Properties

Label 1950.2.a.m.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 $1$

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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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{21} +3.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -7.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +8.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -1.00000 q^{42} +12.0000 q^{43} -3.00000 q^{44} +4.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} -1.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -8.00000 q^{57} +7.00000 q^{58} -9.00000 q^{59} +5.00000 q^{61} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +11.0000 q^{67} +1.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{74} -8.00000 q^{76} -3.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -7.00000 q^{83} +1.00000 q^{84} -12.0000 q^{86} -7.00000 q^{87} +3.00000 q^{88} -8.00000 q^{89} -1.00000 q^{91} -4.00000 q^{92} +1.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} -6.00000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 3.00000 0.639602
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 8.00000 1.29777
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −1.00000 −0.154303
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) −1.00000 −0.138675
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −8.00000 −1.05963
\(58\) 7.00000 0.919145
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −1.00000 −0.127000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 1.00000 0.121268
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −3.00000 −0.341882
\(78\) 1.00000 0.113228
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −7.00000 −0.750479
\(88\) 3.00000 0.319801
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −4.00000 −0.417029
\(93\) 1.00000 0.103695
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 1.00000 0.0995037 0.0497519 0.998762i \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) −1.00000 −0.0924500
\(118\) 9.00000 0.828517
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −5.00000 −0.452679
\(123\) −6.00000 −0.541002
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −10.0000 −0.873704 −0.436852 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) −3.00000 −0.261116
\(133\) −8.00000 −0.693688
\(134\) −11.0000 −0.950255
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 4.00000 0.340503
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −8.00000 −0.671345
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 8.00000 0.648886
\(153\) 1.00000 0.0808452
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 15.0000 1.19713 0.598565 0.801074i \(-0.295738\pi\)
0.598565 + 0.801074i \(0.295738\pi\)
\(158\) 8.00000 0.636446
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 12.0000 0.914991
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) 7.00000 0.530669
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −9.00000 −0.676481
\(178\) 8.00000 0.599625
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 1.00000 0.0741249
\(183\) 5.00000 0.369611
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −3.00000 −0.219382
\(188\) −3.00000 −0.218797
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 3.00000 0.213201
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) −1.00000 −0.0703598
\(203\) −7.00000 −0.491304
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −4.00000 −0.278019
\(208\) −1.00000 −0.0693375
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −5.00000 −0.343401
\(213\) 8.00000 0.548151
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 1.00000 0.0678844
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) 4.00000 0.268462
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) −8.00000 −0.529813
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 7.00000 0.459573
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) −8.00000 −0.519656
\(238\) −1.00000 −0.0648204
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 8.00000 0.509028
\(248\) −1.00000 −0.0635001
\(249\) −7.00000 −0.443607
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 1.00000 0.0629941
\(253\) 12.0000 0.754434
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) −12.0000 −0.747087
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −7.00000 −0.433289
\(262\) 10.0000 0.617802
\(263\) −30.0000 −1.84988 −0.924940 0.380114i \(-0.875885\pi\)
−0.924940 + 0.380114i \(0.875885\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) −8.00000 −0.489592
\(268\) 11.0000 0.671932
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 1.00000 0.0606339
\(273\) −1.00000 −0.0605228
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 14.0000 0.839664
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 3.00000 0.178647
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) −6.00000 −0.354169
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −3.00000 −0.174078
\(298\) −10.0000 −0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −13.0000 −0.748066
\(303\) 1.00000 0.0574485
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −3.00000 −0.170941
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 1.00000 0.0566139
\(313\) −3.00000 −0.169570 −0.0847850 0.996399i \(-0.527020\pi\)
−0.0847850 + 0.996399i \(0.527020\pi\)
\(314\) −15.0000 −0.846499
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −20.0000 −1.12331 −0.561656 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(318\) 5.00000 0.280386
\(319\) 21.0000 1.17577
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 4.00000 0.222911
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −7.00000 −0.384175
\(333\) −4.00000 −0.219199
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) 8.00000 0.432590
\(343\) −13.0000 −0.701934
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −5.00000 −0.268802
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) −7.00000 −0.375239
\(349\) −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
\(352\) 3.00000 0.159901
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 9.00000 0.478345
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 1.00000 0.0529256
\(358\) −10.0000 −0.528516
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −1.00000 −0.0525588
\(363\) −2.00000 −0.104973
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) −5.00000 −0.261354
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) −4.00000 −0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −5.00000 −0.259587
\(372\) 1.00000 0.0518476
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 3.00000 0.155126
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 7.00000 0.360518
\(378\) −1.00000 −0.0514344
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 10.0000 0.511645
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 12.0000 0.609994
\(388\) −6.00000 −0.304604
\(389\) 34.0000 1.72387 0.861934 0.507020i \(-0.169253\pi\)
0.861934 + 0.507020i \(0.169253\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 6.00000 0.303046
\(393\) −10.0000 −0.504433
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 26.0000 1.30326
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) −11.0000 −0.548630
\(403\) −1.00000 −0.0498135
\(404\) 1.00000 0.0497519
\(405\) 0 0
\(406\) 7.00000 0.347404
\(407\) 12.0000 0.594818
\(408\) −1.00000 −0.0495074
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) −9.00000 −0.442861
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −14.0000 −0.685583
\(418\) −24.0000 −1.17388
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −12.0000 −0.584844 −0.292422 0.956289i \(-0.594461\pi\)
−0.292422 + 0.956289i \(0.594461\pi\)
\(422\) −12.0000 −0.584151
\(423\) −3.00000 −0.145865
\(424\) 5.00000 0.242821
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 5.00000 0.241967
\(428\) 18.0000 0.870063
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 32.0000 1.53077
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 1.00000 0.0475651
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 10.0000 0.472984
\(448\) 1.00000 0.0472456
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 2.00000 0.0940721
\(453\) 13.0000 0.610793
\(454\) −3.00000 −0.140797
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −14.0000 −0.654177
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 3.00000 0.139573
\(463\) 11.0000 0.511213 0.255607 0.966781i \(-0.417725\pi\)
0.255607 + 0.966781i \(0.417725\pi\)
\(464\) −7.00000 −0.324967
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 11.0000 0.507933
\(470\) 0 0
\(471\) 15.0000 0.691164
\(472\) 9.00000 0.414259
\(473\) −36.0000 −1.65528
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) −5.00000 −0.228934
\(478\) −3.00000 −0.137217
\(479\) 23.0000 1.05090 0.525448 0.850825i \(-0.323898\pi\)
0.525448 + 0.850825i \(0.323898\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) −4.00000 −0.182006
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 25.0000 1.13286 0.566429 0.824110i \(-0.308325\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) −5.00000 −0.226339
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) −6.00000 −0.270501
\(493\) −7.00000 −0.315264
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 8.00000 0.358849
\(498\) 7.00000 0.313678
\(499\) 25.0000 1.11915 0.559577 0.828778i \(-0.310964\pi\)
0.559577 + 0.828778i \(0.310964\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) −18.0000 −0.803379
\(503\) 22.0000 0.980932 0.490466 0.871460i \(-0.336827\pi\)
0.490466 + 0.871460i \(0.336827\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 1.00000 0.0444116
\(508\) −12.0000 −0.532414
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −8.00000 −0.353209
\(514\) −7.00000 −0.308757
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 9.00000 0.395820
\(518\) 4.00000 0.175750
\(519\) 5.00000 0.219476
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 7.00000 0.306382
\(523\) 2.00000 0.0874539 0.0437269 0.999044i \(-0.486077\pi\)
0.0437269 + 0.999044i \(0.486077\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 1.00000 0.0435607
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −9.00000 −0.390567
\(532\) −8.00000 −0.346844
\(533\) 6.00000 0.259889
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) −11.0000 −0.475128
\(537\) 10.0000 0.431532
\(538\) −1.00000 −0.0431131
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −1.00000 −0.0429537
\(543\) 1.00000 0.0429141
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 1.00000 0.0427960
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) −12.0000 −0.512615
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 56.0000 2.38568
\(552\) 4.00000 0.170251
\(553\) −8.00000 −0.340195
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 18.0000 0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 3.00000 0.125436
\(573\) −10.0000 −0.417756
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 16.0000 0.665512
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −7.00000 −0.290409
\(582\) 6.00000 0.248708
\(583\) 15.0000 0.621237
\(584\) 0 0
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 39.0000 1.60970 0.804851 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) −6.00000 −0.247436
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −4.00000 −0.164399
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −26.0000 −1.06411
\(598\) −4.00000 −0.163572
\(599\) −26.0000 −1.06233 −0.531166 0.847268i \(-0.678246\pi\)
−0.531166 + 0.847268i \(0.678246\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) −12.0000 −0.489083
\(603\) 11.0000 0.447955
\(604\) 13.0000 0.528962
\(605\) 0 0
\(606\) −1.00000 −0.0406222
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 8.00000 0.324443
\(609\) −7.00000 −0.283654
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 1.00000 0.0404226
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) −14.0000 −0.563163
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −18.0000 −0.721734
\(623\) −8.00000 −0.320513
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 3.00000 0.119904
\(627\) 24.0000 0.958468
\(628\) 15.0000 0.598565
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 8.00000 0.318223
\(633\) 12.0000 0.476957
\(634\) 20.0000 0.794301
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) 6.00000 0.237729
\(638\) −21.0000 −0.831398
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) −18.0000 −0.710403
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) 4.00000 0.156652
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 3.00000 0.116952
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) −28.0000 −1.08825
\(663\) −1.00000 −0.0388368
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 28.0000 1.08416
\(668\) −20.0000 −0.773823
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) −15.0000 −0.579069
\(672\) −1.00000 −0.0385758
\(673\) 43.0000 1.65753 0.828764 0.559598i \(-0.189045\pi\)
0.828764 + 0.559598i \(0.189045\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −2.00000 −0.0768095
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 3.00000 0.114876
\(683\) −27.0000 −1.03313 −0.516563 0.856249i \(-0.672789\pi\)
−0.516563 + 0.856249i \(0.672789\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 14.0000 0.534133
\(688\) 12.0000 0.457496
\(689\) 5.00000 0.190485
\(690\) 0 0
\(691\) −43.0000 −1.63580 −0.817899 0.575362i \(-0.804861\pi\)
−0.817899 + 0.575362i \(0.804861\pi\)
\(692\) 5.00000 0.190071
\(693\) −3.00000 −0.113961
\(694\) −14.0000 −0.531433
\(695\) 0 0
\(696\) 7.00000 0.265334
\(697\) −6.00000 −0.227266
\(698\) 28.0000 1.05982
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 1.00000 0.0377426
\(703\) 32.0000 1.20690
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 1.00000 0.0376089
\(708\) −9.00000 −0.338241
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 8.00000 0.299813
\(713\) −4.00000 −0.149801
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 3.00000 0.112037
\(718\) 15.0000 0.559795
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) −45.0000 −1.67473
\(723\) −10.0000 −0.371904
\(724\) 1.00000 0.0371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 1.00000 0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 5.00000 0.184805
\(733\) −28.0000 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −33.0000 −1.21557
\(738\) 6.00000 0.220863
\(739\) −47.0000 −1.72892 −0.864461 0.502699i \(-0.832340\pi\)
−0.864461 + 0.502699i \(0.832340\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 5.00000 0.183556
\(743\) 29.0000 1.06391 0.531953 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 13.0000 0.475964
\(747\) −7.00000 −0.256117
\(748\) −3.00000 −0.109691
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −3.00000 −0.109399
\(753\) 18.0000 0.655956
\(754\) −7.00000 −0.254925
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −43.0000 −1.56286 −0.781431 0.623992i \(-0.785510\pi\)
−0.781431 + 0.623992i \(0.785510\pi\)
\(758\) −11.0000 −0.399538
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 12.0000 0.434714
\(763\) 2.00000 0.0724049
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) 9.00000 0.324971
\(768\) 1.00000 0.0360844
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) −6.00000 −0.215945
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) −4.00000 −0.143499
\(778\) −34.0000 −1.21896
\(779\) 48.0000 1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 4.00000 0.143040
\(783\) −7.00000 −0.250160
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 10.0000 0.356688
\(787\) −7.00000 −0.249523 −0.124762 0.992187i \(-0.539817\pi\)
−0.124762 + 0.992187i \(0.539817\pi\)
\(788\) −6.00000 −0.213741
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 3.00000 0.106600
\(793\) −5.00000 −0.177555
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −26.0000 −0.921546
\(797\) 11.0000 0.389640 0.194820 0.980839i \(-0.437588\pi\)
0.194820 + 0.980839i \(0.437588\pi\)
\(798\) 8.00000 0.283197
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 28.0000 0.988714
\(803\) 0 0
\(804\) 11.0000 0.387940
\(805\) 0 0
\(806\) 1.00000 0.0352235
\(807\) 1.00000 0.0352017
\(808\) −1.00000 −0.0351799
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) −7.00000 −0.245652
\(813\) 1.00000 0.0350715
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) −96.0000 −3.35861
\(818\) 2.00000 0.0699284
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −52.0000 −1.81481 −0.907406 0.420255i \(-0.861941\pi\)
−0.907406 + 0.420255i \(0.861941\pi\)
\(822\) 12.0000 0.418548
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 9.00000 0.313150
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) −4.00000 −0.139010
\(829\) 47.0000 1.63238 0.816189 0.577785i \(-0.196083\pi\)
0.816189 + 0.577785i \(0.196083\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) −1.00000 −0.0346688
\(833\) −6.00000 −0.207888
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 1.00000 0.0345651
\(838\) −6.00000 −0.207267
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 12.0000 0.413547
\(843\) −18.0000 −0.619953
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −2.00000 −0.0687208
\(848\) −5.00000 −0.171701
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 8.00000 0.274075
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) −5.00000 −0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) −3.00000 −0.102418
\(859\) −24.0000 −0.818869 −0.409435 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) −12.0000 −0.408722
\(863\) 9.00000 0.306364 0.153182 0.988198i \(-0.451048\pi\)
0.153182 + 0.988198i \(0.451048\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −10.0000 −0.339814
\(867\) −16.0000 −0.543388
\(868\) 1.00000 0.0339422
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −11.0000 −0.372721
\(872\) −2.00000 −0.0677285
\(873\) −6.00000 −0.203069
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 8.00000 0.269987
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) 6.00000 0.202031
\(883\) −48.0000 −1.61533 −0.807664 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) −1.00000 −0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 4.00000 0.134231
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −24.0000 −0.803579
\(893\) 24.0000 0.803129
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 4.00000 0.133556
\(898\) −36.0000 −1.20134
\(899\) −7.00000 −0.233463
\(900\) 0 0
\(901\) −5.00000 −0.166574
\(902\) −18.0000 −0.599334
\(903\) 12.0000 0.399335
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −13.0000 −0.431896
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 3.00000 0.0995585
\(909\) 1.00000 0.0331679
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) −8.00000 −0.264906
\(913\) 21.0000 0.694999
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −10.0000 −0.330229
\(918\) −1.00000 −0.0330049
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −26.0000 −0.856264
\(923\) −8.00000 −0.263323
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) −11.0000 −0.361482
\(927\) 14.0000 0.459820
\(928\) 7.00000 0.229786
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 48.0000 1.57314
\(932\) −26.0000 −0.851658
\(933\) 18.0000 0.589294
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) −11.0000 −0.359163
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) −15.0000 −0.488726
\(943\) 24.0000 0.781548
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 36.0000 1.17046
\(947\) −47.0000 −1.52729 −0.763647 0.645634i \(-0.776593\pi\)
−0.763647 + 0.645634i \(0.776593\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) −20.0000 −0.648544
\(952\) −1.00000 −0.0324102
\(953\) −57.0000 −1.84641 −0.923206 0.384307i \(-0.874441\pi\)
−0.923206 + 0.384307i \(0.874441\pi\)
\(954\) 5.00000 0.161881
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) 21.0000 0.678834
\(958\) −23.0000 −0.743096
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −4.00000 −0.128965
\(963\) 18.0000 0.580042
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) −7.00000 −0.225105 −0.112552 0.993646i \(-0.535903\pi\)
−0.112552 + 0.993646i \(0.535903\pi\)
\(968\) 2.00000 0.0642824
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 1.00000 0.0320750
\(973\) −14.0000 −0.448819
\(974\) −25.0000 −0.801052
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −52.0000 −1.66363 −0.831814 0.555055i \(-0.812697\pi\)
−0.831814 + 0.555055i \(0.812697\pi\)
\(978\) −4.00000 −0.127906
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −18.0000 −0.574403
\(983\) 11.0000 0.350846 0.175423 0.984493i \(-0.443871\pi\)
0.175423 + 0.984493i \(0.443871\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 7.00000 0.222925
\(987\) −3.00000 −0.0954911
\(988\) 8.00000 0.254514
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 28.0000 0.888553
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −7.00000 −0.221803
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) −25.0000 −0.791361
\(999\) −4.00000 −0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.a.m.1.1 1
3.2 odd 2 5850.2.a.bt.1.1 1
5.2 odd 4 1950.2.e.b.1249.1 2
5.3 odd 4 1950.2.e.b.1249.2 2
5.4 even 2 1950.2.a.q.1.1 yes 1
15.2 even 4 5850.2.e.x.5149.2 2
15.8 even 4 5850.2.e.x.5149.1 2
15.14 odd 2 5850.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.m.1.1 1 1.1 even 1 trivial
1950.2.a.q.1.1 yes 1 5.4 even 2
1950.2.e.b.1249.1 2 5.2 odd 4
1950.2.e.b.1249.2 2 5.3 odd 4
5850.2.a.j.1.1 1 15.14 odd 2
5850.2.a.bt.1.1 1 3.2 odd 2
5850.2.e.x.5149.1 2 15.8 even 4
5850.2.e.x.5149.2 2 15.2 even 4