Properties

Label 1344.4.p.c.223.1
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.1
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -12.5168 q^{5} +(18.2936 + 2.88837i) q^{7} -9.00000 q^{9} -55.4008 q^{11} -92.1564 q^{13} +37.5504i q^{15} +118.220i q^{17} -155.587i q^{19} +(8.66510 - 54.8809i) q^{21} -125.358i q^{23} +31.6706 q^{25} +27.0000i q^{27} -131.643i q^{29} -66.0511 q^{31} +166.202i q^{33} +(-228.978 - 36.1532i) q^{35} +147.117i q^{37} +276.469i q^{39} +20.3097i q^{41} +355.210 q^{43} +112.651 q^{45} -79.5977 q^{47} +(326.315 + 105.678i) q^{49} +354.659 q^{51} +463.012i q^{53} +693.441 q^{55} -466.762 q^{57} +580.211i q^{59} -587.607 q^{61} +(-164.643 - 25.9953i) q^{63} +1153.50 q^{65} +496.198 q^{67} -376.073 q^{69} +232.774i q^{71} -551.640i q^{73} -95.0119i q^{75} +(-1013.48 - 160.018i) q^{77} -437.936i q^{79} +81.0000 q^{81} -191.266i q^{83} -1479.73i q^{85} -394.929 q^{87} +93.7894i q^{89} +(-1685.88 - 266.182i) q^{91} +198.153i q^{93} +1947.46i q^{95} -758.892i q^{97} +498.607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) −12.5168 −1.11954 −0.559769 0.828649i \(-0.689110\pi\)
−0.559769 + 0.828649i \(0.689110\pi\)
\(6\) 0 0
\(7\) 18.2936 + 2.88837i 0.987764 + 0.155957i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −55.4008 −1.51854 −0.759271 0.650774i \(-0.774444\pi\)
−0.759271 + 0.650774i \(0.774444\pi\)
\(12\) 0 0
\(13\) −92.1564 −1.96612 −0.983061 0.183276i \(-0.941330\pi\)
−0.983061 + 0.183276i \(0.941330\pi\)
\(14\) 0 0
\(15\) 37.5504i 0.646365i
\(16\) 0 0
\(17\) 118.220i 1.68662i 0.537430 + 0.843309i \(0.319395\pi\)
−0.537430 + 0.843309i \(0.680605\pi\)
\(18\) 0 0
\(19\) 155.587i 1.87864i −0.343041 0.939321i \(-0.611457\pi\)
0.343041 0.939321i \(-0.388543\pi\)
\(20\) 0 0
\(21\) 8.66510 54.8809i 0.0900419 0.570286i
\(22\) 0 0
\(23\) 125.358i 1.13647i −0.822865 0.568236i \(-0.807626\pi\)
0.822865 0.568236i \(-0.192374\pi\)
\(24\) 0 0
\(25\) 31.6706 0.253365
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 131.643i 0.842947i −0.906841 0.421474i \(-0.861513\pi\)
0.906841 0.421474i \(-0.138487\pi\)
\(30\) 0 0
\(31\) −66.0511 −0.382682 −0.191341 0.981524i \(-0.561284\pi\)
−0.191341 + 0.981524i \(0.561284\pi\)
\(32\) 0 0
\(33\) 166.202i 0.876731i
\(34\) 0 0
\(35\) −228.978 36.1532i −1.10584 0.174600i
\(36\) 0 0
\(37\) 147.117i 0.653671i 0.945081 + 0.326835i \(0.105982\pi\)
−0.945081 + 0.326835i \(0.894018\pi\)
\(38\) 0 0
\(39\) 276.469i 1.13514i
\(40\) 0 0
\(41\) 20.3097i 0.0773622i 0.999252 + 0.0386811i \(0.0123156\pi\)
−0.999252 + 0.0386811i \(0.987684\pi\)
\(42\) 0 0
\(43\) 355.210 1.25974 0.629872 0.776699i \(-0.283107\pi\)
0.629872 + 0.776699i \(0.283107\pi\)
\(44\) 0 0
\(45\) 112.651 0.373179
\(46\) 0 0
\(47\) −79.5977 −0.247032 −0.123516 0.992343i \(-0.539417\pi\)
−0.123516 + 0.992343i \(0.539417\pi\)
\(48\) 0 0
\(49\) 326.315 + 105.678i 0.951355 + 0.308098i
\(50\) 0 0
\(51\) 354.659 0.973769
\(52\) 0 0
\(53\) 463.012i 1.19999i 0.800003 + 0.599996i \(0.204831\pi\)
−0.800003 + 0.599996i \(0.795169\pi\)
\(54\) 0 0
\(55\) 693.441 1.70007
\(56\) 0 0
\(57\) −466.762 −1.08463
\(58\) 0 0
\(59\) 580.211i 1.28029i 0.768255 + 0.640144i \(0.221125\pi\)
−0.768255 + 0.640144i \(0.778875\pi\)
\(60\) 0 0
\(61\) −587.607 −1.23337 −0.616683 0.787212i \(-0.711524\pi\)
−0.616683 + 0.787212i \(0.711524\pi\)
\(62\) 0 0
\(63\) −164.643 25.9953i −0.329255 0.0519857i
\(64\) 0 0
\(65\) 1153.50 2.20115
\(66\) 0 0
\(67\) 496.198 0.904779 0.452390 0.891820i \(-0.350572\pi\)
0.452390 + 0.891820i \(0.350572\pi\)
\(68\) 0 0
\(69\) −376.073 −0.656143
\(70\) 0 0
\(71\) 232.774i 0.389087i 0.980894 + 0.194543i \(0.0623225\pi\)
−0.980894 + 0.194543i \(0.937677\pi\)
\(72\) 0 0
\(73\) 551.640i 0.884447i −0.896905 0.442223i \(-0.854190\pi\)
0.896905 0.442223i \(-0.145810\pi\)
\(74\) 0 0
\(75\) 95.0119i 0.146280i
\(76\) 0 0
\(77\) −1013.48 160.018i −1.49996 0.236828i
\(78\) 0 0
\(79\) 437.936i 0.623692i −0.950133 0.311846i \(-0.899053\pi\)
0.950133 0.311846i \(-0.100947\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 191.266i 0.252941i −0.991970 0.126471i \(-0.959635\pi\)
0.991970 0.126471i \(-0.0403650\pi\)
\(84\) 0 0
\(85\) 1479.73i 1.88823i
\(86\) 0 0
\(87\) −394.929 −0.486676
\(88\) 0 0
\(89\) 93.7894i 0.111704i 0.998439 + 0.0558520i \(0.0177875\pi\)
−0.998439 + 0.0558520i \(0.982213\pi\)
\(90\) 0 0
\(91\) −1685.88 266.182i −1.94207 0.306631i
\(92\) 0 0
\(93\) 198.153i 0.220941i
\(94\) 0 0
\(95\) 1947.46i 2.10321i
\(96\) 0 0
\(97\) 758.892i 0.794369i −0.917739 0.397184i \(-0.869987\pi\)
0.917739 0.397184i \(-0.130013\pi\)
\(98\) 0 0
\(99\) 498.607 0.506181
\(100\) 0 0
\(101\) 803.819 0.791910 0.395955 0.918270i \(-0.370414\pi\)
0.395955 + 0.918270i \(0.370414\pi\)
\(102\) 0 0
\(103\) −946.857 −0.905792 −0.452896 0.891563i \(-0.649609\pi\)
−0.452896 + 0.891563i \(0.649609\pi\)
\(104\) 0 0
\(105\) −108.459 + 686.934i −0.100805 + 0.638456i
\(106\) 0 0
\(107\) 1693.10 1.52970 0.764850 0.644208i \(-0.222813\pi\)
0.764850 + 0.644208i \(0.222813\pi\)
\(108\) 0 0
\(109\) 103.927i 0.0913251i −0.998957 0.0456626i \(-0.985460\pi\)
0.998957 0.0456626i \(-0.0145399\pi\)
\(110\) 0 0
\(111\) 441.350 0.377397
\(112\) 0 0
\(113\) −17.0288 −0.0141764 −0.00708819 0.999975i \(-0.502256\pi\)
−0.00708819 + 0.999975i \(0.502256\pi\)
\(114\) 0 0
\(115\) 1569.08i 1.27232i
\(116\) 0 0
\(117\) 829.408 0.655374
\(118\) 0 0
\(119\) −341.462 + 2162.67i −0.263040 + 1.66598i
\(120\) 0 0
\(121\) 1738.25 1.30597
\(122\) 0 0
\(123\) 60.9292 0.0446651
\(124\) 0 0
\(125\) 1168.19 0.835886
\(126\) 0 0
\(127\) 481.156i 0.336187i 0.985771 + 0.168093i \(0.0537610\pi\)
−0.985771 + 0.168093i \(0.946239\pi\)
\(128\) 0 0
\(129\) 1065.63i 0.727314i
\(130\) 0 0
\(131\) 1599.97i 1.06710i 0.845769 + 0.533550i \(0.179142\pi\)
−0.845769 + 0.533550i \(0.820858\pi\)
\(132\) 0 0
\(133\) 449.393 2846.26i 0.292988 1.85565i
\(134\) 0 0
\(135\) 337.954i 0.215455i
\(136\) 0 0
\(137\) −867.738 −0.541138 −0.270569 0.962701i \(-0.587212\pi\)
−0.270569 + 0.962701i \(0.587212\pi\)
\(138\) 0 0
\(139\) 773.391i 0.471929i 0.971762 + 0.235965i \(0.0758249\pi\)
−0.971762 + 0.235965i \(0.924175\pi\)
\(140\) 0 0
\(141\) 238.793i 0.142624i
\(142\) 0 0
\(143\) 5105.54 2.98564
\(144\) 0 0
\(145\) 1647.75i 0.943711i
\(146\) 0 0
\(147\) 317.033 978.944i 0.177880 0.549265i
\(148\) 0 0
\(149\) 1566.08i 0.861063i 0.902576 + 0.430531i \(0.141674\pi\)
−0.902576 + 0.430531i \(0.858326\pi\)
\(150\) 0 0
\(151\) 168.569i 0.0908474i −0.998968 0.0454237i \(-0.985536\pi\)
0.998968 0.0454237i \(-0.0144638\pi\)
\(152\) 0 0
\(153\) 1063.98i 0.562206i
\(154\) 0 0
\(155\) 826.750 0.428427
\(156\) 0 0
\(157\) 1378.04 0.700506 0.350253 0.936655i \(-0.386096\pi\)
0.350253 + 0.936655i \(0.386096\pi\)
\(158\) 0 0
\(159\) 1389.03 0.692815
\(160\) 0 0
\(161\) 362.079 2293.25i 0.177241 1.12257i
\(162\) 0 0
\(163\) 2078.67 0.998858 0.499429 0.866355i \(-0.333543\pi\)
0.499429 + 0.866355i \(0.333543\pi\)
\(164\) 0 0
\(165\) 2080.32i 0.981534i
\(166\) 0 0
\(167\) −388.656 −0.180090 −0.0900452 0.995938i \(-0.528701\pi\)
−0.0900452 + 0.995938i \(0.528701\pi\)
\(168\) 0 0
\(169\) 6295.81 2.86564
\(170\) 0 0
\(171\) 1400.29i 0.626214i
\(172\) 0 0
\(173\) −2510.22 −1.10317 −0.551586 0.834118i \(-0.685977\pi\)
−0.551586 + 0.834118i \(0.685977\pi\)
\(174\) 0 0
\(175\) 579.371 + 91.4764i 0.250265 + 0.0395141i
\(176\) 0 0
\(177\) 1740.63 0.739175
\(178\) 0 0
\(179\) 1287.12 0.537450 0.268725 0.963217i \(-0.413398\pi\)
0.268725 + 0.963217i \(0.413398\pi\)
\(180\) 0 0
\(181\) 2880.43 1.18288 0.591438 0.806351i \(-0.298561\pi\)
0.591438 + 0.806351i \(0.298561\pi\)
\(182\) 0 0
\(183\) 1762.82i 0.712084i
\(184\) 0 0
\(185\) 1841.43i 0.731809i
\(186\) 0 0
\(187\) 6549.47i 2.56120i
\(188\) 0 0
\(189\) −77.9859 + 493.928i −0.0300140 + 0.190095i
\(190\) 0 0
\(191\) 1264.81i 0.479155i 0.970877 + 0.239577i \(0.0770089\pi\)
−0.970877 + 0.239577i \(0.922991\pi\)
\(192\) 0 0
\(193\) 3174.10 1.18382 0.591908 0.806005i \(-0.298375\pi\)
0.591908 + 0.806005i \(0.298375\pi\)
\(194\) 0 0
\(195\) 3460.51i 1.27083i
\(196\) 0 0
\(197\) 2153.82i 0.778951i 0.921037 + 0.389475i \(0.127344\pi\)
−0.921037 + 0.389475i \(0.872656\pi\)
\(198\) 0 0
\(199\) −5051.78 −1.79955 −0.899776 0.436352i \(-0.856270\pi\)
−0.899776 + 0.436352i \(0.856270\pi\)
\(200\) 0 0
\(201\) 1488.59i 0.522375i
\(202\) 0 0
\(203\) 380.233 2408.23i 0.131464 0.832633i
\(204\) 0 0
\(205\) 254.213i 0.0866099i
\(206\) 0 0
\(207\) 1128.22i 0.378824i
\(208\) 0 0
\(209\) 8619.66i 2.85280i
\(210\) 0 0
\(211\) 1946.49 0.635080 0.317540 0.948245i \(-0.397143\pi\)
0.317540 + 0.948245i \(0.397143\pi\)
\(212\) 0 0
\(213\) 698.321 0.224639
\(214\) 0 0
\(215\) −4446.10 −1.41033
\(216\) 0 0
\(217\) −1208.32 190.780i −0.377999 0.0596820i
\(218\) 0 0
\(219\) −1654.92 −0.510636
\(220\) 0 0
\(221\) 10894.7i 3.31610i
\(222\) 0 0
\(223\) 901.888 0.270829 0.135415 0.990789i \(-0.456763\pi\)
0.135415 + 0.990789i \(0.456763\pi\)
\(224\) 0 0
\(225\) −285.036 −0.0844550
\(226\) 0 0
\(227\) 4975.14i 1.45468i −0.686278 0.727339i \(-0.740757\pi\)
0.686278 0.727339i \(-0.259243\pi\)
\(228\) 0 0
\(229\) −3689.85 −1.06477 −0.532384 0.846503i \(-0.678704\pi\)
−0.532384 + 0.846503i \(0.678704\pi\)
\(230\) 0 0
\(231\) −480.054 + 3040.45i −0.136733 + 0.866003i
\(232\) 0 0
\(233\) 3285.97 0.923909 0.461954 0.886904i \(-0.347148\pi\)
0.461954 + 0.886904i \(0.347148\pi\)
\(234\) 0 0
\(235\) 996.309 0.276562
\(236\) 0 0
\(237\) −1313.81 −0.360089
\(238\) 0 0
\(239\) 2662.25i 0.720529i 0.932850 + 0.360264i \(0.117314\pi\)
−0.932850 + 0.360264i \(0.882686\pi\)
\(240\) 0 0
\(241\) 494.863i 0.132269i 0.997811 + 0.0661347i \(0.0210667\pi\)
−0.997811 + 0.0661347i \(0.978933\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −4084.42 1322.75i −1.06508 0.344927i
\(246\) 0 0
\(247\) 14338.4i 3.69364i
\(248\) 0 0
\(249\) −573.797 −0.146036
\(250\) 0 0
\(251\) 4465.24i 1.12288i 0.827516 + 0.561442i \(0.189753\pi\)
−0.827516 + 0.561442i \(0.810247\pi\)
\(252\) 0 0
\(253\) 6944.91i 1.72578i
\(254\) 0 0
\(255\) −4439.20 −1.09017
\(256\) 0 0
\(257\) 3874.34i 0.940369i −0.882568 0.470185i \(-0.844187\pi\)
0.882568 0.470185i \(-0.155813\pi\)
\(258\) 0 0
\(259\) −424.927 + 2691.30i −0.101945 + 0.645672i
\(260\) 0 0
\(261\) 1184.79i 0.280982i
\(262\) 0 0
\(263\) 4291.91i 1.00628i −0.864206 0.503138i \(-0.832179\pi\)
0.864206 0.503138i \(-0.167821\pi\)
\(264\) 0 0
\(265\) 5795.43i 1.34344i
\(266\) 0 0
\(267\) 281.368 0.0644923
\(268\) 0 0
\(269\) −4934.08 −1.11835 −0.559175 0.829050i \(-0.688882\pi\)
−0.559175 + 0.829050i \(0.688882\pi\)
\(270\) 0 0
\(271\) 3796.02 0.850893 0.425446 0.904984i \(-0.360117\pi\)
0.425446 + 0.904984i \(0.360117\pi\)
\(272\) 0 0
\(273\) −798.545 + 5057.63i −0.177033 + 1.12125i
\(274\) 0 0
\(275\) −1754.58 −0.384745
\(276\) 0 0
\(277\) 6822.56i 1.47988i −0.672671 0.739942i \(-0.734853\pi\)
0.672671 0.739942i \(-0.265147\pi\)
\(278\) 0 0
\(279\) 594.460 0.127561
\(280\) 0 0
\(281\) 7537.89 1.60026 0.800129 0.599827i \(-0.204764\pi\)
0.800129 + 0.599827i \(0.204764\pi\)
\(282\) 0 0
\(283\) 1945.91i 0.408736i −0.978894 0.204368i \(-0.934486\pi\)
0.978894 0.204368i \(-0.0655139\pi\)
\(284\) 0 0
\(285\) 5842.37 1.21429
\(286\) 0 0
\(287\) −58.6620 + 371.539i −0.0120652 + 0.0764155i
\(288\) 0 0
\(289\) −9062.90 −1.84468
\(290\) 0 0
\(291\) −2276.67 −0.458629
\(292\) 0 0
\(293\) −4658.84 −0.928916 −0.464458 0.885595i \(-0.653751\pi\)
−0.464458 + 0.885595i \(0.653751\pi\)
\(294\) 0 0
\(295\) 7262.39i 1.43333i
\(296\) 0 0
\(297\) 1495.82i 0.292244i
\(298\) 0 0
\(299\) 11552.5i 2.23444i
\(300\) 0 0
\(301\) 6498.09 + 1025.98i 1.24433 + 0.196466i
\(302\) 0 0
\(303\) 2411.46i 0.457210i
\(304\) 0 0
\(305\) 7354.96 1.38080
\(306\) 0 0
\(307\) 3147.04i 0.585053i −0.956257 0.292526i \(-0.905504\pi\)
0.956257 0.292526i \(-0.0944959\pi\)
\(308\) 0 0
\(309\) 2840.57i 0.522959i
\(310\) 0 0
\(311\) 9786.74 1.78442 0.892211 0.451619i \(-0.149153\pi\)
0.892211 + 0.451619i \(0.149153\pi\)
\(312\) 0 0
\(313\) 7053.24i 1.27372i −0.770981 0.636858i \(-0.780234\pi\)
0.770981 0.636858i \(-0.219766\pi\)
\(314\) 0 0
\(315\) 2060.80 + 325.378i 0.368613 + 0.0582000i
\(316\) 0 0
\(317\) 7111.48i 1.26000i −0.776595 0.630001i \(-0.783055\pi\)
0.776595 0.630001i \(-0.216945\pi\)
\(318\) 0 0
\(319\) 7293.12i 1.28005i
\(320\) 0 0
\(321\) 5079.29i 0.883173i
\(322\) 0 0
\(323\) 18393.5 3.16855
\(324\) 0 0
\(325\) −2918.65 −0.498147
\(326\) 0 0
\(327\) −311.782 −0.0527266
\(328\) 0 0
\(329\) −1456.13 229.907i −0.244009 0.0385265i
\(330\) 0 0
\(331\) −4383.51 −0.727914 −0.363957 0.931416i \(-0.618575\pi\)
−0.363957 + 0.931416i \(0.618575\pi\)
\(332\) 0 0
\(333\) 1324.05i 0.217890i
\(334\) 0 0
\(335\) −6210.82 −1.01293
\(336\) 0 0
\(337\) 3696.74 0.597549 0.298775 0.954324i \(-0.403422\pi\)
0.298775 + 0.954324i \(0.403422\pi\)
\(338\) 0 0
\(339\) 51.0863i 0.00818474i
\(340\) 0 0
\(341\) 3659.29 0.581119
\(342\) 0 0
\(343\) 5664.25 + 2875.74i 0.891664 + 0.452698i
\(344\) 0 0
\(345\) 4707.23 0.734577
\(346\) 0 0
\(347\) −6007.60 −0.929408 −0.464704 0.885466i \(-0.653839\pi\)
−0.464704 + 0.885466i \(0.653839\pi\)
\(348\) 0 0
\(349\) 1376.99 0.211199 0.105600 0.994409i \(-0.466324\pi\)
0.105600 + 0.994409i \(0.466324\pi\)
\(350\) 0 0
\(351\) 2488.22i 0.378381i
\(352\) 0 0
\(353\) 3533.89i 0.532833i 0.963858 + 0.266417i \(0.0858397\pi\)
−0.963858 + 0.266417i \(0.914160\pi\)
\(354\) 0 0
\(355\) 2913.59i 0.435597i
\(356\) 0 0
\(357\) 6488.01 + 1024.39i 0.961854 + 0.151866i
\(358\) 0 0
\(359\) 4792.38i 0.704547i −0.935897 0.352273i \(-0.885409\pi\)
0.935897 0.352273i \(-0.114591\pi\)
\(360\) 0 0
\(361\) −17348.4 −2.52929
\(362\) 0 0
\(363\) 5214.75i 0.754003i
\(364\) 0 0
\(365\) 6904.78i 0.990172i
\(366\) 0 0
\(367\) −10145.7 −1.44305 −0.721526 0.692387i \(-0.756559\pi\)
−0.721526 + 0.692387i \(0.756559\pi\)
\(368\) 0 0
\(369\) 182.788i 0.0257874i
\(370\) 0 0
\(371\) −1337.35 + 8470.17i −0.187147 + 1.18531i
\(372\) 0 0
\(373\) 8427.27i 1.16983i 0.811094 + 0.584916i \(0.198873\pi\)
−0.811094 + 0.584916i \(0.801127\pi\)
\(374\) 0 0
\(375\) 3504.56i 0.482599i
\(376\) 0 0
\(377\) 12131.7i 1.65734i
\(378\) 0 0
\(379\) 6309.37 0.855121 0.427560 0.903987i \(-0.359373\pi\)
0.427560 + 0.903987i \(0.359373\pi\)
\(380\) 0 0
\(381\) 1443.47 0.194098
\(382\) 0 0
\(383\) 6889.51 0.919158 0.459579 0.888137i \(-0.348000\pi\)
0.459579 + 0.888137i \(0.348000\pi\)
\(384\) 0 0
\(385\) 12685.6 + 2002.91i 1.67926 + 0.265138i
\(386\) 0 0
\(387\) −3196.89 −0.419915
\(388\) 0 0
\(389\) 3025.39i 0.394327i 0.980371 + 0.197164i \(0.0631730\pi\)
−0.980371 + 0.197164i \(0.936827\pi\)
\(390\) 0 0
\(391\) 14819.7 1.91679
\(392\) 0 0
\(393\) 4799.91 0.616090
\(394\) 0 0
\(395\) 5481.56i 0.698247i
\(396\) 0 0
\(397\) −1565.33 −0.197889 −0.0989444 0.995093i \(-0.531547\pi\)
−0.0989444 + 0.995093i \(0.531547\pi\)
\(398\) 0 0
\(399\) −8538.78 1348.18i −1.07136 0.169156i
\(400\) 0 0
\(401\) 2687.03 0.334623 0.167312 0.985904i \(-0.446491\pi\)
0.167312 + 0.985904i \(0.446491\pi\)
\(402\) 0 0
\(403\) 6087.04 0.752399
\(404\) 0 0
\(405\) −1013.86 −0.124393
\(406\) 0 0
\(407\) 8150.38i 0.992627i
\(408\) 0 0
\(409\) 11421.7i 1.38084i 0.723407 + 0.690422i \(0.242575\pi\)
−0.723407 + 0.690422i \(0.757425\pi\)
\(410\) 0 0
\(411\) 2603.21i 0.312426i
\(412\) 0 0
\(413\) −1675.86 + 10614.2i −0.199670 + 1.26462i
\(414\) 0 0
\(415\) 2394.04i 0.283178i
\(416\) 0 0
\(417\) 2320.17 0.272468
\(418\) 0 0
\(419\) 5192.62i 0.605432i 0.953081 + 0.302716i \(0.0978934\pi\)
−0.953081 + 0.302716i \(0.902107\pi\)
\(420\) 0 0
\(421\) 11047.1i 1.27887i −0.768846 0.639434i \(-0.779169\pi\)
0.768846 0.639434i \(-0.220831\pi\)
\(422\) 0 0
\(423\) 716.379 0.0823441
\(424\) 0 0
\(425\) 3744.09i 0.427330i
\(426\) 0 0
\(427\) −10749.5 1697.22i −1.21827 0.192352i
\(428\) 0 0
\(429\) 15316.6i 1.72376i
\(430\) 0 0
\(431\) 7402.32i 0.827279i 0.910441 + 0.413640i \(0.135743\pi\)
−0.910441 + 0.413640i \(0.864257\pi\)
\(432\) 0 0
\(433\) 304.730i 0.0338208i 0.999857 + 0.0169104i \(0.00538300\pi\)
−0.999857 + 0.0169104i \(0.994617\pi\)
\(434\) 0 0
\(435\) 4943.25 0.544852
\(436\) 0 0
\(437\) −19504.1 −2.13502
\(438\) 0 0
\(439\) −7752.82 −0.842875 −0.421437 0.906857i \(-0.638474\pi\)
−0.421437 + 0.906857i \(0.638474\pi\)
\(440\) 0 0
\(441\) −2936.83 951.098i −0.317118 0.102699i
\(442\) 0 0
\(443\) −14980.0 −1.60660 −0.803298 0.595577i \(-0.796923\pi\)
−0.803298 + 0.595577i \(0.796923\pi\)
\(444\) 0 0
\(445\) 1173.94i 0.125057i
\(446\) 0 0
\(447\) 4698.24 0.497135
\(448\) 0 0
\(449\) −17523.6 −1.84185 −0.920925 0.389740i \(-0.872565\pi\)
−0.920925 + 0.389740i \(0.872565\pi\)
\(450\) 0 0
\(451\) 1125.18i 0.117478i
\(452\) 0 0
\(453\) −505.707 −0.0524508
\(454\) 0 0
\(455\) 21101.8 + 3331.75i 2.17422 + 0.343285i
\(456\) 0 0
\(457\) −3264.38 −0.334139 −0.167069 0.985945i \(-0.553430\pi\)
−0.167069 + 0.985945i \(0.553430\pi\)
\(458\) 0 0
\(459\) −3191.93 −0.324590
\(460\) 0 0
\(461\) 4096.53 0.413870 0.206935 0.978355i \(-0.433651\pi\)
0.206935 + 0.978355i \(0.433651\pi\)
\(462\) 0 0
\(463\) 12281.3i 1.23275i −0.787455 0.616373i \(-0.788602\pi\)
0.787455 0.616373i \(-0.211398\pi\)
\(464\) 0 0
\(465\) 2480.25i 0.247352i
\(466\) 0 0
\(467\) 14919.7i 1.47838i 0.673499 + 0.739188i \(0.264791\pi\)
−0.673499 + 0.739188i \(0.735209\pi\)
\(468\) 0 0
\(469\) 9077.26 + 1433.20i 0.893708 + 0.141107i
\(470\) 0 0
\(471\) 4134.12i 0.404438i
\(472\) 0 0
\(473\) −19678.9 −1.91298
\(474\) 0 0
\(475\) 4927.55i 0.475982i
\(476\) 0 0
\(477\) 4167.10i 0.399997i
\(478\) 0 0
\(479\) 3253.39 0.310337 0.155168 0.987888i \(-0.450408\pi\)
0.155168 + 0.987888i \(0.450408\pi\)
\(480\) 0 0
\(481\) 13557.7i 1.28520i
\(482\) 0 0
\(483\) −6879.74 1086.24i −0.648114 0.102330i
\(484\) 0 0
\(485\) 9498.90i 0.889326i
\(486\) 0 0
\(487\) 367.609i 0.0342053i 0.999854 + 0.0171026i \(0.00544420\pi\)
−0.999854 + 0.0171026i \(0.994556\pi\)
\(488\) 0 0
\(489\) 6236.01i 0.576691i
\(490\) 0 0
\(491\) −8656.19 −0.795618 −0.397809 0.917468i \(-0.630229\pi\)
−0.397809 + 0.917468i \(0.630229\pi\)
\(492\) 0 0
\(493\) 15562.8 1.42173
\(494\) 0 0
\(495\) −6240.97 −0.566689
\(496\) 0 0
\(497\) −672.336 + 4258.28i −0.0606809 + 0.384326i
\(498\) 0 0
\(499\) 14675.5 1.31656 0.658280 0.752773i \(-0.271284\pi\)
0.658280 + 0.752773i \(0.271284\pi\)
\(500\) 0 0
\(501\) 1165.97i 0.103975i
\(502\) 0 0
\(503\) 4451.92 0.394635 0.197317 0.980340i \(-0.436777\pi\)
0.197317 + 0.980340i \(0.436777\pi\)
\(504\) 0 0
\(505\) −10061.2 −0.886573
\(506\) 0 0
\(507\) 18887.4i 1.65448i
\(508\) 0 0
\(509\) −17750.3 −1.54572 −0.772858 0.634579i \(-0.781174\pi\)
−0.772858 + 0.634579i \(0.781174\pi\)
\(510\) 0 0
\(511\) 1593.34 10091.5i 0.137936 0.873624i
\(512\) 0 0
\(513\) 4200.86 0.361545
\(514\) 0 0
\(515\) 11851.6 1.01407
\(516\) 0 0
\(517\) 4409.78 0.375129
\(518\) 0 0
\(519\) 7530.67i 0.636916i
\(520\) 0 0
\(521\) 8202.49i 0.689746i −0.938649 0.344873i \(-0.887922\pi\)
0.938649 0.344873i \(-0.112078\pi\)
\(522\) 0 0
\(523\) 14583.3i 1.21928i 0.792678 + 0.609641i \(0.208686\pi\)
−0.792678 + 0.609641i \(0.791314\pi\)
\(524\) 0 0
\(525\) 274.429 1738.11i 0.0228135 0.144490i
\(526\) 0 0
\(527\) 7808.55i 0.645438i
\(528\) 0 0
\(529\) −3547.53 −0.291570
\(530\) 0 0
\(531\) 5221.90i 0.426763i
\(532\) 0 0
\(533\) 1871.67i 0.152104i
\(534\) 0 0
\(535\) −21192.2 −1.71256
\(536\) 0 0
\(537\) 3861.35i 0.310297i
\(538\) 0 0
\(539\) −18078.1 5854.62i −1.44467 0.467860i
\(540\) 0 0
\(541\) 18146.0i 1.44207i 0.692900 + 0.721033i \(0.256333\pi\)
−0.692900 + 0.721033i \(0.743667\pi\)
\(542\) 0 0
\(543\) 8641.28i 0.682933i
\(544\) 0 0
\(545\) 1300.84i 0.102242i
\(546\) 0 0
\(547\) −10223.9 −0.799164 −0.399582 0.916698i \(-0.630845\pi\)
−0.399582 + 0.916698i \(0.630845\pi\)
\(548\) 0 0
\(549\) 5288.46 0.411122
\(550\) 0 0
\(551\) −20482.0 −1.58360
\(552\) 0 0
\(553\) 1264.92 8011.44i 0.0972692 0.616060i
\(554\) 0 0
\(555\) −5524.29 −0.422510
\(556\) 0 0
\(557\) 18894.8i 1.43734i 0.695352 + 0.718669i \(0.255248\pi\)
−0.695352 + 0.718669i \(0.744752\pi\)
\(558\) 0 0
\(559\) −32734.9 −2.47681
\(560\) 0 0
\(561\) −19648.4 −1.47871
\(562\) 0 0
\(563\) 4811.00i 0.360141i 0.983654 + 0.180071i \(0.0576326\pi\)
−0.983654 + 0.180071i \(0.942367\pi\)
\(564\) 0 0
\(565\) 213.146 0.0158710
\(566\) 0 0
\(567\) 1481.78 + 233.958i 0.109752 + 0.0173286i
\(568\) 0 0
\(569\) 15811.2 1.16492 0.582460 0.812859i \(-0.302090\pi\)
0.582460 + 0.812859i \(0.302090\pi\)
\(570\) 0 0
\(571\) −12377.3 −0.907131 −0.453566 0.891223i \(-0.649848\pi\)
−0.453566 + 0.891223i \(0.649848\pi\)
\(572\) 0 0
\(573\) 3794.43 0.276640
\(574\) 0 0
\(575\) 3970.15i 0.287942i
\(576\) 0 0
\(577\) 17225.6i 1.24283i 0.783484 + 0.621413i \(0.213441\pi\)
−0.783484 + 0.621413i \(0.786559\pi\)
\(578\) 0 0
\(579\) 9522.30i 0.683477i
\(580\) 0 0
\(581\) 552.446 3498.95i 0.0394480 0.249846i
\(582\) 0 0
\(583\) 25651.2i 1.82224i
\(584\) 0 0
\(585\) −10381.5 −0.733716
\(586\) 0 0
\(587\) 17524.8i 1.23224i 0.787651 + 0.616121i \(0.211297\pi\)
−0.787651 + 0.616121i \(0.788703\pi\)
\(588\) 0 0
\(589\) 10276.7i 0.718922i
\(590\) 0 0
\(591\) 6461.46 0.449727
\(592\) 0 0
\(593\) 17080.1i 1.18279i 0.806382 + 0.591394i \(0.201422\pi\)
−0.806382 + 0.591394i \(0.798578\pi\)
\(594\) 0 0
\(595\) 4274.02 27069.7i 0.294483 1.86513i
\(596\) 0 0
\(597\) 15155.3i 1.03897i
\(598\) 0 0
\(599\) 18593.0i 1.26826i −0.773225 0.634132i \(-0.781358\pi\)
0.773225 0.634132i \(-0.218642\pi\)
\(600\) 0 0
\(601\) 22600.9i 1.53396i 0.641670 + 0.766981i \(0.278242\pi\)
−0.641670 + 0.766981i \(0.721758\pi\)
\(602\) 0 0
\(603\) −4465.78 −0.301593
\(604\) 0 0
\(605\) −21757.3 −1.46209
\(606\) 0 0
\(607\) −7870.14 −0.526259 −0.263130 0.964761i \(-0.584755\pi\)
−0.263130 + 0.964761i \(0.584755\pi\)
\(608\) 0 0
\(609\) −7224.68 1140.70i −0.480721 0.0759006i
\(610\) 0 0
\(611\) 7335.44 0.485696
\(612\) 0 0
\(613\) 5829.16i 0.384074i −0.981388 0.192037i \(-0.938491\pi\)
0.981388 0.192037i \(-0.0615094\pi\)
\(614\) 0 0
\(615\) −762.640 −0.0500042
\(616\) 0 0
\(617\) 3132.87 0.204416 0.102208 0.994763i \(-0.467409\pi\)
0.102208 + 0.994763i \(0.467409\pi\)
\(618\) 0 0
\(619\) 1967.51i 0.127756i 0.997958 + 0.0638779i \(0.0203468\pi\)
−0.997958 + 0.0638779i \(0.979653\pi\)
\(620\) 0 0
\(621\) 3384.66 0.218714
\(622\) 0 0
\(623\) −270.898 + 1715.75i −0.0174210 + 0.110337i
\(624\) 0 0
\(625\) −18580.8 −1.18917
\(626\) 0 0
\(627\) 25859.0 1.64706
\(628\) 0 0
\(629\) −17392.1 −1.10249
\(630\) 0 0
\(631\) 12821.2i 0.808883i 0.914564 + 0.404441i \(0.132534\pi\)
−0.914564 + 0.404441i \(0.867466\pi\)
\(632\) 0 0
\(633\) 5839.47i 0.366664i
\(634\) 0 0
\(635\) 6022.54i 0.376374i
\(636\) 0 0
\(637\) −30072.0 9738.86i −1.87048 0.605758i
\(638\) 0 0
\(639\) 2094.96i 0.129696i
\(640\) 0 0
\(641\) 7100.25 0.437509 0.218754 0.975780i \(-0.429801\pi\)
0.218754 + 0.975780i \(0.429801\pi\)
\(642\) 0 0
\(643\) 22717.2i 1.39328i −0.717421 0.696640i \(-0.754677\pi\)
0.717421 0.696640i \(-0.245323\pi\)
\(644\) 0 0
\(645\) 13338.3i 0.814256i
\(646\) 0 0
\(647\) 333.262 0.0202502 0.0101251 0.999949i \(-0.496777\pi\)
0.0101251 + 0.999949i \(0.496777\pi\)
\(648\) 0 0
\(649\) 32144.1i 1.94417i
\(650\) 0 0
\(651\) −572.340 + 3624.95i −0.0344574 + 0.218238i
\(652\) 0 0
\(653\) 27002.5i 1.61821i 0.587666 + 0.809103i \(0.300047\pi\)
−0.587666 + 0.809103i \(0.699953\pi\)
\(654\) 0 0
\(655\) 20026.5i 1.19466i
\(656\) 0 0
\(657\) 4964.76i 0.294816i
\(658\) 0 0
\(659\) −17790.1 −1.05160 −0.525799 0.850609i \(-0.676234\pi\)
−0.525799 + 0.850609i \(0.676234\pi\)
\(660\) 0 0
\(661\) 28922.6 1.70190 0.850951 0.525246i \(-0.176027\pi\)
0.850951 + 0.525246i \(0.176027\pi\)
\(662\) 0 0
\(663\) −32684.1 −1.91455
\(664\) 0 0
\(665\) −5624.97 + 35626.1i −0.328011 + 2.07747i
\(666\) 0 0
\(667\) −16502.4 −0.957987
\(668\) 0 0
\(669\) 2705.66i 0.156363i
\(670\) 0 0
\(671\) 32553.9 1.87292
\(672\) 0 0
\(673\) −9655.76 −0.553049 −0.276525 0.961007i \(-0.589183\pi\)
−0.276525 + 0.961007i \(0.589183\pi\)
\(674\) 0 0
\(675\) 855.107i 0.0487601i
\(676\) 0 0
\(677\) −97.6916 −0.00554593 −0.00277296 0.999996i \(-0.500883\pi\)
−0.00277296 + 0.999996i \(0.500883\pi\)
\(678\) 0 0
\(679\) 2191.96 13882.9i 0.123888 0.784649i
\(680\) 0 0
\(681\) −14925.4 −0.839859
\(682\) 0 0
\(683\) 9136.04 0.511831 0.255916 0.966699i \(-0.417623\pi\)
0.255916 + 0.966699i \(0.417623\pi\)
\(684\) 0 0
\(685\) 10861.3 0.605824
\(686\) 0 0
\(687\) 11069.5i 0.614744i
\(688\) 0 0
\(689\) 42669.5i 2.35933i
\(690\) 0 0
\(691\) 20090.0i 1.10602i −0.833174 0.553011i \(-0.813479\pi\)
0.833174 0.553011i \(-0.186521\pi\)
\(692\) 0 0
\(693\) 9121.34 + 1440.16i 0.499987 + 0.0789426i
\(694\) 0 0
\(695\) 9680.39i 0.528342i
\(696\) 0 0
\(697\) −2401.01 −0.130480
\(698\) 0 0
\(699\) 9857.90i 0.533419i
\(700\) 0 0
\(701\) 28733.6i 1.54815i −0.633095 0.774074i \(-0.718216\pi\)
0.633095 0.774074i \(-0.281784\pi\)
\(702\) 0 0
\(703\) 22889.5 1.22801
\(704\) 0 0
\(705\) 2988.93i 0.159673i
\(706\) 0 0
\(707\) 14704.8 + 2321.72i 0.782220 + 0.123504i
\(708\) 0 0
\(709\) 1873.43i 0.0992360i −0.998768 0.0496180i \(-0.984200\pi\)
0.998768 0.0496180i \(-0.0158004\pi\)
\(710\) 0 0
\(711\) 3941.42i 0.207897i
\(712\) 0 0
\(713\) 8280.01i 0.434907i
\(714\) 0 0
\(715\) −63905.1 −3.34254
\(716\) 0 0
\(717\) 7986.74 0.415998
\(718\) 0 0
\(719\) −2073.02 −0.107525 −0.0537625 0.998554i \(-0.517121\pi\)
−0.0537625 + 0.998554i \(0.517121\pi\)
\(720\) 0 0
\(721\) −17321.5 2734.87i −0.894709 0.141265i
\(722\) 0 0
\(723\) 1484.59 0.0763658
\(724\) 0 0
\(725\) 4169.21i 0.213573i
\(726\) 0 0
\(727\) 26274.3 1.34038 0.670191 0.742188i \(-0.266212\pi\)
0.670191 + 0.742188i \(0.266212\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 41992.8i 2.12471i
\(732\) 0 0
\(733\) 18164.4 0.915304 0.457652 0.889131i \(-0.348691\pi\)
0.457652 + 0.889131i \(0.348691\pi\)
\(734\) 0 0
\(735\) −3968.24 + 12253.3i −0.199144 + 0.614923i
\(736\) 0 0
\(737\) −27489.8 −1.37395
\(738\) 0 0
\(739\) 9438.47 0.469824 0.234912 0.972017i \(-0.424520\pi\)
0.234912 + 0.972017i \(0.424520\pi\)
\(740\) 0 0
\(741\) 43015.1 2.13252
\(742\) 0 0
\(743\) 8531.76i 0.421265i 0.977565 + 0.210633i \(0.0675524\pi\)
−0.977565 + 0.210633i \(0.932448\pi\)
\(744\) 0 0
\(745\) 19602.3i 0.963992i
\(746\) 0 0
\(747\) 1721.39i 0.0843138i
\(748\) 0 0
\(749\) 30972.9 + 4890.29i 1.51098 + 0.238568i
\(750\) 0 0
\(751\) 20258.1i 0.984325i 0.870503 + 0.492163i \(0.163794\pi\)
−0.870503 + 0.492163i \(0.836206\pi\)
\(752\) 0 0
\(753\) 13395.7 0.648297
\(754\) 0 0
\(755\) 2109.95i 0.101707i
\(756\) 0 0
\(757\) 30742.9i 1.47605i −0.674772 0.738026i \(-0.735758\pi\)
0.674772 0.738026i \(-0.264242\pi\)
\(758\) 0 0
\(759\) 20834.7 0.996381
\(760\) 0 0
\(761\) 22602.8i 1.07667i 0.842730 + 0.538337i \(0.180947\pi\)
−0.842730 + 0.538337i \(0.819053\pi\)
\(762\) 0 0
\(763\) 300.181 1901.21i 0.0142428 0.0902077i
\(764\) 0 0
\(765\) 13317.6i 0.629411i
\(766\) 0 0
\(767\) 53470.2i 2.51720i
\(768\) 0 0
\(769\) 6329.77i 0.296823i 0.988926 + 0.148412i \(0.0474161\pi\)
−0.988926 + 0.148412i \(0.952584\pi\)
\(770\) 0 0
\(771\) −11623.0 −0.542922
\(772\) 0 0
\(773\) 13945.2 0.648867 0.324433 0.945909i \(-0.394826\pi\)
0.324433 + 0.945909i \(0.394826\pi\)
\(774\) 0 0
\(775\) −2091.88 −0.0969581
\(776\) 0 0
\(777\) 8073.89 + 1274.78i 0.372779 + 0.0588578i
\(778\) 0 0
\(779\) 3159.94 0.145336
\(780\) 0 0
\(781\) 12895.9i 0.590845i
\(782\) 0 0
\(783\) 3554.36 0.162225
\(784\) 0 0
\(785\) −17248.7 −0.784243
\(786\) 0 0
\(787\) 38929.4i 1.76326i 0.471946 + 0.881628i \(0.343552\pi\)
−0.471946 + 0.881628i \(0.656448\pi\)
\(788\) 0 0
\(789\) −12875.7 −0.580973
\(790\) 0 0
\(791\) −311.518 49.1853i −0.0140029 0.00221091i
\(792\) 0 0
\(793\) 54151.7 2.42495
\(794\) 0 0
\(795\) −17386.3 −0.775633
\(796\) 0 0
\(797\) 5612.71 0.249451 0.124726 0.992191i \(-0.460195\pi\)
0.124726 + 0.992191i \(0.460195\pi\)
\(798\) 0 0
\(799\) 9410.02i 0.416649i
\(800\) 0 0
\(801\) 844.104i 0.0372346i
\(802\) 0