Properties

Label 825.6.a.s.1.9
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 1.9
Root \(10.2706\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.2706 q^{2} -9.00000 q^{3} +73.4848 q^{4} -92.4352 q^{6} +129.851 q^{7} +426.072 q^{8} +81.0000 q^{9} -121.000 q^{11} -661.363 q^{12} -26.3756 q^{13} +1333.64 q^{14} +2024.50 q^{16} -1414.41 q^{17} +831.917 q^{18} +2336.09 q^{19} -1168.65 q^{21} -1242.74 q^{22} +939.766 q^{23} -3834.65 q^{24} -270.892 q^{26} -729.000 q^{27} +9542.04 q^{28} +6494.78 q^{29} +3501.81 q^{31} +7158.44 q^{32} +1089.00 q^{33} -14526.8 q^{34} +5952.27 q^{36} +15597.6 q^{37} +23993.0 q^{38} +237.380 q^{39} -12175.4 q^{41} -12002.8 q^{42} +10617.1 q^{43} -8891.66 q^{44} +9651.94 q^{46} -11000.2 q^{47} -18220.5 q^{48} +54.1649 q^{49} +12729.7 q^{51} -1938.20 q^{52} -2759.73 q^{53} -7487.25 q^{54} +55325.7 q^{56} -21024.8 q^{57} +66705.2 q^{58} -22393.5 q^{59} +34019.6 q^{61} +35965.6 q^{62} +10517.9 q^{63} +8737.42 q^{64} +11184.7 q^{66} -10536.4 q^{67} -103938. q^{68} -8457.89 q^{69} +62585.8 q^{71} +34511.9 q^{72} +5720.55 q^{73} +160196. q^{74} +171667. q^{76} -15711.9 q^{77} +2438.03 q^{78} -70532.7 q^{79} +6561.00 q^{81} -125048. q^{82} -25794.2 q^{83} -85878.3 q^{84} +109044. q^{86} -58453.0 q^{87} -51554.8 q^{88} +34850.4 q^{89} -3424.88 q^{91} +69058.4 q^{92} -31516.3 q^{93} -112979. q^{94} -64426.0 q^{96} +18576.6 q^{97} +556.305 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.2706 1.81560 0.907799 0.419405i \(-0.137761\pi\)
0.907799 + 0.419405i \(0.137761\pi\)
\(3\) −9.00000 −0.577350
\(4\) 73.4848 2.29640
\(5\) 0 0
\(6\) −92.4352 −1.04824
\(7\) 129.851 1.00161 0.500805 0.865560i \(-0.333037\pi\)
0.500805 + 0.865560i \(0.333037\pi\)
\(8\) 426.072 2.35374
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −661.363 −1.32583
\(13\) −26.3756 −0.0432856 −0.0216428 0.999766i \(-0.506890\pi\)
−0.0216428 + 0.999766i \(0.506890\pi\)
\(14\) 1333.64 1.81852
\(15\) 0 0
\(16\) 2024.50 1.97705
\(17\) −1414.41 −1.18701 −0.593503 0.804831i \(-0.702256\pi\)
−0.593503 + 0.804831i \(0.702256\pi\)
\(18\) 831.917 0.605200
\(19\) 2336.09 1.48459 0.742294 0.670074i \(-0.233738\pi\)
0.742294 + 0.670074i \(0.233738\pi\)
\(20\) 0 0
\(21\) −1168.65 −0.578280
\(22\) −1242.74 −0.547424
\(23\) 939.766 0.370425 0.185212 0.982699i \(-0.440703\pi\)
0.185212 + 0.982699i \(0.440703\pi\)
\(24\) −3834.65 −1.35893
\(25\) 0 0
\(26\) −270.892 −0.0785893
\(27\) −729.000 −0.192450
\(28\) 9542.04 2.30010
\(29\) 6494.78 1.43407 0.717034 0.697038i \(-0.245499\pi\)
0.717034 + 0.697038i \(0.245499\pi\)
\(30\) 0 0
\(31\) 3501.81 0.654468 0.327234 0.944943i \(-0.393883\pi\)
0.327234 + 0.944943i \(0.393883\pi\)
\(32\) 7158.44 1.23579
\(33\) 1089.00 0.174078
\(34\) −14526.8 −2.15513
\(35\) 0 0
\(36\) 5952.27 0.765466
\(37\) 15597.6 1.87306 0.936531 0.350585i \(-0.114017\pi\)
0.936531 + 0.350585i \(0.114017\pi\)
\(38\) 23993.0 2.69542
\(39\) 237.380 0.0249909
\(40\) 0 0
\(41\) −12175.4 −1.13116 −0.565579 0.824694i \(-0.691347\pi\)
−0.565579 + 0.824694i \(0.691347\pi\)
\(42\) −12002.8 −1.04992
\(43\) 10617.1 0.875658 0.437829 0.899058i \(-0.355748\pi\)
0.437829 + 0.899058i \(0.355748\pi\)
\(44\) −8891.66 −0.692390
\(45\) 0 0
\(46\) 9651.94 0.672543
\(47\) −11000.2 −0.726370 −0.363185 0.931717i \(-0.618311\pi\)
−0.363185 + 0.931717i \(0.618311\pi\)
\(48\) −18220.5 −1.14145
\(49\) 54.1649 0.00322276
\(50\) 0 0
\(51\) 12729.7 0.685319
\(52\) −1938.20 −0.0994010
\(53\) −2759.73 −0.134951 −0.0674755 0.997721i \(-0.521494\pi\)
−0.0674755 + 0.997721i \(0.521494\pi\)
\(54\) −7487.25 −0.349412
\(55\) 0 0
\(56\) 55325.7 2.35753
\(57\) −21024.8 −0.857128
\(58\) 66705.2 2.60369
\(59\) −22393.5 −0.837515 −0.418757 0.908098i \(-0.637534\pi\)
−0.418757 + 0.908098i \(0.637534\pi\)
\(60\) 0 0
\(61\) 34019.6 1.17059 0.585295 0.810820i \(-0.300979\pi\)
0.585295 + 0.810820i \(0.300979\pi\)
\(62\) 35965.6 1.18825
\(63\) 10517.9 0.333870
\(64\) 8737.42 0.266645
\(65\) 0 0
\(66\) 11184.7 0.316055
\(67\) −10536.4 −0.286751 −0.143375 0.989668i \(-0.545796\pi\)
−0.143375 + 0.989668i \(0.545796\pi\)
\(68\) −103938. −2.72584
\(69\) −8457.89 −0.213865
\(70\) 0 0
\(71\) 62585.8 1.47343 0.736716 0.676202i \(-0.236375\pi\)
0.736716 + 0.676202i \(0.236375\pi\)
\(72\) 34511.9 0.784580
\(73\) 5720.55 0.125641 0.0628204 0.998025i \(-0.479990\pi\)
0.0628204 + 0.998025i \(0.479990\pi\)
\(74\) 160196. 3.40073
\(75\) 0 0
\(76\) 171667. 3.40921
\(77\) −15711.9 −0.301997
\(78\) 2438.03 0.0453735
\(79\) −70532.7 −1.27152 −0.635759 0.771887i \(-0.719313\pi\)
−0.635759 + 0.771887i \(0.719313\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −125048. −2.05373
\(83\) −25794.2 −0.410986 −0.205493 0.978659i \(-0.565880\pi\)
−0.205493 + 0.978659i \(0.565880\pi\)
\(84\) −85878.3 −1.32796
\(85\) 0 0
\(86\) 109044. 1.58984
\(87\) −58453.0 −0.827959
\(88\) −51554.8 −0.709679
\(89\) 34850.4 0.466372 0.233186 0.972432i \(-0.425085\pi\)
0.233186 + 0.972432i \(0.425085\pi\)
\(90\) 0 0
\(91\) −3424.88 −0.0433553
\(92\) 69058.4 0.850643
\(93\) −31516.3 −0.377857
\(94\) −112979. −1.31880
\(95\) 0 0
\(96\) −64426.0 −0.713482
\(97\) 18576.6 0.200464 0.100232 0.994964i \(-0.468042\pi\)
0.100232 + 0.994964i \(0.468042\pi\)
\(98\) 556.305 0.00585124
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −73280.9 −0.714805 −0.357403 0.933950i \(-0.616338\pi\)
−0.357403 + 0.933950i \(0.616338\pi\)
\(102\) 130741. 1.24426
\(103\) 151995. 1.41168 0.705842 0.708369i \(-0.250569\pi\)
0.705842 + 0.708369i \(0.250569\pi\)
\(104\) −11237.9 −0.101883
\(105\) 0 0
\(106\) −28344.0 −0.245017
\(107\) 93690.7 0.791110 0.395555 0.918442i \(-0.370552\pi\)
0.395555 + 0.918442i \(0.370552\pi\)
\(108\) −53570.4 −0.441942
\(109\) 139934. 1.12813 0.564063 0.825732i \(-0.309237\pi\)
0.564063 + 0.825732i \(0.309237\pi\)
\(110\) 0 0
\(111\) −140378. −1.08141
\(112\) 262882. 1.98023
\(113\) 18255.0 0.134489 0.0672444 0.997737i \(-0.478579\pi\)
0.0672444 + 0.997737i \(0.478579\pi\)
\(114\) −215937. −1.55620
\(115\) 0 0
\(116\) 477268. 3.29319
\(117\) −2136.42 −0.0144285
\(118\) −229994. −1.52059
\(119\) −183662. −1.18892
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 349401. 2.12532
\(123\) 109578. 0.653074
\(124\) 257330. 1.50292
\(125\) 0 0
\(126\) 108025. 0.606174
\(127\) 174183. 0.958287 0.479143 0.877737i \(-0.340947\pi\)
0.479143 + 0.877737i \(0.340947\pi\)
\(128\) −139332. −0.751667
\(129\) −95553.8 −0.505561
\(130\) 0 0
\(131\) 106563. 0.542536 0.271268 0.962504i \(-0.412557\pi\)
0.271268 + 0.962504i \(0.412557\pi\)
\(132\) 80024.9 0.399752
\(133\) 303343. 1.48698
\(134\) −108215. −0.520625
\(135\) 0 0
\(136\) −602641. −2.79391
\(137\) 31743.1 0.144493 0.0722466 0.997387i \(-0.476983\pi\)
0.0722466 + 0.997387i \(0.476983\pi\)
\(138\) −86867.4 −0.388293
\(139\) 333919. 1.46590 0.732950 0.680283i \(-0.238143\pi\)
0.732950 + 0.680283i \(0.238143\pi\)
\(140\) 0 0
\(141\) 99002.2 0.419370
\(142\) 642793. 2.67516
\(143\) 3191.44 0.0130511
\(144\) 163984. 0.659016
\(145\) 0 0
\(146\) 58753.3 0.228113
\(147\) −487.484 −0.00186066
\(148\) 1.14618e6 4.30130
\(149\) 448303. 1.65427 0.827134 0.562005i \(-0.189970\pi\)
0.827134 + 0.562005i \(0.189970\pi\)
\(150\) 0 0
\(151\) −61013.9 −0.217764 −0.108882 0.994055i \(-0.534727\pi\)
−0.108882 + 0.994055i \(0.534727\pi\)
\(152\) 995345. 3.49434
\(153\) −114567. −0.395669
\(154\) −161370. −0.548305
\(155\) 0 0
\(156\) 17443.8 0.0573892
\(157\) 429707. 1.39131 0.695654 0.718377i \(-0.255115\pi\)
0.695654 + 0.718377i \(0.255115\pi\)
\(158\) −724411. −2.30857
\(159\) 24837.5 0.0779140
\(160\) 0 0
\(161\) 122029. 0.371021
\(162\) 67385.3 0.201733
\(163\) −261760. −0.771675 −0.385837 0.922567i \(-0.626087\pi\)
−0.385837 + 0.922567i \(0.626087\pi\)
\(164\) −894705. −2.59759
\(165\) 0 0
\(166\) −264921. −0.746185
\(167\) −467471. −1.29707 −0.648535 0.761185i \(-0.724618\pi\)
−0.648535 + 0.761185i \(0.724618\pi\)
\(168\) −497932. −1.36112
\(169\) −370597. −0.998126
\(170\) 0 0
\(171\) 189224. 0.494863
\(172\) 780195. 2.01086
\(173\) −231869. −0.589015 −0.294508 0.955649i \(-0.595156\pi\)
−0.294508 + 0.955649i \(0.595156\pi\)
\(174\) −600346. −1.50324
\(175\) 0 0
\(176\) −244964. −0.596103
\(177\) 201542. 0.483539
\(178\) 357933. 0.846744
\(179\) −271629. −0.633642 −0.316821 0.948485i \(-0.602616\pi\)
−0.316821 + 0.948485i \(0.602616\pi\)
\(180\) 0 0
\(181\) −14022.0 −0.0318137 −0.0159069 0.999873i \(-0.505064\pi\)
−0.0159069 + 0.999873i \(0.505064\pi\)
\(182\) −35175.5 −0.0787158
\(183\) −306177. −0.675840
\(184\) 400408. 0.871883
\(185\) 0 0
\(186\) −323691. −0.686037
\(187\) 171144. 0.357896
\(188\) −808351. −1.66804
\(189\) −94661.0 −0.192760
\(190\) 0 0
\(191\) 92573.9 0.183614 0.0918068 0.995777i \(-0.470736\pi\)
0.0918068 + 0.995777i \(0.470736\pi\)
\(192\) −78636.8 −0.153948
\(193\) 475271. 0.918434 0.459217 0.888324i \(-0.348130\pi\)
0.459217 + 0.888324i \(0.348130\pi\)
\(194\) 190792. 0.363962
\(195\) 0 0
\(196\) 3980.30 0.00740074
\(197\) −799351. −1.46748 −0.733740 0.679431i \(-0.762227\pi\)
−0.733740 + 0.679431i \(0.762227\pi\)
\(198\) −100662. −0.182475
\(199\) −248215. −0.444319 −0.222160 0.975010i \(-0.571311\pi\)
−0.222160 + 0.975010i \(0.571311\pi\)
\(200\) 0 0
\(201\) 94827.5 0.165556
\(202\) −752638. −1.29780
\(203\) 843351. 1.43638
\(204\) 935439. 1.57377
\(205\) 0 0
\(206\) 1.56108e6 2.56305
\(207\) 76121.0 0.123475
\(208\) −53397.3 −0.0855777
\(209\) −282667. −0.447620
\(210\) 0 0
\(211\) −699006. −1.08087 −0.540436 0.841385i \(-0.681741\pi\)
−0.540436 + 0.841385i \(0.681741\pi\)
\(212\) −202798. −0.309902
\(213\) −563272. −0.850686
\(214\) 962258. 1.43634
\(215\) 0 0
\(216\) −310607. −0.452978
\(217\) 454712. 0.655522
\(218\) 1.43721e6 2.04822
\(219\) −51484.9 −0.0725387
\(220\) 0 0
\(221\) 37305.9 0.0513803
\(222\) −1.44176e6 −1.96341
\(223\) −877162. −1.18118 −0.590592 0.806970i \(-0.701106\pi\)
−0.590592 + 0.806970i \(0.701106\pi\)
\(224\) 929528. 1.23778
\(225\) 0 0
\(226\) 187490. 0.244178
\(227\) −1.16059e6 −1.49491 −0.747456 0.664311i \(-0.768725\pi\)
−0.747456 + 0.664311i \(0.768725\pi\)
\(228\) −1.54501e6 −1.96831
\(229\) 80586.1 0.101548 0.0507740 0.998710i \(-0.483831\pi\)
0.0507740 + 0.998710i \(0.483831\pi\)
\(230\) 0 0
\(231\) 141407. 0.174358
\(232\) 2.76725e6 3.37542
\(233\) 178613. 0.215538 0.107769 0.994176i \(-0.465629\pi\)
0.107769 + 0.994176i \(0.465629\pi\)
\(234\) −21942.3 −0.0261964
\(235\) 0 0
\(236\) −1.64558e6 −1.92327
\(237\) 634794. 0.734111
\(238\) −1.88631e6 −2.15860
\(239\) 734357. 0.831596 0.415798 0.909457i \(-0.363502\pi\)
0.415798 + 0.909457i \(0.363502\pi\)
\(240\) 0 0
\(241\) 1.75930e6 1.95118 0.975590 0.219598i \(-0.0704745\pi\)
0.975590 + 0.219598i \(0.0704745\pi\)
\(242\) 150372. 0.165054
\(243\) −59049.0 −0.0641500
\(244\) 2.49992e6 2.68814
\(245\) 0 0
\(246\) 1.12543e6 1.18572
\(247\) −61615.8 −0.0642613
\(248\) 1.49203e6 1.54045
\(249\) 232148. 0.237283
\(250\) 0 0
\(251\) −57478.6 −0.0575867 −0.0287934 0.999585i \(-0.509166\pi\)
−0.0287934 + 0.999585i \(0.509166\pi\)
\(252\) 772905. 0.766699
\(253\) −113712. −0.111687
\(254\) 1.78896e6 1.73986
\(255\) 0 0
\(256\) −1.71062e6 −1.63137
\(257\) −1.77829e6 −1.67946 −0.839730 0.543005i \(-0.817287\pi\)
−0.839730 + 0.543005i \(0.817287\pi\)
\(258\) −981393. −0.917896
\(259\) 2.02535e6 1.87608
\(260\) 0 0
\(261\) 526077. 0.478023
\(262\) 1.09447e6 0.985029
\(263\) −2.11350e6 −1.88414 −0.942071 0.335414i \(-0.891124\pi\)
−0.942071 + 0.335414i \(0.891124\pi\)
\(264\) 463993. 0.409734
\(265\) 0 0
\(266\) 3.11551e6 2.69976
\(267\) −313653. −0.269260
\(268\) −774264. −0.658494
\(269\) −1.40580e6 −1.18452 −0.592261 0.805746i \(-0.701765\pi\)
−0.592261 + 0.805746i \(0.701765\pi\)
\(270\) 0 0
\(271\) −1.10334e6 −0.912608 −0.456304 0.889824i \(-0.650827\pi\)
−0.456304 + 0.889824i \(0.650827\pi\)
\(272\) −2.86347e6 −2.34677
\(273\) 30823.9 0.0250312
\(274\) 326020. 0.262342
\(275\) 0 0
\(276\) −621526. −0.491119
\(277\) −1.01726e6 −0.796589 −0.398295 0.917258i \(-0.630398\pi\)
−0.398295 + 0.917258i \(0.630398\pi\)
\(278\) 3.42954e6 2.66148
\(279\) 283647. 0.218156
\(280\) 0 0
\(281\) 811048. 0.612747 0.306373 0.951911i \(-0.400884\pi\)
0.306373 + 0.951911i \(0.400884\pi\)
\(282\) 1.01681e6 0.761407
\(283\) 848509. 0.629782 0.314891 0.949128i \(-0.398032\pi\)
0.314891 + 0.949128i \(0.398032\pi\)
\(284\) 4.59910e6 3.38359
\(285\) 0 0
\(286\) 32778.0 0.0236956
\(287\) −1.58098e6 −1.13298
\(288\) 579834. 0.411929
\(289\) 580701. 0.408985
\(290\) 0 0
\(291\) −167189. −0.115738
\(292\) 420373. 0.288521
\(293\) 2.16475e6 1.47312 0.736559 0.676373i \(-0.236449\pi\)
0.736559 + 0.676373i \(0.236449\pi\)
\(294\) −5006.75 −0.00337822
\(295\) 0 0
\(296\) 6.64569e6 4.40870
\(297\) 88209.0 0.0580259
\(298\) 4.60433e6 3.00349
\(299\) −24786.8 −0.0160341
\(300\) 0 0
\(301\) 1.37864e6 0.877068
\(302\) −626648. −0.395372
\(303\) 659528. 0.412693
\(304\) 4.72942e6 2.93510
\(305\) 0 0
\(306\) −1.17667e6 −0.718376
\(307\) −3.02969e6 −1.83464 −0.917322 0.398145i \(-0.869654\pi\)
−0.917322 + 0.398145i \(0.869654\pi\)
\(308\) −1.15459e6 −0.693505
\(309\) −1.36796e6 −0.815036
\(310\) 0 0
\(311\) 994885. 0.583273 0.291637 0.956529i \(-0.405800\pi\)
0.291637 + 0.956529i \(0.405800\pi\)
\(312\) 101141. 0.0588222
\(313\) −764574. −0.441122 −0.220561 0.975373i \(-0.570789\pi\)
−0.220561 + 0.975373i \(0.570789\pi\)
\(314\) 4.41334e6 2.52606
\(315\) 0 0
\(316\) −5.18308e6 −2.91991
\(317\) 1.77055e6 0.989598 0.494799 0.869007i \(-0.335242\pi\)
0.494799 + 0.869007i \(0.335242\pi\)
\(318\) 255096. 0.141461
\(319\) −785869. −0.432388
\(320\) 0 0
\(321\) −843216. −0.456748
\(322\) 1.25331e6 0.673625
\(323\) −3.30420e6 −1.76222
\(324\) 482134. 0.255155
\(325\) 0 0
\(326\) −2.68843e6 −1.40105
\(327\) −1.25941e6 −0.651324
\(328\) −5.18760e6 −2.66245
\(329\) −1.42839e6 −0.727540
\(330\) 0 0
\(331\) −979878. −0.491589 −0.245794 0.969322i \(-0.579049\pi\)
−0.245794 + 0.969322i \(0.579049\pi\)
\(332\) −1.89548e6 −0.943787
\(333\) 1.26340e6 0.624354
\(334\) −4.80120e6 −2.35496
\(335\) 0 0
\(336\) −2.36594e6 −1.14329
\(337\) −1.82371e6 −0.874745 −0.437372 0.899280i \(-0.644091\pi\)
−0.437372 + 0.899280i \(0.644091\pi\)
\(338\) −3.80625e6 −1.81220
\(339\) −164295. −0.0776472
\(340\) 0 0
\(341\) −423719. −0.197330
\(342\) 1.94344e6 0.898472
\(343\) −2.17536e6 −0.998382
\(344\) 4.52365e6 2.06107
\(345\) 0 0
\(346\) −2.38142e6 −1.06942
\(347\) −1.87237e6 −0.834772 −0.417386 0.908729i \(-0.637054\pi\)
−0.417386 + 0.908729i \(0.637054\pi\)
\(348\) −4.29541e6 −1.90132
\(349\) −4.37905e6 −1.92449 −0.962247 0.272177i \(-0.912256\pi\)
−0.962247 + 0.272177i \(0.912256\pi\)
\(350\) 0 0
\(351\) 19227.8 0.00833031
\(352\) −866172. −0.372604
\(353\) 155327. 0.0663453 0.0331726 0.999450i \(-0.489439\pi\)
0.0331726 + 0.999450i \(0.489439\pi\)
\(354\) 2.06995e6 0.877913
\(355\) 0 0
\(356\) 2.56097e6 1.07098
\(357\) 1.65296e6 0.686422
\(358\) −2.78979e6 −1.15044
\(359\) −3.77688e6 −1.54667 −0.773334 0.633999i \(-0.781412\pi\)
−0.773334 + 0.633999i \(0.781412\pi\)
\(360\) 0 0
\(361\) 2.98123e6 1.20400
\(362\) −144014. −0.0577610
\(363\) −131769. −0.0524864
\(364\) −251677. −0.0995610
\(365\) 0 0
\(366\) −3.14461e6 −1.22705
\(367\) 3.29301e6 1.27623 0.638113 0.769943i \(-0.279715\pi\)
0.638113 + 0.769943i \(0.279715\pi\)
\(368\) 1.90255e6 0.732348
\(369\) −986206. −0.377052
\(370\) 0 0
\(371\) −358352. −0.135168
\(372\) −2.31597e6 −0.867711
\(373\) −3.95094e6 −1.47037 −0.735187 0.677864i \(-0.762906\pi\)
−0.735187 + 0.677864i \(0.762906\pi\)
\(374\) 1.75774e6 0.649796
\(375\) 0 0
\(376\) −4.68690e6 −1.70969
\(377\) −171304. −0.0620745
\(378\) −972224. −0.349975
\(379\) 2.04599e6 0.731654 0.365827 0.930683i \(-0.380786\pi\)
0.365827 + 0.930683i \(0.380786\pi\)
\(380\) 0 0
\(381\) −1.56764e6 −0.553267
\(382\) 950787. 0.333369
\(383\) −3.77153e6 −1.31378 −0.656888 0.753988i \(-0.728127\pi\)
−0.656888 + 0.753988i \(0.728127\pi\)
\(384\) 1.25399e6 0.433975
\(385\) 0 0
\(386\) 4.88131e6 1.66751
\(387\) 859984. 0.291886
\(388\) 1.36510e6 0.460345
\(389\) 3.09884e6 1.03830 0.519152 0.854682i \(-0.326248\pi\)
0.519152 + 0.854682i \(0.326248\pi\)
\(390\) 0 0
\(391\) −1.32921e6 −0.439697
\(392\) 23078.2 0.00758554
\(393\) −959069. −0.313234
\(394\) −8.20980e6 −2.66435
\(395\) 0 0
\(396\) −720224. −0.230797
\(397\) −3.75693e6 −1.19635 −0.598173 0.801367i \(-0.704106\pi\)
−0.598173 + 0.801367i \(0.704106\pi\)
\(398\) −2.54931e6 −0.806705
\(399\) −2.73009e6 −0.858508
\(400\) 0 0
\(401\) −3.88578e6 −1.20675 −0.603376 0.797457i \(-0.706178\pi\)
−0.603376 + 0.797457i \(0.706178\pi\)
\(402\) 973933. 0.300583
\(403\) −92362.2 −0.0283290
\(404\) −5.38503e6 −1.64148
\(405\) 0 0
\(406\) 8.66170e6 2.60788
\(407\) −1.88730e6 −0.564749
\(408\) 5.42377e6 1.61306
\(409\) 1.74268e6 0.515120 0.257560 0.966262i \(-0.417081\pi\)
0.257560 + 0.966262i \(0.417081\pi\)
\(410\) 0 0
\(411\) −285688. −0.0834232
\(412\) 1.11693e7 3.24179
\(413\) −2.90781e6 −0.838863
\(414\) 781807. 0.224181
\(415\) 0 0
\(416\) −188808. −0.0534918
\(417\) −3.00527e6 −0.846337
\(418\) −2.90316e6 −0.812699
\(419\) 2.81726e6 0.783956 0.391978 0.919974i \(-0.371791\pi\)
0.391978 + 0.919974i \(0.371791\pi\)
\(420\) 0 0
\(421\) −3.92334e6 −1.07882 −0.539412 0.842042i \(-0.681353\pi\)
−0.539412 + 0.842042i \(0.681353\pi\)
\(422\) −7.17919e6 −1.96243
\(423\) −891020. −0.242123
\(424\) −1.17584e6 −0.317640
\(425\) 0 0
\(426\) −5.78513e6 −1.54451
\(427\) 4.41747e6 1.17247
\(428\) 6.88484e6 1.81671
\(429\) −28723.0 −0.00753505
\(430\) 0 0
\(431\) 3.54043e6 0.918044 0.459022 0.888425i \(-0.348200\pi\)
0.459022 + 0.888425i \(0.348200\pi\)
\(432\) −1.47586e6 −0.380483
\(433\) −3.93249e6 −1.00797 −0.503985 0.863712i \(-0.668133\pi\)
−0.503985 + 0.863712i \(0.668133\pi\)
\(434\) 4.67016e6 1.19016
\(435\) 0 0
\(436\) 1.02830e7 2.59063
\(437\) 2.19538e6 0.549928
\(438\) −528780. −0.131701
\(439\) −6.90652e6 −1.71040 −0.855201 0.518296i \(-0.826566\pi\)
−0.855201 + 0.518296i \(0.826566\pi\)
\(440\) 0 0
\(441\) 4387.36 0.00107425
\(442\) 383153. 0.0932860
\(443\) 6.56130e6 1.58848 0.794238 0.607607i \(-0.207870\pi\)
0.794238 + 0.607607i \(0.207870\pi\)
\(444\) −1.03156e7 −2.48336
\(445\) 0 0
\(446\) −9.00896e6 −2.14456
\(447\) −4.03472e6 −0.955092
\(448\) 1.13456e6 0.267074
\(449\) −1.77881e6 −0.416403 −0.208201 0.978086i \(-0.566761\pi\)
−0.208201 + 0.978086i \(0.566761\pi\)
\(450\) 0 0
\(451\) 1.47322e6 0.341057
\(452\) 1.34147e6 0.308840
\(453\) 549125. 0.125726
\(454\) −1.19200e7 −2.71416
\(455\) 0 0
\(456\) −8.95811e6 −2.01746
\(457\) 1.32684e6 0.297186 0.148593 0.988898i \(-0.452526\pi\)
0.148593 + 0.988898i \(0.452526\pi\)
\(458\) 827666. 0.184370
\(459\) 1.03111e6 0.228440
\(460\) 0 0
\(461\) 5.96751e6 1.30780 0.653899 0.756582i \(-0.273132\pi\)
0.653899 + 0.756582i \(0.273132\pi\)
\(462\) 1.45233e6 0.316564
\(463\) −2.85291e6 −0.618494 −0.309247 0.950982i \(-0.600077\pi\)
−0.309247 + 0.950982i \(0.600077\pi\)
\(464\) 1.31487e7 2.83522
\(465\) 0 0
\(466\) 1.83446e6 0.391330
\(467\) 4.25236e6 0.902274 0.451137 0.892455i \(-0.351019\pi\)
0.451137 + 0.892455i \(0.351019\pi\)
\(468\) −156994. −0.0331337
\(469\) −1.36816e6 −0.287213
\(470\) 0 0
\(471\) −3.86736e6 −0.803272
\(472\) −9.54126e6 −1.97129
\(473\) −1.28467e6 −0.264021
\(474\) 6.51970e6 1.33285
\(475\) 0 0
\(476\) −1.34964e7 −2.73023
\(477\) −223538. −0.0449837
\(478\) 7.54227e6 1.50984
\(479\) 4.59659e6 0.915371 0.457686 0.889114i \(-0.348678\pi\)
0.457686 + 0.889114i \(0.348678\pi\)
\(480\) 0 0
\(481\) −411394. −0.0810766
\(482\) 1.80690e7 3.54256
\(483\) −1.09826e6 −0.214209
\(484\) 1.07589e6 0.208764
\(485\) 0 0
\(486\) −606467. −0.116471
\(487\) −4.45476e6 −0.851142 −0.425571 0.904925i \(-0.639927\pi\)
−0.425571 + 0.904925i \(0.639927\pi\)
\(488\) 1.44948e7 2.75526
\(489\) 2.35584e6 0.445527
\(490\) 0 0
\(491\) 9.96192e6 1.86483 0.932416 0.361388i \(-0.117697\pi\)
0.932416 + 0.361388i \(0.117697\pi\)
\(492\) 8.05235e6 1.49972
\(493\) −9.18629e6 −1.70225
\(494\) −632829. −0.116673
\(495\) 0 0
\(496\) 7.08941e6 1.29392
\(497\) 8.12680e6 1.47580
\(498\) 2.38429e6 0.430810
\(499\) 1.01344e7 1.82199 0.910993 0.412422i \(-0.135317\pi\)
0.910993 + 0.412422i \(0.135317\pi\)
\(500\) 0 0
\(501\) 4.20724e6 0.748864
\(502\) −590339. −0.104554
\(503\) −3.76588e6 −0.663662 −0.331831 0.943339i \(-0.607666\pi\)
−0.331831 + 0.943339i \(0.607666\pi\)
\(504\) 4.48139e6 0.785843
\(505\) 0 0
\(506\) −1.16788e6 −0.202779
\(507\) 3.33538e6 0.576269
\(508\) 1.27998e7 2.20061
\(509\) 1.22083e6 0.208863 0.104432 0.994532i \(-0.466698\pi\)
0.104432 + 0.994532i \(0.466698\pi\)
\(510\) 0 0
\(511\) 742816. 0.125843
\(512\) −1.31104e7 −2.21025
\(513\) −1.70301e6 −0.285709
\(514\) −1.82641e7 −3.04922
\(515\) 0 0
\(516\) −7.02175e6 −1.16097
\(517\) 1.33103e6 0.219009
\(518\) 2.08015e7 3.40620
\(519\) 2.08682e6 0.340068
\(520\) 0 0
\(521\) 3.00864e6 0.485597 0.242798 0.970077i \(-0.421935\pi\)
0.242798 + 0.970077i \(0.421935\pi\)
\(522\) 5.40312e6 0.867897
\(523\) −2.82585e6 −0.451747 −0.225874 0.974157i \(-0.572524\pi\)
−0.225874 + 0.974157i \(0.572524\pi\)
\(524\) 7.83077e6 1.24588
\(525\) 0 0
\(526\) −2.17069e7 −3.42085
\(527\) −4.95300e6 −0.776858
\(528\) 2.20468e6 0.344160
\(529\) −5.55318e6 −0.862786
\(530\) 0 0
\(531\) −1.81388e6 −0.279172
\(532\) 2.22911e7 3.41470
\(533\) 321133. 0.0489628
\(534\) −3.22140e6 −0.488868
\(535\) 0 0
\(536\) −4.48927e6 −0.674937
\(537\) 2.44467e6 0.365834
\(538\) −1.44384e7 −2.15062
\(539\) −6553.96 −0.000971699 0
\(540\) 0 0
\(541\) 9.06915e6 1.33221 0.666106 0.745857i \(-0.267960\pi\)
0.666106 + 0.745857i \(0.267960\pi\)
\(542\) −1.13319e7 −1.65693
\(543\) 126198. 0.0183677
\(544\) −1.01250e7 −1.46689
\(545\) 0 0
\(546\) 316580. 0.0454466
\(547\) 1.77907e6 0.254229 0.127114 0.991888i \(-0.459428\pi\)
0.127114 + 0.991888i \(0.459428\pi\)
\(548\) 2.33263e6 0.331814
\(549\) 2.75559e6 0.390197
\(550\) 0 0
\(551\) 1.51724e7 2.12900
\(552\) −3.60367e6 −0.503382
\(553\) −9.15871e6 −1.27357
\(554\) −1.04479e7 −1.44629
\(555\) 0 0
\(556\) 2.45380e7 3.36629
\(557\) −361115. −0.0493182 −0.0246591 0.999696i \(-0.507850\pi\)
−0.0246591 + 0.999696i \(0.507850\pi\)
\(558\) 2.91322e6 0.396084
\(559\) −280032. −0.0379034
\(560\) 0 0
\(561\) −1.54029e6 −0.206631
\(562\) 8.32993e6 1.11250
\(563\) 1.24161e7 1.65088 0.825440 0.564489i \(-0.190927\pi\)
0.825440 + 0.564489i \(0.190927\pi\)
\(564\) 7.27516e6 0.963041
\(565\) 0 0
\(566\) 8.71467e6 1.14343
\(567\) 851949. 0.111290
\(568\) 2.66661e7 3.46808
\(569\) −1.47941e7 −1.91561 −0.957806 0.287417i \(-0.907204\pi\)
−0.957806 + 0.287417i \(0.907204\pi\)
\(570\) 0 0
\(571\) 7.97003e6 1.02299 0.511493 0.859287i \(-0.329093\pi\)
0.511493 + 0.859287i \(0.329093\pi\)
\(572\) 234522. 0.0299705
\(573\) −833165. −0.106009
\(574\) −1.62376e7 −2.05703
\(575\) 0 0
\(576\) 707731. 0.0888817
\(577\) −1.48026e7 −1.85096 −0.925480 0.378796i \(-0.876338\pi\)
−0.925480 + 0.378796i \(0.876338\pi\)
\(578\) 5.96413e6 0.742553
\(579\) −4.27744e6 −0.530258
\(580\) 0 0
\(581\) −3.34939e6 −0.411648
\(582\) −1.71713e6 −0.210134
\(583\) 333927. 0.0406893
\(584\) 2.43737e6 0.295726
\(585\) 0 0
\(586\) 2.22332e7 2.67459
\(587\) −1.07889e7 −1.29236 −0.646180 0.763185i \(-0.723635\pi\)
−0.646180 + 0.763185i \(0.723635\pi\)
\(588\) −35822.7 −0.00427282
\(589\) 8.18056e6 0.971616
\(590\) 0 0
\(591\) 7.19416e6 0.847250
\(592\) 3.15772e7 3.70314
\(593\) 9.31223e6 1.08747 0.543734 0.839257i \(-0.317010\pi\)
0.543734 + 0.839257i \(0.317010\pi\)
\(594\) 905957. 0.105352
\(595\) 0 0
\(596\) 3.29434e7 3.79886
\(597\) 2.23393e6 0.256528
\(598\) −254575. −0.0291114
\(599\) −1.19315e7 −1.35871 −0.679354 0.733811i \(-0.737740\pi\)
−0.679354 + 0.733811i \(0.737740\pi\)
\(600\) 0 0
\(601\) −7.25405e6 −0.819208 −0.409604 0.912263i \(-0.634333\pi\)
−0.409604 + 0.912263i \(0.634333\pi\)
\(602\) 1.41594e7 1.59240
\(603\) −853448. −0.0955836
\(604\) −4.48359e6 −0.500073
\(605\) 0 0
\(606\) 6.77374e6 0.749285
\(607\) −1.37134e7 −1.51068 −0.755342 0.655331i \(-0.772529\pi\)
−0.755342 + 0.655331i \(0.772529\pi\)
\(608\) 1.67228e7 1.83464
\(609\) −7.59016e6 −0.829292
\(610\) 0 0
\(611\) 290138. 0.0314414
\(612\) −8.41895e6 −0.908614
\(613\) −1.94436e6 −0.208990 −0.104495 0.994525i \(-0.533323\pi\)
−0.104495 + 0.994525i \(0.533323\pi\)
\(614\) −3.11166e7 −3.33098
\(615\) 0 0
\(616\) −6.69442e6 −0.710822
\(617\) −1.02584e7 −1.08484 −0.542419 0.840108i \(-0.682491\pi\)
−0.542419 + 0.840108i \(0.682491\pi\)
\(618\) −1.40497e7 −1.47978
\(619\) 4.58800e6 0.481279 0.240639 0.970615i \(-0.422643\pi\)
0.240639 + 0.970615i \(0.422643\pi\)
\(620\) 0 0
\(621\) −685089. −0.0712883
\(622\) 1.02180e7 1.05899
\(623\) 4.52534e6 0.467123
\(624\) 480575. 0.0494083
\(625\) 0 0
\(626\) −7.85262e6 −0.800901
\(627\) 2.54401e6 0.258434
\(628\) 3.15769e7 3.19500
\(629\) −2.20613e7 −2.22334
\(630\) 0 0
\(631\) 1.87262e7 1.87230 0.936152 0.351596i \(-0.114361\pi\)
0.936152 + 0.351596i \(0.114361\pi\)
\(632\) −3.00520e7 −2.99282
\(633\) 6.29105e6 0.624042
\(634\) 1.81845e7 1.79671
\(635\) 0 0
\(636\) 1.82518e6 0.178922
\(637\) −1428.63 −0.000139499 0
\(638\) −8.07133e6 −0.785042
\(639\) 5.06945e6 0.491144
\(640\) 0 0
\(641\) −4.84441e6 −0.465689 −0.232845 0.972514i \(-0.574803\pi\)
−0.232845 + 0.972514i \(0.574803\pi\)
\(642\) −8.66032e6 −0.829271
\(643\) −7.56977e6 −0.722030 −0.361015 0.932560i \(-0.617570\pi\)
−0.361015 + 0.932560i \(0.617570\pi\)
\(644\) 8.96728e6 0.852012
\(645\) 0 0
\(646\) −3.39360e7 −3.19948
\(647\) −1.35084e7 −1.26865 −0.634327 0.773065i \(-0.718723\pi\)
−0.634327 + 0.773065i \(0.718723\pi\)
\(648\) 2.79546e6 0.261527
\(649\) 2.70962e6 0.252520
\(650\) 0 0
\(651\) −4.09241e6 −0.378466
\(652\) −1.92354e7 −1.77207
\(653\) −2.14348e6 −0.196714 −0.0983571 0.995151i \(-0.531359\pi\)
−0.0983571 + 0.995151i \(0.531359\pi\)
\(654\) −1.29348e7 −1.18254
\(655\) 0 0
\(656\) −2.46490e7 −2.23635
\(657\) 463364. 0.0418802
\(658\) −1.46704e7 −1.32092
\(659\) 1.27830e7 1.14662 0.573311 0.819338i \(-0.305658\pi\)
0.573311 + 0.819338i \(0.305658\pi\)
\(660\) 0 0
\(661\) 1.67783e7 1.49363 0.746816 0.665030i \(-0.231581\pi\)
0.746816 + 0.665030i \(0.231581\pi\)
\(662\) −1.00639e7 −0.892528
\(663\) −335753. −0.0296644
\(664\) −1.09902e7 −0.967354
\(665\) 0 0
\(666\) 1.29759e7 1.13358
\(667\) 6.10357e6 0.531214
\(668\) −3.43520e7 −2.97859
\(669\) 7.89446e6 0.681957
\(670\) 0 0
\(671\) −4.11637e6 −0.352946
\(672\) −8.36575e6 −0.714631
\(673\) 2.08834e7 1.77731 0.888657 0.458572i \(-0.151639\pi\)
0.888657 + 0.458572i \(0.151639\pi\)
\(674\) −1.87306e7 −1.58819
\(675\) 0 0
\(676\) −2.72333e7 −2.29210
\(677\) 1.10376e7 0.925555 0.462777 0.886475i \(-0.346853\pi\)
0.462777 + 0.886475i \(0.346853\pi\)
\(678\) −1.68741e6 −0.140976
\(679\) 2.41218e6 0.200787
\(680\) 0 0
\(681\) 1.04453e7 0.863088
\(682\) −4.35184e6 −0.358271
\(683\) −6.12820e6 −0.502668 −0.251334 0.967900i \(-0.580869\pi\)
−0.251334 + 0.967900i \(0.580869\pi\)
\(684\) 1.39050e7 1.13640
\(685\) 0 0
\(686\) −2.23423e7 −1.81266
\(687\) −725275. −0.0586288
\(688\) 2.14943e7 1.73122
\(689\) 72789.3 0.00584144
\(690\) 0 0
\(691\) −1.18390e7 −0.943233 −0.471616 0.881804i \(-0.656329\pi\)
−0.471616 + 0.881804i \(0.656329\pi\)
\(692\) −1.70388e7 −1.35261
\(693\) −1.27267e6 −0.100666
\(694\) −1.92303e7 −1.51561
\(695\) 0 0
\(696\) −2.49052e7 −1.94880
\(697\) 1.72210e7 1.34269
\(698\) −4.49754e7 −3.49411
\(699\) −1.60752e6 −0.124441
\(700\) 0 0
\(701\) −2.85962e6 −0.219793 −0.109896 0.993943i \(-0.535052\pi\)
−0.109896 + 0.993943i \(0.535052\pi\)
\(702\) 197480. 0.0151245
\(703\) 3.64373e7 2.78073
\(704\) −1.05723e6 −0.0803965
\(705\) 0 0
\(706\) 1.59530e6 0.120456
\(707\) −9.51557e6 −0.715956
\(708\) 1.48102e7 1.11040
\(709\) −1.53361e7 −1.14577 −0.572887 0.819635i \(-0.694177\pi\)
−0.572887 + 0.819635i \(0.694177\pi\)
\(710\) 0 0
\(711\) −5.71315e6 −0.423839
\(712\) 1.48488e7 1.09772
\(713\) 3.29088e6 0.242431
\(714\) 1.69768e7 1.24627
\(715\) 0 0
\(716\) −1.99606e7 −1.45510
\(717\) −6.60921e6 −0.480122
\(718\) −3.87907e7 −2.80813
\(719\) −4.90410e6 −0.353783 −0.176892 0.984230i \(-0.556604\pi\)
−0.176892 + 0.984230i \(0.556604\pi\)
\(720\) 0 0
\(721\) 1.97367e7 1.41396
\(722\) 3.06190e7 2.18599
\(723\) −1.58337e7 −1.12651
\(724\) −1.03041e6 −0.0730570
\(725\) 0 0
\(726\) −1.35334e6 −0.0952942
\(727\) 1.07476e7 0.754180 0.377090 0.926177i \(-0.376925\pi\)
0.377090 + 0.926177i \(0.376925\pi\)
\(728\) −1.45925e6 −0.102047
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.50169e7 −1.03941
\(732\) −2.24993e7 −1.55200
\(733\) −2.22318e7 −1.52832 −0.764161 0.645025i \(-0.776847\pi\)
−0.764161 + 0.645025i \(0.776847\pi\)
\(734\) 3.38211e7 2.31711
\(735\) 0 0
\(736\) 6.72726e6 0.457766
\(737\) 1.27490e6 0.0864587
\(738\) −1.01289e7 −0.684576
\(739\) 1.16506e7 0.784758 0.392379 0.919804i \(-0.371652\pi\)
0.392379 + 0.919804i \(0.371652\pi\)
\(740\) 0 0
\(741\) 554542. 0.0371013
\(742\) −3.68048e6 −0.245412
\(743\) −8.41242e6 −0.559048 −0.279524 0.960139i \(-0.590177\pi\)
−0.279524 + 0.960139i \(0.590177\pi\)
\(744\) −1.34282e7 −0.889378
\(745\) 0 0
\(746\) −4.05784e7 −2.66961
\(747\) −2.08933e6 −0.136995
\(748\) 1.25765e7 0.821872
\(749\) 1.21658e7 0.792384
\(750\) 0 0
\(751\) 4.28896e6 0.277493 0.138746 0.990328i \(-0.455693\pi\)
0.138746 + 0.990328i \(0.455693\pi\)
\(752\) −2.22700e7 −1.43607
\(753\) 517308. 0.0332477
\(754\) −1.75939e6 −0.112702
\(755\) 0 0
\(756\) −6.95614e6 −0.442654
\(757\) 8.29492e6 0.526105 0.263053 0.964781i \(-0.415271\pi\)
0.263053 + 0.964781i \(0.415271\pi\)
\(758\) 2.10135e7 1.32839
\(759\) 1.02340e6 0.0644827
\(760\) 0 0
\(761\) 6.70114e6 0.419456 0.209728 0.977760i \(-0.432742\pi\)
0.209728 + 0.977760i \(0.432742\pi\)
\(762\) −1.61006e7 −1.00451
\(763\) 1.81705e7 1.12994
\(764\) 6.80277e6 0.421650
\(765\) 0 0
\(766\) −3.87358e7 −2.38529
\(767\) 590642. 0.0362523
\(768\) 1.53955e7 0.941872
\(769\) −8.86059e6 −0.540314 −0.270157 0.962816i \(-0.587076\pi\)
−0.270157 + 0.962816i \(0.587076\pi\)
\(770\) 0 0
\(771\) 1.60046e7 0.969636
\(772\) 3.49252e7 2.10909
\(773\) 4.08253e6 0.245743 0.122871 0.992423i \(-0.460790\pi\)
0.122871 + 0.992423i \(0.460790\pi\)
\(774\) 8.83254e6 0.529948
\(775\) 0 0
\(776\) 7.91497e6 0.471840
\(777\) −1.82282e7 −1.08315
\(778\) 3.18268e7 1.88514
\(779\) −2.84428e7 −1.67930
\(780\) 0 0
\(781\) −7.57288e6 −0.444256
\(782\) −1.36518e7 −0.798313
\(783\) −4.73470e6 −0.275986
\(784\) 109657. 0.00637156
\(785\) 0 0
\(786\) −9.85019e6 −0.568707
\(787\) −1.15210e7 −0.663063 −0.331531 0.943444i \(-0.607565\pi\)
−0.331531 + 0.943444i \(0.607565\pi\)
\(788\) −5.87401e7 −3.36992
\(789\) 1.90215e7 1.08781
\(790\) 0 0
\(791\) 2.37042e6 0.134705
\(792\) −4.17594e6 −0.236560
\(793\) −897286. −0.0506697
\(794\) −3.85858e7 −2.17208
\(795\) 0 0
\(796\) −1.82400e7 −1.02033
\(797\) −1.71525e7 −0.956492 −0.478246 0.878226i \(-0.658727\pi\)
−0.478246 + 0.878226i \(0.658727\pi\)
\(798\) −2.80396e7 −1.55871
\(799\) 1.55589e7 0.862206
\(800\) 0 0
\(801\) 2.82288e6 0.155457
\(802\) −3.99092e7 −2.19098
\(803\) −692186. −0.0378821
\(804\) 6.96838e6 0.380182
\(805\) 0 0
\(806\) −948614. −0.0514342
\(807\) 1.26522e7 0.683884
\(808\) −3.12230e7 −1.68247
\(809\) −6.93330e6 −0.372450 −0.186225 0.982507i \(-0.559625\pi\)
−0.186225 + 0.982507i \(0.559625\pi\)
\(810\) 0 0
\(811\) −2.74832e7 −1.46729 −0.733643 0.679535i \(-0.762182\pi\)
−0.733643 + 0.679535i \(0.762182\pi\)
\(812\) 6.19735e7 3.29849
\(813\) 9.93002e6 0.526895
\(814\) −1.93837e7 −1.02536
\(815\) 0 0
\(816\) 2.57712e7 1.35491
\(817\) 2.48025e7 1.29999
\(818\) 1.78983e7 0.935252
\(819\) −277415. −0.0144518
\(820\) 0 0
\(821\) −1.07753e7 −0.557921 −0.278961 0.960303i \(-0.589990\pi\)
−0.278961 + 0.960303i \(0.589990\pi\)
\(822\) −2.93418e6 −0.151463
\(823\) 9.62133e6 0.495149 0.247574 0.968869i \(-0.420367\pi\)
0.247574 + 0.968869i \(0.420367\pi\)
\(824\) 6.47611e7 3.32274
\(825\) 0 0
\(826\) −2.98649e7 −1.52304
\(827\) 1.14957e7 0.584482 0.292241 0.956345i \(-0.405599\pi\)
0.292241 + 0.956345i \(0.405599\pi\)
\(828\) 5.59373e6 0.283548
\(829\) 3.48689e7 1.76219 0.881094 0.472941i \(-0.156808\pi\)
0.881094 + 0.472941i \(0.156808\pi\)
\(830\) 0 0
\(831\) 9.15538e6 0.459911
\(832\) −230454. −0.0115419
\(833\) −76611.5 −0.00382544
\(834\) −3.08659e7 −1.53661
\(835\) 0 0
\(836\) −2.07717e7 −1.02791
\(837\) −2.55282e6 −0.125952
\(838\) 2.89349e7 1.42335
\(839\) 1.56070e7 0.765445 0.382722 0.923863i \(-0.374987\pi\)
0.382722 + 0.923863i \(0.374987\pi\)
\(840\) 0 0
\(841\) 2.16710e7 1.05655
\(842\) −4.02950e7 −1.95871
\(843\) −7.29944e6 −0.353769
\(844\) −5.13663e7 −2.48211
\(845\) 0 0
\(846\) −9.15129e6 −0.439599
\(847\) 1.90114e6 0.0910555
\(848\) −5.58706e6 −0.266805
\(849\) −7.63658e6 −0.363605
\(850\) 0 0
\(851\) 1.46580e7 0.693828
\(852\) −4.13919e7 −1.95352
\(853\) −5.56534e6 −0.261890 −0.130945 0.991390i \(-0.541801\pi\)
−0.130945 + 0.991390i \(0.541801\pi\)
\(854\) 4.53699e7 2.12874
\(855\) 0 0
\(856\) 3.99190e7 1.86207
\(857\) −1.76227e7 −0.819633 −0.409817 0.912168i \(-0.634407\pi\)
−0.409817 + 0.912168i \(0.634407\pi\)
\(858\) −295002. −0.0136806
\(859\) −7.71598e6 −0.356786 −0.178393 0.983959i \(-0.557090\pi\)
−0.178393 + 0.983959i \(0.557090\pi\)
\(860\) 0 0
\(861\) 1.42288e7 0.654126
\(862\) 3.63623e7 1.66680
\(863\) −2.72946e7 −1.24753 −0.623764 0.781613i \(-0.714397\pi\)
−0.623764 + 0.781613i \(0.714397\pi\)
\(864\) −5.21851e6 −0.237827
\(865\) 0 0
\(866\) −4.03889e7 −1.83007
\(867\) −5.22631e6 −0.236128
\(868\) 3.34144e7 1.50534
\(869\) 8.53445e6 0.383377
\(870\) 0 0
\(871\) 277903. 0.0124122
\(872\) 5.96221e7 2.65532
\(873\) 1.50470e6 0.0668213
\(874\) 2.25478e7 0.998449
\(875\) 0 0
\(876\) −3.78336e6 −0.166578
\(877\) 263106. 0.0115513 0.00577567 0.999983i \(-0.498162\pi\)
0.00577567 + 0.999983i \(0.498162\pi\)
\(878\) −7.09340e7 −3.10540
\(879\) −1.94827e7 −0.850506
\(880\) 0 0
\(881\) 3.47050e7 1.50644 0.753222 0.657767i \(-0.228499\pi\)
0.753222 + 0.657767i \(0.228499\pi\)
\(882\) 45060.7 0.00195041
\(883\) −1.77964e7 −0.768123 −0.384061 0.923308i \(-0.625475\pi\)
−0.384061 + 0.923308i \(0.625475\pi\)
\(884\) 2.74141e6 0.117990
\(885\) 0 0
\(886\) 6.73883e7 2.88403
\(887\) −4.02082e7 −1.71595 −0.857976 0.513690i \(-0.828278\pi\)
−0.857976 + 0.513690i \(0.828278\pi\)
\(888\) −5.98112e7 −2.54536
\(889\) 2.26177e7 0.959830
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −6.44581e7 −2.71247
\(893\) −2.56976e7 −1.07836
\(894\) −4.14389e7 −1.73406
\(895\) 0 0
\(896\) −1.80923e7 −0.752877
\(897\) 223082. 0.00925726
\(898\) −1.82694e7 −0.756021
\(899\) 2.27435e7 0.938552
\(900\) 0 0
\(901\) 3.90339e6 0.160188
\(902\) 1.51308e7 0.619222
\(903\) −1.24077e7 −0.506375
\(904\) 7.77796e6 0.316552
\(905\) 0 0
\(906\) 5.63983e6 0.228268
\(907\) −2.69086e7 −1.08611 −0.543054 0.839698i \(-0.682732\pi\)
−0.543054 + 0.839698i \(0.682732\pi\)
\(908\) −8.52860e7 −3.43291
\(909\) −5.93576e6 −0.238268
\(910\) 0 0
\(911\) −1.43592e7 −0.573236 −0.286618 0.958045i \(-0.592531\pi\)
−0.286618 + 0.958045i \(0.592531\pi\)
\(912\) −4.25647e7 −1.69458
\(913\) 3.12110e6 0.123917
\(914\) 1.36274e7 0.539571
\(915\) 0 0
\(916\) 5.92185e6 0.233195
\(917\) 1.38373e7 0.543410
\(918\) 1.05900e7 0.414755
\(919\) −2.81706e6 −0.110029 −0.0550145 0.998486i \(-0.517521\pi\)
−0.0550145 + 0.998486i \(0.517521\pi\)
\(920\) 0 0
\(921\) 2.72672e7 1.05923
\(922\) 6.12897e7 2.37444
\(923\) −1.65074e6 −0.0637784
\(924\) 1.03913e7 0.400395
\(925\) 0 0
\(926\) −2.93010e7 −1.12294
\(927\) 1.23116e7 0.470561
\(928\) 4.64925e7 1.77220
\(929\) −8.94528e6 −0.340059 −0.170030 0.985439i \(-0.554386\pi\)
−0.170030 + 0.985439i \(0.554386\pi\)
\(930\) 0 0
\(931\) 126534. 0.00478447
\(932\) 1.31253e7 0.494960
\(933\) −8.95397e6 −0.336753
\(934\) 4.36742e7 1.63817
\(935\) 0 0
\(936\) −910270. −0.0339610
\(937\) −3.68278e7 −1.37034 −0.685168 0.728385i \(-0.740271\pi\)
−0.685168 + 0.728385i \(0.740271\pi\)
\(938\) −1.40518e7 −0.521463
\(939\) 6.88117e6 0.254682
\(940\) 0 0
\(941\) 3.22670e7 1.18791 0.593957 0.804497i \(-0.297565\pi\)
0.593957 + 0.804497i \(0.297565\pi\)
\(942\) −3.97201e7 −1.45842
\(943\) −1.14420e7 −0.419009
\(944\) −4.53356e7 −1.65581
\(945\) 0 0
\(946\) −1.31943e7 −0.479356
\(947\) 5.26498e7 1.90775 0.953877 0.300199i \(-0.0970531\pi\)
0.953877 + 0.300199i \(0.0970531\pi\)
\(948\) 4.66477e7 1.68581
\(949\) −150883. −0.00543843
\(950\) 0 0
\(951\) −1.59349e7 −0.571345
\(952\) −7.82533e7 −2.79840
\(953\) 4.80923e7 1.71531 0.857657 0.514222i \(-0.171919\pi\)
0.857657 + 0.514222i \(0.171919\pi\)
\(954\) −2.29586e6 −0.0816723
\(955\) 0 0
\(956\) 5.39640e7 1.90968
\(957\) 7.07282e6 0.249639
\(958\) 4.72097e7 1.66195
\(959\) 4.12186e6 0.144726
\(960\) 0 0
\(961\) −1.63665e7 −0.571671
\(962\) −4.22526e6 −0.147203
\(963\) 7.58895e6 0.263703
\(964\) 1.29282e8 4.48069
\(965\) 0 0
\(966\) −1.12798e7 −0.388918
\(967\) −3.61508e7 −1.24323 −0.621615 0.783323i \(-0.713523\pi\)
−0.621615 + 0.783323i \(0.713523\pi\)
\(968\) 6.23813e6 0.213976
\(969\) 2.97378e7 1.01742
\(970\) 0 0
\(971\) −3.40402e7 −1.15863 −0.579315 0.815104i \(-0.696680\pi\)
−0.579315 + 0.815104i \(0.696680\pi\)
\(972\) −4.33920e6 −0.147314
\(973\) 4.33596e7 1.46826
\(974\) −4.57530e7 −1.54533
\(975\) 0 0
\(976\) 6.88726e7 2.31431
\(977\) −1.92360e7 −0.644730 −0.322365 0.946615i \(-0.604478\pi\)
−0.322365 + 0.946615i \(0.604478\pi\)
\(978\) 2.41958e7 0.808898
\(979\) −4.21689e6 −0.140616
\(980\) 0 0
\(981\) 1.13347e7 0.376042
\(982\) 1.02315e8 3.38578
\(983\) 2.65797e7 0.877336 0.438668 0.898649i \(-0.355450\pi\)
0.438668 + 0.898649i \(0.355450\pi\)
\(984\) 4.66884e7 1.53717
\(985\) 0 0
\(986\) −9.43485e7 −3.09060
\(987\) 1.28555e7 0.420045
\(988\) −4.52782e6 −0.147570
\(989\) 9.97758e6 0.324365
\(990\) 0 0
\(991\) 5.56931e7 1.80143 0.900714 0.434412i \(-0.143044\pi\)
0.900714 + 0.434412i \(0.143044\pi\)
\(992\) 2.50675e7 0.808783
\(993\) 8.81890e6 0.283819
\(994\) 8.34670e7 2.67947
\(995\) 0 0
\(996\) 1.70593e7 0.544896
\(997\) −2.24526e7 −0.715367 −0.357684 0.933843i \(-0.616433\pi\)
−0.357684 + 0.933843i \(0.616433\pi\)
\(998\) 1.04086e8 3.30800
\(999\) −1.13706e7 −0.360471
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.1.9 yes 9
5.4 even 2 825.6.a.r.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.1 9 5.4 even 2
825.6.a.s.1.9 yes 9 1.1 even 1 trivial