Properties

Label 6762.2.a.cp.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.42048.1
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.35854\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.82843 q^{10} -5.10824 q^{11} +1.00000 q^{12} +0.0556701 q^{13} -3.82843 q^{15} +1.00000 q^{16} -6.77276 q^{17} +1.00000 q^{18} +4.16391 q^{19} -3.82843 q^{20} -5.10824 q^{22} +1.00000 q^{23} +1.00000 q^{24} +9.65685 q^{25} +0.0556701 q^{26} +1.00000 q^{27} +2.94433 q^{29} -3.82843 q^{30} -3.21958 q^{31} +1.00000 q^{32} -5.10824 q^{33} -6.77276 q^{34} +1.00000 q^{36} -8.05257 q^{37} +4.16391 q^{38} +0.0556701 q^{39} -3.82843 q^{40} -4.16391 q^{41} +12.5596 q^{43} -5.10824 q^{44} -3.82843 q^{45} +1.00000 q^{46} +11.9969 q^{47} +1.00000 q^{48} +9.65685 q^{50} -6.77276 q^{51} +0.0556701 q^{52} +5.71709 q^{53} +1.00000 q^{54} +19.5565 q^{55} +4.16391 q^{57} +2.94433 q^{58} +8.60118 q^{59} -3.82843 q^{60} -5.77732 q^{61} -3.21958 q^{62} +1.00000 q^{64} -0.213129 q^{65} -5.10824 q^{66} +12.1544 q^{67} -6.77276 q^{68} +1.00000 q^{69} +11.9443 q^{71} +1.00000 q^{72} -2.21958 q^{73} -8.05257 q^{74} +9.65685 q^{75} +4.16391 q^{76} +0.0556701 q^{78} -2.94433 q^{79} -3.82843 q^{80} +1.00000 q^{81} -4.16391 q^{82} -14.7651 q^{83} +25.9290 q^{85} +12.5596 q^{86} +2.94433 q^{87} -5.10824 q^{88} +2.39572 q^{89} -3.82843 q^{90} +1.00000 q^{92} -3.21958 q^{93} +11.9969 q^{94} -15.9412 q^{95} +1.00000 q^{96} +4.60118 q^{97} -5.10824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} - 4q^{10} + 6q^{11} + 4q^{12} + 6q^{13} - 4q^{15} + 4q^{16} - 10q^{17} + 4q^{18} - 4q^{19} - 4q^{20} + 6q^{22} + 4q^{23} + 4q^{24} + 16q^{25} + 6q^{26} + 4q^{27} + 6q^{29} - 4q^{30} + 2q^{31} + 4q^{32} + 6q^{33} - 10q^{34} + 4q^{36} - 4q^{38} + 6q^{39} - 4q^{40} + 4q^{41} + 20q^{43} + 6q^{44} - 4q^{45} + 4q^{46} + 10q^{47} + 4q^{48} + 16q^{50} - 10q^{51} + 6q^{52} + 4q^{54} + 10q^{55} - 4q^{57} + 6q^{58} + 6q^{59} - 4q^{60} + 2q^{62} + 4q^{64} - 14q^{65} + 6q^{66} + 18q^{67} - 10q^{68} + 4q^{69} + 42q^{71} + 4q^{72} + 6q^{73} + 16q^{75} - 4q^{76} + 6q^{78} - 6q^{79} - 4q^{80} + 4q^{81} + 4q^{82} - 10q^{83} + 34q^{85} + 20q^{86} + 6q^{87} + 6q^{88} - 4q^{90} + 4q^{92} + 2q^{93} + 10q^{94} - 20q^{95} + 4q^{96} - 10q^{97} + 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.82843 −1.71212 −0.856062 0.516873i \(-0.827096\pi\)
−0.856062 + 0.516873i \(0.827096\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.82843 −1.21065
\(11\) −5.10824 −1.54019 −0.770096 0.637928i \(-0.779792\pi\)
−0.770096 + 0.637928i \(0.779792\pi\)
\(12\) 1.00000 0.288675
\(13\) 0.0556701 0.0154401 0.00772006 0.999970i \(-0.497543\pi\)
0.00772006 + 0.999970i \(0.497543\pi\)
\(14\) 0 0
\(15\) −3.82843 −0.988496
\(16\) 1.00000 0.250000
\(17\) −6.77276 −1.64263 −0.821317 0.570471i \(-0.806761\pi\)
−0.821317 + 0.570471i \(0.806761\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.16391 0.955267 0.477633 0.878559i \(-0.341495\pi\)
0.477633 + 0.878559i \(0.341495\pi\)
\(20\) −3.82843 −0.856062
\(21\) 0 0
\(22\) −5.10824 −1.08908
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 9.65685 1.93137
\(26\) 0.0556701 0.0109178
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 2.94433 0.546748 0.273374 0.961908i \(-0.411860\pi\)
0.273374 + 0.961908i \(0.411860\pi\)
\(30\) −3.82843 −0.698972
\(31\) −3.21958 −0.578254 −0.289127 0.957291i \(-0.593365\pi\)
−0.289127 + 0.957291i \(0.593365\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.10824 −0.889231
\(34\) −6.77276 −1.16152
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.05257 −1.32383 −0.661917 0.749577i \(-0.730257\pi\)
−0.661917 + 0.749577i \(0.730257\pi\)
\(38\) 4.16391 0.675476
\(39\) 0.0556701 0.00891436
\(40\) −3.82843 −0.605327
\(41\) −4.16391 −0.650294 −0.325147 0.945664i \(-0.605414\pi\)
−0.325147 + 0.945664i \(0.605414\pi\)
\(42\) 0 0
\(43\) 12.5596 1.91533 0.957663 0.287893i \(-0.0929547\pi\)
0.957663 + 0.287893i \(0.0929547\pi\)
\(44\) −5.10824 −0.770096
\(45\) −3.82843 −0.570708
\(46\) 1.00000 0.147442
\(47\) 11.9969 1.74993 0.874964 0.484188i \(-0.160885\pi\)
0.874964 + 0.484188i \(0.160885\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 9.65685 1.36569
\(51\) −6.77276 −0.948376
\(52\) 0.0556701 0.00772006
\(53\) 5.71709 0.785302 0.392651 0.919687i \(-0.371558\pi\)
0.392651 + 0.919687i \(0.371558\pi\)
\(54\) 1.00000 0.136083
\(55\) 19.5565 2.63700
\(56\) 0 0
\(57\) 4.16391 0.551524
\(58\) 2.94433 0.386609
\(59\) 8.60118 1.11978 0.559889 0.828567i \(-0.310844\pi\)
0.559889 + 0.828567i \(0.310844\pi\)
\(60\) −3.82843 −0.494248
\(61\) −5.77732 −0.739710 −0.369855 0.929089i \(-0.620593\pi\)
−0.369855 + 0.929089i \(0.620593\pi\)
\(62\) −3.21958 −0.408887
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.213129 −0.0264354
\(66\) −5.10824 −0.628781
\(67\) 12.1544 1.48489 0.742446 0.669906i \(-0.233666\pi\)
0.742446 + 0.669906i \(0.233666\pi\)
\(68\) −6.77276 −0.821317
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 11.9443 1.41753 0.708766 0.705444i \(-0.249252\pi\)
0.708766 + 0.705444i \(0.249252\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.21958 −0.259782 −0.129891 0.991528i \(-0.541463\pi\)
−0.129891 + 0.991528i \(0.541463\pi\)
\(74\) −8.05257 −0.936092
\(75\) 9.65685 1.11508
\(76\) 4.16391 0.477633
\(77\) 0 0
\(78\) 0.0556701 0.00630340
\(79\) −2.94433 −0.331263 −0.165631 0.986188i \(-0.552966\pi\)
−0.165631 + 0.986188i \(0.552966\pi\)
\(80\) −3.82843 −0.428031
\(81\) 1.00000 0.111111
\(82\) −4.16391 −0.459827
\(83\) −14.7651 −1.62068 −0.810340 0.585960i \(-0.800718\pi\)
−0.810340 + 0.585960i \(0.800718\pi\)
\(84\) 0 0
\(85\) 25.9290 2.81240
\(86\) 12.5596 1.35434
\(87\) 2.94433 0.315665
\(88\) −5.10824 −0.544540
\(89\) 2.39572 0.253945 0.126973 0.991906i \(-0.459474\pi\)
0.126973 + 0.991906i \(0.459474\pi\)
\(90\) −3.82843 −0.403552
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −3.21958 −0.333855
\(94\) 11.9969 1.23739
\(95\) −15.9412 −1.63554
\(96\) 1.00000 0.102062
\(97\) 4.60118 0.467179 0.233590 0.972335i \(-0.424953\pi\)
0.233590 + 0.972335i \(0.424953\pi\)
\(98\) 0 0
\(99\) −5.10824 −0.513398
\(100\) 9.65685 0.965685
\(101\) −14.8875 −1.48136 −0.740678 0.671860i \(-0.765496\pi\)
−0.740678 + 0.671860i \(0.765496\pi\)
\(102\) −6.77276 −0.670603
\(103\) 6.15436 0.606407 0.303204 0.952926i \(-0.401944\pi\)
0.303204 + 0.952926i \(0.401944\pi\)
\(104\) 0.0556701 0.00545891
\(105\) 0 0
\(106\) 5.71709 0.555293
\(107\) 4.12236 0.398523 0.199262 0.979946i \(-0.436146\pi\)
0.199262 + 0.979946i \(0.436146\pi\)
\(108\) 1.00000 0.0962250
\(109\) 20.0960 1.92485 0.962425 0.271549i \(-0.0875358\pi\)
0.962425 + 0.271549i \(0.0875358\pi\)
\(110\) 19.5565 1.86464
\(111\) −8.05257 −0.764316
\(112\) 0 0
\(113\) −5.33238 −0.501629 −0.250814 0.968035i \(-0.580698\pi\)
−0.250814 + 0.968035i \(0.580698\pi\)
\(114\) 4.16391 0.389986
\(115\) −3.82843 −0.357003
\(116\) 2.94433 0.273374
\(117\) 0.0556701 0.00514671
\(118\) 8.60118 0.791803
\(119\) 0 0
\(120\) −3.82843 −0.349486
\(121\) 15.0941 1.37219
\(122\) −5.77732 −0.523054
\(123\) −4.16391 −0.375447
\(124\) −3.21958 −0.289127
\(125\) −17.8284 −1.59462
\(126\) 0 0
\(127\) 8.54861 0.758567 0.379283 0.925281i \(-0.376171\pi\)
0.379283 + 0.925281i \(0.376171\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.5596 1.10581
\(130\) −0.213129 −0.0186927
\(131\) −6.94123 −0.606458 −0.303229 0.952918i \(-0.598065\pi\)
−0.303229 + 0.952918i \(0.598065\pi\)
\(132\) −5.10824 −0.444615
\(133\) 0 0
\(134\) 12.1544 1.04998
\(135\) −3.82843 −0.329499
\(136\) −6.77276 −0.580759
\(137\) 5.27981 0.451085 0.225542 0.974233i \(-0.427585\pi\)
0.225542 + 0.974233i \(0.427585\pi\)
\(138\) 1.00000 0.0851257
\(139\) 9.31371 0.789978 0.394989 0.918686i \(-0.370748\pi\)
0.394989 + 0.918686i \(0.370748\pi\)
\(140\) 0 0
\(141\) 11.9969 1.01032
\(142\) 11.9443 1.00235
\(143\) −0.284376 −0.0237808
\(144\) 1.00000 0.0833333
\(145\) −11.2722 −0.936101
\(146\) −2.21958 −0.183694
\(147\) 0 0
\(148\) −8.05257 −0.661917
\(149\) −11.7190 −0.960056 −0.480028 0.877253i \(-0.659374\pi\)
−0.480028 + 0.877253i \(0.659374\pi\)
\(150\) 9.65685 0.788479
\(151\) 17.5565 1.42873 0.714365 0.699773i \(-0.246716\pi\)
0.714365 + 0.699773i \(0.246716\pi\)
\(152\) 4.16391 0.337738
\(153\) −6.77276 −0.547545
\(154\) 0 0
\(155\) 12.3259 0.990043
\(156\) 0.0556701 0.00445718
\(157\) −9.15892 −0.730962 −0.365481 0.930819i \(-0.619095\pi\)
−0.365481 + 0.930819i \(0.619095\pi\)
\(158\) −2.94433 −0.234238
\(159\) 5.71709 0.453394
\(160\) −3.82843 −0.302664
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.1730 −0.953466 −0.476733 0.879048i \(-0.658179\pi\)
−0.476733 + 0.879048i \(0.658179\pi\)
\(164\) −4.16391 −0.325147
\(165\) 19.5565 1.52247
\(166\) −14.7651 −1.14599
\(167\) 21.0788 1.63113 0.815563 0.578668i \(-0.196427\pi\)
0.815563 + 0.578668i \(0.196427\pi\)
\(168\) 0 0
\(169\) −12.9969 −0.999762
\(170\) 25.9290 1.98866
\(171\) 4.16391 0.318422
\(172\) 12.5596 0.957663
\(173\) 13.4250 1.02069 0.510344 0.859970i \(-0.329518\pi\)
0.510344 + 0.859970i \(0.329518\pi\)
\(174\) 2.94433 0.223209
\(175\) 0 0
\(176\) −5.10824 −0.385048
\(177\) 8.60118 0.646505
\(178\) 2.39572 0.179567
\(179\) 1.99690 0.149255 0.0746277 0.997211i \(-0.476223\pi\)
0.0746277 + 0.997211i \(0.476223\pi\)
\(180\) −3.82843 −0.285354
\(181\) 20.5443 1.52705 0.763523 0.645781i \(-0.223468\pi\)
0.763523 + 0.645781i \(0.223468\pi\)
\(182\) 0 0
\(183\) −5.77732 −0.427072
\(184\) 1.00000 0.0737210
\(185\) 30.8287 2.26657
\(186\) −3.21958 −0.236071
\(187\) 34.5969 2.52997
\(188\) 11.9969 0.874964
\(189\) 0 0
\(190\) −15.9412 −1.15650
\(191\) −1.94743 −0.140911 −0.0704555 0.997515i \(-0.522445\pi\)
−0.0704555 + 0.997515i \(0.522445\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.44037 0.319625 0.159812 0.987147i \(-0.448911\pi\)
0.159812 + 0.987147i \(0.448911\pi\)
\(194\) 4.60118 0.330346
\(195\) −0.213129 −0.0152625
\(196\) 0 0
\(197\) −15.7620 −1.12300 −0.561498 0.827478i \(-0.689775\pi\)
−0.561498 + 0.827478i \(0.689775\pi\)
\(198\) −5.10824 −0.363027
\(199\) 19.8733 1.40878 0.704392 0.709811i \(-0.251220\pi\)
0.704392 + 0.709811i \(0.251220\pi\)
\(200\) 9.65685 0.682843
\(201\) 12.1544 0.857302
\(202\) −14.8875 −1.04748
\(203\) 0 0
\(204\) −6.77276 −0.474188
\(205\) 15.9412 1.11338
\(206\) 6.15436 0.428795
\(207\) 1.00000 0.0695048
\(208\) 0.0556701 0.00386003
\(209\) −21.2703 −1.47129
\(210\) 0 0
\(211\) −10.6427 −0.732676 −0.366338 0.930482i \(-0.619389\pi\)
−0.366338 + 0.930482i \(0.619389\pi\)
\(212\) 5.71709 0.392651
\(213\) 11.9443 0.818412
\(214\) 4.12236 0.281798
\(215\) −48.0836 −3.27928
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 20.0960 1.36107
\(219\) −2.21958 −0.149985
\(220\) 19.5565 1.31850
\(221\) −0.377040 −0.0253625
\(222\) −8.05257 −0.540453
\(223\) −28.5424 −1.91134 −0.955671 0.294438i \(-0.904868\pi\)
−0.955671 + 0.294438i \(0.904868\pi\)
\(224\) 0 0
\(225\) 9.65685 0.643790
\(226\) −5.33238 −0.354705
\(227\) −3.68319 −0.244462 −0.122231 0.992502i \(-0.539005\pi\)
−0.122231 + 0.992502i \(0.539005\pi\)
\(228\) 4.16391 0.275762
\(229\) −12.1639 −0.803814 −0.401907 0.915681i \(-0.631652\pi\)
−0.401907 + 0.915681i \(0.631652\pi\)
\(230\) −3.82843 −0.252439
\(231\) 0 0
\(232\) 2.94433 0.193305
\(233\) −1.49294 −0.0978059 −0.0489030 0.998804i \(-0.515573\pi\)
−0.0489030 + 0.998804i \(0.515573\pi\)
\(234\) 0.0556701 0.00363927
\(235\) −45.9293 −2.99609
\(236\) 8.60118 0.559889
\(237\) −2.94433 −0.191255
\(238\) 0 0
\(239\) 25.1193 1.62483 0.812415 0.583080i \(-0.198153\pi\)
0.812415 + 0.583080i \(0.198153\pi\)
\(240\) −3.82843 −0.247124
\(241\) 11.8471 0.763139 0.381570 0.924340i \(-0.375384\pi\)
0.381570 + 0.924340i \(0.375384\pi\)
\(242\) 15.0941 0.970287
\(243\) 1.00000 0.0641500
\(244\) −5.77732 −0.369855
\(245\) 0 0
\(246\) −4.16391 −0.265481
\(247\) 0.231805 0.0147494
\(248\) −3.21958 −0.204444
\(249\) −14.7651 −0.935700
\(250\) −17.8284 −1.12757
\(251\) 8.09413 0.510897 0.255448 0.966823i \(-0.417777\pi\)
0.255448 + 0.966823i \(0.417777\pi\)
\(252\) 0 0
\(253\) −5.10824 −0.321152
\(254\) 8.54861 0.536388
\(255\) 25.9290 1.62374
\(256\) 1.00000 0.0625000
\(257\) 1.94123 0.121091 0.0605453 0.998165i \(-0.480716\pi\)
0.0605453 + 0.998165i \(0.480716\pi\)
\(258\) 12.5596 0.781928
\(259\) 0 0
\(260\) −0.213129 −0.0132177
\(261\) 2.94433 0.182249
\(262\) −6.94123 −0.428831
\(263\) −29.3075 −1.80718 −0.903589 0.428400i \(-0.859077\pi\)
−0.903589 + 0.428400i \(0.859077\pi\)
\(264\) −5.10824 −0.314391
\(265\) −21.8875 −1.34454
\(266\) 0 0
\(267\) 2.39572 0.146615
\(268\) 12.1544 0.742446
\(269\) 2.62941 0.160318 0.0801590 0.996782i \(-0.474457\pi\)
0.0801590 + 0.996782i \(0.474457\pi\)
\(270\) −3.82843 −0.232991
\(271\) 3.21958 0.195576 0.0977878 0.995207i \(-0.468823\pi\)
0.0977878 + 0.995207i \(0.468823\pi\)
\(272\) −6.77276 −0.410659
\(273\) 0 0
\(274\) 5.27981 0.318965
\(275\) −49.3295 −2.97468
\(276\) 1.00000 0.0601929
\(277\) 14.7701 0.887448 0.443724 0.896163i \(-0.353657\pi\)
0.443724 + 0.896163i \(0.353657\pi\)
\(278\) 9.31371 0.558599
\(279\) −3.21958 −0.192751
\(280\) 0 0
\(281\) 3.44992 0.205805 0.102903 0.994691i \(-0.467187\pi\)
0.102903 + 0.994691i \(0.467187\pi\)
\(282\) 11.9969 0.714405
\(283\) 0.951992 0.0565900 0.0282950 0.999600i \(-0.490992\pi\)
0.0282950 + 0.999600i \(0.490992\pi\)
\(284\) 11.9443 0.708766
\(285\) −15.9412 −0.944277
\(286\) −0.284376 −0.0168155
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 28.8702 1.69825
\(290\) −11.2722 −0.661924
\(291\) 4.60118 0.269726
\(292\) −2.21958 −0.129891
\(293\) −14.5147 −0.847959 −0.423979 0.905672i \(-0.639367\pi\)
−0.423979 + 0.905672i \(0.639367\pi\)
\(294\) 0 0
\(295\) −32.9290 −1.91720
\(296\) −8.05257 −0.468046
\(297\) −5.10824 −0.296410
\(298\) −11.7190 −0.678862
\(299\) 0.0556701 0.00321949
\(300\) 9.65685 0.557539
\(301\) 0 0
\(302\) 17.5565 1.01026
\(303\) −14.8875 −0.855262
\(304\) 4.16391 0.238817
\(305\) 22.1180 1.26648
\(306\) −6.77276 −0.387173
\(307\) 17.2611 0.985145 0.492573 0.870271i \(-0.336057\pi\)
0.492573 + 0.870271i \(0.336057\pi\)
\(308\) 0 0
\(309\) 6.15436 0.350109
\(310\) 12.3259 0.700066
\(311\) 2.37059 0.134424 0.0672119 0.997739i \(-0.478590\pi\)
0.0672119 + 0.997739i \(0.478590\pi\)
\(312\) 0.0556701 0.00315170
\(313\) −5.83798 −0.329982 −0.164991 0.986295i \(-0.552760\pi\)
−0.164991 + 0.986295i \(0.552760\pi\)
\(314\) −9.15892 −0.516868
\(315\) 0 0
\(316\) −2.94433 −0.165631
\(317\) 0.629412 0.0353513 0.0176757 0.999844i \(-0.494373\pi\)
0.0176757 + 0.999844i \(0.494373\pi\)
\(318\) 5.71709 0.320598
\(319\) −15.0403 −0.842098
\(320\) −3.82843 −0.214016
\(321\) 4.12236 0.230087
\(322\) 0 0
\(323\) −28.2012 −1.56915
\(324\) 1.00000 0.0555556
\(325\) 0.537598 0.0298206
\(326\) −12.1730 −0.674202
\(327\) 20.0960 1.11131
\(328\) −4.16391 −0.229914
\(329\) 0 0
\(330\) 19.5565 1.07655
\(331\) 13.7003 0.753037 0.376518 0.926409i \(-0.377121\pi\)
0.376518 + 0.926409i \(0.377121\pi\)
\(332\) −14.7651 −0.810340
\(333\) −8.05257 −0.441278
\(334\) 21.0788 1.15338
\(335\) −46.5321 −2.54232
\(336\) 0 0
\(337\) −19.8318 −1.08031 −0.540153 0.841567i \(-0.681634\pi\)
−0.540153 + 0.841567i \(0.681634\pi\)
\(338\) −12.9969 −0.706938
\(339\) −5.33238 −0.289615
\(340\) 25.9290 1.40620
\(341\) 16.4464 0.890622
\(342\) 4.16391 0.225159
\(343\) 0 0
\(344\) 12.5596 0.677170
\(345\) −3.82843 −0.206116
\(346\) 13.4250 0.721735
\(347\) 5.55774 0.298355 0.149178 0.988810i \(-0.452337\pi\)
0.149178 + 0.988810i \(0.452337\pi\)
\(348\) 2.94433 0.157833
\(349\) −15.5027 −0.829843 −0.414922 0.909857i \(-0.636191\pi\)
−0.414922 + 0.909857i \(0.636191\pi\)
\(350\) 0 0
\(351\) 0.0556701 0.00297145
\(352\) −5.10824 −0.272270
\(353\) 15.9412 0.848466 0.424233 0.905553i \(-0.360544\pi\)
0.424233 + 0.905553i \(0.360544\pi\)
\(354\) 8.60118 0.457148
\(355\) −45.7280 −2.42699
\(356\) 2.39572 0.126973
\(357\) 0 0
\(358\) 1.99690 0.105539
\(359\) 32.4047 1.71026 0.855128 0.518416i \(-0.173478\pi\)
0.855128 + 0.518416i \(0.173478\pi\)
\(360\) −3.82843 −0.201776
\(361\) −1.66184 −0.0874654
\(362\) 20.5443 1.07978
\(363\) 15.0941 0.792236
\(364\) 0 0
\(365\) 8.49751 0.444780
\(366\) −5.77732 −0.301985
\(367\) 10.8669 0.567247 0.283623 0.958936i \(-0.408463\pi\)
0.283623 + 0.958936i \(0.408463\pi\)
\(368\) 1.00000 0.0521286
\(369\) −4.16391 −0.216765
\(370\) 30.8287 1.60271
\(371\) 0 0
\(372\) −3.21958 −0.166928
\(373\) −0.216482 −0.0112090 −0.00560451 0.999984i \(-0.501784\pi\)
−0.00560451 + 0.999984i \(0.501784\pi\)
\(374\) 34.5969 1.78896
\(375\) −17.8284 −0.920656
\(376\) 11.9969 0.618693
\(377\) 0.163911 0.00844186
\(378\) 0 0
\(379\) 8.93667 0.459046 0.229523 0.973303i \(-0.426283\pi\)
0.229523 + 0.973303i \(0.426283\pi\)
\(380\) −15.9412 −0.817768
\(381\) 8.54861 0.437959
\(382\) −1.94743 −0.0996391
\(383\) 26.8287 1.37088 0.685441 0.728128i \(-0.259610\pi\)
0.685441 + 0.728128i \(0.259610\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 4.44037 0.226009
\(387\) 12.5596 0.638442
\(388\) 4.60118 0.233590
\(389\) −3.66598 −0.185873 −0.0929363 0.995672i \(-0.529625\pi\)
−0.0929363 + 0.995672i \(0.529625\pi\)
\(390\) −0.213129 −0.0107922
\(391\) −6.77276 −0.342513
\(392\) 0 0
\(393\) −6.94123 −0.350139
\(394\) −15.7620 −0.794078
\(395\) 11.2722 0.567164
\(396\) −5.10824 −0.256699
\(397\) 19.9572 1.00162 0.500812 0.865556i \(-0.333035\pi\)
0.500812 + 0.865556i \(0.333035\pi\)
\(398\) 19.8733 0.996160
\(399\) 0 0
\(400\) 9.65685 0.482843
\(401\) 0.386165 0.0192842 0.00964209 0.999954i \(-0.496931\pi\)
0.00964209 + 0.999954i \(0.496931\pi\)
\(402\) 12.1544 0.606204
\(403\) −0.179235 −0.00892831
\(404\) −14.8875 −0.740678
\(405\) −3.82843 −0.190236
\(406\) 0 0
\(407\) 41.1345 2.03896
\(408\) −6.77276 −0.335301
\(409\) −15.5314 −0.767978 −0.383989 0.923338i \(-0.625450\pi\)
−0.383989 + 0.923338i \(0.625450\pi\)
\(410\) 15.9412 0.787281
\(411\) 5.27981 0.260434
\(412\) 6.15436 0.303204
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 56.5271 2.77481
\(416\) 0.0556701 0.00272945
\(417\) 9.31371 0.456094
\(418\) −21.2703 −1.04036
\(419\) 0.559628 0.0273396 0.0136698 0.999907i \(-0.495649\pi\)
0.0136698 + 0.999907i \(0.495649\pi\)
\(420\) 0 0
\(421\) 17.2024 0.838392 0.419196 0.907896i \(-0.362312\pi\)
0.419196 + 0.907896i \(0.362312\pi\)
\(422\) −10.6427 −0.518080
\(423\) 11.9969 0.583309
\(424\) 5.71709 0.277646
\(425\) −65.4035 −3.17254
\(426\) 11.9443 0.578705
\(427\) 0 0
\(428\) 4.12236 0.199262
\(429\) −0.284376 −0.0137298
\(430\) −48.0836 −2.31880
\(431\) 18.8501 0.907977 0.453989 0.891007i \(-0.350001\pi\)
0.453989 + 0.891007i \(0.350001\pi\)
\(432\) 1.00000 0.0481125
\(433\) 0.435383 0.0209232 0.0104616 0.999945i \(-0.496670\pi\)
0.0104616 + 0.999945i \(0.496670\pi\)
\(434\) 0 0
\(435\) −11.2722 −0.540458
\(436\) 20.0960 0.962425
\(437\) 4.16391 0.199187
\(438\) −2.21958 −0.106056
\(439\) −34.0048 −1.62296 −0.811481 0.584379i \(-0.801338\pi\)
−0.811481 + 0.584379i \(0.801338\pi\)
\(440\) 19.5565 0.932321
\(441\) 0 0
\(442\) −0.377040 −0.0179340
\(443\) 16.9412 0.804902 0.402451 0.915441i \(-0.368158\pi\)
0.402451 + 0.915441i \(0.368158\pi\)
\(444\) −8.05257 −0.382158
\(445\) −9.17183 −0.434786
\(446\) −28.5424 −1.35152
\(447\) −11.7190 −0.554289
\(448\) 0 0
\(449\) 5.28937 0.249621 0.124810 0.992181i \(-0.460168\pi\)
0.124810 + 0.992181i \(0.460168\pi\)
\(450\) 9.65685 0.455228
\(451\) 21.2703 1.00158
\(452\) −5.33238 −0.250814
\(453\) 17.5565 0.824878
\(454\) −3.68319 −0.172861
\(455\) 0 0
\(456\) 4.16391 0.194993
\(457\) 6.95723 0.325446 0.162723 0.986672i \(-0.447972\pi\)
0.162723 + 0.986672i \(0.447972\pi\)
\(458\) −12.1639 −0.568382
\(459\) −6.77276 −0.316125
\(460\) −3.82843 −0.178501
\(461\) 6.65564 0.309984 0.154992 0.987916i \(-0.450465\pi\)
0.154992 + 0.987916i \(0.450465\pi\)
\(462\) 0 0
\(463\) 19.8604 0.922993 0.461496 0.887142i \(-0.347313\pi\)
0.461496 + 0.887142i \(0.347313\pi\)
\(464\) 2.94433 0.136687
\(465\) 12.3259 0.571601
\(466\) −1.49294 −0.0691592
\(467\) 18.5443 0.858128 0.429064 0.903274i \(-0.358843\pi\)
0.429064 + 0.903274i \(0.358843\pi\)
\(468\) 0.0556701 0.00257335
\(469\) 0 0
\(470\) −45.9293 −2.11856
\(471\) −9.15892 −0.422021
\(472\) 8.60118 0.395902
\(473\) −64.1576 −2.94997
\(474\) −2.94433 −0.135238
\(475\) 40.2103 1.84497
\(476\) 0 0
\(477\) 5.71709 0.261767
\(478\) 25.1193 1.14893
\(479\) 9.28317 0.424159 0.212079 0.977252i \(-0.431976\pi\)
0.212079 + 0.977252i \(0.431976\pi\)
\(480\) −3.82843 −0.174743
\(481\) −0.448288 −0.0204402
\(482\) 11.8471 0.539621
\(483\) 0 0
\(484\) 15.0941 0.686097
\(485\) −17.6153 −0.799869
\(486\) 1.00000 0.0453609
\(487\) 9.86232 0.446904 0.223452 0.974715i \(-0.428267\pi\)
0.223452 + 0.974715i \(0.428267\pi\)
\(488\) −5.77732 −0.261527
\(489\) −12.1730 −0.550484
\(490\) 0 0
\(491\) −41.3368 −1.86551 −0.932753 0.360517i \(-0.882600\pi\)
−0.932753 + 0.360517i \(0.882600\pi\)
\(492\) −4.16391 −0.187724
\(493\) −19.9412 −0.898108
\(494\) 0.231805 0.0104294
\(495\) 19.5565 0.879001
\(496\) −3.21958 −0.144563
\(497\) 0 0
\(498\) −14.7651 −0.661640
\(499\) −4.45449 −0.199410 −0.0997051 0.995017i \(-0.531790\pi\)
−0.0997051 + 0.995017i \(0.531790\pi\)
\(500\) −17.8284 −0.797311
\(501\) 21.0788 0.941732
\(502\) 8.09413 0.361259
\(503\) −9.67596 −0.431430 −0.215715 0.976456i \(-0.569208\pi\)
−0.215715 + 0.976456i \(0.569208\pi\)
\(504\) 0 0
\(505\) 56.9955 2.53627
\(506\) −5.10824 −0.227089
\(507\) −12.9969 −0.577213
\(508\) 8.54861 0.379283
\(509\) 4.58586 0.203265 0.101632 0.994822i \(-0.467593\pi\)
0.101632 + 0.994822i \(0.467593\pi\)
\(510\) 25.9290 1.14816
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.16391 0.183841
\(514\) 1.94123 0.0856240
\(515\) −23.5615 −1.03824
\(516\) 12.5596 0.552907
\(517\) −61.2831 −2.69523
\(518\) 0 0
\(519\) 13.4250 0.589294
\(520\) −0.213129 −0.00934633
\(521\) −22.9305 −1.00460 −0.502301 0.864693i \(-0.667513\pi\)
−0.502301 + 0.864693i \(0.667513\pi\)
\(522\) 2.94433 0.128870
\(523\) 3.05713 0.133679 0.0668396 0.997764i \(-0.478708\pi\)
0.0668396 + 0.997764i \(0.478708\pi\)
\(524\) −6.94123 −0.303229
\(525\) 0 0
\(526\) −29.3075 −1.27787
\(527\) 21.8054 0.949860
\(528\) −5.10824 −0.222308
\(529\) 1.00000 0.0434783
\(530\) −21.8875 −0.950730
\(531\) 8.60118 0.373260
\(532\) 0 0
\(533\) −0.231805 −0.0100406
\(534\) 2.39572 0.103673
\(535\) −15.7821 −0.682321
\(536\) 12.1544 0.524988
\(537\) 1.99690 0.0861726
\(538\) 2.62941 0.113362
\(539\) 0 0
\(540\) −3.82843 −0.164749
\(541\) −39.3350 −1.69114 −0.845571 0.533863i \(-0.820740\pi\)
−0.845571 + 0.533863i \(0.820740\pi\)
\(542\) 3.21958 0.138293
\(543\) 20.5443 0.881640
\(544\) −6.77276 −0.290380
\(545\) −76.9361 −3.29558
\(546\) 0 0
\(547\) −12.4019 −0.530268 −0.265134 0.964212i \(-0.585416\pi\)
−0.265134 + 0.964212i \(0.585416\pi\)
\(548\) 5.27981 0.225542
\(549\) −5.77732 −0.246570
\(550\) −49.3295 −2.10342
\(551\) 12.2599 0.522291
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 14.7701 0.627521
\(555\) 30.8287 1.30860
\(556\) 9.31371 0.394989
\(557\) −31.0838 −1.31706 −0.658531 0.752553i \(-0.728822\pi\)
−0.658531 + 0.752553i \(0.728822\pi\)
\(558\) −3.21958 −0.136296
\(559\) 0.699196 0.0295728
\(560\) 0 0
\(561\) 34.5969 1.46068
\(562\) 3.44992 0.145526
\(563\) −34.1901 −1.44094 −0.720471 0.693485i \(-0.756075\pi\)
−0.720471 + 0.693485i \(0.756075\pi\)
\(564\) 11.9969 0.505161
\(565\) 20.4146 0.858851
\(566\) 0.951992 0.0400152
\(567\) 0 0
\(568\) 11.9443 0.501173
\(569\) −38.4822 −1.61326 −0.806629 0.591059i \(-0.798710\pi\)
−0.806629 + 0.591059i \(0.798710\pi\)
\(570\) −15.9412 −0.667705
\(571\) −27.0815 −1.13332 −0.566662 0.823950i \(-0.691766\pi\)
−0.566662 + 0.823950i \(0.691766\pi\)
\(572\) −0.284376 −0.0118904
\(573\) −1.94743 −0.0813550
\(574\) 0 0
\(575\) 9.65685 0.402719
\(576\) 1.00000 0.0416667
\(577\) −8.33884 −0.347150 −0.173575 0.984821i \(-0.555532\pi\)
−0.173575 + 0.984821i \(0.555532\pi\)
\(578\) 28.8702 1.20084
\(579\) 4.44037 0.184536
\(580\) −11.2722 −0.468051
\(581\) 0 0
\(582\) 4.60118 0.190725
\(583\) −29.2043 −1.20952
\(584\) −2.21958 −0.0918469
\(585\) −0.213129 −0.00881180
\(586\) −14.5147 −0.599597
\(587\) 40.4714 1.67043 0.835217 0.549920i \(-0.185342\pi\)
0.835217 + 0.549920i \(0.185342\pi\)
\(588\) 0 0
\(589\) −13.4061 −0.552387
\(590\) −32.9290 −1.35567
\(591\) −15.7620 −0.648362
\(592\) −8.05257 −0.330959
\(593\) −30.3369 −1.24579 −0.622895 0.782306i \(-0.714043\pi\)
−0.622895 + 0.782306i \(0.714043\pi\)
\(594\) −5.10824 −0.209594
\(595\) 0 0
\(596\) −11.7190 −0.480028
\(597\) 19.8733 0.813362
\(598\) 0.0556701 0.00227652
\(599\) −6.79443 −0.277613 −0.138806 0.990320i \(-0.544327\pi\)
−0.138806 + 0.990320i \(0.544327\pi\)
\(600\) 9.65685 0.394239
\(601\) 21.0251 0.857633 0.428816 0.903392i \(-0.358931\pi\)
0.428816 + 0.903392i \(0.358931\pi\)
\(602\) 0 0
\(603\) 12.1544 0.494964
\(604\) 17.5565 0.714365
\(605\) −57.7868 −2.34937
\(606\) −14.8875 −0.604761
\(607\) 2.43727 0.0989259 0.0494629 0.998776i \(-0.484249\pi\)
0.0494629 + 0.998776i \(0.484249\pi\)
\(608\) 4.16391 0.168869
\(609\) 0 0
\(610\) 22.1180 0.895534
\(611\) 0.667869 0.0270191
\(612\) −6.77276 −0.273772
\(613\) 4.30872 0.174028 0.0870138 0.996207i \(-0.472268\pi\)
0.0870138 + 0.996207i \(0.472268\pi\)
\(614\) 17.2611 0.696603
\(615\) 15.9412 0.642812
\(616\) 0 0
\(617\) −12.0621 −0.485603 −0.242801 0.970076i \(-0.578066\pi\)
−0.242801 + 0.970076i \(0.578066\pi\)
\(618\) 6.15436 0.247565
\(619\) 39.0944 1.57134 0.785668 0.618648i \(-0.212319\pi\)
0.785668 + 0.618648i \(0.212319\pi\)
\(620\) 12.3259 0.495021
\(621\) 1.00000 0.0401286
\(622\) 2.37059 0.0950519
\(623\) 0 0
\(624\) 0.0556701 0.00222859
\(625\) 19.9706 0.798823
\(626\) −5.83798 −0.233333
\(627\) −21.2703 −0.849452
\(628\) −9.15892 −0.365481
\(629\) 54.5381 2.17458
\(630\) 0 0
\(631\) 13.1947 0.525273 0.262636 0.964895i \(-0.415408\pi\)
0.262636 + 0.964895i \(0.415408\pi\)
\(632\) −2.94433 −0.117119
\(633\) −10.6427 −0.423011
\(634\) 0.629412 0.0249971
\(635\) −32.7277 −1.29876
\(636\) 5.71709 0.226697
\(637\) 0 0
\(638\) −15.0403 −0.595453
\(639\) 11.9443 0.472510
\(640\) −3.82843 −0.151332
\(641\) 44.5194 1.75841 0.879206 0.476442i \(-0.158074\pi\)
0.879206 + 0.476442i \(0.158074\pi\)
\(642\) 4.12236 0.162696
\(643\) 4.30348 0.169713 0.0848563 0.996393i \(-0.472957\pi\)
0.0848563 + 0.996393i \(0.472957\pi\)
\(644\) 0 0
\(645\) −48.0836 −1.89329
\(646\) −28.2012 −1.10956
\(647\) 23.6378 0.929296 0.464648 0.885496i \(-0.346181\pi\)
0.464648 + 0.885496i \(0.346181\pi\)
\(648\) 1.00000 0.0392837
\(649\) −43.9369 −1.72468
\(650\) 0.537598 0.0210863
\(651\) 0 0
\(652\) −12.1730 −0.476733
\(653\) −17.0403 −0.666840 −0.333420 0.942778i \(-0.608203\pi\)
−0.333420 + 0.942778i \(0.608203\pi\)
\(654\) 20.0960 0.785816
\(655\) 26.5740 1.03833
\(656\) −4.16391 −0.162573
\(657\) −2.21958 −0.0865941
\(658\) 0 0
\(659\) 0.998791 0.0389074 0.0194537 0.999811i \(-0.493807\pi\)
0.0194537 + 0.999811i \(0.493807\pi\)
\(660\) 19.5565 0.761237
\(661\) 26.3652 1.02549 0.512743 0.858542i \(-0.328629\pi\)
0.512743 + 0.858542i \(0.328629\pi\)
\(662\) 13.7003 0.532477
\(663\) −0.377040 −0.0146430
\(664\) −14.7651 −0.572997
\(665\) 0 0
\(666\) −8.05257 −0.312031
\(667\) 2.94433 0.114005
\(668\) 21.0788 0.815563
\(669\) −28.5424 −1.10351
\(670\) −46.5321 −1.79769
\(671\) 29.5119 1.13930
\(672\) 0 0
\(673\) −20.3388 −0.784005 −0.392002 0.919964i \(-0.628217\pi\)
−0.392002 + 0.919964i \(0.628217\pi\)
\(674\) −19.8318 −0.763892
\(675\) 9.65685 0.371692
\(676\) −12.9969 −0.499881
\(677\) 30.5763 1.17514 0.587572 0.809172i \(-0.300084\pi\)
0.587572 + 0.809172i \(0.300084\pi\)
\(678\) −5.33238 −0.204789
\(679\) 0 0
\(680\) 25.9290 0.994332
\(681\) −3.68319 −0.141140
\(682\) 16.4464 0.629765
\(683\) 40.0360 1.53194 0.765968 0.642878i \(-0.222260\pi\)
0.765968 + 0.642878i \(0.222260\pi\)
\(684\) 4.16391 0.159211
\(685\) −20.2134 −0.772314
\(686\) 0 0
\(687\) −12.1639 −0.464082
\(688\) 12.5596 0.478831
\(689\) 0.318271 0.0121252
\(690\) −3.82843 −0.145746
\(691\) 20.6862 0.786940 0.393470 0.919338i \(-0.371275\pi\)
0.393470 + 0.919338i \(0.371275\pi\)
\(692\) 13.4250 0.510344
\(693\) 0 0
\(694\) 5.55774 0.210969
\(695\) −35.6569 −1.35254
\(696\) 2.94433 0.111605
\(697\) 28.2012 1.06820
\(698\) −15.5027 −0.586788
\(699\) −1.49294 −0.0564683
\(700\) 0 0
\(701\) −18.5954 −0.702339 −0.351170 0.936312i \(-0.614216\pi\)
−0.351170 + 0.936312i \(0.614216\pi\)
\(702\) 0.0556701 0.00210113
\(703\) −33.5302 −1.26462
\(704\) −5.10824 −0.192524
\(705\) −45.9293 −1.72980
\(706\) 15.9412 0.599956
\(707\) 0 0
\(708\) 8.60118 0.323252
\(709\) −19.8886 −0.746930 −0.373465 0.927644i \(-0.621830\pi\)
−0.373465 + 0.927644i \(0.621830\pi\)
\(710\) −45.7280 −1.71614
\(711\) −2.94433 −0.110421
\(712\) 2.39572 0.0897833
\(713\) −3.21958 −0.120574
\(714\) 0 0
\(715\) 1.08871 0.0407156
\(716\) 1.99690 0.0746277
\(717\) 25.1193 0.938096
\(718\) 32.4047 1.20933
\(719\) −28.5412 −1.06441 −0.532204 0.846616i \(-0.678636\pi\)
−0.532204 + 0.846616i \(0.678636\pi\)
\(720\) −3.82843 −0.142677
\(721\) 0 0
\(722\) −1.66184 −0.0618474
\(723\) 11.8471 0.440599
\(724\) 20.5443 0.763523
\(725\) 28.4330 1.05597
\(726\) 15.0941 0.560196
\(727\) −24.4745 −0.907710 −0.453855 0.891076i \(-0.649952\pi\)
−0.453855 + 0.891076i \(0.649952\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.49751 0.314507
\(731\) −85.0633 −3.14618
\(732\) −5.77732 −0.213536
\(733\) 29.9168 1.10500 0.552501 0.833512i \(-0.313674\pi\)
0.552501 + 0.833512i \(0.313674\pi\)
\(734\) 10.8669 0.401104
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −62.0874 −2.28702
\(738\) −4.16391 −0.153276
\(739\) 4.97969 0.183181 0.0915904 0.995797i \(-0.470805\pi\)
0.0915904 + 0.995797i \(0.470805\pi\)
\(740\) 30.8287 1.13328
\(741\) 0.231805 0.00851559
\(742\) 0 0
\(743\) 34.0502 1.24918 0.624589 0.780953i \(-0.285266\pi\)
0.624589 + 0.780953i \(0.285266\pi\)
\(744\) −3.21958 −0.118036
\(745\) 44.8652 1.64374
\(746\) −0.216482 −0.00792598
\(747\) −14.7651 −0.540227
\(748\) 34.5969 1.26499
\(749\) 0 0
\(750\) −17.8284 −0.651002
\(751\) −45.8165 −1.67187 −0.835933 0.548831i \(-0.815073\pi\)
−0.835933 + 0.548831i \(0.815073\pi\)
\(752\) 11.9969 0.437482
\(753\) 8.09413 0.294966
\(754\) 0.163911 0.00596929
\(755\) −67.2139 −2.44616
\(756\) 0 0
\(757\) 17.0256 0.618804 0.309402 0.950931i \(-0.399871\pi\)
0.309402 + 0.950931i \(0.399871\pi\)
\(758\) 8.93667 0.324594
\(759\) −5.10824 −0.185417
\(760\) −15.9412 −0.578249
\(761\) −45.2961 −1.64198 −0.820991 0.570942i \(-0.806578\pi\)
−0.820991 + 0.570942i \(0.806578\pi\)
\(762\) 8.54861 0.309684
\(763\) 0 0
\(764\) −1.94743 −0.0704555
\(765\) 25.9290 0.937465
\(766\) 26.8287 0.969360
\(767\) 0.478829 0.0172895
\(768\) 1.00000 0.0360844
\(769\) 13.8318 0.498787 0.249393 0.968402i \(-0.419769\pi\)
0.249393 + 0.968402i \(0.419769\pi\)
\(770\) 0 0
\(771\) 1.94123 0.0699117
\(772\) 4.44037 0.159812
\(773\) 7.17467 0.258055 0.129028 0.991641i \(-0.458814\pi\)
0.129028 + 0.991641i \(0.458814\pi\)
\(774\) 12.5596 0.451447
\(775\) −31.0910 −1.11682
\(776\) 4.60118 0.165173
\(777\) 0 0
\(778\) −3.66598 −0.131432
\(779\) −17.3382 −0.621204
\(780\) −0.213129 −0.00763124
\(781\) −61.0145 −2.18327
\(782\) −6.77276 −0.242193
\(783\) 2.94433 0.105222
\(784\) 0 0
\(785\) 35.0643 1.25150
\(786\) −6.94123 −0.247585
\(787\) −7.14025 −0.254522 −0.127261 0.991869i \(-0.540619\pi\)
−0.127261 + 0.991869i \(0.540619\pi\)
\(788\) −15.7620 −0.561498
\(789\) −29.3075 −1.04337
\(790\) 11.2722 0.401045
\(791\) 0 0
\(792\) −5.10824 −0.181513
\(793\) −0.321624 −0.0114212
\(794\) 19.9572 0.708256
\(795\) −21.8875 −0.776268
\(796\) 19.8733 0.704392
\(797\) 1.94889 0.0690333 0.0345167 0.999404i \(-0.489011\pi\)
0.0345167 + 0.999404i \(0.489011\pi\)
\(798\) 0 0
\(799\) −81.2521 −2.87449
\(800\) 9.65685 0.341421
\(801\) 2.39572 0.0846485
\(802\) 0.386165 0.0136360
\(803\) 11.3382 0.400115
\(804\) 12.1544 0.428651
\(805\) 0 0
\(806\) −0.179235 −0.00631327
\(807\) 2.62941 0.0925597
\(808\) −14.8875 −0.523739
\(809\) 39.9350 1.40404 0.702020 0.712157i \(-0.252281\pi\)
0.702020 + 0.712157i \(0.252281\pi\)
\(810\) −3.82843 −0.134517
\(811\) −39.4035 −1.38364 −0.691822 0.722068i \(-0.743192\pi\)
−0.691822 + 0.722068i \(0.743192\pi\)
\(812\) 0 0
\(813\) 3.21958 0.112916
\(814\) 41.1345 1.44176
\(815\) 46.6036 1.63245
\(816\) −6.77276 −0.237094
\(817\) 52.2972 1.82965
\(818\) −15.5314 −0.543043
\(819\) 0 0
\(820\) 15.9412 0.556692
\(821\) −39.4978 −1.37848 −0.689241 0.724532i \(-0.742056\pi\)
−0.689241 + 0.724532i \(0.742056\pi\)
\(822\) 5.27981 0.184155
\(823\) 8.81054 0.307116 0.153558 0.988140i \(-0.450927\pi\)
0.153558 + 0.988140i \(0.450927\pi\)
\(824\) 6.15436 0.214397
\(825\) −49.3295 −1.71743
\(826\) 0 0
\(827\) 26.5271 0.922437 0.461219 0.887286i \(-0.347412\pi\)
0.461219 + 0.887286i \(0.347412\pi\)
\(828\) 1.00000 0.0347524
\(829\) −17.1070 −0.594152 −0.297076 0.954854i \(-0.596011\pi\)
−0.297076 + 0.954854i \(0.596011\pi\)
\(830\) 56.5271 1.96208
\(831\) 14.7701 0.512369
\(832\) 0.0556701 0.00193001
\(833\) 0 0
\(834\) 9.31371 0.322507
\(835\) −80.6987 −2.79269
\(836\) −21.2703 −0.735647
\(837\) −3.21958 −0.111285
\(838\) 0.559628 0.0193320
\(839\) −13.4585 −0.464640 −0.232320 0.972639i \(-0.574632\pi\)
−0.232320 + 0.972639i \(0.574632\pi\)
\(840\) 0 0
\(841\) −20.3309 −0.701066
\(842\) 17.2024 0.592833
\(843\) 3.44992 0.118822
\(844\) −10.6427 −0.366338
\(845\) 49.7577 1.71172
\(846\) 11.9969 0.412462
\(847\) 0 0
\(848\) 5.71709 0.196326
\(849\) 0.951992 0.0326723
\(850\) −65.4035 −2.24332
\(851\) −8.05257 −0.276039
\(852\) 11.9443 0.409206
\(853\) 38.3278 1.31232 0.656160 0.754622i \(-0.272180\pi\)
0.656160 + 0.754622i \(0.272180\pi\)
\(854\) 0 0
\(855\) −15.9412 −0.545179
\(856\) 4.12236 0.140899
\(857\) 45.5957 1.55752 0.778759 0.627323i \(-0.215849\pi\)
0.778759 + 0.627323i \(0.215849\pi\)
\(858\) −0.284376 −0.00970845
\(859\) 27.2420 0.929486 0.464743 0.885446i \(-0.346147\pi\)
0.464743 + 0.885446i \(0.346147\pi\)
\(860\) −48.0836 −1.63964
\(861\) 0 0
\(862\) 18.8501 0.642037
\(863\) 56.4633 1.92203 0.961017 0.276489i \(-0.0891709\pi\)
0.961017 + 0.276489i \(0.0891709\pi\)
\(864\) 1.00000 0.0340207
\(865\) −51.3968 −1.74754
\(866\) 0.435383 0.0147949
\(867\) 28.8702 0.980485
\(868\) 0 0
\(869\) 15.0403 0.510209
\(870\) −11.2722 −0.382162
\(871\) 0.676635 0.0229269
\(872\) 20.0960 0.680537
\(873\) 4.60118 0.155726
\(874\) 4.16391 0.140846
\(875\) 0 0
\(876\) −2.21958 −0.0749927
\(877\) 15.1902 0.512938 0.256469 0.966552i \(-0.417441\pi\)
0.256469 + 0.966552i \(0.417441\pi\)
\(878\) −34.0048 −1.14761
\(879\) −14.5147 −0.489569
\(880\) 19.5565 0.659250
\(881\) 2.60885 0.0878942 0.0439471 0.999034i \(-0.486007\pi\)
0.0439471 + 0.999034i \(0.486007\pi\)
\(882\) 0 0
\(883\) −25.7863 −0.867779 −0.433890 0.900966i \(-0.642859\pi\)
−0.433890 + 0.900966i \(0.642859\pi\)
\(884\) −0.377040 −0.0126812
\(885\) −32.9290 −1.10690
\(886\) 16.9412 0.569152
\(887\) 14.2692 0.479111 0.239556 0.970883i \(-0.422998\pi\)
0.239556 + 0.970883i \(0.422998\pi\)
\(888\) −8.05257 −0.270227
\(889\) 0 0
\(890\) −9.17183 −0.307440
\(891\) −5.10824 −0.171133
\(892\) −28.5424 −0.955671
\(893\) 49.9540 1.67165
\(894\) −11.7190 −0.391941
\(895\) −7.64499 −0.255544
\(896\) 0 0
\(897\) 0.0556701 0.00185877
\(898\) 5.28937 0.176508
\(899\) −9.47951 −0.316159
\(900\) 9.65685 0.321895
\(901\) −38.7204 −1.28996
\(902\) 21.2703 0.708222
\(903\) 0 0
\(904\) −5.33238 −0.177352
\(905\) −78.6524 −2.61449
\(906\) 17.5565 0.583277
\(907\) −41.5206 −1.37867 −0.689335 0.724443i \(-0.742097\pi\)
−0.689335 + 0.724443i \(0.742097\pi\)
\(908\) −3.68319 −0.122231
\(909\) −14.8875 −0.493786
\(910\) 0 0
\(911\) 8.50317 0.281723 0.140861 0.990029i \(-0.455013\pi\)
0.140861 + 0.990029i \(0.455013\pi\)
\(912\) 4.16391 0.137881
\(913\) 75.4237 2.49616
\(914\) 6.95723 0.230125
\(915\) 22.1180 0.731200
\(916\) −12.1639 −0.401907
\(917\) 0 0
\(918\) −6.77276 −0.223534
\(919\) 14.5873 0.481191 0.240596 0.970625i \(-0.422657\pi\)
0.240596 + 0.970625i \(0.422657\pi\)
\(920\) −3.82843 −0.126220
\(921\) 17.2611 0.568774
\(922\) 6.65564 0.219192
\(923\) 0.664942 0.0218868
\(924\) 0 0
\(925\) −77.7625 −2.55682
\(926\) 19.8604 0.652654
\(927\) 6.15436 0.202136
\(928\) 2.94433 0.0966524
\(929\) −31.3816 −1.02960 −0.514798 0.857311i \(-0.672133\pi\)
−0.514798 + 0.857311i \(0.672133\pi\)
\(930\) 12.3259 0.404183
\(931\) 0 0
\(932\) −1.49294 −0.0489030
\(933\) 2.37059 0.0776096
\(934\) 18.5443 0.606788
\(935\) −132.452 −4.33163
\(936\) 0.0556701 0.00181964
\(937\) −12.9726 −0.423795 −0.211897 0.977292i \(-0.567964\pi\)
−0.211897 + 0.977292i \(0.567964\pi\)
\(938\) 0 0
\(939\) −5.83798 −0.190515
\(940\) −45.9293 −1.49805
\(941\) 9.88142 0.322125 0.161063 0.986944i \(-0.448508\pi\)
0.161063 + 0.986944i \(0.448508\pi\)
\(942\) −9.15892 −0.298414
\(943\) −4.16391 −0.135596
\(944\) 8.60118 0.279945
\(945\) 0 0
\(946\) −64.1576 −2.08594
\(947\) 26.1607 0.850109 0.425054 0.905168i \(-0.360255\pi\)
0.425054 + 0.905168i \(0.360255\pi\)
\(948\) −2.94433 −0.0956274
\(949\) −0.123564 −0.00401107
\(950\) 40.2103 1.30459
\(951\) 0.629412 0.0204101
\(952\) 0 0
\(953\) 11.8360 0.383405 0.191703 0.981453i \(-0.438599\pi\)
0.191703 + 0.981453i \(0.438599\pi\)
\(954\) 5.71709 0.185098
\(955\) 7.45559 0.241257
\(956\) 25.1193 0.812415
\(957\) −15.0403 −0.486185
\(958\) 9.28317 0.299926
\(959\) 0 0
\(960\) −3.82843 −0.123562
\(961\) −20.6343 −0.665622
\(962\) −0.448288 −0.0144534
\(963\) 4.12236 0.132841
\(964\) 11.8471 0.381570
\(965\) −16.9996 −0.547238
\(966\) 0 0
\(967\) −23.7730 −0.764488 −0.382244 0.924061i \(-0.624849\pi\)
−0.382244 + 0.924061i \(0.624849\pi\)
\(968\) 15.0941 0.485144
\(969\) −28.2012 −0.905952
\(970\) −17.6153 −0.565593
\(971\) 5.57805 0.179008 0.0895041 0.995986i \(-0.471472\pi\)
0.0895041 + 0.995986i \(0.471472\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 9.86232 0.316009
\(975\) 0.537598 0.0172169
\(976\) −5.77732 −0.184927
\(977\) −37.4656 −1.19863 −0.599316 0.800512i \(-0.704561\pi\)
−0.599316 + 0.800512i \(0.704561\pi\)
\(978\) −12.1730 −0.389251
\(979\) −12.2379 −0.391125
\(980\) 0 0
\(981\) 20.0960 0.641616
\(982\) −41.3368 −1.31911
\(983\) 9.25494 0.295187 0.147593 0.989048i \(-0.452847\pi\)
0.147593 + 0.989048i \(0.452847\pi\)
\(984\) −4.16391 −0.132741
\(985\) 60.3437 1.92271
\(986\) −19.9412 −0.635058
\(987\) 0 0
\(988\) 0.231805 0.00737471
\(989\) 12.5596 0.399373
\(990\) 19.5565 0.621547
\(991\) −3.35915 −0.106707 −0.0533535 0.998576i \(-0.516991\pi\)
−0.0533535 + 0.998576i \(0.516991\pi\)
\(992\) −3.21958 −0.102222
\(993\) 13.7003 0.434766
\(994\) 0 0
\(995\) −76.0836 −2.41201
\(996\) −14.7651 −0.467850
\(997\) −3.35337 −0.106202 −0.0531012 0.998589i \(-0.516911\pi\)
−0.0531012 + 0.998589i \(0.516911\pi\)
\(998\) −4.45449 −0.141004
\(999\) −8.05257 −0.254772
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 6762.2.a.cp.1.1 4
7.2 even 3 966.2.i.l.277.4 8
7.4 even 3 966.2.i.l.415.4 yes 8
7.6 odd 2 6762.2.a.cm.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.l.277.4 8 7.2 even 3
966.2.i.l.415.4 yes 8 7.4 even 3
6762.2.a.cm.1.3 4 7.6 odd 2
6762.2.a.cp.1.1 4 1.1 even 1 trivial