Properties

Label 275.2.a.e.1.1
Level $275$
Weight $2$
Character 275.1
Self dual yes
Analytic conductor $2.196$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.19588605559\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.30278 q^{2} +1.30278 q^{3} +3.30278 q^{4} -3.00000 q^{6} -4.30278 q^{7} -3.00000 q^{8} -1.30278 q^{9} +O(q^{10})\) \(q-2.30278 q^{2} +1.30278 q^{3} +3.30278 q^{4} -3.00000 q^{6} -4.30278 q^{7} -3.00000 q^{8} -1.30278 q^{9} -1.00000 q^{11} +4.30278 q^{12} -5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} +3.90833 q^{17} +3.00000 q^{18} -1.00000 q^{19} -5.60555 q^{21} +2.30278 q^{22} +3.69722 q^{23} -3.90833 q^{24} +11.5139 q^{26} -5.60555 q^{27} -14.2111 q^{28} -9.90833 q^{29} -4.21110 q^{31} +5.30278 q^{32} -1.30278 q^{33} -9.00000 q^{34} -4.30278 q^{36} -9.60555 q^{37} +2.30278 q^{38} -6.51388 q^{39} +1.60555 q^{41} +12.9083 q^{42} +7.21110 q^{43} -3.30278 q^{44} -8.51388 q^{46} +3.00000 q^{47} +0.394449 q^{48} +11.5139 q^{49} +5.09167 q^{51} -16.5139 q^{52} -2.30278 q^{53} +12.9083 q^{54} +12.9083 q^{56} -1.30278 q^{57} +22.8167 q^{58} +0.211103 q^{59} +2.90833 q^{61} +9.69722 q^{62} +5.60555 q^{63} -12.8167 q^{64} +3.00000 q^{66} +4.00000 q^{67} +12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} +3.90833 q^{72} -2.90833 q^{73} +22.1194 q^{74} -3.30278 q^{76} +4.30278 q^{77} +15.0000 q^{78} -0.0916731 q^{79} -3.39445 q^{81} -3.69722 q^{82} -14.5139 q^{83} -18.5139 q^{84} -16.6056 q^{86} -12.9083 q^{87} +3.00000 q^{88} +5.30278 q^{89} +21.5139 q^{91} +12.2111 q^{92} -5.48612 q^{93} -6.90833 q^{94} +6.90833 q^{96} -11.6972 q^{97} -26.5139 q^{98} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} + O(q^{10}) \) \( 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 2 q^{11} + 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} - 2 q^{19} - 4 q^{21} + q^{22} + 11 q^{23} + 3 q^{24} + 5 q^{26} - 4 q^{27} - 14 q^{28} - 9 q^{29} + 6 q^{31} + 7 q^{32} + q^{33} - 18 q^{34} - 5 q^{36} - 12 q^{37} + q^{38} + 5 q^{39} - 4 q^{41} + 15 q^{42} - 3 q^{44} + q^{46} + 6 q^{47} + 8 q^{48} + 5 q^{49} + 21 q^{51} - 15 q^{52} - q^{53} + 15 q^{54} + 15 q^{56} + q^{57} + 24 q^{58} - 14 q^{59} - 5 q^{61} + 23 q^{62} + 4 q^{63} - 4 q^{64} + 6 q^{66} + 8 q^{67} + 15 q^{68} - 12 q^{69} + 2 q^{71} - 3 q^{72} + 5 q^{73} + 19 q^{74} - 3 q^{76} + 5 q^{77} + 30 q^{78} - 11 q^{79} - 14 q^{81} - 11 q^{82} - 11 q^{83} - 19 q^{84} - 26 q^{86} - 15 q^{87} + 6 q^{88} + 7 q^{89} + 25 q^{91} + 10 q^{92} - 29 q^{93} - 3 q^{94} + 3 q^{96} - 27 q^{97} - 35 q^{98} - q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30278 −1.62831 −0.814154 0.580649i \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) 1.30278 0.752158 0.376079 0.926588i \(-0.377272\pi\)
0.376079 + 0.926588i \(0.377272\pi\)
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) −4.30278 −1.62630 −0.813148 0.582057i \(-0.802248\pi\)
−0.813148 + 0.582057i \(0.802248\pi\)
\(8\) −3.00000 −1.06066
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 4.30278 1.24210
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 9.90833 2.64811
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) 3.90833 0.947909 0.473954 0.880549i \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 3.00000 0.707107
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) 2.30278 0.490953
\(23\) 3.69722 0.770925 0.385462 0.922724i \(-0.374042\pi\)
0.385462 + 0.922724i \(0.374042\pi\)
\(24\) −3.90833 −0.797784
\(25\) 0 0
\(26\) 11.5139 2.25806
\(27\) −5.60555 −1.07879
\(28\) −14.2111 −2.68565
\(29\) −9.90833 −1.83993 −0.919965 0.392000i \(-0.871783\pi\)
−0.919965 + 0.392000i \(0.871783\pi\)
\(30\) 0 0
\(31\) −4.21110 −0.756336 −0.378168 0.925737i \(-0.623446\pi\)
−0.378168 + 0.925737i \(0.623446\pi\)
\(32\) 5.30278 0.937407
\(33\) −1.30278 −0.226784
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) −4.30278 −0.717129
\(37\) −9.60555 −1.57914 −0.789571 0.613659i \(-0.789697\pi\)
−0.789571 + 0.613659i \(0.789697\pi\)
\(38\) 2.30278 0.373560
\(39\) −6.51388 −1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) 12.9083 1.99180
\(43\) 7.21110 1.09968 0.549841 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) −3.30278 −0.497912
\(45\) 0 0
\(46\) −8.51388 −1.25530
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0.394449 0.0569338
\(49\) 11.5139 1.64484
\(50\) 0 0
\(51\) 5.09167 0.712977
\(52\) −16.5139 −2.29006
\(53\) −2.30278 −0.316311 −0.158155 0.987414i \(-0.550555\pi\)
−0.158155 + 0.987414i \(0.550555\pi\)
\(54\) 12.9083 1.75660
\(55\) 0 0
\(56\) 12.9083 1.72495
\(57\) −1.30278 −0.172557
\(58\) 22.8167 2.99597
\(59\) 0.211103 0.0274832 0.0137416 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) 9.69722 1.23155
\(63\) 5.60555 0.706233
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 12.9083 1.56536
\(69\) 4.81665 0.579857
\(70\) 0 0
\(71\) 4.60555 0.546578 0.273289 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(72\) 3.90833 0.460601
\(73\) −2.90833 −0.340394 −0.170197 0.985410i \(-0.554440\pi\)
−0.170197 + 0.985410i \(0.554440\pi\)
\(74\) 22.1194 2.57133
\(75\) 0 0
\(76\) −3.30278 −0.378854
\(77\) 4.30278 0.490347
\(78\) 15.0000 1.69842
\(79\) −0.0916731 −0.0103140 −0.00515701 0.999987i \(-0.501642\pi\)
−0.00515701 + 0.999987i \(0.501642\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) −3.69722 −0.408290
\(83\) −14.5139 −1.59311 −0.796553 0.604569i \(-0.793345\pi\)
−0.796553 + 0.604569i \(0.793345\pi\)
\(84\) −18.5139 −2.02003
\(85\) 0 0
\(86\) −16.6056 −1.79062
\(87\) −12.9083 −1.38392
\(88\) 3.00000 0.319801
\(89\) 5.30278 0.562093 0.281047 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) 12.2111 1.27310
\(93\) −5.48612 −0.568884
\(94\) −6.90833 −0.712540
\(95\) 0 0
\(96\) 6.90833 0.705078
\(97\) −11.6972 −1.18767 −0.593837 0.804586i \(-0.702387\pi\)
−0.593837 + 0.804586i \(0.702387\pi\)
\(98\) −26.5139 −2.67831
\(99\) 1.30278 0.130934
\(100\) 0 0
\(101\) 17.5139 1.74270 0.871348 0.490666i \(-0.163246\pi\)
0.871348 + 0.490666i \(0.163246\pi\)
\(102\) −11.7250 −1.16095
\(103\) 7.90833 0.779231 0.389615 0.920978i \(-0.372608\pi\)
0.389615 + 0.920978i \(0.372608\pi\)
\(104\) 15.0000 1.47087
\(105\) 0 0
\(106\) 5.30278 0.515051
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −18.5139 −1.78150
\(109\) −6.51388 −0.623916 −0.311958 0.950096i \(-0.600985\pi\)
−0.311958 + 0.950096i \(0.600985\pi\)
\(110\) 0 0
\(111\) −12.5139 −1.18776
\(112\) −1.30278 −0.123101
\(113\) −10.8167 −1.01755 −0.508773 0.860901i \(-0.669901\pi\)
−0.508773 + 0.860901i \(0.669901\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −32.7250 −3.03844
\(117\) 6.51388 0.602208
\(118\) −0.486122 −0.0447511
\(119\) −16.8167 −1.54158
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.69722 −0.606338
\(123\) 2.09167 0.188600
\(124\) −13.9083 −1.24900
\(125\) 0 0
\(126\) −12.9083 −1.14997
\(127\) 17.1194 1.51910 0.759552 0.650447i \(-0.225418\pi\)
0.759552 + 0.650447i \(0.225418\pi\)
\(128\) 18.9083 1.67128
\(129\) 9.39445 0.827135
\(130\) 0 0
\(131\) 0.908327 0.0793609 0.0396804 0.999212i \(-0.487366\pi\)
0.0396804 + 0.999212i \(0.487366\pi\)
\(132\) −4.30278 −0.374509
\(133\) 4.30278 0.373098
\(134\) −9.21110 −0.795718
\(135\) 0 0
\(136\) −11.7250 −1.00541
\(137\) 2.09167 0.178704 0.0893518 0.996000i \(-0.471520\pi\)
0.0893518 + 0.996000i \(0.471520\pi\)
\(138\) −11.0917 −0.944186
\(139\) 8.21110 0.696457 0.348228 0.937410i \(-0.386783\pi\)
0.348228 + 0.937410i \(0.386783\pi\)
\(140\) 0 0
\(141\) 3.90833 0.329141
\(142\) −10.6056 −0.889998
\(143\) 5.00000 0.418121
\(144\) −0.394449 −0.0328707
\(145\) 0 0
\(146\) 6.69722 0.554266
\(147\) 15.0000 1.23718
\(148\) −31.7250 −2.60778
\(149\) 2.78890 0.228475 0.114238 0.993453i \(-0.463557\pi\)
0.114238 + 0.993453i \(0.463557\pi\)
\(150\) 0 0
\(151\) −20.8167 −1.69404 −0.847018 0.531565i \(-0.821604\pi\)
−0.847018 + 0.531565i \(0.821604\pi\)
\(152\) 3.00000 0.243332
\(153\) −5.09167 −0.411637
\(154\) −9.90833 −0.798436
\(155\) 0 0
\(156\) −21.5139 −1.72249
\(157\) −4.78890 −0.382196 −0.191098 0.981571i \(-0.561205\pi\)
−0.191098 + 0.981571i \(0.561205\pi\)
\(158\) 0.211103 0.0167944
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −15.9083 −1.25375
\(162\) 7.81665 0.614134
\(163\) −5.69722 −0.446241 −0.223121 0.974791i \(-0.571624\pi\)
−0.223121 + 0.974791i \(0.571624\pi\)
\(164\) 5.30278 0.414077
\(165\) 0 0
\(166\) 33.4222 2.59407
\(167\) 15.4222 1.19341 0.596703 0.802462i \(-0.296477\pi\)
0.596703 + 0.802462i \(0.296477\pi\)
\(168\) 16.8167 1.29743
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 1.30278 0.0996257
\(172\) 23.8167 1.81600
\(173\) −16.8167 −1.27855 −0.639273 0.768980i \(-0.720765\pi\)
−0.639273 + 0.768980i \(0.720765\pi\)
\(174\) 29.7250 2.25344
\(175\) 0 0
\(176\) −0.302776 −0.0228226
\(177\) 0.275019 0.0206717
\(178\) −12.2111 −0.915261
\(179\) −5.51388 −0.412127 −0.206063 0.978539i \(-0.566065\pi\)
−0.206063 + 0.978539i \(0.566065\pi\)
\(180\) 0 0
\(181\) −9.09167 −0.675779 −0.337889 0.941186i \(-0.609713\pi\)
−0.337889 + 0.941186i \(0.609713\pi\)
\(182\) −49.5416 −3.67227
\(183\) 3.78890 0.280083
\(184\) −11.0917 −0.817689
\(185\) 0 0
\(186\) 12.6333 0.926319
\(187\) −3.90833 −0.285805
\(188\) 9.90833 0.722639
\(189\) 24.1194 1.75443
\(190\) 0 0
\(191\) 6.69722 0.484594 0.242297 0.970202i \(-0.422099\pi\)
0.242297 + 0.970202i \(0.422099\pi\)
\(192\) −16.6972 −1.20502
\(193\) 1.21110 0.0871771 0.0435885 0.999050i \(-0.486121\pi\)
0.0435885 + 0.999050i \(0.486121\pi\)
\(194\) 26.9361 1.93390
\(195\) 0 0
\(196\) 38.0278 2.71627
\(197\) −9.69722 −0.690899 −0.345449 0.938437i \(-0.612273\pi\)
−0.345449 + 0.938437i \(0.612273\pi\)
\(198\) −3.00000 −0.213201
\(199\) −24.5139 −1.73774 −0.868871 0.495038i \(-0.835154\pi\)
−0.868871 + 0.495038i \(0.835154\pi\)
\(200\) 0 0
\(201\) 5.21110 0.367563
\(202\) −40.3305 −2.83765
\(203\) 42.6333 2.99227
\(204\) 16.8167 1.17740
\(205\) 0 0
\(206\) −18.2111 −1.26883
\(207\) −4.81665 −0.334781
\(208\) −1.51388 −0.104969
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 25.2389 1.73751 0.868757 0.495238i \(-0.164919\pi\)
0.868757 + 0.495238i \(0.164919\pi\)
\(212\) −7.60555 −0.522351
\(213\) 6.00000 0.411113
\(214\) −6.90833 −0.472244
\(215\) 0 0
\(216\) 16.8167 1.14423
\(217\) 18.1194 1.23003
\(218\) 15.0000 1.01593
\(219\) −3.78890 −0.256030
\(220\) 0 0
\(221\) −19.5416 −1.31451
\(222\) 28.8167 1.93405
\(223\) −20.6333 −1.38171 −0.690854 0.722994i \(-0.742765\pi\)
−0.690854 + 0.722994i \(0.742765\pi\)
\(224\) −22.8167 −1.52450
\(225\) 0 0
\(226\) 24.9083 1.65688
\(227\) −5.30278 −0.351958 −0.175979 0.984394i \(-0.556309\pi\)
−0.175979 + 0.984394i \(0.556309\pi\)
\(228\) −4.30278 −0.284958
\(229\) 13.7250 0.906972 0.453486 0.891263i \(-0.350180\pi\)
0.453486 + 0.891263i \(0.350180\pi\)
\(230\) 0 0
\(231\) 5.60555 0.368818
\(232\) 29.7250 1.95154
\(233\) −5.09167 −0.333567 −0.166783 0.985994i \(-0.553338\pi\)
−0.166783 + 0.985994i \(0.553338\pi\)
\(234\) −15.0000 −0.980581
\(235\) 0 0
\(236\) 0.697224 0.0453854
\(237\) −0.119429 −0.00775778
\(238\) 38.7250 2.51017
\(239\) −4.11943 −0.266464 −0.133232 0.991085i \(-0.542535\pi\)
−0.133232 + 0.991085i \(0.542535\pi\)
\(240\) 0 0
\(241\) −24.9361 −1.60627 −0.803137 0.595794i \(-0.796837\pi\)
−0.803137 + 0.595794i \(0.796837\pi\)
\(242\) −2.30278 −0.148028
\(243\) 12.3944 0.795104
\(244\) 9.60555 0.614932
\(245\) 0 0
\(246\) −4.81665 −0.307099
\(247\) 5.00000 0.318142
\(248\) 12.6333 0.802216
\(249\) −18.9083 −1.19827
\(250\) 0 0
\(251\) −3.90833 −0.246691 −0.123346 0.992364i \(-0.539362\pi\)
−0.123346 + 0.992364i \(0.539362\pi\)
\(252\) 18.5139 1.16626
\(253\) −3.69722 −0.232443
\(254\) −39.4222 −2.47357
\(255\) 0 0
\(256\) −17.9083 −1.11927
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −21.6333 −1.34683
\(259\) 41.3305 2.56815
\(260\) 0 0
\(261\) 12.9083 0.799005
\(262\) −2.09167 −0.129224
\(263\) 1.18335 0.0729683 0.0364841 0.999334i \(-0.488384\pi\)
0.0364841 + 0.999334i \(0.488384\pi\)
\(264\) 3.90833 0.240541
\(265\) 0 0
\(266\) −9.90833 −0.607519
\(267\) 6.90833 0.422783
\(268\) 13.2111 0.806997
\(269\) −23.7250 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(270\) 0 0
\(271\) 14.2111 0.863263 0.431632 0.902050i \(-0.357938\pi\)
0.431632 + 0.902050i \(0.357938\pi\)
\(272\) 1.18335 0.0717509
\(273\) 28.0278 1.69632
\(274\) −4.81665 −0.290985
\(275\) 0 0
\(276\) 15.9083 0.957569
\(277\) −21.6056 −1.29815 −0.649076 0.760724i \(-0.724844\pi\)
−0.649076 + 0.760724i \(0.724844\pi\)
\(278\) −18.9083 −1.13405
\(279\) 5.48612 0.328446
\(280\) 0 0
\(281\) −22.8167 −1.36113 −0.680564 0.732689i \(-0.738265\pi\)
−0.680564 + 0.732689i \(0.738265\pi\)
\(282\) −9.00000 −0.535942
\(283\) −2.69722 −0.160333 −0.0801667 0.996781i \(-0.525545\pi\)
−0.0801667 + 0.996781i \(0.525545\pi\)
\(284\) 15.2111 0.902613
\(285\) 0 0
\(286\) −11.5139 −0.680830
\(287\) −6.90833 −0.407786
\(288\) −6.90833 −0.407077
\(289\) −1.72498 −0.101469
\(290\) 0 0
\(291\) −15.2389 −0.893318
\(292\) −9.60555 −0.562122
\(293\) −15.2111 −0.888642 −0.444321 0.895868i \(-0.646555\pi\)
−0.444321 + 0.895868i \(0.646555\pi\)
\(294\) −34.5416 −2.01451
\(295\) 0 0
\(296\) 28.8167 1.67493
\(297\) 5.60555 0.325267
\(298\) −6.42221 −0.372028
\(299\) −18.4861 −1.06908
\(300\) 0 0
\(301\) −31.0278 −1.78841
\(302\) 47.9361 2.75841
\(303\) 22.8167 1.31078
\(304\) −0.302776 −0.0173654
\(305\) 0 0
\(306\) 11.7250 0.670273
\(307\) 6.09167 0.347670 0.173835 0.984775i \(-0.444384\pi\)
0.173835 + 0.984775i \(0.444384\pi\)
\(308\) 14.2111 0.809753
\(309\) 10.3028 0.586104
\(310\) 0 0
\(311\) −16.8167 −0.953585 −0.476792 0.879016i \(-0.658201\pi\)
−0.476792 + 0.879016i \(0.658201\pi\)
\(312\) 19.5416 1.10633
\(313\) −21.8167 −1.23315 −0.616575 0.787296i \(-0.711480\pi\)
−0.616575 + 0.787296i \(0.711480\pi\)
\(314\) 11.0278 0.622332
\(315\) 0 0
\(316\) −0.302776 −0.0170325
\(317\) −9.90833 −0.556507 −0.278254 0.960508i \(-0.589756\pi\)
−0.278254 + 0.960508i \(0.589756\pi\)
\(318\) 6.90833 0.387400
\(319\) 9.90833 0.554760
\(320\) 0 0
\(321\) 3.90833 0.218142
\(322\) 36.6333 2.04149
\(323\) −3.90833 −0.217465
\(324\) −11.2111 −0.622839
\(325\) 0 0
\(326\) 13.1194 0.726618
\(327\) −8.48612 −0.469284
\(328\) −4.81665 −0.265955
\(329\) −12.9083 −0.711659
\(330\) 0 0
\(331\) −14.3944 −0.791190 −0.395595 0.918425i \(-0.629462\pi\)
−0.395595 + 0.918425i \(0.629462\pi\)
\(332\) −47.9361 −2.63083
\(333\) 12.5139 0.685756
\(334\) −35.5139 −1.94323
\(335\) 0 0
\(336\) −1.69722 −0.0925912
\(337\) −26.8444 −1.46231 −0.731154 0.682212i \(-0.761018\pi\)
−0.731154 + 0.682212i \(0.761018\pi\)
\(338\) −27.6333 −1.50305
\(339\) −14.0917 −0.765355
\(340\) 0 0
\(341\) 4.21110 0.228044
\(342\) −3.00000 −0.162221
\(343\) −19.4222 −1.04870
\(344\) −21.6333 −1.16639
\(345\) 0 0
\(346\) 38.7250 2.08187
\(347\) −5.51388 −0.296000 −0.148000 0.988987i \(-0.547284\pi\)
−0.148000 + 0.988987i \(0.547284\pi\)
\(348\) −42.6333 −2.28539
\(349\) −26.8167 −1.43546 −0.717731 0.696320i \(-0.754819\pi\)
−0.717731 + 0.696320i \(0.754819\pi\)
\(350\) 0 0
\(351\) 28.0278 1.49601
\(352\) −5.30278 −0.282639
\(353\) 24.6333 1.31110 0.655549 0.755152i \(-0.272437\pi\)
0.655549 + 0.755152i \(0.272437\pi\)
\(354\) −0.633308 −0.0336599
\(355\) 0 0
\(356\) 17.5139 0.928234
\(357\) −21.9083 −1.15951
\(358\) 12.6972 0.671069
\(359\) 15.2111 0.802811 0.401406 0.915900i \(-0.368522\pi\)
0.401406 + 0.915900i \(0.368522\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 20.9361 1.10038
\(363\) 1.30278 0.0683780
\(364\) 71.0555 3.72432
\(365\) 0 0
\(366\) −8.72498 −0.456062
\(367\) 24.3028 1.26859 0.634297 0.773089i \(-0.281290\pi\)
0.634297 + 0.773089i \(0.281290\pi\)
\(368\) 1.11943 0.0583543
\(369\) −2.09167 −0.108888
\(370\) 0 0
\(371\) 9.90833 0.514415
\(372\) −18.1194 −0.939449
\(373\) 1.42221 0.0736390 0.0368195 0.999322i \(-0.488277\pi\)
0.0368195 + 0.999322i \(0.488277\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 49.5416 2.55152
\(378\) −55.5416 −2.85675
\(379\) 24.8167 1.27475 0.637373 0.770555i \(-0.280021\pi\)
0.637373 + 0.770555i \(0.280021\pi\)
\(380\) 0 0
\(381\) 22.3028 1.14261
\(382\) −15.4222 −0.789069
\(383\) 21.6333 1.10541 0.552705 0.833377i \(-0.313596\pi\)
0.552705 + 0.833377i \(0.313596\pi\)
\(384\) 24.6333 1.25706
\(385\) 0 0
\(386\) −2.78890 −0.141951
\(387\) −9.39445 −0.477547
\(388\) −38.6333 −1.96131
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 14.4500 0.730766
\(392\) −34.5416 −1.74462
\(393\) 1.18335 0.0596919
\(394\) 22.3305 1.12500
\(395\) 0 0
\(396\) 4.30278 0.216223
\(397\) −25.3028 −1.26991 −0.634955 0.772549i \(-0.718981\pi\)
−0.634955 + 0.772549i \(0.718981\pi\)
\(398\) 56.4500 2.82958
\(399\) 5.60555 0.280629
\(400\) 0 0
\(401\) −27.2111 −1.35886 −0.679429 0.733741i \(-0.737772\pi\)
−0.679429 + 0.733741i \(0.737772\pi\)
\(402\) −12.0000 −0.598506
\(403\) 21.0555 1.04885
\(404\) 57.8444 2.87787
\(405\) 0 0
\(406\) −98.1749 −4.87234
\(407\) 9.60555 0.476129
\(408\) −15.2750 −0.756226
\(409\) 8.21110 0.406013 0.203006 0.979177i \(-0.434929\pi\)
0.203006 + 0.979177i \(0.434929\pi\)
\(410\) 0 0
\(411\) 2.72498 0.134413
\(412\) 26.1194 1.28681
\(413\) −0.908327 −0.0446958
\(414\) 11.0917 0.545126
\(415\) 0 0
\(416\) −26.5139 −1.29995
\(417\) 10.6972 0.523845
\(418\) −2.30278 −0.112632
\(419\) 13.6056 0.664675 0.332337 0.943161i \(-0.392163\pi\)
0.332337 + 0.943161i \(0.392163\pi\)
\(420\) 0 0
\(421\) 4.30278 0.209704 0.104852 0.994488i \(-0.466563\pi\)
0.104852 + 0.994488i \(0.466563\pi\)
\(422\) −58.1194 −2.82921
\(423\) −3.90833 −0.190029
\(424\) 6.90833 0.335498
\(425\) 0 0
\(426\) −13.8167 −0.669419
\(427\) −12.5139 −0.605589
\(428\) 9.90833 0.478937
\(429\) 6.51388 0.314493
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) −1.69722 −0.0816577
\(433\) −5.00000 −0.240285 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(434\) −41.7250 −2.00286
\(435\) 0 0
\(436\) −21.5139 −1.03033
\(437\) −3.69722 −0.176862
\(438\) 8.72498 0.416896
\(439\) 20.6972 0.987825 0.493912 0.869512i \(-0.335566\pi\)
0.493912 + 0.869512i \(0.335566\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 45.0000 2.14043
\(443\) 1.39445 0.0662523 0.0331261 0.999451i \(-0.489454\pi\)
0.0331261 + 0.999451i \(0.489454\pi\)
\(444\) −41.3305 −1.96146
\(445\) 0 0
\(446\) 47.5139 2.24985
\(447\) 3.63331 0.171850
\(448\) 55.1472 2.60546
\(449\) 41.5139 1.95916 0.979581 0.201052i \(-0.0644361\pi\)
0.979581 + 0.201052i \(0.0644361\pi\)
\(450\) 0 0
\(451\) −1.60555 −0.0756025
\(452\) −35.7250 −1.68036
\(453\) −27.1194 −1.27418
\(454\) 12.2111 0.573095
\(455\) 0 0
\(456\) 3.90833 0.183024
\(457\) 24.3028 1.13684 0.568418 0.822740i \(-0.307556\pi\)
0.568418 + 0.822740i \(0.307556\pi\)
\(458\) −31.6056 −1.47683
\(459\) −21.9083 −1.02259
\(460\) 0 0
\(461\) 17.7889 0.828512 0.414256 0.910161i \(-0.364042\pi\)
0.414256 + 0.910161i \(0.364042\pi\)
\(462\) −12.9083 −0.600550
\(463\) −26.2111 −1.21813 −0.609067 0.793119i \(-0.708456\pi\)
−0.609067 + 0.793119i \(0.708456\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 11.7250 0.543149
\(467\) −24.6333 −1.13989 −0.569947 0.821682i \(-0.693036\pi\)
−0.569947 + 0.821682i \(0.693036\pi\)
\(468\) 21.5139 0.994479
\(469\) −17.2111 −0.794735
\(470\) 0 0
\(471\) −6.23886 −0.287471
\(472\) −0.633308 −0.0291503
\(473\) −7.21110 −0.331567
\(474\) 0.275019 0.0126321
\(475\) 0 0
\(476\) −55.5416 −2.54575
\(477\) 3.00000 0.137361
\(478\) 9.48612 0.433885
\(479\) −13.1833 −0.602362 −0.301181 0.953567i \(-0.597381\pi\)
−0.301181 + 0.953567i \(0.597381\pi\)
\(480\) 0 0
\(481\) 48.0278 2.18988
\(482\) 57.4222 2.61551
\(483\) −20.7250 −0.943019
\(484\) 3.30278 0.150126
\(485\) 0 0
\(486\) −28.5416 −1.29467
\(487\) 10.2111 0.462709 0.231355 0.972869i \(-0.425684\pi\)
0.231355 + 0.972869i \(0.425684\pi\)
\(488\) −8.72498 −0.394961
\(489\) −7.42221 −0.335644
\(490\) 0 0
\(491\) −24.2111 −1.09263 −0.546316 0.837579i \(-0.683970\pi\)
−0.546316 + 0.837579i \(0.683970\pi\)
\(492\) 6.90833 0.311451
\(493\) −38.7250 −1.74409
\(494\) −11.5139 −0.518034
\(495\) 0 0
\(496\) −1.27502 −0.0572501
\(497\) −19.8167 −0.888898
\(498\) 43.5416 1.95115
\(499\) −21.5139 −0.963093 −0.481547 0.876420i \(-0.659925\pi\)
−0.481547 + 0.876420i \(0.659925\pi\)
\(500\) 0 0
\(501\) 20.0917 0.897630
\(502\) 9.00000 0.401690
\(503\) −16.6056 −0.740405 −0.370202 0.928951i \(-0.620712\pi\)
−0.370202 + 0.928951i \(0.620712\pi\)
\(504\) −16.8167 −0.749073
\(505\) 0 0
\(506\) 8.51388 0.378488
\(507\) 15.6333 0.694300
\(508\) 56.5416 2.50863
\(509\) −26.3028 −1.16585 −0.582925 0.812526i \(-0.698092\pi\)
−0.582925 + 0.812526i \(0.698092\pi\)
\(510\) 0 0
\(511\) 12.5139 0.553581
\(512\) 3.42221 0.151242
\(513\) 5.60555 0.247491
\(514\) −41.4500 −1.82828
\(515\) 0 0
\(516\) 31.0278 1.36592
\(517\) −3.00000 −0.131940
\(518\) −95.1749 −4.18175
\(519\) −21.9083 −0.961669
\(520\) 0 0
\(521\) 23.4500 1.02736 0.513681 0.857981i \(-0.328282\pi\)
0.513681 + 0.857981i \(0.328282\pi\)
\(522\) −29.7250 −1.30103
\(523\) 3.57779 0.156446 0.0782230 0.996936i \(-0.475075\pi\)
0.0782230 + 0.996936i \(0.475075\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) −2.72498 −0.118815
\(527\) −16.4584 −0.716938
\(528\) −0.394449 −0.0171662
\(529\) −9.33053 −0.405675
\(530\) 0 0
\(531\) −0.275019 −0.0119348
\(532\) 14.2111 0.616129
\(533\) −8.02776 −0.347721
\(534\) −15.9083 −0.688421
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −7.18335 −0.309984
\(538\) 54.6333 2.35541
\(539\) −11.5139 −0.495938
\(540\) 0 0
\(541\) −6.72498 −0.289130 −0.144565 0.989495i \(-0.546178\pi\)
−0.144565 + 0.989495i \(0.546178\pi\)
\(542\) −32.7250 −1.40566
\(543\) −11.8444 −0.508292
\(544\) 20.7250 0.888576
\(545\) 0 0
\(546\) −64.5416 −2.76213
\(547\) −18.1194 −0.774731 −0.387365 0.921926i \(-0.626615\pi\)
−0.387365 + 0.921926i \(0.626615\pi\)
\(548\) 6.90833 0.295109
\(549\) −3.78890 −0.161706
\(550\) 0 0
\(551\) 9.90833 0.422109
\(552\) −14.4500 −0.615031
\(553\) 0.394449 0.0167737
\(554\) 49.7527 2.11379
\(555\) 0 0
\(556\) 27.1194 1.15012
\(557\) 9.42221 0.399232 0.199616 0.979874i \(-0.436031\pi\)
0.199616 + 0.979874i \(0.436031\pi\)
\(558\) −12.6333 −0.534811
\(559\) −36.0555 −1.52499
\(560\) 0 0
\(561\) −5.09167 −0.214971
\(562\) 52.5416 2.21634
\(563\) −18.9083 −0.796891 −0.398445 0.917192i \(-0.630450\pi\)
−0.398445 + 0.917192i \(0.630450\pi\)
\(564\) 12.9083 0.543539
\(565\) 0 0
\(566\) 6.21110 0.261072
\(567\) 14.6056 0.613375
\(568\) −13.8167 −0.579734
\(569\) 15.1472 0.635003 0.317502 0.948258i \(-0.397156\pi\)
0.317502 + 0.948258i \(0.397156\pi\)
\(570\) 0 0
\(571\) −17.3305 −0.725260 −0.362630 0.931933i \(-0.618121\pi\)
−0.362630 + 0.931933i \(0.618121\pi\)
\(572\) 16.5139 0.690480
\(573\) 8.72498 0.364491
\(574\) 15.9083 0.664001
\(575\) 0 0
\(576\) 16.6972 0.695718
\(577\) 31.3583 1.30546 0.652731 0.757589i \(-0.273623\pi\)
0.652731 + 0.757589i \(0.273623\pi\)
\(578\) 3.97224 0.165224
\(579\) 1.57779 0.0655709
\(580\) 0 0
\(581\) 62.4500 2.59086
\(582\) 35.0917 1.45460
\(583\) 2.30278 0.0953712
\(584\) 8.72498 0.361042
\(585\) 0 0
\(586\) 35.0278 1.44698
\(587\) 37.5416 1.54951 0.774755 0.632262i \(-0.217873\pi\)
0.774755 + 0.632262i \(0.217873\pi\)
\(588\) 49.5416 2.04306
\(589\) 4.21110 0.173515
\(590\) 0 0
\(591\) −12.6333 −0.519665
\(592\) −2.90833 −0.119531
\(593\) 13.6056 0.558713 0.279357 0.960187i \(-0.409879\pi\)
0.279357 + 0.960187i \(0.409879\pi\)
\(594\) −12.9083 −0.529635
\(595\) 0 0
\(596\) 9.21110 0.377301
\(597\) −31.9361 −1.30706
\(598\) 42.5694 1.74079
\(599\) −14.0917 −0.575770 −0.287885 0.957665i \(-0.592952\pi\)
−0.287885 + 0.957665i \(0.592952\pi\)
\(600\) 0 0
\(601\) 8.90833 0.363378 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(602\) 71.4500 2.91208
\(603\) −5.21110 −0.212213
\(604\) −68.7527 −2.79751
\(605\) 0 0
\(606\) −52.5416 −2.13436
\(607\) 7.21110 0.292690 0.146345 0.989234i \(-0.453249\pi\)
0.146345 + 0.989234i \(0.453249\pi\)
\(608\) −5.30278 −0.215056
\(609\) 55.5416 2.25066
\(610\) 0 0
\(611\) −15.0000 −0.606835
\(612\) −16.8167 −0.679773
\(613\) 41.1194 1.66080 0.830399 0.557169i \(-0.188112\pi\)
0.830399 + 0.557169i \(0.188112\pi\)
\(614\) −14.0278 −0.566114
\(615\) 0 0
\(616\) −12.9083 −0.520091
\(617\) −10.6056 −0.426963 −0.213482 0.976947i \(-0.568480\pi\)
−0.213482 + 0.976947i \(0.568480\pi\)
\(618\) −23.7250 −0.954359
\(619\) 17.4222 0.700258 0.350129 0.936702i \(-0.386138\pi\)
0.350129 + 0.936702i \(0.386138\pi\)
\(620\) 0 0
\(621\) −20.7250 −0.831665
\(622\) 38.7250 1.55273
\(623\) −22.8167 −0.914130
\(624\) −1.97224 −0.0789529
\(625\) 0 0
\(626\) 50.2389 2.00795
\(627\) 1.30278 0.0520278
\(628\) −15.8167 −0.631153
\(629\) −37.5416 −1.49688
\(630\) 0 0
\(631\) −39.9361 −1.58983 −0.794915 0.606721i \(-0.792485\pi\)
−0.794915 + 0.606721i \(0.792485\pi\)
\(632\) 0.275019 0.0109397
\(633\) 32.8806 1.30689
\(634\) 22.8167 0.906165
\(635\) 0 0
\(636\) −9.90833 −0.392891
\(637\) −57.5694 −2.28098
\(638\) −22.8167 −0.903320
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 42.2111 1.66724 0.833619 0.552340i \(-0.186265\pi\)
0.833619 + 0.552340i \(0.186265\pi\)
\(642\) −9.00000 −0.355202
\(643\) 22.0000 0.867595 0.433798 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) −52.5416 −2.07043
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) −17.2389 −0.677729 −0.338865 0.940835i \(-0.610043\pi\)
−0.338865 + 0.940835i \(0.610043\pi\)
\(648\) 10.1833 0.400040
\(649\) −0.211103 −0.00828650
\(650\) 0 0
\(651\) 23.6056 0.925174
\(652\) −18.8167 −0.736917
\(653\) 19.1194 0.748201 0.374101 0.927388i \(-0.377951\pi\)
0.374101 + 0.927388i \(0.377951\pi\)
\(654\) 19.5416 0.764138
\(655\) 0 0
\(656\) 0.486122 0.0189799
\(657\) 3.78890 0.147819
\(658\) 29.7250 1.15880
\(659\) −20.0917 −0.782660 −0.391330 0.920250i \(-0.627985\pi\)
−0.391330 + 0.920250i \(0.627985\pi\)
\(660\) 0 0
\(661\) 12.8167 0.498510 0.249255 0.968438i \(-0.419814\pi\)
0.249255 + 0.968438i \(0.419814\pi\)
\(662\) 33.1472 1.28830
\(663\) −25.4584 −0.988721
\(664\) 43.5416 1.68974
\(665\) 0 0
\(666\) −28.8167 −1.11662
\(667\) −36.6333 −1.41845
\(668\) 50.9361 1.97078
\(669\) −26.8806 −1.03926
\(670\) 0 0
\(671\) −2.90833 −0.112275
\(672\) −29.7250 −1.14667
\(673\) 6.02776 0.232353 0.116176 0.993229i \(-0.462936\pi\)
0.116176 + 0.993229i \(0.462936\pi\)
\(674\) 61.8167 2.38109
\(675\) 0 0
\(676\) 39.6333 1.52436
\(677\) 26.2389 1.00844 0.504221 0.863575i \(-0.331780\pi\)
0.504221 + 0.863575i \(0.331780\pi\)
\(678\) 32.4500 1.24623
\(679\) 50.3305 1.93151
\(680\) 0 0
\(681\) −6.90833 −0.264728
\(682\) −9.69722 −0.371326
\(683\) 9.84441 0.376686 0.188343 0.982103i \(-0.439688\pi\)
0.188343 + 0.982103i \(0.439688\pi\)
\(684\) 4.30278 0.164521
\(685\) 0 0
\(686\) 44.7250 1.70761
\(687\) 17.8806 0.682186
\(688\) 2.18335 0.0832393
\(689\) 11.5139 0.438644
\(690\) 0 0
\(691\) −26.5416 −1.00969 −0.504846 0.863210i \(-0.668451\pi\)
−0.504846 + 0.863210i \(0.668451\pi\)
\(692\) −55.5416 −2.11138
\(693\) −5.60555 −0.212937
\(694\) 12.6972 0.481980
\(695\) 0 0
\(696\) 38.7250 1.46787
\(697\) 6.27502 0.237683
\(698\) 61.7527 2.33738
\(699\) −6.63331 −0.250895
\(700\) 0 0
\(701\) −26.7889 −1.01180 −0.505901 0.862591i \(-0.668840\pi\)
−0.505901 + 0.862591i \(0.668840\pi\)
\(702\) −64.5416 −2.43597
\(703\) 9.60555 0.362280
\(704\) 12.8167 0.483046
\(705\) 0 0
\(706\) −56.7250 −2.13487
\(707\) −75.3583 −2.83414
\(708\) 0.908327 0.0341370
\(709\) 11.6333 0.436898 0.218449 0.975848i \(-0.429900\pi\)
0.218449 + 0.975848i \(0.429900\pi\)
\(710\) 0 0
\(711\) 0.119429 0.00447895
\(712\) −15.9083 −0.596190
\(713\) −15.5694 −0.583078
\(714\) 50.4500 1.88804
\(715\) 0 0
\(716\) −18.2111 −0.680581
\(717\) −5.36669 −0.200423
\(718\) −35.0278 −1.30722
\(719\) 28.8167 1.07468 0.537340 0.843366i \(-0.319429\pi\)
0.537340 + 0.843366i \(0.319429\pi\)
\(720\) 0 0
\(721\) −34.0278 −1.26726
\(722\) 41.4500 1.54261
\(723\) −32.4861 −1.20817
\(724\) −30.0278 −1.11597
\(725\) 0 0
\(726\) −3.00000 −0.111340
\(727\) −0.330532 −0.0122588 −0.00612938 0.999981i \(-0.501951\pi\)
−0.00612938 + 0.999981i \(0.501951\pi\)
\(728\) −64.5416 −2.39207
\(729\) 26.3305 0.975205
\(730\) 0 0
\(731\) 28.1833 1.04240
\(732\) 12.5139 0.462526
\(733\) −12.3944 −0.457799 −0.228900 0.973450i \(-0.573513\pi\)
−0.228900 + 0.973450i \(0.573513\pi\)
\(734\) −55.9638 −2.06566
\(735\) 0 0
\(736\) 19.6056 0.722670
\(737\) −4.00000 −0.147342
\(738\) 4.81665 0.177303
\(739\) 9.88057 0.363463 0.181731 0.983348i \(-0.441830\pi\)
0.181731 + 0.983348i \(0.441830\pi\)
\(740\) 0 0
\(741\) 6.51388 0.239293
\(742\) −22.8167 −0.837626
\(743\) −44.3028 −1.62531 −0.812656 0.582744i \(-0.801979\pi\)
−0.812656 + 0.582744i \(0.801979\pi\)
\(744\) 16.4584 0.603393
\(745\) 0 0
\(746\) −3.27502 −0.119907
\(747\) 18.9083 0.691820
\(748\) −12.9083 −0.471975
\(749\) −12.9083 −0.471660
\(750\) 0 0
\(751\) −5.66947 −0.206882 −0.103441 0.994636i \(-0.532985\pi\)
−0.103441 + 0.994636i \(0.532985\pi\)
\(752\) 0.908327 0.0331233
\(753\) −5.09167 −0.185551
\(754\) −114.083 −4.15467
\(755\) 0 0
\(756\) 79.6611 2.89724
\(757\) 23.0555 0.837967 0.418983 0.907994i \(-0.362387\pi\)
0.418983 + 0.907994i \(0.362387\pi\)
\(758\) −57.1472 −2.07568
\(759\) −4.81665 −0.174833
\(760\) 0 0
\(761\) −42.4222 −1.53780 −0.768902 0.639367i \(-0.779197\pi\)
−0.768902 + 0.639367i \(0.779197\pi\)
\(762\) −51.3583 −1.86051
\(763\) 28.0278 1.01467
\(764\) 22.1194 0.800253
\(765\) 0 0
\(766\) −49.8167 −1.79995
\(767\) −1.05551 −0.0381124
\(768\) −23.3305 −0.841868
\(769\) −26.8167 −0.967033 −0.483517 0.875335i \(-0.660641\pi\)
−0.483517 + 0.875335i \(0.660641\pi\)
\(770\) 0 0
\(771\) 23.4500 0.844530
\(772\) 4.00000 0.143963
\(773\) 22.1194 0.795581 0.397790 0.917476i \(-0.369777\pi\)
0.397790 + 0.917476i \(0.369777\pi\)
\(774\) 21.6333 0.777593
\(775\) 0 0
\(776\) 35.0917 1.25972
\(777\) 53.8444 1.93166
\(778\) −27.6333 −0.990702
\(779\) −1.60555 −0.0575248
\(780\) 0 0
\(781\) −4.60555 −0.164800
\(782\) −33.2750 −1.18991
\(783\) 55.5416 1.98490
\(784\) 3.48612 0.124504
\(785\) 0 0
\(786\) −2.72498 −0.0971968
\(787\) 4.21110 0.150110 0.0750548 0.997179i \(-0.476087\pi\)
0.0750548 + 0.997179i \(0.476087\pi\)
\(788\) −32.0278 −1.14094
\(789\) 1.54163 0.0548836
\(790\) 0 0
\(791\) 46.5416 1.65483
\(792\) −3.90833 −0.138876
\(793\) −14.5416 −0.516389
\(794\) 58.2666 2.06780
\(795\) 0 0
\(796\) −80.9638 −2.86969
\(797\) 14.5139 0.514108 0.257054 0.966397i \(-0.417248\pi\)
0.257054 + 0.966397i \(0.417248\pi\)
\(798\) −12.9083 −0.456950
\(799\) 11.7250 0.414800
\(800\) 0 0
\(801\) −6.90833 −0.244094
\(802\) 62.6611 2.21264
\(803\) 2.90833 0.102633
\(804\) 17.2111 0.606989
\(805\) 0 0
\(806\) −48.4861 −1.70785
\(807\) −30.9083 −1.08802
\(808\) −52.5416 −1.84841
\(809\) −3.63331 −0.127740 −0.0638701 0.997958i \(-0.520344\pi\)
−0.0638701 + 0.997958i \(0.520344\pi\)
\(810\) 0 0
\(811\) −54.8722 −1.92682 −0.963411 0.268028i \(-0.913628\pi\)
−0.963411 + 0.268028i \(0.913628\pi\)
\(812\) 140.808 4.94140
\(813\) 18.5139 0.649310
\(814\) −22.1194 −0.775286
\(815\) 0 0
\(816\) 1.54163 0.0539680
\(817\) −7.21110 −0.252285
\(818\) −18.9083 −0.661114
\(819\) −28.0278 −0.979369
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) −6.27502 −0.218866
\(823\) 10.4222 0.363295 0.181648 0.983364i \(-0.441857\pi\)
0.181648 + 0.983364i \(0.441857\pi\)
\(824\) −23.7250 −0.826499
\(825\) 0 0
\(826\) 2.09167 0.0727786
\(827\) 7.81665 0.271812 0.135906 0.990722i \(-0.456606\pi\)
0.135906 + 0.990722i \(0.456606\pi\)
\(828\) −15.9083 −0.552853
\(829\) −38.7527 −1.34594 −0.672969 0.739671i \(-0.734981\pi\)
−0.672969 + 0.739671i \(0.734981\pi\)
\(830\) 0 0
\(831\) −28.1472 −0.976415
\(832\) 64.0833 2.22169
\(833\) 45.0000 1.55916
\(834\) −24.6333 −0.852982
\(835\) 0 0
\(836\) 3.30278 0.114229
\(837\) 23.6056 0.815927
\(838\) −31.3305 −1.08230
\(839\) 16.1194 0.556505 0.278252 0.960508i \(-0.410245\pi\)
0.278252 + 0.960508i \(0.410245\pi\)
\(840\) 0 0
\(841\) 69.1749 2.38534
\(842\) −9.90833 −0.341463
\(843\) −29.7250 −1.02378
\(844\) 83.3583 2.86931
\(845\) 0 0
\(846\) 9.00000 0.309426
\(847\) −4.30278 −0.147845
\(848\) −0.697224 −0.0239428
\(849\) −3.51388 −0.120596
\(850\) 0 0
\(851\) −35.5139 −1.21740
\(852\) 19.8167 0.678907
\(853\) −19.7250 −0.675370 −0.337685 0.941259i \(-0.609644\pi\)
−0.337685 + 0.941259i \(0.609644\pi\)
\(854\) 28.8167 0.986086
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) −15.0000 −0.512092
\(859\) 48.6056 1.65840 0.829200 0.558952i \(-0.188796\pi\)
0.829200 + 0.558952i \(0.188796\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) −75.9916 −2.58828
\(863\) 19.6056 0.667381 0.333690 0.942683i \(-0.391706\pi\)
0.333690 + 0.942683i \(0.391706\pi\)
\(864\) −29.7250 −1.01126
\(865\) 0 0
\(866\) 11.5139 0.391258
\(867\) −2.24726 −0.0763210
\(868\) 59.8444 2.03125
\(869\) 0.0916731 0.00310980
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 19.5416 0.661763
\(873\) 15.2389 0.515757
\(874\) 8.51388 0.287986
\(875\) 0 0
\(876\) −12.5139 −0.422805
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) −47.6611 −1.60848
\(879\) −19.8167 −0.668399
\(880\) 0 0
\(881\) −34.5416 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(882\) 34.5416 1.16308
\(883\) 12.4500 0.418975 0.209487 0.977811i \(-0.432821\pi\)
0.209487 + 0.977811i \(0.432821\pi\)
\(884\) −64.5416 −2.17077
\(885\) 0 0
\(886\) −3.21110 −0.107879
\(887\) −47.2389 −1.58613 −0.793063 0.609140i \(-0.791515\pi\)
−0.793063 + 0.609140i \(0.791515\pi\)
\(888\) 37.5416 1.25981
\(889\) −73.6611 −2.47051
\(890\) 0 0
\(891\) 3.39445 0.113718
\(892\) −68.1472 −2.28174
\(893\) −3.00000 −0.100391
\(894\) −8.36669 −0.279824
\(895\) 0 0
\(896\) −81.3583 −2.71799
\(897\) −24.0833 −0.804117
\(898\) −95.5971 −3.19012
\(899\) 41.7250 1.39161
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 3.69722 0.123104
\(903\) −40.4222 −1.34517
\(904\) 32.4500 1.07927
\(905\) 0 0
\(906\) 62.4500 2.07476
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) −17.5139 −0.581218
\(909\) −22.8167 −0.756781
\(910\) 0 0
\(911\) −39.2111 −1.29912 −0.649561 0.760310i \(-0.725047\pi\)
−0.649561 + 0.760310i \(0.725047\pi\)
\(912\) −0.394449 −0.0130615
\(913\) 14.5139 0.480339
\(914\) −55.9638 −1.85112
\(915\) 0 0
\(916\) 45.3305 1.49776
\(917\) −3.90833 −0.129064
\(918\) 50.4500 1.66510
\(919\) 41.2111 1.35943 0.679714 0.733477i \(-0.262104\pi\)
0.679714 + 0.733477i \(0.262104\pi\)
\(920\) 0 0
\(921\) 7.93608 0.261503
\(922\) −40.9638 −1.34907
\(923\) −23.0278 −0.757968
\(924\) 18.5139 0.609062
\(925\) 0 0
\(926\) 60.3583 1.98350
\(927\) −10.3028 −0.338388
\(928\) −52.5416 −1.72476
\(929\) 46.3944 1.52215 0.761076 0.648662i \(-0.224671\pi\)
0.761076 + 0.648662i \(0.224671\pi\)
\(930\) 0 0
\(931\) −11.5139 −0.377352
\(932\) −16.8167 −0.550848
\(933\) −21.9083 −0.717246
\(934\) 56.7250 1.85610
\(935\) 0 0
\(936\) −19.5416 −0.638738
\(937\) −5.21110 −0.170239 −0.0851196 0.996371i \(-0.527127\pi\)
−0.0851196 + 0.996371i \(0.527127\pi\)
\(938\) 39.6333 1.29407
\(939\) −28.4222 −0.927524
\(940\) 0 0
\(941\) 52.3944 1.70801 0.854005 0.520265i \(-0.174167\pi\)
0.854005 + 0.520265i \(0.174167\pi\)
\(942\) 14.3667 0.468092
\(943\) 5.93608 0.193305
\(944\) 0.0639167 0.00208031
\(945\) 0 0
\(946\) 16.6056 0.539893
\(947\) 36.6333 1.19042 0.595211 0.803569i \(-0.297068\pi\)
0.595211 + 0.803569i \(0.297068\pi\)
\(948\) −0.394449 −0.0128111
\(949\) 14.5416 0.472041
\(950\) 0 0
\(951\) −12.9083 −0.418581
\(952\) 50.4500 1.63509
\(953\) 49.2666 1.59590 0.797951 0.602722i \(-0.205917\pi\)
0.797951 + 0.602722i \(0.205917\pi\)
\(954\) −6.90833 −0.223665
\(955\) 0 0
\(956\) −13.6056 −0.440035
\(957\) 12.9083 0.417267
\(958\) 30.3583 0.980832
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −13.2666 −0.427955
\(962\) −110.597 −3.56580
\(963\) −3.90833 −0.125944
\(964\) −82.3583 −2.65258
\(965\) 0 0
\(966\) 47.7250 1.53553
\(967\) −4.09167 −0.131579 −0.0657897 0.997834i \(-0.520957\pi\)
−0.0657897 + 0.997834i \(0.520957\pi\)
\(968\) −3.00000 −0.0964237
\(969\) −5.09167 −0.163568
\(970\) 0 0
\(971\) −30.3583 −0.974244 −0.487122 0.873334i \(-0.661953\pi\)
−0.487122 + 0.873334i \(0.661953\pi\)
\(972\) 40.9361 1.31303
\(973\) −35.3305 −1.13264
\(974\) −23.5139 −0.753433
\(975\) 0 0
\(976\) 0.880571 0.0281864
\(977\) 15.9722 0.510997 0.255499 0.966809i \(-0.417760\pi\)
0.255499 + 0.966809i \(0.417760\pi\)
\(978\) 17.0917 0.546531
\(979\) −5.30278 −0.169477
\(980\) 0 0
\(981\) 8.48612 0.270941
\(982\) 55.7527 1.77914
\(983\) 48.8444 1.55789 0.778947 0.627089i \(-0.215754\pi\)
0.778947 + 0.627089i \(0.215754\pi\)
\(984\) −6.27502 −0.200040
\(985\) 0 0
\(986\) 89.1749 2.83991
\(987\) −16.8167 −0.535280
\(988\) 16.5139 0.525376
\(989\) 26.6611 0.847773
\(990\) 0 0
\(991\) −6.09167 −0.193508 −0.0967542 0.995308i \(-0.530846\pi\)
−0.0967542 + 0.995308i \(0.530846\pi\)
\(992\) −22.3305 −0.708995
\(993\) −18.7527 −0.595100
\(994\) 45.6333 1.44740
\(995\) 0 0
\(996\) −62.4500 −1.97880
\(997\) −14.2750 −0.452094 −0.226047 0.974116i \(-0.572580\pi\)
−0.226047 + 0.974116i \(0.572580\pi\)
\(998\) 49.5416 1.56821
\(999\) 53.8444 1.70356
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 275.2.a.e.1.1 2
3.2 odd 2 2475.2.a.t.1.2 2
4.3 odd 2 4400.2.a.bs.1.1 2
5.2 odd 4 275.2.b.c.199.1 4
5.3 odd 4 275.2.b.c.199.4 4
5.4 even 2 275.2.a.f.1.2 yes 2
11.10 odd 2 3025.2.a.n.1.2 2
15.2 even 4 2475.2.c.k.199.4 4
15.8 even 4 2475.2.c.k.199.1 4
15.14 odd 2 2475.2.a.o.1.1 2
20.3 even 4 4400.2.b.y.4049.2 4
20.7 even 4 4400.2.b.y.4049.3 4
20.19 odd 2 4400.2.a.bh.1.2 2
55.54 odd 2 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 1.1 even 1 trivial
275.2.a.f.1.2 yes 2 5.4 even 2
275.2.b.c.199.1 4 5.2 odd 4
275.2.b.c.199.4 4 5.3 odd 4
2475.2.a.o.1.1 2 15.14 odd 2
2475.2.a.t.1.2 2 3.2 odd 2
2475.2.c.k.199.1 4 15.8 even 4
2475.2.c.k.199.4 4 15.2 even 4
3025.2.a.h.1.1 2 55.54 odd 2
3025.2.a.n.1.2 2 11.10 odd 2
4400.2.a.bh.1.2 2 20.19 odd 2
4400.2.a.bs.1.1 2 4.3 odd 2
4400.2.b.y.4049.2 4 20.3 even 4
4400.2.b.y.4049.3 4 20.7 even 4