Properties

Label 1600.3.g.k.351.13
Level $1600$
Weight $3$
Character 1600.351
Analytic conductor $43.597$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,3,Mod(351,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{37} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 351.13
Root \(0.265804 - 0.460386i\) of defining polynomial
Character \(\chi\) \(=\) 1600.351
Dual form 1600.3.g.k.351.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.13399 q^{3} -6.19337i q^{7} +17.3578 q^{9} +O(q^{10})\) \(q+5.13399 q^{3} -6.19337i q^{7} +17.3578 q^{9} -20.0431 q^{11} -15.8612i q^{13} +6.98882 q^{17} -10.3923 q^{19} -31.7967i q^{21} -22.3871i q^{23} +42.9089 q^{27} -4.20563i q^{29} +20.7156i q^{31} -102.901 q^{33} -35.4786i q^{37} -81.4313i q^{39} -37.0735 q^{41} +23.8329 q^{43} -48.7515i q^{47} +10.6422 q^{49} +35.8805 q^{51} -77.4691i q^{53} -53.3540 q^{57} +0.497984 q^{59} +60.7490i q^{61} -107.503i q^{63} -82.5209 q^{67} -114.935i q^{69} -28.7156i q^{71} +10.1706 q^{73} +124.134i q^{77} +87.5782i q^{79} +64.0735 q^{81} +103.057 q^{83} -21.5917i q^{87} +49.2844 q^{89} -98.2344 q^{91} +106.354i q^{93} -84.4911 q^{97} -347.904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 96 q^{9} - 48 q^{41} + 352 q^{49} + 480 q^{81} + 1152 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.13399 1.71133 0.855664 0.517531i \(-0.173149\pi\)
0.855664 + 0.517531i \(0.173149\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 6.19337i − 0.884767i −0.896826 0.442383i \(-0.854133\pi\)
0.896826 0.442383i \(-0.145867\pi\)
\(8\) 0 0
\(9\) 17.3578 1.92865
\(10\) 0 0
\(11\) −20.0431 −1.82210 −0.911049 0.412298i \(-0.864726\pi\)
−0.911049 + 0.412298i \(0.864726\pi\)
\(12\) 0 0
\(13\) − 15.8612i − 1.22009i −0.792365 0.610047i \(-0.791151\pi\)
0.792365 0.610047i \(-0.208849\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.98882 0.411107 0.205554 0.978646i \(-0.434100\pi\)
0.205554 + 0.978646i \(0.434100\pi\)
\(18\) 0 0
\(19\) −10.3923 −0.546963 −0.273482 0.961877i \(-0.588175\pi\)
−0.273482 + 0.961877i \(0.588175\pi\)
\(20\) 0 0
\(21\) − 31.7967i − 1.51413i
\(22\) 0 0
\(23\) − 22.3871i − 0.973353i −0.873582 0.486676i \(-0.838209\pi\)
0.873582 0.486676i \(-0.161791\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 42.9089 1.58922
\(28\) 0 0
\(29\) − 4.20563i − 0.145022i −0.997368 0.0725109i \(-0.976899\pi\)
0.997368 0.0725109i \(-0.0231012\pi\)
\(30\) 0 0
\(31\) 20.7156i 0.668246i 0.942529 + 0.334123i \(0.108440\pi\)
−0.942529 + 0.334123i \(0.891560\pi\)
\(32\) 0 0
\(33\) −102.901 −3.11821
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 35.4786i − 0.958882i −0.877574 0.479441i \(-0.840839\pi\)
0.877574 0.479441i \(-0.159161\pi\)
\(38\) 0 0
\(39\) − 81.4313i − 2.08798i
\(40\) 0 0
\(41\) −37.0735 −0.904230 −0.452115 0.891960i \(-0.649330\pi\)
−0.452115 + 0.891960i \(0.649330\pi\)
\(42\) 0 0
\(43\) 23.8329 0.554254 0.277127 0.960833i \(-0.410618\pi\)
0.277127 + 0.960833i \(0.410618\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 48.7515i − 1.03727i −0.854997 0.518633i \(-0.826441\pi\)
0.854997 0.518633i \(-0.173559\pi\)
\(48\) 0 0
\(49\) 10.6422 0.217187
\(50\) 0 0
\(51\) 35.8805 0.703540
\(52\) 0 0
\(53\) − 77.4691i − 1.46168i −0.682549 0.730840i \(-0.739128\pi\)
0.682549 0.730840i \(-0.260872\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −53.3540 −0.936034
\(58\) 0 0
\(59\) 0.497984 0.00844041 0.00422020 0.999991i \(-0.498657\pi\)
0.00422020 + 0.999991i \(0.498657\pi\)
\(60\) 0 0
\(61\) 60.7490i 0.995885i 0.867210 + 0.497943i \(0.165911\pi\)
−0.867210 + 0.497943i \(0.834089\pi\)
\(62\) 0 0
\(63\) − 107.503i − 1.70640i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −82.5209 −1.23165 −0.615827 0.787881i \(-0.711178\pi\)
−0.615827 + 0.787881i \(0.711178\pi\)
\(68\) 0 0
\(69\) − 114.935i − 1.66573i
\(70\) 0 0
\(71\) − 28.7156i − 0.404446i −0.979340 0.202223i \(-0.935183\pi\)
0.979340 0.202223i \(-0.0648165\pi\)
\(72\) 0 0
\(73\) 10.1706 0.139324 0.0696620 0.997571i \(-0.477808\pi\)
0.0696620 + 0.997571i \(0.477808\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 124.134i 1.61213i
\(78\) 0 0
\(79\) 87.5782i 1.10858i 0.832322 + 0.554292i \(0.187011\pi\)
−0.832322 + 0.554292i \(0.812989\pi\)
\(80\) 0 0
\(81\) 64.0735 0.791030
\(82\) 0 0
\(83\) 103.057 1.24165 0.620824 0.783950i \(-0.286798\pi\)
0.620824 + 0.783950i \(0.286798\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 21.5917i − 0.248180i
\(88\) 0 0
\(89\) 49.2844 0.553757 0.276878 0.960905i \(-0.410700\pi\)
0.276878 + 0.960905i \(0.410700\pi\)
\(90\) 0 0
\(91\) −98.2344 −1.07950
\(92\) 0 0
\(93\) 106.354i 1.14359i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −84.4911 −0.871042 −0.435521 0.900179i \(-0.643436\pi\)
−0.435521 + 0.900179i \(0.643436\pi\)
\(98\) 0 0
\(99\) −347.904 −3.51418
\(100\) 0 0
\(101\) − 92.7892i − 0.918705i −0.888254 0.459353i \(-0.848081\pi\)
0.888254 0.459353i \(-0.151919\pi\)
\(102\) 0 0
\(103\) − 42.7284i − 0.414839i −0.978252 0.207419i \(-0.933494\pi\)
0.978252 0.207419i \(-0.0665065\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 29.3440 0.274243 0.137121 0.990554i \(-0.456215\pi\)
0.137121 + 0.990554i \(0.456215\pi\)
\(108\) 0 0
\(109\) − 163.433i − 1.49938i −0.661789 0.749691i \(-0.730202\pi\)
0.661789 0.749691i \(-0.269798\pi\)
\(110\) 0 0
\(111\) − 182.147i − 1.64096i
\(112\) 0 0
\(113\) 120.686 1.06801 0.534007 0.845480i \(-0.320686\pi\)
0.534007 + 0.845480i \(0.320686\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 275.316i − 2.35313i
\(118\) 0 0
\(119\) − 43.2844i − 0.363734i
\(120\) 0 0
\(121\) 280.725 2.32004
\(122\) 0 0
\(123\) −190.335 −1.54744
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 227.564i 1.79184i 0.444215 + 0.895920i \(0.353483\pi\)
−0.444215 + 0.895920i \(0.646517\pi\)
\(128\) 0 0
\(129\) 122.358 0.948510
\(130\) 0 0
\(131\) −34.1430 −0.260634 −0.130317 0.991472i \(-0.541599\pi\)
−0.130317 + 0.991472i \(0.541599\pi\)
\(132\) 0 0
\(133\) 64.3634i 0.483935i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 216.598 1.58100 0.790502 0.612459i \(-0.209820\pi\)
0.790502 + 0.612459i \(0.209820\pi\)
\(138\) 0 0
\(139\) −113.076 −0.813495 −0.406748 0.913541i \(-0.633337\pi\)
−0.406748 + 0.913541i \(0.633337\pi\)
\(140\) 0 0
\(141\) − 250.290i − 1.77510i
\(142\) 0 0
\(143\) 317.908i 2.22313i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.6368 0.371679
\(148\) 0 0
\(149\) − 171.844i − 1.15331i −0.816986 0.576657i \(-0.804357\pi\)
0.816986 0.576657i \(-0.195643\pi\)
\(150\) 0 0
\(151\) 30.5687i 0.202442i 0.994864 + 0.101221i \(0.0322749\pi\)
−0.994864 + 0.101221i \(0.967725\pi\)
\(152\) 0 0
\(153\) 121.311 0.792881
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 265.212i 1.68925i 0.535357 + 0.844626i \(0.320177\pi\)
−0.535357 + 0.844626i \(0.679823\pi\)
\(158\) 0 0
\(159\) − 397.725i − 2.50142i
\(160\) 0 0
\(161\) −138.652 −0.861190
\(162\) 0 0
\(163\) −69.6617 −0.427372 −0.213686 0.976902i \(-0.568547\pi\)
−0.213686 + 0.976902i \(0.568547\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 101.140i 0.605627i 0.953050 + 0.302814i \(0.0979259\pi\)
−0.953050 + 0.302814i \(0.902074\pi\)
\(168\) 0 0
\(169\) −82.5782 −0.488628
\(170\) 0 0
\(171\) −180.388 −1.05490
\(172\) 0 0
\(173\) 65.5284i 0.378777i 0.981902 + 0.189388i \(0.0606505\pi\)
−0.981902 + 0.189388i \(0.939349\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.55664 0.0144443
\(178\) 0 0
\(179\) 47.2688 0.264072 0.132036 0.991245i \(-0.457849\pi\)
0.132036 + 0.991245i \(0.457849\pi\)
\(180\) 0 0
\(181\) 255.613i 1.41223i 0.708099 + 0.706113i \(0.249553\pi\)
−0.708099 + 0.706113i \(0.750447\pi\)
\(182\) 0 0
\(183\) 311.885i 1.70429i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −140.078 −0.749078
\(188\) 0 0
\(189\) − 265.751i − 1.40609i
\(190\) 0 0
\(191\) − 67.7062i − 0.354483i −0.984167 0.177241i \(-0.943283\pi\)
0.984167 0.177241i \(-0.0567173\pi\)
\(192\) 0 0
\(193\) 327.113 1.69488 0.847442 0.530888i \(-0.178142\pi\)
0.847442 + 0.530888i \(0.178142\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 120.542i − 0.611890i −0.952049 0.305945i \(-0.901028\pi\)
0.952049 0.305945i \(-0.0989723\pi\)
\(198\) 0 0
\(199\) 175.578i 0.882302i 0.897433 + 0.441151i \(0.145430\pi\)
−0.897433 + 0.441151i \(0.854570\pi\)
\(200\) 0 0
\(201\) −423.661 −2.10777
\(202\) 0 0
\(203\) −26.0470 −0.128310
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 388.591i − 1.87725i
\(208\) 0 0
\(209\) 208.294 0.996621
\(210\) 0 0
\(211\) 237.053 1.12347 0.561737 0.827316i \(-0.310133\pi\)
0.561737 + 0.827316i \(0.310133\pi\)
\(212\) 0 0
\(213\) − 147.426i − 0.692139i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 128.300 0.591242
\(218\) 0 0
\(219\) 52.2160 0.238429
\(220\) 0 0
\(221\) − 110.851i − 0.501589i
\(222\) 0 0
\(223\) 13.8074i 0.0619165i 0.999521 + 0.0309582i \(0.00985589\pi\)
−0.999521 + 0.0309582i \(0.990144\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.21412 0.00975382 0.00487691 0.999988i \(-0.498448\pi\)
0.00487691 + 0.999988i \(0.498448\pi\)
\(228\) 0 0
\(229\) 241.756i 1.05571i 0.849336 + 0.527853i \(0.177003\pi\)
−0.849336 + 0.527853i \(0.822997\pi\)
\(230\) 0 0
\(231\) 637.303i 2.75889i
\(232\) 0 0
\(233\) −30.5119 −0.130953 −0.0654763 0.997854i \(-0.520857\pi\)
−0.0654763 + 0.997854i \(0.520857\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 449.625i 1.89715i
\(238\) 0 0
\(239\) − 426.441i − 1.78427i −0.451768 0.892135i \(-0.649207\pi\)
0.451768 0.892135i \(-0.350793\pi\)
\(240\) 0 0
\(241\) 147.936 0.613842 0.306921 0.951735i \(-0.400701\pi\)
0.306921 + 0.951735i \(0.400701\pi\)
\(242\) 0 0
\(243\) −57.2280 −0.235506
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 164.835i 0.667347i
\(248\) 0 0
\(249\) 529.092 2.12487
\(250\) 0 0
\(251\) −97.7364 −0.389388 −0.194694 0.980864i \(-0.562371\pi\)
−0.194694 + 0.980864i \(0.562371\pi\)
\(252\) 0 0
\(253\) 448.707i 1.77354i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 354.443 1.37915 0.689577 0.724212i \(-0.257796\pi\)
0.689577 + 0.724212i \(0.257796\pi\)
\(258\) 0 0
\(259\) −219.732 −0.848387
\(260\) 0 0
\(261\) − 73.0006i − 0.279696i
\(262\) 0 0
\(263\) − 44.6039i − 0.169597i −0.996398 0.0847984i \(-0.972975\pi\)
0.996398 0.0847984i \(-0.0270246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 253.025 0.947660
\(268\) 0 0
\(269\) 314.879i 1.17055i 0.810834 + 0.585277i \(0.199014\pi\)
−0.810834 + 0.585277i \(0.800986\pi\)
\(270\) 0 0
\(271\) − 449.009i − 1.65686i −0.560092 0.828431i \(-0.689234\pi\)
0.560092 0.828431i \(-0.310766\pi\)
\(272\) 0 0
\(273\) −504.334 −1.84738
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 330.330i − 1.19253i −0.802789 0.596264i \(-0.796651\pi\)
0.802789 0.596264i \(-0.203349\pi\)
\(278\) 0 0
\(279\) 359.578i 1.28881i
\(280\) 0 0
\(281\) 307.936 1.09586 0.547929 0.836525i \(-0.315416\pi\)
0.547929 + 0.836525i \(0.315416\pi\)
\(282\) 0 0
\(283\) −182.609 −0.645263 −0.322631 0.946525i \(-0.604567\pi\)
−0.322631 + 0.946525i \(0.604567\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 229.610i 0.800033i
\(288\) 0 0
\(289\) −240.156 −0.830991
\(290\) 0 0
\(291\) −433.776 −1.49064
\(292\) 0 0
\(293\) − 259.373i − 0.885231i −0.896711 0.442616i \(-0.854051\pi\)
0.896711 0.442616i \(-0.145949\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −860.027 −2.89571
\(298\) 0 0
\(299\) −355.087 −1.18758
\(300\) 0 0
\(301\) − 147.606i − 0.490385i
\(302\) 0 0
\(303\) − 476.379i − 1.57221i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −209.411 −0.682119 −0.341060 0.940042i \(-0.610786\pi\)
−0.341060 + 0.940042i \(0.610786\pi\)
\(308\) 0 0
\(309\) − 219.367i − 0.709926i
\(310\) 0 0
\(311\) 149.137i 0.479542i 0.970830 + 0.239771i \(0.0770723\pi\)
−0.970830 + 0.239771i \(0.922928\pi\)
\(312\) 0 0
\(313\) −296.601 −0.947606 −0.473803 0.880631i \(-0.657119\pi\)
−0.473803 + 0.880631i \(0.657119\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 23.0450i − 0.0726971i −0.999339 0.0363486i \(-0.988427\pi\)
0.999339 0.0363486i \(-0.0115727\pi\)
\(318\) 0 0
\(319\) 84.2938i 0.264244i
\(320\) 0 0
\(321\) 150.652 0.469320
\(322\) 0 0
\(323\) −72.6300 −0.224861
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 839.060i − 2.56593i
\(328\) 0 0
\(329\) −301.936 −0.917739
\(330\) 0 0
\(331\) 419.665 1.26787 0.633935 0.773386i \(-0.281439\pi\)
0.633935 + 0.773386i \(0.281439\pi\)
\(332\) 0 0
\(333\) − 615.832i − 1.84934i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 52.0477 0.154444 0.0772221 0.997014i \(-0.475395\pi\)
0.0772221 + 0.997014i \(0.475395\pi\)
\(338\) 0 0
\(339\) 619.598 1.82772
\(340\) 0 0
\(341\) − 415.205i − 1.21761i
\(342\) 0 0
\(343\) − 369.386i − 1.07693i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −329.706 −0.950163 −0.475081 0.879942i \(-0.657581\pi\)
−0.475081 + 0.879942i \(0.657581\pi\)
\(348\) 0 0
\(349\) − 517.423i − 1.48259i −0.671180 0.741294i \(-0.734212\pi\)
0.671180 0.741294i \(-0.265788\pi\)
\(350\) 0 0
\(351\) − 680.588i − 1.93900i
\(352\) 0 0
\(353\) −44.4337 −0.125874 −0.0629372 0.998017i \(-0.520047\pi\)
−0.0629372 + 0.998017i \(0.520047\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 222.221i − 0.622469i
\(358\) 0 0
\(359\) 53.1375i 0.148015i 0.997258 + 0.0740076i \(0.0235789\pi\)
−0.997258 + 0.0740076i \(0.976421\pi\)
\(360\) 0 0
\(361\) −253.000 −0.700831
\(362\) 0 0
\(363\) 1441.24 3.97035
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 374.954i 1.02167i 0.859678 + 0.510837i \(0.170664\pi\)
−0.859678 + 0.510837i \(0.829336\pi\)
\(368\) 0 0
\(369\) −643.514 −1.74394
\(370\) 0 0
\(371\) −479.794 −1.29325
\(372\) 0 0
\(373\) − 24.6208i − 0.0660076i −0.999455 0.0330038i \(-0.989493\pi\)
0.999455 0.0330038i \(-0.0105073\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −66.7064 −0.176940
\(378\) 0 0
\(379\) 170.472 0.449793 0.224897 0.974383i \(-0.427796\pi\)
0.224897 + 0.974383i \(0.427796\pi\)
\(380\) 0 0
\(381\) 1168.31i 3.06643i
\(382\) 0 0
\(383\) − 136.084i − 0.355310i −0.984093 0.177655i \(-0.943149\pi\)
0.984093 0.177655i \(-0.0568512\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 413.687 1.06896
\(388\) 0 0
\(389\) − 60.5055i − 0.155541i −0.996971 0.0777705i \(-0.975220\pi\)
0.996971 0.0777705i \(-0.0247801\pi\)
\(390\) 0 0
\(391\) − 156.460i − 0.400152i
\(392\) 0 0
\(393\) −175.290 −0.446030
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 124.216i − 0.312888i −0.987687 0.156444i \(-0.949997\pi\)
0.987687 0.156444i \(-0.0500030\pi\)
\(398\) 0 0
\(399\) 330.441i 0.828172i
\(400\) 0 0
\(401\) 46.7156 0.116498 0.0582489 0.998302i \(-0.481448\pi\)
0.0582489 + 0.998302i \(0.481448\pi\)
\(402\) 0 0
\(403\) 328.575 0.815323
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 711.101i 1.74718i
\(408\) 0 0
\(409\) −582.377 −1.42390 −0.711952 0.702228i \(-0.752189\pi\)
−0.711952 + 0.702228i \(0.752189\pi\)
\(410\) 0 0
\(411\) 1112.01 2.70562
\(412\) 0 0
\(413\) − 3.08420i − 0.00746779i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −580.530 −1.39216
\(418\) 0 0
\(419\) 479.296 1.14391 0.571953 0.820287i \(-0.306186\pi\)
0.571953 + 0.820287i \(0.306186\pi\)
\(420\) 0 0
\(421\) 116.905i 0.277685i 0.990314 + 0.138842i \(0.0443381\pi\)
−0.990314 + 0.138842i \(0.955662\pi\)
\(422\) 0 0
\(423\) − 846.220i − 2.00052i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 376.241 0.881126
\(428\) 0 0
\(429\) 1632.13i 3.80451i
\(430\) 0 0
\(431\) 204.460i 0.474384i 0.971463 + 0.237192i \(0.0762271\pi\)
−0.971463 + 0.237192i \(0.923773\pi\)
\(432\) 0 0
\(433\) 403.364 0.931558 0.465779 0.884901i \(-0.345774\pi\)
0.465779 + 0.884901i \(0.345774\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 232.654i 0.532388i
\(438\) 0 0
\(439\) 114.275i 0.260307i 0.991494 + 0.130154i \(0.0415471\pi\)
−0.991494 + 0.130154i \(0.958453\pi\)
\(440\) 0 0
\(441\) 184.725 0.418878
\(442\) 0 0
\(443\) −750.268 −1.69361 −0.846803 0.531906i \(-0.821476\pi\)
−0.846803 + 0.531906i \(0.821476\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 882.244i − 1.97370i
\(448\) 0 0
\(449\) 554.377 1.23469 0.617346 0.786692i \(-0.288208\pi\)
0.617346 + 0.786692i \(0.288208\pi\)
\(450\) 0 0
\(451\) 743.066 1.64760
\(452\) 0 0
\(453\) 156.939i 0.346445i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 870.823 1.90552 0.952760 0.303724i \(-0.0982301\pi\)
0.952760 + 0.303724i \(0.0982301\pi\)
\(458\) 0 0
\(459\) 299.883 0.653340
\(460\) 0 0
\(461\) 37.3636i 0.0810490i 0.999179 + 0.0405245i \(0.0129029\pi\)
−0.999179 + 0.0405245i \(0.987097\pi\)
\(462\) 0 0
\(463\) − 228.189i − 0.492849i −0.969162 0.246424i \(-0.920744\pi\)
0.969162 0.246424i \(-0.0792557\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 414.441 0.887455 0.443727 0.896162i \(-0.353656\pi\)
0.443727 + 0.896162i \(0.353656\pi\)
\(468\) 0 0
\(469\) 511.082i 1.08973i
\(470\) 0 0
\(471\) 1361.60i 2.89086i
\(472\) 0 0
\(473\) −477.685 −1.00990
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1344.69i − 2.81906i
\(478\) 0 0
\(479\) 871.744i 1.81992i 0.414691 + 0.909962i \(0.363890\pi\)
−0.414691 + 0.909962i \(0.636110\pi\)
\(480\) 0 0
\(481\) −562.735 −1.16993
\(482\) 0 0
\(483\) −711.836 −1.47378
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 316.431i 0.649756i 0.945756 + 0.324878i \(0.105323\pi\)
−0.945756 + 0.324878i \(0.894677\pi\)
\(488\) 0 0
\(489\) −357.642 −0.731375
\(490\) 0 0
\(491\) −372.142 −0.757927 −0.378963 0.925412i \(-0.623719\pi\)
−0.378963 + 0.925412i \(0.623719\pi\)
\(492\) 0 0
\(493\) − 29.3924i − 0.0596195i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −177.847 −0.357840
\(498\) 0 0
\(499\) 198.184 0.397163 0.198582 0.980084i \(-0.436367\pi\)
0.198582 + 0.980084i \(0.436367\pi\)
\(500\) 0 0
\(501\) 519.250i 1.03643i
\(502\) 0 0
\(503\) 638.202i 1.26879i 0.773009 + 0.634395i \(0.218751\pi\)
−0.773009 + 0.634395i \(0.781249\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −423.955 −0.836204
\(508\) 0 0
\(509\) − 58.6571i − 0.115240i −0.998339 0.0576199i \(-0.981649\pi\)
0.998339 0.0576199i \(-0.0183512\pi\)
\(510\) 0 0
\(511\) − 62.9906i − 0.123269i
\(512\) 0 0
\(513\) −445.923 −0.869245
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 977.130i 1.89000i
\(518\) 0 0
\(519\) 336.422i 0.648212i
\(520\) 0 0
\(521\) 171.431 0.329043 0.164521 0.986374i \(-0.447392\pi\)
0.164521 + 0.986374i \(0.447392\pi\)
\(522\) 0 0
\(523\) 407.751 0.779638 0.389819 0.920892i \(-0.372538\pi\)
0.389819 + 0.920892i \(0.372538\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 144.778i 0.274721i
\(528\) 0 0
\(529\) 27.8174 0.0525848
\(530\) 0 0
\(531\) 8.64391 0.0162786
\(532\) 0 0
\(533\) 588.030i 1.10325i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 242.677 0.451913
\(538\) 0 0
\(539\) −213.302 −0.395737
\(540\) 0 0
\(541\) − 144.762i − 0.267582i −0.991010 0.133791i \(-0.957285\pi\)
0.991010 0.133791i \(-0.0427150\pi\)
\(542\) 0 0
\(543\) 1312.31i 2.41678i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −570.627 −1.04319 −0.521597 0.853192i \(-0.674663\pi\)
−0.521597 + 0.853192i \(0.674663\pi\)
\(548\) 0 0
\(549\) 1054.47i 1.92071i
\(550\) 0 0
\(551\) 43.7062i 0.0793216i
\(552\) 0 0
\(553\) 542.404 0.980839
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 824.208i 1.47973i 0.672757 + 0.739864i \(0.265110\pi\)
−0.672757 + 0.739864i \(0.734890\pi\)
\(558\) 0 0
\(559\) − 378.019i − 0.676241i
\(560\) 0 0
\(561\) −719.156 −1.28192
\(562\) 0 0
\(563\) 216.333 0.384251 0.192125 0.981370i \(-0.438462\pi\)
0.192125 + 0.981370i \(0.438462\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 396.831i − 0.699877i
\(568\) 0 0
\(569\) 468.230 0.822899 0.411450 0.911432i \(-0.365023\pi\)
0.411450 + 0.911432i \(0.365023\pi\)
\(570\) 0 0
\(571\) −759.434 −1.33001 −0.665004 0.746840i \(-0.731570\pi\)
−0.665004 + 0.746840i \(0.731570\pi\)
\(572\) 0 0
\(573\) − 347.603i − 0.606636i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −29.2057 −0.0506164 −0.0253082 0.999680i \(-0.508057\pi\)
−0.0253082 + 0.999680i \(0.508057\pi\)
\(578\) 0 0
\(579\) 1679.39 2.90050
\(580\) 0 0
\(581\) − 638.269i − 1.09857i
\(582\) 0 0
\(583\) 1552.72i 2.66332i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1014.74 −1.72869 −0.864345 0.502900i \(-0.832266\pi\)
−0.864345 + 0.502900i \(0.832266\pi\)
\(588\) 0 0
\(589\) − 215.283i − 0.365506i
\(590\) 0 0
\(591\) − 618.863i − 1.04714i
\(592\) 0 0
\(593\) 835.823 1.40948 0.704741 0.709465i \(-0.251063\pi\)
0.704741 + 0.709465i \(0.251063\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 901.416i 1.50991i
\(598\) 0 0
\(599\) − 851.616i − 1.42173i −0.703329 0.710865i \(-0.748304\pi\)
0.703329 0.710865i \(-0.251696\pi\)
\(600\) 0 0
\(601\) −618.377 −1.02891 −0.514456 0.857516i \(-0.672006\pi\)
−0.514456 + 0.857516i \(0.672006\pi\)
\(602\) 0 0
\(603\) −1432.38 −2.37543
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 182.393i 0.300483i 0.988649 + 0.150241i \(0.0480051\pi\)
−0.988649 + 0.150241i \(0.951995\pi\)
\(608\) 0 0
\(609\) −133.725 −0.219581
\(610\) 0 0
\(611\) −773.258 −1.26556
\(612\) 0 0
\(613\) − 638.630i − 1.04181i −0.853614 0.520906i \(-0.825594\pi\)
0.853614 0.520906i \(-0.174406\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 416.661 0.675301 0.337651 0.941271i \(-0.390368\pi\)
0.337651 + 0.941271i \(0.390368\pi\)
\(618\) 0 0
\(619\) 1090.24 1.76129 0.880646 0.473775i \(-0.157109\pi\)
0.880646 + 0.473775i \(0.157109\pi\)
\(620\) 0 0
\(621\) − 960.607i − 1.54687i
\(622\) 0 0
\(623\) − 305.236i − 0.489946i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1069.38 1.70555
\(628\) 0 0
\(629\) − 247.954i − 0.394204i
\(630\) 0 0
\(631\) 574.569i 0.910569i 0.890346 + 0.455284i \(0.150462\pi\)
−0.890346 + 0.455284i \(0.849538\pi\)
\(632\) 0 0
\(633\) 1217.03 1.92263
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 168.798i − 0.264989i
\(638\) 0 0
\(639\) − 498.441i − 0.780032i
\(640\) 0 0
\(641\) 893.367 1.39371 0.696854 0.717213i \(-0.254582\pi\)
0.696854 + 0.717213i \(0.254582\pi\)
\(642\) 0 0
\(643\) −675.801 −1.05101 −0.525506 0.850790i \(-0.676124\pi\)
−0.525506 + 0.850790i \(0.676124\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 737.921i − 1.14053i −0.821462 0.570263i \(-0.806841\pi\)
0.821462 0.570263i \(-0.193159\pi\)
\(648\) 0 0
\(649\) −9.98113 −0.0153792
\(650\) 0 0
\(651\) 658.688 1.01181
\(652\) 0 0
\(653\) 771.853i 1.18201i 0.806668 + 0.591005i \(0.201269\pi\)
−0.806668 + 0.591005i \(0.798731\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 176.540 0.268707
\(658\) 0 0
\(659\) −814.838 −1.23648 −0.618238 0.785991i \(-0.712153\pi\)
−0.618238 + 0.785991i \(0.712153\pi\)
\(660\) 0 0
\(661\) 1038.13i 1.57055i 0.619146 + 0.785276i \(0.287479\pi\)
−0.619146 + 0.785276i \(0.712521\pi\)
\(662\) 0 0
\(663\) − 569.109i − 0.858384i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −94.1519 −0.141157
\(668\) 0 0
\(669\) 70.8869i 0.105959i
\(670\) 0 0
\(671\) − 1217.60i − 1.81460i
\(672\) 0 0
\(673\) −367.795 −0.546501 −0.273250 0.961943i \(-0.588099\pi\)
−0.273250 + 0.961943i \(0.588099\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 463.978i 0.685344i 0.939455 + 0.342672i \(0.111332\pi\)
−0.939455 + 0.342672i \(0.888668\pi\)
\(678\) 0 0
\(679\) 523.284i 0.770669i
\(680\) 0 0
\(681\) 11.3673 0.0166920
\(682\) 0 0
\(683\) 632.563 0.926154 0.463077 0.886318i \(-0.346745\pi\)
0.463077 + 0.886318i \(0.346745\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1241.17i 1.80666i
\(688\) 0 0
\(689\) −1228.75 −1.78339
\(690\) 0 0
\(691\) 436.001 0.630970 0.315485 0.948930i \(-0.397833\pi\)
0.315485 + 0.948930i \(0.397833\pi\)
\(692\) 0 0
\(693\) 2154.70i 3.10923i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −259.100 −0.371736
\(698\) 0 0
\(699\) −156.648 −0.224103
\(700\) 0 0
\(701\) − 364.085i − 0.519380i −0.965692 0.259690i \(-0.916380\pi\)
0.965692 0.259690i \(-0.0836203\pi\)
\(702\) 0 0
\(703\) 368.705i 0.524474i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −574.678 −0.812840
\(708\) 0 0
\(709\) 789.593i 1.11367i 0.830622 + 0.556836i \(0.187985\pi\)
−0.830622 + 0.556836i \(0.812015\pi\)
\(710\) 0 0
\(711\) 1520.17i 2.13807i
\(712\) 0 0
\(713\) 463.763 0.650439
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 2189.34i − 3.05347i
\(718\) 0 0
\(719\) − 168.422i − 0.234245i −0.993118 0.117122i \(-0.962633\pi\)
0.993118 0.117122i \(-0.0373669\pi\)
\(720\) 0 0
\(721\) −264.633 −0.367036
\(722\) 0 0
\(723\) 759.501 1.05049
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 382.568i 0.526229i 0.964765 + 0.263114i \(0.0847496\pi\)
−0.964765 + 0.263114i \(0.915250\pi\)
\(728\) 0 0
\(729\) −870.469 −1.19406
\(730\) 0 0
\(731\) 166.564 0.227858
\(732\) 0 0
\(733\) − 74.5492i − 0.101704i −0.998706 0.0508521i \(-0.983806\pi\)
0.998706 0.0508521i \(-0.0161937\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1653.97 2.24420
\(738\) 0 0
\(739\) −682.206 −0.923148 −0.461574 0.887102i \(-0.652715\pi\)
−0.461574 + 0.887102i \(0.652715\pi\)
\(740\) 0 0
\(741\) 846.259i 1.14205i
\(742\) 0 0
\(743\) 359.274i 0.483545i 0.970333 + 0.241772i \(0.0777287\pi\)
−0.970333 + 0.241772i \(0.922271\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1788.84 2.39470
\(748\) 0 0
\(749\) − 181.738i − 0.242641i
\(750\) 0 0
\(751\) 532.038i 0.708439i 0.935162 + 0.354220i \(0.115253\pi\)
−0.935162 + 0.354220i \(0.884747\pi\)
\(752\) 0 0
\(753\) −501.777 −0.666371
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 444.854i 0.587653i 0.955859 + 0.293827i \(0.0949288\pi\)
−0.955859 + 0.293827i \(0.905071\pi\)
\(758\) 0 0
\(759\) 2303.65i 3.03512i
\(760\) 0 0
\(761\) −244.735 −0.321596 −0.160798 0.986987i \(-0.551407\pi\)
−0.160798 + 0.986987i \(0.551407\pi\)
\(762\) 0 0
\(763\) −1012.20 −1.32660
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 7.89863i − 0.0102981i
\(768\) 0 0
\(769\) 1131.73 1.47168 0.735842 0.677153i \(-0.236787\pi\)
0.735842 + 0.677153i \(0.236787\pi\)
\(770\) 0 0
\(771\) 1819.70 2.36019
\(772\) 0 0
\(773\) − 294.508i − 0.380994i −0.981688 0.190497i \(-0.938990\pi\)
0.981688 0.190497i \(-0.0610099\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1128.10 −1.45187
\(778\) 0 0
\(779\) 385.279 0.494581
\(780\) 0 0
\(781\) 575.550i 0.736939i
\(782\) 0 0
\(783\) − 180.459i − 0.230471i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1017.00 −1.29225 −0.646127 0.763230i \(-0.723612\pi\)
−0.646127 + 0.763230i \(0.723612\pi\)
\(788\) 0 0
\(789\) − 228.996i − 0.290236i
\(790\) 0 0
\(791\) − 747.450i − 0.944943i
\(792\) 0 0
\(793\) 963.553 1.21507
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1105.61i − 1.38722i −0.720353 0.693608i \(-0.756020\pi\)
0.720353 0.693608i \(-0.243980\pi\)
\(798\) 0 0
\(799\) − 340.716i − 0.426428i
\(800\) 0 0
\(801\) 855.469 1.06800
\(802\) 0 0
\(803\) −203.851 −0.253862
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1616.58i 2.00320i
\(808\) 0 0
\(809\) 594.881 0.735329 0.367665 0.929958i \(-0.380157\pi\)
0.367665 + 0.929958i \(0.380157\pi\)
\(810\) 0 0
\(811\) −45.3204 −0.0558822 −0.0279411 0.999610i \(-0.508895\pi\)
−0.0279411 + 0.999610i \(0.508895\pi\)
\(812\) 0 0
\(813\) − 2305.21i − 2.83543i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −247.679 −0.303156
\(818\) 0 0
\(819\) −1705.13 −2.08197
\(820\) 0 0
\(821\) 310.208i 0.377842i 0.981992 + 0.188921i \(0.0604989\pi\)
−0.981992 + 0.188921i \(0.939501\pi\)
\(822\) 0 0
\(823\) − 660.022i − 0.801971i −0.916084 0.400985i \(-0.868668\pi\)
0.916084 0.400985i \(-0.131332\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 168.339 0.203553 0.101777 0.994807i \(-0.467547\pi\)
0.101777 + 0.994807i \(0.467547\pi\)
\(828\) 0 0
\(829\) 597.939i 0.721278i 0.932706 + 0.360639i \(0.117441\pi\)
−0.932706 + 0.360639i \(0.882559\pi\)
\(830\) 0 0
\(831\) − 1695.91i − 2.04081i
\(832\) 0 0
\(833\) 74.3764 0.0892873
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 888.885i 1.06199i
\(838\) 0 0
\(839\) 1113.51i 1.32718i 0.748095 + 0.663592i \(0.230969\pi\)
−0.748095 + 0.663592i \(0.769031\pi\)
\(840\) 0 0
\(841\) 823.313 0.978969
\(842\) 0 0
\(843\) 1580.94 1.87537
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1738.63i − 2.05270i
\(848\) 0 0
\(849\) −937.514 −1.10426
\(850\) 0 0
\(851\) −794.264 −0.933331
\(852\) 0 0
\(853\) − 889.393i − 1.04266i −0.853354 0.521332i \(-0.825435\pi\)
0.853354 0.521332i \(-0.174565\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1246.97 −1.45504 −0.727520 0.686087i \(-0.759327\pi\)
−0.727520 + 0.686087i \(0.759327\pi\)
\(858\) 0 0
\(859\) 977.927 1.13845 0.569224 0.822182i \(-0.307244\pi\)
0.569224 + 0.822182i \(0.307244\pi\)
\(860\) 0 0
\(861\) 1178.81i 1.36912i
\(862\) 0 0
\(863\) − 1366.41i − 1.58332i −0.610961 0.791661i \(-0.709217\pi\)
0.610961 0.791661i \(-0.290783\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1232.96 −1.42210
\(868\) 0 0
\(869\) − 1755.34i − 2.01995i
\(870\) 0 0
\(871\) 1308.88i 1.50273i
\(872\) 0 0
\(873\) −1466.58 −1.67993
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 87.0797i 0.0992927i 0.998767 + 0.0496464i \(0.0158094\pi\)
−0.998767 + 0.0496464i \(0.984191\pi\)
\(878\) 0 0
\(879\) − 1331.62i − 1.51492i
\(880\) 0 0
\(881\) 154.927 0.175853 0.0879265 0.996127i \(-0.471976\pi\)
0.0879265 + 0.996127i \(0.471976\pi\)
\(882\) 0 0
\(883\) 1604.11 1.81666 0.908332 0.418249i \(-0.137356\pi\)
0.908332 + 0.418249i \(0.137356\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 196.767i − 0.221834i −0.993830 0.110917i \(-0.964621\pi\)
0.993830 0.110917i \(-0.0353788\pi\)
\(888\) 0 0
\(889\) 1409.39 1.58536
\(890\) 0 0
\(891\) −1284.23 −1.44133
\(892\) 0 0
\(893\) 506.640i 0.567346i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1823.01 −2.03234
\(898\) 0 0
\(899\) 87.1223 0.0969102
\(900\) 0 0
\(901\) − 541.418i − 0.600907i
\(902\) 0 0
\(903\) − 757.807i − 0.839210i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1079.04 1.18968 0.594838 0.803845i \(-0.297216\pi\)
0.594838 + 0.803845i \(0.297216\pi\)
\(908\) 0 0
\(909\) − 1610.62i − 1.77186i
\(910\) 0 0
\(911\) − 1815.91i − 1.99331i −0.0816953 0.996657i \(-0.526033\pi\)
0.0816953 0.996657i \(-0.473967\pi\)
\(912\) 0 0
\(913\) −2065.58 −2.26241
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 211.460i 0.230600i
\(918\) 0 0
\(919\) − 437.303i − 0.475847i −0.971284 0.237923i \(-0.923533\pi\)
0.971284 0.237923i \(-0.0764667\pi\)
\(920\) 0 0
\(921\) −1075.11 −1.16733
\(922\) 0 0
\(923\) −455.465 −0.493461
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 741.672i − 0.800077i
\(928\) 0 0
\(929\) 164.320 0.176878 0.0884392 0.996082i \(-0.471812\pi\)
0.0884392 + 0.996082i \(0.471812\pi\)
\(930\) 0 0
\(931\) −110.597 −0.118794
\(932\) 0 0
\(933\) 765.670i 0.820654i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1238.62 1.32190 0.660949 0.750431i \(-0.270154\pi\)
0.660949 + 0.750431i \(0.270154\pi\)
\(938\) 0 0
\(939\) −1522.74 −1.62167
\(940\) 0 0
\(941\) − 1846.13i − 1.96188i −0.194299 0.980942i \(-0.562243\pi\)
0.194299 0.980942i \(-0.437757\pi\)
\(942\) 0 0
\(943\) 829.967i 0.880135i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 466.767 0.492890 0.246445 0.969157i \(-0.420737\pi\)
0.246445 + 0.969157i \(0.420737\pi\)
\(948\) 0 0
\(949\) − 161.319i − 0.169988i
\(950\) 0 0
\(951\) − 118.313i − 0.124409i
\(952\) 0 0
\(953\) −1316.75 −1.38168 −0.690842 0.723006i \(-0.742760\pi\)
−0.690842 + 0.723006i \(0.742760\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 432.763i 0.452208i
\(958\) 0 0
\(959\) − 1341.47i − 1.39882i
\(960\) 0 0
\(961\) 531.863 0.553447
\(962\) 0 0
\(963\) 509.347 0.528917
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 572.176i − 0.591702i −0.955234 0.295851i \(-0.904397\pi\)
0.955234 0.295851i \(-0.0956033\pi\)
\(968\) 0 0
\(969\) −372.881 −0.384811
\(970\) 0 0
\(971\) −881.863 −0.908201 −0.454100 0.890951i \(-0.650039\pi\)
−0.454100 + 0.890951i \(0.650039\pi\)
\(972\) 0 0
\(973\) 700.320i 0.719754i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −512.461 −0.524525 −0.262263 0.964997i \(-0.584469\pi\)
−0.262263 + 0.964997i \(0.584469\pi\)
\(978\) 0 0
\(979\) −987.811 −1.00900
\(980\) 0 0
\(981\) − 2836.83i − 2.89178i
\(982\) 0 0
\(983\) − 1665.68i − 1.69448i −0.531209 0.847241i \(-0.678262\pi\)
0.531209 0.847241i \(-0.321738\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1550.14 −1.57055
\(988\) 0 0
\(989\) − 533.550i − 0.539484i
\(990\) 0 0
\(991\) 1696.66i 1.71207i 0.516916 + 0.856036i \(0.327080\pi\)
−0.516916 + 0.856036i \(0.672920\pi\)
\(992\) 0 0
\(993\) 2154.56 2.16974
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1405.45i − 1.40968i −0.709367 0.704840i \(-0.751019\pi\)
0.709367 0.704840i \(-0.248981\pi\)
\(998\) 0 0
\(999\) − 1522.35i − 1.52387i
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1600.3.g.k.351.13 16
4.3 odd 2 inner 1600.3.g.k.351.4 16
5.2 odd 4 320.3.e.b.159.2 yes 16
5.3 odd 4 320.3.e.b.159.16 yes 16
5.4 even 2 inner 1600.3.g.k.351.3 16
8.3 odd 2 inner 1600.3.g.k.351.15 16
8.5 even 2 inner 1600.3.g.k.351.2 16
20.3 even 4 320.3.e.b.159.4 yes 16
20.7 even 4 320.3.e.b.159.14 yes 16
20.19 odd 2 inner 1600.3.g.k.351.14 16
40.3 even 4 320.3.e.b.159.13 yes 16
40.13 odd 4 320.3.e.b.159.1 16
40.19 odd 2 inner 1600.3.g.k.351.1 16
40.27 even 4 320.3.e.b.159.3 yes 16
40.29 even 2 inner 1600.3.g.k.351.16 16
40.37 odd 4 320.3.e.b.159.15 yes 16
80.3 even 4 1280.3.h.n.1279.15 16
80.13 odd 4 1280.3.h.n.1279.3 16
80.27 even 4 1280.3.h.n.1279.13 16
80.37 odd 4 1280.3.h.n.1279.1 16
80.43 even 4 1280.3.h.n.1279.2 16
80.53 odd 4 1280.3.h.n.1279.14 16
80.67 even 4 1280.3.h.n.1279.4 16
80.77 odd 4 1280.3.h.n.1279.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.3.e.b.159.1 16 40.13 odd 4
320.3.e.b.159.2 yes 16 5.2 odd 4
320.3.e.b.159.3 yes 16 40.27 even 4
320.3.e.b.159.4 yes 16 20.3 even 4
320.3.e.b.159.13 yes 16 40.3 even 4
320.3.e.b.159.14 yes 16 20.7 even 4
320.3.e.b.159.15 yes 16 40.37 odd 4
320.3.e.b.159.16 yes 16 5.3 odd 4
1280.3.h.n.1279.1 16 80.37 odd 4
1280.3.h.n.1279.2 16 80.43 even 4
1280.3.h.n.1279.3 16 80.13 odd 4
1280.3.h.n.1279.4 16 80.67 even 4
1280.3.h.n.1279.13 16 80.27 even 4
1280.3.h.n.1279.14 16 80.53 odd 4
1280.3.h.n.1279.15 16 80.3 even 4
1280.3.h.n.1279.16 16 80.77 odd 4
1600.3.g.k.351.1 16 40.19 odd 2 inner
1600.3.g.k.351.2 16 8.5 even 2 inner
1600.3.g.k.351.3 16 5.4 even 2 inner
1600.3.g.k.351.4 16 4.3 odd 2 inner
1600.3.g.k.351.13 16 1.1 even 1 trivial
1600.3.g.k.351.14 16 20.19 odd 2 inner
1600.3.g.k.351.15 16 8.3 odd 2 inner
1600.3.g.k.351.16 16 40.29 even 2 inner