Properties

Label 2898.2.a.j.1.1
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +8.00000 q^{19} +3.00000 q^{20} +1.00000 q^{23} +4.00000 q^{25} -5.00000 q^{26} +1.00000 q^{28} -3.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +3.00000 q^{35} -7.00000 q^{37} -8.00000 q^{38} -3.00000 q^{40} -9.00000 q^{41} -1.00000 q^{43} -1.00000 q^{46} +3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} +5.00000 q^{52} +12.0000 q^{53} -1.00000 q^{56} +3.00000 q^{58} +6.00000 q^{59} +14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -4.00000 q^{67} -3.00000 q^{70} -6.00000 q^{71} -4.00000 q^{73} +7.00000 q^{74} +8.00000 q^{76} -16.0000 q^{79} +3.00000 q^{80} +9.00000 q^{82} +12.0000 q^{83} +1.00000 q^{86} -6.00000 q^{89} +5.00000 q^{91} +1.00000 q^{92} -3.00000 q^{94} +24.0000 q^{95} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 3.00000 0.670820
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −5.00000 −0.980581
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 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
\(52\) 5.00000 0.693375
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 15.0000 1.86052
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 3.00000 0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −15.0000 −1.31559
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) −3.00000 −0.237171
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −5.00000 −0.370625
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) 0 0
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) −24.0000 −1.74114
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.00000 −0.641223 −0.320612 0.947211i \(-0.603888\pi\)
−0.320612 + 0.947211i \(0.603888\pi\)
\(198\) 0 0
\(199\) 17.0000 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) −27.0000 −1.88576
\(206\) 1.00000 0.0696733
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 19.0000 1.28684
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 15.0000 0.997785
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 40.0000 2.54514
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 13.0000 0.815693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 15.0000 0.930261
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −5.00000 −0.299880
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) −11.0000 −0.632979
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 42.0000 2.40491
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.00000 −0.340777
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 27.0000 1.51647 0.758236 0.651981i \(-0.226062\pi\)
0.758236 + 0.651981i \(0.226062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.00000 0.167705
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) −20.0000 −1.10770
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) −27.0000 −1.43706 −0.718532 0.695493i \(-0.755186\pi\)
−0.718532 + 0.695493i \(0.755186\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 15.0000 0.792775
\(359\) −21.0000 −1.10834 −0.554169 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 5.00000 0.262071
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 29.0000 1.51379 0.756894 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 21.0000 1.09174
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 24.0000 1.23117
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.00000 0.0508987
\(387\) 0 0
\(388\) −1.00000 −0.0507673
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 9.00000 0.453413
\(395\) −48.0000 −2.41514
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −17.0000 −0.852133
\(399\) 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
\(403\) 10.0000 0.498135
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 27.0000 1.33343
\(411\) 0 0
\(412\) −1.00000 −0.0492665
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) −5.00000 −0.245145
\(417\) 0 0
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 3.00000 0.144673
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 0 0
\(433\) 17.0000 0.816968 0.408484 0.912766i \(-0.366058\pi\)
0.408484 + 0.912766i \(0.366058\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −19.0000 −0.909935
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.0000 1.28281 0.641404 0.767203i \(-0.278352\pi\)
0.641404 + 0.767203i \(0.278352\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) 0 0
\(454\) −15.0000 −0.703985
\(455\) 15.0000 0.703211
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 4.00000 0.186908
\(459\) 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
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −9.00000 −0.415139
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −35.0000 −1.59586
\(482\) 19.0000 0.865426
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) −31.0000 −1.40474 −0.702372 0.711810i \(-0.747876\pi\)
−0.702372 + 0.711810i \(0.747876\pi\)
\(488\) −14.0000 −0.633750
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −40.0000 −1.79969
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −3.00000 −0.134164
\(501\) 0 0
\(502\) −15.0000 −0.669483
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) −13.0000 −0.576782
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −3.00000 −0.132196
\(516\) 0 0
\(517\) 0 0
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) −15.0000 −0.657794
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −36.0000 −1.56374
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −45.0000 −1.94917
\(534\) 0 0
\(535\) −36.0000 −1.55642
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 28.0000 1.20270
\(543\) 0 0
\(544\) 0 0
\(545\) −57.0000 −2.44161
\(546\) 0 0
\(547\) −10.0000 −0.427569 −0.213785 0.976881i \(-0.568579\pi\)
−0.213785 + 0.976881i \(0.568579\pi\)
\(548\) 9.00000 0.384461
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 5.00000 0.212047
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −45.0000 −1.89316
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −45.0000 −1.88650 −0.943249 0.332086i \(-0.892248\pi\)
−0.943249 + 0.332086i \(0.892248\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) −9.00000 −0.373705
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −5.00000 −0.204465
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) 11.0000 0.447584
\(605\) −33.0000 −1.34164
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −42.0000 −1.70053
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) −1.00000 −0.0403896 −0.0201948 0.999796i \(-0.506429\pi\)
−0.0201948 + 0.999796i \(0.506429\pi\)
\(614\) −11.0000 −0.443924
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) −27.0000 −1.07231
\(635\) −39.0000 −1.54767
\(636\) 0 0
\(637\) 5.00000 0.198107
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 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
\(649\) 0 0
\(650\) −20.0000 −0.784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 0 0
\(655\) 54.0000 2.10995
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) −3.00000 −0.116952
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 16.0000 0.621858
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) 4.00000 0.154074
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −1.00000 −0.0383765
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 27.0000 1.03162
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) −49.0000 −1.86405 −0.932024 0.362397i \(-0.881959\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 3.00000 0.113878
\(695\) 15.0000 0.568982
\(696\) 0 0
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) −56.0000 −2.11208
\(704\) 0 0
\(705\) 0 0
\(706\) 27.0000 1.01616
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 18.0000 0.675528
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 2.00000 0.0749006
\(714\) 0 0
\(715\) 0 0
\(716\) −15.0000 −0.560576
\(717\) 0 0
\(718\) 21.0000 0.783713
\(719\) −45.0000 −1.67822 −0.839108 0.543964i \(-0.816923\pi\)
−0.839108 + 0.543964i \(0.816923\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −5.00000 −0.185312
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 0 0
\(732\) 0 0
\(733\) 8.00000 0.295487 0.147743 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(734\) −29.0000 −1.07041
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) −21.0000 −0.771975
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 3.00000 0.109399
\(753\) 0 0
\(754\) 15.0000 0.546268
\(755\) 33.0000 1.20099
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −29.0000 −1.05333
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −19.0000 −0.687846
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 30.0000 1.08324
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.00000 −0.0359908
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −72.0000 −2.57967
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −9.00000 −0.320612
\(789\) 0 0
\(790\) 48.0000 1.70776
\(791\) −15.0000 −0.533339
\(792\) 0 0
\(793\) 70.0000 2.48577
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 17.0000 0.602549
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) 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
\(806\) −10.0000 −0.352235
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) −3.00000 −0.105279
\(813\) 0 0
\(814\) 0 0
\(815\) 60.0000 2.10171
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) −27.0000 −0.942881
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −13.0000 −0.453152 −0.226576 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 19.0000 0.654783
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) −7.00000 −0.239957
\(852\) 0 0
\(853\) −49.0000 −1.67773 −0.838864 0.544341i \(-0.816780\pi\)
−0.838864 + 0.544341i \(0.816780\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) −3.00000 −0.102299
\(861\) 0 0
\(862\) 15.0000 0.510902
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −17.0000 −0.577684
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 19.0000 0.643421
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −40.0000 −1.35070 −0.675352 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 26.0000 0.874970 0.437485 0.899226i \(-0.355869\pi\)
0.437485 + 0.899226i \(0.355869\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −27.0000 −0.907083
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −45.0000 −1.50418
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 15.0000 0.498893
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 41.0000 1.36138 0.680691 0.732570i \(-0.261680\pi\)
0.680691 + 0.732570i \(0.261680\pi\)
\(908\) 15.0000 0.497792
\(909\) 0 0
\(910\) −15.0000 −0.497245
\(911\) −21.0000 −0.695761 −0.347881 0.937539i \(-0.613099\pi\)
−0.347881 + 0.937539i \(0.613099\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) −30.0000 −0.987462
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 31.0000 1.01872
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −33.0000 −1.07979
\(935\) 0 0
\(936\) 0 0
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 9.00000 0.293548
\(941\) 51.0000 1.66255 0.831276 0.555860i \(-0.187611\pi\)
0.831276 + 0.555860i \(0.187611\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) −32.0000 −1.03822
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 36.0000 1.16493
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 35.0000 1.12845
\(963\) 0 0
\(964\) −19.0000 −0.611949
\(965\) −3.00000 −0.0965734
\(966\) 0 0
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 3.00000 0.0963242
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 31.0000 0.993304
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 0 0
\(985\) −27.0000 −0.860292
\(986\) 0 0
\(987\) 0 0
\(988\) 40.0000 1.27257
\(989\) −1.00000 −0.0317982
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 51.0000 1.61681
\(996\) 0 0
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −14.0000 −0.443162
\(999\) 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.j.1.1 1
3.2 odd 2 966.2.a.i.1.1 1
12.11 even 2 7728.2.a.a.1.1 1
21.20 even 2 6762.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.i.1.1 1 3.2 odd 2
2898.2.a.j.1.1 1 1.1 even 1 trivial
6762.2.a.bd.1.1 1 21.20 even 2
7728.2.a.a.1.1 1 12.11 even 2