Properties

Label 5445.2.a.bm.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: 4.4.27648.1
Defining polynomial: \(x^{4} - 6 x^{2} + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.741964\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} +3.30136 q^{7} +2.55940 q^{8} +O(q^{10})\) \(q-0.741964 q^{2} -1.44949 q^{4} -1.00000 q^{5} +3.30136 q^{7} +2.55940 q^{8} +0.741964 q^{10} -1.81743 q^{13} -2.44949 q^{14} +1.00000 q^{16} -3.30136 q^{17} +1.48393 q^{19} +1.44949 q^{20} +4.89898 q^{23} +1.00000 q^{25} +1.34847 q^{26} -4.78529 q^{28} -8.08665 q^{29} -2.00000 q^{31} -5.86076 q^{32} +2.44949 q^{34} -3.30136 q^{35} +2.89898 q^{37} -1.10102 q^{38} -2.55940 q^{40} +5.11879 q^{41} -3.30136 q^{43} -3.63487 q^{46} -9.79796 q^{47} +3.89898 q^{49} -0.741964 q^{50} +2.63435 q^{52} -1.10102 q^{53} +8.44949 q^{56} +6.00000 q^{58} -1.10102 q^{59} -9.57058 q^{61} +1.48393 q^{62} +2.34847 q^{64} +1.81743 q^{65} -0.898979 q^{67} +4.78529 q^{68} +2.44949 q^{70} -10.8990 q^{71} +11.3880 q^{73} -2.15094 q^{74} -2.15094 q^{76} +17.6572 q^{79} -1.00000 q^{80} -3.79796 q^{82} +4.78529 q^{83} +3.30136 q^{85} +2.44949 q^{86} +15.7980 q^{89} -6.00000 q^{91} -7.10102 q^{92} +7.26973 q^{94} -1.48393 q^{95} -12.6969 q^{97} -2.89290 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 4q^{5} + O(q^{10}) \) \( 4q + 4q^{4} - 4q^{5} + 4q^{16} - 4q^{20} + 4q^{25} - 24q^{26} - 8q^{31} - 8q^{37} - 24q^{38} - 4q^{49} - 24q^{53} + 24q^{56} + 24q^{58} - 24q^{59} - 20q^{64} + 16q^{67} - 24q^{71} - 4q^{80} + 24q^{82} + 24q^{89} - 24q^{91} - 48q^{92} + 8q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.741964 −0.524648 −0.262324 0.964980i \(-0.584489\pi\)
−0.262324 + 0.964980i \(0.584489\pi\)
\(3\) 0 0
\(4\) −1.44949 −0.724745
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.30136 1.24780 0.623898 0.781505i \(-0.285548\pi\)
0.623898 + 0.781505i \(0.285548\pi\)
\(8\) 2.55940 0.904883
\(9\) 0 0
\(10\) 0.741964 0.234630
\(11\) 0 0
\(12\) 0 0
\(13\) −1.81743 −0.504065 −0.252033 0.967719i \(-0.581099\pi\)
−0.252033 + 0.967719i \(0.581099\pi\)
\(14\) −2.44949 −0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.30136 −0.800697 −0.400349 0.916363i \(-0.631111\pi\)
−0.400349 + 0.916363i \(0.631111\pi\)
\(18\) 0 0
\(19\) 1.48393 0.340436 0.170218 0.985406i \(-0.445553\pi\)
0.170218 + 0.985406i \(0.445553\pi\)
\(20\) 1.44949 0.324116
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 1.02151 0.510754 0.859727i \(-0.329366\pi\)
0.510754 + 0.859727i \(0.329366\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.34847 0.264457
\(27\) 0 0
\(28\) −4.78529 −0.904334
\(29\) −8.08665 −1.50165 −0.750826 0.660500i \(-0.770345\pi\)
−0.750826 + 0.660500i \(0.770345\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.86076 −1.03605
\(33\) 0 0
\(34\) 2.44949 0.420084
\(35\) −3.30136 −0.558032
\(36\) 0 0
\(37\) 2.89898 0.476589 0.238295 0.971193i \(-0.423412\pi\)
0.238295 + 0.971193i \(0.423412\pi\)
\(38\) −1.10102 −0.178609
\(39\) 0 0
\(40\) −2.55940 −0.404676
\(41\) 5.11879 0.799421 0.399711 0.916641i \(-0.369111\pi\)
0.399711 + 0.916641i \(0.369111\pi\)
\(42\) 0 0
\(43\) −3.30136 −0.503453 −0.251726 0.967798i \(-0.580998\pi\)
−0.251726 + 0.967798i \(0.580998\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.63487 −0.535932
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 3.89898 0.556997
\(50\) −0.741964 −0.104930
\(51\) 0 0
\(52\) 2.63435 0.365319
\(53\) −1.10102 −0.151237 −0.0756184 0.997137i \(-0.524093\pi\)
−0.0756184 + 0.997137i \(0.524093\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.44949 1.12911
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −1.10102 −0.143341 −0.0716703 0.997428i \(-0.522833\pi\)
−0.0716703 + 0.997428i \(0.522833\pi\)
\(60\) 0 0
\(61\) −9.57058 −1.22539 −0.612693 0.790321i \(-0.709914\pi\)
−0.612693 + 0.790321i \(0.709914\pi\)
\(62\) 1.48393 0.188459
\(63\) 0 0
\(64\) 2.34847 0.293559
\(65\) 1.81743 0.225425
\(66\) 0 0
\(67\) −0.898979 −0.109828 −0.0549139 0.998491i \(-0.517488\pi\)
−0.0549139 + 0.998491i \(0.517488\pi\)
\(68\) 4.78529 0.580301
\(69\) 0 0
\(70\) 2.44949 0.292770
\(71\) −10.8990 −1.29347 −0.646735 0.762714i \(-0.723866\pi\)
−0.646735 + 0.762714i \(0.723866\pi\)
\(72\) 0 0
\(73\) 11.3880 1.33287 0.666433 0.745565i \(-0.267820\pi\)
0.666433 + 0.745565i \(0.267820\pi\)
\(74\) −2.15094 −0.250041
\(75\) 0 0
\(76\) −2.15094 −0.246729
\(77\) 0 0
\(78\) 0 0
\(79\) 17.6572 1.98659 0.993296 0.115595i \(-0.0368774\pi\)
0.993296 + 0.115595i \(0.0368774\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −3.79796 −0.419414
\(83\) 4.78529 0.525254 0.262627 0.964897i \(-0.415411\pi\)
0.262627 + 0.964897i \(0.415411\pi\)
\(84\) 0 0
\(85\) 3.30136 0.358083
\(86\) 2.44949 0.264135
\(87\) 0 0
\(88\) 0 0
\(89\) 15.7980 1.67458 0.837290 0.546759i \(-0.184139\pi\)
0.837290 + 0.546759i \(0.184139\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −7.10102 −0.740333
\(93\) 0 0
\(94\) 7.26973 0.749815
\(95\) −1.48393 −0.152248
\(96\) 0 0
\(97\) −12.6969 −1.28918 −0.644589 0.764529i \(-0.722972\pi\)
−0.644589 + 0.764529i \(0.722972\pi\)
\(98\) −2.89290 −0.292227
\(99\) 0 0
\(100\) −1.44949 −0.144949
\(101\) −8.75366 −0.871022 −0.435511 0.900184i \(-0.643432\pi\)
−0.435511 + 0.900184i \(0.643432\pi\)
\(102\) 0 0
\(103\) 3.10102 0.305553 0.152776 0.988261i \(-0.451179\pi\)
0.152776 + 0.988261i \(0.451179\pi\)
\(104\) −4.65153 −0.456120
\(105\) 0 0
\(106\) 0.816917 0.0793460
\(107\) −8.42015 −0.814007 −0.407003 0.913427i \(-0.633426\pi\)
−0.407003 + 0.913427i \(0.633426\pi\)
\(108\) 0 0
\(109\) 10.2376 0.980583 0.490291 0.871559i \(-0.336890\pi\)
0.490291 + 0.871559i \(0.336890\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.30136 0.311949
\(113\) −10.8990 −1.02529 −0.512645 0.858601i \(-0.671334\pi\)
−0.512645 + 0.858601i \(0.671334\pi\)
\(114\) 0 0
\(115\) −4.89898 −0.456832
\(116\) 11.7215 1.08832
\(117\) 0 0
\(118\) 0.816917 0.0752033
\(119\) −10.8990 −0.999108
\(120\) 0 0
\(121\) 0 0
\(122\) 7.10102 0.642896
\(123\) 0 0
\(124\) 2.89898 0.260336
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.63435 −0.233761 −0.116880 0.993146i \(-0.537289\pi\)
−0.116880 + 0.993146i \(0.537289\pi\)
\(128\) 9.97903 0.882030
\(129\) 0 0
\(130\) −1.34847 −0.118269
\(131\) 16.8403 1.47134 0.735672 0.677338i \(-0.236866\pi\)
0.735672 + 0.677338i \(0.236866\pi\)
\(132\) 0 0
\(133\) 4.89898 0.424795
\(134\) 0.667010 0.0576209
\(135\) 0 0
\(136\) −8.44949 −0.724538
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −18.3242 −1.55424 −0.777121 0.629352i \(-0.783321\pi\)
−0.777121 + 0.629352i \(0.783321\pi\)
\(140\) 4.78529 0.404431
\(141\) 0 0
\(142\) 8.08665 0.678616
\(143\) 0 0
\(144\) 0 0
\(145\) 8.08665 0.671560
\(146\) −8.44949 −0.699285
\(147\) 0 0
\(148\) −4.20204 −0.345406
\(149\) −14.6894 −1.20340 −0.601700 0.798722i \(-0.705510\pi\)
−0.601700 + 0.798722i \(0.705510\pi\)
\(150\) 0 0
\(151\) 14.6894 1.19540 0.597702 0.801718i \(-0.296081\pi\)
0.597702 + 0.801718i \(0.296081\pi\)
\(152\) 3.79796 0.308055
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −11.7980 −0.941580 −0.470790 0.882245i \(-0.656031\pi\)
−0.470790 + 0.882245i \(0.656031\pi\)
\(158\) −13.1010 −1.04226
\(159\) 0 0
\(160\) 5.86076 0.463334
\(161\) 16.1733 1.27463
\(162\) 0 0
\(163\) 22.6969 1.77776 0.888881 0.458139i \(-0.151484\pi\)
0.888881 + 0.458139i \(0.151484\pi\)
\(164\) −7.41964 −0.579376
\(165\) 0 0
\(166\) −3.55051 −0.275573
\(167\) 21.6256 1.67344 0.836719 0.547632i \(-0.184471\pi\)
0.836719 + 0.547632i \(0.184471\pi\)
\(168\) 0 0
\(169\) −9.69694 −0.745918
\(170\) −2.44949 −0.187867
\(171\) 0 0
\(172\) 4.78529 0.364875
\(173\) −22.4425 −1.70627 −0.853136 0.521688i \(-0.825302\pi\)
−0.853136 + 0.521688i \(0.825302\pi\)
\(174\) 0 0
\(175\) 3.30136 0.249559
\(176\) 0 0
\(177\) 0 0
\(178\) −11.7215 −0.878565
\(179\) −2.20204 −0.164588 −0.0822941 0.996608i \(-0.526225\pi\)
−0.0822941 + 0.996608i \(0.526225\pi\)
\(180\) 0 0
\(181\) 12.8990 0.958774 0.479387 0.877604i \(-0.340859\pi\)
0.479387 + 0.877604i \(0.340859\pi\)
\(182\) 4.45178 0.329988
\(183\) 0 0
\(184\) 12.5384 0.924345
\(185\) −2.89898 −0.213137
\(186\) 0 0
\(187\) 0 0
\(188\) 14.2020 1.03579
\(189\) 0 0
\(190\) 1.10102 0.0798764
\(191\) −19.5959 −1.41791 −0.708955 0.705253i \(-0.750833\pi\)
−0.708955 + 0.705253i \(0.750833\pi\)
\(192\) 0 0
\(193\) −17.3237 −1.24699 −0.623494 0.781828i \(-0.714288\pi\)
−0.623494 + 0.781828i \(0.714288\pi\)
\(194\) 9.42067 0.676365
\(195\) 0 0
\(196\) −5.65153 −0.403681
\(197\) 2.63435 0.187690 0.0938448 0.995587i \(-0.470084\pi\)
0.0938448 + 0.995587i \(0.470084\pi\)
\(198\) 0 0
\(199\) 1.79796 0.127454 0.0637270 0.997967i \(-0.479701\pi\)
0.0637270 + 0.997967i \(0.479701\pi\)
\(200\) 2.55940 0.180977
\(201\) 0 0
\(202\) 6.49490 0.456979
\(203\) −26.6969 −1.87376
\(204\) 0 0
\(205\) −5.11879 −0.357512
\(206\) −2.30084 −0.160307
\(207\) 0 0
\(208\) −1.81743 −0.126016
\(209\) 0 0
\(210\) 0 0
\(211\) 4.45178 0.306473 0.153237 0.988190i \(-0.451030\pi\)
0.153237 + 0.988190i \(0.451030\pi\)
\(212\) 1.59592 0.109608
\(213\) 0 0
\(214\) 6.24745 0.427067
\(215\) 3.30136 0.225151
\(216\) 0 0
\(217\) −6.60272 −0.448222
\(218\) −7.59592 −0.514460
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 17.7980 1.19184 0.595920 0.803044i \(-0.296788\pi\)
0.595920 + 0.803044i \(0.296788\pi\)
\(224\) −19.3485 −1.29277
\(225\) 0 0
\(226\) 8.08665 0.537916
\(227\) −11.3880 −0.755849 −0.377924 0.925836i \(-0.623362\pi\)
−0.377924 + 0.925836i \(0.623362\pi\)
\(228\) 0 0
\(229\) 17.5959 1.16277 0.581385 0.813628i \(-0.302511\pi\)
0.581385 + 0.813628i \(0.302511\pi\)
\(230\) 3.63487 0.239676
\(231\) 0 0
\(232\) −20.6969 −1.35882
\(233\) −0.333505 −0.0218486 −0.0109243 0.999940i \(-0.503477\pi\)
−0.0109243 + 0.999940i \(0.503477\pi\)
\(234\) 0 0
\(235\) 9.79796 0.639148
\(236\) 1.59592 0.103885
\(237\) 0 0
\(238\) 8.08665 0.524180
\(239\) 15.5063 1.00302 0.501509 0.865152i \(-0.332778\pi\)
0.501509 + 0.865152i \(0.332778\pi\)
\(240\) 0 0
\(241\) −12.5384 −0.807671 −0.403836 0.914832i \(-0.632323\pi\)
−0.403836 + 0.914832i \(0.632323\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 13.8725 0.888093
\(245\) −3.89898 −0.249097
\(246\) 0 0
\(247\) −2.69694 −0.171602
\(248\) −5.11879 −0.325044
\(249\) 0 0
\(250\) 0.741964 0.0469259
\(251\) −22.8990 −1.44537 −0.722685 0.691177i \(-0.757092\pi\)
−0.722685 + 0.691177i \(0.757092\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.95459 0.122642
\(255\) 0 0
\(256\) −12.1010 −0.756314
\(257\) −13.5959 −0.848090 −0.424045 0.905641i \(-0.639390\pi\)
−0.424045 + 0.905641i \(0.639390\pi\)
\(258\) 0 0
\(259\) 9.57058 0.594687
\(260\) −2.63435 −0.163375
\(261\) 0 0
\(262\) −12.4949 −0.771937
\(263\) 17.3237 1.06823 0.534113 0.845413i \(-0.320646\pi\)
0.534113 + 0.845413i \(0.320646\pi\)
\(264\) 0 0
\(265\) 1.10102 0.0676352
\(266\) −3.63487 −0.222868
\(267\) 0 0
\(268\) 1.30306 0.0795972
\(269\) −31.5959 −1.92644 −0.963219 0.268719i \(-0.913400\pi\)
−0.963219 + 0.268719i \(0.913400\pi\)
\(270\) 0 0
\(271\) 21.9591 1.33392 0.666960 0.745093i \(-0.267595\pi\)
0.666960 + 0.745093i \(0.267595\pi\)
\(272\) −3.30136 −0.200174
\(273\) 0 0
\(274\) 13.3553 0.806826
\(275\) 0 0
\(276\) 0 0
\(277\) −2.48444 −0.149276 −0.0746379 0.997211i \(-0.523780\pi\)
−0.0746379 + 0.997211i \(0.523780\pi\)
\(278\) 13.5959 0.815429
\(279\) 0 0
\(280\) −8.44949 −0.504954
\(281\) −5.11879 −0.305362 −0.152681 0.988276i \(-0.548791\pi\)
−0.152681 + 0.988276i \(0.548791\pi\)
\(282\) 0 0
\(283\) −5.60221 −0.333017 −0.166508 0.986040i \(-0.553249\pi\)
−0.166508 + 0.986040i \(0.553249\pi\)
\(284\) 15.7980 0.937436
\(285\) 0 0
\(286\) 0 0
\(287\) 16.8990 0.997515
\(288\) 0 0
\(289\) −6.10102 −0.358884
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) −16.5068 −0.965987
\(293\) 5.60221 0.327284 0.163642 0.986520i \(-0.447676\pi\)
0.163642 + 0.986520i \(0.447676\pi\)
\(294\) 0 0
\(295\) 1.10102 0.0641039
\(296\) 7.41964 0.431258
\(297\) 0 0
\(298\) 10.8990 0.631361
\(299\) −8.90357 −0.514906
\(300\) 0 0
\(301\) −10.8990 −0.628207
\(302\) −10.8990 −0.627166
\(303\) 0 0
\(304\) 1.48393 0.0851091
\(305\) 9.57058 0.548010
\(306\) 0 0
\(307\) −15.8398 −0.904025 −0.452012 0.892012i \(-0.649294\pi\)
−0.452012 + 0.892012i \(0.649294\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.48393 −0.0842814
\(311\) −13.1010 −0.742891 −0.371445 0.928455i \(-0.621138\pi\)
−0.371445 + 0.928455i \(0.621138\pi\)
\(312\) 0 0
\(313\) −30.8990 −1.74651 −0.873257 0.487260i \(-0.837996\pi\)
−0.873257 + 0.487260i \(0.837996\pi\)
\(314\) 8.75366 0.493998
\(315\) 0 0
\(316\) −25.5940 −1.43977
\(317\) −20.6969 −1.16246 −0.581228 0.813741i \(-0.697428\pi\)
−0.581228 + 0.813741i \(0.697428\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.34847 −0.131283
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −4.89898 −0.272587
\(324\) 0 0
\(325\) −1.81743 −0.100813
\(326\) −16.8403 −0.932698
\(327\) 0 0
\(328\) 13.1010 0.723383
\(329\) −32.3466 −1.78333
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) −6.93623 −0.380675
\(333\) 0 0
\(334\) −16.0454 −0.877966
\(335\) 0.898979 0.0491165
\(336\) 0 0
\(337\) −14.3559 −0.782014 −0.391007 0.920388i \(-0.627873\pi\)
−0.391007 + 0.920388i \(0.627873\pi\)
\(338\) 7.19478 0.391344
\(339\) 0 0
\(340\) −4.78529 −0.259519
\(341\) 0 0
\(342\) 0 0
\(343\) −10.2376 −0.552778
\(344\) −8.44949 −0.455566
\(345\) 0 0
\(346\) 16.6515 0.895192
\(347\) −7.75314 −0.416211 −0.208105 0.978106i \(-0.566730\pi\)
−0.208105 + 0.978106i \(0.566730\pi\)
\(348\) 0 0
\(349\) −9.57058 −0.512301 −0.256151 0.966637i \(-0.582454\pi\)
−0.256151 + 0.966637i \(0.582454\pi\)
\(350\) −2.44949 −0.130931
\(351\) 0 0
\(352\) 0 0
\(353\) 1.10102 0.0586014 0.0293007 0.999571i \(-0.490672\pi\)
0.0293007 + 0.999571i \(0.490672\pi\)
\(354\) 0 0
\(355\) 10.8990 0.578458
\(356\) −22.8990 −1.21364
\(357\) 0 0
\(358\) 1.63383 0.0863508
\(359\) −31.6796 −1.67198 −0.835992 0.548741i \(-0.815107\pi\)
−0.835992 + 0.548741i \(0.815107\pi\)
\(360\) 0 0
\(361\) −16.7980 −0.884103
\(362\) −9.57058 −0.503018
\(363\) 0 0
\(364\) 8.69694 0.455843
\(365\) −11.3880 −0.596076
\(366\) 0 0
\(367\) 15.5959 0.814100 0.407050 0.913406i \(-0.366557\pi\)
0.407050 + 0.913406i \(0.366557\pi\)
\(368\) 4.89898 0.255377
\(369\) 0 0
\(370\) 2.15094 0.111822
\(371\) −3.63487 −0.188713
\(372\) 0 0
\(373\) −15.0229 −0.777855 −0.388927 0.921268i \(-0.627154\pi\)
−0.388927 + 0.921268i \(0.627154\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −25.0769 −1.29324
\(377\) 14.6969 0.756931
\(378\) 0 0
\(379\) 15.5959 0.801108 0.400554 0.916273i \(-0.368818\pi\)
0.400554 + 0.916273i \(0.368818\pi\)
\(380\) 2.15094 0.110341
\(381\) 0 0
\(382\) 14.5395 0.743904
\(383\) −7.10102 −0.362845 −0.181423 0.983405i \(-0.558070\pi\)
−0.181423 + 0.983405i \(0.558070\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.8536 0.654230
\(387\) 0 0
\(388\) 18.4041 0.934326
\(389\) −7.59592 −0.385128 −0.192564 0.981284i \(-0.561680\pi\)
−0.192564 + 0.981284i \(0.561680\pi\)
\(390\) 0 0
\(391\) −16.1733 −0.817919
\(392\) 9.97903 0.504017
\(393\) 0 0
\(394\) −1.95459 −0.0984709
\(395\) −17.6572 −0.888431
\(396\) 0 0
\(397\) −0.696938 −0.0349783 −0.0174892 0.999847i \(-0.505567\pi\)
−0.0174892 + 0.999847i \(0.505567\pi\)
\(398\) −1.33402 −0.0668684
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −27.7980 −1.38816 −0.694082 0.719896i \(-0.744189\pi\)
−0.694082 + 0.719896i \(0.744189\pi\)
\(402\) 0 0
\(403\) 3.63487 0.181066
\(404\) 12.6883 0.631268
\(405\) 0 0
\(406\) 19.8082 0.983063
\(407\) 0 0
\(408\) 0 0
\(409\) −23.4430 −1.15918 −0.579592 0.814907i \(-0.696788\pi\)
−0.579592 + 0.814907i \(0.696788\pi\)
\(410\) 3.79796 0.187568
\(411\) 0 0
\(412\) −4.49490 −0.221448
\(413\) −3.63487 −0.178860
\(414\) 0 0
\(415\) −4.78529 −0.234901
\(416\) 10.6515 0.522234
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −34.6969 −1.69103 −0.845513 0.533955i \(-0.820705\pi\)
−0.845513 + 0.533955i \(0.820705\pi\)
\(422\) −3.30306 −0.160791
\(423\) 0 0
\(424\) −2.81795 −0.136852
\(425\) −3.30136 −0.160139
\(426\) 0 0
\(427\) −31.5959 −1.52903
\(428\) 12.2049 0.589947
\(429\) 0 0
\(430\) −2.44949 −0.118125
\(431\) 13.2054 0.636084 0.318042 0.948077i \(-0.396975\pi\)
0.318042 + 0.948077i \(0.396975\pi\)
\(432\) 0 0
\(433\) 16.6969 0.802404 0.401202 0.915990i \(-0.368593\pi\)
0.401202 + 0.915990i \(0.368593\pi\)
\(434\) 4.89898 0.235159
\(435\) 0 0
\(436\) −14.8393 −0.710672
\(437\) 7.26973 0.347758
\(438\) 0 0
\(439\) −19.9581 −0.952547 −0.476273 0.879297i \(-0.658013\pi\)
−0.476273 + 0.879297i \(0.658013\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.45178 −0.211750
\(443\) −26.6969 −1.26841 −0.634205 0.773165i \(-0.718672\pi\)
−0.634205 + 0.773165i \(0.718672\pi\)
\(444\) 0 0
\(445\) −15.7980 −0.748895
\(446\) −13.2054 −0.625296
\(447\) 0 0
\(448\) 7.75314 0.366302
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 15.7980 0.743073
\(453\) 0 0
\(454\) 8.44949 0.396554
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 26.8943 1.25806 0.629031 0.777380i \(-0.283452\pi\)
0.629031 + 0.777380i \(0.283452\pi\)
\(458\) −13.0555 −0.610045
\(459\) 0 0
\(460\) 7.10102 0.331087
\(461\) −24.9270 −1.16096 −0.580482 0.814273i \(-0.697136\pi\)
−0.580482 + 0.814273i \(0.697136\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −8.08665 −0.375413
\(465\) 0 0
\(466\) 0.247449 0.0114628
\(467\) −4.89898 −0.226698 −0.113349 0.993555i \(-0.536158\pi\)
−0.113349 + 0.993555i \(0.536158\pi\)
\(468\) 0 0
\(469\) −2.96786 −0.137043
\(470\) −7.26973 −0.335328
\(471\) 0 0
\(472\) −2.81795 −0.129707
\(473\) 0 0
\(474\) 0 0
\(475\) 1.48393 0.0680873
\(476\) 15.7980 0.724098
\(477\) 0 0
\(478\) −11.5051 −0.526231
\(479\) 27.0779 1.23722 0.618610 0.785698i \(-0.287696\pi\)
0.618610 + 0.785698i \(0.287696\pi\)
\(480\) 0 0
\(481\) −5.26870 −0.240232
\(482\) 9.30306 0.423743
\(483\) 0 0
\(484\) 0 0
\(485\) 12.6969 0.576538
\(486\) 0 0
\(487\) −0.898979 −0.0407366 −0.0203683 0.999793i \(-0.506484\pi\)
−0.0203683 + 0.999793i \(0.506484\pi\)
\(488\) −24.4949 −1.10883
\(489\) 0 0
\(490\) 2.89290 0.130688
\(491\) 11.8714 0.535750 0.267875 0.963454i \(-0.413679\pi\)
0.267875 + 0.963454i \(0.413679\pi\)
\(492\) 0 0
\(493\) 26.6969 1.20237
\(494\) 2.00103 0.0900306
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −35.9815 −1.61399
\(498\) 0 0
\(499\) −5.79796 −0.259552 −0.129776 0.991543i \(-0.541426\pi\)
−0.129776 + 0.991543i \(0.541426\pi\)
\(500\) 1.44949 0.0648232
\(501\) 0 0
\(502\) 16.9902 0.758310
\(503\) −4.11828 −0.183625 −0.0918125 0.995776i \(-0.529266\pi\)
−0.0918125 + 0.995776i \(0.529266\pi\)
\(504\) 0 0
\(505\) 8.75366 0.389533
\(506\) 0 0
\(507\) 0 0
\(508\) 3.81846 0.169417
\(509\) −9.79796 −0.434287 −0.217143 0.976140i \(-0.569674\pi\)
−0.217143 + 0.976140i \(0.569674\pi\)
\(510\) 0 0
\(511\) 37.5959 1.66315
\(512\) −10.9795 −0.485232
\(513\) 0 0
\(514\) 10.0877 0.444948
\(515\) −3.10102 −0.136647
\(516\) 0 0
\(517\) 0 0
\(518\) −7.10102 −0.312001
\(519\) 0 0
\(520\) 4.65153 0.203983
\(521\) 35.3939 1.55063 0.775317 0.631572i \(-0.217590\pi\)
0.775317 + 0.631572i \(0.217590\pi\)
\(522\) 0 0
\(523\) 36.3150 1.58794 0.793971 0.607955i \(-0.208010\pi\)
0.793971 + 0.607955i \(0.208010\pi\)
\(524\) −24.4099 −1.06635
\(525\) 0 0
\(526\) −12.8536 −0.560442
\(527\) 6.60272 0.287619
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −0.816917 −0.0354846
\(531\) 0 0
\(532\) −7.10102 −0.307868
\(533\) −9.30306 −0.402960
\(534\) 0 0
\(535\) 8.42015 0.364035
\(536\) −2.30084 −0.0993814
\(537\) 0 0
\(538\) 23.4430 1.01070
\(539\) 0 0
\(540\) 0 0
\(541\) 33.6806 1.44804 0.724021 0.689778i \(-0.242292\pi\)
0.724021 + 0.689778i \(0.242292\pi\)
\(542\) −16.2929 −0.699838
\(543\) 0 0
\(544\) 19.3485 0.829559
\(545\) −10.2376 −0.438530
\(546\) 0 0
\(547\) −28.3782 −1.21337 −0.606683 0.794944i \(-0.707500\pi\)
−0.606683 + 0.794944i \(0.707500\pi\)
\(548\) 26.0908 1.11454
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 58.2929 2.47886
\(554\) 1.84337 0.0783171
\(555\) 0 0
\(556\) 26.5608 1.12643
\(557\) 16.5068 0.699416 0.349708 0.936859i \(-0.386281\pi\)
0.349708 + 0.936859i \(0.386281\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) −3.30136 −0.139508
\(561\) 0 0
\(562\) 3.79796 0.160207
\(563\) 37.7989 1.59303 0.796517 0.604617i \(-0.206674\pi\)
0.796517 + 0.604617i \(0.206674\pi\)
\(564\) 0 0
\(565\) 10.8990 0.458524
\(566\) 4.15663 0.174716
\(567\) 0 0
\(568\) −27.8948 −1.17044
\(569\) 30.8627 1.29383 0.646915 0.762562i \(-0.276059\pi\)
0.646915 + 0.762562i \(0.276059\pi\)
\(570\) 0 0
\(571\) 18.3242 0.766845 0.383423 0.923573i \(-0.374745\pi\)
0.383423 + 0.923573i \(0.374745\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12.5384 −0.523344
\(575\) 4.89898 0.204302
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 4.52674 0.188287
\(579\) 0 0
\(580\) −11.7215 −0.486709
\(581\) 15.7980 0.655410
\(582\) 0 0
\(583\) 0 0
\(584\) 29.1464 1.20609
\(585\) 0 0
\(586\) −4.15663 −0.171709
\(587\) 14.6969 0.606608 0.303304 0.952894i \(-0.401910\pi\)
0.303304 + 0.952894i \(0.401910\pi\)
\(588\) 0 0
\(589\) −2.96786 −0.122288
\(590\) −0.816917 −0.0336320
\(591\) 0 0
\(592\) 2.89898 0.119147
\(593\) 9.23707 0.379321 0.189661 0.981850i \(-0.439261\pi\)
0.189661 + 0.981850i \(0.439261\pi\)
\(594\) 0 0
\(595\) 10.8990 0.446815
\(596\) 21.2921 0.872158
\(597\) 0 0
\(598\) 6.60612 0.270144
\(599\) 7.59592 0.310361 0.155180 0.987886i \(-0.450404\pi\)
0.155180 + 0.987886i \(0.450404\pi\)
\(600\) 0 0
\(601\) −22.7760 −0.929053 −0.464527 0.885559i \(-0.653775\pi\)
−0.464527 + 0.885559i \(0.653775\pi\)
\(602\) 8.08665 0.329587
\(603\) 0 0
\(604\) −21.2921 −0.866363
\(605\) 0 0
\(606\) 0 0
\(607\) −15.8398 −0.642917 −0.321459 0.946924i \(-0.604173\pi\)
−0.321459 + 0.946924i \(0.604173\pi\)
\(608\) −8.69694 −0.352707
\(609\) 0 0
\(610\) −7.10102 −0.287512
\(611\) 17.8071 0.720399
\(612\) 0 0
\(613\) −20.2916 −0.819569 −0.409784 0.912182i \(-0.634396\pi\)
−0.409784 + 0.912182i \(0.634396\pi\)
\(614\) 11.7526 0.474294
\(615\) 0 0
\(616\) 0 0
\(617\) −22.8990 −0.921878 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) −2.89898 −0.116426
\(621\) 0 0
\(622\) 9.72048 0.389756
\(623\) 52.1548 2.08954
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.9259 0.916304
\(627\) 0 0
\(628\) 17.1010 0.682405
\(629\) −9.57058 −0.381604
\(630\) 0 0
\(631\) −37.3939 −1.48863 −0.744313 0.667831i \(-0.767223\pi\)
−0.744313 + 0.667831i \(0.767223\pi\)
\(632\) 45.1918 1.79763
\(633\) 0 0
\(634\) 15.3564 0.609880
\(635\) 2.63435 0.104541
\(636\) 0 0
\(637\) −7.08613 −0.280763
\(638\) 0 0
\(639\) 0 0
\(640\) −9.97903 −0.394456
\(641\) −1.59592 −0.0630350 −0.0315175 0.999503i \(-0.510034\pi\)
−0.0315175 + 0.999503i \(0.510034\pi\)
\(642\) 0 0
\(643\) −40.4949 −1.59696 −0.798481 0.602019i \(-0.794363\pi\)
−0.798481 + 0.602019i \(0.794363\pi\)
\(644\) −23.4430 −0.923785
\(645\) 0 0
\(646\) 3.63487 0.143012
\(647\) −31.1010 −1.22271 −0.611354 0.791358i \(-0.709375\pi\)
−0.611354 + 0.791358i \(0.709375\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.34847 0.0528913
\(651\) 0 0
\(652\) −32.8990 −1.28842
\(653\) 37.5959 1.47124 0.735621 0.677393i \(-0.236890\pi\)
0.735621 + 0.677393i \(0.236890\pi\)
\(654\) 0 0
\(655\) −16.8403 −0.658005
\(656\) 5.11879 0.199855
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) 17.5073 0.681988 0.340994 0.940065i \(-0.389236\pi\)
0.340994 + 0.940065i \(0.389236\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 7.41964 0.288372
\(663\) 0 0
\(664\) 12.2474 0.475293
\(665\) −4.89898 −0.189974
\(666\) 0 0
\(667\) −39.6163 −1.53395
\(668\) −31.3461 −1.21282
\(669\) 0 0
\(670\) −0.667010 −0.0257689
\(671\) 0 0
\(672\) 0 0
\(673\) 40.7667 1.57144 0.785721 0.618581i \(-0.212292\pi\)
0.785721 + 0.618581i \(0.212292\pi\)
\(674\) 10.6515 0.410282
\(675\) 0 0
\(676\) 14.0556 0.540600
\(677\) 30.3793 1.16757 0.583785 0.811908i \(-0.301571\pi\)
0.583785 + 0.811908i \(0.301571\pi\)
\(678\) 0 0
\(679\) −41.9172 −1.60863
\(680\) 8.44949 0.324023
\(681\) 0 0
\(682\) 0 0
\(683\) 7.59592 0.290650 0.145325 0.989384i \(-0.453577\pi\)
0.145325 + 0.989384i \(0.453577\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 7.59592 0.290013
\(687\) 0 0
\(688\) −3.30136 −0.125863
\(689\) 2.00103 0.0762332
\(690\) 0 0
\(691\) −37.7980 −1.43790 −0.718951 0.695061i \(-0.755377\pi\)
−0.718951 + 0.695061i \(0.755377\pi\)
\(692\) 32.5302 1.23661
\(693\) 0 0
\(694\) 5.75255 0.218364
\(695\) 18.3242 0.695078
\(696\) 0 0
\(697\) −16.8990 −0.640094
\(698\) 7.10102 0.268778
\(699\) 0 0
\(700\) −4.78529 −0.180867
\(701\) −47.7030 −1.80172 −0.900858 0.434114i \(-0.857062\pi\)
−0.900858 + 0.434114i \(0.857062\pi\)
\(702\) 0 0
\(703\) 4.30188 0.162248
\(704\) 0 0
\(705\) 0 0
\(706\) −0.816917 −0.0307451
\(707\) −28.8990 −1.08686
\(708\) 0 0
\(709\) −22.6969 −0.852401 −0.426201 0.904629i \(-0.640148\pi\)
−0.426201 + 0.904629i \(0.640148\pi\)
\(710\) −8.08665 −0.303486
\(711\) 0 0
\(712\) 40.4332 1.51530
\(713\) −9.79796 −0.366936
\(714\) 0 0
\(715\) 0 0
\(716\) 3.19184 0.119285
\(717\) 0 0
\(718\) 23.5051 0.877203
\(719\) 33.7980 1.26045 0.630226 0.776412i \(-0.282962\pi\)
0.630226 + 0.776412i \(0.282962\pi\)
\(720\) 0 0
\(721\) 10.2376 0.381268
\(722\) 12.4635 0.463843
\(723\) 0 0
\(724\) −18.6969 −0.694866
\(725\) −8.08665 −0.300331
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −15.3564 −0.569145
\(729\) 0 0
\(730\) 8.44949 0.312730
\(731\) 10.8990 0.403113
\(732\) 0 0
\(733\) −43.0676 −1.59074 −0.795369 0.606126i \(-0.792723\pi\)
−0.795369 + 0.606126i \(0.792723\pi\)
\(734\) −11.5716 −0.427116
\(735\) 0 0
\(736\) −28.7117 −1.05833
\(737\) 0 0
\(738\) 0 0
\(739\) −24.2599 −0.892416 −0.446208 0.894929i \(-0.647226\pi\)
−0.446208 + 0.894929i \(0.647226\pi\)
\(740\) 4.20204 0.154470
\(741\) 0 0
\(742\) 2.69694 0.0990077
\(743\) 28.8953 1.06007 0.530033 0.847977i \(-0.322179\pi\)
0.530033 + 0.847977i \(0.322179\pi\)
\(744\) 0 0
\(745\) 14.6894 0.538177
\(746\) 11.1464 0.408100
\(747\) 0 0
\(748\) 0 0
\(749\) −27.7980 −1.01572
\(750\) 0 0
\(751\) −47.1918 −1.72205 −0.861027 0.508559i \(-0.830178\pi\)
−0.861027 + 0.508559i \(0.830178\pi\)
\(752\) −9.79796 −0.357295
\(753\) 0 0
\(754\) −10.9046 −0.397122
\(755\) −14.6894 −0.534601
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −11.5716 −0.420300
\(759\) 0 0
\(760\) −3.79796 −0.137766
\(761\) 21.9591 0.796017 0.398008 0.917382i \(-0.369701\pi\)
0.398008 + 0.917382i \(0.369701\pi\)
\(762\) 0 0
\(763\) 33.7980 1.22357
\(764\) 28.4041 1.02762
\(765\) 0 0
\(766\) 5.26870 0.190366
\(767\) 2.00103 0.0722530
\(768\) 0 0
\(769\) −3.63487 −0.131077 −0.0655383 0.997850i \(-0.520876\pi\)
−0.0655383 + 0.997850i \(0.520876\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 25.1106 0.903749
\(773\) 39.7980 1.43143 0.715717 0.698391i \(-0.246100\pi\)
0.715717 + 0.698391i \(0.246100\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −32.4965 −1.16656
\(777\) 0 0
\(778\) 5.63590 0.202057
\(779\) 7.59592 0.272152
\(780\) 0 0
\(781\) 0 0
\(782\) 12.0000 0.429119
\(783\) 0 0
\(784\) 3.89898 0.139249
\(785\) 11.7980 0.421087
\(786\) 0 0
\(787\) 11.2381 0.400595 0.200298 0.979735i \(-0.435809\pi\)
0.200298 + 0.979735i \(0.435809\pi\)
\(788\) −3.81846 −0.136027
\(789\) 0 0
\(790\) 13.1010 0.466113
\(791\) −35.9815 −1.27935
\(792\) 0 0
\(793\) 17.3939 0.617675
\(794\) 0.517103 0.0183513
\(795\) 0 0
\(796\) −2.60612 −0.0923716
\(797\) −10.8990 −0.386062 −0.193031 0.981193i \(-0.561832\pi\)
−0.193031 + 0.981193i \(0.561832\pi\)
\(798\) 0 0
\(799\) 32.3466 1.14434
\(800\) −5.86076 −0.207209
\(801\) 0 0
\(802\) 20.6251 0.728297
\(803\) 0 0
\(804\) 0 0
\(805\) −16.1733 −0.570034
\(806\) −2.69694 −0.0949956
\(807\) 0 0
\(808\) −22.4041 −0.788173
\(809\) −46.3690 −1.63025 −0.815123 0.579288i \(-0.803331\pi\)
−0.815123 + 0.579288i \(0.803331\pi\)
\(810\) 0 0
\(811\) 2.15094 0.0755296 0.0377648 0.999287i \(-0.487976\pi\)
0.0377648 + 0.999287i \(0.487976\pi\)
\(812\) 38.6969 1.35800
\(813\) 0 0
\(814\) 0 0
\(815\) −22.6969 −0.795039
\(816\) 0 0
\(817\) −4.89898 −0.171394
\(818\) 17.3939 0.608163
\(819\) 0 0
\(820\) 7.41964 0.259105
\(821\) 19.9581 0.696541 0.348271 0.937394i \(-0.386769\pi\)
0.348271 + 0.937394i \(0.386769\pi\)
\(822\) 0 0
\(823\) 48.8990 1.70451 0.852256 0.523126i \(-0.175234\pi\)
0.852256 + 0.523126i \(0.175234\pi\)
\(824\) 7.93674 0.276489
\(825\) 0 0
\(826\) 2.69694 0.0938385
\(827\) 10.0540 0.349611 0.174806 0.984603i \(-0.444070\pi\)
0.174806 + 0.984603i \(0.444070\pi\)
\(828\) 0 0
\(829\) −13.3031 −0.462034 −0.231017 0.972950i \(-0.574205\pi\)
−0.231017 + 0.972950i \(0.574205\pi\)
\(830\) 3.55051 0.123240
\(831\) 0 0
\(832\) −4.26818 −0.147973
\(833\) −12.8719 −0.445986
\(834\) 0 0
\(835\) −21.6256 −0.748385
\(836\) 0 0
\(837\) 0 0
\(838\) 8.90357 0.307569
\(839\) −18.4949 −0.638515 −0.319257 0.947668i \(-0.603433\pi\)
−0.319257 + 0.947668i \(0.603433\pi\)
\(840\) 0 0
\(841\) 36.3939 1.25496
\(842\) 25.7439 0.887192
\(843\) 0 0
\(844\) −6.45281 −0.222115
\(845\) 9.69694 0.333585
\(846\) 0 0
\(847\) 0 0
\(848\) −1.10102 −0.0378092
\(849\) 0 0
\(850\) 2.44949 0.0840168
\(851\) 14.2020 0.486840
\(852\) 0 0
\(853\) 30.5292 1.04530 0.522649 0.852548i \(-0.324944\pi\)
0.522649 + 0.852548i \(0.324944\pi\)
\(854\) 23.4430 0.802204
\(855\) 0 0
\(856\) −21.5505 −0.736581
\(857\) 38.6158 1.31909 0.659545 0.751665i \(-0.270749\pi\)
0.659545 + 0.751665i \(0.270749\pi\)
\(858\) 0 0
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) −4.78529 −0.163177
\(861\) 0 0
\(862\) −9.79796 −0.333720
\(863\) 50.6969 1.72574 0.862872 0.505423i \(-0.168663\pi\)
0.862872 + 0.505423i \(0.168663\pi\)
\(864\) 0 0
\(865\) 22.4425 0.763068
\(866\) −12.3885 −0.420979
\(867\) 0 0
\(868\) 9.57058 0.324847
\(869\) 0 0
\(870\) 0 0
\(871\) 1.63383 0.0553604
\(872\) 26.2020 0.887313
\(873\) 0 0
\(874\) −5.39388 −0.182451
\(875\) −3.30136 −0.111606
\(876\) 0 0
\(877\) −34.8310 −1.17616 −0.588080 0.808803i \(-0.700116\pi\)
−0.588080 + 0.808803i \(0.700116\pi\)
\(878\) 14.8082 0.499751
\(879\) 0 0
\(880\) 0 0
\(881\) −2.20204 −0.0741886 −0.0370943 0.999312i \(-0.511810\pi\)
−0.0370943 + 0.999312i \(0.511810\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −8.69694 −0.292510
\(885\) 0 0
\(886\) 19.8082 0.665468
\(887\) −20.2916 −0.681324 −0.340662 0.940186i \(-0.610651\pi\)
−0.340662 + 0.940186i \(0.610651\pi\)
\(888\) 0 0
\(889\) −8.69694 −0.291686
\(890\) 11.7215 0.392906
\(891\) 0 0
\(892\) −25.7980 −0.863780
\(893\) −14.5395 −0.486545
\(894\) 0 0
\(895\) 2.20204 0.0736061
\(896\) 32.9444 1.10059
\(897\) 0 0
\(898\) 13.3553 0.445674
\(899\) 16.1733 0.539410
\(900\) 0 0
\(901\) 3.63487 0.121095
\(902\) 0 0
\(903\) 0 0
\(904\) −27.8948 −0.927768
\(905\) −12.8990 −0.428777
\(906\) 0 0
\(907\) 49.3939 1.64010 0.820048 0.572294i \(-0.193946\pi\)
0.820048 + 0.572294i \(0.193946\pi\)
\(908\) 16.5068 0.547797
\(909\) 0 0
\(910\) −4.45178 −0.147575
\(911\) −33.7980 −1.11978 −0.559888 0.828568i \(-0.689156\pi\)
−0.559888 + 0.828568i \(0.689156\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −19.9546 −0.660039
\(915\) 0 0
\(916\) −25.5051 −0.842712
\(917\) 55.5959 1.83594
\(918\) 0 0
\(919\) −44.7351 −1.47568 −0.737838 0.674978i \(-0.764153\pi\)
−0.737838 + 0.674978i \(0.764153\pi\)
\(920\) −12.5384 −0.413380
\(921\) 0 0
\(922\) 18.4949 0.609097
\(923\) 19.8082 0.651994
\(924\) 0 0
\(925\) 2.89898 0.0953179
\(926\) −5.93571 −0.195060
\(927\) 0 0
\(928\) 47.3939 1.55578
\(929\) 5.39388 0.176967 0.0884837 0.996078i \(-0.471798\pi\)
0.0884837 + 0.996078i \(0.471798\pi\)
\(930\) 0 0
\(931\) 5.78580 0.189622
\(932\) 0.483412 0.0158347
\(933\) 0 0
\(934\) 3.63487 0.118936
\(935\) 0 0
\(936\) 0 0
\(937\) −22.2926 −0.728268 −0.364134 0.931347i \(-0.618635\pi\)
−0.364134 + 0.931347i \(0.618635\pi\)
\(938\) 2.20204 0.0718992
\(939\) 0 0
\(940\) −14.2020 −0.463220
\(941\) 42.7341 1.39309 0.696546 0.717512i \(-0.254719\pi\)
0.696546 + 0.717512i \(0.254719\pi\)
\(942\) 0 0
\(943\) 25.0769 0.816615
\(944\) −1.10102 −0.0358352
\(945\) 0 0
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) −20.6969 −0.671851
\(950\) −1.10102 −0.0357218
\(951\) 0 0
\(952\) −27.8948 −0.904076
\(953\) 32.6801 1.05861 0.529306 0.848431i \(-0.322452\pi\)
0.529306 + 0.848431i \(0.322452\pi\)
\(954\) 0 0
\(955\) 19.5959 0.634109
\(956\) −22.4762 −0.726932
\(957\) 0 0
\(958\) −20.0908 −0.649105
\(959\) −59.4245 −1.91892
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 3.90918 0.126037
\(963\) 0 0
\(964\) 18.1743 0.585356
\(965\) 17.3237 0.557670
\(966\) 0 0
\(967\) 28.3782 0.912582 0.456291 0.889831i \(-0.349178\pi\)
0.456291 + 0.889831i \(0.349178\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −9.42067 −0.302479
\(971\) 29.3939 0.943294 0.471647 0.881787i \(-0.343660\pi\)
0.471647 + 0.881787i \(0.343660\pi\)
\(972\) 0 0
\(973\) −60.4949 −1.93938
\(974\) 0.667010 0.0213724
\(975\) 0 0
\(976\) −9.57058 −0.306347
\(977\) 45.1918 1.44581 0.722907 0.690945i \(-0.242805\pi\)
0.722907 + 0.690945i \(0.242805\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 5.65153 0.180532
\(981\) 0 0
\(982\) −8.80816 −0.281080
\(983\) 26.6969 0.851500 0.425750 0.904841i \(-0.360010\pi\)
0.425750 + 0.904841i \(0.360010\pi\)
\(984\) 0 0
\(985\) −2.63435 −0.0839374
\(986\) −19.8082 −0.630820
\(987\) 0 0
\(988\) 3.90918 0.124368
\(989\) −16.1733 −0.514281
\(990\) 0 0
\(991\) −53.1918 −1.68969 −0.844847 0.535008i \(-0.820309\pi\)
−0.844847 + 0.535008i \(0.820309\pi\)
\(992\) 11.7215 0.372158
\(993\) 0 0
\(994\) 26.6969 0.846775
\(995\) −1.79796 −0.0569991
\(996\) 0 0
\(997\) 57.6071 1.82443 0.912217 0.409708i \(-0.134369\pi\)
0.912217 + 0.409708i \(0.134369\pi\)
\(998\) 4.30188 0.136173
\(999\) 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.bm.1.2 4
3.2 odd 2 5445.2.a.bn.1.3 yes 4
11.10 odd 2 inner 5445.2.a.bm.1.3 yes 4
33.32 even 2 5445.2.a.bn.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5445.2.a.bm.1.2 4 1.1 even 1 trivial
5445.2.a.bm.1.3 yes 4 11.10 odd 2 inner
5445.2.a.bn.1.2 yes 4 33.32 even 2
5445.2.a.bn.1.3 yes 4 3.2 odd 2