Properties

Label 4032.2.v.e.1583.20
Level 4032
Weight 2
Character 4032.1583
Analytic conductor 32.196
Analytic rank 0
Dimension 40
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.20
Character \(\chi\) = 4032.1583
Dual form 4032.2.v.e.3599.20

$q$-expansion

\(f(q)\) \(=\) \(q+(2.96859 + 2.96859i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(2.96859 + 2.96859i) q^{5} -1.00000 q^{7} +(0.569860 - 0.569860i) q^{11} +(2.53650 + 2.53650i) q^{13} -1.03528i q^{17} +(5.23568 - 5.23568i) q^{19} +8.71217i q^{23} +12.6251i q^{25} +(-6.05233 + 6.05233i) q^{29} -3.00413i q^{31} +(-2.96859 - 2.96859i) q^{35} +(-0.149739 + 0.149739i) q^{37} +8.63459 q^{41} +(-1.73121 - 1.73121i) q^{43} -4.10865 q^{47} +1.00000 q^{49} +(-6.04092 - 6.04092i) q^{53} +3.38337 q^{55} +(6.05121 - 6.05121i) q^{59} +(5.81897 + 5.81897i) q^{61} +15.0597i q^{65} +(-0.0256948 + 0.0256948i) q^{67} +14.5257i q^{71} -5.38496i q^{73} +(-0.569860 + 0.569860i) q^{77} -3.89876i q^{79} +(1.40691 + 1.40691i) q^{83} +(3.07333 - 3.07333i) q^{85} +17.0358 q^{89} +(-2.53650 - 2.53650i) q^{91} +31.0852 q^{95} +3.75210 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{7} + O(q^{10}) \) \( 40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.96859 + 2.96859i 1.32760 + 1.32760i 0.907463 + 0.420133i \(0.138017\pi\)
0.420133 + 0.907463i \(0.361983\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.569860 0.569860i 0.171819 0.171819i −0.615959 0.787778i \(-0.711231\pi\)
0.787778 + 0.615959i \(0.211231\pi\)
\(12\) 0 0
\(13\) 2.53650 + 2.53650i 0.703498 + 0.703498i 0.965160 0.261661i \(-0.0842704\pi\)
−0.261661 + 0.965160i \(0.584270\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.03528i 0.251092i −0.992088 0.125546i \(-0.959932\pi\)
0.992088 0.125546i \(-0.0400683\pi\)
\(18\) 0 0
\(19\) 5.23568 5.23568i 1.20115 1.20115i 0.227329 0.973818i \(-0.427001\pi\)
0.973818 0.227329i \(-0.0729993\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.71217i 1.81661i 0.418304 + 0.908307i \(0.362625\pi\)
−0.418304 + 0.908307i \(0.637375\pi\)
\(24\) 0 0
\(25\) 12.6251i 2.52502i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.05233 + 6.05233i −1.12389 + 1.12389i −0.132739 + 0.991151i \(0.542377\pi\)
−0.991151 + 0.132739i \(0.957623\pi\)
\(30\) 0 0
\(31\) 3.00413i 0.539558i −0.962922 0.269779i \(-0.913049\pi\)
0.962922 0.269779i \(-0.0869506\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.96859 2.96859i −0.501784 0.501784i
\(36\) 0 0
\(37\) −0.149739 + 0.149739i −0.0246169 + 0.0246169i −0.719308 0.694691i \(-0.755541\pi\)
0.694691 + 0.719308i \(0.255541\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.63459 1.34850 0.674248 0.738505i \(-0.264468\pi\)
0.674248 + 0.738505i \(0.264468\pi\)
\(42\) 0 0
\(43\) −1.73121 1.73121i −0.264007 0.264007i 0.562673 0.826680i \(-0.309773\pi\)
−0.826680 + 0.562673i \(0.809773\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.10865 −0.599308 −0.299654 0.954048i \(-0.596871\pi\)
−0.299654 + 0.954048i \(0.596871\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.04092 6.04092i −0.829783 0.829783i 0.157703 0.987487i \(-0.449591\pi\)
−0.987487 + 0.157703i \(0.949591\pi\)
\(54\) 0 0
\(55\) 3.38337 0.456213
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.05121 6.05121i 0.787800 0.787800i −0.193333 0.981133i \(-0.561930\pi\)
0.981133 + 0.193333i \(0.0619298\pi\)
\(60\) 0 0
\(61\) 5.81897 + 5.81897i 0.745042 + 0.745042i 0.973544 0.228501i \(-0.0733826\pi\)
−0.228501 + 0.973544i \(0.573383\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0597i 1.86792i
\(66\) 0 0
\(67\) −0.0256948 + 0.0256948i −0.00313912 + 0.00313912i −0.708675 0.705535i \(-0.750707\pi\)
0.705535 + 0.708675i \(0.250707\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.5257i 1.72388i 0.507008 + 0.861941i \(0.330752\pi\)
−0.507008 + 0.861941i \(0.669248\pi\)
\(72\) 0 0
\(73\) 5.38496i 0.630261i −0.949048 0.315131i \(-0.897952\pi\)
0.949048 0.315131i \(-0.102048\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.569860 + 0.569860i −0.0649416 + 0.0649416i
\(78\) 0 0
\(79\) 3.89876i 0.438645i −0.975652 0.219322i \(-0.929615\pi\)
0.975652 0.219322i \(-0.0703846\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.40691 + 1.40691i 0.154428 + 0.154428i 0.780092 0.625664i \(-0.215172\pi\)
−0.625664 + 0.780092i \(0.715172\pi\)
\(84\) 0 0
\(85\) 3.07333 3.07333i 0.333349 0.333349i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 17.0358 1.80579 0.902897 0.429856i \(-0.141436\pi\)
0.902897 + 0.429856i \(0.141436\pi\)
\(90\) 0 0
\(91\) −2.53650 2.53650i −0.265897 0.265897i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 31.0852 3.18928
\(96\) 0 0
\(97\) 3.75210 0.380968 0.190484 0.981690i \(-0.438994\pi\)
0.190484 + 0.981690i \(0.438994\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.23327 + 1.23327i 0.122715 + 0.122715i 0.765797 0.643082i \(-0.222345\pi\)
−0.643082 + 0.765797i \(0.722345\pi\)
\(102\) 0 0
\(103\) −14.3330 −1.41227 −0.706134 0.708078i \(-0.749562\pi\)
−0.706134 + 0.708078i \(0.749562\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.33395 + 7.33395i −0.709000 + 0.709000i −0.966325 0.257325i \(-0.917159\pi\)
0.257325 + 0.966325i \(0.417159\pi\)
\(108\) 0 0
\(109\) −4.12218 4.12218i −0.394833 0.394833i 0.481573 0.876406i \(-0.340066\pi\)
−0.876406 + 0.481573i \(0.840066\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.1524i 1.14320i −0.820533 0.571599i \(-0.806323\pi\)
0.820533 0.571599i \(-0.193677\pi\)
\(114\) 0 0
\(115\) −25.8629 + 25.8629i −2.41173 + 2.41173i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.03528i 0.0949040i
\(120\) 0 0
\(121\) 10.3505i 0.940956i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −22.6358 + 22.6358i −2.02461 + 2.02461i
\(126\) 0 0
\(127\) 1.32089i 0.117210i −0.998281 0.0586052i \(-0.981335\pi\)
0.998281 0.0586052i \(-0.0186653\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.67723 1.67723i −0.146540 0.146540i 0.630030 0.776571i \(-0.283043\pi\)
−0.776571 + 0.630030i \(0.783043\pi\)
\(132\) 0 0
\(133\) −5.23568 + 5.23568i −0.453991 + 0.453991i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.49554 −0.725823 −0.362911 0.931824i \(-0.618217\pi\)
−0.362911 + 0.931824i \(0.618217\pi\)
\(138\) 0 0
\(139\) 7.80995 + 7.80995i 0.662432 + 0.662432i 0.955953 0.293521i \(-0.0948271\pi\)
−0.293521 + 0.955953i \(0.594827\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.89090 0.241749
\(144\) 0 0
\(145\) −35.9338 −2.98414
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.1113 + 13.1113i 1.07412 + 1.07412i 0.997024 + 0.0770980i \(0.0245654\pi\)
0.0770980 + 0.997024i \(0.475435\pi\)
\(150\) 0 0
\(151\) −13.4389 −1.09364 −0.546822 0.837249i \(-0.684162\pi\)
−0.546822 + 0.837249i \(0.684162\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.91804 8.91804i 0.716314 0.716314i
\(156\) 0 0
\(157\) −5.92492 5.92492i −0.472860 0.472860i 0.429979 0.902839i \(-0.358521\pi\)
−0.902839 + 0.429979i \(0.858521\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.71217i 0.686615i
\(162\) 0 0
\(163\) 1.82469 1.82469i 0.142921 0.142921i −0.632026 0.774947i \(-0.717776\pi\)
0.774947 + 0.632026i \(0.217776\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.49283i 0.425048i 0.977156 + 0.212524i \(0.0681683\pi\)
−0.977156 + 0.212524i \(0.931832\pi\)
\(168\) 0 0
\(169\) 0.132347i 0.0101805i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.4180 11.4180i 0.868093 0.868093i −0.124168 0.992261i \(-0.539626\pi\)
0.992261 + 0.124168i \(0.0396262\pi\)
\(174\) 0 0
\(175\) 12.6251i 0.954368i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.47860 5.47860i −0.409490 0.409490i 0.472071 0.881561i \(-0.343507\pi\)
−0.881561 + 0.472071i \(0.843507\pi\)
\(180\) 0 0
\(181\) −9.42734 + 9.42734i −0.700728 + 0.700728i −0.964567 0.263838i \(-0.915011\pi\)
0.263838 + 0.964567i \(0.415011\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.889027 −0.0653625
\(186\) 0 0
\(187\) −0.589965 0.589965i −0.0431425 0.0431425i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.1433 0.733946 0.366973 0.930232i \(-0.380394\pi\)
0.366973 + 0.930232i \(0.380394\pi\)
\(192\) 0 0
\(193\) −12.6529 −0.910774 −0.455387 0.890294i \(-0.650499\pi\)
−0.455387 + 0.890294i \(0.650499\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.32030 + 4.32030i 0.307808 + 0.307808i 0.844059 0.536250i \(-0.180160\pi\)
−0.536250 + 0.844059i \(0.680160\pi\)
\(198\) 0 0
\(199\) −7.22338 −0.512052 −0.256026 0.966670i \(-0.582413\pi\)
−0.256026 + 0.966670i \(0.582413\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.05233 6.05233i 0.424790 0.424790i
\(204\) 0 0
\(205\) 25.6326 + 25.6326i 1.79026 + 1.79026i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.96721i 0.412760i
\(210\) 0 0
\(211\) −9.51875 + 9.51875i −0.655298 + 0.655298i −0.954264 0.298966i \(-0.903358\pi\)
0.298966 + 0.954264i \(0.403358\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.2785i 0.700989i
\(216\) 0 0
\(217\) 3.00413i 0.203934i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.62599 2.62599i 0.176643 0.176643i
\(222\) 0 0
\(223\) 16.0637i 1.07571i −0.843038 0.537853i \(-0.819235\pi\)
0.843038 0.537853i \(-0.180765\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.5695 + 11.5695i 0.767894 + 0.767894i 0.977735 0.209842i \(-0.0672949\pi\)
−0.209842 + 0.977735i \(0.567295\pi\)
\(228\) 0 0
\(229\) −4.50533 + 4.50533i −0.297721 + 0.297721i −0.840121 0.542400i \(-0.817516\pi\)
0.542400 + 0.840121i \(0.317516\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.2560 1.06496 0.532482 0.846442i \(-0.321260\pi\)
0.532482 + 0.846442i \(0.321260\pi\)
\(234\) 0 0
\(235\) −12.1969 12.1969i −0.795638 0.795638i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.55597 0.294701 0.147351 0.989084i \(-0.452925\pi\)
0.147351 + 0.989084i \(0.452925\pi\)
\(240\) 0 0
\(241\) −22.0079 −1.41765 −0.708827 0.705382i \(-0.750775\pi\)
−0.708827 + 0.705382i \(0.750775\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.96859 + 2.96859i 0.189657 + 0.189657i
\(246\) 0 0
\(247\) 26.5606 1.69001
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.15767 3.15767i 0.199311 0.199311i −0.600394 0.799704i \(-0.704990\pi\)
0.799704 + 0.600394i \(0.204990\pi\)
\(252\) 0 0
\(253\) 4.96472 + 4.96472i 0.312129 + 0.312129i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.930640i 0.0580517i 0.999579 + 0.0290259i \(0.00924052\pi\)
−0.999579 + 0.0290259i \(0.990759\pi\)
\(258\) 0 0
\(259\) 0.149739 0.149739i 0.00930431 0.00930431i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.8926i 0.918319i −0.888354 0.459160i \(-0.848151\pi\)
0.888354 0.459160i \(-0.151849\pi\)
\(264\) 0 0
\(265\) 35.8660i 2.20323i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.20537 + 7.20537i −0.439319 + 0.439319i −0.891783 0.452464i \(-0.850545\pi\)
0.452464 + 0.891783i \(0.350545\pi\)
\(270\) 0 0
\(271\) 25.2262i 1.53238i 0.642611 + 0.766192i \(0.277851\pi\)
−0.642611 + 0.766192i \(0.722149\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.19454 + 7.19454i 0.433847 + 0.433847i
\(276\) 0 0
\(277\) 10.1502 10.1502i 0.609866 0.609866i −0.333045 0.942911i \(-0.608076\pi\)
0.942911 + 0.333045i \(0.108076\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.47979 −0.565517 −0.282758 0.959191i \(-0.591249\pi\)
−0.282758 + 0.959191i \(0.591249\pi\)
\(282\) 0 0
\(283\) −2.74974 2.74974i −0.163455 0.163455i 0.620640 0.784095i \(-0.286873\pi\)
−0.784095 + 0.620640i \(0.786873\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.63459 −0.509683
\(288\) 0 0
\(289\) 15.9282 0.936953
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.497276 + 0.497276i 0.0290512 + 0.0290512i 0.721483 0.692432i \(-0.243461\pi\)
−0.692432 + 0.721483i \(0.743461\pi\)
\(294\) 0 0
\(295\) 35.9272 2.09176
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −22.0984 + 22.0984i −1.27798 + 1.27798i
\(300\) 0 0
\(301\) 1.73121 + 1.73121i 0.0997853 + 0.0997853i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 34.5483i 1.97823i
\(306\) 0 0
\(307\) −11.0456 + 11.0456i −0.630408 + 0.630408i −0.948170 0.317762i \(-0.897069\pi\)
0.317762 + 0.948170i \(0.397069\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.7421i 1.40299i 0.712672 + 0.701497i \(0.247485\pi\)
−0.712672 + 0.701497i \(0.752515\pi\)
\(312\) 0 0
\(313\) 20.1653i 1.13981i −0.821710 0.569906i \(-0.806980\pi\)
0.821710 0.569906i \(-0.193020\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.70522 + 8.70522i −0.488934 + 0.488934i −0.907970 0.419036i \(-0.862368\pi\)
0.419036 + 0.907970i \(0.362368\pi\)
\(318\) 0 0
\(319\) 6.89796i 0.386212i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.42040 5.42040i −0.301599 0.301599i
\(324\) 0 0
\(325\) −32.0235 + 32.0235i −1.77635 + 1.77635i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.10865 0.226517
\(330\) 0 0
\(331\) 2.78793 + 2.78793i 0.153239 + 0.153239i 0.779563 0.626324i \(-0.215441\pi\)
−0.626324 + 0.779563i \(0.715441\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.152555 −0.00833496
\(336\) 0 0
\(337\) −0.465376 −0.0253506 −0.0126753 0.999920i \(-0.504035\pi\)
−0.0126753 + 0.999920i \(0.504035\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.71193 1.71193i −0.0927064 0.0927064i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.3262 + 21.3262i −1.14485 + 1.14485i −0.157299 + 0.987551i \(0.550279\pi\)
−0.987551 + 0.157299i \(0.949721\pi\)
\(348\) 0 0
\(349\) 5.11124 + 5.11124i 0.273598 + 0.273598i 0.830547 0.556949i \(-0.188028\pi\)
−0.556949 + 0.830547i \(0.688028\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.2227i 0.544102i −0.962283 0.272051i \(-0.912298\pi\)
0.962283 0.272051i \(-0.0877019\pi\)
\(354\) 0 0
\(355\) −43.1209 + 43.1209i −2.28862 + 2.28862i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.90071i 0.311427i 0.987802 + 0.155714i \(0.0497677\pi\)
−0.987802 + 0.155714i \(0.950232\pi\)
\(360\) 0 0
\(361\) 35.8247i 1.88551i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.9857 15.9857i 0.836732 0.836732i
\(366\) 0 0
\(367\) 14.1762i 0.739994i −0.929033 0.369997i \(-0.879359\pi\)
0.929033 0.369997i \(-0.120641\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.04092 + 6.04092i 0.313629 + 0.313629i
\(372\) 0 0
\(373\) 21.5106 21.5106i 1.11378 1.11378i 0.121141 0.992635i \(-0.461345\pi\)
0.992635 0.121141i \(-0.0386553\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −30.7035 −1.58131
\(378\) 0 0
\(379\) 8.15510 + 8.15510i 0.418900 + 0.418900i 0.884824 0.465925i \(-0.154278\pi\)
−0.465925 + 0.884824i \(0.654278\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.06121 0.463006 0.231503 0.972834i \(-0.425636\pi\)
0.231503 + 0.972834i \(0.425636\pi\)
\(384\) 0 0
\(385\) −3.38337 −0.172432
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −22.1084 22.1084i −1.12094 1.12094i −0.991600 0.129339i \(-0.958715\pi\)
−0.129339 0.991600i \(1.45871\pi\)
\(390\) 0 0
\(391\) 9.01954 0.456138
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.5738 11.5738i 0.582343 0.582343i
\(396\) 0 0
\(397\) −25.1321 25.1321i −1.26134 1.26134i −0.950443 0.310899i \(-0.899370\pi\)
−0.310899 0.950443i \(-0.600630\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.54588i 0.0771975i −0.999255 0.0385987i \(-0.987711\pi\)
0.999255 0.0385987i \(-0.0122894\pi\)
\(402\) 0 0
\(403\) 7.61997 7.61997i 0.379578 0.379578i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.170660i 0.00845931i
\(408\) 0 0
\(409\) 5.72850i 0.283256i −0.989920 0.141628i \(-0.954766\pi\)
0.989920 0.141628i \(-0.0452337\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.05121 + 6.05121i −0.297761 + 0.297761i
\(414\) 0 0
\(415\) 8.35307i 0.410036i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.2299 + 26.2299i 1.28141 + 1.28141i 0.939863 + 0.341551i \(0.110952\pi\)
0.341551 + 0.939863i \(0.389048\pi\)
\(420\) 0 0
\(421\) 9.38496 9.38496i 0.457395 0.457395i −0.440404 0.897800i \(-0.645165\pi\)
0.897800 + 0.440404i \(0.145165\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.0705 0.634013
\(426\) 0 0
\(427\) −5.81897 5.81897i −0.281599 0.281599i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.9406 0.767830 0.383915 0.923368i \(-0.374576\pi\)
0.383915 + 0.923368i \(0.374576\pi\)
\(432\) 0 0
\(433\) 25.0062 1.20172 0.600862 0.799353i \(-0.294824\pi\)
0.600862 + 0.799353i \(0.294824\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 45.6141 + 45.6141i 2.18202 + 2.18202i
\(438\) 0 0
\(439\) 35.0522 1.67295 0.836474 0.548007i \(-0.184613\pi\)
0.836474 + 0.548007i \(0.184613\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −17.1307 + 17.1307i −0.813904 + 0.813904i −0.985217 0.171313i \(-0.945199\pi\)
0.171313 + 0.985217i \(0.445199\pi\)
\(444\) 0 0
\(445\) 50.5725 + 50.5725i 2.39737 + 2.39737i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.7964i 0.698286i −0.937070 0.349143i \(-0.886473\pi\)
0.937070 0.349143i \(-0.113527\pi\)
\(450\) 0 0
\(451\) 4.92050 4.92050i 0.231697 0.231697i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.0597i 0.706008i
\(456\) 0 0
\(457\) 11.9903i 0.560883i 0.959871 + 0.280442i \(0.0904809\pi\)
−0.959871 + 0.280442i \(0.909519\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.9493 14.9493i 0.696260 0.696260i −0.267342 0.963602i \(-0.586145\pi\)
0.963602 + 0.267342i \(0.0861453\pi\)
\(462\) 0 0
\(463\) 31.4696i 1.46252i −0.682101 0.731258i \(-0.738933\pi\)
0.682101 0.731258i \(-0.261067\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.7699 24.7699i −1.14621 1.14621i −0.987291 0.158922i \(-0.949198\pi\)
−0.158922 0.987291i \(1.44920\pi\)
\(468\) 0 0
\(469\) 0.0256948 0.0256948i 0.00118647 0.00118647i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.97310 −0.0907230
\(474\) 0 0
\(475\) 66.1010 + 66.1010i 3.03292 + 3.03292i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17.7194 0.809620 0.404810 0.914401i \(-0.367338\pi\)
0.404810 + 0.914401i \(0.367338\pi\)
\(480\) 0 0
\(481\) −0.759624 −0.0346359
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.1385 + 11.1385i 0.505772 + 0.505772i
\(486\) 0 0
\(487\) 33.8639 1.53452 0.767260 0.641336i \(-0.221619\pi\)
0.767260 + 0.641336i \(0.221619\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −25.3486 + 25.3486i −1.14397 + 1.14397i −0.156248 + 0.987718i \(0.549940\pi\)
−0.987718 + 0.156248i \(0.950060\pi\)
\(492\) 0 0
\(493\) 6.26586 + 6.26586i 0.282200 + 0.282200i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.5257i 0.651566i
\(498\) 0 0
\(499\) 7.26690 7.26690i 0.325311 0.325311i −0.525489 0.850800i \(-0.676118\pi\)
0.850800 + 0.525489i \(0.176118\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.3031i 0.548567i −0.961649 0.274284i \(-0.911559\pi\)
0.961649 0.274284i \(-0.0884407\pi\)
\(504\) 0 0
\(505\) 7.32217i 0.325832i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0082 14.0082i 0.620901 0.620901i −0.324861 0.945762i \(-0.605318\pi\)
0.945762 + 0.324861i \(0.105318\pi\)
\(510\) 0 0
\(511\) 5.38496i 0.238216i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −42.5487 42.5487i −1.87492 1.87492i
\(516\) 0 0
\(517\) −2.34135 + 2.34135i −0.102973 + 0.102973i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 36.8204 1.61313 0.806566 0.591144i \(-0.201323\pi\)
0.806566 + 0.591144i \(0.201323\pi\)
\(522\) 0 0
\(523\) −2.49783 2.49783i −0.109222 0.109222i 0.650384 0.759606i \(-0.274608\pi\)
−0.759606 + 0.650384i \(0.774608\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.11012 −0.135479
\(528\) 0 0
\(529\) −52.9020 −2.30009
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 21.9016 + 21.9016i 0.948664 + 0.948664i
\(534\) 0 0
\(535\) −43.5431 −1.88253
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.569860 0.569860i 0.0245456 0.0245456i
\(540\) 0 0
\(541\) 14.5049 + 14.5049i 0.623613 + 0.623613i 0.946453 0.322841i \(-0.104638\pi\)
−0.322841 + 0.946453i \(0.604638\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.4742i 1.04836i
\(546\) 0 0
\(547\) −10.3072 + 10.3072i −0.440704 + 0.440704i −0.892249 0.451545i \(-0.850873\pi\)
0.451545 + 0.892249i \(0.350873\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 63.3761i 2.69991i
\(552\) 0 0
\(553\) 3.89876i 0.165792i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.62884 8.62884i 0.365616 0.365616i −0.500260 0.865875i \(-0.666762\pi\)
0.865875 + 0.500260i \(0.166762\pi\)
\(558\) 0 0
\(559\) 8.78243i 0.371457i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.3155 + 10.3155i 0.434748 + 0.434748i 0.890240 0.455492i \(-0.150537\pi\)
−0.455492 + 0.890240i \(0.650537\pi\)
\(564\) 0 0
\(565\) 36.0755 36.0755i 1.51771 1.51771i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.23614 −0.135666 −0.0678330 0.997697i \(-0.521609\pi\)
−0.0678330 + 0.997697i \(0.521609\pi\)
\(570\) 0 0
\(571\) −27.7822 27.7822i −1.16265 1.16265i −0.983893 0.178757i \(-0.942792\pi\)
−0.178757 0.983893i \(-0.557208\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −109.992 −4.58699
\(576\) 0 0
\(577\) 22.0528 0.918072 0.459036 0.888418i \(-0.348195\pi\)
0.459036 + 0.888418i \(0.348195\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.40691 1.40691i −0.0583684 0.0583684i
\(582\) 0 0
\(583\) −6.88495 −0.285146
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.9242 16.9242i 0.698538 0.698538i −0.265557 0.964095i \(-0.585556\pi\)
0.964095 + 0.265557i \(0.0855560\pi\)
\(588\) 0 0
\(589\) −15.7287 15.7287i −0.648088 0.648088i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.14027i 0.375346i 0.982232 + 0.187673i \(0.0600945\pi\)
−0.982232 + 0.187673i \(0.939905\pi\)
\(594\) 0 0
\(595\) −3.07333 + 3.07333i −0.125994 + 0.125994i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 26.4441i 1.08047i −0.841513 0.540237i \(-0.818334\pi\)
0.841513 0.540237i \(-0.181666\pi\)
\(600\) 0 0
\(601\) 28.3857i 1.15788i −0.815371 0.578938i \(-0.803467\pi\)
0.815371 0.578938i \(-0.196533\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −30.7265 + 30.7265i −1.24921 + 1.24921i
\(606\) 0 0
\(607\) 9.53246i 0.386911i −0.981109 0.193455i \(-0.938031\pi\)
0.981109 0.193455i \(-0.0619695\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.4216 10.4216i −0.421612 0.421612i
\(612\) 0 0
\(613\) 1.07498 1.07498i 0.0434181 0.0434181i −0.685064 0.728483i \(-0.740226\pi\)
0.728483 + 0.685064i \(0.240226\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.3828 1.42446 0.712229 0.701948i \(-0.247686\pi\)
0.712229 + 0.701948i \(0.247686\pi\)
\(618\) 0 0
\(619\) 3.97297 + 3.97297i 0.159687 + 0.159687i 0.782428 0.622741i \(-0.213981\pi\)
−0.622741 + 0.782428i \(0.713981\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.0358 −0.682526
\(624\) 0 0
\(625\) −71.2676 −2.85070
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.155022 + 0.155022i 0.00618111 + 0.00618111i
\(630\) 0 0
\(631\) 25.6289 1.02027 0.510135 0.860095i \(-0.329596\pi\)
0.510135 + 0.860095i \(0.329596\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.92120 3.92120i 0.155608 0.155608i
\(636\) 0 0
\(637\) 2.53650 + 2.53650i 0.100500 + 0.100500i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.5770i 0.417766i −0.977941 0.208883i \(-0.933017\pi\)
0.977941 0.208883i \(-0.0669829\pi\)
\(642\) 0 0
\(643\) −8.26445 + 8.26445i −0.325918 + 0.325918i −0.851032 0.525114i \(-0.824023\pi\)
0.525114 + 0.851032i \(0.324023\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.2954i 0.915835i −0.888995 0.457917i \(-0.848596\pi\)
0.888995 0.457917i \(-0.151404\pi\)
\(648\) 0 0
\(649\) 6.89668i 0.270718i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.0365 + 20.0365i −0.784090 + 0.784090i −0.980518 0.196429i \(-0.937066\pi\)
0.196429 + 0.980518i \(0.437066\pi\)
\(654\) 0 0
\(655\) 9.95804i 0.389093i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.7397 24.7397i −0.963723 0.963723i 0.0356416 0.999365i \(-0.488653\pi\)
−0.999365 + 0.0356416i \(0.988653\pi\)
\(660\) 0 0
\(661\) −6.31446 + 6.31446i −0.245604 + 0.245604i −0.819164 0.573560i \(-0.805562\pi\)
0.573560 + 0.819164i \(0.305562\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −31.0852 −1.20543
\(666\) 0 0
\(667\) −52.7290 52.7290i −2.04167 2.04167i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.63199 0.256025
\(672\) 0 0
\(673\) 2.32930 0.0897877 0.0448939 0.998992i \(-0.485705\pi\)
0.0448939 + 0.998992i \(0.485705\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.0208 17.0208i −0.654163 0.654163i 0.299830 0.953993i \(-0.403070\pi\)
−0.953993 + 0.299830i \(0.903070\pi\)
\(678\) 0 0
\(679\) −3.75210 −0.143993
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.18231 5.18231i 0.198296 0.198296i −0.600973 0.799269i \(-0.705220\pi\)
0.799269 + 0.600973i \(0.205220\pi\)
\(684\) 0 0
\(685\) −25.2198 25.2198i −0.963599 0.963599i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 30.6455i 1.16750i
\(690\) 0 0
\(691\) 23.6472 23.6472i 0.899582 0.899582i −0.0958169 0.995399i \(-0.530546\pi\)
0.995399 + 0.0958169i \(0.0305463\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 46.3692i 1.75888i
\(696\) 0 0
\(697\) 8.93922i 0.338597i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.62771 + 4.62771i −0.174786 + 0.174786i −0.789078 0.614292i \(-0.789442\pi\)
0.614292 + 0.789078i \(0.289442\pi\)
\(702\) 0 0
\(703\) 1.56797i 0.0591370i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.23327 1.23327i −0.0463820 0.0463820i
\(708\) 0 0
\(709\) −27.1441 + 27.1441i −1.01942 + 1.01942i −0.0196099 + 0.999808i \(0.506242\pi\)
−0.999808 + 0.0196099i \(0.993758\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 26.1725 0.980168
\(714\) 0 0
\(715\) 8.58190 + 8.58190i 0.320945 + 0.320945i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 39.3061 1.46587 0.732934 0.680299i \(-0.238150\pi\)
0.732934 + 0.680299i \(0.238150\pi\)
\(720\) 0 0
\(721\) 14.3330 0.533787
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −76.4113 76.4113i −2.83784 2.83784i
\(726\) 0 0
\(727\) −36.1990 −1.34255 −0.671274 0.741209i \(-0.734253\pi\)
−0.671274 + 0.741209i \(0.734253\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.79229 + 1.79229i −0.0662902 + 0.0662902i
\(732\) 0 0
\(733\) −3.85554 3.85554i −0.142408 0.142408i 0.632309 0.774716i \(-0.282107\pi\)
−0.774716 + 0.632309i \(0.782107\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.0292849i 0.00107872i
\(738\) 0 0
\(739\) 20.1965 20.1965i 0.742942 0.742942i −0.230201 0.973143i \(-0.573939\pi\)
0.973143 + 0.230201i \(0.0739385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.9110i 1.42751i −0.700398 0.713753i \(-0.746994\pi\)
0.700398 0.713753i \(-0.253006\pi\)
\(744\) 0 0
\(745\) 77.8444i 2.85200i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.33395 7.33395i 0.267977 0.267977i
\(750\) 0 0
\(751\) 1.20106i 0.0438274i −0.999760 0.0219137i \(-0.993024\pi\)
0.999760 0.0219137i \(-0.00697590\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.8947 39.8947i −1.45192 1.45192i
\(756\) 0 0
\(757\) 12.9836 12.9836i 0.471896 0.471896i −0.430632 0.902528i \(-0.641709\pi\)
0.902528 + 0.430632i \(0.141709\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16.4747 −0.597208 −0.298604 0.954377i \(-0.596521\pi\)
−0.298604 + 0.954377i \(0.596521\pi\)
\(762\) 0 0
\(763\) 4.12218 + 4.12218i 0.149233 + 0.149233i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.6978 1.10843
\(768\) 0 0
\(769\) 33.6226 1.21246 0.606231 0.795289i \(-0.292681\pi\)
0.606231 + 0.795289i \(0.292681\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.0820 20.0820i −0.722299 0.722299i 0.246774 0.969073i \(-0.420629\pi\)
−0.969073 + 0.246774i \(0.920629\pi\)
\(774\) 0 0
\(775\) 37.9274 1.36239
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 45.2079 45.2079i 1.61974 1.61974i
\(780\) 0 0
\(781\) 8.27761 + 8.27761i 0.296196 + 0.296196i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 35.1774i 1.25553i
\(786\) 0 0
\(787\) 1.75329 1.75329i 0.0624980 0.0624980i −0.675167 0.737665i \(-0.735928\pi\)
0.737665 + 0.675167i \(0.235928\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.1524i 0.432089i
\(792\) 0 0
\(793\) 29.5196i 1.04827i
\(794\) 0 0