Properties

Label 200.4.c.f.49.1
Level $200$
Weight $4$
Character 200.49
Analytic conductor $11.800$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,4,Mod(49,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not 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: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.4.c.f.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +O(q^{10})\) \(q-4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +36.0000 q^{11} +42.0000i q^{13} -110.000i q^{17} +116.000 q^{19} +64.0000 q^{21} -16.0000i q^{23} -152.000i q^{27} -198.000 q^{29} +240.000 q^{31} -144.000i q^{33} -258.000i q^{37} +168.000 q^{39} +442.000 q^{41} +292.000i q^{43} +392.000i q^{47} +87.0000 q^{49} -440.000 q^{51} -142.000i q^{53} -464.000i q^{57} +348.000 q^{59} -570.000 q^{61} +176.000i q^{63} +692.000i q^{67} -64.0000 q^{69} +168.000 q^{71} +134.000i q^{73} +576.000i q^{77} -784.000 q^{79} -311.000 q^{81} -564.000i q^{83} +792.000i q^{87} -1034.00 q^{89} -672.000 q^{91} -960.000i q^{93} -382.000i q^{97} +396.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 22 q^{9} + 72 q^{11} + 232 q^{19} + 128 q^{21} - 396 q^{29} + 480 q^{31} + 336 q^{39} + 884 q^{41} + 174 q^{49} - 880 q^{51} + 696 q^{59} - 1140 q^{61} - 128 q^{69} + 336 q^{71} - 1568 q^{79} - 622 q^{81} - 2068 q^{89} - 1344 q^{91} + 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 4.00000i − 0.769800i −0.922958 0.384900i \(-0.874236\pi\)
0.922958 0.384900i \(-0.125764\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 16.0000i 0.863919i 0.901893 + 0.431959i \(0.142178\pi\)
−0.901893 + 0.431959i \(0.857822\pi\)
\(8\) 0 0
\(9\) 11.0000 0.407407
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) 42.0000i 0.896054i 0.894020 + 0.448027i \(0.147873\pi\)
−0.894020 + 0.448027i \(0.852127\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 110.000i − 1.56935i −0.619909 0.784674i \(-0.712830\pi\)
0.619909 0.784674i \(-0.287170\pi\)
\(18\) 0 0
\(19\) 116.000 1.40064 0.700322 0.713827i \(-0.253040\pi\)
0.700322 + 0.713827i \(0.253040\pi\)
\(20\) 0 0
\(21\) 64.0000 0.665045
\(22\) 0 0
\(23\) − 16.0000i − 0.145054i −0.997366 0.0725268i \(-0.976894\pi\)
0.997366 0.0725268i \(-0.0231063\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 152.000i − 1.08342i
\(28\) 0 0
\(29\) −198.000 −1.26785 −0.633925 0.773394i \(-0.718557\pi\)
−0.633925 + 0.773394i \(0.718557\pi\)
\(30\) 0 0
\(31\) 240.000 1.39049 0.695246 0.718772i \(-0.255295\pi\)
0.695246 + 0.718772i \(0.255295\pi\)
\(32\) 0 0
\(33\) − 144.000i − 0.759612i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 258.000i − 1.14635i −0.819433 0.573175i \(-0.805712\pi\)
0.819433 0.573175i \(-0.194288\pi\)
\(38\) 0 0
\(39\) 168.000 0.689783
\(40\) 0 0
\(41\) 442.000 1.68363 0.841815 0.539767i \(-0.181488\pi\)
0.841815 + 0.539767i \(0.181488\pi\)
\(42\) 0 0
\(43\) 292.000i 1.03557i 0.855510 + 0.517786i \(0.173244\pi\)
−0.855510 + 0.517786i \(0.826756\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 392.000i 1.21658i 0.793716 + 0.608288i \(0.208143\pi\)
−0.793716 + 0.608288i \(0.791857\pi\)
\(48\) 0 0
\(49\) 87.0000 0.253644
\(50\) 0 0
\(51\) −440.000 −1.20808
\(52\) 0 0
\(53\) − 142.000i − 0.368023i −0.982924 0.184011i \(-0.941092\pi\)
0.982924 0.184011i \(-0.0589083\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 464.000i − 1.07822i
\(58\) 0 0
\(59\) 348.000 0.767894 0.383947 0.923355i \(-0.374565\pi\)
0.383947 + 0.923355i \(0.374565\pi\)
\(60\) 0 0
\(61\) −570.000 −1.19641 −0.598205 0.801343i \(-0.704119\pi\)
−0.598205 + 0.801343i \(0.704119\pi\)
\(62\) 0 0
\(63\) 176.000i 0.351967i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 692.000i 1.26181i 0.775860 + 0.630905i \(0.217316\pi\)
−0.775860 + 0.630905i \(0.782684\pi\)
\(68\) 0 0
\(69\) −64.0000 −0.111662
\(70\) 0 0
\(71\) 168.000 0.280816 0.140408 0.990094i \(-0.455159\pi\)
0.140408 + 0.990094i \(0.455159\pi\)
\(72\) 0 0
\(73\) 134.000i 0.214843i 0.994214 + 0.107421i \(0.0342594\pi\)
−0.994214 + 0.107421i \(0.965741\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 576.000i 0.852484i
\(78\) 0 0
\(79\) −784.000 −1.11654 −0.558271 0.829658i \(-0.688535\pi\)
−0.558271 + 0.829658i \(0.688535\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) − 564.000i − 0.745868i −0.927858 0.372934i \(-0.878352\pi\)
0.927858 0.372934i \(-0.121648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 792.000i 0.975992i
\(88\) 0 0
\(89\) −1034.00 −1.23150 −0.615752 0.787940i \(-0.711148\pi\)
−0.615752 + 0.787940i \(0.711148\pi\)
\(90\) 0 0
\(91\) −672.000 −0.774118
\(92\) 0 0
\(93\) − 960.000i − 1.07040i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 382.000i − 0.399858i −0.979810 0.199929i \(-0.935929\pi\)
0.979810 0.199929i \(-0.0640711\pi\)
\(98\) 0 0
\(99\) 396.000 0.402015
\(100\) 0 0
\(101\) −674.000 −0.664015 −0.332007 0.943277i \(-0.607726\pi\)
−0.332007 + 0.943277i \(0.607726\pi\)
\(102\) 0 0
\(103\) 992.000i 0.948977i 0.880262 + 0.474489i \(0.157367\pi\)
−0.880262 + 0.474489i \(0.842633\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 500.000i − 0.451746i −0.974157 0.225873i \(-0.927477\pi\)
0.974157 0.225873i \(-0.0725234\pi\)
\(108\) 0 0
\(109\) −1046.00 −0.919162 −0.459581 0.888136i \(-0.652000\pi\)
−0.459581 + 0.888136i \(0.652000\pi\)
\(110\) 0 0
\(111\) −1032.00 −0.882460
\(112\) 0 0
\(113\) 558.000i 0.464533i 0.972652 + 0.232266i \(0.0746141\pi\)
−0.972652 + 0.232266i \(0.925386\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 462.000i 0.365059i
\(118\) 0 0
\(119\) 1760.00 1.35579
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) − 1768.00i − 1.29606i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 328.000i − 0.229176i −0.993413 0.114588i \(-0.963445\pi\)
0.993413 0.114588i \(-0.0365547\pi\)
\(128\) 0 0
\(129\) 1168.00 0.797183
\(130\) 0 0
\(131\) −212.000 −0.141393 −0.0706967 0.997498i \(-0.522522\pi\)
−0.0706967 + 0.997498i \(0.522522\pi\)
\(132\) 0 0
\(133\) 1856.00i 1.21004i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1434.00i 0.894269i 0.894467 + 0.447135i \(0.147556\pi\)
−0.894467 + 0.447135i \(0.852444\pi\)
\(138\) 0 0
\(139\) −2196.00 −1.34002 −0.670008 0.742354i \(-0.733709\pi\)
−0.670008 + 0.742354i \(0.733709\pi\)
\(140\) 0 0
\(141\) 1568.00 0.936521
\(142\) 0 0
\(143\) 1512.00i 0.884194i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 348.000i − 0.195255i
\(148\) 0 0
\(149\) 2418.00 1.32946 0.664732 0.747081i \(-0.268546\pi\)
0.664732 + 0.747081i \(0.268546\pi\)
\(150\) 0 0
\(151\) 3672.00 1.97896 0.989481 0.144666i \(-0.0462108\pi\)
0.989481 + 0.144666i \(0.0462108\pi\)
\(152\) 0 0
\(153\) − 1210.00i − 0.639364i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 358.000i 0.181984i 0.995852 + 0.0909921i \(0.0290038\pi\)
−0.995852 + 0.0909921i \(0.970996\pi\)
\(158\) 0 0
\(159\) −568.000 −0.283304
\(160\) 0 0
\(161\) 256.000 0.125314
\(162\) 0 0
\(163\) − 2564.00i − 1.23207i −0.787717 0.616037i \(-0.788737\pi\)
0.787717 0.616037i \(-0.211263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 3056.00i − 1.41605i −0.706187 0.708025i \(-0.749586\pi\)
0.706187 0.708025i \(-0.250414\pi\)
\(168\) 0 0
\(169\) 433.000 0.197087
\(170\) 0 0
\(171\) 1276.00 0.570633
\(172\) 0 0
\(173\) 234.000i 0.102836i 0.998677 + 0.0514182i \(0.0163741\pi\)
−0.998677 + 0.0514182i \(0.983626\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1392.00i − 0.591125i
\(178\) 0 0
\(179\) −524.000 −0.218802 −0.109401 0.993998i \(-0.534893\pi\)
−0.109401 + 0.993998i \(0.534893\pi\)
\(180\) 0 0
\(181\) −1138.00 −0.467331 −0.233665 0.972317i \(-0.575072\pi\)
−0.233665 + 0.972317i \(0.575072\pi\)
\(182\) 0 0
\(183\) 2280.00i 0.920997i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 3960.00i − 1.54858i
\(188\) 0 0
\(189\) 2432.00 0.935989
\(190\) 0 0
\(191\) 1520.00 0.575829 0.287915 0.957656i \(-0.407038\pi\)
0.287915 + 0.957656i \(0.407038\pi\)
\(192\) 0 0
\(193\) 2142.00i 0.798884i 0.916759 + 0.399442i \(0.130796\pi\)
−0.916759 + 0.399442i \(0.869204\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2306.00i − 0.833988i −0.908909 0.416994i \(-0.863084\pi\)
0.908909 0.416994i \(-0.136916\pi\)
\(198\) 0 0
\(199\) −3288.00 −1.17126 −0.585628 0.810580i \(-0.699152\pi\)
−0.585628 + 0.810580i \(0.699152\pi\)
\(200\) 0 0
\(201\) 2768.00 0.971342
\(202\) 0 0
\(203\) − 3168.00i − 1.09532i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 176.000i − 0.0590959i
\(208\) 0 0
\(209\) 4176.00 1.38211
\(210\) 0 0
\(211\) −3876.00 −1.26462 −0.632310 0.774715i \(-0.717893\pi\)
−0.632310 + 0.774715i \(0.717893\pi\)
\(212\) 0 0
\(213\) − 672.000i − 0.216172i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3840.00i 1.20127i
\(218\) 0 0
\(219\) 536.000 0.165386
\(220\) 0 0
\(221\) 4620.00 1.40622
\(222\) 0 0
\(223\) 5688.00i 1.70806i 0.520226 + 0.854028i \(0.325848\pi\)
−0.520226 + 0.854028i \(0.674152\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2796.00i − 0.817520i −0.912642 0.408760i \(-0.865961\pi\)
0.912642 0.408760i \(-0.134039\pi\)
\(228\) 0 0
\(229\) −4446.00 −1.28297 −0.641485 0.767136i \(-0.721681\pi\)
−0.641485 + 0.767136i \(0.721681\pi\)
\(230\) 0 0
\(231\) 2304.00 0.656243
\(232\) 0 0
\(233\) − 2522.00i − 0.709106i −0.935036 0.354553i \(-0.884633\pi\)
0.935036 0.354553i \(-0.115367\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3136.00i 0.859515i
\(238\) 0 0
\(239\) −816.000 −0.220848 −0.110424 0.993885i \(-0.535221\pi\)
−0.110424 + 0.993885i \(0.535221\pi\)
\(240\) 0 0
\(241\) −5422.00 −1.44922 −0.724609 0.689160i \(-0.757980\pi\)
−0.724609 + 0.689160i \(0.757980\pi\)
\(242\) 0 0
\(243\) − 2860.00i − 0.755017i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4872.00i 1.25505i
\(248\) 0 0
\(249\) −2256.00 −0.574169
\(250\) 0 0
\(251\) −5900.00 −1.48368 −0.741842 0.670575i \(-0.766048\pi\)
−0.741842 + 0.670575i \(0.766048\pi\)
\(252\) 0 0
\(253\) − 576.000i − 0.143134i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5250.00i 1.27426i 0.770754 + 0.637132i \(0.219880\pi\)
−0.770754 + 0.637132i \(0.780120\pi\)
\(258\) 0 0
\(259\) 4128.00 0.990353
\(260\) 0 0
\(261\) −2178.00 −0.516532
\(262\) 0 0
\(263\) − 6240.00i − 1.46302i −0.681829 0.731511i \(-0.738815\pi\)
0.681829 0.731511i \(-0.261185\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4136.00i 0.948012i
\(268\) 0 0
\(269\) 714.000 0.161834 0.0809170 0.996721i \(-0.474215\pi\)
0.0809170 + 0.996721i \(0.474215\pi\)
\(270\) 0 0
\(271\) 2144.00 0.480586 0.240293 0.970700i \(-0.422757\pi\)
0.240293 + 0.970700i \(0.422757\pi\)
\(272\) 0 0
\(273\) 2688.00i 0.595916i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4466.00i − 0.968722i −0.874868 0.484361i \(-0.839052\pi\)
0.874868 0.484361i \(-0.160948\pi\)
\(278\) 0 0
\(279\) 2640.00 0.566497
\(280\) 0 0
\(281\) −5302.00 −1.12559 −0.562795 0.826596i \(-0.690274\pi\)
−0.562795 + 0.826596i \(0.690274\pi\)
\(282\) 0 0
\(283\) 6932.00i 1.45606i 0.685546 + 0.728029i \(0.259564\pi\)
−0.685546 + 0.728029i \(0.740436\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7072.00i 1.45452i
\(288\) 0 0
\(289\) −7187.00 −1.46285
\(290\) 0 0
\(291\) −1528.00 −0.307811
\(292\) 0 0
\(293\) 4034.00i 0.804330i 0.915567 + 0.402165i \(0.131742\pi\)
−0.915567 + 0.402165i \(0.868258\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5472.00i − 1.06908i
\(298\) 0 0
\(299\) 672.000 0.129976
\(300\) 0 0
\(301\) −4672.00 −0.894650
\(302\) 0 0
\(303\) 2696.00i 0.511159i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 3836.00i − 0.713134i −0.934270 0.356567i \(-0.883947\pi\)
0.934270 0.356567i \(-0.116053\pi\)
\(308\) 0 0
\(309\) 3968.00 0.730523
\(310\) 0 0
\(311\) 664.000 0.121067 0.0605337 0.998166i \(-0.480720\pi\)
0.0605337 + 0.998166i \(0.480720\pi\)
\(312\) 0 0
\(313\) − 2986.00i − 0.539229i −0.962968 0.269615i \(-0.913104\pi\)
0.962968 0.269615i \(-0.0868963\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2726.00i 0.482989i 0.970402 + 0.241494i \(0.0776375\pi\)
−0.970402 + 0.241494i \(0.922362\pi\)
\(318\) 0 0
\(319\) −7128.00 −1.25107
\(320\) 0 0
\(321\) −2000.00 −0.347754
\(322\) 0 0
\(323\) − 12760.0i − 2.19810i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4184.00i 0.707571i
\(328\) 0 0
\(329\) −6272.00 −1.05102
\(330\) 0 0
\(331\) −9212.00 −1.52972 −0.764860 0.644197i \(-0.777192\pi\)
−0.764860 + 0.644197i \(0.777192\pi\)
\(332\) 0 0
\(333\) − 2838.00i − 0.467031i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3278.00i − 0.529864i −0.964267 0.264932i \(-0.914651\pi\)
0.964267 0.264932i \(-0.0853494\pi\)
\(338\) 0 0
\(339\) 2232.00 0.357598
\(340\) 0 0
\(341\) 8640.00 1.37209
\(342\) 0 0
\(343\) 6880.00i 1.08305i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4956.00i 0.766721i 0.923599 + 0.383360i \(0.125233\pi\)
−0.923599 + 0.383360i \(0.874767\pi\)
\(348\) 0 0
\(349\) −4678.00 −0.717500 −0.358750 0.933434i \(-0.616797\pi\)
−0.358750 + 0.933434i \(0.616797\pi\)
\(350\) 0 0
\(351\) 6384.00 0.970805
\(352\) 0 0
\(353\) − 1890.00i − 0.284970i −0.989797 0.142485i \(-0.954491\pi\)
0.989797 0.142485i \(-0.0455093\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 7040.00i − 1.04369i
\(358\) 0 0
\(359\) 6472.00 0.951474 0.475737 0.879588i \(-0.342181\pi\)
0.475737 + 0.879588i \(0.342181\pi\)
\(360\) 0 0
\(361\) 6597.00 0.961802
\(362\) 0 0
\(363\) 140.000i 0.0202427i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1960.00i 0.278777i 0.990238 + 0.139389i \(0.0445137\pi\)
−0.990238 + 0.139389i \(0.955486\pi\)
\(368\) 0 0
\(369\) 4862.00 0.685923
\(370\) 0 0
\(371\) 2272.00 0.317942
\(372\) 0 0
\(373\) − 8750.00i − 1.21463i −0.794460 0.607316i \(-0.792246\pi\)
0.794460 0.607316i \(-0.207754\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 8316.00i − 1.13606i
\(378\) 0 0
\(379\) 380.000 0.0515021 0.0257510 0.999668i \(-0.491802\pi\)
0.0257510 + 0.999668i \(0.491802\pi\)
\(380\) 0 0
\(381\) −1312.00 −0.176419
\(382\) 0 0
\(383\) 9688.00i 1.29252i 0.763119 + 0.646258i \(0.223667\pi\)
−0.763119 + 0.646258i \(0.776333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3212.00i 0.421900i
\(388\) 0 0
\(389\) −3870.00 −0.504413 −0.252207 0.967673i \(-0.581156\pi\)
−0.252207 + 0.967673i \(0.581156\pi\)
\(390\) 0 0
\(391\) −1760.00 −0.227639
\(392\) 0 0
\(393\) 848.000i 0.108845i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1622.00i 0.205053i 0.994730 + 0.102526i \(0.0326926\pi\)
−0.994730 + 0.102526i \(0.967307\pi\)
\(398\) 0 0
\(399\) 7424.00 0.931491
\(400\) 0 0
\(401\) 9906.00 1.23362 0.616811 0.787112i \(-0.288424\pi\)
0.616811 + 0.787112i \(0.288424\pi\)
\(402\) 0 0
\(403\) 10080.0i 1.24596i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 9288.00i − 1.13118i
\(408\) 0 0
\(409\) 4214.00 0.509459 0.254730 0.967012i \(-0.418014\pi\)
0.254730 + 0.967012i \(0.418014\pi\)
\(410\) 0 0
\(411\) 5736.00 0.688409
\(412\) 0 0
\(413\) 5568.00i 0.663398i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8784.00i 1.03155i
\(418\) 0 0
\(419\) 7012.00 0.817562 0.408781 0.912632i \(-0.365954\pi\)
0.408781 + 0.912632i \(0.365954\pi\)
\(420\) 0 0
\(421\) −1602.00 −0.185455 −0.0927277 0.995692i \(-0.529559\pi\)
−0.0927277 + 0.995692i \(0.529559\pi\)
\(422\) 0 0
\(423\) 4312.00i 0.495642i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 9120.00i − 1.03360i
\(428\) 0 0
\(429\) 6048.00 0.680653
\(430\) 0 0
\(431\) −3584.00 −0.400546 −0.200273 0.979740i \(-0.564183\pi\)
−0.200273 + 0.979740i \(0.564183\pi\)
\(432\) 0 0
\(433\) 3470.00i 0.385121i 0.981285 + 0.192561i \(0.0616792\pi\)
−0.981285 + 0.192561i \(0.938321\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1856.00i − 0.203168i
\(438\) 0 0
\(439\) 3416.00 0.371382 0.185691 0.982608i \(-0.440548\pi\)
0.185691 + 0.982608i \(0.440548\pi\)
\(440\) 0 0
\(441\) 957.000 0.103337
\(442\) 0 0
\(443\) − 9708.00i − 1.04118i −0.853808 0.520588i \(-0.825713\pi\)
0.853808 0.520588i \(-0.174287\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 9672.00i − 1.02342i
\(448\) 0 0
\(449\) 10366.0 1.08954 0.544768 0.838587i \(-0.316618\pi\)
0.544768 + 0.838587i \(0.316618\pi\)
\(450\) 0 0
\(451\) 15912.0 1.66135
\(452\) 0 0
\(453\) − 14688.0i − 1.52340i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16742.0i − 1.71369i −0.515572 0.856847i \(-0.672420\pi\)
0.515572 0.856847i \(-0.327580\pi\)
\(458\) 0 0
\(459\) −16720.0 −1.70027
\(460\) 0 0
\(461\) −1258.00 −0.127095 −0.0635476 0.997979i \(-0.520241\pi\)
−0.0635476 + 0.997979i \(0.520241\pi\)
\(462\) 0 0
\(463\) − 13528.0i − 1.35788i −0.734193 0.678941i \(-0.762439\pi\)
0.734193 0.678941i \(-0.237561\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6916.00i 0.685298i 0.939463 + 0.342649i \(0.111324\pi\)
−0.939463 + 0.342649i \(0.888676\pi\)
\(468\) 0 0
\(469\) −11072.0 −1.09010
\(470\) 0 0
\(471\) 1432.00 0.140091
\(472\) 0 0
\(473\) 10512.0i 1.02187i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1562.00i − 0.149935i
\(478\) 0 0
\(479\) −1728.00 −0.164832 −0.0824158 0.996598i \(-0.526264\pi\)
−0.0824158 + 0.996598i \(0.526264\pi\)
\(480\) 0 0
\(481\) 10836.0 1.02719
\(482\) 0 0
\(483\) − 1024.00i − 0.0964671i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16656.0i 1.54981i 0.632080 + 0.774903i \(0.282201\pi\)
−0.632080 + 0.774903i \(0.717799\pi\)
\(488\) 0 0
\(489\) −10256.0 −0.948451
\(490\) 0 0
\(491\) −1084.00 −0.0996339 −0.0498169 0.998758i \(-0.515864\pi\)
−0.0498169 + 0.998758i \(0.515864\pi\)
\(492\) 0 0
\(493\) 21780.0i 1.98970i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2688.00i 0.242602i
\(498\) 0 0
\(499\) −5804.00 −0.520687 −0.260343 0.965516i \(-0.583836\pi\)
−0.260343 + 0.965516i \(0.583836\pi\)
\(500\) 0 0
\(501\) −12224.0 −1.09008
\(502\) 0 0
\(503\) − 10512.0i − 0.931823i −0.884831 0.465911i \(-0.845727\pi\)
0.884831 0.465911i \(-0.154273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1732.00i − 0.151718i
\(508\) 0 0
\(509\) 4314.00 0.375667 0.187834 0.982201i \(-0.439853\pi\)
0.187834 + 0.982201i \(0.439853\pi\)
\(510\) 0 0
\(511\) −2144.00 −0.185607
\(512\) 0 0
\(513\) − 17632.0i − 1.51749i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 14112.0i 1.20047i
\(518\) 0 0
\(519\) 936.000 0.0791635
\(520\) 0 0
\(521\) −1190.00 −0.100067 −0.0500334 0.998748i \(-0.515933\pi\)
−0.0500334 + 0.998748i \(0.515933\pi\)
\(522\) 0 0
\(523\) 3780.00i 0.316038i 0.987436 + 0.158019i \(0.0505107\pi\)
−0.987436 + 0.158019i \(0.949489\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 26400.0i − 2.18217i
\(528\) 0 0
\(529\) 11911.0 0.978959
\(530\) 0 0
\(531\) 3828.00 0.312846
\(532\) 0 0
\(533\) 18564.0i 1.50862i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2096.00i 0.168434i
\(538\) 0 0
\(539\) 3132.00 0.250287
\(540\) 0 0
\(541\) −11002.0 −0.874331 −0.437165 0.899381i \(-0.644018\pi\)
−0.437165 + 0.899381i \(0.644018\pi\)
\(542\) 0 0
\(543\) 4552.00i 0.359751i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5908.00i 0.461806i 0.972977 + 0.230903i \(0.0741680\pi\)
−0.972977 + 0.230903i \(0.925832\pi\)
\(548\) 0 0
\(549\) −6270.00 −0.487426
\(550\) 0 0
\(551\) −22968.0 −1.77581
\(552\) 0 0
\(553\) − 12544.0i − 0.964602i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14806.0i 1.12630i 0.826354 + 0.563151i \(0.190411\pi\)
−0.826354 + 0.563151i \(0.809589\pi\)
\(558\) 0 0
\(559\) −12264.0 −0.927928
\(560\) 0 0
\(561\) −15840.0 −1.19210
\(562\) 0 0
\(563\) 684.000i 0.0512028i 0.999672 + 0.0256014i \(0.00815007\pi\)
−0.999672 + 0.0256014i \(0.991850\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 4976.00i − 0.368558i
\(568\) 0 0
\(569\) 2582.00 0.190234 0.0951169 0.995466i \(-0.469677\pi\)
0.0951169 + 0.995466i \(0.469677\pi\)
\(570\) 0 0
\(571\) −2540.00 −0.186157 −0.0930785 0.995659i \(-0.529671\pi\)
−0.0930785 + 0.995659i \(0.529671\pi\)
\(572\) 0 0
\(573\) − 6080.00i − 0.443273i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22786.0i 1.64401i 0.569480 + 0.822005i \(0.307144\pi\)
−0.569480 + 0.822005i \(0.692856\pi\)
\(578\) 0 0
\(579\) 8568.00 0.614981
\(580\) 0 0
\(581\) 9024.00 0.644369
\(582\) 0 0
\(583\) − 5112.00i − 0.363152i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7884.00i 0.554357i 0.960818 + 0.277178i \(0.0893993\pi\)
−0.960818 + 0.277178i \(0.910601\pi\)
\(588\) 0 0
\(589\) 27840.0 1.94758
\(590\) 0 0
\(591\) −9224.00 −0.642005
\(592\) 0 0
\(593\) 21902.0i 1.51671i 0.651843 + 0.758354i \(0.273996\pi\)
−0.651843 + 0.758354i \(0.726004\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13152.0i 0.901634i
\(598\) 0 0
\(599\) −15080.0 −1.02863 −0.514317 0.857600i \(-0.671955\pi\)
−0.514317 + 0.857600i \(0.671955\pi\)
\(600\) 0 0
\(601\) −19702.0 −1.33721 −0.668603 0.743619i \(-0.733108\pi\)
−0.668603 + 0.743619i \(0.733108\pi\)
\(602\) 0 0
\(603\) 7612.00i 0.514071i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7320.00i 0.489472i 0.969590 + 0.244736i \(0.0787013\pi\)
−0.969590 + 0.244736i \(0.921299\pi\)
\(608\) 0 0
\(609\) −12672.0 −0.843178
\(610\) 0 0
\(611\) −16464.0 −1.09012
\(612\) 0 0
\(613\) − 24350.0i − 1.60438i −0.597066 0.802192i \(-0.703667\pi\)
0.597066 0.802192i \(-0.296333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19546.0i 1.27535i 0.770305 + 0.637676i \(0.220104\pi\)
−0.770305 + 0.637676i \(0.779896\pi\)
\(618\) 0 0
\(619\) −3476.00 −0.225706 −0.112853 0.993612i \(-0.535999\pi\)
−0.112853 + 0.993612i \(0.535999\pi\)
\(620\) 0 0
\(621\) −2432.00 −0.157154
\(622\) 0 0
\(623\) − 16544.0i − 1.06392i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 16704.0i − 1.06394i
\(628\) 0 0
\(629\) −28380.0 −1.79902
\(630\) 0 0
\(631\) 21880.0 1.38039 0.690197 0.723621i \(-0.257524\pi\)
0.690197 + 0.723621i \(0.257524\pi\)
\(632\) 0 0
\(633\) 15504.0i 0.973505i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3654.00i 0.227279i
\(638\) 0 0
\(639\) 1848.00 0.114406
\(640\) 0 0
\(641\) 20994.0 1.29362 0.646812 0.762649i \(-0.276102\pi\)
0.646812 + 0.762649i \(0.276102\pi\)
\(642\) 0 0
\(643\) 18204.0i 1.11648i 0.829680 + 0.558239i \(0.188523\pi\)
−0.829680 + 0.558239i \(0.811477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2064.00i − 0.125416i −0.998032 0.0627080i \(-0.980026\pi\)
0.998032 0.0627080i \(-0.0199737\pi\)
\(648\) 0 0
\(649\) 12528.0 0.757730
\(650\) 0 0
\(651\) 15360.0 0.924740
\(652\) 0 0
\(653\) − 9942.00i − 0.595805i −0.954596 0.297902i \(-0.903713\pi\)
0.954596 0.297902i \(-0.0962870\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1474.00i 0.0875285i
\(658\) 0 0
\(659\) −24236.0 −1.43263 −0.716313 0.697779i \(-0.754172\pi\)
−0.716313 + 0.697779i \(0.754172\pi\)
\(660\) 0 0
\(661\) 17614.0 1.03647 0.518234 0.855239i \(-0.326590\pi\)
0.518234 + 0.855239i \(0.326590\pi\)
\(662\) 0 0
\(663\) − 18480.0i − 1.08251i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3168.00i 0.183906i
\(668\) 0 0
\(669\) 22752.0 1.31486
\(670\) 0 0
\(671\) −20520.0 −1.18057
\(672\) 0 0
\(673\) − 13058.0i − 0.747918i −0.927445 0.373959i \(-0.878000\pi\)
0.927445 0.373959i \(-0.122000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 33186.0i − 1.88396i −0.335668 0.941980i \(-0.608962\pi\)
0.335668 0.941980i \(-0.391038\pi\)
\(678\) 0 0
\(679\) 6112.00 0.345445
\(680\) 0 0
\(681\) −11184.0 −0.629327
\(682\) 0 0
\(683\) 31716.0i 1.77684i 0.459035 + 0.888418i \(0.348195\pi\)
−0.459035 + 0.888418i \(0.651805\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 17784.0i 0.987630i
\(688\) 0 0
\(689\) 5964.00 0.329768
\(690\) 0 0
\(691\) −2084.00 −0.114731 −0.0573655 0.998353i \(-0.518270\pi\)
−0.0573655 + 0.998353i \(0.518270\pi\)
\(692\) 0 0
\(693\) 6336.00i 0.347308i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 48620.0i − 2.64220i
\(698\) 0 0
\(699\) −10088.0 −0.545870
\(700\) 0 0
\(701\) −7418.00 −0.399678 −0.199839 0.979829i \(-0.564042\pi\)
−0.199839 + 0.979829i \(0.564042\pi\)
\(702\) 0 0
\(703\) − 29928.0i − 1.60563i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 10784.0i − 0.573655i
\(708\) 0 0
\(709\) 18242.0 0.966280 0.483140 0.875543i \(-0.339496\pi\)
0.483140 + 0.875543i \(0.339496\pi\)
\(710\) 0 0
\(711\) −8624.00 −0.454888
\(712\) 0 0
\(713\) − 3840.00i − 0.201696i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3264.00i 0.170009i
\(718\) 0 0
\(719\) −3024.00 −0.156851 −0.0784257 0.996920i \(-0.524989\pi\)
−0.0784257 + 0.996920i \(0.524989\pi\)
\(720\) 0 0
\(721\) −15872.0 −0.819839
\(722\) 0 0
\(723\) 21688.0i 1.11561i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 26176.0i − 1.33537i −0.744444 0.667685i \(-0.767285\pi\)
0.744444 0.667685i \(-0.232715\pi\)
\(728\) 0 0
\(729\) −19837.0 −1.00782
\(730\) 0 0
\(731\) 32120.0 1.62517
\(732\) 0 0
\(733\) 17818.0i 0.897848i 0.893570 + 0.448924i \(0.148193\pi\)
−0.893570 + 0.448924i \(0.851807\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24912.0i 1.24511i
\(738\) 0 0
\(739\) 22052.0 1.09769 0.548847 0.835923i \(-0.315067\pi\)
0.548847 + 0.835923i \(0.315067\pi\)
\(740\) 0 0
\(741\) 19488.0 0.966140
\(742\) 0 0
\(743\) − 15840.0i − 0.782117i −0.920366 0.391059i \(-0.872109\pi\)
0.920366 0.391059i \(-0.127891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6204.00i − 0.303872i
\(748\) 0 0
\(749\) 8000.00 0.390272
\(750\) 0 0
\(751\) 21024.0 1.02154 0.510770 0.859717i \(-0.329360\pi\)
0.510770 + 0.859717i \(0.329360\pi\)
\(752\) 0 0
\(753\) 23600.0i 1.14214i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 38034.0i − 1.82612i −0.407831 0.913058i \(-0.633715\pi\)
0.407831 0.913058i \(-0.366285\pi\)
\(758\) 0 0
\(759\) −2304.00 −0.110184
\(760\) 0 0
\(761\) 37802.0 1.80069 0.900343 0.435182i \(-0.143316\pi\)
0.900343 + 0.435182i \(0.143316\pi\)
\(762\) 0 0
\(763\) − 16736.0i − 0.794081i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14616.0i 0.688075i
\(768\) 0 0
\(769\) −15042.0 −0.705369 −0.352684 0.935742i \(-0.614731\pi\)
−0.352684 + 0.935742i \(0.614731\pi\)
\(770\) 0 0
\(771\) 21000.0 0.980929
\(772\) 0 0
\(773\) − 5950.00i − 0.276852i −0.990373 0.138426i \(-0.955796\pi\)
0.990373 0.138426i \(-0.0442043\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 16512.0i − 0.762374i
\(778\) 0 0
\(779\) 51272.0 2.35816
\(780\) 0 0
\(781\) 6048.00 0.277099
\(782\) 0 0
\(783\) 30096.0i 1.37362i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23364.0i 1.05824i 0.848546 + 0.529121i \(0.177478\pi\)
−0.848546 + 0.529121i \(0.822522\pi\)
\(788\) 0 0
\(789\) −24960.0 −1.12624
\(790\) 0 0
\(791\) −8928.00 −0.401319
\(792\) 0 0
\(793\) − 23940.0i − 1.07205i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19846.0i 0.882034i 0.897499 + 0.441017i \(0.145382\pi\)
−0.897499 + 0.441017i \(0.854618\pi\)
\(798\) 0 0
\(799\) 43120.0 1.90923
\(800\) 0 0
\(801\) −11374.0 −0.501724
\(802\) 0 0
\(803\) 4824.00i 0.211999i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2856.00i − 0.124580i
\(808\) 0 0
\(809\) −24762.0 −1.07613 −0.538063 0.842905i \(-0.680844\pi\)
−0.538063 + 0.842905i \(0.680844\pi\)
\(810\) 0 0
\(811\) 16644.0 0.720653 0.360327 0.932826i \(-0.382665\pi\)
0.360327 + 0.932826i \(0.382665\pi\)
\(812\) 0 0
\(813\) − 8576.00i − 0.369955i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 33872.0i 1.45047i
\(818\) 0 0
\(819\) −7392.00 −0.315381
\(820\) 0 0
\(821\) 3182.00 0.135265 0.0676325 0.997710i \(-0.478455\pi\)
0.0676325 + 0.997710i \(0.478455\pi\)
\(822\) 0 0
\(823\) 7504.00i 0.317829i 0.987292 + 0.158914i \(0.0507994\pi\)
−0.987292 + 0.158914i \(0.949201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12604.0i 0.529969i 0.964253 + 0.264984i \(0.0853668\pi\)
−0.964253 + 0.264984i \(0.914633\pi\)
\(828\) 0 0
\(829\) −12230.0 −0.512383 −0.256191 0.966626i \(-0.582468\pi\)
−0.256191 + 0.966626i \(0.582468\pi\)
\(830\) 0 0
\(831\) −17864.0 −0.745722
\(832\) 0 0
\(833\) − 9570.00i − 0.398056i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 36480.0i − 1.50649i
\(838\) 0 0
\(839\) −9656.00 −0.397333 −0.198666 0.980067i \(-0.563661\pi\)
−0.198666 + 0.980067i \(0.563661\pi\)
\(840\) 0 0
\(841\) 14815.0 0.607446
\(842\) 0 0
\(843\) 21208.0i 0.866480i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 560.000i − 0.0227176i
\(848\) 0 0
\(849\) 27728.0 1.12087
\(850\) 0 0
\(851\) −4128.00 −0.166282
\(852\) 0 0
\(853\) − 5806.00i − 0.233052i −0.993188 0.116526i \(-0.962824\pi\)
0.993188 0.116526i \(-0.0371759\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 39094.0i − 1.55826i −0.626865 0.779128i \(-0.715662\pi\)
0.626865 0.779128i \(-0.284338\pi\)
\(858\) 0 0
\(859\) 18876.0 0.749756 0.374878 0.927074i \(-0.377685\pi\)
0.374878 + 0.927074i \(0.377685\pi\)
\(860\) 0 0
\(861\) 28288.0 1.11969
\(862\) 0 0
\(863\) − 32296.0i − 1.27389i −0.770909 0.636946i \(-0.780197\pi\)
0.770909 0.636946i \(-0.219803\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 28748.0i 1.12611i
\(868\) 0 0
\(869\) −28224.0 −1.10176
\(870\) 0 0
\(871\) −29064.0 −1.13065
\(872\) 0 0
\(873\) − 4202.00i − 0.162905i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 9578.00i − 0.368787i −0.982853 0.184393i \(-0.940968\pi\)
0.982853 0.184393i \(-0.0590321\pi\)
\(878\) 0 0
\(879\) 16136.0 0.619174
\(880\) 0 0
\(881\) −41710.0 −1.59506 −0.797529 0.603281i \(-0.793860\pi\)
−0.797529 + 0.603281i \(0.793860\pi\)
\(882\) 0 0
\(883\) − 2260.00i − 0.0861326i −0.999072 0.0430663i \(-0.986287\pi\)
0.999072 0.0430663i \(-0.0137127\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 33696.0i − 1.27554i −0.770228 0.637768i \(-0.779858\pi\)
0.770228 0.637768i \(-0.220142\pi\)
\(888\) 0 0
\(889\) 5248.00 0.197989
\(890\) 0 0
\(891\) −11196.0 −0.420965
\(892\) 0 0
\(893\) 45472.0i 1.70399i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2688.00i − 0.100055i
\(898\) 0 0
\(899\) −47520.0 −1.76294
\(900\) 0 0
\(901\) −15620.0 −0.577556
\(902\) 0 0
\(903\) 18688.0i 0.688702i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7756.00i 0.283940i 0.989871 + 0.141970i \(0.0453437\pi\)
−0.989871 + 0.141970i \(0.954656\pi\)
\(908\) 0 0
\(909\) −7414.00 −0.270525
\(910\) 0 0
\(911\) −5312.00 −0.193188 −0.0965941 0.995324i \(-0.530795\pi\)
−0.0965941 + 0.995324i \(0.530795\pi\)
\(912\) 0 0
\(913\) − 20304.0i − 0.735996i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3392.00i − 0.122152i
\(918\) 0 0
\(919\) 23576.0 0.846246 0.423123 0.906072i \(-0.360934\pi\)
0.423123 + 0.906072i \(0.360934\pi\)
\(920\) 0 0
\(921\) −15344.0 −0.548971
\(922\) 0 0
\(923\) 7056.00i 0.251626i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 10912.0i 0.386620i
\(928\) 0 0
\(929\) 19038.0 0.672354 0.336177 0.941799i \(-0.390866\pi\)
0.336177 + 0.941799i \(0.390866\pi\)
\(930\) 0 0
\(931\) 10092.0 0.355265
\(932\) 0 0
\(933\) − 2656.00i − 0.0931978i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20570.0i 0.717175i 0.933496 + 0.358587i \(0.116741\pi\)
−0.933496 + 0.358587i \(0.883259\pi\)
\(938\) 0 0
\(939\) −11944.0 −0.415099
\(940\) 0 0
\(941\) −21386.0 −0.740875 −0.370438 0.928857i \(-0.620792\pi\)
−0.370438 + 0.928857i \(0.620792\pi\)
\(942\) 0 0
\(943\) − 7072.00i − 0.244216i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38020.0i 1.30463i 0.757948 + 0.652315i \(0.226202\pi\)
−0.757948 + 0.652315i \(0.773798\pi\)
\(948\) 0 0
\(949\) −5628.00 −0.192511
\(950\) 0 0
\(951\) 10904.0 0.371805
\(952\) 0 0
\(953\) − 20202.0i − 0.686681i −0.939211 0.343340i \(-0.888442\pi\)
0.939211 0.343340i \(-0.111558\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 28512.0i 0.963074i
\(958\) 0 0
\(959\) −22944.0 −0.772576
\(960\) 0 0
\(961\) 27809.0 0.933470
\(962\) 0 0
\(963\) − 5500.00i − 0.184045i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 29840.0i 0.992337i 0.868226 + 0.496168i \(0.165260\pi\)
−0.868226 + 0.496168i \(0.834740\pi\)
\(968\) 0 0
\(969\) −51040.0 −1.69210
\(970\) 0 0
\(971\) −12476.0 −0.412332 −0.206166 0.978517i \(-0.566099\pi\)
−0.206166 + 0.978517i \(0.566099\pi\)
\(972\) 0 0
\(973\) − 35136.0i − 1.15767i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 36974.0i − 1.21075i −0.795940 0.605375i \(-0.793023\pi\)
0.795940 0.605375i \(-0.206977\pi\)
\(978\) 0 0
\(979\) −37224.0 −1.21520
\(980\) 0 0
\(981\) −11506.0 −0.374473
\(982\) 0 0
\(983\) 16368.0i 0.531087i 0.964099 + 0.265543i \(0.0855513\pi\)
−0.964099 + 0.265543i \(0.914449\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 25088.0i 0.809078i
\(988\) 0 0
\(989\) 4672.00 0.150213
\(990\) 0 0
\(991\) −49552.0 −1.58837 −0.794183 0.607678i \(-0.792101\pi\)
−0.794183 + 0.607678i \(0.792101\pi\)
\(992\) 0 0
\(993\) 36848.0i 1.17758i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 24414.0i 0.775526i 0.921759 + 0.387763i \(0.126752\pi\)
−0.921759 + 0.387763i \(0.873248\pi\)
\(998\) 0 0
\(999\) −39216.0 −1.24198
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 200.4.c.f.49.1 2
3.2 odd 2 1800.4.f.d.649.2 2
4.3 odd 2 400.4.c.h.49.2 2
5.2 odd 4 200.4.a.d.1.1 1
5.3 odd 4 40.4.a.b.1.1 1
5.4 even 2 inner 200.4.c.f.49.2 2
15.2 even 4 1800.4.a.h.1.1 1
15.8 even 4 360.4.a.f.1.1 1
15.14 odd 2 1800.4.f.d.649.1 2
20.3 even 4 80.4.a.b.1.1 1
20.7 even 4 400.4.a.p.1.1 1
20.19 odd 2 400.4.c.h.49.1 2
35.13 even 4 1960.4.a.e.1.1 1
40.3 even 4 320.4.a.j.1.1 1
40.13 odd 4 320.4.a.e.1.1 1
40.27 even 4 1600.4.a.q.1.1 1
40.37 odd 4 1600.4.a.bk.1.1 1
60.23 odd 4 720.4.a.d.1.1 1
80.3 even 4 1280.4.d.m.641.1 2
80.13 odd 4 1280.4.d.d.641.2 2
80.43 even 4 1280.4.d.m.641.2 2
80.53 odd 4 1280.4.d.d.641.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.b.1.1 1 5.3 odd 4
80.4.a.b.1.1 1 20.3 even 4
200.4.a.d.1.1 1 5.2 odd 4
200.4.c.f.49.1 2 1.1 even 1 trivial
200.4.c.f.49.2 2 5.4 even 2 inner
320.4.a.e.1.1 1 40.13 odd 4
320.4.a.j.1.1 1 40.3 even 4
360.4.a.f.1.1 1 15.8 even 4
400.4.a.p.1.1 1 20.7 even 4
400.4.c.h.49.1 2 20.19 odd 2
400.4.c.h.49.2 2 4.3 odd 2
720.4.a.d.1.1 1 60.23 odd 4
1280.4.d.d.641.1 2 80.53 odd 4
1280.4.d.d.641.2 2 80.13 odd 4
1280.4.d.m.641.1 2 80.3 even 4
1280.4.d.m.641.2 2 80.43 even 4
1600.4.a.q.1.1 1 40.27 even 4
1600.4.a.bk.1.1 1 40.37 odd 4
1800.4.a.h.1.1 1 15.2 even 4
1800.4.f.d.649.1 2 15.14 odd 2
1800.4.f.d.649.2 2 3.2 odd 2
1960.4.a.e.1.1 1 35.13 even 4