Properties

Label 20.2.e.a.3.1
Level $20$
Weight $2$
Character 20.3
Analytic conductor $0.160$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.159700804043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 3.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 20.3
Dual form 20.2.e.a.7.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\) \(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(-2.00000 - 1.00000i) q^{5} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(3.00000 - 1.00000i) q^{10} +(-1.00000 - 1.00000i) q^{13} -4.00000 q^{16} +(3.00000 - 3.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(-2.00000 + 4.00000i) q^{20} +(3.00000 + 4.00000i) q^{25} +2.00000 q^{26} -4.00000i q^{29} +(4.00000 - 4.00000i) q^{32} +6.00000i q^{34} +6.00000 q^{36} +(-7.00000 + 7.00000i) q^{37} +(-2.00000 - 6.00000i) q^{40} -8.00000 q^{41} +(3.00000 - 6.00000i) q^{45} -7.00000i q^{49} +(-7.00000 - 1.00000i) q^{50} +(-2.00000 + 2.00000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} +12.0000 q^{61} +8.00000i q^{64} +(1.00000 + 3.00000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-6.00000 + 6.00000i) q^{72} +(-11.0000 - 11.0000i) q^{73} -14.0000i q^{74} +(8.00000 + 4.00000i) q^{80} -9.00000 q^{81} +(8.00000 - 8.00000i) q^{82} +(-9.00000 + 3.00000i) q^{85} +16.0000i q^{89} +(3.00000 + 9.00000i) q^{90} +(13.0000 - 13.0000i) q^{97} +(7.00000 + 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 6 q^{10} - 2 q^{13} - 8 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} + 4 q^{26} + 8 q^{32} + 12 q^{36} - 14 q^{37} - 4 q^{40} - 16 q^{41} + 6 q^{45} - 14 q^{50} - 4 q^{52} + 18 q^{53} + 8 q^{58} + 24 q^{61} + 2 q^{65} - 12 q^{68} - 12 q^{72} - 22 q^{73} + 16 q^{80} - 18 q^{81} + 16 q^{82} - 18 q^{85} + 6 q^{90} + 26 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.00000i −0.707107 + 0.707107i
\(3\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 2.00000i 1.00000i
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) 3.00000i 1.00000i
\(10\) 3.00000 1.00000i 0.948683 0.316228i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.00000 1.00000i −0.277350 0.277350i 0.554700 0.832050i \(-0.312833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 3.00000i 0.727607 0.727607i −0.242536 0.970143i \(-0.577979\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −3.00000 3.00000i −0.707107 0.707107i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 + 4.00000i −0.447214 + 0.894427i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i −0.928477 0.371391i \(-0.878881\pi\)
0.928477 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 4.00000 4.00000i 0.707107 0.707107i
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −7.00000 + 7.00000i −1.15079 + 1.15079i −0.164399 + 0.986394i \(0.552568\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.00000 6.00000i −0.316228 0.948683i
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 3.00000 6.00000i 0.447214 0.894427i
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) −7.00000 1.00000i −0.989949 0.141421i
\(51\) 0 0
\(52\) −2.00000 + 2.00000i −0.277350 + 0.277350i
\(53\) 9.00000 + 9.00000i 1.23625 + 1.23625i 0.961524 + 0.274721i \(0.0885855\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 4.00000 + 4.00000i 0.525226 + 0.525226i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 1.00000 + 3.00000i 0.124035 + 0.372104i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −6.00000 6.00000i −0.727607 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −6.00000 + 6.00000i −0.707107 + 0.707107i
\(73\) −11.0000 11.0000i −1.28745 1.28745i −0.936329 0.351123i \(-0.885800\pi\)
−0.351123 0.936329i \(-0.614200\pi\)
\(74\) 14.0000i 1.62747i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 8.00000 + 4.00000i 0.894427 + 0.447214i
\(81\) −9.00000 −1.00000
\(82\) 8.00000 8.00000i 0.883452 0.883452i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −9.00000 + 3.00000i −0.976187 + 0.325396i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000i 1.69600i 0.529999 + 0.847998i \(0.322192\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 3.00000 + 9.00000i 0.316228 + 0.948683i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000 13.0000i 1.31995 1.31995i 0.406138 0.913812i \(-0.366875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 7.00000 + 7.00000i 0.707107 + 0.707107i
\(99\) 0 0
\(100\) 8.00000 6.00000i 0.800000 0.600000i
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 4.00000i 0.392232i
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i 0.957826 + 0.287348i \(0.0927736\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.00000 1.00000i −0.0940721 0.0940721i 0.658505 0.752577i \(-0.271189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 3.00000 3.00000i 0.277350 0.277350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) −12.0000 + 12.0000i −1.08643 + 1.08643i
\(123\) 0 0
\(124\) 0 0
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) −8.00000 8.00000i −0.707107 0.707107i
\(129\) 0 0
\(130\) −4.00000 2.00000i −0.350823 0.175412i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) −7.00000 + 7.00000i −0.598050 + 0.598050i −0.939793 0.341743i \(-0.888983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 12.0000i 1.00000i
\(145\) −4.00000 + 8.00000i −0.332182 + 0.664364i
\(146\) 22.0000 1.82073
\(147\) 0 0
\(148\) 14.0000 + 14.0000i 1.15079 + 1.15079i
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 9.00000 + 9.00000i 0.727607 + 0.727607i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −17.0000 + 17.0000i −1.35675 + 1.35675i −0.478852 + 0.877896i \(0.658947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −12.0000 + 4.00000i −0.948683 + 0.316228i
\(161\) 0 0
\(162\) 9.00000 9.00000i 0.707107 0.707107i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 16.0000i 1.24939i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 6.00000 12.0000i 0.460179 0.920358i
\(171\) 0 0
\(172\) 0 0
\(173\) −11.0000 11.0000i −0.836315 0.836315i 0.152057 0.988372i \(-0.451410\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −16.0000 16.0000i −1.19925 1.19925i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −12.0000 6.00000i −0.894427 0.447214i
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.0000 7.00000i 1.54395 0.514650i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 19.0000 + 19.0000i 1.36765 + 1.36765i 0.863779 + 0.503871i \(0.168091\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 26.0000i 1.86669i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 13.0000 13.0000i 0.926212 0.926212i −0.0712470 0.997459i \(-0.522698\pi\)
0.997459 + 0.0712470i \(0.0226979\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −2.00000 + 14.0000i −0.141421 + 0.989949i
\(201\) 0 0
\(202\) −2.00000 + 2.00000i −0.140720 + 0.140720i
\(203\) 0 0
\(204\) 0 0
\(205\) 16.0000 + 8.00000i 1.11749 + 0.558744i
\(206\) 0 0
\(207\) 0 0
\(208\) 4.00000 + 4.00000i 0.277350 + 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 18.0000 18.0000i 1.23625 1.23625i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 6.00000i −0.406371 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) −12.0000 + 9.00000i −0.800000 + 0.600000i
\(226\) 2.00000 0.133038
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.00000 8.00000i 0.525226 0.525226i
\(233\) −21.0000 21.0000i −1.37576 1.37576i −0.851658 0.524097i \(-0.824403\pi\)
−0.524097 0.851658i \(-0.675597\pi\)
\(234\) 6.00000i 0.392232i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −11.0000 + 11.0000i −0.707107 + 0.707107i
\(243\) 0 0
\(244\) 24.0000i 1.53644i
\(245\) −7.00000 + 14.0000i −0.447214 + 0.894427i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 13.0000 + 9.00000i 0.822192 + 0.569210i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −17.0000 + 17.0000i −1.06043 + 1.06043i −0.0623783 + 0.998053i \(0.519869\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.00000 2.00000i 0.372104 0.124035i
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) −9.00000 27.0000i −0.552866 1.65860i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.0000i 1.58525i 0.609711 + 0.792624i \(0.291286\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −12.0000 + 12.0000i −0.727607 + 0.727607i
\(273\) 0 0
\(274\) 14.0000i 0.845771i
\(275\) 0 0
\(276\) 0 0
\(277\) 23.0000 23.0000i 1.38194 1.38194i 0.540758 0.841178i \(-0.318138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 12.0000 + 12.0000i 0.707107 + 0.707107i
\(289\) 1.00000i 0.0588235i
\(290\) −4.00000 12.0000i −0.234888 0.704664i
\(291\) 0 0
\(292\) −22.0000 + 22.0000i −1.28745 + 1.28745i
\(293\) 19.0000 + 19.0000i 1.10999 + 1.10999i 0.993151 + 0.116841i \(0.0372769\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −28.0000 −1.62747
\(297\) 0 0
\(298\) 14.0000 + 14.0000i 0.810998 + 0.810998i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.0000 12.0000i −1.37424 0.687118i
\(306\) −18.0000 −1.02899
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.00000 1.00000i −0.0565233 0.0565233i 0.678280 0.734803i \(-0.262726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 34.0000i 1.91873i
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 3.00000i 0.168497 0.168497i −0.617822 0.786318i \(-0.711985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.00000 16.0000i 0.447214 0.894427i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) 1.00000 7.00000i 0.0554700 0.388290i
\(326\) 0 0
\(327\) 0 0
\(328\) −16.0000 16.0000i −0.883452 0.883452i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −21.0000 21.0000i −1.15079 1.15079i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.00000 + 7.00000i −0.381314 + 0.381314i −0.871576 0.490261i \(-0.836901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 11.0000 + 11.0000i 0.598321 + 0.598321i
\(339\) 0 0
\(340\) 6.00000 + 18.0000i 0.325396 + 0.976187i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 36.0000i 1.92704i 0.267644 + 0.963518i \(0.413755\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.00000 + 9.00000i 0.479022 + 0.479022i 0.904819 0.425797i \(-0.140006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 32.0000 1.69600
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 18.0000 6.00000i 0.948683 0.316228i
\(361\) −19.0000 −1.00000
\(362\) 18.0000 18.0000i 0.946059 0.946059i
\(363\) 0 0
\(364\) 0 0
\(365\) 11.0000 + 33.0000i 0.575766 + 1.72730i
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 24.0000i 1.24939i
\(370\) −14.0000 + 28.0000i −0.727825 + 1.45565i
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 11.0000i −0.569558 0.569558i 0.362446 0.932005i \(-0.381942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 + 4.00000i −0.206010 + 0.206010i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −38.0000 −1.93415
\(387\) 0 0
\(388\) −26.0000 26.0000i −1.31995 1.31995i
\(389\) 34.0000i 1.72387i −0.507020 0.861934i \(-0.669253\pi\)
0.507020 0.861934i \(-0.330747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 14.0000 14.0000i 0.707107 0.707107i
\(393\) 0 0
\(394\) 26.0000i 1.30986i
\(395\) 0 0
\(396\) 0 0
\(397\) 13.0000 13.0000i 0.652451 0.652451i −0.301131 0.953583i \(-0.597364\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −12.0000 16.0000i −0.600000 0.800000i
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4.00000i 0.199007i
\(405\) 18.0000 + 9.00000i 0.894427 + 0.447214i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000i 0.296681i 0.988936 + 0.148340i \(0.0473931\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) −24.0000 + 8.00000i −1.18528 + 0.395092i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −8.00000 −0.392232
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 36.0000i 1.74831i
\(425\) 21.0000 + 3.00000i 1.01865 + 0.145521i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 29.0000 + 29.0000i 1.39365 + 1.39365i 0.816968 + 0.576683i \(0.195653\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 21.0000 1.00000
\(442\) 6.00000 6.00000i 0.285391 0.285391i
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) 16.0000 32.0000i 0.758473 1.51695i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.0000i 0.660701i −0.943858 0.330350i \(-0.892833\pi\)
0.943858 0.330350i \(-0.107167\pi\)
\(450\) 3.00000 21.0000i 0.141421 0.989949i
\(451\) 0 0
\(452\) −2.00000 + 2.00000i −0.0940721 + 0.0940721i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0000 + 17.0000i −0.795226 + 0.795226i −0.982339 0.187112i \(-0.940087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 4.00000 + 4.00000i 0.186908 + 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 16.0000i 0.742781i
\(465\) 0 0
\(466\) 42.0000 1.94561
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) −6.00000 6.00000i −0.277350 0.277350i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −27.0000 + 27.0000i −1.23625 + 1.23625i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 14.0000 0.638345
\(482\) 8.00000 8.00000i 0.364390 0.364390i
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) −39.0000 + 13.0000i −1.77090 + 0.590300i
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 24.0000 + 24.0000i 1.08643 + 1.08643i
\(489\) 0 0
\(490\) −7.00000 21.0000i −0.316228 0.948683i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −12.0000 12.0000i −0.540453 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −22.0000 + 4.00000i −0.983870 + 0.178885i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) −4.00000 2.00000i −0.177998 0.0889988i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 44.0000i 1.95027i −0.221621 0.975133i \(-0.571135\pi\)
0.221621 0.975133i \(-0.428865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −16.0000 + 16.0000i −0.707107 + 0.707107i
\(513\) 0 0
\(514\) 34.0000i 1.49968i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −4.00000 + 8.00000i −0.175412 + 0.350823i
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −12.0000 + 12.0000i −0.525226 + 0.525226i
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 36.0000 + 18.0000i 1.56374 + 0.781870i
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 + 8.00000i 0.346518 + 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −26.0000 26.0000i −1.12094 1.12094i
\(539\) 0 0
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 24.0000i 1.02899i
\(545\) 6.00000 12.0000i 0.257012 0.514024i
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 14.0000 + 14.0000i 0.598050 + 0.598050i
\(549\) 36.0000i 1.53644i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 46.0000i 1.95435i
\(555\) 0 0
\(556\) 0 0
\(557\) 33.0000 33.0000i 1.39825 1.39825i 0.593199 0.805056i \(-0.297865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −32.0000 + 32.0000i −1.34984 + 1.34984i
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 1.00000 + 3.00000i 0.0420703 + 0.126211i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000i 1.08998i 0.838444 + 0.544988i \(0.183466\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −24.0000 −1.00000
\(577\) 23.0000 23.0000i 0.957503 0.957503i −0.0416305 0.999133i \(-0.513255\pi\)
0.999133 + 0.0416305i \(0.0132552\pi\)
\(578\) 1.00000 + 1.00000i 0.0415945 + 0.0415945i
\(579\) 0 0
\(580\) 16.0000 + 8.00000i 0.664364 + 0.332182i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 44.0000i 1.82073i
\(585\) −9.00000 + 3.00000i −0.372104 + 0.124035i
\(586\) −38.0000 −1.56977
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 28.0000 28.0000i 1.15079 1.15079i
\(593\) −31.0000 31.0000i −1.27302 1.27302i −0.944497 0.328521i \(-0.893450\pi\)
−0.328521 0.944497i \(-0.606550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −28.0000 −1.14692
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −48.0000 −1.95796 −0.978980 0.203954i \(-0.934621\pi\)
−0.978980 + 0.203954i \(0.934621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.0000 11.0000i −0.894427 0.447214i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 36.0000 12.0000i 1.45760 0.485866i
\(611\) 0 0
\(612\) 18.0000 18.0000i 0.727607 0.727607i
\(613\) −1.00000 1.00000i −0.0403896 0.0403896i 0.686624 0.727013i \(-0.259092\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 3.00000i 0.120775 0.120775i −0.644136 0.764911i \(-0.722783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) 34.0000 + 34.0000i 1.35675 + 1.35675i
\(629\) 42.0000i 1.67465i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 6.00000i 0.238290i
\(635\) 0 0
\(636\) 0 0
\(637\) −7.00000 + 7.00000i −0.277350 + 0.277350i
\(638\) 0 0
\(639\) 0 0
\(640\) 8.00000 + 24.0000i 0.316228 + 0.948683i
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) −18.0000 18.0000i −0.707107 0.707107i
\(649\) 0 0
\(650\) 6.00000 + 8.00000i 0.235339 + 0.313786i
\(651\) 0 0
\(652\) 0 0
\(653\) 9.00000 + 9.00000i 0.352197 + 0.352197i 0.860927 0.508729i \(-0.169885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 32.0000 1.24939
\(657\) 33.0000 33.0000i 1.28745 1.28745i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 42.0000 1.62747
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −11.0000 11.0000i −0.424019 0.424019i 0.462566 0.886585i \(-0.346929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 14.0000i 0.539260i
\(675\) 0 0
\(676\) −22.0000 −0.846154
\(677\) −27.0000 + 27.0000i −1.03769 + 1.03769i −0.0384331 + 0.999261i \(0.512237\pi\)
−0.999261 + 0.0384331i \(0.987763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −24.0000 12.0000i −0.920358 0.460179i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) 21.0000 7.00000i 0.802369 0.267456i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.0000i 0.685745i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −22.0000 + 22.0000i −0.836315 + 0.836315i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −24.0000 + 24.0000i −0.909065 + 0.909065i
\(698\) −36.0000 36.0000i −1.36262 1.36262i
\(699\) 0 0
\(700\) 0 0
\(701\) 52.0000 1.96401 0.982006 0.188847i \(-0.0604752\pi\)
0.982006 + 0.188847i \(0.0604752\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000i 1.65245i −0.563337 0.826227i \(-0.690483\pi\)
0.563337 0.826227i \(-0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −32.0000 + 32.0000i −1.19925 + 1.19925i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −12.0000 + 24.0000i −0.447214 + 0.894427i
\(721\) 0 0
\(722\) 19.0000 19.0000i 0.707107 0.707107i
\(723\) 0 0
\(724\) 36.0000i 1.33793i
\(725\) 16.0000 12.0000i 0.594225 0.445669i
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) −44.0000 22.0000i −1.62851 0.814257i
\(731\) 0 0
\(732\) 0 0
\(733\) 29.0000 + 29.0000i 1.07114 + 1.07114i 0.997268 + 0.0738717i \(0.0235355\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 24.0000 + 24.0000i 0.883452 + 0.883452i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −14.0000 42.0000i −0.514650 1.54395i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) −14.0000 + 28.0000i −0.512920 + 1.02584i
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 8.00000i 0.291343i
\(755\) 0 0
\(756\) 0 0
\(757\) −17.0000 + 17.0000i −0.617876 + 0.617876i −0.944986 0.327111i \(-0.893925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −9.00000 27.0000i −0.325396 0.976187i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 24.0000i 0.865462i −0.901523 0.432731i \(-0.857550\pi\)
0.901523 0.432731i \(-0.142450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 38.0000 38.0000i 1.36765 1.36765i
\(773\) 39.0000 + 39.0000i 1.40273 + 1.40273i 0.791285 + 0.611448i \(0.209412\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 52.0000 1.86669
\(777\) 0 0
\(778\) 34.0000 + 34.0000i 1.21896 + 1.21896i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) 51.0000 17.0000i 1.82027 0.606756i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) −26.0000 26.0000i −0.926212 0.926212i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 12.0000i −0.426132 0.426132i
\(794\) 26.0000i 0.922705i
\(795\) 0 0
\(796\) 0 0
\(797\) −37.0000 + 37.0000i −1.31061 + 1.31061i −0.389640 + 0.920967i \(0.627401\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.0000 + 4.00000i 0.989949 + 0.141421i
\(801\) −48.0000 −1.69600