Properties

Label 1805.2.a.k.1.3
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
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 1.3
Root \(1.90211\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +O(q^{10})\) \(q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +0.854102 q^{11} -1.45309 q^{12} -0.726543 q^{13} -0.726543 q^{15} +4.00000 q^{16} +3.85410 q^{17} +2.00000 q^{20} +1.17557 q^{21} -3.47214 q^{23} +1.00000 q^{25} -3.97574 q^{27} -3.23607 q^{28} +2.35114 q^{29} -6.88191 q^{31} +0.620541 q^{33} -1.61803 q^{35} +4.94427 q^{36} +7.77997 q^{37} -0.527864 q^{39} -5.98385 q^{41} -9.61803 q^{43} -1.70820 q^{44} +2.47214 q^{45} -8.70820 q^{47} +2.90617 q^{48} -4.38197 q^{49} +2.80017 q^{51} +1.45309 q^{52} -12.5882 q^{53} -0.854102 q^{55} +2.62866 q^{59} +1.45309 q^{60} -2.76393 q^{61} -4.00000 q^{63} -8.00000 q^{64} +0.726543 q^{65} +3.35520 q^{67} -7.70820 q^{68} -2.52265 q^{69} +13.0373 q^{71} -11.0000 q^{73} +0.726543 q^{75} +1.38197 q^{77} -16.2865 q^{79} -4.00000 q^{80} +4.52786 q^{81} -2.70820 q^{83} -2.35114 q^{84} -3.85410 q^{85} +1.70820 q^{87} +13.3148 q^{89} -1.17557 q^{91} +6.94427 q^{92} -5.00000 q^{93} +7.77997 q^{97} -2.11146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0.726543 0.419470 0.209735 0.977758i \(-0.432740\pi\)
0.209735 + 0.977758i \(0.432740\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.61803 0.611559 0.305780 0.952102i \(-0.401083\pi\)
0.305780 + 0.952102i \(0.401083\pi\)
\(8\) 0 0
\(9\) −2.47214 −0.824045
\(10\) 0 0
\(11\) 0.854102 0.257521 0.128761 0.991676i \(-0.458900\pi\)
0.128761 + 0.991676i \(0.458900\pi\)
\(12\) −1.45309 −0.419470
\(13\) −0.726543 −0.201507 −0.100753 0.994911i \(-0.532125\pi\)
−0.100753 + 0.994911i \(0.532125\pi\)
\(14\) 0 0
\(15\) −0.726543 −0.187592
\(16\) 4.00000 1.00000
\(17\) 3.85410 0.934757 0.467379 0.884057i \(-0.345199\pi\)
0.467379 + 0.884057i \(0.345199\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) 1.17557 0.256531
\(22\) 0 0
\(23\) −3.47214 −0.723990 −0.361995 0.932180i \(-0.617904\pi\)
−0.361995 + 0.932180i \(0.617904\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.97574 −0.765131
\(28\) −3.23607 −0.611559
\(29\) 2.35114 0.436596 0.218298 0.975882i \(-0.429950\pi\)
0.218298 + 0.975882i \(0.429950\pi\)
\(30\) 0 0
\(31\) −6.88191 −1.23603 −0.618014 0.786167i \(-0.712062\pi\)
−0.618014 + 0.786167i \(0.712062\pi\)
\(32\) 0 0
\(33\) 0.620541 0.108022
\(34\) 0 0
\(35\) −1.61803 −0.273498
\(36\) 4.94427 0.824045
\(37\) 7.77997 1.27902 0.639509 0.768783i \(-0.279138\pi\)
0.639509 + 0.768783i \(0.279138\pi\)
\(38\) 0 0
\(39\) −0.527864 −0.0845259
\(40\) 0 0
\(41\) −5.98385 −0.934521 −0.467260 0.884120i \(-0.654759\pi\)
−0.467260 + 0.884120i \(0.654759\pi\)
\(42\) 0 0
\(43\) −9.61803 −1.46674 −0.733368 0.679832i \(-0.762053\pi\)
−0.733368 + 0.679832i \(0.762053\pi\)
\(44\) −1.70820 −0.257521
\(45\) 2.47214 0.368524
\(46\) 0 0
\(47\) −8.70820 −1.27022 −0.635111 0.772421i \(-0.719046\pi\)
−0.635111 + 0.772421i \(0.719046\pi\)
\(48\) 2.90617 0.419470
\(49\) −4.38197 −0.625995
\(50\) 0 0
\(51\) 2.80017 0.392102
\(52\) 1.45309 0.201507
\(53\) −12.5882 −1.72913 −0.864564 0.502522i \(-0.832406\pi\)
−0.864564 + 0.502522i \(0.832406\pi\)
\(54\) 0 0
\(55\) −0.854102 −0.115167
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.62866 0.342222 0.171111 0.985252i \(-0.445264\pi\)
0.171111 + 0.985252i \(0.445264\pi\)
\(60\) 1.45309 0.187592
\(61\) −2.76393 −0.353885 −0.176943 0.984221i \(-0.556621\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) −8.00000 −1.00000
\(65\) 0.726543 0.0901165
\(66\) 0 0
\(67\) 3.35520 0.409903 0.204951 0.978772i \(-0.434296\pi\)
0.204951 + 0.978772i \(0.434296\pi\)
\(68\) −7.70820 −0.934757
\(69\) −2.52265 −0.303692
\(70\) 0 0
\(71\) 13.0373 1.54724 0.773620 0.633650i \(-0.218444\pi\)
0.773620 + 0.633650i \(0.218444\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0.726543 0.0838939
\(76\) 0 0
\(77\) 1.38197 0.157490
\(78\) 0 0
\(79\) −16.2865 −1.83237 −0.916186 0.400754i \(-0.868748\pi\)
−0.916186 + 0.400754i \(0.868748\pi\)
\(80\) −4.00000 −0.447214
\(81\) 4.52786 0.503096
\(82\) 0 0
\(83\) −2.70820 −0.297264 −0.148632 0.988893i \(-0.547487\pi\)
−0.148632 + 0.988893i \(0.547487\pi\)
\(84\) −2.35114 −0.256531
\(85\) −3.85410 −0.418036
\(86\) 0 0
\(87\) 1.70820 0.183139
\(88\) 0 0
\(89\) 13.3148 1.41137 0.705683 0.708528i \(-0.250640\pi\)
0.705683 + 0.708528i \(0.250640\pi\)
\(90\) 0 0
\(91\) −1.17557 −0.123233
\(92\) 6.94427 0.723990
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.77997 0.789936 0.394968 0.918695i \(-0.370756\pi\)
0.394968 + 0.918695i \(0.370756\pi\)
\(98\) 0 0
\(99\) −2.11146 −0.212209
\(100\) −2.00000 −0.200000
\(101\) −13.0902 −1.30252 −0.651260 0.758854i \(-0.725759\pi\)
−0.651260 + 0.758854i \(0.725759\pi\)
\(102\) 0 0
\(103\) 10.8576 1.06984 0.534918 0.844904i \(-0.320343\pi\)
0.534918 + 0.844904i \(0.320343\pi\)
\(104\) 0 0
\(105\) −1.17557 −0.114724
\(106\) 0 0
\(107\) −17.2905 −1.67154 −0.835769 0.549081i \(-0.814978\pi\)
−0.835769 + 0.549081i \(0.814978\pi\)
\(108\) 7.95148 0.765131
\(109\) 13.5923 1.30191 0.650953 0.759118i \(-0.274369\pi\)
0.650953 + 0.759118i \(0.274369\pi\)
\(110\) 0 0
\(111\) 5.65248 0.536509
\(112\) 6.47214 0.611559
\(113\) −8.78402 −0.826331 −0.413166 0.910656i \(-0.635577\pi\)
−0.413166 + 0.910656i \(0.635577\pi\)
\(114\) 0 0
\(115\) 3.47214 0.323778
\(116\) −4.70228 −0.436596
\(117\) 1.79611 0.166051
\(118\) 0 0
\(119\) 6.23607 0.571659
\(120\) 0 0
\(121\) −10.2705 −0.933683
\(122\) 0 0
\(123\) −4.34752 −0.392003
\(124\) 13.7638 1.23603
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.0171 −1.59876 −0.799378 0.600828i \(-0.794838\pi\)
−0.799378 + 0.600828i \(0.794838\pi\)
\(128\) 0 0
\(129\) −6.98791 −0.615251
\(130\) 0 0
\(131\) 0.763932 0.0667451 0.0333725 0.999443i \(-0.489375\pi\)
0.0333725 + 0.999443i \(0.489375\pi\)
\(132\) −1.24108 −0.108022
\(133\) 0 0
\(134\) 0 0
\(135\) 3.97574 0.342177
\(136\) 0 0
\(137\) −5.56231 −0.475220 −0.237610 0.971361i \(-0.576364\pi\)
−0.237610 + 0.971361i \(0.576364\pi\)
\(138\) 0 0
\(139\) −2.61803 −0.222059 −0.111029 0.993817i \(-0.535415\pi\)
−0.111029 + 0.993817i \(0.535415\pi\)
\(140\) 3.23607 0.273498
\(141\) −6.32688 −0.532819
\(142\) 0 0
\(143\) −0.620541 −0.0518923
\(144\) −9.88854 −0.824045
\(145\) −2.35114 −0.195252
\(146\) 0 0
\(147\) −3.18368 −0.262586
\(148\) −15.5599 −1.27902
\(149\) −8.23607 −0.674725 −0.337362 0.941375i \(-0.609535\pi\)
−0.337362 + 0.941375i \(0.609535\pi\)
\(150\) 0 0
\(151\) 7.33094 0.596583 0.298292 0.954475i \(-0.403583\pi\)
0.298292 + 0.954475i \(0.403583\pi\)
\(152\) 0 0
\(153\) −9.52786 −0.770282
\(154\) 0 0
\(155\) 6.88191 0.552768
\(156\) 1.05573 0.0845259
\(157\) −14.0344 −1.12007 −0.560035 0.828469i \(-0.689212\pi\)
−0.560035 + 0.828469i \(0.689212\pi\)
\(158\) 0 0
\(159\) −9.14590 −0.725317
\(160\) 0 0
\(161\) −5.61803 −0.442763
\(162\) 0 0
\(163\) −0.909830 −0.0712634 −0.0356317 0.999365i \(-0.511344\pi\)
−0.0356317 + 0.999365i \(0.511344\pi\)
\(164\) 11.9677 0.934521
\(165\) −0.620541 −0.0483091
\(166\) 0 0
\(167\) 17.2905 1.33798 0.668991 0.743271i \(-0.266727\pi\)
0.668991 + 0.743271i \(0.266727\pi\)
\(168\) 0 0
\(169\) −12.4721 −0.959395
\(170\) 0 0
\(171\) 0 0
\(172\) 19.2361 1.46674
\(173\) 6.60440 0.502123 0.251061 0.967971i \(-0.419220\pi\)
0.251061 + 0.967971i \(0.419220\pi\)
\(174\) 0 0
\(175\) 1.61803 0.122312
\(176\) 3.41641 0.257521
\(177\) 1.90983 0.143552
\(178\) 0 0
\(179\) 4.53077 0.338646 0.169323 0.985561i \(-0.445842\pi\)
0.169323 + 0.985561i \(0.445842\pi\)
\(180\) −4.94427 −0.368524
\(181\) 7.43694 0.552783 0.276392 0.961045i \(-0.410861\pi\)
0.276392 + 0.961045i \(0.410861\pi\)
\(182\) 0 0
\(183\) −2.00811 −0.148444
\(184\) 0 0
\(185\) −7.77997 −0.571994
\(186\) 0 0
\(187\) 3.29180 0.240720
\(188\) 17.4164 1.27022
\(189\) −6.43288 −0.467923
\(190\) 0 0
\(191\) −5.09017 −0.368312 −0.184156 0.982897i \(-0.558955\pi\)
−0.184156 + 0.982897i \(0.558955\pi\)
\(192\) −5.81234 −0.419470
\(193\) −3.91023 −0.281464 −0.140732 0.990048i \(-0.544946\pi\)
−0.140732 + 0.990048i \(0.544946\pi\)
\(194\) 0 0
\(195\) 0.527864 0.0378011
\(196\) 8.76393 0.625995
\(197\) 22.6525 1.61392 0.806961 0.590605i \(-0.201111\pi\)
0.806961 + 0.590605i \(0.201111\pi\)
\(198\) 0 0
\(199\) 12.5623 0.890518 0.445259 0.895402i \(-0.353112\pi\)
0.445259 + 0.895402i \(0.353112\pi\)
\(200\) 0 0
\(201\) 2.43769 0.171942
\(202\) 0 0
\(203\) 3.80423 0.267004
\(204\) −5.60034 −0.392102
\(205\) 5.98385 0.417930
\(206\) 0 0
\(207\) 8.58359 0.596601
\(208\) −2.90617 −0.201507
\(209\) 0 0
\(210\) 0 0
\(211\) −2.80017 −0.192772 −0.0963858 0.995344i \(-0.530728\pi\)
−0.0963858 + 0.995344i \(0.530728\pi\)
\(212\) 25.1765 1.72913
\(213\) 9.47214 0.649020
\(214\) 0 0
\(215\) 9.61803 0.655944
\(216\) 0 0
\(217\) −11.1352 −0.755904
\(218\) 0 0
\(219\) −7.99197 −0.540047
\(220\) 1.70820 0.115167
\(221\) −2.80017 −0.188360
\(222\) 0 0
\(223\) −5.15131 −0.344957 −0.172479 0.985013i \(-0.555178\pi\)
−0.172479 + 0.985013i \(0.555178\pi\)
\(224\) 0 0
\(225\) −2.47214 −0.164809
\(226\) 0 0
\(227\) 26.9726 1.79023 0.895117 0.445830i \(-0.147092\pi\)
0.895117 + 0.445830i \(0.147092\pi\)
\(228\) 0 0
\(229\) −4.85410 −0.320768 −0.160384 0.987055i \(-0.551273\pi\)
−0.160384 + 0.987055i \(0.551273\pi\)
\(230\) 0 0
\(231\) 1.00406 0.0660621
\(232\) 0 0
\(233\) 12.9443 0.848007 0.424004 0.905660i \(-0.360624\pi\)
0.424004 + 0.905660i \(0.360624\pi\)
\(234\) 0 0
\(235\) 8.70820 0.568061
\(236\) −5.25731 −0.342222
\(237\) −11.8328 −0.768624
\(238\) 0 0
\(239\) 3.94427 0.255134 0.127567 0.991830i \(-0.459283\pi\)
0.127567 + 0.991830i \(0.459283\pi\)
\(240\) −2.90617 −0.187592
\(241\) −20.3682 −1.31203 −0.656016 0.754747i \(-0.727760\pi\)
−0.656016 + 0.754747i \(0.727760\pi\)
\(242\) 0 0
\(243\) 15.2169 0.976165
\(244\) 5.52786 0.353885
\(245\) 4.38197 0.279954
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.96763 −0.124693
\(250\) 0 0
\(251\) 6.14590 0.387926 0.193963 0.981009i \(-0.437866\pi\)
0.193963 + 0.981009i \(0.437866\pi\)
\(252\) 8.00000 0.503953
\(253\) −2.96556 −0.186443
\(254\) 0 0
\(255\) −2.80017 −0.175353
\(256\) 16.0000 1.00000
\(257\) −29.6013 −1.84648 −0.923238 0.384228i \(-0.874468\pi\)
−0.923238 + 0.384228i \(0.874468\pi\)
\(258\) 0 0
\(259\) 12.5882 0.782196
\(260\) −1.45309 −0.0901165
\(261\) −5.81234 −0.359775
\(262\) 0 0
\(263\) 10.9098 0.672729 0.336364 0.941732i \(-0.390803\pi\)
0.336364 + 0.941732i \(0.390803\pi\)
\(264\) 0 0
\(265\) 12.5882 0.773290
\(266\) 0 0
\(267\) 9.67376 0.592025
\(268\) −6.71040 −0.409903
\(269\) −19.0866 −1.16373 −0.581867 0.813284i \(-0.697677\pi\)
−0.581867 + 0.813284i \(0.697677\pi\)
\(270\) 0 0
\(271\) −18.4164 −1.11872 −0.559359 0.828926i \(-0.688952\pi\)
−0.559359 + 0.828926i \(0.688952\pi\)
\(272\) 15.4164 0.934757
\(273\) −0.854102 −0.0516926
\(274\) 0 0
\(275\) 0.854102 0.0515043
\(276\) 5.04531 0.303692
\(277\) 25.3607 1.52378 0.761888 0.647709i \(-0.224273\pi\)
0.761888 + 0.647709i \(0.224273\pi\)
\(278\) 0 0
\(279\) 17.0130 1.01854
\(280\) 0 0
\(281\) 4.42477 0.263959 0.131980 0.991252i \(-0.457867\pi\)
0.131980 + 0.991252i \(0.457867\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −26.0746 −1.54724
\(285\) 0 0
\(286\) 0 0
\(287\) −9.68208 −0.571515
\(288\) 0 0
\(289\) −2.14590 −0.126229
\(290\) 0 0
\(291\) 5.65248 0.331354
\(292\) 22.0000 1.28745
\(293\) 6.04937 0.353408 0.176704 0.984264i \(-0.443457\pi\)
0.176704 + 0.984264i \(0.443457\pi\)
\(294\) 0 0
\(295\) −2.62866 −0.153046
\(296\) 0 0
\(297\) −3.39569 −0.197038
\(298\) 0 0
\(299\) 2.52265 0.145889
\(300\) −1.45309 −0.0838939
\(301\) −15.5623 −0.896996
\(302\) 0 0
\(303\) −9.51057 −0.546368
\(304\) 0 0
\(305\) 2.76393 0.158262
\(306\) 0 0
\(307\) 24.0009 1.36981 0.684903 0.728635i \(-0.259845\pi\)
0.684903 + 0.728635i \(0.259845\pi\)
\(308\) −2.76393 −0.157490
\(309\) 7.88854 0.448764
\(310\) 0 0
\(311\) 2.03444 0.115363 0.0576813 0.998335i \(-0.481629\pi\)
0.0576813 + 0.998335i \(0.481629\pi\)
\(312\) 0 0
\(313\) 29.3262 1.65762 0.828808 0.559532i \(-0.189019\pi\)
0.828808 + 0.559532i \(0.189019\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 32.5729 1.83237
\(317\) −8.12299 −0.456233 −0.228116 0.973634i \(-0.573257\pi\)
−0.228116 + 0.973634i \(0.573257\pi\)
\(318\) 0 0
\(319\) 2.00811 0.112433
\(320\) 8.00000 0.447214
\(321\) −12.5623 −0.701160
\(322\) 0 0
\(323\) 0 0
\(324\) −9.05573 −0.503096
\(325\) −0.726543 −0.0403013
\(326\) 0 0
\(327\) 9.87539 0.546110
\(328\) 0 0
\(329\) −14.0902 −0.776816
\(330\) 0 0
\(331\) 17.1845 0.944547 0.472274 0.881452i \(-0.343433\pi\)
0.472274 + 0.881452i \(0.343433\pi\)
\(332\) 5.41641 0.297264
\(333\) −19.2331 −1.05397
\(334\) 0 0
\(335\) −3.35520 −0.183314
\(336\) 4.70228 0.256531
\(337\) −13.0373 −0.710186 −0.355093 0.934831i \(-0.615551\pi\)
−0.355093 + 0.934831i \(0.615551\pi\)
\(338\) 0 0
\(339\) −6.38197 −0.346621
\(340\) 7.70820 0.418036
\(341\) −5.87785 −0.318304
\(342\) 0 0
\(343\) −18.4164 −0.994393
\(344\) 0 0
\(345\) 2.52265 0.135815
\(346\) 0 0
\(347\) −11.2918 −0.606175 −0.303088 0.952963i \(-0.598018\pi\)
−0.303088 + 0.952963i \(0.598018\pi\)
\(348\) −3.41641 −0.183139
\(349\) −30.1246 −1.61253 −0.806267 0.591552i \(-0.798515\pi\)
−0.806267 + 0.591552i \(0.798515\pi\)
\(350\) 0 0
\(351\) 2.88854 0.154179
\(352\) 0 0
\(353\) 17.6180 0.937713 0.468857 0.883274i \(-0.344666\pi\)
0.468857 + 0.883274i \(0.344666\pi\)
\(354\) 0 0
\(355\) −13.0373 −0.691947
\(356\) −26.6296 −1.41137
\(357\) 4.53077 0.239794
\(358\) 0 0
\(359\) −26.5623 −1.40190 −0.700952 0.713208i \(-0.747241\pi\)
−0.700952 + 0.713208i \(0.747241\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −7.46196 −0.391651
\(364\) 2.35114 0.123233
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) 17.6525 0.921452 0.460726 0.887542i \(-0.347589\pi\)
0.460726 + 0.887542i \(0.347589\pi\)
\(368\) −13.8885 −0.723990
\(369\) 14.7929 0.770088
\(370\) 0 0
\(371\) −20.3682 −1.05746
\(372\) 10.0000 0.518476
\(373\) 30.9888 1.60454 0.802271 0.596961i \(-0.203625\pi\)
0.802271 + 0.596961i \(0.203625\pi\)
\(374\) 0 0
\(375\) −0.726543 −0.0375185
\(376\) 0 0
\(377\) −1.70820 −0.0879770
\(378\) 0 0
\(379\) −7.26543 −0.373200 −0.186600 0.982436i \(-0.559747\pi\)
−0.186600 + 0.982436i \(0.559747\pi\)
\(380\) 0 0
\(381\) −13.0902 −0.670630
\(382\) 0 0
\(383\) −20.9888 −1.07248 −0.536238 0.844067i \(-0.680155\pi\)
−0.536238 + 0.844067i \(0.680155\pi\)
\(384\) 0 0
\(385\) −1.38197 −0.0704315
\(386\) 0 0
\(387\) 23.7771 1.20866
\(388\) −15.5599 −0.789936
\(389\) 27.5623 1.39746 0.698732 0.715383i \(-0.253748\pi\)
0.698732 + 0.715383i \(0.253748\pi\)
\(390\) 0 0
\(391\) −13.3820 −0.676755
\(392\) 0 0
\(393\) 0.555029 0.0279975
\(394\) 0 0
\(395\) 16.2865 0.819461
\(396\) 4.22291 0.212209
\(397\) −7.52786 −0.377813 −0.188906 0.981995i \(-0.560494\pi\)
−0.188906 + 0.981995i \(0.560494\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 20.8172 1.03956 0.519782 0.854299i \(-0.326013\pi\)
0.519782 + 0.854299i \(0.326013\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 26.1803 1.30252
\(405\) −4.52786 −0.224991
\(406\) 0 0
\(407\) 6.64488 0.329375
\(408\) 0 0
\(409\) 2.62866 0.129979 0.0649893 0.997886i \(-0.479299\pi\)
0.0649893 + 0.997886i \(0.479299\pi\)
\(410\) 0 0
\(411\) −4.04125 −0.199340
\(412\) −21.7153 −1.06984
\(413\) 4.25325 0.209289
\(414\) 0 0
\(415\) 2.70820 0.132941
\(416\) 0 0
\(417\) −1.90211 −0.0931469
\(418\) 0 0
\(419\) −36.7082 −1.79331 −0.896657 0.442727i \(-0.854011\pi\)
−0.896657 + 0.442727i \(0.854011\pi\)
\(420\) 2.35114 0.114724
\(421\) 2.52265 0.122947 0.0614733 0.998109i \(-0.480420\pi\)
0.0614733 + 0.998109i \(0.480420\pi\)
\(422\) 0 0
\(423\) 21.5279 1.04672
\(424\) 0 0
\(425\) 3.85410 0.186951
\(426\) 0 0
\(427\) −4.47214 −0.216422
\(428\) 34.5811 1.67154
\(429\) −0.450850 −0.0217672
\(430\) 0 0
\(431\) 30.6053 1.47421 0.737103 0.675780i \(-0.236193\pi\)
0.737103 + 0.675780i \(0.236193\pi\)
\(432\) −15.9030 −0.765131
\(433\) 22.5478 1.08358 0.541790 0.840514i \(-0.317747\pi\)
0.541790 + 0.840514i \(0.317747\pi\)
\(434\) 0 0
\(435\) −1.70820 −0.0819021
\(436\) −27.1846 −1.30191
\(437\) 0 0
\(438\) 0 0
\(439\) −28.5972 −1.36487 −0.682435 0.730946i \(-0.739079\pi\)
−0.682435 + 0.730946i \(0.739079\pi\)
\(440\) 0 0
\(441\) 10.8328 0.515848
\(442\) 0 0
\(443\) 0.0557281 0.00264772 0.00132386 0.999999i \(-0.499579\pi\)
0.00132386 + 0.999999i \(0.499579\pi\)
\(444\) −11.3050 −0.536509
\(445\) −13.3148 −0.631182
\(446\) 0 0
\(447\) −5.98385 −0.283027
\(448\) −12.9443 −0.611559
\(449\) 39.1118 1.84580 0.922901 0.385038i \(-0.125812\pi\)
0.922901 + 0.385038i \(0.125812\pi\)
\(450\) 0 0
\(451\) −5.11082 −0.240659
\(452\) 17.5680 0.826331
\(453\) 5.32624 0.250248
\(454\) 0 0
\(455\) 1.17557 0.0551116
\(456\) 0 0
\(457\) 18.6525 0.872526 0.436263 0.899819i \(-0.356302\pi\)
0.436263 + 0.899819i \(0.356302\pi\)
\(458\) 0 0
\(459\) −15.3229 −0.715212
\(460\) −6.94427 −0.323778
\(461\) 20.4164 0.950887 0.475443 0.879746i \(-0.342288\pi\)
0.475443 + 0.879746i \(0.342288\pi\)
\(462\) 0 0
\(463\) −30.7082 −1.42713 −0.713566 0.700588i \(-0.752921\pi\)
−0.713566 + 0.700588i \(0.752921\pi\)
\(464\) 9.40456 0.436596
\(465\) 5.00000 0.231869
\(466\) 0 0
\(467\) −6.94427 −0.321343 −0.160671 0.987008i \(-0.551366\pi\)
−0.160671 + 0.987008i \(0.551366\pi\)
\(468\) −3.59222 −0.166051
\(469\) 5.42882 0.250680
\(470\) 0 0
\(471\) −10.1966 −0.469835
\(472\) 0 0
\(473\) −8.21478 −0.377716
\(474\) 0 0
\(475\) 0 0
\(476\) −12.4721 −0.571659
\(477\) 31.1199 1.42488
\(478\) 0 0
\(479\) 31.8541 1.45545 0.727726 0.685868i \(-0.240577\pi\)
0.727726 + 0.685868i \(0.240577\pi\)
\(480\) 0 0
\(481\) −5.65248 −0.257731
\(482\) 0 0
\(483\) −4.08174 −0.185726
\(484\) 20.5410 0.933683
\(485\) −7.77997 −0.353270
\(486\) 0 0
\(487\) 1.79611 0.0813896 0.0406948 0.999172i \(-0.487043\pi\)
0.0406948 + 0.999172i \(0.487043\pi\)
\(488\) 0 0
\(489\) −0.661030 −0.0298928
\(490\) 0 0
\(491\) −28.3820 −1.28086 −0.640430 0.768016i \(-0.721244\pi\)
−0.640430 + 0.768016i \(0.721244\pi\)
\(492\) 8.69505 0.392003
\(493\) 9.06154 0.408111
\(494\) 0 0
\(495\) 2.11146 0.0949029
\(496\) −27.5276 −1.23603
\(497\) 21.0948 0.946229
\(498\) 0 0
\(499\) −24.1246 −1.07997 −0.539983 0.841676i \(-0.681569\pi\)
−0.539983 + 0.841676i \(0.681569\pi\)
\(500\) 2.00000 0.0894427
\(501\) 12.5623 0.561242
\(502\) 0 0
\(503\) 31.8541 1.42030 0.710152 0.704048i \(-0.248626\pi\)
0.710152 + 0.704048i \(0.248626\pi\)
\(504\) 0 0
\(505\) 13.0902 0.582505
\(506\) 0 0
\(507\) −9.06154 −0.402437
\(508\) 36.0341 1.59876
\(509\) −32.1239 −1.42387 −0.711934 0.702247i \(-0.752180\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(510\) 0 0
\(511\) −17.7984 −0.787354
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.8576 −0.478445
\(516\) 13.9758 0.615251
\(517\) −7.43769 −0.327109
\(518\) 0 0
\(519\) 4.79837 0.210625
\(520\) 0 0
\(521\) 4.76779 0.208881 0.104440 0.994531i \(-0.466695\pi\)
0.104440 + 0.994531i \(0.466695\pi\)
\(522\) 0 0
\(523\) −11.9677 −0.523311 −0.261656 0.965161i \(-0.584268\pi\)
−0.261656 + 0.965161i \(0.584268\pi\)
\(524\) −1.52786 −0.0667451
\(525\) 1.17557 0.0513061
\(526\) 0 0
\(527\) −26.5236 −1.15539
\(528\) 2.48217 0.108022
\(529\) −10.9443 −0.475838
\(530\) 0 0
\(531\) −6.49839 −0.282006
\(532\) 0 0
\(533\) 4.34752 0.188312
\(534\) 0 0
\(535\) 17.2905 0.747535
\(536\) 0 0
\(537\) 3.29180 0.142051
\(538\) 0 0
\(539\) −3.74265 −0.161207
\(540\) −7.95148 −0.342177
\(541\) −11.0557 −0.475323 −0.237661 0.971348i \(-0.576381\pi\)
−0.237661 + 0.971348i \(0.576381\pi\)
\(542\) 0 0
\(543\) 5.40325 0.231876
\(544\) 0 0
\(545\) −13.5923 −0.582230
\(546\) 0 0
\(547\) 9.61657 0.411175 0.205587 0.978639i \(-0.434090\pi\)
0.205587 + 0.978639i \(0.434090\pi\)
\(548\) 11.1246 0.475220
\(549\) 6.83282 0.291617
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −26.3521 −1.12060
\(554\) 0 0
\(555\) −5.65248 −0.239934
\(556\) 5.23607 0.222059
\(557\) 10.3607 0.438996 0.219498 0.975613i \(-0.429558\pi\)
0.219498 + 0.975613i \(0.429558\pi\)
\(558\) 0 0
\(559\) 6.98791 0.295557
\(560\) −6.47214 −0.273498
\(561\) 2.39163 0.100975
\(562\) 0 0
\(563\) 5.98385 0.252189 0.126095 0.992018i \(-0.459756\pi\)
0.126095 + 0.992018i \(0.459756\pi\)
\(564\) 12.6538 0.532819
\(565\) 8.78402 0.369547
\(566\) 0 0
\(567\) 7.32624 0.307673
\(568\) 0 0
\(569\) −17.3560 −0.727603 −0.363802 0.931476i \(-0.618521\pi\)
−0.363802 + 0.931476i \(0.618521\pi\)
\(570\) 0 0
\(571\) 28.5410 1.19440 0.597202 0.802091i \(-0.296279\pi\)
0.597202 + 0.802091i \(0.296279\pi\)
\(572\) 1.24108 0.0518923
\(573\) −3.69822 −0.154496
\(574\) 0 0
\(575\) −3.47214 −0.144798
\(576\) 19.7771 0.824045
\(577\) 28.1246 1.17084 0.585421 0.810729i \(-0.300929\pi\)
0.585421 + 0.810729i \(0.300929\pi\)
\(578\) 0 0
\(579\) −2.84095 −0.118066
\(580\) 4.70228 0.195252
\(581\) −4.38197 −0.181795
\(582\) 0 0
\(583\) −10.7516 −0.445288
\(584\) 0 0
\(585\) −1.79611 −0.0742601
\(586\) 0 0
\(587\) 15.8885 0.655790 0.327895 0.944714i \(-0.393661\pi\)
0.327895 + 0.944714i \(0.393661\pi\)
\(588\) 6.36737 0.262586
\(589\) 0 0
\(590\) 0 0
\(591\) 16.4580 0.676991
\(592\) 31.1199 1.27902
\(593\) 27.2148 1.11758 0.558789 0.829310i \(-0.311266\pi\)
0.558789 + 0.829310i \(0.311266\pi\)
\(594\) 0 0
\(595\) −6.23607 −0.255654
\(596\) 16.4721 0.674725
\(597\) 9.12705 0.373545
\(598\) 0 0
\(599\) 1.55909 0.0637025 0.0318513 0.999493i \(-0.489860\pi\)
0.0318513 + 0.999493i \(0.489860\pi\)
\(600\) 0 0
\(601\) 1.24108 0.0506248 0.0253124 0.999680i \(-0.491942\pi\)
0.0253124 + 0.999680i \(0.491942\pi\)
\(602\) 0 0
\(603\) −8.29451 −0.337778
\(604\) −14.6619 −0.596583
\(605\) 10.2705 0.417556
\(606\) 0 0
\(607\) 26.2866 1.06694 0.533469 0.845819i \(-0.320888\pi\)
0.533469 + 0.845819i \(0.320888\pi\)
\(608\) 0 0
\(609\) 2.76393 0.112000
\(610\) 0 0
\(611\) 6.32688 0.255958
\(612\) 19.0557 0.770282
\(613\) 10.2705 0.414822 0.207411 0.978254i \(-0.433496\pi\)
0.207411 + 0.978254i \(0.433496\pi\)
\(614\) 0 0
\(615\) 4.34752 0.175309
\(616\) 0 0
\(617\) −34.8885 −1.40456 −0.702280 0.711901i \(-0.747835\pi\)
−0.702280 + 0.711901i \(0.747835\pi\)
\(618\) 0 0
\(619\) −32.0902 −1.28981 −0.644906 0.764262i \(-0.723104\pi\)
−0.644906 + 0.764262i \(0.723104\pi\)
\(620\) −13.7638 −0.552768
\(621\) 13.8043 0.553948
\(622\) 0 0
\(623\) 21.5438 0.863134
\(624\) −2.11146 −0.0845259
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 28.0689 1.12007
\(629\) 29.9848 1.19557
\(630\) 0 0
\(631\) −15.5279 −0.618155 −0.309077 0.951037i \(-0.600020\pi\)
−0.309077 + 0.951037i \(0.600020\pi\)
\(632\) 0 0
\(633\) −2.03444 −0.0808618
\(634\) 0 0
\(635\) 18.0171 0.714986
\(636\) 18.2918 0.725317
\(637\) 3.18368 0.126142
\(638\) 0 0
\(639\) −32.2299 −1.27500
\(640\) 0 0
\(641\) 14.3188 0.565561 0.282780 0.959185i \(-0.408743\pi\)
0.282780 + 0.959185i \(0.408743\pi\)
\(642\) 0 0
\(643\) −2.94427 −0.116111 −0.0580554 0.998313i \(-0.518490\pi\)
−0.0580554 + 0.998313i \(0.518490\pi\)
\(644\) 11.2361 0.442763
\(645\) 6.98791 0.275149
\(646\) 0 0
\(647\) 0.639320 0.0251343 0.0125671 0.999921i \(-0.496000\pi\)
0.0125671 + 0.999921i \(0.496000\pi\)
\(648\) 0 0
\(649\) 2.24514 0.0881294
\(650\) 0 0
\(651\) −8.09017 −0.317079
\(652\) 1.81966 0.0712634
\(653\) 6.32624 0.247565 0.123782 0.992309i \(-0.460498\pi\)
0.123782 + 0.992309i \(0.460498\pi\)
\(654\) 0 0
\(655\) −0.763932 −0.0298493
\(656\) −23.9354 −0.934521
\(657\) 27.1935 1.06092
\(658\) 0 0
\(659\) −5.36331 −0.208925 −0.104462 0.994529i \(-0.533312\pi\)
−0.104462 + 0.994529i \(0.533312\pi\)
\(660\) 1.24108 0.0483091
\(661\) −11.2412 −0.437231 −0.218615 0.975811i \(-0.570154\pi\)
−0.218615 + 0.975811i \(0.570154\pi\)
\(662\) 0 0
\(663\) −2.03444 −0.0790112
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.16348 −0.316091
\(668\) −34.5811 −1.33798
\(669\) −3.74265 −0.144699
\(670\) 0 0
\(671\) −2.36068 −0.0911330
\(672\) 0 0
\(673\) −37.2097 −1.43433 −0.717165 0.696904i \(-0.754560\pi\)
−0.717165 + 0.696904i \(0.754560\pi\)
\(674\) 0 0
\(675\) −3.97574 −0.153026
\(676\) 24.9443 0.959395
\(677\) −44.8182 −1.72250 −0.861251 0.508180i \(-0.830319\pi\)
−0.861251 + 0.508180i \(0.830319\pi\)
\(678\) 0 0
\(679\) 12.5882 0.483093
\(680\) 0 0
\(681\) 19.5967 0.750949
\(682\) 0 0
\(683\) 3.42071 0.130890 0.0654449 0.997856i \(-0.479153\pi\)
0.0654449 + 0.997856i \(0.479153\pi\)
\(684\) 0 0
\(685\) 5.56231 0.212525
\(686\) 0 0
\(687\) −3.52671 −0.134552
\(688\) −38.4721 −1.46674
\(689\) 9.14590 0.348431
\(690\) 0 0
\(691\) −11.0557 −0.420580 −0.210290 0.977639i \(-0.567441\pi\)
−0.210290 + 0.977639i \(0.567441\pi\)
\(692\) −13.2088 −0.502123
\(693\) −3.41641 −0.129779
\(694\) 0 0
\(695\) 2.61803 0.0993077
\(696\) 0 0
\(697\) −23.0624 −0.873550
\(698\) 0 0
\(699\) 9.40456 0.355713
\(700\) −3.23607 −0.122312
\(701\) 25.0000 0.944237 0.472118 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −6.83282 −0.257521
\(705\) 6.32688 0.238284
\(706\) 0 0
\(707\) −21.1803 −0.796569
\(708\) −3.81966 −0.143552
\(709\) −34.4721 −1.29463 −0.647314 0.762223i \(-0.724108\pi\)
−0.647314 + 0.762223i \(0.724108\pi\)
\(710\) 0 0
\(711\) 40.2624 1.50996
\(712\) 0 0
\(713\) 23.8949 0.894872
\(714\) 0 0
\(715\) 0.620541 0.0232069
\(716\) −9.06154 −0.338646
\(717\) 2.86568 0.107021
\(718\) 0 0
\(719\) 3.76393 0.140371 0.0701855 0.997534i \(-0.477641\pi\)
0.0701855 + 0.997534i \(0.477641\pi\)
\(720\) 9.88854 0.368524
\(721\) 17.5680 0.654268
\(722\) 0 0
\(723\) −14.7984 −0.550357
\(724\) −14.8739 −0.552783
\(725\) 2.35114 0.0873192
\(726\) 0 0
\(727\) −37.0689 −1.37481 −0.687404 0.726275i \(-0.741250\pi\)
−0.687404 + 0.726275i \(0.741250\pi\)
\(728\) 0 0
\(729\) −2.52786 −0.0936246
\(730\) 0 0
\(731\) −37.0689 −1.37104
\(732\) 4.01623 0.148444
\(733\) −1.20163 −0.0443831 −0.0221915 0.999754i \(-0.507064\pi\)
−0.0221915 + 0.999754i \(0.507064\pi\)
\(734\) 0 0
\(735\) 3.18368 0.117432
\(736\) 0 0
\(737\) 2.86568 0.105559
\(738\) 0 0
\(739\) −11.5836 −0.426109 −0.213055 0.977040i \(-0.568341\pi\)
−0.213055 + 0.977040i \(0.568341\pi\)
\(740\) 15.5599 0.571994
\(741\) 0 0
\(742\) 0 0
\(743\) 36.6952 1.34622 0.673108 0.739544i \(-0.264959\pi\)
0.673108 + 0.739544i \(0.264959\pi\)
\(744\) 0 0
\(745\) 8.23607 0.301746
\(746\) 0 0
\(747\) 6.69505 0.244959
\(748\) −6.58359 −0.240720
\(749\) −27.9767 −1.02225
\(750\) 0 0
\(751\) 27.6992 1.01076 0.505378 0.862898i \(-0.331353\pi\)
0.505378 + 0.862898i \(0.331353\pi\)
\(752\) −34.8328 −1.27022
\(753\) 4.46526 0.162723
\(754\) 0 0
\(755\) −7.33094 −0.266800
\(756\) 12.8658 0.467923
\(757\) −18.7771 −0.682465 −0.341232 0.939979i \(-0.610844\pi\)
−0.341232 + 0.939979i \(0.610844\pi\)
\(758\) 0 0
\(759\) −2.15460 −0.0782072
\(760\) 0 0
\(761\) 19.0689 0.691246 0.345623 0.938373i \(-0.387668\pi\)
0.345623 + 0.938373i \(0.387668\pi\)
\(762\) 0 0
\(763\) 21.9928 0.796193
\(764\) 10.1803 0.368312
\(765\) 9.52786 0.344481
\(766\) 0 0
\(767\) −1.90983 −0.0689600
\(768\) 11.6247 0.419470
\(769\) −26.3607 −0.950590 −0.475295 0.879826i \(-0.657659\pi\)
−0.475295 + 0.879826i \(0.657659\pi\)
\(770\) 0 0
\(771\) −21.5066 −0.774540
\(772\) 7.82045 0.281464
\(773\) 7.95148 0.285995 0.142997 0.989723i \(-0.454326\pi\)
0.142997 + 0.989723i \(0.454326\pi\)
\(774\) 0 0
\(775\) −6.88191 −0.247205
\(776\) 0 0
\(777\) 9.14590 0.328107
\(778\) 0 0
\(779\) 0 0
\(780\) −1.05573 −0.0378011
\(781\) 11.1352 0.398447
\(782\) 0 0
\(783\) −9.34752 −0.334053
\(784\) −17.5279 −0.625995
\(785\) 14.0344 0.500911
\(786\) 0 0
\(787\) −44.1976 −1.57548 −0.787738 0.616011i \(-0.788748\pi\)
−0.787738 + 0.616011i \(0.788748\pi\)
\(788\) −45.3050 −1.61392
\(789\) 7.92646 0.282189
\(790\) 0 0
\(791\) −14.2128 −0.505351
\(792\) 0 0
\(793\) 2.00811 0.0713102
\(794\) 0 0
\(795\) 9.14590 0.324372
\(796\) −25.1246 −0.890518
\(797\) 15.9434 0.564746 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(798\) 0 0
\(799\) −33.5623 −1.18735
\(800\) 0 0
\(801\) −32.9160 −1.16303
\(802\) 0 0
\(803\) −9.39512 −0.331547
\(804\) −4.87539 −0.171942
\(805\) 5.61803 0.198010
\(806\) 0 0
\(807\) −13.8673 −0.488151
\(808\) 0 0
\(809\) 37.7639 1.32771 0.663855 0.747862i \(-0.268919\pi\)
0.663855 + 0.747862i \(0.268919\pi\)
\(810\) 0 0
\(811\) −9.16754 −0.321916 −0.160958 0.986961i \(-0.551458\pi\)
−0.160958 + 0.986961i \(0.551458\pi\)
\(812\) −7.60845 −0.267004
\(813\) −13.3803 −0.469268
\(814\) 0 0
\(815\) 0.909830 0.0318700
\(816\) 11.2007 0.392102
\(817\) 0 0
\(818\) 0 0
\(819\) 2.90617 0.101550
\(820\) −11.9677 −0.417930
\(821\) −31.5967 −1.10273 −0.551367 0.834263i \(-0.685894\pi\)
−0.551367 + 0.834263i \(0.685894\pi\)
\(822\) 0 0
\(823\) −16.8885 −0.588698 −0.294349 0.955698i \(-0.595103\pi\)
−0.294349 + 0.955698i \(0.595103\pi\)
\(824\) 0 0
\(825\) 0.620541 0.0216045
\(826\) 0 0
\(827\) −10.7921 −0.375279 −0.187640 0.982238i \(-0.560084\pi\)
−0.187640 + 0.982238i \(0.560084\pi\)
\(828\) −17.1672 −0.596601
\(829\) 11.9272 0.414249 0.207125 0.978315i \(-0.433589\pi\)
0.207125 + 0.978315i \(0.433589\pi\)
\(830\) 0 0
\(831\) 18.4256 0.639177
\(832\) 5.81234 0.201507
\(833\) −16.8885 −0.585153
\(834\) 0 0
\(835\) −17.2905 −0.598363
\(836\) 0 0
\(837\) 27.3607 0.945723
\(838\) 0 0
\(839\) −18.8496 −0.650761 −0.325381 0.945583i \(-0.605492\pi\)
−0.325381 + 0.945583i \(0.605492\pi\)
\(840\) 0 0
\(841\) −23.4721 −0.809384
\(842\) 0 0
\(843\) 3.21478 0.110723
\(844\) 5.60034 0.192772
\(845\) 12.4721 0.429055
\(846\) 0 0
\(847\) −16.6180 −0.571002
\(848\) −50.3530 −1.72913
\(849\) −17.4370 −0.598437
\(850\) 0 0
\(851\) −27.0131 −0.925997
\(852\) −18.9443 −0.649020
\(853\) 3.27051 0.111980 0.0559901 0.998431i \(-0.482168\pi\)
0.0559901 + 0.998431i \(0.482168\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.237026 −0.00809664 −0.00404832 0.999992i \(-0.501289\pi\)
−0.00404832 + 0.999992i \(0.501289\pi\)
\(858\) 0 0
\(859\) −46.8328 −1.59792 −0.798958 0.601387i \(-0.794615\pi\)
−0.798958 + 0.601387i \(0.794615\pi\)
\(860\) −19.2361 −0.655944
\(861\) −7.03444 −0.239733
\(862\) 0 0
\(863\) 44.3691 1.51034 0.755172 0.655527i \(-0.227554\pi\)
0.755172 + 0.655527i \(0.227554\pi\)
\(864\) 0 0
\(865\) −6.60440 −0.224556
\(866\) 0 0
\(867\) −1.55909 −0.0529493
\(868\) 22.2703 0.755904
\(869\) −13.9103 −0.471875
\(870\) 0 0
\(871\) −2.43769 −0.0825981
\(872\) 0 0
\(873\) −19.2331 −0.650943
\(874\) 0 0
\(875\) −1.61803 −0.0546995
\(876\) 15.9839 0.540047
\(877\) −29.4953 −0.995984 −0.497992 0.867182i \(-0.665929\pi\)
−0.497992 + 0.867182i \(0.665929\pi\)
\(878\) 0 0
\(879\) 4.39512 0.148244
\(880\) −3.41641 −0.115167
\(881\) −19.5967 −0.660231 −0.330116 0.943941i \(-0.607088\pi\)
−0.330116 + 0.943941i \(0.607088\pi\)
\(882\) 0 0
\(883\) −20.9098 −0.703672 −0.351836 0.936062i \(-0.614442\pi\)
−0.351836 + 0.936062i \(0.614442\pi\)
\(884\) 5.60034 0.188360
\(885\) −1.90983 −0.0641982
\(886\) 0 0
\(887\) 49.4800 1.66137 0.830687 0.556739i \(-0.187948\pi\)
0.830687 + 0.556739i \(0.187948\pi\)
\(888\) 0 0
\(889\) −29.1522 −0.977735
\(890\) 0 0
\(891\) 3.86726 0.129558
\(892\) 10.3026 0.344957
\(893\) 0 0
\(894\) 0 0
\(895\) −4.53077 −0.151447
\(896\) 0 0
\(897\) 1.83282 0.0611959
\(898\) 0 0
\(899\) −16.1803 −0.539645
\(900\) 4.94427 0.164809
\(901\) −48.5164 −1.61632
\(902\) 0 0
\(903\) −11.3067 −0.376263
\(904\) 0 0
\(905\) −7.43694 −0.247212
\(906\) 0 0
\(907\) 51.4881 1.70963 0.854817 0.518930i \(-0.173669\pi\)
0.854817 + 0.518930i \(0.173669\pi\)
\(908\) −53.9452 −1.79023
\(909\) 32.3607 1.07334
\(910\) 0 0
\(911\) −1.69011 −0.0559959 −0.0279979 0.999608i \(-0.508913\pi\)
−0.0279979 + 0.999608i \(0.508913\pi\)
\(912\) 0 0
\(913\) −2.31308 −0.0765519
\(914\) 0 0
\(915\) 2.00811 0.0663862
\(916\) 9.70820 0.320768
\(917\) 1.23607 0.0408186
\(918\) 0 0
\(919\) −39.5967 −1.30618 −0.653088 0.757282i \(-0.726527\pi\)
−0.653088 + 0.757282i \(0.726527\pi\)
\(920\) 0 0
\(921\) 17.4377 0.574592
\(922\) 0 0
\(923\) −9.47214 −0.311779
\(924\) −2.00811 −0.0660621
\(925\) 7.77997 0.255804
\(926\) 0 0
\(927\) −26.8416 −0.881593
\(928\) 0 0
\(929\) 43.5410 1.42853 0.714267 0.699873i \(-0.246760\pi\)
0.714267 + 0.699873i \(0.246760\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −25.8885 −0.848007
\(933\) 1.47811 0.0483911
\(934\) 0 0
\(935\) −3.29180 −0.107653
\(936\) 0 0
\(937\) −39.5623 −1.29244 −0.646222 0.763149i \(-0.723652\pi\)
−0.646222 + 0.763149i \(0.723652\pi\)
\(938\) 0 0
\(939\) 21.3068 0.695320
\(940\) −17.4164 −0.568061
\(941\) 49.7980 1.62337 0.811684 0.584097i \(-0.198551\pi\)
0.811684 + 0.584097i \(0.198551\pi\)
\(942\) 0 0
\(943\) 20.7768 0.676584
\(944\) 10.5146 0.342222
\(945\) 6.43288 0.209262
\(946\) 0 0
\(947\) 49.1033 1.59564 0.797822 0.602893i \(-0.205986\pi\)
0.797822 + 0.602893i \(0.205986\pi\)
\(948\) 23.6656 0.768624
\(949\) 7.99197 0.259430
\(950\) 0 0
\(951\) −5.90170 −0.191376
\(952\) 0 0
\(953\) −30.2218 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(954\) 0 0
\(955\) 5.09017 0.164714
\(956\) −7.88854 −0.255134
\(957\) 1.45898 0.0471621
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 5.81234 0.187592
\(961\) 16.3607 0.527764
\(962\) 0 0
\(963\) 42.7445 1.37742
\(964\) 40.7364 1.31203
\(965\) 3.91023 0.125875
\(966\) 0 0
\(967\) 16.0689 0.516740 0.258370 0.966046i \(-0.416815\pi\)
0.258370 + 0.966046i \(0.416815\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −58.1985 −1.86768 −0.933839 0.357694i \(-0.883563\pi\)
−0.933839 + 0.357694i \(0.883563\pi\)
\(972\) −30.4338 −0.976165
\(973\) −4.23607 −0.135802
\(974\) 0 0
\(975\) −0.527864 −0.0169052
\(976\) −11.0557 −0.353885
\(977\) 19.1271 0.611931 0.305966 0.952043i \(-0.401021\pi\)
0.305966 + 0.952043i \(0.401021\pi\)
\(978\) 0 0
\(979\) 11.3722 0.363457
\(980\) −8.76393 −0.279954
\(981\) −33.6020 −1.07283
\(982\) 0 0
\(983\) −24.6215 −0.785303 −0.392651 0.919687i \(-0.628442\pi\)
−0.392651 + 0.919687i \(0.628442\pi\)
\(984\) 0 0
\(985\) −22.6525 −0.721768
\(986\) 0 0
\(987\) −10.2371 −0.325851
\(988\) 0 0
\(989\) 33.3951 1.06190
\(990\) 0 0
\(991\) −28.3602 −0.900891 −0.450445 0.892804i \(-0.648735\pi\)
−0.450445 + 0.892804i \(0.648735\pi\)
\(992\) 0 0
\(993\) 12.4853 0.396209
\(994\) 0 0
\(995\) −12.5623 −0.398252
\(996\) 3.93525 0.124693
\(997\) 4.76393 0.150875 0.0754376 0.997151i \(-0.475965\pi\)
0.0754376 + 0.997151i \(0.475965\pi\)
\(998\) 0 0
\(999\) −30.9311 −0.978617
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 1805.2.a.k.1.3 yes 4
5.4 even 2 9025.2.a.bi.1.2 4
19.18 odd 2 inner 1805.2.a.k.1.2 4
95.94 odd 2 9025.2.a.bi.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.k.1.2 4 19.18 odd 2 inner
1805.2.a.k.1.3 yes 4 1.1 even 1 trivial
9025.2.a.bi.1.2 4 5.4 even 2
9025.2.a.bi.1.3 4 95.94 odd 2