Properties

Label 55.2.a.b.1.2
Level $55$
Weight $2$
Character 55.1
Self dual yes
Analytic conductor $0.439$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.439177211117\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 55.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} -2.82843 q^{3} +3.82843 q^{4} -1.00000 q^{5} -6.82843 q^{6} -2.00000 q^{7} +4.41421 q^{8} +5.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} -2.82843 q^{3} +3.82843 q^{4} -1.00000 q^{5} -6.82843 q^{6} -2.00000 q^{7} +4.41421 q^{8} +5.00000 q^{9} -2.41421 q^{10} +1.00000 q^{11} -10.8284 q^{12} -1.17157 q^{13} -4.82843 q^{14} +2.82843 q^{15} +3.00000 q^{16} +6.82843 q^{17} +12.0711 q^{18} -3.82843 q^{20} +5.65685 q^{21} +2.41421 q^{22} -2.82843 q^{23} -12.4853 q^{24} +1.00000 q^{25} -2.82843 q^{26} -5.65685 q^{27} -7.65685 q^{28} -3.65685 q^{29} +6.82843 q^{30} -1.58579 q^{32} -2.82843 q^{33} +16.4853 q^{34} +2.00000 q^{35} +19.1421 q^{36} -7.65685 q^{37} +3.31371 q^{39} -4.41421 q^{40} +6.00000 q^{41} +13.6569 q^{42} -6.00000 q^{43} +3.82843 q^{44} -5.00000 q^{45} -6.82843 q^{46} +2.82843 q^{47} -8.48528 q^{48} -3.00000 q^{49} +2.41421 q^{50} -19.3137 q^{51} -4.48528 q^{52} +11.6569 q^{53} -13.6569 q^{54} -1.00000 q^{55} -8.82843 q^{56} -8.82843 q^{58} +1.65685 q^{59} +10.8284 q^{60} -9.31371 q^{61} -10.0000 q^{63} -9.82843 q^{64} +1.17157 q^{65} -6.82843 q^{66} +12.4853 q^{67} +26.1421 q^{68} +8.00000 q^{69} +4.82843 q^{70} +11.3137 q^{71} +22.0711 q^{72} -1.17157 q^{73} -18.4853 q^{74} -2.82843 q^{75} -2.00000 q^{77} +8.00000 q^{78} +4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +14.4853 q^{82} -6.00000 q^{83} +21.6569 q^{84} -6.82843 q^{85} -14.4853 q^{86} +10.3431 q^{87} +4.41421 q^{88} -13.3137 q^{89} -12.0711 q^{90} +2.34315 q^{91} -10.8284 q^{92} +6.82843 q^{94} +4.48528 q^{96} +3.65685 q^{97} -7.24264 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} - 16 q^{12} - 8 q^{13} - 4 q^{14} + 6 q^{16} + 8 q^{17} + 10 q^{18} - 2 q^{20} + 2 q^{22} - 8 q^{24} + 2 q^{25} - 4 q^{28} + 4 q^{29} + 8 q^{30} - 6 q^{32} + 16 q^{34} + 4 q^{35} + 10 q^{36} - 4 q^{37} - 16 q^{39} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 12 q^{43} + 2 q^{44} - 10 q^{45} - 8 q^{46} - 6 q^{49} + 2 q^{50} - 16 q^{51} + 8 q^{52} + 12 q^{53} - 16 q^{54} - 2 q^{55} - 12 q^{56} - 12 q^{58} - 8 q^{59} + 16 q^{60} + 4 q^{61} - 20 q^{63} - 14 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} + 24 q^{68} + 16 q^{69} + 4 q^{70} + 30 q^{72} - 8 q^{73} - 20 q^{74} - 4 q^{77} + 16 q^{78} + 8 q^{79} - 6 q^{80} + 2 q^{81} + 12 q^{82} - 12 q^{83} + 32 q^{84} - 8 q^{85} - 12 q^{86} + 32 q^{87} + 6 q^{88} - 4 q^{89} - 10 q^{90} + 16 q^{91} - 16 q^{92} + 8 q^{94} - 8 q^{96} - 4 q^{97} - 6 q^{98} + 10 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) −6.82843 −2.78769
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 4.41421 1.56066
\(9\) 5.00000 1.66667
\(10\) −2.41421 −0.763441
\(11\) 1.00000 0.301511
\(12\) −10.8284 −3.12590
\(13\) −1.17157 −0.324936 −0.162468 0.986714i \(-0.551945\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(14\) −4.82843 −1.29045
\(15\) 2.82843 0.730297
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 12.0711 2.84518
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.82843 −0.856062
\(21\) 5.65685 1.23443
\(22\) 2.41421 0.514712
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) −12.4853 −2.54855
\(25\) 1.00000 0.200000
\(26\) −2.82843 −0.554700
\(27\) −5.65685 −1.08866
\(28\) −7.65685 −1.44701
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 6.82843 1.24669
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.58579 −0.280330
\(33\) −2.82843 −0.492366
\(34\) 16.4853 2.82720
\(35\) 2.00000 0.338062
\(36\) 19.1421 3.19036
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 3.31371 0.530618
\(40\) −4.41421 −0.697948
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 13.6569 2.10730
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 3.82843 0.577157
\(45\) −5.00000 −0.745356
\(46\) −6.82843 −1.00680
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) −8.48528 −1.22474
\(49\) −3.00000 −0.428571
\(50\) 2.41421 0.341421
\(51\) −19.3137 −2.70446
\(52\) −4.48528 −0.621997
\(53\) 11.6569 1.60119 0.800596 0.599204i \(-0.204516\pi\)
0.800596 + 0.599204i \(0.204516\pi\)
\(54\) −13.6569 −1.85846
\(55\) −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 \(-0.465603\pi\)
0.107852 + 0.994167i \(0.465603\pi\)
\(60\) 10.8284 1.39794
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) −10.0000 −1.25988
\(64\) −9.82843 −1.22855
\(65\) 1.17157 0.145316
\(66\) −6.82843 −0.840521
\(67\) 12.4853 1.52532 0.762660 0.646800i \(-0.223893\pi\)
0.762660 + 0.646800i \(0.223893\pi\)
\(68\) 26.1421 3.17020
\(69\) 8.00000 0.963087
\(70\) 4.82843 0.577107
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 22.0711 2.60110
\(73\) −1.17157 −0.137122 −0.0685611 0.997647i \(-0.521841\pi\)
−0.0685611 + 0.997647i \(0.521841\pi\)
\(74\) −18.4853 −2.14887
\(75\) −2.82843 −0.326599
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 8.00000 0.905822
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 14.4853 1.59963
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 21.6569 2.36296
\(85\) −6.82843 −0.740647
\(86\) −14.4853 −1.56199
\(87\) 10.3431 1.10890
\(88\) 4.41421 0.470557
\(89\) −13.3137 −1.41125 −0.705625 0.708585i \(-0.749334\pi\)
−0.705625 + 0.708585i \(0.749334\pi\)
\(90\) −12.0711 −1.27240
\(91\) 2.34315 0.245628
\(92\) −10.8284 −1.12894
\(93\) 0 0
\(94\) 6.82843 0.704298
\(95\) 0 0
\(96\) 4.48528 0.457777
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) −7.24264 −0.731617
\(99\) 5.00000 0.502519
\(100\) 3.82843 0.382843
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) −46.6274 −4.61680
\(103\) 6.82843 0.672825 0.336412 0.941715i \(-0.390786\pi\)
0.336412 + 0.941715i \(0.390786\pi\)
\(104\) −5.17157 −0.507114
\(105\) −5.65685 −0.552052
\(106\) 28.1421 2.73341
\(107\) 7.65685 0.740216 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(108\) −21.6569 −2.08393
\(109\) −7.65685 −0.733394 −0.366697 0.930341i \(-0.619511\pi\)
−0.366697 + 0.930341i \(0.619511\pi\)
\(110\) −2.41421 −0.230186
\(111\) 21.6569 2.05558
\(112\) −6.00000 −0.566947
\(113\) 19.6569 1.84916 0.924581 0.380986i \(-0.124416\pi\)
0.924581 + 0.380986i \(0.124416\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) −14.0000 −1.29987
\(117\) −5.85786 −0.541560
\(118\) 4.00000 0.368230
\(119\) −13.6569 −1.25192
\(120\) 12.4853 1.13975
\(121\) 1.00000 0.0909091
\(122\) −22.4853 −2.03572
\(123\) −16.9706 −1.53018
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −24.1421 −2.15075
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) −20.5563 −1.81694
\(129\) 16.9706 1.49417
\(130\) 2.82843 0.248069
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) −10.8284 −0.942494
\(133\) 0 0
\(134\) 30.1421 2.60388
\(135\) 5.65685 0.486864
\(136\) 30.1421 2.58467
\(137\) −10.9706 −0.937278 −0.468639 0.883390i \(-0.655256\pi\)
−0.468639 + 0.883390i \(0.655256\pi\)
\(138\) 19.3137 1.64409
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 7.65685 0.647122
\(141\) −8.00000 −0.673722
\(142\) 27.3137 2.29212
\(143\) −1.17157 −0.0979718
\(144\) 15.0000 1.25000
\(145\) 3.65685 0.303685
\(146\) −2.82843 −0.234082
\(147\) 8.48528 0.699854
\(148\) −29.3137 −2.40957
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) −6.82843 −0.557539
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 34.1421 2.76023
\(154\) −4.82843 −0.389086
\(155\) 0 0
\(156\) 12.6863 1.01572
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 9.65685 0.768258
\(159\) −32.9706 −2.61474
\(160\) 1.58579 0.125367
\(161\) 5.65685 0.445823
\(162\) 2.41421 0.189679
\(163\) 16.4853 1.29123 0.645613 0.763664i \(-0.276602\pi\)
0.645613 + 0.763664i \(0.276602\pi\)
\(164\) 22.9706 1.79370
\(165\) 2.82843 0.220193
\(166\) −14.4853 −1.12428
\(167\) −22.9706 −1.77752 −0.888758 0.458377i \(-0.848431\pi\)
−0.888758 + 0.458377i \(0.848431\pi\)
\(168\) 24.9706 1.92652
\(169\) −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 \(0.818455\pi\)
\(174\) 24.9706 1.89301
\(175\) −2.00000 −0.151186
\(176\) 3.00000 0.226134
\(177\) −4.68629 −0.352243
\(178\) −32.1421 −2.40915
\(179\) 9.65685 0.721787 0.360894 0.932607i \(-0.382472\pi\)
0.360894 + 0.932607i \(0.382472\pi\)
\(180\) −19.1421 −1.42677
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 5.65685 0.419314
\(183\) 26.3431 1.94734
\(184\) −12.4853 −0.920427
\(185\) 7.65685 0.562943
\(186\) 0 0
\(187\) 6.82843 0.499344
\(188\) 10.8284 0.789744
\(189\) 11.3137 0.822951
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 27.7990 2.00622
\(193\) −1.17157 −0.0843317 −0.0421658 0.999111i \(-0.513426\pi\)
−0.0421658 + 0.999111i \(0.513426\pi\)
\(194\) 8.82843 0.633844
\(195\) −3.31371 −0.237300
\(196\) −11.4853 −0.820377
\(197\) −10.8284 −0.771493 −0.385747 0.922605i \(-0.626056\pi\)
−0.385747 + 0.922605i \(0.626056\pi\)
\(198\) 12.0711 0.857853
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 4.41421 0.312132
\(201\) −35.3137 −2.49084
\(202\) 22.4853 1.58206
\(203\) 7.31371 0.513322
\(204\) −73.9411 −5.17691
\(205\) −6.00000 −0.419058
\(206\) 16.4853 1.14858
\(207\) −14.1421 −0.982946
\(208\) −3.51472 −0.243702
\(209\) 0 0
\(210\) −13.6569 −0.942412
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 44.6274 3.06502
\(213\) −32.0000 −2.19260
\(214\) 18.4853 1.26363
\(215\) 6.00000 0.409197
\(216\) −24.9706 −1.69903
\(217\) 0 0
\(218\) −18.4853 −1.25198
\(219\) 3.31371 0.223920
\(220\) −3.82843 −0.258113
\(221\) −8.00000 −0.538138
\(222\) 52.2843 3.50909
\(223\) −10.8284 −0.725125 −0.362563 0.931959i \(-0.618098\pi\)
−0.362563 + 0.931959i \(0.618098\pi\)
\(224\) 3.17157 0.211910
\(225\) 5.00000 0.333333
\(226\) 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
\(231\) 5.65685 0.372194
\(232\) −16.1421 −1.05978
\(233\) −6.14214 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(234\) −14.1421 −0.924500
\(235\) −2.82843 −0.184506
\(236\) 6.34315 0.412904
\(237\) −11.3137 −0.734904
\(238\) −32.9706 −2.13716
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 8.48528 0.547723
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 2.41421 0.155192
\(243\) 14.1421 0.907218
\(244\) −35.6569 −2.28270
\(245\) 3.00000 0.191663
\(246\) −40.9706 −2.61219
\(247\) 0 0
\(248\) 0 0
\(249\) 16.9706 1.07547
\(250\) −2.41421 −0.152688
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −38.2843 −2.41168
\(253\) −2.82843 −0.177822
\(254\) 10.4853 0.657905
\(255\) 19.3137 1.20947
\(256\) −29.9706 −1.87316
\(257\) −9.31371 −0.580973 −0.290487 0.956879i \(-0.593817\pi\)
−0.290487 + 0.956879i \(0.593817\pi\)
\(258\) 40.9706 2.55072
\(259\) 15.3137 0.951548
\(260\) 4.48528 0.278165
\(261\) −18.2843 −1.13177
\(262\) −27.3137 −1.68745
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) −12.4853 −0.768416
\(265\) −11.6569 −0.716075
\(266\) 0 0
\(267\) 37.6569 2.30456
\(268\) 47.7990 2.91979
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 13.6569 0.831130
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) 20.4853 1.24210
\(273\) −6.62742 −0.401110
\(274\) −26.4853 −1.60003
\(275\) 1.00000 0.0603023
\(276\) 30.6274 1.84355
\(277\) 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 \(-0.327263\pi\)
0.516425 + 0.856333i \(0.327263\pi\)
\(282\) −19.3137 −1.15011
\(283\) 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
\(288\) −7.92893 −0.467217
\(289\) 29.6274 1.74279
\(290\) 8.82843 0.518423
\(291\) −10.3431 −0.606326
\(292\) −4.48528 −0.262481
\(293\) −9.17157 −0.535809 −0.267905 0.963445i \(-0.586331\pi\)
−0.267905 + 0.963445i \(0.586331\pi\)
\(294\) 20.4853 1.19473
\(295\) −1.65685 −0.0964658
\(296\) −33.7990 −1.96453
\(297\) −5.65685 −0.328244
\(298\) 0.828427 0.0479895
\(299\) 3.31371 0.191637
\(300\) −10.8284 −0.625180
\(301\) 12.0000 0.691669
\(302\) −28.9706 −1.66707
\(303\) −26.3431 −1.51337
\(304\) 0 0
\(305\) 9.31371 0.533301
\(306\) 82.4264 4.71200
\(307\) −16.3431 −0.932753 −0.466376 0.884586i \(-0.654441\pi\)
−0.466376 + 0.884586i \(0.654441\pi\)
\(308\) −7.65685 −0.436290
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) 4.68629 0.265735 0.132868 0.991134i \(-0.457581\pi\)
0.132868 + 0.991134i \(0.457581\pi\)
\(312\) 14.6274 0.828114
\(313\) −1.31371 −0.0742552 −0.0371276 0.999311i \(-0.511821\pi\)
−0.0371276 + 0.999311i \(0.511821\pi\)
\(314\) −33.7990 −1.90739
\(315\) 10.0000 0.563436
\(316\) 15.3137 0.861463
\(317\) −1.31371 −0.0737852 −0.0368926 0.999319i \(-0.511746\pi\)
−0.0368926 + 0.999319i \(0.511746\pi\)
\(318\) −79.5980 −4.46363
\(319\) −3.65685 −0.204745
\(320\) 9.82843 0.549426
\(321\) −21.6569 −1.20877
\(322\) 13.6569 0.761067
\(323\) 0 0
\(324\) 3.82843 0.212690
\(325\) −1.17157 −0.0649872
\(326\) 39.7990 2.20426
\(327\) 21.6569 1.19763
\(328\) 26.4853 1.46241
\(329\) −5.65685 −0.311872
\(330\) 6.82843 0.375893
\(331\) −7.31371 −0.401998 −0.200999 0.979591i \(-0.564419\pi\)
−0.200999 + 0.979591i \(0.564419\pi\)
\(332\) −22.9706 −1.26067
\(333\) −38.2843 −2.09797
\(334\) −55.4558 −3.03441
\(335\) −12.4853 −0.682144
\(336\) 16.9706 0.925820
\(337\) −20.4853 −1.11590 −0.557952 0.829873i \(-0.688413\pi\)
−0.557952 + 0.829873i \(0.688413\pi\)
\(338\) −28.0711 −1.52686
\(339\) −55.5980 −3.01967
\(340\) −26.1421 −1.41776
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −26.4853 −1.42799
\(345\) −8.00000 −0.430706
\(346\) −53.4558 −2.87380
\(347\) 10.9706 0.588931 0.294465 0.955662i \(-0.404858\pi\)
0.294465 + 0.955662i \(0.404858\pi\)
\(348\) 39.5980 2.12267
\(349\) 26.9706 1.44370 0.721851 0.692049i \(-0.243292\pi\)
0.721851 + 0.692049i \(0.243292\pi\)
\(350\) −4.82843 −0.258090
\(351\) 6.62742 0.353745
\(352\) −1.58579 −0.0845227
\(353\) 21.3137 1.13441 0.567207 0.823575i \(-0.308024\pi\)
0.567207 + 0.823575i \(0.308024\pi\)
\(354\) −11.3137 −0.601317
\(355\) −11.3137 −0.600469
\(356\) −50.9706 −2.70143
\(357\) 38.6274 2.04438
\(358\) 23.3137 1.23217
\(359\) 0.686292 0.0362211 0.0181105 0.999836i \(-0.494235\pi\)
0.0181105 + 0.999836i \(0.494235\pi\)
\(360\) −22.0711 −1.16325
\(361\) −19.0000 −1.00000
\(362\) 51.4558 2.70446
\(363\) −2.82843 −0.148454
\(364\) 8.97056 0.470185
\(365\) 1.17157 0.0613229
\(366\) 63.5980 3.32432
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) −8.48528 −0.442326
\(369\) 30.0000 1.56174
\(370\) 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
\(375\) 2.82843 0.146059
\(376\) 12.4853 0.643879
\(377\) 4.28427 0.220651
\(378\) 27.3137 1.40487
\(379\) 33.6569 1.72884 0.864418 0.502773i \(-0.167687\pi\)
0.864418 + 0.502773i \(0.167687\pi\)
\(380\) 0 0
\(381\) −12.2843 −0.629342
\(382\) 8.00000 0.409316
\(383\) −5.85786 −0.299323 −0.149661 0.988737i \(-0.547818\pi\)
−0.149661 + 0.988737i \(0.547818\pi\)
\(384\) 58.1421 2.96705
\(385\) 2.00000 0.101929
\(386\) −2.82843 −0.143963
\(387\) −30.0000 −1.52499
\(388\) 14.0000 0.710742
\(389\) 20.6274 1.04585 0.522926 0.852378i \(-0.324840\pi\)
0.522926 + 0.852378i \(0.324840\pi\)
\(390\) −8.00000 −0.405096
\(391\) −19.3137 −0.976736
\(392\) −13.2426 −0.668854
\(393\) 32.0000 1.61419
\(394\) −26.1421 −1.31702
\(395\) −4.00000 −0.201262
\(396\) 19.1421 0.961929
\(397\) −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 \(-0.542357\pi\)
−0.132677 + 0.991159i \(0.542357\pi\)
\(402\) −85.2548 −4.25212
\(403\) 0 0
\(404\) 35.6569 1.77399
\(405\) −1.00000 −0.0496904
\(406\) 17.6569 0.876295
\(407\) −7.65685 −0.379536
\(408\) −85.2548 −4.22074
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) −14.4853 −0.715377
\(411\) 31.0294 1.53057
\(412\) 26.1421 1.28793
\(413\) −3.31371 −0.163057
\(414\) −34.1421 −1.67799
\(415\) 6.00000 0.294528
\(416\) 1.85786 0.0910893
\(417\) 11.3137 0.554035
\(418\) 0 0
\(419\) −25.6569 −1.25342 −0.626710 0.779253i \(-0.715599\pi\)
−0.626710 + 0.779253i \(0.715599\pi\)
\(420\) −21.6569 −1.05675
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −38.6274 −1.88035
\(423\) 14.1421 0.687614
\(424\) 51.4558 2.49892
\(425\) 6.82843 0.331227
\(426\) −77.2548 −3.74301
\(427\) 18.6274 0.901444
\(428\) 29.3137 1.41693
\(429\) 3.31371 0.159987
\(430\) 14.4853 0.698542
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) −16.9706 −0.816497
\(433\) −7.65685 −0.367965 −0.183982 0.982930i \(-0.558899\pi\)
−0.183982 + 0.982930i \(0.558899\pi\)
\(434\) 0 0
\(435\) −10.3431 −0.495916
\(436\) −29.3137 −1.40387
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) −4.41421 −0.210439
\(441\) −15.0000 −0.714286
\(442\) −19.3137 −0.918659
\(443\) −26.8284 −1.27466 −0.637329 0.770592i \(-0.719961\pi\)
−0.637329 + 0.770592i \(0.719961\pi\)
\(444\) 82.9117 3.93481
\(445\) 13.3137 0.631130
\(446\) −26.1421 −1.23787
\(447\) −0.970563 −0.0459060
\(448\) 19.6569 0.928699
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 12.0711 0.569036
\(451\) 6.00000 0.282529
\(452\) 75.2548 3.53969
\(453\) 33.9411 1.59469
\(454\) 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
\(459\) −38.6274 −1.80297
\(460\) 10.8284 0.504878
\(461\) 12.6274 0.588117 0.294059 0.955787i \(-0.404994\pi\)
0.294059 + 0.955787i \(0.404994\pi\)
\(462\) 13.6569 0.635374
\(463\) −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 \(-0.611473\pi\)
−0.343089 + 0.939303i \(0.611473\pi\)
\(468\) −22.4264 −1.03666
\(469\) −24.9706 −1.15303
\(470\) −6.82843 −0.314972
\(471\) 39.5980 1.82458
\(472\) 7.31371 0.336641
\(473\) −6.00000 −0.275880
\(474\) −27.3137 −1.25456
\(475\) 0 0
\(476\) −52.2843 −2.39645
\(477\) 58.2843 2.66865
\(478\) −56.2843 −2.57438
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) −4.48528 −0.204724
\(481\) 8.97056 0.409022
\(482\) 14.4853 0.659786
\(483\) −16.0000 −0.728025
\(484\) 3.82843 0.174019
\(485\) −3.65685 −0.166049
\(486\) 34.1421 1.54872
\(487\) −24.4853 −1.10953 −0.554767 0.832006i \(-0.687193\pi\)
−0.554767 + 0.832006i \(0.687193\pi\)
\(488\) −41.1127 −1.86108
\(489\) −46.6274 −2.10856
\(490\) 7.24264 0.327189
\(491\) −0.686292 −0.0309719 −0.0154860 0.999880i \(-0.504930\pi\)
−0.0154860 + 0.999880i \(0.504930\pi\)
\(492\) −64.9706 −2.92910
\(493\) −24.9706 −1.12462
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 0 0
\(497\) −22.6274 −1.01498
\(498\) 40.9706 1.83593
\(499\) 9.65685 0.432300 0.216150 0.976360i \(-0.430650\pi\)
0.216150 + 0.976360i \(0.430650\pi\)
\(500\) −3.82843 −0.171212
\(501\) 64.9706 2.90267
\(502\) 28.9706 1.29302
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) −44.1421 −1.96625
\(505\) −9.31371 −0.414455
\(506\) −6.82843 −0.303561
\(507\) 32.8873 1.46058
\(508\) 16.6274 0.737722
\(509\) −13.3137 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(510\) 46.6274 2.06470
\(511\) 2.34315 0.103655
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) −22.4853 −0.991783
\(515\) −6.82843 −0.300896
\(516\) 64.9706 2.86017
\(517\) 2.82843 0.124394
\(518\) 36.9706 1.62439
\(519\) 62.6274 2.74904
\(520\) 5.17157 0.226788
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) −44.1421 −1.93205
\(523\) −41.5980 −1.81895 −0.909476 0.415756i \(-0.863517\pi\)
−0.909476 + 0.415756i \(0.863517\pi\)
\(524\) −43.3137 −1.89217
\(525\) 5.65685 0.246885
\(526\) −26.4853 −1.15481
\(527\) 0 0
\(528\) −8.48528 −0.369274
\(529\) −15.0000 −0.652174
\(530\) −28.1421 −1.22242
\(531\) 8.28427 0.359507
\(532\) 0 0
\(533\) −7.02944 −0.304479
\(534\) 90.9117 3.93413
\(535\) −7.65685 −0.331035
\(536\) 55.1127 2.38051
\(537\) −27.3137 −1.17867
\(538\) 41.7990 1.80208
\(539\) −3.00000 −0.129219
\(540\) 21.6569 0.931963
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 17.6569 0.758427
\(543\) −60.2843 −2.58705
\(544\) −10.8284 −0.464265
\(545\) 7.65685 0.327984
\(546\) −16.0000 −0.684737
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) −42.0000 −1.79415
\(549\) −46.5685 −1.98750
\(550\) 2.41421 0.102942
\(551\) 0 0
\(552\) 35.3137 1.50305
\(553\) −8.00000 −0.340195
\(554\) 16.4853 0.700392
\(555\) −21.6569 −0.919282
\(556\) −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
\(561\) −19.3137 −0.815425
\(562\) 41.7990 1.76318
\(563\) 0.343146 0.0144619 0.00723093 0.999974i \(-0.497698\pi\)
0.00723093 + 0.999974i \(0.497698\pi\)
\(564\) −30.6274 −1.28965
\(565\) −19.6569 −0.826970
\(566\) 78.7696 3.31093
\(567\) −2.00000 −0.0839921
\(568\) 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 \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(572\) −4.48528 −0.187539
\(573\) −9.37258 −0.391545
\(574\) −28.9706 −1.20921
\(575\) −2.82843 −0.117954
\(576\) −49.1421 −2.04759
\(577\) −26.9706 −1.12280 −0.561400 0.827545i \(-0.689737\pi\)
−0.561400 + 0.827545i \(0.689737\pi\)
\(578\) 71.5269 2.97513
\(579\) 3.31371 0.137713
\(580\) 14.0000 0.581318
\(581\) 12.0000 0.497844
\(582\) −24.9706 −1.03506
\(583\) 11.6569 0.482778
\(584\) −5.17157 −0.214001
\(585\) 5.85786 0.242193
\(586\) −22.1421 −0.914683
\(587\) −2.14214 −0.0884154 −0.0442077 0.999022i \(-0.514076\pi\)
−0.0442077 + 0.999022i \(0.514076\pi\)
\(588\) 32.4853 1.33967
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 30.6274 1.25984
\(592\) −22.9706 −0.944084
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) −13.6569 −0.560348
\(595\) 13.6569 0.559876
\(596\) 1.31371 0.0538116
\(597\) −29.2548 −1.19732
\(598\) 8.00000 0.327144
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) −12.4853 −0.509709
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) 28.9706 1.18075
\(603\) 62.4264 2.54220
\(604\) −45.9411 −1.86932
\(605\) −1.00000 −0.0406558
\(606\) −63.5980 −2.58349
\(607\) 38.2843 1.55391 0.776955 0.629556i \(-0.216763\pi\)
0.776955 + 0.629556i \(0.216763\pi\)
\(608\) 0 0
\(609\) −20.6863 −0.838251
\(610\) 22.4853 0.910402
\(611\) −3.31371 −0.134058
\(612\) 130.711 5.28367
\(613\) −25.4558 −1.02815 −0.514076 0.857745i \(-0.671865\pi\)
−0.514076 + 0.857745i \(0.671865\pi\)
\(614\) −39.4558 −1.59231
\(615\) 16.9706 0.684319
\(616\) −8.82843 −0.355707
\(617\) 0.343146 0.0138145 0.00690726 0.999976i \(-0.497801\pi\)
0.00690726 + 0.999976i \(0.497801\pi\)
\(618\) −46.6274 −1.87563
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 11.3137 0.453638
\(623\) 26.6274 1.06680
\(624\) 9.94113 0.397964
\(625\) 1.00000 0.0400000
\(626\) −3.17157 −0.126762
\(627\) 0 0
\(628\) −53.5980 −2.13879
\(629\) −52.2843 −2.08471
\(630\) 24.1421 0.961846
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 17.6569 0.702352
\(633\) 45.2548 1.79872
\(634\) −3.17157 −0.125959
\(635\) −4.34315 −0.172352
\(636\) −126.225 −5.00516
\(637\) 3.51472 0.139258
\(638\) −8.82843 −0.349521
\(639\) 56.5685 2.23782
\(640\) 20.5563 0.812561
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −52.2843 −2.06350
\(643\) −1.45584 −0.0574129 −0.0287064 0.999588i \(-0.509139\pi\)
−0.0287064 + 0.999588i \(0.509139\pi\)
\(644\) 21.6569 0.853400
\(645\) −16.9706 −0.668215
\(646\) 0 0
\(647\) 27.1127 1.06591 0.532955 0.846144i \(-0.321081\pi\)
0.532955 + 0.846144i \(0.321081\pi\)
\(648\) 4.41421 0.173407
\(649\) 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 \(-0.426754\pi\)
0.228084 + 0.973641i \(0.426754\pi\)
\(654\) 52.2843 2.04448
\(655\) 11.3137 0.442063
\(656\) 18.0000 0.702782
\(657\) −5.85786 −0.228537
\(658\) −13.6569 −0.532400
\(659\) 45.9411 1.78961 0.894806 0.446455i \(-0.147314\pi\)
0.894806 + 0.446455i \(0.147314\pi\)
\(660\) 10.8284 0.421496
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) −17.6569 −0.686253
\(663\) 22.6274 0.878776
\(664\) −26.4853 −1.02783
\(665\) 0 0
\(666\) −92.4264 −3.58145
\(667\) 10.3431 0.400488
\(668\) −87.9411 −3.40254
\(669\) 30.6274 1.18412
\(670\) −30.1421 −1.16449
\(671\) −9.31371 −0.359552
\(672\) −8.97056 −0.346047
\(673\) −12.4853 −0.481272 −0.240636 0.970615i \(-0.577356\pi\)
−0.240636 + 0.970615i \(0.577356\pi\)
\(674\) −49.4558 −1.90497
\(675\) −5.65685 −0.217732
\(676\) −44.5147 −1.71210
\(677\) 22.8284 0.877368 0.438684 0.898641i \(-0.355445\pi\)
0.438684 + 0.898641i \(0.355445\pi\)
\(678\) −134.225 −5.15490
\(679\) −7.31371 −0.280674
\(680\) −30.1421 −1.15590
\(681\) −71.5980 −2.74364
\(682\) 0 0
\(683\) −7.79899 −0.298420 −0.149210 0.988806i \(-0.547673\pi\)
−0.149210 + 0.988806i \(0.547673\pi\)
\(684\) 0 0
\(685\) 10.9706 0.419164
\(686\) 48.2843 1.84350
\(687\) −3.71573 −0.141764
\(688\) −18.0000 −0.686244
\(689\) −13.6569 −0.520285
\(690\) −19.3137 −0.735260
\(691\) −39.3137 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(692\) −84.7696 −3.22245
\(693\) −10.0000 −0.379869
\(694\) 26.4853 1.00537
\(695\) 4.00000 0.151729
\(696\) 45.6569 1.73062
\(697\) 40.9706 1.55187
\(698\) 65.1127 2.46455
\(699\) 17.3726 0.657091
\(700\) −7.65685 −0.289402
\(701\) −12.6274 −0.476931 −0.238465 0.971151i \(-0.576644\pi\)
−0.238465 + 0.971151i \(0.576644\pi\)
\(702\) 16.0000 0.603881
\(703\) 0 0
\(704\) −9.82843 −0.370423
\(705\) 8.00000 0.301297
\(706\) 51.4558 1.93657
\(707\) −18.6274 −0.700556
\(708\) −17.9411 −0.674269
\(709\) 24.6274 0.924902 0.462451 0.886645i \(-0.346970\pi\)
0.462451 + 0.886645i \(0.346970\pi\)
\(710\) −27.3137 −1.02507
\(711\) 20.0000 0.750059
\(712\) −58.7696 −2.20248
\(713\) 0 0
\(714\) 93.2548 3.48997
\(715\) 1.17157 0.0438143
\(716\) 36.9706 1.38165
\(717\) 65.9411 2.46262
\(718\) 1.65685 0.0618333
\(719\) 18.3431 0.684084 0.342042 0.939685i \(-0.388882\pi\)
0.342042 + 0.939685i \(0.388882\pi\)
\(720\) −15.0000 −0.559017
\(721\) −13.6569 −0.508608
\(722\) −45.8701 −1.70711
\(723\) −16.9706 −0.631142
\(724\) 81.5980 3.03257
\(725\) −3.65685 −0.135812
\(726\) −6.82843 −0.253427
\(727\) −19.5147 −0.723761 −0.361880 0.932225i \(-0.617865\pi\)
−0.361880 + 0.932225i \(0.617865\pi\)
\(728\) 10.3431 0.383342
\(729\) −43.0000 −1.59259
\(730\) 2.82843 0.104685
\(731\) −40.9706 −1.51535
\(732\) 100.853 3.72763
\(733\) −17.4558 −0.644746 −0.322373 0.946613i \(-0.604481\pi\)
−0.322373 + 0.946613i \(0.604481\pi\)
\(734\) −20.4853 −0.756126
\(735\) −8.48528 −0.312984
\(736\) 4.48528 0.165330
\(737\) 12.4853 0.459901
\(738\) 72.4264 2.66605
\(739\) 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 \(-0.863755\pi\)
−0.909787 + 0.415076i \(0.863755\pi\)
\(744\) 0 0
\(745\) −0.343146 −0.0125719
\(746\) 86.4264 3.16430
\(747\) −30.0000 −1.09764
\(748\) 26.1421 0.955851
\(749\) −15.3137 −0.559551
\(750\) 6.82843 0.249339
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.48528 0.309426
\(753\) −33.9411 −1.23688
\(754\) 10.3431 0.376675
\(755\) 12.0000 0.436725
\(756\) 43.3137 1.57530
\(757\) 13.3137 0.483895 0.241947 0.970289i \(-0.422214\pi\)
0.241947 + 0.970289i \(0.422214\pi\)
\(758\) 81.2548 2.95131
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −29.6569 −1.07435
\(763\) 15.3137 0.554393
\(764\) 12.6863 0.458974
\(765\) −34.1421 −1.23441
\(766\) −14.1421 −0.510976
\(767\) −1.94113 −0.0700900
\(768\) 84.7696 3.05886
\(769\) −18.9706 −0.684096 −0.342048 0.939682i \(-0.611121\pi\)
−0.342048 + 0.939682i \(0.611121\pi\)
\(770\) 4.82843 0.174004
\(771\) 26.3431 0.948725
\(772\) −4.48528 −0.161429
\(773\) 26.2843 0.945380 0.472690 0.881229i \(-0.343283\pi\)
0.472690 + 0.881229i \(0.343283\pi\)
\(774\) −72.4264 −2.60331
\(775\) 0 0
\(776\) 16.1421 0.579469
\(777\) −43.3137 −1.55387
\(778\) 49.7990 1.78538
\(779\) 0 0
\(780\) −12.6863 −0.454242
\(781\) 11.3137 0.404836
\(782\) −46.6274 −1.66739
\(783\) 20.6863 0.739268
\(784\) −9.00000 −0.321429
\(785\) 14.0000 0.499681
\(786\) 77.2548 2.75559
\(787\) −14.9706 −0.533643 −0.266821 0.963746i \(-0.585973\pi\)
−0.266821 + 0.963746i \(0.585973\pi\)
\(788\) −41.4558 −1.47680
\(789\) 31.0294 1.10468
\(790\) −9.65685 −0.343575
\(791\) −39.3137 −1.39783
\(792\) 22.0711 0.784261
\(793\) 10.9117 0.387485
\(794\) −22.4853 −0.797973
\(795\) 32.9706 1.16935
\(796\) 39.5980 1.40351
\(797\) 32.6274 1.15572 0.577861 0.816135i \(-0.303887\pi\)
0.577861 + 0.816135i \(0.303887\pi\)
\(798\) 0 0
\(799\) 19.3137 0.683270
\(800\) −1.58579 −0.0560660
\(801\) −66.5685 −2.35208
\(802\) −12.8284 −0.452988
\(803\) −1.17157 −0.0413439
\(804\) −135.196 −4.76799
\(805\) −5.65685 −0.199378
\(806\) 0 0
\(807\) −48.9706 −1.72385
\(808\) 41.1127 1.44634
\(809\) −10.9706 −0.385704 −0.192852 0.981228i \(-0.561774\pi\)
−0.192852 + 0.981228i \(0.561774\pi\)
\(810\) −2.41421 −0.0848268
\(811\) 53.9411 1.89413 0.947065 0.321043i \(-0.104033\pi\)
0.947065 + 0.321043i \(0.104033\pi\)
\(812\) 28.0000 0.982607
\(813\) −20.6863 −0.725500
\(814\) −18.4853 −0.647909
\(815\) −16.4853 −0.577454
\(816\) −57.9411 −2.02835
\(817\) 0 0
\(818\) 2.48528 0.0868958
\(819\) 11.7157 0.409381
\(820\) −22.9706 −0.802167
\(821\) −41.3137 −1.44186 −0.720929 0.693009i \(-0.756285\pi\)
−0.720929 + 0.693009i \(0.756285\pi\)
\(822\) 74.9117 2.61285
\(823\) 19.5147 0.680240 0.340120 0.940382i \(-0.389532\pi\)
0.340120 + 0.940382i \(0.389532\pi\)
\(824\) 30.1421 1.05005
\(825\) −2.82843 −0.0984732
\(826\) −8.00000 −0.278356
\(827\) 22.2843 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(828\) −54.1421 −1.88157
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 14.4853 0.502791
\(831\) −19.3137 −0.669985
\(832\) 11.5147 0.399201
\(833\) −20.4853 −0.709773
\(834\) 27.3137 0.945796
\(835\) 22.9706 0.794929
\(836\) 0 0
\(837\) 0 0
\(838\) −61.9411 −2.13972
\(839\) 26.3431 0.909466 0.454733 0.890628i \(-0.349735\pi\)
0.454733 + 0.890628i \(0.349735\pi\)
\(840\) −24.9706 −0.861566
\(841\) −15.6274 −0.538876
\(842\) −14.4853 −0.499196
\(843\) −48.9706 −1.68664
\(844\) −61.2548 −2.10848
\(845\) 11.6274 0.399995
\(846\) 34.1421 1.17383
\(847\) −2.00000 −0.0687208
\(848\) 34.9706 1.20089
\(849\) −92.2843 −3.16719
\(850\) 16.4853 0.565440
\(851\) 21.6569 0.742387
\(852\) −122.510 −4.19711
\(853\) −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 \(-0.360957\pi\)
0.423056 + 0.906104i \(0.360957\pi\)
\(858\) 8.00000 0.273115
\(859\) 24.2843 0.828569 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(860\) 22.9706 0.783290
\(861\) 33.9411 1.15671
\(862\) −27.3137 −0.930309
\(863\) −9.17157 −0.312204 −0.156102 0.987741i \(-0.549893\pi\)
−0.156102 + 0.987741i \(0.549893\pi\)
\(864\) 8.97056 0.305185
\(865\) 22.1421 0.752855
\(866\) −18.4853 −0.628155
\(867\) −83.7990 −2.84596
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) −24.9706 −0.846581
\(871\) −14.6274 −0.495631
\(872\) −33.7990 −1.14458
\(873\) 18.2843 0.618829
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 12.6863 0.428630
\(877\) −49.4558 −1.67001 −0.835003 0.550246i \(-0.814534\pi\)
−0.835003 + 0.550246i \(0.814534\pi\)
\(878\) −38.6274 −1.30361
\(879\) 25.9411 0.874972
\(880\) −3.00000 −0.101130
\(881\) −7.37258 −0.248389 −0.124194 0.992258i \(-0.539635\pi\)
−0.124194 + 0.992258i \(0.539635\pi\)
\(882\) −36.2132 −1.21936
\(883\) 37.1716 1.25092 0.625462 0.780255i \(-0.284911\pi\)
0.625462 + 0.780255i \(0.284911\pi\)
\(884\) −30.6274 −1.03011
\(885\) 4.68629 0.157528
\(886\) −64.7696 −2.17598
\(887\) 38.2843 1.28546 0.642730 0.766093i \(-0.277802\pi\)
0.642730 + 0.766093i \(0.277802\pi\)
\(888\) 95.5980 3.20806
\(889\) −8.68629 −0.291329
\(890\) 32.1421 1.07741
\(891\) 1.00000 0.0335013
\(892\) −41.4558 −1.38804
\(893\) 0 0
\(894\) −2.34315 −0.0783665
\(895\) −9.65685 −0.322793
\(896\) 41.1127 1.37348
\(897\) −9.37258 −0.312941
\(898\) 69.1127 2.30632
\(899\) 0 0
\(900\) 19.1421 0.638071
\(901\) 79.5980 2.65179
\(902\) 14.4853 0.482307
\(903\) −33.9411 −1.12949
\(904\) 86.7696 2.88591
\(905\) −21.3137 −0.708492
\(906\) 81.9411 2.72231
\(907\) −27.5147 −0.913611 −0.456806 0.889567i \(-0.651007\pi\)
−0.456806 + 0.889567i \(0.651007\pi\)
\(908\) 96.9117 3.21613
\(909\) 46.5685 1.54458
\(910\) −5.65685 −0.187523
\(911\) −9.94113 −0.329364 −0.164682 0.986347i \(-0.552660\pi\)
−0.164682 + 0.986347i \(0.552660\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 1.17157 0.0387522
\(915\) −26.3431 −0.870878
\(916\) 5.02944 0.166177
\(917\) 22.6274 0.747223
\(918\) −93.2548 −3.07787
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 12.4853 0.411628
\(921\) 46.2254 1.52318
\(922\) 30.4853 1.00398
\(923\) −13.2548 −0.436288
\(924\) 21.6569 0.712458
\(925\) −7.65685 −0.251756
\(926\) −14.8284 −0.487292
\(927\) 34.1421 1.12137
\(928\) 5.79899 0.190361
\(929\) 5.31371 0.174337 0.0871686 0.996194i \(-0.472218\pi\)
0.0871686 + 0.996194i \(0.472218\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.5147 −0.770250
\(933\) −13.2548 −0.433944
\(934\) −35.7990 −1.17138
\(935\) −6.82843 −0.223313
\(936\) −25.8579 −0.845191
\(937\) −1.45584 −0.0475604 −0.0237802 0.999717i \(-0.507570\pi\)
−0.0237802 + 0.999717i \(0.507570\pi\)
\(938\) −60.2843 −1.96835
\(939\) 3.71573 0.121258
\(940\) −10.8284 −0.353184
\(941\) −6.68629 −0.217967 −0.108983 0.994044i \(-0.534760\pi\)
−0.108983 + 0.994044i \(0.534760\pi\)
\(942\) 95.5980 3.11475
\(943\) −16.9706 −0.552638
\(944\) 4.97056 0.161778
\(945\) −11.3137 −0.368035
\(946\) −14.4853 −0.470957
\(947\) 41.1716 1.33790 0.668948 0.743309i \(-0.266745\pi\)
0.668948 + 0.743309i \(0.266745\pi\)
\(948\) −43.3137 −1.40676
\(949\) 1.37258 0.0445559
\(950\) 0 0
\(951\) 3.71573 0.120491
\(952\) −60.2843 −1.95382
\(953\) 53.1716 1.72240 0.861198 0.508269i \(-0.169715\pi\)
0.861198 + 0.508269i \(0.169715\pi\)
\(954\) 140.711 4.55568
\(955\) −3.31371 −0.107229
\(956\) −89.2548 −2.88671
\(957\) 10.3431 0.334346
\(958\) −86.9117 −2.80799
\(959\) 21.9411 0.708516
\(960\) −27.7990 −0.897209
\(961\) −31.0000 −1.00000
\(962\) 21.6569 0.698245
\(963\) 38.2843 1.23369
\(964\) 22.9706 0.739832
\(965\) 1.17157 0.0377143
\(966\) −38.6274 −1.24282
\(967\) −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 \(-0.544510\pi\)
−0.139378 + 0.990239i \(0.544510\pi\)
\(972\) 54.1421 1.73661
\(973\) 8.00000 0.256468
\(974\) −59.1127 −1.89409
\(975\) 3.31371 0.106124
\(976\) −27.9411 −0.894374
\(977\) 32.3431 1.03475 0.517374 0.855759i \(-0.326909\pi\)
0.517374 + 0.855759i \(0.326909\pi\)
\(978\) −112.569 −3.59955
\(979\) −13.3137 −0.425508
\(980\) 11.4853 0.366884
\(981\) −38.2843 −1.22232
\(982\) −1.65685 −0.0528723
\(983\) −21.8579 −0.697158 −0.348579 0.937279i \(-0.613336\pi\)
−0.348579 + 0.937279i \(0.613336\pi\)
\(984\) −74.9117 −2.38810
\(985\) 10.8284 0.345022
\(986\) −60.2843 −1.91984
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) −12.0711 −0.383644
\(991\) −57.9411 −1.84056 −0.920280 0.391260i \(-0.872039\pi\)
−0.920280 + 0.391260i \(0.872039\pi\)
\(992\) 0 0
\(993\) 20.6863 0.656460
\(994\) −54.6274 −1.73268
\(995\) −10.3431 −0.327900
\(996\) 64.9706 2.05867
\(997\) −41.4558 −1.31292 −0.656460 0.754361i \(-0.727947\pi\)
−0.656460 + 0.754361i \(0.727947\pi\)
\(998\) 23.3137 0.737983
\(999\) 43.3137 1.37039
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 55.2.a.b.1.2 2
3.2 odd 2 495.2.a.b.1.1 2
4.3 odd 2 880.2.a.m.1.2 2
5.2 odd 4 275.2.b.d.199.4 4
5.3 odd 4 275.2.b.d.199.1 4
5.4 even 2 275.2.a.c.1.1 2
7.6 odd 2 2695.2.a.f.1.2 2
8.3 odd 2 3520.2.a.bo.1.1 2
8.5 even 2 3520.2.a.bn.1.2 2
11.2 odd 10 605.2.g.l.81.2 8
11.3 even 5 605.2.g.f.251.2 8
11.4 even 5 605.2.g.f.511.2 8
11.5 even 5 605.2.g.f.366.1 8
11.6 odd 10 605.2.g.l.366.2 8
11.7 odd 10 605.2.g.l.511.1 8
11.8 odd 10 605.2.g.l.251.1 8
11.9 even 5 605.2.g.f.81.1 8
11.10 odd 2 605.2.a.d.1.1 2
12.11 even 2 7920.2.a.ch.1.2 2
13.12 even 2 9295.2.a.g.1.1 2
15.2 even 4 2475.2.c.l.199.1 4
15.8 even 4 2475.2.c.l.199.4 4
15.14 odd 2 2475.2.a.x.1.2 2
20.3 even 4 4400.2.b.q.4049.3 4
20.7 even 4 4400.2.b.q.4049.2 4
20.19 odd 2 4400.2.a.bn.1.1 2
33.32 even 2 5445.2.a.y.1.2 2
44.43 even 2 9680.2.a.bn.1.2 2
55.54 odd 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 1.1 even 1 trivial
275.2.a.c.1.1 2 5.4 even 2
275.2.b.d.199.1 4 5.3 odd 4
275.2.b.d.199.4 4 5.2 odd 4
495.2.a.b.1.1 2 3.2 odd 2
605.2.a.d.1.1 2 11.10 odd 2
605.2.g.f.81.1 8 11.9 even 5
605.2.g.f.251.2 8 11.3 even 5
605.2.g.f.366.1 8 11.5 even 5
605.2.g.f.511.2 8 11.4 even 5
605.2.g.l.81.2 8 11.2 odd 10
605.2.g.l.251.1 8 11.8 odd 10
605.2.g.l.366.2 8 11.6 odd 10
605.2.g.l.511.1 8 11.7 odd 10
880.2.a.m.1.2 2 4.3 odd 2
2475.2.a.x.1.2 2 15.14 odd 2
2475.2.c.l.199.1 4 15.2 even 4
2475.2.c.l.199.4 4 15.8 even 4
2695.2.a.f.1.2 2 7.6 odd 2
3025.2.a.o.1.2 2 55.54 odd 2
3520.2.a.bn.1.2 2 8.5 even 2
3520.2.a.bo.1.1 2 8.3 odd 2
4400.2.a.bn.1.1 2 20.19 odd 2
4400.2.b.q.4049.2 4 20.7 even 4
4400.2.b.q.4049.3 4 20.3 even 4
5445.2.a.y.1.2 2 33.32 even 2
7920.2.a.ch.1.2 2 12.11 even 2
9295.2.a.g.1.1 2 13.12 even 2
9680.2.a.bn.1.2 2 44.43 even 2