Properties

Label 4025.2.a.i.1.1
Level 4025
Weight 2
Character 4025.1
Self dual yes
Analytic conductor 32.140
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{9} +4.47214 q^{11} -1.61803 q^{12} -0.236068 q^{13} -0.618034 q^{14} +1.85410 q^{16} +1.23607 q^{18} -7.23607 q^{19} +1.00000 q^{21} -2.76393 q^{22} +1.00000 q^{23} +2.23607 q^{24} +0.145898 q^{26} -5.00000 q^{27} -1.61803 q^{28} -1.47214 q^{29} -9.00000 q^{31} -5.61803 q^{32} +4.47214 q^{33} +3.23607 q^{36} +5.70820 q^{37} +4.47214 q^{38} -0.236068 q^{39} -2.23607 q^{41} -0.618034 q^{42} -2.47214 q^{43} -7.23607 q^{44} -0.618034 q^{46} +3.47214 q^{47} +1.85410 q^{48} +1.00000 q^{49} +0.381966 q^{52} -11.2361 q^{53} +3.09017 q^{54} +2.23607 q^{56} -7.23607 q^{57} +0.909830 q^{58} -1.52786 q^{59} +13.4164 q^{61} +5.56231 q^{62} -2.00000 q^{63} -0.236068 q^{64} -2.76393 q^{66} +12.1803 q^{67} +1.00000 q^{69} -10.2361 q^{71} -4.47214 q^{72} +6.70820 q^{73} -3.52786 q^{74} +11.7082 q^{76} +4.47214 q^{77} +0.145898 q^{78} -7.23607 q^{79} +1.00000 q^{81} +1.38197 q^{82} -6.47214 q^{83} -1.61803 q^{84} +1.52786 q^{86} -1.47214 q^{87} +10.0000 q^{88} +8.94427 q^{89} -0.236068 q^{91} -1.61803 q^{92} -9.00000 q^{93} -2.14590 q^{94} -5.61803 q^{96} -3.70820 q^{97} -0.618034 q^{98} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} - 4q^{9} - q^{12} + 4q^{13} + q^{14} - 3q^{16} - 2q^{18} - 10q^{19} + 2q^{21} - 10q^{22} + 2q^{23} + 7q^{26} - 10q^{27} - q^{28} + 6q^{29} - 18q^{31} - 9q^{32} + 2q^{36} - 2q^{37} + 4q^{39} + q^{42} + 4q^{43} - 10q^{44} + q^{46} - 2q^{47} - 3q^{48} + 2q^{49} + 3q^{52} - 18q^{53} - 5q^{54} - 10q^{57} + 13q^{58} - 12q^{59} - 9q^{62} - 4q^{63} + 4q^{64} - 10q^{66} + 2q^{67} + 2q^{69} - 16q^{71} - 16q^{74} + 10q^{76} + 7q^{78} - 10q^{79} + 2q^{81} + 5q^{82} - 4q^{83} - q^{84} + 12q^{86} + 6q^{87} + 20q^{88} + 4q^{91} - q^{92} - 18q^{93} - 11q^{94} - 9q^{96} + 6q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) −1.61803 −0.467086
\(13\) −0.236068 −0.0654735 −0.0327367 0.999464i \(-0.510422\pi\)
−0.0327367 + 0.999464i \(0.510422\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.23607 0.291344
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −2.76393 −0.589272
\(23\) 1.00000 0.208514
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 0.145898 0.0286130
\(27\) −5.00000 −0.962250
\(28\) −1.61803 −0.305780
\(29\) −1.47214 −0.273369 −0.136684 0.990615i \(-0.543645\pi\)
−0.136684 + 0.990615i \(0.543645\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −5.61803 −0.993137
\(33\) 4.47214 0.778499
\(34\) 0 0
\(35\) 0 0
\(36\) 3.23607 0.539345
\(37\) 5.70820 0.938423 0.469211 0.883086i \(-0.344538\pi\)
0.469211 + 0.883086i \(0.344538\pi\)
\(38\) 4.47214 0.725476
\(39\) −0.236068 −0.0378011
\(40\) 0 0
\(41\) −2.23607 −0.349215 −0.174608 0.984638i \(-0.555866\pi\)
−0.174608 + 0.984638i \(0.555866\pi\)
\(42\) −0.618034 −0.0953647
\(43\) −2.47214 −0.376997 −0.188499 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) −7.23607 −1.09088
\(45\) 0 0
\(46\) −0.618034 −0.0911241
\(47\) 3.47214 0.506463 0.253232 0.967406i \(-0.418507\pi\)
0.253232 + 0.967406i \(0.418507\pi\)
\(48\) 1.85410 0.267617
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0.381966 0.0529692
\(53\) −11.2361 −1.54339 −0.771696 0.635991i \(-0.780591\pi\)
−0.771696 + 0.635991i \(0.780591\pi\)
\(54\) 3.09017 0.420519
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) −7.23607 −0.958441
\(58\) 0.909830 0.119467
\(59\) −1.52786 −0.198911 −0.0994555 0.995042i \(-0.531710\pi\)
−0.0994555 + 0.995042i \(0.531710\pi\)
\(60\) 0 0
\(61\) 13.4164 1.71780 0.858898 0.512148i \(-0.171150\pi\)
0.858898 + 0.512148i \(0.171150\pi\)
\(62\) 5.56231 0.706414
\(63\) −2.00000 −0.251976
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) −2.76393 −0.340217
\(67\) 12.1803 1.48807 0.744033 0.668143i \(-0.232911\pi\)
0.744033 + 0.668143i \(0.232911\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.2361 −1.21480 −0.607399 0.794397i \(-0.707787\pi\)
−0.607399 + 0.794397i \(0.707787\pi\)
\(72\) −4.47214 −0.527046
\(73\) 6.70820 0.785136 0.392568 0.919723i \(-0.371587\pi\)
0.392568 + 0.919723i \(0.371587\pi\)
\(74\) −3.52786 −0.410106
\(75\) 0 0
\(76\) 11.7082 1.34302
\(77\) 4.47214 0.509647
\(78\) 0.145898 0.0165197
\(79\) −7.23607 −0.814121 −0.407061 0.913401i \(-0.633446\pi\)
−0.407061 + 0.913401i \(0.633446\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.38197 0.152613
\(83\) −6.47214 −0.710409 −0.355205 0.934789i \(-0.615589\pi\)
−0.355205 + 0.934789i \(0.615589\pi\)
\(84\) −1.61803 −0.176542
\(85\) 0 0
\(86\) 1.52786 0.164754
\(87\) −1.47214 −0.157830
\(88\) 10.0000 1.06600
\(89\) 8.94427 0.948091 0.474045 0.880500i \(-0.342793\pi\)
0.474045 + 0.880500i \(0.342793\pi\)
\(90\) 0 0
\(91\) −0.236068 −0.0247466
\(92\) −1.61803 −0.168692
\(93\) −9.00000 −0.933257
\(94\) −2.14590 −0.221332
\(95\) 0 0
\(96\) −5.61803 −0.573388
\(97\) −3.70820 −0.376511 −0.188256 0.982120i \(-0.560283\pi\)
−0.188256 + 0.982120i \(0.560283\pi\)
\(98\) −0.618034 −0.0624309
\(99\) −8.94427 −0.898933
\(100\) 0 0
\(101\) −13.4164 −1.33498 −0.667491 0.744618i \(-0.732632\pi\)
−0.667491 + 0.744618i \(0.732632\pi\)
\(102\) 0 0
\(103\) 15.7082 1.54778 0.773888 0.633323i \(-0.218309\pi\)
0.773888 + 0.633323i \(0.218309\pi\)
\(104\) −0.527864 −0.0517613
\(105\) 0 0
\(106\) 6.94427 0.674487
\(107\) −11.2361 −1.08623 −0.543116 0.839658i \(-0.682756\pi\)
−0.543116 + 0.839658i \(0.682756\pi\)
\(108\) 8.09017 0.778477
\(109\) −15.4164 −1.47662 −0.738312 0.674459i \(-0.764377\pi\)
−0.738312 + 0.674459i \(0.764377\pi\)
\(110\) 0 0
\(111\) 5.70820 0.541799
\(112\) 1.85410 0.175196
\(113\) −2.47214 −0.232559 −0.116279 0.993217i \(-0.537097\pi\)
−0.116279 + 0.993217i \(0.537097\pi\)
\(114\) 4.47214 0.418854
\(115\) 0 0
\(116\) 2.38197 0.221160
\(117\) 0.472136 0.0436490
\(118\) 0.944272 0.0869273
\(119\) 0 0
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −8.29180 −0.750704
\(123\) −2.23607 −0.201619
\(124\) 14.5623 1.30773
\(125\) 0 0
\(126\) 1.23607 0.110118
\(127\) 2.70820 0.240314 0.120157 0.992755i \(-0.461660\pi\)
0.120157 + 0.992755i \(0.461660\pi\)
\(128\) 11.3820 1.00603
\(129\) −2.47214 −0.217659
\(130\) 0 0
\(131\) 3.94427 0.344613 0.172306 0.985043i \(-0.444878\pi\)
0.172306 + 0.985043i \(0.444878\pi\)
\(132\) −7.23607 −0.629819
\(133\) −7.23607 −0.627447
\(134\) −7.52786 −0.650308
\(135\) 0 0
\(136\) 0 0
\(137\) −15.7082 −1.34204 −0.671021 0.741438i \(-0.734144\pi\)
−0.671021 + 0.741438i \(0.734144\pi\)
\(138\) −0.618034 −0.0526105
\(139\) 2.52786 0.214411 0.107205 0.994237i \(-0.465810\pi\)
0.107205 + 0.994237i \(0.465810\pi\)
\(140\) 0 0
\(141\) 3.47214 0.292407
\(142\) 6.32624 0.530886
\(143\) −1.05573 −0.0882844
\(144\) −3.70820 −0.309017
\(145\) 0 0
\(146\) −4.14590 −0.343117
\(147\) 1.00000 0.0824786
\(148\) −9.23607 −0.759200
\(149\) 15.2361 1.24819 0.624094 0.781350i \(-0.285468\pi\)
0.624094 + 0.781350i \(0.285468\pi\)
\(150\) 0 0
\(151\) −15.1803 −1.23536 −0.617679 0.786430i \(-0.711927\pi\)
−0.617679 + 0.786430i \(0.711927\pi\)
\(152\) −16.1803 −1.31240
\(153\) 0 0
\(154\) −2.76393 −0.222724
\(155\) 0 0
\(156\) 0.381966 0.0305818
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) 4.47214 0.355784
\(159\) −11.2361 −0.891078
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −0.618034 −0.0485573
\(163\) −19.1803 −1.50232 −0.751160 0.660120i \(-0.770505\pi\)
−0.751160 + 0.660120i \(0.770505\pi\)
\(164\) 3.61803 0.282521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −21.8885 −1.69379 −0.846893 0.531763i \(-0.821530\pi\)
−0.846893 + 0.531763i \(0.821530\pi\)
\(168\) 2.23607 0.172516
\(169\) −12.9443 −0.995713
\(170\) 0 0
\(171\) 14.4721 1.10671
\(172\) 4.00000 0.304997
\(173\) 3.52786 0.268219 0.134109 0.990967i \(-0.457183\pi\)
0.134109 + 0.990967i \(0.457183\pi\)
\(174\) 0.909830 0.0689740
\(175\) 0 0
\(176\) 8.29180 0.625018
\(177\) −1.52786 −0.114841
\(178\) −5.52786 −0.414331
\(179\) −18.7082 −1.39832 −0.699158 0.714967i \(-0.746442\pi\)
−0.699158 + 0.714967i \(0.746442\pi\)
\(180\) 0 0
\(181\) −5.05573 −0.375789 −0.187895 0.982189i \(-0.560166\pi\)
−0.187895 + 0.982189i \(0.560166\pi\)
\(182\) 0.145898 0.0108147
\(183\) 13.4164 0.991769
\(184\) 2.23607 0.164845
\(185\) 0 0
\(186\) 5.56231 0.407848
\(187\) 0 0
\(188\) −5.61803 −0.409737
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −24.1803 −1.74963 −0.874814 0.484459i \(-0.839016\pi\)
−0.874814 + 0.484459i \(0.839016\pi\)
\(192\) −0.236068 −0.0170367
\(193\) −4.41641 −0.317900 −0.158950 0.987287i \(-0.550811\pi\)
−0.158950 + 0.987287i \(0.550811\pi\)
\(194\) 2.29180 0.164541
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) 21.4721 1.52983 0.764913 0.644133i \(-0.222782\pi\)
0.764913 + 0.644133i \(0.222782\pi\)
\(198\) 5.52786 0.392848
\(199\) 15.8885 1.12631 0.563155 0.826352i \(-0.309588\pi\)
0.563155 + 0.826352i \(0.309588\pi\)
\(200\) 0 0
\(201\) 12.1803 0.859135
\(202\) 8.29180 0.583409
\(203\) −1.47214 −0.103324
\(204\) 0 0
\(205\) 0 0
\(206\) −9.70820 −0.676403
\(207\) −2.00000 −0.139010
\(208\) −0.437694 −0.0303486
\(209\) −32.3607 −2.23844
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 18.1803 1.24863
\(213\) −10.2361 −0.701364
\(214\) 6.94427 0.474701
\(215\) 0 0
\(216\) −11.1803 −0.760726
\(217\) −9.00000 −0.610960
\(218\) 9.52786 0.645308
\(219\) 6.70820 0.453298
\(220\) 0 0
\(221\) 0 0
\(222\) −3.52786 −0.236775
\(223\) −3.41641 −0.228780 −0.114390 0.993436i \(-0.536491\pi\)
−0.114390 + 0.993436i \(0.536491\pi\)
\(224\) −5.61803 −0.375371
\(225\) 0 0
\(226\) 1.52786 0.101632
\(227\) 5.88854 0.390836 0.195418 0.980720i \(-0.437394\pi\)
0.195418 + 0.980720i \(0.437394\pi\)
\(228\) 11.7082 0.775395
\(229\) 14.7639 0.975628 0.487814 0.872948i \(-0.337794\pi\)
0.487814 + 0.872948i \(0.337794\pi\)
\(230\) 0 0
\(231\) 4.47214 0.294245
\(232\) −3.29180 −0.216117
\(233\) 11.4721 0.751565 0.375782 0.926708i \(-0.377374\pi\)
0.375782 + 0.926708i \(0.377374\pi\)
\(234\) −0.291796 −0.0190753
\(235\) 0 0
\(236\) 2.47214 0.160922
\(237\) −7.23607 −0.470033
\(238\) 0 0
\(239\) −15.7639 −1.01968 −0.509842 0.860268i \(-0.670296\pi\)
−0.509842 + 0.860268i \(0.670296\pi\)
\(240\) 0 0
\(241\) −21.4164 −1.37955 −0.689776 0.724023i \(-0.742291\pi\)
−0.689776 + 0.724023i \(0.742291\pi\)
\(242\) −5.56231 −0.357559
\(243\) 16.0000 1.02640
\(244\) −21.7082 −1.38973
\(245\) 0 0
\(246\) 1.38197 0.0881109
\(247\) 1.70820 0.108690
\(248\) −20.1246 −1.27791
\(249\) −6.47214 −0.410155
\(250\) 0 0
\(251\) 8.29180 0.523374 0.261687 0.965153i \(-0.415721\pi\)
0.261687 + 0.965153i \(0.415721\pi\)
\(252\) 3.23607 0.203853
\(253\) 4.47214 0.281161
\(254\) −1.67376 −0.105021
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −4.23607 −0.264239 −0.132119 0.991234i \(-0.542178\pi\)
−0.132119 + 0.991234i \(0.542178\pi\)
\(258\) 1.52786 0.0951207
\(259\) 5.70820 0.354691
\(260\) 0 0
\(261\) 2.94427 0.182246
\(262\) −2.43769 −0.150601
\(263\) 26.9443 1.66145 0.830727 0.556679i \(-0.187925\pi\)
0.830727 + 0.556679i \(0.187925\pi\)
\(264\) 10.0000 0.615457
\(265\) 0 0
\(266\) 4.47214 0.274204
\(267\) 8.94427 0.547381
\(268\) −19.7082 −1.20387
\(269\) −9.18034 −0.559735 −0.279868 0.960039i \(-0.590291\pi\)
−0.279868 + 0.960039i \(0.590291\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 0 0
\(273\) −0.236068 −0.0142875
\(274\) 9.70820 0.586494
\(275\) 0 0
\(276\) −1.61803 −0.0973942
\(277\) −20.4164 −1.22670 −0.613352 0.789810i \(-0.710179\pi\)
−0.613352 + 0.789810i \(0.710179\pi\)
\(278\) −1.56231 −0.0937009
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) 3.70820 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(282\) −2.14590 −0.127786
\(283\) −18.9443 −1.12612 −0.563060 0.826416i \(-0.690376\pi\)
−0.563060 + 0.826416i \(0.690376\pi\)
\(284\) 16.5623 0.982792
\(285\) 0 0
\(286\) 0.652476 0.0385817
\(287\) −2.23607 −0.131991
\(288\) 11.2361 0.662092
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −3.70820 −0.217379
\(292\) −10.8541 −0.635188
\(293\) 15.7082 0.917683 0.458842 0.888518i \(-0.348265\pi\)
0.458842 + 0.888518i \(0.348265\pi\)
\(294\) −0.618034 −0.0360445
\(295\) 0 0
\(296\) 12.7639 0.741888
\(297\) −22.3607 −1.29750
\(298\) −9.41641 −0.545478
\(299\) −0.236068 −0.0136522
\(300\) 0 0
\(301\) −2.47214 −0.142492
\(302\) 9.38197 0.539871
\(303\) −13.4164 −0.770752
\(304\) −13.4164 −0.769484
\(305\) 0 0
\(306\) 0 0
\(307\) 11.4164 0.651569 0.325784 0.945444i \(-0.394372\pi\)
0.325784 + 0.945444i \(0.394372\pi\)
\(308\) −7.23607 −0.412313
\(309\) 15.7082 0.893609
\(310\) 0 0
\(311\) −2.88854 −0.163794 −0.0818971 0.996641i \(-0.526098\pi\)
−0.0818971 + 0.996641i \(0.526098\pi\)
\(312\) −0.527864 −0.0298844
\(313\) −2.76393 −0.156227 −0.0781133 0.996944i \(-0.524890\pi\)
−0.0781133 + 0.996944i \(0.524890\pi\)
\(314\) 9.52786 0.537688
\(315\) 0 0
\(316\) 11.7082 0.658638
\(317\) −31.3050 −1.75826 −0.879131 0.476581i \(-0.841876\pi\)
−0.879131 + 0.476581i \(0.841876\pi\)
\(318\) 6.94427 0.389415
\(319\) −6.58359 −0.368610
\(320\) 0 0
\(321\) −11.2361 −0.627136
\(322\) −0.618034 −0.0344417
\(323\) 0 0
\(324\) −1.61803 −0.0898908
\(325\) 0 0
\(326\) 11.8541 0.656538
\(327\) −15.4164 −0.852529
\(328\) −5.00000 −0.276079
\(329\) 3.47214 0.191425
\(330\) 0 0
\(331\) 0.708204 0.0389264 0.0194632 0.999811i \(-0.493804\pi\)
0.0194632 + 0.999811i \(0.493804\pi\)
\(332\) 10.4721 0.574733
\(333\) −11.4164 −0.625615
\(334\) 13.5279 0.740212
\(335\) 0 0
\(336\) 1.85410 0.101150
\(337\) 17.5967 0.958556 0.479278 0.877663i \(-0.340899\pi\)
0.479278 + 0.877663i \(0.340899\pi\)
\(338\) 8.00000 0.435143
\(339\) −2.47214 −0.134268
\(340\) 0 0
\(341\) −40.2492 −2.17962
\(342\) −8.94427 −0.483651
\(343\) 1.00000 0.0539949
\(344\) −5.52786 −0.298042
\(345\) 0 0
\(346\) −2.18034 −0.117216
\(347\) 5.88854 0.316114 0.158057 0.987430i \(-0.449477\pi\)
0.158057 + 0.987430i \(0.449477\pi\)
\(348\) 2.38197 0.127687
\(349\) 1.29180 0.0691483 0.0345741 0.999402i \(-0.488993\pi\)
0.0345741 + 0.999402i \(0.488993\pi\)
\(350\) 0 0
\(351\) 1.18034 0.0630019
\(352\) −25.1246 −1.33915
\(353\) 1.76393 0.0938846 0.0469423 0.998898i \(-0.485052\pi\)
0.0469423 + 0.998898i \(0.485052\pi\)
\(354\) 0.944272 0.0501875
\(355\) 0 0
\(356\) −14.4721 −0.767022
\(357\) 0 0
\(358\) 11.5623 0.611087
\(359\) 4.76393 0.251431 0.125715 0.992066i \(-0.459877\pi\)
0.125715 + 0.992066i \(0.459877\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) 3.12461 0.164226
\(363\) 9.00000 0.472377
\(364\) 0.381966 0.0200205
\(365\) 0 0
\(366\) −8.29180 −0.433419
\(367\) 4.58359 0.239262 0.119631 0.992818i \(-0.461829\pi\)
0.119631 + 0.992818i \(0.461829\pi\)
\(368\) 1.85410 0.0966517
\(369\) 4.47214 0.232810
\(370\) 0 0
\(371\) −11.2361 −0.583348
\(372\) 14.5623 0.755020
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 7.76393 0.400394
\(377\) 0.347524 0.0178984
\(378\) 3.09017 0.158941
\(379\) 8.29180 0.425921 0.212960 0.977061i \(-0.431689\pi\)
0.212960 + 0.977061i \(0.431689\pi\)
\(380\) 0 0
\(381\) 2.70820 0.138745
\(382\) 14.9443 0.764615
\(383\) 15.7082 0.802652 0.401326 0.915935i \(-0.368549\pi\)
0.401326 + 0.915935i \(0.368549\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) 2.72949 0.138927
\(387\) 4.94427 0.251331
\(388\) 6.00000 0.304604
\(389\) −7.41641 −0.376027 −0.188013 0.982166i \(-0.560205\pi\)
−0.188013 + 0.982166i \(0.560205\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 2.23607 0.112938
\(393\) 3.94427 0.198962
\(394\) −13.2705 −0.668559
\(395\) 0 0
\(396\) 14.4721 0.727252
\(397\) 25.6525 1.28746 0.643730 0.765252i \(-0.277386\pi\)
0.643730 + 0.765252i \(0.277386\pi\)
\(398\) −9.81966 −0.492215
\(399\) −7.23607 −0.362257
\(400\) 0 0
\(401\) −19.8885 −0.993186 −0.496593 0.867983i \(-0.665416\pi\)
−0.496593 + 0.867983i \(0.665416\pi\)
\(402\) −7.52786 −0.375456
\(403\) 2.12461 0.105834
\(404\) 21.7082 1.08002
\(405\) 0 0
\(406\) 0.909830 0.0451541
\(407\) 25.5279 1.26537
\(408\) 0 0
\(409\) 4.12461 0.203949 0.101974 0.994787i \(-0.467484\pi\)
0.101974 + 0.994787i \(0.467484\pi\)
\(410\) 0 0
\(411\) −15.7082 −0.774829
\(412\) −25.4164 −1.25218
\(413\) −1.52786 −0.0751813
\(414\) 1.23607 0.0607494
\(415\) 0 0
\(416\) 1.32624 0.0650242
\(417\) 2.52786 0.123790
\(418\) 20.0000 0.978232
\(419\) 26.4721 1.29325 0.646624 0.762809i \(-0.276180\pi\)
0.646624 + 0.762809i \(0.276180\pi\)
\(420\) 0 0
\(421\) −8.58359 −0.418339 −0.209169 0.977879i \(-0.567076\pi\)
−0.209169 + 0.977879i \(0.567076\pi\)
\(422\) 7.41641 0.361025
\(423\) −6.94427 −0.337642
\(424\) −25.1246 −1.22016
\(425\) 0 0
\(426\) 6.32624 0.306507
\(427\) 13.4164 0.649265
\(428\) 18.1803 0.878780
\(429\) −1.05573 −0.0509710
\(430\) 0 0
\(431\) 18.1803 0.875716 0.437858 0.899044i \(-0.355737\pi\)
0.437858 + 0.899044i \(0.355737\pi\)
\(432\) −9.27051 −0.446028
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 5.56231 0.266999
\(435\) 0 0
\(436\) 24.9443 1.19461
\(437\) −7.23607 −0.346148
\(438\) −4.14590 −0.198099
\(439\) 30.3050 1.44638 0.723188 0.690651i \(-0.242676\pi\)
0.723188 + 0.690651i \(0.242676\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 7.18034 0.341148 0.170574 0.985345i \(-0.445438\pi\)
0.170574 + 0.985345i \(0.445438\pi\)
\(444\) −9.23607 −0.438324
\(445\) 0 0
\(446\) 2.11146 0.0999803
\(447\) 15.2361 0.720641
\(448\) −0.236068 −0.0111532
\(449\) 32.8328 1.54948 0.774738 0.632282i \(-0.217882\pi\)
0.774738 + 0.632282i \(0.217882\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 4.00000 0.188144
\(453\) −15.1803 −0.713235
\(454\) −3.63932 −0.170802
\(455\) 0 0
\(456\) −16.1803 −0.757714
\(457\) −14.4721 −0.676978 −0.338489 0.940970i \(-0.609916\pi\)
−0.338489 + 0.940970i \(0.609916\pi\)
\(458\) −9.12461 −0.426365
\(459\) 0 0
\(460\) 0 0
\(461\) −25.7639 −1.19995 −0.599973 0.800020i \(-0.704822\pi\)
−0.599973 + 0.800020i \(0.704822\pi\)
\(462\) −2.76393 −0.128590
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.72949 −0.126713
\(465\) 0 0
\(466\) −7.09017 −0.328446
\(467\) 11.5279 0.533446 0.266723 0.963773i \(-0.414059\pi\)
0.266723 + 0.963773i \(0.414059\pi\)
\(468\) −0.763932 −0.0353128
\(469\) 12.1803 0.562436
\(470\) 0 0
\(471\) −15.4164 −0.710351
\(472\) −3.41641 −0.157253
\(473\) −11.0557 −0.508343
\(474\) 4.47214 0.205412
\(475\) 0 0
\(476\) 0 0
\(477\) 22.4721 1.02893
\(478\) 9.74265 0.445618
\(479\) −23.1246 −1.05659 −0.528295 0.849061i \(-0.677169\pi\)
−0.528295 + 0.849061i \(0.677169\pi\)
\(480\) 0 0
\(481\) −1.34752 −0.0614418
\(482\) 13.2361 0.602886
\(483\) 1.00000 0.0455016
\(484\) −14.5623 −0.661923
\(485\) 0 0
\(486\) −9.88854 −0.448553
\(487\) −20.1246 −0.911933 −0.455967 0.889997i \(-0.650706\pi\)
−0.455967 + 0.889997i \(0.650706\pi\)
\(488\) 30.0000 1.35804
\(489\) −19.1803 −0.867365
\(490\) 0 0
\(491\) 11.7639 0.530899 0.265449 0.964125i \(-0.414480\pi\)
0.265449 + 0.964125i \(0.414480\pi\)
\(492\) 3.61803 0.163114
\(493\) 0 0
\(494\) −1.05573 −0.0474995
\(495\) 0 0
\(496\) −16.6869 −0.749265
\(497\) −10.2361 −0.459150
\(498\) 4.00000 0.179244
\(499\) −36.7082 −1.64328 −0.821642 0.570003i \(-0.806942\pi\)
−0.821642 + 0.570003i \(0.806942\pi\)
\(500\) 0 0
\(501\) −21.8885 −0.977908
\(502\) −5.12461 −0.228723
\(503\) 15.5967 0.695425 0.347712 0.937601i \(-0.386959\pi\)
0.347712 + 0.937601i \(0.386959\pi\)
\(504\) −4.47214 −0.199205
\(505\) 0 0
\(506\) −2.76393 −0.122872
\(507\) −12.9443 −0.574875
\(508\) −4.38197 −0.194418
\(509\) −12.8197 −0.568221 −0.284111 0.958791i \(-0.591698\pi\)
−0.284111 + 0.958791i \(0.591698\pi\)
\(510\) 0 0
\(511\) 6.70820 0.296753
\(512\) −18.7082 −0.826794
\(513\) 36.1803 1.59740
\(514\) 2.61803 0.115477
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 15.5279 0.682915
\(518\) −3.52786 −0.155005
\(519\) 3.52786 0.154856
\(520\) 0 0
\(521\) 21.3050 0.933387 0.466693 0.884419i \(-0.345445\pi\)
0.466693 + 0.884419i \(0.345445\pi\)
\(522\) −1.81966 −0.0796444
\(523\) 7.70820 0.337056 0.168528 0.985697i \(-0.446099\pi\)
0.168528 + 0.985697i \(0.446099\pi\)
\(524\) −6.38197 −0.278797
\(525\) 0 0
\(526\) −16.6525 −0.726082
\(527\) 0 0
\(528\) 8.29180 0.360854
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.05573 0.132607
\(532\) 11.7082 0.507615
\(533\) 0.527864 0.0228643
\(534\) −5.52786 −0.239214
\(535\) 0 0
\(536\) 27.2361 1.17642
\(537\) −18.7082 −0.807319
\(538\) 5.67376 0.244613
\(539\) 4.47214 0.192629
\(540\) 0 0
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) 10.4721 0.449817
\(543\) −5.05573 −0.216962
\(544\) 0 0
\(545\) 0 0
\(546\) 0.145898 0.00624386
\(547\) −43.1803 −1.84626 −0.923129 0.384490i \(-0.874377\pi\)
−0.923129 + 0.384490i \(0.874377\pi\)
\(548\) 25.4164 1.08574
\(549\) −26.8328 −1.14520
\(550\) 0 0
\(551\) 10.6525 0.453811
\(552\) 2.23607 0.0951734
\(553\) −7.23607 −0.307709
\(554\) 12.6180 0.536089
\(555\) 0 0
\(556\) −4.09017 −0.173462
\(557\) 26.7639 1.13402 0.567012 0.823709i \(-0.308099\pi\)
0.567012 + 0.823709i \(0.308099\pi\)
\(558\) −11.1246 −0.470942
\(559\) 0.583592 0.0246833
\(560\) 0 0
\(561\) 0 0
\(562\) −2.29180 −0.0966736
\(563\) 9.59675 0.404455 0.202227 0.979339i \(-0.435182\pi\)
0.202227 + 0.979339i \(0.435182\pi\)
\(564\) −5.61803 −0.236562
\(565\) 0 0
\(566\) 11.7082 0.492133
\(567\) 1.00000 0.0419961
\(568\) −22.8885 −0.960382
\(569\) −8.18034 −0.342938 −0.171469 0.985190i \(-0.554851\pi\)
−0.171469 + 0.985190i \(0.554851\pi\)
\(570\) 0 0
\(571\) 16.2918 0.681790 0.340895 0.940101i \(-0.389270\pi\)
0.340895 + 0.940101i \(0.389270\pi\)
\(572\) 1.70820 0.0714236
\(573\) −24.1803 −1.01015
\(574\) 1.38197 0.0576821
\(575\) 0 0
\(576\) 0.472136 0.0196723
\(577\) 15.2918 0.636606 0.318303 0.947989i \(-0.396887\pi\)
0.318303 + 0.947989i \(0.396887\pi\)
\(578\) 10.5066 0.437016
\(579\) −4.41641 −0.183540
\(580\) 0 0
\(581\) −6.47214 −0.268509
\(582\) 2.29180 0.0949980
\(583\) −50.2492 −2.08111
\(584\) 15.0000 0.620704
\(585\) 0 0
\(586\) −9.70820 −0.401042
\(587\) −20.8885 −0.862162 −0.431081 0.902313i \(-0.641868\pi\)
−0.431081 + 0.902313i \(0.641868\pi\)
\(588\) −1.61803 −0.0667266
\(589\) 65.1246 2.68341
\(590\) 0 0
\(591\) 21.4721 0.883246
\(592\) 10.5836 0.434983
\(593\) −34.3607 −1.41102 −0.705512 0.708698i \(-0.749283\pi\)
−0.705512 + 0.708698i \(0.749283\pi\)
\(594\) 13.8197 0.567028
\(595\) 0 0
\(596\) −24.6525 −1.00980
\(597\) 15.8885 0.650275
\(598\) 0.145898 0.00596621
\(599\) −2.47214 −0.101009 −0.0505044 0.998724i \(-0.516083\pi\)
−0.0505044 + 0.998724i \(0.516083\pi\)
\(600\) 0 0
\(601\) 22.2361 0.907028 0.453514 0.891249i \(-0.350170\pi\)
0.453514 + 0.891249i \(0.350170\pi\)
\(602\) 1.52786 0.0622711
\(603\) −24.3607 −0.992044
\(604\) 24.5623 0.999426
\(605\) 0 0
\(606\) 8.29180 0.336831
\(607\) 33.3050 1.35181 0.675903 0.736990i \(-0.263754\pi\)
0.675903 + 0.736990i \(0.263754\pi\)
\(608\) 40.6525 1.64868
\(609\) −1.47214 −0.0596540
\(610\) 0 0
\(611\) −0.819660 −0.0331599
\(612\) 0 0
\(613\) 22.1803 0.895855 0.447928 0.894070i \(-0.352162\pi\)
0.447928 + 0.894070i \(0.352162\pi\)
\(614\) −7.05573 −0.284746
\(615\) 0 0
\(616\) 10.0000 0.402911
\(617\) −35.2361 −1.41855 −0.709275 0.704932i \(-0.750978\pi\)
−0.709275 + 0.704932i \(0.750978\pi\)
\(618\) −9.70820 −0.390521
\(619\) −47.4853 −1.90860 −0.954298 0.298858i \(-0.903394\pi\)
−0.954298 + 0.298858i \(0.903394\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 1.78522 0.0715807
\(623\) 8.94427 0.358345
\(624\) −0.437694 −0.0175218
\(625\) 0 0
\(626\) 1.70820 0.0682736
\(627\) −32.3607 −1.29236
\(628\) 24.9443 0.995385
\(629\) 0 0
\(630\) 0 0
\(631\) 3.41641 0.136005 0.0680025 0.997685i \(-0.478337\pi\)
0.0680025 + 0.997685i \(0.478337\pi\)
\(632\) −16.1803 −0.643619
\(633\) −12.0000 −0.476957
\(634\) 19.3475 0.768388
\(635\) 0 0
\(636\) 18.1803 0.720897
\(637\) −0.236068 −0.00935335
\(638\) 4.06888 0.161089
\(639\) 20.4721 0.809865
\(640\) 0 0
\(641\) 7.05573 0.278685 0.139342 0.990244i \(-0.455501\pi\)
0.139342 + 0.990244i \(0.455501\pi\)
\(642\) 6.94427 0.274069
\(643\) 17.7082 0.698343 0.349172 0.937059i \(-0.386463\pi\)
0.349172 + 0.937059i \(0.386463\pi\)
\(644\) −1.61803 −0.0637595
\(645\) 0 0
\(646\) 0 0
\(647\) 18.5279 0.728405 0.364203 0.931320i \(-0.381342\pi\)
0.364203 + 0.931320i \(0.381342\pi\)
\(648\) 2.23607 0.0878410
\(649\) −6.83282 −0.268211
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 31.0344 1.21540
\(653\) −24.5279 −0.959849 −0.479925 0.877310i \(-0.659336\pi\)
−0.479925 + 0.877310i \(0.659336\pi\)
\(654\) 9.52786 0.372569
\(655\) 0 0
\(656\) −4.14590 −0.161870
\(657\) −13.4164 −0.523424
\(658\) −2.14590 −0.0836558
\(659\) 6.76393 0.263485 0.131743 0.991284i \(-0.457943\pi\)
0.131743 + 0.991284i \(0.457943\pi\)
\(660\) 0 0
\(661\) −41.7082 −1.62226 −0.811131 0.584865i \(-0.801147\pi\)
−0.811131 + 0.584865i \(0.801147\pi\)
\(662\) −0.437694 −0.0170115
\(663\) 0 0
\(664\) −14.4721 −0.561628
\(665\) 0 0
\(666\) 7.05573 0.273404
\(667\) −1.47214 −0.0570013
\(668\) 35.4164 1.37030
\(669\) −3.41641 −0.132086
\(670\) 0 0
\(671\) 60.0000 2.31627
\(672\) −5.61803 −0.216720
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) −10.8754 −0.418904
\(675\) 0 0
\(676\) 20.9443 0.805549
\(677\) −23.2361 −0.893035 −0.446517 0.894775i \(-0.647336\pi\)
−0.446517 + 0.894775i \(0.647336\pi\)
\(678\) 1.52786 0.0586773
\(679\) −3.70820 −0.142308
\(680\) 0 0
\(681\) 5.88854 0.225649
\(682\) 24.8754 0.952528
\(683\) 5.18034 0.198220 0.0991101 0.995076i \(-0.468400\pi\)
0.0991101 + 0.995076i \(0.468400\pi\)
\(684\) −23.4164 −0.895349
\(685\) 0 0
\(686\) −0.618034 −0.0235966
\(687\) 14.7639 0.563279
\(688\) −4.58359 −0.174748
\(689\) 2.65248 0.101051
\(690\) 0 0
\(691\) 42.8328 1.62944 0.814719 0.579857i \(-0.196891\pi\)
0.814719 + 0.579857i \(0.196891\pi\)
\(692\) −5.70820 −0.216993
\(693\) −8.94427 −0.339765
\(694\) −3.63932 −0.138147
\(695\) 0 0
\(696\) −3.29180 −0.124775
\(697\) 0 0
\(698\) −0.798374 −0.0302189
\(699\) 11.4721 0.433916
\(700\) 0 0
\(701\) −22.7639 −0.859782 −0.429891 0.902881i \(-0.641448\pi\)
−0.429891 + 0.902881i \(0.641448\pi\)
\(702\) −0.729490 −0.0275328
\(703\) −41.3050 −1.55785
\(704\) −1.05573 −0.0397892
\(705\) 0 0
\(706\) −1.09017 −0.0410291
\(707\) −13.4164 −0.504576
\(708\) 2.47214 0.0929086
\(709\) 34.2492 1.28626 0.643128 0.765758i \(-0.277636\pi\)
0.643128 + 0.765758i \(0.277636\pi\)
\(710\) 0 0
\(711\) 14.4721 0.542748
\(712\) 20.0000 0.749532
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) 0 0
\(716\) 30.2705 1.13126
\(717\) −15.7639 −0.588715
\(718\) −2.94427 −0.109879
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 15.7082 0.585004
\(722\) −20.6180 −0.767324
\(723\) −21.4164 −0.796485
\(724\) 8.18034 0.304020
\(725\) 0 0
\(726\) −5.56231 −0.206437
\(727\) −3.23607 −0.120019 −0.0600096 0.998198i \(-0.519113\pi\)
−0.0600096 + 0.998198i \(0.519113\pi\)
\(728\) −0.527864 −0.0195639
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) −21.7082 −0.802358
\(733\) −12.9443 −0.478108 −0.239054 0.971006i \(-0.576837\pi\)
−0.239054 + 0.971006i \(0.576837\pi\)
\(734\) −2.83282 −0.104561
\(735\) 0 0
\(736\) −5.61803 −0.207083
\(737\) 54.4721 2.00651
\(738\) −2.76393 −0.101742
\(739\) 8.70820 0.320336 0.160168 0.987090i \(-0.448796\pi\)
0.160168 + 0.987090i \(0.448796\pi\)
\(740\) 0 0
\(741\) 1.70820 0.0627524
\(742\) 6.94427 0.254932
\(743\) 25.5279 0.936527 0.468263 0.883589i \(-0.344880\pi\)
0.468263 + 0.883589i \(0.344880\pi\)
\(744\) −20.1246 −0.737804
\(745\) 0 0
\(746\) 16.0689 0.588324
\(747\) 12.9443 0.473606
\(748\) 0 0
\(749\) −11.2361 −0.410557
\(750\) 0 0
\(751\) −8.58359 −0.313220 −0.156610 0.987661i \(-0.550057\pi\)
−0.156610 + 0.987661i \(0.550057\pi\)
\(752\) 6.43769 0.234759
\(753\) 8.29180 0.302170
\(754\) −0.214782 −0.00782189
\(755\) 0 0
\(756\) 8.09017 0.294237
\(757\) 28.8328 1.04795 0.523973 0.851735i \(-0.324449\pi\)
0.523973 + 0.851735i \(0.324449\pi\)
\(758\) −5.12461 −0.186134
\(759\) 4.47214 0.162328
\(760\) 0 0
\(761\) −31.6525 −1.14740 −0.573701 0.819065i \(-0.694493\pi\)
−0.573701 + 0.819065i \(0.694493\pi\)
\(762\) −1.67376 −0.0606340
\(763\) −15.4164 −0.558111
\(764\) 39.1246 1.41548
\(765\) 0 0
\(766\) −9.70820 −0.350772
\(767\) 0.360680 0.0130234
\(768\) −6.56231 −0.236797
\(769\) −43.2361 −1.55913 −0.779566 0.626320i \(-0.784560\pi\)
−0.779566 + 0.626320i \(0.784560\pi\)
\(770\) 0 0
\(771\) −4.23607 −0.152558
\(772\) 7.14590 0.257186
\(773\) −40.6525 −1.46217 −0.731084 0.682288i \(-0.760985\pi\)
−0.731084 + 0.682288i \(0.760985\pi\)
\(774\) −3.05573 −0.109836
\(775\) 0 0
\(776\) −8.29180 −0.297658
\(777\) 5.70820 0.204781
\(778\) 4.58359 0.164330
\(779\) 16.1803 0.579721
\(780\) 0 0
\(781\) −45.7771 −1.63803
\(782\) 0 0
\(783\) 7.36068 0.263049
\(784\) 1.85410 0.0662179
\(785\) 0 0
\(786\) −2.43769 −0.0869497
\(787\) 0.360680 0.0128568 0.00642842 0.999979i \(-0.497954\pi\)
0.00642842 + 0.999979i \(0.497954\pi\)
\(788\) −34.7426 −1.23766
\(789\) 26.9443 0.959241
\(790\) 0 0
\(791\) −2.47214 −0.0878990
\(792\) −20.0000 −0.710669
\(793\) −3.16718 −0.112470
\(794\) −15.8541 −0.562641
\(795\) 0 0
\(796\) −25.7082 −0.911203
\(797\) 36.7639 1.30225 0.651123 0.758973i \(-0.274298\pi\)
0.651123 + 0.758973i \(0.274298\pi\)
\(798\) 4.47214 0.158312
\(799\) 0 0
\(800\) 0 0
\(801\) −17.8885 −0.632061
\(802\) 12.2918 0.434038
\(803\) 30.0000 1.05868
\(804\) −19.7082 −0.695055
\(805\) 0 0
\(806\) −1.31308 −0.0462514
\(807\) −9.18034 −0.323163
\(808\) −30.0000 −1.05540
\(809\) −32.8328 −1.15434 −0.577170 0.816624i \(-0.695843\pi\)
−0.577170 + 0.816624i \(0.695843\pi\)
\(810\) 0 0
\(811\) −45.3607 −1.59283 −0.796414 0.604751i \(-0.793273\pi\)
−0.796414 + 0.604751i \(0.793273\pi\)
\(812\) 2.38197 0.0835906
\(813\) −16.9443 −0.594262
\(814\) −15.7771 −0.552987
\(815\) 0 0
\(816\) 0 0
\(817\) 17.8885 0.625841
\(818\) −2.54915 −0.0891289
\(819\) 0.472136 0.0164978
\(820\) 0 0
\(821\) 22.5836 0.788173 0.394086 0.919073i \(-0.371061\pi\)
0.394086 + 0.919073i \(0.371061\pi\)
\(822\) 9.70820 0.338612
\(823\) −25.5410 −0.890304 −0.445152 0.895455i \(-0.646850\pi\)
−0.445152 + 0.895455i \(0.646850\pi\)
\(824\) 35.1246 1.22362
\(825\) 0 0
\(826\) 0.944272 0.0328554
\(827\) 17.3050 0.601752 0.300876 0.953663i \(-0.402721\pi\)
0.300876 + 0.953663i \(0.402721\pi\)
\(828\) 3.23607 0.112461
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −20.4164 −0.708237
\(832\) 0.0557281 0.00193202
\(833\) 0 0
\(834\) −1.56231 −0.0540982
\(835\) 0 0
\(836\) 52.3607 1.81093
\(837\) 45.0000 1.55543
\(838\) −16.3607 −0.565170
\(839\) 20.0689 0.692855 0.346427 0.938077i \(-0.387395\pi\)
0.346427 + 0.938077i \(0.387395\pi\)
\(840\) 0 0
\(841\) −26.8328 −0.925270
\(842\) 5.30495 0.182821
\(843\) 3.70820 0.127717
\(844\) 19.4164 0.668340
\(845\) 0 0
\(846\) 4.29180 0.147555
\(847\) 9.00000 0.309244
\(848\) −20.8328 −0.715402
\(849\) −18.9443 −0.650166
\(850\) 0 0
\(851\) 5.70820 0.195675
\(852\) 16.5623 0.567415
\(853\) −44.8328 −1.53505 −0.767523 0.641021i \(-0.778511\pi\)
−0.767523 + 0.641021i \(0.778511\pi\)
\(854\) −8.29180 −0.283739
\(855\) 0 0
\(856\) −25.1246 −0.858742
\(857\) −9.54102 −0.325915 −0.162958 0.986633i \(-0.552103\pi\)
−0.162958 + 0.986633i \(0.552103\pi\)
\(858\) 0.652476 0.0222752
\(859\) 11.5836 0.395227 0.197614 0.980280i \(-0.436681\pi\)
0.197614 + 0.980280i \(0.436681\pi\)
\(860\) 0 0
\(861\) −2.23607 −0.0762050
\(862\) −11.2361 −0.382702
\(863\) −2.23607 −0.0761166 −0.0380583 0.999276i \(-0.512117\pi\)
−0.0380583 + 0.999276i \(0.512117\pi\)
\(864\) 28.0902 0.955647
\(865\) 0 0
\(866\) 8.65248 0.294023
\(867\) −17.0000 −0.577350
\(868\) 14.5623 0.494277
\(869\) −32.3607 −1.09776
\(870\) 0 0
\(871\) −2.87539 −0.0974288
\(872\) −34.4721 −1.16737
\(873\) 7.41641 0.251007
\(874\) 4.47214 0.151272
\(875\) 0 0
\(876\) −10.8541 −0.366726
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) −18.7295 −0.632090
\(879\) 15.7082 0.529825
\(880\) 0 0
\(881\) 25.5967 0.862376 0.431188 0.902262i \(-0.358095\pi\)
0.431188 + 0.902262i \(0.358095\pi\)
\(882\) 1.23607 0.0416206
\(883\) −9.88854 −0.332776 −0.166388 0.986060i \(-0.553210\pi\)
−0.166388 + 0.986060i \(0.553210\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.43769 −0.149087
\(887\) −16.8885 −0.567062 −0.283531 0.958963i \(-0.591506\pi\)
−0.283531 + 0.958963i \(0.591506\pi\)
\(888\) 12.7639 0.428330
\(889\) 2.70820 0.0908302
\(890\) 0 0
\(891\) 4.47214 0.149822
\(892\) 5.52786 0.185087
\(893\) −25.1246 −0.840763
\(894\) −9.41641 −0.314932
\(895\) 0 0
\(896\) 11.3820 0.380245
\(897\) −0.236068 −0.00788208
\(898\) −20.2918 −0.677146
\(899\) 13.2492 0.441886
\(900\) 0 0
\(901\) 0 0
\(902\) 6.18034 0.205783
\(903\) −2.47214 −0.0822675
\(904\) −5.52786 −0.183854
\(905\) 0 0
\(906\) 9.38197 0.311695
\(907\) −12.5410 −0.416418 −0.208209 0.978084i \(-0.566763\pi\)
−0.208209 + 0.978084i \(0.566763\pi\)
\(908\) −9.52786 −0.316193
\(909\) 26.8328 0.889988
\(910\) 0 0
\(911\) −21.7771 −0.721507 −0.360754 0.932661i \(-0.617480\pi\)
−0.360754 + 0.932661i \(0.617480\pi\)
\(912\) −13.4164 −0.444262
\(913\) −28.9443 −0.957916
\(914\) 8.94427 0.295850
\(915\) 0 0
\(916\) −23.8885 −0.789300
\(917\) 3.94427 0.130251
\(918\) 0 0
\(919\) 3.70820 0.122322 0.0611612 0.998128i \(-0.480520\pi\)
0.0611612 + 0.998128i \(0.480520\pi\)
\(920\) 0 0
\(921\) 11.4164 0.376183
\(922\) 15.9230 0.524396
\(923\) 2.41641 0.0795370
\(924\) −7.23607 −0.238049
\(925\) 0 0
\(926\) 0 0
\(927\) −31.4164 −1.03185
\(928\) 8.27051 0.271493
\(929\) 51.6525 1.69466 0.847331 0.531065i \(-0.178208\pi\)
0.847331 + 0.531065i \(0.178208\pi\)
\(930\) 0 0
\(931\) −7.23607 −0.237153
\(932\) −18.5623 −0.608029
\(933\) −2.88854 −0.0945667
\(934\) −7.12461 −0.233124
\(935\) 0 0
\(936\) 1.05573 0.0345076
\(937\) 57.1246 1.86618 0.933090 0.359643i \(-0.117102\pi\)
0.933090 + 0.359643i \(0.117102\pi\)
\(938\) −7.52786 −0.245793
\(939\) −2.76393 −0.0901975
\(940\) 0 0
\(941\) 53.0132 1.72818 0.864090 0.503338i \(-0.167895\pi\)
0.864090 + 0.503338i \(0.167895\pi\)
\(942\) 9.52786 0.310435
\(943\) −2.23607 −0.0728164
\(944\) −2.83282 −0.0922003
\(945\) 0 0
\(946\) 6.83282 0.222154
\(947\) 22.5967 0.734296 0.367148 0.930163i \(-0.380334\pi\)
0.367148 + 0.930163i \(0.380334\pi\)
\(948\) 11.7082 0.380265
\(949\) −1.58359 −0.0514056
\(950\) 0 0
\(951\) −31.3050 −1.01513
\(952\) 0 0
\(953\) −12.1115 −0.392329 −0.196164 0.980571i \(-0.562849\pi\)
−0.196164 + 0.980571i \(0.562849\pi\)
\(954\) −13.8885 −0.449658
\(955\) 0 0
\(956\) 25.5066 0.824942
\(957\) −6.58359 −0.212817
\(958\) 14.2918 0.461747
\(959\) −15.7082 −0.507244
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0.832816 0.0268511
\(963\) 22.4721 0.724154
\(964\) 34.6525 1.11608
\(965\) 0 0
\(966\) −0.618034 −0.0198849
\(967\) 31.0689 0.999108 0.499554 0.866283i \(-0.333497\pi\)
0.499554 + 0.866283i \(0.333497\pi\)
\(968\) 20.1246 0.646830
\(969\) 0 0
\(970\) 0 0
\(971\) 33.7082 1.08175 0.540874 0.841104i \(-0.318094\pi\)
0.540874 + 0.841104i \(0.318094\pi\)
\(972\) −25.8885 −0.830375
\(973\) 2.52786 0.0810396
\(974\) 12.4377 0.398529
\(975\) 0 0
\(976\) 24.8754 0.796242
\(977\) 18.6525 0.596746 0.298373 0.954449i \(-0.403556\pi\)
0.298373 + 0.954449i \(0.403556\pi\)
\(978\) 11.8541 0.379052
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 30.8328 0.984416
\(982\) −7.27051 −0.232011
\(983\) 4.18034 0.133332 0.0666661 0.997775i \(-0.478764\pi\)
0.0666661 + 0.997775i \(0.478764\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) 0 0
\(987\) 3.47214 0.110519
\(988\) −2.76393 −0.0879324
\(989\) −2.47214 −0.0786094
\(990\) 0 0
\(991\) −48.9443 −1.55477 −0.777383 0.629028i \(-0.783453\pi\)
−0.777383 + 0.629028i \(0.783453\pi\)
\(992\) 50.5623 1.60535
\(993\) 0.708204 0.0224742
\(994\) 6.32624 0.200656
\(995\) 0 0
\(996\) 10.4721 0.331822
\(997\) 27.3050 0.864756 0.432378 0.901692i \(-0.357675\pi\)
0.432378 + 0.901692i \(0.357675\pi\)
\(998\) 22.6869 0.718142
\(999\) −28.5410 −0.902998
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 4025.2.a.i.1.1 2
5.4 even 2 161.2.a.b.1.2 2
15.14 odd 2 1449.2.a.i.1.1 2
20.19 odd 2 2576.2.a.s.1.1 2
35.34 odd 2 1127.2.a.d.1.2 2
115.114 odd 2 3703.2.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.b.1.2 2 5.4 even 2
1127.2.a.d.1.2 2 35.34 odd 2
1449.2.a.i.1.1 2 15.14 odd 2
2576.2.a.s.1.1 2 20.19 odd 2
3703.2.a.b.1.2 2 115.114 odd 2
4025.2.a.i.1.1 2 1.1 even 1 trivial