Properties

Label 1850.2.a.p.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +5.00000 q^{19} +8.00000 q^{21} -3.00000 q^{23} +2.00000 q^{24} -2.00000 q^{26} -4.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{36} -1.00000 q^{37} +5.00000 q^{38} -4.00000 q^{39} -9.00000 q^{41} +8.00000 q^{42} +7.00000 q^{43} -3.00000 q^{46} +6.00000 q^{47} +2.00000 q^{48} +9.00000 q^{49} -2.00000 q^{52} -9.00000 q^{53} -4.00000 q^{54} +4.00000 q^{56} +10.0000 q^{57} +6.00000 q^{58} +3.00000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} -6.00000 q^{69} -6.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} -1.00000 q^{74} +5.00000 q^{76} -4.00000 q^{78} -1.00000 q^{79} -11.0000 q^{81} -9.00000 q^{82} +8.00000 q^{84} +7.00000 q^{86} +12.0000 q^{87} -6.00000 q^{89} -8.00000 q^{91} -3.00000 q^{92} -8.00000 q^{93} +6.00000 q^{94} +2.00000 q^{96} +4.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −4.00000 −0.769800
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 5.00000 0.811107
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 8.00000 1.23443
\(43\) 7.00000 1.06749 0.533745 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 2.00000 0.288675
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 10.0000 1.32453
\(58\) 6.00000 0.787839
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −9.00000 −0.993884
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 8.00000 0.872872
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −3.00000 −0.312772
\(93\) −8.00000 −0.829561
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −4.00000 −0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 4.00000 0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 10.0000 0.936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) −18.0000 −1.62301
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.0000 1.23263
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 20.0000 1.73422
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −6.00000 −0.510754
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) 18.0000 1.48461
\(148\) −1.00000 −0.0821995
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) −11.0000 −0.864242
\(163\) 7.00000 0.548282 0.274141 0.961689i \(-0.411606\pi\)
0.274141 + 0.961689i \(0.411606\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 8.00000 0.617213
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) 7.00000 0.533745
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) −8.00000 −0.592999
\(183\) 4.00000 0.295689
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 2.00000 0.144338
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) −13.0000 −0.921546 −0.460773 0.887518i \(-0.652428\pi\)
−0.460773 + 0.887518i \(0.652428\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 3.00000 0.211079
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) −3.00000 −0.208514
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −9.00000 −0.618123
\(213\) −12.0000 −0.822226
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −16.0000 −1.08615
\(218\) −16.0000 −1.08366
\(219\) −22.0000 −1.48662
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 9.00000 0.597351 0.298675 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 10.0000 0.662266
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −11.0000 −0.707107
\(243\) −10.0000 −0.641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −18.0000 −1.14764
\(247\) −10.0000 −0.636285
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 14.0000 0.871602
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 20.0000 1.22628
\(267\) −12.0000 −0.734388
\(268\) −2.00000 −0.122169
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 14.0000 0.839664
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 12.0000 0.714590
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −36.0000 −2.12501
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −11.0000 −0.643726
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 18.0000 1.04978
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 15.0000 0.868927
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 28.0000 1.61389
\(302\) −4.00000 −0.230174
\(303\) 6.00000 0.344691
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 26.0000 1.47909
\(310\) 0 0
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) −4.00000 −0.226455
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −33.0000 −1.85346 −0.926732 0.375722i \(-0.877395\pi\)
−0.926732 + 0.375722i \(0.877395\pi\)
\(318\) −18.0000 −1.00939
\(319\) 0 0
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 7.00000 0.387694
\(327\) −32.0000 −1.76960
\(328\) −9.00000 −0.496942
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) −1.00000 −0.0547997
\(334\) 3.00000 0.164153
\(335\) 0 0
\(336\) 8.00000 0.436436
\(337\) 25.0000 1.36184 0.680918 0.732359i \(-0.261581\pi\)
0.680918 + 0.732359i \(0.261581\pi\)
\(338\) −9.00000 −0.489535
\(339\) −36.0000 −1.95525
\(340\) 0 0
\(341\) 0 0
\(342\) 5.00000 0.270369
\(343\) 8.00000 0.431959
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −15.0000 −0.805242 −0.402621 0.915367i \(-0.631901\pi\)
−0.402621 + 0.915367i \(0.631901\pi\)
\(348\) 12.0000 0.643268
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 5.00000 0.262794
\(363\) −22.0000 −1.15470
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −3.00000 −0.156386
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) −8.00000 −0.414781
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −12.0000 −0.618031
\(378\) −16.0000 −0.822951
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) −9.00000 −0.460480
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 7.00000 0.355830
\(388\) 4.00000 0.203069
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −24.0000 −1.21064
\(394\) −27.0000 −1.36024
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0000 1.25471 0.627357 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −13.0000 −0.651631
\(399\) 40.0000 2.00250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) 3.00000 0.149256
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 0 0
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 13.0000 0.640464
\(413\) 12.0000 0.590481
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 28.0000 1.37117
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 2.00000 0.0973585
\(423\) 6.00000 0.291730
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 8.00000 0.387147
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) −4.00000 −0.192450
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −15.0000 −0.717547
\(438\) −22.0000 −1.05120
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) 30.0000 1.41895
\(448\) 4.00000 0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) −8.00000 −0.375873
\(454\) 9.00000 0.422391
\(455\) 0 0
\(456\) 10.0000 0.468293
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −13.0000 −0.607450
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 21.0000 0.972806
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 3.00000 0.138086
\(473\) 0 0
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 9.00000 0.411650
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 14.0000 0.637683
\(483\) −24.0000 −1.09204
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 2.00000 0.0905357
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) −18.0000 −0.811503
\(493\) 0 0
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) −43.0000 −1.92494 −0.962472 0.271380i \(-0.912520\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) −27.0000 −1.20507
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 16.0000 0.709885
\(509\) 9.00000 0.398918 0.199459 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) −44.0000 −1.94645
\(512\) 1.00000 0.0441942
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) 0 0
\(516\) 14.0000 0.616316
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 20.0000 0.867110
\(533\) 18.0000 0.779667
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) −24.0000 −1.03568
\(538\) −21.0000 −0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) 0 0
\(546\) −16.0000 −0.684737
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 30.0000 1.27804
\(552\) −6.00000 −0.255377
\(553\) −4.00000 −0.170097
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −4.00000 −0.169334
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −45.0000 −1.89652 −0.948262 0.317489i \(-0.897160\pi\)
−0.948262 + 0.317489i \(0.897160\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) 19.0000 0.798630
\(567\) −44.0000 −1.84783
\(568\) −6.00000 −0.251754
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 2.00000 0.0836974 0.0418487 0.999124i \(-0.486675\pi\)
0.0418487 + 0.999124i \(0.486675\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) −36.0000 −1.50261
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 40.0000 1.66522 0.832611 0.553858i \(-0.186845\pi\)
0.832611 + 0.553858i \(0.186845\pi\)
\(578\) −17.0000 −0.707107
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) 3.00000 0.123929
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 18.0000 0.742307
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −54.0000 −2.22126
\(592\) −1.00000 −0.0410997
\(593\) −15.0000 −0.615976 −0.307988 0.951390i \(-0.599656\pi\)
−0.307988 + 0.951390i \(0.599656\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) −26.0000 −1.06411
\(598\) 6.00000 0.245358
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 28.0000 1.14119
\(603\) −2.00000 −0.0814463
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 5.00000 0.202777
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 26.0000 1.04587
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) 12.0000 0.481543
\(622\) 15.0000 0.601445
\(623\) −24.0000 −0.961540
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 4.00000 0.158986
\(634\) −33.0000 −1.31060
\(635\) 0 0
\(636\) −18.0000 −0.713746
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 36.0000 1.42081
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) 0 0
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 7.00000 0.274141
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −32.0000 −1.25130
\(655\) 0 0
\(656\) −9.00000 −0.351391
\(657\) −11.0000 −0.429151
\(658\) 24.0000 0.935617
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −18.0000 −0.696963
\(668\) 3.00000 0.116073
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 8.00000 0.308607
\(673\) 37.0000 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 25.0000 0.962964
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) −36.0000 −1.38257
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 5.00000 0.191180
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −26.0000 −0.991962
\(688\) 7.00000 0.266872
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −15.0000 −0.569392
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) −19.0000 −0.719161
\(699\) 42.0000 1.58859
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 8.00000 0.301941
\(703\) −5.00000 −0.188579
\(704\) 0 0
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 12.0000 0.451306
\(708\) 6.00000 0.225494
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) −6.00000 −0.224860
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 18.0000 0.672222
\(718\) 30.0000 1.11959
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 52.0000 1.93658
\(722\) 6.00000 0.223297
\(723\) 28.0000 1.04133
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) −22.0000 −0.816497
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −8.00000 −0.296500
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 4.00000 0.147844
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 0 0
\(738\) −9.00000 −0.331295
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) −36.0000 −1.32160
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) 72.0000 2.63082
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) 6.00000 0.218797
\(753\) −54.0000 −1.96787
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −16.0000 −0.581914
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 32.0000 1.15924
\(763\) −64.0000 −2.31696
\(764\) −9.00000 −0.325609
\(765\) 0 0
\(766\) 15.0000 0.541972
\(767\) −6.00000 −0.216647
\(768\) 2.00000 0.0721688
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 7.00000 0.251610
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) −8.00000 −0.286998
\(778\) −36.0000 −1.29066
\(779\) −45.0000 −1.61229
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −24.0000 −0.857690
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −27.0000 −0.961835
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) −72.0000 −2.56003
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 25.0000 0.887217
\(795\) 0 0
\(796\) −13.0000 −0.460773
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 40.0000 1.41598
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −42.0000 −1.47847
\(808\) 3.00000 0.105540
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 24.0000 0.842235
\(813\) −32.0000 −1.12229
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 35.0000 1.22449
\(818\) 20.0000 0.699284
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) −12.0000 −0.418548
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −3.00000 −0.104257
\(829\) 56.0000 1.94496 0.972480 0.232986i \(-0.0748495\pi\)
0.972480 + 0.232986i \(0.0748495\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 12.0000 0.414533
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 24.0000 0.826604
\(844\) 2.00000 0.0688428
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) −44.0000 −1.51186
\(848\) −9.00000 −0.309061
\(849\) 38.0000 1.30416
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) −12.0000 −0.411113
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) −72.0000 −2.45375
\(862\) 15.0000 0.510902
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) −34.0000 −1.15470
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −16.0000 −0.541828
\(873\) 4.00000 0.135379
\(874\) −15.0000 −0.507383
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) 25.0000 0.844190 0.422095 0.906552i \(-0.361295\pi\)
0.422095 + 0.906552i \(0.361295\pi\)
\(878\) −7.00000 −0.236239
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −39.0000 −1.31394 −0.656972 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 9.00000 0.303046
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 30.0000 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 30.0000 1.00391
\(894\) 30.0000 1.00335
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 12.0000 0.400668
\(898\) 6.00000 0.200223
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 56.0000 1.86356
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 9.00000 0.298675
\(909\) 3.00000 0.0995037
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 10.0000 0.331133
\(913\) 0 0
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −64.0000 −2.10887
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 13.0000 0.426976
\(928\) 6.00000 0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 45.0000 1.47482
\(932\) 21.0000 0.687878
\(933\) 30.0000 0.982156
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) −8.00000 −0.261209
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 26.0000 0.847126
\(943\) 27.0000 0.879241
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 0 0
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 18.0000 0.580042
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 31.0000 0.996893 0.498446 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −10.0000 −0.320750
\(973\) 56.0000 1.79528
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 14.0000 0.447671
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −18.0000 −0.574403
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 0 0
\(987\) 48.0000 1.52786
\(988\) −10.0000 −0.318142
\(989\) −21.0000 −0.667761
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) −4.00000 −0.127000
\(993\) 16.0000 0.507745
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −43.0000 −1.36114
\(999\) 4.00000 0.126554
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 1850.2.a.p.1.1 yes 1
5.2 odd 4 1850.2.b.f.149.2 2
5.3 odd 4 1850.2.b.f.149.1 2
5.4 even 2 1850.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.a.1.1 1 5.4 even 2
1850.2.a.p.1.1 yes 1 1.1 even 1 trivial
1850.2.b.f.149.1 2 5.3 odd 4
1850.2.b.f.149.2 2 5.2 odd 4