Properties

Label 1344.4.p.c.223.23
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 223.23
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.24

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +1.64166 q^{5} +(-15.2747 + 10.4729i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +1.64166 q^{5} +(-15.2747 + 10.4729i) q^{7} -9.00000 q^{9} +20.3470 q^{11} +13.0999 q^{13} -4.92497i q^{15} +23.9396i q^{17} -87.7589i q^{19} +(31.4188 + 45.8242i) q^{21} +73.6175i q^{23} -122.305 q^{25} +27.0000i q^{27} -58.9537i q^{29} +124.909 q^{31} -61.0410i q^{33} +(-25.0759 + 17.1930i) q^{35} +56.5972i q^{37} -39.2998i q^{39} +135.651i q^{41} +259.929 q^{43} -14.7749 q^{45} -217.682 q^{47} +(123.635 - 319.943i) q^{49} +71.8188 q^{51} +529.342i q^{53} +33.4028 q^{55} -263.277 q^{57} -685.329i q^{59} +149.916 q^{61} +(137.473 - 94.2565i) q^{63} +21.5056 q^{65} -409.156 q^{67} +220.853 q^{69} -885.869i q^{71} -269.426i q^{73} +366.915i q^{75} +(-310.795 + 213.093i) q^{77} -902.587i q^{79} +81.0000 q^{81} -623.963i q^{83} +39.3006i q^{85} -176.861 q^{87} -986.131i q^{89} +(-200.098 + 137.195i) q^{91} -374.726i q^{93} -144.070i q^{95} -179.437i q^{97} -183.123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

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
\(5\) 1.64166 0.146834 0.0734171 0.997301i \(-0.476610\pi\)
0.0734171 + 0.997301i \(0.476610\pi\)
\(6\) 0 0
\(7\) −15.2747 + 10.4729i −0.824758 + 0.565486i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 20.3470 0.557714 0.278857 0.960333i \(-0.410044\pi\)
0.278857 + 0.960333i \(0.410044\pi\)
\(12\) 0 0
\(13\) 13.0999 0.279482 0.139741 0.990188i \(-0.455373\pi\)
0.139741 + 0.990188i \(0.455373\pi\)
\(14\) 0 0
\(15\) 4.92497i 0.0847748i
\(16\) 0 0
\(17\) 23.9396i 0.341542i 0.985311 + 0.170771i \(0.0546258\pi\)
−0.985311 + 0.170771i \(0.945374\pi\)
\(18\) 0 0
\(19\) 87.7589i 1.05965i −0.848108 0.529823i \(-0.822258\pi\)
0.848108 0.529823i \(-0.177742\pi\)
\(20\) 0 0
\(21\) 31.4188 + 45.8242i 0.326483 + 0.476174i
\(22\) 0 0
\(23\) 73.6175i 0.667405i 0.942678 + 0.333703i \(0.108298\pi\)
−0.942678 + 0.333703i \(0.891702\pi\)
\(24\) 0 0
\(25\) −122.305 −0.978440
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 58.9537i 0.377498i −0.982025 0.188749i \(-0.939557\pi\)
0.982025 0.188749i \(-0.0604432\pi\)
\(30\) 0 0
\(31\) 124.909 0.723686 0.361843 0.932239i \(-0.382148\pi\)
0.361843 + 0.932239i \(0.382148\pi\)
\(32\) 0 0
\(33\) 61.0410i 0.321996i
\(34\) 0 0
\(35\) −25.0759 + 17.1930i −0.121103 + 0.0830327i
\(36\) 0 0
\(37\) 56.5972i 0.251474i 0.992064 + 0.125737i \(0.0401295\pi\)
−0.992064 + 0.125737i \(0.959871\pi\)
\(38\) 0 0
\(39\) 39.2998i 0.161359i
\(40\) 0 0
\(41\) 135.651i 0.516711i 0.966050 + 0.258356i \(0.0831806\pi\)
−0.966050 + 0.258356i \(0.916819\pi\)
\(42\) 0 0
\(43\) 259.929 0.921833 0.460917 0.887443i \(-0.347521\pi\)
0.460917 + 0.887443i \(0.347521\pi\)
\(44\) 0 0
\(45\) −14.7749 −0.0489448
\(46\) 0 0
\(47\) −217.682 −0.675580 −0.337790 0.941222i \(-0.609679\pi\)
−0.337790 + 0.941222i \(0.609679\pi\)
\(48\) 0 0
\(49\) 123.635 319.943i 0.360452 0.932778i
\(50\) 0 0
\(51\) 71.8188 0.197189
\(52\) 0 0
\(53\) 529.342i 1.37190i 0.727649 + 0.685950i \(0.240613\pi\)
−0.727649 + 0.685950i \(0.759387\pi\)
\(54\) 0 0
\(55\) 33.4028 0.0818916
\(56\) 0 0
\(57\) −263.277 −0.611787
\(58\) 0 0
\(59\) 685.329i 1.51224i −0.654433 0.756120i \(-0.727092\pi\)
0.654433 0.756120i \(-0.272908\pi\)
\(60\) 0 0
\(61\) 149.916 0.314668 0.157334 0.987545i \(-0.449710\pi\)
0.157334 + 0.987545i \(0.449710\pi\)
\(62\) 0 0
\(63\) 137.473 94.2565i 0.274919 0.188495i
\(64\) 0 0
\(65\) 21.5056 0.0410376
\(66\) 0 0
\(67\) −409.156 −0.746066 −0.373033 0.927818i \(-0.621682\pi\)
−0.373033 + 0.927818i \(0.621682\pi\)
\(68\) 0 0
\(69\) 220.853 0.385326
\(70\) 0 0
\(71\) 885.869i 1.48075i −0.672194 0.740375i \(-0.734648\pi\)
0.672194 0.740375i \(-0.265352\pi\)
\(72\) 0 0
\(73\) 269.426i 0.431971i −0.976397 0.215986i \(-0.930704\pi\)
0.976397 0.215986i \(-0.0692964\pi\)
\(74\) 0 0
\(75\) 366.915i 0.564902i
\(76\) 0 0
\(77\) −310.795 + 213.093i −0.459979 + 0.315379i
\(78\) 0 0
\(79\) 902.587i 1.28543i −0.766105 0.642715i \(-0.777808\pi\)
0.766105 0.642715i \(-0.222192\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 623.963i 0.825166i −0.910920 0.412583i \(-0.864627\pi\)
0.910920 0.412583i \(-0.135373\pi\)
\(84\) 0 0
\(85\) 39.3006i 0.0501500i
\(86\) 0 0
\(87\) −176.861 −0.217948
\(88\) 0 0
\(89\) 986.131i 1.17449i −0.809409 0.587245i \(-0.800212\pi\)
0.809409 0.587245i \(-0.199788\pi\)
\(90\) 0 0
\(91\) −200.098 + 137.195i −0.230505 + 0.158043i
\(92\) 0 0
\(93\) 374.726i 0.417821i
\(94\) 0 0
\(95\) 144.070i 0.155592i
\(96\) 0 0
\(97\) 179.437i 0.187825i −0.995580 0.0939127i \(-0.970063\pi\)
0.995580 0.0939127i \(-0.0299375\pi\)
\(98\) 0 0
\(99\) −183.123 −0.185905
\(100\) 0 0
\(101\) −266.959 −0.263004 −0.131502 0.991316i \(-0.541980\pi\)
−0.131502 + 0.991316i \(0.541980\pi\)
\(102\) 0 0
\(103\) 988.258 0.945398 0.472699 0.881224i \(-0.343280\pi\)
0.472699 + 0.881224i \(0.343280\pi\)
\(104\) 0 0
\(105\) 51.5789 + 75.2276i 0.0479389 + 0.0699187i
\(106\) 0 0
\(107\) −1115.52 −1.00787 −0.503934 0.863742i \(-0.668114\pi\)
−0.503934 + 0.863742i \(0.668114\pi\)
\(108\) 0 0
\(109\) 286.196i 0.251492i −0.992062 0.125746i \(-0.959868\pi\)
0.992062 0.125746i \(-0.0401324\pi\)
\(110\) 0 0
\(111\) 169.792 0.145188
\(112\) 0 0
\(113\) −728.322 −0.606326 −0.303163 0.952939i \(-0.598043\pi\)
−0.303163 + 0.952939i \(0.598043\pi\)
\(114\) 0 0
\(115\) 120.855i 0.0979979i
\(116\) 0 0
\(117\) −117.899 −0.0931607
\(118\) 0 0
\(119\) −250.718 365.671i −0.193137 0.281689i
\(120\) 0 0
\(121\) −916.999 −0.688955
\(122\) 0 0
\(123\) 406.953 0.298323
\(124\) 0 0
\(125\) −405.990 −0.290503
\(126\) 0 0
\(127\) 159.234i 0.111258i 0.998452 + 0.0556288i \(0.0177163\pi\)
−0.998452 + 0.0556288i \(0.982284\pi\)
\(128\) 0 0
\(129\) 779.787i 0.532221i
\(130\) 0 0
\(131\) 748.323i 0.499094i −0.968363 0.249547i \(-0.919718\pi\)
0.968363 0.249547i \(-0.0802817\pi\)
\(132\) 0 0
\(133\) 919.093 + 1340.49i 0.599214 + 0.873951i
\(134\) 0 0
\(135\) 44.3247i 0.0282583i
\(136\) 0 0
\(137\) −1189.48 −0.741780 −0.370890 0.928677i \(-0.620947\pi\)
−0.370890 + 0.928677i \(0.620947\pi\)
\(138\) 0 0
\(139\) 546.158i 0.333270i −0.986019 0.166635i \(-0.946710\pi\)
0.986019 0.166635i \(-0.0532901\pi\)
\(140\) 0 0
\(141\) 653.047i 0.390046i
\(142\) 0 0
\(143\) 266.545 0.155871
\(144\) 0 0
\(145\) 96.7818i 0.0554296i
\(146\) 0 0
\(147\) −959.828 370.905i −0.538539 0.208107i
\(148\) 0 0
\(149\) 2169.41i 1.19279i 0.802693 + 0.596393i \(0.203400\pi\)
−0.802693 + 0.596393i \(0.796600\pi\)
\(150\) 0 0
\(151\) 1873.28i 1.00957i −0.863244 0.504786i \(-0.831571\pi\)
0.863244 0.504786i \(-0.168429\pi\)
\(152\) 0 0
\(153\) 215.457i 0.113847i
\(154\) 0 0
\(155\) 205.057 0.106262
\(156\) 0 0
\(157\) −2457.07 −1.24902 −0.624508 0.781018i \(-0.714701\pi\)
−0.624508 + 0.781018i \(0.714701\pi\)
\(158\) 0 0
\(159\) 1588.02 0.792066
\(160\) 0 0
\(161\) −770.992 1124.49i −0.377408 0.550448i
\(162\) 0 0
\(163\) −1184.76 −0.569308 −0.284654 0.958630i \(-0.591879\pi\)
−0.284654 + 0.958630i \(0.591879\pi\)
\(164\) 0 0
\(165\) 100.208i 0.0472801i
\(166\) 0 0
\(167\) 2493.54 1.15543 0.577713 0.816240i \(-0.303945\pi\)
0.577713 + 0.816240i \(0.303945\pi\)
\(168\) 0 0
\(169\) −2025.39 −0.921890
\(170\) 0 0
\(171\) 789.830i 0.353215i
\(172\) 0 0
\(173\) 336.961 0.148085 0.0740424 0.997255i \(-0.476410\pi\)
0.0740424 + 0.997255i \(0.476410\pi\)
\(174\) 0 0
\(175\) 1868.18 1280.89i 0.806976 0.553294i
\(176\) 0 0
\(177\) −2055.99 −0.873092
\(178\) 0 0
\(179\) 403.536 0.168501 0.0842506 0.996445i \(-0.473150\pi\)
0.0842506 + 0.996445i \(0.473150\pi\)
\(180\) 0 0
\(181\) −626.262 −0.257180 −0.128590 0.991698i \(-0.541045\pi\)
−0.128590 + 0.991698i \(0.541045\pi\)
\(182\) 0 0
\(183\) 449.747i 0.181674i
\(184\) 0 0
\(185\) 92.9132i 0.0369249i
\(186\) 0 0
\(187\) 487.100i 0.190483i
\(188\) 0 0
\(189\) −282.769 412.418i −0.108828 0.158725i
\(190\) 0 0
\(191\) 696.358i 0.263805i −0.991263 0.131902i \(-0.957891\pi\)
0.991263 0.131902i \(-0.0421085\pi\)
\(192\) 0 0
\(193\) −3330.70 −1.24222 −0.621111 0.783723i \(-0.713318\pi\)
−0.621111 + 0.783723i \(0.713318\pi\)
\(194\) 0 0
\(195\) 64.5168i 0.0236930i
\(196\) 0 0
\(197\) 841.118i 0.304199i 0.988365 + 0.152099i \(0.0486034\pi\)
−0.988365 + 0.152099i \(0.951397\pi\)
\(198\) 0 0
\(199\) −719.574 −0.256328 −0.128164 0.991753i \(-0.540908\pi\)
−0.128164 + 0.991753i \(0.540908\pi\)
\(200\) 0 0
\(201\) 1227.47i 0.430741i
\(202\) 0 0
\(203\) 617.419 + 900.503i 0.213470 + 0.311344i
\(204\) 0 0
\(205\) 222.693i 0.0758709i
\(206\) 0 0
\(207\) 662.558i 0.222468i
\(208\) 0 0
\(209\) 1785.63i 0.590979i
\(210\) 0 0
\(211\) −4957.51 −1.61748 −0.808742 0.588163i \(-0.799851\pi\)
−0.808742 + 0.588163i \(0.799851\pi\)
\(212\) 0 0
\(213\) −2657.61 −0.854912
\(214\) 0 0
\(215\) 426.715 0.135357
\(216\) 0 0
\(217\) −1907.95 + 1308.16i −0.596866 + 0.409234i
\(218\) 0 0
\(219\) −808.277 −0.249399
\(220\) 0 0
\(221\) 313.607i 0.0954548i
\(222\) 0 0
\(223\) 3947.42 1.18538 0.592688 0.805432i \(-0.298067\pi\)
0.592688 + 0.805432i \(0.298067\pi\)
\(224\) 0 0
\(225\) 1100.74 0.326147
\(226\) 0 0
\(227\) 2044.37i 0.597751i −0.954292 0.298876i \(-0.903388\pi\)
0.954292 0.298876i \(-0.0966115\pi\)
\(228\) 0 0
\(229\) −4391.90 −1.26736 −0.633679 0.773596i \(-0.718456\pi\)
−0.633679 + 0.773596i \(0.718456\pi\)
\(230\) 0 0
\(231\) 639.279 + 932.386i 0.182084 + 0.265569i
\(232\) 0 0
\(233\) −1970.16 −0.553946 −0.276973 0.960878i \(-0.589331\pi\)
−0.276973 + 0.960878i \(0.589331\pi\)
\(234\) 0 0
\(235\) −357.360 −0.0991983
\(236\) 0 0
\(237\) −2707.76 −0.742144
\(238\) 0 0
\(239\) 2964.89i 0.802437i −0.915982 0.401219i \(-0.868587\pi\)
0.915982 0.401219i \(-0.131413\pi\)
\(240\) 0 0
\(241\) 3658.31i 0.977810i −0.872337 0.488905i \(-0.837396\pi\)
0.872337 0.488905i \(-0.162604\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 202.966 525.236i 0.0529267 0.136964i
\(246\) 0 0
\(247\) 1149.64i 0.296152i
\(248\) 0 0
\(249\) −1871.89 −0.476410
\(250\) 0 0
\(251\) 2719.35i 0.683840i −0.939729 0.341920i \(-0.888923\pi\)
0.939729 0.341920i \(-0.111077\pi\)
\(252\) 0 0
\(253\) 1497.90i 0.372221i
\(254\) 0 0
\(255\) 117.902 0.0289541
\(256\) 0 0
\(257\) 6798.11i 1.65002i −0.565120 0.825009i \(-0.691170\pi\)
0.565120 0.825009i \(-0.308830\pi\)
\(258\) 0 0
\(259\) −592.739 864.507i −0.142205 0.207405i
\(260\) 0 0
\(261\) 530.584i 0.125833i
\(262\) 0 0
\(263\) 7965.94i 1.86768i −0.357686 0.933842i \(-0.616434\pi\)
0.357686 0.933842i \(-0.383566\pi\)
\(264\) 0 0
\(265\) 868.997i 0.201442i
\(266\) 0 0
\(267\) −2958.39 −0.678092
\(268\) 0 0
\(269\) 5718.64 1.29618 0.648089 0.761565i \(-0.275569\pi\)
0.648089 + 0.761565i \(0.275569\pi\)
\(270\) 0 0
\(271\) −1289.85 −0.289124 −0.144562 0.989496i \(-0.546177\pi\)
−0.144562 + 0.989496i \(0.546177\pi\)
\(272\) 0 0
\(273\) 411.584 + 600.294i 0.0912462 + 0.133082i
\(274\) 0 0
\(275\) −2488.54 −0.545690
\(276\) 0 0
\(277\) 4121.16i 0.893922i −0.894553 0.446961i \(-0.852506\pi\)
0.894553 0.446961i \(-0.147494\pi\)
\(278\) 0 0
\(279\) −1124.18 −0.241229
\(280\) 0 0
\(281\) −2432.92 −0.516497 −0.258248 0.966079i \(-0.583145\pi\)
−0.258248 + 0.966079i \(0.583145\pi\)
\(282\) 0 0
\(283\) 0.386542i 8.11927e-5i 1.00000 4.05963e-5i \(1.29222e-5\pi\)
−1.00000 4.05963e-5i \(0.999987\pi\)
\(284\) 0 0
\(285\) −432.210 −0.0898312
\(286\) 0 0
\(287\) −1420.67 2072.04i −0.292193 0.426162i
\(288\) 0 0
\(289\) 4339.89 0.883349
\(290\) 0 0
\(291\) −538.311 −0.108441
\(292\) 0 0
\(293\) 3394.41 0.676805 0.338402 0.941002i \(-0.390114\pi\)
0.338402 + 0.941002i \(0.390114\pi\)
\(294\) 0 0
\(295\) 1125.07i 0.222049i
\(296\) 0 0
\(297\) 549.369i 0.107332i
\(298\) 0 0
\(299\) 964.385i 0.186528i
\(300\) 0 0
\(301\) −3970.35 + 2722.22i −0.760289 + 0.521283i
\(302\) 0 0
\(303\) 800.877i 0.151845i
\(304\) 0 0
\(305\) 246.110 0.0462040
\(306\) 0 0
\(307\) 1720.47i 0.319844i 0.987130 + 0.159922i \(0.0511243\pi\)
−0.987130 + 0.159922i \(0.948876\pi\)
\(308\) 0 0
\(309\) 2964.77i 0.545826i
\(310\) 0 0
\(311\) 4721.33 0.860843 0.430422 0.902628i \(-0.358365\pi\)
0.430422 + 0.902628i \(0.358365\pi\)
\(312\) 0 0
\(313\) 483.679i 0.0873455i −0.999046 0.0436728i \(-0.986094\pi\)
0.999046 0.0436728i \(-0.0139059\pi\)
\(314\) 0 0
\(315\) 225.683 154.737i 0.0403676 0.0276776i
\(316\) 0 0
\(317\) 6443.81i 1.14170i 0.821053 + 0.570852i \(0.193387\pi\)
−0.821053 + 0.570852i \(0.806613\pi\)
\(318\) 0 0
\(319\) 1199.53i 0.210536i
\(320\) 0 0
\(321\) 3346.57i 0.581893i
\(322\) 0 0
\(323\) 2100.91 0.361913
\(324\) 0 0
\(325\) −1602.19 −0.273456
\(326\) 0 0
\(327\) −858.588 −0.145199
\(328\) 0 0
\(329\) 3325.04 2279.78i 0.557190 0.382031i
\(330\) 0 0
\(331\) −2784.85 −0.462444 −0.231222 0.972901i \(-0.574272\pi\)
−0.231222 + 0.972901i \(0.574272\pi\)
\(332\) 0 0
\(333\) 509.375i 0.0838245i
\(334\) 0 0
\(335\) −671.694 −0.109548
\(336\) 0 0
\(337\) −4181.48 −0.675904 −0.337952 0.941163i \(-0.609734\pi\)
−0.337952 + 0.941163i \(0.609734\pi\)
\(338\) 0 0
\(339\) 2184.97i 0.350062i
\(340\) 0 0
\(341\) 2541.52 0.403610
\(342\) 0 0
\(343\) 1462.25 + 6181.86i 0.230187 + 0.973146i
\(344\) 0 0
\(345\) 362.564 0.0565791
\(346\) 0 0
\(347\) 9524.25 1.47345 0.736727 0.676190i \(-0.236370\pi\)
0.736727 + 0.676190i \(0.236370\pi\)
\(348\) 0 0
\(349\) 1248.33 0.191465 0.0957327 0.995407i \(-0.469481\pi\)
0.0957327 + 0.995407i \(0.469481\pi\)
\(350\) 0 0
\(351\) 353.698i 0.0537864i
\(352\) 0 0
\(353\) 605.680i 0.0913233i −0.998957 0.0456616i \(-0.985460\pi\)
0.998957 0.0456616i \(-0.0145396\pi\)
\(354\) 0 0
\(355\) 1454.29i 0.217425i
\(356\) 0 0
\(357\) −1097.01 + 752.154i −0.162633 + 0.111508i
\(358\) 0 0
\(359\) 83.1817i 0.0122289i 0.999981 + 0.00611443i \(0.00194630\pi\)
−0.999981 + 0.00611443i \(0.998054\pi\)
\(360\) 0 0
\(361\) −842.618 −0.122849
\(362\) 0 0
\(363\) 2751.00i 0.397768i
\(364\) 0 0
\(365\) 442.305i 0.0634282i
\(366\) 0 0
\(367\) 4636.32 0.659439 0.329719 0.944079i \(-0.393046\pi\)
0.329719 + 0.944079i \(0.393046\pi\)
\(368\) 0 0
\(369\) 1220.86i 0.172237i
\(370\) 0 0
\(371\) −5543.76 8085.55i −0.775789 1.13148i
\(372\) 0 0
\(373\) 2555.59i 0.354754i 0.984143 + 0.177377i \(0.0567612\pi\)
−0.984143 + 0.177377i \(0.943239\pi\)
\(374\) 0 0
\(375\) 1217.97i 0.167722i
\(376\) 0 0
\(377\) 772.290i 0.105504i
\(378\) 0 0
\(379\) 8981.28 1.21725 0.608624 0.793459i \(-0.291722\pi\)
0.608624 + 0.793459i \(0.291722\pi\)
\(380\) 0 0
\(381\) 477.701 0.0642346
\(382\) 0 0
\(383\) −12164.5 −1.62291 −0.811456 0.584413i \(-0.801325\pi\)
−0.811456 + 0.584413i \(0.801325\pi\)
\(384\) 0 0
\(385\) −510.219 + 349.826i −0.0675407 + 0.0463085i
\(386\) 0 0
\(387\) −2339.36 −0.307278
\(388\) 0 0
\(389\) 3569.55i 0.465252i 0.972566 + 0.232626i \(0.0747319\pi\)
−0.972566 + 0.232626i \(0.925268\pi\)
\(390\) 0 0
\(391\) −1762.38 −0.227947
\(392\) 0 0
\(393\) −2244.97 −0.288152
\(394\) 0 0
\(395\) 1481.74i 0.188745i
\(396\) 0 0
\(397\) −984.957 −0.124518 −0.0622589 0.998060i \(-0.519830\pi\)
−0.0622589 + 0.998060i \(0.519830\pi\)
\(398\) 0 0
\(399\) 4021.48 2757.28i 0.504576 0.345957i
\(400\) 0 0
\(401\) 1700.17 0.211727 0.105863 0.994381i \(-0.466239\pi\)
0.105863 + 0.994381i \(0.466239\pi\)
\(402\) 0 0
\(403\) 1636.30 0.202257
\(404\) 0 0
\(405\) 132.974 0.0163149
\(406\) 0 0
\(407\) 1151.58i 0.140250i
\(408\) 0 0
\(409\) 2739.61i 0.331210i 0.986192 + 0.165605i \(0.0529577\pi\)
−0.986192 + 0.165605i \(0.947042\pi\)
\(410\) 0 0
\(411\) 3568.43i 0.428267i
\(412\) 0 0
\(413\) 7177.41 + 10468.2i 0.855150 + 1.24723i
\(414\) 0 0
\(415\) 1024.33i 0.121163i
\(416\) 0 0
\(417\) −1638.47 −0.192413
\(418\) 0 0
\(419\) 13461.7i 1.56957i −0.619770 0.784783i \(-0.712774\pi\)
0.619770 0.784783i \(-0.287226\pi\)
\(420\) 0 0
\(421\) 5618.17i 0.650387i −0.945647 0.325194i \(-0.894571\pi\)
0.945647 0.325194i \(-0.105429\pi\)
\(422\) 0 0
\(423\) 1959.14 0.225193
\(424\) 0 0
\(425\) 2927.93i 0.334178i
\(426\) 0 0
\(427\) −2289.92 + 1570.06i −0.259525 + 0.177940i
\(428\) 0 0
\(429\) 799.634i 0.0899923i
\(430\) 0 0
\(431\) 7137.27i 0.797657i 0.917026 + 0.398828i \(0.130583\pi\)
−0.917026 + 0.398828i \(0.869417\pi\)
\(432\) 0 0
\(433\) 1458.21i 0.161841i 0.996721 + 0.0809203i \(0.0257859\pi\)
−0.996721 + 0.0809203i \(0.974214\pi\)
\(434\) 0 0
\(435\) −290.346 −0.0320023
\(436\) 0 0
\(437\) 6460.59 0.707213
\(438\) 0 0
\(439\) −3231.27 −0.351299 −0.175650 0.984453i \(-0.556203\pi\)
−0.175650 + 0.984453i \(0.556203\pi\)
\(440\) 0 0
\(441\) −1112.72 + 2879.48i −0.120151 + 0.310926i
\(442\) 0 0
\(443\) 17504.9 1.87739 0.938694 0.344751i \(-0.112037\pi\)
0.938694 + 0.344751i \(0.112037\pi\)
\(444\) 0 0
\(445\) 1618.89i 0.172455i
\(446\) 0 0
\(447\) 6508.23 0.688655
\(448\) 0 0
\(449\) 9069.04 0.953217 0.476608 0.879116i \(-0.341866\pi\)
0.476608 + 0.879116i \(0.341866\pi\)
\(450\) 0 0
\(451\) 2760.10i 0.288177i
\(452\) 0 0
\(453\) −5619.84 −0.582877
\(454\) 0 0
\(455\) −328.492 + 225.227i −0.0338461 + 0.0232062i
\(456\) 0 0
\(457\) −1722.16 −0.176279 −0.0881394 0.996108i \(-0.528092\pi\)
−0.0881394 + 0.996108i \(0.528092\pi\)
\(458\) 0 0
\(459\) −646.370 −0.0657297
\(460\) 0 0
\(461\) 14064.0 1.42088 0.710442 0.703755i \(-0.248495\pi\)
0.710442 + 0.703755i \(0.248495\pi\)
\(462\) 0 0
\(463\) 7092.58i 0.711923i 0.934501 + 0.355961i \(0.115847\pi\)
−0.934501 + 0.355961i \(0.884153\pi\)
\(464\) 0 0
\(465\) 615.172i 0.0613504i
\(466\) 0 0
\(467\) 8397.55i 0.832103i −0.909341 0.416052i \(-0.863414\pi\)
0.909341 0.416052i \(-0.136586\pi\)
\(468\) 0 0
\(469\) 6249.76 4285.07i 0.615324 0.421889i
\(470\) 0 0
\(471\) 7371.21i 0.721120i
\(472\) 0 0
\(473\) 5288.78 0.514119
\(474\) 0 0
\(475\) 10733.3i 1.03680i
\(476\) 0 0
\(477\) 4764.07i 0.457300i
\(478\) 0 0
\(479\) −20229.8 −1.92970 −0.964849 0.262807i \(-0.915352\pi\)
−0.964849 + 0.262807i \(0.915352\pi\)
\(480\) 0 0
\(481\) 741.419i 0.0702824i
\(482\) 0 0
\(483\) −3373.46 + 2312.98i −0.317801 + 0.217897i
\(484\) 0 0
\(485\) 294.574i 0.0275792i
\(486\) 0 0
\(487\) 1879.06i 0.174843i 0.996171 + 0.0874213i \(0.0278626\pi\)
−0.996171 + 0.0874213i \(0.972137\pi\)
\(488\) 0 0
\(489\) 3554.27i 0.328690i
\(490\) 0 0
\(491\) 9249.10 0.850114 0.425057 0.905167i \(-0.360254\pi\)
0.425057 + 0.905167i \(0.360254\pi\)
\(492\) 0 0
\(493\) 1411.33 0.128931
\(494\) 0 0
\(495\) −300.625 −0.0272972
\(496\) 0 0
\(497\) 9277.65 + 13531.4i 0.837343 + 1.22126i
\(498\) 0 0
\(499\) −18142.9 −1.62763 −0.813813 0.581126i \(-0.802612\pi\)
−0.813813 + 0.581126i \(0.802612\pi\)
\(500\) 0 0
\(501\) 7480.63i 0.667086i
\(502\) 0 0
\(503\) −7981.16 −0.707480 −0.353740 0.935344i \(-0.615090\pi\)
−0.353740 + 0.935344i \(0.615090\pi\)
\(504\) 0 0
\(505\) −438.255 −0.0386180
\(506\) 0 0
\(507\) 6076.18i 0.532253i
\(508\) 0 0
\(509\) −9093.08 −0.791834 −0.395917 0.918286i \(-0.629573\pi\)
−0.395917 + 0.918286i \(0.629573\pi\)
\(510\) 0 0
\(511\) 2821.68 + 4115.41i 0.244274 + 0.356272i
\(512\) 0 0
\(513\) 2369.49 0.203929
\(514\) 0 0
\(515\) 1622.38 0.138817
\(516\) 0 0
\(517\) −4429.19 −0.376780
\(518\) 0 0
\(519\) 1010.88i 0.0854968i
\(520\) 0 0
\(521\) 10886.0i 0.915399i −0.889107 0.457699i \(-0.848674\pi\)
0.889107 0.457699i \(-0.151326\pi\)
\(522\) 0 0
\(523\) 16709.0i 1.39701i 0.715607 + 0.698503i \(0.246150\pi\)
−0.715607 + 0.698503i \(0.753850\pi\)
\(524\) 0 0
\(525\) −3842.68 5604.53i −0.319444 0.465908i
\(526\) 0 0
\(527\) 2990.27i 0.247169i
\(528\) 0 0
\(529\) 6747.46 0.554570
\(530\) 0 0
\(531\) 6167.96i 0.504080i
\(532\) 0 0
\(533\) 1777.02i 0.144411i
\(534\) 0 0
\(535\) −1831.31 −0.147990
\(536\) 0 0
\(537\) 1210.61i 0.0972842i
\(538\) 0 0
\(539\) 2515.60 6509.88i 0.201029 0.520223i
\(540\) 0 0
\(541\) 9846.26i 0.782484i −0.920288 0.391242i \(-0.872046\pi\)
0.920288 0.391242i \(-0.127954\pi\)
\(542\) 0 0
\(543\) 1878.78i 0.148483i
\(544\) 0 0
\(545\) 469.836i 0.0369276i
\(546\) 0 0
\(547\) −3956.47 −0.309262 −0.154631 0.987972i \(-0.549419\pi\)
−0.154631 + 0.987972i \(0.549419\pi\)
\(548\) 0 0
\(549\) −1349.24 −0.104889
\(550\) 0 0
\(551\) −5173.71 −0.400014
\(552\) 0 0
\(553\) 9452.74 + 13786.8i 0.726892 + 1.06017i
\(554\) 0 0
\(555\) 278.740 0.0213186
\(556\) 0 0
\(557\) 10754.5i 0.818106i −0.912511 0.409053i \(-0.865859\pi\)
0.912511 0.409053i \(-0.134141\pi\)
\(558\) 0 0
\(559\) 3405.05 0.257636
\(560\) 0 0
\(561\) 1461.30 0.109975
\(562\) 0 0
\(563\) 9499.10i 0.711082i −0.934661 0.355541i \(-0.884297\pi\)
0.934661 0.355541i \(-0.115703\pi\)
\(564\) 0 0
\(565\) −1195.66 −0.0890294
\(566\) 0 0
\(567\) −1237.25 + 848.308i −0.0916398 + 0.0628317i
\(568\) 0 0
\(569\) 21157.1 1.55879 0.779395 0.626533i \(-0.215526\pi\)
0.779395 + 0.626533i \(0.215526\pi\)
\(570\) 0 0
\(571\) 21995.3 1.61204 0.806021 0.591887i \(-0.201617\pi\)
0.806021 + 0.591887i \(0.201617\pi\)
\(572\) 0 0
\(573\) −2089.07 −0.152308
\(574\) 0 0
\(575\) 9003.79i 0.653016i
\(576\) 0 0
\(577\) 7582.38i 0.547068i 0.961862 + 0.273534i \(0.0881927\pi\)
−0.961862 + 0.273534i \(0.911807\pi\)
\(578\) 0 0
\(579\) 9992.09i 0.717197i
\(580\) 0 0
\(581\) 6534.72 + 9530.86i 0.466620 + 0.680563i
\(582\) 0 0
\(583\) 10770.5i 0.765128i
\(584\) 0 0
\(585\) −193.550 −0.0136792
\(586\) 0 0
\(587\) 2177.57i 0.153114i −0.997065 0.0765569i \(-0.975607\pi\)
0.997065 0.0765569i \(-0.0243927\pi\)
\(588\) 0 0
\(589\) 10961.9i 0.766851i
\(590\) 0 0
\(591\) 2523.35 0.175629
\(592\) 0 0
\(593\) 181.962i 0.0126008i 0.999980 + 0.00630042i \(0.00200550\pi\)
−0.999980 + 0.00630042i \(0.997995\pi\)
\(594\) 0 0
\(595\) −411.593 600.307i −0.0283591 0.0413616i
\(596\) 0 0
\(597\) 2158.72i 0.147991i
\(598\) 0 0
\(599\) 255.073i 0.0173990i 0.999962 + 0.00869949i \(0.00276917\pi\)
−0.999962 + 0.00869949i \(0.997231\pi\)
\(600\) 0 0
\(601\) 21122.3i 1.43361i 0.697275 + 0.716804i \(0.254396\pi\)
−0.697275 + 0.716804i \(0.745604\pi\)
\(602\) 0 0
\(603\) 3682.41 0.248689
\(604\) 0 0
\(605\) −1505.40 −0.101162
\(606\) 0 0
\(607\) −20997.8 −1.40408 −0.702038 0.712139i \(-0.747727\pi\)
−0.702038 + 0.712139i \(0.747727\pi\)
\(608\) 0 0
\(609\) 2701.51 1852.26i 0.179755 0.123247i
\(610\) 0 0
\(611\) −2851.63 −0.188813
\(612\) 0 0
\(613\) 13781.4i 0.908032i −0.890993 0.454016i \(-0.849991\pi\)
0.890993 0.454016i \(-0.150009\pi\)
\(614\) 0 0
\(615\) 668.078 0.0438041
\(616\) 0 0
\(617\) 15885.1 1.03648 0.518242 0.855234i \(-0.326586\pi\)
0.518242 + 0.855234i \(0.326586\pi\)
\(618\) 0 0
\(619\) 26764.3i 1.73788i 0.494917 + 0.868940i \(0.335198\pi\)
−0.494917 + 0.868940i \(0.664802\pi\)
\(620\) 0 0
\(621\) −1987.67 −0.128442
\(622\) 0 0
\(623\) 10327.7 + 15062.9i 0.664157 + 0.968671i
\(624\) 0 0
\(625\) 14621.6 0.935784
\(626\) 0 0
\(627\) −5356.89 −0.341202
\(628\) 0 0
\(629\) −1354.91 −0.0858887
\(630\) 0 0
\(631\) 12538.1i 0.791022i 0.918461 + 0.395511i \(0.129432\pi\)
−0.918461 + 0.395511i \(0.870568\pi\)
\(632\) 0 0
\(633\) 14872.5i 0.933855i
\(634\) 0 0
\(635\) 261.407i 0.0163364i
\(636\) 0 0
\(637\) 1619.61 4191.23i 0.100740 0.260695i
\(638\) 0 0
\(639\) 7972.82i 0.493584i
\(640\) 0 0
\(641\) −10139.1 −0.624761 −0.312380 0.949957i \(-0.601126\pi\)
−0.312380 + 0.949957i \(0.601126\pi\)
\(642\) 0 0
\(643\) 14901.0i 0.913902i −0.889492 0.456951i \(-0.848941\pi\)
0.889492 0.456951i \(-0.151059\pi\)
\(644\) 0 0
\(645\) 1280.14i 0.0781482i
\(646\) 0 0
\(647\) −23608.6 −1.43454 −0.717271 0.696794i \(-0.754609\pi\)
−0.717271 + 0.696794i \(0.754609\pi\)
\(648\) 0 0
\(649\) 13944.4i 0.843398i
\(650\) 0 0
\(651\) 3924.49 + 5723.85i 0.236272 + 0.344601i
\(652\) 0 0
\(653\) 1066.91i 0.0639380i 0.999489 + 0.0319690i \(0.0101778\pi\)
−0.999489 + 0.0319690i \(0.989822\pi\)
\(654\) 0 0
\(655\) 1228.49i 0.0732841i
\(656\) 0 0
\(657\) 2424.83i 0.143990i
\(658\) 0 0
\(659\) 1256.55 0.0742766 0.0371383 0.999310i \(-0.488176\pi\)
0.0371383 + 0.999310i \(0.488176\pi\)
\(660\) 0 0
\(661\) −11557.5 −0.680080 −0.340040 0.940411i \(-0.610441\pi\)
−0.340040 + 0.940411i \(0.610441\pi\)
\(662\) 0 0
\(663\) 940.822 0.0551109
\(664\) 0 0
\(665\) 1508.84 + 2200.63i 0.0879852 + 0.128326i
\(666\) 0 0
\(667\) 4340.03 0.251944
\(668\) 0 0
\(669\) 11842.3i 0.684377i
\(670\) 0 0
\(671\) 3050.34 0.175495
\(672\) 0 0
\(673\) 4626.18 0.264972 0.132486 0.991185i \(-0.457704\pi\)
0.132486 + 0.991185i \(0.457704\pi\)
\(674\) 0 0
\(675\) 3302.23i 0.188301i
\(676\) 0 0
\(677\) 33749.5 1.91595 0.957975 0.286850i \(-0.0926082\pi\)
0.957975 + 0.286850i \(0.0926082\pi\)
\(678\) 0 0
\(679\) 1879.23 + 2740.85i 0.106213 + 0.154911i
\(680\) 0 0
\(681\) −6133.11 −0.345112
\(682\) 0 0
\(683\) 22799.7 1.27732 0.638658 0.769491i \(-0.279490\pi\)
0.638658 + 0.769491i \(0.279490\pi\)
\(684\) 0 0
\(685\) −1952.71 −0.108919
\(686\) 0 0
\(687\) 13175.7i 0.731709i
\(688\) 0 0
\(689\) 6934.34i 0.383421i
\(690\) 0 0
\(691\) 22334.8i 1.22960i 0.788682 + 0.614802i \(0.210764\pi\)
−0.788682 + 0.614802i \(0.789236\pi\)
\(692\) 0 0
\(693\) 2797.16 1917.84i 0.153326 0.105126i
\(694\) 0 0
\(695\) 896.603i 0.0489354i
\(696\) 0 0
\(697\) −3247.44 −0.176478
\(698\) 0 0
\(699\) 5910.48i 0.319821i
\(700\) 0 0
\(701\) 8429.88i 0.454197i 0.973872 + 0.227099i \(0.0729240\pi\)
−0.973872 + 0.227099i \(0.927076\pi\)
\(702\) 0 0
\(703\) 4966.91 0.266473
\(704\) 0 0
\(705\) 1072.08i 0.0572722i
\(706\) 0 0
\(707\) 4077.73 2795.85i 0.216915 0.148725i
\(708\) 0 0
\(709\) 6186.00i 0.327673i −0.986488 0.163837i \(-0.947613\pi\)
0.986488 0.163837i \(-0.0523870\pi\)
\(710\) 0 0
\(711\) 8123.29i 0.428477i
\(712\) 0 0
\(713\) 9195.48i 0.482992i
\(714\) 0 0
\(715\) 437.575 0.0228872
\(716\) 0 0
\(717\) −8894.66 −0.463287
\(718\) 0 0
\(719\) −17866.0 −0.926688 −0.463344 0.886178i \(-0.653351\pi\)
−0.463344 + 0.886178i \(0.653351\pi\)
\(720\) 0 0
\(721\) −15095.4 + 10350.0i −0.779725 + 0.534609i
\(722\) 0 0
\(723\) −10974.9 −0.564539
\(724\) 0 0
\(725\) 7210.34i 0.369359i
\(726\) 0 0
\(727\) 27651.7 1.41065 0.705326 0.708883i \(-0.250801\pi\)
0.705326 + 0.708883i \(0.250801\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 6222.60i 0.314844i
\(732\) 0 0
\(733\) −26895.7 −1.35527 −0.677636 0.735397i \(-0.736996\pi\)
−0.677636 + 0.735397i \(0.736996\pi\)
\(734\) 0 0
\(735\) −1575.71 608.899i −0.0790761 0.0305573i
\(736\) 0 0
\(737\) −8325.11 −0.416091
\(738\) 0 0
\(739\) −22507.8 −1.12038 −0.560191 0.828364i \(-0.689272\pi\)
−0.560191 + 0.828364i \(0.689272\pi\)
\(740\) 0 0
\(741\) −3448.91 −0.170983
\(742\) 0 0
\(743\) 19387.3i 0.957268i 0.878014 + 0.478634i \(0.158868\pi\)
−0.878014 + 0.478634i \(0.841132\pi\)
\(744\) 0 0
\(745\) 3561.43i 0.175142i
\(746\) 0 0
\(747\) 5615.66i 0.275055i
\(748\) 0 0
\(749\) 17039.3 11682.8i 0.831247 0.569935i
\(750\) 0 0
\(751\) 34865.6i 1.69409i 0.531518 + 0.847047i \(0.321622\pi\)
−0.531518 + 0.847047i \(0.678378\pi\)
\(752\) 0 0
\(753\) −8158.04 −0.394815
\(754\) 0 0
\(755\) 3075.29i 0.148240i
\(756\) 0 0
\(757\) 19477.5i 0.935165i 0.883949 + 0.467583i \(0.154875\pi\)
−0.883949 + 0.467583i \(0.845125\pi\)
\(758\) 0 0
\(759\) 4493.69 0.214902
\(760\) 0 0
\(761\) 33587.3i 1.59992i −0.600053 0.799960i \(-0.704854\pi\)
0.600053 0.799960i \(-0.295146\pi\)
\(762\) 0 0
\(763\) 2997.31 + 4371.57i 0.142215 + 0.207420i
\(764\) 0 0
\(765\) 353.706i 0.0167167i
\(766\) 0 0
\(767\) 8977.76i 0.422644i
\(768\) 0 0
\(769\) 30658.2i 1.43766i 0.695185 + 0.718831i \(0.255323\pi\)
−0.695185 + 0.718831i \(0.744677\pi\)
\(770\) 0 0
\(771\) −20394.3 −0.952638
\(772\) 0 0
\(773\) 12029.2 0.559714 0.279857 0.960042i \(-0.409713\pi\)
0.279857 + 0.960042i \(0.409713\pi\)
\(774\) 0 0
\(775\) −15277.0 −0.708084
\(776\) 0 0
\(777\) −2593.52 + 1778.22i −0.119745 + 0.0821019i
\(778\) 0 0
\(779\) 11904.6 0.547530
\(780\) 0 0
\(781\) 18024.8i 0.825836i
\(782\) 0 0
\(783\) 1591.75 0.0726495
\(784\) 0 0
\(785\) −4033.67 −0.183398
\(786\) 0 0
\(787\) 33352.6i 1.51066i −0.655342 0.755332i \(-0.727476\pi\)
0.655342 0.755332i \(-0.272524\pi\)
\(788\) 0 0
\(789\) −23897.8 −1.07831
\(790\) 0 0
\(791\) 11124.9 7627.68i 0.500072 0.342868i
\(792\) 0 0
\(793\) 1963.89 0.0879440
\(794\) 0 0
\(795\) 2606.99 0.116302
\(796\) 0 0
\(797\) 44570.0 1.98087 0.990434 0.137989i \(-0.0440638\pi\)
0.990434 + 0.137989i \(0.0440638\pi\)
\(798\) 0 0
\(799\) 5211.23i 0.230739i
\(800\) 0 0
\(801\) 8875.18i 0.391497i