Properties

Label 158.2.a.c.1.1
Level $158$
Weight $2$
Character 158.1
Self dual yes
Analytic conductor $1.262$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 158 = 2 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 158.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.26163635194\)
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 \(\chi\) \(=\) 158.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -3.00000 q^{10} -2.00000 q^{11} -3.00000 q^{12} -5.00000 q^{13} -3.00000 q^{14} +9.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +6.00000 q^{18} -3.00000 q^{20} +9.00000 q^{21} -2.00000 q^{22} -2.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} -5.00000 q^{26} -9.00000 q^{27} -3.00000 q^{28} +6.00000 q^{29} +9.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} +9.00000 q^{35} +6.00000 q^{36} -10.0000 q^{37} +15.0000 q^{39} -3.00000 q^{40} +2.00000 q^{41} +9.00000 q^{42} +4.00000 q^{43} -2.00000 q^{44} -18.0000 q^{45} -2.00000 q^{46} -3.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} -18.0000 q^{51} -5.00000 q^{52} -12.0000 q^{53} -9.00000 q^{54} +6.00000 q^{55} -3.00000 q^{56} +6.00000 q^{58} -1.00000 q^{59} +9.00000 q^{60} +12.0000 q^{61} -10.0000 q^{62} -18.0000 q^{63} +1.00000 q^{64} +15.0000 q^{65} +6.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} +6.00000 q^{69} +9.00000 q^{70} -3.00000 q^{71} +6.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} -12.0000 q^{75} +6.00000 q^{77} +15.0000 q^{78} +1.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +14.0000 q^{83} +9.00000 q^{84} -18.0000 q^{85} +4.00000 q^{86} -18.0000 q^{87} -2.00000 q^{88} -7.00000 q^{89} -18.0000 q^{90} +15.0000 q^{91} -2.00000 q^{92} +30.0000 q^{93} -3.00000 q^{94} -3.00000 q^{96} -11.0000 q^{97} +2.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) −3.00000 −1.22474
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) −3.00000 −0.948683
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.00000 −0.866025
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −3.00000 −0.801784
\(15\) 9.00000 2.32379
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 6.00000 1.41421
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 9.00000 1.96396
\(22\) −2.00000 −0.426401
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −3.00000 −0.612372
\(25\) 4.00000 0.800000
\(26\) −5.00000 −0.980581
\(27\) −9.00000 −1.73205
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 9.00000 1.64317
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 9.00000 1.52128
\(36\) 6.00000 1.00000
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 15.0000 2.40192
\(40\) −3.00000 −0.474342
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 9.00000 1.38873
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) −18.0000 −2.68328
\(46\) −2.00000 −0.294884
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −3.00000 −0.433013
\(49\) 2.00000 0.285714
\(50\) 4.00000 0.565685
\(51\) −18.0000 −2.52050
\(52\) −5.00000 −0.693375
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −9.00000 −1.22474
\(55\) 6.00000 0.809040
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 9.00000 1.16190
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −10.0000 −1.27000
\(63\) −18.0000 −2.26779
\(64\) 1.00000 0.125000
\(65\) 15.0000 1.86052
\(66\) 6.00000 0.738549
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) 6.00000 0.722315
\(70\) 9.00000 1.07571
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 6.00000 0.707107
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −10.0000 −1.16248
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 15.0000 1.69842
\(79\) 1.00000 0.112509
\(80\) −3.00000 −0.335410
\(81\) 9.00000 1.00000
\(82\) 2.00000 0.220863
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 9.00000 0.981981
\(85\) −18.0000 −1.95237
\(86\) 4.00000 0.431331
\(87\) −18.0000 −1.92980
\(88\) −2.00000 −0.213201
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) −18.0000 −1.89737
\(91\) 15.0000 1.57243
\(92\) −2.00000 −0.208514
\(93\) 30.0000 3.11086
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) −11.0000 −1.11688 −0.558440 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 2.00000 0.202031
\(99\) −12.0000 −1.20605
\(100\) 4.00000 0.400000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) −18.0000 −1.78227
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) −5.00000 −0.490290
\(105\) −27.0000 −2.63493
\(106\) −12.0000 −1.16554
\(107\) −1.00000 −0.0966736 −0.0483368 0.998831i \(-0.515392\pi\)
−0.0483368 + 0.998831i \(0.515392\pi\)
\(108\) −9.00000 −0.866025
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 6.00000 0.572078
\(111\) 30.0000 2.84747
\(112\) −3.00000 −0.283473
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) 6.00000 0.557086
\(117\) −30.0000 −2.77350
\(118\) −1.00000 −0.0920575
\(119\) −18.0000 −1.65006
\(120\) 9.00000 0.821584
\(121\) −7.00000 −0.636364
\(122\) 12.0000 1.08643
\(123\) −6.00000 −0.541002
\(124\) −10.0000 −0.898027
\(125\) 3.00000 0.268328
\(126\) −18.0000 −1.60357
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) −12.0000 −1.05654
\(130\) 15.0000 1.31559
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 27.0000 2.32379
\(136\) 6.00000 0.514496
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 6.00000 0.510754
\(139\) −7.00000 −0.593732 −0.296866 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 9.00000 0.760639
\(141\) 9.00000 0.757937
\(142\) −3.00000 −0.251754
\(143\) 10.0000 0.836242
\(144\) 6.00000 0.500000
\(145\) −18.0000 −1.49482
\(146\) −6.00000 −0.496564
\(147\) −6.00000 −0.494872
\(148\) −10.0000 −0.821995
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) −12.0000 −0.979796
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 0 0
\(153\) 36.0000 2.91043
\(154\) 6.00000 0.483494
\(155\) 30.0000 2.40966
\(156\) 15.0000 1.20096
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 1.00000 0.0795557
\(159\) 36.0000 2.85499
\(160\) −3.00000 −0.237171
\(161\) 6.00000 0.472866
\(162\) 9.00000 0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) −18.0000 −1.40130
\(166\) 14.0000 1.08661
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 9.00000 0.694365
\(169\) 12.0000 0.923077
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) −18.0000 −1.36458
\(175\) −12.0000 −0.907115
\(176\) −2.00000 −0.150756
\(177\) 3.00000 0.225494
\(178\) −7.00000 −0.524672
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −18.0000 −1.34164
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 15.0000 1.11187
\(183\) −36.0000 −2.66120
\(184\) −2.00000 −0.147442
\(185\) 30.0000 2.20564
\(186\) 30.0000 2.19971
\(187\) −12.0000 −0.877527
\(188\) −3.00000 −0.218797
\(189\) 27.0000 1.96396
\(190\) 0 0
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) −3.00000 −0.216506
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −11.0000 −0.789754
\(195\) −45.0000 −3.22252
\(196\) 2.00000 0.142857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −12.0000 −0.852803
\(199\) 9.00000 0.637993 0.318997 0.947756i \(-0.396654\pi\)
0.318997 + 0.947756i \(0.396654\pi\)
\(200\) 4.00000 0.282843
\(201\) 24.0000 1.69283
\(202\) 3.00000 0.211079
\(203\) −18.0000 −1.26335
\(204\) −18.0000 −1.26025
\(205\) −6.00000 −0.419058
\(206\) 13.0000 0.905753
\(207\) −12.0000 −0.834058
\(208\) −5.00000 −0.346688
\(209\) 0 0
\(210\) −27.0000 −1.86318
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −12.0000 −0.824163
\(213\) 9.00000 0.616670
\(214\) −1.00000 −0.0683586
\(215\) −12.0000 −0.818393
\(216\) −9.00000 −0.612372
\(217\) 30.0000 2.03653
\(218\) 2.00000 0.135457
\(219\) 18.0000 1.21633
\(220\) 6.00000 0.404520
\(221\) −30.0000 −2.01802
\(222\) 30.0000 2.01347
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −3.00000 −0.200446
\(225\) 24.0000 1.60000
\(226\) 12.0000 0.798228
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 6.00000 0.395628
\(231\) −18.0000 −1.18431
\(232\) 6.00000 0.393919
\(233\) 2.00000 0.131024 0.0655122 0.997852i \(-0.479132\pi\)
0.0655122 + 0.997852i \(0.479132\pi\)
\(234\) −30.0000 −1.96116
\(235\) 9.00000 0.587095
\(236\) −1.00000 −0.0650945
\(237\) −3.00000 −0.194871
\(238\) −18.0000 −1.16677
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 9.00000 0.580948
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 12.0000 0.768221
\(245\) −6.00000 −0.383326
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −10.0000 −0.635001
\(249\) −42.0000 −2.66164
\(250\) 3.00000 0.189737
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) −18.0000 −1.13389
\(253\) 4.00000 0.251478
\(254\) 7.00000 0.439219
\(255\) 54.0000 3.38161
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −12.0000 −0.747087
\(259\) 30.0000 1.86411
\(260\) 15.0000 0.930261
\(261\) 36.0000 2.22834
\(262\) −6.00000 −0.370681
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 6.00000 0.369274
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 21.0000 1.28518
\(268\) −8.00000 −0.488678
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 27.0000 1.64317
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) −45.0000 −2.72352
\(274\) −4.00000 −0.241649
\(275\) −8.00000 −0.482418
\(276\) 6.00000 0.361158
\(277\) −13.0000 −0.781094 −0.390547 0.920583i \(-0.627714\pi\)
−0.390547 + 0.920583i \(0.627714\pi\)
\(278\) −7.00000 −0.419832
\(279\) −60.0000 −3.59211
\(280\) 9.00000 0.537853
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) 9.00000 0.535942
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) −6.00000 −0.354169
\(288\) 6.00000 0.353553
\(289\) 19.0000 1.11765
\(290\) −18.0000 −1.05700
\(291\) 33.0000 1.93449
\(292\) −6.00000 −0.351123
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) −6.00000 −0.349927
\(295\) 3.00000 0.174667
\(296\) −10.0000 −0.581238
\(297\) 18.0000 1.04447
\(298\) 14.0000 0.810998
\(299\) 10.0000 0.578315
\(300\) −12.0000 −0.692820
\(301\) −12.0000 −0.691669
\(302\) 14.0000 0.805609
\(303\) −9.00000 −0.517036
\(304\) 0 0
\(305\) −36.0000 −2.06135
\(306\) 36.0000 2.05798
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 6.00000 0.341882
\(309\) −39.0000 −2.21863
\(310\) 30.0000 1.70389
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 15.0000 0.849208
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −14.0000 −0.790066
\(315\) 54.0000 3.04256
\(316\) 1.00000 0.0562544
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 36.0000 2.01878
\(319\) −12.0000 −0.671871
\(320\) −3.00000 −0.167705
\(321\) 3.00000 0.167444
\(322\) 6.00000 0.334367
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) −20.0000 −1.10940
\(326\) 4.00000 0.221540
\(327\) −6.00000 −0.331801
\(328\) 2.00000 0.110432
\(329\) 9.00000 0.496186
\(330\) −18.0000 −0.990867
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 14.0000 0.768350
\(333\) −60.0000 −3.28798
\(334\) 18.0000 0.984916
\(335\) 24.0000 1.31126
\(336\) 9.00000 0.490990
\(337\) 31.0000 1.68868 0.844339 0.535810i \(-0.179994\pi\)
0.844339 + 0.535810i \(0.179994\pi\)
\(338\) 12.0000 0.652714
\(339\) −36.0000 −1.95525
\(340\) −18.0000 −0.976187
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 4.00000 0.215666
\(345\) −18.0000 −0.969087
\(346\) 4.00000 0.215041
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) −18.0000 −0.964901
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) −12.0000 −0.641427
\(351\) 45.0000 2.40192
\(352\) −2.00000 −0.106600
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 3.00000 0.159448
\(355\) 9.00000 0.477670
\(356\) −7.00000 −0.370999
\(357\) 54.0000 2.85798
\(358\) −4.00000 −0.211407
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) −18.0000 −0.948683
\(361\) −19.0000 −1.00000
\(362\) −18.0000 −0.946059
\(363\) 21.0000 1.10221
\(364\) 15.0000 0.786214
\(365\) 18.0000 0.942163
\(366\) −36.0000 −1.88175
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) −2.00000 −0.104257
\(369\) 12.0000 0.624695
\(370\) 30.0000 1.55963
\(371\) 36.0000 1.86903
\(372\) 30.0000 1.55543
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −12.0000 −0.620505
\(375\) −9.00000 −0.464758
\(376\) −3.00000 −0.154713
\(377\) −30.0000 −1.54508
\(378\) 27.0000 1.38873
\(379\) 27.0000 1.38690 0.693448 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(380\) 0 0
\(381\) −21.0000 −1.07586
\(382\) −21.0000 −1.07445
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −3.00000 −0.153093
\(385\) −18.0000 −0.917365
\(386\) −26.0000 −1.32337
\(387\) 24.0000 1.21999
\(388\) −11.0000 −0.558440
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) −45.0000 −2.27866
\(391\) −12.0000 −0.606866
\(392\) 2.00000 0.101015
\(393\) 18.0000 0.907980
\(394\) −6.00000 −0.302276
\(395\) −3.00000 −0.150946
\(396\) −12.0000 −0.603023
\(397\) 7.00000 0.351320 0.175660 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) 9.00000 0.451129
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 24.0000 1.19701
\(403\) 50.0000 2.49068
\(404\) 3.00000 0.149256
\(405\) −27.0000 −1.34164
\(406\) −18.0000 −0.893325
\(407\) 20.0000 0.991363
\(408\) −18.0000 −0.891133
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) −6.00000 −0.296319
\(411\) 12.0000 0.591916
\(412\) 13.0000 0.640464
\(413\) 3.00000 0.147620
\(414\) −12.0000 −0.589768
\(415\) −42.0000 −2.06170
\(416\) −5.00000 −0.245145
\(417\) 21.0000 1.02837
\(418\) 0 0
\(419\) 27.0000 1.31904 0.659518 0.751689i \(-0.270760\pi\)
0.659518 + 0.751689i \(0.270760\pi\)
\(420\) −27.0000 −1.31747
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 4.00000 0.194717
\(423\) −18.0000 −0.875190
\(424\) −12.0000 −0.582772
\(425\) 24.0000 1.16417
\(426\) 9.00000 0.436051
\(427\) −36.0000 −1.74216
\(428\) −1.00000 −0.0483368
\(429\) −30.0000 −1.44841
\(430\) −12.0000 −0.578691
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −9.00000 −0.433013
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 30.0000 1.44005
\(435\) 54.0000 2.58910
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 18.0000 0.860073
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 6.00000 0.286039
\(441\) 12.0000 0.571429
\(442\) −30.0000 −1.42695
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 30.0000 1.42374
\(445\) 21.0000 0.995495
\(446\) 2.00000 0.0947027
\(447\) −42.0000 −1.98653
\(448\) −3.00000 −0.141737
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 24.0000 1.13137
\(451\) −4.00000 −0.188353
\(452\) 12.0000 0.564433
\(453\) −42.0000 −1.97333
\(454\) 0 0
\(455\) −45.0000 −2.10963
\(456\) 0 0
\(457\) 23.0000 1.07589 0.537947 0.842978i \(-0.319200\pi\)
0.537947 + 0.842978i \(0.319200\pi\)
\(458\) −4.00000 −0.186908
\(459\) −54.0000 −2.52050
\(460\) 6.00000 0.279751
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −18.0000 −0.837436
\(463\) 7.00000 0.325318 0.162659 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 6.00000 0.278543
\(465\) −90.0000 −4.17365
\(466\) 2.00000 0.0926482
\(467\) 2.00000 0.0925490 0.0462745 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) −30.0000 −1.38675
\(469\) 24.0000 1.10822
\(470\) 9.00000 0.415139
\(471\) 42.0000 1.93526
\(472\) −1.00000 −0.0460287
\(473\) −8.00000 −0.367840
\(474\) −3.00000 −0.137795
\(475\) 0 0
\(476\) −18.0000 −0.825029
\(477\) −72.0000 −3.29665
\(478\) −24.0000 −1.09773
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 9.00000 0.410792
\(481\) 50.0000 2.27980
\(482\) −23.0000 −1.04762
\(483\) −18.0000 −0.819028
\(484\) −7.00000 −0.318182
\(485\) 33.0000 1.49845
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 12.0000 0.543214
\(489\) −12.0000 −0.542659
\(490\) −6.00000 −0.271052
\(491\) 23.0000 1.03798 0.518988 0.854782i \(-0.326309\pi\)
0.518988 + 0.854782i \(0.326309\pi\)
\(492\) −6.00000 −0.270501
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 36.0000 1.61808
\(496\) −10.0000 −0.449013
\(497\) 9.00000 0.403705
\(498\) −42.0000 −1.88207
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 3.00000 0.134164
\(501\) −54.0000 −2.41254
\(502\) −7.00000 −0.312425
\(503\) −15.0000 −0.668817 −0.334408 0.942428i \(-0.608537\pi\)
−0.334408 + 0.942428i \(0.608537\pi\)
\(504\) −18.0000 −0.801784
\(505\) −9.00000 −0.400495
\(506\) 4.00000 0.177822
\(507\) −36.0000 −1.59882
\(508\) 7.00000 0.310575
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 54.0000 2.39116
\(511\) 18.0000 0.796273
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −39.0000 −1.71855
\(516\) −12.0000 −0.528271
\(517\) 6.00000 0.263880
\(518\) 30.0000 1.31812
\(519\) −12.0000 −0.526742
\(520\) 15.0000 0.657794
\(521\) 44.0000 1.92767 0.963837 0.266491i \(-0.0858642\pi\)
0.963837 + 0.266491i \(0.0858642\pi\)
\(522\) 36.0000 1.57568
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) −6.00000 −0.262111
\(525\) 36.0000 1.57117
\(526\) 6.00000 0.261612
\(527\) −60.0000 −2.61364
\(528\) 6.00000 0.261116
\(529\) −19.0000 −0.826087
\(530\) 36.0000 1.56374
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 21.0000 0.908759
\(535\) 3.00000 0.129701
\(536\) −8.00000 −0.345547
\(537\) 12.0000 0.517838
\(538\) 5.00000 0.215565
\(539\) −4.00000 −0.172292
\(540\) 27.0000 1.16190
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 8.00000 0.343629
\(543\) 54.0000 2.31736
\(544\) 6.00000 0.257248
\(545\) −6.00000 −0.257012
\(546\) −45.0000 −1.92582
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −4.00000 −0.170872
\(549\) 72.0000 3.07289
\(550\) −8.00000 −0.341121
\(551\) 0 0
\(552\) 6.00000 0.255377
\(553\) −3.00000 −0.127573
\(554\) −13.0000 −0.552317
\(555\) −90.0000 −3.82029
\(556\) −7.00000 −0.296866
\(557\) 45.0000 1.90671 0.953356 0.301849i \(-0.0976040\pi\)
0.953356 + 0.301849i \(0.0976040\pi\)
\(558\) −60.0000 −2.54000
\(559\) −20.0000 −0.845910
\(560\) 9.00000 0.380319
\(561\) 36.0000 1.51992
\(562\) −9.00000 −0.379642
\(563\) −14.0000 −0.590030 −0.295015 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 9.00000 0.378968
\(565\) −36.0000 −1.51453
\(566\) −12.0000 −0.504398
\(567\) −27.0000 −1.13389
\(568\) −3.00000 −0.125877
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) −14.0000 −0.585882 −0.292941 0.956131i \(-0.594634\pi\)
−0.292941 + 0.956131i \(0.594634\pi\)
\(572\) 10.0000 0.418121
\(573\) 63.0000 2.63186
\(574\) −6.00000 −0.250435
\(575\) −8.00000 −0.333623
\(576\) 6.00000 0.250000
\(577\) 36.0000 1.49870 0.749350 0.662174i \(-0.230366\pi\)
0.749350 + 0.662174i \(0.230366\pi\)
\(578\) 19.0000 0.790296
\(579\) 78.0000 3.24157
\(580\) −18.0000 −0.747409
\(581\) −42.0000 −1.74245
\(582\) 33.0000 1.36789
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) 90.0000 3.72104
\(586\) −16.0000 −0.660954
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) −6.00000 −0.247436
\(589\) 0 0
\(590\) 3.00000 0.123508
\(591\) 18.0000 0.740421
\(592\) −10.0000 −0.410997
\(593\) −29.0000 −1.19089 −0.595444 0.803397i \(-0.703024\pi\)
−0.595444 + 0.803397i \(0.703024\pi\)
\(594\) 18.0000 0.738549
\(595\) 54.0000 2.21378
\(596\) 14.0000 0.573462
\(597\) −27.0000 −1.10504
\(598\) 10.0000 0.408930
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) −12.0000 −0.489898
\(601\) −36.0000 −1.46847 −0.734235 0.678895i \(-0.762459\pi\)
−0.734235 + 0.678895i \(0.762459\pi\)
\(602\) −12.0000 −0.489083
\(603\) −48.0000 −1.95471
\(604\) 14.0000 0.569652
\(605\) 21.0000 0.853771
\(606\) −9.00000 −0.365600
\(607\) −41.0000 −1.66414 −0.832069 0.554672i \(-0.812844\pi\)
−0.832069 + 0.554672i \(0.812844\pi\)
\(608\) 0 0
\(609\) 54.0000 2.18819
\(610\) −36.0000 −1.45760
\(611\) 15.0000 0.606835
\(612\) 36.0000 1.45521
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −12.0000 −0.484281
\(615\) 18.0000 0.725830
\(616\) 6.00000 0.241747
\(617\) 29.0000 1.16750 0.583748 0.811935i \(-0.301586\pi\)
0.583748 + 0.811935i \(0.301586\pi\)
\(618\) −39.0000 −1.56881
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 30.0000 1.20483
\(621\) 18.0000 0.722315
\(622\) −28.0000 −1.12270
\(623\) 21.0000 0.841347
\(624\) 15.0000 0.600481
\(625\) −29.0000 −1.16000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −60.0000 −2.39236
\(630\) 54.0000 2.15141
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 1.00000 0.0397779
\(633\) −12.0000 −0.476957
\(634\) −9.00000 −0.357436
\(635\) −21.0000 −0.833360
\(636\) 36.0000 1.42749
\(637\) −10.0000 −0.396214
\(638\) −12.0000 −0.475085
\(639\) −18.0000 −0.712069
\(640\) −3.00000 −0.118585
\(641\) −25.0000 −0.987441 −0.493720 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(642\) 3.00000 0.118401
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 6.00000 0.236433
\(645\) 36.0000 1.41750
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 9.00000 0.353553
\(649\) 2.00000 0.0785069
\(650\) −20.0000 −0.784465
\(651\) −90.0000 −3.52738
\(652\) 4.00000 0.156652
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) −6.00000 −0.234619
\(655\) 18.0000 0.703318
\(656\) 2.00000 0.0780869
\(657\) −36.0000 −1.40449
\(658\) 9.00000 0.350857
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) −18.0000 −0.700649
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −12.0000 −0.466393
\(663\) 90.0000 3.49531
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) −60.0000 −2.32495
\(667\) −12.0000 −0.464642
\(668\) 18.0000 0.696441
\(669\) −6.00000 −0.231973
\(670\) 24.0000 0.927201
\(671\) −24.0000 −0.926510
\(672\) 9.00000 0.347183
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 31.0000 1.19408
\(675\) −36.0000 −1.38564
\(676\) 12.0000 0.461538
\(677\) 17.0000 0.653363 0.326682 0.945134i \(-0.394070\pi\)
0.326682 + 0.945134i \(0.394070\pi\)
\(678\) −36.0000 −1.38257
\(679\) 33.0000 1.26642
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) 20.0000 0.765840
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 15.0000 0.572703
\(687\) 12.0000 0.457829
\(688\) 4.00000 0.152499
\(689\) 60.0000 2.28582
\(690\) −18.0000 −0.685248
\(691\) −5.00000 −0.190209 −0.0951045 0.995467i \(-0.530319\pi\)
−0.0951045 + 0.995467i \(0.530319\pi\)
\(692\) 4.00000 0.152057
\(693\) 36.0000 1.36753
\(694\) 14.0000 0.531433
\(695\) 21.0000 0.796575
\(696\) −18.0000 −0.682288
\(697\) 12.0000 0.454532
\(698\) 20.0000 0.757011
\(699\) −6.00000 −0.226941
\(700\) −12.0000 −0.453557
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 45.0000 1.69842
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) −27.0000 −1.01688
\(706\) −12.0000 −0.451626
\(707\) −9.00000 −0.338480
\(708\) 3.00000 0.112747
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 9.00000 0.337764
\(711\) 6.00000 0.225018
\(712\) −7.00000 −0.262336
\(713\) 20.0000 0.749006
\(714\) 54.0000 2.02090
\(715\) −30.0000 −1.12194
\(716\) −4.00000 −0.149487
\(717\) 72.0000 2.68889
\(718\) −25.0000 −0.932992
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) −18.0000 −0.670820
\(721\) −39.0000 −1.45244
\(722\) −19.0000 −0.707107
\(723\) 69.0000 2.56614
\(724\) −18.0000 −0.668965
\(725\) 24.0000 0.891338
\(726\) 21.0000 0.779383
\(727\) −42.0000 −1.55769 −0.778847 0.627214i \(-0.784195\pi\)
−0.778847 + 0.627214i \(0.784195\pi\)
\(728\) 15.0000 0.555937
\(729\) −27.0000 −1.00000
\(730\) 18.0000 0.666210
\(731\) 24.0000 0.887672
\(732\) −36.0000 −1.33060
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 18.0000 0.663940
\(736\) −2.00000 −0.0737210
\(737\) 16.0000 0.589368
\(738\) 12.0000 0.441726
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 30.0000 1.10282
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 30.0000 1.09985
\(745\) −42.0000 −1.53876
\(746\) 14.0000 0.512576
\(747\) 84.0000 3.07340
\(748\) −12.0000 −0.438763
\(749\) 3.00000 0.109618
\(750\) −9.00000 −0.328634
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) −3.00000 −0.109399
\(753\) 21.0000 0.765283
\(754\) −30.0000 −1.09254
\(755\) −42.0000 −1.52854
\(756\) 27.0000 0.981981
\(757\) 31.0000 1.12671 0.563357 0.826214i \(-0.309510\pi\)
0.563357 + 0.826214i \(0.309510\pi\)
\(758\) 27.0000 0.980684
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) −21.0000 −0.760750
\(763\) −6.00000 −0.217215
\(764\) −21.0000 −0.759753
\(765\) −108.000 −3.90475
\(766\) 6.00000 0.216789
\(767\) 5.00000 0.180540
\(768\) −3.00000 −0.108253
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) −18.0000 −0.648675
\(771\) 6.00000 0.216085
\(772\) −26.0000 −0.935760
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 24.0000 0.862662
\(775\) −40.0000 −1.43684
\(776\) −11.0000 −0.394877
\(777\) −90.0000 −3.22873
\(778\) −9.00000 −0.322666
\(779\) 0 0
\(780\) −45.0000 −1.61126
\(781\) 6.00000 0.214697
\(782\) −12.0000 −0.429119
\(783\) −54.0000 −1.92980
\(784\) 2.00000 0.0714286
\(785\) 42.0000 1.49904
\(786\) 18.0000 0.642039
\(787\) 6.00000 0.213877 0.106938 0.994266i \(-0.465895\pi\)
0.106938 + 0.994266i \(0.465895\pi\)
\(788\) −6.00000 −0.213741
\(789\) −18.0000 −0.640817
\(790\) −3.00000 −0.106735
\(791\) −36.0000 −1.28001
\(792\) −12.0000 −0.426401
\(793\) −60.0000 −2.13066
\(794\) 7.00000 0.248421
\(795\) −108.000 −3.83037
\(796\) 9.00000 0.318997
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 4.00000 0.141421
\(801\) −42.0000 −1.48400
\(802\) −30.0000 −1.05934
\(803\) 12.0000 0.423471
\(804\) 24.0000 0.846415
\(805\) −18.0000 −0.634417
\(806\) 50.0000 1.76117
\(807\) −15.0000 −0.528025
\(808\) 3.00000 0.105540
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) −27.0000 −0.948683
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) −18.0000 −0.631676
\(813\) −24.0000 −0.841717
\(814\) 20.0000 0.701000
\(815\) −12.0000 −0.420342
\(816\) −18.0000 −0.630126
\(817\) 0 0
\(818\) −4.00000 −0.139857
\(819\) 90.0000 3.14485
\(820\) −6.00000 −0.209529
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 12.0000 0.418548
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 13.0000 0.452876
\(825\) 24.0000 0.835573
\(826\) 3.00000 0.104383
\(827\) 37.0000 1.28662 0.643308 0.765607i \(-0.277561\pi\)
0.643308 + 0.765607i \(0.277561\pi\)
\(828\) −12.0000 −0.417029
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) −42.0000 −1.45784
\(831\) 39.0000 1.35290
\(832\) −5.00000 −0.173344
\(833\) 12.0000 0.415775
\(834\) 21.0000 0.727171
\(835\) −54.0000 −1.86875
\(836\) 0 0
\(837\) 90.0000 3.11086
\(838\) 27.0000 0.932700
\(839\) −52.0000 −1.79524 −0.897620 0.440771i \(-0.854705\pi\)
−0.897620 + 0.440771i \(0.854705\pi\)
\(840\) −27.0000 −0.931589
\(841\) 7.00000 0.241379
\(842\) −13.0000 −0.448010
\(843\) 27.0000 0.929929
\(844\) 4.00000 0.137686
\(845\) −36.0000 −1.23844
\(846\) −18.0000 −0.618853
\(847\) 21.0000 0.721569
\(848\) −12.0000 −0.412082
\(849\) 36.0000 1.23552
\(850\) 24.0000 0.823193
\(851\) 20.0000 0.685591
\(852\) 9.00000 0.308335
\(853\) 56.0000 1.91740 0.958702 0.284413i \(-0.0917988\pi\)
0.958702 + 0.284413i \(0.0917988\pi\)
\(854\) −36.0000 −1.23189
\(855\) 0 0
\(856\) −1.00000 −0.0341793
\(857\) −7.00000 −0.239115 −0.119558 0.992827i \(-0.538148\pi\)
−0.119558 + 0.992827i \(0.538148\pi\)
\(858\) −30.0000 −1.02418
\(859\) −41.0000 −1.39890 −0.699451 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) −12.0000 −0.409197
\(861\) 18.0000 0.613438
\(862\) 24.0000 0.817443
\(863\) 54.0000 1.83818 0.919091 0.394046i \(-0.128925\pi\)
0.919091 + 0.394046i \(0.128925\pi\)
\(864\) −9.00000 −0.306186
\(865\) −12.0000 −0.408012
\(866\) 21.0000 0.713609
\(867\) −57.0000 −1.93582
\(868\) 30.0000 1.01827
\(869\) −2.00000 −0.0678454
\(870\) 54.0000 1.83077
\(871\) 40.0000 1.35535
\(872\) 2.00000 0.0677285
\(873\) −66.0000 −2.23376
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 18.0000 0.608164
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −16.0000 −0.539974
\(879\) 48.0000 1.61900
\(880\) 6.00000 0.202260
\(881\) 20.0000 0.673817 0.336909 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) 12.0000 0.404061
\(883\) 37.0000 1.24515 0.622575 0.782560i \(-0.286087\pi\)
0.622575 + 0.782560i \(0.286087\pi\)
\(884\) −30.0000 −1.00901
\(885\) −9.00000 −0.302532
\(886\) −12.0000 −0.403148
\(887\) −10.0000 −0.335767 −0.167884 0.985807i \(-0.553693\pi\)
−0.167884 + 0.985807i \(0.553693\pi\)
\(888\) 30.0000 1.00673
\(889\) −21.0000 −0.704317
\(890\) 21.0000 0.703922
\(891\) −18.0000 −0.603023
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) −42.0000 −1.40469
\(895\) 12.0000 0.401116
\(896\) −3.00000 −0.100223
\(897\) −30.0000 −1.00167
\(898\) 22.0000 0.734150
\(899\) −60.0000 −2.00111
\(900\) 24.0000 0.800000
\(901\) −72.0000 −2.39867
\(902\) −4.00000 −0.133185
\(903\) 36.0000 1.19800
\(904\) 12.0000 0.399114
\(905\) 54.0000 1.79502
\(906\) −42.0000 −1.39536
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) −45.0000 −1.49174
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) 23.0000 0.760772
\(915\) 108.000 3.57037
\(916\) −4.00000 −0.132164
\(917\) 18.0000 0.594412
\(918\) −54.0000 −1.78227
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 6.00000 0.197814
\(921\) 36.0000 1.18624
\(922\) −2.00000 −0.0658665
\(923\) 15.0000 0.493731
\(924\) −18.0000 −0.592157
\(925\) −40.0000 −1.31519
\(926\) 7.00000 0.230034
\(927\) 78.0000 2.56186
\(928\) 6.00000 0.196960
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −90.0000 −2.95122
\(931\) 0 0
\(932\) 2.00000 0.0655122
\(933\) 84.0000 2.75004
\(934\) 2.00000 0.0654420
\(935\) 36.0000 1.17733
\(936\) −30.0000 −0.980581
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 24.0000 0.783628
\(939\) −42.0000 −1.37062
\(940\) 9.00000 0.293548
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 42.0000 1.36843
\(943\) −4.00000 −0.130258
\(944\) −1.00000 −0.0325472
\(945\) −81.0000 −2.63493
\(946\) −8.00000 −0.260102
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) −3.00000 −0.0974355
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) 27.0000 0.875535
\(952\) −18.0000 −0.583383
\(953\) −23.0000 −0.745043 −0.372522 0.928024i \(-0.621507\pi\)
−0.372522 + 0.928024i \(0.621507\pi\)
\(954\) −72.0000 −2.33109
\(955\) 63.0000 2.03863
\(956\) −24.0000 −0.776215
\(957\) 36.0000 1.16371
\(958\) 36.0000 1.16311
\(959\) 12.0000 0.387500
\(960\) 9.00000 0.290474
\(961\) 69.0000 2.22581
\(962\) 50.0000 1.61206
\(963\) −6.00000 −0.193347
\(964\) −23.0000 −0.740780
\(965\) 78.0000 2.51091
\(966\) −18.0000 −0.579141
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 33.0000 1.05957
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 0 0
\(973\) 21.0000 0.673229
\(974\) 2.00000 0.0640841
\(975\) 60.0000 1.92154
\(976\) 12.0000 0.384111
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −12.0000 −0.383718
\(979\) 14.0000 0.447442
\(980\) −6.00000 −0.191663
\(981\) 12.0000 0.383131
\(982\) 23.0000 0.733959
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −6.00000 −0.191273
\(985\) 18.0000 0.573528
\(986\) 36.0000 1.14647
\(987\) −27.0000 −0.859419
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 36.0000 1.14416
\(991\) −35.0000 −1.11181 −0.555906 0.831245i \(-0.687628\pi\)
−0.555906 + 0.831245i \(0.687628\pi\)
\(992\) −10.0000 −0.317500
\(993\) 36.0000 1.14243
\(994\) 9.00000 0.285463
\(995\) −27.0000 −0.855958
\(996\) −42.0000 −1.33082
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 10.0000 0.316544
\(999\) 90.0000 2.84747
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 158.2.a.c.1.1 1
3.2 odd 2 1422.2.a.c.1.1 1
4.3 odd 2 1264.2.a.h.1.1 1
5.4 even 2 3950.2.a.f.1.1 1
7.6 odd 2 7742.2.a.n.1.1 1
8.3 odd 2 5056.2.a.b.1.1 1
8.5 even 2 5056.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
158.2.a.c.1.1 1 1.1 even 1 trivial
1264.2.a.h.1.1 1 4.3 odd 2
1422.2.a.c.1.1 1 3.2 odd 2
3950.2.a.f.1.1 1 5.4 even 2
5056.2.a.b.1.1 1 8.3 odd 2
5056.2.a.v.1.1 1 8.5 even 2
7742.2.a.n.1.1 1 7.6 odd 2