Properties

Label 3240.2.q.q.2161.1
Level $3240$
Weight $2$
Character 3240.2161
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8715302549\)
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 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2161.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.2161
Dual form 3240.2.q.q.1081.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{11} +(-3.00000 - 5.19615i) q^{13} +6.00000 q^{17} -4.00000 q^{19} +(-0.500000 + 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{29} +(4.00000 + 6.92820i) q^{31} -2.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(-6.00000 + 10.3923i) q^{43} +(4.00000 - 6.92820i) q^{47} +(3.50000 + 6.06218i) q^{49} -6.00000 q^{53} -4.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(-7.00000 + 12.1244i) q^{61} +(3.00000 - 5.19615i) q^{65} +(-2.00000 - 3.46410i) q^{67} -8.00000 q^{71} -6.00000 q^{73} +(4.00000 - 6.92820i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(3.00000 + 5.19615i) q^{85} -10.0000 q^{89} +(-2.00000 - 3.46410i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + O(q^{10}) \) \( 2q + q^{5} - 4q^{11} - 6q^{13} + 12q^{17} - 8q^{19} - q^{25} - 2q^{29} + 8q^{31} - 4q^{37} - 6q^{41} - 12q^{43} + 8q^{47} + 7q^{49} - 12q^{53} - 8q^{55} + 12q^{59} - 14q^{61} + 6q^{65} - 4q^{67} - 16q^{71} - 12q^{73} + 8q^{79} - 12q^{83} + 6q^{85} - 20q^{89} - 4q^{95} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(7\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) −3.00000 5.19615i −0.832050 1.44115i −0.896410 0.443227i \(-0.853834\pi\)
0.0643593 0.997927i \(-0.479500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 + 1.73205i −0.185695 + 0.321634i −0.943811 0.330487i \(-0.892787\pi\)
0.758115 + 0.652121i \(0.226120\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 5.19615i −0.468521 0.811503i 0.530831 0.847477i \(-0.321880\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) −6.00000 + 10.3923i −0.914991 + 1.58481i −0.108078 + 0.994142i \(0.534469\pi\)
−0.806914 + 0.590669i \(0.798864\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 6.92820i 0.583460 1.01058i −0.411606 0.911362i \(-0.635032\pi\)
0.995066 0.0992202i \(-0.0316348\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 5.19615i 0.372104 0.644503i
\(66\) 0 0
\(67\) −2.00000 3.46410i −0.244339 0.423207i 0.717607 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 + 10.3923i −0.658586 + 1.14070i 0.322396 + 0.946605i \(0.395512\pi\)
−0.980982 + 0.194099i \(0.937822\pi\)
\(84\) 0 0
\(85\) 3.00000 + 5.19615i 0.325396 + 0.563602i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 3.46410i −0.205196 0.355409i
\(96\) 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
\(101\) 3.00000 5.19615i 0.298511 0.517036i −0.677284 0.735721i \(-0.736843\pi\)
0.975796 + 0.218685i \(0.0701767\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.00000 5.19615i −0.282216 0.488813i 0.689714 0.724082i \(-0.257736\pi\)
−0.971930 + 0.235269i \(0.924403\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 + 3.46410i 0.174741 + 0.302660i 0.940072 0.340977i \(-0.110758\pi\)
−0.765331 + 0.643637i \(0.777425\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00000 1.73205i 0.0854358 0.147979i −0.820141 0.572161i \(-0.806105\pi\)
0.905577 + 0.424182i \(0.139438\pi\)
\(138\) 0 0
\(139\) −2.00000 3.46410i −0.169638 0.293821i 0.768655 0.639664i \(-0.220926\pi\)
−0.938293 + 0.345843i \(0.887593\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.00000 8.66025i −0.409616 0.709476i 0.585231 0.810867i \(-0.301004\pi\)
−0.994847 + 0.101391i \(0.967671\pi\)
\(150\) 0 0
\(151\) −8.00000 + 13.8564i −0.651031 + 1.12762i 0.331842 + 0.943335i \(0.392330\pi\)
−0.982873 + 0.184284i \(0.941004\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 + 6.92820i −0.321288 + 0.556487i
\(156\) 0 0
\(157\) −3.00000 5.19615i −0.239426 0.414698i 0.721124 0.692806i \(-0.243626\pi\)
−0.960550 + 0.278108i \(0.910293\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 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
\(167\) 8.00000 + 13.8564i 0.619059 + 1.07224i 0.989658 + 0.143448i \(0.0458190\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(168\) 0 0
\(169\) −11.5000 + 19.9186i −0.884615 + 1.53220i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.00000 12.1244i 0.532200 0.921798i −0.467093 0.884208i \(-0.654699\pi\)
0.999293 0.0375896i \(-0.0119679\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 1.73205i −0.0735215 0.127343i
\(186\) 0 0
\(187\) −12.0000 + 20.7846i −0.877527 + 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 7.00000 + 12.1244i 0.503871 + 0.872730i 0.999990 + 0.00447566i \(0.00142465\pi\)
−0.496119 + 0.868255i \(0.665242\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 5.19615i 0.209529 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 13.8564i 0.553372 0.958468i
\(210\) 0 0
\(211\) −6.00000 10.3923i −0.413057 0.715436i 0.582165 0.813070i \(-0.302206\pi\)
−0.995222 + 0.0976347i \(0.968872\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.0000 31.1769i −1.21081 2.09719i
\(222\) 0 0
\(223\) −4.00000 + 6.92820i −0.267860 + 0.463947i −0.968309 0.249756i \(-0.919650\pi\)
0.700449 + 0.713702i \(0.252983\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 + 24.2487i −0.929213 + 1.60944i −0.144571 + 0.989494i \(0.546180\pi\)
−0.784642 + 0.619949i \(0.787153\pi\)
\(228\) 0 0
\(229\) 5.00000 + 8.66025i 0.330409 + 0.572286i 0.982592 0.185776i \(-0.0594799\pi\)
−0.652183 + 0.758062i \(0.726147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 7.00000 12.1244i 0.450910 0.780998i −0.547533 0.836784i \(-0.684433\pi\)
0.998443 + 0.0557856i \(0.0177663\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.50000 + 6.06218i −0.223607 + 0.387298i
\(246\) 0 0
\(247\) 12.0000 + 20.7846i 0.763542 + 1.32249i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.00000 + 8.66025i 0.311891 + 0.540212i 0.978772 0.204953i \(-0.0657041\pi\)
−0.666880 + 0.745165i \(0.732371\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) −3.00000 5.19615i −0.184289 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 3.46410i −0.120605 0.208893i
\(276\) 0 0
\(277\) 1.00000 1.73205i 0.0600842 0.104069i −0.834419 0.551131i \(-0.814196\pi\)
0.894503 + 0.447062i \(0.147530\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.0000 + 19.0526i −0.656205 + 1.13658i 0.325385 + 0.945582i \(0.394506\pi\)
−0.981590 + 0.190999i \(0.938827\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.0000 + 19.0526i 0.642627 + 1.11306i 0.984844 + 0.173442i \(0.0554888\pi\)
−0.342217 + 0.939621i \(0.611178\pi\)
\(294\) 0 0
\(295\) −6.00000 + 10.3923i −0.349334 + 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 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
\(311\) 4.00000 + 6.92820i 0.226819 + 0.392862i 0.956864 0.290537i \(-0.0938340\pi\)
−0.730044 + 0.683400i \(0.760501\pi\)
\(312\) 0 0
\(313\) −5.00000 + 8.66025i −0.282617 + 0.489506i −0.972028 0.234863i \(-0.924536\pi\)
0.689412 + 0.724370i \(0.257869\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.00000 12.1244i 0.393159 0.680972i −0.599705 0.800221i \(-0.704715\pi\)
0.992864 + 0.119249i \(0.0380488\pi\)
\(318\) 0 0
\(319\) −4.00000 6.92820i −0.223957 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.0000 24.2487i 0.769510 1.33283i −0.168320 0.985732i \(-0.553834\pi\)
0.937829 0.347097i \(-0.112833\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.00000 3.46410i 0.109272 0.189264i
\(336\) 0 0
\(337\) −1.00000 1.73205i −0.0544735 0.0943508i 0.837503 0.546433i \(-0.184015\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) −7.00000 + 12.1244i −0.374701 + 0.649002i −0.990282 0.139072i \(-0.955588\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.0000 22.5167i 0.691920 1.19844i −0.279288 0.960207i \(-0.590098\pi\)
0.971208 0.238233i \(-0.0765683\pi\)
\(354\) 0 0
\(355\) −4.00000 6.92820i −0.212298 0.367711i
\(356\) 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
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 5.19615i −0.157027 0.271979i
\(366\) 0 0
\(367\) 4.00000 6.92820i 0.208798 0.361649i −0.742538 0.669804i \(-0.766378\pi\)
0.951336 + 0.308155i \(0.0997115\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.00000 + 1.73205i 0.0517780 + 0.0896822i 0.890753 0.454488i \(-0.150178\pi\)
−0.838975 + 0.544170i \(0.816844\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.00000 5.19615i 0.152106 0.263455i −0.779895 0.625910i \(-0.784728\pi\)
0.932002 + 0.362454i \(0.118061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.00000 12.1244i −0.349563 0.605461i 0.636609 0.771187i \(-0.280337\pi\)
−0.986172 + 0.165726i \(0.947003\pi\)
\(402\) 0 0
\(403\) 24.0000 41.5692i 1.19553 2.07071i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 6.92820i 0.198273 0.343418i
\(408\) 0 0
\(409\) −13.0000 22.5167i −0.642809 1.11338i −0.984803 0.173675i \(-0.944436\pi\)
0.341994 0.939702i \(-0.388898\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 + 31.1769i 0.879358 + 1.52309i 0.852047 + 0.523465i \(0.175361\pi\)
0.0273103 + 0.999627i \(0.491306\pi\)
\(420\) 0 0
\(421\) 5.00000 8.66025i 0.243685 0.422075i −0.718076 0.695965i \(-0.754977\pi\)
0.961761 + 0.273890i \(0.0883103\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 + 5.19615i −0.145521 + 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 + 13.8564i −0.381819 + 0.661330i −0.991322 0.131453i \(-0.958036\pi\)
0.609503 + 0.792784i \(0.291369\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 + 3.46410i −0.0950229 + 0.164584i −0.909618 0.415445i \(-0.863626\pi\)
0.814595 + 0.580030i \(0.196959\pi\)
\(444\) 0 0
\(445\) −5.00000 8.66025i −0.237023 0.410535i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.0000 19.0526i 0.514558 0.891241i −0.485299 0.874348i \(-0.661289\pi\)
0.999857 0.0168929i \(-0.00537742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.00000 12.1244i 0.326023 0.564688i −0.655696 0.755025i \(-0.727625\pi\)
0.981719 + 0.190337i \(0.0609581\pi\)
\(462\) 0 0
\(463\) −4.00000 6.92820i −0.185896 0.321981i 0.757982 0.652275i \(-0.226185\pi\)
−0.943878 + 0.330294i \(0.892852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −24.0000 41.5692i −1.10352 1.91135i
\(474\) 0 0
\(475\) 2.00000 3.46410i 0.0917663 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.0000 + 27.7128i −0.731059 + 1.26623i 0.225372 + 0.974273i \(0.427640\pi\)
−0.956431 + 0.291958i \(0.905693\pi\)
\(480\) 0 0
\(481\) 6.00000 + 10.3923i 0.273576 + 0.473848i
\(482\) 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
\(491\) −18.0000 31.1769i −0.812329 1.40699i −0.911230 0.411897i \(-0.864866\pi\)
0.0989017 0.995097i \(-0.468467\pi\)
\(492\) 0 0
\(493\) −6.00000 + 10.3923i −0.270226 + 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.0000 + 17.3205i 0.447661 + 0.775372i 0.998233 0.0594153i \(-0.0189236\pi\)
−0.550572 + 0.834788i \(0.685590\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.00000 1.73205i −0.0443242 0.0767718i 0.843012 0.537895i \(-0.180780\pi\)
−0.887336 + 0.461123i \(0.847447\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.0000 + 27.7128i 0.703679 + 1.21881i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 + 41.5692i 1.04546 + 1.81078i
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −18.0000 + 31.1769i −0.779667 + 1.35042i
\(534\) 0 0
\(535\) 2.00000 + 3.46410i 0.0864675 + 0.149766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.00000 15.5885i −0.385518 0.667736i
\(546\) 0 0
\(547\) −18.0000 + 31.1769i −0.769624 + 1.33303i 0.168142 + 0.985763i \(0.446223\pi\)
−0.937767 + 0.347266i \(0.887110\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.00000 6.92820i 0.170406 0.295151i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 72.0000 3.04528
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00000 + 3.46410i 0.0842900 + 0.145994i 0.905088 0.425223i \(-0.139804\pi\)
−0.820798 + 0.571218i \(0.806471\pi\)
\(564\) 0 0
\(565\) 3.00000 5.19615i 0.126211 0.218604i
\(566\) 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
\(571\) 6.00000 + 10.3923i 0.251092 + 0.434904i 0.963827 0.266529i \(-0.0858769\pi\)
−0.712735 + 0.701434i \(0.752544\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 20.7846i 0.496989 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.00000 + 3.46410i −0.0825488 + 0.142979i −0.904344 0.426804i \(-0.859639\pi\)
0.821795 + 0.569783i \(0.192973\pi\)
\(588\) 0 0
\(589\) −16.0000 27.7128i −0.659269 1.14189i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.0000 34.6410i −0.817178 1.41539i −0.907754 0.419504i \(-0.862204\pi\)
0.0905757 0.995890i \(-0.471129\pi\)
\(600\) 0 0
\(601\) −13.0000 + 22.5167i −0.530281 + 0.918474i 0.469095 + 0.883148i \(0.344580\pi\)
−0.999376 + 0.0353259i \(0.988753\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.50000 4.33013i 0.101639 0.176045i
\(606\) 0 0
\(607\) −20.0000 34.6410i −0.811775 1.40604i −0.911621 0.411033i \(-0.865168\pi\)
0.0998457 0.995003i \(-0.468165\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000 + 1.73205i 0.0402585 + 0.0697297i 0.885453 0.464730i \(-0.153849\pi\)
−0.845194 + 0.534460i \(0.820515\pi\)
\(618\) 0 0
\(619\) 14.0000 24.2487i 0.562708 0.974638i −0.434551 0.900647i \(-0.643093\pi\)
0.997259 0.0739910i \(-0.0235736\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.00000 6.92820i −0.158735 0.274937i
\(636\) 0 0
\(637\) 21.0000 36.3731i 0.832050 1.44115i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 15.5885i 0.355479 0.615707i −0.631721 0.775196i \(-0.717651\pi\)
0.987200 + 0.159489i \(0.0509845\pi\)
\(642\) 0 0
\(643\) −2.00000 3.46410i −0.0788723 0.136611i 0.823891 0.566748i \(-0.191799\pi\)
−0.902764 + 0.430137i \(0.858465\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.00000 + 12.1244i 0.273931 + 0.474463i 0.969865 0.243643i \(-0.0783426\pi\)
−0.695934 + 0.718106i \(0.745009\pi\)
\(654\) 0 0
\(655\) −2.00000 + 3.46410i −0.0781465 + 0.135354i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.00000 3.46410i 0.0779089 0.134942i −0.824439 0.565951i \(-0.808509\pi\)
0.902348 + 0.431009i \(0.141842\pi\)
\(660\) 0 0
\(661\) −3.00000 5.19615i −0.116686 0.202107i 0.801766 0.597638i \(-0.203894\pi\)
−0.918453 + 0.395531i \(0.870561\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −28.0000 48.4974i −1.08093 1.87222i
\(672\) 0 0
\(673\) 7.00000 12.1244i 0.269830 0.467360i −0.698988 0.715134i \(-0.746366\pi\)
0.968818 + 0.247774i \(0.0796991\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.0000 32.9090i 0.730229 1.26479i −0.226556 0.973998i \(-0.572747\pi\)
0.956785 0.290796i \(-0.0939201\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.0000 + 31.1769i 0.685745 + 1.18775i
\(690\) 0 0
\(691\) 10.0000 17.3205i 0.380418 0.658903i −0.610704 0.791859i \(-0.709113\pi\)
0.991122 + 0.132956i \(0.0424468\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 3.46410i 0.0758643 0.131401i
\(696\) 0 0
\(697\) −18.0000 31.1769i −0.681799 1.18091i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 12.0000 + 20.7846i 0.448775 + 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.00000 1.73205i −0.0371391 0.0643268i
\(726\) 0 0
\(727\) 16.0000 27.7128i 0.593407 1.02781i −0.400362 0.916357i \(-0.631116\pi\)
0.993770 0.111454i \(-0.0355509\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −36.0000 + 62.3538i −1.33151 + 2.30624i
\(732\) 0 0
\(733\) 13.0000 + 22.5167i 0.480166 + 0.831672i 0.999741 0.0227529i \(-0.00724310\pi\)
−0.519575 + 0.854425i \(0.673910\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.00000 13.8564i −0.293492 0.508342i 0.681141 0.732152i \(-0.261484\pi\)
−0.974633 + 0.223810i \(0.928151\pi\)
\(744\) 0 0
\(745\) 5.00000 8.66025i 0.183186 0.317287i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 6.92820i −0.145962 0.252814i 0.783769 0.621052i \(-0.213294\pi\)
−0.929731 + 0.368238i \(0.879961\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.0000 + 22.5167i 0.471250 + 0.816228i 0.999459 0.0328858i \(-0.0104698\pi\)
−0.528209 + 0.849114i \(0.677136\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.0000 62.3538i 1.29988 2.25147i
\(768\) 0 0
\(769\) 15.0000 + 25.9808i 0.540914 + 0.936890i 0.998852 + 0.0479061i \(0.0152548\pi\)
−0.457938 + 0.888984i \(0.651412\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 + 20.7846i 0.429945 + 0.744686i
\(780\) 0 0
\(781\) 16.0000 27.7128i 0.572525 0.991642i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.00000 5.19615i 0.107075 0.185459i
\(786\) 0 0
\(787\) −2.00000 3.46410i −0.0712923 0.123482i 0.828176 0.560469i \(-0.189379\pi\)
−0.899468 + 0.436987i \(0.856046\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 84.0000 2.98293
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.00000 1.73205i −0.0354218 0.0613524i 0.847771 0.530362i \(-0.177944\pi\)
−0.883193 + 0.469010i \(0.844611\pi\)
\(798\) 0 0
\(799\) 24.0000 41.5692i 0.849059 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000 20.7846i 0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0