Properties

Label 2394.2.a.s.1.1
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2394.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.23607 q^{10} +5.23607 q^{11} -3.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.70820 q^{17} +1.00000 q^{19} -1.23607 q^{20} -5.23607 q^{22} +0.763932 q^{23} -3.47214 q^{25} +3.23607 q^{26} -1.00000 q^{28} -0.472136 q^{29} -2.00000 q^{31} -1.00000 q^{32} -5.70820 q^{34} +1.23607 q^{35} -1.52786 q^{37} -1.00000 q^{38} +1.23607 q^{40} +6.00000 q^{41} -8.94427 q^{43} +5.23607 q^{44} -0.763932 q^{46} -8.94427 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} +3.52786 q^{53} -6.47214 q^{55} +1.00000 q^{56} +0.472136 q^{58} +8.00000 q^{59} +8.47214 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +13.7082 q^{67} +5.70820 q^{68} -1.23607 q^{70} +1.52786 q^{71} -3.52786 q^{73} +1.52786 q^{74} +1.00000 q^{76} -5.23607 q^{77} +5.23607 q^{79} -1.23607 q^{80} -6.00000 q^{82} -10.0000 q^{83} -7.05573 q^{85} +8.94427 q^{86} -5.23607 q^{88} +6.00000 q^{89} +3.23607 q^{91} +0.763932 q^{92} +8.94427 q^{94} -1.23607 q^{95} +9.23607 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} + 12 q^{41} + 6 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 16 q^{53} - 4 q^{55} + 2 q^{56} - 8 q^{58} + 16 q^{59} + 8 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 14 q^{67} - 2 q^{68} + 2 q^{70} + 12 q^{71} - 16 q^{73} + 12 q^{74} + 2 q^{76} - 6 q^{77} + 6 q^{79} + 2 q^{80} - 12 q^{82} - 20 q^{83} - 32 q^{85} - 6 q^{88} + 12 q^{89} + 2 q^{91} + 6 q^{92} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.23607 0.390879
\(11\) 5.23607 1.57873 0.789367 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.70820 1.38444 0.692221 0.721685i \(-0.256632\pi\)
0.692221 + 0.721685i \(0.256632\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) −1.23607 −0.276393
\(21\) 0 0
\(22\) −5.23607 −1.11633
\(23\) 0.763932 0.159291 0.0796454 0.996823i \(-0.474621\pi\)
0.0796454 + 0.996823i \(0.474621\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 3.23607 0.634645
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −0.472136 −0.0876734 −0.0438367 0.999039i \(-0.513958\pi\)
−0.0438367 + 0.999039i \(0.513958\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.70820 −0.978949
\(35\) 1.23607 0.208934
\(36\) 0 0
\(37\) −1.52786 −0.251179 −0.125590 0.992082i \(-0.540082\pi\)
−0.125590 + 0.992082i \(0.540082\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.23607 0.195440
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.94427 −1.36399 −0.681994 0.731357i \(-0.738887\pi\)
−0.681994 + 0.731357i \(0.738887\pi\)
\(44\) 5.23607 0.789367
\(45\) 0 0
\(46\) −0.763932 −0.112636
\(47\) −8.94427 −1.30466 −0.652328 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 3.47214 0.491034
\(51\) 0 0
\(52\) −3.23607 −0.448762
\(53\) 3.52786 0.484589 0.242295 0.970203i \(-0.422100\pi\)
0.242295 + 0.970203i \(0.422100\pi\)
\(54\) 0 0
\(55\) −6.47214 −0.872703
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 0.472136 0.0619945
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 8.47214 1.08475 0.542373 0.840138i \(-0.317526\pi\)
0.542373 + 0.840138i \(0.317526\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 13.7082 1.67472 0.837362 0.546649i \(-0.184097\pi\)
0.837362 + 0.546649i \(0.184097\pi\)
\(68\) 5.70820 0.692221
\(69\) 0 0
\(70\) −1.23607 −0.147738
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) 0 0
\(73\) −3.52786 −0.412905 −0.206453 0.978457i \(-0.566192\pi\)
−0.206453 + 0.978457i \(0.566192\pi\)
\(74\) 1.52786 0.177611
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −5.23607 −0.596705
\(78\) 0 0
\(79\) 5.23607 0.589104 0.294552 0.955636i \(-0.404830\pi\)
0.294552 + 0.955636i \(0.404830\pi\)
\(80\) −1.23607 −0.138197
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) −7.05573 −0.765301
\(86\) 8.94427 0.964486
\(87\) 0 0
\(88\) −5.23607 −0.558167
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 0.763932 0.0796454
\(93\) 0 0
\(94\) 8.94427 0.922531
\(95\) −1.23607 −0.126818
\(96\) 0 0
\(97\) 9.23607 0.937781 0.468890 0.883256i \(-0.344654\pi\)
0.468890 + 0.883256i \(0.344654\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −3.47214 −0.347214
\(101\) −0.291796 −0.0290348 −0.0145174 0.999895i \(-0.504621\pi\)
−0.0145174 + 0.999895i \(0.504621\pi\)
\(102\) 0 0
\(103\) 10.9443 1.07837 0.539186 0.842187i \(-0.318732\pi\)
0.539186 + 0.842187i \(0.318732\pi\)
\(104\) 3.23607 0.317323
\(105\) 0 0
\(106\) −3.52786 −0.342656
\(107\) 14.4721 1.39907 0.699537 0.714596i \(-0.253390\pi\)
0.699537 + 0.714596i \(0.253390\pi\)
\(108\) 0 0
\(109\) 9.52786 0.912604 0.456302 0.889825i \(-0.349174\pi\)
0.456302 + 0.889825i \(0.349174\pi\)
\(110\) 6.47214 0.617094
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) −0.944272 −0.0880538
\(116\) −0.472136 −0.0438367
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −5.70820 −0.523270
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) −8.47214 −0.767031
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) 3.70820 0.329050 0.164525 0.986373i \(-0.447391\pi\)
0.164525 + 0.986373i \(0.447391\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −7.52786 −0.657713 −0.328856 0.944380i \(-0.606663\pi\)
−0.328856 + 0.944380i \(0.606663\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −13.7082 −1.18421
\(135\) 0 0
\(136\) −5.70820 −0.489474
\(137\) 11.4164 0.975370 0.487685 0.873020i \(-0.337842\pi\)
0.487685 + 0.873020i \(0.337842\pi\)
\(138\) 0 0
\(139\) −13.8885 −1.17801 −0.589005 0.808129i \(-0.700480\pi\)
−0.589005 + 0.808129i \(0.700480\pi\)
\(140\) 1.23607 0.104467
\(141\) 0 0
\(142\) −1.52786 −0.128216
\(143\) −16.9443 −1.41695
\(144\) 0 0
\(145\) 0.583592 0.0484647
\(146\) 3.52786 0.291968
\(147\) 0 0
\(148\) −1.52786 −0.125590
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 5.23607 0.426105 0.213053 0.977041i \(-0.431659\pi\)
0.213053 + 0.977041i \(0.431659\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 5.23607 0.421934
\(155\) 2.47214 0.198567
\(156\) 0 0
\(157\) −13.4164 −1.07075 −0.535373 0.844616i \(-0.679829\pi\)
−0.535373 + 0.844616i \(0.679829\pi\)
\(158\) −5.23607 −0.416559
\(159\) 0 0
\(160\) 1.23607 0.0977198
\(161\) −0.763932 −0.0602063
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) −0.944272 −0.0730700 −0.0365350 0.999332i \(-0.511632\pi\)
−0.0365350 + 0.999332i \(0.511632\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 7.05573 0.541150
\(171\) 0 0
\(172\) −8.94427 −0.681994
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 3.47214 0.262469
\(176\) 5.23607 0.394683
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 21.8885 1.63603 0.818013 0.575199i \(-0.195076\pi\)
0.818013 + 0.575199i \(0.195076\pi\)
\(180\) 0 0
\(181\) −0.763932 −0.0567826 −0.0283913 0.999597i \(-0.509038\pi\)
−0.0283913 + 0.999597i \(0.509038\pi\)
\(182\) −3.23607 −0.239873
\(183\) 0 0
\(184\) −0.763932 −0.0563178
\(185\) 1.88854 0.138849
\(186\) 0 0
\(187\) 29.8885 2.18567
\(188\) −8.94427 −0.652328
\(189\) 0 0
\(190\) 1.23607 0.0896738
\(191\) 12.7639 0.923566 0.461783 0.886993i \(-0.347210\pi\)
0.461783 + 0.886993i \(0.347210\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −9.23607 −0.663111
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.94427 0.494759 0.247379 0.968919i \(-0.420431\pi\)
0.247379 + 0.968919i \(0.420431\pi\)
\(198\) 0 0
\(199\) −11.4164 −0.809288 −0.404644 0.914474i \(-0.632605\pi\)
−0.404644 + 0.914474i \(0.632605\pi\)
\(200\) 3.47214 0.245517
\(201\) 0 0
\(202\) 0.291796 0.0205307
\(203\) 0.472136 0.0331374
\(204\) 0 0
\(205\) −7.41641 −0.517984
\(206\) −10.9443 −0.762524
\(207\) 0 0
\(208\) −3.23607 −0.224381
\(209\) 5.23607 0.362186
\(210\) 0 0
\(211\) 27.5967 1.89984 0.949919 0.312496i \(-0.101165\pi\)
0.949919 + 0.312496i \(0.101165\pi\)
\(212\) 3.52786 0.242295
\(213\) 0 0
\(214\) −14.4721 −0.989295
\(215\) 11.0557 0.753994
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −9.52786 −0.645308
\(219\) 0 0
\(220\) −6.47214 −0.436351
\(221\) −18.4721 −1.24257
\(222\) 0 0
\(223\) 19.5279 1.30768 0.653841 0.756632i \(-0.273156\pi\)
0.653841 + 0.756632i \(0.273156\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) −15.4164 −1.02322 −0.511611 0.859217i \(-0.670951\pi\)
−0.511611 + 0.859217i \(0.670951\pi\)
\(228\) 0 0
\(229\) −18.9443 −1.25187 −0.625936 0.779874i \(-0.715283\pi\)
−0.625936 + 0.779874i \(0.715283\pi\)
\(230\) 0.944272 0.0622634
\(231\) 0 0
\(232\) 0.472136 0.0309972
\(233\) 14.4721 0.948101 0.474051 0.880498i \(-0.342791\pi\)
0.474051 + 0.880498i \(0.342791\pi\)
\(234\) 0 0
\(235\) 11.0557 0.721196
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) 5.70820 0.370008
\(239\) 3.81966 0.247073 0.123537 0.992340i \(-0.460576\pi\)
0.123537 + 0.992340i \(0.460576\pi\)
\(240\) 0 0
\(241\) −5.23607 −0.337285 −0.168642 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(242\) −16.4164 −1.05529
\(243\) 0 0
\(244\) 8.47214 0.542373
\(245\) −1.23607 −0.0789695
\(246\) 0 0
\(247\) −3.23607 −0.205906
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −10.4721 −0.662316
\(251\) 25.4164 1.60427 0.802135 0.597143i \(-0.203698\pi\)
0.802135 + 0.597143i \(0.203698\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −3.70820 −0.232673
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.4721 −0.777990 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(258\) 0 0
\(259\) 1.52786 0.0949369
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 7.52786 0.465073
\(263\) −5.70820 −0.351983 −0.175991 0.984392i \(-0.556313\pi\)
−0.175991 + 0.984392i \(0.556313\pi\)
\(264\) 0 0
\(265\) −4.36068 −0.267874
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) 13.7082 0.837362
\(269\) 10.9443 0.667284 0.333642 0.942700i \(-0.391722\pi\)
0.333642 + 0.942700i \(0.391722\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 5.70820 0.346111
\(273\) 0 0
\(274\) −11.4164 −0.689690
\(275\) −18.1803 −1.09632
\(276\) 0 0
\(277\) 2.94427 0.176904 0.0884521 0.996080i \(-0.471808\pi\)
0.0884521 + 0.996080i \(0.471808\pi\)
\(278\) 13.8885 0.832980
\(279\) 0 0
\(280\) −1.23607 −0.0738692
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 13.8885 0.825588 0.412794 0.910824i \(-0.364553\pi\)
0.412794 + 0.910824i \(0.364553\pi\)
\(284\) 1.52786 0.0906621
\(285\) 0 0
\(286\) 16.9443 1.00194
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 15.5836 0.916682
\(290\) −0.583592 −0.0342697
\(291\) 0 0
\(292\) −3.52786 −0.206453
\(293\) −19.8885 −1.16190 −0.580951 0.813939i \(-0.697319\pi\)
−0.580951 + 0.813939i \(0.697319\pi\)
\(294\) 0 0
\(295\) −9.88854 −0.575733
\(296\) 1.52786 0.0888053
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −2.47214 −0.142967
\(300\) 0 0
\(301\) 8.94427 0.515539
\(302\) −5.23607 −0.301302
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −10.4721 −0.599633
\(306\) 0 0
\(307\) 31.4164 1.79303 0.896515 0.443014i \(-0.146091\pi\)
0.896515 + 0.443014i \(0.146091\pi\)
\(308\) −5.23607 −0.298353
\(309\) 0 0
\(310\) −2.47214 −0.140408
\(311\) −28.3607 −1.60819 −0.804093 0.594503i \(-0.797349\pi\)
−0.804093 + 0.594503i \(0.797349\pi\)
\(312\) 0 0
\(313\) −10.9443 −0.618607 −0.309303 0.950963i \(-0.600096\pi\)
−0.309303 + 0.950963i \(0.600096\pi\)
\(314\) 13.4164 0.757132
\(315\) 0 0
\(316\) 5.23607 0.294552
\(317\) 7.88854 0.443065 0.221532 0.975153i \(-0.428894\pi\)
0.221532 + 0.975153i \(0.428894\pi\)
\(318\) 0 0
\(319\) −2.47214 −0.138413
\(320\) −1.23607 −0.0690983
\(321\) 0 0
\(322\) 0.763932 0.0425723
\(323\) 5.70820 0.317613
\(324\) 0 0
\(325\) 11.2361 0.623265
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 8.94427 0.493114
\(330\) 0 0
\(331\) −19.5967 −1.07713 −0.538567 0.842582i \(-0.681034\pi\)
−0.538567 + 0.842582i \(0.681034\pi\)
\(332\) −10.0000 −0.548821
\(333\) 0 0
\(334\) 0.944272 0.0516683
\(335\) −16.9443 −0.925764
\(336\) 0 0
\(337\) −11.5279 −0.627963 −0.313981 0.949429i \(-0.601663\pi\)
−0.313981 + 0.949429i \(0.601663\pi\)
\(338\) 2.52786 0.137498
\(339\) 0 0
\(340\) −7.05573 −0.382651
\(341\) −10.4721 −0.567098
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.94427 0.482243
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 19.7082 1.05799 0.528996 0.848624i \(-0.322569\pi\)
0.528996 + 0.848624i \(0.322569\pi\)
\(348\) 0 0
\(349\) 28.8328 1.54339 0.771693 0.635996i \(-0.219410\pi\)
0.771693 + 0.635996i \(0.219410\pi\)
\(350\) −3.47214 −0.185593
\(351\) 0 0
\(352\) −5.23607 −0.279083
\(353\) 1.70820 0.0909185 0.0454593 0.998966i \(-0.485525\pi\)
0.0454593 + 0.998966i \(0.485525\pi\)
\(354\) 0 0
\(355\) −1.88854 −0.100233
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −21.8885 −1.15685
\(359\) 29.1246 1.53714 0.768569 0.639767i \(-0.220969\pi\)
0.768569 + 0.639767i \(0.220969\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0.763932 0.0401514
\(363\) 0 0
\(364\) 3.23607 0.169616
\(365\) 4.36068 0.228248
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0.763932 0.0398227
\(369\) 0 0
\(370\) −1.88854 −0.0981807
\(371\) −3.52786 −0.183158
\(372\) 0 0
\(373\) 12.9443 0.670229 0.335114 0.942177i \(-0.391225\pi\)
0.335114 + 0.942177i \(0.391225\pi\)
\(374\) −29.8885 −1.54550
\(375\) 0 0
\(376\) 8.94427 0.461266
\(377\) 1.52786 0.0786890
\(378\) 0 0
\(379\) 13.1246 0.674166 0.337083 0.941475i \(-0.390560\pi\)
0.337083 + 0.941475i \(0.390560\pi\)
\(380\) −1.23607 −0.0634089
\(381\) 0 0
\(382\) −12.7639 −0.653060
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 6.47214 0.329851
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 9.23607 0.468890
\(389\) −29.4164 −1.49147 −0.745736 0.666242i \(-0.767902\pi\)
−0.745736 + 0.666242i \(0.767902\pi\)
\(390\) 0 0
\(391\) 4.36068 0.220529
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −6.94427 −0.349847
\(395\) −6.47214 −0.325649
\(396\) 0 0
\(397\) 1.41641 0.0710875 0.0355437 0.999368i \(-0.488684\pi\)
0.0355437 + 0.999368i \(0.488684\pi\)
\(398\) 11.4164 0.572253
\(399\) 0 0
\(400\) −3.47214 −0.173607
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 6.47214 0.322400
\(404\) −0.291796 −0.0145174
\(405\) 0 0
\(406\) −0.472136 −0.0234317
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 3.12461 0.154502 0.0772511 0.997012i \(-0.475386\pi\)
0.0772511 + 0.997012i \(0.475386\pi\)
\(410\) 7.41641 0.366270
\(411\) 0 0
\(412\) 10.9443 0.539186
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 12.3607 0.606762
\(416\) 3.23607 0.158661
\(417\) 0 0
\(418\) −5.23607 −0.256104
\(419\) 14.3607 0.701565 0.350783 0.936457i \(-0.385916\pi\)
0.350783 + 0.936457i \(0.385916\pi\)
\(420\) 0 0
\(421\) −34.8328 −1.69765 −0.848824 0.528676i \(-0.822689\pi\)
−0.848824 + 0.528676i \(0.822689\pi\)
\(422\) −27.5967 −1.34339
\(423\) 0 0
\(424\) −3.52786 −0.171328
\(425\) −19.8197 −0.961395
\(426\) 0 0
\(427\) −8.47214 −0.409995
\(428\) 14.4721 0.699537
\(429\) 0 0
\(430\) −11.0557 −0.533155
\(431\) 22.8328 1.09982 0.549909 0.835225i \(-0.314662\pi\)
0.549909 + 0.835225i \(0.314662\pi\)
\(432\) 0 0
\(433\) −18.7639 −0.901737 −0.450869 0.892590i \(-0.648886\pi\)
−0.450869 + 0.892590i \(0.648886\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 9.52786 0.456302
\(437\) 0.763932 0.0365438
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 6.47214 0.308547
\(441\) 0 0
\(442\) 18.4721 0.878630
\(443\) −7.70820 −0.366228 −0.183114 0.983092i \(-0.558618\pi\)
−0.183114 + 0.983092i \(0.558618\pi\)
\(444\) 0 0
\(445\) −7.41641 −0.351571
\(446\) −19.5279 −0.924671
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −32.8328 −1.54948 −0.774738 0.632282i \(-0.782118\pi\)
−0.774738 + 0.632282i \(0.782118\pi\)
\(450\) 0 0
\(451\) 31.4164 1.47934
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 15.4164 0.723528
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 35.3050 1.65150 0.825748 0.564039i \(-0.190753\pi\)
0.825748 + 0.564039i \(0.190753\pi\)
\(458\) 18.9443 0.885208
\(459\) 0 0
\(460\) −0.944272 −0.0440269
\(461\) 1.81966 0.0847500 0.0423750 0.999102i \(-0.486508\pi\)
0.0423750 + 0.999102i \(0.486508\pi\)
\(462\) 0 0
\(463\) −41.3050 −1.91960 −0.959802 0.280678i \(-0.909441\pi\)
−0.959802 + 0.280678i \(0.909441\pi\)
\(464\) −0.472136 −0.0219184
\(465\) 0 0
\(466\) −14.4721 −0.670409
\(467\) −19.8885 −0.920332 −0.460166 0.887833i \(-0.652210\pi\)
−0.460166 + 0.887833i \(0.652210\pi\)
\(468\) 0 0
\(469\) −13.7082 −0.632986
\(470\) −11.0557 −0.509963
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −46.8328 −2.15338
\(474\) 0 0
\(475\) −3.47214 −0.159313
\(476\) −5.70820 −0.261635
\(477\) 0 0
\(478\) −3.81966 −0.174707
\(479\) −34.4721 −1.57507 −0.787536 0.616269i \(-0.788644\pi\)
−0.787536 + 0.616269i \(0.788644\pi\)
\(480\) 0 0
\(481\) 4.94427 0.225439
\(482\) 5.23607 0.238496
\(483\) 0 0
\(484\) 16.4164 0.746200
\(485\) −11.4164 −0.518392
\(486\) 0 0
\(487\) −28.0689 −1.27192 −0.635961 0.771721i \(-0.719396\pi\)
−0.635961 + 0.771721i \(0.719396\pi\)
\(488\) −8.47214 −0.383516
\(489\) 0 0
\(490\) 1.23607 0.0558399
\(491\) −0.291796 −0.0131686 −0.00658429 0.999978i \(-0.502096\pi\)
−0.00658429 + 0.999978i \(0.502096\pi\)
\(492\) 0 0
\(493\) −2.69505 −0.121379
\(494\) 3.23607 0.145598
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −1.52786 −0.0685341
\(498\) 0 0
\(499\) 25.3050 1.13281 0.566403 0.824129i \(-0.308335\pi\)
0.566403 + 0.824129i \(0.308335\pi\)
\(500\) 10.4721 0.468328
\(501\) 0 0
\(502\) −25.4164 −1.13439
\(503\) −17.5279 −0.781529 −0.390764 0.920491i \(-0.627789\pi\)
−0.390764 + 0.920491i \(0.627789\pi\)
\(504\) 0 0
\(505\) 0.360680 0.0160500
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 3.70820 0.164525
\(509\) 13.4164 0.594672 0.297336 0.954773i \(-0.403902\pi\)
0.297336 + 0.954773i \(0.403902\pi\)
\(510\) 0 0
\(511\) 3.52786 0.156064
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.4721 0.550122
\(515\) −13.5279 −0.596109
\(516\) 0 0
\(517\) −46.8328 −2.05970
\(518\) −1.52786 −0.0671305
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) −18.3607 −0.804396 −0.402198 0.915553i \(-0.631754\pi\)
−0.402198 + 0.915553i \(0.631754\pi\)
\(522\) 0 0
\(523\) 14.4721 0.632822 0.316411 0.948622i \(-0.397522\pi\)
0.316411 + 0.948622i \(0.397522\pi\)
\(524\) −7.52786 −0.328856
\(525\) 0 0
\(526\) 5.70820 0.248890
\(527\) −11.4164 −0.497307
\(528\) 0 0
\(529\) −22.4164 −0.974626
\(530\) 4.36068 0.189416
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −19.4164 −0.841018
\(534\) 0 0
\(535\) −17.8885 −0.773389
\(536\) −13.7082 −0.592104
\(537\) 0 0
\(538\) −10.9443 −0.471841
\(539\) 5.23607 0.225533
\(540\) 0 0
\(541\) −45.7771 −1.96811 −0.984055 0.177862i \(-0.943082\pi\)
−0.984055 + 0.177862i \(0.943082\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) −5.70820 −0.244737
\(545\) −11.7771 −0.504475
\(546\) 0 0
\(547\) −29.1246 −1.24528 −0.622639 0.782509i \(-0.713940\pi\)
−0.622639 + 0.782509i \(0.713940\pi\)
\(548\) 11.4164 0.487685
\(549\) 0 0
\(550\) 18.1803 0.775212
\(551\) −0.472136 −0.0201137
\(552\) 0 0
\(553\) −5.23607 −0.222660
\(554\) −2.94427 −0.125090
\(555\) 0 0
\(556\) −13.8885 −0.589005
\(557\) −35.8885 −1.52065 −0.760323 0.649545i \(-0.774959\pi\)
−0.760323 + 0.649545i \(0.774959\pi\)
\(558\) 0 0
\(559\) 28.9443 1.22421
\(560\) 1.23607 0.0522334
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) −12.3607 −0.520018
\(566\) −13.8885 −0.583779
\(567\) 0 0
\(568\) −1.52786 −0.0641078
\(569\) 14.9443 0.626496 0.313248 0.949671i \(-0.398583\pi\)
0.313248 + 0.949671i \(0.398583\pi\)
\(570\) 0 0
\(571\) −2.47214 −0.103456 −0.0517278 0.998661i \(-0.516473\pi\)
−0.0517278 + 0.998661i \(0.516473\pi\)
\(572\) −16.9443 −0.708476
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) −2.65248 −0.110616
\(576\) 0 0
\(577\) −5.41641 −0.225488 −0.112744 0.993624i \(-0.535964\pi\)
−0.112744 + 0.993624i \(0.535964\pi\)
\(578\) −15.5836 −0.648192
\(579\) 0 0
\(580\) 0.583592 0.0242323
\(581\) 10.0000 0.414870
\(582\) 0 0
\(583\) 18.4721 0.765038
\(584\) 3.52786 0.145984
\(585\) 0 0
\(586\) 19.8885 0.821588
\(587\) 8.83282 0.364569 0.182285 0.983246i \(-0.441651\pi\)
0.182285 + 0.983246i \(0.441651\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 9.88854 0.407105
\(591\) 0 0
\(592\) −1.52786 −0.0627948
\(593\) −34.6525 −1.42301 −0.711503 0.702683i \(-0.751985\pi\)
−0.711503 + 0.702683i \(0.751985\pi\)
\(594\) 0 0
\(595\) 7.05573 0.289257
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 2.47214 0.101093
\(599\) −12.5836 −0.514152 −0.257076 0.966391i \(-0.582759\pi\)
−0.257076 + 0.966391i \(0.582759\pi\)
\(600\) 0 0
\(601\) −11.7082 −0.477588 −0.238794 0.971070i \(-0.576752\pi\)
−0.238794 + 0.971070i \(0.576752\pi\)
\(602\) −8.94427 −0.364541
\(603\) 0 0
\(604\) 5.23607 0.213053
\(605\) −20.2918 −0.824979
\(606\) 0 0
\(607\) −1.41641 −0.0574902 −0.0287451 0.999587i \(-0.509151\pi\)
−0.0287451 + 0.999587i \(0.509151\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 10.4721 0.424004
\(611\) 28.9443 1.17096
\(612\) 0 0
\(613\) 43.8885 1.77264 0.886321 0.463072i \(-0.153253\pi\)
0.886321 + 0.463072i \(0.153253\pi\)
\(614\) −31.4164 −1.26786
\(615\) 0 0
\(616\) 5.23607 0.210967
\(617\) 23.4164 0.942709 0.471355 0.881944i \(-0.343765\pi\)
0.471355 + 0.881944i \(0.343765\pi\)
\(618\) 0 0
\(619\) −6.11146 −0.245640 −0.122820 0.992429i \(-0.539194\pi\)
−0.122820 + 0.992429i \(0.539194\pi\)
\(620\) 2.47214 0.0992834
\(621\) 0 0
\(622\) 28.3607 1.13716
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 10.9443 0.437421
\(627\) 0 0
\(628\) −13.4164 −0.535373
\(629\) −8.72136 −0.347743
\(630\) 0 0
\(631\) 2.47214 0.0984142 0.0492071 0.998789i \(-0.484331\pi\)
0.0492071 + 0.998789i \(0.484331\pi\)
\(632\) −5.23607 −0.208280
\(633\) 0 0
\(634\) −7.88854 −0.313294
\(635\) −4.58359 −0.181894
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 2.47214 0.0978728
\(639\) 0 0
\(640\) 1.23607 0.0488599
\(641\) −12.1115 −0.478374 −0.239187 0.970974i \(-0.576881\pi\)
−0.239187 + 0.970974i \(0.576881\pi\)
\(642\) 0 0
\(643\) −44.9443 −1.77243 −0.886215 0.463275i \(-0.846674\pi\)
−0.886215 + 0.463275i \(0.846674\pi\)
\(644\) −0.763932 −0.0301031
\(645\) 0 0
\(646\) −5.70820 −0.224586
\(647\) 34.2492 1.34648 0.673238 0.739426i \(-0.264903\pi\)
0.673238 + 0.739426i \(0.264903\pi\)
\(648\) 0 0
\(649\) 41.8885 1.64427
\(650\) −11.2361 −0.440715
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 17.4164 0.681557 0.340778 0.940144i \(-0.389309\pi\)
0.340778 + 0.940144i \(0.389309\pi\)
\(654\) 0 0
\(655\) 9.30495 0.363575
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −8.94427 −0.348684
\(659\) 14.4721 0.563754 0.281877 0.959450i \(-0.409043\pi\)
0.281877 + 0.959450i \(0.409043\pi\)
\(660\) 0 0
\(661\) −40.7639 −1.58553 −0.792767 0.609525i \(-0.791360\pi\)
−0.792767 + 0.609525i \(0.791360\pi\)
\(662\) 19.5967 0.761649
\(663\) 0 0
\(664\) 10.0000 0.388075
\(665\) 1.23607 0.0479327
\(666\) 0 0
\(667\) −0.360680 −0.0139656
\(668\) −0.944272 −0.0365350
\(669\) 0 0
\(670\) 16.9443 0.654614
\(671\) 44.3607 1.71253
\(672\) 0 0
\(673\) −44.2492 −1.70568 −0.852841 0.522171i \(-0.825122\pi\)
−0.852841 + 0.522171i \(0.825122\pi\)
\(674\) 11.5279 0.444037
\(675\) 0 0
\(676\) −2.52786 −0.0972255
\(677\) −19.5279 −0.750517 −0.375258 0.926920i \(-0.622446\pi\)
−0.375258 + 0.926920i \(0.622446\pi\)
\(678\) 0 0
\(679\) −9.23607 −0.354448
\(680\) 7.05573 0.270575
\(681\) 0 0
\(682\) 10.4721 0.400999
\(683\) 45.8885 1.75588 0.877938 0.478774i \(-0.158919\pi\)
0.877938 + 0.478774i \(0.158919\pi\)
\(684\) 0 0
\(685\) −14.1115 −0.539171
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −8.94427 −0.340997
\(689\) −11.4164 −0.434931
\(690\) 0 0
\(691\) 26.8328 1.02077 0.510384 0.859946i \(-0.329503\pi\)
0.510384 + 0.859946i \(0.329503\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −19.7082 −0.748113
\(695\) 17.1672 0.651188
\(696\) 0 0
\(697\) 34.2492 1.29728
\(698\) −28.8328 −1.09134
\(699\) 0 0
\(700\) 3.47214 0.131234
\(701\) −36.4721 −1.37753 −0.688767 0.724983i \(-0.741848\pi\)
−0.688767 + 0.724983i \(0.741848\pi\)
\(702\) 0 0
\(703\) −1.52786 −0.0576245
\(704\) 5.23607 0.197342
\(705\) 0 0
\(706\) −1.70820 −0.0642891
\(707\) 0.291796 0.0109741
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 1.88854 0.0708758
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −1.52786 −0.0572190
\(714\) 0 0
\(715\) 20.9443 0.783271
\(716\) 21.8885 0.818013
\(717\) 0 0
\(718\) −29.1246 −1.08692
\(719\) 2.47214 0.0921951 0.0460976 0.998937i \(-0.485321\pi\)
0.0460976 + 0.998937i \(0.485321\pi\)
\(720\) 0 0
\(721\) −10.9443 −0.407586
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −0.763932 −0.0283913
\(725\) 1.63932 0.0608828
\(726\) 0 0
\(727\) 43.7771 1.62360 0.811801 0.583934i \(-0.198487\pi\)
0.811801 + 0.583934i \(0.198487\pi\)
\(728\) −3.23607 −0.119937
\(729\) 0 0
\(730\) −4.36068 −0.161396
\(731\) −51.0557 −1.88836
\(732\) 0 0
\(733\) −40.8328 −1.50819 −0.754097 0.656763i \(-0.771925\pi\)
−0.754097 + 0.656763i \(0.771925\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −0.763932 −0.0281589
\(737\) 71.7771 2.64394
\(738\) 0 0
\(739\) 23.4164 0.861386 0.430693 0.902498i \(-0.358269\pi\)
0.430693 + 0.902498i \(0.358269\pi\)
\(740\) 1.88854 0.0694243
\(741\) 0 0
\(742\) 3.52786 0.129512
\(743\) 3.41641 0.125336 0.0626679 0.998034i \(-0.480039\pi\)
0.0626679 + 0.998034i \(0.480039\pi\)
\(744\) 0 0
\(745\) −7.41641 −0.271716
\(746\) −12.9443 −0.473923
\(747\) 0 0
\(748\) 29.8885 1.09283
\(749\) −14.4721 −0.528800
\(750\) 0 0
\(751\) −53.0132 −1.93448 −0.967239 0.253868i \(-0.918297\pi\)
−0.967239 + 0.253868i \(0.918297\pi\)
\(752\) −8.94427 −0.326164
\(753\) 0 0
\(754\) −1.52786 −0.0556415
\(755\) −6.47214 −0.235545
\(756\) 0 0
\(757\) 0.111456 0.00405094 0.00202547 0.999998i \(-0.499355\pi\)
0.00202547 + 0.999998i \(0.499355\pi\)
\(758\) −13.1246 −0.476707
\(759\) 0 0
\(760\) 1.23607 0.0448369
\(761\) −2.65248 −0.0961522 −0.0480761 0.998844i \(-0.515309\pi\)
−0.0480761 + 0.998844i \(0.515309\pi\)
\(762\) 0 0
\(763\) −9.52786 −0.344932
\(764\) 12.7639 0.461783
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −25.8885 −0.934781
\(768\) 0 0
\(769\) 6.94427 0.250417 0.125208 0.992130i \(-0.460040\pi\)
0.125208 + 0.992130i \(0.460040\pi\)
\(770\) −6.47214 −0.233240
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −18.5836 −0.668405 −0.334203 0.942501i \(-0.608467\pi\)
−0.334203 + 0.942501i \(0.608467\pi\)
\(774\) 0 0
\(775\) 6.94427 0.249446
\(776\) −9.23607 −0.331556
\(777\) 0 0
\(778\) 29.4164 1.05463
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −4.36068 −0.155938
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 16.5836 0.591894
\(786\) 0 0
\(787\) −7.41641 −0.264366 −0.132183 0.991225i \(-0.542199\pi\)
−0.132183 + 0.991225i \(0.542199\pi\)
\(788\) 6.94427 0.247379
\(789\) 0 0
\(790\) 6.47214 0.230268
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −27.4164 −0.973585
\(794\) −1.41641 −0.0502664
\(795\) 0 0
\(796\) −11.4164 −0.404644
\(797\) 45.4164 1.60873 0.804366 0.594134i \(-0.202505\pi\)
0.804366 + 0.594134i \(0.202505\pi\)
\(798\) 0 0
\(799\) −51.0557 −1.80622
\(800\) 3.47214 0.122759
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −18.4721 −0.651868
\(804\) 0 0
\(805\) 0.944272 0.0332812
\(806\) −6.47214 −0.227971
\(807\) 0 0
\(808\) 0.291796 0.0102653
\(809\) 8.94427 0.314464 0.157232 0.987562i \(-0.449743\pi\)
0.157232 + 0.987562i \(0.449743\pi\)
\(810\) 0 0
\(811\) −23.4164 −0.822261 −0.411131 0.911576i \(-0.634866\pi\)
−0.411131 + 0.911576i \(0.634866\pi\)
\(812\) 0.472136 0.0165687
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −4.94427 −0.173190
\(816\) 0 0
\(817\) −8.94427 −0.312920
\(818\) −3.12461 −0.109249
\(819\) 0 0
\(820\) −7.41641 −0.258992
\(821\) 6.94427 0.242357 0.121178 0.992631i \(-0.461333\pi\)
0.121178 + 0.992631i \(0.461333\pi\)
\(822\) 0 0
\(823\) 29.5279 1.02928 0.514638 0.857407i \(-0.327926\pi\)
0.514638 + 0.857407i \(0.327926\pi\)
\(824\) −10.9443 −0.381262
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) −42.8328 −1.48944 −0.744721 0.667375i \(-0.767418\pi\)
−0.744721 + 0.667375i \(0.767418\pi\)
\(828\) 0 0
\(829\) −23.2361 −0.807022 −0.403511 0.914975i \(-0.632210\pi\)
−0.403511 + 0.914975i \(0.632210\pi\)
\(830\) −12.3607 −0.429045
\(831\) 0 0
\(832\) −3.23607 −0.112190
\(833\) 5.70820 0.197778
\(834\) 0 0
\(835\) 1.16718 0.0403921
\(836\) 5.23607 0.181093
\(837\) 0 0
\(838\) −14.3607 −0.496081
\(839\) 53.8885 1.86044 0.930220 0.367003i \(-0.119616\pi\)
0.930220 + 0.367003i \(0.119616\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) 34.8328 1.20042
\(843\) 0 0
\(844\) 27.5967 0.949919
\(845\) 3.12461 0.107490
\(846\) 0 0
\(847\) −16.4164 −0.564074
\(848\) 3.52786 0.121147
\(849\) 0 0
\(850\) 19.8197 0.679809
\(851\) −1.16718 −0.0400106
\(852\) 0 0
\(853\) 6.94427 0.237767 0.118884 0.992908i \(-0.462068\pi\)
0.118884 + 0.992908i \(0.462068\pi\)
\(854\) 8.47214 0.289911
\(855\) 0 0
\(856\) −14.4721 −0.494647
\(857\) 31.8885 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(858\) 0 0
\(859\) 22.1115 0.754433 0.377217 0.926125i \(-0.376881\pi\)
0.377217 + 0.926125i \(0.376881\pi\)
\(860\) 11.0557 0.376997
\(861\) 0 0
\(862\) −22.8328 −0.777689
\(863\) 48.3607 1.64622 0.823108 0.567884i \(-0.192238\pi\)
0.823108 + 0.567884i \(0.192238\pi\)
\(864\) 0 0
\(865\) 2.47214 0.0840551
\(866\) 18.7639 0.637624
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 27.4164 0.930038
\(870\) 0 0
\(871\) −44.3607 −1.50310
\(872\) −9.52786 −0.322654
\(873\) 0 0
\(874\) −0.763932 −0.0258404
\(875\) −10.4721 −0.354023
\(876\) 0 0
\(877\) −19.4164 −0.655646 −0.327823 0.944739i \(-0.606315\pi\)
−0.327823 + 0.944739i \(0.606315\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) −6.47214 −0.218176
\(881\) 43.5967 1.46881 0.734406 0.678711i \(-0.237461\pi\)
0.734406 + 0.678711i \(0.237461\pi\)
\(882\) 0 0
\(883\) −5.52786 −0.186027 −0.0930137 0.995665i \(-0.529650\pi\)
−0.0930137 + 0.995665i \(0.529650\pi\)
\(884\) −18.4721 −0.621285
\(885\) 0 0
\(886\) 7.70820 0.258962
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) −3.70820 −0.124369
\(890\) 7.41641 0.248599
\(891\) 0 0
\(892\) 19.5279 0.653841
\(893\) −8.94427 −0.299309
\(894\) 0 0
\(895\) −27.0557 −0.904373
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 32.8328 1.09565
\(899\) 0.944272 0.0314932
\(900\) 0 0
\(901\) 20.1378 0.670886
\(902\) −31.4164 −1.04605
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0.944272 0.0313887
\(906\) 0 0
\(907\) 23.2361 0.771541 0.385770 0.922595i \(-0.373936\pi\)
0.385770 + 0.922595i \(0.373936\pi\)
\(908\) −15.4164 −0.511611
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) −42.2492 −1.39978 −0.699890 0.714251i \(-0.746767\pi\)
−0.699890 + 0.714251i \(0.746767\pi\)
\(912\) 0 0
\(913\) −52.3607 −1.73289
\(914\) −35.3050 −1.16778
\(915\) 0 0
\(916\) −18.9443 −0.625936
\(917\) 7.52786 0.248592
\(918\) 0 0
\(919\) −12.3607 −0.407741 −0.203871 0.978998i \(-0.565352\pi\)
−0.203871 + 0.978998i \(0.565352\pi\)
\(920\) 0.944272 0.0311317
\(921\) 0 0
\(922\) −1.81966 −0.0599273
\(923\) −4.94427 −0.162743
\(924\) 0 0
\(925\) 5.30495 0.174426
\(926\) 41.3050 1.35736
\(927\) 0 0
\(928\) 0.472136 0.0154986
\(929\) −42.0689 −1.38024 −0.690118 0.723697i \(-0.742441\pi\)
−0.690118 + 0.723697i \(0.742441\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 14.4721 0.474051
\(933\) 0 0
\(934\) 19.8885 0.650773
\(935\) −36.9443 −1.20821
\(936\) 0 0
\(937\) −46.9443 −1.53360 −0.766801 0.641885i \(-0.778153\pi\)
−0.766801 + 0.641885i \(0.778153\pi\)
\(938\) 13.7082 0.447589
\(939\) 0 0
\(940\) 11.0557 0.360598
\(941\) 35.5279 1.15818 0.579088 0.815265i \(-0.303409\pi\)
0.579088 + 0.815265i \(0.303409\pi\)
\(942\) 0 0
\(943\) 4.58359 0.149262
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 46.8328 1.52267
\(947\) −17.0132 −0.552853 −0.276427 0.961035i \(-0.589150\pi\)
−0.276427 + 0.961035i \(0.589150\pi\)
\(948\) 0 0
\(949\) 11.4164 0.370592
\(950\) 3.47214 0.112651
\(951\) 0 0
\(952\) 5.70820 0.185004
\(953\) 42.9443 1.39110 0.695551 0.718477i \(-0.255160\pi\)
0.695551 + 0.718477i \(0.255160\pi\)
\(954\) 0 0
\(955\) −15.7771 −0.510535
\(956\) 3.81966 0.123537
\(957\) 0 0
\(958\) 34.4721 1.11374
\(959\) −11.4164 −0.368655
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −4.94427 −0.159410
\(963\) 0 0
\(964\) −5.23607 −0.168642
\(965\) 22.2492 0.716228
\(966\) 0 0
\(967\) 30.8328 0.991517 0.495758 0.868461i \(-0.334890\pi\)
0.495758 + 0.868461i \(0.334890\pi\)
\(968\) −16.4164 −0.527643
\(969\) 0 0
\(970\) 11.4164 0.366559
\(971\) 34.4721 1.10626 0.553132 0.833094i \(-0.313433\pi\)
0.553132 + 0.833094i \(0.313433\pi\)
\(972\) 0 0
\(973\) 13.8885 0.445246
\(974\) 28.0689 0.899385
\(975\) 0 0
\(976\) 8.47214 0.271186
\(977\) −27.8885 −0.892234 −0.446117 0.894975i \(-0.647193\pi\)
−0.446117 + 0.894975i \(0.647193\pi\)
\(978\) 0 0
\(979\) 31.4164 1.00407
\(980\) −1.23607 −0.0394847
\(981\) 0 0
\(982\) 0.291796 0.00931159
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) −8.58359 −0.273496
\(986\) 2.69505 0.0858278
\(987\) 0 0
\(988\) −3.23607 −0.102953
\(989\) −6.83282 −0.217271
\(990\) 0 0
\(991\) 29.2361 0.928714 0.464357 0.885648i \(-0.346285\pi\)
0.464357 + 0.885648i \(0.346285\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 1.52786 0.0484609
\(995\) 14.1115 0.447363
\(996\) 0 0
\(997\) 37.1935 1.17793 0.588965 0.808159i \(-0.299536\pi\)
0.588965 + 0.808159i \(0.299536\pi\)
\(998\) −25.3050 −0.801014
\(999\) 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 2394.2.a.s.1.1 2
3.2 odd 2 2394.2.a.v.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2394.2.a.s.1.1 2 1.1 even 1 trivial
2394.2.a.v.1.2 yes 2 3.2 odd 2