Properties

Label 3150.2.b.a.251.3
Level 3150
Weight 2
Character 3150.251
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.3
Root \(0.437016 - 0.437016i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.a.251.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(1.41421 - 2.23607i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(1.41421 - 2.23607i) q^{7} +1.00000i q^{8} -1.41421i q^{11} -0.926210i q^{13} +(-2.23607 - 1.41421i) q^{14} +1.00000 q^{16} +2.23607 q^{17} -7.63441i q^{19} -1.41421 q^{22} +1.00000i q^{23} -0.926210 q^{26} +(-1.41421 + 2.23607i) q^{28} -0.757359i q^{29} +4.08849i q^{31} -1.00000i q^{32} -2.23607i q^{34} -2.82843 q^{37} -7.63441 q^{38} +8.56062 q^{41} -3.58579 q^{43} +1.41421i q^{44} +1.00000 q^{46} -1.30986 q^{47} +(-3.00000 - 6.32456i) q^{49} +0.926210i q^{52} +8.07107i q^{53} +(2.23607 + 1.41421i) q^{56} -0.757359 q^{58} +7.25077 q^{59} +0.926210i q^{61} +4.08849 q^{62} -1.00000 q^{64} -2.23607 q^{68} -15.6569i q^{71} -13.9590i q^{73} +2.82843i q^{74} +7.63441i q^{76} +(-3.16228 - 2.00000i) q^{77} -13.0711 q^{79} -8.56062i q^{82} -14.3426 q^{83} +3.58579i q^{86} +1.41421 q^{88} -2.61972 q^{89} +(-2.07107 - 1.30986i) q^{91} -1.00000i q^{92} +1.30986i q^{94} -0.542561i q^{97} +(-6.32456 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{16} - 40q^{43} + 8q^{46} - 24q^{49} - 40q^{58} - 8q^{64} - 48q^{79} + 40q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.41421 2.23607i 0.534522 0.845154i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 0.926210i 0.256884i −0.991717 0.128442i \(-0.959002\pi\)
0.991717 0.128442i \(-0.0409977\pi\)
\(14\) −2.23607 1.41421i −0.597614 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.23607 0.542326 0.271163 0.962533i \(-0.412592\pi\)
0.271163 + 0.962533i \(0.412592\pi\)
\(18\) 0 0
\(19\) 7.63441i 1.75145i −0.482806 0.875727i \(-0.660382\pi\)
0.482806 0.875727i \(-0.339618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.41421 −0.301511
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.926210 −0.181645
\(27\) 0 0
\(28\) −1.41421 + 2.23607i −0.267261 + 0.422577i
\(29\) 0.757359i 0.140638i −0.997525 0.0703190i \(-0.977598\pi\)
0.997525 0.0703190i \(-0.0224017\pi\)
\(30\) 0 0
\(31\) 4.08849i 0.734314i 0.930159 + 0.367157i \(0.119669\pi\)
−0.930159 + 0.367157i \(0.880331\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.23607i 0.383482i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) −7.63441 −1.23847
\(39\) 0 0
\(40\) 0 0
\(41\) 8.56062 1.33694 0.668472 0.743737i \(-0.266948\pi\)
0.668472 + 0.743737i \(0.266948\pi\)
\(42\) 0 0
\(43\) −3.58579 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −1.30986 −0.191062 −0.0955312 0.995426i \(-0.530455\pi\)
−0.0955312 + 0.995426i \(0.530455\pi\)
\(48\) 0 0
\(49\) −3.00000 6.32456i −0.428571 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.926210i 0.128442i
\(53\) 8.07107i 1.10865i 0.832301 + 0.554323i \(0.187023\pi\)
−0.832301 + 0.554323i \(0.812977\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 + 1.41421i 0.298807 + 0.188982i
\(57\) 0 0
\(58\) −0.757359 −0.0994461
\(59\) 7.25077 0.943969 0.471985 0.881607i \(-0.343538\pi\)
0.471985 + 0.881607i \(0.343538\pi\)
\(60\) 0 0
\(61\) 0.926210i 0.118589i 0.998241 + 0.0592945i \(0.0188851\pi\)
−0.998241 + 0.0592945i \(0.981115\pi\)
\(62\) 4.08849 0.519238
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −2.23607 −0.271163
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6569i 1.85813i −0.369921 0.929063i \(-0.620615\pi\)
0.369921 0.929063i \(-0.379385\pi\)
\(72\) 0 0
\(73\) 13.9590i 1.63377i −0.576798 0.816887i \(-0.695698\pi\)
0.576798 0.816887i \(-0.304302\pi\)
\(74\) 2.82843i 0.328798i
\(75\) 0 0
\(76\) 7.63441i 0.875727i
\(77\) −3.16228 2.00000i −0.360375 0.227921i
\(78\) 0 0
\(79\) −13.0711 −1.47061 −0.735305 0.677736i \(-0.762961\pi\)
−0.735305 + 0.677736i \(0.762961\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.56062i 0.945363i
\(83\) −14.3426 −1.57431 −0.787153 0.616757i \(-0.788446\pi\)
−0.787153 + 0.616757i \(0.788446\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.58579i 0.386665i
\(87\) 0 0
\(88\) 1.41421 0.150756
\(89\) −2.61972 −0.277689 −0.138845 0.990314i \(-0.544339\pi\)
−0.138845 + 0.990314i \(0.544339\pi\)
\(90\) 0 0
\(91\) −2.07107 1.30986i −0.217107 0.137310i
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 1.30986i 0.135102i
\(95\) 0 0
\(96\) 0 0
\(97\) 0.542561i 0.0550887i −0.999621 0.0275444i \(-0.991231\pi\)
0.999621 0.0275444i \(-0.00876875\pi\)
\(98\) −6.32456 + 3.00000i −0.638877 + 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.1065 1.20465 0.602323 0.798252i \(-0.294242\pi\)
0.602323 + 0.798252i \(0.294242\pi\)
\(102\) 0 0
\(103\) 10.4130i 1.02603i −0.858380 0.513014i \(-0.828529\pi\)
0.858380 0.513014i \(-0.171471\pi\)
\(104\) 0.926210 0.0908223
\(105\) 0 0
\(106\) 8.07107 0.783931
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) −7.07107 −0.677285 −0.338643 0.940915i \(-0.609968\pi\)
−0.338643 + 0.940915i \(0.609968\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.41421 2.23607i 0.133631 0.211289i
\(113\) 1.07107i 0.100758i 0.998730 + 0.0503788i \(0.0160429\pi\)
−0.998730 + 0.0503788i \(0.983957\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.757359i 0.0703190i
\(117\) 0 0
\(118\) 7.25077i 0.667487i
\(119\) 3.16228 5.00000i 0.289886 0.458349i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0.926210 0.0838551
\(123\) 0 0
\(124\) 4.08849i 0.367157i
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0000 −0.887357 −0.443678 0.896186i \(-0.646327\pi\)
−0.443678 + 0.896186i \(0.646327\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.6491 1.10516 0.552579 0.833461i \(-0.313644\pi\)
0.552579 + 0.833461i \(0.313644\pi\)
\(132\) 0 0
\(133\) −17.0711 10.7967i −1.48025 0.936192i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.23607i 0.191741i
\(137\) 12.1421i 1.03737i 0.854965 + 0.518686i \(0.173579\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(138\) 0 0
\(139\) 10.7967i 0.915763i −0.889013 0.457882i \(-0.848608\pi\)
0.889013 0.457882i \(-0.151392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −15.6569 −1.31389
\(143\) −1.30986 −0.109536
\(144\) 0 0
\(145\) 0 0
\(146\) −13.9590 −1.15525
\(147\) 0 0
\(148\) 2.82843 0.232495
\(149\) 0.757359i 0.0620453i 0.999519 + 0.0310226i \(0.00987640\pi\)
−0.999519 + 0.0310226i \(0.990124\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 7.63441 0.619233
\(153\) 0 0
\(154\) −2.00000 + 3.16228i −0.161165 + 0.254824i
\(155\) 0 0
\(156\) 0 0
\(157\) 10.7967i 0.861670i −0.902431 0.430835i \(-0.858219\pi\)
0.902431 0.430835i \(-0.141781\pi\)
\(158\) 13.0711i 1.03988i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.23607 + 1.41421i 0.176227 + 0.111456i
\(162\) 0 0
\(163\) −7.92893 −0.621042 −0.310521 0.950567i \(-0.600503\pi\)
−0.310521 + 0.950567i \(0.600503\pi\)
\(164\) −8.56062 −0.668472
\(165\) 0 0
\(166\) 14.3426i 1.11320i
\(167\) −12.1065 −0.936833 −0.468416 0.883508i \(-0.655175\pi\)
−0.468416 + 0.883508i \(0.655175\pi\)
\(168\) 0 0
\(169\) 12.1421 0.934010
\(170\) 0 0
\(171\) 0 0
\(172\) 3.58579 0.273414
\(173\) −10.2541 −0.779607 −0.389804 0.920898i \(-0.627457\pi\)
−0.389804 + 0.920898i \(0.627457\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 2.61972i 0.196356i
\(179\) 11.3137i 0.845626i 0.906217 + 0.422813i \(0.138957\pi\)
−0.906217 + 0.422813i \(0.861043\pi\)
\(180\) 0 0
\(181\) 2.61972i 0.194722i −0.995249 0.0973610i \(-0.968960\pi\)
0.995249 0.0973610i \(-0.0310401\pi\)
\(182\) −1.30986 + 2.07107i −0.0970932 + 0.153518i
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.16228i 0.231249i
\(188\) 1.30986 0.0955312
\(189\) 0 0
\(190\) 0 0
\(191\) 0.656854i 0.0475283i 0.999718 + 0.0237642i \(0.00756508\pi\)
−0.999718 + 0.0237642i \(0.992435\pi\)
\(192\) 0 0
\(193\) −8.48528 −0.610784 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(194\) −0.542561 −0.0389536
\(195\) 0 0
\(196\) 3.00000 + 6.32456i 0.214286 + 0.451754i
\(197\) 5.92893i 0.422419i 0.977441 + 0.211209i \(0.0677402\pi\)
−0.977441 + 0.211209i \(0.932260\pi\)
\(198\) 0 0
\(199\) 21.5934i 1.53071i −0.643606 0.765357i \(-0.722562\pi\)
0.643606 0.765357i \(-0.277438\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.1065i 0.851814i
\(203\) −1.69351 1.07107i −0.118861 0.0751742i
\(204\) 0 0
\(205\) 0 0
\(206\) −10.4130 −0.725511
\(207\) 0 0
\(208\) 0.926210i 0.0642211i
\(209\) −10.7967 −0.746823
\(210\) 0 0
\(211\) 16.2132 1.11616 0.558081 0.829786i \(-0.311538\pi\)
0.558081 + 0.829786i \(0.311538\pi\)
\(212\) 8.07107i 0.554323i
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) 9.14214 + 5.78199i 0.620609 + 0.392507i
\(218\) 7.07107i 0.478913i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.07107i 0.139315i
\(222\) 0 0
\(223\) 13.0328i 0.872738i −0.899768 0.436369i \(-0.856264\pi\)
0.899768 0.436369i \(-0.143736\pi\)
\(224\) −2.23607 1.41421i −0.149404 0.0944911i
\(225\) 0 0
\(226\) 1.07107 0.0712464
\(227\) −25.1393 −1.66855 −0.834277 0.551345i \(-0.814115\pi\)
−0.834277 + 0.551345i \(0.814115\pi\)
\(228\) 0 0
\(229\) 10.7967i 0.713465i 0.934206 + 0.356733i \(0.116109\pi\)
−0.934206 + 0.356733i \(0.883891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.757359 0.0497231
\(233\) 23.0711i 1.51144i −0.654897 0.755718i \(-0.727288\pi\)
0.654897 0.755718i \(-0.272712\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.25077 −0.471985
\(237\) 0 0
\(238\) −5.00000 3.16228i −0.324102 0.204980i
\(239\) 7.17157i 0.463890i −0.972729 0.231945i \(-0.925491\pi\)
0.972729 0.231945i \(-0.0745090\pi\)
\(240\) 0 0
\(241\) 4.47214i 0.288076i −0.989572 0.144038i \(-0.953991\pi\)
0.989572 0.144038i \(-0.0460087\pi\)
\(242\) 9.00000i 0.578542i
\(243\) 0 0
\(244\) 0.926210i 0.0592945i
\(245\) 0 0
\(246\) 0 0
\(247\) −7.07107 −0.449921
\(248\) −4.08849 −0.259619
\(249\) 0 0
\(250\) 0 0
\(251\) 18.8148 1.18758 0.593788 0.804621i \(-0.297632\pi\)
0.593788 + 0.804621i \(0.297632\pi\)
\(252\) 0 0
\(253\) 1.41421 0.0889108
\(254\) 10.0000i 0.627456i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −21.2097 −1.32303 −0.661513 0.749933i \(-0.730086\pi\)
−0.661513 + 0.749933i \(0.730086\pi\)
\(258\) 0 0
\(259\) −4.00000 + 6.32456i −0.248548 + 0.392989i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.6491i 0.781465i
\(263\) 29.2843i 1.80575i 0.429908 + 0.902873i \(0.358546\pi\)
−0.429908 + 0.902873i \(0.641454\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.7967 + 17.0711i −0.661988 + 1.04669i
\(267\) 0 0
\(268\) 0 0
\(269\) −27.9179 −1.70219 −0.851093 0.525014i \(-0.824060\pi\)
−0.851093 + 0.525014i \(0.824060\pi\)
\(270\) 0 0
\(271\) 7.09185i 0.430799i −0.976526 0.215400i \(-0.930895\pi\)
0.976526 0.215400i \(-0.0691054\pi\)
\(272\) 2.23607 0.135582
\(273\) 0 0
\(274\) 12.1421 0.733533
\(275\) 0 0
\(276\) 0 0
\(277\) 21.2132 1.27458 0.637289 0.770625i \(-0.280056\pi\)
0.637289 + 0.770625i \(0.280056\pi\)
\(278\) −10.7967 −0.647543
\(279\) 0 0
\(280\) 0 0
\(281\) 28.4853i 1.69929i −0.527356 0.849645i \(-0.676816\pi\)
0.527356 0.849645i \(-0.323184\pi\)
\(282\) 0 0
\(283\) 7.63441i 0.453819i 0.973916 + 0.226909i \(0.0728621\pi\)
−0.973916 + 0.226909i \(0.927138\pi\)
\(284\) 15.6569i 0.929063i
\(285\) 0 0
\(286\) 1.30986i 0.0774535i
\(287\) 12.1065 19.1421i 0.714627 1.12992i
\(288\) 0 0
\(289\) −12.0000 −0.705882
\(290\) 0 0
\(291\) 0 0
\(292\) 13.9590i 0.816887i
\(293\) 25.2982 1.47794 0.738969 0.673740i \(-0.235313\pi\)
0.738969 + 0.673740i \(0.235313\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) 0.757359 0.0438726
\(299\) 0.926210 0.0535641
\(300\) 0 0
\(301\) −5.07107 + 8.01806i −0.292291 + 0.462153i
\(302\) 14.1421i 0.813788i
\(303\) 0 0
\(304\) 7.63441i 0.437864i
\(305\) 0 0
\(306\) 0 0
\(307\) 3.70484i 0.211446i 0.994396 + 0.105723i \(0.0337157\pi\)
−0.994396 + 0.105723i \(0.966284\pi\)
\(308\) 3.16228 + 2.00000i 0.180187 + 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.23943 −0.297101 −0.148550 0.988905i \(-0.547461\pi\)
−0.148550 + 0.988905i \(0.547461\pi\)
\(312\) 0 0
\(313\) 16.5787i 0.937083i −0.883442 0.468541i \(-0.844780\pi\)
0.883442 0.468541i \(-0.155220\pi\)
\(314\) −10.7967 −0.609293
\(315\) 0 0
\(316\) 13.0711 0.735305
\(317\) 30.0711i 1.68896i 0.535588 + 0.844480i \(0.320090\pi\)
−0.535588 + 0.844480i \(0.679910\pi\)
\(318\) 0 0
\(319\) −1.07107 −0.0599683
\(320\) 0 0
\(321\) 0 0
\(322\) 1.41421 2.23607i 0.0788110 0.124611i
\(323\) 17.0711i 0.949860i
\(324\) 0 0
\(325\) 0 0
\(326\) 7.92893i 0.439143i
\(327\) 0 0
\(328\) 8.56062i 0.472681i
\(329\) −1.85242 + 2.92893i −0.102127 + 0.161477i
\(330\) 0 0
\(331\) 7.92893 0.435814 0.217907 0.975970i \(-0.430077\pi\)
0.217907 + 0.975970i \(0.430077\pi\)
\(332\) 14.3426 0.787153
\(333\) 0 0
\(334\) 12.1065i 0.662441i
\(335\) 0 0
\(336\) 0 0
\(337\) −11.9706 −0.652078 −0.326039 0.945356i \(-0.605714\pi\)
−0.326039 + 0.945356i \(0.605714\pi\)
\(338\) 12.1421i 0.660445i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.78199 0.313113
\(342\) 0 0
\(343\) −18.3848 2.23607i −0.992685 0.120736i
\(344\) 3.58579i 0.193333i
\(345\) 0 0
\(346\) 10.2541i 0.551265i
\(347\) 9.07107i 0.486960i 0.969906 + 0.243480i \(0.0782891\pi\)
−0.969906 + 0.243480i \(0.921711\pi\)
\(348\) 0 0
\(349\) 6.16564i 0.330039i 0.986290 + 0.165020i \(0.0527688\pi\)
−0.986290 + 0.165020i \(0.947231\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.41421 −0.0753778
\(353\) 18.9737 1.00987 0.504933 0.863158i \(-0.331517\pi\)
0.504933 + 0.863158i \(0.331517\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.61972 0.138845
\(357\) 0 0
\(358\) 11.3137 0.597948
\(359\) 16.3137i 0.861005i −0.902589 0.430502i \(-0.858336\pi\)
0.902589 0.430502i \(-0.141664\pi\)
\(360\) 0 0
\(361\) −39.2843 −2.06759
\(362\) −2.61972 −0.137689
\(363\) 0 0
\(364\) 2.07107 + 1.30986i 0.108553 + 0.0686552i
\(365\) 0 0
\(366\) 0 0
\(367\) 24.5967i 1.28394i 0.766730 + 0.641970i \(0.221883\pi\)
−0.766730 + 0.641970i \(0.778117\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) 0 0
\(371\) 18.0475 + 11.4142i 0.936977 + 0.592596i
\(372\) 0 0
\(373\) 7.07107 0.366126 0.183063 0.983101i \(-0.441399\pi\)
0.183063 + 0.983101i \(0.441399\pi\)
\(374\) −3.16228 −0.163517
\(375\) 0 0
\(376\) 1.30986i 0.0675508i
\(377\) −0.701474 −0.0361277
\(378\) 0 0
\(379\) 22.0711 1.13371 0.566857 0.823816i \(-0.308159\pi\)
0.566857 + 0.823816i \(0.308159\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.656854 0.0336076
\(383\) 16.0361 0.819408 0.409704 0.912219i \(-0.365632\pi\)
0.409704 + 0.912219i \(0.365632\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 0.542561i 0.0275444i
\(389\) 22.8284i 1.15745i 0.815524 + 0.578724i \(0.196449\pi\)
−0.815524 + 0.578724i \(0.803551\pi\)
\(390\) 0 0
\(391\) 2.23607i 0.113083i
\(392\) 6.32456 3.00000i 0.319438 0.151523i
\(393\) 0 0
\(394\) 5.92893 0.298695
\(395\) 0 0
\(396\) 0 0
\(397\) 29.6114i 1.48616i 0.669205 + 0.743078i \(0.266635\pi\)
−0.669205 + 0.743078i \(0.733365\pi\)
\(398\) −21.5934 −1.08238
\(399\) 0 0
\(400\) 0 0
\(401\) 9.79899i 0.489338i 0.969607 + 0.244669i \(0.0786793\pi\)
−0.969607 + 0.244669i \(0.921321\pi\)
\(402\) 0 0
\(403\) 3.78680 0.188634
\(404\) −12.1065 −0.602323
\(405\) 0 0
\(406\) −1.07107 + 1.69351i −0.0531562 + 0.0840473i
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 22.9032i 1.13249i 0.824236 + 0.566246i \(0.191605\pi\)
−0.824236 + 0.566246i \(0.808395\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.4130i 0.513014i
\(413\) 10.2541 16.2132i 0.504573 0.797800i
\(414\) 0 0
\(415\) 0 0
\(416\) −0.926210 −0.0454112
\(417\) 0 0
\(418\) 10.7967i 0.528083i
\(419\) 28.8441 1.40913 0.704564 0.709640i \(-0.251142\pi\)
0.704564 + 0.709640i \(0.251142\pi\)
\(420\) 0 0
\(421\) 33.3553 1.62564 0.812820 0.582515i \(-0.197931\pi\)
0.812820 + 0.582515i \(0.197931\pi\)
\(422\) 16.2132i 0.789246i
\(423\) 0 0
\(424\) −8.07107 −0.391966
\(425\) 0 0
\(426\) 0 0
\(427\) 2.07107 + 1.30986i 0.100226 + 0.0633885i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 4.79899i 0.231159i −0.993298 0.115580i \(-0.963127\pi\)
0.993298 0.115580i \(-0.0368725\pi\)
\(432\) 0 0
\(433\) 9.71157i 0.466708i −0.972392 0.233354i \(-0.925030\pi\)
0.972392 0.233354i \(-0.0749701\pi\)
\(434\) 5.78199 9.14214i 0.277545 0.438837i
\(435\) 0 0
\(436\) 7.07107 0.338643
\(437\) 7.63441 0.365204
\(438\) 0 0
\(439\) 4.08849i 0.195133i −0.995229 0.0975664i \(-0.968894\pi\)
0.995229 0.0975664i \(-0.0311058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.07107 −0.0985106
\(443\) 23.0711i 1.09614i −0.836433 0.548070i \(-0.815363\pi\)
0.836433 0.548070i \(-0.184637\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −13.0328 −0.617119
\(447\) 0 0
\(448\) −1.41421 + 2.23607i −0.0668153 + 0.105644i
\(449\) 18.3848i 0.867631i 0.901002 + 0.433816i \(0.142833\pi\)
−0.901002 + 0.433816i \(0.857167\pi\)
\(450\) 0 0
\(451\) 12.1065i 0.570075i
\(452\) 1.07107i 0.0503788i
\(453\) 0 0
\(454\) 25.1393i 1.17985i
\(455\) 0 0
\(456\) 0 0
\(457\) 17.8284 0.833979 0.416989 0.908911i \(-0.363085\pi\)
0.416989 + 0.908911i \(0.363085\pi\)
\(458\) 10.7967 0.504496
\(459\) 0 0
\(460\) 0 0
\(461\) −38.9394 −1.81359 −0.906794 0.421575i \(-0.861477\pi\)
−0.906794 + 0.421575i \(0.861477\pi\)
\(462\) 0 0
\(463\) 38.2843 1.77922 0.889610 0.456720i \(-0.150976\pi\)
0.889610 + 0.456720i \(0.150976\pi\)
\(464\) 0.757359i 0.0351595i
\(465\) 0 0
\(466\) −23.0711 −1.06875
\(467\) 29.6114 1.37025 0.685127 0.728424i \(-0.259747\pi\)
0.685127 + 0.728424i \(0.259747\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 7.25077i 0.333744i
\(473\) 5.07107i 0.233168i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.16228 + 5.00000i −0.144943 + 0.229175i
\(477\) 0 0
\(478\) −7.17157 −0.328020
\(479\) 33.6999 1.53979 0.769895 0.638171i \(-0.220309\pi\)
0.769895 + 0.638171i \(0.220309\pi\)
\(480\) 0 0
\(481\) 2.61972i 0.119449i
\(482\) −4.47214 −0.203700
\(483\) 0 0
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) 29.8995 1.35488 0.677438 0.735580i \(-0.263090\pi\)
0.677438 + 0.735580i \(0.263090\pi\)
\(488\) −0.926210 −0.0419275
\(489\) 0 0
\(490\) 0 0
\(491\) 21.5147i 0.970946i −0.874252 0.485473i \(-0.838647\pi\)
0.874252 0.485473i \(-0.161353\pi\)
\(492\) 0 0
\(493\) 1.69351i 0.0762717i
\(494\) 7.07107i 0.318142i
\(495\) 0 0
\(496\) 4.08849i 0.183579i
\(497\) −35.0098 22.1421i −1.57040 0.993211i
\(498\) 0 0
\(499\) −22.0711 −0.988037 −0.494018 0.869451i \(-0.664472\pi\)
−0.494018 + 0.869451i \(0.664472\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.8148i 0.839744i
\(503\) 21.8181 0.972822 0.486411 0.873730i \(-0.338306\pi\)
0.486411 + 0.873730i \(0.338306\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 0.0628695i
\(507\) 0 0
\(508\) 10.0000 0.443678
\(509\) 0.224736 0.00996125 0.00498063 0.999988i \(-0.498415\pi\)
0.00498063 + 0.999988i \(0.498415\pi\)
\(510\) 0 0
\(511\) −31.2132 19.7410i −1.38079 0.873289i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 21.2097i 0.935521i
\(515\) 0 0
\(516\) 0 0
\(517\) 1.85242i 0.0814693i
\(518\) 6.32456 + 4.00000i 0.277885 + 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 17.5049 0.766903 0.383452 0.923561i \(-0.374735\pi\)
0.383452 + 0.923561i \(0.374735\pi\)
\(522\) 0 0
\(523\) 28.4605i 1.24449i 0.782822 + 0.622245i \(0.213779\pi\)
−0.782822 + 0.622245i \(0.786221\pi\)
\(524\) −12.6491 −0.552579
\(525\) 0 0
\(526\) 29.2843 1.27685
\(527\) 9.14214i 0.398238i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 17.0711 + 10.7967i 0.740125 + 0.468096i
\(533\) 7.92893i 0.343440i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 27.9179i 1.20363i
\(539\) −8.94427 + 4.24264i −0.385257 + 0.182743i
\(540\) 0 0
\(541\) 5.07107 0.218022 0.109011 0.994041i \(-0.465232\pi\)
0.109011 + 0.994041i \(0.465232\pi\)
\(542\) −7.09185 −0.304621
\(543\) 0 0
\(544\) 2.23607i 0.0958706i
\(545\) 0 0
\(546\) 0 0
\(547\) −16.2132 −0.693227 −0.346613 0.938008i \(-0.612668\pi\)
−0.346613 + 0.938008i \(0.612668\pi\)
\(548\) 12.1421i 0.518686i
\(549\) 0 0
\(550\) 0 0
\(551\) −5.78199 −0.246321
\(552\) 0 0
\(553\) −18.4853 + 29.2278i −0.786074 + 1.24289i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 10.7967i 0.457882i
\(557\) 26.1421i 1.10768i −0.832624 0.553839i \(-0.813162\pi\)
0.832624 0.553839i \(-0.186838\pi\)
\(558\) 0 0
\(559\) 3.32119i 0.140471i
\(560\) 0 0
\(561\) 0 0
\(562\) −28.4853 −1.20158
\(563\) −24.3720 −1.02716 −0.513579 0.858042i \(-0.671681\pi\)
−0.513579 + 0.858042i \(0.671681\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.63441 0.320898
\(567\) 0 0
\(568\) 15.6569 0.656947
\(569\) 22.5269i 0.944377i −0.881498 0.472189i \(-0.843464\pi\)
0.881498 0.472189i \(-0.156536\pi\)
\(570\) 0 0
\(571\) −38.2132 −1.59917 −0.799586 0.600551i \(-0.794948\pi\)
−0.799586 + 0.600551i \(0.794948\pi\)
\(572\) 1.30986 0.0547679
\(573\) 0 0
\(574\) −19.1421 12.1065i −0.798977 0.505318i
\(575\) 0 0
\(576\) 0 0
\(577\) 23.4458i 0.976062i 0.872826 + 0.488031i \(0.162285\pi\)
−0.872826 + 0.488031i \(0.837715\pi\)
\(578\) 12.0000i 0.499134i
\(579\) 0 0
\(580\) 0 0
\(581\) −20.2835 + 32.0711i −0.841502 + 1.33053i
\(582\) 0 0
\(583\) 11.4142 0.472728
\(584\) 13.9590 0.577626
\(585\) 0 0
\(586\) 25.2982i 1.04506i
\(587\) 15.4277 0.636771 0.318385 0.947961i \(-0.396859\pi\)
0.318385 + 0.947961i \(0.396859\pi\)
\(588\) 0 0
\(589\) 31.2132 1.28612
\(590\) 0 0
\(591\) 0 0
\(592\) −2.82843 −0.116248
\(593\) 46.8916 1.92561 0.962804 0.270202i \(-0.0870904\pi\)
0.962804 + 0.270202i \(0.0870904\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.757359i 0.0310226i
\(597\) 0 0
\(598\) 0.926210i 0.0378755i
\(599\) 3.68629i 0.150618i −0.997160 0.0753089i \(-0.976006\pi\)
0.997160 0.0753089i \(-0.0239943\pi\)
\(600\) 0 0
\(601\) 12.1065i 0.493836i −0.969036 0.246918i \(-0.920582\pi\)
0.969036 0.246918i \(-0.0794179\pi\)
\(602\) 8.01806 + 5.07107i 0.326792 + 0.206681i
\(603\) 0 0
\(604\) 14.1421 0.575435
\(605\) 0 0
\(606\) 0 0
\(607\) 43.9541i 1.78404i −0.451996 0.892020i \(-0.649288\pi\)
0.451996 0.892020i \(-0.350712\pi\)
\(608\) −7.63441 −0.309616
\(609\) 0 0
\(610\) 0 0
\(611\) 1.21320i 0.0490810i
\(612\) 0 0
\(613\) 15.8579 0.640493 0.320247 0.947334i \(-0.396234\pi\)
0.320247 + 0.947334i \(0.396234\pi\)
\(614\) 3.70484 0.149515
\(615\) 0 0
\(616\) 2.00000 3.16228i 0.0805823 0.127412i
\(617\) 32.0000i 1.28827i 0.764911 + 0.644136i \(0.222783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 18.2064i 0.731776i 0.930659 + 0.365888i \(0.119235\pi\)
−0.930659 + 0.365888i \(0.880765\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.23943i 0.210082i
\(623\) −3.70484 + 5.85786i −0.148431 + 0.234690i
\(624\) 0 0
\(625\) 0 0
\(626\) −16.5787 −0.662618
\(627\) 0 0
\(628\) 10.7967i 0.430835i
\(629\) −6.32456 −0.252177
\(630\) 0 0
\(631\) 10.2843 0.409410 0.204705 0.978824i \(-0.434376\pi\)
0.204705 + 0.978824i \(0.434376\pi\)
\(632\) 13.0711i 0.519939i
\(633\) 0 0
\(634\) 30.0711 1.19427
\(635\) 0 0
\(636\) 0 0
\(637\) −5.85786 + 2.77863i −0.232097 + 0.110093i
\(638\) 1.07107i 0.0424040i
\(639\) 0 0
\(640\) 0 0
\(641\) 28.4853i 1.12510i 0.826763 + 0.562550i \(0.190180\pi\)
−0.826763 + 0.562550i \(0.809820\pi\)
\(642\) 0 0
\(643\) 29.9951i 1.18289i 0.806345 + 0.591446i \(0.201443\pi\)
−0.806345 + 0.591446i \(0.798557\pi\)
\(644\) −2.23607 1.41421i −0.0881134 0.0557278i
\(645\) 0 0
\(646\) −17.0711 −0.671652
\(647\) 37.0869 1.45804 0.729019 0.684493i \(-0.239977\pi\)
0.729019 + 0.684493i \(0.239977\pi\)
\(648\) 0 0
\(649\) 10.2541i 0.402510i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.92893 0.310521
\(653\) 8.14214i 0.318626i −0.987228 0.159313i \(-0.949072\pi\)
0.987228 0.159313i \(-0.0509280\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.56062 0.334236
\(657\) 0 0
\(658\) 2.92893 + 1.85242i 0.114182 + 0.0722148i
\(659\) 48.3848i 1.88480i −0.334483 0.942402i \(-0.608562\pi\)
0.334483 0.942402i \(-0.391438\pi\)
\(660\) 0 0
\(661\) 18.2064i 0.708146i −0.935218 0.354073i \(-0.884796\pi\)
0.935218 0.354073i \(-0.115204\pi\)
\(662\) 7.92893i 0.308167i
\(663\) 0 0
\(664\) 14.3426i 0.556602i
\(665\) 0 0
\(666\) 0 0
\(667\) 0.757359 0.0293251
\(668\) 12.1065 0.468416
\(669\) 0 0
\(670\) 0 0
\(671\) 1.30986 0.0505665
\(672\) 0 0
\(673\) 34.7990 1.34140 0.670701 0.741728i \(-0.265993\pi\)
0.670701 + 0.741728i \(0.265993\pi\)
\(674\) 11.9706i 0.461089i
\(675\) 0 0
\(676\) −12.1421 −0.467005
\(677\) −22.9032 −0.880243 −0.440122 0.897938i \(-0.645065\pi\)
−0.440122 + 0.897938i \(0.645065\pi\)
\(678\) 0 0
\(679\) −1.21320 0.767297i −0.0465585 0.0294462i
\(680\) 0 0
\(681\) 0 0
\(682\) 5.78199i 0.221404i
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.23607 + 18.3848i −0.0853735 + 0.701934i
\(687\) 0 0
\(688\) −3.58579 −0.136707
\(689\) 7.47550 0.284794
\(690\) 0 0
\(691\) 23.9884i 0.912560i −0.889836 0.456280i \(-0.849181\pi\)
0.889836 0.456280i \(-0.150819\pi\)
\(692\) 10.2541 0.389804
\(693\) 0 0
\(694\) 9.07107 0.344333
\(695\) 0 0
\(696\) 0 0
\(697\) 19.1421 0.725060
\(698\) 6.16564 0.233373
\(699\) 0 0
\(700\) 0 0
\(701\) 41.8701i 1.58141i −0.612197 0.790705i \(-0.709714\pi\)
0.612197 0.790705i \(-0.290286\pi\)
\(702\) 0 0
\(703\) 21.5934i 0.814410i
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 18.9737i 0.714083i
\(707\) 17.1212 27.0711i 0.643911 1.01811i
\(708\) 0 0
\(709\) 24.1421 0.906677 0.453338 0.891338i \(-0.350233\pi\)
0.453338 + 0.891338i \(0.350233\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.61972i 0.0981780i
\(713\) −4.08849 −0.153115
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3137i 0.422813i
\(717\) 0 0
\(718\) −16.3137 −0.608822
\(719\) 23.9884 0.894615 0.447307 0.894380i \(-0.352383\pi\)
0.447307 + 0.894380i \(0.352383\pi\)
\(720\) 0 0
\(721\) −23.2843 14.7263i −0.867152 0.548435i
\(722\) 39.2843i 1.46201i
\(723\) 0 0
\(724\) 2.61972i 0.0973610i
\(725\) 0 0
\(726\) 0 0
\(727\) 6.70820i 0.248794i 0.992233 + 0.124397i \(0.0396996\pi\)
−0.992233 + 0.124397i \(0.960300\pi\)
\(728\) 1.30986 2.07107i 0.0485466 0.0767589i
\(729\) 0 0
\(730\) 0 0
\(731\) −8.01806 −0.296559
\(732\) 0 0
\(733\) 39.6408i 1.46417i 0.681214 + 0.732084i \(0.261452\pi\)
−0.681214 + 0.732084i \(0.738548\pi\)
\(734\) 24.5967 0.907883
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −26.3553 −0.969497 −0.484748 0.874654i \(-0.661089\pi\)
−0.484748 + 0.874654i \(0.661089\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.4142 18.0475i 0.419029 0.662543i
\(743\) 35.1421i 1.28924i −0.764503 0.644620i \(-0.777016\pi\)
0.764503 0.644620i \(-0.222984\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07107i 0.258890i
\(747\) 0 0
\(748\) 3.16228i 0.115624i
\(749\) 4.47214 + 2.82843i 0.163408 + 0.103348i
\(750\) 0 0
\(751\) 31.2132 1.13899 0.569493 0.821996i \(-0.307140\pi\)
0.569493 + 0.821996i \(0.307140\pi\)
\(752\) −1.30986 −0.0477656
\(753\) 0 0
\(754\) 0.701474i 0.0255462i
\(755\) 0 0
\(756\) 0 0
\(757\) −44.0416 −1.60072 −0.800360 0.599520i \(-0.795358\pi\)
−0.800360 + 0.599520i \(0.795358\pi\)
\(758\) 22.0711i 0.801657i
\(759\) 0 0
\(760\) 0 0
\(761\) −31.3050 −1.13480 −0.567402 0.823441i \(-0.692051\pi\)
−0.567402 + 0.823441i \(0.692051\pi\)
\(762\) 0 0
\(763\) −10.0000 + 15.8114i −0.362024 + 0.572411i
\(764\) 0.656854i 0.0237642i
\(765\) 0 0
\(766\) 16.0361i 0.579409i
\(767\) 6.71573i 0.242491i
\(768\) 0 0
\(769\) 15.0441i 0.542504i 0.962508 + 0.271252i \(0.0874376\pi\)
−0.962508 + 0.271252i \(0.912562\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.48528 0.305392
\(773\) 17.3460 0.623892 0.311946 0.950100i \(-0.399019\pi\)
0.311946 + 0.950100i \(0.399019\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.542561 0.0194768
\(777\) 0 0
\(778\) 22.8284 0.818439
\(779\) 65.3553i 2.34160i
\(780\) 0 0
\(781\) −22.1421 −0.792308
\(782\) 2.23607 0.0799616
\(783\) 0 0
\(784\) −3.00000 6.32456i −0.107143 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) 33.9247i 1.20928i −0.796497 0.604642i \(-0.793316\pi\)
0.796497 0.604642i \(-0.206684\pi\)
\(788\) 5.92893i 0.211209i
\(789\) 0 0
\(790\) 0 0
\(791\) 2.39498 + 1.51472i 0.0851557 + 0.0538572i
\(792\) 0 0
\(793\) 0.857864 0.0304637
\(794\) 29.6114 1.05087
\(795\) 0 0
\(796\) 21.5934i 0.765357i
\(797\) −11.0214 −0.390399 −0.195199 0.980764i \(-0.562535\pi\)
−0.195199 + 0.980764i \(0.562535\pi\)
\(798\) 0 0
\(799\) −2.92893 −0.103618
\(800\) 0 0
\(801\) 0 0
\(802\) 9.79899 0.346014
\(803\) −19.7410 −0.696643
\(804\)