Properties

Label 5577.2.a.y.1.6
Level $5577$
Weight $2$
Character 5577.1
Self dual yes
Analytic conductor $44.533$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.5325692073\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.53441\) of defining polynomial
Character \(\chi\) \(=\) 5577.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.53441 q^{2} +1.00000 q^{3} +4.42325 q^{4} +3.70100 q^{5} +2.53441 q^{6} -0.957295 q^{7} +6.14151 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.53441 q^{2} +1.00000 q^{3} +4.42325 q^{4} +3.70100 q^{5} +2.53441 q^{6} -0.957295 q^{7} +6.14151 q^{8} +1.00000 q^{9} +9.37985 q^{10} +1.00000 q^{11} +4.42325 q^{12} -2.42618 q^{14} +3.70100 q^{15} +6.71863 q^{16} -2.05542 q^{17} +2.53441 q^{18} +7.67886 q^{19} +16.3704 q^{20} -0.957295 q^{21} +2.53441 q^{22} -4.20997 q^{23} +6.14151 q^{24} +8.69737 q^{25} +1.00000 q^{27} -4.23435 q^{28} +1.97786 q^{29} +9.37985 q^{30} -10.5475 q^{31} +4.74476 q^{32} +1.00000 q^{33} -5.20928 q^{34} -3.54294 q^{35} +4.42325 q^{36} -8.30741 q^{37} +19.4614 q^{38} +22.7297 q^{40} -5.23142 q^{41} -2.42618 q^{42} +5.26116 q^{43} +4.42325 q^{44} +3.70100 q^{45} -10.6698 q^{46} -1.24656 q^{47} +6.71863 q^{48} -6.08359 q^{49} +22.0427 q^{50} -2.05542 q^{51} +2.98297 q^{53} +2.53441 q^{54} +3.70100 q^{55} -5.87924 q^{56} +7.67886 q^{57} +5.01271 q^{58} -12.3548 q^{59} +16.3704 q^{60} -0.183851 q^{61} -26.7317 q^{62} -0.957295 q^{63} -1.41208 q^{64} +2.53441 q^{66} +1.40691 q^{67} -9.09163 q^{68} -4.20997 q^{69} -8.97928 q^{70} +12.1582 q^{71} +6.14151 q^{72} -3.32573 q^{73} -21.0544 q^{74} +8.69737 q^{75} +33.9655 q^{76} -0.957295 q^{77} -3.64911 q^{79} +24.8656 q^{80} +1.00000 q^{81} -13.2586 q^{82} +3.31957 q^{83} -4.23435 q^{84} -7.60710 q^{85} +13.3340 q^{86} +1.97786 q^{87} +6.14151 q^{88} +5.24302 q^{89} +9.37985 q^{90} -18.6217 q^{92} -10.5475 q^{93} -3.15929 q^{94} +28.4194 q^{95} +4.74476 q^{96} +9.82293 q^{97} -15.4183 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + O(q^{10}) \) \( 7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + 7q^{11} + 9q^{12} + 8q^{14} + 6q^{15} + 17q^{16} - 2q^{17} + 3q^{18} + 8q^{19} - 2q^{20} + 6q^{21} + 3q^{22} + 4q^{23} + 15q^{24} + 13q^{25} + 7q^{27} + 12q^{28} - 12q^{29} - 10q^{31} + 33q^{32} + 7q^{33} + 28q^{34} - 4q^{35} + 9q^{36} + 6q^{37} + 16q^{38} - 10q^{40} + 2q^{41} + 8q^{42} - 16q^{43} + 9q^{44} + 6q^{45} - 26q^{46} + 18q^{47} + 17q^{48} + 23q^{49} + 39q^{50} - 2q^{51} + 10q^{53} + 3q^{54} + 6q^{55} + 16q^{56} + 8q^{57} + 10q^{58} + 2q^{59} - 2q^{60} - 10q^{61} - 36q^{62} + 6q^{63} + 29q^{64} + 3q^{66} + 8q^{67} - 10q^{68} + 4q^{69} - 20q^{70} + 36q^{71} + 15q^{72} + 20q^{73} + 13q^{75} + 10q^{76} + 6q^{77} + 6q^{79} - 20q^{80} + 7q^{81} - 10q^{82} + 30q^{83} + 12q^{84} - 40q^{85} + 6q^{86} - 12q^{87} + 15q^{88} + 34q^{89} - 12q^{92} - 10q^{93} + 32q^{94} + 18q^{95} + 33q^{96} + 16q^{97} + q^{98} + 7q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53441 1.79210 0.896050 0.443953i \(-0.146424\pi\)
0.896050 + 0.443953i \(0.146424\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.42325 2.21162
\(5\) 3.70100 1.65514 0.827568 0.561366i \(-0.189724\pi\)
0.827568 + 0.561366i \(0.189724\pi\)
\(6\) 2.53441 1.03467
\(7\) −0.957295 −0.361823 −0.180912 0.983499i \(-0.557905\pi\)
−0.180912 + 0.983499i \(0.557905\pi\)
\(8\) 6.14151 2.17135
\(9\) 1.00000 0.333333
\(10\) 9.37985 2.96617
\(11\) 1.00000 0.301511
\(12\) 4.42325 1.27688
\(13\) 0 0
\(14\) −2.42618 −0.648424
\(15\) 3.70100 0.955593
\(16\) 6.71863 1.67966
\(17\) −2.05542 −0.498512 −0.249256 0.968438i \(-0.580186\pi\)
−0.249256 + 0.968438i \(0.580186\pi\)
\(18\) 2.53441 0.597367
\(19\) 7.67886 1.76165 0.880825 0.473442i \(-0.156988\pi\)
0.880825 + 0.473442i \(0.156988\pi\)
\(20\) 16.3704 3.66054
\(21\) −0.957295 −0.208899
\(22\) 2.53441 0.540339
\(23\) −4.20997 −0.877840 −0.438920 0.898526i \(-0.644639\pi\)
−0.438920 + 0.898526i \(0.644639\pi\)
\(24\) 6.14151 1.25363
\(25\) 8.69737 1.73947
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.23435 −0.800217
\(29\) 1.97786 0.367279 0.183640 0.982994i \(-0.441212\pi\)
0.183640 + 0.982994i \(0.441212\pi\)
\(30\) 9.37985 1.71252
\(31\) −10.5475 −1.89439 −0.947193 0.320664i \(-0.896094\pi\)
−0.947193 + 0.320664i \(0.896094\pi\)
\(32\) 4.74476 0.838763
\(33\) 1.00000 0.174078
\(34\) −5.20928 −0.893384
\(35\) −3.54294 −0.598867
\(36\) 4.42325 0.737208
\(37\) −8.30741 −1.36573 −0.682865 0.730545i \(-0.739266\pi\)
−0.682865 + 0.730545i \(0.739266\pi\)
\(38\) 19.4614 3.15705
\(39\) 0 0
\(40\) 22.7297 3.59388
\(41\) −5.23142 −0.817011 −0.408505 0.912756i \(-0.633950\pi\)
−0.408505 + 0.912756i \(0.633950\pi\)
\(42\) −2.42618 −0.374368
\(43\) 5.26116 0.802320 0.401160 0.916008i \(-0.368607\pi\)
0.401160 + 0.916008i \(0.368607\pi\)
\(44\) 4.42325 0.666830
\(45\) 3.70100 0.551712
\(46\) −10.6698 −1.57318
\(47\) −1.24656 −0.181829 −0.0909145 0.995859i \(-0.528979\pi\)
−0.0909145 + 0.995859i \(0.528979\pi\)
\(48\) 6.71863 0.969751
\(49\) −6.08359 −0.869084
\(50\) 22.0427 3.11731
\(51\) −2.05542 −0.287816
\(52\) 0 0
\(53\) 2.98297 0.409743 0.204871 0.978789i \(-0.434322\pi\)
0.204871 + 0.978789i \(0.434322\pi\)
\(54\) 2.53441 0.344890
\(55\) 3.70100 0.499042
\(56\) −5.87924 −0.785646
\(57\) 7.67886 1.01709
\(58\) 5.01271 0.658202
\(59\) −12.3548 −1.60846 −0.804228 0.594321i \(-0.797421\pi\)
−0.804228 + 0.594321i \(0.797421\pi\)
\(60\) 16.3704 2.11341
\(61\) −0.183851 −0.0235397 −0.0117698 0.999931i \(-0.503747\pi\)
−0.0117698 + 0.999931i \(0.503747\pi\)
\(62\) −26.7317 −3.39493
\(63\) −0.957295 −0.120608
\(64\) −1.41208 −0.176510
\(65\) 0 0
\(66\) 2.53441 0.311965
\(67\) 1.40691 0.171881 0.0859407 0.996300i \(-0.472610\pi\)
0.0859407 + 0.996300i \(0.472610\pi\)
\(68\) −9.09163 −1.10252
\(69\) −4.20997 −0.506821
\(70\) −8.97928 −1.07323
\(71\) 12.1582 1.44291 0.721457 0.692459i \(-0.243473\pi\)
0.721457 + 0.692459i \(0.243473\pi\)
\(72\) 6.14151 0.723784
\(73\) −3.32573 −0.389247 −0.194624 0.980878i \(-0.562349\pi\)
−0.194624 + 0.980878i \(0.562349\pi\)
\(74\) −21.0544 −2.44752
\(75\) 8.69737 1.00429
\(76\) 33.9655 3.89611
\(77\) −0.957295 −0.109094
\(78\) 0 0
\(79\) −3.64911 −0.410557 −0.205279 0.978704i \(-0.565810\pi\)
−0.205279 + 0.978704i \(0.565810\pi\)
\(80\) 24.8656 2.78006
\(81\) 1.00000 0.111111
\(82\) −13.2586 −1.46417
\(83\) 3.31957 0.364370 0.182185 0.983264i \(-0.441683\pi\)
0.182185 + 0.983264i \(0.441683\pi\)
\(84\) −4.23435 −0.462006
\(85\) −7.60710 −0.825106
\(86\) 13.3340 1.43784
\(87\) 1.97786 0.212049
\(88\) 6.14151 0.654687
\(89\) 5.24302 0.555759 0.277879 0.960616i \(-0.410368\pi\)
0.277879 + 0.960616i \(0.410368\pi\)
\(90\) 9.37985 0.988723
\(91\) 0 0
\(92\) −18.6217 −1.94145
\(93\) −10.5475 −1.09372
\(94\) −3.15929 −0.325856
\(95\) 28.4194 2.91577
\(96\) 4.74476 0.484260
\(97\) 9.82293 0.997368 0.498684 0.866784i \(-0.333817\pi\)
0.498684 + 0.866784i \(0.333817\pi\)
\(98\) −15.4183 −1.55749
\(99\) 1.00000 0.100504
\(100\) 38.4706 3.84706
\(101\) −8.42338 −0.838157 −0.419079 0.907950i \(-0.637647\pi\)
−0.419079 + 0.907950i \(0.637647\pi\)
\(102\) −5.20928 −0.515796
\(103\) −13.4029 −1.32063 −0.660314 0.750989i \(-0.729577\pi\)
−0.660314 + 0.750989i \(0.729577\pi\)
\(104\) 0 0
\(105\) −3.54294 −0.345756
\(106\) 7.56008 0.734300
\(107\) −8.05792 −0.778988 −0.389494 0.921029i \(-0.627350\pi\)
−0.389494 + 0.921029i \(0.627350\pi\)
\(108\) 4.42325 0.425627
\(109\) −14.3600 −1.37544 −0.687720 0.725976i \(-0.741388\pi\)
−0.687720 + 0.725976i \(0.741388\pi\)
\(110\) 9.37985 0.894334
\(111\) −8.30741 −0.788504
\(112\) −6.43171 −0.607739
\(113\) 18.6675 1.75609 0.878046 0.478577i \(-0.158847\pi\)
0.878046 + 0.478577i \(0.158847\pi\)
\(114\) 19.4614 1.82273
\(115\) −15.5811 −1.45294
\(116\) 8.74857 0.812284
\(117\) 0 0
\(118\) −31.3121 −2.88251
\(119\) 1.96764 0.180373
\(120\) 22.7297 2.07493
\(121\) 1.00000 0.0909091
\(122\) −0.465954 −0.0421855
\(123\) −5.23142 −0.471701
\(124\) −46.6542 −4.18967
\(125\) 13.6840 1.22393
\(126\) −2.42618 −0.216141
\(127\) 6.60130 0.585770 0.292885 0.956148i \(-0.405385\pi\)
0.292885 + 0.956148i \(0.405385\pi\)
\(128\) −13.0683 −1.15509
\(129\) 5.26116 0.463220
\(130\) 0 0
\(131\) 5.81083 0.507694 0.253847 0.967244i \(-0.418304\pi\)
0.253847 + 0.967244i \(0.418304\pi\)
\(132\) 4.42325 0.384994
\(133\) −7.35093 −0.637406
\(134\) 3.56569 0.308029
\(135\) 3.70100 0.318531
\(136\) −12.6234 −1.08245
\(137\) 5.64208 0.482035 0.241018 0.970521i \(-0.422519\pi\)
0.241018 + 0.970521i \(0.422519\pi\)
\(138\) −10.6698 −0.908274
\(139\) −4.46820 −0.378987 −0.189494 0.981882i \(-0.560685\pi\)
−0.189494 + 0.981882i \(0.560685\pi\)
\(140\) −15.6713 −1.32447
\(141\) −1.24656 −0.104979
\(142\) 30.8139 2.58585
\(143\) 0 0
\(144\) 6.71863 0.559886
\(145\) 7.32005 0.607897
\(146\) −8.42877 −0.697570
\(147\) −6.08359 −0.501766
\(148\) −36.7457 −3.02048
\(149\) −7.39749 −0.606026 −0.303013 0.952986i \(-0.597993\pi\)
−0.303013 + 0.952986i \(0.597993\pi\)
\(150\) 22.0427 1.79978
\(151\) 23.7454 1.93237 0.966187 0.257841i \(-0.0830112\pi\)
0.966187 + 0.257841i \(0.0830112\pi\)
\(152\) 47.1598 3.82516
\(153\) −2.05542 −0.166171
\(154\) −2.42618 −0.195507
\(155\) −39.0362 −3.13547
\(156\) 0 0
\(157\) 12.9704 1.03515 0.517576 0.855637i \(-0.326834\pi\)
0.517576 + 0.855637i \(0.326834\pi\)
\(158\) −9.24836 −0.735760
\(159\) 2.98297 0.236565
\(160\) 17.5603 1.38827
\(161\) 4.03018 0.317623
\(162\) 2.53441 0.199122
\(163\) 18.6880 1.46376 0.731878 0.681436i \(-0.238644\pi\)
0.731878 + 0.681436i \(0.238644\pi\)
\(164\) −23.1399 −1.80692
\(165\) 3.70100 0.288122
\(166\) 8.41316 0.652988
\(167\) 25.0420 1.93781 0.968904 0.247437i \(-0.0795884\pi\)
0.968904 + 0.247437i \(0.0795884\pi\)
\(168\) −5.87924 −0.453593
\(169\) 0 0
\(170\) −19.2795 −1.47867
\(171\) 7.67886 0.587217
\(172\) 23.2714 1.77443
\(173\) −3.16333 −0.240503 −0.120252 0.992743i \(-0.538370\pi\)
−0.120252 + 0.992743i \(0.538370\pi\)
\(174\) 5.01271 0.380013
\(175\) −8.32595 −0.629383
\(176\) 6.71863 0.506436
\(177\) −12.3548 −0.928642
\(178\) 13.2880 0.995976
\(179\) 1.24977 0.0934123 0.0467062 0.998909i \(-0.485128\pi\)
0.0467062 + 0.998909i \(0.485128\pi\)
\(180\) 16.3704 1.22018
\(181\) 16.0712 1.19456 0.597281 0.802032i \(-0.296248\pi\)
0.597281 + 0.802032i \(0.296248\pi\)
\(182\) 0 0
\(183\) −0.183851 −0.0135906
\(184\) −25.8556 −1.90610
\(185\) −30.7457 −2.26047
\(186\) −26.7317 −1.96006
\(187\) −2.05542 −0.150307
\(188\) −5.51383 −0.402137
\(189\) −0.957295 −0.0696329
\(190\) 72.0265 5.22535
\(191\) 15.2909 1.10641 0.553207 0.833044i \(-0.313404\pi\)
0.553207 + 0.833044i \(0.313404\pi\)
\(192\) −1.41208 −0.101908
\(193\) 6.76152 0.486705 0.243353 0.969938i \(-0.421753\pi\)
0.243353 + 0.969938i \(0.421753\pi\)
\(194\) 24.8954 1.78738
\(195\) 0 0
\(196\) −26.9092 −1.92209
\(197\) −11.3320 −0.807374 −0.403687 0.914897i \(-0.632272\pi\)
−0.403687 + 0.914897i \(0.632272\pi\)
\(198\) 2.53441 0.180113
\(199\) 21.5642 1.52864 0.764321 0.644836i \(-0.223074\pi\)
0.764321 + 0.644836i \(0.223074\pi\)
\(200\) 53.4150 3.77701
\(201\) 1.40691 0.0992358
\(202\) −21.3483 −1.50206
\(203\) −1.89339 −0.132890
\(204\) −9.09163 −0.636541
\(205\) −19.3615 −1.35226
\(206\) −33.9685 −2.36670
\(207\) −4.20997 −0.292613
\(208\) 0 0
\(209\) 7.67886 0.531158
\(210\) −8.97928 −0.619629
\(211\) −23.5992 −1.62463 −0.812317 0.583217i \(-0.801794\pi\)
−0.812317 + 0.583217i \(0.801794\pi\)
\(212\) 13.1944 0.906197
\(213\) 12.1582 0.833067
\(214\) −20.4221 −1.39603
\(215\) 19.4715 1.32795
\(216\) 6.14151 0.417877
\(217\) 10.0971 0.685433
\(218\) −36.3942 −2.46493
\(219\) −3.32573 −0.224732
\(220\) 16.3704 1.10369
\(221\) 0 0
\(222\) −21.0544 −1.41308
\(223\) −14.7201 −0.985730 −0.492865 0.870106i \(-0.664050\pi\)
−0.492865 + 0.870106i \(0.664050\pi\)
\(224\) −4.54213 −0.303484
\(225\) 8.69737 0.579825
\(226\) 47.3112 3.14709
\(227\) 0.0358913 0.00238219 0.00119109 0.999999i \(-0.499621\pi\)
0.00119109 + 0.999999i \(0.499621\pi\)
\(228\) 33.9655 2.24942
\(229\) 8.37257 0.553275 0.276637 0.960974i \(-0.410780\pi\)
0.276637 + 0.960974i \(0.410780\pi\)
\(230\) −39.4889 −2.60382
\(231\) −0.957295 −0.0629854
\(232\) 12.1471 0.797493
\(233\) −18.0020 −1.17935 −0.589674 0.807641i \(-0.700744\pi\)
−0.589674 + 0.807641i \(0.700744\pi\)
\(234\) 0 0
\(235\) −4.61350 −0.300952
\(236\) −54.6483 −3.55730
\(237\) −3.64911 −0.237035
\(238\) 4.98682 0.323247
\(239\) −5.74497 −0.371611 −0.185806 0.982587i \(-0.559489\pi\)
−0.185806 + 0.982587i \(0.559489\pi\)
\(240\) 24.8656 1.60507
\(241\) −16.1458 −1.04004 −0.520022 0.854153i \(-0.674076\pi\)
−0.520022 + 0.854153i \(0.674076\pi\)
\(242\) 2.53441 0.162918
\(243\) 1.00000 0.0641500
\(244\) −0.813218 −0.0520609
\(245\) −22.5153 −1.43845
\(246\) −13.2586 −0.845336
\(247\) 0 0
\(248\) −64.7776 −4.11338
\(249\) 3.31957 0.210369
\(250\) 34.6808 2.19341
\(251\) −6.83631 −0.431504 −0.215752 0.976448i \(-0.569220\pi\)
−0.215752 + 0.976448i \(0.569220\pi\)
\(252\) −4.23435 −0.266739
\(253\) −4.20997 −0.264679
\(254\) 16.7304 1.04976
\(255\) −7.60710 −0.476375
\(256\) −30.2963 −1.89352
\(257\) −26.8038 −1.67197 −0.835987 0.548749i \(-0.815104\pi\)
−0.835987 + 0.548749i \(0.815104\pi\)
\(258\) 13.3340 0.830136
\(259\) 7.95264 0.494153
\(260\) 0 0
\(261\) 1.97786 0.122426
\(262\) 14.7270 0.909839
\(263\) −11.9518 −0.736980 −0.368490 0.929632i \(-0.620125\pi\)
−0.368490 + 0.929632i \(0.620125\pi\)
\(264\) 6.14151 0.377984
\(265\) 11.0400 0.678180
\(266\) −18.6303 −1.14230
\(267\) 5.24302 0.320868
\(268\) 6.22311 0.380137
\(269\) 4.59663 0.280261 0.140131 0.990133i \(-0.455248\pi\)
0.140131 + 0.990133i \(0.455248\pi\)
\(270\) 9.37985 0.570840
\(271\) 19.8969 1.20865 0.604326 0.796738i \(-0.293443\pi\)
0.604326 + 0.796738i \(0.293443\pi\)
\(272\) −13.8096 −0.837330
\(273\) 0 0
\(274\) 14.2994 0.863856
\(275\) 8.69737 0.524471
\(276\) −18.6217 −1.12090
\(277\) −11.2319 −0.674857 −0.337428 0.941351i \(-0.609557\pi\)
−0.337428 + 0.941351i \(0.609557\pi\)
\(278\) −11.3243 −0.679183
\(279\) −10.5475 −0.631462
\(280\) −21.7590 −1.30035
\(281\) −30.2946 −1.80722 −0.903612 0.428352i \(-0.859094\pi\)
−0.903612 + 0.428352i \(0.859094\pi\)
\(282\) −3.15929 −0.188133
\(283\) −15.3889 −0.914777 −0.457389 0.889267i \(-0.651215\pi\)
−0.457389 + 0.889267i \(0.651215\pi\)
\(284\) 53.7788 3.19118
\(285\) 28.4194 1.68342
\(286\) 0 0
\(287\) 5.00801 0.295614
\(288\) 4.74476 0.279588
\(289\) −12.7753 −0.751485
\(290\) 18.5520 1.08941
\(291\) 9.82293 0.575831
\(292\) −14.7105 −0.860869
\(293\) 6.95822 0.406503 0.203252 0.979127i \(-0.434849\pi\)
0.203252 + 0.979127i \(0.434849\pi\)
\(294\) −15.4183 −0.899215
\(295\) −45.7250 −2.66221
\(296\) −51.0200 −2.96548
\(297\) 1.00000 0.0580259
\(298\) −18.7483 −1.08606
\(299\) 0 0
\(300\) 38.4706 2.22110
\(301\) −5.03648 −0.290298
\(302\) 60.1807 3.46301
\(303\) −8.42338 −0.483910
\(304\) 51.5914 2.95897
\(305\) −0.680431 −0.0389614
\(306\) −5.20928 −0.297795
\(307\) −12.6233 −0.720447 −0.360224 0.932866i \(-0.617300\pi\)
−0.360224 + 0.932866i \(0.617300\pi\)
\(308\) −4.23435 −0.241275
\(309\) −13.4029 −0.762465
\(310\) −98.9339 −5.61907
\(311\) 2.12116 0.120280 0.0601399 0.998190i \(-0.480845\pi\)
0.0601399 + 0.998190i \(0.480845\pi\)
\(312\) 0 0
\(313\) 21.5919 1.22045 0.610224 0.792229i \(-0.291079\pi\)
0.610224 + 0.792229i \(0.291079\pi\)
\(314\) 32.8724 1.85510
\(315\) −3.54294 −0.199622
\(316\) −16.1409 −0.907999
\(317\) 15.7684 0.885640 0.442820 0.896611i \(-0.353978\pi\)
0.442820 + 0.896611i \(0.353978\pi\)
\(318\) 7.56008 0.423948
\(319\) 1.97786 0.110739
\(320\) −5.22612 −0.292149
\(321\) −8.05792 −0.449749
\(322\) 10.2141 0.569212
\(323\) −15.7833 −0.878205
\(324\) 4.42325 0.245736
\(325\) 0 0
\(326\) 47.3631 2.62320
\(327\) −14.3600 −0.794111
\(328\) −32.1288 −1.77402
\(329\) 1.19332 0.0657900
\(330\) 9.37985 0.516344
\(331\) 11.6602 0.640902 0.320451 0.947265i \(-0.396166\pi\)
0.320451 + 0.947265i \(0.396166\pi\)
\(332\) 14.6833 0.805850
\(333\) −8.30741 −0.455243
\(334\) 63.4668 3.47275
\(335\) 5.20697 0.284487
\(336\) −6.43171 −0.350879
\(337\) −7.63508 −0.415909 −0.207955 0.978138i \(-0.566681\pi\)
−0.207955 + 0.978138i \(0.566681\pi\)
\(338\) 0 0
\(339\) 18.6675 1.01388
\(340\) −33.6481 −1.82482
\(341\) −10.5475 −0.571179
\(342\) 19.4614 1.05235
\(343\) 12.5248 0.676278
\(344\) 32.3115 1.74212
\(345\) −15.5811 −0.838858
\(346\) −8.01717 −0.431006
\(347\) −17.7823 −0.954606 −0.477303 0.878739i \(-0.658386\pi\)
−0.477303 + 0.878739i \(0.658386\pi\)
\(348\) 8.74857 0.468972
\(349\) 12.1663 0.651245 0.325622 0.945500i \(-0.394426\pi\)
0.325622 + 0.945500i \(0.394426\pi\)
\(350\) −21.1014 −1.12792
\(351\) 0 0
\(352\) 4.74476 0.252896
\(353\) −32.9025 −1.75122 −0.875612 0.483014i \(-0.839542\pi\)
−0.875612 + 0.483014i \(0.839542\pi\)
\(354\) −31.3121 −1.66422
\(355\) 44.9975 2.38822
\(356\) 23.1912 1.22913
\(357\) 1.96764 0.104139
\(358\) 3.16744 0.167404
\(359\) 7.29760 0.385153 0.192576 0.981282i \(-0.438316\pi\)
0.192576 + 0.981282i \(0.438316\pi\)
\(360\) 22.7297 1.19796
\(361\) 39.9648 2.10341
\(362\) 40.7310 2.14078
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −12.3085 −0.644257
\(366\) −0.465954 −0.0243558
\(367\) −8.94917 −0.467143 −0.233571 0.972340i \(-0.575041\pi\)
−0.233571 + 0.972340i \(0.575041\pi\)
\(368\) −28.2852 −1.47447
\(369\) −5.23142 −0.272337
\(370\) −77.9222 −4.05098
\(371\) −2.85558 −0.148254
\(372\) −46.6542 −2.41891
\(373\) −18.1913 −0.941908 −0.470954 0.882158i \(-0.656090\pi\)
−0.470954 + 0.882158i \(0.656090\pi\)
\(374\) −5.20928 −0.269366
\(375\) 13.6840 0.706637
\(376\) −7.65574 −0.394815
\(377\) 0 0
\(378\) −2.42618 −0.124789
\(379\) 20.1063 1.03279 0.516396 0.856350i \(-0.327273\pi\)
0.516396 + 0.856350i \(0.327273\pi\)
\(380\) 125.706 6.44859
\(381\) 6.60130 0.338195
\(382\) 38.7536 1.98281
\(383\) −7.99092 −0.408317 −0.204158 0.978938i \(-0.565446\pi\)
−0.204158 + 0.978938i \(0.565446\pi\)
\(384\) −13.0683 −0.666890
\(385\) −3.54294 −0.180565
\(386\) 17.1365 0.872224
\(387\) 5.26116 0.267440
\(388\) 43.4493 2.20580
\(389\) 4.31617 0.218839 0.109419 0.993996i \(-0.465101\pi\)
0.109419 + 0.993996i \(0.465101\pi\)
\(390\) 0 0
\(391\) 8.65326 0.437614
\(392\) −37.3624 −1.88709
\(393\) 5.81083 0.293117
\(394\) −28.7201 −1.44690
\(395\) −13.5054 −0.679528
\(396\) 4.42325 0.222277
\(397\) 23.4560 1.17722 0.588612 0.808416i \(-0.299675\pi\)
0.588612 + 0.808416i \(0.299675\pi\)
\(398\) 54.6525 2.73948
\(399\) −7.35093 −0.368007
\(400\) 58.4344 2.92172
\(401\) −2.38235 −0.118969 −0.0594844 0.998229i \(-0.518946\pi\)
−0.0594844 + 0.998229i \(0.518946\pi\)
\(402\) 3.56569 0.177841
\(403\) 0 0
\(404\) −37.2587 −1.85369
\(405\) 3.70100 0.183904
\(406\) −4.79864 −0.238153
\(407\) −8.30741 −0.411783
\(408\) −12.6234 −0.624951
\(409\) −10.3945 −0.513973 −0.256987 0.966415i \(-0.582730\pi\)
−0.256987 + 0.966415i \(0.582730\pi\)
\(410\) −49.0700 −2.42339
\(411\) 5.64208 0.278303
\(412\) −59.2844 −2.92073
\(413\) 11.8272 0.581977
\(414\) −10.6698 −0.524392
\(415\) 12.2857 0.603082
\(416\) 0 0
\(417\) −4.46820 −0.218808
\(418\) 19.4614 0.951888
\(419\) 13.6447 0.666586 0.333293 0.942823i \(-0.391840\pi\)
0.333293 + 0.942823i \(0.391840\pi\)
\(420\) −15.6713 −0.764682
\(421\) 6.41040 0.312424 0.156212 0.987724i \(-0.450072\pi\)
0.156212 + 0.987724i \(0.450072\pi\)
\(422\) −59.8100 −2.91151
\(423\) −1.24656 −0.0606097
\(424\) 18.3200 0.889696
\(425\) −17.8768 −0.867150
\(426\) 30.8139 1.49294
\(427\) 0.175999 0.00851721
\(428\) −35.6422 −1.72283
\(429\) 0 0
\(430\) 49.3489 2.37982
\(431\) 15.8350 0.762747 0.381374 0.924421i \(-0.375451\pi\)
0.381374 + 0.924421i \(0.375451\pi\)
\(432\) 6.71863 0.323250
\(433\) −10.6273 −0.510716 −0.255358 0.966847i \(-0.582193\pi\)
−0.255358 + 0.966847i \(0.582193\pi\)
\(434\) 25.5901 1.22836
\(435\) 7.32005 0.350970
\(436\) −63.5179 −3.04196
\(437\) −32.3278 −1.54645
\(438\) −8.42877 −0.402742
\(439\) −17.2195 −0.821841 −0.410920 0.911671i \(-0.634793\pi\)
−0.410920 + 0.911671i \(0.634793\pi\)
\(440\) 22.7297 1.08360
\(441\) −6.08359 −0.289695
\(442\) 0 0
\(443\) −25.1359 −1.19424 −0.597120 0.802152i \(-0.703689\pi\)
−0.597120 + 0.802152i \(0.703689\pi\)
\(444\) −36.7457 −1.74387
\(445\) 19.4044 0.919857
\(446\) −37.3068 −1.76653
\(447\) −7.39749 −0.349889
\(448\) 1.35178 0.0638656
\(449\) −2.01596 −0.0951392 −0.0475696 0.998868i \(-0.515148\pi\)
−0.0475696 + 0.998868i \(0.515148\pi\)
\(450\) 22.0427 1.03910
\(451\) −5.23142 −0.246338
\(452\) 82.5710 3.88382
\(453\) 23.7454 1.11566
\(454\) 0.0909634 0.00426912
\(455\) 0 0
\(456\) 47.1598 2.20846
\(457\) 33.7659 1.57950 0.789752 0.613426i \(-0.210209\pi\)
0.789752 + 0.613426i \(0.210209\pi\)
\(458\) 21.2195 0.991524
\(459\) −2.05542 −0.0959388
\(460\) −68.9190 −3.21337
\(461\) 16.7600 0.780589 0.390294 0.920690i \(-0.372373\pi\)
0.390294 + 0.920690i \(0.372373\pi\)
\(462\) −2.42618 −0.112876
\(463\) 27.5446 1.28011 0.640054 0.768330i \(-0.278912\pi\)
0.640054 + 0.768330i \(0.278912\pi\)
\(464\) 13.2885 0.616904
\(465\) −39.0362 −1.81026
\(466\) −45.6244 −2.11351
\(467\) 22.9318 1.06116 0.530578 0.847636i \(-0.321975\pi\)
0.530578 + 0.847636i \(0.321975\pi\)
\(468\) 0 0
\(469\) −1.34683 −0.0621907
\(470\) −11.6925 −0.539336
\(471\) 12.9704 0.597646
\(472\) −75.8770 −3.49252
\(473\) 5.26116 0.241909
\(474\) −9.24836 −0.424791
\(475\) 66.7859 3.06435
\(476\) 8.70337 0.398918
\(477\) 2.98297 0.136581
\(478\) −14.5601 −0.665965
\(479\) −5.71495 −0.261123 −0.130561 0.991440i \(-0.541678\pi\)
−0.130561 + 0.991440i \(0.541678\pi\)
\(480\) 17.5603 0.801516
\(481\) 0 0
\(482\) −40.9202 −1.86386
\(483\) 4.03018 0.183380
\(484\) 4.42325 0.201057
\(485\) 36.3546 1.65078
\(486\) 2.53441 0.114963
\(487\) 10.4987 0.475742 0.237871 0.971297i \(-0.423550\pi\)
0.237871 + 0.971297i \(0.423550\pi\)
\(488\) −1.12912 −0.0511129
\(489\) 18.6880 0.845100
\(490\) −57.0631 −2.57785
\(491\) 13.0458 0.588747 0.294373 0.955691i \(-0.404889\pi\)
0.294373 + 0.955691i \(0.404889\pi\)
\(492\) −23.1399 −1.04323
\(493\) −4.06533 −0.183093
\(494\) 0 0
\(495\) 3.70100 0.166347
\(496\) −70.8647 −3.18192
\(497\) −11.6390 −0.522080
\(498\) 8.41316 0.377003
\(499\) −38.6395 −1.72974 −0.864870 0.501995i \(-0.832600\pi\)
−0.864870 + 0.501995i \(0.832600\pi\)
\(500\) 60.5276 2.70688
\(501\) 25.0420 1.11879
\(502\) −17.3260 −0.773298
\(503\) 0.330867 0.0147526 0.00737631 0.999973i \(-0.497652\pi\)
0.00737631 + 0.999973i \(0.497652\pi\)
\(504\) −5.87924 −0.261882
\(505\) −31.1749 −1.38726
\(506\) −10.6698 −0.474331
\(507\) 0 0
\(508\) 29.1992 1.29550
\(509\) 44.2090 1.95953 0.979764 0.200157i \(-0.0641452\pi\)
0.979764 + 0.200157i \(0.0641452\pi\)
\(510\) −19.2795 −0.853712
\(511\) 3.18370 0.140839
\(512\) −50.6468 −2.23829
\(513\) 7.67886 0.339030
\(514\) −67.9318 −2.99635
\(515\) −49.6041 −2.18582
\(516\) 23.2714 1.02447
\(517\) −1.24656 −0.0548235
\(518\) 20.1553 0.885571
\(519\) −3.16333 −0.138855
\(520\) 0 0
\(521\) 9.66049 0.423234 0.211617 0.977353i \(-0.432127\pi\)
0.211617 + 0.977353i \(0.432127\pi\)
\(522\) 5.01271 0.219401
\(523\) 21.7358 0.950440 0.475220 0.879867i \(-0.342368\pi\)
0.475220 + 0.879867i \(0.342368\pi\)
\(524\) 25.7027 1.12283
\(525\) −8.32595 −0.363374
\(526\) −30.2908 −1.32074
\(527\) 21.6795 0.944375
\(528\) 6.71863 0.292391
\(529\) −5.27614 −0.229398
\(530\) 27.9798 1.21537
\(531\) −12.3548 −0.536152
\(532\) −32.5150 −1.40970
\(533\) 0 0
\(534\) 13.2880 0.575027
\(535\) −29.8223 −1.28933
\(536\) 8.64056 0.373215
\(537\) 1.24977 0.0539316
\(538\) 11.6497 0.502256
\(539\) −6.08359 −0.262039
\(540\) 16.3704 0.704471
\(541\) 18.0139 0.774478 0.387239 0.921979i \(-0.373429\pi\)
0.387239 + 0.921979i \(0.373429\pi\)
\(542\) 50.4270 2.16602
\(543\) 16.0712 0.689681
\(544\) −9.75247 −0.418134
\(545\) −53.1464 −2.27654
\(546\) 0 0
\(547\) 23.2740 0.995125 0.497563 0.867428i \(-0.334228\pi\)
0.497563 + 0.867428i \(0.334228\pi\)
\(548\) 24.9563 1.06608
\(549\) −0.183851 −0.00784656
\(550\) 22.0427 0.939906
\(551\) 15.1877 0.647018
\(552\) −25.8556 −1.10049
\(553\) 3.49328 0.148549
\(554\) −28.4662 −1.20941
\(555\) −30.7457 −1.30508
\(556\) −19.7639 −0.838178
\(557\) 11.5996 0.491491 0.245745 0.969334i \(-0.420967\pi\)
0.245745 + 0.969334i \(0.420967\pi\)
\(558\) −26.7317 −1.13164
\(559\) 0 0
\(560\) −23.8037 −1.00589
\(561\) −2.05542 −0.0867799
\(562\) −76.7790 −3.23873
\(563\) 26.5695 1.11977 0.559885 0.828570i \(-0.310845\pi\)
0.559885 + 0.828570i \(0.310845\pi\)
\(564\) −5.51383 −0.232174
\(565\) 69.0884 2.90657
\(566\) −39.0019 −1.63937
\(567\) −0.957295 −0.0402026
\(568\) 74.6698 3.13308
\(569\) −22.7470 −0.953603 −0.476802 0.879011i \(-0.658204\pi\)
−0.476802 + 0.879011i \(0.658204\pi\)
\(570\) 72.0265 3.01686
\(571\) −29.4622 −1.23296 −0.616478 0.787372i \(-0.711441\pi\)
−0.616478 + 0.787372i \(0.711441\pi\)
\(572\) 0 0
\(573\) 15.2909 0.638788
\(574\) 12.6924 0.529769
\(575\) −36.6157 −1.52698
\(576\) −1.41208 −0.0588368
\(577\) −26.6537 −1.10961 −0.554803 0.831981i \(-0.687207\pi\)
−0.554803 + 0.831981i \(0.687207\pi\)
\(578\) −32.3778 −1.34674
\(579\) 6.76152 0.280999
\(580\) 32.3784 1.34444
\(581\) −3.17781 −0.131838
\(582\) 24.8954 1.03195
\(583\) 2.98297 0.123542
\(584\) −20.4250 −0.845193
\(585\) 0 0
\(586\) 17.6350 0.728495
\(587\) 10.9203 0.450729 0.225364 0.974275i \(-0.427643\pi\)
0.225364 + 0.974275i \(0.427643\pi\)
\(588\) −26.9092 −1.10972
\(589\) −80.9927 −3.33725
\(590\) −115.886 −4.77095
\(591\) −11.3320 −0.466138
\(592\) −55.8144 −2.29396
\(593\) 19.3064 0.792820 0.396410 0.918074i \(-0.370256\pi\)
0.396410 + 0.918074i \(0.370256\pi\)
\(594\) 2.53441 0.103988
\(595\) 7.28224 0.298543
\(596\) −32.7209 −1.34030
\(597\) 21.5642 0.882562
\(598\) 0 0
\(599\) 2.01132 0.0821802 0.0410901 0.999155i \(-0.486917\pi\)
0.0410901 + 0.999155i \(0.486917\pi\)
\(600\) 53.4150 2.18066
\(601\) −40.1233 −1.63667 −0.818333 0.574745i \(-0.805101\pi\)
−0.818333 + 0.574745i \(0.805101\pi\)
\(602\) −12.7645 −0.520243
\(603\) 1.40691 0.0572938
\(604\) 105.032 4.27369
\(605\) 3.70100 0.150467
\(606\) −21.3483 −0.867216
\(607\) −35.4310 −1.43810 −0.719049 0.694959i \(-0.755422\pi\)
−0.719049 + 0.694959i \(0.755422\pi\)
\(608\) 36.4343 1.47761
\(609\) −1.89339 −0.0767242
\(610\) −1.72449 −0.0698227
\(611\) 0 0
\(612\) −9.09163 −0.367507
\(613\) 0.644081 0.0260142 0.0130071 0.999915i \(-0.495860\pi\)
0.0130071 + 0.999915i \(0.495860\pi\)
\(614\) −31.9925 −1.29111
\(615\) −19.3615 −0.780730
\(616\) −5.87924 −0.236881
\(617\) 23.5775 0.949194 0.474597 0.880203i \(-0.342594\pi\)
0.474597 + 0.880203i \(0.342594\pi\)
\(618\) −33.9685 −1.36641
\(619\) −5.86810 −0.235859 −0.117929 0.993022i \(-0.537626\pi\)
−0.117929 + 0.993022i \(0.537626\pi\)
\(620\) −172.667 −6.93447
\(621\) −4.20997 −0.168940
\(622\) 5.37589 0.215553
\(623\) −5.01911 −0.201087
\(624\) 0 0
\(625\) 7.15745 0.286298
\(626\) 54.7229 2.18717
\(627\) 7.67886 0.306664
\(628\) 57.3714 2.28937
\(629\) 17.0752 0.680833
\(630\) −8.97928 −0.357743
\(631\) 11.0386 0.439441 0.219721 0.975563i \(-0.429485\pi\)
0.219721 + 0.975563i \(0.429485\pi\)
\(632\) −22.4111 −0.891465
\(633\) −23.5992 −0.937982
\(634\) 39.9636 1.58716
\(635\) 24.4314 0.969529
\(636\) 13.1944 0.523193
\(637\) 0 0
\(638\) 5.01271 0.198455
\(639\) 12.1582 0.480972
\(640\) −48.3658 −1.91183
\(641\) 11.2924 0.446022 0.223011 0.974816i \(-0.428411\pi\)
0.223011 + 0.974816i \(0.428411\pi\)
\(642\) −20.4221 −0.805996
\(643\) 2.73168 0.107727 0.0538634 0.998548i \(-0.482846\pi\)
0.0538634 + 0.998548i \(0.482846\pi\)
\(644\) 17.8265 0.702463
\(645\) 19.4715 0.766691
\(646\) −40.0013 −1.57383
\(647\) −10.3135 −0.405465 −0.202733 0.979234i \(-0.564982\pi\)
−0.202733 + 0.979234i \(0.564982\pi\)
\(648\) 6.14151 0.241261
\(649\) −12.3548 −0.484968
\(650\) 0 0
\(651\) 10.0971 0.395735
\(652\) 82.6616 3.23728
\(653\) −1.95601 −0.0765446 −0.0382723 0.999267i \(-0.512185\pi\)
−0.0382723 + 0.999267i \(0.512185\pi\)
\(654\) −36.3942 −1.42313
\(655\) 21.5058 0.840303
\(656\) −35.1480 −1.37230
\(657\) −3.32573 −0.129749
\(658\) 3.02437 0.117902
\(659\) −21.0277 −0.819121 −0.409561 0.912283i \(-0.634318\pi\)
−0.409561 + 0.912283i \(0.634318\pi\)
\(660\) 16.3704 0.637218
\(661\) 25.7516 1.00162 0.500810 0.865557i \(-0.333036\pi\)
0.500810 + 0.865557i \(0.333036\pi\)
\(662\) 29.5517 1.14856
\(663\) 0 0
\(664\) 20.3872 0.791176
\(665\) −27.2058 −1.05499
\(666\) −21.0544 −0.815841
\(667\) −8.32673 −0.322412
\(668\) 110.767 4.28570
\(669\) −14.7201 −0.569111
\(670\) 13.1966 0.509830
\(671\) −0.183851 −0.00709748
\(672\) −4.54213 −0.175217
\(673\) −18.2077 −0.701856 −0.350928 0.936403i \(-0.614134\pi\)
−0.350928 + 0.936403i \(0.614134\pi\)
\(674\) −19.3504 −0.745351
\(675\) 8.69737 0.334762
\(676\) 0 0
\(677\) −10.3281 −0.396941 −0.198470 0.980107i \(-0.563597\pi\)
−0.198470 + 0.980107i \(0.563597\pi\)
\(678\) 47.3112 1.81697
\(679\) −9.40344 −0.360871
\(680\) −46.7191 −1.79160
\(681\) 0.0358913 0.00137536
\(682\) −26.7317 −1.02361
\(683\) −33.4285 −1.27911 −0.639553 0.768747i \(-0.720881\pi\)
−0.639553 + 0.768747i \(0.720881\pi\)
\(684\) 33.9655 1.29870
\(685\) 20.8813 0.797834
\(686\) 31.7431 1.21196
\(687\) 8.37257 0.319433
\(688\) 35.3478 1.34762
\(689\) 0 0
\(690\) −39.4889 −1.50332
\(691\) −38.8290 −1.47712 −0.738562 0.674186i \(-0.764495\pi\)
−0.738562 + 0.674186i \(0.764495\pi\)
\(692\) −13.9922 −0.531903
\(693\) −0.957295 −0.0363646
\(694\) −45.0678 −1.71075
\(695\) −16.5368 −0.627276
\(696\) 12.1471 0.460433
\(697\) 10.7528 0.407290
\(698\) 30.8343 1.16710
\(699\) −18.0020 −0.680897
\(700\) −36.8277 −1.39196
\(701\) 47.5275 1.79509 0.897544 0.440925i \(-0.145350\pi\)
0.897544 + 0.440925i \(0.145350\pi\)
\(702\) 0 0
\(703\) −63.7914 −2.40594
\(704\) −1.41208 −0.0532199
\(705\) −4.61350 −0.173755
\(706\) −83.3886 −3.13837
\(707\) 8.06365 0.303265
\(708\) −54.6483 −2.05381
\(709\) −28.9052 −1.08556 −0.542780 0.839875i \(-0.682628\pi\)
−0.542780 + 0.839875i \(0.682628\pi\)
\(710\) 114.042 4.27993
\(711\) −3.64911 −0.136852
\(712\) 32.2001 1.20675
\(713\) 44.4046 1.66297
\(714\) 4.98682 0.186627
\(715\) 0 0
\(716\) 5.52805 0.206593
\(717\) −5.74497 −0.214550
\(718\) 18.4951 0.690233
\(719\) 47.9212 1.78716 0.893579 0.448906i \(-0.148186\pi\)
0.893579 + 0.448906i \(0.148186\pi\)
\(720\) 24.8656 0.926687
\(721\) 12.8305 0.477834
\(722\) 101.287 3.76953
\(723\) −16.1458 −0.600469
\(724\) 71.0869 2.64192
\(725\) 17.2022 0.638873
\(726\) 2.53441 0.0940609
\(727\) 23.4347 0.869146 0.434573 0.900636i \(-0.356899\pi\)
0.434573 + 0.900636i \(0.356899\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −31.1949 −1.15457
\(731\) −10.8139 −0.399966
\(732\) −0.813218 −0.0300574
\(733\) 38.8733 1.43582 0.717910 0.696136i \(-0.245099\pi\)
0.717910 + 0.696136i \(0.245099\pi\)
\(734\) −22.6809 −0.837167
\(735\) −22.5153 −0.830491
\(736\) −19.9753 −0.736299
\(737\) 1.40691 0.0518242
\(738\) −13.2586 −0.488055
\(739\) 29.3240 1.07870 0.539350 0.842082i \(-0.318670\pi\)
0.539350 + 0.842082i \(0.318670\pi\)
\(740\) −135.996 −4.99930
\(741\) 0 0
\(742\) −7.23723 −0.265687
\(743\) 40.8183 1.49748 0.748740 0.662864i \(-0.230659\pi\)
0.748740 + 0.662864i \(0.230659\pi\)
\(744\) −64.7776 −2.37486
\(745\) −27.3781 −1.00305
\(746\) −46.1042 −1.68799
\(747\) 3.31957 0.121457
\(748\) −9.09163 −0.332423
\(749\) 7.71380 0.281856
\(750\) 34.6808 1.26636
\(751\) 4.36042 0.159114 0.0795570 0.996830i \(-0.474649\pi\)
0.0795570 + 0.996830i \(0.474649\pi\)
\(752\) −8.37515 −0.305410
\(753\) −6.83631 −0.249129
\(754\) 0 0
\(755\) 87.8817 3.19834
\(756\) −4.23435 −0.154002
\(757\) 40.1387 1.45887 0.729433 0.684052i \(-0.239784\pi\)
0.729433 + 0.684052i \(0.239784\pi\)
\(758\) 50.9577 1.85087
\(759\) −4.20997 −0.152812
\(760\) 174.538 6.33117
\(761\) −38.2425 −1.38629 −0.693144 0.720799i \(-0.743775\pi\)
−0.693144 + 0.720799i \(0.743775\pi\)
\(762\) 16.7304 0.606079
\(763\) 13.7468 0.497667
\(764\) 67.6357 2.44697
\(765\) −7.60710 −0.275035
\(766\) −20.2523 −0.731745
\(767\) 0 0
\(768\) −30.2963 −1.09323
\(769\) −35.0140 −1.26264 −0.631319 0.775523i \(-0.717486\pi\)
−0.631319 + 0.775523i \(0.717486\pi\)
\(770\) −8.97928 −0.323591
\(771\) −26.8038 −0.965315
\(772\) 29.9079 1.07641
\(773\) −5.69564 −0.204858 −0.102429 0.994740i \(-0.532661\pi\)
−0.102429 + 0.994740i \(0.532661\pi\)
\(774\) 13.3340 0.479279
\(775\) −91.7355 −3.29524
\(776\) 60.3277 2.16564
\(777\) 7.95264 0.285299
\(778\) 10.9390 0.392181
\(779\) −40.1713 −1.43929
\(780\) 0 0
\(781\) 12.1582 0.435055
\(782\) 21.9309 0.784248
\(783\) 1.97786 0.0706830
\(784\) −40.8734 −1.45976
\(785\) 48.0035 1.71332
\(786\) 14.7270 0.525296
\(787\) 31.2092 1.11249 0.556244 0.831019i \(-0.312242\pi\)
0.556244 + 0.831019i \(0.312242\pi\)
\(788\) −50.1244 −1.78561
\(789\) −11.9518 −0.425496
\(790\) −34.2282 −1.21778
\(791\) −17.8703 −0.635395
\(792\) 6.14151 0.218229
\(793\) 0 0
\(794\) 59.4472 2.10970
\(795\) 11.0400 0.391547
\(796\) 95.3836 3.38078
\(797\) −42.7214 −1.51327 −0.756634 0.653839i \(-0.773157\pi\)
−0.756634 + 0.653839i \(0.773157\pi\)
\(798\) −18.6303 −0.659505
\(799\) 2.56220 0.0906440
\(800\) 41.2669 1.45901
\(801\) 5.24302 0.185253
\(802\) −6.03786 −0.213204
\(803\) −3.32573 −0.117362
\(804\) 6.22311 0.219472
\(805\) 14.9157 0.525709
\(806\) 0 0
\(807\) 4.59663 0.161809
\(808\) −51.7323 −1.81993
\(809\) −13.3241 −0.468449 −0.234225 0.972182i \(-0.575255\pi\)
−0.234225 + 0.972182i \(0.575255\pi\)
\(810\) 9.37985 0.329574
\(811\) 41.9849 1.47429 0.737144 0.675735i \(-0.236174\pi\)
0.737144 + 0.675735i \(0.236174\pi\)
\(812\) −8.37496 −0.293903
\(813\) 19.8969 0.697815
\(814\) −21.0544 −0.737956
\(815\) 69.1642 2.42272
\(816\) −13.8096 −0.483433
\(817\) 40.3997 1.41341
\(818\) −26.3439 −0.921092
\(819\) 0 0
\(820\) −85.6406 −2.99070
\(821\) 10.0872 0.352047 0.176023 0.984386i \(-0.443677\pi\)
0.176023 + 0.984386i \(0.443677\pi\)
\(822\) 14.2994 0.498748
\(823\) −1.16139 −0.0404835 −0.0202418 0.999795i \(-0.506444\pi\)
−0.0202418 + 0.999795i \(0.506444\pi\)
\(824\) −82.3142 −2.86755
\(825\) 8.69737 0.302804
\(826\) 29.9749 1.04296
\(827\) −5.42242 −0.188556 −0.0942781 0.995546i \(-0.530054\pi\)
−0.0942781 + 0.995546i \(0.530054\pi\)
\(828\) −18.6217 −0.647151
\(829\) −16.1881 −0.562237 −0.281119 0.959673i \(-0.590705\pi\)
−0.281119 + 0.959673i \(0.590705\pi\)
\(830\) 31.1371 1.08078
\(831\) −11.2319 −0.389629
\(832\) 0 0
\(833\) 12.5043 0.433249
\(834\) −11.3243 −0.392127
\(835\) 92.6803 3.20734
\(836\) 33.9655 1.17472
\(837\) −10.5475 −0.364575
\(838\) 34.5812 1.19459
\(839\) 13.1144 0.452759 0.226379 0.974039i \(-0.427311\pi\)
0.226379 + 0.974039i \(0.427311\pi\)
\(840\) −21.7590 −0.750758
\(841\) −25.0881 −0.865106
\(842\) 16.2466 0.559895
\(843\) −30.2946 −1.04340
\(844\) −104.385 −3.59308
\(845\) 0 0
\(846\) −3.15929 −0.108619
\(847\) −0.957295 −0.0328930
\(848\) 20.0415 0.688227
\(849\) −15.3889 −0.528147
\(850\) −45.3071 −1.55402
\(851\) 34.9739 1.19889
\(852\) 53.7788 1.84243
\(853\) −8.66325 −0.296624 −0.148312 0.988941i \(-0.547384\pi\)
−0.148312 + 0.988941i \(0.547384\pi\)
\(854\) 0.446055 0.0152637
\(855\) 28.4194 0.971924
\(856\) −49.4878 −1.69146
\(857\) −28.1435 −0.961365 −0.480683 0.876895i \(-0.659611\pi\)
−0.480683 + 0.876895i \(0.659611\pi\)
\(858\) 0 0
\(859\) 11.7375 0.400479 0.200240 0.979747i \(-0.435828\pi\)
0.200240 + 0.979747i \(0.435828\pi\)
\(860\) 86.1275 2.93692
\(861\) 5.00801 0.170673
\(862\) 40.1325 1.36692
\(863\) −14.6715 −0.499424 −0.249712 0.968320i \(-0.580336\pi\)
−0.249712 + 0.968320i \(0.580336\pi\)
\(864\) 4.74476 0.161420
\(865\) −11.7075 −0.398066
\(866\) −26.9340 −0.915254
\(867\) −12.7753 −0.433870
\(868\) 44.6618 1.51592
\(869\) −3.64911 −0.123788
\(870\) 18.5520 0.628973
\(871\) 0 0
\(872\) −88.1922 −2.98657
\(873\) 9.82293 0.332456
\(874\) −81.9319 −2.77139
\(875\) −13.0996 −0.442847
\(876\) −14.7105 −0.497023
\(877\) −18.0884 −0.610803 −0.305402 0.952224i \(-0.598791\pi\)
−0.305402 + 0.952224i \(0.598791\pi\)
\(878\) −43.6413 −1.47282
\(879\) 6.95822 0.234695
\(880\) 24.8656 0.838220
\(881\) −31.0695 −1.04676 −0.523379 0.852100i \(-0.675329\pi\)
−0.523379 + 0.852100i \(0.675329\pi\)
\(882\) −15.4183 −0.519162
\(883\) 34.4102 1.15799 0.578997 0.815330i \(-0.303444\pi\)
0.578997 + 0.815330i \(0.303444\pi\)
\(884\) 0 0
\(885\) −45.7250 −1.53703
\(886\) −63.7047 −2.14020
\(887\) −24.0600 −0.807857 −0.403928 0.914791i \(-0.632355\pi\)
−0.403928 + 0.914791i \(0.632355\pi\)
\(888\) −51.0200 −1.71212
\(889\) −6.31938 −0.211945
\(890\) 49.1788 1.64848
\(891\) 1.00000 0.0335013
\(892\) −65.1106 −2.18006
\(893\) −9.57213 −0.320319
\(894\) −18.7483 −0.627036
\(895\) 4.62540 0.154610
\(896\) 12.5102 0.417938
\(897\) 0 0
\(898\) −5.10928 −0.170499
\(899\) −20.8615 −0.695769
\(900\) 38.4706 1.28235
\(901\) −6.13126 −0.204262
\(902\) −13.2586 −0.441462
\(903\) −5.03648 −0.167604
\(904\) 114.647 3.81309
\(905\) 59.4794 1.97716
\(906\) 60.1807 1.99937
\(907\) −7.79308 −0.258765 −0.129382 0.991595i \(-0.541299\pi\)
−0.129382 + 0.991595i \(0.541299\pi\)
\(908\) 0.158756 0.00526851
\(909\) −8.42338 −0.279386
\(910\) 0 0
\(911\) −4.88300 −0.161781 −0.0808904 0.996723i \(-0.525776\pi\)
−0.0808904 + 0.996723i \(0.525776\pi\)
\(912\) 51.5914 1.70836
\(913\) 3.31957 0.109862
\(914\) 85.5768 2.83063
\(915\) −0.680431 −0.0224944
\(916\) 37.0339 1.22364
\(917\) −5.56267 −0.183696
\(918\) −5.20928 −0.171932
\(919\) −53.5402 −1.76613 −0.883065 0.469251i \(-0.844524\pi\)
−0.883065 + 0.469251i \(0.844524\pi\)
\(920\) −95.6914 −3.15485
\(921\) −12.6233 −0.415950
\(922\) 42.4766 1.39889
\(923\) 0 0
\(924\) −4.23435 −0.139300
\(925\) −72.2526 −2.37565
\(926\) 69.8095 2.29408
\(927\) −13.4029 −0.440209
\(928\) 9.38447 0.308060
\(929\) −35.1499 −1.15323 −0.576616 0.817015i \(-0.695627\pi\)
−0.576616 + 0.817015i \(0.695627\pi\)
\(930\) −98.9339 −3.24417
\(931\) −46.7150 −1.53102
\(932\) −79.6272 −2.60828
\(933\) 2.12116 0.0694435
\(934\) 58.1185 1.90170
\(935\) −7.60710 −0.248779
\(936\) 0 0
\(937\) 12.6307 0.412626 0.206313 0.978486i \(-0.433854\pi\)
0.206313 + 0.978486i \(0.433854\pi\)
\(938\) −3.41342 −0.111452
\(939\) 21.5919 0.704626
\(940\) −20.4067 −0.665592
\(941\) −39.6660 −1.29308 −0.646538 0.762882i \(-0.723784\pi\)
−0.646538 + 0.762882i \(0.723784\pi\)
\(942\) 32.8724 1.07104
\(943\) 22.0241 0.717204
\(944\) −83.0072 −2.70165
\(945\) −3.54294 −0.115252
\(946\) 13.3340 0.433524
\(947\) 6.08586 0.197764 0.0988819 0.995099i \(-0.468473\pi\)
0.0988819 + 0.995099i \(0.468473\pi\)
\(948\) −16.1409 −0.524233
\(949\) 0 0
\(950\) 169.263 5.49162
\(951\) 15.7684 0.511324
\(952\) 12.0843 0.391654
\(953\) −29.8978 −0.968484 −0.484242 0.874934i \(-0.660904\pi\)
−0.484242 + 0.874934i \(0.660904\pi\)
\(954\) 7.56008 0.244767
\(955\) 56.5917 1.83127
\(956\) −25.4114 −0.821865
\(957\) 1.97786 0.0639351
\(958\) −14.4840 −0.467958
\(959\) −5.40113 −0.174412
\(960\) −5.22612 −0.168672
\(961\) 80.2496 2.58870
\(962\) 0 0
\(963\) −8.05792 −0.259663
\(964\) −71.4170 −2.30019
\(965\) 25.0244 0.805563
\(966\) 10.2141 0.328635
\(967\) 23.5725 0.758039 0.379020 0.925389i \(-0.376261\pi\)
0.379020 + 0.925389i \(0.376261\pi\)
\(968\) 6.14151 0.197396
\(969\) −15.7833 −0.507032
\(970\) 92.1377 2.95836
\(971\) 31.0731 0.997182 0.498591 0.866837i \(-0.333851\pi\)
0.498591 + 0.866837i \(0.333851\pi\)
\(972\) 4.42325 0.141876
\(973\) 4.27738 0.137126
\(974\) 26.6081 0.852578
\(975\) 0 0
\(976\) −1.23523 −0.0395386
\(977\) 18.8934 0.604454 0.302227 0.953236i \(-0.402270\pi\)
0.302227 + 0.953236i \(0.402270\pi\)
\(978\) 47.3631 1.51450
\(979\) 5.24302 0.167568
\(980\) −99.5909 −3.18132
\(981\) −14.3600 −0.458480
\(982\) 33.0633 1.05509
\(983\) 32.8458 1.04762 0.523809 0.851836i \(-0.324511\pi\)
0.523809 + 0.851836i \(0.324511\pi\)
\(984\) −32.1288 −1.02423
\(985\) −41.9398 −1.33631
\(986\) −10.3032 −0.328122
\(987\) 1.19332 0.0379839
\(988\) 0 0
\(989\) −22.1493 −0.704308
\(990\) 9.37985 0.298111
\(991\) −15.1204 −0.480316 −0.240158 0.970734i \(-0.577199\pi\)
−0.240158 + 0.970734i \(0.577199\pi\)
\(992\) −50.0453 −1.58894
\(993\) 11.6602 0.370025
\(994\) −29.4980 −0.935620
\(995\) 79.8088 2.53011
\(996\) 14.6833 0.465257
\(997\) −2.12151 −0.0671889 −0.0335945 0.999436i \(-0.510695\pi\)
−0.0335945 + 0.999436i \(0.510695\pi\)
\(998\) −97.9284 −3.09987
\(999\) −8.30741 −0.262835
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 5577.2.a.y.1.6 7
13.5 odd 4 429.2.b.b.298.2 14
13.8 odd 4 429.2.b.b.298.13 yes 14
13.12 even 2 5577.2.a.x.1.2 7
39.5 even 4 1287.2.b.c.298.13 14
39.8 even 4 1287.2.b.c.298.2 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.b.b.298.2 14 13.5 odd 4
429.2.b.b.298.13 yes 14 13.8 odd 4
1287.2.b.c.298.2 14 39.8 even 4
1287.2.b.c.298.13 14 39.5 even 4
5577.2.a.x.1.2 7 13.12 even 2
5577.2.a.y.1.6 7 1.1 even 1 trivial