Properties

Label 177.6.a.c.1.2
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
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: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
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 1.2
Root \(8.10567\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.10567 q^{2} +9.00000 q^{3} +5.27923 q^{4} -41.8018 q^{5} -54.9510 q^{6} +208.248 q^{7} +163.148 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-6.10567 q^{2} +9.00000 q^{3} +5.27923 q^{4} -41.8018 q^{5} -54.9510 q^{6} +208.248 q^{7} +163.148 q^{8} +81.0000 q^{9} +255.228 q^{10} +149.105 q^{11} +47.5131 q^{12} -952.113 q^{13} -1271.49 q^{14} -376.216 q^{15} -1165.07 q^{16} +1073.60 q^{17} -494.559 q^{18} -486.976 q^{19} -220.681 q^{20} +1874.23 q^{21} -910.386 q^{22} +2198.23 q^{23} +1468.33 q^{24} -1377.61 q^{25} +5813.29 q^{26} +729.000 q^{27} +1099.39 q^{28} -2548.98 q^{29} +2297.05 q^{30} +1027.11 q^{31} +1892.76 q^{32} +1341.94 q^{33} -6555.08 q^{34} -8705.13 q^{35} +427.618 q^{36} -3164.40 q^{37} +2973.31 q^{38} -8569.02 q^{39} -6819.88 q^{40} +14579.4 q^{41} -11443.4 q^{42} +13357.1 q^{43} +787.159 q^{44} -3385.94 q^{45} -13421.7 q^{46} +15809.5 q^{47} -10485.6 q^{48} +26560.2 q^{49} +8411.26 q^{50} +9662.44 q^{51} -5026.42 q^{52} +746.932 q^{53} -4451.03 q^{54} -6232.85 q^{55} +33975.3 q^{56} -4382.78 q^{57} +15563.2 q^{58} -3481.00 q^{59} -1986.13 q^{60} +31646.6 q^{61} -6271.22 q^{62} +16868.1 q^{63} +25725.5 q^{64} +39800.0 q^{65} -8193.47 q^{66} -36947.6 q^{67} +5667.81 q^{68} +19784.1 q^{69} +53150.6 q^{70} +67026.7 q^{71} +13215.0 q^{72} +15479.2 q^{73} +19320.8 q^{74} -12398.5 q^{75} -2570.86 q^{76} +31050.8 q^{77} +52319.6 q^{78} +45796.7 q^{79} +48701.8 q^{80} +6561.00 q^{81} -89017.1 q^{82} +16456.1 q^{83} +9894.50 q^{84} -44878.6 q^{85} -81554.1 q^{86} -22940.8 q^{87} +24326.2 q^{88} -28480.0 q^{89} +20673.4 q^{90} -198276. q^{91} +11604.9 q^{92} +9244.02 q^{93} -96527.9 q^{94} +20356.4 q^{95} +17034.8 q^{96} +79923.6 q^{97} -162168. q^{98} +12077.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.10567 −1.07934 −0.539670 0.841876i \(-0.681451\pi\)
−0.539670 + 0.841876i \(0.681451\pi\)
\(3\) 9.00000 0.577350
\(4\) 5.27923 0.164976
\(5\) −41.8018 −0.747772 −0.373886 0.927475i \(-0.621975\pi\)
−0.373886 + 0.927475i \(0.621975\pi\)
\(6\) −54.9510 −0.623158
\(7\) 208.248 1.60633 0.803166 0.595755i \(-0.203147\pi\)
0.803166 + 0.595755i \(0.203147\pi\)
\(8\) 163.148 0.901275
\(9\) 81.0000 0.333333
\(10\) 255.228 0.807101
\(11\) 149.105 0.371544 0.185772 0.982593i \(-0.440521\pi\)
0.185772 + 0.982593i \(0.440521\pi\)
\(12\) 47.5131 0.0952489
\(13\) −952.113 −1.56254 −0.781268 0.624196i \(-0.785427\pi\)
−0.781268 + 0.624196i \(0.785427\pi\)
\(14\) −1271.49 −1.73378
\(15\) −376.216 −0.431727
\(16\) −1165.07 −1.13776
\(17\) 1073.60 0.900994 0.450497 0.892778i \(-0.351247\pi\)
0.450497 + 0.892778i \(0.351247\pi\)
\(18\) −494.559 −0.359780
\(19\) −486.976 −0.309473 −0.154737 0.987956i \(-0.549453\pi\)
−0.154737 + 0.987956i \(0.549453\pi\)
\(20\) −220.681 −0.123364
\(21\) 1874.23 0.927417
\(22\) −910.386 −0.401022
\(23\) 2198.23 0.866469 0.433235 0.901281i \(-0.357372\pi\)
0.433235 + 0.901281i \(0.357372\pi\)
\(24\) 1468.33 0.520352
\(25\) −1377.61 −0.440836
\(26\) 5813.29 1.68651
\(27\) 729.000 0.192450
\(28\) 1099.39 0.265006
\(29\) −2548.98 −0.562823 −0.281411 0.959587i \(-0.590803\pi\)
−0.281411 + 0.959587i \(0.590803\pi\)
\(30\) 2297.05 0.465980
\(31\) 1027.11 0.191961 0.0959807 0.995383i \(-0.469401\pi\)
0.0959807 + 0.995383i \(0.469401\pi\)
\(32\) 1892.76 0.326754
\(33\) 1341.94 0.214511
\(34\) −6555.08 −0.972480
\(35\) −8705.13 −1.20117
\(36\) 427.618 0.0549920
\(37\) −3164.40 −0.380003 −0.190001 0.981784i \(-0.560849\pi\)
−0.190001 + 0.981784i \(0.560849\pi\)
\(38\) 2973.31 0.334027
\(39\) −8569.02 −0.902131
\(40\) −6819.88 −0.673949
\(41\) 14579.4 1.35450 0.677252 0.735751i \(-0.263171\pi\)
0.677252 + 0.735751i \(0.263171\pi\)
\(42\) −11443.4 −1.00100
\(43\) 13357.1 1.10164 0.550822 0.834623i \(-0.314314\pi\)
0.550822 + 0.834623i \(0.314314\pi\)
\(44\) 787.159 0.0612958
\(45\) −3385.94 −0.249257
\(46\) −13421.7 −0.935215
\(47\) 15809.5 1.04394 0.521969 0.852965i \(-0.325198\pi\)
0.521969 + 0.852965i \(0.325198\pi\)
\(48\) −10485.6 −0.656885
\(49\) 26560.2 1.58030
\(50\) 8411.26 0.475813
\(51\) 9662.44 0.520189
\(52\) −5026.42 −0.257781
\(53\) 746.932 0.0365251 0.0182625 0.999833i \(-0.494187\pi\)
0.0182625 + 0.999833i \(0.494187\pi\)
\(54\) −4451.03 −0.207719
\(55\) −6232.85 −0.277830
\(56\) 33975.3 1.44775
\(57\) −4382.78 −0.178675
\(58\) 15563.2 0.607478
\(59\) −3481.00 −0.130189
\(60\) −1986.13 −0.0712245
\(61\) 31646.6 1.08894 0.544468 0.838782i \(-0.316732\pi\)
0.544468 + 0.838782i \(0.316732\pi\)
\(62\) −6271.22 −0.207192
\(63\) 16868.1 0.535444
\(64\) 25725.5 0.785080
\(65\) 39800.0 1.16842
\(66\) −8193.47 −0.231530
\(67\) −36947.6 −1.00554 −0.502770 0.864420i \(-0.667686\pi\)
−0.502770 + 0.864420i \(0.667686\pi\)
\(68\) 5667.81 0.148642
\(69\) 19784.1 0.500256
\(70\) 53150.6 1.29647
\(71\) 67026.7 1.57798 0.788991 0.614405i \(-0.210604\pi\)
0.788991 + 0.614405i \(0.210604\pi\)
\(72\) 13215.0 0.300425
\(73\) 15479.2 0.339971 0.169986 0.985447i \(-0.445628\pi\)
0.169986 + 0.985447i \(0.445628\pi\)
\(74\) 19320.8 0.410153
\(75\) −12398.5 −0.254517
\(76\) −2570.86 −0.0510557
\(77\) 31050.8 0.596823
\(78\) 52319.6 0.973706
\(79\) 45796.7 0.825594 0.412797 0.910823i \(-0.364552\pi\)
0.412797 + 0.910823i \(0.364552\pi\)
\(80\) 48701.8 0.850785
\(81\) 6561.00 0.111111
\(82\) −89017.1 −1.46197
\(83\) 16456.1 0.262199 0.131100 0.991369i \(-0.458149\pi\)
0.131100 + 0.991369i \(0.458149\pi\)
\(84\) 9894.50 0.153001
\(85\) −44878.6 −0.673739
\(86\) −81554.1 −1.18905
\(87\) −22940.8 −0.324946
\(88\) 24326.2 0.334863
\(89\) −28480.0 −0.381123 −0.190562 0.981675i \(-0.561031\pi\)
−0.190562 + 0.981675i \(0.561031\pi\)
\(90\) 20673.4 0.269034
\(91\) −198276. −2.50995
\(92\) 11604.9 0.142947
\(93\) 9244.02 0.110829
\(94\) −96527.9 −1.12676
\(95\) 20356.4 0.231416
\(96\) 17034.8 0.188651
\(97\) 79923.6 0.862473 0.431237 0.902239i \(-0.358077\pi\)
0.431237 + 0.902239i \(0.358077\pi\)
\(98\) −162168. −1.70569
\(99\) 12077.5 0.123848
\(100\) −7272.74 −0.0727274
\(101\) 134102. 1.30808 0.654038 0.756462i \(-0.273074\pi\)
0.654038 + 0.756462i \(0.273074\pi\)
\(102\) −58995.7 −0.561461
\(103\) −119371. −1.10868 −0.554339 0.832291i \(-0.687029\pi\)
−0.554339 + 0.832291i \(0.687029\pi\)
\(104\) −155336. −1.40828
\(105\) −78346.1 −0.693497
\(106\) −4560.52 −0.0394230
\(107\) 61222.4 0.516953 0.258476 0.966018i \(-0.416780\pi\)
0.258476 + 0.966018i \(0.416780\pi\)
\(108\) 3848.56 0.0317496
\(109\) 118858. 0.958211 0.479106 0.877757i \(-0.340961\pi\)
0.479106 + 0.877757i \(0.340961\pi\)
\(110\) 38055.7 0.299874
\(111\) −28479.6 −0.219395
\(112\) −242622. −1.82762
\(113\) 248492. 1.83069 0.915347 0.402665i \(-0.131916\pi\)
0.915347 + 0.402665i \(0.131916\pi\)
\(114\) 26759.8 0.192851
\(115\) −91889.8 −0.647922
\(116\) −13456.7 −0.0928522
\(117\) −77121.1 −0.520845
\(118\) 21253.8 0.140518
\(119\) 223576. 1.44730
\(120\) −61378.9 −0.389105
\(121\) −138819. −0.861955
\(122\) −193224. −1.17533
\(123\) 131215. 0.782023
\(124\) 5422.37 0.0316690
\(125\) 188217. 1.07742
\(126\) −102991. −0.577927
\(127\) −184071. −1.01269 −0.506343 0.862332i \(-0.669003\pi\)
−0.506343 + 0.862332i \(0.669003\pi\)
\(128\) −217640. −1.17412
\(129\) 120214. 0.636034
\(130\) −243006. −1.26112
\(131\) −256895. −1.30791 −0.653953 0.756535i \(-0.726891\pi\)
−0.653953 + 0.756535i \(0.726891\pi\)
\(132\) 7084.43 0.0353892
\(133\) −101412. −0.497117
\(134\) 225590. 1.08532
\(135\) −30473.5 −0.143909
\(136\) 175157. 0.812044
\(137\) −118936. −0.541394 −0.270697 0.962665i \(-0.587254\pi\)
−0.270697 + 0.962665i \(0.587254\pi\)
\(138\) −120795. −0.539947
\(139\) −147451. −0.647306 −0.323653 0.946176i \(-0.604911\pi\)
−0.323653 + 0.946176i \(0.604911\pi\)
\(140\) −45956.4 −0.198164
\(141\) 142286. 0.602718
\(142\) −409243. −1.70318
\(143\) −141965. −0.580551
\(144\) −94370.3 −0.379253
\(145\) 106552. 0.420863
\(146\) −94511.0 −0.366945
\(147\) 239042. 0.912389
\(148\) −16705.6 −0.0626913
\(149\) 125781. 0.464139 0.232070 0.972699i \(-0.425450\pi\)
0.232070 + 0.972699i \(0.425450\pi\)
\(150\) 75701.3 0.274711
\(151\) 181920. 0.649288 0.324644 0.945836i \(-0.394755\pi\)
0.324644 + 0.945836i \(0.394755\pi\)
\(152\) −79449.2 −0.278921
\(153\) 86962.0 0.300331
\(154\) −189586. −0.644176
\(155\) −42935.1 −0.143543
\(156\) −45237.8 −0.148830
\(157\) 252024. 0.816004 0.408002 0.912981i \(-0.366226\pi\)
0.408002 + 0.912981i \(0.366226\pi\)
\(158\) −279620. −0.891097
\(159\) 6722.39 0.0210878
\(160\) −79120.7 −0.244338
\(161\) 457776. 1.39184
\(162\) −40059.3 −0.119927
\(163\) −284724. −0.839373 −0.419687 0.907669i \(-0.637860\pi\)
−0.419687 + 0.907669i \(0.637860\pi\)
\(164\) 76968.1 0.223461
\(165\) −56095.6 −0.160405
\(166\) −100475. −0.283002
\(167\) −71081.6 −0.197227 −0.0986134 0.995126i \(-0.531441\pi\)
−0.0986134 + 0.995126i \(0.531441\pi\)
\(168\) 305778. 0.835858
\(169\) 535226. 1.44152
\(170\) 274014. 0.727194
\(171\) −39445.0 −0.103158
\(172\) 70515.2 0.181745
\(173\) −420683. −1.06866 −0.534331 0.845276i \(-0.679436\pi\)
−0.534331 + 0.845276i \(0.679436\pi\)
\(174\) 140069. 0.350727
\(175\) −286885. −0.708130
\(176\) −173717. −0.422727
\(177\) −31329.0 −0.0751646
\(178\) 173890. 0.411362
\(179\) −49547.5 −0.115582 −0.0577909 0.998329i \(-0.518406\pi\)
−0.0577909 + 0.998329i \(0.518406\pi\)
\(180\) −17875.2 −0.0411215
\(181\) −5910.71 −0.0134104 −0.00670522 0.999978i \(-0.502134\pi\)
−0.00670522 + 0.999978i \(0.502134\pi\)
\(182\) 1.21061e6 2.70909
\(183\) 284819. 0.628697
\(184\) 358637. 0.780927
\(185\) 132277. 0.284156
\(186\) −56440.9 −0.119622
\(187\) 160080. 0.334759
\(188\) 83462.2 0.172225
\(189\) 151813. 0.309139
\(190\) −124290. −0.249776
\(191\) 745180. 1.47801 0.739006 0.673699i \(-0.235296\pi\)
0.739006 + 0.673699i \(0.235296\pi\)
\(192\) 231530. 0.453266
\(193\) −29909.0 −0.0577975 −0.0288987 0.999582i \(-0.509200\pi\)
−0.0288987 + 0.999582i \(0.509200\pi\)
\(194\) −487987. −0.930902
\(195\) 358200. 0.674588
\(196\) 140217. 0.260712
\(197\) −500243. −0.918365 −0.459183 0.888342i \(-0.651858\pi\)
−0.459183 + 0.888342i \(0.651858\pi\)
\(198\) −73741.2 −0.133674
\(199\) −517673. −0.926664 −0.463332 0.886185i \(-0.653346\pi\)
−0.463332 + 0.886185i \(0.653346\pi\)
\(200\) −224755. −0.397315
\(201\) −332528. −0.580548
\(202\) −818785. −1.41186
\(203\) −530820. −0.904081
\(204\) 51010.2 0.0858187
\(205\) −609445. −1.01286
\(206\) 728839. 1.19664
\(207\) 178056. 0.288823
\(208\) 1.10927e6 1.77779
\(209\) −72610.5 −0.114983
\(210\) 478356. 0.748519
\(211\) 1.25719e6 1.94400 0.971999 0.234984i \(-0.0755037\pi\)
0.971999 + 0.234984i \(0.0755037\pi\)
\(212\) 3943.23 0.00602576
\(213\) 603240. 0.911048
\(214\) −373804. −0.557968
\(215\) −558350. −0.823779
\(216\) 118935. 0.173451
\(217\) 213894. 0.308354
\(218\) −725707. −1.03424
\(219\) 139313. 0.196282
\(220\) −32904.6 −0.0458353
\(221\) −1.02219e6 −1.40784
\(222\) 173887. 0.236802
\(223\) 967522. 1.30286 0.651431 0.758708i \(-0.274169\pi\)
0.651431 + 0.758708i \(0.274169\pi\)
\(224\) 394163. 0.524876
\(225\) −111587. −0.146945
\(226\) −1.51721e6 −1.97594
\(227\) −374162. −0.481943 −0.240971 0.970532i \(-0.577466\pi\)
−0.240971 + 0.970532i \(0.577466\pi\)
\(228\) −23137.7 −0.0294770
\(229\) 65812.0 0.0829309 0.0414654 0.999140i \(-0.486797\pi\)
0.0414654 + 0.999140i \(0.486797\pi\)
\(230\) 561049. 0.699328
\(231\) 279457. 0.344576
\(232\) −415862. −0.507258
\(233\) −633048. −0.763918 −0.381959 0.924179i \(-0.624751\pi\)
−0.381959 + 0.924179i \(0.624751\pi\)
\(234\) 470876. 0.562170
\(235\) −660866. −0.780628
\(236\) −18377.0 −0.0214780
\(237\) 412170. 0.476657
\(238\) −1.36508e6 −1.56213
\(239\) 728479. 0.824940 0.412470 0.910971i \(-0.364666\pi\)
0.412470 + 0.910971i \(0.364666\pi\)
\(240\) 438316. 0.491201
\(241\) 1.22438e6 1.35792 0.678960 0.734175i \(-0.262431\pi\)
0.678960 + 0.734175i \(0.262431\pi\)
\(242\) 847582. 0.930343
\(243\) 59049.0 0.0641500
\(244\) 167070. 0.179648
\(245\) −1.11026e6 −1.18171
\(246\) −801154. −0.844070
\(247\) 463656. 0.483563
\(248\) 167572. 0.173010
\(249\) 148105. 0.151381
\(250\) −1.14919e6 −1.16290
\(251\) −388944. −0.389675 −0.194838 0.980835i \(-0.562418\pi\)
−0.194838 + 0.980835i \(0.562418\pi\)
\(252\) 89050.5 0.0883354
\(253\) 327767. 0.321931
\(254\) 1.12387e6 1.09303
\(255\) −403907. −0.388983
\(256\) 505621. 0.482198
\(257\) 1.35199e6 1.27685 0.638427 0.769683i \(-0.279585\pi\)
0.638427 + 0.769683i \(0.279585\pi\)
\(258\) −733987. −0.686497
\(259\) −658980. −0.610411
\(260\) 210113. 0.192761
\(261\) −206468. −0.187608
\(262\) 1.56851e6 1.41168
\(263\) −685770. −0.611348 −0.305674 0.952136i \(-0.598882\pi\)
−0.305674 + 0.952136i \(0.598882\pi\)
\(264\) 218936. 0.193333
\(265\) −31223.1 −0.0273125
\(266\) 619186. 0.536559
\(267\) −256320. −0.220041
\(268\) −195055. −0.165890
\(269\) 2.15145e6 1.81281 0.906403 0.422413i \(-0.138817\pi\)
0.906403 + 0.422413i \(0.138817\pi\)
\(270\) 186061. 0.155327
\(271\) −2.36646e6 −1.95738 −0.978692 0.205336i \(-0.934171\pi\)
−0.978692 + 0.205336i \(0.934171\pi\)
\(272\) −1.25082e6 −1.02511
\(273\) −1.78448e6 −1.44912
\(274\) 726186. 0.584348
\(275\) −205409. −0.163790
\(276\) 104445. 0.0825302
\(277\) 2.37254e6 1.85786 0.928930 0.370254i \(-0.120729\pi\)
0.928930 + 0.370254i \(0.120729\pi\)
\(278\) 900285. 0.698663
\(279\) 83196.2 0.0639872
\(280\) −1.42023e6 −1.08259
\(281\) −386861. −0.292273 −0.146137 0.989264i \(-0.546684\pi\)
−0.146137 + 0.989264i \(0.546684\pi\)
\(282\) −868751. −0.650538
\(283\) 19175.2 0.0142323 0.00711613 0.999975i \(-0.497735\pi\)
0.00711613 + 0.999975i \(0.497735\pi\)
\(284\) 353849. 0.260329
\(285\) 183208. 0.133608
\(286\) 866790. 0.626612
\(287\) 3.03613e6 2.17578
\(288\) 153314. 0.108918
\(289\) −267230. −0.188209
\(290\) −650571. −0.454255
\(291\) 719312. 0.497949
\(292\) 81718.4 0.0560870
\(293\) −1.93969e6 −1.31997 −0.659985 0.751279i \(-0.729437\pi\)
−0.659985 + 0.751279i \(0.729437\pi\)
\(294\) −1.45951e6 −0.984779
\(295\) 145512. 0.0973517
\(296\) −516266. −0.342487
\(297\) 108697. 0.0715037
\(298\) −767976. −0.500965
\(299\) −2.09296e6 −1.35389
\(300\) −65454.7 −0.0419892
\(301\) 2.78159e6 1.76961
\(302\) −1.11074e6 −0.700803
\(303\) 1.20692e6 0.755218
\(304\) 567358. 0.352106
\(305\) −1.32288e6 −0.814276
\(306\) −530961. −0.324160
\(307\) −110640. −0.0669988 −0.0334994 0.999439i \(-0.510665\pi\)
−0.0334994 + 0.999439i \(0.510665\pi\)
\(308\) 163924. 0.0984615
\(309\) −1.07434e6 −0.640095
\(310\) 262148. 0.154932
\(311\) 2.13892e6 1.25399 0.626994 0.779024i \(-0.284285\pi\)
0.626994 + 0.779024i \(0.284285\pi\)
\(312\) −1.39802e6 −0.813068
\(313\) 2.94572e6 1.69953 0.849767 0.527158i \(-0.176742\pi\)
0.849767 + 0.527158i \(0.176742\pi\)
\(314\) −1.53877e6 −0.880747
\(315\) −705115. −0.400390
\(316\) 241771. 0.136203
\(317\) 1.02771e6 0.574412 0.287206 0.957869i \(-0.407274\pi\)
0.287206 + 0.957869i \(0.407274\pi\)
\(318\) −41044.7 −0.0227609
\(319\) −380066. −0.209113
\(320\) −1.07537e6 −0.587061
\(321\) 551002. 0.298463
\(322\) −2.79503e6 −1.50227
\(323\) −522819. −0.278834
\(324\) 34637.0 0.0183307
\(325\) 1.31164e6 0.688823
\(326\) 1.73843e6 0.905970
\(327\) 1.06972e6 0.553224
\(328\) 2.37861e6 1.22078
\(329\) 3.29230e6 1.67691
\(330\) 342501. 0.173132
\(331\) −1.99485e6 −1.00078 −0.500391 0.865799i \(-0.666811\pi\)
−0.500391 + 0.865799i \(0.666811\pi\)
\(332\) 86875.4 0.0432565
\(333\) −256316. −0.126668
\(334\) 434001. 0.212875
\(335\) 1.54447e6 0.751915
\(336\) −2.18360e6 −1.05518
\(337\) −2.56238e6 −1.22905 −0.614523 0.788899i \(-0.710652\pi\)
−0.614523 + 0.788899i \(0.710652\pi\)
\(338\) −3.26791e6 −1.55589
\(339\) 2.23643e6 1.05695
\(340\) −236924. −0.111151
\(341\) 153148. 0.0713221
\(342\) 240838. 0.111342
\(343\) 2.03108e6 0.932162
\(344\) 2.17919e6 0.992884
\(345\) −827008. −0.374078
\(346\) 2.56855e6 1.15345
\(347\) −1.22248e6 −0.545026 −0.272513 0.962152i \(-0.587855\pi\)
−0.272513 + 0.962152i \(0.587855\pi\)
\(348\) −121110. −0.0536083
\(349\) −3.21103e6 −1.41117 −0.705587 0.708623i \(-0.749317\pi\)
−0.705587 + 0.708623i \(0.749317\pi\)
\(350\) 1.75163e6 0.764313
\(351\) −694090. −0.300710
\(352\) 282220. 0.121403
\(353\) −4.36299e6 −1.86358 −0.931789 0.363001i \(-0.881752\pi\)
−0.931789 + 0.363001i \(0.881752\pi\)
\(354\) 191285. 0.0811282
\(355\) −2.80183e6 −1.17997
\(356\) −150353. −0.0628761
\(357\) 2.01218e6 0.835597
\(358\) 302521. 0.124752
\(359\) 38987.2 0.0159656 0.00798281 0.999968i \(-0.497459\pi\)
0.00798281 + 0.999968i \(0.497459\pi\)
\(360\) −552411. −0.224650
\(361\) −2.23895e6 −0.904226
\(362\) 36088.9 0.0144744
\(363\) −1.24937e6 −0.497650
\(364\) −1.04674e6 −0.414082
\(365\) −647059. −0.254221
\(366\) −1.73901e6 −0.678578
\(367\) −615070. −0.238374 −0.119187 0.992872i \(-0.538029\pi\)
−0.119187 + 0.992872i \(0.538029\pi\)
\(368\) −2.56108e6 −0.985833
\(369\) 1.18093e6 0.451501
\(370\) −807643. −0.306701
\(371\) 155547. 0.0586715
\(372\) 48801.3 0.0182841
\(373\) −2.59722e6 −0.966579 −0.483289 0.875461i \(-0.660558\pi\)
−0.483289 + 0.875461i \(0.660558\pi\)
\(374\) −977394. −0.361319
\(375\) 1.69395e6 0.622047
\(376\) 2.57930e6 0.940875
\(377\) 2.42692e6 0.879431
\(378\) −926919. −0.333666
\(379\) −294437. −0.105292 −0.0526459 0.998613i \(-0.516765\pi\)
−0.0526459 + 0.998613i \(0.516765\pi\)
\(380\) 107466. 0.0381780
\(381\) −1.65663e6 −0.584675
\(382\) −4.54983e6 −1.59528
\(383\) 2.51121e6 0.874753 0.437377 0.899278i \(-0.355908\pi\)
0.437377 + 0.899278i \(0.355908\pi\)
\(384\) −1.95876e6 −0.677880
\(385\) −1.29798e6 −0.446288
\(386\) 182615. 0.0623831
\(387\) 1.08193e6 0.367214
\(388\) 421935. 0.142287
\(389\) −3.42294e6 −1.14690 −0.573449 0.819241i \(-0.694395\pi\)
−0.573449 + 0.819241i \(0.694395\pi\)
\(390\) −2.18705e6 −0.728111
\(391\) 2.36003e6 0.780684
\(392\) 4.33325e6 1.42429
\(393\) −2.31205e6 −0.755120
\(394\) 3.05432e6 0.991229
\(395\) −1.91438e6 −0.617356
\(396\) 63759.9 0.0204319
\(397\) −4.77198e6 −1.51957 −0.759787 0.650172i \(-0.774697\pi\)
−0.759787 + 0.650172i \(0.774697\pi\)
\(398\) 3.16074e6 1.00019
\(399\) −912705. −0.287011
\(400\) 1.60501e6 0.501565
\(401\) 1.57375e6 0.488737 0.244369 0.969682i \(-0.421419\pi\)
0.244369 + 0.969682i \(0.421419\pi\)
\(402\) 2.03031e6 0.626609
\(403\) −977928. −0.299947
\(404\) 707957. 0.215801
\(405\) −274261. −0.0830858
\(406\) 3.24101e6 0.975811
\(407\) −471828. −0.141188
\(408\) 1.57641e6 0.468834
\(409\) −4.72041e6 −1.39531 −0.697657 0.716432i \(-0.745774\pi\)
−0.697657 + 0.716432i \(0.745774\pi\)
\(410\) 3.72107e6 1.09322
\(411\) −1.07043e6 −0.312574
\(412\) −630186. −0.182905
\(413\) −724911. −0.209127
\(414\) −1.08715e6 −0.311738
\(415\) −687893. −0.196065
\(416\) −1.80212e6 −0.510565
\(417\) −1.32706e6 −0.373722
\(418\) 443336. 0.124106
\(419\) −1.33538e6 −0.371595 −0.185798 0.982588i \(-0.559487\pi\)
−0.185798 + 0.982588i \(0.559487\pi\)
\(420\) −413607. −0.114410
\(421\) −338764. −0.0931521 −0.0465760 0.998915i \(-0.514831\pi\)
−0.0465760 + 0.998915i \(0.514831\pi\)
\(422\) −7.67601e6 −2.09824
\(423\) 1.28057e6 0.347979
\(424\) 121861. 0.0329192
\(425\) −1.47901e6 −0.397191
\(426\) −3.68319e6 −0.983331
\(427\) 6.59033e6 1.74919
\(428\) 323207. 0.0852848
\(429\) −1.27768e6 −0.335181
\(430\) 3.40910e6 0.889138
\(431\) −7.24422e6 −1.87844 −0.939222 0.343311i \(-0.888452\pi\)
−0.939222 + 0.343311i \(0.888452\pi\)
\(432\) −849332. −0.218962
\(433\) 6.16246e6 1.57955 0.789777 0.613394i \(-0.210196\pi\)
0.789777 + 0.613394i \(0.210196\pi\)
\(434\) −1.30597e6 −0.332819
\(435\) 958967. 0.242986
\(436\) 627478. 0.158082
\(437\) −1.07048e6 −0.268149
\(438\) −850599. −0.211856
\(439\) 4.65424e6 1.15262 0.576311 0.817230i \(-0.304492\pi\)
0.576311 + 0.817230i \(0.304492\pi\)
\(440\) −1.01688e6 −0.250402
\(441\) 2.15137e6 0.526768
\(442\) 6.24117e6 1.51953
\(443\) −1.25610e6 −0.304098 −0.152049 0.988373i \(-0.548587\pi\)
−0.152049 + 0.988373i \(0.548587\pi\)
\(444\) −150350. −0.0361949
\(445\) 1.19051e6 0.284993
\(446\) −5.90737e6 −1.40623
\(447\) 1.13203e6 0.267971
\(448\) 5.35728e6 1.26110
\(449\) −4.46355e6 −1.04488 −0.522438 0.852677i \(-0.674977\pi\)
−0.522438 + 0.852677i \(0.674977\pi\)
\(450\) 681312. 0.158604
\(451\) 2.17386e6 0.503258
\(452\) 1.31185e6 0.302021
\(453\) 1.63728e6 0.374867
\(454\) 2.28451e6 0.520180
\(455\) 8.28826e6 1.87687
\(456\) −715043. −0.161035
\(457\) 4.01815e6 0.899986 0.449993 0.893032i \(-0.351427\pi\)
0.449993 + 0.893032i \(0.351427\pi\)
\(458\) −401826. −0.0895106
\(459\) 782658. 0.173396
\(460\) −485107. −0.106891
\(461\) 1.98402e6 0.434804 0.217402 0.976082i \(-0.430242\pi\)
0.217402 + 0.976082i \(0.430242\pi\)
\(462\) −1.70627e6 −0.371915
\(463\) 2.31083e6 0.500975 0.250488 0.968120i \(-0.419409\pi\)
0.250488 + 0.968120i \(0.419409\pi\)
\(464\) 2.96973e6 0.640357
\(465\) −386416. −0.0828749
\(466\) 3.86518e6 0.824528
\(467\) 2.65831e6 0.564044 0.282022 0.959408i \(-0.408995\pi\)
0.282022 + 0.959408i \(0.408995\pi\)
\(468\) −407140. −0.0859270
\(469\) −7.69426e6 −1.61523
\(470\) 4.03503e6 0.842563
\(471\) 2.26821e6 0.471120
\(472\) −567919. −0.117336
\(473\) 1.99161e6 0.409309
\(474\) −2.51658e6 −0.514475
\(475\) 670864. 0.136427
\(476\) 1.18031e6 0.238769
\(477\) 60501.5 0.0121750
\(478\) −4.44785e6 −0.890391
\(479\) −6.22698e6 −1.24005 −0.620024 0.784582i \(-0.712877\pi\)
−0.620024 + 0.784582i \(0.712877\pi\)
\(480\) −712086. −0.141068
\(481\) 3.01287e6 0.593768
\(482\) −7.47567e6 −1.46566
\(483\) 4.11999e6 0.803578
\(484\) −732856. −0.142202
\(485\) −3.34095e6 −0.644934
\(486\) −360534. −0.0692397
\(487\) 8.29735e6 1.58532 0.792660 0.609664i \(-0.208696\pi\)
0.792660 + 0.609664i \(0.208696\pi\)
\(488\) 5.16308e6 0.981431
\(489\) −2.56252e6 −0.484612
\(490\) 6.77890e6 1.27547
\(491\) −6.71368e6 −1.25677 −0.628386 0.777901i \(-0.716284\pi\)
−0.628386 + 0.777901i \(0.716284\pi\)
\(492\) 692713. 0.129015
\(493\) −2.73660e6 −0.507100
\(494\) −2.83093e6 −0.521930
\(495\) −504861. −0.0926101
\(496\) −1.19665e6 −0.218406
\(497\) 1.39582e7 2.53476
\(498\) −904279. −0.163391
\(499\) −1.07848e7 −1.93892 −0.969458 0.245258i \(-0.921127\pi\)
−0.969458 + 0.245258i \(0.921127\pi\)
\(500\) 993641. 0.177748
\(501\) −639734. −0.113869
\(502\) 2.37477e6 0.420592
\(503\) −6.41855e6 −1.13114 −0.565570 0.824700i \(-0.691344\pi\)
−0.565570 + 0.824700i \(0.691344\pi\)
\(504\) 2.75200e6 0.482583
\(505\) −5.60571e6 −0.978143
\(506\) −2.00124e6 −0.347474
\(507\) 4.81703e6 0.832262
\(508\) −971751. −0.167069
\(509\) 2.78903e6 0.477153 0.238577 0.971124i \(-0.423319\pi\)
0.238577 + 0.971124i \(0.423319\pi\)
\(510\) 2.46612e6 0.419845
\(511\) 3.22351e6 0.546107
\(512\) 3.87732e6 0.653667
\(513\) −355005. −0.0595582
\(514\) −8.25482e6 −1.37816
\(515\) 4.98991e6 0.829038
\(516\) 634637. 0.104930
\(517\) 2.35728e6 0.387869
\(518\) 4.02351e6 0.658842
\(519\) −3.78615e6 −0.616992
\(520\) 6.49330e6 1.05307
\(521\) 6.38091e6 1.02988 0.514942 0.857225i \(-0.327813\pi\)
0.514942 + 0.857225i \(0.327813\pi\)
\(522\) 1.26062e6 0.202493
\(523\) −2.42857e6 −0.388237 −0.194119 0.980978i \(-0.562185\pi\)
−0.194119 + 0.980978i \(0.562185\pi\)
\(524\) −1.35621e6 −0.215773
\(525\) −2.58197e6 −0.408839
\(526\) 4.18708e6 0.659853
\(527\) 1.10271e6 0.172956
\(528\) −1.56345e6 −0.244062
\(529\) −1.60414e6 −0.249231
\(530\) 190638. 0.0294794
\(531\) −281961. −0.0433963
\(532\) −535375. −0.0820124
\(533\) −1.38812e7 −2.11646
\(534\) 1.56501e6 0.237500
\(535\) −2.55920e6 −0.386563
\(536\) −6.02793e6 −0.906268
\(537\) −445928. −0.0667312
\(538\) −1.31361e7 −1.95664
\(539\) 3.96025e6 0.587153
\(540\) −160876. −0.0237415
\(541\) 9.78542e6 1.43743 0.718714 0.695305i \(-0.244731\pi\)
0.718714 + 0.695305i \(0.244731\pi\)
\(542\) 1.44488e7 2.11268
\(543\) −53196.4 −0.00774253
\(544\) 2.03208e6 0.294403
\(545\) −4.96846e6 −0.716524
\(546\) 1.08954e7 1.56410
\(547\) 1.26732e7 1.81100 0.905501 0.424345i \(-0.139496\pi\)
0.905501 + 0.424345i \(0.139496\pi\)
\(548\) −627892. −0.0893169
\(549\) 2.56337e6 0.362979
\(550\) 1.25416e6 0.176785
\(551\) 1.24129e6 0.174179
\(552\) 3.22773e6 0.450869
\(553\) 9.53707e6 1.32618
\(554\) −1.44859e7 −2.00526
\(555\) 1.19050e6 0.164057
\(556\) −778426. −0.106790
\(557\) 9.24073e6 1.26203 0.631013 0.775773i \(-0.282640\pi\)
0.631013 + 0.775773i \(0.282640\pi\)
\(558\) −507968. −0.0690639
\(559\) −1.27175e7 −1.72136
\(560\) 1.01420e7 1.36664
\(561\) 1.44072e6 0.193273
\(562\) 2.36205e6 0.315462
\(563\) 1.33652e7 1.77707 0.888535 0.458809i \(-0.151724\pi\)
0.888535 + 0.458809i \(0.151724\pi\)
\(564\) 751160. 0.0994339
\(565\) −1.03874e7 −1.36894
\(566\) −117078. −0.0153615
\(567\) 1.36631e6 0.178481
\(568\) 1.09353e7 1.42220
\(569\) −8.45895e6 −1.09531 −0.547653 0.836705i \(-0.684479\pi\)
−0.547653 + 0.836705i \(0.684479\pi\)
\(570\) −1.11861e6 −0.144208
\(571\) 1.49334e6 0.191677 0.0958385 0.995397i \(-0.469447\pi\)
0.0958385 + 0.995397i \(0.469447\pi\)
\(572\) −749464. −0.0957769
\(573\) 6.70662e6 0.853330
\(574\) −1.85376e7 −2.34841
\(575\) −3.02831e6 −0.381971
\(576\) 2.08377e6 0.261693
\(577\) 2.62239e6 0.327912 0.163956 0.986468i \(-0.447575\pi\)
0.163956 + 0.986468i \(0.447575\pi\)
\(578\) 1.63162e6 0.203142
\(579\) −269181. −0.0333694
\(580\) 562512. 0.0694323
\(581\) 3.42694e6 0.421179
\(582\) −4.39188e6 −0.537457
\(583\) 111371. 0.0135707
\(584\) 2.52541e6 0.306408
\(585\) 3.22380e6 0.389474
\(586\) 1.18431e7 1.42470
\(587\) 9.30156e6 1.11419 0.557097 0.830448i \(-0.311915\pi\)
0.557097 + 0.830448i \(0.311915\pi\)
\(588\) 1.26196e6 0.150522
\(589\) −500179. −0.0594070
\(590\) −888448. −0.105076
\(591\) −4.50219e6 −0.530218
\(592\) 3.68673e6 0.432352
\(593\) −1.04915e7 −1.22518 −0.612592 0.790399i \(-0.709873\pi\)
−0.612592 + 0.790399i \(0.709873\pi\)
\(594\) −663671. −0.0771768
\(595\) −9.34586e6 −1.08225
\(596\) 664025. 0.0765718
\(597\) −4.65905e6 −0.535010
\(598\) 1.27789e7 1.46131
\(599\) −3.77401e6 −0.429770 −0.214885 0.976639i \(-0.568938\pi\)
−0.214885 + 0.976639i \(0.568938\pi\)
\(600\) −2.02280e6 −0.229390
\(601\) −7.42586e6 −0.838611 −0.419305 0.907845i \(-0.637726\pi\)
−0.419305 + 0.907845i \(0.637726\pi\)
\(602\) −1.69835e7 −1.91001
\(603\) −2.99275e6 −0.335180
\(604\) 960396. 0.107117
\(605\) 5.80287e6 0.644546
\(606\) −7.36906e6 −0.815137
\(607\) 9.37397e6 1.03265 0.516323 0.856394i \(-0.327300\pi\)
0.516323 + 0.856394i \(0.327300\pi\)
\(608\) −921729. −0.101122
\(609\) −4.77738e6 −0.521971
\(610\) 8.07709e6 0.878881
\(611\) −1.50525e7 −1.63119
\(612\) 459092. 0.0495475
\(613\) −1.51042e7 −1.62348 −0.811740 0.584019i \(-0.801479\pi\)
−0.811740 + 0.584019i \(0.801479\pi\)
\(614\) 675533. 0.0723145
\(615\) −5.48500e6 −0.584776
\(616\) 5.06588e6 0.537902
\(617\) 1.17572e7 1.24335 0.621674 0.783276i \(-0.286453\pi\)
0.621674 + 0.783276i \(0.286453\pi\)
\(618\) 6.55955e6 0.690880
\(619\) −1.34123e7 −1.40695 −0.703473 0.710722i \(-0.748368\pi\)
−0.703473 + 0.710722i \(0.748368\pi\)
\(620\) −226664. −0.0236812
\(621\) 1.60251e6 0.166752
\(622\) −1.30595e7 −1.35348
\(623\) −5.93090e6 −0.612210
\(624\) 9.98346e6 1.02641
\(625\) −3.56276e6 −0.364827
\(626\) −1.79856e7 −1.83438
\(627\) −653494. −0.0663855
\(628\) 1.33049e6 0.134621
\(629\) −3.39731e6 −0.342381
\(630\) 4.30520e6 0.432158
\(631\) −1.63331e7 −1.63303 −0.816516 0.577323i \(-0.804097\pi\)
−0.816516 + 0.577323i \(0.804097\pi\)
\(632\) 7.47165e6 0.744087
\(633\) 1.13147e7 1.12237
\(634\) −6.27488e6 −0.619986
\(635\) 7.69447e6 0.757259
\(636\) 35489.0 0.00347898
\(637\) −2.52883e7 −2.46928
\(638\) 2.32056e6 0.225705
\(639\) 5.42916e6 0.525994
\(640\) 9.09773e6 0.877977
\(641\) 3.02654e6 0.290938 0.145469 0.989363i \(-0.453531\pi\)
0.145469 + 0.989363i \(0.453531\pi\)
\(642\) −3.36424e6 −0.322143
\(643\) −1.43363e7 −1.36744 −0.683722 0.729743i \(-0.739640\pi\)
−0.683722 + 0.729743i \(0.739640\pi\)
\(644\) 2.41671e6 0.229620
\(645\) −5.02515e6 −0.475609
\(646\) 3.19216e6 0.300957
\(647\) −8.34526e6 −0.783753 −0.391876 0.920018i \(-0.628174\pi\)
−0.391876 + 0.920018i \(0.628174\pi\)
\(648\) 1.07042e6 0.100142
\(649\) −519034. −0.0483709
\(650\) −8.00847e6 −0.743474
\(651\) 1.92505e6 0.178028
\(652\) −1.50312e6 −0.138476
\(653\) −1.91948e7 −1.76157 −0.880787 0.473513i \(-0.842986\pi\)
−0.880787 + 0.473513i \(0.842986\pi\)
\(654\) −6.53136e6 −0.597117
\(655\) 1.07386e7 0.978016
\(656\) −1.69860e7 −1.54110
\(657\) 1.25382e6 0.113324
\(658\) −2.01017e7 −1.80996
\(659\) −1.01224e7 −0.907967 −0.453984 0.891010i \(-0.649998\pi\)
−0.453984 + 0.891010i \(0.649998\pi\)
\(660\) −296142. −0.0264630
\(661\) 1.37317e7 1.22242 0.611211 0.791467i \(-0.290683\pi\)
0.611211 + 0.791467i \(0.290683\pi\)
\(662\) 1.21799e7 1.08019
\(663\) −9.19974e6 −0.812815
\(664\) 2.68478e6 0.236314
\(665\) 4.23919e6 0.371731
\(666\) 1.56498e6 0.136718
\(667\) −5.60324e6 −0.487669
\(668\) −375256. −0.0325377
\(669\) 8.70770e6 0.752208
\(670\) −9.43005e6 −0.811572
\(671\) 4.71866e6 0.404587
\(672\) 3.54747e6 0.303037
\(673\) −1.99795e7 −1.70038 −0.850191 0.526475i \(-0.823513\pi\)
−0.850191 + 0.526475i \(0.823513\pi\)
\(674\) 1.56450e7 1.32656
\(675\) −1.00428e6 −0.0848390
\(676\) 2.82558e6 0.237816
\(677\) −1.45995e7 −1.22424 −0.612118 0.790767i \(-0.709682\pi\)
−0.612118 + 0.790767i \(0.709682\pi\)
\(678\) −1.36549e7 −1.14081
\(679\) 1.66439e7 1.38542
\(680\) −7.32186e6 −0.607224
\(681\) −3.36746e6 −0.278250
\(682\) −935069. −0.0769809
\(683\) 1.29809e7 1.06477 0.532383 0.846504i \(-0.321297\pi\)
0.532383 + 0.846504i \(0.321297\pi\)
\(684\) −208239. −0.0170186
\(685\) 4.97175e6 0.404839
\(686\) −1.24011e7 −1.00612
\(687\) 592308. 0.0478802
\(688\) −1.55619e7 −1.25340
\(689\) −711164. −0.0570718
\(690\) 5.04944e6 0.403757
\(691\) −1.24354e7 −0.990753 −0.495376 0.868678i \(-0.664970\pi\)
−0.495376 + 0.868678i \(0.664970\pi\)
\(692\) −2.22088e6 −0.176303
\(693\) 2.51511e6 0.198941
\(694\) 7.46404e6 0.588268
\(695\) 6.16369e6 0.484038
\(696\) −3.74276e6 −0.292866
\(697\) 1.56525e7 1.22040
\(698\) 1.96055e7 1.52314
\(699\) −5.69743e6 −0.441049
\(700\) −1.51453e6 −0.116824
\(701\) −3.72318e6 −0.286166 −0.143083 0.989711i \(-0.545702\pi\)
−0.143083 + 0.989711i \(0.545702\pi\)
\(702\) 4.23789e6 0.324569
\(703\) 1.54099e6 0.117601
\(704\) 3.83580e6 0.291692
\(705\) −5.94780e6 −0.450696
\(706\) 2.66390e7 2.01144
\(707\) 2.79265e7 2.10120
\(708\) −165393. −0.0124003
\(709\) −9.20964e6 −0.688061 −0.344030 0.938958i \(-0.611792\pi\)
−0.344030 + 0.938958i \(0.611792\pi\)
\(710\) 1.71071e7 1.27359
\(711\) 3.70953e6 0.275198
\(712\) −4.64646e6 −0.343497
\(713\) 2.25783e6 0.166329
\(714\) −1.22857e7 −0.901894
\(715\) 5.93437e6 0.434120
\(716\) −261573. −0.0190682
\(717\) 6.55631e6 0.476279
\(718\) −238043. −0.0172324
\(719\) −4.15670e6 −0.299865 −0.149933 0.988696i \(-0.547906\pi\)
−0.149933 + 0.988696i \(0.547906\pi\)
\(720\) 3.94484e6 0.283595
\(721\) −2.48587e7 −1.78090
\(722\) 1.36703e7 0.975968
\(723\) 1.10194e7 0.783996
\(724\) −31204.0 −0.00221240
\(725\) 3.51151e6 0.248113
\(726\) 7.62823e6 0.537134
\(727\) −1.87342e6 −0.131462 −0.0657309 0.997837i \(-0.520938\pi\)
−0.0657309 + 0.997837i \(0.520938\pi\)
\(728\) −3.23483e7 −2.26216
\(729\) 531441. 0.0370370
\(730\) 3.95073e6 0.274391
\(731\) 1.43402e7 0.992575
\(732\) 1.50363e6 0.103720
\(733\) −1.70190e7 −1.16997 −0.584986 0.811044i \(-0.698900\pi\)
−0.584986 + 0.811044i \(0.698900\pi\)
\(734\) 3.75542e6 0.257287
\(735\) −9.99236e6 −0.682260
\(736\) 4.16072e6 0.283122
\(737\) −5.50907e6 −0.373602
\(738\) −7.21039e6 −0.487324
\(739\) −1.51158e7 −1.01817 −0.509084 0.860717i \(-0.670016\pi\)
−0.509084 + 0.860717i \(0.670016\pi\)
\(740\) 698323. 0.0468789
\(741\) 4.17290e6 0.279185
\(742\) −949719. −0.0633265
\(743\) 1.88824e7 1.25483 0.627416 0.778684i \(-0.284112\pi\)
0.627416 + 0.778684i \(0.284112\pi\)
\(744\) 1.50815e6 0.0998874
\(745\) −5.25786e6 −0.347071
\(746\) 1.58578e7 1.04327
\(747\) 1.33294e6 0.0873997
\(748\) 845098. 0.0552272
\(749\) 1.27494e7 0.830398
\(750\) −1.03427e7 −0.671401
\(751\) −1.00270e7 −0.648739 −0.324369 0.945931i \(-0.605152\pi\)
−0.324369 + 0.945931i \(0.605152\pi\)
\(752\) −1.84191e7 −1.18775
\(753\) −3.50050e6 −0.224979
\(754\) −1.48180e7 −0.949206
\(755\) −7.60456e6 −0.485520
\(756\) 801454. 0.0510005
\(757\) −2.49293e7 −1.58114 −0.790570 0.612372i \(-0.790215\pi\)
−0.790570 + 0.612372i \(0.790215\pi\)
\(758\) 1.79774e6 0.113646
\(759\) 2.94990e6 0.185867
\(760\) 3.32112e6 0.208569
\(761\) −1.02441e7 −0.641229 −0.320614 0.947210i \(-0.603889\pi\)
−0.320614 + 0.947210i \(0.603889\pi\)
\(762\) 1.01149e7 0.631063
\(763\) 2.47519e7 1.53921
\(764\) 3.93398e6 0.243836
\(765\) −3.63516e6 −0.224580
\(766\) −1.53326e7 −0.944157
\(767\) 3.31431e6 0.203425
\(768\) 4.55059e6 0.278397
\(769\) 2.17564e7 1.32670 0.663348 0.748311i \(-0.269135\pi\)
0.663348 + 0.748311i \(0.269135\pi\)
\(770\) 7.92502e6 0.481697
\(771\) 1.21679e7 0.737192
\(772\) −157897. −0.00953519
\(773\) 2.30398e7 1.38685 0.693425 0.720529i \(-0.256101\pi\)
0.693425 + 0.720529i \(0.256101\pi\)
\(774\) −6.60588e6 −0.396349
\(775\) −1.41497e6 −0.0846236
\(776\) 1.30394e7 0.777326
\(777\) −5.93082e6 −0.352421
\(778\) 2.08993e7 1.23789
\(779\) −7.09982e6 −0.419183
\(780\) 1.89102e6 0.111291
\(781\) 9.99401e6 0.586289
\(782\) −1.44096e7 −0.842624
\(783\) −1.85821e6 −0.108315
\(784\) −3.09443e7 −1.79801
\(785\) −1.05350e7 −0.610186
\(786\) 1.41166e7 0.815032
\(787\) 3.85957e6 0.222127 0.111064 0.993813i \(-0.464574\pi\)
0.111064 + 0.993813i \(0.464574\pi\)
\(788\) −2.64090e6 −0.151508
\(789\) −6.17193e6 −0.352962
\(790\) 1.16886e7 0.666338
\(791\) 5.17479e7 2.94071
\(792\) 1.97042e6 0.111621
\(793\) −3.01311e7 −1.70150
\(794\) 2.91361e7 1.64014
\(795\) −281008. −0.0157689
\(796\) −2.73291e6 −0.152877
\(797\) 1.58425e7 0.883444 0.441722 0.897152i \(-0.354368\pi\)
0.441722 + 0.897152i \(0.354368\pi\)
\(798\) 5.57268e6 0.309782
\(799\) 1.69732e7 0.940582
\(800\) −2.60749e6 −0.144045
\(801\) −2.30688e6 −0.127041
\(802\) −9.60881e6 −0.527514
\(803\) 2.30803e6 0.126314
\(804\) −1.75549e6 −0.0957765
\(805\) −1.91359e7 −1.04078
\(806\) 5.97091e6 0.323745
\(807\) 1.93631e7 1.04662
\(808\) 2.18786e7 1.17894
\(809\) 1.45751e7 0.782961 0.391481 0.920186i \(-0.371963\pi\)
0.391481 + 0.920186i \(0.371963\pi\)
\(810\) 1.67455e6 0.0896779
\(811\) 3.03425e7 1.61994 0.809970 0.586472i \(-0.199484\pi\)
0.809970 + 0.586472i \(0.199484\pi\)
\(812\) −2.80232e6 −0.149152
\(813\) −2.12981e7 −1.13010
\(814\) 2.88082e6 0.152390
\(815\) 1.19020e7 0.627660
\(816\) −1.12574e7 −0.591850
\(817\) −6.50458e6 −0.340929
\(818\) 2.88213e7 1.50602
\(819\) −1.60603e7 −0.836651
\(820\) −3.21740e6 −0.167098
\(821\) 3.22288e6 0.166873 0.0834366 0.996513i \(-0.473410\pi\)
0.0834366 + 0.996513i \(0.473410\pi\)
\(822\) 6.53568e6 0.337373
\(823\) −4.75007e6 −0.244456 −0.122228 0.992502i \(-0.539004\pi\)
−0.122228 + 0.992502i \(0.539004\pi\)
\(824\) −1.94751e7 −0.999223
\(825\) −1.84868e6 −0.0945643
\(826\) 4.42607e6 0.225719
\(827\) 1.44496e7 0.734671 0.367336 0.930088i \(-0.380270\pi\)
0.367336 + 0.930088i \(0.380270\pi\)
\(828\) 940001. 0.0476489
\(829\) −2.10004e7 −1.06131 −0.530653 0.847589i \(-0.678053\pi\)
−0.530653 + 0.847589i \(0.678053\pi\)
\(830\) 4.20005e6 0.211621
\(831\) 2.13528e7 1.07264
\(832\) −2.44936e7 −1.22672
\(833\) 2.85151e7 1.42385
\(834\) 8.10257e6 0.403374
\(835\) 2.97133e6 0.147481
\(836\) −383327. −0.0189694
\(837\) 748765. 0.0369430
\(838\) 8.15340e6 0.401078
\(839\) 1.45792e6 0.0715040 0.0357520 0.999361i \(-0.488617\pi\)
0.0357520 + 0.999361i \(0.488617\pi\)
\(840\) −1.27820e7 −0.625031
\(841\) −1.40138e7 −0.683230
\(842\) 2.06838e6 0.100543
\(843\) −3.48175e6 −0.168744
\(844\) 6.63701e6 0.320713
\(845\) −2.23734e7 −1.07793
\(846\) −7.81876e6 −0.375588
\(847\) −2.89087e7 −1.38459
\(848\) −870224. −0.0415568
\(849\) 172577. 0.00821700
\(850\) 9.03036e6 0.428704
\(851\) −6.95607e6 −0.329261
\(852\) 3.18464e6 0.150301
\(853\) −1.47110e7 −0.692262 −0.346131 0.938186i \(-0.612505\pi\)
−0.346131 + 0.938186i \(0.612505\pi\)
\(854\) −4.02384e7 −1.88797
\(855\) 1.64887e6 0.0771386
\(856\) 9.98833e6 0.465917
\(857\) 2.25591e7 1.04923 0.524615 0.851340i \(-0.324209\pi\)
0.524615 + 0.851340i \(0.324209\pi\)
\(858\) 7.80111e6 0.361775
\(859\) 3.74722e7 1.73271 0.866356 0.499427i \(-0.166456\pi\)
0.866356 + 0.499427i \(0.166456\pi\)
\(860\) −2.94766e6 −0.135904
\(861\) 2.73252e7 1.25619
\(862\) 4.42308e7 2.02748
\(863\) 7.18311e6 0.328311 0.164156 0.986434i \(-0.447510\pi\)
0.164156 + 0.986434i \(0.447510\pi\)
\(864\) 1.37982e6 0.0628838
\(865\) 1.75853e7 0.799116
\(866\) −3.76260e7 −1.70488
\(867\) −2.40507e6 −0.108663
\(868\) 1.12920e6 0.0508710
\(869\) 6.82851e6 0.306744
\(870\) −5.85514e6 −0.262264
\(871\) 3.51783e7 1.57119
\(872\) 1.93914e7 0.863612
\(873\) 6.47381e6 0.287491
\(874\) 6.53602e6 0.289424
\(875\) 3.91958e7 1.73069
\(876\) 735465. 0.0323819
\(877\) 4.17457e7 1.83279 0.916394 0.400277i \(-0.131086\pi\)
0.916394 + 0.400277i \(0.131086\pi\)
\(878\) −2.84172e7 −1.24407
\(879\) −1.74572e7 −0.762085
\(880\) 7.26167e6 0.316104
\(881\) −8.72480e6 −0.378718 −0.189359 0.981908i \(-0.560641\pi\)
−0.189359 + 0.981908i \(0.560641\pi\)
\(882\) −1.31356e7 −0.568562
\(883\) 2.98453e6 0.128817 0.0644087 0.997924i \(-0.479484\pi\)
0.0644087 + 0.997924i \(0.479484\pi\)
\(884\) −5.39639e6 −0.232259
\(885\) 1.30961e6 0.0562060
\(886\) 7.66932e6 0.328226
\(887\) −2.69623e7 −1.15066 −0.575332 0.817920i \(-0.695127\pi\)
−0.575332 + 0.817920i \(0.695127\pi\)
\(888\) −4.64640e6 −0.197735
\(889\) −3.83323e7 −1.62671
\(890\) −7.26889e6 −0.307605
\(891\) 978277. 0.0412827
\(892\) 5.10777e6 0.214941
\(893\) −7.69886e6 −0.323071
\(894\) −6.91178e6 −0.289232
\(895\) 2.07117e6 0.0864288
\(896\) −4.53230e7 −1.88603
\(897\) −1.88367e7 −0.781668
\(898\) 2.72530e7 1.12778
\(899\) −2.61809e6 −0.108040
\(900\) −589092. −0.0242425
\(901\) 801910. 0.0329089
\(902\) −1.32729e7 −0.543187
\(903\) 2.50343e7 1.02168
\(904\) 4.05410e7 1.64996
\(905\) 247078. 0.0100280
\(906\) −9.99668e6 −0.404609
\(907\) −1.11838e7 −0.451409 −0.225704 0.974196i \(-0.572468\pi\)
−0.225704 + 0.974196i \(0.572468\pi\)
\(908\) −1.97529e6 −0.0795090
\(909\) 1.08623e7 0.436025
\(910\) −5.06054e7 −2.02579
\(911\) −1.36404e7 −0.544542 −0.272271 0.962221i \(-0.587775\pi\)
−0.272271 + 0.962221i \(0.587775\pi\)
\(912\) 5.10623e6 0.203289
\(913\) 2.45368e6 0.0974185
\(914\) −2.45335e7 −0.971391
\(915\) −1.19059e7 −0.470122
\(916\) 347437. 0.0136816
\(917\) −5.34977e7 −2.10093
\(918\) −4.77865e6 −0.187154
\(919\) 3.56348e7 1.39183 0.695914 0.718125i \(-0.254999\pi\)
0.695914 + 0.718125i \(0.254999\pi\)
\(920\) −1.49917e7 −0.583956
\(921\) −995762. −0.0386818
\(922\) −1.21138e7 −0.469302
\(923\) −6.38170e7 −2.46565
\(924\) 1.47532e6 0.0568468
\(925\) 4.35932e6 0.167519
\(926\) −1.41092e7 −0.540723
\(927\) −9.66904e6 −0.369559
\(928\) −4.82461e6 −0.183905
\(929\) −2.06763e7 −0.786021 −0.393011 0.919534i \(-0.628566\pi\)
−0.393011 + 0.919534i \(0.628566\pi\)
\(930\) 2.35933e6 0.0894502
\(931\) −1.29342e7 −0.489062
\(932\) −3.34201e6 −0.126028
\(933\) 1.92503e7 0.723990
\(934\) −1.62307e7 −0.608795
\(935\) −6.69161e6 −0.250324
\(936\) −1.25822e7 −0.469425
\(937\) 964568. 0.0358909 0.0179454 0.999839i \(-0.494287\pi\)
0.0179454 + 0.999839i \(0.494287\pi\)
\(938\) 4.69786e7 1.74338
\(939\) 2.65114e7 0.981227
\(940\) −3.48887e6 −0.128785
\(941\) 3.97475e7 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(942\) −1.38490e7 −0.508499
\(943\) 3.20489e7 1.17364
\(944\) 4.05559e6 0.148124
\(945\) −6.34604e6 −0.231166
\(946\) −1.21601e7 −0.441784
\(947\) 4.51166e6 0.163479 0.0817395 0.996654i \(-0.473952\pi\)
0.0817395 + 0.996654i \(0.473952\pi\)
\(948\) 2.17594e6 0.0786369
\(949\) −1.47380e7 −0.531217
\(950\) −4.09608e6 −0.147251
\(951\) 9.24942e6 0.331637
\(952\) 3.64760e7 1.30441
\(953\) −2.78914e7 −0.994807 −0.497403 0.867519i \(-0.665713\pi\)
−0.497403 + 0.867519i \(0.665713\pi\)
\(954\) −369402. −0.0131410
\(955\) −3.11498e7 −1.10522
\(956\) 3.84581e6 0.136095
\(957\) −3.42059e6 −0.120732
\(958\) 3.80199e7 1.33843
\(959\) −2.47682e7 −0.869658
\(960\) −9.67834e6 −0.338940
\(961\) −2.75742e7 −0.963151
\(962\) −1.83956e7 −0.640878
\(963\) 4.95902e6 0.172318
\(964\) 6.46379e6 0.224024
\(965\) 1.25025e6 0.0432193
\(966\) −2.51553e7 −0.867334
\(967\) 8.43317e6 0.290018 0.145009 0.989430i \(-0.453679\pi\)
0.145009 + 0.989430i \(0.453679\pi\)
\(968\) −2.26480e7 −0.776859
\(969\) −4.70537e6 −0.160985
\(970\) 2.03987e7 0.696103
\(971\) −7.74992e6 −0.263784 −0.131892 0.991264i \(-0.542105\pi\)
−0.131892 + 0.991264i \(0.542105\pi\)
\(972\) 311733. 0.0105832
\(973\) −3.07063e7 −1.03979
\(974\) −5.06609e7 −1.71110
\(975\) 1.18048e7 0.397692
\(976\) −3.68703e7 −1.23895
\(977\) −2.03500e6 −0.0682069 −0.0341035 0.999418i \(-0.510858\pi\)
−0.0341035 + 0.999418i \(0.510858\pi\)
\(978\) 1.56459e7 0.523062
\(979\) −4.24651e6 −0.141604
\(980\) −5.86133e6 −0.194953
\(981\) 9.62748e6 0.319404
\(982\) 4.09915e7 1.35649
\(983\) 4.87946e7 1.61060 0.805301 0.592866i \(-0.202004\pi\)
0.805301 + 0.592866i \(0.202004\pi\)
\(984\) 2.14075e7 0.704818
\(985\) 2.09110e7 0.686728
\(986\) 1.67088e7 0.547334
\(987\) 2.96307e7 0.968165
\(988\) 2.44775e6 0.0797763
\(989\) 2.93620e7 0.954540
\(990\) 3.08251e6 0.0999579
\(991\) 4.77463e7 1.54439 0.772193 0.635388i \(-0.219160\pi\)
0.772193 + 0.635388i \(0.219160\pi\)
\(992\) 1.94408e6 0.0627242
\(993\) −1.79536e7 −0.577802
\(994\) −8.52240e7 −2.73587
\(995\) 2.16396e7 0.692934
\(996\) 781879. 0.0249742
\(997\) −4.32326e7 −1.37744 −0.688721 0.725026i \(-0.741828\pi\)
−0.688721 + 0.725026i \(0.741828\pi\)
\(998\) 6.58482e7 2.09275
\(999\) −2.30685e6 −0.0731316
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.c.1.2 12
3.2 odd 2 531.6.a.c.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.2 12 1.1 even 1 trivial
531.6.a.c.1.11 12 3.2 odd 2