Properties

Label 2541.2.a.q.1.1
Level 2541
Weight 2
Character 2541.1
Self dual yes
Analytic conductor 20.290
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\)
Character \(\chi\) = 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -2.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +3.61803 q^{10} -0.618034 q^{12} -0.236068 q^{13} +1.61803 q^{14} +2.23607 q^{15} -4.85410 q^{16} -2.00000 q^{17} -1.61803 q^{18} +1.47214 q^{19} -1.38197 q^{20} +1.00000 q^{21} -2.00000 q^{23} -2.23607 q^{24} +0.381966 q^{26} -1.00000 q^{27} -0.618034 q^{28} -5.00000 q^{29} -3.61803 q^{30} +10.4721 q^{31} +3.38197 q^{32} +3.23607 q^{34} +2.23607 q^{35} +0.618034 q^{36} -1.47214 q^{37} -2.38197 q^{38} +0.236068 q^{39} -5.00000 q^{40} -2.47214 q^{41} -1.61803 q^{42} +8.47214 q^{43} -2.23607 q^{45} +3.23607 q^{46} +9.47214 q^{47} +4.85410 q^{48} +1.00000 q^{49} +2.00000 q^{51} -0.145898 q^{52} +6.47214 q^{53} +1.61803 q^{54} -2.23607 q^{56} -1.47214 q^{57} +8.09017 q^{58} +7.94427 q^{59} +1.38197 q^{60} -12.4721 q^{61} -16.9443 q^{62} -1.00000 q^{63} +4.23607 q^{64} +0.527864 q^{65} +9.18034 q^{67} -1.23607 q^{68} +2.00000 q^{69} -3.61803 q^{70} -2.47214 q^{71} +2.23607 q^{72} +10.7082 q^{73} +2.38197 q^{74} +0.909830 q^{76} -0.381966 q^{78} +0.472136 q^{79} +10.8541 q^{80} +1.00000 q^{81} +4.00000 q^{82} -7.52786 q^{83} +0.618034 q^{84} +4.47214 q^{85} -13.7082 q^{86} +5.00000 q^{87} -0.472136 q^{89} +3.61803 q^{90} +0.236068 q^{91} -1.23607 q^{92} -10.4721 q^{93} -15.3262 q^{94} -3.29180 q^{95} -3.38197 q^{96} +9.41641 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} + q^{6} - 2q^{7} + 2q^{9} + 5q^{10} + q^{12} + 4q^{13} + q^{14} - 3q^{16} - 4q^{17} - q^{18} - 6q^{19} - 5q^{20} + 2q^{21} - 4q^{23} + 3q^{26} - 2q^{27} + q^{28} - 10q^{29} - 5q^{30} + 12q^{31} + 9q^{32} + 2q^{34} - q^{36} + 6q^{37} - 7q^{38} - 4q^{39} - 10q^{40} + 4q^{41} - q^{42} + 8q^{43} + 2q^{46} + 10q^{47} + 3q^{48} + 2q^{49} + 4q^{51} - 7q^{52} + 4q^{53} + q^{54} + 6q^{57} + 5q^{58} - 2q^{59} + 5q^{60} - 16q^{61} - 16q^{62} - 2q^{63} + 4q^{64} + 10q^{65} - 4q^{67} + 2q^{68} + 4q^{69} - 5q^{70} + 4q^{71} + 8q^{73} + 7q^{74} + 13q^{76} - 3q^{78} - 8q^{79} + 15q^{80} + 2q^{81} + 8q^{82} - 24q^{83} - q^{84} - 14q^{86} + 10q^{87} + 8q^{89} + 5q^{90} - 4q^{91} + 2q^{92} - 12q^{93} - 15q^{94} - 20q^{95} - 9q^{96} - 8q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 1.61803 0.660560
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 3.61803 1.14412
\(11\) 0 0
\(12\) −0.618034 −0.178411
\(13\) −0.236068 −0.0654735 −0.0327367 0.999464i \(-0.510422\pi\)
−0.0327367 + 0.999464i \(0.510422\pi\)
\(14\) 1.61803 0.432438
\(15\) 2.23607 0.577350
\(16\) −4.85410 −1.21353
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.61803 −0.381374
\(19\) 1.47214 0.337731 0.168866 0.985639i \(-0.445990\pi\)
0.168866 + 0.985639i \(0.445990\pi\)
\(20\) −1.38197 −0.309017
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 0.381966 0.0749097
\(27\) −1.00000 −0.192450
\(28\) −0.618034 −0.116797
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −3.61803 −0.660560
\(31\) 10.4721 1.88085 0.940426 0.340000i \(-0.110427\pi\)
0.940426 + 0.340000i \(0.110427\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) 3.23607 0.554981
\(35\) 2.23607 0.377964
\(36\) 0.618034 0.103006
\(37\) −1.47214 −0.242018 −0.121009 0.992651i \(-0.538613\pi\)
−0.121009 + 0.992651i \(0.538613\pi\)
\(38\) −2.38197 −0.386406
\(39\) 0.236068 0.0378011
\(40\) −5.00000 −0.790569
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) −1.61803 −0.249668
\(43\) 8.47214 1.29199 0.645994 0.763342i \(-0.276443\pi\)
0.645994 + 0.763342i \(0.276443\pi\)
\(44\) 0 0
\(45\) −2.23607 −0.333333
\(46\) 3.23607 0.477132
\(47\) 9.47214 1.38165 0.690827 0.723021i \(-0.257247\pi\)
0.690827 + 0.723021i \(0.257247\pi\)
\(48\) 4.85410 0.700629
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −0.145898 −0.0202324
\(53\) 6.47214 0.889016 0.444508 0.895775i \(-0.353378\pi\)
0.444508 + 0.895775i \(0.353378\pi\)
\(54\) 1.61803 0.220187
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) −1.47214 −0.194989
\(58\) 8.09017 1.06229
\(59\) 7.94427 1.03426 0.517128 0.855908i \(-0.327001\pi\)
0.517128 + 0.855908i \(0.327001\pi\)
\(60\) 1.38197 0.178411
\(61\) −12.4721 −1.59689 −0.798447 0.602066i \(-0.794345\pi\)
−0.798447 + 0.602066i \(0.794345\pi\)
\(62\) −16.9443 −2.15192
\(63\) −1.00000 −0.125988
\(64\) 4.23607 0.529508
\(65\) 0.527864 0.0654735
\(66\) 0 0
\(67\) 9.18034 1.12156 0.560779 0.827966i \(-0.310502\pi\)
0.560779 + 0.827966i \(0.310502\pi\)
\(68\) −1.23607 −0.149895
\(69\) 2.00000 0.240772
\(70\) −3.61803 −0.432438
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 2.23607 0.263523
\(73\) 10.7082 1.25330 0.626650 0.779301i \(-0.284425\pi\)
0.626650 + 0.779301i \(0.284425\pi\)
\(74\) 2.38197 0.276898
\(75\) 0 0
\(76\) 0.909830 0.104365
\(77\) 0 0
\(78\) −0.381966 −0.0432491
\(79\) 0.472136 0.0531194 0.0265597 0.999647i \(-0.491545\pi\)
0.0265597 + 0.999647i \(0.491545\pi\)
\(80\) 10.8541 1.21353
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −7.52786 −0.826290 −0.413145 0.910665i \(-0.635570\pi\)
−0.413145 + 0.910665i \(0.635570\pi\)
\(84\) 0.618034 0.0674330
\(85\) 4.47214 0.485071
\(86\) −13.7082 −1.47819
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −0.472136 −0.0500463 −0.0250232 0.999687i \(-0.507966\pi\)
−0.0250232 + 0.999687i \(0.507966\pi\)
\(90\) 3.61803 0.381374
\(91\) 0.236068 0.0247466
\(92\) −1.23607 −0.128869
\(93\) −10.4721 −1.08591
\(94\) −15.3262 −1.58078
\(95\) −3.29180 −0.337731
\(96\) −3.38197 −0.345170
\(97\) 9.41641 0.956091 0.478046 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(98\) −1.61803 −0.163446
\(99\) 0 0
\(100\) 0 0
\(101\) −9.52786 −0.948058 −0.474029 0.880509i \(-0.657201\pi\)
−0.474029 + 0.880509i \(0.657201\pi\)
\(102\) −3.23607 −0.320418
\(103\) −10.4721 −1.03185 −0.515925 0.856634i \(-0.672552\pi\)
−0.515925 + 0.856634i \(0.672552\pi\)
\(104\) −0.527864 −0.0517613
\(105\) −2.23607 −0.218218
\(106\) −10.4721 −1.01714
\(107\) −19.6525 −1.89988 −0.949938 0.312438i \(-0.898854\pi\)
−0.949938 + 0.312438i \(0.898854\pi\)
\(108\) −0.618034 −0.0594703
\(109\) 10.4721 1.00305 0.501524 0.865144i \(-0.332773\pi\)
0.501524 + 0.865144i \(0.332773\pi\)
\(110\) 0 0
\(111\) 1.47214 0.139729
\(112\) 4.85410 0.458670
\(113\) −5.52786 −0.520018 −0.260009 0.965606i \(-0.583725\pi\)
−0.260009 + 0.965606i \(0.583725\pi\)
\(114\) 2.38197 0.223092
\(115\) 4.47214 0.417029
\(116\) −3.09017 −0.286915
\(117\) −0.236068 −0.0218245
\(118\) −12.8541 −1.18332
\(119\) 2.00000 0.183340
\(120\) 5.00000 0.456435
\(121\) 0 0
\(122\) 20.1803 1.82704
\(123\) 2.47214 0.222905
\(124\) 6.47214 0.581215
\(125\) 11.1803 1.00000
\(126\) 1.61803 0.144146
\(127\) 9.41641 0.835571 0.417786 0.908546i \(-0.362806\pi\)
0.417786 + 0.908546i \(0.362806\pi\)
\(128\) −13.6180 −1.20368
\(129\) −8.47214 −0.745930
\(130\) −0.854102 −0.0749097
\(131\) −19.4164 −1.69642 −0.848210 0.529661i \(-0.822319\pi\)
−0.848210 + 0.529661i \(0.822319\pi\)
\(132\) 0 0
\(133\) −1.47214 −0.127650
\(134\) −14.8541 −1.28320
\(135\) 2.23607 0.192450
\(136\) −4.47214 −0.383482
\(137\) −9.41641 −0.804498 −0.402249 0.915530i \(-0.631771\pi\)
−0.402249 + 0.915530i \(0.631771\pi\)
\(138\) −3.23607 −0.275472
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 1.38197 0.116797
\(141\) −9.47214 −0.797698
\(142\) 4.00000 0.335673
\(143\) 0 0
\(144\) −4.85410 −0.404508
\(145\) 11.1803 0.928477
\(146\) −17.3262 −1.43393
\(147\) −1.00000 −0.0824786
\(148\) −0.909830 −0.0747876
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 3.29180 0.267000
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −23.4164 −1.88085
\(156\) 0.145898 0.0116812
\(157\) 3.52786 0.281554 0.140777 0.990041i \(-0.455040\pi\)
0.140777 + 0.990041i \(0.455040\pi\)
\(158\) −0.763932 −0.0607752
\(159\) −6.47214 −0.513274
\(160\) −7.56231 −0.597853
\(161\) 2.00000 0.157622
\(162\) −1.61803 −0.127125
\(163\) −3.76393 −0.294814 −0.147407 0.989076i \(-0.547093\pi\)
−0.147407 + 0.989076i \(0.547093\pi\)
\(164\) −1.52786 −0.119306
\(165\) 0 0
\(166\) 12.1803 0.945378
\(167\) 5.41641 0.419134 0.209567 0.977794i \(-0.432795\pi\)
0.209567 + 0.977794i \(0.432795\pi\)
\(168\) 2.23607 0.172516
\(169\) −12.9443 −0.995713
\(170\) −7.23607 −0.554981
\(171\) 1.47214 0.112577
\(172\) 5.23607 0.399246
\(173\) 7.52786 0.572333 0.286166 0.958180i \(-0.407619\pi\)
0.286166 + 0.958180i \(0.407619\pi\)
\(174\) −8.09017 −0.613314
\(175\) 0 0
\(176\) 0 0
\(177\) −7.94427 −0.597128
\(178\) 0.763932 0.0572591
\(179\) −26.3607 −1.97029 −0.985145 0.171725i \(-0.945066\pi\)
−0.985145 + 0.171725i \(0.945066\pi\)
\(180\) −1.38197 −0.103006
\(181\) 12.9443 0.962140 0.481070 0.876682i \(-0.340248\pi\)
0.481070 + 0.876682i \(0.340248\pi\)
\(182\) −0.381966 −0.0283132
\(183\) 12.4721 0.921967
\(184\) −4.47214 −0.329690
\(185\) 3.29180 0.242018
\(186\) 16.9443 1.24241
\(187\) 0 0
\(188\) 5.85410 0.426954
\(189\) 1.00000 0.0727393
\(190\) 5.32624 0.386406
\(191\) 15.5279 1.12356 0.561778 0.827288i \(-0.310117\pi\)
0.561778 + 0.827288i \(0.310117\pi\)
\(192\) −4.23607 −0.305712
\(193\) −1.52786 −0.109978 −0.0549890 0.998487i \(-0.517512\pi\)
−0.0549890 + 0.998487i \(0.517512\pi\)
\(194\) −15.2361 −1.09389
\(195\) −0.527864 −0.0378011
\(196\) 0.618034 0.0441453
\(197\) 23.8885 1.70199 0.850994 0.525175i \(-0.176000\pi\)
0.850994 + 0.525175i \(0.176000\pi\)
\(198\) 0 0
\(199\) −11.8885 −0.842757 −0.421378 0.906885i \(-0.638454\pi\)
−0.421378 + 0.906885i \(0.638454\pi\)
\(200\) 0 0
\(201\) −9.18034 −0.647531
\(202\) 15.4164 1.08469
\(203\) 5.00000 0.350931
\(204\) 1.23607 0.0865421
\(205\) 5.52786 0.386083
\(206\) 16.9443 1.18056
\(207\) −2.00000 −0.139010
\(208\) 1.14590 0.0794537
\(209\) 0 0
\(210\) 3.61803 0.249668
\(211\) −26.3607 −1.81474 −0.907372 0.420328i \(-0.861915\pi\)
−0.907372 + 0.420328i \(0.861915\pi\)
\(212\) 4.00000 0.274721
\(213\) 2.47214 0.169388
\(214\) 31.7984 2.17369
\(215\) −18.9443 −1.29199
\(216\) −2.23607 −0.152145
\(217\) −10.4721 −0.710895
\(218\) −16.9443 −1.14761
\(219\) −10.7082 −0.723593
\(220\) 0 0
\(221\) 0.472136 0.0317593
\(222\) −2.38197 −0.159867
\(223\) −23.8885 −1.59970 −0.799848 0.600203i \(-0.795086\pi\)
−0.799848 + 0.600203i \(0.795086\pi\)
\(224\) −3.38197 −0.225967
\(225\) 0 0
\(226\) 8.94427 0.594964
\(227\) −3.52786 −0.234153 −0.117076 0.993123i \(-0.537352\pi\)
−0.117076 + 0.993123i \(0.537352\pi\)
\(228\) −0.909830 −0.0602550
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −7.23607 −0.477132
\(231\) 0 0
\(232\) −11.1803 −0.734025
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) 0.381966 0.0249699
\(235\) −21.1803 −1.38165
\(236\) 4.90983 0.319603
\(237\) −0.472136 −0.0306685
\(238\) −3.23607 −0.209763
\(239\) 26.1246 1.68986 0.844930 0.534876i \(-0.179642\pi\)
0.844930 + 0.534876i \(0.179642\pi\)
\(240\) −10.8541 −0.700629
\(241\) 19.1803 1.23551 0.617757 0.786369i \(-0.288041\pi\)
0.617757 + 0.786369i \(0.288041\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −7.70820 −0.493467
\(245\) −2.23607 −0.142857
\(246\) −4.00000 −0.255031
\(247\) −0.347524 −0.0221124
\(248\) 23.4164 1.48694
\(249\) 7.52786 0.477059
\(250\) −18.0902 −1.14412
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) −0.618034 −0.0389325
\(253\) 0 0
\(254\) −15.2361 −0.955996
\(255\) −4.47214 −0.280056
\(256\) 13.5623 0.847644
\(257\) −13.6525 −0.851618 −0.425809 0.904813i \(-0.640010\pi\)
−0.425809 + 0.904813i \(0.640010\pi\)
\(258\) 13.7082 0.853435
\(259\) 1.47214 0.0914741
\(260\) 0.326238 0.0202324
\(261\) −5.00000 −0.309492
\(262\) 31.4164 1.94091
\(263\) −11.2918 −0.696282 −0.348141 0.937442i \(-0.613187\pi\)
−0.348141 + 0.937442i \(0.613187\pi\)
\(264\) 0 0
\(265\) −14.4721 −0.889016
\(266\) 2.38197 0.146048
\(267\) 0.472136 0.0288943
\(268\) 5.67376 0.346580
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) −3.61803 −0.220187
\(271\) −6.88854 −0.418449 −0.209225 0.977868i \(-0.567094\pi\)
−0.209225 + 0.977868i \(0.567094\pi\)
\(272\) 9.70820 0.588646
\(273\) −0.236068 −0.0142875
\(274\) 15.2361 0.920445
\(275\) 0 0
\(276\) 1.23607 0.0744025
\(277\) −30.3607 −1.82420 −0.912098 0.409972i \(-0.865539\pi\)
−0.912098 + 0.409972i \(0.865539\pi\)
\(278\) −14.4721 −0.867981
\(279\) 10.4721 0.626950
\(280\) 5.00000 0.298807
\(281\) −11.4721 −0.684370 −0.342185 0.939633i \(-0.611167\pi\)
−0.342185 + 0.939633i \(0.611167\pi\)
\(282\) 15.3262 0.912664
\(283\) 18.4164 1.09474 0.547371 0.836890i \(-0.315629\pi\)
0.547371 + 0.836890i \(0.315629\pi\)
\(284\) −1.52786 −0.0906621
\(285\) 3.29180 0.194989
\(286\) 0 0
\(287\) 2.47214 0.145926
\(288\) 3.38197 0.199284
\(289\) −13.0000 −0.764706
\(290\) −18.0902 −1.06229
\(291\) −9.41641 −0.552000
\(292\) 6.61803 0.387291
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.61803 0.0943657
\(295\) −17.7639 −1.03426
\(296\) −3.29180 −0.191332
\(297\) 0 0
\(298\) 33.9787 1.96833
\(299\) 0.472136 0.0273043
\(300\) 0 0
\(301\) −8.47214 −0.488326
\(302\) 16.1803 0.931074
\(303\) 9.52786 0.547361
\(304\) −7.14590 −0.409845
\(305\) 27.8885 1.59689
\(306\) 3.23607 0.184994
\(307\) −29.8885 −1.70583 −0.852915 0.522050i \(-0.825167\pi\)
−0.852915 + 0.522050i \(0.825167\pi\)
\(308\) 0 0
\(309\) 10.4721 0.595739
\(310\) 37.8885 2.15192
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0.527864 0.0298844
\(313\) 14.9443 0.844700 0.422350 0.906433i \(-0.361205\pi\)
0.422350 + 0.906433i \(0.361205\pi\)
\(314\) −5.70820 −0.322133
\(315\) 2.23607 0.125988
\(316\) 0.291796 0.0164148
\(317\) −3.05573 −0.171627 −0.0858134 0.996311i \(-0.527349\pi\)
−0.0858134 + 0.996311i \(0.527349\pi\)
\(318\) 10.4721 0.587248
\(319\) 0 0
\(320\) −9.47214 −0.529508
\(321\) 19.6525 1.09689
\(322\) −3.23607 −0.180339
\(323\) −2.94427 −0.163824
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) 6.09017 0.337303
\(327\) −10.4721 −0.579110
\(328\) −5.52786 −0.305225
\(329\) −9.47214 −0.522216
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) −4.65248 −0.255338
\(333\) −1.47214 −0.0806726
\(334\) −8.76393 −0.479541
\(335\) −20.5279 −1.12156
\(336\) −4.85410 −0.264813
\(337\) 27.5279 1.49954 0.749769 0.661699i \(-0.230165\pi\)
0.749769 + 0.661699i \(0.230165\pi\)
\(338\) 20.9443 1.13922
\(339\) 5.52786 0.300232
\(340\) 2.76393 0.149895
\(341\) 0 0
\(342\) −2.38197 −0.128802
\(343\) −1.00000 −0.0539949
\(344\) 18.9443 1.02141
\(345\) −4.47214 −0.240772
\(346\) −12.1803 −0.654819
\(347\) 4.94427 0.265422 0.132711 0.991155i \(-0.457632\pi\)
0.132711 + 0.991155i \(0.457632\pi\)
\(348\) 3.09017 0.165650
\(349\) −18.1246 −0.970188 −0.485094 0.874462i \(-0.661215\pi\)
−0.485094 + 0.874462i \(0.661215\pi\)
\(350\) 0 0
\(351\) 0.236068 0.0126004
\(352\) 0 0
\(353\) −5.18034 −0.275722 −0.137861 0.990452i \(-0.544023\pi\)
−0.137861 + 0.990452i \(0.544023\pi\)
\(354\) 12.8541 0.683188
\(355\) 5.52786 0.293389
\(356\) −0.291796 −0.0154652
\(357\) −2.00000 −0.105851
\(358\) 42.6525 2.25425
\(359\) −28.9443 −1.52762 −0.763810 0.645441i \(-0.776674\pi\)
−0.763810 + 0.645441i \(0.776674\pi\)
\(360\) −5.00000 −0.263523
\(361\) −16.8328 −0.885938
\(362\) −20.9443 −1.10081
\(363\) 0 0
\(364\) 0.145898 0.00764713
\(365\) −23.9443 −1.25330
\(366\) −20.1803 −1.05484
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 9.70820 0.506075
\(369\) −2.47214 −0.128694
\(370\) −5.32624 −0.276898
\(371\) −6.47214 −0.336017
\(372\) −6.47214 −0.335565
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −11.1803 −0.577350
\(376\) 21.1803 1.09229
\(377\) 1.18034 0.0607906
\(378\) −1.61803 −0.0832227
\(379\) 11.7639 0.604273 0.302136 0.953265i \(-0.402300\pi\)
0.302136 + 0.953265i \(0.402300\pi\)
\(380\) −2.03444 −0.104365
\(381\) −9.41641 −0.482417
\(382\) −25.1246 −1.28549
\(383\) −0.944272 −0.0482500 −0.0241250 0.999709i \(-0.507680\pi\)
−0.0241250 + 0.999709i \(0.507680\pi\)
\(384\) 13.6180 0.694942
\(385\) 0 0
\(386\) 2.47214 0.125828
\(387\) 8.47214 0.430663
\(388\) 5.81966 0.295448
\(389\) −25.5279 −1.29431 −0.647157 0.762357i \(-0.724042\pi\)
−0.647157 + 0.762357i \(0.724042\pi\)
\(390\) 0.854102 0.0432491
\(391\) 4.00000 0.202289
\(392\) 2.23607 0.112938
\(393\) 19.4164 0.979428
\(394\) −38.6525 −1.94728
\(395\) −1.05573 −0.0531194
\(396\) 0 0
\(397\) 26.9443 1.35229 0.676147 0.736767i \(-0.263648\pi\)
0.676147 + 0.736767i \(0.263648\pi\)
\(398\) 19.2361 0.964217
\(399\) 1.47214 0.0736990
\(400\) 0 0
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) 14.8541 0.740855
\(403\) −2.47214 −0.123146
\(404\) −5.88854 −0.292966
\(405\) −2.23607 −0.111111
\(406\) −8.09017 −0.401508
\(407\) 0 0
\(408\) 4.47214 0.221404
\(409\) 7.52786 0.372229 0.186114 0.982528i \(-0.440410\pi\)
0.186114 + 0.982528i \(0.440410\pi\)
\(410\) −8.94427 −0.441726
\(411\) 9.41641 0.464477
\(412\) −6.47214 −0.318859
\(413\) −7.94427 −0.390912
\(414\) 3.23607 0.159044
\(415\) 16.8328 0.826290
\(416\) −0.798374 −0.0391435
\(417\) −8.94427 −0.438003
\(418\) 0 0
\(419\) 17.0000 0.830504 0.415252 0.909706i \(-0.363693\pi\)
0.415252 + 0.909706i \(0.363693\pi\)
\(420\) −1.38197 −0.0674330
\(421\) −37.3607 −1.82085 −0.910424 0.413676i \(-0.864245\pi\)
−0.910424 + 0.413676i \(0.864245\pi\)
\(422\) 42.6525 2.07629
\(423\) 9.47214 0.460551
\(424\) 14.4721 0.702829
\(425\) 0 0
\(426\) −4.00000 −0.193801
\(427\) 12.4721 0.603569
\(428\) −12.1459 −0.587094
\(429\) 0 0
\(430\) 30.6525 1.47819
\(431\) −20.2361 −0.974737 −0.487369 0.873196i \(-0.662043\pi\)
−0.487369 + 0.873196i \(0.662043\pi\)
\(432\) 4.85410 0.233543
\(433\) 6.58359 0.316387 0.158194 0.987408i \(-0.449433\pi\)
0.158194 + 0.987408i \(0.449433\pi\)
\(434\) 16.9443 0.813351
\(435\) −11.1803 −0.536056
\(436\) 6.47214 0.309959
\(437\) −2.94427 −0.140844
\(438\) 17.3262 0.827880
\(439\) −26.8885 −1.28332 −0.641660 0.766989i \(-0.721754\pi\)
−0.641660 + 0.766989i \(0.721754\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −0.763932 −0.0363365
\(443\) 14.0000 0.665160 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) 0.909830 0.0431786
\(445\) 1.05573 0.0500463
\(446\) 38.6525 1.83025
\(447\) 21.0000 0.993266
\(448\) −4.23607 −0.200135
\(449\) 0.472136 0.0222815 0.0111407 0.999938i \(-0.496454\pi\)
0.0111407 + 0.999938i \(0.496454\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.41641 −0.160694
\(453\) 10.0000 0.469841
\(454\) 5.70820 0.267899
\(455\) −0.527864 −0.0247466
\(456\) −3.29180 −0.154152
\(457\) −6.36068 −0.297540 −0.148770 0.988872i \(-0.547531\pi\)
−0.148770 + 0.988872i \(0.547531\pi\)
\(458\) −22.6525 −1.05848
\(459\) 2.00000 0.0933520
\(460\) 2.76393 0.128869
\(461\) −18.9443 −0.882323 −0.441161 0.897428i \(-0.645433\pi\)
−0.441161 + 0.897428i \(0.645433\pi\)
\(462\) 0 0
\(463\) 6.70820 0.311757 0.155878 0.987776i \(-0.450179\pi\)
0.155878 + 0.987776i \(0.450179\pi\)
\(464\) 24.2705 1.12673
\(465\) 23.4164 1.08591
\(466\) 24.1803 1.12013
\(467\) 10.0557 0.465324 0.232662 0.972558i \(-0.425256\pi\)
0.232662 + 0.972558i \(0.425256\pi\)
\(468\) −0.145898 −0.00674414
\(469\) −9.18034 −0.423909
\(470\) 34.2705 1.58078
\(471\) −3.52786 −0.162555
\(472\) 17.7639 0.817651
\(473\) 0 0
\(474\) 0.763932 0.0350886
\(475\) 0 0
\(476\) 1.23607 0.0566551
\(477\) 6.47214 0.296339
\(478\) −42.2705 −1.93341
\(479\) 35.8885 1.63979 0.819895 0.572514i \(-0.194032\pi\)
0.819895 + 0.572514i \(0.194032\pi\)
\(480\) 7.56231 0.345170
\(481\) 0.347524 0.0158457
\(482\) −31.0344 −1.41358
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) −21.0557 −0.956091
\(486\) 1.61803 0.0733955
\(487\) 5.88854 0.266835 0.133418 0.991060i \(-0.457405\pi\)
0.133418 + 0.991060i \(0.457405\pi\)
\(488\) −27.8885 −1.26246
\(489\) 3.76393 0.170211
\(490\) 3.61803 0.163446
\(491\) 12.1246 0.547176 0.273588 0.961847i \(-0.411790\pi\)
0.273588 + 0.961847i \(0.411790\pi\)
\(492\) 1.52786 0.0688814
\(493\) 10.0000 0.450377
\(494\) 0.562306 0.0252993
\(495\) 0 0
\(496\) −50.8328 −2.28246
\(497\) 2.47214 0.110890
\(498\) −12.1803 −0.545814
\(499\) 19.7639 0.884755 0.442378 0.896829i \(-0.354135\pi\)
0.442378 + 0.896829i \(0.354135\pi\)
\(500\) 6.90983 0.309017
\(501\) −5.41641 −0.241987
\(502\) 17.7984 0.794380
\(503\) −9.88854 −0.440908 −0.220454 0.975397i \(-0.570754\pi\)
−0.220454 + 0.975397i \(0.570754\pi\)
\(504\) −2.23607 −0.0996024
\(505\) 21.3050 0.948058
\(506\) 0 0
\(507\) 12.9443 0.574875
\(508\) 5.81966 0.258206
\(509\) 33.4164 1.48116 0.740578 0.671970i \(-0.234552\pi\)
0.740578 + 0.671970i \(0.234552\pi\)
\(510\) 7.23607 0.320418
\(511\) −10.7082 −0.473703
\(512\) 5.29180 0.233867
\(513\) −1.47214 −0.0649964
\(514\) 22.0902 0.974356
\(515\) 23.4164 1.03185
\(516\) −5.23607 −0.230505
\(517\) 0 0
\(518\) −2.38197 −0.104658
\(519\) −7.52786 −0.330437
\(520\) 1.18034 0.0517613
\(521\) 10.1246 0.443567 0.221784 0.975096i \(-0.428812\pi\)
0.221784 + 0.975096i \(0.428812\pi\)
\(522\) 8.09017 0.354097
\(523\) 1.58359 0.0692456 0.0346228 0.999400i \(-0.488977\pi\)
0.0346228 + 0.999400i \(0.488977\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 18.2705 0.796632
\(527\) −20.9443 −0.912347
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 23.4164 1.01714
\(531\) 7.94427 0.344752
\(532\) −0.909830 −0.0394461
\(533\) 0.583592 0.0252782
\(534\) −0.763932 −0.0330586
\(535\) 43.9443 1.89988
\(536\) 20.5279 0.886669
\(537\) 26.3607 1.13755
\(538\) −21.7082 −0.935907
\(539\) 0 0
\(540\) 1.38197 0.0594703
\(541\) 24.3607 1.04735 0.523674 0.851919i \(-0.324561\pi\)
0.523674 + 0.851919i \(0.324561\pi\)
\(542\) 11.1459 0.478757
\(543\) −12.9443 −0.555492
\(544\) −6.76393 −0.290001
\(545\) −23.4164 −1.00305
\(546\) 0.381966 0.0163466
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) −5.81966 −0.248604
\(549\) −12.4721 −0.532298
\(550\) 0 0
\(551\) −7.36068 −0.313576
\(552\) 4.47214 0.190347
\(553\) −0.472136 −0.0200773
\(554\) 49.1246 2.08710
\(555\) −3.29180 −0.139729
\(556\) 5.52786 0.234434
\(557\) −15.8328 −0.670858 −0.335429 0.942066i \(-0.608881\pi\)
−0.335429 + 0.942066i \(0.608881\pi\)
\(558\) −16.9443 −0.717308
\(559\) −2.00000 −0.0845910
\(560\) −10.8541 −0.458670
\(561\) 0 0
\(562\) 18.5623 0.783004
\(563\) −21.8885 −0.922492 −0.461246 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(564\) −5.85410 −0.246502
\(565\) 12.3607 0.520018
\(566\) −29.7984 −1.25252
\(567\) −1.00000 −0.0419961
\(568\) −5.52786 −0.231944
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) −5.32624 −0.223092
\(571\) −44.8328 −1.87619 −0.938097 0.346371i \(-0.887414\pi\)
−0.938097 + 0.346371i \(0.887414\pi\)
\(572\) 0 0
\(573\) −15.5279 −0.648686
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) 26.4721 1.10205 0.551025 0.834489i \(-0.314237\pi\)
0.551025 + 0.834489i \(0.314237\pi\)
\(578\) 21.0344 0.874917
\(579\) 1.52786 0.0634959
\(580\) 6.90983 0.286915
\(581\) 7.52786 0.312308
\(582\) 15.2361 0.631555
\(583\) 0 0
\(584\) 23.9443 0.990821
\(585\) 0.527864 0.0218245
\(586\) 0 0
\(587\) −34.7771 −1.43540 −0.717702 0.696350i \(-0.754806\pi\)
−0.717702 + 0.696350i \(0.754806\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 15.4164 0.635222
\(590\) 28.7426 1.18332
\(591\) −23.8885 −0.982643
\(592\) 7.14590 0.293695
\(593\) −4.11146 −0.168837 −0.0844186 0.996430i \(-0.526903\pi\)
−0.0844186 + 0.996430i \(0.526903\pi\)
\(594\) 0 0
\(595\) −4.47214 −0.183340
\(596\) −12.9787 −0.531629
\(597\) 11.8885 0.486566
\(598\) −0.763932 −0.0312395
\(599\) 4.47214 0.182727 0.0913633 0.995818i \(-0.470878\pi\)
0.0913633 + 0.995818i \(0.470878\pi\)
\(600\) 0 0
\(601\) 18.7082 0.763124 0.381562 0.924343i \(-0.375386\pi\)
0.381562 + 0.924343i \(0.375386\pi\)
\(602\) 13.7082 0.558705
\(603\) 9.18034 0.373852
\(604\) −6.18034 −0.251474
\(605\) 0 0
\(606\) −15.4164 −0.626249
\(607\) 27.9443 1.13422 0.567112 0.823641i \(-0.308061\pi\)
0.567112 + 0.823641i \(0.308061\pi\)
\(608\) 4.97871 0.201914
\(609\) −5.00000 −0.202610
\(610\) −45.1246 −1.82704
\(611\) −2.23607 −0.0904616
\(612\) −1.23607 −0.0499651
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) 48.3607 1.95168
\(615\) −5.52786 −0.222905
\(616\) 0 0
\(617\) 26.8328 1.08025 0.540124 0.841585i \(-0.318377\pi\)
0.540124 + 0.841585i \(0.318377\pi\)
\(618\) −16.9443 −0.681599
\(619\) 5.88854 0.236681 0.118340 0.992973i \(-0.462243\pi\)
0.118340 + 0.992973i \(0.462243\pi\)
\(620\) −14.4721 −0.581215
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) 0.472136 0.0189157
\(624\) −1.14590 −0.0458726
\(625\) −25.0000 −1.00000
\(626\) −24.1803 −0.966441
\(627\) 0 0
\(628\) 2.18034 0.0870050
\(629\) 2.94427 0.117396
\(630\) −3.61803 −0.144146
\(631\) 26.8328 1.06820 0.534099 0.845422i \(-0.320651\pi\)
0.534099 + 0.845422i \(0.320651\pi\)
\(632\) 1.05573 0.0419946
\(633\) 26.3607 1.04774
\(634\) 4.94427 0.196362
\(635\) −21.0557 −0.835571
\(636\) −4.00000 −0.158610
\(637\) −0.236068 −0.00935335
\(638\) 0 0
\(639\) −2.47214 −0.0977962
\(640\) 30.4508 1.20368
\(641\) 15.4164 0.608912 0.304456 0.952526i \(-0.401525\pi\)
0.304456 + 0.952526i \(0.401525\pi\)
\(642\) −31.7984 −1.25498
\(643\) −37.7771 −1.48978 −0.744891 0.667186i \(-0.767499\pi\)
−0.744891 + 0.667186i \(0.767499\pi\)
\(644\) 1.23607 0.0487079
\(645\) 18.9443 0.745930
\(646\) 4.76393 0.187434
\(647\) 50.3050 1.97769 0.988846 0.148942i \(-0.0475869\pi\)
0.988846 + 0.148942i \(0.0475869\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) 10.4721 0.410435
\(652\) −2.32624 −0.0911025
\(653\) 36.9443 1.44574 0.722871 0.690983i \(-0.242822\pi\)
0.722871 + 0.690983i \(0.242822\pi\)
\(654\) 16.9443 0.662573
\(655\) 43.4164 1.69642
\(656\) 12.0000 0.468521
\(657\) 10.7082 0.417767
\(658\) 15.3262 0.597479
\(659\) 24.1246 0.939761 0.469881 0.882730i \(-0.344297\pi\)
0.469881 + 0.882730i \(0.344297\pi\)
\(660\) 0 0
\(661\) −25.3050 −0.984249 −0.492124 0.870525i \(-0.663779\pi\)
−0.492124 + 0.870525i \(0.663779\pi\)
\(662\) 25.8885 1.00619
\(663\) −0.472136 −0.0183362
\(664\) −16.8328 −0.653240
\(665\) 3.29180 0.127650
\(666\) 2.38197 0.0922993
\(667\) 10.0000 0.387202
\(668\) 3.34752 0.129520
\(669\) 23.8885 0.923584
\(670\) 33.2148 1.28320
\(671\) 0 0
\(672\) 3.38197 0.130462
\(673\) −24.9443 −0.961531 −0.480766 0.876849i \(-0.659641\pi\)
−0.480766 + 0.876849i \(0.659641\pi\)
\(674\) −44.5410 −1.71566
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) −17.5279 −0.673651 −0.336825 0.941567i \(-0.609353\pi\)
−0.336825 + 0.941567i \(0.609353\pi\)
\(678\) −8.94427 −0.343503
\(679\) −9.41641 −0.361369
\(680\) 10.0000 0.383482
\(681\) 3.52786 0.135188
\(682\) 0 0
\(683\) 44.9443 1.71974 0.859872 0.510509i \(-0.170543\pi\)
0.859872 + 0.510509i \(0.170543\pi\)
\(684\) 0.909830 0.0347882
\(685\) 21.0557 0.804498
\(686\) 1.61803 0.0617768
\(687\) −14.0000 −0.534133
\(688\) −41.1246 −1.56786
\(689\) −1.52786 −0.0582070
\(690\) 7.23607 0.275472
\(691\) −14.8328 −0.564267 −0.282133 0.959375i \(-0.591042\pi\)
−0.282133 + 0.959375i \(0.591042\pi\)
\(692\) 4.65248 0.176861
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) −20.0000 −0.758643
\(696\) 11.1803 0.423790
\(697\) 4.94427 0.187278
\(698\) 29.3262 1.11001
\(699\) 14.9443 0.565244
\(700\) 0 0
\(701\) 12.1115 0.457443 0.228722 0.973492i \(-0.426545\pi\)
0.228722 + 0.973492i \(0.426545\pi\)
\(702\) −0.381966 −0.0144164
\(703\) −2.16718 −0.0817369
\(704\) 0 0
\(705\) 21.1803 0.797698
\(706\) 8.38197 0.315459
\(707\) 9.52786 0.358332
\(708\) −4.90983 −0.184523
\(709\) −34.5279 −1.29672 −0.648361 0.761333i \(-0.724545\pi\)
−0.648361 + 0.761333i \(0.724545\pi\)
\(710\) −8.94427 −0.335673
\(711\) 0.472136 0.0177065
\(712\) −1.05573 −0.0395651
\(713\) −20.9443 −0.784369
\(714\) 3.23607 0.121107
\(715\) 0 0
\(716\) −16.2918 −0.608853
\(717\) −26.1246 −0.975642
\(718\) 46.8328 1.74779
\(719\) 21.3607 0.796619 0.398309 0.917251i \(-0.369597\pi\)
0.398309 + 0.917251i \(0.369597\pi\)
\(720\) 10.8541 0.404508
\(721\) 10.4721 0.390003
\(722\) 27.2361 1.01362
\(723\) −19.1803 −0.713325
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) −3.52786 −0.130841 −0.0654206 0.997858i \(-0.520839\pi\)
−0.0654206 + 0.997858i \(0.520839\pi\)
\(728\) 0.527864 0.0195639
\(729\) 1.00000 0.0370370
\(730\) 38.7426 1.43393
\(731\) −16.9443 −0.626707
\(732\) 7.70820 0.284903
\(733\) 26.3607 0.973654 0.486827 0.873498i \(-0.338154\pi\)
0.486827 + 0.873498i \(0.338154\pi\)
\(734\) 25.8885 0.955564
\(735\) 2.23607 0.0824786
\(736\) −6.76393 −0.249322
\(737\) 0 0
\(738\) 4.00000 0.147242
\(739\) −51.8885 −1.90875 −0.954375 0.298609i \(-0.903477\pi\)
−0.954375 + 0.298609i \(0.903477\pi\)
\(740\) 2.03444 0.0747876
\(741\) 0.347524 0.0127666
\(742\) 10.4721 0.384444
\(743\) 30.5967 1.12249 0.561243 0.827651i \(-0.310323\pi\)
0.561243 + 0.827651i \(0.310323\pi\)
\(744\) −23.4164 −0.858487
\(745\) 46.9574 1.72039
\(746\) 9.70820 0.355443
\(747\) −7.52786 −0.275430
\(748\) 0 0
\(749\) 19.6525 0.718086
\(750\) 18.0902 0.660560
\(751\) −7.18034 −0.262014 −0.131007 0.991381i \(-0.541821\pi\)
−0.131007 + 0.991381i \(0.541821\pi\)
\(752\) −45.9787 −1.67667
\(753\) 11.0000 0.400862
\(754\) −1.90983 −0.0695519
\(755\) 22.3607 0.813788
\(756\) 0.618034 0.0224777
\(757\) −15.5836 −0.566395 −0.283198 0.959062i \(-0.591395\pi\)
−0.283198 + 0.959062i \(0.591395\pi\)
\(758\) −19.0344 −0.691362
\(759\) 0 0
\(760\) −7.36068 −0.267000
\(761\) −24.4721 −0.887114 −0.443557 0.896246i \(-0.646284\pi\)
−0.443557 + 0.896246i \(0.646284\pi\)
\(762\) 15.2361 0.551945
\(763\) −10.4721 −0.379117
\(764\) 9.59675 0.347198
\(765\) 4.47214 0.161690
\(766\) 1.52786 0.0552040
\(767\) −1.87539 −0.0677163
\(768\) −13.5623 −0.489388
\(769\) 15.1803 0.547417 0.273709 0.961813i \(-0.411750\pi\)
0.273709 + 0.961813i \(0.411750\pi\)
\(770\) 0 0
\(771\) 13.6525 0.491682
\(772\) −0.944272 −0.0339851
\(773\) −19.1803 −0.689869 −0.344934 0.938627i \(-0.612099\pi\)
−0.344934 + 0.938627i \(0.612099\pi\)
\(774\) −13.7082 −0.492731
\(775\) 0 0
\(776\) 21.0557 0.755857
\(777\) −1.47214 −0.0528126
\(778\) 41.3050 1.48085
\(779\) −3.63932 −0.130392
\(780\) −0.326238 −0.0116812
\(781\) 0 0
\(782\) −6.47214 −0.231443
\(783\) 5.00000 0.178685
\(784\) −4.85410 −0.173361
\(785\) −7.88854 −0.281554
\(786\) −31.4164 −1.12059
\(787\) −45.2492 −1.61296 −0.806480 0.591261i \(-0.798630\pi\)
−0.806480 + 0.591261i \(0.798630\pi\)
\(788\) 14.7639 0.525943
\(789\) 11.2918 0.401999
\(790\) 1.70820 0.0607752
\(791\) 5.52786 0.196548
\(792\) 0 0
\(793\) 2.94427 0.104554
\(794\) −43.5967 −1.54719
\(795\) 14.4721 0.513274
\(796\) −7.34752 −0.260426
\(797\) −37.7639 −1.33767 −0.668834 0.743412i \(-0.733206\pi\)
−0.668834 + 0.743412i \(0.733206\pi\)
\(798\) −2.38197 −0.0843207
\(799\) −18.9443 −0.670200
\(800\) 0 0
\(801\) −0.472136 −0.0166821
\(802\) 42.0689 1.48550
\(803\) 0 0
\(804\) −5.67376 −0.200098
\(805\) −4.47214 −0.157622
\(806\) 4.00000 0.140894
\(807\) −13.4164 −0.472280
\(808\) −21.3050 −0.749506
\(809\) −49.3607 −1.73543 −0.867715 0.497063i \(-0.834412\pi\)
−0.867715 + 0.497063i \(0.834412\pi\)
\(810\) 3.61803 0.127125
\(811\) −40.5279 −1.42313 −0.711563 0.702622i \(-0.752012\pi\)
−0.711563 + 0.702622i \(0.752012\pi\)
\(812\) 3.09017 0.108444
\(813\) 6.88854 0.241592
\(814\) 0 0
\(815\) 8.41641 0.294814
\(816\) −9.70820 −0.339855
\(817\) 12.4721 0.436345
\(818\) −12.1803 −0.425876
\(819\) 0.236068 0.00824888
\(820\) 3.41641 0.119306
\(821\) −1.83282 −0.0639657 −0.0319829 0.999488i \(-0.510182\pi\)
−0.0319829 + 0.999488i \(0.510182\pi\)
\(822\) −15.2361 −0.531419
\(823\) −33.0689 −1.15271 −0.576354 0.817200i \(-0.695525\pi\)
−0.576354 + 0.817200i \(0.695525\pi\)
\(824\) −23.4164 −0.815749
\(825\) 0 0
\(826\) 12.8541 0.447251
\(827\) 26.2361 0.912317 0.456159 0.889898i \(-0.349225\pi\)
0.456159 + 0.889898i \(0.349225\pi\)
\(828\) −1.23607 −0.0429563
\(829\) −20.9443 −0.727425 −0.363712 0.931511i \(-0.618491\pi\)
−0.363712 + 0.931511i \(0.618491\pi\)
\(830\) −27.2361 −0.945378
\(831\) 30.3607 1.05320
\(832\) −1.00000 −0.0346688
\(833\) −2.00000 −0.0692959
\(834\) 14.4721 0.501129
\(835\) −12.1115 −0.419134
\(836\) 0 0
\(837\) −10.4721 −0.361970
\(838\) −27.5066 −0.950199
\(839\) 29.2492 1.00980 0.504898 0.863179i \(-0.331530\pi\)
0.504898 + 0.863179i \(0.331530\pi\)
\(840\) −5.00000 −0.172516
\(841\) −4.00000 −0.137931
\(842\) 60.4508 2.08327
\(843\) 11.4721 0.395121
\(844\) −16.2918 −0.560787
\(845\) 28.9443 0.995713
\(846\) −15.3262 −0.526927
\(847\) 0 0
\(848\) −31.4164 −1.07884
\(849\) −18.4164 −0.632049
\(850\) 0 0
\(851\) 2.94427 0.100928
\(852\) 1.52786 0.0523438
\(853\) 29.4164 1.00720 0.503599 0.863937i \(-0.332009\pi\)
0.503599 + 0.863937i \(0.332009\pi\)
\(854\) −20.1803 −0.690557
\(855\) −3.29180 −0.112577
\(856\) −43.9443 −1.50198
\(857\) 14.5836 0.498166 0.249083 0.968482i \(-0.419871\pi\)
0.249083 + 0.968482i \(0.419871\pi\)
\(858\) 0 0
\(859\) 9.05573 0.308977 0.154489 0.987995i \(-0.450627\pi\)
0.154489 + 0.987995i \(0.450627\pi\)
\(860\) −11.7082 −0.399246
\(861\) −2.47214 −0.0842502
\(862\) 32.7426 1.11522
\(863\) 25.8885 0.881256 0.440628 0.897690i \(-0.354756\pi\)
0.440628 + 0.897690i \(0.354756\pi\)
\(864\) −3.38197 −0.115057
\(865\) −16.8328 −0.572333
\(866\) −10.6525 −0.361986
\(867\) 13.0000 0.441503
\(868\) −6.47214 −0.219679
\(869\) 0 0
\(870\) 18.0902 0.613314
\(871\) −2.16718 −0.0734322
\(872\) 23.4164 0.792980
\(873\) 9.41641 0.318697
\(874\) 4.76393 0.161142
\(875\) −11.1803 −0.377964
\(876\) −6.61803 −0.223603
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) 43.5066 1.46828
\(879\) 0 0
\(880\) 0 0
\(881\) 40.2361 1.35559 0.677794 0.735252i \(-0.262936\pi\)
0.677794 + 0.735252i \(0.262936\pi\)
\(882\) −1.61803 −0.0544820
\(883\) −25.1803 −0.847386 −0.423693 0.905806i \(-0.639266\pi\)
−0.423693 + 0.905806i \(0.639266\pi\)
\(884\) 0.291796 0.00981416
\(885\) 17.7639 0.597128
\(886\) −22.6525 −0.761025
\(887\) −16.9443 −0.568933 −0.284466 0.958686i \(-0.591816\pi\)
−0.284466 + 0.958686i \(0.591816\pi\)
\(888\) 3.29180 0.110465
\(889\) −9.41641 −0.315816
\(890\) −1.70820 −0.0572591
\(891\) 0 0
\(892\) −14.7639 −0.494333
\(893\) 13.9443 0.466627
\(894\) −33.9787 −1.13642
\(895\) 58.9443 1.97029
\(896\) 13.6180 0.454947
\(897\) −0.472136 −0.0157642
\(898\) −0.763932 −0.0254927
\(899\) −52.3607 −1.74633
\(900\) 0 0
\(901\) −12.9443 −0.431236
\(902\) 0 0
\(903\) 8.47214 0.281935
\(904\) −12.3607 −0.411110
\(905\) −28.9443 −0.962140
\(906\) −16.1803 −0.537556
\(907\) 55.7771 1.85205 0.926024 0.377465i \(-0.123204\pi\)
0.926024 + 0.377465i \(0.123204\pi\)
\(908\) −2.18034 −0.0723571
\(909\) −9.52786 −0.316019
\(910\) 0.854102 0.0283132
\(911\) −54.2492 −1.79736 −0.898678 0.438608i \(-0.855472\pi\)
−0.898678 + 0.438608i \(0.855472\pi\)
\(912\) 7.14590 0.236624
\(913\) 0 0
\(914\) 10.2918 0.340422
\(915\) −27.8885 −0.921967
\(916\) 8.65248 0.285886
\(917\) 19.4164 0.641186
\(918\) −3.23607 −0.106806
\(919\) −22.4721 −0.741287 −0.370644 0.928775i \(-0.620863\pi\)
−0.370644 + 0.928775i \(0.620863\pi\)
\(920\) 10.0000 0.329690
\(921\) 29.8885 0.984861
\(922\) 30.6525 1.00949
\(923\) 0.583592 0.0192092
\(924\) 0 0
\(925\) 0 0
\(926\) −10.8541 −0.356688
\(927\) −10.4721 −0.343950
\(928\) −16.9098 −0.555092
\(929\) 12.7082 0.416943 0.208471 0.978028i \(-0.433151\pi\)
0.208471 + 0.978028i \(0.433151\pi\)
\(930\) −37.8885 −1.24241
\(931\) 1.47214 0.0482473
\(932\) −9.23607 −0.302537
\(933\) 0 0
\(934\) −16.2705 −0.532387
\(935\) 0 0
\(936\) −0.527864 −0.0172538
\(937\) 38.3607 1.25319 0.626594 0.779346i \(-0.284448\pi\)
0.626594 + 0.779346i \(0.284448\pi\)
\(938\) 14.8541 0.485004
\(939\) −14.9443 −0.487688
\(940\) −13.0902 −0.426954
\(941\) −26.5836 −0.866600 −0.433300 0.901250i \(-0.642651\pi\)
−0.433300 + 0.901250i \(0.642651\pi\)
\(942\) 5.70820 0.185983
\(943\) 4.94427 0.161008
\(944\) −38.5623 −1.25510
\(945\) −2.23607 −0.0727393
\(946\) 0 0
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) −0.291796 −0.00947710
\(949\) −2.52786 −0.0820579
\(950\) 0 0
\(951\) 3.05573 0.0990888
\(952\) 4.47214 0.144943
\(953\) −8.41641 −0.272634 −0.136317 0.990665i \(-0.543527\pi\)
−0.136317 + 0.990665i \(0.543527\pi\)
\(954\) −10.4721 −0.339048
\(955\) −34.7214 −1.12356
\(956\) 16.1459 0.522196
\(957\) 0 0
\(958\) −58.0689 −1.87612
\(959\) 9.41641 0.304072
\(960\) 9.47214 0.305712
\(961\) 78.6656 2.53760
\(962\) −0.562306 −0.0181295
\(963\) −19.6525 −0.633292
\(964\) 11.8541 0.381795
\(965\) 3.41641 0.109978
\(966\) 3.23607 0.104119
\(967\) −13.3050 −0.427858 −0.213929 0.976849i \(-0.568626\pi\)
−0.213929 + 0.976849i \(0.568626\pi\)
\(968\) 0 0
\(969\) 2.94427 0.0945836
\(970\) 34.0689 1.09389
\(971\) 12.0557 0.386887 0.193443 0.981111i \(-0.438034\pi\)
0.193443 + 0.981111i \(0.438034\pi\)
\(972\) −0.618034 −0.0198234
\(973\) −8.94427 −0.286740
\(974\) −9.52786 −0.305292
\(975\) 0 0
\(976\) 60.5410 1.93787
\(977\) 16.4721 0.526990 0.263495 0.964661i \(-0.415125\pi\)
0.263495 + 0.964661i \(0.415125\pi\)
\(978\) −6.09017 −0.194742
\(979\) 0 0
\(980\) −1.38197 −0.0441453
\(981\) 10.4721 0.334350
\(982\) −19.6180 −0.626037
\(983\) −40.0000 −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(984\) 5.52786 0.176222
\(985\) −53.4164 −1.70199
\(986\) −16.1803 −0.515287
\(987\) 9.47214 0.301501
\(988\) −0.214782 −0.00683312
\(989\) −16.9443 −0.538797
\(990\) 0 0
\(991\) −11.1803 −0.355155 −0.177578 0.984107i \(-0.556826\pi\)
−0.177578 + 0.984107i \(0.556826\pi\)
\(992\) 35.4164 1.12447
\(993\) 16.0000 0.507745
\(994\) −4.00000 −0.126872
\(995\) 26.5836 0.842757
\(996\) 4.65248 0.147419
\(997\) 57.1935 1.81134 0.905668 0.423987i \(-0.139370\pi\)
0.905668 + 0.423987i \(0.139370\pi\)
\(998\) −31.9787 −1.01227
\(999\) 1.47214 0.0465763
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 2541.2.a.q.1.1 2
3.2 odd 2 7623.2.a.bn.1.2 2
11.10 odd 2 2541.2.a.y.1.2 yes 2
33.32 even 2 7623.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.q.1.1 2 1.1 even 1 trivial
2541.2.a.y.1.2 yes 2 11.10 odd 2
7623.2.a.y.1.1 2 33.32 even 2
7623.2.a.bn.1.2 2 3.2 odd 2