Properties

Label 2850.2.a.bm.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.42864 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.42864 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.62222 q^{11} -1.00000 q^{12} +5.80642 q^{13} -4.42864 q^{14} +1.00000 q^{16} -3.80642 q^{17} +1.00000 q^{18} +1.00000 q^{19} +4.42864 q^{21} +2.62222 q^{22} -2.62222 q^{23} -1.00000 q^{24} +5.80642 q^{26} -1.00000 q^{27} -4.42864 q^{28} +3.37778 q^{29} -4.42864 q^{31} +1.00000 q^{32} -2.62222 q^{33} -3.80642 q^{34} +1.00000 q^{36} +5.80642 q^{37} +1.00000 q^{38} -5.80642 q^{39} +5.67307 q^{41} +4.42864 q^{42} -10.9906 q^{43} +2.62222 q^{44} -2.62222 q^{46} +2.62222 q^{47} -1.00000 q^{48} +12.6128 q^{49} +3.80642 q^{51} +5.80642 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.42864 q^{56} -1.00000 q^{57} +3.37778 q^{58} -1.05086 q^{59} +4.75557 q^{61} -4.42864 q^{62} -4.42864 q^{63} +1.00000 q^{64} -2.62222 q^{66} +15.6128 q^{67} -3.80642 q^{68} +2.62222 q^{69} +15.6128 q^{71} +1.00000 q^{72} +11.6128 q^{73} +5.80642 q^{74} +1.00000 q^{76} -11.6128 q^{77} -5.80642 q^{78} +4.42864 q^{79} +1.00000 q^{81} +5.67307 q^{82} +11.9081 q^{83} +4.42864 q^{84} -10.9906 q^{86} -3.37778 q^{87} +2.62222 q^{88} +12.4286 q^{89} -25.7146 q^{91} -2.62222 q^{92} +4.42864 q^{93} +2.62222 q^{94} -1.00000 q^{96} +7.37778 q^{97} +12.6128 q^{98} +2.62222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} + 3q^{8} + 3q^{9} + 8q^{11} - 3q^{12} + 4q^{13} + 3q^{16} + 2q^{17} + 3q^{18} + 3q^{19} + 8q^{22} - 8q^{23} - 3q^{24} + 4q^{26} - 3q^{27} + 10q^{29} + 3q^{32} - 8q^{33} + 2q^{34} + 3q^{36} + 4q^{37} + 3q^{38} - 4q^{39} + 4q^{41} - 6q^{43} + 8q^{44} - 8q^{46} + 8q^{47} - 3q^{48} + 11q^{49} - 2q^{51} + 4q^{52} - 18q^{53} - 3q^{54} - 3q^{57} + 10q^{58} + 10q^{59} + 14q^{61} + 3q^{64} - 8q^{66} + 20q^{67} + 2q^{68} + 8q^{69} + 20q^{71} + 3q^{72} + 8q^{73} + 4q^{74} + 3q^{76} - 8q^{77} - 4q^{78} + 3q^{81} + 4q^{82} - 4q^{83} - 6q^{86} - 10q^{87} + 8q^{88} + 24q^{89} - 24q^{91} - 8q^{92} + 8q^{94} - 3q^{96} + 22q^{97} + 11q^{98} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −4.42864 −1.67387 −0.836934 0.547304i \(-0.815654\pi\)
−0.836934 + 0.547304i \(0.815654\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.62222 0.790628 0.395314 0.918546i \(-0.370636\pi\)
0.395314 + 0.918546i \(0.370636\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.80642 1.61041 0.805206 0.592995i \(-0.202055\pi\)
0.805206 + 0.592995i \(0.202055\pi\)
\(14\) −4.42864 −1.18360
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.80642 −0.923193 −0.461597 0.887090i \(-0.652723\pi\)
−0.461597 + 0.887090i \(0.652723\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.42864 0.966408
\(22\) 2.62222 0.559058
\(23\) −2.62222 −0.546770 −0.273385 0.961905i \(-0.588143\pi\)
−0.273385 + 0.961905i \(0.588143\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 5.80642 1.13873
\(27\) −1.00000 −0.192450
\(28\) −4.42864 −0.836934
\(29\) 3.37778 0.627239 0.313619 0.949549i \(-0.398458\pi\)
0.313619 + 0.949549i \(0.398458\pi\)
\(30\) 0 0
\(31\) −4.42864 −0.795407 −0.397704 0.917514i \(-0.630193\pi\)
−0.397704 + 0.917514i \(0.630193\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.62222 −0.456469
\(34\) −3.80642 −0.652796
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.80642 0.954570 0.477285 0.878749i \(-0.341621\pi\)
0.477285 + 0.878749i \(0.341621\pi\)
\(38\) 1.00000 0.162221
\(39\) −5.80642 −0.929772
\(40\) 0 0
\(41\) 5.67307 0.885985 0.442992 0.896525i \(-0.353917\pi\)
0.442992 + 0.896525i \(0.353917\pi\)
\(42\) 4.42864 0.683354
\(43\) −10.9906 −1.67606 −0.838028 0.545627i \(-0.816291\pi\)
−0.838028 + 0.545627i \(0.816291\pi\)
\(44\) 2.62222 0.395314
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) 2.62222 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(48\) −1.00000 −0.144338
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 3.80642 0.533006
\(52\) 5.80642 0.805206
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.42864 −0.591802
\(57\) −1.00000 −0.132453
\(58\) 3.37778 0.443525
\(59\) −1.05086 −0.136810 −0.0684048 0.997658i \(-0.521791\pi\)
−0.0684048 + 0.997658i \(0.521791\pi\)
\(60\) 0 0
\(61\) 4.75557 0.608888 0.304444 0.952530i \(-0.401529\pi\)
0.304444 + 0.952530i \(0.401529\pi\)
\(62\) −4.42864 −0.562438
\(63\) −4.42864 −0.557956
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.62222 −0.322772
\(67\) 15.6128 1.90741 0.953706 0.300739i \(-0.0972333\pi\)
0.953706 + 0.300739i \(0.0972333\pi\)
\(68\) −3.80642 −0.461597
\(69\) 2.62222 0.315678
\(70\) 0 0
\(71\) 15.6128 1.85290 0.926452 0.376413i \(-0.122843\pi\)
0.926452 + 0.376413i \(0.122843\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.6128 1.35918 0.679591 0.733592i \(-0.262157\pi\)
0.679591 + 0.733592i \(0.262157\pi\)
\(74\) 5.80642 0.674983
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −11.6128 −1.32341
\(78\) −5.80642 −0.657448
\(79\) 4.42864 0.498261 0.249130 0.968470i \(-0.419855\pi\)
0.249130 + 0.968470i \(0.419855\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.67307 0.626486
\(83\) 11.9081 1.30709 0.653544 0.756889i \(-0.273282\pi\)
0.653544 + 0.756889i \(0.273282\pi\)
\(84\) 4.42864 0.483204
\(85\) 0 0
\(86\) −10.9906 −1.18515
\(87\) −3.37778 −0.362136
\(88\) 2.62222 0.279529
\(89\) 12.4286 1.31743 0.658717 0.752391i \(-0.271100\pi\)
0.658717 + 0.752391i \(0.271100\pi\)
\(90\) 0 0
\(91\) −25.7146 −2.69562
\(92\) −2.62222 −0.273385
\(93\) 4.42864 0.459229
\(94\) 2.62222 0.270461
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 7.37778 0.749101 0.374550 0.927207i \(-0.377797\pi\)
0.374550 + 0.927207i \(0.377797\pi\)
\(98\) 12.6128 1.27409
\(99\) 2.62222 0.263543
\(100\) 0 0
\(101\) 17.6731 1.75854 0.879268 0.476327i \(-0.158032\pi\)
0.879268 + 0.476327i \(0.158032\pi\)
\(102\) 3.80642 0.376892
\(103\) −1.18421 −0.116684 −0.0583418 0.998297i \(-0.518581\pi\)
−0.0583418 + 0.998297i \(0.518581\pi\)
\(104\) 5.80642 0.569367
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.85728 0.469571 0.234785 0.972047i \(-0.424561\pi\)
0.234785 + 0.972047i \(0.424561\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.04149 −0.578670 −0.289335 0.957228i \(-0.593434\pi\)
−0.289335 + 0.957228i \(0.593434\pi\)
\(110\) 0 0
\(111\) −5.80642 −0.551121
\(112\) −4.42864 −0.418467
\(113\) 9.34614 0.879211 0.439606 0.898191i \(-0.355118\pi\)
0.439606 + 0.898191i \(0.355118\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 3.37778 0.313619
\(117\) 5.80642 0.536804
\(118\) −1.05086 −0.0967391
\(119\) 16.8573 1.54530
\(120\) 0 0
\(121\) −4.12399 −0.374908
\(122\) 4.75557 0.430549
\(123\) −5.67307 −0.511524
\(124\) −4.42864 −0.397704
\(125\) 0 0
\(126\) −4.42864 −0.394535
\(127\) 6.42864 0.570450 0.285225 0.958461i \(-0.407932\pi\)
0.285225 + 0.958461i \(0.407932\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.9906 0.967671
\(130\) 0 0
\(131\) −15.0923 −1.31862 −0.659312 0.751869i \(-0.729152\pi\)
−0.659312 + 0.751869i \(0.729152\pi\)
\(132\) −2.62222 −0.228235
\(133\) −4.42864 −0.384012
\(134\) 15.6128 1.34874
\(135\) 0 0
\(136\) −3.80642 −0.326398
\(137\) −7.53972 −0.644162 −0.322081 0.946712i \(-0.604382\pi\)
−0.322081 + 0.946712i \(0.604382\pi\)
\(138\) 2.62222 0.223218
\(139\) 0.387152 0.0328378 0.0164189 0.999865i \(-0.494773\pi\)
0.0164189 + 0.999865i \(0.494773\pi\)
\(140\) 0 0
\(141\) −2.62222 −0.220830
\(142\) 15.6128 1.31020
\(143\) 15.2257 1.27324
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 11.6128 0.961086
\(147\) −12.6128 −1.04029
\(148\) 5.80642 0.477285
\(149\) −14.5303 −1.19037 −0.595186 0.803588i \(-0.702922\pi\)
−0.595186 + 0.803588i \(0.702922\pi\)
\(150\) 0 0
\(151\) 4.69535 0.382102 0.191051 0.981580i \(-0.438810\pi\)
0.191051 + 0.981580i \(0.438810\pi\)
\(152\) 1.00000 0.0811107
\(153\) −3.80642 −0.307731
\(154\) −11.6128 −0.935790
\(155\) 0 0
\(156\) −5.80642 −0.464886
\(157\) −21.5210 −1.71756 −0.858781 0.512343i \(-0.828778\pi\)
−0.858781 + 0.512343i \(0.828778\pi\)
\(158\) 4.42864 0.352324
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 11.6128 0.915221
\(162\) 1.00000 0.0785674
\(163\) −8.23506 −0.645020 −0.322510 0.946566i \(-0.604527\pi\)
−0.322510 + 0.946566i \(0.604527\pi\)
\(164\) 5.67307 0.442992
\(165\) 0 0
\(166\) 11.9081 0.924250
\(167\) 8.47013 0.655438 0.327719 0.944775i \(-0.393720\pi\)
0.327719 + 0.944775i \(0.393720\pi\)
\(168\) 4.42864 0.341677
\(169\) 20.7146 1.59343
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −10.9906 −0.838028
\(173\) −22.4701 −1.70837 −0.854186 0.519967i \(-0.825944\pi\)
−0.854186 + 0.519967i \(0.825944\pi\)
\(174\) −3.37778 −0.256069
\(175\) 0 0
\(176\) 2.62222 0.197657
\(177\) 1.05086 0.0789871
\(178\) 12.4286 0.931566
\(179\) −2.94914 −0.220429 −0.110215 0.993908i \(-0.535154\pi\)
−0.110215 + 0.993908i \(0.535154\pi\)
\(180\) 0 0
\(181\) 22.8988 1.70205 0.851026 0.525124i \(-0.175981\pi\)
0.851026 + 0.525124i \(0.175981\pi\)
\(182\) −25.7146 −1.90609
\(183\) −4.75557 −0.351542
\(184\) −2.62222 −0.193312
\(185\) 0 0
\(186\) 4.42864 0.324724
\(187\) −9.98126 −0.729902
\(188\) 2.62222 0.191245
\(189\) 4.42864 0.322136
\(190\) 0 0
\(191\) −9.05086 −0.654897 −0.327448 0.944869i \(-0.606189\pi\)
−0.327448 + 0.944869i \(0.606189\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.7239 1.34778 0.673889 0.738833i \(-0.264623\pi\)
0.673889 + 0.738833i \(0.264623\pi\)
\(194\) 7.37778 0.529694
\(195\) 0 0
\(196\) 12.6128 0.900918
\(197\) 0.888922 0.0633331 0.0316665 0.999498i \(-0.489919\pi\)
0.0316665 + 0.999498i \(0.489919\pi\)
\(198\) 2.62222 0.186353
\(199\) 21.7146 1.53930 0.769652 0.638464i \(-0.220430\pi\)
0.769652 + 0.638464i \(0.220430\pi\)
\(200\) 0 0
\(201\) −15.6128 −1.10125
\(202\) 17.6731 1.24347
\(203\) −14.9590 −1.04992
\(204\) 3.80642 0.266503
\(205\) 0 0
\(206\) −1.18421 −0.0825077
\(207\) −2.62222 −0.182257
\(208\) 5.80642 0.402603
\(209\) 2.62222 0.181382
\(210\) 0 0
\(211\) −3.61285 −0.248719 −0.124359 0.992237i \(-0.539688\pi\)
−0.124359 + 0.992237i \(0.539688\pi\)
\(212\) −6.00000 −0.412082
\(213\) −15.6128 −1.06977
\(214\) 4.85728 0.332037
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 19.6128 1.33141
\(218\) −6.04149 −0.409181
\(219\) −11.6128 −0.784724
\(220\) 0 0
\(221\) −22.1017 −1.48672
\(222\) −5.80642 −0.389702
\(223\) −21.3876 −1.43222 −0.716111 0.697987i \(-0.754079\pi\)
−0.716111 + 0.697987i \(0.754079\pi\)
\(224\) −4.42864 −0.295901
\(225\) 0 0
\(226\) 9.34614 0.621696
\(227\) 8.47013 0.562182 0.281091 0.959681i \(-0.409304\pi\)
0.281091 + 0.959681i \(0.409304\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −16.9590 −1.12068 −0.560341 0.828262i \(-0.689330\pi\)
−0.560341 + 0.828262i \(0.689330\pi\)
\(230\) 0 0
\(231\) 11.6128 0.764069
\(232\) 3.37778 0.221762
\(233\) −17.9081 −1.17320 −0.586600 0.809876i \(-0.699534\pi\)
−0.586600 + 0.809876i \(0.699534\pi\)
\(234\) 5.80642 0.379578
\(235\) 0 0
\(236\) −1.05086 −0.0684048
\(237\) −4.42864 −0.287671
\(238\) 16.8573 1.09270
\(239\) −28.2766 −1.82906 −0.914529 0.404520i \(-0.867438\pi\)
−0.914529 + 0.404520i \(0.867438\pi\)
\(240\) 0 0
\(241\) −11.5111 −0.741498 −0.370749 0.928733i \(-0.620899\pi\)
−0.370749 + 0.928733i \(0.620899\pi\)
\(242\) −4.12399 −0.265100
\(243\) −1.00000 −0.0641500
\(244\) 4.75557 0.304444
\(245\) 0 0
\(246\) −5.67307 −0.361702
\(247\) 5.80642 0.369454
\(248\) −4.42864 −0.281219
\(249\) −11.9081 −0.754647
\(250\) 0 0
\(251\) −7.74620 −0.488936 −0.244468 0.969657i \(-0.578613\pi\)
−0.244468 + 0.969657i \(0.578613\pi\)
\(252\) −4.42864 −0.278978
\(253\) −6.87601 −0.432291
\(254\) 6.42864 0.403369
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.8796 −0.865783 −0.432891 0.901446i \(-0.642507\pi\)
−0.432891 + 0.901446i \(0.642507\pi\)
\(258\) 10.9906 0.684247
\(259\) −25.7146 −1.59782
\(260\) 0 0
\(261\) 3.37778 0.209080
\(262\) −15.0923 −0.932408
\(263\) −18.6222 −1.14830 −0.574148 0.818752i \(-0.694666\pi\)
−0.574148 + 0.818752i \(0.694666\pi\)
\(264\) −2.62222 −0.161386
\(265\) 0 0
\(266\) −4.42864 −0.271537
\(267\) −12.4286 −0.760620
\(268\) 15.6128 0.953706
\(269\) 15.8479 0.966264 0.483132 0.875547i \(-0.339499\pi\)
0.483132 + 0.875547i \(0.339499\pi\)
\(270\) 0 0
\(271\) 1.51114 0.0917951 0.0458975 0.998946i \(-0.485385\pi\)
0.0458975 + 0.998946i \(0.485385\pi\)
\(272\) −3.80642 −0.230798
\(273\) 25.7146 1.55632
\(274\) −7.53972 −0.455491
\(275\) 0 0
\(276\) 2.62222 0.157839
\(277\) 1.05086 0.0631398 0.0315699 0.999502i \(-0.489949\pi\)
0.0315699 + 0.999502i \(0.489949\pi\)
\(278\) 0.387152 0.0232199
\(279\) −4.42864 −0.265136
\(280\) 0 0
\(281\) 3.45091 0.205864 0.102932 0.994688i \(-0.467178\pi\)
0.102932 + 0.994688i \(0.467178\pi\)
\(282\) −2.62222 −0.156151
\(283\) 15.5812 0.926206 0.463103 0.886304i \(-0.346736\pi\)
0.463103 + 0.886304i \(0.346736\pi\)
\(284\) 15.6128 0.926452
\(285\) 0 0
\(286\) 15.2257 0.900314
\(287\) −25.1240 −1.48302
\(288\) 1.00000 0.0589256
\(289\) −2.51114 −0.147714
\(290\) 0 0
\(291\) −7.37778 −0.432493
\(292\) 11.6128 0.679591
\(293\) 7.12399 0.416188 0.208094 0.978109i \(-0.433274\pi\)
0.208094 + 0.978109i \(0.433274\pi\)
\(294\) −12.6128 −0.735596
\(295\) 0 0
\(296\) 5.80642 0.337492
\(297\) −2.62222 −0.152156
\(298\) −14.5303 −0.841721
\(299\) −15.2257 −0.880525
\(300\) 0 0
\(301\) 48.6735 2.80550
\(302\) 4.69535 0.270187
\(303\) −17.6731 −1.01529
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −3.80642 −0.217599
\(307\) 1.12399 0.0641492 0.0320746 0.999485i \(-0.489789\pi\)
0.0320746 + 0.999485i \(0.489789\pi\)
\(308\) −11.6128 −0.661703
\(309\) 1.18421 0.0673673
\(310\) 0 0
\(311\) 14.9491 0.847688 0.423844 0.905735i \(-0.360680\pi\)
0.423844 + 0.905735i \(0.360680\pi\)
\(312\) −5.80642 −0.328724
\(313\) 3.14272 0.177637 0.0888185 0.996048i \(-0.471691\pi\)
0.0888185 + 0.996048i \(0.471691\pi\)
\(314\) −21.5210 −1.21450
\(315\) 0 0
\(316\) 4.42864 0.249130
\(317\) 10.5906 0.594826 0.297413 0.954749i \(-0.403876\pi\)
0.297413 + 0.954749i \(0.403876\pi\)
\(318\) 6.00000 0.336463
\(319\) 8.85728 0.495912
\(320\) 0 0
\(321\) −4.85728 −0.271107
\(322\) 11.6128 0.647159
\(323\) −3.80642 −0.211795
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.23506 −0.456098
\(327\) 6.04149 0.334095
\(328\) 5.67307 0.313243
\(329\) −11.6128 −0.640237
\(330\) 0 0
\(331\) −15.1427 −0.832319 −0.416160 0.909292i \(-0.636624\pi\)
−0.416160 + 0.909292i \(0.636624\pi\)
\(332\) 11.9081 0.653544
\(333\) 5.80642 0.318190
\(334\) 8.47013 0.463465
\(335\) 0 0
\(336\) 4.42864 0.241602
\(337\) 11.1111 0.605259 0.302629 0.953108i \(-0.402136\pi\)
0.302629 + 0.953108i \(0.402136\pi\)
\(338\) 20.7146 1.12672
\(339\) −9.34614 −0.507613
\(340\) 0 0
\(341\) −11.6128 −0.628871
\(342\) 1.00000 0.0540738
\(343\) −24.8573 −1.34217
\(344\) −10.9906 −0.592575
\(345\) 0 0
\(346\) −22.4701 −1.20800
\(347\) −14.1936 −0.761951 −0.380976 0.924585i \(-0.624412\pi\)
−0.380976 + 0.924585i \(0.624412\pi\)
\(348\) −3.37778 −0.181068
\(349\) 32.3684 1.73264 0.866321 0.499488i \(-0.166479\pi\)
0.866321 + 0.499488i \(0.166479\pi\)
\(350\) 0 0
\(351\) −5.80642 −0.309924
\(352\) 2.62222 0.139765
\(353\) −11.8064 −0.628393 −0.314196 0.949358i \(-0.601735\pi\)
−0.314196 + 0.949358i \(0.601735\pi\)
\(354\) 1.05086 0.0558523
\(355\) 0 0
\(356\) 12.4286 0.658717
\(357\) −16.8573 −0.892182
\(358\) −2.94914 −0.155867
\(359\) −10.8287 −0.571517 −0.285758 0.958302i \(-0.592245\pi\)
−0.285758 + 0.958302i \(0.592245\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.8988 1.20353
\(363\) 4.12399 0.216453
\(364\) −25.7146 −1.34781
\(365\) 0 0
\(366\) −4.75557 −0.248578
\(367\) −1.46965 −0.0767151 −0.0383576 0.999264i \(-0.512213\pi\)
−0.0383576 + 0.999264i \(0.512213\pi\)
\(368\) −2.62222 −0.136692
\(369\) 5.67307 0.295328
\(370\) 0 0
\(371\) 26.5718 1.37954
\(372\) 4.42864 0.229614
\(373\) 24.3783 1.26226 0.631129 0.775678i \(-0.282592\pi\)
0.631129 + 0.775678i \(0.282592\pi\)
\(374\) −9.98126 −0.516119
\(375\) 0 0
\(376\) 2.62222 0.135230
\(377\) 19.6128 1.01011
\(378\) 4.42864 0.227785
\(379\) 18.9590 0.973858 0.486929 0.873442i \(-0.338117\pi\)
0.486929 + 0.873442i \(0.338117\pi\)
\(380\) 0 0
\(381\) −6.42864 −0.329349
\(382\) −9.05086 −0.463082
\(383\) −26.1017 −1.33374 −0.666868 0.745176i \(-0.732365\pi\)
−0.666868 + 0.745176i \(0.732365\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.7239 0.953023
\(387\) −10.9906 −0.558685
\(388\) 7.37778 0.374550
\(389\) −3.57136 −0.181075 −0.0905376 0.995893i \(-0.528859\pi\)
−0.0905376 + 0.995893i \(0.528859\pi\)
\(390\) 0 0
\(391\) 9.98126 0.504774
\(392\) 12.6128 0.637045
\(393\) 15.0923 0.761308
\(394\) 0.888922 0.0447832
\(395\) 0 0
\(396\) 2.62222 0.131771
\(397\) 31.4193 1.57689 0.788444 0.615107i \(-0.210887\pi\)
0.788444 + 0.615107i \(0.210887\pi\)
\(398\) 21.7146 1.08845
\(399\) 4.42864 0.221709
\(400\) 0 0
\(401\) 12.0415 0.601323 0.300662 0.953731i \(-0.402793\pi\)
0.300662 + 0.953731i \(0.402793\pi\)
\(402\) −15.6128 −0.778698
\(403\) −25.7146 −1.28093
\(404\) 17.6731 0.879268
\(405\) 0 0
\(406\) −14.9590 −0.742402
\(407\) 15.2257 0.754710
\(408\) 3.80642 0.188446
\(409\) 26.4701 1.30886 0.654432 0.756121i \(-0.272908\pi\)
0.654432 + 0.756121i \(0.272908\pi\)
\(410\) 0 0
\(411\) 7.53972 0.371907
\(412\) −1.18421 −0.0583418
\(413\) 4.65386 0.229001
\(414\) −2.62222 −0.128875
\(415\) 0 0
\(416\) 5.80642 0.284683
\(417\) −0.387152 −0.0189589
\(418\) 2.62222 0.128257
\(419\) −25.9684 −1.26864 −0.634319 0.773072i \(-0.718719\pi\)
−0.634319 + 0.773072i \(0.718719\pi\)
\(420\) 0 0
\(421\) −29.2672 −1.42640 −0.713198 0.700963i \(-0.752754\pi\)
−0.713198 + 0.700963i \(0.752754\pi\)
\(422\) −3.61285 −0.175871
\(423\) 2.62222 0.127496
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −15.6128 −0.756445
\(427\) −21.0607 −1.01920
\(428\) 4.85728 0.234785
\(429\) −15.2257 −0.735104
\(430\) 0 0
\(431\) 14.8385 0.714747 0.357374 0.933961i \(-0.383672\pi\)
0.357374 + 0.933961i \(0.383672\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.27607 −0.253552 −0.126776 0.991931i \(-0.540463\pi\)
−0.126776 + 0.991931i \(0.540463\pi\)
\(434\) 19.6128 0.941447
\(435\) 0 0
\(436\) −6.04149 −0.289335
\(437\) −2.62222 −0.125438
\(438\) −11.6128 −0.554883
\(439\) 31.8578 1.52049 0.760244 0.649638i \(-0.225079\pi\)
0.760244 + 0.649638i \(0.225079\pi\)
\(440\) 0 0
\(441\) 12.6128 0.600612
\(442\) −22.1017 −1.05127
\(443\) −4.94914 −0.235141 −0.117570 0.993065i \(-0.537511\pi\)
−0.117570 + 0.993065i \(0.537511\pi\)
\(444\) −5.80642 −0.275561
\(445\) 0 0
\(446\) −21.3876 −1.01273
\(447\) 14.5303 0.687262
\(448\) −4.42864 −0.209234
\(449\) 9.75605 0.460416 0.230208 0.973141i \(-0.426059\pi\)
0.230208 + 0.973141i \(0.426059\pi\)
\(450\) 0 0
\(451\) 14.8760 0.700484
\(452\) 9.34614 0.439606
\(453\) −4.69535 −0.220607
\(454\) 8.47013 0.397523
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −16.9590 −0.792442
\(459\) 3.80642 0.177669
\(460\) 0 0
\(461\) −36.1245 −1.68248 −0.841242 0.540659i \(-0.818175\pi\)
−0.841242 + 0.540659i \(0.818175\pi\)
\(462\) 11.6128 0.540279
\(463\) 35.1209 1.63221 0.816104 0.577905i \(-0.196130\pi\)
0.816104 + 0.577905i \(0.196130\pi\)
\(464\) 3.37778 0.156810
\(465\) 0 0
\(466\) −17.9081 −0.829578
\(467\) −4.56199 −0.211104 −0.105552 0.994414i \(-0.533661\pi\)
−0.105552 + 0.994414i \(0.533661\pi\)
\(468\) 5.80642 0.268402
\(469\) −69.1437 −3.19276
\(470\) 0 0
\(471\) 21.5210 0.991634
\(472\) −1.05086 −0.0483695
\(473\) −28.8198 −1.32514
\(474\) −4.42864 −0.203414
\(475\) 0 0
\(476\) 16.8573 0.772652
\(477\) −6.00000 −0.274721
\(478\) −28.2766 −1.29334
\(479\) 37.4005 1.70887 0.854437 0.519555i \(-0.173902\pi\)
0.854437 + 0.519555i \(0.173902\pi\)
\(480\) 0 0
\(481\) 33.7146 1.53725
\(482\) −11.5111 −0.524318
\(483\) −11.6128 −0.528403
\(484\) −4.12399 −0.187454
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −20.9175 −0.947862 −0.473931 0.880562i \(-0.657165\pi\)
−0.473931 + 0.880562i \(0.657165\pi\)
\(488\) 4.75557 0.215274
\(489\) 8.23506 0.372402
\(490\) 0 0
\(491\) −15.8666 −0.716052 −0.358026 0.933712i \(-0.616550\pi\)
−0.358026 + 0.933712i \(0.616550\pi\)
\(492\) −5.67307 −0.255762
\(493\) −12.8573 −0.579063
\(494\) 5.80642 0.261243
\(495\) 0 0
\(496\) −4.42864 −0.198852
\(497\) −69.1437 −3.10152
\(498\) −11.9081 −0.533616
\(499\) 16.7368 0.749244 0.374622 0.927178i \(-0.377773\pi\)
0.374622 + 0.927178i \(0.377773\pi\)
\(500\) 0 0
\(501\) −8.47013 −0.378417
\(502\) −7.74620 −0.345730
\(503\) 21.9684 0.979521 0.489760 0.871857i \(-0.337084\pi\)
0.489760 + 0.871857i \(0.337084\pi\)
\(504\) −4.42864 −0.197267
\(505\) 0 0
\(506\) −6.87601 −0.305676
\(507\) −20.7146 −0.919966
\(508\) 6.42864 0.285225
\(509\) −15.2573 −0.676270 −0.338135 0.941098i \(-0.609796\pi\)
−0.338135 + 0.941098i \(0.609796\pi\)
\(510\) 0 0
\(511\) −51.4291 −2.27509
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −13.8796 −0.612201
\(515\) 0 0
\(516\) 10.9906 0.483836
\(517\) 6.87601 0.302407
\(518\) −25.7146 −1.12983
\(519\) 22.4701 0.986329
\(520\) 0 0
\(521\) −18.3269 −0.802917 −0.401459 0.915877i \(-0.631497\pi\)
−0.401459 + 0.915877i \(0.631497\pi\)
\(522\) 3.37778 0.147842
\(523\) 24.8573 1.08693 0.543466 0.839431i \(-0.317112\pi\)
0.543466 + 0.839431i \(0.317112\pi\)
\(524\) −15.0923 −0.659312
\(525\) 0 0
\(526\) −18.6222 −0.811967
\(527\) 16.8573 0.734315
\(528\) −2.62222 −0.114117
\(529\) −16.1240 −0.701043
\(530\) 0 0
\(531\) −1.05086 −0.0456032
\(532\) −4.42864 −0.192006
\(533\) 32.9403 1.42680
\(534\) −12.4286 −0.537840
\(535\) 0 0
\(536\) 15.6128 0.674372
\(537\) 2.94914 0.127265
\(538\) 15.8479 0.683252
\(539\) 33.0736 1.42458
\(540\) 0 0
\(541\) −1.34614 −0.0578751 −0.0289376 0.999581i \(-0.509212\pi\)
−0.0289376 + 0.999581i \(0.509212\pi\)
\(542\) 1.51114 0.0649089
\(543\) −22.8988 −0.982680
\(544\) −3.80642 −0.163199
\(545\) 0 0
\(546\) 25.7146 1.10048
\(547\) −8.59057 −0.367306 −0.183653 0.982991i \(-0.558792\pi\)
−0.183653 + 0.982991i \(0.558792\pi\)
\(548\) −7.53972 −0.322081
\(549\) 4.75557 0.202963
\(550\) 0 0
\(551\) 3.37778 0.143898
\(552\) 2.62222 0.111609
\(553\) −19.6128 −0.834023
\(554\) 1.05086 0.0446466
\(555\) 0 0
\(556\) 0.387152 0.0164189
\(557\) −5.86665 −0.248578 −0.124289 0.992246i \(-0.539665\pi\)
−0.124289 + 0.992246i \(0.539665\pi\)
\(558\) −4.42864 −0.187479
\(559\) −63.8163 −2.69914
\(560\) 0 0
\(561\) 9.98126 0.421409
\(562\) 3.45091 0.145568
\(563\) 31.4291 1.32458 0.662290 0.749248i \(-0.269585\pi\)
0.662290 + 0.749248i \(0.269585\pi\)
\(564\) −2.62222 −0.110415
\(565\) 0 0
\(566\) 15.5812 0.654927
\(567\) −4.42864 −0.185985
\(568\) 15.6128 0.655101
\(569\) −7.95851 −0.333638 −0.166819 0.985988i \(-0.553350\pi\)
−0.166819 + 0.985988i \(0.553350\pi\)
\(570\) 0 0
\(571\) −30.8385 −1.29055 −0.645276 0.763949i \(-0.723258\pi\)
−0.645276 + 0.763949i \(0.723258\pi\)
\(572\) 15.2257 0.636618
\(573\) 9.05086 0.378105
\(574\) −25.1240 −1.04865
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −2.51114 −0.104450
\(579\) −18.7239 −0.778140
\(580\) 0 0
\(581\) −52.7368 −2.18789
\(582\) −7.37778 −0.305819
\(583\) −15.7333 −0.651606
\(584\) 11.6128 0.480543
\(585\) 0 0
\(586\) 7.12399 0.294289
\(587\) 17.8064 0.734950 0.367475 0.930033i \(-0.380222\pi\)
0.367475 + 0.930033i \(0.380222\pi\)
\(588\) −12.6128 −0.520145
\(589\) −4.42864 −0.182479
\(590\) 0 0
\(591\) −0.888922 −0.0365654
\(592\) 5.80642 0.238643
\(593\) −13.3176 −0.546887 −0.273443 0.961888i \(-0.588163\pi\)
−0.273443 + 0.961888i \(0.588163\pi\)
\(594\) −2.62222 −0.107591
\(595\) 0 0
\(596\) −14.5303 −0.595186
\(597\) −21.7146 −0.888718
\(598\) −15.2257 −0.622625
\(599\) −43.8163 −1.79028 −0.895142 0.445781i \(-0.852926\pi\)
−0.895142 + 0.445781i \(0.852926\pi\)
\(600\) 0 0
\(601\) −9.73329 −0.397029 −0.198515 0.980098i \(-0.563612\pi\)
−0.198515 + 0.980098i \(0.563612\pi\)
\(602\) 48.6735 1.98379
\(603\) 15.6128 0.635804
\(604\) 4.69535 0.191051
\(605\) 0 0
\(606\) −17.6731 −0.717919
\(607\) −31.9398 −1.29640 −0.648198 0.761472i \(-0.724477\pi\)
−0.648198 + 0.761472i \(0.724477\pi\)
\(608\) 1.00000 0.0405554
\(609\) 14.9590 0.606169
\(610\) 0 0
\(611\) 15.2257 0.615966
\(612\) −3.80642 −0.153866
\(613\) 11.8064 0.476857 0.238428 0.971160i \(-0.423368\pi\)
0.238428 + 0.971160i \(0.423368\pi\)
\(614\) 1.12399 0.0453603
\(615\) 0 0
\(616\) −11.6128 −0.467895
\(617\) 26.5620 1.06935 0.534673 0.845059i \(-0.320435\pi\)
0.534673 + 0.845059i \(0.320435\pi\)
\(618\) 1.18421 0.0476358
\(619\) 7.34614 0.295266 0.147633 0.989042i \(-0.452834\pi\)
0.147633 + 0.989042i \(0.452834\pi\)
\(620\) 0 0
\(621\) 2.62222 0.105226
\(622\) 14.9491 0.599406
\(623\) −55.0420 −2.20521
\(624\) −5.80642 −0.232443
\(625\) 0 0
\(626\) 3.14272 0.125608
\(627\) −2.62222 −0.104721
\(628\) −21.5210 −0.858781
\(629\) −22.1017 −0.881253
\(630\) 0 0
\(631\) −30.7556 −1.22436 −0.612180 0.790718i \(-0.709707\pi\)
−0.612180 + 0.790718i \(0.709707\pi\)
\(632\) 4.42864 0.176162
\(633\) 3.61285 0.143598
\(634\) 10.5906 0.420605
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 73.2355 2.90170
\(638\) 8.85728 0.350663
\(639\) 15.6128 0.617635
\(640\) 0 0
\(641\) −39.3876 −1.55572 −0.777859 0.628439i \(-0.783694\pi\)
−0.777859 + 0.628439i \(0.783694\pi\)
\(642\) −4.85728 −0.191702
\(643\) 24.1146 0.950988 0.475494 0.879719i \(-0.342269\pi\)
0.475494 + 0.879719i \(0.342269\pi\)
\(644\) 11.6128 0.457610
\(645\) 0 0
\(646\) −3.80642 −0.149762
\(647\) −47.6829 −1.87461 −0.937304 0.348512i \(-0.886687\pi\)
−0.937304 + 0.348512i \(0.886687\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.75557 −0.108166
\(650\) 0 0
\(651\) −19.6128 −0.768688
\(652\) −8.23506 −0.322510
\(653\) 21.7462 0.850995 0.425497 0.904960i \(-0.360099\pi\)
0.425497 + 0.904960i \(0.360099\pi\)
\(654\) 6.04149 0.236241
\(655\) 0 0
\(656\) 5.67307 0.221496
\(657\) 11.6128 0.453060
\(658\) −11.6128 −0.452716
\(659\) −16.1936 −0.630812 −0.315406 0.948957i \(-0.602141\pi\)
−0.315406 + 0.948957i \(0.602141\pi\)
\(660\) 0 0
\(661\) −22.5116 −0.875600 −0.437800 0.899072i \(-0.644242\pi\)
−0.437800 + 0.899072i \(0.644242\pi\)
\(662\) −15.1427 −0.588539
\(663\) 22.1017 0.858359
\(664\) 11.9081 0.462125
\(665\) 0 0
\(666\) 5.80642 0.224994
\(667\) −8.85728 −0.342955
\(668\) 8.47013 0.327719
\(669\) 21.3876 0.826893
\(670\) 0 0
\(671\) 12.4701 0.481404
\(672\) 4.42864 0.170838
\(673\) 9.66323 0.372490 0.186245 0.982503i \(-0.440368\pi\)
0.186245 + 0.982503i \(0.440368\pi\)
\(674\) 11.1111 0.427983
\(675\) 0 0
\(676\) 20.7146 0.796714
\(677\) −6.85728 −0.263547 −0.131773 0.991280i \(-0.542067\pi\)
−0.131773 + 0.991280i \(0.542067\pi\)
\(678\) −9.34614 −0.358936
\(679\) −32.6735 −1.25390
\(680\) 0 0
\(681\) −8.47013 −0.324576
\(682\) −11.6128 −0.444679
\(683\) 30.3051 1.15959 0.579797 0.814761i \(-0.303132\pi\)
0.579797 + 0.814761i \(0.303132\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −24.8573 −0.949055
\(687\) 16.9590 0.647026
\(688\) −10.9906 −0.419014
\(689\) −34.8385 −1.32724
\(690\) 0 0
\(691\) 7.61285 0.289606 0.144803 0.989460i \(-0.453745\pi\)
0.144803 + 0.989460i \(0.453745\pi\)
\(692\) −22.4701 −0.854186
\(693\) −11.6128 −0.441136
\(694\) −14.1936 −0.538781
\(695\) 0 0
\(696\) −3.37778 −0.128035
\(697\) −21.5941 −0.817935
\(698\) 32.3684 1.22516
\(699\) 17.9081 0.677348
\(700\) 0 0
\(701\) −32.3654 −1.22242 −0.611211 0.791467i \(-0.709317\pi\)
−0.611211 + 0.791467i \(0.709317\pi\)
\(702\) −5.80642 −0.219149
\(703\) 5.80642 0.218993
\(704\) 2.62222 0.0988285
\(705\) 0 0
\(706\) −11.8064 −0.444341
\(707\) −78.2677 −2.94356
\(708\) 1.05086 0.0394936
\(709\) −49.8992 −1.87401 −0.937003 0.349322i \(-0.886412\pi\)
−0.937003 + 0.349322i \(0.886412\pi\)
\(710\) 0 0
\(711\) 4.42864 0.166087
\(712\) 12.4286 0.465783
\(713\) 11.6128 0.434905
\(714\) −16.8573 −0.630868
\(715\) 0 0
\(716\) −2.94914 −0.110215
\(717\) 28.2766 1.05601
\(718\) −10.8287 −0.404123
\(719\) −26.4415 −0.986103 −0.493052 0.870000i \(-0.664119\pi\)
−0.493052 + 0.870000i \(0.664119\pi\)
\(720\) 0 0
\(721\) 5.24443 0.195313
\(722\) 1.00000 0.0372161
\(723\) 11.5111 0.428104
\(724\) 22.8988 0.851026
\(725\) 0 0
\(726\) 4.12399 0.153055
\(727\) 48.1245 1.78484 0.892419 0.451208i \(-0.149007\pi\)
0.892419 + 0.451208i \(0.149007\pi\)
\(728\) −25.7146 −0.953045
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 41.8350 1.54732
\(732\) −4.75557 −0.175771
\(733\) −14.0286 −0.518157 −0.259079 0.965856i \(-0.583419\pi\)
−0.259079 + 0.965856i \(0.583419\pi\)
\(734\) −1.46965 −0.0542458
\(735\) 0 0
\(736\) −2.62222 −0.0966562
\(737\) 40.9403 1.50805
\(738\) 5.67307 0.208829
\(739\) 10.1847 0.374650 0.187325 0.982298i \(-0.440018\pi\)
0.187325 + 0.982298i \(0.440018\pi\)
\(740\) 0 0
\(741\) −5.80642 −0.213304
\(742\) 26.5718 0.975483
\(743\) −42.9590 −1.57601 −0.788006 0.615667i \(-0.788887\pi\)
−0.788006 + 0.615667i \(0.788887\pi\)
\(744\) 4.42864 0.162362
\(745\) 0 0
\(746\) 24.3783 0.892552
\(747\) 11.9081 0.435696
\(748\) −9.98126 −0.364951
\(749\) −21.5111 −0.786000
\(750\) 0 0
\(751\) −7.18421 −0.262155 −0.131078 0.991372i \(-0.541844\pi\)
−0.131078 + 0.991372i \(0.541844\pi\)
\(752\) 2.62222 0.0956224
\(753\) 7.74620 0.282287
\(754\) 19.6128 0.714258
\(755\) 0 0
\(756\) 4.42864 0.161068
\(757\) 41.2543 1.49941 0.749706 0.661771i \(-0.230195\pi\)
0.749706 + 0.661771i \(0.230195\pi\)
\(758\) 18.9590 0.688621
\(759\) 6.87601 0.249584
\(760\) 0 0
\(761\) 44.3051 1.60606 0.803030 0.595939i \(-0.203220\pi\)
0.803030 + 0.595939i \(0.203220\pi\)
\(762\) −6.42864 −0.232885
\(763\) 26.7556 0.968617
\(764\) −9.05086 −0.327448
\(765\) 0 0
\(766\) −26.1017 −0.943093
\(767\) −6.10171 −0.220320
\(768\) −1.00000 −0.0360844
\(769\) −6.59057 −0.237662 −0.118831 0.992914i \(-0.537915\pi\)
−0.118831 + 0.992914i \(0.537915\pi\)
\(770\) 0 0
\(771\) 13.8796 0.499860
\(772\) 18.7239 0.673889
\(773\) −8.83854 −0.317900 −0.158950 0.987287i \(-0.550811\pi\)
−0.158950 + 0.987287i \(0.550811\pi\)
\(774\) −10.9906 −0.395050
\(775\) 0 0
\(776\) 7.37778 0.264847
\(777\) 25.7146 0.922505
\(778\) −3.57136 −0.128039
\(779\) 5.67307 0.203259
\(780\) 0 0
\(781\) 40.9403 1.46496
\(782\) 9.98126 0.356929
\(783\) −3.37778 −0.120712
\(784\) 12.6128 0.450459
\(785\) 0 0
\(786\) 15.0923 0.538326
\(787\) −16.9403 −0.603855 −0.301927 0.953331i \(-0.597630\pi\)
−0.301927 + 0.953331i \(0.597630\pi\)
\(788\) 0.888922 0.0316665
\(789\) 18.6222 0.662968
\(790\) 0 0
\(791\) −41.3907 −1.47168
\(792\) 2.62222 0.0931764
\(793\) 27.6128 0.980561
\(794\) 31.4193 1.11503
\(795\) 0 0
\(796\) 21.7146 0.769652
\(797\) −40.1847 −1.42341 −0.711707 0.702476i \(-0.752078\pi\)
−0.711707 + 0.702476i \(0.752078\pi\)
\(798\) 4.42864 0.156772
\(799\) −9.98126 −0.353112
\(800\) 0 0
\(801\) 12.4286 0.439144
\(802\) 12.0415 0.425200
\(803\) 30.4514 1.07461
\(804\) −15.6128 −0.550623
\(805\) 0 0
\(806\) −25.7146 −0.905757
\(807\) −15.8479 −0.557873
\(808\) 17.6731 0.621736
\(809\) 29.1052 1.02329 0.511643 0.859198i \(-0.329037\pi\)
0.511643 + 0.859198i \(0.329037\pi\)
\(810\) 0 0
\(811\) 38.7753 1.36158 0.680792 0.732477i \(-0.261636\pi\)
0.680792 + 0.732477i \(0.261636\pi\)
\(812\) −14.9590 −0.524958
\(813\) −1.51114 −0.0529979
\(814\) 15.2257 0.533660
\(815\) 0 0
\(816\) 3.80642 0.133251
\(817\) −10.9906 −0.384514
\(818\) 26.4701 0.925506
\(819\) −25.7146 −0.898539
\(820\) 0 0
\(821\) 2.53035 0.0883098 0.0441549 0.999025i \(-0.485940\pi\)
0.0441549 + 0.999025i \(0.485940\pi\)
\(822\) 7.53972 0.262978
\(823\) −24.0415 −0.838034 −0.419017 0.907978i \(-0.637625\pi\)
−0.419017 + 0.907978i \(0.637625\pi\)
\(824\) −1.18421 −0.0412538
\(825\) 0 0
\(826\) 4.65386 0.161928
\(827\) −6.57184 −0.228525 −0.114263 0.993451i \(-0.536451\pi\)
−0.114263 + 0.993451i \(0.536451\pi\)
\(828\) −2.62222 −0.0911283
\(829\) −28.7338 −0.997965 −0.498983 0.866612i \(-0.666293\pi\)
−0.498983 + 0.866612i \(0.666293\pi\)
\(830\) 0 0
\(831\) −1.05086 −0.0364538
\(832\) 5.80642 0.201302
\(833\) −48.0098 −1.66344
\(834\) −0.387152 −0.0134060
\(835\) 0 0
\(836\) 2.62222 0.0906912
\(837\) 4.42864 0.153076
\(838\) −25.9684 −0.897062
\(839\) −50.5718 −1.74593 −0.872967 0.487780i \(-0.837807\pi\)
−0.872967 + 0.487780i \(0.837807\pi\)
\(840\) 0 0
\(841\) −17.5906 −0.606571
\(842\) −29.2672 −1.00861
\(843\) −3.45091 −0.118856
\(844\) −3.61285 −0.124359
\(845\) 0 0
\(846\) 2.62222 0.0901536
\(847\) 18.2636 0.627546
\(848\) −6.00000 −0.206041
\(849\) −15.5812 −0.534746
\(850\) 0 0
\(851\) −15.2257 −0.521930
\(852\) −15.6128 −0.534887
\(853\) −31.2355 −1.06948 −0.534742 0.845015i \(-0.679591\pi\)
−0.534742 + 0.845015i \(0.679591\pi\)
\(854\) −21.0607 −0.720682
\(855\) 0 0
\(856\) 4.85728 0.166018
\(857\) 27.3274 0.933486 0.466743 0.884393i \(-0.345427\pi\)
0.466743 + 0.884393i \(0.345427\pi\)
\(858\) −15.2257 −0.519797
\(859\) −34.7753 −1.18652 −0.593258 0.805012i \(-0.702159\pi\)
−0.593258 + 0.805012i \(0.702159\pi\)
\(860\) 0 0
\(861\) 25.1240 0.856223
\(862\) 14.8385 0.505403
\(863\) 21.2444 0.723169 0.361584 0.932339i \(-0.382236\pi\)
0.361584 + 0.932339i \(0.382236\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −5.27607 −0.179288
\(867\) 2.51114 0.0852827
\(868\) 19.6128 0.665703
\(869\) 11.6128 0.393939
\(870\) 0 0
\(871\) 90.6548 3.07172
\(872\) −6.04149 −0.204591
\(873\) 7.37778 0.249700
\(874\) −2.62222 −0.0886978
\(875\) 0 0
\(876\) −11.6128 −0.392362
\(877\) 22.3970 0.756293 0.378146 0.925746i \(-0.376562\pi\)
0.378146 + 0.925746i \(0.376562\pi\)
\(878\) 31.8578 1.07515
\(879\) −7.12399 −0.240286
\(880\) 0 0
\(881\) −20.3684 −0.686229 −0.343115 0.939294i \(-0.611482\pi\)
−0.343115 + 0.939294i \(0.611482\pi\)
\(882\) 12.6128 0.424697
\(883\) 21.6829 0.729688 0.364844 0.931069i \(-0.381122\pi\)
0.364844 + 0.931069i \(0.381122\pi\)
\(884\) −22.1017 −0.743361
\(885\) 0 0
\(886\) −4.94914 −0.166270
\(887\) −7.14272 −0.239829 −0.119915 0.992784i \(-0.538262\pi\)
−0.119915 + 0.992784i \(0.538262\pi\)
\(888\) −5.80642 −0.194851
\(889\) −28.4701 −0.954857
\(890\) 0 0
\(891\) 2.62222 0.0878475
\(892\) −21.3876 −0.716111
\(893\) 2.62222 0.0877491
\(894\) 14.5303 0.485968
\(895\) 0 0
\(896\) −4.42864 −0.147950
\(897\) 15.2257 0.508371
\(898\) 9.75605 0.325563
\(899\) −14.9590 −0.498910
\(900\) 0 0
\(901\) 22.8385 0.760862
\(902\) 14.8760 0.495317
\(903\) −48.6735 −1.61975
\(904\) 9.34614 0.310848
\(905\) 0 0
\(906\) −4.69535 −0.155992
\(907\) 15.3461 0.509560 0.254780 0.966999i \(-0.417997\pi\)
0.254780 + 0.966999i \(0.417997\pi\)
\(908\) 8.47013 0.281091
\(909\) 17.6731 0.586179
\(910\) 0 0
\(911\) 12.1204 0.401568 0.200784 0.979636i \(-0.435651\pi\)
0.200784 + 0.979636i \(0.435651\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 31.2257 1.03342
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −16.9590 −0.560341
\(917\) 66.8385 2.20720
\(918\) 3.80642 0.125631
\(919\) −12.2667 −0.404641 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(920\) 0 0
\(921\) −1.12399 −0.0370366
\(922\) −36.1245 −1.18970
\(923\) 90.6548 2.98394
\(924\) 11.6128 0.382035
\(925\) 0 0
\(926\) 35.1209 1.15415
\(927\) −1.18421 −0.0388945
\(928\) 3.37778 0.110881
\(929\) −40.1847 −1.31842 −0.659208 0.751960i \(-0.729108\pi\)
−0.659208 + 0.751960i \(0.729108\pi\)
\(930\) 0 0
\(931\) 12.6128 0.413369
\(932\) −17.9081 −0.586600
\(933\) −14.9491 −0.489413
\(934\) −4.56199 −0.149273
\(935\) 0 0
\(936\) 5.80642 0.189789
\(937\) −24.7368 −0.808117 −0.404059 0.914733i \(-0.632401\pi\)
−0.404059 + 0.914733i \(0.632401\pi\)
\(938\) −69.1437 −2.25762
\(939\) −3.14272 −0.102559
\(940\) 0 0
\(941\) −1.66323 −0.0542196 −0.0271098 0.999632i \(-0.508630\pi\)
−0.0271098 + 0.999632i \(0.508630\pi\)
\(942\) 21.5210 0.701191
\(943\) −14.8760 −0.484430
\(944\) −1.05086 −0.0342024
\(945\) 0 0
\(946\) −28.8198 −0.937013
\(947\) 14.8671 0.483117 0.241558 0.970386i \(-0.422341\pi\)
0.241558 + 0.970386i \(0.422341\pi\)
\(948\) −4.42864 −0.143836
\(949\) 67.4291 2.18884
\(950\) 0 0
\(951\) −10.5906 −0.343423
\(952\) 16.8573 0.546348
\(953\) −8.83854 −0.286308 −0.143154 0.989700i \(-0.545725\pi\)
−0.143154 + 0.989700i \(0.545725\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −28.2766 −0.914529
\(957\) −8.85728 −0.286315
\(958\) 37.4005 1.20836
\(959\) 33.3907 1.07824
\(960\) 0 0
\(961\) −11.3872 −0.367327
\(962\) 33.7146 1.08700
\(963\) 4.85728 0.156524
\(964\) −11.5111 −0.370749
\(965\) 0 0
\(966\) −11.6128 −0.373637
\(967\) 1.55262 0.0499290 0.0249645 0.999688i \(-0.492053\pi\)
0.0249645 + 0.999688i \(0.492053\pi\)
\(968\) −4.12399 −0.132550
\(969\) 3.80642 0.122280
\(970\) 0 0
\(971\) 44.5433 1.42946 0.714731 0.699400i \(-0.246549\pi\)
0.714731 + 0.699400i \(0.246549\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −1.71456 −0.0549662
\(974\) −20.9175 −0.670240
\(975\) 0 0
\(976\) 4.75557 0.152222
\(977\) 15.8350 0.506607 0.253303 0.967387i \(-0.418483\pi\)
0.253303 + 0.967387i \(0.418483\pi\)
\(978\) 8.23506 0.263328
\(979\) 32.5906 1.04160
\(980\) 0 0
\(981\) −6.04149 −0.192890
\(982\) −15.8666 −0.506325
\(983\) 6.87601 0.219311 0.109655 0.993970i \(-0.465025\pi\)
0.109655 + 0.993970i \(0.465025\pi\)
\(984\) −5.67307 −0.180851
\(985\) 0 0
\(986\) −12.8573 −0.409459
\(987\) 11.6128 0.369641
\(988\) 5.80642 0.184727
\(989\) 28.8198 0.916417
\(990\) 0 0
\(991\) −29.8765 −0.949058 −0.474529 0.880240i \(-0.657382\pi\)
−0.474529 + 0.880240i \(0.657382\pi\)
\(992\) −4.42864 −0.140609
\(993\) 15.1427 0.480540
\(994\) −69.1437 −2.19310
\(995\) 0 0
\(996\) −11.9081 −0.377324
\(997\) 5.90813 0.187112 0.0935562 0.995614i \(-0.470177\pi\)
0.0935562 + 0.995614i \(0.470177\pi\)
\(998\) 16.7368 0.529795
\(999\) −5.80642 −0.183707
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 2850.2.a.bm.1.1 3
3.2 odd 2 8550.2.a.ce.1.1 3
5.2 odd 4 570.2.d.c.229.6 yes 6
5.3 odd 4 570.2.d.c.229.3 6
5.4 even 2 2850.2.a.bl.1.3 3
15.2 even 4 1710.2.d.f.1369.1 6
15.8 even 4 1710.2.d.f.1369.4 6
15.14 odd 2 8550.2.a.cq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.c.229.3 6 5.3 odd 4
570.2.d.c.229.6 yes 6 5.2 odd 4
1710.2.d.f.1369.1 6 15.2 even 4
1710.2.d.f.1369.4 6 15.8 even 4
2850.2.a.bl.1.3 3 5.4 even 2
2850.2.a.bm.1.1 3 1.1 even 1 trivial
8550.2.a.ce.1.1 3 3.2 odd 2
8550.2.a.cq.1.3 3 15.14 odd 2