Properties

Label 4560.2.a.bo.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.64575 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -1.64575 q^{7} +1.00000 q^{9} -0.354249 q^{11} -0.354249 q^{13} +1.00000 q^{15} -4.00000 q^{17} +1.00000 q^{19} -1.64575 q^{21} -9.29150 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.93725 q^{29} -6.00000 q^{31} -0.354249 q^{33} -1.64575 q^{35} +3.64575 q^{37} -0.354249 q^{39} -9.64575 q^{41} -5.64575 q^{43} +1.00000 q^{45} +1.29150 q^{47} -4.29150 q^{49} -4.00000 q^{51} +11.2915 q^{53} -0.354249 q^{55} +1.00000 q^{57} -11.2915 q^{59} -11.2915 q^{61} -1.64575 q^{63} -0.354249 q^{65} +6.58301 q^{67} -9.29150 q^{69} -7.29150 q^{71} +10.0000 q^{73} +1.00000 q^{75} +0.583005 q^{77} -6.58301 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} +8.93725 q^{87} -16.9373 q^{89} +0.583005 q^{91} -6.00000 q^{93} +1.00000 q^{95} -2.93725 q^{97} -0.354249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} - 6 q^{39} - 14 q^{41} - 6 q^{43} + 2 q^{45} - 8 q^{47} + 2 q^{49} - 8 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 12 q^{61} + 2 q^{63} - 6 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 20 q^{73} + 2 q^{75} - 20 q^{77} + 8 q^{79} + 2 q^{81} - 12 q^{83} - 8 q^{85} + 2 q^{87} - 18 q^{89} - 20 q^{91} - 12 q^{93} + 2 q^{95} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(7\) −1.64575 −0.622036 −0.311018 0.950404i \(-0.600670\pi\)
−0.311018 + 0.950404i \(0.600670\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.354249 −0.106810 −0.0534050 0.998573i \(-0.517007\pi\)
−0.0534050 + 0.998573i \(0.517007\pi\)
\(12\) 0 0
\(13\) −0.354249 −0.0982509 −0.0491255 0.998793i \(-0.515643\pi\)
−0.0491255 + 0.998793i \(0.515643\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.64575 −0.359132
\(22\) 0 0
\(23\) −9.29150 −1.93741 −0.968706 0.248211i \(-0.920158\pi\)
−0.968706 + 0.248211i \(0.920158\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.93725 1.65961 0.829803 0.558056i \(-0.188453\pi\)
0.829803 + 0.558056i \(0.188453\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −0.354249 −0.0616668
\(34\) 0 0
\(35\) −1.64575 −0.278183
\(36\) 0 0
\(37\) 3.64575 0.599358 0.299679 0.954040i \(-0.403120\pi\)
0.299679 + 0.954040i \(0.403120\pi\)
\(38\) 0 0
\(39\) −0.354249 −0.0567252
\(40\) 0 0
\(41\) −9.64575 −1.50641 −0.753207 0.657784i \(-0.771494\pi\)
−0.753207 + 0.657784i \(0.771494\pi\)
\(42\) 0 0
\(43\) −5.64575 −0.860969 −0.430485 0.902598i \(-0.641657\pi\)
−0.430485 + 0.902598i \(0.641657\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.29150 0.188385 0.0941925 0.995554i \(-0.469973\pi\)
0.0941925 + 0.995554i \(0.469973\pi\)
\(48\) 0 0
\(49\) −4.29150 −0.613072
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 11.2915 1.55101 0.775504 0.631343i \(-0.217496\pi\)
0.775504 + 0.631343i \(0.217496\pi\)
\(54\) 0 0
\(55\) −0.354249 −0.0477669
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −11.2915 −1.47003 −0.735014 0.678052i \(-0.762825\pi\)
−0.735014 + 0.678052i \(0.762825\pi\)
\(60\) 0 0
\(61\) −11.2915 −1.44573 −0.722864 0.690990i \(-0.757175\pi\)
−0.722864 + 0.690990i \(0.757175\pi\)
\(62\) 0 0
\(63\) −1.64575 −0.207345
\(64\) 0 0
\(65\) −0.354249 −0.0439391
\(66\) 0 0
\(67\) 6.58301 0.804242 0.402121 0.915587i \(-0.368273\pi\)
0.402121 + 0.915587i \(0.368273\pi\)
\(68\) 0 0
\(69\) −9.29150 −1.11857
\(70\) 0 0
\(71\) −7.29150 −0.865342 −0.432671 0.901552i \(-0.642429\pi\)
−0.432671 + 0.901552i \(0.642429\pi\)
\(72\) 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
\(77\) 0.583005 0.0664396
\(78\) 0 0
\(79\) −6.58301 −0.740646 −0.370323 0.928903i \(-0.620753\pi\)
−0.370323 + 0.928903i \(0.620753\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 8.93725 0.958174
\(88\) 0 0
\(89\) −16.9373 −1.79535 −0.897673 0.440663i \(-0.854743\pi\)
−0.897673 + 0.440663i \(0.854743\pi\)
\(90\) 0 0
\(91\) 0.583005 0.0611156
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.93725 −0.298233 −0.149116 0.988820i \(-0.547643\pi\)
−0.149116 + 0.988820i \(0.547643\pi\)
\(98\) 0 0
\(99\) −0.354249 −0.0356033
\(100\) 0 0
\(101\) 9.29150 0.924539 0.462270 0.886739i \(-0.347035\pi\)
0.462270 + 0.886739i \(0.347035\pi\)
\(102\) 0 0
\(103\) 10.5830 1.04277 0.521387 0.853320i \(-0.325415\pi\)
0.521387 + 0.853320i \(0.325415\pi\)
\(104\) 0 0
\(105\) −1.64575 −0.160609
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 5.29150 0.506834 0.253417 0.967357i \(-0.418446\pi\)
0.253417 + 0.967357i \(0.418446\pi\)
\(110\) 0 0
\(111\) 3.64575 0.346039
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −9.29150 −0.866437
\(116\) 0 0
\(117\) −0.354249 −0.0327503
\(118\) 0 0
\(119\) 6.58301 0.603463
\(120\) 0 0
\(121\) −10.8745 −0.988592
\(122\) 0 0
\(123\) −9.64575 −0.869728
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) −5.64575 −0.497081
\(130\) 0 0
\(131\) 14.2288 1.24317 0.621586 0.783346i \(-0.286489\pi\)
0.621586 + 0.783346i \(0.286489\pi\)
\(132\) 0 0
\(133\) −1.64575 −0.142705
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −17.8745 −1.51610 −0.758048 0.652199i \(-0.773847\pi\)
−0.758048 + 0.652199i \(0.773847\pi\)
\(140\) 0 0
\(141\) 1.29150 0.108764
\(142\) 0 0
\(143\) 0.125492 0.0104942
\(144\) 0 0
\(145\) 8.93725 0.742199
\(146\) 0 0
\(147\) −4.29150 −0.353957
\(148\) 0 0
\(149\) −20.5830 −1.68623 −0.843113 0.537737i \(-0.819279\pi\)
−0.843113 + 0.537737i \(0.819279\pi\)
\(150\) 0 0
\(151\) 8.58301 0.698475 0.349238 0.937034i \(-0.386441\pi\)
0.349238 + 0.937034i \(0.386441\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −13.2915 −1.06078 −0.530389 0.847755i \(-0.677954\pi\)
−0.530389 + 0.847755i \(0.677954\pi\)
\(158\) 0 0
\(159\) 11.2915 0.895474
\(160\) 0 0
\(161\) 15.2915 1.20514
\(162\) 0 0
\(163\) 2.35425 0.184399 0.0921995 0.995741i \(-0.470610\pi\)
0.0921995 + 0.995741i \(0.470610\pi\)
\(164\) 0 0
\(165\) −0.354249 −0.0275782
\(166\) 0 0
\(167\) 21.2915 1.64759 0.823793 0.566891i \(-0.191854\pi\)
0.823793 + 0.566891i \(0.191854\pi\)
\(168\) 0 0
\(169\) −12.8745 −0.990347
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −1.64575 −0.124407
\(176\) 0 0
\(177\) −11.2915 −0.848721
\(178\) 0 0
\(179\) 7.29150 0.544992 0.272496 0.962157i \(-0.412151\pi\)
0.272496 + 0.962157i \(0.412151\pi\)
\(180\) 0 0
\(181\) 17.2915 1.28527 0.642634 0.766174i \(-0.277842\pi\)
0.642634 + 0.766174i \(0.277842\pi\)
\(182\) 0 0
\(183\) −11.2915 −0.834692
\(184\) 0 0
\(185\) 3.64575 0.268041
\(186\) 0 0
\(187\) 1.41699 0.103621
\(188\) 0 0
\(189\) −1.64575 −0.119711
\(190\) 0 0
\(191\) −6.22876 −0.450697 −0.225349 0.974278i \(-0.572352\pi\)
−0.225349 + 0.974278i \(0.572352\pi\)
\(192\) 0 0
\(193\) −11.6458 −0.838280 −0.419140 0.907922i \(-0.637668\pi\)
−0.419140 + 0.907922i \(0.637668\pi\)
\(194\) 0 0
\(195\) −0.354249 −0.0253683
\(196\) 0 0
\(197\) −25.1660 −1.79300 −0.896502 0.443040i \(-0.853900\pi\)
−0.896502 + 0.443040i \(0.853900\pi\)
\(198\) 0 0
\(199\) −7.29150 −0.516881 −0.258440 0.966027i \(-0.583209\pi\)
−0.258440 + 0.966027i \(0.583209\pi\)
\(200\) 0 0
\(201\) 6.58301 0.464329
\(202\) 0 0
\(203\) −14.7085 −1.03233
\(204\) 0 0
\(205\) −9.64575 −0.673688
\(206\) 0 0
\(207\) −9.29150 −0.645804
\(208\) 0 0
\(209\) −0.354249 −0.0245039
\(210\) 0 0
\(211\) −2.58301 −0.177821 −0.0889107 0.996040i \(-0.528339\pi\)
−0.0889107 + 0.996040i \(0.528339\pi\)
\(212\) 0 0
\(213\) −7.29150 −0.499606
\(214\) 0 0
\(215\) −5.64575 −0.385037
\(216\) 0 0
\(217\) 9.87451 0.670325
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 1.41699 0.0953174
\(222\) 0 0
\(223\) −2.58301 −0.172971 −0.0864854 0.996253i \(-0.527564\pi\)
−0.0864854 + 0.996253i \(0.527564\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 10.7085 0.710748 0.355374 0.934724i \(-0.384354\pi\)
0.355374 + 0.934724i \(0.384354\pi\)
\(228\) 0 0
\(229\) 8.70850 0.575474 0.287737 0.957710i \(-0.407097\pi\)
0.287737 + 0.957710i \(0.407097\pi\)
\(230\) 0 0
\(231\) 0.583005 0.0383589
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 1.29150 0.0842483
\(236\) 0 0
\(237\) −6.58301 −0.427612
\(238\) 0 0
\(239\) −15.6458 −1.01204 −0.506020 0.862522i \(-0.668884\pi\)
−0.506020 + 0.862522i \(0.668884\pi\)
\(240\) 0 0
\(241\) 16.5830 1.06821 0.534103 0.845420i \(-0.320650\pi\)
0.534103 + 0.845420i \(0.320650\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.29150 −0.274174
\(246\) 0 0
\(247\) −0.354249 −0.0225403
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 16.3542 1.03227 0.516136 0.856507i \(-0.327370\pi\)
0.516136 + 0.856507i \(0.327370\pi\)
\(252\) 0 0
\(253\) 3.29150 0.206935
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) −5.41699 −0.337903 −0.168951 0.985624i \(-0.554038\pi\)
−0.168951 + 0.985624i \(0.554038\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 8.93725 0.553202
\(262\) 0 0
\(263\) −4.58301 −0.282600 −0.141300 0.989967i \(-0.545128\pi\)
−0.141300 + 0.989967i \(0.545128\pi\)
\(264\) 0 0
\(265\) 11.2915 0.693631
\(266\) 0 0
\(267\) −16.9373 −1.03654
\(268\) 0 0
\(269\) 4.22876 0.257832 0.128916 0.991656i \(-0.458850\pi\)
0.128916 + 0.991656i \(0.458850\pi\)
\(270\) 0 0
\(271\) 4.70850 0.286021 0.143010 0.989721i \(-0.454322\pi\)
0.143010 + 0.989721i \(0.454322\pi\)
\(272\) 0 0
\(273\) 0.583005 0.0352851
\(274\) 0 0
\(275\) −0.354249 −0.0213620
\(276\) 0 0
\(277\) 0.583005 0.0350294 0.0175147 0.999847i \(-0.494425\pi\)
0.0175147 + 0.999847i \(0.494425\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −32.2288 −1.92261 −0.961303 0.275493i \(-0.911159\pi\)
−0.961303 + 0.275493i \(0.911159\pi\)
\(282\) 0 0
\(283\) 11.5203 0.684808 0.342404 0.939553i \(-0.388759\pi\)
0.342404 + 0.939553i \(0.388759\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 15.8745 0.937043
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.93725 −0.172185
\(292\) 0 0
\(293\) 5.41699 0.316464 0.158232 0.987402i \(-0.449421\pi\)
0.158232 + 0.987402i \(0.449421\pi\)
\(294\) 0 0
\(295\) −11.2915 −0.657417
\(296\) 0 0
\(297\) −0.354249 −0.0205556
\(298\) 0 0
\(299\) 3.29150 0.190353
\(300\) 0 0
\(301\) 9.29150 0.535553
\(302\) 0 0
\(303\) 9.29150 0.533783
\(304\) 0 0
\(305\) −11.2915 −0.646550
\(306\) 0 0
\(307\) 24.4575 1.39586 0.697932 0.716164i \(-0.254104\pi\)
0.697932 + 0.716164i \(0.254104\pi\)
\(308\) 0 0
\(309\) 10.5830 0.602046
\(310\) 0 0
\(311\) −22.9373 −1.30065 −0.650326 0.759655i \(-0.725368\pi\)
−0.650326 + 0.759655i \(0.725368\pi\)
\(312\) 0 0
\(313\) 17.2915 0.977374 0.488687 0.872459i \(-0.337476\pi\)
0.488687 + 0.872459i \(0.337476\pi\)
\(314\) 0 0
\(315\) −1.64575 −0.0927276
\(316\) 0 0
\(317\) 20.4575 1.14901 0.574504 0.818502i \(-0.305195\pi\)
0.574504 + 0.818502i \(0.305195\pi\)
\(318\) 0 0
\(319\) −3.16601 −0.177263
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −0.354249 −0.0196502
\(326\) 0 0
\(327\) 5.29150 0.292621
\(328\) 0 0
\(329\) −2.12549 −0.117182
\(330\) 0 0
\(331\) −11.4170 −0.627535 −0.313767 0.949500i \(-0.601591\pi\)
−0.313767 + 0.949500i \(0.601591\pi\)
\(332\) 0 0
\(333\) 3.64575 0.199786
\(334\) 0 0
\(335\) 6.58301 0.359668
\(336\) 0 0
\(337\) 14.9373 0.813684 0.406842 0.913499i \(-0.366630\pi\)
0.406842 + 0.913499i \(0.366630\pi\)
\(338\) 0 0
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 2.12549 0.115102
\(342\) 0 0
\(343\) 18.5830 1.00339
\(344\) 0 0
\(345\) −9.29150 −0.500238
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) 23.1660 1.24005 0.620024 0.784583i \(-0.287123\pi\)
0.620024 + 0.784583i \(0.287123\pi\)
\(350\) 0 0
\(351\) −0.354249 −0.0189084
\(352\) 0 0
\(353\) 0.583005 0.0310302 0.0155151 0.999880i \(-0.495061\pi\)
0.0155151 + 0.999880i \(0.495061\pi\)
\(354\) 0 0
\(355\) −7.29150 −0.386993
\(356\) 0 0
\(357\) 6.58301 0.348410
\(358\) 0 0
\(359\) 36.1033 1.90546 0.952729 0.303822i \(-0.0982629\pi\)
0.952729 + 0.303822i \(0.0982629\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −10.8745 −0.570764
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −1.64575 −0.0859075 −0.0429538 0.999077i \(-0.513677\pi\)
−0.0429538 + 0.999077i \(0.513677\pi\)
\(368\) 0 0
\(369\) −9.64575 −0.502138
\(370\) 0 0
\(371\) −18.5830 −0.964782
\(372\) 0 0
\(373\) 5.06275 0.262139 0.131070 0.991373i \(-0.458159\pi\)
0.131070 + 0.991373i \(0.458159\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −3.16601 −0.163058
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 18.5830 0.949547 0.474774 0.880108i \(-0.342530\pi\)
0.474774 + 0.880108i \(0.342530\pi\)
\(384\) 0 0
\(385\) 0.583005 0.0297127
\(386\) 0 0
\(387\) −5.64575 −0.286990
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 37.1660 1.87957
\(392\) 0 0
\(393\) 14.2288 0.717746
\(394\) 0 0
\(395\) −6.58301 −0.331227
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −1.64575 −0.0823906
\(400\) 0 0
\(401\) −4.93725 −0.246555 −0.123277 0.992372i \(-0.539340\pi\)
−0.123277 + 0.992372i \(0.539340\pi\)
\(402\) 0 0
\(403\) 2.12549 0.105878
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −1.29150 −0.0640174
\(408\) 0 0
\(409\) −6.70850 −0.331714 −0.165857 0.986150i \(-0.553039\pi\)
−0.165857 + 0.986150i \(0.553039\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 18.5830 0.914410
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −17.8745 −0.875318
\(418\) 0 0
\(419\) −38.9373 −1.90221 −0.951105 0.308869i \(-0.900050\pi\)
−0.951105 + 0.308869i \(0.900050\pi\)
\(420\) 0 0
\(421\) 28.5830 1.39305 0.696525 0.717532i \(-0.254728\pi\)
0.696525 + 0.717532i \(0.254728\pi\)
\(422\) 0 0
\(423\) 1.29150 0.0627950
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 18.5830 0.899295
\(428\) 0 0
\(429\) 0.125492 0.00605882
\(430\) 0 0
\(431\) −3.29150 −0.158546 −0.0792731 0.996853i \(-0.525260\pi\)
−0.0792731 + 0.996853i \(0.525260\pi\)
\(432\) 0 0
\(433\) −27.6458 −1.32857 −0.664285 0.747479i \(-0.731264\pi\)
−0.664285 + 0.747479i \(0.731264\pi\)
\(434\) 0 0
\(435\) 8.93725 0.428509
\(436\) 0 0
\(437\) −9.29150 −0.444473
\(438\) 0 0
\(439\) 1.41699 0.0676295 0.0338147 0.999428i \(-0.489234\pi\)
0.0338147 + 0.999428i \(0.489234\pi\)
\(440\) 0 0
\(441\) −4.29150 −0.204357
\(442\) 0 0
\(443\) −26.7085 −1.26896 −0.634480 0.772940i \(-0.718786\pi\)
−0.634480 + 0.772940i \(0.718786\pi\)
\(444\) 0 0
\(445\) −16.9373 −0.802903
\(446\) 0 0
\(447\) −20.5830 −0.973543
\(448\) 0 0
\(449\) −36.2288 −1.70974 −0.854870 0.518842i \(-0.826363\pi\)
−0.854870 + 0.518842i \(0.826363\pi\)
\(450\) 0 0
\(451\) 3.41699 0.160900
\(452\) 0 0
\(453\) 8.58301 0.403265
\(454\) 0 0
\(455\) 0.583005 0.0273317
\(456\) 0 0
\(457\) 0.125492 0.00587027 0.00293514 0.999996i \(-0.499066\pi\)
0.00293514 + 0.999996i \(0.499066\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −37.7490 −1.75815 −0.879073 0.476686i \(-0.841838\pi\)
−0.879073 + 0.476686i \(0.841838\pi\)
\(462\) 0 0
\(463\) 35.5203 1.65077 0.825383 0.564573i \(-0.190959\pi\)
0.825383 + 0.564573i \(0.190959\pi\)
\(464\) 0 0
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) −19.8745 −0.919683 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(468\) 0 0
\(469\) −10.8340 −0.500267
\(470\) 0 0
\(471\) −13.2915 −0.612440
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 11.2915 0.517002
\(478\) 0 0
\(479\) −4.35425 −0.198951 −0.0994754 0.995040i \(-0.531716\pi\)
−0.0994754 + 0.995040i \(0.531716\pi\)
\(480\) 0 0
\(481\) −1.29150 −0.0588875
\(482\) 0 0
\(483\) 15.2915 0.695787
\(484\) 0 0
\(485\) −2.93725 −0.133374
\(486\) 0 0
\(487\) −17.8745 −0.809971 −0.404986 0.914323i \(-0.632723\pi\)
−0.404986 + 0.914323i \(0.632723\pi\)
\(488\) 0 0
\(489\) 2.35425 0.106463
\(490\) 0 0
\(491\) 5.77124 0.260453 0.130226 0.991484i \(-0.458430\pi\)
0.130226 + 0.991484i \(0.458430\pi\)
\(492\) 0 0
\(493\) −35.7490 −1.61005
\(494\) 0 0
\(495\) −0.354249 −0.0159223
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 31.0405 1.38956 0.694782 0.719220i \(-0.255501\pi\)
0.694782 + 0.719220i \(0.255501\pi\)
\(500\) 0 0
\(501\) 21.2915 0.951234
\(502\) 0 0
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 9.29150 0.413466
\(506\) 0 0
\(507\) −12.8745 −0.571777
\(508\) 0 0
\(509\) −34.1033 −1.51160 −0.755800 0.654802i \(-0.772752\pi\)
−0.755800 + 0.654802i \(0.772752\pi\)
\(510\) 0 0
\(511\) −16.4575 −0.728038
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 10.5830 0.466343
\(516\) 0 0
\(517\) −0.457513 −0.0201214
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.1033 0.968362 0.484181 0.874968i \(-0.339118\pi\)
0.484181 + 0.874968i \(0.339118\pi\)
\(522\) 0 0
\(523\) 23.2915 1.01847 0.509233 0.860629i \(-0.329929\pi\)
0.509233 + 0.860629i \(0.329929\pi\)
\(524\) 0 0
\(525\) −1.64575 −0.0718265
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 63.3320 2.75357
\(530\) 0 0
\(531\) −11.2915 −0.490009
\(532\) 0 0
\(533\) 3.41699 0.148006
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.29150 0.314652
\(538\) 0 0
\(539\) 1.52026 0.0654822
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) 17.2915 0.742049
\(544\) 0 0
\(545\) 5.29150 0.226663
\(546\) 0 0
\(547\) −1.87451 −0.0801482 −0.0400741 0.999197i \(-0.512759\pi\)
−0.0400741 + 0.999197i \(0.512759\pi\)
\(548\) 0 0
\(549\) −11.2915 −0.481910
\(550\) 0 0
\(551\) 8.93725 0.380740
\(552\) 0 0
\(553\) 10.8340 0.460708
\(554\) 0 0
\(555\) 3.64575 0.154754
\(556\) 0 0
\(557\) −11.4170 −0.483754 −0.241877 0.970307i \(-0.577763\pi\)
−0.241877 + 0.970307i \(0.577763\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 1.41699 0.0598256
\(562\) 0 0
\(563\) −8.12549 −0.342449 −0.171224 0.985232i \(-0.554772\pi\)
−0.171224 + 0.985232i \(0.554772\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) −1.64575 −0.0691151
\(568\) 0 0
\(569\) −7.06275 −0.296086 −0.148043 0.988981i \(-0.547297\pi\)
−0.148043 + 0.988981i \(0.547297\pi\)
\(570\) 0 0
\(571\) −16.7085 −0.699229 −0.349614 0.936894i \(-0.613687\pi\)
−0.349614 + 0.936894i \(0.613687\pi\)
\(572\) 0 0
\(573\) −6.22876 −0.260210
\(574\) 0 0
\(575\) −9.29150 −0.387482
\(576\) 0 0
\(577\) −13.2915 −0.553332 −0.276666 0.960966i \(-0.589230\pi\)
−0.276666 + 0.960966i \(0.589230\pi\)
\(578\) 0 0
\(579\) −11.6458 −0.483981
\(580\) 0 0
\(581\) 9.87451 0.409664
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) −0.354249 −0.0146464
\(586\) 0 0
\(587\) 33.2915 1.37409 0.687044 0.726616i \(-0.258908\pi\)
0.687044 + 0.726616i \(0.258908\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −25.1660 −1.03519
\(592\) 0 0
\(593\) 3.41699 0.140319 0.0701596 0.997536i \(-0.477649\pi\)
0.0701596 + 0.997536i \(0.477649\pi\)
\(594\) 0 0
\(595\) 6.58301 0.269877
\(596\) 0 0
\(597\) −7.29150 −0.298421
\(598\) 0 0
\(599\) −9.41699 −0.384768 −0.192384 0.981320i \(-0.561622\pi\)
−0.192384 + 0.981320i \(0.561622\pi\)
\(600\) 0 0
\(601\) 22.7085 0.926299 0.463149 0.886280i \(-0.346719\pi\)
0.463149 + 0.886280i \(0.346719\pi\)
\(602\) 0 0
\(603\) 6.58301 0.268081
\(604\) 0 0
\(605\) −10.8745 −0.442112
\(606\) 0 0
\(607\) 43.0405 1.74696 0.873480 0.486859i \(-0.161858\pi\)
0.873480 + 0.486859i \(0.161858\pi\)
\(608\) 0 0
\(609\) −14.7085 −0.596018
\(610\) 0 0
\(611\) −0.457513 −0.0185090
\(612\) 0 0
\(613\) −30.4575 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(614\) 0 0
\(615\) −9.64575 −0.388954
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −23.2915 −0.936165 −0.468082 0.883685i \(-0.655055\pi\)
−0.468082 + 0.883685i \(0.655055\pi\)
\(620\) 0 0
\(621\) −9.29150 −0.372855
\(622\) 0 0
\(623\) 27.8745 1.11677
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.354249 −0.0141473
\(628\) 0 0
\(629\) −14.5830 −0.581462
\(630\) 0 0
\(631\) 14.5830 0.580540 0.290270 0.956945i \(-0.406255\pi\)
0.290270 + 0.956945i \(0.406255\pi\)
\(632\) 0 0
\(633\) −2.58301 −0.102665
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) 1.52026 0.0602349
\(638\) 0 0
\(639\) −7.29150 −0.288447
\(640\) 0 0
\(641\) −16.9373 −0.668981 −0.334491 0.942399i \(-0.608564\pi\)
−0.334491 + 0.942399i \(0.608564\pi\)
\(642\) 0 0
\(643\) 13.6458 0.538136 0.269068 0.963121i \(-0.413284\pi\)
0.269068 + 0.963121i \(0.413284\pi\)
\(644\) 0 0
\(645\) −5.64575 −0.222301
\(646\) 0 0
\(647\) 3.41699 0.134336 0.0671680 0.997742i \(-0.478604\pi\)
0.0671680 + 0.997742i \(0.478604\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 9.87451 0.387012
\(652\) 0 0
\(653\) 9.41699 0.368515 0.184258 0.982878i \(-0.441012\pi\)
0.184258 + 0.982878i \(0.441012\pi\)
\(654\) 0 0
\(655\) 14.2288 0.555964
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −25.4170 −0.990106 −0.495053 0.868863i \(-0.664851\pi\)
−0.495053 + 0.868863i \(0.664851\pi\)
\(660\) 0 0
\(661\) 1.29150 0.0502336 0.0251168 0.999685i \(-0.492004\pi\)
0.0251168 + 0.999685i \(0.492004\pi\)
\(662\) 0 0
\(663\) 1.41699 0.0550315
\(664\) 0 0
\(665\) −1.64575 −0.0638195
\(666\) 0 0
\(667\) −83.0405 −3.21534
\(668\) 0 0
\(669\) −2.58301 −0.0998648
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −13.0627 −0.503532 −0.251766 0.967788i \(-0.581011\pi\)
−0.251766 + 0.967788i \(0.581011\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 40.4575 1.55491 0.777454 0.628939i \(-0.216511\pi\)
0.777454 + 0.628939i \(0.216511\pi\)
\(678\) 0 0
\(679\) 4.83399 0.185511
\(680\) 0 0
\(681\) 10.7085 0.410351
\(682\) 0 0
\(683\) −19.7490 −0.755675 −0.377838 0.925872i \(-0.623332\pi\)
−0.377838 + 0.925872i \(0.623332\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 8.70850 0.332250
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 43.0405 1.63734 0.818669 0.574265i \(-0.194712\pi\)
0.818669 + 0.574265i \(0.194712\pi\)
\(692\) 0 0
\(693\) 0.583005 0.0221465
\(694\) 0 0
\(695\) −17.8745 −0.678019
\(696\) 0 0
\(697\) 38.5830 1.46144
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 16.5830 0.626331 0.313166 0.949698i \(-0.398610\pi\)
0.313166 + 0.949698i \(0.398610\pi\)
\(702\) 0 0
\(703\) 3.64575 0.137502
\(704\) 0 0
\(705\) 1.29150 0.0486408
\(706\) 0 0
\(707\) −15.2915 −0.575096
\(708\) 0 0
\(709\) −4.70850 −0.176831 −0.0884157 0.996084i \(-0.528180\pi\)
−0.0884157 + 0.996084i \(0.528180\pi\)
\(710\) 0 0
\(711\) −6.58301 −0.246882
\(712\) 0 0
\(713\) 55.7490 2.08782
\(714\) 0 0
\(715\) 0.125492 0.00469314
\(716\) 0 0
\(717\) −15.6458 −0.584301
\(718\) 0 0
\(719\) −32.8118 −1.22367 −0.611836 0.790985i \(-0.709569\pi\)
−0.611836 + 0.790985i \(0.709569\pi\)
\(720\) 0 0
\(721\) −17.4170 −0.648643
\(722\) 0 0
\(723\) 16.5830 0.616729
\(724\) 0 0
\(725\) 8.93725 0.331921
\(726\) 0 0
\(727\) −14.1033 −0.523061 −0.261531 0.965195i \(-0.584227\pi\)
−0.261531 + 0.965195i \(0.584227\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 22.5830 0.835263
\(732\) 0 0
\(733\) 7.41699 0.273953 0.136976 0.990574i \(-0.456262\pi\)
0.136976 + 0.990574i \(0.456262\pi\)
\(734\) 0 0
\(735\) −4.29150 −0.158294
\(736\) 0 0
\(737\) −2.33202 −0.0859011
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −0.354249 −0.0130137
\(742\) 0 0
\(743\) 10.5830 0.388253 0.194126 0.980977i \(-0.437813\pi\)
0.194126 + 0.980977i \(0.437813\pi\)
\(744\) 0 0
\(745\) −20.5830 −0.754103
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 44.5830 1.62686 0.813428 0.581665i \(-0.197599\pi\)
0.813428 + 0.581665i \(0.197599\pi\)
\(752\) 0 0
\(753\) 16.3542 0.595982
\(754\) 0 0
\(755\) 8.58301 0.312368
\(756\) 0 0
\(757\) −23.8745 −0.867734 −0.433867 0.900977i \(-0.642851\pi\)
−0.433867 + 0.900977i \(0.642851\pi\)
\(758\) 0 0
\(759\) 3.29150 0.119474
\(760\) 0 0
\(761\) 25.7490 0.933401 0.466701 0.884415i \(-0.345443\pi\)
0.466701 + 0.884415i \(0.345443\pi\)
\(762\) 0 0
\(763\) −8.70850 −0.315269
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −17.7490 −0.640046 −0.320023 0.947410i \(-0.603691\pi\)
−0.320023 + 0.947410i \(0.603691\pi\)
\(770\) 0 0
\(771\) −5.41699 −0.195088
\(772\) 0 0
\(773\) −8.70850 −0.313223 −0.156611 0.987660i \(-0.550057\pi\)
−0.156611 + 0.987660i \(0.550057\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) −6.00000 −0.215249
\(778\) 0 0
\(779\) −9.64575 −0.345595
\(780\) 0 0
\(781\) 2.58301 0.0924272
\(782\) 0 0
\(783\) 8.93725 0.319391
\(784\) 0 0
\(785\) −13.2915 −0.474394
\(786\) 0 0
\(787\) −37.8745 −1.35008 −0.675040 0.737781i \(-0.735874\pi\)
−0.675040 + 0.737781i \(0.735874\pi\)
\(788\) 0 0
\(789\) −4.58301 −0.163159
\(790\) 0 0
\(791\) 6.58301 0.234065
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 11.2915 0.400468
\(796\) 0 0
\(797\) 40.0000 1.41687 0.708436 0.705775i \(-0.249401\pi\)
0.708436 + 0.705775i \(0.249401\pi\)
\(798\) 0 0
\(799\) −5.16601 −0.182760
\(800\) 0 0
\(801\) −16.9373 −0.598448
\(802\) 0 0
\(803\) −3.54249 −0.125012
\(804\) 0 0
\(805\) 15.2915 0.538955
\(806\) 0 0
\(807\) 4.22876 0.148859
\(808\) 0 0
\(809\) −19.4170 −0.682665 −0.341333 0.939943i \(-0.610878\pi\)
−0.341333 + 0.939943i \(0.610878\pi\)
\(810\) 0 0
\(811\) −38.3320 −1.34602 −0.673010 0.739634i \(-0.734999\pi\)
−0.673010 + 0.739634i \(0.734999\pi\)
\(812\) 0 0
\(813\) 4.70850 0.165134
\(814\) 0 0
\(815\) 2.35425 0.0824657
\(816\) 0 0
\(817\) −5.64575 −0.197520
\(818\) 0 0
\(819\) 0.583005 0.0203719
\(820\) 0 0
\(821\) 47.6235 1.66207 0.831036 0.556218i \(-0.187748\pi\)
0.831036 + 0.556218i \(0.187748\pi\)
\(822\) 0 0
\(823\) −4.22876 −0.147405 −0.0737026 0.997280i \(-0.523482\pi\)
−0.0737026 + 0.997280i \(0.523482\pi\)
\(824\) 0 0
\(825\) −0.354249 −0.0123334
\(826\) 0 0
\(827\) −30.4575 −1.05911 −0.529556 0.848275i \(-0.677641\pi\)
−0.529556 + 0.848275i \(0.677641\pi\)
\(828\) 0 0
\(829\) 2.70850 0.0940700 0.0470350 0.998893i \(-0.485023\pi\)
0.0470350 + 0.998893i \(0.485023\pi\)
\(830\) 0 0
\(831\) 0.583005 0.0202242
\(832\) 0 0
\(833\) 17.1660 0.594767
\(834\) 0 0
\(835\) 21.2915 0.736823
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) 12.4575 0.430081 0.215041 0.976605i \(-0.431012\pi\)
0.215041 + 0.976605i \(0.431012\pi\)
\(840\) 0 0
\(841\) 50.8745 1.75429
\(842\) 0 0
\(843\) −32.2288 −1.11002
\(844\) 0 0
\(845\) −12.8745 −0.442897
\(846\) 0 0
\(847\) 17.8967 0.614939
\(848\) 0 0
\(849\) 11.5203 0.395374
\(850\) 0 0
\(851\) −33.8745 −1.16120
\(852\) 0 0
\(853\) 14.7085 0.503609 0.251805 0.967778i \(-0.418976\pi\)
0.251805 + 0.967778i \(0.418976\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 31.0405 1.06032 0.530162 0.847896i \(-0.322131\pi\)
0.530162 + 0.847896i \(0.322131\pi\)
\(858\) 0 0
\(859\) 34.3320 1.17139 0.585697 0.810530i \(-0.300821\pi\)
0.585697 + 0.810530i \(0.300821\pi\)
\(860\) 0 0
\(861\) 15.8745 0.541002
\(862\) 0 0
\(863\) 20.0000 0.680808 0.340404 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 2.33202 0.0791084
\(870\) 0 0
\(871\) −2.33202 −0.0790175
\(872\) 0 0
\(873\) −2.93725 −0.0994110
\(874\) 0 0
\(875\) −1.64575 −0.0556365
\(876\) 0 0
\(877\) −25.0627 −0.846309 −0.423154 0.906058i \(-0.639077\pi\)
−0.423154 + 0.906058i \(0.639077\pi\)
\(878\) 0 0
\(879\) 5.41699 0.182711
\(880\) 0 0
\(881\) −0.125492 −0.00422794 −0.00211397 0.999998i \(-0.500673\pi\)
−0.00211397 + 0.999998i \(0.500673\pi\)
\(882\) 0 0
\(883\) −10.8118 −0.363845 −0.181922 0.983313i \(-0.558232\pi\)
−0.181922 + 0.983313i \(0.558232\pi\)
\(884\) 0 0
\(885\) −11.2915 −0.379560
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 6.58301 0.220787
\(890\) 0 0
\(891\) −0.354249 −0.0118678
\(892\) 0 0
\(893\) 1.29150 0.0432185
\(894\) 0 0
\(895\) 7.29150 0.243728
\(896\) 0 0
\(897\) 3.29150 0.109900
\(898\) 0 0
\(899\) −53.6235 −1.78844
\(900\) 0 0
\(901\) −45.1660 −1.50470
\(902\) 0 0
\(903\) 9.29150 0.309202
\(904\) 0 0
\(905\) 17.2915 0.574789
\(906\) 0 0
\(907\) 47.0405 1.56195 0.780977 0.624559i \(-0.214721\pi\)
0.780977 + 0.624559i \(0.214721\pi\)
\(908\) 0 0
\(909\) 9.29150 0.308180
\(910\) 0 0
\(911\) −26.5830 −0.880734 −0.440367 0.897818i \(-0.645152\pi\)
−0.440367 + 0.897818i \(0.645152\pi\)
\(912\) 0 0
\(913\) 2.12549 0.0703435
\(914\) 0 0
\(915\) −11.2915 −0.373286
\(916\) 0 0
\(917\) −23.4170 −0.773297
\(918\) 0 0
\(919\) −14.8340 −0.489328 −0.244664 0.969608i \(-0.578678\pi\)
−0.244664 + 0.969608i \(0.578678\pi\)
\(920\) 0 0
\(921\) 24.4575 0.805902
\(922\) 0 0
\(923\) 2.58301 0.0850207
\(924\) 0 0
\(925\) 3.64575 0.119872
\(926\) 0 0
\(927\) 10.5830 0.347591
\(928\) 0 0
\(929\) −11.8745 −0.389590 −0.194795 0.980844i \(-0.562404\pi\)
−0.194795 + 0.980844i \(0.562404\pi\)
\(930\) 0 0
\(931\) −4.29150 −0.140648
\(932\) 0 0
\(933\) −22.9373 −0.750932
\(934\) 0 0
\(935\) 1.41699 0.0463407
\(936\) 0 0
\(937\) 20.1255 0.657471 0.328736 0.944422i \(-0.393378\pi\)
0.328736 + 0.944422i \(0.393378\pi\)
\(938\) 0 0
\(939\) 17.2915 0.564287
\(940\) 0 0
\(941\) −11.7712 −0.383732 −0.191866 0.981421i \(-0.561454\pi\)
−0.191866 + 0.981421i \(0.561454\pi\)
\(942\) 0 0
\(943\) 89.6235 2.91854
\(944\) 0 0
\(945\) −1.64575 −0.0535363
\(946\) 0 0
\(947\) −11.4170 −0.371002 −0.185501 0.982644i \(-0.559391\pi\)
−0.185501 + 0.982644i \(0.559391\pi\)
\(948\) 0 0
\(949\) −3.54249 −0.114994
\(950\) 0 0
\(951\) 20.4575 0.663380
\(952\) 0 0
\(953\) −10.5830 −0.342817 −0.171409 0.985200i \(-0.554832\pi\)
−0.171409 + 0.985200i \(0.554832\pi\)
\(954\) 0 0
\(955\) −6.22876 −0.201558
\(956\) 0 0
\(957\) −3.16601 −0.102343
\(958\) 0 0
\(959\) −9.87451 −0.318864
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −11.6458 −0.374890
\(966\) 0 0
\(967\) 41.6458 1.33924 0.669619 0.742705i \(-0.266458\pi\)
0.669619 + 0.742705i \(0.266458\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 29.4170 0.943066
\(974\) 0 0
\(975\) −0.354249 −0.0113450
\(976\) 0 0
\(977\) −51.0405 −1.63293 −0.816465 0.577394i \(-0.804070\pi\)
−0.816465 + 0.577394i \(0.804070\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 5.29150 0.168945
\(982\) 0 0
\(983\) −34.4575 −1.09902 −0.549512 0.835486i \(-0.685186\pi\)
−0.549512 + 0.835486i \(0.685186\pi\)
\(984\) 0 0
\(985\) −25.1660 −0.801856
\(986\) 0 0
\(987\) −2.12549 −0.0676552
\(988\) 0 0
\(989\) 52.4575 1.66805
\(990\) 0 0
\(991\) 35.7490 1.13560 0.567802 0.823165i \(-0.307794\pi\)
0.567802 + 0.823165i \(0.307794\pi\)
\(992\) 0 0
\(993\) −11.4170 −0.362307
\(994\) 0 0
\(995\) −7.29150 −0.231156
\(996\) 0 0
\(997\) 26.7085 0.845867 0.422933 0.906161i \(-0.361000\pi\)
0.422933 + 0.906161i \(0.361000\pi\)
\(998\) 0 0
\(999\) 3.64575 0.115346
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.bo.1.1 2
4.3 odd 2 285.2.a.d.1.1 2
12.11 even 2 855.2.a.g.1.2 2
20.3 even 4 1425.2.c.i.799.3 4
20.7 even 4 1425.2.c.i.799.2 4
20.19 odd 2 1425.2.a.p.1.2 2
60.59 even 2 4275.2.a.u.1.1 2
76.75 even 2 5415.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.d.1.1 2 4.3 odd 2
855.2.a.g.1.2 2 12.11 even 2
1425.2.a.p.1.2 2 20.19 odd 2
1425.2.c.i.799.2 4 20.7 even 4
1425.2.c.i.799.3 4 20.3 even 4
4275.2.a.u.1.1 2 60.59 even 2
4560.2.a.bo.1.1 2 1.1 even 1 trivial
5415.2.a.s.1.2 2 76.75 even 2