Properties

Label 4140.3.c.a.4049.2
Level $4140$
Weight $3$
Character 4140.4049
Analytic conductor $112.807$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,3,Mod(4049,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4140.4049");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.c (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: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Character \(\chi\) \(=\) 4140.4049
Dual form 4140.3.c.a.4049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.99275 + 0.269219i) q^{5} +13.6272i q^{7} +O(q^{10})\) \(q+(-4.99275 + 0.269219i) q^{5} +13.6272i q^{7} +10.9107i q^{11} -18.5051i q^{13} -6.61399 q^{17} -18.6941 q^{19} +4.79583 q^{23} +(24.8550 - 2.68828i) q^{25} -24.3559i q^{29} -10.8615 q^{31} +(-3.66870 - 68.0372i) q^{35} +40.5811i q^{37} +10.8964i q^{41} +19.6214i q^{43} -55.1384 q^{47} -136.701 q^{49} -83.1263 q^{53} +(-2.93736 - 54.4743i) q^{55} -4.74614i q^{59} +55.7886 q^{61} +(4.98192 + 92.3913i) q^{65} +74.6610i q^{67} -51.0055i q^{71} -129.217i q^{73} -148.682 q^{77} +112.870 q^{79} -156.983 q^{83} +(33.0220 - 1.78061i) q^{85} -119.858i q^{89} +252.173 q^{91} +(93.3348 - 5.03279i) q^{95} -109.487i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −4.99275 + 0.269219i −0.998549 + 0.0538437i
\(6\) 0 0
\(7\) 13.6272i 1.94674i 0.229231 + 0.973372i \(0.426379\pi\)
−0.229231 + 0.973372i \(0.573621\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.9107i 0.991881i 0.868356 + 0.495941i \(0.165177\pi\)
−0.868356 + 0.495941i \(0.834823\pi\)
\(12\) 0 0
\(13\) 18.5051i 1.42347i −0.702448 0.711735i \(-0.747910\pi\)
0.702448 0.711735i \(-0.252090\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.61399 −0.389058 −0.194529 0.980897i \(-0.562318\pi\)
−0.194529 + 0.980897i \(0.562318\pi\)
\(18\) 0 0
\(19\) −18.6941 −0.983899 −0.491949 0.870624i \(-0.663716\pi\)
−0.491949 + 0.870624i \(0.663716\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583 0.208514
\(24\) 0 0
\(25\) 24.8550 2.68828i 0.994202 0.107531i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 24.3559i 0.839860i −0.907556 0.419930i \(-0.862055\pi\)
0.907556 0.419930i \(-0.137945\pi\)
\(30\) 0 0
\(31\) −10.8615 −0.350372 −0.175186 0.984535i \(-0.556053\pi\)
−0.175186 + 0.984535i \(0.556053\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.66870 68.0372i −0.104820 1.94392i
\(36\) 0 0
\(37\) 40.5811i 1.09679i 0.836221 + 0.548393i \(0.184760\pi\)
−0.836221 + 0.548393i \(0.815240\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8964i 0.265765i 0.991132 + 0.132882i \(0.0424233\pi\)
−0.991132 + 0.132882i \(0.957577\pi\)
\(42\) 0 0
\(43\) 19.6214i 0.456310i 0.973625 + 0.228155i \(0.0732694\pi\)
−0.973625 + 0.228155i \(0.926731\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −55.1384 −1.17316 −0.586578 0.809892i \(-0.699525\pi\)
−0.586578 + 0.809892i \(0.699525\pi\)
\(48\) 0 0
\(49\) −136.701 −2.78981
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −83.1263 −1.56842 −0.784210 0.620495i \(-0.786932\pi\)
−0.784210 + 0.620495i \(0.786932\pi\)
\(54\) 0 0
\(55\) −2.93736 54.4743i −0.0534066 0.990443i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.74614i 0.0804431i −0.999191 0.0402215i \(-0.987194\pi\)
0.999191 0.0402215i \(-0.0128064\pi\)
\(60\) 0 0
\(61\) 55.7886 0.914567 0.457284 0.889321i \(-0.348822\pi\)
0.457284 + 0.889321i \(0.348822\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.98192 + 92.3913i 0.0766449 + 1.42140i
\(66\) 0 0
\(67\) 74.6610i 1.11434i 0.830397 + 0.557172i \(0.188114\pi\)
−0.830397 + 0.557172i \(0.811886\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 51.0055i 0.718387i −0.933263 0.359194i \(-0.883052\pi\)
0.933263 0.359194i \(-0.116948\pi\)
\(72\) 0 0
\(73\) 129.217i 1.77010i −0.465496 0.885050i \(-0.654124\pi\)
0.465496 0.885050i \(-0.345876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −148.682 −1.93094
\(78\) 0 0
\(79\) 112.870 1.42873 0.714367 0.699771i \(-0.246715\pi\)
0.714367 + 0.699771i \(0.246715\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −156.983 −1.89136 −0.945680 0.325100i \(-0.894602\pi\)
−0.945680 + 0.325100i \(0.894602\pi\)
\(84\) 0 0
\(85\) 33.0220 1.78061i 0.388494 0.0209484i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 119.858i 1.34672i −0.739313 0.673362i \(-0.764850\pi\)
0.739313 0.673362i \(-0.235150\pi\)
\(90\) 0 0
\(91\) 252.173 2.77113
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 93.3348 5.03279i 0.982472 0.0529768i
\(96\) 0 0
\(97\) 109.487i 1.12873i −0.825524 0.564366i \(-0.809121\pi\)
0.825524 0.564366i \(-0.190879\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 44.7777i 0.443344i −0.975121 0.221672i \(-0.928849\pi\)
0.975121 0.221672i \(-0.0711514\pi\)
\(102\) 0 0
\(103\) 171.082i 1.66099i 0.557024 + 0.830496i \(0.311943\pi\)
−0.557024 + 0.830496i \(0.688057\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 142.700 1.33365 0.666825 0.745215i \(-0.267653\pi\)
0.666825 + 0.745215i \(0.267653\pi\)
\(108\) 0 0
\(109\) −45.6967 −0.419236 −0.209618 0.977783i \(-0.567222\pi\)
−0.209618 + 0.977783i \(0.567222\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −101.184 −0.895436 −0.447718 0.894175i \(-0.647763\pi\)
−0.447718 + 0.894175i \(0.647763\pi\)
\(114\) 0 0
\(115\) −23.9444 + 1.29113i −0.208212 + 0.0112272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 90.1303i 0.757397i
\(120\) 0 0
\(121\) 1.95674 0.0161714
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −123.371 + 20.1133i −0.986970 + 0.160907i
\(126\) 0 0
\(127\) 144.502i 1.13781i 0.822404 + 0.568904i \(0.192633\pi\)
−0.822404 + 0.568904i \(0.807367\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 103.500i 0.790073i 0.918665 + 0.395037i \(0.129268\pi\)
−0.918665 + 0.395037i \(0.870732\pi\)
\(132\) 0 0
\(133\) 254.748i 1.91540i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 94.6970 0.691219 0.345610 0.938378i \(-0.387672\pi\)
0.345610 + 0.938378i \(0.387672\pi\)
\(138\) 0 0
\(139\) −38.3831 −0.276138 −0.138069 0.990423i \(-0.544090\pi\)
−0.138069 + 0.990423i \(0.544090\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 201.904 1.41191
\(144\) 0 0
\(145\) 6.55707 + 121.603i 0.0452212 + 0.838642i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 138.799i 0.931535i −0.884907 0.465767i \(-0.845778\pi\)
0.884907 0.465767i \(-0.154222\pi\)
\(150\) 0 0
\(151\) 43.0538 0.285124 0.142562 0.989786i \(-0.454466\pi\)
0.142562 + 0.989786i \(0.454466\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 54.2289 2.92413i 0.349864 0.0188653i
\(156\) 0 0
\(157\) 56.2911i 0.358542i 0.983800 + 0.179271i \(0.0573739\pi\)
−0.983800 + 0.179271i \(0.942626\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 65.3538i 0.405924i
\(162\) 0 0
\(163\) 275.418i 1.68968i 0.535019 + 0.844840i \(0.320304\pi\)
−0.535019 + 0.844840i \(0.679696\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 103.602 0.620372 0.310186 0.950676i \(-0.399609\pi\)
0.310186 + 0.950676i \(0.399609\pi\)
\(168\) 0 0
\(169\) −173.439 −1.02627
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.9240 0.132509 0.0662543 0.997803i \(-0.478895\pi\)
0.0662543 + 0.997803i \(0.478895\pi\)
\(174\) 0 0
\(175\) 36.6338 + 338.705i 0.209336 + 1.93546i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 38.1958i 0.213385i 0.994292 + 0.106692i \(0.0340260\pi\)
−0.994292 + 0.106692i \(0.965974\pi\)
\(180\) 0 0
\(181\) 273.993 1.51377 0.756887 0.653546i \(-0.226719\pi\)
0.756887 + 0.653546i \(0.226719\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.9252 202.611i −0.0590551 1.09520i
\(186\) 0 0
\(187\) 72.1633i 0.385900i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.7276i 0.103286i 0.998666 + 0.0516428i \(0.0164457\pi\)
−0.998666 + 0.0516428i \(0.983554\pi\)
\(192\) 0 0
\(193\) 91.3081i 0.473099i −0.971619 0.236549i \(-0.923983\pi\)
0.971619 0.236549i \(-0.0760165\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 62.5057 0.317288 0.158644 0.987336i \(-0.449288\pi\)
0.158644 + 0.987336i \(0.449288\pi\)
\(198\) 0 0
\(199\) −132.469 −0.665672 −0.332836 0.942985i \(-0.608006\pi\)
−0.332836 + 0.942985i \(0.608006\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 331.904 1.63499
\(204\) 0 0
\(205\) −2.93350 54.4028i −0.0143098 0.265379i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 203.965i 0.975911i
\(210\) 0 0
\(211\) 166.955 0.791255 0.395628 0.918411i \(-0.370527\pi\)
0.395628 + 0.918411i \(0.370527\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.28243 97.9644i −0.0245695 0.455649i
\(216\) 0 0
\(217\) 148.012i 0.682085i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 122.393i 0.553813i
\(222\) 0 0
\(223\) 293.697i 1.31703i −0.752568 0.658514i \(-0.771185\pi\)
0.752568 0.658514i \(-0.228815\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 388.268 1.71043 0.855216 0.518272i \(-0.173425\pi\)
0.855216 + 0.518272i \(0.173425\pi\)
\(228\) 0 0
\(229\) 97.2207 0.424545 0.212272 0.977211i \(-0.431914\pi\)
0.212272 + 0.977211i \(0.431914\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 261.348 1.12166 0.560832 0.827930i \(-0.310481\pi\)
0.560832 + 0.827930i \(0.310481\pi\)
\(234\) 0 0
\(235\) 275.292 14.8443i 1.17146 0.0631671i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 422.986i 1.76982i −0.465765 0.884909i \(-0.654221\pi\)
0.465765 0.884909i \(-0.345779\pi\)
\(240\) 0 0
\(241\) 344.752 1.43051 0.715253 0.698866i \(-0.246312\pi\)
0.715253 + 0.698866i \(0.246312\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 682.512 36.8024i 2.78576 0.150214i
\(246\) 0 0
\(247\) 345.936i 1.40055i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 252.361i 1.00542i −0.864455 0.502711i \(-0.832336\pi\)
0.864455 0.502711i \(-0.167664\pi\)
\(252\) 0 0
\(253\) 52.3259i 0.206822i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −267.765 −1.04189 −0.520944 0.853591i \(-0.674420\pi\)
−0.520944 + 0.853591i \(0.674420\pi\)
\(258\) 0 0
\(259\) −553.007 −2.13516
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −111.829 −0.425205 −0.212603 0.977139i \(-0.568194\pi\)
−0.212603 + 0.977139i \(0.568194\pi\)
\(264\) 0 0
\(265\) 415.028 22.3791i 1.56615 0.0844496i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 423.875i 1.57574i −0.615839 0.787872i \(-0.711183\pi\)
0.615839 0.787872i \(-0.288817\pi\)
\(270\) 0 0
\(271\) 187.453 0.691708 0.345854 0.938288i \(-0.387589\pi\)
0.345854 + 0.938288i \(0.387589\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 29.3310 + 271.186i 0.106658 + 0.986130i
\(276\) 0 0
\(277\) 227.548i 0.821473i 0.911754 + 0.410736i \(0.134728\pi\)
−0.911754 + 0.410736i \(0.865272\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 268.943i 0.957093i −0.878062 0.478546i \(-0.841164\pi\)
0.878062 0.478546i \(-0.158836\pi\)
\(282\) 0 0
\(283\) 359.867i 1.27161i 0.771848 + 0.635807i \(0.219333\pi\)
−0.771848 + 0.635807i \(0.780667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −148.487 −0.517376
\(288\) 0 0
\(289\) −245.255 −0.848634
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.848990 −0.00289758 −0.00144879 0.999999i \(-0.500461\pi\)
−0.00144879 + 0.999999i \(0.500461\pi\)
\(294\) 0 0
\(295\) 1.27775 + 23.6963i 0.00433135 + 0.0803264i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 88.7474i 0.296814i
\(300\) 0 0
\(301\) −267.384 −0.888320
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −278.538 + 15.0193i −0.913240 + 0.0492437i
\(306\) 0 0
\(307\) 348.180i 1.13414i −0.823670 0.567069i \(-0.808077\pi\)
0.823670 0.567069i \(-0.191923\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 254.597i 0.818638i 0.912391 + 0.409319i \(0.134234\pi\)
−0.912391 + 0.409319i \(0.865766\pi\)
\(312\) 0 0
\(313\) 462.281i 1.47693i −0.674289 0.738467i \(-0.735550\pi\)
0.674289 0.738467i \(-0.264450\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 139.176 0.439042 0.219521 0.975608i \(-0.429551\pi\)
0.219521 + 0.975608i \(0.429551\pi\)
\(318\) 0 0
\(319\) 265.740 0.833042
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 123.643 0.382794
\(324\) 0 0
\(325\) −49.7469 459.945i −0.153067 1.41522i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 751.382i 2.28384i
\(330\) 0 0
\(331\) −12.6330 −0.0381662 −0.0190831 0.999818i \(-0.506075\pi\)
−0.0190831 + 0.999818i \(0.506075\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −20.1001 372.764i −0.0600004 1.11273i
\(336\) 0 0
\(337\) 145.961i 0.433119i −0.976269 0.216560i \(-0.930516\pi\)
0.976269 0.216560i \(-0.0694836\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 118.507i 0.347528i
\(342\) 0 0
\(343\) 1195.12i 3.48431i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −146.681 −0.422710 −0.211355 0.977409i \(-0.567788\pi\)
−0.211355 + 0.977409i \(0.567788\pi\)
\(348\) 0 0
\(349\) 436.675 1.25122 0.625608 0.780137i \(-0.284851\pi\)
0.625608 + 0.780137i \(0.284851\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −461.875 −1.30843 −0.654213 0.756310i \(-0.727000\pi\)
−0.654213 + 0.756310i \(0.727000\pi\)
\(354\) 0 0
\(355\) 13.7316 + 254.658i 0.0386807 + 0.717345i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 380.063i 1.05867i −0.848413 0.529336i \(-0.822441\pi\)
0.848413 0.529336i \(-0.177559\pi\)
\(360\) 0 0
\(361\) −11.5314 −0.0319430
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 34.7877 + 645.149i 0.0953088 + 1.76753i
\(366\) 0 0
\(367\) 435.265i 1.18601i −0.805200 0.593004i \(-0.797942\pi\)
0.805200 0.593004i \(-0.202058\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1132.78i 3.05331i
\(372\) 0 0
\(373\) 158.241i 0.424238i 0.977244 + 0.212119i \(0.0680365\pi\)
−0.977244 + 0.212119i \(0.931964\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −450.709 −1.19552
\(378\) 0 0
\(379\) 245.573 0.647949 0.323975 0.946066i \(-0.394981\pi\)
0.323975 + 0.946066i \(0.394981\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −485.878 −1.26861 −0.634305 0.773083i \(-0.718714\pi\)
−0.634305 + 0.773083i \(0.718714\pi\)
\(384\) 0 0
\(385\) 742.333 40.0280i 1.92814 0.103969i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 370.231i 0.951750i 0.879513 + 0.475875i \(0.157868\pi\)
−0.879513 + 0.475875i \(0.842132\pi\)
\(390\) 0 0
\(391\) −31.7196 −0.0811243
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −563.531 + 30.3867i −1.42666 + 0.0769283i
\(396\) 0 0
\(397\) 336.618i 0.847903i −0.905685 0.423952i \(-0.860643\pi\)
0.905685 0.423952i \(-0.139357\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 216.118i 0.538947i 0.963008 + 0.269473i \(0.0868497\pi\)
−0.963008 + 0.269473i \(0.913150\pi\)
\(402\) 0 0
\(403\) 200.994i 0.498744i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −442.768 −1.08788
\(408\) 0 0
\(409\) 109.277 0.267181 0.133591 0.991037i \(-0.457349\pi\)
0.133591 + 0.991037i \(0.457349\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 64.6766 0.156602
\(414\) 0 0
\(415\) 783.776 42.2627i 1.88862 0.101838i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 414.301i 0.988784i 0.869239 + 0.494392i \(0.164609\pi\)
−0.869239 + 0.494392i \(0.835391\pi\)
\(420\) 0 0
\(421\) −409.425 −0.972506 −0.486253 0.873818i \(-0.661637\pi\)
−0.486253 + 0.873818i \(0.661637\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −164.391 + 17.7803i −0.386803 + 0.0418359i
\(426\) 0 0
\(427\) 760.243i 1.78043i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 80.9100i 0.187726i −0.995585 0.0938631i \(-0.970078\pi\)
0.995585 0.0938631i \(-0.0299216\pi\)
\(432\) 0 0
\(433\) 90.4630i 0.208921i 0.994529 + 0.104461i \(0.0333117\pi\)
−0.994529 + 0.104461i \(0.966688\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −89.6537 −0.205157
\(438\) 0 0
\(439\) 299.175 0.681492 0.340746 0.940155i \(-0.389320\pi\)
0.340746 + 0.940155i \(0.389320\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 489.134 1.10414 0.552069 0.833798i \(-0.313838\pi\)
0.552069 + 0.833798i \(0.313838\pi\)
\(444\) 0 0
\(445\) 32.2681 + 598.423i 0.0725126 + 1.34477i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 316.630i 0.705189i 0.935776 + 0.352594i \(0.114700\pi\)
−0.935776 + 0.352594i \(0.885300\pi\)
\(450\) 0 0
\(451\) −118.887 −0.263607
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1259.04 + 67.8896i −2.76711 + 0.149208i
\(456\) 0 0
\(457\) 689.747i 1.50929i −0.656131 0.754647i \(-0.727808\pi\)
0.656131 0.754647i \(-0.272192\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 835.305i 1.81194i 0.423340 + 0.905971i \(0.360858\pi\)
−0.423340 + 0.905971i \(0.639142\pi\)
\(462\) 0 0
\(463\) 96.5034i 0.208431i 0.994555 + 0.104215i \(0.0332331\pi\)
−0.994555 + 0.104215i \(0.966767\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 624.883 1.33808 0.669040 0.743226i \(-0.266705\pi\)
0.669040 + 0.743226i \(0.266705\pi\)
\(468\) 0 0
\(469\) −1017.42 −2.16934
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −214.083 −0.452606
\(474\) 0 0
\(475\) −464.642 + 50.2549i −0.978194 + 0.105800i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 845.372i 1.76487i −0.470435 0.882435i \(-0.655903\pi\)
0.470435 0.882435i \(-0.344097\pi\)
\(480\) 0 0
\(481\) 750.957 1.56124
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 29.4760 + 546.641i 0.0607752 + 1.12710i
\(486\) 0 0
\(487\) 154.626i 0.317508i 0.987318 + 0.158754i \(0.0507477\pi\)
−0.987318 + 0.158754i \(0.949252\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 494.309i 1.00674i −0.864071 0.503370i \(-0.832093\pi\)
0.864071 0.503370i \(-0.167907\pi\)
\(492\) 0 0
\(493\) 161.090i 0.326755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 695.063 1.39852
\(498\) 0 0
\(499\) 143.978 0.288533 0.144267 0.989539i \(-0.453918\pi\)
0.144267 + 0.989539i \(0.453918\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −567.447 −1.12813 −0.564063 0.825732i \(-0.690763\pi\)
−0.564063 + 0.825732i \(0.690763\pi\)
\(504\) 0 0
\(505\) 12.0550 + 223.564i 0.0238713 + 0.442700i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.9057i 0.0332135i −0.999862 0.0166068i \(-0.994714\pi\)
0.999862 0.0166068i \(-0.00528634\pi\)
\(510\) 0 0
\(511\) 1760.87 3.44593
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −46.0585 854.170i −0.0894340 1.65858i
\(516\) 0 0
\(517\) 601.598i 1.16363i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 946.617i 1.81692i −0.417969 0.908461i \(-0.637258\pi\)
0.417969 0.908461i \(-0.362742\pi\)
\(522\) 0 0
\(523\) 23.3619i 0.0446690i −0.999751 0.0223345i \(-0.992890\pi\)
0.999751 0.0223345i \(-0.00710988\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 71.8381 0.136315
\(528\) 0 0
\(529\) 23.0000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 201.638 0.378308
\(534\) 0 0
\(535\) −712.467 + 38.4176i −1.33171 + 0.0718087i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1491.50i 2.76716i
\(540\) 0 0
\(541\) −157.058 −0.290311 −0.145155 0.989409i \(-0.546368\pi\)
−0.145155 + 0.989409i \(0.546368\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 228.152 12.3024i 0.418628 0.0225732i
\(546\) 0 0
\(547\) 426.038i 0.778862i 0.921056 + 0.389431i \(0.127328\pi\)
−0.921056 + 0.389431i \(0.872672\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 455.312i 0.826338i
\(552\) 0 0
\(553\) 1538.10i 2.78138i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 90.6387 0.162727 0.0813633 0.996685i \(-0.474073\pi\)
0.0813633 + 0.996685i \(0.474073\pi\)
\(558\) 0 0
\(559\) 363.095 0.649544
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 93.6335 0.166312 0.0831559 0.996537i \(-0.473500\pi\)
0.0831559 + 0.996537i \(0.473500\pi\)
\(564\) 0 0
\(565\) 505.187 27.2407i 0.894137 0.0482136i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.9284i 0.0332661i 0.999862 + 0.0166330i \(0.00529471\pi\)
−0.999862 + 0.0166330i \(0.994705\pi\)
\(570\) 0 0
\(571\) −129.940 −0.227566 −0.113783 0.993506i \(-0.536297\pi\)
−0.113783 + 0.993506i \(0.536297\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 119.201 12.8925i 0.207305 0.0224218i
\(576\) 0 0
\(577\) 509.890i 0.883691i −0.897091 0.441846i \(-0.854324\pi\)
0.897091 0.441846i \(-0.145676\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2139.24i 3.68199i
\(582\) 0 0
\(583\) 906.965i 1.55569i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −616.914 −1.05096 −0.525480 0.850806i \(-0.676114\pi\)
−0.525480 + 0.850806i \(0.676114\pi\)
\(588\) 0 0
\(589\) 203.046 0.344731
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1137.91 1.91891 0.959455 0.281862i \(-0.0909521\pi\)
0.959455 + 0.281862i \(0.0909521\pi\)
\(594\) 0 0
\(595\) 24.2647 + 449.998i 0.0407811 + 0.756298i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 384.422i 0.641774i 0.947118 + 0.320887i \(0.103981\pi\)
−0.947118 + 0.320887i \(0.896019\pi\)
\(600\) 0 0
\(601\) −746.538 −1.24216 −0.621080 0.783747i \(-0.713306\pi\)
−0.621080 + 0.783747i \(0.713306\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.76950 + 0.526790i −0.0161479 + 0.000870728i
\(606\) 0 0
\(607\) 922.940i 1.52049i 0.649634 + 0.760247i \(0.274922\pi\)
−0.649634 + 0.760247i \(0.725078\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1020.34i 1.66995i
\(612\) 0 0
\(613\) 791.954i 1.29193i 0.763367 + 0.645965i \(0.223545\pi\)
−0.763367 + 0.645965i \(0.776455\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 347.501 0.563211 0.281606 0.959530i \(-0.409133\pi\)
0.281606 + 0.959530i \(0.409133\pi\)
\(618\) 0 0
\(619\) 745.764 1.20479 0.602394 0.798199i \(-0.294213\pi\)
0.602394 + 0.798199i \(0.294213\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1633.33 2.62173
\(624\) 0 0
\(625\) 610.546 133.635i 0.976874 0.213815i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 268.403i 0.426714i
\(630\) 0 0
\(631\) −1163.24 −1.84349 −0.921744 0.387800i \(-0.873235\pi\)
−0.921744 + 0.387800i \(0.873235\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −38.9026 721.460i −0.0612639 1.13616i
\(636\) 0 0
\(637\) 2529.66i 3.97121i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 164.410i 0.256490i −0.991743 0.128245i \(-0.959066\pi\)
0.991743 0.128245i \(-0.0409344\pi\)
\(642\) 0 0
\(643\) 255.319i 0.397075i 0.980093 + 0.198538i \(0.0636192\pi\)
−0.980093 + 0.198538i \(0.936381\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1238.64 −1.91443 −0.957215 0.289377i \(-0.906552\pi\)
−0.957215 + 0.289377i \(0.906552\pi\)
\(648\) 0 0
\(649\) 51.7837 0.0797900
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1002.98 1.53595 0.767976 0.640478i \(-0.221264\pi\)
0.767976 + 0.640478i \(0.221264\pi\)
\(654\) 0 0
\(655\) −27.8640 516.747i −0.0425405 0.788927i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 615.722i 0.934327i 0.884171 + 0.467164i \(0.154724\pi\)
−0.884171 + 0.467164i \(0.845276\pi\)
\(660\) 0 0
\(661\) −377.413 −0.570972 −0.285486 0.958383i \(-0.592155\pi\)
−0.285486 + 0.958383i \(0.592155\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 68.5829 + 1271.89i 0.103132 + 1.91262i
\(666\) 0 0
\(667\) 116.807i 0.175123i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 608.692i 0.907142i
\(672\) 0 0
\(673\) 954.784i 1.41870i −0.704857 0.709349i \(-0.748989\pi\)
0.704857 0.709349i \(-0.251011\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −140.861 −0.208066 −0.104033 0.994574i \(-0.533175\pi\)
−0.104033 + 0.994574i \(0.533175\pi\)
\(678\) 0 0
\(679\) 1492.00 2.19735
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.4201 −0.0269694 −0.0134847 0.999909i \(-0.504292\pi\)
−0.0134847 + 0.999909i \(0.504292\pi\)
\(684\) 0 0
\(685\) −472.798 + 25.4942i −0.690216 + 0.0372178i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1538.26i 2.23260i
\(690\) 0 0
\(691\) −627.126 −0.907564 −0.453782 0.891113i \(-0.649925\pi\)
−0.453782 + 0.891113i \(0.649925\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 191.637 10.3335i 0.275737 0.0148683i
\(696\) 0 0
\(697\) 72.0685i 0.103398i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 708.842i 1.01119i −0.862772 0.505593i \(-0.831274\pi\)
0.862772 0.505593i \(-0.168726\pi\)
\(702\) 0 0
\(703\) 758.626i 1.07913i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 610.195 0.863076
\(708\) 0 0
\(709\) 294.963 0.416026 0.208013 0.978126i \(-0.433300\pi\)
0.208013 + 0.978126i \(0.433300\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −52.0901 −0.0730577
\(714\) 0 0
\(715\) −1008.05 + 54.3562i −1.40986 + 0.0760227i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 822.939i 1.14456i −0.820058 0.572280i \(-0.806059\pi\)
0.820058 0.572280i \(-0.193941\pi\)
\(720\) 0 0
\(721\) −2331.37 −3.23353
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −65.4756 605.368i −0.0903112 0.834990i
\(726\) 0 0
\(727\) 443.780i 0.610426i 0.952284 + 0.305213i \(0.0987276\pi\)
−0.952284 + 0.305213i \(0.901272\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 129.775i 0.177531i
\(732\) 0 0
\(733\) 130.278i 0.177733i −0.996044 0.0888666i \(-0.971676\pi\)
0.996044 0.0888666i \(-0.0283245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −814.604 −1.10530
\(738\) 0 0
\(739\) −805.386 −1.08983 −0.544916 0.838490i \(-0.683439\pi\)
−0.544916 + 0.838490i \(0.683439\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 284.424 0.382805 0.191402 0.981512i \(-0.438696\pi\)
0.191402 + 0.981512i \(0.438696\pi\)
\(744\) 0 0
\(745\) 37.3672 + 692.987i 0.0501573 + 0.930183i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1944.61i 2.59627i
\(750\) 0 0
\(751\) 72.2500 0.0962050 0.0481025 0.998842i \(-0.484683\pi\)
0.0481025 + 0.998842i \(0.484683\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −214.957 + 11.5909i −0.284711 + 0.0153522i
\(756\) 0 0
\(757\) 1268.09i 1.67515i 0.546321 + 0.837576i \(0.316028\pi\)
−0.546321 + 0.837576i \(0.683972\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 115.653i 0.151976i −0.997109 0.0759878i \(-0.975789\pi\)
0.997109 0.0759878i \(-0.0242110\pi\)
\(762\) 0 0
\(763\) 622.719i 0.816146i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −87.8278 −0.114508
\(768\) 0 0
\(769\) 779.540 1.01371 0.506853 0.862032i \(-0.330809\pi\)
0.506853 + 0.862032i \(0.330809\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −78.4368 −0.101471 −0.0507353 0.998712i \(-0.516156\pi\)
−0.0507353 + 0.998712i \(0.516156\pi\)
\(774\) 0 0
\(775\) −269.964 + 29.1989i −0.348341 + 0.0376760i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 203.697i 0.261486i
\(780\) 0 0
\(781\) 556.506 0.712555
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −15.1546 281.047i −0.0193052 0.358022i
\(786\) 0 0
\(787\) 257.400i 0.327065i 0.986538 + 0.163532i \(0.0522888\pi\)
−0.986538 + 0.163532i \(0.947711\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1378.86i 1.74318i
\(792\) 0 0
\(793\) 1032.37i 1.30186i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1458.64 −1.83017 −0.915083 0.403266i \(-0.867875\pi\)
−0.915083 + 0.403266i \(0.867875\pi\)
\(798\) 0 0
\(799\) 364.685 0.456427
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1409.85 1.75573
\(804\) 0 0
\(805\) −17.5945 326.295i −0.0218565 0.405335i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 275.824i 0.340945i 0.985362 + 0.170472i \(0.0545294\pi\)
−0.985362 + 0.170472i \(0.945471\pi\)
\(810\) 0 0
\(811\) 265.225 0.327034 0.163517 0.986540i \(-0.447716\pi\)
0.163517 + 0.986540i \(0.447716\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −74.1476 1375.09i −0.0909787 1.68723i
\(816\) 0 0
\(817\) 366.803i 0.448963i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 889.794i 1.08379i 0.840445 + 0.541897i \(0.182294\pi\)
−0.840445 + 0.541897i \(0.817706\pi\)
\(822\) 0 0
\(823\) 70.0990i 0.0851750i −0.999093 0.0425875i \(-0.986440\pi\)
0.999093 0.0425875i \(-0.0135601\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 369.845 0.447213 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(828\) 0 0
\(829\) 1366.23 1.64804 0.824020 0.566561i \(-0.191726\pi\)
0.824020 + 0.566561i \(0.191726\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 904.138 1.08540
\(834\) 0 0
\(835\) −517.260 + 27.8916i −0.619472 + 0.0334032i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 916.900i 1.09285i 0.837508 + 0.546425i \(0.184012\pi\)
−0.837508 + 0.546425i \(0.815988\pi\)
\(840\) 0 0
\(841\) 247.788 0.294635
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 865.936 46.6930i 1.02478 0.0552580i
\(846\) 0 0
\(847\) 26.6649i 0.0314815i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 194.620i 0.228696i
\(852\) 0 0
\(853\) 272.464i 0.319419i 0.987164 + 0.159709i \(0.0510557\pi\)
−0.987164 + 0.159709i \(0.948944\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1474.04 1.72000 0.860002 0.510290i \(-0.170462\pi\)
0.860002 + 0.510290i \(0.170462\pi\)
\(858\) 0 0
\(859\) −923.505 −1.07509 −0.537547 0.843234i \(-0.680649\pi\)
−0.537547 + 0.843234i \(0.680649\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1226.67 −1.42140 −0.710699 0.703496i \(-0.751621\pi\)
−0.710699 + 0.703496i \(0.751621\pi\)
\(864\) 0 0
\(865\) −114.454 + 6.17156i −0.132316 + 0.00713475i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1231.49i 1.41713i
\(870\) 0 0
\(871\) 1381.61 1.58623
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −274.089 1681.20i −0.313244 1.92138i
\(876\) 0 0
\(877\) 1.67098i 0.00190534i 1.00000 0.000952669i \(0.000303244\pi\)
−1.00000 0.000952669i \(0.999697\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1360.29i 1.54403i 0.635604 + 0.772015i \(0.280751\pi\)
−0.635604 + 0.772015i \(0.719249\pi\)
\(882\) 0 0
\(883\) 758.529i 0.859037i −0.903058 0.429518i \(-0.858683\pi\)
0.903058 0.429518i \(-0.141317\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1066.21 −1.20204 −0.601018 0.799236i \(-0.705238\pi\)
−0.601018 + 0.799236i \(0.705238\pi\)
\(888\) 0 0
\(889\) −1969.15 −2.21502
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1030.76 1.15427
\(894\) 0 0
\(895\) −10.2830 190.702i −0.0114894 0.213075i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 264.543i 0.294264i
\(900\) 0 0
\(901\) 549.797 0.610207
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1367.98 + 73.7641i −1.51158 + 0.0815073i
\(906\) 0 0
\(907\) 968.211i 1.06749i −0.845646 0.533744i \(-0.820785\pi\)
0.845646 0.533744i \(-0.179215\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 559.212i 0.613844i 0.951735 + 0.306922i \(0.0992990\pi\)
−0.951735 + 0.306922i \(0.900701\pi\)
\(912\) 0 0
\(913\) 1712.79i 1.87600i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1410.41 −1.53807
\(918\) 0 0
\(919\) 1047.46 1.13978 0.569889 0.821722i \(-0.306986\pi\)
0.569889 + 0.821722i \(0.306986\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −943.862 −1.02260
\(924\) 0 0
\(925\) 109.093 + 1008.64i 0.117939 + 1.09043i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1171.93i 1.26149i −0.775990 0.630746i \(-0.782749\pi\)
0.775990 0.630746i \(-0.217251\pi\)
\(930\) 0 0
\(931\) 2555.50 2.74489
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.4277 + 360.293i 0.0207783 + 0.385340i
\(936\) 0 0
\(937\) 694.182i 0.740856i −0.928861 0.370428i \(-0.879211\pi\)
0.928861 0.370428i \(-0.120789\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 643.730i 0.684092i −0.939683 0.342046i \(-0.888880\pi\)
0.939683 0.342046i \(-0.111120\pi\)
\(942\) 0 0
\(943\) 52.2571i 0.0554158i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 668.516 0.705930 0.352965 0.935637i \(-0.385173\pi\)
0.352965 + 0.935637i \(0.385173\pi\)
\(948\) 0 0
\(949\) −2391.18 −2.51968
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1159.32 −1.21650 −0.608249 0.793747i \(-0.708128\pi\)
−0.608249 + 0.793747i \(0.708128\pi\)
\(954\) 0 0
\(955\) −5.31102 98.4947i −0.00556128 0.103136i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1290.46i 1.34563i
\(960\) 0 0
\(961\) −843.027 −0.877239
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 24.5818 + 455.878i 0.0254734 + 0.472412i
\(966\) 0 0
\(967\) 1838.49i 1.90123i −0.310379 0.950613i \(-0.600456\pi\)
0.310379 0.950613i \(-0.399544\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 693.313i 0.714019i 0.934101 + 0.357010i \(0.116204\pi\)
−0.934101 + 0.357010i \(0.883796\pi\)
\(972\) 0 0
\(973\) 523.055i 0.537569i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1157.04 −1.18428 −0.592139 0.805835i \(-0.701717\pi\)
−0.592139 + 0.805835i \(0.701717\pi\)
\(978\) 0 0
\(979\) 1307.74 1.33579
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1346.71 −1.37000 −0.685001 0.728542i \(-0.740198\pi\)
−0.685001 + 0.728542i \(0.740198\pi\)
\(984\) 0 0
\(985\) −312.075 + 16.8277i −0.316828 + 0.0170840i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 94.1007i 0.0951473i
\(990\) 0 0
\(991\) −788.614 −0.795776 −0.397888 0.917434i \(-0.630257\pi\)
−0.397888 + 0.917434i \(0.630257\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 661.383 35.6631i 0.664707 0.0358423i
\(996\) 0 0
\(997\) 569.209i 0.570922i 0.958390 + 0.285461i \(0.0921467\pi\)
−0.958390 + 0.285461i \(0.907853\pi\)
\(998\) 0 0
\(999\) 0 0
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 4140.3.c.a.4049.2 yes 88
3.2 odd 2 inner 4140.3.c.a.4049.87 yes 88
5.4 even 2 inner 4140.3.c.a.4049.88 yes 88
15.14 odd 2 inner 4140.3.c.a.4049.1 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.3.c.a.4049.1 88 15.14 odd 2 inner
4140.3.c.a.4049.2 yes 88 1.1 even 1 trivial
4140.3.c.a.4049.87 yes 88 3.2 odd 2 inner
4140.3.c.a.4049.88 yes 88 5.4 even 2 inner