Properties

Label 177.6.a.b.1.10
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 1.10
Root \(-7.32600\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.32600 q^{2} -9.00000 q^{3} +21.6702 q^{4} +46.7845 q^{5} -65.9340 q^{6} -85.8026 q^{7} -75.6759 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+7.32600 q^{2} -9.00000 q^{3} +21.6702 q^{4} +46.7845 q^{5} -65.9340 q^{6} -85.8026 q^{7} -75.6759 q^{8} +81.0000 q^{9} +342.743 q^{10} -408.472 q^{11} -195.032 q^{12} +27.9269 q^{13} -628.589 q^{14} -421.061 q^{15} -1247.85 q^{16} +1587.94 q^{17} +593.406 q^{18} -2204.29 q^{19} +1013.83 q^{20} +772.223 q^{21} -2992.47 q^{22} -3968.63 q^{23} +681.083 q^{24} -936.208 q^{25} +204.593 q^{26} -729.000 q^{27} -1859.36 q^{28} -2905.16 q^{29} -3084.69 q^{30} +4857.43 q^{31} -6720.11 q^{32} +3676.25 q^{33} +11633.3 q^{34} -4014.23 q^{35} +1755.29 q^{36} +2836.94 q^{37} -16148.6 q^{38} -251.342 q^{39} -3540.46 q^{40} +3698.67 q^{41} +5657.30 q^{42} -8798.36 q^{43} -8851.69 q^{44} +3789.55 q^{45} -29074.2 q^{46} -21002.5 q^{47} +11230.6 q^{48} -9444.92 q^{49} -6858.66 q^{50} -14291.5 q^{51} +605.183 q^{52} -11032.7 q^{53} -5340.65 q^{54} -19110.2 q^{55} +6493.18 q^{56} +19838.6 q^{57} -21283.2 q^{58} -3481.00 q^{59} -9124.48 q^{60} +45308.5 q^{61} +35585.5 q^{62} -6950.01 q^{63} -9300.32 q^{64} +1306.55 q^{65} +26932.2 q^{66} -40826.6 q^{67} +34411.1 q^{68} +35717.7 q^{69} -29408.2 q^{70} -27052.4 q^{71} -6129.75 q^{72} +70459.2 q^{73} +20783.4 q^{74} +8425.88 q^{75} -47767.4 q^{76} +35048.0 q^{77} -1841.33 q^{78} +71673.4 q^{79} -58380.0 q^{80} +6561.00 q^{81} +27096.4 q^{82} +93280.1 q^{83} +16734.2 q^{84} +74291.2 q^{85} -64456.8 q^{86} +26146.4 q^{87} +30911.5 q^{88} -116570. q^{89} +27762.2 q^{90} -2396.20 q^{91} -86001.2 q^{92} -43716.9 q^{93} -153864. q^{94} -103127. q^{95} +60481.0 q^{96} +91107.3 q^{97} -69193.5 q^{98} -33086.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.32600 1.29507 0.647533 0.762038i \(-0.275801\pi\)
0.647533 + 0.762038i \(0.275801\pi\)
\(3\) −9.00000 −0.577350
\(4\) 21.6702 0.677195
\(5\) 46.7845 0.836907 0.418453 0.908238i \(-0.362572\pi\)
0.418453 + 0.908238i \(0.362572\pi\)
\(6\) −65.9340 −0.747706
\(7\) −85.8026 −0.661843 −0.330922 0.943658i \(-0.607360\pi\)
−0.330922 + 0.943658i \(0.607360\pi\)
\(8\) −75.6759 −0.418054
\(9\) 81.0000 0.333333
\(10\) 342.743 1.08385
\(11\) −408.472 −1.01784 −0.508921 0.860813i \(-0.669956\pi\)
−0.508921 + 0.860813i \(0.669956\pi\)
\(12\) −195.032 −0.390979
\(13\) 27.9269 0.0458316 0.0229158 0.999737i \(-0.492705\pi\)
0.0229158 + 0.999737i \(0.492705\pi\)
\(14\) −628.589 −0.857130
\(15\) −421.061 −0.483188
\(16\) −1247.85 −1.21860
\(17\) 1587.94 1.33264 0.666320 0.745666i \(-0.267869\pi\)
0.666320 + 0.745666i \(0.267869\pi\)
\(18\) 593.406 0.431688
\(19\) −2204.29 −1.40083 −0.700413 0.713738i \(-0.747001\pi\)
−0.700413 + 0.713738i \(0.747001\pi\)
\(20\) 1013.83 0.566749
\(21\) 772.223 0.382115
\(22\) −2992.47 −1.31817
\(23\) −3968.63 −1.56430 −0.782152 0.623087i \(-0.785878\pi\)
−0.782152 + 0.623087i \(0.785878\pi\)
\(24\) 681.083 0.241364
\(25\) −936.208 −0.299587
\(26\) 204.593 0.0593549
\(27\) −729.000 −0.192450
\(28\) −1859.36 −0.448197
\(29\) −2905.16 −0.641468 −0.320734 0.947169i \(-0.603930\pi\)
−0.320734 + 0.947169i \(0.603930\pi\)
\(30\) −3084.69 −0.625761
\(31\) 4857.43 0.907826 0.453913 0.891046i \(-0.350028\pi\)
0.453913 + 0.891046i \(0.350028\pi\)
\(32\) −6720.11 −1.16012
\(33\) 3676.25 0.587652
\(34\) 11633.3 1.72586
\(35\) −4014.23 −0.553901
\(36\) 1755.29 0.225732
\(37\) 2836.94 0.340679 0.170340 0.985385i \(-0.445514\pi\)
0.170340 + 0.985385i \(0.445514\pi\)
\(38\) −16148.6 −1.81416
\(39\) −251.342 −0.0264609
\(40\) −3540.46 −0.349872
\(41\) 3698.67 0.343625 0.171813 0.985130i \(-0.445038\pi\)
0.171813 + 0.985130i \(0.445038\pi\)
\(42\) 5657.30 0.494864
\(43\) −8798.36 −0.725656 −0.362828 0.931856i \(-0.618189\pi\)
−0.362828 + 0.931856i \(0.618189\pi\)
\(44\) −8851.69 −0.689278
\(45\) 3789.55 0.278969
\(46\) −29074.2 −2.02588
\(47\) −21002.5 −1.38684 −0.693421 0.720533i \(-0.743897\pi\)
−0.693421 + 0.720533i \(0.743897\pi\)
\(48\) 11230.6 0.703560
\(49\) −9444.92 −0.561963
\(50\) −6858.66 −0.387984
\(51\) −14291.5 −0.769400
\(52\) 605.183 0.0310369
\(53\) −11032.7 −0.539499 −0.269750 0.962930i \(-0.586941\pi\)
−0.269750 + 0.962930i \(0.586941\pi\)
\(54\) −5340.65 −0.249235
\(55\) −19110.2 −0.851840
\(56\) 6493.18 0.276686
\(57\) 19838.6 0.808768
\(58\) −21283.2 −0.830743
\(59\) −3481.00 −0.130189
\(60\) −9124.48 −0.327213
\(61\) 45308.5 1.55903 0.779516 0.626382i \(-0.215465\pi\)
0.779516 + 0.626382i \(0.215465\pi\)
\(62\) 35585.5 1.17569
\(63\) −6950.01 −0.220614
\(64\) −9300.32 −0.283823
\(65\) 1306.55 0.0383568
\(66\) 26932.2 0.761048
\(67\) −40826.6 −1.11111 −0.555553 0.831481i \(-0.687494\pi\)
−0.555553 + 0.831481i \(0.687494\pi\)
\(68\) 34411.1 0.902456
\(69\) 35717.7 0.903152
\(70\) −29408.2 −0.717338
\(71\) −27052.4 −0.636884 −0.318442 0.947942i \(-0.603160\pi\)
−0.318442 + 0.947942i \(0.603160\pi\)
\(72\) −6129.75 −0.139351
\(73\) 70459.2 1.54750 0.773749 0.633492i \(-0.218379\pi\)
0.773749 + 0.633492i \(0.218379\pi\)
\(74\) 20783.4 0.441202
\(75\) 8425.88 0.172966
\(76\) −47767.4 −0.948632
\(77\) 35048.0 0.673653
\(78\) −1841.33 −0.0342686
\(79\) 71673.4 1.29208 0.646042 0.763302i \(-0.276423\pi\)
0.646042 + 0.763302i \(0.276423\pi\)
\(80\) −58380.0 −1.01986
\(81\) 6561.00 0.111111
\(82\) 27096.4 0.445018
\(83\) 93280.1 1.48626 0.743128 0.669149i \(-0.233341\pi\)
0.743128 + 0.669149i \(0.233341\pi\)
\(84\) 16734.2 0.258766
\(85\) 74291.2 1.11530
\(86\) −64456.8 −0.939772
\(87\) 26146.4 0.370352
\(88\) 30911.5 0.425513
\(89\) −116570. −1.55996 −0.779978 0.625806i \(-0.784770\pi\)
−0.779978 + 0.625806i \(0.784770\pi\)
\(90\) 27762.2 0.361283
\(91\) −2396.20 −0.0303333
\(92\) −86001.2 −1.05934
\(93\) −43716.9 −0.524134
\(94\) −153864. −1.79605
\(95\) −103127. −1.17236
\(96\) 60481.0 0.669793
\(97\) 91107.3 0.983159 0.491579 0.870833i \(-0.336420\pi\)
0.491579 + 0.870833i \(0.336420\pi\)
\(98\) −69193.5 −0.727780
\(99\) −33086.2 −0.339281
\(100\) −20287.9 −0.202879
\(101\) 121110. 1.18135 0.590674 0.806910i \(-0.298862\pi\)
0.590674 + 0.806910i \(0.298862\pi\)
\(102\) −104699. −0.996423
\(103\) 99956.2 0.928360 0.464180 0.885741i \(-0.346349\pi\)
0.464180 + 0.885741i \(0.346349\pi\)
\(104\) −2113.39 −0.0191601
\(105\) 36128.1 0.319795
\(106\) −80825.3 −0.698687
\(107\) 191304. 1.61534 0.807669 0.589636i \(-0.200729\pi\)
0.807669 + 0.589636i \(0.200729\pi\)
\(108\) −15797.6 −0.130326
\(109\) −142907. −1.15209 −0.576046 0.817417i \(-0.695405\pi\)
−0.576046 + 0.817417i \(0.695405\pi\)
\(110\) −140001. −1.10319
\(111\) −25532.4 −0.196691
\(112\) 107069. 0.806524
\(113\) −5623.98 −0.0414332 −0.0207166 0.999785i \(-0.506595\pi\)
−0.0207166 + 0.999785i \(0.506595\pi\)
\(114\) 145337. 1.04741
\(115\) −185671. −1.30918
\(116\) −62955.5 −0.434399
\(117\) 2262.08 0.0152772
\(118\) −25501.8 −0.168603
\(119\) −136250. −0.881999
\(120\) 31864.1 0.201999
\(121\) 5798.53 0.0360043
\(122\) 331930. 2.01905
\(123\) −33288.0 −0.198392
\(124\) 105262. 0.614775
\(125\) −190002. −1.08763
\(126\) −50915.7 −0.285710
\(127\) −128360. −0.706190 −0.353095 0.935587i \(-0.614871\pi\)
−0.353095 + 0.935587i \(0.614871\pi\)
\(128\) 146909. 0.792546
\(129\) 79185.2 0.418958
\(130\) 9571.77 0.0496745
\(131\) 175357. 0.892783 0.446391 0.894838i \(-0.352709\pi\)
0.446391 + 0.894838i \(0.352709\pi\)
\(132\) 79665.2 0.397955
\(133\) 189134. 0.927128
\(134\) −299095. −1.43896
\(135\) −34105.9 −0.161063
\(136\) −120169. −0.557115
\(137\) −204387. −0.930360 −0.465180 0.885216i \(-0.654010\pi\)
−0.465180 + 0.885216i \(0.654010\pi\)
\(138\) 261668. 1.16964
\(139\) 2129.04 0.00934644 0.00467322 0.999989i \(-0.498512\pi\)
0.00467322 + 0.999989i \(0.498512\pi\)
\(140\) −86989.3 −0.375099
\(141\) 189023. 0.800694
\(142\) −198186. −0.824807
\(143\) −11407.4 −0.0466493
\(144\) −101076. −0.406201
\(145\) −135916. −0.536849
\(146\) 516184. 2.00411
\(147\) 85004.3 0.324450
\(148\) 61477.1 0.230706
\(149\) −1847.03 −0.00681567 −0.00340784 0.999994i \(-0.501085\pi\)
−0.00340784 + 0.999994i \(0.501085\pi\)
\(150\) 61727.9 0.224003
\(151\) −67707.1 −0.241653 −0.120826 0.992674i \(-0.538554\pi\)
−0.120826 + 0.992674i \(0.538554\pi\)
\(152\) 166811. 0.585621
\(153\) 128623. 0.444213
\(154\) 256761. 0.872424
\(155\) 227253. 0.759766
\(156\) −5446.65 −0.0179192
\(157\) −208769. −0.675953 −0.337976 0.941155i \(-0.609742\pi\)
−0.337976 + 0.941155i \(0.609742\pi\)
\(158\) 525079. 1.67333
\(159\) 99294.0 0.311480
\(160\) −314397. −0.970909
\(161\) 340519. 1.03532
\(162\) 48065.9 0.143896
\(163\) 19124.2 0.0563787 0.0281894 0.999603i \(-0.491026\pi\)
0.0281894 + 0.999603i \(0.491026\pi\)
\(164\) 80150.9 0.232701
\(165\) 171992. 0.491810
\(166\) 683370. 1.92480
\(167\) −313422. −0.869638 −0.434819 0.900518i \(-0.643188\pi\)
−0.434819 + 0.900518i \(0.643188\pi\)
\(168\) −58438.7 −0.159745
\(169\) −370513. −0.997899
\(170\) 544257. 1.44438
\(171\) −178547. −0.466942
\(172\) −190662. −0.491410
\(173\) 620202. 1.57550 0.787750 0.615996i \(-0.211246\pi\)
0.787750 + 0.615996i \(0.211246\pi\)
\(174\) 191549. 0.479630
\(175\) 80329.1 0.198279
\(176\) 509711. 1.24035
\(177\) 31329.0 0.0751646
\(178\) −853993. −2.02025
\(179\) −402042. −0.937861 −0.468931 0.883235i \(-0.655361\pi\)
−0.468931 + 0.883235i \(0.655361\pi\)
\(180\) 82120.3 0.188916
\(181\) 326602. 0.741007 0.370503 0.928831i \(-0.379185\pi\)
0.370503 + 0.928831i \(0.379185\pi\)
\(182\) −17554.6 −0.0392836
\(183\) −407776. −0.900107
\(184\) 300330. 0.653964
\(185\) 132725. 0.285117
\(186\) −320270. −0.678787
\(187\) −648631. −1.35642
\(188\) −455130. −0.939162
\(189\) 62550.1 0.127372
\(190\) −755505. −1.51828
\(191\) 322656. 0.639964 0.319982 0.947424i \(-0.396323\pi\)
0.319982 + 0.947424i \(0.396323\pi\)
\(192\) 83702.9 0.163865
\(193\) −234977. −0.454080 −0.227040 0.973885i \(-0.572905\pi\)
−0.227040 + 0.973885i \(0.572905\pi\)
\(194\) 667451. 1.27325
\(195\) −11758.9 −0.0221453
\(196\) −204674. −0.380559
\(197\) −658997. −1.20981 −0.604906 0.796297i \(-0.706789\pi\)
−0.604906 + 0.796297i \(0.706789\pi\)
\(198\) −242390. −0.439391
\(199\) −339500. −0.607725 −0.303863 0.952716i \(-0.598276\pi\)
−0.303863 + 0.952716i \(0.598276\pi\)
\(200\) 70848.4 0.125243
\(201\) 367439. 0.641498
\(202\) 887254. 1.52992
\(203\) 249270. 0.424551
\(204\) −309700. −0.521033
\(205\) 173040. 0.287583
\(206\) 732279. 1.20229
\(207\) −321459. −0.521435
\(208\) −34848.6 −0.0558505
\(209\) 900390. 1.42582
\(210\) 264674. 0.414156
\(211\) −862627. −1.33388 −0.666940 0.745111i \(-0.732396\pi\)
−0.666940 + 0.745111i \(0.732396\pi\)
\(212\) −239080. −0.365346
\(213\) 243472. 0.367705
\(214\) 1.40149e6 2.09197
\(215\) −411627. −0.607306
\(216\) 55167.7 0.0804545
\(217\) −416780. −0.600839
\(218\) −1.04694e6 −1.49203
\(219\) −634132. −0.893449
\(220\) −414122. −0.576861
\(221\) 44346.4 0.0610770
\(222\) −187051. −0.254728
\(223\) −1.00055e6 −1.34734 −0.673671 0.739032i \(-0.735283\pi\)
−0.673671 + 0.739032i \(0.735283\pi\)
\(224\) 576602. 0.767815
\(225\) −75832.9 −0.0998622
\(226\) −41201.3 −0.0536587
\(227\) −39203.6 −0.0504965 −0.0252482 0.999681i \(-0.508038\pi\)
−0.0252482 + 0.999681i \(0.508038\pi\)
\(228\) 429907. 0.547693
\(229\) −1.53017e6 −1.92819 −0.964096 0.265555i \(-0.914445\pi\)
−0.964096 + 0.265555i \(0.914445\pi\)
\(230\) −1.36022e6 −1.69547
\(231\) −315432. −0.388933
\(232\) 219850. 0.268168
\(233\) −3398.03 −0.00410050 −0.00205025 0.999998i \(-0.500653\pi\)
−0.00205025 + 0.999998i \(0.500653\pi\)
\(234\) 16572.0 0.0197850
\(235\) −982593. −1.16066
\(236\) −75434.1 −0.0881632
\(237\) −645061. −0.745984
\(238\) −998164. −1.14225
\(239\) 514521. 0.582651 0.291326 0.956624i \(-0.405904\pi\)
0.291326 + 0.956624i \(0.405904\pi\)
\(240\) 525420. 0.588814
\(241\) −1.54065e6 −1.70868 −0.854342 0.519711i \(-0.826040\pi\)
−0.854342 + 0.519711i \(0.826040\pi\)
\(242\) 42480.0 0.0466279
\(243\) −59049.0 −0.0641500
\(244\) 981845. 1.05577
\(245\) −441876. −0.470311
\(246\) −243868. −0.256931
\(247\) −61559.0 −0.0642021
\(248\) −367591. −0.379520
\(249\) −839521. −0.858091
\(250\) −1.39195e6 −1.40856
\(251\) 1.00599e6 1.00789 0.503943 0.863737i \(-0.331882\pi\)
0.503943 + 0.863737i \(0.331882\pi\)
\(252\) −150608. −0.149399
\(253\) 1.62108e6 1.59222
\(254\) −940368. −0.914563
\(255\) −668621. −0.643916
\(256\) 1.37387e6 1.31022
\(257\) −821444. −0.775792 −0.387896 0.921703i \(-0.626798\pi\)
−0.387896 + 0.921703i \(0.626798\pi\)
\(258\) 580111. 0.542577
\(259\) −243417. −0.225476
\(260\) 28313.2 0.0259750
\(261\) −235318. −0.213823
\(262\) 1.28467e6 1.15621
\(263\) 336622. 0.300091 0.150045 0.988679i \(-0.452058\pi\)
0.150045 + 0.988679i \(0.452058\pi\)
\(264\) −278203. −0.245670
\(265\) −516158. −0.451511
\(266\) 1.38559e6 1.20069
\(267\) 1.04913e6 0.900642
\(268\) −884721. −0.752436
\(269\) −2.12624e6 −1.79156 −0.895781 0.444496i \(-0.853383\pi\)
−0.895781 + 0.444496i \(0.853383\pi\)
\(270\) −249860. −0.208587
\(271\) −781161. −0.646126 −0.323063 0.946377i \(-0.604713\pi\)
−0.323063 + 0.946377i \(0.604713\pi\)
\(272\) −1.98151e6 −1.62396
\(273\) 21565.8 0.0175129
\(274\) −1.49734e6 −1.20488
\(275\) 382415. 0.304932
\(276\) 774011. 0.611609
\(277\) −22946.2 −0.0179685 −0.00898426 0.999960i \(-0.502860\pi\)
−0.00898426 + 0.999960i \(0.502860\pi\)
\(278\) 15597.3 0.0121043
\(279\) 393452. 0.302609
\(280\) 303781. 0.231561
\(281\) 1.46132e6 1.10403 0.552015 0.833834i \(-0.313859\pi\)
0.552015 + 0.833834i \(0.313859\pi\)
\(282\) 1.38478e6 1.03695
\(283\) 1.25766e6 0.933466 0.466733 0.884398i \(-0.345431\pi\)
0.466733 + 0.884398i \(0.345431\pi\)
\(284\) −586233. −0.431295
\(285\) 928139. 0.676863
\(286\) −83570.4 −0.0604140
\(287\) −317355. −0.227426
\(288\) −544329. −0.386705
\(289\) 1.10171e6 0.775928
\(290\) −995724. −0.695255
\(291\) −819965. −0.567627
\(292\) 1.52687e6 1.04796
\(293\) 1.57391e6 1.07105 0.535525 0.844519i \(-0.320114\pi\)
0.535525 + 0.844519i \(0.320114\pi\)
\(294\) 622741. 0.420184
\(295\) −162857. −0.108956
\(296\) −214688. −0.142422
\(297\) 297776. 0.195884
\(298\) −13531.4 −0.00882675
\(299\) −110832. −0.0716945
\(300\) 182591. 0.117132
\(301\) 754922. 0.480270
\(302\) −496022. −0.312956
\(303\) −1.08999e6 −0.682051
\(304\) 2.75062e6 1.70705
\(305\) 2.11974e6 1.30476
\(306\) 942295. 0.575285
\(307\) 1.48288e6 0.897969 0.448985 0.893539i \(-0.351786\pi\)
0.448985 + 0.893539i \(0.351786\pi\)
\(308\) 759497. 0.456194
\(309\) −899606. −0.535989
\(310\) 1.66485e6 0.983947
\(311\) 762503. 0.447034 0.223517 0.974700i \(-0.428246\pi\)
0.223517 + 0.974700i \(0.428246\pi\)
\(312\) 19020.6 0.0110621
\(313\) −1.35503e6 −0.781786 −0.390893 0.920436i \(-0.627834\pi\)
−0.390893 + 0.920436i \(0.627834\pi\)
\(314\) −1.52944e6 −0.875403
\(315\) −325153. −0.184634
\(316\) 1.55318e6 0.874992
\(317\) −897102. −0.501411 −0.250705 0.968063i \(-0.580663\pi\)
−0.250705 + 0.968063i \(0.580663\pi\)
\(318\) 727428. 0.403387
\(319\) 1.18668e6 0.652914
\(320\) −435111. −0.237534
\(321\) −1.72173e6 −0.932616
\(322\) 2.49464e6 1.34081
\(323\) −3.50028e6 −1.86680
\(324\) 142178. 0.0752438
\(325\) −26145.4 −0.0137305
\(326\) 140104. 0.0730141
\(327\) 1.28616e6 0.665160
\(328\) −279900. −0.143654
\(329\) 1.80207e6 0.917872
\(330\) 1.26001e6 0.636926
\(331\) −1.45033e6 −0.727606 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(332\) 2.02140e6 1.00648
\(333\) 229792. 0.113560
\(334\) −2.29613e6 −1.12624
\(335\) −1.91005e6 −0.929893
\(336\) −963617. −0.465647
\(337\) −2.03832e6 −0.977680 −0.488840 0.872373i \(-0.662580\pi\)
−0.488840 + 0.872373i \(0.662580\pi\)
\(338\) −2.71438e6 −1.29235
\(339\) 50615.9 0.0239214
\(340\) 1.60991e6 0.755272
\(341\) −1.98413e6 −0.924024
\(342\) −1.30804e6 −0.604721
\(343\) 2.25248e6 1.03378
\(344\) 665824. 0.303363
\(345\) 1.67104e6 0.755854
\(346\) 4.54360e6 2.04037
\(347\) 2.94852e6 1.31456 0.657280 0.753646i \(-0.271707\pi\)
0.657280 + 0.753646i \(0.271707\pi\)
\(348\) 566599. 0.250800
\(349\) 4.25738e6 1.87102 0.935511 0.353299i \(-0.114940\pi\)
0.935511 + 0.353299i \(0.114940\pi\)
\(350\) 588491. 0.256785
\(351\) −20358.7 −0.00882029
\(352\) 2.74498e6 1.18082
\(353\) −2.89165e6 −1.23512 −0.617560 0.786524i \(-0.711879\pi\)
−0.617560 + 0.786524i \(0.711879\pi\)
\(354\) 229516. 0.0973431
\(355\) −1.26564e6 −0.533013
\(356\) −2.52610e6 −1.05639
\(357\) 1.22625e6 0.509222
\(358\) −2.94536e6 −1.21459
\(359\) 566386. 0.231940 0.115970 0.993253i \(-0.463002\pi\)
0.115970 + 0.993253i \(0.463002\pi\)
\(360\) −286777. −0.116624
\(361\) 2.38279e6 0.962315
\(362\) 2.39268e6 0.959652
\(363\) −52186.8 −0.0207871
\(364\) −51926.2 −0.0205416
\(365\) 3.29640e6 1.29511
\(366\) −2.98737e6 −1.16570
\(367\) 314932. 0.122054 0.0610270 0.998136i \(-0.480562\pi\)
0.0610270 + 0.998136i \(0.480562\pi\)
\(368\) 4.95225e6 1.90626
\(369\) 299592. 0.114542
\(370\) 972341. 0.369245
\(371\) 946631. 0.357064
\(372\) −947355. −0.354940
\(373\) −1.61192e6 −0.599891 −0.299946 0.953956i \(-0.596969\pi\)
−0.299946 + 0.953956i \(0.596969\pi\)
\(374\) −4.75187e6 −1.75665
\(375\) 1.71002e6 0.627945
\(376\) 1.58938e6 0.579775
\(377\) −81132.2 −0.0293995
\(378\) 458242. 0.164955
\(379\) −3.46978e6 −1.24081 −0.620404 0.784283i \(-0.713031\pi\)
−0.620404 + 0.784283i \(0.713031\pi\)
\(380\) −2.23478e6 −0.793917
\(381\) 1.15524e6 0.407719
\(382\) 2.36377e6 0.828796
\(383\) −632278. −0.220248 −0.110124 0.993918i \(-0.535125\pi\)
−0.110124 + 0.993918i \(0.535125\pi\)
\(384\) −1.32218e6 −0.457576
\(385\) 1.63970e6 0.563784
\(386\) −1.72144e6 −0.588063
\(387\) −712667. −0.241885
\(388\) 1.97432e6 0.665790
\(389\) −1.99683e6 −0.669063 −0.334531 0.942385i \(-0.608578\pi\)
−0.334531 + 0.942385i \(0.608578\pi\)
\(390\) −86145.9 −0.0286796
\(391\) −6.30196e6 −2.08465
\(392\) 714753. 0.234931
\(393\) −1.57822e6 −0.515448
\(394\) −4.82781e6 −1.56679
\(395\) 3.35321e6 1.08135
\(396\) −716987. −0.229759
\(397\) −3.53468e6 −1.12557 −0.562787 0.826602i \(-0.690271\pi\)
−0.562787 + 0.826602i \(0.690271\pi\)
\(398\) −2.48718e6 −0.787044
\(399\) −1.70220e6 −0.535277
\(400\) 1.16825e6 0.365077
\(401\) 3.76927e6 1.17057 0.585283 0.810829i \(-0.300983\pi\)
0.585283 + 0.810829i \(0.300983\pi\)
\(402\) 2.69186e6 0.830782
\(403\) 135653. 0.0416071
\(404\) 2.62449e6 0.800002
\(405\) 306953. 0.0929897
\(406\) 1.82615e6 0.549822
\(407\) −1.15881e6 −0.346758
\(408\) 1.08152e6 0.321651
\(409\) −108345. −0.0320259 −0.0160130 0.999872i \(-0.505097\pi\)
−0.0160130 + 0.999872i \(0.505097\pi\)
\(410\) 1.26769e6 0.372438
\(411\) 1.83948e6 0.537144
\(412\) 2.16607e6 0.628681
\(413\) 298679. 0.0861647
\(414\) −2.35501e6 −0.675292
\(415\) 4.36407e6 1.24386
\(416\) −187672. −0.0531699
\(417\) −19161.3 −0.00539617
\(418\) 6.59626e6 1.84653
\(419\) −4.59866e6 −1.27966 −0.639832 0.768515i \(-0.720996\pi\)
−0.639832 + 0.768515i \(0.720996\pi\)
\(420\) 782904. 0.216563
\(421\) 5.27760e6 1.45121 0.725606 0.688110i \(-0.241560\pi\)
0.725606 + 0.688110i \(0.241560\pi\)
\(422\) −6.31960e6 −1.72746
\(423\) −1.70120e6 −0.462281
\(424\) 834907. 0.225540
\(425\) −1.48665e6 −0.399241
\(426\) 1.78368e6 0.476203
\(427\) −3.88758e6 −1.03183
\(428\) 4.14559e6 1.09390
\(429\) 102666. 0.0269330
\(430\) −3.01558e6 −0.786502
\(431\) −3.25854e6 −0.844948 −0.422474 0.906375i \(-0.638838\pi\)
−0.422474 + 0.906375i \(0.638838\pi\)
\(432\) 909682. 0.234520
\(433\) 2.53904e6 0.650803 0.325402 0.945576i \(-0.394501\pi\)
0.325402 + 0.945576i \(0.394501\pi\)
\(434\) −3.05333e6 −0.778125
\(435\) 1.22325e6 0.309950
\(436\) −3.09683e6 −0.780190
\(437\) 8.74801e6 2.19132
\(438\) −4.64565e6 −1.15707
\(439\) 5.11072e6 1.26567 0.632835 0.774286i \(-0.281891\pi\)
0.632835 + 0.774286i \(0.281891\pi\)
\(440\) 1.44618e6 0.356115
\(441\) −765039. −0.187321
\(442\) 324881. 0.0790987
\(443\) −2.08543e6 −0.504877 −0.252438 0.967613i \(-0.581233\pi\)
−0.252438 + 0.967613i \(0.581233\pi\)
\(444\) −553294. −0.133198
\(445\) −5.45368e6 −1.30554
\(446\) −7.33004e6 −1.74489
\(447\) 16623.3 0.00393503
\(448\) 797992. 0.187847
\(449\) −2.92355e6 −0.684376 −0.342188 0.939632i \(-0.611168\pi\)
−0.342188 + 0.939632i \(0.611168\pi\)
\(450\) −555551. −0.129328
\(451\) −1.51080e6 −0.349757
\(452\) −121873. −0.0280583
\(453\) 609364. 0.139518
\(454\) −287205. −0.0653962
\(455\) −112105. −0.0253862
\(456\) −1.50130e6 −0.338109
\(457\) −8.29298e6 −1.85746 −0.928731 0.370754i \(-0.879099\pi\)
−0.928731 + 0.370754i \(0.879099\pi\)
\(458\) −1.12100e7 −2.49713
\(459\) −1.15761e6 −0.256467
\(460\) −4.02352e6 −0.886568
\(461\) −2.53751e6 −0.556102 −0.278051 0.960566i \(-0.589688\pi\)
−0.278051 + 0.960566i \(0.589688\pi\)
\(462\) −2.31085e6 −0.503694
\(463\) 3.55974e6 0.771731 0.385865 0.922555i \(-0.373903\pi\)
0.385865 + 0.922555i \(0.373903\pi\)
\(464\) 3.62520e6 0.781694
\(465\) −2.04527e6 −0.438651
\(466\) −24893.9 −0.00531042
\(467\) 6.73742e6 1.42956 0.714779 0.699351i \(-0.246527\pi\)
0.714779 + 0.699351i \(0.246527\pi\)
\(468\) 49019.8 0.0103456
\(469\) 3.50302e6 0.735378
\(470\) −7.19848e6 −1.50313
\(471\) 1.87892e6 0.390261
\(472\) 263428. 0.0544260
\(473\) 3.59389e6 0.738604
\(474\) −4.72571e6 −0.966099
\(475\) 2.06367e6 0.419669
\(476\) −2.95256e6 −0.597285
\(477\) −893646. −0.179833
\(478\) 3.76938e6 0.754571
\(479\) 7.29265e6 1.45227 0.726134 0.687554i \(-0.241315\pi\)
0.726134 + 0.687554i \(0.241315\pi\)
\(480\) 2.82957e6 0.560554
\(481\) 79227.0 0.0156139
\(482\) −1.12868e7 −2.21286
\(483\) −3.06467e6 −0.597745
\(484\) 125655. 0.0243819
\(485\) 4.26241e6 0.822812
\(486\) −432593. −0.0830785
\(487\) 1.72557e6 0.329694 0.164847 0.986319i \(-0.447287\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(488\) −3.42876e6 −0.651760
\(489\) −172118. −0.0325503
\(490\) −3.23718e6 −0.609084
\(491\) 8.06758e6 1.51022 0.755109 0.655599i \(-0.227584\pi\)
0.755109 + 0.655599i \(0.227584\pi\)
\(492\) −721358. −0.134350
\(493\) −4.61323e6 −0.854846
\(494\) −450981. −0.0831459
\(495\) −1.54792e6 −0.283947
\(496\) −6.06134e6 −1.10628
\(497\) 2.32117e6 0.421518
\(498\) −6.15033e6 −1.11128
\(499\) −7.26673e6 −1.30644 −0.653218 0.757170i \(-0.726581\pi\)
−0.653218 + 0.757170i \(0.726581\pi\)
\(500\) −4.11738e6 −0.736539
\(501\) 2.82080e6 0.502086
\(502\) 7.36991e6 1.30528
\(503\) −6.90011e6 −1.21601 −0.608004 0.793934i \(-0.708029\pi\)
−0.608004 + 0.793934i \(0.708029\pi\)
\(504\) 525948. 0.0922288
\(505\) 5.66609e6 0.988678
\(506\) 1.18760e7 2.06202
\(507\) 3.33462e6 0.576138
\(508\) −2.78160e6 −0.478228
\(509\) 2.10129e6 0.359495 0.179747 0.983713i \(-0.442472\pi\)
0.179747 + 0.983713i \(0.442472\pi\)
\(510\) −4.89831e6 −0.833914
\(511\) −6.04558e6 −1.02420
\(512\) 5.36385e6 0.904277
\(513\) 1.60693e6 0.269589
\(514\) −6.01790e6 −1.00470
\(515\) 4.67640e6 0.776951
\(516\) 1.71596e6 0.283716
\(517\) 8.57895e6 1.41159
\(518\) −1.78327e6 −0.292006
\(519\) −5.58182e6 −0.909615
\(520\) −98874.2 −0.0160352
\(521\) 5.71866e6 0.922996 0.461498 0.887141i \(-0.347312\pi\)
0.461498 + 0.887141i \(0.347312\pi\)
\(522\) −1.72394e6 −0.276914
\(523\) −548224. −0.0876403 −0.0438202 0.999039i \(-0.513953\pi\)
−0.0438202 + 0.999039i \(0.513953\pi\)
\(524\) 3.80003e6 0.604588
\(525\) −722962. −0.114477
\(526\) 2.46609e6 0.388637
\(527\) 7.71333e6 1.20980
\(528\) −4.58740e6 −0.716114
\(529\) 9.31370e6 1.44705
\(530\) −3.78137e6 −0.584736
\(531\) −281961. −0.0433963
\(532\) 4.09857e6 0.627846
\(533\) 103292. 0.0157489
\(534\) 7.68594e6 1.16639
\(535\) 8.95005e6 1.35189
\(536\) 3.08959e6 0.464503
\(537\) 3.61838e6 0.541475
\(538\) −1.55768e7 −2.32019
\(539\) 3.85799e6 0.571991
\(540\) −739083. −0.109071
\(541\) −1.44297e6 −0.211965 −0.105982 0.994368i \(-0.533799\pi\)
−0.105982 + 0.994368i \(0.533799\pi\)
\(542\) −5.72278e6 −0.836775
\(543\) −2.93942e6 −0.427820
\(544\) −1.06711e7 −1.54602
\(545\) −6.68583e6 −0.964193
\(546\) 157991. 0.0226804
\(547\) 5.77418e6 0.825130 0.412565 0.910928i \(-0.364633\pi\)
0.412565 + 0.910928i \(0.364633\pi\)
\(548\) −4.42910e6 −0.630035
\(549\) 3.66999e6 0.519677
\(550\) 2.80157e6 0.394907
\(551\) 6.40381e6 0.898585
\(552\) −2.70297e6 −0.377566
\(553\) −6.14976e6 −0.855156
\(554\) −168104. −0.0232704
\(555\) −1.19452e6 −0.164612
\(556\) 46136.7 0.00632936
\(557\) −3.90827e6 −0.533761 −0.266880 0.963730i \(-0.585993\pi\)
−0.266880 + 0.963730i \(0.585993\pi\)
\(558\) 2.88243e6 0.391898
\(559\) −245711. −0.0332579
\(560\) 5.00915e6 0.674985
\(561\) 5.83768e6 0.783128
\(562\) 1.07057e7 1.42979
\(563\) −8.26195e6 −1.09853 −0.549264 0.835649i \(-0.685092\pi\)
−0.549264 + 0.835649i \(0.685092\pi\)
\(564\) 4.09617e6 0.542225
\(565\) −263115. −0.0346757
\(566\) 9.21364e6 1.20890
\(567\) −562951. −0.0735381
\(568\) 2.04722e6 0.266252
\(569\) 9.52546e6 1.23340 0.616702 0.787197i \(-0.288468\pi\)
0.616702 + 0.787197i \(0.288468\pi\)
\(570\) 6.79954e6 0.876582
\(571\) −5.73979e6 −0.736725 −0.368363 0.929682i \(-0.620081\pi\)
−0.368363 + 0.929682i \(0.620081\pi\)
\(572\) −247200. −0.0315907
\(573\) −2.90390e6 −0.369484
\(574\) −2.32494e6 −0.294532
\(575\) 3.71547e6 0.468645
\(576\) −753326. −0.0946078
\(577\) 7.36689e6 0.921180 0.460590 0.887613i \(-0.347638\pi\)
0.460590 + 0.887613i \(0.347638\pi\)
\(578\) 8.07111e6 1.00488
\(579\) 2.11479e6 0.262163
\(580\) −2.94534e6 −0.363551
\(581\) −8.00367e6 −0.983669
\(582\) −6.00706e6 −0.735114
\(583\) 4.50654e6 0.549126
\(584\) −5.33206e6 −0.646938
\(585\) 105830. 0.0127856
\(586\) 1.15304e7 1.38708
\(587\) −1.28494e7 −1.53918 −0.769590 0.638539i \(-0.779539\pi\)
−0.769590 + 0.638539i \(0.779539\pi\)
\(588\) 1.84206e6 0.219716
\(589\) −1.07072e7 −1.27171
\(590\) −1.19309e6 −0.141105
\(591\) 5.93098e6 0.698486
\(592\) −3.54007e6 −0.415152
\(593\) −1.31628e7 −1.53714 −0.768568 0.639768i \(-0.779031\pi\)
−0.768568 + 0.639768i \(0.779031\pi\)
\(594\) 2.18151e6 0.253683
\(595\) −6.37437e6 −0.738151
\(596\) −40025.6 −0.00461554
\(597\) 3.05550e6 0.350870
\(598\) −811953. −0.0928491
\(599\) −9.54195e6 −1.08660 −0.543300 0.839538i \(-0.682825\pi\)
−0.543300 + 0.839538i \(0.682825\pi\)
\(600\) −637636. −0.0723093
\(601\) −6.64300e6 −0.750201 −0.375101 0.926984i \(-0.622392\pi\)
−0.375101 + 0.926984i \(0.622392\pi\)
\(602\) 5.53056e6 0.621982
\(603\) −3.30695e6 −0.370369
\(604\) −1.46723e6 −0.163646
\(605\) 271281. 0.0301323
\(606\) −7.98528e6 −0.883301
\(607\) −1.26387e7 −1.39229 −0.696147 0.717899i \(-0.745104\pi\)
−0.696147 + 0.717899i \(0.745104\pi\)
\(608\) 1.48130e7 1.62512
\(609\) −2.24343e6 −0.245115
\(610\) 1.55292e7 1.68976
\(611\) −586536. −0.0635611
\(612\) 2.78730e6 0.300819
\(613\) −1.20014e7 −1.28997 −0.644985 0.764195i \(-0.723136\pi\)
−0.644985 + 0.764195i \(0.723136\pi\)
\(614\) 1.08636e7 1.16293
\(615\) −1.55736e6 −0.166036
\(616\) −2.65229e6 −0.281623
\(617\) 8.14931e6 0.861803 0.430901 0.902399i \(-0.358196\pi\)
0.430901 + 0.902399i \(0.358196\pi\)
\(618\) −6.59051e6 −0.694141
\(619\) 1.19794e6 0.125663 0.0628314 0.998024i \(-0.479987\pi\)
0.0628314 + 0.998024i \(0.479987\pi\)
\(620\) 4.92462e6 0.514509
\(621\) 2.89313e6 0.301051
\(622\) 5.58609e6 0.578938
\(623\) 1.00020e7 1.03245
\(624\) 313637. 0.0322453
\(625\) −5.96349e6 −0.610661
\(626\) −9.92694e6 −1.01246
\(627\) −8.10351e6 −0.823198
\(628\) −4.52407e6 −0.457752
\(629\) 4.50490e6 0.454003
\(630\) −2.38207e6 −0.239113
\(631\) 2.94186e6 0.294136 0.147068 0.989126i \(-0.453016\pi\)
0.147068 + 0.989126i \(0.453016\pi\)
\(632\) −5.42395e6 −0.540161
\(633\) 7.76364e6 0.770116
\(634\) −6.57217e6 −0.649360
\(635\) −6.00528e6 −0.591016
\(636\) 2.15172e6 0.210933
\(637\) −263768. −0.0257557
\(638\) 8.69359e6 0.845566
\(639\) −2.19125e6 −0.212295
\(640\) 6.87308e6 0.663287
\(641\) 1.94544e7 1.87014 0.935068 0.354469i \(-0.115339\pi\)
0.935068 + 0.354469i \(0.115339\pi\)
\(642\) −1.26134e7 −1.20780
\(643\) −1.76373e7 −1.68231 −0.841153 0.540797i \(-0.818123\pi\)
−0.841153 + 0.540797i \(0.818123\pi\)
\(644\) 7.37912e6 0.701116
\(645\) 3.70464e6 0.350628
\(646\) −2.56431e7 −2.41762
\(647\) 1.72256e7 1.61776 0.808878 0.587976i \(-0.200075\pi\)
0.808878 + 0.587976i \(0.200075\pi\)
\(648\) −496509. −0.0464505
\(649\) 1.42189e6 0.132512
\(650\) −191541. −0.0177819
\(651\) 3.75102e6 0.346894
\(652\) 414427. 0.0381794
\(653\) −4.60913e6 −0.422996 −0.211498 0.977378i \(-0.567834\pi\)
−0.211498 + 0.977378i \(0.567834\pi\)
\(654\) 9.42242e6 0.861426
\(655\) 8.20401e6 0.747176
\(656\) −4.61537e6 −0.418743
\(657\) 5.70719e6 0.515833
\(658\) 1.32020e7 1.18870
\(659\) 858283. 0.0769869 0.0384935 0.999259i \(-0.487744\pi\)
0.0384935 + 0.999259i \(0.487744\pi\)
\(660\) 3.72710e6 0.333051
\(661\) 1.26702e7 1.12793 0.563964 0.825799i \(-0.309276\pi\)
0.563964 + 0.825799i \(0.309276\pi\)
\(662\) −1.06251e7 −0.942297
\(663\) −399117. −0.0352628
\(664\) −7.05905e6 −0.621336
\(665\) 8.84852e6 0.775920
\(666\) 1.68346e6 0.147067
\(667\) 1.15295e7 1.00345
\(668\) −6.79193e6 −0.588914
\(669\) 9.00497e6 0.777888
\(670\) −1.39930e7 −1.20427
\(671\) −1.85073e7 −1.58685
\(672\) −5.18942e6 −0.443298
\(673\) −2.56765e6 −0.218524 −0.109262 0.994013i \(-0.534849\pi\)
−0.109262 + 0.994013i \(0.534849\pi\)
\(674\) −1.49327e7 −1.26616
\(675\) 682496. 0.0576555
\(676\) −8.02910e6 −0.675772
\(677\) 1.74857e7 1.46626 0.733131 0.680087i \(-0.238058\pi\)
0.733131 + 0.680087i \(0.238058\pi\)
\(678\) 370812. 0.0309798
\(679\) −7.81724e6 −0.650697
\(680\) −5.62205e6 −0.466254
\(681\) 352832. 0.0291541
\(682\) −1.45357e7 −1.19667
\(683\) 6.06090e6 0.497147 0.248574 0.968613i \(-0.420038\pi\)
0.248574 + 0.968613i \(0.420038\pi\)
\(684\) −3.86916e6 −0.316211
\(685\) −9.56213e6 −0.778625
\(686\) 1.65017e7 1.33881
\(687\) 1.37715e7 1.11324
\(688\) 1.09790e7 0.884286
\(689\) −308109. −0.0247261
\(690\) 1.22420e7 0.978880
\(691\) −7.50002e6 −0.597541 −0.298770 0.954325i \(-0.596576\pi\)
−0.298770 + 0.954325i \(0.596576\pi\)
\(692\) 1.34399e7 1.06692
\(693\) 2.83888e6 0.224551
\(694\) 2.16009e7 1.70244
\(695\) 99606.0 0.00782210
\(696\) −1.97865e6 −0.154827
\(697\) 5.87327e6 0.457929
\(698\) 3.11895e7 2.42310
\(699\) 30582.2 0.00236743
\(700\) 1.74075e6 0.134274
\(701\) −1.97391e7 −1.51717 −0.758583 0.651576i \(-0.774108\pi\)
−0.758583 + 0.651576i \(0.774108\pi\)
\(702\) −149148. −0.0114229
\(703\) −6.25343e6 −0.477232
\(704\) 3.79892e6 0.288888
\(705\) 8.84334e6 0.670106
\(706\) −2.11842e7 −1.59956
\(707\) −1.03916e7 −0.781867
\(708\) 678907. 0.0509011
\(709\) −2.68377e6 −0.200507 −0.100254 0.994962i \(-0.531965\pi\)
−0.100254 + 0.994962i \(0.531965\pi\)
\(710\) −9.27204e6 −0.690287
\(711\) 5.80555e6 0.430694
\(712\) 8.82155e6 0.652146
\(713\) −1.92774e7 −1.42012
\(714\) 8.98348e6 0.659476
\(715\) −533689. −0.0390412
\(716\) −8.71234e6 −0.635115
\(717\) −4.63069e6 −0.336394
\(718\) 4.14934e6 0.300378
\(719\) −1.26395e7 −0.911816 −0.455908 0.890027i \(-0.650685\pi\)
−0.455908 + 0.890027i \(0.650685\pi\)
\(720\) −4.72878e6 −0.339952
\(721\) −8.57650e6 −0.614429
\(722\) 1.74563e7 1.24626
\(723\) 1.38659e7 0.986509
\(724\) 7.07753e6 0.501806
\(725\) 2.71983e6 0.192175
\(726\) −382320. −0.0269207
\(727\) 1.30124e7 0.913105 0.456552 0.889697i \(-0.349084\pi\)
0.456552 + 0.889697i \(0.349084\pi\)
\(728\) 181335. 0.0126810
\(729\) 531441. 0.0370370
\(730\) 2.41494e7 1.67726
\(731\) −1.39713e7 −0.967038
\(732\) −8.83661e6 −0.609548
\(733\) −5.49665e6 −0.377866 −0.188933 0.981990i \(-0.560503\pi\)
−0.188933 + 0.981990i \(0.560503\pi\)
\(734\) 2.30719e6 0.158068
\(735\) 3.97688e6 0.271534
\(736\) 2.66696e7 1.81477
\(737\) 1.66765e7 1.13093
\(738\) 2.19481e6 0.148339
\(739\) −2.85074e6 −0.192020 −0.0960101 0.995380i \(-0.530608\pi\)
−0.0960101 + 0.995380i \(0.530608\pi\)
\(740\) 2.87618e6 0.193080
\(741\) 554031. 0.0370671
\(742\) 6.93502e6 0.462421
\(743\) −8.85765e6 −0.588635 −0.294318 0.955708i \(-0.595092\pi\)
−0.294318 + 0.955708i \(0.595092\pi\)
\(744\) 3.30831e6 0.219116
\(745\) −86412.5 −0.00570409
\(746\) −1.18090e7 −0.776899
\(747\) 7.55569e6 0.495419
\(748\) −1.40560e7 −0.918559
\(749\) −1.64143e7 −1.06910
\(750\) 1.25276e7 0.813230
\(751\) −1.95283e7 −1.26347 −0.631733 0.775186i \(-0.717656\pi\)
−0.631733 + 0.775186i \(0.717656\pi\)
\(752\) 2.62080e7 1.69001
\(753\) −9.05395e6 −0.581903
\(754\) −594374. −0.0380743
\(755\) −3.16764e6 −0.202241
\(756\) 1.35547e6 0.0862555
\(757\) 2.35067e7 1.49091 0.745457 0.666554i \(-0.232231\pi\)
0.745457 + 0.666554i \(0.232231\pi\)
\(758\) −2.54196e7 −1.60693
\(759\) −1.45897e7 −0.919267
\(760\) 7.80419e6 0.490111
\(761\) 1.76666e7 1.10584 0.552920 0.833234i \(-0.313513\pi\)
0.552920 + 0.833234i \(0.313513\pi\)
\(762\) 8.46331e6 0.528023
\(763\) 1.22618e7 0.762504
\(764\) 6.99202e6 0.433380
\(765\) 6.01759e6 0.371765
\(766\) −4.63207e6 −0.285235
\(767\) −97213.6 −0.00596676
\(768\) −1.23648e7 −0.756457
\(769\) −6.78452e6 −0.413717 −0.206858 0.978371i \(-0.566324\pi\)
−0.206858 + 0.978371i \(0.566324\pi\)
\(770\) 1.20125e7 0.730138
\(771\) 7.39300e6 0.447904
\(772\) −5.09201e6 −0.307501
\(773\) −1.32057e7 −0.794901 −0.397451 0.917624i \(-0.630105\pi\)
−0.397451 + 0.917624i \(0.630105\pi\)
\(774\) −5.22100e6 −0.313257
\(775\) −4.54757e6 −0.271973
\(776\) −6.89462e6 −0.411014
\(777\) 2.19075e6 0.130179
\(778\) −1.46288e7 −0.866480
\(779\) −8.15292e6 −0.481360
\(780\) −254819. −0.0149967
\(781\) 1.10502e7 0.648248
\(782\) −4.61682e7 −2.69976
\(783\) 2.11786e6 0.123451
\(784\) 1.17858e7 0.684810
\(785\) −9.76714e6 −0.565710
\(786\) −1.15620e7 −0.667539
\(787\) −5.65588e6 −0.325509 −0.162755 0.986667i \(-0.552038\pi\)
−0.162755 + 0.986667i \(0.552038\pi\)
\(788\) −1.42806e7 −0.819279
\(789\) −3.02959e6 −0.173257
\(790\) 2.45656e7 1.40042
\(791\) 482552. 0.0274223
\(792\) 2.50383e6 0.141838
\(793\) 1.26533e6 0.0714529
\(794\) −2.58951e7 −1.45769
\(795\) 4.64542e6 0.260680
\(796\) −7.35705e6 −0.411548
\(797\) 2.41632e6 0.134744 0.0673718 0.997728i \(-0.478539\pi\)
0.0673718 + 0.997728i \(0.478539\pi\)
\(798\) −1.24703e7 −0.693219
\(799\) −3.33508e7 −1.84816
\(800\) 6.29142e6 0.347555
\(801\) −9.44219e6 −0.519986
\(802\) 2.76136e7 1.51596
\(803\) −2.87806e7 −1.57511
\(804\) 7.96249e6 0.434419
\(805\) 1.59310e7 0.866470
\(806\) 993795. 0.0538839
\(807\) 1.91362e7 1.03436
\(808\) −9.16513e6 −0.493867
\(809\) 1.19990e7 0.644576 0.322288 0.946642i \(-0.395548\pi\)
0.322288 + 0.946642i \(0.395548\pi\)
\(810\) 2.24874e6 0.120428
\(811\) −1.49699e7 −0.799220 −0.399610 0.916685i \(-0.630854\pi\)
−0.399610 + 0.916685i \(0.630854\pi\)
\(812\) 5.40174e6 0.287504
\(813\) 7.03045e6 0.373041
\(814\) −8.48944e6 −0.449074
\(815\) 894719. 0.0471837
\(816\) 1.78336e7 0.937592
\(817\) 1.93941e7 1.01652
\(818\) −793738. −0.0414757
\(819\) −194092. −0.0101111
\(820\) 3.74982e6 0.194749
\(821\) 2.52512e7 1.30745 0.653724 0.756733i \(-0.273206\pi\)
0.653724 + 0.756733i \(0.273206\pi\)
\(822\) 1.34760e7 0.695636
\(823\) 1.71441e6 0.0882299 0.0441150 0.999026i \(-0.485953\pi\)
0.0441150 + 0.999026i \(0.485953\pi\)
\(824\) −7.56427e6 −0.388105
\(825\) −3.44174e6 −0.176053
\(826\) 2.18812e6 0.111589
\(827\) −1.24001e7 −0.630464 −0.315232 0.949015i \(-0.602082\pi\)
−0.315232 + 0.949015i \(0.602082\pi\)
\(828\) −6.96610e6 −0.353113
\(829\) 3.26664e7 1.65088 0.825439 0.564491i \(-0.190928\pi\)
0.825439 + 0.564491i \(0.190928\pi\)
\(830\) 3.19711e7 1.61088
\(831\) 206516. 0.0103741
\(832\) −259729. −0.0130081
\(833\) −1.49980e7 −0.748895
\(834\) −140376. −0.00698839
\(835\) −1.46633e7 −0.727806
\(836\) 1.95117e7 0.965559
\(837\) −3.54107e6 −0.174711
\(838\) −3.36897e7 −1.65725
\(839\) 3.43057e7 1.68252 0.841261 0.540629i \(-0.181814\pi\)
0.841261 + 0.540629i \(0.181814\pi\)
\(840\) −2.73402e6 −0.133692
\(841\) −1.20712e7 −0.588519
\(842\) 3.86636e7 1.87941
\(843\) −1.31519e7 −0.637412
\(844\) −1.86933e7 −0.903297
\(845\) −1.73343e7 −0.835149
\(846\) −1.24630e7 −0.598684
\(847\) −497529. −0.0238292
\(848\) 1.37671e7 0.657435
\(849\) −1.13190e7 −0.538937
\(850\) −1.08912e7 −0.517043
\(851\) −1.12588e7 −0.532926
\(852\) 5.27609e6 0.249008
\(853\) 1.96117e6 0.0922874 0.0461437 0.998935i \(-0.485307\pi\)
0.0461437 + 0.998935i \(0.485307\pi\)
\(854\) −2.84804e7 −1.33629
\(855\) −8.35325e6 −0.390787
\(856\) −1.44771e7 −0.675299
\(857\) −1.73109e7 −0.805134 −0.402567 0.915391i \(-0.631882\pi\)
−0.402567 + 0.915391i \(0.631882\pi\)
\(858\) 752133. 0.0348800
\(859\) 1.08498e7 0.501692 0.250846 0.968027i \(-0.419291\pi\)
0.250846 + 0.968027i \(0.419291\pi\)
\(860\) −8.92005e6 −0.411265
\(861\) 2.85619e6 0.131305
\(862\) −2.38721e7 −1.09426
\(863\) −3.39433e6 −0.155141 −0.0775705 0.996987i \(-0.524716\pi\)
−0.0775705 + 0.996987i \(0.524716\pi\)
\(864\) 4.89896e6 0.223264
\(865\) 2.90159e7 1.31855
\(866\) 1.86010e7 0.842833
\(867\) −9.91537e6 −0.447983
\(868\) −9.03172e6 −0.406885
\(869\) −2.92766e7 −1.31514
\(870\) 8.96151e6 0.401405
\(871\) −1.14016e6 −0.0509238
\(872\) 1.08146e7 0.481637
\(873\) 7.37969e6 0.327720
\(874\) 6.40879e7 2.83790
\(875\) 1.63026e7 0.719843
\(876\) −1.37418e7 −0.605039
\(877\) −2.12843e7 −0.934460 −0.467230 0.884136i \(-0.654748\pi\)
−0.467230 + 0.884136i \(0.654748\pi\)
\(878\) 3.74411e7 1.63913
\(879\) −1.41652e7 −0.618371
\(880\) 2.38466e7 1.03805
\(881\) −4.15740e7 −1.80461 −0.902303 0.431103i \(-0.858124\pi\)
−0.902303 + 0.431103i \(0.858124\pi\)
\(882\) −5.60467e6 −0.242593
\(883\) −3.74730e6 −0.161740 −0.0808699 0.996725i \(-0.525770\pi\)
−0.0808699 + 0.996725i \(0.525770\pi\)
\(884\) 960996. 0.0413610
\(885\) 1.46571e6 0.0629058
\(886\) −1.52778e7 −0.653849
\(887\) −6.86645e6 −0.293038 −0.146519 0.989208i \(-0.546807\pi\)
−0.146519 + 0.989208i \(0.546807\pi\)
\(888\) 1.93219e6 0.0822276
\(889\) 1.10137e7 0.467387
\(890\) −3.99537e7 −1.69076
\(891\) −2.67999e6 −0.113094
\(892\) −2.16822e7 −0.912412
\(893\) 4.62956e7 1.94272
\(894\) 121782. 0.00509612
\(895\) −1.88093e7 −0.784903
\(896\) −1.26052e7 −0.524541
\(897\) 997485. 0.0413929
\(898\) −2.14179e7 −0.886312
\(899\) −1.41116e7 −0.582341
\(900\) −1.64332e6 −0.0676262
\(901\) −1.75193e7 −0.718958
\(902\) −1.10681e7 −0.452958
\(903\) −6.79430e6 −0.277284
\(904\) 425600. 0.0173213
\(905\) 1.52799e7 0.620154
\(906\) 4.46420e6 0.180685
\(907\) −4.68280e7 −1.89011 −0.945056 0.326910i \(-0.893993\pi\)
−0.945056 + 0.326910i \(0.893993\pi\)
\(908\) −849550. −0.0341959
\(909\) 9.80994e6 0.393783
\(910\) −821282. −0.0328767
\(911\) 7.20062e6 0.287458 0.143729 0.989617i \(-0.454091\pi\)
0.143729 + 0.989617i \(0.454091\pi\)
\(912\) −2.47556e7 −0.985566
\(913\) −3.81023e7 −1.51278
\(914\) −6.07543e7 −2.40554
\(915\) −1.90776e7 −0.753306
\(916\) −3.31591e7 −1.30576
\(917\) −1.50461e7 −0.590882
\(918\) −8.48065e6 −0.332141
\(919\) −3.30525e7 −1.29097 −0.645485 0.763773i \(-0.723345\pi\)
−0.645485 + 0.763773i \(0.723345\pi\)
\(920\) 1.40508e7 0.547307
\(921\) −1.33460e7 −0.518443
\(922\) −1.85898e7 −0.720189
\(923\) −755492. −0.0291894
\(924\) −6.83548e6 −0.263384
\(925\) −2.65597e6 −0.102063
\(926\) 2.60786e7 0.999442
\(927\) 8.09645e6 0.309453
\(928\) 1.95230e7 0.744177
\(929\) 2.05879e7 0.782658 0.391329 0.920251i \(-0.372015\pi\)
0.391329 + 0.920251i \(0.372015\pi\)
\(930\) −1.49837e7 −0.568082
\(931\) 2.08193e7 0.787213
\(932\) −73636.0 −0.00277684
\(933\) −6.86252e6 −0.258095
\(934\) 4.93583e7 1.85137
\(935\) −3.03459e7 −1.13520
\(936\) −171185. −0.00638669
\(937\) −2.98879e7 −1.11211 −0.556054 0.831146i \(-0.687685\pi\)
−0.556054 + 0.831146i \(0.687685\pi\)
\(938\) 2.56631e7 0.952363
\(939\) 1.21953e7 0.451364
\(940\) −2.12930e7 −0.785991
\(941\) 3.84112e7 1.41411 0.707056 0.707158i \(-0.250023\pi\)
0.707056 + 0.707158i \(0.250023\pi\)
\(942\) 1.37650e7 0.505414
\(943\) −1.46786e7 −0.537535
\(944\) 4.34376e6 0.158648
\(945\) 2.92638e6 0.106598
\(946\) 2.63288e7 0.956540
\(947\) −827143. −0.0299713 −0.0149857 0.999888i \(-0.504770\pi\)
−0.0149857 + 0.999888i \(0.504770\pi\)
\(948\) −1.39786e7 −0.505177
\(949\) 1.96771e6 0.0709243
\(950\) 1.51185e7 0.543499
\(951\) 8.07392e6 0.289490
\(952\) 1.03108e7 0.368723
\(953\) −1.38653e7 −0.494535 −0.247267 0.968947i \(-0.579533\pi\)
−0.247267 + 0.968947i \(0.579533\pi\)
\(954\) −6.54685e6 −0.232896
\(955\) 1.50953e7 0.535591
\(956\) 1.11498e7 0.394568
\(957\) −1.06801e7 −0.376960
\(958\) 5.34259e7 1.88078
\(959\) 1.75369e7 0.615753
\(960\) 3.91600e6 0.137140
\(961\) −5.03449e6 −0.175852
\(962\) 580417. 0.0202210
\(963\) 1.54956e7 0.538446
\(964\) −3.33863e7 −1.15711
\(965\) −1.09933e7 −0.380023
\(966\) −2.24518e7 −0.774119
\(967\) −1.09394e7 −0.376208 −0.188104 0.982149i \(-0.560234\pi\)
−0.188104 + 0.982149i \(0.560234\pi\)
\(968\) −438809. −0.0150517
\(969\) 3.15026e7 1.07780
\(970\) 3.12264e7 1.06560
\(971\) 4.27950e7 1.45661 0.728307 0.685251i \(-0.240307\pi\)
0.728307 + 0.685251i \(0.240307\pi\)
\(972\) −1.27961e6 −0.0434421
\(973\) −182677. −0.00618588
\(974\) 1.26415e7 0.426975
\(975\) 235309. 0.00792733
\(976\) −5.65381e7 −1.89984
\(977\) 1.09971e7 0.368590 0.184295 0.982871i \(-0.441000\pi\)
0.184295 + 0.982871i \(0.441000\pi\)
\(978\) −1.26094e6 −0.0421547
\(979\) 4.76157e7 1.58779
\(980\) −9.57556e6 −0.318492
\(981\) −1.15755e7 −0.384031
\(982\) 5.91031e7 1.95583
\(983\) 2.08290e7 0.687518 0.343759 0.939058i \(-0.388300\pi\)
0.343759 + 0.939058i \(0.388300\pi\)
\(984\) 2.51910e6 0.0829387
\(985\) −3.08309e7 −1.01250
\(986\) −3.37965e7 −1.10708
\(987\) −1.62186e7 −0.529934
\(988\) −1.33400e6 −0.0434773
\(989\) 3.49175e7 1.13515
\(990\) −1.13401e7 −0.367729
\(991\) 1.73858e7 0.562355 0.281177 0.959656i \(-0.409275\pi\)
0.281177 + 0.959656i \(0.409275\pi\)
\(992\) −3.26425e7 −1.05318
\(993\) 1.30529e7 0.420083
\(994\) 1.70049e7 0.545893
\(995\) −1.58834e7 −0.508610
\(996\) −1.81926e7 −0.581094
\(997\) −1.37910e7 −0.439397 −0.219699 0.975568i \(-0.570507\pi\)
−0.219699 + 0.975568i \(0.570507\pi\)
\(998\) −5.32361e7 −1.69192
\(999\) −2.06813e6 −0.0655637
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.1.10 12
3.2 odd 2 531.6.a.d.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.10 12 1.1 even 1 trivial
531.6.a.d.1.3 12 3.2 odd 2