Properties

Label 950.2.a.f.1.1
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: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} +0.414214 q^{6} +4.41421 q^{7} -1.00000 q^{8} -2.82843 q^{9} -1.41421 q^{11} -0.414214 q^{12} +5.82843 q^{13} -4.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -1.00000 q^{19} -1.82843 q^{21} +1.41421 q^{22} +0.757359 q^{23} +0.414214 q^{24} -5.82843 q^{26} +2.41421 q^{27} +4.41421 q^{28} -0.171573 q^{29} +6.24264 q^{31} -1.00000 q^{32} +0.585786 q^{33} -1.00000 q^{34} -2.82843 q^{36} -8.48528 q^{37} +1.00000 q^{38} -2.41421 q^{39} -4.24264 q^{41} +1.82843 q^{42} +1.75736 q^{43} -1.41421 q^{44} -0.757359 q^{46} -0.414214 q^{48} +12.4853 q^{49} -0.414214 q^{51} +5.82843 q^{52} +5.48528 q^{53} -2.41421 q^{54} -4.41421 q^{56} +0.414214 q^{57} +0.171573 q^{58} +6.89949 q^{59} +14.2426 q^{61} -6.24264 q^{62} -12.4853 q^{63} +1.00000 q^{64} -0.585786 q^{66} +4.75736 q^{67} +1.00000 q^{68} -0.313708 q^{69} -13.4142 q^{71} +2.82843 q^{72} +11.4853 q^{73} +8.48528 q^{74} -1.00000 q^{76} -6.24264 q^{77} +2.41421 q^{78} -6.48528 q^{79} +7.48528 q^{81} +4.24264 q^{82} +14.4853 q^{83} -1.82843 q^{84} -1.75736 q^{86} +0.0710678 q^{87} +1.41421 q^{88} +7.07107 q^{89} +25.7279 q^{91} +0.757359 q^{92} -2.58579 q^{93} +0.414214 q^{96} -0.343146 q^{97} -12.4853 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{12} + 6q^{13} - 6q^{14} + 2q^{16} + 2q^{17} - 2q^{19} + 2q^{21} + 10q^{23} - 2q^{24} - 6q^{26} + 2q^{27} + 6q^{28} - 6q^{29} + 4q^{31} - 2q^{32} + 4q^{33} - 2q^{34} + 2q^{38} - 2q^{39} - 2q^{42} + 12q^{43} - 10q^{46} + 2q^{48} + 8q^{49} + 2q^{51} + 6q^{52} - 6q^{53} - 2q^{54} - 6q^{56} - 2q^{57} + 6q^{58} - 6q^{59} + 20q^{61} - 4q^{62} - 8q^{63} + 2q^{64} - 4q^{66} + 18q^{67} + 2q^{68} + 22q^{69} - 24q^{71} + 6q^{73} - 2q^{76} - 4q^{77} + 2q^{78} + 4q^{79} - 2q^{81} + 12q^{83} + 2q^{84} - 12q^{86} - 14q^{87} + 26q^{91} + 10q^{92} - 8q^{93} - 2q^{96} - 12q^{97} - 8q^{98} + 8q^{99} + O(q^{100}) \)

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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.414214 0.169102
\(7\) 4.41421 1.66842 0.834208 0.551450i \(-0.185925\pi\)
0.834208 + 0.551450i \(0.185925\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) −0.414214 −0.119573
\(13\) 5.82843 1.61651 0.808257 0.588829i \(-0.200411\pi\)
0.808257 + 0.588829i \(0.200411\pi\)
\(14\) −4.41421 −1.17975
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 2.82843 0.666667
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.82843 −0.398996
\(22\) 1.41421 0.301511
\(23\) 0.757359 0.157920 0.0789602 0.996878i \(-0.474840\pi\)
0.0789602 + 0.996878i \(0.474840\pi\)
\(24\) 0.414214 0.0845510
\(25\) 0 0
\(26\) −5.82843 −1.14305
\(27\) 2.41421 0.464616
\(28\) 4.41421 0.834208
\(29\) −0.171573 −0.0318603 −0.0159301 0.999873i \(-0.505071\pi\)
−0.0159301 + 0.999873i \(0.505071\pi\)
\(30\) 0 0
\(31\) 6.24264 1.12121 0.560606 0.828083i \(-0.310568\pi\)
0.560606 + 0.828083i \(0.310568\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.585786 0.101972
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.41421 −0.386584
\(40\) 0 0
\(41\) −4.24264 −0.662589 −0.331295 0.943527i \(-0.607485\pi\)
−0.331295 + 0.943527i \(0.607485\pi\)
\(42\) 1.82843 0.282132
\(43\) 1.75736 0.267995 0.133997 0.990982i \(-0.457219\pi\)
0.133997 + 0.990982i \(0.457219\pi\)
\(44\) −1.41421 −0.213201
\(45\) 0 0
\(46\) −0.757359 −0.111667
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −0.414214 −0.0597866
\(49\) 12.4853 1.78361
\(50\) 0 0
\(51\) −0.414214 −0.0580015
\(52\) 5.82843 0.808257
\(53\) 5.48528 0.753461 0.376731 0.926323i \(-0.377048\pi\)
0.376731 + 0.926323i \(0.377048\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) −4.41421 −0.589874
\(57\) 0.414214 0.0548639
\(58\) 0.171573 0.0225286
\(59\) 6.89949 0.898238 0.449119 0.893472i \(-0.351738\pi\)
0.449119 + 0.893472i \(0.351738\pi\)
\(60\) 0 0
\(61\) 14.2426 1.82358 0.911792 0.410653i \(-0.134699\pi\)
0.911792 + 0.410653i \(0.134699\pi\)
\(62\) −6.24264 −0.792816
\(63\) −12.4853 −1.57300
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −0.585786 −0.0721053
\(67\) 4.75736 0.581204 0.290602 0.956844i \(-0.406144\pi\)
0.290602 + 0.956844i \(0.406144\pi\)
\(68\) 1.00000 0.121268
\(69\) −0.313708 −0.0377661
\(70\) 0 0
\(71\) −13.4142 −1.59197 −0.795987 0.605314i \(-0.793048\pi\)
−0.795987 + 0.605314i \(0.793048\pi\)
\(72\) 2.82843 0.333333
\(73\) 11.4853 1.34425 0.672125 0.740437i \(-0.265382\pi\)
0.672125 + 0.740437i \(0.265382\pi\)
\(74\) 8.48528 0.986394
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −6.24264 −0.711415
\(78\) 2.41421 0.273356
\(79\) −6.48528 −0.729651 −0.364826 0.931076i \(-0.618871\pi\)
−0.364826 + 0.931076i \(0.618871\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 4.24264 0.468521
\(83\) 14.4853 1.58997 0.794983 0.606632i \(-0.207480\pi\)
0.794983 + 0.606632i \(0.207480\pi\)
\(84\) −1.82843 −0.199498
\(85\) 0 0
\(86\) −1.75736 −0.189501
\(87\) 0.0710678 0.00761927
\(88\) 1.41421 0.150756
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 25.7279 2.69702
\(92\) 0.757359 0.0789602
\(93\) −2.58579 −0.268134
\(94\) 0 0
\(95\) 0 0
\(96\) 0.414214 0.0422755
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) −12.4853 −1.26120
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 13.0711 1.30062 0.650310 0.759669i \(-0.274639\pi\)
0.650310 + 0.759669i \(0.274639\pi\)
\(102\) 0.414214 0.0410133
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) −5.82843 −0.571524
\(105\) 0 0
\(106\) −5.48528 −0.532778
\(107\) 19.7279 1.90717 0.953585 0.301124i \(-0.0973617\pi\)
0.953585 + 0.301124i \(0.0973617\pi\)
\(108\) 2.41421 0.232308
\(109\) −17.9706 −1.72127 −0.860634 0.509224i \(-0.829932\pi\)
−0.860634 + 0.509224i \(0.829932\pi\)
\(110\) 0 0
\(111\) 3.51472 0.333602
\(112\) 4.41421 0.417104
\(113\) −10.2426 −0.963547 −0.481773 0.876296i \(-0.660007\pi\)
−0.481773 + 0.876296i \(0.660007\pi\)
\(114\) −0.414214 −0.0387947
\(115\) 0 0
\(116\) −0.171573 −0.0159301
\(117\) −16.4853 −1.52406
\(118\) −6.89949 −0.635150
\(119\) 4.41421 0.404650
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) −14.2426 −1.28947
\(123\) 1.75736 0.158456
\(124\) 6.24264 0.560606
\(125\) 0 0
\(126\) 12.4853 1.11228
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.727922 −0.0640900
\(130\) 0 0
\(131\) −16.9706 −1.48272 −0.741362 0.671105i \(-0.765820\pi\)
−0.741362 + 0.671105i \(0.765820\pi\)
\(132\) 0.585786 0.0509862
\(133\) −4.41421 −0.382761
\(134\) −4.75736 −0.410973
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 13.0000 1.11066 0.555332 0.831628i \(-0.312591\pi\)
0.555332 + 0.831628i \(0.312591\pi\)
\(138\) 0.313708 0.0267046
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.4142 1.12570
\(143\) −8.24264 −0.689284
\(144\) −2.82843 −0.235702
\(145\) 0 0
\(146\) −11.4853 −0.950529
\(147\) −5.17157 −0.426544
\(148\) −8.48528 −0.697486
\(149\) 17.6569 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(150\) 0 0
\(151\) −10.4853 −0.853280 −0.426640 0.904422i \(-0.640303\pi\)
−0.426640 + 0.904422i \(0.640303\pi\)
\(152\) 1.00000 0.0811107
\(153\) −2.82843 −0.228665
\(154\) 6.24264 0.503046
\(155\) 0 0
\(156\) −2.41421 −0.193292
\(157\) 11.6569 0.930318 0.465159 0.885227i \(-0.345997\pi\)
0.465159 + 0.885227i \(0.345997\pi\)
\(158\) 6.48528 0.515941
\(159\) −2.27208 −0.180188
\(160\) 0 0
\(161\) 3.34315 0.263477
\(162\) −7.48528 −0.588099
\(163\) −10.2426 −0.802266 −0.401133 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(164\) −4.24264 −0.331295
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) −18.2426 −1.41166 −0.705829 0.708382i \(-0.749425\pi\)
−0.705829 + 0.708382i \(0.749425\pi\)
\(168\) 1.82843 0.141066
\(169\) 20.9706 1.61312
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 1.75736 0.133997
\(173\) 0.485281 0.0368953 0.0184476 0.999830i \(-0.494128\pi\)
0.0184476 + 0.999830i \(0.494128\pi\)
\(174\) −0.0710678 −0.00538764
\(175\) 0 0
\(176\) −1.41421 −0.106600
\(177\) −2.85786 −0.214810
\(178\) −7.07107 −0.529999
\(179\) −11.6569 −0.871274 −0.435637 0.900122i \(-0.643477\pi\)
−0.435637 + 0.900122i \(0.643477\pi\)
\(180\) 0 0
\(181\) −8.48528 −0.630706 −0.315353 0.948974i \(-0.602123\pi\)
−0.315353 + 0.948974i \(0.602123\pi\)
\(182\) −25.7279 −1.90708
\(183\) −5.89949 −0.436103
\(184\) −0.757359 −0.0558333
\(185\) 0 0
\(186\) 2.58579 0.189599
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) 10.6569 0.775172
\(190\) 0 0
\(191\) 12.5563 0.908546 0.454273 0.890863i \(-0.349899\pi\)
0.454273 + 0.890863i \(0.349899\pi\)
\(192\) −0.414214 −0.0298933
\(193\) −0.343146 −0.0247002 −0.0123501 0.999924i \(-0.503931\pi\)
−0.0123501 + 0.999924i \(0.503931\pi\)
\(194\) 0.343146 0.0246364
\(195\) 0 0
\(196\) 12.4853 0.891806
\(197\) −11.7574 −0.837677 −0.418839 0.908061i \(-0.637563\pi\)
−0.418839 + 0.908061i \(0.637563\pi\)
\(198\) −4.00000 −0.284268
\(199\) 9.24264 0.655193 0.327597 0.944818i \(-0.393761\pi\)
0.327597 + 0.944818i \(0.393761\pi\)
\(200\) 0 0
\(201\) −1.97056 −0.138993
\(202\) −13.0711 −0.919677
\(203\) −0.757359 −0.0531562
\(204\) −0.414214 −0.0290008
\(205\) 0 0
\(206\) −4.24264 −0.295599
\(207\) −2.14214 −0.148889
\(208\) 5.82843 0.404129
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) −19.7279 −1.35813 −0.679063 0.734080i \(-0.737614\pi\)
−0.679063 + 0.734080i \(0.737614\pi\)
\(212\) 5.48528 0.376731
\(213\) 5.55635 0.380715
\(214\) −19.7279 −1.34857
\(215\) 0 0
\(216\) −2.41421 −0.164266
\(217\) 27.5563 1.87065
\(218\) 17.9706 1.21712
\(219\) −4.75736 −0.321473
\(220\) 0 0
\(221\) 5.82843 0.392062
\(222\) −3.51472 −0.235892
\(223\) −15.1716 −1.01596 −0.507982 0.861368i \(-0.669608\pi\)
−0.507982 + 0.861368i \(0.669608\pi\)
\(224\) −4.41421 −0.294937
\(225\) 0 0
\(226\) 10.2426 0.681330
\(227\) −16.7574 −1.11223 −0.556113 0.831107i \(-0.687708\pi\)
−0.556113 + 0.831107i \(0.687708\pi\)
\(228\) 0.414214 0.0274320
\(229\) 14.9706 0.989283 0.494641 0.869097i \(-0.335299\pi\)
0.494641 + 0.869097i \(0.335299\pi\)
\(230\) 0 0
\(231\) 2.58579 0.170132
\(232\) 0.171573 0.0112643
\(233\) −24.9706 −1.63588 −0.817938 0.575306i \(-0.804883\pi\)
−0.817938 + 0.575306i \(0.804883\pi\)
\(234\) 16.4853 1.07768
\(235\) 0 0
\(236\) 6.89949 0.449119
\(237\) 2.68629 0.174493
\(238\) −4.41421 −0.286131
\(239\) −6.89949 −0.446291 −0.223146 0.974785i \(-0.571633\pi\)
−0.223146 + 0.974785i \(0.571633\pi\)
\(240\) 0 0
\(241\) 8.97056 0.577845 0.288922 0.957353i \(-0.406703\pi\)
0.288922 + 0.957353i \(0.406703\pi\)
\(242\) 9.00000 0.578542
\(243\) −10.3431 −0.663513
\(244\) 14.2426 0.911792
\(245\) 0 0
\(246\) −1.75736 −0.112045
\(247\) −5.82843 −0.370854
\(248\) −6.24264 −0.396408
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 3.55635 0.224475 0.112237 0.993681i \(-0.464198\pi\)
0.112237 + 0.993681i \(0.464198\pi\)
\(252\) −12.4853 −0.786499
\(253\) −1.07107 −0.0673375
\(254\) 2.48528 0.155940
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.7279 1.29297 0.646486 0.762926i \(-0.276238\pi\)
0.646486 + 0.762926i \(0.276238\pi\)
\(258\) 0.727922 0.0453184
\(259\) −37.4558 −2.32739
\(260\) 0 0
\(261\) 0.485281 0.0300382
\(262\) 16.9706 1.04844
\(263\) −26.9706 −1.66308 −0.831538 0.555468i \(-0.812539\pi\)
−0.831538 + 0.555468i \(0.812539\pi\)
\(264\) −0.585786 −0.0360527
\(265\) 0 0
\(266\) 4.41421 0.270653
\(267\) −2.92893 −0.179248
\(268\) 4.75736 0.290602
\(269\) −16.6274 −1.01379 −0.506896 0.862007i \(-0.669207\pi\)
−0.506896 + 0.862007i \(0.669207\pi\)
\(270\) 0 0
\(271\) −27.2426 −1.65487 −0.827436 0.561560i \(-0.810202\pi\)
−0.827436 + 0.561560i \(0.810202\pi\)
\(272\) 1.00000 0.0606339
\(273\) −10.6569 −0.644982
\(274\) −13.0000 −0.785359
\(275\) 0 0
\(276\) −0.313708 −0.0188830
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 12.0000 0.719712
\(279\) −17.6569 −1.05709
\(280\) 0 0
\(281\) 4.24264 0.253095 0.126547 0.991961i \(-0.459610\pi\)
0.126547 + 0.991961i \(0.459610\pi\)
\(282\) 0 0
\(283\) 32.1421 1.91065 0.955326 0.295555i \(-0.0955045\pi\)
0.955326 + 0.295555i \(0.0955045\pi\)
\(284\) −13.4142 −0.795987
\(285\) 0 0
\(286\) 8.24264 0.487398
\(287\) −18.7279 −1.10547
\(288\) 2.82843 0.166667
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0.142136 0.00833214
\(292\) 11.4853 0.672125
\(293\) −5.48528 −0.320454 −0.160227 0.987080i \(-0.551223\pi\)
−0.160227 + 0.987080i \(0.551223\pi\)
\(294\) 5.17157 0.301612
\(295\) 0 0
\(296\) 8.48528 0.493197
\(297\) −3.41421 −0.198113
\(298\) −17.6569 −1.02283
\(299\) 4.41421 0.255281
\(300\) 0 0
\(301\) 7.75736 0.447127
\(302\) 10.4853 0.603360
\(303\) −5.41421 −0.311038
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 2.82843 0.161690
\(307\) −17.6569 −1.00773 −0.503865 0.863782i \(-0.668089\pi\)
−0.503865 + 0.863782i \(0.668089\pi\)
\(308\) −6.24264 −0.355707
\(309\) −1.75736 −0.0999727
\(310\) 0 0
\(311\) −4.75736 −0.269765 −0.134883 0.990862i \(-0.543066\pi\)
−0.134883 + 0.990862i \(0.543066\pi\)
\(312\) 2.41421 0.136678
\(313\) −7.97056 −0.450523 −0.225261 0.974298i \(-0.572324\pi\)
−0.225261 + 0.974298i \(0.572324\pi\)
\(314\) −11.6569 −0.657834
\(315\) 0 0
\(316\) −6.48528 −0.364826
\(317\) −9.48528 −0.532746 −0.266373 0.963870i \(-0.585825\pi\)
−0.266373 + 0.963870i \(0.585825\pi\)
\(318\) 2.27208 0.127412
\(319\) 0.242641 0.0135853
\(320\) 0 0
\(321\) −8.17157 −0.456093
\(322\) −3.34315 −0.186306
\(323\) −1.00000 −0.0556415
\(324\) 7.48528 0.415849
\(325\) 0 0
\(326\) 10.2426 0.567287
\(327\) 7.44365 0.411635
\(328\) 4.24264 0.234261
\(329\) 0 0
\(330\) 0 0
\(331\) −19.2426 −1.05767 −0.528836 0.848724i \(-0.677371\pi\)
−0.528836 + 0.848724i \(0.677371\pi\)
\(332\) 14.4853 0.794983
\(333\) 24.0000 1.31519
\(334\) 18.2426 0.998193
\(335\) 0 0
\(336\) −1.82843 −0.0997489
\(337\) 14.1005 0.768103 0.384052 0.923312i \(-0.374528\pi\)
0.384052 + 0.923312i \(0.374528\pi\)
\(338\) −20.9706 −1.14065
\(339\) 4.24264 0.230429
\(340\) 0 0
\(341\) −8.82843 −0.478086
\(342\) −2.82843 −0.152944
\(343\) 24.2132 1.30739
\(344\) −1.75736 −0.0947505
\(345\) 0 0
\(346\) −0.485281 −0.0260889
\(347\) −5.51472 −0.296046 −0.148023 0.988984i \(-0.547291\pi\)
−0.148023 + 0.988984i \(0.547291\pi\)
\(348\) 0.0710678 0.00380963
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 14.0711 0.751058
\(352\) 1.41421 0.0753778
\(353\) −2.51472 −0.133845 −0.0669225 0.997758i \(-0.521318\pi\)
−0.0669225 + 0.997758i \(0.521318\pi\)
\(354\) 2.85786 0.151894
\(355\) 0 0
\(356\) 7.07107 0.374766
\(357\) −1.82843 −0.0967706
\(358\) 11.6569 0.616084
\(359\) −22.7574 −1.20109 −0.600544 0.799592i \(-0.705049\pi\)
−0.600544 + 0.799592i \(0.705049\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 8.48528 0.445976
\(363\) 3.72792 0.195665
\(364\) 25.7279 1.34851
\(365\) 0 0
\(366\) 5.89949 0.308372
\(367\) −25.4558 −1.32878 −0.664392 0.747384i \(-0.731309\pi\)
−0.664392 + 0.747384i \(0.731309\pi\)
\(368\) 0.757359 0.0394801
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 24.2132 1.25709
\(372\) −2.58579 −0.134067
\(373\) 9.00000 0.466002 0.233001 0.972476i \(-0.425145\pi\)
0.233001 + 0.972476i \(0.425145\pi\)
\(374\) 1.41421 0.0731272
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) −10.6569 −0.548129
\(379\) 11.2426 0.577496 0.288748 0.957405i \(-0.406761\pi\)
0.288748 + 0.957405i \(0.406761\pi\)
\(380\) 0 0
\(381\) 1.02944 0.0527397
\(382\) −12.5563 −0.642439
\(383\) 12.2426 0.625570 0.312785 0.949824i \(-0.398738\pi\)
0.312785 + 0.949824i \(0.398738\pi\)
\(384\) 0.414214 0.0211377
\(385\) 0 0
\(386\) 0.343146 0.0174657
\(387\) −4.97056 −0.252668
\(388\) −0.343146 −0.0174206
\(389\) 22.9289 1.16254 0.581272 0.813710i \(-0.302555\pi\)
0.581272 + 0.813710i \(0.302555\pi\)
\(390\) 0 0
\(391\) 0.757359 0.0383013
\(392\) −12.4853 −0.630602
\(393\) 7.02944 0.354588
\(394\) 11.7574 0.592327
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) −9.24264 −0.463292
\(399\) 1.82843 0.0915358
\(400\) 0 0
\(401\) −25.4142 −1.26913 −0.634563 0.772871i \(-0.718820\pi\)
−0.634563 + 0.772871i \(0.718820\pi\)
\(402\) 1.97056 0.0982827
\(403\) 36.3848 1.81245
\(404\) 13.0711 0.650310
\(405\) 0 0
\(406\) 0.757359 0.0375871
\(407\) 12.0000 0.594818
\(408\) 0.414214 0.0205066
\(409\) 25.2132 1.24671 0.623356 0.781938i \(-0.285769\pi\)
0.623356 + 0.781938i \(0.285769\pi\)
\(410\) 0 0
\(411\) −5.38478 −0.265611
\(412\) 4.24264 0.209020
\(413\) 30.4558 1.49863
\(414\) 2.14214 0.105280
\(415\) 0 0
\(416\) −5.82843 −0.285762
\(417\) 4.97056 0.243410
\(418\) −1.41421 −0.0691714
\(419\) 19.4142 0.948446 0.474223 0.880405i \(-0.342729\pi\)
0.474223 + 0.880405i \(0.342729\pi\)
\(420\) 0 0
\(421\) 19.4853 0.949655 0.474827 0.880079i \(-0.342511\pi\)
0.474827 + 0.880079i \(0.342511\pi\)
\(422\) 19.7279 0.960340
\(423\) 0 0
\(424\) −5.48528 −0.266389
\(425\) 0 0
\(426\) −5.55635 −0.269206
\(427\) 62.8701 3.04250
\(428\) 19.7279 0.953585
\(429\) 3.41421 0.164840
\(430\) 0 0
\(431\) −6.38478 −0.307544 −0.153772 0.988106i \(-0.549142\pi\)
−0.153772 + 0.988106i \(0.549142\pi\)
\(432\) 2.41421 0.116154
\(433\) −9.55635 −0.459249 −0.229624 0.973279i \(-0.573750\pi\)
−0.229624 + 0.973279i \(0.573750\pi\)
\(434\) −27.5563 −1.32275
\(435\) 0 0
\(436\) −17.9706 −0.860634
\(437\) −0.757359 −0.0362294
\(438\) 4.75736 0.227315
\(439\) −5.75736 −0.274784 −0.137392 0.990517i \(-0.543872\pi\)
−0.137392 + 0.990517i \(0.543872\pi\)
\(440\) 0 0
\(441\) −35.3137 −1.68161
\(442\) −5.82843 −0.277230
\(443\) 4.24264 0.201574 0.100787 0.994908i \(-0.467864\pi\)
0.100787 + 0.994908i \(0.467864\pi\)
\(444\) 3.51472 0.166801
\(445\) 0 0
\(446\) 15.1716 0.718395
\(447\) −7.31371 −0.345927
\(448\) 4.41421 0.208552
\(449\) 17.3137 0.817084 0.408542 0.912739i \(-0.366037\pi\)
0.408542 + 0.912739i \(0.366037\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −10.2426 −0.481773
\(453\) 4.34315 0.204059
\(454\) 16.7574 0.786462
\(455\) 0 0
\(456\) −0.414214 −0.0193973
\(457\) −3.00000 −0.140334 −0.0701670 0.997535i \(-0.522353\pi\)
−0.0701670 + 0.997535i \(0.522353\pi\)
\(458\) −14.9706 −0.699528
\(459\) 2.41421 0.112686
\(460\) 0 0
\(461\) −3.55635 −0.165636 −0.0828178 0.996565i \(-0.526392\pi\)
−0.0828178 + 0.996565i \(0.526392\pi\)
\(462\) −2.58579 −0.120302
\(463\) 14.1421 0.657241 0.328620 0.944462i \(-0.393416\pi\)
0.328620 + 0.944462i \(0.393416\pi\)
\(464\) −0.171573 −0.00796507
\(465\) 0 0
\(466\) 24.9706 1.15674
\(467\) 0.727922 0.0336842 0.0168421 0.999858i \(-0.494639\pi\)
0.0168421 + 0.999858i \(0.494639\pi\)
\(468\) −16.4853 −0.762032
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) −4.82843 −0.222482
\(472\) −6.89949 −0.317575
\(473\) −2.48528 −0.114273
\(474\) −2.68629 −0.123385
\(475\) 0 0
\(476\) 4.41421 0.202325
\(477\) −15.5147 −0.710370
\(478\) 6.89949 0.315576
\(479\) −31.1127 −1.42158 −0.710788 0.703407i \(-0.751661\pi\)
−0.710788 + 0.703407i \(0.751661\pi\)
\(480\) 0 0
\(481\) −49.4558 −2.25499
\(482\) −8.97056 −0.408598
\(483\) −1.38478 −0.0630095
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 10.3431 0.469175
\(487\) 37.7990 1.71284 0.856418 0.516283i \(-0.172685\pi\)
0.856418 + 0.516283i \(0.172685\pi\)
\(488\) −14.2426 −0.644734
\(489\) 4.24264 0.191859
\(490\) 0 0
\(491\) −33.5563 −1.51438 −0.757188 0.653197i \(-0.773428\pi\)
−0.757188 + 0.653197i \(0.773428\pi\)
\(492\) 1.75736 0.0792279
\(493\) −0.171573 −0.00772725
\(494\) 5.82843 0.262233
\(495\) 0 0
\(496\) 6.24264 0.280303
\(497\) −59.2132 −2.65608
\(498\) 6.00000 0.268866
\(499\) −25.7574 −1.15306 −0.576529 0.817077i \(-0.695593\pi\)
−0.576529 + 0.817077i \(0.695593\pi\)
\(500\) 0 0
\(501\) 7.55635 0.337593
\(502\) −3.55635 −0.158728
\(503\) 14.2721 0.636361 0.318180 0.948030i \(-0.396928\pi\)
0.318180 + 0.948030i \(0.396928\pi\)
\(504\) 12.4853 0.556139
\(505\) 0 0
\(506\) 1.07107 0.0476148
\(507\) −8.68629 −0.385772
\(508\) −2.48528 −0.110267
\(509\) 28.9706 1.28410 0.642049 0.766664i \(-0.278085\pi\)
0.642049 + 0.766664i \(0.278085\pi\)
\(510\) 0 0
\(511\) 50.6985 2.24277
\(512\) −1.00000 −0.0441942
\(513\) −2.41421 −0.106590
\(514\) −20.7279 −0.914269
\(515\) 0 0
\(516\) −0.727922 −0.0320450
\(517\) 0 0
\(518\) 37.4558 1.64572
\(519\) −0.201010 −0.00882337
\(520\) 0 0
\(521\) 23.3137 1.02139 0.510696 0.859761i \(-0.329388\pi\)
0.510696 + 0.859761i \(0.329388\pi\)
\(522\) −0.485281 −0.0212402
\(523\) −2.27208 −0.0993510 −0.0496755 0.998765i \(-0.515819\pi\)
−0.0496755 + 0.998765i \(0.515819\pi\)
\(524\) −16.9706 −0.741362
\(525\) 0 0
\(526\) 26.9706 1.17597
\(527\) 6.24264 0.271934
\(528\) 0.585786 0.0254931
\(529\) −22.4264 −0.975061
\(530\) 0 0
\(531\) −19.5147 −0.846867
\(532\) −4.41421 −0.191380
\(533\) −24.7279 −1.07109
\(534\) 2.92893 0.126747
\(535\) 0 0
\(536\) −4.75736 −0.205487
\(537\) 4.82843 0.208362
\(538\) 16.6274 0.716859
\(539\) −17.6569 −0.760535
\(540\) 0 0
\(541\) 9.75736 0.419502 0.209751 0.977755i \(-0.432735\pi\)
0.209751 + 0.977755i \(0.432735\pi\)
\(542\) 27.2426 1.17017
\(543\) 3.51472 0.150831
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 10.6569 0.456071
\(547\) −17.3137 −0.740281 −0.370140 0.928976i \(-0.620690\pi\)
−0.370140 + 0.928976i \(0.620690\pi\)
\(548\) 13.0000 0.555332
\(549\) −40.2843 −1.71929
\(550\) 0 0
\(551\) 0.171573 0.00730925
\(552\) 0.313708 0.0133523
\(553\) −28.6274 −1.21736
\(554\) 0 0
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 16.0000 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(558\) 17.6569 0.747474
\(559\) 10.2426 0.433218
\(560\) 0 0
\(561\) 0.585786 0.0247319
\(562\) −4.24264 −0.178965
\(563\) 4.97056 0.209484 0.104742 0.994499i \(-0.466598\pi\)
0.104742 + 0.994499i \(0.466598\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −32.1421 −1.35103
\(567\) 33.0416 1.38762
\(568\) 13.4142 0.562848
\(569\) −28.2843 −1.18574 −0.592869 0.805299i \(-0.702005\pi\)
−0.592869 + 0.805299i \(0.702005\pi\)
\(570\) 0 0
\(571\) −2.24264 −0.0938516 −0.0469258 0.998898i \(-0.514942\pi\)
−0.0469258 + 0.998898i \(0.514942\pi\)
\(572\) −8.24264 −0.344642
\(573\) −5.20101 −0.217275
\(574\) 18.7279 0.781688
\(575\) 0 0
\(576\) −2.82843 −0.117851
\(577\) −20.3137 −0.845671 −0.422835 0.906207i \(-0.638965\pi\)
−0.422835 + 0.906207i \(0.638965\pi\)
\(578\) 16.0000 0.665512
\(579\) 0.142136 0.00590695
\(580\) 0 0
\(581\) 63.9411 2.65272
\(582\) −0.142136 −0.00589171
\(583\) −7.75736 −0.321277
\(584\) −11.4853 −0.475264
\(585\) 0 0
\(586\) 5.48528 0.226595
\(587\) 30.2426 1.24825 0.624124 0.781326i \(-0.285456\pi\)
0.624124 + 0.781326i \(0.285456\pi\)
\(588\) −5.17157 −0.213272
\(589\) −6.24264 −0.257224
\(590\) 0 0
\(591\) 4.87006 0.200327
\(592\) −8.48528 −0.348743
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 3.41421 0.140087
\(595\) 0 0
\(596\) 17.6569 0.723253
\(597\) −3.82843 −0.156687
\(598\) −4.41421 −0.180511
\(599\) −15.2132 −0.621595 −0.310797 0.950476i \(-0.600596\pi\)
−0.310797 + 0.950476i \(0.600596\pi\)
\(600\) 0 0
\(601\) −20.2426 −0.825715 −0.412857 0.910796i \(-0.635469\pi\)
−0.412857 + 0.910796i \(0.635469\pi\)
\(602\) −7.75736 −0.316166
\(603\) −13.4558 −0.547964
\(604\) −10.4853 −0.426640
\(605\) 0 0
\(606\) 5.41421 0.219937
\(607\) 3.17157 0.128730 0.0643651 0.997926i \(-0.479498\pi\)
0.0643651 + 0.997926i \(0.479498\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0.313708 0.0127121
\(610\) 0 0
\(611\) 0 0
\(612\) −2.82843 −0.114332
\(613\) −11.6985 −0.472497 −0.236249 0.971693i \(-0.575918\pi\)
−0.236249 + 0.971693i \(0.575918\pi\)
\(614\) 17.6569 0.712573
\(615\) 0 0
\(616\) 6.24264 0.251523
\(617\) −4.48528 −0.180571 −0.0902853 0.995916i \(-0.528778\pi\)
−0.0902853 + 0.995916i \(0.528778\pi\)
\(618\) 1.75736 0.0706914
\(619\) 7.75736 0.311795 0.155897 0.987773i \(-0.450173\pi\)
0.155897 + 0.987773i \(0.450173\pi\)
\(620\) 0 0
\(621\) 1.82843 0.0733723
\(622\) 4.75736 0.190753
\(623\) 31.2132 1.25053
\(624\) −2.41421 −0.0966459
\(625\) 0 0
\(626\) 7.97056 0.318568
\(627\) −0.585786 −0.0233941
\(628\) 11.6569 0.465159
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) −38.9706 −1.55139 −0.775697 0.631106i \(-0.782601\pi\)
−0.775697 + 0.631106i \(0.782601\pi\)
\(632\) 6.48528 0.257971
\(633\) 8.17157 0.324791
\(634\) 9.48528 0.376709
\(635\) 0 0
\(636\) −2.27208 −0.0900938
\(637\) 72.7696 2.88323
\(638\) −0.242641 −0.00960624
\(639\) 37.9411 1.50093
\(640\) 0 0
\(641\) −18.0416 −0.712602 −0.356301 0.934371i \(-0.615962\pi\)
−0.356301 + 0.934371i \(0.615962\pi\)
\(642\) 8.17157 0.322506
\(643\) −14.4853 −0.571244 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(644\) 3.34315 0.131738
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) −18.7574 −0.737428 −0.368714 0.929543i \(-0.620202\pi\)
−0.368714 + 0.929543i \(0.620202\pi\)
\(648\) −7.48528 −0.294050
\(649\) −9.75736 −0.383010
\(650\) 0 0
\(651\) −11.4142 −0.447358
\(652\) −10.2426 −0.401133
\(653\) −28.9706 −1.13371 −0.566853 0.823819i \(-0.691839\pi\)
−0.566853 + 0.823819i \(0.691839\pi\)
\(654\) −7.44365 −0.291070
\(655\) 0 0
\(656\) −4.24264 −0.165647
\(657\) −32.4853 −1.26737
\(658\) 0 0
\(659\) 18.8995 0.736220 0.368110 0.929782i \(-0.380005\pi\)
0.368110 + 0.929782i \(0.380005\pi\)
\(660\) 0 0
\(661\) −18.4558 −0.717849 −0.358925 0.933367i \(-0.616856\pi\)
−0.358925 + 0.933367i \(0.616856\pi\)
\(662\) 19.2426 0.747886
\(663\) −2.41421 −0.0937603
\(664\) −14.4853 −0.562138
\(665\) 0 0
\(666\) −24.0000 −0.929981
\(667\) −0.129942 −0.00503139
\(668\) −18.2426 −0.705829
\(669\) 6.28427 0.242964
\(670\) 0 0
\(671\) −20.1421 −0.777579
\(672\) 1.82843 0.0705331
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) −14.1005 −0.543131
\(675\) 0 0
\(676\) 20.9706 0.806560
\(677\) −40.9411 −1.57350 −0.786748 0.617275i \(-0.788237\pi\)
−0.786748 + 0.617275i \(0.788237\pi\)
\(678\) −4.24264 −0.162938
\(679\) −1.51472 −0.0581296
\(680\) 0 0
\(681\) 6.94113 0.265985
\(682\) 8.82843 0.338058
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.82843 0.108148
\(685\) 0 0
\(686\) −24.2132 −0.924464
\(687\) −6.20101 −0.236583
\(688\) 1.75736 0.0669987
\(689\) 31.9706 1.21798
\(690\) 0 0
\(691\) 22.4853 0.855380 0.427690 0.903925i \(-0.359327\pi\)
0.427690 + 0.903925i \(0.359327\pi\)
\(692\) 0.485281 0.0184476
\(693\) 17.6569 0.670728
\(694\) 5.51472 0.209336
\(695\) 0 0
\(696\) −0.0710678 −0.00269382
\(697\) −4.24264 −0.160701
\(698\) −6.00000 −0.227103
\(699\) 10.3431 0.391214
\(700\) 0 0
\(701\) 16.9706 0.640969 0.320485 0.947254i \(-0.396154\pi\)
0.320485 + 0.947254i \(0.396154\pi\)
\(702\) −14.0711 −0.531078
\(703\) 8.48528 0.320028
\(704\) −1.41421 −0.0533002
\(705\) 0 0
\(706\) 2.51472 0.0946427
\(707\) 57.6985 2.16997
\(708\) −2.85786 −0.107405
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 18.3431 0.687922
\(712\) −7.07107 −0.264999
\(713\) 4.72792 0.177062
\(714\) 1.82843 0.0684272
\(715\) 0 0
\(716\) −11.6569 −0.435637
\(717\) 2.85786 0.106729
\(718\) 22.7574 0.849297
\(719\) −5.10051 −0.190217 −0.0951084 0.995467i \(-0.530320\pi\)
−0.0951084 + 0.995467i \(0.530320\pi\)
\(720\) 0 0
\(721\) 18.7279 0.697464
\(722\) −1.00000 −0.0372161
\(723\) −3.71573 −0.138189
\(724\) −8.48528 −0.315353
\(725\) 0 0
\(726\) −3.72792 −0.138356
\(727\) 9.72792 0.360789 0.180394 0.983594i \(-0.442263\pi\)
0.180394 + 0.983594i \(0.442263\pi\)
\(728\) −25.7279 −0.953540
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 1.75736 0.0649983
\(732\) −5.89949 −0.218052
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 25.4558 0.939592
\(735\) 0 0
\(736\) −0.757359 −0.0279166
\(737\) −6.72792 −0.247826
\(738\) −12.0000 −0.441726
\(739\) 1.27208 0.0467941 0.0233971 0.999726i \(-0.492552\pi\)
0.0233971 + 0.999726i \(0.492552\pi\)
\(740\) 0 0
\(741\) 2.41421 0.0886884
\(742\) −24.2132 −0.888895
\(743\) 6.72792 0.246824 0.123412 0.992356i \(-0.460616\pi\)
0.123412 + 0.992356i \(0.460616\pi\)
\(744\) 2.58579 0.0947995
\(745\) 0 0
\(746\) −9.00000 −0.329513
\(747\) −40.9706 −1.49903
\(748\) −1.41421 −0.0517088
\(749\) 87.0833 3.18195
\(750\) 0 0
\(751\) −26.7279 −0.975316 −0.487658 0.873035i \(-0.662149\pi\)
−0.487658 + 0.873035i \(0.662149\pi\)
\(752\) 0 0
\(753\) −1.47309 −0.0536823
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) 10.6569 0.387586
\(757\) 12.3431 0.448619 0.224310 0.974518i \(-0.427987\pi\)
0.224310 + 0.974518i \(0.427987\pi\)
\(758\) −11.2426 −0.408351
\(759\) 0.443651 0.0161035
\(760\) 0 0
\(761\) 43.9706 1.59393 0.796966 0.604024i \(-0.206437\pi\)
0.796966 + 0.604024i \(0.206437\pi\)
\(762\) −1.02944 −0.0372926
\(763\) −79.3259 −2.87179
\(764\) 12.5563 0.454273
\(765\) 0 0
\(766\) −12.2426 −0.442345
\(767\) 40.2132 1.45201
\(768\) −0.414214 −0.0149466
\(769\) −36.4558 −1.31463 −0.657316 0.753615i \(-0.728308\pi\)
−0.657316 + 0.753615i \(0.728308\pi\)
\(770\) 0 0
\(771\) −8.58579 −0.309210
\(772\) −0.343146 −0.0123501
\(773\) 13.9706 0.502486 0.251243 0.967924i \(-0.419161\pi\)
0.251243 + 0.967924i \(0.419161\pi\)
\(774\) 4.97056 0.178663
\(775\) 0 0
\(776\) 0.343146 0.0123182
\(777\) 15.5147 0.556587
\(778\) −22.9289 −0.822042
\(779\) 4.24264 0.152008
\(780\) 0 0
\(781\) 18.9706 0.678820
\(782\) −0.757359 −0.0270831
\(783\) −0.414214 −0.0148028
\(784\) 12.4853 0.445903
\(785\) 0 0
\(786\) −7.02944 −0.250732
\(787\) −41.1838 −1.46804 −0.734021 0.679126i \(-0.762359\pi\)
−0.734021 + 0.679126i \(0.762359\pi\)
\(788\) −11.7574 −0.418839
\(789\) 11.1716 0.397719
\(790\) 0 0
\(791\) −45.2132 −1.60760
\(792\) −4.00000 −0.142134
\(793\) 83.0122 2.94785
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) 9.24264 0.327597
\(797\) −47.4853 −1.68201 −0.841007 0.541023i \(-0.818037\pi\)
−0.841007 + 0.541023i \(0.818037\pi\)
\(798\) −1.82843 −0.0647256
\(799\) 0 0
\(800\) 0 0
\(801\) −20.0000 −0.706665
\(802\) 25.4142 0.897407
\(803\) −16.2426 −0.573190
\(804\) −1.97056 −0.0694964
\(805\) 0 0
\(806\) −36.3848 −1.28160
\(807\) 6.88730 0.242444
\(808\) −13.0711 −0.459839
\(809\) −28.7990 −1.01252 −0.506259 0.862381i \(-0.668972\pi\)
−0.506259 + 0.862381i \(0.668972\pi\)
\(810\) 0 0
\(811\) 52.6985 1.85049 0.925247 0.379365i \(-0.123858\pi\)
0.925247 + 0.379365i \(0.123858\pi\)
\(812\) −0.757359 −0.0265781
\(813\) 11.2843 0.395757
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) −0.414214 −0.0145004
\(817\) −1.75736 −0.0614822
\(818\) −25.2132 −0.881559
\(819\) −72.7696 −2.54277
\(820\) 0 0
\(821\) 6.34315 0.221377 0.110689 0.993855i \(-0.464694\pi\)
0.110689 + 0.993855i \(0.464694\pi\)
\(822\) 5.38478 0.187816
\(823\) −15.7279 −0.548241 −0.274120 0.961695i \(-0.588387\pi\)
−0.274120 + 0.961695i \(0.588387\pi\)
\(824\) −4.24264 −0.147799
\(825\) 0 0
\(826\) −30.4558 −1.05969
\(827\) −20.6985 −0.719757 −0.359878 0.932999i \(-0.617182\pi\)
−0.359878 + 0.932999i \(0.617182\pi\)
\(828\) −2.14214 −0.0744444
\(829\) 10.4558 0.363146 0.181573 0.983377i \(-0.441881\pi\)
0.181573 + 0.983377i \(0.441881\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.82843 0.202064
\(833\) 12.4853 0.432589
\(834\) −4.97056 −0.172117
\(835\) 0 0
\(836\) 1.41421 0.0489116
\(837\) 15.0711 0.520932
\(838\) −19.4142 −0.670653
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −28.9706 −0.998985
\(842\) −19.4853 −0.671507
\(843\) −1.75736 −0.0605267
\(844\) −19.7279 −0.679063
\(845\) 0 0
\(846\) 0 0
\(847\) −39.7279 −1.36507
\(848\) 5.48528 0.188365
\(849\) −13.3137 −0.456925
\(850\) 0 0
\(851\) −6.42641 −0.220294
\(852\) 5.55635 0.190357
\(853\) −27.9411 −0.956686 −0.478343 0.878173i \(-0.658762\pi\)
−0.478343 + 0.878173i \(0.658762\pi\)
\(854\) −62.8701 −2.15137
\(855\) 0 0
\(856\) −19.7279 −0.674286
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −3.41421 −0.116559
\(859\) 4.72792 0.161315 0.0806573 0.996742i \(-0.474298\pi\)
0.0806573 + 0.996742i \(0.474298\pi\)
\(860\) 0 0
\(861\) 7.75736 0.264370
\(862\) 6.38478 0.217466
\(863\) 0.727922 0.0247788 0.0123894 0.999923i \(-0.496056\pi\)
0.0123894 + 0.999923i \(0.496056\pi\)
\(864\) −2.41421 −0.0821332
\(865\) 0 0
\(866\) 9.55635 0.324738
\(867\) 6.62742 0.225079
\(868\) 27.5563 0.935323
\(869\) 9.17157 0.311124
\(870\) 0 0
\(871\) 27.7279 0.939525
\(872\) 17.9706 0.608560
\(873\) 0.970563 0.0328486
\(874\) 0.757359 0.0256181
\(875\) 0 0
\(876\) −4.75736 −0.160736
\(877\) −18.8579 −0.636785 −0.318392 0.947959i \(-0.603143\pi\)
−0.318392 + 0.947959i \(0.603143\pi\)
\(878\) 5.75736 0.194301
\(879\) 2.27208 0.0766353
\(880\) 0 0
\(881\) −39.5980 −1.33409 −0.667045 0.745018i \(-0.732441\pi\)
−0.667045 + 0.745018i \(0.732441\pi\)
\(882\) 35.3137 1.18907
\(883\) 14.4853 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(884\) 5.82843 0.196031
\(885\) 0 0
\(886\) −4.24264 −0.142534
\(887\) −45.2132 −1.51811 −0.759055 0.651026i \(-0.774339\pi\)
−0.759055 + 0.651026i \(0.774339\pi\)
\(888\) −3.51472 −0.117946
\(889\) −10.9706 −0.367941
\(890\) 0 0
\(891\) −10.5858 −0.354637
\(892\) −15.1716 −0.507982
\(893\) 0 0
\(894\) 7.31371 0.244607
\(895\) 0 0
\(896\) −4.41421 −0.147469
\(897\) −1.82843 −0.0610494
\(898\) −17.3137 −0.577766
\(899\) −1.07107 −0.0357221
\(900\) 0 0
\(901\) 5.48528 0.182741
\(902\) −6.00000 −0.199778
\(903\) −3.21320 −0.106929
\(904\) 10.2426 0.340665
\(905\) 0 0
\(906\) −4.34315 −0.144291
\(907\) 38.6985 1.28496 0.642481 0.766302i \(-0.277905\pi\)
0.642481 + 0.766302i \(0.277905\pi\)
\(908\) −16.7574 −0.556113
\(909\) −36.9706 −1.22624
\(910\) 0 0
\(911\) 40.2843 1.33468 0.667339 0.744754i \(-0.267433\pi\)
0.667339 + 0.744754i \(0.267433\pi\)
\(912\) 0.414214 0.0137160
\(913\) −20.4853 −0.677964
\(914\) 3.00000 0.0992312
\(915\) 0 0
\(916\) 14.9706 0.494641
\(917\) −74.9117 −2.47380
\(918\) −2.41421 −0.0796809
\(919\) −6.21320 −0.204955 −0.102477 0.994735i \(-0.532677\pi\)
−0.102477 + 0.994735i \(0.532677\pi\)
\(920\) 0 0
\(921\) 7.31371 0.240995
\(922\) 3.55635 0.117122
\(923\) −78.1838 −2.57345
\(924\) 2.58579 0.0850661
\(925\) 0 0
\(926\) −14.1421 −0.464739
\(927\) −12.0000 −0.394132
\(928\) 0.171573 0.00563216
\(929\) −12.1716 −0.399336 −0.199668 0.979864i \(-0.563986\pi\)
−0.199668 + 0.979864i \(0.563986\pi\)
\(930\) 0 0
\(931\) −12.4853 −0.409189
\(932\) −24.9706 −0.817938
\(933\) 1.97056 0.0645133
\(934\) −0.727922 −0.0238183
\(935\) 0 0
\(936\) 16.4853 0.538838
\(937\) 27.0000 0.882052 0.441026 0.897494i \(-0.354615\pi\)
0.441026 + 0.897494i \(0.354615\pi\)
\(938\) −21.0000 −0.685674
\(939\) 3.30152 0.107741
\(940\) 0 0
\(941\) 53.1421 1.73238 0.866192 0.499711i \(-0.166561\pi\)
0.866192 + 0.499711i \(0.166561\pi\)
\(942\) 4.82843 0.157319
\(943\) −3.21320 −0.104636
\(944\) 6.89949 0.224559
\(945\) 0 0
\(946\) 2.48528 0.0808035
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 2.68629 0.0872467
\(949\) 66.9411 2.17300
\(950\) 0 0
\(951\) 3.92893 0.127404
\(952\) −4.41421 −0.143065
\(953\) 18.7279 0.606657 0.303328 0.952886i \(-0.401902\pi\)
0.303328 + 0.952886i \(0.401902\pi\)
\(954\) 15.5147 0.502308
\(955\) 0 0
\(956\) −6.89949 −0.223146
\(957\) −0.100505 −0.00324887
\(958\) 31.1127 1.00521
\(959\) 57.3848 1.85305
\(960\) 0 0
\(961\) 7.97056 0.257115
\(962\) 49.4558 1.59452
\(963\) −55.7990 −1.79810
\(964\) 8.97056 0.288922
\(965\) 0 0
\(966\) 1.38478 0.0445544
\(967\) 43.1127 1.38641 0.693205 0.720740i \(-0.256198\pi\)
0.693205 + 0.720740i \(0.256198\pi\)
\(968\) 9.00000 0.289271
\(969\) 0.414214 0.0133065
\(970\) 0 0
\(971\) 41.6569 1.33683 0.668416 0.743788i \(-0.266973\pi\)
0.668416 + 0.743788i \(0.266973\pi\)
\(972\) −10.3431 −0.331757
\(973\) −52.9706 −1.69816
\(974\) −37.7990 −1.21116
\(975\) 0 0
\(976\) 14.2426 0.455896
\(977\) 0.242641 0.00776276 0.00388138 0.999992i \(-0.498765\pi\)
0.00388138 + 0.999992i \(0.498765\pi\)
\(978\) −4.24264 −0.135665
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 50.8284 1.62283
\(982\) 33.5563 1.07083
\(983\) −51.4558 −1.64119 −0.820593 0.571513i \(-0.806357\pi\)
−0.820593 + 0.571513i \(0.806357\pi\)
\(984\) −1.75736 −0.0560226
\(985\) 0 0
\(986\) 0.171573 0.00546399
\(987\) 0 0
\(988\) −5.82843 −0.185427
\(989\) 1.33095 0.0423218
\(990\) 0 0
\(991\) 13.7574 0.437017 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(992\) −6.24264 −0.198204
\(993\) 7.97056 0.252938
\(994\) 59.2132 1.87813
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 48.7279 1.54323 0.771614 0.636091i \(-0.219450\pi\)
0.771614 + 0.636091i \(0.219450\pi\)
\(998\) 25.7574 0.815335
\(999\) −20.4853 −0.648126
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.f.1.1 2
3.2 odd 2 8550.2.a.cb.1.2 2
4.3 odd 2 7600.2.a.v.1.2 2
5.2 odd 4 190.2.b.a.39.2 4
5.3 odd 4 190.2.b.a.39.3 yes 4
5.4 even 2 950.2.a.g.1.2 2
15.2 even 4 1710.2.d.c.1369.3 4
15.8 even 4 1710.2.d.c.1369.1 4
15.14 odd 2 8550.2.a.bn.1.1 2
20.3 even 4 1520.2.d.e.609.3 4
20.7 even 4 1520.2.d.e.609.2 4
20.19 odd 2 7600.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.2 4 5.2 odd 4
190.2.b.a.39.3 yes 4 5.3 odd 4
950.2.a.f.1.1 2 1.1 even 1 trivial
950.2.a.g.1.2 2 5.4 even 2
1520.2.d.e.609.2 4 20.7 even 4
1520.2.d.e.609.3 4 20.3 even 4
1710.2.d.c.1369.1 4 15.8 even 4
1710.2.d.c.1369.3 4 15.2 even 4
7600.2.a.v.1.2 2 4.3 odd 2
7600.2.a.bg.1.1 2 20.19 odd 2
8550.2.a.bn.1.1 2 15.14 odd 2
8550.2.a.cb.1.2 2 3.2 odd 2