Properties

Label 960.3.j.e.319.3
Level $960$
Weight $3$
Character 960.319
Analytic conductor $26.158$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.389136420864.4
Defining polynomial: \(x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.3
Root \(-0.656712 - 1.88911i\) of defining polynomial
Character \(\chi\) \(=\) 960.319
Dual form 960.3.j.e.319.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} +(3.27492 - 3.77822i) q^{5} +9.55505 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(3.27492 - 3.77822i) q^{5} +9.55505 q^{7} +3.00000 q^{9} +9.92480i q^{11} +7.55643i q^{13} +(-5.67232 + 6.54406i) q^{15} +17.1903i q^{17} +26.1762i q^{19} -16.5498 q^{21} -1.67451 q^{23} +(-3.54983 - 24.7467i) q^{25} -5.19615 q^{27} +0.350497 q^{29} +46.0258i q^{31} -17.1903i q^{33} +(31.2920 - 36.1010i) q^{35} -22.6693i q^{37} -13.0881i q^{39} -77.2990 q^{41} +41.7994 q^{43} +(9.82475 - 11.3346i) q^{45} +14.0866 q^{47} +42.2990 q^{49} -29.7744i q^{51} -22.6693i q^{53} +(37.4980 + 32.5029i) q^{55} -45.3386i q^{57} +94.7802i q^{59} -38.0000 q^{61} +28.6652 q^{63} +(28.5498 + 24.7467i) q^{65} +29.8477 q^{67} +2.90033 q^{69} -7.19630i q^{71} +34.3805i q^{73} +(6.14849 + 42.8625i) q^{75} +94.8320i q^{77} -46.0258i q^{79} +9.00000 q^{81} -24.1336 q^{83} +(64.9485 + 56.2967i) q^{85} -0.607078 q^{87} +100.199 q^{89} +72.2021i q^{91} -79.7191i q^{93} +(98.8995 + 85.7250i) q^{95} +131.861i q^{97} +29.7744i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} + 24q^{9} + O(q^{10}) \) \( 8q - 4q^{5} + 24q^{9} - 72q^{21} + 32q^{25} + 184q^{29} - 256q^{41} - 12q^{45} - 24q^{49} - 304q^{61} + 168q^{65} + 144q^{69} + 72q^{81} - 24q^{85} + 560q^{89} + O(q^{100}) \)

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
\(3\) −1.73205 −0.577350
\(4\) 0 0
\(5\) 3.27492 3.77822i 0.654983 0.755643i
\(6\) 0 0
\(7\) 9.55505 1.36501 0.682504 0.730882i \(-0.260891\pi\)
0.682504 + 0.730882i \(0.260891\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 9.92480i 0.902255i 0.892460 + 0.451127i \(0.148978\pi\)
−0.892460 + 0.451127i \(0.851022\pi\)
\(12\) 0 0
\(13\) 7.55643i 0.581264i 0.956835 + 0.290632i \(0.0938655\pi\)
−0.956835 + 0.290632i \(0.906134\pi\)
\(14\) 0 0
\(15\) −5.67232 + 6.54406i −0.378155 + 0.436271i
\(16\) 0 0
\(17\) 17.1903i 1.01119i 0.862770 + 0.505596i \(0.168727\pi\)
−0.862770 + 0.505596i \(0.831273\pi\)
\(18\) 0 0
\(19\) 26.1762i 1.37770i 0.724905 + 0.688849i \(0.241884\pi\)
−0.724905 + 0.688849i \(0.758116\pi\)
\(20\) 0 0
\(21\) −16.5498 −0.788087
\(22\) 0 0
\(23\) −1.67451 −0.0728047 −0.0364023 0.999337i \(-0.511590\pi\)
−0.0364023 + 0.999337i \(0.511590\pi\)
\(24\) 0 0
\(25\) −3.54983 24.7467i −0.141993 0.989868i
\(26\) 0 0
\(27\) −5.19615 −0.192450
\(28\) 0 0
\(29\) 0.350497 0.0120861 0.00604305 0.999982i \(-0.498076\pi\)
0.00604305 + 0.999982i \(0.498076\pi\)
\(30\) 0 0
\(31\) 46.0258i 1.48470i 0.670010 + 0.742352i \(0.266290\pi\)
−0.670010 + 0.742352i \(0.733710\pi\)
\(32\) 0 0
\(33\) 17.1903i 0.520917i
\(34\) 0 0
\(35\) 31.2920 36.1010i 0.894057 1.03146i
\(36\) 0 0
\(37\) 22.6693i 0.612684i −0.951922 0.306342i \(-0.900895\pi\)
0.951922 0.306342i \(-0.0991051\pi\)
\(38\) 0 0
\(39\) 13.0881i 0.335593i
\(40\) 0 0
\(41\) −77.2990 −1.88534 −0.942671 0.333724i \(-0.891695\pi\)
−0.942671 + 0.333724i \(0.891695\pi\)
\(42\) 0 0
\(43\) 41.7994 0.972079 0.486039 0.873937i \(-0.338441\pi\)
0.486039 + 0.873937i \(0.338441\pi\)
\(44\) 0 0
\(45\) 9.82475 11.3346i 0.218328 0.251881i
\(46\) 0 0
\(47\) 14.0866 0.299715 0.149857 0.988708i \(-0.452119\pi\)
0.149857 + 0.988708i \(0.452119\pi\)
\(48\) 0 0
\(49\) 42.2990 0.863245
\(50\) 0 0
\(51\) 29.7744i 0.583812i
\(52\) 0 0
\(53\) 22.6693i 0.427723i −0.976864 0.213861i \(-0.931396\pi\)
0.976864 0.213861i \(-0.0686041\pi\)
\(54\) 0 0
\(55\) 37.4980 + 32.5029i 0.681783 + 0.590962i
\(56\) 0 0
\(57\) 45.3386i 0.795414i
\(58\) 0 0
\(59\) 94.7802i 1.60644i 0.595680 + 0.803222i \(0.296883\pi\)
−0.595680 + 0.803222i \(0.703117\pi\)
\(60\) 0 0
\(61\) −38.0000 −0.622951 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(62\) 0 0
\(63\) 28.6652 0.455002
\(64\) 0 0
\(65\) 28.5498 + 24.7467i 0.439228 + 0.380718i
\(66\) 0 0
\(67\) 29.8477 0.445488 0.222744 0.974877i \(-0.428499\pi\)
0.222744 + 0.974877i \(0.428499\pi\)
\(68\) 0 0
\(69\) 2.90033 0.0420338
\(70\) 0 0
\(71\) 7.19630i 0.101356i −0.998715 0.0506782i \(-0.983862\pi\)
0.998715 0.0506782i \(-0.0161383\pi\)
\(72\) 0 0
\(73\) 34.3805i 0.470966i 0.971878 + 0.235483i \(0.0756672\pi\)
−0.971878 + 0.235483i \(0.924333\pi\)
\(74\) 0 0
\(75\) 6.14849 + 42.8625i 0.0819799 + 0.571500i
\(76\) 0 0
\(77\) 94.8320i 1.23158i
\(78\) 0 0
\(79\) 46.0258i 0.582606i −0.956631 0.291303i \(-0.905911\pi\)
0.956631 0.291303i \(-0.0940887\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −24.1336 −0.290767 −0.145383 0.989375i \(-0.546442\pi\)
−0.145383 + 0.989375i \(0.546442\pi\)
\(84\) 0 0
\(85\) 64.9485 + 56.2967i 0.764100 + 0.662314i
\(86\) 0 0
\(87\) −0.607078 −0.00697791
\(88\) 0 0
\(89\) 100.199 1.12584 0.562918 0.826513i \(-0.309679\pi\)
0.562918 + 0.826513i \(0.309679\pi\)
\(90\) 0 0
\(91\) 72.2021i 0.793430i
\(92\) 0 0
\(93\) 79.7191i 0.857195i
\(94\) 0 0
\(95\) 98.8995 + 85.7250i 1.04105 + 0.902369i
\(96\) 0 0
\(97\) 131.861i 1.35939i 0.733494 + 0.679696i \(0.237888\pi\)
−0.733494 + 0.679696i \(0.762112\pi\)
\(98\) 0 0
\(99\) 29.7744i 0.300752i
\(100\) 0 0
\(101\) −29.4502 −0.291586 −0.145793 0.989315i \(-0.546573\pi\)
−0.145793 + 0.989315i \(0.546573\pi\)
\(102\) 0 0
\(103\) 143.786 1.39598 0.697991 0.716107i \(-0.254078\pi\)
0.697991 + 0.716107i \(0.254078\pi\)
\(104\) 0 0
\(105\) −54.1993 + 62.5289i −0.516184 + 0.595513i
\(106\) 0 0
\(107\) 35.1014 0.328050 0.164025 0.986456i \(-0.447552\pi\)
0.164025 + 0.986456i \(0.447552\pi\)
\(108\) 0 0
\(109\) 151.498 1.38989 0.694947 0.719061i \(-0.255428\pi\)
0.694947 + 0.719061i \(0.255428\pi\)
\(110\) 0 0
\(111\) 39.2644i 0.353733i
\(112\) 0 0
\(113\) 32.3031i 0.285868i −0.989732 0.142934i \(-0.954346\pi\)
0.989732 0.142934i \(-0.0456537\pi\)
\(114\) 0 0
\(115\) −5.48387 + 6.32665i −0.0476858 + 0.0550143i
\(116\) 0 0
\(117\) 22.6693i 0.193755i
\(118\) 0 0
\(119\) 164.254i 1.38028i
\(120\) 0 0
\(121\) 22.4983 0.185937
\(122\) 0 0
\(123\) 133.886 1.08850
\(124\) 0 0
\(125\) −105.124 67.6313i −0.840990 0.541051i
\(126\) 0 0
\(127\) −192.053 −1.51223 −0.756116 0.654438i \(-0.772905\pi\)
−0.756116 + 0.654438i \(0.772905\pi\)
\(128\) 0 0
\(129\) −72.3987 −0.561230
\(130\) 0 0
\(131\) 42.4277i 0.323876i 0.986801 + 0.161938i \(0.0517744\pi\)
−0.986801 + 0.161938i \(0.948226\pi\)
\(132\) 0 0
\(133\) 250.115i 1.88057i
\(134\) 0 0
\(135\) −17.0170 + 19.6322i −0.126052 + 0.145424i
\(136\) 0 0
\(137\) 206.854i 1.50988i −0.655791 0.754942i \(-0.727665\pi\)
0.655791 0.754942i \(-0.272335\pi\)
\(138\) 0 0
\(139\) 46.0258i 0.331121i −0.986200 0.165561i \(-0.947057\pi\)
0.986200 0.165561i \(-0.0529433\pi\)
\(140\) 0 0
\(141\) −24.3987 −0.173040
\(142\) 0 0
\(143\) −74.9961 −0.524448
\(144\) 0 0
\(145\) 1.14785 1.32425i 0.00791619 0.00913277i
\(146\) 0 0
\(147\) −73.2640 −0.498395
\(148\) 0 0
\(149\) −11.6495 −0.0781846 −0.0390923 0.999236i \(-0.512447\pi\)
−0.0390923 + 0.999236i \(0.512447\pi\)
\(150\) 0 0
\(151\) 125.424i 0.830624i 0.909679 + 0.415312i \(0.136328\pi\)
−0.909679 + 0.415312i \(0.863672\pi\)
\(152\) 0 0
\(153\) 51.5708i 0.337064i
\(154\) 0 0
\(155\) 173.896 + 150.731i 1.12191 + 0.972457i
\(156\) 0 0
\(157\) 197.220i 1.25618i −0.778140 0.628090i \(-0.783837\pi\)
0.778140 0.628090i \(-0.216163\pi\)
\(158\) 0 0
\(159\) 39.2644i 0.246946i
\(160\) 0 0
\(161\) −16.0000 −0.0993789
\(162\) 0 0
\(163\) 18.4196 0.113004 0.0565018 0.998402i \(-0.482005\pi\)
0.0565018 + 0.998402i \(0.482005\pi\)
\(164\) 0 0
\(165\) −64.9485 56.2967i −0.393627 0.341192i
\(166\) 0 0
\(167\) 92.8920 0.556240 0.278120 0.960546i \(-0.410289\pi\)
0.278120 + 0.960546i \(0.410289\pi\)
\(168\) 0 0
\(169\) 111.900 0.662132
\(170\) 0 0
\(171\) 78.5287i 0.459232i
\(172\) 0 0
\(173\) 117.501i 0.679198i −0.940570 0.339599i \(-0.889709\pi\)
0.940570 0.339599i \(-0.110291\pi\)
\(174\) 0 0
\(175\) −33.9189 236.456i −0.193822 1.35118i
\(176\) 0 0
\(177\) 164.164i 0.927481i
\(178\) 0 0
\(179\) 231.988i 1.29602i −0.761631 0.648011i \(-0.775601\pi\)
0.761631 0.648011i \(-0.224399\pi\)
\(180\) 0 0
\(181\) 218.096 1.20495 0.602476 0.798137i \(-0.294181\pi\)
0.602476 + 0.798137i \(0.294181\pi\)
\(182\) 0 0
\(183\) 65.8179 0.359661
\(184\) 0 0
\(185\) −85.6495 74.2401i −0.462970 0.401298i
\(186\) 0 0
\(187\) −170.610 −0.912352
\(188\) 0 0
\(189\) −49.6495 −0.262696
\(190\) 0 0
\(191\) 137.208i 0.718366i −0.933267 0.359183i \(-0.883055\pi\)
0.933267 0.359183i \(-0.116945\pi\)
\(192\) 0 0
\(193\) 37.0290i 0.191860i 0.995388 + 0.0959301i \(0.0305825\pi\)
−0.995388 + 0.0959301i \(0.969417\pi\)
\(194\) 0 0
\(195\) −49.4498 42.8625i −0.253589 0.219808i
\(196\) 0 0
\(197\) 194.572i 0.987674i −0.869554 0.493837i \(-0.835594\pi\)
0.869554 0.493837i \(-0.164406\pi\)
\(198\) 0 0
\(199\) 176.037i 0.884610i 0.896865 + 0.442305i \(0.145839\pi\)
−0.896865 + 0.442305i \(0.854161\pi\)
\(200\) 0 0
\(201\) −51.6977 −0.257202
\(202\) 0 0
\(203\) 3.34901 0.0164976
\(204\) 0 0
\(205\) −253.148 + 292.052i −1.23487 + 1.42465i
\(206\) 0 0
\(207\) −5.02352 −0.0242682
\(208\) 0 0
\(209\) −259.794 −1.24303
\(210\) 0 0
\(211\) 20.7193i 0.0981955i 0.998794 + 0.0490978i \(0.0156346\pi\)
−0.998794 + 0.0490978i \(0.984365\pi\)
\(212\) 0 0
\(213\) 12.4644i 0.0585181i
\(214\) 0 0
\(215\) 136.890 157.927i 0.636696 0.734545i
\(216\) 0 0
\(217\) 439.779i 2.02663i
\(218\) 0 0
\(219\) 59.5488i 0.271912i
\(220\) 0 0
\(221\) −129.897 −0.587769
\(222\) 0 0
\(223\) 97.0265 0.435096 0.217548 0.976050i \(-0.430194\pi\)
0.217548 + 0.976050i \(0.430194\pi\)
\(224\) 0 0
\(225\) −10.6495 74.2401i −0.0473311 0.329956i
\(226\) 0 0
\(227\) −407.256 −1.79408 −0.897040 0.441948i \(-0.854287\pi\)
−0.897040 + 0.441948i \(0.854287\pi\)
\(228\) 0 0
\(229\) 7.89702 0.0344848 0.0172424 0.999851i \(-0.494511\pi\)
0.0172424 + 0.999851i \(0.494511\pi\)
\(230\) 0 0
\(231\) 164.254i 0.711055i
\(232\) 0 0
\(233\) 28.1483i 0.120808i −0.998174 0.0604042i \(-0.980761\pi\)
0.998174 0.0604042i \(-0.0192390\pi\)
\(234\) 0 0
\(235\) 46.1324 53.2222i 0.196308 0.226477i
\(236\) 0 0
\(237\) 79.7191i 0.336368i
\(238\) 0 0
\(239\) 296.005i 1.23851i −0.785189 0.619257i \(-0.787434\pi\)
0.785189 0.619257i \(-0.212566\pi\)
\(240\) 0 0
\(241\) 465.794 1.93276 0.966378 0.257127i \(-0.0827758\pi\)
0.966378 + 0.257127i \(0.0827758\pi\)
\(242\) 0 0
\(243\) −15.5885 −0.0641500
\(244\) 0 0
\(245\) 138.526 159.815i 0.565411 0.652305i
\(246\) 0 0
\(247\) −197.799 −0.800806
\(248\) 0 0
\(249\) 41.8007 0.167874
\(250\) 0 0
\(251\) 141.676i 0.564445i −0.959349 0.282223i \(-0.908928\pi\)
0.959349 0.282223i \(-0.0910716\pi\)
\(252\) 0 0
\(253\) 16.6191i 0.0656883i
\(254\) 0 0
\(255\) −112.494 97.5087i −0.441153 0.382387i
\(256\) 0 0
\(257\) 41.7549i 0.162470i −0.996695 0.0812352i \(-0.974113\pi\)
0.996695 0.0812352i \(-0.0258865\pi\)
\(258\) 0 0
\(259\) 216.606i 0.836318i
\(260\) 0 0
\(261\) 1.05149 0.00402870
\(262\) 0 0
\(263\) −203.283 −0.772939 −0.386469 0.922302i \(-0.626306\pi\)
−0.386469 + 0.922302i \(0.626306\pi\)
\(264\) 0 0
\(265\) −85.6495 74.2401i −0.323206 0.280151i
\(266\) 0 0
\(267\) −173.550 −0.650001
\(268\) 0 0
\(269\) 244.048 0.907242 0.453621 0.891195i \(-0.350132\pi\)
0.453621 + 0.891195i \(0.350132\pi\)
\(270\) 0 0
\(271\) 466.585i 1.72172i −0.508845 0.860858i \(-0.669927\pi\)
0.508845 0.860858i \(-0.330073\pi\)
\(272\) 0 0
\(273\) 125.058i 0.458087i
\(274\) 0 0
\(275\) 245.606 35.2314i 0.893113 0.128114i
\(276\) 0 0
\(277\) 494.181i 1.78405i 0.451990 + 0.892023i \(0.350714\pi\)
−0.451990 + 0.892023i \(0.649286\pi\)
\(278\) 0 0
\(279\) 138.078i 0.494902i
\(280\) 0 0
\(281\) −43.4020 −0.154455 −0.0772277 0.997013i \(-0.524607\pi\)
−0.0772277 + 0.997013i \(0.524607\pi\)
\(282\) 0 0
\(283\) 310.785 1.09818 0.549090 0.835763i \(-0.314974\pi\)
0.549090 + 0.835763i \(0.314974\pi\)
\(284\) 0 0
\(285\) −171.299 148.480i −0.601049 0.520983i
\(286\) 0 0
\(287\) −738.596 −2.57351
\(288\) 0 0
\(289\) −6.50497 −0.0225085
\(290\) 0 0
\(291\) 228.390i 0.784845i
\(292\) 0 0
\(293\) 245.207i 0.836886i −0.908243 0.418443i \(-0.862576\pi\)
0.908243 0.418443i \(-0.137424\pi\)
\(294\) 0 0
\(295\) 358.100 + 310.397i 1.21390 + 1.05219i
\(296\) 0 0
\(297\) 51.5708i 0.173639i
\(298\) 0 0
\(299\) 12.6533i 0.0423187i
\(300\) 0 0
\(301\) 399.395 1.32689
\(302\) 0 0
\(303\) 51.0092 0.168347
\(304\) 0 0
\(305\) −124.447 + 143.572i −0.408022 + 0.470729i
\(306\) 0 0
\(307\) −337.514 −1.09939 −0.549697 0.835364i \(-0.685257\pi\)
−0.549697 + 0.835364i \(0.685257\pi\)
\(308\) 0 0
\(309\) −249.045 −0.805970
\(310\) 0 0
\(311\) 427.756i 1.37542i 0.725986 + 0.687710i \(0.241384\pi\)
−0.725986 + 0.687710i \(0.758616\pi\)
\(312\) 0 0
\(313\) 83.8739i 0.267968i −0.990984 0.133984i \(-0.957223\pi\)
0.990984 0.133984i \(-0.0427770\pi\)
\(314\) 0 0
\(315\) 93.8760 108.303i 0.298019 0.343820i
\(316\) 0 0
\(317\) 112.204i 0.353957i 0.984215 + 0.176978i \(0.0566323\pi\)
−0.984215 + 0.176978i \(0.943368\pi\)
\(318\) 0 0
\(319\) 3.47861i 0.0109047i
\(320\) 0 0
\(321\) −60.7974 −0.189400
\(322\) 0 0
\(323\) −449.976 −1.39312
\(324\) 0 0
\(325\) 186.997 26.8241i 0.575374 0.0825356i
\(326\) 0 0
\(327\) −262.403 −0.802455
\(328\) 0 0
\(329\) 134.598 0.409113
\(330\) 0 0
\(331\) 132.621i 0.400666i −0.979728 0.200333i \(-0.935798\pi\)
0.979728 0.200333i \(-0.0642025\pi\)
\(332\) 0 0
\(333\) 68.0079i 0.204228i
\(334\) 0 0
\(335\) 97.7487 112.771i 0.291787 0.336630i
\(336\) 0 0
\(337\) 20.7739i 0.0616437i −0.999525 0.0308219i \(-0.990188\pi\)
0.999525 0.0308219i \(-0.00981246\pi\)
\(338\) 0 0
\(339\) 55.9506i 0.165046i
\(340\) 0 0
\(341\) −456.797 −1.33958
\(342\) 0 0
\(343\) −64.0283 −0.186672
\(344\) 0 0
\(345\) 9.49834 10.9581i 0.0275314 0.0317625i
\(346\) 0 0
\(347\) −8.89616 −0.0256374 −0.0128187 0.999918i \(-0.504080\pi\)
−0.0128187 + 0.999918i \(0.504080\pi\)
\(348\) 0 0
\(349\) −19.4020 −0.0555931 −0.0277965 0.999614i \(-0.508849\pi\)
−0.0277965 + 0.999614i \(0.508849\pi\)
\(350\) 0 0
\(351\) 39.2644i 0.111864i
\(352\) 0 0
\(353\) 80.2902i 0.227451i −0.993512 0.113726i \(-0.963722\pi\)
0.993512 0.113726i \(-0.0362784\pi\)
\(354\) 0 0
\(355\) −27.1892 23.5673i −0.0765892 0.0663867i
\(356\) 0 0
\(357\) 284.496i 0.796907i
\(358\) 0 0
\(359\) 314.115i 0.874972i −0.899225 0.437486i \(-0.855869\pi\)
0.899225 0.437486i \(-0.144131\pi\)
\(360\) 0 0
\(361\) −324.196 −0.898050
\(362\) 0 0
\(363\) −38.9683 −0.107351
\(364\) 0 0
\(365\) 129.897 + 112.593i 0.355882 + 0.308475i
\(366\) 0 0
\(367\) −476.800 −1.29918 −0.649592 0.760283i \(-0.725060\pi\)
−0.649592 + 0.760283i \(0.725060\pi\)
\(368\) 0 0
\(369\) −231.897 −0.628447
\(370\) 0 0
\(371\) 216.606i 0.583844i
\(372\) 0 0
\(373\) 86.1333i 0.230920i −0.993312 0.115460i \(-0.963166\pi\)
0.993312 0.115460i \(-0.0368343\pi\)
\(374\) 0 0
\(375\) 182.080 + 117.141i 0.485546 + 0.312376i
\(376\) 0 0
\(377\) 2.64850i 0.00702521i
\(378\) 0 0
\(379\) 638.035i 1.68347i 0.539891 + 0.841735i \(0.318466\pi\)
−0.539891 + 0.841735i \(0.681534\pi\)
\(380\) 0 0
\(381\) 332.646 0.873087
\(382\) 0 0
\(383\) 216.742 0.565907 0.282953 0.959134i \(-0.408686\pi\)
0.282953 + 0.959134i \(0.408686\pi\)
\(384\) 0 0
\(385\) 358.296 + 310.567i 0.930638 + 0.806667i
\(386\) 0 0
\(387\) 125.398 0.324026
\(388\) 0 0
\(389\) −476.640 −1.22529 −0.612647 0.790356i \(-0.709895\pi\)
−0.612647 + 0.790356i \(0.709895\pi\)
\(390\) 0 0
\(391\) 28.7852i 0.0736195i
\(392\) 0 0
\(393\) 73.4869i 0.186990i
\(394\) 0 0
\(395\) −173.896 150.731i −0.440242 0.381597i
\(396\) 0 0
\(397\) 43.0792i 0.108512i 0.998527 + 0.0542559i \(0.0172787\pi\)
−0.998527 + 0.0542559i \(0.982721\pi\)
\(398\) 0 0
\(399\) 433.213i 1.08575i
\(400\) 0 0
\(401\) −168.694 −0.420684 −0.210342 0.977628i \(-0.567458\pi\)
−0.210342 + 0.977628i \(0.567458\pi\)
\(402\) 0 0
\(403\) −347.791 −0.863005
\(404\) 0 0
\(405\) 29.4743 34.0039i 0.0727759 0.0839604i
\(406\) 0 0
\(407\) 224.988 0.552797
\(408\) 0 0
\(409\) 373.890 0.914157 0.457079 0.889426i \(-0.348896\pi\)
0.457079 + 0.889426i \(0.348896\pi\)
\(410\) 0 0
\(411\) 358.282i 0.871732i
\(412\) 0 0
\(413\) 905.630i 2.19281i
\(414\) 0 0
\(415\) −79.0356 + 91.1820i −0.190447 + 0.219716i
\(416\) 0 0
\(417\) 79.7191i 0.191173i
\(418\) 0 0
\(419\) 87.5839i 0.209031i 0.994523 + 0.104515i \(0.0333291\pi\)
−0.994523 + 0.104515i \(0.966671\pi\)
\(420\) 0 0
\(421\) −70.3023 −0.166989 −0.0834944 0.996508i \(-0.526608\pi\)
−0.0834944 + 0.996508i \(0.526608\pi\)
\(422\) 0 0
\(423\) 42.2597 0.0999048
\(424\) 0 0
\(425\) 425.402 61.0226i 1.00095 0.143583i
\(426\) 0 0
\(427\) −363.092 −0.850332
\(428\) 0 0
\(429\) 129.897 0.302790
\(430\) 0 0
\(431\) 247.370i 0.573944i −0.957939 0.286972i \(-0.907351\pi\)
0.957939 0.286972i \(-0.0926487\pi\)
\(432\) 0 0
\(433\) 636.247i 1.46939i 0.678397 + 0.734696i \(0.262675\pi\)
−0.678397 + 0.734696i \(0.737325\pi\)
\(434\) 0 0
\(435\) −1.98813 + 2.29367i −0.00457041 + 0.00527281i
\(436\) 0 0
\(437\) 43.8323i 0.100303i
\(438\) 0 0
\(439\) 769.786i 1.75350i −0.480947 0.876750i \(-0.659707\pi\)
0.480947 0.876750i \(-0.340293\pi\)
\(440\) 0 0
\(441\) 126.897 0.287748
\(442\) 0 0
\(443\) 612.214 1.38197 0.690986 0.722868i \(-0.257177\pi\)
0.690986 + 0.722868i \(0.257177\pi\)
\(444\) 0 0
\(445\) 328.145 378.575i 0.737403 0.850730i
\(446\) 0 0
\(447\) 20.1775 0.0451399
\(448\) 0 0
\(449\) 175.897 0.391753 0.195876 0.980629i \(-0.437245\pi\)
0.195876 + 0.980629i \(0.437245\pi\)
\(450\) 0 0
\(451\) 767.177i 1.70106i
\(452\) 0 0
\(453\) 217.241i 0.479561i
\(454\) 0 0
\(455\) 272.795 + 236.456i 0.599550 + 0.519683i
\(456\) 0 0
\(457\) 365.357i 0.799469i −0.916631 0.399734i \(-0.869102\pi\)
0.916631 0.399734i \(-0.130898\pi\)
\(458\) 0 0
\(459\) 89.3232i 0.194604i
\(460\) 0 0
\(461\) −308.350 −0.668873 −0.334437 0.942418i \(-0.608546\pi\)
−0.334437 + 0.942418i \(0.608546\pi\)
\(462\) 0 0
\(463\) −92.6302 −0.200065 −0.100033 0.994984i \(-0.531895\pi\)
−0.100033 + 0.994984i \(0.531895\pi\)
\(464\) 0 0
\(465\) −301.196 261.073i −0.647733 0.561448i
\(466\) 0 0
\(467\) −606.103 −1.29786 −0.648932 0.760846i \(-0.724784\pi\)
−0.648932 + 0.760846i \(0.724784\pi\)
\(468\) 0 0
\(469\) 285.196 0.608094
\(470\) 0 0
\(471\) 341.596i 0.725256i
\(472\) 0 0
\(473\) 414.851i 0.877063i
\(474\) 0 0
\(475\) 647.776 92.9214i 1.36374 0.195624i
\(476\) 0 0
\(477\) 68.0079i 0.142574i
\(478\) 0 0
\(479\) 138.947i 0.290078i 0.989426 + 0.145039i \(0.0463307\pi\)
−0.989426 + 0.145039i \(0.953669\pi\)
\(480\) 0 0
\(481\) 171.299 0.356131
\(482\) 0 0
\(483\) 27.7128 0.0573764
\(484\) 0 0
\(485\) 498.199 + 431.834i 1.02722 + 0.890379i
\(486\) 0 0
\(487\) −201.243 −0.413230 −0.206615 0.978422i \(-0.566245\pi\)
−0.206615 + 0.978422i \(0.566245\pi\)
\(488\) 0 0
\(489\) −31.9036 −0.0652426
\(490\) 0 0
\(491\) 347.368i 0.707470i 0.935346 + 0.353735i \(0.115089\pi\)
−0.935346 + 0.353735i \(0.884911\pi\)
\(492\) 0 0
\(493\) 6.02513i 0.0122214i
\(494\) 0 0
\(495\) 112.494 + 97.5087i 0.227261 + 0.196987i
\(496\) 0 0
\(497\) 68.7610i 0.138352i
\(498\) 0 0
\(499\) 672.277i 1.34725i −0.739074 0.673625i \(-0.764736\pi\)
0.739074 0.673625i \(-0.235264\pi\)
\(500\) 0 0
\(501\) −160.894 −0.321145
\(502\) 0 0
\(503\) 436.350 0.867496 0.433748 0.901034i \(-0.357191\pi\)
0.433748 + 0.901034i \(0.357191\pi\)
\(504\) 0 0
\(505\) −96.4469 + 111.269i −0.190984 + 0.220335i
\(506\) 0 0
\(507\) −193.817 −0.382282
\(508\) 0 0
\(509\) 109.547 0.215219 0.107610 0.994193i \(-0.465680\pi\)
0.107610 + 0.994193i \(0.465680\pi\)
\(510\) 0 0
\(511\) 328.508i 0.642872i
\(512\) 0 0
\(513\) 136.016i 0.265138i
\(514\) 0 0
\(515\) 470.888 543.255i 0.914345 1.05486i
\(516\) 0 0
\(517\) 139.807i 0.270419i
\(518\) 0 0
\(519\) 203.518i 0.392135i
\(520\) 0 0
\(521\) −743.100 −1.42629 −0.713147 0.701014i \(-0.752731\pi\)
−0.713147 + 0.701014i \(0.752731\pi\)
\(522\) 0 0
\(523\) 366.211 0.700212 0.350106 0.936710i \(-0.386146\pi\)
0.350106 + 0.936710i \(0.386146\pi\)
\(524\) 0 0
\(525\) 58.7492 + 409.554i 0.111903 + 0.780102i
\(526\) 0 0
\(527\) −791.196 −1.50132
\(528\) 0 0
\(529\) −526.196 −0.994699
\(530\) 0 0
\(531\) 284.341i 0.535481i
\(532\) 0 0
\(533\) 584.105i 1.09588i
\(534\) 0 0
\(535\) 114.954 132.621i 0.214867 0.247889i
\(536\) 0 0
\(537\) 401.815i 0.748259i
\(538\) 0 0
\(539\) 419.809i 0.778867i
\(540\) 0 0
\(541\) −170.688 −0.315504 −0.157752 0.987479i \(-0.550425\pi\)
−0.157752 + 0.987479i \(0.550425\pi\)
\(542\) 0 0
\(543\) −377.754 −0.695680
\(544\) 0 0
\(545\) 496.145 572.393i 0.910357 1.05026i
\(546\) 0 0
\(547\) −507.600 −0.927971 −0.463986 0.885843i \(-0.653581\pi\)
−0.463986 + 0.885843i \(0.653581\pi\)
\(548\) 0 0
\(549\) −114.000 −0.207650
\(550\) 0 0
\(551\) 9.17469i 0.0166510i
\(552\) 0 0
\(553\) 439.779i 0.795261i
\(554\) 0 0
\(555\) 148.349 + 128.588i 0.267296 + 0.231689i
\(556\) 0 0
\(557\) 788.492i 1.41561i −0.706410 0.707803i \(-0.749686\pi\)
0.706410 0.707803i \(-0.250314\pi\)
\(558\) 0 0
\(559\) 315.854i 0.565035i
\(560\) 0 0
\(561\) 295.505 0.526747
\(562\) 0 0
\(563\) 936.102 1.66270 0.831351 0.555747i \(-0.187568\pi\)
0.831351 + 0.555747i \(0.187568\pi\)
\(564\) 0 0
\(565\) −122.048 105.790i −0.216014 0.187239i
\(566\) 0 0
\(567\) 85.9955 0.151667
\(568\) 0 0
\(569\) −95.4983 −0.167835 −0.0839177 0.996473i \(-0.526743\pi\)
−0.0839177 + 0.996473i \(0.526743\pi\)
\(570\) 0 0
\(571\) 889.123i 1.55713i 0.627562 + 0.778566i \(0.284053\pi\)
−0.627562 + 0.778566i \(0.715947\pi\)
\(572\) 0 0
\(573\) 237.651i 0.414749i
\(574\) 0 0
\(575\) 5.94422 + 41.4385i 0.0103378 + 0.0720670i
\(576\) 0 0
\(577\) 522.147i 0.904934i −0.891781 0.452467i \(-0.850544\pi\)
0.891781 0.452467i \(-0.149456\pi\)
\(578\) 0 0
\(579\) 64.1361i 0.110771i
\(580\) 0 0
\(581\) −230.598 −0.396898
\(582\) 0 0
\(583\) 224.988 0.385915
\(584\) 0 0
\(585\) 85.6495 + 74.2401i 0.146409 + 0.126906i
\(586\) 0 0
\(587\) −95.8440 −0.163278 −0.0816388 0.996662i \(-0.526015\pi\)
−0.0816388 + 0.996662i \(0.526015\pi\)
\(588\) 0 0
\(589\) −1204.78 −2.04547
\(590\) 0 0
\(591\) 337.008i 0.570234i
\(592\) 0 0
\(593\) 878.624i 1.48166i −0.671693 0.740829i \(-0.734433\pi\)
0.671693 0.740829i \(-0.265567\pi\)
\(594\) 0 0
\(595\) 620.586 + 537.918i 1.04300 + 0.904063i
\(596\) 0 0
\(597\) 304.906i 0.510730i
\(598\) 0 0
\(599\) 967.652i 1.61545i 0.589563 + 0.807723i \(0.299300\pi\)
−0.589563 + 0.807723i \(0.700700\pi\)
\(600\) 0 0
\(601\) 279.704 0.465398 0.232699 0.972549i \(-0.425244\pi\)
0.232699 + 0.972549i \(0.425244\pi\)
\(602\) 0 0
\(603\) 89.5430 0.148496
\(604\) 0 0
\(605\) 73.6802 85.0036i 0.121785 0.140502i
\(606\) 0 0
\(607\) 747.564 1.23157 0.615786 0.787914i \(-0.288839\pi\)
0.615786 + 0.787914i \(0.288839\pi\)
\(608\) 0 0
\(609\) −5.80066 −0.00952490
\(610\) 0 0
\(611\) 106.444i 0.174213i
\(612\) 0 0
\(613\) 457.152i 0.745761i 0.927879 + 0.372881i \(0.121630\pi\)
−0.927879 + 0.372881i \(0.878370\pi\)
\(614\) 0 0
\(615\) 438.465 505.850i 0.712951 0.822520i
\(616\) 0 0
\(617\) 764.888i 1.23969i 0.784725 + 0.619844i \(0.212804\pi\)
−0.784725 + 0.619844i \(0.787196\pi\)
\(618\) 0 0
\(619\) 365.359i 0.590240i 0.955460 + 0.295120i \(0.0953597\pi\)
−0.955460 + 0.295120i \(0.904640\pi\)
\(620\) 0 0
\(621\) 8.70099 0.0140113
\(622\) 0 0
\(623\) 957.410 1.53677
\(624\) 0 0
\(625\) −599.797 + 175.693i −0.959676 + 0.281109i
\(626\) 0 0
\(627\) 449.976 0.717666
\(628\) 0 0
\(629\) 389.691 0.619541
\(630\) 0 0
\(631\) 62.1578i 0.0985067i 0.998786 + 0.0492534i \(0.0156842\pi\)
−0.998786 + 0.0492534i \(0.984316\pi\)
\(632\) 0 0
\(633\) 35.8868i 0.0566932i
\(634\) 0 0
\(635\) −628.959 + 725.619i −0.990486 + 1.14271i
\(636\) 0 0
\(637\) 319.630i 0.501773i
\(638\) 0 0
\(639\) 21.5889i 0.0337855i
\(640\) 0 0
\(641\) −1111.69 −1.73431 −0.867154 0.498041i \(-0.834053\pi\)
−0.867154 + 0.498041i \(0.834053\pi\)
\(642\) 0 0
\(643\) −468.983 −0.729367 −0.364683 0.931132i \(-0.618823\pi\)
−0.364683 + 0.931132i \(0.618823\pi\)
\(644\) 0 0
\(645\) −237.100 + 273.538i −0.367596 + 0.424090i
\(646\) 0 0
\(647\) −96.7647 −0.149559 −0.0747795 0.997200i \(-0.523825\pi\)
−0.0747795 + 0.997200i \(0.523825\pi\)
\(648\) 0 0
\(649\) −940.675 −1.44942
\(650\) 0 0
\(651\) 761.720i 1.17008i
\(652\) 0 0
\(653\) 920.353i 1.40942i −0.709494 0.704712i \(-0.751076\pi\)
0.709494 0.704712i \(-0.248924\pi\)
\(654\) 0 0
\(655\) 160.301 + 138.947i 0.244734 + 0.212133i
\(656\) 0 0
\(657\) 103.142i 0.156989i
\(658\) 0 0
\(659\) 591.020i 0.896844i 0.893822 + 0.448422i \(0.148014\pi\)
−0.893822 + 0.448422i \(0.851986\pi\)
\(660\) 0 0
\(661\) 306.193 0.463226 0.231613 0.972808i \(-0.425600\pi\)
0.231613 + 0.972808i \(0.425600\pi\)
\(662\) 0 0
\(663\) 224.988 0.339349
\(664\) 0 0
\(665\) 944.990 + 819.107i 1.42104 + 1.23174i
\(666\) 0 0
\(667\) −0.586909 −0.000879924
\(668\) 0 0
\(669\) −168.055 −0.251203
\(670\) 0 0
\(671\) 377.142i 0.562060i
\(672\) 0 0
\(673\) 556.892i 0.827476i 0.910396 + 0.413738i \(0.135777\pi\)
−0.910396 + 0.413738i \(0.864223\pi\)
\(674\) 0 0
\(675\) 18.4455 + 128.588i 0.0273266 + 0.190500i
\(676\) 0 0
\(677\) 58.1920i 0.0859557i −0.999076 0.0429779i \(-0.986316\pi\)
0.999076 0.0429779i \(-0.0136845\pi\)
\(678\) 0 0
\(679\) 1259.94i 1.85558i
\(680\) 0 0
\(681\) 705.389 1.03581
\(682\) 0 0
\(683\) 357.274 0.523096 0.261548 0.965191i \(-0.415767\pi\)
0.261548 + 0.965191i \(0.415767\pi\)
\(684\) 0 0
\(685\) −781.540 677.430i −1.14093 0.988949i
\(686\) 0 0
\(687\) −13.6780 −0.0199098
\(688\) 0 0
\(689\) 171.299 0.248620
\(690\) 0 0
\(691\) 614.707i 0.889590i 0.895632 + 0.444795i \(0.146724\pi\)
−0.895632 + 0.444795i \(0.853276\pi\)
\(692\) 0 0
\(693\) 284.496i 0.410528i
\(694\) 0 0
\(695\) −173.896 150.731i −0.250210 0.216879i
\(696\) 0 0
\(697\) 1328.79i 1.90644i
\(698\) 0 0
\(699\) 48.7543i 0.0697487i
\(700\) 0 0
\(701\) 886.028 1.26395 0.631975 0.774989i \(-0.282245\pi\)
0.631975 + 0.774989i \(0.282245\pi\)
\(702\) 0 0
\(703\) 593.397 0.844093
\(704\) 0 0
\(705\) −79.9036 + 92.1835i −0.113338 + 0.130757i
\(706\) 0 0
\(707\) −281.398 −0.398017
\(708\) 0 0
\(709\) 760.887 1.07318 0.536592 0.843842i \(-0.319712\pi\)
0.536592 + 0.843842i \(0.319712\pi\)
\(710\) 0 0
\(711\) 138.078i 0.194202i
\(712\) 0 0
\(713\) 77.0706i 0.108093i
\(714\) 0 0
\(715\) −245.606 + 283.351i −0.343505 + 0.396296i
\(716\) 0 0
\(717\) 512.695i 0.715056i
\(718\) 0 0
\(719\) 575.877i 0.800942i 0.916309 + 0.400471i \(0.131154\pi\)
−0.916309 + 0.400471i \(0.868846\pi\)
\(720\) 0 0
\(721\) 1373.88 1.90553
\(722\) 0 0
\(723\) −806.779 −1.11588
\(724\) 0 0
\(725\) −1.24421 8.67363i −0.00171615 0.0119636i
\(726\) 0 0
\(727\) −327.332 −0.450250 −0.225125 0.974330i \(-0.572279\pi\)
−0.225125 + 0.974330i \(0.572279\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 718.542i 0.982958i
\(732\) 0 0
\(733\) 947.567i 1.29272i −0.763031 0.646362i \(-0.776290\pi\)
0.763031 0.646362i \(-0.223710\pi\)
\(734\) 0 0
\(735\) −239.934 + 276.807i −0.326440 + 0.376609i
\(736\) 0 0
\(737\) 296.232i 0.401943i
\(738\) 0 0
\(739\) 183.234i 0.247948i 0.992285 + 0.123974i \(0.0395640\pi\)
−0.992285 + 0.123974i \(0.960436\pi\)
\(740\) 0 0
\(741\) 342.598 0.462345
\(742\) 0 0
\(743\) −183.712 −0.247258 −0.123629 0.992329i \(-0.539453\pi\)
−0.123629 + 0.992329i \(0.539453\pi\)
\(744\) 0 0
\(745\) −38.1512 + 44.0143i −0.0512096 + 0.0590797i
\(746\) 0 0
\(747\) −72.4009 −0.0969222
\(748\) 0 0
\(749\) 335.395 0.447791
\(750\) 0 0
\(751\) 345.748i 0.460384i 0.973145 + 0.230192i \(0.0739354\pi\)
−0.973145 + 0.230192i \(0.926065\pi\)
\(752\) 0 0
\(753\) 245.390i 0.325882i
\(754\) 0 0
\(755\) 473.880 + 410.754i 0.627656 + 0.544045i
\(756\) 0 0
\(757\) 549.335i 0.725674i 0.931853 + 0.362837i \(0.118192\pi\)
−0.931853 + 0.362837i \(0.881808\pi\)
\(758\) 0 0
\(759\) 28.7852i 0.0379252i
\(760\) 0 0
\(761\) 251.485 0.330467 0.165233 0.986255i \(-0.447162\pi\)
0.165233 + 0.986255i \(0.447162\pi\)
\(762\) 0 0
\(763\) 1447.57 1.89721
\(764\) 0 0
\(765\) 194.846 + 168.890i 0.254700 + 0.220771i
\(766\) 0 0
\(767\) −716.200 −0.933768
\(768\) 0 0
\(769\) 583.691 0.759026 0.379513 0.925186i \(-0.376092\pi\)
0.379513 + 0.925186i \(0.376092\pi\)
\(770\) 0 0
\(771\) 72.3216i 0.0938024i
\(772\) 0 0
\(773\) 1328.04i 1.71803i 0.511951 + 0.859015i \(0.328923\pi\)
−0.511951 + 0.859015i \(0.671077\pi\)
\(774\) 0 0
\(775\) 1138.99 163.384i 1.46966 0.210818i
\(776\) 0 0
\(777\) 375.173i 0.482848i
\(778\) 0 0
\(779\) 2023.40i 2.59743i
\(780\) 0 0
\(781\) 71.4219 0.0914492
\(782\) 0 0
\(783\) −1.82123 −0.00232597
\(784\) 0 0
\(785\) −745.141 645.880i −0.949224 0.822778i
\(786\) 0 0
\(787\) 1318.83 1.67577 0.837883 0.545850i \(-0.183793\pi\)
0.837883 + 0.545850i \(0.183793\pi\)
\(788\) 0 0
\(789\) 352.096 0.446256
\(790\) 0 0
\(791\) 308.658i 0.390212i
\(792\) 0 0
\(793\) 287.144i 0.362099i
\(794\) 0 0
\(795\) 148.349 + 128.588i 0.186603 + 0.161745i
\(796\) 0 0
\(797\) 1277.40i 1.60276i 0.598154 + 0.801381i \(0.295901\pi\)
−0.598154 + 0.801381i \(0.704099\pi\)
\(798\) 0 0
\(799\) 242.152i 0.303069i
\(800\) 0 0
\(801\) 300.598 0.375278