Properties

Label 720.4.f.j.289.1
Level $720$
Weight $4$
Character 720.289
Analytic conductor $42.481$
Analytic rank $0$
Dimension $4$
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] = [720,4,Mod(289,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("720.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.f (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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.1
Root \(2.70156i\) of defining polynomial
Character \(\chi\) \(=\) 720.289
Dual form 720.4.f.j.289.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-11.1047 - 1.29844i) q^{5} +16.2094i q^{7} +O(q^{10})\) \(q+(-11.1047 - 1.29844i) q^{5} +16.2094i q^{7} -40.2094 q^{11} -19.7906i q^{13} -83.0156i q^{17} -48.8375 q^{19} -1.61250i q^{23} +(121.628 + 28.8375i) q^{25} -24.5344 q^{29} +12.4187 q^{31} +(21.0469 - 180.000i) q^{35} +325.884i q^{37} +242.419 q^{41} +367.350i q^{43} -204.544i q^{47} +80.2562 q^{49} -61.5281i q^{53} +(446.512 + 52.2094i) q^{55} +112.209 q^{59} +477.350 q^{61} +(-25.6969 + 219.769i) q^{65} -558.094i q^{67} +558.281 q^{71} -1011.77i q^{73} -651.769i q^{77} +1150.47 q^{79} +1157.92i q^{83} +(-107.791 + 921.862i) q^{85} +96.9751 q^{89} +320.794 q^{91} +(542.325 + 63.4124i) q^{95} -1152.37i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) −11.1047 1.29844i −0.993233 0.116136i
\(6\) 0 0
\(7\) 16.2094i 0.875224i 0.899164 + 0.437612i \(0.144176\pi\)
−0.899164 + 0.437612i \(0.855824\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −40.2094 −1.10214 −0.551072 0.834458i \(-0.685781\pi\)
−0.551072 + 0.834458i \(0.685781\pi\)
\(12\) 0 0
\(13\) 19.7906i 0.422226i −0.977462 0.211113i \(-0.932291\pi\)
0.977462 0.211113i \(-0.0677087\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 83.0156i 1.18437i −0.805803 0.592184i \(-0.798266\pi\)
0.805803 0.592184i \(-0.201734\pi\)
\(18\) 0 0
\(19\) −48.8375 −0.589689 −0.294844 0.955545i \(-0.595268\pi\)
−0.294844 + 0.955545i \(0.595268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.61250i 0.0146186i −0.999973 0.00730932i \(-0.997673\pi\)
0.999973 0.00730932i \(-0.00232665\pi\)
\(24\) 0 0
\(25\) 121.628 + 28.8375i 0.973025 + 0.230700i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −24.5344 −0.157101 −0.0785504 0.996910i \(-0.525029\pi\)
−0.0785504 + 0.996910i \(0.525029\pi\)
\(30\) 0 0
\(31\) 12.4187 0.0719507 0.0359754 0.999353i \(-0.488546\pi\)
0.0359754 + 0.999353i \(0.488546\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.0469 180.000i 0.101645 0.869302i
\(36\) 0 0
\(37\) 325.884i 1.44797i 0.689813 + 0.723987i \(0.257693\pi\)
−0.689813 + 0.723987i \(0.742307\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 242.419 0.923401 0.461701 0.887036i \(-0.347239\pi\)
0.461701 + 0.887036i \(0.347239\pi\)
\(42\) 0 0
\(43\) 367.350i 1.30280i 0.758735 + 0.651399i \(0.225818\pi\)
−0.758735 + 0.651399i \(0.774182\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 204.544i 0.634804i −0.948291 0.317402i \(-0.897190\pi\)
0.948291 0.317402i \(-0.102810\pi\)
\(48\) 0 0
\(49\) 80.2562 0.233983
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 61.5281i 0.159463i −0.996816 0.0797314i \(-0.974594\pi\)
0.996816 0.0797314i \(-0.0254063\pi\)
\(54\) 0 0
\(55\) 446.512 + 52.2094i 1.09469 + 0.127998i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 112.209 0.247600 0.123800 0.992307i \(-0.460492\pi\)
0.123800 + 0.992307i \(0.460492\pi\)
\(60\) 0 0
\(61\) 477.350 1.00194 0.500970 0.865464i \(-0.332977\pi\)
0.500970 + 0.865464i \(0.332977\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25.6969 + 219.769i −0.0490355 + 0.419369i
\(66\) 0 0
\(67\) 558.094i 1.01764i −0.860872 0.508821i \(-0.830082\pi\)
0.860872 0.508821i \(-0.169918\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 558.281 0.933180 0.466590 0.884474i \(-0.345482\pi\)
0.466590 + 0.884474i \(0.345482\pi\)
\(72\) 0 0
\(73\) 1011.77i 1.62217i −0.584927 0.811086i \(-0.698877\pi\)
0.584927 0.811086i \(-0.301123\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 651.769i 0.964623i
\(78\) 0 0
\(79\) 1150.47 1.63845 0.819227 0.573470i \(-0.194403\pi\)
0.819227 + 0.573470i \(0.194403\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1157.92i 1.53131i 0.643251 + 0.765655i \(0.277585\pi\)
−0.643251 + 0.765655i \(0.722415\pi\)
\(84\) 0 0
\(85\) −107.791 + 921.862i −0.137547 + 1.17635i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 96.9751 0.115498 0.0577491 0.998331i \(-0.481608\pi\)
0.0577491 + 0.998331i \(0.481608\pi\)
\(90\) 0 0
\(91\) 320.794 0.369542
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 542.325 + 63.4124i 0.585699 + 0.0684840i
\(96\) 0 0
\(97\) 1152.37i 1.20625i −0.797648 0.603123i \(-0.793923\pi\)
0.797648 0.603123i \(-0.206077\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1156.49 1.13936 0.569679 0.821867i \(-0.307068\pi\)
0.569679 + 0.821867i \(0.307068\pi\)
\(102\) 0 0
\(103\) 1333.70i 1.27585i −0.770096 0.637927i \(-0.779792\pi\)
0.770096 0.637927i \(-0.220208\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 798.263i 0.721224i −0.932716 0.360612i \(-0.882568\pi\)
0.932716 0.360612i \(-0.117432\pi\)
\(108\) 0 0
\(109\) 985.119 0.865663 0.432831 0.901475i \(-0.357515\pi\)
0.432831 + 0.901475i \(0.357515\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1888.25i 1.57196i 0.618253 + 0.785979i \(0.287841\pi\)
−0.618253 + 0.785979i \(0.712159\pi\)
\(114\) 0 0
\(115\) −2.09373 + 17.9063i −0.00169775 + 0.0145197i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1345.63 1.03659
\(120\) 0 0
\(121\) 285.794 0.214721
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1313.20 478.158i −0.939648 0.342142i
\(126\) 0 0
\(127\) 620.859i 0.433798i 0.976194 + 0.216899i \(0.0695943\pi\)
−0.976194 + 0.216899i \(0.930406\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2588.35 −1.72630 −0.863151 0.504947i \(-0.831512\pi\)
−0.863151 + 0.504947i \(0.831512\pi\)
\(132\) 0 0
\(133\) 791.625i 0.516110i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1656.29i 1.03289i 0.856319 + 0.516447i \(0.172746\pi\)
−0.856319 + 0.516447i \(0.827254\pi\)
\(138\) 0 0
\(139\) −153.256 −0.0935182 −0.0467591 0.998906i \(-0.514889\pi\)
−0.0467591 + 0.998906i \(0.514889\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 795.769i 0.465353i
\(144\) 0 0
\(145\) 272.447 + 31.8564i 0.156038 + 0.0182450i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1483.38 0.815591 0.407795 0.913073i \(-0.366298\pi\)
0.407795 + 0.913073i \(0.366298\pi\)
\(150\) 0 0
\(151\) 394.281 0.212491 0.106246 0.994340i \(-0.466117\pi\)
0.106246 + 0.994340i \(0.466117\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −137.906 16.1250i −0.0714639 0.00835606i
\(156\) 0 0
\(157\) 1727.05i 0.877922i 0.898506 + 0.438961i \(0.144653\pi\)
−0.898506 + 0.438961i \(0.855347\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 26.1376 0.0127946
\(162\) 0 0
\(163\) 2034.28i 0.977529i −0.872416 0.488764i \(-0.837448\pi\)
0.872416 0.488764i \(-0.162552\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 192.900i 0.0893835i 0.999001 + 0.0446918i \(0.0142306\pi\)
−0.999001 + 0.0446918i \(0.985769\pi\)
\(168\) 0 0
\(169\) 1805.33 0.821726
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1239.91i 0.544905i −0.962169 0.272452i \(-0.912165\pi\)
0.962169 0.272452i \(-0.0878347\pi\)
\(174\) 0 0
\(175\) −467.438 + 1971.52i −0.201914 + 0.851615i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2636.86 1.10105 0.550525 0.834818i \(-0.314427\pi\)
0.550525 + 0.834818i \(0.314427\pi\)
\(180\) 0 0
\(181\) 3317.58 1.36240 0.681199 0.732099i \(-0.261459\pi\)
0.681199 + 0.732099i \(0.261459\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 423.141 3618.84i 0.168162 1.43818i
\(186\) 0 0
\(187\) 3338.01i 1.30534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −624.506 −0.236585 −0.118292 0.992979i \(-0.537742\pi\)
−0.118292 + 0.992979i \(0.537742\pi\)
\(192\) 0 0
\(193\) 436.144i 0.162665i 0.996687 + 0.0813324i \(0.0259175\pi\)
−0.996687 + 0.0813324i \(0.974082\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3355.81i 1.21366i −0.794831 0.606831i \(-0.792440\pi\)
0.794831 0.606831i \(-0.207560\pi\)
\(198\) 0 0
\(199\) 3799.77 1.35356 0.676780 0.736185i \(-0.263375\pi\)
0.676780 + 0.736185i \(0.263375\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 397.687i 0.137498i
\(204\) 0 0
\(205\) −2691.98 314.766i −0.917153 0.107240i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1963.72 0.649922
\(210\) 0 0
\(211\) −2365.27 −0.771715 −0.385857 0.922558i \(-0.626094\pi\)
−0.385857 + 0.922558i \(0.626094\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 476.981 4079.31i 0.151302 1.29398i
\(216\) 0 0
\(217\) 201.300i 0.0629730i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1642.93 −0.500070
\(222\) 0 0
\(223\) 3328.58i 0.999545i −0.866157 0.499772i \(-0.833417\pi\)
0.866157 0.499772i \(-0.166583\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 527.100i 0.154118i −0.997027 0.0770592i \(-0.975447\pi\)
0.997027 0.0770592i \(-0.0245530\pi\)
\(228\) 0 0
\(229\) −2566.06 −0.740479 −0.370240 0.928936i \(-0.620724\pi\)
−0.370240 + 0.928936i \(0.620724\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5534.99i 1.55626i 0.628101 + 0.778132i \(0.283832\pi\)
−0.628101 + 0.778132i \(0.716168\pi\)
\(234\) 0 0
\(235\) −265.587 + 2271.39i −0.0737234 + 0.630508i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1010.01 −0.273355 −0.136678 0.990616i \(-0.543642\pi\)
−0.136678 + 0.990616i \(0.543642\pi\)
\(240\) 0 0
\(241\) −4074.29 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −891.220 104.208i −0.232400 0.0271738i
\(246\) 0 0
\(247\) 966.525i 0.248982i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1773.98 0.446107 0.223054 0.974806i \(-0.428398\pi\)
0.223054 + 0.974806i \(0.428398\pi\)
\(252\) 0 0
\(253\) 64.8375i 0.0161119i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 662.784i 0.160869i −0.996760 0.0804345i \(-0.974369\pi\)
0.996760 0.0804345i \(-0.0256308\pi\)
\(258\) 0 0
\(259\) −5282.38 −1.26730
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 712.312i 0.167008i −0.996507 0.0835039i \(-0.973389\pi\)
0.996507 0.0835039i \(-0.0266111\pi\)
\(264\) 0 0
\(265\) −79.8904 + 683.250i −0.0185194 + 0.158384i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3136.41 −0.710894 −0.355447 0.934696i \(-0.615671\pi\)
−0.355447 + 0.934696i \(0.615671\pi\)
\(270\) 0 0
\(271\) 2275.69 0.510105 0.255053 0.966927i \(-0.417907\pi\)
0.255053 + 0.966927i \(0.417907\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4890.59 1159.54i −1.07241 0.254264i
\(276\) 0 0
\(277\) 5171.00i 1.12164i 0.827937 + 0.560821i \(0.189515\pi\)
−0.827937 + 0.560821i \(0.810485\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2240.14 −0.475571 −0.237785 0.971318i \(-0.576422\pi\)
−0.237785 + 0.971318i \(0.576422\pi\)
\(282\) 0 0
\(283\) 225.244i 0.0473123i 0.999720 + 0.0236561i \(0.00753068\pi\)
−0.999720 + 0.0236561i \(0.992469\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3929.46i 0.808183i
\(288\) 0 0
\(289\) −1978.59 −0.402726
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1139.86i 0.227274i 0.993522 + 0.113637i \(0.0362501\pi\)
−0.993522 + 0.113637i \(0.963750\pi\)
\(294\) 0 0
\(295\) −1246.05 145.697i −0.245925 0.0287553i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −31.9123 −0.00617237
\(300\) 0 0
\(301\) −5954.51 −1.14024
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5300.82 619.809i −0.995161 0.116361i
\(306\) 0 0
\(307\) 5244.86i 0.975049i −0.873109 0.487525i \(-0.837900\pi\)
0.873109 0.487525i \(-0.162100\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5188.26 −0.945977 −0.472989 0.881068i \(-0.656825\pi\)
−0.472989 + 0.881068i \(0.656825\pi\)
\(312\) 0 0
\(313\) 486.656i 0.0878832i −0.999034 0.0439416i \(-0.986008\pi\)
0.999034 0.0439416i \(-0.0139915\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4218.87i 0.747493i 0.927531 + 0.373747i \(0.121927\pi\)
−0.927531 + 0.373747i \(0.878073\pi\)
\(318\) 0 0
\(319\) 986.512 0.173148
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4054.27i 0.698408i
\(324\) 0 0
\(325\) 570.712 2407.10i 0.0974074 0.410836i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3315.53 0.555595
\(330\) 0 0
\(331\) −7439.94 −1.23546 −0.617728 0.786392i \(-0.711947\pi\)
−0.617728 + 0.786392i \(0.711947\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −724.650 + 6197.46i −0.118185 + 1.01076i
\(336\) 0 0
\(337\) 6555.39i 1.05963i −0.848113 0.529815i \(-0.822261\pi\)
0.848113 0.529815i \(-0.177739\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −499.350 −0.0793000
\(342\) 0 0
\(343\) 6860.72i 1.08001i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1950.56i 0.301763i 0.988552 + 0.150881i \(0.0482112\pi\)
−0.988552 + 0.150881i \(0.951789\pi\)
\(348\) 0 0
\(349\) 1426.74 0.218830 0.109415 0.993996i \(-0.465102\pi\)
0.109415 + 0.993996i \(0.465102\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7078.96i 1.06735i −0.845689 0.533676i \(-0.820810\pi\)
0.845689 0.533676i \(-0.179190\pi\)
\(354\) 0 0
\(355\) −6199.54 724.893i −0.926866 0.108376i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5409.79 0.795314 0.397657 0.917534i \(-0.369823\pi\)
0.397657 + 0.917534i \(0.369823\pi\)
\(360\) 0 0
\(361\) −4473.90 −0.652267
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1313.72 + 11235.4i −0.188392 + 1.61120i
\(366\) 0 0
\(367\) 4940.09i 0.702645i −0.936255 0.351322i \(-0.885732\pi\)
0.936255 0.351322i \(-0.114268\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 997.332 0.139566
\(372\) 0 0
\(373\) 12891.9i 1.78959i −0.446473 0.894797i \(-0.647320\pi\)
0.446473 0.894797i \(-0.352680\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 485.551i 0.0663320i
\(378\) 0 0
\(379\) −9475.15 −1.28418 −0.642092 0.766627i \(-0.721933\pi\)
−0.642092 + 0.766627i \(0.721933\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5800.97i 0.773931i 0.922094 + 0.386966i \(0.126477\pi\)
−0.922094 + 0.386966i \(0.873523\pi\)
\(384\) 0 0
\(385\) −846.281 + 7237.69i −0.112027 + 0.958095i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 13779.7 1.79603 0.898016 0.439962i \(-0.145008\pi\)
0.898016 + 0.439962i \(0.145008\pi\)
\(390\) 0 0
\(391\) −133.862 −0.0173138
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12775.6 1493.81i −1.62737 0.190283i
\(396\) 0 0
\(397\) 2816.46i 0.356056i −0.984025 0.178028i \(-0.943028\pi\)
0.984025 0.178028i \(-0.0569718\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11986.4 −1.49270 −0.746352 0.665551i \(-0.768196\pi\)
−0.746352 + 0.665551i \(0.768196\pi\)
\(402\) 0 0
\(403\) 245.775i 0.0303794i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13103.6i 1.59588i
\(408\) 0 0
\(409\) 3339.07 0.403683 0.201841 0.979418i \(-0.435307\pi\)
0.201841 + 0.979418i \(0.435307\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1818.84i 0.216706i
\(414\) 0 0
\(415\) 1503.49 12858.4i 0.177840 1.52095i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1688.52 −0.196873 −0.0984363 0.995143i \(-0.531384\pi\)
−0.0984363 + 0.995143i \(0.531384\pi\)
\(420\) 0 0
\(421\) −2664.27 −0.308429 −0.154214 0.988037i \(-0.549285\pi\)
−0.154214 + 0.988037i \(0.549285\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2393.96 10097.0i 0.273233 1.15242i
\(426\) 0 0
\(427\) 7737.54i 0.876923i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12266.0 1.37084 0.685420 0.728148i \(-0.259619\pi\)
0.685420 + 0.728148i \(0.259619\pi\)
\(432\) 0 0
\(433\) 15647.3i 1.73664i 0.496008 + 0.868318i \(0.334799\pi\)
−0.496008 + 0.868318i \(0.665201\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 78.7503i 0.00862045i
\(438\) 0 0
\(439\) 16131.0 1.75373 0.876867 0.480733i \(-0.159629\pi\)
0.876867 + 0.480733i \(0.159629\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10053.7i 1.07825i −0.842225 0.539127i \(-0.818754\pi\)
0.842225 0.539127i \(-0.181246\pi\)
\(444\) 0 0
\(445\) −1076.88 125.916i −0.114717 0.0134135i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7477.71 0.785957 0.392979 0.919548i \(-0.371445\pi\)
0.392979 + 0.919548i \(0.371445\pi\)
\(450\) 0 0
\(451\) −9747.51 −1.01772
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3562.31 416.531i −0.367041 0.0429170i
\(456\) 0 0
\(457\) 1363.46i 0.139562i 0.997562 + 0.0697812i \(0.0222301\pi\)
−0.997562 + 0.0697812i \(0.977770\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5276.77 −0.533109 −0.266555 0.963820i \(-0.585885\pi\)
−0.266555 + 0.963820i \(0.585885\pi\)
\(462\) 0 0
\(463\) 5740.02i 0.576159i 0.957607 + 0.288079i \(0.0930167\pi\)
−0.957607 + 0.288079i \(0.906983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6233.36i 0.617657i 0.951118 + 0.308828i \(0.0999368\pi\)
−0.951118 + 0.308828i \(0.900063\pi\)
\(468\) 0 0
\(469\) 9046.35 0.890664
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14770.9i 1.43587i
\(474\) 0 0
\(475\) −5940.01 1408.35i −0.573782 0.136041i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19688.2 1.87803 0.939013 0.343881i \(-0.111742\pi\)
0.939013 + 0.343881i \(0.111742\pi\)
\(480\) 0 0
\(481\) 6449.46 0.611372
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1496.29 + 12796.8i −0.140088 + 1.19808i
\(486\) 0 0
\(487\) 3955.08i 0.368012i 0.982925 + 0.184006i \(0.0589065\pi\)
−0.982925 + 0.184006i \(0.941093\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13893.5 −1.27699 −0.638497 0.769624i \(-0.720443\pi\)
−0.638497 + 0.769624i \(0.720443\pi\)
\(492\) 0 0
\(493\) 2036.74i 0.186065i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9049.39i 0.816741i
\(498\) 0 0
\(499\) 13523.7 1.21324 0.606618 0.794993i \(-0.292526\pi\)
0.606618 + 0.794993i \(0.292526\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13135.4i 1.16437i −0.813057 0.582184i \(-0.802198\pi\)
0.813057 0.582184i \(-0.197802\pi\)
\(504\) 0 0
\(505\) −12842.5 1501.63i −1.13165 0.132320i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2222.71 −0.193556 −0.0967778 0.995306i \(-0.530854\pi\)
−0.0967778 + 0.995306i \(0.530854\pi\)
\(510\) 0 0
\(511\) 16400.1 1.41976
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1731.72 + 14810.3i −0.148172 + 1.26722i
\(516\) 0 0
\(517\) 8224.57i 0.699645i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4916.42 0.413421 0.206710 0.978402i \(-0.433724\pi\)
0.206710 + 0.978402i \(0.433724\pi\)
\(522\) 0 0
\(523\) 17743.4i 1.48349i 0.670681 + 0.741746i \(0.266002\pi\)
−0.670681 + 0.741746i \(0.733998\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1030.95i 0.0852161i
\(528\) 0 0
\(529\) 12164.4 0.999786
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4797.62i 0.389884i
\(534\) 0 0
\(535\) −1036.49 + 8864.46i −0.0837599 + 0.716344i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3227.05 −0.257883
\(540\) 0 0
\(541\) −12671.3 −1.00699 −0.503495 0.863998i \(-0.667953\pi\)
−0.503495 + 0.863998i \(0.667953\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10939.4 1279.12i −0.859805 0.100534i
\(546\) 0 0
\(547\) 5250.90i 0.410443i −0.978716 0.205221i \(-0.934209\pi\)
0.978716 0.205221i \(-0.0657915\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1198.20 0.0926406
\(552\) 0 0
\(553\) 18648.4i 1.43401i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25830.2i 1.96492i −0.186465 0.982462i \(-0.559703\pi\)
0.186465 0.982462i \(-0.440297\pi\)
\(558\) 0 0
\(559\) 7270.09 0.550075
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2021.14i 0.151298i −0.997135 0.0756490i \(-0.975897\pi\)
0.997135 0.0756490i \(-0.0241029\pi\)
\(564\) 0 0
\(565\) 2451.77 20968.4i 0.182561 1.56132i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8706.51 0.641469 0.320734 0.947169i \(-0.396070\pi\)
0.320734 + 0.947169i \(0.396070\pi\)
\(570\) 0 0
\(571\) 12194.5 0.893740 0.446870 0.894599i \(-0.352539\pi\)
0.446870 + 0.894599i \(0.352539\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 46.5004 196.125i 0.00337252 0.0142243i
\(576\) 0 0
\(577\) 15264.0i 1.10130i −0.834737 0.550649i \(-0.814380\pi\)
0.834737 0.550649i \(-0.185620\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −18769.2 −1.34024
\(582\) 0 0
\(583\) 2474.01i 0.175751i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8456.89i 0.594639i −0.954778 0.297319i \(-0.903907\pi\)
0.954778 0.297319i \(-0.0960926\pi\)
\(588\) 0 0
\(589\) −606.500 −0.0424285
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1225.23i 0.0848467i −0.999100 0.0424234i \(-0.986492\pi\)
0.999100 0.0424234i \(-0.0135078\pi\)
\(594\) 0 0
\(595\) −14942.8 1747.22i −1.02957 0.120385i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16060.0 1.09548 0.547741 0.836648i \(-0.315488\pi\)
0.547741 + 0.836648i \(0.315488\pi\)
\(600\) 0 0
\(601\) 9699.93 0.658350 0.329175 0.944269i \(-0.393229\pi\)
0.329175 + 0.944269i \(0.393229\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3173.65 371.085i −0.213268 0.0249368i
\(606\) 0 0
\(607\) 23661.2i 1.58217i −0.611703 0.791087i \(-0.709515\pi\)
0.611703 0.791087i \(-0.290485\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4048.05 −0.268030
\(612\) 0 0
\(613\) 8085.63i 0.532749i 0.963870 + 0.266375i \(0.0858258\pi\)
−0.963870 + 0.266375i \(0.914174\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11035.1i 0.720029i 0.932947 + 0.360014i \(0.117228\pi\)
−0.932947 + 0.360014i \(0.882772\pi\)
\(618\) 0 0
\(619\) −16826.3 −1.09258 −0.546290 0.837596i \(-0.683960\pi\)
−0.546290 + 0.837596i \(0.683960\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1571.90i 0.101087i
\(624\) 0 0
\(625\) 13961.8 + 7014.90i 0.893555 + 0.448954i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 27053.5 1.71493
\(630\) 0 0
\(631\) 3705.91 0.233803 0.116902 0.993144i \(-0.462704\pi\)
0.116902 + 0.993144i \(0.462704\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 806.147 6894.45i 0.0503795 0.430863i
\(636\) 0 0
\(637\) 1588.32i 0.0987937i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24597.4 1.51566 0.757829 0.652453i \(-0.226260\pi\)
0.757829 + 0.652453i \(0.226260\pi\)
\(642\) 0 0
\(643\) 21479.5i 1.31737i −0.752419 0.658685i \(-0.771113\pi\)
0.752419 0.658685i \(-0.228887\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27119.7i 1.64789i 0.566668 + 0.823946i \(0.308232\pi\)
−0.566668 + 0.823946i \(0.691768\pi\)
\(648\) 0 0
\(649\) −4511.87 −0.272891
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18476.4i 1.10725i 0.832765 + 0.553627i \(0.186757\pi\)
−0.832765 + 0.553627i \(0.813243\pi\)
\(654\) 0 0
\(655\) 28742.8 + 3360.82i 1.71462 + 0.200485i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19273.5 1.13928 0.569641 0.821894i \(-0.307082\pi\)
0.569641 + 0.821894i \(0.307082\pi\)
\(660\) 0 0
\(661\) 25605.3 1.50670 0.753352 0.657618i \(-0.228436\pi\)
0.753352 + 0.657618i \(0.228436\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1027.88 + 8790.75i −0.0599388 + 0.512617i
\(666\) 0 0
\(667\) 39.5616i 0.00229660i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −19193.9 −1.10428
\(672\) 0 0
\(673\) 7855.52i 0.449938i −0.974366 0.224969i \(-0.927772\pi\)
0.974366 0.224969i \(-0.0722280\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6763.09i 0.383939i 0.981401 + 0.191970i \(0.0614875\pi\)
−0.981401 + 0.191970i \(0.938512\pi\)
\(678\) 0 0
\(679\) 18679.3 1.05574
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15608.6i 0.874447i −0.899353 0.437224i \(-0.855962\pi\)
0.899353 0.437224i \(-0.144038\pi\)
\(684\) 0 0
\(685\) 2150.59 18392.6i 0.119956 1.02590i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1217.68 −0.0673293
\(690\) 0 0
\(691\) −6203.15 −0.341504 −0.170752 0.985314i \(-0.554620\pi\)
−0.170752 + 0.985314i \(0.554620\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1701.86 + 198.994i 0.0928854 + 0.0108608i
\(696\) 0 0
\(697\) 20124.5i 1.09365i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 16507.9 0.889435 0.444718 0.895671i \(-0.353304\pi\)
0.444718 + 0.895671i \(0.353304\pi\)
\(702\) 0 0
\(703\) 15915.4i 0.853854i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18746.0i 0.997193i
\(708\) 0 0
\(709\) 25539.6 1.35283 0.676416 0.736520i \(-0.263532\pi\)
0.676416 + 0.736520i \(0.263532\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.0252i 0.00105182i
\(714\) 0 0
\(715\) 1033.26 8836.76i 0.0540442 0.462204i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7353.45 0.381415 0.190708 0.981647i \(-0.438922\pi\)
0.190708 + 0.981647i \(0.438922\pi\)
\(720\) 0 0
\(721\) 21618.4 1.11666
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2984.07 707.510i −0.152863 0.0362431i
\(726\) 0 0
\(727\) 21696.5i 1.10685i 0.832900 + 0.553424i \(0.186679\pi\)
−0.832900 + 0.553424i \(0.813321\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30495.8 1.54299
\(732\) 0 0
\(733\) 90.2714i 0.00454877i −0.999997 0.00227439i \(-0.999276\pi\)
0.999997 0.00227439i \(-0.000723960\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22440.6i 1.12159i
\(738\) 0 0
\(739\) 14273.1 0.710479 0.355239 0.934775i \(-0.384399\pi\)
0.355239 + 0.934775i \(0.384399\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15866.6i 0.783429i −0.920087 0.391715i \(-0.871882\pi\)
0.920087 0.391715i \(-0.128118\pi\)
\(744\) 0 0
\(745\) −16472.4 1926.07i −0.810072 0.0947193i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12939.3 0.631232
\(750\) 0 0
\(751\) −26776.9 −1.30107 −0.650534 0.759477i \(-0.725455\pi\)
−0.650534 + 0.759477i \(0.725455\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4378.37 511.950i −0.211053 0.0246778i
\(756\) 0 0
\(757\) 30478.0i 1.46333i 0.681663 + 0.731666i \(0.261257\pi\)
−0.681663 + 0.731666i \(0.738743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −29104.7 −1.38639 −0.693195 0.720750i \(-0.743798\pi\)
−0.693195 + 0.720750i \(0.743798\pi\)
\(762\) 0 0
\(763\) 15968.2i 0.757649i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2220.69i 0.104543i
\(768\) 0 0
\(769\) −4170.65 −0.195575 −0.0977876 0.995207i \(-0.531177\pi\)
−0.0977876 + 0.995207i \(0.531177\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17738.5i 0.825367i −0.910874 0.412684i \(-0.864591\pi\)
0.910874 0.412684i \(-0.135409\pi\)
\(774\) 0 0
\(775\) 1510.47 + 358.125i 0.0700099 + 0.0165990i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11839.1 −0.544519
\(780\) 0 0
\(781\) −22448.1 −1.02850
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2242.47 19178.4i 0.101958 0.871982i
\(786\) 0 0
\(787\) 3807.92i 0.172475i 0.996275 + 0.0862374i \(0.0274844\pi\)
−0.996275 + 0.0862374i \(0.972516\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −30607.3 −1.37582
\(792\) 0 0
\(793\) 9447.06i 0.423045i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23840.3i 1.