Properties

Label 4010.2.a.i.1.8
Level 4010
Weight 2
Character 4010.1
Self dual yes
Analytic conductor 32.020
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.06458\)
Character \(\chi\) = 4010.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.696154 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.696154 q^{6} +2.65990 q^{7} -1.00000 q^{8} -2.51537 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.696154 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.696154 q^{6} +2.65990 q^{7} -1.00000 q^{8} -2.51537 q^{9} -1.00000 q^{10} -2.85414 q^{11} +0.696154 q^{12} +0.0375050 q^{13} -2.65990 q^{14} +0.696154 q^{15} +1.00000 q^{16} -1.02960 q^{17} +2.51537 q^{18} -7.38755 q^{19} +1.00000 q^{20} +1.85170 q^{21} +2.85414 q^{22} -1.48829 q^{23} -0.696154 q^{24} +1.00000 q^{25} -0.0375050 q^{26} -3.83955 q^{27} +2.65990 q^{28} +9.65453 q^{29} -0.696154 q^{30} +4.28085 q^{31} -1.00000 q^{32} -1.98692 q^{33} +1.02960 q^{34} +2.65990 q^{35} -2.51537 q^{36} -1.90641 q^{37} +7.38755 q^{38} +0.0261093 q^{39} -1.00000 q^{40} +8.37519 q^{41} -1.85170 q^{42} -6.07292 q^{43} -2.85414 q^{44} -2.51537 q^{45} +1.48829 q^{46} -3.89157 q^{47} +0.696154 q^{48} +0.0750867 q^{49} -1.00000 q^{50} -0.716757 q^{51} +0.0375050 q^{52} -9.19937 q^{53} +3.83955 q^{54} -2.85414 q^{55} -2.65990 q^{56} -5.14287 q^{57} -9.65453 q^{58} -10.3821 q^{59} +0.696154 q^{60} -14.5560 q^{61} -4.28085 q^{62} -6.69064 q^{63} +1.00000 q^{64} +0.0375050 q^{65} +1.98692 q^{66} -3.35202 q^{67} -1.02960 q^{68} -1.03608 q^{69} -2.65990 q^{70} +12.4861 q^{71} +2.51537 q^{72} +6.72519 q^{73} +1.90641 q^{74} +0.696154 q^{75} -7.38755 q^{76} -7.59174 q^{77} -0.0261093 q^{78} -6.67283 q^{79} +1.00000 q^{80} +4.87319 q^{81} -8.37519 q^{82} -7.88459 q^{83} +1.85170 q^{84} -1.02960 q^{85} +6.07292 q^{86} +6.72104 q^{87} +2.85414 q^{88} -5.63871 q^{89} +2.51537 q^{90} +0.0997598 q^{91} -1.48829 q^{92} +2.98013 q^{93} +3.89157 q^{94} -7.38755 q^{95} -0.696154 q^{96} +0.00211457 q^{97} -0.0750867 q^{98} +7.17922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} + O(q^{10}) \) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} - 10q^{10} - 11q^{11} - 4q^{12} + 6q^{13} + 3q^{14} - 4q^{15} + 10q^{16} + 9q^{17} - 6q^{18} - 13q^{19} + 10q^{20} - 24q^{21} + 11q^{22} - 3q^{23} + 4q^{24} + 10q^{25} - 6q^{26} - 10q^{27} - 3q^{28} - 4q^{29} + 4q^{30} - 17q^{31} - 10q^{32} - 2q^{33} - 9q^{34} - 3q^{35} + 6q^{36} - 15q^{37} + 13q^{38} - 6q^{39} - 10q^{40} - 11q^{41} + 24q^{42} - 11q^{43} - 11q^{44} + 6q^{45} + 3q^{46} + 3q^{47} - 4q^{48} - 5q^{49} - 10q^{50} - 21q^{51} + 6q^{52} + 25q^{53} + 10q^{54} - 11q^{55} + 3q^{56} + 31q^{57} + 4q^{58} - 46q^{59} - 4q^{60} - 54q^{61} + 17q^{62} - 6q^{63} + 10q^{64} + 6q^{65} + 2q^{66} - 26q^{67} + 9q^{68} - 9q^{69} + 3q^{70} - 16q^{71} - 6q^{72} + 4q^{73} + 15q^{74} - 4q^{75} - 13q^{76} + 11q^{77} + 6q^{78} - 19q^{79} + 10q^{80} - 6q^{81} + 11q^{82} + 19q^{83} - 24q^{84} + 9q^{85} + 11q^{86} + 28q^{87} + 11q^{88} - 30q^{89} - 6q^{90} - 38q^{91} - 3q^{92} - 18q^{93} - 3q^{94} - 13q^{95} + 4q^{96} - 16q^{97} + 5q^{98} - 59q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.696154 0.401925 0.200962 0.979599i \(-0.435593\pi\)
0.200962 + 0.979599i \(0.435593\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.696154 −0.284204
\(7\) 2.65990 1.00535 0.502675 0.864476i \(-0.332349\pi\)
0.502675 + 0.864476i \(0.332349\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.51537 −0.838456
\(10\) −1.00000 −0.316228
\(11\) −2.85414 −0.860556 −0.430278 0.902697i \(-0.641584\pi\)
−0.430278 + 0.902697i \(0.641584\pi\)
\(12\) 0.696154 0.200962
\(13\) 0.0375050 0.0104020 0.00520101 0.999986i \(-0.498344\pi\)
0.00520101 + 0.999986i \(0.498344\pi\)
\(14\) −2.65990 −0.710889
\(15\) 0.696154 0.179746
\(16\) 1.00000 0.250000
\(17\) −1.02960 −0.249713 −0.124857 0.992175i \(-0.539847\pi\)
−0.124857 + 0.992175i \(0.539847\pi\)
\(18\) 2.51537 0.592878
\(19\) −7.38755 −1.69482 −0.847410 0.530939i \(-0.821839\pi\)
−0.847410 + 0.530939i \(0.821839\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.85170 0.404075
\(22\) 2.85414 0.608505
\(23\) −1.48829 −0.310331 −0.155165 0.987889i \(-0.549591\pi\)
−0.155165 + 0.987889i \(0.549591\pi\)
\(24\) −0.696154 −0.142102
\(25\) 1.00000 0.200000
\(26\) −0.0375050 −0.00735534
\(27\) −3.83955 −0.738921
\(28\) 2.65990 0.502675
\(29\) 9.65453 1.79280 0.896401 0.443245i \(-0.146173\pi\)
0.896401 + 0.443245i \(0.146173\pi\)
\(30\) −0.696154 −0.127100
\(31\) 4.28085 0.768863 0.384431 0.923154i \(-0.374398\pi\)
0.384431 + 0.923154i \(0.374398\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.98692 −0.345879
\(34\) 1.02960 0.176574
\(35\) 2.65990 0.449606
\(36\) −2.51537 −0.419228
\(37\) −1.90641 −0.313412 −0.156706 0.987645i \(-0.550087\pi\)
−0.156706 + 0.987645i \(0.550087\pi\)
\(38\) 7.38755 1.19842
\(39\) 0.0261093 0.00418083
\(40\) −1.00000 −0.158114
\(41\) 8.37519 1.30798 0.653992 0.756501i \(-0.273093\pi\)
0.653992 + 0.756501i \(0.273093\pi\)
\(42\) −1.85170 −0.285724
\(43\) −6.07292 −0.926111 −0.463055 0.886329i \(-0.653247\pi\)
−0.463055 + 0.886329i \(0.653247\pi\)
\(44\) −2.85414 −0.430278
\(45\) −2.51537 −0.374969
\(46\) 1.48829 0.219437
\(47\) −3.89157 −0.567644 −0.283822 0.958877i \(-0.591602\pi\)
−0.283822 + 0.958877i \(0.591602\pi\)
\(48\) 0.696154 0.100481
\(49\) 0.0750867 0.0107267
\(50\) −1.00000 −0.141421
\(51\) −0.716757 −0.100366
\(52\) 0.0375050 0.00520101
\(53\) −9.19937 −1.26363 −0.631815 0.775119i \(-0.717690\pi\)
−0.631815 + 0.775119i \(0.717690\pi\)
\(54\) 3.83955 0.522496
\(55\) −2.85414 −0.384852
\(56\) −2.65990 −0.355445
\(57\) −5.14287 −0.681190
\(58\) −9.65453 −1.26770
\(59\) −10.3821 −1.35164 −0.675820 0.737067i \(-0.736210\pi\)
−0.675820 + 0.737067i \(0.736210\pi\)
\(60\) 0.696154 0.0898731
\(61\) −14.5560 −1.86371 −0.931854 0.362834i \(-0.881809\pi\)
−0.931854 + 0.362834i \(0.881809\pi\)
\(62\) −4.28085 −0.543668
\(63\) −6.69064 −0.842941
\(64\) 1.00000 0.125000
\(65\) 0.0375050 0.00465193
\(66\) 1.98692 0.244573
\(67\) −3.35202 −0.409515 −0.204757 0.978813i \(-0.565641\pi\)
−0.204757 + 0.978813i \(0.565641\pi\)
\(68\) −1.02960 −0.124857
\(69\) −1.03608 −0.124730
\(70\) −2.65990 −0.317919
\(71\) 12.4861 1.48182 0.740911 0.671603i \(-0.234394\pi\)
0.740911 + 0.671603i \(0.234394\pi\)
\(72\) 2.51537 0.296439
\(73\) 6.72519 0.787124 0.393562 0.919298i \(-0.371243\pi\)
0.393562 + 0.919298i \(0.371243\pi\)
\(74\) 1.90641 0.221616
\(75\) 0.696154 0.0803850
\(76\) −7.38755 −0.847410
\(77\) −7.59174 −0.865159
\(78\) −0.0261093 −0.00295629
\(79\) −6.67283 −0.750752 −0.375376 0.926873i \(-0.622486\pi\)
−0.375376 + 0.926873i \(0.622486\pi\)
\(80\) 1.00000 0.111803
\(81\) 4.87319 0.541466
\(82\) −8.37519 −0.924885
\(83\) −7.88459 −0.865447 −0.432723 0.901527i \(-0.642447\pi\)
−0.432723 + 0.901527i \(0.642447\pi\)
\(84\) 1.85170 0.202037
\(85\) −1.02960 −0.111675
\(86\) 6.07292 0.654859
\(87\) 6.72104 0.720571
\(88\) 2.85414 0.304252
\(89\) −5.63871 −0.597702 −0.298851 0.954300i \(-0.596603\pi\)
−0.298851 + 0.954300i \(0.596603\pi\)
\(90\) 2.51537 0.265143
\(91\) 0.0997598 0.0104577
\(92\) −1.48829 −0.155165
\(93\) 2.98013 0.309025
\(94\) 3.89157 0.401385
\(95\) −7.38755 −0.757947
\(96\) −0.696154 −0.0710509
\(97\) 0.00211457 0.000214702 0 0.000107351 1.00000i \(-0.499966\pi\)
0.000107351 1.00000i \(0.499966\pi\)
\(98\) −0.0750867 −0.00758490
\(99\) 7.17922 0.721538
\(100\) 1.00000 0.100000
\(101\) −9.15058 −0.910517 −0.455258 0.890359i \(-0.650453\pi\)
−0.455258 + 0.890359i \(0.650453\pi\)
\(102\) 0.716757 0.0709695
\(103\) −3.39406 −0.334427 −0.167213 0.985921i \(-0.553477\pi\)
−0.167213 + 0.985921i \(0.553477\pi\)
\(104\) −0.0375050 −0.00367767
\(105\) 1.85170 0.180708
\(106\) 9.19937 0.893522
\(107\) −17.1126 −1.65434 −0.827171 0.561950i \(-0.810051\pi\)
−0.827171 + 0.561950i \(0.810051\pi\)
\(108\) −3.83955 −0.369461
\(109\) 6.07832 0.582198 0.291099 0.956693i \(-0.405979\pi\)
0.291099 + 0.956693i \(0.405979\pi\)
\(110\) 2.85414 0.272132
\(111\) −1.32716 −0.125968
\(112\) 2.65990 0.251337
\(113\) 8.57048 0.806243 0.403122 0.915146i \(-0.367925\pi\)
0.403122 + 0.915146i \(0.367925\pi\)
\(114\) 5.14287 0.481674
\(115\) −1.48829 −0.138784
\(116\) 9.65453 0.896401
\(117\) −0.0943390 −0.00872165
\(118\) 10.3821 0.955753
\(119\) −2.73862 −0.251049
\(120\) −0.696154 −0.0635499
\(121\) −2.85389 −0.259444
\(122\) 14.5560 1.31784
\(123\) 5.83042 0.525711
\(124\) 4.28085 0.384431
\(125\) 1.00000 0.0894427
\(126\) 6.69064 0.596050
\(127\) −4.07183 −0.361317 −0.180658 0.983546i \(-0.557823\pi\)
−0.180658 + 0.983546i \(0.557823\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.22769 −0.372227
\(130\) −0.0375050 −0.00328941
\(131\) 1.30068 0.113641 0.0568206 0.998384i \(-0.481904\pi\)
0.0568206 + 0.998384i \(0.481904\pi\)
\(132\) −1.98692 −0.172939
\(133\) −19.6502 −1.70389
\(134\) 3.35202 0.289571
\(135\) −3.83955 −0.330456
\(136\) 1.02960 0.0882871
\(137\) 17.9925 1.53720 0.768601 0.639728i \(-0.220953\pi\)
0.768601 + 0.639728i \(0.220953\pi\)
\(138\) 1.03608 0.0881971
\(139\) 4.30048 0.364762 0.182381 0.983228i \(-0.441620\pi\)
0.182381 + 0.983228i \(0.441620\pi\)
\(140\) 2.65990 0.224803
\(141\) −2.70913 −0.228150
\(142\) −12.4861 −1.04781
\(143\) −0.107045 −0.00895152
\(144\) −2.51537 −0.209614
\(145\) 9.65453 0.801765
\(146\) −6.72519 −0.556581
\(147\) 0.0522719 0.00431131
\(148\) −1.90641 −0.156706
\(149\) −11.2429 −0.921058 −0.460529 0.887645i \(-0.652340\pi\)
−0.460529 + 0.887645i \(0.652340\pi\)
\(150\) −0.696154 −0.0568407
\(151\) −18.0929 −1.47238 −0.736189 0.676776i \(-0.763377\pi\)
−0.736189 + 0.676776i \(0.763377\pi\)
\(152\) 7.38755 0.599209
\(153\) 2.58981 0.209374
\(154\) 7.59174 0.611760
\(155\) 4.28085 0.343846
\(156\) 0.0261093 0.00209042
\(157\) −22.1158 −1.76503 −0.882517 0.470281i \(-0.844153\pi\)
−0.882517 + 0.470281i \(0.844153\pi\)
\(158\) 6.67283 0.530862
\(159\) −6.40418 −0.507884
\(160\) −1.00000 −0.0790569
\(161\) −3.95872 −0.311991
\(162\) −4.87319 −0.382874
\(163\) −18.2881 −1.43243 −0.716216 0.697879i \(-0.754127\pi\)
−0.716216 + 0.697879i \(0.754127\pi\)
\(164\) 8.37519 0.653992
\(165\) −1.98692 −0.154682
\(166\) 7.88459 0.611963
\(167\) 17.8347 1.38009 0.690044 0.723768i \(-0.257591\pi\)
0.690044 + 0.723768i \(0.257591\pi\)
\(168\) −1.85170 −0.142862
\(169\) −12.9986 −0.999892
\(170\) 1.02960 0.0789663
\(171\) 18.5824 1.42103
\(172\) −6.07292 −0.463055
\(173\) 13.3119 1.01208 0.506042 0.862509i \(-0.331108\pi\)
0.506042 + 0.862509i \(0.331108\pi\)
\(174\) −6.72104 −0.509521
\(175\) 2.65990 0.201070
\(176\) −2.85414 −0.215139
\(177\) −7.22757 −0.543257
\(178\) 5.63871 0.422639
\(179\) 11.0871 0.828688 0.414344 0.910120i \(-0.364011\pi\)
0.414344 + 0.910120i \(0.364011\pi\)
\(180\) −2.51537 −0.187485
\(181\) −8.93318 −0.663998 −0.331999 0.943280i \(-0.607723\pi\)
−0.331999 + 0.943280i \(0.607723\pi\)
\(182\) −0.0997598 −0.00739469
\(183\) −10.1332 −0.749070
\(184\) 1.48829 0.109718
\(185\) −1.90641 −0.140162
\(186\) −2.98013 −0.218514
\(187\) 2.93861 0.214892
\(188\) −3.89157 −0.283822
\(189\) −10.2128 −0.742874
\(190\) 7.38755 0.535949
\(191\) 6.21680 0.449832 0.224916 0.974378i \(-0.427789\pi\)
0.224916 + 0.974378i \(0.427789\pi\)
\(192\) 0.696154 0.0502406
\(193\) 20.8831 1.50320 0.751598 0.659621i \(-0.229283\pi\)
0.751598 + 0.659621i \(0.229283\pi\)
\(194\) −0.00211457 −0.000151817 0
\(195\) 0.0261093 0.00186972
\(196\) 0.0750867 0.00536333
\(197\) 6.64050 0.473116 0.236558 0.971617i \(-0.423981\pi\)
0.236558 + 0.971617i \(0.423981\pi\)
\(198\) −7.17922 −0.510205
\(199\) −13.3555 −0.946748 −0.473374 0.880861i \(-0.656964\pi\)
−0.473374 + 0.880861i \(0.656964\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.33352 −0.164594
\(202\) 9.15058 0.643833
\(203\) 25.6801 1.80239
\(204\) −0.716757 −0.0501830
\(205\) 8.37519 0.584948
\(206\) 3.39406 0.236475
\(207\) 3.74361 0.260199
\(208\) 0.0375050 0.00260051
\(209\) 21.0851 1.45849
\(210\) −1.85170 −0.127780
\(211\) −25.4796 −1.75409 −0.877045 0.480408i \(-0.840489\pi\)
−0.877045 + 0.480408i \(0.840489\pi\)
\(212\) −9.19937 −0.631815
\(213\) 8.69222 0.595581
\(214\) 17.1126 1.16980
\(215\) −6.07292 −0.414169
\(216\) 3.83955 0.261248
\(217\) 11.3866 0.772976
\(218\) −6.07832 −0.411676
\(219\) 4.68177 0.316365
\(220\) −2.85414 −0.192426
\(221\) −0.0386150 −0.00259753
\(222\) 1.32716 0.0890728
\(223\) 17.6475 1.18177 0.590883 0.806757i \(-0.298780\pi\)
0.590883 + 0.806757i \(0.298780\pi\)
\(224\) −2.65990 −0.177722
\(225\) −2.51537 −0.167691
\(226\) −8.57048 −0.570100
\(227\) 17.6554 1.17183 0.585916 0.810372i \(-0.300735\pi\)
0.585916 + 0.810372i \(0.300735\pi\)
\(228\) −5.14287 −0.340595
\(229\) −17.2580 −1.14044 −0.570219 0.821493i \(-0.693142\pi\)
−0.570219 + 0.821493i \(0.693142\pi\)
\(230\) 1.48829 0.0981352
\(231\) −5.28502 −0.347729
\(232\) −9.65453 −0.633851
\(233\) −17.3621 −1.13743 −0.568715 0.822535i \(-0.692559\pi\)
−0.568715 + 0.822535i \(0.692559\pi\)
\(234\) 0.0943390 0.00616713
\(235\) −3.89157 −0.253858
\(236\) −10.3821 −0.675820
\(237\) −4.64532 −0.301746
\(238\) 2.73862 0.177519
\(239\) 12.6335 0.817193 0.408596 0.912715i \(-0.366018\pi\)
0.408596 + 0.912715i \(0.366018\pi\)
\(240\) 0.696154 0.0449366
\(241\) −5.73290 −0.369288 −0.184644 0.982805i \(-0.559113\pi\)
−0.184644 + 0.982805i \(0.559113\pi\)
\(242\) 2.85389 0.183455
\(243\) 14.9111 0.956550
\(244\) −14.5560 −0.931854
\(245\) 0.0750867 0.00479711
\(246\) −5.83042 −0.371734
\(247\) −0.277070 −0.0176296
\(248\) −4.28085 −0.271834
\(249\) −5.48889 −0.347844
\(250\) −1.00000 −0.0632456
\(251\) 21.4087 1.35130 0.675652 0.737220i \(-0.263862\pi\)
0.675652 + 0.737220i \(0.263862\pi\)
\(252\) −6.69064 −0.421471
\(253\) 4.24780 0.267057
\(254\) 4.07183 0.255490
\(255\) −0.716757 −0.0448851
\(256\) 1.00000 0.0625000
\(257\) 20.1176 1.25490 0.627449 0.778658i \(-0.284099\pi\)
0.627449 + 0.778658i \(0.284099\pi\)
\(258\) 4.22769 0.263204
\(259\) −5.07087 −0.315088
\(260\) 0.0375050 0.00232596
\(261\) −24.2847 −1.50319
\(262\) −1.30068 −0.0803564
\(263\) −27.0762 −1.66959 −0.834795 0.550561i \(-0.814414\pi\)
−0.834795 + 0.550561i \(0.814414\pi\)
\(264\) 1.98692 0.122287
\(265\) −9.19937 −0.565113
\(266\) 19.6502 1.20483
\(267\) −3.92541 −0.240231
\(268\) −3.35202 −0.204757
\(269\) −3.84789 −0.234610 −0.117305 0.993096i \(-0.537425\pi\)
−0.117305 + 0.993096i \(0.537425\pi\)
\(270\) 3.83955 0.233667
\(271\) 0.630614 0.0383071 0.0191535 0.999817i \(-0.493903\pi\)
0.0191535 + 0.999817i \(0.493903\pi\)
\(272\) −1.02960 −0.0624284
\(273\) 0.0694482 0.00420319
\(274\) −17.9925 −1.08697
\(275\) −2.85414 −0.172111
\(276\) −1.03608 −0.0623648
\(277\) 18.0685 1.08563 0.542816 0.839852i \(-0.317358\pi\)
0.542816 + 0.839852i \(0.317358\pi\)
\(278\) −4.30048 −0.257926
\(279\) −10.7679 −0.644658
\(280\) −2.65990 −0.158960
\(281\) −16.3568 −0.975766 −0.487883 0.872909i \(-0.662231\pi\)
−0.487883 + 0.872909i \(0.662231\pi\)
\(282\) 2.70913 0.161326
\(283\) −1.20167 −0.0714321 −0.0357160 0.999362i \(-0.511371\pi\)
−0.0357160 + 0.999362i \(0.511371\pi\)
\(284\) 12.4861 0.740911
\(285\) −5.14287 −0.304638
\(286\) 0.107045 0.00632968
\(287\) 22.2772 1.31498
\(288\) 2.51537 0.148220
\(289\) −15.9399 −0.937643
\(290\) −9.65453 −0.566934
\(291\) 0.00147207 8.62941e−5 0
\(292\) 6.72519 0.393562
\(293\) −17.6326 −1.03011 −0.515054 0.857158i \(-0.672228\pi\)
−0.515054 + 0.857158i \(0.672228\pi\)
\(294\) −0.0522719 −0.00304856
\(295\) −10.3821 −0.604471
\(296\) 1.90641 0.110808
\(297\) 10.9586 0.635883
\(298\) 11.2429 0.651286
\(299\) −0.0558185 −0.00322807
\(300\) 0.696154 0.0401925
\(301\) −16.1534 −0.931065
\(302\) 18.0929 1.04113
\(303\) −6.37021 −0.365959
\(304\) −7.38755 −0.423705
\(305\) −14.5560 −0.833475
\(306\) −2.58981 −0.148050
\(307\) 29.5550 1.68679 0.843397 0.537291i \(-0.180552\pi\)
0.843397 + 0.537291i \(0.180552\pi\)
\(308\) −7.59174 −0.432579
\(309\) −2.36279 −0.134414
\(310\) −4.28085 −0.243136
\(311\) 13.4662 0.763597 0.381799 0.924246i \(-0.375305\pi\)
0.381799 + 0.924246i \(0.375305\pi\)
\(312\) −0.0261093 −0.00147815
\(313\) 18.4655 1.04373 0.521865 0.853028i \(-0.325236\pi\)
0.521865 + 0.853028i \(0.325236\pi\)
\(314\) 22.1158 1.24807
\(315\) −6.69064 −0.376975
\(316\) −6.67283 −0.375376
\(317\) 0.377432 0.0211987 0.0105993 0.999944i \(-0.496626\pi\)
0.0105993 + 0.999944i \(0.496626\pi\)
\(318\) 6.40418 0.359129
\(319\) −27.5554 −1.54281
\(320\) 1.00000 0.0559017
\(321\) −11.9130 −0.664921
\(322\) 3.95872 0.220611
\(323\) 7.60619 0.423220
\(324\) 4.87319 0.270733
\(325\) 0.0375050 0.00208041
\(326\) 18.2881 1.01288
\(327\) 4.23145 0.234000
\(328\) −8.37519 −0.462442
\(329\) −10.3512 −0.570680
\(330\) 1.98692 0.109376
\(331\) 9.69431 0.532848 0.266424 0.963856i \(-0.414158\pi\)
0.266424 + 0.963856i \(0.414158\pi\)
\(332\) −7.88459 −0.432723
\(333\) 4.79533 0.262782
\(334\) −17.8347 −0.975869
\(335\) −3.35202 −0.183141
\(336\) 1.85170 0.101019
\(337\) 9.62228 0.524159 0.262079 0.965046i \(-0.415592\pi\)
0.262079 + 0.965046i \(0.415592\pi\)
\(338\) 12.9986 0.707030
\(339\) 5.96638 0.324049
\(340\) −1.02960 −0.0558376
\(341\) −12.2181 −0.661649
\(342\) −18.5824 −1.00482
\(343\) −18.4196 −0.994565
\(344\) 6.07292 0.327430
\(345\) −1.03608 −0.0557808
\(346\) −13.3119 −0.715651
\(347\) −34.3809 −1.84566 −0.922831 0.385204i \(-0.874131\pi\)
−0.922831 + 0.385204i \(0.874131\pi\)
\(348\) 6.72104 0.360286
\(349\) −26.4606 −1.41641 −0.708203 0.706009i \(-0.750494\pi\)
−0.708203 + 0.706009i \(0.750494\pi\)
\(350\) −2.65990 −0.142178
\(351\) −0.144002 −0.00768628
\(352\) 2.85414 0.152126
\(353\) −22.8742 −1.21747 −0.608734 0.793374i \(-0.708323\pi\)
−0.608734 + 0.793374i \(0.708323\pi\)
\(354\) 7.22757 0.384141
\(355\) 12.4861 0.662691
\(356\) −5.63871 −0.298851
\(357\) −1.90650 −0.100903
\(358\) −11.0871 −0.585971
\(359\) −12.7326 −0.672000 −0.336000 0.941862i \(-0.609074\pi\)
−0.336000 + 0.941862i \(0.609074\pi\)
\(360\) 2.51537 0.132572
\(361\) 35.5759 1.87242
\(362\) 8.93318 0.469518
\(363\) −1.98674 −0.104277
\(364\) 0.0997598 0.00522883
\(365\) 6.72519 0.352013
\(366\) 10.1332 0.529673
\(367\) 6.86945 0.358582 0.179291 0.983796i \(-0.442620\pi\)
0.179291 + 0.983796i \(0.442620\pi\)
\(368\) −1.48829 −0.0775827
\(369\) −21.0667 −1.09669
\(370\) 1.90641 0.0991095
\(371\) −24.4694 −1.27039
\(372\) 2.98013 0.154513
\(373\) 29.5706 1.53111 0.765554 0.643372i \(-0.222465\pi\)
0.765554 + 0.643372i \(0.222465\pi\)
\(374\) −2.93861 −0.151952
\(375\) 0.696154 0.0359492
\(376\) 3.89157 0.200692
\(377\) 0.362094 0.0186488
\(378\) 10.2128 0.525291
\(379\) 26.8611 1.37976 0.689881 0.723922i \(-0.257663\pi\)
0.689881 + 0.723922i \(0.257663\pi\)
\(380\) −7.38755 −0.378973
\(381\) −2.83462 −0.145222
\(382\) −6.21680 −0.318079
\(383\) −15.9093 −0.812925 −0.406463 0.913667i \(-0.633238\pi\)
−0.406463 + 0.913667i \(0.633238\pi\)
\(384\) −0.696154 −0.0355255
\(385\) −7.59174 −0.386911
\(386\) −20.8831 −1.06292
\(387\) 15.2756 0.776504
\(388\) 0.00211457 0.000107351 0
\(389\) −21.0610 −1.06783 −0.533916 0.845537i \(-0.679280\pi\)
−0.533916 + 0.845537i \(0.679280\pi\)
\(390\) −0.0261093 −0.00132210
\(391\) 1.53234 0.0774938
\(392\) −0.0750867 −0.00379245
\(393\) 0.905475 0.0456752
\(394\) −6.64050 −0.334543
\(395\) −6.67283 −0.335746
\(396\) 7.17922 0.360769
\(397\) 22.2540 1.11690 0.558448 0.829540i \(-0.311397\pi\)
0.558448 + 0.829540i \(0.311397\pi\)
\(398\) 13.3555 0.669452
\(399\) −13.6795 −0.684834
\(400\) 1.00000 0.0500000
\(401\) 1.00000 0.0499376
\(402\) 2.33352 0.116386
\(403\) 0.160553 0.00799773
\(404\) −9.15058 −0.455258
\(405\) 4.87319 0.242151
\(406\) −25.6801 −1.27448
\(407\) 5.44116 0.269708
\(408\) 0.716757 0.0354848
\(409\) −23.7118 −1.17247 −0.586235 0.810141i \(-0.699391\pi\)
−0.586235 + 0.810141i \(0.699391\pi\)
\(410\) −8.37519 −0.413621
\(411\) 12.5256 0.617840
\(412\) −3.39406 −0.167213
\(413\) −27.6155 −1.35887
\(414\) −3.74361 −0.183988
\(415\) −7.88459 −0.387040
\(416\) −0.0375050 −0.00183884
\(417\) 2.99380 0.146607
\(418\) −21.0851 −1.03131
\(419\) −33.8273 −1.65257 −0.826287 0.563250i \(-0.809551\pi\)
−0.826287 + 0.563250i \(0.809551\pi\)
\(420\) 1.85170 0.0903538
\(421\) −9.07170 −0.442127 −0.221064 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(422\) 25.4796 1.24033
\(423\) 9.78873 0.475944
\(424\) 9.19937 0.446761
\(425\) −1.02960 −0.0499427
\(426\) −8.69222 −0.421140
\(427\) −38.7176 −1.87368
\(428\) −17.1126 −0.827171
\(429\) −0.0745196 −0.00359784
\(430\) 6.07292 0.292862
\(431\) 13.1477 0.633302 0.316651 0.948542i \(-0.397442\pi\)
0.316651 + 0.948542i \(0.397442\pi\)
\(432\) −3.83955 −0.184730
\(433\) 12.6569 0.608250 0.304125 0.952632i \(-0.401636\pi\)
0.304125 + 0.952632i \(0.401636\pi\)
\(434\) −11.3866 −0.546576
\(435\) 6.72104 0.322249
\(436\) 6.07832 0.291099
\(437\) 10.9948 0.525955
\(438\) −4.68177 −0.223704
\(439\) −22.3147 −1.06502 −0.532510 0.846424i \(-0.678751\pi\)
−0.532510 + 0.846424i \(0.678751\pi\)
\(440\) 2.85414 0.136066
\(441\) −0.188871 −0.00899384
\(442\) 0.0386150 0.00183673
\(443\) −0.221149 −0.0105071 −0.00525356 0.999986i \(-0.501672\pi\)
−0.00525356 + 0.999986i \(0.501672\pi\)
\(444\) −1.32716 −0.0629840
\(445\) −5.63871 −0.267301
\(446\) −17.6475 −0.835634
\(447\) −7.82682 −0.370196
\(448\) 2.65990 0.125669
\(449\) 41.8242 1.97381 0.986904 0.161311i \(-0.0515721\pi\)
0.986904 + 0.161311i \(0.0515721\pi\)
\(450\) 2.51537 0.118576
\(451\) −23.9040 −1.12559
\(452\) 8.57048 0.403122
\(453\) −12.5954 −0.591785
\(454\) −17.6554 −0.828611
\(455\) 0.0997598 0.00467681
\(456\) 5.14287 0.240837
\(457\) −23.0398 −1.07776 −0.538879 0.842383i \(-0.681152\pi\)
−0.538879 + 0.842383i \(0.681152\pi\)
\(458\) 17.2580 0.806411
\(459\) 3.95318 0.184519
\(460\) −1.48829 −0.0693921
\(461\) 10.3050 0.479950 0.239975 0.970779i \(-0.422861\pi\)
0.239975 + 0.970779i \(0.422861\pi\)
\(462\) 5.28502 0.245881
\(463\) −7.58923 −0.352701 −0.176351 0.984327i \(-0.556429\pi\)
−0.176351 + 0.984327i \(0.556429\pi\)
\(464\) 9.65453 0.448200
\(465\) 2.98013 0.138200
\(466\) 17.3621 0.804284
\(467\) −14.0631 −0.650764 −0.325382 0.945583i \(-0.605493\pi\)
−0.325382 + 0.945583i \(0.605493\pi\)
\(468\) −0.0943390 −0.00436082
\(469\) −8.91606 −0.411705
\(470\) 3.89157 0.179505
\(471\) −15.3960 −0.709411
\(472\) 10.3821 0.477877
\(473\) 17.3330 0.796970
\(474\) 4.64532 0.213367
\(475\) −7.38755 −0.338964
\(476\) −2.73862 −0.125525
\(477\) 23.1398 1.05950
\(478\) −12.6335 −0.577843
\(479\) 40.4539 1.84839 0.924193 0.381925i \(-0.124739\pi\)
0.924193 + 0.381925i \(0.124739\pi\)
\(480\) −0.696154 −0.0317749
\(481\) −0.0715000 −0.00326012
\(482\) 5.73290 0.261126
\(483\) −2.75588 −0.125397
\(484\) −2.85389 −0.129722
\(485\) 0.00211457 9.60177e−5 0
\(486\) −14.9111 −0.676383
\(487\) −23.5057 −1.06514 −0.532571 0.846385i \(-0.678774\pi\)
−0.532571 + 0.846385i \(0.678774\pi\)
\(488\) 14.5560 0.658920
\(489\) −12.7313 −0.575730
\(490\) −0.0750867 −0.00339207
\(491\) 29.0932 1.31296 0.656478 0.754345i \(-0.272045\pi\)
0.656478 + 0.754345i \(0.272045\pi\)
\(492\) 5.83042 0.262856
\(493\) −9.94026 −0.447687
\(494\) 0.277070 0.0124660
\(495\) 7.17922 0.322682
\(496\) 4.28085 0.192216
\(497\) 33.2117 1.48975
\(498\) 5.48889 0.245963
\(499\) 1.02863 0.0460480 0.0230240 0.999735i \(-0.492671\pi\)
0.0230240 + 0.999735i \(0.492671\pi\)
\(500\) 1.00000 0.0447214
\(501\) 12.4157 0.554691
\(502\) −21.4087 −0.955517
\(503\) −17.3276 −0.772600 −0.386300 0.922373i \(-0.626247\pi\)
−0.386300 + 0.922373i \(0.626247\pi\)
\(504\) 6.69064 0.298025
\(505\) −9.15058 −0.407195
\(506\) −4.24780 −0.188838
\(507\) −9.04902 −0.401881
\(508\) −4.07183 −0.180658
\(509\) −36.9550 −1.63800 −0.819000 0.573793i \(-0.805471\pi\)
−0.819000 + 0.573793i \(0.805471\pi\)
\(510\) 0.716757 0.0317385
\(511\) 17.8884 0.791335
\(512\) −1.00000 −0.0441942
\(513\) 28.3648 1.25234
\(514\) −20.1176 −0.887347
\(515\) −3.39406 −0.149560
\(516\) −4.22769 −0.186113
\(517\) 11.1071 0.488489
\(518\) 5.07087 0.222801
\(519\) 9.26712 0.406782
\(520\) −0.0375050 −0.00164470
\(521\) 22.6421 0.991966 0.495983 0.868332i \(-0.334808\pi\)
0.495983 + 0.868332i \(0.334808\pi\)
\(522\) 24.2847 1.06291
\(523\) 24.0765 1.05279 0.526397 0.850239i \(-0.323543\pi\)
0.526397 + 0.850239i \(0.323543\pi\)
\(524\) 1.30068 0.0568206
\(525\) 1.85170 0.0808149
\(526\) 27.0762 1.18058
\(527\) −4.40754 −0.191995
\(528\) −1.98692 −0.0864697
\(529\) −20.7850 −0.903695
\(530\) 9.19937 0.399595
\(531\) 26.1149 1.13329
\(532\) −19.6502 −0.851943
\(533\) 0.314112 0.0136057
\(534\) 3.92541 0.169869
\(535\) −17.1126 −0.739844
\(536\) 3.35202 0.144785
\(537\) 7.71832 0.333070
\(538\) 3.84789 0.165894
\(539\) −0.214308 −0.00923089
\(540\) −3.83955 −0.165228
\(541\) 13.0610 0.561537 0.280768 0.959776i \(-0.409411\pi\)
0.280768 + 0.959776i \(0.409411\pi\)
\(542\) −0.630614 −0.0270872
\(543\) −6.21887 −0.266877
\(544\) 1.02960 0.0441435
\(545\) 6.07832 0.260367
\(546\) −0.0694482 −0.00297211
\(547\) −14.7095 −0.628933 −0.314466 0.949269i \(-0.601826\pi\)
−0.314466 + 0.949269i \(0.601826\pi\)
\(548\) 17.9925 0.768601
\(549\) 36.6138 1.56264
\(550\) 2.85414 0.121701
\(551\) −71.3233 −3.03848
\(552\) 1.03608 0.0440986
\(553\) −17.7491 −0.754768
\(554\) −18.0685 −0.767658
\(555\) −1.32716 −0.0563346
\(556\) 4.30048 0.182381
\(557\) −0.463082 −0.0196214 −0.00981071 0.999952i \(-0.503123\pi\)
−0.00981071 + 0.999952i \(0.503123\pi\)
\(558\) 10.7679 0.455842
\(559\) −0.227765 −0.00963343
\(560\) 2.65990 0.112401
\(561\) 2.04572 0.0863706
\(562\) 16.3568 0.689971
\(563\) 23.6664 0.997422 0.498711 0.866768i \(-0.333807\pi\)
0.498711 + 0.866768i \(0.333807\pi\)
\(564\) −2.70913 −0.114075
\(565\) 8.57048 0.360563
\(566\) 1.20167 0.0505101
\(567\) 12.9622 0.544362
\(568\) −12.4861 −0.523903
\(569\) −15.1693 −0.635931 −0.317966 0.948102i \(-0.603000\pi\)
−0.317966 + 0.948102i \(0.603000\pi\)
\(570\) 5.14287 0.215411
\(571\) 27.6020 1.15511 0.577553 0.816353i \(-0.304008\pi\)
0.577553 + 0.816353i \(0.304008\pi\)
\(572\) −0.107045 −0.00447576
\(573\) 4.32785 0.180799
\(574\) −22.2772 −0.929832
\(575\) −1.48829 −0.0620661
\(576\) −2.51537 −0.104807
\(577\) 1.39037 0.0578820 0.0289410 0.999581i \(-0.490787\pi\)
0.0289410 + 0.999581i \(0.490787\pi\)
\(578\) 15.9399 0.663014
\(579\) 14.5378 0.604172
\(580\) 9.65453 0.400883
\(581\) −20.9723 −0.870076
\(582\) −0.00147207 −6.10191e−5 0
\(583\) 26.2563 1.08742
\(584\) −6.72519 −0.278291
\(585\) −0.0943390 −0.00390044
\(586\) 17.6326 0.728396
\(587\) −28.6625 −1.18303 −0.591514 0.806294i \(-0.701470\pi\)
−0.591514 + 0.806294i \(0.701470\pi\)
\(588\) 0.0522719 0.00215566
\(589\) −31.6250 −1.30308
\(590\) 10.3821 0.427426
\(591\) 4.62281 0.190157
\(592\) −1.90641 −0.0783530
\(593\) −29.1603 −1.19747 −0.598735 0.800947i \(-0.704330\pi\)
−0.598735 + 0.800947i \(0.704330\pi\)
\(594\) −10.9586 −0.449637
\(595\) −2.73862 −0.112273
\(596\) −11.2429 −0.460529
\(597\) −9.29751 −0.380522
\(598\) 0.0558185 0.00228259
\(599\) −31.1199 −1.27152 −0.635762 0.771885i \(-0.719314\pi\)
−0.635762 + 0.771885i \(0.719314\pi\)
\(600\) −0.696154 −0.0284204
\(601\) 29.9444 1.22146 0.610729 0.791840i \(-0.290876\pi\)
0.610729 + 0.791840i \(0.290876\pi\)
\(602\) 16.1534 0.658362
\(603\) 8.43157 0.343360
\(604\) −18.0929 −0.736189
\(605\) −2.85389 −0.116027
\(606\) 6.37021 0.258772
\(607\) 1.87477 0.0760944 0.0380472 0.999276i \(-0.487886\pi\)
0.0380472 + 0.999276i \(0.487886\pi\)
\(608\) 7.38755 0.299605
\(609\) 17.8773 0.724426
\(610\) 14.5560 0.589356
\(611\) −0.145953 −0.00590464
\(612\) 2.58981 0.104687
\(613\) −25.8197 −1.04285 −0.521425 0.853297i \(-0.674599\pi\)
−0.521425 + 0.853297i \(0.674599\pi\)
\(614\) −29.5550 −1.19274
\(615\) 5.83042 0.235105
\(616\) 7.59174 0.305880
\(617\) 35.2336 1.41845 0.709226 0.704982i \(-0.249045\pi\)
0.709226 + 0.704982i \(0.249045\pi\)
\(618\) 2.36279 0.0950453
\(619\) −16.0444 −0.644880 −0.322440 0.946590i \(-0.604503\pi\)
−0.322440 + 0.946590i \(0.604503\pi\)
\(620\) 4.28085 0.171923
\(621\) 5.71437 0.229310
\(622\) −13.4662 −0.539945
\(623\) −14.9984 −0.600899
\(624\) 0.0261093 0.00104521
\(625\) 1.00000 0.0400000
\(626\) −18.4655 −0.738029
\(627\) 14.6785 0.586202
\(628\) −22.1158 −0.882517
\(629\) 1.96283 0.0782632
\(630\) 6.69064 0.266561
\(631\) −4.67025 −0.185920 −0.0929599 0.995670i \(-0.529633\pi\)
−0.0929599 + 0.995670i \(0.529633\pi\)
\(632\) 6.67283 0.265431
\(633\) −17.7378 −0.705012
\(634\) −0.377432 −0.0149897
\(635\) −4.07183 −0.161586
\(636\) −6.40418 −0.253942
\(637\) 0.00281613 0.000111579 0
\(638\) 27.5554 1.09093
\(639\) −31.4071 −1.24244
\(640\) −1.00000 −0.0395285
\(641\) −13.1204 −0.518226 −0.259113 0.965847i \(-0.583430\pi\)
−0.259113 + 0.965847i \(0.583430\pi\)
\(642\) 11.9130 0.470170
\(643\) −4.64694 −0.183257 −0.0916286 0.995793i \(-0.529207\pi\)
−0.0916286 + 0.995793i \(0.529207\pi\)
\(644\) −3.95872 −0.155995
\(645\) −4.22769 −0.166465
\(646\) −7.60619 −0.299261
\(647\) 42.6147 1.67536 0.837679 0.546163i \(-0.183912\pi\)
0.837679 + 0.546163i \(0.183912\pi\)
\(648\) −4.87319 −0.191437
\(649\) 29.6321 1.16316
\(650\) −0.0375050 −0.00147107
\(651\) 7.92686 0.310678
\(652\) −18.2881 −0.716216
\(653\) 0.246098 0.00963054 0.00481527 0.999988i \(-0.498467\pi\)
0.00481527 + 0.999988i \(0.498467\pi\)
\(654\) −4.23145 −0.165463
\(655\) 1.30068 0.0508219
\(656\) 8.37519 0.326996
\(657\) −16.9163 −0.659970
\(658\) 10.3512 0.403532
\(659\) 16.4663 0.641438 0.320719 0.947174i \(-0.396076\pi\)
0.320719 + 0.947174i \(0.396076\pi\)
\(660\) −1.98692 −0.0773408
\(661\) 46.2706 1.79972 0.899859 0.436182i \(-0.143670\pi\)
0.899859 + 0.436182i \(0.143670\pi\)
\(662\) −9.69431 −0.376780
\(663\) −0.0268820 −0.00104401
\(664\) 7.88459 0.305982
\(665\) −19.6502 −0.762001
\(666\) −4.79533 −0.185815
\(667\) −14.3688 −0.556361
\(668\) 17.8347 0.690044
\(669\) 12.2854 0.474981
\(670\) 3.35202 0.129500
\(671\) 41.5449 1.60382
\(672\) −1.85170 −0.0714310
\(673\) 28.1747 1.08606 0.543028 0.839715i \(-0.317278\pi\)
0.543028 + 0.839715i \(0.317278\pi\)
\(674\) −9.62228 −0.370636
\(675\) −3.83955 −0.147784
\(676\) −12.9986 −0.499946
\(677\) 29.6039 1.13777 0.568885 0.822417i \(-0.307375\pi\)
0.568885 + 0.822417i \(0.307375\pi\)
\(678\) −5.96638 −0.229137
\(679\) 0.00562455 0.000215851 0
\(680\) 1.02960 0.0394832
\(681\) 12.2909 0.470988
\(682\) 12.2181 0.467857
\(683\) 10.3329 0.395378 0.197689 0.980265i \(-0.436656\pi\)
0.197689 + 0.980265i \(0.436656\pi\)
\(684\) 18.5824 0.710517
\(685\) 17.9925 0.687458
\(686\) 18.4196 0.703264
\(687\) −12.0142 −0.458370
\(688\) −6.07292 −0.231528
\(689\) −0.345023 −0.0131443
\(690\) 1.03608 0.0394430
\(691\) −5.72697 −0.217864 −0.108932 0.994049i \(-0.534743\pi\)
−0.108932 + 0.994049i \(0.534743\pi\)
\(692\) 13.3119 0.506042
\(693\) 19.0960 0.725398
\(694\) 34.3809 1.30508
\(695\) 4.30048 0.163127
\(696\) −6.72104 −0.254760
\(697\) −8.62305 −0.326621
\(698\) 26.4606 1.00155
\(699\) −12.0867 −0.457161
\(700\) 2.65990 0.100535
\(701\) 44.3202 1.67395 0.836976 0.547239i \(-0.184321\pi\)
0.836976 + 0.547239i \(0.184321\pi\)
\(702\) 0.144002 0.00543502
\(703\) 14.0837 0.531177
\(704\) −2.85414 −0.107569
\(705\) −2.70913 −0.102032
\(706\) 22.8742 0.860881
\(707\) −24.3397 −0.915387
\(708\) −7.22757 −0.271629
\(709\) 10.1220 0.380139 0.190069 0.981771i \(-0.439129\pi\)
0.190069 + 0.981771i \(0.439129\pi\)
\(710\) −12.4861 −0.468593
\(711\) 16.7846 0.629473
\(712\) 5.63871 0.211320
\(713\) −6.37116 −0.238602
\(714\) 1.90650 0.0713491
\(715\) −0.107045 −0.00400324
\(716\) 11.0871 0.414344
\(717\) 8.79486 0.328450
\(718\) 12.7326 0.475176
\(719\) 24.9058 0.928828 0.464414 0.885618i \(-0.346265\pi\)
0.464414 + 0.885618i \(0.346265\pi\)
\(720\) −2.51537 −0.0937423
\(721\) −9.02788 −0.336216
\(722\) −35.5759 −1.32400
\(723\) −3.99098 −0.148426
\(724\) −8.93318 −0.331999
\(725\) 9.65453 0.358560
\(726\) 1.98674 0.0737350
\(727\) −35.9493 −1.33329 −0.666643 0.745377i \(-0.732270\pi\)
−0.666643 + 0.745377i \(0.732270\pi\)
\(728\) −0.0997598 −0.00369734
\(729\) −4.23913 −0.157005
\(730\) −6.72519 −0.248911
\(731\) 6.25264 0.231262
\(732\) −10.1332 −0.374535
\(733\) −6.50206 −0.240159 −0.120080 0.992764i \(-0.538315\pi\)
−0.120080 + 0.992764i \(0.538315\pi\)
\(734\) −6.86945 −0.253556
\(735\) 0.0522719 0.00192808
\(736\) 1.48829 0.0548592
\(737\) 9.56714 0.352410
\(738\) 21.0667 0.775476
\(739\) −3.62083 −0.133194 −0.0665971 0.997780i \(-0.521214\pi\)
−0.0665971 + 0.997780i \(0.521214\pi\)
\(740\) −1.90641 −0.0700810
\(741\) −0.192884 −0.00708576
\(742\) 24.4694 0.898301
\(743\) −50.0811 −1.83730 −0.918649 0.395076i \(-0.870718\pi\)
−0.918649 + 0.395076i \(0.870718\pi\)
\(744\) −2.98013 −0.109257
\(745\) −11.2429 −0.411910
\(746\) −29.5706 −1.08266
\(747\) 19.8327 0.725639
\(748\) 2.93861 0.107446
\(749\) −45.5180 −1.66319
\(750\) −0.696154 −0.0254200
\(751\) 1.63536 0.0596753 0.0298376 0.999555i \(-0.490501\pi\)
0.0298376 + 0.999555i \(0.490501\pi\)
\(752\) −3.89157 −0.141911
\(753\) 14.9037 0.543123
\(754\) −0.362094 −0.0131867
\(755\) −18.0929 −0.658468
\(756\) −10.2128 −0.371437
\(757\) 8.12371 0.295261 0.147631 0.989043i \(-0.452835\pi\)
0.147631 + 0.989043i \(0.452835\pi\)
\(758\) −26.8611 −0.975640
\(759\) 2.95712 0.107337
\(760\) 7.38755 0.267975
\(761\) 44.7571 1.62244 0.811222 0.584739i \(-0.198803\pi\)
0.811222 + 0.584739i \(0.198803\pi\)
\(762\) 2.83462 0.102688
\(763\) 16.1678 0.585312
\(764\) 6.21680 0.224916
\(765\) 2.58981 0.0936349
\(766\) 15.9093 0.574825
\(767\) −0.389382 −0.0140598
\(768\) 0.696154 0.0251203
\(769\) −12.7234 −0.458816 −0.229408 0.973330i \(-0.573679\pi\)
−0.229408 + 0.973330i \(0.573679\pi\)
\(770\) 7.59174 0.273587
\(771\) 14.0049 0.504375
\(772\) 20.8831 0.751598
\(773\) 14.5070 0.521781 0.260891 0.965368i \(-0.415984\pi\)
0.260891 + 0.965368i \(0.415984\pi\)
\(774\) −15.2756 −0.549071
\(775\) 4.28085 0.153773
\(776\) −0.00211457 −7.59087e−5 0
\(777\) −3.53010 −0.126642
\(778\) 21.0610 0.755072
\(779\) −61.8721 −2.21680
\(780\) 0.0261093 0.000934862 0
\(781\) −35.6370 −1.27519
\(782\) −1.53234 −0.0547964
\(783\) −37.0690 −1.32474
\(784\) 0.0750867 0.00268167
\(785\) −22.1158 −0.789347
\(786\) −0.905475 −0.0322972
\(787\) −15.4745 −0.551605 −0.275803 0.961214i \(-0.588944\pi\)
−0.275803 + 0.961214i \(0.588944\pi\)
\(788\) 6.64050 0.236558
\(789\) −18.8492 −0.671049
\(790\) 6.67283 0.237409
\(791\) 22.7967 0.810556
\(792\) −7.17922 −0.255102
\(793\) −0.545924 −0.0193863
\(794\) −22.2540 −0.789764
\(795\) −6.40418 −0.227133
\(796\) −13.3555 −0.473374
\(797\) 15.8753 0.562331 0.281166 0.959659i \(-0.409279\pi\)
0.281166 + 0.959659i \(0.409279\pi\)
\(798\) 13.6795 0.484251
\(799\) 4.00674 0.141748
\(800\) −1.00000 −0.0353553
\(801\) 14.1834 0.501147
\(802\) −1.00000 −0.0353112
\(803\) −19.1946 −0.677364
\(804\) −2.33352 −0.0822971
\(805\) −3.95872 −0.139526
\(806\) −0.160553 −0.00565525
\(807\) −2.67872 −0.0942954
\(808\) 9.15058 0.321916
\(809\) 20.3904 0.716888 0.358444 0.933551i \(-0.383307\pi\)
0.358444 + 0.933551i \(0.383307\pi\)
\(810\) −4.87319 −0.171227
\(811\) −41.4037 −1.45388 −0.726941 0.686700i \(-0.759059\pi\)
−0.726941 + 0.686700i \(0.759059\pi\)
\(812\) 25.6801 0.901196
\(813\) 0.439004 0.0153966
\(814\) −5.44116 −0.190713
\(815\) −18.2881 −0.640603
\(816\) −0.716757 −0.0250915
\(817\) 44.8640 1.56959
\(818\) 23.7118 0.829062
\(819\) −0.250933 −0.00876830
\(820\) 8.37519 0.292474
\(821\) −22.2301 −0.775836 −0.387918 0.921694i \(-0.626806\pi\)
−0.387918 + 0.921694i \(0.626806\pi\)
\(822\) −12.5256 −0.436879
\(823\) −30.5016 −1.06322 −0.531610 0.846989i \(-0.678413\pi\)
−0.531610 + 0.846989i \(0.678413\pi\)
\(824\) 3.39406 0.118238
\(825\) −1.98692 −0.0691757
\(826\) 27.6155 0.960866
\(827\) 16.2473 0.564973 0.282487 0.959271i \(-0.408841\pi\)
0.282487 + 0.959271i \(0.408841\pi\)
\(828\) 3.74361 0.130099
\(829\) −13.2892 −0.461552 −0.230776 0.973007i \(-0.574127\pi\)
−0.230776 + 0.973007i \(0.574127\pi\)
\(830\) 7.88459 0.273678
\(831\) 12.5785 0.436342
\(832\) 0.0375050 0.00130025
\(833\) −0.0773089 −0.00267859
\(834\) −2.99380 −0.103667
\(835\) 17.8347 0.617194
\(836\) 21.0851 0.729244
\(837\) −16.4365 −0.568129
\(838\) 33.8273 1.16855
\(839\) −36.4776 −1.25935 −0.629673 0.776860i \(-0.716811\pi\)
−0.629673 + 0.776860i \(0.716811\pi\)
\(840\) −1.85170 −0.0638898
\(841\) 64.2100 2.21414
\(842\) 9.07170 0.312631
\(843\) −11.3869 −0.392185
\(844\) −25.4796 −0.877045
\(845\) −12.9986 −0.447165
\(846\) −9.78873 −0.336544
\(847\) −7.59106 −0.260832
\(848\) −9.19937 −0.315908
\(849\) −0.836550 −0.0287103
\(850\) 1.02960 0.0353148
\(851\) 2.83730 0.0972613
\(852\) 8.69222 0.297791
\(853\) −52.3016 −1.79077 −0.895386 0.445290i \(-0.853101\pi\)
−0.895386 + 0.445290i \(0.853101\pi\)
\(854\) 38.7176 1.32489
\(855\) 18.5824 0.635505
\(856\) 17.1126 0.584898
\(857\) −3.70388 −0.126522 −0.0632611 0.997997i \(-0.520150\pi\)
−0.0632611 + 0.997997i \(0.520150\pi\)
\(858\) 0.0745196 0.00254406
\(859\) −16.4440 −0.561063 −0.280532 0.959845i \(-0.590511\pi\)
−0.280532 + 0.959845i \(0.590511\pi\)
\(860\) −6.07292 −0.207085
\(861\) 15.5084 0.528523
\(862\) −13.1477 −0.447812
\(863\) 46.7402 1.59105 0.795527 0.605918i \(-0.207194\pi\)
0.795527 + 0.605918i \(0.207194\pi\)
\(864\) 3.83955 0.130624
\(865\) 13.3119 0.452618
\(866\) −12.6569 −0.430098
\(867\) −11.0967 −0.376862
\(868\) 11.3866 0.386488
\(869\) 19.0452 0.646064
\(870\) −6.72104 −0.227865
\(871\) −0.125718 −0.00425978
\(872\) −6.07832 −0.205838
\(873\) −0.00531893 −0.000180018 0
\(874\) −10.9948 −0.371906
\(875\) 2.65990 0.0899212
\(876\) 4.68177 0.158182
\(877\) 46.8547 1.58217 0.791085 0.611706i \(-0.209516\pi\)
0.791085 + 0.611706i \(0.209516\pi\)
\(878\) 22.3147 0.753083
\(879\) −12.2750 −0.414026
\(880\) −2.85414 −0.0962130
\(881\) 11.3007 0.380731 0.190366 0.981713i \(-0.439033\pi\)
0.190366 + 0.981713i \(0.439033\pi\)
\(882\) 0.188871 0.00635961
\(883\) 37.3318 1.25632 0.628158 0.778086i \(-0.283809\pi\)
0.628158 + 0.778086i \(0.283809\pi\)
\(884\) −0.0386150 −0.00129876
\(885\) −7.22757 −0.242952
\(886\) 0.221149 0.00742966
\(887\) 37.1233 1.24648 0.623239 0.782031i \(-0.285816\pi\)
0.623239 + 0.782031i \(0.285816\pi\)
\(888\) 1.32716 0.0445364
\(889\) −10.8307 −0.363250
\(890\) 5.63871 0.189010
\(891\) −13.9088 −0.465961
\(892\) 17.6475 0.590883
\(893\) 28.7492 0.962054
\(894\) 7.82682 0.261768
\(895\) 11.0871 0.370601
\(896\) −2.65990 −0.0888611
\(897\) −0.0388583 −0.00129744
\(898\) −41.8242 −1.39569
\(899\) 41.3296 1.37842
\(900\) −2.51537 −0.0838456
\(901\) 9.47163 0.315546
\(902\) 23.9040 0.795915
\(903\) −11.2452 −0.374218
\(904\) −8.57048 −0.285050
\(905\) −8.93318 −0.296949
\(906\) 12.5954 0.418455
\(907\) −37.0817 −1.23128 −0.615639 0.788029i \(-0.711102\pi\)
−0.615639 + 0.788029i \(0.711102\pi\)
\(908\) 17.6554 0.585916
\(909\) 23.0171 0.763429
\(910\) −0.0997598 −0.00330700
\(911\) −28.3851 −0.940440 −0.470220 0.882549i \(-0.655825\pi\)
−0.470220 + 0.882549i \(0.655825\pi\)
\(912\) −5.14287 −0.170298
\(913\) 22.5037 0.744765
\(914\) 23.0398 0.762090
\(915\) −10.1332 −0.334994
\(916\) −17.2580 −0.570219
\(917\) 3.45969 0.114249
\(918\) −3.95318 −0.130474
\(919\) 1.81903 0.0600043 0.0300022 0.999550i \(-0.490449\pi\)
0.0300022 + 0.999550i \(0.490449\pi\)
\(920\) 1.48829 0.0490676
\(921\) 20.5748 0.677964
\(922\) −10.3050 −0.339376
\(923\) 0.468290 0.0154140
\(924\) −5.28502 −0.173864
\(925\) −1.90641 −0.0626824
\(926\) 7.58923 0.249397
\(927\) 8.53732 0.280402
\(928\) −9.65453 −0.316925
\(929\) 29.2371 0.959238 0.479619 0.877477i \(-0.340775\pi\)
0.479619 + 0.877477i \(0.340775\pi\)
\(930\) −2.98013 −0.0977223
\(931\) −0.554707 −0.0181798
\(932\) −17.3621 −0.568715
\(933\) 9.37454 0.306909
\(934\) 14.0631 0.460159
\(935\) 2.93861 0.0961028
\(936\) 0.0943390 0.00308357
\(937\) 10.5322 0.344073 0.172036 0.985091i \(-0.444965\pi\)
0.172036 + 0.985091i \(0.444965\pi\)
\(938\) 8.91606 0.291120
\(939\) 12.8548 0.419501
\(940\) −3.89157 −0.126929
\(941\) −31.2021 −1.01716 −0.508580 0.861015i \(-0.669829\pi\)
−0.508580 + 0.861015i \(0.669829\pi\)
\(942\) 15.3960 0.501629
\(943\) −12.4647 −0.405908
\(944\) −10.3821 −0.337910
\(945\) −10.2128 −0.332223
\(946\) −17.3330 −0.563543
\(947\) 24.7127 0.803053 0.401527 0.915847i \(-0.368480\pi\)
0.401527 + 0.915847i \(0.368480\pi\)
\(948\) −4.64532 −0.150873
\(949\) 0.252229 0.00818769
\(950\) 7.38755 0.239684
\(951\) 0.262751 0.00852027
\(952\) 2.73862 0.0887593
\(953\) −41.9226 −1.35801 −0.679003 0.734136i \(-0.737588\pi\)
−0.679003 + 0.734136i \(0.737588\pi\)
\(954\) −23.1398 −0.749179
\(955\) 6.21680 0.201171
\(956\) 12.6335 0.408596
\(957\) −19.1828 −0.620092
\(958\) −40.4539 −1.30701
\(959\) 47.8583 1.54543
\(960\) 0.696154 0.0224683
\(961\) −12.6743 −0.408850
\(962\) 0.0715000 0.00230525
\(963\) 43.0446 1.38709
\(964\) −5.73290 −0.184644
\(965\) 20.8831 0.672250
\(966\) 2.75588 0.0886689
\(967\) 5.93617 0.190894 0.0954472 0.995434i \(-0.469572\pi\)
0.0954472 + 0.995434i \(0.469572\pi\)
\(968\) 2.85389 0.0917273
\(969\) 5.29508 0.170102
\(970\) −0.00211457 −6.78948e−5 0
\(971\) −41.4457 −1.33006 −0.665029 0.746818i \(-0.731581\pi\)
−0.665029 + 0.746818i \(0.731581\pi\)
\(972\) 14.9111 0.478275
\(973\) 11.4389 0.366713
\(974\) 23.5057 0.753170
\(975\) 0.0261093 0.000836166 0
\(976\) −14.5560 −0.465927
\(977\) 49.8864 1.59601 0.798004 0.602652i \(-0.205889\pi\)
0.798004 + 0.602652i \(0.205889\pi\)
\(978\) 12.7313 0.407102
\(979\) 16.0937 0.514356
\(980\) 0.0750867 0.00239856
\(981\) −15.2892 −0.488147
\(982\) −29.0932 −0.928400
\(983\) −44.4265 −1.41698 −0.708492 0.705719i \(-0.750624\pi\)
−0.708492 + 0.705719i \(0.750624\pi\)
\(984\) −5.83042 −0.185867
\(985\) 6.64050 0.211584
\(986\) 9.94026 0.316562
\(987\) −7.20603 −0.229370
\(988\) −0.277070 −0.00881478
\(989\) 9.03828 0.287401
\(990\) −7.17922 −0.228170
\(991\) −58.1947 −1.84862 −0.924308 0.381647i \(-0.875357\pi\)
−0.924308 + 0.381647i \(0.875357\pi\)
\(992\) −4.28085 −0.135917
\(993\) 6.74873 0.214165
\(994\) −33.2117 −1.05341
\(995\) −13.3555 −0.423399
\(996\) −5.48889 −0.173922
\(997\) 30.3412 0.960914 0.480457 0.877018i \(-0.340471\pi\)
0.480457 + 0.877018i \(0.340471\pi\)
\(998\) −1.02863 −0.0325608
\(999\) 7.31975 0.231587
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 4010.2.a.i.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4010.2.a.i.1.8 10 1.1 even 1 trivial