Properties

Label 980.2.k.c.883.1
Level $980$
Weight $2$
Character 980.883
Analytic conductor $7.825$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
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 883.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 980.883
Dual form 980.2.k.c.687.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\) \(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +(-1.00000 - 3.00000i) q^{10} +(-5.00000 - 5.00000i) q^{13} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} +(-4.00000 - 2.00000i) q^{20} +(-3.00000 - 4.00000i) q^{25} -10.0000 q^{26} +4.00000i q^{29} +(-4.00000 + 4.00000i) q^{32} -10.0000i q^{34} +6.00000 q^{36} +(7.00000 - 7.00000i) q^{37} +(-6.00000 + 2.00000i) q^{40} -10.0000 q^{41} +(6.00000 + 3.00000i) q^{45} +(-7.00000 - 1.00000i) q^{50} +(-10.0000 + 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} +10.0000 q^{61} +8.00000i q^{64} +(-15.0000 + 5.00000i) q^{65} +(-10.0000 - 10.0000i) q^{68} +(6.00000 - 6.00000i) q^{72} +(5.00000 + 5.00000i) q^{73} -14.0000i q^{74} +(-4.00000 + 8.00000i) q^{80} -9.00000 q^{81} +(-10.0000 + 10.0000i) q^{82} +(-5.00000 - 15.0000i) q^{85} -10.0000i q^{89} +(9.00000 - 3.00000i) q^{90} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 2 q^{10} - 10 q^{13} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 8 q^{20} - 6 q^{25} - 20 q^{26} - 8 q^{32} + 12 q^{36} + 14 q^{37} - 12 q^{40} - 20 q^{41} + 12 q^{45} - 14 q^{50} - 20 q^{52} + 18 q^{53} + 8 q^{58} + 20 q^{61} - 30 q^{65} - 20 q^{68} + 12 q^{72} + 10 q^{73} - 8 q^{80} - 18 q^{81} - 20 q^{82} - 10 q^{85} + 18 q^{90} - 10 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
<
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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
\(5\) 1.00000 2.00000i 0.447214 0.894427i
\(6\) 0 0
\(7\) 0 0
\(8\) −2.00000 2.00000i −0.707107 0.707107i
\(9\) 3.00000i 1.00000i
\(10\) −1.00000 3.00000i −0.316228 0.948683i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −5.00000 5.00000i −1.38675 1.38675i −0.832050 0.554700i \(-0.812833\pi\)
−0.554700 0.832050i \(-0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 5.00000 5.00000i 1.21268 1.21268i 0.242536 0.970143i \(-0.422021\pi\)
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 \(0.5\pi\)
\(20\) −4.00000 2.00000i −0.894427 0.447214i
\(21\) 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
\(26\) −10.0000 −1.96116
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.00000 + 4.00000i −0.707107 + 0.707107i
\(33\) 0 0
\(34\) 10.0000i 1.71499i
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 7.00000 7.00000i 1.15079 1.15079i 0.164399 0.986394i \(-0.447432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.00000 + 2.00000i −0.948683 + 0.316228i
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 6.00000 + 3.00000i 0.894427 + 0.447214i
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −7.00000 1.00000i −0.989949 0.141421i
\(51\) 0 0
\(52\) −10.0000 + 10.0000i −1.38675 + 1.38675i
\(53\) 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
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) −15.0000 + 5.00000i −1.86052 + 0.620174i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −10.0000 10.0000i −1.21268 1.21268i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 6.00000 6.00000i 0.707107 0.707107i
\(73\) 5.00000 + 5.00000i 0.585206 + 0.585206i 0.936329 0.351123i \(-0.114200\pi\)
−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 \(0.5\pi\)
\(80\) −4.00000 + 8.00000i −0.447214 + 0.894427i
\(81\) −9.00000 −1.00000
\(82\) −10.0000 + 10.0000i −1.10432 + 1.10432i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −5.00000 15.0000i −0.542326 1.62698i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 9.00000 3.00000i 0.948683 0.316228i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00000 + 5.00000i −0.507673 + 0.507673i −0.913812 0.406138i \(-0.866875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 + 6.00000i −0.800000 + 0.600000i
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 20.0000i 1.96116i
\(105\) 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 \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 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
\(117\) 15.0000 15.0000i 1.38675 1.38675i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 10.0000 10.0000i 0.905357 0.905357i
\(123\) 0 0
\(124\) 0 0
\(125\) −11.0000 + 2.00000i −0.983870 + 0.178885i
\(126\) 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
\(130\) −10.0000 + 20.0000i −0.877058 + 1.75412i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −20.0000 −1.71499
\(137\) 7.00000 7.00000i 0.598050 0.598050i −0.341743 0.939793i \(-0.611017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 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
\(145\) 8.00000 + 4.00000i 0.664364 + 0.332182i
\(146\) 10.0000 0.827606
\(147\) 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
\(153\) 15.0000 + 15.0000i 1.21268 + 1.21268i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.00000 + 5.00000i −0.399043 + 0.399043i −0.877896 0.478852i \(-0.841053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 4.00000 + 12.0000i 0.316228 + 0.948683i
\(161\) 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 \(0.250000\pi\)
\(164\) 20.0000i 1.56174i
\(165\) 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
\(169\) 37.0000i 2.84615i
\(170\) −20.0000 10.0000i −1.53393 0.766965i
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0000 + 15.0000i 1.14043 + 1.14043i 0.988372 + 0.152057i \(0.0485898\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0000 10.0000i −0.749532 0.749532i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 6.00000 12.0000i 0.447214 0.894427i
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.00000 21.0000i −0.514650 1.54395i
\(186\) 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 \(0.331909\pi\)
\(194\) 10.0000i 0.717958i
\(195\) 0 0
\(196\) 0 0
\(197\) 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
\(202\) 20.0000 20.0000i 1.40720 1.40720i
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 + 20.0000i −0.698430 + 1.39686i
\(206\) 0 0
\(207\) 0 0
\(208\) 20.0000 + 20.0000i 1.38675 + 1.38675i
\(209\) 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
\(221\) −50.0000 −3.36336
\(222\) 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
\(229\) 30.0000i 1.98246i 0.132164 + 0.991228i \(0.457808\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 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 \(0.175597\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 30.0000i 1.96116i
\(235\) 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
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 11.0000 11.0000i 0.707107 0.707107i
\(243\) 0 0
\(244\) 20.0000i 1.28037i
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −9.00000 + 13.0000i −0.569210 + 0.822192i
\(251\) 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
\(257\) −15.0000 + 15.0000i −0.935674 + 0.935674i −0.998053 0.0623783i \(-0.980131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 10.0000 + 30.0000i 0.620174 + 1.86052i
\(261\) −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
\(265\) 27.0000 9.00000i 1.65860 0.552866i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.0000i 1.21942i −0.792624 0.609711i \(-0.791286\pi\)
0.792624 0.609711i \(-0.208714\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −20.0000 + 20.0000i −1.21268 + 1.21268i
\(273\) 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 \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 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
\(289\) 33.0000i 1.94118i
\(290\) 12.0000 4.00000i 0.704664 0.234888i
\(291\) 0 0
\(292\) 10.0000 10.0000i 0.585206 0.585206i
\(293\) −15.0000 15.0000i −0.876309 0.876309i 0.116841 0.993151i \(-0.462723\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 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
\(305\) 10.0000 20.0000i 0.572598 1.14520i
\(306\) 30.0000 1.71499
\(307\) 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
\(313\) −25.0000 25.0000i −1.41308 1.41308i −0.734803 0.678280i \(-0.762726\pi\)
−0.678280 0.734803i \(-0.737274\pi\)
\(314\) 10.0000i 0.564333i
\(315\) 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
\(320\) 16.0000 + 8.00000i 0.894427 + 0.447214i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) −5.00000 + 35.0000i −0.277350 + 1.94145i
\(326\) 0 0
\(327\) 0 0
\(328\) 20.0000 + 20.0000i 1.10432 + 1.10432i
\(329\) 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 −0.490261 0.871576i \(-0.663099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 37.0000 + 37.0000i 2.01253 + 2.01253i
\(339\) 0 0
\(340\) −30.0000 + 10.0000i −1.62698 + 0.542326i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 30.0000 1.61281
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.0000 25.0000i −1.33062 1.33062i −0.904819 0.425797i \(-0.859994\pi\)
−0.425797 0.904819i \(-0.640006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −20.0000 −1.06000
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −6.00000 18.0000i −0.316228 0.948683i
\(361\) −19.0000 −1.00000
\(362\) 20.0000 20.0000i 1.05118 1.05118i
\(363\) 0 0
\(364\) 0 0
\(365\) 15.0000 5.00000i 0.785136 0.261712i
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 30.0000i 1.56174i
\(370\) −28.0000 14.0000i −1.45565 0.727825i
\(371\) 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
\(377\) 20.0000 20.0000i 1.03005 1.03005i
\(378\) 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
\(388\) 10.0000 + 10.0000i 0.507673 + 0.507673i
\(389\) 34.0000i 1.72387i 0.507020 + 0.861934i \(0.330747\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 26.0000i 1.30986i
\(395\) 0 0
\(396\) 0 0
\(397\) −25.0000 + 25.0000i −1.25471 + 1.25471i −0.301131 + 0.953583i \(0.597364\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 12.0000 + 16.0000i 0.600000 + 0.800000i
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 40.0000i 1.99007i
\(405\) −9.00000 + 18.0000i −0.447214 + 0.894427i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 40.0000i 1.97787i 0.148340 + 0.988936i \(0.452607\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 10.0000 + 30.0000i 0.493865 + 1.48159i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 40.0000 1.96116
\(417\) 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
\(425\) −35.0000 5.00000i −1.69775 0.242536i
\(426\) 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
\(433\) 5.00000 + 5.00000i 0.240285 + 0.240285i 0.816968 0.576683i \(-0.195653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 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
\(441\) 0 0
\(442\) −50.0000 + 50.0000i −2.37826 + 2.37826i
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −20.0000 10.0000i −0.948091 0.474045i
\(446\) 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 \(0.440087\pi\)
\(458\) 30.0000 + 30.0000i 1.40181 + 1.40181i
\(459\) 0 0
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 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 \(0.750000\pi\)
\(468\) −30.0000 30.0000i −1.38675 1.38675i
\(469\) 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
\(481\) −70.0000 −3.19173
\(482\) −30.0000 + 30.0000i −1.36646 + 1.36646i
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 5.00000 + 15.0000i 0.227038 + 0.681115i
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) −20.0000 20.0000i −0.905357 0.905357i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 20.0000 + 20.0000i 0.900755 + 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 4.00000 + 22.0000i 0.178885 + 0.983870i
\(501\) 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
\(505\) 20.0000 40.0000i 0.889988 1.77998i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.0000i 0.443242i 0.975133 + 0.221621i \(0.0711348\pi\)
−0.975133 + 0.221621i \(0.928865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 16.0000 16.0000i 0.707107 0.707107i
\(513\) 0 0
\(514\) 30.0000i 1.32324i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 40.0000 + 20.0000i 1.75412 + 0.877058i
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) −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
\(530\) 18.0000 36.0000i 0.781870 1.56374i
\(531\) 0 0
\(532\) 0 0
\(533\) 50.0000 + 50.0000i 2.16574 + 2.16574i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −20.0000 20.0000i −0.862261 0.862261i
\(539\) 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
\(544\) 40.0000i 1.71499i
\(545\) −12.0000 6.00000i −0.514024 0.257012i
\(546\) 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
\(549\) 30.0000i 1.28037i
\(550\) 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
\(565\) −3.00000 + 1.00000i −0.126211 + 0.0420703i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000i 1.08998i −0.838444 0.544988i \(-0.816534\pi\)
0.838444 0.544988i \(-0.183466\pi\)
\(570\) 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
\(577\) −25.0000 + 25.0000i −1.04076 + 1.04076i −0.0416305 + 0.999133i \(0.513255\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) −33.0000 33.0000i −1.37262 1.37262i
\(579\) 0 0
\(580\) 8.00000 16.0000i 0.332182 0.664364i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 20.0000i 0.827606i
\(585\) −15.0000 45.0000i −0.620174 1.86052i
\(586\) −30.0000 −1.23929
\(587\) 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
\(593\) −15.0000 15.0000i −0.615976 0.615976i 0.328521 0.944497i \(-0.393450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 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
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 22.0000i 0.447214 0.894427i
\(606\) 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
\(610\) −10.0000 30.0000i −0.404888 1.21466i
\(611\) 0 0
\(612\) 30.0000 30.0000i 1.21268 1.21268i
\(613\) 1.00000 + 1.00000i 0.0403896 + 0.0403896i 0.727013 0.686624i \(-0.240908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 + 3.00000i −0.120775 + 0.120775i −0.764911 0.644136i \(-0.777217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(618\) 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
\(626\) −50.0000 −1.99840
\(627\) 0 0
\(628\) 10.0000 + 10.0000i 0.399043 + 0.399043i
\(629\) 70.0000i 2.79108i
\(630\) 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
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 24.0000 8.00000i 0.948683 0.316228i
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 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
\(650\) 30.0000 + 40.0000i 1.17670 + 1.56893i
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 9.00000i −0.352197 0.352197i 0.508729 0.860927i \(-0.330115\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 40.0000 1.56174
\(657\) −15.0000 + 15.0000i −0.585206 + 0.585206i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) 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 0.886585 0.462566i \(-0.153071\pi\)
−0.462566 + 0.886585i \(0.653071\pi\)
\(674\) 14.0000i 0.539260i
\(675\) 0 0
\(676\) 74.0000 2.84615
\(677\) 25.0000 25.0000i 0.960828 0.960828i −0.0384331 0.999261i \(-0.512237\pi\)
0.999261 + 0.0384331i \(0.0122367\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −20.0000 + 40.0000i −0.766965 + 1.53393i
\(681\) 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
\(685\) −7.00000 21.0000i −0.267456 0.802369i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 90.0000i 3.42873i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 30.0000 30.0000i 1.14043 1.14043i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −50.0000 + 50.0000i −1.89389 + 1.89389i
\(698\) 10.0000 + 10.0000i 0.378506 + 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) −52.0000 −1.96401 −0.982006 0.188847i \(-0.939525\pi\)
−0.982006 + 0.188847i \(0.939525\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −50.0000 −1.88177
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000i 1.65245i 0.563337 + 0.826227i \(0.309517\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −20.0000 + 20.0000i −0.749532 + 0.749532i
\(713\) 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 \(0.5\pi\)
\(720\) −24.0000 12.0000i −0.894427 0.447214i
\(721\) 0 0
\(722\) −19.0000 + 19.0000i −0.707107 + 0.707107i
\(723\) 0 0
\(724\) 40.0000i 1.48659i
\(725\) 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
\(730\) 10.0000 20.0000i 0.370117 0.740233i
\(731\) 0 0
\(732\) 0 0
\(733\) 25.0000 + 25.0000i 0.923396 + 0.923396i 0.997268 0.0738717i \(-0.0235355\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −30.0000 30.0000i −1.10432 1.10432i
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −42.0000 + 14.0000i −1.54395 + 0.514650i
\(741\) 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
\(745\) −28.0000 14.0000i −1.02584 0.512920i
\(746\) −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
\(754\) 40.0000i 1.45671i
\(755\) 0 0
\(756\) 0 0
\(757\) 17.0000 17.0000i 0.617876 0.617876i −0.327111 0.944986i \(-0.606075\pi\)
0.944986 + 0.327111i \(0.106075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 45.0000 15.0000i 1.62698 0.542326i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 50.0000i 1.80305i 0.432731 + 0.901523i \(0.357550\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 38.0000 38.0000i 1.36765 1.36765i
\(773\) −5.00000 5.00000i −0.179838 0.179838i 0.611448 0.791285i \(-0.290588\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 20.0000 0.717958
\(777\) 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
\(784\) 0 0
\(785\) 5.00000 + 15.0000i 0.178458 + 0.535373i
\(786\) 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
\(793\) −50.0000 50.0000i −1.77555 1.77555i
\(794\) 50.0000i 1.77443i
\(795\) 0 0
\(796\) 0 0
\(797\) −15.0000 + 15.0000i −0.531327 + 0.531327i −0.920967 0.389640i \(-0.872599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.0000 + 4.00000i 0.989949 + 0.141421i
\(801\) 30.0000 1.06000
\(802\) −2.00000 + 2.00000i −0.0706225 + 0.0706225i
\(803\) 0 0
\(804\)