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Properties

Label 4410.2.a.bs.1.2
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +3.41421 q^{11} +1.17157 q^{13} +1.00000 q^{16} -3.41421 q^{17} +4.82843 q^{19} +1.00000 q^{20} -3.41421 q^{22} -3.65685 q^{23} +1.00000 q^{25} -1.17157 q^{26} +5.41421 q^{29} +7.41421 q^{31} -1.00000 q^{32} +3.41421 q^{34} -3.41421 q^{37} -4.82843 q^{38} -1.00000 q^{40} +3.65685 q^{41} -7.07107 q^{43} +3.41421 q^{44} +3.65685 q^{46} -2.58579 q^{47} -1.00000 q^{50} +1.17157 q^{52} +6.48528 q^{53} +3.41421 q^{55} -5.41421 q^{58} +0.828427 q^{59} -1.65685 q^{61} -7.41421 q^{62} +1.00000 q^{64} +1.17157 q^{65} +5.89949 q^{67} -3.41421 q^{68} -10.4853 q^{71} +12.8284 q^{73} +3.41421 q^{74} +4.82843 q^{76} +13.6569 q^{79} +1.00000 q^{80} -3.65685 q^{82} -0.485281 q^{83} -3.41421 q^{85} +7.07107 q^{86} -3.41421 q^{88} -14.4853 q^{89} -3.65685 q^{92} +2.58579 q^{94} +4.82843 q^{95} +14.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} - 2q^{8} - 2q^{10} + 4q^{11} + 8q^{13} + 2q^{16} - 4q^{17} + 4q^{19} + 2q^{20} - 4q^{22} + 4q^{23} + 2q^{25} - 8q^{26} + 8q^{29} + 12q^{31} - 2q^{32} + 4q^{34} - 4q^{37} - 4q^{38} - 2q^{40} - 4q^{41} + 4q^{44} - 4q^{46} - 8q^{47} - 2q^{50} + 8q^{52} - 4q^{53} + 4q^{55} - 8q^{58} - 4q^{59} + 8q^{61} - 12q^{62} + 2q^{64} + 8q^{65} - 8q^{67} - 4q^{68} - 4q^{71} + 20q^{73} + 4q^{74} + 4q^{76} + 16q^{79} + 2q^{80} + 4q^{82} + 16q^{83} - 4q^{85} - 4q^{88} - 12q^{89} + 4q^{92} + 8q^{94} + 4q^{95} + 12q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 3.41421 1.02942 0.514712 − 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 − 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.41421 −0.828068 −0.414034 − 0.910261i \(-0.635881\pi\)
−0.414034 + 0.910261i \(0.635881\pi\)
\(18\) 0 0
\(19\) 4.82843 1.10772 0.553859 − 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.41421 −0.727913
\(23\) −3.65685 −0.762507 −0.381253 − 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.17157 −0.229764
\(27\) 0 0
\(28\) 0 0
\(29\) 5.41421 1.00539 0.502697 − 0.864463i \(-0.332341\pi\)
0.502697 + 0.864463i \(0.332341\pi\)
\(30\) 0 0
\(31\) 7.41421 1.33163 0.665816 − 0.746116i \(-0.268084\pi\)
0.665816 + 0.746116i \(0.268084\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.41421 0.585533
\(35\) 0 0
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 − 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) −4.82843 −0.783274
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.65685 0.571105 0.285552 − 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) −7.07107 −1.07833 −0.539164 − 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(44\) 3.41421 0.514712
\(45\) 0 0
\(46\) 3.65685 0.539174
\(47\) −2.58579 −0.377176 −0.188588 − 0.982056i \(-0.560391\pi\)
−0.188588 + 0.982056i \(0.560391\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 1.17157 0.162468
\(53\) 6.48528 0.890822 0.445411 − 0.895326i \(-0.353058\pi\)
0.445411 + 0.895326i \(0.353058\pi\)
\(54\) 0 0
\(55\) 3.41421 0.460372
\(56\) 0 0
\(57\) 0 0
\(58\) −5.41421 −0.710921
\(59\) 0.828427 0.107852 0.0539260 − 0.998545i \(-0.482826\pi\)
0.0539260 + 0.998545i \(0.482826\pi\)
\(60\) 0 0
\(61\) −1.65685 −0.212138 −0.106069 − 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) −7.41421 −0.941606
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.17157 0.145316
\(66\) 0 0
\(67\) 5.89949 0.720738 0.360369 − 0.932810i \(-0.382651\pi\)
0.360369 + 0.932810i \(0.382651\pi\)
\(68\) −3.41421 −0.414034
\(69\) 0 0
\(70\) 0 0
\(71\) −10.4853 −1.24437 −0.622187 − 0.782869i \(-0.713756\pi\)
−0.622187 + 0.782869i \(0.713756\pi\)
\(72\) 0 0
\(73\) 12.8284 1.50145 0.750727 − 0.660613i \(-0.229703\pi\)
0.750727 + 0.660613i \(0.229703\pi\)
\(74\) 3.41421 0.396894
\(75\) 0 0
\(76\) 4.82843 0.553859
\(77\) 0 0
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 − 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −3.65685 −0.403832
\(83\) −0.485281 −0.0532666 −0.0266333 − 0.999645i \(-0.508479\pi\)
−0.0266333 + 0.999645i \(0.508479\pi\)
\(84\) 0 0
\(85\) −3.41421 −0.370323
\(86\) 7.07107 0.762493
\(87\) 0 0
\(88\) −3.41421 −0.363956
\(89\) −14.4853 −1.53544 −0.767718 − 0.640787i \(-0.778608\pi\)
−0.767718 + 0.640787i \(0.778608\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.65685 −0.381253
\(93\) 0 0
\(94\) 2.58579 0.266704
\(95\) 4.82843 0.495386
\(96\) 0 0
\(97\) 14.4853 1.47076 0.735379 − 0.677656i \(-0.237004\pi\)
0.735379 + 0.677656i \(0.237004\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −9.65685 −0.960893 −0.480446 − 0.877024i \(-0.659525\pi\)
−0.480446 + 0.877024i \(0.659525\pi\)
\(102\) 0 0
\(103\) 5.65685 0.557386 0.278693 − 0.960380i \(-0.410099\pi\)
0.278693 + 0.960380i \(0.410099\pi\)
\(104\) −1.17157 −0.114882
\(105\) 0 0
\(106\) −6.48528 −0.629906
\(107\) −10.8284 −1.04682 −0.523412 − 0.852080i \(-0.675341\pi\)
−0.523412 + 0.852080i \(0.675341\pi\)
\(108\) 0 0
\(109\) 14.4853 1.38744 0.693719 − 0.720246i \(-0.255971\pi\)
0.693719 + 0.720246i \(0.255971\pi\)
\(110\) −3.41421 −0.325532
\(111\) 0 0
\(112\) 0 0
\(113\) −4.34315 −0.408569 −0.204284 − 0.978912i \(-0.565487\pi\)
−0.204284 + 0.978912i \(0.565487\pi\)
\(114\) 0 0
\(115\) −3.65685 −0.341003
\(116\) 5.41421 0.502697
\(117\) 0 0
\(118\) −0.828427 −0.0762629
\(119\) 0 0
\(120\) 0 0
\(121\) 0.656854 0.0597140
\(122\) 1.65685 0.150005
\(123\) 0 0
\(124\) 7.41421 0.665816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.1421 −1.25491 −0.627456 − 0.778652i \(-0.715904\pi\)
−0.627456 + 0.778652i \(0.715904\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −1.17157 −0.102754
\(131\) 6.48528 0.566622 0.283311 − 0.959028i \(-0.408567\pi\)
0.283311 + 0.959028i \(0.408567\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −5.89949 −0.509639
\(135\) 0 0
\(136\) 3.41421 0.292766
\(137\) 4.48528 0.383203 0.191602 − 0.981473i \(-0.438632\pi\)
0.191602 + 0.981473i \(0.438632\pi\)
\(138\) 0 0
\(139\) −1.31371 −0.111427 −0.0557137 − 0.998447i \(-0.517743\pi\)
−0.0557137 + 0.998447i \(0.517743\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.4853 0.879905
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 5.41421 0.449626
\(146\) −12.8284 −1.06169
\(147\) 0 0
\(148\) −3.41421 −0.280647
\(149\) −12.7279 −1.04271 −0.521356 − 0.853339i \(-0.674574\pi\)
−0.521356 + 0.853339i \(0.674574\pi\)
\(150\) 0 0
\(151\) 3.17157 0.258099 0.129049 − 0.991638i \(-0.458807\pi\)
0.129049 + 0.991638i \(0.458807\pi\)
\(152\) −4.82843 −0.391637
\(153\) 0 0
\(154\) 0 0
\(155\) 7.41421 0.595524
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −13.6569 −1.08648
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 7.07107 0.553849 0.276924 − 0.960892i \(-0.410685\pi\)
0.276924 + 0.960892i \(0.410685\pi\)
\(164\) 3.65685 0.285552
\(165\) 0 0
\(166\) 0.485281 0.0376651
\(167\) 6.58579 0.509623 0.254812 − 0.966991i \(-0.417987\pi\)
0.254812 + 0.966991i \(0.417987\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 3.41421 0.261858
\(171\) 0 0
\(172\) −7.07107 −0.539164
\(173\) 20.1421 1.53138 0.765689 − 0.643211i \(-0.222398\pi\)
0.765689 + 0.643211i \(0.222398\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.41421 0.257356
\(177\) 0 0
\(178\) 14.4853 1.08572
\(179\) 10.7279 0.801843 0.400921 − 0.916113i \(-0.368690\pi\)
0.400921 + 0.916113i \(0.368690\pi\)
\(180\) 0 0
\(181\) 3.51472 0.261247 0.130623 − 0.991432i \(-0.458302\pi\)
0.130623 + 0.991432i \(0.458302\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.65685 0.269587
\(185\) −3.41421 −0.251018
\(186\) 0 0
\(187\) −11.6569 −0.852434
\(188\) −2.58579 −0.188588
\(189\) 0 0
\(190\) −4.82843 −0.350291
\(191\) −9.31371 −0.673916 −0.336958 − 0.941520i \(-0.609398\pi\)
−0.336958 + 0.941520i \(0.609398\pi\)
\(192\) 0 0
\(193\) 20.1421 1.44986 0.724931 − 0.688821i \(-0.241871\pi\)
0.724931 + 0.688821i \(0.241871\pi\)
\(194\) −14.4853 −1.03998
\(195\) 0 0
\(196\) 0 0
\(197\) −0.343146 −0.0244481 −0.0122241 − 0.999925i \(-0.503891\pi\)
−0.0122241 + 0.999925i \(0.503891\pi\)
\(198\) 0 0
\(199\) 18.2426 1.29319 0.646593 − 0.762835i \(-0.276193\pi\)
0.646593 + 0.762835i \(0.276193\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 9.65685 0.679454
\(203\) 0 0
\(204\) 0 0
\(205\) 3.65685 0.255406
\(206\) −5.65685 −0.394132
\(207\) 0 0
\(208\) 1.17157 0.0812340
\(209\) 16.4853 1.14031
\(210\) 0 0
\(211\) −23.7990 −1.63839 −0.819195 − 0.573515i \(-0.805579\pi\)
−0.819195 + 0.573515i \(0.805579\pi\)
\(212\) 6.48528 0.445411
\(213\) 0 0
\(214\) 10.8284 0.740216
\(215\) −7.07107 −0.482243
\(216\) 0 0
\(217\) 0 0
\(218\) −14.4853 −0.981067
\(219\) 0 0
\(220\) 3.41421 0.230186
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 − 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.34315 0.288902
\(227\) 24.9706 1.65735 0.828677 − 0.559727i \(-0.189094\pi\)
0.828677 + 0.559727i \(0.189094\pi\)
\(228\) 0 0
\(229\) −5.17157 −0.341747 −0.170874 − 0.985293i \(-0.554659\pi\)
−0.170874 + 0.985293i \(0.554659\pi\)
\(230\) 3.65685 0.241126
\(231\) 0 0
\(232\) −5.41421 −0.355461
\(233\) −23.6569 −1.54981 −0.774906 − 0.632076i \(-0.782203\pi\)
−0.774906 + 0.632076i \(0.782203\pi\)
\(234\) 0 0
\(235\) −2.58579 −0.168678
\(236\) 0.828427 0.0539260
\(237\) 0 0
\(238\) 0 0
\(239\) 19.6569 1.27150 0.635748 − 0.771897i \(-0.280692\pi\)
0.635748 + 0.771897i \(0.280692\pi\)
\(240\) 0 0
\(241\) 19.0711 1.22848 0.614238 − 0.789121i \(-0.289464\pi\)
0.614238 + 0.789121i \(0.289464\pi\)
\(242\) −0.656854 −0.0422242
\(243\) 0 0
\(244\) −1.65685 −0.106069
\(245\) 0 0
\(246\) 0 0
\(247\) 5.65685 0.359937
\(248\) −7.41421 −0.470803
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 14.3431 0.905331 0.452666 − 0.891680i \(-0.350473\pi\)
0.452666 + 0.891680i \(0.350473\pi\)
\(252\) 0 0
\(253\) −12.4853 −0.784943
\(254\) 14.1421 0.887357
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.89949 −0.243244 −0.121622 − 0.992577i \(-0.538810\pi\)
−0.121622 + 0.992577i \(0.538810\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.17157 0.0726579
\(261\) 0 0
\(262\) −6.48528 −0.400662
\(263\) 10.0000 0.616626 0.308313 − 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) 0 0
\(265\) 6.48528 0.398388
\(266\) 0 0
\(267\) 0 0
\(268\) 5.89949 0.360369
\(269\) 14.9706 0.912771 0.456386 − 0.889782i \(-0.349144\pi\)
0.456386 + 0.889782i \(0.349144\pi\)
\(270\) 0 0
\(271\) 1.75736 0.106752 0.0533760 − 0.998574i \(-0.483002\pi\)
0.0533760 + 0.998574i \(0.483002\pi\)
\(272\) −3.41421 −0.207017
\(273\) 0 0
\(274\) −4.48528 −0.270966
\(275\) 3.41421 0.205885
\(276\) 0 0
\(277\) −31.4142 −1.88750 −0.943749 − 0.330664i \(-0.892727\pi\)
−0.943749 + 0.330664i \(0.892727\pi\)
\(278\) 1.31371 0.0787910
\(279\) 0 0
\(280\) 0 0
\(281\) 21.4558 1.27995 0.639974 − 0.768396i \(-0.278945\pi\)
0.639974 + 0.768396i \(0.278945\pi\)
\(282\) 0 0
\(283\) −14.9706 −0.889908 −0.444954 − 0.895554i \(-0.646780\pi\)
−0.444954 + 0.895554i \(0.646780\pi\)
\(284\) −10.4853 −0.622187
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −5.34315 −0.314303
\(290\) −5.41421 −0.317934
\(291\) 0 0
\(292\) 12.8284 0.750727
\(293\) −25.3137 −1.47884 −0.739421 − 0.673243i \(-0.764901\pi\)
−0.739421 + 0.673243i \(0.764901\pi\)
\(294\) 0 0
\(295\) 0.828427 0.0482329
\(296\) 3.41421 0.198447
\(297\) 0 0
\(298\) 12.7279 0.737309
\(299\) −4.28427 −0.247766
\(300\) 0 0
\(301\) 0 0
\(302\) −3.17157 −0.182504
\(303\) 0 0
\(304\) 4.82843 0.276929
\(305\) −1.65685 −0.0948712
\(306\) 0 0
\(307\) 26.6274 1.51971 0.759853 − 0.650094i \(-0.225271\pi\)
0.759853 + 0.650094i \(0.225271\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −7.41421 −0.421099
\(311\) −21.4558 −1.21665 −0.608325 − 0.793688i \(-0.708158\pi\)
−0.608325 + 0.793688i \(0.708158\pi\)
\(312\) 0 0
\(313\) 7.65685 0.432791 0.216395 − 0.976306i \(-0.430570\pi\)
0.216395 + 0.976306i \(0.430570\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 13.6569 0.768258
\(317\) −14.9706 −0.840831 −0.420415 − 0.907332i \(-0.638116\pi\)
−0.420415 + 0.907332i \(0.638116\pi\)
\(318\) 0 0
\(319\) 18.4853 1.03498
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −16.4853 −0.917266
\(324\) 0 0
\(325\) 1.17157 0.0649872
\(326\) −7.07107 −0.391630
\(327\) 0 0
\(328\) −3.65685 −0.201916
\(329\) 0 0
\(330\) 0 0
\(331\) 15.7990 0.868391 0.434196 − 0.900819i \(-0.357033\pi\)
0.434196 + 0.900819i \(0.357033\pi\)
\(332\) −0.485281 −0.0266333
\(333\) 0 0
\(334\) −6.58579 −0.360358
\(335\) 5.89949 0.322324
\(336\) 0 0
\(337\) −26.4853 −1.44275 −0.721373 − 0.692547i \(-0.756488\pi\)
−0.721373 + 0.692547i \(0.756488\pi\)
\(338\) 11.6274 0.632448
\(339\) 0 0
\(340\) −3.41421 −0.185162
\(341\) 25.3137 1.37081
\(342\) 0 0
\(343\) 0 0
\(344\) 7.07107 0.381246
\(345\) 0 0
\(346\) −20.1421 −1.08285
\(347\) −17.6569 −0.947870 −0.473935 − 0.880560i \(-0.657167\pi\)
−0.473935 + 0.880560i \(0.657167\pi\)
\(348\) 0 0
\(349\) −20.4853 −1.09655 −0.548276 − 0.836297i \(-0.684716\pi\)
−0.548276 + 0.836297i \(0.684716\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.41421 −0.181978
\(353\) −11.8995 −0.633346 −0.316673 − 0.948535i \(-0.602566\pi\)
−0.316673 + 0.948535i \(0.602566\pi\)
\(354\) 0 0
\(355\) −10.4853 −0.556501
\(356\) −14.4853 −0.767718
\(357\) 0 0
\(358\) −10.7279 −0.566988
\(359\) 32.8284 1.73262 0.866309 − 0.499508i \(-0.166486\pi\)
0.866309 + 0.499508i \(0.166486\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) −3.51472 −0.184730
\(363\) 0 0
\(364\) 0 0
\(365\) 12.8284 0.671471
\(366\) 0 0
\(367\) −0.485281 −0.0253315 −0.0126657 − 0.999920i \(-0.504032\pi\)
−0.0126657 + 0.999920i \(0.504032\pi\)
\(368\) −3.65685 −0.190627
\(369\) 0 0
\(370\) 3.41421 0.177497
\(371\) 0 0
\(372\) 0 0
\(373\) 21.7574 1.12655 0.563277 − 0.826268i \(-0.309541\pi\)
0.563277 + 0.826268i \(0.309541\pi\)
\(374\) 11.6569 0.602762
\(375\) 0 0
\(376\) 2.58579 0.133352
\(377\) 6.34315 0.326689
\(378\) 0 0
\(379\) 4.48528 0.230393 0.115197 − 0.993343i \(-0.463250\pi\)
0.115197 + 0.993343i \(0.463250\pi\)
\(380\) 4.82843 0.247693
\(381\) 0 0
\(382\) 9.31371 0.476531
\(383\) 29.6985 1.51752 0.758761 − 0.651369i \(-0.225805\pi\)
0.758761 + 0.651369i \(0.225805\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −20.1421 −1.02521
\(387\) 0 0
\(388\) 14.4853 0.735379
\(389\) −16.0416 −0.813343 −0.406671 − 0.913574i \(-0.633311\pi\)
−0.406671 + 0.913574i \(0.633311\pi\)
\(390\) 0 0
\(391\) 12.4853 0.631408
\(392\) 0 0
\(393\) 0 0
\(394\) 0.343146 0.0172874
\(395\) 13.6569 0.687151
\(396\) 0 0
\(397\) 8.34315 0.418730 0.209365 − 0.977838i \(-0.432860\pi\)
0.209365 + 0.977838i \(0.432860\pi\)
\(398\) −18.2426 −0.914421
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −4.97056 −0.248218 −0.124109 − 0.992269i \(-0.539607\pi\)
−0.124109 + 0.992269i \(0.539607\pi\)
\(402\) 0 0
\(403\) 8.68629 0.432695
\(404\) −9.65685 −0.480446
\(405\) 0 0
\(406\) 0 0
\(407\) −11.6569 −0.577809
\(408\) 0 0
\(409\) 26.8701 1.32864 0.664319 − 0.747449i \(-0.268721\pi\)
0.664319 + 0.747449i \(0.268721\pi\)
\(410\) −3.65685 −0.180599
\(411\) 0 0
\(412\) 5.65685 0.278693
\(413\) 0 0
\(414\) 0 0
\(415\) −0.485281 −0.0238215
\(416\) −1.17157 −0.0574411
\(417\) 0 0
\(418\) −16.4853 −0.806321
\(419\) 40.2843 1.96802 0.984008 − 0.178126i \(-0.0570034\pi\)
0.984008 + 0.178126i \(0.0570034\pi\)
\(420\) 0 0
\(421\) 8.82843 0.430271 0.215136 − 0.976584i \(-0.430981\pi\)
0.215136 + 0.976584i \(0.430981\pi\)
\(422\) 23.7990 1.15852
\(423\) 0 0
\(424\) −6.48528 −0.314953
\(425\) −3.41421 −0.165614
\(426\) 0 0
\(427\) 0 0
\(428\) −10.8284 −0.523412
\(429\) 0 0
\(430\) 7.07107 0.340997
\(431\) 16.1421 0.777539 0.388770 − 0.921335i \(-0.372900\pi\)
0.388770 + 0.921335i \(0.372900\pi\)
\(432\) 0 0
\(433\) 32.8284 1.57763 0.788817 − 0.614628i \(-0.210694\pi\)
0.788817 + 0.614628i \(0.210694\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.4853 0.693719
\(437\) −17.6569 −0.844642
\(438\) 0 0
\(439\) 14.2426 0.679764 0.339882 − 0.940468i \(-0.389613\pi\)
0.339882 + 0.940468i \(0.389613\pi\)
\(440\) −3.41421 −0.162766
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 0.970563 0.0461128 0.0230564 − 0.999734i \(-0.492660\pi\)
0.0230564 + 0.999734i \(0.492660\pi\)
\(444\) 0 0
\(445\) −14.4853 −0.686668
\(446\) 10.8284 0.512741
\(447\) 0 0
\(448\) 0 0
\(449\) 38.8284 1.83243 0.916213 − 0.400691i \(-0.131230\pi\)
0.916213 + 0.400691i \(0.131230\pi\)
\(450\) 0 0
\(451\) 12.4853 0.587909
\(452\) −4.34315 −0.204284
\(453\) 0 0
\(454\) −24.9706 −1.17193
\(455\) 0 0
\(456\) 0 0
\(457\) 0.142136 0.00664882 0.00332441 − 0.999994i \(-0.498942\pi\)
0.00332441 + 0.999994i \(0.498942\pi\)
\(458\) 5.17157 0.241652
\(459\) 0 0
\(460\) −3.65685 −0.170502
\(461\) 2.68629 0.125113 0.0625565 − 0.998041i \(-0.480075\pi\)
0.0625565 + 0.998041i \(0.480075\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 5.41421 0.251349
\(465\) 0 0
\(466\) 23.6569 1.09588
\(467\) 38.8284 1.79677 0.898383 − 0.439214i \(-0.144743\pi\)
0.898383 + 0.439214i \(0.144743\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 2.58579 0.119273
\(471\) 0 0
\(472\) −0.828427 −0.0381314
\(473\) −24.1421 −1.11006
\(474\) 0 0
\(475\) 4.82843 0.221543
\(476\) 0 0
\(477\) 0 0
\(478\) −19.6569 −0.899084
\(479\) 30.8284 1.40859 0.704293 − 0.709909i \(-0.251264\pi\)
0.704293 + 0.709909i \(0.251264\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −19.0711 −0.868663
\(483\) 0 0
\(484\) 0.656854 0.0298570
\(485\) 14.4853 0.657743
\(486\) 0 0
\(487\) −9.17157 −0.415604 −0.207802 − 0.978171i \(-0.566631\pi\)
−0.207802 + 0.978171i \(0.566631\pi\)
\(488\) 1.65685 0.0750023
\(489\) 0 0
\(490\) 0 0
\(491\) −18.9289 −0.854251 −0.427125 − 0.904192i \(-0.640474\pi\)
−0.427125 + 0.904192i \(0.640474\pi\)
\(492\) 0 0
\(493\) −18.4853 −0.832535
\(494\) −5.65685 −0.254514
\(495\) 0 0
\(496\) 7.41421 0.332908
\(497\) 0 0
\(498\) 0 0
\(499\) −31.1127 −1.39280 −0.696398 − 0.717656i \(-0.745215\pi\)
−0.696398 + 0.717656i \(0.745215\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −14.3431 −0.640166
\(503\) 25.4142 1.13316 0.566582 − 0.824005i \(-0.308265\pi\)
0.566582 + 0.824005i \(0.308265\pi\)
\(504\) 0 0
\(505\) −9.65685 −0.429724
\(506\) 12.4853 0.555038
\(507\) 0 0
\(508\) −14.1421 −0.627456
\(509\) −35.5980 −1.57785 −0.788926 − 0.614488i \(-0.789363\pi\)
−0.788926 + 0.614488i \(0.789363\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 3.89949 0.171999
\(515\) 5.65685 0.249271
\(516\) 0 0
\(517\) −8.82843 −0.388274
\(518\) 0 0
\(519\) 0 0
\(520\) −1.17157 −0.0513769
\(521\) 20.8284 0.912510 0.456255 − 0.889849i \(-0.349190\pi\)
0.456255 + 0.889849i \(0.349190\pi\)
\(522\) 0 0
\(523\) 24.6274 1.07688 0.538441 − 0.842663i \(-0.319014\pi\)
0.538441 + 0.842663i \(0.319014\pi\)
\(524\) 6.48528 0.283311
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) −25.3137 −1.10268
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) −6.48528 −0.281703
\(531\) 0 0
\(532\) 0 0
\(533\) 4.28427 0.185572
\(534\) 0 0
\(535\) −10.8284 −0.468154
\(536\) −5.89949 −0.254819
\(537\) 0 0
\(538\) −14.9706 −0.645427
\(539\) 0 0
\(540\) 0 0
\(541\) −43.9411 −1.88918 −0.944588 − 0.328258i \(-0.893539\pi\)
−0.944588 + 0.328258i \(0.893539\pi\)
\(542\) −1.75736 −0.0754850
\(543\) 0 0
\(544\) 3.41421 0.146383
\(545\) 14.4853 0.620481
\(546\) 0 0
\(547\) −7.27208 −0.310932 −0.155466 − 0.987841i \(-0.549688\pi\)
−0.155466 + 0.987841i \(0.549688\pi\)
\(548\) 4.48528 0.191602
\(549\) 0 0
\(550\) −3.41421 −0.145583
\(551\) 26.1421 1.11369
\(552\) 0 0
\(553\) 0 0
\(554\) 31.4142 1.33466
\(555\) 0 0
\(556\) −1.31371 −0.0557137
\(557\) 6.68629 0.283307 0.141654 − 0.989916i \(-0.454758\pi\)
0.141654 + 0.989916i \(0.454758\pi\)
\(558\) 0 0
\(559\) −8.28427 −0.350387
\(560\) 0 0
\(561\) 0 0
\(562\) −21.4558 −0.905060
\(563\) −15.7990 −0.665848 −0.332924 − 0.942954i \(-0.608035\pi\)
−0.332924 + 0.942954i \(0.608035\pi\)
\(564\) 0 0
\(565\) −4.34315 −0.182718
\(566\) 14.9706 0.629260
\(567\) 0 0
\(568\) 10.4853 0.439953
\(569\) 0.686292 0.0287708 0.0143854 − 0.999897i \(-0.495421\pi\)
0.0143854 + 0.999897i \(0.495421\pi\)
\(570\) 0 0
\(571\) −4.68629 −0.196115 −0.0980576 − 0.995181i \(-0.531263\pi\)
−0.0980576 + 0.995181i \(0.531263\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) −3.65685 −0.152501
\(576\) 0 0
\(577\) −36.6274 −1.52482 −0.762410 − 0.647095i \(-0.775984\pi\)
−0.762410 + 0.647095i \(0.775984\pi\)
\(578\) 5.34315 0.222246
\(579\) 0 0
\(580\) 5.41421 0.224813
\(581\) 0 0
\(582\) 0 0
\(583\) 22.1421 0.917034
\(584\) −12.8284 −0.530844
\(585\) 0 0
\(586\) 25.3137 1.04570
\(587\) −10.1421 −0.418611 −0.209305 − 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(588\) 0 0
\(589\) 35.7990 1.47507
\(590\) −0.828427 −0.0341058
\(591\) 0 0
\(592\) −3.41421 −0.140323
\(593\) 39.6985 1.63022 0.815111 − 0.579305i \(-0.196676\pi\)
0.815111 + 0.579305i \(0.196676\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.7279 −0.521356
\(597\) 0 0
\(598\) 4.28427 0.175197
\(599\) 32.6274 1.33312 0.666560 − 0.745451i \(-0.267766\pi\)
0.666560 + 0.745451i \(0.267766\pi\)
\(600\) 0 0
\(601\) −46.3848 −1.89207 −0.946037 − 0.324058i \(-0.894953\pi\)
−0.946037 + 0.324058i \(0.894953\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.17157 0.129049
\(605\) 0.656854 0.0267049
\(606\) 0 0
\(607\) −31.1127 −1.26283 −0.631413 − 0.775447i \(-0.717525\pi\)
−0.631413 + 0.775447i \(0.717525\pi\)
\(608\) −4.82843 −0.195819
\(609\) 0 0
\(610\) 1.65685 0.0670841
\(611\) −3.02944 −0.122558
\(612\) 0 0
\(613\) 19.4142 0.784133 0.392066 − 0.919937i \(-0.371760\pi\)
0.392066 + 0.919937i \(0.371760\pi\)
\(614\) −26.6274 −1.07460
\(615\) 0 0
\(616\) 0 0
\(617\) 39.1127 1.57462 0.787309 − 0.616559i \(-0.211474\pi\)
0.787309 + 0.616559i \(0.211474\pi\)
\(618\) 0 0
\(619\) 0.828427 0.0332973 0.0166486 − 0.999861i \(-0.494700\pi\)
0.0166486 + 0.999861i \(0.494700\pi\)
\(620\) 7.41421 0.297762
\(621\) 0 0
\(622\) 21.4558 0.860301
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −7.65685 −0.306029
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 11.6569 0.464789
\(630\) 0 0
\(631\) 15.4558 0.615287 0.307644 − 0.951502i \(-0.400460\pi\)
0.307644 + 0.951502i \(0.400460\pi\)
\(632\) −13.6569 −0.543240
\(633\) 0 0
\(634\) 14.9706 0.594557
\(635\) −14.1421 −0.561214
\(636\) 0 0
\(637\) 0 0
\(638\) −18.4853 −0.731839
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.6274 −0.735739 −0.367869 − 0.929877i \(-0.619913\pi\)
−0.367869 + 0.929877i \(0.619913\pi\)
\(642\) 0 0
\(643\) 6.97056 0.274892 0.137446 − 0.990509i \(-0.456111\pi\)
0.137446 + 0.990509i \(0.456111\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.4853 0.648605
\(647\) 13.6152 0.535270 0.267635 − 0.963520i \(-0.413758\pi\)
0.267635 + 0.963520i \(0.413758\pi\)
\(648\) 0 0
\(649\) 2.82843 0.111025
\(650\) −1.17157 −0.0459529
\(651\) 0 0
\(652\) 7.07107 0.276924
\(653\) −45.1127 −1.76540 −0.882698 − 0.469940i \(-0.844275\pi\)
−0.882698 + 0.469940i \(0.844275\pi\)
\(654\) 0 0
\(655\) 6.48528 0.253401
\(656\) 3.65685 0.142776
\(657\) 0 0
\(658\) 0 0
\(659\) 28.5858 1.11354 0.556772 − 0.830665i \(-0.312040\pi\)
0.556772 + 0.830665i \(0.312040\pi\)
\(660\) 0 0
\(661\) −22.3431 −0.869048 −0.434524 − 0.900660i \(-0.643083\pi\)
−0.434524 + 0.900660i \(0.643083\pi\)
\(662\) −15.7990 −0.614045
\(663\) 0 0
\(664\) 0.485281 0.0188326
\(665\) 0 0
\(666\) 0 0
\(667\) −19.7990 −0.766620
\(668\) 6.58579 0.254812
\(669\) 0 0
\(670\) −5.89949 −0.227917
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) 10.9706 0.422884 0.211442 − 0.977391i \(-0.432184\pi\)
0.211442 + 0.977391i \(0.432184\pi\)
\(674\) 26.4853 1.02017
\(675\) 0 0
\(676\) −11.6274 −0.447208
\(677\) 15.4558 0.594016 0.297008 − 0.954875i \(-0.404011\pi\)
0.297008 + 0.954875i \(0.404011\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.41421 0.130929
\(681\) 0 0
\(682\) −25.3137 −0.969312
\(683\) −26.3431 −1.00799 −0.503996 − 0.863706i \(-0.668137\pi\)
−0.503996 + 0.863706i \(0.668137\pi\)
\(684\) 0 0
\(685\) 4.48528 0.171374
\(686\) 0 0
\(687\) 0 0
\(688\) −7.07107 −0.269582
\(689\) 7.59798 0.289460
\(690\) 0 0
\(691\) −1.02944 −0.0391616 −0.0195808 − 0.999808i \(-0.506233\pi\)
−0.0195808 + 0.999808i \(0.506233\pi\)
\(692\) 20.1421 0.765689
\(693\) 0 0
\(694\) 17.6569 0.670245
\(695\) −1.31371 −0.0498318
\(696\) 0 0
\(697\) −12.4853 −0.472914
\(698\) 20.4853 0.775379
\(699\) 0 0
\(700\) 0 0
\(701\) −7.75736 −0.292991 −0.146496 − 0.989211i \(-0.546799\pi\)
−0.146496 + 0.989211i \(0.546799\pi\)
\(702\) 0 0
\(703\) −16.4853 −0.621754
\(704\) 3.41421 0.128678
\(705\) 0 0
\(706\) 11.8995 0.447843
\(707\) 0 0
\(708\) 0 0
\(709\) 21.1127 0.792904 0.396452 − 0.918055i \(-0.370241\pi\)
0.396452 + 0.918055i \(0.370241\pi\)
\(710\) 10.4853 0.393506
\(711\) 0 0
\(712\) 14.4853 0.542859
\(713\) −27.1127 −1.01538
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 10.7279 0.400921
\(717\) 0 0
\(718\) −32.8284 −1.22515
\(719\) −33.4558 −1.24769 −0.623846 − 0.781547i \(-0.714431\pi\)
−0.623846 + 0.781547i \(0.714431\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −4.31371 −0.160540
\(723\) 0 0
\(724\) 3.51472 0.130623
\(725\) 5.41421 0.201079
\(726\) 0 0
\(727\) 13.4558 0.499050 0.249525 − 0.968368i \(-0.419726\pi\)
0.249525 + 0.968368i \(0.419726\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −12.8284 −0.474801
\(731\) 24.1421 0.892929
\(732\) 0 0
\(733\) −6.97056 −0.257464 −0.128732 − 0.991679i \(-0.541091\pi\)
−0.128732 + 0.991679i \(0.541091\pi\)
\(734\) 0.485281 0.0179121
\(735\) 0 0
\(736\) 3.65685 0.134793
\(737\) 20.1421 0.741945
\(738\) 0 0
\(739\) −2.82843 −0.104045 −0.0520227 − 0.998646i \(-0.516567\pi\)
−0.0520227 + 0.998646i \(0.516567\pi\)
\(740\) −3.41421 −0.125509
\(741\) 0 0
\(742\) 0 0
\(743\) −45.6569 −1.67499 −0.837494 − 0.546447i \(-0.815980\pi\)
−0.837494 + 0.546447i \(0.815980\pi\)
\(744\) 0 0
\(745\) −12.7279 −0.466315
\(746\) −21.7574 −0.796594
\(747\) 0 0
\(748\) −11.6569 −0.426217
\(749\) 0 0
\(750\) 0 0
\(751\) −41.7990 −1.52527 −0.762633 − 0.646831i \(-0.776094\pi\)
−0.762633 + 0.646831i \(0.776094\pi\)
\(752\) −2.58579 −0.0942939
\(753\) 0 0
\(754\) −6.34315 −0.231004
\(755\) 3.17157 0.115425
\(756\) 0 0
\(757\) 21.3553 0.776173 0.388087 − 0.921623i \(-0.373136\pi\)
0.388087 + 0.921623i \(0.373136\pi\)
\(758\) −4.48528 −0.162913
\(759\) 0 0
\(760\) −4.82843 −0.175145
\(761\) 8.62742 0.312744 0.156372 − 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −9.31371 −0.336958
\(765\) 0 0
\(766\) −29.6985 −1.07305
\(767\) 0.970563 0.0350450
\(768\) 0 0
\(769\) −12.9289 −0.466229 −0.233115 − 0.972449i \(-0.574892\pi\)
−0.233115 + 0.972449i \(0.574892\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.1421 0.724931
\(773\) −18.9706 −0.682324 −0.341162 − 0.940005i \(-0.610820\pi\)
−0.341162 + 0.940005i \(0.610820\pi\)
\(774\) 0 0
\(775\) 7.41421 0.266326
\(776\) −14.4853 −0.519991
\(777\) 0 0
\(778\) 16.0416 0.575120
\(779\) 17.6569 0.632622
\(780\) 0 0
\(781\) −35.7990 −1.28099
\(782\) −12.4853 −0.446473
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −3.02944 −0.107988 −0.0539939 − 0.998541i \(-0.517195\pi\)
−0.0539939 + 0.998541i \(0.517195\pi\)
\(788\) −0.343146 −0.0122241
\(789\) 0 0
\(790\) −13.6569 −0.485889
\(791\) 0 0
\(792\) 0 0
\(793\) −1.94113 −0.0689314
\(794\) −8.34315 −0.296087
\(795\) 0 0
\(796\) 18.2426 0.646593
\(797\) −7.85786 −0.278340 −0.139170 − 0.990269i \(-0.544443\pi\)
−0.139170 + 0.990269i \(0.544443\pi\)
\(798\) 0 0
\(799\) 8.82843 0.312327
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 4.97056 0.175517
\(803\) 43.7990 1.54563
\(804\) 0 0
\(805\) 0 0
\(806\) −8.68629 −0.305962
\(807\) 0 0
\(808\) 9.65685 0.339727
\(809\) 16.4853 0.579592 0.289796 − 0.957088i \(-0.406413\pi\)
0.289796 + 0.957088i \(0.406413\pi\)
\(810\) 0 0
\(811\) −24.6274 −0.864786 −0.432393 − 0.901685i \(-0.642331\pi\)
−0.432393 + 0.901685i \(0.642331\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 11.6569 0.408573
\(815\) 7.07107 0.247689
\(816\) 0 0
\(817\) −34.1421 −1.19448
\(818\) −26.8701 −0.939490
\(819\) 0 0
\(820\) 3.65685 0.127703
\(821\) −28.9289 −1.00963 −0.504813 − 0.863229i \(-0.668439\pi\)
−0.504813 + 0.863229i \(0.668439\pi\)
\(822\) 0 0
\(823\) 10.3431 0.360539 0.180270 − 0.983617i \(-0.442303\pi\)
0.180270 + 0.983617i \(0.442303\pi\)
\(824\) −5.65685 −0.197066
\(825\) 0 0
\(826\) 0 0
\(827\) −23.7990 −0.827572 −0.413786 − 0.910374i \(-0.635794\pi\)
−0.413786 + 0.910374i \(0.635794\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 − 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 0.485281 0.0168444
\(831\) 0 0
\(832\) 1.17157 0.0406170
\(833\) 0 0
\(834\) 0 0
\(835\) 6.58579 0.227911
\(836\) 16.4853 0.570155
\(837\) 0 0
\(838\) −40.2843 −1.39160
\(839\) 24.2843 0.838386 0.419193 − 0.907897i \(-0.362313\pi\)
0.419193 + 0.907897i \(0.362313\pi\)
\(840\) 0 0
\(841\) 0.313708 0.0108175
\(842\) −8.82843 −0.304248
\(843\) 0 0
\(844\) −23.7990 −0.819195
\(845\) −11.6274 −0.399995
\(846\) 0 0
\(847\) 0 0
\(848\) 6.48528 0.222705
\(849\) 0 0
\(850\) 3.41421 0.117107
\(851\) 12.4853 0.427990
\(852\) 0 0
\(853\) −51.6569 −1.76870 −0.884349 − 0.466827i \(-0.845397\pi\)
−0.884349 + 0.466827i \(0.845397\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10.8284 0.370108
\(857\) −51.8995 −1.77285 −0.886426 − 0.462869i \(-0.846820\pi\)
−0.886426 + 0.462869i \(0.846820\pi\)
\(858\) 0 0
\(859\) 30.9706 1.05670 0.528351 − 0.849026i \(-0.322811\pi\)
0.528351 + 0.849026i \(0.322811\pi\)
\(860\) −7.07107 −0.241121
\(861\) 0 0
\(862\) −16.1421 −0.549803
\(863\) −28.9706 −0.986169 −0.493085 − 0.869981i \(-0.664131\pi\)
−0.493085 + 0.869981i \(0.664131\pi\)
\(864\) 0 0
\(865\) 20.1421 0.684853
\(866\) −32.8284 −1.11556
\(867\) 0 0
\(868\) 0 0
\(869\) 46.6274 1.58173
\(870\) 0 0
\(871\) 6.91169 0.234194
\(872\) −14.4853 −0.490534
\(873\) 0 0
\(874\) 17.6569 0.597252
\(875\) 0 0
\(876\) 0 0
\(877\) −41.5563 −1.40326 −0.701629 − 0.712542i \(-0.747544\pi\)
−0.701629 + 0.712542i \(0.747544\pi\)
\(878\) −14.2426 −0.480666
\(879\) 0 0
\(880\) 3.41421 0.115093
\(881\) −52.1421 −1.75671 −0.878357 − 0.478006i \(-0.841360\pi\)
−0.878357 + 0.478006i \(0.841360\pi\)
\(882\) 0 0
\(883\) 20.5269 0.690786 0.345393 − 0.938458i \(-0.387746\pi\)
0.345393 + 0.938458i \(0.387746\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −0.970563 −0.0326067
\(887\) −45.8995 −1.54115 −0.770577 − 0.637347i \(-0.780032\pi\)
−0.770577 + 0.637347i \(0.780032\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 14.4853 0.485548
\(891\) 0 0
\(892\) −10.8284 −0.362563
\(893\) −12.4853 −0.417804
\(894\) 0 0
\(895\) 10.7279 0.358595
\(896\) 0 0
\(897\) 0 0
\(898\) −38.8284 −1.29572
\(899\) 40.1421 1.33882
\(900\) 0 0
\(901\) −22.1421 −0.737661
\(902\) −12.4853 −0.415714
\(903\) 0 0
\(904\) 4.34315 0.144451
\(905\) 3.51472 0.116833
\(906\) 0 0
\(907\) −56.3259 −1.87027 −0.935135 − 0.354290i \(-0.884722\pi\)
−0.935135 + 0.354290i \(0.884722\pi\)
\(908\) 24.9706 0.828677
\(909\) 0 0
\(910\) 0 0
\(911\) −25.5980 −0.848099 −0.424049 − 0.905639i \(-0.639392\pi\)
−0.424049 + 0.905639i \(0.639392\pi\)
\(912\) 0 0
\(913\) −1.65685 −0.0548339
\(914\) −0.142136 −0.00470143
\(915\) 0 0
\(916\) −5.17157 −0.170874
\(917\) 0 0
\(918\) 0 0
\(919\) −19.4558 −0.641789 −0.320895 − 0.947115i \(-0.603984\pi\)
−0.320895 + 0.947115i \(0.603984\pi\)
\(920\) 3.65685 0.120563
\(921\) 0 0
\(922\) −2.68629 −0.0884683
\(923\) −12.2843 −0.404342
\(924\) 0 0
\(925\) −3.41421 −0.112259
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −5.41421 −0.177730
\(929\) −52.9117 −1.73598 −0.867988 − 0.496585i \(-0.834587\pi\)
−0.867988 + 0.496585i \(0.834587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.6569 −0.774906
\(933\) 0 0
\(934\) −38.8284 −1.27050
\(935\) −11.6569 −0.381220
\(936\) 0 0
\(937\) −18.9706 −0.619741 −0.309871 − 0.950779i \(-0.600286\pi\)
−0.309871 + 0.950779i \(0.600286\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.58579 −0.0843391
\(941\) −56.0000 −1.82555 −0.912774 − 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) −13.3726 −0.435471
\(944\) 0.828427 0.0269630
\(945\) 0 0
\(946\) 24.1421 0.784929
\(947\) 38.6274 1.25522 0.627611 − 0.778527i \(-0.284033\pi\)
0.627611 + 0.778527i \(0.284033\pi\)
\(948\) 0 0
\(949\) 15.0294 0.487876
\(950\) −4.82843 −0.156655
\(951\) 0 0
\(952\) 0 0
\(953\) −45.1716 −1.46325 −0.731625 − 0.681707i \(-0.761238\pi\)
−0.731625 + 0.681707i \(0.761238\pi\)
\(954\) 0 0
\(955\) −9.31371 −0.301385
\(956\) 19.6569 0.635748
\(957\) 0 0
\(958\) −30.8284 −0.996021
\(959\) 0 0
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) 19.0711 0.614238
\(965\) 20.1421 0.648398
\(966\) 0 0
\(967\) 40.9706 1.31752 0.658762 − 0.752351i \(-0.271080\pi\)
0.658762 + 0.752351i \(0.271080\pi\)
\(968\) −0.656854 −0.0211121
\(969\) 0 0
\(970\) −14.4853 −0.465094
\(971\) −9.65685 −0.309903 −0.154952 − 0.987922i \(-0.549522\pi\)
−0.154952 + 0.987922i \(0.549522\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 9.17157 0.293876
\(975\) 0 0
\(976\) −1.65685 −0.0530346
\(977\) 29.3137 0.937829 0.468914 − 0.883244i \(-0.344645\pi\)
0.468914 + 0.883244i \(0.344645\pi\)
\(978\) 0 0
\(979\) −49.4558 −1.58062
\(980\) 0 0
\(981\) 0 0
\(982\) 18.9289 0.604046
\(983\) −14.3848 −0.458803 −0.229402 − 0.973332i \(-0.573677\pi\)
−0.229402 + 0.973332i \(0.573677\pi\)
\(984\) 0 0
\(985\) −0.343146 −0.0109335
\(986\) 18.4853 0.588691
\(987\) 0 0
\(988\) 5.65685 0.179969
\(989\) 25.8579 0.822232
\(990\) 0 0
\(991\) −12.1421 −0.385708 −0.192854 − 0.981227i \(-0.561774\pi\)
−0.192854 + 0.981227i \(0.561774\pi\)
\(992\) −7.41421 −0.235402
\(993\) 0 0
\(994\) 0 0
\(995\) 18.2426 0.578331
\(996\) 0 0
\(997\) −8.34315 −0.264230 −0.132115 − 0.991234i \(-0.542177\pi\)
−0.132115 + 0.991234i \(0.542177\pi\)
\(998\) 31.1127 0.984855
\(999\) 0 0
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 4410.2.a.bs.1.2 yes 2
3.2 odd 2 4410.2.a.bu.1.1 yes 2
7.6 odd 2 4410.2.a.bp.1.2 ✓ 2
21.20 even 2 4410.2.a.bx.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.a.bp.1.2 ✓ 2 7.6 odd 2
4410.2.a.bs.1.2 yes 2 1.1 even 1 trivial
4410.2.a.bu.1.1 yes 2 3.2 odd 2
4410.2.a.bx.1.1 yes 2 21.20 even 2