Properties

Label 4560.2.d.j.2431.7
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.7
Root \(0.738386i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.j.2431.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +0.238175i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +0.238175i q^{7} +1.00000 q^{9} +5.02175i q^{11} +5.35047i q^{13} -1.00000 q^{15} +4.36324 q^{17} +(1.72588 - 4.00267i) q^{19} -0.238175i q^{21} +5.83099i q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.48585i q^{29} -5.65870 q^{31} -5.02175i q^{33} +0.238175i q^{35} +6.51714i q^{37} -5.35047i q^{39} -5.59282i q^{41} -6.99046i q^{43} +1.00000 q^{45} -0.328717i q^{47} +6.94327 q^{49} -4.36324 q^{51} -9.29509i q^{53} +5.02175i q^{55} +(-1.72588 + 4.00267i) q^{57} -1.71543 q^{59} -13.9433 q^{61} +0.238175i q^{63} +5.35047i q^{65} -10.6486 q^{67} -5.83099i q^{69} -3.29546 q^{71} +8.74722 q^{73} -1.00000 q^{75} -1.19605 q^{77} -7.52352 q^{79} +1.00000 q^{81} +7.08101i q^{83} +4.36324 q^{85} -8.48585i q^{87} +5.67099i q^{89} -1.27435 q^{91} +5.65870 q^{93} +(1.72588 - 4.00267i) q^{95} -10.4546i q^{97} +5.02175i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} + 12q^{5} + 12q^{9} - 12q^{15} + 4q^{17} + 12q^{25} - 12q^{27} + 12q^{31} + 12q^{45} - 28q^{49} - 4q^{51} - 52q^{59} - 56q^{61} + 32q^{67} - 8q^{71} + 32q^{73} - 12q^{75} + 24q^{77} + 28q^{79} + 12q^{81} + 4q^{85} - 32q^{91} - 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.238175i 0.0900215i 0.998987 + 0.0450108i \(0.0143322\pi\)
−0.998987 + 0.0450108i \(0.985668\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.02175i 1.51412i 0.653348 + 0.757058i \(0.273364\pi\)
−0.653348 + 0.757058i \(0.726636\pi\)
\(12\) 0 0
\(13\) 5.35047i 1.48395i 0.670426 + 0.741977i \(0.266111\pi\)
−0.670426 + 0.741977i \(0.733889\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 4.36324 1.05824 0.529121 0.848547i \(-0.322522\pi\)
0.529121 + 0.848547i \(0.322522\pi\)
\(18\) 0 0
\(19\) 1.72588 4.00267i 0.395944 0.918275i
\(20\) 0 0
\(21\) 0.238175i 0.0519739i
\(22\) 0 0
\(23\) 5.83099i 1.21585i 0.793996 + 0.607923i \(0.207997\pi\)
−0.793996 + 0.607923i \(0.792003\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.48585i 1.57578i 0.615814 + 0.787892i \(0.288827\pi\)
−0.615814 + 0.787892i \(0.711173\pi\)
\(30\) 0 0
\(31\) −5.65870 −1.01633 −0.508166 0.861259i \(-0.669677\pi\)
−0.508166 + 0.861259i \(0.669677\pi\)
\(32\) 0 0
\(33\) 5.02175i 0.874175i
\(34\) 0 0
\(35\) 0.238175i 0.0402588i
\(36\) 0 0
\(37\) 6.51714i 1.07141i 0.844405 + 0.535706i \(0.179954\pi\)
−0.844405 + 0.535706i \(0.820046\pi\)
\(38\) 0 0
\(39\) 5.35047i 0.856761i
\(40\) 0 0
\(41\) 5.59282i 0.873451i −0.899595 0.436726i \(-0.856138\pi\)
0.899595 0.436726i \(-0.143862\pi\)
\(42\) 0 0
\(43\) 6.99046i 1.06604i −0.846104 0.533018i \(-0.821058\pi\)
0.846104 0.533018i \(-0.178942\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0.328717i 0.0479484i −0.999713 0.0239742i \(-0.992368\pi\)
0.999713 0.0239742i \(-0.00763195\pi\)
\(48\) 0 0
\(49\) 6.94327 0.991896
\(50\) 0 0
\(51\) −4.36324 −0.610976
\(52\) 0 0
\(53\) 9.29509i 1.27678i −0.769713 0.638390i \(-0.779601\pi\)
0.769713 0.638390i \(-0.220399\pi\)
\(54\) 0 0
\(55\) 5.02175i 0.677133i
\(56\) 0 0
\(57\) −1.72588 + 4.00267i −0.228598 + 0.530166i
\(58\) 0 0
\(59\) −1.71543 −0.223330 −0.111665 0.993746i \(-0.535618\pi\)
−0.111665 + 0.993746i \(0.535618\pi\)
\(60\) 0 0
\(61\) −13.9433 −1.78525 −0.892627 0.450797i \(-0.851140\pi\)
−0.892627 + 0.450797i \(0.851140\pi\)
\(62\) 0 0
\(63\) 0.238175i 0.0300072i
\(64\) 0 0
\(65\) 5.35047i 0.663644i
\(66\) 0 0
\(67\) −10.6486 −1.30094 −0.650469 0.759533i \(-0.725428\pi\)
−0.650469 + 0.759533i \(0.725428\pi\)
\(68\) 0 0
\(69\) 5.83099i 0.701969i
\(70\) 0 0
\(71\) −3.29546 −0.391099 −0.195550 0.980694i \(-0.562649\pi\)
−0.195550 + 0.980694i \(0.562649\pi\)
\(72\) 0 0
\(73\) 8.74722 1.02378 0.511892 0.859050i \(-0.328945\pi\)
0.511892 + 0.859050i \(0.328945\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −1.19605 −0.136303
\(78\) 0 0
\(79\) −7.52352 −0.846462 −0.423231 0.906022i \(-0.639104\pi\)
−0.423231 + 0.906022i \(0.639104\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.08101i 0.777241i 0.921398 + 0.388621i \(0.127048\pi\)
−0.921398 + 0.388621i \(0.872952\pi\)
\(84\) 0 0
\(85\) 4.36324 0.473260
\(86\) 0 0
\(87\) 8.48585i 0.909779i
\(88\) 0 0
\(89\) 5.67099i 0.601123i 0.953762 + 0.300562i \(0.0971741\pi\)
−0.953762 + 0.300562i \(0.902826\pi\)
\(90\) 0 0
\(91\) −1.27435 −0.133588
\(92\) 0 0
\(93\) 5.65870 0.586780
\(94\) 0 0
\(95\) 1.72588 4.00267i 0.177072 0.410665i
\(96\) 0 0
\(97\) 10.4546i 1.06150i −0.847528 0.530750i \(-0.821910\pi\)
0.847528 0.530750i \(-0.178090\pi\)
\(98\) 0 0
\(99\) 5.02175i 0.504705i
\(100\) 0 0
\(101\) 8.07846 0.803837 0.401918 0.915676i \(-0.368344\pi\)
0.401918 + 0.915676i \(0.368344\pi\)
\(102\) 0 0
\(103\) 10.6486 1.04924 0.524621 0.851336i \(-0.324207\pi\)
0.524621 + 0.851336i \(0.324207\pi\)
\(104\) 0 0
\(105\) 0.238175i 0.0232435i
\(106\) 0 0
\(107\) −15.3532 −1.48425 −0.742124 0.670263i \(-0.766181\pi\)
−0.742124 + 0.670263i \(0.766181\pi\)
\(108\) 0 0
\(109\) 11.2935i 1.08172i 0.841112 + 0.540862i \(0.181902\pi\)
−0.841112 + 0.540862i \(0.818098\pi\)
\(110\) 0 0
\(111\) 6.51714i 0.618580i
\(112\) 0 0
\(113\) 8.62931i 0.811778i −0.913922 0.405889i \(-0.866962\pi\)
0.913922 0.405889i \(-0.133038\pi\)
\(114\) 0 0
\(115\) 5.83099i 0.543743i
\(116\) 0 0
\(117\) 5.35047i 0.494651i
\(118\) 0 0
\(119\) 1.03921i 0.0952645i
\(120\) 0 0
\(121\) −14.2180 −1.29255
\(122\) 0 0
\(123\) 5.59282i 0.504287i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 3.90142 0.346195 0.173098 0.984905i \(-0.444622\pi\)
0.173098 + 0.984905i \(0.444622\pi\)
\(128\) 0 0
\(129\) 6.99046i 0.615476i
\(130\) 0 0
\(131\) 9.53844i 0.833377i −0.909049 0.416689i \(-0.863191\pi\)
0.909049 0.416689i \(-0.136809\pi\)
\(132\) 0 0
\(133\) 0.953333 + 0.411061i 0.0826645 + 0.0356435i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −20.4270 −1.74520 −0.872600 0.488436i \(-0.837568\pi\)
−0.872600 + 0.488436i \(0.837568\pi\)
\(138\) 0 0
\(139\) 17.7567i 1.50610i 0.657962 + 0.753051i \(0.271419\pi\)
−0.657962 + 0.753051i \(0.728581\pi\)
\(140\) 0 0
\(141\) 0.328717i 0.0276830i
\(142\) 0 0
\(143\) −26.8687 −2.24688
\(144\) 0 0
\(145\) 8.48585i 0.704712i
\(146\) 0 0
\(147\) −6.94327 −0.572671
\(148\) 0 0
\(149\) 12.6129 1.03329 0.516643 0.856201i \(-0.327181\pi\)
0.516643 + 0.856201i \(0.327181\pi\)
\(150\) 0 0
\(151\) −19.5026 −1.58710 −0.793548 0.608507i \(-0.791769\pi\)
−0.793548 + 0.608507i \(0.791769\pi\)
\(152\) 0 0
\(153\) 4.36324 0.352747
\(154\) 0 0
\(155\) −5.65870 −0.454518
\(156\) 0 0
\(157\) 19.0895 1.52351 0.761754 0.647866i \(-0.224338\pi\)
0.761754 + 0.647866i \(0.224338\pi\)
\(158\) 0 0
\(159\) 9.29509i 0.737149i
\(160\) 0 0
\(161\) −1.38879 −0.109452
\(162\) 0 0
\(163\) 5.51818i 0.432218i 0.976369 + 0.216109i \(0.0693366\pi\)
−0.976369 + 0.216109i \(0.930663\pi\)
\(164\) 0 0
\(165\) 5.02175i 0.390943i
\(166\) 0 0
\(167\) 11.5513 0.893869 0.446934 0.894567i \(-0.352516\pi\)
0.446934 + 0.894567i \(0.352516\pi\)
\(168\) 0 0
\(169\) −15.6275 −1.20212
\(170\) 0 0
\(171\) 1.72588 4.00267i 0.131981 0.306092i
\(172\) 0 0
\(173\) 8.05025i 0.612049i −0.952024 0.306025i \(-0.901001\pi\)
0.952024 0.306025i \(-0.0989990\pi\)
\(174\) 0 0
\(175\) 0.238175i 0.0180043i
\(176\) 0 0
\(177\) 1.71543 0.128940
\(178\) 0 0
\(179\) −2.79703 −0.209060 −0.104530 0.994522i \(-0.533334\pi\)
−0.104530 + 0.994522i \(0.533334\pi\)
\(180\) 0 0
\(181\) 7.70499i 0.572707i 0.958124 + 0.286354i \(0.0924432\pi\)
−0.958124 + 0.286354i \(0.907557\pi\)
\(182\) 0 0
\(183\) 13.9433 1.03072
\(184\) 0 0
\(185\) 6.51714i 0.479150i
\(186\) 0 0
\(187\) 21.9111i 1.60230i
\(188\) 0 0
\(189\) 0.238175i 0.0173246i
\(190\) 0 0
\(191\) 0.0287172i 0.00207791i −0.999999 0.00103895i \(-0.999669\pi\)
0.999999 0.00103895i \(-0.000330709\pi\)
\(192\) 0 0
\(193\) 4.18380i 0.301156i −0.988598 0.150578i \(-0.951886\pi\)
0.988598 0.150578i \(-0.0481135\pi\)
\(194\) 0 0
\(195\) 5.35047i 0.383155i
\(196\) 0 0
\(197\) 0.155471 0.0110768 0.00553841 0.999985i \(-0.498237\pi\)
0.00553841 + 0.999985i \(0.498237\pi\)
\(198\) 0 0
\(199\) 1.74291i 0.123551i −0.998090 0.0617757i \(-0.980324\pi\)
0.998090 0.0617757i \(-0.0196764\pi\)
\(200\) 0 0
\(201\) 10.6486 0.751097
\(202\) 0 0
\(203\) −2.02111 −0.141854
\(204\) 0 0
\(205\) 5.59282i 0.390619i
\(206\) 0 0
\(207\) 5.83099i 0.405282i
\(208\) 0 0
\(209\) 20.1004 + 8.66694i 1.39037 + 0.599505i
\(210\) 0 0
\(211\) 18.7478 1.29065 0.645327 0.763906i \(-0.276721\pi\)
0.645327 + 0.763906i \(0.276721\pi\)
\(212\) 0 0
\(213\) 3.29546 0.225801
\(214\) 0 0
\(215\) 6.99046i 0.476746i
\(216\) 0 0
\(217\) 1.34776i 0.0914918i
\(218\) 0 0
\(219\) −8.74722 −0.591082
\(220\) 0 0
\(221\) 23.3454i 1.57038i
\(222\) 0 0
\(223\) −13.8236 −0.925695 −0.462848 0.886438i \(-0.653172\pi\)
−0.462848 + 0.886438i \(0.653172\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 18.1423 1.20414 0.602072 0.798442i \(-0.294342\pi\)
0.602072 + 0.798442i \(0.294342\pi\)
\(228\) 0 0
\(229\) −5.92237 −0.391361 −0.195681 0.980668i \(-0.562692\pi\)
−0.195681 + 0.980668i \(0.562692\pi\)
\(230\) 0 0
\(231\) 1.19605 0.0786945
\(232\) 0 0
\(233\) 20.8767 1.36768 0.683839 0.729633i \(-0.260309\pi\)
0.683839 + 0.729633i \(0.260309\pi\)
\(234\) 0 0
\(235\) 0.328717i 0.0214432i
\(236\) 0 0
\(237\) 7.52352 0.488705
\(238\) 0 0
\(239\) 3.58436i 0.231853i −0.993258 0.115927i \(-0.963016\pi\)
0.993258 0.115927i \(-0.0369837\pi\)
\(240\) 0 0
\(241\) 16.4985i 1.06276i 0.847133 + 0.531380i \(0.178326\pi\)
−0.847133 + 0.531380i \(0.821674\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 6.94327 0.443589
\(246\) 0 0
\(247\) 21.4161 + 9.23427i 1.36268 + 0.587562i
\(248\) 0 0
\(249\) 7.08101i 0.448741i
\(250\) 0 0
\(251\) 14.9787i 0.945450i 0.881210 + 0.472725i \(0.156730\pi\)
−0.881210 + 0.472725i \(0.843270\pi\)
\(252\) 0 0
\(253\) −29.2818 −1.84093
\(254\) 0 0
\(255\) −4.36324 −0.273237
\(256\) 0 0
\(257\) 13.8363i 0.863086i 0.902092 + 0.431543i \(0.142031\pi\)
−0.902092 + 0.431543i \(0.857969\pi\)
\(258\) 0 0
\(259\) −1.55222 −0.0964501
\(260\) 0 0
\(261\) 8.48585i 0.525261i
\(262\) 0 0
\(263\) 29.2462i 1.80340i −0.432364 0.901699i \(-0.642320\pi\)
0.432364 0.901699i \(-0.357680\pi\)
\(264\) 0 0
\(265\) 9.29509i 0.570993i
\(266\) 0 0
\(267\) 5.67099i 0.347059i
\(268\) 0 0
\(269\) 24.8947i 1.51786i 0.651174 + 0.758928i \(0.274277\pi\)
−0.651174 + 0.758928i \(0.725723\pi\)
\(270\) 0 0
\(271\) 14.9419i 0.907655i 0.891090 + 0.453827i \(0.149942\pi\)
−0.891090 + 0.453827i \(0.850058\pi\)
\(272\) 0 0
\(273\) 1.27435 0.0771269
\(274\) 0 0
\(275\) 5.02175i 0.302823i
\(276\) 0 0
\(277\) −16.1722 −0.971691 −0.485845 0.874045i \(-0.661488\pi\)
−0.485845 + 0.874045i \(0.661488\pi\)
\(278\) 0 0
\(279\) −5.65870 −0.338778
\(280\) 0 0
\(281\) 12.0827i 0.720792i 0.932799 + 0.360396i \(0.117358\pi\)
−0.932799 + 0.360396i \(0.882642\pi\)
\(282\) 0 0
\(283\) 5.35350i 0.318232i −0.987260 0.159116i \(-0.949136\pi\)
0.987260 0.159116i \(-0.0508645\pi\)
\(284\) 0 0
\(285\) −1.72588 + 4.00267i −0.102232 + 0.237098i
\(286\) 0 0
\(287\) 1.33207 0.0786294
\(288\) 0 0
\(289\) 2.03787 0.119875
\(290\) 0 0
\(291\) 10.4546i 0.612857i
\(292\) 0 0
\(293\) 1.93582i 0.113092i 0.998400 + 0.0565460i \(0.0180087\pi\)
−0.998400 + 0.0565460i \(0.981991\pi\)
\(294\) 0 0
\(295\) −1.71543 −0.0998761
\(296\) 0 0
\(297\) 5.02175i 0.291392i
\(298\) 0 0
\(299\) −31.1985 −1.80426
\(300\) 0 0
\(301\) 1.66495 0.0959662
\(302\) 0 0
\(303\) −8.07846 −0.464095
\(304\) 0 0
\(305\) −13.9433 −0.798389
\(306\) 0 0
\(307\) −17.6518 −1.00744 −0.503720 0.863867i \(-0.668036\pi\)
−0.503720 + 0.863867i \(0.668036\pi\)
\(308\) 0 0
\(309\) −10.6486 −0.605780
\(310\) 0 0
\(311\) 8.01253i 0.454349i 0.973854 + 0.227175i \(0.0729488\pi\)
−0.973854 + 0.227175i \(0.927051\pi\)
\(312\) 0 0
\(313\) −4.84580 −0.273901 −0.136950 0.990578i \(-0.543730\pi\)
−0.136950 + 0.990578i \(0.543730\pi\)
\(314\) 0 0
\(315\) 0.238175i 0.0134196i
\(316\) 0 0
\(317\) 0.756756i 0.0425036i −0.999774 0.0212518i \(-0.993235\pi\)
0.999774 0.0212518i \(-0.00676517\pi\)
\(318\) 0 0
\(319\) −42.6139 −2.38592
\(320\) 0 0
\(321\) 15.3532 0.856931
\(322\) 0 0
\(323\) 7.53043 17.4646i 0.419004 0.971756i
\(324\) 0 0
\(325\) 5.35047i 0.296791i
\(326\) 0 0
\(327\) 11.2935i 0.624533i
\(328\) 0 0
\(329\) 0.0782921 0.00431638
\(330\) 0 0
\(331\) 20.0857 1.10401 0.552006 0.833840i \(-0.313863\pi\)
0.552006 + 0.833840i \(0.313863\pi\)
\(332\) 0 0
\(333\) 6.51714i 0.357137i
\(334\) 0 0
\(335\) −10.6486 −0.581797
\(336\) 0 0
\(337\) 9.90316i 0.539460i 0.962936 + 0.269730i \(0.0869344\pi\)
−0.962936 + 0.269730i \(0.913066\pi\)
\(338\) 0 0
\(339\) 8.62931i 0.468680i
\(340\) 0 0
\(341\) 28.4166i 1.53885i
\(342\) 0 0
\(343\) 3.32093i 0.179314i
\(344\) 0 0
\(345\) 5.83099i 0.313930i
\(346\) 0 0
\(347\) 4.39677i 0.236031i −0.993012 0.118016i \(-0.962347\pi\)
0.993012 0.118016i \(-0.0376533\pi\)
\(348\) 0 0
\(349\) −30.8909 −1.65355 −0.826776 0.562531i \(-0.809828\pi\)
−0.826776 + 0.562531i \(0.809828\pi\)
\(350\) 0 0
\(351\) 5.35047i 0.285587i
\(352\) 0 0
\(353\) −31.2106 −1.66118 −0.830588 0.556888i \(-0.811995\pi\)
−0.830588 + 0.556888i \(0.811995\pi\)
\(354\) 0 0
\(355\) −3.29546 −0.174905
\(356\) 0 0
\(357\) 1.03921i 0.0550010i
\(358\) 0 0
\(359\) 16.6056i 0.876408i 0.898876 + 0.438204i \(0.144385\pi\)
−0.898876 + 0.438204i \(0.855615\pi\)
\(360\) 0 0
\(361\) −13.0427 13.8162i −0.686457 0.727171i
\(362\) 0 0
\(363\) 14.2180 0.746251
\(364\) 0 0
\(365\) 8.74722 0.457850
\(366\) 0 0
\(367\) 16.0432i 0.837449i 0.908113 + 0.418724i \(0.137523\pi\)
−0.908113 + 0.418724i \(0.862477\pi\)
\(368\) 0 0
\(369\) 5.59282i 0.291150i
\(370\) 0 0
\(371\) 2.21385 0.114938
\(372\) 0 0
\(373\) 2.83604i 0.146844i −0.997301 0.0734222i \(-0.976608\pi\)
0.997301 0.0734222i \(-0.0233921\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −45.4033 −2.33839
\(378\) 0 0
\(379\) −27.8447 −1.43029 −0.715143 0.698978i \(-0.753638\pi\)
−0.715143 + 0.698978i \(0.753638\pi\)
\(380\) 0 0
\(381\) −3.90142 −0.199876
\(382\) 0 0
\(383\) 14.3563 0.733574 0.366787 0.930305i \(-0.380458\pi\)
0.366787 + 0.930305i \(0.380458\pi\)
\(384\) 0 0
\(385\) −1.19605 −0.0609565
\(386\) 0 0
\(387\) 6.99046i 0.355345i
\(388\) 0 0
\(389\) 17.6300 0.893877 0.446938 0.894565i \(-0.352514\pi\)
0.446938 + 0.894565i \(0.352514\pi\)
\(390\) 0 0
\(391\) 25.4420i 1.28666i
\(392\) 0 0
\(393\) 9.53844i 0.481151i
\(394\) 0 0
\(395\) −7.52352 −0.378549
\(396\) 0 0
\(397\) 20.5833 1.03305 0.516524 0.856273i \(-0.327226\pi\)
0.516524 + 0.856273i \(0.327226\pi\)
\(398\) 0 0
\(399\) −0.953333 0.411061i −0.0477264 0.0205788i
\(400\) 0 0
\(401\) 38.6997i 1.93257i −0.257472 0.966286i \(-0.582890\pi\)
0.257472 0.966286i \(-0.417110\pi\)
\(402\) 0 0
\(403\) 30.2767i 1.50819i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −32.7275 −1.62224
\(408\) 0 0
\(409\) 22.3093i 1.10312i 0.834134 + 0.551562i \(0.185968\pi\)
−0.834134 + 0.551562i \(0.814032\pi\)
\(410\) 0 0
\(411\) 20.4270 1.00759
\(412\) 0 0
\(413\) 0.408571i 0.0201045i
\(414\) 0 0
\(415\) 7.08101i 0.347593i
\(416\) 0 0
\(417\) 17.7567i 0.869549i
\(418\) 0 0
\(419\) 16.0263i 0.782936i 0.920192 + 0.391468i \(0.128033\pi\)
−0.920192 + 0.391468i \(0.871967\pi\)
\(420\) 0 0
\(421\) 32.4549i 1.58175i 0.611975 + 0.790877i \(0.290375\pi\)
−0.611975 + 0.790877i \(0.709625\pi\)
\(422\) 0 0
\(423\) 0.328717i 0.0159828i
\(424\) 0 0
\(425\) 4.36324 0.211648
\(426\) 0 0
\(427\) 3.32093i 0.160711i
\(428\) 0 0
\(429\) 26.8687 1.29723
\(430\) 0 0
\(431\) −10.3195 −0.497073 −0.248536 0.968623i \(-0.579950\pi\)
−0.248536 + 0.968623i \(0.579950\pi\)
\(432\) 0 0
\(433\) 5.28376i 0.253921i 0.991908 + 0.126961i \(0.0405222\pi\)
−0.991908 + 0.126961i \(0.959478\pi\)
\(434\) 0 0
\(435\) 8.48585i 0.406866i
\(436\) 0 0
\(437\) 23.3395 + 10.0636i 1.11648 + 0.481407i
\(438\) 0 0
\(439\) −29.5247 −1.40914 −0.704569 0.709635i \(-0.748860\pi\)
−0.704569 + 0.709635i \(0.748860\pi\)
\(440\) 0 0
\(441\) 6.94327 0.330632
\(442\) 0 0
\(443\) 6.29905i 0.299277i 0.988741 + 0.149638i \(0.0478110\pi\)
−0.988741 + 0.149638i \(0.952189\pi\)
\(444\) 0 0
\(445\) 5.67099i 0.268831i
\(446\) 0 0
\(447\) −12.6129 −0.596568
\(448\) 0 0
\(449\) 30.4136i 1.43531i −0.696401 0.717653i \(-0.745216\pi\)
0.696401 0.717653i \(-0.254784\pi\)
\(450\) 0 0
\(451\) 28.0857 1.32251
\(452\) 0 0
\(453\) 19.5026 0.916311
\(454\) 0 0
\(455\) −1.27435 −0.0597422
\(456\) 0 0
\(457\) −28.1217 −1.31548 −0.657739 0.753246i \(-0.728487\pi\)
−0.657739 + 0.753246i \(0.728487\pi\)
\(458\) 0 0
\(459\) −4.36324 −0.203659
\(460\) 0 0
\(461\) −3.95794 −0.184339 −0.0921697 0.995743i \(-0.529380\pi\)
−0.0921697 + 0.995743i \(0.529380\pi\)
\(462\) 0 0
\(463\) 12.2869i 0.571019i 0.958376 + 0.285510i \(0.0921629\pi\)
−0.958376 + 0.285510i \(0.907837\pi\)
\(464\) 0 0
\(465\) 5.65870 0.262416
\(466\) 0 0
\(467\) 31.5931i 1.46195i 0.682403 + 0.730976i \(0.260935\pi\)
−0.682403 + 0.730976i \(0.739065\pi\)
\(468\) 0 0
\(469\) 2.53624i 0.117112i
\(470\) 0 0
\(471\) −19.0895 −0.879598
\(472\) 0 0
\(473\) 35.1044 1.61410
\(474\) 0 0
\(475\) 1.72588 4.00267i 0.0791888 0.183655i
\(476\) 0 0
\(477\) 9.29509i 0.425593i
\(478\) 0 0
\(479\) 24.6890i 1.12807i −0.825752 0.564034i \(-0.809249\pi\)
0.825752 0.564034i \(-0.190751\pi\)
\(480\) 0 0
\(481\) −34.8698 −1.58992
\(482\) 0 0
\(483\) 1.38879 0.0631923
\(484\) 0 0
\(485\) 10.4546i 0.474717i
\(486\) 0 0
\(487\) 20.9253 0.948216 0.474108 0.880467i \(-0.342771\pi\)
0.474108 + 0.880467i \(0.342771\pi\)
\(488\) 0 0
\(489\) 5.51818i 0.249541i
\(490\) 0 0
\(491\) 9.15466i 0.413144i 0.978431 + 0.206572i \(0.0662308\pi\)
−0.978431 + 0.206572i \(0.933769\pi\)
\(492\) 0 0
\(493\) 37.0258i 1.66756i
\(494\) 0 0
\(495\) 5.02175i 0.225711i
\(496\) 0 0
\(497\) 0.784895i 0.0352073i
\(498\) 0 0
\(499\) 32.8579i 1.47092i 0.677566 + 0.735462i \(0.263035\pi\)
−0.677566 + 0.735462i \(0.736965\pi\)
\(500\) 0 0
\(501\) −11.5513 −0.516075
\(502\) 0 0
\(503\) 39.3433i 1.75423i −0.480278 0.877116i \(-0.659464\pi\)
0.480278 0.877116i \(-0.340536\pi\)
\(504\) 0 0
\(505\) 8.07846 0.359487
\(506\) 0 0
\(507\) 15.6275 0.694043
\(508\) 0 0
\(509\) 1.73039i 0.0766983i 0.999264 + 0.0383492i \(0.0122099\pi\)
−0.999264 + 0.0383492i \(0.987790\pi\)
\(510\) 0 0
\(511\) 2.08337i 0.0921626i
\(512\) 0 0
\(513\) −1.72588 + 4.00267i −0.0761994 + 0.176722i
\(514\) 0 0
\(515\) 10.6486 0.469235
\(516\) 0 0
\(517\) 1.65074 0.0725994
\(518\) 0 0
\(519\) 8.05025i 0.353367i
\(520\) 0 0
\(521\) 17.5057i 0.766939i −0.923554 0.383470i \(-0.874729\pi\)
0.923554 0.383470i \(-0.125271\pi\)
\(522\) 0 0
\(523\) 17.0971 0.747604 0.373802 0.927509i \(-0.378054\pi\)
0.373802 + 0.927509i \(0.378054\pi\)
\(524\) 0 0
\(525\) 0.238175i 0.0103948i
\(526\) 0 0
\(527\) −24.6903 −1.07553
\(528\) 0 0
\(529\) −11.0005 −0.478280
\(530\) 0 0
\(531\) −1.71543 −0.0744433
\(532\) 0 0
\(533\) 29.9242 1.29616
\(534\) 0 0
\(535\) −15.3532 −0.663776
\(536\) 0 0
\(537\) 2.79703 0.120701
\(538\) 0 0
\(539\) 34.8674i 1.50185i
\(540\) 0 0
\(541\) 29.4743 1.26720 0.633600 0.773661i \(-0.281577\pi\)
0.633600 + 0.773661i \(0.281577\pi\)
\(542\) 0 0
\(543\) 7.70499i 0.330653i
\(544\) 0 0
\(545\) 11.2935i 0.483761i
\(546\) 0 0
\(547\) 26.8488 1.14797 0.573986 0.818865i \(-0.305396\pi\)
0.573986 + 0.818865i \(0.305396\pi\)
\(548\) 0 0
\(549\) −13.9433 −0.595084
\(550\) 0 0
\(551\) 33.9660 + 14.6456i 1.44700 + 0.623922i
\(552\) 0 0
\(553\) 1.79191i 0.0761998i
\(554\) 0 0
\(555\) 6.51714i 0.276637i
\(556\) 0 0
\(557\) 20.7543 0.879387 0.439694 0.898148i \(-0.355087\pi\)
0.439694 + 0.898148i \(0.355087\pi\)
\(558\) 0 0
\(559\) 37.4023 1.58195
\(560\) 0 0
\(561\) 21.9111i 0.925088i
\(562\) 0 0
\(563\) 26.5147 1.11746 0.558730 0.829350i \(-0.311289\pi\)
0.558730 + 0.829350i \(0.311289\pi\)
\(564\) 0 0
\(565\) 8.62931i 0.363038i
\(566\) 0 0
\(567\) 0.238175i 0.0100024i
\(568\) 0 0
\(569\) 26.1478i 1.09617i −0.836421 0.548087i \(-0.815356\pi\)
0.836421 0.548087i \(-0.184644\pi\)
\(570\) 0 0
\(571\) 31.8025i 1.33089i −0.746445 0.665447i \(-0.768241\pi\)
0.746445 0.665447i \(-0.231759\pi\)
\(572\) 0 0
\(573\) 0.0287172i 0.00119968i
\(574\) 0 0
\(575\) 5.83099i 0.243169i
\(576\) 0 0
\(577\) 20.2797 0.844255 0.422127 0.906537i \(-0.361284\pi\)
0.422127 + 0.906537i \(0.361284\pi\)
\(578\) 0 0
\(579\) 4.18380i 0.173873i
\(580\) 0 0
\(581\) −1.68652 −0.0699685
\(582\) 0 0
\(583\) 46.6777 1.93319
\(584\) 0 0
\(585\) 5.35047i 0.221215i
\(586\) 0 0
\(587\) 43.6829i 1.80299i −0.432794 0.901493i \(-0.642472\pi\)
0.432794 0.901493i \(-0.357528\pi\)
\(588\) 0 0
\(589\) −9.76624 + 22.6499i −0.402411 + 0.933273i
\(590\) 0 0
\(591\) −0.155471 −0.00639521
\(592\) 0 0
\(593\) −18.1504 −0.745346 −0.372673 0.927963i \(-0.621559\pi\)
−0.372673 + 0.927963i \(0.621559\pi\)
\(594\) 0 0
\(595\) 1.03921i 0.0426036i
\(596\) 0 0
\(597\) 1.74291i 0.0713325i
\(598\) 0 0
\(599\) −5.53126 −0.226001 −0.113001 0.993595i \(-0.536046\pi\)
−0.113001 + 0.993595i \(0.536046\pi\)
\(600\) 0 0
\(601\) 30.5360i 1.24559i 0.782386 + 0.622794i \(0.214003\pi\)
−0.782386 + 0.622794i \(0.785997\pi\)
\(602\) 0 0
\(603\) −10.6486 −0.433646
\(604\) 0 0
\(605\) −14.2180 −0.578044
\(606\) 0 0
\(607\) 35.7869 1.45255 0.726273 0.687406i \(-0.241251\pi\)
0.726273 + 0.687406i \(0.241251\pi\)
\(608\) 0 0
\(609\) 2.02111 0.0818997
\(610\) 0 0
\(611\) 1.75879 0.0711531
\(612\) 0 0
\(613\) −31.9162 −1.28908 −0.644542 0.764569i \(-0.722952\pi\)
−0.644542 + 0.764569i \(0.722952\pi\)
\(614\) 0 0
\(615\) 5.59282i 0.225524i
\(616\) 0 0
\(617\) 6.75535 0.271960 0.135980 0.990712i \(-0.456582\pi\)
0.135980 + 0.990712i \(0.456582\pi\)
\(618\) 0 0
\(619\) 34.1162i 1.37125i −0.727956 0.685624i \(-0.759529\pi\)
0.727956 0.685624i \(-0.240471\pi\)
\(620\) 0 0
\(621\) 5.83099i 0.233990i
\(622\) 0 0
\(623\) −1.35068 −0.0541140
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −20.1004 8.66694i −0.802733 0.346124i
\(628\) 0 0
\(629\) 28.4359i 1.13381i
\(630\) 0 0
\(631\) 25.8948i 1.03086i 0.856933 + 0.515428i \(0.172367\pi\)
−0.856933 + 0.515428i \(0.827633\pi\)
\(632\) 0 0
\(633\) −18.7478 −0.745160
\(634\) 0 0
\(635\) 3.90142 0.154823
\(636\) 0 0
\(637\) 37.1498i 1.47193i
\(638\) 0 0
\(639\) −3.29546 −0.130366
\(640\) 0 0
\(641\) 18.7724i 0.741464i 0.928740 + 0.370732i \(0.120893\pi\)
−0.928740 + 0.370732i \(0.879107\pi\)
\(642\) 0 0
\(643\) 4.49043i 0.177085i −0.996072 0.0885427i \(-0.971779\pi\)
0.996072 0.0885427i \(-0.0282210\pi\)
\(644\) 0 0
\(645\) 6.99046i 0.275249i
\(646\) 0 0
\(647\) 13.8250i 0.543515i 0.962366 + 0.271757i \(0.0876049\pi\)
−0.962366 + 0.271757i \(0.912395\pi\)
\(648\) 0 0
\(649\) 8.61446i 0.338147i
\(650\) 0 0
\(651\) 1.34776i 0.0528228i
\(652\) 0 0
\(653\) 26.7480 1.04673 0.523366 0.852108i \(-0.324676\pi\)
0.523366 + 0.852108i \(0.324676\pi\)
\(654\) 0 0
\(655\) 9.53844i 0.372698i
\(656\) 0 0
\(657\) 8.74722 0.341262
\(658\) 0 0
\(659\) −9.18851 −0.357934 −0.178967 0.983855i \(-0.557275\pi\)
−0.178967 + 0.983855i \(0.557275\pi\)
\(660\) 0 0
\(661\) 33.0472i 1.28539i −0.766123 0.642694i \(-0.777817\pi\)
0.766123 0.642694i \(-0.222183\pi\)
\(662\) 0 0
\(663\) 23.3454i 0.906660i
\(664\) 0 0
\(665\) 0.953333 + 0.411061i 0.0369687 + 0.0159402i
\(666\) 0 0
\(667\) −49.4809 −1.91591
\(668\) 0 0
\(669\) 13.8236 0.534450
\(670\) 0 0
\(671\) 70.0197i 2.70308i
\(672\) 0 0
\(673\) 27.8213i 1.07243i −0.844080 0.536217i \(-0.819853\pi\)
0.844080 0.536217i \(-0.180147\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 37.1408i 1.42744i 0.700433 + 0.713718i \(0.252990\pi\)
−0.700433 + 0.713718i \(0.747010\pi\)
\(678\) 0 0
\(679\) 2.49001 0.0955579
\(680\) 0 0
\(681\) −18.1423 −0.695213
\(682\) 0 0
\(683\) 36.0081 1.37781 0.688906 0.724851i \(-0.258091\pi\)
0.688906 + 0.724851i \(0.258091\pi\)
\(684\) 0 0
\(685\) −20.4270 −0.780477
\(686\) 0 0
\(687\) 5.92237 0.225953
\(688\) 0 0
\(689\) 49.7331 1.89468
\(690\) 0 0
\(691\) 0.782984i 0.0297861i −0.999889 0.0148931i \(-0.995259\pi\)
0.999889 0.0148931i \(-0.00474078\pi\)
\(692\) 0 0
\(693\) −1.19605 −0.0454343
\(694\) 0 0
\(695\) 17.7567i 0.673550i
\(696\) 0 0
\(697\) 24.4028i 0.924322i
\(698\) 0 0
\(699\) −20.8767 −0.789629
\(700\) 0 0
\(701\) −12.3563 −0.466692 −0.233346 0.972394i \(-0.574967\pi\)
−0.233346 + 0.972394i \(0.574967\pi\)
\(702\) 0 0
\(703\) 26.0859 + 11.2478i 0.983850 + 0.424219i
\(704\) 0 0
\(705\) 0.328717i 0.0123802i
\(706\) 0 0
\(707\) 1.92408i 0.0723626i
\(708\) 0 0
\(709\) −5.63067 −0.211464 −0.105732 0.994395i \(-0.533719\pi\)
−0.105732 + 0.994395i \(0.533719\pi\)
\(710\) 0 0
\(711\) −7.52352 −0.282154
\(712\) 0 0
\(713\) 32.9958i 1.23570i
\(714\) 0 0
\(715\) −26.8687 −1.00483
\(716\) 0 0
\(717\) 3.58436i 0.133860i
\(718\) 0 0
\(719\) 25.8084i 0.962489i −0.876586 0.481245i \(-0.840185\pi\)
0.876586 0.481245i \(-0.159815\pi\)
\(720\) 0 0
\(721\) 2.53624i 0.0944543i
\(722\) 0 0
\(723\) 16.4985i 0.613585i
\(724\) 0 0
\(725\) 8.48585i 0.315157i
\(726\) 0 0
\(727\) 6.92021i 0.256656i 0.991732 + 0.128328i \(0.0409611\pi\)
−0.991732 + 0.128328i \(0.959039\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.5011i 1.12812i
\(732\) 0 0
\(733\) 30.3083 1.11946 0.559732 0.828674i \(-0.310904\pi\)
0.559732 + 0.828674i \(0.310904\pi\)
\(734\) 0 0
\(735\) −6.94327 −0.256106
\(736\) 0 0
\(737\) 53.4748i 1.96977i
\(738\) 0 0
\(739\) 37.7028i 1.38692i 0.720496 + 0.693460i \(0.243914\pi\)
−0.720496 + 0.693460i \(0.756086\pi\)
\(740\) 0 0
\(741\) −21.4161 9.23427i −0.786742 0.339229i
\(742\) 0 0
\(743\) 23.4967 0.862010 0.431005 0.902349i \(-0.358159\pi\)
0.431005 + 0.902349i \(0.358159\pi\)
\(744\) 0 0
\(745\) 12.6129 0.462100
\(746\) 0 0
\(747\) 7.08101i 0.259080i
\(748\) 0 0
\(749\) 3.65674i 0.133614i
\(750\) 0 0
\(751\) 31.7453 1.15840 0.579201 0.815185i \(-0.303365\pi\)
0.579201 + 0.815185i \(0.303365\pi\)
\(752\) 0 0
\(753\) 14.9787i 0.545856i
\(754\) 0 0
\(755\) −19.5026 −0.709771
\(756\) 0 0
\(757\) −50.7634 −1.84503 −0.922514 0.385964i \(-0.873869\pi\)
−0.922514 + 0.385964i \(0.873869\pi\)
\(758\) 0 0
\(759\) 29.2818 1.06286
\(760\) 0 0
\(761\) 9.70934 0.351963 0.175982 0.984393i \(-0.443690\pi\)
0.175982 + 0.984393i \(0.443690\pi\)
\(762\) 0 0
\(763\) −2.68983 −0.0973784
\(764\) 0 0
\(765\) 4.36324 0.157753
\(766\) 0 0
\(767\) 9.17835i 0.331411i
\(768\) 0 0
\(769\) −38.8126 −1.39962 −0.699809 0.714330i \(-0.746732\pi\)
−0.699809 + 0.714330i \(0.746732\pi\)
\(770\) 0 0
\(771\) 13.8363i 0.498303i
\(772\) 0 0
\(773\) 26.6751i 0.959438i −0.877422 0.479719i \(-0.840739\pi\)
0.877422 0.479719i \(-0.159261\pi\)
\(774\) 0 0
\(775\) −5.65870 −0.203267
\(776\) 0 0
\(777\) 1.55222 0.0556855
\(778\) 0 0
\(779\) −22.3862 9.65253i −0.802068 0.345838i
\(780\) 0 0
\(781\) 16.5490i 0.592169i
\(782\) 0 0
\(783\) 8.48585i 0.303260i
\(784\) 0 0
\(785\) 19.0895 0.681334
\(786\) 0 0
\(787\) 17.1398 0.610966 0.305483 0.952198i \(-0.401182\pi\)
0.305483 + 0.952198i \(0.401182\pi\)
\(788\) 0 0
\(789\) 29.2462i 1.04119i
\(790\) 0 0
\(791\) 2.05528 0.0730774
\(792\) 0 0
\(793\) 74.6031i 2.64923i
\(794\) 0 0
\(795\) 9.29509i 0.329663i
\(796\) 0 0
\(797\) 5.75453i 0.203836i 0.994793 + 0.101918i \(0.0324979\pi\)
−0.994793 + 0.101918i \(0.967502\pi\)
\(798\) 0 0
\(799\) 1.43427i 0.0507409i
\(800\) 0 0
\(801\) 5.67099i 0.200374i
\(802\) 0 0
\(803\) 43.9264i 1.55013i
\(804\) 0