Properties

Label 80.6.a.a.1.1
Level $80$
Weight $6$
Character 80.1
Self dual yes
Analytic conductor $12.831$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8307055850\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 80.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} +O(q^{10})\) \(q-24.0000 q^{3} +25.0000 q^{5} +172.000 q^{7} +333.000 q^{9} -132.000 q^{11} -946.000 q^{13} -600.000 q^{15} -222.000 q^{17} -500.000 q^{19} -4128.00 q^{21} -3564.00 q^{23} +625.000 q^{25} -2160.00 q^{27} +2190.00 q^{29} -2312.00 q^{31} +3168.00 q^{33} +4300.00 q^{35} -11242.0 q^{37} +22704.0 q^{39} +1242.00 q^{41} -20624.0 q^{43} +8325.00 q^{45} -6588.00 q^{47} +12777.0 q^{49} +5328.00 q^{51} -21066.0 q^{53} -3300.00 q^{55} +12000.0 q^{57} -7980.00 q^{59} +16622.0 q^{61} +57276.0 q^{63} -23650.0 q^{65} -1808.00 q^{67} +85536.0 q^{69} +24528.0 q^{71} +20474.0 q^{73} -15000.0 q^{75} -22704.0 q^{77} +46240.0 q^{79} -29079.0 q^{81} +51576.0 q^{83} -5550.00 q^{85} -52560.0 q^{87} -110310. q^{89} -162712. q^{91} +55488.0 q^{93} -12500.0 q^{95} -78382.0 q^{97} -43956.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −24.0000 −1.53960 −0.769800 0.638285i \(-0.779644\pi\)
−0.769800 + 0.638285i \(0.779644\pi\)
\(4\) 0 0
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) 172.000 1.32673 0.663366 0.748295i \(-0.269127\pi\)
0.663366 + 0.748295i \(0.269127\pi\)
\(8\) 0 0
\(9\) 333.000 1.37037
\(10\) 0 0
\(11\) −132.000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) −946.000 −1.55250 −0.776252 0.630423i \(-0.782882\pi\)
−0.776252 + 0.630423i \(0.782882\pi\)
\(14\) 0 0
\(15\) −600.000 −0.688530
\(16\) 0 0
\(17\) −222.000 −0.186308 −0.0931538 0.995652i \(-0.529695\pi\)
−0.0931538 + 0.995652i \(0.529695\pi\)
\(18\) 0 0
\(19\) −500.000 −0.317750 −0.158875 0.987299i \(-0.550787\pi\)
−0.158875 + 0.987299i \(0.550787\pi\)
\(20\) 0 0
\(21\) −4128.00 −2.04264
\(22\) 0 0
\(23\) −3564.00 −1.40481 −0.702406 0.711777i \(-0.747891\pi\)
−0.702406 + 0.711777i \(0.747891\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −2160.00 −0.570222
\(28\) 0 0
\(29\) 2190.00 0.483559 0.241779 0.970331i \(-0.422269\pi\)
0.241779 + 0.970331i \(0.422269\pi\)
\(30\) 0 0
\(31\) −2312.00 −0.432099 −0.216050 0.976382i \(-0.569317\pi\)
−0.216050 + 0.976382i \(0.569317\pi\)
\(32\) 0 0
\(33\) 3168.00 0.506408
\(34\) 0 0
\(35\) 4300.00 0.593333
\(36\) 0 0
\(37\) −11242.0 −1.35002 −0.675009 0.737810i \(-0.735860\pi\)
−0.675009 + 0.737810i \(0.735860\pi\)
\(38\) 0 0
\(39\) 22704.0 2.39024
\(40\) 0 0
\(41\) 1242.00 0.115388 0.0576942 0.998334i \(-0.481625\pi\)
0.0576942 + 0.998334i \(0.481625\pi\)
\(42\) 0 0
\(43\) −20624.0 −1.70099 −0.850495 0.525983i \(-0.823697\pi\)
−0.850495 + 0.525983i \(0.823697\pi\)
\(44\) 0 0
\(45\) 8325.00 0.612848
\(46\) 0 0
\(47\) −6588.00 −0.435020 −0.217510 0.976058i \(-0.569793\pi\)
−0.217510 + 0.976058i \(0.569793\pi\)
\(48\) 0 0
\(49\) 12777.0 0.760219
\(50\) 0 0
\(51\) 5328.00 0.286839
\(52\) 0 0
\(53\) −21066.0 −1.03013 −0.515065 0.857151i \(-0.672232\pi\)
−0.515065 + 0.857151i \(0.672232\pi\)
\(54\) 0 0
\(55\) −3300.00 −0.147098
\(56\) 0 0
\(57\) 12000.0 0.489209
\(58\) 0 0
\(59\) −7980.00 −0.298451 −0.149225 0.988803i \(-0.547678\pi\)
−0.149225 + 0.988803i \(0.547678\pi\)
\(60\) 0 0
\(61\) 16622.0 0.571951 0.285975 0.958237i \(-0.407682\pi\)
0.285975 + 0.958237i \(0.407682\pi\)
\(62\) 0 0
\(63\) 57276.0 1.81811
\(64\) 0 0
\(65\) −23650.0 −0.694301
\(66\) 0 0
\(67\) −1808.00 −0.0492052 −0.0246026 0.999697i \(-0.507832\pi\)
−0.0246026 + 0.999697i \(0.507832\pi\)
\(68\) 0 0
\(69\) 85536.0 2.16285
\(70\) 0 0
\(71\) 24528.0 0.577452 0.288726 0.957412i \(-0.406768\pi\)
0.288726 + 0.957412i \(0.406768\pi\)
\(72\) 0 0
\(73\) 20474.0 0.449672 0.224836 0.974397i \(-0.427815\pi\)
0.224836 + 0.974397i \(0.427815\pi\)
\(74\) 0 0
\(75\) −15000.0 −0.307920
\(76\) 0 0
\(77\) −22704.0 −0.436391
\(78\) 0 0
\(79\) 46240.0 0.833585 0.416793 0.909002i \(-0.363154\pi\)
0.416793 + 0.909002i \(0.363154\pi\)
\(80\) 0 0
\(81\) −29079.0 −0.492455
\(82\) 0 0
\(83\) 51576.0 0.821774 0.410887 0.911686i \(-0.365219\pi\)
0.410887 + 0.911686i \(0.365219\pi\)
\(84\) 0 0
\(85\) −5550.00 −0.0833193
\(86\) 0 0
\(87\) −52560.0 −0.744487
\(88\) 0 0
\(89\) −110310. −1.47618 −0.738091 0.674701i \(-0.764272\pi\)
−0.738091 + 0.674701i \(0.764272\pi\)
\(90\) 0 0
\(91\) −162712. −2.05976
\(92\) 0 0
\(93\) 55488.0 0.665260
\(94\) 0 0
\(95\) −12500.0 −0.142102
\(96\) 0 0
\(97\) −78382.0 −0.845838 −0.422919 0.906168i \(-0.638994\pi\)
−0.422919 + 0.906168i \(0.638994\pi\)
\(98\) 0 0
\(99\) −43956.0 −0.450744
\(100\) 0 0
\(101\) 141942. 1.38455 0.692273 0.721636i \(-0.256609\pi\)
0.692273 + 0.721636i \(0.256609\pi\)
\(102\) 0 0
\(103\) 436.000 0.00404943 0.00202471 0.999998i \(-0.499356\pi\)
0.00202471 + 0.999998i \(0.499356\pi\)
\(104\) 0 0
\(105\) −103200. −0.913496
\(106\) 0 0
\(107\) −232968. −1.96715 −0.983574 0.180508i \(-0.942226\pi\)
−0.983574 + 0.180508i \(0.942226\pi\)
\(108\) 0 0
\(109\) −174850. −1.40961 −0.704806 0.709400i \(-0.748966\pi\)
−0.704806 + 0.709400i \(0.748966\pi\)
\(110\) 0 0
\(111\) 269808. 2.07849
\(112\) 0 0
\(113\) 182994. 1.34816 0.674079 0.738659i \(-0.264541\pi\)
0.674079 + 0.738659i \(0.264541\pi\)
\(114\) 0 0
\(115\) −89100.0 −0.628251
\(116\) 0 0
\(117\) −315018. −2.12751
\(118\) 0 0
\(119\) −38184.0 −0.247180
\(120\) 0 0
\(121\) −143627. −0.891811
\(122\) 0 0
\(123\) −29808.0 −0.177652
\(124\) 0 0
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) 122452. 0.673685 0.336842 0.941561i \(-0.390641\pi\)
0.336842 + 0.941561i \(0.390641\pi\)
\(128\) 0 0
\(129\) 494976. 2.61885
\(130\) 0 0
\(131\) 241908. 1.23161 0.615803 0.787900i \(-0.288832\pi\)
0.615803 + 0.787900i \(0.288832\pi\)
\(132\) 0 0
\(133\) −86000.0 −0.421570
\(134\) 0 0
\(135\) −54000.0 −0.255011
\(136\) 0 0
\(137\) 277098. 1.26134 0.630670 0.776051i \(-0.282780\pi\)
0.630670 + 0.776051i \(0.282780\pi\)
\(138\) 0 0
\(139\) 193540. 0.849638 0.424819 0.905278i \(-0.360338\pi\)
0.424819 + 0.905278i \(0.360338\pi\)
\(140\) 0 0
\(141\) 158112. 0.669757
\(142\) 0 0
\(143\) 124872. 0.510652
\(144\) 0 0
\(145\) 54750.0 0.216254
\(146\) 0 0
\(147\) −306648. −1.17043
\(148\) 0 0
\(149\) 140550. 0.518639 0.259320 0.965792i \(-0.416502\pi\)
0.259320 + 0.965792i \(0.416502\pi\)
\(150\) 0 0
\(151\) −433952. −1.54881 −0.774407 0.632688i \(-0.781952\pi\)
−0.774407 + 0.632688i \(0.781952\pi\)
\(152\) 0 0
\(153\) −73926.0 −0.255310
\(154\) 0 0
\(155\) −57800.0 −0.193241
\(156\) 0 0
\(157\) −555922. −1.79997 −0.899984 0.435923i \(-0.856422\pi\)
−0.899984 + 0.435923i \(0.856422\pi\)
\(158\) 0 0
\(159\) 505584. 1.58599
\(160\) 0 0
\(161\) −613008. −1.86381
\(162\) 0 0
\(163\) 66616.0 0.196386 0.0981928 0.995167i \(-0.468694\pi\)
0.0981928 + 0.995167i \(0.468694\pi\)
\(164\) 0 0
\(165\) 79200.0 0.226472
\(166\) 0 0
\(167\) 205692. 0.570724 0.285362 0.958420i \(-0.407886\pi\)
0.285362 + 0.958420i \(0.407886\pi\)
\(168\) 0 0
\(169\) 523623. 1.41027
\(170\) 0 0
\(171\) −166500. −0.435436
\(172\) 0 0
\(173\) 433854. 1.10212 0.551059 0.834466i \(-0.314224\pi\)
0.551059 + 0.834466i \(0.314224\pi\)
\(174\) 0 0
\(175\) 107500. 0.265346
\(176\) 0 0
\(177\) 191520. 0.459495
\(178\) 0 0
\(179\) 489180. 1.14113 0.570566 0.821252i \(-0.306724\pi\)
0.570566 + 0.821252i \(0.306724\pi\)
\(180\) 0 0
\(181\) 719462. 1.63234 0.816172 0.577810i \(-0.196092\pi\)
0.816172 + 0.577810i \(0.196092\pi\)
\(182\) 0 0
\(183\) −398928. −0.880576
\(184\) 0 0
\(185\) −281050. −0.603746
\(186\) 0 0
\(187\) 29304.0 0.0612806
\(188\) 0 0
\(189\) −371520. −0.756533
\(190\) 0 0
\(191\) 185928. 0.368775 0.184387 0.982854i \(-0.440970\pi\)
0.184387 + 0.982854i \(0.440970\pi\)
\(192\) 0 0
\(193\) −591406. −1.14286 −0.571429 0.820651i \(-0.693611\pi\)
−0.571429 + 0.820651i \(0.693611\pi\)
\(194\) 0 0
\(195\) 567600. 1.06895
\(196\) 0 0
\(197\) 449478. 0.825169 0.412584 0.910919i \(-0.364626\pi\)
0.412584 + 0.910919i \(0.364626\pi\)
\(198\) 0 0
\(199\) −157160. −0.281326 −0.140663 0.990058i \(-0.544923\pi\)
−0.140663 + 0.990058i \(0.544923\pi\)
\(200\) 0 0
\(201\) 43392.0 0.0757564
\(202\) 0 0
\(203\) 376680. 0.641553
\(204\) 0 0
\(205\) 31050.0 0.0516032
\(206\) 0 0
\(207\) −1.18681e6 −1.92511
\(208\) 0 0
\(209\) 66000.0 0.104515
\(210\) 0 0
\(211\) −253052. −0.391294 −0.195647 0.980674i \(-0.562681\pi\)
−0.195647 + 0.980674i \(0.562681\pi\)
\(212\) 0 0
\(213\) −588672. −0.889046
\(214\) 0 0
\(215\) −515600. −0.760706
\(216\) 0 0
\(217\) −397664. −0.573280
\(218\) 0 0
\(219\) −491376. −0.692315
\(220\) 0 0
\(221\) 210012. 0.289243
\(222\) 0 0
\(223\) −1.07344e6 −1.44550 −0.722749 0.691111i \(-0.757122\pi\)
−0.722749 + 0.691111i \(0.757122\pi\)
\(224\) 0 0
\(225\) 208125. 0.274074
\(226\) 0 0
\(227\) 626832. 0.807396 0.403698 0.914892i \(-0.367725\pi\)
0.403698 + 0.914892i \(0.367725\pi\)
\(228\) 0 0
\(229\) −116650. −0.146993 −0.0734964 0.997295i \(-0.523416\pi\)
−0.0734964 + 0.997295i \(0.523416\pi\)
\(230\) 0 0
\(231\) 544896. 0.671868
\(232\) 0 0
\(233\) −743046. −0.896656 −0.448328 0.893869i \(-0.647980\pi\)
−0.448328 + 0.893869i \(0.647980\pi\)
\(234\) 0 0
\(235\) −164700. −0.194547
\(236\) 0 0
\(237\) −1.10976e6 −1.28339
\(238\) 0 0
\(239\) −978720. −1.10832 −0.554158 0.832411i \(-0.686960\pi\)
−0.554158 + 0.832411i \(0.686960\pi\)
\(240\) 0 0
\(241\) −1.13280e6 −1.25635 −0.628174 0.778073i \(-0.716197\pi\)
−0.628174 + 0.778073i \(0.716197\pi\)
\(242\) 0 0
\(243\) 1.22278e6 1.32841
\(244\) 0 0
\(245\) 319425. 0.339980
\(246\) 0 0
\(247\) 473000. 0.493309
\(248\) 0 0
\(249\) −1.23782e6 −1.26520
\(250\) 0 0
\(251\) −905652. −0.907355 −0.453677 0.891166i \(-0.649888\pi\)
−0.453677 + 0.891166i \(0.649888\pi\)
\(252\) 0 0
\(253\) 470448. 0.462073
\(254\) 0 0
\(255\) 133200. 0.128278
\(256\) 0 0
\(257\) 1.93994e6 1.83212 0.916062 0.401036i \(-0.131350\pi\)
0.916062 + 0.401036i \(0.131350\pi\)
\(258\) 0 0
\(259\) −1.93362e6 −1.79111
\(260\) 0 0
\(261\) 729270. 0.662654
\(262\) 0 0
\(263\) 805476. 0.718064 0.359032 0.933325i \(-0.383107\pi\)
0.359032 + 0.933325i \(0.383107\pi\)
\(264\) 0 0
\(265\) −526650. −0.460689
\(266\) 0 0
\(267\) 2.64744e6 2.27273
\(268\) 0 0
\(269\) −858690. −0.723529 −0.361764 0.932270i \(-0.617825\pi\)
−0.361764 + 0.932270i \(0.617825\pi\)
\(270\) 0 0
\(271\) 383608. 0.317296 0.158648 0.987335i \(-0.449287\pi\)
0.158648 + 0.987335i \(0.449287\pi\)
\(272\) 0 0
\(273\) 3.90509e6 3.17120
\(274\) 0 0
\(275\) −82500.0 −0.0657843
\(276\) 0 0
\(277\) 2.01076e6 1.57456 0.787282 0.616593i \(-0.211488\pi\)
0.787282 + 0.616593i \(0.211488\pi\)
\(278\) 0 0
\(279\) −769896. −0.592136
\(280\) 0 0
\(281\) 202602. 0.153066 0.0765329 0.997067i \(-0.475615\pi\)
0.0765329 + 0.997067i \(0.475615\pi\)
\(282\) 0 0
\(283\) 221536. 0.164429 0.0822145 0.996615i \(-0.473801\pi\)
0.0822145 + 0.996615i \(0.473801\pi\)
\(284\) 0 0
\(285\) 300000. 0.218781
\(286\) 0 0
\(287\) 213624. 0.153089
\(288\) 0 0
\(289\) −1.37057e6 −0.965289
\(290\) 0 0
\(291\) 1.88117e6 1.30225
\(292\) 0 0
\(293\) −322506. −0.219467 −0.109733 0.993961i \(-0.535000\pi\)
−0.109733 + 0.993961i \(0.535000\pi\)
\(294\) 0 0
\(295\) −199500. −0.133471
\(296\) 0 0
\(297\) 285120. 0.187558
\(298\) 0 0
\(299\) 3.37154e6 2.18098
\(300\) 0 0
\(301\) −3.54733e6 −2.25676
\(302\) 0 0
\(303\) −3.40661e6 −2.13165
\(304\) 0 0
\(305\) 415550. 0.255784
\(306\) 0 0
\(307\) −1.44301e6 −0.873822 −0.436911 0.899505i \(-0.643927\pi\)
−0.436911 + 0.899505i \(0.643927\pi\)
\(308\) 0 0
\(309\) −10464.0 −0.00623450
\(310\) 0 0
\(311\) −171312. −0.100435 −0.0502177 0.998738i \(-0.515992\pi\)
−0.0502177 + 0.998738i \(0.515992\pi\)
\(312\) 0 0
\(313\) −1.02689e6 −0.592463 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(314\) 0 0
\(315\) 1.43190e6 0.813086
\(316\) 0 0
\(317\) 752958. 0.420845 0.210423 0.977610i \(-0.432516\pi\)
0.210423 + 0.977610i \(0.432516\pi\)
\(318\) 0 0
\(319\) −289080. −0.159053
\(320\) 0 0
\(321\) 5.59123e6 3.02862
\(322\) 0 0
\(323\) 111000. 0.0591993
\(324\) 0 0
\(325\) −591250. −0.310501
\(326\) 0 0
\(327\) 4.19640e6 2.17024
\(328\) 0 0
\(329\) −1.13314e6 −0.577155
\(330\) 0 0
\(331\) −1.99413e6 −1.00042 −0.500212 0.865903i \(-0.666745\pi\)
−0.500212 + 0.865903i \(0.666745\pi\)
\(332\) 0 0
\(333\) −3.74359e6 −1.85002
\(334\) 0 0
\(335\) −45200.0 −0.0220053
\(336\) 0 0
\(337\) −987022. −0.473426 −0.236713 0.971580i \(-0.576070\pi\)
−0.236713 + 0.971580i \(0.576070\pi\)
\(338\) 0 0
\(339\) −4.39186e6 −2.07562
\(340\) 0 0
\(341\) 305184. 0.142127
\(342\) 0 0
\(343\) −693160. −0.318125
\(344\) 0 0
\(345\) 2.13840e6 0.967256
\(346\) 0 0
\(347\) −2.20601e6 −0.983520 −0.491760 0.870731i \(-0.663646\pi\)
−0.491760 + 0.870731i \(0.663646\pi\)
\(348\) 0 0
\(349\) 2.74187e6 1.20499 0.602495 0.798123i \(-0.294173\pi\)
0.602495 + 0.798123i \(0.294173\pi\)
\(350\) 0 0
\(351\) 2.04336e6 0.885273
\(352\) 0 0
\(353\) −2.38957e6 −1.02066 −0.510331 0.859978i \(-0.670477\pi\)
−0.510331 + 0.859978i \(0.670477\pi\)
\(354\) 0 0
\(355\) 613200. 0.258245
\(356\) 0 0
\(357\) 916416. 0.380559
\(358\) 0 0
\(359\) 279480. 0.114450 0.0572248 0.998361i \(-0.481775\pi\)
0.0572248 + 0.998361i \(0.481775\pi\)
\(360\) 0 0
\(361\) −2.22610e6 −0.899035
\(362\) 0 0
\(363\) 3.44705e6 1.37303
\(364\) 0 0
\(365\) 511850. 0.201099
\(366\) 0 0
\(367\) 2.47637e6 0.959734 0.479867 0.877341i \(-0.340685\pi\)
0.479867 + 0.877341i \(0.340685\pi\)
\(368\) 0 0
\(369\) 413586. 0.158125
\(370\) 0 0
\(371\) −3.62335e6 −1.36671
\(372\) 0 0
\(373\) 2.74525e6 1.02167 0.510835 0.859679i \(-0.329336\pi\)
0.510835 + 0.859679i \(0.329336\pi\)
\(374\) 0 0
\(375\) −375000. −0.137706
\(376\) 0 0
\(377\) −2.07174e6 −0.750727
\(378\) 0 0
\(379\) 1.18906e6 0.425212 0.212606 0.977138i \(-0.431805\pi\)
0.212606 + 0.977138i \(0.431805\pi\)
\(380\) 0 0
\(381\) −2.93885e6 −1.03721
\(382\) 0 0
\(383\) −3.25760e6 −1.13475 −0.567377 0.823458i \(-0.692042\pi\)
−0.567377 + 0.823458i \(0.692042\pi\)
\(384\) 0 0
\(385\) −567600. −0.195160
\(386\) 0 0
\(387\) −6.86779e6 −2.33099
\(388\) 0 0
\(389\) 1.98351e6 0.664600 0.332300 0.943174i \(-0.392175\pi\)
0.332300 + 0.943174i \(0.392175\pi\)
\(390\) 0 0
\(391\) 791208. 0.261727
\(392\) 0 0
\(393\) −5.80579e6 −1.89618
\(394\) 0 0
\(395\) 1.15600e6 0.372791
\(396\) 0 0
\(397\) 4.97416e6 1.58396 0.791978 0.610549i \(-0.209051\pi\)
0.791978 + 0.610549i \(0.209051\pi\)
\(398\) 0 0
\(399\) 2.06400e6 0.649049
\(400\) 0 0
\(401\) −1.34264e6 −0.416963 −0.208482 0.978026i \(-0.566852\pi\)
−0.208482 + 0.978026i \(0.566852\pi\)
\(402\) 0 0
\(403\) 2.18715e6 0.670836
\(404\) 0 0
\(405\) −726975. −0.220233
\(406\) 0 0
\(407\) 1.48394e6 0.444050
\(408\) 0 0
\(409\) −1.09423e6 −0.323445 −0.161722 0.986836i \(-0.551705\pi\)
−0.161722 + 0.986836i \(0.551705\pi\)
\(410\) 0 0
\(411\) −6.65035e6 −1.94196
\(412\) 0 0
\(413\) −1.37256e6 −0.395964
\(414\) 0 0
\(415\) 1.28940e6 0.367509
\(416\) 0 0
\(417\) −4.64496e6 −1.30810
\(418\) 0 0
\(419\) 954060. 0.265485 0.132743 0.991151i \(-0.457622\pi\)
0.132743 + 0.991151i \(0.457622\pi\)
\(420\) 0 0
\(421\) −1.59390e6 −0.438284 −0.219142 0.975693i \(-0.570326\pi\)
−0.219142 + 0.975693i \(0.570326\pi\)
\(422\) 0 0
\(423\) −2.19380e6 −0.596138
\(424\) 0 0
\(425\) −138750. −0.0372615
\(426\) 0 0
\(427\) 2.85898e6 0.758826
\(428\) 0 0
\(429\) −2.99693e6 −0.786200
\(430\) 0 0
\(431\) 2.64665e6 0.686283 0.343141 0.939284i \(-0.388509\pi\)
0.343141 + 0.939284i \(0.388509\pi\)
\(432\) 0 0
\(433\) 3.72355e6 0.954416 0.477208 0.878790i \(-0.341649\pi\)
0.477208 + 0.878790i \(0.341649\pi\)
\(434\) 0 0
\(435\) −1.31400e6 −0.332945
\(436\) 0 0
\(437\) 1.78200e6 0.446379
\(438\) 0 0
\(439\) 2.58340e6 0.639780 0.319890 0.947455i \(-0.396354\pi\)
0.319890 + 0.947455i \(0.396354\pi\)
\(440\) 0 0
\(441\) 4.25474e6 1.04178
\(442\) 0 0
\(443\) −7.56206e6 −1.83076 −0.915379 0.402593i \(-0.868109\pi\)
−0.915379 + 0.402593i \(0.868109\pi\)
\(444\) 0 0
\(445\) −2.75775e6 −0.660169
\(446\) 0 0
\(447\) −3.37320e6 −0.798497
\(448\) 0 0
\(449\) 4.30773e6 1.00840 0.504200 0.863587i \(-0.331788\pi\)
0.504200 + 0.863587i \(0.331788\pi\)
\(450\) 0 0
\(451\) −163944. −0.0379537
\(452\) 0 0
\(453\) 1.04148e7 2.38456
\(454\) 0 0
\(455\) −4.06780e6 −0.921152
\(456\) 0 0
\(457\) −2.24354e6 −0.502509 −0.251254 0.967921i \(-0.580843\pi\)
−0.251254 + 0.967921i \(0.580843\pi\)
\(458\) 0 0
\(459\) 479520. 0.106237
\(460\) 0 0
\(461\) 1.65670e6 0.363071 0.181536 0.983384i \(-0.441893\pi\)
0.181536 + 0.983384i \(0.441893\pi\)
\(462\) 0 0
\(463\) 2.89160e6 0.626881 0.313441 0.949608i \(-0.398518\pi\)
0.313441 + 0.949608i \(0.398518\pi\)
\(464\) 0 0
\(465\) 1.38720e6 0.297514
\(466\) 0 0
\(467\) 6.52699e6 1.38491 0.692454 0.721462i \(-0.256530\pi\)
0.692454 + 0.721462i \(0.256530\pi\)
\(468\) 0 0
\(469\) −310976. −0.0652822
\(470\) 0 0
\(471\) 1.33421e7 2.77123
\(472\) 0 0
\(473\) 2.72237e6 0.559492
\(474\) 0 0
\(475\) −312500. −0.0635501
\(476\) 0 0
\(477\) −7.01498e6 −1.41166
\(478\) 0 0
\(479\) 5.96232e6 1.18734 0.593672 0.804707i \(-0.297678\pi\)
0.593672 + 0.804707i \(0.297678\pi\)
\(480\) 0 0
\(481\) 1.06349e7 2.09591
\(482\) 0 0
\(483\) 1.47122e7 2.86952
\(484\) 0 0
\(485\) −1.95955e6 −0.378270
\(486\) 0 0
\(487\) −2.99191e6 −0.571644 −0.285822 0.958283i \(-0.592267\pi\)
−0.285822 + 0.958283i \(0.592267\pi\)
\(488\) 0 0
\(489\) −1.59878e6 −0.302355
\(490\) 0 0
\(491\) 1.20419e6 0.225419 0.112710 0.993628i \(-0.464047\pi\)
0.112710 + 0.993628i \(0.464047\pi\)
\(492\) 0 0
\(493\) −486180. −0.0900907
\(494\) 0 0
\(495\) −1.09890e6 −0.201579
\(496\) 0 0
\(497\) 4.21882e6 0.766125
\(498\) 0 0
\(499\) −9.20546e6 −1.65499 −0.827493 0.561477i \(-0.810233\pi\)
−0.827493 + 0.561477i \(0.810233\pi\)
\(500\) 0 0
\(501\) −4.93661e6 −0.878687
\(502\) 0 0
\(503\) 3.35956e6 0.592055 0.296027 0.955179i \(-0.404338\pi\)
0.296027 + 0.955179i \(0.404338\pi\)
\(504\) 0 0
\(505\) 3.54855e6 0.619188
\(506\) 0 0
\(507\) −1.25670e7 −2.17125
\(508\) 0 0
\(509\) −2.53701e6 −0.434038 −0.217019 0.976167i \(-0.569633\pi\)
−0.217019 + 0.976167i \(0.569633\pi\)
\(510\) 0 0
\(511\) 3.52153e6 0.596594
\(512\) 0 0
\(513\) 1.08000e6 0.181188
\(514\) 0 0
\(515\) 10900.0 0.00181096
\(516\) 0 0
\(517\) 869616. 0.143087
\(518\) 0 0
\(519\) −1.04125e7 −1.69682
\(520\) 0 0
\(521\) −9.31580e6 −1.50358 −0.751789 0.659404i \(-0.770809\pi\)
−0.751789 + 0.659404i \(0.770809\pi\)
\(522\) 0 0
\(523\) 5.02802e6 0.803790 0.401895 0.915686i \(-0.368352\pi\)
0.401895 + 0.915686i \(0.368352\pi\)
\(524\) 0 0
\(525\) −2.58000e6 −0.408528
\(526\) 0 0
\(527\) 513264. 0.0805034
\(528\) 0 0
\(529\) 6.26575e6 0.973496
\(530\) 0 0
\(531\) −2.65734e6 −0.408988
\(532\) 0 0
\(533\) −1.17493e6 −0.179141
\(534\) 0 0
\(535\) −5.82420e6 −0.879735
\(536\) 0 0
\(537\) −1.17403e7 −1.75689
\(538\) 0 0
\(539\) −1.68656e6 −0.250052
\(540\) 0 0
\(541\) 134222. 0.0197165 0.00985827 0.999951i \(-0.496862\pi\)
0.00985827 + 0.999951i \(0.496862\pi\)
\(542\) 0 0
\(543\) −1.72671e7 −2.51316
\(544\) 0 0
\(545\) −4.37125e6 −0.630397
\(546\) 0 0
\(547\) −605648. −0.0865470 −0.0432735 0.999063i \(-0.513779\pi\)
−0.0432735 + 0.999063i \(0.513779\pi\)
\(548\) 0 0
\(549\) 5.53513e6 0.783784
\(550\) 0 0
\(551\) −1.09500e6 −0.153651
\(552\) 0 0
\(553\) 7.95328e6 1.10594
\(554\) 0 0
\(555\) 6.74520e6 0.929528
\(556\) 0 0
\(557\) −7.06240e6 −0.964527 −0.482264 0.876026i \(-0.660185\pi\)
−0.482264 + 0.876026i \(0.660185\pi\)
\(558\) 0 0
\(559\) 1.95103e7 2.64079
\(560\) 0 0
\(561\) −703296. −0.0943476
\(562\) 0 0
\(563\) 1.03029e7 1.36990 0.684952 0.728588i \(-0.259823\pi\)
0.684952 + 0.728588i \(0.259823\pi\)
\(564\) 0 0
\(565\) 4.57485e6 0.602915
\(566\) 0 0
\(567\) −5.00159e6 −0.653357
\(568\) 0 0
\(569\) 1.04769e6 0.135660 0.0678300 0.997697i \(-0.478392\pi\)
0.0678300 + 0.997697i \(0.478392\pi\)
\(570\) 0 0
\(571\) −1.40765e7 −1.80677 −0.903385 0.428830i \(-0.858926\pi\)
−0.903385 + 0.428830i \(0.858926\pi\)
\(572\) 0 0
\(573\) −4.46227e6 −0.567766
\(574\) 0 0
\(575\) −2.22750e6 −0.280962
\(576\) 0 0
\(577\) 1.62682e6 0.203423 0.101711 0.994814i \(-0.467568\pi\)
0.101711 + 0.994814i \(0.467568\pi\)
\(578\) 0 0
\(579\) 1.41937e7 1.75955
\(580\) 0 0
\(581\) 8.87107e6 1.09027
\(582\) 0 0
\(583\) 2.78071e6 0.338832
\(584\) 0 0
\(585\) −7.87545e6 −0.951449
\(586\) 0 0
\(587\) −6.96089e6 −0.833814 −0.416907 0.908949i \(-0.636886\pi\)
−0.416907 + 0.908949i \(0.636886\pi\)
\(588\) 0 0
\(589\) 1.15600e6 0.137300
\(590\) 0 0
\(591\) −1.07875e7 −1.27043
\(592\) 0 0
\(593\) −1.13639e7 −1.32706 −0.663529 0.748150i \(-0.730942\pi\)
−0.663529 + 0.748150i \(0.730942\pi\)
\(594\) 0 0
\(595\) −954600. −0.110542
\(596\) 0 0
\(597\) 3.77184e6 0.433129
\(598\) 0 0
\(599\) −1.48688e7 −1.69321 −0.846603 0.532224i \(-0.821356\pi\)
−0.846603 + 0.532224i \(0.821356\pi\)
\(600\) 0 0
\(601\) −1.23612e6 −0.139596 −0.0697981 0.997561i \(-0.522236\pi\)
−0.0697981 + 0.997561i \(0.522236\pi\)
\(602\) 0 0
\(603\) −602064. −0.0674294
\(604\) 0 0
\(605\) −3.59068e6 −0.398830
\(606\) 0 0
\(607\) 1.24498e7 1.37149 0.685743 0.727844i \(-0.259478\pi\)
0.685743 + 0.727844i \(0.259478\pi\)
\(608\) 0 0
\(609\) −9.04032e6 −0.987735
\(610\) 0 0
\(611\) 6.23225e6 0.675370
\(612\) 0 0
\(613\) −8.73491e6 −0.938873 −0.469437 0.882966i \(-0.655543\pi\)
−0.469437 + 0.882966i \(0.655543\pi\)
\(614\) 0 0
\(615\) −745200. −0.0794484
\(616\) 0 0
\(617\) 1.25495e7 1.32713 0.663565 0.748119i \(-0.269043\pi\)
0.663565 + 0.748119i \(0.269043\pi\)
\(618\) 0 0
\(619\) 1.46658e7 1.53843 0.769216 0.638988i \(-0.220647\pi\)
0.769216 + 0.638988i \(0.220647\pi\)
\(620\) 0 0
\(621\) 7.69824e6 0.801055
\(622\) 0 0
\(623\) −1.89733e7 −1.95850
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) −1.58400e6 −0.160911
\(628\) 0 0
\(629\) 2.49572e6 0.251519
\(630\) 0 0
\(631\) 196288. 0.0196255 0.00981274 0.999952i \(-0.496876\pi\)
0.00981274 + 0.999952i \(0.496876\pi\)
\(632\) 0 0
\(633\) 6.07325e6 0.602437
\(634\) 0 0
\(635\) 3.06130e6 0.301281
\(636\) 0 0
\(637\) −1.20870e7 −1.18024
\(638\) 0 0
\(639\) 8.16782e6 0.791324
\(640\) 0 0
\(641\) −1.11596e7 −1.07276 −0.536381 0.843976i \(-0.680209\pi\)
−0.536381 + 0.843976i \(0.680209\pi\)
\(642\) 0 0
\(643\) 2.25158e6 0.214763 0.107381 0.994218i \(-0.465753\pi\)
0.107381 + 0.994218i \(0.465753\pi\)
\(644\) 0 0
\(645\) 1.23744e7 1.17118
\(646\) 0 0
\(647\) −8.05319e6 −0.756323 −0.378161 0.925740i \(-0.623444\pi\)
−0.378161 + 0.925740i \(0.623444\pi\)
\(648\) 0 0
\(649\) 1.05336e6 0.0981669
\(650\) 0 0
\(651\) 9.54394e6 0.882623
\(652\) 0 0
\(653\) −416466. −0.0382205 −0.0191103 0.999817i \(-0.506083\pi\)
−0.0191103 + 0.999817i \(0.506083\pi\)
\(654\) 0 0
\(655\) 6.04770e6 0.550791
\(656\) 0 0
\(657\) 6.81784e6 0.616217
\(658\) 0 0
\(659\) −1.31721e7 −1.18152 −0.590761 0.806847i \(-0.701172\pi\)
−0.590761 + 0.806847i \(0.701172\pi\)
\(660\) 0 0
\(661\) −1.69494e6 −0.150886 −0.0754432 0.997150i \(-0.524037\pi\)
−0.0754432 + 0.997150i \(0.524037\pi\)
\(662\) 0 0
\(663\) −5.04029e6 −0.445319
\(664\) 0 0
\(665\) −2.15000e6 −0.188532
\(666\) 0 0
\(667\) −7.80516e6 −0.679309
\(668\) 0 0
\(669\) 2.57627e7 2.22549
\(670\) 0 0
\(671\) −2.19410e6 −0.188127
\(672\) 0 0
\(673\) −8.91605e6 −0.758813 −0.379406 0.925230i \(-0.623872\pi\)
−0.379406 + 0.925230i \(0.623872\pi\)
\(674\) 0 0
\(675\) −1.35000e6 −0.114044
\(676\) 0 0
\(677\) −1.42894e7 −1.19824 −0.599118 0.800661i \(-0.704482\pi\)
−0.599118 + 0.800661i \(0.704482\pi\)
\(678\) 0 0
\(679\) −1.34817e7 −1.12220
\(680\) 0 0
\(681\) −1.50440e7 −1.24307
\(682\) 0 0
\(683\) 5.33314e6 0.437452 0.218726 0.975786i \(-0.429810\pi\)
0.218726 + 0.975786i \(0.429810\pi\)
\(684\) 0 0
\(685\) 6.92745e6 0.564088
\(686\) 0 0
\(687\) 2.79960e6 0.226310
\(688\) 0 0
\(689\) 1.99284e7 1.59928
\(690\) 0 0
\(691\) −698252. −0.0556310 −0.0278155 0.999613i \(-0.508855\pi\)
−0.0278155 + 0.999613i \(0.508855\pi\)
\(692\) 0 0
\(693\) −7.56043e6 −0.598017
\(694\) 0 0
\(695\) 4.83850e6 0.379969
\(696\) 0 0
\(697\) −275724. −0.0214977
\(698\) 0 0
\(699\) 1.78331e7 1.38049
\(700\) 0 0
\(701\) 1.79880e7 1.38257 0.691285 0.722582i \(-0.257045\pi\)
0.691285 + 0.722582i \(0.257045\pi\)
\(702\) 0 0
\(703\) 5.62100e6 0.428968
\(704\) 0 0
\(705\) 3.95280e6 0.299524
\(706\) 0 0
\(707\) 2.44140e7 1.83692
\(708\) 0 0
\(709\) −1.39464e7 −1.04195 −0.520975 0.853572i \(-0.674432\pi\)
−0.520975 + 0.853572i \(0.674432\pi\)
\(710\) 0 0
\(711\) 1.53979e7 1.14232
\(712\) 0 0
\(713\) 8.23997e6 0.607018
\(714\) 0 0
\(715\) 3.12180e6 0.228370
\(716\) 0 0
\(717\) 2.34893e7 1.70636
\(718\) 0 0
\(719\) −6.22272e6 −0.448909 −0.224454 0.974485i \(-0.572060\pi\)
−0.224454 + 0.974485i \(0.572060\pi\)
\(720\) 0 0
\(721\) 74992.0 0.00537250
\(722\) 0 0
\(723\) 2.71872e7 1.93427
\(724\) 0 0
\(725\) 1.36875e6 0.0967117
\(726\) 0 0
\(727\) 7.76729e6 0.545047 0.272523 0.962149i \(-0.412142\pi\)
0.272523 + 0.962149i \(0.412142\pi\)
\(728\) 0 0
\(729\) −2.22804e7 −1.55276
\(730\) 0 0
\(731\) 4.57853e6 0.316907
\(732\) 0 0
\(733\) 2.42083e7 1.66420 0.832099 0.554627i \(-0.187139\pi\)
0.832099 + 0.554627i \(0.187139\pi\)
\(734\) 0 0
\(735\) −7.66620e6 −0.523434
\(736\) 0 0
\(737\) 238656. 0.0161847
\(738\) 0 0
\(739\) −1.26850e7 −0.854434 −0.427217 0.904149i \(-0.640506\pi\)
−0.427217 + 0.904149i \(0.640506\pi\)
\(740\) 0 0
\(741\) −1.13520e7 −0.759498
\(742\) 0 0
\(743\) −1.97632e7 −1.31337 −0.656684 0.754166i \(-0.728041\pi\)
−0.656684 + 0.754166i \(0.728041\pi\)
\(744\) 0 0
\(745\) 3.51375e6 0.231942
\(746\) 0 0
\(747\) 1.71748e7 1.12613
\(748\) 0 0
\(749\) −4.00705e7 −2.60988
\(750\) 0 0
\(751\) 9.01761e6 0.583434 0.291717 0.956505i \(-0.405774\pi\)
0.291717 + 0.956505i \(0.405774\pi\)
\(752\) 0 0
\(753\) 2.17356e7 1.39696
\(754\) 0 0
\(755\) −1.08488e7 −0.692651
\(756\) 0 0
\(757\) −1.12556e6 −0.0713887 −0.0356944 0.999363i \(-0.511364\pi\)
−0.0356944 + 0.999363i \(0.511364\pi\)
\(758\) 0 0
\(759\) −1.12908e7 −0.711407
\(760\) 0 0
\(761\) 2.25747e7 1.41306 0.706529 0.707684i \(-0.250260\pi\)
0.706529 + 0.707684i \(0.250260\pi\)
\(762\) 0 0
\(763\) −3.00742e7 −1.87018
\(764\) 0 0
\(765\) −1.84815e6 −0.114178
\(766\) 0 0
\(767\) 7.54908e6 0.463346
\(768\) 0 0
\(769\) −632350. −0.0385604 −0.0192802 0.999814i \(-0.506137\pi\)
−0.0192802 + 0.999814i \(0.506137\pi\)
\(770\) 0 0
\(771\) −4.65585e7 −2.82074
\(772\) 0 0
\(773\) −1.25867e7 −0.757643 −0.378822 0.925470i \(-0.623671\pi\)
−0.378822 + 0.925470i \(0.623671\pi\)
\(774\) 0 0
\(775\) −1.44500e6 −0.0864199
\(776\) 0 0
\(777\) 4.64070e7 2.75760
\(778\) 0 0
\(779\) −621000. −0.0366647
\(780\) 0 0
\(781\) −3.23770e6 −0.189937
\(782\) 0 0
\(783\) −4.73040e6 −0.275736
\(784\) 0 0
\(785\) −1.38981e7 −0.804970
\(786\) 0 0
\(787\) −2.15792e7 −1.24194 −0.620968 0.783836i \(-0.713260\pi\)
−0.620968 + 0.783836i \(0.713260\pi\)
\(788\) 0 0
\(789\) −1.93314e7 −1.10553
\(790\) 0 0
\(791\) 3.14750e7 1.78864
\(792\) 0 0
\(793\) −1.57244e7 −0.887956
\(794\) 0 0
\(795\) 1.26396e7 0.709276
\(796\) 0 0
\(797\) −3.09760e7 −1.72735 −0.863673 0.504052i \(-0.831842\pi\)
−0.863673 + 0.504052i \(0.831842\pi\)
\(798\) 0 0
\(799\) 1.46254e6 0.0810475
\(800\) 0 0
\(801\) −3.67332e7 −2.02292
\(802\) 0 0
\(803\) −2.70257e6 −0.147907
\(804\) 0 0
\(805\) −1.53252e7 −0.833521
\(806\) 0 0
\(807\) 2.06086e7 1.11395
\(808\) 0 0
\(809\) 4.24929e6 0.228268 0.114134 0.993465i \(-0.463591\pi\)
0.114134 + 0.993465i \(0.463591\pi\)
\(810\) 0 0
\(811\) −3.42333e6 −0.182767 −0.0913833 0.995816i \(-0.529129\pi\)
−0.0913833 + 0.995816i \(0.529129\pi\)
\(812\) 0 0
\(813\) −9.20659e6 −0.488509
\(814\) 0 0
\(815\) 1.66540e6 0.0878263
\(816\) 0 0
\(817\) 1.03120e7 0.540490
\(818\) 0 0
\(819\) −5.41831e7 −2.82263
\(820\) 0 0
\(821\) 3.10571e7 1.60806 0.804030 0.594588i \(-0.202685\pi\)
0.804030 + 0.594588i \(0.202685\pi\)
\(822\) 0 0
\(823\) 3.11904e7 1.60517 0.802584 0.596538i \(-0.203458\pi\)
0.802584 + 0.596538i \(0.203458\pi\)
\(824\) 0 0
\(825\) 1.98000e6 0.101282
\(826\) 0 0
\(827\) 8.28487e6 0.421233 0.210616 0.977569i \(-0.432453\pi\)
0.210616 + 0.977569i \(0.432453\pi\)
\(828\) 0 0
\(829\) −1.81689e7 −0.918208 −0.459104 0.888383i \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(830\) 0 0
\(831\) −4.82582e7 −2.42420
\(832\) 0 0
\(833\) −2.83649e6 −0.141635
\(834\) 0 0
\(835\) 5.14230e6 0.255236
\(836\) 0 0
\(837\) 4.99392e6 0.246393
\(838\) 0 0
\(839\) 1.02743e7 0.503902 0.251951 0.967740i \(-0.418928\pi\)
0.251951 + 0.967740i \(0.418928\pi\)
\(840\) 0 0
\(841\) −1.57150e7 −0.766171
\(842\) 0 0
\(843\) −4.86245e6 −0.235660
\(844\) 0 0
\(845\) 1.30906e7 0.630691
\(846\) 0 0
\(847\) −2.47038e7 −1.18319
\(848\) 0 0
\(849\) −5.31686e6 −0.253155
\(850\) 0 0
\(851\) 4.00665e7 1.89652
\(852\) 0 0
\(853\) 6.28597e6 0.295801 0.147901 0.989002i \(-0.452748\pi\)
0.147901 + 0.989002i \(0.452748\pi\)
\(854\) 0 0
\(855\) −4.16250e6 −0.194733
\(856\) 0 0
\(857\) 1.54050e7 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(858\) 0 0
\(859\) −1.43526e7 −0.663664 −0.331832 0.943338i \(-0.607667\pi\)
−0.331832 + 0.943338i \(0.607667\pi\)
\(860\) 0 0
\(861\) −5.12698e6 −0.235697
\(862\) 0 0
\(863\) −1.33278e7 −0.609158 −0.304579 0.952487i \(-0.598516\pi\)
−0.304579 + 0.952487i \(0.598516\pi\)
\(864\) 0 0
\(865\) 1.08464e7 0.492882
\(866\) 0 0
\(867\) 3.28938e7 1.48616
\(868\) 0 0
\(869\) −6.10368e6 −0.274184
\(870\) 0 0
\(871\) 1.71037e6 0.0763913
\(872\) 0 0
\(873\) −2.61012e7 −1.15911
\(874\) 0 0
\(875\) 2.68750e6 0.118667
\(876\) 0 0
\(877\) 3.24846e7 1.42620 0.713098 0.701065i \(-0.247292\pi\)
0.713098 + 0.701065i \(0.247292\pi\)
\(878\) 0 0
\(879\) 7.74014e6 0.337891
\(880\) 0 0
\(881\) 1.54600e7 0.671073 0.335537 0.942027i \(-0.391082\pi\)
0.335537 + 0.942027i \(0.391082\pi\)
\(882\) 0 0
\(883\) 1.69478e6 0.0731494 0.0365747 0.999331i \(-0.488355\pi\)
0.0365747 + 0.999331i \(0.488355\pi\)
\(884\) 0 0
\(885\) 4.78800e6 0.205492
\(886\) 0 0
\(887\) 2.87257e6 0.122592 0.0612960 0.998120i \(-0.480477\pi\)
0.0612960 + 0.998120i \(0.480477\pi\)
\(888\) 0 0
\(889\) 2.10617e7 0.893799
\(890\) 0 0
\(891\) 3.83843e6 0.161979
\(892\) 0 0
\(893\) 3.29400e6 0.138228
\(894\) 0 0
\(895\) 1.22295e7 0.510330
\(896\) 0 0
\(897\) −8.09171e7 −3.35783
\(898\) 0 0
\(899\) −5.06328e6 −0.208945
\(900\) 0 0
\(901\) 4.67665e6 0.191921
\(902\) 0 0
\(903\) 8.51359e7 3.47451
\(904\) 0 0
\(905\) 1.79865e7 0.730006
\(906\) 0 0
\(907\) −3.95422e7 −1.59603 −0.798017 0.602635i \(-0.794118\pi\)
−0.798017 + 0.602635i \(0.794118\pi\)
\(908\) 0 0
\(909\) 4.72667e7 1.89734
\(910\) 0 0
\(911\) −1.13178e7 −0.451819 −0.225909 0.974148i \(-0.572535\pi\)
−0.225909 + 0.974148i \(0.572535\pi\)
\(912\) 0 0
\(913\) −6.80803e6 −0.270299
\(914\) 0 0
\(915\) −9.97320e6 −0.393806
\(916\) 0 0
\(917\) 4.16082e7 1.63401
\(918\) 0 0
\(919\) −8.51348e6 −0.332520 −0.166260 0.986082i \(-0.553169\pi\)
−0.166260 + 0.986082i \(0.553169\pi\)
\(920\) 0 0
\(921\) 3.46322e7 1.34534
\(922\) 0 0
\(923\) −2.32035e7 −0.896497
\(924\) 0 0
\(925\) −7.02625e6 −0.270003
\(926\) 0 0
\(927\) 145188. 0.00554921
\(928\) 0 0
\(929\) −7.54587e6 −0.286860 −0.143430 0.989660i \(-0.545813\pi\)
−0.143430 + 0.989660i \(0.545813\pi\)
\(930\) 0 0
\(931\) −6.38850e6 −0.241560
\(932\) 0 0
\(933\) 4.11149e6 0.154630
\(934\) 0 0
\(935\) 732600. 0.0274055
\(936\) 0 0
\(937\) −1.84500e7 −0.686512 −0.343256 0.939242i \(-0.611530\pi\)
−0.343256 + 0.939242i \(0.611530\pi\)
\(938\) 0 0
\(939\) 2.46453e7 0.912157
\(940\) 0 0
\(941\) 6.75046e6 0.248519 0.124259 0.992250i \(-0.460344\pi\)
0.124259 + 0.992250i \(0.460344\pi\)
\(942\) 0 0
\(943\) −4.42649e6 −0.162099
\(944\) 0 0
\(945\) −9.28800e6 −0.338332
\(946\) 0 0
\(947\) −6.45677e6 −0.233959 −0.116980 0.993134i \(-0.537321\pi\)
−0.116980 + 0.993134i \(0.537321\pi\)
\(948\) 0 0
\(949\) −1.93684e7 −0.698117
\(950\) 0 0
\(951\) −1.80710e7 −0.647934
\(952\) 0 0
\(953\) −3.96648e7 −1.41473 −0.707364 0.706849i \(-0.750116\pi\)
−0.707364 + 0.706849i \(0.750116\pi\)
\(954\) 0 0
\(955\) 4.64820e6 0.164921
\(956\) 0 0
\(957\) 6.93792e6 0.244878
\(958\) 0 0
\(959\) 4.76609e7 1.67346
\(960\) 0 0
\(961\) −2.32838e7 −0.813290
\(962\) 0 0
\(963\) −7.75783e7 −2.69572
\(964\) 0 0
\(965\) −1.47851e7 −0.511102
\(966\) 0 0
\(967\) 3.43015e7 1.17963 0.589816 0.807538i \(-0.299200\pi\)
0.589816 + 0.807538i \(0.299200\pi\)
\(968\) 0 0
\(969\) −2.66400e6 −0.0911433
\(970\) 0 0
\(971\) 5.77115e6 0.196433 0.0982164 0.995165i \(-0.468686\pi\)
0.0982164 + 0.995165i \(0.468686\pi\)
\(972\) 0 0
\(973\) 3.32889e7 1.12724
\(974\) 0 0
\(975\) 1.41900e7 0.478047
\(976\) 0 0
\(977\) 7.08746e6 0.237549 0.118775 0.992921i \(-0.462103\pi\)
0.118775 + 0.992921i \(0.462103\pi\)
\(978\) 0 0
\(979\) 1.45609e7 0.485548
\(980\) 0 0
\(981\) −5.82251e7 −1.93169
\(982\) 0 0
\(983\) −4.59362e7 −1.51625 −0.758126 0.652108i \(-0.773885\pi\)
−0.758126 + 0.652108i \(0.773885\pi\)
\(984\) 0 0
\(985\) 1.12369e7 0.369027
\(986\) 0 0
\(987\) 2.71953e7 0.888588
\(988\) 0 0
\(989\) 7.35039e7 2.38957
\(990\) 0 0
\(991\) 4.50298e7 1.45652 0.728260 0.685301i \(-0.240329\pi\)
0.728260 + 0.685301i \(0.240329\pi\)
\(992\) 0 0
\(993\) 4.78592e7 1.54025
\(994\) 0 0
\(995\) −3.92900e6 −0.125813
\(996\) 0 0
\(997\) −2.37364e7 −0.756271 −0.378136 0.925750i \(-0.623435\pi\)
−0.378136 + 0.925750i \(0.623435\pi\)
\(998\) 0 0
\(999\) 2.42827e7 0.769810
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 80.6.a.a.1.1 1
3.2 odd 2 720.6.a.j.1.1 1
4.3 odd 2 10.6.a.b.1.1 1
5.2 odd 4 400.6.c.b.49.2 2
5.3 odd 4 400.6.c.b.49.1 2
5.4 even 2 400.6.a.n.1.1 1
8.3 odd 2 320.6.a.b.1.1 1
8.5 even 2 320.6.a.o.1.1 1
12.11 even 2 90.6.a.d.1.1 1
20.3 even 4 50.6.b.a.49.2 2
20.7 even 4 50.6.b.a.49.1 2
20.19 odd 2 50.6.a.d.1.1 1
28.27 even 2 490.6.a.a.1.1 1
60.23 odd 4 450.6.c.h.199.1 2
60.47 odd 4 450.6.c.h.199.2 2
60.59 even 2 450.6.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.b.1.1 1 4.3 odd 2
50.6.a.d.1.1 1 20.19 odd 2
50.6.b.a.49.1 2 20.7 even 4
50.6.b.a.49.2 2 20.3 even 4
80.6.a.a.1.1 1 1.1 even 1 trivial
90.6.a.d.1.1 1 12.11 even 2
320.6.a.b.1.1 1 8.3 odd 2
320.6.a.o.1.1 1 8.5 even 2
400.6.a.n.1.1 1 5.4 even 2
400.6.c.b.49.1 2 5.3 odd 4
400.6.c.b.49.2 2 5.2 odd 4
450.6.a.l.1.1 1 60.59 even 2
450.6.c.h.199.1 2 60.23 odd 4
450.6.c.h.199.2 2 60.47 odd 4
490.6.a.a.1.1 1 28.27 even 2
720.6.a.j.1.1 1 3.2 odd 2