Properties

Label 1805.2.a.b.1.1
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -4.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +4.00000 q^{16} +6.00000 q^{17} +2.00000 q^{20} -8.00000 q^{21} +1.00000 q^{25} -4.00000 q^{27} +8.00000 q^{28} +3.00000 q^{29} +7.00000 q^{31} +6.00000 q^{33} +4.00000 q^{35} -2.00000 q^{36} -8.00000 q^{37} -4.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +6.00000 q^{47} +8.00000 q^{48} +9.00000 q^{49} +12.0000 q^{51} +4.00000 q^{52} +6.00000 q^{53} -3.00000 q^{55} +15.0000 q^{59} +4.00000 q^{60} +5.00000 q^{61} -4.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} -2.00000 q^{67} -12.0000 q^{68} +3.00000 q^{71} +8.00000 q^{73} +2.00000 q^{75} -12.0000 q^{77} -5.00000 q^{79} -4.00000 q^{80} -11.0000 q^{81} +12.0000 q^{83} +16.0000 q^{84} -6.00000 q^{85} +6.00000 q^{87} +15.0000 q^{89} +8.00000 q^{91} +14.0000 q^{93} -8.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −4.00000 −1.15470
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 4.00000 1.00000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 8.00000 1.51186
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 8.00000 1.15470
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 4.00000 0.554700
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 4.00000 0.516398
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −12.0000 −1.45521
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) −4.00000 −0.447214
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 16.0000 1.74574
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 14.0000 1.45173
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) −2.00000 −0.200000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 8.00000 0.769800
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) −16.0000 −1.51865
\(112\) −16.0000 −1.51186
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 12.0000 1.08200
\(124\) −14.0000 −1.25724
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −12.0000 −1.04447
\(133\) 0 0
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −8.00000 −0.676123
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −6.00000 −0.501745
\(144\) 4.00000 0.333333
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 18.0000 1.48461
\(148\) 16.0000 1.31519
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −7.00000 −0.562254
\(156\) 8.00000 0.640513
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −12.0000 −0.937043
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 12.0000 0.904534
\(177\) 30.0000 2.25494
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) −12.0000 −0.875190
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) −16.0000 −1.15470
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) −18.0000 −1.28571
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −19.0000 −1.34687 −0.673437 0.739244i \(-0.735183\pi\)
−0.673437 + 0.739244i \(0.735183\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) −24.0000 −1.68034
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) −12.0000 −0.824163
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −28.0000 −1.90076
\(218\) 0 0
\(219\) 16.0000 1.08118
\(220\) 6.00000 0.404520
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) −24.0000 −1.57908
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −30.0000 −1.95283
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) −8.00000 −0.516398
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) −10.0000 −0.640184
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 8.00000 0.503953
\(253\) 0 0
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 16.0000 1.00000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 32.0000 1.98838
\(260\) −4.00000 −0.248069
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 30.0000 1.83597
\(268\) 4.00000 0.244339
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 24.0000 1.45521
\(273\) 16.0000 0.968364
\(274\) 0 0
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 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
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −16.0000 −0.936329
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 0 0
\(297\) −12.0000 −0.696311
\(298\) 0 0
\(299\) 0 0
\(300\) −4.00000 −0.230940
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) 30.0000 1.72345
\(304\) 0 0
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) 24.0000 1.36753
\(309\) 32.0000 1.82042
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 10.0000 0.562544
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 22.0000 1.22222
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) −22.0000 −1.21660
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −24.0000 −1.31717
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) −32.0000 −1.74574
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 12.0000 0.650791
\(341\) 21.0000 1.13721
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −12.0000 −0.643268
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) −30.0000 −1.59000
\(357\) −48.0000 −2.54043
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −4.00000 −0.209946
\(364\) −16.0000 −0.838628
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) −28.0000 −1.45173
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 37.0000 1.90056 0.950281 0.311393i \(-0.100796\pi\)
0.950281 + 0.311393i \(0.100796\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 16.0000 0.812277
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) 5.00000 0.251577
\(396\) −6.00000 −0.301511
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 33.0000 1.64794 0.823971 0.566632i \(-0.191754\pi\)
0.823971 + 0.566632i \(0.191754\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) −30.0000 −1.49256
\(405\) 11.0000 0.546594
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −23.0000 −1.13728 −0.568638 0.822588i \(-0.692530\pi\)
−0.568638 + 0.822588i \(0.692530\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) −32.0000 −1.57653
\(413\) −60.0000 −2.95241
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −32.0000 −1.56705
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) −16.0000 −0.780720
\(421\) 19.0000 0.926003 0.463002 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) −16.0000 −0.769800
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 22.0000 1.05361
\(437\) 0 0
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 32.0000 1.51865
\(445\) −15.0000 −0.711068
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 32.0000 1.51186
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 12.0000 0.564433
\(453\) −34.0000 −1.59746
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) −24.0000 −1.12022
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 12.0000 0.557086
\(465\) −14.0000 −0.649234
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 4.00000 0.184900
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 48.0000 2.20008
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −24.0000 −1.08200
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 28.0000 1.25724
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 4.00000 0.177471
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 16.0000 0.704361
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 45.0000 1.97149 0.985743 0.168259i \(-0.0538144\pi\)
0.985743 + 0.168259i \(0.0538144\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −24.0000 −1.04844
\(525\) −8.00000 −0.349149
\(526\) 0 0
\(527\) 42.0000 1.82955
\(528\) 24.0000 1.04447
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 15.0000 0.650945
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 27.0000 1.16297
\(540\) −8.00000 −0.344265
\(541\) −37.0000 −1.59075 −0.795377 0.606115i \(-0.792727\pi\)
−0.795377 + 0.606115i \(0.792727\pi\)
\(542\) 0 0
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) 11.0000 0.471188
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 24.0000 1.02523
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 32.0000 1.35710
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 16.0000 0.676123
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) −24.0000 −1.01058
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 44.0000 1.84783
\(568\) 0 0
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 12.0000 0.501745
\(573\) −30.0000 −1.25327
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 0 0
\(579\) 32.0000 1.32987
\(580\) 6.00000 0.249136
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −36.0000 −1.48461
\(589\) 0 0
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) −32.0000 −1.31519
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) −6.00000 −0.245770
\(597\) −38.0000 −1.55524
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 31.0000 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 34.0000 1.38344
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) −12.0000 −0.485071
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 14.0000 0.562254
\(621\) 0 0
\(622\) 0 0
\(623\) −60.0000 −2.40385
\(624\) −16.0000 −0.640513
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) 14.0000 0.556450
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) −24.0000 −0.951662
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 45.0000 1.76640
\(650\) 0 0
\(651\) −56.0000 −2.19481
\(652\) 20.0000 0.783260
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 24.0000 0.937043
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 12.0000 0.467099
\(661\) −5.00000 −0.194477 −0.0972387 0.995261i \(-0.531001\pi\)
−0.0972387 + 0.995261i \(0.531001\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 15.0000 0.579069
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 18.0000 0.692308
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −16.0000 −0.609994
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) 0 0
\(693\) −12.0000 −0.455842
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 8.00000 0.302372
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −24.0000 −0.904534
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) −60.0000 −2.25653
\(708\) −60.0000 −2.25494
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) −18.0000 −0.672692
\(717\) −6.00000 −0.224074
\(718\) 0 0
\(719\) 15.0000 0.559406 0.279703 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) −4.00000 −0.149071
\(721\) −64.0000 −2.38348
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 4.00000 0.148659
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −20.0000 −0.739221
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 0 0
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) −16.0000 −0.588172
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) −36.0000 −1.31629
\(749\) 0 0
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 24.0000 0.875190
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) 17.0000 0.618693
\(756\) −32.0000 −1.16383
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 44.0000 1.59291
\(764\) 30.0000 1.08536
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −30.0000 −1.08324
\(768\) 32.0000 1.15470
\(769\) 29.0000 1.04577 0.522883 0.852404i \(-0.324856\pi\)
0.522883 + 0.852404i \(0.324856\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) −32.0000 −1.15171
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) 0 0
\(777\) 64.0000 2.29599
\(778\) 0 0
\(779\) 0 0
\(780\) −8.00000 −0.286446
\(781\) 9.00000 0.322045
\(782\) 0 0
\(783\) −12.0000 −0.428845
\(784\) 36.0000 1.28571
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) 36.0000 1.28245
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 38.0000 1.34687
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 15.0000 0.529999
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 42.0000 1.47847
\(808\) 0 0
\(809\) −27.0000 −0.949269 −0.474635 0.880183i \(-0.657420\pi\)
−0.474635 + 0.880183i \(0.657420\pi\)
\(810\) 0 0
\(811\) 43.0000 1.50993 0.754967 0.655763i \(-0.227653\pi\)
0.754967 + 0.655763i \(0.227653\pi\)
\(812\) 24.0000 0.842235
\(813\) 22.0000 0.771574
\(814\) 0 0
\(815\) 10.0000 0.350285
\(816\) 48.0000 1.68034
\(817\) 0 0
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 12.0000 0.419058
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 16.0000 0.555034
\(832\) 16.0000 0.554700
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −28.0000 −0.967822
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) −14.0000 −0.481900
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 8.00000 0.274883
\(848\) 24.0000 0.824163
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) −12.0000 −0.411113
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) −7.00000 −0.238837 −0.119418 0.992844i \(-0.538103\pi\)
−0.119418 + 0.992844i \(0.538103\pi\)
\(860\) −8.00000 −0.272798
\(861\) −48.0000 −1.63584
\(862\) 0 0
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 38.0000 1.29055
\(868\) 56.0000 1.90076
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) −32.0000 −1.08118
\(877\) 52.0000 1.75592 0.877958 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) 0 0
\(879\) −48.0000 −1.61900
\(880\) −12.0000 −0.404520
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 24.0000 0.807207
\(885\) −30.0000 −1.00844
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.0000 0.700389
\(900\) −2.00000 −0.0666667
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 32.0000 1.06489
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −48.0000 −1.59294
\(909\) 15.0000 0.497519
\(910\) 0 0
\(911\) −21.0000 −0.695761 −0.347881 0.937539i \(-0.613099\pi\)
−0.347881 + 0.937539i \(0.613099\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) −10.0000 −0.330590
\(916\) 14.0000 0.462573
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 48.0000 1.57908
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −12.0000 −0.393073
\(933\) 48.0000 1.57145
\(934\) 0 0
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 12.0000 0.391397
\(941\) 9.00000 0.293392 0.146696 0.989182i \(-0.453136\pi\)
0.146696 + 0.989182i \(0.453136\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 60.0000 1.95283
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 20.0000 0.649570
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −48.0000 −1.55651
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) 15.0000 0.485389
\(956\) 6.00000 0.194054
\(957\) 18.0000 0.581857
\(958\) 0 0
\(959\) 48.0000 1.55000
\(960\) 16.0000 0.516398
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 20.0000 0.641500
\(973\) 64.0000 2.05175
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 20.0000 0.640184
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 18.0000 0.574989
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) −48.0000 −1.52786
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 19.0000 0.602340
\(996\) −48.0000 −1.52094
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 0 0
\(999\) 32.0000 1.01244
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 1805.2.a.b.1.1 1
5.4 even 2 9025.2.a.e.1.1 1
19.8 odd 6 95.2.e.a.26.1 yes 2
19.12 odd 6 95.2.e.a.11.1 2
19.18 odd 2 1805.2.a.a.1.1 1
57.8 even 6 855.2.k.b.406.1 2
57.50 even 6 855.2.k.b.676.1 2
76.27 even 6 1520.2.q.c.881.1 2
76.31 even 6 1520.2.q.c.961.1 2
95.8 even 12 475.2.j.a.349.1 4
95.12 even 12 475.2.j.a.49.1 4
95.27 even 12 475.2.j.a.349.2 4
95.69 odd 6 475.2.e.b.201.1 2
95.84 odd 6 475.2.e.b.26.1 2
95.88 even 12 475.2.j.a.49.2 4
95.94 odd 2 9025.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.a.11.1 2 19.12 odd 6
95.2.e.a.26.1 yes 2 19.8 odd 6
475.2.e.b.26.1 2 95.84 odd 6
475.2.e.b.201.1 2 95.69 odd 6
475.2.j.a.49.1 4 95.12 even 12
475.2.j.a.49.2 4 95.88 even 12
475.2.j.a.349.1 4 95.8 even 12
475.2.j.a.349.2 4 95.27 even 12
855.2.k.b.406.1 2 57.8 even 6
855.2.k.b.676.1 2 57.50 even 6
1520.2.q.c.881.1 2 76.27 even 6
1520.2.q.c.961.1 2 76.31 even 6
1805.2.a.a.1.1 1 19.18 odd 2
1805.2.a.b.1.1 1 1.1 even 1 trivial
9025.2.a.e.1.1 1 5.4 even 2
9025.2.a.g.1.1 1 95.94 odd 2