# Properties

 Label 69.6.a.b.1.1 Level $69$ Weight $6$ Character 69.1 Self dual yes Analytic conductor $11.066$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 69.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.0664835671$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.5333.1 Defining polynomial: $$x^{3} - x^{2} - 11 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.714018$$ of defining polynomial Character $$\chi$$ $$=$$ 69.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-10.2042 q^{2} +9.00000 q^{3} +72.1256 q^{4} -55.5168 q^{5} -91.8378 q^{6} +2.50462 q^{7} -409.450 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-10.2042 q^{2} +9.00000 q^{3} +72.1256 q^{4} -55.5168 q^{5} -91.8378 q^{6} +2.50462 q^{7} -409.450 q^{8} +81.0000 q^{9} +566.504 q^{10} +228.550 q^{11} +649.131 q^{12} +658.703 q^{13} -25.5577 q^{14} -499.651 q^{15} +1870.09 q^{16} -1443.81 q^{17} -826.540 q^{18} -982.167 q^{19} -4004.18 q^{20} +22.5416 q^{21} -2332.17 q^{22} +529.000 q^{23} -3685.05 q^{24} -42.8875 q^{25} -6721.54 q^{26} +729.000 q^{27} +180.648 q^{28} -7157.56 q^{29} +5098.54 q^{30} -9259.98 q^{31} -5980.33 q^{32} +2056.95 q^{33} +14732.9 q^{34} -139.049 q^{35} +5842.18 q^{36} +2422.50 q^{37} +10022.2 q^{38} +5928.33 q^{39} +22731.3 q^{40} -4075.27 q^{41} -230.019 q^{42} -10417.3 q^{43} +16484.3 q^{44} -4496.86 q^{45} -5398.02 q^{46} +9358.08 q^{47} +16830.8 q^{48} -16800.7 q^{49} +437.632 q^{50} -12994.3 q^{51} +47509.4 q^{52} -34280.3 q^{53} -7438.86 q^{54} -12688.4 q^{55} -1025.52 q^{56} -8839.51 q^{57} +73037.2 q^{58} -7268.79 q^{59} -36037.6 q^{60} +26611.7 q^{61} +94490.6 q^{62} +202.874 q^{63} +1181.70 q^{64} -36569.1 q^{65} -20989.5 q^{66} +53450.8 q^{67} -104136. q^{68} +4761.00 q^{69} +1418.88 q^{70} +21673.7 q^{71} -33165.4 q^{72} -82856.3 q^{73} -24719.7 q^{74} -385.987 q^{75} -70839.4 q^{76} +572.432 q^{77} -60493.9 q^{78} -23960.0 q^{79} -103821. q^{80} +6561.00 q^{81} +41584.9 q^{82} +81187.7 q^{83} +1625.83 q^{84} +80155.6 q^{85} +106300. q^{86} -64418.0 q^{87} -93579.7 q^{88} +100115. q^{89} +45886.8 q^{90} +1649.80 q^{91} +38154.5 q^{92} -83339.8 q^{93} -95491.7 q^{94} +54526.8 q^{95} -53822.9 q^{96} +36122.1 q^{97} +171438. q^{98} +18512.6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + O(q^{10})$$ $$3q - 8q^{2} + 27q^{3} + 22q^{4} - 56q^{5} - 72q^{6} - 114q^{7} - 510q^{8} + 243q^{9} + 282q^{10} - 376q^{11} + 198q^{12} - 858q^{13} + 588q^{14} - 504q^{15} + 2738q^{16} - 2548q^{17} - 648q^{18} - 2846q^{19} - 4618q^{20} - 1026q^{21} - 5050q^{22} + 1587q^{23} - 4590q^{24} + 753q^{25} - 7788q^{26} + 2187q^{27} + 4736q^{28} - 16370q^{29} + 2538q^{30} - 14756q^{31} - 3878q^{32} - 3384q^{33} + 16520q^{34} - 18520q^{35} + 1782q^{36} + 15874q^{37} + 12438q^{38} - 7722q^{39} + 38270q^{40} + 12606q^{41} + 5292q^{42} + 3154q^{43} + 27114q^{44} - 4536q^{45} - 4232q^{46} + 29928q^{47} + 24642q^{48} + 4471q^{49} + 1452q^{50} - 22932q^{51} + 86856q^{52} - 44084q^{53} - 5832q^{54} + 38360q^{55} - 35704q^{56} - 25614q^{57} + 73316q^{58} - 29300q^{59} - 41562q^{60} + 54010q^{61} + 99908q^{62} - 9234q^{63} - 1582q^{64} - 51216q^{65} - 45450q^{66} + 43390q^{67} - 69840q^{68} + 14283q^{69} - 2476q^{70} + 23424q^{71} - 41310q^{72} - 91402q^{73} - 2294q^{74} + 6777q^{75} - 14274q^{76} - 97208q^{77} - 70092q^{78} - 49398q^{79} - 52626q^{80} + 19683q^{81} + 40152q^{82} - 103936q^{83} + 42624q^{84} + 5888q^{85} + 133634q^{86} - 147330q^{87} + 48898q^{88} + 96112q^{89} + 22842q^{90} + 129228q^{91} + 11638q^{92} - 132804q^{93} - 133688q^{94} - 55928q^{95} - 34902q^{96} - 135318q^{97} + 108440q^{98} - 30456q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.2042 −1.80386 −0.901932 0.431878i $$-0.857851\pi$$
−0.901932 + 0.431878i $$0.857851\pi$$
$$3$$ 9.00000 0.577350
$$4$$ 72.1256 2.25393
$$5$$ −55.5168 −0.993114 −0.496557 0.868004i $$-0.665403\pi$$
−0.496557 + 0.868004i $$0.665403\pi$$
$$6$$ −91.8378 −1.04146
$$7$$ 2.50462 0.0193196 0.00965978 0.999953i $$-0.496925\pi$$
0.00965978 + 0.999953i $$0.496925\pi$$
$$8$$ −409.450 −2.26191
$$9$$ 81.0000 0.333333
$$10$$ 566.504 1.79144
$$11$$ 228.550 0.569508 0.284754 0.958601i $$-0.408088\pi$$
0.284754 + 0.958601i $$0.408088\pi$$
$$12$$ 649.131 1.30130
$$13$$ 658.703 1.08101 0.540507 0.841339i $$-0.318232\pi$$
0.540507 + 0.841339i $$0.318232\pi$$
$$14$$ −25.5577 −0.0348499
$$15$$ −499.651 −0.573375
$$16$$ 1870.09 1.82626
$$17$$ −1443.81 −1.21168 −0.605839 0.795587i $$-0.707162\pi$$
−0.605839 + 0.795587i $$0.707162\pi$$
$$18$$ −826.540 −0.601288
$$19$$ −982.167 −0.624168 −0.312084 0.950055i $$-0.601027\pi$$
−0.312084 + 0.950055i $$0.601027\pi$$
$$20$$ −4004.18 −2.23841
$$21$$ 22.5416 0.0111542
$$22$$ −2332.17 −1.02731
$$23$$ 529.000 0.208514
$$24$$ −3685.05 −1.30592
$$25$$ −42.8875 −0.0137240
$$26$$ −6721.54 −1.95000
$$27$$ 729.000 0.192450
$$28$$ 180.648 0.0435449
$$29$$ −7157.56 −1.58041 −0.790205 0.612842i $$-0.790026\pi$$
−0.790205 + 0.612842i $$0.790026\pi$$
$$30$$ 5098.54 1.03429
$$31$$ −9259.98 −1.73064 −0.865318 0.501223i $$-0.832884\pi$$
−0.865318 + 0.501223i $$0.832884\pi$$
$$32$$ −5980.33 −1.03240
$$33$$ 2056.95 0.328805
$$34$$ 14732.9 2.18570
$$35$$ −139.049 −0.0191865
$$36$$ 5842.18 0.751309
$$37$$ 2422.50 0.290911 0.145455 0.989365i $$-0.453535\pi$$
0.145455 + 0.989365i $$0.453535\pi$$
$$38$$ 10022.2 1.12591
$$39$$ 5928.33 0.624124
$$40$$ 22731.3 2.24634
$$41$$ −4075.27 −0.378614 −0.189307 0.981918i $$-0.560624\pi$$
−0.189307 + 0.981918i $$0.560624\pi$$
$$42$$ −230.019 −0.0201206
$$43$$ −10417.3 −0.859178 −0.429589 0.903024i $$-0.641342\pi$$
−0.429589 + 0.903024i $$0.641342\pi$$
$$44$$ 16484.3 1.28363
$$45$$ −4496.86 −0.331038
$$46$$ −5398.02 −0.376132
$$47$$ 9358.08 0.617934 0.308967 0.951073i $$-0.400017\pi$$
0.308967 + 0.951073i $$0.400017\pi$$
$$48$$ 16830.8 1.05439
$$49$$ −16800.7 −0.999627
$$50$$ 437.632 0.0247562
$$51$$ −12994.3 −0.699563
$$52$$ 47509.4 2.43653
$$53$$ −34280.3 −1.67631 −0.838157 0.545428i $$-0.816367\pi$$
−0.838157 + 0.545428i $$0.816367\pi$$
$$54$$ −7438.86 −0.347154
$$55$$ −12688.4 −0.565586
$$56$$ −1025.52 −0.0436991
$$57$$ −8839.51 −0.360364
$$58$$ 73037.2 2.85085
$$59$$ −7268.79 −0.271852 −0.135926 0.990719i $$-0.543401\pi$$
−0.135926 + 0.990719i $$0.543401\pi$$
$$60$$ −36037.6 −1.29234
$$61$$ 26611.7 0.915687 0.457844 0.889033i $$-0.348622\pi$$
0.457844 + 0.889033i $$0.348622\pi$$
$$62$$ 94490.6 3.12183
$$63$$ 202.874 0.00643985
$$64$$ 1181.70 0.0360625
$$65$$ −36569.1 −1.07357
$$66$$ −20989.5 −0.593120
$$67$$ 53450.8 1.45468 0.727339 0.686278i $$-0.240757\pi$$
0.727339 + 0.686278i $$0.240757\pi$$
$$68$$ −104136. −2.73103
$$69$$ 4761.00 0.120386
$$70$$ 1418.88 0.0346099
$$71$$ 21673.7 0.510255 0.255128 0.966907i $$-0.417882\pi$$
0.255128 + 0.966907i $$0.417882\pi$$
$$72$$ −33165.4 −0.753970
$$73$$ −82856.3 −1.81978 −0.909889 0.414851i $$-0.863834\pi$$
−0.909889 + 0.414851i $$0.863834\pi$$
$$74$$ −24719.7 −0.524764
$$75$$ −385.987 −0.00792355
$$76$$ −70839.4 −1.40683
$$77$$ 572.432 0.0110026
$$78$$ −60493.9 −1.12584
$$79$$ −23960.0 −0.431936 −0.215968 0.976400i $$-0.569291\pi$$
−0.215968 + 0.976400i $$0.569291\pi$$
$$80$$ −103821. −1.81368
$$81$$ 6561.00 0.111111
$$82$$ 41584.9 0.682968
$$83$$ 81187.7 1.29359 0.646793 0.762666i $$-0.276110\pi$$
0.646793 + 0.762666i $$0.276110\pi$$
$$84$$ 1625.83 0.0251406
$$85$$ 80155.6 1.20334
$$86$$ 106300. 1.54984
$$87$$ −64418.0 −0.912451
$$88$$ −93579.7 −1.28818
$$89$$ 100115. 1.33975 0.669876 0.742473i $$-0.266347\pi$$
0.669876 + 0.742473i $$0.266347\pi$$
$$90$$ 45886.8 0.597148
$$91$$ 1649.80 0.0208847
$$92$$ 38154.5 0.469976
$$93$$ −83339.8 −0.999183
$$94$$ −95491.7 −1.11467
$$95$$ 54526.8 0.619870
$$96$$ −53822.9 −0.596059
$$97$$ 36122.1 0.389802 0.194901 0.980823i $$-0.437562\pi$$
0.194901 + 0.980823i $$0.437562\pi$$
$$98$$ 171438. 1.80319
$$99$$ 18512.6 0.189836
$$100$$ −3093.29 −0.0309329
$$101$$ −41151.8 −0.401408 −0.200704 0.979652i $$-0.564323\pi$$
−0.200704 + 0.979652i $$0.564323\pi$$
$$102$$ 132596. 1.26192
$$103$$ −172534. −1.60244 −0.801221 0.598369i $$-0.795816\pi$$
−0.801221 + 0.598369i $$0.795816\pi$$
$$104$$ −269706. −2.44516
$$105$$ −1251.44 −0.0110774
$$106$$ 349803. 3.02384
$$107$$ 178228. 1.50493 0.752464 0.658633i $$-0.228865\pi$$
0.752464 + 0.658633i $$0.228865\pi$$
$$108$$ 52579.6 0.433768
$$109$$ −138352. −1.11537 −0.557686 0.830052i $$-0.688311\pi$$
−0.557686 + 0.830052i $$0.688311\pi$$
$$110$$ 129475. 1.02024
$$111$$ 21802.5 0.167958
$$112$$ 4683.86 0.0352825
$$113$$ −13523.5 −0.0996307 −0.0498153 0.998758i $$-0.515863\pi$$
−0.0498153 + 0.998758i $$0.515863\pi$$
$$114$$ 90200.0 0.650047
$$115$$ −29368.4 −0.207079
$$116$$ −516243. −3.56213
$$117$$ 53355.0 0.360338
$$118$$ 74172.1 0.490383
$$119$$ −3616.20 −0.0234091
$$120$$ 204582. 1.29692
$$121$$ −108816. −0.675661
$$122$$ −271551. −1.65178
$$123$$ −36677.4 −0.218593
$$124$$ −667882. −3.90073
$$125$$ 175871. 1.00674
$$126$$ −2070.17 −0.0116166
$$127$$ 158567. 0.872376 0.436188 0.899856i $$-0.356328\pi$$
0.436188 + 0.899856i $$0.356328\pi$$
$$128$$ 179312. 0.967353
$$129$$ −93755.6 −0.496047
$$130$$ 373158. 1.93658
$$131$$ −383514. −1.95255 −0.976277 0.216526i $$-0.930527\pi$$
−0.976277 + 0.216526i $$0.930527\pi$$
$$132$$ 148359. 0.741103
$$133$$ −2459.96 −0.0120587
$$134$$ −545422. −2.62404
$$135$$ −40471.7 −0.191125
$$136$$ 591167. 2.74071
$$137$$ −33171.1 −0.150994 −0.0754968 0.997146i $$-0.524054\pi$$
−0.0754968 + 0.997146i $$0.524054\pi$$
$$138$$ −48582.2 −0.217160
$$139$$ −128488. −0.564060 −0.282030 0.959406i $$-0.591008\pi$$
−0.282030 + 0.959406i $$0.591008\pi$$
$$140$$ −10029.0 −0.0432450
$$141$$ 84222.7 0.356764
$$142$$ −221163. −0.920431
$$143$$ 150547. 0.615646
$$144$$ 151477. 0.608752
$$145$$ 397365. 1.56953
$$146$$ 845482. 3.28263
$$147$$ −151207. −0.577135
$$148$$ 174725. 0.655692
$$149$$ −184228. −0.679814 −0.339907 0.940459i $$-0.610396\pi$$
−0.339907 + 0.940459i $$0.610396\pi$$
$$150$$ 3938.69 0.0142930
$$151$$ −19798.1 −0.0706610 −0.0353305 0.999376i $$-0.511248\pi$$
−0.0353305 + 0.999376i $$0.511248\pi$$
$$152$$ 402148. 1.41181
$$153$$ −116948. −0.403893
$$154$$ −5841.21 −0.0198473
$$155$$ 514084. 1.71872
$$156$$ 427585. 1.40673
$$157$$ 193317. 0.625923 0.312962 0.949766i $$-0.398679\pi$$
0.312962 + 0.949766i $$0.398679\pi$$
$$158$$ 244493. 0.779154
$$159$$ −308523. −0.967821
$$160$$ 332008. 1.02530
$$161$$ 1324.95 0.00402841
$$162$$ −66949.7 −0.200429
$$163$$ 106286. 0.313334 0.156667 0.987651i $$-0.449925\pi$$
0.156667 + 0.987651i $$0.449925\pi$$
$$164$$ −293931. −0.853368
$$165$$ −114195. −0.326541
$$166$$ −828456. −2.33345
$$167$$ −77715.5 −0.215634 −0.107817 0.994171i $$-0.534386\pi$$
−0.107817 + 0.994171i $$0.534386\pi$$
$$168$$ −9229.66 −0.0252297
$$169$$ 62597.3 0.168593
$$170$$ −817923. −2.17065
$$171$$ −79555.5 −0.208056
$$172$$ −751353. −1.93652
$$173$$ 218524. 0.555115 0.277558 0.960709i $$-0.410475\pi$$
0.277558 + 0.960709i $$0.410475\pi$$
$$174$$ 657334. 1.64594
$$175$$ −107.417 −0.000265142 0
$$176$$ 427408. 1.04007
$$177$$ −65419.1 −0.156954
$$178$$ −1.02159e6 −2.41673
$$179$$ −545493. −1.27250 −0.636248 0.771485i $$-0.719514\pi$$
−0.636248 + 0.771485i $$0.719514\pi$$
$$180$$ −324339. −0.746135
$$181$$ 510878. 1.15910 0.579549 0.814937i $$-0.303229\pi$$
0.579549 + 0.814937i $$0.303229\pi$$
$$182$$ −16834.9 −0.0376732
$$183$$ 239505. 0.528672
$$184$$ −216599. −0.471641
$$185$$ −134490. −0.288908
$$186$$ 850416. 1.80239
$$187$$ −329982. −0.690060
$$188$$ 674957. 1.39278
$$189$$ 1825.87 0.00371805
$$190$$ −556402. −1.11816
$$191$$ −57685.2 −0.114415 −0.0572073 0.998362i $$-0.518220\pi$$
−0.0572073 + 0.998362i $$0.518220\pi$$
$$192$$ 10635.3 0.0208207
$$193$$ −822091. −1.58864 −0.794322 0.607497i $$-0.792174\pi$$
−0.794322 + 0.607497i $$0.792174\pi$$
$$194$$ −368597. −0.703149
$$195$$ −329122. −0.619827
$$196$$ −1.21176e6 −2.25308
$$197$$ 971571. 1.78365 0.891824 0.452383i $$-0.149426\pi$$
0.891824 + 0.452383i $$0.149426\pi$$
$$198$$ −188906. −0.342438
$$199$$ 806110. 1.44298 0.721492 0.692423i $$-0.243457\pi$$
0.721492 + 0.692423i $$0.243457\pi$$
$$200$$ 17560.3 0.0310425
$$201$$ 481057. 0.839859
$$202$$ 419921. 0.724085
$$203$$ −17927.0 −0.0305329
$$204$$ −937220. −1.57676
$$205$$ 226246. 0.376007
$$206$$ 1.76057e6 2.89059
$$207$$ 42849.0 0.0695048
$$208$$ 1.23183e6 1.97421
$$209$$ −224474. −0.355468
$$210$$ 12769.9 0.0199820
$$211$$ −821913. −1.27092 −0.635462 0.772132i $$-0.719190\pi$$
−0.635462 + 0.772132i $$0.719190\pi$$
$$212$$ −2.47249e6 −3.77829
$$213$$ 195063. 0.294596
$$214$$ −1.81867e6 −2.71469
$$215$$ 578334. 0.853262
$$216$$ −298489. −0.435305
$$217$$ −23192.8 −0.0334351
$$218$$ 1.41177e6 2.01198
$$219$$ −745707. −1.05065
$$220$$ −915156. −1.27479
$$221$$ −951042. −1.30984
$$222$$ −222477. −0.302973
$$223$$ 511948. 0.689388 0.344694 0.938715i $$-0.387983\pi$$
0.344694 + 0.938715i $$0.387983\pi$$
$$224$$ −14978.5 −0.0199456
$$225$$ −3473.89 −0.00457467
$$226$$ 137996. 0.179720
$$227$$ −1.45074e6 −1.86864 −0.934321 0.356433i $$-0.883993\pi$$
−0.934321 + 0.356433i $$0.883993\pi$$
$$228$$ −637555. −0.812233
$$229$$ 621169. 0.782747 0.391373 0.920232i $$-0.372000\pi$$
0.391373 + 0.920232i $$0.372000\pi$$
$$230$$ 299681. 0.373542
$$231$$ 5151.89 0.00635238
$$232$$ 2.93066e6 3.57475
$$233$$ 490334. 0.591701 0.295851 0.955234i $$-0.404397\pi$$
0.295851 + 0.955234i $$0.404397\pi$$
$$234$$ −544445. −0.650001
$$235$$ −519530. −0.613679
$$236$$ −524266. −0.612733
$$237$$ −215640. −0.249378
$$238$$ 36900.4 0.0422268
$$239$$ 109503. 0.124003 0.0620014 0.998076i $$-0.480252\pi$$
0.0620014 + 0.998076i $$0.480252\pi$$
$$240$$ −934390. −1.04713
$$241$$ 1.36171e6 1.51022 0.755111 0.655597i $$-0.227583\pi$$
0.755111 + 0.655597i $$0.227583\pi$$
$$242$$ 1.11038e6 1.21880
$$243$$ 59049.0 0.0641500
$$244$$ 1.91938e6 2.06389
$$245$$ 932722. 0.992744
$$246$$ 374264. 0.394312
$$247$$ −646957. −0.674735
$$248$$ 3.79150e6 3.91455
$$249$$ 730690. 0.746852
$$250$$ −1.79462e6 −1.81603
$$251$$ 8042.27 0.00805739 0.00402869 0.999992i $$-0.498718\pi$$
0.00402869 + 0.999992i $$0.498718\pi$$
$$252$$ 14632.4 0.0145150
$$253$$ 120903. 0.118751
$$254$$ −1.61805e6 −1.57365
$$255$$ 721400. 0.694746
$$256$$ −1.86755e6 −1.78104
$$257$$ 1.16730e6 1.10243 0.551214 0.834364i $$-0.314165\pi$$
0.551214 + 0.834364i $$0.314165\pi$$
$$258$$ 956700. 0.894801
$$259$$ 6067.46 0.00562027
$$260$$ −2.63757e6 −2.41975
$$261$$ −579762. −0.526804
$$262$$ 3.91345e6 3.52214
$$263$$ −558051. −0.497490 −0.248745 0.968569i $$-0.580018\pi$$
−0.248745 + 0.968569i $$0.580018\pi$$
$$264$$ −842218. −0.743729
$$265$$ 1.90313e6 1.66477
$$266$$ 25101.9 0.0217522
$$267$$ 901036. 0.773506
$$268$$ 3.85517e6 3.27874
$$269$$ −1.21947e6 −1.02752 −0.513759 0.857935i $$-0.671747\pi$$
−0.513759 + 0.857935i $$0.671747\pi$$
$$270$$ 412981. 0.344763
$$271$$ 1.40581e6 1.16280 0.581399 0.813618i $$-0.302505\pi$$
0.581399 + 0.813618i $$0.302505\pi$$
$$272$$ −2.70005e6 −2.21283
$$273$$ 14848.2 0.0120578
$$274$$ 338484. 0.272372
$$275$$ −9801.94 −0.00781592
$$276$$ 343390. 0.271341
$$277$$ 1.87799e6 1.47060 0.735300 0.677742i $$-0.237041\pi$$
0.735300 + 0.677742i $$0.237041\pi$$
$$278$$ 1.31112e6 1.01749
$$279$$ −750058. −0.576879
$$280$$ 56933.4 0.0433982
$$281$$ −2.11759e6 −1.59984 −0.799920 0.600106i $$-0.795125\pi$$
−0.799920 + 0.600106i $$0.795125\pi$$
$$282$$ −859425. −0.643555
$$283$$ 1.16042e6 0.861288 0.430644 0.902522i $$-0.358286\pi$$
0.430644 + 0.902522i $$0.358286\pi$$
$$284$$ 1.56323e6 1.15008
$$285$$ 490741. 0.357882
$$286$$ −1.53621e6 −1.11054
$$287$$ −10207.0 −0.00731466
$$288$$ −484406. −0.344135
$$289$$ 664726. 0.468164
$$290$$ −4.05479e6 −2.83122
$$291$$ 325099. 0.225052
$$292$$ −5.97606e6 −4.10165
$$293$$ −1.27325e6 −0.866449 −0.433225 0.901286i $$-0.642624\pi$$
−0.433225 + 0.901286i $$0.642624\pi$$
$$294$$ 1.54294e6 1.04107
$$295$$ 403540. 0.269980
$$296$$ −991893. −0.658015
$$297$$ 166613. 0.109602
$$298$$ 1.87990e6 1.22629
$$299$$ 348454. 0.225407
$$300$$ −27839.6 −0.0178591
$$301$$ −26091.4 −0.0165990
$$302$$ 202023. 0.127463
$$303$$ −370367. −0.231753
$$304$$ −1.83674e6 −1.13989
$$305$$ −1.47739e6 −0.909382
$$306$$ 1.19337e6 0.728568
$$307$$ −2.26175e6 −1.36961 −0.684807 0.728724i $$-0.740114\pi$$
−0.684807 + 0.728724i $$0.740114\pi$$
$$308$$ 41287.0 0.0247991
$$309$$ −1.55281e6 −0.925170
$$310$$ −5.24582e6 −3.10034
$$311$$ 1.20263e6 0.705066 0.352533 0.935799i $$-0.385320\pi$$
0.352533 + 0.935799i $$0.385320\pi$$
$$312$$ −2.42735e6 −1.41171
$$313$$ −484311. −0.279424 −0.139712 0.990192i $$-0.544618\pi$$
−0.139712 + 0.990192i $$0.544618\pi$$
$$314$$ −1.97265e6 −1.12908
$$315$$ −11262.9 −0.00639551
$$316$$ −1.72813e6 −0.973551
$$317$$ 796624. 0.445251 0.222626 0.974904i $$-0.428537\pi$$
0.222626 + 0.974904i $$0.428537\pi$$
$$318$$ 3.14823e6 1.74582
$$319$$ −1.63586e6 −0.900056
$$320$$ −65603.9 −0.0358142
$$321$$ 1.60405e6 0.868871
$$322$$ −13520.0 −0.00726670
$$323$$ 1.41806e6 0.756291
$$324$$ 473216. 0.250436
$$325$$ −28250.1 −0.0148358
$$326$$ −1.08456e6 −0.565212
$$327$$ −1.24517e6 −0.643961
$$328$$ 1.66862e6 0.856391
$$329$$ 23438.5 0.0119382
$$330$$ 1.16527e6 0.589036
$$331$$ −262856. −0.131871 −0.0659353 0.997824i $$-0.521003\pi$$
−0.0659353 + 0.997824i $$0.521003\pi$$
$$332$$ 5.85572e6 2.91565
$$333$$ 196223. 0.0969703
$$334$$ 793024. 0.388974
$$335$$ −2.96741e6 −1.44466
$$336$$ 42154.7 0.0203703
$$337$$ 1.23123e6 0.590558 0.295279 0.955411i $$-0.404587\pi$$
0.295279 + 0.955411i $$0.404587\pi$$
$$338$$ −638755. −0.304118
$$339$$ −121712. −0.0575218
$$340$$ 5.78127e6 2.71223
$$341$$ −2.11637e6 −0.985611
$$342$$ 811800. 0.375305
$$343$$ −84174.7 −0.0386319
$$344$$ 4.26535e6 1.94339
$$345$$ −264315. −0.119557
$$346$$ −2.22986e6 −1.00135
$$347$$ −3.70344e6 −1.65113 −0.825567 0.564305i $$-0.809144\pi$$
−0.825567 + 0.564305i $$0.809144\pi$$
$$348$$ −4.64619e6 −2.05660
$$349$$ 1.62884e6 0.715837 0.357918 0.933753i $$-0.383487\pi$$
0.357918 + 0.933753i $$0.383487\pi$$
$$350$$ 1096.10 0.000478279 0
$$351$$ 480195. 0.208041
$$352$$ −1.36680e6 −0.587962
$$353$$ 2.56589e6 1.09598 0.547989 0.836486i $$-0.315394\pi$$
0.547989 + 0.836486i $$0.315394\pi$$
$$354$$ 667549. 0.283123
$$355$$ −1.20326e6 −0.506742
$$356$$ 7.22086e6 3.01970
$$357$$ −32545.8 −0.0135152
$$358$$ 5.56631e6 2.29541
$$359$$ −3.46895e6 −1.42057 −0.710284 0.703916i $$-0.751433\pi$$
−0.710284 + 0.703916i $$0.751433\pi$$
$$360$$ 1.84124e6 0.748779
$$361$$ −1.51145e6 −0.610414
$$362$$ −5.21310e6 −2.09086
$$363$$ −979343. −0.390093
$$364$$ 118993. 0.0470726
$$365$$ 4.59992e6 1.80725
$$366$$ −2.44395e6 −0.953653
$$367$$ 1.64135e6 0.636115 0.318058 0.948071i $$-0.396969\pi$$
0.318058 + 0.948071i $$0.396969\pi$$
$$368$$ 989275. 0.380801
$$369$$ −330097. −0.126205
$$370$$ 1.37236e6 0.521151
$$371$$ −85859.4 −0.0323857
$$372$$ −6.01094e6 −2.25208
$$373$$ 1.58607e6 0.590268 0.295134 0.955456i $$-0.404636\pi$$
0.295134 + 0.955456i $$0.404636\pi$$
$$374$$ 3.36721e6 1.24477
$$375$$ 1.58284e6 0.581244
$$376$$ −3.83166e6 −1.39771
$$377$$ −4.71471e6 −1.70845
$$378$$ −18631.5 −0.00670686
$$379$$ −966030. −0.345456 −0.172728 0.984970i $$-0.555258\pi$$
−0.172728 + 0.984970i $$0.555258\pi$$
$$380$$ 3.93278e6 1.39714
$$381$$ 1.42710e6 0.503667
$$382$$ 588631. 0.206388
$$383$$ −3.10322e6 −1.08097 −0.540487 0.841352i $$-0.681760\pi$$
−0.540487 + 0.841352i $$0.681760\pi$$
$$384$$ 1.61381e6 0.558501
$$385$$ −31779.6 −0.0109269
$$386$$ 8.38878e6 2.86570
$$387$$ −843800. −0.286393
$$388$$ 2.60533e6 0.878584
$$389$$ 624568. 0.209269 0.104635 0.994511i $$-0.466633\pi$$
0.104635 + 0.994511i $$0.466633\pi$$
$$390$$ 3.35842e6 1.11808
$$391$$ −763775. −0.252652
$$392$$ 6.87905e6 2.26107
$$393$$ −3.45163e6 −1.12731
$$394$$ −9.91410e6 −3.21746
$$395$$ 1.33018e6 0.428962
$$396$$ 1.33523e6 0.427876
$$397$$ 2.80814e6 0.894216 0.447108 0.894480i $$-0.352454\pi$$
0.447108 + 0.894480i $$0.352454\pi$$
$$398$$ −8.22570e6 −2.60295
$$399$$ −22139.6 −0.00696207
$$400$$ −80203.3 −0.0250635
$$401$$ 3.81287e6 1.18411 0.592053 0.805899i $$-0.298318\pi$$
0.592053 + 0.805899i $$0.298318\pi$$
$$402$$ −4.90880e6 −1.51499
$$403$$ −6.09958e6 −1.87084
$$404$$ −2.96810e6 −0.904743
$$405$$ −364246. −0.110346
$$406$$ 182931. 0.0550771
$$407$$ 553663. 0.165676
$$408$$ 5.32050e6 1.58235
$$409$$ −3.86268e6 −1.14178 −0.570888 0.821028i $$-0.693401\pi$$
−0.570888 + 0.821028i $$0.693401\pi$$
$$410$$ −2.30866e6 −0.678266
$$411$$ −298540. −0.0871762
$$412$$ −1.24441e7 −3.61178
$$413$$ −18205.6 −0.00525206
$$414$$ −437240. −0.125377
$$415$$ −4.50728e6 −1.28468
$$416$$ −3.93926e6 −1.11604
$$417$$ −1.15639e6 −0.325660
$$418$$ 2.29058e6 0.641217
$$419$$ −4.00521e6 −1.11453 −0.557263 0.830336i $$-0.688149\pi$$
−0.557263 + 0.830336i $$0.688149\pi$$
$$420$$ −90260.7 −0.0249675
$$421$$ 4.89697e6 1.34655 0.673275 0.739392i $$-0.264887\pi$$
0.673275 + 0.739392i $$0.264887\pi$$
$$422$$ 8.38696e6 2.29257
$$423$$ 758005. 0.205978
$$424$$ 1.40361e7 3.79168
$$425$$ 61921.3 0.0166291
$$426$$ −1.99047e6 −0.531411
$$427$$ 66652.2 0.0176907
$$428$$ 1.28548e7 3.39200
$$429$$ 1.35492e6 0.355443
$$430$$ −5.90143e6 −1.53917
$$431$$ 1.81874e6 0.471605 0.235803 0.971801i $$-0.424228\pi$$
0.235803 + 0.971801i $$0.424228\pi$$
$$432$$ 1.36329e6 0.351463
$$433$$ 545672. 0.139866 0.0699330 0.997552i $$-0.477721\pi$$
0.0699330 + 0.997552i $$0.477721\pi$$
$$434$$ 236663. 0.0603124
$$435$$ 3.57628e6 0.906168
$$436$$ −9.97874e6 −2.51397
$$437$$ −519566. −0.130148
$$438$$ 7.60934e6 1.89523
$$439$$ 2.81114e6 0.696179 0.348090 0.937461i $$-0.386830\pi$$
0.348090 + 0.937461i $$0.386830\pi$$
$$440$$ 5.19525e6 1.27931
$$441$$ −1.36086e6 −0.333209
$$442$$ 9.70462e6 2.36278
$$443$$ −2.81428e6 −0.681332 −0.340666 0.940184i $$-0.610652\pi$$
−0.340666 + 0.940184i $$0.610652\pi$$
$$444$$ 1.57252e6 0.378564
$$445$$ −5.55807e6 −1.33053
$$446$$ −5.22402e6 −1.24356
$$447$$ −1.65805e6 −0.392491
$$448$$ 2959.70 0.000696712 0
$$449$$ 4.00644e6 0.937871 0.468936 0.883232i $$-0.344638\pi$$
0.468936 + 0.883232i $$0.344638\pi$$
$$450$$ 35448.2 0.00825207
$$451$$ −931403. −0.215624
$$452$$ −975391. −0.224560
$$453$$ −178182. −0.0407962
$$454$$ 1.48037e7 3.37078
$$455$$ −91591.8 −0.0207409
$$456$$ 3.61933e6 0.815110
$$457$$ −5.43801e6 −1.21801 −0.609003 0.793168i $$-0.708430\pi$$
−0.609003 + 0.793168i $$0.708430\pi$$
$$458$$ −6.33853e6 −1.41197
$$459$$ −1.05254e6 −0.233188
$$460$$ −2.11821e6 −0.466740
$$461$$ 2.50966e6 0.550001 0.275000 0.961444i $$-0.411322\pi$$
0.275000 + 0.961444i $$0.411322\pi$$
$$462$$ −52570.9 −0.0114588
$$463$$ 108662. 0.0235573 0.0117787 0.999931i $$-0.496251\pi$$
0.0117787 + 0.999931i $$0.496251\pi$$
$$464$$ −1.33852e7 −2.88623
$$465$$ 4.62676e6 0.992303
$$466$$ −5.00347e6 −1.06735
$$467$$ 3.09125e6 0.655906 0.327953 0.944694i $$-0.393641\pi$$
0.327953 + 0.944694i $$0.393641\pi$$
$$468$$ 3.84826e6 0.812176
$$469$$ 133874. 0.0281037
$$470$$ 5.30139e6 1.10699
$$471$$ 1.73985e6 0.361377
$$472$$ 2.97620e6 0.614904
$$473$$ −2.38087e6 −0.489309
$$474$$ 2.20043e6 0.449845
$$475$$ 42122.7 0.00856608
$$476$$ −260820. −0.0527624
$$477$$ −2.77671e6 −0.558772
$$478$$ −1.11739e6 −0.223684
$$479$$ −7.74743e6 −1.54283 −0.771417 0.636330i $$-0.780452\pi$$
−0.771417 + 0.636330i $$0.780452\pi$$
$$480$$ 2.98808e6 0.591955
$$481$$ 1.59571e6 0.314479
$$482$$ −1.38951e7 −2.72424
$$483$$ 11924.5 0.00232580
$$484$$ −7.84841e6 −1.52289
$$485$$ −2.00538e6 −0.387118
$$486$$ −602548. −0.115718
$$487$$ 6.60929e6 1.26279 0.631397 0.775460i $$-0.282482\pi$$
0.631397 + 0.775460i $$0.282482\pi$$
$$488$$ −1.08961e7 −2.07120
$$489$$ 956575. 0.180903
$$490$$ −9.51768e6 −1.79077
$$491$$ 1.68453e6 0.315337 0.157669 0.987492i $$-0.449602\pi$$
0.157669 + 0.987492i $$0.449602\pi$$
$$492$$ −2.64538e6 −0.492692
$$493$$ 1.03341e7 1.91495
$$494$$ 6.60168e6 1.21713
$$495$$ −1.02776e6 −0.188529
$$496$$ −1.73170e7 −3.16058
$$497$$ 54284.5 0.00985791
$$498$$ −7.45610e6 −1.34722
$$499$$ 6.29003e6 1.13084 0.565420 0.824803i $$-0.308714\pi$$
0.565420 + 0.824803i $$0.308714\pi$$
$$500$$ 1.26848e7 2.26913
$$501$$ −699440. −0.124496
$$502$$ −82064.9 −0.0145344
$$503$$ 3.68350e6 0.649143 0.324572 0.945861i $$-0.394780\pi$$
0.324572 + 0.945861i $$0.394780\pi$$
$$504$$ −83066.9 −0.0145664
$$505$$ 2.28462e6 0.398644
$$506$$ −1.23372e6 −0.214210
$$507$$ 563376. 0.0973371
$$508$$ 1.14367e7 1.96627
$$509$$ 3.33612e6 0.570752 0.285376 0.958416i $$-0.407882\pi$$
0.285376 + 0.958416i $$0.407882\pi$$
$$510$$ −7.36131e6 −1.25323
$$511$$ −207524. −0.0351573
$$512$$ 1.33189e7 2.24539
$$513$$ −716000. −0.120121
$$514$$ −1.19114e7 −1.98863
$$515$$ 9.57854e6 1.59141
$$516$$ −6.76218e6 −1.11805
$$517$$ 2.13879e6 0.351918
$$518$$ −61913.5 −0.0101382
$$519$$ 1.96671e6 0.320496
$$520$$ 1.49732e7 2.42832
$$521$$ −987399. −0.159367 −0.0796835 0.996820i $$-0.525391\pi$$
−0.0796835 + 0.996820i $$0.525391\pi$$
$$522$$ 5.91601e6 0.950282
$$523$$ 6.57834e6 1.05163 0.525814 0.850600i $$-0.323761\pi$$
0.525814 + 0.850600i $$0.323761\pi$$
$$524$$ −2.76612e7 −4.40091
$$525$$ −966.753 −0.000153080 0
$$526$$ 5.69446e6 0.897404
$$527$$ 1.33696e7 2.09697
$$528$$ 3.84667e6 0.600483
$$529$$ 279841. 0.0434783
$$530$$ −1.94200e7 −3.00302
$$531$$ −588772. −0.0906172
$$532$$ −177426. −0.0271793
$$533$$ −2.68439e6 −0.409287
$$534$$ −9.19435e6 −1.39530
$$535$$ −9.89463e6 −1.49457
$$536$$ −2.18854e7 −3.29035
$$537$$ −4.90943e6 −0.734676
$$538$$ 1.24437e7 1.85350
$$539$$ −3.83981e6 −0.569295
$$540$$ −2.91905e6 −0.430781
$$541$$ −6.47470e6 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$542$$ −1.43452e7 −2.09753
$$543$$ 4.59790e6 0.669206
$$544$$ 8.63445e6 1.25094
$$545$$ 7.68087e6 1.10769
$$546$$ −151514. −0.0217506
$$547$$ −495799. −0.0708496 −0.0354248 0.999372i $$-0.511278\pi$$
−0.0354248 + 0.999372i $$0.511278\pi$$
$$548$$ −2.39249e6 −0.340328
$$549$$ 2.15554e6 0.305229
$$550$$ 100021. 0.0140989
$$551$$ 7.02992e6 0.986442
$$552$$ −1.94939e6 −0.272302
$$553$$ −60010.8 −0.00834481
$$554$$ −1.91634e7 −2.65276
$$555$$ −1.21041e6 −0.166801
$$556$$ −9.26727e6 −1.27135
$$557$$ 3.94887e6 0.539306 0.269653 0.962958i $$-0.413091\pi$$
0.269653 + 0.962958i $$0.413091\pi$$
$$558$$ 7.65374e6 1.04061
$$559$$ −6.86190e6 −0.928784
$$560$$ −260033. −0.0350395
$$561$$ −2.96984e6 −0.398406
$$562$$ 2.16083e7 2.88590
$$563$$ −2.70237e6 −0.359314 −0.179657 0.983729i $$-0.557499\pi$$
−0.179657 + 0.983729i $$0.557499\pi$$
$$564$$ 6.07462e6 0.804120
$$565$$ 750781. 0.0989447
$$566$$ −1.18411e7 −1.55365
$$567$$ 16432.8 0.00214662
$$568$$ −8.87430e6 −1.15415
$$569$$ 1.33108e7 1.72355 0.861773 0.507294i $$-0.169354\pi$$
0.861773 + 0.507294i $$0.169354\pi$$
$$570$$ −5.00762e6 −0.645571
$$571$$ −1.10053e7 −1.41257 −0.706286 0.707926i $$-0.749631\pi$$
−0.706286 + 0.707926i $$0.749631\pi$$
$$572$$ 1.08583e7 1.38762
$$573$$ −519167. −0.0660572
$$574$$ 104154. 0.0131947
$$575$$ −22687.5 −0.00286165
$$576$$ 95717.3 0.0120208
$$577$$ −480544. −0.0600888 −0.0300444 0.999549i $$-0.509565\pi$$
−0.0300444 + 0.999549i $$0.509565\pi$$
$$578$$ −6.78300e6 −0.844505
$$579$$ −7.39882e6 −0.917204
$$580$$ 2.86602e7 3.53760
$$581$$ 203345. 0.0249915
$$582$$ −3.31737e6 −0.405963
$$583$$ −7.83477e6 −0.954674
$$584$$ 3.39255e7 4.11618
$$585$$ −2.96210e6 −0.357857
$$586$$ 1.29924e7 1.56296
$$587$$ 9.12805e6 1.09341 0.546705 0.837325i $$-0.315882\pi$$
0.546705 + 0.837325i $$0.315882\pi$$
$$588$$ −1.09059e7 −1.30082
$$589$$ 9.09485e6 1.08021
$$590$$ −4.11780e6 −0.487007
$$591$$ 8.74414e6 1.02979
$$592$$ 4.53029e6 0.531278
$$593$$ −2.66114e6 −0.310765 −0.155382 0.987854i $$-0.549661\pi$$
−0.155382 + 0.987854i $$0.549661\pi$$
$$594$$ −1.70015e6 −0.197707
$$595$$ 200760. 0.0232479
$$596$$ −1.32876e7 −1.53225
$$597$$ 7.25499e6 0.833107
$$598$$ −3.55569e6 −0.406604
$$599$$ 6.44915e6 0.734405 0.367202 0.930141i $$-0.380316\pi$$
0.367202 + 0.930141i $$0.380316\pi$$
$$600$$ 158042. 0.0179224
$$601$$ −1.48491e6 −0.167693 −0.0838463 0.996479i $$-0.526720\pi$$
−0.0838463 + 0.996479i $$0.526720\pi$$
$$602$$ 266241. 0.0299423
$$603$$ 4.32951e6 0.484893
$$604$$ −1.42795e6 −0.159265
$$605$$ 6.04111e6 0.671009
$$606$$ 3.77929e6 0.418051
$$607$$ −6.56656e6 −0.723379 −0.361690 0.932299i $$-0.617800\pi$$
−0.361690 + 0.932299i $$0.617800\pi$$
$$608$$ 5.87368e6 0.644394
$$609$$ −161343. −0.0176281
$$610$$ 1.50756e7 1.64040
$$611$$ 6.16420e6 0.667996
$$612$$ −8.43498e6 −0.910344
$$613$$ −9.33817e6 −1.00372 −0.501858 0.864950i $$-0.667350\pi$$
−0.501858 + 0.864950i $$0.667350\pi$$
$$614$$ 2.30793e7 2.47060
$$615$$ 2.03621e6 0.217088
$$616$$ −234382. −0.0248870
$$617$$ 7.56128e6 0.799617 0.399809 0.916599i $$-0.369077\pi$$
0.399809 + 0.916599i $$0.369077\pi$$
$$618$$ 1.58452e7 1.66888
$$619$$ 1.16180e7 1.21872 0.609359 0.792895i $$-0.291427\pi$$
0.609359 + 0.792895i $$0.291427\pi$$
$$620$$ 3.70786e7 3.87387
$$621$$ 385641. 0.0401286
$$622$$ −1.22718e7 −1.27184
$$623$$ 250751. 0.0258834
$$624$$ 1.10865e7 1.13981
$$625$$ −9.62976e6 −0.986088
$$626$$ 4.94201e6 0.504043
$$627$$ −2.02027e6 −0.205230
$$628$$ 1.39431e7 1.41078
$$629$$ −3.49763e6 −0.352491
$$630$$ 114929. 0.0115366
$$631$$ −8.43872e6 −0.843730 −0.421865 0.906659i $$-0.638624\pi$$
−0.421865 + 0.906659i $$0.638624\pi$$
$$632$$ 9.81042e6 0.977001
$$633$$ −7.39722e6 −0.733768
$$634$$ −8.12891e6 −0.803173
$$635$$ −8.80313e6 −0.866369
$$636$$ −2.22524e7 −2.18140
$$637$$ −1.10667e7 −1.08061
$$638$$ 1.66926e7 1.62358
$$639$$ 1.75557e6 0.170085
$$640$$ −9.95483e6 −0.960692
$$641$$ 1.32010e7 1.26900 0.634500 0.772923i $$-0.281206\pi$$
0.634500 + 0.772923i $$0.281206\pi$$
$$642$$ −1.63680e7 −1.56733
$$643$$ −1.57321e7 −1.50058 −0.750290 0.661109i $$-0.770086\pi$$
−0.750290 + 0.661109i $$0.770086\pi$$
$$644$$ 95562.5 0.00907973
$$645$$ 5.20501e6 0.492631
$$646$$ −1.44702e7 −1.36425
$$647$$ −6.22066e6 −0.584219 −0.292109 0.956385i $$-0.594357\pi$$
−0.292109 + 0.956385i $$0.594357\pi$$
$$648$$ −2.68640e6 −0.251323
$$649$$ −1.66128e6 −0.154822
$$650$$ 288270. 0.0267618
$$651$$ −208735. −0.0193038
$$652$$ 7.66595e6 0.706231
$$653$$ −1.57628e7 −1.44660 −0.723302 0.690532i $$-0.757376\pi$$
−0.723302 + 0.690532i $$0.757376\pi$$
$$654$$ 1.27060e7 1.16162
$$655$$ 2.12915e7 1.93911
$$656$$ −7.62110e6 −0.691446
$$657$$ −6.71136e6 −0.606593
$$658$$ −239171. −0.0215349
$$659$$ −5.77008e6 −0.517569 −0.258784 0.965935i $$-0.583322\pi$$
−0.258784 + 0.965935i $$0.583322\pi$$
$$660$$ −8.23640e6 −0.736000
$$661$$ −1.48192e7 −1.31924 −0.659618 0.751601i $$-0.729282\pi$$
−0.659618 + 0.751601i $$0.729282\pi$$
$$662$$ 2.68223e6 0.237877
$$663$$ −8.55938e6 −0.756238
$$664$$ −3.32423e7 −2.92598
$$665$$ 136569. 0.0119756
$$666$$ −2.00230e6 −0.174921
$$667$$ −3.78635e6 −0.329538
$$668$$ −5.60528e6 −0.486022
$$669$$ 4.60753e6 0.398018
$$670$$ 3.02801e7 2.60597
$$671$$ 6.08209e6 0.521491
$$672$$ −134806. −0.0115156
$$673$$ −1.74615e7 −1.48609 −0.743045 0.669242i $$-0.766619\pi$$
−0.743045 + 0.669242i $$0.766619\pi$$
$$674$$ −1.25637e7 −1.06529
$$675$$ −31265.0 −0.00264118
$$676$$ 4.51487e6 0.379995
$$677$$ −2.10975e7 −1.76913 −0.884564 0.466418i $$-0.845544\pi$$
−0.884564 + 0.466418i $$0.845544\pi$$
$$678$$ 1.24197e6 0.103762
$$679$$ 90472.2 0.00753080
$$680$$ −3.28197e7 −2.72184
$$681$$ −1.30567e7 −1.07886
$$682$$ 2.15958e7 1.77791
$$683$$ −1.43654e7 −1.17833 −0.589163 0.808014i $$-0.700542\pi$$
−0.589163 + 0.808014i $$0.700542\pi$$
$$684$$ −5.73799e6 −0.468943
$$685$$ 1.84155e6 0.149954
$$686$$ 858935. 0.0696867
$$687$$ 5.59052e6 0.451919
$$688$$ −1.94812e7 −1.56908
$$689$$ −2.25806e7 −1.81212
$$690$$ 2.69713e6 0.215664
$$691$$ 1.64154e7 1.30785 0.653923 0.756561i $$-0.273122\pi$$
0.653923 + 0.756561i $$0.273122\pi$$
$$692$$ 1.57612e7 1.25119
$$693$$ 46367.0 0.00366755
$$694$$ 3.77907e7 2.97842
$$695$$ 7.13324e6 0.560176
$$696$$ 2.63759e7 2.06388
$$697$$ 5.88391e6 0.458758
$$698$$ −1.66210e7 −1.29127
$$699$$ 4.41301e6 0.341619
$$700$$ −7747.52 −0.000597609 0
$$701$$ −1.07940e7 −0.829635 −0.414818 0.909905i $$-0.636155\pi$$
−0.414818 + 0.909905i $$0.636155\pi$$
$$702$$ −4.90000e6 −0.375278
$$703$$ −2.37930e6 −0.181577
$$704$$ 270077. 0.0205379
$$705$$ −4.67577e6 −0.354308
$$706$$ −2.61829e7 −1.97700
$$707$$ −103070. −0.00775502
$$708$$ −4.71839e6 −0.353762
$$709$$ 2.44390e7 1.82586 0.912930 0.408115i $$-0.133814\pi$$
0.912930 + 0.408115i $$0.133814\pi$$
$$710$$ 1.22783e7 0.914093
$$711$$ −1.94076e6 −0.143979
$$712$$ −4.09921e7 −3.03040
$$713$$ −4.89853e6 −0.360863
$$714$$ 332103. 0.0243797
$$715$$ −8.35787e6 −0.611407
$$716$$ −3.93440e7 −2.86811
$$717$$ 985527. 0.0715930
$$718$$ 3.53978e7 2.56251
$$719$$ −6.41280e6 −0.462621 −0.231310 0.972880i $$-0.574301\pi$$
−0.231310 + 0.972880i $$0.574301\pi$$
$$720$$ −8.40951e6 −0.604560
$$721$$ −432133. −0.0309585
$$722$$ 1.54231e7 1.10110
$$723$$ 1.22554e7 0.871928
$$724$$ 3.68474e7 2.61252
$$725$$ 306970. 0.0216896
$$726$$ 9.99341e6 0.703675
$$727$$ −1.58788e7 −1.11424 −0.557122 0.830431i $$-0.688095\pi$$
−0.557122 + 0.830431i $$0.688095\pi$$
$$728$$ −675512. −0.0472394
$$729$$ 531441. 0.0370370
$$730$$ −4.69385e7 −3.26003
$$731$$ 1.50406e7 1.04105
$$732$$ 1.72744e7 1.19159
$$733$$ 2.52517e7 1.73592 0.867962 0.496631i $$-0.165430\pi$$
0.867962 + 0.496631i $$0.165430\pi$$
$$734$$ −1.67487e7 −1.14747
$$735$$ 8.39450e6 0.573161
$$736$$ −3.16359e6 −0.215271
$$737$$ 1.22162e7 0.828450
$$738$$ 3.36837e6 0.227656
$$739$$ −1.56260e7 −1.05253 −0.526267 0.850319i $$-0.676409\pi$$
−0.526267 + 0.850319i $$0.676409\pi$$
$$740$$ −9.70015e6 −0.651177
$$741$$ −5.82261e6 −0.389558
$$742$$ 876126. 0.0584193
$$743$$ 1.42696e7 0.948284 0.474142 0.880448i $$-0.342758\pi$$
0.474142 + 0.880448i $$0.342758\pi$$
$$744$$ 3.41235e7 2.26006
$$745$$ 1.02278e7 0.675133
$$746$$ −1.61845e7 −1.06476
$$747$$ 6.57621e6 0.431195
$$748$$ −2.38002e7 −1.55534
$$749$$ 446393. 0.0290746
$$750$$ −1.61516e7 −1.04848
$$751$$ 2.22359e7 1.43865 0.719323 0.694676i $$-0.244452\pi$$
0.719323 + 0.694676i $$0.244452\pi$$
$$752$$ 1.75004e7 1.12851
$$753$$ 72380.4 0.00465193
$$754$$ 4.81098e7 3.08181
$$755$$ 1.09912e6 0.0701745
$$756$$ 131692. 0.00838021
$$757$$ −1.09113e7 −0.692048 −0.346024 0.938226i $$-0.612468\pi$$
−0.346024 + 0.938226i $$0.612468\pi$$
$$758$$ 9.85756e6 0.623155
$$759$$ 1.08813e6 0.0685607
$$760$$ −2.23260e7 −1.40209
$$761$$ 2.42444e6 0.151758 0.0758788 0.997117i $$-0.475824\pi$$
0.0758788 + 0.997117i $$0.475824\pi$$
$$762$$ −1.45624e7 −0.908546
$$763$$ −346520. −0.0215485
$$764$$ −4.16058e6 −0.257882
$$765$$ 6.49260e6 0.401112
$$766$$ 3.16659e7 1.94993
$$767$$ −4.78798e6 −0.293876
$$768$$ −1.68080e7 −1.02828
$$769$$ 2.00363e6 0.122181 0.0610903 0.998132i $$-0.480542\pi$$
0.0610903 + 0.998132i $$0.480542\pi$$
$$770$$ 324285. 0.0197106
$$771$$ 1.05057e7 0.636487
$$772$$ −5.92938e7 −3.58069
$$773$$ 3.91181e6 0.235466 0.117733 0.993045i $$-0.462437\pi$$
0.117733 + 0.993045i $$0.462437\pi$$
$$774$$ 8.61030e6 0.516614
$$775$$ 397137. 0.0237512
$$776$$ −1.47902e7 −0.881697
$$777$$ 54607.1 0.00324487
$$778$$ −6.37322e6 −0.377494
$$779$$ 4.00260e6 0.236319
$$780$$ −2.37381e7 −1.39704
$$781$$ 4.95353e6 0.290594
$$782$$ 7.79371e6 0.455751
$$783$$ −5.21786e6 −0.304150
$$784$$ −3.14188e7 −1.82557
$$785$$ −1.07323e7 −0.621614
$$786$$ 3.52211e7 2.03351
$$787$$ −2.37516e7 −1.36696 −0.683479 0.729970i $$-0.739534\pi$$
−0.683479 + 0.729970i $$0.739534\pi$$
$$788$$ 7.00752e7 4.02021
$$789$$ −5.02246e6 −0.287226
$$790$$ −1.35734e7 −0.773789
$$791$$ −33871.3 −0.00192482
$$792$$ −7.57996e6 −0.429392
$$793$$ 1.75292e7 0.989872
$$794$$ −2.86548e7 −1.61304
$$795$$ 1.71282e7 0.961157
$$796$$ 5.81412e7 3.25238
$$797$$ −2.50486e7 −1.39681 −0.698405 0.715703i $$-0.746106\pi$$
−0.698405 + 0.715703i $$0.746106\pi$$
$$798$$ 225917. 0.0125586
$$799$$ −1.35113e7 −0.748737
$$800$$ 256481. 0.0141687
$$801$$ 8.10932e6 0.446584
$$802$$ −3.89072e7 −2.13597
$$803$$ −1.89368e7 −1.03638
$$804$$ 3.46965e7 1.89298
$$805$$ −73556.7 −0.00400067
$$806$$ 6.22413e7 3.37475
$$807$$ −1.09752e7 −0.593237
$$808$$ 1.68496e7 0.907949
$$809$$ −2.26709e7 −1.21786 −0.608931 0.793223i $$-0.708402\pi$$
−0.608931 + 0.793223i $$0.708402\pi$$
$$810$$ 3.71683e6 0.199049
$$811$$ −1.66885e7 −0.890974 −0.445487 0.895288i $$-0.646969\pi$$
−0.445487 + 0.895288i $$0.646969\pi$$
$$812$$ −1.29300e6 −0.0688188
$$813$$ 1.26523e7 0.671342
$$814$$ −5.64969e6 −0.298857
$$815$$ −5.90066e6 −0.311176
$$816$$ −2.43004e7 −1.27758
$$817$$ 1.02315e7 0.536272
$$818$$ 3.94156e7 2.05961
$$819$$ 133634. 0.00696158
$$820$$ 1.63181e7 0.847492
$$821$$ −2.41094e7 −1.24833 −0.624164 0.781293i $$-0.714560\pi$$
−0.624164 + 0.781293i $$0.714560\pi$$
$$822$$ 3.04636e6 0.157254
$$823$$ −9.03070e6 −0.464753 −0.232376 0.972626i $$-0.574650\pi$$
−0.232376 + 0.972626i $$0.574650\pi$$
$$824$$ 7.06441e7 3.62458
$$825$$ −88217.4 −0.00451252
$$826$$ 185773. 0.00947399
$$827$$ 1.22307e7 0.621852 0.310926 0.950434i $$-0.399361\pi$$
0.310926 + 0.950434i $$0.399361\pi$$
$$828$$ 3.09051e6 0.156659
$$829$$ 2.01683e7 1.01925 0.509627 0.860395i $$-0.329783\pi$$
0.509627 + 0.860395i $$0.329783\pi$$
$$830$$ 4.59932e7 2.31739
$$831$$ 1.69019e7 0.849051
$$832$$ 778387. 0.0389841
$$833$$ 2.42570e7 1.21123
$$834$$ 1.18000e7 0.587447
$$835$$ 4.31452e6 0.214149
$$836$$ −1.61904e7 −0.801199
$$837$$ −6.75053e6 −0.333061
$$838$$ 4.08700e7 2.01045
$$839$$ 1.80995e7 0.887690 0.443845 0.896103i $$-0.353614\pi$$
0.443845 + 0.896103i $$0.353614\pi$$
$$840$$ 512401. 0.0250560
$$841$$ 3.07195e7 1.49770
$$842$$ −4.99697e7 −2.42899
$$843$$ −1.90583e7 −0.923668
$$844$$ −5.92810e7 −2.86457
$$845$$ −3.47520e6 −0.167432
$$846$$ −7.73483e6 −0.371556
$$847$$ −272543. −0.0130535
$$848$$ −6.41072e7 −3.06138
$$849$$ 1.04438e7 0.497265
$$850$$ −631857. −0.0299966
$$851$$ 1.28150e6 0.0606591
$$852$$ 1.40691e7 0.663998
$$853$$ −1.82781e7 −0.860121 −0.430060 0.902800i $$-0.641508\pi$$
−0.430060 + 0.902800i $$0.641508\pi$$
$$854$$ −680132. −0.0319116
$$855$$ 4.41667e6 0.206623
$$856$$ −7.29753e7 −3.40402
$$857$$ −1.47267e7 −0.684940 −0.342470 0.939529i $$-0.611264\pi$$
−0.342470 + 0.939529i $$0.611264\pi$$
$$858$$ −1.38259e7 −0.641172
$$859$$ −1.82990e7 −0.846142 −0.423071 0.906096i $$-0.639048\pi$$
−0.423071 + 0.906096i $$0.639048\pi$$
$$860$$ 4.17127e7 1.92319
$$861$$ −91863.1 −0.00422312
$$862$$ −1.85588e7 −0.850712
$$863$$ −3.92569e6 −0.179427 −0.0897137 0.995968i $$-0.528595\pi$$
−0.0897137 + 0.995968i $$0.528595\pi$$
$$864$$ −4.35966e6 −0.198686
$$865$$ −1.21317e7 −0.551293
$$866$$ −5.56814e6 −0.252299
$$867$$ 5.98254e6 0.270295
$$868$$ −1.67279e6 −0.0753603
$$869$$ −5.47606e6 −0.245991
$$870$$ −3.64931e7 −1.63460
$$871$$ 3.52082e7 1.57253
$$872$$ 5.66483e7 2.52287
$$873$$ 2.92589e6 0.129934
$$874$$ 5.30176e6 0.234769
$$875$$ 440490. 0.0194499
$$876$$ −5.37846e7 −2.36809
$$877$$ −3.49463e7 −1.53427 −0.767135 0.641486i $$-0.778318\pi$$
−0.767135 + 0.641486i $$0.778318\pi$$
$$878$$ −2.86854e7 −1.25581
$$879$$ −1.14592e7 −0.500245
$$880$$ −2.37283e7 −1.03290
$$881$$ −3.45541e7 −1.49989 −0.749945 0.661501i $$-0.769920\pi$$
−0.749945 + 0.661501i $$0.769920\pi$$
$$882$$ 1.38865e7 0.601064
$$883$$ 2.29538e6 0.0990722 0.0495361 0.998772i $$-0.484226\pi$$
0.0495361 + 0.998772i $$0.484226\pi$$
$$884$$ −6.85945e7 −2.95229
$$885$$ 3.63186e6 0.155873
$$886$$ 2.87175e7 1.22903
$$887$$ 2.48902e7 1.06223 0.531116 0.847299i $$-0.321773\pi$$
0.531116 + 0.847299i $$0.321773\pi$$
$$888$$ −8.92704e6 −0.379905
$$889$$ 397151. 0.0168539
$$890$$ 5.67156e7 2.40009
$$891$$ 1.49952e6 0.0632786
$$892$$ 3.69246e7 1.55383
$$893$$ −9.19120e6 −0.385695
$$894$$ 1.69191e7 0.708000
$$895$$ 3.02840e7 1.26373
$$896$$ 449109. 0.0186888
$$897$$ 3.13609e6 0.130139
$$898$$ −4.08826e7 −1.69179
$$899$$ 6.62789e7 2.73512
$$900$$ −250556. −0.0103110
$$901$$ 4.94943e7 2.03115
$$902$$ 9.50422e6 0.388956
$$903$$ −234822. −0.00958341
$$904$$ 5.53719e6 0.225356
$$905$$ −2.83623e7 −1.15112
$$906$$ 1.81821e6 0.0735908
$$907$$ −1.12581e7 −0.454411 −0.227205 0.973847i $$-0.572959\pi$$
−0.227205 + 0.973847i $$0.572959\pi$$
$$908$$ −1.04636e8 −4.21178
$$909$$ −3.33330e6 −0.133803
$$910$$ 934621. 0.0374138
$$911$$ −2.07869e7 −0.829837 −0.414919 0.909859i $$-0.636190\pi$$
−0.414919 + 0.909859i $$0.636190\pi$$
$$912$$ −1.65306e7 −0.658116
$$913$$ 1.85555e7 0.736707
$$914$$ 5.54905e7 2.19712
$$915$$ −1.32965e7 −0.525032
$$916$$ 4.48022e7 1.76425
$$917$$ −960558. −0.0377225
$$918$$ 1.07403e7 0.420639
$$919$$ −1.17836e7 −0.460244 −0.230122 0.973162i $$-0.573913\pi$$
−0.230122 + 0.973162i $$0.573913\pi$$
$$920$$ 1.20249e7 0.468394
$$921$$ −2.03557e7 −0.790747
$$922$$ −2.56091e7 −0.992126
$$923$$ 1.42766e7 0.551594
$$924$$ 371583. 0.0143178
$$925$$ −103895. −0.00399246
$$926$$ −1.10881e6 −0.0424942
$$927$$ −1.39753e7 −0.534147
$$928$$ 4.28045e7 1.63162
$$929$$ −2.82219e7 −1.07287 −0.536435 0.843942i $$-0.680229\pi$$
−0.536435 + 0.843942i $$0.680229\pi$$
$$930$$ −4.72123e7 −1.78998
$$931$$ 1.65011e7 0.623935
$$932$$ 3.53657e7 1.33365
$$933$$ 1.08236e7 0.407070
$$934$$ −3.15437e7 −1.18316
$$935$$ 1.83196e7 0.685308
$$936$$ −2.18462e7 −0.815053
$$937$$ 1.34325e7 0.499814 0.249907 0.968270i $$-0.419600\pi$$
0.249907 + 0.968270i $$0.419600\pi$$
$$938$$ −1.36608e6 −0.0506953
$$939$$ −4.35880e6 −0.161325
$$940$$ −3.74715e7 −1.38319
$$941$$ 4.06264e7 1.49566 0.747832 0.663888i $$-0.231095\pi$$
0.747832 + 0.663888i $$0.231095\pi$$
$$942$$ −1.77538e7 −0.651875
$$943$$ −2.15582e6 −0.0789465
$$944$$ −1.35933e7 −0.496470
$$945$$ −101366. −0.00369245
$$946$$ 2.42949e7 0.882646
$$947$$ −3.34341e7 −1.21148 −0.605738 0.795664i $$-0.707122\pi$$
−0.605738 + 0.795664i $$0.707122\pi$$
$$948$$ −1.55532e7 −0.562080
$$949$$ −5.45778e7 −1.96721
$$950$$ −429828. −0.0154520
$$951$$ 7.16962e6 0.257066
$$952$$ 1.48065e6 0.0529493
$$953$$ −8.95158e6 −0.319277 −0.159638 0.987176i $$-0.551033\pi$$
−0.159638 + 0.987176i $$0.551033\pi$$
$$954$$ 2.83341e7 1.00795
$$955$$ 3.20250e6 0.113627
$$956$$ 7.89797e6 0.279493
$$957$$ −1.47227e7 −0.519648
$$958$$ 7.90563e7 2.78306
$$959$$ −83081.1 −0.00291713
$$960$$ −590435. −0.0206773
$$961$$ 5.71181e7 1.99510
$$962$$ −1.62830e7 −0.567277
$$963$$ 1.44365e7 0.501643
$$964$$ 9.82140e7 3.40393
$$965$$ 4.56399e7 1.57771
$$966$$ −121680. −0.00419543
$$967$$ −2.94663e7 −1.01335 −0.506675 0.862137i $$-0.669126\pi$$
−0.506675 + 0.862137i $$0.669126\pi$$
$$968$$ 4.45546e7 1.52829
$$969$$ 1.27626e7 0.436645
$$970$$ 2.04633e7 0.698307
$$971$$ −3.03617e7 −1.03342 −0.516712 0.856159i $$-0.672844\pi$$
−0.516712 + 0.856159i $$0.672844\pi$$
$$972$$ 4.25895e6 0.144589
$$973$$ −321814. −0.0108974
$$974$$ −6.74425e7 −2.27791
$$975$$ −254251. −0.00856548
$$976$$ 4.97661e7 1.67228
$$977$$ 3.77587e6 0.126555 0.0632777 0.997996i $$-0.479845\pi$$
0.0632777 + 0.997996i $$0.479845\pi$$
$$978$$ −9.76107e6 −0.326325
$$979$$ 2.28813e7 0.762999
$$980$$ 6.72732e7 2.23757
$$981$$ −1.12065e7 −0.371791
$$982$$ −1.71893e7 −0.568826
$$983$$ 1.74788e7 0.576938 0.288469 0.957489i $$-0.406854\pi$$
0.288469 + 0.957489i $$0.406854\pi$$
$$984$$ 1.50176e7 0.494438
$$985$$ −5.39385e7 −1.77137
$$986$$ −1.05452e8 −3.45431
$$987$$ 210946. 0.00689253
$$988$$ −4.66622e7 −1.52080
$$989$$ −5.51074e6 −0.179151
$$990$$ 1.04874e7 0.340080
$$991$$ 4.83380e7 1.56352 0.781762 0.623576i $$-0.214321\pi$$
0.781762 + 0.623576i $$0.214321\pi$$
$$992$$ 5.53777e7 1.78672
$$993$$ −2.36570e6 −0.0761355
$$994$$ −553930. −0.0177823
$$995$$ −4.47526e7 −1.43305
$$996$$ 5.27014e7 1.68335
$$997$$ −1.74335e7 −0.555452 −0.277726 0.960660i $$-0.589581\pi$$
−0.277726 + 0.960660i $$0.589581\pi$$
$$998$$ −6.41847e7 −2.03988
$$999$$ 1.76601e6 0.0559858
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 69.6.a.b.1.1 3
3.2 odd 2 207.6.a.c.1.3 3
4.3 odd 2 1104.6.a.i.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.1 3 1.1 even 1 trivial
207.6.a.c.1.3 3 3.2 odd 2
1104.6.a.i.1.2 3 4.3 odd 2