Properties

Label 1950.2.a.h.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -2.00000 q^{21} -4.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} -1.00000 q^{32} +4.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +6.00000 q^{38} +1.00000 q^{39} -2.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -8.00000 q^{51} +1.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +2.00000 q^{56} -6.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} -8.00000 q^{68} -6.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{83} -2.00000 q^{84} -4.00000 q^{86} -4.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} -2.00000 q^{91} -6.00000 q^{92} -1.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −4.00000 −0.852803
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.00000 0.973329
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 1.00000 0.138675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −6.00000 −0.794719
\(58\) 4.00000 0.525226
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −8.00000 −0.970143
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −8.00000 −0.911685
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −4.00000 −0.428845
\(88\) −4.00000 −0.426401
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 3.00000 0.303046
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 8.00000 0.792118
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −2.00000 −0.188982
\(113\) −20.0000 −1.88144 −0.940721 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 4.00000 0.348155
\(133\) 12.0000 1.04053
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 6.00000 0.510754
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 6.00000 0.486664
\(153\) −8.00000 −0.646762
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 2.00000 0.154303
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 4.00000 0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000 0.300658
\(178\) 14.0000 1.04934
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 2.00000 0.148250
\(183\) −10.0000 −0.739221
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) −32.0000 −2.34007
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −4.00000 −0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 16.0000 1.12576
\(203\) 8.00000 0.561490
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) −6.00000 −0.417029
\(208\) 1.00000 0.0693375
\(209\) −24.0000 −1.66011
\(210\) 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
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −2.00000 −0.134231
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −6.00000 −0.397360
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 4.00000 0.262613
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000 0.519656
\(238\) −16.0000 −1.03713
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) −2.00000 −0.125988
\(253\) −24.0000 −1.50887
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) −10.0000 −0.617802
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −12.0000 −0.735767
\(267\) −14.0000 −0.856786
\(268\) −12.0000 −0.733017
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −8.00000 −0.485071
\(273\) −2.00000 −0.121046
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 8.00000 0.468165
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) −16.0000 −0.919176
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −8.00000 −0.455842
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −10.0000 −0.560772
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −12.0000 −0.668734
\(323\) 48.0000 2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 12.0000 0.663602
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) −4.00000 −0.218870
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 20.0000 1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) −4.00000 −0.214423
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −4.00000 −0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 16.0000 0.846810
\(358\) 10.0000 0.528516
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −10.0000 −0.525588
\(363\) 5.00000 0.262432
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) −6.00000 −0.312772
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 32.0000 1.65468
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 2.00000 0.102869
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 4.00000 0.203331
\(388\) 16.0000 0.812277
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 3.00000 0.151523
\(393\) 10.0000 0.504433
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −16.0000 −0.796030
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 8.00000 0.396545
\(408\) 8.00000 0.396059
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 12.0000 0.591198
\(413\) −8.00000 −0.393654
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) −8.00000 −0.391762
\(418\) 24.0000 1.17388
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 20.0000 0.967868
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 36.0000 1.72211
\(438\) −8.00000 −0.382255
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000 0.380521
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −10.0000 −0.472984
\(448\) −2.00000 −0.0944911
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −20.0000 −0.940721
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −4.00000 −0.186908
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 8.00000 0.372194
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 1.00000 0.0462250
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −4.00000 −0.184115
\(473\) 16.0000 0.735681
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 10.0000 0.457869
\(478\) −16.0000 −0.731823
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 12.0000 0.546019
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 10.0000 0.452679
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 32.0000 1.44121
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 12.0000 0.537733
\(499\) 38.0000 1.70111 0.850557 0.525883i \(-0.176265\pi\)
0.850557 + 0.525883i \(0.176265\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) −6.00000 −0.267793
\(503\) −10.0000 −0.445878 −0.222939 0.974832i \(-0.571565\pi\)
−0.222939 + 0.974832i \(0.571565\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 1.00000 0.0444116
\(508\) −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
\(511\) −16.0000 −0.707798
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 4.00000 0.175750
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 4.00000 0.175075
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 12.0000 0.520266
\(533\) −2.00000 −0.0866296
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) −10.0000 −0.431532
\(538\) 24.0000 1.03471
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −16.0000 −0.687259
\(543\) 10.0000 0.429141
\(544\) 8.00000 0.342997
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 6.00000 0.255377
\(553\) −16.0000 −0.680389
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −32.0000 −1.35104
\(562\) 6.00000 0.253095
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −2.00000 −0.0839921
\(568\) 8.00000 0.335673
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) −47.0000 −1.95494
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) −16.0000 −0.663221
\(583\) 40.0000 1.65663
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 6.00000 0.245358
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 8.00000 0.326056
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 16.0000 0.649956
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 6.00000 0.243332
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) −8.00000 −0.323381
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) −12.0000 −0.482711
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) −24.0000 −0.962312
\(623\) 28.0000 1.12180
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) −24.0000 −0.958468
\(628\) −22.0000 −0.877896
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) −8.00000 −0.318223
\(633\) −20.0000 −0.794929
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) −3.00000 −0.118864
\(638\) 16.0000 0.633446
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 48.0000 1.86698 0.933492 0.358599i \(-0.116745\pi\)
0.933492 + 0.358599i \(0.116745\pi\)
\(662\) −10.0000 −0.388661
\(663\) −8.00000 −0.310694
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 24.0000 0.929284
\(668\) 4.00000 0.154765
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 2.00000 0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 20.0000 0.768095
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 4.00000 0.152610
\(688\) 4.00000 0.152499
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −22.0000 −0.836315
\(693\) −8.00000 −0.303895
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 16.0000 0.606043
\(698\) 28.0000 1.05982
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −12.0000 −0.452589
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000 1.20348
\(708\) 4.00000 0.150329
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 16.0000 0.597531
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) −17.0000 −0.632674
\(723\) 2.00000 0.0743808
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) −10.0000 −0.369611
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −48.0000 −1.76810
\(738\) 2.00000 0.0736210
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 20.0000 0.734223
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) −12.0000 −0.439057
\(748\) −32.0000 −1.17004
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 18.0000 0.653789
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 4.00000 0.144905
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 4.00000 0.144432
\(768\) 1.00000 0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 4.00000 0.143963
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) −4.00000 −0.143499
\(778\) −8.00000 −0.286814
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) −48.0000 −1.71648
\(783\) −4.00000 −0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 18.0000 0.641223
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) −4.00000 −0.142134
\(793\) −10.0000 −0.355110
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0000 0.354218 0.177109 0.984191i \(-0.443325\pi\)
0.177109 + 0.984191i \(0.443325\pi\)
\(798\) −12.0000 −0.424795
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) −30.0000 −1.05934
\(803\) 32.0000 1.12926
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 16.0000 0.562878
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 8.00000 0.280745
\(813\) 16.0000 0.561144
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −24.0000 −0.839654
\(818\) 26.0000 0.909069
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) −6.00000 −0.209274
\(823\) 52.0000 1.81261 0.906303 0.422628i \(-0.138892\pi\)
0.906303 + 0.422628i \(0.138892\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 1.00000 0.0346688
\(833\) 24.0000 0.831551
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) −10.0000 −0.345444
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 8.00000 0.275698
\(843\) −6.00000 −0.206651
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) −8.00000 −0.274075
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −4.00000 −0.136637 −0.0683187 0.997664i \(-0.521763\pi\)
−0.0683187 + 0.997664i \(0.521763\pi\)
\(858\) −4.00000 −0.136558
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 47.0000 1.59620
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) −12.0000 −0.406371
\(873\) 16.0000 0.541518
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 32.0000 1.07995
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000 0.101015
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) −6.00000 −0.200334
\(898\) 6.00000 0.200223
\(899\) 0 0
\(900\) 0 0
\(901\) −80.0000 −2.66519
\(902\) 8.00000 0.266371
\(903\) −8.00000 −0.266223
\(904\) 20.0000 0.665190
\(905\) 0 0
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 4.00000 0.132745
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) −6.00000 −0.198680
\(913\) −48.0000 −1.58857
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) −20.0000 −0.660458
\(918\) 8.00000 0.264039
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 6.00000 0.197599
\(923\) −8.00000 −0.263323
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 12.0000 0.394132
\(928\) 4.00000 0.131306
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) −24.0000 −0.786146
\(933\) 24.0000 0.785725
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) −24.0000 −0.783628
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 22.0000 0.716799
\(943\) 12.0000 0.390774
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 8.00000 0.259828
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) −16.0000 −0.518563
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −16.0000 −0.517207
\(958\) −32.0000 −1.03387
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −2.00000 −0.0644826
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) −5.00000 −0.160706
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) 26.0000 0.833094
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −16.0000 −0.511624
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) −42.0000 −1.34027
\(983\) 52.0000 1.65854 0.829271 0.558846i \(-0.188756\pi\)
0.829271 + 0.558846i \(0.188756\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) −38.0000 −1.20287
\(999\) 2.00000 0.0632772
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 1950.2.a.h.1.1 1
3.2 odd 2 5850.2.a.bi.1.1 1
5.2 odd 4 1950.2.e.f.1249.1 2
5.3 odd 4 1950.2.e.f.1249.2 2
5.4 even 2 390.2.a.e.1.1 1
15.2 even 4 5850.2.e.i.5149.2 2
15.8 even 4 5850.2.e.i.5149.1 2
15.14 odd 2 1170.2.a.e.1.1 1
20.19 odd 2 3120.2.a.o.1.1 1
60.59 even 2 9360.2.a.bh.1.1 1
65.34 odd 4 5070.2.b.e.1351.2 2
65.44 odd 4 5070.2.b.e.1351.1 2
65.64 even 2 5070.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 5.4 even 2
1170.2.a.e.1.1 1 15.14 odd 2
1950.2.a.h.1.1 1 1.1 even 1 trivial
1950.2.e.f.1249.1 2 5.2 odd 4
1950.2.e.f.1249.2 2 5.3 odd 4
3120.2.a.o.1.1 1 20.19 odd 2
5070.2.a.e.1.1 1 65.64 even 2
5070.2.b.e.1351.1 2 65.44 odd 4
5070.2.b.e.1351.2 2 65.34 odd 4
5850.2.a.bi.1.1 1 3.2 odd 2
5850.2.e.i.5149.1 2 15.8 even 4
5850.2.e.i.5149.2 2 15.2 even 4
9360.2.a.bh.1.1 1 60.59 even 2