Properties

Label 804.2.s.a.125.1
Level 804
Weight 2
Character 804.125
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 125.1
Root \(-0.995472 - 0.0950560i\)
Character \(\chi\) = 804.125
Dual form 804.2.s.a.521.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.487975 + 1.66189i) q^{3} +(1.18913 - 1.85033i) q^{7} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\) \(q+(-0.487975 + 1.66189i) q^{3} +(1.18913 - 1.85033i) q^{7} +(-2.52376 - 1.62192i) q^{9} +(4.69878 - 0.675582i) q^{13} +(5.63639 - 3.62229i) q^{19} +(2.49477 + 2.87912i) q^{21} +(0.711574 + 4.94911i) q^{25} +(3.92699 - 3.40276i) q^{27} +(3.74401 + 0.538308i) q^{31} -8.28083 q^{37} +(-1.17014 + 8.13852i) q^{39} +(8.54831 + 3.90388i) q^{43} +(0.898232 + 1.96685i) q^{49} +(3.26943 + 11.1347i) q^{57} +(-2.04302 + 1.77029i) q^{61} +(-6.00217 + 2.74110i) q^{63} +(0.575902 - 8.16507i) q^{67} +(11.1461 + 12.8633i) q^{73} +(-8.57211 - 1.23248i) q^{75} +(6.35886 - 0.914266i) q^{79} +(3.73874 + 8.18669i) q^{81} +(4.33742 - 9.49763i) q^{91} +(-2.72159 + 5.95946i) q^{93} -17.2394i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 6q^{9} + O(q^{10}) \) \( 20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\) \(1\)

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
\(3\) −0.487975 + 1.66189i −0.281733 + 0.959493i
\(4\) 0 0
\(5\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(6\) 0 0
\(7\) 1.18913 1.85033i 0.449450 0.699358i −0.540410 0.841402i \(-0.681731\pi\)
0.989860 + 0.142044i \(0.0453673\pi\)
\(8\) 0 0
\(9\) −2.52376 1.62192i −0.841254 0.540641i
\(10\) 0 0
\(11\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(12\) 0 0
\(13\) 4.69878 0.675582i 1.30321 0.187373i 0.544475 0.838777i \(-0.316729\pi\)
0.758731 + 0.651404i \(0.225820\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(18\) 0 0
\(19\) 5.63639 3.62229i 1.29308 0.831010i 0.300637 0.953739i \(-0.402801\pi\)
0.992440 + 0.122728i \(0.0391644\pi\)
\(20\) 0 0
\(21\) 2.49477 + 2.87912i 0.544404 + 0.628276i
\(22\) 0 0
\(23\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(24\) 0 0
\(25\) 0.711574 + 4.94911i 0.142315 + 0.989821i
\(26\) 0 0
\(27\) 3.92699 3.40276i 0.755750 0.654861i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.74401 + 0.538308i 0.672445 + 0.0966829i 0.470075 0.882626i \(-0.344227\pi\)
0.202369 + 0.979309i \(0.435136\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.28083 −1.36136 −0.680680 0.732581i \(-0.738316\pi\)
−0.680680 + 0.732581i \(0.738316\pi\)
\(38\) 0 0
\(39\) −1.17014 + 8.13852i −0.187373 + 1.30321i
\(40\) 0 0
\(41\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(42\) 0 0
\(43\) 8.54831 + 3.90388i 1.30361 + 0.595337i 0.941567 0.336826i \(-0.109354\pi\)
0.362039 + 0.932163i \(0.382081\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(48\) 0 0
\(49\) 0.898232 + 1.96685i 0.128319 + 0.280979i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.26943 + 11.1347i 0.433047 + 1.47482i
\(58\) 0 0
\(59\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(60\) 0 0
\(61\) −2.04302 + 1.77029i −0.261582 + 0.226662i −0.775771 0.631015i \(-0.782639\pi\)
0.514188 + 0.857677i \(0.328093\pi\)
\(62\) 0 0
\(63\) −6.00217 + 2.74110i −0.756203 + 0.345346i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.575902 8.16507i 0.0703577 0.997522i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(72\) 0 0
\(73\) 11.1461 + 12.8633i 1.30455 + 1.50553i 0.719348 + 0.694650i \(0.244441\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −8.57211 1.23248i −0.989821 0.142315i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.35886 0.914266i 0.715428 0.102863i 0.225018 0.974355i \(-0.427756\pi\)
0.490410 + 0.871492i \(0.336847\pi\)
\(80\) 0 0
\(81\) 3.73874 + 8.18669i 0.415415 + 0.909632i
\(82\) 0 0
\(83\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(90\) 0 0
\(91\) 4.33742 9.49763i 0.454685 0.995622i
\(92\) 0 0
\(93\) −2.72159 + 5.95946i −0.282216 + 0.617967i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.2394i 1.75039i −0.483767 0.875197i \(-0.660732\pi\)
0.483767 0.875197i \(-0.339268\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(102\) 0 0
\(103\) 2.12793 14.8000i 0.209671 1.45829i −0.564562 0.825391i \(-0.690955\pi\)
0.774233 0.632901i \(-0.218136\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(108\) 0 0
\(109\) −14.9686 + 2.15216i −1.43373 + 0.206139i −0.814999 0.579462i \(-0.803263\pi\)
−0.618733 + 0.785602i \(0.712354\pi\)
\(110\) 0 0
\(111\) 4.04084 13.7618i 0.383540 1.30622i
\(112\) 0 0
\(113\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −12.9543 5.91604i −1.19763 0.546938i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.56546 + 10.8880i 0.142315 + 0.989821i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −6.25912 4.02249i −0.555407 0.356938i 0.232631 0.972565i \(-0.425267\pi\)
−0.788038 + 0.615627i \(0.788903\pi\)
\(128\) 0 0
\(129\) −10.6592 + 12.3014i −0.938489 + 1.08307i
\(130\) 0 0
\(131\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(132\) 0 0
\(133\) 14.7366i 1.27782i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(138\) 0 0
\(139\) −9.15302 7.93114i −0.776349 0.672710i 0.173704 0.984798i \(-0.444426\pi\)
−0.950053 + 0.312087i \(0.898972\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.70701 + 0.532988i −0.305749 + 0.0439601i
\(148\) 0 0
\(149\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(150\) 0 0
\(151\) 8.70103 19.0526i 0.708080 1.55048i −0.121811 0.992553i \(-0.538870\pi\)
0.829891 0.557926i \(-0.188403\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −22.0823 6.48395i −1.76236 0.517476i −0.769699 0.638407i \(-0.779594\pi\)
−0.992661 + 0.120931i \(0.961412\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −18.7341 −1.46737 −0.733683 0.679492i \(-0.762200\pi\)
−0.733683 + 0.679492i \(0.762200\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(168\) 0 0
\(169\) 9.14867 2.68629i 0.703744 0.206638i
\(170\) 0 0
\(171\) −20.1000 −1.53708
\(172\) 0 0
\(173\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(174\) 0 0
\(175\) 10.0036 + 4.56850i 0.756203 + 0.345346i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(180\) 0 0
\(181\) 25.8091 + 7.57824i 1.91838 + 0.563286i 0.966282 + 0.257485i \(0.0828937\pi\)
0.952095 + 0.305802i \(0.0989245\pi\)
\(182\) 0 0
\(183\) −1.94508 4.25914i −0.143785 0.314844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.62650 11.3125i −0.118310 0.822867i
\(190\) 0 0
\(191\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(192\) 0 0
\(193\) −2.37170 + 16.4956i −0.170719 + 1.18738i 0.706651 + 0.707562i \(0.250205\pi\)
−0.877370 + 0.479814i \(0.840704\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(198\) 0 0
\(199\) 12.4360 + 7.99212i 0.881563 + 0.566546i 0.901269 0.433260i \(-0.142637\pi\)
−0.0197060 + 0.999806i \(0.506273\pi\)
\(200\) 0 0
\(201\) 13.2884 + 4.94144i 0.937293 + 0.348542i
\(202\) 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
\(211\) −7.53028 + 2.21109i −0.518406 + 0.152218i −0.530460 0.847710i \(-0.677981\pi\)
0.0120548 + 0.999927i \(0.496163\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.44818 6.28753i 0.369846 0.426825i
\(218\) 0 0
\(219\) −26.8164 + 12.2466i −1.81209 + 0.827552i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −22.9683 + 6.74410i −1.53807 + 0.451618i −0.937509 0.347960i \(-0.886874\pi\)
−0.600562 + 0.799578i \(0.705056\pi\)
\(224\) 0 0
\(225\) 6.23123 13.6445i 0.415415 0.909632i
\(226\) 0 0
\(227\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(228\) 0 0
\(229\) −18.6266 2.67810i −1.23088 0.176974i −0.503978 0.863716i \(-0.668131\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.58356 + 11.0139i −0.102863 + 0.715428i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −16.6586 19.2250i −1.07307 1.23839i −0.969842 0.243734i \(-0.921628\pi\)
−0.103230 0.994657i \(-0.532918\pi\)
\(242\) 0 0
\(243\) −15.4298 + 2.21847i −0.989821 + 0.142315i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 24.0370 20.8282i 1.52944 1.32526i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(258\) 0 0
\(259\) −9.84701 + 15.3222i −0.611864 + 0.952078i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(264\) 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
\(271\) −0.159744 + 0.544037i −0.00970373 + 0.0330479i −0.964203 0.265167i \(-0.914573\pi\)
0.954499 + 0.298215i \(0.0963911\pi\)
\(272\) 0 0
\(273\) 13.6675 + 11.8429i 0.827193 + 0.716766i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.81883 + 4.38219i 0.409704 + 0.263301i 0.729219 0.684281i \(-0.239884\pi\)
−0.319515 + 0.947581i \(0.603520\pi\)
\(278\) 0 0
\(279\) −8.57590 7.43106i −0.513426 0.444886i
\(280\) 0 0
\(281\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(282\) 0 0
\(283\) −24.2361 + 15.5756i −1.44069 + 0.925874i −0.441091 + 0.897462i \(0.645408\pi\)
−0.999596 + 0.0284112i \(0.990955\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −11.1326 12.8477i −0.654861 0.755750i
\(290\) 0 0
\(291\) 28.6500 + 8.41239i 1.67949 + 0.493143i
\(292\) 0 0
\(293\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 17.3885 11.1749i 1.00226 0.644113i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.09006 + 21.4919i −0.176359 + 1.22661i 0.688741 + 0.725007i \(0.258164\pi\)
−0.865100 + 0.501599i \(0.832745\pi\)
\(308\) 0 0
\(309\) 23.5577 + 10.7584i 1.34015 + 0.612026i
\(310\) 0 0
\(311\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(312\) 0 0
\(313\) 9.91913 + 33.7814i 0.560662 + 1.90944i 0.375915 + 0.926654i \(0.377328\pi\)
0.184747 + 0.982786i \(0.440853\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 6.68705 + 22.7740i 0.370931 + 1.26328i
\(326\) 0 0
\(327\) 3.72765 25.9264i 0.206139 1.43373i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 23.3178 10.6489i 1.28166 0.585316i 0.346008 0.938231i \(-0.387537\pi\)
0.935655 + 0.352915i \(0.114810\pi\)
\(332\) 0 0
\(333\) 20.8988 + 13.4309i 1.14525 + 0.736007i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −17.1286 + 26.6526i −0.933052 + 1.45186i −0.0413966 + 0.999143i \(0.513181\pi\)
−0.891656 + 0.452715i \(0.850456\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.9471 + 2.86797i 1.07704 + 0.154856i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(348\) 0 0
\(349\) −12.6726 27.7491i −0.678349 1.48538i −0.864383 0.502834i \(-0.832291\pi\)
0.186034 0.982543i \(-0.440437\pi\)
\(350\) 0 0
\(351\) 16.1532 18.6418i 0.862194 0.995025i
\(352\) 0 0
\(353\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(360\) 0 0
\(361\) 10.7551 23.5503i 0.566056 1.23949i
\(362\) 0 0
\(363\) −18.8586 2.71146i −0.989821 0.142315i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.88190 + 26.8433i 0.411432 + 1.40121i 0.861292 + 0.508110i \(0.169656\pi\)
−0.449860 + 0.893099i \(0.648526\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.7160i 0.761964i −0.924582 0.380982i \(-0.875586\pi\)
0.924582 0.380982i \(-0.124414\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 7.81271 26.6076i 0.401312 1.36674i −0.472865 0.881135i \(-0.656780\pi\)
0.874177 0.485608i \(-0.161402\pi\)
\(380\) 0 0
\(381\) 9.73924 8.43910i 0.498956 0.432348i
\(382\) 0 0
\(383\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −15.2421 23.7172i −0.774799 1.20561i
\(388\) 0 0
\(389\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −21.6303 + 24.9628i −1.08560 + 1.25284i −0.120007 + 0.992773i \(0.538292\pi\)
−0.965589 + 0.260072i \(0.916254\pi\)
\(398\) 0 0
\(399\) 24.4905 + 7.19107i 1.22606 + 0.360004i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 17.9559 0.894450
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −20.4322 + 31.7931i −1.01031 + 1.57207i −0.205387 + 0.978681i \(0.565845\pi\)
−0.804918 + 0.593386i \(0.797791\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 17.6471 11.3411i 0.864184 0.555377i
\(418\) 0 0
\(419\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(420\) 0 0
\(421\) 26.2114 16.8450i 1.27746 0.820976i 0.286890 0.957963i \(-0.407378\pi\)
0.990573 + 0.136988i \(0.0437421\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.846188 + 5.88537i 0.0409499 + 0.284813i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −7.81519 1.12366i −0.375574 0.0539994i −0.0480569 0.998845i \(-0.515303\pi\)
−0.327517 + 0.944845i \(0.606212\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 39.9911 1.90867 0.954336 0.298734i \(-0.0965643\pi\)
0.954336 + 0.298734i \(0.0965643\pi\)
\(440\) 0 0
\(441\) 0.923162 6.42073i 0.0439601 0.305749i
\(442\) 0 0
\(443\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 27.4175 + 23.7574i 1.28818 + 1.11622i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00116 13.9184i −0.0936102 0.651073i −0.981563 0.191139i \(-0.938782\pi\)
0.887953 0.459935i \(-0.152127\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(462\) 0 0
\(463\) −16.0546 + 13.9114i −0.746119 + 0.646516i −0.942574 0.333997i \(-0.891603\pi\)
0.196455 + 0.980513i \(0.437057\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(468\) 0 0
\(469\) −14.4232 10.7750i −0.666003 0.497541i
\(470\) 0 0
\(471\) 21.5512 33.5344i 0.993028 1.54518i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21.9378 + 25.3176i 1.00658 + 1.16165i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(480\) 0 0
\(481\) −38.9098 + 5.59438i −1.77413 + 0.255082i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −35.0979 + 16.0287i −1.59044 + 0.726329i −0.996915 0.0784867i \(-0.974991\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(488\) 0 0
\(489\) 9.14177 31.1340i 0.413405 1.40793i
\(490\) 0 0
\(491\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 27.2373i 1.21931i 0.792666 + 0.609656i \(0.208692\pi\)
−0.792666 + 0.609656i \(0.791308\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 16.5149i 0.733454i
\(508\) 0 0
\(509\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(510\) 0 0
\(511\) 37.0555 5.32778i 1.63924 0.235687i
\(512\) 0 0
\(513\) 9.80829 33.4040i 0.433047 1.47482i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(522\) 0 0
\(523\) 4.42270 + 30.7606i 0.193391 + 1.34507i 0.822951 + 0.568113i \(0.192326\pi\)
−0.629559 + 0.776952i \(0.716765\pi\)
\(524\) 0 0
\(525\) −12.4739 + 14.3956i −0.544404 + 0.628276i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.3488 + 12.4347i 0.841254 + 0.540641i
\(530\) 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
\(541\) −33.7980 29.2861i −1.45309 1.25911i −0.906833 0.421491i \(-0.861507\pi\)
−0.546257 0.837618i \(-0.683948\pi\)
\(542\) 0 0
\(543\) −25.1884 + 39.1940i −1.08094 + 1.68197i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19.8853 + 17.2307i 0.850233 + 0.736731i 0.966549 0.256481i \(-0.0825631\pi\)
−0.116316 + 0.993212i \(0.537109\pi\)
\(548\) 0 0
\(549\) 8.02737 1.15416i 0.342600 0.0492584i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.86984 12.8532i 0.249611 0.546572i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(558\) 0 0
\(559\) 42.8040 + 12.5684i 1.81042 + 0.531586i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 19.5939 + 2.81718i 0.822867 + 0.118310i
\(568\) 0 0
\(569\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(570\) 0 0
\(571\) −26.9035 + 7.89957i −1.12587 + 0.330587i −0.791085 0.611706i \(-0.790483\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −39.8769 18.2112i −1.66010 0.758141i −0.999976 0.00698200i \(-0.997778\pi\)
−0.660121 0.751159i \(-0.729495\pi\)
\(578\) 0 0
\(579\) −26.2565 11.9909i −1.09118 0.498326i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(588\) 0 0
\(589\) 23.0526 10.5278i 0.949867 0.433790i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.3505 + 16.7673i −0.791962 + 0.686239i
\(598\) 0 0
\(599\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(600\) 0 0
\(601\) 40.7523 + 26.1899i 1.66232 + 1.06831i 0.914659 + 0.404226i \(0.132459\pi\)
0.747660 + 0.664082i \(0.231177\pi\)
\(602\) 0 0
\(603\) −14.6965 + 19.6726i −0.598490 + 0.801131i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.1034 + 37.4513i 0.694207 + 1.52010i 0.846857 + 0.531821i \(0.178492\pi\)
−0.152650 + 0.988280i \(0.548781\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −38.3420 + 11.2582i −1.54862 + 0.454715i −0.940687 0.339275i \(-0.889818\pi\)
−0.607930 + 0.793990i \(0.708000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(618\) 0 0
\(619\) −5.13567 + 5.92688i −0.206420 + 0.238221i −0.849514 0.527566i \(-0.823105\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.9873 + 7.04331i −0.959493 + 0.281733i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −10.7596 1.54700i −0.428333 0.0615849i −0.0752227 0.997167i \(-0.523967\pi\)
−0.353110 + 0.935582i \(0.614876\pi\)
\(632\) 0 0
\(633\) 13.5934i 0.540291i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.54936 + 8.63497i 0.219874 + 0.342130i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −29.4531 33.9907i −1.16152 1.34046i −0.929969 0.367638i \(-0.880167\pi\)
−0.231548 0.972824i \(-0.574379\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.79061 + 12.1224i 0.305338 + 0.475116i
\(652\) 0 0
\(653\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.26684 50.5420i −0.283507 1.97183i
\(658\) 0 0
\(659\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(660\) 0 0
\(661\) −10.9078 + 16.9728i −0.424263 + 0.660166i −0.985923 0.167203i \(-0.946527\pi\)
0.561660 + 0.827368i \(0.310163\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 41.4618i 1.60300i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.08952 + 27.5504i −0.311828 + 1.06199i 0.643255 + 0.765652i \(0.277583\pi\)
−0.955083 + 0.296337i \(0.904235\pi\)
\(674\) 0 0
\(675\) 19.6349 + 17.0138i 0.755750 + 0.654861i
\(676\) 0 0
\(677\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(678\) 0 0
\(679\) −31.8985 20.4999i −1.22415 0.786715i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.5400 29.6485i 0.516583 1.13116i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −34.4268 39.7306i −1.30966 1.51142i −0.662356 0.749189i \(-0.730443\pi\)
−0.647300 0.762235i \(-0.724102\pi\)
\(692\) 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 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(702\) 0 0
\(703\) −46.6740 + 29.9956i −1.76034 + 1.13130i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.94859 34.4182i 0.185848 1.29260i −0.656770 0.754091i \(-0.728078\pi\)
0.842618 0.538512i \(-0.181013\pi\)
\(710\) 0 0
\(711\) −17.5311 8.00619i −0.657468 0.300256i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(720\) 0 0
\(721\) −24.8545 21.5366i −0.925631 0.802064i
\(722\) 0 0
\(723\) 40.0788 18.3034i 1.49055 0.680710i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7.62719 25.9758i −0.282877 0.963390i −0.971256 0.238039i \(-0.923495\pi\)
0.688379 0.725351i \(-0.258323\pi\)
\(728\) 0 0
\(729\) 3.84250 26.7252i 0.142315 0.989821i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −40.4942 + 18.4931i −1.49569 + 0.683058i −0.984334 0.176312i \(-0.943583\pi\)
−0.511354 + 0.859370i \(0.670856\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 19.5386 30.4027i 0.718739 1.11838i −0.269132 0.963103i \(-0.586737\pi\)
0.987871 0.155277i \(-0.0496269\pi\)
\(740\) 0 0
\(741\) 22.8847 + 50.1105i 0.840690 + 1.84085i
\(742\) 0 0
\(743\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15.5186 33.9810i −0.566282 1.23998i −0.948753 0.316017i \(-0.897654\pi\)
0.382472 0.923967i \(-0.375073\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.4835 49.3261i 0.526410 1.79279i −0.0789989 0.996875i \(-0.525172\pi\)
0.605409 0.795914i \(-0.293009\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(762\) 0 0
\(763\) −13.8175 + 30.2560i −0.500226 + 1.09534i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.66455 + 5.66892i 0.0600250 + 0.204427i 0.984046 0.177916i \(-0.0569356\pi\)
−0.924021 + 0.382343i \(0.875117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(774\) 0 0
\(775\) 18.9126i 0.679360i
\(776\) 0 0
\(777\) −20.6588 23.8415i −0.741131 0.855310i
\(778\) 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
\(787\) −50.8693 23.2313i −1.81330 0.828105i −0.940268 0.340435i \(-0.889426\pi\)
−0.873028 0.487670i \(-0.837847\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −8.40373 + 9.69842i −0.298425 + 0.344401i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0