Properties

Label 4005.2.a.o.1.5
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.07810\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.83770 q^{2} +1.37716 q^{4} -1.00000 q^{5} -4.72699 q^{7} -1.14460 q^{8} +O(q^{10})\) \(q+1.83770 q^{2} +1.37716 q^{4} -1.00000 q^{5} -4.72699 q^{7} -1.14460 q^{8} -1.83770 q^{10} +4.62787 q^{11} -4.47978 q^{13} -8.68681 q^{14} -4.85775 q^{16} +5.66969 q^{17} +1.34308 q^{19} -1.37716 q^{20} +8.50466 q^{22} -4.44187 q^{23} +1.00000 q^{25} -8.23250 q^{26} -6.50981 q^{28} +4.27246 q^{29} +9.40035 q^{31} -6.63791 q^{32} +10.4192 q^{34} +4.72699 q^{35} +6.89093 q^{37} +2.46817 q^{38} +1.14460 q^{40} -5.69631 q^{41} -5.04294 q^{43} +6.37330 q^{44} -8.16284 q^{46} +8.76271 q^{47} +15.3444 q^{49} +1.83770 q^{50} -6.16935 q^{52} +8.23538 q^{53} -4.62787 q^{55} +5.41053 q^{56} +7.85151 q^{58} +7.52173 q^{59} -6.94612 q^{61} +17.2751 q^{62} -2.48300 q^{64} +4.47978 q^{65} -6.38703 q^{67} +7.80805 q^{68} +8.68681 q^{70} -5.84983 q^{71} +11.0282 q^{73} +12.6635 q^{74} +1.84962 q^{76} -21.8759 q^{77} +7.89145 q^{79} +4.85775 q^{80} -10.4681 q^{82} +15.3880 q^{83} -5.66969 q^{85} -9.26742 q^{86} -5.29707 q^{88} -1.00000 q^{89} +21.1759 q^{91} -6.11715 q^{92} +16.1033 q^{94} -1.34308 q^{95} -0.674067 q^{97} +28.1985 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83770 1.29945 0.649727 0.760168i \(-0.274883\pi\)
0.649727 + 0.760168i \(0.274883\pi\)
\(3\) 0 0
\(4\) 1.37716 0.688578
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.72699 −1.78663 −0.893317 0.449426i \(-0.851628\pi\)
−0.893317 + 0.449426i \(0.851628\pi\)
\(8\) −1.14460 −0.404678
\(9\) 0 0
\(10\) −1.83770 −0.581133
\(11\) 4.62787 1.39536 0.697678 0.716412i \(-0.254217\pi\)
0.697678 + 0.716412i \(0.254217\pi\)
\(12\) 0 0
\(13\) −4.47978 −1.24247 −0.621233 0.783626i \(-0.713368\pi\)
−0.621233 + 0.783626i \(0.713368\pi\)
\(14\) −8.68681 −2.32165
\(15\) 0 0
\(16\) −4.85775 −1.21444
\(17\) 5.66969 1.37510 0.687551 0.726136i \(-0.258686\pi\)
0.687551 + 0.726136i \(0.258686\pi\)
\(18\) 0 0
\(19\) 1.34308 0.308123 0.154061 0.988061i \(-0.450765\pi\)
0.154061 + 0.988061i \(0.450765\pi\)
\(20\) −1.37716 −0.307942
\(21\) 0 0
\(22\) 8.50466 1.81320
\(23\) −4.44187 −0.926193 −0.463097 0.886308i \(-0.653262\pi\)
−0.463097 + 0.886308i \(0.653262\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −8.23250 −1.61453
\(27\) 0 0
\(28\) −6.50981 −1.23024
\(29\) 4.27246 0.793376 0.396688 0.917954i \(-0.370160\pi\)
0.396688 + 0.917954i \(0.370160\pi\)
\(30\) 0 0
\(31\) 9.40035 1.68835 0.844176 0.536066i \(-0.180090\pi\)
0.844176 + 0.536066i \(0.180090\pi\)
\(32\) −6.63791 −1.17343
\(33\) 0 0
\(34\) 10.4192 1.78688
\(35\) 4.72699 0.799007
\(36\) 0 0
\(37\) 6.89093 1.13286 0.566431 0.824109i \(-0.308324\pi\)
0.566431 + 0.824109i \(0.308324\pi\)
\(38\) 2.46817 0.400391
\(39\) 0 0
\(40\) 1.14460 0.180978
\(41\) −5.69631 −0.889614 −0.444807 0.895626i \(-0.646728\pi\)
−0.444807 + 0.895626i \(0.646728\pi\)
\(42\) 0 0
\(43\) −5.04294 −0.769041 −0.384520 0.923117i \(-0.625633\pi\)
−0.384520 + 0.923117i \(0.625633\pi\)
\(44\) 6.37330 0.960812
\(45\) 0 0
\(46\) −8.16284 −1.20354
\(47\) 8.76271 1.27817 0.639087 0.769135i \(-0.279313\pi\)
0.639087 + 0.769135i \(0.279313\pi\)
\(48\) 0 0
\(49\) 15.3444 2.19206
\(50\) 1.83770 0.259891
\(51\) 0 0
\(52\) −6.16935 −0.855535
\(53\) 8.23538 1.13122 0.565608 0.824674i \(-0.308642\pi\)
0.565608 + 0.824674i \(0.308642\pi\)
\(54\) 0 0
\(55\) −4.62787 −0.624022
\(56\) 5.41053 0.723012
\(57\) 0 0
\(58\) 7.85151 1.03095
\(59\) 7.52173 0.979246 0.489623 0.871934i \(-0.337134\pi\)
0.489623 + 0.871934i \(0.337134\pi\)
\(60\) 0 0
\(61\) −6.94612 −0.889360 −0.444680 0.895689i \(-0.646683\pi\)
−0.444680 + 0.895689i \(0.646683\pi\)
\(62\) 17.2751 2.19393
\(63\) 0 0
\(64\) −2.48300 −0.310376
\(65\) 4.47978 0.555648
\(66\) 0 0
\(67\) −6.38703 −0.780300 −0.390150 0.920751i \(-0.627577\pi\)
−0.390150 + 0.920751i \(0.627577\pi\)
\(68\) 7.80805 0.946866
\(69\) 0 0
\(70\) 8.68681 1.03827
\(71\) −5.84983 −0.694247 −0.347123 0.937820i \(-0.612841\pi\)
−0.347123 + 0.937820i \(0.612841\pi\)
\(72\) 0 0
\(73\) 11.0282 1.29076 0.645378 0.763863i \(-0.276700\pi\)
0.645378 + 0.763863i \(0.276700\pi\)
\(74\) 12.6635 1.47210
\(75\) 0 0
\(76\) 1.84962 0.212166
\(77\) −21.8759 −2.49299
\(78\) 0 0
\(79\) 7.89145 0.887857 0.443929 0.896062i \(-0.353584\pi\)
0.443929 + 0.896062i \(0.353584\pi\)
\(80\) 4.85775 0.543113
\(81\) 0 0
\(82\) −10.4681 −1.15601
\(83\) 15.3880 1.68905 0.844527 0.535513i \(-0.179882\pi\)
0.844527 + 0.535513i \(0.179882\pi\)
\(84\) 0 0
\(85\) −5.66969 −0.614965
\(86\) −9.26742 −0.999332
\(87\) 0 0
\(88\) −5.29707 −0.564670
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) 21.1759 2.21983
\(92\) −6.11715 −0.637757
\(93\) 0 0
\(94\) 16.1033 1.66093
\(95\) −1.34308 −0.137797
\(96\) 0 0
\(97\) −0.674067 −0.0684412 −0.0342206 0.999414i \(-0.510895\pi\)
−0.0342206 + 0.999414i \(0.510895\pi\)
\(98\) 28.1985 2.84848
\(99\) 0 0
\(100\) 1.37716 0.137716
\(101\) −13.4407 −1.33740 −0.668699 0.743533i \(-0.733149\pi\)
−0.668699 + 0.743533i \(0.733149\pi\)
\(102\) 0 0
\(103\) −11.4024 −1.12352 −0.561758 0.827302i \(-0.689875\pi\)
−0.561758 + 0.827302i \(0.689875\pi\)
\(104\) 5.12756 0.502799
\(105\) 0 0
\(106\) 15.1342 1.46996
\(107\) 8.51924 0.823586 0.411793 0.911277i \(-0.364903\pi\)
0.411793 + 0.911277i \(0.364903\pi\)
\(108\) 0 0
\(109\) 12.7180 1.21816 0.609081 0.793108i \(-0.291538\pi\)
0.609081 + 0.793108i \(0.291538\pi\)
\(110\) −8.50466 −0.810887
\(111\) 0 0
\(112\) 22.9626 2.16976
\(113\) −8.25522 −0.776586 −0.388293 0.921536i \(-0.626935\pi\)
−0.388293 + 0.921536i \(0.626935\pi\)
\(114\) 0 0
\(115\) 4.44187 0.414206
\(116\) 5.88384 0.546301
\(117\) 0 0
\(118\) 13.8227 1.27248
\(119\) −26.8006 −2.45681
\(120\) 0 0
\(121\) 10.4172 0.947018
\(122\) −12.7649 −1.15568
\(123\) 0 0
\(124\) 12.9457 1.16256
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.77097 0.157149 0.0785743 0.996908i \(-0.474963\pi\)
0.0785743 + 0.996908i \(0.474963\pi\)
\(128\) 8.71279 0.770109
\(129\) 0 0
\(130\) 8.23250 0.722038
\(131\) 7.28762 0.636722 0.318361 0.947969i \(-0.396868\pi\)
0.318361 + 0.947969i \(0.396868\pi\)
\(132\) 0 0
\(133\) −6.34870 −0.550503
\(134\) −11.7375 −1.01396
\(135\) 0 0
\(136\) −6.48955 −0.556474
\(137\) −0.754924 −0.0644975 −0.0322488 0.999480i \(-0.510267\pi\)
−0.0322488 + 0.999480i \(0.510267\pi\)
\(138\) 0 0
\(139\) −6.29913 −0.534285 −0.267142 0.963657i \(-0.586079\pi\)
−0.267142 + 0.963657i \(0.586079\pi\)
\(140\) 6.50981 0.550179
\(141\) 0 0
\(142\) −10.7502 −0.902141
\(143\) −20.7318 −1.73368
\(144\) 0 0
\(145\) −4.27246 −0.354808
\(146\) 20.2666 1.67728
\(147\) 0 0
\(148\) 9.48989 0.780064
\(149\) 17.7115 1.45098 0.725490 0.688233i \(-0.241613\pi\)
0.725490 + 0.688233i \(0.241613\pi\)
\(150\) 0 0
\(151\) 10.3991 0.846263 0.423132 0.906068i \(-0.360931\pi\)
0.423132 + 0.906068i \(0.360931\pi\)
\(152\) −1.53729 −0.124690
\(153\) 0 0
\(154\) −40.2014 −3.23953
\(155\) −9.40035 −0.755054
\(156\) 0 0
\(157\) −12.0575 −0.962292 −0.481146 0.876641i \(-0.659779\pi\)
−0.481146 + 0.876641i \(0.659779\pi\)
\(158\) 14.5021 1.15373
\(159\) 0 0
\(160\) 6.63791 0.524773
\(161\) 20.9967 1.65477
\(162\) 0 0
\(163\) 6.76614 0.529965 0.264982 0.964253i \(-0.414634\pi\)
0.264982 + 0.964253i \(0.414634\pi\)
\(164\) −7.84471 −0.612569
\(165\) 0 0
\(166\) 28.2786 2.19485
\(167\) 5.08722 0.393661 0.196830 0.980438i \(-0.436935\pi\)
0.196830 + 0.980438i \(0.436935\pi\)
\(168\) 0 0
\(169\) 7.06839 0.543722
\(170\) −10.4192 −0.799118
\(171\) 0 0
\(172\) −6.94491 −0.529545
\(173\) −1.63396 −0.124228 −0.0621139 0.998069i \(-0.519784\pi\)
−0.0621139 + 0.998069i \(0.519784\pi\)
\(174\) 0 0
\(175\) −4.72699 −0.357327
\(176\) −22.4811 −1.69457
\(177\) 0 0
\(178\) −1.83770 −0.137742
\(179\) 8.89727 0.665014 0.332507 0.943101i \(-0.392106\pi\)
0.332507 + 0.943101i \(0.392106\pi\)
\(180\) 0 0
\(181\) −7.14852 −0.531345 −0.265673 0.964063i \(-0.585594\pi\)
−0.265673 + 0.964063i \(0.585594\pi\)
\(182\) 38.9150 2.88457
\(183\) 0 0
\(184\) 5.08417 0.374810
\(185\) −6.89093 −0.506631
\(186\) 0 0
\(187\) 26.2386 1.91876
\(188\) 12.0676 0.880122
\(189\) 0 0
\(190\) −2.46817 −0.179060
\(191\) 8.74055 0.632444 0.316222 0.948685i \(-0.397586\pi\)
0.316222 + 0.948685i \(0.397586\pi\)
\(192\) 0 0
\(193\) 3.38192 0.243436 0.121718 0.992565i \(-0.461160\pi\)
0.121718 + 0.992565i \(0.461160\pi\)
\(194\) −1.23874 −0.0889361
\(195\) 0 0
\(196\) 21.1317 1.50941
\(197\) −25.4934 −1.81633 −0.908164 0.418614i \(-0.862516\pi\)
−0.908164 + 0.418614i \(0.862516\pi\)
\(198\) 0 0
\(199\) −7.06669 −0.500945 −0.250472 0.968124i \(-0.580586\pi\)
−0.250472 + 0.968124i \(0.580586\pi\)
\(200\) −1.14460 −0.0809356
\(201\) 0 0
\(202\) −24.7000 −1.73789
\(203\) −20.1959 −1.41747
\(204\) 0 0
\(205\) 5.69631 0.397847
\(206\) −20.9543 −1.45995
\(207\) 0 0
\(208\) 21.7616 1.50890
\(209\) 6.21558 0.429941
\(210\) 0 0
\(211\) −14.2330 −0.979842 −0.489921 0.871767i \(-0.662974\pi\)
−0.489921 + 0.871767i \(0.662974\pi\)
\(212\) 11.3414 0.778931
\(213\) 0 0
\(214\) 15.6559 1.07021
\(215\) 5.04294 0.343925
\(216\) 0 0
\(217\) −44.4354 −3.01647
\(218\) 23.3719 1.58295
\(219\) 0 0
\(220\) −6.37330 −0.429688
\(221\) −25.3990 −1.70852
\(222\) 0 0
\(223\) −3.11207 −0.208400 −0.104200 0.994556i \(-0.533228\pi\)
−0.104200 + 0.994556i \(0.533228\pi\)
\(224\) 31.3773 2.09649
\(225\) 0 0
\(226\) −15.1707 −1.00914
\(227\) 3.57158 0.237054 0.118527 0.992951i \(-0.462183\pi\)
0.118527 + 0.992951i \(0.462183\pi\)
\(228\) 0 0
\(229\) 5.99340 0.396055 0.198027 0.980196i \(-0.436547\pi\)
0.198027 + 0.980196i \(0.436547\pi\)
\(230\) 8.16284 0.538242
\(231\) 0 0
\(232\) −4.89027 −0.321062
\(233\) 4.75231 0.311334 0.155667 0.987810i \(-0.450247\pi\)
0.155667 + 0.987810i \(0.450247\pi\)
\(234\) 0 0
\(235\) −8.76271 −0.571616
\(236\) 10.3586 0.674288
\(237\) 0 0
\(238\) −49.2516 −3.19250
\(239\) 15.9181 1.02965 0.514827 0.857294i \(-0.327856\pi\)
0.514827 + 0.857294i \(0.327856\pi\)
\(240\) 0 0
\(241\) 12.3726 0.796987 0.398493 0.917171i \(-0.369533\pi\)
0.398493 + 0.917171i \(0.369533\pi\)
\(242\) 19.1437 1.23061
\(243\) 0 0
\(244\) −9.56590 −0.612394
\(245\) −15.3444 −0.980321
\(246\) 0 0
\(247\) −6.01668 −0.382832
\(248\) −10.7597 −0.683239
\(249\) 0 0
\(250\) −1.83770 −0.116227
\(251\) 28.0777 1.77225 0.886125 0.463447i \(-0.153388\pi\)
0.886125 + 0.463447i \(0.153388\pi\)
\(252\) 0 0
\(253\) −20.5564 −1.29237
\(254\) 3.25453 0.204207
\(255\) 0 0
\(256\) 20.9775 1.31110
\(257\) 8.05379 0.502382 0.251191 0.967938i \(-0.419178\pi\)
0.251191 + 0.967938i \(0.419178\pi\)
\(258\) 0 0
\(259\) −32.5734 −2.02401
\(260\) 6.16935 0.382607
\(261\) 0 0
\(262\) 13.3925 0.827391
\(263\) −28.3461 −1.74790 −0.873949 0.486018i \(-0.838449\pi\)
−0.873949 + 0.486018i \(0.838449\pi\)
\(264\) 0 0
\(265\) −8.23538 −0.505895
\(266\) −11.6670 −0.715352
\(267\) 0 0
\(268\) −8.79594 −0.537298
\(269\) 4.54674 0.277220 0.138610 0.990347i \(-0.455737\pi\)
0.138610 + 0.990347i \(0.455737\pi\)
\(270\) 0 0
\(271\) 12.5691 0.763521 0.381760 0.924261i \(-0.375318\pi\)
0.381760 + 0.924261i \(0.375318\pi\)
\(272\) −27.5420 −1.66998
\(273\) 0 0
\(274\) −1.38733 −0.0838115
\(275\) 4.62787 0.279071
\(276\) 0 0
\(277\) 16.6171 0.998426 0.499213 0.866479i \(-0.333622\pi\)
0.499213 + 0.866479i \(0.333622\pi\)
\(278\) −11.5759 −0.694278
\(279\) 0 0
\(280\) −5.41053 −0.323341
\(281\) 32.3979 1.93270 0.966348 0.257237i \(-0.0828121\pi\)
0.966348 + 0.257237i \(0.0828121\pi\)
\(282\) 0 0
\(283\) 1.59565 0.0948517 0.0474258 0.998875i \(-0.484898\pi\)
0.0474258 + 0.998875i \(0.484898\pi\)
\(284\) −8.05612 −0.478043
\(285\) 0 0
\(286\) −38.0990 −2.25284
\(287\) 26.9264 1.58942
\(288\) 0 0
\(289\) 15.1454 0.890908
\(290\) −7.85151 −0.461057
\(291\) 0 0
\(292\) 15.1876 0.888786
\(293\) −12.2986 −0.718491 −0.359246 0.933243i \(-0.616966\pi\)
−0.359246 + 0.933243i \(0.616966\pi\)
\(294\) 0 0
\(295\) −7.52173 −0.437932
\(296\) −7.88737 −0.458444
\(297\) 0 0
\(298\) 32.5484 1.88548
\(299\) 19.8986 1.15076
\(300\) 0 0
\(301\) 23.8379 1.37399
\(302\) 19.1104 1.09968
\(303\) 0 0
\(304\) −6.52433 −0.374196
\(305\) 6.94612 0.397734
\(306\) 0 0
\(307\) −25.7367 −1.46887 −0.734436 0.678678i \(-0.762553\pi\)
−0.734436 + 0.678678i \(0.762553\pi\)
\(308\) −30.1265 −1.71662
\(309\) 0 0
\(310\) −17.2751 −0.981157
\(311\) −8.88817 −0.504002 −0.252001 0.967727i \(-0.581089\pi\)
−0.252001 + 0.967727i \(0.581089\pi\)
\(312\) 0 0
\(313\) 12.2041 0.689818 0.344909 0.938636i \(-0.387910\pi\)
0.344909 + 0.938636i \(0.387910\pi\)
\(314\) −22.1581 −1.25045
\(315\) 0 0
\(316\) 10.8678 0.611359
\(317\) −14.6948 −0.825341 −0.412671 0.910880i \(-0.635404\pi\)
−0.412671 + 0.910880i \(0.635404\pi\)
\(318\) 0 0
\(319\) 19.7724 1.10704
\(320\) 2.48300 0.138804
\(321\) 0 0
\(322\) 38.5857 2.15030
\(323\) 7.61483 0.423700
\(324\) 0 0
\(325\) −4.47978 −0.248493
\(326\) 12.4342 0.688664
\(327\) 0 0
\(328\) 6.52001 0.360007
\(329\) −41.4213 −2.28363
\(330\) 0 0
\(331\) −18.4337 −1.01321 −0.506603 0.862179i \(-0.669099\pi\)
−0.506603 + 0.862179i \(0.669099\pi\)
\(332\) 21.1917 1.16305
\(333\) 0 0
\(334\) 9.34880 0.511544
\(335\) 6.38703 0.348961
\(336\) 0 0
\(337\) −31.8582 −1.73543 −0.867714 0.497064i \(-0.834411\pi\)
−0.867714 + 0.497064i \(0.834411\pi\)
\(338\) 12.9896 0.706542
\(339\) 0 0
\(340\) −7.80805 −0.423451
\(341\) 43.5036 2.35585
\(342\) 0 0
\(343\) −39.4441 −2.12978
\(344\) 5.77216 0.311214
\(345\) 0 0
\(346\) −3.00274 −0.161428
\(347\) −5.94469 −0.319128 −0.159564 0.987188i \(-0.551009\pi\)
−0.159564 + 0.987188i \(0.551009\pi\)
\(348\) 0 0
\(349\) 12.4803 0.668053 0.334026 0.942564i \(-0.391593\pi\)
0.334026 + 0.942564i \(0.391593\pi\)
\(350\) −8.68681 −0.464330
\(351\) 0 0
\(352\) −30.7194 −1.63735
\(353\) −17.7880 −0.946761 −0.473380 0.880858i \(-0.656966\pi\)
−0.473380 + 0.880858i \(0.656966\pi\)
\(354\) 0 0
\(355\) 5.84983 0.310476
\(356\) −1.37716 −0.0729891
\(357\) 0 0
\(358\) 16.3506 0.864154
\(359\) −6.80968 −0.359401 −0.179700 0.983721i \(-0.557513\pi\)
−0.179700 + 0.983721i \(0.557513\pi\)
\(360\) 0 0
\(361\) −17.1961 −0.905060
\(362\) −13.1369 −0.690458
\(363\) 0 0
\(364\) 29.1625 1.52853
\(365\) −11.0282 −0.577244
\(366\) 0 0
\(367\) −10.9282 −0.570447 −0.285224 0.958461i \(-0.592068\pi\)
−0.285224 + 0.958461i \(0.592068\pi\)
\(368\) 21.5775 1.12480
\(369\) 0 0
\(370\) −12.6635 −0.658343
\(371\) −38.9285 −2.02107
\(372\) 0 0
\(373\) −6.11907 −0.316834 −0.158417 0.987372i \(-0.550639\pi\)
−0.158417 + 0.987372i \(0.550639\pi\)
\(374\) 48.2188 2.49334
\(375\) 0 0
\(376\) −10.0298 −0.517249
\(377\) −19.1397 −0.985742
\(378\) 0 0
\(379\) 28.8933 1.48415 0.742074 0.670318i \(-0.233842\pi\)
0.742074 + 0.670318i \(0.233842\pi\)
\(380\) −1.84962 −0.0948837
\(381\) 0 0
\(382\) 16.0626 0.821832
\(383\) 15.9953 0.817322 0.408661 0.912686i \(-0.365996\pi\)
0.408661 + 0.912686i \(0.365996\pi\)
\(384\) 0 0
\(385\) 21.8759 1.11490
\(386\) 6.21497 0.316334
\(387\) 0 0
\(388\) −0.928296 −0.0471271
\(389\) −30.0561 −1.52390 −0.761952 0.647633i \(-0.775759\pi\)
−0.761952 + 0.647633i \(0.775759\pi\)
\(390\) 0 0
\(391\) −25.1840 −1.27361
\(392\) −17.5633 −0.887080
\(393\) 0 0
\(394\) −46.8493 −2.36023
\(395\) −7.89145 −0.397062
\(396\) 0 0
\(397\) 25.5045 1.28004 0.640018 0.768360i \(-0.278927\pi\)
0.640018 + 0.768360i \(0.278927\pi\)
\(398\) −12.9865 −0.650954
\(399\) 0 0
\(400\) −4.85775 −0.242888
\(401\) 19.4677 0.972172 0.486086 0.873911i \(-0.338424\pi\)
0.486086 + 0.873911i \(0.338424\pi\)
\(402\) 0 0
\(403\) −42.1114 −2.09772
\(404\) −18.5099 −0.920903
\(405\) 0 0
\(406\) −37.1140 −1.84194
\(407\) 31.8903 1.58075
\(408\) 0 0
\(409\) 23.2110 1.14771 0.573854 0.818957i \(-0.305448\pi\)
0.573854 + 0.818957i \(0.305448\pi\)
\(410\) 10.4681 0.516984
\(411\) 0 0
\(412\) −15.7029 −0.773628
\(413\) −35.5552 −1.74956
\(414\) 0 0
\(415\) −15.3880 −0.755368
\(416\) 29.7363 1.45794
\(417\) 0 0
\(418\) 11.4224 0.558688
\(419\) 22.3096 1.08990 0.544948 0.838470i \(-0.316549\pi\)
0.544948 + 0.838470i \(0.316549\pi\)
\(420\) 0 0
\(421\) −29.2036 −1.42330 −0.711648 0.702536i \(-0.752051\pi\)
−0.711648 + 0.702536i \(0.752051\pi\)
\(422\) −26.1561 −1.27326
\(423\) 0 0
\(424\) −9.42623 −0.457778
\(425\) 5.66969 0.275021
\(426\) 0 0
\(427\) 32.8343 1.58896
\(428\) 11.7323 0.567104
\(429\) 0 0
\(430\) 9.26742 0.446915
\(431\) −14.8466 −0.715134 −0.357567 0.933888i \(-0.616394\pi\)
−0.357567 + 0.933888i \(0.616394\pi\)
\(432\) 0 0
\(433\) −13.0139 −0.625408 −0.312704 0.949851i \(-0.601235\pi\)
−0.312704 + 0.949851i \(0.601235\pi\)
\(434\) −81.6590 −3.91976
\(435\) 0 0
\(436\) 17.5147 0.838800
\(437\) −5.96576 −0.285381
\(438\) 0 0
\(439\) −12.1861 −0.581609 −0.290804 0.956783i \(-0.593923\pi\)
−0.290804 + 0.956783i \(0.593923\pi\)
\(440\) 5.29707 0.252528
\(441\) 0 0
\(442\) −46.6758 −2.22014
\(443\) 1.82580 0.0867465 0.0433732 0.999059i \(-0.486190\pi\)
0.0433732 + 0.999059i \(0.486190\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −5.71906 −0.270806
\(447\) 0 0
\(448\) 11.7371 0.554528
\(449\) 10.9646 0.517449 0.258725 0.965951i \(-0.416698\pi\)
0.258725 + 0.965951i \(0.416698\pi\)
\(450\) 0 0
\(451\) −26.3618 −1.24133
\(452\) −11.3687 −0.534740
\(453\) 0 0
\(454\) 6.56351 0.308041
\(455\) −21.1759 −0.992740
\(456\) 0 0
\(457\) 4.14517 0.193903 0.0969514 0.995289i \(-0.469091\pi\)
0.0969514 + 0.995289i \(0.469091\pi\)
\(458\) 11.0141 0.514655
\(459\) 0 0
\(460\) 6.11715 0.285213
\(461\) 35.6431 1.66006 0.830031 0.557717i \(-0.188322\pi\)
0.830031 + 0.557717i \(0.188322\pi\)
\(462\) 0 0
\(463\) −3.08964 −0.143588 −0.0717939 0.997419i \(-0.522872\pi\)
−0.0717939 + 0.997419i \(0.522872\pi\)
\(464\) −20.7545 −0.963506
\(465\) 0 0
\(466\) 8.73335 0.404564
\(467\) 5.13078 0.237424 0.118712 0.992929i \(-0.462123\pi\)
0.118712 + 0.992929i \(0.462123\pi\)
\(468\) 0 0
\(469\) 30.1914 1.39411
\(470\) −16.1033 −0.742789
\(471\) 0 0
\(472\) −8.60939 −0.396279
\(473\) −23.3381 −1.07309
\(474\) 0 0
\(475\) 1.34308 0.0616245
\(476\) −36.9086 −1.69170
\(477\) 0 0
\(478\) 29.2527 1.33799
\(479\) −6.76545 −0.309121 −0.154561 0.987983i \(-0.549396\pi\)
−0.154561 + 0.987983i \(0.549396\pi\)
\(480\) 0 0
\(481\) −30.8698 −1.40754
\(482\) 22.7371 1.03565
\(483\) 0 0
\(484\) 14.3461 0.652096
\(485\) 0.674067 0.0306078
\(486\) 0 0
\(487\) −0.747648 −0.0338792 −0.0169396 0.999857i \(-0.505392\pi\)
−0.0169396 + 0.999857i \(0.505392\pi\)
\(488\) 7.95055 0.359905
\(489\) 0 0
\(490\) −28.1985 −1.27388
\(491\) −22.1110 −0.997857 −0.498928 0.866643i \(-0.666273\pi\)
−0.498928 + 0.866643i \(0.666273\pi\)
\(492\) 0 0
\(493\) 24.2235 1.09097
\(494\) −11.0569 −0.497472
\(495\) 0 0
\(496\) −45.6646 −2.05040
\(497\) 27.6521 1.24036
\(498\) 0 0
\(499\) −35.1425 −1.57319 −0.786597 0.617467i \(-0.788159\pi\)
−0.786597 + 0.617467i \(0.788159\pi\)
\(500\) −1.37716 −0.0615883
\(501\) 0 0
\(502\) 51.5985 2.30295
\(503\) 14.8767 0.663320 0.331660 0.943399i \(-0.392391\pi\)
0.331660 + 0.943399i \(0.392391\pi\)
\(504\) 0 0
\(505\) 13.4407 0.598103
\(506\) −37.7766 −1.67937
\(507\) 0 0
\(508\) 2.43891 0.108209
\(509\) 13.5925 0.602476 0.301238 0.953549i \(-0.402600\pi\)
0.301238 + 0.953549i \(0.402600\pi\)
\(510\) 0 0
\(511\) −52.1303 −2.30611
\(512\) 21.1249 0.933599
\(513\) 0 0
\(514\) 14.8005 0.652822
\(515\) 11.4024 0.502451
\(516\) 0 0
\(517\) 40.5527 1.78351
\(518\) −59.8602 −2.63011
\(519\) 0 0
\(520\) −5.12756 −0.224858
\(521\) −21.2471 −0.930851 −0.465425 0.885087i \(-0.654099\pi\)
−0.465425 + 0.885087i \(0.654099\pi\)
\(522\) 0 0
\(523\) 37.9681 1.66023 0.830115 0.557592i \(-0.188275\pi\)
0.830115 + 0.557592i \(0.188275\pi\)
\(524\) 10.0362 0.438433
\(525\) 0 0
\(526\) −52.0918 −2.27131
\(527\) 53.2971 2.32166
\(528\) 0 0
\(529\) −3.26981 −0.142166
\(530\) −15.1342 −0.657387
\(531\) 0 0
\(532\) −8.74316 −0.379064
\(533\) 25.5182 1.10532
\(534\) 0 0
\(535\) −8.51924 −0.368319
\(536\) 7.31061 0.315770
\(537\) 0 0
\(538\) 8.35556 0.360234
\(539\) 71.0121 3.05871
\(540\) 0 0
\(541\) 24.0786 1.03522 0.517609 0.855617i \(-0.326822\pi\)
0.517609 + 0.855617i \(0.326822\pi\)
\(542\) 23.0984 0.992159
\(543\) 0 0
\(544\) −37.6349 −1.61358
\(545\) −12.7180 −0.544779
\(546\) 0 0
\(547\) −7.90482 −0.337986 −0.168993 0.985617i \(-0.554052\pi\)
−0.168993 + 0.985617i \(0.554052\pi\)
\(548\) −1.03965 −0.0444116
\(549\) 0 0
\(550\) 8.50466 0.362640
\(551\) 5.73823 0.244457
\(552\) 0 0
\(553\) −37.3028 −1.58628
\(554\) 30.5374 1.29741
\(555\) 0 0
\(556\) −8.67488 −0.367897
\(557\) 28.0206 1.18727 0.593635 0.804734i \(-0.297692\pi\)
0.593635 + 0.804734i \(0.297692\pi\)
\(558\) 0 0
\(559\) 22.5912 0.955507
\(560\) −22.9626 −0.970345
\(561\) 0 0
\(562\) 59.5378 2.51145
\(563\) 41.0773 1.73120 0.865601 0.500735i \(-0.166937\pi\)
0.865601 + 0.500735i \(0.166937\pi\)
\(564\) 0 0
\(565\) 8.25522 0.347300
\(566\) 2.93234 0.123255
\(567\) 0 0
\(568\) 6.69572 0.280946
\(569\) 30.8521 1.29339 0.646693 0.762751i \(-0.276152\pi\)
0.646693 + 0.762751i \(0.276152\pi\)
\(570\) 0 0
\(571\) 27.2466 1.14023 0.570117 0.821564i \(-0.306898\pi\)
0.570117 + 0.821564i \(0.306898\pi\)
\(572\) −28.5510 −1.19378
\(573\) 0 0
\(574\) 49.4828 2.06537
\(575\) −4.44187 −0.185239
\(576\) 0 0
\(577\) 26.1936 1.09045 0.545227 0.838288i \(-0.316443\pi\)
0.545227 + 0.838288i \(0.316443\pi\)
\(578\) 27.8328 1.15769
\(579\) 0 0
\(580\) −5.88384 −0.244313
\(581\) −72.7390 −3.01772
\(582\) 0 0
\(583\) 38.1123 1.57845
\(584\) −12.6229 −0.522341
\(585\) 0 0
\(586\) −22.6012 −0.933645
\(587\) 16.6599 0.687627 0.343814 0.939038i \(-0.388281\pi\)
0.343814 + 0.939038i \(0.388281\pi\)
\(588\) 0 0
\(589\) 12.6254 0.520219
\(590\) −13.8227 −0.569072
\(591\) 0 0
\(592\) −33.4744 −1.37579
\(593\) 30.7704 1.26359 0.631794 0.775136i \(-0.282319\pi\)
0.631794 + 0.775136i \(0.282319\pi\)
\(594\) 0 0
\(595\) 26.8006 1.09872
\(596\) 24.3914 0.999113
\(597\) 0 0
\(598\) 36.5677 1.49536
\(599\) 24.7821 1.01257 0.506285 0.862366i \(-0.331019\pi\)
0.506285 + 0.862366i \(0.331019\pi\)
\(600\) 0 0
\(601\) 2.25624 0.0920340 0.0460170 0.998941i \(-0.485347\pi\)
0.0460170 + 0.998941i \(0.485347\pi\)
\(602\) 43.8070 1.78544
\(603\) 0 0
\(604\) 14.3211 0.582719
\(605\) −10.4172 −0.423519
\(606\) 0 0
\(607\) 20.1084 0.816175 0.408088 0.912943i \(-0.366196\pi\)
0.408088 + 0.912943i \(0.366196\pi\)
\(608\) −8.91521 −0.361560
\(609\) 0 0
\(610\) 12.7649 0.516837
\(611\) −39.2550 −1.58809
\(612\) 0 0
\(613\) −14.9346 −0.603204 −0.301602 0.953434i \(-0.597521\pi\)
−0.301602 + 0.953434i \(0.597521\pi\)
\(614\) −47.2964 −1.90873
\(615\) 0 0
\(616\) 25.0392 1.00886
\(617\) −17.1073 −0.688714 −0.344357 0.938839i \(-0.611903\pi\)
−0.344357 + 0.938839i \(0.611903\pi\)
\(618\) 0 0
\(619\) 39.8548 1.60190 0.800950 0.598732i \(-0.204328\pi\)
0.800950 + 0.598732i \(0.204328\pi\)
\(620\) −12.9457 −0.519914
\(621\) 0 0
\(622\) −16.3338 −0.654927
\(623\) 4.72699 0.189383
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.4276 0.896387
\(627\) 0 0
\(628\) −16.6050 −0.662613
\(629\) 39.0695 1.55780
\(630\) 0 0
\(631\) −1.26371 −0.0503077 −0.0251538 0.999684i \(-0.508008\pi\)
−0.0251538 + 0.999684i \(0.508008\pi\)
\(632\) −9.03257 −0.359296
\(633\) 0 0
\(634\) −27.0047 −1.07249
\(635\) −1.77097 −0.0702790
\(636\) 0 0
\(637\) −68.7397 −2.72356
\(638\) 36.3358 1.43855
\(639\) 0 0
\(640\) −8.71279 −0.344403
\(641\) −10.4319 −0.412037 −0.206018 0.978548i \(-0.566051\pi\)
−0.206018 + 0.978548i \(0.566051\pi\)
\(642\) 0 0
\(643\) 35.6759 1.40692 0.703460 0.710735i \(-0.251637\pi\)
0.703460 + 0.710735i \(0.251637\pi\)
\(644\) 28.9157 1.13944
\(645\) 0 0
\(646\) 13.9938 0.550579
\(647\) 27.5263 1.08217 0.541085 0.840968i \(-0.318014\pi\)
0.541085 + 0.840968i \(0.318014\pi\)
\(648\) 0 0
\(649\) 34.8096 1.36640
\(650\) −8.23250 −0.322905
\(651\) 0 0
\(652\) 9.31803 0.364922
\(653\) 1.79316 0.0701718 0.0350859 0.999384i \(-0.488830\pi\)
0.0350859 + 0.999384i \(0.488830\pi\)
\(654\) 0 0
\(655\) −7.28762 −0.284751
\(656\) 27.6713 1.08038
\(657\) 0 0
\(658\) −76.1200 −2.96747
\(659\) −5.13756 −0.200131 −0.100065 0.994981i \(-0.531905\pi\)
−0.100065 + 0.994981i \(0.531905\pi\)
\(660\) 0 0
\(661\) −18.0419 −0.701748 −0.350874 0.936423i \(-0.614115\pi\)
−0.350874 + 0.936423i \(0.614115\pi\)
\(662\) −33.8756 −1.31661
\(663\) 0 0
\(664\) −17.6132 −0.683523
\(665\) 6.34870 0.246192
\(666\) 0 0
\(667\) −18.9777 −0.734819
\(668\) 7.00590 0.271066
\(669\) 0 0
\(670\) 11.7375 0.453458
\(671\) −32.1458 −1.24097
\(672\) 0 0
\(673\) −1.02851 −0.0396461 −0.0198231 0.999804i \(-0.506310\pi\)
−0.0198231 + 0.999804i \(0.506310\pi\)
\(674\) −58.5460 −2.25511
\(675\) 0 0
\(676\) 9.73428 0.374395
\(677\) −36.0528 −1.38562 −0.692812 0.721119i \(-0.743628\pi\)
−0.692812 + 0.721119i \(0.743628\pi\)
\(678\) 0 0
\(679\) 3.18631 0.122279
\(680\) 6.48955 0.248863
\(681\) 0 0
\(682\) 79.9467 3.06132
\(683\) −14.0836 −0.538895 −0.269447 0.963015i \(-0.586841\pi\)
−0.269447 + 0.963015i \(0.586841\pi\)
\(684\) 0 0
\(685\) 0.754924 0.0288442
\(686\) −72.4866 −2.76755
\(687\) 0 0
\(688\) 24.4973 0.933952
\(689\) −36.8926 −1.40550
\(690\) 0 0
\(691\) 15.6187 0.594163 0.297081 0.954852i \(-0.403987\pi\)
0.297081 + 0.954852i \(0.403987\pi\)
\(692\) −2.25022 −0.0855406
\(693\) 0 0
\(694\) −10.9246 −0.414692
\(695\) 6.29913 0.238939
\(696\) 0 0
\(697\) −32.2963 −1.22331
\(698\) 22.9350 0.868103
\(699\) 0 0
\(700\) −6.50981 −0.246048
\(701\) −23.8768 −0.901814 −0.450907 0.892571i \(-0.648899\pi\)
−0.450907 + 0.892571i \(0.648899\pi\)
\(702\) 0 0
\(703\) 9.25504 0.349060
\(704\) −11.4910 −0.433084
\(705\) 0 0
\(706\) −32.6891 −1.23027
\(707\) 63.5340 2.38944
\(708\) 0 0
\(709\) −13.7608 −0.516798 −0.258399 0.966038i \(-0.583195\pi\)
−0.258399 + 0.966038i \(0.583195\pi\)
\(710\) 10.7502 0.403450
\(711\) 0 0
\(712\) 1.14460 0.0428958
\(713\) −41.7551 −1.56374
\(714\) 0 0
\(715\) 20.7318 0.775326
\(716\) 12.2529 0.457914
\(717\) 0 0
\(718\) −12.5142 −0.467025
\(719\) 18.7708 0.700032 0.350016 0.936744i \(-0.386176\pi\)
0.350016 + 0.936744i \(0.386176\pi\)
\(720\) 0 0
\(721\) 53.8992 2.00731
\(722\) −31.6014 −1.17608
\(723\) 0 0
\(724\) −9.84462 −0.365873
\(725\) 4.27246 0.158675
\(726\) 0 0
\(727\) 26.9412 0.999192 0.499596 0.866258i \(-0.333482\pi\)
0.499596 + 0.866258i \(0.333482\pi\)
\(728\) −24.2379 −0.898318
\(729\) 0 0
\(730\) −20.2666 −0.750101
\(731\) −28.5919 −1.05751
\(732\) 0 0
\(733\) 17.5711 0.649004 0.324502 0.945885i \(-0.394803\pi\)
0.324502 + 0.945885i \(0.394803\pi\)
\(734\) −20.0828 −0.741269
\(735\) 0 0
\(736\) 29.4847 1.08682
\(737\) −29.5584 −1.08880
\(738\) 0 0
\(739\) 11.5994 0.426689 0.213345 0.976977i \(-0.431564\pi\)
0.213345 + 0.976977i \(0.431564\pi\)
\(740\) −9.48989 −0.348855
\(741\) 0 0
\(742\) −71.5392 −2.62628
\(743\) −8.10055 −0.297180 −0.148590 0.988899i \(-0.547474\pi\)
−0.148590 + 0.988899i \(0.547474\pi\)
\(744\) 0 0
\(745\) −17.7115 −0.648898
\(746\) −11.2450 −0.411710
\(747\) 0 0
\(748\) 36.1347 1.32121
\(749\) −40.2704 −1.47145
\(750\) 0 0
\(751\) 2.23998 0.0817381 0.0408690 0.999165i \(-0.486987\pi\)
0.0408690 + 0.999165i \(0.486987\pi\)
\(752\) −42.5671 −1.55226
\(753\) 0 0
\(754\) −35.1730 −1.28093
\(755\) −10.3991 −0.378461
\(756\) 0 0
\(757\) −13.5325 −0.491846 −0.245923 0.969289i \(-0.579091\pi\)
−0.245923 + 0.969289i \(0.579091\pi\)
\(758\) 53.0973 1.92858
\(759\) 0 0
\(760\) 1.53729 0.0557633
\(761\) −37.5201 −1.36010 −0.680051 0.733165i \(-0.738042\pi\)
−0.680051 + 0.733165i \(0.738042\pi\)
\(762\) 0 0
\(763\) −60.1178 −2.17641
\(764\) 12.0371 0.435487
\(765\) 0 0
\(766\) 29.3946 1.06207
\(767\) −33.6957 −1.21668
\(768\) 0 0
\(769\) 7.95922 0.287017 0.143508 0.989649i \(-0.454162\pi\)
0.143508 + 0.989649i \(0.454162\pi\)
\(770\) 40.2014 1.44876
\(771\) 0 0
\(772\) 4.65744 0.167625
\(773\) −40.5378 −1.45804 −0.729021 0.684491i \(-0.760024\pi\)
−0.729021 + 0.684491i \(0.760024\pi\)
\(774\) 0 0
\(775\) 9.40035 0.337670
\(776\) 0.771539 0.0276966
\(777\) 0 0
\(778\) −55.2342 −1.98024
\(779\) −7.65057 −0.274110
\(780\) 0 0
\(781\) −27.0722 −0.968721
\(782\) −46.2808 −1.65500
\(783\) 0 0
\(784\) −74.5395 −2.66213
\(785\) 12.0575 0.430350
\(786\) 0 0
\(787\) −39.0735 −1.39282 −0.696409 0.717645i \(-0.745220\pi\)
−0.696409 + 0.717645i \(0.745220\pi\)
\(788\) −35.1084 −1.25068
\(789\) 0 0
\(790\) −14.5021 −0.515963
\(791\) 39.0224 1.38748
\(792\) 0 0
\(793\) 31.1171 1.10500
\(794\) 46.8698 1.66335
\(795\) 0 0
\(796\) −9.73194 −0.344939
\(797\) 1.92101 0.0680456 0.0340228 0.999421i \(-0.489168\pi\)
0.0340228 + 0.999421i \(0.489168\pi\)
\(798\) 0 0
\(799\) 49.6819 1.75762
\(800\) −6.63791 −0.234685
\(801\) 0 0
\(802\) 35.7759 1.26329
\(803\) 51.0372 1.80106
\(804\) 0 0
\(805\) −20.9967 −0.740035
\(806\) −77.3884 −2.72589
\(807\) 0 0
\(808\) 15.3842 0.541216
\(809\) 35.3476 1.24276 0.621378 0.783511i \(-0.286573\pi\)
0.621378 + 0.783511i \(0.286573\pi\)
\(810\) 0 0
\(811\) −42.5750 −1.49501 −0.747506 0.664255i \(-0.768749\pi\)
−0.747506 + 0.664255i \(0.768749\pi\)
\(812\) −27.8129 −0.976040
\(813\) 0 0
\(814\) 58.6050 2.05410
\(815\) −6.76614 −0.237007
\(816\) 0 0
\(817\) −6.77304 −0.236959
\(818\) 42.6549 1.49139
\(819\) 0 0
\(820\) 7.84471 0.273949
\(821\) 15.8671 0.553766 0.276883 0.960904i \(-0.410699\pi\)
0.276883 + 0.960904i \(0.410699\pi\)
\(822\) 0 0
\(823\) −13.2314 −0.461217 −0.230609 0.973047i \(-0.574072\pi\)
−0.230609 + 0.973047i \(0.574072\pi\)
\(824\) 13.0513 0.454662
\(825\) 0 0
\(826\) −65.3399 −2.27346
\(827\) 26.0034 0.904228 0.452114 0.891960i \(-0.350670\pi\)
0.452114 + 0.891960i \(0.350670\pi\)
\(828\) 0 0
\(829\) −4.11404 −0.142886 −0.0714432 0.997445i \(-0.522760\pi\)
−0.0714432 + 0.997445i \(0.522760\pi\)
\(830\) −28.2786 −0.981565
\(831\) 0 0
\(832\) 11.1233 0.385631
\(833\) 86.9983 3.01431
\(834\) 0 0
\(835\) −5.08722 −0.176050
\(836\) 8.55983 0.296048
\(837\) 0 0
\(838\) 40.9985 1.41627
\(839\) −34.8946 −1.20469 −0.602347 0.798234i \(-0.705768\pi\)
−0.602347 + 0.798234i \(0.705768\pi\)
\(840\) 0 0
\(841\) −10.7461 −0.370555
\(842\) −53.6675 −1.84951
\(843\) 0 0
\(844\) −19.6011 −0.674698
\(845\) −7.06839 −0.243160
\(846\) 0 0
\(847\) −49.2420 −1.69198
\(848\) −40.0054 −1.37379
\(849\) 0 0
\(850\) 10.4192 0.357376
\(851\) −30.6086 −1.04925
\(852\) 0 0
\(853\) 2.21046 0.0756847 0.0378423 0.999284i \(-0.487952\pi\)
0.0378423 + 0.999284i \(0.487952\pi\)
\(854\) 60.3397 2.06478
\(855\) 0 0
\(856\) −9.75115 −0.333287
\(857\) −0.0315501 −0.00107773 −0.000538865 1.00000i \(-0.500172\pi\)
−0.000538865 1.00000i \(0.500172\pi\)
\(858\) 0 0
\(859\) 47.3550 1.61573 0.807867 0.589365i \(-0.200622\pi\)
0.807867 + 0.589365i \(0.200622\pi\)
\(860\) 6.94491 0.236820
\(861\) 0 0
\(862\) −27.2836 −0.929283
\(863\) −2.08780 −0.0710696 −0.0355348 0.999368i \(-0.511313\pi\)
−0.0355348 + 0.999368i \(0.511313\pi\)
\(864\) 0 0
\(865\) 1.63396 0.0555564
\(866\) −23.9157 −0.812689
\(867\) 0 0
\(868\) −61.1944 −2.07707
\(869\) 36.5206 1.23888
\(870\) 0 0
\(871\) 28.6125 0.969496
\(872\) −14.5570 −0.492964
\(873\) 0 0
\(874\) −10.9633 −0.370839
\(875\) 4.72699 0.159801
\(876\) 0 0
\(877\) −49.8979 −1.68493 −0.842467 0.538748i \(-0.818898\pi\)
−0.842467 + 0.538748i \(0.818898\pi\)
\(878\) −22.3944 −0.755773
\(879\) 0 0
\(880\) 22.4811 0.757836
\(881\) −35.1975 −1.18583 −0.592917 0.805264i \(-0.702024\pi\)
−0.592917 + 0.805264i \(0.702024\pi\)
\(882\) 0 0
\(883\) −47.7239 −1.60604 −0.803019 0.595953i \(-0.796774\pi\)
−0.803019 + 0.595953i \(0.796774\pi\)
\(884\) −34.9783 −1.17645
\(885\) 0 0
\(886\) 3.35528 0.112723
\(887\) 20.4982 0.688263 0.344132 0.938921i \(-0.388173\pi\)
0.344132 + 0.938921i \(0.388173\pi\)
\(888\) 0 0
\(889\) −8.37138 −0.280767
\(890\) 1.83770 0.0616000
\(891\) 0 0
\(892\) −4.28581 −0.143499
\(893\) 11.7690 0.393834
\(894\) 0 0
\(895\) −8.89727 −0.297403
\(896\) −41.1853 −1.37590
\(897\) 0 0
\(898\) 20.1496 0.672401
\(899\) 40.1626 1.33950
\(900\) 0 0
\(901\) 46.6921 1.55554
\(902\) −48.4452 −1.61305
\(903\) 0 0
\(904\) 9.44895 0.314267
\(905\) 7.14852 0.237625
\(906\) 0 0
\(907\) 16.1289 0.535551 0.267775 0.963481i \(-0.413712\pi\)
0.267775 + 0.963481i \(0.413712\pi\)
\(908\) 4.91862 0.163230
\(909\) 0 0
\(910\) −38.9150 −1.29002
\(911\) 5.86877 0.194441 0.0972204 0.995263i \(-0.469005\pi\)
0.0972204 + 0.995263i \(0.469005\pi\)
\(912\) 0 0
\(913\) 71.2138 2.35683
\(914\) 7.61759 0.251968
\(915\) 0 0
\(916\) 8.25384 0.272715
\(917\) −34.4485 −1.13759
\(918\) 0 0
\(919\) −52.0408 −1.71667 −0.858333 0.513093i \(-0.828500\pi\)
−0.858333 + 0.513093i \(0.828500\pi\)
\(920\) −5.08417 −0.167620
\(921\) 0 0
\(922\) 65.5014 2.15717
\(923\) 26.2059 0.862578
\(924\) 0 0
\(925\) 6.89093 0.226572
\(926\) −5.67784 −0.186585
\(927\) 0 0
\(928\) −28.3602 −0.930969
\(929\) 9.81663 0.322073 0.161037 0.986948i \(-0.448516\pi\)
0.161037 + 0.986948i \(0.448516\pi\)
\(930\) 0 0
\(931\) 20.6087 0.675424
\(932\) 6.54468 0.214378
\(933\) 0 0
\(934\) 9.42886 0.308522
\(935\) −26.2386 −0.858095
\(936\) 0 0
\(937\) 37.5704 1.22737 0.613686 0.789550i \(-0.289686\pi\)
0.613686 + 0.789550i \(0.289686\pi\)
\(938\) 55.4829 1.81158
\(939\) 0 0
\(940\) −12.0676 −0.393603
\(941\) 0.0461835 0.00150554 0.000752769 1.00000i \(-0.499760\pi\)
0.000752769 1.00000i \(0.499760\pi\)
\(942\) 0 0
\(943\) 25.3023 0.823955
\(944\) −36.5387 −1.18923
\(945\) 0 0
\(946\) −42.8885 −1.39442
\(947\) 26.0877 0.847735 0.423867 0.905724i \(-0.360672\pi\)
0.423867 + 0.905724i \(0.360672\pi\)
\(948\) 0 0
\(949\) −49.4040 −1.60372
\(950\) 2.46817 0.0800782
\(951\) 0 0
\(952\) 30.6760 0.994216
\(953\) −35.0541 −1.13551 −0.567757 0.823196i \(-0.692189\pi\)
−0.567757 + 0.823196i \(0.692189\pi\)
\(954\) 0 0
\(955\) −8.74055 −0.282838
\(956\) 21.9217 0.708997
\(957\) 0 0
\(958\) −12.4329 −0.401688
\(959\) 3.56852 0.115234
\(960\) 0 0
\(961\) 57.3665 1.85053
\(962\) −56.7296 −1.82904
\(963\) 0 0
\(964\) 17.0389 0.548788
\(965\) −3.38192 −0.108868
\(966\) 0 0
\(967\) −14.8997 −0.479141 −0.239570 0.970879i \(-0.577007\pi\)
−0.239570 + 0.970879i \(0.577007\pi\)
\(968\) −11.9236 −0.383238
\(969\) 0 0
\(970\) 1.23874 0.0397734
\(971\) 9.37217 0.300767 0.150384 0.988628i \(-0.451949\pi\)
0.150384 + 0.988628i \(0.451949\pi\)
\(972\) 0 0
\(973\) 29.7759 0.954572
\(974\) −1.37396 −0.0440244
\(975\) 0 0
\(976\) 33.7426 1.08007
\(977\) 10.5242 0.336698 0.168349 0.985727i \(-0.446156\pi\)
0.168349 + 0.985727i \(0.446156\pi\)
\(978\) 0 0
\(979\) −4.62787 −0.147907
\(980\) −21.1317 −0.675027
\(981\) 0 0
\(982\) −40.6335 −1.29667
\(983\) 1.32593 0.0422905 0.0211453 0.999776i \(-0.493269\pi\)
0.0211453 + 0.999776i \(0.493269\pi\)
\(984\) 0 0
\(985\) 25.4934 0.812287
\(986\) 44.5157 1.41767
\(987\) 0 0
\(988\) −8.28590 −0.263610
\(989\) 22.4001 0.712280
\(990\) 0 0
\(991\) −3.04920 −0.0968609 −0.0484305 0.998827i \(-0.515422\pi\)
−0.0484305 + 0.998827i \(0.515422\pi\)
\(992\) −62.3986 −1.98116
\(993\) 0 0
\(994\) 50.8163 1.61180
\(995\) 7.06669 0.224029
\(996\) 0 0
\(997\) −25.7222 −0.814630 −0.407315 0.913288i \(-0.633535\pi\)
−0.407315 + 0.913288i \(0.633535\pi\)
\(998\) −64.5815 −2.04429
\(999\) 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 4005.2.a.o.1.5 7
3.2 odd 2 445.2.a.f.1.3 7
12.11 even 2 7120.2.a.bj.1.1 7
15.14 odd 2 2225.2.a.k.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.3 7 3.2 odd 2
2225.2.a.k.1.5 7 15.14 odd 2
4005.2.a.o.1.5 7 1.1 even 1 trivial
7120.2.a.bj.1.1 7 12.11 even 2