Properties

Label 6762.2.a.cn.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: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.18398\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.18398 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.18398 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.18398 q^{10} -1.67440 q^{11} -1.00000 q^{12} -2.27259 q^{13} -3.18398 q^{15} +1.00000 q^{16} -0.367959 q^{17} +1.00000 q^{18} +3.41421 q^{19} +3.18398 q^{20} -1.67440 q^{22} -1.00000 q^{23} -1.00000 q^{24} +5.13772 q^{25} -2.27259 q^{26} -1.00000 q^{27} +3.44417 q^{29} -3.18398 q^{30} +3.41421 q^{31} +1.00000 q^{32} +1.67440 q^{33} -0.367959 q^{34} +1.00000 q^{36} -0.0953661 q^{37} +3.41421 q^{38} +2.27259 q^{39} +3.18398 q^{40} +0.681153 q^{41} -7.94699 q^{43} -1.67440 q^{44} +3.18398 q^{45} -1.00000 q^{46} +1.64445 q^{47} -1.00000 q^{48} +5.13772 q^{50} +0.367959 q^{51} -2.27259 q^{52} +9.69921 q^{53} -1.00000 q^{54} -5.33125 q^{55} -3.41421 q^{57} +3.44417 q^{58} +3.15403 q^{59} -3.18398 q^{60} +13.9594 q^{61} +3.41421 q^{62} +1.00000 q^{64} -7.23589 q^{65} +1.67440 q^{66} +5.19639 q^{67} -0.367959 q^{68} +1.00000 q^{69} +1.67440 q^{71} +1.00000 q^{72} +11.1310 q^{73} -0.0953661 q^{74} -5.13772 q^{75} +3.41421 q^{76} +2.27259 q^{78} -4.32560 q^{79} +3.18398 q^{80} +1.00000 q^{81} +0.681153 q^{82} -4.74547 q^{83} -1.17157 q^{85} -7.94699 q^{86} -3.44417 q^{87} -1.67440 q^{88} +11.3912 q^{89} +3.18398 q^{90} -1.00000 q^{92} -3.41421 q^{93} +1.64445 q^{94} +10.8708 q^{95} -1.00000 q^{96} +8.29014 q^{97} -1.67440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + 6q^{10} - 4q^{12} + 10q^{13} - 6q^{15} + 4q^{16} + 12q^{17} + 4q^{18} + 8q^{19} + 6q^{20} - 4q^{23} - 4q^{24} + 6q^{25} + 10q^{26} - 4q^{27} + 6q^{29} - 6q^{30} + 8q^{31} + 4q^{32} + 12q^{34} + 4q^{36} - 6q^{37} + 8q^{38} - 10q^{39} + 6q^{40} + 14q^{41} - 6q^{43} + 6q^{45} - 4q^{46} + 2q^{47} - 4q^{48} + 6q^{50} - 12q^{51} + 10q^{52} - 4q^{53} - 4q^{54} + 8q^{55} - 8q^{57} + 6q^{58} + 8q^{59} - 6q^{60} + 12q^{61} + 8q^{62} + 4q^{64} + 6q^{65} - 4q^{67} + 12q^{68} + 4q^{69} + 4q^{72} + 12q^{73} - 6q^{74} - 6q^{75} + 8q^{76} - 10q^{78} - 24q^{79} + 6q^{80} + 4q^{81} + 14q^{82} + 16q^{83} - 16q^{85} - 6q^{86} - 6q^{87} + 12q^{89} + 6q^{90} - 4q^{92} - 8q^{93} + 2q^{94} + 12q^{95} - 4q^{96} + 30q^{97} + O(q^{100}) \)

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
\(5\) 3.18398 1.42392 0.711959 0.702221i \(-0.247808\pi\)
0.711959 + 0.702221i \(0.247808\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.18398 1.00686
\(11\) −1.67440 −0.504850 −0.252425 0.967616i \(-0.581228\pi\)
−0.252425 + 0.967616i \(0.581228\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.27259 −0.630304 −0.315152 0.949041i \(-0.602055\pi\)
−0.315152 + 0.949041i \(0.602055\pi\)
\(14\) 0 0
\(15\) −3.18398 −0.822100
\(16\) 1.00000 0.250000
\(17\) −0.367959 −0.0892431 −0.0446215 0.999004i \(-0.514208\pi\)
−0.0446215 + 0.999004i \(0.514208\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 3.18398 0.711959
\(21\) 0 0
\(22\) −1.67440 −0.356983
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 5.13772 1.02754
\(26\) −2.27259 −0.445692
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.44417 0.639565 0.319783 0.947491i \(-0.396390\pi\)
0.319783 + 0.947491i \(0.396390\pi\)
\(30\) −3.18398 −0.581312
\(31\) 3.41421 0.613211 0.306605 0.951837i \(-0.400807\pi\)
0.306605 + 0.951837i \(0.400807\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.67440 0.291476
\(34\) −0.367959 −0.0631044
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.0953661 −0.0156781 −0.00783905 0.999969i \(-0.502495\pi\)
−0.00783905 + 0.999969i \(0.502495\pi\)
\(38\) 3.41421 0.553859
\(39\) 2.27259 0.363906
\(40\) 3.18398 0.503431
\(41\) 0.681153 0.106378 0.0531891 0.998584i \(-0.483061\pi\)
0.0531891 + 0.998584i \(0.483061\pi\)
\(42\) 0 0
\(43\) −7.94699 −1.21190 −0.605952 0.795501i \(-0.707208\pi\)
−0.605952 + 0.795501i \(0.707208\pi\)
\(44\) −1.67440 −0.252425
\(45\) 3.18398 0.474640
\(46\) −1.00000 −0.147442
\(47\) 1.64445 0.239867 0.119934 0.992782i \(-0.461732\pi\)
0.119934 + 0.992782i \(0.461732\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 5.13772 0.726584
\(51\) 0.367959 0.0515245
\(52\) −2.27259 −0.315152
\(53\) 9.69921 1.33229 0.666145 0.745823i \(-0.267943\pi\)
0.666145 + 0.745823i \(0.267943\pi\)
\(54\) −1.00000 −0.136083
\(55\) −5.33125 −0.718866
\(56\) 0 0
\(57\) −3.41421 −0.452224
\(58\) 3.44417 0.452241
\(59\) 3.15403 0.410619 0.205310 0.978697i \(-0.434180\pi\)
0.205310 + 0.978697i \(0.434180\pi\)
\(60\) −3.18398 −0.411050
\(61\) 13.9594 1.78732 0.893659 0.448747i \(-0.148130\pi\)
0.893659 + 0.448747i \(0.148130\pi\)
\(62\) 3.41421 0.433606
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.23589 −0.897501
\(66\) 1.67440 0.206104
\(67\) 5.19639 0.634840 0.317420 0.948285i \(-0.397184\pi\)
0.317420 + 0.948285i \(0.397184\pi\)
\(68\) −0.367959 −0.0446215
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.67440 0.198715 0.0993573 0.995052i \(-0.468321\pi\)
0.0993573 + 0.995052i \(0.468321\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.1310 1.30278 0.651391 0.758742i \(-0.274186\pi\)
0.651391 + 0.758742i \(0.274186\pi\)
\(74\) −0.0953661 −0.0110861
\(75\) −5.13772 −0.593253
\(76\) 3.41421 0.391637
\(77\) 0 0
\(78\) 2.27259 0.257320
\(79\) −4.32560 −0.486668 −0.243334 0.969943i \(-0.578241\pi\)
−0.243334 + 0.969943i \(0.578241\pi\)
\(80\) 3.18398 0.355980
\(81\) 1.00000 0.111111
\(82\) 0.681153 0.0752207
\(83\) −4.74547 −0.520883 −0.260441 0.965490i \(-0.583868\pi\)
−0.260441 + 0.965490i \(0.583868\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) −7.94699 −0.856946
\(87\) −3.44417 −0.369253
\(88\) −1.67440 −0.178492
\(89\) 11.3912 1.20746 0.603730 0.797189i \(-0.293680\pi\)
0.603730 + 0.797189i \(0.293680\pi\)
\(90\) 3.18398 0.335621
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −3.41421 −0.354037
\(94\) 1.64445 0.169612
\(95\) 10.8708 1.11532
\(96\) −1.00000 −0.102062
\(97\) 8.29014 0.841736 0.420868 0.907122i \(-0.361726\pi\)
0.420868 + 0.907122i \(0.361726\pi\)
\(98\) 0 0
\(99\) −1.67440 −0.168283
\(100\) 5.13772 0.513772
\(101\) 10.1597 1.01093 0.505463 0.862848i \(-0.331322\pi\)
0.505463 + 0.862848i \(0.331322\pi\)
\(102\) 0.367959 0.0364333
\(103\) 8.27259 0.815123 0.407561 0.913178i \(-0.366379\pi\)
0.407561 + 0.913178i \(0.366379\pi\)
\(104\) −2.27259 −0.222846
\(105\) 0 0
\(106\) 9.69921 0.942071
\(107\) −0.283242 −0.0273820 −0.0136910 0.999906i \(-0.504358\pi\)
−0.0136910 + 0.999906i \(0.504358\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −15.1857 −1.45453 −0.727265 0.686357i \(-0.759209\pi\)
−0.727265 + 0.686357i \(0.759209\pi\)
\(110\) −5.33125 −0.508315
\(111\) 0.0953661 0.00905175
\(112\) 0 0
\(113\) 7.57903 0.712975 0.356488 0.934300i \(-0.383974\pi\)
0.356488 + 0.934300i \(0.383974\pi\)
\(114\) −3.41421 −0.319770
\(115\) −3.18398 −0.296908
\(116\) 3.44417 0.319783
\(117\) −2.27259 −0.210101
\(118\) 3.15403 0.290352
\(119\) 0 0
\(120\) −3.18398 −0.290656
\(121\) −8.19639 −0.745126
\(122\) 13.9594 1.26382
\(123\) −0.681153 −0.0614175
\(124\) 3.41421 0.306605
\(125\) 0.438512 0.0392217
\(126\) 0 0
\(127\) 5.62139 0.498818 0.249409 0.968398i \(-0.419764\pi\)
0.249409 + 0.968398i \(0.419764\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.94699 0.699694
\(130\) −7.23589 −0.634629
\(131\) −6.23309 −0.544588 −0.272294 0.962214i \(-0.587782\pi\)
−0.272294 + 0.962214i \(0.587782\pi\)
\(132\) 1.67440 0.145738
\(133\) 0 0
\(134\) 5.19639 0.448899
\(135\) −3.18398 −0.274033
\(136\) −0.367959 −0.0315522
\(137\) 7.51913 0.642403 0.321201 0.947011i \(-0.395913\pi\)
0.321201 + 0.947011i \(0.395913\pi\)
\(138\) 1.00000 0.0851257
\(139\) 4.46736 0.378917 0.189458 0.981889i \(-0.439327\pi\)
0.189458 + 0.981889i \(0.439327\pi\)
\(140\) 0 0
\(141\) −1.64445 −0.138487
\(142\) 1.67440 0.140512
\(143\) 3.80523 0.318209
\(144\) 1.00000 0.0833333
\(145\) 10.9662 0.910689
\(146\) 11.1310 0.921206
\(147\) 0 0
\(148\) −0.0953661 −0.00783905
\(149\) −16.8956 −1.38414 −0.692071 0.721830i \(-0.743301\pi\)
−0.692071 + 0.721830i \(0.743301\pi\)
\(150\) −5.13772 −0.419493
\(151\) −5.81212 −0.472984 −0.236492 0.971633i \(-0.575998\pi\)
−0.236492 + 0.971633i \(0.575998\pi\)
\(152\) 3.41421 0.276929
\(153\) −0.367959 −0.0297477
\(154\) 0 0
\(155\) 10.8708 0.873163
\(156\) 2.27259 0.181953
\(157\) 13.0136 1.03860 0.519301 0.854591i \(-0.326192\pi\)
0.519301 + 0.854591i \(0.326192\pi\)
\(158\) −4.32560 −0.344126
\(159\) −9.69921 −0.769198
\(160\) 3.18398 0.251716
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0.681153 0.0531891
\(165\) 5.33125 0.415038
\(166\) −4.74547 −0.368320
\(167\) −0.685564 −0.0530505 −0.0265253 0.999648i \(-0.508444\pi\)
−0.0265253 + 0.999648i \(0.508444\pi\)
\(168\) 0 0
\(169\) −7.83532 −0.602717
\(170\) −1.17157 −0.0898555
\(171\) 3.41421 0.261091
\(172\) −7.94699 −0.605952
\(173\) −2.26974 −0.172565 −0.0862824 0.996271i \(-0.527499\pi\)
−0.0862824 + 0.996271i \(0.527499\pi\)
\(174\) −3.44417 −0.261102
\(175\) 0 0
\(176\) −1.67440 −0.126213
\(177\) −3.15403 −0.237071
\(178\) 11.3912 0.853803
\(179\) 5.32840 0.398263 0.199132 0.979973i \(-0.436188\pi\)
0.199132 + 0.979973i \(0.436188\pi\)
\(180\) 3.18398 0.237320
\(181\) 1.41017 0.104817 0.0524087 0.998626i \(-0.483310\pi\)
0.0524087 + 0.998626i \(0.483310\pi\)
\(182\) 0 0
\(183\) −13.9594 −1.03191
\(184\) −1.00000 −0.0737210
\(185\) −0.303644 −0.0223243
\(186\) −3.41421 −0.250342
\(187\) 0.616110 0.0450544
\(188\) 1.64445 0.119934
\(189\) 0 0
\(190\) 10.8708 0.788650
\(191\) −10.3329 −0.747660 −0.373830 0.927497i \(-0.621956\pi\)
−0.373830 + 0.927497i \(0.621956\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 1.70986 0.123079 0.0615393 0.998105i \(-0.480399\pi\)
0.0615393 + 0.998105i \(0.480399\pi\)
\(194\) 8.29014 0.595197
\(195\) 7.23589 0.518173
\(196\) 0 0
\(197\) −6.81778 −0.485747 −0.242873 0.970058i \(-0.578090\pi\)
−0.242873 + 0.970058i \(0.578090\pi\)
\(198\) −1.67440 −0.118994
\(199\) 13.4690 0.954791 0.477395 0.878689i \(-0.341581\pi\)
0.477395 + 0.878689i \(0.341581\pi\)
\(200\) 5.13772 0.363292
\(201\) −5.19639 −0.366525
\(202\) 10.1597 0.714833
\(203\) 0 0
\(204\) 0.367959 0.0257623
\(205\) 2.16878 0.151474
\(206\) 8.27259 0.576379
\(207\) −1.00000 −0.0695048
\(208\) −2.27259 −0.157576
\(209\) −5.71676 −0.395436
\(210\) 0 0
\(211\) −6.44292 −0.443549 −0.221775 0.975098i \(-0.571185\pi\)
−0.221775 + 0.975098i \(0.571185\pi\)
\(212\) 9.69921 0.666145
\(213\) −1.67440 −0.114728
\(214\) −0.283242 −0.0193620
\(215\) −25.3031 −1.72565
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −15.1857 −1.02851
\(219\) −11.1310 −0.752161
\(220\) −5.33125 −0.359433
\(221\) 0.836220 0.0562503
\(222\) 0.0953661 0.00640056
\(223\) −3.85962 −0.258459 −0.129230 0.991615i \(-0.541250\pi\)
−0.129230 + 0.991615i \(0.541250\pi\)
\(224\) 0 0
\(225\) 5.13772 0.342515
\(226\) 7.57903 0.504150
\(227\) 6.05476 0.401869 0.200934 0.979605i \(-0.435602\pi\)
0.200934 + 0.979605i \(0.435602\pi\)
\(228\) −3.41421 −0.226112
\(229\) 14.0943 0.931375 0.465688 0.884949i \(-0.345807\pi\)
0.465688 + 0.884949i \(0.345807\pi\)
\(230\) −3.18398 −0.209945
\(231\) 0 0
\(232\) 3.44417 0.226121
\(233\) −1.69921 −0.111319 −0.0556596 0.998450i \(-0.517726\pi\)
−0.0556596 + 0.998450i \(0.517726\pi\)
\(234\) −2.27259 −0.148564
\(235\) 5.23589 0.341552
\(236\) 3.15403 0.205310
\(237\) 4.32560 0.280978
\(238\) 0 0
\(239\) −0.0982225 −0.00635349 −0.00317674 0.999995i \(-0.501011\pi\)
−0.00317674 + 0.999995i \(0.501011\pi\)
\(240\) −3.18398 −0.205525
\(241\) −22.9775 −1.48011 −0.740054 0.672547i \(-0.765200\pi\)
−0.740054 + 0.672547i \(0.765200\pi\)
\(242\) −8.19639 −0.526884
\(243\) −1.00000 −0.0641500
\(244\) 13.9594 0.893659
\(245\) 0 0
\(246\) −0.681153 −0.0434287
\(247\) −7.75912 −0.493701
\(248\) 3.41421 0.216803
\(249\) 4.74547 0.300732
\(250\) 0.438512 0.0277339
\(251\) −0.128174 −0.00809030 −0.00404515 0.999992i \(-0.501288\pi\)
−0.00404515 + 0.999992i \(0.501288\pi\)
\(252\) 0 0
\(253\) 1.67440 0.105269
\(254\) 5.62139 0.352717
\(255\) 1.17157 0.0733667
\(256\) 1.00000 0.0625000
\(257\) −10.4757 −0.653458 −0.326729 0.945118i \(-0.605946\pi\)
−0.326729 + 0.945118i \(0.605946\pi\)
\(258\) 7.94699 0.494758
\(259\) 0 0
\(260\) −7.23589 −0.448751
\(261\) 3.44417 0.213188
\(262\) −6.23309 −0.385082
\(263\) 4.48814 0.276750 0.138375 0.990380i \(-0.455812\pi\)
0.138375 + 0.990380i \(0.455812\pi\)
\(264\) 1.67440 0.103052
\(265\) 30.8821 1.89707
\(266\) 0 0
\(267\) −11.3912 −0.697128
\(268\) 5.19639 0.317420
\(269\) −30.0594 −1.83275 −0.916376 0.400318i \(-0.868900\pi\)
−0.916376 + 0.400318i \(0.868900\pi\)
\(270\) −3.18398 −0.193771
\(271\) −8.14442 −0.494738 −0.247369 0.968921i \(-0.579566\pi\)
−0.247369 + 0.968921i \(0.579566\pi\)
\(272\) −0.367959 −0.0223108
\(273\) 0 0
\(274\) 7.51913 0.454247
\(275\) −8.60260 −0.518757
\(276\) 1.00000 0.0601929
\(277\) 23.3561 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(278\) 4.46736 0.267935
\(279\) 3.41421 0.204404
\(280\) 0 0
\(281\) −11.1434 −0.664758 −0.332379 0.943146i \(-0.607851\pi\)
−0.332379 + 0.943146i \(0.607851\pi\)
\(282\) −1.64445 −0.0979254
\(283\) −29.9707 −1.78157 −0.890787 0.454422i \(-0.849846\pi\)
−0.890787 + 0.454422i \(0.849846\pi\)
\(284\) 1.67440 0.0993573
\(285\) −10.8708 −0.643930
\(286\) 3.80523 0.225008
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.8646 −0.992036
\(290\) 10.9662 0.643955
\(291\) −8.29014 −0.485976
\(292\) 11.1310 0.651391
\(293\) 2.47002 0.144300 0.0721500 0.997394i \(-0.477014\pi\)
0.0721500 + 0.997394i \(0.477014\pi\)
\(294\) 0 0
\(295\) 10.0424 0.584689
\(296\) −0.0953661 −0.00554304
\(297\) 1.67440 0.0971585
\(298\) −16.8956 −0.978736
\(299\) 2.27259 0.131427
\(300\) −5.13772 −0.296627
\(301\) 0 0
\(302\) −5.81212 −0.334450
\(303\) −10.1597 −0.583658
\(304\) 3.41421 0.195819
\(305\) 44.4464 2.54500
\(306\) −0.367959 −0.0210348
\(307\) 34.4787 1.96780 0.983901 0.178715i \(-0.0571940\pi\)
0.983901 + 0.178715i \(0.0571940\pi\)
\(308\) 0 0
\(309\) −8.27259 −0.470611
\(310\) 10.8708 0.617419
\(311\) −7.85962 −0.445678 −0.222839 0.974855i \(-0.571532\pi\)
−0.222839 + 0.974855i \(0.571532\pi\)
\(312\) 2.27259 0.128660
\(313\) 13.0982 0.740352 0.370176 0.928962i \(-0.379297\pi\)
0.370176 + 0.928962i \(0.379297\pi\)
\(314\) 13.0136 0.734403
\(315\) 0 0
\(316\) −4.32560 −0.243334
\(317\) −17.3956 −0.977032 −0.488516 0.872555i \(-0.662462\pi\)
−0.488516 + 0.872555i \(0.662462\pi\)
\(318\) −9.69921 −0.543905
\(319\) −5.76691 −0.322885
\(320\) 3.18398 0.177990
\(321\) 0.283242 0.0158090
\(322\) 0 0
\(323\) −1.25629 −0.0699018
\(324\) 1.00000 0.0555556
\(325\) −11.6760 −0.647665
\(326\) 8.48528 0.469956
\(327\) 15.1857 0.839773
\(328\) 0.681153 0.0376104
\(329\) 0 0
\(330\) 5.33125 0.293476
\(331\) −7.96491 −0.437791 −0.218896 0.975748i \(-0.570245\pi\)
−0.218896 + 0.975748i \(0.570245\pi\)
\(332\) −4.74547 −0.260441
\(333\) −0.0953661 −0.00522603
\(334\) −0.685564 −0.0375124
\(335\) 16.5452 0.903960
\(336\) 0 0
\(337\) 2.71514 0.147903 0.0739517 0.997262i \(-0.476439\pi\)
0.0739517 + 0.997262i \(0.476439\pi\)
\(338\) −7.83532 −0.426185
\(339\) −7.57903 −0.411637
\(340\) −1.17157 −0.0635375
\(341\) −5.71676 −0.309580
\(342\) 3.41421 0.184620
\(343\) 0 0
\(344\) −7.94699 −0.428473
\(345\) 3.18398 0.171420
\(346\) −2.26974 −0.122022
\(347\) 0.459285 0.0246557 0.0123279 0.999924i \(-0.496076\pi\)
0.0123279 + 0.999924i \(0.496076\pi\)
\(348\) −3.44417 −0.184627
\(349\) 3.49313 0.186983 0.0934916 0.995620i \(-0.470197\pi\)
0.0934916 + 0.995620i \(0.470197\pi\)
\(350\) 0 0
\(351\) 2.27259 0.121302
\(352\) −1.67440 −0.0892458
\(353\) 33.8811 1.80331 0.901653 0.432460i \(-0.142354\pi\)
0.901653 + 0.432460i \(0.142354\pi\)
\(354\) −3.15403 −0.167635
\(355\) 5.33125 0.282954
\(356\) 11.3912 0.603730
\(357\) 0 0
\(358\) 5.32840 0.281615
\(359\) −5.85044 −0.308775 −0.154387 0.988010i \(-0.549340\pi\)
−0.154387 + 0.988010i \(0.549340\pi\)
\(360\) 3.18398 0.167810
\(361\) −7.34315 −0.386481
\(362\) 1.41017 0.0741171
\(363\) 8.19639 0.430199
\(364\) 0 0
\(365\) 35.4408 1.85506
\(366\) −13.9594 −0.729669
\(367\) −16.5072 −0.861671 −0.430835 0.902431i \(-0.641781\pi\)
−0.430835 + 0.902431i \(0.641781\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0.681153 0.0354594
\(370\) −0.303644 −0.0157857
\(371\) 0 0
\(372\) −3.41421 −0.177019
\(373\) 5.31619 0.275262 0.137631 0.990484i \(-0.456051\pi\)
0.137631 + 0.990484i \(0.456051\pi\)
\(374\) 0.616110 0.0318583
\(375\) −0.438512 −0.0226447
\(376\) 1.64445 0.0848059
\(377\) −7.82718 −0.403121
\(378\) 0 0
\(379\) −13.0722 −0.671472 −0.335736 0.941956i \(-0.608985\pi\)
−0.335736 + 0.941956i \(0.608985\pi\)
\(380\) 10.8708 0.557660
\(381\) −5.62139 −0.287993
\(382\) −10.3329 −0.528676
\(383\) −1.81654 −0.0928206 −0.0464103 0.998922i \(-0.514778\pi\)
−0.0464103 + 0.998922i \(0.514778\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 1.70986 0.0870297
\(387\) −7.94699 −0.403968
\(388\) 8.29014 0.420868
\(389\) 24.5101 1.24271 0.621356 0.783529i \(-0.286582\pi\)
0.621356 + 0.783529i \(0.286582\pi\)
\(390\) 7.23589 0.366403
\(391\) 0.367959 0.0186085
\(392\) 0 0
\(393\) 6.23309 0.314418
\(394\) −6.81778 −0.343475
\(395\) −13.7726 −0.692976
\(396\) −1.67440 −0.0841417
\(397\) 29.3385 1.47246 0.736229 0.676732i \(-0.236605\pi\)
0.736229 + 0.676732i \(0.236605\pi\)
\(398\) 13.4690 0.675139
\(399\) 0 0
\(400\) 5.13772 0.256886
\(401\) −7.65114 −0.382080 −0.191040 0.981582i \(-0.561186\pi\)
−0.191040 + 0.981582i \(0.561186\pi\)
\(402\) −5.19639 −0.259172
\(403\) −7.75912 −0.386509
\(404\) 10.1597 0.505463
\(405\) 3.18398 0.158213
\(406\) 0 0
\(407\) 0.159681 0.00791509
\(408\) 0.367959 0.0182167
\(409\) −14.4798 −0.715979 −0.357989 0.933726i \(-0.616538\pi\)
−0.357989 + 0.933726i \(0.616538\pi\)
\(410\) 2.16878 0.107108
\(411\) −7.51913 −0.370891
\(412\) 8.27259 0.407561
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −15.1095 −0.741695
\(416\) −2.27259 −0.111423
\(417\) −4.46736 −0.218768
\(418\) −5.71676 −0.279616
\(419\) 12.4838 0.609874 0.304937 0.952373i \(-0.401365\pi\)
0.304937 + 0.952373i \(0.401365\pi\)
\(420\) 0 0
\(421\) −25.0301 −1.21989 −0.609946 0.792443i \(-0.708809\pi\)
−0.609946 + 0.792443i \(0.708809\pi\)
\(422\) −6.44292 −0.313637
\(423\) 1.64445 0.0799558
\(424\) 9.69921 0.471035
\(425\) −1.89047 −0.0917013
\(426\) −1.67440 −0.0811249
\(427\) 0 0
\(428\) −0.283242 −0.0136910
\(429\) −3.80523 −0.183718
\(430\) −25.3031 −1.22022
\(431\) −35.8866 −1.72859 −0.864297 0.502981i \(-0.832237\pi\)
−0.864297 + 0.502981i \(0.832237\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.7538 1.28571 0.642854 0.765989i \(-0.277750\pi\)
0.642854 + 0.765989i \(0.277750\pi\)
\(434\) 0 0
\(435\) −10.9662 −0.525787
\(436\) −15.1857 −0.727265
\(437\) −3.41421 −0.163324
\(438\) −11.1310 −0.531858
\(439\) −20.5046 −0.978630 −0.489315 0.872107i \(-0.662753\pi\)
−0.489315 + 0.872107i \(0.662753\pi\)
\(440\) −5.33125 −0.254158
\(441\) 0 0
\(442\) 0.836220 0.0397749
\(443\) 16.0916 0.764536 0.382268 0.924052i \(-0.375143\pi\)
0.382268 + 0.924052i \(0.375143\pi\)
\(444\) 0.0953661 0.00452588
\(445\) 36.2692 1.71933
\(446\) −3.85962 −0.182758
\(447\) 16.8956 0.799134
\(448\) 0 0
\(449\) 37.7122 1.77975 0.889874 0.456206i \(-0.150792\pi\)
0.889874 + 0.456206i \(0.150792\pi\)
\(450\) 5.13772 0.242195
\(451\) −1.14052 −0.0537051
\(452\) 7.57903 0.356488
\(453\) 5.81212 0.273077
\(454\) 6.05476 0.284164
\(455\) 0 0
\(456\) −3.41421 −0.159885
\(457\) 1.65934 0.0776206 0.0388103 0.999247i \(-0.487643\pi\)
0.0388103 + 0.999247i \(0.487643\pi\)
\(458\) 14.0943 0.658582
\(459\) 0.367959 0.0171748
\(460\) −3.18398 −0.148454
\(461\) −15.0455 −0.700740 −0.350370 0.936611i \(-0.613944\pi\)
−0.350370 + 0.936611i \(0.613944\pi\)
\(462\) 0 0
\(463\) 5.98935 0.278349 0.139174 0.990268i \(-0.455555\pi\)
0.139174 + 0.990268i \(0.455555\pi\)
\(464\) 3.44417 0.159891
\(465\) −10.8708 −0.504121
\(466\) −1.69921 −0.0787145
\(467\) −4.07635 −0.188631 −0.0943155 0.995542i \(-0.530066\pi\)
−0.0943155 + 0.995542i \(0.530066\pi\)
\(468\) −2.27259 −0.105051
\(469\) 0 0
\(470\) 5.23589 0.241513
\(471\) −13.0136 −0.599637
\(472\) 3.15403 0.145176
\(473\) 13.3064 0.611831
\(474\) 4.32560 0.198681
\(475\) 17.5413 0.804850
\(476\) 0 0
\(477\) 9.69921 0.444096
\(478\) −0.0982225 −0.00449259
\(479\) 29.1999 1.33418 0.667089 0.744978i \(-0.267540\pi\)
0.667089 + 0.744978i \(0.267540\pi\)
\(480\) −3.18398 −0.145328
\(481\) 0.216728 0.00988196
\(482\) −22.9775 −1.04659
\(483\) 0 0
\(484\) −8.19639 −0.372563
\(485\) 26.3956 1.19856
\(486\) −1.00000 −0.0453609
\(487\) 28.3413 1.28427 0.642134 0.766593i \(-0.278049\pi\)
0.642134 + 0.766593i \(0.278049\pi\)
\(488\) 13.9594 0.631912
\(489\) −8.48528 −0.383718
\(490\) 0 0
\(491\) 34.3680 1.55100 0.775502 0.631345i \(-0.217497\pi\)
0.775502 + 0.631345i \(0.217497\pi\)
\(492\) −0.681153 −0.0307087
\(493\) −1.26731 −0.0570768
\(494\) −7.75912 −0.349099
\(495\) −5.33125 −0.239622
\(496\) 3.41421 0.153303
\(497\) 0 0
\(498\) 4.74547 0.212649
\(499\) 14.8071 0.662858 0.331429 0.943480i \(-0.392469\pi\)
0.331429 + 0.943480i \(0.392469\pi\)
\(500\) 0.438512 0.0196108
\(501\) 0.685564 0.0306287
\(502\) −0.128174 −0.00572070
\(503\) −14.4319 −0.643487 −0.321743 0.946827i \(-0.604269\pi\)
−0.321743 + 0.946827i \(0.604269\pi\)
\(504\) 0 0
\(505\) 32.3482 1.43948
\(506\) 1.67440 0.0744361
\(507\) 7.83532 0.347979
\(508\) 5.62139 0.249409
\(509\) 4.19229 0.185820 0.0929099 0.995675i \(-0.470383\pi\)
0.0929099 + 0.995675i \(0.470383\pi\)
\(510\) 1.17157 0.0518781
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −3.41421 −0.150741
\(514\) −10.4757 −0.462065
\(515\) 26.3398 1.16067
\(516\) 7.94699 0.349847
\(517\) −2.75346 −0.121097
\(518\) 0 0
\(519\) 2.26974 0.0996304
\(520\) −7.23589 −0.317315
\(521\) 12.1116 0.530619 0.265310 0.964163i \(-0.414526\pi\)
0.265310 + 0.964163i \(0.414526\pi\)
\(522\) 3.44417 0.150747
\(523\) −37.0187 −1.61872 −0.809358 0.587316i \(-0.800185\pi\)
−0.809358 + 0.587316i \(0.800185\pi\)
\(524\) −6.23309 −0.272294
\(525\) 0 0
\(526\) 4.48814 0.195692
\(527\) −1.25629 −0.0547248
\(528\) 1.67440 0.0728689
\(529\) 1.00000 0.0434783
\(530\) 30.8821 1.34143
\(531\) 3.15403 0.136873
\(532\) 0 0
\(533\) −1.54798 −0.0670506
\(534\) −11.3912 −0.492944
\(535\) −0.901837 −0.0389898
\(536\) 5.19639 0.224450
\(537\) −5.32840 −0.229937
\(538\) −30.0594 −1.29595
\(539\) 0 0
\(540\) −3.18398 −0.137017
\(541\) 38.0634 1.63647 0.818237 0.574881i \(-0.194952\pi\)
0.818237 + 0.574881i \(0.194952\pi\)
\(542\) −8.14442 −0.349833
\(543\) −1.41017 −0.0605164
\(544\) −0.367959 −0.0157761
\(545\) −48.3511 −2.07113
\(546\) 0 0
\(547\) −12.9999 −0.555837 −0.277919 0.960605i \(-0.589645\pi\)
−0.277919 + 0.960605i \(0.589645\pi\)
\(548\) 7.51913 0.321201
\(549\) 13.9594 0.595773
\(550\) −8.60260 −0.366816
\(551\) 11.7591 0.500955
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 23.3561 0.992304
\(555\) 0.303644 0.0128890
\(556\) 4.46736 0.189458
\(557\) −1.65685 −0.0702032 −0.0351016 0.999384i \(-0.511175\pi\)
−0.0351016 + 0.999384i \(0.511175\pi\)
\(558\) 3.41421 0.144535
\(559\) 18.0603 0.763868
\(560\) 0 0
\(561\) −0.616110 −0.0260122
\(562\) −11.1434 −0.470055
\(563\) 5.67901 0.239342 0.119671 0.992814i \(-0.461816\pi\)
0.119671 + 0.992814i \(0.461816\pi\)
\(564\) −1.64445 −0.0692437
\(565\) 24.1315 1.01522
\(566\) −29.9707 −1.25976
\(567\) 0 0
\(568\) 1.67440 0.0702562
\(569\) 0.205421 0.00861171 0.00430585 0.999991i \(-0.498629\pi\)
0.00430585 + 0.999991i \(0.498629\pi\)
\(570\) −10.8708 −0.455327
\(571\) 23.3137 0.975648 0.487824 0.872942i \(-0.337791\pi\)
0.487824 + 0.872942i \(0.337791\pi\)
\(572\) 3.80523 0.159105
\(573\) 10.3329 0.431662
\(574\) 0 0
\(575\) −5.13772 −0.214258
\(576\) 1.00000 0.0416667
\(577\) 25.8574 1.07646 0.538229 0.842798i \(-0.319093\pi\)
0.538229 + 0.842798i \(0.319093\pi\)
\(578\) −16.8646 −0.701475
\(579\) −1.70986 −0.0710594
\(580\) 10.9662 0.455345
\(581\) 0 0
\(582\) −8.29014 −0.343637
\(583\) −16.2404 −0.672607
\(584\) 11.1310 0.460603
\(585\) −7.23589 −0.299167
\(586\) 2.47002 0.102036
\(587\) −14.4078 −0.594675 −0.297337 0.954772i \(-0.596099\pi\)
−0.297337 + 0.954772i \(0.596099\pi\)
\(588\) 0 0
\(589\) 11.6569 0.480312
\(590\) 10.0424 0.413437
\(591\) 6.81778 0.280446
\(592\) −0.0953661 −0.00391952
\(593\) −27.0987 −1.11281 −0.556406 0.830911i \(-0.687820\pi\)
−0.556406 + 0.830911i \(0.687820\pi\)
\(594\) 1.67440 0.0687014
\(595\) 0 0
\(596\) −16.8956 −0.692071
\(597\) −13.4690 −0.551249
\(598\) 2.27259 0.0929332
\(599\) −0.895598 −0.0365932 −0.0182966 0.999833i \(-0.505824\pi\)
−0.0182966 + 0.999833i \(0.505824\pi\)
\(600\) −5.13772 −0.209747
\(601\) −7.77813 −0.317277 −0.158638 0.987337i \(-0.550710\pi\)
−0.158638 + 0.987337i \(0.550710\pi\)
\(602\) 0 0
\(603\) 5.19639 0.211613
\(604\) −5.81212 −0.236492
\(605\) −26.0971 −1.06100
\(606\) −10.1597 −0.412709
\(607\) −9.07731 −0.368436 −0.184218 0.982885i \(-0.558975\pi\)
−0.184218 + 0.982885i \(0.558975\pi\)
\(608\) 3.41421 0.138465
\(609\) 0 0
\(610\) 44.4464 1.79958
\(611\) −3.73716 −0.151189
\(612\) −0.367959 −0.0148738
\(613\) 29.6632 1.19809 0.599043 0.800717i \(-0.295548\pi\)
0.599043 + 0.800717i \(0.295548\pi\)
\(614\) 34.4787 1.39145
\(615\) −2.16878 −0.0874535
\(616\) 0 0
\(617\) 32.6274 1.31353 0.656765 0.754095i \(-0.271924\pi\)
0.656765 + 0.754095i \(0.271924\pi\)
\(618\) −8.27259 −0.332772
\(619\) −41.7967 −1.67995 −0.839976 0.542624i \(-0.817431\pi\)
−0.839976 + 0.542624i \(0.817431\pi\)
\(620\) 10.8708 0.436581
\(621\) 1.00000 0.0401286
\(622\) −7.85962 −0.315142
\(623\) 0 0
\(624\) 2.27259 0.0909765
\(625\) −24.2924 −0.971696
\(626\) 13.0982 0.523508
\(627\) 5.71676 0.228305
\(628\) 13.0136 0.519301
\(629\) 0.0350908 0.00139916
\(630\) 0 0
\(631\) −49.8551 −1.98470 −0.992351 0.123449i \(-0.960604\pi\)
−0.992351 + 0.123449i \(0.960604\pi\)
\(632\) −4.32560 −0.172063
\(633\) 6.44292 0.256083
\(634\) −17.3956 −0.690866
\(635\) 17.8984 0.710276
\(636\) −9.69921 −0.384599
\(637\) 0 0
\(638\) −5.76691 −0.228314
\(639\) 1.67440 0.0662382
\(640\) 3.18398 0.125858
\(641\) 19.0644 0.752998 0.376499 0.926417i \(-0.377128\pi\)
0.376499 + 0.926417i \(0.377128\pi\)
\(642\) 0.283242 0.0111787
\(643\) −17.7862 −0.701420 −0.350710 0.936484i \(-0.614060\pi\)
−0.350710 + 0.936484i \(0.614060\pi\)
\(644\) 0 0
\(645\) 25.3031 0.996307
\(646\) −1.25629 −0.0494281
\(647\) −5.02092 −0.197393 −0.0986963 0.995118i \(-0.531467\pi\)
−0.0986963 + 0.995118i \(0.531467\pi\)
\(648\) 1.00000 0.0392837
\(649\) −5.28110 −0.207301
\(650\) −11.6760 −0.457969
\(651\) 0 0
\(652\) 8.48528 0.332309
\(653\) −34.1205 −1.33524 −0.667618 0.744504i \(-0.732686\pi\)
−0.667618 + 0.744504i \(0.732686\pi\)
\(654\) 15.1857 0.593809
\(655\) −19.8460 −0.775449
\(656\) 0.681153 0.0265945
\(657\) 11.1310 0.434261
\(658\) 0 0
\(659\) 2.22738 0.0867663 0.0433832 0.999059i \(-0.486186\pi\)
0.0433832 + 0.999059i \(0.486186\pi\)
\(660\) 5.33125 0.207519
\(661\) 18.0730 0.702957 0.351478 0.936196i \(-0.385679\pi\)
0.351478 + 0.936196i \(0.385679\pi\)
\(662\) −7.96491 −0.309565
\(663\) −0.836220 −0.0324761
\(664\) −4.74547 −0.184160
\(665\) 0 0
\(666\) −0.0953661 −0.00369536
\(667\) −3.44417 −0.133359
\(668\) −0.685564 −0.0265253
\(669\) 3.85962 0.149222
\(670\) 16.5452 0.639196
\(671\) −23.3736 −0.902328
\(672\) 0 0
\(673\) −4.32523 −0.166725 −0.0833627 0.996519i \(-0.526566\pi\)
−0.0833627 + 0.996519i \(0.526566\pi\)
\(674\) 2.71514 0.104583
\(675\) −5.13772 −0.197751
\(676\) −7.83532 −0.301359
\(677\) 18.9553 0.728511 0.364256 0.931299i \(-0.381323\pi\)
0.364256 + 0.931299i \(0.381323\pi\)
\(678\) −7.57903 −0.291071
\(679\) 0 0
\(680\) −1.17157 −0.0449278
\(681\) −6.05476 −0.232019
\(682\) −5.71676 −0.218906
\(683\) −45.9659 −1.75884 −0.879419 0.476049i \(-0.842069\pi\)
−0.879419 + 0.476049i \(0.842069\pi\)
\(684\) 3.41421 0.130546
\(685\) 23.9408 0.914729
\(686\) 0 0
\(687\) −14.0943 −0.537730
\(688\) −7.94699 −0.302976
\(689\) −22.0424 −0.839747
\(690\) 3.18398 0.121212
\(691\) −33.3093 −1.26715 −0.633573 0.773683i \(-0.718412\pi\)
−0.633573 + 0.773683i \(0.718412\pi\)
\(692\) −2.26974 −0.0862824
\(693\) 0 0
\(694\) 0.459285 0.0174342
\(695\) 14.2240 0.539547
\(696\) −3.44417 −0.130551
\(697\) −0.250636 −0.00949352
\(698\) 3.49313 0.132217
\(699\) 1.69921 0.0642701
\(700\) 0 0
\(701\) −30.9763 −1.16996 −0.584979 0.811049i \(-0.698897\pi\)
−0.584979 + 0.811049i \(0.698897\pi\)
\(702\) 2.27259 0.0857735
\(703\) −0.325600 −0.0122802
\(704\) −1.67440 −0.0631063
\(705\) −5.23589 −0.197195
\(706\) 33.8811 1.27513
\(707\) 0 0
\(708\) −3.15403 −0.118536
\(709\) 18.8908 0.709459 0.354730 0.934969i \(-0.384573\pi\)
0.354730 + 0.934969i \(0.384573\pi\)
\(710\) 5.33125 0.200078
\(711\) −4.32560 −0.162223
\(712\) 11.3912 0.426902
\(713\) −3.41421 −0.127863
\(714\) 0 0
\(715\) 12.1158 0.453104
\(716\) 5.32840 0.199132
\(717\) 0.0982225 0.00366819
\(718\) −5.85044 −0.218337
\(719\) 19.2609 0.718310 0.359155 0.933278i \(-0.383065\pi\)
0.359155 + 0.933278i \(0.383065\pi\)
\(720\) 3.18398 0.118660
\(721\) 0 0
\(722\) −7.34315 −0.273284
\(723\) 22.9775 0.854541
\(724\) 1.41017 0.0524087
\(725\) 17.6952 0.657182
\(726\) 8.19639 0.304196
\(727\) −19.7281 −0.731673 −0.365837 0.930679i \(-0.619217\pi\)
−0.365837 + 0.930679i \(0.619217\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 35.4408 1.31172
\(731\) 2.92416 0.108154
\(732\) −13.9594 −0.515954
\(733\) −11.8070 −0.436101 −0.218050 0.975937i \(-0.569970\pi\)
−0.218050 + 0.975937i \(0.569970\pi\)
\(734\) −16.5072 −0.609293
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −8.70083 −0.320499
\(738\) 0.681153 0.0250736
\(739\) −19.4273 −0.714644 −0.357322 0.933981i \(-0.616310\pi\)
−0.357322 + 0.933981i \(0.616310\pi\)
\(740\) −0.303644 −0.0111622
\(741\) 7.75912 0.285038
\(742\) 0 0
\(743\) 51.9765 1.90683 0.953416 0.301657i \(-0.0975398\pi\)
0.953416 + 0.301657i \(0.0975398\pi\)
\(744\) −3.41421 −0.125171
\(745\) −53.7952 −1.97090
\(746\) 5.31619 0.194640
\(747\) −4.74547 −0.173628
\(748\) 0.616110 0.0225272
\(749\) 0 0
\(750\) −0.438512 −0.0160122
\(751\) −5.61698 −0.204967 −0.102483 0.994735i \(-0.532679\pi\)
−0.102483 + 0.994735i \(0.532679\pi\)
\(752\) 1.64445 0.0599668
\(753\) 0.128174 0.00467093
\(754\) −7.82718 −0.285049
\(755\) −18.5057 −0.673491
\(756\) 0 0
\(757\) −26.2810 −0.955201 −0.477600 0.878577i \(-0.658493\pi\)
−0.477600 + 0.878577i \(0.658493\pi\)
\(758\) −13.0722 −0.474802
\(759\) −1.67440 −0.0607769
\(760\) 10.8708 0.394325
\(761\) −5.14280 −0.186427 −0.0932133 0.995646i \(-0.529714\pi\)
−0.0932133 + 0.995646i \(0.529714\pi\)
\(762\) −5.62139 −0.203642
\(763\) 0 0
\(764\) −10.3329 −0.373830
\(765\) −1.17157 −0.0423583
\(766\) −1.81654 −0.0656341
\(767\) −7.16782 −0.258815
\(768\) −1.00000 −0.0360844
\(769\) −29.0158 −1.04634 −0.523168 0.852230i \(-0.675250\pi\)
−0.523168 + 0.852230i \(0.675250\pi\)
\(770\) 0 0
\(771\) 10.4757 0.377274
\(772\) 1.70986 0.0615393
\(773\) 7.93092 0.285255 0.142628 0.989776i \(-0.454445\pi\)
0.142628 + 0.989776i \(0.454445\pi\)
\(774\) −7.94699 −0.285649
\(775\) 17.5413 0.630102
\(776\) 8.29014 0.297599
\(777\) 0 0
\(778\) 24.5101 0.878729
\(779\) 2.32560 0.0833233
\(780\) 7.23589 0.259086
\(781\) −2.80361 −0.100321
\(782\) 0.367959 0.0131582
\(783\) −3.44417 −0.123084
\(784\) 0 0
\(785\) 41.4352 1.47889
\(786\) 6.23309 0.222327
\(787\) −38.7584 −1.38159 −0.690795 0.723051i \(-0.742739\pi\)
−0.690795 + 0.723051i \(0.742739\pi\)
\(788\) −6.81778 −0.242873
\(789\) −4.48814 −0.159782
\(790\) −13.7726 −0.490008
\(791\) 0 0
\(792\) −1.67440 −0.0594972
\(793\) −31.7240 −1.12655
\(794\) 29.3385 1.04119
\(795\) −30.8821 −1.09527
\(796\) 13.4690 0.477395
\(797\) −45.8149 −1.62285 −0.811424 0.584458i \(-0.801307\pi\)
−0.811424 + 0.584458i \(0.801307\pi\)
\(798\) 0 0
\(799\) −0.605089 −0.0214065
\(800\) 5.13772 0.181646
\(801\) 11.3912 0.402487
\(802\) −7.65114 −0.270171
\(803\) −18.6377 −0.657710
\(804\) −5.19639 −0.183262
\(805\) 0 0
\(806\) −7.75912 −0.273303
\(807\) 30.0594 1.05814
\(808\) 10.1597 0.357416
\(809\) 42.9477 1.50996 0.754981 0.655747i \(-0.227646\pi\)
0.754981 + 0.655747i \(0.227646\pi\)
\(810\) 3.18398 0.111874
\(811\) −53.0023 −1.86116 −0.930581 0.366087i \(-0.880697\pi\)
−0.930581 + 0.366087i \(0.880697\pi\)
\(812\) 0 0
\(813\) 8.14442 0.285637
\(814\) 0.159681 0.00559682
\(815\) 27.0170 0.946363
\(816\) 0.367959 0.0128811
\(817\) −27.1327 −0.949254
\(818\) −14.4798 −0.506273
\(819\) 0 0
\(820\) 2.16878 0.0757369
\(821\) −21.1741 −0.738980 −0.369490 0.929235i \(-0.620468\pi\)
−0.369490 + 0.929235i \(0.620468\pi\)
\(822\) −7.51913 −0.262260
\(823\) −7.06490 −0.246267 −0.123133 0.992390i \(-0.539294\pi\)
−0.123133 + 0.992390i \(0.539294\pi\)
\(824\) 8.27259 0.288189
\(825\) 8.60260 0.299504
\(826\) 0 0
\(827\) −13.5994 −0.472899 −0.236449 0.971644i \(-0.575984\pi\)
−0.236449 + 0.971644i \(0.575984\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −15.9075 −0.552490 −0.276245 0.961087i \(-0.589090\pi\)
−0.276245 + 0.961087i \(0.589090\pi\)
\(830\) −15.1095 −0.524457
\(831\) −23.3561 −0.810213
\(832\) −2.27259 −0.0787880
\(833\) 0 0
\(834\) −4.46736 −0.154692
\(835\) −2.18282 −0.0755396
\(836\) −5.71676 −0.197718
\(837\) −3.41421 −0.118012
\(838\) 12.4838 0.431246
\(839\) −29.2794 −1.01084 −0.505419 0.862874i \(-0.668662\pi\)
−0.505419 + 0.862874i \(0.668662\pi\)
\(840\) 0 0
\(841\) −17.1377 −0.590956
\(842\) −25.0301 −0.862594
\(843\) 11.1434 0.383798
\(844\) −6.44292 −0.221775
\(845\) −24.9475 −0.858220
\(846\) 1.64445 0.0565373
\(847\) 0 0
\(848\) 9.69921 0.333072
\(849\) 29.9707 1.02859
\(850\) −1.89047 −0.0648426
\(851\) 0.0953661 0.00326911
\(852\) −1.67440 −0.0573640
\(853\) 6.23750 0.213568 0.106784 0.994282i \(-0.465945\pi\)
0.106784 + 0.994282i \(0.465945\pi\)
\(854\) 0 0
\(855\) 10.8708 0.371773
\(856\) −0.283242 −0.00968101
\(857\) −3.57976 −0.122282 −0.0611412 0.998129i \(-0.519474\pi\)
−0.0611412 + 0.998129i \(0.519474\pi\)
\(858\) −3.80523 −0.129908
\(859\) −31.3206 −1.06865 −0.534323 0.845281i \(-0.679433\pi\)
−0.534323 + 0.845281i \(0.679433\pi\)
\(860\) −25.3031 −0.862827
\(861\) 0 0
\(862\) −35.8866 −1.22230
\(863\) 25.4232 0.865417 0.432709 0.901534i \(-0.357558\pi\)
0.432709 + 0.901534i \(0.357558\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −7.22679 −0.245718
\(866\) 26.7538 0.909132
\(867\) 16.8646 0.572752
\(868\) 0 0
\(869\) 7.24278 0.245695
\(870\) −10.9662 −0.371787
\(871\) −11.8093 −0.400142
\(872\) −15.1857 −0.514254
\(873\) 8.29014 0.280579
\(874\) −3.41421 −0.115487
\(875\) 0 0
\(876\) −11.1310 −0.376081
\(877\) 25.4583 0.859667 0.429833 0.902908i \(-0.358572\pi\)
0.429833 + 0.902908i \(0.358572\pi\)
\(878\) −20.5046 −0.691996
\(879\) −2.47002 −0.0833117
\(880\) −5.33125 −0.179717
\(881\) 41.3523 1.39319 0.696597 0.717462i \(-0.254696\pi\)
0.696597 + 0.717462i \(0.254696\pi\)
\(882\) 0 0
\(883\) −28.9342 −0.973713 −0.486857 0.873482i \(-0.661857\pi\)
−0.486857 + 0.873482i \(0.661857\pi\)
\(884\) 0.836220 0.0281251
\(885\) −10.0424 −0.337570
\(886\) 16.0916 0.540608
\(887\) 27.6880 0.929672 0.464836 0.885397i \(-0.346113\pi\)
0.464836 + 0.885397i \(0.346113\pi\)
\(888\) 0.0953661 0.00320028
\(889\) 0 0
\(890\) 36.2692 1.21575
\(891\) −1.67440 −0.0560945
\(892\) −3.85962 −0.129230
\(893\) 5.61450 0.187882
\(894\) 16.8956 0.565073
\(895\) 16.9655 0.567094
\(896\) 0 0
\(897\) −2.27259 −0.0758797
\(898\) 37.7122 1.25847
\(899\) 11.7591 0.392188
\(900\) 5.13772 0.171257
\(901\) −3.56891 −0.118898
\(902\) −1.14052 −0.0379752
\(903\) 0 0
\(904\) 7.57903 0.252075
\(905\) 4.48996 0.149251
\(906\) 5.81212 0.193095
\(907\) 31.9831 1.06198 0.530991 0.847378i \(-0.321820\pi\)
0.530991 + 0.847378i \(0.321820\pi\)
\(908\) 6.05476 0.200934
\(909\) 10.1597 0.336975
\(910\) 0 0
\(911\) −38.7589 −1.28414 −0.642070 0.766646i \(-0.721924\pi\)
−0.642070 + 0.766646i \(0.721924\pi\)
\(912\) −3.41421 −0.113056
\(913\) 7.94581 0.262968
\(914\) 1.65934 0.0548861
\(915\) −44.4464 −1.46935
\(916\) 14.0943 0.465688
\(917\) 0 0
\(918\) 0.367959 0.0121444
\(919\) −52.0882 −1.71823 −0.859116 0.511781i \(-0.828986\pi\)
−0.859116 + 0.511781i \(0.828986\pi\)
\(920\) −3.18398 −0.104973
\(921\) −34.4787 −1.13611
\(922\) −15.0455 −0.495498
\(923\) −3.80523 −0.125251
\(924\) 0 0
\(925\) −0.489965 −0.0161099
\(926\) 5.98935 0.196822
\(927\) 8.27259 0.271708
\(928\) 3.44417 0.113060
\(929\) 33.3337 1.09364 0.546821 0.837249i \(-0.315838\pi\)
0.546821 + 0.837249i \(0.315838\pi\)
\(930\) −10.8708 −0.356467
\(931\) 0 0
\(932\) −1.69921 −0.0556596
\(933\) 7.85962 0.257312
\(934\) −4.07635 −0.133382
\(935\) 1.96168 0.0641538
\(936\) −2.27259 −0.0742820
\(937\) 5.66127 0.184945 0.0924727 0.995715i \(-0.470523\pi\)
0.0924727 + 0.995715i \(0.470523\pi\)
\(938\) 0 0
\(939\) −13.0982 −0.427442
\(940\) 5.23589 0.170776
\(941\) 24.5250 0.799492 0.399746 0.916626i \(-0.369098\pi\)
0.399746 + 0.916626i \(0.369098\pi\)
\(942\) −13.0136 −0.424008
\(943\) −0.681153 −0.0221814
\(944\) 3.15403 0.102655
\(945\) 0 0
\(946\) 13.3064 0.432630
\(947\) 21.9392 0.712928 0.356464 0.934309i \(-0.383982\pi\)
0.356464 + 0.934309i \(0.383982\pi\)
\(948\) 4.32560 0.140489
\(949\) −25.2962 −0.821148
\(950\) 17.5413 0.569115
\(951\) 17.3956 0.564090
\(952\) 0 0
\(953\) 27.6897 0.896959 0.448479 0.893793i \(-0.351966\pi\)
0.448479 + 0.893793i \(0.351966\pi\)
\(954\) 9.69921 0.314024
\(955\) −32.8996 −1.06461
\(956\) −0.0982225 −0.00317674
\(957\) 5.76691 0.186418
\(958\) 29.1999 0.943406
\(959\) 0 0
\(960\) −3.18398 −0.102762
\(961\) −19.3431 −0.623972
\(962\) 0.216728 0.00698760
\(963\) −0.283242 −0.00912735
\(964\) −22.9775 −0.740054
\(965\) 5.44417 0.175254
\(966\) 0 0
\(967\) −3.84510 −0.123650 −0.0618251 0.998087i \(-0.519692\pi\)
−0.0618251 + 0.998087i \(0.519692\pi\)
\(968\) −8.19639 −0.263442
\(969\) 1.25629 0.0403578
\(970\) 26.3956 0.847513
\(971\) 23.3583 0.749605 0.374803 0.927105i \(-0.377711\pi\)
0.374803 + 0.927105i \(0.377711\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 28.3413 0.908114
\(975\) 11.6760 0.373930
\(976\) 13.9594 0.446829
\(977\) 15.8034 0.505596 0.252798 0.967519i \(-0.418649\pi\)
0.252798 + 0.967519i \(0.418649\pi\)
\(978\) −8.48528 −0.271329
\(979\) −19.0734 −0.609587
\(980\) 0 0
\(981\) −15.1857 −0.484843
\(982\) 34.3680 1.09673
\(983\) 17.6528 0.563037 0.281519 0.959556i \(-0.409162\pi\)
0.281519 + 0.959556i \(0.409162\pi\)
\(984\) −0.681153 −0.0217144
\(985\) −21.7077 −0.691664
\(986\) −1.26731 −0.0403594
\(987\) 0 0
\(988\) −7.75912 −0.246850
\(989\) 7.94699 0.252700
\(990\) −5.33125 −0.169438
\(991\) −15.0224 −0.477202 −0.238601 0.971118i \(-0.576689\pi\)
−0.238601 + 0.971118i \(0.576689\pi\)
\(992\) 3.41421 0.108401
\(993\) 7.96491 0.252759
\(994\) 0 0
\(995\) 42.8849 1.35954
\(996\) 4.74547 0.150366
\(997\) −33.9954 −1.07665 −0.538323 0.842739i \(-0.680942\pi\)
−0.538323 + 0.842739i \(0.680942\pi\)
\(998\) 14.8071 0.468711
\(999\) 0.0953661 0.00301725
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.cn.1.3 4
7.6 odd 2 6762.2.a.co.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.3 4 1.1 even 1 trivial
6762.2.a.co.1.2 yes 4 7.6 odd 2