Properties

Label 912.2.ch.c.719.1
Level $912$
Weight $2$
Character 912.719
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ch (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 719.1
Root \(-0.766044 - 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 912.719
Dual form 912.2.ch.c.671.1

$q$-expansion

\(f(q)\) \(=\) \(q+(0.592396 + 1.62760i) q^{3} +(-4.00387 + 2.31164i) q^{7} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})\) \(q+(0.592396 + 1.62760i) q^{3} +(-4.00387 + 2.31164i) q^{7} +(-2.29813 + 1.92836i) q^{9} +(6.73783 + 2.45237i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(-6.13429 - 5.14728i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(-9.12108 + 5.26606i) q^{31} -12.1138 q^{37} +12.4192i q^{39} +(2.77584 + 0.489456i) q^{43} +(7.18732 - 12.4488i) q^{49} +(2.15523 - 7.23567i) q^{57} +(1.60694 + 9.11343i) q^{61} +(4.74376 - 13.0334i) q^{63} +(5.60994 + 6.68566i) q^{67} +(11.1912 - 4.07326i) q^{73} -8.66025i q^{75} +(6.03462 + 16.5800i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-32.6464 + 5.75643i) q^{91} +(-13.9743 - 11.7258i) q^{93} +(-3.83022 - 3.21394i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + O(q^{10}) \) \( 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} - 27 q^{27} + 15 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 15 q^{67} + 21 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.592396 + 1.62760i 0.342020 + 0.939693i
\(4\) 0 0
\(5\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(6\) 0 0
\(7\) −4.00387 + 2.31164i −1.51332 + 0.873716i −0.513442 + 0.858124i \(0.671630\pi\)
−0.999878 + 0.0155920i \(0.995037\pi\)
\(8\) 0 0
\(9\) −2.29813 + 1.92836i −0.766044 + 0.642788i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 6.73783 + 2.45237i 1.86874 + 0.680165i 0.970725 + 0.240192i \(0.0772105\pi\)
0.898011 + 0.439972i \(0.145012\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(18\) 0 0
\(19\) −3.50000 2.59808i −0.802955 0.596040i
\(20\) 0 0
\(21\) −6.13429 5.14728i −1.33861 1.12323i
\(22\) 0 0
\(23\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(24\) 0 0
\(25\) −4.69846 1.71010i −0.939693 0.342020i
\(26\) 0 0
\(27\) −4.50000 2.59808i −0.866025 0.500000i
\(28\) 0 0
\(29\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(30\) 0 0
\(31\) −9.12108 + 5.26606i −1.63819 + 0.945812i −0.656740 + 0.754117i \(0.728065\pi\)
−0.981455 + 0.191695i \(0.938602\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.1138 −1.99150 −0.995749 0.0921098i \(-0.970639\pi\)
−0.995749 + 0.0921098i \(0.970639\pi\)
\(38\) 0 0
\(39\) 12.4192i 1.98867i
\(40\) 0 0
\(41\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(42\) 0 0
\(43\) 2.77584 + 0.489456i 0.423312 + 0.0746414i 0.381246 0.924473i \(-0.375495\pi\)
0.0420659 + 0.999115i \(0.486606\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(48\) 0 0
\(49\) 7.18732 12.4488i 1.02676 1.77840i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.15523 7.23567i 0.285467 0.958388i
\(58\) 0 0
\(59\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(60\) 0 0
\(61\) 1.60694 + 9.11343i 0.205748 + 1.16686i 0.896258 + 0.443533i \(0.146275\pi\)
−0.690510 + 0.723323i \(0.742614\pi\)
\(62\) 0 0
\(63\) 4.74376 13.0334i 0.597657 1.64205i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.60994 + 6.68566i 0.685363 + 0.816784i 0.990787 0.135433i \(-0.0432425\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(72\) 0 0
\(73\) 11.1912 4.07326i 1.30983 0.476739i 0.409644 0.912245i \(-0.365653\pi\)
0.900186 + 0.435506i \(0.143431\pi\)
\(74\) 0 0
\(75\) 8.66025i 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.03462 + 16.5800i 0.678947 + 1.86539i 0.453930 + 0.891038i \(0.350022\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.56283 8.86327i 0.173648 0.984808i
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(90\) 0 0
\(91\) −32.6464 + 5.75643i −3.42227 + 0.603438i
\(92\) 0 0
\(93\) −13.9743 11.7258i −1.44907 1.21591i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.83022 3.21394i −0.388900 0.326326i 0.427284 0.904117i \(-0.359470\pi\)
−0.816185 + 0.577791i \(0.803915\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(102\) 0 0
\(103\) 16.5364 + 9.54731i 1.62938 + 0.940724i 0.984277 + 0.176631i \(0.0565198\pi\)
0.645105 + 0.764094i \(0.276813\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(108\) 0 0
\(109\) −2.95202 + 16.7417i −0.282752 + 1.60357i 0.430454 + 0.902613i \(0.358354\pi\)
−0.713206 + 0.700954i \(0.752758\pi\)
\(110\) 0 0
\(111\) −7.17617 19.7164i −0.681132 1.87140i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −20.2135 + 7.35710i −1.86874 + 0.680165i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.70115 21.1587i 0.683367 1.87753i 0.299991 0.953942i \(-0.403016\pi\)
0.383375 0.923593i \(-0.374762\pi\)
\(128\) 0 0
\(129\) 0.847763 + 4.80790i 0.0746414 + 0.423312i
\(130\) 0 0
\(131\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(132\) 0 0
\(133\) 20.0194 + 2.31164i 1.73590 + 0.200444i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(138\) 0 0
\(139\) 4.36050 11.9804i 0.369853 1.01616i −0.605564 0.795796i \(-0.707053\pi\)
0.975417 0.220366i \(-0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 24.5194 + 4.32342i 2.02232 + 0.356590i
\(148\) 0 0
\(149\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(150\) 0 0
\(151\) 8.66025i 0.704761i 0.935857 + 0.352381i \(0.114628\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.51889 + 19.9566i −0.280838 + 1.59271i 0.438948 + 0.898513i \(0.355351\pi\)
−0.719785 + 0.694197i \(0.755760\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.4696 6.04460i −0.820039 0.473450i 0.0303908 0.999538i \(-0.490325\pi\)
−0.850430 + 0.526088i \(0.823658\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(168\) 0 0
\(169\) 29.4256 + 24.6910i 2.26351 + 1.89931i
\(170\) 0 0
\(171\) 13.0535 0.778544i 0.998226 0.0595368i
\(172\) 0 0
\(173\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(174\) 0 0
\(175\) 22.7652 4.01411i 1.72088 0.303438i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 5.36231 4.49951i 0.398577 0.334446i −0.421366 0.906891i \(-0.638449\pi\)
0.819943 + 0.572444i \(0.194005\pi\)
\(182\) 0 0
\(183\) −13.8810 + 8.01422i −1.02612 + 0.592428i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 24.0232 1.74743
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −18.1789 + 6.61656i −1.30854 + 0.476271i −0.899770 0.436365i \(-0.856266\pi\)
−0.408773 + 0.912636i \(0.634043\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(198\) 0 0
\(199\) −9.86319 11.7545i −0.699183 0.833254i 0.293251 0.956036i \(-0.405263\pi\)
−0.992434 + 0.122782i \(0.960818\pi\)
\(200\) 0 0
\(201\) −7.55825 + 13.0913i −0.533118 + 0.923387i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5.32934 + 6.35127i −0.366887 + 0.437239i −0.917630 0.397436i \(-0.869900\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.3464 42.1692i 1.65274 2.86263i
\(218\) 0 0
\(219\) 13.2592 + 15.8017i 0.895976 + 1.06778i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.186137 + 0.0328209i 0.0124646 + 0.00219785i 0.179877 0.983689i \(-0.442430\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) 14.0954 5.13030i 0.939693 0.342020i
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 3.49289 0.230817 0.115408 0.993318i \(-0.463182\pi\)
0.115408 + 0.993318i \(0.463182\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −23.4106 + 19.6438i −1.52068 + 1.27600i
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −3.45558 1.25773i −0.222594 0.0810175i 0.228316 0.973587i \(-0.426678\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 15.3516 2.70691i 0.984808 0.173648i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −17.2110 26.0887i −1.09511 1.65998i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(258\) 0 0
\(259\) 48.5021 28.0027i 3.01377 1.74000i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(270\) 0 0
\(271\) 15.3516 + 2.70691i 0.932545 + 0.164433i 0.619224 0.785214i \(-0.287447\pi\)
0.313321 + 0.949647i \(0.398558\pi\)
\(272\) 0 0
\(273\) −28.7087 49.7250i −1.73753 3.00949i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −15.5000 + 26.8468i −0.931305 + 1.61307i −0.150210 + 0.988654i \(0.547995\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 10.8066 29.6909i 0.646974 1.77755i
\(280\) 0 0
\(281\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(282\) 0 0
\(283\) −14.4734 + 17.2488i −0.860356 + 1.02533i 0.139030 + 0.990288i \(0.455602\pi\)
−0.999386 + 0.0350443i \(0.988843\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.95202 + 16.7417i 0.173648 + 0.984808i
\(290\) 0 0
\(291\) 2.96198 8.13798i 0.173634 0.477057i
\(292\) 0 0
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.2456 + 4.45702i −0.705823 + 0.256898i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.592396 1.62760i −0.0338098 0.0928918i 0.921639 0.388048i \(-0.126851\pi\)
−0.955449 + 0.295156i \(0.904628\pi\)
\(308\) 0 0
\(309\) −5.74304 + 32.5704i −0.326710 + 1.85287i
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 26.8116 22.4976i 1.51548 1.27164i 0.663345 0.748314i \(-0.269136\pi\)
0.852134 0.523324i \(-0.175308\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −27.4636 23.0447i −1.52341 1.27829i
\(326\) 0 0
\(327\) −28.9975 + 5.11305i −1.60357 + 0.282752i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 29.3041 + 16.9187i 1.61070 + 0.929938i 0.989208 + 0.146518i \(0.0468065\pi\)
0.621492 + 0.783420i \(0.286527\pi\)
\(332\) 0 0
\(333\) 27.8391 23.3598i 1.52558 1.28011i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.34436 + 7.62424i −0.0732319 + 0.415319i 0.926049 + 0.377403i \(0.123183\pi\)
−0.999281 + 0.0379157i \(0.987928\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 34.0949i 1.84095i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(348\) 0 0
\(349\) −16.4957 28.5714i −0.882996 1.52939i −0.847994 0.530006i \(-0.822190\pi\)
−0.0350017 0.999387i \(-0.511144\pi\)
\(350\) 0 0
\(351\) −23.9488 28.5410i −1.27829 1.52341i
\(352\) 0 0
\(353\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(360\) 0 0
\(361\) 5.50000 + 18.1865i 0.289474 + 0.957186i
\(362\) 0 0
\(363\) −12.2467 + 14.5951i −0.642788 + 0.766044i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.10426 + 25.0137i −0.475238 + 1.30571i 0.438254 + 0.898851i \(0.355597\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −19.0000 32.9090i −0.983783 1.70396i −0.647225 0.762299i \(-0.724071\pi\)
−0.336557 0.941663i \(-0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.4953i 0.744576i 0.928117 + 0.372288i \(0.121427\pi\)
−0.928117 + 0.372288i \(0.878573\pi\)
\(380\) 0 0
\(381\) 39.0000 1.99803
\(382\) 0 0
\(383\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.32311 + 4.22800i −0.372255 + 0.214921i
\(388\) 0 0
\(389\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.1573 + 9.36208i 0.559968 + 0.469869i 0.878300 0.478110i \(-0.158678\pi\)
−0.318332 + 0.947979i \(0.603123\pi\)
\(398\) 0 0
\(399\) 8.09698 + 33.9528i 0.405356 + 1.69977i
\(400\) 0 0
\(401\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(402\) 0 0
\(403\) −74.3706 + 13.1135i −3.70466 + 0.653232i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −29.1097 + 24.4259i −1.43938 + 1.20778i −0.499486 + 0.866322i \(0.666478\pi\)
−0.939895 + 0.341463i \(0.889078\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.0823 1.08138
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 38.5274 14.0228i 1.87771 0.683431i 0.924419 0.381377i \(-0.124550\pi\)
0.953291 0.302053i \(-0.0976721\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −27.5009 32.7743i −1.33086 1.58606i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(432\) 0 0
\(433\) 2.14249 + 12.1507i 0.102962 + 0.583924i 0.992015 + 0.126121i \(0.0402527\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 14.2447 16.9762i 0.679862 0.810228i −0.310228 0.950662i \(-0.600405\pi\)
0.990090 + 0.140434i \(0.0448499\pi\)
\(440\) 0 0
\(441\) 7.48839 + 42.4688i 0.356590 + 2.02232i
\(442\) 0 0
\(443\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −14.0954 + 5.13030i −0.662259 + 0.241043i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.82058 0.225497 0.112749 0.993624i \(-0.464034\pi\)
0.112749 + 0.993624i \(0.464034\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(462\) 0 0
\(463\) 11.3270 6.53964i 0.526409 0.303923i −0.213144 0.977021i \(-0.568370\pi\)
0.739553 + 0.673098i \(0.235037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) −37.9163 13.8004i −1.75081 0.637243i
\(470\) 0 0
\(471\) −34.5658 + 6.09489i −1.59271 + 0.280838i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 12.0016 + 18.1923i 0.550673 + 0.834721i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(480\) 0 0
\(481\) −81.6207 29.7075i −3.72158 1.35455i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 31.5000 18.1865i 1.42740 0.824110i 0.430486 0.902597i \(-0.358342\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 3.63604 20.6210i 0.164427 0.932514i
\(490\) 0 0
\(491\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −36.3666 6.41242i −1.62799 0.287059i −0.716258 0.697835i \(-0.754147\pi\)
−0.911736 + 0.410776i \(0.865258\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −22.7554 + 62.5199i −1.01060 + 2.77660i
\(508\) 0 0
\(509\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(510\) 0 0
\(511\) −35.3922 + 42.1788i −1.56566 + 1.86588i
\(512\) 0 0
\(513\) 9.00000 + 20.7846i 0.397360 + 0.917663i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −18.8692 22.4874i −0.825091 0.983306i 0.174908 0.984585i \(-0.444037\pi\)
−0.999999 + 0.00127919i \(0.999593\pi\)
\(524\) 0 0
\(525\) 20.0194 + 34.6745i 0.873716 + 1.51332i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.6129 7.86646i 0.939693 0.342020i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −31.2977 + 26.2619i −1.34559 + 1.12909i −0.365444 + 0.930834i \(0.619083\pi\)
−0.980151 + 0.198254i \(0.936473\pi\)
\(542\) 0 0
\(543\) 10.5000 + 6.06218i 0.450598 + 0.260153i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.96720 1.58116i 0.383410 0.0676055i 0.0213785 0.999771i \(-0.493195\pi\)
0.362031 + 0.932166i \(0.382083\pi\)
\(548\) 0 0
\(549\) −21.2670 17.8451i −0.907653 0.761611i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −62.4887 52.4342i −2.65729 2.22973i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(558\) 0 0
\(559\) 17.5028 + 10.1053i 0.740291 + 0.427407i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 14.2313 + 39.1001i 0.597657 + 1.64205i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 44.7821i 1.87407i −0.349231 0.937037i \(-0.613557\pi\)
0.349231 0.937037i \(-0.386443\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.50000 9.52628i −0.228968 0.396584i 0.728535 0.685009i \(-0.240202\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) −21.5382 25.6682i −0.895096 1.06673i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(588\) 0 0
\(589\) 45.6054 + 5.26606i 1.87914 + 0.216984i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.2886 23.0166i 0.543868 0.942007i
\(598\) 0 0
\(599\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(600\) 0 0
\(601\) 18.2114 + 31.5431i 0.742859 + 1.28667i 0.951188 + 0.308611i \(0.0998642\pi\)
−0.208329 + 0.978059i \(0.566802\pi\)
\(602\) 0 0
\(603\) −25.7848 4.54655i −1.05004 0.185150i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 35.4771i 1.43997i −0.693990 0.719985i \(-0.744149\pi\)
0.693990 0.719985i \(-0.255851\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.16146 46.2860i 0.329638 1.86947i −0.145204 0.989402i \(-0.546384\pi\)
0.474843 0.880071i \(-0.342505\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(618\) 0 0
\(619\) 21.5315 + 12.4312i 0.865425 + 0.499653i 0.865825 0.500347i \(-0.166794\pi\)
−0.000400419 1.00000i \(0.500127\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.1511 + 16.0697i 0.766044 + 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −9.55959 + 1.68561i −0.380561 + 0.0671032i −0.360657 0.932699i \(-0.617447\pi\)
−0.0199047 + 0.999802i \(0.506336\pi\)
\(632\) 0 0
\(633\) −13.4944 4.91155i −0.536353 0.195217i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 78.9559 66.2519i 3.12835 2.62500i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(642\) 0 0
\(643\) −5.34096 14.6742i −0.210627 0.578692i 0.788723 0.614749i \(-0.210743\pi\)
−0.999350 + 0.0360565i \(0.988520\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 83.0572 + 14.6452i 3.25527 + 0.573991i
\(652\) 0 0
\(653\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −17.8641 + 30.9416i −0.696946 + 1.20715i
\(658\) 0 0
\(659\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(660\) 0 0
\(661\) 6.59863 + 37.4227i 0.256657 + 1.45557i 0.791783 + 0.610802i \(0.209153\pi\)
−0.535126 + 0.844772i \(0.679736\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.0568475 + 0.322398i 0.00219785 + 0.0124646i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.8619 44.7941i 0.996903 1.72669i 0.430346 0.902664i \(-0.358391\pi\)
0.566557 0.824023i \(-0.308275\pi\)
\(674\) 0 0
\(675\) 16.7001 + 19.9024i 0.642788 + 0.766044i
\(676\) 0 0
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 22.7652 + 4.01411i 0.873647 + 0.154048i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.06917 + 5.68501i 0.0789439 + 0.216897i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −16.5000 + 9.52628i −0.627690 + 0.362397i −0.779857 0.625958i \(-0.784708\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(702\) 0 0
\(703\) 42.3983 + 31.4726i 1.59908 + 1.18701i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −48.4702 17.6417i −1.82034 0.662548i −0.995228 0.0975728i \(-0.968892\pi\)
−0.825108 0.564975i \(-0.808886\pi\)
\(710\) 0 0
\(711\) −45.8405 26.4661i −1.71915 0.992554i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(720\) 0 0
\(721\) −88.2796 −3.28770
\(722\) 0 0
\(723\) 6.36937i 0.236879i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −37.3417 6.58434i −1.38493 0.244200i −0.568991 0.822344i \(-0.692666\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.5000 + 23.3827i 0.500000 + 0.866025i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.50000 + 6.06218i −0.129275 + 0.223912i −0.923396 0.383849i \(-0.874598\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 27.8684 33.2123i 1.02516 1.22173i 0.0503375 0.998732i \(-0.483970\pi\)
0.974818 0.223001i \(-0.0715853\pi\)
\(740\) 0 0
\(741\) 32.2661 43.4673i 1.18532 1.59681i
\(742\) 0 0
\(743\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 34.8505 + 41.5332i 1.27171 + 1.51557i 0.748056 + 0.663636i \(0.230988\pi\)
0.523655 + 0.851930i \(0.324568\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.5775 + 3.84991i −0.384447 + 0.139927i −0.527011 0.849858i \(-0.676688\pi\)
0.142564 + 0.989786i \(0.454465\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −26.8813 73.8557i −0.973168 2.67376i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −41.5474 + 34.8624i −1.49824 + 1.25717i −0.614745 + 0.788726i \(0.710741\pi\)
−0.883493 + 0.468445i \(0.844814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(774\) 0 0
\(775\) 51.8606 9.14442i 1.86289 0.328477i
\(776\) 0 0
\(777\) 74.3096 + 62.3531i 2.66584 + 2.23691i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 48.2816 + 27.8754i 1.72105 + 0.993650i 0.916783 + 0.399385i \(0.130776\pi\)
0.804269 + 0.594265i \(0.202557\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −11.5222 + 65.3455i −0.409165 + 2.32049i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\)