Properties

Label 5520.2.a.bw.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.48361\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.13555 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -3.13555 q^{7} +1.00000 q^{9} -2.65194 q^{11} +0.651938 q^{13} +1.00000 q^{15} +4.48361 q^{17} +0.651938 q^{19} +3.13555 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.48361 q^{29} +3.13555 q^{31} +2.65194 q^{33} +3.13555 q^{35} -6.10277 q^{37} -0.651938 q^{39} +0.820265 q^{41} -0.696123 q^{43} -1.00000 q^{45} +11.8903 q^{47} +2.83167 q^{49} -4.48361 q^{51} +9.45083 q^{53} +2.65194 q^{55} -0.651938 q^{57} -12.7547 q^{59} +2.65194 q^{61} -3.13555 q^{63} -0.651938 q^{65} +6.10277 q^{67} +1.00000 q^{69} -8.48361 q^{71} +6.31528 q^{73} -1.00000 q^{75} +8.31528 q^{77} +8.00000 q^{79} +1.00000 q^{81} -5.51639 q^{83} -4.48361 q^{85} -2.48361 q^{87} -11.2383 q^{89} -2.04419 q^{91} -3.13555 q^{93} -0.651938 q^{95} -5.93445 q^{97} -2.65194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} + 3 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 2 q^{31} + 4 q^{33} - 2 q^{35} + 8 q^{37} + 2 q^{39} + 2 q^{41} - 10 q^{43} - 3 q^{45} - 6 q^{47} + 5 q^{49} - 6 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} + 3 q^{69} - 18 q^{71} + 8 q^{73} - 3 q^{75} + 14 q^{77} + 24 q^{79} + 3 q^{81} - 24 q^{83} - 6 q^{85} + 4 q^{89} - 18 q^{91} + 2 q^{93} + 2 q^{95} + 12 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.13555 −1.18513 −0.592563 0.805524i \(-0.701884\pi\)
−0.592563 + 0.805524i \(0.701884\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.65194 −0.799589 −0.399795 0.916605i \(-0.630919\pi\)
−0.399795 + 0.916605i \(0.630919\pi\)
\(12\) 0 0
\(13\) 0.651938 0.180815 0.0904076 0.995905i \(-0.471183\pi\)
0.0904076 + 0.995905i \(0.471183\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.48361 1.08744 0.543718 0.839268i \(-0.317016\pi\)
0.543718 + 0.839268i \(0.317016\pi\)
\(18\) 0 0
\(19\) 0.651938 0.149565 0.0747825 0.997200i \(-0.476174\pi\)
0.0747825 + 0.997200i \(0.476174\pi\)
\(20\) 0 0
\(21\) 3.13555 0.684233
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.48361 0.461195 0.230598 0.973049i \(-0.425932\pi\)
0.230598 + 0.973049i \(0.425932\pi\)
\(30\) 0 0
\(31\) 3.13555 0.563161 0.281581 0.959538i \(-0.409141\pi\)
0.281581 + 0.959538i \(0.409141\pi\)
\(32\) 0 0
\(33\) 2.65194 0.461643
\(34\) 0 0
\(35\) 3.13555 0.530005
\(36\) 0 0
\(37\) −6.10277 −1.00329 −0.501645 0.865074i \(-0.667272\pi\)
−0.501645 + 0.865074i \(0.667272\pi\)
\(38\) 0 0
\(39\) −0.651938 −0.104394
\(40\) 0 0
\(41\) 0.820265 0.128104 0.0640519 0.997947i \(-0.479598\pi\)
0.0640519 + 0.997947i \(0.479598\pi\)
\(42\) 0 0
\(43\) −0.696123 −0.106158 −0.0530789 0.998590i \(-0.516903\pi\)
−0.0530789 + 0.998590i \(0.516903\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 11.8903 1.73437 0.867186 0.497984i \(-0.165926\pi\)
0.867186 + 0.497984i \(0.165926\pi\)
\(48\) 0 0
\(49\) 2.83167 0.404525
\(50\) 0 0
\(51\) −4.48361 −0.627831
\(52\) 0 0
\(53\) 9.45083 1.29817 0.649086 0.760715i \(-0.275152\pi\)
0.649086 + 0.760715i \(0.275152\pi\)
\(54\) 0 0
\(55\) 2.65194 0.357587
\(56\) 0 0
\(57\) −0.651938 −0.0863513
\(58\) 0 0
\(59\) −12.7547 −1.66052 −0.830261 0.557375i \(-0.811809\pi\)
−0.830261 + 0.557375i \(0.811809\pi\)
\(60\) 0 0
\(61\) 2.65194 0.339546 0.169773 0.985483i \(-0.445697\pi\)
0.169773 + 0.985483i \(0.445697\pi\)
\(62\) 0 0
\(63\) −3.13555 −0.395042
\(64\) 0 0
\(65\) −0.651938 −0.0808630
\(66\) 0 0
\(67\) 6.10277 0.745572 0.372786 0.927917i \(-0.378402\pi\)
0.372786 + 0.927917i \(0.378402\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.48361 −1.00682 −0.503410 0.864048i \(-0.667921\pi\)
−0.503410 + 0.864048i \(0.667921\pi\)
\(72\) 0 0
\(73\) 6.31528 0.739148 0.369574 0.929201i \(-0.379504\pi\)
0.369574 + 0.929201i \(0.379504\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.31528 0.947615
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.51639 −0.605502 −0.302751 0.953070i \(-0.597905\pi\)
−0.302751 + 0.953070i \(0.597905\pi\)
\(84\) 0 0
\(85\) −4.48361 −0.486316
\(86\) 0 0
\(87\) −2.48361 −0.266271
\(88\) 0 0
\(89\) −11.2383 −1.19126 −0.595630 0.803259i \(-0.703098\pi\)
−0.595630 + 0.803259i \(0.703098\pi\)
\(90\) 0 0
\(91\) −2.04419 −0.214289
\(92\) 0 0
\(93\) −3.13555 −0.325141
\(94\) 0 0
\(95\) −0.651938 −0.0668875
\(96\) 0 0
\(97\) −5.93445 −0.602552 −0.301276 0.953537i \(-0.597413\pi\)
−0.301276 + 0.953537i \(0.597413\pi\)
\(98\) 0 0
\(99\) −2.65194 −0.266530
\(100\) 0 0
\(101\) 8.75471 0.871126 0.435563 0.900158i \(-0.356549\pi\)
0.435563 + 0.900158i \(0.356549\pi\)
\(102\) 0 0
\(103\) 3.66335 0.360960 0.180480 0.983579i \(-0.442235\pi\)
0.180480 + 0.983579i \(0.442235\pi\)
\(104\) 0 0
\(105\) −3.13555 −0.305998
\(106\) 0 0
\(107\) −3.24529 −0.313734 −0.156867 0.987620i \(-0.550139\pi\)
−0.156867 + 0.987620i \(0.550139\pi\)
\(108\) 0 0
\(109\) 3.61916 0.346653 0.173326 0.984864i \(-0.444548\pi\)
0.173326 + 0.984864i \(0.444548\pi\)
\(110\) 0 0
\(111\) 6.10277 0.579250
\(112\) 0 0
\(113\) 3.51639 0.330794 0.165397 0.986227i \(-0.447109\pi\)
0.165397 + 0.986227i \(0.447109\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0.651938 0.0602717
\(118\) 0 0
\(119\) −14.0586 −1.28875
\(120\) 0 0
\(121\) −3.96722 −0.360657
\(122\) 0 0
\(123\) −0.820265 −0.0739608
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.89026 −0.877619 −0.438809 0.898580i \(-0.644600\pi\)
−0.438809 + 0.898580i \(0.644600\pi\)
\(128\) 0 0
\(129\) 0.696123 0.0612902
\(130\) 0 0
\(131\) −21.9344 −1.91642 −0.958211 0.286063i \(-0.907653\pi\)
−0.958211 + 0.286063i \(0.907653\pi\)
\(132\) 0 0
\(133\) −2.04419 −0.177253
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 6.33665 0.541377 0.270688 0.962667i \(-0.412749\pi\)
0.270688 + 0.962667i \(0.412749\pi\)
\(138\) 0 0
\(139\) 7.13555 0.605229 0.302615 0.953113i \(-0.402140\pi\)
0.302615 + 0.953113i \(0.402140\pi\)
\(140\) 0 0
\(141\) −11.8903 −1.00134
\(142\) 0 0
\(143\) −1.72890 −0.144578
\(144\) 0 0
\(145\) −2.48361 −0.206253
\(146\) 0 0
\(147\) −2.83167 −0.233552
\(148\) 0 0
\(149\) −14.3153 −1.17275 −0.586377 0.810038i \(-0.699446\pi\)
−0.586377 + 0.810038i \(0.699446\pi\)
\(150\) 0 0
\(151\) −11.5750 −0.941958 −0.470979 0.882144i \(-0.656099\pi\)
−0.470979 + 0.882144i \(0.656099\pi\)
\(152\) 0 0
\(153\) 4.48361 0.362479
\(154\) 0 0
\(155\) −3.13555 −0.251853
\(156\) 0 0
\(157\) 13.7661 1.09866 0.549328 0.835607i \(-0.314884\pi\)
0.549328 + 0.835607i \(0.314884\pi\)
\(158\) 0 0
\(159\) −9.45083 −0.749500
\(160\) 0 0
\(161\) 3.13555 0.247116
\(162\) 0 0
\(163\) −6.27110 −0.491190 −0.245595 0.969372i \(-0.578983\pi\)
−0.245595 + 0.969372i \(0.578983\pi\)
\(164\) 0 0
\(165\) −2.65194 −0.206453
\(166\) 0 0
\(167\) −13.2825 −1.02783 −0.513916 0.857841i \(-0.671806\pi\)
−0.513916 + 0.857841i \(0.671806\pi\)
\(168\) 0 0
\(169\) −12.5750 −0.967306
\(170\) 0 0
\(171\) 0.651938 0.0498550
\(172\) 0 0
\(173\) −8.27110 −0.628840 −0.314420 0.949284i \(-0.601810\pi\)
−0.314420 + 0.949284i \(0.601810\pi\)
\(174\) 0 0
\(175\) −3.13555 −0.237025
\(176\) 0 0
\(177\) 12.7547 0.958703
\(178\) 0 0
\(179\) −5.64053 −0.421593 −0.210797 0.977530i \(-0.567606\pi\)
−0.210797 + 0.977530i \(0.567606\pi\)
\(180\) 0 0
\(181\) 0.359470 0.0267192 0.0133596 0.999911i \(-0.495747\pi\)
0.0133596 + 0.999911i \(0.495747\pi\)
\(182\) 0 0
\(183\) −2.65194 −0.196037
\(184\) 0 0
\(185\) 6.10277 0.448685
\(186\) 0 0
\(187\) −11.8903 −0.869502
\(188\) 0 0
\(189\) 3.13555 0.228078
\(190\) 0 0
\(191\) 19.1286 1.38410 0.692048 0.721852i \(-0.256709\pi\)
0.692048 + 0.721852i \(0.256709\pi\)
\(192\) 0 0
\(193\) −10.9672 −0.789438 −0.394719 0.918802i \(-0.629158\pi\)
−0.394719 + 0.918802i \(0.629158\pi\)
\(194\) 0 0
\(195\) 0.651938 0.0466863
\(196\) 0 0
\(197\) −13.2383 −0.943191 −0.471596 0.881815i \(-0.656322\pi\)
−0.471596 + 0.881815i \(0.656322\pi\)
\(198\) 0 0
\(199\) 2.35947 0.167258 0.0836292 0.996497i \(-0.473349\pi\)
0.0836292 + 0.996497i \(0.473349\pi\)
\(200\) 0 0
\(201\) −6.10277 −0.430456
\(202\) 0 0
\(203\) −7.78749 −0.546575
\(204\) 0 0
\(205\) −0.820265 −0.0572898
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −1.72890 −0.119591
\(210\) 0 0
\(211\) −15.1355 −1.04197 −0.520987 0.853565i \(-0.674436\pi\)
−0.520987 + 0.853565i \(0.674436\pi\)
\(212\) 0 0
\(213\) 8.48361 0.581287
\(214\) 0 0
\(215\) 0.696123 0.0474752
\(216\) 0 0
\(217\) −9.83167 −0.667417
\(218\) 0 0
\(219\) −6.31528 −0.426747
\(220\) 0 0
\(221\) 2.92304 0.196625
\(222\) 0 0
\(223\) 12.6078 0.844278 0.422139 0.906531i \(-0.361279\pi\)
0.422139 + 0.906531i \(0.361279\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.54220 0.301476 0.150738 0.988574i \(-0.451835\pi\)
0.150738 + 0.988574i \(0.451835\pi\)
\(228\) 0 0
\(229\) 1.57498 0.104077 0.0520387 0.998645i \(-0.483428\pi\)
0.0520387 + 0.998645i \(0.483428\pi\)
\(230\) 0 0
\(231\) −8.31528 −0.547106
\(232\) 0 0
\(233\) −6.33665 −0.415128 −0.207564 0.978221i \(-0.566554\pi\)
−0.207564 + 0.978221i \(0.566554\pi\)
\(234\) 0 0
\(235\) −11.8903 −0.775635
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 7.09136 0.458702 0.229351 0.973344i \(-0.426340\pi\)
0.229351 + 0.973344i \(0.426340\pi\)
\(240\) 0 0
\(241\) −5.68472 −0.366185 −0.183092 0.983096i \(-0.558611\pi\)
−0.183092 + 0.983096i \(0.558611\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.83167 −0.180909
\(246\) 0 0
\(247\) 0.425024 0.0270436
\(248\) 0 0
\(249\) 5.51639 0.349587
\(250\) 0 0
\(251\) −27.9344 −1.76321 −0.881603 0.471991i \(-0.843535\pi\)
−0.881603 + 0.471991i \(0.843535\pi\)
\(252\) 0 0
\(253\) 2.65194 0.166726
\(254\) 0 0
\(255\) 4.48361 0.280775
\(256\) 0 0
\(257\) 7.61916 0.475270 0.237635 0.971354i \(-0.423628\pi\)
0.237635 + 0.971354i \(0.423628\pi\)
\(258\) 0 0
\(259\) 19.1355 1.18903
\(260\) 0 0
\(261\) 2.48361 0.153732
\(262\) 0 0
\(263\) 5.85304 0.360914 0.180457 0.983583i \(-0.442242\pi\)
0.180457 + 0.983583i \(0.442242\pi\)
\(264\) 0 0
\(265\) −9.45083 −0.580560
\(266\) 0 0
\(267\) 11.2383 0.687774
\(268\) 0 0
\(269\) −15.7875 −0.962580 −0.481290 0.876561i \(-0.659832\pi\)
−0.481290 + 0.876561i \(0.659832\pi\)
\(270\) 0 0
\(271\) −3.56057 −0.216289 −0.108145 0.994135i \(-0.534491\pi\)
−0.108145 + 0.994135i \(0.534491\pi\)
\(272\) 0 0
\(273\) 2.04419 0.123720
\(274\) 0 0
\(275\) −2.65194 −0.159918
\(276\) 0 0
\(277\) −20.8789 −1.25449 −0.627244 0.778823i \(-0.715817\pi\)
−0.627244 + 0.778823i \(0.715817\pi\)
\(278\) 0 0
\(279\) 3.13555 0.187720
\(280\) 0 0
\(281\) 11.5536 0.689230 0.344615 0.938744i \(-0.388009\pi\)
0.344615 + 0.938744i \(0.388009\pi\)
\(282\) 0 0
\(283\) −7.83167 −0.465545 −0.232772 0.972531i \(-0.574780\pi\)
−0.232772 + 0.972531i \(0.574780\pi\)
\(284\) 0 0
\(285\) 0.651938 0.0386175
\(286\) 0 0
\(287\) −2.57198 −0.151819
\(288\) 0 0
\(289\) 3.10277 0.182516
\(290\) 0 0
\(291\) 5.93445 0.347883
\(292\) 0 0
\(293\) 3.42802 0.200267 0.100133 0.994974i \(-0.468073\pi\)
0.100133 + 0.994974i \(0.468073\pi\)
\(294\) 0 0
\(295\) 12.7547 0.742608
\(296\) 0 0
\(297\) 2.65194 0.153881
\(298\) 0 0
\(299\) −0.651938 −0.0377026
\(300\) 0 0
\(301\) 2.18273 0.125810
\(302\) 0 0
\(303\) −8.75471 −0.502945
\(304\) 0 0
\(305\) −2.65194 −0.151850
\(306\) 0 0
\(307\) 6.67475 0.380948 0.190474 0.981692i \(-0.438997\pi\)
0.190474 + 0.981692i \(0.438997\pi\)
\(308\) 0 0
\(309\) −3.66335 −0.208401
\(310\) 0 0
\(311\) 14.2055 0.805523 0.402761 0.915305i \(-0.368050\pi\)
0.402761 + 0.915305i \(0.368050\pi\)
\(312\) 0 0
\(313\) 1.83167 0.103532 0.0517661 0.998659i \(-0.483515\pi\)
0.0517661 + 0.998659i \(0.483515\pi\)
\(314\) 0 0
\(315\) 3.13555 0.176668
\(316\) 0 0
\(317\) −16.9230 −0.950493 −0.475246 0.879853i \(-0.657641\pi\)
−0.475246 + 0.879853i \(0.657641\pi\)
\(318\) 0 0
\(319\) −6.58638 −0.368767
\(320\) 0 0
\(321\) 3.24529 0.181134
\(322\) 0 0
\(323\) 2.92304 0.162642
\(324\) 0 0
\(325\) 0.651938 0.0361630
\(326\) 0 0
\(327\) −3.61916 −0.200140
\(328\) 0 0
\(329\) −37.2825 −2.05545
\(330\) 0 0
\(331\) 18.3739 1.00992 0.504960 0.863143i \(-0.331507\pi\)
0.504960 + 0.863143i \(0.331507\pi\)
\(332\) 0 0
\(333\) −6.10277 −0.334430
\(334\) 0 0
\(335\) −6.10277 −0.333430
\(336\) 0 0
\(337\) 26.8133 1.46061 0.730307 0.683119i \(-0.239377\pi\)
0.730307 + 0.683119i \(0.239377\pi\)
\(338\) 0 0
\(339\) −3.51639 −0.190984
\(340\) 0 0
\(341\) −8.31528 −0.450298
\(342\) 0 0
\(343\) 13.0700 0.705713
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −28.4538 −1.52748 −0.763741 0.645523i \(-0.776639\pi\)
−0.763741 + 0.645523i \(0.776639\pi\)
\(348\) 0 0
\(349\) −4.79890 −0.256879 −0.128440 0.991717i \(-0.540997\pi\)
−0.128440 + 0.991717i \(0.540997\pi\)
\(350\) 0 0
\(351\) −0.651938 −0.0347979
\(352\) 0 0
\(353\) −20.7919 −1.10664 −0.553321 0.832968i \(-0.686640\pi\)
−0.553321 + 0.832968i \(0.686640\pi\)
\(354\) 0 0
\(355\) 8.48361 0.450263
\(356\) 0 0
\(357\) 14.0586 0.744059
\(358\) 0 0
\(359\) −2.04419 −0.107888 −0.0539440 0.998544i \(-0.517179\pi\)
−0.0539440 + 0.998544i \(0.517179\pi\)
\(360\) 0 0
\(361\) −18.5750 −0.977630
\(362\) 0 0
\(363\) 3.96722 0.208225
\(364\) 0 0
\(365\) −6.31528 −0.330557
\(366\) 0 0
\(367\) −0.527797 −0.0275508 −0.0137754 0.999905i \(-0.504385\pi\)
−0.0137754 + 0.999905i \(0.504385\pi\)
\(368\) 0 0
\(369\) 0.820265 0.0427013
\(370\) 0 0
\(371\) −29.6336 −1.53850
\(372\) 0 0
\(373\) 29.4439 1.52455 0.762273 0.647256i \(-0.224083\pi\)
0.762273 + 0.647256i \(0.224083\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 1.61916 0.0833911
\(378\) 0 0
\(379\) −7.32669 −0.376347 −0.188173 0.982136i \(-0.560257\pi\)
−0.188173 + 0.982136i \(0.560257\pi\)
\(380\) 0 0
\(381\) 9.89026 0.506693
\(382\) 0 0
\(383\) 5.42802 0.277359 0.138679 0.990337i \(-0.455714\pi\)
0.138679 + 0.990337i \(0.455714\pi\)
\(384\) 0 0
\(385\) −8.31528 −0.423786
\(386\) 0 0
\(387\) −0.696123 −0.0353859
\(388\) 0 0
\(389\) −28.3883 −1.43934 −0.719671 0.694315i \(-0.755708\pi\)
−0.719671 + 0.694315i \(0.755708\pi\)
\(390\) 0 0
\(391\) −4.48361 −0.226746
\(392\) 0 0
\(393\) 21.9344 1.10645
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −14.6078 −0.733142 −0.366571 0.930390i \(-0.619468\pi\)
−0.366571 + 0.930390i \(0.619468\pi\)
\(398\) 0 0
\(399\) 2.04419 0.102337
\(400\) 0 0
\(401\) −19.5750 −0.977528 −0.488764 0.872416i \(-0.662552\pi\)
−0.488764 + 0.872416i \(0.662552\pi\)
\(402\) 0 0
\(403\) 2.04419 0.101828
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 16.1842 0.802220
\(408\) 0 0
\(409\) −10.5278 −0.520566 −0.260283 0.965532i \(-0.583816\pi\)
−0.260283 + 0.965532i \(0.583816\pi\)
\(410\) 0 0
\(411\) −6.33665 −0.312564
\(412\) 0 0
\(413\) 39.9930 1.96793
\(414\) 0 0
\(415\) 5.51639 0.270789
\(416\) 0 0
\(417\) −7.13555 −0.349429
\(418\) 0 0
\(419\) 11.1941 0.546869 0.273435 0.961891i \(-0.411840\pi\)
0.273435 + 0.961891i \(0.411840\pi\)
\(420\) 0 0
\(421\) −23.5308 −1.14682 −0.573410 0.819268i \(-0.694380\pi\)
−0.573410 + 0.819268i \(0.694380\pi\)
\(422\) 0 0
\(423\) 11.8903 0.578124
\(424\) 0 0
\(425\) 4.48361 0.217487
\(426\) 0 0
\(427\) −8.31528 −0.402405
\(428\) 0 0
\(429\) 1.72890 0.0834721
\(430\) 0 0
\(431\) −27.2383 −1.31202 −0.656012 0.754751i \(-0.727758\pi\)
−0.656012 + 0.754751i \(0.727758\pi\)
\(432\) 0 0
\(433\) 13.8317 0.664708 0.332354 0.943155i \(-0.392157\pi\)
0.332354 + 0.943155i \(0.392157\pi\)
\(434\) 0 0
\(435\) 2.48361 0.119080
\(436\) 0 0
\(437\) −0.651938 −0.0311864
\(438\) 0 0
\(439\) 33.4211 1.59510 0.797550 0.603253i \(-0.206129\pi\)
0.797550 + 0.603253i \(0.206129\pi\)
\(440\) 0 0
\(441\) 2.83167 0.134842
\(442\) 0 0
\(443\) 31.1286 1.47896 0.739482 0.673177i \(-0.235071\pi\)
0.739482 + 0.673177i \(0.235071\pi\)
\(444\) 0 0
\(445\) 11.2383 0.532748
\(446\) 0 0
\(447\) 14.3153 0.677090
\(448\) 0 0
\(449\) −23.0030 −1.08558 −0.542789 0.839869i \(-0.682632\pi\)
−0.542789 + 0.839869i \(0.682632\pi\)
\(450\) 0 0
\(451\) −2.17529 −0.102431
\(452\) 0 0
\(453\) 11.5750 0.543840
\(454\) 0 0
\(455\) 2.04419 0.0958329
\(456\) 0 0
\(457\) −0.233880 −0.0109405 −0.00547023 0.999985i \(-0.501741\pi\)
−0.00547023 + 0.999985i \(0.501741\pi\)
\(458\) 0 0
\(459\) −4.48361 −0.209277
\(460\) 0 0
\(461\) 13.5094 0.629197 0.314598 0.949225i \(-0.398130\pi\)
0.314598 + 0.949225i \(0.398130\pi\)
\(462\) 0 0
\(463\) 27.3997 1.27337 0.636686 0.771123i \(-0.280305\pi\)
0.636686 + 0.771123i \(0.280305\pi\)
\(464\) 0 0
\(465\) 3.13555 0.145408
\(466\) 0 0
\(467\) −9.17974 −0.424787 −0.212394 0.977184i \(-0.568126\pi\)
−0.212394 + 0.977184i \(0.568126\pi\)
\(468\) 0 0
\(469\) −19.1355 −0.883598
\(470\) 0 0
\(471\) −13.7661 −0.634309
\(472\) 0 0
\(473\) 1.84608 0.0848827
\(474\) 0 0
\(475\) 0.651938 0.0299130
\(476\) 0 0
\(477\) 9.45083 0.432724
\(478\) 0 0
\(479\) 2.04419 0.0934012 0.0467006 0.998909i \(-0.485129\pi\)
0.0467006 + 0.998909i \(0.485129\pi\)
\(480\) 0 0
\(481\) −3.97863 −0.181410
\(482\) 0 0
\(483\) −3.13555 −0.142672
\(484\) 0 0
\(485\) 5.93445 0.269469
\(486\) 0 0
\(487\) −29.0402 −1.31594 −0.657969 0.753045i \(-0.728584\pi\)
−0.657969 + 0.753045i \(0.728584\pi\)
\(488\) 0 0
\(489\) 6.27110 0.283589
\(490\) 0 0
\(491\) −37.9275 −1.71164 −0.855822 0.517271i \(-0.826948\pi\)
−0.855822 + 0.517271i \(0.826948\pi\)
\(492\) 0 0
\(493\) 11.1355 0.501520
\(494\) 0 0
\(495\) 2.65194 0.119196
\(496\) 0 0
\(497\) 26.6008 1.19321
\(498\) 0 0
\(499\) 12.7761 0.571936 0.285968 0.958239i \(-0.407685\pi\)
0.285968 + 0.958239i \(0.407685\pi\)
\(500\) 0 0
\(501\) 13.2825 0.593419
\(502\) 0 0
\(503\) −38.2641 −1.70611 −0.853057 0.521818i \(-0.825254\pi\)
−0.853057 + 0.521818i \(0.825254\pi\)
\(504\) 0 0
\(505\) −8.75471 −0.389580
\(506\) 0 0
\(507\) 12.5750 0.558474
\(508\) 0 0
\(509\) −28.0000 −1.24108 −0.620539 0.784176i \(-0.713086\pi\)
−0.620539 + 0.784176i \(0.713086\pi\)
\(510\) 0 0
\(511\) −19.8019 −0.875984
\(512\) 0 0
\(513\) −0.651938 −0.0287838
\(514\) 0 0
\(515\) −3.66335 −0.161426
\(516\) 0 0
\(517\) −31.5322 −1.38679
\(518\) 0 0
\(519\) 8.27110 0.363061
\(520\) 0 0
\(521\) −24.7691 −1.08515 −0.542577 0.840006i \(-0.682551\pi\)
−0.542577 + 0.840006i \(0.682551\pi\)
\(522\) 0 0
\(523\) 0.271100 0.0118544 0.00592718 0.999982i \(-0.498113\pi\)
0.00592718 + 0.999982i \(0.498113\pi\)
\(524\) 0 0
\(525\) 3.13555 0.136847
\(526\) 0 0
\(527\) 14.0586 0.612402
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.7547 −0.553507
\(532\) 0 0
\(533\) 0.534762 0.0231631
\(534\) 0 0
\(535\) 3.24529 0.140306
\(536\) 0 0
\(537\) 5.64053 0.243407
\(538\) 0 0
\(539\) −7.50942 −0.323454
\(540\) 0 0
\(541\) −40.1400 −1.72575 −0.862877 0.505415i \(-0.831340\pi\)
−0.862877 + 0.505415i \(0.831340\pi\)
\(542\) 0 0
\(543\) −0.359470 −0.0154263
\(544\) 0 0
\(545\) −3.61916 −0.155028
\(546\) 0 0
\(547\) 4.67331 0.199816 0.0999081 0.994997i \(-0.468145\pi\)
0.0999081 + 0.994997i \(0.468145\pi\)
\(548\) 0 0
\(549\) 2.65194 0.113182
\(550\) 0 0
\(551\) 1.61916 0.0689786
\(552\) 0 0
\(553\) −25.0844 −1.06670
\(554\) 0 0
\(555\) −6.10277 −0.259048
\(556\) 0 0
\(557\) −21.5680 −0.913866 −0.456933 0.889501i \(-0.651052\pi\)
−0.456933 + 0.889501i \(0.651052\pi\)
\(558\) 0 0
\(559\) −0.453830 −0.0191949
\(560\) 0 0
\(561\) 11.8903 0.502007
\(562\) 0 0
\(563\) −31.9047 −1.34462 −0.672311 0.740269i \(-0.734698\pi\)
−0.672311 + 0.740269i \(0.734698\pi\)
\(564\) 0 0
\(565\) −3.51639 −0.147936
\(566\) 0 0
\(567\) −3.13555 −0.131681
\(568\) 0 0
\(569\) 15.4439 0.647441 0.323720 0.946153i \(-0.395066\pi\)
0.323720 + 0.946153i \(0.395066\pi\)
\(570\) 0 0
\(571\) 28.6380 1.19846 0.599232 0.800576i \(-0.295473\pi\)
0.599232 + 0.800576i \(0.295473\pi\)
\(572\) 0 0
\(573\) −19.1286 −0.799108
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −1.12115 −0.0466740 −0.0233370 0.999728i \(-0.507429\pi\)
−0.0233370 + 0.999728i \(0.507429\pi\)
\(578\) 0 0
\(579\) 10.9672 0.455782
\(580\) 0 0
\(581\) 17.2969 0.717597
\(582\) 0 0
\(583\) −25.0630 −1.03800
\(584\) 0 0
\(585\) −0.651938 −0.0269543
\(586\) 0 0
\(587\) −28.2055 −1.16417 −0.582084 0.813129i \(-0.697762\pi\)
−0.582084 + 0.813129i \(0.697762\pi\)
\(588\) 0 0
\(589\) 2.04419 0.0842292
\(590\) 0 0
\(591\) 13.2383 0.544552
\(592\) 0 0
\(593\) 19.5308 0.802033 0.401017 0.916071i \(-0.368657\pi\)
0.401017 + 0.916071i \(0.368657\pi\)
\(594\) 0 0
\(595\) 14.0586 0.576346
\(596\) 0 0
\(597\) −2.35947 −0.0965667
\(598\) 0 0
\(599\) 32.9017 1.34433 0.672163 0.740403i \(-0.265365\pi\)
0.672163 + 0.740403i \(0.265365\pi\)
\(600\) 0 0
\(601\) −19.2011 −0.783229 −0.391615 0.920129i \(-0.628083\pi\)
−0.391615 + 0.920129i \(0.628083\pi\)
\(602\) 0 0
\(603\) 6.10277 0.248524
\(604\) 0 0
\(605\) 3.96722 0.161291
\(606\) 0 0
\(607\) 23.2825 0.945008 0.472504 0.881329i \(-0.343350\pi\)
0.472504 + 0.881329i \(0.343350\pi\)
\(608\) 0 0
\(609\) 7.78749 0.315565
\(610\) 0 0
\(611\) 7.75172 0.313601
\(612\) 0 0
\(613\) −5.86889 −0.237042 −0.118521 0.992952i \(-0.537815\pi\)
−0.118521 + 0.992952i \(0.537815\pi\)
\(614\) 0 0
\(615\) 0.820265 0.0330763
\(616\) 0 0
\(617\) 33.2314 1.33784 0.668922 0.743333i \(-0.266756\pi\)
0.668922 + 0.743333i \(0.266756\pi\)
\(618\) 0 0
\(619\) 19.3267 0.776805 0.388403 0.921490i \(-0.373027\pi\)
0.388403 + 0.921490i \(0.373027\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 35.2383 1.41179
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.72890 0.0690456
\(628\) 0 0
\(629\) −27.3625 −1.09101
\(630\) 0 0
\(631\) −9.39968 −0.374196 −0.187098 0.982341i \(-0.559908\pi\)
−0.187098 + 0.982341i \(0.559908\pi\)
\(632\) 0 0
\(633\) 15.1355 0.601584
\(634\) 0 0
\(635\) 9.89026 0.392483
\(636\) 0 0
\(637\) 1.84608 0.0731442
\(638\) 0 0
\(639\) −8.48361 −0.335606
\(640\) 0 0
\(641\) −42.0303 −1.66010 −0.830048 0.557693i \(-0.811687\pi\)
−0.830048 + 0.557693i \(0.811687\pi\)
\(642\) 0 0
\(643\) 31.9488 1.25994 0.629970 0.776620i \(-0.283067\pi\)
0.629970 + 0.776620i \(0.283067\pi\)
\(644\) 0 0
\(645\) −0.696123 −0.0274098
\(646\) 0 0
\(647\) −20.7691 −0.816518 −0.408259 0.912866i \(-0.633864\pi\)
−0.408259 + 0.912866i \(0.633864\pi\)
\(648\) 0 0
\(649\) 33.8247 1.32774
\(650\) 0 0
\(651\) 9.83167 0.385334
\(652\) 0 0
\(653\) 13.3481 0.522350 0.261175 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(654\) 0 0
\(655\) 21.9344 0.857050
\(656\) 0 0
\(657\) 6.31528 0.246383
\(658\) 0 0
\(659\) 18.5636 0.723134 0.361567 0.932346i \(-0.382242\pi\)
0.361567 + 0.932346i \(0.382242\pi\)
\(660\) 0 0
\(661\) 23.1728 0.901316 0.450658 0.892697i \(-0.351189\pi\)
0.450658 + 0.892697i \(0.351189\pi\)
\(662\) 0 0
\(663\) −2.92304 −0.113521
\(664\) 0 0
\(665\) 2.04419 0.0792701
\(666\) 0 0
\(667\) −2.48361 −0.0961658
\(668\) 0 0
\(669\) −12.6078 −0.487444
\(670\) 0 0
\(671\) −7.03278 −0.271497
\(672\) 0 0
\(673\) 51.0630 1.96834 0.984168 0.177240i \(-0.0567170\pi\)
0.984168 + 0.177240i \(0.0567170\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 5.87586 0.225828 0.112914 0.993605i \(-0.463982\pi\)
0.112914 + 0.993605i \(0.463982\pi\)
\(678\) 0 0
\(679\) 18.6078 0.714100
\(680\) 0 0
\(681\) −4.54220 −0.174057
\(682\) 0 0
\(683\) −19.2597 −0.736952 −0.368476 0.929637i \(-0.620120\pi\)
−0.368476 + 0.929637i \(0.620120\pi\)
\(684\) 0 0
\(685\) −6.33665 −0.242111
\(686\) 0 0
\(687\) −1.57498 −0.0600891
\(688\) 0 0
\(689\) 6.16136 0.234729
\(690\) 0 0
\(691\) −8.11718 −0.308792 −0.154396 0.988009i \(-0.549343\pi\)
−0.154396 + 0.988009i \(0.549343\pi\)
\(692\) 0 0
\(693\) 8.31528 0.315872
\(694\) 0 0
\(695\) −7.13555 −0.270667
\(696\) 0 0
\(697\) 3.67775 0.139305
\(698\) 0 0
\(699\) 6.33665 0.239674
\(700\) 0 0
\(701\) −18.9459 −0.715575 −0.357788 0.933803i \(-0.616469\pi\)
−0.357788 + 0.933803i \(0.616469\pi\)
\(702\) 0 0
\(703\) −3.97863 −0.150057
\(704\) 0 0
\(705\) 11.8903 0.447813
\(706\) 0 0
\(707\) −27.4508 −1.03239
\(708\) 0 0
\(709\) −6.06700 −0.227851 −0.113926 0.993489i \(-0.536343\pi\)
−0.113926 + 0.993489i \(0.536343\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −3.13555 −0.117427
\(714\) 0 0
\(715\) 1.72890 0.0646572
\(716\) 0 0
\(717\) −7.09136 −0.264832
\(718\) 0 0
\(719\) 19.4737 0.726245 0.363122 0.931741i \(-0.381711\pi\)
0.363122 + 0.931741i \(0.381711\pi\)
\(720\) 0 0
\(721\) −11.4866 −0.427784
\(722\) 0 0
\(723\) 5.68472 0.211417
\(724\) 0 0
\(725\) 2.48361 0.0922390
\(726\) 0 0
\(727\) 41.5238 1.54003 0.770017 0.638024i \(-0.220248\pi\)
0.770017 + 0.638024i \(0.220248\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.12115 −0.115440
\(732\) 0 0
\(733\) −13.8833 −0.512791 −0.256396 0.966572i \(-0.582535\pi\)
−0.256396 + 0.966572i \(0.582535\pi\)
\(734\) 0 0
\(735\) 2.83167 0.104448
\(736\) 0 0
\(737\) −16.1842 −0.596152
\(738\) 0 0
\(739\) 23.7661 0.874251 0.437125 0.899401i \(-0.355997\pi\)
0.437125 + 0.899401i \(0.355997\pi\)
\(740\) 0 0
\(741\) −0.425024 −0.0156136
\(742\) 0 0
\(743\) 41.9344 1.53843 0.769213 0.638993i \(-0.220649\pi\)
0.769213 + 0.638993i \(0.220649\pi\)
\(744\) 0 0
\(745\) 14.3153 0.524471
\(746\) 0 0
\(747\) −5.51639 −0.201834
\(748\) 0 0
\(749\) 10.1758 0.371814
\(750\) 0 0
\(751\) −7.89026 −0.287920 −0.143960 0.989584i \(-0.545984\pi\)
−0.143960 + 0.989584i \(0.545984\pi\)
\(752\) 0 0
\(753\) 27.9344 1.01799
\(754\) 0 0
\(755\) 11.5750 0.421257
\(756\) 0 0
\(757\) −26.5566 −0.965216 −0.482608 0.875836i \(-0.660310\pi\)
−0.482608 + 0.875836i \(0.660310\pi\)
\(758\) 0 0
\(759\) −2.65194 −0.0962593
\(760\) 0 0
\(761\) −18.1241 −0.657000 −0.328500 0.944504i \(-0.606543\pi\)
−0.328500 + 0.944504i \(0.606543\pi\)
\(762\) 0 0
\(763\) −11.3481 −0.410827
\(764\) 0 0
\(765\) −4.48361 −0.162105
\(766\) 0 0
\(767\) −8.31528 −0.300248
\(768\) 0 0
\(769\) 7.86744 0.283707 0.141854 0.989888i \(-0.454694\pi\)
0.141854 + 0.989888i \(0.454694\pi\)
\(770\) 0 0
\(771\) −7.61916 −0.274397
\(772\) 0 0
\(773\) −7.17277 −0.257987 −0.128993 0.991645i \(-0.541175\pi\)
−0.128993 + 0.991645i \(0.541175\pi\)
\(774\) 0 0
\(775\) 3.13555 0.112632
\(776\) 0 0
\(777\) −19.1355 −0.686484
\(778\) 0 0
\(779\) 0.534762 0.0191598
\(780\) 0 0
\(781\) 22.4980 0.805042
\(782\) 0 0
\(783\) −2.48361 −0.0887570
\(784\) 0 0
\(785\) −13.7661 −0.491334
\(786\) 0 0
\(787\) 6.01440 0.214390 0.107195 0.994238i \(-0.465813\pi\)
0.107195 + 0.994238i \(0.465813\pi\)
\(788\) 0 0
\(789\) −5.85304 −0.208374
\(790\) 0 0
\(791\) −11.0258 −0.392033
\(792\) 0 0
\(793\) 1.72890 0.0613950
\(794\) 0 0
\(795\) 9.45083 0.335187
\(796\) 0 0
\(797\) −1.45083 −0.0513912 −0.0256956 0.999670i \(-0.508180\pi\)
−0.0256956 + 0.999670i \(0.508180\pi\)
\(798\) 0 0
\(799\) 53.3113 1.88602
\(800\) 0 0
\(801\) −11.2383 −0.397087
\(802\) 0 0
\(803\) −16.7477 −0.591015
\(804\) 0 0
\(805\) −3.13555 −0.110514
\(806\) 0 0
\(807\) 15.7875 0.555746
\(808\) 0 0
\(809\) 9.11418 0.320438 0.160219 0.987082i \(-0.448780\pi\)
0.160219 + 0.987082i \(0.448780\pi\)
\(810\) 0 0
\(811\) −19.0044 −0.667336 −0.333668 0.942691i \(-0.608286\pi\)
−0.333668 + 0.942691i \(0.608286\pi\)
\(812\) 0 0
\(813\) 3.56057 0.124875
\(814\) 0 0
\(815\) 6.27110 0.219667
\(816\) 0 0
\(817\) −0.453830 −0.0158775
\(818\) 0 0
\(819\) −2.04419 −0.0714296
\(820\) 0 0
\(821\) −55.5610 −1.93909 −0.969547 0.244906i \(-0.921243\pi\)
−0.969547 + 0.244906i \(0.921243\pi\)
\(822\) 0 0
\(823\) −5.66335 −0.197412 −0.0987059 0.995117i \(-0.531470\pi\)
−0.0987059 + 0.995117i \(0.531470\pi\)
\(824\) 0 0
\(825\) 2.65194 0.0923286
\(826\) 0 0
\(827\) −46.0586 −1.60161 −0.800807 0.598922i \(-0.795596\pi\)
−0.800807 + 0.598922i \(0.795596\pi\)
\(828\) 0 0
\(829\) −0.710526 −0.0246776 −0.0123388 0.999924i \(-0.503928\pi\)
−0.0123388 + 0.999924i \(0.503928\pi\)
\(830\) 0 0
\(831\) 20.8789 0.724279
\(832\) 0 0
\(833\) 12.6961 0.439895
\(834\) 0 0
\(835\) 13.2825 0.459660
\(836\) 0 0
\(837\) −3.13555 −0.108380
\(838\) 0 0
\(839\) 10.9444 0.377843 0.188921 0.981992i \(-0.439501\pi\)
0.188921 + 0.981992i \(0.439501\pi\)
\(840\) 0 0
\(841\) −22.8317 −0.787299
\(842\) 0 0
\(843\) −11.5536 −0.397927
\(844\) 0 0
\(845\) 12.5750 0.432592
\(846\) 0 0
\(847\) 12.4394 0.427424
\(848\) 0 0
\(849\) 7.83167 0.268782
\(850\) 0 0
\(851\) 6.10277 0.209200
\(852\) 0 0
\(853\) −1.81727 −0.0622222 −0.0311111 0.999516i \(-0.509905\pi\)
−0.0311111 + 0.999516i \(0.509905\pi\)
\(854\) 0 0
\(855\) −0.651938 −0.0222958
\(856\) 0 0
\(857\) −27.8033 −0.949744 −0.474872 0.880055i \(-0.657506\pi\)
−0.474872 + 0.880055i \(0.657506\pi\)
\(858\) 0 0
\(859\) 21.4066 0.730385 0.365193 0.930932i \(-0.381003\pi\)
0.365193 + 0.930932i \(0.381003\pi\)
\(860\) 0 0
\(861\) 2.57198 0.0876529
\(862\) 0 0
\(863\) 18.2284 0.620501 0.310250 0.950655i \(-0.399587\pi\)
0.310250 + 0.950655i \(0.399587\pi\)
\(864\) 0 0
\(865\) 8.27110 0.281226
\(866\) 0 0
\(867\) −3.10277 −0.105376
\(868\) 0 0
\(869\) −21.2155 −0.719687
\(870\) 0 0
\(871\) 3.97863 0.134811
\(872\) 0 0
\(873\) −5.93445 −0.200851
\(874\) 0 0
\(875\) 3.13555 0.106001
\(876\) 0 0
\(877\) −47.2671 −1.59610 −0.798049 0.602593i \(-0.794134\pi\)
−0.798049 + 0.602593i \(0.794134\pi\)
\(878\) 0 0
\(879\) −3.42802 −0.115624
\(880\) 0 0
\(881\) −2.71749 −0.0915546 −0.0457773 0.998952i \(-0.514576\pi\)
−0.0457773 + 0.998952i \(0.514576\pi\)
\(882\) 0 0
\(883\) −20.6380 −0.694524 −0.347262 0.937768i \(-0.612889\pi\)
−0.347262 + 0.937768i \(0.612889\pi\)
\(884\) 0 0
\(885\) −12.7547 −0.428745
\(886\) 0 0
\(887\) 48.1172 1.61562 0.807808 0.589445i \(-0.200654\pi\)
0.807808 + 0.589445i \(0.200654\pi\)
\(888\) 0 0
\(889\) 31.0114 1.04009
\(890\) 0 0
\(891\) −2.65194 −0.0888433
\(892\) 0 0
\(893\) 7.75172 0.259401
\(894\) 0 0
\(895\) 5.64053 0.188542
\(896\) 0 0
\(897\) 0.651938 0.0217676
\(898\) 0 0
\(899\) 7.78749 0.259727
\(900\) 0 0
\(901\) 42.3739 1.41168
\(902\) 0 0
\(903\) −2.18273 −0.0726367
\(904\) 0 0
\(905\) −0.359470 −0.0119492
\(906\) 0 0
\(907\) −2.30832 −0.0766465 −0.0383232 0.999265i \(-0.512202\pi\)
−0.0383232 + 0.999265i \(0.512202\pi\)
\(908\) 0 0
\(909\) 8.75471 0.290375
\(910\) 0 0
\(911\) −50.4766 −1.67237 −0.836183 0.548451i \(-0.815218\pi\)
−0.836183 + 0.548451i \(0.815218\pi\)
\(912\) 0 0
\(913\) 14.6291 0.484153
\(914\) 0 0
\(915\) 2.65194 0.0876704
\(916\) 0 0
\(917\) 68.7766 2.27120
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −6.67475 −0.219941
\(922\) 0 0
\(923\) −5.53079 −0.182048
\(924\) 0 0
\(925\) −6.10277 −0.200658
\(926\) 0 0
\(927\) 3.66335 0.120320
\(928\) 0 0
\(929\) −39.6336 −1.30034 −0.650168 0.759791i \(-0.725301\pi\)
−0.650168 + 0.759791i \(0.725301\pi\)
\(930\) 0 0
\(931\) 1.84608 0.0605027
\(932\) 0 0
\(933\) −14.2055 −0.465069
\(934\) 0 0
\(935\) 11.8903 0.388853
\(936\) 0 0
\(937\) −42.4766 −1.38765 −0.693826 0.720143i \(-0.744076\pi\)
−0.693826 + 0.720143i \(0.744076\pi\)
\(938\) 0 0
\(939\) −1.83167 −0.0597744
\(940\) 0 0
\(941\) 12.2497 0.399330 0.199665 0.979864i \(-0.436015\pi\)
0.199665 + 0.979864i \(0.436015\pi\)
\(942\) 0 0
\(943\) −0.820265 −0.0267115
\(944\) 0 0
\(945\) −3.13555 −0.101999
\(946\) 0 0
\(947\) −29.3039 −0.952248 −0.476124 0.879378i \(-0.657959\pi\)
−0.476124 + 0.879378i \(0.657959\pi\)
\(948\) 0 0
\(949\) 4.11718 0.133649
\(950\) 0 0
\(951\) 16.9230 0.548767
\(952\) 0 0
\(953\) −5.32669 −0.172549 −0.0862743 0.996271i \(-0.527496\pi\)
−0.0862743 + 0.996271i \(0.527496\pi\)
\(954\) 0 0
\(955\) −19.1286 −0.618986
\(956\) 0 0
\(957\) 6.58638 0.212908
\(958\) 0 0
\(959\) −19.8689 −0.641600
\(960\) 0 0
\(961\) −21.1683 −0.682849
\(962\) 0 0
\(963\) −3.24529 −0.104578
\(964\) 0 0
\(965\) 10.9672 0.353047
\(966\) 0 0
\(967\) 10.1097 0.325107 0.162554 0.986700i \(-0.448027\pi\)
0.162554 + 0.986700i \(0.448027\pi\)
\(968\) 0 0
\(969\) −2.92304 −0.0939015
\(970\) 0 0
\(971\) 20.7249 0.665095 0.332547 0.943087i \(-0.392092\pi\)
0.332547 + 0.943087i \(0.392092\pi\)
\(972\) 0 0
\(973\) −22.3739 −0.717273
\(974\) 0 0
\(975\) −0.651938 −0.0208787
\(976\) 0 0
\(977\) −6.37242 −0.203872 −0.101936 0.994791i \(-0.532504\pi\)
−0.101936 + 0.994791i \(0.532504\pi\)
\(978\) 0 0
\(979\) 29.8033 0.952519
\(980\) 0 0
\(981\) 3.61916 0.115551
\(982\) 0 0
\(983\) −26.6008 −0.848433 −0.424217 0.905561i \(-0.639451\pi\)
−0.424217 + 0.905561i \(0.639451\pi\)
\(984\) 0 0
\(985\) 13.2383 0.421808
\(986\) 0 0
\(987\) 37.2825 1.18672
\(988\) 0 0
\(989\) 0.696123 0.0221354
\(990\) 0 0
\(991\) 51.0332 1.62112 0.810562 0.585652i \(-0.199162\pi\)
0.810562 + 0.585652i \(0.199162\pi\)
\(992\) 0 0
\(993\) −18.3739 −0.583077
\(994\) 0 0
\(995\) −2.35947 −0.0748002
\(996\) 0 0
\(997\) 0.0883702 0.00279871 0.00139936 0.999999i \(-0.499555\pi\)
0.00139936 + 0.999999i \(0.499555\pi\)
\(998\) 0 0
\(999\) 6.10277 0.193083
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 5520.2.a.bw.1.1 3
4.3 odd 2 2760.2.a.s.1.3 3
12.11 even 2 8280.2.a.bn.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.s.1.3 3 4.3 odd 2
5520.2.a.bw.1.1 3 1.1 even 1 trivial
8280.2.a.bn.1.3 3 12.11 even 2