Properties

Label 1520.2.d.g.609.4
Level $1520$
Weight $2$
Character 1520.609
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.g.609.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} -0.828427i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{3} +(1.00000 + 2.00000i) q^{5} -0.828427i q^{7} +1.00000 q^{9} +0.828427 q^{11} +3.41421i q^{13} +(-2.82843 + 1.41421i) q^{15} +2.82843i q^{17} -1.00000 q^{19} +1.17157 q^{21} -3.65685i q^{23} +(-3.00000 + 4.00000i) q^{25} +5.65685i q^{27} +7.65685 q^{29} -1.17157 q^{31} +1.17157i q^{33} +(1.65685 - 0.828427i) q^{35} +3.41421i q^{37} -4.82843 q^{39} -4.82843 q^{41} -3.17157i q^{43} +(1.00000 + 2.00000i) q^{45} +4.82843i q^{47} +6.31371 q^{49} -4.00000 q^{51} -7.89949i q^{53} +(0.828427 + 1.65685i) q^{55} -1.41421i q^{57} -1.17157 q^{59} +5.65685 q^{61} -0.828427i q^{63} +(-6.82843 + 3.41421i) q^{65} +9.89949i q^{67} +5.17157 q^{69} -8.48528 q^{71} +6.82843i q^{73} +(-5.65685 - 4.24264i) q^{75} -0.686292i q^{77} -8.48528 q^{79} -5.00000 q^{81} +4.34315i q^{83} +(-5.65685 + 2.82843i) q^{85} +10.8284i q^{87} -9.31371 q^{89} +2.82843 q^{91} -1.65685i q^{93} +(-1.00000 - 2.00000i) q^{95} -13.0711i q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{5} + 4 q^{9} - 8 q^{11} - 4 q^{19} + 16 q^{21} - 12 q^{25} + 8 q^{29} - 16 q^{31} - 16 q^{35} - 8 q^{39} - 8 q^{41} + 4 q^{45} - 20 q^{49} - 16 q^{51} - 8 q^{55} - 16 q^{59} - 16 q^{65} + 32 q^{69} - 20 q^{81} + 8 q^{89} - 4 q^{95} - 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.41421i 0.816497i 0.912871 + 0.408248i \(0.133860\pi\)
−0.912871 + 0.408248i \(0.866140\pi\)
\(4\) 0 0
\(5\) 1.00000 + 2.00000i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 0.828427i 0.313116i −0.987669 0.156558i \(-0.949960\pi\)
0.987669 0.156558i \(-0.0500398\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 3.41421i 0.946932i 0.880812 + 0.473466i \(0.156997\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(14\) 0 0
\(15\) −2.82843 + 1.41421i −0.730297 + 0.365148i
\(16\) 0 0
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.17157 0.255658
\(22\) 0 0
\(23\) 3.65685i 0.762507i −0.924471 0.381253i \(-0.875493\pi\)
0.924471 0.381253i \(-0.124507\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 0 0
\(33\) 1.17157i 0.203945i
\(34\) 0 0
\(35\) 1.65685 0.828427i 0.280059 0.140030i
\(36\) 0 0
\(37\) 3.41421i 0.561293i 0.959811 + 0.280647i \(0.0905489\pi\)
−0.959811 + 0.280647i \(0.909451\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) −4.82843 −0.754074 −0.377037 0.926198i \(-0.623057\pi\)
−0.377037 + 0.926198i \(0.623057\pi\)
\(42\) 0 0
\(43\) 3.17157i 0.483660i −0.970319 0.241830i \(-0.922252\pi\)
0.970319 0.241830i \(-0.0777477\pi\)
\(44\) 0 0
\(45\) 1.00000 + 2.00000i 0.149071 + 0.298142i
\(46\) 0 0
\(47\) 4.82843i 0.704298i 0.935944 + 0.352149i \(0.114549\pi\)
−0.935944 + 0.352149i \(0.885451\pi\)
\(48\) 0 0
\(49\) 6.31371 0.901958
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 7.89949i 1.08508i −0.840030 0.542540i \(-0.817463\pi\)
0.840030 0.542540i \(-0.182537\pi\)
\(54\) 0 0
\(55\) 0.828427 + 1.65685i 0.111705 + 0.223410i
\(56\) 0 0
\(57\) 1.41421i 0.187317i
\(58\) 0 0
\(59\) −1.17157 −0.152526 −0.0762629 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) 0 0
\(63\) 0.828427i 0.104372i
\(64\) 0 0
\(65\) −6.82843 + 3.41421i −0.846962 + 0.423481i
\(66\) 0 0
\(67\) 9.89949i 1.20942i 0.796447 + 0.604708i \(0.206710\pi\)
−0.796447 + 0.604708i \(0.793290\pi\)
\(68\) 0 0
\(69\) 5.17157 0.622584
\(70\) 0 0
\(71\) −8.48528 −1.00702 −0.503509 0.863990i \(-0.667958\pi\)
−0.503509 + 0.863990i \(0.667958\pi\)
\(72\) 0 0
\(73\) 6.82843i 0.799207i 0.916688 + 0.399603i \(0.130852\pi\)
−0.916688 + 0.399603i \(0.869148\pi\)
\(74\) 0 0
\(75\) −5.65685 4.24264i −0.653197 0.489898i
\(76\) 0 0
\(77\) 0.686292i 0.0782102i
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 4.34315i 0.476722i 0.971177 + 0.238361i \(0.0766102\pi\)
−0.971177 + 0.238361i \(0.923390\pi\)
\(84\) 0 0
\(85\) −5.65685 + 2.82843i −0.613572 + 0.306786i
\(86\) 0 0
\(87\) 10.8284i 1.16093i
\(88\) 0 0
\(89\) −9.31371 −0.987251 −0.493626 0.869675i \(-0.664329\pi\)
−0.493626 + 0.869675i \(0.664329\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 1.65685i 0.171808i
\(94\) 0 0
\(95\) −1.00000 2.00000i −0.102598 0.205196i
\(96\) 0 0
\(97\) 13.0711i 1.32717i −0.748103 0.663583i \(-0.769035\pi\)
0.748103 0.663583i \(-0.230965\pi\)
\(98\) 0 0
\(99\) 0.828427 0.0832601
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i −0.873004 0.487713i \(-0.837831\pi\)
0.873004 0.487713i \(-0.162169\pi\)
\(104\) 0 0
\(105\) 1.17157 + 2.34315i 0.114334 + 0.228668i
\(106\) 0 0
\(107\) 7.75736i 0.749932i 0.927039 + 0.374966i \(0.122346\pi\)
−0.927039 + 0.374966i \(0.877654\pi\)
\(108\) 0 0
\(109\) −6.48528 −0.621177 −0.310589 0.950544i \(-0.600526\pi\)
−0.310589 + 0.950544i \(0.600526\pi\)
\(110\) 0 0
\(111\) −4.82843 −0.458294
\(112\) 0 0
\(113\) 8.58579i 0.807683i 0.914829 + 0.403841i \(0.132325\pi\)
−0.914829 + 0.403841i \(0.867675\pi\)
\(114\) 0 0
\(115\) 7.31371 3.65685i 0.682007 0.341003i
\(116\) 0 0
\(117\) 3.41421i 0.315644i
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) 6.82843i 0.615699i
\(124\) 0 0
\(125\) −11.0000 2.00000i −0.983870 0.178885i
\(126\) 0 0
\(127\) 12.7279i 1.12942i −0.825289 0.564710i \(-0.808988\pi\)
0.825289 0.564710i \(-0.191012\pi\)
\(128\) 0 0
\(129\) 4.48528 0.394907
\(130\) 0 0
\(131\) 6.34315 0.554203 0.277102 0.960841i \(-0.410626\pi\)
0.277102 + 0.960841i \(0.410626\pi\)
\(132\) 0 0
\(133\) 0.828427i 0.0718337i
\(134\) 0 0
\(135\) −11.3137 + 5.65685i −0.973729 + 0.486864i
\(136\) 0 0
\(137\) 2.82843i 0.241649i 0.992674 + 0.120824i \(0.0385538\pi\)
−0.992674 + 0.120824i \(0.961446\pi\)
\(138\) 0 0
\(139\) 20.8284 1.76664 0.883322 0.468767i \(-0.155301\pi\)
0.883322 + 0.468767i \(0.155301\pi\)
\(140\) 0 0
\(141\) −6.82843 −0.575057
\(142\) 0 0
\(143\) 2.82843i 0.236525i
\(144\) 0 0
\(145\) 7.65685 + 15.3137i 0.635867 + 1.27173i
\(146\) 0 0
\(147\) 8.92893i 0.736446i
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 13.6569 1.11138 0.555690 0.831390i \(-0.312454\pi\)
0.555690 + 0.831390i \(0.312454\pi\)
\(152\) 0 0
\(153\) 2.82843i 0.228665i
\(154\) 0 0
\(155\) −1.17157 2.34315i −0.0941030 0.188206i
\(156\) 0 0
\(157\) 12.4853i 0.996434i −0.867052 0.498217i \(-0.833988\pi\)
0.867052 0.498217i \(-0.166012\pi\)
\(158\) 0 0
\(159\) 11.1716 0.885963
\(160\) 0 0
\(161\) −3.02944 −0.238753
\(162\) 0 0
\(163\) 2.48528i 0.194662i 0.995252 + 0.0973311i \(0.0310306\pi\)
−0.995252 + 0.0973311i \(0.968969\pi\)
\(164\) 0 0
\(165\) −2.34315 + 1.17157i −0.182414 + 0.0912068i
\(166\) 0 0
\(167\) 11.0711i 0.856705i 0.903612 + 0.428352i \(0.140906\pi\)
−0.903612 + 0.428352i \(0.859094\pi\)
\(168\) 0 0
\(169\) 1.34315 0.103319
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 11.4142i 0.867807i −0.900959 0.433903i \(-0.857136\pi\)
0.900959 0.433903i \(-0.142864\pi\)
\(174\) 0 0
\(175\) 3.31371 + 2.48528i 0.250493 + 0.187870i
\(176\) 0 0
\(177\) 1.65685i 0.124537i
\(178\) 0 0
\(179\) 17.1716 1.28346 0.641732 0.766929i \(-0.278216\pi\)
0.641732 + 0.766929i \(0.278216\pi\)
\(180\) 0 0
\(181\) 25.3137 1.88155 0.940777 0.339027i \(-0.110098\pi\)
0.940777 + 0.339027i \(0.110098\pi\)
\(182\) 0 0
\(183\) 8.00000i 0.591377i
\(184\) 0 0
\(185\) −6.82843 + 3.41421i −0.502036 + 0.251018i
\(186\) 0 0
\(187\) 2.34315i 0.171348i
\(188\) 0 0
\(189\) 4.68629 0.340878
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 0 0
\(193\) 13.0711i 0.940876i 0.882433 + 0.470438i \(0.155904\pi\)
−0.882433 + 0.470438i \(0.844096\pi\)
\(194\) 0 0
\(195\) −4.82843 9.65685i −0.345771 0.691542i
\(196\) 0 0
\(197\) 21.6569i 1.54299i −0.636237 0.771493i \(-0.719510\pi\)
0.636237 0.771493i \(-0.280490\pi\)
\(198\) 0 0
\(199\) 11.3137 0.802008 0.401004 0.916076i \(-0.368661\pi\)
0.401004 + 0.916076i \(0.368661\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) 6.34315i 0.445202i
\(204\) 0 0
\(205\) −4.82843 9.65685i −0.337232 0.674464i
\(206\) 0 0
\(207\) 3.65685i 0.254169i
\(208\) 0 0
\(209\) −0.828427 −0.0573035
\(210\) 0 0
\(211\) −22.6274 −1.55774 −0.778868 0.627188i \(-0.784206\pi\)
−0.778868 + 0.627188i \(0.784206\pi\)
\(212\) 0 0
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) 6.34315 3.17157i 0.432599 0.216299i
\(216\) 0 0
\(217\) 0.970563i 0.0658861i
\(218\) 0 0
\(219\) −9.65685 −0.652550
\(220\) 0 0
\(221\) −9.65685 −0.649590
\(222\) 0 0
\(223\) 9.89949i 0.662919i 0.943469 + 0.331460i \(0.107541\pi\)
−0.943469 + 0.331460i \(0.892459\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 19.0711i 1.26579i 0.774237 + 0.632896i \(0.218134\pi\)
−0.774237 + 0.632896i \(0.781866\pi\)
\(228\) 0 0
\(229\) 19.3137 1.27629 0.638143 0.769918i \(-0.279703\pi\)
0.638143 + 0.769918i \(0.279703\pi\)
\(230\) 0 0
\(231\) 0.970563 0.0638583
\(232\) 0 0
\(233\) 4.97056i 0.325632i −0.986656 0.162816i \(-0.947942\pi\)
0.986656 0.162816i \(-0.0520578\pi\)
\(234\) 0 0
\(235\) −9.65685 + 4.82843i −0.629944 + 0.314972i
\(236\) 0 0
\(237\) 12.0000i 0.779484i
\(238\) 0 0
\(239\) 24.9706 1.61521 0.807606 0.589723i \(-0.200763\pi\)
0.807606 + 0.589723i \(0.200763\pi\)
\(240\) 0 0
\(241\) 8.82843 0.568689 0.284344 0.958722i \(-0.408224\pi\)
0.284344 + 0.958722i \(0.408224\pi\)
\(242\) 0 0
\(243\) 9.89949i 0.635053i
\(244\) 0 0
\(245\) 6.31371 + 12.6274i 0.403368 + 0.806736i
\(246\) 0 0
\(247\) 3.41421i 0.217241i
\(248\) 0 0
\(249\) −6.14214 −0.389242
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 3.02944i 0.190459i
\(254\) 0 0
\(255\) −4.00000 8.00000i −0.250490 0.500979i
\(256\) 0 0
\(257\) 1.75736i 0.109621i 0.998497 + 0.0548105i \(0.0174555\pi\)
−0.998497 + 0.0548105i \(0.982545\pi\)
\(258\) 0 0
\(259\) 2.82843 0.175750
\(260\) 0 0
\(261\) 7.65685 0.473947
\(262\) 0 0
\(263\) 17.3137i 1.06761i −0.845608 0.533805i \(-0.820762\pi\)
0.845608 0.533805i \(-0.179238\pi\)
\(264\) 0 0
\(265\) 15.7990 7.89949i 0.970524 0.485262i
\(266\) 0 0
\(267\) 13.1716i 0.806087i
\(268\) 0 0
\(269\) 8.14214 0.496435 0.248217 0.968704i \(-0.420155\pi\)
0.248217 + 0.968704i \(0.420155\pi\)
\(270\) 0 0
\(271\) −24.8284 −1.50822 −0.754110 0.656748i \(-0.771931\pi\)
−0.754110 + 0.656748i \(0.771931\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) −2.48528 + 3.31371i −0.149868 + 0.199824i
\(276\) 0 0
\(277\) 14.8284i 0.890954i −0.895293 0.445477i \(-0.853034\pi\)
0.895293 0.445477i \(-0.146966\pi\)
\(278\) 0 0
\(279\) −1.17157 −0.0701402
\(280\) 0 0
\(281\) −5.51472 −0.328981 −0.164490 0.986379i \(-0.552598\pi\)
−0.164490 + 0.986379i \(0.552598\pi\)
\(282\) 0 0
\(283\) 30.9706i 1.84101i −0.390732 0.920504i \(-0.627778\pi\)
0.390732 0.920504i \(-0.372222\pi\)
\(284\) 0 0
\(285\) 2.82843 1.41421i 0.167542 0.0837708i
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) 18.4853 1.08363
\(292\) 0 0
\(293\) 18.2426i 1.06575i −0.846195 0.532873i \(-0.821112\pi\)
0.846195 0.532873i \(-0.178888\pi\)
\(294\) 0 0
\(295\) −1.17157 2.34315i −0.0682116 0.136423i
\(296\) 0 0
\(297\) 4.68629i 0.271926i
\(298\) 0 0
\(299\) 12.4853 0.722042
\(300\) 0 0
\(301\) −2.62742 −0.151442
\(302\) 0 0
\(303\) 11.3137i 0.649956i
\(304\) 0 0
\(305\) 5.65685 + 11.3137i 0.323911 + 0.647821i
\(306\) 0 0
\(307\) 5.41421i 0.309005i 0.987992 + 0.154503i \(0.0493775\pi\)
−0.987992 + 0.154503i \(0.950622\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 28.8284 1.63471 0.817355 0.576134i \(-0.195439\pi\)
0.817355 + 0.576134i \(0.195439\pi\)
\(312\) 0 0
\(313\) 12.9706i 0.733140i −0.930391 0.366570i \(-0.880532\pi\)
0.930391 0.366570i \(-0.119468\pi\)
\(314\) 0 0
\(315\) 1.65685 0.828427i 0.0933532 0.0466766i
\(316\) 0 0
\(317\) 14.7279i 0.827203i −0.910458 0.413601i \(-0.864271\pi\)
0.910458 0.413601i \(-0.135729\pi\)
\(318\) 0 0
\(319\) 6.34315 0.355148
\(320\) 0 0
\(321\) −10.9706 −0.612317
\(322\) 0 0
\(323\) 2.82843i 0.157378i
\(324\) 0 0
\(325\) −13.6569 10.2426i −0.757546 0.568159i
\(326\) 0 0
\(327\) 9.17157i 0.507189i
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 2.34315 0.128791 0.0643955 0.997924i \(-0.479488\pi\)
0.0643955 + 0.997924i \(0.479488\pi\)
\(332\) 0 0
\(333\) 3.41421i 0.187098i
\(334\) 0 0
\(335\) −19.7990 + 9.89949i −1.08173 + 0.540867i
\(336\) 0 0
\(337\) 3.89949i 0.212419i −0.994344 0.106210i \(-0.966129\pi\)
0.994344 0.106210i \(-0.0338714\pi\)
\(338\) 0 0
\(339\) −12.1421 −0.659470
\(340\) 0 0
\(341\) −0.970563 −0.0525589
\(342\) 0 0
\(343\) 11.0294i 0.595534i
\(344\) 0 0
\(345\) 5.17157 + 10.3431i 0.278428 + 0.556856i
\(346\) 0 0
\(347\) 18.0000i 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) 16.6274 0.890045 0.445023 0.895519i \(-0.353196\pi\)
0.445023 + 0.895519i \(0.353196\pi\)
\(350\) 0 0
\(351\) −19.3137 −1.03089
\(352\) 0 0
\(353\) 14.1421i 0.752710i 0.926476 + 0.376355i \(0.122823\pi\)
−0.926476 + 0.376355i \(0.877177\pi\)
\(354\) 0 0
\(355\) −8.48528 16.9706i −0.450352 0.900704i
\(356\) 0 0
\(357\) 3.31371i 0.175380i
\(358\) 0 0
\(359\) 6.48528 0.342280 0.171140 0.985247i \(-0.445255\pi\)
0.171140 + 0.985247i \(0.445255\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 14.5858i 0.765555i
\(364\) 0 0
\(365\) −13.6569 + 6.82843i −0.714832 + 0.357416i
\(366\) 0 0
\(367\) 8.14214i 0.425016i −0.977159 0.212508i \(-0.931837\pi\)
0.977159 0.212508i \(-0.0681632\pi\)
\(368\) 0 0
\(369\) −4.82843 −0.251358
\(370\) 0 0
\(371\) −6.54416 −0.339756
\(372\) 0 0
\(373\) 8.10051i 0.419428i 0.977763 + 0.209714i \(0.0672533\pi\)
−0.977763 + 0.209714i \(0.932747\pi\)
\(374\) 0 0
\(375\) 2.82843 15.5563i 0.146059 0.803326i
\(376\) 0 0
\(377\) 26.1421i 1.34639i
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) 10.5858i 0.540908i 0.962733 + 0.270454i \(0.0871739\pi\)
−0.962733 + 0.270454i \(0.912826\pi\)
\(384\) 0 0
\(385\) 1.37258 0.686292i 0.0699533 0.0349767i
\(386\) 0 0
\(387\) 3.17157i 0.161220i
\(388\) 0 0
\(389\) −13.3137 −0.675032 −0.337516 0.941320i \(-0.609587\pi\)
−0.337516 + 0.941320i \(0.609587\pi\)
\(390\) 0 0
\(391\) 10.3431 0.523075
\(392\) 0 0
\(393\) 8.97056i 0.452505i
\(394\) 0 0
\(395\) −8.48528 16.9706i −0.426941 0.853882i
\(396\) 0 0
\(397\) 18.8284i 0.944972i −0.881338 0.472486i \(-0.843357\pi\)
0.881338 0.472486i \(-0.156643\pi\)
\(398\) 0 0
\(399\) −1.17157 −0.0586520
\(400\) 0 0
\(401\) 6.97056 0.348093 0.174047 0.984737i \(-0.444316\pi\)
0.174047 + 0.984737i \(0.444316\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) −5.00000 10.0000i −0.248452 0.496904i
\(406\) 0 0
\(407\) 2.82843i 0.140200i
\(408\) 0 0
\(409\) 31.1716 1.54134 0.770668 0.637237i \(-0.219923\pi\)
0.770668 + 0.637237i \(0.219923\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) 0.970563i 0.0477583i
\(414\) 0 0
\(415\) −8.68629 + 4.34315i −0.426393 + 0.213197i
\(416\) 0 0
\(417\) 29.4558i 1.44246i
\(418\) 0 0
\(419\) 34.6274 1.69166 0.845830 0.533453i \(-0.179106\pi\)
0.845830 + 0.533453i \(0.179106\pi\)
\(420\) 0 0
\(421\) 37.7990 1.84221 0.921105 0.389314i \(-0.127288\pi\)
0.921105 + 0.389314i \(0.127288\pi\)
\(422\) 0 0
\(423\) 4.82843i 0.234766i
\(424\) 0 0
\(425\) −11.3137 8.48528i −0.548795 0.411597i
\(426\) 0 0
\(427\) 4.68629i 0.226786i
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −37.9411 −1.82756 −0.913780 0.406210i \(-0.866850\pi\)
−0.913780 + 0.406210i \(0.866850\pi\)
\(432\) 0 0
\(433\) 6.44365i 0.309662i −0.987941 0.154831i \(-0.950517\pi\)
0.987941 0.154831i \(-0.0494833\pi\)
\(434\) 0 0
\(435\) −21.6569 + 10.8284i −1.03837 + 0.519183i
\(436\) 0 0
\(437\) 3.65685i 0.174931i
\(438\) 0 0
\(439\) 11.5147 0.549568 0.274784 0.961506i \(-0.411394\pi\)
0.274784 + 0.961506i \(0.411394\pi\)
\(440\) 0 0
\(441\) 6.31371 0.300653
\(442\) 0 0
\(443\) 28.6274i 1.36013i 0.733152 + 0.680065i \(0.238048\pi\)
−0.733152 + 0.680065i \(0.761952\pi\)
\(444\) 0 0
\(445\) −9.31371 18.6274i −0.441512 0.883024i
\(446\) 0 0
\(447\) 5.65685i 0.267560i
\(448\) 0 0
\(449\) 9.51472 0.449027 0.224514 0.974471i \(-0.427921\pi\)
0.224514 + 0.974471i \(0.427921\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 19.3137i 0.907437i
\(454\) 0 0
\(455\) 2.82843 + 5.65685i 0.132599 + 0.265197i
\(456\) 0 0
\(457\) 24.9706i 1.16807i −0.811727 0.584037i \(-0.801472\pi\)
0.811727 0.584037i \(-0.198528\pi\)
\(458\) 0 0
\(459\) −16.0000 −0.746816
\(460\) 0 0
\(461\) 17.3137 0.806380 0.403190 0.915116i \(-0.367901\pi\)
0.403190 + 0.915116i \(0.367901\pi\)
\(462\) 0 0
\(463\) 2.97056i 0.138054i 0.997615 + 0.0690269i \(0.0219894\pi\)
−0.997615 + 0.0690269i \(0.978011\pi\)
\(464\) 0 0
\(465\) 3.31371 1.65685i 0.153670 0.0768348i
\(466\) 0 0
\(467\) 8.34315i 0.386075i 0.981191 + 0.193037i \(0.0618339\pi\)
−0.981191 + 0.193037i \(0.938166\pi\)
\(468\) 0 0
\(469\) 8.20101 0.378687
\(470\) 0 0
\(471\) 17.6569 0.813585
\(472\) 0 0
\(473\) 2.62742i 0.120809i
\(474\) 0 0
\(475\) 3.00000 4.00000i 0.137649 0.183533i
\(476\) 0 0
\(477\) 7.89949i 0.361693i
\(478\) 0 0
\(479\) −7.17157 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(480\) 0 0
\(481\) −11.6569 −0.531507
\(482\) 0 0
\(483\) 4.28427i 0.194941i
\(484\) 0 0
\(485\) 26.1421 13.0711i 1.18705 0.593527i
\(486\) 0 0
\(487\) 32.7279i 1.48304i −0.670929 0.741522i \(-0.734104\pi\)
0.670929 0.741522i \(-0.265896\pi\)
\(488\) 0 0
\(489\) −3.51472 −0.158941
\(490\) 0 0
\(491\) −17.6569 −0.796843 −0.398421 0.917203i \(-0.630442\pi\)
−0.398421 + 0.917203i \(0.630442\pi\)
\(492\) 0 0
\(493\) 21.6569i 0.975376i
\(494\) 0 0
\(495\) 0.828427 + 1.65685i 0.0372350 + 0.0744701i
\(496\) 0 0
\(497\) 7.02944i 0.315313i
\(498\) 0 0
\(499\) −35.4558 −1.58722 −0.793611 0.608426i \(-0.791801\pi\)
−0.793611 + 0.608426i \(0.791801\pi\)
\(500\) 0 0
\(501\) −15.6569 −0.699497
\(502\) 0 0
\(503\) 26.0000i 1.15928i 0.814872 + 0.579641i \(0.196807\pi\)
−0.814872 + 0.579641i \(0.803193\pi\)
\(504\) 0 0
\(505\) −8.00000 16.0000i −0.355995 0.711991i
\(506\) 0 0
\(507\) 1.89949i 0.0843595i
\(508\) 0 0
\(509\) −23.9411 −1.06117 −0.530586 0.847631i \(-0.678028\pi\)
−0.530586 + 0.847631i \(0.678028\pi\)
\(510\) 0 0
\(511\) 5.65685 0.250244
\(512\) 0 0
\(513\) 5.65685i 0.249756i
\(514\) 0 0
\(515\) 19.7990 9.89949i 0.872448 0.436224i
\(516\) 0 0
\(517\) 4.00000i 0.175920i
\(518\) 0 0
\(519\) 16.1421 0.708561
\(520\) 0 0
\(521\) −35.6569 −1.56216 −0.781078 0.624434i \(-0.785330\pi\)
−0.781078 + 0.624434i \(0.785330\pi\)
\(522\) 0 0
\(523\) 37.4142i 1.63601i −0.575212 0.818005i \(-0.695080\pi\)
0.575212 0.818005i \(-0.304920\pi\)
\(524\) 0 0
\(525\) −3.51472 + 4.68629i −0.153395 + 0.204527i
\(526\) 0 0
\(527\) 3.31371i 0.144347i
\(528\) 0 0
\(529\) 9.62742 0.418583
\(530\) 0 0
\(531\) −1.17157 −0.0508419
\(532\) 0 0
\(533\) 16.4853i 0.714057i
\(534\) 0 0
\(535\) −15.5147 + 7.75736i −0.670760 + 0.335380i
\(536\) 0 0
\(537\) 24.2843i 1.04794i
\(538\) 0 0
\(539\) 5.23045 0.225291
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 0 0
\(543\) 35.7990i 1.53628i
\(544\) 0 0
\(545\) −6.48528 12.9706i −0.277799 0.555598i
\(546\) 0 0
\(547\) 0.242641i 0.0103746i 0.999987 + 0.00518728i \(0.00165117\pi\)
−0.999987 + 0.00518728i \(0.998349\pi\)
\(548\) 0 0
\(549\) 5.65685 0.241429
\(550\) 0 0
\(551\) −7.65685 −0.326193
\(552\) 0 0
\(553\) 7.02944i 0.298922i
\(554\) 0 0
\(555\) −4.82843 9.65685i −0.204955 0.409911i
\(556\) 0 0
\(557\) 9.17157i 0.388612i −0.980941 0.194306i \(-0.937755\pi\)
0.980941 0.194306i \(-0.0622455\pi\)
\(558\) 0 0
\(559\) 10.8284 0.457994
\(560\) 0 0
\(561\) −3.31371 −0.139905
\(562\) 0 0
\(563\) 42.5858i 1.79478i 0.441241 + 0.897388i \(0.354538\pi\)
−0.441241 + 0.897388i \(0.645462\pi\)
\(564\) 0 0
\(565\) −17.1716 + 8.58579i −0.722414 + 0.361207i
\(566\) 0 0
\(567\) 4.14214i 0.173953i
\(568\) 0 0
\(569\) 39.4558 1.65408 0.827038 0.562147i \(-0.190024\pi\)
0.827038 + 0.562147i \(0.190024\pi\)
\(570\) 0 0
\(571\) −3.45584 −0.144623 −0.0723113 0.997382i \(-0.523038\pi\)
−0.0723113 + 0.997382i \(0.523038\pi\)
\(572\) 0 0
\(573\) 4.68629i 0.195773i
\(574\) 0 0
\(575\) 14.6274 + 10.9706i 0.610005 + 0.457504i
\(576\) 0 0
\(577\) 17.6569i 0.735064i −0.930011 0.367532i \(-0.880203\pi\)
0.930011 0.367532i \(-0.119797\pi\)
\(578\) 0 0
\(579\) −18.4853 −0.768222
\(580\) 0 0
\(581\) 3.59798 0.149269
\(582\) 0 0
\(583\) 6.54416i 0.271031i
\(584\) 0 0
\(585\) −6.82843 + 3.41421i −0.282321 + 0.141160i
\(586\) 0 0
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) 1.17157 0.0482738
\(590\) 0 0
\(591\) 30.6274 1.25984
\(592\) 0 0
\(593\) 30.6274i 1.25772i 0.777520 + 0.628859i \(0.216478\pi\)
−0.777520 + 0.628859i \(0.783522\pi\)
\(594\) 0 0
\(595\) 2.34315 + 4.68629i 0.0960596 + 0.192119i
\(596\) 0 0
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) −5.45584 −0.222920 −0.111460 0.993769i \(-0.535553\pi\)
−0.111460 + 0.993769i \(0.535553\pi\)
\(600\) 0 0
\(601\) −40.1421 −1.63743 −0.818716 0.574199i \(-0.805314\pi\)
−0.818716 + 0.574199i \(0.805314\pi\)
\(602\) 0 0
\(603\) 9.89949i 0.403139i
\(604\) 0 0
\(605\) −10.3137 20.6274i −0.419312 0.838624i
\(606\) 0 0
\(607\) 1.21320i 0.0492424i 0.999697 + 0.0246212i \(0.00783796\pi\)
−0.999697 + 0.0246212i \(0.992162\pi\)
\(608\) 0 0
\(609\) 8.97056 0.363506
\(610\) 0 0
\(611\) −16.4853 −0.666923
\(612\) 0 0
\(613\) 7.51472i 0.303517i −0.988418 0.151758i \(-0.951506\pi\)
0.988418 0.151758i \(-0.0484935\pi\)
\(614\) 0 0
\(615\) 13.6569 6.82843i 0.550698 0.275349i
\(616\) 0 0
\(617\) 1.17157i 0.0471657i 0.999722 + 0.0235829i \(0.00750736\pi\)
−0.999722 + 0.0235829i \(0.992493\pi\)
\(618\) 0 0
\(619\) 36.8284 1.48026 0.740130 0.672464i \(-0.234764\pi\)
0.740130 + 0.672464i \(0.234764\pi\)
\(620\) 0 0
\(621\) 20.6863 0.830112
\(622\) 0 0
\(623\) 7.71573i 0.309124i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 1.17157i 0.0467881i
\(628\) 0 0
\(629\) −9.65685 −0.385044
\(630\) 0 0
\(631\) −17.7990 −0.708567 −0.354283 0.935138i \(-0.615275\pi\)
−0.354283 + 0.935138i \(0.615275\pi\)
\(632\) 0 0
\(633\) 32.0000i 1.27189i
\(634\) 0 0
\(635\) 25.4558 12.7279i 1.01018 0.505092i
\(636\) 0 0
\(637\) 21.5563i 0.854094i
\(638\) 0 0
\(639\) −8.48528 −0.335673
\(640\) 0 0
\(641\) −40.1421 −1.58552 −0.792760 0.609535i \(-0.791356\pi\)
−0.792760 + 0.609535i \(0.791356\pi\)
\(642\) 0 0
\(643\) 22.9706i 0.905871i 0.891543 + 0.452935i \(0.149623\pi\)
−0.891543 + 0.452935i \(0.850377\pi\)
\(644\) 0 0
\(645\) 4.48528 + 8.97056i 0.176608 + 0.353216i
\(646\) 0 0
\(647\) 46.2843i 1.81962i −0.415022 0.909811i \(-0.636226\pi\)
0.415022 0.909811i \(-0.363774\pi\)
\(648\) 0 0
\(649\) −0.970563 −0.0380979
\(650\) 0 0
\(651\) −1.37258 −0.0537958
\(652\) 0 0
\(653\) 2.82843i 0.110685i 0.998467 + 0.0553425i \(0.0176251\pi\)
−0.998467 + 0.0553425i \(0.982375\pi\)
\(654\) 0 0
\(655\) 6.34315 + 12.6863i 0.247847 + 0.495694i
\(656\) 0 0
\(657\) 6.82843i 0.266402i
\(658\) 0 0
\(659\) −15.5147 −0.604368 −0.302184 0.953250i \(-0.597716\pi\)
−0.302184 + 0.953250i \(0.597716\pi\)
\(660\) 0 0
\(661\) −23.4558 −0.912327 −0.456163 0.889896i \(-0.650777\pi\)
−0.456163 + 0.889896i \(0.650777\pi\)
\(662\) 0 0
\(663\) 13.6569i 0.530388i
\(664\) 0 0
\(665\) −1.65685 + 0.828427i −0.0642501 + 0.0321250i
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 4.68629 0.180912
\(672\) 0 0
\(673\) 30.2426i 1.16577i −0.812555 0.582884i \(-0.801924\pi\)
0.812555 0.582884i \(-0.198076\pi\)
\(674\) 0 0
\(675\) −22.6274 16.9706i −0.870930 0.653197i
\(676\) 0 0
\(677\) 31.8995i 1.22600i 0.790084 + 0.612999i \(0.210037\pi\)
−0.790084 + 0.612999i \(0.789963\pi\)
\(678\) 0 0
\(679\) −10.8284 −0.415557
\(680\) 0 0
\(681\) −26.9706 −1.03351
\(682\) 0 0
\(683\) 20.9289i 0.800823i 0.916335 + 0.400412i \(0.131133\pi\)
−0.916335 + 0.400412i \(0.868867\pi\)
\(684\) 0 0
\(685\) −5.65685 + 2.82843i −0.216137 + 0.108069i
\(686\) 0 0
\(687\) 27.3137i 1.04208i
\(688\) 0 0
\(689\) 26.9706 1.02750
\(690\) 0 0
\(691\) 34.7696 1.32270 0.661348 0.750079i \(-0.269985\pi\)
0.661348 + 0.750079i \(0.269985\pi\)
\(692\) 0 0
\(693\) 0.686292i 0.0260701i
\(694\) 0 0
\(695\) 20.8284 + 41.6569i 0.790067 + 1.58013i
\(696\) 0 0
\(697\) 13.6569i 0.517290i
\(698\) 0 0
\(699\) 7.02944 0.265878
\(700\) 0 0
\(701\) 8.68629 0.328077 0.164038 0.986454i \(-0.447548\pi\)
0.164038 + 0.986454i \(0.447548\pi\)
\(702\) 0 0
\(703\) 3.41421i 0.128770i
\(704\) 0 0
\(705\) −6.82843 13.6569i −0.257173 0.514347i
\(706\) 0 0
\(707\) 6.62742i 0.249250i
\(708\) 0 0
\(709\) −10.6863 −0.401332 −0.200666 0.979660i \(-0.564311\pi\)
−0.200666 + 0.979660i \(0.564311\pi\)
\(710\) 0 0
\(711\) −8.48528 −0.318223
\(712\) 0 0
\(713\) 4.28427i 0.160447i
\(714\) 0 0
\(715\) −5.65685 + 2.82843i −0.211554 + 0.105777i
\(716\) 0 0
\(717\) 35.3137i 1.31881i
\(718\) 0 0
\(719\) −14.4853 −0.540210 −0.270105 0.962831i \(-0.587058\pi\)
−0.270105 + 0.962831i \(0.587058\pi\)
\(720\) 0 0
\(721\) −8.20101 −0.305422
\(722\) 0 0
\(723\) 12.4853i 0.464333i
\(724\) 0 0
\(725\) −22.9706 + 30.6274i −0.853105 + 1.13747i
\(726\) 0 0
\(727\) 19.4558i 0.721577i −0.932648 0.360789i \(-0.882508\pi\)
0.932648 0.360789i \(-0.117492\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) 8.97056 0.331788
\(732\) 0 0
\(733\) 35.3137i 1.30434i 0.758072 + 0.652171i \(0.226142\pi\)
−0.758072 + 0.652171i \(0.773858\pi\)
\(734\) 0 0
\(735\) −17.8579 + 8.92893i −0.658697 + 0.329349i
\(736\) 0 0
\(737\) 8.20101i 0.302088i
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 4.82843 0.177377
\(742\) 0 0
\(743\) 21.8995i 0.803415i −0.915768 0.401707i \(-0.868417\pi\)
0.915768 0.401707i \(-0.131583\pi\)
\(744\) 0 0
\(745\) −4.00000 8.00000i −0.146549 0.293097i
\(746\) 0 0
\(747\) 4.34315i 0.158907i
\(748\) 0 0
\(749\) 6.42641 0.234816
\(750\) 0 0
\(751\) 35.5980 1.29899 0.649494 0.760366i \(-0.274981\pi\)
0.649494 + 0.760366i \(0.274981\pi\)
\(752\) 0 0
\(753\) 5.65685i 0.206147i
\(754\) 0 0
\(755\) 13.6569 + 27.3137i 0.497024 + 0.994048i
\(756\) 0 0
\(757\) 6.62742i 0.240878i −0.992721 0.120439i \(-0.961570\pi\)
0.992721 0.120439i \(-0.0384301\pi\)
\(758\) 0 0
\(759\) 4.28427 0.155509
\(760\) 0 0
\(761\) 32.2843 1.17030 0.585152 0.810924i \(-0.301035\pi\)
0.585152 + 0.810924i \(0.301035\pi\)
\(762\) 0 0
\(763\) 5.37258i 0.194501i
\(764\) 0 0
\(765\) −5.65685 + 2.82843i −0.204524 + 0.102262i
\(766\) 0 0
\(767\) 4.00000i 0.144432i
\(768\) 0 0
\(769\) −20.2843 −0.731470 −0.365735 0.930719i \(-0.619182\pi\)
−0.365735 + 0.930719i \(0.619182\pi\)
\(770\) 0 0
\(771\) −2.48528 −0.0895052
\(772\) 0 0
\(773\) 1.07107i 0.0385236i 0.999814 + 0.0192618i \(0.00613161\pi\)
−0.999814 + 0.0192618i \(0.993868\pi\)
\(774\) 0 0
\(775\) 3.51472 4.68629i 0.126252 0.168337i
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) 4.82843 0.172996
\(780\) 0 0
\(781\) −7.02944 −0.251533
\(782\) 0 0
\(783\) 43.3137i 1.54791i
\(784\) 0 0
\(785\) 24.9706 12.4853i 0.891238 0.445619i
\(786\) 0 0
\(787\) 36.5269i 1.30204i 0.759059 + 0.651022i \(0.225659\pi\)
−0.759059 + 0.651022i \(0.774341\pi\)
\(788\) 0 0
\(789\) 24.4853 0.871699
\(790\) 0 0
\(791\) 7.11270 0.252898
\(792\) 0 0
\(793\) 19.3137i 0.685850i
\(794\) 0 0
\(795\) 11.1716 + 22.3431i 0.396215 + 0.792430i
\(796\) 0 0
\(797\) 10.0416i 0.355693i −0.984058 0.177846i \(-0.943087\pi\)
0.984058 0.177846i \(-0.0569130\pi\)
\(798\) 0 0
\(799\) −13.6569 −0.483145
\(800\) 0 0
\(801\) −9.31371 −0.329084
\(802\) 0