Properties

Label 8001.2.a.l.1.1
Level 8001
Weight 2
Character 8001.1
Self dual yes
Analytic conductor 63.888
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
Defining polynomial: \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.301070\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} +O(q^{10})\) \(q-2.09968 q^{2} +2.40865 q^{4} +3.29054 q^{5} +1.00000 q^{7} -0.858029 q^{8} -6.90907 q^{10} +3.62521 q^{11} -2.16007 q^{13} -2.09968 q^{14} -3.01571 q^{16} -2.19233 q^{17} -6.66601 q^{19} +7.92575 q^{20} -7.61177 q^{22} -2.36522 q^{23} +5.82765 q^{25} +4.53544 q^{26} +2.40865 q^{28} -2.06180 q^{29} -2.19207 q^{31} +8.04808 q^{32} +4.60319 q^{34} +3.29054 q^{35} -10.1454 q^{37} +13.9965 q^{38} -2.82338 q^{40} +0.647495 q^{41} +3.67859 q^{43} +8.73185 q^{44} +4.96620 q^{46} +8.59064 q^{47} +1.00000 q^{49} -12.2362 q^{50} -5.20284 q^{52} -2.84924 q^{53} +11.9289 q^{55} -0.858029 q^{56} +4.32912 q^{58} +0.592789 q^{59} -13.5269 q^{61} +4.60265 q^{62} -10.8670 q^{64} -7.10778 q^{65} +0.664083 q^{67} -5.28055 q^{68} -6.90907 q^{70} +5.65049 q^{71} -3.24624 q^{73} +21.3021 q^{74} -16.0561 q^{76} +3.62521 q^{77} +4.01101 q^{79} -9.92332 q^{80} -1.35953 q^{82} +3.40033 q^{83} -7.21395 q^{85} -7.72385 q^{86} -3.11054 q^{88} -6.31252 q^{89} -2.16007 q^{91} -5.69698 q^{92} -18.0376 q^{94} -21.9348 q^{95} -0.0689323 q^{97} -2.09968 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.09968 −1.48470 −0.742348 0.670014i \(-0.766288\pi\)
−0.742348 + 0.670014i \(0.766288\pi\)
\(3\) 0 0
\(4\) 2.40865 1.20432
\(5\) 3.29054 1.47157 0.735787 0.677213i \(-0.236812\pi\)
0.735787 + 0.677213i \(0.236812\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −0.858029 −0.303359
\(9\) 0 0
\(10\) −6.90907 −2.18484
\(11\) 3.62521 1.09304 0.546521 0.837445i \(-0.315952\pi\)
0.546521 + 0.837445i \(0.315952\pi\)
\(12\) 0 0
\(13\) −2.16007 −0.599095 −0.299547 0.954081i \(-0.596836\pi\)
−0.299547 + 0.954081i \(0.596836\pi\)
\(14\) −2.09968 −0.561163
\(15\) 0 0
\(16\) −3.01571 −0.753928
\(17\) −2.19233 −0.531718 −0.265859 0.964012i \(-0.585656\pi\)
−0.265859 + 0.964012i \(0.585656\pi\)
\(18\) 0 0
\(19\) −6.66601 −1.52929 −0.764643 0.644454i \(-0.777085\pi\)
−0.764643 + 0.644454i \(0.777085\pi\)
\(20\) 7.92575 1.77225
\(21\) 0 0
\(22\) −7.61177 −1.62284
\(23\) −2.36522 −0.493183 −0.246591 0.969120i \(-0.579311\pi\)
−0.246591 + 0.969120i \(0.579311\pi\)
\(24\) 0 0
\(25\) 5.82765 1.16553
\(26\) 4.53544 0.889474
\(27\) 0 0
\(28\) 2.40865 0.455192
\(29\) −2.06180 −0.382867 −0.191433 0.981506i \(-0.561314\pi\)
−0.191433 + 0.981506i \(0.561314\pi\)
\(30\) 0 0
\(31\) −2.19207 −0.393708 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(32\) 8.04808 1.42271
\(33\) 0 0
\(34\) 4.60319 0.789440
\(35\) 3.29054 0.556203
\(36\) 0 0
\(37\) −10.1454 −1.66789 −0.833947 0.551845i \(-0.813924\pi\)
−0.833947 + 0.551845i \(0.813924\pi\)
\(38\) 13.9965 2.27053
\(39\) 0 0
\(40\) −2.82338 −0.446416
\(41\) 0.647495 0.101122 0.0505608 0.998721i \(-0.483899\pi\)
0.0505608 + 0.998721i \(0.483899\pi\)
\(42\) 0 0
\(43\) 3.67859 0.560979 0.280490 0.959857i \(-0.409503\pi\)
0.280490 + 0.959857i \(0.409503\pi\)
\(44\) 8.73185 1.31638
\(45\) 0 0
\(46\) 4.96620 0.732227
\(47\) 8.59064 1.25307 0.626537 0.779392i \(-0.284472\pi\)
0.626537 + 0.779392i \(0.284472\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −12.2362 −1.73046
\(51\) 0 0
\(52\) −5.20284 −0.721504
\(53\) −2.84924 −0.391373 −0.195686 0.980667i \(-0.562693\pi\)
−0.195686 + 0.980667i \(0.562693\pi\)
\(54\) 0 0
\(55\) 11.9289 1.60849
\(56\) −0.858029 −0.114659
\(57\) 0 0
\(58\) 4.32912 0.568441
\(59\) 0.592789 0.0771745 0.0385873 0.999255i \(-0.487714\pi\)
0.0385873 + 0.999255i \(0.487714\pi\)
\(60\) 0 0
\(61\) −13.5269 −1.73195 −0.865973 0.500091i \(-0.833300\pi\)
−0.865973 + 0.500091i \(0.833300\pi\)
\(62\) 4.60265 0.584537
\(63\) 0 0
\(64\) −10.8670 −1.35837
\(65\) −7.10778 −0.881612
\(66\) 0 0
\(67\) 0.664083 0.0811306 0.0405653 0.999177i \(-0.487084\pi\)
0.0405653 + 0.999177i \(0.487084\pi\)
\(68\) −5.28055 −0.640361
\(69\) 0 0
\(70\) −6.90907 −0.825792
\(71\) 5.65049 0.670589 0.335295 0.942113i \(-0.391164\pi\)
0.335295 + 0.942113i \(0.391164\pi\)
\(72\) 0 0
\(73\) −3.24624 −0.379943 −0.189972 0.981790i \(-0.560840\pi\)
−0.189972 + 0.981790i \(0.560840\pi\)
\(74\) 21.3021 2.47632
\(75\) 0 0
\(76\) −16.0561 −1.84176
\(77\) 3.62521 0.413131
\(78\) 0 0
\(79\) 4.01101 0.451274 0.225637 0.974211i \(-0.427554\pi\)
0.225637 + 0.974211i \(0.427554\pi\)
\(80\) −9.92332 −1.10946
\(81\) 0 0
\(82\) −1.35953 −0.150135
\(83\) 3.40033 0.373235 0.186617 0.982433i \(-0.440248\pi\)
0.186617 + 0.982433i \(0.440248\pi\)
\(84\) 0 0
\(85\) −7.21395 −0.782463
\(86\) −7.72385 −0.832884
\(87\) 0 0
\(88\) −3.11054 −0.331584
\(89\) −6.31252 −0.669126 −0.334563 0.942373i \(-0.608589\pi\)
−0.334563 + 0.942373i \(0.608589\pi\)
\(90\) 0 0
\(91\) −2.16007 −0.226436
\(92\) −5.69698 −0.593952
\(93\) 0 0
\(94\) −18.0376 −1.86043
\(95\) −21.9348 −2.25046
\(96\) 0 0
\(97\) −0.0689323 −0.00699902 −0.00349951 0.999994i \(-0.501114\pi\)
−0.00349951 + 0.999994i \(0.501114\pi\)
\(98\) −2.09968 −0.212100
\(99\) 0 0
\(100\) 14.0368 1.40368
\(101\) −6.53614 −0.650370 −0.325185 0.945650i \(-0.605427\pi\)
−0.325185 + 0.945650i \(0.605427\pi\)
\(102\) 0 0
\(103\) 1.42084 0.140000 0.0700000 0.997547i \(-0.477700\pi\)
0.0700000 + 0.997547i \(0.477700\pi\)
\(104\) 1.85340 0.181741
\(105\) 0 0
\(106\) 5.98248 0.581070
\(107\) −17.1418 −1.65716 −0.828581 0.559869i \(-0.810851\pi\)
−0.828581 + 0.559869i \(0.810851\pi\)
\(108\) 0 0
\(109\) 14.3137 1.37100 0.685500 0.728073i \(-0.259584\pi\)
0.685500 + 0.728073i \(0.259584\pi\)
\(110\) −25.0468 −2.38812
\(111\) 0 0
\(112\) −3.01571 −0.284958
\(113\) −9.27873 −0.872870 −0.436435 0.899736i \(-0.643759\pi\)
−0.436435 + 0.899736i \(0.643759\pi\)
\(114\) 0 0
\(115\) −7.78285 −0.725755
\(116\) −4.96615 −0.461095
\(117\) 0 0
\(118\) −1.24467 −0.114581
\(119\) −2.19233 −0.200971
\(120\) 0 0
\(121\) 2.14215 0.194741
\(122\) 28.4022 2.57141
\(123\) 0 0
\(124\) −5.27993 −0.474152
\(125\) 2.72342 0.243590
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 6.72095 0.594054
\(129\) 0 0
\(130\) 14.9241 1.30893
\(131\) −17.8118 −1.55623 −0.778113 0.628124i \(-0.783823\pi\)
−0.778113 + 0.628124i \(0.783823\pi\)
\(132\) 0 0
\(133\) −6.66601 −0.578016
\(134\) −1.39436 −0.120454
\(135\) 0 0
\(136\) 1.88108 0.161302
\(137\) −7.51288 −0.641869 −0.320934 0.947101i \(-0.603997\pi\)
−0.320934 + 0.947101i \(0.603997\pi\)
\(138\) 0 0
\(139\) −4.67226 −0.396296 −0.198148 0.980172i \(-0.563493\pi\)
−0.198148 + 0.980172i \(0.563493\pi\)
\(140\) 7.92575 0.669848
\(141\) 0 0
\(142\) −11.8642 −0.995622
\(143\) −7.83069 −0.654836
\(144\) 0 0
\(145\) −6.78443 −0.563417
\(146\) 6.81605 0.564101
\(147\) 0 0
\(148\) −24.4367 −2.00868
\(149\) 12.9632 1.06199 0.530994 0.847375i \(-0.321819\pi\)
0.530994 + 0.847375i \(0.321819\pi\)
\(150\) 0 0
\(151\) 10.9338 0.889780 0.444890 0.895585i \(-0.353243\pi\)
0.444890 + 0.895585i \(0.353243\pi\)
\(152\) 5.71963 0.463923
\(153\) 0 0
\(154\) −7.61177 −0.613374
\(155\) −7.21310 −0.579370
\(156\) 0 0
\(157\) −20.5157 −1.63733 −0.818665 0.574271i \(-0.805285\pi\)
−0.818665 + 0.574271i \(0.805285\pi\)
\(158\) −8.42183 −0.670005
\(159\) 0 0
\(160\) 26.4825 2.09363
\(161\) −2.36522 −0.186406
\(162\) 0 0
\(163\) 1.33340 0.104440 0.0522199 0.998636i \(-0.483370\pi\)
0.0522199 + 0.998636i \(0.483370\pi\)
\(164\) 1.55959 0.121783
\(165\) 0 0
\(166\) −7.13960 −0.554140
\(167\) −0.822111 −0.0636169 −0.0318084 0.999494i \(-0.510127\pi\)
−0.0318084 + 0.999494i \(0.510127\pi\)
\(168\) 0 0
\(169\) −8.33411 −0.641086
\(170\) 15.1470 1.16172
\(171\) 0 0
\(172\) 8.86042 0.675601
\(173\) 22.8603 1.73804 0.869020 0.494777i \(-0.164750\pi\)
0.869020 + 0.494777i \(0.164750\pi\)
\(174\) 0 0
\(175\) 5.82765 0.440529
\(176\) −10.9326 −0.824075
\(177\) 0 0
\(178\) 13.2543 0.993449
\(179\) −1.52134 −0.113710 −0.0568552 0.998382i \(-0.518107\pi\)
−0.0568552 + 0.998382i \(0.518107\pi\)
\(180\) 0 0
\(181\) 14.4514 1.07417 0.537084 0.843529i \(-0.319526\pi\)
0.537084 + 0.843529i \(0.319526\pi\)
\(182\) 4.53544 0.336189
\(183\) 0 0
\(184\) 2.02943 0.149611
\(185\) −33.3838 −2.45443
\(186\) 0 0
\(187\) −7.94766 −0.581190
\(188\) 20.6918 1.50911
\(189\) 0 0
\(190\) 46.0559 3.34125
\(191\) −6.46436 −0.467744 −0.233872 0.972267i \(-0.575140\pi\)
−0.233872 + 0.972267i \(0.575140\pi\)
\(192\) 0 0
\(193\) −21.4158 −1.54154 −0.770771 0.637112i \(-0.780129\pi\)
−0.770771 + 0.637112i \(0.780129\pi\)
\(194\) 0.144736 0.0103914
\(195\) 0 0
\(196\) 2.40865 0.172046
\(197\) −15.2826 −1.08884 −0.544419 0.838814i \(-0.683250\pi\)
−0.544419 + 0.838814i \(0.683250\pi\)
\(198\) 0 0
\(199\) −9.68248 −0.686373 −0.343186 0.939267i \(-0.611506\pi\)
−0.343186 + 0.939267i \(0.611506\pi\)
\(200\) −5.00030 −0.353574
\(201\) 0 0
\(202\) 13.7238 0.965603
\(203\) −2.06180 −0.144710
\(204\) 0 0
\(205\) 2.13061 0.148808
\(206\) −2.98332 −0.207857
\(207\) 0 0
\(208\) 6.51414 0.451674
\(209\) −24.1657 −1.67157
\(210\) 0 0
\(211\) −22.7122 −1.56357 −0.781785 0.623549i \(-0.785690\pi\)
−0.781785 + 0.623549i \(0.785690\pi\)
\(212\) −6.86281 −0.471340
\(213\) 0 0
\(214\) 35.9923 2.46038
\(215\) 12.1045 0.825523
\(216\) 0 0
\(217\) −2.19207 −0.148808
\(218\) −30.0541 −2.03552
\(219\) 0 0
\(220\) 28.7325 1.93715
\(221\) 4.73558 0.318550
\(222\) 0 0
\(223\) 6.33989 0.424551 0.212275 0.977210i \(-0.431913\pi\)
0.212275 + 0.977210i \(0.431913\pi\)
\(224\) 8.04808 0.537735
\(225\) 0 0
\(226\) 19.4824 1.29595
\(227\) 15.7349 1.04436 0.522181 0.852835i \(-0.325119\pi\)
0.522181 + 0.852835i \(0.325119\pi\)
\(228\) 0 0
\(229\) −0.365275 −0.0241380 −0.0120690 0.999927i \(-0.503842\pi\)
−0.0120690 + 0.999927i \(0.503842\pi\)
\(230\) 16.3415 1.07753
\(231\) 0 0
\(232\) 1.76908 0.116146
\(233\) 4.21293 0.275998 0.137999 0.990432i \(-0.455933\pi\)
0.137999 + 0.990432i \(0.455933\pi\)
\(234\) 0 0
\(235\) 28.2678 1.84399
\(236\) 1.42782 0.0929431
\(237\) 0 0
\(238\) 4.60319 0.298380
\(239\) 3.38821 0.219165 0.109582 0.993978i \(-0.465049\pi\)
0.109582 + 0.993978i \(0.465049\pi\)
\(240\) 0 0
\(241\) −10.0609 −0.648077 −0.324039 0.946044i \(-0.605041\pi\)
−0.324039 + 0.946044i \(0.605041\pi\)
\(242\) −4.49782 −0.289131
\(243\) 0 0
\(244\) −32.5816 −2.08582
\(245\) 3.29054 0.210225
\(246\) 0 0
\(247\) 14.3990 0.916188
\(248\) 1.88086 0.119435
\(249\) 0 0
\(250\) −5.71830 −0.361657
\(251\) −1.13195 −0.0714479 −0.0357240 0.999362i \(-0.511374\pi\)
−0.0357240 + 0.999362i \(0.511374\pi\)
\(252\) 0 0
\(253\) −8.57442 −0.539069
\(254\) 2.09968 0.131746
\(255\) 0 0
\(256\) 7.62208 0.476380
\(257\) 29.2462 1.82433 0.912165 0.409824i \(-0.134410\pi\)
0.912165 + 0.409824i \(0.134410\pi\)
\(258\) 0 0
\(259\) −10.1454 −0.630404
\(260\) −17.1202 −1.06175
\(261\) 0 0
\(262\) 37.3991 2.31052
\(263\) −26.3656 −1.62577 −0.812887 0.582422i \(-0.802105\pi\)
−0.812887 + 0.582422i \(0.802105\pi\)
\(264\) 0 0
\(265\) −9.37552 −0.575934
\(266\) 13.9965 0.858179
\(267\) 0 0
\(268\) 1.59954 0.0977076
\(269\) 22.2274 1.35523 0.677613 0.735419i \(-0.263014\pi\)
0.677613 + 0.735419i \(0.263014\pi\)
\(270\) 0 0
\(271\) −26.8490 −1.63096 −0.815482 0.578783i \(-0.803528\pi\)
−0.815482 + 0.578783i \(0.803528\pi\)
\(272\) 6.61144 0.400877
\(273\) 0 0
\(274\) 15.7746 0.952981
\(275\) 21.1265 1.27397
\(276\) 0 0
\(277\) 2.10995 0.126775 0.0633873 0.997989i \(-0.479810\pi\)
0.0633873 + 0.997989i \(0.479810\pi\)
\(278\) 9.81024 0.588379
\(279\) 0 0
\(280\) −2.82338 −0.168729
\(281\) −19.6051 −1.16954 −0.584772 0.811198i \(-0.698816\pi\)
−0.584772 + 0.811198i \(0.698816\pi\)
\(282\) 0 0
\(283\) −5.28771 −0.314322 −0.157161 0.987573i \(-0.550234\pi\)
−0.157161 + 0.987573i \(0.550234\pi\)
\(284\) 13.6100 0.807607
\(285\) 0 0
\(286\) 16.4419 0.972232
\(287\) 0.647495 0.0382204
\(288\) 0 0
\(289\) −12.1937 −0.717276
\(290\) 14.2451 0.836503
\(291\) 0 0
\(292\) −7.81904 −0.457575
\(293\) −9.05623 −0.529071 −0.264535 0.964376i \(-0.585219\pi\)
−0.264535 + 0.964376i \(0.585219\pi\)
\(294\) 0 0
\(295\) 1.95059 0.113568
\(296\) 8.70505 0.505971
\(297\) 0 0
\(298\) −27.2186 −1.57673
\(299\) 5.10903 0.295463
\(300\) 0 0
\(301\) 3.67859 0.212030
\(302\) −22.9575 −1.32105
\(303\) 0 0
\(304\) 20.1027 1.15297
\(305\) −44.5109 −2.54869
\(306\) 0 0
\(307\) 23.4922 1.34077 0.670386 0.742012i \(-0.266128\pi\)
0.670386 + 0.742012i \(0.266128\pi\)
\(308\) 8.73185 0.497544
\(309\) 0 0
\(310\) 15.1452 0.860189
\(311\) 0.106610 0.00604531 0.00302265 0.999995i \(-0.499038\pi\)
0.00302265 + 0.999995i \(0.499038\pi\)
\(312\) 0 0
\(313\) 23.6312 1.33572 0.667858 0.744289i \(-0.267211\pi\)
0.667858 + 0.744289i \(0.267211\pi\)
\(314\) 43.0763 2.43094
\(315\) 0 0
\(316\) 9.66112 0.543480
\(317\) −8.87083 −0.498236 −0.249118 0.968473i \(-0.580141\pi\)
−0.249118 + 0.968473i \(0.580141\pi\)
\(318\) 0 0
\(319\) −7.47446 −0.418489
\(320\) −35.7582 −1.99894
\(321\) 0 0
\(322\) 4.96620 0.276756
\(323\) 14.6141 0.813150
\(324\) 0 0
\(325\) −12.5881 −0.698263
\(326\) −2.79971 −0.155061
\(327\) 0 0
\(328\) −0.555570 −0.0306762
\(329\) 8.59064 0.473617
\(330\) 0 0
\(331\) −27.3847 −1.50520 −0.752599 0.658480i \(-0.771200\pi\)
−0.752599 + 0.658480i \(0.771200\pi\)
\(332\) 8.19020 0.449496
\(333\) 0 0
\(334\) 1.72617 0.0944518
\(335\) 2.18519 0.119390
\(336\) 0 0
\(337\) 18.4898 1.00720 0.503601 0.863936i \(-0.332008\pi\)
0.503601 + 0.863936i \(0.332008\pi\)
\(338\) 17.4990 0.951818
\(339\) 0 0
\(340\) −17.3759 −0.942339
\(341\) −7.94673 −0.430339
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −3.15634 −0.170178
\(345\) 0 0
\(346\) −47.9994 −2.58046
\(347\) −24.2408 −1.30132 −0.650658 0.759371i \(-0.725507\pi\)
−0.650658 + 0.759371i \(0.725507\pi\)
\(348\) 0 0
\(349\) −7.74084 −0.414358 −0.207179 0.978303i \(-0.566428\pi\)
−0.207179 + 0.978303i \(0.566428\pi\)
\(350\) −12.2362 −0.654052
\(351\) 0 0
\(352\) 29.1760 1.55508
\(353\) 10.6201 0.565253 0.282626 0.959230i \(-0.408794\pi\)
0.282626 + 0.959230i \(0.408794\pi\)
\(354\) 0 0
\(355\) 18.5932 0.986822
\(356\) −15.2046 −0.805845
\(357\) 0 0
\(358\) 3.19433 0.168825
\(359\) −11.5939 −0.611900 −0.305950 0.952048i \(-0.598974\pi\)
−0.305950 + 0.952048i \(0.598974\pi\)
\(360\) 0 0
\(361\) 25.4357 1.33872
\(362\) −30.3434 −1.59481
\(363\) 0 0
\(364\) −5.20284 −0.272703
\(365\) −10.6819 −0.559115
\(366\) 0 0
\(367\) −19.5505 −1.02053 −0.510263 0.860018i \(-0.670452\pi\)
−0.510263 + 0.860018i \(0.670452\pi\)
\(368\) 7.13282 0.371824
\(369\) 0 0
\(370\) 70.0953 3.64408
\(371\) −2.84924 −0.147925
\(372\) 0 0
\(373\) 3.95621 0.204845 0.102423 0.994741i \(-0.467341\pi\)
0.102423 + 0.994741i \(0.467341\pi\)
\(374\) 16.6875 0.862891
\(375\) 0 0
\(376\) −7.37102 −0.380131
\(377\) 4.45362 0.229373
\(378\) 0 0
\(379\) −26.2433 −1.34803 −0.674014 0.738718i \(-0.735431\pi\)
−0.674014 + 0.738718i \(0.735431\pi\)
\(380\) −52.8331 −2.71028
\(381\) 0 0
\(382\) 13.5731 0.694459
\(383\) −2.25979 −0.115470 −0.0577349 0.998332i \(-0.518388\pi\)
−0.0577349 + 0.998332i \(0.518388\pi\)
\(384\) 0 0
\(385\) 11.9289 0.607953
\(386\) 44.9663 2.28872
\(387\) 0 0
\(388\) −0.166034 −0.00842909
\(389\) 4.79264 0.242997 0.121498 0.992592i \(-0.461230\pi\)
0.121498 + 0.992592i \(0.461230\pi\)
\(390\) 0 0
\(391\) 5.18535 0.262234
\(392\) −0.858029 −0.0433370
\(393\) 0 0
\(394\) 32.0885 1.61659
\(395\) 13.1984 0.664083
\(396\) 0 0
\(397\) 34.5066 1.73184 0.865919 0.500185i \(-0.166735\pi\)
0.865919 + 0.500185i \(0.166735\pi\)
\(398\) 20.3301 1.01906
\(399\) 0 0
\(400\) −17.5745 −0.878725
\(401\) 2.10653 0.105195 0.0525975 0.998616i \(-0.483250\pi\)
0.0525975 + 0.998616i \(0.483250\pi\)
\(402\) 0 0
\(403\) 4.73502 0.235868
\(404\) −15.7433 −0.783257
\(405\) 0 0
\(406\) 4.32912 0.214850
\(407\) −36.7792 −1.82308
\(408\) 0 0
\(409\) 36.8469 1.82196 0.910981 0.412449i \(-0.135327\pi\)
0.910981 + 0.412449i \(0.135327\pi\)
\(410\) −4.47359 −0.220935
\(411\) 0 0
\(412\) 3.42231 0.168605
\(413\) 0.592789 0.0291692
\(414\) 0 0
\(415\) 11.1889 0.549243
\(416\) −17.3844 −0.852340
\(417\) 0 0
\(418\) 50.7401 2.48178
\(419\) 23.4163 1.14396 0.571981 0.820267i \(-0.306175\pi\)
0.571981 + 0.820267i \(0.306175\pi\)
\(420\) 0 0
\(421\) 7.19583 0.350703 0.175352 0.984506i \(-0.443894\pi\)
0.175352 + 0.984506i \(0.443894\pi\)
\(422\) 47.6882 2.32143
\(423\) 0 0
\(424\) 2.44473 0.118727
\(425\) −12.7761 −0.619734
\(426\) 0 0
\(427\) −13.5269 −0.654614
\(428\) −41.2886 −1.99576
\(429\) 0 0
\(430\) −25.4156 −1.22565
\(431\) 18.8941 0.910098 0.455049 0.890466i \(-0.349622\pi\)
0.455049 + 0.890466i \(0.349622\pi\)
\(432\) 0 0
\(433\) 18.2660 0.877810 0.438905 0.898534i \(-0.355367\pi\)
0.438905 + 0.898534i \(0.355367\pi\)
\(434\) 4.60265 0.220934
\(435\) 0 0
\(436\) 34.4766 1.65113
\(437\) 15.7666 0.754218
\(438\) 0 0
\(439\) −31.3238 −1.49501 −0.747503 0.664259i \(-0.768747\pi\)
−0.747503 + 0.664259i \(0.768747\pi\)
\(440\) −10.2353 −0.487951
\(441\) 0 0
\(442\) −9.94319 −0.472950
\(443\) 7.54570 0.358507 0.179254 0.983803i \(-0.442632\pi\)
0.179254 + 0.983803i \(0.442632\pi\)
\(444\) 0 0
\(445\) −20.7716 −0.984669
\(446\) −13.3117 −0.630329
\(447\) 0 0
\(448\) −10.8670 −0.513415
\(449\) 24.4439 1.15358 0.576790 0.816892i \(-0.304305\pi\)
0.576790 + 0.816892i \(0.304305\pi\)
\(450\) 0 0
\(451\) 2.34730 0.110530
\(452\) −22.3492 −1.05122
\(453\) 0 0
\(454\) −33.0382 −1.55056
\(455\) −7.10778 −0.333218
\(456\) 0 0
\(457\) 10.3236 0.482915 0.241458 0.970411i \(-0.422375\pi\)
0.241458 + 0.970411i \(0.422375\pi\)
\(458\) 0.766959 0.0358377
\(459\) 0 0
\(460\) −18.7462 −0.874044
\(461\) 0.224984 0.0104785 0.00523927 0.999986i \(-0.498332\pi\)
0.00523927 + 0.999986i \(0.498332\pi\)
\(462\) 0 0
\(463\) −21.4269 −0.995795 −0.497897 0.867236i \(-0.665894\pi\)
−0.497897 + 0.867236i \(0.665894\pi\)
\(464\) 6.21779 0.288654
\(465\) 0 0
\(466\) −8.84580 −0.409774
\(467\) −14.8070 −0.685186 −0.342593 0.939484i \(-0.611305\pi\)
−0.342593 + 0.939484i \(0.611305\pi\)
\(468\) 0 0
\(469\) 0.664083 0.0306645
\(470\) −59.3534 −2.73777
\(471\) 0 0
\(472\) −0.508630 −0.0234116
\(473\) 13.3357 0.613174
\(474\) 0 0
\(475\) −38.8472 −1.78243
\(476\) −5.28055 −0.242034
\(477\) 0 0
\(478\) −7.11414 −0.325393
\(479\) 10.2074 0.466390 0.233195 0.972430i \(-0.425082\pi\)
0.233195 + 0.972430i \(0.425082\pi\)
\(480\) 0 0
\(481\) 21.9147 0.999226
\(482\) 21.1246 0.962198
\(483\) 0 0
\(484\) 5.15968 0.234531
\(485\) −0.226825 −0.0102996
\(486\) 0 0
\(487\) 16.1543 0.732023 0.366011 0.930610i \(-0.380723\pi\)
0.366011 + 0.930610i \(0.380723\pi\)
\(488\) 11.6065 0.525402
\(489\) 0 0
\(490\) −6.90907 −0.312120
\(491\) 12.4633 0.562459 0.281229 0.959641i \(-0.409258\pi\)
0.281229 + 0.959641i \(0.409258\pi\)
\(492\) 0 0
\(493\) 4.52015 0.203577
\(494\) −30.2333 −1.36026
\(495\) 0 0
\(496\) 6.61066 0.296827
\(497\) 5.65049 0.253459
\(498\) 0 0
\(499\) −13.4987 −0.604283 −0.302142 0.953263i \(-0.597702\pi\)
−0.302142 + 0.953263i \(0.597702\pi\)
\(500\) 6.55976 0.293361
\(501\) 0 0
\(502\) 2.37673 0.106078
\(503\) −22.8347 −1.01815 −0.509074 0.860723i \(-0.670012\pi\)
−0.509074 + 0.860723i \(0.670012\pi\)
\(504\) 0 0
\(505\) −21.5074 −0.957068
\(506\) 18.0035 0.800354
\(507\) 0 0
\(508\) −2.40865 −0.106866
\(509\) 27.5341 1.22043 0.610214 0.792237i \(-0.291084\pi\)
0.610214 + 0.792237i \(0.291084\pi\)
\(510\) 0 0
\(511\) −3.24624 −0.143605
\(512\) −29.4458 −1.30133
\(513\) 0 0
\(514\) −61.4077 −2.70858
\(515\) 4.67534 0.206020
\(516\) 0 0
\(517\) 31.1429 1.36966
\(518\) 21.3021 0.935959
\(519\) 0 0
\(520\) 6.09869 0.267445
\(521\) −21.7185 −0.951504 −0.475752 0.879580i \(-0.657824\pi\)
−0.475752 + 0.879580i \(0.657824\pi\)
\(522\) 0 0
\(523\) −25.3129 −1.10685 −0.553427 0.832898i \(-0.686680\pi\)
−0.553427 + 0.832898i \(0.686680\pi\)
\(524\) −42.9024 −1.87420
\(525\) 0 0
\(526\) 55.3593 2.41378
\(527\) 4.80575 0.209342
\(528\) 0 0
\(529\) −17.4057 −0.756771
\(530\) 19.6856 0.855087
\(531\) 0 0
\(532\) −16.0561 −0.696119
\(533\) −1.39863 −0.0605815
\(534\) 0 0
\(535\) −56.4058 −2.43864
\(536\) −0.569803 −0.0246117
\(537\) 0 0
\(538\) −46.6703 −2.01210
\(539\) 3.62521 0.156149
\(540\) 0 0
\(541\) −0.966528 −0.0415543 −0.0207771 0.999784i \(-0.506614\pi\)
−0.0207771 + 0.999784i \(0.506614\pi\)
\(542\) 56.3744 2.42149
\(543\) 0 0
\(544\) −17.6441 −0.756483
\(545\) 47.0997 2.01753
\(546\) 0 0
\(547\) −6.92682 −0.296170 −0.148085 0.988975i \(-0.547311\pi\)
−0.148085 + 0.988975i \(0.547311\pi\)
\(548\) −18.0959 −0.773018
\(549\) 0 0
\(550\) −44.3588 −1.89146
\(551\) 13.7440 0.585513
\(552\) 0 0
\(553\) 4.01101 0.170566
\(554\) −4.43021 −0.188222
\(555\) 0 0
\(556\) −11.2538 −0.477268
\(557\) 41.5856 1.76204 0.881019 0.473080i \(-0.156858\pi\)
0.881019 + 0.473080i \(0.156858\pi\)
\(558\) 0 0
\(559\) −7.94600 −0.336080
\(560\) −9.92332 −0.419337
\(561\) 0 0
\(562\) 41.1645 1.73642
\(563\) 31.7414 1.33774 0.668871 0.743378i \(-0.266778\pi\)
0.668871 + 0.743378i \(0.266778\pi\)
\(564\) 0 0
\(565\) −30.5320 −1.28449
\(566\) 11.1025 0.466673
\(567\) 0 0
\(568\) −4.84828 −0.203429
\(569\) 0.333729 0.0139906 0.00699532 0.999976i \(-0.497773\pi\)
0.00699532 + 0.999976i \(0.497773\pi\)
\(570\) 0 0
\(571\) 12.9921 0.543704 0.271852 0.962339i \(-0.412364\pi\)
0.271852 + 0.962339i \(0.412364\pi\)
\(572\) −18.8614 −0.788634
\(573\) 0 0
\(574\) −1.35953 −0.0567457
\(575\) −13.7837 −0.574819
\(576\) 0 0
\(577\) 32.0896 1.33591 0.667954 0.744203i \(-0.267170\pi\)
0.667954 + 0.744203i \(0.267170\pi\)
\(578\) 25.6028 1.06494
\(579\) 0 0
\(580\) −16.3413 −0.678536
\(581\) 3.40033 0.141070
\(582\) 0 0
\(583\) −10.3291 −0.427787
\(584\) 2.78537 0.115259
\(585\) 0 0
\(586\) 19.0152 0.785509
\(587\) 40.9497 1.69017 0.845087 0.534629i \(-0.179549\pi\)
0.845087 + 0.534629i \(0.179549\pi\)
\(588\) 0 0
\(589\) 14.6124 0.602093
\(590\) −4.09562 −0.168614
\(591\) 0 0
\(592\) 30.5956 1.25747
\(593\) −42.4563 −1.74347 −0.871735 0.489978i \(-0.837005\pi\)
−0.871735 + 0.489978i \(0.837005\pi\)
\(594\) 0 0
\(595\) −7.21395 −0.295743
\(596\) 31.2238 1.27898
\(597\) 0 0
\(598\) −10.7273 −0.438673
\(599\) −26.8144 −1.09560 −0.547802 0.836608i \(-0.684535\pi\)
−0.547802 + 0.836608i \(0.684535\pi\)
\(600\) 0 0
\(601\) −32.3520 −1.31967 −0.659833 0.751412i \(-0.729373\pi\)
−0.659833 + 0.751412i \(0.729373\pi\)
\(602\) −7.72385 −0.314801
\(603\) 0 0
\(604\) 26.3357 1.07158
\(605\) 7.04882 0.286575
\(606\) 0 0
\(607\) −5.75375 −0.233538 −0.116769 0.993159i \(-0.537254\pi\)
−0.116769 + 0.993159i \(0.537254\pi\)
\(608\) −53.6486 −2.17574
\(609\) 0 0
\(610\) 93.4585 3.78402
\(611\) −18.5564 −0.750710
\(612\) 0 0
\(613\) −28.0175 −1.13162 −0.565808 0.824537i \(-0.691436\pi\)
−0.565808 + 0.824537i \(0.691436\pi\)
\(614\) −49.3261 −1.99064
\(615\) 0 0
\(616\) −3.11054 −0.125327
\(617\) 1.13222 0.0455814 0.0227907 0.999740i \(-0.492745\pi\)
0.0227907 + 0.999740i \(0.492745\pi\)
\(618\) 0 0
\(619\) 14.6756 0.589862 0.294931 0.955519i \(-0.404703\pi\)
0.294931 + 0.955519i \(0.404703\pi\)
\(620\) −17.3738 −0.697750
\(621\) 0 0
\(622\) −0.223847 −0.00897545
\(623\) −6.31252 −0.252906
\(624\) 0 0
\(625\) −20.1767 −0.807070
\(626\) −49.6180 −1.98313
\(627\) 0 0
\(628\) −49.4151 −1.97188
\(629\) 22.2421 0.886849
\(630\) 0 0
\(631\) −33.4295 −1.33081 −0.665404 0.746483i \(-0.731741\pi\)
−0.665404 + 0.746483i \(0.731741\pi\)
\(632\) −3.44157 −0.136898
\(633\) 0 0
\(634\) 18.6259 0.739729
\(635\) −3.29054 −0.130581
\(636\) 0 0
\(637\) −2.16007 −0.0855850
\(638\) 15.6940 0.621329
\(639\) 0 0
\(640\) 22.1156 0.874194
\(641\) −13.4509 −0.531278 −0.265639 0.964073i \(-0.585583\pi\)
−0.265639 + 0.964073i \(0.585583\pi\)
\(642\) 0 0
\(643\) 11.7618 0.463838 0.231919 0.972735i \(-0.425499\pi\)
0.231919 + 0.972735i \(0.425499\pi\)
\(644\) −5.69698 −0.224493
\(645\) 0 0
\(646\) −30.6849 −1.20728
\(647\) −8.12298 −0.319347 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(648\) 0 0
\(649\) 2.14898 0.0843550
\(650\) 26.4310 1.03671
\(651\) 0 0
\(652\) 3.21169 0.125779
\(653\) 29.8203 1.16696 0.583479 0.812128i \(-0.301691\pi\)
0.583479 + 0.812128i \(0.301691\pi\)
\(654\) 0 0
\(655\) −58.6105 −2.29010
\(656\) −1.95266 −0.0762384
\(657\) 0 0
\(658\) −18.0376 −0.703178
\(659\) −7.08303 −0.275916 −0.137958 0.990438i \(-0.544054\pi\)
−0.137958 + 0.990438i \(0.544054\pi\)
\(660\) 0 0
\(661\) 36.2548 1.41015 0.705075 0.709133i \(-0.250913\pi\)
0.705075 + 0.709133i \(0.250913\pi\)
\(662\) 57.4990 2.23476
\(663\) 0 0
\(664\) −2.91758 −0.113224
\(665\) −21.9348 −0.850594
\(666\) 0 0
\(667\) 4.87661 0.188823
\(668\) −1.98018 −0.0766153
\(669\) 0 0
\(670\) −4.58820 −0.177258
\(671\) −49.0379 −1.89309
\(672\) 0 0
\(673\) 22.9499 0.884656 0.442328 0.896853i \(-0.354153\pi\)
0.442328 + 0.896853i \(0.354153\pi\)
\(674\) −38.8226 −1.49539
\(675\) 0 0
\(676\) −20.0739 −0.772075
\(677\) −3.57237 −0.137297 −0.0686486 0.997641i \(-0.521869\pi\)
−0.0686486 + 0.997641i \(0.521869\pi\)
\(678\) 0 0
\(679\) −0.0689323 −0.00264538
\(680\) 6.18978 0.237367
\(681\) 0 0
\(682\) 16.6856 0.638923
\(683\) 22.1918 0.849145 0.424573 0.905394i \(-0.360424\pi\)
0.424573 + 0.905394i \(0.360424\pi\)
\(684\) 0 0
\(685\) −24.7214 −0.944558
\(686\) −2.09968 −0.0801661
\(687\) 0 0
\(688\) −11.0936 −0.422938
\(689\) 6.15454 0.234469
\(690\) 0 0
\(691\) −27.5328 −1.04740 −0.523698 0.851904i \(-0.675448\pi\)
−0.523698 + 0.851904i \(0.675448\pi\)
\(692\) 55.0625 2.09316
\(693\) 0 0
\(694\) 50.8979 1.93206
\(695\) −15.3742 −0.583178
\(696\) 0 0
\(697\) −1.41952 −0.0537683
\(698\) 16.2533 0.615195
\(699\) 0 0
\(700\) 14.0368 0.530540
\(701\) 33.7129 1.27332 0.636659 0.771146i \(-0.280316\pi\)
0.636659 + 0.771146i \(0.280316\pi\)
\(702\) 0 0
\(703\) 67.6293 2.55069
\(704\) −39.3950 −1.48475
\(705\) 0 0
\(706\) −22.2989 −0.839229
\(707\) −6.53614 −0.245817
\(708\) 0 0
\(709\) −43.1638 −1.62105 −0.810525 0.585704i \(-0.800818\pi\)
−0.810525 + 0.585704i \(0.800818\pi\)
\(710\) −39.0396 −1.46513
\(711\) 0 0
\(712\) 5.41633 0.202986
\(713\) 5.18474 0.194170
\(714\) 0 0
\(715\) −25.7672 −0.963639
\(716\) −3.66438 −0.136944
\(717\) 0 0
\(718\) 24.3434 0.908486
\(719\) −13.6734 −0.509931 −0.254965 0.966950i \(-0.582064\pi\)
−0.254965 + 0.966950i \(0.582064\pi\)
\(720\) 0 0
\(721\) 1.42084 0.0529150
\(722\) −53.4067 −1.98759
\(723\) 0 0
\(724\) 34.8085 1.29365
\(725\) −12.0154 −0.446243
\(726\) 0 0
\(727\) −32.4355 −1.20297 −0.601483 0.798886i \(-0.705423\pi\)
−0.601483 + 0.798886i \(0.705423\pi\)
\(728\) 1.85340 0.0686916
\(729\) 0 0
\(730\) 22.4285 0.830116
\(731\) −8.06468 −0.298283
\(732\) 0 0
\(733\) −42.8281 −1.58189 −0.790946 0.611886i \(-0.790411\pi\)
−0.790946 + 0.611886i \(0.790411\pi\)
\(734\) 41.0497 1.51517
\(735\) 0 0
\(736\) −19.0355 −0.701657
\(737\) 2.40744 0.0886792
\(738\) 0 0
\(739\) 38.7706 1.42620 0.713099 0.701063i \(-0.247291\pi\)
0.713099 + 0.701063i \(0.247291\pi\)
\(740\) −80.4099 −2.95593
\(741\) 0 0
\(742\) 5.98248 0.219624
\(743\) −13.9130 −0.510418 −0.255209 0.966886i \(-0.582144\pi\)
−0.255209 + 0.966886i \(0.582144\pi\)
\(744\) 0 0
\(745\) 42.6560 1.56280
\(746\) −8.30678 −0.304133
\(747\) 0 0
\(748\) −19.1431 −0.699942
\(749\) −17.1418 −0.626348
\(750\) 0 0
\(751\) −0.457421 −0.0166915 −0.00834576 0.999965i \(-0.502657\pi\)
−0.00834576 + 0.999965i \(0.502657\pi\)
\(752\) −25.9069 −0.944727
\(753\) 0 0
\(754\) −9.35118 −0.340550
\(755\) 35.9781 1.30938
\(756\) 0 0
\(757\) −20.9802 −0.762539 −0.381269 0.924464i \(-0.624513\pi\)
−0.381269 + 0.924464i \(0.624513\pi\)
\(758\) 55.1025 2.00141
\(759\) 0 0
\(760\) 18.8207 0.682697
\(761\) 30.4760 1.10476 0.552378 0.833594i \(-0.313721\pi\)
0.552378 + 0.833594i \(0.313721\pi\)
\(762\) 0 0
\(763\) 14.3137 0.518189
\(764\) −15.5704 −0.563316
\(765\) 0 0
\(766\) 4.74483 0.171438
\(767\) −1.28046 −0.0462348
\(768\) 0 0
\(769\) −21.8499 −0.787927 −0.393963 0.919126i \(-0.628896\pi\)
−0.393963 + 0.919126i \(0.628896\pi\)
\(770\) −25.0468 −0.902626
\(771\) 0 0
\(772\) −51.5831 −1.85652
\(773\) −19.4525 −0.699659 −0.349830 0.936813i \(-0.613761\pi\)
−0.349830 + 0.936813i \(0.613761\pi\)
\(774\) 0 0
\(775\) −12.7746 −0.458879
\(776\) 0.0591460 0.00212322
\(777\) 0 0
\(778\) −10.0630 −0.360776
\(779\) −4.31620 −0.154644
\(780\) 0 0
\(781\) 20.4842 0.732982
\(782\) −10.8876 −0.389338
\(783\) 0 0
\(784\) −3.01571 −0.107704
\(785\) −67.5077 −2.40945
\(786\) 0 0
\(787\) 41.2974 1.47209 0.736047 0.676930i \(-0.236690\pi\)
0.736047 + 0.676930i \(0.236690\pi\)
\(788\) −36.8103 −1.31131
\(789\) 0 0
\(790\) −27.7124 −0.985962
\(791\) −9.27873 −0.329914
\(792\) 0 0
\(793\) 29.2191 1.03760
\(794\) −72.4528 −2.57125
\(795\) 0 0
\(796\) −23.3217 −0.826615
\(797\) 35.1784 1.24608 0.623042 0.782188i \(-0.285897\pi\)
0.623042 + 0.782188i \(0.285897\pi\)
\(798\) 0 0
\(799\) −18.8335 −0.666282
\(800\) 46.9014 1.65822
\(801\) 0 0
\(802\) −4.42303 −0.156183
\(803\) −11.7683 −0.415294
\(804\) 0 0
\(805\) −7.78285 −0.274310
\(806\) −9.94203 −0.350193
\(807\) 0 0
\(808\) 5.60820 0.197296
\(809\) −32.9258 −1.15761 −0.578804 0.815466i \(-0.696480\pi\)
−0.578804 + 0.815466i \(0.696480\pi\)
\(810\) 0 0
\(811\) −46.5481 −1.63453 −0.817263 0.576265i \(-0.804510\pi\)
−0.817263 + 0.576265i \(0.804510\pi\)
\(812\) −4.96615 −0.174278
\(813\) 0 0
\(814\) 77.2245 2.70672
\(815\) 4.38760 0.153691
\(816\) 0 0
\(817\) −24.5215 −0.857899
\(818\) −77.3666 −2.70506
\(819\) 0 0
\(820\) 5.13188 0.179213
\(821\) −24.6736 −0.861114 −0.430557 0.902563i \(-0.641683\pi\)
−0.430557 + 0.902563i \(0.641683\pi\)
\(822\) 0 0
\(823\) −35.8137 −1.24839 −0.624193 0.781270i \(-0.714572\pi\)
−0.624193 + 0.781270i \(0.714572\pi\)
\(824\) −1.21913 −0.0424703
\(825\) 0 0
\(826\) −1.24467 −0.0433075
\(827\) 43.4022 1.50924 0.754621 0.656161i \(-0.227821\pi\)
0.754621 + 0.656161i \(0.227821\pi\)
\(828\) 0 0
\(829\) −12.6369 −0.438896 −0.219448 0.975624i \(-0.570426\pi\)
−0.219448 + 0.975624i \(0.570426\pi\)
\(830\) −23.4931 −0.815459
\(831\) 0 0
\(832\) 23.4733 0.813792
\(833\) −2.19233 −0.0759598
\(834\) 0 0
\(835\) −2.70519 −0.0936170
\(836\) −58.2066 −2.01312
\(837\) 0 0
\(838\) −49.1668 −1.69844
\(839\) −28.0483 −0.968336 −0.484168 0.874975i \(-0.660878\pi\)
−0.484168 + 0.874975i \(0.660878\pi\)
\(840\) 0 0
\(841\) −24.7490 −0.853413
\(842\) −15.1089 −0.520688
\(843\) 0 0
\(844\) −54.7056 −1.88304
\(845\) −27.4237 −0.943405
\(846\) 0 0
\(847\) 2.14215 0.0736050
\(848\) 8.59247 0.295067
\(849\) 0 0
\(850\) 26.8258 0.920117
\(851\) 23.9961 0.822576
\(852\) 0 0
\(853\) 19.5717 0.670121 0.335060 0.942197i \(-0.391243\pi\)
0.335060 + 0.942197i \(0.391243\pi\)
\(854\) 28.4022 0.971903
\(855\) 0 0
\(856\) 14.7082 0.502715
\(857\) −5.60926 −0.191609 −0.0958043 0.995400i \(-0.530542\pi\)
−0.0958043 + 0.995400i \(0.530542\pi\)
\(858\) 0 0
\(859\) 9.59070 0.327231 0.163615 0.986524i \(-0.447684\pi\)
0.163615 + 0.986524i \(0.447684\pi\)
\(860\) 29.1556 0.994197
\(861\) 0 0
\(862\) −39.6716 −1.35122
\(863\) 30.6668 1.04391 0.521955 0.852973i \(-0.325203\pi\)
0.521955 + 0.852973i \(0.325203\pi\)
\(864\) 0 0
\(865\) 75.2229 2.55765
\(866\) −38.3528 −1.30328
\(867\) 0 0
\(868\) −5.27993 −0.179213
\(869\) 14.5408 0.493262
\(870\) 0 0
\(871\) −1.43446 −0.0486049
\(872\) −12.2815 −0.415905
\(873\) 0 0
\(874\) −33.1047 −1.11978
\(875\) 2.72342 0.0920684
\(876\) 0 0
\(877\) −3.44789 −0.116427 −0.0582136 0.998304i \(-0.518540\pi\)
−0.0582136 + 0.998304i \(0.518540\pi\)
\(878\) 65.7700 2.21963
\(879\) 0 0
\(880\) −35.9741 −1.21269
\(881\) −37.1051 −1.25010 −0.625052 0.780583i \(-0.714922\pi\)
−0.625052 + 0.780583i \(0.714922\pi\)
\(882\) 0 0
\(883\) −18.6304 −0.626964 −0.313482 0.949594i \(-0.601496\pi\)
−0.313482 + 0.949594i \(0.601496\pi\)
\(884\) 11.4063 0.383637
\(885\) 0 0
\(886\) −15.8435 −0.532274
\(887\) 4.44415 0.149220 0.0746100 0.997213i \(-0.476229\pi\)
0.0746100 + 0.997213i \(0.476229\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 43.6137 1.46193
\(891\) 0 0
\(892\) 15.2706 0.511297
\(893\) −57.2653 −1.91631
\(894\) 0 0
\(895\) −5.00604 −0.167333
\(896\) 6.72095 0.224531
\(897\) 0 0
\(898\) −51.3244 −1.71272
\(899\) 4.51962 0.150738
\(900\) 0 0
\(901\) 6.24647 0.208100
\(902\) −4.92858 −0.164104
\(903\) 0 0
\(904\) 7.96142 0.264793
\(905\) 47.5531 1.58072
\(906\) 0 0
\(907\) 48.7902 1.62005 0.810026 0.586393i \(-0.199453\pi\)
0.810026 + 0.586393i \(0.199453\pi\)
\(908\) 37.8998 1.25775
\(909\) 0 0
\(910\) 14.9241 0.494728
\(911\) −33.0894 −1.09630 −0.548150 0.836380i \(-0.684668\pi\)
−0.548150 + 0.836380i \(0.684668\pi\)
\(912\) 0 0
\(913\) 12.3269 0.407961
\(914\) −21.6761 −0.716983
\(915\) 0 0
\(916\) −0.879818 −0.0290700
\(917\) −17.8118 −0.588198
\(918\) 0 0
\(919\) 4.96601 0.163814 0.0819068 0.996640i \(-0.473899\pi\)
0.0819068 + 0.996640i \(0.473899\pi\)
\(920\) 6.67792 0.220164
\(921\) 0 0
\(922\) −0.472394 −0.0155575
\(923\) −12.2054 −0.401747
\(924\) 0 0
\(925\) −59.1238 −1.94398
\(926\) 44.9897 1.47845
\(927\) 0 0
\(928\) −16.5935 −0.544709
\(929\) −2.64029 −0.0866251 −0.0433125 0.999062i \(-0.513791\pi\)
−0.0433125 + 0.999062i \(0.513791\pi\)
\(930\) 0 0
\(931\) −6.66601 −0.218470
\(932\) 10.1475 0.332391
\(933\) 0 0
\(934\) 31.0899 1.01729
\(935\) −26.1521 −0.855265
\(936\) 0 0
\(937\) −25.1677 −0.822194 −0.411097 0.911592i \(-0.634854\pi\)
−0.411097 + 0.911592i \(0.634854\pi\)
\(938\) −1.39436 −0.0455275
\(939\) 0 0
\(940\) 68.0873 2.22076
\(941\) −35.0361 −1.14215 −0.571073 0.820899i \(-0.693473\pi\)
−0.571073 + 0.820899i \(0.693473\pi\)
\(942\) 0 0
\(943\) −1.53147 −0.0498715
\(944\) −1.78768 −0.0581840
\(945\) 0 0
\(946\) −28.0006 −0.910377
\(947\) −34.0698 −1.10712 −0.553560 0.832809i \(-0.686731\pi\)
−0.553560 + 0.832809i \(0.686731\pi\)
\(948\) 0 0
\(949\) 7.01209 0.227622
\(950\) 81.5665 2.64637
\(951\) 0 0
\(952\) 1.88108 0.0609663
\(953\) 26.4808 0.857796 0.428898 0.903353i \(-0.358902\pi\)
0.428898 + 0.903353i \(0.358902\pi\)
\(954\) 0 0
\(955\) −21.2712 −0.688321
\(956\) 8.16100 0.263945
\(957\) 0 0
\(958\) −21.4323 −0.692447
\(959\) −7.51288 −0.242604
\(960\) 0 0
\(961\) −26.1948 −0.844994
\(962\) −46.0139 −1.48355
\(963\) 0 0
\(964\) −24.2331 −0.780495
\(965\) −70.4695 −2.26849
\(966\) 0 0
\(967\) −10.2513 −0.329659 −0.164829 0.986322i \(-0.552707\pi\)
−0.164829 + 0.986322i \(0.552707\pi\)
\(968\) −1.83802 −0.0590763
\(969\) 0 0
\(970\) 0.476259 0.0152917
\(971\) −10.1088 −0.324407 −0.162203 0.986757i \(-0.551860\pi\)
−0.162203 + 0.986757i \(0.551860\pi\)
\(972\) 0 0
\(973\) −4.67226 −0.149786
\(974\) −33.9189 −1.08683
\(975\) 0 0
\(976\) 40.7933 1.30576
\(977\) −10.4900 −0.335604 −0.167802 0.985821i \(-0.553667\pi\)
−0.167802 + 0.985821i \(0.553667\pi\)
\(978\) 0 0
\(979\) −22.8842 −0.731383
\(980\) 7.92575 0.253179
\(981\) 0 0
\(982\) −26.1688 −0.835081
\(983\) −15.4551 −0.492940 −0.246470 0.969150i \(-0.579271\pi\)
−0.246470 + 0.969150i \(0.579271\pi\)
\(984\) 0 0
\(985\) −50.2879 −1.60230
\(986\) −9.49085 −0.302250
\(987\) 0 0
\(988\) 34.6822 1.10339
\(989\) −8.70067 −0.276665
\(990\) 0 0
\(991\) 58.7140 1.86511 0.932556 0.361026i \(-0.117574\pi\)
0.932556 + 0.361026i \(0.117574\pi\)
\(992\) −17.6420 −0.560134
\(993\) 0 0
\(994\) −11.8642 −0.376310
\(995\) −31.8606 −1.01005
\(996\) 0 0
\(997\) −9.14112 −0.289502 −0.144751 0.989468i \(-0.546238\pi\)
−0.144751 + 0.989468i \(0.546238\pi\)
\(998\) 28.3428 0.897177
\(999\) 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 8001.2.a.l.1.1 7
3.2 odd 2 2667.2.a.j.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.j.1.7 7 3.2 odd 2
8001.2.a.l.1.1 7 1.1 even 1 trivial