Properties

Label 8624.2.a.bf.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} +O(q^{10})\) \(q-3.23607 q^{3} -3.23607 q^{5} +7.47214 q^{9} -1.00000 q^{11} -1.23607 q^{13} +10.4721 q^{15} +6.47214 q^{17} -2.76393 q^{19} -4.00000 q^{23} +5.47214 q^{25} -14.4721 q^{27} -4.47214 q^{29} +2.00000 q^{31} +3.23607 q^{33} -10.9443 q^{37} +4.00000 q^{39} -6.47214 q^{41} +1.52786 q^{43} -24.1803 q^{45} -2.00000 q^{47} -20.9443 q^{51} -0.472136 q^{53} +3.23607 q^{55} +8.94427 q^{57} +7.23607 q^{59} +5.23607 q^{61} +4.00000 q^{65} +15.4164 q^{67} +12.9443 q^{69} +2.47214 q^{71} +4.94427 q^{73} -17.7082 q^{75} +24.4164 q^{81} +10.1803 q^{83} -20.9443 q^{85} +14.4721 q^{87} -10.0000 q^{89} -6.47214 q^{93} +8.94427 q^{95} -3.52786 q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −3.23607 −1.86834 −0.934172 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 0 0
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.23607 −0.342824 −0.171412 0.985199i \(-0.554833\pi\)
−0.171412 + 0.985199i \(0.554833\pi\)
\(14\) 0 0
\(15\) 10.4721 2.70389
\(16\) 0 0
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) 0 0
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −14.4721 −2.78516
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 3.23607 0.563327
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.47214 −1.01078 −0.505389 0.862892i \(-0.668651\pi\)
−0.505389 + 0.862892i \(0.668651\pi\)
\(42\) 0 0
\(43\) 1.52786 0.232997 0.116499 0.993191i \(-0.462833\pi\)
0.116499 + 0.993191i \(0.462833\pi\)
\(44\) 0 0
\(45\) −24.1803 −3.60459
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −20.9443 −2.93278
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) 3.23607 0.436351
\(56\) 0 0
\(57\) 8.94427 1.18470
\(58\) 0 0
\(59\) 7.23607 0.942056 0.471028 0.882118i \(-0.343883\pi\)
0.471028 + 0.882118i \(0.343883\pi\)
\(60\) 0 0
\(61\) 5.23607 0.670410 0.335205 0.942145i \(-0.391194\pi\)
0.335205 + 0.942145i \(0.391194\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 15.4164 1.88341 0.941707 0.336434i \(-0.109221\pi\)
0.941707 + 0.336434i \(0.109221\pi\)
\(68\) 0 0
\(69\) 12.9443 1.55831
\(70\) 0 0
\(71\) 2.47214 0.293389 0.146694 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(72\) 0 0
\(73\) 4.94427 0.578683 0.289342 0.957226i \(-0.406564\pi\)
0.289342 + 0.957226i \(0.406564\pi\)
\(74\) 0 0
\(75\) −17.7082 −2.04477
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 10.1803 1.11744 0.558719 0.829357i \(-0.311293\pi\)
0.558719 + 0.829357i \(0.311293\pi\)
\(84\) 0 0
\(85\) −20.9443 −2.27173
\(86\) 0 0
\(87\) 14.4721 1.55158
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.47214 −0.671129
\(94\) 0 0
\(95\) 8.94427 0.917663
\(96\) 0 0
\(97\) −3.52786 −0.358200 −0.179100 0.983831i \(-0.557319\pi\)
−0.179100 + 0.983831i \(0.557319\pi\)
\(98\) 0 0
\(99\) −7.47214 −0.750978
\(100\) 0 0
\(101\) 14.1803 1.41100 0.705498 0.708712i \(-0.250723\pi\)
0.705498 + 0.708712i \(0.250723\pi\)
\(102\) 0 0
\(103\) 2.94427 0.290108 0.145054 0.989424i \(-0.453664\pi\)
0.145054 + 0.989424i \(0.453664\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.47214 0.625685 0.312842 0.949805i \(-0.398719\pi\)
0.312842 + 0.949805i \(0.398719\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 35.4164 3.36158
\(112\) 0 0
\(113\) 8.47214 0.796992 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(114\) 0 0
\(115\) 12.9443 1.20706
\(116\) 0 0
\(117\) −9.23607 −0.853875
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 20.9443 1.88848
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −4.94427 −0.435319
\(130\) 0 0
\(131\) 9.23607 0.806959 0.403480 0.914989i \(-0.367801\pi\)
0.403480 + 0.914989i \(0.367801\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 46.8328 4.03073
\(136\) 0 0
\(137\) 15.8885 1.35745 0.678725 0.734393i \(-0.262533\pi\)
0.678725 + 0.734393i \(0.262533\pi\)
\(138\) 0 0
\(139\) −8.29180 −0.703301 −0.351650 0.936131i \(-0.614379\pi\)
−0.351650 + 0.936131i \(0.614379\pi\)
\(140\) 0 0
\(141\) 6.47214 0.545052
\(142\) 0 0
\(143\) 1.23607 0.103365
\(144\) 0 0
\(145\) 14.4721 1.20185
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.3607 1.83186 0.915929 0.401340i \(-0.131455\pi\)
0.915929 + 0.401340i \(0.131455\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 48.3607 3.90973
\(154\) 0 0
\(155\) −6.47214 −0.519854
\(156\) 0 0
\(157\) −18.6525 −1.48863 −0.744315 0.667829i \(-0.767224\pi\)
−0.744315 + 0.667829i \(0.767224\pi\)
\(158\) 0 0
\(159\) 1.52786 0.121168
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −7.41641 −0.580898 −0.290449 0.956890i \(-0.593805\pi\)
−0.290449 + 0.956890i \(0.593805\pi\)
\(164\) 0 0
\(165\) −10.4721 −0.815255
\(166\) 0 0
\(167\) −15.4164 −1.19296 −0.596479 0.802629i \(-0.703434\pi\)
−0.596479 + 0.802629i \(0.703434\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 0 0
\(171\) −20.6525 −1.57933
\(172\) 0 0
\(173\) −1.23607 −0.0939765 −0.0469883 0.998895i \(-0.514962\pi\)
−0.0469883 + 0.998895i \(0.514962\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −23.4164 −1.76008
\(178\) 0 0
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) −4.76393 −0.354100 −0.177050 0.984202i \(-0.556655\pi\)
−0.177050 + 0.984202i \(0.556655\pi\)
\(182\) 0 0
\(183\) −16.9443 −1.25256
\(184\) 0 0
\(185\) 35.4164 2.60387
\(186\) 0 0
\(187\) −6.47214 −0.473289
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) 0 0
\(193\) 2.94427 0.211933 0.105967 0.994370i \(-0.466206\pi\)
0.105967 + 0.994370i \(0.466206\pi\)
\(194\) 0 0
\(195\) −12.9443 −0.926959
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 1.05573 0.0748386 0.0374193 0.999300i \(-0.488086\pi\)
0.0374193 + 0.999300i \(0.488086\pi\)
\(200\) 0 0
\(201\) −49.8885 −3.51887
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 20.9443 1.46281
\(206\) 0 0
\(207\) −29.8885 −2.07740
\(208\) 0 0
\(209\) 2.76393 0.191185
\(210\) 0 0
\(211\) 22.4721 1.54705 0.773523 0.633768i \(-0.218493\pi\)
0.773523 + 0.633768i \(0.218493\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −4.94427 −0.337197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 8.47214 0.567336 0.283668 0.958923i \(-0.408449\pi\)
0.283668 + 0.958923i \(0.408449\pi\)
\(224\) 0 0
\(225\) 40.8885 2.72590
\(226\) 0 0
\(227\) −14.7639 −0.979917 −0.489958 0.871746i \(-0.662988\pi\)
−0.489958 + 0.871746i \(0.662988\pi\)
\(228\) 0 0
\(229\) −12.7639 −0.843464 −0.421732 0.906720i \(-0.638578\pi\)
−0.421732 + 0.906720i \(0.638578\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 0 0
\(235\) 6.47214 0.422196
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 11.4164 0.735395 0.367698 0.929945i \(-0.380146\pi\)
0.367698 + 0.929945i \(0.380146\pi\)
\(242\) 0 0
\(243\) −35.5967 −2.28353
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.41641 0.217381
\(248\) 0 0
\(249\) −32.9443 −2.08776
\(250\) 0 0
\(251\) 24.7639 1.56309 0.781543 0.623852i \(-0.214433\pi\)
0.781543 + 0.623852i \(0.214433\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 67.7771 4.24437
\(256\) 0 0
\(257\) 10.9443 0.682685 0.341342 0.939939i \(-0.389118\pi\)
0.341342 + 0.939939i \(0.389118\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −33.4164 −2.06842
\(262\) 0 0
\(263\) −12.9443 −0.798178 −0.399089 0.916912i \(-0.630674\pi\)
−0.399089 + 0.916912i \(0.630674\pi\)
\(264\) 0 0
\(265\) 1.52786 0.0938559
\(266\) 0 0
\(267\) 32.3607 1.98044
\(268\) 0 0
\(269\) 27.2361 1.66061 0.830306 0.557307i \(-0.188166\pi\)
0.830306 + 0.557307i \(0.188166\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.47214 −0.329982
\(276\) 0 0
\(277\) 12.4721 0.749378 0.374689 0.927151i \(-0.377749\pi\)
0.374689 + 0.927151i \(0.377749\pi\)
\(278\) 0 0
\(279\) 14.9443 0.894690
\(280\) 0 0
\(281\) −24.8328 −1.48140 −0.740701 0.671835i \(-0.765506\pi\)
−0.740701 + 0.671835i \(0.765506\pi\)
\(282\) 0 0
\(283\) −16.6525 −0.989887 −0.494943 0.868925i \(-0.664811\pi\)
−0.494943 + 0.868925i \(0.664811\pi\)
\(284\) 0 0
\(285\) −28.9443 −1.71451
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 11.4164 0.669242
\(292\) 0 0
\(293\) −4.65248 −0.271801 −0.135900 0.990723i \(-0.543393\pi\)
−0.135900 + 0.990723i \(0.543393\pi\)
\(294\) 0 0
\(295\) −23.4164 −1.36336
\(296\) 0 0
\(297\) 14.4721 0.839759
\(298\) 0 0
\(299\) 4.94427 0.285935
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −45.8885 −2.63623
\(304\) 0 0
\(305\) −16.9443 −0.970226
\(306\) 0 0
\(307\) 32.0689 1.83027 0.915134 0.403150i \(-0.132085\pi\)
0.915134 + 0.403150i \(0.132085\pi\)
\(308\) 0 0
\(309\) −9.52786 −0.542021
\(310\) 0 0
\(311\) 5.41641 0.307136 0.153568 0.988138i \(-0.450924\pi\)
0.153568 + 0.988138i \(0.450924\pi\)
\(312\) 0 0
\(313\) −28.4721 −1.60934 −0.804670 0.593722i \(-0.797658\pi\)
−0.804670 + 0.593722i \(0.797658\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.0557 −0.733283 −0.366641 0.930362i \(-0.619492\pi\)
−0.366641 + 0.930362i \(0.619492\pi\)
\(318\) 0 0
\(319\) 4.47214 0.250392
\(320\) 0 0
\(321\) −20.9443 −1.16900
\(322\) 0 0
\(323\) −17.8885 −0.995345
\(324\) 0 0
\(325\) −6.76393 −0.375195
\(326\) 0 0
\(327\) 32.3607 1.78955
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.944272 −0.0519019 −0.0259509 0.999663i \(-0.508261\pi\)
−0.0259509 + 0.999663i \(0.508261\pi\)
\(332\) 0 0
\(333\) −81.7771 −4.48136
\(334\) 0 0
\(335\) −49.8885 −2.72570
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −27.4164 −1.48905
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −41.8885 −2.25520
\(346\) 0 0
\(347\) 6.47214 0.347442 0.173721 0.984795i \(-0.444421\pi\)
0.173721 + 0.984795i \(0.444421\pi\)
\(348\) 0 0
\(349\) −8.29180 −0.443850 −0.221925 0.975064i \(-0.571234\pi\)
−0.221925 + 0.975064i \(0.571234\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) 0 0
\(353\) 34.9443 1.85990 0.929948 0.367691i \(-0.119852\pi\)
0.929948 + 0.367691i \(0.119852\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.8328 1.41618 0.708091 0.706121i \(-0.249557\pi\)
0.708091 + 0.706121i \(0.249557\pi\)
\(360\) 0 0
\(361\) −11.3607 −0.597931
\(362\) 0 0
\(363\) −3.23607 −0.169850
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) 21.4164 1.11793 0.558964 0.829192i \(-0.311199\pi\)
0.558964 + 0.829192i \(0.311199\pi\)
\(368\) 0 0
\(369\) −48.3607 −2.51756
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 4.94427 0.255321
\(376\) 0 0
\(377\) 5.52786 0.284699
\(378\) 0 0
\(379\) −5.52786 −0.283947 −0.141974 0.989870i \(-0.545345\pi\)
−0.141974 + 0.989870i \(0.545345\pi\)
\(380\) 0 0
\(381\) −38.8328 −1.98947
\(382\) 0 0
\(383\) 11.8885 0.607476 0.303738 0.952756i \(-0.401765\pi\)
0.303738 + 0.952756i \(0.401765\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.4164 0.580329
\(388\) 0 0
\(389\) 6.58359 0.333801 0.166901 0.985974i \(-0.446624\pi\)
0.166901 + 0.985974i \(0.446624\pi\)
\(390\) 0 0
\(391\) −25.8885 −1.30924
\(392\) 0 0
\(393\) −29.8885 −1.50768
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.2918 0.516530 0.258265 0.966074i \(-0.416849\pi\)
0.258265 + 0.966074i \(0.416849\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.3607 −1.51614 −0.758070 0.652173i \(-0.773857\pi\)
−0.758070 + 0.652173i \(0.773857\pi\)
\(402\) 0 0
\(403\) −2.47214 −0.123146
\(404\) 0 0
\(405\) −79.0132 −3.92620
\(406\) 0 0
\(407\) 10.9443 0.542487
\(408\) 0 0
\(409\) 23.4164 1.15787 0.578933 0.815375i \(-0.303469\pi\)
0.578933 + 0.815375i \(0.303469\pi\)
\(410\) 0 0
\(411\) −51.4164 −2.53618
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −32.9443 −1.61717
\(416\) 0 0
\(417\) 26.8328 1.31401
\(418\) 0 0
\(419\) 12.7639 0.623559 0.311779 0.950155i \(-0.399075\pi\)
0.311779 + 0.950155i \(0.399075\pi\)
\(420\) 0 0
\(421\) 7.52786 0.366886 0.183443 0.983030i \(-0.441276\pi\)
0.183443 + 0.983030i \(0.441276\pi\)
\(422\) 0 0
\(423\) −14.9443 −0.726615
\(424\) 0 0
\(425\) 35.4164 1.71795
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −40.9443 −1.97222 −0.986108 0.166105i \(-0.946881\pi\)
−0.986108 + 0.166105i \(0.946881\pi\)
\(432\) 0 0
\(433\) −19.5279 −0.938449 −0.469225 0.883079i \(-0.655467\pi\)
−0.469225 + 0.883079i \(0.655467\pi\)
\(434\) 0 0
\(435\) −46.8328 −2.24546
\(436\) 0 0
\(437\) 11.0557 0.528867
\(438\) 0 0
\(439\) −8.94427 −0.426887 −0.213443 0.976955i \(-0.568468\pi\)
−0.213443 + 0.976955i \(0.568468\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.05573 0.335228 0.167614 0.985853i \(-0.446394\pi\)
0.167614 + 0.985853i \(0.446394\pi\)
\(444\) 0 0
\(445\) 32.3607 1.53404
\(446\) 0 0
\(447\) −72.3607 −3.42254
\(448\) 0 0
\(449\) 1.05573 0.0498229 0.0249114 0.999690i \(-0.492070\pi\)
0.0249114 + 0.999690i \(0.492070\pi\)
\(450\) 0 0
\(451\) 6.47214 0.304761
\(452\) 0 0
\(453\) 38.8328 1.82452
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.05573 0.423609 0.211805 0.977312i \(-0.432066\pi\)
0.211805 + 0.977312i \(0.432066\pi\)
\(458\) 0 0
\(459\) −93.6656 −4.37194
\(460\) 0 0
\(461\) −29.2361 −1.36166 −0.680830 0.732442i \(-0.738381\pi\)
−0.680830 + 0.732442i \(0.738381\pi\)
\(462\) 0 0
\(463\) 21.5279 1.00048 0.500242 0.865885i \(-0.333244\pi\)
0.500242 + 0.865885i \(0.333244\pi\)
\(464\) 0 0
\(465\) 20.9443 0.971267
\(466\) 0 0
\(467\) 13.1246 0.607335 0.303667 0.952778i \(-0.401789\pi\)
0.303667 + 0.952778i \(0.401789\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 60.3607 2.78127
\(472\) 0 0
\(473\) −1.52786 −0.0702513
\(474\) 0 0
\(475\) −15.1246 −0.693965
\(476\) 0 0
\(477\) −3.52786 −0.161530
\(478\) 0 0
\(479\) −32.3607 −1.47860 −0.739299 0.673378i \(-0.764843\pi\)
−0.739299 + 0.673378i \(0.764843\pi\)
\(480\) 0 0
\(481\) 13.5279 0.616818
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.4164 0.518392
\(486\) 0 0
\(487\) 0.944272 0.0427890 0.0213945 0.999771i \(-0.493189\pi\)
0.0213945 + 0.999771i \(0.493189\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) −0.944272 −0.0426144 −0.0213072 0.999773i \(-0.506783\pi\)
−0.0213072 + 0.999773i \(0.506783\pi\)
\(492\) 0 0
\(493\) −28.9443 −1.30358
\(494\) 0 0
\(495\) 24.1803 1.08683
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −12.3607 −0.553340 −0.276670 0.960965i \(-0.589231\pi\)
−0.276670 + 0.960965i \(0.589231\pi\)
\(500\) 0 0
\(501\) 49.8885 2.22886
\(502\) 0 0
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) −45.8885 −2.04201
\(506\) 0 0
\(507\) 37.1246 1.64876
\(508\) 0 0
\(509\) 34.0689 1.51008 0.755038 0.655681i \(-0.227618\pi\)
0.755038 + 0.655681i \(0.227618\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 40.0000 1.76604
\(514\) 0 0
\(515\) −9.52786 −0.419848
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −34.3607 −1.50537 −0.752684 0.658382i \(-0.771241\pi\)
−0.752684 + 0.658382i \(0.771241\pi\)
\(522\) 0 0
\(523\) −27.7082 −1.21160 −0.605798 0.795619i \(-0.707146\pi\)
−0.605798 + 0.795619i \(0.707146\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.9443 0.563861
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 54.0689 2.34639
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −20.9443 −0.905500
\(536\) 0 0
\(537\) 28.9443 1.24904
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.05573 −0.389336 −0.194668 0.980869i \(-0.562363\pi\)
−0.194668 + 0.980869i \(0.562363\pi\)
\(542\) 0 0
\(543\) 15.4164 0.661581
\(544\) 0 0
\(545\) 32.3607 1.38618
\(546\) 0 0
\(547\) −16.9443 −0.724485 −0.362242 0.932084i \(-0.617989\pi\)
−0.362242 + 0.932084i \(0.617989\pi\)
\(548\) 0 0
\(549\) 39.1246 1.66980
\(550\) 0 0
\(551\) 12.3607 0.526583
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −114.610 −4.86492
\(556\) 0 0
\(557\) −28.8328 −1.22169 −0.610843 0.791752i \(-0.709169\pi\)
−0.610843 + 0.791752i \(0.709169\pi\)
\(558\) 0 0
\(559\) −1.88854 −0.0798769
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 0 0
\(563\) 26.7639 1.12797 0.563983 0.825787i \(-0.309268\pi\)
0.563983 + 0.825787i \(0.309268\pi\)
\(564\) 0 0
\(565\) −27.4164 −1.15342
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.8328 0.705668 0.352834 0.935686i \(-0.385218\pi\)
0.352834 + 0.935686i \(0.385218\pi\)
\(570\) 0 0
\(571\) 45.8885 1.92038 0.960188 0.279355i \(-0.0901206\pi\)
0.960188 + 0.279355i \(0.0901206\pi\)
\(572\) 0 0
\(573\) 20.9443 0.874960
\(574\) 0 0
\(575\) −21.8885 −0.912815
\(576\) 0 0
\(577\) −9.05573 −0.376995 −0.188497 0.982074i \(-0.560362\pi\)
−0.188497 + 0.982074i \(0.560362\pi\)
\(578\) 0 0
\(579\) −9.52786 −0.395965
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.472136 0.0195539
\(584\) 0 0
\(585\) 29.8885 1.23574
\(586\) 0 0
\(587\) −28.1803 −1.16313 −0.581564 0.813501i \(-0.697559\pi\)
−0.581564 + 0.813501i \(0.697559\pi\)
\(588\) 0 0
\(589\) −5.52786 −0.227772
\(590\) 0 0
\(591\) −58.2492 −2.39605
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.41641 −0.139824
\(598\) 0 0
\(599\) −12.3607 −0.505044 −0.252522 0.967591i \(-0.581260\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(600\) 0 0
\(601\) −18.8328 −0.768207 −0.384103 0.923290i \(-0.625489\pi\)
−0.384103 + 0.923290i \(0.625489\pi\)
\(602\) 0 0
\(603\) 115.193 4.69104
\(604\) 0 0
\(605\) −3.23607 −0.131565
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.47214 0.100012
\(612\) 0 0
\(613\) 19.5279 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(614\) 0 0
\(615\) −67.7771 −2.73304
\(616\) 0 0
\(617\) −5.41641 −0.218056 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(618\) 0 0
\(619\) −48.5410 −1.95103 −0.975514 0.219937i \(-0.929415\pi\)
−0.975514 + 0.219937i \(0.929415\pi\)
\(620\) 0 0
\(621\) 57.8885 2.32299
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) −8.94427 −0.357200
\(628\) 0 0
\(629\) −70.8328 −2.82429
\(630\) 0 0
\(631\) 4.58359 0.182470 0.0912350 0.995829i \(-0.470919\pi\)
0.0912350 + 0.995829i \(0.470919\pi\)
\(632\) 0 0
\(633\) −72.7214 −2.89041
\(634\) 0 0
\(635\) −38.8328 −1.54103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 18.4721 0.730746
\(640\) 0 0
\(641\) 36.4721 1.44056 0.720281 0.693682i \(-0.244013\pi\)
0.720281 + 0.693682i \(0.244013\pi\)
\(642\) 0 0
\(643\) −23.2361 −0.916341 −0.458171 0.888864i \(-0.651495\pi\)
−0.458171 + 0.888864i \(0.651495\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 24.8328 0.976279 0.488139 0.872766i \(-0.337676\pi\)
0.488139 + 0.872766i \(0.337676\pi\)
\(648\) 0 0
\(649\) −7.23607 −0.284041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.63932 0.0641516 0.0320758 0.999485i \(-0.489788\pi\)
0.0320758 + 0.999485i \(0.489788\pi\)
\(654\) 0 0
\(655\) −29.8885 −1.16784
\(656\) 0 0
\(657\) 36.9443 1.44133
\(658\) 0 0
\(659\) −43.4164 −1.69126 −0.845632 0.533767i \(-0.820776\pi\)
−0.845632 + 0.533767i \(0.820776\pi\)
\(660\) 0 0
\(661\) −37.1246 −1.44398 −0.721990 0.691903i \(-0.756772\pi\)
−0.721990 + 0.691903i \(0.756772\pi\)
\(662\) 0 0
\(663\) 25.8885 1.00543
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 17.8885 0.692647
\(668\) 0 0
\(669\) −27.4164 −1.05998
\(670\) 0 0
\(671\) −5.23607 −0.202136
\(672\) 0 0
\(673\) 31.8885 1.22921 0.614607 0.788834i \(-0.289315\pi\)
0.614607 + 0.788834i \(0.289315\pi\)
\(674\) 0 0
\(675\) −79.1935 −3.04816
\(676\) 0 0
\(677\) −32.0689 −1.23251 −0.616254 0.787548i \(-0.711350\pi\)
−0.616254 + 0.787548i \(0.711350\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 47.7771 1.83082
\(682\) 0 0
\(683\) −15.0557 −0.576091 −0.288046 0.957617i \(-0.593006\pi\)
−0.288046 + 0.957617i \(0.593006\pi\)
\(684\) 0 0
\(685\) −51.4164 −1.96452
\(686\) 0 0
\(687\) 41.3050 1.57588
\(688\) 0 0
\(689\) 0.583592 0.0222331
\(690\) 0 0
\(691\) −18.6525 −0.709574 −0.354787 0.934947i \(-0.615447\pi\)
−0.354787 + 0.934947i \(0.615447\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 26.8328 1.01783
\(696\) 0 0
\(697\) −41.8885 −1.58664
\(698\) 0 0
\(699\) −9.52786 −0.360377
\(700\) 0 0
\(701\) 46.7214 1.76464 0.882321 0.470649i \(-0.155980\pi\)
0.882321 + 0.470649i \(0.155980\pi\)
\(702\) 0 0
\(703\) 30.2492 1.14087
\(704\) 0 0
\(705\) −20.9443 −0.788807
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.47214 −0.167955 −0.0839773 0.996468i \(-0.526762\pi\)
−0.0839773 + 0.996468i \(0.526762\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 64.7214 2.41706
\(718\) 0 0
\(719\) −36.8328 −1.37363 −0.686816 0.726831i \(-0.740992\pi\)
−0.686816 + 0.726831i \(0.740992\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −36.9443 −1.37397
\(724\) 0 0
\(725\) −24.4721 −0.908872
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 41.9443 1.55349
\(730\) 0 0
\(731\) 9.88854 0.365741
\(732\) 0 0
\(733\) −8.87539 −0.327820 −0.163910 0.986475i \(-0.552411\pi\)
−0.163910 + 0.986475i \(0.552411\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.4164 −0.567871
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −11.0557 −0.406142
\(742\) 0 0
\(743\) 13.8885 0.509521 0.254761 0.967004i \(-0.418003\pi\)
0.254761 + 0.967004i \(0.418003\pi\)
\(744\) 0 0
\(745\) −72.3607 −2.65109
\(746\) 0 0
\(747\) 76.0689 2.78321
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.944272 −0.0344570 −0.0172285 0.999852i \(-0.505484\pi\)
−0.0172285 + 0.999852i \(0.505484\pi\)
\(752\) 0 0
\(753\) −80.1378 −2.92038
\(754\) 0 0
\(755\) 38.8328 1.41327
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) 0 0
\(759\) −12.9443 −0.469847
\(760\) 0 0
\(761\) −15.4164 −0.558844 −0.279422 0.960168i \(-0.590143\pi\)
−0.279422 + 0.960168i \(0.590143\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −156.498 −5.65821
\(766\) 0 0
\(767\) −8.94427 −0.322959
\(768\) 0 0
\(769\) 16.5836 0.598020 0.299010 0.954250i \(-0.403344\pi\)
0.299010 + 0.954250i \(0.403344\pi\)
\(770\) 0 0
\(771\) −35.4164 −1.27549
\(772\) 0 0
\(773\) −2.29180 −0.0824302 −0.0412151 0.999150i \(-0.513123\pi\)
−0.0412151 + 0.999150i \(0.513123\pi\)
\(774\) 0 0
\(775\) 10.9443 0.393130
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.8885 0.640924
\(780\) 0 0
\(781\) −2.47214 −0.0884600
\(782\) 0 0
\(783\) 64.7214 2.31295
\(784\) 0 0
\(785\) 60.3607 2.15437
\(786\) 0 0
\(787\) −5.81966 −0.207448 −0.103724 0.994606i \(-0.533076\pi\)
−0.103724 + 0.994606i \(0.533076\pi\)
\(788\) 0 0
\(789\) 41.8885 1.49127
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.47214 −0.229832
\(794\) 0 0
\(795\) −4.94427 −0.175355
\(796\) 0 0
\(797\) −7.59675 −0.269091 −0.134545 0.990907i \(-0.542957\pi\)
−0.134545 + 0.990907i \(0.542957\pi\)
\(798\) 0 0
\(799\) −12.9443 −0.457935
\(800\) 0 0
\(801\) −74.7214 −2.64015
\(802\) 0 0
\(803\) −4.94427 −0.174480
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −88.1378 −3.10260
\(808\) 0 0
\(809\) −38.9443 −1.36921 −0.684604 0.728915i \(-0.740025\pi\)
−0.684604 + 0.728915i \(0.740025\pi\)
\(810\) 0 0
\(811\) 9.23607 0.324322 0.162161 0.986764i \(-0.448154\pi\)
0.162161 + 0.986764i \(0.448154\pi\)
\(812\) 0 0
\(813\) 54.8328 1.92307
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) −4.22291 −0.147741
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.4164 0.887039 0.443519 0.896265i \(-0.353730\pi\)
0.443519 + 0.896265i \(0.353730\pi\)
\(822\) 0 0
\(823\) −34.2492 −1.19385 −0.596926 0.802296i \(-0.703612\pi\)
−0.596926 + 0.802296i \(0.703612\pi\)
\(824\) 0 0
\(825\) 17.7082 0.616521
\(826\) 0 0
\(827\) 0.944272 0.0328356 0.0164178 0.999865i \(-0.494774\pi\)
0.0164178 + 0.999865i \(0.494774\pi\)
\(828\) 0 0
\(829\) 1.70820 0.0593284 0.0296642 0.999560i \(-0.490556\pi\)
0.0296642 + 0.999560i \(0.490556\pi\)
\(830\) 0 0
\(831\) −40.3607 −1.40010
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 49.8885 1.72646
\(836\) 0 0
\(837\) −28.9443 −1.00046
\(838\) 0 0
\(839\) −36.8328 −1.27161 −0.635805 0.771850i \(-0.719332\pi\)
−0.635805 + 0.771850i \(0.719332\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 80.3607 2.76777
\(844\) 0 0
\(845\) 37.1246 1.27713
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 53.8885 1.84945
\(850\) 0 0
\(851\) 43.7771 1.50066
\(852\) 0 0
\(853\) 34.5410 1.18266 0.591331 0.806429i \(-0.298603\pi\)
0.591331 + 0.806429i \(0.298603\pi\)
\(854\) 0 0
\(855\) 66.8328 2.28563
\(856\) 0 0
\(857\) 37.5279 1.28193 0.640964 0.767571i \(-0.278535\pi\)
0.640964 + 0.767571i \(0.278535\pi\)
\(858\) 0 0
\(859\) −25.1246 −0.857241 −0.428620 0.903485i \(-0.641000\pi\)
−0.428620 + 0.903485i \(0.641000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −27.4164 −0.933265 −0.466633 0.884451i \(-0.654533\pi\)
−0.466633 + 0.884451i \(0.654533\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) −80.5410 −2.73532
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −19.0557 −0.645679
\(872\) 0 0
\(873\) −26.3607 −0.892174
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.9443 0.909843 0.454922 0.890531i \(-0.349667\pi\)
0.454922 + 0.890531i \(0.349667\pi\)
\(878\) 0 0
\(879\) 15.0557 0.507817
\(880\) 0 0
\(881\) 24.8328 0.836639 0.418319 0.908300i \(-0.362619\pi\)
0.418319 + 0.908300i \(0.362619\pi\)
\(882\) 0 0
\(883\) −50.8328 −1.71066 −0.855330 0.518083i \(-0.826646\pi\)
−0.855330 + 0.518083i \(0.826646\pi\)
\(884\) 0 0
\(885\) 75.7771 2.54722
\(886\) 0 0
\(887\) 0.360680 0.0121104 0.00605522 0.999982i \(-0.498073\pi\)
0.00605522 + 0.999982i \(0.498073\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −24.4164 −0.817980
\(892\) 0 0
\(893\) 5.52786 0.184983
\(894\) 0 0
\(895\) 28.9443 0.967500
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) −8.94427 −0.298308
\(900\) 0 0
\(901\) −3.05573 −0.101801
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.4164 0.512459
\(906\) 0 0
\(907\) −20.3607 −0.676065 −0.338033 0.941134i \(-0.609761\pi\)
−0.338033 + 0.941134i \(0.609761\pi\)
\(908\) 0 0
\(909\) 105.957 3.51439
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) −10.1803 −0.336920
\(914\) 0 0
\(915\) 54.8328 1.81272
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 57.8885 1.90957 0.954783 0.297302i \(-0.0960869\pi\)
0.954783 + 0.297302i \(0.0960869\pi\)
\(920\) 0 0
\(921\) −103.777 −3.41957
\(922\) 0 0
\(923\) −3.05573 −0.100581
\(924\) 0 0
\(925\) −59.8885 −1.96912
\(926\) 0 0
\(927\) 22.0000 0.722575
\(928\) 0 0
\(929\) 40.2492 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −17.5279 −0.573837
\(934\) 0 0
\(935\) 20.9443 0.684951
\(936\) 0 0
\(937\) 20.9443 0.684220 0.342110 0.939660i \(-0.388859\pi\)
0.342110 + 0.939660i \(0.388859\pi\)
\(938\) 0 0
\(939\) 92.1378 3.00680
\(940\) 0 0
\(941\) 34.1803 1.11425 0.557124 0.830430i \(-0.311905\pi\)
0.557124 + 0.830430i \(0.311905\pi\)
\(942\) 0 0
\(943\) 25.8885 0.843047
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.944272 0.0306847 0.0153424 0.999882i \(-0.495116\pi\)
0.0153424 + 0.999882i \(0.495116\pi\)
\(948\) 0 0
\(949\) −6.11146 −0.198386
\(950\) 0 0
\(951\) 42.2492 1.37002
\(952\) 0 0
\(953\) 5.05573 0.163771 0.0818855 0.996642i \(-0.473906\pi\)
0.0818855 + 0.996642i \(0.473906\pi\)
\(954\) 0 0
\(955\) 20.9443 0.677741
\(956\) 0 0
\(957\) −14.4721 −0.467818
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 48.3607 1.55840
\(964\) 0 0
\(965\) −9.52786 −0.306713
\(966\) 0 0
\(967\) −10.1115 −0.325163 −0.162581 0.986695i \(-0.551982\pi\)
−0.162581 + 0.986695i \(0.551982\pi\)
\(968\) 0 0
\(969\) 57.8885 1.85965
\(970\) 0 0
\(971\) −16.5410 −0.530827 −0.265413 0.964135i \(-0.585508\pi\)
−0.265413 + 0.964135i \(0.585508\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 21.8885 0.700994
\(976\) 0 0
\(977\) 24.8328 0.794472 0.397236 0.917716i \(-0.369969\pi\)
0.397236 + 0.917716i \(0.369969\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −74.7214 −2.38567
\(982\) 0 0
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) −58.2492 −1.85597
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.11146 −0.194333
\(990\) 0 0
\(991\) −44.3607 −1.40916 −0.704582 0.709623i \(-0.748865\pi\)
−0.704582 + 0.709623i \(0.748865\pi\)
\(992\) 0 0
\(993\) 3.05573 0.0969706
\(994\) 0 0
\(995\) −3.41641 −0.108307
\(996\) 0 0
\(997\) −1.81966 −0.0576292 −0.0288146 0.999585i \(-0.509173\pi\)
−0.0288146 + 0.999585i \(0.509173\pi\)
\(998\) 0 0
\(999\) 158.387 5.01114
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 8624.2.a.bf.1.1 2
4.3 odd 2 1078.2.a.w.1.2 2
7.6 odd 2 1232.2.a.p.1.2 2
12.11 even 2 9702.2.a.cu.1.2 2
28.3 even 6 1078.2.e.q.177.2 4
28.11 odd 6 1078.2.e.n.177.1 4
28.19 even 6 1078.2.e.q.67.2 4
28.23 odd 6 1078.2.e.n.67.1 4
28.27 even 2 154.2.a.d.1.1 2
56.13 odd 2 4928.2.a.bk.1.1 2
56.27 even 2 4928.2.a.bt.1.2 2
84.83 odd 2 1386.2.a.m.1.1 2
140.27 odd 4 3850.2.c.q.1849.4 4
140.83 odd 4 3850.2.c.q.1849.1 4
140.139 even 2 3850.2.a.bj.1.2 2
308.307 odd 2 1694.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 2 28.27 even 2
1078.2.a.w.1.2 2 4.3 odd 2
1078.2.e.n.67.1 4 28.23 odd 6
1078.2.e.n.177.1 4 28.11 odd 6
1078.2.e.q.67.2 4 28.19 even 6
1078.2.e.q.177.2 4 28.3 even 6
1232.2.a.p.1.2 2 7.6 odd 2
1386.2.a.m.1.1 2 84.83 odd 2
1694.2.a.l.1.1 2 308.307 odd 2
3850.2.a.bj.1.2 2 140.139 even 2
3850.2.c.q.1849.1 4 140.83 odd 4
3850.2.c.q.1849.4 4 140.27 odd 4
4928.2.a.bk.1.1 2 56.13 odd 2
4928.2.a.bt.1.2 2 56.27 even 2
8624.2.a.bf.1.1 2 1.1 even 1 trivial
9702.2.a.cu.1.2 2 12.11 even 2