Properties

Label 450.6.a.bf.1.1
Level $450$
Weight $6$
Character 450.1
Self dual yes
Analytic conductor $72.173$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{4081}) \)
Defining polynomial: \(x^{2} - x - 1020\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(32.4414\) of defining polynomial
Character \(\chi\) \(=\) 450.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +O(q^{10})\) \(q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +113.296 q^{11} -61.7038 q^{13} -566.592 q^{14} +256.000 q^{16} +1670.48 q^{17} -662.241 q^{19} +453.185 q^{22} +86.4812 q^{23} -246.815 q^{26} -2266.37 q^{28} -3233.30 q^{29} -3812.24 q^{31} +1024.00 q^{32} +6681.92 q^{34} -10249.6 q^{37} -2648.96 q^{38} -13459.4 q^{41} +4697.57 q^{43} +1812.74 q^{44} +345.925 q^{46} -15264.8 q^{47} +3257.19 q^{49} -987.260 q^{52} +498.632 q^{53} -9065.48 q^{56} -12933.2 q^{58} -15294.3 q^{59} -31811.3 q^{61} -15249.0 q^{62} +4096.00 q^{64} -49274.9 q^{67} +26727.7 q^{68} +21057.7 q^{71} -39447.8 q^{73} -40998.2 q^{74} -10595.8 q^{76} -16048.2 q^{77} +72949.3 q^{79} -53837.8 q^{82} +100897. q^{83} +18790.3 q^{86} +7250.96 q^{88} -146714. q^{89} +8740.22 q^{91} +1383.70 q^{92} -61059.2 q^{94} +43544.9 q^{97} +13028.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} + O(q^{10}) \) \( 2q + 8q^{2} + 32q^{4} + 100q^{7} + 128q^{8} - 540q^{11} - 890q^{13} + 400q^{14} + 512q^{16} - 492q^{17} + 592q^{19} - 2160q^{22} - 3660q^{23} - 3560q^{26} + 1600q^{28} - 5700q^{29} - 5708q^{31} + 2048q^{32} - 1968q^{34} - 11300q^{37} + 2368q^{38} - 15420q^{41} - 6320q^{43} - 8640q^{44} - 14640q^{46} + 7800q^{47} + 44844q^{49} - 14240q^{52} + 27828q^{53} + 6400q^{56} - 22800q^{58} - 50520q^{59} - 29126q^{61} - 22832q^{62} + 8192q^{64} - 97400q^{67} - 7872q^{68} - 6180q^{71} - 32900q^{73} - 45200q^{74} + 9472q^{76} - 173916q^{77} + 7912q^{79} - 61680q^{82} + 163464q^{83} - 25280q^{86} - 34560q^{88} - 164640q^{89} - 191416q^{91} - 58560q^{92} + 31200q^{94} - 52430q^{97} + 179376q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −141.648 −1.09261 −0.546306 0.837586i \(-0.683966\pi\)
−0.546306 + 0.837586i \(0.683966\pi\)
\(8\) 64.0000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 113.296 0.282315 0.141157 0.989987i \(-0.454918\pi\)
0.141157 + 0.989987i \(0.454918\pi\)
\(12\) 0 0
\(13\) −61.7038 −0.101264 −0.0506318 0.998717i \(-0.516123\pi\)
−0.0506318 + 0.998717i \(0.516123\pi\)
\(14\) −566.592 −0.772593
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1670.48 1.40191 0.700954 0.713207i \(-0.252758\pi\)
0.700954 + 0.713207i \(0.252758\pi\)
\(18\) 0 0
\(19\) −662.241 −0.420854 −0.210427 0.977610i \(-0.567485\pi\)
−0.210427 + 0.977610i \(0.567485\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 453.185 0.199627
\(23\) 86.4812 0.0340880 0.0170440 0.999855i \(-0.494574\pi\)
0.0170440 + 0.999855i \(0.494574\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −246.815 −0.0716042
\(27\) 0 0
\(28\) −2266.37 −0.546306
\(29\) −3233.30 −0.713922 −0.356961 0.934119i \(-0.616187\pi\)
−0.356961 + 0.934119i \(0.616187\pi\)
\(30\) 0 0
\(31\) −3812.24 −0.712486 −0.356243 0.934393i \(-0.615942\pi\)
−0.356243 + 0.934393i \(0.615942\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 6681.92 0.991298
\(35\) 0 0
\(36\) 0 0
\(37\) −10249.6 −1.23084 −0.615419 0.788200i \(-0.711013\pi\)
−0.615419 + 0.788200i \(0.711013\pi\)
\(38\) −2648.96 −0.297589
\(39\) 0 0
\(40\) 0 0
\(41\) −13459.4 −1.25045 −0.625227 0.780443i \(-0.714993\pi\)
−0.625227 + 0.780443i \(0.714993\pi\)
\(42\) 0 0
\(43\) 4697.57 0.387438 0.193719 0.981057i \(-0.437945\pi\)
0.193719 + 0.981057i \(0.437945\pi\)
\(44\) 1812.74 0.141157
\(45\) 0 0
\(46\) 345.925 0.0241039
\(47\) −15264.8 −1.00797 −0.503984 0.863713i \(-0.668133\pi\)
−0.503984 + 0.863713i \(0.668133\pi\)
\(48\) 0 0
\(49\) 3257.19 0.193800
\(50\) 0 0
\(51\) 0 0
\(52\) −987.260 −0.0506318
\(53\) 498.632 0.0243832 0.0121916 0.999926i \(-0.496119\pi\)
0.0121916 + 0.999926i \(0.496119\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9065.48 −0.386296
\(57\) 0 0
\(58\) −12933.2 −0.504819
\(59\) −15294.3 −0.572005 −0.286002 0.958229i \(-0.592327\pi\)
−0.286002 + 0.958229i \(0.592327\pi\)
\(60\) 0 0
\(61\) −31811.3 −1.09460 −0.547302 0.836935i \(-0.684345\pi\)
−0.547302 + 0.836935i \(0.684345\pi\)
\(62\) −15249.0 −0.503803
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −49274.9 −1.34103 −0.670516 0.741895i \(-0.733927\pi\)
−0.670516 + 0.741895i \(0.733927\pi\)
\(68\) 26727.7 0.700954
\(69\) 0 0
\(70\) 0 0
\(71\) 21057.7 0.495752 0.247876 0.968792i \(-0.420267\pi\)
0.247876 + 0.968792i \(0.420267\pi\)
\(72\) 0 0
\(73\) −39447.8 −0.866394 −0.433197 0.901299i \(-0.642615\pi\)
−0.433197 + 0.901299i \(0.642615\pi\)
\(74\) −40998.2 −0.870333
\(75\) 0 0
\(76\) −10595.8 −0.210427
\(77\) −16048.2 −0.308460
\(78\) 0 0
\(79\) 72949.3 1.31508 0.657542 0.753418i \(-0.271596\pi\)
0.657542 + 0.753418i \(0.271596\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −53837.8 −0.884204
\(83\) 100897. 1.60762 0.803808 0.594889i \(-0.202804\pi\)
0.803808 + 0.594889i \(0.202804\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 18790.3 0.273960
\(87\) 0 0
\(88\) 7250.96 0.0998134
\(89\) −146714. −1.96334 −0.981671 0.190585i \(-0.938962\pi\)
−0.981671 + 0.190585i \(0.938962\pi\)
\(90\) 0 0
\(91\) 8740.22 0.110642
\(92\) 1383.70 0.0170440
\(93\) 0 0
\(94\) −61059.2 −0.712741
\(95\) 0 0
\(96\) 0 0
\(97\) 43544.9 0.469903 0.234951 0.972007i \(-0.424507\pi\)
0.234951 + 0.972007i \(0.424507\pi\)
\(98\) 13028.8 0.137037
\(99\) 0 0
\(100\) 0 0
\(101\) −49966.3 −0.487386 −0.243693 0.969852i \(-0.578359\pi\)
−0.243693 + 0.969852i \(0.578359\pi\)
\(102\) 0 0
\(103\) 50579.6 0.469766 0.234883 0.972024i \(-0.424529\pi\)
0.234883 + 0.972024i \(0.424529\pi\)
\(104\) −3949.04 −0.0358021
\(105\) 0 0
\(106\) 1994.53 0.0172415
\(107\) −191553. −1.61744 −0.808721 0.588193i \(-0.799840\pi\)
−0.808721 + 0.588193i \(0.799840\pi\)
\(108\) 0 0
\(109\) 71531.6 0.576676 0.288338 0.957529i \(-0.406897\pi\)
0.288338 + 0.957529i \(0.406897\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −36261.9 −0.273153
\(113\) 163395. 1.20377 0.601885 0.798583i \(-0.294417\pi\)
0.601885 + 0.798583i \(0.294417\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −51732.7 −0.356961
\(117\) 0 0
\(118\) −61177.2 −0.404468
\(119\) −236621. −1.53174
\(120\) 0 0
\(121\) −148215. −0.920298
\(122\) −127245. −0.774002
\(123\) 0 0
\(124\) −60995.8 −0.356243
\(125\) 0 0
\(126\) 0 0
\(127\) 34350.7 0.188984 0.0944922 0.995526i \(-0.469877\pi\)
0.0944922 + 0.995526i \(0.469877\pi\)
\(128\) 16384.0 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −349293. −1.77833 −0.889164 0.457588i \(-0.848713\pi\)
−0.889164 + 0.457588i \(0.848713\pi\)
\(132\) 0 0
\(133\) 93805.1 0.459830
\(134\) −197100. −0.948253
\(135\) 0 0
\(136\) 106911. 0.495649
\(137\) −274334. −1.24876 −0.624379 0.781122i \(-0.714648\pi\)
−0.624379 + 0.781122i \(0.714648\pi\)
\(138\) 0 0
\(139\) −47842.2 −0.210027 −0.105013 0.994471i \(-0.533489\pi\)
−0.105013 + 0.994471i \(0.533489\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 84230.6 0.350549
\(143\) −6990.80 −0.0285882
\(144\) 0 0
\(145\) 0 0
\(146\) −157791. −0.612633
\(147\) 0 0
\(148\) −163993. −0.615419
\(149\) −335645. −1.23855 −0.619276 0.785174i \(-0.712574\pi\)
−0.619276 + 0.785174i \(0.712574\pi\)
\(150\) 0 0
\(151\) 336370. 1.20054 0.600268 0.799799i \(-0.295061\pi\)
0.600268 + 0.799799i \(0.295061\pi\)
\(152\) −42383.4 −0.148794
\(153\) 0 0
\(154\) −64192.8 −0.218114
\(155\) 0 0
\(156\) 0 0
\(157\) 381573. 1.23546 0.617729 0.786391i \(-0.288053\pi\)
0.617729 + 0.786391i \(0.288053\pi\)
\(158\) 291797. 0.929905
\(159\) 0 0
\(160\) 0 0
\(161\) −12249.9 −0.0372450
\(162\) 0 0
\(163\) 579006. 1.70692 0.853462 0.521155i \(-0.174499\pi\)
0.853462 + 0.521155i \(0.174499\pi\)
\(164\) −215351. −0.625227
\(165\) 0 0
\(166\) 403587. 1.13676
\(167\) 128108. 0.355455 0.177728 0.984080i \(-0.443125\pi\)
0.177728 + 0.984080i \(0.443125\pi\)
\(168\) 0 0
\(169\) −367486. −0.989746
\(170\) 0 0
\(171\) 0 0
\(172\) 75161.2 0.193719
\(173\) 487187. 1.23760 0.618800 0.785548i \(-0.287619\pi\)
0.618800 + 0.785548i \(0.287619\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 29003.8 0.0705787
\(177\) 0 0
\(178\) −586855. −1.38829
\(179\) −848235. −1.97872 −0.989359 0.145497i \(-0.953522\pi\)
−0.989359 + 0.145497i \(0.953522\pi\)
\(180\) 0 0
\(181\) −575423. −1.30554 −0.652771 0.757555i \(-0.726394\pi\)
−0.652771 + 0.757555i \(0.726394\pi\)
\(182\) 34960.9 0.0782355
\(183\) 0 0
\(184\) 5534.79 0.0120519
\(185\) 0 0
\(186\) 0 0
\(187\) 189259. 0.395779
\(188\) −244237. −0.503984
\(189\) 0 0
\(190\) 0 0
\(191\) 18362.7 0.0364211 0.0182105 0.999834i \(-0.494203\pi\)
0.0182105 + 0.999834i \(0.494203\pi\)
\(192\) 0 0
\(193\) 349513. 0.675414 0.337707 0.941251i \(-0.390349\pi\)
0.337707 + 0.941251i \(0.390349\pi\)
\(194\) 174180. 0.332272
\(195\) 0 0
\(196\) 52115.0 0.0968998
\(197\) −371791. −0.682547 −0.341274 0.939964i \(-0.610858\pi\)
−0.341274 + 0.939964i \(0.610858\pi\)
\(198\) 0 0
\(199\) 300569. 0.538036 0.269018 0.963135i \(-0.413301\pi\)
0.269018 + 0.963135i \(0.413301\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −199865. −0.344634
\(203\) 457990. 0.780039
\(204\) 0 0
\(205\) 0 0
\(206\) 202318. 0.332175
\(207\) 0 0
\(208\) −15796.2 −0.0253159
\(209\) −75029.4 −0.118813
\(210\) 0 0
\(211\) 59499.2 0.0920036 0.0460018 0.998941i \(-0.485352\pi\)
0.0460018 + 0.998941i \(0.485352\pi\)
\(212\) 7978.11 0.0121916
\(213\) 0 0
\(214\) −766210. −1.14370
\(215\) 0 0
\(216\) 0 0
\(217\) 539997. 0.778470
\(218\) 286127. 0.407772
\(219\) 0 0
\(220\) 0 0
\(221\) −103075. −0.141962
\(222\) 0 0
\(223\) 866124. 1.16632 0.583160 0.812357i \(-0.301816\pi\)
0.583160 + 0.812357i \(0.301816\pi\)
\(224\) −145048. −0.193148
\(225\) 0 0
\(226\) 653581. 0.851194
\(227\) −737058. −0.949374 −0.474687 0.880155i \(-0.657439\pi\)
−0.474687 + 0.880155i \(0.657439\pi\)
\(228\) 0 0
\(229\) 846501. 1.06669 0.533346 0.845897i \(-0.320935\pi\)
0.533346 + 0.845897i \(0.320935\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −206931. −0.252409
\(233\) −590760. −0.712888 −0.356444 0.934317i \(-0.616011\pi\)
−0.356444 + 0.934317i \(0.616011\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −244709. −0.286002
\(237\) 0 0
\(238\) −946482. −1.08310
\(239\) 1.60375e6 1.81611 0.908054 0.418853i \(-0.137568\pi\)
0.908054 + 0.418853i \(0.137568\pi\)
\(240\) 0 0
\(241\) −347071. −0.384925 −0.192463 0.981304i \(-0.561647\pi\)
−0.192463 + 0.981304i \(0.561647\pi\)
\(242\) −592860. −0.650749
\(243\) 0 0
\(244\) −508981. −0.547302
\(245\) 0 0
\(246\) 0 0
\(247\) 40862.7 0.0426172
\(248\) −243983. −0.251902
\(249\) 0 0
\(250\) 0 0
\(251\) 536438. 0.537446 0.268723 0.963217i \(-0.413398\pi\)
0.268723 + 0.963217i \(0.413398\pi\)
\(252\) 0 0
\(253\) 9797.99 0.00962356
\(254\) 137403. 0.133632
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 763565. 0.721130 0.360565 0.932734i \(-0.382584\pi\)
0.360565 + 0.932734i \(0.382584\pi\)
\(258\) 0 0
\(259\) 1.45183e6 1.34483
\(260\) 0 0
\(261\) 0 0
\(262\) −1.39717e6 −1.25747
\(263\) −341099. −0.304082 −0.152041 0.988374i \(-0.548585\pi\)
−0.152041 + 0.988374i \(0.548585\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 375221. 0.325149
\(267\) 0 0
\(268\) −788399. −0.670516
\(269\) 750163. 0.632084 0.316042 0.948745i \(-0.397646\pi\)
0.316042 + 0.948745i \(0.397646\pi\)
\(270\) 0 0
\(271\) −1.64462e6 −1.36033 −0.680163 0.733061i \(-0.738091\pi\)
−0.680163 + 0.733061i \(0.738091\pi\)
\(272\) 427643. 0.350477
\(273\) 0 0
\(274\) −1.09734e6 −0.883005
\(275\) 0 0
\(276\) 0 0
\(277\) −1.04631e6 −0.819337 −0.409668 0.912235i \(-0.634356\pi\)
−0.409668 + 0.912235i \(0.634356\pi\)
\(278\) −191369. −0.148511
\(279\) 0 0
\(280\) 0 0
\(281\) −54503.6 −0.0411774 −0.0205887 0.999788i \(-0.506554\pi\)
−0.0205887 + 0.999788i \(0.506554\pi\)
\(282\) 0 0
\(283\) −150130. −0.111429 −0.0557147 0.998447i \(-0.517744\pi\)
−0.0557147 + 0.998447i \(0.517744\pi\)
\(284\) 336923. 0.247876
\(285\) 0 0
\(286\) −27963.2 −0.0202149
\(287\) 1.90650e6 1.36626
\(288\) 0 0
\(289\) 1.37065e6 0.965344
\(290\) 0 0
\(291\) 0 0
\(292\) −631164. −0.433197
\(293\) −1.54001e6 −1.04799 −0.523993 0.851723i \(-0.675558\pi\)
−0.523993 + 0.851723i \(0.675558\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −655972. −0.435167
\(297\) 0 0
\(298\) −1.34258e6 −0.875788
\(299\) −5336.21 −0.00345188
\(300\) 0 0
\(301\) −665402. −0.423319
\(302\) 1.34548e6 0.848907
\(303\) 0 0
\(304\) −169534. −0.105214
\(305\) 0 0
\(306\) 0 0
\(307\) 914773. 0.553946 0.276973 0.960878i \(-0.410669\pi\)
0.276973 + 0.960878i \(0.410669\pi\)
\(308\) −256771. −0.154230
\(309\) 0 0
\(310\) 0 0
\(311\) 1.59808e6 0.936910 0.468455 0.883487i \(-0.344811\pi\)
0.468455 + 0.883487i \(0.344811\pi\)
\(312\) 0 0
\(313\) 2.38412e6 1.37552 0.687760 0.725938i \(-0.258594\pi\)
0.687760 + 0.725938i \(0.258594\pi\)
\(314\) 1.52629e6 0.873601
\(315\) 0 0
\(316\) 1.16719e6 0.657542
\(317\) 2.00013e6 1.11792 0.558960 0.829194i \(-0.311200\pi\)
0.558960 + 0.829194i \(0.311200\pi\)
\(318\) 0 0
\(319\) −366320. −0.201551
\(320\) 0 0
\(321\) 0 0
\(322\) −48999.6 −0.0263362
\(323\) −1.10626e6 −0.589999
\(324\) 0 0
\(325\) 0 0
\(326\) 2.31602e6 1.20698
\(327\) 0 0
\(328\) −861404. −0.442102
\(329\) 2.16223e6 1.10132
\(330\) 0 0
\(331\) −1.48362e6 −0.744306 −0.372153 0.928171i \(-0.621380\pi\)
−0.372153 + 0.928171i \(0.621380\pi\)
\(332\) 1.61435e6 0.803808
\(333\) 0 0
\(334\) 512432. 0.251345
\(335\) 0 0
\(336\) 0 0
\(337\) −2.42490e6 −1.16311 −0.581553 0.813508i \(-0.697555\pi\)
−0.581553 + 0.813508i \(0.697555\pi\)
\(338\) −1.46994e6 −0.699856
\(339\) 0 0
\(340\) 0 0
\(341\) −431912. −0.201145
\(342\) 0 0
\(343\) 1.91931e6 0.880864
\(344\) 300645. 0.136980
\(345\) 0 0
\(346\) 1.94875e6 0.875116
\(347\) 1.26681e6 0.564793 0.282397 0.959298i \(-0.408871\pi\)
0.282397 + 0.959298i \(0.408871\pi\)
\(348\) 0 0
\(349\) 2.54257e6 1.11740 0.558701 0.829369i \(-0.311300\pi\)
0.558701 + 0.829369i \(0.311300\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 116015. 0.0499067
\(353\) −1.28509e6 −0.548906 −0.274453 0.961601i \(-0.588497\pi\)
−0.274453 + 0.961601i \(0.588497\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.34742e6 −0.981671
\(357\) 0 0
\(358\) −3.39294e6 −1.39916
\(359\) 4.16251e6 1.70459 0.852294 0.523063i \(-0.175211\pi\)
0.852294 + 0.523063i \(0.175211\pi\)
\(360\) 0 0
\(361\) −2.03754e6 −0.822882
\(362\) −2.30169e6 −0.923158
\(363\) 0 0
\(364\) 139844. 0.0553209
\(365\) 0 0
\(366\) 0 0
\(367\) 1.13066e6 0.438195 0.219098 0.975703i \(-0.429689\pi\)
0.219098 + 0.975703i \(0.429689\pi\)
\(368\) 22139.2 0.00852201
\(369\) 0 0
\(370\) 0 0
\(371\) −70630.3 −0.0266413
\(372\) 0 0
\(373\) 312371. 0.116251 0.0581257 0.998309i \(-0.481488\pi\)
0.0581257 + 0.998309i \(0.481488\pi\)
\(374\) 757037. 0.279858
\(375\) 0 0
\(376\) −976948. −0.356371
\(377\) 199507. 0.0722943
\(378\) 0 0
\(379\) −2.19990e6 −0.786692 −0.393346 0.919390i \(-0.628683\pi\)
−0.393346 + 0.919390i \(0.628683\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 73450.8 0.0257536
\(383\) −3.93331e6 −1.37013 −0.685065 0.728482i \(-0.740226\pi\)
−0.685065 + 0.728482i \(0.740226\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.39805e6 0.477590
\(387\) 0 0
\(388\) 696719. 0.234951
\(389\) 4.20593e6 1.40925 0.704625 0.709580i \(-0.251115\pi\)
0.704625 + 0.709580i \(0.251115\pi\)
\(390\) 0 0
\(391\) 144465. 0.0477883
\(392\) 208460. 0.0685185
\(393\) 0 0
\(394\) −1.48716e6 −0.482634
\(395\) 0 0
\(396\) 0 0
\(397\) 2.39484e6 0.762606 0.381303 0.924450i \(-0.375475\pi\)
0.381303 + 0.924450i \(0.375475\pi\)
\(398\) 1.20228e6 0.380449
\(399\) 0 0
\(400\) 0 0
\(401\) 4.42161e6 1.37315 0.686577 0.727057i \(-0.259112\pi\)
0.686577 + 0.727057i \(0.259112\pi\)
\(402\) 0 0
\(403\) 235230. 0.0721488
\(404\) −799460. −0.243693
\(405\) 0 0
\(406\) 1.83196e6 0.551571
\(407\) −1.16124e6 −0.347484
\(408\) 0 0
\(409\) −2.61075e6 −0.771717 −0.385858 0.922558i \(-0.626095\pi\)
−0.385858 + 0.922558i \(0.626095\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 809273. 0.234883
\(413\) 2.16641e6 0.624979
\(414\) 0 0
\(415\) 0 0
\(416\) −63184.7 −0.0179010
\(417\) 0 0
\(418\) −300117. −0.0840138
\(419\) 1.01624e6 0.282789 0.141395 0.989953i \(-0.454841\pi\)
0.141395 + 0.989953i \(0.454841\pi\)
\(420\) 0 0
\(421\) 5.96915e6 1.64137 0.820687 0.571379i \(-0.193591\pi\)
0.820687 + 0.571379i \(0.193591\pi\)
\(422\) 237997. 0.0650564
\(423\) 0 0
\(424\) 31912.4 0.00862076
\(425\) 0 0
\(426\) 0 0
\(427\) 4.50601e6 1.19598
\(428\) −3.06484e6 −0.808721
\(429\) 0 0
\(430\) 0 0
\(431\) −2.97145e6 −0.770504 −0.385252 0.922811i \(-0.625885\pi\)
−0.385252 + 0.922811i \(0.625885\pi\)
\(432\) 0 0
\(433\) 4.95237e6 1.26938 0.634692 0.772765i \(-0.281127\pi\)
0.634692 + 0.772765i \(0.281127\pi\)
\(434\) 2.15999e6 0.550461
\(435\) 0 0
\(436\) 1.14451e6 0.288338
\(437\) −57271.3 −0.0143461
\(438\) 0 0
\(439\) −1.12115e6 −0.277652 −0.138826 0.990317i \(-0.544333\pi\)
−0.138826 + 0.990317i \(0.544333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −412300. −0.100382
\(443\) 2.66233e6 0.644544 0.322272 0.946647i \(-0.395553\pi\)
0.322272 + 0.946647i \(0.395553\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.46449e6 0.824713
\(447\) 0 0
\(448\) −580191. −0.136576
\(449\) −3.88781e6 −0.910101 −0.455050 0.890466i \(-0.650379\pi\)
−0.455050 + 0.890466i \(0.650379\pi\)
\(450\) 0 0
\(451\) −1.52490e6 −0.353022
\(452\) 2.61432e6 0.601885
\(453\) 0 0
\(454\) −2.94823e6 −0.671309
\(455\) 0 0
\(456\) 0 0
\(457\) −2.52630e6 −0.565841 −0.282920 0.959143i \(-0.591303\pi\)
−0.282920 + 0.959143i \(0.591303\pi\)
\(458\) 3.38600e6 0.754265
\(459\) 0 0
\(460\) 0 0
\(461\) 4.63589e6 1.01597 0.507985 0.861366i \(-0.330390\pi\)
0.507985 + 0.861366i \(0.330390\pi\)
\(462\) 0 0
\(463\) −3.62380e6 −0.785619 −0.392810 0.919620i \(-0.628497\pi\)
−0.392810 + 0.919620i \(0.628497\pi\)
\(464\) −827724. −0.178480
\(465\) 0 0
\(466\) −2.36304e6 −0.504088
\(467\) 3.72757e6 0.790922 0.395461 0.918483i \(-0.370585\pi\)
0.395461 + 0.918483i \(0.370585\pi\)
\(468\) 0 0
\(469\) 6.97970e6 1.46523
\(470\) 0 0
\(471\) 0 0
\(472\) −978835. −0.202234
\(473\) 532217. 0.109380
\(474\) 0 0
\(475\) 0 0
\(476\) −3.78593e6 −0.765870
\(477\) 0 0
\(478\) 6.41500e6 1.28418
\(479\) −6.88374e6 −1.37084 −0.685418 0.728150i \(-0.740381\pi\)
−0.685418 + 0.728150i \(0.740381\pi\)
\(480\) 0 0
\(481\) 632436. 0.124639
\(482\) −1.38829e6 −0.272183
\(483\) 0 0
\(484\) −2.37144e6 −0.460149
\(485\) 0 0
\(486\) 0 0
\(487\) 8.58990e6 1.64122 0.820608 0.571492i \(-0.193635\pi\)
0.820608 + 0.571492i \(0.193635\pi\)
\(488\) −2.03593e6 −0.387001
\(489\) 0 0
\(490\) 0 0
\(491\) −1.46999e6 −0.275176 −0.137588 0.990490i \(-0.543935\pi\)
−0.137588 + 0.990490i \(0.543935\pi\)
\(492\) 0 0
\(493\) −5.40116e6 −1.00085
\(494\) 163451. 0.0301349
\(495\) 0 0
\(496\) −975934. −0.178121
\(497\) −2.98278e6 −0.541664
\(498\) 0 0
\(499\) −3.13043e6 −0.562798 −0.281399 0.959591i \(-0.590798\pi\)
−0.281399 + 0.959591i \(0.590798\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.14575e6 0.380032
\(503\) 2.29993e6 0.405317 0.202658 0.979249i \(-0.435042\pi\)
0.202658 + 0.979249i \(0.435042\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 39192.0 0.00680488
\(507\) 0 0
\(508\) 549611. 0.0944922
\(509\) −5.16574e6 −0.883767 −0.441884 0.897072i \(-0.645690\pi\)
−0.441884 + 0.897072i \(0.645690\pi\)
\(510\) 0 0
\(511\) 5.58770e6 0.946632
\(512\) 262144. 0.0441942
\(513\) 0 0
\(514\) 3.05426e6 0.509916
\(515\) 0 0
\(516\) 0 0
\(517\) −1.72945e6 −0.284564
\(518\) 5.80732e6 0.950936
\(519\) 0 0
\(520\) 0 0
\(521\) −1.12853e7 −1.82146 −0.910729 0.413004i \(-0.864480\pi\)
−0.910729 + 0.413004i \(0.864480\pi\)
\(522\) 0 0
\(523\) 4.86022e6 0.776966 0.388483 0.921456i \(-0.372999\pi\)
0.388483 + 0.921456i \(0.372999\pi\)
\(524\) −5.58869e6 −0.889164
\(525\) 0 0
\(526\) −1.36440e6 −0.215019
\(527\) −6.36828e6 −0.998839
\(528\) 0 0
\(529\) −6.42886e6 −0.998838
\(530\) 0 0
\(531\) 0 0
\(532\) 1.50088e6 0.229915
\(533\) 830498. 0.126625
\(534\) 0 0
\(535\) 0 0
\(536\) −3.15360e6 −0.474126
\(537\) 0 0
\(538\) 3.00065e6 0.446951
\(539\) 369027. 0.0547125
\(540\) 0 0
\(541\) −5.23864e6 −0.769530 −0.384765 0.923015i \(-0.625717\pi\)
−0.384765 + 0.923015i \(0.625717\pi\)
\(542\) −6.57849e6 −0.961896
\(543\) 0 0
\(544\) 1.71057e6 0.247825
\(545\) 0 0
\(546\) 0 0
\(547\) −2.88586e6 −0.412389 −0.206194 0.978511i \(-0.566108\pi\)
−0.206194 + 0.978511i \(0.566108\pi\)
\(548\) −4.38934e6 −0.624379
\(549\) 0 0
\(550\) 0 0
\(551\) 2.14122e6 0.300457
\(552\) 0 0
\(553\) −1.03331e7 −1.43688
\(554\) −4.18525e6 −0.579359
\(555\) 0 0
\(556\) −765476. −0.105013
\(557\) −3.27936e6 −0.447869 −0.223935 0.974604i \(-0.571890\pi\)
−0.223935 + 0.974604i \(0.571890\pi\)
\(558\) 0 0
\(559\) −289858. −0.0392334
\(560\) 0 0
\(561\) 0 0
\(562\) −218014. −0.0291168
\(563\) 664023. 0.0882901 0.0441451 0.999025i \(-0.485944\pi\)
0.0441451 + 0.999025i \(0.485944\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −600518. −0.0787925
\(567\) 0 0
\(568\) 1.34769e6 0.175275
\(569\) −3.00851e6 −0.389557 −0.194778 0.980847i \(-0.562399\pi\)
−0.194778 + 0.980847i \(0.562399\pi\)
\(570\) 0 0
\(571\) 1.07571e7 1.38072 0.690359 0.723467i \(-0.257453\pi\)
0.690359 + 0.723467i \(0.257453\pi\)
\(572\) −111853. −0.0142941
\(573\) 0 0
\(574\) 7.62602e6 0.966091
\(575\) 0 0
\(576\) 0 0
\(577\) −1.53859e7 −1.92390 −0.961952 0.273220i \(-0.911911\pi\)
−0.961952 + 0.273220i \(0.911911\pi\)
\(578\) 5.48260e6 0.682601
\(579\) 0 0
\(580\) 0 0
\(581\) −1.42918e7 −1.75650
\(582\) 0 0
\(583\) 56493.1 0.00688374
\(584\) −2.52466e6 −0.306316
\(585\) 0 0
\(586\) −6.16005e6 −0.741038
\(587\) −1.16896e7 −1.40025 −0.700125 0.714020i \(-0.746872\pi\)
−0.700125 + 0.714020i \(0.746872\pi\)
\(588\) 0 0
\(589\) 2.52462e6 0.299853
\(590\) 0 0
\(591\) 0 0
\(592\) −2.62389e6 −0.307709
\(593\) 2.29301e6 0.267774 0.133887 0.990997i \(-0.457254\pi\)
0.133887 + 0.990997i \(0.457254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.37031e6 −0.619276
\(597\) 0 0
\(598\) −21344.9 −0.00244084
\(599\) −1.58740e7 −1.80767 −0.903834 0.427884i \(-0.859259\pi\)
−0.903834 + 0.427884i \(0.859259\pi\)
\(600\) 0 0
\(601\) −1.16506e7 −1.31571 −0.657855 0.753144i \(-0.728536\pi\)
−0.657855 + 0.753144i \(0.728536\pi\)
\(602\) −2.66161e6 −0.299332
\(603\) 0 0
\(604\) 5.38193e6 0.600268
\(605\) 0 0
\(606\) 0 0
\(607\) −9.40797e6 −1.03639 −0.518196 0.855262i \(-0.673396\pi\)
−0.518196 + 0.855262i \(0.673396\pi\)
\(608\) −678134. −0.0743972
\(609\) 0 0
\(610\) 0 0
\(611\) 941896. 0.102070
\(612\) 0 0
\(613\) −9.30509e6 −1.00016 −0.500080 0.865979i \(-0.666696\pi\)
−0.500080 + 0.865979i \(0.666696\pi\)
\(614\) 3.65909e6 0.391699
\(615\) 0 0
\(616\) −1.02708e6 −0.109057
\(617\) 268346. 0.0283780 0.0141890 0.999899i \(-0.495483\pi\)
0.0141890 + 0.999899i \(0.495483\pi\)
\(618\) 0 0
\(619\) −7.97678e6 −0.836760 −0.418380 0.908272i \(-0.637402\pi\)
−0.418380 + 0.908272i \(0.637402\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.39233e6 0.662496
\(623\) 2.07817e7 2.14517
\(624\) 0 0
\(625\) 0 0
\(626\) 9.53647e6 0.972640
\(627\) 0 0
\(628\) 6.10516e6 0.617729
\(629\) −1.71217e7 −1.72552
\(630\) 0 0
\(631\) 2.94679e6 0.294629 0.147314 0.989090i \(-0.452937\pi\)
0.147314 + 0.989090i \(0.452937\pi\)
\(632\) 4.66876e6 0.464952
\(633\) 0 0
\(634\) 8.00053e6 0.790489
\(635\) 0 0
\(636\) 0 0
\(637\) −200981. −0.0196248
\(638\) −1.46528e6 −0.142518
\(639\) 0 0
\(640\) 0 0
\(641\) 1.28989e7 1.23996 0.619981 0.784617i \(-0.287140\pi\)
0.619981 + 0.784617i \(0.287140\pi\)
\(642\) 0 0
\(643\) −9.20262e6 −0.877777 −0.438889 0.898542i \(-0.644628\pi\)
−0.438889 + 0.898542i \(0.644628\pi\)
\(644\) −195998. −0.0186225
\(645\) 0 0
\(646\) −4.42504e6 −0.417192
\(647\) −177907. −0.0167083 −0.00835417 0.999965i \(-0.502659\pi\)
−0.00835417 + 0.999965i \(0.502659\pi\)
\(648\) 0 0
\(649\) −1.73279e6 −0.161485
\(650\) 0 0
\(651\) 0 0
\(652\) 9.26410e6 0.853462
\(653\) 1.11867e7 1.02664 0.513321 0.858197i \(-0.328415\pi\)
0.513321 + 0.858197i \(0.328415\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.44562e6 −0.312613
\(657\) 0 0
\(658\) 8.64893e6 0.778749
\(659\) 2.34591e6 0.210425 0.105213 0.994450i \(-0.466448\pi\)
0.105213 + 0.994450i \(0.466448\pi\)
\(660\) 0 0
\(661\) −3.73844e6 −0.332803 −0.166401 0.986058i \(-0.553215\pi\)
−0.166401 + 0.986058i \(0.553215\pi\)
\(662\) −5.93446e6 −0.526304
\(663\) 0 0
\(664\) 6.45740e6 0.568378
\(665\) 0 0
\(666\) 0 0
\(667\) −279619. −0.0243362
\(668\) 2.04973e6 0.177728
\(669\) 0 0
\(670\) 0 0
\(671\) −3.60410e6 −0.309023
\(672\) 0 0
\(673\) −6.11773e6 −0.520658 −0.260329 0.965520i \(-0.583831\pi\)
−0.260329 + 0.965520i \(0.583831\pi\)
\(674\) −9.69961e6 −0.822440
\(675\) 0 0
\(676\) −5.87977e6 −0.494873
\(677\) −6.50221e6 −0.545242 −0.272621 0.962121i \(-0.587891\pi\)
−0.272621 + 0.962121i \(0.587891\pi\)
\(678\) 0 0
\(679\) −6.16805e6 −0.513421
\(680\) 0 0
\(681\) 0 0
\(682\) −1.72765e6 −0.142231
\(683\) −1.37285e7 −1.12609 −0.563044 0.826427i \(-0.690370\pi\)
−0.563044 + 0.826427i \(0.690370\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7.67722e6 0.622865
\(687\) 0 0
\(688\) 1.20258e6 0.0968595
\(689\) −30767.5 −0.00246913
\(690\) 0 0
\(691\) 1.36316e7 1.08606 0.543028 0.839715i \(-0.317278\pi\)
0.543028 + 0.839715i \(0.317278\pi\)
\(692\) 7.79499e6 0.618800
\(693\) 0 0
\(694\) 5.06726e6 0.399369
\(695\) 0 0
\(696\) 0 0
\(697\) −2.24837e7 −1.75302
\(698\) 1.01703e7 0.790123
\(699\) 0 0
\(700\) 0 0
\(701\) 2.08567e6 0.160306 0.0801531 0.996783i \(-0.474459\pi\)
0.0801531 + 0.996783i \(0.474459\pi\)
\(702\) 0 0
\(703\) 6.78767e6 0.518003
\(704\) 464061. 0.0352894
\(705\) 0 0
\(706\) −5.14037e6 −0.388135
\(707\) 7.07763e6 0.532524
\(708\) 0 0
\(709\) 1.23349e7 0.921554 0.460777 0.887516i \(-0.347571\pi\)
0.460777 + 0.887516i \(0.347571\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −9.38968e6 −0.694146
\(713\) −329687. −0.0242872
\(714\) 0 0
\(715\) 0 0
\(716\) −1.35718e7 −0.989359
\(717\) 0 0
\(718\) 1.66500e7 1.20533
\(719\) −2.83517e6 −0.204530 −0.102265 0.994757i \(-0.532609\pi\)
−0.102265 + 0.994757i \(0.532609\pi\)
\(720\) 0 0
\(721\) −7.16450e6 −0.513272
\(722\) −8.15015e6 −0.581865
\(723\) 0 0
\(724\) −9.20677e6 −0.652771
\(725\) 0 0
\(726\) 0 0
\(727\) 1.19423e7 0.838017 0.419009 0.907982i \(-0.362378\pi\)
0.419009 + 0.907982i \(0.362378\pi\)
\(728\) 559374. 0.0391178
\(729\) 0 0
\(730\) 0 0
\(731\) 7.84721e6 0.543152
\(732\) 0 0
\(733\) −2.50999e7 −1.72549 −0.862744 0.505642i \(-0.831256\pi\)
−0.862744 + 0.505642i \(0.831256\pi\)
\(734\) 4.52265e6 0.309851
\(735\) 0 0
\(736\) 88556.7 0.00602597
\(737\) −5.58267e6 −0.378593
\(738\) 0 0
\(739\) −1.66189e6 −0.111942 −0.0559708 0.998432i \(-0.517825\pi\)
−0.0559708 + 0.998432i \(0.517825\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −282521. −0.0188383
\(743\) 988158. 0.0656681 0.0328340 0.999461i \(-0.489547\pi\)
0.0328340 + 0.999461i \(0.489547\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.24948e6 0.0822022
\(747\) 0 0
\(748\) 3.02815e6 0.197890
\(749\) 2.71331e7 1.76724
\(750\) 0 0
\(751\) 2.42260e7 1.56741 0.783704 0.621135i \(-0.213328\pi\)
0.783704 + 0.621135i \(0.213328\pi\)
\(752\) −3.90779e6 −0.251992
\(753\) 0 0
\(754\) 798026. 0.0511198
\(755\) 0 0
\(756\) 0 0
\(757\) 8.03239e6 0.509454 0.254727 0.967013i \(-0.418014\pi\)
0.254727 + 0.967013i \(0.418014\pi\)
\(758\) −8.79960e6 −0.556275
\(759\) 0 0
\(760\) 0 0
\(761\) −8.44956e6 −0.528899 −0.264449 0.964400i \(-0.585190\pi\)
−0.264449 + 0.964400i \(0.585190\pi\)
\(762\) 0 0
\(763\) −1.01323e7 −0.630083
\(764\) 293803. 0.0182105
\(765\) 0 0
\(766\) −1.57333e7 −0.968828
\(767\) 943716. 0.0579232
\(768\) 0 0
\(769\) −1.68674e7 −1.02857 −0.514285 0.857620i \(-0.671943\pi\)
−0.514285 + 0.857620i \(0.671943\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.59220e6 0.337707
\(773\) −2.37205e7 −1.42782 −0.713912 0.700236i \(-0.753078\pi\)
−0.713912 + 0.700236i \(0.753078\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.78687e6 0.166136
\(777\) 0 0
\(778\) 1.68237e7 0.996490
\(779\) 8.91339e6 0.526259
\(780\) 0 0
\(781\) 2.38575e6 0.139958
\(782\) 577861. 0.0337914
\(783\) 0 0
\(784\) 833840. 0.0484499
\(785\) 0 0
\(786\) 0 0
\(787\) 2.21145e7 1.27274 0.636370 0.771384i \(-0.280435\pi\)
0.636370 + 0.771384i \(0.280435\pi\)
\(788\) −5.94865e6 −0.341274
\(789\) 0 0
\(790\) 0 0
\(791\) −2.31446e7 −1.31525
\(792\) 0 0
\(793\) 1.96288e6 0.110844
\(794\) 9.57936e6 0.539244
\(795\) 0 0
\(796\) 4.80910e6 0.269018
\(797\) 1.86780e6 0.104156 0.0520779 0.998643i \(-0.483416\pi\)
0.0520779 + 0.998643i \(0.483416\pi\)
\(798\) 0 0
\(799\) −2.54996e7 −1.41308
\(800\) 0 0
\(801\) 0 0
\(802\) 1.76864e7 0.970967
\(803\) −4.46928e6 −0.244596
\(804\) 0 0
\(805\) 0 0
\(806\) 940918. 0.0510169
\(807\) 0 0
\(808\) −3.19784e6 −0.172317
\(809\) −203197. −0.0109156 −0.00545778 0.999985i \(-0.501737\pi\)
−0.00545778 + 0.999985i \(0.501737\pi\)
\(810\) 0 0
\(811\) 2.27341e7 1.21374 0.606870 0.794801i \(-0.292425\pi\)
0.606870 + 0.794801i \(0.292425\pi\)
\(812\) 7.32785e6 0.390019
\(813\) 0 0
\(814\) −4.64494e6 −0.245708
\(815\) 0 0
\(816\) 0 0
\(817\) −3.11092e6 −0.163055
\(818\) −1.04430e7 −0.545686
\(819\) 0 0
\(820\) 0 0
\(821\) 2.56056e7 1.32579 0.662897 0.748710i \(-0.269327\pi\)
0.662897 + 0.748710i \(0.269327\pi\)
\(822\) 0 0
\(823\) 7.10878e6 0.365844 0.182922 0.983127i \(-0.441444\pi\)
0.182922 + 0.983127i \(0.441444\pi\)
\(824\) 3.23709e6 0.166087
\(825\) 0 0
\(826\) 8.66563e6 0.441927
\(827\) 2.69334e7 1.36939 0.684695 0.728830i \(-0.259935\pi\)
0.684695 + 0.728830i \(0.259935\pi\)
\(828\) 0 0
\(829\) 2.89570e7 1.46341 0.731707 0.681619i \(-0.238724\pi\)
0.731707 + 0.681619i \(0.238724\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −252739. −0.0126579
\(833\) 5.44107e6 0.271689
\(834\) 0 0
\(835\) 0 0
\(836\) −1.20047e6 −0.0594067
\(837\) 0 0
\(838\) 4.06498e6 0.199962
\(839\) 1.95591e7 0.959279 0.479639 0.877466i \(-0.340767\pi\)
0.479639 + 0.877466i \(0.340767\pi\)
\(840\) 0 0
\(841\) −1.00569e7 −0.490316
\(842\) 2.38766e7 1.16063
\(843\) 0 0
\(844\) 951987. 0.0460018
\(845\) 0 0
\(846\) 0 0
\(847\) 2.09944e7 1.00553
\(848\) 127650. 0.00609580
\(849\) 0 0
\(850\) 0 0
\(851\) −886393. −0.0419568
\(852\) 0 0
\(853\) −1.74251e7 −0.819981 −0.409990 0.912090i \(-0.634468\pi\)
−0.409990 + 0.912090i \(0.634468\pi\)
\(854\) 1.80241e7 0.845684
\(855\) 0 0
\(856\) −1.22594e7 −0.571852
\(857\) 2.03324e7 0.945663 0.472831 0.881153i \(-0.343232\pi\)
0.472831 + 0.881153i \(0.343232\pi\)
\(858\) 0 0
\(859\) −1.19607e7 −0.553061 −0.276530 0.961005i \(-0.589185\pi\)
−0.276530 + 0.961005i \(0.589185\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.18858e7 −0.544829
\(863\) −3.87101e7 −1.76928 −0.884640 0.466274i \(-0.845596\pi\)
−0.884640 + 0.466274i \(0.845596\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.98095e7 0.897590
\(867\) 0 0
\(868\) 8.63995e6 0.389235
\(869\) 8.26488e6 0.371268
\(870\) 0 0
\(871\) 3.04045e6 0.135798
\(872\) 4.57802e6 0.203886
\(873\) 0 0
\(874\) −229085. −0.0101442
\(875\) 0 0
\(876\) 0 0
\(877\) −3.76410e7 −1.65258 −0.826289 0.563247i \(-0.809552\pi\)
−0.826289 + 0.563247i \(0.809552\pi\)
\(878\) −4.48459e6 −0.196330
\(879\) 0 0
\(880\) 0 0
\(881\) 2.92452e6 0.126945 0.0634723 0.997984i \(-0.479783\pi\)
0.0634723 + 0.997984i \(0.479783\pi\)
\(882\) 0 0
\(883\) 3.90249e7 1.68438 0.842190 0.539181i \(-0.181266\pi\)
0.842190 + 0.539181i \(0.181266\pi\)
\(884\) −1.64920e6 −0.0709811
\(885\) 0 0
\(886\) 1.06493e7 0.455761
\(887\) 1.16063e7 0.495318 0.247659 0.968847i \(-0.420339\pi\)
0.247659 + 0.968847i \(0.420339\pi\)
\(888\) 0 0
\(889\) −4.86571e6 −0.206487
\(890\) 0 0
\(891\) 0 0
\(892\) 1.38580e7 0.583160
\(893\) 1.01090e7 0.424208
\(894\) 0 0
\(895\) 0 0
\(896\) −2.32076e6 −0.0965741
\(897\) 0 0
\(898\) −1.55513e7 −0.643538
\(899\) 1.23261e7 0.508659
\(900\) 0 0
\(901\) 832955. 0.0341830
\(902\) −6.09962e6 −0.249624
\(903\) 0 0
\(904\) 1.04573e7 0.425597
\(905\) 0 0
\(906\) 0 0
\(907\) 3.80327e7 1.53511 0.767554 0.640984i \(-0.221473\pi\)
0.767554 + 0.640984i \(0.221473\pi\)
\(908\) −1.17929e7 −0.474687
\(909\) 0 0
\(910\) 0 0
\(911\) 4.76995e7 1.90422 0.952112 0.305748i \(-0.0989066\pi\)
0.952112 + 0.305748i \(0.0989066\pi\)
\(912\) 0 0
\(913\) 1.14312e7 0.453854
\(914\) −1.01052e7 −0.400110
\(915\) 0 0
\(916\) 1.35440e7 0.533346
\(917\) 4.94767e7 1.94302
\(918\) 0 0
\(919\) −3.06445e7 −1.19692 −0.598459 0.801154i \(-0.704220\pi\)
−0.598459 + 0.801154i \(0.704220\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.85436e7 0.718400
\(923\) −1.29934e6 −0.0502016
\(924\) 0 0
\(925\) 0 0
\(926\) −1.44952e7 −0.555517
\(927\) 0 0
\(928\) −3.31090e6 −0.126205
\(929\) 4.20442e7 1.59833 0.799165 0.601112i \(-0.205275\pi\)
0.799165 + 0.601112i \(0.205275\pi\)
\(930\) 0 0
\(931\) −2.15704e6 −0.0815614
\(932\) −9.45216e6 −0.356444
\(933\) 0 0
\(934\) 1.49103e7 0.559267
\(935\) 0 0
\(936\) 0 0
\(937\) −2.09921e7 −0.781101 −0.390551 0.920582i \(-0.627715\pi\)
−0.390551 + 0.920582i \(0.627715\pi\)
\(938\) 2.79188e7 1.03607
\(939\) 0 0
\(940\) 0 0
\(941\) 1.06080e7 0.390536 0.195268 0.980750i \(-0.437442\pi\)
0.195268 + 0.980750i \(0.437442\pi\)
\(942\) 0 0
\(943\) −1.16399e6 −0.0426255
\(944\) −3.91534e6 −0.143001
\(945\) 0 0
\(946\) 2.12887e6 0.0773430
\(947\) −2.66948e7 −0.967280 −0.483640 0.875267i \(-0.660686\pi\)
−0.483640 + 0.875267i \(0.660686\pi\)
\(948\) 0 0
\(949\) 2.43408e6 0.0877341
\(950\) 0 0
\(951\) 0 0
\(952\) −1.51437e7 −0.541552
\(953\) −2.68015e7 −0.955932 −0.477966 0.878378i \(-0.658626\pi\)
−0.477966 + 0.878378i \(0.658626\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2.56600e7 0.908054
\(957\) 0 0
\(958\) −2.75350e7 −0.969328
\(959\) 3.88589e7 1.36441
\(960\) 0 0
\(961\) −1.40960e7 −0.492364
\(962\) 2.52974e6 0.0881331
\(963\) 0 0
\(964\) −5.55314e6 −0.192463
\(965\) 0 0
\(966\) 0 0
\(967\) 3.43270e7 1.18051 0.590255 0.807217i \(-0.299027\pi\)
0.590255 + 0.807217i \(0.299027\pi\)
\(968\) −9.48576e6 −0.325375
\(969\) 0 0
\(970\) 0 0
\(971\) −1.43606e7 −0.488791 −0.244396 0.969676i \(-0.578590\pi\)
−0.244396 + 0.969676i \(0.578590\pi\)
\(972\) 0 0
\(973\) 6.77676e6 0.229477
\(974\) 3.43596e7 1.16051
\(975\) 0 0
\(976\) −8.14370e6 −0.273651
\(977\) −1.16248e7 −0.389627 −0.194813 0.980840i \(-0.562410\pi\)
−0.194813 + 0.980840i \(0.562410\pi\)
\(978\) 0 0
\(979\) −1.66221e7 −0.554280
\(980\) 0 0
\(981\) 0 0
\(982\) −5.87996e6 −0.194579
\(983\) −3.11250e6 −0.102737 −0.0513683 0.998680i \(-0.516358\pi\)
−0.0513683 + 0.998680i \(0.516358\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.16046e7 −0.707709
\(987\) 0 0
\(988\) 653804. 0.0213086
\(989\) 406251. 0.0132070
\(990\) 0 0
\(991\) −2.33325e7 −0.754704 −0.377352 0.926070i \(-0.623165\pi\)
−0.377352 + 0.926070i \(0.623165\pi\)
\(992\) −3.90373e6 −0.125951
\(993\) 0 0
\(994\) −1.19311e7 −0.383014
\(995\) 0 0
\(996\) 0 0
\(997\) −5.35497e7 −1.70616 −0.853079 0.521782i \(-0.825267\pi\)
−0.853079 + 0.521782i \(0.825267\pi\)
\(998\) −1.25217e7 −0.397958
\(999\) 0 0
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 450.6.a.bf.1.1 yes 2
3.2 odd 2 450.6.a.ba.1.1 yes 2
5.2 odd 4 450.6.c.p.199.3 4
5.3 odd 4 450.6.c.p.199.2 4
5.4 even 2 450.6.a.y.1.2 2
15.2 even 4 450.6.c.q.199.1 4
15.8 even 4 450.6.c.q.199.4 4
15.14 odd 2 450.6.a.bd.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.6.a.y.1.2 2 5.4 even 2
450.6.a.ba.1.1 yes 2 3.2 odd 2
450.6.a.bd.1.2 yes 2 15.14 odd 2
450.6.a.bf.1.1 yes 2 1.1 even 1 trivial
450.6.c.p.199.2 4 5.3 odd 4
450.6.c.p.199.3 4 5.2 odd 4
450.6.c.q.199.1 4 15.2 even 4
450.6.c.q.199.4 4 15.8 even 4