Properties

Label 4864.2.a.bm.1.4
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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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: \(8\)
Coefficient field: 8.8.34309996544.1
Defining polynomial: \(x^{8} - 4 x^{7} - 8 x^{6} + 28 x^{5} + 31 x^{4} - 36 x^{3} - 22 x^{2} + 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.79159\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.65222 q^{3} +4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} +O(q^{10})\) \(q-1.65222 q^{3} +4.30533 q^{5} -2.73423 q^{7} -0.270160 q^{9} -1.72984 q^{11} -0.546394 q^{13} -7.11337 q^{15} -3.82843 q^{17} +1.00000 q^{19} +4.51756 q^{21} +0.546394 q^{23} +13.5359 q^{25} +5.40303 q^{27} -0.0738068 q^{29} -1.49730 q^{31} +2.85808 q^{33} -11.7718 q^{35} +8.56253 q^{37} +0.902764 q^{39} +1.90604 q^{41} +9.07622 q^{43} -1.16313 q^{45} -5.33004 q^{47} +0.476019 q^{49} +6.32541 q^{51} -10.1336 q^{53} -7.44754 q^{55} -1.65222 q^{57} -6.03429 q^{59} -1.31074 q^{61} +0.738679 q^{63} -2.35241 q^{65} -9.02097 q^{67} -0.902764 q^{69} -14.6512 q^{71} +8.50162 q^{73} -22.3643 q^{75} +4.72978 q^{77} +7.68543 q^{79} -8.11654 q^{81} -16.8061 q^{83} -16.4827 q^{85} +0.121945 q^{87} +14.0166 q^{89} +1.49397 q^{91} +2.47387 q^{93} +4.30533 q^{95} +7.30445 q^{97} +0.467333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} + 4q^{9} - 20q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 24q^{33} - 12q^{35} + 8q^{41} - 28q^{43} + 8q^{49} - 12q^{51} - 4q^{57} - 36q^{59} + 8q^{65} - 28q^{67} - 8q^{73} - 68q^{75} - 32q^{81} - 40q^{83} - 8q^{89} + 12q^{91} + 40q^{97} - 60q^{99} + O(q^{100}) \)

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
\(3\) −1.65222 −0.953911 −0.476956 0.878927i \(-0.658260\pi\)
−0.476956 + 0.878927i \(0.658260\pi\)
\(4\) 0 0
\(5\) 4.30533 1.92540 0.962702 0.270564i \(-0.0872101\pi\)
0.962702 + 0.270564i \(0.0872101\pi\)
\(6\) 0 0
\(7\) −2.73423 −1.03344 −0.516721 0.856154i \(-0.672848\pi\)
−0.516721 + 0.856154i \(0.672848\pi\)
\(8\) 0 0
\(9\) −0.270160 −0.0900532
\(10\) 0 0
\(11\) −1.72984 −0.521567 −0.260783 0.965397i \(-0.583981\pi\)
−0.260783 + 0.965397i \(0.583981\pi\)
\(12\) 0 0
\(13\) −0.546394 −0.151542 −0.0757712 0.997125i \(-0.524142\pi\)
−0.0757712 + 0.997125i \(0.524142\pi\)
\(14\) 0 0
\(15\) −7.11337 −1.83666
\(16\) 0 0
\(17\) −3.82843 −0.928530 −0.464265 0.885696i \(-0.653681\pi\)
−0.464265 + 0.885696i \(0.653681\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.51756 0.985812
\(22\) 0 0
\(23\) 0.546394 0.113931 0.0569655 0.998376i \(-0.481857\pi\)
0.0569655 + 0.998376i \(0.481857\pi\)
\(24\) 0 0
\(25\) 13.5359 2.70718
\(26\) 0 0
\(27\) 5.40303 1.03981
\(28\) 0 0
\(29\) −0.0738068 −0.0137056 −0.00685279 0.999977i \(-0.502181\pi\)
−0.00685279 + 0.999977i \(0.502181\pi\)
\(30\) 0 0
\(31\) −1.49730 −0.268922 −0.134461 0.990919i \(-0.542930\pi\)
−0.134461 + 0.990919i \(0.542930\pi\)
\(32\) 0 0
\(33\) 2.85808 0.497528
\(34\) 0 0
\(35\) −11.7718 −1.98979
\(36\) 0 0
\(37\) 8.56253 1.40767 0.703836 0.710363i \(-0.251469\pi\)
0.703836 + 0.710363i \(0.251469\pi\)
\(38\) 0 0
\(39\) 0.902764 0.144558
\(40\) 0 0
\(41\) 1.90604 0.297674 0.148837 0.988862i \(-0.452447\pi\)
0.148837 + 0.988862i \(0.452447\pi\)
\(42\) 0 0
\(43\) 9.07622 1.38411 0.692056 0.721844i \(-0.256705\pi\)
0.692056 + 0.721844i \(0.256705\pi\)
\(44\) 0 0
\(45\) −1.16313 −0.173389
\(46\) 0 0
\(47\) −5.33004 −0.777467 −0.388733 0.921350i \(-0.627087\pi\)
−0.388733 + 0.921350i \(0.627087\pi\)
\(48\) 0 0
\(49\) 0.476019 0.0680027
\(50\) 0 0
\(51\) 6.32541 0.885735
\(52\) 0 0
\(53\) −10.1336 −1.39196 −0.695981 0.718060i \(-0.745030\pi\)
−0.695981 + 0.718060i \(0.745030\pi\)
\(54\) 0 0
\(55\) −7.44754 −1.00423
\(56\) 0 0
\(57\) −1.65222 −0.218842
\(58\) 0 0
\(59\) −6.03429 −0.785597 −0.392799 0.919625i \(-0.628493\pi\)
−0.392799 + 0.919625i \(0.628493\pi\)
\(60\) 0 0
\(61\) −1.31074 −0.167823 −0.0839116 0.996473i \(-0.526741\pi\)
−0.0839116 + 0.996473i \(0.526741\pi\)
\(62\) 0 0
\(63\) 0.738679 0.0930648
\(64\) 0 0
\(65\) −2.35241 −0.291780
\(66\) 0 0
\(67\) −9.02097 −1.10209 −0.551043 0.834477i \(-0.685770\pi\)
−0.551043 + 0.834477i \(0.685770\pi\)
\(68\) 0 0
\(69\) −0.902764 −0.108680
\(70\) 0 0
\(71\) −14.6512 −1.73878 −0.869388 0.494129i \(-0.835487\pi\)
−0.869388 + 0.494129i \(0.835487\pi\)
\(72\) 0 0
\(73\) 8.50162 0.995039 0.497520 0.867453i \(-0.334244\pi\)
0.497520 + 0.867453i \(0.334244\pi\)
\(74\) 0 0
\(75\) −22.3643 −2.58241
\(76\) 0 0
\(77\) 4.72978 0.539009
\(78\) 0 0
\(79\) 7.68543 0.864679 0.432339 0.901711i \(-0.357688\pi\)
0.432339 + 0.901711i \(0.357688\pi\)
\(80\) 0 0
\(81\) −8.11654 −0.901837
\(82\) 0 0
\(83\) −16.8061 −1.84471 −0.922353 0.386349i \(-0.873736\pi\)
−0.922353 + 0.386349i \(0.873736\pi\)
\(84\) 0 0
\(85\) −16.4827 −1.78780
\(86\) 0 0
\(87\) 0.121945 0.0130739
\(88\) 0 0
\(89\) 14.0166 1.48575 0.742876 0.669429i \(-0.233461\pi\)
0.742876 + 0.669429i \(0.233461\pi\)
\(90\) 0 0
\(91\) 1.49397 0.156610
\(92\) 0 0
\(93\) 2.47387 0.256528
\(94\) 0 0
\(95\) 4.30533 0.441718
\(96\) 0 0
\(97\) 7.30445 0.741654 0.370827 0.928702i \(-0.379074\pi\)
0.370827 + 0.928702i \(0.379074\pi\)
\(98\) 0 0
\(99\) 0.467333 0.0469687
\(100\) 0 0
\(101\) −9.85107 −0.980218 −0.490109 0.871661i \(-0.663043\pi\)
−0.490109 + 0.871661i \(0.663043\pi\)
\(102\) 0 0
\(103\) −12.8457 −1.26572 −0.632860 0.774266i \(-0.718119\pi\)
−0.632860 + 0.774266i \(0.718119\pi\)
\(104\) 0 0
\(105\) 19.4496 1.89809
\(106\) 0 0
\(107\) −7.81048 −0.755067 −0.377534 0.925996i \(-0.623228\pi\)
−0.377534 + 0.925996i \(0.623228\pi\)
\(108\) 0 0
\(109\) −16.4792 −1.57842 −0.789210 0.614123i \(-0.789510\pi\)
−0.789210 + 0.614123i \(0.789510\pi\)
\(110\) 0 0
\(111\) −14.1472 −1.34279
\(112\) 0 0
\(113\) −14.8673 −1.39860 −0.699301 0.714827i \(-0.746505\pi\)
−0.699301 + 0.714827i \(0.746505\pi\)
\(114\) 0 0
\(115\) 2.35241 0.219363
\(116\) 0 0
\(117\) 0.147614 0.0136469
\(118\) 0 0
\(119\) 10.4678 0.959582
\(120\) 0 0
\(121\) −8.00765 −0.727968
\(122\) 0 0
\(123\) −3.14921 −0.283955
\(124\) 0 0
\(125\) 36.7499 3.28701
\(126\) 0 0
\(127\) 10.1080 0.896937 0.448468 0.893799i \(-0.351970\pi\)
0.448468 + 0.893799i \(0.351970\pi\)
\(128\) 0 0
\(129\) −14.9959 −1.32032
\(130\) 0 0
\(131\) −0.356822 −0.0311757 −0.0155879 0.999879i \(-0.504962\pi\)
−0.0155879 + 0.999879i \(0.504962\pi\)
\(132\) 0 0
\(133\) −2.73423 −0.237088
\(134\) 0 0
\(135\) 23.2619 2.00206
\(136\) 0 0
\(137\) −8.68953 −0.742397 −0.371198 0.928554i \(-0.621053\pi\)
−0.371198 + 0.928554i \(0.621053\pi\)
\(138\) 0 0
\(139\) −14.4673 −1.22710 −0.613552 0.789655i \(-0.710260\pi\)
−0.613552 + 0.789655i \(0.710260\pi\)
\(140\) 0 0
\(141\) 8.80642 0.741634
\(142\) 0 0
\(143\) 0.945174 0.0790394
\(144\) 0 0
\(145\) −0.317763 −0.0263888
\(146\) 0 0
\(147\) −0.786489 −0.0648685
\(148\) 0 0
\(149\) −9.77380 −0.800701 −0.400350 0.916362i \(-0.631112\pi\)
−0.400350 + 0.916362i \(0.631112\pi\)
\(150\) 0 0
\(151\) 8.61067 0.700726 0.350363 0.936614i \(-0.386058\pi\)
0.350363 + 0.936614i \(0.386058\pi\)
\(152\) 0 0
\(153\) 1.03429 0.0836171
\(154\) 0 0
\(155\) −6.44636 −0.517784
\(156\) 0 0
\(157\) −18.8862 −1.50728 −0.753641 0.657286i \(-0.771704\pi\)
−0.753641 + 0.657286i \(0.771704\pi\)
\(158\) 0 0
\(159\) 16.7430 1.32781
\(160\) 0 0
\(161\) −1.49397 −0.117741
\(162\) 0 0
\(163\) 20.6867 1.62031 0.810155 0.586216i \(-0.199383\pi\)
0.810155 + 0.586216i \(0.199383\pi\)
\(164\) 0 0
\(165\) 12.3050 0.957943
\(166\) 0 0
\(167\) 20.3835 1.57732 0.788661 0.614829i \(-0.210775\pi\)
0.788661 + 0.614829i \(0.210775\pi\)
\(168\) 0 0
\(169\) −12.7015 −0.977035
\(170\) 0 0
\(171\) −0.270160 −0.0206596
\(172\) 0 0
\(173\) −8.96704 −0.681751 −0.340876 0.940108i \(-0.610724\pi\)
−0.340876 + 0.940108i \(0.610724\pi\)
\(174\) 0 0
\(175\) −37.0103 −2.79771
\(176\) 0 0
\(177\) 9.96999 0.749390
\(178\) 0 0
\(179\) −10.6702 −0.797526 −0.398763 0.917054i \(-0.630560\pi\)
−0.398763 + 0.917054i \(0.630560\pi\)
\(180\) 0 0
\(181\) 14.4424 1.07350 0.536749 0.843742i \(-0.319652\pi\)
0.536749 + 0.843742i \(0.319652\pi\)
\(182\) 0 0
\(183\) 2.16564 0.160088
\(184\) 0 0
\(185\) 36.8646 2.71034
\(186\) 0 0
\(187\) 6.62257 0.484290
\(188\) 0 0
\(189\) −14.7731 −1.07459
\(190\) 0 0
\(191\) 21.8574 1.58154 0.790772 0.612111i \(-0.209679\pi\)
0.790772 + 0.612111i \(0.209679\pi\)
\(192\) 0 0
\(193\) 0.601599 0.0433040 0.0216520 0.999766i \(-0.493107\pi\)
0.0216520 + 0.999766i \(0.493107\pi\)
\(194\) 0 0
\(195\) 3.88670 0.278333
\(196\) 0 0
\(197\) −6.28441 −0.447746 −0.223873 0.974618i \(-0.571870\pi\)
−0.223873 + 0.974618i \(0.571870\pi\)
\(198\) 0 0
\(199\) −24.0262 −1.70317 −0.851586 0.524214i \(-0.824359\pi\)
−0.851586 + 0.524214i \(0.824359\pi\)
\(200\) 0 0
\(201\) 14.9047 1.05129
\(202\) 0 0
\(203\) 0.201805 0.0141639
\(204\) 0 0
\(205\) 8.20616 0.573143
\(206\) 0 0
\(207\) −0.147614 −0.0102598
\(208\) 0 0
\(209\) −1.72984 −0.119656
\(210\) 0 0
\(211\) −6.25382 −0.430531 −0.215265 0.976556i \(-0.569062\pi\)
−0.215265 + 0.976556i \(0.569062\pi\)
\(212\) 0 0
\(213\) 24.2070 1.65864
\(214\) 0 0
\(215\) 39.0762 2.66497
\(216\) 0 0
\(217\) 4.09396 0.277916
\(218\) 0 0
\(219\) −14.0466 −0.949179
\(220\) 0 0
\(221\) 2.09183 0.140712
\(222\) 0 0
\(223\) 12.3449 0.826674 0.413337 0.910578i \(-0.364363\pi\)
0.413337 + 0.910578i \(0.364363\pi\)
\(224\) 0 0
\(225\) −3.65685 −0.243790
\(226\) 0 0
\(227\) −2.30886 −0.153244 −0.0766222 0.997060i \(-0.524414\pi\)
−0.0766222 + 0.997060i \(0.524414\pi\)
\(228\) 0 0
\(229\) −2.11976 −0.140078 −0.0700388 0.997544i \(-0.522312\pi\)
−0.0700388 + 0.997544i \(0.522312\pi\)
\(230\) 0 0
\(231\) −7.81466 −0.514167
\(232\) 0 0
\(233\) −25.2134 −1.65178 −0.825891 0.563829i \(-0.809328\pi\)
−0.825891 + 0.563829i \(0.809328\pi\)
\(234\) 0 0
\(235\) −22.9476 −1.49694
\(236\) 0 0
\(237\) −12.6981 −0.824827
\(238\) 0 0
\(239\) 22.7927 1.47434 0.737170 0.675707i \(-0.236162\pi\)
0.737170 + 0.675707i \(0.236162\pi\)
\(240\) 0 0
\(241\) 2.98064 0.192000 0.0960000 0.995381i \(-0.469395\pi\)
0.0960000 + 0.995381i \(0.469395\pi\)
\(242\) 0 0
\(243\) −2.79877 −0.179541
\(244\) 0 0
\(245\) 2.04942 0.130933
\(246\) 0 0
\(247\) −0.546394 −0.0347662
\(248\) 0 0
\(249\) 27.7674 1.75969
\(250\) 0 0
\(251\) −8.84638 −0.558378 −0.279189 0.960236i \(-0.590066\pi\)
−0.279189 + 0.960236i \(0.590066\pi\)
\(252\) 0 0
\(253\) −0.945174 −0.0594226
\(254\) 0 0
\(255\) 27.2330 1.70540
\(256\) 0 0
\(257\) −6.04194 −0.376886 −0.188443 0.982084i \(-0.560344\pi\)
−0.188443 + 0.982084i \(0.560344\pi\)
\(258\) 0 0
\(259\) −23.4119 −1.45475
\(260\) 0 0
\(261\) 0.0199396 0.00123423
\(262\) 0 0
\(263\) 13.1317 0.809735 0.404868 0.914375i \(-0.367318\pi\)
0.404868 + 0.914375i \(0.367318\pi\)
\(264\) 0 0
\(265\) −43.6287 −2.68009
\(266\) 0 0
\(267\) −23.1585 −1.41728
\(268\) 0 0
\(269\) −13.4659 −0.821028 −0.410514 0.911854i \(-0.634651\pi\)
−0.410514 + 0.911854i \(0.634651\pi\)
\(270\) 0 0
\(271\) −3.02026 −0.183468 −0.0917339 0.995784i \(-0.529241\pi\)
−0.0917339 + 0.995784i \(0.529241\pi\)
\(272\) 0 0
\(273\) −2.46837 −0.149392
\(274\) 0 0
\(275\) −23.4150 −1.41197
\(276\) 0 0
\(277\) −8.16722 −0.490721 −0.245360 0.969432i \(-0.578906\pi\)
−0.245360 + 0.969432i \(0.578906\pi\)
\(278\) 0 0
\(279\) 0.404509 0.0242173
\(280\) 0 0
\(281\) −20.4956 −1.22266 −0.611332 0.791374i \(-0.709366\pi\)
−0.611332 + 0.791374i \(0.709366\pi\)
\(282\) 0 0
\(283\) 19.9690 1.18703 0.593515 0.804823i \(-0.297740\pi\)
0.593515 + 0.804823i \(0.297740\pi\)
\(284\) 0 0
\(285\) −7.11337 −0.421360
\(286\) 0 0
\(287\) −5.21157 −0.307629
\(288\) 0 0
\(289\) −2.34315 −0.137832
\(290\) 0 0
\(291\) −12.0686 −0.707472
\(292\) 0 0
\(293\) 10.8432 0.633465 0.316733 0.948515i \(-0.397414\pi\)
0.316733 + 0.948515i \(0.397414\pi\)
\(294\) 0 0
\(295\) −25.9796 −1.51259
\(296\) 0 0
\(297\) −9.34638 −0.542332
\(298\) 0 0
\(299\) −0.298546 −0.0172654
\(300\) 0 0
\(301\) −24.8165 −1.43040
\(302\) 0 0
\(303\) 16.2762 0.935041
\(304\) 0 0
\(305\) −5.64318 −0.323127
\(306\) 0 0
\(307\) 5.36573 0.306238 0.153119 0.988208i \(-0.451068\pi\)
0.153119 + 0.988208i \(0.451068\pi\)
\(308\) 0 0
\(309\) 21.2239 1.20739
\(310\) 0 0
\(311\) −10.3271 −0.585598 −0.292799 0.956174i \(-0.594587\pi\)
−0.292799 + 0.956174i \(0.594587\pi\)
\(312\) 0 0
\(313\) −29.2020 −1.65060 −0.825298 0.564698i \(-0.808993\pi\)
−0.825298 + 0.564698i \(0.808993\pi\)
\(314\) 0 0
\(315\) 3.18026 0.179187
\(316\) 0 0
\(317\) 11.5115 0.646551 0.323276 0.946305i \(-0.395216\pi\)
0.323276 + 0.946305i \(0.395216\pi\)
\(318\) 0 0
\(319\) 0.127674 0.00714837
\(320\) 0 0
\(321\) 12.9047 0.720267
\(322\) 0 0
\(323\) −3.82843 −0.213019
\(324\) 0 0
\(325\) −7.39594 −0.410253
\(326\) 0 0
\(327\) 27.2273 1.50567
\(328\) 0 0
\(329\) 14.5736 0.803467
\(330\) 0 0
\(331\) −27.0629 −1.48751 −0.743756 0.668451i \(-0.766958\pi\)
−0.743756 + 0.668451i \(0.766958\pi\)
\(332\) 0 0
\(333\) −2.31325 −0.126765
\(334\) 0 0
\(335\) −38.8383 −2.12196
\(336\) 0 0
\(337\) −26.1851 −1.42639 −0.713197 0.700963i \(-0.752754\pi\)
−0.713197 + 0.700963i \(0.752754\pi\)
\(338\) 0 0
\(339\) 24.5642 1.33414
\(340\) 0 0
\(341\) 2.59008 0.140261
\(342\) 0 0
\(343\) 17.8381 0.963165
\(344\) 0 0
\(345\) −3.88670 −0.209253
\(346\) 0 0
\(347\) −27.2734 −1.46411 −0.732056 0.681244i \(-0.761439\pi\)
−0.732056 + 0.681244i \(0.761439\pi\)
\(348\) 0 0
\(349\) 6.11780 0.327478 0.163739 0.986504i \(-0.447644\pi\)
0.163739 + 0.986504i \(0.447644\pi\)
\(350\) 0 0
\(351\) −2.95218 −0.157576
\(352\) 0 0
\(353\) −8.51030 −0.452958 −0.226479 0.974016i \(-0.572721\pi\)
−0.226479 + 0.974016i \(0.572721\pi\)
\(354\) 0 0
\(355\) −63.0783 −3.34785
\(356\) 0 0
\(357\) −17.2951 −0.915356
\(358\) 0 0
\(359\) −13.0097 −0.686628 −0.343314 0.939221i \(-0.611550\pi\)
−0.343314 + 0.939221i \(0.611550\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 13.2304 0.694417
\(364\) 0 0
\(365\) 36.6023 1.91585
\(366\) 0 0
\(367\) −26.1953 −1.36738 −0.683692 0.729771i \(-0.739627\pi\)
−0.683692 + 0.729771i \(0.739627\pi\)
\(368\) 0 0
\(369\) −0.514936 −0.0268065
\(370\) 0 0
\(371\) 27.7077 1.43851
\(372\) 0 0
\(373\) 33.4436 1.73165 0.865823 0.500351i \(-0.166796\pi\)
0.865823 + 0.500351i \(0.166796\pi\)
\(374\) 0 0
\(375\) −60.7191 −3.13552
\(376\) 0 0
\(377\) 0.0403276 0.00207698
\(378\) 0 0
\(379\) 28.6269 1.47046 0.735231 0.677816i \(-0.237073\pi\)
0.735231 + 0.677816i \(0.237073\pi\)
\(380\) 0 0
\(381\) −16.7006 −0.855598
\(382\) 0 0
\(383\) 20.1066 1.02740 0.513701 0.857969i \(-0.328274\pi\)
0.513701 + 0.857969i \(0.328274\pi\)
\(384\) 0 0
\(385\) 20.3633 1.03781
\(386\) 0 0
\(387\) −2.45203 −0.124644
\(388\) 0 0
\(389\) −25.6141 −1.29868 −0.649342 0.760496i \(-0.724956\pi\)
−0.649342 + 0.760496i \(0.724956\pi\)
\(390\) 0 0
\(391\) −2.09183 −0.105788
\(392\) 0 0
\(393\) 0.589550 0.0297389
\(394\) 0 0
\(395\) 33.0884 1.66486
\(396\) 0 0
\(397\) 4.40161 0.220911 0.110455 0.993881i \(-0.464769\pi\)
0.110455 + 0.993881i \(0.464769\pi\)
\(398\) 0 0
\(399\) 4.51756 0.226161
\(400\) 0 0
\(401\) −9.38031 −0.468430 −0.234215 0.972185i \(-0.575252\pi\)
−0.234215 + 0.972185i \(0.575252\pi\)
\(402\) 0 0
\(403\) 0.818114 0.0407532
\(404\) 0 0
\(405\) −34.9444 −1.73640
\(406\) 0 0
\(407\) −14.8118 −0.734194
\(408\) 0 0
\(409\) −0.216515 −0.0107060 −0.00535299 0.999986i \(-0.501704\pi\)
−0.00535299 + 0.999986i \(0.501704\pi\)
\(410\) 0 0
\(411\) 14.3570 0.708181
\(412\) 0 0
\(413\) 16.4991 0.811869
\(414\) 0 0
\(415\) −72.3557 −3.55180
\(416\) 0 0
\(417\) 23.9033 1.17055
\(418\) 0 0
\(419\) 18.7827 0.917593 0.458796 0.888541i \(-0.348281\pi\)
0.458796 + 0.888541i \(0.348281\pi\)
\(420\) 0 0
\(421\) −2.19130 −0.106798 −0.0533988 0.998573i \(-0.517005\pi\)
−0.0533988 + 0.998573i \(0.517005\pi\)
\(422\) 0 0
\(423\) 1.43996 0.0700134
\(424\) 0 0
\(425\) −51.8212 −2.51370
\(426\) 0 0
\(427\) 3.58387 0.173436
\(428\) 0 0
\(429\) −1.56164 −0.0753966
\(430\) 0 0
\(431\) 20.2673 0.976240 0.488120 0.872777i \(-0.337683\pi\)
0.488120 + 0.872777i \(0.337683\pi\)
\(432\) 0 0
\(433\) 10.8766 0.522696 0.261348 0.965245i \(-0.415833\pi\)
0.261348 + 0.965245i \(0.415833\pi\)
\(434\) 0 0
\(435\) 0.525015 0.0251725
\(436\) 0 0
\(437\) 0.546394 0.0261376
\(438\) 0 0
\(439\) 10.3917 0.495970 0.247985 0.968764i \(-0.420232\pi\)
0.247985 + 0.968764i \(0.420232\pi\)
\(440\) 0 0
\(441\) −0.128601 −0.00612386
\(442\) 0 0
\(443\) 26.2254 1.24601 0.623004 0.782219i \(-0.285912\pi\)
0.623004 + 0.782219i \(0.285912\pi\)
\(444\) 0 0
\(445\) 60.3460 2.86067
\(446\) 0 0
\(447\) 16.1485 0.763797
\(448\) 0 0
\(449\) 26.6928 1.25971 0.629855 0.776713i \(-0.283114\pi\)
0.629855 + 0.776713i \(0.283114\pi\)
\(450\) 0 0
\(451\) −3.29715 −0.155257
\(452\) 0 0
\(453\) −14.2267 −0.668431
\(454\) 0 0
\(455\) 6.43203 0.301538
\(456\) 0 0
\(457\) 14.1819 0.663401 0.331700 0.943385i \(-0.392378\pi\)
0.331700 + 0.943385i \(0.392378\pi\)
\(458\) 0 0
\(459\) −20.6851 −0.965499
\(460\) 0 0
\(461\) 34.9375 1.62720 0.813600 0.581425i \(-0.197505\pi\)
0.813600 + 0.581425i \(0.197505\pi\)
\(462\) 0 0
\(463\) 36.9587 1.71762 0.858808 0.512298i \(-0.171206\pi\)
0.858808 + 0.512298i \(0.171206\pi\)
\(464\) 0 0
\(465\) 10.6508 0.493920
\(466\) 0 0
\(467\) −5.58065 −0.258242 −0.129121 0.991629i \(-0.541215\pi\)
−0.129121 + 0.991629i \(0.541215\pi\)
\(468\) 0 0
\(469\) 24.6654 1.13894
\(470\) 0 0
\(471\) 31.2042 1.43781
\(472\) 0 0
\(473\) −15.7004 −0.721906
\(474\) 0 0
\(475\) 13.5359 0.621070
\(476\) 0 0
\(477\) 2.73770 0.125351
\(478\) 0 0
\(479\) −35.0828 −1.60297 −0.801487 0.598012i \(-0.795958\pi\)
−0.801487 + 0.598012i \(0.795958\pi\)
\(480\) 0 0
\(481\) −4.67851 −0.213322
\(482\) 0 0
\(483\) 2.46837 0.112315
\(484\) 0 0
\(485\) 31.4481 1.42798
\(486\) 0 0
\(487\) −30.5808 −1.38575 −0.692874 0.721059i \(-0.743656\pi\)
−0.692874 + 0.721059i \(0.743656\pi\)
\(488\) 0 0
\(489\) −34.1791 −1.54563
\(490\) 0 0
\(491\) −12.7282 −0.574417 −0.287208 0.957868i \(-0.592727\pi\)
−0.287208 + 0.957868i \(0.592727\pi\)
\(492\) 0 0
\(493\) 0.282564 0.0127260
\(494\) 0 0
\(495\) 2.01202 0.0904338
\(496\) 0 0
\(497\) 40.0597 1.79693
\(498\) 0 0
\(499\) −13.4286 −0.601148 −0.300574 0.953758i \(-0.597178\pi\)
−0.300574 + 0.953758i \(0.597178\pi\)
\(500\) 0 0
\(501\) −33.6781 −1.50462
\(502\) 0 0
\(503\) −30.3977 −1.35537 −0.677683 0.735354i \(-0.737016\pi\)
−0.677683 + 0.735354i \(0.737016\pi\)
\(504\) 0 0
\(505\) −42.4122 −1.88732
\(506\) 0 0
\(507\) 20.9856 0.932005
\(508\) 0 0
\(509\) −21.5175 −0.953745 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(510\) 0 0
\(511\) −23.2454 −1.02832
\(512\) 0 0
\(513\) 5.40303 0.238550
\(514\) 0 0
\(515\) −55.3049 −2.43702
\(516\) 0 0
\(517\) 9.22013 0.405501
\(518\) 0 0
\(519\) 14.8155 0.650330
\(520\) 0 0
\(521\) −14.0851 −0.617081 −0.308540 0.951211i \(-0.599840\pi\)
−0.308540 + 0.951211i \(0.599840\pi\)
\(522\) 0 0
\(523\) 22.8507 0.999190 0.499595 0.866259i \(-0.333482\pi\)
0.499595 + 0.866259i \(0.333482\pi\)
\(524\) 0 0
\(525\) 61.1492 2.66877
\(526\) 0 0
\(527\) 5.73229 0.249703
\(528\) 0 0
\(529\) −22.7015 −0.987020
\(530\) 0 0
\(531\) 1.63022 0.0707455
\(532\) 0 0
\(533\) −1.04145 −0.0451103
\(534\) 0 0
\(535\) −33.6267 −1.45381
\(536\) 0 0
\(537\) 17.6295 0.760769
\(538\) 0 0
\(539\) −0.823436 −0.0354679
\(540\) 0 0
\(541\) −36.2741 −1.55955 −0.779774 0.626062i \(-0.784666\pi\)
−0.779774 + 0.626062i \(0.784666\pi\)
\(542\) 0 0
\(543\) −23.8621 −1.02402
\(544\) 0 0
\(545\) −70.9484 −3.03910
\(546\) 0 0
\(547\) 22.1698 0.947913 0.473957 0.880548i \(-0.342825\pi\)
0.473957 + 0.880548i \(0.342825\pi\)
\(548\) 0 0
\(549\) 0.354109 0.0151130
\(550\) 0 0
\(551\) −0.0738068 −0.00314427
\(552\) 0 0
\(553\) −21.0138 −0.893596
\(554\) 0 0
\(555\) −60.9085 −2.58542
\(556\) 0 0
\(557\) −15.3385 −0.649915 −0.324957 0.945729i \(-0.605350\pi\)
−0.324957 + 0.945729i \(0.605350\pi\)
\(558\) 0 0
\(559\) −4.95919 −0.209752
\(560\) 0 0
\(561\) −10.9420 −0.461970
\(562\) 0 0
\(563\) −8.47021 −0.356977 −0.178488 0.983942i \(-0.557121\pi\)
−0.178488 + 0.983942i \(0.557121\pi\)
\(564\) 0 0
\(565\) −64.0089 −2.69287
\(566\) 0 0
\(567\) 22.1925 0.931997
\(568\) 0 0
\(569\) 25.7940 1.08134 0.540670 0.841235i \(-0.318171\pi\)
0.540670 + 0.841235i \(0.318171\pi\)
\(570\) 0 0
\(571\) −12.3855 −0.518318 −0.259159 0.965835i \(-0.583445\pi\)
−0.259159 + 0.965835i \(0.583445\pi\)
\(572\) 0 0
\(573\) −36.1133 −1.50865
\(574\) 0 0
\(575\) 7.39594 0.308432
\(576\) 0 0
\(577\) 26.8555 1.11801 0.559005 0.829164i \(-0.311183\pi\)
0.559005 + 0.829164i \(0.311183\pi\)
\(578\) 0 0
\(579\) −0.993976 −0.0413082
\(580\) 0 0
\(581\) 45.9517 1.90640
\(582\) 0 0
\(583\) 17.5296 0.726001
\(584\) 0 0
\(585\) 0.635526 0.0262758
\(586\) 0 0
\(587\) −7.02574 −0.289983 −0.144992 0.989433i \(-0.546316\pi\)
−0.144992 + 0.989433i \(0.546316\pi\)
\(588\) 0 0
\(589\) −1.49730 −0.0616950
\(590\) 0 0
\(591\) 10.3833 0.427110
\(592\) 0 0
\(593\) 12.2750 0.504074 0.252037 0.967718i \(-0.418900\pi\)
0.252037 + 0.967718i \(0.418900\pi\)
\(594\) 0 0
\(595\) 45.0674 1.84758
\(596\) 0 0
\(597\) 39.6967 1.62468
\(598\) 0 0
\(599\) −1.13112 −0.0462163 −0.0231081 0.999733i \(-0.507356\pi\)
−0.0231081 + 0.999733i \(0.507356\pi\)
\(600\) 0 0
\(601\) −15.4182 −0.628921 −0.314460 0.949271i \(-0.601824\pi\)
−0.314460 + 0.949271i \(0.601824\pi\)
\(602\) 0 0
\(603\) 2.43710 0.0992464
\(604\) 0 0
\(605\) −34.4756 −1.40163
\(606\) 0 0
\(607\) −41.2922 −1.67600 −0.838000 0.545671i \(-0.816275\pi\)
−0.838000 + 0.545671i \(0.816275\pi\)
\(608\) 0 0
\(609\) −0.333426 −0.0135111
\(610\) 0 0
\(611\) 2.91230 0.117819
\(612\) 0 0
\(613\) −19.0972 −0.771329 −0.385664 0.922639i \(-0.626028\pi\)
−0.385664 + 0.922639i \(0.626028\pi\)
\(614\) 0 0
\(615\) −13.5584 −0.546728
\(616\) 0 0
\(617\) 40.7423 1.64022 0.820112 0.572202i \(-0.193911\pi\)
0.820112 + 0.572202i \(0.193911\pi\)
\(618\) 0 0
\(619\) −35.8186 −1.43967 −0.719835 0.694145i \(-0.755782\pi\)
−0.719835 + 0.694145i \(0.755782\pi\)
\(620\) 0 0
\(621\) 2.95218 0.118467
\(622\) 0 0
\(623\) −38.3245 −1.53544
\(624\) 0 0
\(625\) 90.5412 3.62165
\(626\) 0 0
\(627\) 2.85808 0.114141
\(628\) 0 0
\(629\) −32.7810 −1.30707
\(630\) 0 0
\(631\) 7.14251 0.284339 0.142169 0.989842i \(-0.454592\pi\)
0.142169 + 0.989842i \(0.454592\pi\)
\(632\) 0 0
\(633\) 10.3327 0.410688
\(634\) 0 0
\(635\) 43.5182 1.72697
\(636\) 0 0
\(637\) −0.260094 −0.0103053
\(638\) 0 0
\(639\) 3.95816 0.156582
\(640\) 0 0
\(641\) −9.03591 −0.356897 −0.178449 0.983949i \(-0.557108\pi\)
−0.178449 + 0.983949i \(0.557108\pi\)
\(642\) 0 0
\(643\) −33.5484 −1.32302 −0.661510 0.749936i \(-0.730084\pi\)
−0.661510 + 0.749936i \(0.730084\pi\)
\(644\) 0 0
\(645\) −64.5626 −2.54215
\(646\) 0 0
\(647\) 29.9641 1.17801 0.589004 0.808130i \(-0.299520\pi\)
0.589004 + 0.808130i \(0.299520\pi\)
\(648\) 0 0
\(649\) 10.4384 0.409741
\(650\) 0 0
\(651\) −6.76413 −0.265107
\(652\) 0 0
\(653\) −5.44946 −0.213254 −0.106627 0.994299i \(-0.534005\pi\)
−0.106627 + 0.994299i \(0.534005\pi\)
\(654\) 0 0
\(655\) −1.53624 −0.0600258
\(656\) 0 0
\(657\) −2.29679 −0.0896065
\(658\) 0 0
\(659\) −12.1018 −0.471420 −0.235710 0.971823i \(-0.575741\pi\)
−0.235710 + 0.971823i \(0.575741\pi\)
\(660\) 0 0
\(661\) 33.3474 1.29706 0.648531 0.761188i \(-0.275384\pi\)
0.648531 + 0.761188i \(0.275384\pi\)
\(662\) 0 0
\(663\) −3.45617 −0.134226
\(664\) 0 0
\(665\) −11.7718 −0.456490
\(666\) 0 0
\(667\) −0.0403276 −0.00156149
\(668\) 0 0
\(669\) −20.3965 −0.788574
\(670\) 0 0
\(671\) 2.26737 0.0875309
\(672\) 0 0
\(673\) 37.9052 1.46114 0.730570 0.682838i \(-0.239255\pi\)
0.730570 + 0.682838i \(0.239255\pi\)
\(674\) 0 0
\(675\) 73.1349 2.81496
\(676\) 0 0
\(677\) 47.5427 1.82721 0.913607 0.406598i \(-0.133285\pi\)
0.913607 + 0.406598i \(0.133285\pi\)
\(678\) 0 0
\(679\) −19.9720 −0.766457
\(680\) 0 0
\(681\) 3.81475 0.146182
\(682\) 0 0
\(683\) −29.7347 −1.13777 −0.568883 0.822418i \(-0.692624\pi\)
−0.568883 + 0.822418i \(0.692624\pi\)
\(684\) 0 0
\(685\) −37.4113 −1.42941
\(686\) 0 0
\(687\) 3.50231 0.133622
\(688\) 0 0
\(689\) 5.53696 0.210941
\(690\) 0 0
\(691\) 2.82857 0.107604 0.0538019 0.998552i \(-0.482866\pi\)
0.0538019 + 0.998552i \(0.482866\pi\)
\(692\) 0 0
\(693\) −1.27780 −0.0485395
\(694\) 0 0
\(695\) −62.2867 −2.36267
\(696\) 0 0
\(697\) −7.29715 −0.276399
\(698\) 0 0
\(699\) 41.6581 1.57565
\(700\) 0 0
\(701\) −38.0093 −1.43559 −0.717796 0.696253i \(-0.754849\pi\)
−0.717796 + 0.696253i \(0.754849\pi\)
\(702\) 0 0
\(703\) 8.56253 0.322942
\(704\) 0 0
\(705\) 37.9146 1.42795
\(706\) 0 0
\(707\) 26.9351 1.01300
\(708\) 0 0
\(709\) −0.225499 −0.00846877 −0.00423439 0.999991i \(-0.501348\pi\)
−0.00423439 + 0.999991i \(0.501348\pi\)
\(710\) 0 0
\(711\) −2.07629 −0.0778671
\(712\) 0 0
\(713\) −0.818114 −0.0306386
\(714\) 0 0
\(715\) 4.06929 0.152183
\(716\) 0 0
\(717\) −37.6587 −1.40639
\(718\) 0 0
\(719\) 26.7607 0.998006 0.499003 0.866600i \(-0.333700\pi\)
0.499003 + 0.866600i \(0.333700\pi\)
\(720\) 0 0
\(721\) 35.1230 1.30805
\(722\) 0 0
\(723\) −4.92468 −0.183151
\(724\) 0 0
\(725\) −0.999041 −0.0371035
\(726\) 0 0
\(727\) 10.0434 0.372487 0.186244 0.982504i \(-0.440369\pi\)
0.186244 + 0.982504i \(0.440369\pi\)
\(728\) 0 0
\(729\) 28.9738 1.07310
\(730\) 0 0
\(731\) −34.7477 −1.28519
\(732\) 0 0
\(733\) 1.32974 0.0491152 0.0245576 0.999698i \(-0.492182\pi\)
0.0245576 + 0.999698i \(0.492182\pi\)
\(734\) 0 0
\(735\) −3.38610 −0.124898
\(736\) 0 0
\(737\) 15.6048 0.574812
\(738\) 0 0
\(739\) 13.0730 0.480898 0.240449 0.970662i \(-0.422705\pi\)
0.240449 + 0.970662i \(0.422705\pi\)
\(740\) 0 0
\(741\) 0.902764 0.0331639
\(742\) 0 0
\(743\) 11.0577 0.405666 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(744\) 0 0
\(745\) −42.0795 −1.54167
\(746\) 0 0
\(747\) 4.54032 0.166122
\(748\) 0 0
\(749\) 21.3557 0.780319
\(750\) 0 0
\(751\) 43.0977 1.57266 0.786330 0.617807i \(-0.211979\pi\)
0.786330 + 0.617807i \(0.211979\pi\)
\(752\) 0 0
\(753\) 14.6162 0.532643
\(754\) 0 0
\(755\) 37.0718 1.34918
\(756\) 0 0
\(757\) −7.53688 −0.273933 −0.136966 0.990576i \(-0.543735\pi\)
−0.136966 + 0.990576i \(0.543735\pi\)
\(758\) 0 0
\(759\) 1.56164 0.0566839
\(760\) 0 0
\(761\) −32.3702 −1.17342 −0.586710 0.809797i \(-0.699577\pi\)
−0.586710 + 0.809797i \(0.699577\pi\)
\(762\) 0 0
\(763\) 45.0579 1.63121
\(764\) 0 0
\(765\) 4.45295 0.160997
\(766\) 0 0
\(767\) 3.29710 0.119051
\(768\) 0 0
\(769\) 11.1972 0.403780 0.201890 0.979408i \(-0.435292\pi\)
0.201890 + 0.979408i \(0.435292\pi\)
\(770\) 0 0
\(771\) 9.98263 0.359516
\(772\) 0 0
\(773\) 47.2345 1.69891 0.849453 0.527664i \(-0.176932\pi\)
0.849453 + 0.527664i \(0.176932\pi\)
\(774\) 0 0
\(775\) −20.2673 −0.728022
\(776\) 0 0
\(777\) 38.6817 1.38770
\(778\) 0 0
\(779\) 1.90604 0.0682911
\(780\) 0 0
\(781\) 25.3442 0.906888
\(782\) 0 0
\(783\) −0.398780 −0.0142512
\(784\) 0 0
\(785\) −81.3114 −2.90213
\(786\) 0 0
\(787\) 35.6791 1.27182 0.635911 0.771762i \(-0.280624\pi\)
0.635911 + 0.771762i \(0.280624\pi\)
\(788\) 0 0
\(789\) −21.6965 −0.772415
\(790\) 0 0
\(791\) 40.6508 1.44537
\(792\) 0 0
\(793\) 0.716181 0.0254323
\(794\) 0 0
\(795\) 72.0843 2.55657
\(796\) 0 0
\(797\) −12.4881 −0.442351 −0.221175 0.975234i \(-0.570989\pi\)
−0.221175 + 0.975234i \(0.570989\pi\)
\(798\) 0 0
\(799\) 20.4057 0.721901
\(800\) 0 0
\(801\) −3.78671 −0.133797
\(802\) 0 0
\(803\) −14.7064 −0.518979
\(804\) 0 0
\(805\) −6.43203 −0.226699
\(806\) 0 0
\(807\) 22.2486 0.783188
\(808\) 0 0
\(809\) 25.1660 0.884789 0.442394 0.896821i \(-0.354129\pi\)
0.442394 + 0.896821i \(0.354129\pi\)
\(810\) 0 0
\(811\) 2.99454 0.105152 0.0525762 0.998617i \(-0.483257\pi\)
0.0525762 + 0.998617i \(0.483257\pi\)
\(812\) 0 0
\(813\) 4.99015 0.175012
\(814\) 0 0
\(815\) 89.0633 3.11975
\(816\) 0 0
\(817\) 9.07622 0.317537
\(818\) 0 0
\(819\) −0.403609 −0.0141033
\(820\) 0 0
\(821\) −3.63700 −0.126932 −0.0634660 0.997984i \(-0.520215\pi\)
−0.0634660 + 0.997984i \(0.520215\pi\)
\(822\) 0 0
\(823\) 16.7620 0.584287 0.292144 0.956374i \(-0.405631\pi\)
0.292144 + 0.956374i \(0.405631\pi\)
\(824\) 0 0
\(825\) 38.6867 1.34690
\(826\) 0 0
\(827\) −1.31834 −0.0458432 −0.0229216 0.999737i \(-0.507297\pi\)
−0.0229216 + 0.999737i \(0.507297\pi\)
\(828\) 0 0
\(829\) −18.2603 −0.634205 −0.317102 0.948391i \(-0.602710\pi\)
−0.317102 + 0.948391i \(0.602710\pi\)
\(830\) 0 0
\(831\) 13.4941 0.468104
\(832\) 0 0
\(833\) −1.82240 −0.0631425
\(834\) 0 0
\(835\) 87.7577 3.03698
\(836\) 0 0
\(837\) −8.08994 −0.279629
\(838\) 0 0
\(839\) −24.0265 −0.829486 −0.414743 0.909939i \(-0.636129\pi\)
−0.414743 + 0.909939i \(0.636129\pi\)
\(840\) 0 0
\(841\) −28.9946 −0.999812
\(842\) 0 0
\(843\) 33.8633 1.16631
\(844\) 0 0
\(845\) −54.6840 −1.88119
\(846\) 0 0
\(847\) 21.8948 0.752313
\(848\) 0 0
\(849\) −32.9932 −1.13232
\(850\) 0 0
\(851\) 4.67851 0.160377
\(852\) 0 0
\(853\) 11.1739 0.382586 0.191293 0.981533i \(-0.438732\pi\)
0.191293 + 0.981533i \(0.438732\pi\)
\(854\) 0 0
\(855\) −1.16313 −0.0397781
\(856\) 0 0
\(857\) 28.6182 0.977578 0.488789 0.872402i \(-0.337439\pi\)
0.488789 + 0.872402i \(0.337439\pi\)
\(858\) 0 0
\(859\) −16.1420 −0.550758 −0.275379 0.961336i \(-0.588803\pi\)
−0.275379 + 0.961336i \(0.588803\pi\)
\(860\) 0 0
\(861\) 8.61067 0.293451
\(862\) 0 0
\(863\) 3.17360 0.108031 0.0540154 0.998540i \(-0.482798\pi\)
0.0540154 + 0.998540i \(0.482798\pi\)
\(864\) 0 0
\(865\) −38.6061 −1.31265
\(866\) 0 0
\(867\) 3.87140 0.131480
\(868\) 0 0
\(869\) −13.2946 −0.450988
\(870\) 0 0
\(871\) 4.92900 0.167013
\(872\) 0 0
\(873\) −1.97337 −0.0667883
\(874\) 0 0
\(875\) −100.483 −3.39694
\(876\) 0 0
\(877\) 30.9086 1.04371 0.521855 0.853034i \(-0.325240\pi\)
0.521855 + 0.853034i \(0.325240\pi\)
\(878\) 0 0
\(879\) −17.9153 −0.604270
\(880\) 0 0
\(881\) −13.2076 −0.444976 −0.222488 0.974935i \(-0.571418\pi\)
−0.222488 + 0.974935i \(0.571418\pi\)
\(882\) 0 0
\(883\) 2.47616 0.0833294 0.0416647 0.999132i \(-0.486734\pi\)
0.0416647 + 0.999132i \(0.486734\pi\)
\(884\) 0 0
\(885\) 42.9241 1.44288
\(886\) 0 0
\(887\) −27.5413 −0.924745 −0.462372 0.886686i \(-0.653002\pi\)
−0.462372 + 0.886686i \(0.653002\pi\)
\(888\) 0 0
\(889\) −27.6375 −0.926932
\(890\) 0 0
\(891\) 14.0403 0.470368
\(892\) 0 0
\(893\) −5.33004 −0.178363
\(894\) 0 0
\(895\) −45.9387 −1.53556
\(896\) 0 0
\(897\) 0.493265 0.0164696
\(898\) 0 0
\(899\) 0.110511 0.00368574
\(900\) 0 0
\(901\) 38.7959 1.29248
\(902\) 0 0
\(903\) 41.0024 1.36447
\(904\) 0 0
\(905\) 62.1795 2.06692
\(906\) 0 0
\(907\) −10.7841 −0.358079 −0.179039 0.983842i \(-0.557299\pi\)
−0.179039 + 0.983842i \(0.557299\pi\)
\(908\) 0 0
\(909\) 2.66136 0.0882718
\(910\) 0 0
\(911\) 14.3560 0.475634 0.237817 0.971310i \(-0.423568\pi\)
0.237817 + 0.971310i \(0.423568\pi\)
\(912\) 0 0
\(913\) 29.0718 0.962137
\(914\) 0 0
\(915\) 9.32379 0.308235
\(916\) 0 0
\(917\) 0.975635 0.0322183
\(918\) 0 0
\(919\) 32.1519 1.06059 0.530297 0.847812i \(-0.322081\pi\)
0.530297 + 0.847812i \(0.322081\pi\)
\(920\) 0 0
\(921\) −8.86537 −0.292124
\(922\) 0 0
\(923\) 8.00532 0.263498
\(924\) 0 0
\(925\) 115.902 3.81082
\(926\) 0 0
\(927\) 3.47038 0.113982
\(928\) 0 0
\(929\) 14.8371 0.486790 0.243395 0.969927i \(-0.421739\pi\)
0.243395 + 0.969927i \(0.421739\pi\)
\(930\) 0 0
\(931\) 0.476019 0.0156009
\(932\) 0 0
\(933\) 17.0627 0.558608
\(934\) 0 0
\(935\) 28.5124 0.932454
\(936\) 0 0
\(937\) −18.1301 −0.592284 −0.296142 0.955144i \(-0.595700\pi\)
−0.296142 + 0.955144i \(0.595700\pi\)
\(938\) 0 0
\(939\) 48.2482 1.57452
\(940\) 0 0
\(941\) −30.2219 −0.985206 −0.492603 0.870254i \(-0.663954\pi\)
−0.492603 + 0.870254i \(0.663954\pi\)
\(942\) 0 0
\(943\) 1.04145 0.0339143
\(944\) 0 0
\(945\) −63.6033 −2.06902
\(946\) 0 0
\(947\) −37.5706 −1.22088 −0.610441 0.792062i \(-0.709008\pi\)
−0.610441 + 0.792062i \(0.709008\pi\)
\(948\) 0 0
\(949\) −4.64523 −0.150791
\(950\) 0 0
\(951\) −19.0196 −0.616752
\(952\) 0 0
\(953\) 46.0319 1.49112 0.745559 0.666440i \(-0.232183\pi\)
0.745559 + 0.666440i \(0.232183\pi\)
\(954\) 0 0
\(955\) 94.1033 3.04511
\(956\) 0 0
\(957\) −0.210946 −0.00681891
\(958\) 0 0
\(959\) 23.7592 0.767224
\(960\) 0 0
\(961\) −28.7581 −0.927681
\(962\) 0 0
\(963\) 2.11008 0.0679962
\(964\) 0 0
\(965\) 2.59008 0.0833778
\(966\) 0 0
\(967\) 55.6384 1.78921 0.894605 0.446858i \(-0.147457\pi\)
0.894605 + 0.446858i \(0.147457\pi\)
\(968\) 0 0
\(969\) 6.32541 0.203202
\(970\) 0 0
\(971\) 3.24836 0.104245 0.0521224 0.998641i \(-0.483401\pi\)
0.0521224 + 0.998641i \(0.483401\pi\)
\(972\) 0 0
\(973\) 39.5570 1.26814
\(974\) 0 0
\(975\) 12.2197 0.391345
\(976\) 0 0
\(977\) 37.9387 1.21377 0.606884 0.794791i \(-0.292419\pi\)
0.606884 + 0.794791i \(0.292419\pi\)
\(978\) 0 0
\(979\) −24.2464 −0.774918
\(980\) 0 0
\(981\) 4.45201 0.142142
\(982\) 0 0
\(983\) 40.6822 1.29756 0.648780 0.760976i \(-0.275280\pi\)
0.648780 + 0.760976i \(0.275280\pi\)
\(984\) 0 0
\(985\) −27.0565 −0.862092
\(986\) 0 0
\(987\) −24.0788 −0.766436
\(988\) 0 0
\(989\) 4.95919 0.157693
\(990\) 0 0
\(991\) −38.5615 −1.22494 −0.612472 0.790492i \(-0.709825\pi\)
−0.612472 + 0.790492i \(0.709825\pi\)
\(992\) 0 0
\(993\) 44.7140 1.41895
\(994\) 0 0
\(995\) −103.441 −3.27930
\(996\) 0 0
\(997\) −53.2516 −1.68649 −0.843247 0.537526i \(-0.819359\pi\)
−0.843247 + 0.537526i \(0.819359\pi\)
\(998\) 0 0
\(999\) 46.2636 1.46372
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.bm.1.4 8
4.3 odd 2 4864.2.a.br.1.6 8
8.3 odd 2 inner 4864.2.a.bm.1.3 8
8.5 even 2 4864.2.a.br.1.5 8
16.3 odd 4 2432.2.c.i.1217.11 yes 16
16.5 even 4 2432.2.c.i.1217.12 yes 16
16.11 odd 4 2432.2.c.i.1217.6 yes 16
16.13 even 4 2432.2.c.i.1217.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.i.1217.5 16 16.13 even 4
2432.2.c.i.1217.6 yes 16 16.11 odd 4
2432.2.c.i.1217.11 yes 16 16.3 odd 4
2432.2.c.i.1217.12 yes 16 16.5 even 4
4864.2.a.bm.1.3 8 8.3 odd 2 inner
4864.2.a.bm.1.4 8 1.1 even 1 trivial
4864.2.a.br.1.5 8 8.5 even 2
4864.2.a.br.1.6 8 4.3 odd 2