Properties

Label 585.2.a.l.1.2
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
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 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} -0.438447 q^{7} +6.56155 q^{8} +O(q^{10})\) \(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{5} -0.438447 q^{7} +6.56155 q^{8} +2.56155 q^{10} -1.56155 q^{11} -1.00000 q^{13} -1.12311 q^{14} +7.68466 q^{16} +1.56155 q^{17} -5.12311 q^{19} +4.56155 q^{20} -4.00000 q^{22} +2.43845 q^{23} +1.00000 q^{25} -2.56155 q^{26} -2.00000 q^{28} -7.12311 q^{29} +6.00000 q^{31} +6.56155 q^{32} +4.00000 q^{34} -0.438447 q^{35} -10.6847 q^{37} -13.1231 q^{38} +6.56155 q^{40} +3.56155 q^{41} +3.12311 q^{43} -7.12311 q^{44} +6.24621 q^{46} +11.1231 q^{47} -6.80776 q^{49} +2.56155 q^{50} -4.56155 q^{52} -4.68466 q^{53} -1.56155 q^{55} -2.87689 q^{56} -18.2462 q^{58} +12.0000 q^{59} -6.68466 q^{61} +15.3693 q^{62} +1.43845 q^{64} -1.00000 q^{65} -11.3693 q^{67} +7.12311 q^{68} -1.12311 q^{70} +10.4384 q^{71} -6.00000 q^{73} -27.3693 q^{74} -23.3693 q^{76} +0.684658 q^{77} +4.68466 q^{79} +7.68466 q^{80} +9.12311 q^{82} -16.4924 q^{83} +1.56155 q^{85} +8.00000 q^{86} -10.2462 q^{88} +10.6847 q^{89} +0.438447 q^{91} +11.1231 q^{92} +28.4924 q^{94} -5.12311 q^{95} +16.9309 q^{97} -17.4384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 0 0
\(4\) 4.56155 2.28078
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.438447 −0.165717 −0.0828587 0.996561i \(-0.526405\pi\)
−0.0828587 + 0.996561i \(0.526405\pi\)
\(8\) 6.56155 2.31986
\(9\) 0 0
\(10\) 2.56155 0.810034
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −1.12311 −0.300163
\(15\) 0 0
\(16\) 7.68466 1.92116
\(17\) 1.56155 0.378732 0.189366 0.981907i \(-0.439357\pi\)
0.189366 + 0.981907i \(0.439357\pi\)
\(18\) 0 0
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 4.56155 1.01999
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 2.43845 0.508451 0.254226 0.967145i \(-0.418179\pi\)
0.254226 + 0.967145i \(0.418179\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.56155 −0.502362
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −7.12311 −1.32273 −0.661364 0.750065i \(-0.730022\pi\)
−0.661364 + 0.750065i \(0.730022\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 6.56155 1.15993
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −0.438447 −0.0741111
\(36\) 0 0
\(37\) −10.6847 −1.75655 −0.878274 0.478159i \(-0.841304\pi\)
−0.878274 + 0.478159i \(0.841304\pi\)
\(38\) −13.1231 −2.12885
\(39\) 0 0
\(40\) 6.56155 1.03747
\(41\) 3.56155 0.556221 0.278111 0.960549i \(-0.410292\pi\)
0.278111 + 0.960549i \(0.410292\pi\)
\(42\) 0 0
\(43\) 3.12311 0.476269 0.238135 0.971232i \(-0.423464\pi\)
0.238135 + 0.971232i \(0.423464\pi\)
\(44\) −7.12311 −1.07385
\(45\) 0 0
\(46\) 6.24621 0.920954
\(47\) 11.1231 1.62247 0.811236 0.584719i \(-0.198795\pi\)
0.811236 + 0.584719i \(0.198795\pi\)
\(48\) 0 0
\(49\) −6.80776 −0.972538
\(50\) 2.56155 0.362258
\(51\) 0 0
\(52\) −4.56155 −0.632574
\(53\) −4.68466 −0.643487 −0.321744 0.946827i \(-0.604269\pi\)
−0.321744 + 0.946827i \(0.604269\pi\)
\(54\) 0 0
\(55\) −1.56155 −0.210560
\(56\) −2.87689 −0.384441
\(57\) 0 0
\(58\) −18.2462 −2.39584
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −6.68466 −0.855883 −0.427941 0.903806i \(-0.640761\pi\)
−0.427941 + 0.903806i \(0.640761\pi\)
\(62\) 15.3693 1.95191
\(63\) 0 0
\(64\) 1.43845 0.179806
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −11.3693 −1.38898 −0.694492 0.719501i \(-0.744371\pi\)
−0.694492 + 0.719501i \(0.744371\pi\)
\(68\) 7.12311 0.863803
\(69\) 0 0
\(70\) −1.12311 −0.134237
\(71\) 10.4384 1.23882 0.619408 0.785069i \(-0.287373\pi\)
0.619408 + 0.785069i \(0.287373\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −27.3693 −3.18162
\(75\) 0 0
\(76\) −23.3693 −2.68064
\(77\) 0.684658 0.0780241
\(78\) 0 0
\(79\) 4.68466 0.527065 0.263533 0.964650i \(-0.415112\pi\)
0.263533 + 0.964650i \(0.415112\pi\)
\(80\) 7.68466 0.859171
\(81\) 0 0
\(82\) 9.12311 1.00748
\(83\) −16.4924 −1.81028 −0.905139 0.425115i \(-0.860234\pi\)
−0.905139 + 0.425115i \(0.860234\pi\)
\(84\) 0 0
\(85\) 1.56155 0.169374
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −10.2462 −1.09225
\(89\) 10.6847 1.13257 0.566286 0.824209i \(-0.308380\pi\)
0.566286 + 0.824209i \(0.308380\pi\)
\(90\) 0 0
\(91\) 0.438447 0.0459618
\(92\) 11.1231 1.15966
\(93\) 0 0
\(94\) 28.4924 2.93877
\(95\) −5.12311 −0.525620
\(96\) 0 0
\(97\) 16.9309 1.71907 0.859535 0.511077i \(-0.170753\pi\)
0.859535 + 0.511077i \(0.170753\pi\)
\(98\) −17.4384 −1.76155
\(99\) 0 0
\(100\) 4.56155 0.456155
\(101\) 10.2462 1.01954 0.509768 0.860312i \(-0.329731\pi\)
0.509768 + 0.860312i \(0.329731\pi\)
\(102\) 0 0
\(103\) −15.1231 −1.49012 −0.745062 0.666995i \(-0.767580\pi\)
−0.745062 + 0.666995i \(0.767580\pi\)
\(104\) −6.56155 −0.643413
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 10.9309 1.05673 0.528364 0.849018i \(-0.322806\pi\)
0.528364 + 0.849018i \(0.322806\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −3.36932 −0.318371
\(113\) −4.87689 −0.458780 −0.229390 0.973335i \(-0.573673\pi\)
−0.229390 + 0.973335i \(0.573673\pi\)
\(114\) 0 0
\(115\) 2.43845 0.227386
\(116\) −32.4924 −3.01685
\(117\) 0 0
\(118\) 30.7386 2.82972
\(119\) −0.684658 −0.0627625
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −17.1231 −1.55025
\(123\) 0 0
\(124\) 27.3693 2.45784
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 1.75379 0.155624 0.0778118 0.996968i \(-0.475207\pi\)
0.0778118 + 0.996968i \(0.475207\pi\)
\(128\) −9.43845 −0.834249
\(129\) 0 0
\(130\) −2.56155 −0.224663
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.24621 0.194771
\(134\) −29.1231 −2.51585
\(135\) 0 0
\(136\) 10.2462 0.878605
\(137\) 1.12311 0.0959534 0.0479767 0.998848i \(-0.484723\pi\)
0.0479767 + 0.998848i \(0.484723\pi\)
\(138\) 0 0
\(139\) 3.31534 0.281204 0.140602 0.990066i \(-0.455096\pi\)
0.140602 + 0.990066i \(0.455096\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 26.7386 2.24386
\(143\) 1.56155 0.130584
\(144\) 0 0
\(145\) −7.12311 −0.591542
\(146\) −15.3693 −1.27197
\(147\) 0 0
\(148\) −48.7386 −4.00629
\(149\) −17.8078 −1.45887 −0.729434 0.684051i \(-0.760217\pi\)
−0.729434 + 0.684051i \(0.760217\pi\)
\(150\) 0 0
\(151\) 11.3693 0.925222 0.462611 0.886561i \(-0.346913\pi\)
0.462611 + 0.886561i \(0.346913\pi\)
\(152\) −33.6155 −2.72658
\(153\) 0 0
\(154\) 1.75379 0.141324
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 3.36932 0.268901 0.134450 0.990920i \(-0.457073\pi\)
0.134450 + 0.990920i \(0.457073\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 6.56155 0.518736
\(161\) −1.06913 −0.0842593
\(162\) 0 0
\(163\) 16.0540 1.25744 0.628722 0.777630i \(-0.283578\pi\)
0.628722 + 0.777630i \(0.283578\pi\)
\(164\) 16.2462 1.26862
\(165\) 0 0
\(166\) −42.2462 −3.27894
\(167\) 4.87689 0.377385 0.188693 0.982036i \(-0.439575\pi\)
0.188693 + 0.982036i \(0.439575\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 14.2462 1.08626
\(173\) 12.8769 0.979012 0.489506 0.872000i \(-0.337177\pi\)
0.489506 + 0.872000i \(0.337177\pi\)
\(174\) 0 0
\(175\) −0.438447 −0.0331435
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 27.3693 2.05142
\(179\) −4.87689 −0.364516 −0.182258 0.983251i \(-0.558341\pi\)
−0.182258 + 0.983251i \(0.558341\pi\)
\(180\) 0 0
\(181\) 13.3153 0.989722 0.494861 0.868972i \(-0.335219\pi\)
0.494861 + 0.868972i \(0.335219\pi\)
\(182\) 1.12311 0.0832501
\(183\) 0 0
\(184\) 16.0000 1.17954
\(185\) −10.6847 −0.785552
\(186\) 0 0
\(187\) −2.43845 −0.178317
\(188\) 50.7386 3.70050
\(189\) 0 0
\(190\) −13.1231 −0.952050
\(191\) 19.6155 1.41933 0.709665 0.704539i \(-0.248846\pi\)
0.709665 + 0.704539i \(0.248846\pi\)
\(192\) 0 0
\(193\) 19.5616 1.40807 0.704036 0.710165i \(-0.251379\pi\)
0.704036 + 0.710165i \(0.251379\pi\)
\(194\) 43.3693 3.11374
\(195\) 0 0
\(196\) −31.0540 −2.21814
\(197\) −3.36932 −0.240054 −0.120027 0.992771i \(-0.538298\pi\)
−0.120027 + 0.992771i \(0.538298\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 6.56155 0.463972
\(201\) 0 0
\(202\) 26.2462 1.84668
\(203\) 3.12311 0.219199
\(204\) 0 0
\(205\) 3.56155 0.248750
\(206\) −38.7386 −2.69905
\(207\) 0 0
\(208\) −7.68466 −0.532835
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 6.24621 0.430007 0.215003 0.976613i \(-0.431024\pi\)
0.215003 + 0.976613i \(0.431024\pi\)
\(212\) −21.3693 −1.46765
\(213\) 0 0
\(214\) 28.0000 1.91404
\(215\) 3.12311 0.212994
\(216\) 0 0
\(217\) −2.63068 −0.178582
\(218\) −5.12311 −0.346980
\(219\) 0 0
\(220\) −7.12311 −0.480240
\(221\) −1.56155 −0.105041
\(222\) 0 0
\(223\) −15.3693 −1.02921 −0.514603 0.857429i \(-0.672061\pi\)
−0.514603 + 0.857429i \(0.672061\pi\)
\(224\) −2.87689 −0.192221
\(225\) 0 0
\(226\) −12.4924 −0.830984
\(227\) 5.75379 0.381892 0.190946 0.981601i \(-0.438844\pi\)
0.190946 + 0.981601i \(0.438844\pi\)
\(228\) 0 0
\(229\) 17.1231 1.13153 0.565763 0.824568i \(-0.308582\pi\)
0.565763 + 0.824568i \(0.308582\pi\)
\(230\) 6.24621 0.411863
\(231\) 0 0
\(232\) −46.7386 −3.06854
\(233\) −27.8078 −1.82175 −0.910874 0.412685i \(-0.864591\pi\)
−0.910874 + 0.412685i \(0.864591\pi\)
\(234\) 0 0
\(235\) 11.1231 0.725591
\(236\) 54.7386 3.56318
\(237\) 0 0
\(238\) −1.75379 −0.113681
\(239\) 22.9309 1.48327 0.741637 0.670801i \(-0.234050\pi\)
0.741637 + 0.670801i \(0.234050\pi\)
\(240\) 0 0
\(241\) 24.7386 1.59356 0.796778 0.604272i \(-0.206536\pi\)
0.796778 + 0.604272i \(0.206536\pi\)
\(242\) −21.9309 −1.40977
\(243\) 0 0
\(244\) −30.4924 −1.95208
\(245\) −6.80776 −0.434932
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 39.3693 2.49995
\(249\) 0 0
\(250\) 2.56155 0.162007
\(251\) 26.2462 1.65665 0.828323 0.560251i \(-0.189295\pi\)
0.828323 + 0.560251i \(0.189295\pi\)
\(252\) 0 0
\(253\) −3.80776 −0.239392
\(254\) 4.49242 0.281880
\(255\) 0 0
\(256\) −27.0540 −1.69087
\(257\) −12.8769 −0.803239 −0.401619 0.915807i \(-0.631552\pi\)
−0.401619 + 0.915807i \(0.631552\pi\)
\(258\) 0 0
\(259\) 4.68466 0.291091
\(260\) −4.56155 −0.282895
\(261\) 0 0
\(262\) 0 0
\(263\) −2.24621 −0.138507 −0.0692537 0.997599i \(-0.522062\pi\)
−0.0692537 + 0.997599i \(0.522062\pi\)
\(264\) 0 0
\(265\) −4.68466 −0.287776
\(266\) 5.75379 0.352787
\(267\) 0 0
\(268\) −51.8617 −3.16796
\(269\) −0.876894 −0.0534652 −0.0267326 0.999643i \(-0.508510\pi\)
−0.0267326 + 0.999643i \(0.508510\pi\)
\(270\) 0 0
\(271\) −19.3693 −1.17660 −0.588301 0.808642i \(-0.700203\pi\)
−0.588301 + 0.808642i \(0.700203\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 2.87689 0.173800
\(275\) −1.56155 −0.0941652
\(276\) 0 0
\(277\) −12.2462 −0.735804 −0.367902 0.929865i \(-0.619924\pi\)
−0.367902 + 0.929865i \(0.619924\pi\)
\(278\) 8.49242 0.509342
\(279\) 0 0
\(280\) −2.87689 −0.171927
\(281\) 4.24621 0.253308 0.126654 0.991947i \(-0.459576\pi\)
0.126654 + 0.991947i \(0.459576\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 47.6155 2.82546
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −1.56155 −0.0921755
\(288\) 0 0
\(289\) −14.5616 −0.856562
\(290\) −18.2462 −1.07145
\(291\) 0 0
\(292\) −27.3693 −1.60167
\(293\) −20.2462 −1.18280 −0.591398 0.806380i \(-0.701424\pi\)
−0.591398 + 0.806380i \(0.701424\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −70.1080 −4.07494
\(297\) 0 0
\(298\) −45.6155 −2.64244
\(299\) −2.43845 −0.141019
\(300\) 0 0
\(301\) −1.36932 −0.0789261
\(302\) 29.1231 1.67585
\(303\) 0 0
\(304\) −39.3693 −2.25799
\(305\) −6.68466 −0.382762
\(306\) 0 0
\(307\) 7.56155 0.431561 0.215780 0.976442i \(-0.430770\pi\)
0.215780 + 0.976442i \(0.430770\pi\)
\(308\) 3.12311 0.177955
\(309\) 0 0
\(310\) 15.3693 0.872919
\(311\) 2.63068 0.149172 0.0745862 0.997215i \(-0.476236\pi\)
0.0745862 + 0.997215i \(0.476236\pi\)
\(312\) 0 0
\(313\) 29.1231 1.64614 0.823068 0.567943i \(-0.192261\pi\)
0.823068 + 0.567943i \(0.192261\pi\)
\(314\) 8.63068 0.487058
\(315\) 0 0
\(316\) 21.3693 1.20212
\(317\) 1.50758 0.0846740 0.0423370 0.999103i \(-0.486520\pi\)
0.0423370 + 0.999103i \(0.486520\pi\)
\(318\) 0 0
\(319\) 11.1231 0.622774
\(320\) 1.43845 0.0804116
\(321\) 0 0
\(322\) −2.73863 −0.152618
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 41.1231 2.27760
\(327\) 0 0
\(328\) 23.3693 1.29035
\(329\) −4.87689 −0.268872
\(330\) 0 0
\(331\) 29.1231 1.60075 0.800375 0.599499i \(-0.204634\pi\)
0.800375 + 0.599499i \(0.204634\pi\)
\(332\) −75.2311 −4.12884
\(333\) 0 0
\(334\) 12.4924 0.683555
\(335\) −11.3693 −0.621172
\(336\) 0 0
\(337\) −30.4924 −1.66103 −0.830514 0.556998i \(-0.811953\pi\)
−0.830514 + 0.556998i \(0.811953\pi\)
\(338\) 2.56155 0.139330
\(339\) 0 0
\(340\) 7.12311 0.386305
\(341\) −9.36932 −0.507377
\(342\) 0 0
\(343\) 6.05398 0.326884
\(344\) 20.4924 1.10488
\(345\) 0 0
\(346\) 32.9848 1.77328
\(347\) −26.0540 −1.39865 −0.699325 0.714804i \(-0.746516\pi\)
−0.699325 + 0.714804i \(0.746516\pi\)
\(348\) 0 0
\(349\) −23.3693 −1.25093 −0.625465 0.780252i \(-0.715091\pi\)
−0.625465 + 0.780252i \(0.715091\pi\)
\(350\) −1.12311 −0.0600325
\(351\) 0 0
\(352\) −10.2462 −0.546125
\(353\) −22.4924 −1.19715 −0.598575 0.801066i \(-0.704266\pi\)
−0.598575 + 0.801066i \(0.704266\pi\)
\(354\) 0 0
\(355\) 10.4384 0.554015
\(356\) 48.7386 2.58314
\(357\) 0 0
\(358\) −12.4924 −0.660245
\(359\) −14.2462 −0.751886 −0.375943 0.926643i \(-0.622681\pi\)
−0.375943 + 0.926643i \(0.622681\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 34.1080 1.79267
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −1.75379 −0.0915470 −0.0457735 0.998952i \(-0.514575\pi\)
−0.0457735 + 0.998952i \(0.514575\pi\)
\(368\) 18.7386 0.976819
\(369\) 0 0
\(370\) −27.3693 −1.42286
\(371\) 2.05398 0.106637
\(372\) 0 0
\(373\) 12.2462 0.634085 0.317042 0.948411i \(-0.397310\pi\)
0.317042 + 0.948411i \(0.397310\pi\)
\(374\) −6.24621 −0.322984
\(375\) 0 0
\(376\) 72.9848 3.76391
\(377\) 7.12311 0.366859
\(378\) 0 0
\(379\) −30.4924 −1.56629 −0.783145 0.621839i \(-0.786386\pi\)
−0.783145 + 0.621839i \(0.786386\pi\)
\(380\) −23.3693 −1.19882
\(381\) 0 0
\(382\) 50.2462 2.57082
\(383\) 3.50758 0.179229 0.0896144 0.995977i \(-0.471437\pi\)
0.0896144 + 0.995977i \(0.471437\pi\)
\(384\) 0 0
\(385\) 0.684658 0.0348934
\(386\) 50.1080 2.55043
\(387\) 0 0
\(388\) 77.2311 3.92081
\(389\) 37.8617 1.91967 0.959833 0.280571i \(-0.0905239\pi\)
0.959833 + 0.280571i \(0.0905239\pi\)
\(390\) 0 0
\(391\) 3.80776 0.192567
\(392\) −44.6695 −2.25615
\(393\) 0 0
\(394\) −8.63068 −0.434808
\(395\) 4.68466 0.235711
\(396\) 0 0
\(397\) −4.43845 −0.222759 −0.111380 0.993778i \(-0.535527\pi\)
−0.111380 + 0.993778i \(0.535527\pi\)
\(398\) −20.4924 −1.02719
\(399\) 0 0
\(400\) 7.68466 0.384233
\(401\) 3.75379 0.187455 0.0937276 0.995598i \(-0.470122\pi\)
0.0937276 + 0.995598i \(0.470122\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 46.7386 2.32533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 16.6847 0.827028
\(408\) 0 0
\(409\) −6.87689 −0.340041 −0.170020 0.985441i \(-0.554383\pi\)
−0.170020 + 0.985441i \(0.554383\pi\)
\(410\) 9.12311 0.450558
\(411\) 0 0
\(412\) −68.9848 −3.39864
\(413\) −5.26137 −0.258895
\(414\) 0 0
\(415\) −16.4924 −0.809581
\(416\) −6.56155 −0.321707
\(417\) 0 0
\(418\) 20.4924 1.00232
\(419\) 7.61553 0.372043 0.186021 0.982546i \(-0.440441\pi\)
0.186021 + 0.982546i \(0.440441\pi\)
\(420\) 0 0
\(421\) 39.3693 1.91874 0.959372 0.282146i \(-0.0910462\pi\)
0.959372 + 0.282146i \(0.0910462\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) −30.7386 −1.49280
\(425\) 1.56155 0.0757464
\(426\) 0 0
\(427\) 2.93087 0.141835
\(428\) 49.8617 2.41016
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −3.50758 −0.168954 −0.0844770 0.996425i \(-0.526922\pi\)
−0.0844770 + 0.996425i \(0.526922\pi\)
\(432\) 0 0
\(433\) −9.61553 −0.462093 −0.231046 0.972943i \(-0.574215\pi\)
−0.231046 + 0.972943i \(0.574215\pi\)
\(434\) −6.73863 −0.323465
\(435\) 0 0
\(436\) −9.12311 −0.436918
\(437\) −12.4924 −0.597594
\(438\) 0 0
\(439\) 22.0540 1.05258 0.526289 0.850306i \(-0.323583\pi\)
0.526289 + 0.850306i \(0.323583\pi\)
\(440\) −10.2462 −0.488469
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −7.80776 −0.370958 −0.185479 0.982648i \(-0.559384\pi\)
−0.185479 + 0.982648i \(0.559384\pi\)
\(444\) 0 0
\(445\) 10.6847 0.506501
\(446\) −39.3693 −1.86419
\(447\) 0 0
\(448\) −0.630683 −0.0297970
\(449\) −35.1771 −1.66011 −0.830055 0.557682i \(-0.811691\pi\)
−0.830055 + 0.557682i \(0.811691\pi\)
\(450\) 0 0
\(451\) −5.56155 −0.261883
\(452\) −22.2462 −1.04637
\(453\) 0 0
\(454\) 14.7386 0.691718
\(455\) 0.438447 0.0205547
\(456\) 0 0
\(457\) −13.3153 −0.622865 −0.311433 0.950268i \(-0.600809\pi\)
−0.311433 + 0.950268i \(0.600809\pi\)
\(458\) 43.8617 2.04952
\(459\) 0 0
\(460\) 11.1231 0.518617
\(461\) 8.05398 0.375111 0.187556 0.982254i \(-0.439944\pi\)
0.187556 + 0.982254i \(0.439944\pi\)
\(462\) 0 0
\(463\) −28.9309 −1.34453 −0.672266 0.740310i \(-0.734679\pi\)
−0.672266 + 0.740310i \(0.734679\pi\)
\(464\) −54.7386 −2.54118
\(465\) 0 0
\(466\) −71.2311 −3.29971
\(467\) 6.93087 0.320722 0.160361 0.987058i \(-0.448734\pi\)
0.160361 + 0.987058i \(0.448734\pi\)
\(468\) 0 0
\(469\) 4.98485 0.230179
\(470\) 28.4924 1.31426
\(471\) 0 0
\(472\) 78.7386 3.62424
\(473\) −4.87689 −0.224240
\(474\) 0 0
\(475\) −5.12311 −0.235064
\(476\) −3.12311 −0.143147
\(477\) 0 0
\(478\) 58.7386 2.68664
\(479\) 26.0540 1.19044 0.595218 0.803564i \(-0.297066\pi\)
0.595218 + 0.803564i \(0.297066\pi\)
\(480\) 0 0
\(481\) 10.6847 0.487178
\(482\) 63.3693 2.88639
\(483\) 0 0
\(484\) −39.0540 −1.77518
\(485\) 16.9309 0.768791
\(486\) 0 0
\(487\) −32.0540 −1.45250 −0.726252 0.687428i \(-0.758740\pi\)
−0.726252 + 0.687428i \(0.758740\pi\)
\(488\) −43.8617 −1.98553
\(489\) 0 0
\(490\) −17.4384 −0.787789
\(491\) −27.6155 −1.24627 −0.623136 0.782114i \(-0.714142\pi\)
−0.623136 + 0.782114i \(0.714142\pi\)
\(492\) 0 0
\(493\) −11.1231 −0.500959
\(494\) 13.1231 0.590436
\(495\) 0 0
\(496\) 46.1080 2.07031
\(497\) −4.57671 −0.205293
\(498\) 0 0
\(499\) 34.9848 1.56614 0.783068 0.621936i \(-0.213653\pi\)
0.783068 + 0.621936i \(0.213653\pi\)
\(500\) 4.56155 0.203999
\(501\) 0 0
\(502\) 67.2311 3.00067
\(503\) 13.7538 0.613251 0.306626 0.951830i \(-0.400800\pi\)
0.306626 + 0.951830i \(0.400800\pi\)
\(504\) 0 0
\(505\) 10.2462 0.455950
\(506\) −9.75379 −0.433609
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −23.5616 −1.04435 −0.522174 0.852839i \(-0.674879\pi\)
−0.522174 + 0.852839i \(0.674879\pi\)
\(510\) 0 0
\(511\) 2.63068 0.116375
\(512\) −50.4233 −2.22842
\(513\) 0 0
\(514\) −32.9848 −1.45490
\(515\) −15.1231 −0.666404
\(516\) 0 0
\(517\) −17.3693 −0.763902
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) −6.56155 −0.287743
\(521\) −25.8617 −1.13302 −0.566512 0.824054i \(-0.691707\pi\)
−0.566512 + 0.824054i \(0.691707\pi\)
\(522\) 0 0
\(523\) −30.7386 −1.34411 −0.672053 0.740503i \(-0.734587\pi\)
−0.672053 + 0.740503i \(0.734587\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −5.75379 −0.250877
\(527\) 9.36932 0.408134
\(528\) 0 0
\(529\) −17.0540 −0.741477
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 10.2462 0.444230
\(533\) −3.56155 −0.154268
\(534\) 0 0
\(535\) 10.9309 0.472583
\(536\) −74.6004 −3.22225
\(537\) 0 0
\(538\) −2.24621 −0.0968410
\(539\) 10.6307 0.457896
\(540\) 0 0
\(541\) 18.8769 0.811581 0.405791 0.913966i \(-0.366996\pi\)
0.405791 + 0.913966i \(0.366996\pi\)
\(542\) −49.6155 −2.13117
\(543\) 0 0
\(544\) 10.2462 0.439303
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −5.36932 −0.229575 −0.114788 0.993390i \(-0.536619\pi\)
−0.114788 + 0.993390i \(0.536619\pi\)
\(548\) 5.12311 0.218848
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 36.4924 1.55463
\(552\) 0 0
\(553\) −2.05398 −0.0873439
\(554\) −31.3693 −1.33275
\(555\) 0 0
\(556\) 15.1231 0.641363
\(557\) 6.49242 0.275093 0.137546 0.990495i \(-0.456078\pi\)
0.137546 + 0.990495i \(0.456078\pi\)
\(558\) 0 0
\(559\) −3.12311 −0.132093
\(560\) −3.36932 −0.142380
\(561\) 0 0
\(562\) 10.8769 0.458814
\(563\) −19.3153 −0.814045 −0.407022 0.913418i \(-0.633433\pi\)
−0.407022 + 0.913418i \(0.633433\pi\)
\(564\) 0 0
\(565\) −4.87689 −0.205172
\(566\) −10.2462 −0.430680
\(567\) 0 0
\(568\) 68.4924 2.87388
\(569\) −32.8769 −1.37827 −0.689136 0.724632i \(-0.742010\pi\)
−0.689136 + 0.724632i \(0.742010\pi\)
\(570\) 0 0
\(571\) −22.0540 −0.922930 −0.461465 0.887158i \(-0.652676\pi\)
−0.461465 + 0.887158i \(0.652676\pi\)
\(572\) 7.12311 0.297832
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 2.43845 0.101690
\(576\) 0 0
\(577\) 24.4384 1.01739 0.508693 0.860948i \(-0.330129\pi\)
0.508693 + 0.860948i \(0.330129\pi\)
\(578\) −37.3002 −1.55148
\(579\) 0 0
\(580\) −32.4924 −1.34917
\(581\) 7.23106 0.299995
\(582\) 0 0
\(583\) 7.31534 0.302970
\(584\) −39.3693 −1.62911
\(585\) 0 0
\(586\) −51.8617 −2.14239
\(587\) −32.4924 −1.34111 −0.670553 0.741862i \(-0.733943\pi\)
−0.670553 + 0.741862i \(0.733943\pi\)
\(588\) 0 0
\(589\) −30.7386 −1.26656
\(590\) 30.7386 1.26549
\(591\) 0 0
\(592\) −82.1080 −3.37462
\(593\) 24.2462 0.995673 0.497836 0.867271i \(-0.334128\pi\)
0.497836 + 0.867271i \(0.334128\pi\)
\(594\) 0 0
\(595\) −0.684658 −0.0280683
\(596\) −81.2311 −3.32735
\(597\) 0 0
\(598\) −6.24621 −0.255427
\(599\) −9.36932 −0.382820 −0.191410 0.981510i \(-0.561306\pi\)
−0.191410 + 0.981510i \(0.561306\pi\)
\(600\) 0 0
\(601\) −28.5464 −1.16443 −0.582216 0.813034i \(-0.697814\pi\)
−0.582216 + 0.813034i \(0.697814\pi\)
\(602\) −3.50758 −0.142958
\(603\) 0 0
\(604\) 51.8617 2.11022
\(605\) −8.56155 −0.348077
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) −33.6155 −1.36329
\(609\) 0 0
\(610\) −17.1231 −0.693294
\(611\) −11.1231 −0.449993
\(612\) 0 0
\(613\) −32.5464 −1.31454 −0.657268 0.753657i \(-0.728288\pi\)
−0.657268 + 0.753657i \(0.728288\pi\)
\(614\) 19.3693 0.781682
\(615\) 0 0
\(616\) 4.49242 0.181005
\(617\) 0.738634 0.0297363 0.0148681 0.999889i \(-0.495267\pi\)
0.0148681 + 0.999889i \(0.495267\pi\)
\(618\) 0 0
\(619\) −8.63068 −0.346896 −0.173448 0.984843i \(-0.555491\pi\)
−0.173448 + 0.984843i \(0.555491\pi\)
\(620\) 27.3693 1.09918
\(621\) 0 0
\(622\) 6.73863 0.270195
\(623\) −4.68466 −0.187687
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 74.6004 2.98163
\(627\) 0 0
\(628\) 15.3693 0.613303
\(629\) −16.6847 −0.665261
\(630\) 0 0
\(631\) −32.7386 −1.30330 −0.651652 0.758518i \(-0.725924\pi\)
−0.651652 + 0.758518i \(0.725924\pi\)
\(632\) 30.7386 1.22272
\(633\) 0 0
\(634\) 3.86174 0.153369
\(635\) 1.75379 0.0695970
\(636\) 0 0
\(637\) 6.80776 0.269733
\(638\) 28.4924 1.12803
\(639\) 0 0
\(640\) −9.43845 −0.373087
\(641\) −32.9848 −1.30282 −0.651412 0.758725i \(-0.725823\pi\)
−0.651412 + 0.758725i \(0.725823\pi\)
\(642\) 0 0
\(643\) −10.6847 −0.421362 −0.210681 0.977555i \(-0.567568\pi\)
−0.210681 + 0.977555i \(0.567568\pi\)
\(644\) −4.87689 −0.192177
\(645\) 0 0
\(646\) −20.4924 −0.806264
\(647\) 31.8078 1.25049 0.625246 0.780428i \(-0.284999\pi\)
0.625246 + 0.780428i \(0.284999\pi\)
\(648\) 0 0
\(649\) −18.7386 −0.735556
\(650\) −2.56155 −0.100472
\(651\) 0 0
\(652\) 73.2311 2.86795
\(653\) −33.3693 −1.30584 −0.652921 0.757426i \(-0.726457\pi\)
−0.652921 + 0.757426i \(0.726457\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 27.3693 1.06859
\(657\) 0 0
\(658\) −12.4924 −0.487005
\(659\) 0.876894 0.0341590 0.0170795 0.999854i \(-0.494563\pi\)
0.0170795 + 0.999854i \(0.494563\pi\)
\(660\) 0 0
\(661\) 6.49242 0.252526 0.126263 0.991997i \(-0.459702\pi\)
0.126263 + 0.991997i \(0.459702\pi\)
\(662\) 74.6004 2.89943
\(663\) 0 0
\(664\) −108.216 −4.19959
\(665\) 2.24621 0.0871043
\(666\) 0 0
\(667\) −17.3693 −0.672543
\(668\) 22.2462 0.860732
\(669\) 0 0
\(670\) −29.1231 −1.12512
\(671\) 10.4384 0.402972
\(672\) 0 0
\(673\) 16.7386 0.645227 0.322613 0.946531i \(-0.395439\pi\)
0.322613 + 0.946531i \(0.395439\pi\)
\(674\) −78.1080 −3.00861
\(675\) 0 0
\(676\) 4.56155 0.175444
\(677\) −14.4384 −0.554915 −0.277457 0.960738i \(-0.589492\pi\)
−0.277457 + 0.960738i \(0.589492\pi\)
\(678\) 0 0
\(679\) −7.42329 −0.284880
\(680\) 10.2462 0.392924
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) 32.4924 1.24329 0.621644 0.783300i \(-0.286465\pi\)
0.621644 + 0.783300i \(0.286465\pi\)
\(684\) 0 0
\(685\) 1.12311 0.0429117
\(686\) 15.5076 0.592082
\(687\) 0 0
\(688\) 24.0000 0.914991
\(689\) 4.68466 0.178471
\(690\) 0 0
\(691\) −21.6155 −0.822293 −0.411147 0.911569i \(-0.634872\pi\)
−0.411147 + 0.911569i \(0.634872\pi\)
\(692\) 58.7386 2.23291
\(693\) 0 0
\(694\) −66.7386 −2.53336
\(695\) 3.31534 0.125758
\(696\) 0 0
\(697\) 5.56155 0.210659
\(698\) −59.8617 −2.26580
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −48.9848 −1.85013 −0.925066 0.379806i \(-0.875991\pi\)
−0.925066 + 0.379806i \(0.875991\pi\)
\(702\) 0 0
\(703\) 54.7386 2.06451
\(704\) −2.24621 −0.0846573
\(705\) 0 0
\(706\) −57.6155 −2.16839
\(707\) −4.49242 −0.168955
\(708\) 0 0
\(709\) 9.12311 0.342625 0.171313 0.985217i \(-0.445199\pi\)
0.171313 + 0.985217i \(0.445199\pi\)
\(710\) 26.7386 1.00348
\(711\) 0 0
\(712\) 70.1080 2.62741
\(713\) 14.6307 0.547923
\(714\) 0 0
\(715\) 1.56155 0.0583988
\(716\) −22.2462 −0.831380
\(717\) 0 0
\(718\) −36.4924 −1.36189
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 6.63068 0.246940
\(722\) 18.5616 0.690789
\(723\) 0 0
\(724\) 60.7386 2.25733
\(725\) −7.12311 −0.264546
\(726\) 0 0
\(727\) 12.8769 0.477578 0.238789 0.971072i \(-0.423250\pi\)
0.238789 + 0.971072i \(0.423250\pi\)
\(728\) 2.87689 0.106625
\(729\) 0 0
\(730\) −15.3693 −0.568844
\(731\) 4.87689 0.180378
\(732\) 0 0
\(733\) −16.4384 −0.607168 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(734\) −4.49242 −0.165818
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) 17.7538 0.653969
\(738\) 0 0
\(739\) 48.7386 1.79288 0.896440 0.443166i \(-0.146145\pi\)
0.896440 + 0.443166i \(0.146145\pi\)
\(740\) −48.7386 −1.79167
\(741\) 0 0
\(742\) 5.26137 0.193151
\(743\) 18.7386 0.687454 0.343727 0.939070i \(-0.388311\pi\)
0.343727 + 0.939070i \(0.388311\pi\)
\(744\) 0 0
\(745\) −17.8078 −0.652426
\(746\) 31.3693 1.14851
\(747\) 0 0
\(748\) −11.1231 −0.406701
\(749\) −4.79261 −0.175118
\(750\) 0 0
\(751\) 14.0540 0.512837 0.256418 0.966566i \(-0.417458\pi\)
0.256418 + 0.966566i \(0.417458\pi\)
\(752\) 85.4773 3.11704
\(753\) 0 0
\(754\) 18.2462 0.664488
\(755\) 11.3693 0.413772
\(756\) 0 0
\(757\) −14.4924 −0.526736 −0.263368 0.964695i \(-0.584833\pi\)
−0.263368 + 0.964695i \(0.584833\pi\)
\(758\) −78.1080 −2.83701
\(759\) 0 0
\(760\) −33.6155 −1.21936
\(761\) 45.2311 1.63962 0.819812 0.572632i \(-0.194078\pi\)
0.819812 + 0.572632i \(0.194078\pi\)
\(762\) 0 0
\(763\) 0.876894 0.0317457
\(764\) 89.4773 3.23717
\(765\) 0 0
\(766\) 8.98485 0.324636
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 1.75379 0.0632022
\(771\) 0 0
\(772\) 89.2311 3.21150
\(773\) −9.12311 −0.328135 −0.164068 0.986449i \(-0.552462\pi\)
−0.164068 + 0.986449i \(0.552462\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 111.093 3.98800
\(777\) 0 0
\(778\) 96.9848 3.47708
\(779\) −18.2462 −0.653738
\(780\) 0 0
\(781\) −16.3002 −0.583267
\(782\) 9.75379 0.348795
\(783\) 0 0
\(784\) −52.3153 −1.86841
\(785\) 3.36932 0.120256
\(786\) 0 0
\(787\) 11.3693 0.405272 0.202636 0.979254i \(-0.435049\pi\)
0.202636 + 0.979254i \(0.435049\pi\)
\(788\) −15.3693 −0.547509
\(789\) 0 0
\(790\) 12.0000 0.426941
\(791\) 2.13826 0.0760278
\(792\) 0 0
\(793\) 6.68466 0.237379
\(794\) −11.3693 −0.403482
\(795\) 0 0
\(796\) −36.4924 −1.29344
\(797\) 4.68466 0.165939 0.0829696 0.996552i \(-0.473560\pi\)
0.0829696 + 0.996552i \(0.473560\pi\)
\(798\) 0 0
\(799\) 17.3693 0.614482
\(800\) 6.56155 0.231986
\(801\) 0 0
\(802\) 9.61553 0.339536
\(803\) 9.36932 0.330636
\(804\) 0 0
\(805\) −1.06913 −0.0376819
\(806\) −15.3693 −0.541361
\(807\) 0 0
\(808\) 67.2311 2.36518
\(809\) 7.50758 0.263952 0.131976 0.991253i \(-0.457868\pi\)
0.131976 + 0.991253i \(0.457868\pi\)
\(810\) 0 0
\(811\) 4.24621 0.149105 0.0745523 0.997217i \(-0.476247\pi\)
0.0745523 + 0.997217i \(0.476247\pi\)
\(812\) 14.2462 0.499944
\(813\) 0 0
\(814\) 42.7386 1.49799
\(815\) 16.0540 0.562346
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −17.6155 −0.615912
\(819\) 0 0
\(820\) 16.2462 0.567342
\(821\) 48.5464 1.69428 0.847140 0.531369i \(-0.178322\pi\)
0.847140 + 0.531369i \(0.178322\pi\)
\(822\) 0 0
\(823\) −29.7538 −1.03715 −0.518576 0.855032i \(-0.673538\pi\)
−0.518576 + 0.855032i \(0.673538\pi\)
\(824\) −99.2311 −3.45688
\(825\) 0 0
\(826\) −13.4773 −0.468934
\(827\) 7.12311 0.247695 0.123847 0.992301i \(-0.460477\pi\)
0.123847 + 0.992301i \(0.460477\pi\)
\(828\) 0 0
\(829\) 0.738634 0.0256538 0.0128269 0.999918i \(-0.495917\pi\)
0.0128269 + 0.999918i \(0.495917\pi\)
\(830\) −42.2462 −1.46639
\(831\) 0 0
\(832\) −1.43845 −0.0498692
\(833\) −10.6307 −0.368331
\(834\) 0 0
\(835\) 4.87689 0.168772
\(836\) 36.4924 1.26212
\(837\) 0 0
\(838\) 19.5076 0.673878
\(839\) −44.7926 −1.54641 −0.773206 0.634155i \(-0.781348\pi\)
−0.773206 + 0.634155i \(0.781348\pi\)
\(840\) 0 0
\(841\) 21.7386 0.749608
\(842\) 100.847 3.47540
\(843\) 0 0
\(844\) 28.4924 0.980750
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 3.75379 0.128982
\(848\) −36.0000 −1.23625
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) −26.0540 −0.893119
\(852\) 0 0
\(853\) 41.4233 1.41831 0.709153 0.705054i \(-0.249077\pi\)
0.709153 + 0.705054i \(0.249077\pi\)
\(854\) 7.50758 0.256904
\(855\) 0 0
\(856\) 71.7235 2.45146
\(857\) 2.43845 0.0832958 0.0416479 0.999132i \(-0.486739\pi\)
0.0416479 + 0.999132i \(0.486739\pi\)
\(858\) 0 0
\(859\) 3.80776 0.129919 0.0649596 0.997888i \(-0.479308\pi\)
0.0649596 + 0.997888i \(0.479308\pi\)
\(860\) 14.2462 0.485792
\(861\) 0 0
\(862\) −8.98485 −0.306025
\(863\) −9.36932 −0.318935 −0.159468 0.987203i \(-0.550978\pi\)
−0.159468 + 0.987203i \(0.550978\pi\)
\(864\) 0 0
\(865\) 12.8769 0.437828
\(866\) −24.6307 −0.836985
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −7.31534 −0.248156
\(870\) 0 0
\(871\) 11.3693 0.385235
\(872\) −13.1231 −0.444404
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) −0.438447 −0.0148222
\(876\) 0 0
\(877\) 46.9848 1.58657 0.793283 0.608853i \(-0.208370\pi\)
0.793283 + 0.608853i \(0.208370\pi\)
\(878\) 56.4924 1.90653
\(879\) 0 0
\(880\) −12.0000 −0.404520
\(881\) −21.3693 −0.719951 −0.359975 0.932962i \(-0.617215\pi\)
−0.359975 + 0.932962i \(0.617215\pi\)
\(882\) 0 0
\(883\) −56.1080 −1.88818 −0.944091 0.329684i \(-0.893058\pi\)
−0.944091 + 0.329684i \(0.893058\pi\)
\(884\) −7.12311 −0.239576
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 5.56155 0.186739 0.0933693 0.995632i \(-0.470236\pi\)
0.0933693 + 0.995632i \(0.470236\pi\)
\(888\) 0 0
\(889\) −0.768944 −0.0257895
\(890\) 27.3693 0.917422
\(891\) 0 0
\(892\) −70.1080 −2.34739
\(893\) −56.9848 −1.90693
\(894\) 0 0
\(895\) −4.87689 −0.163017
\(896\) 4.13826 0.138250
\(897\) 0 0
\(898\) −90.1080 −3.00694
\(899\) −42.7386 −1.42541
\(900\) 0 0
\(901\) −7.31534 −0.243709
\(902\) −14.2462 −0.474347
\(903\) 0 0
\(904\) −32.0000 −1.06430
\(905\) 13.3153 0.442617
\(906\) 0 0
\(907\) −2.63068 −0.0873504 −0.0436752 0.999046i \(-0.513907\pi\)
−0.0436752 + 0.999046i \(0.513907\pi\)
\(908\) 26.2462 0.871011
\(909\) 0 0
\(910\) 1.12311 0.0372306
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 25.7538 0.852326
\(914\) −34.1080 −1.12819
\(915\) 0 0
\(916\) 78.1080 2.58076
\(917\) 0 0
\(918\) 0 0
\(919\) 1.94602 0.0641934 0.0320967 0.999485i \(-0.489782\pi\)
0.0320967 + 0.999485i \(0.489782\pi\)
\(920\) 16.0000 0.527504
\(921\) 0 0
\(922\) 20.6307 0.679435
\(923\) −10.4384 −0.343586
\(924\) 0 0
\(925\) −10.6847 −0.351309
\(926\) −74.1080 −2.43534
\(927\) 0 0
\(928\) −46.7386 −1.53427
\(929\) 39.6695 1.30151 0.650757 0.759286i \(-0.274452\pi\)
0.650757 + 0.759286i \(0.274452\pi\)
\(930\) 0 0
\(931\) 34.8769 1.14304
\(932\) −126.847 −4.15500
\(933\) 0 0
\(934\) 17.7538 0.580922
\(935\) −2.43845 −0.0797458
\(936\) 0 0
\(937\) −27.3693 −0.894117 −0.447058 0.894505i \(-0.647528\pi\)
−0.447058 + 0.894505i \(0.647528\pi\)
\(938\) 12.7689 0.416921
\(939\) 0 0
\(940\) 50.7386 1.65491
\(941\) −41.8078 −1.36289 −0.681447 0.731867i \(-0.738649\pi\)
−0.681447 + 0.731867i \(0.738649\pi\)
\(942\) 0 0
\(943\) 8.68466 0.282811
\(944\) 92.2159 3.00137
\(945\) 0 0
\(946\) −12.4924 −0.406164
\(947\) 60.6004 1.96925 0.984624 0.174688i \(-0.0558918\pi\)
0.984624 + 0.174688i \(0.0558918\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) −13.1231 −0.425770
\(951\) 0 0
\(952\) −4.49242 −0.145600
\(953\) 29.5616 0.957593 0.478796 0.877926i \(-0.341073\pi\)
0.478796 + 0.877926i \(0.341073\pi\)
\(954\) 0 0
\(955\) 19.6155 0.634744
\(956\) 104.600 3.38302
\(957\) 0 0
\(958\) 66.7386 2.15623
\(959\) −0.492423 −0.0159012
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 27.3693 0.882422
\(963\) 0 0
\(964\) 112.847 3.63454
\(965\) 19.5616 0.629709
\(966\) 0 0
\(967\) −7.36932 −0.236981 −0.118491 0.992955i \(-0.537806\pi\)
−0.118491 + 0.992955i \(0.537806\pi\)
\(968\) −56.1771 −1.80560
\(969\) 0 0
\(970\) 43.3693 1.39250
\(971\) −24.4924 −0.785999 −0.393000 0.919539i \(-0.628563\pi\)
−0.393000 + 0.919539i \(0.628563\pi\)
\(972\) 0 0
\(973\) −1.45360 −0.0466003
\(974\) −82.1080 −2.63091
\(975\) 0 0
\(976\) −51.3693 −1.64429
\(977\) −43.4773 −1.39096 −0.695481 0.718545i \(-0.744808\pi\)
−0.695481 + 0.718545i \(0.744808\pi\)
\(978\) 0 0
\(979\) −16.6847 −0.533244
\(980\) −31.0540 −0.991983
\(981\) 0 0
\(982\) −70.7386 −2.25736
\(983\) 42.7386 1.36315 0.681575 0.731748i \(-0.261295\pi\)
0.681575 + 0.731748i \(0.261295\pi\)
\(984\) 0 0
\(985\) −3.36932 −0.107355
\(986\) −28.4924 −0.907384
\(987\) 0 0
\(988\) 23.3693 0.743477
\(989\) 7.61553 0.242160
\(990\) 0 0
\(991\) 10.0540 0.319375 0.159688 0.987168i \(-0.448951\pi\)
0.159688 + 0.987168i \(0.448951\pi\)
\(992\) 39.3693 1.24998
\(993\) 0 0
\(994\) −11.7235 −0.371846
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 28.2462 0.894566 0.447283 0.894392i \(-0.352392\pi\)
0.447283 + 0.894392i \(0.352392\pi\)
\(998\) 89.6155 2.83673
\(999\) 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 585.2.a.l.1.2 yes 2
3.2 odd 2 585.2.a.j.1.1 2
4.3 odd 2 9360.2.a.cw.1.1 2
5.2 odd 4 2925.2.c.p.2224.4 4
5.3 odd 4 2925.2.c.p.2224.1 4
5.4 even 2 2925.2.a.x.1.1 2
12.11 even 2 9360.2.a.cl.1.1 2
13.12 even 2 7605.2.a.bd.1.1 2
15.2 even 4 2925.2.c.o.2224.1 4
15.8 even 4 2925.2.c.o.2224.4 4
15.14 odd 2 2925.2.a.bc.1.2 2
39.38 odd 2 7605.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.1 2 3.2 odd 2
585.2.a.l.1.2 yes 2 1.1 even 1 trivial
2925.2.a.x.1.1 2 5.4 even 2
2925.2.a.bc.1.2 2 15.14 odd 2
2925.2.c.o.2224.1 4 15.2 even 4
2925.2.c.o.2224.4 4 15.8 even 4
2925.2.c.p.2224.1 4 5.3 odd 4
2925.2.c.p.2224.4 4 5.2 odd 4
7605.2.a.bd.1.1 2 13.12 even 2
7605.2.a.bi.1.2 2 39.38 odd 2
9360.2.a.cl.1.1 2 12.11 even 2
9360.2.a.cw.1.1 2 4.3 odd 2