Properties

Label 5445.2.a.cb.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 3x^{6} + 22x^{5} - 3x^{4} - 32x^{3} + 9x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.75229\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.75229 q^{2} +5.57509 q^{4} +1.00000 q^{5} -2.73187 q^{7} -9.83966 q^{8} +O(q^{10})\) \(q-2.75229 q^{2} +5.57509 q^{4} +1.00000 q^{5} -2.73187 q^{7} -9.83966 q^{8} -2.75229 q^{10} +2.77482 q^{13} +7.51889 q^{14} +15.9314 q^{16} +3.78118 q^{17} +4.25186 q^{19} +5.57509 q^{20} -1.37658 q^{23} +1.00000 q^{25} -7.63712 q^{26} -15.2304 q^{28} -8.86735 q^{29} +5.46722 q^{31} -24.1685 q^{32} -10.4069 q^{34} -2.73187 q^{35} -2.30142 q^{37} -11.7023 q^{38} -9.83966 q^{40} -8.61928 q^{41} -12.3981 q^{43} +3.78875 q^{46} -5.59315 q^{47} +0.463108 q^{49} -2.75229 q^{50} +15.4699 q^{52} -0.543807 q^{53} +26.8807 q^{56} +24.4055 q^{58} +0.389279 q^{59} +14.3414 q^{61} -15.0474 q^{62} +34.6558 q^{64} +2.77482 q^{65} +3.65454 q^{67} +21.0804 q^{68} +7.51889 q^{70} -9.16556 q^{71} -8.16025 q^{73} +6.33417 q^{74} +23.7045 q^{76} +3.44708 q^{79} +15.9314 q^{80} +23.7227 q^{82} +4.65472 q^{83} +3.78118 q^{85} +34.1231 q^{86} +6.62318 q^{89} -7.58046 q^{91} -7.67457 q^{92} +15.3940 q^{94} +4.25186 q^{95} -16.5938 q^{97} -1.27461 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 6 q^{4} + 8 q^{5} - 8 q^{7} - 12 q^{8} - 4 q^{10} - 6 q^{13} + 14 q^{14} + 14 q^{16} - 8 q^{17} + 2 q^{19} + 6 q^{20} + 4 q^{23} + 8 q^{25} - 2 q^{26} - 24 q^{28} - 22 q^{29} + 10 q^{31} - 28 q^{32} - 2 q^{34} - 8 q^{35} - 14 q^{37} - 20 q^{38} - 12 q^{40} - 22 q^{41} - 14 q^{43} + 2 q^{46} + 10 q^{47} - 4 q^{50} + 10 q^{52} - 18 q^{53} + 34 q^{56} + 12 q^{58} + 2 q^{59} + 14 q^{61} - 30 q^{62} + 30 q^{64} - 6 q^{65} + 10 q^{67} - 6 q^{68} + 14 q^{70} - 2 q^{71} - 16 q^{73} - 24 q^{74} + 22 q^{76} + 16 q^{79} + 14 q^{80} + 10 q^{82} - 46 q^{83} - 8 q^{85} - 28 q^{86} + 38 q^{89} + 8 q^{91} - 24 q^{92} + 10 q^{94} + 2 q^{95} - 4 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75229 −1.94616 −0.973081 0.230465i \(-0.925975\pi\)
−0.973081 + 0.230465i \(0.925975\pi\)
\(3\) 0 0
\(4\) 5.57509 2.78754
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.73187 −1.03255 −0.516275 0.856423i \(-0.672682\pi\)
−0.516275 + 0.856423i \(0.672682\pi\)
\(8\) −9.83966 −3.47885
\(9\) 0 0
\(10\) −2.75229 −0.870350
\(11\) 0 0
\(12\) 0 0
\(13\) 2.77482 0.769598 0.384799 0.923000i \(-0.374271\pi\)
0.384799 + 0.923000i \(0.374271\pi\)
\(14\) 7.51889 2.00951
\(15\) 0 0
\(16\) 15.9314 3.98285
\(17\) 3.78118 0.917071 0.458536 0.888676i \(-0.348374\pi\)
0.458536 + 0.888676i \(0.348374\pi\)
\(18\) 0 0
\(19\) 4.25186 0.975444 0.487722 0.872999i \(-0.337828\pi\)
0.487722 + 0.872999i \(0.337828\pi\)
\(20\) 5.57509 1.24663
\(21\) 0 0
\(22\) 0 0
\(23\) −1.37658 −0.287038 −0.143519 0.989648i \(-0.545842\pi\)
−0.143519 + 0.989648i \(0.545842\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.63712 −1.49776
\(27\) 0 0
\(28\) −15.2304 −2.87828
\(29\) −8.86735 −1.64663 −0.823313 0.567588i \(-0.807877\pi\)
−0.823313 + 0.567588i \(0.807877\pi\)
\(30\) 0 0
\(31\) 5.46722 0.981942 0.490971 0.871176i \(-0.336642\pi\)
0.490971 + 0.871176i \(0.336642\pi\)
\(32\) −24.1685 −4.27243
\(33\) 0 0
\(34\) −10.4069 −1.78477
\(35\) −2.73187 −0.461770
\(36\) 0 0
\(37\) −2.30142 −0.378351 −0.189176 0.981943i \(-0.560582\pi\)
−0.189176 + 0.981943i \(0.560582\pi\)
\(38\) −11.7023 −1.89837
\(39\) 0 0
\(40\) −9.83966 −1.55579
\(41\) −8.61928 −1.34611 −0.673053 0.739594i \(-0.735017\pi\)
−0.673053 + 0.739594i \(0.735017\pi\)
\(42\) 0 0
\(43\) −12.3981 −1.89069 −0.945346 0.326070i \(-0.894276\pi\)
−0.945346 + 0.326070i \(0.894276\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.78875 0.558621
\(47\) −5.59315 −0.815845 −0.407923 0.913017i \(-0.633747\pi\)
−0.407923 + 0.913017i \(0.633747\pi\)
\(48\) 0 0
\(49\) 0.463108 0.0661582
\(50\) −2.75229 −0.389232
\(51\) 0 0
\(52\) 15.4699 2.14529
\(53\) −0.543807 −0.0746976 −0.0373488 0.999302i \(-0.511891\pi\)
−0.0373488 + 0.999302i \(0.511891\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 26.8807 3.59208
\(57\) 0 0
\(58\) 24.4055 3.20460
\(59\) 0.389279 0.0506797 0.0253399 0.999679i \(-0.491933\pi\)
0.0253399 + 0.999679i \(0.491933\pi\)
\(60\) 0 0
\(61\) 14.3414 1.83622 0.918112 0.396320i \(-0.129713\pi\)
0.918112 + 0.396320i \(0.129713\pi\)
\(62\) −15.0474 −1.91102
\(63\) 0 0
\(64\) 34.6558 4.33198
\(65\) 2.77482 0.344175
\(66\) 0 0
\(67\) 3.65454 0.446473 0.223237 0.974764i \(-0.428338\pi\)
0.223237 + 0.974764i \(0.428338\pi\)
\(68\) 21.0804 2.55638
\(69\) 0 0
\(70\) 7.51889 0.898679
\(71\) −9.16556 −1.08775 −0.543876 0.839166i \(-0.683044\pi\)
−0.543876 + 0.839166i \(0.683044\pi\)
\(72\) 0 0
\(73\) −8.16025 −0.955085 −0.477542 0.878609i \(-0.658472\pi\)
−0.477542 + 0.878609i \(0.658472\pi\)
\(74\) 6.33417 0.736333
\(75\) 0 0
\(76\) 23.7045 2.71909
\(77\) 0 0
\(78\) 0 0
\(79\) 3.44708 0.387827 0.193913 0.981019i \(-0.437882\pi\)
0.193913 + 0.981019i \(0.437882\pi\)
\(80\) 15.9314 1.78119
\(81\) 0 0
\(82\) 23.7227 2.61974
\(83\) 4.65472 0.510922 0.255461 0.966819i \(-0.417773\pi\)
0.255461 + 0.966819i \(0.417773\pi\)
\(84\) 0 0
\(85\) 3.78118 0.410127
\(86\) 34.1231 3.67959
\(87\) 0 0
\(88\) 0 0
\(89\) 6.62318 0.702056 0.351028 0.936365i \(-0.385832\pi\)
0.351028 + 0.936365i \(0.385832\pi\)
\(90\) 0 0
\(91\) −7.58046 −0.794648
\(92\) −7.67457 −0.800129
\(93\) 0 0
\(94\) 15.3940 1.58777
\(95\) 4.25186 0.436232
\(96\) 0 0
\(97\) −16.5938 −1.68484 −0.842421 0.538820i \(-0.818870\pi\)
−0.842421 + 0.538820i \(0.818870\pi\)
\(98\) −1.27461 −0.128755
\(99\) 0 0
\(100\) 5.57509 0.557509
\(101\) −0.539052 −0.0536376 −0.0268188 0.999640i \(-0.508538\pi\)
−0.0268188 + 0.999640i \(0.508538\pi\)
\(102\) 0 0
\(103\) −6.11063 −0.602098 −0.301049 0.953609i \(-0.597337\pi\)
−0.301049 + 0.953609i \(0.597337\pi\)
\(104\) −27.3033 −2.67731
\(105\) 0 0
\(106\) 1.49671 0.145374
\(107\) −9.87705 −0.954851 −0.477425 0.878672i \(-0.658430\pi\)
−0.477425 + 0.878672i \(0.658430\pi\)
\(108\) 0 0
\(109\) 5.44683 0.521712 0.260856 0.965378i \(-0.415995\pi\)
0.260856 + 0.965378i \(0.415995\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −43.5225 −4.11249
\(113\) 0.777605 0.0731510 0.0365755 0.999331i \(-0.488355\pi\)
0.0365755 + 0.999331i \(0.488355\pi\)
\(114\) 0 0
\(115\) −1.37658 −0.128367
\(116\) −49.4362 −4.59004
\(117\) 0 0
\(118\) −1.07141 −0.0986310
\(119\) −10.3297 −0.946921
\(120\) 0 0
\(121\) 0 0
\(122\) −39.4716 −3.57359
\(123\) 0 0
\(124\) 30.4802 2.73721
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.33061 −0.384279 −0.192140 0.981368i \(-0.561543\pi\)
−0.192140 + 0.981368i \(0.561543\pi\)
\(128\) −47.0458 −4.15830
\(129\) 0 0
\(130\) −7.63712 −0.669819
\(131\) 11.7312 1.02496 0.512478 0.858700i \(-0.328728\pi\)
0.512478 + 0.858700i \(0.328728\pi\)
\(132\) 0 0
\(133\) −11.6155 −1.00719
\(134\) −10.0583 −0.868909
\(135\) 0 0
\(136\) −37.2056 −3.19035
\(137\) 13.5841 1.16057 0.580285 0.814413i \(-0.302941\pi\)
0.580285 + 0.814413i \(0.302941\pi\)
\(138\) 0 0
\(139\) 12.0322 1.02056 0.510278 0.860009i \(-0.329542\pi\)
0.510278 + 0.860009i \(0.329542\pi\)
\(140\) −15.2304 −1.28720
\(141\) 0 0
\(142\) 25.2263 2.11694
\(143\) 0 0
\(144\) 0 0
\(145\) −8.86735 −0.736393
\(146\) 22.4594 1.85875
\(147\) 0 0
\(148\) −12.8306 −1.05467
\(149\) 13.7934 1.13000 0.564999 0.825092i \(-0.308877\pi\)
0.564999 + 0.825092i \(0.308877\pi\)
\(150\) 0 0
\(151\) 17.0063 1.38395 0.691976 0.721921i \(-0.256741\pi\)
0.691976 + 0.721921i \(0.256741\pi\)
\(152\) −41.8369 −3.39342
\(153\) 0 0
\(154\) 0 0
\(155\) 5.46722 0.439138
\(156\) 0 0
\(157\) −3.61081 −0.288174 −0.144087 0.989565i \(-0.546025\pi\)
−0.144087 + 0.989565i \(0.546025\pi\)
\(158\) −9.48735 −0.754773
\(159\) 0 0
\(160\) −24.1685 −1.91069
\(161\) 3.76065 0.296380
\(162\) 0 0
\(163\) −3.62347 −0.283812 −0.141906 0.989880i \(-0.545323\pi\)
−0.141906 + 0.989880i \(0.545323\pi\)
\(164\) −48.0532 −3.75233
\(165\) 0 0
\(166\) −12.8111 −0.994337
\(167\) −2.27168 −0.175788 −0.0878939 0.996130i \(-0.528014\pi\)
−0.0878939 + 0.996130i \(0.528014\pi\)
\(168\) 0 0
\(169\) −5.30035 −0.407719
\(170\) −10.4069 −0.798173
\(171\) 0 0
\(172\) −69.1204 −5.27038
\(173\) 19.2705 1.46511 0.732554 0.680709i \(-0.238328\pi\)
0.732554 + 0.680709i \(0.238328\pi\)
\(174\) 0 0
\(175\) −2.73187 −0.206510
\(176\) 0 0
\(177\) 0 0
\(178\) −18.2289 −1.36631
\(179\) −7.26883 −0.543298 −0.271649 0.962396i \(-0.587569\pi\)
−0.271649 + 0.962396i \(0.587569\pi\)
\(180\) 0 0
\(181\) −8.11886 −0.603470 −0.301735 0.953392i \(-0.597566\pi\)
−0.301735 + 0.953392i \(0.597566\pi\)
\(182\) 20.8636 1.54651
\(183\) 0 0
\(184\) 13.5451 0.998560
\(185\) −2.30142 −0.169204
\(186\) 0 0
\(187\) 0 0
\(188\) −31.1823 −2.27420
\(189\) 0 0
\(190\) −11.7023 −0.848977
\(191\) −20.0211 −1.44868 −0.724339 0.689444i \(-0.757855\pi\)
−0.724339 + 0.689444i \(0.757855\pi\)
\(192\) 0 0
\(193\) 19.5724 1.40885 0.704425 0.709779i \(-0.251205\pi\)
0.704425 + 0.709779i \(0.251205\pi\)
\(194\) 45.6708 3.27897
\(195\) 0 0
\(196\) 2.58186 0.184419
\(197\) −22.9044 −1.63187 −0.815937 0.578141i \(-0.803778\pi\)
−0.815937 + 0.578141i \(0.803778\pi\)
\(198\) 0 0
\(199\) −17.3671 −1.23112 −0.615560 0.788090i \(-0.711070\pi\)
−0.615560 + 0.788090i \(0.711070\pi\)
\(200\) −9.83966 −0.695769
\(201\) 0 0
\(202\) 1.48363 0.104388
\(203\) 24.2244 1.70022
\(204\) 0 0
\(205\) −8.61928 −0.601997
\(206\) 16.8182 1.17178
\(207\) 0 0
\(208\) 44.2069 3.06520
\(209\) 0 0
\(210\) 0 0
\(211\) −16.6106 −1.14352 −0.571760 0.820421i \(-0.693739\pi\)
−0.571760 + 0.820421i \(0.693739\pi\)
\(212\) −3.03177 −0.208223
\(213\) 0 0
\(214\) 27.1845 1.85829
\(215\) −12.3981 −0.845543
\(216\) 0 0
\(217\) −14.9357 −1.01390
\(218\) −14.9912 −1.01533
\(219\) 0 0
\(220\) 0 0
\(221\) 10.4921 0.705776
\(222\) 0 0
\(223\) −8.96047 −0.600037 −0.300019 0.953933i \(-0.596993\pi\)
−0.300019 + 0.953933i \(0.596993\pi\)
\(224\) 66.0252 4.41149
\(225\) 0 0
\(226\) −2.14019 −0.142364
\(227\) −1.06154 −0.0704571 −0.0352286 0.999379i \(-0.511216\pi\)
−0.0352286 + 0.999379i \(0.511216\pi\)
\(228\) 0 0
\(229\) −5.85354 −0.386813 −0.193406 0.981119i \(-0.561954\pi\)
−0.193406 + 0.981119i \(0.561954\pi\)
\(230\) 3.78875 0.249823
\(231\) 0 0
\(232\) 87.2518 5.72836
\(233\) −1.87722 −0.122981 −0.0614904 0.998108i \(-0.519585\pi\)
−0.0614904 + 0.998108i \(0.519585\pi\)
\(234\) 0 0
\(235\) −5.59315 −0.364857
\(236\) 2.17026 0.141272
\(237\) 0 0
\(238\) 28.4303 1.84286
\(239\) −2.04058 −0.131994 −0.0659970 0.997820i \(-0.521023\pi\)
−0.0659970 + 0.997820i \(0.521023\pi\)
\(240\) 0 0
\(241\) −15.3922 −0.991500 −0.495750 0.868465i \(-0.665107\pi\)
−0.495750 + 0.868465i \(0.665107\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 79.9544 5.11856
\(245\) 0.463108 0.0295869
\(246\) 0 0
\(247\) 11.7982 0.750700
\(248\) −53.7956 −3.41603
\(249\) 0 0
\(250\) −2.75229 −0.174070
\(251\) 13.4172 0.846889 0.423444 0.905922i \(-0.360821\pi\)
0.423444 + 0.905922i \(0.360821\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 11.9191 0.747869
\(255\) 0 0
\(256\) 60.1719 3.76074
\(257\) 9.61357 0.599678 0.299839 0.953990i \(-0.403067\pi\)
0.299839 + 0.953990i \(0.403067\pi\)
\(258\) 0 0
\(259\) 6.28718 0.390666
\(260\) 15.4699 0.959402
\(261\) 0 0
\(262\) −32.2875 −1.99473
\(263\) 23.4178 1.44400 0.722002 0.691891i \(-0.243222\pi\)
0.722002 + 0.691891i \(0.243222\pi\)
\(264\) 0 0
\(265\) −0.543807 −0.0334058
\(266\) 31.9693 1.96016
\(267\) 0 0
\(268\) 20.3744 1.24456
\(269\) 9.46210 0.576914 0.288457 0.957493i \(-0.406858\pi\)
0.288457 + 0.957493i \(0.406858\pi\)
\(270\) 0 0
\(271\) −25.5459 −1.55180 −0.775902 0.630853i \(-0.782705\pi\)
−0.775902 + 0.630853i \(0.782705\pi\)
\(272\) 60.2396 3.65256
\(273\) 0 0
\(274\) −37.3874 −2.25866
\(275\) 0 0
\(276\) 0 0
\(277\) 4.65032 0.279411 0.139705 0.990193i \(-0.455384\pi\)
0.139705 + 0.990193i \(0.455384\pi\)
\(278\) −33.1160 −1.98617
\(279\) 0 0
\(280\) 26.8807 1.60643
\(281\) −10.8043 −0.644533 −0.322266 0.946649i \(-0.604445\pi\)
−0.322266 + 0.946649i \(0.604445\pi\)
\(282\) 0 0
\(283\) −2.00667 −0.119284 −0.0596420 0.998220i \(-0.518996\pi\)
−0.0596420 + 0.998220i \(0.518996\pi\)
\(284\) −51.0988 −3.03215
\(285\) 0 0
\(286\) 0 0
\(287\) 23.5467 1.38992
\(288\) 0 0
\(289\) −2.70266 −0.158980
\(290\) 24.4055 1.43314
\(291\) 0 0
\(292\) −45.4941 −2.66234
\(293\) 0.602557 0.0352018 0.0176009 0.999845i \(-0.494397\pi\)
0.0176009 + 0.999845i \(0.494397\pi\)
\(294\) 0 0
\(295\) 0.389279 0.0226647
\(296\) 22.6452 1.31623
\(297\) 0 0
\(298\) −37.9633 −2.19916
\(299\) −3.81978 −0.220903
\(300\) 0 0
\(301\) 33.8700 1.95223
\(302\) −46.8062 −2.69339
\(303\) 0 0
\(304\) 67.7382 3.88505
\(305\) 14.3414 0.821185
\(306\) 0 0
\(307\) −30.7715 −1.75622 −0.878112 0.478454i \(-0.841197\pi\)
−0.878112 + 0.478454i \(0.841197\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −15.0474 −0.854633
\(311\) −1.44287 −0.0818175 −0.0409087 0.999163i \(-0.513025\pi\)
−0.0409087 + 0.999163i \(0.513025\pi\)
\(312\) 0 0
\(313\) 23.4992 1.32826 0.664128 0.747619i \(-0.268803\pi\)
0.664128 + 0.747619i \(0.268803\pi\)
\(314\) 9.93800 0.560834
\(315\) 0 0
\(316\) 19.2178 1.08108
\(317\) −14.3181 −0.804184 −0.402092 0.915599i \(-0.631717\pi\)
−0.402092 + 0.915599i \(0.631717\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 34.6558 1.93732
\(321\) 0 0
\(322\) −10.3504 −0.576804
\(323\) 16.0771 0.894552
\(324\) 0 0
\(325\) 2.77482 0.153920
\(326\) 9.97282 0.552344
\(327\) 0 0
\(328\) 84.8108 4.68289
\(329\) 15.2798 0.842400
\(330\) 0 0
\(331\) −4.93282 −0.271132 −0.135566 0.990768i \(-0.543285\pi\)
−0.135566 + 0.990768i \(0.543285\pi\)
\(332\) 25.9505 1.42422
\(333\) 0 0
\(334\) 6.25231 0.342111
\(335\) 3.65454 0.199669
\(336\) 0 0
\(337\) −7.69258 −0.419042 −0.209521 0.977804i \(-0.567190\pi\)
−0.209521 + 0.977804i \(0.567190\pi\)
\(338\) 14.5881 0.793487
\(339\) 0 0
\(340\) 21.0804 1.14325
\(341\) 0 0
\(342\) 0 0
\(343\) 17.8579 0.964238
\(344\) 121.993 6.57743
\(345\) 0 0
\(346\) −53.0379 −2.85134
\(347\) −17.1661 −0.921522 −0.460761 0.887524i \(-0.652424\pi\)
−0.460761 + 0.887524i \(0.652424\pi\)
\(348\) 0 0
\(349\) −24.2805 −1.29971 −0.649853 0.760060i \(-0.725170\pi\)
−0.649853 + 0.760060i \(0.725170\pi\)
\(350\) 7.51889 0.401901
\(351\) 0 0
\(352\) 0 0
\(353\) −5.84602 −0.311152 −0.155576 0.987824i \(-0.549723\pi\)
−0.155576 + 0.987824i \(0.549723\pi\)
\(354\) 0 0
\(355\) −9.16556 −0.486457
\(356\) 36.9248 1.95701
\(357\) 0 0
\(358\) 20.0059 1.05735
\(359\) −8.14440 −0.429845 −0.214922 0.976631i \(-0.568950\pi\)
−0.214922 + 0.976631i \(0.568950\pi\)
\(360\) 0 0
\(361\) −0.921675 −0.0485092
\(362\) 22.3454 1.17445
\(363\) 0 0
\(364\) −42.2617 −2.21512
\(365\) −8.16025 −0.427127
\(366\) 0 0
\(367\) 22.6747 1.18361 0.591806 0.806081i \(-0.298415\pi\)
0.591806 + 0.806081i \(0.298415\pi\)
\(368\) −21.9309 −1.14323
\(369\) 0 0
\(370\) 6.33417 0.329298
\(371\) 1.48561 0.0771290
\(372\) 0 0
\(373\) 14.0566 0.727825 0.363913 0.931433i \(-0.381441\pi\)
0.363913 + 0.931433i \(0.381441\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 55.0347 2.83820
\(377\) −24.6053 −1.26724
\(378\) 0 0
\(379\) −25.2171 −1.29532 −0.647659 0.761931i \(-0.724252\pi\)
−0.647659 + 0.761931i \(0.724252\pi\)
\(380\) 23.7045 1.21601
\(381\) 0 0
\(382\) 55.1039 2.81936
\(383\) −24.5315 −1.25350 −0.626750 0.779220i \(-0.715615\pi\)
−0.626750 + 0.779220i \(0.715615\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −53.8688 −2.74185
\(387\) 0 0
\(388\) −92.5116 −4.69657
\(389\) 24.1607 1.22500 0.612499 0.790472i \(-0.290164\pi\)
0.612499 + 0.790472i \(0.290164\pi\)
\(390\) 0 0
\(391\) −5.20511 −0.263234
\(392\) −4.55682 −0.230154
\(393\) 0 0
\(394\) 63.0396 3.17589
\(395\) 3.44708 0.173441
\(396\) 0 0
\(397\) −29.4680 −1.47896 −0.739479 0.673179i \(-0.764928\pi\)
−0.739479 + 0.673179i \(0.764928\pi\)
\(398\) 47.7992 2.39596
\(399\) 0 0
\(400\) 15.9314 0.796571
\(401\) −4.57792 −0.228610 −0.114305 0.993446i \(-0.536464\pi\)
−0.114305 + 0.993446i \(0.536464\pi\)
\(402\) 0 0
\(403\) 15.1706 0.755701
\(404\) −3.00526 −0.149517
\(405\) 0 0
\(406\) −66.6726 −3.30891
\(407\) 0 0
\(408\) 0 0
\(409\) 19.0113 0.940050 0.470025 0.882653i \(-0.344245\pi\)
0.470025 + 0.882653i \(0.344245\pi\)
\(410\) 23.7227 1.17158
\(411\) 0 0
\(412\) −34.0673 −1.67838
\(413\) −1.06346 −0.0523293
\(414\) 0 0
\(415\) 4.65472 0.228491
\(416\) −67.0633 −3.28805
\(417\) 0 0
\(418\) 0 0
\(419\) −2.54104 −0.124138 −0.0620690 0.998072i \(-0.519770\pi\)
−0.0620690 + 0.998072i \(0.519770\pi\)
\(420\) 0 0
\(421\) 2.05374 0.100093 0.0500466 0.998747i \(-0.484063\pi\)
0.0500466 + 0.998747i \(0.484063\pi\)
\(422\) 45.7171 2.22547
\(423\) 0 0
\(424\) 5.35088 0.259862
\(425\) 3.78118 0.183414
\(426\) 0 0
\(427\) −39.1788 −1.89599
\(428\) −55.0654 −2.66169
\(429\) 0 0
\(430\) 34.1231 1.64556
\(431\) −37.8911 −1.82515 −0.912576 0.408907i \(-0.865910\pi\)
−0.912576 + 0.408907i \(0.865910\pi\)
\(432\) 0 0
\(433\) 9.10654 0.437633 0.218816 0.975766i \(-0.429780\pi\)
0.218816 + 0.975766i \(0.429780\pi\)
\(434\) 41.1074 1.97322
\(435\) 0 0
\(436\) 30.3665 1.45429
\(437\) −5.85304 −0.279989
\(438\) 0 0
\(439\) −31.0675 −1.48277 −0.741385 0.671079i \(-0.765831\pi\)
−0.741385 + 0.671079i \(0.765831\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −28.8773 −1.37355
\(443\) −14.9113 −0.708458 −0.354229 0.935159i \(-0.615257\pi\)
−0.354229 + 0.935159i \(0.615257\pi\)
\(444\) 0 0
\(445\) 6.62318 0.313969
\(446\) 24.6618 1.16777
\(447\) 0 0
\(448\) −94.6752 −4.47298
\(449\) 25.6099 1.20861 0.604303 0.796755i \(-0.293452\pi\)
0.604303 + 0.796755i \(0.293452\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.33522 0.203911
\(453\) 0 0
\(454\) 2.92167 0.137121
\(455\) −7.58046 −0.355377
\(456\) 0 0
\(457\) −8.02615 −0.375447 −0.187724 0.982222i \(-0.560111\pi\)
−0.187724 + 0.982222i \(0.560111\pi\)
\(458\) 16.1106 0.752800
\(459\) 0 0
\(460\) −7.67457 −0.357829
\(461\) −16.8858 −0.786449 −0.393225 0.919442i \(-0.628640\pi\)
−0.393225 + 0.919442i \(0.628640\pi\)
\(462\) 0 0
\(463\) −4.70959 −0.218873 −0.109437 0.993994i \(-0.534905\pi\)
−0.109437 + 0.993994i \(0.534905\pi\)
\(464\) −141.269 −6.55827
\(465\) 0 0
\(466\) 5.16665 0.239340
\(467\) 15.4149 0.713314 0.356657 0.934235i \(-0.383916\pi\)
0.356657 + 0.934235i \(0.383916\pi\)
\(468\) 0 0
\(469\) −9.98373 −0.461006
\(470\) 15.3940 0.710071
\(471\) 0 0
\(472\) −3.83037 −0.176307
\(473\) 0 0
\(474\) 0 0
\(475\) 4.25186 0.195089
\(476\) −57.5889 −2.63958
\(477\) 0 0
\(478\) 5.61626 0.256882
\(479\) 9.84576 0.449864 0.224932 0.974374i \(-0.427784\pi\)
0.224932 + 0.974374i \(0.427784\pi\)
\(480\) 0 0
\(481\) −6.38604 −0.291178
\(482\) 42.3638 1.92962
\(483\) 0 0
\(484\) 0 0
\(485\) −16.5938 −0.753484
\(486\) 0 0
\(487\) −8.17219 −0.370317 −0.185159 0.982709i \(-0.559280\pi\)
−0.185159 + 0.982709i \(0.559280\pi\)
\(488\) −141.114 −6.38794
\(489\) 0 0
\(490\) −1.27461 −0.0575808
\(491\) −38.5774 −1.74097 −0.870487 0.492192i \(-0.836196\pi\)
−0.870487 + 0.492192i \(0.836196\pi\)
\(492\) 0 0
\(493\) −33.5291 −1.51007
\(494\) −32.4720 −1.46098
\(495\) 0 0
\(496\) 87.1006 3.91093
\(497\) 25.0391 1.12316
\(498\) 0 0
\(499\) −2.34961 −0.105183 −0.0525916 0.998616i \(-0.516748\pi\)
−0.0525916 + 0.998616i \(0.516748\pi\)
\(500\) 5.57509 0.249325
\(501\) 0 0
\(502\) −36.9281 −1.64818
\(503\) −6.35233 −0.283237 −0.141618 0.989921i \(-0.545231\pi\)
−0.141618 + 0.989921i \(0.545231\pi\)
\(504\) 0 0
\(505\) −0.539052 −0.0239875
\(506\) 0 0
\(507\) 0 0
\(508\) −24.1435 −1.07119
\(509\) 30.0048 1.32994 0.664970 0.746870i \(-0.268444\pi\)
0.664970 + 0.746870i \(0.268444\pi\)
\(510\) 0 0
\(511\) 22.2927 0.986172
\(512\) −71.5187 −3.16071
\(513\) 0 0
\(514\) −26.4593 −1.16707
\(515\) −6.11063 −0.269267
\(516\) 0 0
\(517\) 0 0
\(518\) −17.3041 −0.760300
\(519\) 0 0
\(520\) −27.3033 −1.19733
\(521\) −23.9434 −1.04898 −0.524489 0.851417i \(-0.675744\pi\)
−0.524489 + 0.851417i \(0.675744\pi\)
\(522\) 0 0
\(523\) 16.9932 0.743060 0.371530 0.928421i \(-0.378833\pi\)
0.371530 + 0.928421i \(0.378833\pi\)
\(524\) 65.4022 2.85711
\(525\) 0 0
\(526\) −64.4526 −2.81026
\(527\) 20.6726 0.900511
\(528\) 0 0
\(529\) −21.1050 −0.917609
\(530\) 1.49671 0.0650131
\(531\) 0 0
\(532\) −64.7576 −2.80760
\(533\) −23.9170 −1.03596
\(534\) 0 0
\(535\) −9.87705 −0.427022
\(536\) −35.9594 −1.55321
\(537\) 0 0
\(538\) −26.0424 −1.12277
\(539\) 0 0
\(540\) 0 0
\(541\) −1.24537 −0.0535427 −0.0267714 0.999642i \(-0.508523\pi\)
−0.0267714 + 0.999642i \(0.508523\pi\)
\(542\) 70.3097 3.02006
\(543\) 0 0
\(544\) −91.3855 −3.91812
\(545\) 5.44683 0.233317
\(546\) 0 0
\(547\) −7.71616 −0.329919 −0.164960 0.986300i \(-0.552749\pi\)
−0.164960 + 0.986300i \(0.552749\pi\)
\(548\) 75.7327 3.23514
\(549\) 0 0
\(550\) 0 0
\(551\) −37.7027 −1.60619
\(552\) 0 0
\(553\) −9.41696 −0.400450
\(554\) −12.7990 −0.543779
\(555\) 0 0
\(556\) 67.0804 2.84484
\(557\) −10.9219 −0.462775 −0.231388 0.972862i \(-0.574327\pi\)
−0.231388 + 0.972862i \(0.574327\pi\)
\(558\) 0 0
\(559\) −34.4025 −1.45507
\(560\) −43.5225 −1.83916
\(561\) 0 0
\(562\) 29.7366 1.25436
\(563\) −12.2033 −0.514309 −0.257154 0.966370i \(-0.582785\pi\)
−0.257154 + 0.966370i \(0.582785\pi\)
\(564\) 0 0
\(565\) 0.777605 0.0327141
\(566\) 5.52293 0.232146
\(567\) 0 0
\(568\) 90.1860 3.78412
\(569\) −6.66849 −0.279558 −0.139779 0.990183i \(-0.544639\pi\)
−0.139779 + 0.990183i \(0.544639\pi\)
\(570\) 0 0
\(571\) 9.35392 0.391450 0.195725 0.980659i \(-0.437294\pi\)
0.195725 + 0.980659i \(0.437294\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −64.8074 −2.70501
\(575\) −1.37658 −0.0574075
\(576\) 0 0
\(577\) −10.9547 −0.456048 −0.228024 0.973656i \(-0.573226\pi\)
−0.228024 + 0.973656i \(0.573226\pi\)
\(578\) 7.43851 0.309401
\(579\) 0 0
\(580\) −49.4362 −2.05273
\(581\) −12.7161 −0.527553
\(582\) 0 0
\(583\) 0 0
\(584\) 80.2941 3.32259
\(585\) 0 0
\(586\) −1.65841 −0.0685083
\(587\) −23.9718 −0.989420 −0.494710 0.869058i \(-0.664726\pi\)
−0.494710 + 0.869058i \(0.664726\pi\)
\(588\) 0 0
\(589\) 23.2459 0.957830
\(590\) −1.07141 −0.0441091
\(591\) 0 0
\(592\) −36.6649 −1.50692
\(593\) 2.17705 0.0894006 0.0447003 0.999000i \(-0.485767\pi\)
0.0447003 + 0.999000i \(0.485767\pi\)
\(594\) 0 0
\(595\) −10.3297 −0.423476
\(596\) 76.8993 3.14992
\(597\) 0 0
\(598\) 10.5131 0.429914
\(599\) −28.7881 −1.17625 −0.588124 0.808770i \(-0.700133\pi\)
−0.588124 + 0.808770i \(0.700133\pi\)
\(600\) 0 0
\(601\) 10.4372 0.425744 0.212872 0.977080i \(-0.431718\pi\)
0.212872 + 0.977080i \(0.431718\pi\)
\(602\) −93.2199 −3.79936
\(603\) 0 0
\(604\) 94.8115 3.85782
\(605\) 0 0
\(606\) 0 0
\(607\) −29.3274 −1.19036 −0.595180 0.803592i \(-0.702919\pi\)
−0.595180 + 0.803592i \(0.702919\pi\)
\(608\) −102.761 −4.16751
\(609\) 0 0
\(610\) −39.4716 −1.59816
\(611\) −15.5200 −0.627873
\(612\) 0 0
\(613\) 27.5911 1.11439 0.557197 0.830381i \(-0.311877\pi\)
0.557197 + 0.830381i \(0.311877\pi\)
\(614\) 84.6921 3.41790
\(615\) 0 0
\(616\) 0 0
\(617\) 46.9079 1.88844 0.944220 0.329315i \(-0.106818\pi\)
0.944220 + 0.329315i \(0.106818\pi\)
\(618\) 0 0
\(619\) −19.1067 −0.767964 −0.383982 0.923341i \(-0.625448\pi\)
−0.383982 + 0.923341i \(0.625448\pi\)
\(620\) 30.4802 1.22412
\(621\) 0 0
\(622\) 3.97118 0.159230
\(623\) −18.0937 −0.724907
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −64.6767 −2.58500
\(627\) 0 0
\(628\) −20.1306 −0.803298
\(629\) −8.70209 −0.346975
\(630\) 0 0
\(631\) −30.8127 −1.22663 −0.613317 0.789837i \(-0.710165\pi\)
−0.613317 + 0.789837i \(0.710165\pi\)
\(632\) −33.9181 −1.34919
\(633\) 0 0
\(634\) 39.4075 1.56507
\(635\) −4.33061 −0.171855
\(636\) 0 0
\(637\) 1.28504 0.0509152
\(638\) 0 0
\(639\) 0 0
\(640\) −47.0458 −1.85965
\(641\) −5.27921 −0.208516 −0.104258 0.994550i \(-0.533247\pi\)
−0.104258 + 0.994550i \(0.533247\pi\)
\(642\) 0 0
\(643\) −43.7670 −1.72600 −0.863001 0.505203i \(-0.831418\pi\)
−0.863001 + 0.505203i \(0.831418\pi\)
\(644\) 20.9659 0.826173
\(645\) 0 0
\(646\) −44.2487 −1.74094
\(647\) 31.4016 1.23453 0.617263 0.786757i \(-0.288242\pi\)
0.617263 + 0.786757i \(0.288242\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −7.63712 −0.299552
\(651\) 0 0
\(652\) −20.2011 −0.791138
\(653\) 32.2029 1.26020 0.630099 0.776514i \(-0.283014\pi\)
0.630099 + 0.776514i \(0.283014\pi\)
\(654\) 0 0
\(655\) 11.7312 0.458375
\(656\) −137.317 −5.36134
\(657\) 0 0
\(658\) −42.0543 −1.63945
\(659\) −20.7286 −0.807472 −0.403736 0.914875i \(-0.632289\pi\)
−0.403736 + 0.914875i \(0.632289\pi\)
\(660\) 0 0
\(661\) −43.6387 −1.69735 −0.848673 0.528917i \(-0.822598\pi\)
−0.848673 + 0.528917i \(0.822598\pi\)
\(662\) 13.5765 0.527667
\(663\) 0 0
\(664\) −45.8009 −1.77742
\(665\) −11.6155 −0.450431
\(666\) 0 0
\(667\) 12.2067 0.472643
\(668\) −12.6648 −0.490016
\(669\) 0 0
\(670\) −10.0583 −0.388588
\(671\) 0 0
\(672\) 0 0
\(673\) 28.7584 1.10855 0.554277 0.832333i \(-0.312995\pi\)
0.554277 + 0.832333i \(0.312995\pi\)
\(674\) 21.1722 0.815523
\(675\) 0 0
\(676\) −29.5499 −1.13653
\(677\) 13.9609 0.536559 0.268280 0.963341i \(-0.413545\pi\)
0.268280 + 0.963341i \(0.413545\pi\)
\(678\) 0 0
\(679\) 45.3320 1.73968
\(680\) −37.2056 −1.42677
\(681\) 0 0
\(682\) 0 0
\(683\) 12.1334 0.464272 0.232136 0.972683i \(-0.425429\pi\)
0.232136 + 0.972683i \(0.425429\pi\)
\(684\) 0 0
\(685\) 13.5841 0.519023
\(686\) −49.1502 −1.87656
\(687\) 0 0
\(688\) −197.519 −7.53035
\(689\) −1.50897 −0.0574871
\(690\) 0 0
\(691\) −23.9834 −0.912373 −0.456187 0.889884i \(-0.650785\pi\)
−0.456187 + 0.889884i \(0.650785\pi\)
\(692\) 107.435 4.08405
\(693\) 0 0
\(694\) 47.2459 1.79343
\(695\) 12.0322 0.456407
\(696\) 0 0
\(697\) −32.5911 −1.23447
\(698\) 66.8270 2.52944
\(699\) 0 0
\(700\) −15.2304 −0.575655
\(701\) 42.7155 1.61334 0.806671 0.591001i \(-0.201267\pi\)
0.806671 + 0.591001i \(0.201267\pi\)
\(702\) 0 0
\(703\) −9.78532 −0.369060
\(704\) 0 0
\(705\) 0 0
\(706\) 16.0899 0.605552
\(707\) 1.47262 0.0553835
\(708\) 0 0
\(709\) −4.14512 −0.155673 −0.0778366 0.996966i \(-0.524801\pi\)
−0.0778366 + 0.996966i \(0.524801\pi\)
\(710\) 25.2263 0.946724
\(711\) 0 0
\(712\) −65.1699 −2.44234
\(713\) −7.52609 −0.281854
\(714\) 0 0
\(715\) 0 0
\(716\) −40.5244 −1.51447
\(717\) 0 0
\(718\) 22.4157 0.836547
\(719\) 26.3332 0.982062 0.491031 0.871142i \(-0.336620\pi\)
0.491031 + 0.871142i \(0.336620\pi\)
\(720\) 0 0
\(721\) 16.6934 0.621696
\(722\) 2.53671 0.0944067
\(723\) 0 0
\(724\) −45.2634 −1.68220
\(725\) −8.86735 −0.329325
\(726\) 0 0
\(727\) 14.3772 0.533220 0.266610 0.963804i \(-0.414096\pi\)
0.266610 + 0.963804i \(0.414096\pi\)
\(728\) 74.5892 2.76446
\(729\) 0 0
\(730\) 22.4594 0.831258
\(731\) −46.8794 −1.73390
\(732\) 0 0
\(733\) −34.4717 −1.27324 −0.636621 0.771177i \(-0.719668\pi\)
−0.636621 + 0.771177i \(0.719668\pi\)
\(734\) −62.4074 −2.30350
\(735\) 0 0
\(736\) 33.2700 1.22635
\(737\) 0 0
\(738\) 0 0
\(739\) 5.01642 0.184532 0.0922659 0.995734i \(-0.470589\pi\)
0.0922659 + 0.995734i \(0.470589\pi\)
\(740\) −12.8306 −0.471663
\(741\) 0 0
\(742\) −4.08882 −0.150105
\(743\) −38.6622 −1.41838 −0.709190 0.705018i \(-0.750939\pi\)
−0.709190 + 0.705018i \(0.750939\pi\)
\(744\) 0 0
\(745\) 13.7934 0.505350
\(746\) −38.6879 −1.41647
\(747\) 0 0
\(748\) 0 0
\(749\) 26.9828 0.985930
\(750\) 0 0
\(751\) −5.20080 −0.189780 −0.0948899 0.995488i \(-0.530250\pi\)
−0.0948899 + 0.995488i \(0.530250\pi\)
\(752\) −89.1068 −3.24939
\(753\) 0 0
\(754\) 67.7210 2.46625
\(755\) 17.0063 0.618922
\(756\) 0 0
\(757\) 6.19362 0.225111 0.112556 0.993645i \(-0.464096\pi\)
0.112556 + 0.993645i \(0.464096\pi\)
\(758\) 69.4048 2.52090
\(759\) 0 0
\(760\) −41.8369 −1.51758
\(761\) 11.8213 0.428520 0.214260 0.976777i \(-0.431266\pi\)
0.214260 + 0.976777i \(0.431266\pi\)
\(762\) 0 0
\(763\) −14.8800 −0.538693
\(764\) −111.620 −4.03825
\(765\) 0 0
\(766\) 67.5177 2.43951
\(767\) 1.08018 0.0390030
\(768\) 0 0
\(769\) −2.73424 −0.0985992 −0.0492996 0.998784i \(-0.515699\pi\)
−0.0492996 + 0.998784i \(0.515699\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 109.118 3.92723
\(773\) −1.54037 −0.0554032 −0.0277016 0.999616i \(-0.508819\pi\)
−0.0277016 + 0.999616i \(0.508819\pi\)
\(774\) 0 0
\(775\) 5.46722 0.196388
\(776\) 163.277 5.86130
\(777\) 0 0
\(778\) −66.4973 −2.38404
\(779\) −36.6480 −1.31305
\(780\) 0 0
\(781\) 0 0
\(782\) 14.3260 0.512296
\(783\) 0 0
\(784\) 7.37796 0.263498
\(785\) −3.61081 −0.128875
\(786\) 0 0
\(787\) 47.4293 1.69067 0.845336 0.534235i \(-0.179400\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(788\) −127.694 −4.54892
\(789\) 0 0
\(790\) −9.48735 −0.337545
\(791\) −2.12432 −0.0755320
\(792\) 0 0
\(793\) 39.7948 1.41315
\(794\) 81.1045 2.87829
\(795\) 0 0
\(796\) −96.8230 −3.43180
\(797\) 51.5701 1.82671 0.913353 0.407170i \(-0.133484\pi\)
0.913353 + 0.407170i \(0.133484\pi\)
\(798\) 0 0
\(799\) −21.1487 −0.748188
\(800\) −24.1685 −0.854485
\(801\) 0 0
\(802\) 12.5997 0.444912
\(803\) 0 0
\(804\) 0 0
\(805\) 3.76065 0.132545
\(806\) −41.7538 −1.47072
\(807\) 0 0
\(808\) 5.30409 0.186597
\(809\) −27.5009 −0.966881 −0.483440 0.875377i \(-0.660613\pi\)
−0.483440 + 0.875377i \(0.660613\pi\)
\(810\) 0 0
\(811\) −17.5146 −0.615020 −0.307510 0.951545i \(-0.599496\pi\)
−0.307510 + 0.951545i \(0.599496\pi\)
\(812\) 135.053 4.73944
\(813\) 0 0
\(814\) 0 0
\(815\) −3.62347 −0.126925
\(816\) 0 0
\(817\) −52.7150 −1.84426
\(818\) −52.3247 −1.82949
\(819\) 0 0
\(820\) −48.0532 −1.67809
\(821\) −12.7216 −0.443986 −0.221993 0.975048i \(-0.571256\pi\)
−0.221993 + 0.975048i \(0.571256\pi\)
\(822\) 0 0
\(823\) 12.5648 0.437981 0.218991 0.975727i \(-0.429724\pi\)
0.218991 + 0.975727i \(0.429724\pi\)
\(824\) 60.1266 2.09461
\(825\) 0 0
\(826\) 2.92694 0.101841
\(827\) −27.6527 −0.961579 −0.480790 0.876836i \(-0.659650\pi\)
−0.480790 + 0.876836i \(0.659650\pi\)
\(828\) 0 0
\(829\) 31.8545 1.10635 0.553176 0.833064i \(-0.313416\pi\)
0.553176 + 0.833064i \(0.313416\pi\)
\(830\) −12.8111 −0.444681
\(831\) 0 0
\(832\) 96.1638 3.33388
\(833\) 1.75109 0.0606718
\(834\) 0 0
\(835\) −2.27168 −0.0786147
\(836\) 0 0
\(837\) 0 0
\(838\) 6.99367 0.241592
\(839\) −23.1323 −0.798617 −0.399308 0.916817i \(-0.630750\pi\)
−0.399308 + 0.916817i \(0.630750\pi\)
\(840\) 0 0
\(841\) 49.6299 1.71138
\(842\) −5.65248 −0.194797
\(843\) 0 0
\(844\) −92.6054 −3.18761
\(845\) −5.30035 −0.182337
\(846\) 0 0
\(847\) 0 0
\(848\) −8.66361 −0.297510
\(849\) 0 0
\(850\) −10.4069 −0.356954
\(851\) 3.16810 0.108601
\(852\) 0 0
\(853\) 14.0442 0.480863 0.240432 0.970666i \(-0.422711\pi\)
0.240432 + 0.970666i \(0.422711\pi\)
\(854\) 107.831 3.68991
\(855\) 0 0
\(856\) 97.1869 3.32178
\(857\) −26.1013 −0.891603 −0.445801 0.895132i \(-0.647081\pi\)
−0.445801 + 0.895132i \(0.647081\pi\)
\(858\) 0 0
\(859\) 1.98117 0.0675967 0.0337983 0.999429i \(-0.489240\pi\)
0.0337983 + 0.999429i \(0.489240\pi\)
\(860\) −69.1204 −2.35699
\(861\) 0 0
\(862\) 104.287 3.55204
\(863\) 19.4162 0.660935 0.330467 0.943817i \(-0.392794\pi\)
0.330467 + 0.943817i \(0.392794\pi\)
\(864\) 0 0
\(865\) 19.2705 0.655216
\(866\) −25.0638 −0.851704
\(867\) 0 0
\(868\) −83.2680 −2.82630
\(869\) 0 0
\(870\) 0 0
\(871\) 10.1407 0.343605
\(872\) −53.5950 −1.81495
\(873\) 0 0
\(874\) 16.1093 0.544904
\(875\) −2.73187 −0.0923540
\(876\) 0 0
\(877\) 10.2208 0.345132 0.172566 0.984998i \(-0.444794\pi\)
0.172566 + 0.984998i \(0.444794\pi\)
\(878\) 85.5067 2.88571
\(879\) 0 0
\(880\) 0 0
\(881\) −26.4278 −0.890377 −0.445188 0.895437i \(-0.646863\pi\)
−0.445188 + 0.895437i \(0.646863\pi\)
\(882\) 0 0
\(883\) 44.1883 1.48705 0.743527 0.668706i \(-0.233152\pi\)
0.743527 + 0.668706i \(0.233152\pi\)
\(884\) 58.4945 1.96738
\(885\) 0 0
\(886\) 41.0402 1.37877
\(887\) −8.80191 −0.295539 −0.147770 0.989022i \(-0.547209\pi\)
−0.147770 + 0.989022i \(0.547209\pi\)
\(888\) 0 0
\(889\) 11.8306 0.396787
\(890\) −18.2289 −0.611034
\(891\) 0 0
\(892\) −49.9554 −1.67263
\(893\) −23.7813 −0.795811
\(894\) 0 0
\(895\) −7.26883 −0.242970
\(896\) 128.523 4.29365
\(897\) 0 0
\(898\) −70.4858 −2.35214
\(899\) −48.4798 −1.61689
\(900\) 0 0
\(901\) −2.05623 −0.0685031
\(902\) 0 0
\(903\) 0 0
\(904\) −7.65138 −0.254481
\(905\) −8.11886 −0.269880
\(906\) 0 0
\(907\) 18.8345 0.625389 0.312695 0.949854i \(-0.398768\pi\)
0.312695 + 0.949854i \(0.398768\pi\)
\(908\) −5.91820 −0.196402
\(909\) 0 0
\(910\) 20.8636 0.691622
\(911\) −22.0961 −0.732077 −0.366038 0.930600i \(-0.619286\pi\)
−0.366038 + 0.930600i \(0.619286\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 22.0903 0.730681
\(915\) 0 0
\(916\) −32.6340 −1.07826
\(917\) −32.0480 −1.05832
\(918\) 0 0
\(919\) −4.04875 −0.133556 −0.0667779 0.997768i \(-0.521272\pi\)
−0.0667779 + 0.997768i \(0.521272\pi\)
\(920\) 13.5451 0.446569
\(921\) 0 0
\(922\) 46.4745 1.53056
\(923\) −25.4328 −0.837132
\(924\) 0 0
\(925\) −2.30142 −0.0756703
\(926\) 12.9621 0.425962
\(927\) 0 0
\(928\) 214.311 7.03509
\(929\) 40.2897 1.32186 0.660931 0.750447i \(-0.270162\pi\)
0.660931 + 0.750447i \(0.270162\pi\)
\(930\) 0 0
\(931\) 1.96907 0.0645336
\(932\) −10.4657 −0.342814
\(933\) 0 0
\(934\) −42.4261 −1.38822
\(935\) 0 0
\(936\) 0 0
\(937\) −36.9767 −1.20798 −0.603988 0.796994i \(-0.706422\pi\)
−0.603988 + 0.796994i \(0.706422\pi\)
\(938\) 27.4781 0.897191
\(939\) 0 0
\(940\) −31.1823 −1.01705
\(941\) −43.1362 −1.40620 −0.703100 0.711091i \(-0.748201\pi\)
−0.703100 + 0.711091i \(0.748201\pi\)
\(942\) 0 0
\(943\) 11.8652 0.386383
\(944\) 6.20176 0.201850
\(945\) 0 0
\(946\) 0 0
\(947\) 42.5933 1.38410 0.692048 0.721851i \(-0.256709\pi\)
0.692048 + 0.721851i \(0.256709\pi\)
\(948\) 0 0
\(949\) −22.6433 −0.735031
\(950\) −11.7023 −0.379674
\(951\) 0 0
\(952\) 101.641 3.29419
\(953\) −27.3349 −0.885466 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(954\) 0 0
\(955\) −20.0211 −0.647869
\(956\) −11.3764 −0.367939
\(957\) 0 0
\(958\) −27.0984 −0.875509
\(959\) −37.1101 −1.19835
\(960\) 0 0
\(961\) −1.10947 −0.0357893
\(962\) 17.5762 0.566680
\(963\) 0 0
\(964\) −85.8130 −2.76385
\(965\) 19.5724 0.630057
\(966\) 0 0
\(967\) −20.7000 −0.665668 −0.332834 0.942985i \(-0.608005\pi\)
−0.332834 + 0.942985i \(0.608005\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 45.6708 1.46640
\(971\) −27.5250 −0.883318 −0.441659 0.897183i \(-0.645610\pi\)
−0.441659 + 0.897183i \(0.645610\pi\)
\(972\) 0 0
\(973\) −32.8703 −1.05377
\(974\) 22.4922 0.720697
\(975\) 0 0
\(976\) 228.478 7.31341
\(977\) −36.4948 −1.16757 −0.583786 0.811908i \(-0.698429\pi\)
−0.583786 + 0.811908i \(0.698429\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.58186 0.0824746
\(981\) 0 0
\(982\) 106.176 3.38822
\(983\) −49.1047 −1.56620 −0.783098 0.621899i \(-0.786362\pi\)
−0.783098 + 0.621899i \(0.786362\pi\)
\(984\) 0 0
\(985\) −22.9044 −0.729796
\(986\) 92.2816 2.93885
\(987\) 0 0
\(988\) 65.7758 2.09261
\(989\) 17.0670 0.542699
\(990\) 0 0
\(991\) −5.08586 −0.161558 −0.0807788 0.996732i \(-0.525741\pi\)
−0.0807788 + 0.996732i \(0.525741\pi\)
\(992\) −132.135 −4.19528
\(993\) 0 0
\(994\) −68.9148 −2.18585
\(995\) −17.3671 −0.550573
\(996\) 0 0
\(997\) 18.3350 0.580677 0.290338 0.956924i \(-0.406232\pi\)
0.290338 + 0.956924i \(0.406232\pi\)
\(998\) 6.46681 0.204703
\(999\) 0 0
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 5445.2.a.cb.1.1 8
3.2 odd 2 5445.2.a.cc.1.8 8
11.7 odd 10 495.2.n.g.181.4 16
11.8 odd 10 495.2.n.g.361.4 yes 16
11.10 odd 2 5445.2.a.cd.1.8 8
33.8 even 10 495.2.n.h.361.1 yes 16
33.29 even 10 495.2.n.h.181.1 yes 16
33.32 even 2 5445.2.a.ca.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.n.g.181.4 16 11.7 odd 10
495.2.n.g.361.4 yes 16 11.8 odd 10
495.2.n.h.181.1 yes 16 33.29 even 10
495.2.n.h.361.1 yes 16 33.8 even 10
5445.2.a.ca.1.1 8 33.32 even 2
5445.2.a.cb.1.1 8 1.1 even 1 trivial
5445.2.a.cc.1.8 8 3.2 odd 2
5445.2.a.cd.1.8 8 11.10 odd 2