Properties

Label 177.6.a.a.1.3
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + 13849341 x^{3} - 23890558 x^{2} - 74443300 x - 14846072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-5.70379\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.70379 q^{2} +9.00000 q^{3} +12.9408 q^{4} +105.016 q^{5} -60.3341 q^{6} -129.262 q^{7} +127.769 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-6.70379 q^{2} +9.00000 q^{3} +12.9408 q^{4} +105.016 q^{5} -60.3341 q^{6} -129.262 q^{7} +127.769 q^{8} +81.0000 q^{9} -704.003 q^{10} -188.535 q^{11} +116.467 q^{12} -958.805 q^{13} +866.546 q^{14} +945.141 q^{15} -1270.64 q^{16} +82.6984 q^{17} -543.007 q^{18} -2066.89 q^{19} +1358.99 q^{20} -1163.36 q^{21} +1263.90 q^{22} -589.875 q^{23} +1149.92 q^{24} +7903.29 q^{25} +6427.63 q^{26} +729.000 q^{27} -1672.76 q^{28} -1121.65 q^{29} -6336.03 q^{30} +940.683 q^{31} +4429.51 q^{32} -1696.82 q^{33} -554.393 q^{34} -13574.5 q^{35} +1048.21 q^{36} +575.784 q^{37} +13856.0 q^{38} -8629.25 q^{39} +13417.7 q^{40} -5521.52 q^{41} +7798.92 q^{42} -16502.9 q^{43} -2439.80 q^{44} +8506.27 q^{45} +3954.40 q^{46} +23876.5 q^{47} -11435.8 q^{48} -98.3079 q^{49} -52982.0 q^{50} +744.286 q^{51} -12407.7 q^{52} -39649.3 q^{53} -4887.06 q^{54} -19799.2 q^{55} -16515.7 q^{56} -18602.0 q^{57} +7519.30 q^{58} +3481.00 q^{59} +12230.9 q^{60} +1264.98 q^{61} -6306.14 q^{62} -10470.2 q^{63} +10966.0 q^{64} -100690. q^{65} +11375.1 q^{66} -49177.7 q^{67} +1070.19 q^{68} -5308.87 q^{69} +91000.9 q^{70} -53621.6 q^{71} +10349.3 q^{72} +30870.6 q^{73} -3859.93 q^{74} +71129.6 q^{75} -26747.3 q^{76} +24370.5 q^{77} +57848.7 q^{78} -26724.2 q^{79} -133437. q^{80} +6561.00 q^{81} +37015.1 q^{82} +27418.4 q^{83} -15054.8 q^{84} +8684.63 q^{85} +110632. q^{86} -10094.8 q^{87} -24088.9 q^{88} -30163.0 q^{89} -57024.2 q^{90} +123937. q^{91} -7633.46 q^{92} +8466.14 q^{93} -160063. q^{94} -217056. q^{95} +39865.6 q^{96} -80237.8 q^{97} +659.035 q^{98} -15271.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} + O(q^{10}) \) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} - 399q^{10} - 698q^{11} + 1350q^{12} - 1556q^{13} - 1679q^{14} - 1728q^{15} - 2662q^{16} - 4793q^{17} - 486q^{18} - 3753q^{19} - 11023q^{20} - 3339q^{21} - 9534q^{22} - 7323q^{23} - 5589q^{24} + 7867q^{25} - 4844q^{26} + 8019q^{27} + 3650q^{28} - 15467q^{29} - 3591q^{30} - 5151q^{31} - 15368q^{32} - 6282q^{33} + 8452q^{34} - 23285q^{35} + 12150q^{36} + 8623q^{37} + 15205q^{38} - 14004q^{39} + 41530q^{40} - 6369q^{41} - 15111q^{42} - 20506q^{43} - 55632q^{44} - 15552q^{45} - 45191q^{46} - 47899q^{47} - 23958q^{48} - 10322q^{49} - 102147q^{50} - 43137q^{51} - 292q^{52} - 80048q^{53} - 4374q^{54} - 2114q^{55} - 108126q^{56} - 33777q^{57} - 58294q^{58} + 38291q^{59} - 99207q^{60} - 82527q^{61} - 67438q^{62} - 30051q^{63} - 51411q^{64} - 167646q^{65} - 85806q^{66} - 166976q^{67} - 136533q^{68} - 65907q^{69} + 76140q^{70} - 183560q^{71} - 50301q^{72} - 36809q^{73} - 116686q^{74} + 70803q^{75} + 55580q^{76} - 164885q^{77} - 43596q^{78} - 281518q^{79} - 32683q^{80} + 72171q^{81} + 178815q^{82} - 254691q^{83} + 32850q^{84} + 4763q^{85} + 349324q^{86} - 139203q^{87} + 251285q^{88} - 89687q^{89} - 32319q^{90} + 34897q^{91} - 20240q^{92} - 46359q^{93} + 96548q^{94} - 155113q^{95} - 138312q^{96} - 45828q^{97} + 465864q^{98} - 56538q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.70379 −1.18507 −0.592537 0.805543i \(-0.701874\pi\)
−0.592537 + 0.805543i \(0.701874\pi\)
\(3\) 9.00000 0.577350
\(4\) 12.9408 0.404401
\(5\) 105.016 1.87858 0.939289 0.343128i \(-0.111487\pi\)
0.939289 + 0.343128i \(0.111487\pi\)
\(6\) −60.3341 −0.684203
\(7\) −129.262 −0.997071 −0.498536 0.866869i \(-0.666129\pi\)
−0.498536 + 0.866869i \(0.666129\pi\)
\(8\) 127.769 0.705829
\(9\) 81.0000 0.333333
\(10\) −704.003 −2.22625
\(11\) −188.535 −0.469798 −0.234899 0.972020i \(-0.575476\pi\)
−0.234899 + 0.972020i \(0.575476\pi\)
\(12\) 116.467 0.233481
\(13\) −958.805 −1.57352 −0.786760 0.617260i \(-0.788243\pi\)
−0.786760 + 0.617260i \(0.788243\pi\)
\(14\) 866.546 1.18160
\(15\) 945.141 1.08460
\(16\) −1270.64 −1.24086
\(17\) 82.6984 0.0694025 0.0347012 0.999398i \(-0.488952\pi\)
0.0347012 + 0.999398i \(0.488952\pi\)
\(18\) −543.007 −0.395025
\(19\) −2066.89 −1.31351 −0.656755 0.754104i \(-0.728072\pi\)
−0.656755 + 0.754104i \(0.728072\pi\)
\(20\) 1358.99 0.759698
\(21\) −1163.36 −0.575659
\(22\) 1263.90 0.556745
\(23\) −589.875 −0.232509 −0.116255 0.993219i \(-0.537089\pi\)
−0.116255 + 0.993219i \(0.537089\pi\)
\(24\) 1149.92 0.407511
\(25\) 7903.29 2.52905
\(26\) 6427.63 1.86474
\(27\) 729.000 0.192450
\(28\) −1672.76 −0.403216
\(29\) −1121.65 −0.247664 −0.123832 0.992303i \(-0.539518\pi\)
−0.123832 + 0.992303i \(0.539518\pi\)
\(30\) −6336.03 −1.28533
\(31\) 940.683 0.175808 0.0879041 0.996129i \(-0.471983\pi\)
0.0879041 + 0.996129i \(0.471983\pi\)
\(32\) 4429.51 0.764682
\(33\) −1696.82 −0.271238
\(34\) −554.393 −0.0822471
\(35\) −13574.5 −1.87307
\(36\) 1048.21 0.134800
\(37\) 575.784 0.0691441 0.0345720 0.999402i \(-0.488993\pi\)
0.0345720 + 0.999402i \(0.488993\pi\)
\(38\) 13856.0 1.55661
\(39\) −8629.25 −0.908472
\(40\) 13417.7 1.32596
\(41\) −5521.52 −0.512979 −0.256489 0.966547i \(-0.582566\pi\)
−0.256489 + 0.966547i \(0.582566\pi\)
\(42\) 7798.92 0.682199
\(43\) −16502.9 −1.36110 −0.680551 0.732701i \(-0.738259\pi\)
−0.680551 + 0.732701i \(0.738259\pi\)
\(44\) −2439.80 −0.189987
\(45\) 8506.27 0.626192
\(46\) 3954.40 0.275541
\(47\) 23876.5 1.57662 0.788309 0.615279i \(-0.210957\pi\)
0.788309 + 0.615279i \(0.210957\pi\)
\(48\) −11435.8 −0.716411
\(49\) −98.3079 −0.00584922
\(50\) −52982.0 −2.99711
\(51\) 744.286 0.0400695
\(52\) −12407.7 −0.636332
\(53\) −39649.3 −1.93886 −0.969428 0.245376i \(-0.921089\pi\)
−0.969428 + 0.245376i \(0.921089\pi\)
\(54\) −4887.06 −0.228068
\(55\) −19799.2 −0.882551
\(56\) −16515.7 −0.703762
\(57\) −18602.0 −0.758356
\(58\) 7519.30 0.293500
\(59\) 3481.00 0.130189
\(60\) 12230.9 0.438612
\(61\) 1264.98 0.0435270 0.0217635 0.999763i \(-0.493072\pi\)
0.0217635 + 0.999763i \(0.493072\pi\)
\(62\) −6306.14 −0.208346
\(63\) −10470.2 −0.332357
\(64\) 10966.0 0.334655
\(65\) −100690. −2.95598
\(66\) 11375.1 0.321437
\(67\) −49177.7 −1.33838 −0.669192 0.743089i \(-0.733360\pi\)
−0.669192 + 0.743089i \(0.733360\pi\)
\(68\) 1070.19 0.0280664
\(69\) −5308.87 −0.134239
\(70\) 91000.9 2.21973
\(71\) −53621.6 −1.26239 −0.631195 0.775624i \(-0.717435\pi\)
−0.631195 + 0.775624i \(0.717435\pi\)
\(72\) 10349.3 0.235276
\(73\) 30870.6 0.678012 0.339006 0.940784i \(-0.389909\pi\)
0.339006 + 0.940784i \(0.389909\pi\)
\(74\) −3859.93 −0.0819409
\(75\) 71129.6 1.46015
\(76\) −26747.3 −0.531184
\(77\) 24370.5 0.468422
\(78\) 57848.7 1.07661
\(79\) −26724.2 −0.481768 −0.240884 0.970554i \(-0.577437\pi\)
−0.240884 + 0.970554i \(0.577437\pi\)
\(80\) −133437. −2.33105
\(81\) 6561.00 0.111111
\(82\) 37015.1 0.607918
\(83\) 27418.4 0.436864 0.218432 0.975852i \(-0.429906\pi\)
0.218432 + 0.975852i \(0.429906\pi\)
\(84\) −15054.8 −0.232797
\(85\) 8684.63 0.130378
\(86\) 110632. 1.61301
\(87\) −10094.8 −0.142989
\(88\) −24088.9 −0.331597
\(89\) −30163.0 −0.403645 −0.201822 0.979422i \(-0.564686\pi\)
−0.201822 + 0.979422i \(0.564686\pi\)
\(90\) −57024.2 −0.742084
\(91\) 123937. 1.56891
\(92\) −7633.46 −0.0940269
\(93\) 8466.14 0.101503
\(94\) −160063. −1.86841
\(95\) −217056. −2.46753
\(96\) 39865.6 0.441490
\(97\) −80237.8 −0.865864 −0.432932 0.901427i \(-0.642521\pi\)
−0.432932 + 0.901427i \(0.642521\pi\)
\(98\) 659.035 0.00693176
\(99\) −15271.4 −0.156599
\(100\) 102275. 1.02275
\(101\) 94103.2 0.917912 0.458956 0.888459i \(-0.348224\pi\)
0.458956 + 0.888459i \(0.348224\pi\)
\(102\) −4989.54 −0.0474854
\(103\) −35090.7 −0.325911 −0.162955 0.986633i \(-0.552103\pi\)
−0.162955 + 0.986633i \(0.552103\pi\)
\(104\) −122505. −1.11064
\(105\) −122171. −1.08142
\(106\) 265800. 2.29769
\(107\) −160286. −1.35343 −0.676715 0.736245i \(-0.736597\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(108\) 9433.86 0.0778269
\(109\) 240043. 1.93518 0.967592 0.252518i \(-0.0812588\pi\)
0.967592 + 0.252518i \(0.0812588\pi\)
\(110\) 132729. 1.04589
\(111\) 5182.05 0.0399204
\(112\) 164246. 1.23723
\(113\) 185172. 1.36421 0.682103 0.731257i \(-0.261066\pi\)
0.682103 + 0.731257i \(0.261066\pi\)
\(114\) 124704. 0.898708
\(115\) −61946.1 −0.436787
\(116\) −14515.1 −0.100155
\(117\) −77663.2 −0.524506
\(118\) −23335.9 −0.154284
\(119\) −10689.8 −0.0691992
\(120\) 120759. 0.765541
\(121\) −125505. −0.779290
\(122\) −8480.15 −0.0515827
\(123\) −49693.7 −0.296168
\(124\) 12173.2 0.0710969
\(125\) 501795. 2.87244
\(126\) 70190.2 0.393868
\(127\) −102539. −0.564132 −0.282066 0.959395i \(-0.591020\pi\)
−0.282066 + 0.959395i \(0.591020\pi\)
\(128\) −215258. −1.16127
\(129\) −148527. −0.785832
\(130\) 675002. 3.50305
\(131\) 166552. 0.847952 0.423976 0.905673i \(-0.360634\pi\)
0.423976 + 0.905673i \(0.360634\pi\)
\(132\) −21958.2 −0.109689
\(133\) 267171. 1.30966
\(134\) 329677. 1.58609
\(135\) 76556.4 0.361532
\(136\) 10566.3 0.0489863
\(137\) 373498. 1.70015 0.850073 0.526664i \(-0.176558\pi\)
0.850073 + 0.526664i \(0.176558\pi\)
\(138\) 35589.6 0.159084
\(139\) −2560.95 −0.0112425 −0.00562126 0.999984i \(-0.501789\pi\)
−0.00562126 + 0.999984i \(0.501789\pi\)
\(140\) −175666. −0.757473
\(141\) 214889. 0.910261
\(142\) 359468. 1.49603
\(143\) 180769. 0.739236
\(144\) −102922. −0.413620
\(145\) −117791. −0.465255
\(146\) −206950. −0.803495
\(147\) −884.771 −0.00337705
\(148\) 7451.11 0.0279619
\(149\) −529574. −1.95416 −0.977082 0.212863i \(-0.931721\pi\)
−0.977082 + 0.212863i \(0.931721\pi\)
\(150\) −476838. −1.73038
\(151\) −477364. −1.70376 −0.851878 0.523740i \(-0.824536\pi\)
−0.851878 + 0.523740i \(0.824536\pi\)
\(152\) −264084. −0.927115
\(153\) 6698.57 0.0231342
\(154\) −163375. −0.555115
\(155\) 98786.4 0.330269
\(156\) −111670. −0.367386
\(157\) −210317. −0.680965 −0.340483 0.940251i \(-0.610590\pi\)
−0.340483 + 0.940251i \(0.610590\pi\)
\(158\) 179154. 0.570930
\(159\) −356843. −1.11940
\(160\) 465168. 1.43651
\(161\) 76248.5 0.231828
\(162\) −43983.6 −0.131675
\(163\) 82201.3 0.242331 0.121166 0.992632i \(-0.461337\pi\)
0.121166 + 0.992632i \(0.461337\pi\)
\(164\) −71453.0 −0.207449
\(165\) −178192. −0.509541
\(166\) −183807. −0.517716
\(167\) −713691. −1.98024 −0.990122 0.140206i \(-0.955223\pi\)
−0.990122 + 0.140206i \(0.955223\pi\)
\(168\) −148641. −0.406317
\(169\) 548014. 1.47596
\(170\) −58219.9 −0.154507
\(171\) −167418. −0.437837
\(172\) −213562. −0.550430
\(173\) 465524. 1.18257 0.591286 0.806462i \(-0.298621\pi\)
0.591286 + 0.806462i \(0.298621\pi\)
\(174\) 67673.7 0.169452
\(175\) −1.02160e6 −2.52164
\(176\) 239561. 0.582954
\(177\) 31329.0 0.0751646
\(178\) 202206. 0.478349
\(179\) −40052.8 −0.0934331 −0.0467165 0.998908i \(-0.514876\pi\)
−0.0467165 + 0.998908i \(0.514876\pi\)
\(180\) 110078. 0.253233
\(181\) 346928. 0.787123 0.393562 0.919298i \(-0.371243\pi\)
0.393562 + 0.919298i \(0.371243\pi\)
\(182\) −830849. −1.85927
\(183\) 11384.8 0.0251303
\(184\) −75367.6 −0.164112
\(185\) 60466.3 0.129893
\(186\) −56755.3 −0.120288
\(187\) −15591.6 −0.0326051
\(188\) 308982. 0.637585
\(189\) −94232.1 −0.191886
\(190\) 1.45510e6 2.92421
\(191\) 176540. 0.350155 0.175078 0.984555i \(-0.443982\pi\)
0.175078 + 0.984555i \(0.443982\pi\)
\(192\) 98693.9 0.193213
\(193\) −438623. −0.847614 −0.423807 0.905753i \(-0.639307\pi\)
−0.423807 + 0.905753i \(0.639307\pi\)
\(194\) 537897. 1.02611
\(195\) −906206. −1.70663
\(196\) −1272.18 −0.00236543
\(197\) −670092. −1.23018 −0.615091 0.788456i \(-0.710881\pi\)
−0.615091 + 0.788456i \(0.710881\pi\)
\(198\) 102376. 0.185582
\(199\) −236725. −0.423752 −0.211876 0.977297i \(-0.567957\pi\)
−0.211876 + 0.977297i \(0.567957\pi\)
\(200\) 1.00979e6 1.78508
\(201\) −442599. −0.772717
\(202\) −630848. −1.08779
\(203\) 144987. 0.246938
\(204\) 9631.67 0.0162041
\(205\) −579847. −0.963670
\(206\) 235241. 0.386228
\(207\) −47779.9 −0.0775031
\(208\) 1.21830e6 1.95252
\(209\) 389682. 0.617084
\(210\) 819008. 1.28156
\(211\) −739722. −1.14383 −0.571916 0.820312i \(-0.693800\pi\)
−0.571916 + 0.820312i \(0.693800\pi\)
\(212\) −513094. −0.784075
\(213\) −482594. −0.728841
\(214\) 1.07452e6 1.60391
\(215\) −1.73307e6 −2.55693
\(216\) 93143.4 0.135837
\(217\) −121595. −0.175293
\(218\) −1.60920e6 −2.29334
\(219\) 277835. 0.391451
\(220\) −256217. −0.356904
\(221\) −79291.7 −0.109206
\(222\) −34739.4 −0.0473086
\(223\) 126063. 0.169757 0.0848783 0.996391i \(-0.472950\pi\)
0.0848783 + 0.996391i \(0.472950\pi\)
\(224\) −572568. −0.762443
\(225\) 640166. 0.843017
\(226\) −1.24136e6 −1.61668
\(227\) 411990. 0.530667 0.265333 0.964157i \(-0.414518\pi\)
0.265333 + 0.964157i \(0.414518\pi\)
\(228\) −240725. −0.306679
\(229\) 750528. 0.945754 0.472877 0.881128i \(-0.343215\pi\)
0.472877 + 0.881128i \(0.343215\pi\)
\(230\) 415274. 0.517624
\(231\) 219334. 0.270443
\(232\) −143312. −0.174808
\(233\) −149621. −0.180553 −0.0902764 0.995917i \(-0.528775\pi\)
−0.0902764 + 0.995917i \(0.528775\pi\)
\(234\) 520638. 0.621579
\(235\) 2.50741e6 2.96180
\(236\) 45047.0 0.0526485
\(237\) −240518. −0.278149
\(238\) 71662.0 0.0820062
\(239\) 171621. 0.194346 0.0971730 0.995268i \(-0.469020\pi\)
0.0971730 + 0.995268i \(0.469020\pi\)
\(240\) −1.20094e6 −1.34583
\(241\) −267062. −0.296189 −0.148094 0.988973i \(-0.547314\pi\)
−0.148094 + 0.988973i \(0.547314\pi\)
\(242\) 841362. 0.923516
\(243\) 59049.0 0.0641500
\(244\) 16369.8 0.0176023
\(245\) −10323.9 −0.0109882
\(246\) 333136. 0.350982
\(247\) 1.98175e6 2.06683
\(248\) 120190. 0.124091
\(249\) 246765. 0.252224
\(250\) −3.36393e6 −3.40406
\(251\) 1.10039e6 1.10246 0.551231 0.834352i \(-0.314158\pi\)
0.551231 + 0.834352i \(0.314158\pi\)
\(252\) −135493. −0.134405
\(253\) 111212. 0.109232
\(254\) 687402. 0.668538
\(255\) 78161.7 0.0752737
\(256\) 1.09213e6 1.04154
\(257\) −567643. −0.536096 −0.268048 0.963406i \(-0.586379\pi\)
−0.268048 + 0.963406i \(0.586379\pi\)
\(258\) 995691. 0.931269
\(259\) −74427.0 −0.0689416
\(260\) −1.30301e6 −1.19540
\(261\) −90853.6 −0.0825545
\(262\) −1.11653e6 −1.00489
\(263\) 1.09213e6 0.973613 0.486807 0.873510i \(-0.338162\pi\)
0.486807 + 0.873510i \(0.338162\pi\)
\(264\) −216800. −0.191448
\(265\) −4.16379e6 −3.64229
\(266\) −1.79106e6 −1.55205
\(267\) −271467. −0.233044
\(268\) −636399. −0.541244
\(269\) 1.75427e6 1.47814 0.739069 0.673630i \(-0.235266\pi\)
0.739069 + 0.673630i \(0.235266\pi\)
\(270\) −513218. −0.428443
\(271\) 1.21698e6 1.00661 0.503306 0.864108i \(-0.332117\pi\)
0.503306 + 0.864108i \(0.332117\pi\)
\(272\) −105080. −0.0861188
\(273\) 1.11543e6 0.905811
\(274\) −2.50385e6 −2.01480
\(275\) −1.49005e6 −1.18814
\(276\) −68701.2 −0.0542864
\(277\) 2.13269e6 1.67004 0.835022 0.550217i \(-0.185455\pi\)
0.835022 + 0.550217i \(0.185455\pi\)
\(278\) 17168.1 0.0133232
\(279\) 76195.3 0.0586027
\(280\) −1.73440e6 −1.32207
\(281\) 948331. 0.716464 0.358232 0.933633i \(-0.383380\pi\)
0.358232 + 0.933633i \(0.383380\pi\)
\(282\) −1.44057e6 −1.07873
\(283\) 1.97106e6 1.46297 0.731484 0.681859i \(-0.238828\pi\)
0.731484 + 0.681859i \(0.238828\pi\)
\(284\) −693907. −0.510511
\(285\) −1.95350e6 −1.42463
\(286\) −1.21184e6 −0.876049
\(287\) 713724. 0.511476
\(288\) 358791. 0.254894
\(289\) −1.41302e6 −0.995183
\(290\) 789644. 0.551362
\(291\) −722140. −0.499907
\(292\) 399490. 0.274189
\(293\) 588573. 0.400526 0.200263 0.979742i \(-0.435820\pi\)
0.200263 + 0.979742i \(0.435820\pi\)
\(294\) 5931.32 0.00400205
\(295\) 365559. 0.244570
\(296\) 73567.2 0.0488039
\(297\) −137442. −0.0904126
\(298\) 3.55015e6 2.31583
\(299\) 565575. 0.365858
\(300\) 920475. 0.590485
\(301\) 2.13321e6 1.35711
\(302\) 3.20015e6 2.01908
\(303\) 846929. 0.529957
\(304\) 2.62628e6 1.62988
\(305\) 132842. 0.0817687
\(306\) −44905.8 −0.0274157
\(307\) −1.76408e6 −1.06825 −0.534125 0.845405i \(-0.679359\pi\)
−0.534125 + 0.845405i \(0.679359\pi\)
\(308\) 315374. 0.189430
\(309\) −315816. −0.188165
\(310\) −662244. −0.391393
\(311\) −1.05327e6 −0.617501 −0.308750 0.951143i \(-0.599911\pi\)
−0.308750 + 0.951143i \(0.599911\pi\)
\(312\) −1.10255e6 −0.641226
\(313\) 1.11290e6 0.642089 0.321045 0.947064i \(-0.395966\pi\)
0.321045 + 0.947064i \(0.395966\pi\)
\(314\) 1.40992e6 0.806994
\(315\) −1.09954e6 −0.624358
\(316\) −345833. −0.194827
\(317\) −878252. −0.490875 −0.245438 0.969412i \(-0.578932\pi\)
−0.245438 + 0.969412i \(0.578932\pi\)
\(318\) 2.39220e6 1.32657
\(319\) 211470. 0.116352
\(320\) 1.15160e6 0.628676
\(321\) −1.44257e6 −0.781403
\(322\) −511154. −0.274734
\(323\) −170929. −0.0911609
\(324\) 84904.7 0.0449334
\(325\) −7.57771e6 −3.97951
\(326\) −551060. −0.287181
\(327\) 2.16038e6 1.11728
\(328\) −705478. −0.362076
\(329\) −3.08633e6 −1.57200
\(330\) 1.19456e6 0.603844
\(331\) 2.13208e6 1.06963 0.534815 0.844969i \(-0.320381\pi\)
0.534815 + 0.844969i \(0.320381\pi\)
\(332\) 354816. 0.176668
\(333\) 46638.5 0.0230480
\(334\) 4.78443e6 2.34674
\(335\) −5.16443e6 −2.51426
\(336\) 1.47821e6 0.714313
\(337\) 1.12736e6 0.540739 0.270370 0.962757i \(-0.412854\pi\)
0.270370 + 0.962757i \(0.412854\pi\)
\(338\) −3.67377e6 −1.74912
\(339\) 1.66655e6 0.787624
\(340\) 112386. 0.0527249
\(341\) −177352. −0.0825943
\(342\) 1.12234e6 0.518869
\(343\) 2.18522e6 1.00290
\(344\) −2.10856e6 −0.960705
\(345\) −557515. −0.252179
\(346\) −3.12078e6 −1.40143
\(347\) −2.16376e6 −0.964683 −0.482342 0.875983i \(-0.660214\pi\)
−0.482342 + 0.875983i \(0.660214\pi\)
\(348\) −130636. −0.0578247
\(349\) 1.74582e6 0.767250 0.383625 0.923489i \(-0.374676\pi\)
0.383625 + 0.923489i \(0.374676\pi\)
\(350\) 6.84856e6 2.98834
\(351\) −698969. −0.302824
\(352\) −835120. −0.359246
\(353\) 3.74324e6 1.59886 0.799431 0.600758i \(-0.205134\pi\)
0.799431 + 0.600758i \(0.205134\pi\)
\(354\) −210023. −0.0890756
\(355\) −5.63110e6 −2.37150
\(356\) −390334. −0.163234
\(357\) −96207.9 −0.0399522
\(358\) 268506. 0.110725
\(359\) −2.31593e6 −0.948395 −0.474197 0.880419i \(-0.657262\pi\)
−0.474197 + 0.880419i \(0.657262\pi\)
\(360\) 1.08684e6 0.441985
\(361\) 1.79594e6 0.725310
\(362\) −2.32573e6 −0.932800
\(363\) −1.12955e6 −0.449923
\(364\) 1.60385e6 0.634468
\(365\) 3.24189e6 1.27370
\(366\) −76321.3 −0.0297813
\(367\) −3.24429e6 −1.25735 −0.628673 0.777670i \(-0.716402\pi\)
−0.628673 + 0.777670i \(0.716402\pi\)
\(368\) 749519. 0.288512
\(369\) −447243. −0.170993
\(370\) −405353. −0.153932
\(371\) 5.12515e6 1.93318
\(372\) 109559. 0.0410478
\(373\) −2.30146e6 −0.856508 −0.428254 0.903658i \(-0.640871\pi\)
−0.428254 + 0.903658i \(0.640871\pi\)
\(374\) 104523. 0.0386395
\(375\) 4.51616e6 1.65841
\(376\) 3.05068e6 1.11282
\(377\) 1.07544e6 0.389703
\(378\) 631712. 0.227400
\(379\) 2.81081e6 1.00515 0.502577 0.864532i \(-0.332385\pi\)
0.502577 + 0.864532i \(0.332385\pi\)
\(380\) −2.80888e6 −0.997871
\(381\) −922853. −0.325702
\(382\) −1.18349e6 −0.414960
\(383\) 580596. 0.202245 0.101122 0.994874i \(-0.467757\pi\)
0.101122 + 0.994874i \(0.467757\pi\)
\(384\) −1.93732e6 −0.670462
\(385\) 2.55928e6 0.879967
\(386\) 2.94044e6 1.00449
\(387\) −1.33674e6 −0.453700
\(388\) −1.03834e6 −0.350156
\(389\) −4.35788e6 −1.46016 −0.730082 0.683360i \(-0.760518\pi\)
−0.730082 + 0.683360i \(0.760518\pi\)
\(390\) 6.07502e6 2.02249
\(391\) −48781.7 −0.0161367
\(392\) −12560.7 −0.00412855
\(393\) 1.49897e6 0.489565
\(394\) 4.49216e6 1.45786
\(395\) −2.80646e6 −0.905037
\(396\) −197624. −0.0633288
\(397\) 1.87737e6 0.597826 0.298913 0.954280i \(-0.403376\pi\)
0.298913 + 0.954280i \(0.403376\pi\)
\(398\) 1.58696e6 0.502178
\(399\) 2.40454e6 0.756135
\(400\) −1.00422e7 −3.13820
\(401\) −4.27006e6 −1.32609 −0.663046 0.748579i \(-0.730737\pi\)
−0.663046 + 0.748579i \(0.730737\pi\)
\(402\) 2.96709e6 0.915727
\(403\) −901932. −0.276637
\(404\) 1.21777e6 0.371204
\(405\) 689008. 0.208731
\(406\) −971961. −0.292640
\(407\) −108556. −0.0324837
\(408\) 95096.5 0.0282823
\(409\) −265237. −0.0784017 −0.0392008 0.999231i \(-0.512481\pi\)
−0.0392008 + 0.999231i \(0.512481\pi\)
\(410\) 3.88717e6 1.14202
\(411\) 3.36148e6 0.981580
\(412\) −454102. −0.131799
\(413\) −449961. −0.129808
\(414\) 320306. 0.0918469
\(415\) 2.87936e6 0.820683
\(416\) −4.24704e6 −1.20324
\(417\) −23048.5 −0.00649087
\(418\) −2.61235e6 −0.731291
\(419\) −5.27516e6 −1.46791 −0.733957 0.679196i \(-0.762328\pi\)
−0.733957 + 0.679196i \(0.762328\pi\)
\(420\) −1.58099e6 −0.437327
\(421\) 5.84793e6 1.60804 0.804020 0.594602i \(-0.202690\pi\)
0.804020 + 0.594602i \(0.202690\pi\)
\(422\) 4.95894e6 1.35553
\(423\) 1.93400e6 0.525540
\(424\) −5.06594e6 −1.36850
\(425\) 653589. 0.175522
\(426\) 3.23521e6 0.863731
\(427\) −163514. −0.0433995
\(428\) −2.07423e6 −0.547328
\(429\) 1.62692e6 0.426798
\(430\) 1.16181e7 3.03016
\(431\) 7.13139e6 1.84919 0.924594 0.380955i \(-0.124405\pi\)
0.924594 + 0.380955i \(0.124405\pi\)
\(432\) −926298. −0.238804
\(433\) −3.38539e6 −0.867739 −0.433869 0.900976i \(-0.642852\pi\)
−0.433869 + 0.900976i \(0.642852\pi\)
\(434\) 815145. 0.207735
\(435\) −1.06012e6 −0.268615
\(436\) 3.10635e6 0.782590
\(437\) 1.21921e6 0.305403
\(438\) −1.86255e6 −0.463898
\(439\) 2.34305e6 0.580257 0.290129 0.956988i \(-0.406302\pi\)
0.290129 + 0.956988i \(0.406302\pi\)
\(440\) −2.52971e6 −0.622931
\(441\) −7962.94 −0.00194974
\(442\) 531555. 0.129417
\(443\) −999893. −0.242072 −0.121036 0.992648i \(-0.538622\pi\)
−0.121036 + 0.992648i \(0.538622\pi\)
\(444\) 67060.0 0.0161438
\(445\) −3.16759e6 −0.758278
\(446\) −845102. −0.201174
\(447\) −4.76617e6 −1.12824
\(448\) −1.41749e6 −0.333675
\(449\) 7.27002e6 1.70184 0.850921 0.525293i \(-0.176044\pi\)
0.850921 + 0.525293i \(0.176044\pi\)
\(450\) −4.29154e6 −0.999038
\(451\) 1.04100e6 0.240996
\(452\) 2.39628e6 0.551685
\(453\) −4.29628e6 −0.983664
\(454\) −2.76189e6 −0.628879
\(455\) 1.30153e7 2.94732
\(456\) −2.37676e6 −0.535270
\(457\) −6.00303e6 −1.34456 −0.672279 0.740297i \(-0.734685\pi\)
−0.672279 + 0.740297i \(0.734685\pi\)
\(458\) −5.03138e6 −1.12079
\(459\) 60287.1 0.0133565
\(460\) −801633. −0.176637
\(461\) −2.14478e6 −0.470035 −0.235018 0.971991i \(-0.575515\pi\)
−0.235018 + 0.971991i \(0.575515\pi\)
\(462\) −1.47037e6 −0.320496
\(463\) −4.81062e6 −1.04291 −0.521457 0.853278i \(-0.674611\pi\)
−0.521457 + 0.853278i \(0.674611\pi\)
\(464\) 1.42521e6 0.307316
\(465\) 889078. 0.190681
\(466\) 1.00303e6 0.213968
\(467\) −6.00558e6 −1.27427 −0.637137 0.770750i \(-0.719882\pi\)
−0.637137 + 0.770750i \(0.719882\pi\)
\(468\) −1.00503e6 −0.212111
\(469\) 6.35681e6 1.33446
\(470\) −1.68092e7 −3.50995
\(471\) −1.89285e6 −0.393155
\(472\) 444763. 0.0918912
\(473\) 3.11139e6 0.639442
\(474\) 1.61238e6 0.329627
\(475\) −1.63352e7 −3.32194
\(476\) −138334. −0.0279842
\(477\) −3.21159e6 −0.646285
\(478\) −1.15051e6 −0.230314
\(479\) 4.87817e6 0.971445 0.485723 0.874113i \(-0.338556\pi\)
0.485723 + 0.874113i \(0.338556\pi\)
\(480\) 4.18651e6 0.829372
\(481\) −552064. −0.108800
\(482\) 1.79032e6 0.351006
\(483\) 686236. 0.133846
\(484\) −1.62414e6 −0.315145
\(485\) −8.42622e6 −1.62659
\(486\) −395852. −0.0760225
\(487\) −3.07733e6 −0.587966 −0.293983 0.955811i \(-0.594981\pi\)
−0.293983 + 0.955811i \(0.594981\pi\)
\(488\) 161625. 0.0307226
\(489\) 739812. 0.139910
\(490\) 69209.0 0.0130218
\(491\) 8.06992e6 1.51066 0.755328 0.655347i \(-0.227478\pi\)
0.755328 + 0.655347i \(0.227478\pi\)
\(492\) −643077. −0.119771
\(493\) −92758.6 −0.0171885
\(494\) −1.32852e7 −2.44935
\(495\) −1.60373e6 −0.294184
\(496\) −1.19527e6 −0.218153
\(497\) 6.93124e6 1.25869
\(498\) −1.65426e6 −0.298904
\(499\) 4.28183e6 0.769800 0.384900 0.922958i \(-0.374236\pi\)
0.384900 + 0.922958i \(0.374236\pi\)
\(500\) 6.49364e6 1.16162
\(501\) −6.42322e6 −1.14329
\(502\) −7.37681e6 −1.30650
\(503\) −2.87943e6 −0.507442 −0.253721 0.967277i \(-0.581655\pi\)
−0.253721 + 0.967277i \(0.581655\pi\)
\(504\) −1.33777e6 −0.234587
\(505\) 9.88231e6 1.72437
\(506\) −745544. −0.129448
\(507\) 4.93213e6 0.852147
\(508\) −1.32694e6 −0.228135
\(509\) 3.51526e6 0.601400 0.300700 0.953719i \(-0.402780\pi\)
0.300700 + 0.953719i \(0.402780\pi\)
\(510\) −523979. −0.0892049
\(511\) −3.99040e6 −0.676026
\(512\) −433178. −0.0730284
\(513\) −1.50676e6 −0.252785
\(514\) 3.80536e6 0.635313
\(515\) −3.68507e6 −0.612249
\(516\) −1.92205e6 −0.317791
\(517\) −4.50157e6 −0.740692
\(518\) 498943. 0.0817009
\(519\) 4.18972e6 0.682758
\(520\) −1.28650e7 −2.08642
\(521\) 3.02436e6 0.488134 0.244067 0.969758i \(-0.421518\pi\)
0.244067 + 0.969758i \(0.421518\pi\)
\(522\) 609063. 0.0978332
\(523\) −4.60919e6 −0.736836 −0.368418 0.929660i \(-0.620100\pi\)
−0.368418 + 0.929660i \(0.620100\pi\)
\(524\) 2.15532e6 0.342912
\(525\) −9.19436e6 −1.45587
\(526\) −7.32144e6 −1.15380
\(527\) 77793.0 0.0122015
\(528\) 2.15605e6 0.336568
\(529\) −6.08839e6 −0.945939
\(530\) 2.79132e7 4.31638
\(531\) 281961. 0.0433963
\(532\) 3.45741e6 0.529629
\(533\) 5.29407e6 0.807182
\(534\) 1.81986e6 0.276175
\(535\) −1.68325e7 −2.54252
\(536\) −6.28337e6 −0.944671
\(537\) −360476. −0.0539436
\(538\) −1.17602e7 −1.75170
\(539\) 18534.5 0.00274795
\(540\) 990703. 0.146204
\(541\) 3.48957e6 0.512600 0.256300 0.966597i \(-0.417496\pi\)
0.256300 + 0.966597i \(0.417496\pi\)
\(542\) −8.15841e6 −1.19291
\(543\) 3.12235e6 0.454446
\(544\) 366314. 0.0530708
\(545\) 2.52082e7 3.63539
\(546\) −7.47764e6 −1.07345
\(547\) −5.80736e6 −0.829871 −0.414935 0.909851i \(-0.636196\pi\)
−0.414935 + 0.909851i \(0.636196\pi\)
\(548\) 4.83336e6 0.687540
\(549\) 102463. 0.0145090
\(550\) 9.98898e6 1.40804
\(551\) 2.31833e6 0.325309
\(552\) −678308. −0.0947500
\(553\) 3.45443e6 0.480356
\(554\) −1.42971e7 −1.97913
\(555\) 544197. 0.0749935
\(556\) −33140.8 −0.00454648
\(557\) −1.35811e7 −1.85480 −0.927399 0.374075i \(-0.877960\pi\)
−0.927399 + 0.374075i \(0.877960\pi\)
\(558\) −510797. −0.0694486
\(559\) 1.58231e7 2.14172
\(560\) 1.72484e7 2.32423
\(561\) −140324. −0.0188246
\(562\) −6.35742e6 −0.849063
\(563\) 4.82039e6 0.640930 0.320465 0.947260i \(-0.396161\pi\)
0.320465 + 0.947260i \(0.396161\pi\)
\(564\) 2.78084e6 0.368110
\(565\) 1.94460e7 2.56276
\(566\) −1.32136e7 −1.73372
\(567\) −848089. −0.110786
\(568\) −6.85116e6 −0.891032
\(569\) −897706. −0.116239 −0.0581197 0.998310i \(-0.518511\pi\)
−0.0581197 + 0.998310i \(0.518511\pi\)
\(570\) 1.30959e7 1.68829
\(571\) 6.48530e6 0.832415 0.416207 0.909270i \(-0.363359\pi\)
0.416207 + 0.909270i \(0.363359\pi\)
\(572\) 2.33929e6 0.298947
\(573\) 1.58886e6 0.202162
\(574\) −4.78466e6 −0.606137
\(575\) −4.66195e6 −0.588028
\(576\) 888245. 0.111552
\(577\) −3.92319e6 −0.490569 −0.245285 0.969451i \(-0.578881\pi\)
−0.245285 + 0.969451i \(0.578881\pi\)
\(578\) 9.47258e6 1.17937
\(579\) −3.94761e6 −0.489370
\(580\) −1.52431e6 −0.188149
\(581\) −3.54416e6 −0.435585
\(582\) 4.84107e6 0.592426
\(583\) 7.47529e6 0.910870
\(584\) 3.94430e6 0.478561
\(585\) −8.15585e6 −0.985326
\(586\) −3.94567e6 −0.474653
\(587\) 1.83842e6 0.220217 0.110108 0.993920i \(-0.464880\pi\)
0.110108 + 0.993920i \(0.464880\pi\)
\(588\) −11449.7 −0.00136568
\(589\) −1.94429e6 −0.230926
\(590\) −2.45063e6 −0.289833
\(591\) −6.03083e6 −0.710245
\(592\) −731615. −0.0857982
\(593\) 2.60549e6 0.304265 0.152133 0.988360i \(-0.451386\pi\)
0.152133 + 0.988360i \(0.451386\pi\)
\(594\) 921384. 0.107146
\(595\) −1.12259e6 −0.129996
\(596\) −6.85312e6 −0.790265
\(597\) −2.13053e6 −0.244653
\(598\) −3.79150e6 −0.433569
\(599\) −790975. −0.0900732 −0.0450366 0.998985i \(-0.514340\pi\)
−0.0450366 + 0.998985i \(0.514340\pi\)
\(600\) 9.08814e6 1.03062
\(601\) 1.28795e7 1.45449 0.727246 0.686377i \(-0.240800\pi\)
0.727246 + 0.686377i \(0.240800\pi\)
\(602\) −1.43006e7 −1.60828
\(603\) −3.98339e6 −0.446128
\(604\) −6.17749e6 −0.689000
\(605\) −1.31800e7 −1.46396
\(606\) −5.67763e6 −0.628038
\(607\) 5.79335e6 0.638202 0.319101 0.947721i \(-0.396619\pi\)
0.319101 + 0.947721i \(0.396619\pi\)
\(608\) −9.15532e6 −1.00442
\(609\) 1.30488e6 0.142570
\(610\) −890548. −0.0969020
\(611\) −2.28929e7 −2.48084
\(612\) 86685.0 0.00935546
\(613\) 6.59875e6 0.709268 0.354634 0.935005i \(-0.384605\pi\)
0.354634 + 0.935005i \(0.384605\pi\)
\(614\) 1.18260e7 1.26596
\(615\) −5.21862e6 −0.556375
\(616\) 3.11379e6 0.330626
\(617\) −2.10218e6 −0.222309 −0.111155 0.993803i \(-0.535455\pi\)
−0.111155 + 0.993803i \(0.535455\pi\)
\(618\) 2.11717e6 0.222989
\(619\) 8.05878e6 0.845362 0.422681 0.906278i \(-0.361089\pi\)
0.422681 + 0.906278i \(0.361089\pi\)
\(620\) 1.27838e6 0.133561
\(621\) −430019. −0.0447464
\(622\) 7.06088e6 0.731784
\(623\) 3.89893e6 0.402463
\(624\) 1.09647e7 1.12729
\(625\) 2.79986e7 2.86705
\(626\) −7.46065e6 −0.760923
\(627\) 3.50714e6 0.356274
\(628\) −2.72167e6 −0.275383
\(629\) 47616.4 0.00479877
\(630\) 7.37107e6 0.739911
\(631\) 2.29082e6 0.229043 0.114521 0.993421i \(-0.463467\pi\)
0.114521 + 0.993421i \(0.463467\pi\)
\(632\) −3.41452e6 −0.340046
\(633\) −6.65750e6 −0.660392
\(634\) 5.88762e6 0.581723
\(635\) −1.07682e7 −1.05977
\(636\) −4.61785e6 −0.452686
\(637\) 94258.1 0.00920386
\(638\) −1.41765e6 −0.137885
\(639\) −4.34335e6 −0.420797
\(640\) −2.26055e7 −2.18154
\(641\) −1.22229e7 −1.17498 −0.587489 0.809232i \(-0.699884\pi\)
−0.587489 + 0.809232i \(0.699884\pi\)
\(642\) 9.67071e6 0.926021
\(643\) 1.94577e7 1.85594 0.927969 0.372656i \(-0.121553\pi\)
0.927969 + 0.372656i \(0.121553\pi\)
\(644\) 986718. 0.0937515
\(645\) −1.55976e7 −1.47625
\(646\) 1.14587e6 0.108032
\(647\) −1.97484e7 −1.85469 −0.927346 0.374205i \(-0.877916\pi\)
−0.927346 + 0.374205i \(0.877916\pi\)
\(648\) 838291. 0.0784255
\(649\) −656291. −0.0611625
\(650\) 5.07994e7 4.71602
\(651\) −1.09435e6 −0.101206
\(652\) 1.06375e6 0.0979989
\(653\) 4.80387e6 0.440868 0.220434 0.975402i \(-0.429253\pi\)
0.220434 + 0.975402i \(0.429253\pi\)
\(654\) −1.44828e7 −1.32406
\(655\) 1.74905e7 1.59294
\(656\) 7.01588e6 0.636535
\(657\) 2.50052e6 0.226004
\(658\) 2.06901e7 1.86294
\(659\) −6.39074e6 −0.573242 −0.286621 0.958044i \(-0.592532\pi\)
−0.286621 + 0.958044i \(0.592532\pi\)
\(660\) −2.30596e6 −0.206059
\(661\) −2.04223e6 −0.181803 −0.0909017 0.995860i \(-0.528975\pi\)
−0.0909017 + 0.995860i \(0.528975\pi\)
\(662\) −1.42930e7 −1.26759
\(663\) −713625. −0.0630502
\(664\) 3.50321e6 0.308352
\(665\) 2.80571e7 2.46030
\(666\) −312655. −0.0273136
\(667\) 661633. 0.0575841
\(668\) −9.23574e6 −0.800812
\(669\) 1.13457e6 0.0980090
\(670\) 3.46212e7 2.97958
\(671\) −238493. −0.0204489
\(672\) −5.15311e6 −0.440197
\(673\) −7.12679e6 −0.606535 −0.303268 0.952905i \(-0.598078\pi\)
−0.303268 + 0.952905i \(0.598078\pi\)
\(674\) −7.55758e6 −0.640816
\(675\) 5.76150e6 0.486716
\(676\) 7.09175e6 0.596880
\(677\) 9.46228e6 0.793458 0.396729 0.917936i \(-0.370145\pi\)
0.396729 + 0.917936i \(0.370145\pi\)
\(678\) −1.11722e7 −0.933393
\(679\) 1.03717e7 0.863328
\(680\) 1.10962e6 0.0920245
\(681\) 3.70791e6 0.306380
\(682\) 1.18893e6 0.0978803
\(683\) −2.26865e7 −1.86087 −0.930433 0.366462i \(-0.880569\pi\)
−0.930433 + 0.366462i \(0.880569\pi\)
\(684\) −2.16653e6 −0.177061
\(685\) 3.92231e7 3.19386
\(686\) −1.46492e7 −1.18851
\(687\) 6.75475e6 0.546031
\(688\) 2.09693e7 1.68894
\(689\) 3.80159e7 3.05083
\(690\) 3.73746e6 0.298851
\(691\) 8.35889e6 0.665969 0.332984 0.942932i \(-0.391944\pi\)
0.332984 + 0.942932i \(0.391944\pi\)
\(692\) 6.02427e6 0.478232
\(693\) 1.97401e6 0.156141
\(694\) 1.45054e7 1.14322
\(695\) −268940. −0.0211199
\(696\) −1.28981e6 −0.100926
\(697\) −456621. −0.0356020
\(698\) −1.17036e7 −0.909248
\(699\) −1.34659e6 −0.104242
\(700\) −1.32203e7 −1.01975
\(701\) −4.42447e6 −0.340068 −0.170034 0.985438i \(-0.554388\pi\)
−0.170034 + 0.985438i \(0.554388\pi\)
\(702\) 4.68574e6 0.358869
\(703\) −1.19008e6 −0.0908215
\(704\) −2.06748e6 −0.157220
\(705\) 2.25667e7 1.71000
\(706\) −2.50939e7 −1.89477
\(707\) −1.21640e7 −0.915223
\(708\) 405423. 0.0303966
\(709\) −5.65728e6 −0.422661 −0.211330 0.977415i \(-0.567780\pi\)
−0.211330 + 0.977415i \(0.567780\pi\)
\(710\) 3.77497e7 2.81040
\(711\) −2.16466e6 −0.160589
\(712\) −3.85389e6 −0.284904
\(713\) −554885. −0.0408770
\(714\) 644958. 0.0473463
\(715\) 1.89835e7 1.38871
\(716\) −518316. −0.0377844
\(717\) 1.54459e6 0.112206
\(718\) 1.55255e7 1.12392
\(719\) −797010. −0.0574965 −0.0287483 0.999587i \(-0.509152\pi\)
−0.0287483 + 0.999587i \(0.509152\pi\)
\(720\) −1.08084e7 −0.777018
\(721\) 4.53590e6 0.324956
\(722\) −1.20396e7 −0.859546
\(723\) −2.40355e6 −0.171005
\(724\) 4.48953e6 0.318313
\(725\) −8.86472e6 −0.626354
\(726\) 7.57226e6 0.533192
\(727\) −2.01191e7 −1.41180 −0.705900 0.708312i \(-0.749457\pi\)
−0.705900 + 0.708312i \(0.749457\pi\)
\(728\) 1.58353e7 1.10738
\(729\) 531441. 0.0370370
\(730\) −2.17330e7 −1.50943
\(731\) −1.36477e6 −0.0944638
\(732\) 147329. 0.0101627
\(733\) −2.35216e7 −1.61699 −0.808493 0.588505i \(-0.799717\pi\)
−0.808493 + 0.588505i \(0.799717\pi\)
\(734\) 2.17491e7 1.49005
\(735\) −92914.8 −0.00634405
\(736\) −2.61286e6 −0.177796
\(737\) 9.27173e6 0.628770
\(738\) 2.99823e6 0.202639
\(739\) −2.16190e7 −1.45621 −0.728107 0.685464i \(-0.759600\pi\)
−0.728107 + 0.685464i \(0.759600\pi\)
\(740\) 782483. 0.0525286
\(741\) 1.78357e7 1.19329
\(742\) −3.43579e7 −2.29096
\(743\) 2.43927e6 0.162102 0.0810510 0.996710i \(-0.474172\pi\)
0.0810510 + 0.996710i \(0.474172\pi\)
\(744\) 1.08171e6 0.0716437
\(745\) −5.56136e7 −3.67105
\(746\) 1.54285e7 1.01503
\(747\) 2.22089e6 0.145621
\(748\) −201768. −0.0131855
\(749\) 2.07189e7 1.34947
\(750\) −3.02754e7 −1.96533
\(751\) −7.79043e6 −0.504036 −0.252018 0.967723i \(-0.581094\pi\)
−0.252018 + 0.967723i \(0.581094\pi\)
\(752\) −3.03385e7 −1.95636
\(753\) 9.90355e6 0.636507
\(754\) −7.20955e6 −0.461827
\(755\) −5.01307e7 −3.20064
\(756\) −1.21944e6 −0.0775990
\(757\) 2.03779e7 1.29247 0.646235 0.763138i \(-0.276343\pi\)
0.646235 + 0.763138i \(0.276343\pi\)
\(758\) −1.88431e7 −1.19118
\(759\) 1.00091e6 0.0630653
\(760\) −2.77330e7 −1.74166
\(761\) −2.59966e7 −1.62725 −0.813626 0.581389i \(-0.802510\pi\)
−0.813626 + 0.581389i \(0.802510\pi\)
\(762\) 6.18662e6 0.385981
\(763\) −3.10284e7 −1.92952
\(764\) 2.28458e6 0.141603
\(765\) 703455. 0.0434593
\(766\) −3.89219e6 −0.239675
\(767\) −3.33760e6 −0.204855
\(768\) 9.82921e6 0.601334
\(769\) 2.44602e7 1.49157 0.745785 0.666187i \(-0.232075\pi\)
0.745785 + 0.666187i \(0.232075\pi\)
\(770\) −1.71569e7 −1.04283
\(771\) −5.10879e6 −0.309515
\(772\) −5.67614e6 −0.342776
\(773\) 1.92647e7 1.15962 0.579808 0.814753i \(-0.303128\pi\)
0.579808 + 0.814753i \(0.303128\pi\)
\(774\) 8.96122e6 0.537669
\(775\) 7.43449e6 0.444628
\(776\) −1.02519e7 −0.611152
\(777\) −669843. −0.0398034
\(778\) 2.92143e7 1.73040
\(779\) 1.14124e7 0.673803
\(780\) −1.17270e7 −0.690164
\(781\) 1.01096e7 0.593068
\(782\) 327022. 0.0191232
\(783\) −817682. −0.0476629
\(784\) 124914. 0.00725807
\(785\) −2.20866e7 −1.27925
\(786\) −1.00488e7 −0.580171
\(787\) −2.18742e7 −1.25891 −0.629455 0.777037i \(-0.716722\pi\)
−0.629455 + 0.777037i \(0.716722\pi\)
\(788\) −8.67154e6 −0.497486
\(789\) 9.82920e6 0.562116
\(790\) 1.88139e7 1.07254
\(791\) −2.39357e7 −1.36021
\(792\) −1.95120e6 −0.110532
\(793\) −1.21287e6 −0.0684905
\(794\) −1.25855e7 −0.708468
\(795\) −3.74742e7 −2.10288
\(796\) −3.06342e6 −0.171366
\(797\) −2.81755e7 −1.57118 −0.785591 0.618746i \(-0.787641\pi\)
−0.785591 + 0.618746i \(0.787641\pi\)
\(798\) −1.61195e7 −0.896075
\(799\) 1.97455e6 0.109421
\(800\) 3.50077e7 1.93392
\(801\) −2.44320e6 −0.134548
\(802\) 2.86256e7 1.57152
\(803\) −5.82019e6 −0.318529
\(804\) −5.72759e6 −0.312487
\(805\) 8.00728e6 0.435507
\(806\) 6.04636e6 0.327836
\(807\) 1.57884e7 0.853403
\(808\) 1.20234e7 0.647889
\(809\) 1.83693e7 0.986785 0.493392 0.869807i \(-0.335757\pi\)
0.493392 + 0.869807i \(0.335757\pi\)
\(810\) −4.61896e6 −0.247361
\(811\) 2.97018e7 1.58574 0.792868 0.609393i \(-0.208587\pi\)
0.792868 + 0.609393i \(0.208587\pi\)
\(812\) 1.87625e6 0.0998619
\(813\) 1.09529e7 0.581167
\(814\) 727734. 0.0384956
\(815\) 8.63242e6 0.455238
\(816\) −945720. −0.0497207
\(817\) 3.41098e7 1.78782
\(818\) 1.77809e6 0.0929118
\(819\) 1.00389e7 0.522970
\(820\) −7.50369e6 −0.389709
\(821\) −2.76967e6 −0.143407 −0.0717034 0.997426i \(-0.522844\pi\)
−0.0717034 + 0.997426i \(0.522844\pi\)
\(822\) −2.25346e7 −1.16325
\(823\) 1.00360e7 0.516489 0.258244 0.966080i \(-0.416856\pi\)
0.258244 + 0.966080i \(0.416856\pi\)
\(824\) −4.48349e6 −0.230037
\(825\) −1.34104e7 −0.685975
\(826\) 3.01645e6 0.153832
\(827\) −3.34891e7 −1.70271 −0.851354 0.524591i \(-0.824218\pi\)
−0.851354 + 0.524591i \(0.824218\pi\)
\(828\) −618311. −0.0313423
\(829\) 1.64842e7 0.833072 0.416536 0.909119i \(-0.363244\pi\)
0.416536 + 0.909119i \(0.363244\pi\)
\(830\) −1.93026e7 −0.972570
\(831\) 1.91942e7 0.964200
\(832\) −1.05142e7 −0.526587
\(833\) −8129.90 −0.000405950 0
\(834\) 154513. 0.00769216
\(835\) −7.49487e7 −3.72004
\(836\) 5.04280e6 0.249549
\(837\) 685758. 0.0338343
\(838\) 3.53636e7 1.73959
\(839\) −2.49583e7 −1.22408 −0.612042 0.790826i \(-0.709651\pi\)
−0.612042 + 0.790826i \(0.709651\pi\)
\(840\) −1.56096e7 −0.763298
\(841\) −1.92531e7 −0.938663
\(842\) −3.92033e7 −1.90565
\(843\) 8.53498e6 0.413651
\(844\) −9.57260e6 −0.462566
\(845\) 5.75501e7 2.77271
\(846\) −1.29651e7 −0.622803
\(847\) 1.62231e7 0.777008
\(848\) 5.03800e7 2.40585
\(849\) 1.77396e7 0.844645
\(850\) −4.38153e6 −0.208007
\(851\) −339640. −0.0160766
\(852\) −6.24516e6 −0.294744
\(853\) −2.83557e7 −1.33434 −0.667171 0.744904i \(-0.732495\pi\)
−0.667171 + 0.744904i \(0.732495\pi\)
\(854\) 1.09616e6 0.0514316
\(855\) −1.75815e7 −0.822510
\(856\) −2.04795e7 −0.955291
\(857\) −1.22086e7 −0.567822 −0.283911 0.958851i \(-0.591632\pi\)
−0.283911 + 0.958851i \(0.591632\pi\)
\(858\) −1.09065e7 −0.505787
\(859\) 1.60844e7 0.743740 0.371870 0.928285i \(-0.378717\pi\)
0.371870 + 0.928285i \(0.378717\pi\)
\(860\) −2.24273e7 −1.03403
\(861\) 6.42352e6 0.295301
\(862\) −4.78073e7 −2.19142
\(863\) 656036. 0.0299848 0.0149924 0.999888i \(-0.495228\pi\)
0.0149924 + 0.999888i \(0.495228\pi\)
\(864\) 3.22912e6 0.147163
\(865\) 4.88874e7 2.22155
\(866\) 2.26949e7 1.02833
\(867\) −1.27172e7 −0.574569
\(868\) −1.57353e6 −0.0708887
\(869\) 5.03846e6 0.226333
\(870\) 7.10680e6 0.318329
\(871\) 4.71518e7 2.10597
\(872\) 3.06700e7 1.36591
\(873\) −6.49926e6 −0.288621
\(874\) −8.17331e6 −0.361926
\(875\) −6.48631e7 −2.86403
\(876\) 3.59541e6 0.158303
\(877\) −9.37944e6 −0.411792 −0.205896 0.978574i \(-0.566011\pi\)
−0.205896 + 0.978574i \(0.566011\pi\)
\(878\) −1.57073e7 −0.687648
\(879\) 5.29716e6 0.231244
\(880\) 2.51576e7 1.09512
\(881\) 950816. 0.0412721 0.0206361 0.999787i \(-0.493431\pi\)
0.0206361 + 0.999787i \(0.493431\pi\)
\(882\) 53381.9 0.00231059
\(883\) −1.58113e7 −0.682442 −0.341221 0.939983i \(-0.610840\pi\)
−0.341221 + 0.939983i \(0.610840\pi\)
\(884\) −1.02610e6 −0.0441630
\(885\) 3.29004e6 0.141203
\(886\) 6.70308e6 0.286873
\(887\) 2.23299e7 0.952969 0.476484 0.879183i \(-0.341911\pi\)
0.476484 + 0.879183i \(0.341911\pi\)
\(888\) 662105. 0.0281770
\(889\) 1.32544e7 0.562480
\(890\) 2.12348e7 0.898616
\(891\) −1.23698e6 −0.0521998
\(892\) 1.63136e6 0.0686497
\(893\) −4.93502e7 −2.07091
\(894\) 3.19514e7 1.33704
\(895\) −4.20617e6 −0.175521
\(896\) 2.78247e7 1.15787
\(897\) 5.09018e6 0.211228
\(898\) −4.87367e7 −2.01681
\(899\) −1.05512e6 −0.0435413
\(900\) 8.28428e6 0.340917
\(901\) −3.27893e6 −0.134561
\(902\) −6.97866e6 −0.285598
\(903\) 1.91989e7 0.783531
\(904\) 2.36592e7 0.962896
\(905\) 3.64329e7 1.47867
\(906\) 2.88014e7 1.16572
\(907\) −4.23650e7 −1.70997 −0.854986 0.518651i \(-0.826434\pi\)
−0.854986 + 0.518651i \(0.826434\pi\)
\(908\) 5.33148e6 0.214602
\(909\) 7.62236e6 0.305971
\(910\) −8.72521e7 −3.49279
\(911\) 6.22591e6 0.248546 0.124273 0.992248i \(-0.460340\pi\)
0.124273 + 0.992248i \(0.460340\pi\)
\(912\) 2.36365e7 0.941014
\(913\) −5.16933e6 −0.205238
\(914\) 4.02430e7 1.59340
\(915\) 1.19558e6 0.0472092
\(916\) 9.71245e6 0.382463
\(917\) −2.15288e7 −0.845468
\(918\) −404152. −0.0158285
\(919\) 5.05205e7 1.97324 0.986618 0.163051i \(-0.0521336\pi\)
0.986618 + 0.163051i \(0.0521336\pi\)
\(920\) −7.91478e6 −0.308297
\(921\) −1.58767e7 −0.616755
\(922\) 1.43782e7 0.557026
\(923\) 5.14126e7 1.98639
\(924\) 2.83836e6 0.109367
\(925\) 4.55058e6 0.174869
\(926\) 3.22494e7 1.23593
\(927\) −2.84235e6 −0.108637
\(928\) −4.96836e6 −0.189384
\(929\) −3.87776e7 −1.47415 −0.737075 0.675811i \(-0.763794\pi\)
−0.737075 + 0.675811i \(0.763794\pi\)
\(930\) −5.96019e6 −0.225971
\(931\) 203192. 0.00768301
\(932\) −1.93622e6 −0.0730156
\(933\) −9.47940e6 −0.356514
\(934\) 4.02602e7 1.51011
\(935\) −1.63736e6 −0.0612512
\(936\) −9.92293e6 −0.370212
\(937\) 2.07364e7 0.771586 0.385793 0.922585i \(-0.373928\pi\)
0.385793 + 0.922585i \(0.373928\pi\)
\(938\) −4.26147e7 −1.58144
\(939\) 1.00161e7 0.370710
\(940\) 3.24479e7 1.19775
\(941\) 1.54725e7 0.569623 0.284811 0.958584i \(-0.408069\pi\)
0.284811 + 0.958584i \(0.408069\pi\)
\(942\) 1.26893e7 0.465918
\(943\) 3.25701e6 0.119272
\(944\) −4.42310e6 −0.161546
\(945\) −9.89584e6 −0.360473
\(946\) −2.08581e7 −0.757787
\(947\) 3.92696e7 1.42292 0.711462 0.702724i \(-0.248033\pi\)
0.711462 + 0.702724i \(0.248033\pi\)
\(948\) −3.11250e6 −0.112483
\(949\) −2.95989e7 −1.06687
\(950\) 1.09508e8 3.93674
\(951\) −7.90427e6 −0.283407
\(952\) −1.36582e6 −0.0488428
\(953\) 1.72996e7 0.617027 0.308513 0.951220i \(-0.400169\pi\)
0.308513 + 0.951220i \(0.400169\pi\)
\(954\) 2.15298e7 0.765896
\(955\) 1.85395e7 0.657794
\(956\) 2.22092e6 0.0785936
\(957\) 1.90323e6 0.0671757
\(958\) −3.27023e7 −1.15123
\(959\) −4.82791e7 −1.69517
\(960\) 1.03644e7 0.362966
\(961\) −2.77443e7 −0.969092
\(962\) 3.70092e6 0.128936
\(963\) −1.29832e7 −0.451143
\(964\) −3.45599e6 −0.119779
\(965\) −4.60623e7 −1.59231
\(966\) −4.60038e6 −0.158618
\(967\) −9.19283e6 −0.316143 −0.158071 0.987428i \(-0.550528\pi\)
−0.158071 + 0.987428i \(0.550528\pi\)
\(968\) −1.60357e7 −0.550046
\(969\) −1.53836e6 −0.0526318
\(970\) 5.64876e7 1.92763
\(971\) 2.71756e7 0.924979 0.462489 0.886625i \(-0.346956\pi\)
0.462489 + 0.886625i \(0.346956\pi\)
\(972\) 764142. 0.0259423
\(973\) 331033. 0.0112096
\(974\) 2.06298e7 0.696783
\(975\) −6.81994e7 −2.29757
\(976\) −1.60733e6 −0.0540109
\(977\) −2.95289e7 −0.989715 −0.494858 0.868974i \(-0.664780\pi\)
−0.494858 + 0.868974i \(0.664780\pi\)
\(978\) −4.95954e6 −0.165804
\(979\) 5.68679e6 0.189631
\(980\) −133599. −0.00444364
\(981\) 1.94435e7 0.645061
\(982\) −5.40991e7 −1.79024
\(983\) −3.03956e7 −1.00329 −0.501645 0.865074i \(-0.667272\pi\)
−0.501645 + 0.865074i \(0.667272\pi\)
\(984\) −6.34931e6 −0.209044
\(985\) −7.03702e7 −2.31099
\(986\) 621834. 0.0203696
\(987\) −2.77770e7 −0.907595
\(988\) 2.56454e7 0.835829
\(989\) 9.73467e6 0.316469
\(990\) 1.07511e7 0.348630
\(991\) −1.19636e7 −0.386969 −0.193485 0.981103i \(-0.561979\pi\)
−0.193485 + 0.981103i \(0.561979\pi\)
\(992\) 4.16677e6 0.134437
\(993\) 1.91887e7 0.617551
\(994\) −4.64656e7 −1.49164
\(995\) −2.48599e7 −0.796051
\(996\) 3.19334e6 0.101999
\(997\) 6.46829e6 0.206088 0.103044 0.994677i \(-0.467142\pi\)
0.103044 + 0.994677i \(0.467142\pi\)
\(998\) −2.87045e7 −0.912271
\(999\) 419746. 0.0133068
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.6.a.a.1.3 11
3.2 odd 2 531.6.a.b.1.9 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.a.1.3 11 1.1 even 1 trivial
531.6.a.b.1.9 11 3.2 odd 2