Properties

Label 950.2.a.h.1.2
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.56155 q^{6} +2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.56155 q^{3} +1.00000 q^{4} +2.56155 q^{6} +2.56155 q^{7} +1.00000 q^{8} +3.56155 q^{9} +4.00000 q^{11} +2.56155 q^{12} -5.68466 q^{13} +2.56155 q^{14} +1.00000 q^{16} -3.43845 q^{17} +3.56155 q^{18} -1.00000 q^{19} +6.56155 q^{21} +4.00000 q^{22} -7.68466 q^{23} +2.56155 q^{24} -5.68466 q^{26} +1.43845 q^{27} +2.56155 q^{28} -5.68466 q^{29} -5.12311 q^{31} +1.00000 q^{32} +10.2462 q^{33} -3.43845 q^{34} +3.56155 q^{36} +6.00000 q^{37} -1.00000 q^{38} -14.5616 q^{39} +12.2462 q^{41} +6.56155 q^{42} +2.87689 q^{43} +4.00000 q^{44} -7.68466 q^{46} -6.24621 q^{47} +2.56155 q^{48} -0.438447 q^{49} -8.80776 q^{51} -5.68466 q^{52} +4.56155 q^{53} +1.43845 q^{54} +2.56155 q^{56} -2.56155 q^{57} -5.68466 q^{58} +2.56155 q^{59} +11.1231 q^{61} -5.12311 q^{62} +9.12311 q^{63} +1.00000 q^{64} +10.2462 q^{66} +2.56155 q^{67} -3.43845 q^{68} -19.6847 q^{69} +10.2462 q^{71} +3.56155 q^{72} +1.68466 q^{73} +6.00000 q^{74} -1.00000 q^{76} +10.2462 q^{77} -14.5616 q^{78} -5.12311 q^{79} -7.00000 q^{81} +12.2462 q^{82} -2.87689 q^{83} +6.56155 q^{84} +2.87689 q^{86} -14.5616 q^{87} +4.00000 q^{88} +2.00000 q^{89} -14.5616 q^{91} -7.68466 q^{92} -13.1231 q^{93} -6.24621 q^{94} +2.56155 q^{96} -6.00000 q^{97} -0.438447 q^{98} +14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} + q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} + 2q^{4} + q^{6} + q^{7} + 2q^{8} + 3q^{9} + 8q^{11} + q^{12} + q^{13} + q^{14} + 2q^{16} - 11q^{17} + 3q^{18} - 2q^{19} + 9q^{21} + 8q^{22} - 3q^{23} + q^{24} + q^{26} + 7q^{27} + q^{28} + q^{29} - 2q^{31} + 2q^{32} + 4q^{33} - 11q^{34} + 3q^{36} + 12q^{37} - 2q^{38} - 25q^{39} + 8q^{41} + 9q^{42} + 14q^{43} + 8q^{44} - 3q^{46} + 4q^{47} + q^{48} - 5q^{49} + 3q^{51} + q^{52} + 5q^{53} + 7q^{54} + q^{56} - q^{57} + q^{58} + q^{59} + 14q^{61} - 2q^{62} + 10q^{63} + 2q^{64} + 4q^{66} + q^{67} - 11q^{68} - 27q^{69} + 4q^{71} + 3q^{72} - 9q^{73} + 12q^{74} - 2q^{76} + 4q^{77} - 25q^{78} - 2q^{79} - 14q^{81} + 8q^{82} - 14q^{83} + 9q^{84} + 14q^{86} - 25q^{87} + 8q^{88} + 4q^{89} - 25q^{91} - 3q^{92} - 18q^{93} + 4q^{94} + q^{96} - 12q^{97} - 5q^{98} + 12q^{99} + O(q^{100}) \)

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
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.56155 1.04575
\(7\) 2.56155 0.968176 0.484088 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 2.56155 0.739457
\(13\) −5.68466 −1.57664 −0.788320 0.615265i \(-0.789049\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 2.56155 0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.43845 −0.833946 −0.416973 0.908919i \(-0.636909\pi\)
−0.416973 + 0.908919i \(0.636909\pi\)
\(18\) 3.56155 0.839466
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 6.56155 1.43185
\(22\) 4.00000 0.852803
\(23\) −7.68466 −1.60236 −0.801181 0.598422i \(-0.795795\pi\)
−0.801181 + 0.598422i \(0.795795\pi\)
\(24\) 2.56155 0.522875
\(25\) 0 0
\(26\) −5.68466 −1.11485
\(27\) 1.43845 0.276829
\(28\) 2.56155 0.484088
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 0 0
\(31\) −5.12311 −0.920137 −0.460068 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) 1.00000 0.176777
\(33\) 10.2462 1.78364
\(34\) −3.43845 −0.589689
\(35\) 0 0
\(36\) 3.56155 0.593592
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.00000 −0.162221
\(39\) −14.5616 −2.33171
\(40\) 0 0
\(41\) 12.2462 1.91254 0.956268 0.292490i \(-0.0944840\pi\)
0.956268 + 0.292490i \(0.0944840\pi\)
\(42\) 6.56155 1.01247
\(43\) 2.87689 0.438722 0.219361 0.975644i \(-0.429603\pi\)
0.219361 + 0.975644i \(0.429603\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −7.68466 −1.13304
\(47\) −6.24621 −0.911104 −0.455552 0.890209i \(-0.650558\pi\)
−0.455552 + 0.890209i \(0.650558\pi\)
\(48\) 2.56155 0.369728
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) −8.80776 −1.23333
\(52\) −5.68466 −0.788320
\(53\) 4.56155 0.626577 0.313289 0.949658i \(-0.398569\pi\)
0.313289 + 0.949658i \(0.398569\pi\)
\(54\) 1.43845 0.195748
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) −2.56155 −0.339286
\(58\) −5.68466 −0.746432
\(59\) 2.56155 0.333486 0.166743 0.986000i \(-0.446675\pi\)
0.166743 + 0.986000i \(0.446675\pi\)
\(60\) 0 0
\(61\) 11.1231 1.42417 0.712084 0.702094i \(-0.247752\pi\)
0.712084 + 0.702094i \(0.247752\pi\)
\(62\) −5.12311 −0.650635
\(63\) 9.12311 1.14940
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.2462 1.26122
\(67\) 2.56155 0.312943 0.156472 0.987682i \(-0.449988\pi\)
0.156472 + 0.987682i \(0.449988\pi\)
\(68\) −3.43845 −0.416973
\(69\) −19.6847 −2.36975
\(70\) 0 0
\(71\) 10.2462 1.21600 0.608001 0.793936i \(-0.291972\pi\)
0.608001 + 0.793936i \(0.291972\pi\)
\(72\) 3.56155 0.419733
\(73\) 1.68466 0.197174 0.0985872 0.995128i \(-0.468568\pi\)
0.0985872 + 0.995128i \(0.468568\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 10.2462 1.16766
\(78\) −14.5616 −1.64877
\(79\) −5.12311 −0.576394 −0.288197 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 12.2462 1.35237
\(83\) −2.87689 −0.315780 −0.157890 0.987457i \(-0.550469\pi\)
−0.157890 + 0.987457i \(0.550469\pi\)
\(84\) 6.56155 0.715924
\(85\) 0 0
\(86\) 2.87689 0.310223
\(87\) −14.5616 −1.56116
\(88\) 4.00000 0.426401
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −14.5616 −1.52647
\(92\) −7.68466 −0.801181
\(93\) −13.1231 −1.36080
\(94\) −6.24621 −0.644247
\(95\) 0 0
\(96\) 2.56155 0.261437
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −0.438447 −0.0442899
\(99\) 14.2462 1.43180
\(100\) 0 0
\(101\) −17.3693 −1.72831 −0.864156 0.503224i \(-0.832147\pi\)
−0.864156 + 0.503224i \(0.832147\pi\)
\(102\) −8.80776 −0.872099
\(103\) −2.24621 −0.221326 −0.110663 0.993858i \(-0.535297\pi\)
−0.110663 + 0.993858i \(0.535297\pi\)
\(104\) −5.68466 −0.557427
\(105\) 0 0
\(106\) 4.56155 0.443057
\(107\) 5.43845 0.525755 0.262877 0.964829i \(-0.415329\pi\)
0.262877 + 0.964829i \(0.415329\pi\)
\(108\) 1.43845 0.138415
\(109\) −0.561553 −0.0537870 −0.0268935 0.999638i \(-0.508561\pi\)
−0.0268935 + 0.999638i \(0.508561\pi\)
\(110\) 0 0
\(111\) 15.3693 1.45879
\(112\) 2.56155 0.242044
\(113\) −8.87689 −0.835068 −0.417534 0.908661i \(-0.637106\pi\)
−0.417534 + 0.908661i \(0.637106\pi\)
\(114\) −2.56155 −0.239911
\(115\) 0 0
\(116\) −5.68466 −0.527807
\(117\) −20.2462 −1.87176
\(118\) 2.56155 0.235810
\(119\) −8.80776 −0.807406
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 11.1231 1.00704
\(123\) 31.3693 2.82848
\(124\) −5.12311 −0.460068
\(125\) 0 0
\(126\) 9.12311 0.812751
\(127\) −13.1231 −1.16449 −0.582244 0.813014i \(-0.697825\pi\)
−0.582244 + 0.813014i \(0.697825\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.36932 0.648832
\(130\) 0 0
\(131\) −16.4924 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(132\) 10.2462 0.891818
\(133\) −2.56155 −0.222115
\(134\) 2.56155 0.221284
\(135\) 0 0
\(136\) −3.43845 −0.294844
\(137\) 14.8078 1.26511 0.632556 0.774514i \(-0.282006\pi\)
0.632556 + 0.774514i \(0.282006\pi\)
\(138\) −19.6847 −1.67567
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 10.2462 0.859843
\(143\) −22.7386 −1.90150
\(144\) 3.56155 0.296796
\(145\) 0 0
\(146\) 1.68466 0.139423
\(147\) −1.12311 −0.0926322
\(148\) 6.00000 0.493197
\(149\) 13.3693 1.09526 0.547629 0.836722i \(-0.315531\pi\)
0.547629 + 0.836722i \(0.315531\pi\)
\(150\) 0 0
\(151\) 5.12311 0.416912 0.208456 0.978032i \(-0.433156\pi\)
0.208456 + 0.978032i \(0.433156\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −12.2462 −0.990048
\(154\) 10.2462 0.825663
\(155\) 0 0
\(156\) −14.5616 −1.16586
\(157\) 20.2462 1.61582 0.807912 0.589303i \(-0.200598\pi\)
0.807912 + 0.589303i \(0.200598\pi\)
\(158\) −5.12311 −0.407572
\(159\) 11.6847 0.926654
\(160\) 0 0
\(161\) −19.6847 −1.55137
\(162\) −7.00000 −0.549972
\(163\) 15.3693 1.20382 0.601909 0.798565i \(-0.294407\pi\)
0.601909 + 0.798565i \(0.294407\pi\)
\(164\) 12.2462 0.956268
\(165\) 0 0
\(166\) −2.87689 −0.223290
\(167\) 7.36932 0.570255 0.285127 0.958490i \(-0.407964\pi\)
0.285127 + 0.958490i \(0.407964\pi\)
\(168\) 6.56155 0.506235
\(169\) 19.3153 1.48580
\(170\) 0 0
\(171\) −3.56155 −0.272359
\(172\) 2.87689 0.219361
\(173\) −20.2462 −1.53929 −0.769645 0.638471i \(-0.779567\pi\)
−0.769645 + 0.638471i \(0.779567\pi\)
\(174\) −14.5616 −1.10391
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 6.56155 0.493197
\(178\) 2.00000 0.149906
\(179\) −22.2462 −1.66276 −0.831380 0.555704i \(-0.812449\pi\)
−0.831380 + 0.555704i \(0.812449\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −14.5616 −1.07937
\(183\) 28.4924 2.10622
\(184\) −7.68466 −0.566521
\(185\) 0 0
\(186\) −13.1231 −0.962233
\(187\) −13.7538 −1.00578
\(188\) −6.24621 −0.455552
\(189\) 3.68466 0.268019
\(190\) 0 0
\(191\) −3.68466 −0.266613 −0.133306 0.991075i \(-0.542559\pi\)
−0.133306 + 0.991075i \(0.542559\pi\)
\(192\) 2.56155 0.184864
\(193\) 14.4924 1.04319 0.521594 0.853194i \(-0.325338\pi\)
0.521594 + 0.853194i \(0.325338\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −0.438447 −0.0313177
\(197\) 20.2462 1.44248 0.721241 0.692684i \(-0.243572\pi\)
0.721241 + 0.692684i \(0.243572\pi\)
\(198\) 14.2462 1.01243
\(199\) 16.8078 1.19147 0.595735 0.803181i \(-0.296861\pi\)
0.595735 + 0.803181i \(0.296861\pi\)
\(200\) 0 0
\(201\) 6.56155 0.462816
\(202\) −17.3693 −1.22210
\(203\) −14.5616 −1.02202
\(204\) −8.80776 −0.616667
\(205\) 0 0
\(206\) −2.24621 −0.156501
\(207\) −27.3693 −1.90230
\(208\) −5.68466 −0.394160
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 8.31534 0.572452 0.286226 0.958162i \(-0.407599\pi\)
0.286226 + 0.958162i \(0.407599\pi\)
\(212\) 4.56155 0.313289
\(213\) 26.2462 1.79836
\(214\) 5.43845 0.371765
\(215\) 0 0
\(216\) 1.43845 0.0978739
\(217\) −13.1231 −0.890854
\(218\) −0.561553 −0.0380332
\(219\) 4.31534 0.291604
\(220\) 0 0
\(221\) 19.5464 1.31483
\(222\) 15.3693 1.03152
\(223\) 23.3693 1.56493 0.782463 0.622698i \(-0.213963\pi\)
0.782463 + 0.622698i \(0.213963\pi\)
\(224\) 2.56155 0.171151
\(225\) 0 0
\(226\) −8.87689 −0.590482
\(227\) −25.9309 −1.72109 −0.860546 0.509373i \(-0.829878\pi\)
−0.860546 + 0.509373i \(0.829878\pi\)
\(228\) −2.56155 −0.169643
\(229\) −14.4924 −0.957686 −0.478843 0.877900i \(-0.658944\pi\)
−0.478843 + 0.877900i \(0.658944\pi\)
\(230\) 0 0
\(231\) 26.2462 1.72687
\(232\) −5.68466 −0.373216
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) −20.2462 −1.32354
\(235\) 0 0
\(236\) 2.56155 0.166743
\(237\) −13.1231 −0.852437
\(238\) −8.80776 −0.570923
\(239\) −1.43845 −0.0930454 −0.0465227 0.998917i \(-0.514814\pi\)
−0.0465227 + 0.998917i \(0.514814\pi\)
\(240\) 0 0
\(241\) 23.1231 1.48949 0.744745 0.667349i \(-0.232571\pi\)
0.744745 + 0.667349i \(0.232571\pi\)
\(242\) 5.00000 0.321412
\(243\) −22.2462 −1.42710
\(244\) 11.1231 0.712084
\(245\) 0 0
\(246\) 31.3693 2.00003
\(247\) 5.68466 0.361706
\(248\) −5.12311 −0.325318
\(249\) −7.36932 −0.467011
\(250\) 0 0
\(251\) 6.24621 0.394257 0.197129 0.980378i \(-0.436838\pi\)
0.197129 + 0.980378i \(0.436838\pi\)
\(252\) 9.12311 0.574702
\(253\) −30.7386 −1.93252
\(254\) −13.1231 −0.823417
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 7.36932 0.458794
\(259\) 15.3693 0.955003
\(260\) 0 0
\(261\) −20.2462 −1.25321
\(262\) −16.4924 −1.01891
\(263\) 22.2462 1.37176 0.685880 0.727715i \(-0.259417\pi\)
0.685880 + 0.727715i \(0.259417\pi\)
\(264\) 10.2462 0.630611
\(265\) 0 0
\(266\) −2.56155 −0.157059
\(267\) 5.12311 0.313529
\(268\) 2.56155 0.156472
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −21.9309 −1.33221 −0.666103 0.745860i \(-0.732039\pi\)
−0.666103 + 0.745860i \(0.732039\pi\)
\(272\) −3.43845 −0.208486
\(273\) −37.3002 −2.25751
\(274\) 14.8078 0.894570
\(275\) 0 0
\(276\) −19.6847 −1.18488
\(277\) −0.876894 −0.0526875 −0.0263437 0.999653i \(-0.508386\pi\)
−0.0263437 + 0.999653i \(0.508386\pi\)
\(278\) 16.4924 0.989150
\(279\) −18.2462 −1.09237
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −16.0000 −0.952786
\(283\) 21.1231 1.25564 0.627819 0.778359i \(-0.283948\pi\)
0.627819 + 0.778359i \(0.283948\pi\)
\(284\) 10.2462 0.608001
\(285\) 0 0
\(286\) −22.7386 −1.34456
\(287\) 31.3693 1.85167
\(288\) 3.56155 0.209867
\(289\) −5.17708 −0.304534
\(290\) 0 0
\(291\) −15.3693 −0.900965
\(292\) 1.68466 0.0985872
\(293\) 22.1771 1.29560 0.647799 0.761811i \(-0.275689\pi\)
0.647799 + 0.761811i \(0.275689\pi\)
\(294\) −1.12311 −0.0655009
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 5.75379 0.333869
\(298\) 13.3693 0.774464
\(299\) 43.6847 2.52635
\(300\) 0 0
\(301\) 7.36932 0.424760
\(302\) 5.12311 0.294802
\(303\) −44.4924 −2.55602
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −12.2462 −0.700069
\(307\) −32.4924 −1.85444 −0.927220 0.374516i \(-0.877809\pi\)
−0.927220 + 0.374516i \(0.877809\pi\)
\(308\) 10.2462 0.583832
\(309\) −5.75379 −0.327322
\(310\) 0 0
\(311\) −3.68466 −0.208938 −0.104469 0.994528i \(-0.533314\pi\)
−0.104469 + 0.994528i \(0.533314\pi\)
\(312\) −14.5616 −0.824386
\(313\) −5.05398 −0.285668 −0.142834 0.989747i \(-0.545621\pi\)
−0.142834 + 0.989747i \(0.545621\pi\)
\(314\) 20.2462 1.14256
\(315\) 0 0
\(316\) −5.12311 −0.288197
\(317\) −13.0540 −0.733184 −0.366592 0.930382i \(-0.619476\pi\)
−0.366592 + 0.930382i \(0.619476\pi\)
\(318\) 11.6847 0.655243
\(319\) −22.7386 −1.27312
\(320\) 0 0
\(321\) 13.9309 0.777545
\(322\) −19.6847 −1.09698
\(323\) 3.43845 0.191320
\(324\) −7.00000 −0.388889
\(325\) 0 0
\(326\) 15.3693 0.851228
\(327\) −1.43845 −0.0795463
\(328\) 12.2462 0.676184
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −2.56155 −0.140796 −0.0703978 0.997519i \(-0.522427\pi\)
−0.0703978 + 0.997519i \(0.522427\pi\)
\(332\) −2.87689 −0.157890
\(333\) 21.3693 1.17103
\(334\) 7.36932 0.403231
\(335\) 0 0
\(336\) 6.56155 0.357962
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 19.3153 1.05062
\(339\) −22.7386 −1.23499
\(340\) 0 0
\(341\) −20.4924 −1.10973
\(342\) −3.56155 −0.192587
\(343\) −19.0540 −1.02882
\(344\) 2.87689 0.155112
\(345\) 0 0
\(346\) −20.2462 −1.08844
\(347\) 8.63068 0.463319 0.231660 0.972797i \(-0.425584\pi\)
0.231660 + 0.972797i \(0.425584\pi\)
\(348\) −14.5616 −0.780581
\(349\) 3.75379 0.200936 0.100468 0.994940i \(-0.467966\pi\)
0.100468 + 0.994940i \(0.467966\pi\)
\(350\) 0 0
\(351\) −8.17708 −0.436460
\(352\) 4.00000 0.213201
\(353\) 3.93087 0.209219 0.104610 0.994513i \(-0.466641\pi\)
0.104610 + 0.994513i \(0.466641\pi\)
\(354\) 6.56155 0.348743
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) −22.5616 −1.19408
\(358\) −22.2462 −1.17575
\(359\) 1.43845 0.0759183 0.0379592 0.999279i \(-0.487914\pi\)
0.0379592 + 0.999279i \(0.487914\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −18.0000 −0.946059
\(363\) 12.8078 0.672233
\(364\) −14.5616 −0.763233
\(365\) 0 0
\(366\) 28.4924 1.48932
\(367\) −6.24621 −0.326050 −0.163025 0.986622i \(-0.552125\pi\)
−0.163025 + 0.986622i \(0.552125\pi\)
\(368\) −7.68466 −0.400591
\(369\) 43.6155 2.27053
\(370\) 0 0
\(371\) 11.6847 0.606637
\(372\) −13.1231 −0.680401
\(373\) 23.4384 1.21360 0.606798 0.794856i \(-0.292454\pi\)
0.606798 + 0.794856i \(0.292454\pi\)
\(374\) −13.7538 −0.711191
\(375\) 0 0
\(376\) −6.24621 −0.322124
\(377\) 32.3153 1.66432
\(378\) 3.68466 0.189518
\(379\) 10.5616 0.542511 0.271255 0.962507i \(-0.412561\pi\)
0.271255 + 0.962507i \(0.412561\pi\)
\(380\) 0 0
\(381\) −33.6155 −1.72218
\(382\) −3.68466 −0.188524
\(383\) −13.7538 −0.702786 −0.351393 0.936228i \(-0.614292\pi\)
−0.351393 + 0.936228i \(0.614292\pi\)
\(384\) 2.56155 0.130719
\(385\) 0 0
\(386\) 14.4924 0.737645
\(387\) 10.2462 0.520844
\(388\) −6.00000 −0.304604
\(389\) −7.12311 −0.361156 −0.180578 0.983561i \(-0.557797\pi\)
−0.180578 + 0.983561i \(0.557797\pi\)
\(390\) 0 0
\(391\) 26.4233 1.33628
\(392\) −0.438447 −0.0221449
\(393\) −42.2462 −2.13104
\(394\) 20.2462 1.01999
\(395\) 0 0
\(396\) 14.2462 0.715899
\(397\) 7.12311 0.357498 0.178749 0.983895i \(-0.442795\pi\)
0.178749 + 0.983895i \(0.442795\pi\)
\(398\) 16.8078 0.842497
\(399\) −6.56155 −0.328489
\(400\) 0 0
\(401\) −3.75379 −0.187455 −0.0937276 0.995598i \(-0.529878\pi\)
−0.0937276 + 0.995598i \(0.529878\pi\)
\(402\) 6.56155 0.327261
\(403\) 29.1231 1.45073
\(404\) −17.3693 −0.864156
\(405\) 0 0
\(406\) −14.5616 −0.722678
\(407\) 24.0000 1.18964
\(408\) −8.80776 −0.436049
\(409\) 24.7386 1.22325 0.611623 0.791149i \(-0.290517\pi\)
0.611623 + 0.791149i \(0.290517\pi\)
\(410\) 0 0
\(411\) 37.9309 1.87099
\(412\) −2.24621 −0.110663
\(413\) 6.56155 0.322873
\(414\) −27.3693 −1.34513
\(415\) 0 0
\(416\) −5.68466 −0.278713
\(417\) 42.2462 2.06881
\(418\) −4.00000 −0.195646
\(419\) 23.8617 1.16572 0.582861 0.812572i \(-0.301933\pi\)
0.582861 + 0.812572i \(0.301933\pi\)
\(420\) 0 0
\(421\) −23.9309 −1.16632 −0.583160 0.812358i \(-0.698184\pi\)
−0.583160 + 0.812358i \(0.698184\pi\)
\(422\) 8.31534 0.404784
\(423\) −22.2462 −1.08165
\(424\) 4.56155 0.221529
\(425\) 0 0
\(426\) 26.2462 1.27163
\(427\) 28.4924 1.37884
\(428\) 5.43845 0.262877
\(429\) −58.2462 −2.81215
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.43845 0.0692073
\(433\) −14.6307 −0.703106 −0.351553 0.936168i \(-0.614346\pi\)
−0.351553 + 0.936168i \(0.614346\pi\)
\(434\) −13.1231 −0.629929
\(435\) 0 0
\(436\) −0.561553 −0.0268935
\(437\) 7.68466 0.367607
\(438\) 4.31534 0.206195
\(439\) −13.1231 −0.626332 −0.313166 0.949698i \(-0.601390\pi\)
−0.313166 + 0.949698i \(0.601390\pi\)
\(440\) 0 0
\(441\) −1.56155 −0.0743597
\(442\) 19.5464 0.929727
\(443\) −2.24621 −0.106721 −0.0533604 0.998575i \(-0.516993\pi\)
−0.0533604 + 0.998575i \(0.516993\pi\)
\(444\) 15.3693 0.729396
\(445\) 0 0
\(446\) 23.3693 1.10657
\(447\) 34.2462 1.61979
\(448\) 2.56155 0.121022
\(449\) −28.7386 −1.35626 −0.678130 0.734942i \(-0.737209\pi\)
−0.678130 + 0.734942i \(0.737209\pi\)
\(450\) 0 0
\(451\) 48.9848 2.30661
\(452\) −8.87689 −0.417534
\(453\) 13.1231 0.616577
\(454\) −25.9309 −1.21700
\(455\) 0 0
\(456\) −2.56155 −0.119956
\(457\) −6.31534 −0.295419 −0.147710 0.989031i \(-0.547190\pi\)
−0.147710 + 0.989031i \(0.547190\pi\)
\(458\) −14.4924 −0.677186
\(459\) −4.94602 −0.230861
\(460\) 0 0
\(461\) 3.75379 0.174831 0.0874157 0.996172i \(-0.472139\pi\)
0.0874157 + 0.996172i \(0.472139\pi\)
\(462\) 26.2462 1.22108
\(463\) 30.2462 1.40566 0.702830 0.711358i \(-0.251919\pi\)
0.702830 + 0.711358i \(0.251919\pi\)
\(464\) −5.68466 −0.263904
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −18.2462 −0.844334 −0.422167 0.906518i \(-0.638730\pi\)
−0.422167 + 0.906518i \(0.638730\pi\)
\(468\) −20.2462 −0.935881
\(469\) 6.56155 0.302984
\(470\) 0 0
\(471\) 51.8617 2.38966
\(472\) 2.56155 0.117905
\(473\) 11.5076 0.529119
\(474\) −13.1231 −0.602764
\(475\) 0 0
\(476\) −8.80776 −0.403703
\(477\) 16.2462 0.743863
\(478\) −1.43845 −0.0657930
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −34.1080 −1.55519
\(482\) 23.1231 1.05323
\(483\) −50.4233 −2.29434
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −22.2462 −1.00911
\(487\) 17.6155 0.798236 0.399118 0.916900i \(-0.369316\pi\)
0.399118 + 0.916900i \(0.369316\pi\)
\(488\) 11.1231 0.503519
\(489\) 39.3693 1.78034
\(490\) 0 0
\(491\) −1.12311 −0.0506850 −0.0253425 0.999679i \(-0.508068\pi\)
−0.0253425 + 0.999679i \(0.508068\pi\)
\(492\) 31.3693 1.41424
\(493\) 19.5464 0.880325
\(494\) 5.68466 0.255765
\(495\) 0 0
\(496\) −5.12311 −0.230034
\(497\) 26.2462 1.17730
\(498\) −7.36932 −0.330227
\(499\) 42.1080 1.88501 0.942505 0.334191i \(-0.108463\pi\)
0.942505 + 0.334191i \(0.108463\pi\)
\(500\) 0 0
\(501\) 18.8769 0.843357
\(502\) 6.24621 0.278782
\(503\) 7.05398 0.314521 0.157261 0.987557i \(-0.449734\pi\)
0.157261 + 0.987557i \(0.449734\pi\)
\(504\) 9.12311 0.406375
\(505\) 0 0
\(506\) −30.7386 −1.36650
\(507\) 49.4773 2.19736
\(508\) −13.1231 −0.582244
\(509\) 2.49242 0.110475 0.0552373 0.998473i \(-0.482408\pi\)
0.0552373 + 0.998473i \(0.482408\pi\)
\(510\) 0 0
\(511\) 4.31534 0.190899
\(512\) 1.00000 0.0441942
\(513\) −1.43845 −0.0635090
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 7.36932 0.324416
\(517\) −24.9848 −1.09883
\(518\) 15.3693 0.675289
\(519\) −51.8617 −2.27648
\(520\) 0 0
\(521\) −3.12311 −0.136826 −0.0684129 0.997657i \(-0.521794\pi\)
−0.0684129 + 0.997657i \(0.521794\pi\)
\(522\) −20.2462 −0.886153
\(523\) 31.6847 1.38547 0.692737 0.721191i \(-0.256405\pi\)
0.692737 + 0.721191i \(0.256405\pi\)
\(524\) −16.4924 −0.720475
\(525\) 0 0
\(526\) 22.2462 0.969981
\(527\) 17.6155 0.767344
\(528\) 10.2462 0.445909
\(529\) 36.0540 1.56756
\(530\) 0 0
\(531\) 9.12311 0.395909
\(532\) −2.56155 −0.111057
\(533\) −69.6155 −3.01538
\(534\) 5.12311 0.221698
\(535\) 0 0
\(536\) 2.56155 0.110642
\(537\) −56.9848 −2.45908
\(538\) −26.0000 −1.12094
\(539\) −1.75379 −0.0755410
\(540\) 0 0
\(541\) −0.384472 −0.0165297 −0.00826487 0.999966i \(-0.502631\pi\)
−0.00826487 + 0.999966i \(0.502631\pi\)
\(542\) −21.9309 −0.942012
\(543\) −46.1080 −1.97868
\(544\) −3.43845 −0.147422
\(545\) 0 0
\(546\) −37.3002 −1.59630
\(547\) −16.4924 −0.705165 −0.352583 0.935781i \(-0.614696\pi\)
−0.352583 + 0.935781i \(0.614696\pi\)
\(548\) 14.8078 0.632556
\(549\) 39.6155 1.69075
\(550\) 0 0
\(551\) 5.68466 0.242175
\(552\) −19.6847 −0.837835
\(553\) −13.1231 −0.558051
\(554\) −0.876894 −0.0372557
\(555\) 0 0
\(556\) 16.4924 0.699435
\(557\) −39.6155 −1.67856 −0.839282 0.543697i \(-0.817024\pi\)
−0.839282 + 0.543697i \(0.817024\pi\)
\(558\) −18.2462 −0.772424
\(559\) −16.3542 −0.691707
\(560\) 0 0
\(561\) −35.2311 −1.48746
\(562\) 2.00000 0.0843649
\(563\) 8.49242 0.357913 0.178956 0.983857i \(-0.442728\pi\)
0.178956 + 0.983857i \(0.442728\pi\)
\(564\) −16.0000 −0.673722
\(565\) 0 0
\(566\) 21.1231 0.887870
\(567\) −17.9309 −0.753026
\(568\) 10.2462 0.429921
\(569\) −3.12311 −0.130927 −0.0654637 0.997855i \(-0.520853\pi\)
−0.0654637 + 0.997855i \(0.520853\pi\)
\(570\) 0 0
\(571\) 21.6155 0.904582 0.452291 0.891870i \(-0.350607\pi\)
0.452291 + 0.891870i \(0.350607\pi\)
\(572\) −22.7386 −0.950750
\(573\) −9.43845 −0.394297
\(574\) 31.3693 1.30933
\(575\) 0 0
\(576\) 3.56155 0.148398
\(577\) 10.3153 0.429433 0.214717 0.976676i \(-0.431117\pi\)
0.214717 + 0.976676i \(0.431117\pi\)
\(578\) −5.17708 −0.215338
\(579\) 37.1231 1.54278
\(580\) 0 0
\(581\) −7.36932 −0.305731
\(582\) −15.3693 −0.637079
\(583\) 18.2462 0.755681
\(584\) 1.68466 0.0697117
\(585\) 0 0
\(586\) 22.1771 0.916127
\(587\) 7.36932 0.304164 0.152082 0.988368i \(-0.451402\pi\)
0.152082 + 0.988368i \(0.451402\pi\)
\(588\) −1.12311 −0.0463161
\(589\) 5.12311 0.211094
\(590\) 0 0
\(591\) 51.8617 2.13331
\(592\) 6.00000 0.246598
\(593\) −7.75379 −0.318410 −0.159205 0.987246i \(-0.550893\pi\)
−0.159205 + 0.987246i \(0.550893\pi\)
\(594\) 5.75379 0.236081
\(595\) 0 0
\(596\) 13.3693 0.547629
\(597\) 43.0540 1.76208
\(598\) 43.6847 1.78640
\(599\) −11.8617 −0.484658 −0.242329 0.970194i \(-0.577911\pi\)
−0.242329 + 0.970194i \(0.577911\pi\)
\(600\) 0 0
\(601\) 18.0000 0.734235 0.367118 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) 7.36932 0.300351
\(603\) 9.12311 0.371522
\(604\) 5.12311 0.208456
\(605\) 0 0
\(606\) −44.4924 −1.80738
\(607\) 21.1231 0.857360 0.428680 0.903456i \(-0.358979\pi\)
0.428680 + 0.903456i \(0.358979\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −37.3002 −1.51148
\(610\) 0 0
\(611\) 35.5076 1.43648
\(612\) −12.2462 −0.495024
\(613\) −5.36932 −0.216865 −0.108432 0.994104i \(-0.534583\pi\)
−0.108432 + 0.994104i \(0.534583\pi\)
\(614\) −32.4924 −1.31129
\(615\) 0 0
\(616\) 10.2462 0.412832
\(617\) −12.2462 −0.493014 −0.246507 0.969141i \(-0.579283\pi\)
−0.246507 + 0.969141i \(0.579283\pi\)
\(618\) −5.75379 −0.231451
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −11.0540 −0.443581
\(622\) −3.68466 −0.147741
\(623\) 5.12311 0.205253
\(624\) −14.5616 −0.582929
\(625\) 0 0
\(626\) −5.05398 −0.201997
\(627\) −10.2462 −0.409194
\(628\) 20.2462 0.807912
\(629\) −20.6307 −0.822599
\(630\) 0 0
\(631\) −20.4924 −0.815790 −0.407895 0.913029i \(-0.633737\pi\)
−0.407895 + 0.913029i \(0.633737\pi\)
\(632\) −5.12311 −0.203786
\(633\) 21.3002 0.846606
\(634\) −13.0540 −0.518440
\(635\) 0 0
\(636\) 11.6847 0.463327
\(637\) 2.49242 0.0987534
\(638\) −22.7386 −0.900231
\(639\) 36.4924 1.44362
\(640\) 0 0
\(641\) 21.8617 0.863487 0.431743 0.901996i \(-0.357899\pi\)
0.431743 + 0.901996i \(0.357899\pi\)
\(642\) 13.9309 0.549808
\(643\) 33.6155 1.32567 0.662834 0.748767i \(-0.269354\pi\)
0.662834 + 0.748767i \(0.269354\pi\)
\(644\) −19.6847 −0.775684
\(645\) 0 0
\(646\) 3.43845 0.135284
\(647\) −5.43845 −0.213807 −0.106904 0.994269i \(-0.534094\pi\)
−0.106904 + 0.994269i \(0.534094\pi\)
\(648\) −7.00000 −0.274986
\(649\) 10.2462 0.402199
\(650\) 0 0
\(651\) −33.6155 −1.31750
\(652\) 15.3693 0.601909
\(653\) 7.12311 0.278749 0.139374 0.990240i \(-0.455491\pi\)
0.139374 + 0.990240i \(0.455491\pi\)
\(654\) −1.43845 −0.0562477
\(655\) 0 0
\(656\) 12.2462 0.478134
\(657\) 6.00000 0.234082
\(658\) −16.0000 −0.623745
\(659\) −14.0691 −0.548056 −0.274028 0.961722i \(-0.588356\pi\)
−0.274028 + 0.961722i \(0.588356\pi\)
\(660\) 0 0
\(661\) −10.8078 −0.420373 −0.210187 0.977661i \(-0.567407\pi\)
−0.210187 + 0.977661i \(0.567407\pi\)
\(662\) −2.56155 −0.0995576
\(663\) 50.0691 1.94452
\(664\) −2.87689 −0.111645
\(665\) 0 0
\(666\) 21.3693 0.828044
\(667\) 43.6847 1.69148
\(668\) 7.36932 0.285127
\(669\) 59.8617 2.31439
\(670\) 0 0
\(671\) 44.4924 1.71761
\(672\) 6.56155 0.253117
\(673\) −21.3693 −0.823727 −0.411863 0.911246i \(-0.635122\pi\)
−0.411863 + 0.911246i \(0.635122\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 19.3153 0.742898
\(677\) −40.5616 −1.55891 −0.779454 0.626460i \(-0.784503\pi\)
−0.779454 + 0.626460i \(0.784503\pi\)
\(678\) −22.7386 −0.873272
\(679\) −15.3693 −0.589820
\(680\) 0 0
\(681\) −66.4233 −2.54535
\(682\) −20.4924 −0.784695
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −3.56155 −0.136179
\(685\) 0 0
\(686\) −19.0540 −0.727484
\(687\) −37.1231 −1.41633
\(688\) 2.87689 0.109681
\(689\) −25.9309 −0.987887
\(690\) 0 0
\(691\) −17.1231 −0.651394 −0.325697 0.945474i \(-0.605599\pi\)
−0.325697 + 0.945474i \(0.605599\pi\)
\(692\) −20.2462 −0.769645
\(693\) 36.4924 1.38623
\(694\) 8.63068 0.327616
\(695\) 0 0
\(696\) −14.5616 −0.551954
\(697\) −42.1080 −1.59495
\(698\) 3.75379 0.142083
\(699\) −25.6155 −0.968868
\(700\) 0 0
\(701\) −20.8769 −0.788509 −0.394255 0.919001i \(-0.628997\pi\)
−0.394255 + 0.919001i \(0.628997\pi\)
\(702\) −8.17708 −0.308624
\(703\) −6.00000 −0.226294
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 3.93087 0.147940
\(707\) −44.4924 −1.67331
\(708\) 6.56155 0.246598
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −18.2462 −0.684286
\(712\) 2.00000 0.0749532
\(713\) 39.3693 1.47439
\(714\) −22.5616 −0.844345
\(715\) 0 0
\(716\) −22.2462 −0.831380
\(717\) −3.68466 −0.137606
\(718\) 1.43845 0.0536824
\(719\) −25.4384 −0.948694 −0.474347 0.880338i \(-0.657316\pi\)
−0.474347 + 0.880338i \(0.657316\pi\)
\(720\) 0 0
\(721\) −5.75379 −0.214282
\(722\) 1.00000 0.0372161
\(723\) 59.2311 2.20283
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 12.8078 0.475341
\(727\) 24.3153 0.901806 0.450903 0.892573i \(-0.351102\pi\)
0.450903 + 0.892573i \(0.351102\pi\)
\(728\) −14.5616 −0.539687
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −9.89205 −0.365871
\(732\) 28.4924 1.05311
\(733\) 36.8769 1.36208 0.681040 0.732247i \(-0.261528\pi\)
0.681040 + 0.732247i \(0.261528\pi\)
\(734\) −6.24621 −0.230552
\(735\) 0 0
\(736\) −7.68466 −0.283260
\(737\) 10.2462 0.377424
\(738\) 43.6155 1.60551
\(739\) 25.1231 0.924168 0.462084 0.886836i \(-0.347102\pi\)
0.462084 + 0.886836i \(0.347102\pi\)
\(740\) 0 0
\(741\) 14.5616 0.534932
\(742\) 11.6847 0.428957
\(743\) −18.8769 −0.692526 −0.346263 0.938137i \(-0.612550\pi\)
−0.346263 + 0.938137i \(0.612550\pi\)
\(744\) −13.1231 −0.481116
\(745\) 0 0
\(746\) 23.4384 0.858143
\(747\) −10.2462 −0.374889
\(748\) −13.7538 −0.502888
\(749\) 13.9309 0.509023
\(750\) 0 0
\(751\) −34.8769 −1.27268 −0.636338 0.771410i \(-0.719552\pi\)
−0.636338 + 0.771410i \(0.719552\pi\)
\(752\) −6.24621 −0.227776
\(753\) 16.0000 0.583072
\(754\) 32.3153 1.17686
\(755\) 0 0
\(756\) 3.68466 0.134010
\(757\) −42.4924 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(758\) 10.5616 0.383613
\(759\) −78.7386 −2.85803
\(760\) 0 0
\(761\) 41.5464 1.50606 0.753028 0.657989i \(-0.228593\pi\)
0.753028 + 0.657989i \(0.228593\pi\)
\(762\) −33.6155 −1.21776
\(763\) −1.43845 −0.0520753
\(764\) −3.68466 −0.133306
\(765\) 0 0
\(766\) −13.7538 −0.496945
\(767\) −14.5616 −0.525787
\(768\) 2.56155 0.0924321
\(769\) 27.4384 0.989456 0.494728 0.869048i \(-0.335268\pi\)
0.494728 + 0.869048i \(0.335268\pi\)
\(770\) 0 0
\(771\) −35.8617 −1.29153
\(772\) 14.4924 0.521594
\(773\) −10.8078 −0.388728 −0.194364 0.980929i \(-0.562264\pi\)
−0.194364 + 0.980929i \(0.562264\pi\)
\(774\) 10.2462 0.368292
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 39.3693 1.41237
\(778\) −7.12311 −0.255376
\(779\) −12.2462 −0.438766
\(780\) 0 0
\(781\) 40.9848 1.46655
\(782\) 26.4233 0.944895
\(783\) −8.17708 −0.292225
\(784\) −0.438447 −0.0156588
\(785\) 0 0
\(786\) −42.2462 −1.50687
\(787\) −11.1922 −0.398960 −0.199480 0.979902i \(-0.563925\pi\)
−0.199480 + 0.979902i \(0.563925\pi\)
\(788\) 20.2462 0.721241
\(789\) 56.9848 2.02871
\(790\) 0 0
\(791\) −22.7386 −0.808493
\(792\) 14.2462 0.506217
\(793\) −63.2311 −2.24540
\(794\) 7.12311 0.252790
\(795\) 0 0
\(796\) 16.8078 0.595735
\(797\) 11.3002 0.400273 0.200137 0.979768i \(-0.435861\pi\)
0.200137 + 0.979768i \(0.435861\pi\)
\(798\) −6.56155 −0.232276
\(799\) 21.4773 0.759811
\(800\) 0 0
\(801\) 7.12311 0.251683
\(802\) −3.75379 −0.132551
\(803\) 6.73863 0.237801
\(804\) 6.56155 0.231408
\(805\) 0 0
\(806\) 29.1231 1.02582
\(807\) −66.6004 −2.34444
\(808\) −17.3693 −0.611050
\(809\) −12.5616 −0.441641 −0.220820 0.975315i \(-0.570873\pi\)
−0.220820 + 0.975315i \(0.570873\pi\)
\(810\) 0 0
\(811\) −20.8078 −0.730659 −0.365330 0.930878i \(-0.619044\pi\)
−0.365330 + 0.930878i \(0.619044\pi\)
\(812\) −14.5616 −0.511010
\(813\) −56.1771 −1.97022
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −8.80776 −0.308333
\(817\) −2.87689 −0.100650
\(818\) 24.7386 0.864966
\(819\) −51.8617 −1.81220
\(820\) 0 0
\(821\) −17.3693 −0.606193 −0.303097 0.952960i \(-0.598021\pi\)
−0.303097 + 0.952960i \(0.598021\pi\)
\(822\) 37.9309 1.32299
\(823\) 29.4384 1.02616 0.513080 0.858341i \(-0.328504\pi\)
0.513080 + 0.858341i \(0.328504\pi\)
\(824\) −2.24621 −0.0782505
\(825\) 0 0
\(826\) 6.56155 0.228306
\(827\) 10.5616 0.367261 0.183631 0.982995i \(-0.441215\pi\)
0.183631 + 0.982995i \(0.441215\pi\)
\(828\) −27.3693 −0.951150
\(829\) −21.0540 −0.731235 −0.365617 0.930765i \(-0.619142\pi\)
−0.365617 + 0.930765i \(0.619142\pi\)
\(830\) 0 0
\(831\) −2.24621 −0.0779202
\(832\) −5.68466 −0.197080
\(833\) 1.50758 0.0522345
\(834\) 42.2462 1.46287
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) −7.36932 −0.254721
\(838\) 23.8617 0.824290
\(839\) 20.4924 0.707477 0.353738 0.935344i \(-0.384910\pi\)
0.353738 + 0.935344i \(0.384910\pi\)
\(840\) 0 0
\(841\) 3.31534 0.114322
\(842\) −23.9309 −0.824712
\(843\) 5.12311 0.176449
\(844\) 8.31534 0.286226
\(845\) 0 0
\(846\) −22.2462 −0.764840
\(847\) 12.8078 0.440080
\(848\) 4.56155 0.156644
\(849\) 54.1080 1.85698
\(850\) 0 0
\(851\) −46.1080 −1.58056
\(852\) 26.2462 0.899180
\(853\) 24.7386 0.847035 0.423517 0.905888i \(-0.360795\pi\)
0.423517 + 0.905888i \(0.360795\pi\)
\(854\) 28.4924 0.974991
\(855\) 0 0
\(856\) 5.43845 0.185882
\(857\) −14.6307 −0.499775 −0.249887 0.968275i \(-0.580394\pi\)
−0.249887 + 0.968275i \(0.580394\pi\)
\(858\) −58.2462 −1.98849
\(859\) −52.9848 −1.80782 −0.903910 0.427723i \(-0.859316\pi\)
−0.903910 + 0.427723i \(0.859316\pi\)
\(860\) 0 0
\(861\) 80.3542 2.73846
\(862\) −16.0000 −0.544962
\(863\) −2.24621 −0.0764619 −0.0382310 0.999269i \(-0.512172\pi\)
−0.0382310 + 0.999269i \(0.512172\pi\)
\(864\) 1.43845 0.0489370
\(865\) 0 0
\(866\) −14.6307 −0.497171
\(867\) −13.2614 −0.450380
\(868\) −13.1231 −0.445427
\(869\) −20.4924 −0.695158
\(870\) 0 0
\(871\) −14.5616 −0.493399
\(872\) −0.561553 −0.0190166
\(873\) −21.3693 −0.723242
\(874\) 7.68466 0.259937
\(875\) 0 0
\(876\) 4.31534 0.145802
\(877\) 3.93087 0.132736 0.0663680 0.997795i \(-0.478859\pi\)
0.0663680 + 0.997795i \(0.478859\pi\)
\(878\) −13.1231 −0.442883
\(879\) 56.8078 1.91608
\(880\) 0 0
\(881\) 42.9848 1.44820 0.724098 0.689697i \(-0.242256\pi\)
0.724098 + 0.689697i \(0.242256\pi\)
\(882\) −1.56155 −0.0525802
\(883\) 6.38447 0.214855 0.107427 0.994213i \(-0.465739\pi\)
0.107427 + 0.994213i \(0.465739\pi\)
\(884\) 19.5464 0.657417
\(885\) 0 0
\(886\) −2.24621 −0.0754629
\(887\) −28.4924 −0.956682 −0.478341 0.878174i \(-0.658762\pi\)
−0.478341 + 0.878174i \(0.658762\pi\)
\(888\) 15.3693 0.515761
\(889\) −33.6155 −1.12743
\(890\) 0 0
\(891\) −28.0000 −0.938035
\(892\) 23.3693 0.782463
\(893\) 6.24621 0.209021
\(894\) 34.2462 1.14536
\(895\) 0 0
\(896\) 2.56155 0.0855755
\(897\) 111.901 3.73625
\(898\) −28.7386 −0.959021
\(899\) 29.1231 0.971310
\(900\) 0 0
\(901\) −15.6847 −0.522532
\(902\) 48.9848 1.63102
\(903\) 18.8769 0.628184
\(904\) −8.87689 −0.295241
\(905\) 0 0
\(906\) 13.1231 0.435986
\(907\) 20.1771 0.669969 0.334984 0.942224i \(-0.391269\pi\)
0.334984 + 0.942224i \(0.391269\pi\)
\(908\) −25.9309 −0.860546
\(909\) −61.8617 −2.05182
\(910\) 0 0
\(911\) 4.49242 0.148841 0.0744203 0.997227i \(-0.476289\pi\)
0.0744203 + 0.997227i \(0.476289\pi\)
\(912\) −2.56155 −0.0848215
\(913\) −11.5076 −0.380845
\(914\) −6.31534 −0.208893
\(915\) 0 0
\(916\) −14.4924 −0.478843
\(917\) −42.2462 −1.39509
\(918\) −4.94602 −0.163243
\(919\) −2.06913 −0.0682543 −0.0341272 0.999417i \(-0.510865\pi\)
−0.0341272 + 0.999417i \(0.510865\pi\)
\(920\) 0 0
\(921\) −83.2311 −2.74256
\(922\) 3.75379 0.123624
\(923\) −58.2462 −1.91720
\(924\) 26.2462 0.863437
\(925\) 0 0
\(926\) 30.2462 0.993952
\(927\) −8.00000 −0.262754
\(928\) −5.68466 −0.186608
\(929\) −19.3002 −0.633219 −0.316609 0.948556i \(-0.602544\pi\)
−0.316609 + 0.948556i \(0.602544\pi\)
\(930\) 0 0
\(931\) 0.438447 0.0143695
\(932\) −10.0000 −0.327561
\(933\) −9.43845 −0.309001
\(934\) −18.2462 −0.597034
\(935\) 0 0
\(936\) −20.2462 −0.661768
\(937\) −40.5616 −1.32509 −0.662544 0.749023i \(-0.730523\pi\)
−0.662544 + 0.749023i \(0.730523\pi\)
\(938\) 6.56155 0.214242
\(939\) −12.9460 −0.422478
\(940\) 0 0
\(941\) 54.8078 1.78668 0.893341 0.449379i \(-0.148355\pi\)
0.893341 + 0.449379i \(0.148355\pi\)
\(942\) 51.8617 1.68975
\(943\) −94.1080 −3.06458
\(944\) 2.56155 0.0833714
\(945\) 0 0
\(946\) 11.5076 0.374144
\(947\) 34.2462 1.11285 0.556426 0.830897i \(-0.312172\pi\)
0.556426 + 0.830897i \(0.312172\pi\)
\(948\) −13.1231 −0.426219
\(949\) −9.57671 −0.310873
\(950\) 0 0
\(951\) −33.4384 −1.08432
\(952\) −8.80776 −0.285461
\(953\) −44.1080 −1.42880 −0.714398 0.699739i \(-0.753300\pi\)
−0.714398 + 0.699739i \(0.753300\pi\)
\(954\) 16.2462 0.525991
\(955\) 0 0
\(956\) −1.43845 −0.0465227
\(957\) −58.2462 −1.88283
\(958\) 32.0000 1.03387
\(959\) 37.9309 1.22485
\(960\) 0 0
\(961\) −4.75379 −0.153348
\(962\) −34.1080 −1.09968
\(963\) 19.3693 0.624168
\(964\) 23.1231 0.744745
\(965\) 0 0
\(966\) −50.4233 −1.62234
\(967\) 15.5076 0.498690 0.249345 0.968415i \(-0.419785\pi\)
0.249345 + 0.968415i \(0.419785\pi\)
\(968\) 5.00000 0.160706
\(969\) 8.80776 0.282946
\(970\) 0 0
\(971\) −56.4924 −1.81293 −0.906464 0.422283i \(-0.861229\pi\)
−0.906464 + 0.422283i \(0.861229\pi\)
\(972\) −22.2462 −0.713548
\(973\) 42.2462 1.35435
\(974\) 17.6155 0.564438
\(975\) 0 0
\(976\) 11.1231 0.356042
\(977\) −28.7386 −0.919430 −0.459715 0.888066i \(-0.652048\pi\)
−0.459715 + 0.888066i \(0.652048\pi\)
\(978\) 39.3693 1.25889
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) −1.12311 −0.0358397
\(983\) −18.8769 −0.602079 −0.301040 0.953612i \(-0.597334\pi\)
−0.301040 + 0.953612i \(0.597334\pi\)
\(984\) 31.3693 1.00002
\(985\) 0 0
\(986\) 19.5464 0.622484
\(987\) −40.9848 −1.30456
\(988\) 5.68466 0.180853
\(989\) −22.1080 −0.702992
\(990\) 0 0
\(991\) 2.87689 0.0913876 0.0456938 0.998955i \(-0.485450\pi\)
0.0456938 + 0.998955i \(0.485450\pi\)
\(992\) −5.12311 −0.162659
\(993\) −6.56155 −0.208225
\(994\) 26.2462 0.832479
\(995\) 0 0
\(996\) −7.36932 −0.233506
\(997\) 16.7386 0.530118 0.265059 0.964232i \(-0.414609\pi\)
0.265059 + 0.964232i \(0.414609\pi\)
\(998\) 42.1080 1.33290
\(999\) 8.63068 0.273063
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 950.2.a.h.1.2 2
3.2 odd 2 8550.2.a.br.1.2 2
4.3 odd 2 7600.2.a.y.1.1 2
5.2 odd 4 950.2.b.f.799.3 4
5.3 odd 4 950.2.b.f.799.2 4
5.4 even 2 190.2.a.d.1.1 2
15.14 odd 2 1710.2.a.w.1.1 2
20.19 odd 2 1520.2.a.n.1.2 2
35.34 odd 2 9310.2.a.bc.1.2 2
40.19 odd 2 6080.2.a.bb.1.1 2
40.29 even 2 6080.2.a.bh.1.2 2
95.94 odd 2 3610.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.d.1.1 2 5.4 even 2
950.2.a.h.1.2 2 1.1 even 1 trivial
950.2.b.f.799.2 4 5.3 odd 4
950.2.b.f.799.3 4 5.2 odd 4
1520.2.a.n.1.2 2 20.19 odd 2
1710.2.a.w.1.1 2 15.14 odd 2
3610.2.a.t.1.2 2 95.94 odd 2
6080.2.a.bb.1.1 2 40.19 odd 2
6080.2.a.bh.1.2 2 40.29 even 2
7600.2.a.y.1.1 2 4.3 odd 2
8550.2.a.br.1.2 2 3.2 odd 2
9310.2.a.bc.1.2 2 35.34 odd 2