Properties

Label 5586.2.a.bw.1.1
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.29268.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.85121\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46130 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46130 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.46130 q^{10} +0.278197 q^{11} -1.00000 q^{12} +6.92261 q^{13} +3.46130 q^{15} +1.00000 q^{16} +5.79751 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.46130 q^{20} -0.278197 q^{22} +3.09233 q^{23} +1.00000 q^{24} +6.98062 q^{25} -6.92261 q^{26} -1.00000 q^{27} -8.55364 q^{29} -3.46130 q^{30} +7.34959 q^{31} -1.00000 q^{32} -0.278197 q^{33} -5.79751 q^{34} +1.00000 q^{36} -5.24112 q^{37} -1.00000 q^{38} -6.92261 q^{39} +3.46130 q^{40} +8.88553 q^{41} +10.3705 q^{43} +0.278197 q^{44} -3.46130 q^{45} -3.09233 q^{46} -1.79320 q^{47} -1.00000 q^{48} -6.98062 q^{50} -5.79751 q^{51} +6.92261 q^{52} +8.38823 q^{53} +1.00000 q^{54} -0.962923 q^{55} -1.00000 q^{57} +8.55364 q^{58} -7.79476 q^{59} +3.46130 q^{60} +9.12941 q^{61} -7.34959 q^{62} +1.00000 q^{64} -23.9612 q^{65} +0.278197 q^{66} -13.7378 q^{67} +5.79751 q^{68} -3.09233 q^{69} +0.684727 q^{71} -1.00000 q^{72} -1.92536 q^{73} +5.24112 q^{74} -6.98062 q^{75} +1.00000 q^{76} +6.92261 q^{78} +1.40761 q^{79} -3.46130 q^{80} +1.00000 q^{81} -8.88553 q^{82} -0.146030 q^{83} -20.0670 q^{85} -10.3705 q^{86} +8.55364 q^{87} -0.278197 q^{88} -13.7007 q^{89} +3.46130 q^{90} +3.09233 q^{92} -7.34959 q^{93} +1.79320 q^{94} -3.46130 q^{95} +1.00000 q^{96} +4.09233 q^{97} +0.278197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{11} - 4q^{12} + 4q^{16} + 10q^{17} - 4q^{18} + 4q^{19} + 2q^{22} - 5q^{23} + 4q^{24} + 4q^{25} - 4q^{27} - 3q^{29} + 9q^{31} - 4q^{32} + 2q^{33} - 10q^{34} + 4q^{36} - 14q^{37} - 4q^{38} + 4q^{41} + 21q^{43} - 2q^{44} + 5q^{46} + 7q^{47} - 4q^{48} - 4q^{50} - 10q^{51} - 7q^{53} + 4q^{54} - 4q^{57} + 3q^{58} + 7q^{59} + 23q^{61} - 9q^{62} + 4q^{64} - 48q^{65} - 2q^{66} + 6q^{67} + 10q^{68} + 5q^{69} + 2q^{71} - 4q^{72} - 5q^{73} + 14q^{74} - 4q^{75} + 4q^{76} - 11q^{79} + 4q^{81} - 4q^{82} + 14q^{83} + 6q^{85} - 21q^{86} + 3q^{87} + 2q^{88} + 10q^{89} - 5q^{92} - 9q^{93} - 7q^{94} + 4q^{96} - q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) −3.46130 −1.54794 −0.773971 0.633221i \(-0.781732\pi\)
−0.773971 + 0.633221i \(0.781732\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.46130 1.09456
\(11\) 0.278197 0.0838795 0.0419397 0.999120i \(-0.486646\pi\)
0.0419397 + 0.999120i \(0.486646\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.92261 1.91999 0.959993 0.280025i \(-0.0903427\pi\)
0.959993 + 0.280025i \(0.0903427\pi\)
\(14\) 0 0
\(15\) 3.46130 0.893705
\(16\) 1.00000 0.250000
\(17\) 5.79751 1.40610 0.703052 0.711139i \(-0.251820\pi\)
0.703052 + 0.711139i \(0.251820\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −3.46130 −0.773971
\(21\) 0 0
\(22\) −0.278197 −0.0593118
\(23\) 3.09233 0.644796 0.322398 0.946604i \(-0.395511\pi\)
0.322398 + 0.946604i \(0.395511\pi\)
\(24\) 1.00000 0.204124
\(25\) 6.98062 1.39612
\(26\) −6.92261 −1.35763
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.55364 −1.58837 −0.794185 0.607676i \(-0.792102\pi\)
−0.794185 + 0.607676i \(0.792102\pi\)
\(30\) −3.46130 −0.631945
\(31\) 7.34959 1.32003 0.660013 0.751254i \(-0.270551\pi\)
0.660013 + 0.751254i \(0.270551\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.278197 −0.0484278
\(34\) −5.79751 −0.994265
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.24112 −0.861635 −0.430817 0.902439i \(-0.641775\pi\)
−0.430817 + 0.902439i \(0.641775\pi\)
\(38\) −1.00000 −0.162221
\(39\) −6.92261 −1.10850
\(40\) 3.46130 0.547280
\(41\) 8.88553 1.38769 0.693843 0.720126i \(-0.255916\pi\)
0.693843 + 0.720126i \(0.255916\pi\)
\(42\) 0 0
\(43\) 10.3705 1.58149 0.790745 0.612145i \(-0.209693\pi\)
0.790745 + 0.612145i \(0.209693\pi\)
\(44\) 0.278197 0.0419397
\(45\) −3.46130 −0.515981
\(46\) −3.09233 −0.455939
\(47\) −1.79320 −0.261565 −0.130782 0.991411i \(-0.541749\pi\)
−0.130782 + 0.991411i \(0.541749\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −6.98062 −0.987209
\(51\) −5.79751 −0.811814
\(52\) 6.92261 0.959993
\(53\) 8.38823 1.15221 0.576106 0.817375i \(-0.304572\pi\)
0.576106 + 0.817375i \(0.304572\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.962923 −0.129841
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 8.55364 1.12315
\(59\) −7.79476 −1.01479 −0.507395 0.861713i \(-0.669392\pi\)
−0.507395 + 0.861713i \(0.669392\pi\)
\(60\) 3.46130 0.446852
\(61\) 9.12941 1.16890 0.584450 0.811429i \(-0.301310\pi\)
0.584450 + 0.811429i \(0.301310\pi\)
\(62\) −7.34959 −0.933399
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −23.9612 −2.97203
\(66\) 0.278197 0.0342437
\(67\) −13.7378 −1.67834 −0.839171 0.543868i \(-0.816959\pi\)
−0.839171 + 0.543868i \(0.816959\pi\)
\(68\) 5.79751 0.703052
\(69\) −3.09233 −0.372273
\(70\) 0 0
\(71\) 0.684727 0.0812621 0.0406311 0.999174i \(-0.487063\pi\)
0.0406311 + 0.999174i \(0.487063\pi\)
\(72\) −1.00000 −0.117851
\(73\) −1.92536 −0.225347 −0.112673 0.993632i \(-0.535941\pi\)
−0.112673 + 0.993632i \(0.535941\pi\)
\(74\) 5.24112 0.609268
\(75\) −6.98062 −0.806053
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 6.92261 0.783831
\(79\) 1.40761 0.158368 0.0791840 0.996860i \(-0.474769\pi\)
0.0791840 + 0.996860i \(0.474769\pi\)
\(80\) −3.46130 −0.386985
\(81\) 1.00000 0.111111
\(82\) −8.88553 −0.981243
\(83\) −0.146030 −0.0160289 −0.00801443 0.999968i \(-0.502551\pi\)
−0.00801443 + 0.999968i \(0.502551\pi\)
\(84\) 0 0
\(85\) −20.0670 −2.17657
\(86\) −10.3705 −1.11828
\(87\) 8.55364 0.917046
\(88\) −0.278197 −0.0296559
\(89\) −13.7007 −1.45228 −0.726138 0.687549i \(-0.758687\pi\)
−0.726138 + 0.687549i \(0.758687\pi\)
\(90\) 3.46130 0.364853
\(91\) 0 0
\(92\) 3.09233 0.322398
\(93\) −7.34959 −0.762117
\(94\) 1.79320 0.184954
\(95\) −3.46130 −0.355122
\(96\) 1.00000 0.102062
\(97\) 4.09233 0.415513 0.207757 0.978181i \(-0.433384\pi\)
0.207757 + 0.978181i \(0.433384\pi\)
\(98\) 0 0
\(99\) 0.278197 0.0279598
\(100\) 6.98062 0.698062
\(101\) −2.45699 −0.244479 −0.122240 0.992501i \(-0.539008\pi\)
−0.122240 + 0.992501i \(0.539008\pi\)
\(102\) 5.79751 0.574039
\(103\) 17.3468 1.70923 0.854617 0.519259i \(-0.173792\pi\)
0.854617 + 0.519259i \(0.173792\pi\)
\(104\) −6.92261 −0.678817
\(105\) 0 0
\(106\) −8.38823 −0.814737
\(107\) −11.4779 −1.10961 −0.554806 0.831979i \(-0.687208\pi\)
−0.554806 + 0.831979i \(0.687208\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.6814 −1.78935 −0.894675 0.446718i \(-0.852593\pi\)
−0.894675 + 0.446718i \(0.852593\pi\)
\(110\) 0.962923 0.0918112
\(111\) 5.24112 0.497465
\(112\) 0 0
\(113\) 3.29913 0.310356 0.155178 0.987886i \(-0.450405\pi\)
0.155178 + 0.987886i \(0.450405\pi\)
\(114\) 1.00000 0.0936586
\(115\) −10.7035 −0.998106
\(116\) −8.55364 −0.794185
\(117\) 6.92261 0.639995
\(118\) 7.79476 0.717565
\(119\) 0 0
\(120\) −3.46130 −0.315972
\(121\) −10.9226 −0.992964
\(122\) −9.12941 −0.826538
\(123\) −8.88553 −0.801181
\(124\) 7.34959 0.660013
\(125\) −6.85553 −0.613177
\(126\) 0 0
\(127\) −3.45699 −0.306758 −0.153379 0.988167i \(-0.549016\pi\)
−0.153379 + 0.988167i \(0.549016\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.3705 −0.913074
\(130\) 23.9612 2.10154
\(131\) 18.4956 1.61597 0.807985 0.589203i \(-0.200558\pi\)
0.807985 + 0.589203i \(0.200558\pi\)
\(132\) −0.278197 −0.0242139
\(133\) 0 0
\(134\) 13.7378 1.18677
\(135\) 3.46130 0.297902
\(136\) −5.79751 −0.497133
\(137\) −12.1680 −1.03959 −0.519793 0.854292i \(-0.673991\pi\)
−0.519793 + 0.854292i \(0.673991\pi\)
\(138\) 3.09233 0.263237
\(139\) −2.91829 −0.247526 −0.123763 0.992312i \(-0.539496\pi\)
−0.123763 + 0.992312i \(0.539496\pi\)
\(140\) 0 0
\(141\) 1.79320 0.151015
\(142\) −0.684727 −0.0574610
\(143\) 1.92585 0.161047
\(144\) 1.00000 0.0833333
\(145\) 29.6067 2.45870
\(146\) 1.92536 0.159344
\(147\) 0 0
\(148\) −5.24112 −0.430817
\(149\) −4.16493 −0.341204 −0.170602 0.985340i \(-0.554571\pi\)
−0.170602 + 0.985340i \(0.554571\pi\)
\(150\) 6.98062 0.569965
\(151\) 19.2754 1.56861 0.784306 0.620374i \(-0.213019\pi\)
0.784306 + 0.620374i \(0.213019\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.79751 0.468701
\(154\) 0 0
\(155\) −25.4392 −2.04332
\(156\) −6.92261 −0.554252
\(157\) −21.4569 −1.71244 −0.856222 0.516608i \(-0.827194\pi\)
−0.856222 + 0.516608i \(0.827194\pi\)
\(158\) −1.40761 −0.111983
\(159\) −8.38823 −0.665230
\(160\) 3.46130 0.273640
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 15.9446 1.24888 0.624439 0.781073i \(-0.285327\pi\)
0.624439 + 0.781073i \(0.285327\pi\)
\(164\) 8.88553 0.693843
\(165\) 0.962923 0.0749635
\(166\) 0.146030 0.0113341
\(167\) 17.6073 1.36250 0.681248 0.732053i \(-0.261437\pi\)
0.681248 + 0.732053i \(0.261437\pi\)
\(168\) 0 0
\(169\) 34.9225 2.68634
\(170\) 20.0670 1.53907
\(171\) 1.00000 0.0764719
\(172\) 10.3705 0.790745
\(173\) 10.0106 0.761094 0.380547 0.924762i \(-0.375736\pi\)
0.380547 + 0.924762i \(0.375736\pi\)
\(174\) −8.55364 −0.648449
\(175\) 0 0
\(176\) 0.278197 0.0209699
\(177\) 7.79476 0.585890
\(178\) 13.7007 1.02691
\(179\) 6.25882 0.467806 0.233903 0.972260i \(-0.424850\pi\)
0.233903 + 0.972260i \(0.424850\pi\)
\(180\) −3.46130 −0.257990
\(181\) −4.25019 −0.315914 −0.157957 0.987446i \(-0.550491\pi\)
−0.157957 + 0.987446i \(0.550491\pi\)
\(182\) 0 0
\(183\) −9.12941 −0.674865
\(184\) −3.09233 −0.227970
\(185\) 18.1411 1.33376
\(186\) 7.34959 0.538898
\(187\) 1.61285 0.117943
\(188\) −1.79320 −0.130782
\(189\) 0 0
\(190\) 3.46130 0.251109
\(191\) −10.9479 −0.792159 −0.396080 0.918216i \(-0.629630\pi\)
−0.396080 + 0.918216i \(0.629630\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 21.2899 1.53248 0.766240 0.642554i \(-0.222125\pi\)
0.766240 + 0.642554i \(0.222125\pi\)
\(194\) −4.09233 −0.293812
\(195\) 23.9612 1.71590
\(196\) 0 0
\(197\) −6.63103 −0.472441 −0.236221 0.971699i \(-0.575909\pi\)
−0.236221 + 0.971699i \(0.575909\pi\)
\(198\) −0.278197 −0.0197706
\(199\) −7.87047 −0.557923 −0.278961 0.960302i \(-0.589990\pi\)
−0.278961 + 0.960302i \(0.589990\pi\)
\(200\) −6.98062 −0.493604
\(201\) 13.7378 0.968991
\(202\) 2.45699 0.172873
\(203\) 0 0
\(204\) −5.79751 −0.405907
\(205\) −30.7555 −2.14806
\(206\) −17.3468 −1.20861
\(207\) 3.09233 0.214932
\(208\) 6.92261 0.479996
\(209\) 0.278197 0.0192433
\(210\) 0 0
\(211\) 14.9123 1.02660 0.513302 0.858208i \(-0.328422\pi\)
0.513302 + 0.858208i \(0.328422\pi\)
\(212\) 8.38823 0.576106
\(213\) −0.684727 −0.0469167
\(214\) 11.4779 0.784615
\(215\) −35.8955 −2.44806
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 18.6814 1.26526
\(219\) 1.92536 0.130104
\(220\) −0.962923 −0.0649203
\(221\) 40.1339 2.69970
\(222\) −5.24112 −0.351761
\(223\) 3.92861 0.263079 0.131539 0.991311i \(-0.458008\pi\)
0.131539 + 0.991311i \(0.458008\pi\)
\(224\) 0 0
\(225\) 6.98062 0.465375
\(226\) −3.29913 −0.219455
\(227\) −7.03971 −0.467242 −0.233621 0.972328i \(-0.575057\pi\)
−0.233621 + 0.972328i \(0.575057\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 5.88397 0.388824 0.194412 0.980920i \(-0.437720\pi\)
0.194412 + 0.980920i \(0.437720\pi\)
\(230\) 10.7035 0.705768
\(231\) 0 0
\(232\) 8.55364 0.561574
\(233\) 2.80658 0.183865 0.0919326 0.995765i \(-0.470696\pi\)
0.0919326 + 0.995765i \(0.470696\pi\)
\(234\) −6.92261 −0.452545
\(235\) 6.20680 0.404887
\(236\) −7.79476 −0.507395
\(237\) −1.40761 −0.0914338
\(238\) 0 0
\(239\) −12.4016 −0.802193 −0.401097 0.916036i \(-0.631371\pi\)
−0.401097 + 0.916036i \(0.631371\pi\)
\(240\) 3.46130 0.223426
\(241\) −15.1965 −0.978892 −0.489446 0.872034i \(-0.662801\pi\)
−0.489446 + 0.872034i \(0.662801\pi\)
\(242\) 10.9226 0.702132
\(243\) −1.00000 −0.0641500
\(244\) 9.12941 0.584450
\(245\) 0 0
\(246\) 8.88553 0.566521
\(247\) 6.92261 0.440475
\(248\) −7.34959 −0.466700
\(249\) 0.146030 0.00925427
\(250\) 6.85553 0.433582
\(251\) 5.91198 0.373161 0.186581 0.982440i \(-0.440259\pi\)
0.186581 + 0.982440i \(0.440259\pi\)
\(252\) 0 0
\(253\) 0.860277 0.0540851
\(254\) 3.45699 0.216911
\(255\) 20.0670 1.25664
\(256\) 1.00000 0.0625000
\(257\) 1.40485 0.0876320 0.0438160 0.999040i \(-0.486048\pi\)
0.0438160 + 0.999040i \(0.486048\pi\)
\(258\) 10.3705 0.645641
\(259\) 0 0
\(260\) −23.9612 −1.48601
\(261\) −8.55364 −0.529457
\(262\) −18.4956 −1.14266
\(263\) −10.7781 −0.664608 −0.332304 0.943172i \(-0.607826\pi\)
−0.332304 + 0.943172i \(0.607826\pi\)
\(264\) 0.278197 0.0171218
\(265\) −29.0342 −1.78356
\(266\) 0 0
\(267\) 13.7007 0.838472
\(268\) −13.7378 −0.839171
\(269\) 22.4462 1.36857 0.684286 0.729214i \(-0.260114\pi\)
0.684286 + 0.729214i \(0.260114\pi\)
\(270\) −3.46130 −0.210648
\(271\) −2.27651 −0.138288 −0.0691442 0.997607i \(-0.522027\pi\)
−0.0691442 + 0.997607i \(0.522027\pi\)
\(272\) 5.79751 0.351526
\(273\) 0 0
\(274\) 12.1680 0.735098
\(275\) 1.94199 0.117106
\(276\) −3.09233 −0.186137
\(277\) 27.4521 1.64943 0.824717 0.565545i \(-0.191334\pi\)
0.824717 + 0.565545i \(0.191334\pi\)
\(278\) 2.91829 0.175027
\(279\) 7.34959 0.440009
\(280\) 0 0
\(281\) 8.14279 0.485758 0.242879 0.970057i \(-0.421908\pi\)
0.242879 + 0.970057i \(0.421908\pi\)
\(282\) −1.79320 −0.106783
\(283\) −8.18586 −0.486599 −0.243300 0.969951i \(-0.578230\pi\)
−0.243300 + 0.969951i \(0.578230\pi\)
\(284\) 0.684727 0.0406311
\(285\) 3.46130 0.205030
\(286\) −1.92585 −0.113878
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 16.6112 0.977127
\(290\) −29.6067 −1.73857
\(291\) −4.09233 −0.239897
\(292\) −1.92536 −0.112673
\(293\) −19.3421 −1.12998 −0.564988 0.825099i \(-0.691119\pi\)
−0.564988 + 0.825099i \(0.691119\pi\)
\(294\) 0 0
\(295\) 26.9800 1.57084
\(296\) 5.24112 0.304634
\(297\) −0.278197 −0.0161426
\(298\) 4.16493 0.241268
\(299\) 21.4070 1.23800
\(300\) −6.98062 −0.403026
\(301\) 0 0
\(302\) −19.2754 −1.10918
\(303\) 2.45699 0.141150
\(304\) 1.00000 0.0573539
\(305\) −31.5997 −1.80939
\(306\) −5.79751 −0.331422
\(307\) −11.7007 −0.667797 −0.333898 0.942609i \(-0.608364\pi\)
−0.333898 + 0.942609i \(0.608364\pi\)
\(308\) 0 0
\(309\) −17.3468 −0.986827
\(310\) 25.4392 1.44485
\(311\) 13.8555 0.785675 0.392837 0.919608i \(-0.371494\pi\)
0.392837 + 0.919608i \(0.371494\pi\)
\(312\) 6.92261 0.391915
\(313\) −1.97199 −0.111463 −0.0557317 0.998446i \(-0.517749\pi\)
−0.0557317 + 0.998446i \(0.517749\pi\)
\(314\) 21.4569 1.21088
\(315\) 0 0
\(316\) 1.40761 0.0791840
\(317\) 19.9005 1.11772 0.558861 0.829261i \(-0.311239\pi\)
0.558861 + 0.829261i \(0.311239\pi\)
\(318\) 8.38823 0.470388
\(319\) −2.37959 −0.133232
\(320\) −3.46130 −0.193493
\(321\) 11.4779 0.640635
\(322\) 0 0
\(323\) 5.79751 0.322582
\(324\) 1.00000 0.0555556
\(325\) 48.3241 2.68054
\(326\) −15.9446 −0.883091
\(327\) 18.6814 1.03308
\(328\) −8.88553 −0.490621
\(329\) 0 0
\(330\) −0.962923 −0.0530072
\(331\) −4.66762 −0.256556 −0.128278 0.991738i \(-0.540945\pi\)
−0.128278 + 0.991738i \(0.540945\pi\)
\(332\) −0.146030 −0.00801443
\(333\) −5.24112 −0.287212
\(334\) −17.6073 −0.963430
\(335\) 47.5508 2.59798
\(336\) 0 0
\(337\) −11.4639 −0.624480 −0.312240 0.950003i \(-0.601079\pi\)
−0.312240 + 0.950003i \(0.601079\pi\)
\(338\) −34.9225 −1.89953
\(339\) −3.29913 −0.179184
\(340\) −20.0670 −1.08828
\(341\) 2.04463 0.110723
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −10.3705 −0.559141
\(345\) 10.7035 0.576257
\(346\) −10.0106 −0.538174
\(347\) 7.84258 0.421012 0.210506 0.977593i \(-0.432489\pi\)
0.210506 + 0.977593i \(0.432489\pi\)
\(348\) 8.55364 0.458523
\(349\) 21.2991 1.14012 0.570058 0.821604i \(-0.306921\pi\)
0.570058 + 0.821604i \(0.306921\pi\)
\(350\) 0 0
\(351\) −6.92261 −0.369501
\(352\) −0.278197 −0.0148279
\(353\) 8.71701 0.463959 0.231980 0.972721i \(-0.425480\pi\)
0.231980 + 0.972721i \(0.425480\pi\)
\(354\) −7.79476 −0.414287
\(355\) −2.37005 −0.125789
\(356\) −13.7007 −0.726138
\(357\) 0 0
\(358\) −6.25882 −0.330789
\(359\) 1.17128 0.0618180 0.0309090 0.999522i \(-0.490160\pi\)
0.0309090 + 0.999522i \(0.490160\pi\)
\(360\) 3.46130 0.182427
\(361\) 1.00000 0.0526316
\(362\) 4.25019 0.223385
\(363\) 10.9226 0.573288
\(364\) 0 0
\(365\) 6.66427 0.348824
\(366\) 9.12941 0.477202
\(367\) 5.01295 0.261674 0.130837 0.991404i \(-0.458234\pi\)
0.130837 + 0.991404i \(0.458234\pi\)
\(368\) 3.09233 0.161199
\(369\) 8.88553 0.462562
\(370\) −18.1411 −0.943111
\(371\) 0 0
\(372\) −7.34959 −0.381059
\(373\) 1.86584 0.0966096 0.0483048 0.998833i \(-0.484618\pi\)
0.0483048 + 0.998833i \(0.484618\pi\)
\(374\) −1.61285 −0.0833985
\(375\) 6.85553 0.354018
\(376\) 1.79320 0.0924771
\(377\) −59.2135 −3.04965
\(378\) 0 0
\(379\) −11.8065 −0.606457 −0.303228 0.952918i \(-0.598065\pi\)
−0.303228 + 0.952918i \(0.598065\pi\)
\(380\) −3.46130 −0.177561
\(381\) 3.45699 0.177107
\(382\) 10.9479 0.560141
\(383\) 5.52639 0.282385 0.141193 0.989982i \(-0.454906\pi\)
0.141193 + 0.989982i \(0.454906\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −21.2899 −1.08363
\(387\) 10.3705 0.527164
\(388\) 4.09233 0.207757
\(389\) 1.26421 0.0640979 0.0320490 0.999486i \(-0.489797\pi\)
0.0320490 + 0.999486i \(0.489797\pi\)
\(390\) −23.9612 −1.21332
\(391\) 17.9278 0.906650
\(392\) 0 0
\(393\) −18.4956 −0.932981
\(394\) 6.63103 0.334066
\(395\) −4.87215 −0.245144
\(396\) 0.278197 0.0139799
\(397\) −1.40485 −0.0705073 −0.0352536 0.999378i \(-0.511224\pi\)
−0.0352536 + 0.999378i \(0.511224\pi\)
\(398\) 7.87047 0.394511
\(399\) 0 0
\(400\) 6.98062 0.349031
\(401\) −4.77119 −0.238262 −0.119131 0.992879i \(-0.538011\pi\)
−0.119131 + 0.992879i \(0.538011\pi\)
\(402\) −13.7378 −0.685180
\(403\) 50.8783 2.53443
\(404\) −2.45699 −0.122240
\(405\) −3.46130 −0.171994
\(406\) 0 0
\(407\) −1.45806 −0.0722735
\(408\) 5.79751 0.287020
\(409\) −18.0890 −0.894442 −0.447221 0.894424i \(-0.647586\pi\)
−0.447221 + 0.894424i \(0.647586\pi\)
\(410\) 30.7555 1.51891
\(411\) 12.1680 0.600205
\(412\) 17.3468 0.854617
\(413\) 0 0
\(414\) −3.09233 −0.151980
\(415\) 0.505454 0.0248118
\(416\) −6.92261 −0.339409
\(417\) 2.91829 0.142909
\(418\) −0.278197 −0.0136071
\(419\) 2.70518 0.132157 0.0660784 0.997814i \(-0.478951\pi\)
0.0660784 + 0.997814i \(0.478951\pi\)
\(420\) 0 0
\(421\) 6.14064 0.299276 0.149638 0.988741i \(-0.452189\pi\)
0.149638 + 0.988741i \(0.452189\pi\)
\(422\) −14.9123 −0.725919
\(423\) −1.79320 −0.0871883
\(424\) −8.38823 −0.407368
\(425\) 40.4702 1.96310
\(426\) 0.684727 0.0331751
\(427\) 0 0
\(428\) −11.4779 −0.554806
\(429\) −1.92585 −0.0929808
\(430\) 35.8955 1.73104
\(431\) 21.2765 1.02485 0.512427 0.858731i \(-0.328747\pi\)
0.512427 + 0.858731i \(0.328747\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −9.42099 −0.452744 −0.226372 0.974041i \(-0.572686\pi\)
−0.226372 + 0.974041i \(0.572686\pi\)
\(434\) 0 0
\(435\) −29.6067 −1.41953
\(436\) −18.6814 −0.894675
\(437\) 3.09233 0.147926
\(438\) −1.92536 −0.0919975
\(439\) 12.5924 0.601002 0.300501 0.953781i \(-0.402846\pi\)
0.300501 + 0.953781i \(0.402846\pi\)
\(440\) 0.962923 0.0459056
\(441\) 0 0
\(442\) −40.1339 −1.90898
\(443\) −23.2947 −1.10676 −0.553382 0.832927i \(-0.686663\pi\)
−0.553382 + 0.832927i \(0.686663\pi\)
\(444\) 5.24112 0.248733
\(445\) 47.4224 2.24804
\(446\) −3.92861 −0.186025
\(447\) 4.16493 0.196994
\(448\) 0 0
\(449\) −0.486074 −0.0229393 −0.0114696 0.999934i \(-0.503651\pi\)
−0.0114696 + 0.999934i \(0.503651\pi\)
\(450\) −6.98062 −0.329070
\(451\) 2.47193 0.116398
\(452\) 3.29913 0.155178
\(453\) −19.2754 −0.905639
\(454\) 7.03971 0.330390
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 14.4597 0.676398 0.338199 0.941075i \(-0.390182\pi\)
0.338199 + 0.941075i \(0.390182\pi\)
\(458\) −5.88397 −0.274940
\(459\) −5.79751 −0.270605
\(460\) −10.7035 −0.499053
\(461\) 18.0188 0.839218 0.419609 0.907705i \(-0.362167\pi\)
0.419609 + 0.907705i \(0.362167\pi\)
\(462\) 0 0
\(463\) 23.3408 1.08474 0.542370 0.840140i \(-0.317527\pi\)
0.542370 + 0.840140i \(0.317527\pi\)
\(464\) −8.55364 −0.397093
\(465\) 25.4392 1.17971
\(466\) −2.80658 −0.130012
\(467\) −8.14879 −0.377081 −0.188540 0.982065i \(-0.560376\pi\)
−0.188540 + 0.982065i \(0.560376\pi\)
\(468\) 6.92261 0.319998
\(469\) 0 0
\(470\) −6.20680 −0.286298
\(471\) 21.4569 0.988680
\(472\) 7.79476 0.358783
\(473\) 2.88505 0.132655
\(474\) 1.40761 0.0646535
\(475\) 6.98062 0.320293
\(476\) 0 0
\(477\) 8.38823 0.384070
\(478\) 12.4016 0.567236
\(479\) 27.1627 1.24109 0.620547 0.784170i \(-0.286911\pi\)
0.620547 + 0.784170i \(0.286911\pi\)
\(480\) −3.46130 −0.157986
\(481\) −36.2822 −1.65433
\(482\) 15.1965 0.692181
\(483\) 0 0
\(484\) −10.9226 −0.496482
\(485\) −14.1648 −0.643191
\(486\) 1.00000 0.0453609
\(487\) −11.9423 −0.541159 −0.270580 0.962698i \(-0.587215\pi\)
−0.270580 + 0.962698i \(0.587215\pi\)
\(488\) −9.12941 −0.413269
\(489\) −15.9446 −0.721041
\(490\) 0 0
\(491\) −7.00863 −0.316295 −0.158148 0.987415i \(-0.550552\pi\)
−0.158148 + 0.987415i \(0.550552\pi\)
\(492\) −8.88553 −0.400591
\(493\) −49.5898 −2.23341
\(494\) −6.92261 −0.311463
\(495\) −0.962923 −0.0432802
\(496\) 7.34959 0.330006
\(497\) 0 0
\(498\) −0.146030 −0.00654376
\(499\) −9.12604 −0.408538 −0.204269 0.978915i \(-0.565482\pi\)
−0.204269 + 0.978915i \(0.565482\pi\)
\(500\) −6.85553 −0.306589
\(501\) −17.6073 −0.786638
\(502\) −5.91198 −0.263865
\(503\) −24.1190 −1.07541 −0.537706 0.843133i \(-0.680709\pi\)
−0.537706 + 0.843133i \(0.680709\pi\)
\(504\) 0 0
\(505\) 8.50438 0.378440
\(506\) −0.860277 −0.0382440
\(507\) −34.9225 −1.55096
\(508\) −3.45699 −0.153379
\(509\) −37.5061 −1.66243 −0.831215 0.555951i \(-0.812354\pi\)
−0.831215 + 0.555951i \(0.812354\pi\)
\(510\) −20.0670 −0.888580
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −1.40485 −0.0619651
\(515\) −60.0427 −2.64580
\(516\) −10.3705 −0.456537
\(517\) −0.498862 −0.0219399
\(518\) 0 0
\(519\) −10.0106 −0.439418
\(520\) 23.9612 1.05077
\(521\) 42.2049 1.84903 0.924516 0.381143i \(-0.124469\pi\)
0.924516 + 0.381143i \(0.124469\pi\)
\(522\) 8.55364 0.374382
\(523\) 19.0938 0.834915 0.417458 0.908696i \(-0.362921\pi\)
0.417458 + 0.908696i \(0.362921\pi\)
\(524\) 18.4956 0.807985
\(525\) 0 0
\(526\) 10.7781 0.469949
\(527\) 42.6094 1.85609
\(528\) −0.278197 −0.0121070
\(529\) −13.4375 −0.584238
\(530\) 29.0342 1.26116
\(531\) −7.79476 −0.338264
\(532\) 0 0
\(533\) 61.5110 2.66434
\(534\) −13.7007 −0.592889
\(535\) 39.7286 1.71762
\(536\) 13.7378 0.593383
\(537\) −6.25882 −0.270088
\(538\) −22.4462 −0.967726
\(539\) 0 0
\(540\) 3.46130 0.148951
\(541\) 0.786124 0.0337981 0.0168991 0.999857i \(-0.494621\pi\)
0.0168991 + 0.999857i \(0.494621\pi\)
\(542\) 2.27651 0.0977846
\(543\) 4.25019 0.182393
\(544\) −5.79751 −0.248566
\(545\) 64.6619 2.76981
\(546\) 0 0
\(547\) 0.670738 0.0286787 0.0143394 0.999897i \(-0.495435\pi\)
0.0143394 + 0.999897i \(0.495435\pi\)
\(548\) −12.1680 −0.519793
\(549\) 9.12941 0.389634
\(550\) −1.94199 −0.0828066
\(551\) −8.55364 −0.364397
\(552\) 3.09233 0.131618
\(553\) 0 0
\(554\) −27.4521 −1.16633
\(555\) −18.1411 −0.770047
\(556\) −2.91829 −0.123763
\(557\) 6.15298 0.260710 0.130355 0.991467i \(-0.458388\pi\)
0.130355 + 0.991467i \(0.458388\pi\)
\(558\) −7.34959 −0.311133
\(559\) 71.7911 3.03644
\(560\) 0 0
\(561\) −1.61285 −0.0680946
\(562\) −8.14279 −0.343483
\(563\) −23.6754 −0.997798 −0.498899 0.866660i \(-0.666262\pi\)
−0.498899 + 0.866660i \(0.666262\pi\)
\(564\) 1.79320 0.0755073
\(565\) −11.4193 −0.480414
\(566\) 8.18586 0.344078
\(567\) 0 0
\(568\) −0.684727 −0.0287305
\(569\) 36.6691 1.53725 0.768623 0.639702i \(-0.220942\pi\)
0.768623 + 0.639702i \(0.220942\pi\)
\(570\) −3.46130 −0.144978
\(571\) −17.7280 −0.741893 −0.370947 0.928654i \(-0.620967\pi\)
−0.370947 + 0.928654i \(0.620967\pi\)
\(572\) 1.92585 0.0805237
\(573\) 10.9479 0.457353
\(574\) 0 0
\(575\) 21.5864 0.900215
\(576\) 1.00000 0.0416667
\(577\) −1.23206 −0.0512911 −0.0256456 0.999671i \(-0.508164\pi\)
−0.0256456 + 0.999671i \(0.508164\pi\)
\(578\) −16.6112 −0.690933
\(579\) −21.2899 −0.884778
\(580\) 29.6067 1.22935
\(581\) 0 0
\(582\) 4.09233 0.169633
\(583\) 2.33358 0.0966469
\(584\) 1.92536 0.0796722
\(585\) −23.9612 −0.990675
\(586\) 19.3421 0.799014
\(587\) −10.0714 −0.415691 −0.207845 0.978162i \(-0.566645\pi\)
−0.207845 + 0.978162i \(0.566645\pi\)
\(588\) 0 0
\(589\) 7.34959 0.302835
\(590\) −26.9800 −1.11075
\(591\) 6.63103 0.272764
\(592\) −5.24112 −0.215409
\(593\) 28.2634 1.16064 0.580320 0.814388i \(-0.302927\pi\)
0.580320 + 0.814388i \(0.302927\pi\)
\(594\) 0.278197 0.0114146
\(595\) 0 0
\(596\) −4.16493 −0.170602
\(597\) 7.87047 0.322117
\(598\) −21.4070 −0.875397
\(599\) 4.27975 0.174866 0.0874330 0.996170i \(-0.472134\pi\)
0.0874330 + 0.996170i \(0.472134\pi\)
\(600\) 6.98062 0.284983
\(601\) −47.6366 −1.94314 −0.971569 0.236757i \(-0.923915\pi\)
−0.971569 + 0.236757i \(0.923915\pi\)
\(602\) 0 0
\(603\) −13.7378 −0.559447
\(604\) 19.2754 0.784306
\(605\) 37.8065 1.53705
\(606\) −2.45699 −0.0998083
\(607\) −20.6931 −0.839905 −0.419953 0.907546i \(-0.637953\pi\)
−0.419953 + 0.907546i \(0.637953\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 31.5997 1.27943
\(611\) −12.4136 −0.502201
\(612\) 5.79751 0.234351
\(613\) 1.67398 0.0676114 0.0338057 0.999428i \(-0.489237\pi\)
0.0338057 + 0.999428i \(0.489237\pi\)
\(614\) 11.7007 0.472203
\(615\) 30.7555 1.24018
\(616\) 0 0
\(617\) −13.5252 −0.544504 −0.272252 0.962226i \(-0.587768\pi\)
−0.272252 + 0.962226i \(0.587768\pi\)
\(618\) 17.3468 0.697792
\(619\) 45.1559 1.81497 0.907485 0.420085i \(-0.138000\pi\)
0.907485 + 0.420085i \(0.138000\pi\)
\(620\) −25.4392 −1.02166
\(621\) −3.09233 −0.124091
\(622\) −13.8555 −0.555556
\(623\) 0 0
\(624\) −6.92261 −0.277126
\(625\) −11.1740 −0.446962
\(626\) 1.97199 0.0788165
\(627\) −0.278197 −0.0111101
\(628\) −21.4569 −0.856222
\(629\) −30.3855 −1.21155
\(630\) 0 0
\(631\) −3.01542 −0.120042 −0.0600210 0.998197i \(-0.519117\pi\)
−0.0600210 + 0.998197i \(0.519117\pi\)
\(632\) −1.40761 −0.0559915
\(633\) −14.9123 −0.592710
\(634\) −19.9005 −0.790349
\(635\) 11.9657 0.474844
\(636\) −8.38823 −0.332615
\(637\) 0 0
\(638\) 2.37959 0.0942090
\(639\) 0.684727 0.0270874
\(640\) 3.46130 0.136820
\(641\) 15.9962 0.631811 0.315905 0.948791i \(-0.397692\pi\)
0.315905 + 0.948791i \(0.397692\pi\)
\(642\) −11.4779 −0.452998
\(643\) −20.9315 −0.825460 −0.412730 0.910854i \(-0.635425\pi\)
−0.412730 + 0.910854i \(0.635425\pi\)
\(644\) 0 0
\(645\) 35.8955 1.41339
\(646\) −5.79751 −0.228100
\(647\) 43.4198 1.70701 0.853504 0.521086i \(-0.174473\pi\)
0.853504 + 0.521086i \(0.174473\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.16848 −0.0851201
\(650\) −48.3241 −1.89543
\(651\) 0 0
\(652\) 15.9446 0.624439
\(653\) −6.06220 −0.237232 −0.118616 0.992940i \(-0.537846\pi\)
−0.118616 + 0.992940i \(0.537846\pi\)
\(654\) −18.6814 −0.730499
\(655\) −64.0190 −2.50143
\(656\) 8.88553 0.346922
\(657\) −1.92536 −0.0751156
\(658\) 0 0
\(659\) −0.570382 −0.0222189 −0.0111095 0.999938i \(-0.503536\pi\)
−0.0111095 + 0.999938i \(0.503536\pi\)
\(660\) 0.962923 0.0374817
\(661\) −17.6396 −0.686101 −0.343051 0.939317i \(-0.611460\pi\)
−0.343051 + 0.939317i \(0.611460\pi\)
\(662\) 4.66762 0.181412
\(663\) −40.1339 −1.55867
\(664\) 0.146030 0.00566706
\(665\) 0 0
\(666\) 5.24112 0.203089
\(667\) −26.4507 −1.02417
\(668\) 17.6073 0.681248
\(669\) −3.92861 −0.151889
\(670\) −47.5508 −1.83705
\(671\) 2.53977 0.0980468
\(672\) 0 0
\(673\) −39.1296 −1.50833 −0.754167 0.656682i \(-0.771959\pi\)
−0.754167 + 0.656682i \(0.771959\pi\)
\(674\) 11.4639 0.441574
\(675\) −6.98062 −0.268684
\(676\) 34.9225 1.34317
\(677\) 35.4149 1.36110 0.680552 0.732700i \(-0.261740\pi\)
0.680552 + 0.732700i \(0.261740\pi\)
\(678\) 3.29913 0.126702
\(679\) 0 0
\(680\) 20.0670 0.769533
\(681\) 7.03971 0.269762
\(682\) −2.04463 −0.0782930
\(683\) −36.2898 −1.38859 −0.694295 0.719691i \(-0.744284\pi\)
−0.694295 + 0.719691i \(0.744284\pi\)
\(684\) 1.00000 0.0382360
\(685\) 42.1173 1.60922
\(686\) 0 0
\(687\) −5.88397 −0.224488
\(688\) 10.3705 0.395373
\(689\) 58.0684 2.21223
\(690\) −10.7035 −0.407475
\(691\) 33.7147 1.28257 0.641284 0.767303i \(-0.278402\pi\)
0.641284 + 0.767303i \(0.278402\pi\)
\(692\) 10.0106 0.380547
\(693\) 0 0
\(694\) −7.84258 −0.297700
\(695\) 10.1011 0.383156
\(696\) −8.55364 −0.324225
\(697\) 51.5140 1.95123
\(698\) −21.2991 −0.806184
\(699\) −2.80658 −0.106155
\(700\) 0 0
\(701\) 12.6976 0.479583 0.239791 0.970824i \(-0.422921\pi\)
0.239791 + 0.970824i \(0.422921\pi\)
\(702\) 6.92261 0.261277
\(703\) −5.24112 −0.197673
\(704\) 0.278197 0.0104849
\(705\) −6.20680 −0.233762
\(706\) −8.71701 −0.328069
\(707\) 0 0
\(708\) 7.79476 0.292945
\(709\) −21.2791 −0.799155 −0.399578 0.916699i \(-0.630843\pi\)
−0.399578 + 0.916699i \(0.630843\pi\)
\(710\) 2.37005 0.0889463
\(711\) 1.40761 0.0527893
\(712\) 13.7007 0.513457
\(713\) 22.7274 0.851147
\(714\) 0 0
\(715\) −6.66594 −0.249292
\(716\) 6.25882 0.233903
\(717\) 12.4016 0.463146
\(718\) −1.17128 −0.0437119
\(719\) 48.2232 1.79842 0.899210 0.437516i \(-0.144142\pi\)
0.899210 + 0.437516i \(0.144142\pi\)
\(720\) −3.46130 −0.128995
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 15.1965 0.565163
\(724\) −4.25019 −0.157957
\(725\) −59.7097 −2.21756
\(726\) −10.9226 −0.405376
\(727\) 49.2945 1.82823 0.914116 0.405452i \(-0.132886\pi\)
0.914116 + 0.405452i \(0.132886\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.66427 −0.246656
\(731\) 60.1233 2.22374
\(732\) −9.12941 −0.337433
\(733\) −0.924289 −0.0341394 −0.0170697 0.999854i \(-0.505434\pi\)
−0.0170697 + 0.999854i \(0.505434\pi\)
\(734\) −5.01295 −0.185031
\(735\) 0 0
\(736\) −3.09233 −0.113985
\(737\) −3.82182 −0.140778
\(738\) −8.88553 −0.327081
\(739\) 2.35176 0.0865108 0.0432554 0.999064i \(-0.486227\pi\)
0.0432554 + 0.999064i \(0.486227\pi\)
\(740\) 18.1411 0.666880
\(741\) −6.92261 −0.254308
\(742\) 0 0
\(743\) −50.4616 −1.85126 −0.925628 0.378433i \(-0.876463\pi\)
−0.925628 + 0.378433i \(0.876463\pi\)
\(744\) 7.34959 0.269449
\(745\) 14.4161 0.528164
\(746\) −1.86584 −0.0683133
\(747\) −0.146030 −0.00534296
\(748\) 1.61285 0.0589716
\(749\) 0 0
\(750\) −6.85553 −0.250328
\(751\) −6.22282 −0.227074 −0.113537 0.993534i \(-0.536218\pi\)
−0.113537 + 0.993534i \(0.536218\pi\)
\(752\) −1.79320 −0.0653912
\(753\) −5.91198 −0.215445
\(754\) 59.2135 2.15643
\(755\) −66.7181 −2.42812
\(756\) 0 0
\(757\) 19.9762 0.726047 0.363023 0.931780i \(-0.381745\pi\)
0.363023 + 0.931780i \(0.381745\pi\)
\(758\) 11.8065 0.428830
\(759\) −0.860277 −0.0312261
\(760\) 3.46130 0.125555
\(761\) 15.3054 0.554822 0.277411 0.960751i \(-0.410524\pi\)
0.277411 + 0.960751i \(0.410524\pi\)
\(762\) −3.45699 −0.125233
\(763\) 0 0
\(764\) −10.9479 −0.396080
\(765\) −20.0670 −0.725522
\(766\) −5.52639 −0.199677
\(767\) −53.9600 −1.94838
\(768\) −1.00000 −0.0360844
\(769\) −18.0461 −0.650761 −0.325380 0.945583i \(-0.605492\pi\)
−0.325380 + 0.945583i \(0.605492\pi\)
\(770\) 0 0
\(771\) −1.40485 −0.0505943
\(772\) 21.2899 0.766240
\(773\) −40.2781 −1.44870 −0.724351 0.689432i \(-0.757860\pi\)
−0.724351 + 0.689432i \(0.757860\pi\)
\(774\) −10.3705 −0.372761
\(775\) 51.3047 1.84292
\(776\) −4.09233 −0.146906
\(777\) 0 0
\(778\) −1.26421 −0.0453241
\(779\) 8.88553 0.318357
\(780\) 23.9612 0.857950
\(781\) 0.190489 0.00681622
\(782\) −17.9278 −0.641098
\(783\) 8.55364 0.305682
\(784\) 0 0
\(785\) 74.2687 2.65076
\(786\) 18.4956 0.659717
\(787\) −22.2856 −0.794395 −0.397198 0.917733i \(-0.630017\pi\)
−0.397198 + 0.917733i \(0.630017\pi\)
\(788\) −6.63103 −0.236221
\(789\) 10.7781 0.383712
\(790\) 4.87215 0.173343
\(791\) 0 0
\(792\) −0.278197 −0.00988529
\(793\) 63.1993 2.24427
\(794\) 1.40485 0.0498562
\(795\) 29.0342 1.02974
\(796\) −7.87047 −0.278961
\(797\) −11.6159 −0.411457 −0.205728 0.978609i \(-0.565956\pi\)
−0.205728 + 0.978609i \(0.565956\pi\)
\(798\) 0 0
\(799\) −10.3961 −0.367787
\(800\) −6.98062 −0.246802
\(801\) −13.7007 −0.484092
\(802\) 4.77119 0.168476
\(803\) −0.535630 −0.0189020
\(804\) 13.7378 0.484495
\(805\) 0 0
\(806\) −50.8783 −1.79211
\(807\) −22.4462 −0.790145
\(808\) 2.45699 0.0864365
\(809\) −21.9957 −0.773327 −0.386664 0.922221i \(-0.626373\pi\)
−0.386664 + 0.922221i \(0.626373\pi\)
\(810\) 3.46130 0.121618
\(811\) 24.2399 0.851177 0.425588 0.904917i \(-0.360067\pi\)
0.425588 + 0.904917i \(0.360067\pi\)
\(812\) 0 0
\(813\) 2.27651 0.0798408
\(814\) 1.45806 0.0511051
\(815\) −55.1892 −1.93319
\(816\) −5.79751 −0.202954
\(817\) 10.3705 0.362819
\(818\) 18.0890 0.632466
\(819\) 0 0
\(820\) −30.7555 −1.07403
\(821\) −13.4332 −0.468821 −0.234410 0.972138i \(-0.575316\pi\)
−0.234410 + 0.972138i \(0.575316\pi\)
\(822\) −12.1680 −0.424409
\(823\) 9.53653 0.332423 0.166211 0.986090i \(-0.446847\pi\)
0.166211 + 0.986090i \(0.446847\pi\)
\(824\) −17.3468 −0.604306
\(825\) −1.94199 −0.0676113
\(826\) 0 0
\(827\) 46.7140 1.62440 0.812202 0.583376i \(-0.198268\pi\)
0.812202 + 0.583376i \(0.198268\pi\)
\(828\) 3.09233 0.107466
\(829\) 17.3276 0.601812 0.300906 0.953654i \(-0.402711\pi\)
0.300906 + 0.953654i \(0.402711\pi\)
\(830\) −0.505454 −0.0175446
\(831\) −27.4521 −0.952302
\(832\) 6.92261 0.239998
\(833\) 0 0
\(834\) −2.91829 −0.101052
\(835\) −60.9443 −2.10907
\(836\) 0.278197 0.00962164
\(837\) −7.34959 −0.254039
\(838\) −2.70518 −0.0934490
\(839\) −26.5740 −0.917435 −0.458718 0.888582i \(-0.651691\pi\)
−0.458718 + 0.888582i \(0.651691\pi\)
\(840\) 0 0
\(841\) 44.1647 1.52292
\(842\) −6.14064 −0.211620
\(843\) −8.14279 −0.280453
\(844\) 14.9123 0.513302
\(845\) −120.877 −4.15831
\(846\) 1.79320 0.0616514
\(847\) 0 0
\(848\) 8.38823 0.288053
\(849\) 8.18586 0.280938
\(850\) −40.4702 −1.38812
\(851\) −16.2073 −0.555579
\(852\) −0.684727 −0.0234583
\(853\) −4.35464 −0.149100 −0.0745500 0.997217i \(-0.523752\pi\)
−0.0745500 + 0.997217i \(0.523752\pi\)
\(854\) 0 0
\(855\) −3.46130 −0.118374
\(856\) 11.4779 0.392307
\(857\) −44.7011 −1.52696 −0.763480 0.645832i \(-0.776511\pi\)
−0.763480 + 0.645832i \(0.776511\pi\)
\(858\) 1.92585 0.0657473
\(859\) 5.42050 0.184945 0.0924726 0.995715i \(-0.470523\pi\)
0.0924726 + 0.995715i \(0.470523\pi\)
\(860\) −35.8955 −1.22403
\(861\) 0 0
\(862\) −21.2765 −0.724681
\(863\) −41.7588 −1.42148 −0.710742 0.703453i \(-0.751641\pi\)
−0.710742 + 0.703453i \(0.751641\pi\)
\(864\) 1.00000 0.0340207
\(865\) −34.6498 −1.17813
\(866\) 9.42099 0.320138
\(867\) −16.6112 −0.564145
\(868\) 0 0
\(869\) 0.391591 0.0132838
\(870\) 29.6067 1.00376
\(871\) −95.1015 −3.22239
\(872\) 18.6814 0.632631
\(873\) 4.09233 0.138504
\(874\) −3.09233 −0.104600
\(875\) 0 0
\(876\) 1.92536 0.0650520
\(877\) 35.9401 1.21361 0.606805 0.794851i \(-0.292451\pi\)
0.606805 + 0.794851i \(0.292451\pi\)
\(878\) −12.5924 −0.424973
\(879\) 19.3421 0.652392
\(880\) −0.962923 −0.0324601
\(881\) 49.3541 1.66278 0.831391 0.555688i \(-0.187545\pi\)
0.831391 + 0.555688i \(0.187545\pi\)
\(882\) 0 0
\(883\) 12.0927 0.406951 0.203476 0.979080i \(-0.434776\pi\)
0.203476 + 0.979080i \(0.434776\pi\)
\(884\) 40.1339 1.34985
\(885\) −26.9800 −0.906923
\(886\) 23.2947 0.782601
\(887\) 27.5621 0.925445 0.462722 0.886503i \(-0.346873\pi\)
0.462722 + 0.886503i \(0.346873\pi\)
\(888\) −5.24112 −0.175880
\(889\) 0 0
\(890\) −47.4224 −1.58960
\(891\) 0.278197 0.00931994
\(892\) 3.92861 0.131539
\(893\) −1.79320 −0.0600071
\(894\) −4.16493 −0.139296
\(895\) −21.6637 −0.724136
\(896\) 0 0
\(897\) −21.4070 −0.714759
\(898\) 0.486074 0.0162205
\(899\) −62.8657 −2.09669
\(900\) 6.98062 0.232687
\(901\) 48.6309 1.62013
\(902\) −2.47193 −0.0823061
\(903\) 0 0
\(904\) −3.29913 −0.109728
\(905\) 14.7112 0.489016
\(906\) 19.2754 0.640384
\(907\) 40.3679 1.34039 0.670197 0.742183i \(-0.266210\pi\)
0.670197 + 0.742183i \(0.266210\pi\)
\(908\) −7.03971 −0.233621
\(909\) −2.45699 −0.0814931
\(910\) 0 0
\(911\) −6.02988 −0.199779 −0.0998894 0.994999i \(-0.531849\pi\)
−0.0998894 + 0.994999i \(0.531849\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −0.0406251 −0.00134449
\(914\) −14.4597 −0.478286
\(915\) 31.5997 1.04465
\(916\) 5.88397 0.194412
\(917\) 0 0
\(918\) 5.79751 0.191346
\(919\) 51.5149 1.69932 0.849659 0.527332i \(-0.176807\pi\)
0.849659 + 0.527332i \(0.176807\pi\)
\(920\) 10.7035 0.352884
\(921\) 11.7007 0.385553
\(922\) −18.0188 −0.593417
\(923\) 4.74009 0.156022
\(924\) 0 0
\(925\) −36.5863 −1.20295
\(926\) −23.3408 −0.767027
\(927\) 17.3468 0.569745
\(928\) 8.55364 0.280787
\(929\) −8.68537 −0.284958 −0.142479 0.989798i \(-0.545507\pi\)
−0.142479 + 0.989798i \(0.545507\pi\)
\(930\) −25.4392 −0.834183
\(931\) 0 0
\(932\) 2.80658 0.0919326
\(933\) −13.8555 −0.453610
\(934\) 8.14879 0.266636
\(935\) −5.58256 −0.182569
\(936\) −6.92261 −0.226272
\(937\) −59.2801 −1.93660 −0.968298 0.249799i \(-0.919635\pi\)
−0.968298 + 0.249799i \(0.919635\pi\)
\(938\) 0 0
\(939\) 1.97199 0.0643534
\(940\) 6.20680 0.202444
\(941\) −14.2282 −0.463825 −0.231913 0.972737i \(-0.574498\pi\)
−0.231913 + 0.972737i \(0.574498\pi\)
\(942\) −21.4569 −0.699102
\(943\) 27.4770 0.894775
\(944\) −7.79476 −0.253698
\(945\) 0 0
\(946\) −2.88505 −0.0938010
\(947\) 15.6081 0.507194 0.253597 0.967310i \(-0.418386\pi\)
0.253597 + 0.967310i \(0.418386\pi\)
\(948\) −1.40761 −0.0457169
\(949\) −13.3285 −0.432663
\(950\) −6.98062 −0.226481
\(951\) −19.9005 −0.645317
\(952\) 0 0
\(953\) 9.11279 0.295192 0.147596 0.989048i \(-0.452846\pi\)
0.147596 + 0.989048i \(0.452846\pi\)
\(954\) −8.38823 −0.271579
\(955\) 37.8939 1.22622
\(956\) −12.4016 −0.401097
\(957\) 2.37959 0.0769213
\(958\) −27.1627 −0.877585
\(959\) 0 0
\(960\) 3.46130 0.111713
\(961\) 23.0165 0.742468
\(962\) 36.2822 1.16979
\(963\) −11.4779 −0.369871
\(964\) −15.1965 −0.489446
\(965\) −73.6908 −2.37219
\(966\) 0 0
\(967\) −4.04494 −0.130077 −0.0650383 0.997883i \(-0.520717\pi\)
−0.0650383 + 0.997883i \(0.520717\pi\)
\(968\) 10.9226 0.351066
\(969\) −5.79751 −0.186243
\(970\) 14.1648 0.454804
\(971\) 18.0859 0.580403 0.290201 0.956966i \(-0.406278\pi\)
0.290201 + 0.956966i \(0.406278\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 11.9423 0.382657
\(975\) −48.3241 −1.54761
\(976\) 9.12941 0.292225
\(977\) −43.1513 −1.38053 −0.690266 0.723556i \(-0.742507\pi\)
−0.690266 + 0.723556i \(0.742507\pi\)
\(978\) 15.9446 0.509853
\(979\) −3.81150 −0.121816
\(980\) 0 0
\(981\) −18.6814 −0.596450
\(982\) 7.00863 0.223654
\(983\) −31.5917 −1.00762 −0.503809 0.863815i \(-0.668068\pi\)
−0.503809 + 0.863815i \(0.668068\pi\)
\(984\) 8.88553 0.283260
\(985\) 22.9520 0.731312
\(986\) 49.5898 1.57926
\(987\) 0 0
\(988\) 6.92261 0.220237
\(989\) 32.0691 1.01974
\(990\) 0.962923 0.0306037
\(991\) 36.7951 1.16883 0.584417 0.811453i \(-0.301323\pi\)
0.584417 + 0.811453i \(0.301323\pi\)
\(992\) −7.34959 −0.233350
\(993\) 4.66762 0.148123
\(994\) 0 0
\(995\) 27.2421 0.863632
\(996\) 0.146030 0.00462714
\(997\) −42.3123 −1.34005 −0.670023 0.742341i \(-0.733716\pi\)
−0.670023 + 0.742341i \(0.733716\pi\)
\(998\) 9.12604 0.288880
\(999\) 5.24112 0.165822
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 5586.2.a.bw.1.1 4
7.2 even 3 798.2.j.l.571.4 yes 8
7.4 even 3 798.2.j.l.457.4 8
7.6 odd 2 5586.2.a.bz.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.l.457.4 8 7.4 even 3
798.2.j.l.571.4 yes 8 7.2 even 3
5586.2.a.bw.1.1 4 1.1 even 1 trivial
5586.2.a.bz.1.4 4 7.6 odd 2