Properties

Label 495.2.a.b.1.1
Level $495$
Weight $2$
Character 495.1
Self dual yes
Analytic conductor $3.953$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.95259490005\)
Analytic rank: \(1\)
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.00000 q^{7} -4.41421 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} -2.00000 q^{7} -4.41421 q^{8} -2.41421 q^{10} -1.00000 q^{11} -1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} -6.82843 q^{17} +3.82843 q^{20} +2.41421 q^{22} +2.82843 q^{23} +1.00000 q^{25} +2.82843 q^{26} -7.65685 q^{28} +3.65685 q^{29} +1.58579 q^{32} +16.4853 q^{34} -2.00000 q^{35} -7.65685 q^{37} -4.41421 q^{40} -6.00000 q^{41} -6.00000 q^{43} -3.82843 q^{44} -6.82843 q^{46} -2.82843 q^{47} -3.00000 q^{49} -2.41421 q^{50} -4.48528 q^{52} -11.6569 q^{53} -1.00000 q^{55} +8.82843 q^{56} -8.82843 q^{58} -1.65685 q^{59} -9.31371 q^{61} -9.82843 q^{64} -1.17157 q^{65} +12.4853 q^{67} -26.1421 q^{68} +4.82843 q^{70} -11.3137 q^{71} -1.17157 q^{73} +18.4853 q^{74} +2.00000 q^{77} +4.00000 q^{79} +3.00000 q^{80} +14.4853 q^{82} +6.00000 q^{83} -6.82843 q^{85} +14.4853 q^{86} +4.41421 q^{88} +13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} +6.82843 q^{94} +3.65685 q^{97} +7.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{11} - 8 q^{13} + 4 q^{14} + 6 q^{16} - 8 q^{17} + 2 q^{20} + 2 q^{22} + 2 q^{25} - 4 q^{28} - 4 q^{29} + 6 q^{32} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 6 q^{40} - 12 q^{41} - 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 2 q^{50} + 8 q^{52} - 12 q^{53} - 2 q^{55} + 12 q^{56} - 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{65} + 8 q^{67} - 24 q^{68} + 4 q^{70} - 8 q^{73} + 20 q^{74} + 4 q^{77} + 8 q^{79} + 6 q^{80} + 12 q^{82} + 12 q^{83} - 8 q^{85} + 12 q^{86} + 6 q^{88} + 4 q^{89} + 16 q^{91} + 16 q^{92} + 8 q^{94} - 4 q^{97} + 6 q^{98} + O(q^{100}) \)

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 \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) −2.41421 −0.763441
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.17157 −0.324936 −0.162468 0.986714i \(-0.551945\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(14\) 4.82843 1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.82843 0.856062
\(21\) 0 0
\(22\) 2.41421 0.514712
\(23\) 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 \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 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 \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −3.82843 −0.577157
\(45\) 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
\(55\) −1.00000 −0.134840
\(56\) 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 \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 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 \(-0.521841\pi\)
−0.0685611 + 0.997647i \(0.521841\pi\)
\(74\) 18.4853 2.14887
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 14.4853 1.59963
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) 14.4853 1.56199
\(87\) 0 0
\(88\) 4.41421 0.470557
\(89\) 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 \(-0.653361\pi\)
−0.463374 + 0.886163i \(0.653361\pi\)
\(102\) 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 \(-0.620679\pi\)
−0.370108 + 0.928989i \(0.620679\pi\)
\(108\) 0 0
\(109\) −7.65685 −0.733394 −0.366697 0.930341i \(-0.619511\pi\)
−0.366697 + 0.930341i \(0.619511\pi\)
\(110\) 2.41421 0.230186
\(111\) 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
\(121\) 1.00000 0.0909091
\(122\) 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 \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) 20.5563 1.81694
\(129\) 0 0
\(130\) 2.82843 0.248069
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 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 \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −7.65685 −0.647122
\(141\) 0 0
\(142\) 27.3137 2.29212
\(143\) 1.17157 0.0979718
\(144\) 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 \(-0.504474\pi\)
−0.0140558 + 0.999901i \(0.504474\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.82843 −0.389086
\(155\) 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 \(-0.151569\pi\)
0.888758 + 0.458377i \(0.151569\pi\)
\(168\) 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 \(-0.181545\pi\)
0.841718 + 0.539918i \(0.181545\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −3.00000 −0.226134
\(177\) 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
\(187\) 6.82843 0.499344
\(188\) −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 \(-0.513426\pi\)
−0.0421658 + 0.999111i \(0.513426\pi\)
\(194\) −8.82843 −0.633844
\(195\) 0 0
\(196\) −11.4853 −0.820377
\(197\) 10.8284 0.771493 0.385747 0.922605i \(-0.373944\pi\)
0.385747 + 0.922605i \(0.373944\pi\)
\(198\) 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 \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −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
\(220\) −3.82843 −0.258113
\(221\) 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 \(-0.817483\pi\)
−0.840065 + 0.542486i \(0.817483\pi\)
\(228\) 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 \(-0.435518\pi\)
0.201192 + 0.979552i \(0.435518\pi\)
\(234\) 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 \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −2.41421 −0.155192
\(243\) 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
\(253\) −2.82843 −0.177822
\(254\) −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 \(-0.390169\pi\)
0.338237 + 0.941061i \(0.390169\pi\)
\(264\) 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 \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) −20.4853 −1.24210
\(273\) 0 0
\(274\) −26.4853 −1.60003
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 6.82843 0.410280 0.205140 0.978733i \(-0.434235\pi\)
0.205140 + 0.978733i \(0.434235\pi\)
\(278\) 9.65685 0.579180
\(279\) 0 0
\(280\) 8.82843 0.527599
\(281\) −17.3137 −1.03285 −0.516425 0.856333i \(-0.672737\pi\)
−0.516425 + 0.856333i \(0.672737\pi\)
\(282\) 0 0
\(283\) 32.6274 1.93950 0.969749 0.244103i \(-0.0784935\pi\)
0.969749 + 0.244103i \(0.0784935\pi\)
\(284\) −43.3137 −2.57020
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) 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 \(-0.413669\pi\)
0.267905 + 0.963445i \(0.413669\pi\)
\(294\) 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 \(-0.654441\pi\)
−0.466376 + 0.884586i \(0.654441\pi\)
\(308\) 7.65685 0.436290
\(309\) 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
\(319\) −3.65685 −0.204745
\(320\) −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 \(-0.688413\pi\)
−0.557952 + 0.829873i \(0.688413\pi\)
\(338\) 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 \(-0.595142\pi\)
−0.294465 + 0.955662i \(0.595142\pi\)
\(348\) 0 0
\(349\) 26.9706 1.44370 0.721851 0.692049i \(-0.243292\pi\)
0.721851 + 0.692049i \(0.243292\pi\)
\(350\) 4.82843 0.258090
\(351\) 0 0
\(352\) −1.58579 −0.0845227
\(353\) −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 \(-0.505765\pi\)
−0.0181105 + 0.999836i \(0.505765\pi\)
\(360\) 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 \(-0.122547\pi\)
0.926801 + 0.375554i \(0.122547\pi\)
\(374\) −16.4853 −0.852434
\(375\) 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
\(385\) 2.00000 0.101929
\(386\) 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
\(407\) 7.65685 0.379536
\(408\) 0 0
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) 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 \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 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 \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 4.41421 0.210439
\(441\) 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
\(451\) 6.00000 0.282529
\(452\) −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 \(-0.496387\pi\)
0.0113503 + 0.999936i \(0.496387\pi\)
\(458\) −3.17157 −0.148198
\(459\) 0 0
\(460\) 10.8284 0.504878
\(461\) −12.6274 −0.588117 −0.294059 0.955787i \(-0.595006\pi\)
−0.294059 + 0.955787i \(0.595006\pi\)
\(462\) 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
\(473\) 6.00000 0.275880
\(474\) 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 \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) −14.4853 −0.659786
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) 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 \(-0.495070\pi\)
0.0154860 + 0.999880i \(0.495070\pi\)
\(492\) 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 \(-0.620879\pi\)
−0.370690 + 0.928757i \(0.620879\pi\)
\(504\) 0 0
\(505\) −9.31371 −0.414455
\(506\) 6.82843 0.303561
\(507\) 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
\(517\) 2.82843 0.124394
\(518\) −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 \(-0.863517\pi\)
−0.909476 + 0.415756i \(0.863517\pi\)
\(524\) 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
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) −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 \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 42.0000 1.79415
\(549\) 0 0
\(550\) 2.41421 0.102942
\(551\) 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 \(-0.566971\pi\)
−0.208846 + 0.977949i \(0.566971\pi\)
\(558\) 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 \(-0.502302\pi\)
−0.00723093 + 0.999974i \(0.502302\pi\)
\(564\) 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 \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) −21.9411 −0.918208 −0.459104 0.888383i \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(572\) 4.48528 0.187539
\(573\) 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
\(583\) 11.6569 0.482778
\(584\) 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 \(-0.522991\pi\)
−0.0721661 + 0.997393i \(0.522991\pi\)
\(594\) 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 \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) −28.9706 −1.18075
\(603\) 0 0
\(604\) −45.9411 −1.86932
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 38.2843 1.55391 0.776955 0.629556i \(-0.216763\pi\)
0.776955 + 0.629556i \(0.216763\pi\)
\(608\) 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 \(-0.671865\pi\)
−0.514076 + 0.857745i \(0.671865\pi\)
\(614\) 39.4558 1.59231
\(615\) 0 0
\(616\) −8.82843 −0.355707
\(617\) −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
\(638\) 8.82843 0.349521
\(639\) 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
\(649\) 1.65685 0.0650372
\(650\) 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 \(-0.852686\pi\)
−0.894806 + 0.446455i \(0.852686\pi\)
\(660\) 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
\(671\) 9.31371 0.359552
\(672\) 0 0
\(673\) −12.4853 −0.481272 −0.240636 0.970615i \(-0.577356\pi\)
−0.240636 + 0.970615i \(0.577356\pi\)
\(674\) 49.4558 1.90497
\(675\) 0 0
\(676\) −44.5147 −1.71210
\(677\) −22.8284 −0.877368 −0.438684 0.898641i \(-0.644555\pi\)
−0.438684 + 0.898641i \(0.644555\pi\)
\(678\) 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 \(-0.423356\pi\)
0.238465 + 0.971151i \(0.423356\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 9.82843 0.370423
\(705\) 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
\(715\) 1.17157 0.0438143
\(716\) −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 \(-0.604481\pi\)
−0.322373 + 0.946613i \(0.604481\pi\)
\(734\) 20.4853 0.756126
\(735\) 0 0
\(736\) 4.48528 0.165330
\(737\) −12.4853 −0.459901
\(738\) 0 0
\(739\) 29.9411 1.10140 0.550701 0.834703i \(-0.314360\pi\)
0.550701 + 0.834703i \(0.314360\pi\)
\(740\) −29.3137 −1.07759
\(741\) 0 0
\(742\) −56.2843 −2.06626
\(743\) 49.5980 1.81957 0.909787 0.415076i \(-0.136245\pi\)
0.909787 + 0.415076i \(0.136245\pi\)
\(744\) 0 0
\(745\) −0.343146 −0.0125719
\(746\) −86.4264 −3.16430
\(747\) 0 0
\(748\) 26.1421 0.955851
\(749\) 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 \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 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 \(-0.611121\pi\)
−0.342048 + 0.939682i \(0.611121\pi\)
\(770\) −4.82843 −0.174004
\(771\) 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
\(781\) 11.3137 0.404836
\(782\) 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 \(-0.585973\pi\)
−0.266821 + 0.963746i \(0.585973\pi\)
\(788\) 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
\(803\) 1.17157 0.0413439
\(804\) 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 \(-0.438226\pi\)
0.192852 + 0.981228i \(0.438226\pi\)
\(810\) 0 0
\(811\) 53.9411 1.89413 0.947065 0.321043i \(-0.104033\pi\)
0.947065 + 0.321043i \(0.104033\pi\)
\(812\) −28.0000 −0.982607
\(813\) 0 0
\(814\) −18.4853 −0.647909
\(815\) 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 \(-0.243715\pi\)
0.720929 + 0.693009i \(0.243715\pi\)
\(822\) 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 \(-0.626644\pi\)
−0.387450 + 0.921891i \(0.626644\pi\)
\(828\) 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
\(847\) −2.00000 −0.0687208
\(848\) −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 \(-0.585572\pi\)
−0.265607 + 0.964081i \(0.585572\pi\)
\(854\) −44.9706 −1.53886
\(855\) 0 0
\(856\) 33.7990 1.15523
\(857\) −24.7696 −0.846112 −0.423056 0.906104i \(-0.639043\pi\)
−0.423056 + 0.906104i \(0.639043\pi\)
\(858\) 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
\(869\) −4.00000 −0.135691
\(870\) 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 \(-0.814534\pi\)
−0.835003 + 0.550246i \(0.814534\pi\)
\(878\) 38.6274 1.30361
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) 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 \(-0.722198\pi\)
−0.642730 + 0.766093i \(0.722198\pi\)
\(888\) 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
\(902\) −14.4853 −0.482307
\(903\) 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
\(913\) −6.00000 −0.198571
\(914\) −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 \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −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
\(935\) 6.82843 0.223313
\(936\) 0 0
\(937\) −1.45584 −0.0475604 −0.0237802 0.999717i \(-0.507570\pi\)
−0.0237802 + 0.999717i \(0.507570\pi\)
\(938\) 60.2843 1.96835
\(939\) 0 0
\(940\) −10.8284 −0.353184
\(941\) 6.68629 0.217967 0.108983 0.994044i \(-0.465240\pi\)
0.108983 + 0.994044i \(0.465240\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) −4.97056 −0.161778
\(945\) 0 0
\(946\) −14.4853 −0.470957
\(947\) −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 \(-0.830285\pi\)
−0.861198 + 0.508269i \(0.830285\pi\)
\(954\) 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 \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) −4.41421 −0.141878
\(969\) 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
\(979\) −13.3137 −0.425508
\(980\) −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 \(-0.727947\pi\)
−0.656460 + 0.754361i \(0.727947\pi\)
\(998\) −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 495.2.a.b.1.1 2
3.2 odd 2 55.2.a.b.1.2 2
4.3 odd 2 7920.2.a.ch.1.2 2
5.2 odd 4 2475.2.c.l.199.1 4
5.3 odd 4 2475.2.c.l.199.4 4
5.4 even 2 2475.2.a.x.1.2 2
11.10 odd 2 5445.2.a.y.1.2 2
12.11 even 2 880.2.a.m.1.2 2
15.2 even 4 275.2.b.d.199.4 4
15.8 even 4 275.2.b.d.199.1 4
15.14 odd 2 275.2.a.c.1.1 2
21.20 even 2 2695.2.a.f.1.2 2
24.5 odd 2 3520.2.a.bn.1.2 2
24.11 even 2 3520.2.a.bo.1.1 2
33.2 even 10 605.2.g.l.81.2 8
33.5 odd 10 605.2.g.f.366.1 8
33.8 even 10 605.2.g.l.251.1 8
33.14 odd 10 605.2.g.f.251.2 8
33.17 even 10 605.2.g.l.366.2 8
33.20 odd 10 605.2.g.f.81.1 8
33.26 odd 10 605.2.g.f.511.2 8
33.29 even 10 605.2.g.l.511.1 8
33.32 even 2 605.2.a.d.1.1 2
39.38 odd 2 9295.2.a.g.1.1 2
60.23 odd 4 4400.2.b.q.4049.3 4
60.47 odd 4 4400.2.b.q.4049.2 4
60.59 even 2 4400.2.a.bn.1.1 2
132.131 odd 2 9680.2.a.bn.1.2 2
165.164 even 2 3025.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 3.2 odd 2
275.2.a.c.1.1 2 15.14 odd 2
275.2.b.d.199.1 4 15.8 even 4
275.2.b.d.199.4 4 15.2 even 4
495.2.a.b.1.1 2 1.1 even 1 trivial
605.2.a.d.1.1 2 33.32 even 2
605.2.g.f.81.1 8 33.20 odd 10
605.2.g.f.251.2 8 33.14 odd 10
605.2.g.f.366.1 8 33.5 odd 10
605.2.g.f.511.2 8 33.26 odd 10
605.2.g.l.81.2 8 33.2 even 10
605.2.g.l.251.1 8 33.8 even 10
605.2.g.l.366.2 8 33.17 even 10
605.2.g.l.511.1 8 33.29 even 10
880.2.a.m.1.2 2 12.11 even 2
2475.2.a.x.1.2 2 5.4 even 2
2475.2.c.l.199.1 4 5.2 odd 4
2475.2.c.l.199.4 4 5.3 odd 4
2695.2.a.f.1.2 2 21.20 even 2
3025.2.a.o.1.2 2 165.164 even 2
3520.2.a.bn.1.2 2 24.5 odd 2
3520.2.a.bo.1.1 2 24.11 even 2
4400.2.a.bn.1.1 2 60.59 even 2
4400.2.b.q.4049.2 4 60.47 odd 4
4400.2.b.q.4049.3 4 60.23 odd 4
5445.2.a.y.1.2 2 11.10 odd 2
7920.2.a.ch.1.2 2 4.3 odd 2
9295.2.a.g.1.1 2 39.38 odd 2
9680.2.a.bn.1.2 2 132.131 odd 2