Properties

Label 405.4.a.k.1.1
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.23336\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +O(q^{10})\) \(q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +26.1668 q^{10} -6.08311 q^{11} -64.4507 q^{13} -172.795 q^{14} +156.794 q^{16} +76.9845 q^{17} -118.716 q^{19} -96.9405 q^{20} +31.8351 q^{22} -77.6004 q^{23} +25.0000 q^{25} +337.294 q^{26} +640.157 q^{28} -62.8149 q^{29} -106.914 q^{31} -343.774 q^{32} -402.888 q^{34} -165.090 q^{35} +108.268 q^{37} +621.285 q^{38} +297.990 q^{40} -142.766 q^{41} +339.568 q^{43} -117.940 q^{44} +406.111 q^{46} +598.346 q^{47} +747.191 q^{49} -130.834 q^{50} -1249.58 q^{52} -488.041 q^{53} +30.4155 q^{55} -1967.81 q^{56} +328.733 q^{58} -242.826 q^{59} -499.339 q^{61} +559.519 q^{62} +544.744 q^{64} +322.254 q^{65} -921.635 q^{67} +1492.58 q^{68} +863.977 q^{70} +60.6882 q^{71} -338.439 q^{73} -566.606 q^{74} -2301.68 q^{76} -200.852 q^{77} -556.311 q^{79} -783.969 q^{80} +747.147 q^{82} +64.8420 q^{83} -384.923 q^{85} -1777.08 q^{86} +362.542 q^{88} -941.159 q^{89} -2128.04 q^{91} -1504.52 q^{92} -3131.36 q^{94} +593.581 q^{95} -1042.17 q^{97} -3910.32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.23336 −1.85027 −0.925137 0.379634i \(-0.876050\pi\)
−0.925137 + 0.379634i \(0.876050\pi\)
\(3\) 0 0
\(4\) 19.3881 2.42351
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 33.0180 1.78281 0.891403 0.453211i \(-0.149722\pi\)
0.891403 + 0.453211i \(0.149722\pi\)
\(8\) −59.5981 −2.63389
\(9\) 0 0
\(10\) 26.1668 0.827468
\(11\) −6.08311 −0.166739 −0.0833694 0.996519i \(-0.526568\pi\)
−0.0833694 + 0.996519i \(0.526568\pi\)
\(12\) 0 0
\(13\) −64.4507 −1.37503 −0.687516 0.726169i \(-0.741299\pi\)
−0.687516 + 0.726169i \(0.741299\pi\)
\(14\) −172.795 −3.29868
\(15\) 0 0
\(16\) 156.794 2.44990
\(17\) 76.9845 1.09832 0.549162 0.835716i \(-0.314947\pi\)
0.549162 + 0.835716i \(0.314947\pi\)
\(18\) 0 0
\(19\) −118.716 −1.43344 −0.716720 0.697361i \(-0.754357\pi\)
−0.716720 + 0.697361i \(0.754357\pi\)
\(20\) −96.9405 −1.08383
\(21\) 0 0
\(22\) 31.8351 0.308512
\(23\) −77.6004 −0.703513 −0.351757 0.936092i \(-0.614416\pi\)
−0.351757 + 0.936092i \(0.614416\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 337.294 2.54419
\(27\) 0 0
\(28\) 640.157 4.32065
\(29\) −62.8149 −0.402222 −0.201111 0.979568i \(-0.564455\pi\)
−0.201111 + 0.979568i \(0.564455\pi\)
\(30\) 0 0
\(31\) −106.914 −0.619429 −0.309714 0.950830i \(-0.600233\pi\)
−0.309714 + 0.950830i \(0.600233\pi\)
\(32\) −343.774 −1.89910
\(33\) 0 0
\(34\) −402.888 −2.03220
\(35\) −165.090 −0.797295
\(36\) 0 0
\(37\) 108.268 0.481058 0.240529 0.970642i \(-0.422679\pi\)
0.240529 + 0.970642i \(0.422679\pi\)
\(38\) 621.285 2.65226
\(39\) 0 0
\(40\) 297.990 1.17791
\(41\) −142.766 −0.543813 −0.271906 0.962324i \(-0.587654\pi\)
−0.271906 + 0.962324i \(0.587654\pi\)
\(42\) 0 0
\(43\) 339.568 1.20427 0.602136 0.798394i \(-0.294317\pi\)
0.602136 + 0.798394i \(0.294317\pi\)
\(44\) −117.940 −0.404093
\(45\) 0 0
\(46\) 406.111 1.30169
\(47\) 598.346 1.85697 0.928486 0.371368i \(-0.121111\pi\)
0.928486 + 0.371368i \(0.121111\pi\)
\(48\) 0 0
\(49\) 747.191 2.17840
\(50\) −130.834 −0.370055
\(51\) 0 0
\(52\) −1249.58 −3.33241
\(53\) −488.041 −1.26486 −0.632431 0.774617i \(-0.717943\pi\)
−0.632431 + 0.774617i \(0.717943\pi\)
\(54\) 0 0
\(55\) 30.4155 0.0745678
\(56\) −1967.81 −4.69571
\(57\) 0 0
\(58\) 328.733 0.744221
\(59\) −242.826 −0.535819 −0.267909 0.963444i \(-0.586333\pi\)
−0.267909 + 0.963444i \(0.586333\pi\)
\(60\) 0 0
\(61\) −499.339 −1.04809 −0.524047 0.851689i \(-0.675578\pi\)
−0.524047 + 0.851689i \(0.675578\pi\)
\(62\) 559.519 1.14611
\(63\) 0 0
\(64\) 544.744 1.06395
\(65\) 322.254 0.614933
\(66\) 0 0
\(67\) −921.635 −1.68053 −0.840266 0.542175i \(-0.817601\pi\)
−0.840266 + 0.542175i \(0.817601\pi\)
\(68\) 1492.58 2.66180
\(69\) 0 0
\(70\) 863.977 1.47521
\(71\) 60.6882 0.101442 0.0507209 0.998713i \(-0.483848\pi\)
0.0507209 + 0.998713i \(0.483848\pi\)
\(72\) 0 0
\(73\) −338.439 −0.542621 −0.271311 0.962492i \(-0.587457\pi\)
−0.271311 + 0.962492i \(0.587457\pi\)
\(74\) −566.606 −0.890090
\(75\) 0 0
\(76\) −2301.68 −3.47396
\(77\) −200.852 −0.297263
\(78\) 0 0
\(79\) −556.311 −0.792276 −0.396138 0.918191i \(-0.629650\pi\)
−0.396138 + 0.918191i \(0.629650\pi\)
\(80\) −783.969 −1.09563
\(81\) 0 0
\(82\) 747.147 1.00620
\(83\) 64.8420 0.0857510 0.0428755 0.999080i \(-0.486348\pi\)
0.0428755 + 0.999080i \(0.486348\pi\)
\(84\) 0 0
\(85\) −384.923 −0.491185
\(86\) −1777.08 −2.22823
\(87\) 0 0
\(88\) 362.542 0.439171
\(89\) −941.159 −1.12093 −0.560465 0.828178i \(-0.689377\pi\)
−0.560465 + 0.828178i \(0.689377\pi\)
\(90\) 0 0
\(91\) −2128.04 −2.45142
\(92\) −1504.52 −1.70497
\(93\) 0 0
\(94\) −3131.36 −3.43591
\(95\) 593.581 0.641054
\(96\) 0 0
\(97\) −1042.17 −1.09089 −0.545445 0.838147i \(-0.683639\pi\)
−0.545445 + 0.838147i \(0.683639\pi\)
\(98\) −3910.32 −4.03064
\(99\) 0 0
\(100\) 484.703 0.484703
\(101\) −682.243 −0.672135 −0.336068 0.941838i \(-0.609097\pi\)
−0.336068 + 0.941838i \(0.609097\pi\)
\(102\) 0 0
\(103\) 1213.66 1.16103 0.580514 0.814250i \(-0.302852\pi\)
0.580514 + 0.814250i \(0.302852\pi\)
\(104\) 3841.14 3.62168
\(105\) 0 0
\(106\) 2554.10 2.34034
\(107\) 1311.93 1.18532 0.592658 0.805454i \(-0.298079\pi\)
0.592658 + 0.805454i \(0.298079\pi\)
\(108\) 0 0
\(109\) −294.780 −0.259035 −0.129518 0.991577i \(-0.541343\pi\)
−0.129518 + 0.991577i \(0.541343\pi\)
\(110\) −159.176 −0.137971
\(111\) 0 0
\(112\) 5177.02 4.36770
\(113\) −1585.12 −1.31961 −0.659805 0.751437i \(-0.729361\pi\)
−0.659805 + 0.751437i \(0.729361\pi\)
\(114\) 0 0
\(115\) 388.002 0.314621
\(116\) −1217.86 −0.974790
\(117\) 0 0
\(118\) 1270.80 0.991412
\(119\) 2541.88 1.95810
\(120\) 0 0
\(121\) −1294.00 −0.972198
\(122\) 2613.22 1.93926
\(123\) 0 0
\(124\) −2072.86 −1.50119
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −149.176 −0.104230 −0.0521149 0.998641i \(-0.516596\pi\)
−0.0521149 + 0.998641i \(0.516596\pi\)
\(128\) −100.655 −0.0695057
\(129\) 0 0
\(130\) −1686.47 −1.13779
\(131\) −1171.77 −0.781513 −0.390757 0.920494i \(-0.627787\pi\)
−0.390757 + 0.920494i \(0.627787\pi\)
\(132\) 0 0
\(133\) −3919.77 −2.55555
\(134\) 4823.25 3.10944
\(135\) 0 0
\(136\) −4588.13 −2.89286
\(137\) 1158.63 0.722541 0.361270 0.932461i \(-0.382343\pi\)
0.361270 + 0.932461i \(0.382343\pi\)
\(138\) 0 0
\(139\) 1586.52 0.968110 0.484055 0.875038i \(-0.339163\pi\)
0.484055 + 0.875038i \(0.339163\pi\)
\(140\) −3200.79 −1.93226
\(141\) 0 0
\(142\) −317.603 −0.187695
\(143\) 392.061 0.229271
\(144\) 0 0
\(145\) 314.075 0.179879
\(146\) 1771.18 1.00400
\(147\) 0 0
\(148\) 2099.11 1.16585
\(149\) −517.109 −0.284317 −0.142158 0.989844i \(-0.545404\pi\)
−0.142158 + 0.989844i \(0.545404\pi\)
\(150\) 0 0
\(151\) 555.586 0.299423 0.149712 0.988730i \(-0.452165\pi\)
0.149712 + 0.988730i \(0.452165\pi\)
\(152\) 7075.25 3.77552
\(153\) 0 0
\(154\) 1051.13 0.550018
\(155\) 534.569 0.277017
\(156\) 0 0
\(157\) 1054.03 0.535802 0.267901 0.963446i \(-0.413670\pi\)
0.267901 + 0.963446i \(0.413670\pi\)
\(158\) 2911.38 1.46593
\(159\) 0 0
\(160\) 1718.87 0.849304
\(161\) −2562.21 −1.25423
\(162\) 0 0
\(163\) −635.916 −0.305575 −0.152788 0.988259i \(-0.548825\pi\)
−0.152788 + 0.988259i \(0.548825\pi\)
\(164\) −2767.96 −1.31794
\(165\) 0 0
\(166\) −339.342 −0.158663
\(167\) −1365.11 −0.632549 −0.316274 0.948668i \(-0.602432\pi\)
−0.316274 + 0.948668i \(0.602432\pi\)
\(168\) 0 0
\(169\) 1956.90 0.890713
\(170\) 2014.44 0.908827
\(171\) 0 0
\(172\) 6583.59 2.91857
\(173\) −1134.93 −0.498770 −0.249385 0.968404i \(-0.580228\pi\)
−0.249385 + 0.968404i \(0.580228\pi\)
\(174\) 0 0
\(175\) 825.451 0.356561
\(176\) −953.793 −0.408494
\(177\) 0 0
\(178\) 4925.43 2.07403
\(179\) −1813.33 −0.757179 −0.378589 0.925565i \(-0.623591\pi\)
−0.378589 + 0.925565i \(0.623591\pi\)
\(180\) 0 0
\(181\) 2334.37 0.958634 0.479317 0.877642i \(-0.340884\pi\)
0.479317 + 0.877642i \(0.340884\pi\)
\(182\) 11136.8 4.53579
\(183\) 0 0
\(184\) 4624.84 1.85298
\(185\) −541.340 −0.215136
\(186\) 0 0
\(187\) −468.305 −0.183133
\(188\) 11600.8 4.50039
\(189\) 0 0
\(190\) −3106.42 −1.18612
\(191\) −442.644 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(192\) 0 0
\(193\) −3702.54 −1.38090 −0.690452 0.723378i \(-0.742589\pi\)
−0.690452 + 0.723378i \(0.742589\pi\)
\(194\) 5454.05 2.01844
\(195\) 0 0
\(196\) 14486.6 5.27938
\(197\) −4491.92 −1.62455 −0.812274 0.583276i \(-0.801771\pi\)
−0.812274 + 0.583276i \(0.801771\pi\)
\(198\) 0 0
\(199\) −2934.08 −1.04518 −0.522591 0.852584i \(-0.675034\pi\)
−0.522591 + 0.852584i \(0.675034\pi\)
\(200\) −1489.95 −0.526778
\(201\) 0 0
\(202\) 3570.42 1.24363
\(203\) −2074.03 −0.717084
\(204\) 0 0
\(205\) 713.830 0.243200
\(206\) −6351.54 −2.14822
\(207\) 0 0
\(208\) −10105.5 −3.36869
\(209\) 722.163 0.239010
\(210\) 0 0
\(211\) −673.940 −0.219886 −0.109943 0.993938i \(-0.535067\pi\)
−0.109943 + 0.993938i \(0.535067\pi\)
\(212\) −9462.20 −3.06541
\(213\) 0 0
\(214\) −6865.79 −2.19316
\(215\) −1697.84 −0.538567
\(216\) 0 0
\(217\) −3530.08 −1.10432
\(218\) 1542.69 0.479286
\(219\) 0 0
\(220\) 589.700 0.180716
\(221\) −4961.71 −1.51023
\(222\) 0 0
\(223\) 178.516 0.0536068 0.0268034 0.999641i \(-0.491467\pi\)
0.0268034 + 0.999641i \(0.491467\pi\)
\(224\) −11350.7 −3.38573
\(225\) 0 0
\(226\) 8295.53 2.44164
\(227\) −5052.11 −1.47718 −0.738590 0.674154i \(-0.764508\pi\)
−0.738590 + 0.674154i \(0.764508\pi\)
\(228\) 0 0
\(229\) 1308.97 0.377725 0.188863 0.982004i \(-0.439520\pi\)
0.188863 + 0.982004i \(0.439520\pi\)
\(230\) −2030.56 −0.582134
\(231\) 0 0
\(232\) 3743.65 1.05941
\(233\) 6591.30 1.85326 0.926632 0.375970i \(-0.122691\pi\)
0.926632 + 0.375970i \(0.122691\pi\)
\(234\) 0 0
\(235\) −2991.73 −0.830463
\(236\) −4707.95 −1.29856
\(237\) 0 0
\(238\) −13302.6 −3.62302
\(239\) −1584.50 −0.428841 −0.214421 0.976741i \(-0.568786\pi\)
−0.214421 + 0.976741i \(0.568786\pi\)
\(240\) 0 0
\(241\) −1674.47 −0.447560 −0.223780 0.974640i \(-0.571840\pi\)
−0.223780 + 0.974640i \(0.571840\pi\)
\(242\) 6771.95 1.79883
\(243\) 0 0
\(244\) −9681.23 −2.54007
\(245\) −3735.95 −0.974210
\(246\) 0 0
\(247\) 7651.34 1.97102
\(248\) 6371.86 1.63151
\(249\) 0 0
\(250\) 654.171 0.165494
\(251\) 1200.75 0.301955 0.150978 0.988537i \(-0.451758\pi\)
0.150978 + 0.988537i \(0.451758\pi\)
\(252\) 0 0
\(253\) 472.052 0.117303
\(254\) 780.690 0.192854
\(255\) 0 0
\(256\) −3831.19 −0.935350
\(257\) 5206.69 1.26375 0.631876 0.775070i \(-0.282285\pi\)
0.631876 + 0.775070i \(0.282285\pi\)
\(258\) 0 0
\(259\) 3574.80 0.857634
\(260\) 6247.89 1.49030
\(261\) 0 0
\(262\) 6132.31 1.44601
\(263\) −7683.56 −1.80148 −0.900739 0.434362i \(-0.856974\pi\)
−0.900739 + 0.434362i \(0.856974\pi\)
\(264\) 0 0
\(265\) 2440.21 0.565663
\(266\) 20513.6 4.72846
\(267\) 0 0
\(268\) −17868.7 −4.07279
\(269\) 624.277 0.141497 0.0707487 0.997494i \(-0.477461\pi\)
0.0707487 + 0.997494i \(0.477461\pi\)
\(270\) 0 0
\(271\) −1462.51 −0.327828 −0.163914 0.986475i \(-0.552412\pi\)
−0.163914 + 0.986475i \(0.552412\pi\)
\(272\) 12070.7 2.69078
\(273\) 0 0
\(274\) −6063.51 −1.33690
\(275\) −152.078 −0.0333477
\(276\) 0 0
\(277\) −5898.04 −1.27935 −0.639673 0.768647i \(-0.720930\pi\)
−0.639673 + 0.768647i \(0.720930\pi\)
\(278\) −8302.86 −1.79127
\(279\) 0 0
\(280\) 9839.06 2.09999
\(281\) 5272.91 1.11941 0.559707 0.828691i \(-0.310914\pi\)
0.559707 + 0.828691i \(0.310914\pi\)
\(282\) 0 0
\(283\) −1119.26 −0.235100 −0.117550 0.993067i \(-0.537504\pi\)
−0.117550 + 0.993067i \(0.537504\pi\)
\(284\) 1176.63 0.245845
\(285\) 0 0
\(286\) −2051.80 −0.424214
\(287\) −4713.86 −0.969513
\(288\) 0 0
\(289\) 1013.62 0.206313
\(290\) −1643.67 −0.332826
\(291\) 0 0
\(292\) −6561.70 −1.31505
\(293\) 940.740 0.187572 0.0937860 0.995592i \(-0.470103\pi\)
0.0937860 + 0.995592i \(0.470103\pi\)
\(294\) 0 0
\(295\) 1214.13 0.239626
\(296\) −6452.57 −1.26705
\(297\) 0 0
\(298\) 2706.22 0.526064
\(299\) 5001.40 0.967353
\(300\) 0 0
\(301\) 11211.9 2.14698
\(302\) −2907.58 −0.554015
\(303\) 0 0
\(304\) −18613.9 −3.51179
\(305\) 2496.69 0.468722
\(306\) 0 0
\(307\) 1931.88 0.359148 0.179574 0.983744i \(-0.442528\pi\)
0.179574 + 0.983744i \(0.442528\pi\)
\(308\) −3894.15 −0.720421
\(309\) 0 0
\(310\) −2797.59 −0.512557
\(311\) 6298.16 1.14835 0.574174 0.818734i \(-0.305323\pi\)
0.574174 + 0.818734i \(0.305323\pi\)
\(312\) 0 0
\(313\) −1818.81 −0.328450 −0.164225 0.986423i \(-0.552512\pi\)
−0.164225 + 0.986423i \(0.552512\pi\)
\(314\) −5516.13 −0.991380
\(315\) 0 0
\(316\) −10785.8 −1.92009
\(317\) −3158.64 −0.559643 −0.279821 0.960052i \(-0.590275\pi\)
−0.279821 + 0.960052i \(0.590275\pi\)
\(318\) 0 0
\(319\) 382.110 0.0670660
\(320\) −2723.72 −0.475815
\(321\) 0 0
\(322\) 13409.0 2.32066
\(323\) −9139.31 −1.57438
\(324\) 0 0
\(325\) −1611.27 −0.275006
\(326\) 3327.98 0.565398
\(327\) 0 0
\(328\) 8508.59 1.43234
\(329\) 19756.2 3.31062
\(330\) 0 0
\(331\) −953.090 −0.158268 −0.0791338 0.996864i \(-0.525215\pi\)
−0.0791338 + 0.996864i \(0.525215\pi\)
\(332\) 1257.16 0.207819
\(333\) 0 0
\(334\) 7144.14 1.17039
\(335\) 4608.17 0.751556
\(336\) 0 0
\(337\) −568.893 −0.0919572 −0.0459786 0.998942i \(-0.514641\pi\)
−0.0459786 + 0.998942i \(0.514641\pi\)
\(338\) −10241.1 −1.64806
\(339\) 0 0
\(340\) −7462.92 −1.19039
\(341\) 650.368 0.103283
\(342\) 0 0
\(343\) 13345.6 2.10086
\(344\) −20237.6 −3.17192
\(345\) 0 0
\(346\) 5939.50 0.922860
\(347\) −8426.55 −1.30363 −0.651817 0.758376i \(-0.725993\pi\)
−0.651817 + 0.758376i \(0.725993\pi\)
\(348\) 0 0
\(349\) 7143.96 1.09572 0.547861 0.836569i \(-0.315442\pi\)
0.547861 + 0.836569i \(0.315442\pi\)
\(350\) −4319.89 −0.659736
\(351\) 0 0
\(352\) 2091.21 0.316654
\(353\) −5300.69 −0.799228 −0.399614 0.916684i \(-0.630856\pi\)
−0.399614 + 0.916684i \(0.630856\pi\)
\(354\) 0 0
\(355\) −303.441 −0.0453661
\(356\) −18247.3 −2.71659
\(357\) 0 0
\(358\) 9489.84 1.40099
\(359\) 9536.35 1.40198 0.700988 0.713173i \(-0.252743\pi\)
0.700988 + 0.713173i \(0.252743\pi\)
\(360\) 0 0
\(361\) 7234.52 1.05475
\(362\) −12216.6 −1.77373
\(363\) 0 0
\(364\) −41258.6 −5.94104
\(365\) 1692.20 0.242668
\(366\) 0 0
\(367\) −730.208 −0.103860 −0.0519299 0.998651i \(-0.516537\pi\)
−0.0519299 + 0.998651i \(0.516537\pi\)
\(368\) −12167.3 −1.72354
\(369\) 0 0
\(370\) 2833.03 0.398060
\(371\) −16114.2 −2.25500
\(372\) 0 0
\(373\) 12868.9 1.78640 0.893201 0.449657i \(-0.148454\pi\)
0.893201 + 0.449657i \(0.148454\pi\)
\(374\) 2450.81 0.338846
\(375\) 0 0
\(376\) −35660.3 −4.89106
\(377\) 4048.47 0.553068
\(378\) 0 0
\(379\) −4000.88 −0.542246 −0.271123 0.962545i \(-0.587395\pi\)
−0.271123 + 0.962545i \(0.587395\pi\)
\(380\) 11508.4 1.55360
\(381\) 0 0
\(382\) 2316.52 0.310270
\(383\) 65.3199 0.00871459 0.00435730 0.999991i \(-0.498613\pi\)
0.00435730 + 0.999991i \(0.498613\pi\)
\(384\) 0 0
\(385\) 1004.26 0.132940
\(386\) 19376.7 2.55505
\(387\) 0 0
\(388\) −20205.7 −2.64379
\(389\) −5919.77 −0.771579 −0.385790 0.922587i \(-0.626071\pi\)
−0.385790 + 0.922587i \(0.626071\pi\)
\(390\) 0 0
\(391\) −5974.03 −0.772685
\(392\) −44531.2 −5.73766
\(393\) 0 0
\(394\) 23507.8 3.00586
\(395\) 2781.55 0.354317
\(396\) 0 0
\(397\) −10228.7 −1.29310 −0.646551 0.762871i \(-0.723789\pi\)
−0.646551 + 0.762871i \(0.723789\pi\)
\(398\) 15355.1 1.93387
\(399\) 0 0
\(400\) 3919.84 0.489980
\(401\) −12245.1 −1.52492 −0.762460 0.647036i \(-0.776008\pi\)
−0.762460 + 0.647036i \(0.776008\pi\)
\(402\) 0 0
\(403\) 6890.67 0.851734
\(404\) −13227.4 −1.62893
\(405\) 0 0
\(406\) 10854.1 1.32680
\(407\) −658.606 −0.0802110
\(408\) 0 0
\(409\) 9832.95 1.18877 0.594386 0.804180i \(-0.297395\pi\)
0.594386 + 0.804180i \(0.297395\pi\)
\(410\) −3735.73 −0.449987
\(411\) 0 0
\(412\) 23530.6 2.81377
\(413\) −8017.65 −0.955262
\(414\) 0 0
\(415\) −324.210 −0.0383490
\(416\) 22156.5 2.61132
\(417\) 0 0
\(418\) −3779.34 −0.442234
\(419\) −12315.1 −1.43588 −0.717941 0.696104i \(-0.754915\pi\)
−0.717941 + 0.696104i \(0.754915\pi\)
\(420\) 0 0
\(421\) −4318.42 −0.499921 −0.249961 0.968256i \(-0.580418\pi\)
−0.249961 + 0.968256i \(0.580418\pi\)
\(422\) 3526.97 0.406849
\(423\) 0 0
\(424\) 29086.3 3.33150
\(425\) 1924.61 0.219665
\(426\) 0 0
\(427\) −16487.2 −1.86855
\(428\) 25435.8 2.87263
\(429\) 0 0
\(430\) 8885.42 0.996496
\(431\) −5265.39 −0.588457 −0.294228 0.955735i \(-0.595063\pi\)
−0.294228 + 0.955735i \(0.595063\pi\)
\(432\) 0 0
\(433\) 5855.07 0.649831 0.324916 0.945743i \(-0.394664\pi\)
0.324916 + 0.945743i \(0.394664\pi\)
\(434\) 18474.2 2.04330
\(435\) 0 0
\(436\) −5715.23 −0.627775
\(437\) 9212.42 1.00844
\(438\) 0 0
\(439\) 4022.69 0.437341 0.218671 0.975799i \(-0.429828\pi\)
0.218671 + 0.975799i \(0.429828\pi\)
\(440\) −1812.71 −0.196403
\(441\) 0 0
\(442\) 25966.4 2.79434
\(443\) −7502.23 −0.804608 −0.402304 0.915506i \(-0.631791\pi\)
−0.402304 + 0.915506i \(0.631791\pi\)
\(444\) 0 0
\(445\) 4705.80 0.501295
\(446\) −934.240 −0.0991873
\(447\) 0 0
\(448\) 17986.4 1.89682
\(449\) 18574.2 1.95227 0.976135 0.217166i \(-0.0696813\pi\)
0.976135 + 0.217166i \(0.0696813\pi\)
\(450\) 0 0
\(451\) 868.461 0.0906746
\(452\) −30732.5 −3.19809
\(453\) 0 0
\(454\) 26439.5 2.73319
\(455\) 10640.2 1.09631
\(456\) 0 0
\(457\) 14145.6 1.44792 0.723962 0.689840i \(-0.242319\pi\)
0.723962 + 0.689840i \(0.242319\pi\)
\(458\) −6850.31 −0.698895
\(459\) 0 0
\(460\) 7522.62 0.762487
\(461\) 4910.75 0.496131 0.248066 0.968743i \(-0.420205\pi\)
0.248066 + 0.968743i \(0.420205\pi\)
\(462\) 0 0
\(463\) 9255.45 0.929022 0.464511 0.885567i \(-0.346230\pi\)
0.464511 + 0.885567i \(0.346230\pi\)
\(464\) −9848.98 −0.985404
\(465\) 0 0
\(466\) −34494.7 −3.42904
\(467\) 2825.10 0.279936 0.139968 0.990156i \(-0.455300\pi\)
0.139968 + 0.990156i \(0.455300\pi\)
\(468\) 0 0
\(469\) −30430.6 −2.99606
\(470\) 15656.8 1.53658
\(471\) 0 0
\(472\) 14472.0 1.41129
\(473\) −2065.63 −0.200799
\(474\) 0 0
\(475\) −2967.90 −0.286688
\(476\) 49282.2 4.74547
\(477\) 0 0
\(478\) 8292.29 0.793474
\(479\) −9173.48 −0.875046 −0.437523 0.899207i \(-0.644144\pi\)
−0.437523 + 0.899207i \(0.644144\pi\)
\(480\) 0 0
\(481\) −6977.95 −0.661470
\(482\) 8763.10 0.828109
\(483\) 0 0
\(484\) −25088.1 −2.35613
\(485\) 5210.85 0.487861
\(486\) 0 0
\(487\) 13663.7 1.27138 0.635690 0.771945i \(-0.280716\pi\)
0.635690 + 0.771945i \(0.280716\pi\)
\(488\) 29759.6 2.76056
\(489\) 0 0
\(490\) 19551.6 1.80255
\(491\) 5850.31 0.537721 0.268860 0.963179i \(-0.413353\pi\)
0.268860 + 0.963179i \(0.413353\pi\)
\(492\) 0 0
\(493\) −4835.78 −0.441770
\(494\) −40042.2 −3.64694
\(495\) 0 0
\(496\) −16763.4 −1.51754
\(497\) 2003.81 0.180851
\(498\) 0 0
\(499\) 3842.97 0.344760 0.172380 0.985031i \(-0.444854\pi\)
0.172380 + 0.985031i \(0.444854\pi\)
\(500\) −2423.51 −0.216766
\(501\) 0 0
\(502\) −6283.97 −0.558700
\(503\) −4242.88 −0.376104 −0.188052 0.982159i \(-0.560217\pi\)
−0.188052 + 0.982159i \(0.560217\pi\)
\(504\) 0 0
\(505\) 3411.21 0.300588
\(506\) −2470.42 −0.217042
\(507\) 0 0
\(508\) −2892.23 −0.252602
\(509\) −7365.60 −0.641404 −0.320702 0.947180i \(-0.603919\pi\)
−0.320702 + 0.947180i \(0.603919\pi\)
\(510\) 0 0
\(511\) −11174.6 −0.967388
\(512\) 20855.3 1.80016
\(513\) 0 0
\(514\) −27248.5 −2.33829
\(515\) −6068.32 −0.519227
\(516\) 0 0
\(517\) −3639.80 −0.309629
\(518\) −18708.2 −1.58686
\(519\) 0 0
\(520\) −19205.7 −1.61966
\(521\) 8717.87 0.733084 0.366542 0.930401i \(-0.380542\pi\)
0.366542 + 0.930401i \(0.380542\pi\)
\(522\) 0 0
\(523\) 23515.4 1.96607 0.983037 0.183407i \(-0.0587127\pi\)
0.983037 + 0.183407i \(0.0587127\pi\)
\(524\) −22718.4 −1.89401
\(525\) 0 0
\(526\) 40210.9 3.33323
\(527\) −8230.71 −0.680333
\(528\) 0 0
\(529\) −6145.18 −0.505069
\(530\) −12770.5 −1.04663
\(531\) 0 0
\(532\) −75997.0 −6.19340
\(533\) 9201.38 0.747760
\(534\) 0 0
\(535\) −6559.63 −0.530089
\(536\) 54927.7 4.42633
\(537\) 0 0
\(538\) −3267.07 −0.261809
\(539\) −4545.24 −0.363224
\(540\) 0 0
\(541\) 3803.94 0.302300 0.151150 0.988511i \(-0.451702\pi\)
0.151150 + 0.988511i \(0.451702\pi\)
\(542\) 7653.87 0.606572
\(543\) 0 0
\(544\) −26465.3 −2.08583
\(545\) 1473.90 0.115844
\(546\) 0 0
\(547\) 17335.9 1.35508 0.677540 0.735486i \(-0.263046\pi\)
0.677540 + 0.735486i \(0.263046\pi\)
\(548\) 22463.6 1.75109
\(549\) 0 0
\(550\) 795.878 0.0617025
\(551\) 7457.14 0.576561
\(552\) 0 0
\(553\) −18368.3 −1.41248
\(554\) 30866.6 2.36714
\(555\) 0 0
\(556\) 30759.7 2.34623
\(557\) −7797.62 −0.593170 −0.296585 0.955006i \(-0.595848\pi\)
−0.296585 + 0.955006i \(0.595848\pi\)
\(558\) 0 0
\(559\) −21885.4 −1.65591
\(560\) −25885.1 −1.95330
\(561\) 0 0
\(562\) −27595.1 −2.07122
\(563\) 11535.6 0.863532 0.431766 0.901986i \(-0.357891\pi\)
0.431766 + 0.901986i \(0.357891\pi\)
\(564\) 0 0
\(565\) 7925.62 0.590147
\(566\) 5857.52 0.435000
\(567\) 0 0
\(568\) −3616.90 −0.267186
\(569\) 10077.7 0.742496 0.371248 0.928534i \(-0.378930\pi\)
0.371248 + 0.928534i \(0.378930\pi\)
\(570\) 0 0
\(571\) −16339.3 −1.19751 −0.598756 0.800932i \(-0.704338\pi\)
−0.598756 + 0.800932i \(0.704338\pi\)
\(572\) 7601.31 0.555641
\(573\) 0 0
\(574\) 24669.3 1.79386
\(575\) −1940.01 −0.140703
\(576\) 0 0
\(577\) 18648.6 1.34550 0.672748 0.739871i \(-0.265114\pi\)
0.672748 + 0.739871i \(0.265114\pi\)
\(578\) −5304.63 −0.381736
\(579\) 0 0
\(580\) 6089.31 0.435939
\(581\) 2140.95 0.152877
\(582\) 0 0
\(583\) 2968.81 0.210901
\(584\) 20170.3 1.42920
\(585\) 0 0
\(586\) −4923.24 −0.347060
\(587\) 18097.1 1.27248 0.636241 0.771491i \(-0.280488\pi\)
0.636241 + 0.771491i \(0.280488\pi\)
\(588\) 0 0
\(589\) 12692.4 0.887913
\(590\) −6354.00 −0.443373
\(591\) 0 0
\(592\) 16975.7 1.17855
\(593\) −27154.1 −1.88041 −0.940207 0.340602i \(-0.889369\pi\)
−0.940207 + 0.340602i \(0.889369\pi\)
\(594\) 0 0
\(595\) −12709.4 −0.875688
\(596\) −10025.8 −0.689046
\(597\) 0 0
\(598\) −26174.2 −1.78987
\(599\) −3457.46 −0.235840 −0.117920 0.993023i \(-0.537623\pi\)
−0.117920 + 0.993023i \(0.537623\pi\)
\(600\) 0 0
\(601\) −2938.43 −0.199436 −0.0997179 0.995016i \(-0.531794\pi\)
−0.0997179 + 0.995016i \(0.531794\pi\)
\(602\) −58675.9 −3.97251
\(603\) 0 0
\(604\) 10771.8 0.725656
\(605\) 6469.98 0.434780
\(606\) 0 0
\(607\) −7785.94 −0.520628 −0.260314 0.965524i \(-0.583826\pi\)
−0.260314 + 0.965524i \(0.583826\pi\)
\(608\) 40811.5 2.72225
\(609\) 0 0
\(610\) −13066.1 −0.867264
\(611\) −38563.8 −2.55339
\(612\) 0 0
\(613\) 23893.4 1.57430 0.787151 0.616760i \(-0.211555\pi\)
0.787151 + 0.616760i \(0.211555\pi\)
\(614\) −10110.3 −0.664522
\(615\) 0 0
\(616\) 11970.4 0.782957
\(617\) 4880.46 0.318444 0.159222 0.987243i \(-0.449101\pi\)
0.159222 + 0.987243i \(0.449101\pi\)
\(618\) 0 0
\(619\) 15883.5 1.03136 0.515681 0.856780i \(-0.327539\pi\)
0.515681 + 0.856780i \(0.327539\pi\)
\(620\) 10364.3 0.671354
\(621\) 0 0
\(622\) −32960.6 −2.12476
\(623\) −31075.2 −1.99840
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 9518.48 0.607723
\(627\) 0 0
\(628\) 20435.7 1.29852
\(629\) 8334.96 0.528357
\(630\) 0 0
\(631\) −12862.4 −0.811479 −0.405740 0.913989i \(-0.632986\pi\)
−0.405740 + 0.913989i \(0.632986\pi\)
\(632\) 33155.0 2.08677
\(633\) 0 0
\(634\) 16530.3 1.03549
\(635\) 745.878 0.0466130
\(636\) 0 0
\(637\) −48157.0 −2.99537
\(638\) −1999.72 −0.124090
\(639\) 0 0
\(640\) 503.275 0.0310839
\(641\) 29485.7 1.81687 0.908435 0.418025i \(-0.137278\pi\)
0.908435 + 0.418025i \(0.137278\pi\)
\(642\) 0 0
\(643\) −14114.6 −0.865673 −0.432836 0.901473i \(-0.642487\pi\)
−0.432836 + 0.901473i \(0.642487\pi\)
\(644\) −49676.5 −3.03964
\(645\) 0 0
\(646\) 47829.3 2.91303
\(647\) 13313.3 0.808962 0.404481 0.914546i \(-0.367452\pi\)
0.404481 + 0.914546i \(0.367452\pi\)
\(648\) 0 0
\(649\) 1477.14 0.0893418
\(650\) 8432.35 0.508837
\(651\) 0 0
\(652\) −12329.2 −0.740566
\(653\) −11506.2 −0.689541 −0.344771 0.938687i \(-0.612043\pi\)
−0.344771 + 0.938687i \(0.612043\pi\)
\(654\) 0 0
\(655\) 5858.86 0.349503
\(656\) −22384.8 −1.33229
\(657\) 0 0
\(658\) −103391. −6.12556
\(659\) 25771.6 1.52340 0.761699 0.647931i \(-0.224365\pi\)
0.761699 + 0.647931i \(0.224365\pi\)
\(660\) 0 0
\(661\) 7882.88 0.463855 0.231928 0.972733i \(-0.425497\pi\)
0.231928 + 0.972733i \(0.425497\pi\)
\(662\) 4987.87 0.292838
\(663\) 0 0
\(664\) −3864.46 −0.225859
\(665\) 19598.9 1.14287
\(666\) 0 0
\(667\) 4874.46 0.282968
\(668\) −26467.0 −1.53299
\(669\) 0 0
\(670\) −24116.2 −1.39058
\(671\) 3037.53 0.174758
\(672\) 0 0
\(673\) −19631.3 −1.12441 −0.562207 0.826997i \(-0.690048\pi\)
−0.562207 + 0.826997i \(0.690048\pi\)
\(674\) 2977.23 0.170146
\(675\) 0 0
\(676\) 37940.5 2.15865
\(677\) −598.326 −0.0339668 −0.0169834 0.999856i \(-0.505406\pi\)
−0.0169834 + 0.999856i \(0.505406\pi\)
\(678\) 0 0
\(679\) −34410.4 −1.94485
\(680\) 22940.7 1.29373
\(681\) 0 0
\(682\) −3403.61 −0.191101
\(683\) 19041.9 1.06679 0.533394 0.845867i \(-0.320916\pi\)
0.533394 + 0.845867i \(0.320916\pi\)
\(684\) 0 0
\(685\) −5793.13 −0.323130
\(686\) −69842.4 −3.88716
\(687\) 0 0
\(688\) 53242.2 2.95035
\(689\) 31454.6 1.73922
\(690\) 0 0
\(691\) −27170.8 −1.49584 −0.747921 0.663787i \(-0.768948\pi\)
−0.747921 + 0.663787i \(0.768948\pi\)
\(692\) −22004.1 −1.20877
\(693\) 0 0
\(694\) 44099.2 2.41208
\(695\) −7932.62 −0.432952
\(696\) 0 0
\(697\) −10990.8 −0.597282
\(698\) −37386.9 −2.02739
\(699\) 0 0
\(700\) 16003.9 0.864131
\(701\) 27091.0 1.45965 0.729824 0.683635i \(-0.239602\pi\)
0.729824 + 0.683635i \(0.239602\pi\)
\(702\) 0 0
\(703\) −12853.2 −0.689568
\(704\) −3313.74 −0.177402
\(705\) 0 0
\(706\) 27740.5 1.47879
\(707\) −22526.3 −1.19829
\(708\) 0 0
\(709\) −4954.60 −0.262445 −0.131223 0.991353i \(-0.541890\pi\)
−0.131223 + 0.991353i \(0.541890\pi\)
\(710\) 1588.02 0.0839398
\(711\) 0 0
\(712\) 56091.3 2.95240
\(713\) 8296.56 0.435776
\(714\) 0 0
\(715\) −1960.30 −0.102533
\(716\) −35157.1 −1.83503
\(717\) 0 0
\(718\) −49907.2 −2.59404
\(719\) 23414.3 1.21447 0.607235 0.794522i \(-0.292279\pi\)
0.607235 + 0.794522i \(0.292279\pi\)
\(720\) 0 0
\(721\) 40072.8 2.06989
\(722\) −37860.9 −1.95157
\(723\) 0 0
\(724\) 45259.1 2.32326
\(725\) −1570.37 −0.0804444
\(726\) 0 0
\(727\) 30955.7 1.57921 0.789603 0.613618i \(-0.210286\pi\)
0.789603 + 0.613618i \(0.210286\pi\)
\(728\) 126827. 6.45676
\(729\) 0 0
\(730\) −8855.88 −0.449001
\(731\) 26141.5 1.32268
\(732\) 0 0
\(733\) 15572.7 0.784709 0.392354 0.919814i \(-0.371661\pi\)
0.392354 + 0.919814i \(0.371661\pi\)
\(734\) 3821.45 0.192169
\(735\) 0 0
\(736\) 26677.0 1.33604
\(737\) 5606.40 0.280210
\(738\) 0 0
\(739\) −30909.9 −1.53862 −0.769310 0.638876i \(-0.779400\pi\)
−0.769310 + 0.638876i \(0.779400\pi\)
\(740\) −10495.6 −0.521384
\(741\) 0 0
\(742\) 84331.3 4.17237
\(743\) −35023.2 −1.72931 −0.864656 0.502365i \(-0.832463\pi\)
−0.864656 + 0.502365i \(0.832463\pi\)
\(744\) 0 0
\(745\) 2585.55 0.127150
\(746\) −67347.8 −3.30533
\(747\) 0 0
\(748\) −9079.55 −0.443825
\(749\) 43317.3 2.11319
\(750\) 0 0
\(751\) −3610.78 −0.175445 −0.0877226 0.996145i \(-0.527959\pi\)
−0.0877226 + 0.996145i \(0.527959\pi\)
\(752\) 93816.8 4.54940
\(753\) 0 0
\(754\) −21187.1 −1.02333
\(755\) −2777.93 −0.133906
\(756\) 0 0
\(757\) −19982.2 −0.959399 −0.479700 0.877433i \(-0.659254\pi\)
−0.479700 + 0.877433i \(0.659254\pi\)
\(758\) 20938.1 1.00330
\(759\) 0 0
\(760\) −35376.3 −1.68846
\(761\) −35157.3 −1.67471 −0.837354 0.546662i \(-0.815898\pi\)
−0.837354 + 0.546662i \(0.815898\pi\)
\(762\) 0 0
\(763\) −9733.07 −0.461810
\(764\) −8582.02 −0.406396
\(765\) 0 0
\(766\) −341.843 −0.0161244
\(767\) 15650.3 0.736768
\(768\) 0 0
\(769\) −39041.2 −1.83077 −0.915384 0.402583i \(-0.868113\pi\)
−0.915384 + 0.402583i \(0.868113\pi\)
\(770\) −5255.67 −0.245975
\(771\) 0 0
\(772\) −71785.2 −3.34664
\(773\) 18244.4 0.848907 0.424453 0.905450i \(-0.360466\pi\)
0.424453 + 0.905450i \(0.360466\pi\)
\(774\) 0 0
\(775\) −2672.85 −0.123886
\(776\) 62111.3 2.87328
\(777\) 0 0
\(778\) 30980.3 1.42763
\(779\) 16948.6 0.779522
\(780\) 0 0
\(781\) −369.173 −0.0169143
\(782\) 31264.3 1.42968
\(783\) 0 0
\(784\) 117155. 5.33686
\(785\) −5270.16 −0.239618
\(786\) 0 0
\(787\) 42888.2 1.94256 0.971282 0.237930i \(-0.0764690\pi\)
0.971282 + 0.237930i \(0.0764690\pi\)
\(788\) −87089.8 −3.93711
\(789\) 0 0
\(790\) −14556.9 −0.655583
\(791\) −52337.7 −2.35261
\(792\) 0 0
\(793\) 32182.7 1.44116
\(794\) 53530.3 2.39259
\(795\) 0 0
\(796\) −56886.2 −2.53301
\(797\) 11473.1 0.509911 0.254956 0.966953i \(-0.417939\pi\)
0.254956 + 0.966953i \(0.417939\pi\)
\(798\) 0 0
\(799\) 46063.3 2.03955
\(800\) −8594.35 −0.379820
\(801\) 0 0
\(802\) 64083.2 2.82152
\(803\) 2058.76 0.0904759
\(804\) 0 0
\(805\) 12811.1 0.560908
\(806\) −36061.4 −1.57594
\(807\) 0 0
\(808\) 40660.4 1.77033
\(809\) −6454.45 −0.280502 −0.140251 0.990116i \(-0.544791\pi\)
−0.140251 + 0.990116i \(0.544791\pi\)
\(810\) 0 0
\(811\) −11250.0 −0.487102 −0.243551 0.969888i \(-0.578312\pi\)
−0.243551 + 0.969888i \(0.578312\pi\)
\(812\) −40211.4 −1.73786
\(813\) 0 0
\(814\) 3446.73 0.148412
\(815\) 3179.58 0.136658
\(816\) 0 0
\(817\) −40312.2 −1.72625
\(818\) −51459.4 −2.19955
\(819\) 0 0
\(820\) 13839.8 0.589399
\(821\) −532.438 −0.0226336 −0.0113168 0.999936i \(-0.503602\pi\)
−0.0113168 + 0.999936i \(0.503602\pi\)
\(822\) 0 0
\(823\) −33016.7 −1.39841 −0.699204 0.714922i \(-0.746462\pi\)
−0.699204 + 0.714922i \(0.746462\pi\)
\(824\) −72332.0 −3.05802
\(825\) 0 0
\(826\) 41959.3 1.76750
\(827\) −36231.3 −1.52344 −0.761720 0.647906i \(-0.775645\pi\)
−0.761720 + 0.647906i \(0.775645\pi\)
\(828\) 0 0
\(829\) 36588.3 1.53289 0.766445 0.642310i \(-0.222024\pi\)
0.766445 + 0.642310i \(0.222024\pi\)
\(830\) 1696.71 0.0709562
\(831\) 0 0
\(832\) −35109.2 −1.46297
\(833\) 57522.1 2.39259
\(834\) 0 0
\(835\) 6825.57 0.282885
\(836\) 14001.4 0.579243
\(837\) 0 0
\(838\) 64449.7 2.65677
\(839\) 15326.8 0.630677 0.315339 0.948979i \(-0.397882\pi\)
0.315339 + 0.948979i \(0.397882\pi\)
\(840\) 0 0
\(841\) −20443.3 −0.838218
\(842\) 22599.9 0.924992
\(843\) 0 0
\(844\) −13066.4 −0.532896
\(845\) −9784.48 −0.398339
\(846\) 0 0
\(847\) −42725.2 −1.73324
\(848\) −76521.8 −3.09879
\(849\) 0 0
\(850\) −10072.2 −0.406440
\(851\) −8401.64 −0.338431
\(852\) 0 0
\(853\) 4573.30 0.183572 0.0917859 0.995779i \(-0.470742\pi\)
0.0917859 + 0.995779i \(0.470742\pi\)
\(854\) 86283.4 3.45733
\(855\) 0 0
\(856\) −78188.3 −3.12199
\(857\) 19569.2 0.780011 0.390006 0.920813i \(-0.372473\pi\)
0.390006 + 0.920813i \(0.372473\pi\)
\(858\) 0 0
\(859\) 21897.0 0.869750 0.434875 0.900491i \(-0.356792\pi\)
0.434875 + 0.900491i \(0.356792\pi\)
\(860\) −32917.9 −1.30522
\(861\) 0 0
\(862\) 27555.7 1.08881
\(863\) 13335.5 0.526008 0.263004 0.964795i \(-0.415287\pi\)
0.263004 + 0.964795i \(0.415287\pi\)
\(864\) 0 0
\(865\) 5674.65 0.223057
\(866\) −30641.7 −1.20237
\(867\) 0 0
\(868\) −68441.7 −2.67634
\(869\) 3384.10 0.132103
\(870\) 0 0
\(871\) 59400.0 2.31078
\(872\) 17568.4 0.682270
\(873\) 0 0
\(874\) −48211.9 −1.86590
\(875\) −4127.26 −0.159459
\(876\) 0 0
\(877\) −21811.8 −0.839830 −0.419915 0.907563i \(-0.637940\pi\)
−0.419915 + 0.907563i \(0.637940\pi\)
\(878\) −21052.2 −0.809201
\(879\) 0 0
\(880\) 4768.97 0.182684
\(881\) 3646.54 0.139450 0.0697248 0.997566i \(-0.477788\pi\)
0.0697248 + 0.997566i \(0.477788\pi\)
\(882\) 0 0
\(883\) 9019.80 0.343760 0.171880 0.985118i \(-0.445016\pi\)
0.171880 + 0.985118i \(0.445016\pi\)
\(884\) −96198.1 −3.66006
\(885\) 0 0
\(886\) 39261.9 1.48875
\(887\) −19731.9 −0.746938 −0.373469 0.927643i \(-0.621832\pi\)
−0.373469 + 0.927643i \(0.621832\pi\)
\(888\) 0 0
\(889\) −4925.49 −0.185822
\(890\) −24627.2 −0.927533
\(891\) 0 0
\(892\) 3461.09 0.129917
\(893\) −71033.3 −2.66186
\(894\) 0 0
\(895\) 9066.67 0.338621
\(896\) −3323.43 −0.123915
\(897\) 0 0
\(898\) −97205.4 −3.61223
\(899\) 6715.78 0.249148
\(900\) 0 0
\(901\) −37571.6 −1.38923
\(902\) −4544.98 −0.167773
\(903\) 0 0
\(904\) 94470.4 3.47571
\(905\) −11671.9 −0.428714
\(906\) 0 0
\(907\) 7706.14 0.282115 0.141057 0.990001i \(-0.454950\pi\)
0.141057 + 0.990001i \(0.454950\pi\)
\(908\) −97950.7 −3.57997
\(909\) 0 0
\(910\) −55684.0 −2.02847
\(911\) −24765.1 −0.900665 −0.450332 0.892861i \(-0.648694\pi\)
−0.450332 + 0.892861i \(0.648694\pi\)
\(912\) 0 0
\(913\) −394.441 −0.0142980
\(914\) −74028.8 −2.67905
\(915\) 0 0
\(916\) 25378.4 0.915422
\(917\) −38689.6 −1.39329
\(918\) 0 0
\(919\) −4899.44 −0.175862 −0.0879312 0.996127i \(-0.528026\pi\)
−0.0879312 + 0.996127i \(0.528026\pi\)
\(920\) −23124.2 −0.828676
\(921\) 0 0
\(922\) −25699.7 −0.917978
\(923\) −3911.40 −0.139486
\(924\) 0 0
\(925\) 2706.70 0.0962117
\(926\) −48437.1 −1.71894
\(927\) 0 0
\(928\) 21594.1 0.763860
\(929\) −13356.2 −0.471692 −0.235846 0.971790i \(-0.575786\pi\)
−0.235846 + 0.971790i \(0.575786\pi\)
\(930\) 0 0
\(931\) −88703.6 −3.12260
\(932\) 127793. 4.49141
\(933\) 0 0
\(934\) −14784.8 −0.517958
\(935\) 2341.53 0.0818996
\(936\) 0 0
\(937\) 4960.13 0.172935 0.0864677 0.996255i \(-0.472442\pi\)
0.0864677 + 0.996255i \(0.472442\pi\)
\(938\) 159254. 5.54353
\(939\) 0 0
\(940\) −58003.9 −2.01264
\(941\) −55574.3 −1.92526 −0.962631 0.270817i \(-0.912706\pi\)
−0.962631 + 0.270817i \(0.912706\pi\)
\(942\) 0 0
\(943\) 11078.7 0.382579
\(944\) −38073.7 −1.31270
\(945\) 0 0
\(946\) 10810.2 0.371533
\(947\) 20945.5 0.718732 0.359366 0.933197i \(-0.382993\pi\)
0.359366 + 0.933197i \(0.382993\pi\)
\(948\) 0 0
\(949\) 21812.7 0.746121
\(950\) 15532.1 0.530451
\(951\) 0 0
\(952\) −151491. −5.15741
\(953\) 26967.8 0.916656 0.458328 0.888783i \(-0.348449\pi\)
0.458328 + 0.888783i \(0.348449\pi\)
\(954\) 0 0
\(955\) 2213.22 0.0749927
\(956\) −30720.5 −1.03930
\(957\) 0 0
\(958\) 48008.2 1.61907
\(959\) 38255.6 1.28815
\(960\) 0 0
\(961\) −18360.4 −0.616308
\(962\) 36518.2 1.22390
\(963\) 0 0
\(964\) −32464.8 −1.08467
\(965\) 18512.7 0.617559
\(966\) 0 0
\(967\) −18506.5 −0.615439 −0.307719 0.951477i \(-0.599566\pi\)
−0.307719 + 0.951477i \(0.599566\pi\)
\(968\) 77119.7 2.56066
\(969\) 0 0
\(970\) −27270.3 −0.902676
\(971\) 56274.4 1.85987 0.929934 0.367726i \(-0.119864\pi\)
0.929934 + 0.367726i \(0.119864\pi\)
\(972\) 0 0
\(973\) 52383.9 1.72595
\(974\) −71507.2 −2.35240
\(975\) 0 0
\(976\) −78293.2 −2.56773
\(977\) −23165.7 −0.758583 −0.379292 0.925277i \(-0.623832\pi\)
−0.379292 + 0.925277i \(0.623832\pi\)
\(978\) 0 0
\(979\) 5725.17 0.186902
\(980\) −72433.1 −2.36101
\(981\) 0 0
\(982\) −30616.8 −0.994931
\(983\) −3286.86 −0.106647 −0.0533237 0.998577i \(-0.516982\pi\)
−0.0533237 + 0.998577i \(0.516982\pi\)
\(984\) 0 0
\(985\) 22459.6 0.726520
\(986\) 25307.4 0.817395
\(987\) 0 0
\(988\) 148345. 4.77680
\(989\) −26350.6 −0.847221
\(990\) 0 0
\(991\) 35155.7 1.12690 0.563450 0.826150i \(-0.309474\pi\)
0.563450 + 0.826150i \(0.309474\pi\)
\(992\) 36754.2 1.17636
\(993\) 0 0
\(994\) −10486.6 −0.334624
\(995\) 14670.4 0.467419
\(996\) 0 0
\(997\) −13641.5 −0.433329 −0.216665 0.976246i \(-0.569518\pi\)
−0.216665 + 0.976246i \(0.569518\pi\)
\(998\) −20111.7 −0.637900
\(999\) 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 405.4.a.k.1.1 6
3.2 odd 2 405.4.a.l.1.6 yes 6
5.4 even 2 2025.4.a.z.1.6 6
9.2 odd 6 405.4.e.w.271.1 12
9.4 even 3 405.4.e.x.136.6 12
9.5 odd 6 405.4.e.w.136.1 12
9.7 even 3 405.4.e.x.271.6 12
15.14 odd 2 2025.4.a.y.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.k.1.1 6 1.1 even 1 trivial
405.4.a.l.1.6 yes 6 3.2 odd 2
405.4.e.w.136.1 12 9.5 odd 6
405.4.e.w.271.1 12 9.2 odd 6
405.4.e.x.136.6 12 9.4 even 3
405.4.e.x.271.6 12 9.7 even 3
2025.4.a.y.1.1 6 15.14 odd 2
2025.4.a.z.1.6 6 5.4 even 2