Properties

Label 4864.2.a.w.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.46410 q^{5} -1.73205 q^{7} -2.00000 q^{9} -5.19615 q^{13} +3.46410 q^{15} -3.00000 q^{17} -1.00000 q^{19} -1.73205 q^{21} +1.73205 q^{23} +7.00000 q^{25} -5.00000 q^{27} -1.73205 q^{29} -3.46410 q^{31} -6.00000 q^{35} -5.19615 q^{39} +10.0000 q^{43} -6.92820 q^{45} +10.3923 q^{47} -4.00000 q^{49} -3.00000 q^{51} -5.19615 q^{53} -1.00000 q^{57} -9.00000 q^{59} -10.3923 q^{61} +3.46410 q^{63} -18.0000 q^{65} -13.0000 q^{67} +1.73205 q^{69} +3.46410 q^{71} +1.00000 q^{73} +7.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -10.3923 q^{85} -1.73205 q^{87} -18.0000 q^{89} +9.00000 q^{91} -3.46410 q^{93} -3.46410 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} - 6q^{17} - 2q^{19} + 14q^{25} - 10q^{27} - 12q^{35} + 20q^{43} - 8q^{49} - 6q^{51} - 2q^{57} - 18q^{59} - 36q^{65} - 26q^{67} + 2q^{73} + 14q^{75} + 2q^{81} - 36q^{89} + 18q^{91} - 28q^{97} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −5.19615 −1.44115 −0.720577 0.693375i \(-0.756123\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) 1.73205 0.361158 0.180579 0.983561i \(-0.442203\pi\)
0.180579 + 0.983561i \(0.442203\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −5.19615 −0.832050
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) −6.92820 −1.03280
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −5.19615 −0.713746 −0.356873 0.934153i \(-0.616157\pi\)
−0.356873 + 0.934153i \(0.616157\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −10.3923 −1.33060 −0.665299 0.746577i \(-0.731696\pi\)
−0.665299 + 0.746577i \(0.731696\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) −18.0000 −2.23263
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 1.73205 0.208514
\(70\) 0 0
\(71\) 3.46410 0.411113 0.205557 0.978645i \(-0.434100\pi\)
0.205557 + 0.978645i \(0.434100\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −10.3923 −1.12720
\(86\) 0 0
\(87\) −1.73205 −0.185695
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) 0 0
\(95\) −3.46410 −0.355409
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −13.8564 −1.36531 −0.682656 0.730740i \(-0.739175\pi\)
−0.682656 + 0.730740i \(0.739175\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −5.19615 −0.497701 −0.248851 0.968542i \(-0.580053\pi\)
−0.248851 + 0.968542i \(0.580053\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 0 0
\(117\) 10.3923 0.960769
\(118\) 0 0
\(119\) 5.19615 0.476331
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 1.73205 0.150188
\(134\) 0 0
\(135\) −17.3205 −1.49071
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 10.3923 0.875190
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) −4.00000 −0.329914
\(148\) 0 0
\(149\) 20.7846 1.70274 0.851371 0.524564i \(-0.175772\pi\)
0.851371 + 0.524564i \(0.175772\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 0 0
\(157\) 3.46410 0.276465 0.138233 0.990400i \(-0.455858\pi\)
0.138233 + 0.990400i \(0.455858\pi\)
\(158\) 0 0
\(159\) −5.19615 −0.412082
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) 14.0000 1.07692
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 20.7846 1.58022 0.790112 0.612962i \(-0.210022\pi\)
0.790112 + 0.612962i \(0.210022\pi\)
\(174\) 0 0
\(175\) −12.1244 −0.916515
\(176\) 0 0
\(177\) −9.00000 −0.676481
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −13.8564 −1.02994 −0.514969 0.857209i \(-0.672197\pi\)
−0.514969 + 0.857209i \(0.672197\pi\)
\(182\) 0 0
\(183\) −10.3923 −0.768221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.66025 0.629941
\(190\) 0 0
\(191\) −1.73205 −0.125327 −0.0626634 0.998035i \(-0.519959\pi\)
−0.0626634 + 0.998035i \(0.519959\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) −18.0000 −1.28901
\(196\) 0 0
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) 22.5167 1.59616 0.798082 0.602549i \(-0.205848\pi\)
0.798082 + 0.602549i \(0.205848\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) 0 0
\(213\) 3.46410 0.237356
\(214\) 0 0
\(215\) 34.6410 2.36250
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 15.5885 1.04859
\(222\) 0 0
\(223\) −6.92820 −0.463947 −0.231973 0.972722i \(-0.574518\pi\)
−0.231973 + 0.972722i \(0.574518\pi\)
\(224\) 0 0
\(225\) −14.0000 −0.933333
\(226\) 0 0
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 0 0
\(229\) 3.46410 0.228914 0.114457 0.993428i \(-0.463487\pi\)
0.114457 + 0.993428i \(0.463487\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 36.0000 2.34838
\(236\) 0 0
\(237\) 10.3923 0.675053
\(238\) 0 0
\(239\) −19.0526 −1.23241 −0.616204 0.787587i \(-0.711330\pi\)
−0.616204 + 0.787587i \(0.711330\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −13.8564 −0.885253
\(246\) 0 0
\(247\) 5.19615 0.330623
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −10.3923 −0.650791
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) 0 0
\(263\) 24.2487 1.49524 0.747620 0.664127i \(-0.231197\pi\)
0.747620 + 0.664127i \(0.231197\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 27.7128 1.68968 0.844840 0.535019i \(-0.179696\pi\)
0.844840 + 0.535019i \(0.179696\pi\)
\(270\) 0 0
\(271\) 22.5167 1.36779 0.683895 0.729581i \(-0.260285\pi\)
0.683895 + 0.729581i \(0.260285\pi\)
\(272\) 0 0
\(273\) 9.00000 0.544705
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3923 0.624413 0.312207 0.950014i \(-0.398932\pi\)
0.312207 + 0.950014i \(0.398932\pi\)
\(278\) 0 0
\(279\) 6.92820 0.414781
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) −3.46410 −0.205196
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 1.73205 0.101187 0.0505937 0.998719i \(-0.483889\pi\)
0.0505937 + 0.998719i \(0.483889\pi\)
\(294\) 0 0
\(295\) −31.1769 −1.81519
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.00000 −0.520483
\(300\) 0 0
\(301\) −17.3205 −0.998337
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −36.0000 −2.06135
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −13.8564 −0.788263
\(310\) 0 0
\(311\) 1.73205 0.0982156 0.0491078 0.998793i \(-0.484362\pi\)
0.0491078 + 0.998793i \(0.484362\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 12.0000 0.676123
\(316\) 0 0
\(317\) 8.66025 0.486408 0.243204 0.969975i \(-0.421801\pi\)
0.243204 + 0.969975i \(0.421801\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −36.3731 −2.01761
\(326\) 0 0
\(327\) −5.19615 −0.287348
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −45.0333 −2.46043
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 19.0526 1.02874
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) −10.3923 −0.556287 −0.278144 0.960539i \(-0.589719\pi\)
−0.278144 + 0.960539i \(0.589719\pi\)
\(350\) 0 0
\(351\) 25.9808 1.38675
\(352\) 0 0
\(353\) 27.0000 1.43706 0.718532 0.695493i \(-0.244814\pi\)
0.718532 + 0.695493i \(0.244814\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 5.19615 0.275010
\(358\) 0 0
\(359\) 8.66025 0.457071 0.228535 0.973536i \(-0.426606\pi\)
0.228535 + 0.973536i \(0.426606\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 3.46410 0.181319
\(366\) 0 0
\(367\) 31.1769 1.62742 0.813711 0.581270i \(-0.197444\pi\)
0.813711 + 0.581270i \(0.197444\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) −12.1244 −0.627775 −0.313888 0.949460i \(-0.601632\pi\)
−0.313888 + 0.949460i \(0.601632\pi\)
\(374\) 0 0
\(375\) 6.92820 0.357771
\(376\) 0 0
\(377\) 9.00000 0.463524
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) −10.3923 −0.532414
\(382\) 0 0
\(383\) 34.6410 1.77007 0.885037 0.465521i \(-0.154133\pi\)
0.885037 + 0.465521i \(0.154133\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.0000 −1.01666
\(388\) 0 0
\(389\) −27.7128 −1.40510 −0.702548 0.711637i \(-0.747954\pi\)
−0.702548 + 0.711637i \(0.747954\pi\)
\(390\) 0 0
\(391\) −5.19615 −0.262781
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) 36.0000 1.81136
\(396\) 0 0
\(397\) −17.3205 −0.869291 −0.434646 0.900602i \(-0.643126\pi\)
−0.434646 + 0.900602i \(0.643126\pi\)
\(398\) 0 0
\(399\) 1.73205 0.0867110
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 0 0
\(413\) 15.5885 0.767058
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 8.66025 0.422075 0.211037 0.977478i \(-0.432316\pi\)
0.211037 + 0.977478i \(0.432316\pi\)
\(422\) 0 0
\(423\) −20.7846 −1.01058
\(424\) 0 0
\(425\) −21.0000 −1.01865
\(426\) 0 0
\(427\) 18.0000 0.871081
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.2487 −1.16802 −0.584010 0.811747i \(-0.698517\pi\)
−0.584010 + 0.811747i \(0.698517\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 0 0
\(437\) −1.73205 −0.0828552
\(438\) 0 0
\(439\) −34.6410 −1.65333 −0.826663 0.562698i \(-0.809764\pi\)
−0.826663 + 0.562698i \(0.809764\pi\)
\(440\) 0 0
\(441\) 8.00000 0.380952
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) −62.3538 −2.95585
\(446\) 0 0
\(447\) 20.7846 0.983078
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.8564 0.651031
\(454\) 0 0
\(455\) 31.1769 1.46160
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −13.8564 −0.645357 −0.322679 0.946509i \(-0.604583\pi\)
−0.322679 + 0.946509i \(0.604583\pi\)
\(462\) 0 0
\(463\) 10.3923 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 22.5167 1.03972
\(470\) 0 0
\(471\) 3.46410 0.159617
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) 10.3923 0.475831
\(478\) 0 0
\(479\) −31.1769 −1.42451 −0.712255 0.701921i \(-0.752326\pi\)
−0.712255 + 0.701921i \(0.752326\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) 0 0
\(485\) −48.4974 −2.20215
\(486\) 0 0
\(487\) −34.6410 −1.56973 −0.784867 0.619664i \(-0.787269\pi\)
−0.784867 + 0.619664i \(0.787269\pi\)
\(488\) 0 0
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 5.19615 0.234023
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) −13.8564 −0.619059
\(502\) 0 0
\(503\) 5.19615 0.231685 0.115842 0.993268i \(-0.463043\pi\)
0.115842 + 0.993268i \(0.463043\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.0000 0.621762
\(508\) 0 0
\(509\) 27.7128 1.22835 0.614174 0.789170i \(-0.289489\pi\)
0.614174 + 0.789170i \(0.289489\pi\)
\(510\) 0 0
\(511\) −1.73205 −0.0766214
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) −48.0000 −2.11513
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 20.7846 0.912343
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 7.00000 0.306089 0.153044 0.988219i \(-0.451092\pi\)
0.153044 + 0.988219i \(0.451092\pi\)
\(524\) 0 0
\(525\) −12.1244 −0.529150
\(526\) 0 0
\(527\) 10.3923 0.452696
\(528\) 0 0
\(529\) −20.0000 −0.869565
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −10.3923 −0.449299
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) −13.8564 −0.594635
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 20.7846 0.887066
\(550\) 0 0
\(551\) 1.73205 0.0737878
\(552\) 0 0
\(553\) −18.0000 −0.765438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.2487 −1.02745 −0.513725 0.857955i \(-0.671735\pi\)
−0.513725 + 0.857955i \(0.671735\pi\)
\(558\) 0 0
\(559\) −51.9615 −2.19774
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 20.7846 0.874415
\(566\) 0 0
\(567\) −1.73205 −0.0727393
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) −1.73205 −0.0723575
\(574\) 0 0
\(575\) 12.1244 0.505621
\(576\) 0 0
\(577\) −35.0000 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 36.0000 1.48842
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 3.46410 0.142736
\(590\) 0 0
\(591\) 13.8564 0.569976
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) 22.5167 0.921546
\(598\) 0 0
\(599\) −27.7128 −1.13231 −0.566157 0.824297i \(-0.691571\pi\)
−0.566157 + 0.824297i \(0.691571\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 26.0000 1.05880
\(604\) 0 0
\(605\) −38.1051 −1.54919
\(606\) 0 0
\(607\) −3.46410 −0.140604 −0.0703018 0.997526i \(-0.522396\pi\)
−0.0703018 + 0.997526i \(0.522396\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) −54.0000 −2.18461
\(612\) 0 0
\(613\) 6.92820 0.279827 0.139914 0.990164i \(-0.455317\pi\)
0.139914 + 0.990164i \(0.455317\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −8.66025 −0.347524
\(622\) 0 0
\(623\) 31.1769 1.24908
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −31.1769 −1.24113 −0.620567 0.784154i \(-0.713097\pi\)
−0.620567 + 0.784154i \(0.713097\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) −36.0000 −1.42862
\(636\) 0 0
\(637\) 20.7846 0.823516
\(638\) 0 0
\(639\) −6.92820 −0.274075
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) 22.0000 0.867595 0.433798 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) 0 0
\(645\) 34.6410 1.36399
\(646\) 0 0
\(647\) 22.5167 0.885221 0.442611 0.896714i \(-0.354052\pi\)
0.442611 + 0.896714i \(0.354052\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) 3.46410 0.135561 0.0677804 0.997700i \(-0.478408\pi\)
0.0677804 + 0.997700i \(0.478408\pi\)
\(654\) 0 0
\(655\) 62.3538 2.43637
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 51.0000 1.98668 0.993339 0.115229i \(-0.0367601\pi\)
0.993339 + 0.115229i \(0.0367601\pi\)
\(660\) 0 0
\(661\) −19.0526 −0.741059 −0.370529 0.928821i \(-0.620824\pi\)
−0.370529 + 0.928821i \(0.620824\pi\)
\(662\) 0 0
\(663\) 15.5885 0.605406
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) −6.92820 −0.267860
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) −35.0000 −1.34715
\(676\) 0 0
\(677\) 36.3731 1.39793 0.698965 0.715156i \(-0.253644\pi\)
0.698965 + 0.715156i \(0.253644\pi\)
\(678\) 0 0
\(679\) 24.2487 0.930580
\(680\) 0 0
\(681\) 15.0000 0.574801
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −72.7461 −2.77949
\(686\) 0 0
\(687\) 3.46410 0.132164
\(688\) 0 0
\(689\) 27.0000 1.02862
\(690\) 0 0
\(691\) 34.0000 1.29342 0.646710 0.762736i \(-0.276144\pi\)
0.646710 + 0.762736i \(0.276144\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −76.2102 −2.89082
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24.2487 −0.910679 −0.455340 0.890318i \(-0.650482\pi\)
−0.455340 + 0.890318i \(0.650482\pi\)
\(710\) 0 0
\(711\) −20.7846 −0.779484
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −19.0526 −0.711531
\(718\) 0 0
\(719\) −8.66025 −0.322973 −0.161486 0.986875i \(-0.551629\pi\)
−0.161486 + 0.986875i \(0.551629\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 8.00000 0.297523
\(724\) 0 0
\(725\) −12.1244 −0.450287
\(726\) 0 0
\(727\) −19.0526 −0.706620 −0.353310 0.935506i \(-0.614944\pi\)
−0.353310 + 0.935506i \(0.614944\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) 0 0
\(733\) 31.1769 1.15155 0.575773 0.817610i \(-0.304701\pi\)
0.575773 + 0.817610i \(0.304701\pi\)
\(734\) 0 0
\(735\) −13.8564 −0.511101
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 5.19615 0.190885
\(742\) 0 0
\(743\) 17.3205 0.635428 0.317714 0.948187i \(-0.397085\pi\)
0.317714 + 0.948187i \(0.397085\pi\)
\(744\) 0 0
\(745\) 72.0000 2.63788
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.19615 0.189863
\(750\) 0 0
\(751\) 20.7846 0.758441 0.379221 0.925306i \(-0.376192\pi\)
0.379221 + 0.925306i \(0.376192\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 48.0000 1.74690
\(756\) 0 0
\(757\) 51.9615 1.88857 0.944287 0.329124i \(-0.106753\pi\)
0.944287 + 0.329124i \(0.106753\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 0 0
\(763\) 9.00000 0.325822
\(764\) 0 0
\(765\) 20.7846 0.751469
\(766\) 0 0
\(767\) 46.7654 1.68860
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −32.9090 −1.18365 −0.591827 0.806065i \(-0.701593\pi\)
−0.591827 + 0.806065i \(0.701593\pi\)
\(774\) 0 0
\(775\) −24.2487 −0.871039
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.66025 0.309492
\(784\) 0 0
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) −53.0000 −1.88925 −0.944623 0.328158i \(-0.893572\pi\)
−0.944623 + 0.328158i \(0.893572\pi\)
\(788\) 0 0
\(789\) 24.2487 0.863277
\(790\) 0 0
\(791\) −10.3923 −0.369508
\(792\) 0 0
\(793\) 54.0000 1.91760
\(794\) 0 0
\(795\) −18.0000 −0.638394
\(796\) 0 0
\(797\) −29.4449 −1.04299 −0.521495 0.853254i \(-0.674626\pi\)
−0.521495 + 0.853254i \(0.674626\pi\)
\(798\) 0 0
\(799\) −31.1769 −1.10296
\(800\) 0 0
\(801\) 36.0000 1.27200
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) 27.7128 0.975537
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) 31.0000 1.08856 0.544279 0.838905i \(-0.316803\pi\)
0.544279 + 0.838905i \(0.316803\pi\)
\(812\) 0 0
\(813\) 22.5167 0.789694
\(814\) 0 0
\(815\) −48.4974 −1.69879
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) 0 0
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) 10.3923 0.362694 0.181347 0.983419i \(-0.441954\pi\)
0.181347 + 0.983419i \(0.441954\pi\)
\(822\) 0 0
\(823\) 8.66025 0.301877 0.150939 0.988543i \(-0.451770\pi\)
0.150939 + 0.988543i \(0.451770\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) 0 0
\(829\) 22.5167 0.782036 0.391018 0.920383i \(-0.372123\pi\)
0.391018 + 0.920383i \(0.372123\pi\)
\(830\) 0 0
\(831\) 10.3923 0.360505
\(832\) 0 0
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 17.3205 0.598684
\(838\) 0 0
\(839\) 38.1051 1.31553 0.657767 0.753221i \(-0.271501\pi\)
0.657767 + 0.753221i \(0.271501\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) 0 0
\(843\) −24.0000 −0.826604
\(844\) 0 0
\(845\) 48.4974 1.66836
\(846\) 0 0
\(847\) 19.0526 0.654654
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −13.8564 −0.474434 −0.237217 0.971457i \(-0.576235\pi\)
−0.237217 + 0.971457i \(0.576235\pi\)
\(854\) 0 0
\(855\) 6.92820 0.236940
\(856\) 0 0
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45.0333 −1.53295 −0.766476 0.642273i \(-0.777992\pi\)
−0.766476 + 0.642273i \(0.777992\pi\)
\(864\) 0 0
\(865\) 72.0000 2.44807
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 67.5500 2.28884
\(872\) 0 0
\(873\) 28.0000 0.947656
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 5.19615 0.175462 0.0877308 0.996144i \(-0.472038\pi\)
0.0877308 + 0.996144i \(0.472038\pi\)
\(878\) 0 0
\(879\) 1.73205 0.0584206
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) −31.1769 −1.04800
\(886\) 0 0
\(887\) −51.9615 −1.74470 −0.872349 0.488884i \(-0.837404\pi\)
−0.872349 + 0.488884i \(0.837404\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.3923 −0.347765
\(894\) 0 0
\(895\) 41.5692 1.38951
\(896\) 0 0
\(897\) −9.00000 −0.300501
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 15.5885 0.519327
\(902\) 0 0
\(903\) −17.3205 −0.576390
\(904\) 0 0
\(905\) −48.0000 −1.59557
\(906\) 0 0
\(907\) 19.0000 0.630885 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1769 −1.03294 −0.516469 0.856306i \(-0.672754\pi\)
−0.516469 + 0.856306i \(0.672754\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −36.0000 −1.19012
\(916\) 0 0
\(917\) −31.1769 −1.02955
\(918\) 0 0
\(919\) −32.9090 −1.08557 −0.542783 0.839873i \(-0.682630\pi\)
−0.542783 + 0.839873i \(0.682630\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 27.7128 0.910208
\(928\) 0 0
\(929\) 21.0000 0.688988 0.344494 0.938789i \(-0.388051\pi\)
0.344494 + 0.938789i \(0.388051\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 1.73205 0.0567048
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 59.0000 1.92745 0.963723 0.266904i \(-0.0860008\pi\)
0.963723 + 0.266904i \(0.0860008\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 43.3013 1.41158 0.705791 0.708421i \(-0.250592\pi\)
0.705791 + 0.708421i \(0.250592\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) −5.19615 −0.168674
\(950\) 0 0
\(951\) 8.66025 0.280828
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.3731 1.17455
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −55.4256 −1.78421
\(966\) 0 0
\(967\) 24.2487 0.779786 0.389893 0.920860i \(-0.372512\pi\)
0.389893 + 0.920860i \(0.372512\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) 38.1051 1.22159
\(974\) 0 0
\(975\) −36.3731 −1.16487
\(976\) 0 0
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.3923 0.331801
\(982\) 0 0
\(983\) 45.0333 1.43634 0.718170 0.695868i \(-0.244980\pi\)
0.718170 + 0.695868i \(0.244980\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) −18.0000 −0.572946
\(988\) 0 0
\(989\) 17.3205 0.550760
\(990\) 0 0
\(991\) −31.1769 −0.990367 −0.495184 0.868788i \(-0.664899\pi\)
−0.495184 + 0.868788i \(0.664899\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) 78.0000 2.47277
\(996\) 0 0
\(997\) 38.1051 1.20680 0.603401 0.797438i \(-0.293812\pi\)
0.603401 + 0.797438i \(0.293812\pi\)
\(998\) 0 0
\(999\) 0 0
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.w.1.2 2
4.3 odd 2 4864.2.a.t.1.2 2
8.3 odd 2 inner 4864.2.a.w.1.1 2
8.5 even 2 4864.2.a.t.1.1 2
16.3 odd 4 1216.2.c.g.609.1 4
16.5 even 4 1216.2.c.g.609.2 yes 4
16.11 odd 4 1216.2.c.g.609.4 yes 4
16.13 even 4 1216.2.c.g.609.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.g.609.1 4 16.3 odd 4
1216.2.c.g.609.2 yes 4 16.5 even 4
1216.2.c.g.609.3 yes 4 16.13 even 4
1216.2.c.g.609.4 yes 4 16.11 odd 4
4864.2.a.t.1.1 2 8.5 even 2
4864.2.a.t.1.2 2 4.3 odd 2
4864.2.a.w.1.1 2 8.3 odd 2 inner
4864.2.a.w.1.2 2 1.1 even 1 trivial