Properties

Label 1152.3.m.f.415.6
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.6
Root \(-1.96679 - 0.362960i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.f.991.6

$q$-expansion

\(f(q)\) \(=\) \(q+(1.69930 + 1.69930i) q^{5} -5.74280 q^{7} +O(q^{10})\) \(q+(1.69930 + 1.69930i) q^{5} -5.74280 q^{7} +(-5.59560 + 5.59560i) q^{11} +(13.5782 - 13.5782i) q^{13} -19.7023 q^{17} +(21.6943 + 21.6943i) q^{19} -24.9257 q^{23} -19.2247i q^{25} +(1.50581 - 1.50581i) q^{29} +2.20037i q^{31} +(-9.75877 - 9.75877i) q^{35} +(-27.6956 - 27.6956i) q^{37} +51.3127i q^{41} +(-21.4400 + 21.4400i) q^{43} -76.5216i q^{47} -16.0202 q^{49} +(-56.5145 - 56.5145i) q^{53} -19.0173 q^{55} +(-48.0041 + 48.0041i) q^{59} +(51.5587 - 51.5587i) q^{61} +46.1469 q^{65} +(-63.4445 - 63.4445i) q^{67} -43.4856 q^{71} -73.9992i q^{73} +(32.1344 - 32.1344i) q^{77} +4.12659i q^{79} +(38.4428 + 38.4428i) q^{83} +(-33.4803 - 33.4803i) q^{85} -52.9839i q^{89} +(-77.9767 + 77.9767i) q^{91} +73.7305i q^{95} +23.1008 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 32q^{11} + 32q^{19} + 128q^{23} + 32q^{29} + 96q^{35} + 96q^{37} - 160q^{43} + 112q^{49} - 160q^{53} - 256q^{55} - 128q^{59} + 32q^{61} + 32q^{65} - 320q^{67} - 512q^{71} + 224q^{77} - 160q^{83} - 160q^{85} + 480q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) 1.69930 + 1.69930i 0.339861 + 0.339861i 0.856315 0.516454i \(-0.172748\pi\)
−0.516454 + 0.856315i \(0.672748\pi\)
\(6\) 0 0
\(7\) −5.74280 −0.820400 −0.410200 0.911996i \(-0.634541\pi\)
−0.410200 + 0.911996i \(0.634541\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.59560 + 5.59560i −0.508691 + 0.508691i −0.914125 0.405434i \(-0.867121\pi\)
0.405434 + 0.914125i \(0.367121\pi\)
\(12\) 0 0
\(13\) 13.5782 13.5782i 1.04447 1.04447i 0.0455110 0.998964i \(-0.485508\pi\)
0.998964 0.0455110i \(-0.0144916\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −19.7023 −1.15896 −0.579481 0.814986i \(-0.696745\pi\)
−0.579481 + 0.814986i \(0.696745\pi\)
\(18\) 0 0
\(19\) 21.6943 + 21.6943i 1.14181 + 1.14181i 0.988120 + 0.153687i \(0.0491147\pi\)
0.153687 + 0.988120i \(0.450885\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −24.9257 −1.08373 −0.541863 0.840467i \(-0.682281\pi\)
−0.541863 + 0.840467i \(0.682281\pi\)
\(24\) 0 0
\(25\) 19.2247i 0.768989i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.50581 1.50581i 0.0519245 0.0519245i −0.680668 0.732592i \(-0.738310\pi\)
0.732592 + 0.680668i \(0.238310\pi\)
\(30\) 0 0
\(31\) 2.20037i 0.0709796i 0.999370 + 0.0354898i \(0.0112991\pi\)
−0.999370 + 0.0354898i \(0.988701\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.75877 9.75877i −0.278822 0.278822i
\(36\) 0 0
\(37\) −27.6956 27.6956i −0.748530 0.748530i 0.225673 0.974203i \(-0.427542\pi\)
−0.974203 + 0.225673i \(0.927542\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 51.3127i 1.25153i 0.780012 + 0.625764i \(0.215213\pi\)
−0.780012 + 0.625764i \(0.784787\pi\)
\(42\) 0 0
\(43\) −21.4400 + 21.4400i −0.498606 + 0.498606i −0.911004 0.412398i \(-0.864691\pi\)
0.412398 + 0.911004i \(0.364691\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 76.5216i 1.62812i −0.580781 0.814060i \(-0.697253\pi\)
0.580781 0.814060i \(-0.302747\pi\)
\(48\) 0 0
\(49\) −16.0202 −0.326944
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −56.5145 56.5145i −1.06631 1.06631i −0.997639 0.0686712i \(-0.978124\pi\)
−0.0686712 0.997639i \(-0.521876\pi\)
\(54\) 0 0
\(55\) −19.0173 −0.345768
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −48.0041 + 48.0041i −0.813628 + 0.813628i −0.985176 0.171547i \(-0.945123\pi\)
0.171547 + 0.985176i \(0.445123\pi\)
\(60\) 0 0
\(61\) 51.5587 51.5587i 0.845224 0.845224i −0.144308 0.989533i \(-0.546096\pi\)
0.989533 + 0.144308i \(0.0460957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 46.1469 0.709952
\(66\) 0 0
\(67\) −63.4445 63.4445i −0.946934 0.946934i 0.0517277 0.998661i \(-0.483527\pi\)
−0.998661 + 0.0517277i \(0.983527\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −43.4856 −0.612473 −0.306237 0.951955i \(-0.599070\pi\)
−0.306237 + 0.951955i \(0.599070\pi\)
\(72\) 0 0
\(73\) 73.9992i 1.01369i −0.862038 0.506844i \(-0.830812\pi\)
0.862038 0.506844i \(-0.169188\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 32.1344 32.1344i 0.417330 0.417330i
\(78\) 0 0
\(79\) 4.12659i 0.0522354i 0.999659 + 0.0261177i \(0.00831446\pi\)
−0.999659 + 0.0261177i \(0.991686\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 38.4428 + 38.4428i 0.463166 + 0.463166i 0.899692 0.436526i \(-0.143791\pi\)
−0.436526 + 0.899692i \(0.643791\pi\)
\(84\) 0 0
\(85\) −33.4803 33.4803i −0.393886 0.393886i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 52.9839i 0.595325i −0.954671 0.297662i \(-0.903793\pi\)
0.954671 0.297662i \(-0.0962070\pi\)
\(90\) 0 0
\(91\) −77.9767 + 77.9767i −0.856887 + 0.856887i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 73.7305i 0.776111i
\(96\) 0 0
\(97\) 23.1008 0.238153 0.119077 0.992885i \(-0.462007\pi\)
0.119077 + 0.992885i \(0.462007\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.1216 + 16.1216i 0.159619 + 0.159619i 0.782398 0.622779i \(-0.213996\pi\)
−0.622779 + 0.782398i \(0.713996\pi\)
\(102\) 0 0
\(103\) −98.8380 −0.959592 −0.479796 0.877380i \(-0.659289\pi\)
−0.479796 + 0.877380i \(0.659289\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.6655 15.6655i 0.146406 0.146406i −0.630104 0.776511i \(-0.716988\pi\)
0.776511 + 0.630104i \(0.216988\pi\)
\(108\) 0 0
\(109\) −84.6938 + 84.6938i −0.777008 + 0.777008i −0.979321 0.202313i \(-0.935154\pi\)
0.202313 + 0.979321i \(0.435154\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −63.8537 −0.565077 −0.282538 0.959256i \(-0.591176\pi\)
−0.282538 + 0.959256i \(0.591176\pi\)
\(114\) 0 0
\(115\) −42.3563 42.3563i −0.368316 0.368316i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 113.147 0.950812
\(120\) 0 0
\(121\) 58.3785i 0.482467i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 75.1513 75.1513i 0.601210 0.601210i
\(126\) 0 0
\(127\) 36.8901i 0.290473i −0.989397 0.145237i \(-0.953606\pi\)
0.989397 0.145237i \(-0.0463944\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −40.4136 40.4136i −0.308500 0.308500i 0.535827 0.844328i \(-0.320000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(132\) 0 0
\(133\) −124.586 124.586i −0.936738 0.936738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 253.499i 1.85036i −0.379531 0.925179i \(-0.623915\pi\)
0.379531 0.925179i \(-0.376085\pi\)
\(138\) 0 0
\(139\) −67.8065 + 67.8065i −0.487816 + 0.487816i −0.907617 0.419800i \(-0.862100\pi\)
0.419800 + 0.907617i \(0.362100\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 151.956i 1.06263i
\(144\) 0 0
\(145\) 5.11766 0.0352942
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −43.9337 43.9337i −0.294857 0.294857i 0.544138 0.838996i \(-0.316857\pi\)
−0.838996 + 0.544138i \(0.816857\pi\)
\(150\) 0 0
\(151\) −223.084 −1.47738 −0.738688 0.674047i \(-0.764554\pi\)
−0.738688 + 0.674047i \(0.764554\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.73909 + 3.73909i −0.0241232 + 0.0241232i
\(156\) 0 0
\(157\) 78.8526 78.8526i 0.502246 0.502246i −0.409889 0.912135i \(-0.634433\pi\)
0.912135 + 0.409889i \(0.134433\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 143.143 0.889089
\(162\) 0 0
\(163\) −52.2425 52.2425i −0.320506 0.320506i 0.528455 0.848961i \(-0.322772\pi\)
−0.848961 + 0.528455i \(0.822772\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −96.5201 −0.577965 −0.288982 0.957334i \(-0.593317\pi\)
−0.288982 + 0.957334i \(0.593317\pi\)
\(168\) 0 0
\(169\) 199.734i 1.18186i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −46.3076 + 46.3076i −0.267674 + 0.267674i −0.828162 0.560488i \(-0.810614\pi\)
0.560488 + 0.828162i \(0.310614\pi\)
\(174\) 0 0
\(175\) 110.404i 0.630879i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −93.5440 93.5440i −0.522592 0.522592i 0.395761 0.918353i \(-0.370481\pi\)
−0.918353 + 0.395761i \(0.870481\pi\)
\(180\) 0 0
\(181\) 115.810 + 115.810i 0.639836 + 0.639836i 0.950515 0.310679i \(-0.100556\pi\)
−0.310679 + 0.950515i \(0.600556\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 94.1266i 0.508792i
\(186\) 0 0
\(187\) 110.246 110.246i 0.589553 0.589553i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 35.2964i 0.184798i 0.995722 + 0.0923991i \(0.0294535\pi\)
−0.995722 + 0.0923991i \(0.970546\pi\)
\(192\) 0 0
\(193\) −364.339 −1.88777 −0.943884 0.330277i \(-0.892858\pi\)
−0.943884 + 0.330277i \(0.892858\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 130.582 + 130.582i 0.662851 + 0.662851i 0.956051 0.293200i \(-0.0947203\pi\)
−0.293200 + 0.956051i \(0.594720\pi\)
\(198\) 0 0
\(199\) −12.7493 −0.0640670 −0.0320335 0.999487i \(-0.510198\pi\)
−0.0320335 + 0.999487i \(0.510198\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.64756 + 8.64756i −0.0425988 + 0.0425988i
\(204\) 0 0
\(205\) −87.1958 + 87.1958i −0.425346 + 0.425346i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −242.786 −1.16165
\(210\) 0 0
\(211\) −8.59499 8.59499i −0.0407345 0.0407345i 0.686446 0.727181i \(-0.259170\pi\)
−0.727181 + 0.686446i \(0.759170\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −72.8663 −0.338913
\(216\) 0 0
\(217\) 12.6363i 0.0582317i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −267.522 + 267.522i −1.21051 + 1.21051i
\(222\) 0 0
\(223\) 50.5909i 0.226865i 0.993546 + 0.113433i \(0.0361846\pi\)
−0.993546 + 0.113433i \(0.963815\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 31.7175 + 31.7175i 0.139725 + 0.139725i 0.773509 0.633785i \(-0.218499\pi\)
−0.633785 + 0.773509i \(0.718499\pi\)
\(228\) 0 0
\(229\) 169.826 + 169.826i 0.741599 + 0.741599i 0.972886 0.231287i \(-0.0742936\pi\)
−0.231287 + 0.972886i \(0.574294\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 363.082i 1.55829i 0.626844 + 0.779145i \(0.284346\pi\)
−0.626844 + 0.779145i \(0.715654\pi\)
\(234\) 0 0
\(235\) 130.033 130.033i 0.553334 0.553334i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 27.6282i 0.115599i −0.998328 0.0577996i \(-0.981592\pi\)
0.998328 0.0577996i \(-0.0184084\pi\)
\(240\) 0 0
\(241\) 368.121 1.52747 0.763737 0.645527i \(-0.223362\pi\)
0.763737 + 0.645527i \(0.223362\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −27.2233 27.2233i −0.111115 0.111115i
\(246\) 0 0
\(247\) 589.139 2.38518
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 329.839 329.839i 1.31410 1.31410i 0.395734 0.918365i \(-0.370490\pi\)
0.918365 0.395734i \(-0.129510\pi\)
\(252\) 0 0
\(253\) 139.474 139.474i 0.551281 0.551281i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −23.6762 −0.0921252 −0.0460626 0.998939i \(-0.514667\pi\)
−0.0460626 + 0.998939i \(0.514667\pi\)
\(258\) 0 0
\(259\) 159.050 + 159.050i 0.614094 + 0.614094i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −243.854 −0.927202 −0.463601 0.886044i \(-0.653443\pi\)
−0.463601 + 0.886044i \(0.653443\pi\)
\(264\) 0 0
\(265\) 192.071i 0.724794i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 234.293 234.293i 0.870976 0.870976i −0.121603 0.992579i \(-0.538803\pi\)
0.992579 + 0.121603i \(0.0388035\pi\)
\(270\) 0 0
\(271\) 30.9533i 0.114219i 0.998368 + 0.0571094i \(0.0181884\pi\)
−0.998368 + 0.0571094i \(0.981812\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 107.574 + 107.574i 0.391178 + 0.391178i
\(276\) 0 0
\(277\) 41.4479 + 41.4479i 0.149631 + 0.149631i 0.777953 0.628322i \(-0.216258\pi\)
−0.628322 + 0.777953i \(0.716258\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 93.3971i 0.332374i −0.986094 0.166187i \(-0.946854\pi\)
0.986094 0.166187i \(-0.0531455\pi\)
\(282\) 0 0
\(283\) −40.0982 + 40.0982i −0.141690 + 0.141690i −0.774394 0.632704i \(-0.781945\pi\)
0.632704 + 0.774394i \(0.281945\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 294.678i 1.02675i
\(288\) 0 0
\(289\) 99.1824 0.343192
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 141.326 + 141.326i 0.482340 + 0.482340i 0.905878 0.423538i \(-0.139212\pi\)
−0.423538 + 0.905878i \(0.639212\pi\)
\(294\) 0 0
\(295\) −163.147 −0.553041
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −338.445 + 338.445i −1.13192 + 1.13192i
\(300\) 0 0
\(301\) 123.126 123.126i 0.409056 0.409056i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 175.228 0.574517
\(306\) 0 0
\(307\) 285.548 + 285.548i 0.930125 + 0.930125i 0.997713 0.0675885i \(-0.0215305\pi\)
−0.0675885 + 0.997713i \(0.521530\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 365.454 1.17509 0.587547 0.809190i \(-0.300094\pi\)
0.587547 + 0.809190i \(0.300094\pi\)
\(312\) 0 0
\(313\) 461.508i 1.47447i 0.675638 + 0.737234i \(0.263868\pi\)
−0.675638 + 0.737234i \(0.736132\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −319.216 + 319.216i −1.00699 + 1.00699i −0.00701388 + 0.999975i \(0.502233\pi\)
−0.999975 + 0.00701388i \(0.997767\pi\)
\(318\) 0 0
\(319\) 16.8518i 0.0528270i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −427.429 427.429i −1.32331 1.32331i
\(324\) 0 0
\(325\) −261.037 261.037i −0.803190 0.803190i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 439.448i 1.33571i
\(330\) 0 0
\(331\) 85.7864 85.7864i 0.259173 0.259173i −0.565544 0.824718i \(-0.691334\pi\)
0.824718 + 0.565544i \(0.191334\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 215.623i 0.643651i
\(336\) 0 0
\(337\) 258.256 0.766339 0.383170 0.923678i \(-0.374832\pi\)
0.383170 + 0.923678i \(0.374832\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.3124 12.3124i −0.0361067 0.0361067i
\(342\) 0 0
\(343\) 373.398 1.08862
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.7237 27.7237i 0.0798953 0.0798953i −0.666030 0.745925i \(-0.732008\pi\)
0.745925 + 0.666030i \(0.232008\pi\)
\(348\) 0 0
\(349\) −321.089 + 321.089i −0.920027 + 0.920027i −0.997031 0.0770037i \(-0.975465\pi\)
0.0770037 + 0.997031i \(0.475465\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 241.363 0.683748 0.341874 0.939746i \(-0.388938\pi\)
0.341874 + 0.939746i \(0.388938\pi\)
\(354\) 0 0
\(355\) −73.8953 73.8953i −0.208156 0.208156i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 363.821 1.01343 0.506714 0.862114i \(-0.330860\pi\)
0.506714 + 0.862114i \(0.330860\pi\)
\(360\) 0 0
\(361\) 580.287i 1.60744i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 125.747 125.747i 0.344513 0.344513i
\(366\) 0 0
\(367\) 411.402i 1.12099i −0.828159 0.560493i \(-0.810612\pi\)
0.828159 0.560493i \(-0.189388\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 324.551 + 324.551i 0.874801 + 0.874801i
\(372\) 0 0
\(373\) 225.677 + 225.677i 0.605033 + 0.605033i 0.941644 0.336611i \(-0.109281\pi\)
−0.336611 + 0.941644i \(0.609281\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.8923i 0.108468i
\(378\) 0 0
\(379\) 157.180 157.180i 0.414724 0.414724i −0.468656 0.883381i \(-0.655262\pi\)
0.883381 + 0.468656i \(0.155262\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 703.356i 1.83644i 0.396072 + 0.918219i \(0.370373\pi\)
−0.396072 + 0.918219i \(0.629627\pi\)
\(384\) 0 0
\(385\) 109.212 0.283668
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.7401 + 10.7401i 0.0276095 + 0.0276095i 0.720777 0.693167i \(-0.243785\pi\)
−0.693167 + 0.720777i \(0.743785\pi\)
\(390\) 0 0
\(391\) 491.095 1.25600
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.01234 + 7.01234i −0.0177528 + 0.0177528i
\(396\) 0 0
\(397\) −365.020 + 365.020i −0.919446 + 0.919446i −0.996989 0.0775433i \(-0.975292\pi\)
0.0775433 + 0.996989i \(0.475292\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −341.735 −0.852207 −0.426104 0.904674i \(-0.640114\pi\)
−0.426104 + 0.904674i \(0.640114\pi\)
\(402\) 0 0
\(403\) 29.8770 + 29.8770i 0.0741364 + 0.0741364i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 309.947 0.761541
\(408\) 0 0
\(409\) 368.259i 0.900389i 0.892931 + 0.450194i \(0.148645\pi\)
−0.892931 + 0.450194i \(0.851355\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 275.678 275.678i 0.667501 0.667501i
\(414\) 0 0
\(415\) 130.652i 0.314824i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 407.140 + 407.140i 0.971694 + 0.971694i 0.999610 0.0279165i \(-0.00888725\pi\)
−0.0279165 + 0.999610i \(0.508887\pi\)
\(420\) 0 0
\(421\) −57.5576 57.5576i −0.136716 0.136716i 0.635437 0.772153i \(-0.280820\pi\)
−0.772153 + 0.635437i \(0.780820\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 378.772i 0.891229i
\(426\) 0 0
\(427\) −296.091 + 296.091i −0.693422 + 0.693422i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 796.565i 1.84818i −0.382177 0.924089i \(-0.624826\pi\)
0.382177 0.924089i \(-0.375174\pi\)
\(432\) 0 0
\(433\) −335.804 −0.775529 −0.387764 0.921758i \(-0.626753\pi\)
−0.387764 + 0.921758i \(0.626753\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −540.746 540.746i −1.23741 1.23741i
\(438\) 0 0
\(439\) −285.630 −0.650638 −0.325319 0.945604i \(-0.605472\pi\)
−0.325319 + 0.945604i \(0.605472\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 111.596 111.596i 0.251909 0.251909i −0.569844 0.821753i \(-0.692996\pi\)
0.821753 + 0.569844i \(0.192996\pi\)
\(444\) 0 0
\(445\) 90.0358 90.0358i 0.202328 0.202328i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 99.6741 0.221991 0.110996 0.993821i \(-0.464596\pi\)
0.110996 + 0.993821i \(0.464596\pi\)
\(450\) 0 0
\(451\) −287.125 287.125i −0.636641 0.636641i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −265.012 −0.582445
\(456\) 0 0
\(457\) 32.1643i 0.0703813i −0.999381 0.0351907i \(-0.988796\pi\)
0.999381 0.0351907i \(-0.0112039\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 165.361 165.361i 0.358701 0.358701i −0.504633 0.863334i \(-0.668372\pi\)
0.863334 + 0.504633i \(0.168372\pi\)
\(462\) 0 0
\(463\) 923.215i 1.99398i −0.0774991 0.996992i \(-0.524693\pi\)
0.0774991 0.996992i \(-0.475307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 507.842 + 507.842i 1.08746 + 1.08746i 0.995790 + 0.0916660i \(0.0292192\pi\)
0.0916660 + 0.995790i \(0.470781\pi\)
\(468\) 0 0
\(469\) 364.349 + 364.349i 0.776864 + 0.776864i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 239.940i 0.507272i
\(474\) 0 0
\(475\) 417.068 417.068i 0.878037 0.878037i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 52.3866i 0.109367i −0.998504 0.0546833i \(-0.982585\pi\)
0.998504 0.0546833i \(-0.0174149\pi\)
\(480\) 0 0
\(481\) −752.112 −1.56364
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 39.2554 + 39.2554i 0.0809389 + 0.0809389i
\(486\) 0 0
\(487\) −715.733 −1.46968 −0.734839 0.678241i \(-0.762742\pi\)
−0.734839 + 0.678241i \(0.762742\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −22.3258 + 22.3258i −0.0454701 + 0.0454701i −0.729476 0.684006i \(-0.760236\pi\)
0.684006 + 0.729476i \(0.260236\pi\)
\(492\) 0 0
\(493\) −29.6680 + 29.6680i −0.0601784 + 0.0601784i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 249.729 0.502473
\(498\) 0 0
\(499\) −84.0984 84.0984i −0.168534 0.168534i 0.617801 0.786335i \(-0.288024\pi\)
−0.786335 + 0.617801i \(0.788024\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −327.870 −0.651829 −0.325914 0.945399i \(-0.605672\pi\)
−0.325914 + 0.945399i \(0.605672\pi\)
\(504\) 0 0
\(505\) 54.7909i 0.108497i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.6224 34.6224i 0.0680205 0.0680205i −0.672278 0.740299i \(-0.734684\pi\)
0.740299 + 0.672278i \(0.234684\pi\)
\(510\) 0 0
\(511\) 424.963i 0.831630i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −167.956 167.956i −0.326128 0.326128i
\(516\) 0 0
\(517\) 428.184 + 428.184i 0.828210 + 0.828210i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 235.719i 0.452436i −0.974077 0.226218i \(-0.927364\pi\)
0.974077 0.226218i \(-0.0726362\pi\)
\(522\) 0 0
\(523\) 185.851 185.851i 0.355356 0.355356i −0.506742 0.862098i \(-0.669150\pi\)
0.862098 + 0.506742i \(0.169150\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 43.3524i 0.0822626i
\(528\) 0 0
\(529\) 92.2900 0.174461
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 696.732 + 696.732i 1.30719 + 1.30719i
\(534\) 0 0
\(535\) 53.2408 0.0995155
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 89.6428 89.6428i 0.166313 0.166313i
\(540\) 0 0
\(541\) 315.952 315.952i 0.584015 0.584015i −0.351989 0.936004i \(-0.614494\pi\)
0.936004 + 0.351989i \(0.114494\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −287.841 −0.528149
\(546\) 0 0
\(547\) 550.957 + 550.957i 1.00723 + 1.00723i 0.999974 + 0.00725954i \(0.00231080\pi\)
0.00725954 + 0.999974i \(0.497689\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 65.3350 0.118575
\(552\) 0 0
\(553\) 23.6982i 0.0428539i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.35545 2.35545i 0.00422882 0.00422882i −0.704989 0.709218i \(-0.749048\pi\)
0.709218 + 0.704989i \(0.249048\pi\)
\(558\) 0 0
\(559\) 582.233i 1.04156i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 269.210 + 269.210i 0.478170 + 0.478170i 0.904546 0.426376i \(-0.140210\pi\)
−0.426376 + 0.904546i \(0.640210\pi\)
\(564\) 0 0
\(565\) −108.507 108.507i −0.192047 0.192047i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 342.558i 0.602035i 0.953619 + 0.301018i \(0.0973263\pi\)
−0.953619 + 0.301018i \(0.902674\pi\)
\(570\) 0 0
\(571\) −153.948 + 153.948i −0.269610 + 0.269610i −0.828943 0.559333i \(-0.811057\pi\)
0.559333 + 0.828943i \(0.311057\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 479.190i 0.833373i
\(576\) 0 0
\(577\) 563.693 0.976938 0.488469 0.872581i \(-0.337556\pi\)
0.488469 + 0.872581i \(0.337556\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −220.769 220.769i −0.379981 0.379981i
\(582\) 0 0
\(583\) 632.465 1.08484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 176.603 176.603i 0.300857 0.300857i −0.540492 0.841349i \(-0.681762\pi\)
0.841349 + 0.540492i \(0.181762\pi\)
\(588\) 0 0
\(589\) −47.7355 + 47.7355i −0.0810450 + 0.0810450i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 996.597 1.68060 0.840301 0.542120i \(-0.182378\pi\)
0.840301 + 0.542120i \(0.182378\pi\)
\(594\) 0 0
\(595\) 192.271 + 192.271i 0.323144 + 0.323144i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 854.031 1.42576 0.712880 0.701286i \(-0.247390\pi\)
0.712880 + 0.701286i \(0.247390\pi\)
\(600\) 0 0
\(601\) 345.733i 0.575263i −0.957741 0.287631i \(-0.907132\pi\)
0.957741 0.287631i \(-0.0928678\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −99.2029 + 99.2029i −0.163972 + 0.163972i
\(606\) 0 0
\(607\) 526.354i 0.867141i 0.901120 + 0.433570i \(0.142746\pi\)
−0.901120 + 0.433570i \(0.857254\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1039.02 1039.02i −1.70053 1.70053i
\(612\) 0 0
\(613\) −410.567 410.567i −0.669767 0.669767i 0.287895 0.957662i \(-0.407045\pi\)
−0.957662 + 0.287895i \(0.907045\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 514.755i 0.834287i −0.908841 0.417144i \(-0.863031\pi\)
0.908841 0.417144i \(-0.136969\pi\)
\(618\) 0 0
\(619\) −314.214 + 314.214i −0.507615 + 0.507615i −0.913794 0.406179i \(-0.866861\pi\)
0.406179 + 0.913794i \(0.366861\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 304.276i 0.488404i
\(624\) 0 0
\(625\) −225.209 −0.360334
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 545.669 + 545.669i 0.867518 + 0.867518i
\(630\) 0 0
\(631\) −230.081 −0.364629 −0.182315 0.983240i \(-0.558359\pi\)
−0.182315 + 0.983240i \(0.558359\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 62.6875 62.6875i 0.0987205 0.0987205i
\(636\) 0 0
\(637\) −217.526 + 217.526i −0.341484 + 0.341484i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −746.825 −1.16509 −0.582547 0.812797i \(-0.697944\pi\)
−0.582547 + 0.812797i \(0.697944\pi\)
\(642\) 0 0
\(643\) −548.092 548.092i −0.852398 0.852398i 0.138030 0.990428i \(-0.455923\pi\)
−0.990428 + 0.138030i \(0.955923\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1055.00 −1.63060 −0.815302 0.579036i \(-0.803429\pi\)
−0.815302 + 0.579036i \(0.803429\pi\)
\(648\) 0 0
\(649\) 537.223i 0.827771i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −854.888 + 854.888i −1.30917 + 1.30917i −0.387155 + 0.922015i \(0.626542\pi\)
−0.922015 + 0.387155i \(0.873458\pi\)
\(654\) 0 0
\(655\) 137.350i 0.209694i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −768.766 768.766i −1.16656 1.16656i −0.983009 0.183556i \(-0.941239\pi\)
−0.183556 0.983009i \(-0.558761\pi\)
\(660\) 0 0
\(661\) −312.323 312.323i −0.472500 0.472500i 0.430223 0.902723i \(-0.358435\pi\)
−0.902723 + 0.430223i \(0.858435\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 423.420i 0.636721i
\(666\) 0 0
\(667\) −37.5333 + 37.5333i −0.0562719 + 0.0562719i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 577.004i 0.859916i
\(672\) 0 0
\(673\) 740.565 1.10039 0.550197 0.835035i \(-0.314553\pi\)
0.550197 + 0.835035i \(0.314553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −547.118 547.118i −0.808151 0.808151i 0.176203 0.984354i \(-0.443619\pi\)
−0.984354 + 0.176203i \(0.943619\pi\)
\(678\) 0 0
\(679\) −132.664 −0.195381
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −407.623 + 407.623i −0.596813 + 0.596813i −0.939463 0.342650i \(-0.888676\pi\)
0.342650 + 0.939463i \(0.388676\pi\)
\(684\) 0 0
\(685\) 430.772 430.772i 0.628864 0.628864i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1534.73 −2.22747
\(690\) 0 0
\(691\) −17.6037 17.6037i −0.0254757 0.0254757i 0.694254 0.719730i \(-0.255734\pi\)
−0.719730 + 0.694254i \(0.755734\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −230.448 −0.331579
\(696\) 0 0
\(697\) 1010.98i 1.45047i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 164.273 164.273i 0.234341 0.234341i −0.580161 0.814502i \(-0.697010\pi\)
0.814502 + 0.580161i \(0.197010\pi\)
\(702\) 0 0
\(703\) 1201.68i 1.70935i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −92.5829 92.5829i −0.130952 0.130952i
\(708\) 0 0
\(709\) −422.796 422.796i −0.596327 0.596327i 0.343006 0.939333i \(-0.388555\pi\)
−0.939333 + 0.343006i \(0.888555\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 54.8457i 0.0769224i
\(714\) 0 0
\(715\) −258.220 + 258.220i −0.361146 + 0.361146i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1029.00i 1.43115i −0.698534 0.715577i \(-0.746164\pi\)
0.698534 0.715577i \(-0.253836\pi\)
\(720\) 0 0
\(721\) 567.607 0.787250
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −28.9488 28.9488i −0.0399293 0.0399293i
\(726\) 0 0
\(727\) −475.001 −0.653372 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 422.419 422.419i 0.577865 0.577865i
\(732\) 0 0
\(733\) 344.939 344.939i 0.470586 0.470586i −0.431519 0.902104i \(-0.642022\pi\)
0.902104 + 0.431519i \(0.142022\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 710.021 0.963393
\(738\) 0 0
\(739\) −363.340 363.340i −0.491665 0.491665i 0.417166 0.908831i \(-0.363024\pi\)
−0.908831 + 0.417166i \(0.863024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −271.667 −0.365636 −0.182818 0.983147i \(-0.558522\pi\)
−0.182818 + 0.983147i \(0.558522\pi\)
\(744\) 0 0
\(745\) 149.314i 0.200421i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −89.9637 + 89.9637i −0.120112 + 0.120112i
\(750\) 0 0
\(751\) 1105.27i 1.47173i 0.677128 + 0.735866i \(0.263224\pi\)
−0.677128 + 0.735866i \(0.736776\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −379.087 379.087i −0.502102 0.502102i
\(756\) 0 0
\(757\) −554.565 554.565i −0.732583 0.732583i 0.238548 0.971131i \(-0.423329\pi\)
−0.971131 + 0.238548i \(0.923329\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 188.496i 0.247695i 0.992301 + 0.123847i \(0.0395234\pi\)
−0.992301 + 0.123847i \(0.960477\pi\)
\(762\) 0 0
\(763\) 486.380 486.380i 0.637457 0.637457i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1303.62i 1.69963i
\(768\) 0 0
\(769\) −593.354 −0.771592 −0.385796 0.922584i \(-0.626073\pi\)
−0.385796 + 0.922584i \(0.626073\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 514.720 + 514.720i 0.665873 + 0.665873i 0.956758 0.290885i \(-0.0939498\pi\)
−0.290885 + 0.956758i \(0.593950\pi\)
\(774\) 0 0
\(775\) 42.3015 0.0545826
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1113.19 + 1113.19i −1.42900 + 1.42900i
\(780\) 0 0
\(781\) 243.328 243.328i 0.311560 0.311560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 267.989 0.341387
\(786\) 0 0
\(787\) −96.1835 96.1835i −0.122215 0.122215i 0.643354 0.765569i \(-0.277542\pi\)
−0.765569 + 0.643354i \(0.777542\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 366.699 0.463589
\(792\) 0 0
\(793\) 1400.15i 1.76563i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −664.410 + 664.410i −0.833639 + 0.833639i −0.988013 0.154374i \(-0.950664\pi\)
0.154374 + 0.988013i \(0.450664\pi\)
\(798\) 0 0
\(799\) 1507.66i 1.88693i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 414.070 + 414.070i 0.515654 + 0.515654i
\(804\) 0