Properties

Label 3120.2.l.i.1249.2
Level $3120$
Weight $2$
Character 3120.1249
Analytic conductor $24.913$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \( x^{2} + 1 \) Copy content Toggle raw display
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 1249.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.i.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} -1.00000i q^{13} +(1.00000 + 2.00000i) q^{15} -4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000i q^{27} +10.0000 q^{29} -4.00000 q^{31} +6.00000i q^{33} +(4.00000 + 8.00000i) q^{35} -6.00000i q^{37} +1.00000 q^{39} +10.0000 q^{41} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} +6.00000i q^{53} +(12.0000 - 6.00000i) q^{55} +2.00000i q^{57} -6.00000 q^{59} -6.00000 q^{61} -4.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} +12.0000i q^{67} +6.00000 q^{69} -2.00000i q^{73} +(4.00000 + 3.00000i) q^{75} +24.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(-4.00000 - 8.00000i) q^{85} +10.0000i q^{87} -14.0000 q^{89} +4.00000 q^{91} -4.00000i q^{93} +(4.00000 - 2.00000i) q^{95} +14.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 20 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{39} + 20 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} + 24 q^{55} - 12 q^{59} - 12 q^{61} - 2 q^{65} + 12 q^{69} + 8 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) 4.00000i 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 4.00000 + 8.00000i 0.676123 + 1.35225i
\(36\) 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
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 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
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 12.0000 6.00000i 1.61808 0.809040i
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) −1.00000 2.00000i −0.124035 0.248069i
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 4.00000 + 3.00000i 0.461880 + 0.346410i
\(76\) 0 0
\(77\) 24.0000i 2.73505i
\(78\) 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
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) −4.00000 8.00000i −0.433861 0.867722i
\(86\) 0 0
\(87\) 10.0000i 1.07211i
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 4.00000 2.00000i 0.410391 0.205196i
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 2.00000i 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 0 0
\(105\) −8.00000 + 4.00000i −0.780720 + 0.390360i
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) −6.00000 12.0000i −0.559503 1.11901i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 2.00000 11.0000i 0.178885 0.983870i
\(126\) 0 0
\(127\) 10.0000i 0.887357i 0.896186 + 0.443678i \(0.146327\pi\)
−0.896186 + 0.443678i \(0.853673\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 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 \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 20.0000 10.0000i 1.66091 0.830455i
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) −8.00000 + 4.00000i −0.642575 + 0.321288i
\(156\) 0 0
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) 4.00000i 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 6.00000 + 12.0000i 0.467099 + 0.934199i
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 0 0
\(175\) 16.0000 + 12.0000i 1.20949 + 0.907115i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) −6.00000 12.0000i −0.441129 0.882258i
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0 0
\(195\) 2.00000 1.00000i 0.143223 0.0716115i
\(196\) 0 0
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 40.0000i 2.80745i
\(204\) 0 0
\(205\) 20.0000 10.0000i 1.39686 0.698430i
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) −24.0000 −1.57908
\(232\) 0 0
\(233\) 12.0000i 0.786146i 0.919507 + 0.393073i \(0.128588\pi\)
−0.919507 + 0.393073i \(0.871412\pi\)
\(234\) 0 0
\(235\) −8.00000 16.0000i −0.521862 1.04372i
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −18.0000 + 9.00000i −1.14998 + 0.574989i
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 36.0000i 2.26330i
\(254\) 0 0
\(255\) 8.00000 4.00000i 0.500979 0.250490i
\(256\) 0 0
\(257\) 28.0000i 1.74659i 0.487190 + 0.873296i \(0.338022\pi\)
−0.487190 + 0.873296i \(0.661978\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 10.0000i 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) 0 0
\(265\) 6.00000 + 12.0000i 0.368577 + 0.737154i
\(266\) 0 0
\(267\) 14.0000i 0.856786i
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 0 0
\(275\) 18.0000 24.0000i 1.08544 1.44725i
\(276\) 0 0
\(277\) 14.0000i 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 0 0
\(285\) 2.00000 + 4.00000i 0.118470 + 0.236940i
\(286\) 0 0
\(287\) 40.0000i 2.36113i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 18.0000i 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) −12.0000 + 6.00000i −0.698667 + 0.349334i
\(296\) 0 0
\(297\) 6.00000i 0.348155i
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 10.0000i 0.574485i
\(304\) 0 0
\(305\) −12.0000 + 6.00000i −0.687118 + 0.343559i
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) 0 0
\(315\) −4.00000 8.00000i −0.225374 0.450749i
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) 60.0000 3.35936
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) −4.00000 3.00000i −0.221880 0.166410i
\(326\) 0 0
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 12.0000 + 24.0000i 0.655630 + 1.31126i
\(336\) 0 0
\(337\) 16.0000i 0.871576i 0.900049 + 0.435788i \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) 0 0
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 12.0000 6.00000i 0.646058 0.323029i
\(346\) 0 0
\(347\) 16.0000i 0.858925i −0.903085 0.429463i \(-0.858703\pi\)
0.903085 0.429463i \(-0.141297\pi\)
\(348\) 0 0
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) −2.00000 4.00000i −0.104685 0.209370i
\(366\) 0 0
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 18.0000i 0.932005i 0.884783 + 0.466002i \(0.154306\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(374\) 0 0
\(375\) 11.0000 + 2.00000i 0.568038 + 0.103280i
\(376\) 0 0
\(377\) 10.0000i 0.515026i
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −10.0000 −0.512316
\(382\) 0 0
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 0 0
\(385\) 24.0000 + 48.0000i 1.22315 + 2.44631i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) 8.00000i 0.403547i
\(394\) 0 0
\(395\) −16.0000 + 8.00000i −0.805047 + 0.402524i
\(396\) 0 0
\(397\) 10.0000i 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 36.0000i 1.78445i
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 24.0000i 1.18096i
\(414\) 0 0
\(415\) −4.00000 8.00000i −0.196352 0.392705i
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) −16.0000 12.0000i −0.776114 0.582086i
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) 0 0
\(435\) 10.0000 + 20.0000i 0.479463 + 0.958927i
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −28.0000 + 14.0000i −1.32733 + 0.663664i
\(446\) 0 0
\(447\) 16.0000i 0.756774i
\(448\) 0 0
\(449\) −38.0000 −1.79333 −0.896665 0.442709i \(-0.854018\pi\)
−0.896665 + 0.442709i \(0.854018\pi\)
\(450\) 0 0
\(451\) 60.0000 2.82529
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) 8.00000 4.00000i 0.375046 0.187523i
\(456\) 0 0
\(457\) 14.0000i 0.654892i 0.944870 + 0.327446i \(0.106188\pi\)
−0.944870 + 0.327446i \(0.893812\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 0 0
\(465\) −4.00000 8.00000i −0.185496 0.370991i
\(466\) 0 0
\(467\) 8.00000i 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 0 0
\(469\) −48.0000 −2.21643
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 8.00000i 0.275299 0.367065i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 24.0000i 1.09204i
\(484\) 0 0
\(485\) 14.0000 + 28.0000i 0.635707 + 1.27141i
\(486\) 0 0
\(487\) 32.0000i 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 40.0000i 1.80151i
\(494\) 0 0
\(495\) −12.0000 + 6.00000i −0.539360 + 0.269680i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −18.0000 −0.805791 −0.402895 0.915246i \(-0.631996\pi\)
−0.402895 + 0.915246i \(0.631996\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) 14.0000i 0.624229i −0.950044 0.312115i \(-0.898963\pi\)
0.950044 0.312115i \(-0.101037\pi\)
\(504\) 0 0
\(505\) 20.0000 10.0000i 0.889988 0.444994i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −16.0000 −0.709188 −0.354594 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 2.00000i 0.0883022i
\(514\) 0 0
\(515\) −2.00000 4.00000i −0.0881305 0.176261i
\(516\) 0 0
\(517\) 48.0000i 2.11104i
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) −12.0000 + 16.0000i −0.523723 + 0.698297i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 24.0000i 1.03568i
\(538\) 0 0
\(539\) −54.0000 −2.32594
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 0 0
\(543\) 6.00000i 0.257485i
\(544\) 0 0
\(545\) −8.00000 + 4.00000i −0.342682 + 0.171341i
\(546\) 0 0
\(547\) 36.0000i 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 12.0000 6.00000i 0.509372 0.254686i
\(556\) 0 0
\(557\) 18.0000i 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 4.00000 + 8.00000i 0.168281 + 0.336563i
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 18.0000i −1.00087 0.750652i
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 0 0
\(585\) 1.00000 + 2.00000i 0.0413449 + 0.0826898i
\(586\) 0 0
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 14.0000i 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) 32.0000 16.0000i 1.31187 0.655936i
\(596\) 0 0
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 50.0000 25.0000i 2.03279 1.01639i
\(606\) 0 0
\(607\) 42.0000i 1.70473i −0.522949 0.852364i \(-0.675168\pi\)
0.522949 0.852364i \(-0.324832\pi\)
\(608\) 0 0
\(609\) −40.0000 −1.62088
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 38.0000i 1.53481i 0.641165 + 0.767403i \(0.278451\pi\)
−0.641165 + 0.767403i \(0.721549\pi\)
\(614\) 0 0
\(615\) 10.0000 + 20.0000i 0.403239 + 0.806478i
\(616\) 0 0
\(617\) 18.0000i 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 56.0000i 2.24359i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 12.0000i 0.479234i
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 10.0000 + 20.0000i 0.396838 + 0.793676i
\(636\) 0 0
\(637\) 9.00000i 0.356593i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 0 0
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) 0 0
\(655\) 16.0000 8.00000i 0.625172 0.312586i
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 4.00000i 0.155347i
\(664\) 0 0
\(665\) 8.00000 + 16.0000i 0.310227 + 0.620453i
\(666\) 0 0
\(667\) 60.0000i 2.32321i
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) 44.0000i 1.69608i −0.529936 0.848038i \(-0.677784\pi\)
0.529936 0.848038i \(-0.322216\pi\)
\(674\) 0 0
\(675\) −4.00000 3.00000i −0.153960 0.115470i
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 20.0000i 0.765279i −0.923898 0.382639i \(-0.875015\pi\)
0.923898 0.382639i \(-0.124985\pi\)
\(684\) 0 0
\(685\) −14.0000 28.0000i −0.534913 1.06983i
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 0 0
\(693\) 24.0000i 0.911685i
\(694\) 0 0
\(695\) 32.0000 16.0000i 1.21383 0.606915i
\(696\) 0 0
\(697\) 40.0000i 1.51511i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 12.0000i 0.452589i
\(704\) 0 0
\(705\) 16.0000 8.00000i 0.602595 0.301297i
\(706\) 0 0
\(707\) 40.0000i 1.50435i
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 24.0000i 0.898807i
\(714\) 0 0
\(715\) −6.00000 12.0000i −0.224387 0.448775i
\(716\) 0 0
\(717\) 4.00000i 0.149383i
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 30.0000 40.0000i 1.11417 1.48556i
\(726\) 0 0
\(727\) 38.0000i 1.40934i −0.709534 0.704671i \(-0.751095\pi\)
0.709534 0.704671i \(-0.248905\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 18.0000i 0.664845i 0.943131 + 0.332423i \(0.107866\pi\)
−0.943131 + 0.332423i \(0.892134\pi\)
\(734\) 0 0
\(735\) −9.00000 18.0000i −0.331970 0.663940i
\(736\) 0 0
\(737\) 72.0000i 2.65215i
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 36.0000i 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) −32.0000 + 16.0000i −1.17239 + 0.586195i
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 20.0000i 0.728841i
\(754\) 0 0
\(755\) −32.0000 + 16.0000i −1.16460 + 0.582300i
\(756\) 0 0
\(757\) 22.0000i 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 16.0000i 0.579239i
\(764\) 0 0
\(765\) 4.00000 + 8.00000i 0.144620 + 0.289241i
\(766\) 0 0
\(767\) 6.00000i 0.216647i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −28.0000 −1.00840
\(772\) 0 0
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) −12.0000 + 16.0000i −0.431053 + 0.574737i
\(776\) 0 0
\(777\) 24.0000i 0.860995i
\(778\) 0 0
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 10.0000i 0.357371i
\(784\) 0 0
\(785\) 6.00000 + 12.0000i 0.214149 + 0.428298i
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) 10.0000 0.356009
\(790\) 0 0
\(791\) −16.0000 −0.568895
\(792\) 0 0
\(793\) 6.00000i 0.213066i
\(794\) 0 0
\(795\) −12.0000 + 6.00000i −0.425596 + 0.212798i
\(796\) 0 0
\(797\) 30.0000i 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0