Properties

Label 825.6.a.s.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.18565\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.18565 q^{2} -9.00000 q^{3} +35.0048 q^{4} +73.6708 q^{6} +163.661 q^{7} -24.5962 q^{8} +81.0000 q^{9} -121.000 q^{11} -315.043 q^{12} -728.536 q^{13} -1339.68 q^{14} -918.818 q^{16} -1475.58 q^{17} -663.037 q^{18} -341.835 q^{19} -1472.95 q^{21} +990.463 q^{22} -2943.46 q^{23} +221.366 q^{24} +5963.54 q^{26} -729.000 q^{27} +5728.94 q^{28} -2823.82 q^{29} +5163.10 q^{31} +8308.19 q^{32} +1089.00 q^{33} +12078.5 q^{34} +2835.39 q^{36} -4254.93 q^{37} +2798.14 q^{38} +6556.83 q^{39} +5874.23 q^{41} +12057.1 q^{42} -1662.76 q^{43} -4235.58 q^{44} +24094.1 q^{46} +5981.04 q^{47} +8269.36 q^{48} +9978.09 q^{49} +13280.2 q^{51} -25502.3 q^{52} -27557.3 q^{53} +5967.34 q^{54} -4025.45 q^{56} +3076.51 q^{57} +23114.8 q^{58} +8248.03 q^{59} -2342.27 q^{61} -42263.3 q^{62} +13256.6 q^{63} -38605.8 q^{64} -8914.17 q^{66} -49456.4 q^{67} -51652.2 q^{68} +26491.1 q^{69} -24643.4 q^{71} -1992.29 q^{72} +23000.9 q^{73} +34829.4 q^{74} -11965.9 q^{76} -19803.0 q^{77} -53671.9 q^{78} +87214.6 q^{79} +6561.00 q^{81} -48084.4 q^{82} -90598.7 q^{83} -51560.4 q^{84} +13610.8 q^{86} +25414.4 q^{87} +2976.14 q^{88} +86239.2 q^{89} -119233. q^{91} -103035. q^{92} -46467.9 q^{93} -48958.7 q^{94} -74773.8 q^{96} -13088.6 q^{97} -81677.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.18565 −1.44703 −0.723516 0.690308i \(-0.757475\pi\)
−0.723516 + 0.690308i \(0.757475\pi\)
\(3\) −9.00000 −0.577350
\(4\) 35.0048 1.09390
\(5\) 0 0
\(6\) 73.6708 0.835444
\(7\) 163.661 1.26241 0.631206 0.775615i \(-0.282560\pi\)
0.631206 + 0.775615i \(0.282560\pi\)
\(8\) −24.5962 −0.135876
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −315.043 −0.631563
\(13\) −728.536 −1.19562 −0.597809 0.801638i \(-0.703962\pi\)
−0.597809 + 0.801638i \(0.703962\pi\)
\(14\) −1339.68 −1.82675
\(15\) 0 0
\(16\) −918.818 −0.897283
\(17\) −1475.58 −1.23834 −0.619169 0.785258i \(-0.712531\pi\)
−0.619169 + 0.785258i \(0.712531\pi\)
\(18\) −663.037 −0.482344
\(19\) −341.835 −0.217236 −0.108618 0.994084i \(-0.534643\pi\)
−0.108618 + 0.994084i \(0.534643\pi\)
\(20\) 0 0
\(21\) −1472.95 −0.728854
\(22\) 990.463 0.436296
\(23\) −2943.46 −1.16021 −0.580107 0.814541i \(-0.696989\pi\)
−0.580107 + 0.814541i \(0.696989\pi\)
\(24\) 221.366 0.0784480
\(25\) 0 0
\(26\) 5963.54 1.73010
\(27\) −729.000 −0.192450
\(28\) 5728.94 1.38095
\(29\) −2823.82 −0.623508 −0.311754 0.950163i \(-0.600916\pi\)
−0.311754 + 0.950163i \(0.600916\pi\)
\(30\) 0 0
\(31\) 5163.10 0.964953 0.482476 0.875909i \(-0.339737\pi\)
0.482476 + 0.875909i \(0.339737\pi\)
\(32\) 8308.19 1.43427
\(33\) 1089.00 0.174078
\(34\) 12078.5 1.79191
\(35\) 0 0
\(36\) 2835.39 0.364633
\(37\) −4254.93 −0.510962 −0.255481 0.966814i \(-0.582234\pi\)
−0.255481 + 0.966814i \(0.582234\pi\)
\(38\) 2798.14 0.314348
\(39\) 6556.83 0.690291
\(40\) 0 0
\(41\) 5874.23 0.545747 0.272873 0.962050i \(-0.412026\pi\)
0.272873 + 0.962050i \(0.412026\pi\)
\(42\) 12057.1 1.05468
\(43\) −1662.76 −0.137138 −0.0685691 0.997646i \(-0.521843\pi\)
−0.0685691 + 0.997646i \(0.521843\pi\)
\(44\) −4235.58 −0.329823
\(45\) 0 0
\(46\) 24094.1 1.67886
\(47\) 5981.04 0.394941 0.197471 0.980309i \(-0.436727\pi\)
0.197471 + 0.980309i \(0.436727\pi\)
\(48\) 8269.36 0.518047
\(49\) 9978.09 0.593686
\(50\) 0 0
\(51\) 13280.2 0.714955
\(52\) −25502.3 −1.30789
\(53\) −27557.3 −1.34755 −0.673777 0.738935i \(-0.735329\pi\)
−0.673777 + 0.738935i \(0.735329\pi\)
\(54\) 5967.34 0.278481
\(55\) 0 0
\(56\) −4025.45 −0.171532
\(57\) 3076.51 0.125421
\(58\) 23114.8 0.902236
\(59\) 8248.03 0.308475 0.154238 0.988034i \(-0.450708\pi\)
0.154238 + 0.988034i \(0.450708\pi\)
\(60\) 0 0
\(61\) −2342.27 −0.0805959 −0.0402980 0.999188i \(-0.512831\pi\)
−0.0402980 + 0.999188i \(0.512831\pi\)
\(62\) −42263.3 −1.39632
\(63\) 13256.6 0.420804
\(64\) −38605.8 −1.17815
\(65\) 0 0
\(66\) −8914.17 −0.251896
\(67\) −49456.4 −1.34597 −0.672985 0.739656i \(-0.734988\pi\)
−0.672985 + 0.739656i \(0.734988\pi\)
\(68\) −51652.2 −1.35462
\(69\) 26491.1 0.669849
\(70\) 0 0
\(71\) −24643.4 −0.580169 −0.290085 0.957001i \(-0.593683\pi\)
−0.290085 + 0.957001i \(0.593683\pi\)
\(72\) −1992.29 −0.0452920
\(73\) 23000.9 0.505170 0.252585 0.967575i \(-0.418719\pi\)
0.252585 + 0.967575i \(0.418719\pi\)
\(74\) 34829.4 0.739378
\(75\) 0 0
\(76\) −11965.9 −0.237635
\(77\) −19803.0 −0.380632
\(78\) −53671.9 −0.998873
\(79\) 87214.6 1.57225 0.786124 0.618068i \(-0.212085\pi\)
0.786124 + 0.618068i \(0.212085\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −48084.4 −0.789713
\(83\) −90598.7 −1.44353 −0.721766 0.692137i \(-0.756669\pi\)
−0.721766 + 0.692137i \(0.756669\pi\)
\(84\) −51560.4 −0.797294
\(85\) 0 0
\(86\) 13610.8 0.198443
\(87\) 25414.4 0.359983
\(88\) 2976.14 0.0409681
\(89\) 86239.2 1.15406 0.577032 0.816722i \(-0.304211\pi\)
0.577032 + 0.816722i \(0.304211\pi\)
\(90\) 0 0
\(91\) −119233. −1.50936
\(92\) −103035. −1.26916
\(93\) −46467.9 −0.557116
\(94\) −48958.7 −0.571492
\(95\) 0 0
\(96\) −74773.8 −0.828078
\(97\) −13088.6 −0.141242 −0.0706211 0.997503i \(-0.522498\pi\)
−0.0706211 + 0.997503i \(0.522498\pi\)
\(98\) −81677.1 −0.859083
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 34292.9 0.334503 0.167252 0.985914i \(-0.446511\pi\)
0.167252 + 0.985914i \(0.446511\pi\)
\(102\) −108707. −1.03456
\(103\) 32773.7 0.304391 0.152196 0.988350i \(-0.451366\pi\)
0.152196 + 0.988350i \(0.451366\pi\)
\(104\) 17919.2 0.162456
\(105\) 0 0
\(106\) 225574. 1.94995
\(107\) −203291. −1.71656 −0.858278 0.513186i \(-0.828465\pi\)
−0.858278 + 0.513186i \(0.828465\pi\)
\(108\) −25518.5 −0.210521
\(109\) 203531. 1.64084 0.820418 0.571764i \(-0.193741\pi\)
0.820418 + 0.571764i \(0.193741\pi\)
\(110\) 0 0
\(111\) 38294.4 0.295004
\(112\) −150375. −1.13274
\(113\) 140927. 1.03824 0.519120 0.854702i \(-0.326260\pi\)
0.519120 + 0.854702i \(0.326260\pi\)
\(114\) −25183.2 −0.181489
\(115\) 0 0
\(116\) −98847.3 −0.682055
\(117\) −59011.4 −0.398540
\(118\) −67515.4 −0.446373
\(119\) −241495. −1.56329
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 19173.0 0.116625
\(123\) −52868.1 −0.315087
\(124\) 180733. 1.05556
\(125\) 0 0
\(126\) −108514. −0.608917
\(127\) 2686.76 0.0147815 0.00739076 0.999973i \(-0.497647\pi\)
0.00739076 + 0.999973i \(0.497647\pi\)
\(128\) 50150.9 0.270554
\(129\) 14964.8 0.0791767
\(130\) 0 0
\(131\) −280946. −1.43036 −0.715179 0.698941i \(-0.753655\pi\)
−0.715179 + 0.698941i \(0.753655\pi\)
\(132\) 38120.2 0.190424
\(133\) −55945.2 −0.274242
\(134\) 404832. 1.94766
\(135\) 0 0
\(136\) 36293.5 0.168260
\(137\) 272454. 1.24020 0.620100 0.784523i \(-0.287092\pi\)
0.620100 + 0.784523i \(0.287092\pi\)
\(138\) −216847. −0.969293
\(139\) 324210. 1.42328 0.711638 0.702547i \(-0.247954\pi\)
0.711638 + 0.702547i \(0.247954\pi\)
\(140\) 0 0
\(141\) −53829.4 −0.228019
\(142\) 201722. 0.839523
\(143\) 88152.9 0.360493
\(144\) −74424.2 −0.299094
\(145\) 0 0
\(146\) −188277. −0.730997
\(147\) −89802.8 −0.342765
\(148\) −148943. −0.558941
\(149\) −325211. −1.20005 −0.600024 0.799982i \(-0.704843\pi\)
−0.600024 + 0.799982i \(0.704843\pi\)
\(150\) 0 0
\(151\) −399144. −1.42458 −0.712290 0.701885i \(-0.752342\pi\)
−0.712290 + 0.701885i \(0.752342\pi\)
\(152\) 8407.83 0.0295172
\(153\) −119522. −0.412779
\(154\) 162101. 0.550786
\(155\) 0 0
\(156\) 229520. 0.755109
\(157\) 82871.8 0.268323 0.134161 0.990959i \(-0.457166\pi\)
0.134161 + 0.990959i \(0.457166\pi\)
\(158\) −713907. −2.27509
\(159\) 248015. 0.778011
\(160\) 0 0
\(161\) −481730. −1.46467
\(162\) −53706.0 −0.160781
\(163\) 311672. 0.918818 0.459409 0.888225i \(-0.348061\pi\)
0.459409 + 0.888225i \(0.348061\pi\)
\(164\) 205626. 0.596992
\(165\) 0 0
\(166\) 741609. 2.08884
\(167\) 129998. 0.360700 0.180350 0.983602i \(-0.442277\pi\)
0.180350 + 0.983602i \(0.442277\pi\)
\(168\) 36229.0 0.0990338
\(169\) 159472. 0.429505
\(170\) 0 0
\(171\) −27688.6 −0.0724121
\(172\) −58204.5 −0.150015
\(173\) −744520. −1.89130 −0.945651 0.325183i \(-0.894574\pi\)
−0.945651 + 0.325183i \(0.894574\pi\)
\(174\) −208033. −0.520906
\(175\) 0 0
\(176\) 111177. 0.270541
\(177\) −74232.3 −0.178098
\(178\) −705924. −1.66997
\(179\) 309968. 0.723077 0.361539 0.932357i \(-0.382252\pi\)
0.361539 + 0.932357i \(0.382252\pi\)
\(180\) 0 0
\(181\) −656347. −1.48915 −0.744573 0.667541i \(-0.767347\pi\)
−0.744573 + 0.667541i \(0.767347\pi\)
\(182\) 976002. 2.18410
\(183\) 21080.5 0.0465321
\(184\) 72397.7 0.157645
\(185\) 0 0
\(186\) 380370. 0.806164
\(187\) 178545. 0.373373
\(188\) 209365. 0.432026
\(189\) −119309. −0.242951
\(190\) 0 0
\(191\) −336716. −0.667853 −0.333926 0.942599i \(-0.608374\pi\)
−0.333926 + 0.942599i \(0.608374\pi\)
\(192\) 347452. 0.680208
\(193\) 599122. 1.15777 0.578884 0.815410i \(-0.303488\pi\)
0.578884 + 0.815410i \(0.303488\pi\)
\(194\) 107139. 0.204382
\(195\) 0 0
\(196\) 349281. 0.649433
\(197\) 365021. 0.670120 0.335060 0.942197i \(-0.391243\pi\)
0.335060 + 0.942197i \(0.391243\pi\)
\(198\) 80227.5 0.145432
\(199\) 333867. 0.597641 0.298820 0.954309i \(-0.403407\pi\)
0.298820 + 0.954309i \(0.403407\pi\)
\(200\) 0 0
\(201\) 445108. 0.777096
\(202\) −280709. −0.484037
\(203\) −462151. −0.787125
\(204\) 464870. 0.782089
\(205\) 0 0
\(206\) −268274. −0.440463
\(207\) −238420. −0.386738
\(208\) 669392. 1.07281
\(209\) 41362.0 0.0654992
\(210\) 0 0
\(211\) −980083. −1.51550 −0.757751 0.652543i \(-0.773702\pi\)
−0.757751 + 0.652543i \(0.773702\pi\)
\(212\) −964636. −1.47409
\(213\) 221791. 0.334961
\(214\) 1.66406e6 2.48391
\(215\) 0 0
\(216\) 17930.6 0.0261493
\(217\) 845000. 1.21817
\(218\) −1.66604e6 −2.37434
\(219\) −207008. −0.291660
\(220\) 0 0
\(221\) 1.07501e6 1.48058
\(222\) −313464. −0.426880
\(223\) 80967.8 0.109031 0.0545155 0.998513i \(-0.482639\pi\)
0.0545155 + 0.998513i \(0.482639\pi\)
\(224\) 1.35973e6 1.81064
\(225\) 0 0
\(226\) −1.15358e6 −1.50236
\(227\) −524287. −0.675312 −0.337656 0.941270i \(-0.609634\pi\)
−0.337656 + 0.941270i \(0.609634\pi\)
\(228\) 107693. 0.137198
\(229\) 816517. 1.02891 0.514454 0.857518i \(-0.327995\pi\)
0.514454 + 0.857518i \(0.327995\pi\)
\(230\) 0 0
\(231\) 178227. 0.219758
\(232\) 69455.2 0.0847198
\(233\) 338979. 0.409056 0.204528 0.978861i \(-0.434434\pi\)
0.204528 + 0.978861i \(0.434434\pi\)
\(234\) 483047. 0.576699
\(235\) 0 0
\(236\) 288721. 0.337441
\(237\) −784931. −0.907738
\(238\) 1.97679e6 2.26214
\(239\) 98246.5 0.111256 0.0556278 0.998452i \(-0.482284\pi\)
0.0556278 + 0.998452i \(0.482284\pi\)
\(240\) 0 0
\(241\) 1.00307e6 1.11247 0.556233 0.831026i \(-0.312246\pi\)
0.556233 + 0.831026i \(0.312246\pi\)
\(242\) −119846. −0.131548
\(243\) −59049.0 −0.0641500
\(244\) −81990.8 −0.0881639
\(245\) 0 0
\(246\) 432759. 0.455941
\(247\) 249039. 0.259732
\(248\) −126992. −0.131114
\(249\) 815388. 0.833424
\(250\) 0 0
\(251\) 989640. 0.991501 0.495750 0.868465i \(-0.334893\pi\)
0.495750 + 0.868465i \(0.334893\pi\)
\(252\) 464044. 0.460318
\(253\) 356158. 0.349817
\(254\) −21992.8 −0.0213893
\(255\) 0 0
\(256\) 824867. 0.786655
\(257\) 1.15250e6 1.08845 0.544225 0.838939i \(-0.316824\pi\)
0.544225 + 0.838939i \(0.316824\pi\)
\(258\) −122497. −0.114571
\(259\) −696369. −0.645045
\(260\) 0 0
\(261\) −228729. −0.207836
\(262\) 2.29973e6 2.06977
\(263\) 1.03041e6 0.918589 0.459294 0.888284i \(-0.348102\pi\)
0.459294 + 0.888284i \(0.348102\pi\)
\(264\) −26785.2 −0.0236530
\(265\) 0 0
\(266\) 457948. 0.396836
\(267\) −776153. −0.666299
\(268\) −1.73121e6 −1.47236
\(269\) −2.24263e6 −1.88963 −0.944814 0.327607i \(-0.893758\pi\)
−0.944814 + 0.327607i \(0.893758\pi\)
\(270\) 0 0
\(271\) 2.17227e6 1.79676 0.898382 0.439216i \(-0.144744\pi\)
0.898382 + 0.439216i \(0.144744\pi\)
\(272\) 1.35579e6 1.11114
\(273\) 1.07310e6 0.871432
\(274\) −2.23021e6 −1.79461
\(275\) 0 0
\(276\) 927315. 0.732748
\(277\) −1.04100e6 −0.815178 −0.407589 0.913165i \(-0.633630\pi\)
−0.407589 + 0.913165i \(0.633630\pi\)
\(278\) −2.65387e6 −2.05952
\(279\) 418211. 0.321651
\(280\) 0 0
\(281\) −1.34424e6 −1.01557 −0.507786 0.861483i \(-0.669536\pi\)
−0.507786 + 0.861483i \(0.669536\pi\)
\(282\) 440628. 0.329951
\(283\) −828093. −0.614629 −0.307314 0.951608i \(-0.599430\pi\)
−0.307314 + 0.951608i \(0.599430\pi\)
\(284\) −862637. −0.634647
\(285\) 0 0
\(286\) −721588. −0.521644
\(287\) 961385. 0.688958
\(288\) 672964. 0.478091
\(289\) 757468. 0.533482
\(290\) 0 0
\(291\) 117798. 0.0815462
\(292\) 805142. 0.552606
\(293\) 1.31305e6 0.893538 0.446769 0.894649i \(-0.352575\pi\)
0.446769 + 0.894649i \(0.352575\pi\)
\(294\) 735094. 0.495992
\(295\) 0 0
\(296\) 104655. 0.0694274
\(297\) 88209.0 0.0580259
\(298\) 2.66206e6 1.73651
\(299\) 2.14441e6 1.38717
\(300\) 0 0
\(301\) −272130. −0.173125
\(302\) 3.26725e6 2.06141
\(303\) −308636. −0.193126
\(304\) 314084. 0.194922
\(305\) 0 0
\(306\) 978362. 0.597305
\(307\) −138234. −0.0837082 −0.0418541 0.999124i \(-0.513326\pi\)
−0.0418541 + 0.999124i \(0.513326\pi\)
\(308\) −693201. −0.416373
\(309\) −294963. −0.175740
\(310\) 0 0
\(311\) 1.17643e6 0.689710 0.344855 0.938656i \(-0.387928\pi\)
0.344855 + 0.938656i \(0.387928\pi\)
\(312\) −161273. −0.0937939
\(313\) −919942. −0.530762 −0.265381 0.964144i \(-0.585498\pi\)
−0.265381 + 0.964144i \(0.585498\pi\)
\(314\) −678359. −0.388271
\(315\) 0 0
\(316\) 3.05293e6 1.71988
\(317\) 1.96510e6 1.09834 0.549171 0.835710i \(-0.314944\pi\)
0.549171 + 0.835710i \(0.314944\pi\)
\(318\) −2.03016e6 −1.12581
\(319\) 341682. 0.187995
\(320\) 0 0
\(321\) 1.82962e6 0.991054
\(322\) 3.94327e6 2.11942
\(323\) 504403. 0.269012
\(324\) 229666. 0.121544
\(325\) 0 0
\(326\) −2.55124e6 −1.32956
\(327\) −1.83178e6 −0.947337
\(328\) −144484. −0.0741539
\(329\) 978866. 0.498579
\(330\) 0 0
\(331\) 1.98383e6 0.995255 0.497628 0.867391i \(-0.334205\pi\)
0.497628 + 0.867391i \(0.334205\pi\)
\(332\) −3.17139e6 −1.57908
\(333\) −344650. −0.170321
\(334\) −1.06412e6 −0.521945
\(335\) 0 0
\(336\) 1.35338e6 0.653989
\(337\) 3.28452e6 1.57543 0.787713 0.616043i \(-0.211265\pi\)
0.787713 + 0.616043i \(0.211265\pi\)
\(338\) −1.30538e6 −0.621507
\(339\) −1.26834e6 −0.599428
\(340\) 0 0
\(341\) −624735. −0.290944
\(342\) 226649. 0.104783
\(343\) −1.11763e6 −0.512936
\(344\) 40897.5 0.0186338
\(345\) 0 0
\(346\) 6.09437e6 2.73677
\(347\) −2.23106e6 −0.994691 −0.497345 0.867553i \(-0.665692\pi\)
−0.497345 + 0.867553i \(0.665692\pi\)
\(348\) 889625. 0.393785
\(349\) 1.14634e6 0.503790 0.251895 0.967755i \(-0.418946\pi\)
0.251895 + 0.967755i \(0.418946\pi\)
\(350\) 0 0
\(351\) 531103. 0.230097
\(352\) −1.00529e6 −0.432449
\(353\) −1.17280e6 −0.500941 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(354\) 607639. 0.257714
\(355\) 0 0
\(356\) 3.01879e6 1.26243
\(357\) 2.17346e6 0.902568
\(358\) −2.53729e6 −1.04632
\(359\) 718129. 0.294081 0.147040 0.989130i \(-0.453025\pi\)
0.147040 + 0.989130i \(0.453025\pi\)
\(360\) 0 0
\(361\) −2.35925e6 −0.952808
\(362\) 5.37263e6 2.15484
\(363\) −131769. −0.0524864
\(364\) −4.17374e6 −1.65109
\(365\) 0 0
\(366\) −172557. −0.0673334
\(367\) −933545. −0.361801 −0.180901 0.983501i \(-0.557901\pi\)
−0.180901 + 0.983501i \(0.557901\pi\)
\(368\) 2.70450e6 1.04104
\(369\) 475812. 0.181916
\(370\) 0 0
\(371\) −4.51006e6 −1.70117
\(372\) −1.62660e6 −0.609429
\(373\) 4.48359e6 1.66860 0.834302 0.551307i \(-0.185871\pi\)
0.834302 + 0.551307i \(0.185871\pi\)
\(374\) −1.46150e6 −0.540283
\(375\) 0 0
\(376\) −147111. −0.0536630
\(377\) 2.05726e6 0.745478
\(378\) 976623. 0.351558
\(379\) 2.61145e6 0.933863 0.466931 0.884294i \(-0.345360\pi\)
0.466931 + 0.884294i \(0.345360\pi\)
\(380\) 0 0
\(381\) −24180.8 −0.00853411
\(382\) 2.75624e6 0.966404
\(383\) −2.87502e6 −1.00148 −0.500741 0.865597i \(-0.666939\pi\)
−0.500741 + 0.865597i \(0.666939\pi\)
\(384\) −451358. −0.156204
\(385\) 0 0
\(386\) −4.90420e6 −1.67533
\(387\) −134683. −0.0457127
\(388\) −458164. −0.154505
\(389\) 231232. 0.0774771 0.0387386 0.999249i \(-0.487666\pi\)
0.0387386 + 0.999249i \(0.487666\pi\)
\(390\) 0 0
\(391\) 4.34329e6 1.43674
\(392\) −245423. −0.0806677
\(393\) 2.52852e6 0.825818
\(394\) −2.98794e6 −0.969685
\(395\) 0 0
\(396\) −343082. −0.109941
\(397\) 2.01862e6 0.642804 0.321402 0.946943i \(-0.395846\pi\)
0.321402 + 0.946943i \(0.395846\pi\)
\(398\) −2.73291e6 −0.864805
\(399\) 503507. 0.158334
\(400\) 0 0
\(401\) −4.53630e6 −1.40877 −0.704386 0.709817i \(-0.748777\pi\)
−0.704386 + 0.709817i \(0.748777\pi\)
\(402\) −3.64349e6 −1.12448
\(403\) −3.76150e6 −1.15372
\(404\) 1.20041e6 0.365913
\(405\) 0 0
\(406\) 3.78300e6 1.13899
\(407\) 514847. 0.154061
\(408\) −326642. −0.0971452
\(409\) 629053. 0.185943 0.0929714 0.995669i \(-0.470363\pi\)
0.0929714 + 0.995669i \(0.470363\pi\)
\(410\) 0 0
\(411\) −2.45209e6 −0.716030
\(412\) 1.14724e6 0.332973
\(413\) 1.34988e6 0.389423
\(414\) 1.95162e6 0.559622
\(415\) 0 0
\(416\) −6.05282e6 −1.71484
\(417\) −2.91789e6 −0.821729
\(418\) −338575. −0.0947794
\(419\) 1.00216e6 0.278870 0.139435 0.990231i \(-0.455471\pi\)
0.139435 + 0.990231i \(0.455471\pi\)
\(420\) 0 0
\(421\) −1.43349e6 −0.394175 −0.197087 0.980386i \(-0.563148\pi\)
−0.197087 + 0.980386i \(0.563148\pi\)
\(422\) 8.02261e6 2.19298
\(423\) 484464. 0.131647
\(424\) 677803. 0.183100
\(425\) 0 0
\(426\) −1.81550e6 −0.484699
\(427\) −383340. −0.101745
\(428\) −7.11614e6 −1.87774
\(429\) −793376. −0.208131
\(430\) 0 0
\(431\) −4.80577e6 −1.24615 −0.623074 0.782163i \(-0.714116\pi\)
−0.623074 + 0.782163i \(0.714116\pi\)
\(432\) 669818. 0.172682
\(433\) 1.53507e6 0.393467 0.196733 0.980457i \(-0.436967\pi\)
0.196733 + 0.980457i \(0.436967\pi\)
\(434\) −6.91687e6 −1.76273
\(435\) 0 0
\(436\) 7.12458e6 1.79491
\(437\) 1.00618e6 0.252040
\(438\) 1.69450e6 0.422042
\(439\) 7.14894e6 1.77044 0.885218 0.465176i \(-0.154009\pi\)
0.885218 + 0.465176i \(0.154009\pi\)
\(440\) 0 0
\(441\) 808225. 0.197895
\(442\) −8.79966e6 −2.14245
\(443\) −712048. −0.172385 −0.0861926 0.996278i \(-0.527470\pi\)
−0.0861926 + 0.996278i \(0.527470\pi\)
\(444\) 1.34049e6 0.322705
\(445\) 0 0
\(446\) −662773. −0.157771
\(447\) 2.92689e6 0.692849
\(448\) −6.31828e6 −1.48732
\(449\) −3.18417e6 −0.745385 −0.372693 0.927955i \(-0.621565\pi\)
−0.372693 + 0.927955i \(0.621565\pi\)
\(450\) 0 0
\(451\) −710782. −0.164549
\(452\) 4.93311e6 1.13573
\(453\) 3.59229e6 0.822481
\(454\) 4.29163e6 0.977197
\(455\) 0 0
\(456\) −75670.4 −0.0170418
\(457\) −8.26972e6 −1.85225 −0.926127 0.377213i \(-0.876883\pi\)
−0.926127 + 0.377213i \(0.876883\pi\)
\(458\) −6.68372e6 −1.48886
\(459\) 1.07570e6 0.238318
\(460\) 0 0
\(461\) 7.66057e6 1.67884 0.839419 0.543485i \(-0.182896\pi\)
0.839419 + 0.543485i \(0.182896\pi\)
\(462\) −1.45891e6 −0.317997
\(463\) −203047. −0.0440194 −0.0220097 0.999758i \(-0.507006\pi\)
−0.0220097 + 0.999758i \(0.507006\pi\)
\(464\) 2.59458e6 0.559463
\(465\) 0 0
\(466\) −2.77476e6 −0.591917
\(467\) −3.28726e6 −0.697497 −0.348748 0.937216i \(-0.613393\pi\)
−0.348748 + 0.937216i \(0.613393\pi\)
\(468\) −2.06568e6 −0.435962
\(469\) −8.09411e6 −1.69917
\(470\) 0 0
\(471\) −745846. −0.154916
\(472\) −202870. −0.0419144
\(473\) 201194. 0.0413487
\(474\) 6.42517e6 1.31353
\(475\) 0 0
\(476\) −8.45348e6 −1.71009
\(477\) −2.23214e6 −0.449185
\(478\) −804211. −0.160990
\(479\) −3.08986e6 −0.615319 −0.307659 0.951497i \(-0.599546\pi\)
−0.307659 + 0.951497i \(0.599546\pi\)
\(480\) 0 0
\(481\) 3.09987e6 0.610916
\(482\) −8.21074e6 −1.60977
\(483\) 4.33557e6 0.845627
\(484\) 512505. 0.0994454
\(485\) 0 0
\(486\) 483354. 0.0928271
\(487\) −2.37051e6 −0.452918 −0.226459 0.974021i \(-0.572715\pi\)
−0.226459 + 0.974021i \(0.572715\pi\)
\(488\) 57611.0 0.0109510
\(489\) −2.80505e6 −0.530480
\(490\) 0 0
\(491\) −7.56399e6 −1.41595 −0.707974 0.706239i \(-0.750391\pi\)
−0.707974 + 0.706239i \(0.750391\pi\)
\(492\) −1.85064e6 −0.344674
\(493\) 4.16676e6 0.772114
\(494\) −2.03855e6 −0.375840
\(495\) 0 0
\(496\) −4.74395e6 −0.865836
\(497\) −4.03318e6 −0.732413
\(498\) −6.67448e6 −1.20599
\(499\) −1.56759e6 −0.281826 −0.140913 0.990022i \(-0.545004\pi\)
−0.140913 + 0.990022i \(0.545004\pi\)
\(500\) 0 0
\(501\) −1.16999e6 −0.208251
\(502\) −8.10084e6 −1.43473
\(503\) 2.34418e6 0.413115 0.206557 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(504\) −326061. −0.0571772
\(505\) 0 0
\(506\) −2.91538e6 −0.506197
\(507\) −1.43525e6 −0.247975
\(508\) 94049.3 0.0161695
\(509\) 8.79759e6 1.50511 0.752557 0.658527i \(-0.228820\pi\)
0.752557 + 0.658527i \(0.228820\pi\)
\(510\) 0 0
\(511\) 3.76436e6 0.637734
\(512\) −8.35690e6 −1.40887
\(513\) 249198. 0.0418071
\(514\) −9.43397e6 −1.57502
\(515\) 0 0
\(516\) 523841. 0.0866114
\(517\) −723706. −0.119079
\(518\) 5.70023e6 0.933400
\(519\) 6.70068e6 1.09194
\(520\) 0 0
\(521\) −3.43077e6 −0.553729 −0.276865 0.960909i \(-0.589295\pi\)
−0.276865 + 0.960909i \(0.589295\pi\)
\(522\) 1.87230e6 0.300745
\(523\) 1.24453e7 1.98953 0.994763 0.102212i \(-0.0325920\pi\)
0.994763 + 0.102212i \(0.0325920\pi\)
\(524\) −9.83446e6 −1.56467
\(525\) 0 0
\(526\) −8.43458e6 −1.32923
\(527\) −7.61854e6 −1.19494
\(528\) −1.00059e6 −0.156197
\(529\) 2.22758e6 0.346095
\(530\) 0 0
\(531\) 668090. 0.102825
\(532\) −1.95835e6 −0.299993
\(533\) −4.27959e6 −0.652505
\(534\) 6.35331e6 0.964155
\(535\) 0 0
\(536\) 1.21644e6 0.182885
\(537\) −2.78972e6 −0.417469
\(538\) 1.83574e7 2.73435
\(539\) −1.20735e6 −0.179003
\(540\) 0 0
\(541\) 4.87769e6 0.716508 0.358254 0.933624i \(-0.383372\pi\)
0.358254 + 0.933624i \(0.383372\pi\)
\(542\) −1.77814e7 −2.59997
\(543\) 5.90713e6 0.859759
\(544\) −1.22594e7 −1.77611
\(545\) 0 0
\(546\) −8.78402e6 −1.26099
\(547\) 3.49347e6 0.499217 0.249608 0.968347i \(-0.419698\pi\)
0.249608 + 0.968347i \(0.419698\pi\)
\(548\) 9.53719e6 1.35665
\(549\) −189724. −0.0268653
\(550\) 0 0
\(551\) 965280. 0.135449
\(552\) −651580. −0.0910164
\(553\) 1.42737e7 1.98483
\(554\) 8.52128e6 1.17959
\(555\) 0 0
\(556\) 1.13489e7 1.55692
\(557\) 1.34838e7 1.84151 0.920757 0.390137i \(-0.127572\pi\)
0.920757 + 0.390137i \(0.127572\pi\)
\(558\) −3.42333e6 −0.465439
\(559\) 1.21138e6 0.163965
\(560\) 0 0
\(561\) −1.60690e6 −0.215567
\(562\) 1.10035e7 1.46956
\(563\) 2.07215e6 0.275519 0.137759 0.990466i \(-0.456010\pi\)
0.137759 + 0.990466i \(0.456010\pi\)
\(564\) −1.88429e6 −0.249430
\(565\) 0 0
\(566\) 6.77847e6 0.889387
\(567\) 1.07378e6 0.140268
\(568\) 606133. 0.0788310
\(569\) 1.11686e7 1.44617 0.723085 0.690759i \(-0.242723\pi\)
0.723085 + 0.690759i \(0.242723\pi\)
\(570\) 0 0
\(571\) 1.04065e7 1.33572 0.667861 0.744286i \(-0.267210\pi\)
0.667861 + 0.744286i \(0.267210\pi\)
\(572\) 3.08577e6 0.394343
\(573\) 3.03045e6 0.385585
\(574\) −7.86956e6 −0.996944
\(575\) 0 0
\(576\) −3.12707e6 −0.392718
\(577\) 4.15724e6 0.519835 0.259918 0.965631i \(-0.416305\pi\)
0.259918 + 0.965631i \(0.416305\pi\)
\(578\) −6.20037e6 −0.771965
\(579\) −5.39209e6 −0.668438
\(580\) 0 0
\(581\) −1.48275e7 −1.82233
\(582\) −964249. −0.118000
\(583\) 3.33443e6 0.406303
\(584\) −565734. −0.0686405
\(585\) 0 0
\(586\) −1.07482e7 −1.29298
\(587\) −8.38623e6 −1.00455 −0.502275 0.864708i \(-0.667504\pi\)
−0.502275 + 0.864708i \(0.667504\pi\)
\(588\) −3.14353e6 −0.374951
\(589\) −1.76493e6 −0.209623
\(590\) 0 0
\(591\) −3.28519e6 −0.386894
\(592\) 3.90951e6 0.458477
\(593\) −2.10957e6 −0.246353 −0.123177 0.992385i \(-0.539308\pi\)
−0.123177 + 0.992385i \(0.539308\pi\)
\(594\) −722048. −0.0839653
\(595\) 0 0
\(596\) −1.13839e7 −1.31273
\(597\) −3.00480e6 −0.345048
\(598\) −1.75534e7 −2.00728
\(599\) 3.77702e6 0.430113 0.215057 0.976602i \(-0.431006\pi\)
0.215057 + 0.976602i \(0.431006\pi\)
\(600\) 0 0
\(601\) 7.88038e6 0.889940 0.444970 0.895545i \(-0.353214\pi\)
0.444970 + 0.895545i \(0.353214\pi\)
\(602\) 2.22756e6 0.250517
\(603\) −4.00597e6 −0.448657
\(604\) −1.39719e7 −1.55835
\(605\) 0 0
\(606\) 2.52638e6 0.279459
\(607\) 5.43393e6 0.598608 0.299304 0.954158i \(-0.403246\pi\)
0.299304 + 0.954158i \(0.403246\pi\)
\(608\) −2.84003e6 −0.311576
\(609\) 4.15936e6 0.454447
\(610\) 0 0
\(611\) −4.35741e6 −0.472199
\(612\) −4.18383e6 −0.451539
\(613\) −1.36825e7 −1.47067 −0.735333 0.677706i \(-0.762974\pi\)
−0.735333 + 0.677706i \(0.762974\pi\)
\(614\) 1.13153e6 0.121128
\(615\) 0 0
\(616\) 487079. 0.0517187
\(617\) −1.24288e7 −1.31437 −0.657184 0.753730i \(-0.728253\pi\)
−0.657184 + 0.753730i \(0.728253\pi\)
\(618\) 2.41446e6 0.254302
\(619\) 1.29365e7 1.35704 0.678518 0.734583i \(-0.262622\pi\)
0.678518 + 0.734583i \(0.262622\pi\)
\(620\) 0 0
\(621\) 2.14578e6 0.223283
\(622\) −9.62988e6 −0.998033
\(623\) 1.41140e7 1.45690
\(624\) −6.02453e6 −0.619386
\(625\) 0 0
\(626\) 7.53032e6 0.768029
\(627\) −372258. −0.0378160
\(628\) 2.90091e6 0.293518
\(629\) 6.27848e6 0.632744
\(630\) 0 0
\(631\) 572856. 0.0572759 0.0286379 0.999590i \(-0.490883\pi\)
0.0286379 + 0.999590i \(0.490883\pi\)
\(632\) −2.14514e6 −0.213631
\(633\) 8.82075e6 0.874976
\(634\) −1.60856e7 −1.58933
\(635\) 0 0
\(636\) 8.68172e6 0.851066
\(637\) −7.26940e6 −0.709823
\(638\) −2.79689e6 −0.272034
\(639\) −1.99612e6 −0.193390
\(640\) 0 0
\(641\) 2.13431e6 0.205170 0.102585 0.994724i \(-0.467289\pi\)
0.102585 + 0.994724i \(0.467289\pi\)
\(642\) −1.49766e7 −1.43409
\(643\) −4.15866e6 −0.396667 −0.198333 0.980135i \(-0.563553\pi\)
−0.198333 + 0.980135i \(0.563553\pi\)
\(644\) −1.68629e7 −1.60220
\(645\) 0 0
\(646\) −4.12887e6 −0.389269
\(647\) 3.44527e6 0.323565 0.161783 0.986826i \(-0.448276\pi\)
0.161783 + 0.986826i \(0.448276\pi\)
\(648\) −161375. −0.0150973
\(649\) −998012. −0.0930088
\(650\) 0 0
\(651\) −7.60500e6 −0.703310
\(652\) 1.09100e7 1.00509
\(653\) 2.88090e6 0.264390 0.132195 0.991224i \(-0.457798\pi\)
0.132195 + 0.991224i \(0.457798\pi\)
\(654\) 1.49943e7 1.37083
\(655\) 0 0
\(656\) −5.39735e6 −0.489689
\(657\) 1.86307e6 0.168390
\(658\) −8.01265e6 −0.721459
\(659\) 1.31524e7 1.17975 0.589876 0.807494i \(-0.299177\pi\)
0.589876 + 0.807494i \(0.299177\pi\)
\(660\) 0 0
\(661\) −4.14045e6 −0.368591 −0.184295 0.982871i \(-0.559000\pi\)
−0.184295 + 0.982871i \(0.559000\pi\)
\(662\) −1.62389e7 −1.44017
\(663\) −9.67510e6 −0.854814
\(664\) 2.22838e6 0.196141
\(665\) 0 0
\(666\) 2.82118e6 0.246459
\(667\) 8.31179e6 0.723402
\(668\) 4.55057e6 0.394570
\(669\) −728710. −0.0629491
\(670\) 0 0
\(671\) 283415. 0.0243006
\(672\) −1.22376e7 −1.04538
\(673\) 8.46081e6 0.720070 0.360035 0.932939i \(-0.382765\pi\)
0.360035 + 0.932939i \(0.382765\pi\)
\(674\) −2.68860e7 −2.27969
\(675\) 0 0
\(676\) 5.58229e6 0.469835
\(677\) 7.85095e6 0.658340 0.329170 0.944271i \(-0.393231\pi\)
0.329170 + 0.944271i \(0.393231\pi\)
\(678\) 1.03822e7 0.867391
\(679\) −2.14210e6 −0.178306
\(680\) 0 0
\(681\) 4.71858e6 0.389891
\(682\) 5.11386e6 0.421005
\(683\) 1.84419e7 1.51271 0.756354 0.654163i \(-0.226979\pi\)
0.756354 + 0.654163i \(0.226979\pi\)
\(684\) −969234. −0.0792116
\(685\) 0 0
\(686\) 9.14853e6 0.742234
\(687\) −7.34866e6 −0.594040
\(688\) 1.52777e6 0.123052
\(689\) 2.00765e7 1.61116
\(690\) 0 0
\(691\) −1.74029e7 −1.38652 −0.693259 0.720688i \(-0.743826\pi\)
−0.693259 + 0.720688i \(0.743826\pi\)
\(692\) −2.60618e7 −2.06890
\(693\) −1.60405e6 −0.126877
\(694\) 1.82627e7 1.43935
\(695\) 0 0
\(696\) −625097. −0.0489130
\(697\) −8.66787e6 −0.675819
\(698\) −9.38353e6 −0.729000
\(699\) −3.05081e6 −0.236169
\(700\) 0 0
\(701\) −4.85133e6 −0.372877 −0.186438 0.982467i \(-0.559695\pi\)
−0.186438 + 0.982467i \(0.559695\pi\)
\(702\) −4.34742e6 −0.332958
\(703\) 1.45448e6 0.110999
\(704\) 4.67130e6 0.355227
\(705\) 0 0
\(706\) 9.60010e6 0.724877
\(707\) 5.61242e6 0.422281
\(708\) −2.59848e6 −0.194822
\(709\) −7.10939e6 −0.531150 −0.265575 0.964090i \(-0.585562\pi\)
−0.265575 + 0.964090i \(0.585562\pi\)
\(710\) 0 0
\(711\) 7.06438e6 0.524083
\(712\) −2.12115e6 −0.156809
\(713\) −1.51973e7 −1.11955
\(714\) −1.77911e7 −1.30604
\(715\) 0 0
\(716\) 1.08504e7 0.790974
\(717\) −884218. −0.0642335
\(718\) −5.87835e6 −0.425544
\(719\) 1.99693e6 0.144059 0.0720297 0.997402i \(-0.477052\pi\)
0.0720297 + 0.997402i \(0.477052\pi\)
\(720\) 0 0
\(721\) 5.36379e6 0.384267
\(722\) 1.93120e7 1.37874
\(723\) −9.02760e6 −0.642283
\(724\) −2.29753e7 −1.62898
\(725\) 0 0
\(726\) 1.07861e6 0.0759495
\(727\) −1.47012e7 −1.03161 −0.515806 0.856705i \(-0.672507\pi\)
−0.515806 + 0.856705i \(0.672507\pi\)
\(728\) 2.93268e6 0.205086
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.45353e6 0.169823
\(732\) 737917. 0.0509014
\(733\) 1.03899e7 0.714249 0.357125 0.934057i \(-0.383757\pi\)
0.357125 + 0.934057i \(0.383757\pi\)
\(734\) 7.64166e6 0.523538
\(735\) 0 0
\(736\) −2.44548e7 −1.66406
\(737\) 5.98422e6 0.405825
\(738\) −3.89483e6 −0.263238
\(739\) 1.40217e7 0.944470 0.472235 0.881473i \(-0.343447\pi\)
0.472235 + 0.881473i \(0.343447\pi\)
\(740\) 0 0
\(741\) −2.24135e6 −0.149956
\(742\) 3.69178e7 2.46165
\(743\) 1.62282e7 1.07845 0.539224 0.842162i \(-0.318717\pi\)
0.539224 + 0.842162i \(0.318717\pi\)
\(744\) 1.14293e6 0.0756986
\(745\) 0 0
\(746\) −3.67010e7 −2.41452
\(747\) −7.33849e6 −0.481178
\(748\) 6.24992e6 0.408433
\(749\) −3.32708e7 −2.16700
\(750\) 0 0
\(751\) −2.21572e7 −1.43356 −0.716779 0.697300i \(-0.754385\pi\)
−0.716779 + 0.697300i \(0.754385\pi\)
\(752\) −5.49549e6 −0.354374
\(753\) −8.90676e6 −0.572443
\(754\) −1.68400e7 −1.07873
\(755\) 0 0
\(756\) −4.17640e6 −0.265765
\(757\) 2.95113e7 1.87176 0.935878 0.352325i \(-0.114609\pi\)
0.935878 + 0.352325i \(0.114609\pi\)
\(758\) −2.13764e7 −1.35133
\(759\) −3.20542e6 −0.201967
\(760\) 0 0
\(761\) 1.82482e7 1.14224 0.571122 0.820865i \(-0.306508\pi\)
0.571122 + 0.820865i \(0.306508\pi\)
\(762\) 197936. 0.0123491
\(763\) 3.33103e7 2.07141
\(764\) −1.17867e7 −0.730564
\(765\) 0 0
\(766\) 2.35339e7 1.44918
\(767\) −6.00899e6 −0.368819
\(768\) −7.42380e6 −0.454175
\(769\) −1.89729e7 −1.15696 −0.578480 0.815697i \(-0.696354\pi\)
−0.578480 + 0.815697i \(0.696354\pi\)
\(770\) 0 0
\(771\) −1.03725e7 −0.628417
\(772\) 2.09721e7 1.26648
\(773\) 9.39706e6 0.565644 0.282822 0.959172i \(-0.408729\pi\)
0.282822 + 0.959172i \(0.408729\pi\)
\(774\) 1.10247e6 0.0661477
\(775\) 0 0
\(776\) 321930. 0.0191914
\(777\) 6.26732e6 0.372417
\(778\) −1.89278e6 −0.112112
\(779\) −2.00802e6 −0.118556
\(780\) 0 0
\(781\) 2.98185e6 0.174928
\(782\) −3.55527e7 −2.07900
\(783\) 2.05857e6 0.119994
\(784\) −9.16804e6 −0.532705
\(785\) 0 0
\(786\) −2.06975e7 −1.19498
\(787\) 2.29536e7 1.32103 0.660517 0.750811i \(-0.270337\pi\)
0.660517 + 0.750811i \(0.270337\pi\)
\(788\) 1.27775e7 0.733044
\(789\) −9.27370e6 −0.530347
\(790\) 0 0
\(791\) 2.30643e7 1.31069
\(792\) 241067. 0.0136560
\(793\) 1.70643e6 0.0963620
\(794\) −1.65237e7 −0.930157
\(795\) 0 0
\(796\) 1.16869e7 0.653759
\(797\) 1.05979e7 0.590984 0.295492 0.955345i \(-0.404516\pi\)
0.295492 + 0.955345i \(0.404516\pi\)
\(798\) −4.12153e6 −0.229114
\(799\) −8.82549e6 −0.489071
\(800\) 0 0
\(801\) 6.98538e6 0.384688
\(802\) 3.71325e7 2.03854
\(803\) −2.78311e6 −0.152315
\(804\) 1.55809e7 0.850065
\(805\) 0 0
\(806\) 3.07903e7 1.66946
\(807\) 2.01836e7 1.09098
\(808\) −843473. −0.0454509
\(809\) −1.95635e7 −1.05093 −0.525467 0.850814i \(-0.676109\pi\)
−0.525467 + 0.850814i \(0.676109\pi\)
\(810\) 0 0
\(811\) −3.23306e7 −1.72609 −0.863043 0.505131i \(-0.831444\pi\)
−0.863043 + 0.505131i \(0.831444\pi\)
\(812\) −1.61775e7 −0.861036
\(813\) −1.95504e7 −1.03736
\(814\) −4.21436e6 −0.222931
\(815\) 0 0
\(816\) −1.22021e7 −0.641517
\(817\) 568389. 0.0297914
\(818\) −5.14921e6 −0.269065
\(819\) −9.65790e6 −0.503122
\(820\) 0 0
\(821\) 3.15924e7 1.63578 0.817890 0.575375i \(-0.195144\pi\)
0.817890 + 0.575375i \(0.195144\pi\)
\(822\) 2.00719e7 1.03612
\(823\) −1.52405e7 −0.784333 −0.392167 0.919894i \(-0.628274\pi\)
−0.392167 + 0.919894i \(0.628274\pi\)
\(824\) −806107. −0.0413594
\(825\) 0 0
\(826\) −1.10497e7 −0.563507
\(827\) −2.87105e7 −1.45975 −0.729873 0.683583i \(-0.760421\pi\)
−0.729873 + 0.683583i \(0.760421\pi\)
\(828\) −8.34584e6 −0.423052
\(829\) −2.45816e7 −1.24229 −0.621146 0.783695i \(-0.713333\pi\)
−0.621146 + 0.783695i \(0.713333\pi\)
\(830\) 0 0
\(831\) 9.36903e6 0.470643
\(832\) 2.81257e7 1.40862
\(833\) −1.47234e7 −0.735185
\(834\) 2.38848e7 1.18907
\(835\) 0 0
\(836\) 1.44787e6 0.0716495
\(837\) −3.76390e6 −0.185705
\(838\) −8.20334e6 −0.403534
\(839\) −1.59309e6 −0.0781333 −0.0390667 0.999237i \(-0.512438\pi\)
−0.0390667 + 0.999237i \(0.512438\pi\)
\(840\) 0 0
\(841\) −1.25372e7 −0.611238
\(842\) 1.17340e7 0.570383
\(843\) 1.20981e7 0.586341
\(844\) −3.43076e7 −1.65781
\(845\) 0 0
\(846\) −3.96565e6 −0.190497
\(847\) 2.39617e6 0.114765
\(848\) 2.53201e7 1.20914
\(849\) 7.45283e6 0.354856
\(850\) 0 0
\(851\) 1.25242e7 0.592825
\(852\) 7.76373e6 0.366414
\(853\) −9.71355e6 −0.457094 −0.228547 0.973533i \(-0.573397\pi\)
−0.228547 + 0.973533i \(0.573397\pi\)
\(854\) 3.13789e6 0.147229
\(855\) 0 0
\(856\) 5.00017e6 0.233239
\(857\) 2.81098e7 1.30739 0.653696 0.756758i \(-0.273218\pi\)
0.653696 + 0.756758i \(0.273218\pi\)
\(858\) 6.49429e6 0.301171
\(859\) 1.96952e7 0.910703 0.455352 0.890312i \(-0.349514\pi\)
0.455352 + 0.890312i \(0.349514\pi\)
\(860\) 0 0
\(861\) −8.65247e6 −0.397770
\(862\) 3.93383e7 1.80322
\(863\) 9.07936e6 0.414981 0.207490 0.978237i \(-0.433470\pi\)
0.207490 + 0.978237i \(0.433470\pi\)
\(864\) −6.05667e6 −0.276026
\(865\) 0 0
\(866\) −1.25655e7 −0.569359
\(867\) −6.81722e6 −0.308006
\(868\) 2.95791e7 1.33255
\(869\) −1.05530e7 −0.474051
\(870\) 0 0
\(871\) 3.60308e7 1.60927
\(872\) −5.00609e6 −0.222950
\(873\) −1.06018e6 −0.0470807
\(874\) −8.23619e6 −0.364710
\(875\) 0 0
\(876\) −7.24628e6 −0.319047
\(877\) −1.35513e7 −0.594954 −0.297477 0.954729i \(-0.596145\pi\)
−0.297477 + 0.954729i \(0.596145\pi\)
\(878\) −5.85187e7 −2.56188
\(879\) −1.18175e7 −0.515884
\(880\) 0 0
\(881\) 3.41684e7 1.48315 0.741574 0.670871i \(-0.234080\pi\)
0.741574 + 0.670871i \(0.234080\pi\)
\(882\) −6.61584e6 −0.286361
\(883\) 4.27723e7 1.84612 0.923061 0.384654i \(-0.125679\pi\)
0.923061 + 0.384654i \(0.125679\pi\)
\(884\) 3.76305e7 1.61961
\(885\) 0 0
\(886\) 5.82857e6 0.249447
\(887\) 1.84459e7 0.787210 0.393605 0.919280i \(-0.371228\pi\)
0.393605 + 0.919280i \(0.371228\pi\)
\(888\) −941896. −0.0400839
\(889\) 439719. 0.0186604
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 2.83426e6 0.119269
\(893\) −2.04453e6 −0.0857955
\(894\) −2.39585e7 −1.00257
\(895\) 0 0
\(896\) 8.20777e6 0.341551
\(897\) −1.92997e7 −0.800885
\(898\) 2.60645e7 1.07860
\(899\) −1.45797e7 −0.601656
\(900\) 0 0
\(901\) 4.06628e7 1.66873
\(902\) 5.81821e6 0.238107
\(903\) 2.44917e6 0.0999537
\(904\) −3.46626e6 −0.141072
\(905\) 0 0
\(906\) −2.94052e7 −1.19016
\(907\) 4.12614e7 1.66543 0.832713 0.553704i \(-0.186786\pi\)
0.832713 + 0.553704i \(0.186786\pi\)
\(908\) −1.83525e7 −0.738723
\(909\) 2.77772e6 0.111501
\(910\) 0 0
\(911\) 2.05996e7 0.822360 0.411180 0.911554i \(-0.365117\pi\)
0.411180 + 0.911554i \(0.365117\pi\)
\(912\) −2.82675e6 −0.112538
\(913\) 1.09624e7 0.435241
\(914\) 6.76930e7 2.68027
\(915\) 0 0
\(916\) 2.85820e7 1.12552
\(917\) −4.59801e7 −1.80570
\(918\) −8.80526e6 −0.344854
\(919\) 5.10704e6 0.199471 0.0997356 0.995014i \(-0.468200\pi\)
0.0997356 + 0.995014i \(0.468200\pi\)
\(920\) 0 0
\(921\) 1.24410e6 0.0483290
\(922\) −6.27067e7 −2.42933
\(923\) 1.79536e7 0.693661
\(924\) 6.23881e6 0.240393
\(925\) 0 0
\(926\) 1.66207e6 0.0636975
\(927\) 2.65467e6 0.101464
\(928\) −2.34609e7 −0.894281
\(929\) −213931. −0.00813271 −0.00406636 0.999992i \(-0.501294\pi\)
−0.00406636 + 0.999992i \(0.501294\pi\)
\(930\) 0 0
\(931\) −3.41086e6 −0.128970
\(932\) 1.18659e7 0.447466
\(933\) −1.05879e7 −0.398205
\(934\) 2.69084e7 1.00930
\(935\) 0 0
\(936\) 1.45146e6 0.0541520
\(937\) 1.71430e7 0.637878 0.318939 0.947775i \(-0.396674\pi\)
0.318939 + 0.947775i \(0.396674\pi\)
\(938\) 6.62555e7 2.45875
\(939\) 8.27948e6 0.306436
\(940\) 0 0
\(941\) 3.27424e7 1.20542 0.602708 0.797962i \(-0.294088\pi\)
0.602708 + 0.797962i \(0.294088\pi\)
\(942\) 6.10523e6 0.224169
\(943\) −1.72905e7 −0.633183
\(944\) −7.57844e6 −0.276789
\(945\) 0 0
\(946\) −1.64690e6 −0.0598329
\(947\) −2.30829e7 −0.836403 −0.418201 0.908354i \(-0.637339\pi\)
−0.418201 + 0.908354i \(0.637339\pi\)
\(948\) −2.74763e7 −0.992975
\(949\) −1.67570e7 −0.603991
\(950\) 0 0
\(951\) −1.76859e7 −0.634128
\(952\) 5.93985e6 0.212414
\(953\) 2.57216e7 0.917414 0.458707 0.888587i \(-0.348313\pi\)
0.458707 + 0.888587i \(0.348313\pi\)
\(954\) 1.82715e7 0.649984
\(955\) 0 0
\(956\) 3.43910e6 0.121703
\(957\) −3.07514e6 −0.108539
\(958\) 2.52925e7 0.890385
\(959\) 4.45902e7 1.56564
\(960\) 0 0
\(961\) −1.97158e6 −0.0688663
\(962\) −2.53745e7 −0.884014
\(963\) −1.64665e7 −0.572185
\(964\) 3.51121e7 1.21693
\(965\) 0 0
\(966\) −3.54895e7 −1.22365
\(967\) 1.86464e6 0.0641253 0.0320627 0.999486i \(-0.489792\pi\)
0.0320627 + 0.999486i \(0.489792\pi\)
\(968\) −360113. −0.0123524
\(969\) −4.53963e6 −0.155314
\(970\) 0 0
\(971\) 2.90344e7 0.988245 0.494122 0.869392i \(-0.335489\pi\)
0.494122 + 0.869392i \(0.335489\pi\)
\(972\) −2.06700e6 −0.0701737
\(973\) 5.30606e7 1.79676
\(974\) 1.94042e7 0.655387
\(975\) 0 0
\(976\) 2.15212e6 0.0723174
\(977\) −5.86267e6 −0.196498 −0.0982492 0.995162i \(-0.531324\pi\)
−0.0982492 + 0.995162i \(0.531324\pi\)
\(978\) 2.29612e7 0.767621
\(979\) −1.04349e7 −0.347963
\(980\) 0 0
\(981\) 1.64860e7 0.546945
\(982\) 6.19161e7 2.04892
\(983\) −2.93292e7 −0.968093 −0.484047 0.875042i \(-0.660833\pi\)
−0.484047 + 0.875042i \(0.660833\pi\)
\(984\) 1.30035e6 0.0428128
\(985\) 0 0
\(986\) −3.41076e7 −1.11727
\(987\) −8.80980e6 −0.287855
\(988\) 8.71756e6 0.284120
\(989\) 4.89426e6 0.159109
\(990\) 0 0
\(991\) 2.75751e7 0.891935 0.445968 0.895049i \(-0.352860\pi\)
0.445968 + 0.895049i \(0.352860\pi\)
\(992\) 4.28960e7 1.38401
\(993\) −1.78545e7 −0.574611
\(994\) 3.30141e7 1.05982
\(995\) 0 0
\(996\) 2.85425e7 0.911682
\(997\) 1.69502e7 0.540052 0.270026 0.962853i \(-0.412968\pi\)
0.270026 + 0.962853i \(0.412968\pi\)
\(998\) 1.28317e7 0.407811
\(999\) 3.10185e6 0.0983347
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 825.6.a.s.1.2 yes 9
5.4 even 2 825.6.a.r.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.8 9 5.4 even 2
825.6.a.s.1.2 yes 9 1.1 even 1 trivial