Properties

Label 1815.2.a.d.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -1.00000 q^{30} +5.00000 q^{32} -2.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +7.00000 q^{64} +2.00000 q^{65} +12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{71} -3.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} -2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} +1.00000 q^{90} +8.00000 q^{94} -4.00000 q^{95} -5.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) −3.00000 −0.474342
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −3.00000 −0.353553
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −5.00000 −0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 2.00000 0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 3.00000 0.273861
\(121\) 0 0
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 2.00000 0.166091
\(146\) −10.0000 −0.827606
\(147\) 7.00000 0.577350
\(148\) 10.0000 0.821995
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 12.0000 0.973329
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 5.00000 0.395285
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −7.00000 −0.505181
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) 7.00000 0.500000
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −3.00000 −0.212132
\(201\) −12.0000 −0.846415
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −10.0000 −0.698430
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 8.00000 0.548151
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 1.00000 0.0645497
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −7.00000 −0.447214
\(246\) 10.0000 0.637577
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 2.00000 0.123797
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −8.00000 −0.476393
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 8.00000 0.474713
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 7.00000 0.408248
\(295\) −4.00000 −0.232889
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) 2.00000 0.114520
\(306\) −2.00000 −0.114332
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 6.00000 0.339683
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 14.0000 0.774202
\(328\) 30.0000 1.65647
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −9.00000 −0.489535
\(339\) −2.00000 −0.108625
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 2.00000 0.107211
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) −2.00000 −0.104542
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) −10.0000 −0.519875
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −24.0000 −1.23771
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 4.00000 0.205196
\(381\) −8.00000 −0.409852
\(382\) 16.0000 0.818631
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) 21.0000 1.06066
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −10.0000 −0.493865
\(411\) 6.00000 0.295958
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 10.0000 0.490290
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) 30.0000 1.45693
\(425\) −2.00000 −0.0970143
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) −4.00000 −0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) 8.00000 0.378811
\(447\) 22.0000 1.04056
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) −8.00000 −0.375873
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 6.00000 0.280362
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) −14.0000 −0.645086
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 16.0000 0.731823
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −5.00000 −0.228218
\(481\) −20.0000 −0.911922
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −6.00000 −0.271607
\(489\) 4.00000 0.180886
\(490\) −7.00000 −0.316228
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −10.0000 −0.450835
\(493\) −4.00000 −0.180151
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) −6.00000 −0.263117
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 2.00000 0.0875376
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −10.0000 −0.434372
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 6.00000 0.259645
\(535\) 12.0000 0.518805
\(536\) −36.0000 −1.55496
\(537\) −20.0000 −0.863064
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) −10.0000 −0.428746
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 10.0000 0.424476
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 8.00000 0.336861
\(565\) 2.00000 0.0841406
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) 24.0000 1.00702
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 4.00000 0.167542
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 2.00000 0.0826898
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −7.00000 −0.288675
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 6.00000 0.246807
\(592\) 10.0000 0.410997
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 3.00000 0.122474
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −20.0000 −0.811107
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 16.0000 0.647291
\(612\) 2.00000 0.0808452
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −28.0000 −1.12999
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 16.0000 0.643614
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) −2.00000 −0.0794301
\(635\) 8.00000 0.317470
\(636\) −10.0000 −0.396526
\(637\) −14.0000 −0.554700
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) −3.00000 −0.118585
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −12.0000 −0.473602
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 8.00000 0.314756
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 46.0000 1.80012 0.900060 0.435767i \(-0.143523\pi\)
0.900060 + 0.435767i \(0.143523\pi\)
\(654\) 14.0000 0.547443
\(655\) 12.0000 0.468879
\(656\) 10.0000 0.390434
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000 0.466393
\(663\) 4.00000 0.155347
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 0 0
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 30.0000 1.15642 0.578208 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) 14.0000 0.539260
\(675\) −1.00000 −0.0384900
\(676\) 9.00000 0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 4.00000 0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 20.0000 0.757554
\(698\) 2.00000 0.0757011
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −16.0000 −0.597531
\(718\) 24.0000 0.895672
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −14.0000 −0.520666
\(724\) 10.0000 0.371647
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −24.0000 −0.885856
\(735\) 7.00000 0.258199
\(736\) 0 0
\(737\) 0 0
\(738\) −10.0000 −0.368105
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 10.0000 0.367607
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −22.0000 −0.806018
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) 4.00000 0.145671
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) −8.00000 −0.288863
\(768\) 17.0000 0.613435
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 40.0000 1.43315
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 7.00000 0.250000
\(785\) 14.0000 0.499681
\(786\) −12.0000 −0.428026
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 6.00000 0.213741
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −2.00000 −0.0709773
\(795\) 10.0000 0.354663
\(796\) 8.00000 0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 18.0000 0.633238
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 1.00000 0.0351364
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) −2.00000 −0.0700140
\(817\) 16.0000 0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 6.00000 0.209274
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 48.0000 1.67216
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −12.0000 −0.416526
\(831\) 6.00000 0.208138
\(832\) 14.0000 0.485363
\(833\) 14.0000 0.485071
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 4.00000 0.138178
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) −6.00000 −0.206651
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 10.0000 0.343401
\(849\) −12.0000 −0.411839
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) −36.0000 −1.23045
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 0 0
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) −5.00000 −0.170103
\(865\) 18.0000 0.612018
\(866\) −14.0000 −0.475739
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) 42.0000 1.42230
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) −40.0000 −1.34993
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) −7.00000 −0.235702
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 4.00000 0.134535
\(885\) 4.00000 0.134459
\(886\) −12.0000 −0.403148
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −30.0000 −1.00673
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) 22.0000 0.735790
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 2.00000 0.0667409
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −10.0000 −0.332411
\(906\) −8.00000 −0.265782
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −20.0000 −0.663723
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) −2.00000 −0.0661180
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 18.0000 0.592798
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 24.0000 0.788689
\(927\) −16.0000 −0.525509
\(928\) 10.0000 0.328266
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) −6.00000 −0.196537
\(933\) 24.0000 0.785725
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 54.0000 1.76410 0.882052 0.471153i \(-0.156162\pi\)
0.882052 + 0.471153i \(0.156162\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) −8.00000 −0.260931
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) −14.0000 −0.456145
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) −10.0000 −0.323762
\(955\) 16.0000 0.517748
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −7.00000 −0.225924
\(961\) −31.0000 −1.00000
\(962\) −20.0000 −0.644826
\(963\) 12.0000 0.386695
\(964\) −14.0000 −0.450910
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 2.00000 0.0642161
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) 7.00000 0.223607
\(981\) −14.0000 −0.446986
\(982\) −28.0000 −0.893516
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −30.0000 −0.956365
\(985\) −6.00000 −0.191176
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) −12.0000 −0.380235
\(997\) −54.0000 −1.71020 −0.855099 0.518465i \(-0.826503\pi\)
−0.855099 + 0.518465i \(0.826503\pi\)
\(998\) 4.00000 0.126618
\(999\) 10.0000 0.316386
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 1815.2.a.d.1.1 1
3.2 odd 2 5445.2.a.c.1.1 1
5.4 even 2 9075.2.a.g.1.1 1
11.10 odd 2 15.2.a.a.1.1 1
33.32 even 2 45.2.a.a.1.1 1
44.43 even 2 240.2.a.d.1.1 1
55.32 even 4 75.2.b.b.49.1 2
55.43 even 4 75.2.b.b.49.2 2
55.54 odd 2 75.2.a.b.1.1 1
77.10 even 6 735.2.i.d.226.1 2
77.32 odd 6 735.2.i.e.226.1 2
77.54 even 6 735.2.i.d.361.1 2
77.65 odd 6 735.2.i.e.361.1 2
77.76 even 2 735.2.a.c.1.1 1
88.21 odd 2 960.2.a.l.1.1 1
88.43 even 2 960.2.a.a.1.1 1
99.32 even 6 405.2.e.c.136.1 2
99.43 odd 6 405.2.e.f.271.1 2
99.65 even 6 405.2.e.c.271.1 2
99.76 odd 6 405.2.e.f.136.1 2
132.131 odd 2 720.2.a.c.1.1 1
143.142 odd 2 2535.2.a.j.1.1 1
165.32 odd 4 225.2.b.b.199.2 2
165.98 odd 4 225.2.b.b.199.1 2
165.164 even 2 225.2.a.b.1.1 1
176.21 odd 4 3840.2.k.m.1921.2 2
176.43 even 4 3840.2.k.r.1921.1 2
176.109 odd 4 3840.2.k.m.1921.1 2
176.131 even 4 3840.2.k.r.1921.2 2
187.186 odd 2 4335.2.a.c.1.1 1
209.208 even 2 5415.2.a.j.1.1 1
220.43 odd 4 1200.2.f.h.49.2 2
220.87 odd 4 1200.2.f.h.49.1 2
220.219 even 2 1200.2.a.e.1.1 1
231.230 odd 2 2205.2.a.i.1.1 1
253.252 even 2 7935.2.a.d.1.1 1
264.131 odd 2 2880.2.a.bc.1.1 1
264.197 even 2 2880.2.a.y.1.1 1
385.384 even 2 3675.2.a.j.1.1 1
429.428 even 2 7605.2.a.g.1.1 1
440.43 odd 4 4800.2.f.c.3649.1 2
440.109 odd 2 4800.2.a.t.1.1 1
440.197 even 4 4800.2.f.bf.3649.1 2
440.219 even 2 4800.2.a.bz.1.1 1
440.307 odd 4 4800.2.f.c.3649.2 2
440.373 even 4 4800.2.f.bf.3649.2 2
660.263 even 4 3600.2.f.e.2449.1 2
660.527 even 4 3600.2.f.e.2449.2 2
660.659 odd 2 3600.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 11.10 odd 2
45.2.a.a.1.1 1 33.32 even 2
75.2.a.b.1.1 1 55.54 odd 2
75.2.b.b.49.1 2 55.32 even 4
75.2.b.b.49.2 2 55.43 even 4
225.2.a.b.1.1 1 165.164 even 2
225.2.b.b.199.1 2 165.98 odd 4
225.2.b.b.199.2 2 165.32 odd 4
240.2.a.d.1.1 1 44.43 even 2
405.2.e.c.136.1 2 99.32 even 6
405.2.e.c.271.1 2 99.65 even 6
405.2.e.f.136.1 2 99.76 odd 6
405.2.e.f.271.1 2 99.43 odd 6
720.2.a.c.1.1 1 132.131 odd 2
735.2.a.c.1.1 1 77.76 even 2
735.2.i.d.226.1 2 77.10 even 6
735.2.i.d.361.1 2 77.54 even 6
735.2.i.e.226.1 2 77.32 odd 6
735.2.i.e.361.1 2 77.65 odd 6
960.2.a.a.1.1 1 88.43 even 2
960.2.a.l.1.1 1 88.21 odd 2
1200.2.a.e.1.1 1 220.219 even 2
1200.2.f.h.49.1 2 220.87 odd 4
1200.2.f.h.49.2 2 220.43 odd 4
1815.2.a.d.1.1 1 1.1 even 1 trivial
2205.2.a.i.1.1 1 231.230 odd 2
2535.2.a.j.1.1 1 143.142 odd 2
2880.2.a.y.1.1 1 264.197 even 2
2880.2.a.bc.1.1 1 264.131 odd 2
3600.2.a.u.1.1 1 660.659 odd 2
3600.2.f.e.2449.1 2 660.263 even 4
3600.2.f.e.2449.2 2 660.527 even 4
3675.2.a.j.1.1 1 385.384 even 2
3840.2.k.m.1921.1 2 176.109 odd 4
3840.2.k.m.1921.2 2 176.21 odd 4
3840.2.k.r.1921.1 2 176.43 even 4
3840.2.k.r.1921.2 2 176.131 even 4
4335.2.a.c.1.1 1 187.186 odd 2
4800.2.a.t.1.1 1 440.109 odd 2
4800.2.a.bz.1.1 1 440.219 even 2
4800.2.f.c.3649.1 2 440.43 odd 4
4800.2.f.c.3649.2 2 440.307 odd 4
4800.2.f.bf.3649.1 2 440.197 even 4
4800.2.f.bf.3649.2 2 440.373 even 4
5415.2.a.j.1.1 1 209.208 even 2
5445.2.a.c.1.1 1 3.2 odd 2
7605.2.a.g.1.1 1 429.428 even 2
7935.2.a.d.1.1 1 253.252 even 2
9075.2.a.g.1.1 1 5.4 even 2