Properties

Label 2898.2.a.k.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.1406465058\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2898.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -4.00000 q^{11} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{35} -5.00000 q^{37} -3.00000 q^{40} -5.00000 q^{41} -7.00000 q^{43} -4.00000 q^{44} +1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} +3.00000 q^{52} -12.0000 q^{53} +12.0000 q^{55} +1.00000 q^{56} -1.00000 q^{58} +2.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -9.00000 q^{65} -12.0000 q^{67} -3.00000 q^{70} -10.0000 q^{71} -5.00000 q^{74} -4.00000 q^{77} +4.00000 q^{79} -3.00000 q^{80} -5.00000 q^{82} -4.00000 q^{83} -7.00000 q^{86} -4.00000 q^{88} -10.0000 q^{89} +3.00000 q^{91} +1.00000 q^{92} +3.00000 q^{94} +19.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.00000 −1.11631
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) −5.00000 −0.552158
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) 19.0000 1.92916 0.964579 0.263795i \(-0.0849741\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −7.00000 −0.658505 −0.329252 0.944242i \(-0.606797\pi\)
−0.329252 + 0.944242i \(0.606797\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −9.00000 −0.789352
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0000 1.45241 0.726204 0.687479i \(-0.241283\pi\)
0.726204 + 0.687479i \(0.241283\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) −5.00000 −0.410997
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −24.0000 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −3.00000 −0.237171
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) −7.00000 −0.533745
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 3.00000 0.222375
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) 15.0000 1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 19.0000 1.36412
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) 0 0
\(199\) −23.0000 −1.63043 −0.815213 0.579161i \(-0.803380\pi\)
−0.815213 + 0.579161i \(0.803380\pi\)
\(200\) 4.00000 0.282843
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) −1.00000 −0.0701862
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) −1.00000 −0.0696733
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) 21.0000 1.43219
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −1.00000 −0.0677285
\(219\) 0 0
\(220\) 12.0000 0.809040
\(221\) 0 0
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −7.00000 −0.465633
\(227\) 5.00000 0.331862 0.165931 0.986137i \(-0.446937\pi\)
0.165931 + 0.986137i \(0.446937\pi\)
\(228\) 0 0
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 9.00000 0.579741 0.289870 0.957066i \(-0.406388\pi\)
0.289870 + 0.957066i \(0.406388\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) −9.00000 −0.558156
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) −16.0000 −0.964836
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) 7.00000 0.419832
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −5.00000 −0.295141
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 3.00000 0.176166
\(291\) 0 0
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) −5.00000 −0.290619
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) −17.0000 −0.978240
\(303\) 0 0
\(304\) 0 0
\(305\) 18.0000 1.03068
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 6.00000 0.340777
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 33.0000 1.85346 0.926732 0.375722i \(-0.122605\pi\)
0.926732 + 0.375722i \(0.122605\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) 12.0000 0.665640
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −5.00000 −0.276079
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 36.0000 1.96689
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) −4.00000 −0.217571
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 17.0000 0.904819 0.452409 0.891810i \(-0.350565\pi\)
0.452409 + 0.891810i \(0.350565\pi\)
\(354\) 0 0
\(355\) 30.0000 1.59223
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −9.00000 −0.475665
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) 3.00000 0.157243
\(365\) 0 0
\(366\) 0 0
\(367\) 21.0000 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 15.0000 0.779813
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) −1.00000 −0.0508987
\(387\) 0 0
\(388\) 19.0000 0.964579
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 21.0000 1.05796
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −23.0000 −1.15289
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 15.0000 0.740797
\(411\) 0 0
\(412\) −1.00000 −0.0492665
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 3.00000 0.147087
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) −6.00000 −0.290360
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 21.0000 1.01271
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) −27.0000 −1.29754 −0.648769 0.760986i \(-0.724716\pi\)
−0.648769 + 0.760986i \(0.724716\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) 0 0
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 12.0000 0.572078
\(441\) 0 0
\(442\) 0 0
\(443\) 29.0000 1.37783 0.688916 0.724841i \(-0.258087\pi\)
0.688916 + 0.724841i \(0.258087\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) 2.00000 0.0947027
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) −7.00000 −0.329252
\(453\) 0 0
\(454\) 5.00000 0.234662
\(455\) −9.00000 −0.421927
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 16.0000 0.747631
\(459\) 0 0
\(460\) −3.00000 −0.139876
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 5.00000 0.232370 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) −9.00000 −0.415139
\(471\) 0 0
\(472\) 2.00000 0.0920575
\(473\) 28.0000 1.28744
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) 9.00000 0.409939
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −57.0000 −2.58824
\(486\) 0 0
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) −19.0000 −0.848012
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 42.0000 1.86898
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 7.00000 0.310575
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 3.00000 0.132196
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) −5.00000 −0.219687
\(519\) 0 0
\(520\) −9.00000 −0.394676
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 21.0000 0.915644
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 36.0000 1.56374
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 17.0000 0.726204
\(549\) 0 0
\(550\) −16.0000 −0.682242
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −24.0000 −1.01966
\(555\) 0 0
\(556\) 7.00000 0.296866
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) −21.0000 −0.888205
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −1.00000 −0.0421825
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) 0 0
\(565\) 21.0000 0.883477
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) 19.0000 0.796521 0.398261 0.917272i \(-0.369614\pi\)
0.398261 + 0.917272i \(0.369614\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) −5.00000 −0.208696
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 3.00000 0.124568
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 48.0000 1.98796
\(584\) 0 0
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) −5.00000 −0.205499
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 0 0
\(598\) 3.00000 0.122679
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) −7.00000 −0.285299
\(603\) 0 0
\(604\) −17.0000 −0.691720
\(605\) −15.0000 −0.609837
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 18.0000 0.728799
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 21.0000 0.848182 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(614\) 25.0000 1.00892
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) −24.0000 −0.957704
\(629\) 0 0
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 33.0000 1.31060
\(635\) −21.0000 −0.833360
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −1.00000 −0.0394976 −0.0197488 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 12.0000 0.470679
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −13.0000 −0.508729 −0.254365 0.967108i \(-0.581866\pi\)
−0.254365 + 0.967108i \(0.581866\pi\)
\(654\) 0 0
\(655\) −54.0000 −2.10995
\(656\) −5.00000 −0.195217
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 36.0000 1.39080
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 19.0000 0.729153
\(680\) 0 0
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −51.0000 −1.94861
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −7.00000 −0.266872
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 21.0000 0.798878 0.399439 0.916760i \(-0.369205\pi\)
0.399439 + 0.916760i \(0.369205\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) −21.0000 −0.796575
\(696\) 0 0
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 17.0000 0.639803
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 30.0000 1.12588
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) −2.00000 −0.0749006
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) −9.00000 −0.335877
\(719\) 43.0000 1.60363 0.801815 0.597573i \(-0.203868\pi\)
0.801815 + 0.597573i \(0.203868\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 3.00000 0.111187
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 21.0000 0.775124
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 15.0000 0.551411
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 36.0000 1.31894
\(746\) −2.00000 −0.0732252
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) −3.00000 −0.109254
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −5.00000 −0.181608
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) −1.00000 −0.0362024
\(764\) 20.0000 0.723575
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 12.0000 0.432450
\(771\) 0 0
\(772\) −1.00000 −0.0359908
\(773\) −35.0000 −1.25886 −0.629431 0.777056i \(-0.716712\pi\)
−0.629431 + 0.777056i \(0.716712\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 19.0000 0.682060
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 72.0000 2.56979
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 21.0000 0.748094
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) −7.00000 −0.248891
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −23.0000 −0.815213
\(797\) −47.0000 −1.66483 −0.832413 0.554156i \(-0.813041\pi\)
−0.832413 + 0.554156i \(0.813041\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 0.141421
\(801\) 0 0
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) −6.00000 −0.211341
\(807\) 0 0
\(808\) −14.0000 −0.492518
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 23.0000 0.807639 0.403820 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) −1.00000 −0.0350931
\(813\) 0 0
\(814\) 20.0000 0.701000
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 15.0000 0.523823
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) 31.0000 1.08059 0.540296 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 0 0
\(826\) 2.00000 0.0695889
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −1.00000 −0.0344623
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) −5.00000 −0.171398
\(852\) 0 0
\(853\) 41.0000 1.40381 0.701907 0.712269i \(-0.252332\pi\)
0.701907 + 0.712269i \(0.252332\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) 21.0000 0.716094
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −27.0000 −0.917497
\(867\) 0 0
\(868\) −2.00000 −0.0678844
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −36.0000 −1.21981
\(872\) −1.00000 −0.0338643
\(873\) 0 0
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) 14.0000 0.472477
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 0 0
\(883\) 6.00000 0.201916 0.100958 0.994891i \(-0.467809\pi\)
0.100958 + 0.994891i \(0.467809\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.0000 0.974274
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 30.0000 1.00560
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) 27.0000 0.902510
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) 20.0000 0.665927
\(903\) 0 0
\(904\) −7.00000 −0.232817
\(905\) −54.0000 −1.79502
\(906\) 0 0
\(907\) −49.0000 −1.62702 −0.813509 0.581552i \(-0.802446\pi\)
−0.813509 + 0.581552i \(0.802446\pi\)
\(908\) 5.00000 0.165931
\(909\) 0 0
\(910\) −9.00000 −0.298347
\(911\) 55.0000 1.82223 0.911116 0.412151i \(-0.135222\pi\)
0.911116 + 0.412151i \(0.135222\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) −4.00000 −0.132308
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) 60.0000 1.97922 0.989609 0.143787i \(-0.0459280\pi\)
0.989609 + 0.143787i \(0.0459280\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 5.00000 0.164310
\(927\) 0 0
\(928\) −1.00000 −0.0328266
\(929\) −21.0000 −0.688988 −0.344494 0.938789i \(-0.611949\pi\)
−0.344494 + 0.938789i \(0.611949\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 0 0
\(937\) −53.0000 −1.73143 −0.865717 0.500533i \(-0.833137\pi\)
−0.865717 + 0.500533i \(0.833137\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) −9.00000 −0.293548
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 0 0
\(943\) −5.00000 −0.162822
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 28.0000 0.910359
\(947\) −3.00000 −0.0974869 −0.0487435 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) −60.0000 −1.94155
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 6.00000 0.193851
\(959\) 17.0000 0.548959
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −15.0000 −0.483619
\(963\) 0 0
\(964\) 9.00000 0.289870
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −57.0000 −1.83016
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 7.00000 0.224410
\(974\) 13.0000 0.416547
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) −2.00000 −0.0637901 −0.0318950 0.999491i \(-0.510154\pi\)
−0.0318950 + 0.999491i \(0.510154\pi\)
\(984\) 0 0
\(985\) −63.0000 −2.00735
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.00000 −0.222587
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) −10.0000 −0.317181
\(995\) 69.0000 2.18745
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) −10.0000 −0.316544
\(999\) 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 2898.2.a.k.1.1 1
3.2 odd 2 966.2.a.d.1.1 1
12.11 even 2 7728.2.a.u.1.1 1
21.20 even 2 6762.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.d.1.1 1 3.2 odd 2
2898.2.a.k.1.1 1 1.1 even 1 trivial
6762.2.a.l.1.1 1 21.20 even 2
7728.2.a.u.1.1 1 12.11 even 2