Properties

Label 1008.2.s.k.865.1
Level $1008$
Weight $2$
Character 1008.865
Analytic conductor $8.049$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.865
Dual form 1008.2.s.k.289.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +(-2.50000 + 4.33013i) q^{11} +(-2.00000 + 3.46410i) q^{17} +(4.00000 + 6.92820i) q^{19} +(2.00000 + 3.46410i) q^{23} +(2.00000 - 3.46410i) q^{25} +5.00000 q^{29} +(1.50000 - 2.59808i) q^{31} +(2.00000 - 1.73205i) q^{35} +(2.00000 + 3.46410i) q^{37} -2.00000 q^{43} +(3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-4.50000 + 7.79423i) q^{53} -5.00000 q^{55} +(5.50000 - 9.52628i) q^{59} +(3.00000 + 5.19615i) q^{61} +(-1.00000 + 1.73205i) q^{67} +2.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(12.5000 + 4.33013i) q^{77} +(1.50000 + 2.59808i) q^{79} -7.00000 q^{83} -4.00000 q^{85} +(-3.00000 - 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - q^{7} + O(q^{10}) \) \( 2q + q^{5} - q^{7} - 5q^{11} - 4q^{17} + 8q^{19} + 4q^{23} + 4q^{25} + 10q^{29} + 3q^{31} + 4q^{35} + 4q^{37} - 4q^{43} + 6q^{47} - 13q^{49} - 9q^{53} - 10q^{55} + 11q^{59} + 6q^{61} - 2q^{67} + 4q^{71} - 10q^{73} + 25q^{77} + 3q^{79} - 14q^{83} - 8q^{85} - 6q^{89} - 8q^{95} + 14q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i 0.955901 0.293691i \(-0.0948835\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.50000 + 4.33013i −0.753778 + 1.30558i 0.192201 + 0.981356i \(0.438437\pi\)
−0.945979 + 0.324227i \(0.894896\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 + 3.46410i −0.485071 + 0.840168i −0.999853 0.0171533i \(-0.994540\pi\)
0.514782 + 0.857321i \(0.327873\pi\)
\(18\) 0 0
\(19\) 4.00000 + 6.92820i 0.917663 + 1.58944i 0.802955 + 0.596040i \(0.203260\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) 2.00000 3.46410i 0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 1.50000 2.59808i 0.269408 0.466628i −0.699301 0.714827i \(-0.746505\pi\)
0.968709 + 0.248199i \(0.0798387\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 1.73205i 0.338062 0.292770i
\(36\) 0 0
\(37\) 2.00000 + 3.46410i 0.328798 + 0.569495i 0.982274 0.187453i \(-0.0600231\pi\)
−0.653476 + 0.756948i \(0.726690\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 + 7.79423i −0.618123 + 1.07062i 0.371706 + 0.928351i \(0.378773\pi\)
−0.989828 + 0.142269i \(0.954560\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.50000 9.52628i 0.716039 1.24022i −0.246518 0.969138i \(-0.579287\pi\)
0.962557 0.271078i \(-0.0873801\pi\)
\(60\) 0 0
\(61\) 3.00000 + 5.19615i 0.384111 + 0.665299i 0.991645 0.128994i \(-0.0411748\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.5000 + 4.33013i 1.42451 + 0.493464i
\(78\) 0 0
\(79\) 1.50000 + 2.59808i 0.168763 + 0.292306i 0.937985 0.346675i \(-0.112689\pi\)
−0.769222 + 0.638982i \(0.779356\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 5.19615i −0.317999 0.550791i 0.662071 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.00000 8.66025i 0.497519 0.861727i −0.502477 0.864590i \(-0.667578\pi\)
0.999996 + 0.00286291i \(0.000911295\pi\)
\(102\) 0 0
\(103\) 4.00000 + 6.92820i 0.394132 + 0.682656i 0.992990 0.118199i \(-0.0377120\pi\)
−0.598858 + 0.800855i \(0.704379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.50000 2.59808i −0.145010 0.251166i 0.784366 0.620298i \(-0.212988\pi\)
−0.929377 + 0.369132i \(0.879655\pi\)
\(108\) 0 0
\(109\) 1.00000 1.73205i 0.0957826 0.165900i −0.814152 0.580651i \(-0.802798\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −2.00000 + 3.46410i −0.186501 + 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.0000 + 3.46410i 0.916698 + 0.317554i
\(120\) 0 0
\(121\) −7.00000 12.1244i −0.636364 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.500000 0.866025i −0.0436852 0.0756650i 0.843356 0.537355i \(-0.180577\pi\)
−0.887041 + 0.461690i \(0.847243\pi\)
\(132\) 0 0
\(133\) 16.0000 13.8564i 1.38738 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.00000 + 1.73205i −0.0854358 + 0.147979i −0.905577 0.424182i \(-0.860562\pi\)
0.820141 + 0.572161i \(0.193895\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.50000 + 4.33013i 0.207614 + 0.359597i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 15.5885i −0.737309 1.27706i −0.953703 0.300750i \(-0.902763\pi\)
0.216394 0.976306i \(-0.430570\pi\)
\(150\) 0 0
\(151\) 9.50000 16.4545i 0.773099 1.33905i −0.162758 0.986666i \(-0.552039\pi\)
0.935857 0.352381i \(-0.114628\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 2.00000 3.46410i 0.159617 0.276465i −0.775113 0.631822i \(-0.782307\pi\)
0.934731 + 0.355357i \(0.115641\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 6.92820i 0.630488 0.546019i
\(162\) 0 0
\(163\) −2.00000 3.46410i −0.156652 0.271329i 0.777007 0.629492i \(-0.216737\pi\)
−0.933659 + 0.358162i \(0.883403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.0000 + 19.0526i 0.836315 + 1.44854i 0.892956 + 0.450145i \(0.148628\pi\)
−0.0566411 + 0.998395i \(0.518039\pi\)
\(174\) 0 0
\(175\) −10.0000 3.46410i −0.755929 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 + 10.3923i −0.448461 + 0.776757i −0.998286 0.0585225i \(-0.981361\pi\)
0.549825 + 0.835280i \(0.314694\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 + 3.46410i −0.147043 + 0.254686i
\(186\) 0 0
\(187\) −10.0000 17.3205i −0.731272 1.26660i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.0000 20.7846i −0.868290 1.50392i −0.863743 0.503932i \(-0.831886\pi\)
−0.00454614 0.999990i \(-0.501447\pi\)
\(192\) 0 0
\(193\) −2.50000 + 4.33013i −0.179954 + 0.311689i −0.941865 0.335993i \(-0.890928\pi\)
0.761911 + 0.647682i \(0.224262\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −2.00000 + 3.46410i −0.141776 + 0.245564i −0.928166 0.372168i \(-0.878615\pi\)
0.786389 + 0.617731i \(0.211948\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.50000 12.9904i −0.175466 0.911746i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −40.0000 −2.76686
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 1.73205i −0.0681994 0.118125i
\(216\) 0 0
\(217\) −7.50000 2.59808i −0.509133 0.176369i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 + 2.59808i −0.0995585 + 0.172440i −0.911502 0.411296i \(-0.865076\pi\)
0.811943 + 0.583736i \(0.198410\pi\)
\(228\) 0 0
\(229\) 10.0000 + 17.3205i 0.660819 + 1.14457i 0.980401 + 0.197013i \(0.0631241\pi\)
−0.319582 + 0.947559i \(0.603543\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 3.46410i −0.131024 0.226941i 0.793047 0.609160i \(-0.208493\pi\)
−0.924072 + 0.382219i \(0.875160\pi\)
\(234\) 0 0
\(235\) −3.00000 + 5.19615i −0.195698 + 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 12.5000 21.6506i 0.805196 1.39464i −0.110963 0.993825i \(-0.535394\pi\)
0.916159 0.400815i \(-0.131273\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.50000 4.33013i −0.351382 0.276642i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.00000 5.19615i −0.187135 0.324127i 0.757159 0.653231i \(-0.226587\pi\)
−0.944294 + 0.329104i \(0.893253\pi\)
\(258\) 0 0
\(259\) 8.00000 6.92820i 0.497096 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.0000 25.9808i 0.924940 1.60204i 0.133281 0.991078i \(-0.457449\pi\)
0.791658 0.610964i \(-0.209218\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.5000 26.8468i 0.945052 1.63688i 0.189404 0.981899i \(-0.439344\pi\)
0.755648 0.654978i \(-0.227322\pi\)
\(270\) 0 0
\(271\) 7.50000 + 12.9904i 0.455593 + 0.789109i 0.998722 0.0505395i \(-0.0160941\pi\)
−0.543130 + 0.839649i \(0.682761\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 + 17.3205i 0.603023 + 1.04447i
\(276\) 0 0
\(277\) 8.00000 13.8564i 0.480673 0.832551i −0.519081 0.854725i \(-0.673726\pi\)
0.999754 + 0.0221745i \(0.00705893\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 5.00000 8.66025i 0.297219 0.514799i −0.678280 0.734804i \(-0.737274\pi\)
0.975499 + 0.220005i \(0.0706075\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.0000 1.22683 0.613417 0.789760i \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 + 5.19615i 0.0576390 + 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.00000 + 5.19615i −0.171780 + 0.297531i
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 27.7128i 0.907277 1.57145i 0.0894452 0.995992i \(-0.471491\pi\)
0.817832 0.575458i \(-0.195176\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.0282617 0.0489506i 0.851549 0.524276i \(-0.175664\pi\)
−0.879810 + 0.475325i \(0.842331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.50000 + 2.59808i 0.0842484 + 0.145922i 0.905071 0.425261i \(-0.139818\pi\)
−0.820822 + 0.571184i \(0.806484\pi\)
\(318\) 0 0
\(319\) −12.5000 + 21.6506i −0.699866 + 1.21220i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 10.3923i 0.661581 0.572946i
\(330\) 0 0
\(331\) −2.00000 3.46410i −0.109930 0.190404i 0.805812 0.592172i \(-0.201729\pi\)
−0.915742 + 0.401768i \(0.868396\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.50000 + 12.9904i 0.406148 + 0.703469i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 20.7846i 0.638696 1.10625i −0.347024 0.937856i \(-0.612808\pi\)
0.985719 0.168397i \(-0.0538590\pi\)
\(354\) 0 0
\(355\) 1.00000 + 1.73205i 0.0530745 + 0.0919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.00000 8.66025i −0.263890 0.457071i 0.703382 0.710812i \(-0.251672\pi\)
−0.967272 + 0.253741i \(0.918339\pi\)
\(360\) 0 0
\(361\) −22.5000 + 38.9711i −1.18421 + 2.05111i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 8.50000 14.7224i 0.443696 0.768505i −0.554264 0.832341i \(-0.687000\pi\)
0.997960 + 0.0638362i \(0.0203335\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.5000 + 7.79423i 1.16814 + 0.404656i
\(372\) 0 0
\(373\) 16.0000 + 27.7128i 0.828449 + 1.43492i 0.899255 + 0.437425i \(0.144109\pi\)
−0.0708063 + 0.997490i \(0.522557\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.0000 + 29.4449i 0.868659 + 1.50456i 0.863367 + 0.504576i \(0.168351\pi\)
0.00529229 + 0.999986i \(0.498315\pi\)
\(384\) 0 0
\(385\) 2.50000 + 12.9904i 0.127412 + 0.662051i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.00000 + 1.73205i −0.0507020 + 0.0878185i −0.890263 0.455448i \(-0.849479\pi\)
0.839561 + 0.543266i \(0.182813\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.50000 + 2.59808i −0.0754732 + 0.130723i
\(396\) 0 0
\(397\) −18.0000 31.1769i −0.903394 1.56472i −0.823058 0.567957i \(-0.807734\pi\)
−0.0803356 0.996768i \(-0.525599\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 + 20.7846i 0.599251 + 1.03793i 0.992932 + 0.118686i \(0.0378683\pi\)
−0.393680 + 0.919247i \(0.628798\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) 12.5000 21.6506i 0.618085 1.07056i −0.371750 0.928333i \(-0.621242\pi\)
0.989835 0.142222i \(-0.0454247\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.5000 9.52628i −1.35319 0.468758i
\(414\) 0 0
\(415\) −3.50000 6.06218i −0.171808 0.297581i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 + 13.8564i 0.388057 + 0.672134i
\(426\) 0 0
\(427\) 12.0000 10.3923i 0.580721 0.502919i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 + 10.3923i −0.289010 + 0.500580i −0.973574 0.228373i \(-0.926659\pi\)
0.684564 + 0.728953i \(0.259993\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.0000 + 27.7128i −0.765384 + 1.32568i
\(438\) 0 0
\(439\) 7.50000 + 12.9904i 0.357955 + 0.619997i 0.987619 0.156871i \(-0.0501406\pi\)
−0.629664 + 0.776868i \(0.716807\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.50000 14.7224i −0.403847 0.699484i 0.590339 0.807155i \(-0.298994\pi\)
−0.994187 + 0.107671i \(0.965661\pi\)
\(444\) 0 0
\(445\) 3.00000 5.19615i 0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.5000 26.8468i −0.725059 1.25584i −0.958950 0.283577i \(-0.908479\pi\)
0.233890 0.972263i \(-0.424854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.0000 + 17.3205i 0.462745 + 0.801498i 0.999097 0.0424970i \(-0.0135313\pi\)
−0.536352 + 0.843995i \(0.680198\pi\)
\(468\) 0 0
\(469\) 5.00000 + 1.73205i 0.230879 + 0.0799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.00000 8.66025i 0.229900 0.398199i
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.0000 + 32.9090i −0.868132 + 1.50365i −0.00422900 + 0.999991i \(0.501346\pi\)
−0.863903 + 0.503658i \(0.831987\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.50000 + 6.06218i 0.158927 + 0.275269i
\(486\) 0 0
\(487\) 2.50000 4.33013i 0.113286 0.196217i −0.803807 0.594890i \(-0.797196\pi\)
0.917093 + 0.398673i \(0.130529\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) −10.0000 + 17.3205i −0.450377 + 0.780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.00000 5.19615i −0.0448561 0.233079i
\(498\) 0 0
\(499\) 5.00000 + 8.66025i 0.223831 + 0.387686i 0.955968 0.293471i \(-0.0948104\pi\)
−0.732137 + 0.681157i \(0.761477\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.50000 + 12.9904i 0.332432 + 0.575789i 0.982988 0.183669i \(-0.0587976\pi\)
−0.650556 + 0.759458i \(0.725464\pi\)
\(510\) 0 0
\(511\) 25.0000 + 8.66025i 1.10593 + 0.383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.00000 + 6.92820i −0.176261 + 0.305293i
\(516\) 0 0
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 + 15.5885i −0.394297 + 0.682943i −0.993011 0.118020i \(-0.962345\pi\)
0.598714 + 0.800963i \(0.295679\pi\)
\(522\) 0 0
\(523\) 4.00000 + 6.92820i 0.174908 + 0.302949i 0.940129 0.340818i \(-0.110704\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 + 10.3923i 0.261364 + 0.452696i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.50000 2.59808i 0.0648507 0.112325i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.00000 34.6410i 0.215365 1.49209i
\(540\) 0 0
\(541\) 9.00000 + 15.5885i 0.386940 + 0.670200i 0.992036 0.125952i \(-0.0401986\pi\)
−0.605096 + 0.796152i \(0.706865\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.0000 + 34.6410i 0.852029 + 1.47576i
\(552\) 0 0
\(553\) 6.00000 5.19615i 0.255146 0.220963i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.5000 + 19.9186i −0.487271 + 0.843978i −0.999893 0.0146368i \(-0.995341\pi\)
0.512622 + 0.858614i \(0.328674\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.50000 + 14.7224i −0.358232 + 0.620477i −0.987666 0.156578i \(-0.949954\pi\)
0.629433 + 0.777055i \(0.283287\pi\)
\(564\) 0 0
\(565\) −8.00000 13.8564i −0.336563 0.582943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 + 20.7846i 0.503066 + 0.871336i 0.999994 + 0.00354413i \(0.00112814\pi\)
−0.496928 + 0.867792i \(0.665539\pi\)
\(570\) 0 0
\(571\) −15.0000 + 25.9808i −0.627730 + 1.08726i 0.360276 + 0.932846i \(0.382683\pi\)
−0.988006 + 0.154415i \(0.950651\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) −15.5000 + 26.8468i −0.645273 + 1.11765i 0.338965 + 0.940799i \(0.389923\pi\)
−0.984238 + 0.176847i \(0.943410\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.50000 + 18.1865i 0.145204 + 0.754505i
\(582\) 0 0
\(583\) −22.5000 38.9711i −0.931855 1.61402i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.0000 + 31.1769i 0.739171 + 1.28028i 0.952869 + 0.303383i \(0.0981160\pi\)
−0.213697 + 0.976900i \(0.568551\pi\)
\(594\) 0 0
\(595\) 2.00000 + 10.3923i 0.0819920 + 0.426043i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.0000 25.9808i 0.612883 1.06155i −0.377869 0.925859i \(-0.623343\pi\)
0.990752 0.135686i \(-0.0433238\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 12.1244i 0.284590 0.492925i
\(606\) 0 0
\(607\) −13.5000 23.3827i −0.547948 0.949074i −0.998415 0.0562808i \(-0.982076\pi\)
0.450467 0.892793i \(-0.351258\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −6.00000 + 10.3923i −0.242338 + 0.419741i −0.961380 0.275225i \(-0.911248\pi\)
0.719042 + 0.694967i \(0.244581\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) 5.00000 8.66025i 0.200967 0.348085i −0.747873 0.663842i \(-0.768925\pi\)
0.948840 + 0.315757i \(0.102258\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 + 10.3923i −0.480770 + 0.416359i
\(624\) 0 0
\(625\) −5.50000 9.52628i −0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 19.0000 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.50000 7.79423i −0.178577 0.309305i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.0000 22.5167i 0.513469 0.889355i −0.486409 0.873731i \(-0.661693\pi\)
0.999878 0.0156233i \(-0.00497325\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 15.5885i 0.353827 0.612845i −0.633090 0.774078i \(-0.718214\pi\)
0.986916 + 0.161233i \(0.0515470\pi\)
\(648\) 0 0
\(649\) 27.5000 + 47.6314i 1.07947 + 1.86970i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.5000 33.7750i −0.763094 1.32172i −0.941248 0.337715i \(-0.890346\pi\)
0.178154 0.984003i \(-0.442987\pi\)
\(654\) 0 0
\(655\) 0.500000 0.866025i 0.0195366 0.0338384i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −5.00000 + 8.66025i −0.194477 + 0.336845i −0.946729 0.322031i \(-0.895634\pi\)
0.752252 + 0.658876i \(0.228968\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.0000 + 6.92820i 0.775567 + 0.268664i
\(666\) 0 0
\(667\) 10.0000 + 17.3205i 0.387202 + 0.670653i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.5000 23.3827i −0.518847 0.898670i −0.999760 0.0219013i \(-0.993028\pi\)
0.480913 0.876768i \(-0.340305\pi\)
\(678\) 0 0
\(679\) −3.50000 18.1865i −0.134318 0.697935i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.50000 7.79423i 0.172188 0.298238i −0.766997 0.641651i \(-0.778250\pi\)
0.939184 + 0.343413i \(0.111583\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 4.00000 + 6.92820i 0.152167 + 0.263561i 0.932024 0.362397i \(-0.118041\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.00000 + 12.1244i 0.265525 + 0.459903i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −16.0000 + 27.7128i −0.603451 + 1.04521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.0000 8.66025i −0.940222 0.325702i
\(708\) 0 0
\(709\) −19.0000 32.9090i −0.713560 1.23592i −0.963512 0.267664i \(-0.913748\pi\)
0.249952 0.968258i \(-0.419585\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.00000 + 5.19615i 0.111881 + 0.193784i 0.916529 0.399969i \(-0.130979\pi\)
−0.804648 + 0.593753i \(0.797646\pi\)
\(720\) 0 0
\(721\) 16.0000 13.8564i 0.595871 0.516040i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.0000 17.3205i 0.371391 0.643268i
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.00000 6.92820i 0.147945 0.256249i
\(732\) 0 0
\(733\) 3.00000 + 5.19615i 0.110808 + 0.191924i 0.916096 0.400959i \(-0.131323\pi\)
−0.805289 + 0.592883i \(0.797990\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 8.66025i −0.184177 0.319005i
\(738\) 0 0
\(739\) −15.0000 + 25.9808i −0.551784 + 0.955718i 0.446362 + 0.894852i \(0.352719\pi\)
−0.998146 + 0.0608653i \(0.980614\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 9.00000 15.5885i 0.329734 0.571117i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 + 5.19615i −0.219235 + 0.189863i
\(750\) 0 0
\(751\) 22.5000 + 38.9711i 0.821037 + 1.42208i 0.904911 + 0.425601i \(0.139937\pi\)
−0.0838743 + 0.996476i \(0.526729\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.0000 0.691481
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.00000 + 6.92820i 0.145000 + 0.251147i 0.929373 0.369142i \(-0.120348\pi\)
−0.784373 + 0.620289i \(0.787015\pi\)
\(762\) 0 0
\(763\) −5.00000 1.73205i −0.181012 0.0627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.00000 8.66025i 0.179838 0.311488i −0.761987 0.647592i \(-0.775776\pi\)
0.941825 + 0.336104i \(0.109109\pi\)
\(774\) 0 0
\(775\) −6.00000 10.3923i −0.215526 0.373303i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −5.00000 + 8.66025i −0.178914 + 0.309888i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −9.00000 + 15.5885i −0.320815 + 0.555668i −0.980656 0.195737i \(-0.937290\pi\)
0.659841 + 0.751405i \(0.270624\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.00000 + 41.5692i 0.284447 + 1.47803i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −25.0000 43.3013i −0.882231 1.52807i
\(804\) 0 0
\(805\) 10.0000 + 3.46410i 0.352454 + 0.122094i