Properties

Label 804.2.s.a.137.1
Level 804
Weight 2
Character 804.137
Analytic conductor 6.420
Analytic rank 0
Dimension 20
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.s (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 137.1
Root \(0.0475819 - 0.998867i\)
Character \(\chi\) = 804.137
Dual form 804.2.s.a.581.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +O(q^{10})\) \(q+(-1.71442 - 0.246497i) q^{3} +(-1.43029 + 4.87111i) q^{7} +(2.87848 + 0.845198i) q^{9} +(-5.44973 - 4.72222i) q^{13} +(2.19755 - 0.645259i) q^{19} +(3.65282 - 7.99857i) q^{21} +(3.27430 - 3.77875i) q^{25} +(-4.72659 - 2.15856i) q^{27} +(-0.512845 + 0.444383i) q^{31} -11.7846 q^{37} +(8.17912 + 9.43921i) q^{39} +(-6.94805 - 10.8114i) q^{43} +(-15.7932 - 10.1497i) q^{49} +(-3.92658 + 0.564557i) q^{57} +(-11.7083 - 5.34702i) q^{61} +(-8.23410 + 12.8125i) q^{63} +(-3.92989 + 7.18025i) q^{67} +(6.96168 - 15.2440i) q^{73} +(-6.54498 + 5.67126i) q^{75} +(10.8567 + 9.40741i) q^{79} +(7.57128 + 4.86577i) q^{81} +(30.7971 - 19.7921i) q^{91} +(0.988771 - 0.635445i) q^{93} +15.8295i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{22}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.71442 0.246497i −0.989821 0.142315i
\(4\) 0 0
\(5\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(6\) 0 0
\(7\) −1.43029 + 4.87111i −0.540597 + 1.84111i 0.000273951 1.00000i \(0.499913\pi\)
−0.540871 + 0.841105i \(0.681905\pi\)
\(8\) 0 0
\(9\) 2.87848 + 0.845198i 0.959493 + 0.281733i
\(10\) 0 0
\(11\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(12\) 0 0
\(13\) −5.44973 4.72222i −1.51148 1.30971i −0.758731 0.651404i \(-0.774180\pi\)
−0.752753 0.658303i \(-0.771274\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(18\) 0 0
\(19\) 2.19755 0.645259i 0.504152 0.148032i −0.0197599 0.999805i \(-0.506290\pi\)
0.523912 + 0.851772i \(0.324472\pi\)
\(20\) 0 0
\(21\) 3.65282 7.99857i 0.797111 1.74543i
\(22\) 0 0
\(23\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(24\) 0 0
\(25\) 3.27430 3.77875i 0.654861 0.755750i
\(26\) 0 0
\(27\) −4.72659 2.15856i −0.909632 0.415415i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −0.512845 + 0.444383i −0.0921097 + 0.0798135i −0.699699 0.714438i \(-0.746682\pi\)
0.607589 + 0.794252i \(0.292137\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.7846 −1.93738 −0.968688 0.248283i \(-0.920134\pi\)
−0.968688 + 0.248283i \(0.920134\pi\)
\(38\) 0 0
\(39\) 8.17912 + 9.43921i 1.30971 + 1.51148i
\(40\) 0 0
\(41\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(42\) 0 0
\(43\) −6.94805 10.8114i −1.05957 1.64872i −0.697529 0.716557i \(-0.745717\pi\)
−0.362039 0.932163i \(-0.617919\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(48\) 0 0
\(49\) −15.7932 10.1497i −2.25617 1.44995i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.92658 + 0.564557i −0.520088 + 0.0747774i
\(58\) 0 0
\(59\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(60\) 0 0
\(61\) −11.7083 5.34702i −1.49910 0.684616i −0.514188 0.857677i \(-0.671907\pi\)
−0.984911 + 0.173061i \(0.944634\pi\)
\(62\) 0 0
\(63\) −8.23410 + 12.8125i −1.03740 + 1.61422i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.92989 + 7.18025i −0.480112 + 0.877207i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(72\) 0 0
\(73\) 6.96168 15.2440i 0.814803 1.78417i 0.229598 0.973286i \(-0.426259\pi\)
0.585206 0.810885i \(-0.301014\pi\)
\(74\) 0 0
\(75\) −6.54498 + 5.67126i −0.755750 + 0.654861i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.8567 + 9.40741i 1.22148 + 1.05842i 0.996461 + 0.0840621i \(0.0267894\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 7.57128 + 4.86577i 0.841254 + 0.540641i
\(82\) 0 0
\(83\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(90\) 0 0
\(91\) 30.7971 19.7921i 3.22841 2.07478i
\(92\) 0 0
\(93\) 0.988771 0.635445i 0.102531 0.0658926i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.8295i 1.60724i 0.595143 + 0.803620i \(0.297095\pi\)
−0.595143 + 0.803620i \(0.702905\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(102\) 0 0
\(103\) −13.2058 15.2403i −1.30121 1.50167i −0.736644 0.676281i \(-0.763591\pi\)
−0.564562 0.825391i \(-0.690955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(108\) 0 0
\(109\) 7.90023 + 6.84559i 0.756705 + 0.655688i 0.945239 0.326378i \(-0.105828\pi\)
−0.188534 + 0.982067i \(0.560374\pi\)
\(110\) 0 0
\(111\) 20.2038 + 2.90486i 1.91766 + 0.275717i
\(112\) 0 0
\(113\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −11.6957 18.1989i −1.08127 1.68249i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.20347 8.31325i 0.654861 0.755750i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.03085 1.47719i −0.446415 0.131079i 0.0507955 0.998709i \(-0.483824\pi\)
−0.497211 + 0.867630i \(0.665643\pi\)
\(128\) 0 0
\(129\) 9.24692 + 20.2479i 0.814146 + 1.78273i
\(130\) 0 0
\(131\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(132\) 0 0
\(133\) 11.6274i 1.00822i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(138\) 0 0
\(139\) −12.1638 + 5.55500i −1.03172 + 0.471169i −0.858013 0.513629i \(-0.828301\pi\)
−0.173704 + 0.984798i \(0.555574\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 24.5743 + 21.2938i 2.02686 + 1.75628i
\(148\) 0 0
\(149\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(150\) 0 0
\(151\) −19.9537 + 12.8235i −1.62381 + 1.04356i −0.670352 + 0.742043i \(0.733857\pi\)
−0.953460 + 0.301518i \(0.902507\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.07835 7.50008i 0.0860615 0.598571i −0.900460 0.434939i \(-0.856770\pi\)
0.986521 0.163632i \(-0.0523210\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.8399 1.55398 0.776989 0.629514i \(-0.216746\pi\)
0.776989 + 0.629514i \(0.216746\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(168\) 0 0
\(169\) 5.55012 + 38.6020i 0.426933 + 2.96938i
\(170\) 0 0
\(171\) 6.87097 0.525436
\(172\) 0 0
\(173\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(174\) 0 0
\(175\) 13.7235 + 21.3542i 1.03740 + 1.61422i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(180\) 0 0
\(181\) 1.50254 10.4504i 0.111683 0.776773i −0.854599 0.519288i \(-0.826197\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) 18.7550 + 12.0531i 1.38641 + 0.890991i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 17.2749 19.9364i 1.25657 1.45016i
\(190\) 0 0
\(191\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(192\) 0 0
\(193\) −3.85622 4.45032i −0.277577 0.320341i 0.599793 0.800155i \(-0.295250\pi\)
−0.877370 + 0.479814i \(0.840704\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(198\) 0 0
\(199\) −20.8040 6.10860i −1.47476 0.433027i −0.557114 0.830436i \(-0.688091\pi\)
−0.917642 + 0.397409i \(0.869909\pi\)
\(200\) 0 0
\(201\) 8.50739 11.3413i 0.600065 0.799951i
\(202\) 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
\(211\) 2.19318 + 15.2539i 0.150985 + 1.05012i 0.914574 + 0.404419i \(0.132526\pi\)
−0.763589 + 0.645703i \(0.776565\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.43112 3.13372i −0.0971508 0.212731i
\(218\) 0 0
\(219\) −15.6928 + 24.4185i −1.06042 + 1.65005i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.896827 + 6.23757i 0.0600560 + 0.417698i 0.997565 + 0.0697384i \(0.0222164\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 12.6188 8.10961i 0.841254 0.540641i
\(226\) 0 0
\(227\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(228\) 0 0
\(229\) −2.27894 + 1.97471i −0.150596 + 0.130493i −0.726900 0.686743i \(-0.759040\pi\)
0.576304 + 0.817235i \(0.304494\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −16.2941 18.8044i −1.05842 1.22148i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 4.89188 10.7117i 0.315114 0.690003i −0.684111 0.729378i \(-0.739809\pi\)
0.999225 + 0.0393750i \(0.0125367\pi\)
\(242\) 0 0
\(243\) −11.7810 10.2083i −0.755750 0.654861i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.0231 6.86082i −0.955897 0.436544i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(258\) 0 0
\(259\) 16.8553 57.4040i 1.04734 3.56691i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −24.2580 3.48777i −1.47357 0.211867i −0.641748 0.766915i \(-0.721791\pi\)
−0.831818 + 0.555048i \(0.812700\pi\)
\(272\) 0 0
\(273\) −57.6779 + 26.3406i −3.49082 + 1.59421i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 31.7684 + 9.32805i 1.90878 + 0.560468i 0.983408 + 0.181408i \(0.0580655\pi\)
0.925372 + 0.379060i \(0.123753\pi\)
\(278\) 0 0
\(279\) −1.85181 + 0.845691i −0.110865 + 0.0506302i
\(280\) 0 0
\(281\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(282\) 0 0
\(283\) −21.8251 + 6.40842i −1.29737 + 0.380941i −0.856274 0.516522i \(-0.827227\pi\)
−0.441091 + 0.897462i \(0.645408\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.06206 15.4637i 0.415415 0.909632i
\(290\) 0 0
\(291\) 3.90191 27.1384i 0.228734 1.59088i
\(292\) 0 0
\(293\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 62.6011 18.3813i 3.60827 1.05948i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.8525 + 22.9110i 1.13304 + 1.30760i 0.945603 + 0.325322i \(0.105473\pi\)
0.187437 + 0.982277i \(0.439982\pi\)
\(308\) 0 0
\(309\) 18.8836 + 29.3835i 1.07425 + 1.67157i
\(310\) 0 0
\(311\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(312\) 0 0
\(313\) −20.1846 + 2.90210i −1.14090 + 0.164037i −0.686753 0.726891i \(-0.740965\pi\)
−0.454146 + 0.890927i \(0.650056\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −35.6881 + 5.13118i −1.97962 + 0.284627i
\(326\) 0 0
\(327\) −11.8569 13.6836i −0.655688 0.756705i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −18.4571 + 28.7198i −1.01449 + 1.57858i −0.216166 + 0.976357i \(0.569355\pi\)
−0.798327 + 0.602224i \(0.794281\pi\)
\(332\) 0 0
\(333\) −33.9217 9.96031i −1.85890 0.545822i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.68281 + 15.9482i −0.255089 + 0.868753i 0.727993 + 0.685585i \(0.240453\pi\)
−0.983082 + 0.183168i \(0.941365\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 45.1716 39.1414i 2.43904 2.11344i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(348\) 0 0
\(349\) 3.09026 + 1.98599i 0.165418 + 0.106308i 0.620730 0.784024i \(-0.286836\pi\)
−0.455313 + 0.890332i \(0.650472\pi\)
\(350\) 0 0
\(351\) 15.5654 + 34.0835i 0.830821 + 1.81925i
\(352\) 0 0
\(353\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(360\) 0 0
\(361\) −11.5710 + 7.43620i −0.608998 + 0.391379i
\(362\) 0 0
\(363\) −14.3990 + 12.4768i −0.755750 + 0.654861i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −19.2698 + 2.77058i −1.00588 + 0.144623i −0.625513 0.780214i \(-0.715110\pi\)
−0.380363 + 0.924837i \(0.624201\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 17.3534i 0.898525i 0.893400 + 0.449262i \(0.148313\pi\)
−0.893400 + 0.449262i \(0.851687\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −38.4204 5.52401i −1.97352 0.283750i −0.997944 0.0640964i \(-0.979583\pi\)
−0.975578 0.219653i \(-0.929507\pi\)
\(380\) 0 0
\(381\) 8.26087 + 3.77261i 0.423217 + 0.193277i
\(382\) 0 0
\(383\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.8621 36.9928i −0.552150 1.88045i
\(388\) 0 0
\(389\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.21394 + 15.7963i 0.362057 + 0.792794i 0.999747 + 0.0225011i \(0.00716292\pi\)
−0.637690 + 0.770293i \(0.720110\pi\)
\(398\) 0 0
\(399\) 2.86611 19.9343i 0.143485 0.997961i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 4.89334 0.243755
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 11.3601 38.6888i 0.561719 1.91304i 0.205387 0.978681i \(-0.434155\pi\)
0.356332 0.934359i \(-0.384027\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.2231 6.52529i 1.08827 0.319545i
\(418\) 0 0
\(419\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(420\) 0 0
\(421\) −11.2961 + 3.31684i −0.550539 + 0.161653i −0.545159 0.838333i \(-0.683531\pi\)
−0.00537983 + 0.999986i \(0.501712\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 42.7922 49.3848i 2.07086 2.38990i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −21.7154 + 18.8165i −1.04357 + 0.904262i −0.995517 0.0945826i \(-0.969848\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 6.69959 0.319754 0.159877 0.987137i \(-0.448890\pi\)
0.159877 + 0.987137i \(0.448890\pi\)
\(440\) 0 0
\(441\) −36.8819 42.5640i −1.75628 2.02686i
\(442\) 0 0
\(443\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 37.3701 17.0663i 1.75580 0.801847i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13.9527 + 16.1023i −0.652679 + 0.753232i −0.981563 0.191139i \(-0.938782\pi\)
0.328884 + 0.944370i \(0.393327\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(462\) 0 0
\(463\) −38.3831 17.5290i −1.78381 0.814640i −0.973516 0.228618i \(-0.926579\pi\)
−0.810296 0.586021i \(-0.800693\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(468\) 0 0
\(469\) −29.3549 29.4127i −1.35548 1.35815i
\(470\) 0 0
\(471\) −3.69749 + 12.5925i −0.170371 + 0.580231i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.75717 10.4168i 0.218274 0.477954i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(480\) 0 0
\(481\) 64.2229 + 55.6494i 2.92831 + 2.53740i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.4364 + 22.4635i −0.654176 + 1.01792i 0.342739 + 0.939430i \(0.388645\pi\)
−0.996915 + 0.0784867i \(0.974991\pi\)
\(488\) 0 0
\(489\) −34.0139 4.89046i −1.53816 0.221154i
\(490\) 0 0
\(491\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 3.76745i 0.168654i −0.996438 0.0843271i \(-0.973126\pi\)
0.996438 0.0843271i \(-0.0268741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 67.5481i 2.99992i
\(508\) 0 0
\(509\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(510\) 0 0
\(511\) 64.2978 + 55.7143i 2.84437 + 2.46466i
\(512\) 0 0
\(513\) −11.7797 1.69367i −0.520088 0.0747774i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(522\) 0 0
\(523\) 3.28177 3.78737i 0.143502 0.165610i −0.679449 0.733723i \(-0.737781\pi\)
0.822951 + 0.568113i \(0.192326\pi\)
\(524\) 0 0
\(525\) −18.2641 39.9928i −0.797111 1.74543i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −22.0683 6.47985i −0.959493 0.281733i
\(530\) 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
\(541\) −17.3204 + 7.90996i −0.744662 + 0.340076i −0.751352 0.659902i \(-0.770598\pi\)
0.00668980 + 0.999978i \(0.497871\pi\)
\(542\) 0 0
\(543\) −5.15198 + 17.5460i −0.221093 + 0.752972i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.2272 17.4578i 1.63448 0.746441i 0.634822 0.772658i \(-0.281073\pi\)
0.999656 + 0.0262168i \(0.00834602\pi\)
\(548\) 0 0
\(549\) −29.1829 25.2871i −1.24550 1.07923i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −61.3528 + 39.4290i −2.60898 + 1.67669i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(558\) 0 0
\(559\) −13.1887 + 91.7293i −0.557822 + 3.87974i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −34.5308 + 29.9211i −1.45016 + 1.25657i
\(568\) 0 0
\(569\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(570\) 0 0
\(571\) −6.66773 46.3751i −0.279036 1.94074i −0.334790 0.942293i \(-0.608665\pi\)
0.0557537 0.998445i \(-0.482244\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −25.7341 40.0430i −1.07132 1.66701i −0.649568 0.760303i \(-0.725050\pi\)
−0.421756 0.906709i \(-0.638586\pi\)
\(578\) 0 0
\(579\) 5.51420 + 8.58027i 0.229162 + 0.356584i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(588\) 0 0
\(589\) −0.840261 + 1.30747i −0.0346223 + 0.0538734i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 34.1610 + 15.6008i 1.39812 + 0.638499i
\(598\) 0 0
\(599\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(600\) 0 0
\(601\) 46.6443 + 13.6960i 1.90266 + 0.558672i 0.988001 + 0.154446i \(0.0493591\pi\)
0.914659 + 0.404226i \(0.132459\pi\)
\(602\) 0 0
\(603\) −17.3808 + 17.3467i −0.707802 + 0.706411i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.47026 3.51552i −0.222031 0.142691i 0.424897 0.905242i \(-0.360310\pi\)
−0.646928 + 0.762551i \(0.723947\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.92868 + 34.2797i 0.199067 + 1.38454i 0.806998 + 0.590554i \(0.201091\pi\)
−0.607930 + 0.793990i \(0.708000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(618\) 0 0
\(619\) −8.87674 19.4373i −0.356786 0.781253i −0.999880 0.0154656i \(-0.995077\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.55787 24.7455i −0.142315 0.989821i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 30.3065 26.2607i 1.20648 1.04542i 0.208759 0.977967i \(-0.433058\pi\)
0.997722 0.0674546i \(-0.0214878\pi\)
\(632\) 0 0
\(633\) 26.6922i 1.06092i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 38.1397 + 129.892i 1.51115 + 5.14650i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 6.97674 15.2769i 0.275136 0.602463i −0.720738 0.693207i \(-0.756197\pi\)
0.995874 + 0.0907437i \(0.0289244\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 1.68109 + 5.72528i 0.0658873 + 0.224391i
\(652\) 0 0
\(653\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 32.9232 37.9954i 1.28446 1.48234i
\(658\) 0 0
\(659\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(660\) 0 0
\(661\) 13.9982 47.6736i 0.544468 1.85429i 0.0251892 0.999683i \(-0.491981\pi\)
0.519279 0.854605i \(-0.326201\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 10.9149i 0.421994i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 39.3211 + 5.65352i 1.51572 + 0.217927i 0.849403 0.527744i \(-0.176962\pi\)
0.666314 + 0.745672i \(0.267871\pi\)
\(674\) 0 0
\(675\) −23.6329 + 10.7928i −0.909632 + 0.415415i
\(676\) 0 0
\(677\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(678\) 0 0
\(679\) −77.1071 22.6407i −2.95910 0.868869i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.39382 2.82373i 0.167635 0.107732i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 18.4896 40.4866i 0.703378 1.54018i −0.132448 0.991190i \(-0.542284\pi\)
0.835827 0.548994i \(-0.184989\pi\)
\(692\) 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 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(702\) 0 0
\(703\) −25.8972 + 7.60411i −0.976732 + 0.286794i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 33.4861 + 38.6450i 1.25760 + 1.45134i 0.839884 + 0.542766i \(0.182623\pi\)
0.417714 + 0.908579i \(0.362832\pi\)
\(710\) 0 0
\(711\) 23.2998 + 36.2551i 0.873809 + 1.35967i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(720\) 0 0
\(721\) 93.1252 42.5289i 3.46816 1.58386i
\(722\) 0 0
\(723\) −11.0272 + 17.1586i −0.410104 + 0.638135i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 49.5068 7.11800i 1.83611 0.263992i 0.864850 0.502031i \(-0.167414\pi\)
0.971256 + 0.238039i \(0.0765045\pi\)
\(728\) 0 0
\(729\) 17.6812 + 20.4052i 0.654861 + 0.755750i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.16602 + 4.92643i −0.116940 + 0.181962i −0.894789 0.446489i \(-0.852674\pi\)
0.777849 + 0.628451i \(0.216311\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 9.87827 33.6423i 0.363378 1.23755i −0.551621 0.834095i \(-0.685990\pi\)
0.914999 0.403456i \(-0.132191\pi\)
\(740\) 0 0
\(741\) 24.0648 + 15.4655i 0.884041 + 0.568139i
\(742\) 0 0
\(743\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.9231 18.5878i −1.05542 0.678278i −0.106667 0.994295i \(-0.534018\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −43.3512 6.23296i −1.57563 0.226541i −0.701654 0.712517i \(-0.747555\pi\)
−0.873972 + 0.485977i \(0.838464\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(762\) 0 0
\(763\) −44.6452 + 28.6917i −1.61626 + 1.03871i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 34.4305 4.95036i 1.24160 0.178515i 0.509953 0.860202i \(-0.329663\pi\)
0.731643 + 0.681688i \(0.238754\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(774\) 0 0
\(775\) 3.39296i 0.121879i
\(776\) 0 0
\(777\) −43.0470 + 94.2599i −1.54430 + 3.38155i
\(778\) 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
\(787\) 24.1074 + 37.5119i 0.859337 + 1.33715i 0.940268 + 0.340435i \(0.110574\pi\)
−0.0809308 + 0.996720i \(0.525789\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 38.5575 + 84.4292i 1.36922 + 2.99817i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0