Properties

Label 6762.2.a.cm.1.3
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 1.3
Root \(-2.35854\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.82843 1.71212 0.856062 0.516873i \(-0.172904\pi\)
0.856062 + 0.516873i \(0.172904\pi\)
\(6\) −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 \(-0.502457\pi\)
−0.00772006 + 0.999970i \(0.502457\pi\)
\(14\) 0 0
\(15\) −3.82843 −0.988496
\(16\) 1.00000 0.250000
\(17\) 6.77276 1.64263 0.821317 0.570471i \(-0.193239\pi\)
0.821317 + 0.570471i \(0.193239\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.16391 −0.955267 −0.477633 0.878559i \(-0.658505\pi\)
−0.477633 + 0.878559i \(0.658505\pi\)
\(20\) 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 \(-0.406635\pi\)
0.289127 + 0.957291i \(0.406635\pi\)
\(32\) 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 \(-0.394586\pi\)
0.325147 + 0.945664i \(0.394586\pi\)
\(42\) 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 \(-0.839115\pi\)
−0.874964 + 0.484188i \(0.839115\pi\)
\(48\) −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 \(-0.689156\pi\)
−0.559889 + 0.828567i \(0.689156\pi\)
\(60\) −3.82843 −0.494248
\(61\) 5.77732 0.739710 0.369855 0.929089i \(-0.379407\pi\)
0.369855 + 0.929089i \(0.379407\pi\)
\(62\) 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 \(-0.458537\pi\)
0.129891 + 0.991528i \(0.458537\pi\)
\(74\) −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 \(-0.199282\pi\)
0.810340 + 0.585960i \(0.199282\pi\)
\(84\) 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 \(-0.540526\pi\)
−0.126973 + 0.991906i \(0.540526\pi\)
\(90\) 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 \(-0.575047\pi\)
−0.233590 + 0.972335i \(0.575047\pi\)
\(98\) 0 0
\(99\) −5.10824 −0.513398
\(100\) 9.65685 0.965685
\(101\) 14.8875 1.48136 0.740678 0.671860i \(-0.234504\pi\)
0.740678 + 0.671860i \(0.234504\pi\)
\(102\) −6.77276 −0.670603
\(103\) −6.15436 −0.606407 −0.303204 0.952926i \(-0.598056\pi\)
−0.303204 + 0.952926i \(0.598056\pi\)
\(104\) −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 \(-0.401935\pi\)
0.303229 + 0.952918i \(0.401935\pi\)
\(132\) 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 \(-0.629252\pi\)
−0.394989 + 0.918686i \(0.629252\pi\)
\(140\) 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 \(-0.380905\pi\)
0.365481 + 0.930819i \(0.380905\pi\)
\(158\) −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 \(-0.803573\pi\)
−0.815563 + 0.578668i \(0.803573\pi\)
\(168\) 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 \(-0.670482\pi\)
−0.510344 + 0.859970i \(0.670482\pi\)
\(174\) −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 \(-0.776532\pi\)
−0.763523 + 0.645781i \(0.776532\pi\)
\(182\) 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 \(-0.748780\pi\)
−0.704392 + 0.709811i \(0.748780\pi\)
\(200\) 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 \(-0.0951323\pi\)
0.955671 + 0.294438i \(0.0951323\pi\)
\(224\) 0 0
\(225\) 9.65685 0.643790
\(226\) −5.33238 −0.354705
\(227\) 3.68319 0.244462 0.122231 0.992502i \(-0.460995\pi\)
0.122231 + 0.992502i \(0.460995\pi\)
\(228\) 4.16391 0.275762
\(229\) 12.1639 0.803814 0.401907 0.915681i \(-0.368348\pi\)
0.401907 + 0.915681i \(0.368348\pi\)
\(230\) 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 \(-0.624616\pi\)
−0.381570 + 0.924340i \(0.624616\pi\)
\(242\) 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 \(-0.582223\pi\)
−0.255448 + 0.966823i \(0.582223\pi\)
\(252\) 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 \(-0.519284\pi\)
−0.0605453 + 0.998165i \(0.519284\pi\)
\(258\) −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 \(-0.525543\pi\)
−0.0801590 + 0.996782i \(0.525543\pi\)
\(270\) −3.82843 −0.232991
\(271\) −3.21958 −0.195576 −0.0977878 0.995207i \(-0.531177\pi\)
−0.0977878 + 0.995207i \(0.531177\pi\)
\(272\) 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 \(-0.509008\pi\)
−0.0282950 + 0.999600i \(0.509008\pi\)
\(284\) 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 \(-0.360633\pi\)
0.423979 + 0.905672i \(0.360633\pi\)
\(294\) 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 \(-0.663943\pi\)
−0.492573 + 0.870271i \(0.663943\pi\)
\(308\) 0 0
\(309\) 6.15436 0.350109
\(310\) 12.3259 0.700066
\(311\) −2.37059 −0.134424 −0.0672119 0.997739i \(-0.521410\pi\)
−0.0672119 + 0.997739i \(0.521410\pi\)
\(312\) 0.0556701 0.00315170
\(313\) 5.83798 0.329982 0.164991 0.986295i \(-0.447240\pi\)
0.164991 + 0.986295i \(0.447240\pi\)
\(314\) 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 \(-0.363809\pi\)
0.414922 + 0.909857i \(0.363809\pi\)
\(350\) 0 0
\(351\) 0.0556701 0.00297145
\(352\) −5.10824 −0.272270
\(353\) −15.9412 −0.848466 −0.424233 0.905553i \(-0.639456\pi\)
−0.424233 + 0.905553i \(0.639456\pi\)
\(354\) 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 \(-0.591537\pi\)
−0.283623 + 0.958936i \(0.591537\pi\)
\(368\) 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 \(-0.740390\pi\)
−0.685441 + 0.728128i \(0.740390\pi\)
\(384\) −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 \(-0.666965\pi\)
−0.500812 + 0.865556i \(0.666965\pi\)
\(398\) −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 \(-0.374550\pi\)
0.383989 + 0.923338i \(0.374550\pi\)
\(410\) 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 \(-0.504351\pi\)
−0.0136698 + 0.999907i \(0.504351\pi\)
\(420\) 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 \(-0.503330\pi\)
−0.0104616 + 0.999945i \(0.503330\pi\)
\(434\) 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 \(-0.198662\pi\)
0.811481 + 0.584379i \(0.198662\pi\)
\(440\) −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 \(-0.549535\pi\)
−0.154992 + 0.987916i \(0.549535\pi\)
\(462\) 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 \(-0.641157\pi\)
−0.429064 + 0.903274i \(0.641157\pi\)
\(468\) −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 \(-0.568024\pi\)
−0.212079 + 0.977252i \(0.568024\pi\)
\(480\) −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 \(-0.430792\pi\)
0.215715 + 0.976456i \(0.430792\pi\)
\(504\) 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 \(-0.532407\pi\)
−0.101632 + 0.994822i \(0.532407\pi\)
\(510\) −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 \(-0.332487\pi\)
0.502301 + 0.864693i \(0.332487\pi\)
\(522\) 2.94433 0.128870
\(523\) −3.05713 −0.133679 −0.0668396 0.997764i \(-0.521292\pi\)
−0.0668396 + 0.997764i \(0.521292\pi\)
\(524\) 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 \(-0.243925\pi\)
0.720471 + 0.693485i \(0.243925\pi\)
\(564\) 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 \(-0.444468\pi\)
0.173575 + 0.984821i \(0.444468\pi\)
\(578\) 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 \(-0.814658\pi\)
−0.835217 + 0.549920i \(0.814658\pi\)
\(588\) 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 \(-0.285957\pi\)
0.622895 + 0.782306i \(0.285957\pi\)
\(594\) 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 \(-0.641069\pi\)
−0.428816 + 0.903392i \(0.641069\pi\)
\(602\) 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 \(-0.515751\pi\)
−0.0494629 + 0.998776i \(0.515751\pi\)
\(608\) −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 \(-0.787681\pi\)
−0.785668 + 0.618648i \(0.787681\pi\)
\(620\) 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 \(-0.527043\pi\)
−0.0848563 + 0.996393i \(0.527043\pi\)
\(644\) 0 0
\(645\) −48.0836 −1.89329
\(646\) −28.2012 −1.10956
\(647\) −23.6378 −0.929296 −0.464648 0.885496i \(-0.653819\pi\)
−0.464648 + 0.885496i \(0.653819\pi\)
\(648\) 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 \(-0.671371\pi\)
−0.512743 + 0.858542i \(0.671371\pi\)
\(662\) 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 \(-0.699916\pi\)
−0.587572 + 0.809172i \(0.699916\pi\)
\(678\) 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 \(-0.628725\pi\)
−0.393470 + 0.919338i \(0.628725\pi\)
\(692\) −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 \(-0.321364\pi\)
0.532204 + 0.846616i \(0.321364\pi\)
\(720\) 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 \(-0.350048\pi\)
0.453855 + 0.891076i \(0.350048\pi\)
\(728\) 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 \(-0.686326\pi\)
−0.552501 + 0.833512i \(0.686326\pi\)
\(734\) −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 \(-0.193422\pi\)
0.820991 + 0.570942i \(0.193422\pi\)
\(762\) −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 \(-0.580231\pi\)
−0.249393 + 0.968402i \(0.580231\pi\)
\(770\) 0 0
\(771\) 1.94123 0.0699117
\(772\) 4.44037 0.159812
\(773\) −7.17467 −0.258055 −0.129028 0.991641i \(-0.541186\pi\)
−0.129028 + 0.991641i \(0.541186\pi\)
\(774\) 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 \(-0.459381\pi\)
0.127261 + 0.991869i \(0.459381\pi\)
\(788\) −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 \(-0.510989\pi\)
−0.0345167 + 0.999404i \(0.510989\pi\)
\(798\) 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 \(-0.256808\pi\)
0.691822 + 0.722068i \(0.256808\pi\)
\(812\) 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 \(-0.403989\pi\)
0.297076 + 0.954854i \(0.403989\pi\)
\(830\) 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 \(-0.425368\pi\)
0.232320 + 0.972639i \(0.425368\pi\)
\(840\) 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 \(-0.727820\pi\)
−0.656160 + 0.754622i \(0.727820\pi\)
\(854\) 0 0
\(855\) −15.9412 −0.545179
\(856\) 4.12236 0.140899
\(857\) −45.5957 −1.55752 −0.778759 0.627323i \(-0.784151\pi\)
−0.778759 + 0.627323i \(0.784151\pi\)
\(858\) −0.284376 −0.00970845
\(859\) −27.2420 −0.929486 −0.464743 0.885446i \(-0.653853\pi\)
−0.464743 + 0.885446i \(0.653853\pi\)
\(860\) 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 \(-0.513993\pi\)
−0.0439471 + 0.999034i \(0.513993\pi\)
\(882\) 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 \(-0.577002\pi\)
−0.239556 + 0.970883i \(0.577002\pi\)
\(888\) 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 \(-0.327867\pi\)
0.514798 + 0.857311i \(0.327867\pi\)
\(930\) −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 \(-0.432036\pi\)
0.211897 + 0.977292i \(0.432036\pi\)
\(938\) 0 0
\(939\) −5.83798 −0.190515
\(940\) −45.9293 −1.49805
\(941\) −9.88142 −0.322125 −0.161063 0.986944i \(-0.551492\pi\)
−0.161063 + 0.986944i \(0.551492\pi\)
\(942\) −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 \(-0.528528\pi\)
−0.0895041 + 0.995986i \(0.528528\pi\)
\(972\) −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 \(-0.547153\pi\)
−0.147593 + 0.989048i \(0.547153\pi\)
\(984\) −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 \(-0.483089\pi\)
0.0531012 + 0.998589i \(0.483089\pi\)
\(998\) −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.cm.1.3 4
7.3 odd 6 966.2.i.l.415.4 yes 8
7.5 odd 6 966.2.i.l.277.4 8
7.6 odd 2 6762.2.a.cp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.l.277.4 8 7.5 odd 6
966.2.i.l.415.4 yes 8 7.3 odd 6
6762.2.a.cm.1.3 4 1.1 even 1 trivial
6762.2.a.cp.1.1 4 7.6 odd 2