Properties

Label 4560.2.a.bl.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.60555 q^{7} +1.00000 q^{9} +4.60555 q^{11} +4.60555 q^{13} -1.00000 q^{15} +2.00000 q^{17} +1.00000 q^{19} +2.60555 q^{21} +2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.60555 q^{29} -4.00000 q^{31} +4.60555 q^{33} -2.60555 q^{35} +3.39445 q^{37} +4.60555 q^{39} +6.60555 q^{41} -10.6056 q^{43} -1.00000 q^{45} -6.00000 q^{47} -0.211103 q^{49} +2.00000 q^{51} -4.60555 q^{55} +1.00000 q^{57} -5.21110 q^{59} -7.21110 q^{61} +2.60555 q^{63} -4.60555 q^{65} -4.00000 q^{67} +2.00000 q^{69} +9.21110 q^{71} +6.00000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} +11.2111 q^{83} -2.00000 q^{85} +2.60555 q^{87} +6.60555 q^{89} +12.0000 q^{91} -4.00000 q^{93} -1.00000 q^{95} +16.6056 q^{97} +4.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 8 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 2 q^{45} - 12 q^{47} + 14 q^{49} + 4 q^{51} - 2 q^{55} + 2 q^{57} + 4 q^{59} - 2 q^{63} - 2 q^{65} - 8 q^{67} + 4 q^{69} + 4 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} - 2 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93} - 2 q^{95} + 26 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.60555 0.984806 0.492403 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.60555 1.38863 0.694313 0.719673i \(-0.255708\pi\)
0.694313 + 0.719673i \(0.255708\pi\)
\(12\) 0 0
\(13\) 4.60555 1.27735 0.638675 0.769477i \(-0.279483\pi\)
0.638675 + 0.769477i \(0.279483\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.60555 0.568578
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.60555 0.483839 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 4.60555 0.801724
\(34\) 0 0
\(35\) −2.60555 −0.440419
\(36\) 0 0
\(37\) 3.39445 0.558044 0.279022 0.960285i \(-0.409990\pi\)
0.279022 + 0.960285i \(0.409990\pi\)
\(38\) 0 0
\(39\) 4.60555 0.737478
\(40\) 0 0
\(41\) 6.60555 1.03161 0.515807 0.856705i \(-0.327492\pi\)
0.515807 + 0.856705i \(0.327492\pi\)
\(42\) 0 0
\(43\) −10.6056 −1.61733 −0.808666 0.588268i \(-0.799810\pi\)
−0.808666 + 0.588268i \(0.799810\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −4.60555 −0.621012
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 2.60555 0.328269
\(64\) 0 0
\(65\) −4.60555 −0.571248
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 9.21110 1.09316 0.546578 0.837408i \(-0.315930\pi\)
0.546578 + 0.837408i \(0.315930\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.2111 1.23058 0.615289 0.788301i \(-0.289039\pi\)
0.615289 + 0.788301i \(0.289039\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 2.60555 0.279344
\(88\) 0 0
\(89\) 6.60555 0.700187 0.350094 0.936715i \(-0.386150\pi\)
0.350094 + 0.936715i \(0.386150\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 16.6056 1.68604 0.843019 0.537884i \(-0.180776\pi\)
0.843019 + 0.537884i \(0.180776\pi\)
\(98\) 0 0
\(99\) 4.60555 0.462875
\(100\) 0 0
\(101\) −7.21110 −0.717532 −0.358766 0.933428i \(-0.616802\pi\)
−0.358766 + 0.933428i \(0.616802\pi\)
\(102\) 0 0
\(103\) −18.4222 −1.81519 −0.907597 0.419843i \(-0.862085\pi\)
−0.907597 + 0.419843i \(0.862085\pi\)
\(104\) 0 0
\(105\) −2.60555 −0.254276
\(106\) 0 0
\(107\) −18.4222 −1.78094 −0.890471 0.455040i \(-0.849625\pi\)
−0.890471 + 0.455040i \(0.849625\pi\)
\(108\) 0 0
\(109\) −11.2111 −1.07383 −0.536914 0.843637i \(-0.680410\pi\)
−0.536914 + 0.843637i \(0.680410\pi\)
\(110\) 0 0
\(111\) 3.39445 0.322187
\(112\) 0 0
\(113\) −1.21110 −0.113931 −0.0569655 0.998376i \(-0.518142\pi\)
−0.0569655 + 0.998376i \(0.518142\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) 4.60555 0.425783
\(118\) 0 0
\(119\) 5.21110 0.477701
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 6.60555 0.595603
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.4222 −1.63471 −0.817353 0.576137i \(-0.804559\pi\)
−0.817353 + 0.576137i \(0.804559\pi\)
\(128\) 0 0
\(129\) −10.6056 −0.933767
\(130\) 0 0
\(131\) 15.3944 1.34502 0.672510 0.740088i \(-0.265216\pi\)
0.672510 + 0.740088i \(0.265216\pi\)
\(132\) 0 0
\(133\) 2.60555 0.225930
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −12.4222 −1.06130 −0.530650 0.847591i \(-0.678052\pi\)
−0.530650 + 0.847591i \(0.678052\pi\)
\(138\) 0 0
\(139\) −9.21110 −0.781276 −0.390638 0.920544i \(-0.627745\pi\)
−0.390638 + 0.920544i \(0.627745\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 21.2111 1.77376
\(144\) 0 0
\(145\) −2.60555 −0.216379
\(146\) 0 0
\(147\) −0.211103 −0.0174114
\(148\) 0 0
\(149\) 16.4222 1.34536 0.672680 0.739934i \(-0.265143\pi\)
0.672680 + 0.739934i \(0.265143\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 15.2111 1.21398 0.606989 0.794710i \(-0.292377\pi\)
0.606989 + 0.794710i \(0.292377\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.21110 0.410692
\(162\) 0 0
\(163\) 17.0278 1.33372 0.666858 0.745184i \(-0.267639\pi\)
0.666858 + 0.745184i \(0.267639\pi\)
\(164\) 0 0
\(165\) −4.60555 −0.358542
\(166\) 0 0
\(167\) 6.78890 0.525341 0.262670 0.964886i \(-0.415397\pi\)
0.262670 + 0.964886i \(0.415397\pi\)
\(168\) 0 0
\(169\) 8.21110 0.631623
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −9.21110 −0.700307 −0.350154 0.936692i \(-0.613871\pi\)
−0.350154 + 0.936692i \(0.613871\pi\)
\(174\) 0 0
\(175\) 2.60555 0.196961
\(176\) 0 0
\(177\) −5.21110 −0.391690
\(178\) 0 0
\(179\) 21.2111 1.58539 0.792696 0.609617i \(-0.208677\pi\)
0.792696 + 0.609617i \(0.208677\pi\)
\(180\) 0 0
\(181\) −0.788897 −0.0586383 −0.0293191 0.999570i \(-0.509334\pi\)
−0.0293191 + 0.999570i \(0.509334\pi\)
\(182\) 0 0
\(183\) −7.21110 −0.533060
\(184\) 0 0
\(185\) −3.39445 −0.249565
\(186\) 0 0
\(187\) 9.21110 0.673583
\(188\) 0 0
\(189\) 2.60555 0.189526
\(190\) 0 0
\(191\) 5.81665 0.420878 0.210439 0.977607i \(-0.432511\pi\)
0.210439 + 0.977607i \(0.432511\pi\)
\(192\) 0 0
\(193\) −0.605551 −0.0435885 −0.0217943 0.999762i \(-0.506938\pi\)
−0.0217943 + 0.999762i \(0.506938\pi\)
\(194\) 0 0
\(195\) −4.60555 −0.329810
\(196\) 0 0
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) −17.2111 −1.22006 −0.610031 0.792377i \(-0.708843\pi\)
−0.610031 + 0.792377i \(0.708843\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.78890 0.476487
\(204\) 0 0
\(205\) −6.60555 −0.461352
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 4.60555 0.318573
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 9.21110 0.631134
\(214\) 0 0
\(215\) 10.6056 0.723293
\(216\) 0 0
\(217\) −10.4222 −0.707505
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 9.21110 0.619606
\(222\) 0 0
\(223\) 10.4222 0.697922 0.348961 0.937137i \(-0.386534\pi\)
0.348961 + 0.937137i \(0.386534\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 13.2111 0.876852 0.438426 0.898767i \(-0.355536\pi\)
0.438426 + 0.898767i \(0.355536\pi\)
\(228\) 0 0
\(229\) −3.21110 −0.212196 −0.106098 0.994356i \(-0.533836\pi\)
−0.106098 + 0.994356i \(0.533836\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) −16.4222 −1.07585 −0.537927 0.842991i \(-0.680792\pi\)
−0.537927 + 0.842991i \(0.680792\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) −25.8167 −1.66994 −0.834970 0.550295i \(-0.814515\pi\)
−0.834970 + 0.550295i \(0.814515\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.211103 0.0134868
\(246\) 0 0
\(247\) 4.60555 0.293044
\(248\) 0 0
\(249\) 11.2111 0.710475
\(250\) 0 0
\(251\) 6.18335 0.390289 0.195145 0.980774i \(-0.437482\pi\)
0.195145 + 0.980774i \(0.437482\pi\)
\(252\) 0 0
\(253\) 9.21110 0.579097
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) 2.78890 0.173967 0.0869833 0.996210i \(-0.472277\pi\)
0.0869833 + 0.996210i \(0.472277\pi\)
\(258\) 0 0
\(259\) 8.84441 0.549565
\(260\) 0 0
\(261\) 2.60555 0.161280
\(262\) 0 0
\(263\) −7.21110 −0.444656 −0.222328 0.974972i \(-0.571366\pi\)
−0.222328 + 0.974972i \(0.571366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.60555 0.404253
\(268\) 0 0
\(269\) −17.3944 −1.06056 −0.530279 0.847823i \(-0.677913\pi\)
−0.530279 + 0.847823i \(0.677913\pi\)
\(270\) 0 0
\(271\) −11.6333 −0.706673 −0.353337 0.935496i \(-0.614953\pi\)
−0.353337 + 0.935496i \(0.614953\pi\)
\(272\) 0 0
\(273\) 12.0000 0.726273
\(274\) 0 0
\(275\) 4.60555 0.277725
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 18.2389 1.08804 0.544020 0.839073i \(-0.316902\pi\)
0.544020 + 0.839073i \(0.316902\pi\)
\(282\) 0 0
\(283\) 18.6056 1.10599 0.552993 0.833186i \(-0.313486\pi\)
0.552993 + 0.833186i \(0.313486\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 17.2111 1.01594
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 16.6056 0.973435
\(292\) 0 0
\(293\) 19.6333 1.14699 0.573495 0.819209i \(-0.305587\pi\)
0.573495 + 0.819209i \(0.305587\pi\)
\(294\) 0 0
\(295\) 5.21110 0.303402
\(296\) 0 0
\(297\) 4.60555 0.267241
\(298\) 0 0
\(299\) 9.21110 0.532692
\(300\) 0 0
\(301\) −27.6333 −1.59276
\(302\) 0 0
\(303\) −7.21110 −0.414267
\(304\) 0 0
\(305\) 7.21110 0.412907
\(306\) 0 0
\(307\) 11.6333 0.663948 0.331974 0.943289i \(-0.392285\pi\)
0.331974 + 0.943289i \(0.392285\pi\)
\(308\) 0 0
\(309\) −18.4222 −1.04800
\(310\) 0 0
\(311\) −16.6056 −0.941614 −0.470807 0.882236i \(-0.656037\pi\)
−0.470807 + 0.882236i \(0.656037\pi\)
\(312\) 0 0
\(313\) 0.788897 0.0445911 0.0222956 0.999751i \(-0.492903\pi\)
0.0222956 + 0.999751i \(0.492903\pi\)
\(314\) 0 0
\(315\) −2.60555 −0.146806
\(316\) 0 0
\(317\) −6.42221 −0.360707 −0.180353 0.983602i \(-0.557724\pi\)
−0.180353 + 0.983602i \(0.557724\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −18.4222 −1.02823
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 0 0
\(325\) 4.60555 0.255470
\(326\) 0 0
\(327\) −11.2111 −0.619975
\(328\) 0 0
\(329\) −15.6333 −0.861892
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 3.39445 0.186015
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 3.02776 0.164932 0.0824662 0.996594i \(-0.473720\pi\)
0.0824662 + 0.996594i \(0.473720\pi\)
\(338\) 0 0
\(339\) −1.21110 −0.0657781
\(340\) 0 0
\(341\) −18.4222 −0.997618
\(342\) 0 0
\(343\) −18.7889 −1.01451
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) −21.6333 −1.16134 −0.580668 0.814140i \(-0.697209\pi\)
−0.580668 + 0.814140i \(0.697209\pi\)
\(348\) 0 0
\(349\) 34.8444 1.86518 0.932589 0.360939i \(-0.117544\pi\)
0.932589 + 0.360939i \(0.117544\pi\)
\(350\) 0 0
\(351\) 4.60555 0.245826
\(352\) 0 0
\(353\) 0.422205 0.0224717 0.0112359 0.999937i \(-0.496423\pi\)
0.0112359 + 0.999937i \(0.496423\pi\)
\(354\) 0 0
\(355\) −9.21110 −0.488875
\(356\) 0 0
\(357\) 5.21110 0.275801
\(358\) 0 0
\(359\) 13.8167 0.729215 0.364608 0.931161i \(-0.381203\pi\)
0.364608 + 0.931161i \(0.381203\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 10.2111 0.535944
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) −25.0278 −1.30644 −0.653219 0.757169i \(-0.726582\pi\)
−0.653219 + 0.757169i \(0.726582\pi\)
\(368\) 0 0
\(369\) 6.60555 0.343871
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.6056 0.859803 0.429901 0.902876i \(-0.358548\pi\)
0.429901 + 0.902876i \(0.358548\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −26.4222 −1.35722 −0.678609 0.734500i \(-0.737417\pi\)
−0.678609 + 0.734500i \(0.737417\pi\)
\(380\) 0 0
\(381\) −18.4222 −0.943798
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) −10.6056 −0.539110
\(388\) 0 0
\(389\) 36.4222 1.84668 0.923340 0.383984i \(-0.125448\pi\)
0.923340 + 0.383984i \(0.125448\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 15.3944 0.776547
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −7.57779 −0.380319 −0.190159 0.981753i \(-0.560900\pi\)
−0.190159 + 0.981753i \(0.560900\pi\)
\(398\) 0 0
\(399\) 2.60555 0.130441
\(400\) 0 0
\(401\) 25.3944 1.26814 0.634069 0.773276i \(-0.281383\pi\)
0.634069 + 0.773276i \(0.281383\pi\)
\(402\) 0 0
\(403\) −18.4222 −0.917675
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 15.6333 0.774914
\(408\) 0 0
\(409\) −12.7889 −0.632370 −0.316185 0.948698i \(-0.602402\pi\)
−0.316185 + 0.948698i \(0.602402\pi\)
\(410\) 0 0
\(411\) −12.4222 −0.612742
\(412\) 0 0
\(413\) −13.5778 −0.668120
\(414\) 0 0
\(415\) −11.2111 −0.550331
\(416\) 0 0
\(417\) −9.21110 −0.451070
\(418\) 0 0
\(419\) 7.02776 0.343328 0.171664 0.985156i \(-0.445086\pi\)
0.171664 + 0.985156i \(0.445086\pi\)
\(420\) 0 0
\(421\) −8.42221 −0.410473 −0.205237 0.978712i \(-0.565796\pi\)
−0.205237 + 0.978712i \(0.565796\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −18.7889 −0.909258
\(428\) 0 0
\(429\) 21.2111 1.02408
\(430\) 0 0
\(431\) 9.21110 0.443683 0.221842 0.975083i \(-0.428793\pi\)
0.221842 + 0.975083i \(0.428793\pi\)
\(432\) 0 0
\(433\) −16.6056 −0.798012 −0.399006 0.916948i \(-0.630645\pi\)
−0.399006 + 0.916948i \(0.630645\pi\)
\(434\) 0 0
\(435\) −2.60555 −0.124927
\(436\) 0 0
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) −34.4222 −1.64288 −0.821441 0.570293i \(-0.806830\pi\)
−0.821441 + 0.570293i \(0.806830\pi\)
\(440\) 0 0
\(441\) −0.211103 −0.0100525
\(442\) 0 0
\(443\) −18.8444 −0.895325 −0.447662 0.894203i \(-0.647743\pi\)
−0.447662 + 0.894203i \(0.647743\pi\)
\(444\) 0 0
\(445\) −6.60555 −0.313133
\(446\) 0 0
\(447\) 16.4222 0.776744
\(448\) 0 0
\(449\) −18.2389 −0.860745 −0.430372 0.902651i \(-0.641618\pi\)
−0.430372 + 0.902651i \(0.641618\pi\)
\(450\) 0 0
\(451\) 30.4222 1.43253
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) 32.0555 1.49949 0.749747 0.661725i \(-0.230175\pi\)
0.749747 + 0.661725i \(0.230175\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −22.8444 −1.06397 −0.531985 0.846754i \(-0.678554\pi\)
−0.531985 + 0.846754i \(0.678554\pi\)
\(462\) 0 0
\(463\) −33.0278 −1.53493 −0.767465 0.641091i \(-0.778482\pi\)
−0.767465 + 0.641091i \(0.778482\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −22.8444 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(468\) 0 0
\(469\) −10.4222 −0.481253
\(470\) 0 0
\(471\) 15.2111 0.700891
\(472\) 0 0
\(473\) −48.8444 −2.24587
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.6056 −1.30702 −0.653510 0.756917i \(-0.726704\pi\)
−0.653510 + 0.756917i \(0.726704\pi\)
\(480\) 0 0
\(481\) 15.6333 0.712817
\(482\) 0 0
\(483\) 5.21110 0.237113
\(484\) 0 0
\(485\) −16.6056 −0.754019
\(486\) 0 0
\(487\) −0.366692 −0.0166164 −0.00830821 0.999965i \(-0.502645\pi\)
−0.00830821 + 0.999965i \(0.502645\pi\)
\(488\) 0 0
\(489\) 17.0278 0.770022
\(490\) 0 0
\(491\) 12.2389 0.552332 0.276166 0.961110i \(-0.410936\pi\)
0.276166 + 0.961110i \(0.410936\pi\)
\(492\) 0 0
\(493\) 5.21110 0.234696
\(494\) 0 0
\(495\) −4.60555 −0.207004
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 27.6333 1.23704 0.618518 0.785770i \(-0.287733\pi\)
0.618518 + 0.785770i \(0.287733\pi\)
\(500\) 0 0
\(501\) 6.78890 0.303306
\(502\) 0 0
\(503\) −12.7889 −0.570229 −0.285114 0.958494i \(-0.592032\pi\)
−0.285114 + 0.958494i \(0.592032\pi\)
\(504\) 0 0
\(505\) 7.21110 0.320890
\(506\) 0 0
\(507\) 8.21110 0.364668
\(508\) 0 0
\(509\) −35.4500 −1.57129 −0.785646 0.618676i \(-0.787669\pi\)
−0.785646 + 0.618676i \(0.787669\pi\)
\(510\) 0 0
\(511\) 15.6333 0.691577
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 18.4222 0.811779
\(516\) 0 0
\(517\) −27.6333 −1.21531
\(518\) 0 0
\(519\) −9.21110 −0.404323
\(520\) 0 0
\(521\) −4.18335 −0.183276 −0.0916379 0.995792i \(-0.529210\pi\)
−0.0916379 + 0.995792i \(0.529210\pi\)
\(522\) 0 0
\(523\) 25.2111 1.10240 0.551202 0.834372i \(-0.314169\pi\)
0.551202 + 0.834372i \(0.314169\pi\)
\(524\) 0 0
\(525\) 2.60555 0.113716
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −5.21110 −0.226143
\(532\) 0 0
\(533\) 30.4222 1.31773
\(534\) 0 0
\(535\) 18.4222 0.796461
\(536\) 0 0
\(537\) 21.2111 0.915327
\(538\) 0 0
\(539\) −0.972244 −0.0418775
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) −0.788897 −0.0338548
\(544\) 0 0
\(545\) 11.2111 0.480231
\(546\) 0 0
\(547\) 22.7889 0.974383 0.487191 0.873295i \(-0.338021\pi\)
0.487191 + 0.873295i \(0.338021\pi\)
\(548\) 0 0
\(549\) −7.21110 −0.307762
\(550\) 0 0
\(551\) 2.60555 0.111000
\(552\) 0 0
\(553\) −20.8444 −0.886394
\(554\) 0 0
\(555\) −3.39445 −0.144086
\(556\) 0 0
\(557\) −4.42221 −0.187375 −0.0936874 0.995602i \(-0.529865\pi\)
−0.0936874 + 0.995602i \(0.529865\pi\)
\(558\) 0 0
\(559\) −48.8444 −2.06590
\(560\) 0 0
\(561\) 9.21110 0.388893
\(562\) 0 0
\(563\) −31.6333 −1.33318 −0.666592 0.745422i \(-0.732248\pi\)
−0.666592 + 0.745422i \(0.732248\pi\)
\(564\) 0 0
\(565\) 1.21110 0.0509515
\(566\) 0 0
\(567\) 2.60555 0.109423
\(568\) 0 0
\(569\) 39.4500 1.65383 0.826914 0.562328i \(-0.190094\pi\)
0.826914 + 0.562328i \(0.190094\pi\)
\(570\) 0 0
\(571\) −7.63331 −0.319444 −0.159722 0.987162i \(-0.551060\pi\)
−0.159722 + 0.987162i \(0.551060\pi\)
\(572\) 0 0
\(573\) 5.81665 0.242994
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −23.2111 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(578\) 0 0
\(579\) −0.605551 −0.0251659
\(580\) 0 0
\(581\) 29.2111 1.21188
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −4.60555 −0.190416
\(586\) 0 0
\(587\) −12.4222 −0.512719 −0.256360 0.966581i \(-0.582523\pi\)
−0.256360 + 0.966581i \(0.582523\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 0 0
\(593\) −3.57779 −0.146922 −0.0734612 0.997298i \(-0.523405\pi\)
−0.0734612 + 0.997298i \(0.523405\pi\)
\(594\) 0 0
\(595\) −5.21110 −0.213634
\(596\) 0 0
\(597\) −17.2111 −0.704404
\(598\) 0 0
\(599\) −36.8444 −1.50542 −0.752711 0.658351i \(-0.771254\pi\)
−0.752711 + 0.658351i \(0.771254\pi\)
\(600\) 0 0
\(601\) 4.78890 0.195343 0.0976716 0.995219i \(-0.468861\pi\)
0.0976716 + 0.995219i \(0.468861\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) −10.2111 −0.415140
\(606\) 0 0
\(607\) 23.6333 0.959246 0.479623 0.877475i \(-0.340773\pi\)
0.479623 + 0.877475i \(0.340773\pi\)
\(608\) 0 0
\(609\) 6.78890 0.275100
\(610\) 0 0
\(611\) −27.6333 −1.11792
\(612\) 0 0
\(613\) 4.78890 0.193422 0.0967109 0.995313i \(-0.469168\pi\)
0.0967109 + 0.995313i \(0.469168\pi\)
\(614\) 0 0
\(615\) −6.60555 −0.266362
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 4.36669 0.175512 0.0877561 0.996142i \(-0.472030\pi\)
0.0877561 + 0.996142i \(0.472030\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) 17.2111 0.689548
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.60555 0.183928
\(628\) 0 0
\(629\) 6.78890 0.270691
\(630\) 0 0
\(631\) −36.8444 −1.46675 −0.733376 0.679823i \(-0.762057\pi\)
−0.733376 + 0.679823i \(0.762057\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 18.4222 0.731063
\(636\) 0 0
\(637\) −0.972244 −0.0385217
\(638\) 0 0
\(639\) 9.21110 0.364386
\(640\) 0 0
\(641\) 35.4500 1.40019 0.700095 0.714050i \(-0.253141\pi\)
0.700095 + 0.714050i \(0.253141\pi\)
\(642\) 0 0
\(643\) −5.39445 −0.212736 −0.106368 0.994327i \(-0.533922\pi\)
−0.106368 + 0.994327i \(0.533922\pi\)
\(644\) 0 0
\(645\) 10.6056 0.417593
\(646\) 0 0
\(647\) 2.36669 0.0930443 0.0465221 0.998917i \(-0.485186\pi\)
0.0465221 + 0.998917i \(0.485186\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −10.4222 −0.408478
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) −15.3944 −0.601511
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −31.2111 −1.21397 −0.606986 0.794713i \(-0.707621\pi\)
−0.606986 + 0.794713i \(0.707621\pi\)
\(662\) 0 0
\(663\) 9.21110 0.357730
\(664\) 0 0
\(665\) −2.60555 −0.101039
\(666\) 0 0
\(667\) 5.21110 0.201775
\(668\) 0 0
\(669\) 10.4222 0.402946
\(670\) 0 0
\(671\) −33.2111 −1.28210
\(672\) 0 0
\(673\) 36.6056 1.41104 0.705520 0.708690i \(-0.250713\pi\)
0.705520 + 0.708690i \(0.250713\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 10.4222 0.400558 0.200279 0.979739i \(-0.435815\pi\)
0.200279 + 0.979739i \(0.435815\pi\)
\(678\) 0 0
\(679\) 43.2666 1.66042
\(680\) 0 0
\(681\) 13.2111 0.506251
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 12.4222 0.474628
\(686\) 0 0
\(687\) −3.21110 −0.122511
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.0555 1.59987 0.799934 0.600089i \(-0.204868\pi\)
0.799934 + 0.600089i \(0.204868\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 9.21110 0.349397
\(696\) 0 0
\(697\) 13.2111 0.500406
\(698\) 0 0
\(699\) −16.4222 −0.621145
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 3.39445 0.128024
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) −18.7889 −0.706629
\(708\) 0 0
\(709\) −25.6333 −0.962679 −0.481340 0.876534i \(-0.659850\pi\)
−0.481340 + 0.876534i \(0.659850\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −21.2111 −0.793250
\(716\) 0 0
\(717\) −25.8167 −0.964141
\(718\) 0 0
\(719\) −16.2389 −0.605607 −0.302804 0.953053i \(-0.597923\pi\)
−0.302804 + 0.953053i \(0.597923\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) 2.60555 0.0967677
\(726\) 0 0
\(727\) 17.3944 0.645124 0.322562 0.946548i \(-0.395456\pi\)
0.322562 + 0.946548i \(0.395456\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.2111 −0.784521
\(732\) 0 0
\(733\) −24.4222 −0.902055 −0.451027 0.892510i \(-0.648942\pi\)
−0.451027 + 0.892510i \(0.648942\pi\)
\(734\) 0 0
\(735\) 0.211103 0.00778663
\(736\) 0 0
\(737\) −18.4222 −0.678591
\(738\) 0 0
\(739\) −38.4222 −1.41338 −0.706692 0.707521i \(-0.749813\pi\)
−0.706692 + 0.707521i \(0.749813\pi\)
\(740\) 0 0
\(741\) 4.60555 0.169189
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) −16.4222 −0.601663
\(746\) 0 0
\(747\) 11.2111 0.410193
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −40.8444 −1.49043 −0.745217 0.666822i \(-0.767654\pi\)
−0.745217 + 0.666822i \(0.767654\pi\)
\(752\) 0 0
\(753\) 6.18335 0.225334
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −36.0555 −1.31046 −0.655230 0.755429i \(-0.727428\pi\)
−0.655230 + 0.755429i \(0.727428\pi\)
\(758\) 0 0
\(759\) 9.21110 0.334342
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) −29.2111 −1.05751
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −20.4222 −0.736444 −0.368222 0.929738i \(-0.620033\pi\)
−0.368222 + 0.929738i \(0.620033\pi\)
\(770\) 0 0
\(771\) 2.78890 0.100440
\(772\) 0 0
\(773\) −43.2666 −1.55619 −0.778096 0.628145i \(-0.783814\pi\)
−0.778096 + 0.628145i \(0.783814\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 8.84441 0.317291
\(778\) 0 0
\(779\) 6.60555 0.236668
\(780\) 0 0
\(781\) 42.4222 1.51799
\(782\) 0 0
\(783\) 2.60555 0.0931148
\(784\) 0 0
\(785\) −15.2111 −0.542908
\(786\) 0 0
\(787\) 19.6333 0.699852 0.349926 0.936777i \(-0.386207\pi\)
0.349926 + 0.936777i \(0.386207\pi\)
\(788\) 0 0
\(789\) −7.21110 −0.256722
\(790\) 0 0
\(791\) −3.15559 −0.112200
\(792\) 0 0
\(793\) −33.2111 −1.17936
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.2111 0.609649 0.304824 0.952409i \(-0.401402\pi\)
0.304824 + 0.952409i \(0.401402\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 6.60555 0.233396
\(802\) 0 0
\(803\) 27.6333 0.975158
\(804\) 0 0
\(805\) −5.21110 −0.183667
\(806\) 0 0
\(807\) −17.3944 −0.612314
\(808\) 0 0
\(809\) −48.4222 −1.70243 −0.851217 0.524814i \(-0.824135\pi\)
−0.851217 + 0.524814i \(0.824135\pi\)
\(810\) 0 0
\(811\) 40.8444 1.43424 0.717121 0.696949i \(-0.245460\pi\)
0.717121 + 0.696949i \(0.245460\pi\)
\(812\) 0 0
\(813\) −11.6333 −0.407998
\(814\) 0 0
\(815\) −17.0278 −0.596456
\(816\) 0 0
\(817\) −10.6056 −0.371041
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) 43.2111 1.50808 0.754039 0.656830i \(-0.228103\pi\)
0.754039 + 0.656830i \(0.228103\pi\)
\(822\) 0 0
\(823\) 47.8167 1.66678 0.833392 0.552683i \(-0.186396\pi\)
0.833392 + 0.552683i \(0.186396\pi\)
\(824\) 0 0
\(825\) 4.60555 0.160345
\(826\) 0 0
\(827\) 27.6333 0.960904 0.480452 0.877021i \(-0.340473\pi\)
0.480452 + 0.877021i \(0.340473\pi\)
\(828\) 0 0
\(829\) 15.2111 0.528303 0.264152 0.964481i \(-0.414908\pi\)
0.264152 + 0.964481i \(0.414908\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −0.422205 −0.0146285
\(834\) 0 0
\(835\) −6.78890 −0.234939
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 0 0
\(843\) 18.2389 0.628180
\(844\) 0 0
\(845\) −8.21110 −0.282471
\(846\) 0 0
\(847\) 26.6056 0.914178
\(848\) 0 0
\(849\) 18.6056 0.638541
\(850\) 0 0
\(851\) 6.78890 0.232720
\(852\) 0 0
\(853\) 29.6333 1.01463 0.507313 0.861762i \(-0.330639\pi\)
0.507313 + 0.861762i \(0.330639\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 6.42221 0.219378 0.109689 0.993966i \(-0.465014\pi\)
0.109689 + 0.993966i \(0.465014\pi\)
\(858\) 0 0
\(859\) −8.84441 −0.301767 −0.150884 0.988552i \(-0.548212\pi\)
−0.150884 + 0.988552i \(0.548212\pi\)
\(860\) 0 0
\(861\) 17.2111 0.586553
\(862\) 0 0
\(863\) −6.42221 −0.218614 −0.109307 0.994008i \(-0.534863\pi\)
−0.109307 + 0.994008i \(0.534863\pi\)
\(864\) 0 0
\(865\) 9.21110 0.313187
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −36.8444 −1.24986
\(870\) 0 0
\(871\) −18.4222 −0.624213
\(872\) 0 0
\(873\) 16.6056 0.562013
\(874\) 0 0
\(875\) −2.60555 −0.0880837
\(876\) 0 0
\(877\) 27.0278 0.912662 0.456331 0.889810i \(-0.349163\pi\)
0.456331 + 0.889810i \(0.349163\pi\)
\(878\) 0 0
\(879\) 19.6333 0.662215
\(880\) 0 0
\(881\) 59.2111 1.99487 0.997436 0.0715590i \(-0.0227974\pi\)
0.997436 + 0.0715590i \(0.0227974\pi\)
\(882\) 0 0
\(883\) −27.8167 −0.936105 −0.468052 0.883701i \(-0.655044\pi\)
−0.468052 + 0.883701i \(0.655044\pi\)
\(884\) 0 0
\(885\) 5.21110 0.175169
\(886\) 0 0
\(887\) 20.8444 0.699887 0.349943 0.936771i \(-0.386201\pi\)
0.349943 + 0.936771i \(0.386201\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 4.60555 0.154292
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) −21.2111 −0.709009
\(896\) 0 0
\(897\) 9.21110 0.307550
\(898\) 0 0
\(899\) −10.4222 −0.347600
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −27.6333 −0.919579
\(904\) 0 0
\(905\) 0.788897 0.0262238
\(906\) 0 0
\(907\) −30.0555 −0.997977 −0.498988 0.866609i \(-0.666295\pi\)
−0.498988 + 0.866609i \(0.666295\pi\)
\(908\) 0 0
\(909\) −7.21110 −0.239177
\(910\) 0 0
\(911\) −10.4222 −0.345303 −0.172652 0.984983i \(-0.555233\pi\)
−0.172652 + 0.984983i \(0.555233\pi\)
\(912\) 0 0
\(913\) 51.6333 1.70881
\(914\) 0 0
\(915\) 7.21110 0.238392
\(916\) 0 0
\(917\) 40.1110 1.32458
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 11.6333 0.383331
\(922\) 0 0
\(923\) 42.4222 1.39634
\(924\) 0 0
\(925\) 3.39445 0.111609
\(926\) 0 0
\(927\) −18.4222 −0.605065
\(928\) 0 0
\(929\) −45.6333 −1.49718 −0.748590 0.663033i \(-0.769269\pi\)
−0.748590 + 0.663033i \(0.769269\pi\)
\(930\) 0 0
\(931\) −0.211103 −0.00691861
\(932\) 0 0
\(933\) −16.6056 −0.543641
\(934\) 0 0
\(935\) −9.21110 −0.301235
\(936\) 0 0
\(937\) −19.2111 −0.627599 −0.313800 0.949489i \(-0.601602\pi\)
−0.313800 + 0.949489i \(0.601602\pi\)
\(938\) 0 0
\(939\) 0.788897 0.0257447
\(940\) 0 0
\(941\) −30.2389 −0.985759 −0.492879 0.870098i \(-0.664056\pi\)
−0.492879 + 0.870098i \(0.664056\pi\)
\(942\) 0 0
\(943\) 13.2111 0.430213
\(944\) 0 0
\(945\) −2.60555 −0.0847586
\(946\) 0 0
\(947\) 7.21110 0.234329 0.117165 0.993113i \(-0.462619\pi\)
0.117165 + 0.993113i \(0.462619\pi\)
\(948\) 0 0
\(949\) 27.6333 0.897015
\(950\) 0 0
\(951\) −6.42221 −0.208254
\(952\) 0 0
\(953\) 5.21110 0.168804 0.0844021 0.996432i \(-0.473102\pi\)
0.0844021 + 0.996432i \(0.473102\pi\)
\(954\) 0 0
\(955\) −5.81665 −0.188222
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) −32.3667 −1.04518
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −18.4222 −0.593647
\(964\) 0 0
\(965\) 0.605551 0.0194934
\(966\) 0 0
\(967\) 26.2389 0.843785 0.421892 0.906646i \(-0.361366\pi\)
0.421892 + 0.906646i \(0.361366\pi\)
\(968\) 0 0
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −24.0000 −0.769405
\(974\) 0 0
\(975\) 4.60555 0.147496
\(976\) 0 0
\(977\) 6.42221 0.205465 0.102732 0.994709i \(-0.467242\pi\)
0.102732 + 0.994709i \(0.467242\pi\)
\(978\) 0 0
\(979\) 30.4222 0.972298
\(980\) 0 0
\(981\) −11.2111 −0.357943
\(982\) 0 0
\(983\) 9.21110 0.293789 0.146894 0.989152i \(-0.453072\pi\)
0.146894 + 0.989152i \(0.453072\pi\)
\(984\) 0 0
\(985\) −14.0000 −0.446077
\(986\) 0 0
\(987\) −15.6333 −0.497614
\(988\) 0 0
\(989\) −21.2111 −0.674474
\(990\) 0 0
\(991\) −50.4222 −1.60171 −0.800857 0.598856i \(-0.795622\pi\)
−0.800857 + 0.598856i \(0.795622\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 17.2111 0.545629
\(996\) 0 0
\(997\) −61.6333 −1.95195 −0.975973 0.217891i \(-0.930082\pi\)
−0.975973 + 0.217891i \(0.930082\pi\)
\(998\) 0 0
\(999\) 3.39445 0.107396
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 4560.2.a.bl.1.2 2
4.3 odd 2 1140.2.a.e.1.1 2
12.11 even 2 3420.2.a.i.1.1 2
20.3 even 4 5700.2.f.n.3649.2 4
20.7 even 4 5700.2.f.n.3649.3 4
20.19 odd 2 5700.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.e.1.1 2 4.3 odd 2
3420.2.a.i.1.1 2 12.11 even 2
4560.2.a.bl.1.2 2 1.1 even 1 trivial
5700.2.a.u.1.2 2 20.19 odd 2
5700.2.f.n.3649.2 4 20.3 even 4
5700.2.f.n.3649.3 4 20.7 even 4