Properties

Label 1950.2.a.x.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +7.00000 q^{19} -4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +5.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +9.00000 q^{37} +7.00000 q^{38} +1.00000 q^{39} -5.00000 q^{41} -4.00000 q^{42} -10.0000 q^{43} +4.00000 q^{44} +4.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -4.00000 q^{51} +1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} +7.00000 q^{57} +5.00000 q^{58} -6.00000 q^{59} +4.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -7.00000 q^{67} -4.00000 q^{68} +4.00000 q^{69} -15.0000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +9.00000 q^{74} +7.00000 q^{76} -16.0000 q^{77} +1.00000 q^{78} +7.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -10.0000 q^{86} +5.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -4.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +3.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 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
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 9.00000 1.47959 0.739795 0.672832i \(-0.234922\pi\)
0.739795 + 0.672832i \(0.234922\pi\)
\(38\) 7.00000 1.13555
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) −4.00000 −0.617213
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 1.00000 0.138675
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 7.00000 0.927173
\(58\) 5.00000 0.656532
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000 0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −4.00000 −0.485071
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 9.00000 1.04623
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) −16.0000 −1.82337
\(78\) 1.00000 0.113228
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 5.00000 0.536056
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −4.00000 −0.396059
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 5.00000 0.483368 0.241684 0.970355i \(-0.422300\pi\)
0.241684 + 0.970355i \(0.422300\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) −4.00000 −0.377964
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 7.00000 0.655610
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 1.00000 0.0924500
\(118\) −6.00000 −0.552345
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 4.00000 0.362143
\(123\) −5.00000 −0.450835
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 17.0000 1.48530 0.742648 0.669681i \(-0.233569\pi\)
0.742648 + 0.669681i \(0.233569\pi\)
\(132\) 4.00000 0.348155
\(133\) −28.0000 −2.42791
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −19.0000 −1.62328 −0.811640 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) 4.00000 0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −15.0000 −1.25877
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 9.00000 0.742307
\(148\) 9.00000 0.739795
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 7.00000 0.567775
\(153\) −4.00000 −0.323381
\(154\) −16.0000 −1.28932
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 7.00000 0.556890
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) −10.0000 −0.762493
\(173\) 13.0000 0.988372 0.494186 0.869356i \(-0.335466\pi\)
0.494186 + 0.869356i \(0.335466\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −6.00000 −0.450988
\(178\) 14.0000 1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −4.00000 −0.297318 −0.148659 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) −4.00000 −0.296500
\(183\) 4.00000 0.295689
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −16.0000 −1.17004
\(188\) 3.00000 0.218797
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 1.00000 0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −16.0000 −1.13995 −0.569976 0.821661i \(-0.693048\pi\)
−0.569976 + 0.821661i \(0.693048\pi\)
\(198\) 4.00000 0.284268
\(199\) −3.00000 −0.212664 −0.106332 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) −10.0000 −0.703598
\(203\) −20.0000 −1.40372
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) 1.00000 0.0693375
\(209\) 28.0000 1.93680
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 9.00000 0.618123
\(213\) −15.0000 −1.02778
\(214\) 5.00000 0.341793
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) −11.0000 −0.745014
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 9.00000 0.604040
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) 7.00000 0.463586
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 5.00000 0.328266
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 7.00000 0.454699
\(238\) 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
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) 7.00000 0.445399
\(248\) 4.00000 0.254000
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) −4.00000 −0.251976
\(253\) 16.0000 1.00591
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) −10.0000 −0.622573
\(259\) −36.0000 −2.23693
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 17.0000 1.05026
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −28.0000 −1.71679
\(267\) 14.0000 0.856786
\(268\) −7.00000 −0.427593
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −4.00000 −0.242536
\(273\) −4.00000 −0.242091
\(274\) −19.0000 −1.14783
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −4.00000 −0.239904
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 17.0000 1.01413 0.507067 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) 3.00000 0.178647
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −15.0000 −0.890086
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 20.0000 1.18056
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 12.0000 0.702247
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 9.00000 0.523114
\(297\) 4.00000 0.232104
\(298\) −4.00000 −0.231714
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 40.0000 2.30556
\(302\) −4.00000 −0.230174
\(303\) −10.0000 −0.574485
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 21.0000 1.19853 0.599267 0.800549i \(-0.295459\pi\)
0.599267 + 0.800549i \(0.295459\pi\)
\(308\) −16.0000 −0.911685
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 1.00000 0.0566139
\(313\) −23.0000 −1.30004 −0.650018 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 9.00000 0.504695
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) −16.0000 −0.891645
\(323\) −28.0000 −1.55796
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −24.0000 −1.32924
\(327\) −11.0000 −0.608301
\(328\) −5.00000 −0.276079
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 6.00000 0.329293
\(333\) 9.00000 0.493197
\(334\) 9.00000 0.492458
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 1.00000 0.0543928
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 7.00000 0.378517
\(343\) −8.00000 −0.431959
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) 11.0000 0.590511 0.295255 0.955418i \(-0.404595\pi\)
0.295255 + 0.955418i \(0.404595\pi\)
\(348\) 5.00000 0.268028
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.00000 0.213201
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 16.0000 0.846810
\(358\) −4.00000 −0.211407
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −4.00000 −0.210235
\(363\) 5.00000 0.262432
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 4.00000 0.208514
\(369\) −5.00000 −0.260290
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 5.00000 0.257513
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 10.0000 0.511645
\(383\) 5.00000 0.255488 0.127744 0.991807i \(-0.459226\pi\)
0.127744 + 0.991807i \(0.459226\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −12.0000 −0.610784
\(387\) −10.0000 −0.508329
\(388\) −16.0000 −0.812277
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 9.00000 0.454569
\(393\) 17.0000 0.857537
\(394\) −16.0000 −0.806068
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) −15.0000 −0.752828 −0.376414 0.926451i \(-0.622843\pi\)
−0.376414 + 0.926451i \(0.622843\pi\)
\(398\) −3.00000 −0.150376
\(399\) −28.0000 −1.40175
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −7.00000 −0.349128
\(403\) 4.00000 0.199254
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 36.0000 1.78445
\(408\) −4.00000 −0.198030
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 0 0
\(411\) −19.0000 −0.937201
\(412\) 8.00000 0.394132
\(413\) 24.0000 1.18096
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) −4.00000 −0.195881
\(418\) 28.0000 1.36952
\(419\) 23.0000 1.12362 0.561812 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −16.0000 −0.778868
\(423\) 3.00000 0.145865
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) −15.0000 −0.726752
\(427\) −16.0000 −0.774294
\(428\) 5.00000 0.241684
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 19.0000 0.915198 0.457599 0.889159i \(-0.348710\pi\)
0.457599 + 0.889159i \(0.348710\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) 28.0000 1.33942
\(438\) 12.0000 0.573382
\(439\) −7.00000 −0.334092 −0.167046 0.985949i \(-0.553423\pi\)
−0.167046 + 0.985949i \(0.553423\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −4.00000 −0.190261
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) 9.00000 0.427121
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) −4.00000 −0.189194
\(448\) −4.00000 −0.188982
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) −14.0000 −0.658505
\(453\) −4.00000 −0.187936
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 7.00000 0.327805
\(457\) 4.00000 0.187112 0.0935561 0.995614i \(-0.470177\pi\)
0.0935561 + 0.995614i \(0.470177\pi\)
\(458\) −17.0000 −0.794358
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) −16.0000 −0.744387
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 1.00000 0.0462250
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) −6.00000 −0.276172
\(473\) −40.0000 −1.83920
\(474\) 7.00000 0.321521
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 9.00000 0.412082
\(478\) 16.0000 0.731823
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) 9.00000 0.410365
\(482\) 8.00000 0.364390
\(483\) −16.0000 −0.728025
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 4.00000 0.181071
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) −5.00000 −0.225417
\(493\) −20.0000 −0.900755
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 60.0000 2.69137
\(498\) 6.00000 0.268866
\(499\) −23.0000 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) −5.00000 −0.223161
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 1.00000 0.0444116
\(508\) −5.00000 −0.221839
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) −48.0000 −2.12339
\(512\) 1.00000 0.0441942
\(513\) 7.00000 0.309058
\(514\) 8.00000 0.352865
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 12.0000 0.527759
\(518\) −36.0000 −1.58175
\(519\) 13.0000 0.570637
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 5.00000 0.218844
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 17.0000 0.742648
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) −16.0000 −0.696971
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −28.0000 −1.21395
\(533\) −5.00000 −0.216574
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) −4.00000 −0.172613
\(538\) 5.00000 0.215565
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 32.0000 1.37452
\(543\) −4.00000 −0.171656
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 6.00000 0.256541 0.128271 0.991739i \(-0.459057\pi\)
0.128271 + 0.991739i \(0.459057\pi\)
\(548\) −19.0000 −0.811640
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 35.0000 1.49105
\(552\) 4.00000 0.170251
\(553\) −28.0000 −1.19068
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 4.00000 0.169334
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 17.0000 0.717102
\(563\) 39.0000 1.64365 0.821827 0.569737i \(-0.192955\pi\)
0.821827 + 0.569737i \(0.192955\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) −4.00000 −0.167984
\(568\) −15.0000 −0.629386
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 4.00000 0.167248
\(573\) 10.0000 0.417756
\(574\) 20.0000 0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) −16.0000 −0.663221
\(583\) 36.0000 1.49097
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 9.00000 0.371154
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 9.00000 0.369898
\(593\) 29.0000 1.19089 0.595444 0.803397i \(-0.296976\pi\)
0.595444 + 0.803397i \(0.296976\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) −3.00000 −0.122782
\(598\) 4.00000 0.163572
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 40.0000 1.63028
\(603\) −7.00000 −0.285062
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −3.00000 −0.121766 −0.0608831 0.998145i \(-0.519392\pi\)
−0.0608831 + 0.998145i \(0.519392\pi\)
\(608\) 7.00000 0.283887
\(609\) −20.0000 −0.810441
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) −4.00000 −0.161690
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 21.0000 0.847491
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) 7.00000 0.281809 0.140905 0.990023i \(-0.454999\pi\)
0.140905 + 0.990023i \(0.454999\pi\)
\(618\) 8.00000 0.321807
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 2.00000 0.0801927
\(623\) −56.0000 −2.24359
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −23.0000 −0.919265
\(627\) 28.0000 1.11821
\(628\) −14.0000 −0.558661
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 7.00000 0.278445
\(633\) −16.0000 −0.635943
\(634\) −28.0000 −1.11202
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 9.00000 0.356593
\(638\) 20.0000 0.791808
\(639\) −15.0000 −0.593391
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 5.00000 0.197334
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −28.0000 −1.10165
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) −24.0000 −0.939913
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −11.0000 −0.430134
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) 12.0000 0.468165
\(658\) −12.0000 −0.467809
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) −28.0000 −1.08825
\(663\) −4.00000 −0.155347
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 9.00000 0.348743
\(667\) 20.0000 0.774403
\(668\) 9.00000 0.348220
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) −4.00000 −0.154303
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −14.0000 −0.538064 −0.269032 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) −14.0000 −0.537667
\(679\) 64.0000 2.45609
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 16.0000 0.612672
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 7.00000 0.267652
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) −17.0000 −0.648590
\(688\) −10.0000 −0.381246
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 3.00000 0.114125 0.0570627 0.998371i \(-0.481827\pi\)
0.0570627 + 0.998371i \(0.481827\pi\)
\(692\) 13.0000 0.494186
\(693\) −16.0000 −0.607790
\(694\) 11.0000 0.417554
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) 20.0000 0.757554
\(698\) −34.0000 −1.28692
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 1.00000 0.0377426
\(703\) 63.0000 2.37609
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) 40.0000 1.50435
\(708\) −6.00000 −0.225494
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 7.00000 0.262521
\(712\) 14.0000 0.524672
\(713\) 16.0000 0.599205
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 16.0000 0.597531
\(718\) −15.0000 −0.559795
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 30.0000 1.11648
\(723\) 8.00000 0.297523
\(724\) −4.00000 −0.148659
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 4.00000 0.147844
\(733\) −3.00000 −0.110808 −0.0554038 0.998464i \(-0.517645\pi\)
−0.0554038 + 0.998464i \(0.517645\pi\)
\(734\) 7.00000 0.258375
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −28.0000 −1.03139
\(738\) −5.00000 −0.184053
\(739\) −37.0000 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) 7.00000 0.257151
\(742\) −36.0000 −1.32160
\(743\) −21.0000 −0.770415 −0.385208 0.922830i \(-0.625870\pi\)
−0.385208 + 0.922830i \(0.625870\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 6.00000 0.219529
\(748\) −16.0000 −0.585018
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 3.00000 0.109399
\(753\) −5.00000 −0.182210
\(754\) 5.00000 0.182089
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) −20.0000 −0.726433
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 35.0000 1.26875 0.634375 0.773026i \(-0.281258\pi\)
0.634375 + 0.773026i \(0.281258\pi\)
\(762\) −5.00000 −0.181131
\(763\) 44.0000 1.59291
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) 5.00000 0.180657
\(767\) −6.00000 −0.216647
\(768\) 1.00000 0.0360844
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) −12.0000 −0.431889
\(773\) −8.00000 −0.287740 −0.143870 0.989597i \(-0.545955\pi\)
−0.143870 + 0.989597i \(0.545955\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) −36.0000 −1.29149
\(778\) 5.00000 0.179259
\(779\) −35.0000 −1.25401
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) −16.0000 −0.572159
\(783\) 5.00000 0.178685
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 17.0000 0.606370
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −16.0000 −0.569976
\(789\) 30.0000 1.06803
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) 4.00000 0.142134
\(793\) 4.00000 0.142044
\(794\) −15.0000 −0.532330
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) −28.0000 −0.991189
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) −30.0000 −1.05934
\(803\) 48.0000 1.69388
\(804\) −7.00000 −0.246871
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 5.00000 0.176008
\(808\) −10.0000 −0.351799
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −20.0000 −0.701862
\(813\) 32.0000 1.12229
\(814\) 36.0000 1.26180
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) −70.0000 −2.44899
\(818\) −24.0000 −0.839140
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −19.0000 −0.662701
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 4.00000 0.139010
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 1.00000 0.0346688
\(833\) −36.0000 −1.24733
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 28.0000 0.968400
\(837\) 4.00000 0.138260
\(838\) 23.0000 0.794522
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 14.0000 0.482472
\(843\) 17.0000 0.585511
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) −20.0000 −0.687208
\(848\) 9.00000 0.309061
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) −15.0000 −0.513892
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) 5.00000 0.170896
\(857\) −50.0000 −1.70797 −0.853984 0.520300i \(-0.825820\pi\)
−0.853984 + 0.520300i \(0.825820\pi\)
\(858\) 4.00000 0.136558
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 19.0000 0.647143
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −19.0000 −0.645646
\(867\) −1.00000 −0.0339618
\(868\) −16.0000 −0.543075
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) −11.0000 −0.372507
\(873\) −16.0000 −0.541518
\(874\) 28.0000 0.947114
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 43.0000 1.45201 0.726003 0.687691i \(-0.241376\pi\)
0.726003 + 0.687691i \(0.241376\pi\)
\(878\) −7.00000 −0.236239
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 9.00000 0.303046
\(883\) 40.0000 1.34611 0.673054 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 9.00000 0.302020
\(889\) 20.0000 0.670778
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) −2.00000 −0.0669650
\(893\) 21.0000 0.702738
\(894\) −4.00000 −0.133780
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 4.00000 0.133556
\(898\) −27.0000 −0.901002
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −20.0000 −0.665927
\(903\) 40.0000 1.33112
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) 50.0000 1.66022 0.830111 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) 6.00000 0.199117
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 7.00000 0.231793
\(913\) 24.0000 0.794284
\(914\) 4.00000 0.132308
\(915\) 0 0
\(916\) −17.0000 −0.561696
\(917\) −68.0000 −2.24556
\(918\) −4.00000 −0.132020
\(919\) 29.0000 0.956622 0.478311 0.878191i \(-0.341249\pi\)
0.478311 + 0.878191i \(0.341249\pi\)
\(920\) 0 0
\(921\) 21.0000 0.691974
\(922\) −24.0000 −0.790398
\(923\) −15.0000 −0.493731
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 8.00000 0.262754
\(928\) 5.00000 0.164133
\(929\) 11.0000 0.360898 0.180449 0.983584i \(-0.442245\pi\)
0.180449 + 0.983584i \(0.442245\pi\)
\(930\) 0 0
\(931\) 63.0000 2.06474
\(932\) 8.00000 0.262049
\(933\) 2.00000 0.0654771
\(934\) 3.00000 0.0981630
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 28.0000 0.914232
\(939\) −23.0000 −0.750577
\(940\) 0 0
\(941\) 52.0000 1.69515 0.847576 0.530674i \(-0.178061\pi\)
0.847576 + 0.530674i \(0.178061\pi\)
\(942\) −14.0000 −0.456145
\(943\) −20.0000 −0.651290
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 7.00000 0.227349
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −28.0000 −0.907962
\(952\) 16.0000 0.518563
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 20.0000 0.646508
\(958\) 15.0000 0.484628
\(959\) 76.0000 2.45417
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 9.00000 0.290172
\(963\) 5.00000 0.161123
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 5.00000 0.160706
\(969\) −28.0000 −0.899490
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −24.0000 −0.767435
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) −40.0000 −1.27645
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −5.00000 −0.159394
\(985\) 0 0
\(986\) −20.0000 −0.636930
\(987\) −12.0000 −0.381964
\(988\) 7.00000 0.222700
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 4.00000 0.127000
\(993\) −28.0000 −0.888553
\(994\) 60.0000 1.90308
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −23.0000 −0.728052
\(999\) 9.00000 0.284747
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.x.1.1 yes 1
3.2 odd 2 5850.2.a.a.1.1 1
5.2 odd 4 1950.2.e.n.1249.2 2
5.3 odd 4 1950.2.e.n.1249.1 2
5.4 even 2 1950.2.a.e.1.1 1
15.2 even 4 5850.2.e.c.5149.1 2
15.8 even 4 5850.2.e.c.5149.2 2
15.14 odd 2 5850.2.a.by.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.e.1.1 1 5.4 even 2
1950.2.a.x.1.1 yes 1 1.1 even 1 trivial
1950.2.e.n.1249.1 2 5.3 odd 4
1950.2.e.n.1249.2 2 5.2 odd 4
5850.2.a.a.1.1 1 3.2 odd 2
5850.2.a.by.1.1 1 15.14 odd 2
5850.2.e.c.5149.1 2 15.2 even 4
5850.2.e.c.5149.2 2 15.8 even 4