Properties

Label 3120.2.l.h.1249.1
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 1249.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.h.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +2.00000i q^{33} +(-4.00000 - 8.00000i) q^{35} -6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -10.0000i q^{53} +(-4.00000 + 2.00000i) q^{55} -2.00000i q^{57} -14.0000 q^{59} +10.0000 q^{61} +4.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} -10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +(4.00000 + 8.00000i) q^{85} -2.00000i q^{87} +18.0000 q^{89} -4.00000 q^{91} -4.00000i q^{93} +(4.00000 - 2.00000i) q^{95} +6.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 4 q^{29} + 8 q^{31} - 8 q^{35} - 2 q^{39} - 12 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 28 q^{59} + 20 q^{61} - 2 q^{65} - 12 q^{69} - 16 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} + 8 q^{85} + 36 q^{89} - 8 q^{91} + 8 q^{95} + 4 q^{99}+O(q^{100}) \) 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 \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 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
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 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
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 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
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 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
\(53\) 10.0000i 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) −4.00000 + 2.00000i −0.539360 + 0.269680i
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) −1.00000 2.00000i −0.124035 0.248069i
\(66\) 0 0
\(67\) 4.00000i 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) −4.00000 3.00000i −0.461880 0.346410i
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 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 \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 4.00000 + 8.00000i 0.433861 + 0.867722i
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 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
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) −8.00000 + 4.00000i −0.780720 + 0.390360i
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 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
\(121\) −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
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 1.00000 + 2.00000i 0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 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
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 4.00000 2.00000i 0.332182 0.166091i
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) 8.00000 4.00000i 0.642575 0.321288i
\(156\) 0 0
\(157\) 10.0000i 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 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
\(165\) 2.00000 + 4.00000i 0.155700 + 0.311400i
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 10.0000i 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 0 0
\(175\) −16.0000 12.0000i −1.20949 0.907115i
\(176\) 0 0
\(177\) 14.0000i 1.05230i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) −6.00000 12.0000i −0.441129 0.882258i
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 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
\(193\) 6.00000i 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 0 0
\(195\) −2.00000 + 1.00000i −0.143223 + 0.0716115i
\(196\) 0 0
\(197\) 26.0000i 1.85242i 0.377004 + 0.926212i \(0.376954\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 8.00000i 0.561490i
\(204\) 0 0
\(205\) −12.0000 + 6.00000i −0.838116 + 0.419058i
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 0 0
\(213\) 8.00000i 0.548151i
\(214\) 0 0
\(215\) 8.00000 + 16.0000i 0.545595 + 1.09119i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 16.0000i 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) 12.0000i 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) 4.00000i 0.262049i 0.991379 + 0.131024i \(0.0418266\pi\)
−0.991379 + 0.131024i \(0.958173\pi\)
\(234\) 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
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 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
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) 0 0
\(255\) 8.00000 4.00000i 0.500979 0.250490i
\(256\) 0 0
\(257\) 12.0000i 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 10.0000i 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) 0 0
\(265\) −10.0000 20.0000i −0.614295 1.22859i
\(266\) 0 0
\(267\) 18.0000i 1.10158i
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 4.00000i 0.242091i
\(274\) 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
\(279\) −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
\(283\) 4.00000i 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) −2.00000 4.00000i −0.118470 0.236940i
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) −28.0000 + 14.0000i −1.63022 + 0.815112i
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 0 0
\(303\) 14.0000i 0.804279i
\(304\) 0 0
\(305\) 20.0000 10.0000i 1.14520 0.572598i
\(306\) 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
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i −0.974106 0.226093i \(-0.927405\pi\)
0.974106 0.226093i \(-0.0725954\pi\)
\(314\) 0 0
\(315\) 4.00000 + 8.00000i 0.225374 + 0.450749i
\(316\) 0 0
\(317\) 30.0000i 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 0 0
\(325\) −4.00000 3.00000i −0.221880 0.166410i
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) −4.00000 8.00000i −0.218543 0.437087i
\(336\) 0 0
\(337\) 32.0000i 1.74315i −0.490261 0.871576i \(-0.663099\pi\)
0.490261 0.871576i \(-0.336901\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) −12.0000 + 6.00000i −0.646058 + 0.323029i
\(346\) 0 0
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 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
\(355\) −16.0000 + 8.00000i −0.849192 + 0.424596i
\(356\) 0 0
\(357\) 16.0000i 0.846810i
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) −10.0000 20.0000i −0.523424 1.04685i
\(366\) 0 0
\(367\) 22.0000i 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −40.0000 −2.07670
\(372\) 0 0
\(373\) 2.00000i 0.103556i 0.998659 + 0.0517780i \(0.0164888\pi\)
−0.998659 + 0.0517780i \(0.983511\pi\)
\(374\) 0 0
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) 12.0000i 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 8.00000 + 16.0000i 0.407718 + 0.815436i
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 + 8.00000i −0.805047 + 0.402524i
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 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
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) 2.00000 1.00000i 0.0993808 0.0496904i
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 56.0000i 2.75558i
\(414\) 0 0
\(415\) 12.0000 + 24.0000i 0.589057 + 1.17811i
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) 16.0000 + 12.0000i 0.776114 + 0.582086i
\(426\) 0 0
\(427\) 40.0000i 1.93574i
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −2.00000 4.00000i −0.0958927 0.191785i
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 8.00000i 0.380091i 0.981775 + 0.190046i \(0.0608636\pi\)
−0.981775 + 0.190046i \(0.939136\pi\)
\(444\) 0 0
\(445\) 36.0000 18.0000i 1.70656 0.853282i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) −8.00000 + 4.00000i −0.375046 + 0.187523i
\(456\) 0 0
\(457\) 26.0000i 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 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
\(463\) 36.0000i 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 0 0
\(465\) −4.00000 8.00000i −0.185496 0.370991i
\(466\) 0 0
\(467\) 16.0000i 0.740392i 0.928954 + 0.370196i \(0.120709\pi\)
−0.928954 + 0.370196i \(0.879291\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 6.00000 8.00000i 0.275299 0.367065i
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) 24.0000i 1.09204i
\(484\) 0 0
\(485\) 6.00000 + 12.0000i 0.272446 + 0.544892i
\(486\) 0 0
\(487\) 40.0000i 1.81257i 0.422664 + 0.906287i \(0.361095\pi\)
−0.422664 + 0.906287i \(0.638905\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 8.00000i 0.360302i
\(494\) 0 0
\(495\) 4.00000 2.00000i 0.179787 0.0898933i
\(496\) 0 0
\(497\) 32.0000i 1.43540i
\(498\) 0 0
\(499\) 30.0000 1.34298 0.671492 0.741012i \(-0.265654\pi\)
0.671492 + 0.741012i \(0.265654\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.0000i 0.802580i 0.915951 + 0.401290i \(0.131438\pi\)
−0.915951 + 0.401290i \(0.868562\pi\)
\(504\) 0 0
\(505\) −28.0000 + 14.0000i −1.24598 + 0.622992i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) 16.0000 0.709188 0.354594 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) −40.0000 −1.76950
\(512\) 0 0
\(513\) 2.00000i 0.0883022i
\(514\) 0 0
\(515\) 6.00000 + 12.0000i 0.264392 + 0.528783i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 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
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 8.00000 + 16.0000i 0.345870 + 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 6.00000i 0.257485i
\(544\) 0 0
\(545\) −24.0000 + 12.0000i −1.02805 + 0.514024i
\(546\) 0 0
\(547\) 44.0000i 1.88130i −0.339372 0.940652i \(-0.610215\pi\)
0.339372 0.940652i \(-0.389785\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 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
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 16.0000i 0.674320i −0.941447 0.337160i \(-0.890534\pi\)
0.941447 0.337160i \(-0.109466\pi\)
\(564\) 0 0
\(565\) 12.0000 + 24.0000i 0.504844 + 1.00969i
\(566\) 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
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 18.0000i −1.00087 0.750652i
\(576\) 0 0
\(577\) 26.0000i 1.08239i 0.840896 + 0.541197i \(0.182029\pi\)
−0.840896 + 0.541197i \(0.817971\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 20.0000i 0.828315i
\(584\) 0 0
\(585\) 1.00000 + 2.00000i 0.0413449 + 0.0826898i
\(586\) 0 0
\(587\) 36.0000i 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) 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
\(597\) 24.0000i 0.982255i
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) −14.0000 + 7.00000i −0.569181 + 0.284590i
\(606\) 0 0
\(607\) 18.0000i 0.730597i −0.930890 0.365299i \(-0.880967\pi\)
0.930890 0.365299i \(-0.119033\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 26.0000i 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) 6.00000 + 12.0000i 0.241943 + 0.483887i
\(616\) 0 0
\(617\) 34.0000i 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 0 0
\(619\) 42.0000 1.68812 0.844061 0.536247i \(-0.180158\pi\)
0.844061 + 0.536247i \(0.180158\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 72.0000i 2.88462i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 28.0000i 1.11290i
\(634\) 0 0
\(635\) 18.0000 + 36.0000i 0.714308 + 1.42862i
\(636\) 0 0
\(637\) 9.00000i 0.356593i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 16.0000 8.00000i 0.629999 0.315000i
\(646\) 0 0
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 0 0
\(649\) 28.0000 1.09910
\(650\) 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
\(655\) 0 0
\(656\) 0 0
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 24.0000 0.933492 0.466746 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) 0 0
\(663\) 4.00000i 0.155347i
\(664\) 0 0
\(665\) −8.00000 16.0000i −0.310227 0.620453i
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 12.0000i 0.462566i −0.972887 0.231283i \(-0.925708\pi\)
0.972887 0.231283i \(-0.0742923\pi\)
\(674\) 0 0
\(675\) 4.00000 + 3.00000i 0.153960 + 0.115470i
\(676\) 0 0
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 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 \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 2.00000 + 4.00000i 0.0764161 + 0.152832i
\(686\) 0 0
\(687\) 16.0000i 0.610438i
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) 0 0
\(693\) 8.00000i 0.303895i
\(694\) 0 0
\(695\) 32.0000 16.0000i 1.21383 0.606915i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 12.0000i 0.452589i
\(704\) 0 0
\(705\) −16.0000 + 8.00000i −0.602595 + 0.301297i
\(706\) 0 0
\(707\) 56.0000i 2.10610i
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 24.0000i 0.898807i
\(714\) 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 \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 30.0000i 1.11571i
\(724\) 0 0
\(725\) 6.00000 8.00000i 0.222834 0.297113i
\(726\) 0 0
\(727\) 14.0000i 0.519231i −0.965712 0.259616i \(-0.916404\pi\)
0.965712 0.259616i \(-0.0835959\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) 0 0
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 0 0
\(735\) 9.00000 + 18.0000i 0.331970 + 0.663940i
\(736\) 0 0
\(737\) 8.00000i 0.294684i
\(738\) 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
\(743\) 4.00000i 0.146746i −0.997305 0.0733729i \(-0.976624\pi\)
0.997305 0.0733729i \(-0.0233763\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) −16.0000 + 8.00000i −0.582300 + 0.291150i
\(756\) 0 0
\(757\) 22.0000i 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) 48.0000i 1.73772i
\(764\) 0 0
\(765\) −4.00000 8.00000i −0.144620 0.289241i
\(766\) 0 0
\(767\) 14.0000i 0.505511i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 0 0
\(773\) 14.0000i 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 12.0000 16.0000i 0.431053 0.574737i
\(776\) 0 0
\(777\) 24.0000i 0.860995i
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) 0 0
\(785\) −10.0000 20.0000i −0.356915 0.713831i
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) −10.0000 −0.356009
\(790\) 0 0
\(791\) 48.0000 1.70668
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 0 0
\(795\) −20.0000 + 10.0000i −0.709327 + 0.354663i
\(796\) 0 0
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 0 0