Properties

Label 177.6.a.b.1.8
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 1.8
Root \(-1.65902\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.65902 q^{2} -9.00000 q^{3} -29.2477 q^{4} +59.9319 q^{5} -14.9312 q^{6} -87.2478 q^{7} -101.611 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.65902 q^{2} -9.00000 q^{3} -29.2477 q^{4} +59.9319 q^{5} -14.9312 q^{6} -87.2478 q^{7} -101.611 q^{8} +81.0000 q^{9} +99.4282 q^{10} +655.885 q^{11} +263.229 q^{12} +139.966 q^{13} -144.746 q^{14} -539.387 q^{15} +767.351 q^{16} -1518.98 q^{17} +134.380 q^{18} +393.396 q^{19} -1752.87 q^{20} +785.230 q^{21} +1088.12 q^{22} -4746.68 q^{23} +914.499 q^{24} +466.837 q^{25} +232.206 q^{26} -729.000 q^{27} +2551.79 q^{28} +2791.29 q^{29} -894.853 q^{30} -1808.12 q^{31} +4524.60 q^{32} -5902.96 q^{33} -2520.02 q^{34} -5228.93 q^{35} -2369.06 q^{36} +6710.46 q^{37} +652.650 q^{38} -1259.69 q^{39} -6089.74 q^{40} -13926.6 q^{41} +1302.71 q^{42} -19381.1 q^{43} -19183.1 q^{44} +4854.49 q^{45} -7874.82 q^{46} -15752.6 q^{47} -6906.16 q^{48} -9194.83 q^{49} +774.491 q^{50} +13670.8 q^{51} -4093.68 q^{52} -25608.5 q^{53} -1209.42 q^{54} +39308.4 q^{55} +8865.33 q^{56} -3540.56 q^{57} +4630.80 q^{58} -3481.00 q^{59} +15775.8 q^{60} +11066.2 q^{61} -2999.70 q^{62} -7067.07 q^{63} -17048.8 q^{64} +8388.43 q^{65} -9793.12 q^{66} -34547.3 q^{67} +44426.7 q^{68} +42720.1 q^{69} -8674.88 q^{70} +43063.2 q^{71} -8230.49 q^{72} -34956.0 q^{73} +11132.8 q^{74} -4201.53 q^{75} -11505.9 q^{76} -57224.5 q^{77} -2089.85 q^{78} +91122.1 q^{79} +45988.8 q^{80} +6561.00 q^{81} -23104.4 q^{82} -109543. q^{83} -22966.1 q^{84} -91035.5 q^{85} -32153.5 q^{86} -25121.6 q^{87} -66645.1 q^{88} +136931. q^{89} +8053.68 q^{90} -12211.7 q^{91} +138829. q^{92} +16273.0 q^{93} -26133.8 q^{94} +23577.0 q^{95} -40721.4 q^{96} -139282. q^{97} -15254.4 q^{98} +53126.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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 (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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.65902 0.293276 0.146638 0.989190i \(-0.453155\pi\)
0.146638 + 0.989190i \(0.453155\pi\)
\(3\) −9.00000 −0.577350
\(4\) −29.2477 −0.913989
\(5\) 59.9319 1.07210 0.536048 0.844188i \(-0.319917\pi\)
0.536048 + 0.844188i \(0.319917\pi\)
\(6\) −14.9312 −0.169323
\(7\) −87.2478 −0.672991 −0.336495 0.941685i \(-0.609242\pi\)
−0.336495 + 0.941685i \(0.609242\pi\)
\(8\) −101.611 −0.561327
\(9\) 81.0000 0.333333
\(10\) 99.4282 0.314419
\(11\) 655.885 1.63435 0.817176 0.576388i \(-0.195538\pi\)
0.817176 + 0.576388i \(0.195538\pi\)
\(12\) 263.229 0.527692
\(13\) 139.966 0.229702 0.114851 0.993383i \(-0.463361\pi\)
0.114851 + 0.993383i \(0.463361\pi\)
\(14\) −144.746 −0.197372
\(15\) −539.387 −0.618974
\(16\) 767.351 0.749366
\(17\) −1518.98 −1.27477 −0.637383 0.770548i \(-0.719983\pi\)
−0.637383 + 0.770548i \(0.719983\pi\)
\(18\) 134.380 0.0977586
\(19\) 393.396 0.250003 0.125002 0.992157i \(-0.460106\pi\)
0.125002 + 0.992157i \(0.460106\pi\)
\(20\) −1752.87 −0.979883
\(21\) 785.230 0.388551
\(22\) 1088.12 0.479316
\(23\) −4746.68 −1.87098 −0.935492 0.353348i \(-0.885043\pi\)
−0.935492 + 0.353348i \(0.885043\pi\)
\(24\) 914.499 0.324082
\(25\) 466.837 0.149388
\(26\) 232.206 0.0673659
\(27\) −729.000 −0.192450
\(28\) 2551.79 0.615106
\(29\) 2791.29 0.616326 0.308163 0.951334i \(-0.400286\pi\)
0.308163 + 0.951334i \(0.400286\pi\)
\(30\) −894.853 −0.181530
\(31\) −1808.12 −0.337926 −0.168963 0.985622i \(-0.554042\pi\)
−0.168963 + 0.985622i \(0.554042\pi\)
\(32\) 4524.60 0.781097
\(33\) −5902.96 −0.943594
\(34\) −2520.02 −0.373858
\(35\) −5228.93 −0.721510
\(36\) −2369.06 −0.304663
\(37\) 6710.46 0.805838 0.402919 0.915236i \(-0.367996\pi\)
0.402919 + 0.915236i \(0.367996\pi\)
\(38\) 652.650 0.0733199
\(39\) −1259.69 −0.132618
\(40\) −6089.74 −0.601795
\(41\) −13926.6 −1.29385 −0.646925 0.762553i \(-0.723945\pi\)
−0.646925 + 0.762553i \(0.723945\pi\)
\(42\) 1302.71 0.113953
\(43\) −19381.1 −1.59848 −0.799239 0.601014i \(-0.794764\pi\)
−0.799239 + 0.601014i \(0.794764\pi\)
\(44\) −19183.1 −1.49378
\(45\) 4854.49 0.357365
\(46\) −7874.82 −0.548714
\(47\) −15752.6 −1.04018 −0.520088 0.854113i \(-0.674101\pi\)
−0.520088 + 0.854113i \(0.674101\pi\)
\(48\) −6906.16 −0.432647
\(49\) −9194.83 −0.547083
\(50\) 774.491 0.0438118
\(51\) 13670.8 0.735986
\(52\) −4093.68 −0.209945
\(53\) −25608.5 −1.25226 −0.626131 0.779718i \(-0.715363\pi\)
−0.626131 + 0.779718i \(0.715363\pi\)
\(54\) −1209.42 −0.0564409
\(55\) 39308.4 1.75218
\(56\) 8865.33 0.377768
\(57\) −3540.56 −0.144339
\(58\) 4630.80 0.180753
\(59\) −3481.00 −0.130189
\(60\) 15775.8 0.565736
\(61\) 11066.2 0.380778 0.190389 0.981709i \(-0.439025\pi\)
0.190389 + 0.981709i \(0.439025\pi\)
\(62\) −2999.70 −0.0991056
\(63\) −7067.07 −0.224330
\(64\) −17048.8 −0.520289
\(65\) 8388.43 0.246262
\(66\) −9793.12 −0.276733
\(67\) −34547.3 −0.940216 −0.470108 0.882609i \(-0.655785\pi\)
−0.470108 + 0.882609i \(0.655785\pi\)
\(68\) 44426.7 1.16512
\(69\) 42720.1 1.08021
\(70\) −8674.88 −0.211601
\(71\) 43063.2 1.01382 0.506910 0.861999i \(-0.330788\pi\)
0.506910 + 0.861999i \(0.330788\pi\)
\(72\) −8230.49 −0.187109
\(73\) −34956.0 −0.767741 −0.383871 0.923387i \(-0.625409\pi\)
−0.383871 + 0.923387i \(0.625409\pi\)
\(74\) 11132.8 0.236333
\(75\) −4201.53 −0.0862491
\(76\) −11505.9 −0.228500
\(77\) −57224.5 −1.09990
\(78\) −2089.85 −0.0388937
\(79\) 91122.1 1.64269 0.821345 0.570431i \(-0.193224\pi\)
0.821345 + 0.570431i \(0.193224\pi\)
\(80\) 45988.8 0.803391
\(81\) 6561.00 0.111111
\(82\) −23104.4 −0.379455
\(83\) −109543. −1.74538 −0.872692 0.488271i \(-0.837628\pi\)
−0.872692 + 0.488271i \(0.837628\pi\)
\(84\) −22966.1 −0.355132
\(85\) −91035.5 −1.36667
\(86\) −32153.5 −0.468795
\(87\) −25121.6 −0.355836
\(88\) −66645.1 −0.917406
\(89\) 136931. 1.83243 0.916215 0.400687i \(-0.131228\pi\)
0.916215 + 0.400687i \(0.131228\pi\)
\(90\) 8053.68 0.104806
\(91\) −12211.7 −0.154587
\(92\) 138829. 1.71006
\(93\) 16273.0 0.195102
\(94\) −26133.8 −0.305058
\(95\) 23577.0 0.268027
\(96\) −40721.4 −0.450967
\(97\) −139282. −1.50302 −0.751511 0.659721i \(-0.770675\pi\)
−0.751511 + 0.659721i \(0.770675\pi\)
\(98\) −15254.4 −0.160446
\(99\) 53126.7 0.544784
\(100\) −13653.9 −0.136539
\(101\) −79019.0 −0.770776 −0.385388 0.922755i \(-0.625932\pi\)
−0.385388 + 0.922755i \(0.625932\pi\)
\(102\) 22680.2 0.215847
\(103\) −8298.49 −0.0770736 −0.0385368 0.999257i \(-0.512270\pi\)
−0.0385368 + 0.999257i \(0.512270\pi\)
\(104\) −14222.1 −0.128938
\(105\) 47060.3 0.416564
\(106\) −42485.0 −0.367258
\(107\) −34725.6 −0.293218 −0.146609 0.989195i \(-0.546836\pi\)
−0.146609 + 0.989195i \(0.546836\pi\)
\(108\) 21321.5 0.175897
\(109\) 211277. 1.70328 0.851639 0.524130i \(-0.175609\pi\)
0.851639 + 0.524130i \(0.175609\pi\)
\(110\) 65213.4 0.513872
\(111\) −60394.1 −0.465251
\(112\) −66949.6 −0.504316
\(113\) 175697. 1.29440 0.647201 0.762319i \(-0.275939\pi\)
0.647201 + 0.762319i \(0.275939\pi\)
\(114\) −5873.85 −0.0423312
\(115\) −284478. −2.00587
\(116\) −81638.8 −0.563315
\(117\) 11337.2 0.0765672
\(118\) −5775.04 −0.0381812
\(119\) 132528. 0.857905
\(120\) 54807.7 0.347447
\(121\) 269134. 1.67111
\(122\) 18359.0 0.111673
\(123\) 125339. 0.747005
\(124\) 52883.2 0.308861
\(125\) −159309. −0.911937
\(126\) −11724.4 −0.0657906
\(127\) 156670. 0.861941 0.430970 0.902366i \(-0.358171\pi\)
0.430970 + 0.902366i \(0.358171\pi\)
\(128\) −173071. −0.933686
\(129\) 174430. 0.922881
\(130\) 13916.6 0.0722226
\(131\) 21735.5 0.110660 0.0553300 0.998468i \(-0.482379\pi\)
0.0553300 + 0.998468i \(0.482379\pi\)
\(132\) 172648. 0.862435
\(133\) −34322.9 −0.168250
\(134\) −57314.6 −0.275742
\(135\) −43690.4 −0.206325
\(136\) 154345. 0.715559
\(137\) 293416. 1.33562 0.667810 0.744332i \(-0.267232\pi\)
0.667810 + 0.744332i \(0.267232\pi\)
\(138\) 70873.4 0.316800
\(139\) −128147. −0.562564 −0.281282 0.959625i \(-0.590760\pi\)
−0.281282 + 0.959625i \(0.590760\pi\)
\(140\) 152934. 0.659453
\(141\) 141773. 0.600546
\(142\) 71442.7 0.297329
\(143\) 91801.5 0.375413
\(144\) 62155.4 0.249789
\(145\) 167288. 0.660760
\(146\) −57992.7 −0.225160
\(147\) 82753.5 0.315859
\(148\) −196265. −0.736528
\(149\) −217922. −0.804146 −0.402073 0.915608i \(-0.631710\pi\)
−0.402073 + 0.915608i \(0.631710\pi\)
\(150\) −6970.42 −0.0252948
\(151\) −394958. −1.40964 −0.704821 0.709385i \(-0.748973\pi\)
−0.704821 + 0.709385i \(0.748973\pi\)
\(152\) −39973.3 −0.140333
\(153\) −123037. −0.424922
\(154\) −94936.4 −0.322575
\(155\) −108364. −0.362289
\(156\) 36843.1 0.121212
\(157\) 20578.0 0.0666277 0.0333139 0.999445i \(-0.489394\pi\)
0.0333139 + 0.999445i \(0.489394\pi\)
\(158\) 151173. 0.481761
\(159\) 230477. 0.722994
\(160\) 271168. 0.837411
\(161\) 414137. 1.25916
\(162\) 10884.8 0.0325862
\(163\) −406284. −1.19773 −0.598867 0.800848i \(-0.704382\pi\)
−0.598867 + 0.800848i \(0.704382\pi\)
\(164\) 407319. 1.18257
\(165\) −353776. −1.01162
\(166\) −181734. −0.511879
\(167\) −439495. −1.21945 −0.609723 0.792614i \(-0.708719\pi\)
−0.609723 + 0.792614i \(0.708719\pi\)
\(168\) −79788.0 −0.218104
\(169\) −351703. −0.947237
\(170\) −151030. −0.400811
\(171\) 31865.0 0.0833344
\(172\) 566851. 1.46099
\(173\) 586270. 1.48930 0.744651 0.667454i \(-0.232616\pi\)
0.744651 + 0.667454i \(0.232616\pi\)
\(174\) −41677.2 −0.104358
\(175\) −40730.5 −0.100537
\(176\) 503294. 1.22473
\(177\) 31329.0 0.0751646
\(178\) 227171. 0.537407
\(179\) −548999. −1.28068 −0.640338 0.768093i \(-0.721206\pi\)
−0.640338 + 0.768093i \(0.721206\pi\)
\(180\) −141982. −0.326628
\(181\) −634515. −1.43961 −0.719806 0.694175i \(-0.755769\pi\)
−0.719806 + 0.694175i \(0.755769\pi\)
\(182\) −20259.5 −0.0453366
\(183\) −99595.4 −0.219842
\(184\) 482314. 1.05023
\(185\) 402171. 0.863935
\(186\) 26997.3 0.0572186
\(187\) −996277. −2.08342
\(188\) 460726. 0.950710
\(189\) 63603.6 0.129517
\(190\) 39114.6 0.0786059
\(191\) −408074. −0.809385 −0.404692 0.914453i \(-0.632621\pi\)
−0.404692 + 0.914453i \(0.632621\pi\)
\(192\) 153439. 0.300389
\(193\) −29069.5 −0.0561751 −0.0280875 0.999605i \(-0.508942\pi\)
−0.0280875 + 0.999605i \(0.508942\pi\)
\(194\) −231071. −0.440800
\(195\) −75495.9 −0.142179
\(196\) 268927. 0.500028
\(197\) 658204. 1.20836 0.604178 0.796849i \(-0.293502\pi\)
0.604178 + 0.796849i \(0.293502\pi\)
\(198\) 88138.1 0.159772
\(199\) 483961. 0.866319 0.433159 0.901317i \(-0.357399\pi\)
0.433159 + 0.901317i \(0.357399\pi\)
\(200\) −47435.7 −0.0838553
\(201\) 310926. 0.542834
\(202\) −131094. −0.226050
\(203\) −243534. −0.414782
\(204\) −399840. −0.672683
\(205\) −834645. −1.38713
\(206\) −13767.3 −0.0226038
\(207\) −384481. −0.623661
\(208\) 107403. 0.172131
\(209\) 258022. 0.408593
\(210\) 78073.9 0.122168
\(211\) 167978. 0.259745 0.129872 0.991531i \(-0.458543\pi\)
0.129872 + 0.991531i \(0.458543\pi\)
\(212\) 748990. 1.14455
\(213\) −387569. −0.585329
\(214\) −57610.4 −0.0859936
\(215\) −1.16154e6 −1.71372
\(216\) 74074.4 0.108027
\(217\) 157754. 0.227421
\(218\) 350512. 0.499530
\(219\) 314604. 0.443256
\(220\) −1.14968e6 −1.60148
\(221\) −212606. −0.292816
\(222\) −100195. −0.136447
\(223\) −391141. −0.526709 −0.263355 0.964699i \(-0.584829\pi\)
−0.263355 + 0.964699i \(0.584829\pi\)
\(224\) −394761. −0.525671
\(225\) 37813.8 0.0497959
\(226\) 291485. 0.379617
\(227\) 1.24360e6 1.60183 0.800913 0.598780i \(-0.204348\pi\)
0.800913 + 0.598780i \(0.204348\pi\)
\(228\) 103553. 0.131925
\(229\) 330176. 0.416061 0.208031 0.978122i \(-0.433295\pi\)
0.208031 + 0.978122i \(0.433295\pi\)
\(230\) −471953. −0.588274
\(231\) 515020. 0.635030
\(232\) −283626. −0.345960
\(233\) −857543. −1.03482 −0.517412 0.855737i \(-0.673104\pi\)
−0.517412 + 0.855737i \(0.673104\pi\)
\(234\) 18808.7 0.0224553
\(235\) −944082. −1.11517
\(236\) 101811. 0.118991
\(237\) −820099. −0.948408
\(238\) 219866. 0.251603
\(239\) −1.03288e6 −1.16965 −0.584825 0.811160i \(-0.698837\pi\)
−0.584825 + 0.811160i \(0.698837\pi\)
\(240\) −413899. −0.463838
\(241\) 136392. 0.151267 0.0756336 0.997136i \(-0.475902\pi\)
0.0756336 + 0.997136i \(0.475902\pi\)
\(242\) 446498. 0.490096
\(243\) −59049.0 −0.0641500
\(244\) −323659. −0.348027
\(245\) −551064. −0.586525
\(246\) 207940. 0.219078
\(247\) 55062.0 0.0574261
\(248\) 183724. 0.189687
\(249\) 985891. 1.00770
\(250\) −264296. −0.267449
\(251\) −494314. −0.495243 −0.247621 0.968857i \(-0.579649\pi\)
−0.247621 + 0.968857i \(0.579649\pi\)
\(252\) 206695. 0.205035
\(253\) −3.11327e6 −3.05785
\(254\) 259919. 0.252786
\(255\) 819319. 0.789047
\(256\) 258434. 0.246462
\(257\) 259654. 0.245223 0.122612 0.992455i \(-0.460873\pi\)
0.122612 + 0.992455i \(0.460873\pi\)
\(258\) 289382. 0.270659
\(259\) −585473. −0.542322
\(260\) −245342. −0.225081
\(261\) 226095. 0.205442
\(262\) 36059.6 0.0324539
\(263\) −696202. −0.620649 −0.310324 0.950631i \(-0.600438\pi\)
−0.310324 + 0.950631i \(0.600438\pi\)
\(264\) 599806. 0.529664
\(265\) −1.53477e6 −1.34254
\(266\) −56942.3 −0.0493436
\(267\) −1.23238e6 −1.05795
\(268\) 1.01043e6 0.859347
\(269\) 403193. 0.339729 0.169865 0.985467i \(-0.445667\pi\)
0.169865 + 0.985467i \(0.445667\pi\)
\(270\) −72483.1 −0.0605100
\(271\) −83968.2 −0.0694531 −0.0347265 0.999397i \(-0.511056\pi\)
−0.0347265 + 0.999397i \(0.511056\pi\)
\(272\) −1.16559e6 −0.955265
\(273\) 109905. 0.0892509
\(274\) 486783. 0.391705
\(275\) 306191. 0.244152
\(276\) −1.24946e6 −0.987303
\(277\) −679016. −0.531717 −0.265859 0.964012i \(-0.585655\pi\)
−0.265859 + 0.964012i \(0.585655\pi\)
\(278\) −212598. −0.164986
\(279\) −146457. −0.112642
\(280\) 531316. 0.405003
\(281\) 1.00480e6 0.759125 0.379563 0.925166i \(-0.376074\pi\)
0.379563 + 0.925166i \(0.376074\pi\)
\(282\) 235204. 0.176126
\(283\) 535938. 0.397785 0.198893 0.980021i \(-0.436265\pi\)
0.198893 + 0.980021i \(0.436265\pi\)
\(284\) −1.25950e6 −0.926621
\(285\) −212193. −0.154746
\(286\) 152300. 0.110100
\(287\) 1.21506e6 0.870749
\(288\) 366493. 0.260366
\(289\) 887447. 0.625026
\(290\) 277533. 0.193785
\(291\) 1.25354e6 0.867770
\(292\) 1.02238e6 0.701707
\(293\) 869160. 0.591467 0.295734 0.955270i \(-0.404436\pi\)
0.295734 + 0.955270i \(0.404436\pi\)
\(294\) 137289. 0.0926337
\(295\) −208623. −0.139575
\(296\) −681856. −0.452339
\(297\) −478140. −0.314531
\(298\) −361536. −0.235836
\(299\) −664373. −0.429768
\(300\) 122885. 0.0788307
\(301\) 1.69095e6 1.07576
\(302\) −655243. −0.413414
\(303\) 711171. 0.445008
\(304\) 301872. 0.187344
\(305\) 663216. 0.408231
\(306\) −204121. −0.124619
\(307\) 693445. 0.419919 0.209960 0.977710i \(-0.432667\pi\)
0.209960 + 0.977710i \(0.432667\pi\)
\(308\) 1.67368e6 1.00530
\(309\) 74686.4 0.0444985
\(310\) −179778. −0.106251
\(311\) 1.08592e6 0.636647 0.318324 0.947982i \(-0.396880\pi\)
0.318324 + 0.947982i \(0.396880\pi\)
\(312\) 127999. 0.0744422
\(313\) 2.46914e6 1.42458 0.712288 0.701888i \(-0.247659\pi\)
0.712288 + 0.701888i \(0.247659\pi\)
\(314\) 34139.3 0.0195403
\(315\) −423543. −0.240503
\(316\) −2.66511e6 −1.50140
\(317\) 2.16299e6 1.20894 0.604471 0.796627i \(-0.293385\pi\)
0.604471 + 0.796627i \(0.293385\pi\)
\(318\) 382365. 0.212037
\(319\) 1.83077e6 1.00729
\(320\) −1.02177e6 −0.557799
\(321\) 312530. 0.169289
\(322\) 687061. 0.369280
\(323\) −597561. −0.318695
\(324\) −191894. −0.101554
\(325\) 65341.2 0.0343146
\(326\) −674032. −0.351266
\(327\) −1.90149e6 −0.983387
\(328\) 1.41509e6 0.726273
\(329\) 1.37438e6 0.700029
\(330\) −586921. −0.296684
\(331\) −1.30386e6 −0.654128 −0.327064 0.945002i \(-0.606059\pi\)
−0.327064 + 0.945002i \(0.606059\pi\)
\(332\) 3.20389e6 1.59526
\(333\) 543547. 0.268613
\(334\) −729130. −0.357634
\(335\) −2.07049e6 −1.00800
\(336\) 602547. 0.291167
\(337\) 3.43860e6 1.64933 0.824664 0.565623i \(-0.191364\pi\)
0.824664 + 0.565623i \(0.191364\pi\)
\(338\) −583481. −0.277802
\(339\) −1.58128e6 −0.747323
\(340\) 2.66258e6 1.24912
\(341\) −1.18592e6 −0.552291
\(342\) 52864.7 0.0244400
\(343\) 2.26860e6 1.04117
\(344\) 1.96933e6 0.897268
\(345\) 2.56030e6 1.15809
\(346\) 972633. 0.436776
\(347\) −337924. −0.150659 −0.0753296 0.997159i \(-0.524001\pi\)
−0.0753296 + 0.997159i \(0.524001\pi\)
\(348\) 734749. 0.325230
\(349\) −1.09909e6 −0.483025 −0.241513 0.970398i \(-0.577644\pi\)
−0.241513 + 0.970398i \(0.577644\pi\)
\(350\) −67572.6 −0.0294849
\(351\) −102035. −0.0442061
\(352\) 2.96762e6 1.27659
\(353\) −2.32702e6 −0.993945 −0.496973 0.867766i \(-0.665555\pi\)
−0.496973 + 0.867766i \(0.665555\pi\)
\(354\) 51975.4 0.0220440
\(355\) 2.58086e6 1.08691
\(356\) −4.00492e6 −1.67482
\(357\) −1.19275e6 −0.495312
\(358\) −910800. −0.375591
\(359\) −2.78944e6 −1.14230 −0.571151 0.820845i \(-0.693503\pi\)
−0.571151 + 0.820845i \(0.693503\pi\)
\(360\) −493269. −0.200598
\(361\) −2.32134e6 −0.937498
\(362\) −1.05267e6 −0.422203
\(363\) −2.42220e6 −0.964816
\(364\) 357164. 0.141291
\(365\) −2.09498e6 −0.823091
\(366\) −165231. −0.0644745
\(367\) −1.60366e6 −0.621510 −0.310755 0.950490i \(-0.600582\pi\)
−0.310755 + 0.950490i \(0.600582\pi\)
\(368\) −3.64237e6 −1.40205
\(369\) −1.12805e6 −0.431283
\(370\) 667209. 0.253371
\(371\) 2.23429e6 0.842761
\(372\) −475948. −0.178321
\(373\) −1.49956e6 −0.558075 −0.279037 0.960280i \(-0.590015\pi\)
−0.279037 + 0.960280i \(0.590015\pi\)
\(374\) −1.65284e6 −0.611015
\(375\) 1.43378e6 0.526507
\(376\) 1.60063e6 0.583879
\(377\) 390686. 0.141571
\(378\) 105520. 0.0379842
\(379\) 2.95105e6 1.05531 0.527653 0.849460i \(-0.323072\pi\)
0.527653 + 0.849460i \(0.323072\pi\)
\(380\) −689571. −0.244974
\(381\) −1.41003e6 −0.497642
\(382\) −677001. −0.237373
\(383\) 1.98411e6 0.691145 0.345572 0.938392i \(-0.387685\pi\)
0.345572 + 0.938392i \(0.387685\pi\)
\(384\) 1.55764e6 0.539064
\(385\) −3.42957e6 −1.17920
\(386\) −48226.8 −0.0164748
\(387\) −1.56987e6 −0.532826
\(388\) 4.07367e6 1.37375
\(389\) 1.68710e6 0.565284 0.282642 0.959225i \(-0.408789\pi\)
0.282642 + 0.959225i \(0.408789\pi\)
\(390\) −125249. −0.0416978
\(391\) 7.21011e6 2.38507
\(392\) 934295. 0.307092
\(393\) −195619. −0.0638896
\(394\) 1.09197e6 0.354381
\(395\) 5.46112e6 1.76112
\(396\) −1.55383e6 −0.497927
\(397\) 233554. 0.0743722 0.0371861 0.999308i \(-0.488161\pi\)
0.0371861 + 0.999308i \(0.488161\pi\)
\(398\) 802900. 0.254070
\(399\) 308906. 0.0971391
\(400\) 358228. 0.111946
\(401\) −4.29311e6 −1.33325 −0.666625 0.745394i \(-0.732262\pi\)
−0.666625 + 0.745394i \(0.732262\pi\)
\(402\) 515832. 0.159200
\(403\) −253075. −0.0776222
\(404\) 2.31112e6 0.704481
\(405\) 393213. 0.119122
\(406\) −404027. −0.121645
\(407\) 4.40129e6 1.31702
\(408\) −1.38911e6 −0.413128
\(409\) 1.98371e6 0.586368 0.293184 0.956056i \(-0.405285\pi\)
0.293184 + 0.956056i \(0.405285\pi\)
\(410\) −1.38469e6 −0.406812
\(411\) −2.64075e6 −0.771120
\(412\) 242711. 0.0704445
\(413\) 303709. 0.0876159
\(414\) −637861. −0.182905
\(415\) −6.56515e6 −1.87122
\(416\) 633290. 0.179419
\(417\) 1.15332e6 0.324797
\(418\) 428063. 0.119831
\(419\) −76952.5 −0.0214135 −0.0107068 0.999943i \(-0.503408\pi\)
−0.0107068 + 0.999943i \(0.503408\pi\)
\(420\) −1.37640e6 −0.380735
\(421\) 1.97318e6 0.542577 0.271288 0.962498i \(-0.412550\pi\)
0.271288 + 0.962498i \(0.412550\pi\)
\(422\) 278679. 0.0761769
\(423\) −1.27596e6 −0.346725
\(424\) 2.60211e6 0.702928
\(425\) −709116. −0.190434
\(426\) −642984. −0.171663
\(427\) −965497. −0.256260
\(428\) 1.01564e6 0.267998
\(429\) −826214. −0.216745
\(430\) −1.92702e6 −0.502592
\(431\) 2.53130e6 0.656372 0.328186 0.944613i \(-0.393563\pi\)
0.328186 + 0.944613i \(0.393563\pi\)
\(432\) −559399. −0.144216
\(433\) 896878. 0.229887 0.114943 0.993372i \(-0.463331\pi\)
0.114943 + 0.993372i \(0.463331\pi\)
\(434\) 261717. 0.0666972
\(435\) −1.50559e6 −0.381490
\(436\) −6.17935e6 −1.55678
\(437\) −1.86732e6 −0.467752
\(438\) 521934. 0.129996
\(439\) −5.15275e6 −1.27608 −0.638040 0.770003i \(-0.720254\pi\)
−0.638040 + 0.770003i \(0.720254\pi\)
\(440\) −3.99417e6 −0.983546
\(441\) −744781. −0.182361
\(442\) −352717. −0.0858757
\(443\) 1.77270e6 0.429166 0.214583 0.976706i \(-0.431161\pi\)
0.214583 + 0.976706i \(0.431161\pi\)
\(444\) 1.76639e6 0.425235
\(445\) 8.20655e6 1.96454
\(446\) −648910. −0.154471
\(447\) 1.96130e6 0.464274
\(448\) 1.48747e6 0.350150
\(449\) 652208. 0.152676 0.0763379 0.997082i \(-0.475677\pi\)
0.0763379 + 0.997082i \(0.475677\pi\)
\(450\) 62733.7 0.0146039
\(451\) −9.13422e6 −2.11461
\(452\) −5.13874e6 −1.18307
\(453\) 3.55462e6 0.813857
\(454\) 2.06315e6 0.469777
\(455\) −731872. −0.165732
\(456\) 359760. 0.0810215
\(457\) 6.01636e6 1.34755 0.673773 0.738939i \(-0.264673\pi\)
0.673773 + 0.738939i \(0.264673\pi\)
\(458\) 547769. 0.122021
\(459\) 1.10734e6 0.245329
\(460\) 8.32030e6 1.83335
\(461\) 8.80449e6 1.92953 0.964765 0.263111i \(-0.0847487\pi\)
0.964765 + 0.263111i \(0.0847487\pi\)
\(462\) 854428. 0.186239
\(463\) −6.15435e6 −1.33423 −0.667113 0.744956i \(-0.732470\pi\)
−0.667113 + 0.744956i \(0.732470\pi\)
\(464\) 2.14190e6 0.461853
\(465\) 975275. 0.209168
\(466\) −1.42268e6 −0.303489
\(467\) 5.99157e6 1.27130 0.635651 0.771977i \(-0.280732\pi\)
0.635651 + 0.771977i \(0.280732\pi\)
\(468\) −331588. −0.0699816
\(469\) 3.01418e6 0.632756
\(470\) −1.56625e6 −0.327052
\(471\) −185202. −0.0384675
\(472\) 353708. 0.0730785
\(473\) −1.27117e7 −2.61248
\(474\) −1.36056e6 −0.278145
\(475\) 183652. 0.0373474
\(476\) −3.87613e6 −0.784116
\(477\) −2.07429e6 −0.417421
\(478\) −1.71357e6 −0.343030
\(479\) −2.22116e6 −0.442325 −0.221163 0.975237i \(-0.570985\pi\)
−0.221163 + 0.975237i \(0.570985\pi\)
\(480\) −2.44051e6 −0.483479
\(481\) 939236. 0.185102
\(482\) 226276. 0.0443630
\(483\) −3.72723e6 −0.726974
\(484\) −7.87154e6 −1.52738
\(485\) −8.34743e6 −1.61138
\(486\) −97963.3 −0.0188136
\(487\) 8.07055e6 1.54199 0.770993 0.636844i \(-0.219760\pi\)
0.770993 + 0.636844i \(0.219760\pi\)
\(488\) −1.12444e6 −0.213741
\(489\) 3.65655e6 0.691512
\(490\) −914225. −0.172014
\(491\) −4.90913e6 −0.918968 −0.459484 0.888186i \(-0.651966\pi\)
−0.459484 + 0.888186i \(0.651966\pi\)
\(492\) −3.66587e6 −0.682754
\(493\) −4.23992e6 −0.785670
\(494\) 91348.8 0.0168417
\(495\) 3.18398e6 0.584061
\(496\) −1.38746e6 −0.253230
\(497\) −3.75717e6 −0.682292
\(498\) 1.63561e6 0.295533
\(499\) −6.59947e6 −1.18647 −0.593236 0.805029i \(-0.702150\pi\)
−0.593236 + 0.805029i \(0.702150\pi\)
\(500\) 4.65941e6 0.833501
\(501\) 3.95546e6 0.704048
\(502\) −820075. −0.145243
\(503\) 1.46824e6 0.258749 0.129374 0.991596i \(-0.458703\pi\)
0.129374 + 0.991596i \(0.458703\pi\)
\(504\) 718092. 0.125923
\(505\) −4.73576e6 −0.826345
\(506\) −5.16498e6 −0.896793
\(507\) 3.16532e6 0.546888
\(508\) −4.58224e6 −0.787805
\(509\) −6.43314e6 −1.10060 −0.550299 0.834968i \(-0.685486\pi\)
−0.550299 + 0.834968i \(0.685486\pi\)
\(510\) 1.35927e6 0.231408
\(511\) 3.04983e6 0.516683
\(512\) 5.96703e6 1.00597
\(513\) −286785. −0.0481131
\(514\) 430770. 0.0719181
\(515\) −497344. −0.0826302
\(516\) −5.10166e6 −0.843504
\(517\) −1.03319e7 −1.70002
\(518\) −971309. −0.159050
\(519\) −5.27643e6 −0.859849
\(520\) −852356. −0.138233
\(521\) −508460. −0.0820658 −0.0410329 0.999158i \(-0.513065\pi\)
−0.0410329 + 0.999158i \(0.513065\pi\)
\(522\) 375095. 0.0602511
\(523\) 1.57290e6 0.251447 0.125723 0.992065i \(-0.459875\pi\)
0.125723 + 0.992065i \(0.459875\pi\)
\(524\) −635712. −0.101142
\(525\) 366574. 0.0580448
\(526\) −1.15501e6 −0.182021
\(527\) 2.74649e6 0.430777
\(528\) −4.52964e6 −0.707097
\(529\) 1.60946e7 2.50058
\(530\) −2.54621e6 −0.393736
\(531\) −281961. −0.0433963
\(532\) 1.00386e6 0.153779
\(533\) −1.94924e6 −0.297199
\(534\) −2.04454e6 −0.310272
\(535\) −2.08117e6 −0.314357
\(536\) 3.51039e6 0.527768
\(537\) 4.94099e6 0.739399
\(538\) 668905. 0.0996343
\(539\) −6.03075e6 −0.894127
\(540\) 1.27784e6 0.188579
\(541\) −2.60509e6 −0.382675 −0.191337 0.981524i \(-0.561282\pi\)
−0.191337 + 0.981524i \(0.561282\pi\)
\(542\) −139305. −0.0203689
\(543\) 5.71064e6 0.831161
\(544\) −6.87278e6 −0.995716
\(545\) 1.26622e7 1.82607
\(546\) 182335. 0.0261751
\(547\) 6.69896e6 0.957281 0.478640 0.878011i \(-0.341130\pi\)
0.478640 + 0.878011i \(0.341130\pi\)
\(548\) −8.58174e6 −1.22074
\(549\) 896359. 0.126926
\(550\) 507977. 0.0716039
\(551\) 1.09808e6 0.154083
\(552\) −4.34083e6 −0.606352
\(553\) −7.95020e6 −1.10552
\(554\) −1.12650e6 −0.155940
\(555\) −3.61954e6 −0.498793
\(556\) 3.74800e6 0.514178
\(557\) 7.81333e6 1.06708 0.533541 0.845774i \(-0.320861\pi\)
0.533541 + 0.845774i \(0.320861\pi\)
\(558\) −242975. −0.0330352
\(559\) −2.71269e6 −0.367173
\(560\) −4.01242e6 −0.540675
\(561\) 8.96649e6 1.20286
\(562\) 1.66698e6 0.222633
\(563\) −4.79892e6 −0.638076 −0.319038 0.947742i \(-0.603360\pi\)
−0.319038 + 0.947742i \(0.603360\pi\)
\(564\) −4.14653e6 −0.548893
\(565\) 1.05299e7 1.38772
\(566\) 889131. 0.116661
\(567\) −572433. −0.0747768
\(568\) −4.37570e6 −0.569084
\(569\) 1.34812e7 1.74561 0.872807 0.488065i \(-0.162297\pi\)
0.872807 + 0.488065i \(0.162297\pi\)
\(570\) −352031. −0.0453831
\(571\) −2.16268e6 −0.277589 −0.138795 0.990321i \(-0.544323\pi\)
−0.138795 + 0.990321i \(0.544323\pi\)
\(572\) −2.68498e6 −0.343124
\(573\) 3.67266e6 0.467299
\(574\) 2.01581e6 0.255370
\(575\) −2.21592e6 −0.279502
\(576\) −1.38096e6 −0.173430
\(577\) 1.07356e7 1.34241 0.671207 0.741270i \(-0.265776\pi\)
0.671207 + 0.741270i \(0.265776\pi\)
\(578\) 1.47229e6 0.183305
\(579\) 261625. 0.0324327
\(580\) −4.89277e6 −0.603927
\(581\) 9.55742e6 1.17463
\(582\) 2.07964e6 0.254496
\(583\) −1.67963e7 −2.04664
\(584\) 3.55191e6 0.430954
\(585\) 679463. 0.0820873
\(586\) 1.44195e6 0.173463
\(587\) 1.12328e7 1.34553 0.672766 0.739855i \(-0.265106\pi\)
0.672766 + 0.739855i \(0.265106\pi\)
\(588\) −2.42035e6 −0.288691
\(589\) −711305. −0.0844827
\(590\) −346109. −0.0409339
\(591\) −5.92384e6 −0.697645
\(592\) 5.14928e6 0.603868
\(593\) 3.19409e6 0.373001 0.186501 0.982455i \(-0.440285\pi\)
0.186501 + 0.982455i \(0.440285\pi\)
\(594\) −793243. −0.0922444
\(595\) 7.94264e6 0.919756
\(596\) 6.37370e6 0.734981
\(597\) −4.35565e6 −0.500169
\(598\) −1.10221e6 −0.126041
\(599\) −2.10973e6 −0.240248 −0.120124 0.992759i \(-0.538329\pi\)
−0.120124 + 0.992759i \(0.538329\pi\)
\(600\) 426922. 0.0484139
\(601\) −1.29771e6 −0.146552 −0.0732758 0.997312i \(-0.523345\pi\)
−0.0732758 + 0.997312i \(0.523345\pi\)
\(602\) 2.80532e6 0.315494
\(603\) −2.79833e6 −0.313405
\(604\) 1.15516e7 1.28840
\(605\) 1.61297e7 1.79159
\(606\) 1.17985e6 0.130510
\(607\) −1.34730e7 −1.48420 −0.742101 0.670288i \(-0.766171\pi\)
−0.742101 + 0.670288i \(0.766171\pi\)
\(608\) 1.77996e6 0.195277
\(609\) 2.19181e6 0.239474
\(610\) 1.10029e6 0.119724
\(611\) −2.20482e6 −0.238930
\(612\) 3.59856e6 0.388374
\(613\) −4.89540e6 −0.526183 −0.263092 0.964771i \(-0.584742\pi\)
−0.263092 + 0.964771i \(0.584742\pi\)
\(614\) 1.15044e6 0.123152
\(615\) 7.51181e6 0.800860
\(616\) 5.81463e6 0.617406
\(617\) −1.41804e7 −1.49960 −0.749798 0.661666i \(-0.769849\pi\)
−0.749798 + 0.661666i \(0.769849\pi\)
\(618\) 123906. 0.0130503
\(619\) 1.40027e7 1.46887 0.734436 0.678678i \(-0.237447\pi\)
0.734436 + 0.678678i \(0.237447\pi\)
\(620\) 3.16939e6 0.331128
\(621\) 3.46033e6 0.360071
\(622\) 1.80157e6 0.186713
\(623\) −1.19469e7 −1.23321
\(624\) −966627. −0.0993796
\(625\) −1.10066e7 −1.12707
\(626\) 4.09635e6 0.417793
\(627\) −2.32220e6 −0.235902
\(628\) −601860. −0.0608970
\(629\) −1.01931e7 −1.02725
\(630\) −702666. −0.0705338
\(631\) −131565. −0.0131543 −0.00657714 0.999978i \(-0.502094\pi\)
−0.00657714 + 0.999978i \(0.502094\pi\)
\(632\) −9.25900e6 −0.922086
\(633\) −1.51180e6 −0.149964
\(634\) 3.58843e6 0.354553
\(635\) 9.38956e6 0.924083
\(636\) −6.74091e6 −0.660809
\(637\) −1.28696e6 −0.125666
\(638\) 3.03727e6 0.295415
\(639\) 3.48812e6 0.337940
\(640\) −1.03725e7 −1.00100
\(641\) 1.53623e7 1.47676 0.738382 0.674383i \(-0.235590\pi\)
0.738382 + 0.674383i \(0.235590\pi\)
\(642\) 518493. 0.0496484
\(643\) −1.11320e7 −1.06181 −0.530904 0.847432i \(-0.678148\pi\)
−0.530904 + 0.847432i \(0.678148\pi\)
\(644\) −1.21125e7 −1.15085
\(645\) 1.04539e7 0.989417
\(646\) −991364. −0.0934656
\(647\) 584009. 0.0548477 0.0274239 0.999624i \(-0.491270\pi\)
0.0274239 + 0.999624i \(0.491270\pi\)
\(648\) −666670. −0.0623696
\(649\) −2.28313e6 −0.212775
\(650\) 108402. 0.0100636
\(651\) −1.41979e6 −0.131302
\(652\) 1.18829e7 1.09472
\(653\) 1.13214e7 1.03900 0.519502 0.854469i \(-0.326117\pi\)
0.519502 + 0.854469i \(0.326117\pi\)
\(654\) −3.15461e6 −0.288404
\(655\) 1.30265e6 0.118638
\(656\) −1.06866e7 −0.969567
\(657\) −2.83144e6 −0.255914
\(658\) 2.28012e6 0.205302
\(659\) −4.47059e6 −0.401007 −0.200503 0.979693i \(-0.564258\pi\)
−0.200503 + 0.979693i \(0.564258\pi\)
\(660\) 1.03471e7 0.924612
\(661\) −7.44213e6 −0.662512 −0.331256 0.943541i \(-0.607472\pi\)
−0.331256 + 0.943541i \(0.607472\pi\)
\(662\) −2.16314e6 −0.191840
\(663\) 1.91345e6 0.169057
\(664\) 1.11308e7 0.979730
\(665\) −2.05704e6 −0.180380
\(666\) 901755. 0.0787776
\(667\) −1.32494e7 −1.15314
\(668\) 1.28542e7 1.11456
\(669\) 3.52027e6 0.304096
\(670\) −3.43498e6 −0.295622
\(671\) 7.25812e6 0.622326
\(672\) 3.55285e6 0.303497
\(673\) −5.93148e6 −0.504807 −0.252404 0.967622i \(-0.581221\pi\)
−0.252404 + 0.967622i \(0.581221\pi\)
\(674\) 5.70470e6 0.483708
\(675\) −340324. −0.0287497
\(676\) 1.02865e7 0.865765
\(677\) −4.33126e6 −0.363197 −0.181599 0.983373i \(-0.558127\pi\)
−0.181599 + 0.983373i \(0.558127\pi\)
\(678\) −2.62337e6 −0.219172
\(679\) 1.21520e7 1.01152
\(680\) 9.25020e6 0.767148
\(681\) −1.11924e7 −0.924815
\(682\) −1.96746e6 −0.161974
\(683\) 2.27665e7 1.86743 0.933715 0.358016i \(-0.116547\pi\)
0.933715 + 0.358016i \(0.116547\pi\)
\(684\) −931978. −0.0761667
\(685\) 1.75850e7 1.43191
\(686\) 3.76365e6 0.305351
\(687\) −2.97159e6 −0.240213
\(688\) −1.48721e7 −1.19784
\(689\) −3.58432e6 −0.287647
\(690\) 4.24758e6 0.339640
\(691\) −2.18683e7 −1.74229 −0.871145 0.491026i \(-0.836622\pi\)
−0.871145 + 0.491026i \(0.836622\pi\)
\(692\) −1.71470e7 −1.36121
\(693\) −4.63518e6 −0.366635
\(694\) −560623. −0.0441847
\(695\) −7.68011e6 −0.603122
\(696\) 2.55263e6 0.199740
\(697\) 2.11542e7 1.64936
\(698\) −1.82341e6 −0.141660
\(699\) 7.71789e6 0.597456
\(700\) 1.19127e6 0.0918894
\(701\) −4.41044e6 −0.338990 −0.169495 0.985531i \(-0.554214\pi\)
−0.169495 + 0.985531i \(0.554214\pi\)
\(702\) −169278. −0.0129646
\(703\) 2.63987e6 0.201462
\(704\) −1.11821e7 −0.850336
\(705\) 8.49674e6 0.643842
\(706\) −3.86056e6 −0.291500
\(707\) 6.89423e6 0.518725
\(708\) −916300. −0.0686996
\(709\) −4.71774e6 −0.352467 −0.176233 0.984348i \(-0.556391\pi\)
−0.176233 + 0.984348i \(0.556391\pi\)
\(710\) 4.28170e6 0.318765
\(711\) 7.38089e6 0.547564
\(712\) −1.39137e7 −1.02859
\(713\) 8.58254e6 0.632255
\(714\) −1.97879e6 −0.145263
\(715\) 5.50184e6 0.402479
\(716\) 1.60569e7 1.17052
\(717\) 9.29594e6 0.675298
\(718\) −4.62773e6 −0.335009
\(719\) 3.92428e6 0.283098 0.141549 0.989931i \(-0.454792\pi\)
0.141549 + 0.989931i \(0.454792\pi\)
\(720\) 3.72509e6 0.267797
\(721\) 724024. 0.0518698
\(722\) −3.85114e6 −0.274946
\(723\) −1.22752e6 −0.0873342
\(724\) 1.85581e7 1.31579
\(725\) 1.30308e6 0.0920715
\(726\) −4.01848e6 −0.282957
\(727\) −1.54526e7 −1.08434 −0.542170 0.840269i \(-0.682397\pi\)
−0.542170 + 0.840269i \(0.682397\pi\)
\(728\) 1.24084e6 0.0867738
\(729\) 531441. 0.0370370
\(730\) −3.47561e6 −0.241393
\(731\) 2.94395e7 2.03768
\(732\) 2.91293e6 0.200934
\(733\) −2.46334e7 −1.69342 −0.846711 0.532053i \(-0.821421\pi\)
−0.846711 + 0.532053i \(0.821421\pi\)
\(734\) −2.66051e6 −0.182274
\(735\) 4.95958e6 0.338631
\(736\) −2.14768e7 −1.46142
\(737\) −2.26591e7 −1.53664
\(738\) −1.87146e6 −0.126485
\(739\) −1.82294e7 −1.22789 −0.613946 0.789348i \(-0.710419\pi\)
−0.613946 + 0.789348i \(0.710419\pi\)
\(740\) −1.17626e7 −0.789628
\(741\) −495558. −0.0331550
\(742\) 3.70672e6 0.247161
\(743\) 2.17675e7 1.44656 0.723281 0.690553i \(-0.242633\pi\)
0.723281 + 0.690553i \(0.242633\pi\)
\(744\) −1.65352e6 −0.109516
\(745\) −1.30605e7 −0.862121
\(746\) −2.48780e6 −0.163670
\(747\) −8.87302e6 −0.581795
\(748\) 2.91388e7 1.90422
\(749\) 3.02973e6 0.197333
\(750\) 2.37867e6 0.154412
\(751\) 2.75126e7 1.78005 0.890024 0.455915i \(-0.150688\pi\)
0.890024 + 0.455915i \(0.150688\pi\)
\(752\) −1.20877e7 −0.779473
\(753\) 4.44882e6 0.285929
\(754\) 648155. 0.0415193
\(755\) −2.36706e7 −1.51127
\(756\) −1.86026e6 −0.118377
\(757\) −1.19675e7 −0.759039 −0.379519 0.925184i \(-0.623911\pi\)
−0.379519 + 0.925184i \(0.623911\pi\)
\(758\) 4.89585e6 0.309496
\(759\) 2.80195e7 1.76545
\(760\) −2.39568e6 −0.150451
\(761\) −1.99524e7 −1.24892 −0.624460 0.781057i \(-0.714681\pi\)
−0.624460 + 0.781057i \(0.714681\pi\)
\(762\) −2.33927e6 −0.145946
\(763\) −1.84334e7 −1.14629
\(764\) 1.19352e7 0.739769
\(765\) −7.37387e6 −0.455556
\(766\) 3.29168e6 0.202696
\(767\) −487221. −0.0299046
\(768\) −2.32590e6 −0.142295
\(769\) −2.53565e7 −1.54623 −0.773113 0.634268i \(-0.781302\pi\)
−0.773113 + 0.634268i \(0.781302\pi\)
\(770\) −5.68972e6 −0.345831
\(771\) −2.33688e6 −0.141580
\(772\) 850214. 0.0513434
\(773\) 1.62164e7 0.976123 0.488062 0.872809i \(-0.337704\pi\)
0.488062 + 0.872809i \(0.337704\pi\)
\(774\) −2.60444e6 −0.156265
\(775\) −844095. −0.0504821
\(776\) 1.41526e7 0.843686
\(777\) 5.26925e6 0.313110
\(778\) 2.79893e6 0.165784
\(779\) −5.47865e6 −0.323467
\(780\) 2.20808e6 0.129950
\(781\) 2.82445e7 1.65694
\(782\) 1.19617e7 0.699482
\(783\) −2.03485e6 −0.118612
\(784\) −7.05566e6 −0.409966
\(785\) 1.23328e6 0.0714312
\(786\) −324536. −0.0187373
\(787\) −1.75632e7 −1.01080 −0.505401 0.862885i \(-0.668655\pi\)
−0.505401 + 0.862885i \(0.668655\pi\)
\(788\) −1.92509e7 −1.10442
\(789\) 6.26582e6 0.358332
\(790\) 9.06010e6 0.516494
\(791\) −1.53292e7 −0.871121
\(792\) −5.39825e6 −0.305802
\(793\) 1.54888e6 0.0874654
\(794\) 387470. 0.0218115
\(795\) 1.38129e7 0.775118
\(796\) −1.41547e7 −0.791806
\(797\) 2.00493e7 1.11803 0.559015 0.829158i \(-0.311179\pi\)
0.559015 + 0.829158i \(0.311179\pi\)
\(798\) 512481. 0.0284885
\(799\) 2.39279e7 1.32598
\(800\) 2.11225e6 0.116686
\(801\) 1.10914e7 0.610810
\(802\) −7.12235e6 −0.391010
\(803\) −2.29271e7 −1.25476
\(804\) −9.09386e6 −0.496144
\(805\) 2.48200e7 1.34993
\(806\) −419855. −0.0227647
\(807\) −3.62874e6 −0.196143
\(808\) 8.02920e6 0.432657
\(809\) −6.09192e6 −0.327253 −0.163626 0.986522i \(-0.552319\pi\)
−0.163626 + 0.986522i \(0.552319\pi\)
\(810\) 652348. 0.0349355
\(811\) 3.13530e6 0.167389 0.0836945 0.996491i \(-0.473328\pi\)
0.0836945 + 0.996491i \(0.473328\pi\)
\(812\) 7.12280e6 0.379106
\(813\) 755713. 0.0400987
\(814\) 7.30182e6 0.386251
\(815\) −2.43494e7 −1.28409
\(816\) 1.04903e7 0.551523
\(817\) −7.62443e6 −0.399624
\(818\) 3.29101e6 0.171967
\(819\) −989149. −0.0515290
\(820\) 2.44114e7 1.26782
\(821\) −1.57429e7 −0.815128 −0.407564 0.913177i \(-0.633622\pi\)
−0.407564 + 0.913177i \(0.633622\pi\)
\(822\) −4.38105e6 −0.226151
\(823\) 1.59042e7 0.818486 0.409243 0.912425i \(-0.365793\pi\)
0.409243 + 0.912425i \(0.365793\pi\)
\(824\) 843217. 0.0432635
\(825\) −2.75572e6 −0.140961
\(826\) 503859. 0.0256956
\(827\) −1.65484e7 −0.841378 −0.420689 0.907205i \(-0.638212\pi\)
−0.420689 + 0.907205i \(0.638212\pi\)
\(828\) 1.12452e7 0.570020
\(829\) −1.93567e7 −0.978240 −0.489120 0.872217i \(-0.662682\pi\)
−0.489120 + 0.872217i \(0.662682\pi\)
\(830\) −1.08917e7 −0.548783
\(831\) 6.11115e6 0.306987
\(832\) −2.38626e6 −0.119511
\(833\) 1.39668e7 0.697403
\(834\) 1.91339e6 0.0952549
\(835\) −2.63398e7 −1.30736
\(836\) −7.54655e6 −0.373450
\(837\) 1.31812e6 0.0650340
\(838\) −127666. −0.00628006
\(839\) 1.52692e7 0.748878 0.374439 0.927252i \(-0.377835\pi\)
0.374439 + 0.927252i \(0.377835\pi\)
\(840\) −4.78185e6 −0.233828
\(841\) −1.27198e7 −0.620143
\(842\) 3.27354e6 0.159125
\(843\) −9.04319e6 −0.438281
\(844\) −4.91297e6 −0.237404
\(845\) −2.10782e7 −1.01553
\(846\) −2.11684e6 −0.101686
\(847\) −2.34813e7 −1.12464
\(848\) −1.96507e7 −0.938403
\(849\) −4.82345e6 −0.229661
\(850\) −1.17644e6 −0.0558498
\(851\) −3.18524e7 −1.50771
\(852\) 1.13355e7 0.534985
\(853\) −1.98871e7 −0.935836 −0.467918 0.883772i \(-0.654996\pi\)
−0.467918 + 0.883772i \(0.654996\pi\)
\(854\) −1.60178e6 −0.0751549
\(855\) 1.90973e6 0.0893424
\(856\) 3.52850e6 0.164591
\(857\) 3.45630e6 0.160753 0.0803765 0.996765i \(-0.474388\pi\)
0.0803765 + 0.996765i \(0.474388\pi\)
\(858\) −1.37070e6 −0.0635661
\(859\) −3.93767e7 −1.82077 −0.910387 0.413758i \(-0.864216\pi\)
−0.910387 + 0.413758i \(0.864216\pi\)
\(860\) 3.39725e7 1.56632
\(861\) −1.09355e7 −0.502727
\(862\) 4.19947e6 0.192498
\(863\) 9.80271e6 0.448043 0.224021 0.974584i \(-0.428081\pi\)
0.224021 + 0.974584i \(0.428081\pi\)
\(864\) −3.29843e6 −0.150322
\(865\) 3.51363e7 1.59667
\(866\) 1.48794e6 0.0674202
\(867\) −7.98703e6 −0.360859
\(868\) −4.61394e6 −0.207861
\(869\) 5.97656e7 2.68474
\(870\) −2.49780e6 −0.111882
\(871\) −4.83545e6 −0.215969
\(872\) −2.14680e7 −0.956095
\(873\) −1.12818e7 −0.501007
\(874\) −3.09792e6 −0.137180
\(875\) 1.38993e7 0.613725
\(876\) −9.20144e6 −0.405131
\(877\) 8.26623e6 0.362918 0.181459 0.983399i \(-0.441918\pi\)
0.181459 + 0.983399i \(0.441918\pi\)
\(878\) −8.54850e6 −0.374243
\(879\) −7.82244e6 −0.341484
\(880\) 3.01634e7 1.31303
\(881\) 1.37572e7 0.597160 0.298580 0.954385i \(-0.403487\pi\)
0.298580 + 0.954385i \(0.403487\pi\)
\(882\) −1.23561e6 −0.0534821
\(883\) −1.63401e7 −0.705267 −0.352634 0.935762i \(-0.614714\pi\)
−0.352634 + 0.935762i \(0.614714\pi\)
\(884\) 6.21822e6 0.267630
\(885\) 1.87761e6 0.0805836
\(886\) 2.94094e6 0.125864
\(887\) −5.51374e6 −0.235308 −0.117654 0.993055i \(-0.537537\pi\)
−0.117654 + 0.993055i \(0.537537\pi\)
\(888\) 6.13671e6 0.261158
\(889\) −1.36691e7 −0.580078
\(890\) 1.36148e7 0.576152
\(891\) 4.30326e6 0.181595
\(892\) 1.14400e7 0.481407
\(893\) −6.19699e6 −0.260047
\(894\) 3.25382e6 0.136160
\(895\) −3.29026e7 −1.37301
\(896\) 1.51001e7 0.628362
\(897\) 5.97936e6 0.248127
\(898\) 1.08202e6 0.0447761
\(899\) −5.04698e6 −0.208273
\(900\) −1.10596e6 −0.0455129
\(901\) 3.88989e7 1.59634
\(902\) −1.51538e7 −0.620163
\(903\) −1.52186e7 −0.621091
\(904\) −1.78528e7 −0.726582
\(905\) −3.80277e7 −1.54340
\(906\) 5.89718e6 0.238684
\(907\) −1.18836e7 −0.479655 −0.239828 0.970816i \(-0.577091\pi\)
−0.239828 + 0.970816i \(0.577091\pi\)
\(908\) −3.63723e7 −1.46405
\(909\) −6.40054e6 −0.256925
\(910\) −1.21419e6 −0.0486052
\(911\) 425792. 0.0169981 0.00849907 0.999964i \(-0.497295\pi\)
0.00849907 + 0.999964i \(0.497295\pi\)
\(912\) −2.71685e6 −0.108163
\(913\) −7.18479e7 −2.85257
\(914\) 9.98125e6 0.395202
\(915\) −5.96895e6 −0.235692
\(916\) −9.65689e6 −0.380276
\(917\) −1.89637e6 −0.0744732
\(918\) 1.83709e6 0.0719489
\(919\) −3.84199e7 −1.50061 −0.750304 0.661093i \(-0.770093\pi\)
−0.750304 + 0.661093i \(0.770093\pi\)
\(920\) 2.89060e7 1.12595
\(921\) −6.24100e6 −0.242440
\(922\) 1.46068e7 0.565885
\(923\) 6.02739e6 0.232876
\(924\) −1.50631e7 −0.580411
\(925\) 3.13269e6 0.120382
\(926\) −1.02102e7 −0.391296
\(927\) −672177. −0.0256912
\(928\) 1.26295e7 0.481410
\(929\) 3.79880e7 1.44413 0.722066 0.691824i \(-0.243193\pi\)
0.722066 + 0.691824i \(0.243193\pi\)
\(930\) 1.61800e6 0.0613438
\(931\) −3.61721e6 −0.136773
\(932\) 2.50811e7 0.945818
\(933\) −9.77332e6 −0.367568
\(934\) 9.94012e6 0.372842
\(935\) −5.97088e7 −2.23362
\(936\) −1.15199e6 −0.0429792
\(937\) 2.79681e7 1.04067 0.520337 0.853961i \(-0.325806\pi\)
0.520337 + 0.853961i \(0.325806\pi\)
\(938\) 5.00057e6 0.185572
\(939\) −2.22223e7 −0.822479
\(940\) 2.76122e7 1.01925
\(941\) −6.99862e6 −0.257655 −0.128827 0.991667i \(-0.541121\pi\)
−0.128827 + 0.991667i \(0.541121\pi\)
\(942\) −307254. −0.0112816
\(943\) 6.61049e7 2.42077
\(944\) −2.67115e6 −0.0975591
\(945\) 3.81189e6 0.138855
\(946\) −2.10890e7 −0.766176
\(947\) −1.69519e7 −0.614248 −0.307124 0.951670i \(-0.599367\pi\)
−0.307124 + 0.951670i \(0.599367\pi\)
\(948\) 2.39860e7 0.866835
\(949\) −4.89265e6 −0.176351
\(950\) 304681. 0.0109531
\(951\) −1.94669e7 −0.697983
\(952\) −1.34663e7 −0.481565
\(953\) −2.60411e7 −0.928810 −0.464405 0.885623i \(-0.653732\pi\)
−0.464405 + 0.885623i \(0.653732\pi\)
\(954\) −3.44129e6 −0.122419
\(955\) −2.44566e7 −0.867737
\(956\) 3.02094e7 1.06905
\(957\) −1.64769e7 −0.581561
\(958\) −3.68495e6 −0.129723
\(959\) −2.55999e7 −0.898860
\(960\) 9.19592e6 0.322046
\(961\) −2.53599e7 −0.885806
\(962\) 1.55821e6 0.0542860
\(963\) −2.81277e6 −0.0977392
\(964\) −3.98913e6 −0.138257
\(965\) −1.74219e6 −0.0602250
\(966\) −6.18355e6 −0.213204
\(967\) 3.96216e7 1.36259 0.681296 0.732008i \(-0.261417\pi\)
0.681296 + 0.732008i \(0.261417\pi\)
\(968\) −2.73470e7 −0.938038
\(969\) 5.37805e6 0.183999
\(970\) −1.38485e7 −0.472579
\(971\) 1.53561e7 0.522675 0.261338 0.965247i \(-0.415836\pi\)
0.261338 + 0.965247i \(0.415836\pi\)
\(972\) 1.72705e6 0.0586324
\(973\) 1.11806e7 0.378600
\(974\) 1.33892e7 0.452227
\(975\) −588071. −0.0198115
\(976\) 8.49162e6 0.285342
\(977\) 1.61769e7 0.542200 0.271100 0.962551i \(-0.412613\pi\)
0.271100 + 0.962551i \(0.412613\pi\)
\(978\) 6.06629e6 0.202804
\(979\) 8.98111e7 2.99484
\(980\) 1.61173e7 0.536078
\(981\) 1.71134e7 0.567759
\(982\) −8.14433e6 −0.269511
\(983\) −2.91065e7 −0.960739 −0.480370 0.877066i \(-0.659497\pi\)
−0.480370 + 0.877066i \(0.659497\pi\)
\(984\) −1.27358e7 −0.419314
\(985\) 3.94474e7 1.29547
\(986\) −7.03410e6 −0.230418
\(987\) −1.23694e7 −0.404162
\(988\) −1.61043e6 −0.0524869
\(989\) 9.19957e7 2.99073
\(990\) 5.28229e6 0.171291
\(991\) 423539. 0.0136996 0.00684982 0.999977i \(-0.497820\pi\)
0.00684982 + 0.999977i \(0.497820\pi\)
\(992\) −8.18100e6 −0.263953
\(993\) 1.17348e7 0.377661
\(994\) −6.23322e6 −0.200100
\(995\) 2.90047e7 0.928776
\(996\) −2.88350e7 −0.921025
\(997\) 4.48353e7 1.42851 0.714254 0.699887i \(-0.246766\pi\)
0.714254 + 0.699887i \(0.246766\pi\)
\(998\) −1.09486e7 −0.347963
\(999\) −4.89193e6 −0.155084
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.b.1.8 12
3.2 odd 2 531.6.a.d.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.8 12 1.1 even 1 trivial
531.6.a.d.1.5 12 3.2 odd 2