Properties

Label 177.6.a.b.1.12
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 1.12
Root \(-10.7661\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.7661 q^{2} -9.00000 q^{3} +83.9098 q^{4} -60.5969 q^{5} -96.8953 q^{6} -233.164 q^{7} +558.868 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.7661 q^{2} -9.00000 q^{3} +83.9098 q^{4} -60.5969 q^{5} -96.8953 q^{6} -233.164 q^{7} +558.868 q^{8} +81.0000 q^{9} -652.394 q^{10} +441.792 q^{11} -755.188 q^{12} -615.817 q^{13} -2510.28 q^{14} +545.372 q^{15} +3331.74 q^{16} -1491.95 q^{17} +872.057 q^{18} -1189.53 q^{19} -5084.67 q^{20} +2098.48 q^{21} +4756.40 q^{22} -4791.05 q^{23} -5029.81 q^{24} +546.980 q^{25} -6629.97 q^{26} -729.000 q^{27} -19564.7 q^{28} -4769.56 q^{29} +5871.55 q^{30} +6722.80 q^{31} +17986.2 q^{32} -3976.13 q^{33} -16062.6 q^{34} +14129.0 q^{35} +6796.69 q^{36} +3091.95 q^{37} -12806.6 q^{38} +5542.35 q^{39} -33865.6 q^{40} +16633.3 q^{41} +22592.5 q^{42} +16690.1 q^{43} +37070.7 q^{44} -4908.35 q^{45} -51581.1 q^{46} -5195.02 q^{47} -29985.6 q^{48} +37558.5 q^{49} +5888.86 q^{50} +13427.6 q^{51} -51673.0 q^{52} -4994.59 q^{53} -7848.52 q^{54} -26771.2 q^{55} -130308. q^{56} +10705.8 q^{57} -51349.7 q^{58} -3481.00 q^{59} +45762.0 q^{60} -41246.4 q^{61} +72378.6 q^{62} -18886.3 q^{63} +87026.0 q^{64} +37316.6 q^{65} -42807.6 q^{66} +41332.6 q^{67} -125190. q^{68} +43119.5 q^{69} +152115. q^{70} -3791.51 q^{71} +45268.3 q^{72} -15511.7 q^{73} +33288.3 q^{74} -4922.82 q^{75} -99813.1 q^{76} -103010. q^{77} +59669.7 q^{78} +6475.81 q^{79} -201893. q^{80} +6561.00 q^{81} +179076. q^{82} -20354.7 q^{83} +176083. q^{84} +90407.8 q^{85} +179688. q^{86} +42926.0 q^{87} +246903. q^{88} +35567.6 q^{89} -52843.9 q^{90} +143586. q^{91} -402016. q^{92} -60505.2 q^{93} -55930.3 q^{94} +72081.7 q^{95} -161875. q^{96} -18405.4 q^{97} +404360. q^{98} +35785.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.7661 1.90320 0.951601 0.307335i \(-0.0994373\pi\)
0.951601 + 0.307335i \(0.0994373\pi\)
\(3\) −9.00000 −0.577350
\(4\) 83.9098 2.62218
\(5\) −60.5969 −1.08399 −0.541995 0.840382i \(-0.682331\pi\)
−0.541995 + 0.840382i \(0.682331\pi\)
\(6\) −96.8953 −1.09881
\(7\) −233.164 −1.79853 −0.899263 0.437409i \(-0.855896\pi\)
−0.899263 + 0.437409i \(0.855896\pi\)
\(8\) 558.868 3.08734
\(9\) 81.0000 0.333333
\(10\) −652.394 −2.06305
\(11\) 441.792 1.10087 0.550435 0.834878i \(-0.314462\pi\)
0.550435 + 0.834878i \(0.314462\pi\)
\(12\) −755.188 −1.51392
\(13\) −615.817 −1.01063 −0.505316 0.862934i \(-0.668624\pi\)
−0.505316 + 0.862934i \(0.668624\pi\)
\(14\) −2510.28 −3.42296
\(15\) 545.372 0.625842
\(16\) 3331.74 3.25365
\(17\) −1491.95 −1.25208 −0.626042 0.779789i \(-0.715326\pi\)
−0.626042 + 0.779789i \(0.715326\pi\)
\(18\) 872.057 0.634401
\(19\) −1189.53 −0.755946 −0.377973 0.925817i \(-0.623379\pi\)
−0.377973 + 0.925817i \(0.623379\pi\)
\(20\) −5084.67 −2.84242
\(21\) 2098.48 1.03838
\(22\) 4756.40 2.09518
\(23\) −4791.05 −1.88848 −0.944238 0.329265i \(-0.893199\pi\)
−0.944238 + 0.329265i \(0.893199\pi\)
\(24\) −5029.81 −1.78247
\(25\) 546.980 0.175033
\(26\) −6629.97 −1.92344
\(27\) −729.000 −0.192450
\(28\) −19564.7 −4.71606
\(29\) −4769.56 −1.05313 −0.526566 0.850134i \(-0.676521\pi\)
−0.526566 + 0.850134i \(0.676521\pi\)
\(30\) 5871.55 1.19110
\(31\) 6722.80 1.25645 0.628226 0.778031i \(-0.283781\pi\)
0.628226 + 0.778031i \(0.283781\pi\)
\(32\) 17986.2 3.10501
\(33\) −3976.13 −0.635588
\(34\) −16062.6 −2.38297
\(35\) 14129.0 1.94958
\(36\) 6796.69 0.874060
\(37\) 3091.95 0.371302 0.185651 0.982616i \(-0.440561\pi\)
0.185651 + 0.982616i \(0.440561\pi\)
\(38\) −12806.6 −1.43872
\(39\) 5542.35 0.583489
\(40\) −33865.6 −3.34664
\(41\) 16633.3 1.54532 0.772661 0.634819i \(-0.218925\pi\)
0.772661 + 0.634819i \(0.218925\pi\)
\(42\) 22592.5 1.97625
\(43\) 16690.1 1.37654 0.688268 0.725456i \(-0.258371\pi\)
0.688268 + 0.725456i \(0.258371\pi\)
\(44\) 37070.7 2.88668
\(45\) −4908.35 −0.361330
\(46\) −51581.1 −3.59415
\(47\) −5195.02 −0.343038 −0.171519 0.985181i \(-0.554868\pi\)
−0.171519 + 0.985181i \(0.554868\pi\)
\(48\) −29985.6 −1.87849
\(49\) 37558.5 2.23469
\(50\) 5888.86 0.333124
\(51\) 13427.6 0.722891
\(52\) −51673.0 −2.65006
\(53\) −4994.59 −0.244236 −0.122118 0.992516i \(-0.538969\pi\)
−0.122118 + 0.992516i \(0.538969\pi\)
\(54\) −7848.52 −0.366272
\(55\) −26771.2 −1.19333
\(56\) −130308. −5.55265
\(57\) 10705.8 0.436446
\(58\) −51349.7 −2.00432
\(59\) −3481.00 −0.130189
\(60\) 45762.0 1.64107
\(61\) −41246.4 −1.41926 −0.709630 0.704575i \(-0.751138\pi\)
−0.709630 + 0.704575i \(0.751138\pi\)
\(62\) 72378.6 2.39128
\(63\) −18886.3 −0.599508
\(64\) 87026.0 2.65582
\(65\) 37316.6 1.09551
\(66\) −42807.6 −1.20965
\(67\) 41332.6 1.12488 0.562439 0.826838i \(-0.309863\pi\)
0.562439 + 0.826838i \(0.309863\pi\)
\(68\) −125190. −3.28319
\(69\) 43119.5 1.09031
\(70\) 152115. 3.71045
\(71\) −3791.51 −0.0892618 −0.0446309 0.999004i \(-0.514211\pi\)
−0.0446309 + 0.999004i \(0.514211\pi\)
\(72\) 45268.3 1.02911
\(73\) −15511.7 −0.340685 −0.170343 0.985385i \(-0.554487\pi\)
−0.170343 + 0.985385i \(0.554487\pi\)
\(74\) 33288.3 0.706663
\(75\) −4922.82 −0.101056
\(76\) −99813.1 −1.98223
\(77\) −103010. −1.97994
\(78\) 59669.7 1.11050
\(79\) 6475.81 0.116742 0.0583709 0.998295i \(-0.481409\pi\)
0.0583709 + 0.998295i \(0.481409\pi\)
\(80\) −201893. −3.52692
\(81\) 6561.00 0.111111
\(82\) 179076. 2.94106
\(83\) −20354.7 −0.324318 −0.162159 0.986765i \(-0.551846\pi\)
−0.162159 + 0.986765i \(0.551846\pi\)
\(84\) 176083. 2.72282
\(85\) 90407.8 1.35725
\(86\) 179688. 2.61983
\(87\) 42926.0 0.608026
\(88\) 246903. 3.39876
\(89\) 35567.6 0.475970 0.237985 0.971269i \(-0.423513\pi\)
0.237985 + 0.971269i \(0.423513\pi\)
\(90\) −52843.9 −0.687684
\(91\) 143586. 1.81765
\(92\) −402016. −4.95192
\(93\) −60505.2 −0.725413
\(94\) −55930.3 −0.652871
\(95\) 72081.7 0.819438
\(96\) −161875. −1.79268
\(97\) −18405.4 −0.198616 −0.0993082 0.995057i \(-0.531663\pi\)
−0.0993082 + 0.995057i \(0.531663\pi\)
\(98\) 404360. 4.25307
\(99\) 35785.2 0.366957
\(100\) 45896.9 0.458969
\(101\) 77102.1 0.752078 0.376039 0.926604i \(-0.377286\pi\)
0.376039 + 0.926604i \(0.377286\pi\)
\(102\) 144563. 1.37581
\(103\) −192261. −1.78566 −0.892829 0.450395i \(-0.851283\pi\)
−0.892829 + 0.450395i \(0.851283\pi\)
\(104\) −344160. −3.12016
\(105\) −127161. −1.12559
\(106\) −53772.4 −0.464831
\(107\) −52637.5 −0.444463 −0.222232 0.974994i \(-0.571334\pi\)
−0.222232 + 0.974994i \(0.571334\pi\)
\(108\) −61170.2 −0.504639
\(109\) −192595. −1.55267 −0.776334 0.630322i \(-0.782923\pi\)
−0.776334 + 0.630322i \(0.782923\pi\)
\(110\) −288223. −2.27115
\(111\) −27827.5 −0.214371
\(112\) −776841. −5.85177
\(113\) 84620.6 0.623419 0.311709 0.950177i \(-0.399098\pi\)
0.311709 + 0.950177i \(0.399098\pi\)
\(114\) 115260. 0.830645
\(115\) 290323. 2.04709
\(116\) −400212. −2.76150
\(117\) −49881.1 −0.336877
\(118\) −37476.9 −0.247776
\(119\) 347870. 2.25190
\(120\) 304791. 1.93218
\(121\) 34129.4 0.211916
\(122\) −444065. −2.70114
\(123\) −149700. −0.892192
\(124\) 564109. 3.29464
\(125\) 156220. 0.894255
\(126\) −203332. −1.14099
\(127\) −42822.7 −0.235594 −0.117797 0.993038i \(-0.537583\pi\)
−0.117797 + 0.993038i \(0.537583\pi\)
\(128\) 361377. 1.94955
\(129\) −150211. −0.794744
\(130\) 401755. 2.08499
\(131\) −165092. −0.840519 −0.420260 0.907404i \(-0.638061\pi\)
−0.420260 + 0.907404i \(0.638061\pi\)
\(132\) −333636. −1.66663
\(133\) 277355. 1.35959
\(134\) 444993. 2.14087
\(135\) 44175.1 0.208614
\(136\) −833805. −3.86560
\(137\) 76275.0 0.347201 0.173600 0.984816i \(-0.444460\pi\)
0.173600 + 0.984816i \(0.444460\pi\)
\(138\) 464230. 2.07508
\(139\) −401114. −1.76088 −0.880441 0.474155i \(-0.842754\pi\)
−0.880441 + 0.474155i \(0.842754\pi\)
\(140\) 1.18556e6 5.11216
\(141\) 46755.2 0.198053
\(142\) −40819.9 −0.169883
\(143\) −272063. −1.11258
\(144\) 269871. 1.08455
\(145\) 289020. 1.14158
\(146\) −167002. −0.648393
\(147\) −338026. −1.29020
\(148\) 259444. 0.973621
\(149\) 97892.0 0.361228 0.180614 0.983554i \(-0.442192\pi\)
0.180614 + 0.983554i \(0.442192\pi\)
\(150\) −52999.7 −0.192329
\(151\) 165925. 0.592202 0.296101 0.955157i \(-0.404313\pi\)
0.296101 + 0.955157i \(0.404313\pi\)
\(152\) −664789. −2.33386
\(153\) −120848. −0.417361
\(154\) −1.10902e6 −3.76823
\(155\) −407381. −1.36198
\(156\) 465057. 1.53001
\(157\) −176014. −0.569899 −0.284950 0.958542i \(-0.591977\pi\)
−0.284950 + 0.958542i \(0.591977\pi\)
\(158\) 69719.4 0.222183
\(159\) 44951.3 0.141010
\(160\) −1.08990e6 −3.36580
\(161\) 1.11710e6 3.39647
\(162\) 70636.6 0.211467
\(163\) −139166. −0.410263 −0.205132 0.978734i \(-0.565762\pi\)
−0.205132 + 0.978734i \(0.565762\pi\)
\(164\) 1.39570e6 4.05211
\(165\) 240941. 0.688971
\(166\) −219142. −0.617242
\(167\) −20633.9 −0.0572518 −0.0286259 0.999590i \(-0.509113\pi\)
−0.0286259 + 0.999590i \(0.509113\pi\)
\(168\) 1.17277e6 3.20583
\(169\) 7937.10 0.0213769
\(170\) 973343. 2.58311
\(171\) −96351.8 −0.251982
\(172\) 1.40046e6 3.60953
\(173\) −460281. −1.16925 −0.584626 0.811303i \(-0.698759\pi\)
−0.584626 + 0.811303i \(0.698759\pi\)
\(174\) 462147. 1.15720
\(175\) −127536. −0.314802
\(176\) 1.47193e6 3.58185
\(177\) 31329.0 0.0751646
\(178\) 382925. 0.905866
\(179\) −51633.5 −0.120448 −0.0602239 0.998185i \(-0.519181\pi\)
−0.0602239 + 0.998185i \(0.519181\pi\)
\(180\) −411858. −0.947472
\(181\) 312077. 0.708052 0.354026 0.935236i \(-0.384812\pi\)
0.354026 + 0.935236i \(0.384812\pi\)
\(182\) 1.54587e6 3.45935
\(183\) 371218. 0.819410
\(184\) −2.67756e6 −5.83036
\(185\) −187362. −0.402488
\(186\) −651407. −1.38061
\(187\) −659134. −1.37838
\(188\) −435913. −0.899508
\(189\) 169977. 0.346126
\(190\) 776042. 1.55956
\(191\) −818682. −1.62380 −0.811899 0.583798i \(-0.801566\pi\)
−0.811899 + 0.583798i \(0.801566\pi\)
\(192\) −783234. −1.53334
\(193\) −29498.1 −0.0570034 −0.0285017 0.999594i \(-0.509074\pi\)
−0.0285017 + 0.999594i \(0.509074\pi\)
\(194\) −198155. −0.378007
\(195\) −335849. −0.632496
\(196\) 3.15152e6 5.85977
\(197\) 608501. 1.11711 0.558555 0.829467i \(-0.311356\pi\)
0.558555 + 0.829467i \(0.311356\pi\)
\(198\) 385268. 0.698393
\(199\) −452321. −0.809681 −0.404840 0.914387i \(-0.632673\pi\)
−0.404840 + 0.914387i \(0.632673\pi\)
\(200\) 305689. 0.540387
\(201\) −371993. −0.649449
\(202\) 830092. 1.43136
\(203\) 1.11209e6 1.89409
\(204\) 1.12671e6 1.89555
\(205\) −1.00793e6 −1.67511
\(206\) −2.06991e6 −3.39847
\(207\) −388075. −0.629492
\(208\) −2.05174e6 −3.28824
\(209\) −525524. −0.832199
\(210\) −1.36903e6 −2.14223
\(211\) 29005.8 0.0448516 0.0224258 0.999749i \(-0.492861\pi\)
0.0224258 + 0.999749i \(0.492861\pi\)
\(212\) −419095. −0.640431
\(213\) 34123.5 0.0515353
\(214\) −566703. −0.845903
\(215\) −1.01137e6 −1.49215
\(216\) −407415. −0.594158
\(217\) −1.56752e6 −2.25976
\(218\) −2.07350e6 −2.95504
\(219\) 139606. 0.196695
\(220\) −2.24637e6 −3.12913
\(221\) 918771. 1.26540
\(222\) −299595. −0.407992
\(223\) 917713. 1.23579 0.617895 0.786260i \(-0.287986\pi\)
0.617895 + 0.786260i \(0.287986\pi\)
\(224\) −4.19373e6 −5.58444
\(225\) 44305.3 0.0583445
\(226\) 911037. 1.18649
\(227\) −888321. −1.14421 −0.572104 0.820181i \(-0.693873\pi\)
−0.572104 + 0.820181i \(0.693873\pi\)
\(228\) 898318. 1.14444
\(229\) −613349. −0.772892 −0.386446 0.922312i \(-0.626297\pi\)
−0.386446 + 0.922312i \(0.626297\pi\)
\(230\) 3.12565e6 3.89602
\(231\) 927091. 1.14312
\(232\) −2.66555e6 −3.25138
\(233\) −460846. −0.556116 −0.278058 0.960564i \(-0.589691\pi\)
−0.278058 + 0.960564i \(0.589691\pi\)
\(234\) −537027. −0.641146
\(235\) 314802. 0.371850
\(236\) −292090. −0.341379
\(237\) −58282.3 −0.0674009
\(238\) 3.74522e6 4.28583
\(239\) 1.00128e6 1.13387 0.566934 0.823763i \(-0.308129\pi\)
0.566934 + 0.823763i \(0.308129\pi\)
\(240\) 1.81703e6 2.03627
\(241\) −350750. −0.389005 −0.194502 0.980902i \(-0.562309\pi\)
−0.194502 + 0.980902i \(0.562309\pi\)
\(242\) 367441. 0.403320
\(243\) −59049.0 −0.0641500
\(244\) −3.46098e6 −3.72156
\(245\) −2.27593e6 −2.42238
\(246\) −1.61169e6 −1.69802
\(247\) 732531. 0.763983
\(248\) 3.75716e6 3.87909
\(249\) 183193. 0.187245
\(250\) 1.68189e6 1.70195
\(251\) 1.91311e6 1.91671 0.958353 0.285585i \(-0.0921878\pi\)
0.958353 + 0.285585i \(0.0921878\pi\)
\(252\) −1.58474e6 −1.57202
\(253\) −2.11665e6 −2.07897
\(254\) −461035. −0.448383
\(255\) −813670. −0.783606
\(256\) 1.10580e6 1.05457
\(257\) 834372. 0.788001 0.394001 0.919110i \(-0.371091\pi\)
0.394001 + 0.919110i \(0.371091\pi\)
\(258\) −1.61719e6 −1.51256
\(259\) −720930. −0.667796
\(260\) 3.13122e6 2.87264
\(261\) −386334. −0.351044
\(262\) −1.77740e6 −1.59968
\(263\) −295137. −0.263108 −0.131554 0.991309i \(-0.541997\pi\)
−0.131554 + 0.991309i \(0.541997\pi\)
\(264\) −2.22213e6 −1.96227
\(265\) 302656. 0.264750
\(266\) 2.98605e6 2.58757
\(267\) −320108. −0.274801
\(268\) 3.46821e6 2.94964
\(269\) 308327. 0.259795 0.129897 0.991527i \(-0.458535\pi\)
0.129897 + 0.991527i \(0.458535\pi\)
\(270\) 475595. 0.397034
\(271\) 1.89619e6 1.56840 0.784202 0.620506i \(-0.213073\pi\)
0.784202 + 0.620506i \(0.213073\pi\)
\(272\) −4.97080e6 −4.07384
\(273\) −1.29228e6 −1.04942
\(274\) 821187. 0.660793
\(275\) 241651. 0.192689
\(276\) 3.61814e6 2.85899
\(277\) −37822.4 −0.0296176 −0.0148088 0.999890i \(-0.504714\pi\)
−0.0148088 + 0.999890i \(0.504714\pi\)
\(278\) −4.31845e6 −3.35132
\(279\) 544547. 0.418817
\(280\) 7.89625e6 6.01902
\(281\) 1.67537e6 1.26574 0.632871 0.774258i \(-0.281876\pi\)
0.632871 + 0.774258i \(0.281876\pi\)
\(282\) 503373. 0.376935
\(283\) 23322.7 0.0173106 0.00865530 0.999963i \(-0.497245\pi\)
0.00865530 + 0.999963i \(0.497245\pi\)
\(284\) −318144. −0.234061
\(285\) −648735. −0.473103
\(286\) −2.92907e6 −2.11746
\(287\) −3.87829e6 −2.77930
\(288\) 1.45688e6 1.03500
\(289\) 806072. 0.567714
\(290\) 3.11163e6 2.17267
\(291\) 165648. 0.114671
\(292\) −1.30159e6 −0.893338
\(293\) −1.04372e6 −0.710259 −0.355130 0.934817i \(-0.615563\pi\)
−0.355130 + 0.934817i \(0.615563\pi\)
\(294\) −3.63924e6 −2.45551
\(295\) 210938. 0.141123
\(296\) 1.72799e6 1.14633
\(297\) −322067. −0.211863
\(298\) 1.05392e6 0.687490
\(299\) 2.95041e6 1.90855
\(300\) −413072. −0.264986
\(301\) −3.89153e6 −2.47574
\(302\) 1.78637e6 1.12708
\(303\) −693919. −0.434212
\(304\) −3.96319e6 −2.45958
\(305\) 2.49941e6 1.53846
\(306\) −1.30107e6 −0.794323
\(307\) −99932.3 −0.0605146 −0.0302573 0.999542i \(-0.509633\pi\)
−0.0302573 + 0.999542i \(0.509633\pi\)
\(308\) −8.64355e6 −5.19177
\(309\) 1.73035e6 1.03095
\(310\) −4.38592e6 −2.59213
\(311\) −1.52909e6 −0.896460 −0.448230 0.893918i \(-0.647945\pi\)
−0.448230 + 0.893918i \(0.647945\pi\)
\(312\) 3.09744e6 1.80143
\(313\) −975618. −0.562884 −0.281442 0.959578i \(-0.590813\pi\)
−0.281442 + 0.959578i \(0.590813\pi\)
\(314\) −1.89499e6 −1.08463
\(315\) 1.14445e6 0.649861
\(316\) 543383. 0.306118
\(317\) 302184. 0.168897 0.0844487 0.996428i \(-0.473087\pi\)
0.0844487 + 0.996428i \(0.473087\pi\)
\(318\) 483952. 0.268370
\(319\) −2.10715e6 −1.15936
\(320\) −5.27350e6 −2.87888
\(321\) 473737. 0.256611
\(322\) 1.20269e7 6.46417
\(323\) 1.77472e6 0.946508
\(324\) 550532. 0.291353
\(325\) −336839. −0.176894
\(326\) −1.49828e6 −0.780814
\(327\) 1.73335e6 0.896433
\(328\) 9.29582e6 4.77093
\(329\) 1.21129e6 0.616963
\(330\) 2.59400e6 1.31125
\(331\) −1.49800e6 −0.751521 −0.375760 0.926717i \(-0.622618\pi\)
−0.375760 + 0.926717i \(0.622618\pi\)
\(332\) −1.70796e6 −0.850419
\(333\) 250448. 0.123767
\(334\) −222147. −0.108962
\(335\) −2.50463e6 −1.21936
\(336\) 6.99157e6 3.37852
\(337\) −1.15647e6 −0.554703 −0.277352 0.960769i \(-0.589457\pi\)
−0.277352 + 0.960769i \(0.589457\pi\)
\(338\) 85451.9 0.0406846
\(339\) −761585. −0.359931
\(340\) 7.58609e6 3.55894
\(341\) 2.97008e6 1.38319
\(342\) −1.03734e6 −0.479573
\(343\) −4.83850e6 −2.22063
\(344\) 9.32755e6 4.24983
\(345\) −2.61290e6 −1.18189
\(346\) −4.95545e6 −2.22532
\(347\) −3.05494e6 −1.36200 −0.681002 0.732281i \(-0.738456\pi\)
−0.681002 + 0.732281i \(0.738456\pi\)
\(348\) 3.60191e6 1.59435
\(349\) −1.18451e6 −0.520566 −0.260283 0.965532i \(-0.583816\pi\)
−0.260283 + 0.965532i \(0.583816\pi\)
\(350\) −1.37307e6 −0.599132
\(351\) 448930. 0.194496
\(352\) 7.94615e6 3.41822
\(353\) 1.37958e6 0.589266 0.294633 0.955610i \(-0.404803\pi\)
0.294633 + 0.955610i \(0.404803\pi\)
\(354\) 337292. 0.143053
\(355\) 229753. 0.0967589
\(356\) 2.98447e6 1.24808
\(357\) −3.13083e6 −1.30014
\(358\) −555893. −0.229237
\(359\) −1.97393e6 −0.808341 −0.404171 0.914684i \(-0.632440\pi\)
−0.404171 + 0.914684i \(0.632440\pi\)
\(360\) −2.74312e6 −1.11555
\(361\) −1.06112e6 −0.428545
\(362\) 3.35986e6 1.34757
\(363\) −307164. −0.122350
\(364\) 1.20483e7 4.76620
\(365\) 939963. 0.369299
\(366\) 3.99659e6 1.55950
\(367\) −2.33800e6 −0.906107 −0.453054 0.891483i \(-0.649665\pi\)
−0.453054 + 0.891483i \(0.649665\pi\)
\(368\) −1.59625e7 −6.14443
\(369\) 1.34730e6 0.515107
\(370\) −2.01717e6 −0.766016
\(371\) 1.16456e6 0.439265
\(372\) −5.07698e6 −1.90216
\(373\) −1.13024e6 −0.420627 −0.210313 0.977634i \(-0.567448\pi\)
−0.210313 + 0.977634i \(0.567448\pi\)
\(374\) −7.09633e6 −2.62334
\(375\) −1.40598e6 −0.516298
\(376\) −2.90333e6 −1.05907
\(377\) 2.93717e6 1.06433
\(378\) 1.82999e6 0.658748
\(379\) −3.32442e6 −1.18883 −0.594413 0.804160i \(-0.702615\pi\)
−0.594413 + 0.804160i \(0.702615\pi\)
\(380\) 6.04836e6 2.14871
\(381\) 385404. 0.136020
\(382\) −8.81405e6 −3.09042
\(383\) −844054. −0.294018 −0.147009 0.989135i \(-0.546965\pi\)
−0.147009 + 0.989135i \(0.546965\pi\)
\(384\) −3.25239e6 −1.12558
\(385\) 6.24209e6 2.14624
\(386\) −317581. −0.108489
\(387\) 1.35190e6 0.458845
\(388\) −1.54439e6 −0.520808
\(389\) 2.14732e6 0.719485 0.359743 0.933052i \(-0.382865\pi\)
0.359743 + 0.933052i \(0.382865\pi\)
\(390\) −3.61580e6 −1.20377
\(391\) 7.14803e6 2.36453
\(392\) 2.09902e7 6.89925
\(393\) 1.48583e6 0.485274
\(394\) 6.55121e6 2.12609
\(395\) −392413. −0.126547
\(396\) 3.00272e6 0.962227
\(397\) 3.41321e6 1.08689 0.543446 0.839444i \(-0.317119\pi\)
0.543446 + 0.839444i \(0.317119\pi\)
\(398\) −4.86975e6 −1.54099
\(399\) −2.49620e6 −0.784959
\(400\) 1.82239e6 0.569497
\(401\) −5.32467e6 −1.65361 −0.826803 0.562491i \(-0.809843\pi\)
−0.826803 + 0.562491i \(0.809843\pi\)
\(402\) −4.00493e6 −1.23603
\(403\) −4.14001e6 −1.26981
\(404\) 6.46962e6 1.97208
\(405\) −397576. −0.120443
\(406\) 1.19729e7 3.60483
\(407\) 1.36600e6 0.408756
\(408\) 7.50425e6 2.23181
\(409\) 5.35983e6 1.58432 0.792160 0.610314i \(-0.208957\pi\)
0.792160 + 0.610314i \(0.208957\pi\)
\(410\) −1.08515e7 −3.18808
\(411\) −686475. −0.200456
\(412\) −1.61326e7 −4.68232
\(413\) 811644. 0.234148
\(414\) −4.17807e6 −1.19805
\(415\) 1.23343e6 0.351557
\(416\) −1.10762e7 −3.13803
\(417\) 3.61002e6 1.01665
\(418\) −5.65787e6 −1.58384
\(419\) 6.25129e6 1.73954 0.869770 0.493457i \(-0.164267\pi\)
0.869770 + 0.493457i \(0.164267\pi\)
\(420\) −1.06701e7 −2.95150
\(421\) 1.98238e6 0.545106 0.272553 0.962141i \(-0.412132\pi\)
0.272553 + 0.962141i \(0.412132\pi\)
\(422\) 312280. 0.0853618
\(423\) −420796. −0.114346
\(424\) −2.79131e6 −0.754040
\(425\) −816069. −0.219157
\(426\) 367379. 0.0980822
\(427\) 9.61719e6 2.55257
\(428\) −4.41680e6 −1.16546
\(429\) 2.44857e6 0.642346
\(430\) −1.08885e7 −2.83987
\(431\) 2.80610e6 0.727628 0.363814 0.931472i \(-0.381474\pi\)
0.363814 + 0.931472i \(0.381474\pi\)
\(432\) −2.42883e6 −0.626165
\(433\) 171431. 0.0439408 0.0219704 0.999759i \(-0.493006\pi\)
0.0219704 + 0.999759i \(0.493006\pi\)
\(434\) −1.68761e7 −4.30078
\(435\) −2.60118e6 −0.659094
\(436\) −1.61606e7 −4.07138
\(437\) 5.69909e6 1.42759
\(438\) 1.50301e6 0.374350
\(439\) 5.47874e6 1.35681 0.678406 0.734688i \(-0.262671\pi\)
0.678406 + 0.734688i \(0.262671\pi\)
\(440\) −1.49616e7 −3.68422
\(441\) 3.04224e6 0.744897
\(442\) 9.89161e6 2.40830
\(443\) −2.02077e6 −0.489223 −0.244611 0.969621i \(-0.578660\pi\)
−0.244611 + 0.969621i \(0.578660\pi\)
\(444\) −2.33500e6 −0.562120
\(445\) −2.15528e6 −0.515946
\(446\) 9.88023e6 2.35196
\(447\) −881028. −0.208555
\(448\) −2.02913e7 −4.77656
\(449\) 8.24886e6 1.93098 0.965491 0.260436i \(-0.0838662\pi\)
0.965491 + 0.260436i \(0.0838662\pi\)
\(450\) 476997. 0.111041
\(451\) 7.34846e6 1.70120
\(452\) 7.10049e6 1.63472
\(453\) −1.49333e6 −0.341908
\(454\) −9.56379e6 −2.17766
\(455\) −8.70088e6 −1.97031
\(456\) 5.98310e6 1.34746
\(457\) −5.70265e6 −1.27728 −0.638640 0.769505i \(-0.720503\pi\)
−0.638640 + 0.769505i \(0.720503\pi\)
\(458\) −6.60340e6 −1.47097
\(459\) 1.08764e6 0.240964
\(460\) 2.43609e7 5.36783
\(461\) 3.39304e6 0.743596 0.371798 0.928314i \(-0.378741\pi\)
0.371798 + 0.928314i \(0.378741\pi\)
\(462\) 9.98119e6 2.17559
\(463\) −1.50634e6 −0.326565 −0.163283 0.986579i \(-0.552208\pi\)
−0.163283 + 0.986579i \(0.552208\pi\)
\(464\) −1.58909e7 −3.42652
\(465\) 3.66643e6 0.786340
\(466\) −4.96153e6 −1.05840
\(467\) −2.86503e6 −0.607908 −0.303954 0.952687i \(-0.598307\pi\)
−0.303954 + 0.952687i \(0.598307\pi\)
\(468\) −4.18551e6 −0.883353
\(469\) −9.63728e6 −2.02312
\(470\) 3.38920e6 0.707705
\(471\) 1.58413e6 0.329032
\(472\) −1.94542e6 −0.401937
\(473\) 7.37355e6 1.51539
\(474\) −627475. −0.128277
\(475\) −650648. −0.132316
\(476\) 2.91897e7 5.90490
\(477\) −404562. −0.0814121
\(478\) 1.07800e7 2.15798
\(479\) −7.27328e6 −1.44841 −0.724205 0.689585i \(-0.757793\pi\)
−0.724205 + 0.689585i \(0.757793\pi\)
\(480\) 9.80914e6 1.94325
\(481\) −1.90407e6 −0.375250
\(482\) −3.77622e6 −0.740355
\(483\) −1.00539e7 −1.96095
\(484\) 2.86379e6 0.555683
\(485\) 1.11531e6 0.215298
\(486\) −635730. −0.122091
\(487\) −1.00706e7 −1.92413 −0.962066 0.272816i \(-0.912045\pi\)
−0.962066 + 0.272816i \(0.912045\pi\)
\(488\) −2.30513e7 −4.38173
\(489\) 1.25249e6 0.236866
\(490\) −2.45029e7 −4.61029
\(491\) 2.30880e6 0.432198 0.216099 0.976371i \(-0.430667\pi\)
0.216099 + 0.976371i \(0.430667\pi\)
\(492\) −1.25613e7 −2.33949
\(493\) 7.11596e6 1.31861
\(494\) 7.88654e6 1.45402
\(495\) −2.16847e6 −0.397777
\(496\) 2.23986e7 4.08805
\(497\) 884043. 0.160540
\(498\) 1.97228e6 0.356365
\(499\) −530048. −0.0952937 −0.0476468 0.998864i \(-0.515172\pi\)
−0.0476468 + 0.998864i \(0.515172\pi\)
\(500\) 1.31084e7 2.34490
\(501\) 185705. 0.0330544
\(502\) 2.05968e7 3.64788
\(503\) 1.05998e6 0.186800 0.0934002 0.995629i \(-0.470226\pi\)
0.0934002 + 0.995629i \(0.470226\pi\)
\(504\) −1.05549e7 −1.85088
\(505\) −4.67214e6 −0.815244
\(506\) −2.27881e7 −3.95670
\(507\) −71433.9 −0.0123420
\(508\) −3.59324e6 −0.617770
\(509\) 604951. 0.103497 0.0517483 0.998660i \(-0.483521\pi\)
0.0517483 + 0.998660i \(0.483521\pi\)
\(510\) −8.76008e6 −1.49136
\(511\) 3.61678e6 0.612731
\(512\) 341148. 0.0575132
\(513\) 867166. 0.145482
\(514\) 8.98297e6 1.49973
\(515\) 1.16504e7 1.93564
\(516\) −1.26042e7 −2.08396
\(517\) −2.29512e6 −0.377641
\(518\) −7.76164e6 −1.27095
\(519\) 4.14253e6 0.675067
\(520\) 2.08550e7 3.38222
\(521\) 786658. 0.126967 0.0634836 0.997983i \(-0.479779\pi\)
0.0634836 + 0.997983i \(0.479779\pi\)
\(522\) −4.15933e6 −0.668108
\(523\) 4.91796e6 0.786196 0.393098 0.919497i \(-0.371403\pi\)
0.393098 + 0.919497i \(0.371403\pi\)
\(524\) −1.38528e7 −2.20399
\(525\) 1.14782e6 0.181751
\(526\) −3.17749e6 −0.500749
\(527\) −1.00301e7 −1.57318
\(528\) −1.32474e7 −2.06798
\(529\) 1.65178e7 2.56634
\(530\) 3.25844e6 0.503872
\(531\) −281961. −0.0433963
\(532\) 2.32728e7 3.56508
\(533\) −1.02431e7 −1.56175
\(534\) −3.44633e6 −0.523002
\(535\) 3.18967e6 0.481793
\(536\) 2.30995e7 3.47288
\(537\) 464701. 0.0695406
\(538\) 3.31949e6 0.494442
\(539\) 1.65930e7 2.46011
\(540\) 3.70672e6 0.547023
\(541\) 6.04007e6 0.887255 0.443628 0.896211i \(-0.353691\pi\)
0.443628 + 0.896211i \(0.353691\pi\)
\(542\) 2.04146e7 2.98499
\(543\) −2.80869e6 −0.408794
\(544\) −2.68345e7 −3.88774
\(545\) 1.16706e7 1.68308
\(546\) −1.39128e7 −1.99726
\(547\) −7.80143e6 −1.11482 −0.557412 0.830236i \(-0.688206\pi\)
−0.557412 + 0.830236i \(0.688206\pi\)
\(548\) 6.40022e6 0.910423
\(549\) −3.34096e6 −0.473087
\(550\) 2.60165e6 0.366727
\(551\) 5.67352e6 0.796112
\(552\) 2.40981e7 3.36616
\(553\) −1.50993e6 −0.209963
\(554\) −407201. −0.0563682
\(555\) 1.68626e6 0.232376
\(556\) −3.36573e7 −4.61735
\(557\) 4.58897e6 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(558\) 5.86267e6 0.797094
\(559\) −1.02780e7 −1.39117
\(560\) 4.70741e7 6.34325
\(561\) 5.93221e6 0.795809
\(562\) 1.80373e7 2.40896
\(563\) −1.33156e7 −1.77048 −0.885239 0.465137i \(-0.846005\pi\)
−0.885239 + 0.465137i \(0.846005\pi\)
\(564\) 3.92321e6 0.519331
\(565\) −5.12774e6 −0.675780
\(566\) 251095. 0.0329456
\(567\) −1.52979e6 −0.199836
\(568\) −2.11895e6 −0.275581
\(569\) 8.48086e6 1.09814 0.549072 0.835775i \(-0.314981\pi\)
0.549072 + 0.835775i \(0.314981\pi\)
\(570\) −6.98437e6 −0.900410
\(571\) −8.94164e6 −1.14770 −0.573848 0.818962i \(-0.694550\pi\)
−0.573848 + 0.818962i \(0.694550\pi\)
\(572\) −2.28287e7 −2.91737
\(573\) 7.36814e6 0.937500
\(574\) −4.17542e7 −5.28957
\(575\) −2.62061e6 −0.330546
\(576\) 7.04910e6 0.885274
\(577\) 6.27318e6 0.784419 0.392210 0.919876i \(-0.371711\pi\)
0.392210 + 0.919876i \(0.371711\pi\)
\(578\) 8.67828e6 1.08047
\(579\) 265483. 0.0329109
\(580\) 2.42516e7 2.99344
\(581\) 4.74599e6 0.583293
\(582\) 1.78339e6 0.218243
\(583\) −2.20657e6 −0.268873
\(584\) −8.66901e6 −1.05181
\(585\) 3.02264e6 0.365172
\(586\) −1.12369e7 −1.35177
\(587\) 8.73012e6 1.04574 0.522872 0.852411i \(-0.324861\pi\)
0.522872 + 0.852411i \(0.324861\pi\)
\(588\) −2.83637e7 −3.38314
\(589\) −7.99696e6 −0.949810
\(590\) 2.27098e6 0.268586
\(591\) −5.47651e6 −0.644964
\(592\) 1.03015e7 1.20809
\(593\) 9.23757e6 1.07875 0.539375 0.842065i \(-0.318660\pi\)
0.539375 + 0.842065i \(0.318660\pi\)
\(594\) −3.46741e6 −0.403218
\(595\) −2.10798e7 −2.44104
\(596\) 8.21409e6 0.947205
\(597\) 4.07089e6 0.467469
\(598\) 3.17645e7 3.63236
\(599\) −2.25499e6 −0.256790 −0.128395 0.991723i \(-0.540983\pi\)
−0.128395 + 0.991723i \(0.540983\pi\)
\(600\) −2.75120e6 −0.311993
\(601\) −1.51694e7 −1.71310 −0.856549 0.516066i \(-0.827396\pi\)
−0.856549 + 0.516066i \(0.827396\pi\)
\(602\) −4.18968e7 −4.71183
\(603\) 3.34794e6 0.374960
\(604\) 1.39227e7 1.55286
\(605\) −2.06813e6 −0.229715
\(606\) −7.47083e6 −0.826394
\(607\) 1.16631e7 1.28481 0.642407 0.766363i \(-0.277936\pi\)
0.642407 + 0.766363i \(0.277936\pi\)
\(608\) −2.13951e7 −2.34722
\(609\) −1.00088e7 −1.09355
\(610\) 2.69089e7 2.92801
\(611\) 3.19918e6 0.346685
\(612\) −1.01404e7 −1.09440
\(613\) 1.66308e6 0.178756 0.0893780 0.995998i \(-0.471512\pi\)
0.0893780 + 0.995998i \(0.471512\pi\)
\(614\) −1.07589e6 −0.115172
\(615\) 9.07133e6 0.967127
\(616\) −5.75690e7 −6.11275
\(617\) 1.02503e7 1.08399 0.541994 0.840382i \(-0.317669\pi\)
0.541994 + 0.840382i \(0.317669\pi\)
\(618\) 1.86292e7 1.96211
\(619\) −1.17846e7 −1.23619 −0.618097 0.786102i \(-0.712096\pi\)
−0.618097 + 0.786102i \(0.712096\pi\)
\(620\) −3.41832e7 −3.57136
\(621\) 3.49268e6 0.363437
\(622\) −1.64624e7 −1.70615
\(623\) −8.29308e6 −0.856043
\(624\) 1.84656e7 1.89847
\(625\) −1.11757e7 −1.14440
\(626\) −1.05036e7 −1.07128
\(627\) 4.72972e6 0.480470
\(628\) −1.47693e7 −1.49438
\(629\) −4.61304e6 −0.464901
\(630\) 1.23213e7 1.23682
\(631\) −1.77856e7 −1.77826 −0.889130 0.457654i \(-0.848690\pi\)
−0.889130 + 0.457654i \(0.848690\pi\)
\(632\) 3.61912e6 0.360421
\(633\) −261052. −0.0258951
\(634\) 3.25335e6 0.321446
\(635\) 2.59492e6 0.255382
\(636\) 3.77185e6 0.369753
\(637\) −2.31291e7 −2.25845
\(638\) −2.26859e7 −2.20650
\(639\) −307112. −0.0297539
\(640\) −2.18983e7 −2.11330
\(641\) −9.51708e6 −0.914869 −0.457434 0.889243i \(-0.651232\pi\)
−0.457434 + 0.889243i \(0.651232\pi\)
\(642\) 5.10032e6 0.488383
\(643\) −383483. −0.0365779 −0.0182890 0.999833i \(-0.505822\pi\)
−0.0182890 + 0.999833i \(0.505822\pi\)
\(644\) 9.37357e7 8.90615
\(645\) 9.10231e6 0.861494
\(646\) 1.91069e7 1.80140
\(647\) 6.28292e6 0.590066 0.295033 0.955487i \(-0.404669\pi\)
0.295033 + 0.955487i \(0.404669\pi\)
\(648\) 3.66673e6 0.343037
\(649\) −1.53788e6 −0.143321
\(650\) −3.62646e6 −0.336666
\(651\) 1.41076e7 1.30467
\(652\) −1.16773e7 −1.07578
\(653\) 4.44773e6 0.408184 0.204092 0.978952i \(-0.434576\pi\)
0.204092 + 0.978952i \(0.434576\pi\)
\(654\) 1.86615e7 1.70609
\(655\) 1.00041e7 0.911114
\(656\) 5.54178e7 5.02793
\(657\) −1.25645e6 −0.113562
\(658\) 1.30409e7 1.17420
\(659\) −3.86787e6 −0.346943 −0.173471 0.984839i \(-0.555498\pi\)
−0.173471 + 0.984839i \(0.555498\pi\)
\(660\) 2.02173e7 1.80661
\(661\) 8.16662e6 0.727007 0.363504 0.931593i \(-0.381580\pi\)
0.363504 + 0.931593i \(0.381580\pi\)
\(662\) −1.61276e7 −1.43030
\(663\) −8.26893e6 −0.730577
\(664\) −1.13756e7 −1.00128
\(665\) −1.68069e7 −1.47378
\(666\) 2.69635e6 0.235554
\(667\) 2.28512e7 1.98881
\(668\) −1.73138e6 −0.150125
\(669\) −8.25942e6 −0.713484
\(670\) −2.69652e7 −2.32068
\(671\) −1.82224e7 −1.56242
\(672\) 3.77435e7 3.22418
\(673\) 1.90914e7 1.62480 0.812402 0.583098i \(-0.198160\pi\)
0.812402 + 0.583098i \(0.198160\pi\)
\(674\) −1.24507e7 −1.05571
\(675\) −398748. −0.0336852
\(676\) 666000. 0.0560541
\(677\) 1.41093e7 1.18313 0.591567 0.806256i \(-0.298509\pi\)
0.591567 + 0.806256i \(0.298509\pi\)
\(678\) −8.19933e6 −0.685022
\(679\) 4.29147e6 0.357217
\(680\) 5.05260e7 4.19027
\(681\) 7.99489e6 0.660609
\(682\) 3.19763e7 2.63249
\(683\) 7.16115e6 0.587396 0.293698 0.955898i \(-0.405114\pi\)
0.293698 + 0.955898i \(0.405114\pi\)
\(684\) −8.08486e6 −0.660742
\(685\) −4.62202e6 −0.376362
\(686\) −5.20919e7 −4.22630
\(687\) 5.52014e6 0.446229
\(688\) 5.56070e7 4.47876
\(689\) 3.07575e6 0.246833
\(690\) −2.81309e7 −2.24937
\(691\) 9.02513e6 0.719048 0.359524 0.933136i \(-0.382939\pi\)
0.359524 + 0.933136i \(0.382939\pi\)
\(692\) −3.86221e7 −3.06599
\(693\) −8.34382e6 −0.659981
\(694\) −3.28899e7 −2.59217
\(695\) 2.43062e7 1.90878
\(696\) 2.39900e7 1.87718
\(697\) −2.48161e7 −1.93487
\(698\) −1.27526e7 −0.990743
\(699\) 4.14761e6 0.321074
\(700\) −1.07015e7 −0.825468
\(701\) −5.85795e6 −0.450247 −0.225123 0.974330i \(-0.572278\pi\)
−0.225123 + 0.974330i \(0.572278\pi\)
\(702\) 4.83325e6 0.370166
\(703\) −3.67796e6 −0.280684
\(704\) 3.84474e7 2.92372
\(705\) −2.83322e6 −0.214688
\(706\) 1.48528e7 1.12149
\(707\) −1.79774e7 −1.35263
\(708\) 2.62881e6 0.197095
\(709\) −941213. −0.0703189 −0.0351595 0.999382i \(-0.511194\pi\)
−0.0351595 + 0.999382i \(0.511194\pi\)
\(710\) 2.47356e6 0.184152
\(711\) 524540. 0.0389139
\(712\) 1.98776e7 1.46948
\(713\) −3.22093e7 −2.37278
\(714\) −3.37070e7 −2.47442
\(715\) 1.64862e7 1.20602
\(716\) −4.33255e6 −0.315836
\(717\) −9.01156e6 −0.654639
\(718\) −2.12516e7 −1.53844
\(719\) −1.49350e7 −1.07741 −0.538706 0.842494i \(-0.681087\pi\)
−0.538706 + 0.842494i \(0.681087\pi\)
\(720\) −1.63533e7 −1.17564
\(721\) 4.48284e7 3.21155
\(722\) −1.14242e7 −0.815609
\(723\) 3.15675e6 0.224592
\(724\) 2.61863e7 1.85664
\(725\) −2.60885e6 −0.184333
\(726\) −3.30697e6 −0.232857
\(727\) −6.59529e6 −0.462805 −0.231402 0.972858i \(-0.574331\pi\)
−0.231402 + 0.972858i \(0.574331\pi\)
\(728\) 8.02457e7 5.61169
\(729\) 531441. 0.0370370
\(730\) 1.01198e7 0.702851
\(731\) −2.49009e7 −1.72354
\(732\) 3.11488e7 2.14864
\(733\) −1.07918e7 −0.741877 −0.370939 0.928657i \(-0.620964\pi\)
−0.370939 + 0.928657i \(0.620964\pi\)
\(734\) −2.51712e7 −1.72451
\(735\) 2.04833e7 1.39856
\(736\) −8.61726e7 −5.86374
\(737\) 1.82604e7 1.23835
\(738\) 1.45052e7 0.980353
\(739\) −2.62963e7 −1.77126 −0.885631 0.464389i \(-0.846274\pi\)
−0.885631 + 0.464389i \(0.846274\pi\)
\(740\) −1.57215e7 −1.05540
\(741\) −6.59278e6 −0.441086
\(742\) 1.25378e7 0.836010
\(743\) −1.79796e7 −1.19484 −0.597419 0.801929i \(-0.703807\pi\)
−0.597419 + 0.801929i \(0.703807\pi\)
\(744\) −3.38144e7 −2.23959
\(745\) −5.93195e6 −0.391567
\(746\) −1.21683e7 −0.800538
\(747\) −1.64873e6 −0.108106
\(748\) −5.53078e7 −3.61437
\(749\) 1.22732e7 0.799378
\(750\) −1.51370e7 −0.982621
\(751\) −3.69885e6 −0.239313 −0.119657 0.992815i \(-0.538179\pi\)
−0.119657 + 0.992815i \(0.538179\pi\)
\(752\) −1.73084e7 −1.11613
\(753\) −1.72180e7 −1.10661
\(754\) 3.16220e7 2.02563
\(755\) −1.00545e7 −0.641941
\(756\) 1.42627e7 0.907605
\(757\) 1.89384e7 1.20117 0.600583 0.799563i \(-0.294935\pi\)
0.600583 + 0.799563i \(0.294935\pi\)
\(758\) −3.57912e7 −2.26258
\(759\) 1.90498e7 1.20029
\(760\) 4.02841e7 2.52988
\(761\) −1.99494e7 −1.24873 −0.624365 0.781132i \(-0.714642\pi\)
−0.624365 + 0.781132i \(0.714642\pi\)
\(762\) 4.14931e6 0.258874
\(763\) 4.49062e7 2.79251
\(764\) −6.86954e7 −4.25789
\(765\) 7.32303e6 0.452415
\(766\) −9.08721e6 −0.559575
\(767\) 2.14366e6 0.131573
\(768\) −9.95220e6 −0.608858
\(769\) 9.25313e6 0.564251 0.282126 0.959377i \(-0.408960\pi\)
0.282126 + 0.959377i \(0.408960\pi\)
\(770\) 6.72032e7 4.08473
\(771\) −7.50935e6 −0.454953
\(772\) −2.47518e6 −0.149473
\(773\) 2.38372e7 1.43485 0.717424 0.696637i \(-0.245321\pi\)
0.717424 + 0.696637i \(0.245321\pi\)
\(774\) 1.45547e7 0.873276
\(775\) 3.67723e6 0.219921
\(776\) −1.02862e7 −0.613196
\(777\) 6.48837e6 0.385552
\(778\) 2.31183e7 1.36933
\(779\) −1.97858e7 −1.16818
\(780\) −2.81810e7 −1.65852
\(781\) −1.67506e6 −0.0982657
\(782\) 7.69567e7 4.50018
\(783\) 3.47701e6 0.202675
\(784\) 1.25135e8 7.27090
\(785\) 1.06659e7 0.617765
\(786\) 1.59966e7 0.923575
\(787\) 1.76172e7 1.01391 0.506956 0.861972i \(-0.330771\pi\)
0.506956 + 0.861972i \(0.330771\pi\)
\(788\) 5.10592e7 2.92926
\(789\) 2.65624e6 0.151906
\(790\) −4.22478e6 −0.240844
\(791\) −1.97305e7 −1.12123
\(792\) 1.99992e7 1.13292
\(793\) 2.54002e7 1.43435
\(794\) 3.67471e7 2.06858
\(795\) −2.72391e6 −0.152853
\(796\) −3.79541e7 −2.12313
\(797\) −3.37738e7 −1.88336 −0.941682 0.336504i \(-0.890756\pi\)
−0.941682 + 0.336504i \(0.890756\pi\)
\(798\) −2.68744e7 −1.49394
\(799\) 7.75073e6 0.429512
\(800\) 9.83806e6 0.543481
\(801\) 2.88097e6 0.158657
\(802\) −5.73262e7 −3.14715
\(803\) −6.85296e6 −0.375050
\(804\) −3.12139e7 −1.70297
\(805\) −6.76928e7 −3.68174
\(806\) −4.45719e7 −2.41671
\(807\) −2.77494e6 −0.149993
\(808\) 4.30899e7 2.32192
\(809\) −3.55067e7 −1.90739 −0.953695 0.300776i \(-0.902755\pi\)
−0.953695 + 0.300776i \(0.902755\pi\)
\(810\) −4.28036e6 −0.229228
\(811\) 6.18290e6 0.330096 0.165048 0.986286i \(-0.447222\pi\)
0.165048 + 0.986286i \(0.447222\pi\)
\(812\) 9.33151e7 4.96663
\(813\) −1.70657e7 −0.905518
\(814\) 1.47065e7 0.777945
\(815\) 8.43300e6 0.444721
\(816\) 4.47372e7 2.35203
\(817\) −1.98533e7 −1.04059
\(818\) 5.77047e7 3.01528
\(819\) 1.16305e7 0.605882
\(820\) −8.45748e7 −4.39245
\(821\) 2.77015e7 1.43432 0.717159 0.696909i \(-0.245442\pi\)
0.717159 + 0.696909i \(0.245442\pi\)
\(822\) −7.39068e6 −0.381509
\(823\) 1.16521e7 0.599660 0.299830 0.953993i \(-0.403070\pi\)
0.299830 + 0.953993i \(0.403070\pi\)
\(824\) −1.07449e8 −5.51293
\(825\) −2.17486e6 −0.111249
\(826\) 8.73827e6 0.445631
\(827\) 1.70280e7 0.865764 0.432882 0.901450i \(-0.357497\pi\)
0.432882 + 0.901450i \(0.357497\pi\)
\(828\) −3.25633e7 −1.65064
\(829\) 4.16981e6 0.210732 0.105366 0.994434i \(-0.466399\pi\)
0.105366 + 0.994434i \(0.466399\pi\)
\(830\) 1.32793e7 0.669084
\(831\) 340401. 0.0170997
\(832\) −5.35920e7 −2.68406
\(833\) −5.60355e7 −2.79802
\(834\) 3.88660e7 1.93488
\(835\) 1.25035e6 0.0620604
\(836\) −4.40966e7 −2.18218
\(837\) −4.90092e6 −0.241804
\(838\) 6.73023e7 3.31070
\(839\) 2.01150e6 0.0986539 0.0493269 0.998783i \(-0.484292\pi\)
0.0493269 + 0.998783i \(0.484292\pi\)
\(840\) −7.10662e7 −3.47508
\(841\) 2.23752e6 0.109088
\(842\) 2.13425e7 1.03745
\(843\) −1.50783e7 −0.730776
\(844\) 2.43387e6 0.117609
\(845\) −480963. −0.0231724
\(846\) −4.53035e6 −0.217624
\(847\) −7.95774e6 −0.381137
\(848\) −1.66406e7 −0.794659
\(849\) −209904. −0.00999428
\(850\) −8.78591e6 −0.417099
\(851\) −1.48137e7 −0.701195
\(852\) 2.86330e6 0.135135
\(853\) 1.07451e7 0.505634 0.252817 0.967514i \(-0.418643\pi\)
0.252817 + 0.967514i \(0.418643\pi\)
\(854\) 1.03540e8 4.85807
\(855\) 5.83862e6 0.273146
\(856\) −2.94174e7 −1.37221
\(857\) −2.35618e6 −0.109586 −0.0547931 0.998498i \(-0.517450\pi\)
−0.0547931 + 0.998498i \(0.517450\pi\)
\(858\) 2.63616e7 1.22251
\(859\) 3.64884e6 0.168722 0.0843611 0.996435i \(-0.473115\pi\)
0.0843611 + 0.996435i \(0.473115\pi\)
\(860\) −8.48636e7 −3.91269
\(861\) 3.49046e7 1.60463
\(862\) 3.02108e7 1.38482
\(863\) 3.09382e7 1.41406 0.707029 0.707184i \(-0.250035\pi\)
0.707029 + 0.707184i \(0.250035\pi\)
\(864\) −1.31119e7 −0.597560
\(865\) 2.78916e7 1.26746
\(866\) 1.84564e6 0.0836283
\(867\) −7.25465e6 −0.327770
\(868\) −1.31530e8 −5.92550
\(869\) 2.86096e6 0.128518
\(870\) −2.80047e7 −1.25439
\(871\) −2.54533e7 −1.13684
\(872\) −1.07635e8 −4.79361
\(873\) −1.49083e6 −0.0662055
\(874\) 6.13572e7 2.71698
\(875\) −3.64249e7 −1.60834
\(876\) 1.17143e7 0.515769
\(877\) 1.80640e7 0.793075 0.396538 0.918018i \(-0.370212\pi\)
0.396538 + 0.918018i \(0.370212\pi\)
\(878\) 5.89849e7 2.58229
\(879\) 9.39352e6 0.410068
\(880\) −8.91946e7 −3.88268
\(881\) −2.41533e7 −1.04842 −0.524211 0.851588i \(-0.675640\pi\)
−0.524211 + 0.851588i \(0.675640\pi\)
\(882\) 3.27531e7 1.41769
\(883\) −3.60339e7 −1.55528 −0.777642 0.628708i \(-0.783584\pi\)
−0.777642 + 0.628708i \(0.783584\pi\)
\(884\) 7.70938e7 3.31810
\(885\) −1.89844e6 −0.0814776
\(886\) −2.17559e7 −0.931090
\(887\) −6.17290e6 −0.263439 −0.131719 0.991287i \(-0.542050\pi\)
−0.131719 + 0.991287i \(0.542050\pi\)
\(888\) −1.55519e7 −0.661837
\(889\) 9.98471e6 0.423722
\(890\) −2.32041e7 −0.981950
\(891\) 2.89860e6 0.122319
\(892\) 7.70051e7 3.24047
\(893\) 6.17962e6 0.259318
\(894\) −9.48527e6 −0.396923
\(895\) 3.12883e6 0.130564
\(896\) −8.42600e7 −3.50632
\(897\) −2.65537e7 −1.10190
\(898\) 8.88084e7 3.67505
\(899\) −3.20648e7 −1.32321
\(900\) 3.71765e6 0.152990
\(901\) 7.45170e6 0.305804
\(902\) 7.91146e7 3.23773
\(903\) 3.50238e7 1.42937
\(904\) 4.72917e7 1.92470
\(905\) −1.89109e7 −0.767521
\(906\) −1.60774e7 −0.650720
\(907\) −3.72640e7 −1.50408 −0.752041 0.659116i \(-0.770931\pi\)
−0.752041 + 0.659116i \(0.770931\pi\)
\(908\) −7.45388e7 −3.00032
\(909\) 6.24527e6 0.250693
\(910\) −9.36749e7 −3.74990
\(911\) −502464. −0.0200590 −0.0100295 0.999950i \(-0.503193\pi\)
−0.0100295 + 0.999950i \(0.503193\pi\)
\(912\) 3.56687e7 1.42004
\(913\) −8.99257e6 −0.357032
\(914\) −6.13956e7 −2.43092
\(915\) −2.24946e7 −0.888232
\(916\) −5.14659e7 −2.02666
\(917\) 3.84935e7 1.51170
\(918\) 1.17096e7 0.458603
\(919\) −4.73641e7 −1.84995 −0.924976 0.380025i \(-0.875916\pi\)
−0.924976 + 0.380025i \(0.875916\pi\)
\(920\) 1.62252e8 6.32005
\(921\) 899391. 0.0349381
\(922\) 3.65300e7 1.41521
\(923\) 2.33487e6 0.0902109
\(924\) 7.77919e7 2.99747
\(925\) 1.69123e6 0.0649903
\(926\) −1.62174e7 −0.621520
\(927\) −1.55732e7 −0.595220
\(928\) −8.57860e7 −3.26999
\(929\) −2.27180e7 −0.863635 −0.431817 0.901961i \(-0.642127\pi\)
−0.431817 + 0.901961i \(0.642127\pi\)
\(930\) 3.94732e7 1.49656
\(931\) −4.46769e7 −1.68931
\(932\) −3.86694e7 −1.45824
\(933\) 1.37618e7 0.517572
\(934\) −3.08454e7 −1.15697
\(935\) 3.99415e7 1.49415
\(936\) −2.78770e7 −1.04005
\(937\) 1.02685e7 0.382082 0.191041 0.981582i \(-0.438814\pi\)
0.191041 + 0.981582i \(0.438814\pi\)
\(938\) −1.03756e8 −3.85041
\(939\) 8.78056e6 0.324981
\(940\) 2.64149e7 0.975057
\(941\) 2.02021e7 0.743741 0.371871 0.928285i \(-0.378716\pi\)
0.371871 + 0.928285i \(0.378716\pi\)
\(942\) 1.70549e7 0.626214
\(943\) −7.96910e7 −2.91830
\(944\) −1.15978e7 −0.423589
\(945\) −1.03000e7 −0.375197
\(946\) 7.93847e7 2.88409
\(947\) 4.57833e7 1.65895 0.829473 0.558547i \(-0.188641\pi\)
0.829473 + 0.558547i \(0.188641\pi\)
\(948\) −4.89045e6 −0.176737
\(949\) 9.55238e6 0.344307
\(950\) −7.00496e6 −0.251824
\(951\) −2.71965e6 −0.0975129
\(952\) 1.94413e8 6.95239
\(953\) 1.44135e7 0.514087 0.257043 0.966400i \(-0.417252\pi\)
0.257043 + 0.966400i \(0.417252\pi\)
\(954\) −4.35557e6 −0.154944
\(955\) 4.96096e7 1.76018
\(956\) 8.40175e7 2.97321
\(957\) 1.89644e7 0.669358
\(958\) −7.83051e7 −2.75662
\(959\) −1.77846e7 −0.624449
\(960\) 4.74615e7 1.66212
\(961\) 1.65669e7 0.578672
\(962\) −2.04995e7 −0.714176
\(963\) −4.26364e6 −0.148154
\(964\) −2.94313e7 −1.02004
\(965\) 1.78749e6 0.0617911
\(966\) −1.08242e8 −3.73209
\(967\) −3.29498e7 −1.13315 −0.566574 0.824011i \(-0.691732\pi\)
−0.566574 + 0.824011i \(0.691732\pi\)
\(968\) 1.90738e7 0.654257
\(969\) −1.59725e7 −0.546467
\(970\) 1.20076e7 0.409756
\(971\) 4.05011e7 1.37854 0.689269 0.724506i \(-0.257932\pi\)
0.689269 + 0.724506i \(0.257932\pi\)
\(972\) −4.95479e6 −0.168213
\(973\) 9.35253e7 3.16699
\(974\) −1.08422e8 −3.66201
\(975\) 3.03155e6 0.102130
\(976\) −1.37422e8 −4.61777
\(977\) −8.62455e6 −0.289068 −0.144534 0.989500i \(-0.546168\pi\)
−0.144534 + 0.989500i \(0.546168\pi\)
\(978\) 1.34845e7 0.450803
\(979\) 1.57135e7 0.523981
\(980\) −1.90972e8 −6.35193
\(981\) −1.56002e7 −0.517556
\(982\) 2.48569e7 0.822561
\(983\) 5.74597e7 1.89662 0.948309 0.317349i \(-0.102793\pi\)
0.948309 + 0.317349i \(0.102793\pi\)
\(984\) −8.36623e7 −2.75450
\(985\) −3.68733e7 −1.21094
\(986\) 7.66115e7 2.50958
\(987\) −1.09016e7 −0.356204
\(988\) 6.14665e7 2.00330
\(989\) −7.99631e7 −2.59955
\(990\) −2.33460e7 −0.757051
\(991\) −447099. −0.0144617 −0.00723086 0.999974i \(-0.502302\pi\)
−0.00723086 + 0.999974i \(0.502302\pi\)
\(992\) 1.20917e8 3.90130
\(993\) 1.34820e7 0.433891
\(994\) 9.51773e6 0.305539
\(995\) 2.74092e7 0.877685
\(996\) 1.53717e7 0.490990
\(997\) 2.65036e7 0.844437 0.422218 0.906494i \(-0.361252\pi\)
0.422218 + 0.906494i \(0.361252\pi\)
\(998\) −5.70657e6 −0.181363
\(999\) −2.25403e6 −0.0714571
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.1.12 12
3.2 odd 2 531.6.a.d.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.12 12 1.1 even 1 trivial
531.6.a.d.1.1 12 3.2 odd 2