Properties

Label 3120.2.l.g.1249.1
Level $3120$
Weight $2$
Character 3120.1249
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.g.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} -2.00000i q^{17} -4.00000 q^{19} +2.00000 q^{21} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} -4.00000 q^{29} -8.00000 q^{31} +2.00000i q^{33} +(2.00000 + 4.00000i) q^{35} -6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} +2.00000i q^{53} +(-4.00000 + 2.00000i) q^{55} +4.00000i q^{57} +10.0000 q^{59} -14.0000 q^{61} -2.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} -16.0000i q^{67} +4.00000 q^{71} +8.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} -4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +(-2.00000 - 4.00000i) q^{85} +4.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} +8.00000i q^{93} +(-8.00000 + 4.00000i) q^{95} -12.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{9} - 4q^{11} - 2q^{15} - 8q^{19} + 4q^{21} + 6q^{25} - 8q^{29} - 16q^{31} + 4q^{35} - 2q^{39} - 12q^{41} - 4q^{45} + 6q^{49} - 4q^{51} - 8q^{55} + 20q^{59} - 28q^{61} - 2q^{65} + 8q^{71} - 8q^{75} - 16q^{79} + 2q^{81} - 4q^{85} - 12q^{89} + 4q^{91} - 16q^{95} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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 \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) −1.00000 2.00000i −0.258199 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) 2.00000 + 4.00000i 0.338062 + 0.676123i
\(36\) 0 0
\(37\) 6.00000i 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −2.00000 + 1.00000i −0.298142 + 0.149071i
\(46\) 0 0
\(47\) 8.00000i 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) −4.00000 + 2.00000i −0.539360 + 0.269680i
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −1.00000 2.00000i −0.124035 0.248069i
\(66\) 0 0
\(67\) 16.0000i 1.95471i −0.211604 0.977356i \(-0.567869\pi\)
0.211604 0.977356i \(-0.432131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) −4.00000 3.00000i −0.461880 0.346410i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) −2.00000 4.00000i −0.216930 0.433861i
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) −8.00000 + 4.00000i −0.820783 + 0.410391i
\(96\) 0 0
\(97\) 12.0000i 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 4.00000 2.00000i 0.390360 0.195180i
\(106\) 0 0
\(107\) 4.00000i 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 1.00000 + 2.00000i 0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 14.0000i 1.19610i 0.801459 + 0.598050i \(0.204058\pi\)
−0.801459 + 0.598050i \(0.795942\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) −8.00000 + 4.00000i −0.664364 + 0.332182i
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) −16.0000 + 8.00000i −1.28515 + 0.642575i
\(156\) 0 0
\(157\) 10.0000i 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) 2.00000 + 4.00000i 0.155700 + 0.311400i
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) −6.00000 12.0000i −0.441129 0.882258i
\(186\) 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 \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −2.00000 + 1.00000i −0.143223 + 0.0716115i
\(196\) 0 0
\(197\) 10.0000i 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) 8.00000i 0.561490i
\(204\) 0 0
\(205\) −12.0000 + 6.00000i −0.838116 + 0.419058i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 4.00000i 0.274075i
\(214\) 0 0
\(215\) −4.00000 8.00000i −0.272798 0.545595i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 14.0000i 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 0 0
\(235\) −8.00000 16.0000i −0.521862 1.04372i
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 6.00000 3.00000i 0.383326 0.191663i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.00000 + 2.00000i −0.250490 + 0.125245i
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 2.00000 + 4.00000i 0.122859 + 0.245718i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) −6.00000 + 8.00000i −0.361814 + 0.482418i
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 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
\(285\) 4.00000 + 8.00000i 0.236940 + 0.473879i
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) 26.0000i 1.51894i 0.650545 + 0.759468i \(0.274541\pi\)
−0.650545 + 0.759468i \(0.725459\pi\)
\(294\) 0 0
\(295\) 20.0000 10.0000i 1.16445 0.582223i
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 4.00000i 0.229794i
\(304\) 0 0
\(305\) −28.0000 + 14.0000i −1.60328 + 0.801638i
\(306\) 0 0
\(307\) 28.0000i 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i −0.824931 0.565233i \(-0.808786\pi\)
0.824931 0.565233i \(-0.191214\pi\)
\(314\) 0 0
\(315\) −2.00000 4.00000i −0.112687 0.225374i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) −4.00000 3.00000i −0.221880 0.166410i
\(326\) 0 0
\(327\) 18.0000i 0.995402i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) −16.0000 32.0000i −0.874173 1.74835i
\(336\) 0 0
\(337\) 20.0000i 1.08947i −0.838608 0.544735i \(-0.816630\pi\)
0.838608 0.544735i \(-0.183370\pi\)
\(338\) 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
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 34.0000i 1.80964i −0.425797 0.904819i \(-0.640006\pi\)
0.425797 0.904819i \(-0.359994\pi\)
\(354\) 0 0
\(355\) 8.00000 4.00000i 0.424596 0.212298i
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 8.00000 + 16.0000i 0.418739 + 0.837478i
\(366\) 0 0
\(367\) 10.0000i 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 22.0000i 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 0 0
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 4.00000i 0.206010i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) −4.00000 8.00000i −0.203859 0.407718i
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) −16.0000 + 8.00000i −0.805047 + 0.402524i
\(396\) 0 0
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 20.0000i 0.984136i
\(414\) 0 0
\(415\) −12.0000 24.0000i −0.589057 1.17811i
\(416\) 0 0
\(417\) 4.00000i 0.195881i
\(418\) 0 0
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) −8.00000 6.00000i −0.388057 0.291043i
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) 12.0000i 0.576683i 0.957528 + 0.288342i \(0.0931039\pi\)
−0.957528 + 0.288342i \(0.906896\pi\)
\(434\) 0 0
\(435\) 4.00000 + 8.00000i 0.191785 + 0.383571i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 4.00000i 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 0 0
\(445\) −12.0000 + 6.00000i −0.568855 + 0.284427i
\(446\) 0 0
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) 4.00000 2.00000i 0.187523 0.0937614i
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 8.00000 + 16.0000i 0.370991 + 0.741982i
\(466\) 0 0
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) 0 0
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.0000 24.0000i −0.544892 1.08978i
\(486\) 0 0
\(487\) 26.0000i 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 4.00000 2.00000i 0.179787 0.0898933i
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 8.00000 4.00000i 0.355995 0.177998i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) 40.0000 1.77297 0.886484 0.462758i \(-0.153140\pi\)
0.886484 + 0.462758i \(0.153140\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) −6.00000 12.0000i −0.264392 0.528783i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 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
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) −4.00000 8.00000i −0.172935 0.345870i
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 6.00000i 0.257485i
\(544\) 0 0
\(545\) 36.0000 18.0000i 1.54207 0.771035i
\(546\) 0 0
\(547\) 20.0000i 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −12.0000 + 6.00000i −0.509372 + 0.254686i
\(556\) 0 0
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 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
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.0000i 1.16566i −0.812596 0.582828i \(-0.801946\pi\)
0.812596 0.582828i \(-0.198054\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 1.00000 + 2.00000i 0.0413449 + 0.0826898i
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 0 0
\(595\) 8.00000 4.00000i 0.327968 0.163984i
\(596\) 0 0
\(597\) 24.0000i 0.982255i
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) 16.0000i 0.651570i
\(604\) 0 0
\(605\) −14.0000 + 7.00000i −0.569181 + 0.284590i
\(606\) 0 0
\(607\) 18.0000i 0.730597i 0.930890 + 0.365299i \(0.119033\pi\)
−0.930890 + 0.365299i \(0.880967\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 2.00000i 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 6.00000 + 12.0000i 0.241943 + 0.483887i
\(616\) 0 0
\(617\) 46.0000i 1.85189i −0.377658 0.925945i \(-0.623271\pi\)
0.377658 0.925945i \(-0.376729\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 6.00000 + 12.0000i 0.238103 + 0.476205i
\(636\) 0 0
\(637\) 3.00000i 0.118864i
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) 32.0000i 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) 0 0
\(645\) −8.00000 + 4.00000i −0.315000 + 0.157500i
\(646\) 0 0
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) 34.0000i 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 0 0
\(655\) 12.0000 6.00000i 0.468879 0.234439i
\(656\) 0 0
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 2.00000i 0.0776736i
\(664\) 0 0
\(665\) −8.00000 16.0000i −0.310227 0.620453i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 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
\(677\) 10.0000i 0.384331i 0.981363 + 0.192166i \(0.0615511\pi\)
−0.981363 + 0.192166i \(0.938449\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 14.0000 + 28.0000i 0.534913 + 1.06983i
\(686\) 0 0
\(687\) 22.0000i 0.839352i
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 8.00000 4.00000i 0.303457 0.151729i
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 24.0000i 0.905177i
\(704\) 0 0
\(705\) −16.0000 + 8.00000i −0.602595 + 0.301297i
\(706\) 0 0
\(707\) 8.00000i 0.300871i
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 + 4.00000i 0.0747958 + 0.149592i
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) −12.0000 + 16.0000i −0.445669 + 0.594225i
\(726\) 0 0
\(727\) 34.0000i 1.26099i 0.776193 + 0.630495i \(0.217148\pi\)
−0.776193 + 0.630495i \(0.782852\pi\)
\(728\) 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 \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) −3.00000 6.00000i −0.110657 0.221313i
\(736\) 0 0
\(737\) 32.0000i 1.17874i
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 16.0000i 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) 0 0
\(745\) −24.0000 + 12.0000i −0.879292 + 0.439646i
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 32.0000 16.0000i 1.16460 0.582300i
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) 36.0000i 1.30329i
\(764\) 0 0
\(765\) 2.00000 + 4.00000i 0.0723102 + 0.144620i
\(766\) 0 0
\(767\) 10.0000i 0.361079i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 22.0000i 0.791285i 0.918405 + 0.395643i \(0.129478\pi\)
−0.918405 + 0.395643i \(0.870522\pi\)
\(774\) 0 0
\(775\) −24.0000 + 32.0000i −0.862105 + 1.14947i
\(776\) 0 0
\(777\) 12.0000i 0.430498i
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 4.00000i 0.142948i
\(784\) 0 0
\(785\) −10.0000 20.0000i −0.356915 0.713831i
\(786\) 0 0
\(787\) 8.00000i 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) 0 0
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 14.0000i 0.497155i
\(794\) 0 0
\(795\) 4.00000 2.00000i 0.141865 0.0709327i
\(796\) 0 0
\(797\) 14.0000i 0.495905i −0.968772 0.247953i \(-0.920242\pi\)
0.968772 0.247953i \(-0.0797578\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0