Properties

Label 4560.2.a.bq.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
Defining polynomial: \(x^{3} - x^{2} - 6 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.17741\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.91852 q^{7} +1.00000 q^{9} -3.43630 q^{11} +4.91852 q^{13} +1.00000 q^{15} -4.35482 q^{17} +1.00000 q^{19} +2.91852 q^{21} -6.35482 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.27334 q^{29} +2.35482 q^{31} +3.43630 q^{33} +2.91852 q^{35} +4.91852 q^{37} -4.91852 q^{39} +1.43630 q^{41} -6.91852 q^{43} -1.00000 q^{45} +6.35482 q^{47} +1.51777 q^{49} +4.35482 q^{51} -4.35482 q^{53} +3.43630 q^{55} -1.00000 q^{57} -9.83705 q^{59} -6.19186 q^{61} -2.91852 q^{63} -4.91852 q^{65} +8.70964 q^{67} +6.35482 q^{69} -6.87259 q^{71} +10.0000 q^{73} -1.00000 q^{75} +10.0289 q^{77} -8.70964 q^{79} +1.00000 q^{81} +16.9015 q^{83} +4.35482 q^{85} +7.27334 q^{87} -4.40075 q^{89} -14.3548 q^{91} -2.35482 q^{93} -1.00000 q^{95} +12.9185 q^{97} -3.43630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{9} - 4 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 12 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} - 2 q^{51} + 2 q^{53} + 4 q^{55} - 3 q^{57} - 12 q^{59} + 14 q^{61} - 6 q^{65} - 4 q^{67} + 4 q^{69} - 8 q^{71} + 30 q^{73} - 3 q^{75} - 20 q^{77} + 4 q^{79} + 3 q^{81} - 12 q^{83} - 2 q^{85} - 2 q^{87} - 2 q^{89} - 28 q^{91} + 8 q^{93} - 3 q^{95} + 30 q^{97} - 4 q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.91852 −1.10310 −0.551549 0.834143i \(-0.685963\pi\)
−0.551549 + 0.834143i \(0.685963\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.43630 −1.03608 −0.518041 0.855356i \(-0.673339\pi\)
−0.518041 + 0.855356i \(0.673339\pi\)
\(12\) 0 0
\(13\) 4.91852 1.36415 0.682076 0.731281i \(-0.261077\pi\)
0.682076 + 0.731281i \(0.261077\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.35482 −1.05620 −0.528099 0.849183i \(-0.677095\pi\)
−0.528099 + 0.849183i \(0.677095\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.91852 0.636874
\(22\) 0 0
\(23\) −6.35482 −1.32507 −0.662536 0.749030i \(-0.730520\pi\)
−0.662536 + 0.749030i \(0.730520\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.27334 −1.35063 −0.675313 0.737531i \(-0.735991\pi\)
−0.675313 + 0.737531i \(0.735991\pi\)
\(30\) 0 0
\(31\) 2.35482 0.422938 0.211469 0.977385i \(-0.432175\pi\)
0.211469 + 0.977385i \(0.432175\pi\)
\(32\) 0 0
\(33\) 3.43630 0.598182
\(34\) 0 0
\(35\) 2.91852 0.493320
\(36\) 0 0
\(37\) 4.91852 0.808600 0.404300 0.914626i \(-0.367515\pi\)
0.404300 + 0.914626i \(0.367515\pi\)
\(38\) 0 0
\(39\) −4.91852 −0.787594
\(40\) 0 0
\(41\) 1.43630 0.224312 0.112156 0.993691i \(-0.464224\pi\)
0.112156 + 0.993691i \(0.464224\pi\)
\(42\) 0 0
\(43\) −6.91852 −1.05506 −0.527532 0.849535i \(-0.676883\pi\)
−0.527532 + 0.849535i \(0.676883\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.35482 0.926946 0.463473 0.886111i \(-0.346603\pi\)
0.463473 + 0.886111i \(0.346603\pi\)
\(48\) 0 0
\(49\) 1.51777 0.216825
\(50\) 0 0
\(51\) 4.35482 0.609797
\(52\) 0 0
\(53\) −4.35482 −0.598180 −0.299090 0.954225i \(-0.596683\pi\)
−0.299090 + 0.954225i \(0.596683\pi\)
\(54\) 0 0
\(55\) 3.43630 0.463350
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −9.83705 −1.28067 −0.640337 0.768094i \(-0.721205\pi\)
−0.640337 + 0.768094i \(0.721205\pi\)
\(60\) 0 0
\(61\) −6.19186 −0.792787 −0.396394 0.918081i \(-0.629738\pi\)
−0.396394 + 0.918081i \(0.629738\pi\)
\(62\) 0 0
\(63\) −2.91852 −0.367699
\(64\) 0 0
\(65\) −4.91852 −0.610068
\(66\) 0 0
\(67\) 8.70964 1.06405 0.532026 0.846728i \(-0.321431\pi\)
0.532026 + 0.846728i \(0.321431\pi\)
\(68\) 0 0
\(69\) 6.35482 0.765030
\(70\) 0 0
\(71\) −6.87259 −0.815627 −0.407813 0.913065i \(-0.633709\pi\)
−0.407813 + 0.913065i \(0.633709\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 10.0289 1.14290
\(78\) 0 0
\(79\) −8.70964 −0.979911 −0.489955 0.871747i \(-0.662987\pi\)
−0.489955 + 0.871747i \(0.662987\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.9015 1.85518 0.927591 0.373599i \(-0.121876\pi\)
0.927591 + 0.373599i \(0.121876\pi\)
\(84\) 0 0
\(85\) 4.35482 0.472346
\(86\) 0 0
\(87\) 7.27334 0.779784
\(88\) 0 0
\(89\) −4.40075 −0.466478 −0.233239 0.972419i \(-0.574933\pi\)
−0.233239 + 0.972419i \(0.574933\pi\)
\(90\) 0 0
\(91\) −14.3548 −1.50479
\(92\) 0 0
\(93\) −2.35482 −0.244183
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 12.9185 1.31168 0.655839 0.754901i \(-0.272315\pi\)
0.655839 + 0.754901i \(0.272315\pi\)
\(98\) 0 0
\(99\) −3.43630 −0.345361
\(100\) 0 0
\(101\) 7.83705 0.779815 0.389908 0.920854i \(-0.372507\pi\)
0.389908 + 0.920854i \(0.372507\pi\)
\(102\) 0 0
\(103\) −8.70964 −0.858186 −0.429093 0.903260i \(-0.641167\pi\)
−0.429093 + 0.903260i \(0.641167\pi\)
\(104\) 0 0
\(105\) −2.91852 −0.284819
\(106\) 0 0
\(107\) −8.70964 −0.841993 −0.420996 0.907062i \(-0.638319\pi\)
−0.420996 + 0.907062i \(0.638319\pi\)
\(108\) 0 0
\(109\) 4.16295 0.398739 0.199369 0.979924i \(-0.436111\pi\)
0.199369 + 0.979924i \(0.436111\pi\)
\(110\) 0 0
\(111\) −4.91852 −0.466846
\(112\) 0 0
\(113\) −6.19186 −0.582482 −0.291241 0.956650i \(-0.594068\pi\)
−0.291241 + 0.956650i \(0.594068\pi\)
\(114\) 0 0
\(115\) 6.35482 0.592590
\(116\) 0 0
\(117\) 4.91852 0.454718
\(118\) 0 0
\(119\) 12.7096 1.16509
\(120\) 0 0
\(121\) 0.808135 0.0734669
\(122\) 0 0
\(123\) −1.43630 −0.129507
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.709639 −0.0629703 −0.0314851 0.999504i \(-0.510024\pi\)
−0.0314851 + 0.999504i \(0.510024\pi\)
\(128\) 0 0
\(129\) 6.91852 0.609142
\(130\) 0 0
\(131\) 9.27334 0.810216 0.405108 0.914269i \(-0.367234\pi\)
0.405108 + 0.914269i \(0.367234\pi\)
\(132\) 0 0
\(133\) −2.91852 −0.253068
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 13.6741 1.16826 0.584128 0.811661i \(-0.301437\pi\)
0.584128 + 0.811661i \(0.301437\pi\)
\(138\) 0 0
\(139\) −5.12741 −0.434901 −0.217450 0.976071i \(-0.569774\pi\)
−0.217450 + 0.976071i \(0.569774\pi\)
\(140\) 0 0
\(141\) −6.35482 −0.535172
\(142\) 0 0
\(143\) −16.9015 −1.41337
\(144\) 0 0
\(145\) 7.27334 0.604018
\(146\) 0 0
\(147\) −1.51777 −0.125184
\(148\) 0 0
\(149\) 18.3837 1.50605 0.753027 0.657990i \(-0.228593\pi\)
0.753027 + 0.657990i \(0.228593\pi\)
\(150\) 0 0
\(151\) 22.0289 1.79269 0.896344 0.443360i \(-0.146214\pi\)
0.896344 + 0.443360i \(0.146214\pi\)
\(152\) 0 0
\(153\) −4.35482 −0.352066
\(154\) 0 0
\(155\) −2.35482 −0.189144
\(156\) 0 0
\(157\) 4.87259 0.388875 0.194438 0.980915i \(-0.437712\pi\)
0.194438 + 0.980915i \(0.437712\pi\)
\(158\) 0 0
\(159\) 4.35482 0.345360
\(160\) 0 0
\(161\) 18.5467 1.46168
\(162\) 0 0
\(163\) −11.6282 −0.910788 −0.455394 0.890290i \(-0.650502\pi\)
−0.455394 + 0.890290i \(0.650502\pi\)
\(164\) 0 0
\(165\) −3.43630 −0.267515
\(166\) 0 0
\(167\) 20.1919 1.56249 0.781247 0.624222i \(-0.214584\pi\)
0.781247 + 0.624222i \(0.214584\pi\)
\(168\) 0 0
\(169\) 11.1919 0.860913
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 21.1563 1.60848 0.804242 0.594301i \(-0.202571\pi\)
0.804242 + 0.594301i \(0.202571\pi\)
\(174\) 0 0
\(175\) −2.91852 −0.220620
\(176\) 0 0
\(177\) 9.83705 0.739398
\(178\) 0 0
\(179\) −8.80150 −0.657855 −0.328927 0.944355i \(-0.606687\pi\)
−0.328927 + 0.944355i \(0.606687\pi\)
\(180\) 0 0
\(181\) −4.87259 −0.362177 −0.181089 0.983467i \(-0.557962\pi\)
−0.181089 + 0.983467i \(0.557962\pi\)
\(182\) 0 0
\(183\) 6.19186 0.457716
\(184\) 0 0
\(185\) −4.91852 −0.361617
\(186\) 0 0
\(187\) 14.9645 1.09431
\(188\) 0 0
\(189\) 2.91852 0.212291
\(190\) 0 0
\(191\) 24.1459 1.74714 0.873569 0.486700i \(-0.161799\pi\)
0.873569 + 0.486700i \(0.161799\pi\)
\(192\) 0 0
\(193\) 2.37184 0.170729 0.0853643 0.996350i \(-0.472795\pi\)
0.0853643 + 0.996350i \(0.472795\pi\)
\(194\) 0 0
\(195\) 4.91852 0.352223
\(196\) 0 0
\(197\) −24.0289 −1.71199 −0.855994 0.516985i \(-0.827054\pi\)
−0.855994 + 0.516985i \(0.827054\pi\)
\(198\) 0 0
\(199\) 26.5467 1.88184 0.940922 0.338623i \(-0.109961\pi\)
0.940922 + 0.338623i \(0.109961\pi\)
\(200\) 0 0
\(201\) −8.70964 −0.614331
\(202\) 0 0
\(203\) 21.2274 1.48987
\(204\) 0 0
\(205\) −1.43630 −0.100315
\(206\) 0 0
\(207\) −6.35482 −0.441690
\(208\) 0 0
\(209\) −3.43630 −0.237694
\(210\) 0 0
\(211\) −21.4193 −1.47456 −0.737282 0.675585i \(-0.763891\pi\)
−0.737282 + 0.675585i \(0.763891\pi\)
\(212\) 0 0
\(213\) 6.87259 0.470902
\(214\) 0 0
\(215\) 6.91852 0.471839
\(216\) 0 0
\(217\) −6.87259 −0.466542
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) −21.4193 −1.44082
\(222\) 0 0
\(223\) 16.7096 1.11896 0.559480 0.828844i \(-0.311001\pi\)
0.559480 + 0.828844i \(0.311001\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.51777 0.299855 0.149928 0.988697i \(-0.452096\pi\)
0.149928 + 0.988697i \(0.452096\pi\)
\(228\) 0 0
\(229\) −15.2274 −1.00626 −0.503128 0.864212i \(-0.667818\pi\)
−0.503128 + 0.864212i \(0.667818\pi\)
\(230\) 0 0
\(231\) −10.0289 −0.659854
\(232\) 0 0
\(233\) 15.7452 1.03150 0.515751 0.856739i \(-0.327513\pi\)
0.515751 + 0.856739i \(0.327513\pi\)
\(234\) 0 0
\(235\) −6.35482 −0.414543
\(236\) 0 0
\(237\) 8.70964 0.565752
\(238\) 0 0
\(239\) 3.43630 0.222276 0.111138 0.993805i \(-0.464551\pi\)
0.111138 + 0.993805i \(0.464551\pi\)
\(240\) 0 0
\(241\) 15.7452 1.01424 0.507118 0.861876i \(-0.330711\pi\)
0.507118 + 0.861876i \(0.330711\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.51777 −0.0969670
\(246\) 0 0
\(247\) 4.91852 0.312958
\(248\) 0 0
\(249\) −16.9015 −1.07109
\(250\) 0 0
\(251\) 2.40075 0.151534 0.0757669 0.997126i \(-0.475859\pi\)
0.0757669 + 0.997126i \(0.475859\pi\)
\(252\) 0 0
\(253\) 21.8370 1.37288
\(254\) 0 0
\(255\) −4.35482 −0.272709
\(256\) 0 0
\(257\) −30.1919 −1.88332 −0.941658 0.336570i \(-0.890733\pi\)
−0.941658 + 0.336570i \(0.890733\pi\)
\(258\) 0 0
\(259\) −14.3548 −0.891965
\(260\) 0 0
\(261\) −7.27334 −0.450209
\(262\) 0 0
\(263\) 27.1563 1.67453 0.837265 0.546797i \(-0.184153\pi\)
0.837265 + 0.546797i \(0.184153\pi\)
\(264\) 0 0
\(265\) 4.35482 0.267514
\(266\) 0 0
\(267\) 4.40075 0.269321
\(268\) 0 0
\(269\) −0.400748 −0.0244341 −0.0122170 0.999925i \(-0.503889\pi\)
−0.0122170 + 0.999925i \(0.503889\pi\)
\(270\) 0 0
\(271\) 7.90814 0.480385 0.240193 0.970725i \(-0.422789\pi\)
0.240193 + 0.970725i \(0.422789\pi\)
\(272\) 0 0
\(273\) 14.3548 0.868793
\(274\) 0 0
\(275\) −3.43630 −0.207216
\(276\) 0 0
\(277\) 11.7452 0.705700 0.352850 0.935680i \(-0.385213\pi\)
0.352850 + 0.935680i \(0.385213\pi\)
\(278\) 0 0
\(279\) 2.35482 0.140979
\(280\) 0 0
\(281\) 17.8200 1.06305 0.531527 0.847041i \(-0.321618\pi\)
0.531527 + 0.847041i \(0.321618\pi\)
\(282\) 0 0
\(283\) −10.5926 −0.629665 −0.314833 0.949147i \(-0.601948\pi\)
−0.314833 + 0.949147i \(0.601948\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −4.19186 −0.247438
\(288\) 0 0
\(289\) 1.96445 0.115556
\(290\) 0 0
\(291\) −12.9185 −0.757297
\(292\) 0 0
\(293\) 2.51777 0.147090 0.0735450 0.997292i \(-0.476569\pi\)
0.0735450 + 0.997292i \(0.476569\pi\)
\(294\) 0 0
\(295\) 9.83705 0.572735
\(296\) 0 0
\(297\) 3.43630 0.199394
\(298\) 0 0
\(299\) −31.2563 −1.80760
\(300\) 0 0
\(301\) 20.1919 1.16384
\(302\) 0 0
\(303\) −7.83705 −0.450226
\(304\) 0 0
\(305\) 6.19186 0.354545
\(306\) 0 0
\(307\) 13.1274 0.749221 0.374610 0.927182i \(-0.377776\pi\)
0.374610 + 0.927182i \(0.377776\pi\)
\(308\) 0 0
\(309\) 8.70964 0.495474
\(310\) 0 0
\(311\) −24.5297 −1.39095 −0.695475 0.718550i \(-0.744806\pi\)
−0.695475 + 0.718550i \(0.744806\pi\)
\(312\) 0 0
\(313\) 3.12741 0.176771 0.0883857 0.996086i \(-0.471829\pi\)
0.0883857 + 0.996086i \(0.471829\pi\)
\(314\) 0 0
\(315\) 2.91852 0.164440
\(316\) 0 0
\(317\) −22.9934 −1.29144 −0.645718 0.763576i \(-0.723442\pi\)
−0.645718 + 0.763576i \(0.723442\pi\)
\(318\) 0 0
\(319\) 24.9934 1.39936
\(320\) 0 0
\(321\) 8.70964 0.486125
\(322\) 0 0
\(323\) −4.35482 −0.242309
\(324\) 0 0
\(325\) 4.91852 0.272831
\(326\) 0 0
\(327\) −4.16295 −0.230212
\(328\) 0 0
\(329\) −18.5467 −1.02251
\(330\) 0 0
\(331\) −4.60963 −0.253368 −0.126684 0.991943i \(-0.540433\pi\)
−0.126684 + 0.991943i \(0.540433\pi\)
\(332\) 0 0
\(333\) 4.91852 0.269533
\(334\) 0 0
\(335\) −8.70964 −0.475858
\(336\) 0 0
\(337\) −25.3023 −1.37830 −0.689151 0.724618i \(-0.742016\pi\)
−0.689151 + 0.724618i \(0.742016\pi\)
\(338\) 0 0
\(339\) 6.19186 0.336296
\(340\) 0 0
\(341\) −8.09186 −0.438199
\(342\) 0 0
\(343\) 16.0000 0.863919
\(344\) 0 0
\(345\) −6.35482 −0.342132
\(346\) 0 0
\(347\) 12.1919 0.654494 0.327247 0.944939i \(-0.393879\pi\)
0.327247 + 0.944939i \(0.393879\pi\)
\(348\) 0 0
\(349\) −2.38373 −0.127598 −0.0637990 0.997963i \(-0.520322\pi\)
−0.0637990 + 0.997963i \(0.520322\pi\)
\(350\) 0 0
\(351\) −4.91852 −0.262531
\(352\) 0 0
\(353\) −6.38373 −0.339772 −0.169886 0.985464i \(-0.554340\pi\)
−0.169886 + 0.985464i \(0.554340\pi\)
\(354\) 0 0
\(355\) 6.87259 0.364759
\(356\) 0 0
\(357\) −12.7096 −0.672665
\(358\) 0 0
\(359\) −9.27334 −0.489428 −0.244714 0.969595i \(-0.578694\pi\)
−0.244714 + 0.969595i \(0.578694\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −0.808135 −0.0424161
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 35.2104 1.83797 0.918984 0.394295i \(-0.129011\pi\)
0.918984 + 0.394295i \(0.129011\pi\)
\(368\) 0 0
\(369\) 1.43630 0.0747706
\(370\) 0 0
\(371\) 12.7096 0.659852
\(372\) 0 0
\(373\) 17.6282 0.912752 0.456376 0.889787i \(-0.349147\pi\)
0.456376 + 0.889787i \(0.349147\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −35.7741 −1.84246
\(378\) 0 0
\(379\) −27.7741 −1.42666 −0.713330 0.700829i \(-0.752814\pi\)
−0.713330 + 0.700829i \(0.752814\pi\)
\(380\) 0 0
\(381\) 0.709639 0.0363559
\(382\) 0 0
\(383\) 34.4548 1.76056 0.880280 0.474455i \(-0.157355\pi\)
0.880280 + 0.474455i \(0.157355\pi\)
\(384\) 0 0
\(385\) −10.0289 −0.511121
\(386\) 0 0
\(387\) −6.91852 −0.351688
\(388\) 0 0
\(389\) −16.6385 −0.843608 −0.421804 0.906687i \(-0.638603\pi\)
−0.421804 + 0.906687i \(0.638603\pi\)
\(390\) 0 0
\(391\) 27.6741 1.39954
\(392\) 0 0
\(393\) −9.27334 −0.467778
\(394\) 0 0
\(395\) 8.70964 0.438229
\(396\) 0 0
\(397\) −5.29036 −0.265516 −0.132758 0.991149i \(-0.542383\pi\)
−0.132758 + 0.991149i \(0.542383\pi\)
\(398\) 0 0
\(399\) 2.91852 0.146109
\(400\) 0 0
\(401\) 13.2022 0.659289 0.329644 0.944105i \(-0.393071\pi\)
0.329644 + 0.944105i \(0.393071\pi\)
\(402\) 0 0
\(403\) 11.5822 0.576952
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −16.9015 −0.837776
\(408\) 0 0
\(409\) −1.19850 −0.0592622 −0.0296311 0.999561i \(-0.509433\pi\)
−0.0296311 + 0.999561i \(0.509433\pi\)
\(410\) 0 0
\(411\) −13.6741 −0.674493
\(412\) 0 0
\(413\) 28.7096 1.41271
\(414\) 0 0
\(415\) −16.9015 −0.829662
\(416\) 0 0
\(417\) 5.12741 0.251090
\(418\) 0 0
\(419\) 2.30889 0.112797 0.0563983 0.998408i \(-0.482038\pi\)
0.0563983 + 0.998408i \(0.482038\pi\)
\(420\) 0 0
\(421\) 36.4548 1.77670 0.888350 0.459167i \(-0.151852\pi\)
0.888350 + 0.459167i \(0.151852\pi\)
\(422\) 0 0
\(423\) 6.35482 0.308982
\(424\) 0 0
\(425\) −4.35482 −0.211240
\(426\) 0 0
\(427\) 18.0711 0.874522
\(428\) 0 0
\(429\) 16.9015 0.816012
\(430\) 0 0
\(431\) 19.5822 0.943243 0.471621 0.881801i \(-0.343669\pi\)
0.471621 + 0.881801i \(0.343669\pi\)
\(432\) 0 0
\(433\) 9.72002 0.467114 0.233557 0.972343i \(-0.424963\pi\)
0.233557 + 0.972343i \(0.424963\pi\)
\(434\) 0 0
\(435\) −7.27334 −0.348730
\(436\) 0 0
\(437\) −6.35482 −0.303992
\(438\) 0 0
\(439\) −13.0355 −0.622153 −0.311076 0.950385i \(-0.600690\pi\)
−0.311076 + 0.950385i \(0.600690\pi\)
\(440\) 0 0
\(441\) 1.51777 0.0722750
\(442\) 0 0
\(443\) 5.31927 0.252726 0.126363 0.991984i \(-0.459670\pi\)
0.126363 + 0.991984i \(0.459670\pi\)
\(444\) 0 0
\(445\) 4.40075 0.208615
\(446\) 0 0
\(447\) −18.3837 −0.869521
\(448\) 0 0
\(449\) −1.85406 −0.0874987 −0.0437494 0.999043i \(-0.513930\pi\)
−0.0437494 + 0.999043i \(0.513930\pi\)
\(450\) 0 0
\(451\) −4.93554 −0.232406
\(452\) 0 0
\(453\) −22.0289 −1.03501
\(454\) 0 0
\(455\) 14.3548 0.672964
\(456\) 0 0
\(457\) −25.5822 −1.19669 −0.598343 0.801240i \(-0.704174\pi\)
−0.598343 + 0.801240i \(0.704174\pi\)
\(458\) 0 0
\(459\) 4.35482 0.203266
\(460\) 0 0
\(461\) −11.7452 −0.547028 −0.273514 0.961868i \(-0.588186\pi\)
−0.273514 + 0.961868i \(0.588186\pi\)
\(462\) 0 0
\(463\) 7.72002 0.358780 0.179390 0.983778i \(-0.442588\pi\)
0.179390 + 0.983778i \(0.442588\pi\)
\(464\) 0 0
\(465\) 2.35482 0.109202
\(466\) 0 0
\(467\) 2.68073 0.124049 0.0620247 0.998075i \(-0.480244\pi\)
0.0620247 + 0.998075i \(0.480244\pi\)
\(468\) 0 0
\(469\) −25.4193 −1.17375
\(470\) 0 0
\(471\) −4.87259 −0.224517
\(472\) 0 0
\(473\) 23.7741 1.09313
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) −4.35482 −0.199393
\(478\) 0 0
\(479\) −12.4718 −0.569853 −0.284927 0.958549i \(-0.591969\pi\)
−0.284927 + 0.958549i \(0.591969\pi\)
\(480\) 0 0
\(481\) 24.1919 1.10305
\(482\) 0 0
\(483\) −18.5467 −0.843903
\(484\) 0 0
\(485\) −12.9185 −0.586600
\(486\) 0 0
\(487\) 33.8370 1.53330 0.766651 0.642064i \(-0.221921\pi\)
0.766651 + 0.642064i \(0.221921\pi\)
\(488\) 0 0
\(489\) 11.6282 0.525844
\(490\) 0 0
\(491\) 14.7267 0.664605 0.332302 0.943173i \(-0.392175\pi\)
0.332302 + 0.943173i \(0.392175\pi\)
\(492\) 0 0
\(493\) 31.6741 1.42653
\(494\) 0 0
\(495\) 3.43630 0.154450
\(496\) 0 0
\(497\) 20.0578 0.899716
\(498\) 0 0
\(499\) −7.19850 −0.322249 −0.161125 0.986934i \(-0.551512\pi\)
−0.161125 + 0.986934i \(0.551512\pi\)
\(500\) 0 0
\(501\) −20.1919 −0.902106
\(502\) 0 0
\(503\) −21.6111 −0.963593 −0.481797 0.876283i \(-0.660016\pi\)
−0.481797 + 0.876283i \(0.660016\pi\)
\(504\) 0 0
\(505\) −7.83705 −0.348744
\(506\) 0 0
\(507\) −11.1919 −0.497048
\(508\) 0 0
\(509\) −4.72666 −0.209505 −0.104753 0.994498i \(-0.533405\pi\)
−0.104753 + 0.994498i \(0.533405\pi\)
\(510\) 0 0
\(511\) −29.1852 −1.29108
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 8.70964 0.383793
\(516\) 0 0
\(517\) −21.8370 −0.960392
\(518\) 0 0
\(519\) −21.1563 −0.928659
\(520\) 0 0
\(521\) 1.43630 0.0629253 0.0314627 0.999505i \(-0.489983\pi\)
0.0314627 + 0.999505i \(0.489983\pi\)
\(522\) 0 0
\(523\) 14.1630 0.619303 0.309651 0.950850i \(-0.399788\pi\)
0.309651 + 0.950850i \(0.399788\pi\)
\(524\) 0 0
\(525\) 2.91852 0.127375
\(526\) 0 0
\(527\) −10.2548 −0.446707
\(528\) 0 0
\(529\) 17.3837 0.755814
\(530\) 0 0
\(531\) −9.83705 −0.426891
\(532\) 0 0
\(533\) 7.06446 0.305996
\(534\) 0 0
\(535\) 8.70964 0.376551
\(536\) 0 0
\(537\) 8.80150 0.379813
\(538\) 0 0
\(539\) −5.21552 −0.224648
\(540\) 0 0
\(541\) −19.4193 −0.834900 −0.417450 0.908700i \(-0.637076\pi\)
−0.417450 + 0.908700i \(0.637076\pi\)
\(542\) 0 0
\(543\) 4.87259 0.209103
\(544\) 0 0
\(545\) −4.16295 −0.178321
\(546\) 0 0
\(547\) 13.1274 0.561287 0.280644 0.959812i \(-0.409452\pi\)
0.280644 + 0.959812i \(0.409452\pi\)
\(548\) 0 0
\(549\) −6.19186 −0.264262
\(550\) 0 0
\(551\) −7.27334 −0.309855
\(552\) 0 0
\(553\) 25.4193 1.08094
\(554\) 0 0
\(555\) 4.91852 0.208780
\(556\) 0 0
\(557\) −0.254813 −0.0107968 −0.00539839 0.999985i \(-0.501718\pi\)
−0.00539839 + 0.999985i \(0.501718\pi\)
\(558\) 0 0
\(559\) −34.0289 −1.43927
\(560\) 0 0
\(561\) −14.9645 −0.631800
\(562\) 0 0
\(563\) 10.4467 0.440275 0.220137 0.975469i \(-0.429349\pi\)
0.220137 + 0.975469i \(0.429349\pi\)
\(564\) 0 0
\(565\) 6.19186 0.260494
\(566\) 0 0
\(567\) −2.91852 −0.122566
\(568\) 0 0
\(569\) −12.6926 −0.532102 −0.266051 0.963959i \(-0.585719\pi\)
−0.266051 + 0.963959i \(0.585719\pi\)
\(570\) 0 0
\(571\) −29.3274 −1.22731 −0.613657 0.789573i \(-0.710302\pi\)
−0.613657 + 0.789573i \(0.710302\pi\)
\(572\) 0 0
\(573\) −24.1459 −1.00871
\(574\) 0 0
\(575\) −6.35482 −0.265014
\(576\) 0 0
\(577\) 41.6401 1.73350 0.866749 0.498745i \(-0.166205\pi\)
0.866749 + 0.498745i \(0.166205\pi\)
\(578\) 0 0
\(579\) −2.37184 −0.0985703
\(580\) 0 0
\(581\) −49.3274 −2.04645
\(582\) 0 0
\(583\) 14.9645 0.619764
\(584\) 0 0
\(585\) −4.91852 −0.203356
\(586\) 0 0
\(587\) 10.0289 0.413937 0.206969 0.978348i \(-0.433640\pi\)
0.206969 + 0.978348i \(0.433640\pi\)
\(588\) 0 0
\(589\) 2.35482 0.0970286
\(590\) 0 0
\(591\) 24.0289 0.988417
\(592\) 0 0
\(593\) −32.4548 −1.33276 −0.666380 0.745612i \(-0.732157\pi\)
−0.666380 + 0.745612i \(0.732157\pi\)
\(594\) 0 0
\(595\) −12.7096 −0.521044
\(596\) 0 0
\(597\) −26.5467 −1.08648
\(598\) 0 0
\(599\) −2.07110 −0.0846227 −0.0423114 0.999104i \(-0.513472\pi\)
−0.0423114 + 0.999104i \(0.513472\pi\)
\(600\) 0 0
\(601\) −9.58223 −0.390867 −0.195434 0.980717i \(-0.562611\pi\)
−0.195434 + 0.980717i \(0.562611\pi\)
\(602\) 0 0
\(603\) 8.70964 0.354684
\(604\) 0 0
\(605\) −0.808135 −0.0328554
\(606\) 0 0
\(607\) 24.8015 1.00666 0.503331 0.864094i \(-0.332108\pi\)
0.503331 + 0.864094i \(0.332108\pi\)
\(608\) 0 0
\(609\) −21.2274 −0.860178
\(610\) 0 0
\(611\) 31.2563 1.26450
\(612\) 0 0
\(613\) 25.5822 1.03326 0.516628 0.856210i \(-0.327187\pi\)
0.516628 + 0.856210i \(0.327187\pi\)
\(614\) 0 0
\(615\) 1.43630 0.0579171
\(616\) 0 0
\(617\) −43.7030 −1.75942 −0.879708 0.475514i \(-0.842262\pi\)
−0.879708 + 0.475514i \(0.842262\pi\)
\(618\) 0 0
\(619\) 1.83705 0.0738371 0.0369185 0.999318i \(-0.488246\pi\)
0.0369185 + 0.999318i \(0.488246\pi\)
\(620\) 0 0
\(621\) 6.35482 0.255010
\(622\) 0 0
\(623\) 12.8437 0.514571
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.43630 0.137232
\(628\) 0 0
\(629\) −21.4193 −0.854043
\(630\) 0 0
\(631\) 43.0223 1.71269 0.856345 0.516404i \(-0.172730\pi\)
0.856345 + 0.516404i \(0.172730\pi\)
\(632\) 0 0
\(633\) 21.4193 0.851340
\(634\) 0 0
\(635\) 0.709639 0.0281612
\(636\) 0 0
\(637\) 7.46521 0.295782
\(638\) 0 0
\(639\) −6.87259 −0.271876
\(640\) 0 0
\(641\) 17.5282 0.692320 0.346160 0.938175i \(-0.387485\pi\)
0.346160 + 0.938175i \(0.387485\pi\)
\(642\) 0 0
\(643\) 47.6860 1.88055 0.940276 0.340414i \(-0.110567\pi\)
0.940276 + 0.340414i \(0.110567\pi\)
\(644\) 0 0
\(645\) −6.91852 −0.272417
\(646\) 0 0
\(647\) −21.6111 −0.849622 −0.424811 0.905282i \(-0.639659\pi\)
−0.424811 + 0.905282i \(0.639659\pi\)
\(648\) 0 0
\(649\) 33.8030 1.32688
\(650\) 0 0
\(651\) 6.87259 0.269358
\(652\) 0 0
\(653\) 28.0289 1.09686 0.548428 0.836198i \(-0.315226\pi\)
0.548428 + 0.836198i \(0.315226\pi\)
\(654\) 0 0
\(655\) −9.27334 −0.362339
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −9.74519 −0.379619 −0.189809 0.981821i \(-0.560787\pi\)
−0.189809 + 0.981821i \(0.560787\pi\)
\(660\) 0 0
\(661\) −16.5467 −0.643591 −0.321796 0.946809i \(-0.604286\pi\)
−0.321796 + 0.946809i \(0.604286\pi\)
\(662\) 0 0
\(663\) 21.4193 0.831856
\(664\) 0 0
\(665\) 2.91852 0.113175
\(666\) 0 0
\(667\) 46.2208 1.78968
\(668\) 0 0
\(669\) −16.7096 −0.646032
\(670\) 0 0
\(671\) 21.2771 0.821393
\(672\) 0 0
\(673\) −24.2667 −0.935413 −0.467706 0.883884i \(-0.654920\pi\)
−0.467706 + 0.883884i \(0.654920\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −24.6807 −0.948557 −0.474279 0.880375i \(-0.657291\pi\)
−0.474279 + 0.880375i \(0.657291\pi\)
\(678\) 0 0
\(679\) −37.7030 −1.44691
\(680\) 0 0
\(681\) −4.51777 −0.173121
\(682\) 0 0
\(683\) −2.96445 −0.113432 −0.0567158 0.998390i \(-0.518063\pi\)
−0.0567158 + 0.998390i \(0.518063\pi\)
\(684\) 0 0
\(685\) −13.6741 −0.522460
\(686\) 0 0
\(687\) 15.2274 0.580962
\(688\) 0 0
\(689\) −21.4193 −0.816009
\(690\) 0 0
\(691\) −39.5822 −1.50578 −0.752890 0.658147i \(-0.771341\pi\)
−0.752890 + 0.658147i \(0.771341\pi\)
\(692\) 0 0
\(693\) 10.0289 0.380967
\(694\) 0 0
\(695\) 5.12741 0.194494
\(696\) 0 0
\(697\) −6.25481 −0.236918
\(698\) 0 0
\(699\) −15.7452 −0.595538
\(700\) 0 0
\(701\) −44.3126 −1.67367 −0.836833 0.547459i \(-0.815595\pi\)
−0.836833 + 0.547459i \(0.815595\pi\)
\(702\) 0 0
\(703\) 4.91852 0.185506
\(704\) 0 0
\(705\) 6.35482 0.239336
\(706\) 0 0
\(707\) −22.8726 −0.860212
\(708\) 0 0
\(709\) 48.3208 1.81473 0.907363 0.420349i \(-0.138092\pi\)
0.907363 + 0.420349i \(0.138092\pi\)
\(710\) 0 0
\(711\) −8.70964 −0.326637
\(712\) 0 0
\(713\) −14.9645 −0.560423
\(714\) 0 0
\(715\) 16.9015 0.632080
\(716\) 0 0
\(717\) −3.43630 −0.128331
\(718\) 0 0
\(719\) 12.4718 0.465121 0.232561 0.972582i \(-0.425290\pi\)
0.232561 + 0.972582i \(0.425290\pi\)
\(720\) 0 0
\(721\) 25.4193 0.946663
\(722\) 0 0
\(723\) −15.7452 −0.585570
\(724\) 0 0
\(725\) −7.27334 −0.270125
\(726\) 0 0
\(727\) −7.72002 −0.286320 −0.143160 0.989700i \(-0.545726\pi\)
−0.143160 + 0.989700i \(0.545726\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.1289 1.11436
\(732\) 0 0
\(733\) −0.964452 −0.0356228 −0.0178114 0.999841i \(-0.505670\pi\)
−0.0178114 + 0.999841i \(0.505670\pi\)
\(734\) 0 0
\(735\) 1.51777 0.0559839
\(736\) 0 0
\(737\) −29.9289 −1.10245
\(738\) 0 0
\(739\) 5.41928 0.199351 0.0996757 0.995020i \(-0.468219\pi\)
0.0996757 + 0.995020i \(0.468219\pi\)
\(740\) 0 0
\(741\) −4.91852 −0.180686
\(742\) 0 0
\(743\) 9.03555 0.331482 0.165741 0.986169i \(-0.446998\pi\)
0.165741 + 0.986169i \(0.446998\pi\)
\(744\) 0 0
\(745\) −18.3837 −0.673528
\(746\) 0 0
\(747\) 16.9015 0.618394
\(748\) 0 0
\(749\) 25.4193 0.928800
\(750\) 0 0
\(751\) −22.0289 −0.803846 −0.401923 0.915673i \(-0.631658\pi\)
−0.401923 + 0.915673i \(0.631658\pi\)
\(752\) 0 0
\(753\) −2.40075 −0.0874881
\(754\) 0 0
\(755\) −22.0289 −0.801714
\(756\) 0 0
\(757\) −39.0015 −1.41753 −0.708767 0.705443i \(-0.750748\pi\)
−0.708767 + 0.705443i \(0.750748\pi\)
\(758\) 0 0
\(759\) −21.8370 −0.792635
\(760\) 0 0
\(761\) −46.3837 −1.68141 −0.840704 0.541494i \(-0.817859\pi\)
−0.840704 + 0.541494i \(0.817859\pi\)
\(762\) 0 0
\(763\) −12.1497 −0.439848
\(764\) 0 0
\(765\) 4.35482 0.157449
\(766\) 0 0
\(767\) −48.3837 −1.74704
\(768\) 0 0
\(769\) −43.7452 −1.57749 −0.788746 0.614719i \(-0.789269\pi\)
−0.788746 + 0.614719i \(0.789269\pi\)
\(770\) 0 0
\(771\) 30.1919 1.08733
\(772\) 0 0
\(773\) −3.13555 −0.112778 −0.0563890 0.998409i \(-0.517959\pi\)
−0.0563890 + 0.998409i \(0.517959\pi\)
\(774\) 0 0
\(775\) 2.35482 0.0845876
\(776\) 0 0
\(777\) 14.3548 0.514976
\(778\) 0 0
\(779\) 1.43630 0.0514607
\(780\) 0 0
\(781\) 23.6163 0.845057
\(782\) 0 0
\(783\) 7.27334 0.259928
\(784\) 0 0
\(785\) −4.87259 −0.173910
\(786\) 0 0
\(787\) 23.5822 0.840616 0.420308 0.907382i \(-0.361922\pi\)
0.420308 + 0.907382i \(0.361922\pi\)
\(788\) 0 0
\(789\) −27.1563 −0.966790
\(790\) 0 0
\(791\) 18.0711 0.642534
\(792\) 0 0
\(793\) −30.4548 −1.08148
\(794\) 0 0
\(795\) −4.35482 −0.154450
\(796\) 0 0
\(797\) 7.22741 0.256008 0.128004 0.991774i \(-0.459143\pi\)
0.128004 + 0.991774i \(0.459143\pi\)
\(798\) 0 0
\(799\) −27.6741 −0.979039
\(800\) 0 0
\(801\) −4.40075 −0.155493
\(802\) 0 0
\(803\) −34.3630 −1.21264
\(804\) 0 0
\(805\) −18.5467 −0.653685
\(806\) 0 0
\(807\) 0.400748 0.0141070
\(808\) 0 0
\(809\) 42.3837 1.49013 0.745066 0.666990i \(-0.232418\pi\)
0.745066 + 0.666990i \(0.232418\pi\)
\(810\) 0 0
\(811\) −41.0934 −1.44298 −0.721492 0.692423i \(-0.756543\pi\)
−0.721492 + 0.692423i \(0.756543\pi\)
\(812\) 0 0
\(813\) −7.90814 −0.277351
\(814\) 0 0
\(815\) 11.6282 0.407317
\(816\) 0 0
\(817\) −6.91852 −0.242048
\(818\) 0 0
\(819\) −14.3548 −0.501598
\(820\) 0 0
\(821\) 13.9660 0.487415 0.243708 0.969849i \(-0.421636\pi\)
0.243708 + 0.969849i \(0.421636\pi\)
\(822\) 0 0
\(823\) −22.4089 −0.781125 −0.390563 0.920576i \(-0.627719\pi\)
−0.390563 + 0.920576i \(0.627719\pi\)
\(824\) 0 0
\(825\) 3.43630 0.119636
\(826\) 0 0
\(827\) −35.8660 −1.24718 −0.623591 0.781751i \(-0.714327\pi\)
−0.623591 + 0.781751i \(0.714327\pi\)
\(828\) 0 0
\(829\) −38.2919 −1.32993 −0.664966 0.746874i \(-0.731554\pi\)
−0.664966 + 0.746874i \(0.731554\pi\)
\(830\) 0 0
\(831\) −11.7452 −0.407436
\(832\) 0 0
\(833\) −6.60963 −0.229010
\(834\) 0 0
\(835\) −20.1919 −0.698768
\(836\) 0 0
\(837\) −2.35482 −0.0813945
\(838\) 0 0
\(839\) −1.12741 −0.0389224 −0.0194612 0.999811i \(-0.506195\pi\)
−0.0194612 + 0.999811i \(0.506195\pi\)
\(840\) 0 0
\(841\) 23.9015 0.824190
\(842\) 0 0
\(843\) −17.8200 −0.613754
\(844\) 0 0
\(845\) −11.1919 −0.385012
\(846\) 0 0
\(847\) −2.35856 −0.0810411
\(848\) 0 0
\(849\) 10.5926 0.363538
\(850\) 0 0
\(851\) −31.2563 −1.07145
\(852\) 0 0
\(853\) −42.6756 −1.46118 −0.730592 0.682814i \(-0.760756\pi\)
−0.730592 + 0.682814i \(0.760756\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 57.4482 1.96239 0.981196 0.193012i \(-0.0618257\pi\)
0.981196 + 0.193012i \(0.0618257\pi\)
\(858\) 0 0
\(859\) 6.45483 0.220236 0.110118 0.993919i \(-0.464877\pi\)
0.110118 + 0.993919i \(0.464877\pi\)
\(860\) 0 0
\(861\) 4.19186 0.142858
\(862\) 0 0
\(863\) 17.8030 0.606021 0.303011 0.952987i \(-0.402008\pi\)
0.303011 + 0.952987i \(0.402008\pi\)
\(864\) 0 0
\(865\) −21.1563 −0.719336
\(866\) 0 0
\(867\) −1.96445 −0.0667163
\(868\) 0 0
\(869\) 29.9289 1.01527
\(870\) 0 0
\(871\) 42.8386 1.45153
\(872\) 0 0
\(873\) 12.9185 0.437226
\(874\) 0 0
\(875\) 2.91852 0.0986641
\(876\) 0 0
\(877\) −39.4312 −1.33150 −0.665748 0.746177i \(-0.731887\pi\)
−0.665748 + 0.746177i \(0.731887\pi\)
\(878\) 0 0
\(879\) −2.51777 −0.0849224
\(880\) 0 0
\(881\) 14.8015 0.498675 0.249338 0.968417i \(-0.419787\pi\)
0.249338 + 0.968417i \(0.419787\pi\)
\(882\) 0 0
\(883\) −1.55706 −0.0523994 −0.0261997 0.999657i \(-0.508341\pi\)
−0.0261997 + 0.999657i \(0.508341\pi\)
\(884\) 0 0
\(885\) −9.83705 −0.330669
\(886\) 0 0
\(887\) −4.32591 −0.145250 −0.0726249 0.997359i \(-0.523138\pi\)
−0.0726249 + 0.997359i \(0.523138\pi\)
\(888\) 0 0
\(889\) 2.07110 0.0694624
\(890\) 0 0
\(891\) −3.43630 −0.115120
\(892\) 0 0
\(893\) 6.35482 0.212656
\(894\) 0 0
\(895\) 8.80150 0.294202
\(896\) 0 0
\(897\) 31.2563 1.04362
\(898\) 0 0
\(899\) −17.1274 −0.571231
\(900\) 0 0
\(901\) 18.9645 0.631797
\(902\) 0 0
\(903\) −20.1919 −0.671943
\(904\) 0 0
\(905\) 4.87259 0.161970
\(906\) 0 0
\(907\) −26.2208 −0.870647 −0.435323 0.900274i \(-0.643366\pi\)
−0.435323 + 0.900274i \(0.643366\pi\)
\(908\) 0 0
\(909\) 7.83705 0.259938
\(910\) 0 0
\(911\) 32.3837 1.07292 0.536460 0.843925i \(-0.319761\pi\)
0.536460 + 0.843925i \(0.319761\pi\)
\(912\) 0 0
\(913\) −58.0786 −1.92212
\(914\) 0 0
\(915\) −6.19186 −0.204697
\(916\) 0 0
\(917\) −27.0645 −0.893747
\(918\) 0 0
\(919\) 6.78074 0.223676 0.111838 0.993726i \(-0.464326\pi\)
0.111838 + 0.993726i \(0.464326\pi\)
\(920\) 0 0
\(921\) −13.1274 −0.432563
\(922\) 0 0
\(923\) −33.8030 −1.11264
\(924\) 0 0
\(925\) 4.91852 0.161720
\(926\) 0 0
\(927\) −8.70964 −0.286062
\(928\) 0 0
\(929\) 26.4756 0.868636 0.434318 0.900760i \(-0.356989\pi\)
0.434318 + 0.900760i \(0.356989\pi\)
\(930\) 0 0
\(931\) 1.51777 0.0497430
\(932\) 0 0
\(933\) 24.5297 0.803065
\(934\) 0 0
\(935\) −14.9645 −0.489390
\(936\) 0 0
\(937\) 37.9660 1.24029 0.620147 0.784486i \(-0.287073\pi\)
0.620147 + 0.784486i \(0.287073\pi\)
\(938\) 0 0
\(939\) −3.12741 −0.102059
\(940\) 0 0
\(941\) 34.4378 1.12264 0.561320 0.827599i \(-0.310294\pi\)
0.561320 + 0.827599i \(0.310294\pi\)
\(942\) 0 0
\(943\) −9.12741 −0.297229
\(944\) 0 0
\(945\) −2.91852 −0.0949395
\(946\) 0 0
\(947\) 0.517774 0.0168254 0.00841270 0.999965i \(-0.497322\pi\)
0.00841270 + 0.999965i \(0.497322\pi\)
\(948\) 0 0
\(949\) 49.1852 1.59662
\(950\) 0 0
\(951\) 22.9934 0.745611
\(952\) 0 0
\(953\) −29.3563 −0.950945 −0.475472 0.879731i \(-0.657723\pi\)
−0.475472 + 0.879731i \(0.657723\pi\)
\(954\) 0 0
\(955\) −24.1459 −0.781344
\(956\) 0 0
\(957\) −24.9934 −0.807921
\(958\) 0 0
\(959\) −39.9081 −1.28870
\(960\) 0 0
\(961\) −25.4548 −0.821123
\(962\) 0 0
\(963\) −8.70964 −0.280664
\(964\) 0 0
\(965\) −2.37184 −0.0763522
\(966\) 0 0
\(967\) 42.2667 1.35921 0.679603 0.733580i \(-0.262152\pi\)
0.679603 + 0.733580i \(0.262152\pi\)
\(968\) 0 0
\(969\) 4.35482 0.139897
\(970\) 0 0
\(971\) −43.3482 −1.39111 −0.695555 0.718473i \(-0.744841\pi\)
−0.695555 + 0.718473i \(0.744841\pi\)
\(972\) 0 0
\(973\) 14.9645 0.479738
\(974\) 0 0
\(975\) −4.91852 −0.157519
\(976\) 0 0
\(977\) 54.8096 1.75352 0.876758 0.480932i \(-0.159702\pi\)
0.876758 + 0.480932i \(0.159702\pi\)
\(978\) 0 0
\(979\) 15.1223 0.483310
\(980\) 0 0
\(981\) 4.16295 0.132913
\(982\) 0 0
\(983\) 6.64669 0.211996 0.105998 0.994366i \(-0.466196\pi\)
0.105998 + 0.994366i \(0.466196\pi\)
\(984\) 0 0
\(985\) 24.0289 0.765625
\(986\) 0 0
\(987\) 18.5467 0.590347
\(988\) 0 0
\(989\) 43.9660 1.39804
\(990\) 0 0
\(991\) 23.2904 0.739843 0.369921 0.929063i \(-0.379385\pi\)
0.369921 + 0.929063i \(0.379385\pi\)
\(992\) 0 0
\(993\) 4.60963 0.146282
\(994\) 0 0
\(995\) −26.5467 −0.841586
\(996\) 0 0
\(997\) 4.22077 0.133673 0.0668366 0.997764i \(-0.478709\pi\)
0.0668366 + 0.997764i \(0.478709\pi\)
\(998\) 0 0
\(999\) −4.91852 −0.155615
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 4560.2.a.bq.1.1 3
4.3 odd 2 2280.2.a.t.1.3 3
12.11 even 2 6840.2.a.bn.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.t.1.3 3 4.3 odd 2
4560.2.a.bq.1.1 3 1.1 even 1 trivial
6840.2.a.bn.1.3 3 12.11 even 2