Properties

Label 960.4.f.p.769.1
Level $960$
Weight $4$
Character 960.769
Analytic conductor $56.642$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.1
Root \(2.70156i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.p.769.3

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +(-11.1047 - 1.29844i) q^{5} +16.2094i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +(-11.1047 - 1.29844i) q^{5} +16.2094i q^{7} -9.00000 q^{9} -40.2094 q^{11} +19.7906i q^{13} +(-3.89531 + 33.3141i) q^{15} +83.0156i q^{17} +48.8375 q^{19} +48.6281 q^{21} +1.61250i q^{23} +(121.628 + 28.8375i) q^{25} +27.0000i q^{27} -24.5344 q^{29} +12.4187 q^{31} +120.628i q^{33} +(21.0469 - 180.000i) q^{35} -325.884i q^{37} +59.3719 q^{39} -242.419 q^{41} -367.350i q^{43} +(99.9422 + 11.6859i) q^{45} +204.544i q^{47} +80.2562 q^{49} +249.047 q^{51} -61.5281i q^{53} +(446.512 + 52.2094i) q^{55} -146.512i q^{57} +112.209 q^{59} -477.350 q^{61} -145.884i q^{63} +(25.6969 - 219.769i) q^{65} +558.094i q^{67} +4.83749 q^{69} -558.281 q^{71} -1011.77i q^{73} +(86.5125 - 364.884i) q^{75} -651.769i q^{77} +1150.47 q^{79} +81.0000 q^{81} +1157.92i q^{83} +(107.791 - 921.862i) q^{85} +73.6032i q^{87} -96.9751 q^{89} -320.794 q^{91} -37.2562i q^{93} +(-542.325 - 63.4124i) q^{95} -1152.37i q^{97} +361.884 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 36 q^{9} - 84 q^{11} - 54 q^{15} - 112 q^{19} - 36 q^{21} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 468 q^{39} - 816 q^{41} + 54 q^{45} - 140 q^{49} + 612 q^{51} + 864 q^{55} + 372 q^{59} - 680 q^{61} + 948 q^{65} - 288 q^{69} + 72 q^{71} - 576 q^{75} + 760 q^{79} + 324 q^{81} + 508 q^{85} - 2232 q^{89} + 1944 q^{91} - 2784 q^{95} + 756 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) −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
\(9\) −9.00000 −0.333333
\(10\) 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 \(0.0677087\pi\)
−0.977462 + 0.211113i \(0.932291\pi\)
\(14\) 0 0
\(15\) −3.89531 + 33.3141i −0.0670510 + 0.573444i
\(16\) 0 0
\(17\) 83.0156i 1.18437i 0.805803 + 0.592184i \(0.201734\pi\)
−0.805803 + 0.592184i \(0.798266\pi\)
\(18\) 0 0
\(19\) 48.8375 0.589689 0.294844 0.955545i \(-0.404732\pi\)
0.294844 + 0.955545i \(0.404732\pi\)
\(20\) 0 0
\(21\) 48.6281 0.505311
\(22\) 0 0
\(23\) 1.61250i 0.0146186i 0.999973 + 0.00730932i \(0.00232665\pi\)
−0.999973 + 0.00730932i \(0.997673\pi\)
\(24\) 0 0
\(25\) 121.628 + 28.8375i 0.973025 + 0.230700i
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 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
\(33\) 120.628i 0.636323i
\(34\) 0 0
\(35\) 21.0469 180.000i 0.101645 0.869302i
\(36\) 0 0
\(37\) 325.884i 1.44797i −0.689813 0.723987i \(-0.742307\pi\)
0.689813 0.723987i \(-0.257693\pi\)
\(38\) 0 0
\(39\) 59.3719 0.243772
\(40\) 0 0
\(41\) −242.419 −0.923401 −0.461701 0.887036i \(-0.652761\pi\)
−0.461701 + 0.887036i \(0.652761\pi\)
\(42\) 0 0
\(43\) 367.350i 1.30280i −0.758735 0.651399i \(-0.774182\pi\)
0.758735 0.651399i \(-0.225818\pi\)
\(44\) 0 0
\(45\) 99.9422 + 11.6859i 0.331078 + 0.0387119i
\(46\) 0 0
\(47\) 204.544i 0.634804i 0.948291 + 0.317402i \(0.102810\pi\)
−0.948291 + 0.317402i \(0.897190\pi\)
\(48\) 0 0
\(49\) 80.2562 0.233983
\(50\) 0 0
\(51\) 249.047 0.683795
\(52\) 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
\(57\) 146.512i 0.340457i
\(58\) 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 \(-0.667023\pi\)
−0.500970 + 0.865464i \(0.667023\pi\)
\(62\) 0 0
\(63\) 145.884i 0.291741i
\(64\) 0 0
\(65\) 25.6969 219.769i 0.0490355 0.419369i
\(66\) 0 0
\(67\) 558.094i 1.01764i 0.860872 + 0.508821i \(0.169918\pi\)
−0.860872 + 0.508821i \(0.830082\pi\)
\(68\) 0 0
\(69\) 4.83749 0.00844008
\(70\) 0 0
\(71\) −558.281 −0.933180 −0.466590 0.884474i \(-0.654518\pi\)
−0.466590 + 0.884474i \(0.654518\pi\)
\(72\) 0 0
\(73\) 1011.77i 1.62217i −0.584927 0.811086i \(-0.698877\pi\)
0.584927 0.811086i \(-0.301123\pi\)
\(74\) 0 0
\(75\) 86.5125 364.884i 0.133195 0.561776i
\(76\) 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
\(81\) 81.0000 0.111111
\(82\) 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
\(87\) 73.6032i 0.0907022i
\(88\) 0 0
\(89\) −96.9751 −0.115498 −0.0577491 0.998331i \(-0.518392\pi\)
−0.0577491 + 0.998331i \(0.518392\pi\)
\(90\) 0 0
\(91\) −320.794 −0.369542
\(92\) 0 0
\(93\) 37.2562i 0.0415408i
\(94\) 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
\(99\) 361.884 0.367381
\(100\) 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
\(105\) −540.000 63.1406i −0.501891 0.0586847i
\(106\) 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 \(-0.642485\pi\)
−0.432831 + 0.901475i \(0.642485\pi\)
\(110\) 0 0
\(111\) −977.653 −0.835988
\(112\) 0 0
\(113\) 1888.25i 1.57196i −0.618253 0.785979i \(-0.712159\pi\)
0.618253 0.785979i \(-0.287841\pi\)
\(114\) 0 0
\(115\) 2.09373 17.9063i 0.00169775 0.0145197i
\(116\) 0 0
\(117\) 178.116i 0.140742i
\(118\) 0 0
\(119\) −1345.63 −1.03659
\(120\) 0 0
\(121\) 285.794 0.214721
\(122\) 0 0
\(123\) 727.256i 0.533126i
\(124\) 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
\(129\) −1102.05 −0.752171
\(130\) 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
\(135\) 35.0578 299.827i 0.0223503 0.191148i
\(136\) 0 0
\(137\) 1656.29i 1.03289i −0.856319 0.516447i \(-0.827254\pi\)
0.856319 0.516447i \(-0.172746\pi\)
\(138\) 0 0
\(139\) 153.256 0.0935182 0.0467591 0.998906i \(-0.485111\pi\)
0.0467591 + 0.998906i \(0.485111\pi\)
\(140\) 0 0
\(141\) 613.631 0.366504
\(142\) 0 0
\(143\) 795.769i 0.465353i
\(144\) 0 0
\(145\) 272.447 + 31.8564i 0.156038 + 0.0182450i
\(146\) 0 0
\(147\) 240.769i 0.135090i
\(148\) 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
\(153\) 747.141i 0.394789i
\(154\) 0 0
\(155\) −137.906 16.1250i −0.0714639 0.00835606i
\(156\) 0 0
\(157\) 1727.05i 0.877922i −0.898506 0.438961i \(-0.855347\pi\)
0.898506 0.438961i \(-0.144653\pi\)
\(158\) 0 0
\(159\) −184.584 −0.0920659
\(160\) 0 0
\(161\) −26.1376 −0.0127946
\(162\) 0 0
\(163\) 2034.28i 0.977529i 0.872416 + 0.488764i \(0.162552\pi\)
−0.872416 + 0.488764i \(0.837448\pi\)
\(164\) 0 0
\(165\) 156.628 1339.54i 0.0738999 0.632017i
\(166\) 0 0
\(167\) 192.900i 0.0893835i −0.999001 0.0446918i \(-0.985769\pi\)
0.999001 0.0446918i \(-0.0142306\pi\)
\(168\) 0 0
\(169\) 1805.33 0.821726
\(170\) 0 0
\(171\) −439.537 −0.196563
\(172\) 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
\(177\) 336.628i 0.142952i
\(178\) 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 \(-0.738541\pi\)
−0.681199 + 0.732099i \(0.738541\pi\)
\(182\) 0 0
\(183\) 1432.05i 0.578471i
\(184\) 0 0
\(185\) −423.141 + 3618.84i −0.168162 + 1.43818i
\(186\) 0 0
\(187\) 3338.01i 1.30534i
\(188\) 0 0
\(189\) −437.653 −0.168437
\(190\) 0 0
\(191\) 624.506 0.236585 0.118292 0.992979i \(-0.462258\pi\)
0.118292 + 0.992979i \(0.462258\pi\)
\(192\) 0 0
\(193\) 436.144i 0.162665i 0.996687 + 0.0813324i \(0.0259175\pi\)
−0.996687 + 0.0813324i \(0.974082\pi\)
\(194\) 0 0
\(195\) −659.306 77.0907i −0.242123 0.0283107i
\(196\) 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
\(201\) 1674.28 0.587536
\(202\) 0 0
\(203\) 397.687i 0.137498i
\(204\) 0 0
\(205\) 2691.98 + 314.766i 0.917153 + 0.107240i
\(206\) 0 0
\(207\) 14.5125i 0.00487288i
\(208\) 0 0
\(209\) −1963.72 −0.649922
\(210\) 0 0
\(211\) 2365.27 0.771715 0.385857 0.922558i \(-0.373906\pi\)
0.385857 + 0.922558i \(0.373906\pi\)
\(212\) 0 0
\(213\) 1674.84i 0.538772i
\(214\) 0 0
\(215\) −476.981 + 4079.31i −0.151302 + 1.29398i
\(216\) 0 0
\(217\) 201.300i 0.0629730i
\(218\) 0 0
\(219\) −3035.31 −0.936562
\(220\) 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
\(225\) −1094.65 259.537i −0.324342 0.0769000i
\(226\) 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 \(-0.379276\pi\)
0.370240 + 0.928936i \(0.379276\pi\)
\(230\) 0 0
\(231\) −1955.31 −0.556925
\(232\) 0 0
\(233\) 5534.99i 1.55626i −0.628101 0.778132i \(-0.716168\pi\)
0.628101 0.778132i \(-0.283832\pi\)
\(234\) 0 0
\(235\) 265.587 2271.39i 0.0737234 0.630508i
\(236\) 0 0
\(237\) 3451.41i 0.945962i
\(238\) 0 0
\(239\) 1010.01 0.273355 0.136678 0.990616i \(-0.456358\pi\)
0.136678 + 0.990616i \(0.456358\pi\)
\(240\) 0 0
\(241\) −4074.29 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) −891.220 104.208i −0.232400 0.0271738i
\(246\) 0 0
\(247\) 966.525i 0.248982i
\(248\) 0 0
\(249\) 3473.77 0.884103
\(250\) 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
\(255\) −2765.59 323.372i −0.679168 0.0794131i
\(256\) 0 0
\(257\) 662.784i 0.160869i 0.996760 + 0.0804345i \(0.0256308\pi\)
−0.996760 + 0.0804345i \(0.974369\pi\)
\(258\) 0 0
\(259\) 5282.38 1.26730
\(260\) 0 0
\(261\) 220.810 0.0523669
\(262\) 0 0
\(263\) 712.312i 0.167008i 0.996507 + 0.0835039i \(0.0266111\pi\)
−0.996507 + 0.0835039i \(0.973389\pi\)
\(264\) 0 0
\(265\) −79.8904 + 683.250i −0.0185194 + 0.158384i
\(266\) 0 0
\(267\) 290.925i 0.0666829i
\(268\) 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
\(273\) 962.381i 0.213355i
\(274\) 0 0
\(275\) −4890.59 1159.54i −1.07241 0.254264i
\(276\) 0 0
\(277\) 5171.00i 1.12164i −0.827937 0.560821i \(-0.810485\pi\)
0.827937 0.560821i \(-0.189515\pi\)
\(278\) 0 0
\(279\) −111.769 −0.0239836
\(280\) 0 0
\(281\) 2240.14 0.475571 0.237785 0.971318i \(-0.423578\pi\)
0.237785 + 0.971318i \(0.423578\pi\)
\(282\) 0 0
\(283\) 225.244i 0.0473123i −0.999720 0.0236561i \(-0.992469\pi\)
0.999720 0.0236561i \(-0.00753068\pi\)
\(284\) 0 0
\(285\) −190.237 + 1626.98i −0.0395393 + 0.338153i
\(286\) 0 0
\(287\) 3929.46i 0.808183i
\(288\) 0 0
\(289\) −1978.59 −0.402726
\(290\) 0 0
\(291\) −3457.12 −0.696427
\(292\) 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
\(297\) 1085.65i 0.212108i
\(298\) 0 0
\(299\) −31.9123 −0.00617237
\(300\) 0 0
\(301\) 5954.51 1.14024
\(302\) 0 0
\(303\) 3469.47i 0.657808i
\(304\) 0 0
\(305\) 5300.82 + 619.809i 0.995161 + 0.116361i
\(306\) 0 0
\(307\) 5244.86i 0.975049i 0.873109 + 0.487525i \(0.162100\pi\)
−0.873109 + 0.487525i \(0.837900\pi\)
\(308\) 0 0
\(309\) −4001.09 −0.736615
\(310\) 0 0
\(311\) 5188.26 0.945977 0.472989 0.881068i \(-0.343175\pi\)
0.472989 + 0.881068i \(0.343175\pi\)
\(312\) 0 0
\(313\) 486.656i 0.0878832i −0.999034 0.0439416i \(-0.986008\pi\)
0.999034 0.0439416i \(-0.0139915\pi\)
\(314\) 0 0
\(315\) −189.422 + 1620.00i −0.0338816 + 0.289767i
\(316\) 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
\(321\) −2394.79 −0.416399
\(322\) 0 0
\(323\) 4054.27i 0.698408i
\(324\) 0 0
\(325\) −570.712 + 2407.10i −0.0974074 + 0.410836i
\(326\) 0 0
\(327\) 2955.36i 0.499791i
\(328\) 0 0
\(329\) −3315.53 −0.555595
\(330\) 0 0
\(331\) 7439.94 1.23546 0.617728 0.786392i \(-0.288053\pi\)
0.617728 + 0.786392i \(0.288053\pi\)
\(332\) 0 0
\(333\) 2932.96i 0.482658i
\(334\) 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
\(339\) −5664.74 −0.907571
\(340\) 0 0
\(341\) −499.350 −0.0793000
\(342\) 0 0
\(343\) 6860.72i 1.08001i
\(344\) 0 0
\(345\) −53.7188 6.28118i −0.00838297 0.000980195i
\(346\) 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 \(-0.534898\pi\)
−0.109415 + 0.993996i \(0.534898\pi\)
\(350\) 0 0
\(351\) −534.347 −0.0812573
\(352\) 0 0
\(353\) 7078.96i 1.06735i 0.845689 + 0.533676i \(0.179190\pi\)
−0.845689 + 0.533676i \(0.820810\pi\)
\(354\) 0 0
\(355\) 6199.54 + 724.893i 0.926866 + 0.108376i
\(356\) 0 0
\(357\) 4036.89i 0.598474i
\(358\) 0 0
\(359\) −5409.79 −0.795314 −0.397657 0.917534i \(-0.630177\pi\)
−0.397657 + 0.917534i \(0.630177\pi\)
\(360\) 0 0
\(361\) −4473.90 −0.652267
\(362\) 0 0
\(363\) 857.381i 0.123969i
\(364\) 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
\(369\) 2181.77 0.307800
\(370\) 0 0
\(371\) 997.332 0.139566
\(372\) 0 0
\(373\) 12891.9i 1.78959i 0.446473 + 0.894797i \(0.352680\pi\)
−0.446473 + 0.894797i \(0.647320\pi\)
\(374\) 0 0
\(375\) −1434.47 + 3939.60i −0.197536 + 0.542506i
\(376\) 0 0
\(377\) 485.551i 0.0663320i
\(378\) 0 0
\(379\) 9475.15 1.28418 0.642092 0.766627i \(-0.278067\pi\)
0.642092 + 0.766627i \(0.278067\pi\)
\(380\) 0 0
\(381\) 1862.58 0.250453
\(382\) 0 0
\(383\) 5800.97i 0.773931i −0.922094 0.386966i \(-0.873523\pi\)
0.922094 0.386966i \(-0.126477\pi\)
\(384\) 0 0
\(385\) −846.281 + 7237.69i −0.112027 + 0.958095i
\(386\) 0 0
\(387\) 3306.15i 0.434266i
\(388\) 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
\(393\) 7765.06i 0.996680i
\(394\) 0 0
\(395\) −12775.6 1493.81i −1.62737 0.190283i
\(396\) 0 0
\(397\) 2816.46i 0.356056i 0.984025 + 0.178028i \(0.0569718\pi\)
−0.984025 + 0.178028i \(0.943028\pi\)
\(398\) 0 0
\(399\) 2374.88 0.297976
\(400\) 0 0
\(401\) 11986.4 1.49270 0.746352 0.665551i \(-0.231804\pi\)
0.746352 + 0.665551i \(0.231804\pi\)
\(402\) 0 0
\(403\) 245.775i 0.0303794i
\(404\) 0 0
\(405\) −899.480 105.173i −0.110359 0.0129040i
\(406\) 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
\(411\) −4968.87 −0.596342
\(412\) 0 0
\(413\) 1818.84i 0.216706i
\(414\) 0 0
\(415\) 1503.49 12858.4i 0.177840 1.52095i
\(416\) 0 0
\(417\) 459.769i 0.0539927i
\(418\) 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 \(-0.450715\pi\)
0.154214 + 0.988037i \(0.450715\pi\)
\(422\) 0 0
\(423\) 1840.89i 0.211601i
\(424\) 0 0
\(425\) −2393.96 + 10097.0i −0.273233 + 1.15242i
\(426\) 0 0
\(427\) 7737.54i 0.876923i
\(428\) 0 0
\(429\) −2387.31 −0.268672
\(430\) 0 0
\(431\) −12266.0 −1.37084 −0.685420 0.728148i \(-0.740381\pi\)
−0.685420 + 0.728148i \(0.740381\pi\)
\(432\) 0 0
\(433\) 15647.3i 1.73664i 0.496008 + 0.868318i \(0.334799\pi\)
−0.496008 + 0.868318i \(0.665201\pi\)
\(434\) 0 0
\(435\) 95.5691 817.340i 0.0105338 0.0900884i
\(436\) 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
\(441\) −722.306 −0.0779944
\(442\) 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
\(447\) 4450.13i 0.470882i
\(448\) 0 0
\(449\) −7477.71 −0.785957 −0.392979 0.919548i \(-0.628555\pi\)
−0.392979 + 0.919548i \(0.628555\pi\)
\(450\) 0 0
\(451\) 9747.51 1.01772
\(452\) 0 0
\(453\) 1182.84i 0.122682i
\(454\) 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
\(459\) −2241.42 −0.227932
\(460\) 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
\(465\) −48.3749 + 413.719i −0.00482437 + 0.0412597i
\(466\) 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
\(471\) −5181.16 −0.506869
\(472\) 0 0
\(473\) 14770.9i 1.43587i
\(474\) 0 0
\(475\) 5940.01 + 1408.35i 0.573782 + 0.136041i
\(476\) 0 0
\(477\) 553.753i 0.0531543i
\(478\) 0 0
\(479\) −19688.2 −1.87803 −0.939013 0.343881i \(-0.888258\pi\)
−0.939013 + 0.343881i \(0.888258\pi\)
\(480\) 0 0
\(481\) 6449.46 0.611372
\(482\) 0 0
\(483\) 78.4127i 0.00738696i
\(484\) 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
\(489\) 6102.84 0.564377
\(490\) 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
\(495\) −4018.61 469.884i −0.364895 0.0426661i
\(496\) 0 0
\(497\) 9049.39i 0.816741i
\(498\) 0 0
\(499\) −13523.7 −1.21324 −0.606618 0.794993i \(-0.707474\pi\)
−0.606618 + 0.794993i \(0.707474\pi\)
\(500\) 0 0
\(501\) −578.700 −0.0516056
\(502\) 0 0
\(503\) 13135.4i 1.16437i 0.813057 + 0.582184i \(0.197802\pi\)
−0.813057 + 0.582184i \(0.802198\pi\)
\(504\) 0 0
\(505\) −12842.5 1501.63i −1.13165 0.132320i
\(506\) 0 0
\(507\) 5415.99i 0.474423i
\(508\) 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
\(513\) 1318.61i 0.113486i
\(514\) 0 0
\(515\) −1731.72 + 14810.3i −0.148172 + 1.26722i
\(516\) 0 0
\(517\) 8224.57i 0.699645i
\(518\) 0 0
\(519\) −3719.73 −0.314601
\(520\) 0 0
\(521\) −4916.42 −0.413421 −0.206710 0.978402i \(-0.566276\pi\)
−0.206710 + 0.978402i \(0.566276\pi\)
\(522\) 0 0
\(523\) 17743.4i 1.48349i −0.670681 0.741746i \(-0.733998\pi\)
0.670681 0.741746i \(-0.266002\pi\)
\(524\) 0 0
\(525\) 5914.55 + 1402.31i 0.491680 + 0.116575i
\(526\) 0 0
\(527\) 1030.95i 0.0852161i
\(528\) 0 0
\(529\) 12164.4 0.999786
\(530\) 0 0
\(531\) −1009.88 −0.0825334
\(532\) 0 0
\(533\) 4797.62i 0.389884i
\(534\) 0 0
\(535\) −1036.49 + 8864.46i −0.0837599 + 0.716344i
\(536\) 0 0
\(537\) 7910.58i 0.635692i
\(538\) 0 0
\(539\) −3227.05 −0.257883
\(540\) 0 0
\(541\) 12671.3 1.00699 0.503495 0.863998i \(-0.332047\pi\)
0.503495 + 0.863998i \(0.332047\pi\)
\(542\) 0 0
\(543\) 9952.74i 0.786580i
\(544\) 0 0
\(545\) 10939.4 + 1279.12i 0.859805 + 0.100534i
\(546\) 0 0
\(547\) 5250.90i 0.410443i 0.978716 + 0.205221i \(0.0657915\pi\)
−0.978716 + 0.205221i \(0.934209\pi\)
\(548\) 0 0
\(549\) 4296.15 0.333980
\(550\) 0 0
\(551\) −1198.20 −0.0926406
\(552\) 0 0
\(553\) 18648.4i 1.43401i
\(554\) 0 0
\(555\) 10856.5 + 1269.42i 0.830332 + 0.0970882i
\(556\) 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
\(561\) −10014.0 −0.753640
\(562\) 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
\(567\) 1312.96i 0.0972471i
\(568\) 0 0
\(569\) −8706.51 −0.641469 −0.320734 0.947169i \(-0.603930\pi\)
−0.320734 + 0.947169i \(0.603930\pi\)
\(570\) 0 0
\(571\) −12194.5 −0.893740 −0.446870 0.894599i \(-0.647461\pi\)
−0.446870 + 0.894599i \(0.647461\pi\)
\(572\) 0 0
\(573\) 1873.52i 0.136592i
\(574\) 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
\(579\) 1308.43 0.0939146
\(580\) 0 0
\(581\) −18769.2 −1.34024
\(582\) 0 0
\(583\) 2474.01i 0.175751i
\(584\) 0 0
\(585\) −231.272 + 1977.92i −0.0163452 + 0.139790i
\(586\) 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
\(591\) −10067.4 −0.700708
\(592\) 0 0
\(593\) 1225.23i 0.0848467i 0.999100 + 0.0424234i \(0.0135078\pi\)
−0.999100 + 0.0424234i \(0.986492\pi\)
\(594\) 0 0
\(595\) 14942.8 + 1747.22i 1.02957 + 0.120385i
\(596\) 0 0
\(597\) 11399.3i 0.781478i
\(598\) 0 0
\(599\) −16060.0 −1.09548 −0.547741 0.836648i \(-0.684512\pi\)
−0.547741 + 0.836648i \(0.684512\pi\)
\(600\) 0 0
\(601\) 9699.93 0.658350 0.329175 0.944269i \(-0.393229\pi\)
0.329175 + 0.944269i \(0.393229\pi\)
\(602\) 0 0
\(603\) 5022.84i 0.339214i
\(604\) 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
\(609\) −1193.06 −0.0793847
\(610\) 0 0
\(611\) −4048.05 −0.268030
\(612\) 0 0
\(613\) 8085.63i 0.532749i −0.963870 0.266375i \(-0.914174\pi\)
0.963870 0.266375i \(-0.0858258\pi\)
\(614\) 0 0
\(615\) 944.297 8075.95i 0.0619150 0.529518i
\(616\) 0 0
\(617\) 11035.1i 0.720029i −0.932947 0.360014i \(-0.882772\pi\)
0.932947 0.360014i \(-0.117228\pi\)
\(618\) 0 0
\(619\) 16826.3 1.09258 0.546290 0.837596i \(-0.316040\pi\)
0.546290 + 0.837596i \(0.316040\pi\)
\(620\) 0 0
\(621\) −43.5374 −0.00281336
\(622\) 0 0
\(623\) 1571.90i 0.101087i
\(624\) 0 0
\(625\) 13961.8 + 7014.90i 0.893555 + 0.448954i
\(626\) 0 0
\(627\) 5891.17i 0.375233i
\(628\) 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
\(633\) 7095.81i 0.445550i
\(634\) 0 0
\(635\) 806.147 6894.45i 0.0503795 0.430863i
\(636\) 0 0
\(637\) 1588.32i 0.0987937i
\(638\) 0 0
\(639\) 5024.53 0.311060
\(640\) 0 0
\(641\) −24597.4 −1.51566 −0.757829 0.652453i \(-0.773740\pi\)
−0.757829 + 0.652453i \(0.773740\pi\)
\(642\) 0 0
\(643\) 21479.5i 1.31737i 0.752419 + 0.658685i \(0.228887\pi\)
−0.752419 + 0.658685i \(0.771113\pi\)
\(644\) 0 0
\(645\) 12237.9 + 1430.94i 0.747081 + 0.0873540i
\(646\) 0 0
\(647\) 27119.7i 1.64789i −0.566668 0.823946i \(-0.691768\pi\)
0.566668 0.823946i \(-0.308232\pi\)
\(648\) 0 0
\(649\) −4511.87 −0.272891
\(650\) 0 0
\(651\) 603.900 0.0363575
\(652\) 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
\(657\) 9105.92i 0.540724i
\(658\) 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 \(-0.771564\pi\)
−0.753352 + 0.657618i \(0.771564\pi\)
\(662\) 0 0
\(663\) 4928.79i 0.288716i
\(664\) 0 0
\(665\) 1027.88 8790.75i 0.0599388 0.512617i
\(666\) 0 0
\(667\) 39.5616i 0.00229660i
\(668\) 0 0
\(669\) −9985.75 −0.577087
\(670\) 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
\(675\) −778.612 + 3283.96i −0.0443982 + 0.187259i
\(676\) 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
\(681\) −1581.30 −0.0889803
\(682\) 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
\(687\) 7698.17i 0.427516i
\(688\) 0 0
\(689\) 1217.68 0.0673293
\(690\) 0 0
\(691\) 6203.15 0.341504 0.170752 0.985314i \(-0.445380\pi\)
0.170752 + 0.985314i \(0.445380\pi\)
\(692\) 0 0
\(693\) 5865.92i 0.321541i
\(694\) 0 0
\(695\) −1701.86 198.994i −0.0928854 0.0108608i
\(696\) 0 0
\(697\) 20124.5i 1.09365i
\(698\) 0 0
\(699\) −16605.0 −0.898509
\(700\) 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
\(705\) −6814.18 796.762i −0.364024 0.0425642i
\(706\) 0 0
\(707\) 18746.0i 0.997193i
\(708\) 0 0
\(709\) −25539.6 −1.35283 −0.676416 0.736520i \(-0.736468\pi\)
−0.676416 + 0.736520i \(0.736468\pi\)
\(710\) 0 0
\(711\) −10354.2 −0.546151
\(712\) 0 0
\(713\) 20.0252i 0.00105182i
\(714\) 0 0
\(715\) −1033.26 + 8836.76i −0.0540442 + 0.462204i
\(716\) 0 0
\(717\) 3030.02i 0.157822i
\(718\) 0 0
\(719\) −7353.45 −0.381415 −0.190708 0.981647i \(-0.561078\pi\)
−0.190708 + 0.981647i \(0.561078\pi\)
\(720\) 0 0
\(721\) 21618.4 1.11666
\(722\) 0 0
\(723\) 12222.9i 0.628732i
\(724\) 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
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 30495.8 1.54299
\(732\) 0 0
\(733\) 90.2714i 0.00454877i 0.999997 + 0.00227439i \(0.000723960\pi\)
−0.999997 + 0.00227439i \(0.999276\pi\)
\(734\) 0 0
\(735\) −312.623 + 2673.66i −0.0156888 + 0.134176i
\(736\) 0 0
\(737\) 22440.6i 1.12159i
\(738\) 0 0
\(739\) −14273.1 −0.710479 −0.355239 0.934775i \(-0.615601\pi\)
−0.355239 + 0.934775i \(0.615601\pi\)
\(740\) 0 0
\(741\) 2899.57 0.143750
\(742\) 0 0
\(743\) 15866.6i 0.783429i 0.920087 + 0.391715i \(0.128118\pi\)
−0.920087 + 0.391715i \(0.871882\pi\)
\(744\) 0 0
\(745\) −16472.4 1926.07i −0.810072 0.0947193i
\(746\) 0 0
\(747\) 10421.3i 0.510437i
\(748\) 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
\(753\) 5321.95i 0.257560i
\(754\) 0 0
\(755\) −4378.37 511.950i −0.211053 0.0246778i
\(756\) 0 0
\(757\) 30478.0i 1.46333i −0.681663 0.731666i \(-0.738743\pi\)
0.681663 0.731666i \(-0.261257\pi\)
\(758\) 0 0
\(759\) −194.512 −0.00930218
\(760\) 0 0
\(761\) 29104.7 1.38639 0.693195 0.720750i \(-0.256202\pi\)
0.693195 + 0.720750i \(0.256202\pi\)
\(762\) 0 0
\(763\) 15968.2i 0.757649i
\(764\) 0 0
\(765\) −970.116 + 8296.76i −0.0458492 + 0.392118i
\(766\) 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
\(771\) 1988.35 0.0928778
\(772\) 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
\(777\) 15847.1i 0.731677i
\(778\) 0 0
\(779\) −11839.1 −0.544519
\(780\) 0 0
\(781\) 22448.1 1.02850
\(782\) 0 0
\(783\) 662.429i 0.0302341i
\(784\) 0 0
\(785\) −2242.47 + 19178.4i −0.101958 + 0.871982i
\(786\) 0 0
\(787\) 3807.92i 0.172475i −0.996275 0.0862374i \(-0.972516\pi\)
0.996275 0.0862374i \(-0.0274844\pi\)
\(788\) 0 0
\(789\) 2136.94 0.0964220
\(790\) 0 0
\(791\) 30607.3 1.37582
\(792\) 0 0
\(793\) 9447.06i 0.423045i
\(794\) 0 0
\(795\) 2049.75 + 239.671i 0.0914430 + 0.0106922i
\(796\) 0 0
\(797\) 23840.3i