Properties

Label 4005.2.a.o.1.2
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 1.2
Root \(-1.96388\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +O(q^{10})\) \(q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +0.856822 q^{10} +4.74359 q^{11} +6.25246 q^{13} +0.497280 q^{14} +0.134107 q^{16} +5.87935 q^{17} -5.41445 q^{19} +1.26586 q^{20} -4.06441 q^{22} +8.56239 q^{23} +1.00000 q^{25} -5.35724 q^{26} +0.734675 q^{28} +3.96542 q^{29} +5.68893 q^{31} -5.71142 q^{32} -5.03755 q^{34} +0.580377 q^{35} +0.495300 q^{37} +4.63922 q^{38} -2.79826 q^{40} -11.8503 q^{41} -4.78767 q^{43} -6.00471 q^{44} -7.33644 q^{46} +3.78707 q^{47} -6.66316 q^{49} -0.856822 q^{50} -7.91471 q^{52} +7.91925 q^{53} -4.74359 q^{55} -1.62404 q^{56} -3.39766 q^{58} -10.6416 q^{59} +9.77193 q^{61} -4.87439 q^{62} +4.62545 q^{64} -6.25246 q^{65} +1.35061 q^{67} -7.44242 q^{68} -0.497280 q^{70} +10.7514 q^{71} +1.31208 q^{73} -0.424384 q^{74} +6.85392 q^{76} -2.75307 q^{77} +0.492431 q^{79} -0.134107 q^{80} +10.1536 q^{82} -1.62413 q^{83} -5.87935 q^{85} +4.10218 q^{86} +13.2738 q^{88} -1.00000 q^{89} -3.62878 q^{91} -10.8388 q^{92} -3.24485 q^{94} +5.41445 q^{95} -9.61533 q^{97} +5.70914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.856822 −0.605864 −0.302932 0.953012i \(-0.597966\pi\)
−0.302932 + 0.953012i \(0.597966\pi\)
\(3\) 0 0
\(4\) −1.26586 −0.632928
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.580377 −0.219362 −0.109681 0.993967i \(-0.534983\pi\)
−0.109681 + 0.993967i \(0.534983\pi\)
\(8\) 2.79826 0.989333
\(9\) 0 0
\(10\) 0.856822 0.270951
\(11\) 4.74359 1.43025 0.715123 0.698998i \(-0.246371\pi\)
0.715123 + 0.698998i \(0.246371\pi\)
\(12\) 0 0
\(13\) 6.25246 1.73412 0.867060 0.498204i \(-0.166007\pi\)
0.867060 + 0.498204i \(0.166007\pi\)
\(14\) 0.497280 0.132904
\(15\) 0 0
\(16\) 0.134107 0.0335267
\(17\) 5.87935 1.42595 0.712976 0.701189i \(-0.247347\pi\)
0.712976 + 0.701189i \(0.247347\pi\)
\(18\) 0 0
\(19\) −5.41445 −1.24216 −0.621080 0.783747i \(-0.713306\pi\)
−0.621080 + 0.783747i \(0.713306\pi\)
\(20\) 1.26586 0.283054
\(21\) 0 0
\(22\) −4.06441 −0.866535
\(23\) 8.56239 1.78538 0.892691 0.450669i \(-0.148815\pi\)
0.892691 + 0.450669i \(0.148815\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.35724 −1.05064
\(27\) 0 0
\(28\) 0.734675 0.138840
\(29\) 3.96542 0.736361 0.368180 0.929754i \(-0.379981\pi\)
0.368180 + 0.929754i \(0.379981\pi\)
\(30\) 0 0
\(31\) 5.68893 1.02176 0.510881 0.859652i \(-0.329319\pi\)
0.510881 + 0.859652i \(0.329319\pi\)
\(32\) −5.71142 −1.00965
\(33\) 0 0
\(34\) −5.03755 −0.863933
\(35\) 0.580377 0.0981017
\(36\) 0 0
\(37\) 0.495300 0.0814269 0.0407134 0.999171i \(-0.487037\pi\)
0.0407134 + 0.999171i \(0.487037\pi\)
\(38\) 4.63922 0.752581
\(39\) 0 0
\(40\) −2.79826 −0.442443
\(41\) −11.8503 −1.85071 −0.925356 0.379098i \(-0.876234\pi\)
−0.925356 + 0.379098i \(0.876234\pi\)
\(42\) 0 0
\(43\) −4.78767 −0.730113 −0.365056 0.930985i \(-0.618950\pi\)
−0.365056 + 0.930985i \(0.618950\pi\)
\(44\) −6.00471 −0.905244
\(45\) 0 0
\(46\) −7.33644 −1.08170
\(47\) 3.78707 0.552401 0.276201 0.961100i \(-0.410925\pi\)
0.276201 + 0.961100i \(0.410925\pi\)
\(48\) 0 0
\(49\) −6.66316 −0.951880
\(50\) −0.856822 −0.121173
\(51\) 0 0
\(52\) −7.91471 −1.09757
\(53\) 7.91925 1.08779 0.543897 0.839152i \(-0.316948\pi\)
0.543897 + 0.839152i \(0.316948\pi\)
\(54\) 0 0
\(55\) −4.74359 −0.639626
\(56\) −1.62404 −0.217022
\(57\) 0 0
\(58\) −3.39766 −0.446135
\(59\) −10.6416 −1.38542 −0.692711 0.721216i \(-0.743584\pi\)
−0.692711 + 0.721216i \(0.743584\pi\)
\(60\) 0 0
\(61\) 9.77193 1.25117 0.625584 0.780157i \(-0.284861\pi\)
0.625584 + 0.780157i \(0.284861\pi\)
\(62\) −4.87439 −0.619049
\(63\) 0 0
\(64\) 4.62545 0.578182
\(65\) −6.25246 −0.775522
\(66\) 0 0
\(67\) 1.35061 0.165004 0.0825019 0.996591i \(-0.473709\pi\)
0.0825019 + 0.996591i \(0.473709\pi\)
\(68\) −7.44242 −0.902525
\(69\) 0 0
\(70\) −0.497280 −0.0594363
\(71\) 10.7514 1.27595 0.637976 0.770056i \(-0.279772\pi\)
0.637976 + 0.770056i \(0.279772\pi\)
\(72\) 0 0
\(73\) 1.31208 0.153567 0.0767834 0.997048i \(-0.475535\pi\)
0.0767834 + 0.997048i \(0.475535\pi\)
\(74\) −0.424384 −0.0493336
\(75\) 0 0
\(76\) 6.85392 0.786198
\(77\) −2.75307 −0.313742
\(78\) 0 0
\(79\) 0.492431 0.0554028 0.0277014 0.999616i \(-0.491181\pi\)
0.0277014 + 0.999616i \(0.491181\pi\)
\(80\) −0.134107 −0.0149936
\(81\) 0 0
\(82\) 10.1536 1.12128
\(83\) −1.62413 −0.178272 −0.0891359 0.996019i \(-0.528411\pi\)
−0.0891359 + 0.996019i \(0.528411\pi\)
\(84\) 0 0
\(85\) −5.87935 −0.637705
\(86\) 4.10218 0.442349
\(87\) 0 0
\(88\) 13.2738 1.41499
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −3.62878 −0.380400
\(92\) −10.8388 −1.13002
\(93\) 0 0
\(94\) −3.24485 −0.334680
\(95\) 5.41445 0.555511
\(96\) 0 0
\(97\) −9.61533 −0.976289 −0.488144 0.872763i \(-0.662326\pi\)
−0.488144 + 0.872763i \(0.662326\pi\)
\(98\) 5.70914 0.576710
\(99\) 0 0
\(100\) −1.26586 −0.126586
\(101\) −4.72415 −0.470070 −0.235035 0.971987i \(-0.575521\pi\)
−0.235035 + 0.971987i \(0.575521\pi\)
\(102\) 0 0
\(103\) 5.17345 0.509755 0.254877 0.966973i \(-0.417965\pi\)
0.254877 + 0.966973i \(0.417965\pi\)
\(104\) 17.4960 1.71562
\(105\) 0 0
\(106\) −6.78539 −0.659055
\(107\) −5.68158 −0.549259 −0.274630 0.961550i \(-0.588555\pi\)
−0.274630 + 0.961550i \(0.588555\pi\)
\(108\) 0 0
\(109\) −6.48821 −0.621458 −0.310729 0.950499i \(-0.600573\pi\)
−0.310729 + 0.950499i \(0.600573\pi\)
\(110\) 4.06441 0.387526
\(111\) 0 0
\(112\) −0.0778326 −0.00735449
\(113\) −9.49005 −0.892749 −0.446375 0.894846i \(-0.647285\pi\)
−0.446375 + 0.894846i \(0.647285\pi\)
\(114\) 0 0
\(115\) −8.56239 −0.798447
\(116\) −5.01966 −0.466064
\(117\) 0 0
\(118\) 9.11797 0.839377
\(119\) −3.41224 −0.312800
\(120\) 0 0
\(121\) 11.5017 1.04561
\(122\) −8.37280 −0.758038
\(123\) 0 0
\(124\) −7.20137 −0.646702
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.22863 −0.730173 −0.365087 0.930974i \(-0.618961\pi\)
−0.365087 + 0.930974i \(0.618961\pi\)
\(128\) 7.45965 0.659346
\(129\) 0 0
\(130\) 5.35724 0.469861
\(131\) 4.92961 0.430702 0.215351 0.976537i \(-0.430910\pi\)
0.215351 + 0.976537i \(0.430910\pi\)
\(132\) 0 0
\(133\) 3.14242 0.272483
\(134\) −1.15723 −0.0999699
\(135\) 0 0
\(136\) 16.4519 1.41074
\(137\) 11.9905 1.02442 0.512211 0.858860i \(-0.328827\pi\)
0.512211 + 0.858860i \(0.328827\pi\)
\(138\) 0 0
\(139\) 18.0960 1.53488 0.767440 0.641120i \(-0.221530\pi\)
0.767440 + 0.641120i \(0.221530\pi\)
\(140\) −0.734675 −0.0620913
\(141\) 0 0
\(142\) −9.21200 −0.773054
\(143\) 29.6591 2.48022
\(144\) 0 0
\(145\) −3.96542 −0.329311
\(146\) −1.12421 −0.0930406
\(147\) 0 0
\(148\) −0.626979 −0.0515374
\(149\) 19.9604 1.63522 0.817609 0.575773i \(-0.195299\pi\)
0.817609 + 0.575773i \(0.195299\pi\)
\(150\) 0 0
\(151\) −9.01450 −0.733589 −0.366795 0.930302i \(-0.619545\pi\)
−0.366795 + 0.930302i \(0.619545\pi\)
\(152\) −15.1510 −1.22891
\(153\) 0 0
\(154\) 2.35889 0.190085
\(155\) −5.68893 −0.456946
\(156\) 0 0
\(157\) −9.23610 −0.737121 −0.368561 0.929604i \(-0.620149\pi\)
−0.368561 + 0.929604i \(0.620149\pi\)
\(158\) −0.421925 −0.0335666
\(159\) 0 0
\(160\) 5.71142 0.451527
\(161\) −4.96942 −0.391645
\(162\) 0 0
\(163\) −19.2838 −1.51043 −0.755213 0.655479i \(-0.772467\pi\)
−0.755213 + 0.655479i \(0.772467\pi\)
\(164\) 15.0008 1.17137
\(165\) 0 0
\(166\) 1.39159 0.108009
\(167\) 24.8161 1.92033 0.960163 0.279439i \(-0.0901485\pi\)
0.960163 + 0.279439i \(0.0901485\pi\)
\(168\) 0 0
\(169\) 26.0932 2.00717
\(170\) 5.03755 0.386363
\(171\) 0 0
\(172\) 6.06050 0.462109
\(173\) 17.0185 1.29389 0.646945 0.762537i \(-0.276046\pi\)
0.646945 + 0.762537i \(0.276046\pi\)
\(174\) 0 0
\(175\) −0.580377 −0.0438724
\(176\) 0.636148 0.0479515
\(177\) 0 0
\(178\) 0.856822 0.0642215
\(179\) 6.10678 0.456442 0.228221 0.973609i \(-0.426709\pi\)
0.228221 + 0.973609i \(0.426709\pi\)
\(180\) 0 0
\(181\) −1.02757 −0.0763785 −0.0381892 0.999271i \(-0.512159\pi\)
−0.0381892 + 0.999271i \(0.512159\pi\)
\(182\) 3.10922 0.230471
\(183\) 0 0
\(184\) 23.9598 1.76634
\(185\) −0.495300 −0.0364152
\(186\) 0 0
\(187\) 27.8892 2.03946
\(188\) −4.79389 −0.349630
\(189\) 0 0
\(190\) −4.63922 −0.336564
\(191\) 11.3023 0.817808 0.408904 0.912577i \(-0.365911\pi\)
0.408904 + 0.912577i \(0.365911\pi\)
\(192\) 0 0
\(193\) −19.0759 −1.37311 −0.686555 0.727077i \(-0.740878\pi\)
−0.686555 + 0.727077i \(0.740878\pi\)
\(194\) 8.23862 0.591499
\(195\) 0 0
\(196\) 8.43461 0.602472
\(197\) −20.1996 −1.43916 −0.719580 0.694409i \(-0.755666\pi\)
−0.719580 + 0.694409i \(0.755666\pi\)
\(198\) 0 0
\(199\) −11.2701 −0.798915 −0.399457 0.916752i \(-0.630801\pi\)
−0.399457 + 0.916752i \(0.630801\pi\)
\(200\) 2.79826 0.197867
\(201\) 0 0
\(202\) 4.04775 0.284799
\(203\) −2.30144 −0.161530
\(204\) 0 0
\(205\) 11.8503 0.827664
\(206\) −4.43272 −0.308842
\(207\) 0 0
\(208\) 0.838497 0.0581393
\(209\) −25.6839 −1.77660
\(210\) 0 0
\(211\) 6.09599 0.419665 0.209833 0.977737i \(-0.432708\pi\)
0.209833 + 0.977737i \(0.432708\pi\)
\(212\) −10.0246 −0.688495
\(213\) 0 0
\(214\) 4.86810 0.332777
\(215\) 4.78767 0.326516
\(216\) 0 0
\(217\) −3.30172 −0.224136
\(218\) 5.55924 0.376519
\(219\) 0 0
\(220\) 6.00471 0.404837
\(221\) 36.7604 2.47277
\(222\) 0 0
\(223\) 7.65809 0.512824 0.256412 0.966568i \(-0.417460\pi\)
0.256412 + 0.966568i \(0.417460\pi\)
\(224\) 3.31478 0.221478
\(225\) 0 0
\(226\) 8.13128 0.540885
\(227\) 29.5274 1.95980 0.979900 0.199490i \(-0.0639285\pi\)
0.979900 + 0.199490i \(0.0639285\pi\)
\(228\) 0 0
\(229\) 7.14724 0.472303 0.236151 0.971716i \(-0.424114\pi\)
0.236151 + 0.971716i \(0.424114\pi\)
\(230\) 7.33644 0.483751
\(231\) 0 0
\(232\) 11.0963 0.728506
\(233\) 10.9605 0.718045 0.359023 0.933329i \(-0.383110\pi\)
0.359023 + 0.933329i \(0.383110\pi\)
\(234\) 0 0
\(235\) −3.78707 −0.247041
\(236\) 13.4708 0.876873
\(237\) 0 0
\(238\) 2.92368 0.189514
\(239\) −19.7441 −1.27714 −0.638571 0.769563i \(-0.720474\pi\)
−0.638571 + 0.769563i \(0.720474\pi\)
\(240\) 0 0
\(241\) −19.2285 −1.23862 −0.619309 0.785148i \(-0.712587\pi\)
−0.619309 + 0.785148i \(0.712587\pi\)
\(242\) −9.85487 −0.633495
\(243\) 0 0
\(244\) −12.3699 −0.791899
\(245\) 6.66316 0.425694
\(246\) 0 0
\(247\) −33.8536 −2.15405
\(248\) 15.9191 1.01086
\(249\) 0 0
\(250\) 0.856822 0.0541902
\(251\) 2.44743 0.154481 0.0772403 0.997013i \(-0.475389\pi\)
0.0772403 + 0.997013i \(0.475389\pi\)
\(252\) 0 0
\(253\) 40.6165 2.55354
\(254\) 7.05047 0.442386
\(255\) 0 0
\(256\) −15.6425 −0.977656
\(257\) −20.5627 −1.28267 −0.641333 0.767263i \(-0.721618\pi\)
−0.641333 + 0.767263i \(0.721618\pi\)
\(258\) 0 0
\(259\) −0.287461 −0.0178620
\(260\) 7.91471 0.490850
\(261\) 0 0
\(262\) −4.22379 −0.260947
\(263\) −10.4549 −0.644679 −0.322340 0.946624i \(-0.604469\pi\)
−0.322340 + 0.946624i \(0.604469\pi\)
\(264\) 0 0
\(265\) −7.91925 −0.486476
\(266\) −2.69250 −0.165088
\(267\) 0 0
\(268\) −1.70968 −0.104436
\(269\) −26.2193 −1.59862 −0.799308 0.600921i \(-0.794801\pi\)
−0.799308 + 0.600921i \(0.794801\pi\)
\(270\) 0 0
\(271\) 20.6659 1.25536 0.627681 0.778470i \(-0.284004\pi\)
0.627681 + 0.778470i \(0.284004\pi\)
\(272\) 0.788461 0.0478075
\(273\) 0 0
\(274\) −10.2738 −0.620660
\(275\) 4.74359 0.286049
\(276\) 0 0
\(277\) 0.433915 0.0260714 0.0130357 0.999915i \(-0.495850\pi\)
0.0130357 + 0.999915i \(0.495850\pi\)
\(278\) −15.5050 −0.929930
\(279\) 0 0
\(280\) 1.62404 0.0970552
\(281\) −16.7226 −0.997585 −0.498793 0.866721i \(-0.666223\pi\)
−0.498793 + 0.866721i \(0.666223\pi\)
\(282\) 0 0
\(283\) −10.4735 −0.622587 −0.311293 0.950314i \(-0.600762\pi\)
−0.311293 + 0.950314i \(0.600762\pi\)
\(284\) −13.6097 −0.807586
\(285\) 0 0
\(286\) −25.4126 −1.50268
\(287\) 6.87767 0.405976
\(288\) 0 0
\(289\) 17.5668 1.03334
\(290\) 3.39766 0.199518
\(291\) 0 0
\(292\) −1.66090 −0.0971968
\(293\) −7.94600 −0.464210 −0.232105 0.972691i \(-0.574561\pi\)
−0.232105 + 0.972691i \(0.574561\pi\)
\(294\) 0 0
\(295\) 10.6416 0.619579
\(296\) 1.38598 0.0805583
\(297\) 0 0
\(298\) −17.1025 −0.990721
\(299\) 53.5360 3.09607
\(300\) 0 0
\(301\) 2.77865 0.160159
\(302\) 7.72382 0.444456
\(303\) 0 0
\(304\) −0.726115 −0.0416456
\(305\) −9.77193 −0.559539
\(306\) 0 0
\(307\) −2.11312 −0.120602 −0.0603011 0.998180i \(-0.519206\pi\)
−0.0603011 + 0.998180i \(0.519206\pi\)
\(308\) 3.48500 0.198576
\(309\) 0 0
\(310\) 4.87439 0.276847
\(311\) 8.66507 0.491351 0.245675 0.969352i \(-0.420990\pi\)
0.245675 + 0.969352i \(0.420990\pi\)
\(312\) 0 0
\(313\) 16.0660 0.908105 0.454052 0.890975i \(-0.349978\pi\)
0.454052 + 0.890975i \(0.349978\pi\)
\(314\) 7.91369 0.446595
\(315\) 0 0
\(316\) −0.623347 −0.0350660
\(317\) 1.70619 0.0958291 0.0479146 0.998851i \(-0.484742\pi\)
0.0479146 + 0.998851i \(0.484742\pi\)
\(318\) 0 0
\(319\) 18.8104 1.05318
\(320\) −4.62545 −0.258571
\(321\) 0 0
\(322\) 4.25790 0.237284
\(323\) −31.8335 −1.77126
\(324\) 0 0
\(325\) 6.25246 0.346824
\(326\) 16.5228 0.915113
\(327\) 0 0
\(328\) −33.1603 −1.83097
\(329\) −2.19793 −0.121176
\(330\) 0 0
\(331\) −7.90149 −0.434305 −0.217153 0.976138i \(-0.569677\pi\)
−0.217153 + 0.976138i \(0.569677\pi\)
\(332\) 2.05592 0.112833
\(333\) 0 0
\(334\) −21.2630 −1.16346
\(335\) −1.35061 −0.0737919
\(336\) 0 0
\(337\) 7.99703 0.435626 0.217813 0.975991i \(-0.430108\pi\)
0.217813 + 0.975991i \(0.430108\pi\)
\(338\) −22.3572 −1.21607
\(339\) 0 0
\(340\) 7.44242 0.403622
\(341\) 26.9859 1.46137
\(342\) 0 0
\(343\) 7.92979 0.428168
\(344\) −13.3971 −0.722325
\(345\) 0 0
\(346\) −14.5818 −0.783922
\(347\) −6.24790 −0.335405 −0.167702 0.985838i \(-0.553635\pi\)
−0.167702 + 0.985838i \(0.553635\pi\)
\(348\) 0 0
\(349\) 7.07360 0.378641 0.189320 0.981915i \(-0.439372\pi\)
0.189320 + 0.981915i \(0.439372\pi\)
\(350\) 0.497280 0.0265807
\(351\) 0 0
\(352\) −27.0926 −1.44404
\(353\) −17.2887 −0.920184 −0.460092 0.887871i \(-0.652184\pi\)
−0.460092 + 0.887871i \(0.652184\pi\)
\(354\) 0 0
\(355\) −10.7514 −0.570623
\(356\) 1.26586 0.0670903
\(357\) 0 0
\(358\) −5.23242 −0.276542
\(359\) −12.3384 −0.651193 −0.325597 0.945509i \(-0.605565\pi\)
−0.325597 + 0.945509i \(0.605565\pi\)
\(360\) 0 0
\(361\) 10.3163 0.542962
\(362\) 0.880442 0.0462750
\(363\) 0 0
\(364\) 4.59352 0.240766
\(365\) −1.31208 −0.0686771
\(366\) 0 0
\(367\) 30.6605 1.60047 0.800233 0.599689i \(-0.204709\pi\)
0.800233 + 0.599689i \(0.204709\pi\)
\(368\) 1.14828 0.0598580
\(369\) 0 0
\(370\) 0.424384 0.0220627
\(371\) −4.59616 −0.238621
\(372\) 0 0
\(373\) −5.45234 −0.282312 −0.141156 0.989987i \(-0.545082\pi\)
−0.141156 + 0.989987i \(0.545082\pi\)
\(374\) −23.8961 −1.23564
\(375\) 0 0
\(376\) 10.5972 0.546509
\(377\) 24.7936 1.27694
\(378\) 0 0
\(379\) −35.5232 −1.82471 −0.912353 0.409405i \(-0.865736\pi\)
−0.912353 + 0.409405i \(0.865736\pi\)
\(380\) −6.85392 −0.351599
\(381\) 0 0
\(382\) −9.68408 −0.495481
\(383\) 12.8946 0.658882 0.329441 0.944176i \(-0.393140\pi\)
0.329441 + 0.944176i \(0.393140\pi\)
\(384\) 0 0
\(385\) 2.75307 0.140310
\(386\) 16.3446 0.831919
\(387\) 0 0
\(388\) 12.1716 0.617921
\(389\) 28.2948 1.43460 0.717300 0.696764i \(-0.245378\pi\)
0.717300 + 0.696764i \(0.245378\pi\)
\(390\) 0 0
\(391\) 50.3413 2.54587
\(392\) −18.6452 −0.941727
\(393\) 0 0
\(394\) 17.3074 0.871936
\(395\) −0.492431 −0.0247769
\(396\) 0 0
\(397\) −32.7862 −1.64549 −0.822745 0.568411i \(-0.807558\pi\)
−0.822745 + 0.568411i \(0.807558\pi\)
\(398\) 9.65645 0.484034
\(399\) 0 0
\(400\) 0.134107 0.00670534
\(401\) −19.8756 −0.992538 −0.496269 0.868169i \(-0.665297\pi\)
−0.496269 + 0.868169i \(0.665297\pi\)
\(402\) 0 0
\(403\) 35.5698 1.77186
\(404\) 5.98009 0.297521
\(405\) 0 0
\(406\) 1.97193 0.0978650
\(407\) 2.34950 0.116461
\(408\) 0 0
\(409\) 24.6891 1.22080 0.610398 0.792095i \(-0.291009\pi\)
0.610398 + 0.792095i \(0.291009\pi\)
\(410\) −10.1536 −0.501452
\(411\) 0 0
\(412\) −6.54884 −0.322638
\(413\) 6.17616 0.303909
\(414\) 0 0
\(415\) 1.62413 0.0797256
\(416\) −35.7104 −1.75085
\(417\) 0 0
\(418\) 22.0066 1.07638
\(419\) 23.6692 1.15632 0.578159 0.815924i \(-0.303771\pi\)
0.578159 + 0.815924i \(0.303771\pi\)
\(420\) 0 0
\(421\) 15.9257 0.776169 0.388085 0.921624i \(-0.373137\pi\)
0.388085 + 0.921624i \(0.373137\pi\)
\(422\) −5.22317 −0.254260
\(423\) 0 0
\(424\) 22.1601 1.07619
\(425\) 5.87935 0.285190
\(426\) 0 0
\(427\) −5.67141 −0.274459
\(428\) 7.19207 0.347642
\(429\) 0 0
\(430\) −4.10218 −0.197825
\(431\) −7.78225 −0.374858 −0.187429 0.982278i \(-0.560015\pi\)
−0.187429 + 0.982278i \(0.560015\pi\)
\(432\) 0 0
\(433\) −3.59930 −0.172971 −0.0864856 0.996253i \(-0.527564\pi\)
−0.0864856 + 0.996253i \(0.527564\pi\)
\(434\) 2.82899 0.135796
\(435\) 0 0
\(436\) 8.21314 0.393338
\(437\) −46.3606 −2.21773
\(438\) 0 0
\(439\) 30.2840 1.44538 0.722689 0.691173i \(-0.242906\pi\)
0.722689 + 0.691173i \(0.242906\pi\)
\(440\) −13.2738 −0.632803
\(441\) 0 0
\(442\) −31.4971 −1.49816
\(443\) −15.1094 −0.717870 −0.358935 0.933362i \(-0.616860\pi\)
−0.358935 + 0.933362i \(0.616860\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −6.56162 −0.310702
\(447\) 0 0
\(448\) −2.68451 −0.126831
\(449\) −18.3101 −0.864106 −0.432053 0.901848i \(-0.642211\pi\)
−0.432053 + 0.901848i \(0.642211\pi\)
\(450\) 0 0
\(451\) −56.2132 −2.64698
\(452\) 12.0130 0.565046
\(453\) 0 0
\(454\) −25.2997 −1.18737
\(455\) 3.62878 0.170120
\(456\) 0 0
\(457\) −21.0509 −0.984720 −0.492360 0.870392i \(-0.663866\pi\)
−0.492360 + 0.870392i \(0.663866\pi\)
\(458\) −6.12391 −0.286151
\(459\) 0 0
\(460\) 10.8388 0.505360
\(461\) 38.8056 1.80735 0.903677 0.428214i \(-0.140857\pi\)
0.903677 + 0.428214i \(0.140857\pi\)
\(462\) 0 0
\(463\) −12.1126 −0.562921 −0.281460 0.959573i \(-0.590819\pi\)
−0.281460 + 0.959573i \(0.590819\pi\)
\(464\) 0.531791 0.0246878
\(465\) 0 0
\(466\) −9.39118 −0.435038
\(467\) −2.25635 −0.104411 −0.0522056 0.998636i \(-0.516625\pi\)
−0.0522056 + 0.998636i \(0.516625\pi\)
\(468\) 0 0
\(469\) −0.783866 −0.0361955
\(470\) 3.24485 0.149674
\(471\) 0 0
\(472\) −29.7780 −1.37064
\(473\) −22.7107 −1.04424
\(474\) 0 0
\(475\) −5.41445 −0.248432
\(476\) 4.31941 0.197980
\(477\) 0 0
\(478\) 16.9172 0.773775
\(479\) −0.237254 −0.0108404 −0.00542022 0.999985i \(-0.501725\pi\)
−0.00542022 + 0.999985i \(0.501725\pi\)
\(480\) 0 0
\(481\) 3.09684 0.141204
\(482\) 16.4754 0.750434
\(483\) 0 0
\(484\) −14.5595 −0.661793
\(485\) 9.61533 0.436610
\(486\) 0 0
\(487\) 1.69314 0.0767237 0.0383619 0.999264i \(-0.487786\pi\)
0.0383619 + 0.999264i \(0.487786\pi\)
\(488\) 27.3444 1.23782
\(489\) 0 0
\(490\) −5.70914 −0.257913
\(491\) 18.2064 0.821641 0.410821 0.911716i \(-0.365242\pi\)
0.410821 + 0.911716i \(0.365242\pi\)
\(492\) 0 0
\(493\) 23.3141 1.05002
\(494\) 29.0065 1.30506
\(495\) 0 0
\(496\) 0.762924 0.0342563
\(497\) −6.23985 −0.279896
\(498\) 0 0
\(499\) 20.0254 0.896462 0.448231 0.893918i \(-0.352054\pi\)
0.448231 + 0.893918i \(0.352054\pi\)
\(500\) 1.26586 0.0566108
\(501\) 0 0
\(502\) −2.09701 −0.0935943
\(503\) −33.4990 −1.49365 −0.746824 0.665021i \(-0.768422\pi\)
−0.746824 + 0.665021i \(0.768422\pi\)
\(504\) 0 0
\(505\) 4.72415 0.210222
\(506\) −34.8011 −1.54710
\(507\) 0 0
\(508\) 10.4163 0.462147
\(509\) 15.2860 0.677541 0.338770 0.940869i \(-0.389989\pi\)
0.338770 + 0.940869i \(0.389989\pi\)
\(510\) 0 0
\(511\) −0.761499 −0.0336867
\(512\) −1.51647 −0.0670192
\(513\) 0 0
\(514\) 17.6186 0.777122
\(515\) −5.17345 −0.227969
\(516\) 0 0
\(517\) 17.9643 0.790070
\(518\) 0.246303 0.0108219
\(519\) 0 0
\(520\) −17.4960 −0.767249
\(521\) 23.7850 1.04204 0.521021 0.853544i \(-0.325551\pi\)
0.521021 + 0.853544i \(0.325551\pi\)
\(522\) 0 0
\(523\) 26.9989 1.18058 0.590290 0.807192i \(-0.299014\pi\)
0.590290 + 0.807192i \(0.299014\pi\)
\(524\) −6.24018 −0.272603
\(525\) 0 0
\(526\) 8.95802 0.390588
\(527\) 33.4472 1.45698
\(528\) 0 0
\(529\) 50.3145 2.18759
\(530\) 6.78539 0.294738
\(531\) 0 0
\(532\) −3.97786 −0.172462
\(533\) −74.0938 −3.20936
\(534\) 0 0
\(535\) 5.68158 0.245636
\(536\) 3.77936 0.163244
\(537\) 0 0
\(538\) 22.4652 0.968545
\(539\) −31.6073 −1.36142
\(540\) 0 0
\(541\) −21.0584 −0.905370 −0.452685 0.891671i \(-0.649534\pi\)
−0.452685 + 0.891671i \(0.649534\pi\)
\(542\) −17.7070 −0.760580
\(543\) 0 0
\(544\) −33.5794 −1.43971
\(545\) 6.48821 0.277924
\(546\) 0 0
\(547\) 12.6410 0.540491 0.270245 0.962792i \(-0.412895\pi\)
0.270245 + 0.962792i \(0.412895\pi\)
\(548\) −15.1783 −0.648385
\(549\) 0 0
\(550\) −4.06441 −0.173307
\(551\) −21.4706 −0.914678
\(552\) 0 0
\(553\) −0.285796 −0.0121533
\(554\) −0.371788 −0.0157957
\(555\) 0 0
\(556\) −22.9069 −0.971470
\(557\) −7.81870 −0.331289 −0.165644 0.986186i \(-0.552970\pi\)
−0.165644 + 0.986186i \(0.552970\pi\)
\(558\) 0 0
\(559\) −29.9347 −1.26610
\(560\) 0.0778326 0.00328903
\(561\) 0 0
\(562\) 14.3283 0.604401
\(563\) 20.7792 0.875737 0.437869 0.899039i \(-0.355734\pi\)
0.437869 + 0.899039i \(0.355734\pi\)
\(564\) 0 0
\(565\) 9.49005 0.399250
\(566\) 8.97394 0.377203
\(567\) 0 0
\(568\) 30.0851 1.26234
\(569\) 19.4719 0.816307 0.408153 0.912913i \(-0.366173\pi\)
0.408153 + 0.912913i \(0.366173\pi\)
\(570\) 0 0
\(571\) −9.07799 −0.379902 −0.189951 0.981794i \(-0.560833\pi\)
−0.189951 + 0.981794i \(0.560833\pi\)
\(572\) −37.5442 −1.56980
\(573\) 0 0
\(574\) −5.89294 −0.245966
\(575\) 8.56239 0.357076
\(576\) 0 0
\(577\) 9.99714 0.416186 0.208093 0.978109i \(-0.433274\pi\)
0.208093 + 0.978109i \(0.433274\pi\)
\(578\) −15.0516 −0.626063
\(579\) 0 0
\(580\) 5.01966 0.208430
\(581\) 0.942611 0.0391061
\(582\) 0 0
\(583\) 37.5657 1.55581
\(584\) 3.67152 0.151929
\(585\) 0 0
\(586\) 6.80830 0.281248
\(587\) 29.0697 1.19984 0.599918 0.800062i \(-0.295200\pi\)
0.599918 + 0.800062i \(0.295200\pi\)
\(588\) 0 0
\(589\) −30.8024 −1.26919
\(590\) −9.11797 −0.375381
\(591\) 0 0
\(592\) 0.0664232 0.00272998
\(593\) 7.26916 0.298509 0.149254 0.988799i \(-0.452313\pi\)
0.149254 + 0.988799i \(0.452313\pi\)
\(594\) 0 0
\(595\) 3.41224 0.139888
\(596\) −25.2670 −1.03498
\(597\) 0 0
\(598\) −45.8708 −1.87580
\(599\) −15.9436 −0.651438 −0.325719 0.945467i \(-0.605606\pi\)
−0.325719 + 0.945467i \(0.605606\pi\)
\(600\) 0 0
\(601\) −7.56369 −0.308530 −0.154265 0.988030i \(-0.549301\pi\)
−0.154265 + 0.988030i \(0.549301\pi\)
\(602\) −2.38081 −0.0970346
\(603\) 0 0
\(604\) 11.4111 0.464310
\(605\) −11.5017 −0.467609
\(606\) 0 0
\(607\) −1.88969 −0.0767000 −0.0383500 0.999264i \(-0.512210\pi\)
−0.0383500 + 0.999264i \(0.512210\pi\)
\(608\) 30.9242 1.25414
\(609\) 0 0
\(610\) 8.37280 0.339005
\(611\) 23.6785 0.957930
\(612\) 0 0
\(613\) −12.4920 −0.504548 −0.252274 0.967656i \(-0.581178\pi\)
−0.252274 + 0.967656i \(0.581178\pi\)
\(614\) 1.81057 0.0730686
\(615\) 0 0
\(616\) −7.70381 −0.310395
\(617\) 10.0989 0.406567 0.203284 0.979120i \(-0.434839\pi\)
0.203284 + 0.979120i \(0.434839\pi\)
\(618\) 0 0
\(619\) −30.5158 −1.22653 −0.613266 0.789876i \(-0.710145\pi\)
−0.613266 + 0.789876i \(0.710145\pi\)
\(620\) 7.20137 0.289214
\(621\) 0 0
\(622\) −7.42442 −0.297692
\(623\) 0.580377 0.0232523
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.7657 −0.550188
\(627\) 0 0
\(628\) 11.6916 0.466545
\(629\) 2.91204 0.116111
\(630\) 0 0
\(631\) −15.5337 −0.618386 −0.309193 0.950999i \(-0.600059\pi\)
−0.309193 + 0.950999i \(0.600059\pi\)
\(632\) 1.37795 0.0548118
\(633\) 0 0
\(634\) −1.46190 −0.0580595
\(635\) 8.22863 0.326543
\(636\) 0 0
\(637\) −41.6611 −1.65067
\(638\) −16.1171 −0.638083
\(639\) 0 0
\(640\) −7.45965 −0.294869
\(641\) 17.5661 0.693818 0.346909 0.937899i \(-0.387231\pi\)
0.346909 + 0.937899i \(0.387231\pi\)
\(642\) 0 0
\(643\) −40.7523 −1.60712 −0.803558 0.595227i \(-0.797062\pi\)
−0.803558 + 0.595227i \(0.797062\pi\)
\(644\) 6.29057 0.247883
\(645\) 0 0
\(646\) 27.2756 1.07314
\(647\) 20.1397 0.791775 0.395887 0.918299i \(-0.370437\pi\)
0.395887 + 0.918299i \(0.370437\pi\)
\(648\) 0 0
\(649\) −50.4795 −1.98149
\(650\) −5.35724 −0.210128
\(651\) 0 0
\(652\) 24.4106 0.955992
\(653\) 31.5360 1.23410 0.617050 0.786924i \(-0.288328\pi\)
0.617050 + 0.786924i \(0.288328\pi\)
\(654\) 0 0
\(655\) −4.92961 −0.192616
\(656\) −1.58921 −0.0620483
\(657\) 0 0
\(658\) 1.88324 0.0734162
\(659\) −7.91412 −0.308290 −0.154145 0.988048i \(-0.549262\pi\)
−0.154145 + 0.988048i \(0.549262\pi\)
\(660\) 0 0
\(661\) −39.7578 −1.54640 −0.773200 0.634163i \(-0.781345\pi\)
−0.773200 + 0.634163i \(0.781345\pi\)
\(662\) 6.77017 0.263130
\(663\) 0 0
\(664\) −4.54474 −0.176370
\(665\) −3.14242 −0.121858
\(666\) 0 0
\(667\) 33.9535 1.31469
\(668\) −31.4136 −1.21543
\(669\) 0 0
\(670\) 1.15723 0.0447079
\(671\) 46.3540 1.78948
\(672\) 0 0
\(673\) 21.6297 0.833763 0.416882 0.908961i \(-0.363123\pi\)
0.416882 + 0.908961i \(0.363123\pi\)
\(674\) −6.85202 −0.263930
\(675\) 0 0
\(676\) −33.0303 −1.27039
\(677\) 16.7735 0.644658 0.322329 0.946628i \(-0.395534\pi\)
0.322329 + 0.946628i \(0.395534\pi\)
\(678\) 0 0
\(679\) 5.58052 0.214161
\(680\) −16.4519 −0.630903
\(681\) 0 0
\(682\) −23.1221 −0.885392
\(683\) −24.7665 −0.947663 −0.473832 0.880615i \(-0.657129\pi\)
−0.473832 + 0.880615i \(0.657129\pi\)
\(684\) 0 0
\(685\) −11.9905 −0.458135
\(686\) −6.79442 −0.259412
\(687\) 0 0
\(688\) −0.642059 −0.0244783
\(689\) 49.5148 1.88636
\(690\) 0 0
\(691\) 47.4194 1.80392 0.901959 0.431822i \(-0.142129\pi\)
0.901959 + 0.431822i \(0.142129\pi\)
\(692\) −21.5429 −0.818940
\(693\) 0 0
\(694\) 5.35333 0.203210
\(695\) −18.0960 −0.686420
\(696\) 0 0
\(697\) −69.6723 −2.63903
\(698\) −6.06081 −0.229405
\(699\) 0 0
\(700\) 0.734675 0.0277681
\(701\) −34.9075 −1.31844 −0.659218 0.751952i \(-0.729113\pi\)
−0.659218 + 0.751952i \(0.729113\pi\)
\(702\) 0 0
\(703\) −2.68178 −0.101145
\(704\) 21.9413 0.826942
\(705\) 0 0
\(706\) 14.8133 0.557507
\(707\) 2.74179 0.103116
\(708\) 0 0
\(709\) 48.6918 1.82866 0.914330 0.404970i \(-0.132718\pi\)
0.914330 + 0.404970i \(0.132718\pi\)
\(710\) 9.21200 0.345720
\(711\) 0 0
\(712\) −2.79826 −0.104869
\(713\) 48.7108 1.82423
\(714\) 0 0
\(715\) −29.6591 −1.10919
\(716\) −7.73030 −0.288895
\(717\) 0 0
\(718\) 10.5718 0.394535
\(719\) −35.5414 −1.32547 −0.662736 0.748853i \(-0.730605\pi\)
−0.662736 + 0.748853i \(0.730605\pi\)
\(720\) 0 0
\(721\) −3.00255 −0.111821
\(722\) −8.83921 −0.328961
\(723\) 0 0
\(724\) 1.30075 0.0483421
\(725\) 3.96542 0.147272
\(726\) 0 0
\(727\) 2.22393 0.0824812 0.0412406 0.999149i \(-0.486869\pi\)
0.0412406 + 0.999149i \(0.486869\pi\)
\(728\) −10.1543 −0.376342
\(729\) 0 0
\(730\) 1.12421 0.0416090
\(731\) −28.1484 −1.04111
\(732\) 0 0
\(733\) −9.92710 −0.366666 −0.183333 0.983051i \(-0.558689\pi\)
−0.183333 + 0.983051i \(0.558689\pi\)
\(734\) −26.2706 −0.969665
\(735\) 0 0
\(736\) −48.9034 −1.80260
\(737\) 6.40676 0.235996
\(738\) 0 0
\(739\) −10.1642 −0.373896 −0.186948 0.982370i \(-0.559860\pi\)
−0.186948 + 0.982370i \(0.559860\pi\)
\(740\) 0.626979 0.0230482
\(741\) 0 0
\(742\) 3.93809 0.144572
\(743\) 43.3395 1.58997 0.794985 0.606629i \(-0.207478\pi\)
0.794985 + 0.606629i \(0.207478\pi\)
\(744\) 0 0
\(745\) −19.9604 −0.731292
\(746\) 4.67168 0.171042
\(747\) 0 0
\(748\) −35.3038 −1.29083
\(749\) 3.29746 0.120487
\(750\) 0 0
\(751\) −3.92124 −0.143088 −0.0715440 0.997437i \(-0.522793\pi\)
−0.0715440 + 0.997437i \(0.522793\pi\)
\(752\) 0.507872 0.0185202
\(753\) 0 0
\(754\) −21.2437 −0.773651
\(755\) 9.01450 0.328071
\(756\) 0 0
\(757\) 7.19956 0.261672 0.130836 0.991404i \(-0.458234\pi\)
0.130836 + 0.991404i \(0.458234\pi\)
\(758\) 30.4371 1.10552
\(759\) 0 0
\(760\) 15.1510 0.549585
\(761\) −40.8629 −1.48128 −0.740639 0.671903i \(-0.765477\pi\)
−0.740639 + 0.671903i \(0.765477\pi\)
\(762\) 0 0
\(763\) 3.76561 0.136324
\(764\) −14.3071 −0.517614
\(765\) 0 0
\(766\) −11.0484 −0.399193
\(767\) −66.5363 −2.40249
\(768\) 0 0
\(769\) 51.3277 1.85092 0.925462 0.378840i \(-0.123677\pi\)
0.925462 + 0.378840i \(0.123677\pi\)
\(770\) −2.35889 −0.0850086
\(771\) 0 0
\(772\) 24.1473 0.869081
\(773\) 2.66946 0.0960137 0.0480069 0.998847i \(-0.484713\pi\)
0.0480069 + 0.998847i \(0.484713\pi\)
\(774\) 0 0
\(775\) 5.68893 0.204352
\(776\) −26.9062 −0.965875
\(777\) 0 0
\(778\) −24.2436 −0.869173
\(779\) 64.1631 2.29888
\(780\) 0 0
\(781\) 51.0001 1.82493
\(782\) −43.1335 −1.54245
\(783\) 0 0
\(784\) −0.893576 −0.0319134
\(785\) 9.23610 0.329651
\(786\) 0 0
\(787\) −16.3284 −0.582043 −0.291022 0.956716i \(-0.593995\pi\)
−0.291022 + 0.956716i \(0.593995\pi\)
\(788\) 25.5698 0.910886
\(789\) 0 0
\(790\) 0.421925 0.0150114
\(791\) 5.50781 0.195835
\(792\) 0 0
\(793\) 61.0986 2.16967
\(794\) 28.0919 0.996944
\(795\) 0 0
\(796\) 14.2663 0.505656
\(797\) −34.8557 −1.23465 −0.617325 0.786708i \(-0.711784\pi\)
−0.617325 + 0.786708i \(0.711784\pi\)
\(798\) 0 0
\(799\) 22.2655 0.787698
\(800\) −5.71142 −0.201929
\(801\) 0 0
\(802\) 17.0298 0.601344
\(803\) 6.22395 0.219638
\(804\) 0 0
\(805\) 4.96942 0.175149
\(806\) −30.4769 −1.07350
\(807\) 0 0
\(808\) −13.2194 −0.465056
\(809\) 41.4204 1.45626 0.728132 0.685437i \(-0.240389\pi\)
0.728132 + 0.685437i \(0.240389\pi\)
\(810\) 0 0
\(811\) 7.34996 0.258092 0.129046 0.991639i \(-0.458808\pi\)
0.129046 + 0.991639i \(0.458808\pi\)
\(812\) 2.91330 0.102237
\(813\) 0 0
\(814\) −2.01310 −0.0705593
\(815\) 19.2838 0.675483
\(816\) 0 0
\(817\) 25.9226 0.906917
\(818\) −21.1541 −0.739637
\(819\) 0 0
\(820\) −15.0008 −0.523852
\(821\) 38.8242 1.35498 0.677488 0.735534i \(-0.263069\pi\)
0.677488 + 0.735534i \(0.263069\pi\)
\(822\) 0 0
\(823\) −21.4901 −0.749099 −0.374550 0.927207i \(-0.622203\pi\)
−0.374550 + 0.927207i \(0.622203\pi\)
\(824\) 14.4766 0.504317
\(825\) 0 0
\(826\) −5.29187 −0.184128
\(827\) −19.6349 −0.682771 −0.341386 0.939923i \(-0.610896\pi\)
−0.341386 + 0.939923i \(0.610896\pi\)
\(828\) 0 0
\(829\) −20.8272 −0.723358 −0.361679 0.932303i \(-0.617796\pi\)
−0.361679 + 0.932303i \(0.617796\pi\)
\(830\) −1.39159 −0.0483029
\(831\) 0 0
\(832\) 28.9204 1.00264
\(833\) −39.1751 −1.35734
\(834\) 0 0
\(835\) −24.8161 −0.858796
\(836\) 32.5122 1.12446
\(837\) 0 0
\(838\) −20.2803 −0.700572
\(839\) −22.3217 −0.770630 −0.385315 0.922785i \(-0.625907\pi\)
−0.385315 + 0.922785i \(0.625907\pi\)
\(840\) 0 0
\(841\) −13.2754 −0.457773
\(842\) −13.6455 −0.470253
\(843\) 0 0
\(844\) −7.71665 −0.265618
\(845\) −26.0932 −0.897634
\(846\) 0 0
\(847\) −6.67530 −0.229366
\(848\) 1.06203 0.0364701
\(849\) 0 0
\(850\) −5.03755 −0.172787
\(851\) 4.24096 0.145378
\(852\) 0 0
\(853\) −4.79245 −0.164090 −0.0820452 0.996629i \(-0.526145\pi\)
−0.0820452 + 0.996629i \(0.526145\pi\)
\(854\) 4.85938 0.166285
\(855\) 0 0
\(856\) −15.8985 −0.543400
\(857\) −49.1822 −1.68003 −0.840016 0.542561i \(-0.817455\pi\)
−0.840016 + 0.542561i \(0.817455\pi\)
\(858\) 0 0
\(859\) 45.6426 1.55731 0.778653 0.627454i \(-0.215903\pi\)
0.778653 + 0.627454i \(0.215903\pi\)
\(860\) −6.06050 −0.206661
\(861\) 0 0
\(862\) 6.66800 0.227113
\(863\) −16.9728 −0.577761 −0.288880 0.957365i \(-0.593283\pi\)
−0.288880 + 0.957365i \(0.593283\pi\)
\(864\) 0 0
\(865\) −17.0185 −0.578645
\(866\) 3.08396 0.104797
\(867\) 0 0
\(868\) 4.17951 0.141862
\(869\) 2.33589 0.0792397
\(870\) 0 0
\(871\) 8.44465 0.286136
\(872\) −18.1557 −0.614829
\(873\) 0 0
\(874\) 39.7228 1.34364
\(875\) 0.580377 0.0196203
\(876\) 0 0
\(877\) −6.47951 −0.218797 −0.109399 0.993998i \(-0.534893\pi\)
−0.109399 + 0.993998i \(0.534893\pi\)
\(878\) −25.9480 −0.875703
\(879\) 0 0
\(880\) −0.636148 −0.0214446
\(881\) 13.0221 0.438724 0.219362 0.975644i \(-0.429602\pi\)
0.219362 + 0.975644i \(0.429602\pi\)
\(882\) 0 0
\(883\) 24.1962 0.814268 0.407134 0.913369i \(-0.366528\pi\)
0.407134 + 0.913369i \(0.366528\pi\)
\(884\) −46.5334 −1.56509
\(885\) 0 0
\(886\) 12.9461 0.434932
\(887\) 18.6517 0.626264 0.313132 0.949710i \(-0.398622\pi\)
0.313132 + 0.949710i \(0.398622\pi\)
\(888\) 0 0
\(889\) 4.77571 0.160172
\(890\) −0.856822 −0.0287207
\(891\) 0 0
\(892\) −9.69404 −0.324581
\(893\) −20.5049 −0.686171
\(894\) 0 0
\(895\) −6.10678 −0.204127
\(896\) −4.32941 −0.144636
\(897\) 0 0
\(898\) 15.6885 0.523531
\(899\) 22.5590 0.752385
\(900\) 0 0
\(901\) 46.5601 1.55114
\(902\) 48.1647 1.60371
\(903\) 0 0
\(904\) −26.5556 −0.883226
\(905\) 1.02757 0.0341575
\(906\) 0 0
\(907\) −19.1208 −0.634896 −0.317448 0.948276i \(-0.602826\pi\)
−0.317448 + 0.948276i \(0.602826\pi\)
\(908\) −37.3774 −1.24041
\(909\) 0 0
\(910\) −3.10922 −0.103070
\(911\) −6.04981 −0.200439 −0.100220 0.994965i \(-0.531955\pi\)
−0.100220 + 0.994965i \(0.531955\pi\)
\(912\) 0 0
\(913\) −7.70423 −0.254973
\(914\) 18.0369 0.596607
\(915\) 0 0
\(916\) −9.04738 −0.298934
\(917\) −2.86103 −0.0944796
\(918\) 0 0
\(919\) 19.9158 0.656962 0.328481 0.944511i \(-0.393463\pi\)
0.328481 + 0.944511i \(0.393463\pi\)
\(920\) −23.9598 −0.789930
\(921\) 0 0
\(922\) −33.2494 −1.09501
\(923\) 67.2224 2.21265
\(924\) 0 0
\(925\) 0.495300 0.0162854
\(926\) 10.3783 0.341054
\(927\) 0 0
\(928\) −22.6482 −0.743464
\(929\) −0.333759 −0.0109503 −0.00547514 0.999985i \(-0.501743\pi\)
−0.00547514 + 0.999985i \(0.501743\pi\)
\(930\) 0 0
\(931\) 36.0774 1.18239
\(932\) −13.8744 −0.454471
\(933\) 0 0
\(934\) 1.93329 0.0632590
\(935\) −27.8892 −0.912075
\(936\) 0 0
\(937\) −14.6611 −0.478957 −0.239479 0.970902i \(-0.576977\pi\)
−0.239479 + 0.970902i \(0.576977\pi\)
\(938\) 0.671633 0.0219296
\(939\) 0 0
\(940\) 4.79389 0.156360
\(941\) 4.63624 0.151137 0.0755686 0.997141i \(-0.475923\pi\)
0.0755686 + 0.997141i \(0.475923\pi\)
\(942\) 0 0
\(943\) −101.467 −3.30423
\(944\) −1.42711 −0.0464486
\(945\) 0 0
\(946\) 19.4591 0.632669
\(947\) −10.9543 −0.355965 −0.177983 0.984034i \(-0.556957\pi\)
−0.177983 + 0.984034i \(0.556957\pi\)
\(948\) 0 0
\(949\) 8.20369 0.266303
\(950\) 4.63922 0.150516
\(951\) 0 0
\(952\) −9.54833 −0.309463
\(953\) 26.4968 0.858315 0.429158 0.903230i \(-0.358810\pi\)
0.429158 + 0.903230i \(0.358810\pi\)
\(954\) 0 0
\(955\) −11.3023 −0.365735
\(956\) 24.9932 0.808339
\(957\) 0 0
\(958\) 0.203285 0.00656783
\(959\) −6.95904 −0.224719
\(960\) 0 0
\(961\) 1.36388 0.0439962
\(962\) −2.65344 −0.0855504
\(963\) 0 0
\(964\) 24.3405 0.783956
\(965\) 19.0759 0.614074
\(966\) 0 0
\(967\) −59.1812 −1.90314 −0.951569 0.307435i \(-0.900529\pi\)
−0.951569 + 0.307435i \(0.900529\pi\)
\(968\) 32.1846 1.03445
\(969\) 0 0
\(970\) −8.23862 −0.264526
\(971\) 5.45699 0.175123 0.0875616 0.996159i \(-0.472093\pi\)
0.0875616 + 0.996159i \(0.472093\pi\)
\(972\) 0 0
\(973\) −10.5025 −0.336695
\(974\) −1.45072 −0.0464842
\(975\) 0 0
\(976\) 1.31048 0.0419475
\(977\) −17.0837 −0.546555 −0.273278 0.961935i \(-0.588108\pi\)
−0.273278 + 0.961935i \(0.588108\pi\)
\(978\) 0 0
\(979\) −4.74359 −0.151606
\(980\) −8.43461 −0.269434
\(981\) 0 0
\(982\) −15.5996 −0.497803
\(983\) −42.1851 −1.34549 −0.672747 0.739872i \(-0.734886\pi\)
−0.672747 + 0.739872i \(0.734886\pi\)
\(984\) 0 0
\(985\) 20.1996 0.643612
\(986\) −19.9760 −0.636167
\(987\) 0 0
\(988\) 42.8538 1.36336
\(989\) −40.9939 −1.30353
\(990\) 0 0
\(991\) −11.7224 −0.372375 −0.186188 0.982514i \(-0.559613\pi\)
−0.186188 + 0.982514i \(0.559613\pi\)
\(992\) −32.4918 −1.03162
\(993\) 0 0
\(994\) 5.34644 0.169579
\(995\) 11.2701 0.357286
\(996\) 0 0
\(997\) 0.629575 0.0199388 0.00996942 0.999950i \(-0.496827\pi\)
0.00996942 + 0.999950i \(0.496827\pi\)
\(998\) −17.1582 −0.543134
\(999\) 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.1.2 7
3.2 odd 2 445.2.a.f.1.6 7
12.11 even 2 7120.2.a.bj.1.4 7
15.14 odd 2 2225.2.a.k.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.6 7 3.2 odd 2
2225.2.a.k.1.2 7 15.14 odd 2
4005.2.a.o.1.2 7 1.1 even 1 trivial
7120.2.a.bj.1.4 7 12.11 even 2