Properties

Label 5520.2.be.c.1471.7
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $32$
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: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.7
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.c.1471.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{5} +3.67125 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{5} +3.67125 q^{7} -1.00000 q^{9} -1.00786 q^{11} -0.593836 q^{13} -1.00000 q^{15} -5.94819i q^{17} +0.286037 q^{19} -3.67125i q^{21} +(-2.84988 - 3.85723i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -6.36715 q^{29} +1.69759i q^{31} +1.00786i q^{33} -3.67125i q^{35} +10.8469i q^{37} +0.593836i q^{39} -9.08618 q^{41} -2.53039 q^{43} +1.00000i q^{45} -9.21965i q^{47} +6.47806 q^{49} -5.94819 q^{51} -9.48440i q^{53} +1.00786i q^{55} -0.286037i q^{57} +7.46964i q^{59} -10.2566i q^{61} -3.67125 q^{63} +0.593836i q^{65} -6.43360 q^{67} +(-3.85723 + 2.84988i) q^{69} +2.38489i q^{71} +10.8439 q^{73} +1.00000i q^{75} -3.70011 q^{77} -13.4756 q^{79} +1.00000 q^{81} -1.86819 q^{83} -5.94819 q^{85} +6.36715i q^{87} +14.2439i q^{89} -2.18012 q^{91} +1.69759 q^{93} -0.286037i q^{95} -0.822365i q^{97} +1.00786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 8q^{7} - 32q^{9} + O(q^{10}) \) \( 32q - 8q^{7} - 32q^{9} + 8q^{11} - 8q^{13} - 32q^{15} - 32q^{25} + 4q^{29} + 20q^{41} + 52q^{49} - 4q^{51} + 8q^{63} + 32q^{67} - 40q^{73} - 24q^{77} + 32q^{79} + 32q^{81} - 4q^{85} - 48q^{91} - 8q^{99} + O(q^{100}) \)

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
\(7\) 3.67125 1.38760 0.693801 0.720167i \(-0.255935\pi\)
0.693801 + 0.720167i \(0.255935\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00786 −0.303882 −0.151941 0.988390i \(-0.548552\pi\)
−0.151941 + 0.988390i \(0.548552\pi\)
\(12\) 0 0
\(13\) −0.593836 −0.164701 −0.0823503 0.996603i \(-0.526243\pi\)
−0.0823503 + 0.996603i \(0.526243\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.94819i 1.44265i −0.692597 0.721325i \(-0.743534\pi\)
0.692597 0.721325i \(-0.256466\pi\)
\(18\) 0 0
\(19\) 0.286037 0.0656214 0.0328107 0.999462i \(-0.489554\pi\)
0.0328107 + 0.999462i \(0.489554\pi\)
\(20\) 0 0
\(21\) 3.67125i 0.801132i
\(22\) 0 0
\(23\) −2.84988 3.85723i −0.594241 0.804287i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.36715 −1.18235 −0.591175 0.806543i \(-0.701336\pi\)
−0.591175 + 0.806543i \(0.701336\pi\)
\(30\) 0 0
\(31\) 1.69759i 0.304895i 0.988312 + 0.152448i \(0.0487156\pi\)
−0.988312 + 0.152448i \(0.951284\pi\)
\(32\) 0 0
\(33\) 1.00786i 0.175446i
\(34\) 0 0
\(35\) 3.67125i 0.620554i
\(36\) 0 0
\(37\) 10.8469i 1.78322i 0.452806 + 0.891609i \(0.350423\pi\)
−0.452806 + 0.891609i \(0.649577\pi\)
\(38\) 0 0
\(39\) 0.593836i 0.0950899i
\(40\) 0 0
\(41\) −9.08618 −1.41902 −0.709511 0.704694i \(-0.751084\pi\)
−0.709511 + 0.704694i \(0.751084\pi\)
\(42\) 0 0
\(43\) −2.53039 −0.385881 −0.192941 0.981210i \(-0.561802\pi\)
−0.192941 + 0.981210i \(0.561802\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 9.21965i 1.34482i −0.740177 0.672412i \(-0.765258\pi\)
0.740177 0.672412i \(-0.234742\pi\)
\(48\) 0 0
\(49\) 6.47806 0.925437
\(50\) 0 0
\(51\) −5.94819 −0.832914
\(52\) 0 0
\(53\) 9.48440i 1.30278i −0.758742 0.651391i \(-0.774186\pi\)
0.758742 0.651391i \(-0.225814\pi\)
\(54\) 0 0
\(55\) 1.00786i 0.135900i
\(56\) 0 0
\(57\) 0.286037i 0.0378865i
\(58\) 0 0
\(59\) 7.46964i 0.972465i 0.873830 + 0.486232i \(0.161629\pi\)
−0.873830 + 0.486232i \(0.838371\pi\)
\(60\) 0 0
\(61\) 10.2566i 1.31322i −0.754231 0.656609i \(-0.771990\pi\)
0.754231 0.656609i \(-0.228010\pi\)
\(62\) 0 0
\(63\) −3.67125 −0.462534
\(64\) 0 0
\(65\) 0.593836i 0.0736563i
\(66\) 0 0
\(67\) −6.43360 −0.785990 −0.392995 0.919541i \(-0.628561\pi\)
−0.392995 + 0.919541i \(0.628561\pi\)
\(68\) 0 0
\(69\) −3.85723 + 2.84988i −0.464356 + 0.343085i
\(70\) 0 0
\(71\) 2.38489i 0.283035i 0.989936 + 0.141517i \(0.0451981\pi\)
−0.989936 + 0.141517i \(0.954802\pi\)
\(72\) 0 0
\(73\) 10.8439 1.26919 0.634593 0.772846i \(-0.281168\pi\)
0.634593 + 0.772846i \(0.281168\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) −3.70011 −0.421667
\(78\) 0 0
\(79\) −13.4756 −1.51612 −0.758061 0.652183i \(-0.773853\pi\)
−0.758061 + 0.652183i \(0.773853\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.86819 −0.205061 −0.102530 0.994730i \(-0.532694\pi\)
−0.102530 + 0.994730i \(0.532694\pi\)
\(84\) 0 0
\(85\) −5.94819 −0.645172
\(86\) 0 0
\(87\) 6.36715i 0.682631i
\(88\) 0 0
\(89\) 14.2439i 1.50985i 0.655810 + 0.754926i \(0.272327\pi\)
−0.655810 + 0.754926i \(0.727673\pi\)
\(90\) 0 0
\(91\) −2.18012 −0.228539
\(92\) 0 0
\(93\) 1.69759 0.176031
\(94\) 0 0
\(95\) 0.286037i 0.0293468i
\(96\) 0 0
\(97\) 0.822365i 0.0834985i −0.999128 0.0417492i \(-0.986707\pi\)
0.999128 0.0417492i \(-0.0132931\pi\)
\(98\) 0 0
\(99\) 1.00786 0.101294
\(100\) 0 0
\(101\) 17.9340 1.78450 0.892250 0.451542i \(-0.149126\pi\)
0.892250 + 0.451542i \(0.149126\pi\)
\(102\) 0 0
\(103\) −5.71186 −0.562806 −0.281403 0.959590i \(-0.590800\pi\)
−0.281403 + 0.959590i \(0.590800\pi\)
\(104\) 0 0
\(105\) −3.67125 −0.358277
\(106\) 0 0
\(107\) −0.528453 −0.0510875 −0.0255437 0.999674i \(-0.508132\pi\)
−0.0255437 + 0.999674i \(0.508132\pi\)
\(108\) 0 0
\(109\) 6.39885i 0.612898i 0.951887 + 0.306449i \(0.0991410\pi\)
−0.951887 + 0.306449i \(0.900859\pi\)
\(110\) 0 0
\(111\) 10.8469 1.02954
\(112\) 0 0
\(113\) 6.58160i 0.619145i −0.950876 0.309572i \(-0.899814\pi\)
0.950876 0.309572i \(-0.100186\pi\)
\(114\) 0 0
\(115\) −3.85723 + 2.84988i −0.359688 + 0.265752i
\(116\) 0 0
\(117\) 0.593836 0.0549002
\(118\) 0 0
\(119\) 21.8373i 2.00182i
\(120\) 0 0
\(121\) −9.98421 −0.907656
\(122\) 0 0
\(123\) 9.08618i 0.819273i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 17.3310i 1.53788i −0.639324 0.768938i \(-0.720786\pi\)
0.639324 0.768938i \(-0.279214\pi\)
\(128\) 0 0
\(129\) 2.53039i 0.222789i
\(130\) 0 0
\(131\) 18.1819i 1.58856i −0.607552 0.794280i \(-0.707848\pi\)
0.607552 0.794280i \(-0.292152\pi\)
\(132\) 0 0
\(133\) 1.05011 0.0910563
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 19.6173i 1.67602i −0.545657 0.838009i \(-0.683720\pi\)
0.545657 0.838009i \(-0.316280\pi\)
\(138\) 0 0
\(139\) 11.6864i 0.991224i −0.868544 0.495612i \(-0.834944\pi\)
0.868544 0.495612i \(-0.165056\pi\)
\(140\) 0 0
\(141\) −9.21965 −0.776434
\(142\) 0 0
\(143\) 0.598506 0.0500496
\(144\) 0 0
\(145\) 6.36715i 0.528763i
\(146\) 0 0
\(147\) 6.47806i 0.534301i
\(148\) 0 0
\(149\) 20.0940i 1.64616i 0.567922 + 0.823082i \(0.307747\pi\)
−0.567922 + 0.823082i \(0.692253\pi\)
\(150\) 0 0
\(151\) 18.0658i 1.47017i −0.677972 0.735087i \(-0.737141\pi\)
0.677972 0.735087i \(-0.262859\pi\)
\(152\) 0 0
\(153\) 5.94819i 0.480883i
\(154\) 0 0
\(155\) 1.69759 0.136353
\(156\) 0 0
\(157\) 16.5993i 1.32477i 0.749164 + 0.662384i \(0.230455\pi\)
−0.749164 + 0.662384i \(0.769545\pi\)
\(158\) 0 0
\(159\) −9.48440 −0.752162
\(160\) 0 0
\(161\) −10.4626 14.1608i −0.824569 1.11603i
\(162\) 0 0
\(163\) 2.12245i 0.166243i 0.996539 + 0.0831216i \(0.0264890\pi\)
−0.996539 + 0.0831216i \(0.973511\pi\)
\(164\) 0 0
\(165\) 1.00786 0.0784620
\(166\) 0 0
\(167\) 16.0048i 1.23849i −0.785199 0.619243i \(-0.787440\pi\)
0.785199 0.619243i \(-0.212560\pi\)
\(168\) 0 0
\(169\) −12.6474 −0.972874
\(170\) 0 0
\(171\) −0.286037 −0.0218738
\(172\) 0 0
\(173\) 9.91583 0.753887 0.376943 0.926236i \(-0.376975\pi\)
0.376943 + 0.926236i \(0.376975\pi\)
\(174\) 0 0
\(175\) −3.67125 −0.277520
\(176\) 0 0
\(177\) 7.46964 0.561453
\(178\) 0 0
\(179\) 0.533246i 0.0398567i 0.999801 + 0.0199284i \(0.00634381\pi\)
−0.999801 + 0.0199284i \(0.993656\pi\)
\(180\) 0 0
\(181\) 1.19483i 0.0888110i 0.999014 + 0.0444055i \(0.0141394\pi\)
−0.999014 + 0.0444055i \(0.985861\pi\)
\(182\) 0 0
\(183\) −10.2566 −0.758187
\(184\) 0 0
\(185\) 10.8469 0.797479
\(186\) 0 0
\(187\) 5.99497i 0.438395i
\(188\) 0 0
\(189\) 3.67125i 0.267044i
\(190\) 0 0
\(191\) −12.5425 −0.907545 −0.453772 0.891118i \(-0.649922\pi\)
−0.453772 + 0.891118i \(0.649922\pi\)
\(192\) 0 0
\(193\) 21.5783 1.55324 0.776620 0.629969i \(-0.216932\pi\)
0.776620 + 0.629969i \(0.216932\pi\)
\(194\) 0 0
\(195\) 0.593836 0.0425255
\(196\) 0 0
\(197\) 1.21949 0.0868848 0.0434424 0.999056i \(-0.486168\pi\)
0.0434424 + 0.999056i \(0.486168\pi\)
\(198\) 0 0
\(199\) 16.5570 1.17370 0.586848 0.809697i \(-0.300368\pi\)
0.586848 + 0.809697i \(0.300368\pi\)
\(200\) 0 0
\(201\) 6.43360i 0.453791i
\(202\) 0 0
\(203\) −23.3754 −1.64063
\(204\) 0 0
\(205\) 9.08618i 0.634606i
\(206\) 0 0
\(207\) 2.84988 + 3.85723i 0.198080 + 0.268096i
\(208\) 0 0
\(209\) −0.288286 −0.0199412
\(210\) 0 0
\(211\) 14.6512i 1.00863i 0.863520 + 0.504315i \(0.168255\pi\)
−0.863520 + 0.504315i \(0.831745\pi\)
\(212\) 0 0
\(213\) 2.38489 0.163410
\(214\) 0 0
\(215\) 2.53039i 0.172571i
\(216\) 0 0
\(217\) 6.23226i 0.423073i
\(218\) 0 0
\(219\) 10.8439i 0.732765i
\(220\) 0 0
\(221\) 3.53225i 0.237605i
\(222\) 0 0
\(223\) 5.18336i 0.347104i −0.984825 0.173552i \(-0.944476\pi\)
0.984825 0.173552i \(-0.0555244\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.09848 −0.272026 −0.136013 0.990707i \(-0.543429\pi\)
−0.136013 + 0.990707i \(0.543429\pi\)
\(228\) 0 0
\(229\) 3.06167i 0.202321i −0.994870 0.101160i \(-0.967744\pi\)
0.994870 0.101160i \(-0.0322555\pi\)
\(230\) 0 0
\(231\) 3.70011i 0.243450i
\(232\) 0 0
\(233\) −20.2216 −1.32476 −0.662381 0.749167i \(-0.730454\pi\)
−0.662381 + 0.749167i \(0.730454\pi\)
\(234\) 0 0
\(235\) −9.21965 −0.601423
\(236\) 0 0
\(237\) 13.4756i 0.875334i
\(238\) 0 0
\(239\) 3.16908i 0.204991i 0.994733 + 0.102495i \(0.0326827\pi\)
−0.994733 + 0.102495i \(0.967317\pi\)
\(240\) 0 0
\(241\) 13.3177i 0.857870i 0.903335 + 0.428935i \(0.141111\pi\)
−0.903335 + 0.428935i \(0.858889\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.47806i 0.413868i
\(246\) 0 0
\(247\) −0.169859 −0.0108079
\(248\) 0 0
\(249\) 1.86819i 0.118392i
\(250\) 0 0
\(251\) −8.62068 −0.544132 −0.272066 0.962279i \(-0.587707\pi\)
−0.272066 + 0.962279i \(0.587707\pi\)
\(252\) 0 0
\(253\) 2.87229 + 3.88756i 0.180579 + 0.244409i
\(254\) 0 0
\(255\) 5.94819i 0.372490i
\(256\) 0 0
\(257\) 22.9596 1.43218 0.716092 0.698006i \(-0.245929\pi\)
0.716092 + 0.698006i \(0.245929\pi\)
\(258\) 0 0
\(259\) 39.8216i 2.47440i
\(260\) 0 0
\(261\) 6.36715 0.394117
\(262\) 0 0
\(263\) 6.66741 0.411130 0.205565 0.978643i \(-0.434097\pi\)
0.205565 + 0.978643i \(0.434097\pi\)
\(264\) 0 0
\(265\) −9.48440 −0.582622
\(266\) 0 0
\(267\) 14.2439 0.871713
\(268\) 0 0
\(269\) −8.78049 −0.535356 −0.267678 0.963508i \(-0.586256\pi\)
−0.267678 + 0.963508i \(0.586256\pi\)
\(270\) 0 0
\(271\) 10.2578i 0.623117i −0.950227 0.311558i \(-0.899149\pi\)
0.950227 0.311558i \(-0.100851\pi\)
\(272\) 0 0
\(273\) 2.18012i 0.131947i
\(274\) 0 0
\(275\) 1.00786 0.0607764
\(276\) 0 0
\(277\) −18.6778 −1.12224 −0.561120 0.827734i \(-0.689630\pi\)
−0.561120 + 0.827734i \(0.689630\pi\)
\(278\) 0 0
\(279\) 1.69759i 0.101632i
\(280\) 0 0
\(281\) 20.9324i 1.24872i 0.781135 + 0.624362i \(0.214641\pi\)
−0.781135 + 0.624362i \(0.785359\pi\)
\(282\) 0 0
\(283\) −13.7104 −0.814996 −0.407498 0.913206i \(-0.633599\pi\)
−0.407498 + 0.913206i \(0.633599\pi\)
\(284\) 0 0
\(285\) −0.286037 −0.0169434
\(286\) 0 0
\(287\) −33.3576 −1.96904
\(288\) 0 0
\(289\) −18.3810 −1.08124
\(290\) 0 0
\(291\) −0.822365 −0.0482079
\(292\) 0 0
\(293\) 10.4892i 0.612786i 0.951905 + 0.306393i \(0.0991222\pi\)
−0.951905 + 0.306393i \(0.900878\pi\)
\(294\) 0 0
\(295\) 7.46964 0.434899
\(296\) 0 0
\(297\) 1.00786i 0.0584821i
\(298\) 0 0
\(299\) 1.69236 + 2.29056i 0.0978718 + 0.132467i
\(300\) 0 0
\(301\) −9.28970 −0.535449
\(302\) 0 0
\(303\) 17.9340i 1.03028i
\(304\) 0 0
\(305\) −10.2566 −0.587289
\(306\) 0 0
\(307\) 4.93404i 0.281600i −0.990038 0.140800i \(-0.955033\pi\)
0.990038 0.140800i \(-0.0449675\pi\)
\(308\) 0 0
\(309\) 5.71186i 0.324936i
\(310\) 0 0
\(311\) 4.34926i 0.246624i −0.992368 0.123312i \(-0.960648\pi\)
0.992368 0.123312i \(-0.0393515\pi\)
\(312\) 0 0
\(313\) 5.19385i 0.293574i −0.989168 0.146787i \(-0.953107\pi\)
0.989168 0.146787i \(-0.0468931\pi\)
\(314\) 0 0
\(315\) 3.67125i 0.206851i
\(316\) 0 0
\(317\) 19.0034 1.06734 0.533668 0.845694i \(-0.320813\pi\)
0.533668 + 0.845694i \(0.320813\pi\)
\(318\) 0 0
\(319\) 6.41722 0.359295
\(320\) 0 0
\(321\) 0.528453i 0.0294954i
\(322\) 0 0
\(323\) 1.70140i 0.0946686i
\(324\) 0 0
\(325\) 0.593836 0.0329401
\(326\) 0 0
\(327\) 6.39885 0.353857
\(328\) 0 0
\(329\) 33.8476i 1.86608i
\(330\) 0 0
\(331\) 20.5124i 1.12746i −0.825958 0.563732i \(-0.809365\pi\)
0.825958 0.563732i \(-0.190635\pi\)
\(332\) 0 0
\(333\) 10.8469i 0.594406i
\(334\) 0 0
\(335\) 6.43360i 0.351505i
\(336\) 0 0
\(337\) 13.4104i 0.730512i 0.930907 + 0.365256i \(0.119019\pi\)
−0.930907 + 0.365256i \(0.880981\pi\)
\(338\) 0 0
\(339\) −6.58160 −0.357463
\(340\) 0 0
\(341\) 1.71093i 0.0926523i
\(342\) 0 0
\(343\) −1.91617 −0.103464
\(344\) 0 0
\(345\) 2.84988 + 3.85723i 0.153432 + 0.207666i
\(346\) 0 0
\(347\) 26.1151i 1.40193i 0.713196 + 0.700965i \(0.247247\pi\)
−0.713196 + 0.700965i \(0.752753\pi\)
\(348\) 0 0
\(349\) 5.72337 0.306365 0.153182 0.988198i \(-0.451048\pi\)
0.153182 + 0.988198i \(0.451048\pi\)
\(350\) 0 0
\(351\) 0.593836i 0.0316966i
\(352\) 0 0
\(353\) 21.2032 1.12853 0.564267 0.825592i \(-0.309159\pi\)
0.564267 + 0.825592i \(0.309159\pi\)
\(354\) 0 0
\(355\) 2.38489 0.126577
\(356\) 0 0
\(357\) −21.8373 −1.15575
\(358\) 0 0
\(359\) −5.32343 −0.280960 −0.140480 0.990084i \(-0.544865\pi\)
−0.140480 + 0.990084i \(0.544865\pi\)
\(360\) 0 0
\(361\) −18.9182 −0.995694
\(362\) 0 0
\(363\) 9.98421i 0.524035i
\(364\) 0 0
\(365\) 10.8439i 0.567597i
\(366\) 0 0
\(367\) −28.3069 −1.47761 −0.738803 0.673921i \(-0.764609\pi\)
−0.738803 + 0.673921i \(0.764609\pi\)
\(368\) 0 0
\(369\) 9.08618 0.473008
\(370\) 0 0
\(371\) 34.8196i 1.80774i
\(372\) 0 0
\(373\) 15.9995i 0.828425i 0.910180 + 0.414212i \(0.135943\pi\)
−0.910180 + 0.414212i \(0.864057\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 3.78105 0.194734
\(378\) 0 0
\(379\) 32.1655 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(380\) 0 0
\(381\) −17.3310 −0.887893
\(382\) 0 0
\(383\) −28.8992 −1.47668 −0.738339 0.674430i \(-0.764389\pi\)
−0.738339 + 0.674430i \(0.764389\pi\)
\(384\) 0 0
\(385\) 3.70011i 0.188575i
\(386\) 0 0
\(387\) 2.53039 0.128627
\(388\) 0 0
\(389\) 17.2346i 0.873831i 0.899503 + 0.436915i \(0.143929\pi\)
−0.899503 + 0.436915i \(0.856071\pi\)
\(390\) 0 0
\(391\) −22.9435 + 16.9516i −1.16030 + 0.857281i
\(392\) 0 0
\(393\) −18.1819 −0.917155
\(394\) 0 0
\(395\) 13.4756i 0.678031i
\(396\) 0 0
\(397\) −32.6669 −1.63950 −0.819751 0.572719i \(-0.805888\pi\)
−0.819751 + 0.572719i \(0.805888\pi\)
\(398\) 0 0
\(399\) 1.05011i 0.0525714i
\(400\) 0 0
\(401\) 38.6626i 1.93072i −0.260926 0.965359i \(-0.584028\pi\)
0.260926 0.965359i \(-0.415972\pi\)
\(402\) 0 0
\(403\) 1.00809i 0.0502165i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 10.9322i 0.541888i
\(408\) 0 0
\(409\) 22.5284 1.11396 0.556978 0.830527i \(-0.311961\pi\)
0.556978 + 0.830527i \(0.311961\pi\)
\(410\) 0 0
\(411\) −19.6173 −0.967649
\(412\) 0 0
\(413\) 27.4229i 1.34939i
\(414\) 0 0
\(415\) 1.86819i 0.0917059i
\(416\) 0 0
\(417\) −11.6864 −0.572283
\(418\) 0 0
\(419\) 14.3376 0.700438 0.350219 0.936668i \(-0.386107\pi\)
0.350219 + 0.936668i \(0.386107\pi\)
\(420\) 0 0
\(421\) 1.89252i 0.0922360i −0.998936 0.0461180i \(-0.985315\pi\)
0.998936 0.0461180i \(-0.0146850\pi\)
\(422\) 0 0
\(423\) 9.21965i 0.448275i
\(424\) 0 0
\(425\) 5.94819i 0.288530i
\(426\) 0 0
\(427\) 37.6544i 1.82222i
\(428\) 0 0
\(429\) 0.598506i 0.0288961i
\(430\) 0 0
\(431\) −15.9441 −0.768002 −0.384001 0.923333i \(-0.625454\pi\)
−0.384001 + 0.923333i \(0.625454\pi\)
\(432\) 0 0
\(433\) 11.9290i 0.573273i −0.958039 0.286636i \(-0.907463\pi\)
0.958039 0.286636i \(-0.0925371\pi\)
\(434\) 0 0
\(435\) 6.36715 0.305282
\(436\) 0 0
\(437\) −0.815170 1.10331i −0.0389949 0.0527784i
\(438\) 0 0
\(439\) 15.4107i 0.735514i 0.929922 + 0.367757i \(0.119874\pi\)
−0.929922 + 0.367757i \(0.880126\pi\)
\(440\) 0 0
\(441\) −6.47806 −0.308479
\(442\) 0 0
\(443\) 9.58510i 0.455402i −0.973731 0.227701i \(-0.926879\pi\)
0.973731 0.227701i \(-0.0731209\pi\)
\(444\) 0 0
\(445\) 14.2439 0.675226
\(446\) 0 0
\(447\) 20.0940 0.950414
\(448\) 0 0
\(449\) −9.57007 −0.451640 −0.225820 0.974169i \(-0.572506\pi\)
−0.225820 + 0.974169i \(0.572506\pi\)
\(450\) 0 0
\(451\) 9.15762 0.431216
\(452\) 0 0
\(453\) −18.0658 −0.848806
\(454\) 0 0
\(455\) 2.18012i 0.102206i
\(456\) 0 0
\(457\) 27.1216i 1.26870i −0.773047 0.634348i \(-0.781269\pi\)
0.773047 0.634348i \(-0.218731\pi\)
\(458\) 0 0
\(459\) 5.94819 0.277638
\(460\) 0 0
\(461\) −38.2112 −1.77967 −0.889837 0.456278i \(-0.849182\pi\)
−0.889837 + 0.456278i \(0.849182\pi\)
\(462\) 0 0
\(463\) 32.4925i 1.51005i 0.655694 + 0.755027i \(0.272376\pi\)
−0.655694 + 0.755027i \(0.727624\pi\)
\(464\) 0 0
\(465\) 1.69759i 0.0787237i
\(466\) 0 0
\(467\) 19.9305 0.922274 0.461137 0.887329i \(-0.347442\pi\)
0.461137 + 0.887329i \(0.347442\pi\)
\(468\) 0 0
\(469\) −23.6193 −1.09064
\(470\) 0 0
\(471\) 16.5993 0.764855
\(472\) 0 0
\(473\) 2.55029 0.117262
\(474\) 0 0
\(475\) −0.286037 −0.0131243
\(476\) 0 0
\(477\) 9.48440i 0.434261i
\(478\) 0 0
\(479\) −25.0942 −1.14658 −0.573292 0.819351i \(-0.694334\pi\)
−0.573292 + 0.819351i \(0.694334\pi\)
\(480\) 0 0
\(481\) 6.44128i 0.293697i
\(482\) 0 0
\(483\) −14.1608 + 10.4626i −0.644340 + 0.476065i
\(484\) 0 0
\(485\) −0.822365 −0.0373417
\(486\) 0 0
\(487\) 40.1634i 1.81998i −0.414635 0.909988i \(-0.636091\pi\)
0.414635 0.909988i \(-0.363909\pi\)
\(488\) 0 0
\(489\) 2.12245 0.0959806
\(490\) 0 0
\(491\) 19.1707i 0.865161i 0.901595 + 0.432581i \(0.142397\pi\)
−0.901595 + 0.432581i \(0.857603\pi\)
\(492\) 0 0
\(493\) 37.8731i 1.70572i
\(494\) 0 0
\(495\) 1.00786i 0.0453001i
\(496\) 0 0
\(497\) 8.75554i 0.392740i
\(498\) 0 0
\(499\) 14.0542i 0.629154i −0.949232 0.314577i \(-0.898137\pi\)
0.949232 0.314577i \(-0.101863\pi\)
\(500\) 0 0
\(501\) −16.0048 −0.715040
\(502\) 0 0
\(503\) −28.6280 −1.27646 −0.638230 0.769846i \(-0.720333\pi\)
−0.638230 + 0.769846i \(0.720333\pi\)
\(504\) 0 0
\(505\) 17.9340i 0.798053i
\(506\) 0 0
\(507\) 12.6474i 0.561689i
\(508\) 0 0
\(509\) 16.8077 0.744989 0.372494 0.928034i \(-0.378503\pi\)
0.372494 + 0.928034i \(0.378503\pi\)
\(510\) 0 0
\(511\) 39.8108 1.76112
\(512\) 0 0
\(513\) 0.286037i 0.0126288i
\(514\) 0 0
\(515\) 5.71186i 0.251694i
\(516\) 0 0
\(517\) 9.29214i 0.408668i
\(518\) 0 0
\(519\) 9.91583i 0.435257i
\(520\) 0 0
\(521\) 18.7758i 0.822585i −0.911503 0.411292i \(-0.865077\pi\)
0.911503 0.411292i \(-0.134923\pi\)
\(522\) 0 0
\(523\) −2.25654 −0.0986715 −0.0493357 0.998782i \(-0.515710\pi\)
−0.0493357 + 0.998782i \(0.515710\pi\)
\(524\) 0 0
\(525\) 3.67125i 0.160226i
\(526\) 0 0
\(527\) 10.0976 0.439857
\(528\) 0 0
\(529\) −6.75640 + 21.9852i −0.293756 + 0.955880i
\(530\) 0 0
\(531\) 7.46964i 0.324155i
\(532\) 0 0
\(533\) 5.39570 0.233714
\(534\) 0 0
\(535\) 0.528453i 0.0228470i
\(536\) 0 0
\(537\) 0.533246 0.0230113
\(538\) 0 0
\(539\) −6.52900 −0.281224
\(540\) 0 0
\(541\) −4.49190 −0.193122 −0.0965608 0.995327i \(-0.530784\pi\)
−0.0965608 + 0.995327i \(0.530784\pi\)
\(542\) 0 0
\(543\) 1.19483 0.0512751
\(544\) 0 0
\(545\) 6.39885 0.274096
\(546\) 0 0
\(547\) 38.1115i 1.62953i −0.579793 0.814764i \(-0.696866\pi\)
0.579793 0.814764i \(-0.303134\pi\)
\(548\) 0 0
\(549\) 10.2566i 0.437740i
\(550\) 0 0
\(551\) −1.82124 −0.0775875
\(552\) 0 0
\(553\) −49.4722 −2.10377
\(554\) 0 0
\(555\) 10.8469i 0.460425i
\(556\) 0 0
\(557\) 15.9889i 0.677472i −0.940881 0.338736i \(-0.890001\pi\)
0.940881 0.338736i \(-0.109999\pi\)
\(558\) 0 0
\(559\) 1.50264 0.0635549
\(560\) 0 0
\(561\) 5.99497 0.253108
\(562\) 0 0
\(563\) 0.649938 0.0273916 0.0136958 0.999906i \(-0.495640\pi\)
0.0136958 + 0.999906i \(0.495640\pi\)
\(564\) 0 0
\(565\) −6.58160 −0.276890
\(566\) 0 0
\(567\) 3.67125 0.154178
\(568\) 0 0
\(569\) 34.2077i 1.43406i −0.697042 0.717030i \(-0.745501\pi\)
0.697042 0.717030i \(-0.254499\pi\)
\(570\) 0 0
\(571\) −6.84324 −0.286381 −0.143190 0.989695i \(-0.545736\pi\)
−0.143190 + 0.989695i \(0.545736\pi\)
\(572\) 0 0
\(573\) 12.5425i 0.523971i
\(574\) 0 0
\(575\) 2.84988 + 3.85723i 0.118848 + 0.160857i
\(576\) 0 0
\(577\) 29.3607 1.22230 0.611150 0.791514i \(-0.290707\pi\)
0.611150 + 0.791514i \(0.290707\pi\)
\(578\) 0 0
\(579\) 21.5783i 0.896764i
\(580\) 0 0
\(581\) −6.85860 −0.284543
\(582\) 0 0
\(583\) 9.55898i 0.395892i
\(584\) 0 0
\(585\) 0.593836i 0.0245521i
\(586\) 0 0
\(587\) 31.8191i 1.31332i −0.754188 0.656658i \(-0.771970\pi\)
0.754188 0.656658i \(-0.228030\pi\)
\(588\) 0 0
\(589\) 0.485572i 0.0200077i
\(590\) 0 0
\(591\) 1.21949i 0.0501630i
\(592\) 0 0
\(593\) 7.54885 0.309994 0.154997 0.987915i \(-0.450463\pi\)
0.154997 + 0.987915i \(0.450463\pi\)
\(594\) 0 0
\(595\) −21.8373 −0.895242
\(596\) 0 0
\(597\) 16.5570i 0.677634i
\(598\) 0 0
\(599\) 16.0867i 0.657284i −0.944455 0.328642i \(-0.893409\pi\)
0.944455 0.328642i \(-0.106591\pi\)
\(600\) 0 0
\(601\) 15.1192 0.616723 0.308362 0.951269i \(-0.400219\pi\)
0.308362 + 0.951269i \(0.400219\pi\)
\(602\) 0 0
\(603\) 6.43360 0.261997
\(604\) 0 0
\(605\) 9.98421i 0.405916i
\(606\) 0 0
\(607\) 46.4376i 1.88484i 0.334429 + 0.942421i \(0.391457\pi\)
−0.334429 + 0.942421i \(0.608543\pi\)
\(608\) 0 0
\(609\) 23.3754i 0.947219i
\(610\) 0 0
\(611\) 5.47496i 0.221493i
\(612\) 0 0
\(613\) 20.0498i 0.809805i 0.914360 + 0.404903i \(0.132695\pi\)
−0.914360 + 0.404903i \(0.867305\pi\)
\(614\) 0 0
\(615\) 9.08618 0.366390
\(616\) 0 0
\(617\) 1.64661i 0.0662900i 0.999451 + 0.0331450i \(0.0105523\pi\)
−0.999451 + 0.0331450i \(0.989448\pi\)
\(618\) 0 0
\(619\) 8.64294 0.347389 0.173695 0.984800i \(-0.444429\pi\)
0.173695 + 0.984800i \(0.444429\pi\)
\(620\) 0 0
\(621\) 3.85723 2.84988i 0.154785 0.114362i
\(622\) 0 0
\(623\) 52.2929i 2.09507i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.288286i 0.0115130i
\(628\) 0 0
\(629\) 64.5194 2.57256
\(630\) 0 0
\(631\) 42.8648 1.70642 0.853209 0.521569i \(-0.174653\pi\)
0.853209 + 0.521569i \(0.174653\pi\)
\(632\) 0 0
\(633\) 14.6512 0.582333
\(634\) 0 0
\(635\) −17.3310 −0.687759
\(636\) 0 0
\(637\) −3.84691 −0.152420
\(638\) 0 0
\(639\) 2.38489i 0.0943450i
\(640\) 0 0
\(641\) 20.3332i 0.803111i −0.915835 0.401556i \(-0.868470\pi\)
0.915835 0.401556i \(-0.131530\pi\)
\(642\) 0 0
\(643\) 41.0171 1.61756 0.808779 0.588113i \(-0.200129\pi\)
0.808779 + 0.588113i \(0.200129\pi\)
\(644\) 0 0
\(645\) 2.53039 0.0996341
\(646\) 0 0
\(647\) 3.97272i 0.156184i −0.996946 0.0780918i \(-0.975117\pi\)
0.996946 0.0780918i \(-0.0248827\pi\)
\(648\) 0 0
\(649\) 7.52838i 0.295515i
\(650\) 0 0
\(651\) 6.23226 0.244261
\(652\) 0 0
\(653\) 20.7527 0.812115 0.406057 0.913848i \(-0.366903\pi\)
0.406057 + 0.913848i \(0.366903\pi\)
\(654\) 0 0
\(655\) −18.1819 −0.710425
\(656\) 0 0
\(657\) −10.8439 −0.423062
\(658\) 0 0
\(659\) −35.5149 −1.38346 −0.691732 0.722155i \(-0.743152\pi\)
−0.691732 + 0.722155i \(0.743152\pi\)
\(660\) 0 0
\(661\) 6.05928i 0.235679i −0.993033 0.117839i \(-0.962403\pi\)
0.993033 0.117839i \(-0.0375968\pi\)
\(662\) 0 0
\(663\) 3.53225 0.137181
\(664\) 0 0
\(665\) 1.05011i 0.0407216i
\(666\) 0 0
\(667\) 18.1456 + 24.5596i 0.702601 + 0.950950i
\(668\) 0 0
\(669\) −5.18336 −0.200400
\(670\) 0 0
\(671\) 10.3372i 0.399064i
\(672\) 0 0
\(673\) 47.9256 1.84740 0.923698 0.383121i \(-0.125151\pi\)
0.923698 + 0.383121i \(0.125151\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 9.09024i 0.349366i 0.984625 + 0.174683i \(0.0558901\pi\)
−0.984625 + 0.174683i \(0.944110\pi\)
\(678\) 0 0
\(679\) 3.01910i 0.115863i
\(680\) 0 0
\(681\) 4.09848i 0.157054i
\(682\) 0 0
\(683\) 22.5635i 0.863370i 0.902024 + 0.431685i \(0.142081\pi\)
−0.902024 + 0.431685i \(0.857919\pi\)
\(684\) 0 0
\(685\) −19.6173 −0.749538
\(686\) 0 0
\(687\) −3.06167 −0.116810
\(688\) 0 0
\(689\) 5.63218i 0.214569i
\(690\) 0 0
\(691\) 28.3258i 1.07756i −0.842445 0.538782i \(-0.818885\pi\)
0.842445 0.538782i \(-0.181115\pi\)
\(692\) 0 0
\(693\) 3.70011 0.140556
\(694\) 0 0
\(695\) −11.6864 −0.443289
\(696\) 0 0
\(697\) 54.0464i 2.04715i
\(698\) 0 0
\(699\) 20.2216i 0.764851i
\(700\) 0 0
\(701\) 19.9040i 0.751764i −0.926667 0.375882i \(-0.877340\pi\)
0.926667 0.375882i \(-0.122660\pi\)
\(702\) 0 0
\(703\) 3.10261i 0.117017i
\(704\) 0 0
\(705\) 9.21965i 0.347232i
\(706\) 0 0
\(707\) 65.8402 2.47617
\(708\) 0 0
\(709\) 41.9327i 1.57482i −0.616432 0.787409i \(-0.711422\pi\)
0.616432 0.787409i \(-0.288578\pi\)
\(710\) 0 0
\(711\) 13.4756 0.505374
\(712\) 0 0
\(713\) 6.54797 4.83791i 0.245224 0.181181i
\(714\) 0 0
\(715\) 0.598506i 0.0223828i
\(716\) 0 0
\(717\) 3.16908 0.118352
\(718\) 0 0
\(719\) 44.5808i 1.66258i −0.555835 0.831292i \(-0.687602\pi\)
0.555835 0.831292i \(-0.312398\pi\)
\(720\) 0 0
\(721\) −20.9696 −0.780950
\(722\) 0 0
\(723\) 13.3177 0.495291
\(724\) 0 0
\(725\) 6.36715 0.236470
\(726\) 0 0
\(727\) −2.26736 −0.0840916 −0.0420458 0.999116i \(-0.513388\pi\)
−0.0420458 + 0.999116i \(0.513388\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 15.0513i 0.556691i
\(732\) 0 0
\(733\) 40.9074i 1.51095i 0.655179 + 0.755474i \(0.272593\pi\)
−0.655179 + 0.755474i \(0.727407\pi\)
\(734\) 0 0
\(735\) −6.47806 −0.238947
\(736\) 0 0
\(737\) 6.48419 0.238848
\(738\) 0 0
\(739\) 37.9120i 1.39461i −0.716773 0.697307i \(-0.754382\pi\)
0.716773 0.697307i \(-0.245618\pi\)
\(740\) 0 0
\(741\) 0.169859i 0.00623993i
\(742\) 0 0
\(743\) −34.3878 −1.26157 −0.630784 0.775959i \(-0.717266\pi\)
−0.630784 + 0.775959i \(0.717266\pi\)
\(744\) 0 0
\(745\) 20.0940 0.736187
\(746\) 0 0
\(747\) 1.86819 0.0683536
\(748\) 0 0
\(749\) −1.94008 −0.0708890
\(750\) 0 0
\(751\) 15.5273 0.566599 0.283300 0.959031i \(-0.408571\pi\)
0.283300 + 0.959031i \(0.408571\pi\)
\(752\) 0 0
\(753\) 8.62068i 0.314155i
\(754\) 0 0
\(755\) −18.0658 −0.657482
\(756\) 0 0
\(757\) 21.2158i 0.771102i −0.922687 0.385551i \(-0.874011\pi\)
0.922687 0.385551i \(-0.125989\pi\)
\(758\) 0 0
\(759\) 3.88756 2.87229i 0.141109 0.104257i
\(760\) 0 0
\(761\) 38.2514 1.38661 0.693306 0.720643i \(-0.256153\pi\)
0.693306 + 0.720643i \(0.256153\pi\)
\(762\) 0 0
\(763\) 23.4918i 0.850459i
\(764\) 0 0
\(765\) 5.94819 0.215057
\(766\) 0 0
\(767\) 4.43575i 0.160165i
\(768\) 0 0
\(769\) 19.2419i 0.693880i −0.937887 0.346940i \(-0.887221\pi\)
0.937887 0.346940i \(-0.112779\pi\)
\(770\) 0 0
\(771\) 22.9596i 0.826871i
\(772\) 0 0
\(773\) 3.65019i 0.131288i −0.997843 0.0656441i \(-0.979090\pi\)
0.997843 0.0656441i \(-0.0209102\pi\)
\(774\) 0 0
\(775\) 1.69759i 0.0609791i
\(776\) 0 0
\(777\) 39.8216 1.42859
\(778\) 0 0
\(779\) −2.59898 −0.0931182
\(780\) 0 0
\(781\) 2.40365i 0.0860093i
\(782\) 0 0
\(783\) 6.36715i 0.227544i
\(784\) 0 0
\(785\) 16.5993 0.592454
\(786\) 0 0
\(787\) −17.4135 −0.620724 −0.310362 0.950618i \(-0.600450\pi\)
−0.310362 + 0.950618i \(0.600450\pi\)
\(788\) 0 0
\(789\) 6.66741i 0.237366i
\(790\) 0 0
\(791\) 24.1627i 0.859126i
\(792\) 0 0
\(793\) 6.09072i 0.216288i
\(794\) 0 0
\(795\) 9.48440i 0.336377i
\(796\) 0 0
\(797\) 14.0360i 0.497182i 0.968609 + 0.248591i \(0.0799675\pi\)
−0.968609 + 0.248591i \(0.920033\pi\)
\(798\) 0 0
\(799\) −54.8402 −1.94011
\(800\) 0 0
\(801\) 14.2439i 0.503284i
\(802\) 0 0
\(803\) −10.9292 −0.385683
\(804\) 0 0
\(805\) −14.1608 + 10.4626i −0.499104 + 0.368758i
\(806\) 0 0