Properties

Label 5445.2.a.y.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +2.41421 q^{10} +1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843 q^{17} +3.82843 q^{20} +2.82843 q^{23} +1.00000 q^{25} +2.82843 q^{26} +7.65685 q^{28} -3.65685 q^{29} -1.58579 q^{32} +16.4853 q^{34} +2.00000 q^{35} -7.65685 q^{37} +4.41421 q^{40} +6.00000 q^{41} +6.00000 q^{43} +6.82843 q^{46} -2.82843 q^{47} -3.00000 q^{49} +2.41421 q^{50} +4.48528 q^{52} -11.6569 q^{53} +8.82843 q^{56} -8.82843 q^{58} -1.65685 q^{59} +9.31371 q^{61} -9.82843 q^{64} +1.17157 q^{65} +12.4853 q^{67} +26.1421 q^{68} +4.82843 q^{70} -11.3137 q^{71} +1.17157 q^{73} -18.4853 q^{74} -4.00000 q^{79} +3.00000 q^{80} +14.4853 q^{82} -6.00000 q^{83} +6.82843 q^{85} +14.4853 q^{86} +13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} -6.82843 q^{94} +3.65685 q^{97} -7.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{20} + 2 q^{25} + 4 q^{28} + 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{35} - 4 q^{37} + 6 q^{40} + 12 q^{41} + 12 q^{43} + 8 q^{46} - 6 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} - 12 q^{58} + 8 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} + 8 q^{67} + 24 q^{68} + 4 q^{70} + 8 q^{73} - 20 q^{74} - 8 q^{79} + 6 q^{80} + 12 q^{82} - 12 q^{83} + 8 q^{85} + 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{92} - 8 q^{94} - 4 q^{97} - 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 2.41421 0.763441
\(11\) 0 0
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 4.82843 1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.82843 0.554700
\(27\) 0 0
\(28\) 7.65685 1.44701
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 16.4853 2.82720
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 4.41421 0.697948
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.82843 1.00680
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 2.41421 0.341421
\(51\) 0 0
\(52\) 4.48528 0.621997
\(53\) −11.6569 −1.60119 −0.800596 0.599204i \(-0.795484\pi\)
−0.800596 + 0.599204i \(0.795484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.82843 1.17975
\(57\) 0 0
\(58\) −8.82843 −1.15923
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) 9.31371 1.19250 0.596249 0.802799i \(-0.296657\pi\)
0.596249 + 0.802799i \(0.296657\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 1.17157 0.145316
\(66\) 0 0
\(67\) 12.4853 1.52532 0.762660 0.646800i \(-0.223893\pi\)
0.762660 + 0.646800i \(0.223893\pi\)
\(68\) 26.1421 3.17020
\(69\) 0 0
\(70\) 4.82843 0.577107
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) 1.17157 0.137122 0.0685611 0.997647i \(-0.478159\pi\)
0.0685611 + 0.997647i \(0.478159\pi\)
\(74\) −18.4853 −2.14887
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 14.4853 1.59963
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 14.4853 1.56199
\(87\) 0 0
\(88\) 0 0
\(89\) 13.3137 1.41125 0.705625 0.708585i \(-0.250666\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(90\) 0 0
\(91\) 2.34315 0.245628
\(92\) 10.8284 1.12894
\(93\) 0 0
\(94\) −6.82843 −0.704298
\(95\) 0 0
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) −7.24264 −0.731617
\(99\) 0 0
\(100\) 3.82843 0.382843
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 0 0
\(103\) 6.82843 0.672825 0.336412 0.941715i \(-0.390786\pi\)
0.336412 + 0.941715i \(0.390786\pi\)
\(104\) 5.17157 0.507114
\(105\) 0 0
\(106\) −28.1421 −2.73341
\(107\) 7.65685 0.740216 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(108\) 0 0
\(109\) 7.65685 0.733394 0.366697 0.930341i \(-0.380489\pi\)
0.366697 + 0.930341i \(0.380489\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.00000 0.566947
\(113\) −19.6569 −1.84916 −0.924581 0.380986i \(-0.875584\pi\)
−0.924581 + 0.380986i \(0.875584\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) −14.0000 −1.29987
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) 0 0
\(122\) 22.4853 2.03572
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.34315 −0.385392 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(128\) −20.5563 −1.81694
\(129\) 0 0
\(130\) 2.82843 0.248069
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 30.1421 2.60388
\(135\) 0 0
\(136\) 30.1421 2.58467
\(137\) 10.9706 0.937278 0.468639 0.883390i \(-0.344744\pi\)
0.468639 + 0.883390i \(0.344744\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 7.65685 0.647122
\(141\) 0 0
\(142\) −27.3137 −2.29212
\(143\) 0 0
\(144\) 0 0
\(145\) −3.65685 −0.303685
\(146\) 2.82843 0.234082
\(147\) 0 0
\(148\) −29.3137 −2.40957
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −9.65685 −0.768258
\(159\) 0 0
\(160\) −1.58579 −0.125367
\(161\) 5.65685 0.445823
\(162\) 0 0
\(163\) 16.4853 1.29123 0.645613 0.763664i \(-0.276602\pi\)
0.645613 + 0.763664i \(0.276602\pi\)
\(164\) 22.9706 1.79370
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) −22.9706 −1.77752 −0.888758 0.458377i \(-0.848431\pi\)
−0.888758 + 0.458377i \(0.848431\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 16.4853 1.26436
\(171\) 0 0
\(172\) 22.9706 1.75149
\(173\) −22.1421 −1.68344 −0.841718 0.539918i \(-0.818455\pi\)
−0.841718 + 0.539918i \(0.818455\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 32.1421 2.40915
\(179\) −9.65685 −0.721787 −0.360894 0.932607i \(-0.617528\pi\)
−0.360894 + 0.932607i \(0.617528\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 5.65685 0.419314
\(183\) 0 0
\(184\) 12.4853 0.920427
\(185\) −7.65685 −0.562943
\(186\) 0 0
\(187\) 0 0
\(188\) −10.8284 −0.789744
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) 1.17157 0.0843317 0.0421658 0.999111i \(-0.486574\pi\)
0.0421658 + 0.999111i \(0.486574\pi\)
\(194\) 8.82843 0.633844
\(195\) 0 0
\(196\) −11.4853 −0.820377
\(197\) −10.8284 −0.771493 −0.385747 0.922605i \(-0.626056\pi\)
−0.385747 + 0.922605i \(0.626056\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 4.41421 0.312132
\(201\) 0 0
\(202\) 22.4853 1.58206
\(203\) −7.31371 −0.513322
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 16.4853 1.14858
\(207\) 0 0
\(208\) 3.51472 0.243702
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −44.6274 −3.06502
\(213\) 0 0
\(214\) 18.4853 1.26363
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 18.4853 1.25198
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) −3.17157 −0.211910
\(225\) 0 0
\(226\) −47.4558 −3.15672
\(227\) 25.3137 1.68013 0.840065 0.542486i \(-0.182517\pi\)
0.840065 + 0.542486i \(0.182517\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 6.82843 0.450253
\(231\) 0 0
\(232\) −16.1421 −1.05978
\(233\) −6.14214 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(234\) 0 0
\(235\) −2.82843 −0.184506
\(236\) −6.34315 −0.412904
\(237\) 0 0
\(238\) 32.9706 2.13716
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 35.6569 2.28270
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 2.41421 0.152688
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −10.4853 −0.657905
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 9.31371 0.580973 0.290487 0.956879i \(-0.406183\pi\)
0.290487 + 0.956879i \(0.406183\pi\)
\(258\) 0 0
\(259\) −15.3137 −0.951548
\(260\) 4.48528 0.278165
\(261\) 0 0
\(262\) −27.3137 −1.68745
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) 0 0
\(265\) −11.6569 −0.716075
\(266\) 0 0
\(267\) 0 0
\(268\) 47.7990 2.91979
\(269\) −17.3137 −1.05564 −0.527818 0.849358i \(-0.676990\pi\)
−0.527818 + 0.849358i \(0.676990\pi\)
\(270\) 0 0
\(271\) −7.31371 −0.444276 −0.222138 0.975015i \(-0.571304\pi\)
−0.222138 + 0.975015i \(0.571304\pi\)
\(272\) 20.4853 1.24210
\(273\) 0 0
\(274\) 26.4853 1.60003
\(275\) 0 0
\(276\) 0 0
\(277\) −6.82843 −0.410280 −0.205140 0.978733i \(-0.565765\pi\)
−0.205140 + 0.978733i \(0.565765\pi\)
\(278\) 9.65685 0.579180
\(279\) 0 0
\(280\) 8.82843 0.527599
\(281\) 17.3137 1.03285 0.516425 0.856333i \(-0.327263\pi\)
0.516425 + 0.856333i \(0.327263\pi\)
\(282\) 0 0
\(283\) −32.6274 −1.93950 −0.969749 0.244103i \(-0.921507\pi\)
−0.969749 + 0.244103i \(0.921507\pi\)
\(284\) −43.3137 −2.57020
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) −8.82843 −0.518423
\(291\) 0 0
\(292\) 4.48528 0.262481
\(293\) −9.17157 −0.535809 −0.267905 0.963445i \(-0.586331\pi\)
−0.267905 + 0.963445i \(0.586331\pi\)
\(294\) 0 0
\(295\) −1.65685 −0.0964658
\(296\) −33.7990 −1.96453
\(297\) 0 0
\(298\) 0.828427 0.0479895
\(299\) 3.31371 0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 28.9706 1.66707
\(303\) 0 0
\(304\) 0 0
\(305\) 9.31371 0.533301
\(306\) 0 0
\(307\) 16.3431 0.932753 0.466376 0.884586i \(-0.345559\pi\)
0.466376 + 0.884586i \(0.345559\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) −1.31371 −0.0742552 −0.0371276 0.999311i \(-0.511821\pi\)
−0.0371276 + 0.999311i \(0.511821\pi\)
\(314\) −33.7990 −1.90739
\(315\) 0 0
\(316\) −15.3137 −0.861463
\(317\) 1.31371 0.0737852 0.0368926 0.999319i \(-0.488254\pi\)
0.0368926 + 0.999319i \(0.488254\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −9.82843 −0.549426
\(321\) 0 0
\(322\) 13.6569 0.761067
\(323\) 0 0
\(324\) 0 0
\(325\) 1.17157 0.0649872
\(326\) 39.7990 2.20426
\(327\) 0 0
\(328\) 26.4853 1.46241
\(329\) −5.65685 −0.311872
\(330\) 0 0
\(331\) −7.31371 −0.401998 −0.200999 0.979591i \(-0.564419\pi\)
−0.200999 + 0.979591i \(0.564419\pi\)
\(332\) −22.9706 −1.26067
\(333\) 0 0
\(334\) −55.4558 −3.03441
\(335\) 12.4853 0.682144
\(336\) 0 0
\(337\) 20.4853 1.11590 0.557952 0.829873i \(-0.311587\pi\)
0.557952 + 0.829873i \(0.311587\pi\)
\(338\) −28.0711 −1.52686
\(339\) 0 0
\(340\) 26.1421 1.41776
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 26.4853 1.42799
\(345\) 0 0
\(346\) −53.4558 −2.87380
\(347\) 10.9706 0.588931 0.294465 0.955662i \(-0.404858\pi\)
0.294465 + 0.955662i \(0.404858\pi\)
\(348\) 0 0
\(349\) −26.9706 −1.44370 −0.721851 0.692049i \(-0.756708\pi\)
−0.721851 + 0.692049i \(0.756708\pi\)
\(350\) 4.82843 0.258090
\(351\) 0 0
\(352\) 0 0
\(353\) −21.3137 −1.13441 −0.567207 0.823575i \(-0.691976\pi\)
−0.567207 + 0.823575i \(0.691976\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 50.9706 2.70143
\(357\) 0 0
\(358\) −23.3137 −1.23217
\(359\) 0.686292 0.0362211 0.0181105 0.999836i \(-0.494235\pi\)
0.0181105 + 0.999836i \(0.494235\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 51.4558 2.70446
\(363\) 0 0
\(364\) 8.97056 0.470185
\(365\) 1.17157 0.0613229
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 8.48528 0.442326
\(369\) 0 0
\(370\) −18.4853 −0.961004
\(371\) −23.3137 −1.21039
\(372\) 0 0
\(373\) −35.7990 −1.85360 −0.926801 0.375554i \(-0.877453\pi\)
−0.926801 + 0.375554i \(0.877453\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.4853 −0.643879
\(377\) −4.28427 −0.220651
\(378\) 0 0
\(379\) 33.6569 1.72884 0.864418 0.502773i \(-0.167687\pi\)
0.864418 + 0.502773i \(0.167687\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 5.85786 0.299323 0.149661 0.988737i \(-0.452182\pi\)
0.149661 + 0.988737i \(0.452182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.82843 0.143963
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) 0 0
\(391\) 19.3137 0.976736
\(392\) −13.2426 −0.668854
\(393\) 0 0
\(394\) −26.1421 −1.31702
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −9.31371 −0.467442 −0.233721 0.972304i \(-0.575090\pi\)
−0.233721 + 0.972304i \(0.575090\pi\)
\(398\) 24.9706 1.25166
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 35.6569 1.77399
\(405\) 0 0
\(406\) −17.6569 −0.876295
\(407\) 0 0
\(408\) 0 0
\(409\) −1.02944 −0.0509024 −0.0254512 0.999676i \(-0.508102\pi\)
−0.0254512 + 0.999676i \(0.508102\pi\)
\(410\) 14.4853 0.715377
\(411\) 0 0
\(412\) 26.1421 1.28793
\(413\) −3.31371 −0.163057
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) −1.85786 −0.0910893
\(417\) 0 0
\(418\) 0 0
\(419\) 25.6569 1.25342 0.626710 0.779253i \(-0.284401\pi\)
0.626710 + 0.779253i \(0.284401\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 38.6274 1.88035
\(423\) 0 0
\(424\) −51.4558 −2.49892
\(425\) 6.82843 0.331227
\(426\) 0 0
\(427\) 18.6274 0.901444
\(428\) 29.3137 1.41693
\(429\) 0 0
\(430\) 14.4853 0.698542
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) −7.65685 −0.367965 −0.183982 0.982930i \(-0.558899\pi\)
−0.183982 + 0.982930i \(0.558899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 29.3137 1.40387
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 19.3137 0.918659
\(443\) 26.8284 1.27466 0.637329 0.770592i \(-0.280039\pi\)
0.637329 + 0.770592i \(0.280039\pi\)
\(444\) 0 0
\(445\) 13.3137 0.631130
\(446\) −26.1421 −1.23787
\(447\) 0 0
\(448\) −19.6569 −0.928699
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −75.2548 −3.53969
\(453\) 0 0
\(454\) 61.1127 2.86816
\(455\) 2.34315 0.109848
\(456\) 0 0
\(457\) −0.485281 −0.0227005 −0.0113503 0.999936i \(-0.503613\pi\)
−0.0113503 + 0.999936i \(0.503613\pi\)
\(458\) 3.17157 0.148198
\(459\) 0 0
\(460\) 10.8284 0.504878
\(461\) 12.6274 0.588117 0.294059 0.955787i \(-0.404994\pi\)
0.294059 + 0.955787i \(0.404994\pi\)
\(462\) 0 0
\(463\) −6.14214 −0.285449 −0.142725 0.989762i \(-0.545586\pi\)
−0.142725 + 0.989762i \(0.545586\pi\)
\(464\) −10.9706 −0.509296
\(465\) 0 0
\(466\) −14.8284 −0.686914
\(467\) 14.8284 0.686178 0.343089 0.939303i \(-0.388527\pi\)
0.343089 + 0.939303i \(0.388527\pi\)
\(468\) 0 0
\(469\) 24.9706 1.15303
\(470\) −6.82843 −0.314972
\(471\) 0 0
\(472\) −7.31371 −0.336641
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 52.2843 2.39645
\(477\) 0 0
\(478\) −56.2843 −2.57438
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −8.97056 −0.409022
\(482\) −14.4853 −0.659786
\(483\) 0 0
\(484\) 0 0
\(485\) 3.65685 0.166049
\(486\) 0 0
\(487\) −24.4853 −1.10953 −0.554767 0.832006i \(-0.687193\pi\)
−0.554767 + 0.832006i \(0.687193\pi\)
\(488\) 41.1127 1.86108
\(489\) 0 0
\(490\) −7.24264 −0.327189
\(491\) −0.686292 −0.0309719 −0.0154860 0.999880i \(-0.504930\pi\)
−0.0154860 + 0.999880i \(0.504930\pi\)
\(492\) 0 0
\(493\) −24.9706 −1.12462
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −22.6274 −1.01498
\(498\) 0 0
\(499\) 9.65685 0.432300 0.216150 0.976360i \(-0.430650\pi\)
0.216150 + 0.976360i \(0.430650\pi\)
\(500\) 3.82843 0.171212
\(501\) 0 0
\(502\) −28.9706 −1.29302
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) 0 0
\(505\) 9.31371 0.414455
\(506\) 0 0
\(507\) 0 0
\(508\) −16.6274 −0.737722
\(509\) 13.3137 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(510\) 0 0
\(511\) 2.34315 0.103655
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) 22.4853 0.991783
\(515\) 6.82843 0.300896
\(516\) 0 0
\(517\) 0 0
\(518\) −36.9706 −1.62439
\(519\) 0 0
\(520\) 5.17157 0.226788
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0 0
\(523\) 41.5980 1.81895 0.909476 0.415756i \(-0.136483\pi\)
0.909476 + 0.415756i \(0.136483\pi\)
\(524\) −43.3137 −1.89217
\(525\) 0 0
\(526\) −26.4853 −1.15481
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) −28.1421 −1.22242
\(531\) 0 0
\(532\) 0 0
\(533\) 7.02944 0.304479
\(534\) 0 0
\(535\) 7.65685 0.331035
\(536\) 55.1127 2.38051
\(537\) 0 0
\(538\) −41.7990 −1.80208
\(539\) 0 0
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) −17.6569 −0.758427
\(543\) 0 0
\(544\) −10.8284 −0.464265
\(545\) 7.65685 0.327984
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) 42.0000 1.79415
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −16.4853 −0.700392
\(555\) 0 0
\(556\) 15.3137 0.649446
\(557\) 9.85786 0.417691 0.208846 0.977949i \(-0.433029\pi\)
0.208846 + 0.977949i \(0.433029\pi\)
\(558\) 0 0
\(559\) 7.02944 0.297314
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) 41.7990 1.76318
\(563\) 0.343146 0.0144619 0.00723093 0.999974i \(-0.497698\pi\)
0.00723093 + 0.999974i \(0.497698\pi\)
\(564\) 0 0
\(565\) −19.6569 −0.826970
\(566\) −78.7696 −3.31093
\(567\) 0 0
\(568\) −49.9411 −2.09548
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 0 0
\(571\) 21.9411 0.918208 0.459104 0.888383i \(-0.348171\pi\)
0.459104 + 0.888383i \(0.348171\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 28.9706 1.20921
\(575\) 2.82843 0.117954
\(576\) 0 0
\(577\) −26.9706 −1.12280 −0.561400 0.827545i \(-0.689737\pi\)
−0.561400 + 0.827545i \(0.689737\pi\)
\(578\) 71.5269 2.97513
\(579\) 0 0
\(580\) −14.0000 −0.581318
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 5.17157 0.214001
\(585\) 0 0
\(586\) −22.1421 −0.914683
\(587\) 2.14214 0.0884154 0.0442077 0.999022i \(-0.485924\pi\)
0.0442077 + 0.999022i \(0.485924\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −22.9706 −0.944084
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) 0 0
\(595\) 13.6569 0.559876
\(596\) 1.31371 0.0538116
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) −23.9411 −0.976579 −0.488289 0.872682i \(-0.662379\pi\)
−0.488289 + 0.872682i \(0.662379\pi\)
\(602\) 28.9706 1.18075
\(603\) 0 0
\(604\) 45.9411 1.86932
\(605\) 0 0
\(606\) 0 0
\(607\) −38.2843 −1.55391 −0.776955 0.629556i \(-0.783237\pi\)
−0.776955 + 0.629556i \(0.783237\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 22.4853 0.910402
\(611\) −3.31371 −0.134058
\(612\) 0 0
\(613\) 25.4558 1.02815 0.514076 0.857745i \(-0.328135\pi\)
0.514076 + 0.857745i \(0.328135\pi\)
\(614\) 39.4558 1.59231
\(615\) 0 0
\(616\) 0 0
\(617\) −0.343146 −0.0138145 −0.00690726 0.999976i \(-0.502199\pi\)
−0.00690726 + 0.999976i \(0.502199\pi\)
\(618\) 0 0
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11.3137 −0.453638
\(623\) 26.6274 1.06680
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −3.17157 −0.126762
\(627\) 0 0
\(628\) −53.5980 −2.13879
\(629\) −52.2843 −2.08471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −17.6569 −0.702352
\(633\) 0 0
\(634\) 3.17157 0.125959
\(635\) −4.34315 −0.172352
\(636\) 0 0
\(637\) −3.51472 −0.139258
\(638\) 0 0
\(639\) 0 0
\(640\) −20.5563 −0.812561
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −1.45584 −0.0574129 −0.0287064 0.999588i \(-0.509139\pi\)
−0.0287064 + 0.999588i \(0.509139\pi\)
\(644\) 21.6569 0.853400
\(645\) 0 0
\(646\) 0 0
\(647\) −27.1127 −1.06591 −0.532955 0.846144i \(-0.678919\pi\)
−0.532955 + 0.846144i \(0.678919\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.82843 0.110940
\(651\) 0 0
\(652\) 63.1127 2.47168
\(653\) −11.6569 −0.456168 −0.228084 0.973641i \(-0.573246\pi\)
−0.228084 + 0.973641i \(0.573246\pi\)
\(654\) 0 0
\(655\) −11.3137 −0.442063
\(656\) 18.0000 0.702782
\(657\) 0 0
\(658\) −13.6569 −0.532400
\(659\) 45.9411 1.78961 0.894806 0.446455i \(-0.147314\pi\)
0.894806 + 0.446455i \(0.147314\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) −17.6569 −0.686253
\(663\) 0 0
\(664\) −26.4853 −1.02783
\(665\) 0 0
\(666\) 0 0
\(667\) −10.3431 −0.400488
\(668\) −87.9411 −3.40254
\(669\) 0 0
\(670\) 30.1421 1.16449
\(671\) 0 0
\(672\) 0 0
\(673\) 12.4853 0.481272 0.240636 0.970615i \(-0.422644\pi\)
0.240636 + 0.970615i \(0.422644\pi\)
\(674\) 49.4558 1.90497
\(675\) 0 0
\(676\) −44.5147 −1.71210
\(677\) 22.8284 0.877368 0.438684 0.898641i \(-0.355445\pi\)
0.438684 + 0.898641i \(0.355445\pi\)
\(678\) 0 0
\(679\) 7.31371 0.280674
\(680\) 30.1421 1.15590
\(681\) 0 0
\(682\) 0 0
\(683\) 7.79899 0.298420 0.149210 0.988806i \(-0.452327\pi\)
0.149210 + 0.988806i \(0.452327\pi\)
\(684\) 0 0
\(685\) 10.9706 0.419164
\(686\) −48.2843 −1.84350
\(687\) 0 0
\(688\) 18.0000 0.686244
\(689\) −13.6569 −0.520285
\(690\) 0 0
\(691\) −39.3137 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(692\) −84.7696 −3.22245
\(693\) 0 0
\(694\) 26.4853 1.00537
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 40.9706 1.55187
\(698\) −65.1127 −2.46455
\(699\) 0 0
\(700\) 7.65685 0.289402
\(701\) −12.6274 −0.476931 −0.238465 0.971151i \(-0.576644\pi\)
−0.238465 + 0.971151i \(0.576644\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −51.4558 −1.93657
\(707\) 18.6274 0.700556
\(708\) 0 0
\(709\) 24.6274 0.924902 0.462451 0.886645i \(-0.346970\pi\)
0.462451 + 0.886645i \(0.346970\pi\)
\(710\) −27.3137 −1.02507
\(711\) 0 0
\(712\) 58.7696 2.20248
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −36.9706 −1.38165
\(717\) 0 0
\(718\) 1.65685 0.0618333
\(719\) −18.3431 −0.684084 −0.342042 0.939685i \(-0.611118\pi\)
−0.342042 + 0.939685i \(0.611118\pi\)
\(720\) 0 0
\(721\) 13.6569 0.508608
\(722\) −45.8701 −1.70711
\(723\) 0 0
\(724\) 81.5980 3.03257
\(725\) −3.65685 −0.135812
\(726\) 0 0
\(727\) −19.5147 −0.723761 −0.361880 0.932225i \(-0.617865\pi\)
−0.361880 + 0.932225i \(0.617865\pi\)
\(728\) 10.3431 0.383342
\(729\) 0 0
\(730\) 2.82843 0.104685
\(731\) 40.9706 1.51535
\(732\) 0 0
\(733\) 17.4558 0.644746 0.322373 0.946613i \(-0.395519\pi\)
0.322373 + 0.946613i \(0.395519\pi\)
\(734\) −20.4853 −0.756126
\(735\) 0 0
\(736\) −4.48528 −0.165330
\(737\) 0 0
\(738\) 0 0
\(739\) −29.9411 −1.10140 −0.550701 0.834703i \(-0.685640\pi\)
−0.550701 + 0.834703i \(0.685640\pi\)
\(740\) −29.3137 −1.07759
\(741\) 0 0
\(742\) −56.2843 −2.06626
\(743\) −49.5980 −1.81957 −0.909787 0.415076i \(-0.863755\pi\)
−0.909787 + 0.415076i \(0.863755\pi\)
\(744\) 0 0
\(745\) 0.343146 0.0125719
\(746\) −86.4264 −3.16430
\(747\) 0 0
\(748\) 0 0
\(749\) 15.3137 0.559551
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −8.48528 −0.309426
\(753\) 0 0
\(754\) −10.3431 −0.376675
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 13.3137 0.483895 0.241947 0.970289i \(-0.422214\pi\)
0.241947 + 0.970289i \(0.422214\pi\)
\(758\) 81.2548 2.95131
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 15.3137 0.554393
\(764\) −12.6863 −0.458974
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) −1.94113 −0.0700900
\(768\) 0 0
\(769\) 18.9706 0.684096 0.342048 0.939682i \(-0.388879\pi\)
0.342048 + 0.939682i \(0.388879\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.48528 0.161429
\(773\) −26.2843 −0.945380 −0.472690 0.881229i \(-0.656717\pi\)
−0.472690 + 0.881229i \(0.656717\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.1421 0.579469
\(777\) 0 0
\(778\) −49.7990 −1.78538
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 46.6274 1.66739
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 14.9706 0.533643 0.266821 0.963746i \(-0.414027\pi\)
0.266821 + 0.963746i \(0.414027\pi\)
\(788\) −41.4558 −1.47680
\(789\) 0 0
\(790\) −9.65685 −0.343575
\(791\) −39.3137 −1.39783
\(792\) 0 0
\(793\) 10.9117 0.387485
\(794\) −22.4853 −0.797973
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) −32.6274 −1.15572 −0.577861 0.816135i \(-0.696113\pi\)
−0.577861 + 0.816135i \(0.696113\pi\)
\(798\) 0 0
\(799\) −19.3137 −0.683270
\(800\) −1.58579 −0.0560660
\(801\) 0 0
\(802\) 12.8284 0.452988
\(803\) 0 0
\(804\) 0 0
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) 0 0
\(808\) 41.1127 1.44634
\(809\) −10.9706 −0.385704 −0.192852 0.981228i \(-0.561774\pi\)
−0.192852 + 0.981228i \(0.561774\pi\)
\(810\) 0 0
\(811\) −53.9411 −1.89413 −0.947065 0.321043i \(-0.895967\pi\)
−0.947065 + 0.321043i \(0.895967\pi\)
\(812\) −28.0000 −0.982607
\(813\) 0 0
\(814\) 0 0
\(815\) 16.4853 0.577454
\(816\) 0 0
\(817\) 0 0
\(818\) −2.48528 −0.0868958
\(819\) 0 0
\(820\) 22.9706 0.802167
\(821\) −41.3137 −1.44186 −0.720929 0.693009i \(-0.756285\pi\)
−0.720929 + 0.693009i \(0.756285\pi\)
\(822\) 0 0
\(823\) 19.5147 0.680240 0.340120 0.940382i \(-0.389532\pi\)
0.340120 + 0.940382i \(0.389532\pi\)
\(824\) 30.1421 1.05005
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 22.2843 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) −14.4853 −0.502791
\(831\) 0 0
\(832\) −11.5147 −0.399201
\(833\) −20.4853 −0.709773
\(834\) 0 0
\(835\) −22.9706 −0.794929
\(836\) 0 0
\(837\) 0 0
\(838\) 61.9411 2.13972
\(839\) −26.3431 −0.909466 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) −14.4853 −0.499196
\(843\) 0 0
\(844\) 61.2548 2.10848
\(845\) −11.6274 −0.399995
\(846\) 0 0
\(847\) 0 0
\(848\) −34.9706 −1.20089
\(849\) 0 0
\(850\) 16.4853 0.565440
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) 15.5147 0.531214 0.265607 0.964081i \(-0.414428\pi\)
0.265607 + 0.964081i \(0.414428\pi\)
\(854\) 44.9706 1.53886
\(855\) 0 0
\(856\) 33.7990 1.15523
\(857\) 24.7696 0.846112 0.423056 0.906104i \(-0.360957\pi\)
0.423056 + 0.906104i \(0.360957\pi\)
\(858\) 0 0
\(859\) 24.2843 0.828569 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(860\) 22.9706 0.783290
\(861\) 0 0
\(862\) −27.3137 −0.930309
\(863\) 9.17157 0.312204 0.156102 0.987741i \(-0.450107\pi\)
0.156102 + 0.987741i \(0.450107\pi\)
\(864\) 0 0
\(865\) −22.1421 −0.752855
\(866\) −18.4853 −0.628155
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 14.6274 0.495631
\(872\) 33.7990 1.14458
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 49.4558 1.67001 0.835003 0.550246i \(-0.185466\pi\)
0.835003 + 0.550246i \(0.185466\pi\)
\(878\) 38.6274 1.30361
\(879\) 0 0
\(880\) 0 0
\(881\) 7.37258 0.248389 0.124194 0.992258i \(-0.460365\pi\)
0.124194 + 0.992258i \(0.460365\pi\)
\(882\) 0 0
\(883\) 37.1716 1.25092 0.625462 0.780255i \(-0.284911\pi\)
0.625462 + 0.780255i \(0.284911\pi\)
\(884\) 30.6274 1.03011
\(885\) 0 0
\(886\) 64.7696 2.17598
\(887\) 38.2843 1.28546 0.642730 0.766093i \(-0.277802\pi\)
0.642730 + 0.766093i \(0.277802\pi\)
\(888\) 0 0
\(889\) −8.68629 −0.291329
\(890\) 32.1421 1.07741
\(891\) 0 0
\(892\) −41.4558 −1.38804
\(893\) 0 0
\(894\) 0 0
\(895\) −9.65685 −0.322793
\(896\) −41.1127 −1.37348
\(897\) 0 0
\(898\) −69.1127 −2.30632
\(899\) 0 0
\(900\) 0 0
\(901\) −79.5980 −2.65179
\(902\) 0 0
\(903\) 0 0
\(904\) −86.7696 −2.88591
\(905\) 21.3137 0.708492
\(906\) 0 0
\(907\) −27.5147 −0.913611 −0.456806 0.889567i \(-0.651007\pi\)
−0.456806 + 0.889567i \(0.651007\pi\)
\(908\) 96.9117 3.21613
\(909\) 0 0
\(910\) 5.65685 0.187523
\(911\) 9.94113 0.329364 0.164682 0.986347i \(-0.447340\pi\)
0.164682 + 0.986347i \(0.447340\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.17157 −0.0387522
\(915\) 0 0
\(916\) 5.02944 0.166177
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 12.4853 0.411628
\(921\) 0 0
\(922\) 30.4853 1.00398
\(923\) −13.2548 −0.436288
\(924\) 0 0
\(925\) −7.65685 −0.251756
\(926\) −14.8284 −0.487292
\(927\) 0 0
\(928\) 5.79899 0.190361
\(929\) −5.31371 −0.174337 −0.0871686 0.996194i \(-0.527782\pi\)
−0.0871686 + 0.996194i \(0.527782\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.5147 −0.770250
\(933\) 0 0
\(934\) 35.7990 1.17138
\(935\) 0 0
\(936\) 0 0
\(937\) 1.45584 0.0475604 0.0237802 0.999717i \(-0.492430\pi\)
0.0237802 + 0.999717i \(0.492430\pi\)
\(938\) 60.2843 1.96835
\(939\) 0 0
\(940\) −10.8284 −0.353184
\(941\) −6.68629 −0.217967 −0.108983 0.994044i \(-0.534760\pi\)
−0.108983 + 0.994044i \(0.534760\pi\)
\(942\) 0 0
\(943\) 16.9706 0.552638
\(944\) −4.97056 −0.161778
\(945\) 0 0
\(946\) 0 0
\(947\) −41.1716 −1.33790 −0.668948 0.743309i \(-0.733255\pi\)
−0.668948 + 0.743309i \(0.733255\pi\)
\(948\) 0 0
\(949\) 1.37258 0.0445559
\(950\) 0 0
\(951\) 0 0
\(952\) 60.2843 1.95382
\(953\) 53.1716 1.72240 0.861198 0.508269i \(-0.169715\pi\)
0.861198 + 0.508269i \(0.169715\pi\)
\(954\) 0 0
\(955\) −3.31371 −0.107229
\(956\) −89.2548 −2.88671
\(957\) 0 0
\(958\) −86.9117 −2.80799
\(959\) 21.9411 0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −21.6569 −0.698245
\(963\) 0 0
\(964\) −22.9706 −0.739832
\(965\) 1.17157 0.0377143
\(966\) 0 0
\(967\) 14.9706 0.481421 0.240710 0.970597i \(-0.422620\pi\)
0.240710 + 0.970597i \(0.422620\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 8.82843 0.283464
\(971\) 8.68629 0.278756 0.139378 0.990239i \(-0.455490\pi\)
0.139378 + 0.990239i \(0.455490\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) −59.1127 −1.89409
\(975\) 0 0
\(976\) 27.9411 0.894374
\(977\) −32.3431 −1.03475 −0.517374 0.855759i \(-0.673091\pi\)
−0.517374 + 0.855759i \(0.673091\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −11.4853 −0.366884
\(981\) 0 0
\(982\) −1.65685 −0.0528723
\(983\) 21.8579 0.697158 0.348579 0.937279i \(-0.386664\pi\)
0.348579 + 0.937279i \(0.386664\pi\)
\(984\) 0 0
\(985\) −10.8284 −0.345022
\(986\) −60.2843 −1.91984
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) −57.9411 −1.84056 −0.920280 0.391260i \(-0.872039\pi\)
−0.920280 + 0.391260i \(0.872039\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −54.6274 −1.73268
\(995\) 10.3431 0.327900
\(996\) 0 0
\(997\) 41.4558 1.31292 0.656460 0.754361i \(-0.272053\pi\)
0.656460 + 0.754361i \(0.272053\pi\)
\(998\) 23.3137 0.737983
\(999\) 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.y.1.2 2
3.2 odd 2 605.2.a.d.1.1 2
11.10 odd 2 495.2.a.b.1.1 2
12.11 even 2 9680.2.a.bn.1.2 2
15.14 odd 2 3025.2.a.o.1.2 2
33.2 even 10 605.2.g.f.81.1 8
33.5 odd 10 605.2.g.l.366.2 8
33.8 even 10 605.2.g.f.251.2 8
33.14 odd 10 605.2.g.l.251.1 8
33.17 even 10 605.2.g.f.366.1 8
33.20 odd 10 605.2.g.l.81.2 8
33.26 odd 10 605.2.g.l.511.1 8
33.29 even 10 605.2.g.f.511.2 8
33.32 even 2 55.2.a.b.1.2 2
44.43 even 2 7920.2.a.ch.1.2 2
55.32 even 4 2475.2.c.l.199.1 4
55.43 even 4 2475.2.c.l.199.4 4
55.54 odd 2 2475.2.a.x.1.2 2
132.131 odd 2 880.2.a.m.1.2 2
165.32 odd 4 275.2.b.d.199.4 4
165.98 odd 4 275.2.b.d.199.1 4
165.164 even 2 275.2.a.c.1.1 2
231.230 odd 2 2695.2.a.f.1.2 2
264.131 odd 2 3520.2.a.bo.1.1 2
264.197 even 2 3520.2.a.bn.1.2 2
429.428 even 2 9295.2.a.g.1.1 2
660.263 even 4 4400.2.b.q.4049.3 4
660.527 even 4 4400.2.b.q.4049.2 4
660.659 odd 2 4400.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 33.32 even 2
275.2.a.c.1.1 2 165.164 even 2
275.2.b.d.199.1 4 165.98 odd 4
275.2.b.d.199.4 4 165.32 odd 4
495.2.a.b.1.1 2 11.10 odd 2
605.2.a.d.1.1 2 3.2 odd 2
605.2.g.f.81.1 8 33.2 even 10
605.2.g.f.251.2 8 33.8 even 10
605.2.g.f.366.1 8 33.17 even 10
605.2.g.f.511.2 8 33.29 even 10
605.2.g.l.81.2 8 33.20 odd 10
605.2.g.l.251.1 8 33.14 odd 10
605.2.g.l.366.2 8 33.5 odd 10
605.2.g.l.511.1 8 33.26 odd 10
880.2.a.m.1.2 2 132.131 odd 2
2475.2.a.x.1.2 2 55.54 odd 2
2475.2.c.l.199.1 4 55.32 even 4
2475.2.c.l.199.4 4 55.43 even 4
2695.2.a.f.1.2 2 231.230 odd 2
3025.2.a.o.1.2 2 15.14 odd 2
3520.2.a.bn.1.2 2 264.197 even 2
3520.2.a.bo.1.1 2 264.131 odd 2
4400.2.a.bn.1.1 2 660.659 odd 2
4400.2.b.q.4049.2 4 660.527 even 4
4400.2.b.q.4049.3 4 660.263 even 4
5445.2.a.y.1.2 2 1.1 even 1 trivial
7920.2.a.ch.1.2 2 44.43 even 2
9295.2.a.g.1.1 2 429.428 even 2
9680.2.a.bn.1.2 2 12.11 even 2