Properties

Label 2004.2.a.b.1.4
Level 2004
Weight 2
Character 2004.1
Self dual yes
Analytic conductor 16.002
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0020205651\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.07823\) of defining polynomial
Character \(\chi\) \(=\) 2004.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +3.80906 q^{11} -2.93060 q^{13} -0.614948 q^{15} +3.15646 q^{17} -5.58256 q^{19} -3.46328 q^{21} -0.574360 q^{23} -4.62184 q^{25} +1.00000 q^{27} -3.25128 q^{29} -1.74532 q^{31} +3.80906 q^{33} +2.12974 q^{35} -4.35964 q^{37} -2.93060 q^{39} -1.51130 q^{41} +9.97984 q^{43} -0.614948 q^{45} -8.29318 q^{47} +4.99433 q^{49} +3.15646 q^{51} +0.465048 q^{53} -2.34237 q^{55} -5.58256 q^{57} -9.58387 q^{59} -6.22999 q^{61} -3.46328 q^{63} +1.80217 q^{65} +3.41132 q^{67} -0.574360 q^{69} -8.58319 q^{71} -9.90115 q^{73} -4.62184 q^{75} -13.1918 q^{77} +10.2406 q^{79} +1.00000 q^{81} -8.39859 q^{83} -1.94106 q^{85} -3.25128 q^{87} -7.12992 q^{89} +10.1495 q^{91} -1.74532 q^{93} +3.43299 q^{95} +4.70410 q^{97} +3.80906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} - 7q^{5} - 2q^{7} + 5q^{9} - 5q^{11} - 8q^{13} - 7q^{15} - 7q^{17} + 2q^{19} - 2q^{21} - 13q^{23} + 2q^{25} + 5q^{27} - 11q^{29} - 12q^{31} - 5q^{33} - 12q^{35} - 7q^{37} - 8q^{39} - 12q^{41} - 7q^{45} - 19q^{47} - 9q^{49} - 7q^{51} - 21q^{53} - q^{55} + 2q^{57} - 7q^{59} - 6q^{61} - 2q^{63} + 14q^{65} + 10q^{67} - 13q^{69} - 35q^{71} - 8q^{73} + 2q^{75} - 6q^{77} + 5q^{81} - 11q^{83} + 5q^{85} - 11q^{87} - 32q^{89} + 5q^{91} - 12q^{93} - 19q^{95} + 11q^{97} - 5q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.614948 −0.275013 −0.137507 0.990501i \(-0.543909\pi\)
−0.137507 + 0.990501i \(0.543909\pi\)
\(6\) 0 0
\(7\) −3.46328 −1.30900 −0.654499 0.756063i \(-0.727120\pi\)
−0.654499 + 0.756063i \(0.727120\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.80906 1.14847 0.574237 0.818689i \(-0.305299\pi\)
0.574237 + 0.818689i \(0.305299\pi\)
\(12\) 0 0
\(13\) −2.93060 −0.812801 −0.406401 0.913695i \(-0.633216\pi\)
−0.406401 + 0.913695i \(0.633216\pi\)
\(14\) 0 0
\(15\) −0.614948 −0.158779
\(16\) 0 0
\(17\) 3.15646 0.765555 0.382778 0.923841i \(-0.374968\pi\)
0.382778 + 0.923841i \(0.374968\pi\)
\(18\) 0 0
\(19\) −5.58256 −1.28073 −0.640363 0.768072i \(-0.721216\pi\)
−0.640363 + 0.768072i \(0.721216\pi\)
\(20\) 0 0
\(21\) −3.46328 −0.755750
\(22\) 0 0
\(23\) −0.574360 −0.119762 −0.0598811 0.998206i \(-0.519072\pi\)
−0.0598811 + 0.998206i \(0.519072\pi\)
\(24\) 0 0
\(25\) −4.62184 −0.924368
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.25128 −0.603748 −0.301874 0.953348i \(-0.597612\pi\)
−0.301874 + 0.953348i \(0.597612\pi\)
\(30\) 0 0
\(31\) −1.74532 −0.313468 −0.156734 0.987641i \(-0.550097\pi\)
−0.156734 + 0.987641i \(0.550097\pi\)
\(32\) 0 0
\(33\) 3.80906 0.663072
\(34\) 0 0
\(35\) 2.12974 0.359992
\(36\) 0 0
\(37\) −4.35964 −0.716720 −0.358360 0.933583i \(-0.616664\pi\)
−0.358360 + 0.933583i \(0.616664\pi\)
\(38\) 0 0
\(39\) −2.93060 −0.469271
\(40\) 0 0
\(41\) −1.51130 −0.236026 −0.118013 0.993012i \(-0.537652\pi\)
−0.118013 + 0.993012i \(0.537652\pi\)
\(42\) 0 0
\(43\) 9.97984 1.52191 0.760956 0.648804i \(-0.224730\pi\)
0.760956 + 0.648804i \(0.224730\pi\)
\(44\) 0 0
\(45\) −0.614948 −0.0916711
\(46\) 0 0
\(47\) −8.29318 −1.20968 −0.604842 0.796345i \(-0.706764\pi\)
−0.604842 + 0.796345i \(0.706764\pi\)
\(48\) 0 0
\(49\) 4.99433 0.713476
\(50\) 0 0
\(51\) 3.15646 0.441993
\(52\) 0 0
\(53\) 0.465048 0.0638792 0.0319396 0.999490i \(-0.489832\pi\)
0.0319396 + 0.999490i \(0.489832\pi\)
\(54\) 0 0
\(55\) −2.34237 −0.315845
\(56\) 0 0
\(57\) −5.58256 −0.739428
\(58\) 0 0
\(59\) −9.58387 −1.24771 −0.623857 0.781539i \(-0.714435\pi\)
−0.623857 + 0.781539i \(0.714435\pi\)
\(60\) 0 0
\(61\) −6.22999 −0.797668 −0.398834 0.917023i \(-0.630585\pi\)
−0.398834 + 0.917023i \(0.630585\pi\)
\(62\) 0 0
\(63\) −3.46328 −0.436333
\(64\) 0 0
\(65\) 1.80217 0.223531
\(66\) 0 0
\(67\) 3.41132 0.416759 0.208380 0.978048i \(-0.433181\pi\)
0.208380 + 0.978048i \(0.433181\pi\)
\(68\) 0 0
\(69\) −0.574360 −0.0691448
\(70\) 0 0
\(71\) −8.58319 −1.01864 −0.509319 0.860578i \(-0.670103\pi\)
−0.509319 + 0.860578i \(0.670103\pi\)
\(72\) 0 0
\(73\) −9.90115 −1.15884 −0.579421 0.815028i \(-0.696721\pi\)
−0.579421 + 0.815028i \(0.696721\pi\)
\(74\) 0 0
\(75\) −4.62184 −0.533684
\(76\) 0 0
\(77\) −13.1918 −1.50335
\(78\) 0 0
\(79\) 10.2406 1.15216 0.576078 0.817395i \(-0.304583\pi\)
0.576078 + 0.817395i \(0.304583\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.39859 −0.921865 −0.460932 0.887435i \(-0.652485\pi\)
−0.460932 + 0.887435i \(0.652485\pi\)
\(84\) 0 0
\(85\) −1.94106 −0.210538
\(86\) 0 0
\(87\) −3.25128 −0.348574
\(88\) 0 0
\(89\) −7.12992 −0.755770 −0.377885 0.925853i \(-0.623348\pi\)
−0.377885 + 0.925853i \(0.623348\pi\)
\(90\) 0 0
\(91\) 10.1495 1.06396
\(92\) 0 0
\(93\) −1.74532 −0.180981
\(94\) 0 0
\(95\) 3.43299 0.352217
\(96\) 0 0
\(97\) 4.70410 0.477629 0.238815 0.971065i \(-0.423241\pi\)
0.238815 + 0.971065i \(0.423241\pi\)
\(98\) 0 0
\(99\) 3.80906 0.382825
\(100\) 0 0
\(101\) −6.74351 −0.671004 −0.335502 0.942039i \(-0.608906\pi\)
−0.335502 + 0.942039i \(0.608906\pi\)
\(102\) 0 0
\(103\) −3.37572 −0.332620 −0.166310 0.986074i \(-0.553185\pi\)
−0.166310 + 0.986074i \(0.553185\pi\)
\(104\) 0 0
\(105\) 2.12974 0.207841
\(106\) 0 0
\(107\) −6.87891 −0.665010 −0.332505 0.943102i \(-0.607894\pi\)
−0.332505 + 0.943102i \(0.607894\pi\)
\(108\) 0 0
\(109\) −12.2964 −1.17778 −0.588891 0.808213i \(-0.700435\pi\)
−0.588891 + 0.808213i \(0.700435\pi\)
\(110\) 0 0
\(111\) −4.35964 −0.413798
\(112\) 0 0
\(113\) 5.04942 0.475009 0.237505 0.971386i \(-0.423671\pi\)
0.237505 + 0.971386i \(0.423671\pi\)
\(114\) 0 0
\(115\) 0.353202 0.0329362
\(116\) 0 0
\(117\) −2.93060 −0.270934
\(118\) 0 0
\(119\) −10.9317 −1.00211
\(120\) 0 0
\(121\) 3.50891 0.318992
\(122\) 0 0
\(123\) −1.51130 −0.136269
\(124\) 0 0
\(125\) 5.91693 0.529227
\(126\) 0 0
\(127\) 8.43089 0.748121 0.374060 0.927404i \(-0.377965\pi\)
0.374060 + 0.927404i \(0.377965\pi\)
\(128\) 0 0
\(129\) 9.97984 0.878676
\(130\) 0 0
\(131\) 7.65327 0.668669 0.334335 0.942454i \(-0.391488\pi\)
0.334335 + 0.942454i \(0.391488\pi\)
\(132\) 0 0
\(133\) 19.3340 1.67647
\(134\) 0 0
\(135\) −0.614948 −0.0529263
\(136\) 0 0
\(137\) −5.27810 −0.450938 −0.225469 0.974250i \(-0.572391\pi\)
−0.225469 + 0.974250i \(0.572391\pi\)
\(138\) 0 0
\(139\) 5.53478 0.469454 0.234727 0.972061i \(-0.424580\pi\)
0.234727 + 0.972061i \(0.424580\pi\)
\(140\) 0 0
\(141\) −8.29318 −0.698412
\(142\) 0 0
\(143\) −11.1628 −0.933481
\(144\) 0 0
\(145\) 1.99937 0.166039
\(146\) 0 0
\(147\) 4.99433 0.411926
\(148\) 0 0
\(149\) 13.1803 1.07977 0.539886 0.841738i \(-0.318467\pi\)
0.539886 + 0.841738i \(0.318467\pi\)
\(150\) 0 0
\(151\) −2.29363 −0.186653 −0.0933266 0.995636i \(-0.529750\pi\)
−0.0933266 + 0.995636i \(0.529750\pi\)
\(152\) 0 0
\(153\) 3.15646 0.255185
\(154\) 0 0
\(155\) 1.07328 0.0862080
\(156\) 0 0
\(157\) −14.6203 −1.16682 −0.583412 0.812177i \(-0.698283\pi\)
−0.583412 + 0.812177i \(0.698283\pi\)
\(158\) 0 0
\(159\) 0.465048 0.0368807
\(160\) 0 0
\(161\) 1.98917 0.156769
\(162\) 0 0
\(163\) 15.6158 1.22312 0.611560 0.791198i \(-0.290542\pi\)
0.611560 + 0.791198i \(0.290542\pi\)
\(164\) 0 0
\(165\) −2.34237 −0.182353
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −4.41160 −0.339354
\(170\) 0 0
\(171\) −5.58256 −0.426909
\(172\) 0 0
\(173\) 13.8874 1.05584 0.527919 0.849295i \(-0.322972\pi\)
0.527919 + 0.849295i \(0.322972\pi\)
\(174\) 0 0
\(175\) 16.0067 1.21000
\(176\) 0 0
\(177\) −9.58387 −0.720368
\(178\) 0 0
\(179\) 3.21586 0.240364 0.120182 0.992752i \(-0.461652\pi\)
0.120182 + 0.992752i \(0.461652\pi\)
\(180\) 0 0
\(181\) 7.33183 0.544971 0.272485 0.962160i \(-0.412154\pi\)
0.272485 + 0.962160i \(0.412154\pi\)
\(182\) 0 0
\(183\) −6.22999 −0.460534
\(184\) 0 0
\(185\) 2.68095 0.197107
\(186\) 0 0
\(187\) 12.0231 0.879220
\(188\) 0 0
\(189\) −3.46328 −0.251917
\(190\) 0 0
\(191\) 2.33599 0.169026 0.0845130 0.996422i \(-0.473067\pi\)
0.0845130 + 0.996422i \(0.473067\pi\)
\(192\) 0 0
\(193\) 9.73301 0.700597 0.350299 0.936638i \(-0.386080\pi\)
0.350299 + 0.936638i \(0.386080\pi\)
\(194\) 0 0
\(195\) 1.80217 0.129056
\(196\) 0 0
\(197\) −19.0432 −1.35677 −0.678386 0.734705i \(-0.737320\pi\)
−0.678386 + 0.734705i \(0.737320\pi\)
\(198\) 0 0
\(199\) 5.26986 0.373570 0.186785 0.982401i \(-0.440193\pi\)
0.186785 + 0.982401i \(0.440193\pi\)
\(200\) 0 0
\(201\) 3.41132 0.240616
\(202\) 0 0
\(203\) 11.2601 0.790305
\(204\) 0 0
\(205\) 0.929372 0.0649102
\(206\) 0 0
\(207\) −0.574360 −0.0399208
\(208\) 0 0
\(209\) −21.2643 −1.47088
\(210\) 0 0
\(211\) −1.49086 −0.102635 −0.0513176 0.998682i \(-0.516342\pi\)
−0.0513176 + 0.998682i \(0.516342\pi\)
\(212\) 0 0
\(213\) −8.58319 −0.588110
\(214\) 0 0
\(215\) −6.13709 −0.418546
\(216\) 0 0
\(217\) 6.04453 0.410330
\(218\) 0 0
\(219\) −9.90115 −0.669058
\(220\) 0 0
\(221\) −9.25032 −0.622244
\(222\) 0 0
\(223\) −0.343506 −0.0230029 −0.0115014 0.999934i \(-0.503661\pi\)
−0.0115014 + 0.999934i \(0.503661\pi\)
\(224\) 0 0
\(225\) −4.62184 −0.308123
\(226\) 0 0
\(227\) 8.12470 0.539256 0.269628 0.962965i \(-0.413099\pi\)
0.269628 + 0.962965i \(0.413099\pi\)
\(228\) 0 0
\(229\) 5.95619 0.393596 0.196798 0.980444i \(-0.436946\pi\)
0.196798 + 0.980444i \(0.436946\pi\)
\(230\) 0 0
\(231\) −13.1918 −0.867959
\(232\) 0 0
\(233\) 0.870260 0.0570126 0.0285063 0.999594i \(-0.490925\pi\)
0.0285063 + 0.999594i \(0.490925\pi\)
\(234\) 0 0
\(235\) 5.09988 0.332679
\(236\) 0 0
\(237\) 10.2406 0.665197
\(238\) 0 0
\(239\) 13.0177 0.842046 0.421023 0.907050i \(-0.361671\pi\)
0.421023 + 0.907050i \(0.361671\pi\)
\(240\) 0 0
\(241\) −2.82934 −0.182254 −0.0911270 0.995839i \(-0.529047\pi\)
−0.0911270 + 0.995839i \(0.529047\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.07126 −0.196215
\(246\) 0 0
\(247\) 16.3602 1.04098
\(248\) 0 0
\(249\) −8.39859 −0.532239
\(250\) 0 0
\(251\) −16.4580 −1.03882 −0.519410 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(252\) 0 0
\(253\) −2.18777 −0.137544
\(254\) 0 0
\(255\) −1.94106 −0.121554
\(256\) 0 0
\(257\) 24.5773 1.53309 0.766545 0.642191i \(-0.221974\pi\)
0.766545 + 0.642191i \(0.221974\pi\)
\(258\) 0 0
\(259\) 15.0987 0.938185
\(260\) 0 0
\(261\) −3.25128 −0.201249
\(262\) 0 0
\(263\) 24.6483 1.51988 0.759939 0.649994i \(-0.225229\pi\)
0.759939 + 0.649994i \(0.225229\pi\)
\(264\) 0 0
\(265\) −0.285980 −0.0175676
\(266\) 0 0
\(267\) −7.12992 −0.436344
\(268\) 0 0
\(269\) −4.52549 −0.275924 −0.137962 0.990438i \(-0.544055\pi\)
−0.137962 + 0.990438i \(0.544055\pi\)
\(270\) 0 0
\(271\) −22.1048 −1.34277 −0.671385 0.741109i \(-0.734300\pi\)
−0.671385 + 0.741109i \(0.734300\pi\)
\(272\) 0 0
\(273\) 10.1495 0.614275
\(274\) 0 0
\(275\) −17.6048 −1.06161
\(276\) 0 0
\(277\) −14.4555 −0.868548 −0.434274 0.900781i \(-0.642995\pi\)
−0.434274 + 0.900781i \(0.642995\pi\)
\(278\) 0 0
\(279\) −1.74532 −0.104489
\(280\) 0 0
\(281\) 3.60180 0.214866 0.107433 0.994212i \(-0.465737\pi\)
0.107433 + 0.994212i \(0.465737\pi\)
\(282\) 0 0
\(283\) −22.0928 −1.31328 −0.656639 0.754205i \(-0.728023\pi\)
−0.656639 + 0.754205i \(0.728023\pi\)
\(284\) 0 0
\(285\) 3.43299 0.203352
\(286\) 0 0
\(287\) 5.23407 0.308957
\(288\) 0 0
\(289\) −7.03673 −0.413926
\(290\) 0 0
\(291\) 4.70410 0.275759
\(292\) 0 0
\(293\) −0.403322 −0.0235623 −0.0117812 0.999931i \(-0.503750\pi\)
−0.0117812 + 0.999931i \(0.503750\pi\)
\(294\) 0 0
\(295\) 5.89358 0.343138
\(296\) 0 0
\(297\) 3.80906 0.221024
\(298\) 0 0
\(299\) 1.68322 0.0973430
\(300\) 0 0
\(301\) −34.5630 −1.99218
\(302\) 0 0
\(303\) −6.74351 −0.387404
\(304\) 0 0
\(305\) 3.83112 0.219369
\(306\) 0 0
\(307\) 28.5615 1.63009 0.815045 0.579397i \(-0.196712\pi\)
0.815045 + 0.579397i \(0.196712\pi\)
\(308\) 0 0
\(309\) −3.37572 −0.192038
\(310\) 0 0
\(311\) −14.0740 −0.798061 −0.399031 0.916938i \(-0.630653\pi\)
−0.399031 + 0.916938i \(0.630653\pi\)
\(312\) 0 0
\(313\) 13.3132 0.752504 0.376252 0.926517i \(-0.377213\pi\)
0.376252 + 0.926517i \(0.377213\pi\)
\(314\) 0 0
\(315\) 2.12974 0.119997
\(316\) 0 0
\(317\) −23.0614 −1.29526 −0.647630 0.761955i \(-0.724240\pi\)
−0.647630 + 0.761955i \(0.724240\pi\)
\(318\) 0 0
\(319\) −12.3843 −0.693388
\(320\) 0 0
\(321\) −6.87891 −0.383944
\(322\) 0 0
\(323\) −17.6211 −0.980467
\(324\) 0 0
\(325\) 13.5447 0.751327
\(326\) 0 0
\(327\) −12.2964 −0.679993
\(328\) 0 0
\(329\) 28.7216 1.58347
\(330\) 0 0
\(331\) 31.1825 1.71394 0.856972 0.515363i \(-0.172343\pi\)
0.856972 + 0.515363i \(0.172343\pi\)
\(332\) 0 0
\(333\) −4.35964 −0.238907
\(334\) 0 0
\(335\) −2.09779 −0.114614
\(336\) 0 0
\(337\) −0.817442 −0.0445289 −0.0222645 0.999752i \(-0.507088\pi\)
−0.0222645 + 0.999752i \(0.507088\pi\)
\(338\) 0 0
\(339\) 5.04942 0.274247
\(340\) 0 0
\(341\) −6.64802 −0.360010
\(342\) 0 0
\(343\) 6.94619 0.375059
\(344\) 0 0
\(345\) 0.353202 0.0190157
\(346\) 0 0
\(347\) −9.97471 −0.535471 −0.267735 0.963492i \(-0.586275\pi\)
−0.267735 + 0.963492i \(0.586275\pi\)
\(348\) 0 0
\(349\) −13.6700 −0.731740 −0.365870 0.930666i \(-0.619229\pi\)
−0.365870 + 0.930666i \(0.619229\pi\)
\(350\) 0 0
\(351\) −2.93060 −0.156424
\(352\) 0 0
\(353\) −0.0887759 −0.00472506 −0.00236253 0.999997i \(-0.500752\pi\)
−0.00236253 + 0.999997i \(0.500752\pi\)
\(354\) 0 0
\(355\) 5.27822 0.280139
\(356\) 0 0
\(357\) −10.9317 −0.578569
\(358\) 0 0
\(359\) −6.69628 −0.353416 −0.176708 0.984263i \(-0.556545\pi\)
−0.176708 + 0.984263i \(0.556545\pi\)
\(360\) 0 0
\(361\) 12.1650 0.640261
\(362\) 0 0
\(363\) 3.50891 0.184170
\(364\) 0 0
\(365\) 6.08870 0.318697
\(366\) 0 0
\(367\) 26.5833 1.38764 0.693819 0.720150i \(-0.255927\pi\)
0.693819 + 0.720150i \(0.255927\pi\)
\(368\) 0 0
\(369\) −1.51130 −0.0786752
\(370\) 0 0
\(371\) −1.61059 −0.0836178
\(372\) 0 0
\(373\) 25.1865 1.30411 0.652053 0.758173i \(-0.273908\pi\)
0.652053 + 0.758173i \(0.273908\pi\)
\(374\) 0 0
\(375\) 5.91693 0.305549
\(376\) 0 0
\(377\) 9.52820 0.490727
\(378\) 0 0
\(379\) −15.4647 −0.794370 −0.397185 0.917739i \(-0.630013\pi\)
−0.397185 + 0.917739i \(0.630013\pi\)
\(380\) 0 0
\(381\) 8.43089 0.431928
\(382\) 0 0
\(383\) 30.0700 1.53651 0.768253 0.640147i \(-0.221126\pi\)
0.768253 + 0.640147i \(0.221126\pi\)
\(384\) 0 0
\(385\) 8.11230 0.413441
\(386\) 0 0
\(387\) 9.97984 0.507304
\(388\) 0 0
\(389\) −26.8151 −1.35958 −0.679790 0.733407i \(-0.737929\pi\)
−0.679790 + 0.733407i \(0.737929\pi\)
\(390\) 0 0
\(391\) −1.81295 −0.0916846
\(392\) 0 0
\(393\) 7.65327 0.386056
\(394\) 0 0
\(395\) −6.29743 −0.316858
\(396\) 0 0
\(397\) −10.7692 −0.540490 −0.270245 0.962792i \(-0.587105\pi\)
−0.270245 + 0.962792i \(0.587105\pi\)
\(398\) 0 0
\(399\) 19.3340 0.967910
\(400\) 0 0
\(401\) 17.6845 0.883121 0.441561 0.897231i \(-0.354425\pi\)
0.441561 + 0.897231i \(0.354425\pi\)
\(402\) 0 0
\(403\) 5.11483 0.254788
\(404\) 0 0
\(405\) −0.614948 −0.0305570
\(406\) 0 0
\(407\) −16.6061 −0.823134
\(408\) 0 0
\(409\) −20.1733 −0.997506 −0.498753 0.866744i \(-0.666209\pi\)
−0.498753 + 0.866744i \(0.666209\pi\)
\(410\) 0 0
\(411\) −5.27810 −0.260349
\(412\) 0 0
\(413\) 33.1917 1.63325
\(414\) 0 0
\(415\) 5.16470 0.253525
\(416\) 0 0
\(417\) 5.53478 0.271039
\(418\) 0 0
\(419\) 7.91248 0.386550 0.193275 0.981145i \(-0.438089\pi\)
0.193275 + 0.981145i \(0.438089\pi\)
\(420\) 0 0
\(421\) 7.95900 0.387898 0.193949 0.981012i \(-0.437870\pi\)
0.193949 + 0.981012i \(0.437870\pi\)
\(422\) 0 0
\(423\) −8.29318 −0.403228
\(424\) 0 0
\(425\) −14.5887 −0.707654
\(426\) 0 0
\(427\) 21.5762 1.04415
\(428\) 0 0
\(429\) −11.1628 −0.538945
\(430\) 0 0
\(431\) 6.43209 0.309823 0.154912 0.987928i \(-0.450491\pi\)
0.154912 + 0.987928i \(0.450491\pi\)
\(432\) 0 0
\(433\) 25.3350 1.21752 0.608761 0.793354i \(-0.291667\pi\)
0.608761 + 0.793354i \(0.291667\pi\)
\(434\) 0 0
\(435\) 1.99937 0.0958625
\(436\) 0 0
\(437\) 3.20640 0.153383
\(438\) 0 0
\(439\) −5.63416 −0.268904 −0.134452 0.990920i \(-0.542927\pi\)
−0.134452 + 0.990920i \(0.542927\pi\)
\(440\) 0 0
\(441\) 4.99433 0.237825
\(442\) 0 0
\(443\) 34.5091 1.63958 0.819789 0.572666i \(-0.194091\pi\)
0.819789 + 0.572666i \(0.194091\pi\)
\(444\) 0 0
\(445\) 4.38453 0.207847
\(446\) 0 0
\(447\) 13.1803 0.623406
\(448\) 0 0
\(449\) −37.4839 −1.76897 −0.884487 0.466565i \(-0.845491\pi\)
−0.884487 + 0.466565i \(0.845491\pi\)
\(450\) 0 0
\(451\) −5.75663 −0.271069
\(452\) 0 0
\(453\) −2.29363 −0.107764
\(454\) 0 0
\(455\) −6.24141 −0.292602
\(456\) 0 0
\(457\) 8.87783 0.415287 0.207644 0.978205i \(-0.433421\pi\)
0.207644 + 0.978205i \(0.433421\pi\)
\(458\) 0 0
\(459\) 3.15646 0.147331
\(460\) 0 0
\(461\) 9.67760 0.450731 0.225365 0.974274i \(-0.427642\pi\)
0.225365 + 0.974274i \(0.427642\pi\)
\(462\) 0 0
\(463\) 8.66347 0.402626 0.201313 0.979527i \(-0.435479\pi\)
0.201313 + 0.979527i \(0.435479\pi\)
\(464\) 0 0
\(465\) 1.07328 0.0497722
\(466\) 0 0
\(467\) 31.1541 1.44164 0.720819 0.693124i \(-0.243766\pi\)
0.720819 + 0.693124i \(0.243766\pi\)
\(468\) 0 0
\(469\) −11.8144 −0.545537
\(470\) 0 0
\(471\) −14.6203 −0.673666
\(472\) 0 0
\(473\) 38.0138 1.74787
\(474\) 0 0
\(475\) 25.8017 1.18386
\(476\) 0 0
\(477\) 0.465048 0.0212931
\(478\) 0 0
\(479\) 6.78094 0.309829 0.154915 0.987928i \(-0.450490\pi\)
0.154915 + 0.987928i \(0.450490\pi\)
\(480\) 0 0
\(481\) 12.7763 0.582551
\(482\) 0 0
\(483\) 1.98917 0.0905104
\(484\) 0 0
\(485\) −2.89278 −0.131354
\(486\) 0 0
\(487\) 3.19422 0.144744 0.0723719 0.997378i \(-0.476943\pi\)
0.0723719 + 0.997378i \(0.476943\pi\)
\(488\) 0 0
\(489\) 15.6158 0.706169
\(490\) 0 0
\(491\) 36.1947 1.63344 0.816722 0.577031i \(-0.195789\pi\)
0.816722 + 0.577031i \(0.195789\pi\)
\(492\) 0 0
\(493\) −10.2626 −0.462202
\(494\) 0 0
\(495\) −2.34237 −0.105282
\(496\) 0 0
\(497\) 29.7260 1.33339
\(498\) 0 0
\(499\) −19.5019 −0.873026 −0.436513 0.899698i \(-0.643787\pi\)
−0.436513 + 0.899698i \(0.643787\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −23.7924 −1.06085 −0.530426 0.847731i \(-0.677968\pi\)
−0.530426 + 0.847731i \(0.677968\pi\)
\(504\) 0 0
\(505\) 4.14691 0.184535
\(506\) 0 0
\(507\) −4.41160 −0.195926
\(508\) 0 0
\(509\) −17.6743 −0.783399 −0.391700 0.920093i \(-0.628113\pi\)
−0.391700 + 0.920093i \(0.628113\pi\)
\(510\) 0 0
\(511\) 34.2905 1.51692
\(512\) 0 0
\(513\) −5.58256 −0.246476
\(514\) 0 0
\(515\) 2.07589 0.0914748
\(516\) 0 0
\(517\) −31.5892 −1.38929
\(518\) 0 0
\(519\) 13.8874 0.609588
\(520\) 0 0
\(521\) −41.0601 −1.79887 −0.899437 0.437050i \(-0.856023\pi\)
−0.899437 + 0.437050i \(0.856023\pi\)
\(522\) 0 0
\(523\) 2.50620 0.109588 0.0547942 0.998498i \(-0.482550\pi\)
0.0547942 + 0.998498i \(0.482550\pi\)
\(524\) 0 0
\(525\) 16.0067 0.698591
\(526\) 0 0
\(527\) −5.50904 −0.239977
\(528\) 0 0
\(529\) −22.6701 −0.985657
\(530\) 0 0
\(531\) −9.58387 −0.415904
\(532\) 0 0
\(533\) 4.42902 0.191842
\(534\) 0 0
\(535\) 4.23018 0.182886
\(536\) 0 0
\(537\) 3.21586 0.138774
\(538\) 0 0
\(539\) 19.0237 0.819409
\(540\) 0 0
\(541\) 36.2521 1.55860 0.779299 0.626652i \(-0.215575\pi\)
0.779299 + 0.626652i \(0.215575\pi\)
\(542\) 0 0
\(543\) 7.33183 0.314639
\(544\) 0 0
\(545\) 7.56165 0.323906
\(546\) 0 0
\(547\) 10.5354 0.450461 0.225230 0.974306i \(-0.427686\pi\)
0.225230 + 0.974306i \(0.427686\pi\)
\(548\) 0 0
\(549\) −6.22999 −0.265889
\(550\) 0 0
\(551\) 18.1505 0.773236
\(552\) 0 0
\(553\) −35.4660 −1.50817
\(554\) 0 0
\(555\) 2.68095 0.113800
\(556\) 0 0
\(557\) −17.9650 −0.761200 −0.380600 0.924740i \(-0.624283\pi\)
−0.380600 + 0.924740i \(0.624283\pi\)
\(558\) 0 0
\(559\) −29.2469 −1.23701
\(560\) 0 0
\(561\) 12.0231 0.507618
\(562\) 0 0
\(563\) −9.50941 −0.400774 −0.200387 0.979717i \(-0.564220\pi\)
−0.200387 + 0.979717i \(0.564220\pi\)
\(564\) 0 0
\(565\) −3.10513 −0.130634
\(566\) 0 0
\(567\) −3.46328 −0.145444
\(568\) 0 0
\(569\) −18.4813 −0.774778 −0.387389 0.921916i \(-0.626623\pi\)
−0.387389 + 0.921916i \(0.626623\pi\)
\(570\) 0 0
\(571\) 12.7478 0.533478 0.266739 0.963769i \(-0.414054\pi\)
0.266739 + 0.963769i \(0.414054\pi\)
\(572\) 0 0
\(573\) 2.33599 0.0975872
\(574\) 0 0
\(575\) 2.65460 0.110704
\(576\) 0 0
\(577\) −36.1464 −1.50480 −0.752398 0.658709i \(-0.771103\pi\)
−0.752398 + 0.658709i \(0.771103\pi\)
\(578\) 0 0
\(579\) 9.73301 0.404490
\(580\) 0 0
\(581\) 29.0867 1.20672
\(582\) 0 0
\(583\) 1.77139 0.0733636
\(584\) 0 0
\(585\) 1.80217 0.0745104
\(586\) 0 0
\(587\) −44.8985 −1.85316 −0.926580 0.376098i \(-0.877266\pi\)
−0.926580 + 0.376098i \(0.877266\pi\)
\(588\) 0 0
\(589\) 9.74334 0.401468
\(590\) 0 0
\(591\) −19.0432 −0.783333
\(592\) 0 0
\(593\) −24.0796 −0.988832 −0.494416 0.869225i \(-0.664618\pi\)
−0.494416 + 0.869225i \(0.664618\pi\)
\(594\) 0 0
\(595\) 6.72245 0.275594
\(596\) 0 0
\(597\) 5.26986 0.215681
\(598\) 0 0
\(599\) 4.07062 0.166321 0.0831604 0.996536i \(-0.473499\pi\)
0.0831604 + 0.996536i \(0.473499\pi\)
\(600\) 0 0
\(601\) −13.0899 −0.533949 −0.266974 0.963704i \(-0.586024\pi\)
−0.266974 + 0.963704i \(0.586024\pi\)
\(602\) 0 0
\(603\) 3.41132 0.138920
\(604\) 0 0
\(605\) −2.15780 −0.0877269
\(606\) 0 0
\(607\) 29.3314 1.19053 0.595263 0.803531i \(-0.297048\pi\)
0.595263 + 0.803531i \(0.297048\pi\)
\(608\) 0 0
\(609\) 11.2601 0.456283
\(610\) 0 0
\(611\) 24.3040 0.983233
\(612\) 0 0
\(613\) −0.550896 −0.0222505 −0.0111252 0.999938i \(-0.503541\pi\)
−0.0111252 + 0.999938i \(0.503541\pi\)
\(614\) 0 0
\(615\) 0.929372 0.0374759
\(616\) 0 0
\(617\) −47.2384 −1.90175 −0.950874 0.309578i \(-0.899812\pi\)
−0.950874 + 0.309578i \(0.899812\pi\)
\(618\) 0 0
\(619\) −2.51048 −0.100905 −0.0504524 0.998726i \(-0.516066\pi\)
−0.0504524 + 0.998726i \(0.516066\pi\)
\(620\) 0 0
\(621\) −0.574360 −0.0230483
\(622\) 0 0
\(623\) 24.6929 0.989301
\(624\) 0 0
\(625\) 19.4706 0.778823
\(626\) 0 0
\(627\) −21.2643 −0.849214
\(628\) 0 0
\(629\) −13.7610 −0.548689
\(630\) 0 0
\(631\) 42.8135 1.70438 0.852190 0.523233i \(-0.175274\pi\)
0.852190 + 0.523233i \(0.175274\pi\)
\(632\) 0 0
\(633\) −1.49086 −0.0592564
\(634\) 0 0
\(635\) −5.18456 −0.205743
\(636\) 0 0
\(637\) −14.6364 −0.579915
\(638\) 0 0
\(639\) −8.58319 −0.339546
\(640\) 0 0
\(641\) −38.7594 −1.53090 −0.765452 0.643493i \(-0.777485\pi\)
−0.765452 + 0.643493i \(0.777485\pi\)
\(642\) 0 0
\(643\) 30.1097 1.18741 0.593705 0.804682i \(-0.297664\pi\)
0.593705 + 0.804682i \(0.297664\pi\)
\(644\) 0 0
\(645\) −6.13709 −0.241648
\(646\) 0 0
\(647\) 14.3050 0.562389 0.281195 0.959651i \(-0.409269\pi\)
0.281195 + 0.959651i \(0.409269\pi\)
\(648\) 0 0
\(649\) −36.5055 −1.43297
\(650\) 0 0
\(651\) 6.04453 0.236904
\(652\) 0 0
\(653\) 21.0547 0.823933 0.411966 0.911199i \(-0.364842\pi\)
0.411966 + 0.911199i \(0.364842\pi\)
\(654\) 0 0
\(655\) −4.70637 −0.183893
\(656\) 0 0
\(657\) −9.90115 −0.386281
\(658\) 0 0
\(659\) −4.02197 −0.156674 −0.0783368 0.996927i \(-0.524961\pi\)
−0.0783368 + 0.996927i \(0.524961\pi\)
\(660\) 0 0
\(661\) 19.5974 0.762252 0.381126 0.924523i \(-0.375536\pi\)
0.381126 + 0.924523i \(0.375536\pi\)
\(662\) 0 0
\(663\) −9.25032 −0.359253
\(664\) 0 0
\(665\) −11.8894 −0.461051
\(666\) 0 0
\(667\) 1.86741 0.0723062
\(668\) 0 0
\(669\) −0.343506 −0.0132807
\(670\) 0 0
\(671\) −23.7304 −0.916101
\(672\) 0 0
\(673\) −18.6277 −0.718046 −0.359023 0.933329i \(-0.616890\pi\)
−0.359023 + 0.933329i \(0.616890\pi\)
\(674\) 0 0
\(675\) −4.62184 −0.177895
\(676\) 0 0
\(677\) −19.2425 −0.739549 −0.369774 0.929122i \(-0.620565\pi\)
−0.369774 + 0.929122i \(0.620565\pi\)
\(678\) 0 0
\(679\) −16.2916 −0.625216
\(680\) 0 0
\(681\) 8.12470 0.311339
\(682\) 0 0
\(683\) 36.3752 1.39186 0.695929 0.718111i \(-0.254993\pi\)
0.695929 + 0.718111i \(0.254993\pi\)
\(684\) 0 0
\(685\) 3.24576 0.124014
\(686\) 0 0
\(687\) 5.95619 0.227243
\(688\) 0 0
\(689\) −1.36287 −0.0519211
\(690\) 0 0
\(691\) −31.4677 −1.19709 −0.598543 0.801091i \(-0.704254\pi\)
−0.598543 + 0.801091i \(0.704254\pi\)
\(692\) 0 0
\(693\) −13.1918 −0.501117
\(694\) 0 0
\(695\) −3.40360 −0.129106
\(696\) 0 0
\(697\) −4.77037 −0.180691
\(698\) 0 0
\(699\) 0.870260 0.0329162
\(700\) 0 0
\(701\) −38.1706 −1.44168 −0.720841 0.693100i \(-0.756244\pi\)
−0.720841 + 0.693100i \(0.756244\pi\)
\(702\) 0 0
\(703\) 24.3379 0.917922
\(704\) 0 0
\(705\) 5.09988 0.192072
\(706\) 0 0
\(707\) 23.3547 0.878343
\(708\) 0 0
\(709\) 12.0061 0.450900 0.225450 0.974255i \(-0.427615\pi\)
0.225450 + 0.974255i \(0.427615\pi\)
\(710\) 0 0
\(711\) 10.2406 0.384052
\(712\) 0 0
\(713\) 1.00244 0.0375417
\(714\) 0 0
\(715\) 6.86455 0.256720
\(716\) 0 0
\(717\) 13.0177 0.486155
\(718\) 0 0
\(719\) 18.4358 0.687539 0.343769 0.939054i \(-0.388296\pi\)
0.343769 + 0.939054i \(0.388296\pi\)
\(720\) 0 0
\(721\) 11.6911 0.435399
\(722\) 0 0
\(723\) −2.82934 −0.105224
\(724\) 0 0
\(725\) 15.0269 0.558085
\(726\) 0 0
\(727\) −14.4468 −0.535802 −0.267901 0.963446i \(-0.586330\pi\)
−0.267901 + 0.963446i \(0.586330\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.5010 1.16511
\(732\) 0 0
\(733\) 44.5104 1.64403 0.822014 0.569467i \(-0.192850\pi\)
0.822014 + 0.569467i \(0.192850\pi\)
\(734\) 0 0
\(735\) −3.07126 −0.113285
\(736\) 0 0
\(737\) 12.9939 0.478637
\(738\) 0 0
\(739\) 29.3056 1.07802 0.539011 0.842298i \(-0.318798\pi\)
0.539011 + 0.842298i \(0.318798\pi\)
\(740\) 0 0
\(741\) 16.3602 0.601008
\(742\) 0 0
\(743\) 9.42691 0.345840 0.172920 0.984936i \(-0.444680\pi\)
0.172920 + 0.984936i \(0.444680\pi\)
\(744\) 0 0
\(745\) −8.10520 −0.296951
\(746\) 0 0
\(747\) −8.39859 −0.307288
\(748\) 0 0
\(749\) 23.8236 0.870496
\(750\) 0 0
\(751\) 32.0724 1.17034 0.585170 0.810911i \(-0.301028\pi\)
0.585170 + 0.810911i \(0.301028\pi\)
\(752\) 0 0
\(753\) −16.4580 −0.599763
\(754\) 0 0
\(755\) 1.41047 0.0513321
\(756\) 0 0
\(757\) −17.9811 −0.653533 −0.326767 0.945105i \(-0.605959\pi\)
−0.326767 + 0.945105i \(0.605959\pi\)
\(758\) 0 0
\(759\) −2.18777 −0.0794110
\(760\) 0 0
\(761\) 34.0598 1.23467 0.617333 0.786702i \(-0.288213\pi\)
0.617333 + 0.786702i \(0.288213\pi\)
\(762\) 0 0
\(763\) 42.5859 1.54171
\(764\) 0 0
\(765\) −1.94106 −0.0701793
\(766\) 0 0
\(767\) 28.0865 1.01414
\(768\) 0 0
\(769\) 1.20670 0.0435147 0.0217574 0.999763i \(-0.493074\pi\)
0.0217574 + 0.999763i \(0.493074\pi\)
\(770\) 0 0
\(771\) 24.5773 0.885129
\(772\) 0 0
\(773\) −15.9957 −0.575327 −0.287663 0.957732i \(-0.592878\pi\)
−0.287663 + 0.957732i \(0.592878\pi\)
\(774\) 0 0
\(775\) 8.06658 0.289760
\(776\) 0 0
\(777\) 15.0987 0.541661
\(778\) 0 0
\(779\) 8.43693 0.302284
\(780\) 0 0
\(781\) −32.6938 −1.16988
\(782\) 0 0
\(783\) −3.25128 −0.116191
\(784\) 0 0
\(785\) 8.99070 0.320892
\(786\) 0 0
\(787\) −32.7244 −1.16650 −0.583250 0.812292i \(-0.698219\pi\)
−0.583250 + 0.812292i \(0.698219\pi\)
\(788\) 0 0
\(789\) 24.6483 0.877502
\(790\) 0 0
\(791\) −17.4876 −0.621786
\(792\) 0 0
\(793\) 18.2576 0.648346
\(794\) 0 0
\(795\) −0.285980 −0.0101427
\(796\) 0 0
\(797\) −31.2345 −1.10638 −0.553191 0.833054i \(-0.686590\pi\)
−0.553191 + 0.833054i \(0.686590\pi\)
\(798\) 0 0
\(799\) −26.1771 −0.926080
\(800\) 0 0
\(801\) −7.12992 −0.251923
\(802\) 0 0
\(803\) −37.7140 −1.33090
\(804\) 0 0
\(805\) −1.22324 −0.0431135
\(806\) 0 0
\(807\) −4.52549 −0.159305
\(808\) 0 0
\(809\) −46.8880 −1.64849 −0.824247 0.566230i \(-0.808401\pi\)
−0.824247 + 0.566230i \(0.808401\pi\)
\(810\) 0 0
\(811\) −40.2325 −1.41275 −0.706377 0.707836i \(-0.749672\pi\)
−0.706377 + 0.707836i \(0.749672\pi\)
\(812\) 0 0
\(813\) −22.1048 −0.775248
\(814\) 0 0
\(815\) −9.60288 −0.336374
\(816\) 0 0
\(817\) −55.7130 −1.94915
\(818\) 0 0
\(819\) 10.1495 0.354652
\(820\) 0 0
\(821\) 39.3186 1.37223 0.686115 0.727494i \(-0.259315\pi\)
0.686115 + 0.727494i \(0.259315\pi\)
\(822\) 0 0
\(823\) −4.64345 −0.161860 −0.0809302 0.996720i \(-0.525789\pi\)
−0.0809302 + 0.996720i \(0.525789\pi\)
\(824\) 0 0
\(825\) −17.6048 −0.612922
\(826\) 0 0
\(827\) 19.5624 0.680251 0.340125 0.940380i \(-0.389530\pi\)
0.340125 + 0.940380i \(0.389530\pi\)
\(828\) 0 0
\(829\) 3.01629 0.104760 0.0523801 0.998627i \(-0.483319\pi\)
0.0523801 + 0.998627i \(0.483319\pi\)
\(830\) 0 0
\(831\) −14.4555 −0.501457
\(832\) 0 0
\(833\) 15.7644 0.546205
\(834\) 0 0
\(835\) 0.614948 0.0212812
\(836\) 0 0
\(837\) −1.74532 −0.0603270
\(838\) 0 0
\(839\) −25.4908 −0.880041 −0.440020 0.897988i \(-0.645029\pi\)
−0.440020 + 0.897988i \(0.645029\pi\)
\(840\) 0 0
\(841\) −18.4292 −0.635489
\(842\) 0 0
\(843\) 3.60180 0.124053
\(844\) 0 0
\(845\) 2.71291 0.0933268
\(846\) 0 0
\(847\) −12.1523 −0.417559
\(848\) 0 0
\(849\) −22.0928 −0.758222
\(850\) 0 0
\(851\) 2.50400 0.0858360
\(852\) 0 0
\(853\) 22.8504 0.782382 0.391191 0.920310i \(-0.372063\pi\)
0.391191 + 0.920310i \(0.372063\pi\)
\(854\) 0 0
\(855\) 3.43299 0.117406
\(856\) 0 0
\(857\) 13.9103 0.475167 0.237584 0.971367i \(-0.423645\pi\)
0.237584 + 0.971367i \(0.423645\pi\)
\(858\) 0 0
\(859\) −33.0864 −1.12889 −0.564447 0.825469i \(-0.690910\pi\)
−0.564447 + 0.825469i \(0.690910\pi\)
\(860\) 0 0
\(861\) 5.23407 0.178377
\(862\) 0 0
\(863\) −48.6677 −1.65667 −0.828334 0.560234i \(-0.810711\pi\)
−0.828334 + 0.560234i \(0.810711\pi\)
\(864\) 0 0
\(865\) −8.54002 −0.290369
\(866\) 0 0
\(867\) −7.03673 −0.238980
\(868\) 0 0
\(869\) 39.0069 1.32322
\(870\) 0 0
\(871\) −9.99721 −0.338742
\(872\) 0 0
\(873\) 4.70410 0.159210
\(874\) 0 0
\(875\) −20.4920 −0.692757
\(876\) 0 0
\(877\) 21.5965 0.729264 0.364632 0.931152i \(-0.381195\pi\)
0.364632 + 0.931152i \(0.381195\pi\)
\(878\) 0 0
\(879\) −0.403322 −0.0136037
\(880\) 0 0
\(881\) −1.38305 −0.0465960 −0.0232980 0.999729i \(-0.507417\pi\)
−0.0232980 + 0.999729i \(0.507417\pi\)
\(882\) 0 0
\(883\) −24.0085 −0.807951 −0.403976 0.914770i \(-0.632372\pi\)
−0.403976 + 0.914770i \(0.632372\pi\)
\(884\) 0 0
\(885\) 5.89358 0.198111
\(886\) 0 0
\(887\) −38.6184 −1.29668 −0.648339 0.761352i \(-0.724536\pi\)
−0.648339 + 0.761352i \(0.724536\pi\)
\(888\) 0 0
\(889\) −29.1986 −0.979289
\(890\) 0 0
\(891\) 3.80906 0.127608
\(892\) 0 0
\(893\) 46.2972 1.54928
\(894\) 0 0
\(895\) −1.97759 −0.0661034
\(896\) 0 0
\(897\) 1.68322 0.0562010
\(898\) 0 0
\(899\) 5.67452 0.189256
\(900\) 0 0
\(901\) 1.46791 0.0489030
\(902\) 0 0
\(903\) −34.5630 −1.15019
\(904\) 0 0
\(905\) −4.50870 −0.149874
\(906\) 0 0
\(907\) 33.3349 1.10687 0.553433 0.832893i \(-0.313317\pi\)
0.553433 + 0.832893i \(0.313317\pi\)
\(908\) 0 0
\(909\) −6.74351 −0.223668
\(910\) 0 0
\(911\) −10.7100 −0.354838 −0.177419 0.984135i \(-0.556775\pi\)
−0.177419 + 0.984135i \(0.556775\pi\)
\(912\) 0 0
\(913\) −31.9907 −1.05874
\(914\) 0 0
\(915\) 3.83112 0.126653
\(916\) 0 0
\(917\) −26.5054 −0.875287
\(918\) 0 0
\(919\) −56.1186 −1.85118 −0.925591 0.378525i \(-0.876431\pi\)
−0.925591 + 0.378525i \(0.876431\pi\)
\(920\) 0 0
\(921\) 28.5615 0.941133
\(922\) 0 0
\(923\) 25.1539 0.827950
\(924\) 0 0
\(925\) 20.1495 0.662513
\(926\) 0 0
\(927\) −3.37572 −0.110873
\(928\) 0 0
\(929\) 29.8268 0.978587 0.489293 0.872119i \(-0.337255\pi\)
0.489293 + 0.872119i \(0.337255\pi\)
\(930\) 0 0
\(931\) −27.8812 −0.913768
\(932\) 0 0
\(933\) −14.0740 −0.460761
\(934\) 0 0
\(935\) −7.39361 −0.241797
\(936\) 0 0
\(937\) 23.8975 0.780696 0.390348 0.920667i \(-0.372355\pi\)
0.390348 + 0.920667i \(0.372355\pi\)
\(938\) 0 0
\(939\) 13.3132 0.434458
\(940\) 0 0
\(941\) 15.7999 0.515061 0.257530 0.966270i \(-0.417091\pi\)
0.257530 + 0.966270i \(0.417091\pi\)
\(942\) 0 0
\(943\) 0.868031 0.0282670
\(944\) 0 0
\(945\) 2.12974 0.0692805
\(946\) 0 0
\(947\) −50.5289 −1.64197 −0.820985 0.570950i \(-0.806575\pi\)
−0.820985 + 0.570950i \(0.806575\pi\)
\(948\) 0 0
\(949\) 29.0163 0.941909
\(950\) 0 0
\(951\) −23.0614 −0.747818
\(952\) 0 0
\(953\) −14.2502 −0.461609 −0.230804 0.973000i \(-0.574136\pi\)
−0.230804 + 0.973000i \(0.574136\pi\)
\(954\) 0 0
\(955\) −1.43651 −0.0464844
\(956\) 0 0
\(957\) −12.3843 −0.400328
\(958\) 0 0
\(959\) 18.2795 0.590277
\(960\) 0 0
\(961\) −27.9539 −0.901738
\(962\) 0 0
\(963\) −6.87891 −0.221670
\(964\) 0 0
\(965\) −5.98530 −0.192673
\(966\) 0 0
\(967\) 25.5040 0.820152 0.410076 0.912051i \(-0.365502\pi\)
0.410076 + 0.912051i \(0.365502\pi\)
\(968\) 0 0
\(969\) −17.6211 −0.566073
\(970\) 0 0
\(971\) −6.48205 −0.208019 −0.104009 0.994576i \(-0.533167\pi\)
−0.104009 + 0.994576i \(0.533167\pi\)
\(972\) 0 0
\(973\) −19.1685 −0.614514
\(974\) 0 0
\(975\) 13.5447 0.433779
\(976\) 0 0
\(977\) −17.9008 −0.572698 −0.286349 0.958125i \(-0.592442\pi\)
−0.286349 + 0.958125i \(0.592442\pi\)
\(978\) 0 0
\(979\) −27.1582 −0.867981
\(980\) 0 0
\(981\) −12.2964 −0.392594
\(982\) 0 0
\(983\) −54.4682 −1.73727 −0.868634 0.495455i \(-0.835001\pi\)
−0.868634 + 0.495455i \(0.835001\pi\)
\(984\) 0 0
\(985\) 11.7106 0.373130
\(986\) 0 0
\(987\) 28.7216 0.914220
\(988\) 0 0
\(989\) −5.73202 −0.182268
\(990\) 0 0
\(991\) −39.3981 −1.25152 −0.625760 0.780016i \(-0.715211\pi\)
−0.625760 + 0.780016i \(0.715211\pi\)
\(992\) 0 0
\(993\) 31.1825 0.989546
\(994\) 0 0
\(995\) −3.24069 −0.102737
\(996\) 0 0
\(997\) 14.2166 0.450245 0.225122 0.974331i \(-0.427722\pi\)
0.225122 + 0.974331i \(0.427722\pi\)
\(998\) 0 0
\(999\) −4.35964 −0.137933
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 2004.2.a.b.1.4 5
3.2 odd 2 6012.2.a.f.1.2 5
4.3 odd 2 8016.2.a.q.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.b.1.4 5 1.1 even 1 trivial
6012.2.a.f.1.2 5 3.2 odd 2
8016.2.a.q.1.4 5 4.3 odd 2