$$ −2.15773e8 −0.0173459
$$767$$ 6.13478e8 0.0490925
$$768$$ 0 0
$$769$$ 1.85468e10 1.47071 0.735354 0.677683i $$-0.237016\pi$$
0.735354 + 0.677683i $$0.237016\pi$$
$$770$$ 1.08710e10 0.858125
$$771$$ 0 0
$$772$$ −7.91355e9 −0.619029
$$773$$ −9.18489e9 −0.715230 −0.357615 0.933869i $$-0.616410\pi$$
−0.357615 + 0.933869i $$0.616410\pi$$
$$774$$ 0 0
$$775$$ 9.81163e9 0.757156
$$776$$ 3.56220e9 0.273654
$$777$$ 0 0
$$778$$ 6.83530e9 0.520390
$$779$$ −2.60537e10 −1.97464
$$780$$ 0 0
$$781$$ 1.51921e9 0.114114
$$782$$ −6.76196e8 −0.0505649
$$783$$ 0 0
$$784$$ −3.99123e8 −0.0295801
$$785$$ 7.22025e9 0.532732
$$786$$ 0 0
$$787$$ −1.00821e10 −0.737291 −0.368646 0.929570i $$-0.620178\pi$$
−0.368646 + 0.929570i $$0.620178\pi$$
$$788$$ −1.59923e10 −1.16431
$$789$$ 0 0
$$790$$ 2.61043e9 0.188372
$$791$$ −1.22478e10 −0.879916
$$792$$ 0 0
$$793$$ 2.13467e8 0.0152011
$$794$$ −3.01745e9 −0.213928
$$795$$ 0 0
$$796$$ −8.32975e9 −0.585378
$$797$$ 1.03385e10 0.723361 0.361681 0.932302i $$-0.382203\pi$$
0.361681 + 0.932302i $$0.382203\pi$$
$$798$$ 0 0
$$799$$ 7.84025e9 0.543771
$$800$$ 1.35857e10 0.938139
$$801$$ 0 0
$$802$$ −3.71150e9 −0.254062
$$803$$ −2.95117e10 −2.01136
$$804$$ 0 0
$$805$$ −5.52760e9 −0.373466
$$806$$ 2.49265e9 0.167683
$$807$$ 0 0
$$808$$ 2.00384e10 1.33636
$$809$$ −2.64371e10 −1.75547 −0.877735 0.479146i $$-0.840947\pi$$
−0.877735 + 0.479146i $$0.840947\pi$$
$$810$$ 0 0
$$811$$ 2.22378e10 1.46392 0.731961 0.681346i $$-0.238605\pi$$
0.731961 + 0.681346i $$0.238605\pi$$
$$812$$ 2.05372e8 0.0134616
$$813$$ 0 0
$$814$$ −3.03229e9 −0.197054
$$815$$ 3.47530e10 2.24875
$$816$$ 0 0
$$817$$ −1.14211e10 −0.732710
$$818$$ 1.00524e10 0.642147
$$819$$ 0 0
$$820$$ 1.94626e10 1.23269
$$821$$ −1.19117e10 −0.751229 −0.375615 0.926776i $$-0.622568\pi$$
−0.375615 + 0.926776i $$0.622568\pi$$
$$822$$ 0 0
$$823$$ 1.73151e9 0.108274 0.0541372 0.998534i $$-0.482759\pi$$
0.0541372 + 0.998534i $$0.482759\pi$$
$$824$$ 1.85147e10 1.15285
$$825$$ 0 0
$$826$$ 1.04714e9 0.0646510
$$827$$ −2.70946e10 −1.66576 −0.832882 0.553451i $$-0.813311\pi$$
−0.832882 + 0.553451i $$0.813311\pi$$
$$828$$ 0 0
$$829$$ 1.95311e10 1.19065 0.595326 0.803484i $$-0.297023\pi$$
0.595326 + 0.803484i $$0.297023\pi$$
$$830$$ 1.14871e10 0.697331
$$831$$ 0 0
$$832$$ 1.99859e9 0.120307
$$833$$ −6.98589e8 −0.0418759
$$834$$ 0 0
$$835$$ 2.67255e10 1.58863
$$836$$ −2.40637e10 −1.42442
$$837$$ 0 0
$$838$$ 4.19045e8 0.0245984
$$839$$ 2.20343e10 1.28805 0.644024 0.765005i $$-0.277263\pi$$
0.644024 + 0.765005i $$0.277263\pi$$
$$840$$ 0 0
$$841$$ −1.72429e10 −0.999596
$$842$$ −1.21645e10 −0.702268
$$843$$ 0 0
$$844$$ 1.12562e10 0.644453
$$845$$ −2.07665e10 −1.18403
$$846$$ 0 0
$$847$$ 9.37053e9 0.529873
$$848$$ 3.80212e8 0.0214112
$$849$$ 0 0
$$850$$ 2.82929e9 0.158020
$$851$$ 1.54184e9 0.0857604
$$852$$ 0 0
$$853$$ 2.56385e10 1.41440 0.707199 0.707014i $$-0.249958\pi$$
0.707199 + 0.707014i $$0.249958\pi$$
$$854$$ 3.64366e8 0.0200187
$$855$$ 0 0
$$856$$ −1.17354e10 −0.639502
$$857$$ 1.90552e10 1.03414 0.517072 0.855942i $$-0.327022\pi$$
0.517072 + 0.855942i $$0.327022\pi$$
$$858$$ 0 0
$$859$$ 1.94648e10 1.04779 0.523896 0.851782i $$-0.324478\pi$$
0.523896 + 0.851782i $$0.324478\pi$$
$$860$$ 8.53182e9 0.457401
$$861$$ 0 0
$$862$$ 1.32409e10 0.704115
$$863$$ 6.07650e9 0.321822 0.160911 0.986969i $$-0.448557\pi$$
0.160911 + 0.986969i $$0.448557\pi$$
$$864$$ 0 0
$$865$$ 3.92159e10 2.06018
$$866$$ −6.72987e9 −0.352123
$$867$$ 0 0
$$868$$ −1.07979e10 −0.560428
$$869$$ 6.21663e9 0.321356
$$870$$ 0 0
$$871$$ −2.72594e9 −0.139783
$$872$$ 3.30095e10 1.68590
$$873$$ 0 0
$$874$$ −4.82126e9 −0.244270
$$875$$ −2.42092e9 −0.122167
$$876$$ 0 0
$$877$$ −1.82857e10 −0.915404 −0.457702 0.889106i $$-0.651327\pi$$
−0.457702 + 0.889106i $$0.651327\pi$$
$$878$$ 1.94420e9 0.0969415
$$879$$ 0 0
$$880$$ 8.10198e9 0.400776
$$881$$ −1.41923e10 −0.699259 −0.349629 0.936888i $$-0.613692\pi$$
−0.349629 + 0.936888i $$0.613692\pi$$
$$882$$ 0 0
$$883$$ 1.61031e10 0.787131 0.393566 0.919297i $$-0.371241\pi$$
0.393566 + 0.919297i $$0.371241\pi$$
$$884$$ −1.82418e9 −0.0888149
$$885$$ 0 0
$$886$$ 1.37270e10 0.663068
$$887$$ 1.49039e9 0.0717077 0.0358539 0.999357i $$-0.488585\pi$$
0.0358539 + 0.999357i $$0.488585\pi$$
$$888$$ 0 0
$$889$$ 1.45250e10 0.693363
$$890$$ 2.00624e10 0.953933
$$891$$ 0 0
$$892$$ 5.03884e9 0.237713
$$893$$ 5.59007e10 2.62686
$$894$$ 0 0
$$895$$ −6.14736e9 −0.286621
$$896$$ −1.74312e10 −0.809561
$$897$$ 0 0
$$898$$ −1.74338e10 −0.803385
$$899$$ −3.66078e8 −0.0168041
$$900$$ 0 0
$$901$$ 6.65490e8 0.0303113
$$902$$ −1.82631e10 −0.828613
$$903$$ 0 0
$$904$$ 1.91049e10 0.860111
$$905$$ 3.20213e10 1.43605
$$906$$ 0 0
$$907$$ −1.44651e10 −0.643717 −0.321858 0.946788i $$-0.604307\pi$$
−0.321858 + 0.946788i $$0.604307\pi$$
$$908$$ 1.51463e10 0.671436
$$909$$ 0 0
$$910$$ 5.87574e9 0.258475
$$911$$ 2.79187e10 1.22343 0.611717 0.791077i $$-0.290479\pi$$
0.611717 + 0.791077i $$0.290479\pi$$
$$912$$ 0 0
$$913$$ 2.73562e10 1.18962
$$914$$ −9.69353e9 −0.419923
$$915$$ 0 0
$$916$$ −1.08458e10 −0.466259
$$917$$ 1.55024e10 0.663905
$$918$$ 0 0
$$919$$ 1.89312e10 0.804590 0.402295 0.915510i $$-0.368213\pi$$
0.402295 + 0.915510i $$0.368213\pi$$
$$920$$ 8.62228e9 0.365061
$$921$$ 0 0
$$922$$ −2.99575e9 −0.125877
$$923$$ 8.21133e8 0.0343722
$$924$$ 0 0
$$925$$ −6.45127e9 −0.268009
$$926$$ −6.81005e9 −0.281846
$$927$$ 0 0
$$928$$ −5.06891e8 −0.0208208
$$929$$ 3.58709e10 1.46787 0.733934 0.679221i $$-0.237682\pi$$
0.733934 + 0.679221i $$0.237682\pi$$
$$930$$ 0 0
$$931$$ −4.98092e9 −0.202295
$$932$$ −1.03146e10 −0.417348
$$933$$ 0 0
$$934$$ −1.83082e10 −0.735245
$$935$$ 1.41810e10 0.567369
$$936$$ 0 0
$$937$$ 1.18752e10 0.471577 0.235789 0.971804i $$-0.424233\pi$$
0.235789 + 0.971804i $$0.424233\pi$$
$$938$$ −4.65289e9 −0.184083
$$939$$ 0 0
$$940$$ −4.17590e10 −1.63984
$$941$$ 2.99086e10 1.17013 0.585063 0.810988i $$-0.301070\pi$$
0.585063 + 0.810988i $$0.301070\pi$$
$$942$$ 0 0
$$943$$ 9.28632e9 0.360623
$$944$$ 7.80420e8 0.0301944
$$945$$ 0 0
$$946$$ −8.00599e9 −0.307465
$$947$$ 1.97326e10 0.755021 0.377510 0.926005i $$-0.376780\pi$$
0.377510 + 0.926005i $$0.376780\pi$$
$$948$$ 0 0
$$949$$ −1.59510e10 −0.605838
$$950$$ 2.01728e10 0.763366
$$951$$ 0 0
$$952$$ −7.45427e9 −0.280011
$$953$$ −3.05905e10 −1.14488 −0.572442 0.819945i $$-0.694004\pi$$
−0.572442 + 0.819945i $$0.694004\pi$$
$$954$$ 0 0
$$955$$ 1.13600e10 0.422053
$$956$$ 2.72481e10 1.00863
$$957$$ 0 0
$$958$$ −1.11784e10 −0.410772
$$959$$ 4.06157e10 1.48706
$$960$$ 0 0
$$961$$ −8.26528e9 −0.300418
$$962$$ −1.63895e9 −0.0593545
$$963$$ 0 0
$$964$$ −1.26017e9 −0.0453062
$$965$$ 3.32510e10 1.19113
$$966$$ 0 0
$$967$$ 1.80021e10 0.640221 0.320111 0.947380i $$-0.396280\pi$$
0.320111 + 0.947380i $$0.396280\pi$$
$$968$$ −1.46167e10 −0.517947
$$969$$ 0 0
$$970$$ −6.25203e9 −0.219948
$$971$$ 6.56410e9 0.230095 0.115048 0.993360i $$-0.463298\pi$$
0.115048 + 0.993360i $$0.463298\pi$$
$$972$$ 0 0
$$973$$ 2.34235e10 0.815188
$$974$$ −2.38115e9 −0.0825716
$$975$$ 0 0
$$976$$ 2.71557e8 0.00934947
$$977$$ −1.67236e10 −0.573718 −0.286859 0.957973i $$-0.592611\pi$$
−0.286859 + 0.957973i $$0.592611\pi$$
$$978$$ 0 0
$$979$$ 4.77779e10 1.62737
$$980$$ 3.72085e9 0.126285
$$981$$ 0 0
$$982$$ −5.17655e9 −0.174442
$$983$$ 6.60882e9 0.221915 0.110957 0.993825i $$-0.464608\pi$$
0.110957 + 0.993825i $$0.464608\pi$$
$$984$$ 0 0
$$985$$ 6.71959e10 2.24035
$$986$$ −1.05563e8 −0.00350704
$$987$$ 0 0
$$988$$ −1.30064e10 −0.429049
$$989$$ 4.07084e9 0.133813
$$990$$ 0 0
$$991$$ −2.45768e10 −0.802173 −0.401086 0.916040i $$-0.631367\pi$$
−0.401086 + 0.916040i $$0.631367\pi$$
$$992$$ 2.66509e10 0.866803
$$993$$ 0 0
$$994$$ 1.40159e9 0.0452655
$$995$$ 3.49997e10 1.12638
$$996$$ 0 0
$$997$$ 6.18690e10 1.97715 0.988575 0.150727i $$-0.0481614\pi$$
0.988575 + 0.150727i $$0.0481614\pi$$
$$998$$ 1.75830e10 0.559932
$$999$$ 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.1.10 17
3.2 odd 2 177.8.a.b.1.8 17

By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.8 17 3.2 odd 2
531.8.a.d.1.10 17 1.1 even 1 trivial