Properties

Label 2160.2.q.a.721.1
Level $2160$
Weight $2$
Character 2160.721
Analytic conductor $17.248$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 721.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2160.721
Dual form 2160.2.q.a.1441.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.50000 + 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{11} +(1.00000 + 1.73205i) q^{13} -4.00000 q^{17} +8.00000 q^{19} +(-1.50000 - 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{29} +3.00000 q^{35} -4.00000 q^{37} +(2.50000 + 4.33013i) q^{41} +(-4.00000 + 6.92820i) q^{43} +(-3.50000 + 6.06218i) q^{47} +(-1.00000 - 1.73205i) q^{49} +2.00000 q^{53} -2.00000 q^{55} +(7.00000 + 12.1244i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-1.50000 - 2.59808i) q^{67} +2.00000 q^{71} +4.00000 q^{73} +(3.00000 + 5.19615i) q^{77} +(-3.00000 + 5.19615i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(2.00000 + 3.46410i) q^{85} +15.0000 q^{89} -6.00000 q^{91} +(-4.00000 - 6.92820i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - 3 q^{7} + 2 q^{11} + 2 q^{13} - 8 q^{17} + 16 q^{19} - 3 q^{23} - q^{25} - q^{29} + 6 q^{35} - 8 q^{37} + 5 q^{41} - 8 q^{43} - 7 q^{47} - 2 q^{49} + 4 q^{53} - 4 q^{55} + 14 q^{59} - 7 q^{61} + 2 q^{65} - 3 q^{67} + 4 q^{71} + 8 q^{73} + 6 q^{77} - 6 q^{79} - 9 q^{83} + 4 q^{85} + 30 q^{89} - 12 q^{91} - 8 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(6\) 0 0
\(7\) −1.50000 + 2.59808i −0.566947 + 0.981981i 0.429919 + 0.902867i \(0.358542\pi\)
−0.996866 + 0.0791130i \(0.974791\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.50000 2.59808i −0.312772 0.541736i 0.666190 0.745782i \(-0.267924\pi\)
−0.978961 + 0.204046i \(0.934591\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 + 0.866025i −0.0928477 + 0.160817i −0.908708 0.417432i \(-0.862930\pi\)
0.815861 + 0.578249i \(0.196264\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.50000 + 4.33013i 0.390434 + 0.676252i 0.992507 0.122189i \(-0.0389915\pi\)
−0.602072 + 0.798441i \(0.705658\pi\)
\(42\) 0 0
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.50000 + 6.06218i −0.510527 + 0.884260i 0.489398 + 0.872060i \(0.337217\pi\)
−0.999926 + 0.0121990i \(0.996117\pi\)
\(48\) 0 0
\(49\) −1.00000 1.73205i −0.142857 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 + 12.1244i 0.911322 + 1.57846i 0.812198 + 0.583382i \(0.198271\pi\)
0.0991242 + 0.995075i \(0.468396\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 1.73205i 0.124035 0.214834i
\(66\) 0 0
\(67\) −1.50000 2.59808i −0.183254 0.317406i 0.759733 0.650236i \(-0.225330\pi\)
−0.942987 + 0.332830i \(0.891996\pi\)
\(68\) 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
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 + 5.19615i 0.341882 + 0.592157i
\(78\) 0 0
\(79\) −3.00000 + 5.19615i −0.337526 + 0.584613i −0.983967 0.178352i \(-0.942924\pi\)
0.646440 + 0.762964i \(0.276257\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) 0 0
\(85\) 2.00000 + 3.46410i 0.216930 + 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 6.92820i −0.410391 0.710819i
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 + 15.5885i −0.895533 + 1.55111i −0.0623905 + 0.998052i \(0.519872\pi\)
−0.833143 + 0.553058i \(0.813461\pi\)
\(102\) 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
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.00000 6.92820i −0.376288 0.651751i 0.614231 0.789127i \(-0.289466\pi\)
−0.990519 + 0.137376i \(0.956133\pi\)
\(114\) 0 0
\(115\) −1.50000 + 2.59808i −0.139876 + 0.242272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 10.3923i 0.550019 0.952661i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) −12.0000 + 20.7846i −1.04053 + 1.80225i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 10.3923i 0.512615 0.887875i −0.487278 0.873247i \(-0.662010\pi\)
0.999893 0.0146279i \(-0.00465636\pi\)
\(138\) 0 0
\(139\) −8.00000 13.8564i −0.678551 1.17529i −0.975417 0.220366i \(-0.929275\pi\)
0.296866 0.954919i \(-0.404058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.50000 + 14.7224i 0.696347 + 1.20611i 0.969724 + 0.244202i \(0.0785259\pi\)
−0.273377 + 0.961907i \(0.588141\pi\)
\(150\) 0 0
\(151\) −1.00000 + 1.73205i −0.0813788 + 0.140952i −0.903842 0.427865i \(-0.859266\pi\)
0.822464 + 0.568818i \(0.192599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.00000 12.1244i −0.558661 0.967629i −0.997609 0.0691164i \(-0.977982\pi\)
0.438948 0.898513i \(-0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.50000 + 7.79423i 0.348220 + 0.603136i 0.985933 0.167139i \(-0.0534527\pi\)
−0.637713 + 0.770274i \(0.720119\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −1.50000 2.59808i −0.113389 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000 + 3.46410i 0.147043 + 0.254686i
\(186\) 0 0
\(187\) −4.00000 + 6.92820i −0.292509 + 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 + 6.92820i −0.289430 + 0.501307i −0.973674 0.227946i \(-0.926799\pi\)
0.684244 + 0.729253i \(0.260132\pi\)
\(192\) 0 0
\(193\) 5.00000 + 8.66025i 0.359908 + 0.623379i 0.987945 0.154805i \(-0.0494748\pi\)
−0.628037 + 0.778183i \(0.716141\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.50000 2.59808i −0.105279 0.182349i
\(204\) 0 0
\(205\) 2.50000 4.33013i 0.174608 0.302429i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 13.8564i 0.553372 0.958468i
\(210\) 0 0
\(211\) −11.0000 19.0526i −0.757271 1.31163i −0.944237 0.329266i \(-0.893199\pi\)
0.186966 0.982366i \(-0.440135\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 6.92820i −0.269069 0.466041i
\(222\) 0 0
\(223\) −9.50000 + 16.4545i −0.636167 + 1.10187i 0.350100 + 0.936713i \(0.386148\pi\)
−0.986267 + 0.165161i \(0.947186\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.00000 3.46410i 0.132745 0.229920i −0.791989 0.610535i \(-0.790954\pi\)
0.924734 + 0.380615i \(0.124288\pi\)
\(228\) 0 0
\(229\) −7.50000 12.9904i −0.495614 0.858429i 0.504373 0.863486i \(-0.331724\pi\)
−0.999987 + 0.00505719i \(0.998390\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 + 6.92820i 0.258738 + 0.448148i 0.965904 0.258900i \(-0.0833599\pi\)
−0.707166 + 0.707048i \(0.750027\pi\)
\(240\) 0 0
\(241\) 5.50000 9.52628i 0.354286 0.613642i −0.632709 0.774389i \(-0.718057\pi\)
0.986996 + 0.160748i \(0.0513906\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 + 1.73205i −0.0638877 + 0.110657i
\(246\) 0 0
\(247\) 8.00000 + 13.8564i 0.509028 + 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 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
\(259\) 6.00000 10.3923i 0.372822 0.645746i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00000 13.8564i 0.493301 0.854423i −0.506669 0.862141i \(-0.669123\pi\)
0.999970 + 0.00771799i \(0.00245674\pi\)
\(264\) 0 0
\(265\) −1.00000 1.73205i −0.0614295 0.106399i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.0000 −1.52428 −0.762138 0.647414i \(-0.775850\pi\)
−0.762138 + 0.647414i \(0.775850\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 + 1.73205i 0.0603023 + 0.104447i
\(276\) 0 0
\(277\) −6.00000 + 10.3923i −0.360505 + 0.624413i −0.988044 0.154172i \(-0.950729\pi\)
0.627539 + 0.778585i \(0.284062\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.50000 + 12.9904i −0.447412 + 0.774941i −0.998217 0.0596933i \(-0.980988\pi\)
0.550804 + 0.834634i \(0.314321\pi\)
\(282\) 0 0
\(283\) 10.5000 + 18.1865i 0.624160 + 1.08108i 0.988703 + 0.149890i \(0.0478921\pi\)
−0.364542 + 0.931187i \(0.618775\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −15.0000 −0.885422
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00000 + 10.3923i 0.350524 + 0.607125i 0.986341 0.164714i \(-0.0526703\pi\)
−0.635818 + 0.771839i \(0.719337\pi\)
\(294\) 0 0
\(295\) 7.00000 12.1244i 0.407556 0.705907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 5.19615i 0.173494 0.300501i
\(300\) 0 0
\(301\) −12.0000 20.7846i −0.691669 1.19800i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −7.00000 + 12.1244i −0.395663 + 0.685309i −0.993186 0.116543i \(-0.962819\pi\)
0.597522 + 0.801852i \(0.296152\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.0000 + 29.4449i −0.954815 + 1.65379i −0.220024 + 0.975494i \(0.570614\pi\)
−0.734791 + 0.678294i \(0.762720\pi\)
\(318\) 0 0
\(319\) 1.00000 + 1.73205i 0.0559893 + 0.0969762i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.5000 18.1865i −0.578884 1.00266i
\(330\) 0 0
\(331\) −3.00000 + 5.19615i −0.164895 + 0.285606i −0.936618 0.350352i \(-0.886062\pi\)
0.771723 + 0.635959i \(0.219395\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.50000 + 2.59808i −0.0819538 + 0.141948i
\(336\) 0 0
\(337\) 4.00000 + 6.92820i 0.217894 + 0.377403i 0.954164 0.299285i \(-0.0967480\pi\)
−0.736270 + 0.676688i \(0.763415\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 3.46410i −0.107366 0.185963i 0.807337 0.590091i \(-0.200908\pi\)
−0.914702 + 0.404128i \(0.867575\pi\)
\(348\) 0 0
\(349\) 2.50000 4.33013i 0.133822 0.231786i −0.791325 0.611396i \(-0.790608\pi\)
0.925147 + 0.379610i \(0.123942\pi\)
\(350\) 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
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.00000 3.46410i −0.104685 0.181319i
\(366\) 0 0
\(367\) 12.0000 20.7846i 0.626395 1.08495i −0.361874 0.932227i \(-0.617863\pi\)
0.988269 0.152721i \(-0.0488036\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.00000 + 5.19615i −0.155752 + 0.269771i
\(372\) 0 0
\(373\) −5.00000 8.66025i −0.258890 0.448411i 0.707055 0.707159i \(-0.250023\pi\)
−0.965945 + 0.258748i \(0.916690\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 31.1769i −0.919757 1.59307i −0.799783 0.600289i \(-0.795052\pi\)
−0.119974 0.992777i \(-0.538281\pi\)
\(384\) 0 0
\(385\) 3.00000 5.19615i 0.152894 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.5000 28.5788i 0.836583 1.44900i −0.0561516 0.998422i \(-0.517883\pi\)
0.892735 0.450582i \(-0.148784\pi\)
\(390\) 0 0
\(391\) 6.00000 + 10.3923i 0.303433 + 0.525561i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 34.0000 1.70641 0.853206 0.521575i \(-0.174655\pi\)
0.853206 + 0.521575i \(0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 15.5885i −0.449439 0.778450i 0.548911 0.835881i \(-0.315043\pi\)
−0.998350 + 0.0574304i \(0.981709\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 + 6.92820i −0.198273 + 0.343418i
\(408\) 0 0
\(409\) −7.00000 12.1244i −0.346128 0.599511i 0.639430 0.768849i \(-0.279170\pi\)
−0.985558 + 0.169338i \(0.945837\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −42.0000 −2.06668
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13.0000 22.5167i −0.635092 1.10001i −0.986496 0.163787i \(-0.947629\pi\)
0.351404 0.936224i \(-0.385704\pi\)
\(420\) 0 0
\(421\) −17.0000 + 29.4449i −0.828529 + 1.43505i 0.0706626 + 0.997500i \(0.477489\pi\)
−0.899192 + 0.437555i \(0.855845\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 3.46410i 0.0970143 0.168034i
\(426\) 0 0
\(427\) −10.5000 18.1865i −0.508131 0.880108i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0000 20.7846i −0.574038 0.994263i
\(438\) 0 0
\(439\) 14.0000 24.2487i 0.668184 1.15733i −0.310228 0.950662i \(-0.600405\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.50000 + 12.9904i −0.356336 + 0.617192i −0.987346 0.158583i \(-0.949307\pi\)
0.631010 + 0.775775i \(0.282641\pi\)
\(444\) 0 0
\(445\) −7.50000 12.9904i −0.355534 0.615803i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 + 5.19615i 0.140642 + 0.243599i
\(456\) 0 0
\(457\) 10.0000 17.3205i 0.467780 0.810219i −0.531542 0.847032i \(-0.678387\pi\)
0.999322 + 0.0368128i \(0.0117205\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.50000 7.79423i 0.209586 0.363013i −0.741998 0.670402i \(-0.766122\pi\)
0.951584 + 0.307388i \(0.0994551\pi\)
\(462\) 0 0
\(463\) −18.0000 31.1769i −0.836531 1.44891i −0.892778 0.450497i \(-0.851247\pi\)
0.0562469 0.998417i \(-0.482087\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 + 13.8564i 0.367840 + 0.637118i
\(474\) 0 0
\(475\) −4.00000 + 6.92820i −0.183533 + 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.00000 15.5885i 0.411220 0.712255i −0.583803 0.811895i \(-0.698436\pi\)
0.995023 + 0.0996406i \(0.0317693\pi\)
\(480\) 0 0
\(481\) −4.00000 6.92820i −0.182384 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.0000 17.3205i −0.451294 0.781664i 0.547173 0.837020i \(-0.315704\pi\)
−0.998467 + 0.0553560i \(0.982371\pi\)
\(492\) 0 0
\(493\) 2.00000 3.46410i 0.0900755 0.156015i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.00000 + 5.19615i −0.134568 + 0.233079i
\(498\) 0 0
\(499\) −16.0000 27.7128i −0.716258 1.24060i −0.962472 0.271380i \(-0.912520\pi\)
0.246214 0.969216i \(-0.420813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.00000 −0.312115 −0.156057 0.987748i \(-0.549878\pi\)
−0.156057 + 0.987748i \(0.549878\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 21.5000 + 37.2391i 0.952971 + 1.65059i 0.738945 + 0.673766i \(0.235324\pi\)
0.214026 + 0.976828i \(0.431342\pi\)
\(510\) 0 0
\(511\) −6.00000 + 10.3923i −0.265424 + 0.459728i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000 6.92820i 0.176261 0.305293i
\(516\) 0 0
\(517\) 7.00000 + 12.1244i 0.307860 + 0.533229i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.00000 + 8.66025i −0.216574 + 0.375117i
\(534\) 0 0
\(535\) −1.50000 2.59808i −0.0648507 0.112325i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.50000 4.33013i −0.107088 0.185482i
\(546\) 0 0
\(547\) −14.5000 + 25.1147i −0.619975 + 1.07383i 0.369514 + 0.929225i \(0.379524\pi\)
−0.989490 + 0.144604i \(0.953809\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.00000 + 6.92820i −0.170406 + 0.295151i
\(552\) 0 0
\(553\) −9.00000 15.5885i −0.382719 0.662889i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.5000 + 18.1865i 0.442522 + 0.766471i 0.997876 0.0651433i \(-0.0207504\pi\)
−0.555354 + 0.831614i \(0.687417\pi\)
\(564\) 0 0
\(565\) −4.00000 + 6.92820i −0.168281 + 0.291472i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 16.0000 + 27.7128i 0.669579 + 1.15975i 0.978022 + 0.208502i \(0.0668588\pi\)
−0.308443 + 0.951243i \(0.599808\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.5000 23.3827i −0.560074 0.970077i
\(582\) 0 0
\(583\) 2.00000 3.46410i 0.0828315 0.143468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.5000 28.5788i 0.681028 1.17957i −0.293640 0.955916i \(-0.594867\pi\)
0.974668 0.223659i \(-0.0718001\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.00000 + 8.66025i 0.204294 + 0.353848i 0.949908 0.312531i \(-0.101177\pi\)
−0.745613 + 0.666379i \(0.767843\pi\)
\(600\) 0 0
\(601\) −1.00000 + 1.73205i −0.0407909 + 0.0706518i −0.885700 0.464258i \(-0.846321\pi\)
0.844909 + 0.534910i \(0.179654\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.50000 6.06218i 0.142295 0.246463i
\(606\) 0 0
\(607\) −20.5000 35.5070i −0.832069 1.44119i −0.896394 0.443257i \(-0.853823\pi\)
0.0643251 0.997929i \(-0.479511\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −14.0000 −0.566379
\(612\) 0 0
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 31.1769i −0.724653 1.25514i −0.959117 0.283011i \(-0.908667\pi\)
0.234464 0.972125i \(-0.424666\pi\)
\(618\) 0 0
\(619\) −2.00000 + 3.46410i −0.0803868 + 0.139234i −0.903416 0.428765i \(-0.858949\pi\)
0.823029 + 0.567999i \(0.192282\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −22.5000 + 38.9711i −0.901443 + 1.56135i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.50000 4.33013i −0.0992095 0.171836i
\(636\) 0 0
\(637\) 2.00000 3.46410i 0.0792429 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5000 28.5788i 0.651711 1.12880i −0.330997 0.943632i \(-0.607385\pi\)
0.982708 0.185164i \(-0.0592817\pi\)
\(642\) 0 0
\(643\) −4.50000 7.79423i −0.177463 0.307374i 0.763548 0.645751i \(-0.223456\pi\)
−0.941011 + 0.338377i \(0.890122\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.0000 −0.668339 −0.334169 0.942513i \(-0.608456\pi\)
−0.334169 + 0.942513i \(0.608456\pi\)
\(648\) 0 0
\(649\) 28.0000 1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.00000 3.46410i −0.0782660 0.135561i 0.824236 0.566247i \(-0.191605\pi\)
−0.902502 + 0.430686i \(0.858272\pi\)
\(654\) 0 0
\(655\) 3.00000 5.19615i 0.117220 0.203030i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 6.92820i 0.155818 0.269884i −0.777539 0.628835i \(-0.783532\pi\)
0.933357 + 0.358951i \(0.116865\pi\)
\(660\) 0 0
\(661\) 7.00000 + 12.1244i 0.272268 + 0.471583i 0.969442 0.245319i \(-0.0788928\pi\)
−0.697174 + 0.716902i \(0.745559\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.00000 + 12.1244i 0.270232 + 0.468056i
\(672\) 0 0
\(673\) −3.00000 + 5.19615i −0.115642 + 0.200297i −0.918036 0.396497i \(-0.870226\pi\)
0.802395 + 0.596794i \(0.203559\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.0000 + 36.3731i −0.807096 + 1.39793i 0.107772 + 0.994176i \(0.465628\pi\)
−0.914867 + 0.403755i \(0.867705\pi\)
\(678\) 0 0
\(679\) −3.00000 5.19615i −0.115129 0.199410i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.00000 + 3.46410i 0.0761939 + 0.131972i
\(690\) 0 0
\(691\) −7.00000 + 12.1244i −0.266293 + 0.461232i −0.967901 0.251330i \(-0.919132\pi\)
0.701609 + 0.712562i \(0.252465\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 + 13.8564i −0.303457 + 0.525603i
\(696\) 0 0
\(697\) −10.0000 17.3205i −0.378777 0.656061i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −23.0000 −0.868698 −0.434349 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.0000 46.7654i −1.01544 1.75879i
\(708\) 0 0
\(709\) 20.5000 35.5070i 0.769894 1.33349i −0.167727 0.985834i \(-0.553643\pi\)
0.937620 0.347661i \(-0.113024\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −2.00000 3.46410i −0.0747958 0.129550i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.500000 0.866025i −0.0185695 0.0321634i
\(726\) 0 0
\(727\) 11.5000 19.9186i 0.426511 0.738739i −0.570049 0.821611i \(-0.693076\pi\)
0.996560 + 0.0828714i \(0.0264091\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.0000 27.7128i 0.591781 1.02500i
\(732\) 0 0
\(733\) 17.0000 + 29.4449i 0.627909 + 1.08757i 0.987971 + 0.154642i \(0.0494225\pi\)
−0.360061 + 0.932929i \(0.617244\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.5000 + 25.1147i 0.531953 + 0.921370i 0.999304 + 0.0372984i \(0.0118752\pi\)
−0.467351 + 0.884072i \(0.654791\pi\)
\(744\) 0 0
\(745\) 8.50000 14.7224i 0.311416 0.539388i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.50000 + 7.79423i −0.164426 + 0.284795i
\(750\) 0 0
\(751\) 5.00000 + 8.66025i 0.182453 + 0.316017i 0.942715 0.333599i \(-0.108263\pi\)
−0.760263 + 0.649616i \(0.774930\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.50000 + 12.9904i 0.271875 + 0.470901i 0.969342 0.245716i \(-0.0790230\pi\)
−0.697467 + 0.716617i \(0.745690\pi\)
\(762\) 0 0
\(763\) −7.50000 + 12.9904i −0.271518 + 0.470283i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.0000 + 24.2487i −0.505511 + 0.875570i
\(768\) 0 0
\(769\) −2.50000 4.33013i −0.0901523 0.156148i 0.817423 0.576038i \(-0.195402\pi\)
−0.907575 + 0.419890i \(0.862069\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.0000 + 34.6410i 0.716574 + 1.24114i
\(780\) 0 0
\(781\) 2.00000 3.46410i 0.0715656 0.123955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.00000 + 12.1244i −0.249841 + 0.432737i
\(786\) 0 0
\(787\) 14.0000 + 24.2487i 0.499046 + 0.864373i 0.999999 0.00110111i \(-0.000350496\pi\)
−0.500953 + 0.865474i \(0.667017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.0000 + 22.5167i 0.460484 + 0.797581i 0.998985 0.0450436i \(-0.0143427\pi\)
−0.538501 + 0.842625i \(0.681009\pi\)
\(798\) 0 0
\(799\) 14.0000 24.2487i 0.495284 0.857858i
\(800\) 0 0
\(801\) 0 0