↑

Properties

Label 960.2.f.i.769.2
Level $960$
Weight $2$
Character 960.769
Analytic conductor $7.666$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.2.f.i.769.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +6.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +2.00000i q^{17} +2.00000 q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} +8.00000 q^{31} +2.00000i q^{33} +(2.00000 - 4.00000i) q^{35} -2.00000i q^{37} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} +(4.00000 + 2.00000i) q^{55} -10.0000 q^{59} -2.00000 q^{61} +2.00000i q^{63} +(-6.00000 + 12.0000i) q^{65} -8.00000i q^{67} +4.00000 q^{69} -12.0000 q^{71} +4.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -4.00000i q^{77} +1.00000 q^{81} +4.00000i q^{83} +(-2.00000 + 4.00000i) q^{85} +10.0000 q^{89} +12.0000 q^{91} +8.00000i q^{93} -8.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} + 4q^{11} - 2q^{15} + 4q^{21} + 6q^{25} + 16q^{31} + 4q^{35} - 12q^{39} + 4q^{41} - 4q^{45} + 6q^{49} - 4q^{51} + 8q^{55} - 20q^{59} - 4q^{61} - 12q^{65} + 8q^{69} - 24q^{71} - 8q^{75} + 2q^{81} - 4q^{85} + 20q^{89} + 24q^{91} - 4q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.00000 + 2.00000i −0.258199 + 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 2.00000 − 4.00000i 0.338062 − 0.676123i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) −2.00000 − 1.00000i −0.298142 − 0.149071i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 4.00000 + 2.00000i 0.539360 + 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −6.00000 + 12.0000i −0.744208 + 1.48842i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) −2.00000 + 4.00000i −0.216930 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 − 0.406138i \(-0.866875\pi\)
0.913812 − 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 − 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 − 0.689730i \(-0.757729\pi\)
0.724066 − 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 4.00000 + 2.00000i 0.390360 + 0.195180i
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 − 0.282216i \(-0.908930\pi\)
0.959351 − 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 4.00000 − 8.00000i 0.373002 − 0.746004i
\(116\) 0 0
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 − 0.0887357i \(-0.971717\pi\)
0.996055 − 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 − 2.00000i 0.0860663 − 0.172133i
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 − 0.768922i \(-0.720793\pi\)
0.639343 − 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) −20.0000 −1.63846 −0.819232 − 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 16.0000 + 8.00000i 1.28515 + 0.642575i
\(156\) 0 0
\(157\) − 22.0000i − 1.75579i −0.478852 − 0.877896i \(-0.658947\pi\)
0.478852 − 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) − 16.0000i − 1.25322i −0.779334 − 0.626608i \(-0.784443\pi\)
0.779334 − 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) −2.00000 + 4.00000i −0.155700 + 0.311400i
\(166\) 0 0
\(167\) − 12.0000i − 0.928588i −0.885681 − 0.464294i \(-0.846308\pi\)
0.885681 − 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 − 0.532200i \(-0.821365\pi\)
0.846619 − 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 8.00000 − 6.00000i 0.604743 − 0.453557i
\(176\) 0 0
\(177\) − 10.0000i − 0.751646i
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 − 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) − 2.00000i − 0.147844i
\(184\) 0 0
\(185\) 2.00000 − 4.00000i 0.147043 − 0.294086i
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 0 0
\(195\) −12.0000 − 6.00000i −0.859338 − 0.429669i
\(196\) 0 0
\(197\) − 22.0000i − 1.56744i −0.621117 − 0.783718i \(-0.713321\pi\)
0.621117 − 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 + 2.00000i 0.279372 + 0.139686i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) − 12.0000i − 0.822226i
\(214\) 0 0
\(215\) −4.00000 + 8.00000i −0.272798 + 0.545595i
\(216\) 0 0
\(217\) − 16.0000i − 1.08615i
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 26.0000i 1.74109i 0.492090 + 0.870544i \(0.336233\pi\)
−0.492090 + 0.870544i \(0.663767\pi\)
\(224\) 0 0
\(225\) −3.00000 − 4.00000i −0.200000 − 0.266667i
\(226\) 0 0
\(227\) − 28.0000i − 1.85843i −0.369546 − 0.929213i \(-0.620487\pi\)
0.369546 − 0.929213i \(-0.379513\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 14.0000i 0.917170i 0.888650 + 0.458585i \(0.151644\pi\)
−0.888650 + 0.458585i \(0.848356\pi\)
\(234\) 0 0
\(235\) −8.00000 + 16.0000i −0.521862 + 1.04372i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.00000 + 3.00000i 0.383326 + 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 − 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 0 0
\(255\) −4.00000 − 2.00000i −0.250490 − 0.125245i
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 − 0.561405i \(-0.810261\pi\)
0.827541 − 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 4.00000i − 0.246651i −0.992366 − 0.123325i \(-0.960644\pi\)
0.992366 − 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) −6.00000 + 12.0000i −0.368577 + 0.737154i
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 0 0
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 12.0000i 0.726273i
\(274\) 0 0
\(275\) 6.00000 + 8.00000i 0.361814 + 0.482418i
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 − 0.0600842i \(-0.980863\pi\)
0.998193 − 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 − 0.475551i \(-0.842249\pi\)
0.879688 − 0.475551i \(-0.157751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4.00000i − 0.236113i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) −20.0000 − 10.0000i −1.16445 − 0.582223i
\(296\) 0 0
\(297\) − 2.00000i − 0.116052i
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 8.00000i 0.459588i
\(304\) 0 0
\(305\) −4.00000 − 2.00000i −0.229039 − 0.114520i
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) −2.00000 + 4.00000i −0.112687 + 0.225374i
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 − 0.0561656i \(-0.982113\pi\)
0.998421 − 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −24.0000 + 18.0000i −1.33128 + 0.998460i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 8.00000 − 16.0000i 0.437087 − 0.874173i
\(336\) 0 0
\(337\) − 28.0000i − 1.52526i −0.646837 − 0.762629i \(-0.723908\pi\)
0.646837 − 0.762629i \(-0.276092\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 8.00000 + 4.00000i 0.430706 + 0.215353i
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) −24.0000 − 12.0000i −1.27379 − 0.636894i
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) −4.00000 + 8.00000i −0.209370 + 0.418739i
\(366\) 0 0
\(367\) − 2.00000i − 0.104399i −0.998637 − 0.0521996i \(-0.983377\pi\)
0.998637 − 0.0521996i \(-0.0166232\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 0 0
\(375\) −11.0000 + 2.00000i −0.568038 + 0.103280i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 0 0
\(385\) 4.00000 − 8.00000i 0.203859 − 0.407718i
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) 20.0000 1.01404 0.507020 − 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) − 18.0000i − 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 − 0.0501886i \(-0.984018\pi\)
0.998740 − 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 − 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 48.0000i 2.39105i
\(404\) 0 0
\(405\) 2.00000 + 1.00000i 0.0993808 + 0.0496904i
\(406\) 0 0
\(407\) − 4.00000i − 0.198273i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 20.0000i 0.984136i
\(414\) 0 0
\(415\) −4.00000 + 8.00000i −0.196352 + 0.392705i
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 − 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) − 8.00000i − 0.388973i
\(424\) 0 0
\(425\) −8.00000 + 6.00000i −0.388057 + 0.291043i
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 36.0000i − 1.71041i −0.518289 − 0.855206i \(-0.673431\pi\)
0.518289 − 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) 20.0000 + 10.0000i 0.948091 + 0.474045i
\(446\) 0 0
\(447\) − 20.0000i − 0.945968i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 24.0000 + 12.0000i 1.12514 + 0.562569i
\(456\) 0 0
\(457\) 32.0000i 1.49690i 0.663193 + 0.748448i \(0.269201\pi\)
−0.663193 + 0.748448i \(0.730799\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i 0.990233 + 0.139422i \(0.0445244\pi\)
−0.990233 + 0.139422i \(0.955476\pi\)
\(464\) 0 0
\(465\) −8.00000 + 16.0000i −0.370991 + 0.741982i
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 − 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) − 8.00000i − 0.364013i
\(484\) 0 0
\(485\) 8.00000 − 16.0000i 0.363261 − 0.726523i
\(486\) 0 0
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 − 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 − 2.00000i −0.179787 − 0.0898933i
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 24.0000i − 1.07011i −0.844818 − 0.535054i \(-0.820291\pi\)
0.844818 − 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 16.0000 + 8.00000i 0.711991 + 0.355995i
\(506\) 0 0
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.0000 − 28.0000i 0.616914 − 1.23383i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 − 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 − 0.349816i \(-0.886244\pi\)
0.936819 − 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 6.00000 + 8.00000i 0.261861 + 0.349149i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −12.0000 + 24.0000i −0.518805 + 1.03761i
\(536\) 0 0
\(537\) − 10.0000i − 0.431532i
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) − 2.00000i − 0.0858282i
\(544\) 0 0
\(545\) 20.0000 + 10.0000i 0.856706 + 0.428353i
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 − 0.598597i \(-0.795725\pi\)
0.801050 − 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.00000 + 2.00000i 0.169791 + 0.0848953i
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 0 0
\(565\) 6.00000 − 12.0000i 0.252422 − 0.504844i
\(566\) 0 0
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 − 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) − 12.0000i − 0.501307i
\(574\) 0 0
\(575\) 16.0000 − 12.0000i 0.667246 − 0.500435i
\(576\) 0 0
\(577\) 32.0000i 1.33218i 0.745873 + 0.666089i \(0.232033\pi\)
−0.745873 + 0.666089i \(0.767967\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) 6.00000 − 12.0000i 0.248069 − 0.496139i
\(586\) 0 0
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 0 0
\(593\) − 6.00000i − 0.246390i −0.992382 − 0.123195i \(-0.960686\pi\)
0.992382 − 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 8.00000 + 4.00000i 0.327968 + 0.163984i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) −14.0000 − 7.00000i −0.569181 − 0.284590i
\(606\) 0 0
\(607\) − 22.0000i − 0.892952i −0.894795 − 0.446476i \(-0.852679\pi\)
0.894795 − 0.446476i \(-0.147321\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) −2.00000 + 4.00000i −0.0806478 + 0.161296i
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) − 20.0000i − 0.801283i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 − 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) 2.00000 − 4.00000i 0.0793676 − 0.158735i
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 24.0000i 0.946468i 0.880937 + 0.473234i \(0.156913\pi\)
−0.880937 + 0.473234i \(0.843087\pi\)
\(644\) 0 0
\(645\) −8.00000 − 4.00000i −0.315000 − 0.157500i
\(646\) 0 0
\(647\) 48.0000i 1.88707i 0.331266 + 0.943537i \(0.392524\pi\)
−0.331266 + 0.943537i \(0.607476\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) 26.0000i 1.01746i 0.860927 + 0.508729i \(0.169885\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 0 0
\(655\) −36.0000 − 18.0000i −1.40664 − 0.703318i
\(656\) 0 0
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) 50.0000 1.94772 0.973862 − 0.227142i \(-0.0729380\pi\)
0.973862 + 0.227142i \(0.0729380\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) − 12.0000i − 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) − 36.0000i − 1.38770i −0.720121 − 0.693849i \(-0.755914\pi\)
0.720121 − 0.693849i \(-0.244086\pi\)
\(674\) 0 0
\(675\) 4.00000 − 3.00000i 0.153960 − 0.115470i
\(676\) 0 0
\(677\) − 2.00000i − 0.0768662i −0.999261 − 0.0384331i \(-0.987763\pi\)
0.999261 − 0.0384331i \(-0.0122367\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) 18.0000 − 36.0000i 0.687745 − 1.37549i
\(686\) 0 0
\(687\) − 10.0000i − 0.381524i
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 − 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 40.0000 + 20.0000i 1.51729 + 0.758643i
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 − 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −16.0000 − 8.00000i −0.602595 − 0.301297i
\(706\) 0 0
\(707\) − 16.0000i − 0.601742i
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 − 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 32.0000i − 1.19841i
\(714\) 0 0
\(715\) −12.0000 + 24.0000i −0.448775 + 0.897549i
\(716\) 0 0
\(717\) 20.0000i 0.746914i
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) 22.0000i 0.818189i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000i 0.667583i 0.942647 + 0.333792i \(0.108328\pi\)
−0.942647 + 0.333792i \(0.891672\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) − 14.0000i − 0.517102i −0.965998 − 0.258551i \(-0.916755\pi\)
0.965998 − 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) −3.00000 + 6.00000i −0.110657 + 0.221313i
\(736\) 0 0
\(737\) − 16.0000i − 0.589368i
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 − 0.440237i \(-0.854894\pi\)
0.897881 − 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) −40.0000 − 20.0000i −1.46549 − 0.732743i
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 16.0000 + 8.00000i 0.582300 + 0.291150i
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 − 0.0363456i \(-0.988428\pi\)
0.999339 − 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) − 20.0000i − 0.724049i
\(764\) 0 0
\(765\) 2.00000 − 4.00000i 0.0723102 − 0.144620i
\(766\) 0 0
\(767\) − 60.0000i − 2.16647i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) − 54.0000i − 1.94225i −0.238581 − 0.971123i \(-0.576682\pi\)
0.238581 − 0.971123i \(-0.423318\pi\)
\(774\) 0 0
\(775\) 24.0000 + 32.0000i 0.862105 + 1.14947i
\(776\) 0 0
\(777\) − 4.00000i − 0.143499i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.0000 − 44.0000i 0.785214 − 1.57043i
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) − 12.0000i − 0.426132i
\(794\) 0 0
\(795\) −12.0000 − 6.00000i −0.425596 − 0.212798i
\(796\) 0 0
\(797\) − 2.00000i − 0.0708436i −0.999372 − 0.0354218i \(-0.988723\pi\)
0.999372 − 0.0354218i \(-0.0112775\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 8.00000i 0.282314i
\(804\) 0