# Properties

 Label 177.6.a.b.1.1 Level $177$ Weight $6$ Character 177.1 Self dual yes Analytic conductor $28.388$ Analytic rank $1$ Dimension $12$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 177.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3879361069$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$10.9029$$ of defining polynomial Character $$\chi$$ $$=$$ 177.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-10.9029 q^{2} -9.00000 q^{3} +86.8728 q^{4} +88.1143 q^{5} +98.1259 q^{6} +61.7114 q^{7} -598.272 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-10.9029 q^{2} -9.00000 q^{3} +86.8728 q^{4} +88.1143 q^{5} +98.1259 q^{6} +61.7114 q^{7} -598.272 q^{8} +81.0000 q^{9} -960.699 q^{10} +423.027 q^{11} -781.856 q^{12} -1047.72 q^{13} -672.832 q^{14} -793.028 q^{15} +3742.96 q^{16} -2034.81 q^{17} -883.133 q^{18} -1532.11 q^{19} +7654.74 q^{20} -555.403 q^{21} -4612.21 q^{22} +2986.67 q^{23} +5384.45 q^{24} +4639.12 q^{25} +11423.2 q^{26} -729.000 q^{27} +5361.05 q^{28} +115.477 q^{29} +8646.29 q^{30} -3626.28 q^{31} -21664.3 q^{32} -3807.24 q^{33} +22185.3 q^{34} +5437.66 q^{35} +7036.70 q^{36} -15040.3 q^{37} +16704.4 q^{38} +9429.51 q^{39} -52716.3 q^{40} -829.256 q^{41} +6055.49 q^{42} -12294.1 q^{43} +36749.5 q^{44} +7137.25 q^{45} -32563.3 q^{46} -16104.8 q^{47} -33686.6 q^{48} -12998.7 q^{49} -50579.8 q^{50} +18313.3 q^{51} -91018.7 q^{52} +34428.4 q^{53} +7948.20 q^{54} +37274.7 q^{55} -36920.2 q^{56} +13788.9 q^{57} -1259.04 q^{58} -3481.00 q^{59} -68892.6 q^{60} +16418.9 q^{61} +39536.9 q^{62} +4998.63 q^{63} +116429. q^{64} -92319.3 q^{65} +41509.9 q^{66} +9818.25 q^{67} -176770. q^{68} -26880.0 q^{69} -59286.1 q^{70} +3381.64 q^{71} -48460.0 q^{72} +27012.0 q^{73} +163982. q^{74} -41752.1 q^{75} -133098. q^{76} +26105.6 q^{77} -102809. q^{78} +20426.3 q^{79} +329808. q^{80} +6561.00 q^{81} +9041.29 q^{82} -54511.3 q^{83} -48249.4 q^{84} -179296. q^{85} +134041. q^{86} -1039.30 q^{87} -253085. q^{88} -69700.2 q^{89} -77816.6 q^{90} -64656.5 q^{91} +259460. q^{92} +32636.5 q^{93} +175589. q^{94} -135000. q^{95} +194979. q^{96} +48526.4 q^{97} +141723. q^{98} +34265.1 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10})$$ $$12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.9029 −1.92738 −0.963688 0.267032i $$-0.913957\pi$$
−0.963688 + 0.267032i $$0.913957\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 86.8728 2.71478
$$5$$ 88.1143 1.57624 0.788118 0.615524i $$-0.211056\pi$$
0.788118 + 0.615524i $$0.211056\pi$$
$$6$$ 98.1259 1.11277
$$7$$ 61.7114 0.476015 0.238007 0.971263i $$-0.423506\pi$$
0.238007 + 0.971263i $$0.423506\pi$$
$$8$$ −598.272 −3.30502
$$9$$ 81.0000 0.333333
$$10$$ −960.699 −3.03800
$$11$$ 423.027 1.05411 0.527055 0.849831i $$-0.323296\pi$$
0.527055 + 0.849831i $$0.323296\pi$$
$$12$$ −781.856 −1.56738
$$13$$ −1047.72 −1.71944 −0.859722 0.510762i $$-0.829363\pi$$
−0.859722 + 0.510762i $$0.829363\pi$$
$$14$$ −672.832 −0.917459
$$15$$ −793.028 −0.910040
$$16$$ 3742.96 3.65523
$$17$$ −2034.81 −1.70766 −0.853831 0.520550i $$-0.825727\pi$$
−0.853831 + 0.520550i $$0.825727\pi$$
$$18$$ −883.133 −0.642458
$$19$$ −1532.11 −0.973654 −0.486827 0.873498i $$-0.661846\pi$$
−0.486827 + 0.873498i $$0.661846\pi$$
$$20$$ 7654.74 4.27913
$$21$$ −555.403 −0.274827
$$22$$ −4612.21 −2.03167
$$23$$ 2986.67 1.17725 0.588623 0.808408i $$-0.299670\pi$$
0.588623 + 0.808408i $$0.299670\pi$$
$$24$$ 5384.45 1.90815
$$25$$ 4639.12 1.48452
$$26$$ 11423.2 3.31402
$$27$$ −729.000 −0.192450
$$28$$ 5361.05 1.29227
$$29$$ 115.477 0.0254978 0.0127489 0.999919i $$-0.495942\pi$$
0.0127489 + 0.999919i $$0.495942\pi$$
$$30$$ 8646.29 1.75399
$$31$$ −3626.28 −0.677730 −0.338865 0.940835i $$-0.610043\pi$$
−0.338865 + 0.940835i $$0.610043\pi$$
$$32$$ −21664.3 −3.73999
$$33$$ −3807.24 −0.608591
$$34$$ 22185.3 3.29131
$$35$$ 5437.66 0.750312
$$36$$ 7036.70 0.904925
$$37$$ −15040.3 −1.80614 −0.903070 0.429493i $$-0.858692\pi$$
−0.903070 + 0.429493i $$0.858692\pi$$
$$38$$ 16704.4 1.87660
$$39$$ 9429.51 0.992722
$$40$$ −52716.3 −5.20949
$$41$$ −829.256 −0.0770423 −0.0385211 0.999258i $$-0.512265\pi$$
−0.0385211 + 0.999258i $$0.512265\pi$$
$$42$$ 6055.49 0.529695
$$43$$ −12294.1 −1.01397 −0.506985 0.861955i $$-0.669240\pi$$
−0.506985 + 0.861955i $$0.669240\pi$$
$$44$$ 36749.5 2.86167
$$45$$ 7137.25 0.525412
$$46$$ −32563.3 −2.26900
$$47$$ −16104.8 −1.06343 −0.531717 0.846922i $$-0.678453\pi$$
−0.531717 + 0.846922i $$0.678453\pi$$
$$48$$ −33686.6 −2.11035
$$49$$ −12998.7 −0.773410
$$50$$ −50579.8 −2.86123
$$51$$ 18313.3 0.985919
$$52$$ −91018.7 −4.66791
$$53$$ 34428.4 1.68356 0.841778 0.539824i $$-0.181509\pi$$
0.841778 + 0.539824i $$0.181509\pi$$
$$54$$ 7948.20 0.370924
$$55$$ 37274.7 1.66153
$$56$$ −36920.2 −1.57324
$$57$$ 13788.9 0.562139
$$58$$ −1259.04 −0.0491438
$$59$$ −3481.00 −0.130189
$$60$$ −68892.6 −2.47056
$$61$$ 16418.9 0.564961 0.282481 0.959273i $$-0.408843\pi$$
0.282481 + 0.959273i $$0.408843\pi$$
$$62$$ 39536.9 1.30624
$$63$$ 4998.63 0.158672
$$64$$ 116429. 3.55313
$$65$$ −92319.3 −2.71025
$$66$$ 41509.9 1.17298
$$67$$ 9818.25 0.267206 0.133603 0.991035i $$-0.457345\pi$$
0.133603 + 0.991035i $$0.457345\pi$$
$$68$$ −176770. −4.63592
$$69$$ −26880.0 −0.679683
$$70$$ −59286.1 −1.44613
$$71$$ 3381.64 0.0796125 0.0398062 0.999207i $$-0.487326\pi$$
0.0398062 + 0.999207i $$0.487326\pi$$
$$72$$ −48460.0 −1.10167
$$73$$ 27012.0 0.593267 0.296634 0.954991i $$-0.404136\pi$$
0.296634 + 0.954991i $$0.404136\pi$$
$$74$$ 163982. 3.48111
$$75$$ −41752.1 −0.857087
$$76$$ −133098. −2.64325
$$77$$ 26105.6 0.501772
$$78$$ −102809. −1.91335
$$79$$ 20426.3 0.368233 0.184117 0.982904i $$-0.441058\pi$$
0.184117 + 0.982904i $$0.441058\pi$$
$$80$$ 329808. 5.76151
$$81$$ 6561.00 0.111111
$$82$$ 9041.29 0.148489
$$83$$ −54511.3 −0.868543 −0.434272 0.900782i $$-0.642994\pi$$
−0.434272 + 0.900782i $$0.642994\pi$$
$$84$$ −48249.4 −0.746095
$$85$$ −179296. −2.69168
$$86$$ 134041. 1.95430
$$87$$ −1039.30 −0.0147211
$$88$$ −253085. −3.48385
$$89$$ −69700.2 −0.932736 −0.466368 0.884591i $$-0.654438\pi$$
−0.466368 + 0.884591i $$0.654438\pi$$
$$90$$ −77816.6 −1.01267
$$91$$ −64656.5 −0.818481
$$92$$ 259460. 3.19596
$$93$$ 32636.5 0.391287
$$94$$ 175589. 2.04964
$$95$$ −135000. −1.53471
$$96$$ 194979. 2.15928
$$97$$ 48526.4 0.523660 0.261830 0.965114i $$-0.415674\pi$$
0.261830 + 0.965114i $$0.415674\pi$$
$$98$$ 141723. 1.49065
$$99$$ 34265.1 0.351370
$$100$$ 403014. 4.03014
$$101$$ 107128. 1.04496 0.522478 0.852653i $$-0.325008\pi$$
0.522478 + 0.852653i $$0.325008\pi$$
$$102$$ −199668. −1.90024
$$103$$ −93523.6 −0.868617 −0.434308 0.900764i $$-0.643007\pi$$
−0.434308 + 0.900764i $$0.643007\pi$$
$$104$$ 626823. 5.68279
$$105$$ −48938.9 −0.433193
$$106$$ −375369. −3.24484
$$107$$ 51795.7 0.437356 0.218678 0.975797i $$-0.429826\pi$$
0.218678 + 0.975797i $$0.429826\pi$$
$$108$$ −63330.3 −0.522459
$$109$$ −147885. −1.19223 −0.596113 0.802901i $$-0.703289\pi$$
−0.596113 + 0.802901i $$0.703289\pi$$
$$110$$ −406401. −3.20238
$$111$$ 135363. 1.04278
$$112$$ 230983. 1.73995
$$113$$ −83662.7 −0.616362 −0.308181 0.951328i $$-0.599720\pi$$
−0.308181 + 0.951328i $$0.599720\pi$$
$$114$$ −150339. −1.08345
$$115$$ 263168. 1.85562
$$116$$ 10031.9 0.0692208
$$117$$ −84865.6 −0.573148
$$118$$ 37952.9 0.250923
$$119$$ −125571. −0.812873
$$120$$ 474447. 3.00770
$$121$$ 17900.5 0.111148
$$122$$ −179013. −1.08889
$$123$$ 7463.31 0.0444804
$$124$$ −315025. −1.83988
$$125$$ 133416. 0.763716
$$126$$ −54499.4 −0.305820
$$127$$ −155048. −0.853016 −0.426508 0.904484i $$-0.640256\pi$$
−0.426508 + 0.904484i $$0.640256\pi$$
$$128$$ −576153. −3.10823
$$129$$ 110647. 0.585416
$$130$$ 1.00655e6 5.22367
$$131$$ −370672. −1.88717 −0.943587 0.331124i $$-0.892572\pi$$
−0.943587 + 0.331124i $$0.892572\pi$$
$$132$$ −330746. −1.65219
$$133$$ −94548.4 −0.463474
$$134$$ −107047. −0.515007
$$135$$ −64235.3 −0.303347
$$136$$ 1.21737e6 5.64385
$$137$$ 194058. 0.883345 0.441673 0.897176i $$-0.354385\pi$$
0.441673 + 0.897176i $$0.354385\pi$$
$$138$$ 293070. 1.31001
$$139$$ −80499.1 −0.353390 −0.176695 0.984266i $$-0.556541\pi$$
−0.176695 + 0.984266i $$0.556541\pi$$
$$140$$ 472385. 2.03693
$$141$$ 144943. 0.613973
$$142$$ −36869.6 −0.153443
$$143$$ −443215. −1.81248
$$144$$ 303180. 1.21841
$$145$$ 10175.2 0.0401905
$$146$$ −294509. −1.14345
$$147$$ 116988. 0.446528
$$148$$ −1.30659e6 −4.90327
$$149$$ 229972. 0.848611 0.424305 0.905519i $$-0.360518\pi$$
0.424305 + 0.905519i $$0.360518\pi$$
$$150$$ 455218. 1.65193
$$151$$ −103481. −0.369333 −0.184667 0.982801i $$-0.559121\pi$$
−0.184667 + 0.982801i $$0.559121\pi$$
$$152$$ 916616. 3.21794
$$153$$ −164820. −0.569221
$$154$$ −284626. −0.967103
$$155$$ −319527. −1.06826
$$156$$ 819168. 2.69502
$$157$$ 222025. 0.718875 0.359438 0.933169i $$-0.382969\pi$$
0.359438 + 0.933169i $$0.382969\pi$$
$$158$$ −222706. −0.709723
$$159$$ −309856. −0.972001
$$160$$ −1.90894e6 −5.89510
$$161$$ 184312. 0.560387
$$162$$ −71533.8 −0.214153
$$163$$ −1674.56 −0.00493664 −0.00246832 0.999997i $$-0.500786\pi$$
−0.00246832 + 0.999997i $$0.500786\pi$$
$$164$$ −72039.9 −0.209153
$$165$$ −335472. −0.959282
$$166$$ 594331. 1.67401
$$167$$ −633831. −1.75866 −0.879330 0.476212i $$-0.842009\pi$$
−0.879330 + 0.476212i $$0.842009\pi$$
$$168$$ 332282. 0.908309
$$169$$ 726431. 1.95649
$$170$$ 1.95484e6 5.18788
$$171$$ −124101. −0.324551
$$172$$ −1.06802e6 −2.75270
$$173$$ 304757. 0.774175 0.387087 0.922043i $$-0.373481\pi$$
0.387087 + 0.922043i $$0.373481\pi$$
$$174$$ 11331.3 0.0283732
$$175$$ 286287. 0.706653
$$176$$ 1.58337e6 3.85302
$$177$$ 31329.0 0.0751646
$$178$$ 759933. 1.79773
$$179$$ 67132.3 0.156603 0.0783013 0.996930i $$-0.475050\pi$$
0.0783013 + 0.996930i $$0.475050\pi$$
$$180$$ 620034. 1.42638
$$181$$ 111781. 0.253613 0.126807 0.991927i $$-0.459527\pi$$
0.126807 + 0.991927i $$0.459527\pi$$
$$182$$ 704942. 1.57752
$$183$$ −147770. −0.326180
$$184$$ −1.78684e6 −3.89082
$$185$$ −1.32526e6 −2.84690
$$186$$ −355832. −0.754158
$$187$$ −860780. −1.80006
$$188$$ −1.39907e6 −2.88698
$$189$$ −44987.6 −0.0916091
$$190$$ 1.47189e6 2.95796
$$191$$ −731353. −1.45059 −0.725294 0.688440i $$-0.758296\pi$$
−0.725294 + 0.688440i $$0.758296\pi$$
$$192$$ −1.04786e6 −2.05140
$$193$$ 685155. 1.32402 0.662011 0.749494i $$-0.269703\pi$$
0.662011 + 0.749494i $$0.269703\pi$$
$$194$$ −529078. −1.00929
$$195$$ 830874. 1.56476
$$196$$ −1.12923e6 −2.09963
$$197$$ −723750. −1.32869 −0.664344 0.747427i $$-0.731289\pi$$
−0.664344 + 0.747427i $$0.731289\pi$$
$$198$$ −373589. −0.677222
$$199$$ −186876. −0.334519 −0.167260 0.985913i $$-0.553492\pi$$
−0.167260 + 0.985913i $$0.553492\pi$$
$$200$$ −2.77546e6 −4.90636
$$201$$ −88364.2 −0.154272
$$202$$ −1.16800e6 −2.01402
$$203$$ 7126.28 0.0121373
$$204$$ 1.59093e6 2.67655
$$205$$ −73069.3 −0.121437
$$206$$ 1.01968e6 1.67415
$$207$$ 241920. 0.392415
$$208$$ −3.92158e6 −6.28497
$$209$$ −648121. −1.02634
$$210$$ 533575. 0.834925
$$211$$ −143708. −0.222216 −0.111108 0.993808i $$-0.535440\pi$$
−0.111108 + 0.993808i $$0.535440\pi$$
$$212$$ 2.99089e6 4.57048
$$213$$ −30434.7 −0.0459643
$$214$$ −564723. −0.842948
$$215$$ −1.08329e6 −1.59826
$$216$$ 436140. 0.636051
$$217$$ −223783. −0.322609
$$218$$ 1.61237e6 2.29787
$$219$$ −243108. −0.342523
$$220$$ 3.23816e6 4.51067
$$221$$ 2.13192e6 2.93623
$$222$$ −1.47584e6 −2.00982
$$223$$ 299009. 0.402645 0.201322 0.979525i $$-0.435476\pi$$
0.201322 + 0.979525i $$0.435476\pi$$
$$224$$ −1.33694e6 −1.78029
$$225$$ 375769. 0.494840
$$226$$ 912164. 1.18796
$$227$$ 106091. 0.136651 0.0683255 0.997663i $$-0.478234\pi$$
0.0683255 + 0.997663i $$0.478234\pi$$
$$228$$ 1.19789e6 1.52608
$$229$$ −222769. −0.280715 −0.140358 0.990101i $$-0.544825\pi$$
−0.140358 + 0.990101i $$0.544825\pi$$
$$230$$ −2.86929e6 −3.57647
$$231$$ −234950. −0.289698
$$232$$ −69086.9 −0.0842706
$$233$$ −877575. −1.05900 −0.529498 0.848311i $$-0.677620\pi$$
−0.529498 + 0.848311i $$0.677620\pi$$
$$234$$ 925279. 1.10467
$$235$$ −1.41906e6 −1.67622
$$236$$ −302404. −0.353434
$$237$$ −183837. −0.212599
$$238$$ 1.36909e6 1.56671
$$239$$ 1.55403e6 1.75980 0.879902 0.475156i $$-0.157608\pi$$
0.879902 + 0.475156i $$0.157608\pi$$
$$240$$ −2.96827e6 −3.32641
$$241$$ −65896.5 −0.0730836 −0.0365418 0.999332i $$-0.511634\pi$$
−0.0365418 + 0.999332i $$0.511634\pi$$
$$242$$ −195167. −0.214223
$$243$$ −59049.0 −0.0641500
$$244$$ 1.42635e6 1.53374
$$245$$ −1.14537e6 −1.21908
$$246$$ −81371.6 −0.0857304
$$247$$ 1.60522e6 1.67414
$$248$$ 2.16950e6 2.23991
$$249$$ 490602. 0.501454
$$250$$ −1.45462e6 −1.47197
$$251$$ −24842.7 −0.0248894 −0.0124447 0.999923i $$-0.503961\pi$$
−0.0124447 + 0.999923i $$0.503961\pi$$
$$252$$ 434245. 0.430758
$$253$$ 1.26344e6 1.24095
$$254$$ 1.69047e6 1.64408
$$255$$ 1.61366e6 1.55404
$$256$$ 2.55600e6 2.43759
$$257$$ 501967. 0.474070 0.237035 0.971501i $$-0.423824\pi$$
0.237035 + 0.971501i $$0.423824\pi$$
$$258$$ −1.20637e6 −1.12832
$$259$$ −928157. −0.859750
$$260$$ −8.02004e6 −7.35772
$$261$$ 9353.68 0.00849926
$$262$$ 4.04140e6 3.63729
$$263$$ 1.26902e6 1.13130 0.565652 0.824644i $$-0.308624\pi$$
0.565652 + 0.824644i $$0.308624\pi$$
$$264$$ 2.27776e6 2.01140
$$265$$ 3.03363e6 2.65368
$$266$$ 1.03085e6 0.893288
$$267$$ 627302. 0.538516
$$268$$ 852939. 0.725406
$$269$$ −1.04097e6 −0.877121 −0.438560 0.898702i $$-0.644511\pi$$
−0.438560 + 0.898702i $$0.644511\pi$$
$$270$$ 700350. 0.584663
$$271$$ −2.07017e6 −1.71231 −0.856155 0.516718i $$-0.827153\pi$$
−0.856155 + 0.516718i $$0.827153\pi$$
$$272$$ −7.61622e6 −6.24191
$$273$$ 581908. 0.472550
$$274$$ −2.11579e6 −1.70254
$$275$$ 1.96247e6 1.56485
$$276$$ −2.33514e6 −1.84519
$$277$$ 1.93149e6 1.51249 0.756247 0.654287i $$-0.227031\pi$$
0.756247 + 0.654287i $$0.227031\pi$$
$$278$$ 877672. 0.681115
$$279$$ −293728. −0.225910
$$280$$ −3.25320e6 −2.47979
$$281$$ 859707. 0.649508 0.324754 0.945799i $$-0.394718\pi$$
0.324754 + 0.945799i $$0.394718\pi$$
$$282$$ −1.58030e6 −1.18336
$$283$$ −789427. −0.585930 −0.292965 0.956123i $$-0.594642\pi$$
−0.292965 + 0.956123i $$0.594642\pi$$
$$284$$ 293772. 0.216130
$$285$$ 1.21500e6 0.886064
$$286$$ 4.83232e6 3.49334
$$287$$ −51174.6 −0.0366733
$$288$$ −1.75481e6 −1.24666
$$289$$ 2.72060e6 1.91611
$$290$$ −110939. −0.0774622
$$291$$ −436738. −0.302335
$$292$$ 2.34661e6 1.61059
$$293$$ −1.83732e6 −1.25030 −0.625152 0.780503i $$-0.714963\pi$$
−0.625152 + 0.780503i $$0.714963\pi$$
$$294$$ −1.27551e6 −0.860628
$$295$$ −306726. −0.205208
$$296$$ 8.99818e6 5.96933
$$297$$ −308386. −0.202864
$$298$$ −2.50735e6 −1.63559
$$299$$ −3.12920e6 −2.02421
$$300$$ −3.62712e6 −2.32680
$$301$$ −758686. −0.482665
$$302$$ 1.12824e6 0.711843
$$303$$ −964148. −0.603305
$$304$$ −5.73461e6 −3.55893
$$305$$ 1.44674e6 0.890512
$$306$$ 1.79701e6 1.09710
$$307$$ −443750. −0.268715 −0.134358 0.990933i $$-0.542897\pi$$
−0.134358 + 0.990933i $$0.542897\pi$$
$$308$$ 2.26787e6 1.36220
$$309$$ 841713. 0.501496
$$310$$ 3.48376e6 2.05894
$$311$$ −651682. −0.382063 −0.191031 0.981584i $$-0.561183\pi$$
−0.191031 + 0.981584i $$0.561183\pi$$
$$312$$ −5.64141e6 −3.28096
$$313$$ −1.28063e6 −0.738859 −0.369430 0.929259i $$-0.620447\pi$$
−0.369430 + 0.929259i $$0.620447\pi$$
$$314$$ −2.42072e6 −1.38554
$$315$$ 440450. 0.250104
$$316$$ 1.77449e6 0.999670
$$317$$ 1.85275e6 1.03554 0.517772 0.855518i $$-0.326761\pi$$
0.517772 + 0.855518i $$0.326761\pi$$
$$318$$ 3.37832e6 1.87341
$$319$$ 48850.0 0.0268775
$$320$$ 1.02591e7 5.60057
$$321$$ −466162. −0.252507
$$322$$ −2.00953e6 −1.08008
$$323$$ 3.11755e6 1.66267
$$324$$ 569973. 0.301642
$$325$$ −4.86052e6 −2.55255
$$326$$ 18257.5 0.00951475
$$327$$ 1.33097e6 0.688332
$$328$$ 496121. 0.254626
$$329$$ −993849. −0.506210
$$330$$ 3.65761e6 1.84890
$$331$$ −646915. −0.324547 −0.162273 0.986746i $$-0.551883\pi$$
−0.162273 + 0.986746i $$0.551883\pi$$
$$332$$ −4.73555e6 −2.35790
$$333$$ −1.21826e6 −0.602047
$$334$$ 6.91058e6 3.38960
$$335$$ 865128. 0.421180
$$336$$ −2.07885e6 −1.00456
$$337$$ −1.93315e6 −0.927235 −0.463618 0.886035i $$-0.653449\pi$$
−0.463618 + 0.886035i $$0.653449\pi$$
$$338$$ −7.92019e6 −3.77089
$$339$$ 752964. 0.355857
$$340$$ −1.55759e7 −7.30731
$$341$$ −1.53401e6 −0.714402
$$342$$ 1.35305e6 0.625532
$$343$$ −1.83935e6 −0.844169
$$344$$ 7.35521e6 3.35119
$$345$$ −2.36851e6 −1.07134
$$346$$ −3.32273e6 −1.49213
$$347$$ −2.55520e6 −1.13920 −0.569601 0.821921i $$-0.692902\pi$$
−0.569601 + 0.821921i $$0.692902\pi$$
$$348$$ −90286.7 −0.0399646
$$349$$ −106267. −0.0467019 −0.0233509 0.999727i $$-0.507434\pi$$
−0.0233509 + 0.999727i $$0.507434\pi$$
$$350$$ −3.12135e6 −1.36199
$$351$$ 763790. 0.330907
$$352$$ −9.16459e6 −3.94236
$$353$$ 421809. 0.180169 0.0900844 0.995934i $$-0.471286\pi$$
0.0900844 + 0.995934i $$0.471286\pi$$
$$354$$ −341576. −0.144870
$$355$$ 297970. 0.125488
$$356$$ −6.05505e6 −2.53217
$$357$$ 1.13014e6 0.469312
$$358$$ −731936. −0.301832
$$359$$ 2.33843e6 0.957607 0.478804 0.877922i $$-0.341071\pi$$
0.478804 + 0.877922i $$0.341071\pi$$
$$360$$ −4.27002e6 −1.73650
$$361$$ −128752. −0.0519980
$$362$$ −1.21874e6 −0.488808
$$363$$ −161104. −0.0641712
$$364$$ −5.61689e6 −2.22199
$$365$$ 2.38015e6 0.935129
$$366$$ 1.61112e6 0.628672
$$367$$ 1.76169e6 0.682755 0.341377 0.939926i $$-0.389107\pi$$
0.341377 + 0.939926i $$0.389107\pi$$
$$368$$ 1.11790e7 4.30311
$$369$$ −67169.8 −0.0256808
$$370$$ 1.44492e7 5.48705
$$371$$ 2.12463e6 0.801397
$$372$$ 2.83522e6 1.06226
$$373$$ 555814. 0.206851 0.103425 0.994637i $$-0.467020\pi$$
0.103425 + 0.994637i $$0.467020\pi$$
$$374$$ 9.38498e6 3.46940
$$375$$ −1.20074e6 −0.440932
$$376$$ 9.63504e6 3.51467
$$377$$ −120988. −0.0438420
$$378$$ 490495. 0.176565
$$379$$ 1.23190e6 0.440532 0.220266 0.975440i $$-0.429307\pi$$
0.220266 + 0.975440i $$0.429307\pi$$
$$380$$ −1.17279e7 −4.16639
$$381$$ 1.39543e6 0.492489
$$382$$ 7.97386e6 2.79583
$$383$$ 2.07159e6 0.721617 0.360808 0.932640i $$-0.382501\pi$$
0.360808 + 0.932640i $$0.382501\pi$$
$$384$$ 5.18538e6 1.79454
$$385$$ 2.30027e6 0.790911
$$386$$ −7.47016e6 −2.55189
$$387$$ −995822. −0.337990
$$388$$ 4.21563e6 1.42162
$$389$$ −4.06653e6 −1.36254 −0.681271 0.732031i $$-0.738573\pi$$
−0.681271 + 0.732031i $$0.738573\pi$$
$$390$$ −9.05892e6 −3.01589
$$391$$ −6.07731e6 −2.01034
$$392$$ 7.77676e6 2.55613
$$393$$ 3.33605e6 1.08956
$$394$$ 7.89096e6 2.56088
$$395$$ 1.79985e6 0.580422
$$396$$ 2.97671e6 0.953891
$$397$$ −4.63853e6 −1.47708 −0.738541 0.674209i $$-0.764485\pi$$
−0.738541 + 0.674209i $$0.764485\pi$$
$$398$$ 2.03749e6 0.644744
$$399$$ 850936. 0.267587
$$400$$ 1.73640e7 5.42626
$$401$$ 2.63194e6 0.817363 0.408681 0.912677i $$-0.365989\pi$$
0.408681 + 0.912677i $$0.365989\pi$$
$$402$$ 963425. 0.297340
$$403$$ 3.79933e6 1.16532
$$404$$ 9.30647e6 2.83682
$$405$$ 578118. 0.175137
$$406$$ −77697.0 −0.0233932
$$407$$ −6.36244e6 −1.90387
$$408$$ −1.09563e7 −3.25848
$$409$$ 3.04976e6 0.901482 0.450741 0.892655i $$-0.351160\pi$$
0.450741 + 0.892655i $$0.351160\pi$$
$$410$$ 796666. 0.234054
$$411$$ −1.74652e6 −0.510000
$$412$$ −8.12466e6 −2.35810
$$413$$ −214817. −0.0619719
$$414$$ −2.63763e6 −0.756332
$$415$$ −4.80322e6 −1.36903
$$416$$ 2.26982e7 6.43070
$$417$$ 724492. 0.204030
$$418$$ 7.06639e6 1.97814
$$419$$ 1.34161e6 0.373329 0.186664 0.982424i $$-0.440232\pi$$
0.186664 + 0.982424i $$0.440232\pi$$
$$420$$ −4.25146e6 −1.17602
$$421$$ 2.54019e6 0.698492 0.349246 0.937031i $$-0.386438\pi$$
0.349246 + 0.937031i $$0.386438\pi$$
$$422$$ 1.56683e6 0.428293
$$423$$ −1.30449e6 −0.354478
$$424$$ −2.05976e7 −5.56418
$$425$$ −9.43974e6 −2.53506
$$426$$ 331826. 0.0885904
$$427$$ 1.01323e6 0.268930
$$428$$ 4.49964e6 1.18732
$$429$$ 3.98893e6 1.04644
$$430$$ 1.18109e7 3.08044
$$431$$ 4.17832e6 1.08345 0.541725 0.840556i $$-0.317772\pi$$
0.541725 + 0.840556i $$0.317772\pi$$
$$432$$ −2.72862e6 −0.703450
$$433$$ −74580.5 −0.0191164 −0.00955819 0.999954i $$-0.503043\pi$$
−0.00955819 + 0.999954i $$0.503043\pi$$
$$434$$ 2.43988e6 0.621790
$$435$$ −91576.9 −0.0232040
$$436$$ −1.28472e7 −3.23663
$$437$$ −4.57589e6 −1.14623
$$438$$ 2.65058e6 0.660170
$$439$$ 7.12201e6 1.76377 0.881884 0.471466i $$-0.156275\pi$$
0.881884 + 0.471466i $$0.156275\pi$$
$$440$$ −2.23004e7 −5.49137
$$441$$ −1.05289e6 −0.257803
$$442$$ −2.32441e7 −5.65922
$$443$$ −7.36242e6 −1.78243 −0.891213 0.453585i $$-0.850145\pi$$
−0.891213 + 0.453585i $$0.850145\pi$$
$$444$$ 1.17593e7 2.83090
$$445$$ −6.14158e6 −1.47021
$$446$$ −3.26006e6 −0.776047
$$447$$ −2.06974e6 −0.489946
$$448$$ 7.18500e6 1.69134
$$449$$ −5.60487e6 −1.31205 −0.656023 0.754741i $$-0.727763\pi$$
−0.656023 + 0.754741i $$0.727763\pi$$
$$450$$ −4.09696e6 −0.953742
$$451$$ −350798. −0.0812111
$$452$$ −7.26801e6 −1.67328
$$453$$ 931329. 0.213235
$$454$$ −1.15669e6 −0.263378
$$455$$ −5.69716e6 −1.29012
$$456$$ −8.24954e6 −1.85788
$$457$$ −2.27985e6 −0.510641 −0.255320 0.966856i $$-0.582181\pi$$
−0.255320 + 0.966856i $$0.582181\pi$$
$$458$$ 2.42883e6 0.541044
$$459$$ 1.48338e6 0.328640
$$460$$ 2.28621e7 5.03759
$$461$$ 5.70375e6 1.24999 0.624997 0.780627i $$-0.285100\pi$$
0.624997 + 0.780627i $$0.285100\pi$$
$$462$$ 2.56163e6 0.558357
$$463$$ 3.15681e6 0.684378 0.342189 0.939631i $$-0.388832\pi$$
0.342189 + 0.939631i $$0.388832\pi$$
$$464$$ 432227. 0.0932003
$$465$$ 2.87574e6 0.616761
$$466$$ 9.56810e6 2.04108
$$467$$ 4.10668e6 0.871362 0.435681 0.900101i $$-0.356508\pi$$
0.435681 + 0.900101i $$0.356508\pi$$
$$468$$ −7.37251e6 −1.55597
$$469$$ 605898. 0.127194
$$470$$ 1.54719e7 3.23071
$$471$$ −1.99823e6 −0.415043
$$472$$ 2.08259e6 0.430277
$$473$$ −5.20073e6 −1.06884
$$474$$ 2.00435e6 0.409759
$$475$$ −7.10762e6 −1.44541
$$476$$ −1.09087e7 −2.20677
$$477$$ 2.78870e6 0.561185
$$478$$ −1.69434e7 −3.39180
$$479$$ −3.96139e6 −0.788876 −0.394438 0.918923i $$-0.629061\pi$$
−0.394438 + 0.918923i $$0.629061\pi$$
$$480$$ 1.71804e7 3.40354
$$481$$ 1.57580e7 3.10556
$$482$$ 718462. 0.140860
$$483$$ −1.65880e6 −0.323539
$$484$$ 1.55506e6 0.301741
$$485$$ 4.27587e6 0.825411
$$486$$ 643804. 0.123641
$$487$$ −9.18635e6 −1.75517 −0.877587 0.479417i $$-0.840848\pi$$
−0.877587 + 0.479417i $$0.840848\pi$$
$$488$$ −9.82295e6 −1.86721
$$489$$ 15071.0 0.00285017
$$490$$ 1.24878e7 2.34962
$$491$$ 3.85969e6 0.722518 0.361259 0.932465i $$-0.382347\pi$$
0.361259 + 0.932465i $$0.382347\pi$$
$$492$$ 648359. 0.120754
$$493$$ −234975. −0.0435416
$$494$$ −1.75015e7 −3.22670
$$495$$ 3.01925e6 0.553842
$$496$$ −1.35730e7 −2.47726
$$497$$ 208686. 0.0378967
$$498$$ −5.34897e6 −0.966490
$$499$$ −3.44774e6 −0.619845 −0.309922 0.950762i $$-0.600303\pi$$
−0.309922 + 0.950762i $$0.600303\pi$$
$$500$$ 1.15902e7 2.07332
$$501$$ 5.70448e6 1.01536
$$502$$ 270857. 0.0479712
$$503$$ 5.17527e6 0.912038 0.456019 0.889970i $$-0.349275\pi$$
0.456019 + 0.889970i $$0.349275\pi$$
$$504$$ −2.99054e6 −0.524412
$$505$$ 9.43946e6 1.64710
$$506$$ −1.37751e7 −2.39177
$$507$$ −6.53788e6 −1.12958
$$508$$ −1.34695e7 −2.31575
$$509$$ −86953.7 −0.0148763 −0.00743813 0.999972i $$-0.502368\pi$$
−0.00743813 + 0.999972i $$0.502368\pi$$
$$510$$ −1.75936e7 −2.99522
$$511$$ 1.66695e6 0.282404
$$512$$ −9.43086e6 −1.58992
$$513$$ 1.11690e6 0.187380
$$514$$ −5.47289e6 −0.913710
$$515$$ −8.24077e6 −1.36915
$$516$$ 9.61221e6 1.58927
$$517$$ −6.81275e6 −1.12098
$$518$$ 1.01196e7 1.65706
$$519$$ −2.74282e6 −0.446970
$$520$$ 5.52321e7 8.95742
$$521$$ −6.15153e6 −0.992861 −0.496431 0.868076i $$-0.665356\pi$$
−0.496431 + 0.868076i $$0.665356\pi$$
$$522$$ −101982. −0.0163813
$$523$$ 1.72407e6 0.275613 0.137807 0.990459i $$-0.455995\pi$$
0.137807 + 0.990459i $$0.455995\pi$$
$$524$$ −3.22014e7 −5.12326
$$525$$ −2.57658e6 −0.407986
$$526$$ −1.38360e7 −2.18045
$$527$$ 7.37879e6 1.15733
$$528$$ −1.42503e7 −2.22454
$$529$$ 2.48384e6 0.385909
$$530$$ −3.30754e7 −5.11464
$$531$$ −281961. −0.0433963
$$532$$ −8.21369e6 −1.25823
$$533$$ 868831. 0.132470
$$534$$ −6.83940e6 −1.03792
$$535$$ 4.56394e6 0.689375
$$536$$ −5.87398e6 −0.883122
$$537$$ −604191. −0.0904146
$$538$$ 1.13496e7 1.69054
$$539$$ −5.49879e6 −0.815259
$$540$$ −5.58030e6 −0.823518
$$541$$ −8.39290e6 −1.23287 −0.616437 0.787404i $$-0.711425\pi$$
−0.616437 + 0.787404i $$0.711425\pi$$
$$542$$ 2.25708e7 3.30027
$$543$$ −1.00603e6 −0.146424
$$544$$ 4.40829e7 6.38664
$$545$$ −1.30308e7 −1.87923
$$546$$ −6.34448e6 −0.910782
$$547$$ −1.01323e7 −1.44790 −0.723952 0.689850i $$-0.757676\pi$$
−0.723952 + 0.689850i $$0.757676\pi$$
$$548$$ 1.68584e7 2.39808
$$549$$ 1.32993e6 0.188320
$$550$$ −2.13966e7 −3.01605
$$551$$ −176924. −0.0248260
$$552$$ 1.60816e7 2.24637
$$553$$ 1.26054e6 0.175284
$$554$$ −2.10588e7 −2.91514
$$555$$ 1.19274e7 1.64366
$$556$$ −6.99318e6 −0.959374
$$557$$ 9.82722e6 1.34212 0.671062 0.741401i $$-0.265838\pi$$
0.671062 + 0.741401i $$0.265838\pi$$
$$558$$ 3.20249e6 0.435413
$$559$$ 1.28808e7 1.74347
$$560$$ 2.03529e7 2.74256
$$561$$ 7.74702e6 1.03927
$$562$$ −9.37328e6 −1.25185
$$563$$ 9.96382e6 1.32481 0.662407 0.749144i $$-0.269535\pi$$
0.662407 + 0.749144i $$0.269535\pi$$
$$564$$ 1.25916e7 1.66680
$$565$$ −7.37187e6 −0.971531
$$566$$ 8.60703e6 1.12931
$$567$$ 404889. 0.0528905
$$568$$ −2.02314e6 −0.263121
$$569$$ 1.30624e7 1.69138 0.845692 0.533672i $$-0.179188\pi$$
0.845692 + 0.533672i $$0.179188\pi$$
$$570$$ −1.32470e7 −1.70778
$$571$$ 1.30362e7 1.67325 0.836625 0.547776i $$-0.184525\pi$$
0.836625 + 0.547776i $$0.184525\pi$$
$$572$$ −3.85033e7 −4.92049
$$573$$ 6.58218e6 0.837497
$$574$$ 557951. 0.0706832
$$575$$ 1.38555e7 1.74764
$$576$$ 9.43075e6 1.18438
$$577$$ −1.37814e7 −1.72327 −0.861637 0.507524i $$-0.830561\pi$$
−0.861637 + 0.507524i $$0.830561\pi$$
$$578$$ −2.96624e7 −3.69307
$$579$$ −6.16639e6 −0.764425
$$580$$ 883949. 0.109108
$$581$$ −3.36397e6 −0.413440
$$582$$ 4.76170e6 0.582713
$$583$$ 1.45641e7 1.77465
$$584$$ −1.61606e7 −1.96076
$$585$$ −7.47787e6 −0.903417
$$586$$ 2.00321e7 2.40981
$$587$$ −4.70405e6 −0.563478 −0.281739 0.959491i $$-0.590911\pi$$
−0.281739 + 0.959491i $$0.590911\pi$$
$$588$$ 1.01631e7 1.21222
$$589$$ 5.55584e6 0.659874
$$590$$ 3.34419e6 0.395514
$$591$$ 6.51375e6 0.767118
$$592$$ −5.62951e7 −6.60187
$$593$$ 4.77416e6 0.557519 0.278760 0.960361i $$-0.410077\pi$$
0.278760 + 0.960361i $$0.410077\pi$$
$$594$$ 3.36230e6 0.390994
$$595$$ −1.10646e7 −1.28128
$$596$$ 1.99783e7 2.30379
$$597$$ 1.68188e6 0.193135
$$598$$ 3.41173e7 3.90141
$$599$$ −1.34470e6 −0.153129 −0.0765645 0.997065i $$-0.524395\pi$$
−0.0765645 + 0.997065i $$0.524395\pi$$
$$600$$ 2.49791e7 2.83269
$$601$$ −1.25261e7 −1.41458 −0.707292 0.706921i $$-0.750084\pi$$
−0.707292 + 0.706921i $$0.750084\pi$$
$$602$$ 8.27187e6 0.930277
$$603$$ 795278. 0.0890688
$$604$$ −8.98969e6 −1.00266
$$605$$ 1.57728e6 0.175195
$$606$$ 1.05120e7 1.16280
$$607$$ −1.01433e7 −1.11740 −0.558698 0.829371i $$-0.688699\pi$$
−0.558698 + 0.829371i $$0.688699\pi$$
$$608$$ 3.31920e7 3.64146
$$609$$ −64136.5 −0.00700749
$$610$$ −1.57736e7 −1.71635
$$611$$ 1.68734e7 1.82851
$$612$$ −1.43184e7 −1.54531
$$613$$ 5.39122e6 0.579476 0.289738 0.957106i $$-0.406432\pi$$
0.289738 + 0.957106i $$0.406432\pi$$
$$614$$ 4.83815e6 0.517915
$$615$$ 657624. 0.0701116
$$616$$ −1.56182e7 −1.65837
$$617$$ 1.83062e6 0.193591 0.0967957 0.995304i $$-0.469141\pi$$
0.0967957 + 0.995304i $$0.469141\pi$$
$$618$$ −9.17709e6 −0.966572
$$619$$ 1.04088e7 1.09188 0.545939 0.837825i $$-0.316173\pi$$
0.545939 + 0.837825i $$0.316173\pi$$
$$620$$ −2.77582e7 −2.90009
$$621$$ −2.17728e6 −0.226561
$$622$$ 7.10521e6 0.736378
$$623$$ −4.30130e6 −0.443996
$$624$$ 3.52943e7 3.62863
$$625$$ −2.74143e6 −0.280722
$$626$$ 1.39625e7 1.42406
$$627$$ 5.83309e6 0.592557
$$628$$ 1.92880e7 1.95159
$$629$$ 3.06041e7 3.08428
$$630$$ −4.80218e6 −0.482044
$$631$$ 4.99395e6 0.499310 0.249655 0.968335i $$-0.419683\pi$$
0.249655 + 0.968335i $$0.419683\pi$$
$$632$$ −1.22205e7 −1.21702
$$633$$ 1.29337e6 0.128296
$$634$$ −2.02003e7 −1.99588
$$635$$ −1.36620e7 −1.34455
$$636$$ −2.69181e7 −2.63877
$$637$$ 1.36190e7 1.32984
$$638$$ −532606. −0.0518030
$$639$$ 273913. 0.0265375
$$640$$ −5.07673e7 −4.89930
$$641$$ 6.06889e6 0.583397 0.291699 0.956510i $$-0.405780\pi$$
0.291699 + 0.956510i $$0.405780\pi$$
$$642$$ 5.08251e6 0.486676
$$643$$ −1.56458e7 −1.49235 −0.746176 0.665748i $$-0.768112\pi$$
−0.746176 + 0.665748i $$0.768112\pi$$
$$644$$ 1.60117e7 1.52132
$$645$$ 9.74957e6 0.922754
$$646$$ −3.39902e7 −3.20459
$$647$$ 1.46905e7 1.37967 0.689837 0.723964i $$-0.257682\pi$$
0.689837 + 0.723964i $$0.257682\pi$$
$$648$$ −3.92526e6 −0.367224
$$649$$ −1.47256e6 −0.137233
$$650$$ 5.29936e7 4.91972
$$651$$ 2.01404e6 0.186259
$$652$$ −145474. −0.0134019
$$653$$ 1.07967e7 0.990852 0.495426 0.868650i $$-0.335012\pi$$
0.495426 + 0.868650i $$0.335012\pi$$
$$654$$ −1.45114e7 −1.32667
$$655$$ −3.26615e7 −2.97463
$$656$$ −3.10387e6 −0.281608
$$657$$ 2.18798e6 0.197756
$$658$$ 1.08358e7 0.975657
$$659$$ −1.33499e7 −1.19747 −0.598734 0.800948i $$-0.704329\pi$$
−0.598734 + 0.800948i $$0.704329\pi$$
$$660$$ −2.91434e7 −2.60424
$$661$$ −1.52627e7 −1.35871 −0.679354 0.733810i $$-0.737740\pi$$
−0.679354 + 0.733810i $$0.737740\pi$$
$$662$$ 7.05324e6 0.625524
$$663$$ −1.91873e7 −1.69523
$$664$$ 3.26126e7 2.87055
$$665$$ −8.33106e6 −0.730544
$$666$$ 1.32826e7 1.16037
$$667$$ 344893. 0.0300172
$$668$$ −5.50627e7 −4.77437
$$669$$ −2.69108e6 −0.232467
$$670$$ −9.43238e6 −0.811773
$$671$$ 6.94562e6 0.595531
$$672$$ 1.20324e7 1.02785
$$673$$ 700630. 0.0596281 0.0298141 0.999555i $$-0.490508\pi$$
0.0298141 + 0.999555i $$0.490508\pi$$
$$674$$ 2.10769e7 1.78713
$$675$$ −3.38192e6 −0.285696
$$676$$ 6.31071e7 5.31143
$$677$$ 1.76900e7 1.48339 0.741697 0.670735i $$-0.234021\pi$$
0.741697 + 0.670735i $$0.234021\pi$$
$$678$$ −8.20948e6 −0.685869
$$679$$ 2.99464e6 0.249270
$$680$$ 1.07268e8 8.89605
$$681$$ −954816. −0.0788954
$$682$$ 1.67251e7 1.37692
$$683$$ −1.72148e7 −1.41205 −0.706024 0.708187i $$-0.749513\pi$$
−0.706024 + 0.708187i $$0.749513\pi$$
$$684$$ −1.07810e7 −0.881084
$$685$$ 1.70993e7 1.39236
$$686$$ 2.00542e7 1.62703
$$687$$ 2.00492e6 0.162071
$$688$$ −4.60163e7 −3.70630
$$689$$ −3.60715e7 −2.89478
$$690$$ 2.58236e7 2.06488
$$691$$ −1.54571e7 −1.23149 −0.615746 0.787944i $$-0.711145\pi$$
−0.615746 + 0.787944i $$0.711145\pi$$
$$692$$ 2.64751e7 2.10171
$$693$$ 2.11455e6 0.167257
$$694$$ 2.78590e7 2.19567
$$695$$ −7.09312e6 −0.557025
$$696$$ 621783. 0.0486536
$$697$$ 1.68738e6 0.131562
$$698$$ 1.15862e6 0.0900120
$$699$$ 7.89818e6 0.611412
$$700$$ 2.48706e7 1.91841
$$701$$ 90599.9 0.00696358 0.00348179 0.999994i $$-0.498892\pi$$
0.00348179 + 0.999994i $$0.498892\pi$$
$$702$$ −8.32751e6 −0.637783
$$703$$ 2.30433e7 1.75856
$$704$$ 4.92526e7 3.74539
$$705$$ 1.27715e7 0.967767
$$706$$ −4.59894e6 −0.347253
$$707$$ 6.61099e6 0.497414
$$708$$ 2.72164e6 0.204055
$$709$$ 7.81917e6 0.584178 0.292089 0.956391i $$-0.405650\pi$$
0.292089 + 0.956391i $$0.405650\pi$$
$$710$$ −3.24874e6 −0.241863
$$711$$ 1.65453e6 0.122744
$$712$$ 4.16997e7 3.08271
$$713$$ −1.08305e7 −0.797855
$$714$$ −1.23218e7 −0.904541
$$715$$ −3.90535e7 −2.85690
$$716$$ 5.83197e6 0.425141
$$717$$ −1.39863e7 −1.01602
$$718$$ −2.54956e7 −1.84567
$$719$$ 2.06917e7 1.49270 0.746352 0.665552i $$-0.231804\pi$$
0.746352 + 0.665552i $$0.231804\pi$$
$$720$$ 2.67145e7 1.92050
$$721$$ −5.77148e6 −0.413475
$$722$$ 1.40377e6 0.100220
$$723$$ 593069. 0.0421948
$$724$$ 9.71073e6 0.688503
$$725$$ 535714. 0.0378519
$$726$$ 1.75650e6 0.123682
$$727$$ −1.54163e7 −1.08179 −0.540896 0.841089i $$-0.681915\pi$$
−0.540896 + 0.841089i $$0.681915\pi$$
$$728$$ 3.86822e7 2.70509
$$729$$ 531441. 0.0370370
$$730$$ −2.59505e7 −1.80234
$$731$$ 2.50162e7 1.73152
$$732$$ −1.28372e7 −0.885507
$$733$$ 1.32653e7 0.911924 0.455962 0.889999i $$-0.349295\pi$$
0.455962 + 0.889999i $$0.349295\pi$$
$$734$$ −1.92075e7 −1.31592
$$735$$ 1.03083e7 0.703834
$$736$$ −6.47042e7 −4.40289
$$737$$ 4.15338e6 0.281665
$$738$$ 732344. 0.0494965
$$739$$ 1.43430e7 0.966114 0.483057 0.875589i $$-0.339526\pi$$
0.483057 + 0.875589i $$0.339526\pi$$
$$740$$ −1.15129e8 −7.72871
$$741$$ −1.44470e7 −0.966567
$$742$$ −2.31646e7 −1.54459
$$743$$ 6.61514e6 0.439609 0.219805 0.975544i $$-0.429458\pi$$
0.219805 + 0.975544i $$0.429458\pi$$
$$744$$ −1.95255e7 −1.29321
$$745$$ 2.02638e7 1.33761
$$746$$ −6.05998e6 −0.398679
$$747$$ −4.41542e6 −0.289514
$$748$$ −7.47784e7 −4.88677
$$749$$ 3.19639e6 0.208188
$$750$$ 1.30915e7 0.849841
$$751$$ 5.45848e6 0.353160 0.176580 0.984286i $$-0.443497\pi$$
0.176580 + 0.984286i $$0.443497\pi$$
$$752$$ −6.02795e7 −3.88710
$$753$$ 223584. 0.0143699
$$754$$ 1.31912e6 0.0845000
$$755$$ −9.11815e6 −0.582156
$$756$$ −3.90820e6 −0.248698
$$757$$ 1.40547e7 0.891416 0.445708 0.895178i $$-0.352952\pi$$
0.445708 + 0.895178i $$0.352952\pi$$
$$758$$ −1.34313e7 −0.849070
$$759$$ −1.13710e7 −0.716461
$$760$$ 8.07669e7 5.07224
$$761$$ −7.51836e6 −0.470610 −0.235305 0.971922i $$-0.575609\pi$$
−0.235305 + 0.971922i $$0.575609\pi$$
$$762$$ −1.52142e7 −0.949212
$$763$$ −9.12621e6 −0.567517
$$764$$ −6.35347e7 −3.93802
$$765$$ −1.45230e7 −0.897226
$$766$$ −2.25863e7 −1.39083
$$767$$ 3.64712e6 0.223853
$$768$$ −2.30040e7 −1.40734
$$769$$ 1.72875e7 1.05418 0.527091 0.849809i $$-0.323283\pi$$
0.527091 + 0.849809i $$0.323283\pi$$
$$770$$ −2.50796e7 −1.52438
$$771$$ −4.51770e6 −0.273704
$$772$$ 5.95213e7 3.59442
$$773$$ 7.13686e6 0.429594 0.214797 0.976659i $$-0.431091\pi$$
0.214797 + 0.976659i $$0.431091\pi$$
$$774$$ 1.08573e7 0.651434
$$775$$ −1.68227e7 −1.00610
$$776$$ −2.90320e7 −1.73070
$$777$$ 8.35342e6 0.496377
$$778$$ 4.43369e7 2.62613
$$779$$ 1.27051e6 0.0750125
$$780$$ 7.21804e7 4.24798
$$781$$ 1.43052e6 0.0839203
$$782$$ 6.62602e7 3.87468
$$783$$ −84183.1 −0.00490705
$$784$$ −4.86536e7 −2.82699
$$785$$ 1.95636e7 1.13312
$$786$$ −3.63726e7 −2.09999
$$787$$ −5.86391e6 −0.337482 −0.168741 0.985660i $$-0.553970\pi$$
−0.168741 + 0.985660i $$0.553970\pi$$
$$788$$ −6.28742e7 −3.60709
$$789$$ −1.14212e7 −0.653159
$$790$$ −1.96236e7 −1.11869
$$791$$ −5.16294e6 −0.293397
$$792$$ −2.04999e7 −1.16128
$$793$$ −1.72024e7 −0.971419
$$794$$ 5.05734e7 2.84689
$$795$$ −2.73027e7 −1.53210
$$796$$ −1.62344e7 −0.908144
$$797$$ 1.96169e7 1.09392 0.546958 0.837160i $$-0.315786\pi$$
0.546958 + 0.837160i $$0.315786\pi$$
$$798$$ −9.27765e6 −0.515740
$$799$$ 3.27702e7 1.81599
$$800$$ −1.00503e8 −5.55209
$$801$$ −5.64571e6 −0.310912
$$802$$ −2.86957e7 −1.57536
$$803$$ 1.14268e7 0.625369
$$804$$ −7.67645e6 −0.418813
$$805$$ 1.62405e7 0.883302
$$806$$ −4.14237e7 −2.24601
$$807$$ 9.36877e6 0.506406
$$808$$ −6.40914e7 −3.45360
$$809$$ −2.49811e7 −1.34196 −0.670982 0.741474i $$-0.734127\pi$$
−0.670982 + 0.741474i $$0.734127\pi$$
$$810$$ −6.30315e6 −0.337555
$$811$$ −1.44411e7 −0.770987 −0.385493 0.922711i $$-0.625969\pi$$
−0.385493 + 0.922711i $$0.625969\pi$$
$$812$$ 619080. 0.0329501
$$813$$ 1.86315e7 0.988603
$$814$$ 6.93689e7 3.66947
$$815$$ −147552. −0.00778130
$$816$$ 6.85460e7 3.60377
$$817$$ 1.88359e7 0.987257
$$818$$ −3.32512e7 −1.73749
$$819$$ −5.23718e6 −0.272827
$$820$$ −6.34774e6 −0.329674
$$821$$ −625911. −0.0324082 −0.0162041 0.999869i $$-0.505158\pi$$
−0.0162041 + 0.999869i $$0.505158\pi$$
$$822$$ 1.90421e7 0.982961
$$823$$ 2.15073e7 1.10684 0.553422 0.832901i $$-0.313322\pi$$
0.553422 + 0.832901i $$0.313322\pi$$
$$824$$ 5.59526e7 2.87079
$$825$$ −1.76622e7 −0.903464
$$826$$ 2.34213e6 0.119443
$$827$$ 2.06626e7 1.05056 0.525280 0.850929i $$-0.323960\pi$$
0.525280 + 0.850929i $$0.323960\pi$$
$$828$$ 2.10163e7 1.06532
$$829$$ −5.93386e6 −0.299882 −0.149941 0.988695i $$-0.547908\pi$$
−0.149941 + 0.988695i $$0.547908\pi$$
$$830$$ 5.23690e7 2.63863
$$831$$ −1.73834e7 −0.873238
$$832$$ −1.21985e8 −6.10941
$$833$$ 2.64499e7 1.32072
$$834$$ −7.89905e6 −0.393242
$$835$$ −5.58495e7 −2.77206
$$836$$ −5.63041e7 −2.78628
$$837$$ 2.64355e6 0.130429
$$838$$ −1.46274e7 −0.719544
$$839$$ −925271. −0.0453800 −0.0226900 0.999743i $$-0.507223\pi$$
−0.0226900 + 0.999743i $$0.507223\pi$$
$$840$$ 2.92788e7 1.43171
$$841$$ −2.04978e7 −0.999350
$$842$$ −2.76954e7 −1.34626
$$843$$ −7.73736e6 −0.374994
$$844$$ −1.24843e7 −0.603265
$$845$$ 6.40089e7 3.08389
$$846$$ 1.42227e7 0.683212
$$847$$ 1.10466e6 0.0529080
$$848$$ 1.28864e8 6.15379
$$849$$ 7.10485e6 0.338287
$$850$$ 1.02920e8 4.88601
$$851$$ −4.49203e7 −2.12627
$$852$$ −2.64395e6 −0.124783
$$853$$ 1.48698e7 0.699732 0.349866 0.936800i $$-0.386227\pi$$
0.349866 + 0.936800i $$0.386227\pi$$
$$854$$ −1.10471e7 −0.518329
$$855$$ −1.09350e7 −0.511569
$$856$$ −3.09879e7 −1.44547
$$857$$ −4.11203e7 −1.91251 −0.956256 0.292531i $$-0.905503\pi$$
−0.956256 + 0.292531i $$0.905503\pi$$
$$858$$ −4.34909e7 −2.01688
$$859$$ 1.38706e7 0.641378 0.320689 0.947185i $$-0.396086\pi$$
0.320689 + 0.947185i $$0.396086\pi$$
$$860$$ −9.41081e7 −4.33891
$$861$$ 460571. 0.0211733
$$862$$ −4.55557e7 −2.08821
$$863$$ 1.87918e7 0.858898 0.429449 0.903091i $$-0.358708\pi$$
0.429449 + 0.903091i $$0.358708\pi$$
$$864$$ 1.57933e7 0.719761
$$865$$ 2.68535e7 1.22028
$$866$$ 813143. 0.0368445
$$867$$ −2.44854e7 −1.10627
$$868$$ −1.94406e7 −0.875812
$$869$$ 8.64088e6 0.388158
$$870$$ 998452. 0.0447228
$$871$$ −1.02868e7 −0.459447
$$872$$ 8.84756e7 3.94033
$$873$$ 3.93064e6 0.174553
$$874$$ 4.98904e7 2.20922
$$875$$ 8.23327e6 0.363540
$$876$$ −2.11195e7 −0.929873
$$877$$ 4.12113e7 1.80933 0.904665 0.426124i $$-0.140121\pi$$
0.904665 + 0.426124i $$0.140121\pi$$
$$878$$ −7.76505e7 −3.39944
$$879$$ 1.65359e7 0.721863
$$880$$ 1.39518e8 6.07326
$$881$$ 3.73000e7 1.61908 0.809542 0.587062i $$-0.199716\pi$$
0.809542 + 0.587062i $$0.199716\pi$$
$$882$$ 1.14796e7 0.496884
$$883$$ −3.73701e7 −1.61295 −0.806477 0.591265i $$-0.798629\pi$$
−0.806477 + 0.591265i $$0.798629\pi$$
$$884$$ 1.85206e8 7.97121
$$885$$ 2.76053e6 0.118477
$$886$$ 8.02716e7 3.43540
$$887$$ −2.39606e6 −0.102256 −0.0511280 0.998692i $$-0.516282\pi$$
−0.0511280 + 0.998692i $$0.516282\pi$$
$$888$$ −8.09836e7 −3.44639
$$889$$ −9.56824e6 −0.406048
$$890$$ 6.69609e7 2.83365
$$891$$ 2.77548e6 0.117123
$$892$$ 2.59758e7 1.09309
$$893$$ 2.46742e7 1.03542
$$894$$ 2.25662e7 0.944309
$$895$$ 5.91531e6 0.246843
$$896$$ −3.55552e7 −1.47956
$$897$$ 2.81628e7 1.16868
$$898$$ 6.11092e7 2.52881
$$899$$ −418753. −0.0172806
$$900$$ 3.26441e7 1.34338
$$901$$ −7.00554e7 −2.87494
$$902$$ 3.82470e6 0.156524
$$903$$ 6.82818e6 0.278667
$$904$$ 5.00530e7 2.03709
$$905$$ 9.84950e6 0.399754
$$906$$ −1.01542e7 −0.410983
$$907$$ −6.32877e6 −0.255447 −0.127724 0.991810i $$-0.540767\pi$$
−0.127724 + 0.991810i $$0.540767\pi$$
$$908$$ 9.21640e6 0.370977
$$909$$ 8.67733e6 0.348318
$$910$$ 6.21155e7 2.48654
$$911$$ −1.22155e6 −0.0487659 −0.0243830 0.999703i $$-0.507762\pi$$
−0.0243830 + 0.999703i $$0.507762\pi$$
$$912$$ 5.16115e7 2.05475
$$913$$ −2.30597e7 −0.915540
$$914$$ 2.48569e7 0.984197
$$915$$ −1.30206e7 −0.514137
$$916$$ −1.93526e7 −0.762080
$$917$$ −2.28747e7 −0.898323
$$918$$ −1.61731e7 −0.633412
$$919$$ −2.44744e7 −0.955924 −0.477962 0.878381i $$-0.658624\pi$$
−0.477962 + 0.878381i $$0.658624\pi$$
$$920$$ −1.57446e8 −6.13285
$$921$$ 3.99375e6 0.155143
$$922$$ −6.21873e7 −2.40921
$$923$$ −3.54302e6 −0.136889
$$924$$ −2.04108e7 −0.786466
$$925$$ −6.97737e7 −2.68125
$$926$$ −3.44184e7 −1.31905
$$927$$ −7.57541e6 −0.289539
$$928$$ −2.50174e6 −0.0953614
$$929$$ 471008. 0.0179056 0.00895280 0.999960i $$-0.497150\pi$$
0.00895280 + 0.999960i $$0.497150\pi$$
$$930$$ −3.13538e7 −1.18873
$$931$$ 1.99154e7 0.753034
$$932$$ −7.62374e7 −2.87494
$$933$$ 5.86513e6 0.220584
$$934$$ −4.47746e7 −1.67944
$$935$$ −7.58470e7 −2.83733
$$936$$ 5.07727e7 1.89426
$$937$$ 9.92896e6 0.369449 0.184725 0.982790i $$-0.440861\pi$$
0.184725 + 0.982790i $$0.440861\pi$$
$$938$$ −6.60604e6 −0.245151
$$939$$ 1.15256e7 0.426581
$$940$$ −1.23278e8 −4.55057
$$941$$ −2.51107e7 −0.924452 −0.462226 0.886762i $$-0.652949\pi$$
−0.462226 + 0.886762i $$0.652949\pi$$
$$942$$ 2.17865e7 0.799943
$$943$$ −2.47671e6 −0.0906978
$$944$$ −1.30292e7 −0.475871
$$945$$ −3.96405e6 −0.144398
$$946$$ 5.67029e7 2.06005
$$947$$ 1.98803e7 0.720356 0.360178 0.932884i $$-0.382716\pi$$
0.360178 + 0.932884i $$0.382716\pi$$
$$948$$ −1.59704e7 −0.577160
$$949$$ −2.83011e7 −1.02009
$$950$$ 7.74936e7 2.78584
$$951$$ −1.66748e7 −0.597872
$$952$$ 7.51257e7 2.68656
$$953$$ 3.07521e7 1.09684 0.548418 0.836204i $$-0.315230\pi$$
0.548418 + 0.836204i $$0.315230\pi$$
$$954$$ −3.04049e7 −1.08161
$$955$$ −6.44427e7 −2.28647
$$956$$ 1.35003e8 4.77747
$$957$$ −439650. −0.0155177
$$958$$ 4.31905e7 1.52046
$$959$$ 1.19756e7 0.420486
$$960$$ −9.23315e7 −3.23349
$$961$$ −1.54793e7 −0.540682
$$962$$ −1.71808e8 −5.98558
$$963$$ 4.19546e6 0.145785
$$964$$ −5.72462e6 −0.198406
$$965$$ 6.03719e7 2.08697
$$966$$ 1.80857e7 0.623582
$$967$$ 3.47128e7 1.19378 0.596888 0.802324i $$-0.296404\pi$$
0.596888 + 0.802324i $$0.296404\pi$$
$$968$$ −1.07093e7 −0.367345
$$969$$ −2.80579e7 −0.959944
$$970$$ −4.66193e7 −1.59088
$$971$$ 6.63760e6 0.225924 0.112962 0.993599i $$-0.463966\pi$$
0.112962 + 0.993599i $$0.463966\pi$$
$$972$$ −5.12975e6 −0.174153
$$973$$ −4.96771e6 −0.168219
$$974$$ 1.00158e8 3.38288
$$975$$ 4.37446e7 1.47371
$$976$$ 6.14551e7 2.06506
$$977$$ 2.24812e7 0.753500 0.376750 0.926315i $$-0.377042\pi$$
0.376750 + 0.926315i $$0.377042\pi$$
$$978$$ −164318. −0.00549334
$$979$$ −2.94850e7 −0.983207
$$980$$ −9.95016e7 −3.30952
$$981$$ −1.19787e7 −0.397409
$$982$$ −4.20818e7 −1.39256
$$983$$ −4.47702e7 −1.47776 −0.738882 0.673835i $$-0.764646\pi$$
−0.738882 + 0.673835i $$0.764646\pi$$
$$984$$ −4.46509e6 −0.147008
$$985$$ −6.37727e7 −2.09432
$$986$$ 2.56190e6 0.0839210
$$987$$ 8.94464e6 0.292260
$$988$$ 1.39450e8 4.54493
$$989$$ −3.67184e7 −1.19369
$$990$$ −3.29185e7 −1.06746
$$991$$ −173229. −0.00560320 −0.00280160 0.999996i $$-0.500892\pi$$
−0.00280160 + 0.999996i $$0.500892\pi$$
$$992$$ 7.85608e7 2.53470
$$993$$ 5.82224e6 0.187377
$$994$$ −2.27528e6 −0.0730412
$$995$$ −1.64664e7 −0.527281
$$996$$ 4.26200e7 1.36133
$$997$$ 1.72883e7 0.550825 0.275413 0.961326i $$-0.411186\pi$$
0.275413 + 0.961326i $$0.411186\pi$$
$$998$$ 3.75903e7 1.19467
$$999$$ 1.09644e7 0.347592
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 177.6.a.b.1.1 12
3.2 odd 2 531.6.a.d.1.12 12

By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.1 12 1.1 even 1 trivial
531.6.a.d.1.12 12 3.2 odd 2