Properties

Label 177.6.a.b.1.6
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 1.6
Root \(1.62334\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.62334 q^{2} -9.00000 q^{3} -29.3648 q^{4} +103.513 q^{5} +14.6101 q^{6} -137.577 q^{7} +99.6159 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.62334 q^{2} -9.00000 q^{3} -29.3648 q^{4} +103.513 q^{5} +14.6101 q^{6} -137.577 q^{7} +99.6159 q^{8} +81.0000 q^{9} -168.037 q^{10} -636.961 q^{11} +264.283 q^{12} +834.694 q^{13} +223.334 q^{14} -931.615 q^{15} +777.962 q^{16} +444.945 q^{17} -131.491 q^{18} -513.762 q^{19} -3039.63 q^{20} +1238.19 q^{21} +1034.01 q^{22} +3568.14 q^{23} -896.543 q^{24} +7589.90 q^{25} -1354.99 q^{26} -729.000 q^{27} +4039.91 q^{28} -6879.19 q^{29} +1512.33 q^{30} -3639.93 q^{31} -4450.61 q^{32} +5732.65 q^{33} -722.298 q^{34} -14241.0 q^{35} -2378.55 q^{36} -7911.65 q^{37} +834.010 q^{38} -7512.25 q^{39} +10311.5 q^{40} -19911.3 q^{41} -2010.01 q^{42} -2563.64 q^{43} +18704.2 q^{44} +8384.54 q^{45} -5792.30 q^{46} -22255.9 q^{47} -7001.66 q^{48} +2120.41 q^{49} -12321.0 q^{50} -4004.51 q^{51} -24510.6 q^{52} +29943.8 q^{53} +1183.42 q^{54} -65933.7 q^{55} -13704.9 q^{56} +4623.85 q^{57} +11167.3 q^{58} -3481.00 q^{59} +27356.7 q^{60} -34505.5 q^{61} +5908.84 q^{62} -11143.7 q^{63} -17669.9 q^{64} +86401.6 q^{65} -9306.05 q^{66} +49378.2 q^{67} -13065.7 q^{68} -32113.2 q^{69} +23118.0 q^{70} -32409.2 q^{71} +8068.89 q^{72} -7831.86 q^{73} +12843.3 q^{74} -68309.1 q^{75} +15086.5 q^{76} +87631.2 q^{77} +12194.9 q^{78} -69257.0 q^{79} +80529.0 q^{80} +6561.00 q^{81} +32322.8 q^{82} -30470.2 q^{83} -36359.2 q^{84} +46057.5 q^{85} +4161.67 q^{86} +61912.7 q^{87} -63451.5 q^{88} +20158.9 q^{89} -13611.0 q^{90} -114835. q^{91} -104777. q^{92} +32759.3 q^{93} +36128.9 q^{94} -53180.9 q^{95} +40055.5 q^{96} -91622.0 q^{97} -3442.14 q^{98} -51593.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.62334 −0.286969 −0.143484 0.989653i \(-0.545831\pi\)
−0.143484 + 0.989653i \(0.545831\pi\)
\(3\) −9.00000 −0.577350
\(4\) −29.3648 −0.917649
\(5\) 103.513 1.85169 0.925847 0.377899i \(-0.123353\pi\)
0.925847 + 0.377899i \(0.123353\pi\)
\(6\) 14.6101 0.165682
\(7\) −137.577 −1.06121 −0.530604 0.847620i \(-0.678035\pi\)
−0.530604 + 0.847620i \(0.678035\pi\)
\(8\) 99.6159 0.550305
\(9\) 81.0000 0.333333
\(10\) −168.037 −0.531378
\(11\) −636.961 −1.58720 −0.793599 0.608441i \(-0.791795\pi\)
−0.793599 + 0.608441i \(0.791795\pi\)
\(12\) 264.283 0.529805
\(13\) 834.694 1.36984 0.684919 0.728619i \(-0.259838\pi\)
0.684919 + 0.728619i \(0.259838\pi\)
\(14\) 223.334 0.304534
\(15\) −931.615 −1.06908
\(16\) 777.962 0.759728
\(17\) 444.945 0.373408 0.186704 0.982416i \(-0.440219\pi\)
0.186704 + 0.982416i \(0.440219\pi\)
\(18\) −131.491 −0.0956563
\(19\) −513.762 −0.326496 −0.163248 0.986585i \(-0.552197\pi\)
−0.163248 + 0.986585i \(0.552197\pi\)
\(20\) −3039.63 −1.69920
\(21\) 1238.19 0.612689
\(22\) 1034.01 0.455477
\(23\) 3568.14 1.40644 0.703221 0.710971i \(-0.251745\pi\)
0.703221 + 0.710971i \(0.251745\pi\)
\(24\) −896.543 −0.317719
\(25\) 7589.90 2.42877
\(26\) −1354.99 −0.393101
\(27\) −729.000 −0.192450
\(28\) 4039.91 0.973816
\(29\) −6879.19 −1.51895 −0.759473 0.650538i \(-0.774543\pi\)
−0.759473 + 0.650538i \(0.774543\pi\)
\(30\) 1512.33 0.306791
\(31\) −3639.93 −0.680281 −0.340140 0.940375i \(-0.610475\pi\)
−0.340140 + 0.940375i \(0.610475\pi\)
\(32\) −4450.61 −0.768324
\(33\) 5732.65 0.916370
\(34\) −722.298 −0.107157
\(35\) −14241.0 −1.96503
\(36\) −2378.55 −0.305883
\(37\) −7911.65 −0.950086 −0.475043 0.879963i \(-0.657568\pi\)
−0.475043 + 0.879963i \(0.657568\pi\)
\(38\) 834.010 0.0936941
\(39\) −7512.25 −0.790876
\(40\) 10311.5 1.01900
\(41\) −19911.3 −1.84986 −0.924931 0.380135i \(-0.875878\pi\)
−0.924931 + 0.380135i \(0.875878\pi\)
\(42\) −2010.01 −0.175823
\(43\) −2563.64 −0.211440 −0.105720 0.994396i \(-0.533715\pi\)
−0.105720 + 0.994396i \(0.533715\pi\)
\(44\) 18704.2 1.45649
\(45\) 8384.54 0.617231
\(46\) −5792.30 −0.403605
\(47\) −22255.9 −1.46961 −0.734803 0.678281i \(-0.762725\pi\)
−0.734803 + 0.678281i \(0.762725\pi\)
\(48\) −7001.66 −0.438629
\(49\) 2120.41 0.126162
\(50\) −12321.0 −0.696981
\(51\) −4004.51 −0.215587
\(52\) −24510.6 −1.25703
\(53\) 29943.8 1.46426 0.732129 0.681166i \(-0.238527\pi\)
0.732129 + 0.681166i \(0.238527\pi\)
\(54\) 1183.42 0.0552272
\(55\) −65933.7 −2.93901
\(56\) −13704.9 −0.583989
\(57\) 4623.85 0.188502
\(58\) 11167.3 0.435890
\(59\) −3481.00 −0.130189
\(60\) 27356.7 0.981036
\(61\) −34505.5 −1.18731 −0.593655 0.804720i \(-0.702316\pi\)
−0.593655 + 0.804720i \(0.702316\pi\)
\(62\) 5908.84 0.195219
\(63\) −11143.7 −0.353736
\(64\) −17669.9 −0.539243
\(65\) 86401.6 2.53652
\(66\) −9306.05 −0.262970
\(67\) 49378.2 1.34384 0.671921 0.740623i \(-0.265470\pi\)
0.671921 + 0.740623i \(0.265470\pi\)
\(68\) −13065.7 −0.342658
\(69\) −32113.2 −0.812010
\(70\) 23118.0 0.563903
\(71\) −32409.2 −0.762996 −0.381498 0.924370i \(-0.624592\pi\)
−0.381498 + 0.924370i \(0.624592\pi\)
\(72\) 8068.89 0.183435
\(73\) −7831.86 −0.172012 −0.0860058 0.996295i \(-0.527410\pi\)
−0.0860058 + 0.996295i \(0.527410\pi\)
\(74\) 12843.3 0.272645
\(75\) −68309.1 −1.40225
\(76\) 15086.5 0.299609
\(77\) 87631.2 1.68435
\(78\) 12194.9 0.226957
\(79\) −69257.0 −1.24852 −0.624261 0.781216i \(-0.714600\pi\)
−0.624261 + 0.781216i \(0.714600\pi\)
\(80\) 80529.0 1.40678
\(81\) 6561.00 0.111111
\(82\) 32322.8 0.530853
\(83\) −30470.2 −0.485489 −0.242745 0.970090i \(-0.578048\pi\)
−0.242745 + 0.970090i \(0.578048\pi\)
\(84\) −36359.2 −0.562233
\(85\) 46057.5 0.691438
\(86\) 4161.67 0.0606766
\(87\) 61912.7 0.876964
\(88\) −63451.5 −0.873444
\(89\) 20158.9 0.269769 0.134884 0.990861i \(-0.456934\pi\)
0.134884 + 0.990861i \(0.456934\pi\)
\(90\) −13611.0 −0.177126
\(91\) −114835. −1.45368
\(92\) −104777. −1.29062
\(93\) 32759.3 0.392760
\(94\) 36128.9 0.421731
\(95\) −53180.9 −0.604570
\(96\) 40055.5 0.443592
\(97\) −91622.0 −0.988713 −0.494357 0.869259i \(-0.664596\pi\)
−0.494357 + 0.869259i \(0.664596\pi\)
\(98\) −3442.14 −0.0362046
\(99\) −51593.9 −0.529066
\(100\) −222876. −2.22876
\(101\) −95282.8 −0.929418 −0.464709 0.885464i \(-0.653841\pi\)
−0.464709 + 0.885464i \(0.653841\pi\)
\(102\) 6500.68 0.0618669
\(103\) −9335.09 −0.0867013 −0.0433506 0.999060i \(-0.513803\pi\)
−0.0433506 + 0.999060i \(0.513803\pi\)
\(104\) 83148.9 0.753829
\(105\) 128169. 1.13451
\(106\) −48609.0 −0.420196
\(107\) −191457. −1.61663 −0.808315 0.588750i \(-0.799620\pi\)
−0.808315 + 0.588750i \(0.799620\pi\)
\(108\) 21406.9 0.176602
\(109\) −49018.5 −0.395179 −0.197589 0.980285i \(-0.563311\pi\)
−0.197589 + 0.980285i \(0.563311\pi\)
\(110\) 107033. 0.843403
\(111\) 71204.9 0.548533
\(112\) −107030. −0.806230
\(113\) 194538. 1.43321 0.716603 0.697481i \(-0.245696\pi\)
0.716603 + 0.697481i \(0.245696\pi\)
\(114\) −7506.09 −0.0540943
\(115\) 369348. 2.60430
\(116\) 202006. 1.39386
\(117\) 67610.2 0.456613
\(118\) 5650.85 0.0373602
\(119\) −61214.2 −0.396264
\(120\) −92803.7 −0.588318
\(121\) 244669. 1.51920
\(122\) 56014.2 0.340721
\(123\) 179201. 1.06802
\(124\) 106886. 0.624259
\(125\) 462175. 2.64564
\(126\) 18090.1 0.101511
\(127\) −14170.2 −0.0779593 −0.0389796 0.999240i \(-0.512411\pi\)
−0.0389796 + 0.999240i \(0.512411\pi\)
\(128\) 171104. 0.923070
\(129\) 23072.8 0.122075
\(130\) −140259. −0.727902
\(131\) 194270. 0.989069 0.494534 0.869158i \(-0.335339\pi\)
0.494534 + 0.869158i \(0.335339\pi\)
\(132\) −168338. −0.840906
\(133\) 70681.7 0.346480
\(134\) −80157.6 −0.385641
\(135\) −75460.8 −0.356359
\(136\) 44323.6 0.205489
\(137\) −224935. −1.02389 −0.511947 0.859017i \(-0.671076\pi\)
−0.511947 + 0.859017i \(0.671076\pi\)
\(138\) 52130.7 0.233021
\(139\) −94653.9 −0.415529 −0.207765 0.978179i \(-0.566619\pi\)
−0.207765 + 0.978179i \(0.566619\pi\)
\(140\) 418183. 1.80321
\(141\) 200303. 0.848477
\(142\) 52611.2 0.218956
\(143\) −531668. −2.17420
\(144\) 63014.9 0.253243
\(145\) −712085. −2.81262
\(146\) 12713.8 0.0493620
\(147\) −19083.7 −0.0728397
\(148\) 232324. 0.871846
\(149\) 31008.8 0.114425 0.0572123 0.998362i \(-0.481779\pi\)
0.0572123 + 0.998362i \(0.481779\pi\)
\(150\) 110889. 0.402402
\(151\) −74478.6 −0.265821 −0.132910 0.991128i \(-0.542432\pi\)
−0.132910 + 0.991128i \(0.542432\pi\)
\(152\) −51178.8 −0.179672
\(153\) 36040.6 0.124469
\(154\) −142255. −0.483355
\(155\) −376779. −1.25967
\(156\) 220595. 0.725747
\(157\) 438750. 1.42059 0.710293 0.703906i \(-0.248562\pi\)
0.710293 + 0.703906i \(0.248562\pi\)
\(158\) 112428. 0.358287
\(159\) −269494. −0.845390
\(160\) −460695. −1.42270
\(161\) −490893. −1.49253
\(162\) −10650.7 −0.0318854
\(163\) 489187. 1.44214 0.721068 0.692864i \(-0.243651\pi\)
0.721068 + 0.692864i \(0.243651\pi\)
\(164\) 584690. 1.69752
\(165\) 593403. 1.69684
\(166\) 49463.5 0.139320
\(167\) 277073. 0.768781 0.384391 0.923171i \(-0.374412\pi\)
0.384391 + 0.923171i \(0.374412\pi\)
\(168\) 123344. 0.337166
\(169\) 325422. 0.876455
\(170\) −74767.1 −0.198421
\(171\) −41614.7 −0.108832
\(172\) 75280.8 0.194027
\(173\) −402349. −1.02209 −0.511043 0.859555i \(-0.670741\pi\)
−0.511043 + 0.859555i \(0.670741\pi\)
\(174\) −100506. −0.251661
\(175\) −1.04420e6 −2.57743
\(176\) −495532. −1.20584
\(177\) 31329.0 0.0751646
\(178\) −32724.7 −0.0774152
\(179\) −492255. −1.14831 −0.574153 0.818748i \(-0.694668\pi\)
−0.574153 + 0.818748i \(0.694668\pi\)
\(180\) −246210. −0.566401
\(181\) 83424.0 0.189276 0.0946378 0.995512i \(-0.469831\pi\)
0.0946378 + 0.995512i \(0.469831\pi\)
\(182\) 186416. 0.417162
\(183\) 310550. 0.685494
\(184\) 355443. 0.773973
\(185\) −818958. −1.75927
\(186\) −53179.6 −0.112710
\(187\) −283413. −0.592673
\(188\) 653540. 1.34858
\(189\) 100294. 0.204230
\(190\) 86330.8 0.173493
\(191\) 761817. 1.51101 0.755505 0.655143i \(-0.227392\pi\)
0.755505 + 0.655143i \(0.227392\pi\)
\(192\) 159029. 0.311332
\(193\) −245032. −0.473510 −0.236755 0.971569i \(-0.576084\pi\)
−0.236755 + 0.971569i \(0.576084\pi\)
\(194\) 148734. 0.283730
\(195\) −777614. −1.46446
\(196\) −62265.3 −0.115773
\(197\) 40850.3 0.0749945 0.0374973 0.999297i \(-0.488061\pi\)
0.0374973 + 0.999297i \(0.488061\pi\)
\(198\) 83754.4 0.151826
\(199\) −167233. −0.299357 −0.149679 0.988735i \(-0.547824\pi\)
−0.149679 + 0.988735i \(0.547824\pi\)
\(200\) 756075. 1.33656
\(201\) −444404. −0.775867
\(202\) 154676. 0.266714
\(203\) 946418. 1.61192
\(204\) 117591. 0.197834
\(205\) −2.06107e6 −3.42538
\(206\) 15154.0 0.0248806
\(207\) 289019. 0.468814
\(208\) 649360. 1.04070
\(209\) 327246. 0.518214
\(210\) −208062. −0.325569
\(211\) −489590. −0.757054 −0.378527 0.925590i \(-0.623569\pi\)
−0.378527 + 0.925590i \(0.623569\pi\)
\(212\) −879293. −1.34367
\(213\) 291683. 0.440516
\(214\) 310799. 0.463923
\(215\) −265370. −0.391521
\(216\) −72620.0 −0.105906
\(217\) 500770. 0.721919
\(218\) 79573.7 0.113404
\(219\) 70486.8 0.0993110
\(220\) 1.93613e6 2.69698
\(221\) 371393. 0.511509
\(222\) −115590. −0.157412
\(223\) −856174. −1.15292 −0.576461 0.817125i \(-0.695567\pi\)
−0.576461 + 0.817125i \(0.695567\pi\)
\(224\) 612301. 0.815351
\(225\) 614782. 0.809590
\(226\) −315802. −0.411286
\(227\) −1.00899e6 −1.29964 −0.649821 0.760087i \(-0.725156\pi\)
−0.649821 + 0.760087i \(0.725156\pi\)
\(228\) −135778. −0.172979
\(229\) −507970. −0.640102 −0.320051 0.947400i \(-0.603700\pi\)
−0.320051 + 0.947400i \(0.603700\pi\)
\(230\) −599577. −0.747353
\(231\) −788681. −0.972459
\(232\) −685277. −0.835885
\(233\) −156420. −0.188756 −0.0943782 0.995536i \(-0.530086\pi\)
−0.0943782 + 0.995536i \(0.530086\pi\)
\(234\) −109754. −0.131034
\(235\) −2.30377e6 −2.72126
\(236\) 102219. 0.119468
\(237\) 623313. 0.720834
\(238\) 99371.5 0.113715
\(239\) 518961. 0.587678 0.293839 0.955855i \(-0.405067\pi\)
0.293839 + 0.955855i \(0.405067\pi\)
\(240\) −724761. −0.812207
\(241\) 151673. 0.168216 0.0841079 0.996457i \(-0.473196\pi\)
0.0841079 + 0.996457i \(0.473196\pi\)
\(242\) −397181. −0.435963
\(243\) −59049.0 −0.0641500
\(244\) 1.01325e6 1.08953
\(245\) 219489. 0.233614
\(246\) −290905. −0.306488
\(247\) −428834. −0.447246
\(248\) −362595. −0.374362
\(249\) 274232. 0.280297
\(250\) −750267. −0.759217
\(251\) 698841. 0.700155 0.350077 0.936721i \(-0.386155\pi\)
0.350077 + 0.936721i \(0.386155\pi\)
\(252\) 327233. 0.324605
\(253\) −2.27276e6 −2.23230
\(254\) 23003.1 0.0223719
\(255\) −414518. −0.399202
\(256\) 287678. 0.274351
\(257\) 460742. 0.435136 0.217568 0.976045i \(-0.430188\pi\)
0.217568 + 0.976045i \(0.430188\pi\)
\(258\) −37455.0 −0.0350316
\(259\) 1.08846e6 1.00824
\(260\) −2.53716e6 −2.32763
\(261\) −557215. −0.506316
\(262\) −315366. −0.283832
\(263\) 1.07274e6 0.956325 0.478163 0.878271i \(-0.341303\pi\)
0.478163 + 0.878271i \(0.341303\pi\)
\(264\) 571063. 0.504283
\(265\) 3.09957e6 2.71136
\(266\) −114741. −0.0994290
\(267\) −181430. −0.155751
\(268\) −1.44998e6 −1.23317
\(269\) 581921. 0.490324 0.245162 0.969482i \(-0.421159\pi\)
0.245162 + 0.969482i \(0.421159\pi\)
\(270\) 122499. 0.102264
\(271\) −1.54968e6 −1.28179 −0.640897 0.767627i \(-0.721438\pi\)
−0.640897 + 0.767627i \(0.721438\pi\)
\(272\) 346150. 0.283689
\(273\) 1.03351e6 0.839284
\(274\) 365146. 0.293826
\(275\) −4.83447e6 −3.85494
\(276\) 942997. 0.745140
\(277\) 1.52988e6 1.19800 0.599000 0.800749i \(-0.295565\pi\)
0.599000 + 0.800749i \(0.295565\pi\)
\(278\) 153655. 0.119244
\(279\) −294834. −0.226760
\(280\) −1.41863e6 −1.08137
\(281\) 942600. 0.712134 0.356067 0.934460i \(-0.384118\pi\)
0.356067 + 0.934460i \(0.384118\pi\)
\(282\) −325161. −0.243487
\(283\) −675204. −0.501151 −0.250576 0.968097i \(-0.580620\pi\)
−0.250576 + 0.968097i \(0.580620\pi\)
\(284\) 951688. 0.700162
\(285\) 478628. 0.349049
\(286\) 863079. 0.623929
\(287\) 2.73933e6 1.96309
\(288\) −360499. −0.256108
\(289\) −1.22188e6 −0.860566
\(290\) 1.15596e6 0.807136
\(291\) 824598. 0.570834
\(292\) 229981. 0.157846
\(293\) −475173. −0.323357 −0.161679 0.986843i \(-0.551691\pi\)
−0.161679 + 0.986843i \(0.551691\pi\)
\(294\) 30979.3 0.0209027
\(295\) −360328. −0.241070
\(296\) −788127. −0.522838
\(297\) 464345. 0.305457
\(298\) −50337.9 −0.0328363
\(299\) 2.97830e6 1.92660
\(300\) 2.00588e6 1.28677
\(301\) 352698. 0.224381
\(302\) 120904. 0.0762823
\(303\) 857545. 0.536600
\(304\) −399687. −0.248048
\(305\) −3.57177e6 −2.19853
\(306\) −58506.1 −0.0357189
\(307\) 1.72181e6 1.04265 0.521325 0.853358i \(-0.325438\pi\)
0.521325 + 0.853358i \(0.325438\pi\)
\(308\) −2.57327e6 −1.54564
\(309\) 84015.8 0.0500570
\(310\) 611641. 0.361487
\(311\) 234169. 0.137287 0.0686435 0.997641i \(-0.478133\pi\)
0.0686435 + 0.997641i \(0.478133\pi\)
\(312\) −748340. −0.435224
\(313\) −425464. −0.245472 −0.122736 0.992439i \(-0.539167\pi\)
−0.122736 + 0.992439i \(0.539167\pi\)
\(314\) −712240. −0.407664
\(315\) −1.15352e6 −0.655011
\(316\) 2.03372e6 1.14570
\(317\) −3.34935e6 −1.87203 −0.936014 0.351963i \(-0.885514\pi\)
−0.936014 + 0.351963i \(0.885514\pi\)
\(318\) 437481. 0.242601
\(319\) 4.38178e6 2.41087
\(320\) −1.82906e6 −0.998513
\(321\) 1.72311e6 0.933362
\(322\) 796887. 0.428309
\(323\) −228596. −0.121916
\(324\) −192662. −0.101961
\(325\) 6.33525e6 3.32702
\(326\) −794118. −0.413848
\(327\) 441166. 0.228157
\(328\) −1.98348e6 −1.01799
\(329\) 3.06190e6 1.55956
\(330\) −963295. −0.486939
\(331\) −3.15018e6 −1.58039 −0.790197 0.612853i \(-0.790022\pi\)
−0.790197 + 0.612853i \(0.790022\pi\)
\(332\) 894750. 0.445509
\(333\) −640844. −0.316695
\(334\) −449784. −0.220616
\(335\) 5.11127e6 2.48838
\(336\) 963266. 0.465477
\(337\) 1.88668e6 0.904950 0.452475 0.891777i \(-0.350541\pi\)
0.452475 + 0.891777i \(0.350541\pi\)
\(338\) −528270. −0.251515
\(339\) −1.75084e6 −0.827462
\(340\) −1.35247e6 −0.634497
\(341\) 2.31849e6 1.07974
\(342\) 67554.8 0.0312314
\(343\) 2.02054e6 0.927324
\(344\) −255380. −0.116356
\(345\) −3.32413e6 −1.50359
\(346\) 653149. 0.293307
\(347\) −161465. −0.0719872 −0.0359936 0.999352i \(-0.511460\pi\)
−0.0359936 + 0.999352i \(0.511460\pi\)
\(348\) −1.81805e6 −0.804745
\(349\) −2.48443e6 −1.09185 −0.545925 0.837834i \(-0.683821\pi\)
−0.545925 + 0.837834i \(0.683821\pi\)
\(350\) 1.69509e6 0.739642
\(351\) −608492. −0.263625
\(352\) 2.83486e6 1.21948
\(353\) 503098. 0.214890 0.107445 0.994211i \(-0.465733\pi\)
0.107445 + 0.994211i \(0.465733\pi\)
\(354\) −50857.6 −0.0215699
\(355\) −3.35477e6 −1.41283
\(356\) −591961. −0.247553
\(357\) 550928. 0.228783
\(358\) 799097. 0.329528
\(359\) 1.49405e6 0.611826 0.305913 0.952059i \(-0.401038\pi\)
0.305913 + 0.952059i \(0.401038\pi\)
\(360\) 835234. 0.339666
\(361\) −2.21215e6 −0.893400
\(362\) −135426. −0.0543162
\(363\) −2.20202e6 −0.877111
\(364\) 3.37209e6 1.33397
\(365\) −810698. −0.318513
\(366\) −504128. −0.196715
\(367\) 2.46095e6 0.953758 0.476879 0.878969i \(-0.341768\pi\)
0.476879 + 0.878969i \(0.341768\pi\)
\(368\) 2.77587e6 1.06851
\(369\) −1.61281e6 −0.616621
\(370\) 1.32945e6 0.504855
\(371\) −4.11958e6 −1.55388
\(372\) −961970. −0.360416
\(373\) −4.12944e6 −1.53681 −0.768404 0.639965i \(-0.778949\pi\)
−0.768404 + 0.639965i \(0.778949\pi\)
\(374\) 460076. 0.170079
\(375\) −4.15957e6 −1.52746
\(376\) −2.21704e6 −0.808732
\(377\) −5.74203e6 −2.08071
\(378\) −162811. −0.0586075
\(379\) 24757.7 0.00885345 0.00442672 0.999990i \(-0.498591\pi\)
0.00442672 + 0.999990i \(0.498591\pi\)
\(380\) 1.56165e6 0.554783
\(381\) 127532. 0.0450098
\(382\) −1.23669e6 −0.433613
\(383\) 721550. 0.251344 0.125672 0.992072i \(-0.459891\pi\)
0.125672 + 0.992072i \(0.459891\pi\)
\(384\) −1.53993e6 −0.532935
\(385\) 9.07095e6 3.11890
\(386\) 397770. 0.135883
\(387\) −207655. −0.0704799
\(388\) 2.69046e6 0.907292
\(389\) 2.77630e6 0.930234 0.465117 0.885249i \(-0.346012\pi\)
0.465117 + 0.885249i \(0.346012\pi\)
\(390\) 1.26233e6 0.420254
\(391\) 1.58762e6 0.525177
\(392\) 211226. 0.0694277
\(393\) −1.74843e6 −0.571039
\(394\) −66313.9 −0.0215211
\(395\) −7.16899e6 −2.31188
\(396\) 1.51504e6 0.485497
\(397\) 2.50395e6 0.797350 0.398675 0.917092i \(-0.369470\pi\)
0.398675 + 0.917092i \(0.369470\pi\)
\(398\) 271477. 0.0859063
\(399\) −636136. −0.200040
\(400\) 5.90465e6 1.84520
\(401\) −3.85075e6 −1.19587 −0.597935 0.801544i \(-0.704012\pi\)
−0.597935 + 0.801544i \(0.704012\pi\)
\(402\) 721419. 0.222650
\(403\) −3.03823e6 −0.931874
\(404\) 2.79796e6 0.852879
\(405\) 679148. 0.205744
\(406\) −1.53636e6 −0.462570
\(407\) 5.03942e6 1.50798
\(408\) −398913. −0.118639
\(409\) 420063. 0.124167 0.0620835 0.998071i \(-0.480225\pi\)
0.0620835 + 0.998071i \(0.480225\pi\)
\(410\) 3.34582e6 0.982977
\(411\) 2.02441e6 0.591145
\(412\) 274123. 0.0795613
\(413\) 478905. 0.138157
\(414\) −469176. −0.134535
\(415\) −3.15405e6 −0.898978
\(416\) −3.71490e6 −1.05248
\(417\) 851885. 0.239906
\(418\) −531232. −0.148711
\(419\) −3.16478e6 −0.880661 −0.440330 0.897836i \(-0.645139\pi\)
−0.440330 + 0.897836i \(0.645139\pi\)
\(420\) −3.76365e6 −1.04108
\(421\) −6.88209e6 −1.89241 −0.946205 0.323567i \(-0.895118\pi\)
−0.946205 + 0.323567i \(0.895118\pi\)
\(422\) 794772. 0.217251
\(423\) −1.80273e6 −0.489869
\(424\) 2.98288e6 0.805789
\(425\) 3.37709e6 0.906923
\(426\) −473500. −0.126414
\(427\) 4.74717e6 1.25998
\(428\) 5.62208e6 1.48350
\(429\) 4.78501e6 1.25528
\(430\) 430786. 0.112354
\(431\) 6.64605e6 1.72334 0.861668 0.507472i \(-0.169420\pi\)
0.861668 + 0.507472i \(0.169420\pi\)
\(432\) −567134. −0.146210
\(433\) −5.40532e6 −1.38548 −0.692742 0.721186i \(-0.743597\pi\)
−0.692742 + 0.721186i \(0.743597\pi\)
\(434\) −812920. −0.207168
\(435\) 6.40876e6 1.62387
\(436\) 1.43942e6 0.362635
\(437\) −1.83317e6 −0.459197
\(438\) −114424. −0.0284992
\(439\) 3.49764e6 0.866190 0.433095 0.901348i \(-0.357421\pi\)
0.433095 + 0.901348i \(0.357421\pi\)
\(440\) −6.56804e6 −1.61735
\(441\) 171753. 0.0420540
\(442\) −602898. −0.146787
\(443\) 7.83503e6 1.89684 0.948421 0.317013i \(-0.102680\pi\)
0.948421 + 0.317013i \(0.102680\pi\)
\(444\) −2.09091e6 −0.503360
\(445\) 2.08670e6 0.499529
\(446\) 1.38986e6 0.330853
\(447\) −279079. −0.0660631
\(448\) 2.43097e6 0.572249
\(449\) 3.72658e6 0.872357 0.436178 0.899860i \(-0.356332\pi\)
0.436178 + 0.899860i \(0.356332\pi\)
\(450\) −998001. −0.232327
\(451\) 1.26827e7 2.93610
\(452\) −5.71257e6 −1.31518
\(453\) 670307. 0.153472
\(454\) 1.63794e6 0.372957
\(455\) −1.18869e7 −2.69177
\(456\) 460610. 0.103734
\(457\) −1.25040e6 −0.280065 −0.140033 0.990147i \(-0.544721\pi\)
−0.140033 + 0.990147i \(0.544721\pi\)
\(458\) 824608. 0.183689
\(459\) −324365. −0.0718625
\(460\) −1.08458e7 −2.38983
\(461\) 3.23924e6 0.709890 0.354945 0.934887i \(-0.384500\pi\)
0.354945 + 0.934887i \(0.384500\pi\)
\(462\) 1.28030e6 0.279065
\(463\) 481992. 0.104493 0.0522465 0.998634i \(-0.483362\pi\)
0.0522465 + 0.998634i \(0.483362\pi\)
\(464\) −5.35175e6 −1.15399
\(465\) 3.39101e6 0.727272
\(466\) 253923. 0.0541672
\(467\) 4.61594e6 0.979418 0.489709 0.871886i \(-0.337103\pi\)
0.489709 + 0.871886i \(0.337103\pi\)
\(468\) −1.98536e6 −0.419010
\(469\) −6.79330e6 −1.42610
\(470\) 3.73981e6 0.780917
\(471\) −3.94875e6 −0.820176
\(472\) −346763. −0.0716437
\(473\) 1.63294e6 0.335597
\(474\) −1.01185e6 −0.206857
\(475\) −3.89940e6 −0.792983
\(476\) 1.79754e6 0.363631
\(477\) 2.42545e6 0.488086
\(478\) −842450. −0.168645
\(479\) −6.61817e6 −1.31795 −0.658975 0.752165i \(-0.729010\pi\)
−0.658975 + 0.752165i \(0.729010\pi\)
\(480\) 4.14625e6 0.821396
\(481\) −6.60381e6 −1.30146
\(482\) −246218. −0.0482727
\(483\) 4.41804e6 0.861711
\(484\) −7.18464e6 −1.39409
\(485\) −9.48405e6 −1.83079
\(486\) 95856.7 0.0184091
\(487\) 5.72990e6 1.09477 0.547387 0.836879i \(-0.315623\pi\)
0.547387 + 0.836879i \(0.315623\pi\)
\(488\) −3.43730e6 −0.653383
\(489\) −4.40269e6 −0.832618
\(490\) −356306. −0.0670398
\(491\) −3.96105e6 −0.741492 −0.370746 0.928734i \(-0.620898\pi\)
−0.370746 + 0.928734i \(0.620898\pi\)
\(492\) −5.26221e6 −0.980066
\(493\) −3.06086e6 −0.567188
\(494\) 696144. 0.128346
\(495\) −5.34063e6 −0.979669
\(496\) −2.83172e6 −0.516829
\(497\) 4.45876e6 0.809697
\(498\) −445171. −0.0804366
\(499\) −2.33209e6 −0.419269 −0.209635 0.977780i \(-0.567227\pi\)
−0.209635 + 0.977780i \(0.567227\pi\)
\(500\) −1.35717e7 −2.42777
\(501\) −2.49366e6 −0.443856
\(502\) −1.13446e6 −0.200923
\(503\) −6.88900e6 −1.21405 −0.607024 0.794683i \(-0.707637\pi\)
−0.607024 + 0.794683i \(0.707637\pi\)
\(504\) −1.11009e6 −0.194663
\(505\) −9.86299e6 −1.72100
\(506\) 3.68947e6 0.640601
\(507\) −2.92880e6 −0.506022
\(508\) 416106. 0.0715392
\(509\) 1.16204e6 0.198804 0.0994019 0.995047i \(-0.468307\pi\)
0.0994019 + 0.995047i \(0.468307\pi\)
\(510\) 672904. 0.114559
\(511\) 1.07748e6 0.182540
\(512\) −5.94232e6 −1.00180
\(513\) 374532. 0.0628342
\(514\) −747941. −0.124870
\(515\) −966301. −0.160544
\(516\) −677527. −0.112022
\(517\) 1.41762e7 2.33256
\(518\) −1.76694e6 −0.289333
\(519\) 3.62114e6 0.590101
\(520\) 8.60697e6 1.39586
\(521\) 2.98863e6 0.482367 0.241184 0.970480i \(-0.422464\pi\)
0.241184 + 0.970480i \(0.422464\pi\)
\(522\) 904550. 0.145297
\(523\) 9.74843e6 1.55840 0.779202 0.626772i \(-0.215624\pi\)
0.779202 + 0.626772i \(0.215624\pi\)
\(524\) −5.70468e6 −0.907618
\(525\) 9.39776e6 1.48808
\(526\) −1.74143e6 −0.274436
\(527\) −1.61957e6 −0.254023
\(528\) 4.45978e6 0.696192
\(529\) 6.29525e6 0.978078
\(530\) −5.03166e6 −0.778075
\(531\) −281961. −0.0433963
\(532\) −2.07555e6 −0.317947
\(533\) −1.66198e7 −2.53401
\(534\) 294523. 0.0446957
\(535\) −1.98182e7 −2.99350
\(536\) 4.91885e6 0.739523
\(537\) 4.43029e6 0.662974
\(538\) −944656. −0.140708
\(539\) −1.35062e6 −0.200244
\(540\) 2.21589e6 0.327012
\(541\) −977703. −0.143620 −0.0718098 0.997418i \(-0.522877\pi\)
−0.0718098 + 0.997418i \(0.522877\pi\)
\(542\) 2.51566e6 0.367835
\(543\) −750816. −0.109278
\(544\) −1.98028e6 −0.286899
\(545\) −5.07404e6 −0.731750
\(546\) −1.67774e6 −0.240848
\(547\) 3.99104e6 0.570319 0.285159 0.958480i \(-0.407953\pi\)
0.285159 + 0.958480i \(0.407953\pi\)
\(548\) 6.60515e6 0.939575
\(549\) −2.79495e6 −0.395770
\(550\) 7.84800e6 1.10625
\(551\) 3.53427e6 0.495930
\(552\) −3.19899e6 −0.446853
\(553\) 9.52817e6 1.32494
\(554\) −2.48351e6 −0.343789
\(555\) 7.37062e6 1.01571
\(556\) 2.77949e6 0.381310
\(557\) −2.64820e6 −0.361670 −0.180835 0.983513i \(-0.557880\pi\)
−0.180835 + 0.983513i \(0.557880\pi\)
\(558\) 478616. 0.0650731
\(559\) −2.13986e6 −0.289638
\(560\) −1.10789e7 −1.49289
\(561\) 2.55072e6 0.342180
\(562\) −1.53016e6 −0.204360
\(563\) 719804. 0.0957069 0.0478534 0.998854i \(-0.484762\pi\)
0.0478534 + 0.998854i \(0.484762\pi\)
\(564\) −5.88186e6 −0.778604
\(565\) 2.01372e7 2.65386
\(566\) 1.09609e6 0.143815
\(567\) −902642. −0.117912
\(568\) −3.22847e6 −0.419881
\(569\) 5.99643e6 0.776448 0.388224 0.921565i \(-0.373089\pi\)
0.388224 + 0.921565i \(0.373089\pi\)
\(570\) −776977. −0.100166
\(571\) −1.31757e7 −1.69115 −0.845577 0.533854i \(-0.820743\pi\)
−0.845577 + 0.533854i \(0.820743\pi\)
\(572\) 1.56123e7 1.99516
\(573\) −6.85635e6 −0.872382
\(574\) −4.44687e6 −0.563345
\(575\) 2.70818e7 3.41592
\(576\) −1.43126e6 −0.179748
\(577\) −9.18759e6 −1.14885 −0.574424 0.818558i \(-0.694774\pi\)
−0.574424 + 0.818558i \(0.694774\pi\)
\(578\) 1.98353e6 0.246956
\(579\) 2.20529e6 0.273381
\(580\) 2.09102e7 2.58100
\(581\) 4.19199e6 0.515205
\(582\) −1.33860e6 −0.163812
\(583\) −1.90731e7 −2.32407
\(584\) −780178. −0.0946590
\(585\) 6.99853e6 0.845507
\(586\) 771368. 0.0927935
\(587\) −4.96246e6 −0.594431 −0.297215 0.954810i \(-0.596058\pi\)
−0.297215 + 0.954810i \(0.596058\pi\)
\(588\) 560387. 0.0668413
\(589\) 1.87005e6 0.222109
\(590\) 584935. 0.0691796
\(591\) −367652. −0.0432981
\(592\) −6.15497e6 −0.721807
\(593\) 1.07757e7 1.25837 0.629186 0.777255i \(-0.283389\pi\)
0.629186 + 0.777255i \(0.283389\pi\)
\(594\) −753790. −0.0876565
\(595\) −6.33645e6 −0.733759
\(596\) −910567. −0.105002
\(597\) 1.50510e6 0.172834
\(598\) −4.83480e6 −0.552873
\(599\) 9.20870e6 1.04865 0.524326 0.851518i \(-0.324317\pi\)
0.524326 + 0.851518i \(0.324317\pi\)
\(600\) −6.80468e6 −0.771666
\(601\) −6.22345e6 −0.702822 −0.351411 0.936221i \(-0.614298\pi\)
−0.351411 + 0.936221i \(0.614298\pi\)
\(602\) −572549. −0.0643905
\(603\) 3.99963e6 0.447947
\(604\) 2.18705e6 0.243930
\(605\) 2.53263e7 2.81309
\(606\) −1.39209e6 −0.153987
\(607\) −1.27596e7 −1.40561 −0.702805 0.711382i \(-0.748069\pi\)
−0.702805 + 0.711382i \(0.748069\pi\)
\(608\) 2.28655e6 0.250855
\(609\) −8.51776e6 −0.930642
\(610\) 5.79819e6 0.630911
\(611\) −1.85769e7 −2.01312
\(612\) −1.05832e6 −0.114219
\(613\) 8.11365e6 0.872097 0.436049 0.899923i \(-0.356378\pi\)
0.436049 + 0.899923i \(0.356378\pi\)
\(614\) −2.79508e6 −0.299208
\(615\) 1.85496e7 1.97764
\(616\) 8.72946e6 0.926906
\(617\) −1.42296e7 −1.50480 −0.752399 0.658707i \(-0.771104\pi\)
−0.752399 + 0.658707i \(0.771104\pi\)
\(618\) −136386. −0.0143648
\(619\) −1.08278e6 −0.113583 −0.0567916 0.998386i \(-0.518087\pi\)
−0.0567916 + 0.998386i \(0.518087\pi\)
\(620\) 1.10640e7 1.15594
\(621\) −2.60117e6 −0.270670
\(622\) −380137. −0.0393971
\(623\) −2.77340e6 −0.286281
\(624\) −5.84424e6 −0.600851
\(625\) 2.41226e7 2.47015
\(626\) 690673. 0.0704428
\(627\) −2.94522e6 −0.299191
\(628\) −1.28838e7 −1.30360
\(629\) −3.52025e6 −0.354770
\(630\) 1.87255e6 0.187968
\(631\) −5.94918e6 −0.594817 −0.297409 0.954750i \(-0.596122\pi\)
−0.297409 + 0.954750i \(0.596122\pi\)
\(632\) −6.89910e6 −0.687068
\(633\) 4.40631e6 0.437085
\(634\) 5.43714e6 0.537214
\(635\) −1.46680e6 −0.144357
\(636\) 7.91364e6 0.775771
\(637\) 1.76989e6 0.172822
\(638\) −7.11312e6 −0.691845
\(639\) −2.62514e6 −0.254332
\(640\) 1.77114e7 1.70924
\(641\) 1.50581e7 1.44752 0.723760 0.690052i \(-0.242412\pi\)
0.723760 + 0.690052i \(0.242412\pi\)
\(642\) −2.79719e6 −0.267846
\(643\) −7.34341e6 −0.700439 −0.350219 0.936668i \(-0.613893\pi\)
−0.350219 + 0.936668i \(0.613893\pi\)
\(644\) 1.44150e7 1.36962
\(645\) 2.38833e6 0.226045
\(646\) 371089. 0.0349862
\(647\) 1.71639e7 1.61196 0.805980 0.591943i \(-0.201639\pi\)
0.805980 + 0.591943i \(0.201639\pi\)
\(648\) 653580. 0.0611451
\(649\) 2.21726e6 0.206636
\(650\) −1.02843e7 −0.954751
\(651\) −4.50693e6 −0.416800
\(652\) −1.43649e7 −1.32337
\(653\) −1.67889e7 −1.54078 −0.770390 0.637573i \(-0.779938\pi\)
−0.770390 + 0.637573i \(0.779938\pi\)
\(654\) −716163. −0.0654738
\(655\) 2.01094e7 1.83145
\(656\) −1.54902e7 −1.40539
\(657\) −634381. −0.0573372
\(658\) −4.97051e6 −0.447544
\(659\) 1.22668e7 1.10031 0.550157 0.835061i \(-0.314568\pi\)
0.550157 + 0.835061i \(0.314568\pi\)
\(660\) −1.74251e7 −1.55710
\(661\) −2.16263e6 −0.192522 −0.0962608 0.995356i \(-0.530688\pi\)
−0.0962608 + 0.995356i \(0.530688\pi\)
\(662\) 5.11382e6 0.453524
\(663\) −3.34254e6 −0.295320
\(664\) −3.03532e6 −0.267167
\(665\) 7.31647e6 0.641575
\(666\) 1.04031e6 0.0908817
\(667\) −2.45459e7 −2.13631
\(668\) −8.13618e6 −0.705471
\(669\) 7.70557e6 0.665640
\(670\) −8.29734e6 −0.714088
\(671\) 2.19787e7 1.88450
\(672\) −5.51071e6 −0.470743
\(673\) −3.23411e6 −0.275243 −0.137622 0.990485i \(-0.543946\pi\)
−0.137622 + 0.990485i \(0.543946\pi\)
\(674\) −3.06273e6 −0.259692
\(675\) −5.53304e6 −0.467417
\(676\) −9.55593e6 −0.804278
\(677\) 2.05131e7 1.72012 0.860060 0.510193i \(-0.170426\pi\)
0.860060 + 0.510193i \(0.170426\pi\)
\(678\) 2.84222e6 0.237456
\(679\) 1.26051e7 1.04923
\(680\) 4.58806e6 0.380502
\(681\) 9.08094e6 0.750349
\(682\) −3.76370e6 −0.309852
\(683\) −1.31062e6 −0.107504 −0.0537519 0.998554i \(-0.517118\pi\)
−0.0537519 + 0.998554i \(0.517118\pi\)
\(684\) 1.22201e6 0.0998695
\(685\) −2.32836e7 −1.89594
\(686\) −3.28002e6 −0.266113
\(687\) 4.57173e6 0.369563
\(688\) −1.99442e6 −0.160637
\(689\) 2.49939e7 2.00580
\(690\) 5.39620e6 0.431484
\(691\) 2.35958e7 1.87992 0.939959 0.341287i \(-0.110863\pi\)
0.939959 + 0.341287i \(0.110863\pi\)
\(692\) 1.18149e7 0.937916
\(693\) 7.09812e6 0.561449
\(694\) 262113. 0.0206581
\(695\) −9.79789e6 −0.769432
\(696\) 6.16750e6 0.482598
\(697\) −8.85942e6 −0.690754
\(698\) 4.03307e6 0.313327
\(699\) 1.40778e6 0.108979
\(700\) 3.06626e7 2.36517
\(701\) 1.46503e7 1.12604 0.563018 0.826445i \(-0.309640\pi\)
0.563018 + 0.826445i \(0.309640\pi\)
\(702\) 987790. 0.0756523
\(703\) 4.06470e6 0.310199
\(704\) 1.12551e7 0.855886
\(705\) 2.07340e7 1.57112
\(706\) −816699. −0.0616667
\(707\) 1.31087e7 0.986306
\(708\) −919969. −0.0689747
\(709\) −4.71244e6 −0.352071 −0.176035 0.984384i \(-0.556327\pi\)
−0.176035 + 0.984384i \(0.556327\pi\)
\(710\) 5.44593e6 0.405440
\(711\) −5.60982e6 −0.416174
\(712\) 2.00815e6 0.148455
\(713\) −1.29877e7 −0.956775
\(714\) −894343. −0.0656536
\(715\) −5.50345e7 −4.02596
\(716\) 1.44549e7 1.05374
\(717\) −4.67065e6 −0.339296
\(718\) −2.42535e6 −0.175575
\(719\) −7.34095e6 −0.529578 −0.264789 0.964306i \(-0.585302\pi\)
−0.264789 + 0.964306i \(0.585302\pi\)
\(720\) 6.52285e6 0.468928
\(721\) 1.28429e6 0.0920081
\(722\) 3.59107e6 0.256378
\(723\) −1.36506e6 −0.0971195
\(724\) −2.44973e6 −0.173689
\(725\) −5.22124e7 −3.68917
\(726\) 3.57463e6 0.251703
\(727\) 3.78991e6 0.265946 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(728\) −1.14394e7 −0.799970
\(729\) 531441. 0.0370370
\(730\) 1.31604e6 0.0914033
\(731\) −1.14068e6 −0.0789533
\(732\) −9.11922e6 −0.629043
\(733\) 1.24229e7 0.854008 0.427004 0.904250i \(-0.359569\pi\)
0.427004 + 0.904250i \(0.359569\pi\)
\(734\) −3.99496e6 −0.273699
\(735\) −1.97540e6 −0.134877
\(736\) −1.58804e7 −1.08060
\(737\) −3.14520e7 −2.13294
\(738\) 2.61815e6 0.176951
\(739\) −5.27123e6 −0.355059 −0.177530 0.984115i \(-0.556811\pi\)
−0.177530 + 0.984115i \(0.556811\pi\)
\(740\) 2.40485e7 1.61439
\(741\) 3.85951e6 0.258218
\(742\) 6.68748e6 0.445916
\(743\) −1.42533e7 −0.947202 −0.473601 0.880740i \(-0.657046\pi\)
−0.473601 + 0.880740i \(0.657046\pi\)
\(744\) 3.26335e6 0.216138
\(745\) 3.20981e6 0.211879
\(746\) 6.70350e6 0.441016
\(747\) −2.46808e6 −0.161830
\(748\) 8.32235e6 0.543866
\(749\) 2.63400e7 1.71558
\(750\) 6.75240e6 0.438334
\(751\) −5.30422e6 −0.343180 −0.171590 0.985168i \(-0.554890\pi\)
−0.171590 + 0.985168i \(0.554890\pi\)
\(752\) −1.73143e7 −1.11650
\(753\) −6.28957e6 −0.404235
\(754\) 9.32126e6 0.597099
\(755\) −7.70949e6 −0.492219
\(756\) −2.94510e6 −0.187411
\(757\) 1.55013e7 0.983169 0.491585 0.870830i \(-0.336418\pi\)
0.491585 + 0.870830i \(0.336418\pi\)
\(758\) −40190.2 −0.00254066
\(759\) 2.04549e7 1.28882
\(760\) −5.29767e6 −0.332698
\(761\) −1.61620e7 −1.01165 −0.505827 0.862635i \(-0.668813\pi\)
−0.505827 + 0.862635i \(0.668813\pi\)
\(762\) −207028. −0.0129164
\(763\) 6.74381e6 0.419367
\(764\) −2.23706e7 −1.38658
\(765\) 3.73066e6 0.230479
\(766\) −1.17132e6 −0.0721280
\(767\) −2.90557e6 −0.178338
\(768\) −2.58910e6 −0.158397
\(769\) −5.54537e6 −0.338154 −0.169077 0.985603i \(-0.554079\pi\)
−0.169077 + 0.985603i \(0.554079\pi\)
\(770\) −1.47252e7 −0.895026
\(771\) −4.14668e6 −0.251226
\(772\) 7.19531e6 0.434516
\(773\) 124974. 0.00752266 0.00376133 0.999993i \(-0.498803\pi\)
0.00376133 + 0.999993i \(0.498803\pi\)
\(774\) 337095. 0.0202255
\(775\) −2.76267e7 −1.65225
\(776\) −9.12701e6 −0.544094
\(777\) −9.79615e6 −0.582107
\(778\) −4.50688e6 −0.266948
\(779\) 1.02296e7 0.603972
\(780\) 2.28345e7 1.34386
\(781\) 2.06434e7 1.21103
\(782\) −2.57726e6 −0.150709
\(783\) 5.01493e6 0.292321
\(784\) 1.64960e6 0.0958490
\(785\) 4.54162e7 2.63049
\(786\) 2.83829e6 0.163870
\(787\) −2.36036e6 −0.135845 −0.0679223 0.997691i \(-0.521637\pi\)
−0.0679223 + 0.997691i \(0.521637\pi\)
\(788\) −1.19956e6 −0.0688186
\(789\) −9.65467e6 −0.552135
\(790\) 1.16377e7 0.663437
\(791\) −2.67640e7 −1.52093
\(792\) −5.13957e6 −0.291148
\(793\) −2.88016e7 −1.62642
\(794\) −4.06476e6 −0.228814
\(795\) −2.78961e7 −1.56540
\(796\) 4.91077e6 0.274705
\(797\) 3.55059e6 0.197995 0.0989976 0.995088i \(-0.468436\pi\)
0.0989976 + 0.995088i \(0.468436\pi\)
\(798\) 1.03267e6 0.0574053
\(799\) −9.90266e6 −0.548763
\(800\) −3.37797e7 −1.86608
\(801\) 1.63287e6 0.0899229
\(802\) 6.25108e6 0.343178
\(803\) 4.98859e6 0.273017
\(804\) 1.30498e7 0.711974
\(805\) −5.08137e7 −2.76370
\(806\) 4.93208e6 0.267419
\(807\) −5.23729e6 −0.283089
\(808\) −9.49168e6 −0.511464
\(809\) 1.25623e7 0.674837 0.337418 0.941355i \(-0.390446\pi\)
0.337418 + 0.941355i \(0.390446\pi\)
\(810\) −1.10249e6 −0.0590420
\(811\) −1.43135e7 −0.764176 −0.382088 0.924126i \(-0.624795\pi\)
−0.382088 + 0.924126i \(0.624795\pi\)
\(812\) −2.77914e7 −1.47918
\(813\) 1.39471e7 0.740045
\(814\) −8.18069e6 −0.432742
\(815\) 5.06372e7 2.67039
\(816\) −3.11535e6 −0.163788
\(817\) 1.31710e6 0.0690342
\(818\) −681906. −0.0356321
\(819\) −9.30161e6 −0.484561
\(820\) 6.05229e7 3.14329
\(821\) −4.11110e6 −0.212863 −0.106431 0.994320i \(-0.533942\pi\)
−0.106431 + 0.994320i \(0.533942\pi\)
\(822\) −3.28631e6 −0.169640
\(823\) −1.57208e7 −0.809050 −0.404525 0.914527i \(-0.632563\pi\)
−0.404525 + 0.914527i \(0.632563\pi\)
\(824\) −929924. −0.0477122
\(825\) 4.35103e7 2.22565
\(826\) −777426. −0.0396469
\(827\) −2.20932e7 −1.12330 −0.561649 0.827376i \(-0.689833\pi\)
−0.561649 + 0.827376i \(0.689833\pi\)
\(828\) −8.48697e6 −0.430207
\(829\) 5.93151e6 0.299763 0.149882 0.988704i \(-0.452111\pi\)
0.149882 + 0.988704i \(0.452111\pi\)
\(830\) 5.12010e6 0.257979
\(831\) −1.37689e7 −0.691666
\(832\) −1.47490e7 −0.738676
\(833\) 943465. 0.0471100
\(834\) −1.38290e6 −0.0688455
\(835\) 2.86806e7 1.42355
\(836\) −9.60951e6 −0.475538
\(837\) 2.65351e6 0.130920
\(838\) 5.13752e6 0.252722
\(839\) 3.16728e7 1.55340 0.776698 0.629874i \(-0.216893\pi\)
0.776698 + 0.629874i \(0.216893\pi\)
\(840\) 1.27677e7 0.624328
\(841\) 2.68122e7 1.30720
\(842\) 1.11720e7 0.543063
\(843\) −8.48340e6 −0.411151
\(844\) 1.43767e7 0.694710
\(845\) 3.36853e7 1.62293
\(846\) 2.92644e6 0.140577
\(847\) −3.36608e7 −1.61219
\(848\) 2.32952e7 1.11244
\(849\) 6.07683e6 0.289340
\(850\) −5.48217e6 −0.260259
\(851\) −2.82299e7 −1.33624
\(852\) −8.56519e6 −0.404239
\(853\) 1.13914e7 0.536049 0.268024 0.963412i \(-0.413629\pi\)
0.268024 + 0.963412i \(0.413629\pi\)
\(854\) −7.70627e6 −0.361576
\(855\) −4.30765e6 −0.201523
\(856\) −1.90721e7 −0.889641
\(857\) −1.71933e7 −0.799664 −0.399832 0.916588i \(-0.630932\pi\)
−0.399832 + 0.916588i \(0.630932\pi\)
\(858\) −7.76771e6 −0.360226
\(859\) −1.03966e7 −0.480738 −0.240369 0.970682i \(-0.577268\pi\)
−0.240369 + 0.970682i \(0.577268\pi\)
\(860\) 7.79253e6 0.359279
\(861\) −2.46540e7 −1.13339
\(862\) −1.07888e7 −0.494544
\(863\) 1.64704e7 0.752797 0.376399 0.926458i \(-0.377162\pi\)
0.376399 + 0.926458i \(0.377162\pi\)
\(864\) 3.24449e6 0.147864
\(865\) −4.16482e7 −1.89259
\(866\) 8.77467e6 0.397591
\(867\) 1.09969e7 0.496848
\(868\) −1.47050e7 −0.662469
\(869\) 4.41140e7 1.98165
\(870\) −1.04036e7 −0.466000
\(871\) 4.12157e7 1.84084
\(872\) −4.88302e6 −0.217469
\(873\) −7.42138e6 −0.329571
\(874\) 2.97586e6 0.131775
\(875\) −6.35846e7 −2.80758
\(876\) −2.06983e6 −0.0911326
\(877\) −7.65036e6 −0.335879 −0.167940 0.985797i \(-0.553711\pi\)
−0.167940 + 0.985797i \(0.553711\pi\)
\(878\) −5.67785e6 −0.248570
\(879\) 4.27656e6 0.186690
\(880\) −5.12939e7 −2.23285
\(881\) 1.06801e7 0.463591 0.231795 0.972765i \(-0.425540\pi\)
0.231795 + 0.972765i \(0.425540\pi\)
\(882\) −278814. −0.0120682
\(883\) −1.47279e7 −0.635679 −0.317840 0.948144i \(-0.602957\pi\)
−0.317840 + 0.948144i \(0.602957\pi\)
\(884\) −1.09059e7 −0.469386
\(885\) 3.24295e6 0.139182
\(886\) −1.27189e7 −0.544335
\(887\) −9.04933e6 −0.386196 −0.193098 0.981180i \(-0.561853\pi\)
−0.193098 + 0.981180i \(0.561853\pi\)
\(888\) 7.09314e6 0.301860
\(889\) 1.94950e6 0.0827310
\(890\) −3.38743e6 −0.143349
\(891\) −4.17910e6 −0.176355
\(892\) 2.51414e7 1.05798
\(893\) 1.14342e7 0.479820
\(894\) 453041. 0.0189581
\(895\) −5.09547e7 −2.12631
\(896\) −2.35399e7 −0.979569
\(897\) −2.68047e7 −1.11232
\(898\) −6.04951e6 −0.250339
\(899\) 2.50398e7 1.03331
\(900\) −1.80529e7 −0.742919
\(901\) 1.33234e7 0.546766
\(902\) −2.05884e7 −0.842569
\(903\) −3.17428e6 −0.129547
\(904\) 1.93791e7 0.788702
\(905\) 8.63545e6 0.350480
\(906\) −1.08814e6 −0.0440416
\(907\) 6.23712e6 0.251748 0.125874 0.992046i \(-0.459827\pi\)
0.125874 + 0.992046i \(0.459827\pi\)
\(908\) 2.96289e7 1.19261
\(909\) −7.71791e6 −0.309806
\(910\) 1.92964e7 0.772455
\(911\) 304116. 0.0121407 0.00607035 0.999982i \(-0.498068\pi\)
0.00607035 + 0.999982i \(0.498068\pi\)
\(912\) 3.59718e6 0.143211
\(913\) 1.94083e7 0.770568
\(914\) 2.02983e6 0.0803699
\(915\) 3.21459e7 1.26932
\(916\) 1.49164e7 0.587389
\(917\) −2.67270e7 −1.04961
\(918\) 526555. 0.0206223
\(919\) 4.04845e7 1.58125 0.790623 0.612303i \(-0.209757\pi\)
0.790623 + 0.612303i \(0.209757\pi\)
\(920\) 3.67929e7 1.43316
\(921\) −1.54963e7 −0.601974
\(922\) −5.25839e6 −0.203716
\(923\) −2.70518e7 −1.04518
\(924\) 2.31594e7 0.892376
\(925\) −6.00487e7 −2.30754
\(926\) −782438. −0.0299863
\(927\) −756142. −0.0289004
\(928\) 3.06166e7 1.16704
\(929\) −3.99001e7 −1.51682 −0.758411 0.651776i \(-0.774024\pi\)
−0.758411 + 0.651776i \(0.774024\pi\)
\(930\) −5.50477e6 −0.208704
\(931\) −1.08938e6 −0.0411914
\(932\) 4.59323e6 0.173212
\(933\) −2.10752e6 −0.0792626
\(934\) −7.49325e6 −0.281062
\(935\) −2.93369e7 −1.09745
\(936\) 6.73506e6 0.251276
\(937\) −8.96793e6 −0.333690 −0.166845 0.985983i \(-0.553358\pi\)
−0.166845 + 0.985983i \(0.553358\pi\)
\(938\) 1.10278e7 0.409245
\(939\) 3.82917e6 0.141723
\(940\) 6.76498e7 2.49716
\(941\) −1.34693e7 −0.495873 −0.247937 0.968776i \(-0.579752\pi\)
−0.247937 + 0.968776i \(0.579752\pi\)
\(942\) 6.41016e6 0.235365
\(943\) −7.10461e7 −2.60172
\(944\) −2.70809e6 −0.0989082
\(945\) 1.03817e7 0.378171
\(946\) −2.65082e6 −0.0963058
\(947\) 7.49930e6 0.271735 0.135868 0.990727i \(-0.456618\pi\)
0.135868 + 0.990727i \(0.456618\pi\)
\(948\) −1.83034e7 −0.661473
\(949\) −6.53721e6 −0.235628
\(950\) 6.33006e6 0.227561
\(951\) 3.01441e7 1.08082
\(952\) −6.09791e6 −0.218066
\(953\) −2.02266e6 −0.0721423 −0.0360711 0.999349i \(-0.511484\pi\)
−0.0360711 + 0.999349i \(0.511484\pi\)
\(954\) −3.93733e6 −0.140065
\(955\) 7.88578e7 2.79793
\(956\) −1.52392e7 −0.539283
\(957\) −3.94360e7 −1.39192
\(958\) 1.07435e7 0.378211
\(959\) 3.09458e7 1.08656
\(960\) 1.64616e7 0.576492
\(961\) −1.53801e7 −0.537218
\(962\) 1.07202e7 0.373480
\(963\) −1.55080e7 −0.538877
\(964\) −4.45385e6 −0.154363
\(965\) −2.53639e7 −0.876796
\(966\) −7.17198e6 −0.247284
\(967\) 3.81493e6 0.131196 0.0655979 0.997846i \(-0.479105\pi\)
0.0655979 + 0.997846i \(0.479105\pi\)
\(968\) 2.43729e7 0.836024
\(969\) 2.05736e6 0.0703884
\(970\) 1.53958e7 0.525381
\(971\) −3.88611e7 −1.32272 −0.661359 0.750070i \(-0.730020\pi\)
−0.661359 + 0.750070i \(0.730020\pi\)
\(972\) 1.73396e6 0.0588672
\(973\) 1.30222e7 0.440963
\(974\) −9.30158e6 −0.314166
\(975\) −5.70172e7 −1.92086
\(976\) −2.68440e7 −0.902033
\(977\) 3.83275e7 1.28462 0.642310 0.766445i \(-0.277976\pi\)
0.642310 + 0.766445i \(0.277976\pi\)
\(978\) 7.14706e6 0.238935
\(979\) −1.28404e7 −0.428176
\(980\) −6.44525e6 −0.214375
\(981\) −3.97050e6 −0.131726
\(982\) 6.43013e6 0.212785
\(983\) 3.97346e6 0.131155 0.0655776 0.997847i \(-0.479111\pi\)
0.0655776 + 0.997847i \(0.479111\pi\)
\(984\) 1.78513e7 0.587736
\(985\) 4.22853e6 0.138867
\(986\) 4.96883e6 0.162765
\(987\) −2.75571e7 −0.900411
\(988\) 1.25926e7 0.410415
\(989\) −9.14743e6 −0.297378
\(990\) 8.66966e6 0.281134
\(991\) 4.07143e7 1.31693 0.658465 0.752612i \(-0.271206\pi\)
0.658465 + 0.752612i \(0.271206\pi\)
\(992\) 1.61999e7 0.522676
\(993\) 2.83516e7 0.912441
\(994\) −7.23808e6 −0.232358
\(995\) −1.73108e7 −0.554318
\(996\) −8.05275e6 −0.257215
\(997\) 1.23365e7 0.393055 0.196527 0.980498i \(-0.437034\pi\)
0.196527 + 0.980498i \(0.437034\pi\)
\(998\) 3.78577e6 0.120317
\(999\) 5.76760e6 0.182844
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.1.6 12
3.2 odd 2 531.6.a.d.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.6 12 1.1 even 1 trivial
531.6.a.d.1.7 12 3.2 odd 2