Properties

Label 7800.2.a.z.1.2
Level $7800$
Weight $2$
Character 7800.1
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7800.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.41421 q^{7} +1.00000 q^{9} +0.414214 q^{11} -1.00000 q^{13} +3.82843 q^{17} -2.00000 q^{19} -2.41421 q^{21} -0.828427 q^{23} -1.00000 q^{27} -8.65685 q^{29} +4.41421 q^{31} -0.414214 q^{33} -7.65685 q^{37} +1.00000 q^{39} +5.65685 q^{41} -10.4853 q^{43} -2.07107 q^{47} -1.17157 q^{49} -3.82843 q^{51} -5.82843 q^{53} +2.00000 q^{57} -6.41421 q^{59} -1.82843 q^{61} +2.41421 q^{63} +0.757359 q^{67} +0.828427 q^{69} -7.65685 q^{71} +9.31371 q^{73} +1.00000 q^{77} -8.82843 q^{79} +1.00000 q^{81} +13.7279 q^{83} +8.65685 q^{87} -2.41421 q^{91} -4.41421 q^{93} +14.9706 q^{97} +0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 10 q^{47} - 8 q^{49} - 2 q^{51} - 6 q^{53} + 4 q^{57} - 10 q^{59} + 2 q^{61} + 2 q^{63} + 10 q^{67} - 4 q^{69} - 4 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{87} - 2 q^{91} - 6 q^{93} - 4 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.414214 0.124890 0.0624450 0.998048i \(-0.480110\pi\)
0.0624450 + 0.998048i \(0.480110\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.82843 0.928530 0.464265 0.885696i \(-0.346319\pi\)
0.464265 + 0.885696i \(0.346319\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −2.41421 −0.526825
\(22\) 0 0
\(23\) −0.828427 −0.172739 −0.0863695 0.996263i \(-0.527527\pi\)
−0.0863695 + 0.996263i \(0.527527\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) 4.41421 0.792816 0.396408 0.918074i \(-0.370257\pi\)
0.396408 + 0.918074i \(0.370257\pi\)
\(32\) 0 0
\(33\) −0.414214 −0.0721053
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) −10.4853 −1.59899 −0.799495 0.600672i \(-0.794900\pi\)
−0.799495 + 0.600672i \(0.794900\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.07107 −0.302096 −0.151048 0.988526i \(-0.548265\pi\)
−0.151048 + 0.988526i \(0.548265\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) −3.82843 −0.536087
\(52\) 0 0
\(53\) −5.82843 −0.800596 −0.400298 0.916385i \(-0.631093\pi\)
−0.400298 + 0.916385i \(0.631093\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −6.41421 −0.835059 −0.417530 0.908663i \(-0.637104\pi\)
−0.417530 + 0.908663i \(0.637104\pi\)
\(60\) 0 0
\(61\) −1.82843 −0.234106 −0.117053 0.993126i \(-0.537345\pi\)
−0.117053 + 0.993126i \(0.537345\pi\)
\(62\) 0 0
\(63\) 2.41421 0.304162
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.757359 0.0925262 0.0462631 0.998929i \(-0.485269\pi\)
0.0462631 + 0.998929i \(0.485269\pi\)
\(68\) 0 0
\(69\) 0.828427 0.0997309
\(70\) 0 0
\(71\) −7.65685 −0.908701 −0.454351 0.890823i \(-0.650129\pi\)
−0.454351 + 0.890823i \(0.650129\pi\)
\(72\) 0 0
\(73\) 9.31371 1.09009 0.545044 0.838408i \(-0.316513\pi\)
0.545044 + 0.838408i \(0.316513\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −8.82843 −0.993276 −0.496638 0.867958i \(-0.665432\pi\)
−0.496638 + 0.867958i \(0.665432\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.7279 1.50684 0.753418 0.657542i \(-0.228404\pi\)
0.753418 + 0.657542i \(0.228404\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.65685 0.928112
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −2.41421 −0.253078
\(92\) 0 0
\(93\) −4.41421 −0.457733
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.9706 1.52003 0.760015 0.649905i \(-0.225191\pi\)
0.760015 + 0.649905i \(0.225191\pi\)
\(98\) 0 0
\(99\) 0.414214 0.0416300
\(100\) 0 0
\(101\) −6.17157 −0.614094 −0.307047 0.951694i \(-0.599341\pi\)
−0.307047 + 0.951694i \(0.599341\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.4853 1.78704 0.893520 0.449024i \(-0.148228\pi\)
0.893520 + 0.449024i \(0.148228\pi\)
\(108\) 0 0
\(109\) −19.3137 −1.84992 −0.924959 0.380067i \(-0.875901\pi\)
−0.924959 + 0.380067i \(0.875901\pi\)
\(110\) 0 0
\(111\) 7.65685 0.726756
\(112\) 0 0
\(113\) −13.3137 −1.25245 −0.626224 0.779643i \(-0.715401\pi\)
−0.626224 + 0.779643i \(0.715401\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 9.24264 0.847271
\(120\) 0 0
\(121\) −10.8284 −0.984402
\(122\) 0 0
\(123\) −5.65685 −0.510061
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.1716 0.991317 0.495658 0.868518i \(-0.334927\pi\)
0.495658 + 0.868518i \(0.334927\pi\)
\(128\) 0 0
\(129\) 10.4853 0.923178
\(130\) 0 0
\(131\) −17.6569 −1.54269 −0.771343 0.636419i \(-0.780415\pi\)
−0.771343 + 0.636419i \(0.780415\pi\)
\(132\) 0 0
\(133\) −4.82843 −0.418678
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16.9706 −1.44989 −0.724947 0.688805i \(-0.758136\pi\)
−0.724947 + 0.688805i \(0.758136\pi\)
\(138\) 0 0
\(139\) −3.17157 −0.269009 −0.134505 0.990913i \(-0.542944\pi\)
−0.134505 + 0.990913i \(0.542944\pi\)
\(140\) 0 0
\(141\) 2.07107 0.174415
\(142\) 0 0
\(143\) −0.414214 −0.0346383
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.17157 0.0966297
\(148\) 0 0
\(149\) 17.3137 1.41839 0.709197 0.705010i \(-0.249058\pi\)
0.709197 + 0.705010i \(0.249058\pi\)
\(150\) 0 0
\(151\) −7.24264 −0.589398 −0.294699 0.955590i \(-0.595219\pi\)
−0.294699 + 0.955590i \(0.595219\pi\)
\(152\) 0 0
\(153\) 3.82843 0.309510
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.3137 −1.46159 −0.730797 0.682595i \(-0.760851\pi\)
−0.730797 + 0.682595i \(0.760851\pi\)
\(158\) 0 0
\(159\) 5.82843 0.462224
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −2.68629 −0.210407 −0.105203 0.994451i \(-0.533549\pi\)
−0.105203 + 0.994451i \(0.533549\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 25.3137 1.95883 0.979417 0.201848i \(-0.0646948\pi\)
0.979417 + 0.201848i \(0.0646948\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 3.48528 0.264981 0.132491 0.991184i \(-0.457703\pi\)
0.132491 + 0.991184i \(0.457703\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.41421 0.482122
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −3.34315 −0.248494 −0.124247 0.992251i \(-0.539652\pi\)
−0.124247 + 0.992251i \(0.539652\pi\)
\(182\) 0 0
\(183\) 1.82843 0.135161
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.58579 0.115964
\(188\) 0 0
\(189\) −2.41421 −0.175608
\(190\) 0 0
\(191\) 14.4853 1.04812 0.524059 0.851682i \(-0.324417\pi\)
0.524059 + 0.851682i \(0.324417\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.00000 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 0 0
\(199\) −0.828427 −0.0587256 −0.0293628 0.999569i \(-0.509348\pi\)
−0.0293628 + 0.999569i \(0.509348\pi\)
\(200\) 0 0
\(201\) −0.757359 −0.0534200
\(202\) 0 0
\(203\) −20.8995 −1.46686
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.828427 −0.0575797
\(208\) 0 0
\(209\) −0.828427 −0.0573035
\(210\) 0 0
\(211\) 10.4853 0.721837 0.360918 0.932597i \(-0.382463\pi\)
0.360918 + 0.932597i \(0.382463\pi\)
\(212\) 0 0
\(213\) 7.65685 0.524639
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 10.6569 0.723434
\(218\) 0 0
\(219\) −9.31371 −0.629362
\(220\) 0 0
\(221\) −3.82843 −0.257528
\(222\) 0 0
\(223\) −23.6569 −1.58418 −0.792090 0.610404i \(-0.791007\pi\)
−0.792090 + 0.610404i \(0.791007\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.5858 0.768976 0.384488 0.923130i \(-0.374378\pi\)
0.384488 + 0.923130i \(0.374378\pi\)
\(228\) 0 0
\(229\) −4.34315 −0.287003 −0.143502 0.989650i \(-0.545836\pi\)
−0.143502 + 0.989650i \(0.545836\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.82843 0.573468
\(238\) 0 0
\(239\) −5.24264 −0.339118 −0.169559 0.985520i \(-0.554234\pi\)
−0.169559 + 0.985520i \(0.554234\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −13.7279 −0.869972
\(250\) 0 0
\(251\) −8.00000 −0.504956 −0.252478 0.967603i \(-0.581245\pi\)
−0.252478 + 0.967603i \(0.581245\pi\)
\(252\) 0 0
\(253\) −0.343146 −0.0215734
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.00000 0.0623783 0.0311891 0.999514i \(-0.490071\pi\)
0.0311891 + 0.999514i \(0.490071\pi\)
\(258\) 0 0
\(259\) −18.4853 −1.14862
\(260\) 0 0
\(261\) −8.65685 −0.535846
\(262\) 0 0
\(263\) −3.31371 −0.204332 −0.102166 0.994767i \(-0.532577\pi\)
−0.102166 + 0.994767i \(0.532577\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.3137 0.750780 0.375390 0.926867i \(-0.377509\pi\)
0.375390 + 0.926867i \(0.377509\pi\)
\(270\) 0 0
\(271\) −4.89949 −0.297623 −0.148812 0.988866i \(-0.547545\pi\)
−0.148812 + 0.988866i \(0.547545\pi\)
\(272\) 0 0
\(273\) 2.41421 0.146115
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.31371 −0.559607 −0.279803 0.960057i \(-0.590269\pi\)
−0.279803 + 0.960057i \(0.590269\pi\)
\(278\) 0 0
\(279\) 4.41421 0.264272
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 27.4558 1.63208 0.816040 0.577995i \(-0.196165\pi\)
0.816040 + 0.577995i \(0.196165\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.6569 0.806139
\(288\) 0 0
\(289\) −2.34315 −0.137832
\(290\) 0 0
\(291\) −14.9706 −0.877590
\(292\) 0 0
\(293\) 5.31371 0.310430 0.155215 0.987881i \(-0.450393\pi\)
0.155215 + 0.987881i \(0.450393\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.414214 −0.0240351
\(298\) 0 0
\(299\) 0.828427 0.0479092
\(300\) 0 0
\(301\) −25.3137 −1.45906
\(302\) 0 0
\(303\) 6.17157 0.354548
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.4853 −1.04820 −0.524102 0.851655i \(-0.675599\pi\)
−0.524102 + 0.851655i \(0.675599\pi\)
\(312\) 0 0
\(313\) −19.8284 −1.12077 −0.560384 0.828233i \(-0.689347\pi\)
−0.560384 + 0.828233i \(0.689347\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.97056 0.166843 0.0834217 0.996514i \(-0.473415\pi\)
0.0834217 + 0.996514i \(0.473415\pi\)
\(318\) 0 0
\(319\) −3.58579 −0.200765
\(320\) 0 0
\(321\) −18.4853 −1.03175
\(322\) 0 0
\(323\) −7.65685 −0.426039
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 19.3137 1.06805
\(328\) 0 0
\(329\) −5.00000 −0.275659
\(330\) 0 0
\(331\) 18.9706 1.04272 0.521358 0.853338i \(-0.325426\pi\)
0.521358 + 0.853338i \(0.325426\pi\)
\(332\) 0 0
\(333\) −7.65685 −0.419593
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.48528 −0.407749 −0.203875 0.978997i \(-0.565353\pi\)
−0.203875 + 0.978997i \(0.565353\pi\)
\(338\) 0 0
\(339\) 13.3137 0.723101
\(340\) 0 0
\(341\) 1.82843 0.0990149
\(342\) 0 0
\(343\) −19.7279 −1.06521
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.5147 0.725508 0.362754 0.931885i \(-0.381837\pi\)
0.362754 + 0.931885i \(0.381837\pi\)
\(348\) 0 0
\(349\) −3.65685 −0.195747 −0.0978735 0.995199i \(-0.531204\pi\)
−0.0978735 + 0.995199i \(0.531204\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 0.686292 0.0365276 0.0182638 0.999833i \(-0.494186\pi\)
0.0182638 + 0.999833i \(0.494186\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −9.24264 −0.489172
\(358\) 0 0
\(359\) −1.10051 −0.0580824 −0.0290412 0.999578i \(-0.509245\pi\)
−0.0290412 + 0.999578i \(0.509245\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 10.8284 0.568345
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.65685 −0.504084 −0.252042 0.967716i \(-0.581102\pi\)
−0.252042 + 0.967716i \(0.581102\pi\)
\(368\) 0 0
\(369\) 5.65685 0.294484
\(370\) 0 0
\(371\) −14.0711 −0.730533
\(372\) 0 0
\(373\) −32.6569 −1.69091 −0.845454 0.534048i \(-0.820670\pi\)
−0.845454 + 0.534048i \(0.820670\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.65685 0.445851
\(378\) 0 0
\(379\) 0.0710678 0.00365051 0.00182525 0.999998i \(-0.499419\pi\)
0.00182525 + 0.999998i \(0.499419\pi\)
\(380\) 0 0
\(381\) −11.1716 −0.572337
\(382\) 0 0
\(383\) 3.65685 0.186857 0.0934283 0.995626i \(-0.470217\pi\)
0.0934283 + 0.995626i \(0.470217\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.4853 −0.532997
\(388\) 0 0
\(389\) −16.6274 −0.843044 −0.421522 0.906818i \(-0.638504\pi\)
−0.421522 + 0.906818i \(0.638504\pi\)
\(390\) 0 0
\(391\) −3.17157 −0.160393
\(392\) 0 0
\(393\) 17.6569 0.890670
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 34.2843 1.72068 0.860339 0.509722i \(-0.170252\pi\)
0.860339 + 0.509722i \(0.170252\pi\)
\(398\) 0 0
\(399\) 4.82843 0.241724
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 0 0
\(403\) −4.41421 −0.219888
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.17157 −0.157209
\(408\) 0 0
\(409\) 19.3137 0.955001 0.477501 0.878631i \(-0.341543\pi\)
0.477501 + 0.878631i \(0.341543\pi\)
\(410\) 0 0
\(411\) 16.9706 0.837096
\(412\) 0 0
\(413\) −15.4853 −0.761981
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.17157 0.155313
\(418\) 0 0
\(419\) 2.34315 0.114470 0.0572351 0.998361i \(-0.481772\pi\)
0.0572351 + 0.998361i \(0.481772\pi\)
\(420\) 0 0
\(421\) −11.3137 −0.551396 −0.275698 0.961244i \(-0.588909\pi\)
−0.275698 + 0.961244i \(0.588909\pi\)
\(422\) 0 0
\(423\) −2.07107 −0.100699
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.41421 −0.213619
\(428\) 0 0
\(429\) 0.414214 0.0199984
\(430\) 0 0
\(431\) 6.97056 0.335760 0.167880 0.985807i \(-0.446308\pi\)
0.167880 + 0.985807i \(0.446308\pi\)
\(432\) 0 0
\(433\) −25.3137 −1.21650 −0.608250 0.793746i \(-0.708128\pi\)
−0.608250 + 0.793746i \(0.708128\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.65685 0.0792581
\(438\) 0 0
\(439\) −18.4853 −0.882254 −0.441127 0.897445i \(-0.645421\pi\)
−0.441127 + 0.897445i \(0.645421\pi\)
\(440\) 0 0
\(441\) −1.17157 −0.0557892
\(442\) 0 0
\(443\) 23.4558 1.11442 0.557210 0.830371i \(-0.311872\pi\)
0.557210 + 0.830371i \(0.311872\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −17.3137 −0.818910
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 2.34315 0.110334
\(452\) 0 0
\(453\) 7.24264 0.340289
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.6569 −1.38729 −0.693645 0.720317i \(-0.743996\pi\)
−0.693645 + 0.720317i \(0.743996\pi\)
\(458\) 0 0
\(459\) −3.82843 −0.178696
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −23.7279 −1.10273 −0.551365 0.834264i \(-0.685893\pi\)
−0.551365 + 0.834264i \(0.685893\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) 1.82843 0.0844289
\(470\) 0 0
\(471\) 18.3137 0.843851
\(472\) 0 0
\(473\) −4.34315 −0.199698
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.82843 −0.266865
\(478\) 0 0
\(479\) 21.0416 0.961417 0.480708 0.876881i \(-0.340380\pi\)
0.480708 + 0.876881i \(0.340380\pi\)
\(480\) 0 0
\(481\) 7.65685 0.349123
\(482\) 0 0
\(483\) 2.00000 0.0910032
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.24264 −0.0563094 −0.0281547 0.999604i \(-0.508963\pi\)
−0.0281547 + 0.999604i \(0.508963\pi\)
\(488\) 0 0
\(489\) 2.68629 0.121478
\(490\) 0 0
\(491\) −22.4853 −1.01475 −0.507373 0.861726i \(-0.669383\pi\)
−0.507373 + 0.861726i \(0.669383\pi\)
\(492\) 0 0
\(493\) −33.1421 −1.49265
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.4853 −0.829178
\(498\) 0 0
\(499\) −24.4142 −1.09293 −0.546465 0.837482i \(-0.684027\pi\)
−0.546465 + 0.837482i \(0.684027\pi\)
\(500\) 0 0
\(501\) −25.3137 −1.13093
\(502\) 0 0
\(503\) 20.2843 0.904431 0.452215 0.891909i \(-0.350634\pi\)
0.452215 + 0.891909i \(0.350634\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) 22.4853 0.994690
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.857864 −0.0377288
\(518\) 0 0
\(519\) −3.48528 −0.152987
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 10.4853 0.458489 0.229245 0.973369i \(-0.426374\pi\)
0.229245 + 0.973369i \(0.426374\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.8995 0.736154
\(528\) 0 0
\(529\) −22.3137 −0.970161
\(530\) 0 0
\(531\) −6.41421 −0.278353
\(532\) 0 0
\(533\) −5.65685 −0.245026
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) −0.485281 −0.0209025
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 3.34315 0.143468
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22.6274 0.967478 0.483739 0.875212i \(-0.339278\pi\)
0.483739 + 0.875212i \(0.339278\pi\)
\(548\) 0 0
\(549\) −1.82843 −0.0780354
\(550\) 0 0
\(551\) 17.3137 0.737589
\(552\) 0 0
\(553\) −21.3137 −0.906351
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) 10.4853 0.443480
\(560\) 0 0
\(561\) −1.58579 −0.0669520
\(562\) 0 0
\(563\) 10.4853 0.441902 0.220951 0.975285i \(-0.429084\pi\)
0.220951 + 0.975285i \(0.429084\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.41421 0.101387
\(568\) 0 0
\(569\) 21.6863 0.909137 0.454568 0.890712i \(-0.349794\pi\)
0.454568 + 0.890712i \(0.349794\pi\)
\(570\) 0 0
\(571\) −18.4853 −0.773585 −0.386792 0.922167i \(-0.626417\pi\)
−0.386792 + 0.922167i \(0.626417\pi\)
\(572\) 0 0
\(573\) −14.4853 −0.605131
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −42.2843 −1.76032 −0.880159 0.474680i \(-0.842564\pi\)
−0.880159 + 0.474680i \(0.842564\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 33.1421 1.37497
\(582\) 0 0
\(583\) −2.41421 −0.0999865
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −11.7279 −0.484063 −0.242032 0.970268i \(-0.577814\pi\)
−0.242032 + 0.970268i \(0.577814\pi\)
\(588\) 0 0
\(589\) −8.82843 −0.363769
\(590\) 0 0
\(591\) 4.00000 0.164538
\(592\) 0 0
\(593\) 16.6274 0.682806 0.341403 0.939917i \(-0.389098\pi\)
0.341403 + 0.939917i \(0.389098\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.828427 0.0339053
\(598\) 0 0
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) −27.1421 −1.10715 −0.553575 0.832799i \(-0.686737\pi\)
−0.553575 + 0.832799i \(0.686737\pi\)
\(602\) 0 0
\(603\) 0.757359 0.0308421
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.1421 0.655189 0.327595 0.944818i \(-0.393762\pi\)
0.327595 + 0.944818i \(0.393762\pi\)
\(608\) 0 0
\(609\) 20.8995 0.846890
\(610\) 0 0
\(611\) 2.07107 0.0837864
\(612\) 0 0
\(613\) −24.3431 −0.983210 −0.491605 0.870818i \(-0.663590\pi\)
−0.491605 + 0.870818i \(0.663590\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.9706 −1.56890 −0.784448 0.620195i \(-0.787054\pi\)
−0.784448 + 0.620195i \(0.787054\pi\)
\(618\) 0 0
\(619\) −44.6274 −1.79373 −0.896864 0.442307i \(-0.854160\pi\)
−0.896864 + 0.442307i \(0.854160\pi\)
\(620\) 0 0
\(621\) 0.828427 0.0332436
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.828427 0.0330842
\(628\) 0 0
\(629\) −29.3137 −1.16881
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 0 0
\(633\) −10.4853 −0.416753
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.17157 0.0464194
\(638\) 0 0
\(639\) −7.65685 −0.302900
\(640\) 0 0
\(641\) −18.5147 −0.731287 −0.365644 0.930755i \(-0.619151\pi\)
−0.365644 + 0.930755i \(0.619151\pi\)
\(642\) 0 0
\(643\) −40.6274 −1.60219 −0.801094 0.598538i \(-0.795749\pi\)
−0.801094 + 0.598538i \(0.795749\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.9411 −0.862595 −0.431297 0.902210i \(-0.641944\pi\)
−0.431297 + 0.902210i \(0.641944\pi\)
\(648\) 0 0
\(649\) −2.65685 −0.104291
\(650\) 0 0
\(651\) −10.6569 −0.417675
\(652\) 0 0
\(653\) 23.3431 0.913488 0.456744 0.889598i \(-0.349016\pi\)
0.456744 + 0.889598i \(0.349016\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.31371 0.363362
\(658\) 0 0
\(659\) −3.85786 −0.150281 −0.0751405 0.997173i \(-0.523941\pi\)
−0.0751405 + 0.997173i \(0.523941\pi\)
\(660\) 0 0
\(661\) 45.9411 1.78690 0.893451 0.449160i \(-0.148277\pi\)
0.893451 + 0.449160i \(0.148277\pi\)
\(662\) 0 0
\(663\) 3.82843 0.148684
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.17157 0.277684
\(668\) 0 0
\(669\) 23.6569 0.914627
\(670\) 0 0
\(671\) −0.757359 −0.0292375
\(672\) 0 0
\(673\) 7.14214 0.275309 0.137655 0.990480i \(-0.456044\pi\)
0.137655 + 0.990480i \(0.456044\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) 36.1421 1.38701
\(680\) 0 0
\(681\) −11.5858 −0.443968
\(682\) 0 0
\(683\) −43.1838 −1.65238 −0.826190 0.563391i \(-0.809497\pi\)
−0.826190 + 0.563391i \(0.809497\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 4.34315 0.165701
\(688\) 0 0
\(689\) 5.82843 0.222045
\(690\) 0 0
\(691\) 9.24264 0.351607 0.175803 0.984425i \(-0.443748\pi\)
0.175803 + 0.984425i \(0.443748\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 21.6569 0.820312
\(698\) 0 0
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −32.3137 −1.22047 −0.610236 0.792220i \(-0.708925\pi\)
−0.610236 + 0.792220i \(0.708925\pi\)
\(702\) 0 0
\(703\) 15.3137 0.577567
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.8995 −0.560353
\(708\) 0 0
\(709\) 34.6274 1.30046 0.650230 0.759737i \(-0.274673\pi\)
0.650230 + 0.759737i \(0.274673\pi\)
\(710\) 0 0
\(711\) −8.82843 −0.331092
\(712\) 0 0
\(713\) −3.65685 −0.136950
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.24264 0.195790
\(718\) 0 0
\(719\) −42.6274 −1.58973 −0.794867 0.606783i \(-0.792460\pi\)
−0.794867 + 0.606783i \(0.792460\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.686292 0.0254531 0.0127266 0.999919i \(-0.495949\pi\)
0.0127266 + 0.999919i \(0.495949\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.1421 −1.48471
\(732\) 0 0
\(733\) −28.6863 −1.05955 −0.529776 0.848137i \(-0.677724\pi\)
−0.529776 + 0.848137i \(0.677724\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.313708 0.0115556
\(738\) 0 0
\(739\) 34.5563 1.27118 0.635588 0.772028i \(-0.280758\pi\)
0.635588 + 0.772028i \(0.280758\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −18.4142 −0.675552 −0.337776 0.941227i \(-0.609675\pi\)
−0.337776 + 0.941227i \(0.609675\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 13.7279 0.502278
\(748\) 0 0
\(749\) 44.6274 1.63065
\(750\) 0 0
\(751\) 50.4853 1.84223 0.921117 0.389286i \(-0.127278\pi\)
0.921117 + 0.389286i \(0.127278\pi\)
\(752\) 0 0
\(753\) 8.00000 0.291536
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −30.7990 −1.11941 −0.559704 0.828692i \(-0.689085\pi\)
−0.559704 + 0.828692i \(0.689085\pi\)
\(758\) 0 0
\(759\) 0.343146 0.0124554
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) −46.6274 −1.68803
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.41421 0.231604
\(768\) 0 0
\(769\) −17.6569 −0.636722 −0.318361 0.947969i \(-0.603132\pi\)
−0.318361 + 0.947969i \(0.603132\pi\)
\(770\) 0 0
\(771\) −1.00000 −0.0360141
\(772\) 0 0
\(773\) −28.6274 −1.02966 −0.514828 0.857293i \(-0.672144\pi\)
−0.514828 + 0.857293i \(0.672144\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 18.4853 0.663156
\(778\) 0 0
\(779\) −11.3137 −0.405356
\(780\) 0 0
\(781\) −3.17157 −0.113488
\(782\) 0 0
\(783\) 8.65685 0.309371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.38478 −0.120654 −0.0603271 0.998179i \(-0.519214\pi\)
−0.0603271 + 0.998179i \(0.519214\pi\)
\(788\) 0 0
\(789\) 3.31371 0.117971
\(790\) 0 0
\(791\) −32.1421 −1.14284
\(792\) 0 0
\(793\) 1.82843 0.0649294
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.34315 0.189264 0.0946320 0.995512i \(-0.469833\pi\)
0.0946320 + 0.995512i \(0.469833\pi\)
\(798\) 0 0
\(799\) −7.92893 −0.280505
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.85786 0.136141
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −12.3137 −0.433463
\(808\) 0 0
\(809\) 41.3137 1.45251 0.726256 0.687424i \(-0.241259\pi\)
0.726256 + 0.687424i \(0.241259\pi\)
\(810\) 0 0
\(811\) 28.2721 0.992767 0.496383 0.868103i \(-0.334661\pi\)
0.496383 + 0.868103i \(0.334661\pi\)
\(812\) 0 0
\(813\) 4.89949 0.171833
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 20.9706 0.733667
\(818\) 0 0
\(819\) −2.41421 −0.0843594
\(820\) 0 0
\(821\) 33.3137 1.16266 0.581328 0.813669i \(-0.302533\pi\)
0.581328 + 0.813669i \(0.302533\pi\)
\(822\) 0 0
\(823\) 49.9411 1.74084 0.870419 0.492311i \(-0.163848\pi\)
0.870419 + 0.492311i \(0.163848\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.3848 0.604528 0.302264 0.953224i \(-0.402258\pi\)
0.302264 + 0.953224i \(0.402258\pi\)
\(828\) 0 0
\(829\) −3.82843 −0.132967 −0.0664834 0.997788i \(-0.521178\pi\)
−0.0664834 + 0.997788i \(0.521178\pi\)
\(830\) 0 0
\(831\) 9.31371 0.323089
\(832\) 0 0
\(833\) −4.48528 −0.155406
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.41421 −0.152578
\(838\) 0 0
\(839\) 7.65685 0.264344 0.132172 0.991227i \(-0.457805\pi\)
0.132172 + 0.991227i \(0.457805\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) 0 0
\(843\) 14.0000 0.482186
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −26.1421 −0.898254
\(848\) 0 0
\(849\) −27.4558 −0.942282
\(850\) 0 0
\(851\) 6.34315 0.217440
\(852\) 0 0
\(853\) −18.3431 −0.628057 −0.314029 0.949413i \(-0.601679\pi\)
−0.314029 + 0.949413i \(0.601679\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.6274 0.431344 0.215672 0.976466i \(-0.430806\pi\)
0.215672 + 0.976466i \(0.430806\pi\)
\(858\) 0 0
\(859\) −39.1716 −1.33652 −0.668258 0.743929i \(-0.732960\pi\)
−0.668258 + 0.743929i \(0.732960\pi\)
\(860\) 0 0
\(861\) −13.6569 −0.465424
\(862\) 0 0
\(863\) 29.5858 1.00711 0.503556 0.863963i \(-0.332025\pi\)
0.503556 + 0.863963i \(0.332025\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.34315 0.0795774
\(868\) 0 0
\(869\) −3.65685 −0.124050
\(870\) 0 0
\(871\) −0.757359 −0.0256621
\(872\) 0 0
\(873\) 14.9706 0.506677
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.3431 −0.484334 −0.242167 0.970235i \(-0.577858\pi\)
−0.242167 + 0.970235i \(0.577858\pi\)
\(878\) 0 0
\(879\) −5.31371 −0.179227
\(880\) 0 0
\(881\) −54.5980 −1.83945 −0.919726 0.392560i \(-0.871589\pi\)
−0.919726 + 0.392560i \(0.871589\pi\)
\(882\) 0 0
\(883\) −3.31371 −0.111515 −0.0557576 0.998444i \(-0.517757\pi\)
−0.0557576 + 0.998444i \(0.517757\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.17157 0.106491 0.0532455 0.998581i \(-0.483043\pi\)
0.0532455 + 0.998581i \(0.483043\pi\)
\(888\) 0 0
\(889\) 26.9706 0.904564
\(890\) 0 0
\(891\) 0.414214 0.0138767
\(892\) 0 0
\(893\) 4.14214 0.138611
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.828427 −0.0276604
\(898\) 0 0
\(899\) −38.2132 −1.27448
\(900\) 0 0
\(901\) −22.3137 −0.743377
\(902\) 0 0
\(903\) 25.3137 0.842387
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 33.9411 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(908\) 0 0
\(909\) −6.17157 −0.204698
\(910\) 0 0
\(911\) −20.8284 −0.690077 −0.345038 0.938589i \(-0.612134\pi\)
−0.345038 + 0.938589i \(0.612134\pi\)
\(912\) 0 0
\(913\) 5.68629 0.188189
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −42.6274 −1.40768
\(918\) 0 0
\(919\) 17.7990 0.587135 0.293567 0.955938i \(-0.405158\pi\)
0.293567 + 0.955938i \(0.405158\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 0 0
\(923\) 7.65685 0.252028
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 2.34315 0.0767935
\(932\) 0 0
\(933\) 18.4853 0.605181
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 46.5980 1.52229 0.761145 0.648582i \(-0.224638\pi\)
0.761145 + 0.648582i \(0.224638\pi\)
\(938\) 0 0
\(939\) 19.8284 0.647076
\(940\) 0 0
\(941\) 45.3137 1.47718 0.738592 0.674152i \(-0.235491\pi\)
0.738592 + 0.674152i \(0.235491\pi\)
\(942\) 0 0
\(943\) −4.68629 −0.152607
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.4142 0.663373 0.331686 0.943390i \(-0.392382\pi\)
0.331686 + 0.943390i \(0.392382\pi\)
\(948\) 0 0
\(949\) −9.31371 −0.302336
\(950\) 0 0
\(951\) −2.97056 −0.0963271
\(952\) 0 0
\(953\) 34.6569 1.12265 0.561323 0.827597i \(-0.310293\pi\)
0.561323 + 0.827597i \(0.310293\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.58579 0.115912
\(958\) 0 0
\(959\) −40.9706 −1.32301
\(960\) 0 0
\(961\) −11.5147 −0.371443
\(962\) 0 0
\(963\) 18.4853 0.595680
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.0122 0.772180 0.386090 0.922461i \(-0.373825\pi\)
0.386090 + 0.922461i \(0.373825\pi\)
\(968\) 0 0
\(969\) 7.65685 0.245974
\(970\) 0 0
\(971\) −28.8284 −0.925148 −0.462574 0.886581i \(-0.653074\pi\)
−0.462574 + 0.886581i \(0.653074\pi\)
\(972\) 0 0
\(973\) −7.65685 −0.245467
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.9706 1.05482 0.527411 0.849610i \(-0.323163\pi\)
0.527411 + 0.849610i \(0.323163\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −19.3137 −0.616639
\(982\) 0 0
\(983\) 8.75736 0.279316 0.139658 0.990200i \(-0.455400\pi\)
0.139658 + 0.990200i \(0.455400\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 5.00000 0.159152
\(988\) 0 0
\(989\) 8.68629 0.276208
\(990\) 0 0
\(991\) 15.8579 0.503742 0.251871 0.967761i \(-0.418954\pi\)
0.251871 + 0.967761i \(0.418954\pi\)
\(992\) 0 0
\(993\) −18.9706 −0.602013
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −58.7990 −1.86218 −0.931091 0.364786i \(-0.881142\pi\)
−0.931091 + 0.364786i \(0.881142\pi\)
\(998\) 0 0
\(999\) 7.65685 0.242252
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7800.2.a.z.1.2 2
5.4 even 2 7800.2.a.ba.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.z.1.2 2 1.1 even 1 trivial
7800.2.a.ba.1.1 yes 2 5.4 even 2