Properties

Label 5520.2.be.a.1471.6
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.6
Root \(1.30491 - 1.30491i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.a.1471.14

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} +0.482745 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} +0.482745 q^{7} -1.00000 q^{9} +1.10189 q^{11} +2.25517 q^{13} +1.00000 q^{15} +6.03534i q^{17} -1.48254 q^{19} -0.482745i q^{21} +(-4.55260 - 1.50793i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -7.68679 q^{29} +5.12465i q^{31} -1.10189i q^{33} +0.482745i q^{35} +5.83765i q^{37} -2.25517i q^{39} +1.27767 q^{41} -8.46629 q^{43} -1.00000i q^{45} -11.2632i q^{47} -6.76696 q^{49} +6.03534 q^{51} -8.81467i q^{53} +1.10189i q^{55} +1.48254i q^{57} -8.08002i q^{59} -0.0392191i q^{61} -0.482745 q^{63} +2.25517i q^{65} -8.39370 q^{67} +(-1.50793 + 4.55260i) q^{69} +1.25598i q^{71} -3.47480 q^{73} +1.00000i q^{75} +0.531931 q^{77} -3.61831 q^{79} +1.00000 q^{81} +9.12164 q^{83} -6.03534 q^{85} +7.68679i q^{87} -6.85106i q^{89} +1.08867 q^{91} +5.12465 q^{93} -1.48254i q^{95} -0.330286i q^{97} -1.10189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q - 8q^{7} - 16q^{9} + 8q^{11} + 8q^{13} + 16q^{15} + 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} - 4q^{51} + 8q^{63} - 16q^{67} + 40q^{73} + 24q^{77} + 32q^{79} + 16q^{81} + 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 0.482745 0.182460 0.0912302 0.995830i \(-0.470920\pi\)
0.0912302 + 0.995830i \(0.470920\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.10189 0.332232 0.166116 0.986106i \(-0.446877\pi\)
0.166116 + 0.986106i \(0.446877\pi\)
\(12\) 0 0
\(13\) 2.25517 0.625471 0.312735 0.949840i \(-0.398755\pi\)
0.312735 + 0.949840i \(0.398755\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 6.03534i 1.46379i 0.681420 + 0.731893i \(0.261363\pi\)
−0.681420 + 0.731893i \(0.738637\pi\)
\(18\) 0 0
\(19\) −1.48254 −0.340118 −0.170059 0.985434i \(-0.554396\pi\)
−0.170059 + 0.985434i \(0.554396\pi\)
\(20\) 0 0
\(21\) 0.482745i 0.105344i
\(22\) 0 0
\(23\) −4.55260 1.50793i −0.949283 0.314424i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −7.68679 −1.42740 −0.713700 0.700451i \(-0.752982\pi\)
−0.713700 + 0.700451i \(0.752982\pi\)
\(30\) 0 0
\(31\) 5.12465i 0.920414i 0.887812 + 0.460207i \(0.152225\pi\)
−0.887812 + 0.460207i \(0.847775\pi\)
\(32\) 0 0
\(33\) 1.10189i 0.191814i
\(34\) 0 0
\(35\) 0.482745i 0.0815987i
\(36\) 0 0
\(37\) 5.83765i 0.959705i 0.877349 + 0.479852i \(0.159310\pi\)
−0.877349 + 0.479852i \(0.840690\pi\)
\(38\) 0 0
\(39\) 2.25517i 0.361116i
\(40\) 0 0
\(41\) 1.27767 0.199538 0.0997689 0.995011i \(-0.468190\pi\)
0.0997689 + 0.995011i \(0.468190\pi\)
\(42\) 0 0
\(43\) −8.46629 −1.29110 −0.645548 0.763720i \(-0.723371\pi\)
−0.645548 + 0.763720i \(0.723371\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 11.2632i 1.64290i −0.570278 0.821452i \(-0.693164\pi\)
0.570278 0.821452i \(-0.306836\pi\)
\(48\) 0 0
\(49\) −6.76696 −0.966708
\(50\) 0 0
\(51\) 6.03534 0.845117
\(52\) 0 0
\(53\) 8.81467i 1.21079i −0.795926 0.605394i \(-0.793015\pi\)
0.795926 0.605394i \(-0.206985\pi\)
\(54\) 0 0
\(55\) 1.10189i 0.148579i
\(56\) 0 0
\(57\) 1.48254i 0.196367i
\(58\) 0 0
\(59\) 8.08002i 1.05193i −0.850507 0.525964i \(-0.823705\pi\)
0.850507 0.525964i \(-0.176295\pi\)
\(60\) 0 0
\(61\) 0.0392191i 0.00502149i −0.999997 0.00251075i \(-0.999201\pi\)
0.999997 0.00251075i \(-0.000799197\pi\)
\(62\) 0 0
\(63\) −0.482745 −0.0608201
\(64\) 0 0
\(65\) 2.25517i 0.279719i
\(66\) 0 0
\(67\) −8.39370 −1.02545 −0.512727 0.858552i \(-0.671365\pi\)
−0.512727 + 0.858552i \(0.671365\pi\)
\(68\) 0 0
\(69\) −1.50793 + 4.55260i −0.181533 + 0.548069i
\(70\) 0 0
\(71\) 1.25598i 0.149058i 0.997219 + 0.0745289i \(0.0237453\pi\)
−0.997219 + 0.0745289i \(0.976255\pi\)
\(72\) 0 0
\(73\) −3.47480 −0.406694 −0.203347 0.979107i \(-0.565182\pi\)
−0.203347 + 0.979107i \(0.565182\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 0.531931 0.0606192
\(78\) 0 0
\(79\) −3.61831 −0.407092 −0.203546 0.979065i \(-0.565247\pi\)
−0.203546 + 0.979065i \(0.565247\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.12164 1.00123 0.500615 0.865670i \(-0.333107\pi\)
0.500615 + 0.865670i \(0.333107\pi\)
\(84\) 0 0
\(85\) −6.03534 −0.654625
\(86\) 0 0
\(87\) 7.68679i 0.824110i
\(88\) 0 0
\(89\) 6.85106i 0.726211i −0.931748 0.363105i \(-0.881717\pi\)
0.931748 0.363105i \(-0.118283\pi\)
\(90\) 0 0
\(91\) 1.08867 0.114124
\(92\) 0 0
\(93\) 5.12465 0.531401
\(94\) 0 0
\(95\) 1.48254i 0.152105i
\(96\) 0 0
\(97\) 0.330286i 0.0335355i −0.999859 0.0167677i \(-0.994662\pi\)
0.999859 0.0167677i \(-0.00533759\pi\)
\(98\) 0 0
\(99\) −1.10189 −0.110744
\(100\) 0 0
\(101\) −3.89119 −0.387188 −0.193594 0.981082i \(-0.562014\pi\)
−0.193594 + 0.981082i \(0.562014\pi\)
\(102\) 0 0
\(103\) 11.2588 1.10937 0.554683 0.832062i \(-0.312840\pi\)
0.554683 + 0.832062i \(0.312840\pi\)
\(104\) 0 0
\(105\) 0.482745 0.0471111
\(106\) 0 0
\(107\) −0.801493 −0.0774833 −0.0387416 0.999249i \(-0.512335\pi\)
−0.0387416 + 0.999249i \(0.512335\pi\)
\(108\) 0 0
\(109\) 1.25885i 0.120576i 0.998181 + 0.0602880i \(0.0192019\pi\)
−0.998181 + 0.0602880i \(0.980798\pi\)
\(110\) 0 0
\(111\) 5.83765 0.554086
\(112\) 0 0
\(113\) 8.70667i 0.819054i 0.912298 + 0.409527i \(0.134306\pi\)
−0.912298 + 0.409527i \(0.865694\pi\)
\(114\) 0 0
\(115\) 1.50793 4.55260i 0.140615 0.424532i
\(116\) 0 0
\(117\) −2.25517 −0.208490
\(118\) 0 0
\(119\) 2.91353i 0.267083i
\(120\) 0 0
\(121\) −9.78584 −0.889622
\(122\) 0 0
\(123\) 1.27767i 0.115203i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 5.51836i 0.489676i 0.969564 + 0.244838i \(0.0787347\pi\)
−0.969564 + 0.244838i \(0.921265\pi\)
\(128\) 0 0
\(129\) 8.46629i 0.745415i
\(130\) 0 0
\(131\) 14.5197i 1.26859i −0.773089 0.634297i \(-0.781289\pi\)
0.773089 0.634297i \(-0.218711\pi\)
\(132\) 0 0
\(133\) −0.715687 −0.0620580
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 4.50448i 0.384844i 0.981312 + 0.192422i \(0.0616342\pi\)
−0.981312 + 0.192422i \(0.938366\pi\)
\(138\) 0 0
\(139\) 2.13377i 0.180984i −0.995897 0.0904921i \(-0.971156\pi\)
0.995897 0.0904921i \(-0.0288440\pi\)
\(140\) 0 0
\(141\) −11.2632 −0.948531
\(142\) 0 0
\(143\) 2.48494 0.207801
\(144\) 0 0
\(145\) 7.68679i 0.638353i
\(146\) 0 0
\(147\) 6.76696i 0.558129i
\(148\) 0 0
\(149\) 11.9315i 0.977468i 0.872433 + 0.488734i \(0.162541\pi\)
−0.872433 + 0.488734i \(0.837459\pi\)
\(150\) 0 0
\(151\) 9.63555i 0.784130i −0.919937 0.392065i \(-0.871761\pi\)
0.919937 0.392065i \(-0.128239\pi\)
\(152\) 0 0
\(153\) 6.03534i 0.487929i
\(154\) 0 0
\(155\) −5.12465 −0.411622
\(156\) 0 0
\(157\) 0.935197i 0.0746368i −0.999303 0.0373184i \(-0.988118\pi\)
0.999303 0.0373184i \(-0.0118816\pi\)
\(158\) 0 0
\(159\) −8.81467 −0.699049
\(160\) 0 0
\(161\) −2.19774 0.727943i −0.173206 0.0573700i
\(162\) 0 0
\(163\) 13.8165i 1.08220i −0.840960 0.541098i \(-0.818009\pi\)
0.840960 0.541098i \(-0.181991\pi\)
\(164\) 0 0
\(165\) 1.10189 0.0857820
\(166\) 0 0
\(167\) 6.69307i 0.517925i −0.965887 0.258963i \(-0.916619\pi\)
0.965887 0.258963i \(-0.0833807\pi\)
\(168\) 0 0
\(169\) −7.91423 −0.608787
\(170\) 0 0
\(171\) 1.48254 0.113373
\(172\) 0 0
\(173\) −24.6878 −1.87698 −0.938490 0.345305i \(-0.887775\pi\)
−0.938490 + 0.345305i \(0.887775\pi\)
\(174\) 0 0
\(175\) −0.482745 −0.0364921
\(176\) 0 0
\(177\) −8.08002 −0.607331
\(178\) 0 0
\(179\) 4.65679i 0.348065i 0.984740 + 0.174032i \(0.0556797\pi\)
−0.984740 + 0.174032i \(0.944320\pi\)
\(180\) 0 0
\(181\) 10.4913i 0.779809i 0.920855 + 0.389905i \(0.127492\pi\)
−0.920855 + 0.389905i \(0.872508\pi\)
\(182\) 0 0
\(183\) −0.0392191 −0.00289916
\(184\) 0 0
\(185\) −5.83765 −0.429193
\(186\) 0 0
\(187\) 6.65028i 0.486317i
\(188\) 0 0
\(189\) 0.482745i 0.0351145i
\(190\) 0 0
\(191\) −10.1702 −0.735886 −0.367943 0.929848i \(-0.619938\pi\)
−0.367943 + 0.929848i \(0.619938\pi\)
\(192\) 0 0
\(193\) −13.3593 −0.961626 −0.480813 0.876823i \(-0.659658\pi\)
−0.480813 + 0.876823i \(0.659658\pi\)
\(194\) 0 0
\(195\) 2.25517 0.161496
\(196\) 0 0
\(197\) −2.65615 −0.189243 −0.0946216 0.995513i \(-0.530164\pi\)
−0.0946216 + 0.995513i \(0.530164\pi\)
\(198\) 0 0
\(199\) 8.43038 0.597614 0.298807 0.954314i \(-0.403411\pi\)
0.298807 + 0.954314i \(0.403411\pi\)
\(200\) 0 0
\(201\) 8.39370i 0.592046i
\(202\) 0 0
\(203\) −3.71075 −0.260444
\(204\) 0 0
\(205\) 1.27767i 0.0892360i
\(206\) 0 0
\(207\) 4.55260 + 1.50793i 0.316428 + 0.104808i
\(208\) 0 0
\(209\) −1.63359 −0.112998
\(210\) 0 0
\(211\) 0.771524i 0.0531139i 0.999647 + 0.0265570i \(0.00845434\pi\)
−0.999647 + 0.0265570i \(0.991546\pi\)
\(212\) 0 0
\(213\) 1.25598 0.0860586
\(214\) 0 0
\(215\) 8.46629i 0.577396i
\(216\) 0 0
\(217\) 2.47390i 0.167939i
\(218\) 0 0
\(219\) 3.47480i 0.234805i
\(220\) 0 0
\(221\) 13.6107i 0.915555i
\(222\) 0 0
\(223\) 19.0727i 1.27720i 0.769538 + 0.638601i \(0.220487\pi\)
−0.769538 + 0.638601i \(0.779513\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.95607 −0.461691 −0.230845 0.972990i \(-0.574149\pi\)
−0.230845 + 0.972990i \(0.574149\pi\)
\(228\) 0 0
\(229\) 1.27163i 0.0840316i 0.999117 + 0.0420158i \(0.0133780\pi\)
−0.999117 + 0.0420158i \(0.986622\pi\)
\(230\) 0 0
\(231\) 0.531931i 0.0349985i
\(232\) 0 0
\(233\) −9.90098 −0.648635 −0.324317 0.945948i \(-0.605135\pi\)
−0.324317 + 0.945948i \(0.605135\pi\)
\(234\) 0 0
\(235\) 11.2632 0.734729
\(236\) 0 0
\(237\) 3.61831i 0.235035i
\(238\) 0 0
\(239\) 10.3420i 0.668968i 0.942401 + 0.334484i \(0.108562\pi\)
−0.942401 + 0.334484i \(0.891438\pi\)
\(240\) 0 0
\(241\) 10.9888i 0.707852i 0.935273 + 0.353926i \(0.115153\pi\)
−0.935273 + 0.353926i \(0.884847\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.76696i 0.432325i
\(246\) 0 0
\(247\) −3.34337 −0.212734
\(248\) 0 0
\(249\) 9.12164i 0.578061i
\(250\) 0 0
\(251\) 8.22629 0.519239 0.259619 0.965711i \(-0.416403\pi\)
0.259619 + 0.965711i \(0.416403\pi\)
\(252\) 0 0
\(253\) −5.01646 1.66157i −0.315382 0.104462i
\(254\) 0 0
\(255\) 6.03534i 0.377948i
\(256\) 0 0
\(257\) −30.1958 −1.88357 −0.941783 0.336223i \(-0.890851\pi\)
−0.941783 + 0.336223i \(0.890851\pi\)
\(258\) 0 0
\(259\) 2.81810i 0.175108i
\(260\) 0 0
\(261\) 7.68679 0.475800
\(262\) 0 0
\(263\) −19.7029 −1.21493 −0.607466 0.794346i \(-0.707814\pi\)
−0.607466 + 0.794346i \(0.707814\pi\)
\(264\) 0 0
\(265\) 8.81467 0.541481
\(266\) 0 0
\(267\) −6.85106 −0.419278
\(268\) 0 0
\(269\) 12.1993 0.743805 0.371902 0.928272i \(-0.378706\pi\)
0.371902 + 0.928272i \(0.378706\pi\)
\(270\) 0 0
\(271\) 17.7526i 1.07840i 0.842179 + 0.539198i \(0.181273\pi\)
−0.842179 + 0.539198i \(0.818727\pi\)
\(272\) 0 0
\(273\) 1.08867i 0.0658893i
\(274\) 0 0
\(275\) −1.10189 −0.0664465
\(276\) 0 0
\(277\) −17.0510 −1.02449 −0.512246 0.858839i \(-0.671187\pi\)
−0.512246 + 0.858839i \(0.671187\pi\)
\(278\) 0 0
\(279\) 5.12465i 0.306805i
\(280\) 0 0
\(281\) 19.4880i 1.16256i −0.813705 0.581278i \(-0.802553\pi\)
0.813705 0.581278i \(-0.197447\pi\)
\(282\) 0 0
\(283\) 4.52826 0.269177 0.134589 0.990902i \(-0.457029\pi\)
0.134589 + 0.990902i \(0.457029\pi\)
\(284\) 0 0
\(285\) −1.48254 −0.0878180
\(286\) 0 0
\(287\) 0.616786 0.0364077
\(288\) 0 0
\(289\) −19.4254 −1.14267
\(290\) 0 0
\(291\) −0.330286 −0.0193617
\(292\) 0 0
\(293\) 14.3433i 0.837942i −0.908000 0.418971i \(-0.862391\pi\)
0.908000 0.418971i \(-0.137609\pi\)
\(294\) 0 0
\(295\) 8.08002 0.470437
\(296\) 0 0
\(297\) 1.10189i 0.0639381i
\(298\) 0 0
\(299\) −10.2669 3.40062i −0.593748 0.196663i
\(300\) 0 0
\(301\) −4.08705 −0.235574
\(302\) 0 0
\(303\) 3.89119i 0.223543i
\(304\) 0 0
\(305\) 0.0392191 0.00224568
\(306\) 0 0
\(307\) 20.1488i 1.14995i 0.818170 + 0.574976i \(0.194989\pi\)
−0.818170 + 0.574976i \(0.805011\pi\)
\(308\) 0 0
\(309\) 11.2588i 0.640493i
\(310\) 0 0
\(311\) 7.50388i 0.425506i 0.977106 + 0.212753i \(0.0682430\pi\)
−0.977106 + 0.212753i \(0.931757\pi\)
\(312\) 0 0
\(313\) 27.7341i 1.56763i −0.620997 0.783813i \(-0.713272\pi\)
0.620997 0.783813i \(-0.286728\pi\)
\(314\) 0 0
\(315\) 0.482745i 0.0271996i
\(316\) 0 0
\(317\) 23.7305 1.33284 0.666419 0.745577i \(-0.267826\pi\)
0.666419 + 0.745577i \(0.267826\pi\)
\(318\) 0 0
\(319\) −8.46999 −0.474228
\(320\) 0 0
\(321\) 0.801493i 0.0447350i
\(322\) 0 0
\(323\) 8.94763i 0.497859i
\(324\) 0 0
\(325\) −2.25517 −0.125094
\(326\) 0 0
\(327\) 1.25885 0.0696146
\(328\) 0 0
\(329\) 5.43724i 0.299765i
\(330\) 0 0
\(331\) 14.4675i 0.795205i 0.917558 + 0.397602i \(0.130158\pi\)
−0.917558 + 0.397602i \(0.869842\pi\)
\(332\) 0 0
\(333\) 5.83765i 0.319902i
\(334\) 0 0
\(335\) 8.39370i 0.458597i
\(336\) 0 0
\(337\) 2.09258i 0.113990i 0.998374 + 0.0569950i \(0.0181519\pi\)
−0.998374 + 0.0569950i \(0.981848\pi\)
\(338\) 0 0
\(339\) 8.70667 0.472881
\(340\) 0 0
\(341\) 5.64680i 0.305791i
\(342\) 0 0
\(343\) −6.64593 −0.358846
\(344\) 0 0
\(345\) −4.55260 1.50793i −0.245104 0.0811840i
\(346\) 0 0
\(347\) 35.3150i 1.89581i −0.318552 0.947905i \(-0.603196\pi\)
0.318552 0.947905i \(-0.396804\pi\)
\(348\) 0 0
\(349\) 25.4994 1.36495 0.682477 0.730907i \(-0.260903\pi\)
0.682477 + 0.730907i \(0.260903\pi\)
\(350\) 0 0
\(351\) 2.25517i 0.120372i
\(352\) 0 0
\(353\) −22.8722 −1.21737 −0.608683 0.793414i \(-0.708302\pi\)
−0.608683 + 0.793414i \(0.708302\pi\)
\(354\) 0 0
\(355\) −1.25598 −0.0666607
\(356\) 0 0
\(357\) 2.91353 0.154200
\(358\) 0 0
\(359\) 29.5592 1.56008 0.780038 0.625732i \(-0.215200\pi\)
0.780038 + 0.625732i \(0.215200\pi\)
\(360\) 0 0
\(361\) −16.8021 −0.884320
\(362\) 0 0
\(363\) 9.78584i 0.513623i
\(364\) 0 0
\(365\) 3.47480i 0.181879i
\(366\) 0 0
\(367\) −13.9579 −0.728598 −0.364299 0.931282i \(-0.618691\pi\)
−0.364299 + 0.931282i \(0.618691\pi\)
\(368\) 0 0
\(369\) −1.27767 −0.0665126
\(370\) 0 0
\(371\) 4.25524i 0.220921i
\(372\) 0 0
\(373\) 18.8723i 0.977170i 0.872516 + 0.488585i \(0.162487\pi\)
−0.872516 + 0.488585i \(0.837513\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −17.3350 −0.892797
\(378\) 0 0
\(379\) 15.7908 0.811121 0.405560 0.914068i \(-0.367076\pi\)
0.405560 + 0.914068i \(0.367076\pi\)
\(380\) 0 0
\(381\) 5.51836 0.282714
\(382\) 0 0
\(383\) 12.4419 0.635749 0.317875 0.948133i \(-0.397031\pi\)
0.317875 + 0.948133i \(0.397031\pi\)
\(384\) 0 0
\(385\) 0.531931i 0.0271097i
\(386\) 0 0
\(387\) 8.46629 0.430365
\(388\) 0 0
\(389\) 13.3784i 0.678312i 0.940730 + 0.339156i \(0.110142\pi\)
−0.940730 + 0.339156i \(0.889858\pi\)
\(390\) 0 0
\(391\) 9.10085 27.4765i 0.460250 1.38955i
\(392\) 0 0
\(393\) −14.5197 −0.732424
\(394\) 0 0
\(395\) 3.61831i 0.182057i
\(396\) 0 0
\(397\) 22.8329 1.14595 0.572976 0.819572i \(-0.305789\pi\)
0.572976 + 0.819572i \(0.305789\pi\)
\(398\) 0 0
\(399\) 0.715687i 0.0358292i
\(400\) 0 0
\(401\) 24.0495i 1.20098i 0.799634 + 0.600488i \(0.205027\pi\)
−0.799634 + 0.600488i \(0.794973\pi\)
\(402\) 0 0
\(403\) 11.5569i 0.575692i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 6.43245i 0.318845i
\(408\) 0 0
\(409\) −19.5572 −0.967042 −0.483521 0.875333i \(-0.660642\pi\)
−0.483521 + 0.875333i \(0.660642\pi\)
\(410\) 0 0
\(411\) 4.50448 0.222190
\(412\) 0 0
\(413\) 3.90059i 0.191935i
\(414\) 0 0
\(415\) 9.12164i 0.447764i
\(416\) 0 0
\(417\) −2.13377 −0.104491
\(418\) 0 0
\(419\) −26.0411 −1.27219 −0.636095 0.771611i \(-0.719451\pi\)
−0.636095 + 0.771611i \(0.719451\pi\)
\(420\) 0 0
\(421\) 37.9967i 1.85185i 0.377711 + 0.925924i \(0.376711\pi\)
−0.377711 + 0.925924i \(0.623289\pi\)
\(422\) 0 0
\(423\) 11.2632i 0.547635i
\(424\) 0 0
\(425\) 6.03534i 0.292757i
\(426\) 0 0
\(427\) 0.0189328i 0.000916224i
\(428\) 0 0
\(429\) 2.48494i 0.119974i
\(430\) 0 0
\(431\) 41.2449 1.98670 0.993348 0.115152i \(-0.0367354\pi\)
0.993348 + 0.115152i \(0.0367354\pi\)
\(432\) 0 0
\(433\) 2.54638i 0.122371i −0.998126 0.0611856i \(-0.980512\pi\)
0.998126 0.0611856i \(-0.0194882\pi\)
\(434\) 0 0
\(435\) −7.68679 −0.368553
\(436\) 0 0
\(437\) 6.74940 + 2.23556i 0.322868 + 0.106941i
\(438\) 0 0
\(439\) 9.22800i 0.440428i −0.975452 0.220214i \(-0.929324\pi\)
0.975452 0.220214i \(-0.0706756\pi\)
\(440\) 0 0
\(441\) 6.76696 0.322236
\(442\) 0 0
\(443\) 10.8713i 0.516511i −0.966077 0.258256i \(-0.916852\pi\)
0.966077 0.258256i \(-0.0831477\pi\)
\(444\) 0 0
\(445\) 6.85106 0.324771
\(446\) 0 0
\(447\) 11.9315 0.564341
\(448\) 0 0
\(449\) −12.3548 −0.583061 −0.291530 0.956562i \(-0.594164\pi\)
−0.291530 + 0.956562i \(0.594164\pi\)
\(450\) 0 0
\(451\) 1.40785 0.0662929
\(452\) 0 0
\(453\) −9.63555 −0.452718
\(454\) 0 0
\(455\) 1.08867i 0.0510376i
\(456\) 0 0
\(457\) 36.0224i 1.68506i 0.538651 + 0.842529i \(0.318934\pi\)
−0.538651 + 0.842529i \(0.681066\pi\)
\(458\) 0 0
\(459\) −6.03534 −0.281706
\(460\) 0 0
\(461\) −1.41328 −0.0658229 −0.0329115 0.999458i \(-0.510478\pi\)
−0.0329115 + 0.999458i \(0.510478\pi\)
\(462\) 0 0
\(463\) 22.3322i 1.03787i 0.854815 + 0.518934i \(0.173671\pi\)
−0.854815 + 0.518934i \(0.826329\pi\)
\(464\) 0 0
\(465\) 5.12465i 0.237650i
\(466\) 0 0
\(467\) −13.6777 −0.632930 −0.316465 0.948604i \(-0.602496\pi\)
−0.316465 + 0.948604i \(0.602496\pi\)
\(468\) 0 0
\(469\) −4.05201 −0.187105
\(470\) 0 0
\(471\) −0.935197 −0.0430916
\(472\) 0 0
\(473\) −9.32891 −0.428944
\(474\) 0 0
\(475\) 1.48254 0.0680235
\(476\) 0 0
\(477\) 8.81467i 0.403596i
\(478\) 0 0
\(479\) −19.9153 −0.909954 −0.454977 0.890503i \(-0.650353\pi\)
−0.454977 + 0.890503i \(0.650353\pi\)
\(480\) 0 0
\(481\) 13.1649i 0.600267i
\(482\) 0 0
\(483\) −0.727943 + 2.19774i −0.0331226 + 0.100001i
\(484\) 0 0
\(485\) 0.330286 0.0149975
\(486\) 0 0
\(487\) 36.4307i 1.65083i 0.564524 + 0.825417i \(0.309060\pi\)
−0.564524 + 0.825417i \(0.690940\pi\)
\(488\) 0 0
\(489\) −13.8165 −0.624806
\(490\) 0 0
\(491\) 18.4791i 0.833948i −0.908918 0.416974i \(-0.863091\pi\)
0.908918 0.416974i \(-0.136909\pi\)
\(492\) 0 0
\(493\) 46.3924i 2.08941i
\(494\) 0 0
\(495\) 1.10189i 0.0495263i
\(496\) 0 0
\(497\) 0.606320i 0.0271971i
\(498\) 0 0
\(499\) 11.1690i 0.499993i 0.968247 + 0.249997i \(0.0804295\pi\)
−0.968247 + 0.249997i \(0.919571\pi\)
\(500\) 0 0
\(501\) −6.69307 −0.299024
\(502\) 0 0
\(503\) −6.27100 −0.279610 −0.139805 0.990179i \(-0.544648\pi\)
−0.139805 + 0.990179i \(0.544648\pi\)
\(504\) 0 0
\(505\) 3.89119i 0.173156i
\(506\) 0 0
\(507\) 7.91423i 0.351483i
\(508\) 0 0
\(509\) 39.1536 1.73545 0.867726 0.497043i \(-0.165581\pi\)
0.867726 + 0.497043i \(0.165581\pi\)
\(510\) 0 0
\(511\) −1.67744 −0.0742056
\(512\) 0 0
\(513\) 1.48254i 0.0654557i
\(514\) 0 0
\(515\) 11.2588i 0.496124i
\(516\) 0 0
\(517\) 12.4108i 0.545826i
\(518\) 0 0
\(519\) 24.6878i 1.08368i
\(520\) 0 0
\(521\) 37.4123i 1.63906i 0.573035 + 0.819531i \(0.305766\pi\)
−0.573035 + 0.819531i \(0.694234\pi\)
\(522\) 0 0
\(523\) −14.7986 −0.647098 −0.323549 0.946211i \(-0.604876\pi\)
−0.323549 + 0.946211i \(0.604876\pi\)
\(524\) 0 0
\(525\) 0.482745i 0.0210687i
\(526\) 0 0
\(527\) −30.9290 −1.34729
\(528\) 0 0
\(529\) 18.4523 + 13.7300i 0.802275 + 0.596955i
\(530\) 0 0
\(531\) 8.08002i 0.350643i
\(532\) 0 0
\(533\) 2.88135 0.124805
\(534\) 0 0
\(535\) 0.801493i 0.0346516i
\(536\) 0 0
\(537\) 4.65679 0.200955
\(538\) 0 0
\(539\) −7.45644 −0.321172
\(540\) 0 0
\(541\) 1.29246 0.0555670 0.0277835 0.999614i \(-0.491155\pi\)
0.0277835 + 0.999614i \(0.491155\pi\)
\(542\) 0 0
\(543\) 10.4913 0.450223
\(544\) 0 0
\(545\) −1.25885 −0.0539232
\(546\) 0 0
\(547\) 12.2259i 0.522741i −0.965239 0.261370i \(-0.915826\pi\)
0.965239 0.261370i \(-0.0841744\pi\)
\(548\) 0 0
\(549\) 0.0392191i 0.00167383i
\(550\) 0 0
\(551\) 11.3960 0.485484
\(552\) 0 0
\(553\) −1.74672 −0.0742781
\(554\) 0 0
\(555\) 5.83765i 0.247795i
\(556\) 0 0
\(557\) 17.1020i 0.724637i −0.932054 0.362318i \(-0.881985\pi\)
0.932054 0.362318i \(-0.118015\pi\)
\(558\) 0 0
\(559\) −19.0929 −0.807543
\(560\) 0 0
\(561\) 6.65028 0.280775
\(562\) 0 0
\(563\) 23.9801 1.01064 0.505321 0.862931i \(-0.331374\pi\)
0.505321 + 0.862931i \(0.331374\pi\)
\(564\) 0 0
\(565\) −8.70667 −0.366292
\(566\) 0 0
\(567\) 0.482745 0.0202734
\(568\) 0 0
\(569\) 2.83032i 0.118653i 0.998239 + 0.0593265i \(0.0188953\pi\)
−0.998239 + 0.0593265i \(0.981105\pi\)
\(570\) 0 0
\(571\) −7.44436 −0.311537 −0.155768 0.987794i \(-0.549785\pi\)
−0.155768 + 0.987794i \(0.549785\pi\)
\(572\) 0 0
\(573\) 10.1702i 0.424864i
\(574\) 0 0
\(575\) 4.55260 + 1.50793i 0.189857 + 0.0628849i
\(576\) 0 0
\(577\) −10.4658 −0.435695 −0.217848 0.975983i \(-0.569904\pi\)
−0.217848 + 0.975983i \(0.569904\pi\)
\(578\) 0 0
\(579\) 13.3593i 0.555195i
\(580\) 0 0
\(581\) 4.40342 0.182685
\(582\) 0 0
\(583\) 9.71280i 0.402263i
\(584\) 0 0
\(585\) 2.25517i 0.0932396i
\(586\) 0 0
\(587\) 5.27783i 0.217839i −0.994051 0.108920i \(-0.965261\pi\)
0.994051 0.108920i \(-0.0347391\pi\)
\(588\) 0 0
\(589\) 7.59749i 0.313049i
\(590\) 0 0
\(591\) 2.65615i 0.109260i
\(592\) 0 0
\(593\) 18.1506 0.745357 0.372679 0.927960i \(-0.378439\pi\)
0.372679 + 0.927960i \(0.378439\pi\)
\(594\) 0 0
\(595\) −2.91353 −0.119443
\(596\) 0 0
\(597\) 8.43038i 0.345033i
\(598\) 0 0
\(599\) 15.0499i 0.614921i −0.951561 0.307460i \(-0.900521\pi\)
0.951561 0.307460i \(-0.0994792\pi\)
\(600\) 0 0
\(601\) 17.8769 0.729213 0.364607 0.931162i \(-0.381203\pi\)
0.364607 + 0.931162i \(0.381203\pi\)
\(602\) 0 0
\(603\) 8.39370 0.341818
\(604\) 0 0
\(605\) 9.78584i 0.397851i
\(606\) 0 0
\(607\) 16.8849i 0.685336i −0.939457 0.342668i \(-0.888669\pi\)
0.939457 0.342668i \(-0.111331\pi\)
\(608\) 0 0
\(609\) 3.71075i 0.150367i
\(610\) 0 0
\(611\) 25.4003i 1.02759i
\(612\) 0 0
\(613\) 25.0275i 1.01085i −0.862871 0.505425i \(-0.831336\pi\)
0.862871 0.505425i \(-0.168664\pi\)
\(614\) 0 0
\(615\) 1.27767 0.0515204
\(616\) 0 0
\(617\) 32.3884i 1.30391i 0.758259 + 0.651953i \(0.226050\pi\)
−0.758259 + 0.651953i \(0.773950\pi\)
\(618\) 0 0
\(619\) 34.1700 1.37341 0.686703 0.726938i \(-0.259057\pi\)
0.686703 + 0.726938i \(0.259057\pi\)
\(620\) 0 0
\(621\) 1.50793 4.55260i 0.0605110 0.182690i
\(622\) 0 0
\(623\) 3.30731i 0.132505i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.63359i 0.0652394i
\(628\) 0 0
\(629\) −35.2323 −1.40480
\(630\) 0 0
\(631\) 8.12167 0.323319 0.161659 0.986847i \(-0.448315\pi\)
0.161659 + 0.986847i \(0.448315\pi\)
\(632\) 0 0
\(633\) 0.771524 0.0306653
\(634\) 0 0
\(635\) −5.51836 −0.218990
\(636\) 0 0
\(637\) −15.2606 −0.604648
\(638\) 0 0
\(639\) 1.25598i 0.0496860i
\(640\) 0 0
\(641\) 32.8333i 1.29684i 0.761285 + 0.648418i \(0.224569\pi\)
−0.761285 + 0.648418i \(0.775431\pi\)
\(642\) 0 0
\(643\) −11.4559 −0.451776 −0.225888 0.974153i \(-0.572528\pi\)
−0.225888 + 0.974153i \(0.572528\pi\)
\(644\) 0 0
\(645\) −8.46629 −0.333360
\(646\) 0 0
\(647\) 11.8499i 0.465867i −0.972493 0.232933i \(-0.925168\pi\)
0.972493 0.232933i \(-0.0748324\pi\)
\(648\) 0 0
\(649\) 8.90329i 0.349485i
\(650\) 0 0
\(651\) 2.47390 0.0969597
\(652\) 0 0
\(653\) 19.4675 0.761821 0.380911 0.924612i \(-0.375611\pi\)
0.380911 + 0.924612i \(0.375611\pi\)
\(654\) 0 0
\(655\) 14.5197 0.567333
\(656\) 0 0
\(657\) 3.47480 0.135565
\(658\) 0 0
\(659\) −38.7108 −1.50796 −0.753980 0.656897i \(-0.771869\pi\)
−0.753980 + 0.656897i \(0.771869\pi\)
\(660\) 0 0
\(661\) 23.6248i 0.918900i −0.888204 0.459450i \(-0.848047\pi\)
0.888204 0.459450i \(-0.151953\pi\)
\(662\) 0 0
\(663\) 13.6107 0.528596
\(664\) 0 0
\(665\) 0.715687i 0.0277532i
\(666\) 0 0
\(667\) 34.9949 + 11.5911i 1.35501 + 0.448809i
\(668\) 0 0
\(669\) 19.0727 0.737393
\(670\) 0 0
\(671\) 0.0432152i 0.00166830i
\(672\) 0 0
\(673\) −41.6743 −1.60643 −0.803213 0.595692i \(-0.796878\pi\)
−0.803213 + 0.595692i \(0.796878\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 2.61727i 0.100590i 0.998734 + 0.0502948i \(0.0160161\pi\)
−0.998734 + 0.0502948i \(0.983984\pi\)
\(678\) 0 0
\(679\) 0.159444i 0.00611890i
\(680\) 0 0
\(681\) 6.95607i 0.266557i
\(682\) 0 0
\(683\) 17.4757i 0.668688i −0.942451 0.334344i \(-0.891485\pi\)
0.942451 0.334344i \(-0.108515\pi\)
\(684\) 0 0
\(685\) −4.50448 −0.172107
\(686\) 0 0
\(687\) 1.27163 0.0485157
\(688\) 0 0
\(689\) 19.8786i 0.757313i
\(690\) 0 0
\(691\) 33.9569i 1.29178i 0.763430 + 0.645890i \(0.223514\pi\)
−0.763430 + 0.645890i \(0.776486\pi\)
\(692\) 0 0
\(693\) −0.531931 −0.0202064
\(694\) 0 0
\(695\) 2.13377 0.0809386
\(696\) 0 0
\(697\) 7.71115i 0.292081i
\(698\) 0 0
\(699\) 9.90098i 0.374489i
\(700\) 0 0
\(701\) 17.9294i 0.677183i 0.940933 + 0.338592i \(0.109951\pi\)
−0.940933 + 0.338592i \(0.890049\pi\)
\(702\) 0 0
\(703\) 8.65455i 0.326412i
\(704\) 0 0
\(705\) 11.2632i 0.424196i
\(706\) 0 0
\(707\) −1.87845 −0.0706464
\(708\) 0 0
\(709\) 23.2328i 0.872526i 0.899819 + 0.436263i \(0.143698\pi\)
−0.899819 + 0.436263i \(0.856302\pi\)
\(710\) 0 0
\(711\) 3.61831 0.135697
\(712\) 0 0
\(713\) 7.72759 23.3305i 0.289401 0.873733i
\(714\) 0 0
\(715\) 2.48494i 0.0929317i
\(716\) 0 0
\(717\) 10.3420 0.386229
\(718\) 0 0
\(719\) 34.5036i 1.28677i −0.765544 0.643384i \(-0.777530\pi\)
0.765544 0.643384i \(-0.222470\pi\)
\(720\) 0 0
\(721\) 5.43514 0.202415
\(722\) 0 0
\(723\) 10.9888 0.408678
\(724\) 0 0
\(725\) 7.68679 0.285480
\(726\) 0 0
\(727\) −45.7234 −1.69579 −0.847893 0.530168i \(-0.822129\pi\)
−0.847893 + 0.530168i \(0.822129\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 51.0969i 1.88989i
\(732\) 0 0
\(733\) 4.56648i 0.168667i −0.996438 0.0843333i \(-0.973124\pi\)
0.996438 0.0843333i \(-0.0268760\pi\)
\(734\) 0 0
\(735\) −6.76696 −0.249603
\(736\) 0 0
\(737\) −9.24893 −0.340689
\(738\) 0 0
\(739\) 8.71551i 0.320605i 0.987068 + 0.160303i \(0.0512470\pi\)
−0.987068 + 0.160303i \(0.948753\pi\)
\(740\) 0 0
\(741\) 3.34337i 0.122822i
\(742\) 0 0
\(743\) −39.8622 −1.46240 −0.731202 0.682161i \(-0.761040\pi\)
−0.731202 + 0.682161i \(0.761040\pi\)
\(744\) 0 0
\(745\) −11.9315 −0.437137
\(746\) 0 0
\(747\) −9.12164 −0.333743
\(748\) 0 0
\(749\) −0.386917 −0.0141376
\(750\) 0 0
\(751\) −6.71643 −0.245086 −0.122543 0.992463i \(-0.539105\pi\)
−0.122543 + 0.992463i \(0.539105\pi\)
\(752\) 0 0
\(753\) 8.22629i 0.299783i
\(754\) 0 0
\(755\) 9.63555 0.350674
\(756\) 0 0
\(757\) 39.8926i 1.44992i −0.688790 0.724961i \(-0.741858\pi\)
0.688790 0.724961i \(-0.258142\pi\)
\(758\) 0 0
\(759\) −1.66157 + 5.01646i −0.0603111 + 0.182086i
\(760\) 0 0
\(761\) 2.24308 0.0813116 0.0406558 0.999173i \(-0.487055\pi\)
0.0406558 + 0.999173i \(0.487055\pi\)
\(762\) 0 0
\(763\) 0.607703i 0.0220003i
\(764\) 0 0
\(765\) 6.03534 0.218208
\(766\) 0 0
\(767\) 18.2218i 0.657950i
\(768\) 0 0
\(769\) 26.5172i 0.956233i 0.878296 + 0.478117i \(0.158680\pi\)
−0.878296 + 0.478117i \(0.841320\pi\)
\(770\) 0 0
\(771\) 30.1958i 1.08748i
\(772\) 0 0
\(773\) 49.8502i 1.79299i −0.443057 0.896493i \(-0.646106\pi\)
0.443057 0.896493i \(-0.353894\pi\)
\(774\) 0 0
\(775\) 5.12465i 0.184083i
\(776\) 0 0
\(777\) 2.81810 0.101099
\(778\) 0 0
\(779\) −1.89419 −0.0678663
\(780\) 0 0
\(781\) 1.38396i 0.0495218i
\(782\) 0 0
\(783\) 7.68679i 0.274703i
\(784\) 0 0
\(785\) 0.935197 0.0333786
\(786\) 0 0
\(787\) 25.8578 0.921732 0.460866 0.887470i \(-0.347539\pi\)
0.460866 + 0.887470i \(0.347539\pi\)
\(788\) 0 0
\(789\) 19.7029i 0.701441i
\(790\) 0 0
\(791\) 4.20310i 0.149445i
\(792\) 0 0
\(793\) 0.0884456i 0.00314080i
\(794\) 0 0
\(795\) 8.81467i 0.312624i
\(796\) 0 0
\(797\) 5.23771i 0.185529i 0.995688 + 0.0927647i \(0.0295704\pi\)
−0.995688 + 0.0927647i \(0.970430\pi\)
\(798\) 0 0
\(799\) 67.9772 2.40486
\(800\) 0 0
\(801\) 6.85106i 0.242070i
\(802\) 0 0
\(803\) −3.82884 −0.135117
\(804\) 0 0