Properties

Label 3528.2.s.bd.3313.1
Level $3528$
Weight $2$
Character 3528.3313
Analytic conductor $28.171$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3313.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3528.3313
Dual form 3528.2.s.bd.361.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.70711 + 2.95680i) q^{5} +O(q^{10})\) \(q+(-1.70711 + 2.95680i) q^{5} +(2.41421 + 4.18154i) q^{11} -1.41421 q^{13} +(3.12132 + 5.40629i) q^{17} +(0.585786 - 1.01461i) q^{19} +(-0.414214 + 0.717439i) q^{23} +(-3.32843 - 5.76500i) q^{25} +8.48528 q^{29} +(5.41421 + 9.37769i) q^{31} +(4.82843 - 8.36308i) q^{37} -3.41421 q^{41} -8.00000 q^{43} +(-0.585786 + 1.01461i) q^{47} +(4.65685 + 8.06591i) q^{53} -16.4853 q^{55} +(-5.41421 - 9.37769i) q^{59} +(-2.94975 + 5.10911i) q^{61} +(2.41421 - 4.18154i) q^{65} +(4.00000 + 6.92820i) q^{67} -4.82843 q^{71} +(1.53553 + 2.65962i) q^{73} +(6.82843 - 11.8272i) q^{79} -7.31371 q^{83} -21.3137 q^{85} +(-7.36396 + 12.7548i) q^{89} +(2.00000 + 3.46410i) q^{95} +16.2426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{11} + 4q^{17} + 8q^{19} + 4q^{23} - 2q^{25} + 16q^{31} + 8q^{37} - 8q^{41} - 32q^{43} - 8q^{47} - 4q^{53} - 32q^{55} - 16q^{59} + 8q^{61} + 4q^{65} + 16q^{67} - 8q^{71} - 8q^{73} + 16q^{79} + 16q^{83} - 40q^{85} - 4q^{89} + 8q^{95} + 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.70711 + 2.95680i −0.763441 + 1.32232i 0.177625 + 0.984098i \(0.443158\pi\)
−0.941067 + 0.338221i \(0.890175\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.41421 + 4.18154i 0.727913 + 1.26078i 0.957764 + 0.287556i \(0.0928428\pi\)
−0.229851 + 0.973226i \(0.573824\pi\)
\(12\) 0 0
\(13\) −1.41421 −0.392232 −0.196116 0.980581i \(-0.562833\pi\)
−0.196116 + 0.980581i \(0.562833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.12132 + 5.40629i 0.757031 + 1.31122i 0.944358 + 0.328919i \(0.106684\pi\)
−0.187327 + 0.982298i \(0.559982\pi\)
\(18\) 0 0
\(19\) 0.585786 1.01461i 0.134389 0.232768i −0.790975 0.611848i \(-0.790426\pi\)
0.925364 + 0.379080i \(0.123760\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.414214 + 0.717439i −0.0863695 + 0.149596i −0.905974 0.423333i \(-0.860860\pi\)
0.819604 + 0.572930i \(0.194193\pi\)
\(24\) 0 0
\(25\) −3.32843 5.76500i −0.665685 1.15300i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48528 1.57568 0.787839 0.615882i \(-0.211200\pi\)
0.787839 + 0.615882i \(0.211200\pi\)
\(30\) 0 0
\(31\) 5.41421 + 9.37769i 0.972421 + 1.68428i 0.688194 + 0.725526i \(0.258404\pi\)
0.284227 + 0.958757i \(0.408263\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.82843 8.36308i 0.793789 1.37488i −0.129817 0.991538i \(-0.541439\pi\)
0.923606 0.383344i \(-0.125228\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.41421 −0.533211 −0.266605 0.963806i \(-0.585902\pi\)
−0.266605 + 0.963806i \(0.585902\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.585786 + 1.01461i −0.0854457 + 0.147996i −0.905581 0.424173i \(-0.860565\pi\)
0.820135 + 0.572170i \(0.193898\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.65685 + 8.06591i 0.639668 + 1.10794i 0.985506 + 0.169643i \(0.0542615\pi\)
−0.345837 + 0.938294i \(0.612405\pi\)
\(54\) 0 0
\(55\) −16.4853 −2.22287
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.41421 9.37769i −0.704871 1.22087i −0.966738 0.255768i \(-0.917672\pi\)
0.261868 0.965104i \(-0.415662\pi\)
\(60\) 0 0
\(61\) −2.94975 + 5.10911i −0.377676 + 0.654155i −0.990724 0.135891i \(-0.956610\pi\)
0.613047 + 0.790046i \(0.289944\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.41421 4.18154i 0.299446 0.518656i
\(66\) 0 0
\(67\) 4.00000 + 6.92820i 0.488678 + 0.846415i 0.999915 0.0130248i \(-0.00414604\pi\)
−0.511237 + 0.859440i \(0.670813\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.82843 −0.573029 −0.286514 0.958076i \(-0.592497\pi\)
−0.286514 + 0.958076i \(0.592497\pi\)
\(72\) 0 0
\(73\) 1.53553 + 2.65962i 0.179721 + 0.311285i 0.941785 0.336216i \(-0.109147\pi\)
−0.762064 + 0.647501i \(0.775814\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.82843 11.8272i 0.768258 1.33066i −0.170249 0.985401i \(-0.554457\pi\)
0.938507 0.345261i \(-0.112210\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.31371 −0.802784 −0.401392 0.915906i \(-0.631473\pi\)
−0.401392 + 0.915906i \(0.631473\pi\)
\(84\) 0 0
\(85\) −21.3137 −2.31180
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.36396 + 12.7548i −0.780578 + 1.35200i 0.151027 + 0.988530i \(0.451742\pi\)
−0.931605 + 0.363472i \(0.881591\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 + 3.46410i 0.205196 + 0.355409i
\(96\) 0 0
\(97\) 16.2426 1.64919 0.824595 0.565723i \(-0.191403\pi\)
0.824595 + 0.565723i \(0.191403\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.292893 0.507306i −0.0291440 0.0504788i 0.851086 0.525027i \(-0.175945\pi\)
−0.880230 + 0.474548i \(0.842611\pi\)
\(102\) 0 0
\(103\) 2.58579 4.47871i 0.254785 0.441301i −0.710052 0.704149i \(-0.751329\pi\)
0.964837 + 0.262848i \(0.0846619\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.24264 + 2.15232i −0.120131 + 0.208072i −0.919819 0.392343i \(-0.871665\pi\)
0.799688 + 0.600415i \(0.204998\pi\)
\(108\) 0 0
\(109\) −5.65685 9.79796i −0.541828 0.938474i −0.998799 0.0489926i \(-0.984399\pi\)
0.456971 0.889482i \(-0.348934\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.41421 2.44949i −0.131876 0.228416i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.15685 + 10.6640i −0.559714 + 0.969453i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 7.31371 0.648987 0.324493 0.945888i \(-0.394806\pi\)
0.324493 + 0.945888i \(0.394806\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.65685 + 13.2621i −0.668982 + 1.15871i 0.309207 + 0.950995i \(0.399937\pi\)
−0.978189 + 0.207717i \(0.933397\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.24264 10.8126i −0.533345 0.923780i −0.999242 0.0389412i \(-0.987602\pi\)
0.465897 0.884839i \(-0.345732\pi\)
\(138\) 0 0
\(139\) −9.65685 −0.819084 −0.409542 0.912291i \(-0.634311\pi\)
−0.409542 + 0.912291i \(0.634311\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.41421 5.91359i −0.285511 0.494519i
\(144\) 0 0
\(145\) −14.4853 + 25.0892i −1.20294 + 2.08355i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.00000 + 8.66025i −0.409616 + 0.709476i −0.994847 0.101391i \(-0.967671\pi\)
0.585231 + 0.810867i \(0.301004\pi\)
\(150\) 0 0
\(151\) 0.828427 + 1.43488i 0.0674164 + 0.116769i 0.897763 0.440478i \(-0.145191\pi\)
−0.830347 + 0.557247i \(0.811858\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −36.9706 −2.96955
\(156\) 0 0
\(157\) 2.94975 + 5.10911i 0.235415 + 0.407752i 0.959393 0.282072i \(-0.0910216\pi\)
−0.723978 + 0.689823i \(0.757688\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.17157 2.02922i 0.0917647 0.158941i −0.816489 0.577361i \(-0.804083\pi\)
0.908254 + 0.418420i \(0.137416\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.82843 0.528400 0.264200 0.964468i \(-0.414892\pi\)
0.264200 + 0.964468i \(0.414892\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.292893 0.507306i 0.0222683 0.0385698i −0.854677 0.519161i \(-0.826245\pi\)
0.876945 + 0.480591i \(0.159578\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.8995 18.8785i −0.814667 1.41104i −0.909567 0.415557i \(-0.863587\pi\)
0.0949006 0.995487i \(-0.469747\pi\)
\(180\) 0 0
\(181\) −9.89949 −0.735824 −0.367912 0.929861i \(-0.619927\pi\)
−0.367912 + 0.929861i \(0.619927\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 16.4853 + 28.5533i 1.21202 + 2.09928i
\(186\) 0 0
\(187\) −15.0711 + 26.1039i −1.10211 + 1.90890i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.4142 18.0379i 0.753546 1.30518i −0.192548 0.981288i \(-0.561675\pi\)
0.946094 0.323892i \(-0.104992\pi\)
\(192\) 0 0
\(193\) 10.3137 + 17.8639i 0.742397 + 1.28587i 0.951401 + 0.307955i \(0.0996445\pi\)
−0.209004 + 0.977915i \(0.567022\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −2.82843 4.89898i −0.200502 0.347279i 0.748188 0.663486i \(-0.230924\pi\)
−0.948690 + 0.316207i \(0.897591\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 5.82843 10.0951i 0.407075 0.705075i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −25.6569 −1.76629 −0.883145 0.469099i \(-0.844579\pi\)
−0.883145 + 0.469099i \(0.844579\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.6569 23.6544i 0.931390 1.61321i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.41421 7.64564i −0.296932 0.514302i
\(222\) 0 0
\(223\) 2.34315 0.156909 0.0784543 0.996918i \(-0.475002\pi\)
0.0784543 + 0.996918i \(0.475002\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −9.89949 17.1464i −0.657053 1.13805i −0.981375 0.192102i \(-0.938470\pi\)
0.324322 0.945947i \(-0.394864\pi\)
\(228\) 0 0
\(229\) 0.464466 0.804479i 0.0306928 0.0531615i −0.850271 0.526345i \(-0.823562\pi\)
0.880964 + 0.473184i \(0.156895\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.75736 + 9.97204i −0.377177 + 0.653290i −0.990650 0.136426i \(-0.956439\pi\)
0.613473 + 0.789716i \(0.289772\pi\)
\(234\) 0 0
\(235\) −2.00000 3.46410i −0.130466 0.225973i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.82843 0.571063 0.285532 0.958369i \(-0.407830\pi\)
0.285532 + 0.958369i \(0.407830\pi\)
\(240\) 0 0
\(241\) −1.05025 1.81909i −0.0676527 0.117178i 0.830215 0.557443i \(-0.188218\pi\)
−0.897868 + 0.440265i \(0.854884\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.828427 + 1.43488i −0.0527116 + 0.0912991i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.8787 18.8424i 0.678593 1.17536i −0.296811 0.954936i \(-0.595923\pi\)
0.975405 0.220422i \(-0.0707435\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.58579 + 16.6031i 0.591085 + 1.02379i 0.994087 + 0.108590i \(0.0346335\pi\)
−0.403002 + 0.915199i \(0.632033\pi\)
\(264\) 0 0
\(265\) −31.7990 −1.95340
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.02082 + 15.6245i 0.550009 + 0.952643i 0.998273 + 0.0587422i \(0.0187090\pi\)
−0.448264 + 0.893901i \(0.647958\pi\)
\(270\) 0 0
\(271\) −9.41421 + 16.3059i −0.571873 + 0.990513i 0.424501 + 0.905427i \(0.360450\pi\)
−0.996374 + 0.0850852i \(0.972884\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.0711 27.8359i 0.969122 1.67857i
\(276\) 0 0
\(277\) 3.00000 + 5.19615i 0.180253 + 0.312207i 0.941966 0.335707i \(-0.108975\pi\)
−0.761714 + 0.647913i \(0.775642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.48528 −0.267569 −0.133785 0.991010i \(-0.542713\pi\)
−0.133785 + 0.991010i \(0.542713\pi\)
\(282\) 0 0
\(283\) −4.58579 7.94282i −0.272597 0.472151i 0.696929 0.717140i \(-0.254549\pi\)
−0.969526 + 0.244989i \(0.921216\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.9853 + 19.0271i −0.646193 + 1.11924i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.0711 −0.763620 −0.381810 0.924241i \(-0.624699\pi\)
−0.381810 + 0.924241i \(0.624699\pi\)
\(294\) 0 0
\(295\) 36.9706 2.15251
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.585786 1.01461i 0.0338769 0.0586765i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10.0711 17.4436i −0.576668 0.998818i
\(306\) 0 0
\(307\) 28.4853 1.62574 0.812870 0.582445i \(-0.197904\pi\)
0.812870 + 0.582445i \(0.197904\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.07107 1.85514i −0.0607347 0.105196i 0.834059 0.551675i \(-0.186011\pi\)
−0.894794 + 0.446479i \(0.852678\pi\)
\(312\) 0 0
\(313\) −7.29289 + 12.6317i −0.412219 + 0.713984i −0.995132 0.0985506i \(-0.968579\pi\)
0.582913 + 0.812534i \(0.301913\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.656854 + 1.13770i −0.0368926 + 0.0638999i −0.883882 0.467710i \(-0.845079\pi\)
0.846990 + 0.531610i \(0.178413\pi\)
\(318\) 0 0
\(319\) 20.4853 + 35.4815i 1.14696 + 1.98659i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.31371 0.406946
\(324\) 0 0
\(325\) 4.70711 + 8.15295i 0.261103 + 0.452244i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.6569 27.1185i 0.860579 1.49057i −0.0107928 0.999942i \(-0.503436\pi\)
0.871371 0.490624i \(-0.163231\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −27.3137 −1.49231
\(336\) 0 0
\(337\) 16.9706 0.924445 0.462223 0.886764i \(-0.347052\pi\)
0.462223 + 0.886764i \(0.347052\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −26.1421 + 45.2795i −1.41568 + 2.45202i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0711 + 20.9077i 0.648009 + 1.12238i 0.983598 + 0.180376i \(0.0577314\pi\)
−0.335589 + 0.942009i \(0.608935\pi\)
\(348\) 0 0
\(349\) 6.38478 0.341769 0.170885 0.985291i \(-0.445337\pi\)
0.170885 + 0.985291i \(0.445337\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.94975 3.37706i −0.103775 0.179743i 0.809462 0.587172i \(-0.199759\pi\)
−0.913237 + 0.407429i \(0.866425\pi\)
\(354\) 0 0
\(355\) 8.24264 14.2767i 0.437474 0.757727i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.72792 13.3852i 0.407864 0.706441i −0.586786 0.809742i \(-0.699607\pi\)
0.994650 + 0.103301i \(0.0329404\pi\)
\(360\) 0 0
\(361\) 8.81371 + 15.2658i 0.463879 + 0.803463i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.4853 −0.548825
\(366\) 0 0
\(367\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −18.6569 + 32.3146i −0.966015 + 1.67319i −0.259153 + 0.965836i \(0.583443\pi\)
−0.706862 + 0.707351i \(0.749890\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.48528 7.76874i 0.229187 0.396964i −0.728380 0.685173i \(-0.759727\pi\)
0.957567 + 0.288209i \(0.0930599\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.07107 + 12.2474i 0.358517 + 0.620970i 0.987713 0.156276i \(-0.0499491\pi\)
−0.629196 + 0.777247i \(0.716616\pi\)
\(390\) 0 0
\(391\) −5.17157 −0.261538
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 23.3137 + 40.3805i 1.17304 + 2.03176i
\(396\) 0 0
\(397\) 16.3640 28.3432i 0.821284 1.42251i −0.0834430 0.996513i \(-0.526592\pi\)
0.904727 0.425992i \(-0.140075\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.24264 3.88437i 0.111992 0.193976i −0.804581 0.593843i \(-0.797610\pi\)
0.916573 + 0.399867i \(0.130944\pi\)
\(402\) 0 0
\(403\) −7.65685 13.2621i −0.381415 0.660630i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 46.6274 2.31124
\(408\) 0 0
\(409\) 3.87868 + 6.71807i 0.191788 + 0.332187i 0.945843 0.324625i \(-0.105238\pi\)
−0.754055 + 0.656812i \(0.771905\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.4853 21.6251i 0.612878 1.06154i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.17157 0.252648 0.126324 0.991989i \(-0.459682\pi\)
0.126324 + 0.991989i \(0.459682\pi\)
\(420\) 0 0
\(421\) 29.3137 1.42866 0.714331 0.699808i \(-0.246731\pi\)
0.714331 + 0.699808i \(0.246731\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 20.7782 35.9889i 1.00789 1.74572i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.75736 11.7041i −0.325491 0.563766i 0.656121 0.754656i \(-0.272196\pi\)
−0.981612 + 0.190890i \(0.938863\pi\)
\(432\) 0 0
\(433\) 10.3848 0.499061 0.249530 0.968367i \(-0.419724\pi\)
0.249530 + 0.968367i \(0.419724\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.485281 + 0.840532i 0.0232142 + 0.0402081i
\(438\) 0 0
\(439\) 9.65685 16.7262i 0.460897 0.798296i −0.538109 0.842875i \(-0.680861\pi\)
0.999006 + 0.0445789i \(0.0141946\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.7574 18.6323i 0.511098 0.885247i −0.488820 0.872385i \(-0.662572\pi\)
0.999917 0.0128621i \(-0.00409425\pi\)
\(444\) 0 0
\(445\) −25.1421 43.5475i −1.19185 2.06435i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −8.24264 14.2767i −0.388131 0.672262i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.3137 21.3280i 0.576011 0.997680i −0.419920 0.907561i \(-0.637942\pi\)
0.995931 0.0901192i \(-0.0287248\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.75736 0.454446 0.227223 0.973843i \(-0.427035\pi\)
0.227223 + 0.973843i \(0.427035\pi\)
\(462\) 0 0
\(463\) −12.9706 −0.602793 −0.301397 0.953499i \(-0.597453\pi\)
−0.301397 + 0.953499i \(0.597453\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.58579 4.47871i 0.119656 0.207250i −0.799975 0.600033i \(-0.795154\pi\)
0.919631 + 0.392783i \(0.128488\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −19.3137 33.4523i −0.888045 1.53814i
\(474\) 0 0
\(475\) −7.79899 −0.357842
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.55635 16.5521i −0.436641 0.756284i 0.560787 0.827960i \(-0.310499\pi\)
−0.997428 + 0.0716760i \(0.977165\pi\)
\(480\) 0 0
\(481\) −6.82843 + 11.8272i −0.311349 + 0.539273i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −27.7279 + 48.0262i −1.25906 + 2.18076i
\(486\) 0 0
\(487\) −6.48528 11.2328i −0.293876 0.509008i 0.680847 0.732426i \(-0.261612\pi\)
−0.974723 + 0.223418i \(0.928279\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.14214 0.186932 0.0934660 0.995622i \(-0.470205\pi\)
0.0934660 + 0.995622i \(0.470205\pi\)
\(492\) 0 0
\(493\) 26.4853 + 45.8739i 1.19284 + 2.06605i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.1421 + 24.4949i −0.633089 + 1.09654i 0.353828 + 0.935311i \(0.384880\pi\)
−0.986917 + 0.161232i \(0.948453\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.6274 1.36561 0.682805 0.730601i \(-0.260760\pi\)
0.682805 + 0.730601i \(0.260760\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.4645 + 19.8570i −0.508154 + 0.880148i 0.491802 + 0.870707i \(0.336338\pi\)
−0.999955 + 0.00944061i \(0.996995\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.82843 + 15.2913i 0.389027 + 0.673814i
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.63604 4.56575i −0.115487 0.200029i 0.802487 0.596669i \(-0.203510\pi\)
−0.917974 + 0.396640i \(0.870176\pi\)
\(522\) 0 0
\(523\) −0.828427 + 1.43488i −0.0362246 + 0.0627428i −0.883569 0.468300i \(-0.844867\pi\)
0.847345 + 0.531043i \(0.178200\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −33.7990 + 58.5416i −1.47231 + 2.55011i
\(528\) 0 0
\(529\) 11.1569 + 19.3242i 0.485081 + 0.840184i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.82843 0.209142
\(534\) 0 0
\(535\) −4.24264 7.34847i −0.183425 0.317702i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4.31371 7.47156i 0.185461 0.321228i −0.758271 0.651940i \(-0.773956\pi\)
0.943732 + 0.330712i \(0.107289\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 38.6274 1.65462
\(546\) 0 0
\(547\) −4.97056 −0.212526 −0.106263 0.994338i \(-0.533889\pi\)
−0.106263 + 0.994338i \(0.533889\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.97056 8.60927i 0.211753 0.366767i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.9706 + 38.0541i 0.930923 + 1.61241i 0.781748 + 0.623594i \(0.214328\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(558\) 0 0
\(559\) 11.3137 0.478519
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.4142 23.2341i −0.565342 0.979201i −0.997018 0.0771719i \(-0.975411\pi\)
0.431676 0.902029i \(-0.357922\pi\)
\(564\) 0 0
\(565\) 10.2426 17.7408i 0.430911 0.746360i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.41421 5.91359i 0.143131 0.247911i −0.785543 0.618807i \(-0.787616\pi\)
0.928674 + 0.370897i \(0.120950\pi\)
\(570\) 0 0
\(571\) −20.1421 34.8872i −0.842922 1.45998i −0.887414 0.460974i \(-0.847500\pi\)
0.0444914 0.999010i \(-0.485833\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.51472 0.229980
\(576\) 0 0
\(577\) −4.70711 8.15295i −0.195959 0.339412i 0.751255 0.660012i \(-0.229449\pi\)
−0.947215 + 0.320600i \(0.896115\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −22.4853 + 38.9456i −0.931245 + 1.61296i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26.8284 −1.10733 −0.553664 0.832740i \(-0.686771\pi\)
−0.553664 + 0.832740i \(0.686771\pi\)
\(588\) 0 0
\(589\) 12.6863 0.522730
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.5355 + 25.1763i −0.596903 + 1.03387i 0.396372 + 0.918090i \(0.370269\pi\)
−0.993275 + 0.115776i \(0.963064\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.07107 + 13.9795i 0.329775 + 0.571187i 0.982467 0.186436i \(-0.0596938\pi\)
−0.652692 + 0.757623i \(0.726360\pi\)
\(600\) 0 0
\(601\) 12.2426 0.499388 0.249694 0.968325i \(-0.419670\pi\)
0.249694 + 0.968325i \(0.419670\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.0208 36.4091i −0.854618 1.48024i
\(606\) 0 0
\(607\) 15.7990 27.3647i 0.641261 1.11070i −0.343890 0.939010i \(-0.611745\pi\)
0.985152 0.171687i \(-0.0549218\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.828427 1.43488i 0.0335146 0.0580489i
\(612\) 0 0
\(613\) 1.17157 + 2.02922i 0.0473194 + 0.0819596i 0.888715 0.458460i \(-0.151599\pi\)
−0.841396 + 0.540420i \(0.818265\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.4558 −0.863780 −0.431890 0.901926i \(-0.642153\pi\)
−0.431890 + 0.901926i \(0.642153\pi\)
\(618\) 0 0
\(619\) −20.1421 34.8872i −0.809581 1.40224i −0.913155 0.407613i \(-0.866361\pi\)
0.103574 0.994622i \(-0.466972\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.98528 12.0989i 0.279411 0.483954i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 60.2843 2.40369
\(630\) 0 0
\(631\) −8.28427 −0.329792 −0.164896 0.986311i \(-0.552729\pi\)
−0.164896 + 0.986311i \(0.552729\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.4853 + 21.6251i −0.495463 + 0.858168i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.5858 + 21.7992i 0.497109 + 0.861017i 0.999994 0.00333540i \(-0.00106169\pi\)
−0.502886 + 0.864353i \(0.667728\pi\)
\(642\) 0 0
\(643\) −19.5147 −0.769585 −0.384793 0.923003i \(-0.625727\pi\)
−0.384793 + 0.923003i \(0.625727\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.41421 12.8418i −0.291483 0.504863i 0.682678 0.730720i \(-0.260815\pi\)
−0.974161 + 0.225857i \(0.927482\pi\)
\(648\) 0 0
\(649\) 26.1421 45.2795i 1.02617 1.77738i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.41421 2.44949i 0.0553425 0.0958559i −0.837027 0.547162i \(-0.815708\pi\)
0.892369 + 0.451306i \(0.149042\pi\)
\(654\) 0 0
\(655\) −26.1421 45.2795i −1.02146 1.76922i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −25.5147 −0.993912 −0.496956 0.867776i \(-0.665549\pi\)
−0.496956 + 0.867776i \(0.665549\pi\)
\(660\) 0 0
\(661\) −11.6777 20.2263i −0.454209 0.786713i 0.544434 0.838804i \(-0.316745\pi\)
−0.998642 + 0.0520914i \(0.983411\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.51472 + 6.08767i −0.136090 + 0.235716i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −28.4853 −1.09966
\(672\) 0 0
\(673\) 15.3137 0.590300 0.295150 0.955451i \(-0.404630\pi\)
0.295150 + 0.955451i \(0.404630\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.778175 1.34784i 0.0299077 0.0518016i −0.850684 0.525677i \(-0.823812\pi\)
0.880592 + 0.473876i \(0.157145\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.07107 + 10.5154i 0.232303 + 0.402361i 0.958485 0.285141i \(-0.0920406\pi\)
−0.726182 + 0.687502i \(0.758707\pi\)
\(684\) 0 0
\(685\) 42.6274 1.62871
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.58579 11.4069i −0.250898 0.434569i
\(690\) 0 0
\(691\) 14.0000 24.2487i 0.532585 0.922464i −0.466691 0.884420i \(-0.654554\pi\)
0.999276 0.0380440i \(-0.0121127\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.4853 28.5533i 0.625322 1.08309i
\(696\) 0 0
\(697\) −10.6569 18.4582i −0.403657 0.699155i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.8284 0.408984 0.204492 0.978868i \(-0.434446\pi\)
0.204492 + 0.978868i \(0.434446\pi\)
\(702\) 0 0
\(703\) −5.65685 9.79796i −0.213352 0.369537i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 6.92820i 0.150223 0.260194i −0.781086 0.624423i \(-0.785334\pi\)
0.931309 + 0.364229i \(0.118667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.97056 −0.335950
\(714\) 0 0
\(715\) 23.3137 0.871883
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.6569 23.6544i 0.509315 0.882159i −0.490627 0.871370i \(-0.663232\pi\)
0.999942 0.0107893i \(-0.00343441\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −28.2426 48.9177i −1.04891 1.81676i
\(726\) 0 0
\(727\) 25.4558 0.944105 0.472052 0.881570i \(-0.343513\pi\)
0.472052 + 0.881570i \(0.343513\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.9706 43.2503i −0.923570 1.59967i
\(732\) 0 0
\(733\) 10.1213 17.5306i 0.373839 0.647509i −0.616313 0.787501i \(-0.711374\pi\)
0.990153 + 0.139992i \(0.0447078\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.3137 + 33.4523i −0.711430 + 1.23223i
\(738\) 0 0
\(739\) 18.1421 + 31.4231i 0.667369 + 1.15592i 0.978637 + 0.205595i \(0.0659130\pi\)
−0.311268 + 0.950322i \(0.600754\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.8284 0.617375 0.308688 0.951163i \(-0.400110\pi\)
0.308688 + 0.951163i \(0.400110\pi\)
\(744\) 0 0
\(745\) −17.0711 29.5680i −0.625436 1.08329i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −9.17157 + 15.8856i −0.334675 + 0.579675i −0.983422 0.181329i \(-0.941960\pi\)
0.648747 + 0.761004i \(0.275293\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) 11.3137 0.411204 0.205602 0.978636i \(-0.434085\pi\)
0.205602 + 0.978636i \(0.434085\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.12132 15.7986i 0.330648 0.572698i −0.651991 0.758226i \(-0.726066\pi\)
0.982639 + 0.185528i \(0.0593995\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.65685 + 13.2621i 0.276473 + 0.478865i
\(768\) 0 0
\(769\) −0.928932 −0.0334982 −0.0167491 0.999860i \(-0.505332\pi\)
−0.0167491 + 0.999860i \(0.505332\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.292893 + 0.507306i 0.0105346 + 0.0182465i 0.871245 0.490849i \(-0.163313\pi\)
−0.860710 + 0.509096i \(0.829980\pi\)
\(774\) 0 0
\(775\) 36.0416 62.4259i 1.29465 2.24241i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 + 3.46410i −0.0716574 + 0.124114i
\(780\) 0 0
\(781\) −11.6569 20.1903i −0.417115 0.722464i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −20.1421 −0.718904
\(786\) 0 0
\(787\) 17.3137 + 29.9882i 0.617167 + 1.06896i 0.990000 + 0.141066i \(0.0450531\pi\)
−0.372833 + 0.927898i \(0.621614\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.17157 7.22538i 0.148137 0.256581i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.786797 −0.0278698 −0.0139349 0.999903i \(-0.504436\pi\)
−0.0139349 + 0.999903i \(0.504436\pi\)
\(798\) 0 0
\(799\) −7.31371 −0.258740
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.41421 + 12.8418i −0.261642 + 0.453177i
\(804\) 0 0
\(805\) 0 0