Properties

Label 4864.2.a.bf.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.91729\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.91729 q^{3} +3.91729 q^{5} -0.593272 q^{7} +5.51056 q^{9} +O(q^{10})\) \(q-2.91729 q^{3} +3.91729 q^{5} -0.593272 q^{7} +5.51056 q^{9} +3.51056 q^{11} -2.51056 q^{13} -11.4278 q^{15} +3.00000 q^{17} -1.00000 q^{19} +1.73074 q^{21} -7.32401 q^{23} +10.3451 q^{25} -7.32401 q^{27} -5.73074 q^{29} -4.40673 q^{31} -10.2413 q^{33} -2.32401 q^{35} +3.42784 q^{37} +7.32401 q^{39} -10.2413 q^{41} -4.69710 q^{43} +21.5864 q^{45} -2.08271 q^{47} -6.64803 q^{49} -8.75186 q^{51} -1.08271 q^{53} +13.7519 q^{55} +2.91729 q^{57} -8.51056 q^{59} -7.10383 q^{61} -3.26926 q^{63} -9.83457 q^{65} +12.9173 q^{67} +21.3662 q^{69} -12.8557 q^{71} -16.2288 q^{73} -30.1797 q^{75} -2.08271 q^{77} -4.40673 q^{79} +4.83457 q^{81} +13.4615 q^{83} +11.7519 q^{85} +16.7182 q^{87} +0.241300 q^{89} +1.48944 q^{91} +12.8557 q^{93} -3.91729 q^{95} +15.4615 q^{97} +19.3451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{3} + 2q^{5} - 3q^{7} + 8q^{9} + O(q^{10}) \) \( 3q + q^{3} + 2q^{5} - 3q^{7} + 8q^{9} + 2q^{11} + q^{13} - 16q^{15} + 9q^{17} - 3q^{19} - 7q^{21} - 11q^{23} + 3q^{25} - 11q^{27} - 5q^{29} - 12q^{31} - 10q^{33} + 4q^{35} - 8q^{37} + 11q^{39} - 10q^{41} - 8q^{43} + 16q^{45} - 16q^{47} + 2q^{49} + 3q^{51} - 13q^{53} + 12q^{55} - q^{57} - 17q^{59} - 14q^{61} - 22q^{63} - 10q^{65} + 29q^{67} + 19q^{69} - 2q^{71} - 17q^{73} - 43q^{75} - 16q^{77} - 12q^{79} - 5q^{81} + 16q^{83} + 6q^{85} + 27q^{87} - 20q^{89} + 13q^{91} + 2q^{93} - 2q^{95} + 22q^{97} + 30q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.91729 −1.68430 −0.842148 0.539247i \(-0.818709\pi\)
−0.842148 + 0.539247i \(0.818709\pi\)
\(4\) 0 0
\(5\) 3.91729 1.75186 0.875932 0.482435i \(-0.160248\pi\)
0.875932 + 0.482435i \(0.160248\pi\)
\(6\) 0 0
\(7\) −0.593272 −0.224236 −0.112118 0.993695i \(-0.535763\pi\)
−0.112118 + 0.993695i \(0.535763\pi\)
\(8\) 0 0
\(9\) 5.51056 1.83685
\(10\) 0 0
\(11\) 3.51056 1.05847 0.529236 0.848474i \(-0.322478\pi\)
0.529236 + 0.848474i \(0.322478\pi\)
\(12\) 0 0
\(13\) −2.51056 −0.696303 −0.348152 0.937438i \(-0.613191\pi\)
−0.348152 + 0.937438i \(0.613191\pi\)
\(14\) 0 0
\(15\) −11.4278 −2.95066
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.73074 0.377679
\(22\) 0 0
\(23\) −7.32401 −1.52716 −0.763581 0.645712i \(-0.776561\pi\)
−0.763581 + 0.645712i \(0.776561\pi\)
\(24\) 0 0
\(25\) 10.3451 2.06903
\(26\) 0 0
\(27\) −7.32401 −1.40951
\(28\) 0 0
\(29\) −5.73074 −1.06417 −0.532086 0.846690i \(-0.678592\pi\)
−0.532086 + 0.846690i \(0.678592\pi\)
\(30\) 0 0
\(31\) −4.40673 −0.791472 −0.395736 0.918364i \(-0.629510\pi\)
−0.395736 + 0.918364i \(0.629510\pi\)
\(32\) 0 0
\(33\) −10.2413 −1.78278
\(34\) 0 0
\(35\) −2.32401 −0.392830
\(36\) 0 0
\(37\) 3.42784 0.563534 0.281767 0.959483i \(-0.409079\pi\)
0.281767 + 0.959483i \(0.409079\pi\)
\(38\) 0 0
\(39\) 7.32401 1.17278
\(40\) 0 0
\(41\) −10.2413 −1.59942 −0.799711 0.600385i \(-0.795014\pi\)
−0.799711 + 0.600385i \(0.795014\pi\)
\(42\) 0 0
\(43\) −4.69710 −0.716301 −0.358151 0.933664i \(-0.616593\pi\)
−0.358151 + 0.933664i \(0.616593\pi\)
\(44\) 0 0
\(45\) 21.5864 3.21791
\(46\) 0 0
\(47\) −2.08271 −0.303795 −0.151898 0.988396i \(-0.548538\pi\)
−0.151898 + 0.988396i \(0.548538\pi\)
\(48\) 0 0
\(49\) −6.64803 −0.949718
\(50\) 0 0
\(51\) −8.75186 −1.22551
\(52\) 0 0
\(53\) −1.08271 −0.148722 −0.0743611 0.997231i \(-0.523692\pi\)
−0.0743611 + 0.997231i \(0.523692\pi\)
\(54\) 0 0
\(55\) 13.7519 1.85430
\(56\) 0 0
\(57\) 2.91729 0.386404
\(58\) 0 0
\(59\) −8.51056 −1.10798 −0.553990 0.832523i \(-0.686895\pi\)
−0.553990 + 0.832523i \(0.686895\pi\)
\(60\) 0 0
\(61\) −7.10383 −0.909552 −0.454776 0.890606i \(-0.650281\pi\)
−0.454776 + 0.890606i \(0.650281\pi\)
\(62\) 0 0
\(63\) −3.26926 −0.411888
\(64\) 0 0
\(65\) −9.83457 −1.21983
\(66\) 0 0
\(67\) 12.9173 1.57810 0.789049 0.614330i \(-0.210574\pi\)
0.789049 + 0.614330i \(0.210574\pi\)
\(68\) 0 0
\(69\) 21.3662 2.57219
\(70\) 0 0
\(71\) −12.8557 −1.52569 −0.762845 0.646582i \(-0.776198\pi\)
−0.762845 + 0.646582i \(0.776198\pi\)
\(72\) 0 0
\(73\) −16.2288 −1.89943 −0.949717 0.313109i \(-0.898629\pi\)
−0.949717 + 0.313109i \(0.898629\pi\)
\(74\) 0 0
\(75\) −30.1797 −3.48485
\(76\) 0 0
\(77\) −2.08271 −0.237347
\(78\) 0 0
\(79\) −4.40673 −0.495796 −0.247898 0.968786i \(-0.579740\pi\)
−0.247898 + 0.968786i \(0.579740\pi\)
\(80\) 0 0
\(81\) 4.83457 0.537175
\(82\) 0 0
\(83\) 13.4615 1.47759 0.738795 0.673930i \(-0.235395\pi\)
0.738795 + 0.673930i \(0.235395\pi\)
\(84\) 0 0
\(85\) 11.7519 1.27467
\(86\) 0 0
\(87\) 16.7182 1.79238
\(88\) 0 0
\(89\) 0.241300 0.0255778 0.0127889 0.999918i \(-0.495929\pi\)
0.0127889 + 0.999918i \(0.495929\pi\)
\(90\) 0 0
\(91\) 1.48944 0.156136
\(92\) 0 0
\(93\) 12.8557 1.33307
\(94\) 0 0
\(95\) −3.91729 −0.401905
\(96\) 0 0
\(97\) 15.4615 1.56988 0.784938 0.619574i \(-0.212695\pi\)
0.784938 + 0.619574i \(0.212695\pi\)
\(98\) 0 0
\(99\) 19.3451 1.94426
\(100\) 0 0
\(101\) 15.8346 1.57560 0.787799 0.615932i \(-0.211220\pi\)
0.787799 + 0.615932i \(0.211220\pi\)
\(102\) 0 0
\(103\) −13.4615 −1.32640 −0.663200 0.748442i \(-0.730802\pi\)
−0.663200 + 0.748442i \(0.730802\pi\)
\(104\) 0 0
\(105\) 6.77981 0.661642
\(106\) 0 0
\(107\) −0.302899 −0.0292824 −0.0146412 0.999893i \(-0.504661\pi\)
−0.0146412 + 0.999893i \(0.504661\pi\)
\(108\) 0 0
\(109\) −5.29037 −0.506726 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 2.77981 0.261503 0.130751 0.991415i \(-0.458261\pi\)
0.130751 + 0.991415i \(0.458261\pi\)
\(114\) 0 0
\(115\) −28.6903 −2.67538
\(116\) 0 0
\(117\) −13.8346 −1.27901
\(118\) 0 0
\(119\) −1.77981 −0.163155
\(120\) 0 0
\(121\) 1.32401 0.120365
\(122\) 0 0
\(123\) 29.8768 2.69390
\(124\) 0 0
\(125\) 20.9384 1.87279
\(126\) 0 0
\(127\) 14.2835 1.26746 0.633729 0.773555i \(-0.281523\pi\)
0.633729 + 0.773555i \(0.281523\pi\)
\(128\) 0 0
\(129\) 13.7028 1.20646
\(130\) 0 0
\(131\) 16.3662 1.42993 0.714963 0.699163i \(-0.246444\pi\)
0.714963 + 0.699163i \(0.246444\pi\)
\(132\) 0 0
\(133\) 0.593272 0.0514432
\(134\) 0 0
\(135\) −28.6903 −2.46926
\(136\) 0 0
\(137\) −4.66914 −0.398912 −0.199456 0.979907i \(-0.563917\pi\)
−0.199456 + 0.979907i \(0.563917\pi\)
\(138\) 0 0
\(139\) −6.48944 −0.550427 −0.275214 0.961383i \(-0.588749\pi\)
−0.275214 + 0.961383i \(0.588749\pi\)
\(140\) 0 0
\(141\) 6.07587 0.511681
\(142\) 0 0
\(143\) −8.81346 −0.737018
\(144\) 0 0
\(145\) −22.4490 −1.86428
\(146\) 0 0
\(147\) 19.3942 1.59961
\(148\) 0 0
\(149\) −6.73074 −0.551404 −0.275702 0.961243i \(-0.588910\pi\)
−0.275702 + 0.961243i \(0.588910\pi\)
\(150\) 0 0
\(151\) −18.2077 −1.48172 −0.740859 0.671660i \(-0.765581\pi\)
−0.740859 + 0.671660i \(0.765581\pi\)
\(152\) 0 0
\(153\) 16.5317 1.33651
\(154\) 0 0
\(155\) −17.2624 −1.38655
\(156\) 0 0
\(157\) 16.0422 1.28031 0.640155 0.768246i \(-0.278870\pi\)
0.640155 + 0.768246i \(0.278870\pi\)
\(158\) 0 0
\(159\) 3.15859 0.250492
\(160\) 0 0
\(161\) 4.34513 0.342444
\(162\) 0 0
\(163\) −24.0422 −1.88313 −0.941566 0.336827i \(-0.890646\pi\)
−0.941566 + 0.336827i \(0.890646\pi\)
\(164\) 0 0
\(165\) −40.1181 −3.12319
\(166\) 0 0
\(167\) 19.6355 1.51944 0.759720 0.650250i \(-0.225336\pi\)
0.759720 + 0.650250i \(0.225336\pi\)
\(168\) 0 0
\(169\) −6.69710 −0.515162
\(170\) 0 0
\(171\) −5.51056 −0.421403
\(172\) 0 0
\(173\) 8.37309 0.636594 0.318297 0.947991i \(-0.396889\pi\)
0.318297 + 0.947991i \(0.396889\pi\)
\(174\) 0 0
\(175\) −6.13747 −0.463949
\(176\) 0 0
\(177\) 24.8277 1.86617
\(178\) 0 0
\(179\) 10.2413 0.765471 0.382735 0.923858i \(-0.374982\pi\)
0.382735 + 0.923858i \(0.374982\pi\)
\(180\) 0 0
\(181\) −24.5248 −1.82292 −0.911458 0.411393i \(-0.865042\pi\)
−0.911458 + 0.411393i \(0.865042\pi\)
\(182\) 0 0
\(183\) 20.7239 1.53195
\(184\) 0 0
\(185\) 13.4278 0.987235
\(186\) 0 0
\(187\) 10.5317 0.770152
\(188\) 0 0
\(189\) 4.34513 0.316062
\(190\) 0 0
\(191\) 9.28353 0.671733 0.335866 0.941910i \(-0.390971\pi\)
0.335866 + 0.941910i \(0.390971\pi\)
\(192\) 0 0
\(193\) −8.03364 −0.578274 −0.289137 0.957288i \(-0.593368\pi\)
−0.289137 + 0.957288i \(0.593368\pi\)
\(194\) 0 0
\(195\) 28.6903 2.05455
\(196\) 0 0
\(197\) 5.62691 0.400901 0.200451 0.979704i \(-0.435759\pi\)
0.200451 + 0.979704i \(0.435759\pi\)
\(198\) 0 0
\(199\) 18.0970 1.28286 0.641431 0.767181i \(-0.278341\pi\)
0.641431 + 0.767181i \(0.278341\pi\)
\(200\) 0 0
\(201\) −37.6834 −2.65798
\(202\) 0 0
\(203\) 3.39989 0.238625
\(204\) 0 0
\(205\) −40.1181 −2.80197
\(206\) 0 0
\(207\) −40.3594 −2.80517
\(208\) 0 0
\(209\) −3.51056 −0.242830
\(210\) 0 0
\(211\) 0.302899 0.0208524 0.0104262 0.999946i \(-0.496681\pi\)
0.0104262 + 0.999946i \(0.496681\pi\)
\(212\) 0 0
\(213\) 37.5037 2.56971
\(214\) 0 0
\(215\) −18.3999 −1.25486
\(216\) 0 0
\(217\) 2.61439 0.177476
\(218\) 0 0
\(219\) 47.3440 3.19921
\(220\) 0 0
\(221\) −7.53167 −0.506635
\(222\) 0 0
\(223\) 0.847099 0.0567259 0.0283630 0.999598i \(-0.490971\pi\)
0.0283630 + 0.999598i \(0.490971\pi\)
\(224\) 0 0
\(225\) 57.0074 3.80050
\(226\) 0 0
\(227\) 12.5528 0.833158 0.416579 0.909100i \(-0.363229\pi\)
0.416579 + 0.909100i \(0.363229\pi\)
\(228\) 0 0
\(229\) −14.7730 −0.976226 −0.488113 0.872781i \(-0.662315\pi\)
−0.488113 + 0.872781i \(0.662315\pi\)
\(230\) 0 0
\(231\) 6.07587 0.399763
\(232\) 0 0
\(233\) 7.71822 0.505637 0.252819 0.967514i \(-0.418642\pi\)
0.252819 + 0.967514i \(0.418642\pi\)
\(234\) 0 0
\(235\) −8.15859 −0.532207
\(236\) 0 0
\(237\) 12.8557 0.835067
\(238\) 0 0
\(239\) −27.6144 −1.78623 −0.893113 0.449832i \(-0.851484\pi\)
−0.893113 + 0.449832i \(0.851484\pi\)
\(240\) 0 0
\(241\) −1.42784 −0.0919755 −0.0459877 0.998942i \(-0.514644\pi\)
−0.0459877 + 0.998942i \(0.514644\pi\)
\(242\) 0 0
\(243\) 7.86821 0.504746
\(244\) 0 0
\(245\) −26.0422 −1.66378
\(246\) 0 0
\(247\) 2.51056 0.159743
\(248\) 0 0
\(249\) −39.2710 −2.48870
\(250\) 0 0
\(251\) 20.5317 1.29595 0.647974 0.761663i \(-0.275617\pi\)
0.647974 + 0.761663i \(0.275617\pi\)
\(252\) 0 0
\(253\) −25.7114 −1.61646
\(254\) 0 0
\(255\) −34.2835 −2.14692
\(256\) 0 0
\(257\) −25.5037 −1.59088 −0.795439 0.606034i \(-0.792760\pi\)
−0.795439 + 0.606034i \(0.792760\pi\)
\(258\) 0 0
\(259\) −2.03364 −0.126364
\(260\) 0 0
\(261\) −31.5796 −1.95473
\(262\) 0 0
\(263\) −13.7519 −0.847976 −0.423988 0.905668i \(-0.639370\pi\)
−0.423988 + 0.905668i \(0.639370\pi\)
\(264\) 0 0
\(265\) −4.24130 −0.260541
\(266\) 0 0
\(267\) −0.703942 −0.0430806
\(268\) 0 0
\(269\) −20.8220 −1.26954 −0.634771 0.772700i \(-0.718906\pi\)
−0.634771 + 0.772700i \(0.718906\pi\)
\(270\) 0 0
\(271\) −14.8277 −0.900720 −0.450360 0.892847i \(-0.648704\pi\)
−0.450360 + 0.892847i \(0.648704\pi\)
\(272\) 0 0
\(273\) −4.34513 −0.262979
\(274\) 0 0
\(275\) 36.3172 2.19001
\(276\) 0 0
\(277\) 15.9173 0.956377 0.478189 0.878257i \(-0.341294\pi\)
0.478189 + 0.878257i \(0.341294\pi\)
\(278\) 0 0
\(279\) −24.2835 −1.45382
\(280\) 0 0
\(281\) 4.64803 0.277278 0.138639 0.990343i \(-0.455727\pi\)
0.138639 + 0.990343i \(0.455727\pi\)
\(282\) 0 0
\(283\) −3.99316 −0.237369 −0.118684 0.992932i \(-0.537868\pi\)
−0.118684 + 0.992932i \(0.537868\pi\)
\(284\) 0 0
\(285\) 11.4278 0.676927
\(286\) 0 0
\(287\) 6.07587 0.358647
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −45.1056 −2.64414
\(292\) 0 0
\(293\) −3.11636 −0.182059 −0.0910297 0.995848i \(-0.529016\pi\)
−0.0910297 + 0.995848i \(0.529016\pi\)
\(294\) 0 0
\(295\) −33.3383 −1.94103
\(296\) 0 0
\(297\) −25.7114 −1.49193
\(298\) 0 0
\(299\) 18.3874 1.06337
\(300\) 0 0
\(301\) 2.78666 0.160620
\(302\) 0 0
\(303\) −46.1940 −2.65377
\(304\) 0 0
\(305\) −27.8277 −1.59341
\(306\) 0 0
\(307\) −23.7028 −1.35279 −0.676395 0.736539i \(-0.736459\pi\)
−0.676395 + 0.736539i \(0.736459\pi\)
\(308\) 0 0
\(309\) 39.2710 2.23405
\(310\) 0 0
\(311\) −1.90301 −0.107910 −0.0539550 0.998543i \(-0.517183\pi\)
−0.0539550 + 0.998543i \(0.517183\pi\)
\(312\) 0 0
\(313\) 31.6834 1.79085 0.895426 0.445210i \(-0.146871\pi\)
0.895426 + 0.445210i \(0.146871\pi\)
\(314\) 0 0
\(315\) −12.8066 −0.721571
\(316\) 0 0
\(317\) −12.1797 −0.684080 −0.342040 0.939685i \(-0.611118\pi\)
−0.342040 + 0.939685i \(0.611118\pi\)
\(318\) 0 0
\(319\) −20.1181 −1.12640
\(320\) 0 0
\(321\) 0.883644 0.0493202
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −25.9720 −1.44067
\(326\) 0 0
\(327\) 15.4335 0.853476
\(328\) 0 0
\(329\) 1.23562 0.0681217
\(330\) 0 0
\(331\) 7.08271 0.389301 0.194651 0.980873i \(-0.437643\pi\)
0.194651 + 0.980873i \(0.437643\pi\)
\(332\) 0 0
\(333\) 18.8893 1.03513
\(334\) 0 0
\(335\) 50.6007 2.76461
\(336\) 0 0
\(337\) −9.02112 −0.491411 −0.245706 0.969344i \(-0.579020\pi\)
−0.245706 + 0.969344i \(0.579020\pi\)
\(338\) 0 0
\(339\) −8.10951 −0.440448
\(340\) 0 0
\(341\) −15.4701 −0.837751
\(342\) 0 0
\(343\) 8.09699 0.437196
\(344\) 0 0
\(345\) 83.6977 4.50613
\(346\) 0 0
\(347\) −28.0354 −1.50502 −0.752509 0.658582i \(-0.771157\pi\)
−0.752509 + 0.658582i \(0.771157\pi\)
\(348\) 0 0
\(349\) −4.39989 −0.235521 −0.117760 0.993042i \(-0.537571\pi\)
−0.117760 + 0.993042i \(0.537571\pi\)
\(350\) 0 0
\(351\) 18.3874 0.981445
\(352\) 0 0
\(353\) −0.950928 −0.0506128 −0.0253064 0.999680i \(-0.508056\pi\)
−0.0253064 + 0.999680i \(0.508056\pi\)
\(354\) 0 0
\(355\) −50.3594 −2.67280
\(356\) 0 0
\(357\) 5.19223 0.274802
\(358\) 0 0
\(359\) −12.3856 −0.653688 −0.326844 0.945078i \(-0.605985\pi\)
−0.326844 + 0.945078i \(0.605985\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.86253 −0.202730
\(364\) 0 0
\(365\) −63.5727 −3.32755
\(366\) 0 0
\(367\) 20.4826 1.06918 0.534592 0.845111i \(-0.320465\pi\)
0.534592 + 0.845111i \(0.320465\pi\)
\(368\) 0 0
\(369\) −56.4353 −2.93790
\(370\) 0 0
\(371\) 0.642343 0.0333488
\(372\) 0 0
\(373\) 4.88364 0.252865 0.126433 0.991975i \(-0.459647\pi\)
0.126433 + 0.991975i \(0.459647\pi\)
\(374\) 0 0
\(375\) −61.0833 −3.15433
\(376\) 0 0
\(377\) 14.3874 0.740987
\(378\) 0 0
\(379\) −0.179702 −0.00923065 −0.00461532 0.999989i \(-0.501469\pi\)
−0.00461532 + 0.999989i \(0.501469\pi\)
\(380\) 0 0
\(381\) −41.6691 −2.13477
\(382\) 0 0
\(383\) −7.86821 −0.402047 −0.201023 0.979586i \(-0.564427\pi\)
−0.201023 + 0.979586i \(0.564427\pi\)
\(384\) 0 0
\(385\) −8.15859 −0.415800
\(386\) 0 0
\(387\) −25.8836 −1.31574
\(388\) 0 0
\(389\) −31.8614 −1.61544 −0.807718 0.589569i \(-0.799298\pi\)
−0.807718 + 0.589569i \(0.799298\pi\)
\(390\) 0 0
\(391\) −21.9720 −1.11117
\(392\) 0 0
\(393\) −47.7450 −2.40842
\(394\) 0 0
\(395\) −17.2624 −0.868566
\(396\) 0 0
\(397\) −2.93840 −0.147474 −0.0737371 0.997278i \(-0.523493\pi\)
−0.0737371 + 0.997278i \(0.523493\pi\)
\(398\) 0 0
\(399\) −1.73074 −0.0866455
\(400\) 0 0
\(401\) 4.77981 0.238693 0.119346 0.992853i \(-0.461920\pi\)
0.119346 + 0.992853i \(0.461920\pi\)
\(402\) 0 0
\(403\) 11.0633 0.551104
\(404\) 0 0
\(405\) 18.9384 0.941057
\(406\) 0 0
\(407\) 12.0336 0.596485
\(408\) 0 0
\(409\) −17.8009 −0.880199 −0.440100 0.897949i \(-0.645057\pi\)
−0.440100 + 0.897949i \(0.645057\pi\)
\(410\) 0 0
\(411\) 13.6212 0.671886
\(412\) 0 0
\(413\) 5.04907 0.248449
\(414\) 0 0
\(415\) 52.7325 2.58854
\(416\) 0 0
\(417\) 18.9316 0.927082
\(418\) 0 0
\(419\) 16.9230 0.826741 0.413371 0.910563i \(-0.364351\pi\)
0.413371 + 0.910563i \(0.364351\pi\)
\(420\) 0 0
\(421\) −11.9384 −0.581842 −0.290921 0.956747i \(-0.593962\pi\)
−0.290921 + 0.956747i \(0.593962\pi\)
\(422\) 0 0
\(423\) −11.4769 −0.558027
\(424\) 0 0
\(425\) 31.0354 1.50544
\(426\) 0 0
\(427\) 4.21450 0.203954
\(428\) 0 0
\(429\) 25.7114 1.24136
\(430\) 0 0
\(431\) 29.1729 1.40521 0.702604 0.711581i \(-0.252021\pi\)
0.702604 + 0.711581i \(0.252021\pi\)
\(432\) 0 0
\(433\) −5.91044 −0.284038 −0.142019 0.989864i \(-0.545359\pi\)
−0.142019 + 0.989864i \(0.545359\pi\)
\(434\) 0 0
\(435\) 65.4900 3.14001
\(436\) 0 0
\(437\) 7.32401 0.350355
\(438\) 0 0
\(439\) 37.7450 1.80147 0.900736 0.434368i \(-0.143028\pi\)
0.900736 + 0.434368i \(0.143028\pi\)
\(440\) 0 0
\(441\) −36.6343 −1.74449
\(442\) 0 0
\(443\) 9.95093 0.472783 0.236391 0.971658i \(-0.424035\pi\)
0.236391 + 0.971658i \(0.424035\pi\)
\(444\) 0 0
\(445\) 0.945243 0.0448088
\(446\) 0 0
\(447\) 19.6355 0.928727
\(448\) 0 0
\(449\) 21.6269 1.02064 0.510319 0.859985i \(-0.329527\pi\)
0.510319 + 0.859985i \(0.329527\pi\)
\(450\) 0 0
\(451\) −35.9527 −1.69295
\(452\) 0 0
\(453\) 53.1169 2.49565
\(454\) 0 0
\(455\) 5.83457 0.273529
\(456\) 0 0
\(457\) 5.49628 0.257105 0.128553 0.991703i \(-0.458967\pi\)
0.128553 + 0.991703i \(0.458967\pi\)
\(458\) 0 0
\(459\) −21.9720 −1.02557
\(460\) 0 0
\(461\) −5.42100 −0.252481 −0.126241 0.992000i \(-0.540291\pi\)
−0.126241 + 0.992000i \(0.540291\pi\)
\(462\) 0 0
\(463\) −7.31149 −0.339794 −0.169897 0.985462i \(-0.554343\pi\)
−0.169897 + 0.985462i \(0.554343\pi\)
\(464\) 0 0
\(465\) 50.3594 2.33536
\(466\) 0 0
\(467\) −6.97204 −0.322628 −0.161314 0.986903i \(-0.551573\pi\)
−0.161314 + 0.986903i \(0.551573\pi\)
\(468\) 0 0
\(469\) −7.66346 −0.353866
\(470\) 0 0
\(471\) −46.7998 −2.15642
\(472\) 0 0
\(473\) −16.4894 −0.758185
\(474\) 0 0
\(475\) −10.3451 −0.474667
\(476\) 0 0
\(477\) −5.96636 −0.273181
\(478\) 0 0
\(479\) −25.2288 −1.15273 −0.576366 0.817192i \(-0.695530\pi\)
−0.576366 + 0.817192i \(0.695530\pi\)
\(480\) 0 0
\(481\) −8.60580 −0.392391
\(482\) 0 0
\(483\) −12.6760 −0.576777
\(484\) 0 0
\(485\) 60.5671 2.75021
\(486\) 0 0
\(487\) 1.06335 0.0481848 0.0240924 0.999710i \(-0.492330\pi\)
0.0240924 + 0.999710i \(0.492330\pi\)
\(488\) 0 0
\(489\) 70.1381 3.17175
\(490\) 0 0
\(491\) 18.6903 0.843480 0.421740 0.906717i \(-0.361420\pi\)
0.421740 + 0.906717i \(0.361420\pi\)
\(492\) 0 0
\(493\) −17.1922 −0.774299
\(494\) 0 0
\(495\) 75.7804 3.40608
\(496\) 0 0
\(497\) 7.62691 0.342114
\(498\) 0 0
\(499\) −15.8836 −0.711050 −0.355525 0.934667i \(-0.615698\pi\)
−0.355525 + 0.934667i \(0.615698\pi\)
\(500\) 0 0
\(501\) −57.2824 −2.55919
\(502\) 0 0
\(503\) 21.2430 0.947181 0.473590 0.880745i \(-0.342958\pi\)
0.473590 + 0.880745i \(0.342958\pi\)
\(504\) 0 0
\(505\) 62.0285 2.76023
\(506\) 0 0
\(507\) 19.5374 0.867685
\(508\) 0 0
\(509\) −20.6230 −0.914097 −0.457049 0.889442i \(-0.651093\pi\)
−0.457049 + 0.889442i \(0.651093\pi\)
\(510\) 0 0
\(511\) 9.62807 0.425921
\(512\) 0 0
\(513\) 7.32401 0.323363
\(514\) 0 0
\(515\) −52.7325 −2.32367
\(516\) 0 0
\(517\) −7.31149 −0.321559
\(518\) 0 0
\(519\) −24.4267 −1.07221
\(520\) 0 0
\(521\) −5.26242 −0.230551 −0.115275 0.993334i \(-0.536775\pi\)
−0.115275 + 0.993334i \(0.536775\pi\)
\(522\) 0 0
\(523\) −2.67599 −0.117013 −0.0585063 0.998287i \(-0.518634\pi\)
−0.0585063 + 0.998287i \(0.518634\pi\)
\(524\) 0 0
\(525\) 17.9048 0.781428
\(526\) 0 0
\(527\) −13.2202 −0.575880
\(528\) 0 0
\(529\) 30.6412 1.33223
\(530\) 0 0
\(531\) −46.8979 −2.03520
\(532\) 0 0
\(533\) 25.7114 1.11368
\(534\) 0 0
\(535\) −1.18654 −0.0512987
\(536\) 0 0
\(537\) −29.8768 −1.28928
\(538\) 0 0
\(539\) −23.3383 −1.00525
\(540\) 0 0
\(541\) 22.5653 0.970159 0.485079 0.874470i \(-0.338791\pi\)
0.485079 + 0.874470i \(0.338791\pi\)
\(542\) 0 0
\(543\) 71.5459 3.07033
\(544\) 0 0
\(545\) −20.7239 −0.887714
\(546\) 0 0
\(547\) 8.08446 0.345667 0.172833 0.984951i \(-0.444708\pi\)
0.172833 + 0.984951i \(0.444708\pi\)
\(548\) 0 0
\(549\) −39.1461 −1.67071
\(550\) 0 0
\(551\) 5.73074 0.244138
\(552\) 0 0
\(553\) 2.61439 0.111175
\(554\) 0 0
\(555\) −39.1729 −1.66280
\(556\) 0 0
\(557\) −1.12888 −0.0478323 −0.0239162 0.999714i \(-0.507613\pi\)
−0.0239162 + 0.999714i \(0.507613\pi\)
\(558\) 0 0
\(559\) 11.7923 0.498763
\(560\) 0 0
\(561\) −30.7239 −1.29716
\(562\) 0 0
\(563\) 35.4701 1.49489 0.747443 0.664326i \(-0.231281\pi\)
0.747443 + 0.664326i \(0.231281\pi\)
\(564\) 0 0
\(565\) 10.8893 0.458118
\(566\) 0 0
\(567\) −2.86821 −0.120454
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −12.2499 −0.512642 −0.256321 0.966592i \(-0.582510\pi\)
−0.256321 + 0.966592i \(0.582510\pi\)
\(572\) 0 0
\(573\) −27.0827 −1.13140
\(574\) 0 0
\(575\) −75.7679 −3.15974
\(576\) 0 0
\(577\) −22.8768 −0.952374 −0.476187 0.879344i \(-0.657981\pi\)
−0.476187 + 0.879344i \(0.657981\pi\)
\(578\) 0 0
\(579\) 23.4364 0.973985
\(580\) 0 0
\(581\) −7.98632 −0.331328
\(582\) 0 0
\(583\) −3.80093 −0.157418
\(584\) 0 0
\(585\) −54.1940 −2.24065
\(586\) 0 0
\(587\) −29.2219 −1.20612 −0.603059 0.797697i \(-0.706052\pi\)
−0.603059 + 0.797697i \(0.706052\pi\)
\(588\) 0 0
\(589\) 4.40673 0.181576
\(590\) 0 0
\(591\) −16.4153 −0.675236
\(592\) 0 0
\(593\) −1.35197 −0.0555188 −0.0277594 0.999615i \(-0.508837\pi\)
−0.0277594 + 0.999615i \(0.508837\pi\)
\(594\) 0 0
\(595\) −6.97204 −0.285826
\(596\) 0 0
\(597\) −52.7941 −2.16072
\(598\) 0 0
\(599\) −41.1814 −1.68263 −0.841314 0.540546i \(-0.818218\pi\)
−0.841314 + 0.540546i \(0.818218\pi\)
\(600\) 0 0
\(601\) −5.25383 −0.214308 −0.107154 0.994242i \(-0.534174\pi\)
−0.107154 + 0.994242i \(0.534174\pi\)
\(602\) 0 0
\(603\) 71.1814 2.89873
\(604\) 0 0
\(605\) 5.18654 0.210863
\(606\) 0 0
\(607\) −9.17286 −0.372315 −0.186157 0.982520i \(-0.559603\pi\)
−0.186157 + 0.982520i \(0.559603\pi\)
\(608\) 0 0
\(609\) −9.91844 −0.401916
\(610\) 0 0
\(611\) 5.22877 0.211534
\(612\) 0 0
\(613\) 28.3577 1.14535 0.572677 0.819781i \(-0.305905\pi\)
0.572677 + 0.819781i \(0.305905\pi\)
\(614\) 0 0
\(615\) 117.036 4.71935
\(616\) 0 0
\(617\) 19.4546 0.783214 0.391607 0.920132i \(-0.371919\pi\)
0.391607 + 0.920132i \(0.371919\pi\)
\(618\) 0 0
\(619\) −25.2538 −1.01504 −0.507519 0.861641i \(-0.669437\pi\)
−0.507519 + 0.861641i \(0.669437\pi\)
\(620\) 0 0
\(621\) 53.6412 2.15255
\(622\) 0 0
\(623\) −0.143157 −0.00573545
\(624\) 0 0
\(625\) 30.2961 1.21184
\(626\) 0 0
\(627\) 10.2413 0.408998
\(628\) 0 0
\(629\) 10.2835 0.410031
\(630\) 0 0
\(631\) 23.6287 0.940642 0.470321 0.882495i \(-0.344138\pi\)
0.470321 + 0.882495i \(0.344138\pi\)
\(632\) 0 0
\(633\) −0.883644 −0.0351217
\(634\) 0 0
\(635\) 55.9527 2.22041
\(636\) 0 0
\(637\) 16.6903 0.661292
\(638\) 0 0
\(639\) −70.8420 −2.80247
\(640\) 0 0
\(641\) 26.9230 1.06339 0.531697 0.846935i \(-0.321555\pi\)
0.531697 + 0.846935i \(0.321555\pi\)
\(642\) 0 0
\(643\) −13.4124 −0.528934 −0.264467 0.964395i \(-0.585196\pi\)
−0.264467 + 0.964395i \(0.585196\pi\)
\(644\) 0 0
\(645\) 53.6777 2.11356
\(646\) 0 0
\(647\) −6.55104 −0.257548 −0.128774 0.991674i \(-0.541104\pi\)
−0.128774 + 0.991674i \(0.541104\pi\)
\(648\) 0 0
\(649\) −29.8768 −1.17277
\(650\) 0 0
\(651\) −7.62691 −0.298922
\(652\) 0 0
\(653\) 2.52308 0.0987359 0.0493680 0.998781i \(-0.484279\pi\)
0.0493680 + 0.998781i \(0.484279\pi\)
\(654\) 0 0
\(655\) 64.1113 2.50503
\(656\) 0 0
\(657\) −89.4296 −3.48898
\(658\) 0 0
\(659\) 5.17227 0.201483 0.100742 0.994913i \(-0.467878\pi\)
0.100742 + 0.994913i \(0.467878\pi\)
\(660\) 0 0
\(661\) −41.6412 −1.61965 −0.809827 0.586668i \(-0.800439\pi\)
−0.809827 + 0.586668i \(0.800439\pi\)
\(662\) 0 0
\(663\) 21.9720 0.853323
\(664\) 0 0
\(665\) 2.32401 0.0901214
\(666\) 0 0
\(667\) 41.9720 1.62516
\(668\) 0 0
\(669\) −2.47123 −0.0955433
\(670\) 0 0
\(671\) −24.9384 −0.962736
\(672\) 0 0
\(673\) −16.7325 −0.644990 −0.322495 0.946571i \(-0.604522\pi\)
−0.322495 + 0.946571i \(0.604522\pi\)
\(674\) 0 0
\(675\) −75.7679 −2.91631
\(676\) 0 0
\(677\) −1.45580 −0.0559510 −0.0279755 0.999609i \(-0.508906\pi\)
−0.0279755 + 0.999609i \(0.508906\pi\)
\(678\) 0 0
\(679\) −9.17286 −0.352022
\(680\) 0 0
\(681\) −36.6201 −1.40328
\(682\) 0 0
\(683\) 8.81346 0.337238 0.168619 0.985681i \(-0.446069\pi\)
0.168619 + 0.985681i \(0.446069\pi\)
\(684\) 0 0
\(685\) −18.2904 −0.698839
\(686\) 0 0
\(687\) 43.0970 1.64425
\(688\) 0 0
\(689\) 2.71822 0.103556
\(690\) 0 0
\(691\) 12.6834 0.482500 0.241250 0.970463i \(-0.422443\pi\)
0.241250 + 0.970463i \(0.422443\pi\)
\(692\) 0 0
\(693\) −11.4769 −0.435972
\(694\) 0 0
\(695\) −25.4210 −0.964274
\(696\) 0 0
\(697\) −30.7239 −1.16375
\(698\) 0 0
\(699\) −22.5162 −0.851643
\(700\) 0 0
\(701\) 1.46149 0.0551996 0.0275998 0.999619i \(-0.491214\pi\)
0.0275998 + 0.999619i \(0.491214\pi\)
\(702\) 0 0
\(703\) −3.42784 −0.129284
\(704\) 0 0
\(705\) 23.8009 0.896395
\(706\) 0 0
\(707\) −9.39420 −0.353305
\(708\) 0 0
\(709\) 19.7114 0.740276 0.370138 0.928977i \(-0.379310\pi\)
0.370138 + 0.928977i \(0.379310\pi\)
\(710\) 0 0
\(711\) −24.2835 −0.910704
\(712\) 0 0
\(713\) 32.2749 1.20871
\(714\) 0 0
\(715\) −34.5248 −1.29116
\(716\) 0 0
\(717\) 80.5591 3.00853
\(718\) 0 0
\(719\) −20.5933 −0.767999 −0.384000 0.923333i \(-0.625454\pi\)
−0.384000 + 0.923333i \(0.625454\pi\)
\(720\) 0 0
\(721\) 7.98632 0.297426
\(722\) 0 0
\(723\) 4.16543 0.154914
\(724\) 0 0
\(725\) −59.2853 −2.20180
\(726\) 0 0
\(727\) −4.66056 −0.172850 −0.0864252 0.996258i \(-0.527544\pi\)
−0.0864252 + 0.996258i \(0.527544\pi\)
\(728\) 0 0
\(729\) −37.4575 −1.38732
\(730\) 0 0
\(731\) −14.0913 −0.521186
\(732\) 0 0
\(733\) −34.4826 −1.27364 −0.636822 0.771011i \(-0.719751\pi\)
−0.636822 + 0.771011i \(0.719751\pi\)
\(734\) 0 0
\(735\) 75.9726 2.80229
\(736\) 0 0
\(737\) 45.3469 1.67037
\(738\) 0 0
\(739\) −12.4222 −0.456956 −0.228478 0.973549i \(-0.573375\pi\)
−0.228478 + 0.973549i \(0.573375\pi\)
\(740\) 0 0
\(741\) −7.32401 −0.269054
\(742\) 0 0
\(743\) −6.77981 −0.248727 −0.124364 0.992237i \(-0.539689\pi\)
−0.124364 + 0.992237i \(0.539689\pi\)
\(744\) 0 0
\(745\) −26.3662 −0.965984
\(746\) 0 0
\(747\) 74.1803 2.71411
\(748\) 0 0
\(749\) 0.179702 0.00656615
\(750\) 0 0
\(751\) 40.4826 1.47723 0.738616 0.674127i \(-0.235480\pi\)
0.738616 + 0.674127i \(0.235480\pi\)
\(752\) 0 0
\(753\) −59.8968 −2.18276
\(754\) 0 0
\(755\) −71.3246 −2.59577
\(756\) 0 0
\(757\) 23.5191 0.854818 0.427409 0.904058i \(-0.359426\pi\)
0.427409 + 0.904058i \(0.359426\pi\)
\(758\) 0 0
\(759\) 75.0074 2.72260
\(760\) 0 0
\(761\) −2.32692 −0.0843507 −0.0421753 0.999110i \(-0.513429\pi\)
−0.0421753 + 0.999110i \(0.513429\pi\)
\(762\) 0 0
\(763\) 3.13863 0.113626
\(764\) 0 0
\(765\) 64.7593 2.34138
\(766\) 0 0
\(767\) 21.3662 0.771490
\(768\) 0 0
\(769\) 34.8768 1.25769 0.628845 0.777531i \(-0.283528\pi\)
0.628845 + 0.777531i \(0.283528\pi\)
\(770\) 0 0
\(771\) 74.4016 2.67951
\(772\) 0 0
\(773\) 5.94699 0.213898 0.106949 0.994264i \(-0.465892\pi\)
0.106949 + 0.994264i \(0.465892\pi\)
\(774\) 0 0
\(775\) −45.5882 −1.63758
\(776\) 0 0
\(777\) 5.93272 0.212835
\(778\) 0 0
\(779\) 10.2413 0.366933
\(780\) 0 0
\(781\) −45.1306 −1.61490
\(782\) 0 0
\(783\) 41.9720 1.49996
\(784\) 0 0
\(785\) 62.8420 2.24293
\(786\) 0 0
\(787\) −0.277845 −0.00990411 −0.00495206 0.999988i \(-0.501576\pi\)
−0.00495206 + 0.999988i \(0.501576\pi\)
\(788\) 0 0
\(789\) 40.1181 1.42824
\(790\) 0 0
\(791\) −1.64919 −0.0586383
\(792\) 0 0
\(793\) 17.8346 0.633324
\(794\) 0 0
\(795\) 12.3731 0.438828
\(796\) 0 0
\(797\) −5.01543 −0.177656 −0.0888278 0.996047i \(-0.528312\pi\)
−0.0888278 + 0.996047i \(0.528312\pi\)
\(798\) 0 0
\(799\) −6.24814 −0.221043
\(800\) 0 0
\(801\) 1.32970 0.0469826
\(802\) 0 0
\(803\) −56.9720 −2.01050
\(804\) 0 0
\(805\) 17.0211 0.599915
\(806\) 0 0
\(807\) 60.7439 2.13829
\(808\) 0 0
\(809\) −18.9190 −0.665158 −0.332579 0.943075i \(-0.607919\pi\)
−0.332579 + 0.943075i \(0.607919\pi\)
\(810\) 0 0
\(811\) 45.3103 1.59106 0.795530 0.605914i \(-0.207192\pi\)
0.795530 + 0.605914i \(0.207192\pi\)
\(812\) 0 0
\(813\) 43.2567 1.51708
\(814\) 0 0
\(815\) −94.1803 −3.29899
\(816\) 0 0
\(817\) 4.69710 0.164331
\(818\) 0 0
\(819\) 8.20766 0.286799
\(820\) 0 0
\(821\) −12.6498 −0.441480 −0.220740 0.975333i \(-0.570847\pi\)
−0.220740 + 0.975333i \(0.570847\pi\)
\(822\) 0 0
\(823\) −43.9065 −1.53048 −0.765242 0.643742i \(-0.777381\pi\)
−0.765242 + 0.643742i \(0.777381\pi\)
\(824\) 0 0
\(825\) −105.948 −3.68862
\(826\) 0 0
\(827\) −13.0405 −0.453462 −0.226731 0.973957i \(-0.572804\pi\)
−0.226731 + 0.973957i \(0.572804\pi\)
\(828\) 0 0
\(829\) 49.4507 1.71749 0.858747 0.512400i \(-0.171243\pi\)
0.858747 + 0.512400i \(0.171243\pi\)
\(830\) 0 0
\(831\) −46.4353 −1.61082
\(832\) 0 0
\(833\) −19.9441 −0.691022
\(834\) 0 0
\(835\) 76.9179 2.66185
\(836\) 0 0
\(837\) 32.2749 1.11559
\(838\) 0 0
\(839\) 15.2288 0.525756 0.262878 0.964829i \(-0.415328\pi\)
0.262878 + 0.964829i \(0.415328\pi\)
\(840\) 0 0
\(841\) 3.84141 0.132463
\(842\) 0 0
\(843\) −13.5596 −0.467018
\(844\) 0 0
\(845\) −26.2345 −0.902493
\(846\) 0 0
\(847\) −0.785500 −0.0269901
\(848\) 0 0
\(849\) 11.6492 0.399799
\(850\) 0 0
\(851\) −25.1056 −0.860608
\(852\) 0 0
\(853\) −15.4193 −0.527945 −0.263973 0.964530i \(-0.585033\pi\)
−0.263973 + 0.964530i \(0.585033\pi\)
\(854\) 0 0
\(855\) −21.5864 −0.738240
\(856\) 0 0
\(857\) −14.3422 −0.489921 −0.244961 0.969533i \(-0.578775\pi\)
−0.244961 + 0.969533i \(0.578775\pi\)
\(858\) 0 0
\(859\) 46.2681 1.57865 0.789324 0.613977i \(-0.210431\pi\)
0.789324 + 0.613977i \(0.210431\pi\)
\(860\) 0 0
\(861\) −17.7251 −0.604068
\(862\) 0 0
\(863\) −25.9527 −0.883439 −0.441720 0.897153i \(-0.645631\pi\)
−0.441720 + 0.897153i \(0.645631\pi\)
\(864\) 0 0
\(865\) 32.7998 1.11523
\(866\) 0 0
\(867\) 23.3383 0.792610
\(868\) 0 0
\(869\) −15.4701 −0.524786
\(870\) 0 0
\(871\) −32.4296 −1.09883
\(872\) 0 0
\(873\) 85.2014 2.88363
\(874\) 0 0
\(875\) −12.4222 −0.419946
\(876\) 0 0
\(877\) 21.3526 0.721025 0.360512 0.932754i \(-0.382602\pi\)
0.360512 + 0.932754i \(0.382602\pi\)
\(878\) 0 0
\(879\) 9.09130 0.306642
\(880\) 0 0
\(881\) 53.2642 1.79452 0.897258 0.441507i \(-0.145556\pi\)
0.897258 + 0.441507i \(0.145556\pi\)
\(882\) 0 0
\(883\) −38.1586 −1.28414 −0.642069 0.766647i \(-0.721924\pi\)
−0.642069 + 0.766647i \(0.721924\pi\)
\(884\) 0 0
\(885\) 97.2573 3.26927
\(886\) 0 0
\(887\) 11.1615 0.374766 0.187383 0.982287i \(-0.439999\pi\)
0.187383 + 0.982287i \(0.439999\pi\)
\(888\) 0 0
\(889\) −8.47401 −0.284209
\(890\) 0 0
\(891\) 16.9720 0.568585
\(892\) 0 0
\(893\) 2.08271 0.0696954
\(894\) 0 0
\(895\) 40.1181 1.34100
\(896\) 0 0
\(897\) −53.6412 −1.79103
\(898\) 0 0
\(899\) 25.2538 0.842262
\(900\) 0 0
\(901\) −3.24814 −0.108211
\(902\) 0 0
\(903\) −8.12947 −0.270532
\(904\) 0 0
\(905\) −96.0708 −3.19350
\(906\) 0 0
\(907\) 32.7941 1.08891 0.544455 0.838790i \(-0.316737\pi\)
0.544455 + 0.838790i \(0.316737\pi\)
\(908\) 0 0
\(909\) 87.2573 2.89414
\(910\) 0 0
\(911\) −46.8979 −1.55380 −0.776899 0.629626i \(-0.783208\pi\)
−0.776899 + 0.629626i \(0.783208\pi\)
\(912\) 0 0
\(913\) 47.2573 1.56399
\(914\) 0 0
\(915\) 81.1814 2.68378
\(916\) 0 0
\(917\) −9.70963 −0.320640
\(918\) 0 0
\(919\) 45.1871 1.49059 0.745293 0.666737i \(-0.232310\pi\)
0.745293 + 0.666737i \(0.232310\pi\)
\(920\) 0 0
\(921\) 69.1478 2.27850
\(922\) 0 0
\(923\) 32.2749 1.06234
\(924\) 0 0
\(925\) 35.4615 1.16597
\(926\) 0 0
\(927\) −74.1803 −2.43640
\(928\) 0 0
\(929\) −51.0776 −1.67580 −0.837901 0.545822i \(-0.816217\pi\)
−0.837901 + 0.545822i \(0.816217\pi\)
\(930\) 0 0
\(931\) 6.64803 0.217880
\(932\) 0 0
\(933\) 5.55163 0.181752
\(934\) 0 0
\(935\) 41.2556 1.34920
\(936\) 0 0
\(937\) 24.5037 0.800502 0.400251 0.916406i \(-0.368923\pi\)
0.400251 + 0.916406i \(0.368923\pi\)
\(938\) 0 0
\(939\) −92.4296 −3.01633
\(940\) 0 0
\(941\) 55.1871 1.79905 0.899525 0.436870i \(-0.143913\pi\)
0.899525 + 0.436870i \(0.143913\pi\)
\(942\) 0 0
\(943\) 75.0074 2.44258
\(944\) 0 0
\(945\) 17.0211 0.553697
\(946\) 0 0
\(947\) 29.9019 0.971680 0.485840 0.874048i \(-0.338514\pi\)
0.485840 + 0.874048i \(0.338514\pi\)
\(948\) 0 0
\(949\) 40.7433 1.32258
\(950\) 0 0
\(951\) 35.5317 1.15219
\(952\) 0 0
\(953\) 26.0086 0.842501 0.421250 0.906944i \(-0.361591\pi\)
0.421250 + 0.906944i \(0.361591\pi\)
\(954\) 0 0
\(955\) 36.3662 1.17678
\(956\) 0 0
\(957\) 58.6903 1.89719
\(958\) 0 0
\(959\) 2.77007 0.0894502
\(960\) 0 0
\(961\) −11.5807 −0.373572
\(962\) 0 0
\(963\) −1.66914 −0.0537874
\(964\) 0 0
\(965\) −31.4701 −1.01306
\(966\) 0 0
\(967\) −50.8420 −1.63497 −0.817484 0.575951i \(-0.804632\pi\)
−0.817484 + 0.575951i \(0.804632\pi\)
\(968\) 0 0
\(969\) 8.75186 0.281150
\(970\) 0 0
\(971\) 22.1267 0.710079 0.355040 0.934851i \(-0.384467\pi\)
0.355040 + 0.934851i \(0.384467\pi\)
\(972\) 0 0
\(973\) 3.85000 0.123425
\(974\) 0 0
\(975\) 75.7679 2.42651
\(976\) 0 0
\(977\) 43.1814 1.38150 0.690748 0.723095i \(-0.257281\pi\)
0.690748 + 0.723095i \(0.257281\pi\)
\(978\) 0 0
\(979\) 0.847099 0.0270734
\(980\) 0 0
\(981\) −29.1529 −0.930780
\(982\) 0 0
\(983\) −34.8842 −1.11263 −0.556317 0.830970i \(-0.687786\pi\)
−0.556317 + 0.830970i \(0.687786\pi\)
\(984\) 0 0
\(985\) 22.0422 0.702324
\(986\) 0 0
\(987\) −3.60464 −0.114737
\(988\) 0 0
\(989\) 34.4016 1.09391
\(990\) 0 0
\(991\) −7.36056 −0.233816 −0.116908 0.993143i \(-0.537298\pi\)
−0.116908 + 0.993143i \(0.537298\pi\)
\(992\) 0 0
\(993\) −20.6623 −0.655698
\(994\) 0 0
\(995\) 70.8911 2.24740
\(996\) 0 0
\(997\) 37.2978 1.18123 0.590617 0.806952i \(-0.298885\pi\)
0.590617 + 0.806952i \(0.298885\pi\)
\(998\) 0 0
\(999\) −25.1056 −0.794305
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 4864.2.a.bf.1.1 3
4.3 odd 2 4864.2.a.bd.1.3 3
8.3 odd 2 4864.2.a.be.1.1 3
8.5 even 2 4864.2.a.bc.1.3 3
16.3 odd 4 2432.2.c.f.1217.6 yes 6
16.5 even 4 2432.2.c.g.1217.6 yes 6
16.11 odd 4 2432.2.c.f.1217.1 6
16.13 even 4 2432.2.c.g.1217.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.f.1217.1 6 16.11 odd 4
2432.2.c.f.1217.6 yes 6 16.3 odd 4
2432.2.c.g.1217.1 yes 6 16.13 even 4
2432.2.c.g.1217.6 yes 6 16.5 even 4
4864.2.a.bc.1.3 3 8.5 even 2
4864.2.a.bd.1.3 3 4.3 odd 2
4864.2.a.be.1.1 3 8.3 odd 2
4864.2.a.bf.1.1 3 1.1 even 1 trivial