Properties

Label 1078.2.a.f.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} +3.00000 q^{12} +1.00000 q^{13} +12.0000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{18} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} -2.00000 q^{23} -3.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} +9.00000 q^{27} +1.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +2.00000 q^{34} +6.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} +3.00000 q^{39} -4.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +24.0000 q^{45} +2.00000 q^{46} -2.00000 q^{47} +3.00000 q^{48} -11.0000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -12.0000 q^{53} -9.00000 q^{54} -4.00000 q^{55} -18.0000 q^{57} -1.00000 q^{58} -9.00000 q^{59} +12.0000 q^{60} +5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +3.00000 q^{66} -9.00000 q^{67} -2.00000 q^{68} -6.00000 q^{69} +4.00000 q^{71} -6.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +33.0000 q^{75} -6.00000 q^{76} -3.00000 q^{78} -15.0000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -8.00000 q^{85} -4.00000 q^{86} +3.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -24.0000 q^{90} -2.00000 q^{92} -12.0000 q^{93} +2.00000 q^{94} -24.0000 q^{95} -3.00000 q^{96} +5.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −3.00000 −1.22474
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) −4.00000 −1.26491
\(11\) −1.00000 −0.301511
\(12\) 3.00000 0.866025
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 12.0000 3.09839
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −6.00000 −1.41421
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −3.00000 −0.612372
\(25\) 11.0000 2.20000
\(26\) −1.00000 −0.196116
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) −12.0000 −2.19089
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 3.00000 0.480384
\(40\) −4.00000 −0.632456
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 24.0000 3.57771
\(46\) 2.00000 0.294884
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) −11.0000 −1.55563
\(51\) −6.00000 −0.840168
\(52\) 1.00000 0.138675
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −9.00000 −1.22474
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) −1.00000 −0.131306
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 12.0000 1.54919
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 3.00000 0.369274
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −2.00000 −0.242536
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −6.00000 −0.707107
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 33.0000 3.81051
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 4.00000 0.447214
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) −4.00000 −0.431331
\(87\) 3.00000 0.321634
\(88\) 1.00000 0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −24.0000 −2.52982
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −12.0000 −1.24434
\(94\) 2.00000 0.206284
\(95\) −24.0000 −2.46235
\(96\) −3.00000 −0.306186
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 11.0000 1.10000
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 6.00000 0.594089
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 9.00000 0.866025
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 4.00000 0.381385
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 17.0000 1.59923 0.799613 0.600516i \(-0.205038\pi\)
0.799613 + 0.600516i \(0.205038\pi\)
\(114\) 18.0000 1.68585
\(115\) −8.00000 −0.746004
\(116\) 1.00000 0.0928477
\(117\) 6.00000 0.554700
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) −12.0000 −1.09545
\(121\) 1.00000 0.0909091
\(122\) −5.00000 −0.452679
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) −4.00000 −0.350823
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) 36.0000 3.09839
\(136\) 2.00000 0.171499
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 6.00000 0.510754
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −4.00000 −0.335673
\(143\) −1.00000 −0.0836242
\(144\) 6.00000 0.500000
\(145\) 4.00000 0.332182
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −33.0000 −2.69444
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 6.00000 0.486664
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 3.00000 0.240192
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 15.0000 1.19334
\(159\) −36.0000 −2.85499
\(160\) −4.00000 −0.316228
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 13.0000 1.01824 0.509119 0.860696i \(-0.329971\pi\)
0.509119 + 0.860696i \(0.329971\pi\)
\(164\) 2.00000 0.156174
\(165\) −12.0000 −0.934199
\(166\) −6.00000 −0.465690
\(167\) 17.0000 1.31550 0.657750 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 8.00000 0.613572
\(171\) −36.0000 −2.75299
\(172\) 4.00000 0.304997
\(173\) 5.00000 0.380143 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −27.0000 −2.02944
\(178\) 6.00000 0.449719
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 24.0000 1.78885
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 2.00000 0.147442
\(185\) −8.00000 −0.588172
\(186\) 12.0000 0.879883
\(187\) 2.00000 0.146254
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) 14.0000 1.01300 0.506502 0.862239i \(-0.330938\pi\)
0.506502 + 0.862239i \(0.330938\pi\)
\(192\) 3.00000 0.216506
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) −5.00000 −0.358979
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 6.00000 0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −11.0000 −0.777817
\(201\) −27.0000 −1.90443
\(202\) −15.0000 −1.05540
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 8.00000 0.558744
\(206\) 12.0000 0.836080
\(207\) −12.0000 −0.834058
\(208\) 1.00000 0.0693375
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −14.0000 −0.963800 −0.481900 0.876226i \(-0.660053\pi\)
−0.481900 + 0.876226i \(0.660053\pi\)
\(212\) −12.0000 −0.824163
\(213\) 12.0000 0.822226
\(214\) −8.00000 −0.546869
\(215\) 16.0000 1.09119
\(216\) −9.00000 −0.612372
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 6.00000 0.405442
\(220\) −4.00000 −0.269680
\(221\) −2.00000 −0.134535
\(222\) 6.00000 0.402694
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 66.0000 4.40000
\(226\) −17.0000 −1.13082
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) −18.0000 −1.19208
\(229\) 16.0000 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −6.00000 −0.392232
\(235\) −8.00000 −0.521862
\(236\) −9.00000 −0.585850
\(237\) −45.0000 −2.92306
\(238\) 0 0
\(239\) 19.0000 1.22901 0.614504 0.788914i \(-0.289356\pi\)
0.614504 + 0.788914i \(0.289356\pi\)
\(240\) 12.0000 0.774597
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −6.00000 −0.381771
\(248\) 4.00000 0.254000
\(249\) 18.0000 1.14070
\(250\) −24.0000 −1.51789
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −5.00000 −0.313728
\(255\) −24.0000 −1.50294
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) −18.0000 −1.11204
\(263\) 3.00000 0.184988 0.0924940 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 3.00000 0.184637
\(265\) −48.0000 −2.94862
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) −9.00000 −0.549762
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) −36.0000 −2.19089
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 9.00000 0.543710
\(275\) −11.0000 −0.663325
\(276\) −6.00000 −0.361158
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) 8.00000 0.479808
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 6.00000 0.357295
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) 4.00000 0.237356
\(285\) −72.0000 −4.26491
\(286\) 1.00000 0.0591312
\(287\) 0 0
\(288\) −6.00000 −0.353553
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 15.0000 0.879316
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) 2.00000 0.116248
\(297\) −9.00000 −0.522233
\(298\) 10.0000 0.579284
\(299\) −2.00000 −0.115663
\(300\) 33.0000 1.90526
\(301\) 0 0
\(302\) −9.00000 −0.517892
\(303\) 45.0000 2.58518
\(304\) −6.00000 −0.344124
\(305\) 20.0000 1.14520
\(306\) 12.0000 0.685994
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) −36.0000 −2.04797
\(310\) 16.0000 0.908739
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) −3.00000 −0.169842
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −15.0000 −0.843816
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 36.0000 2.01878
\(319\) −1.00000 −0.0559893
\(320\) 4.00000 0.223607
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 9.00000 0.500000
\(325\) 11.0000 0.610170
\(326\) −13.0000 −0.720003
\(327\) 30.0000 1.65900
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 12.0000 0.660578
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 6.00000 0.329293
\(333\) −12.0000 −0.657596
\(334\) −17.0000 −0.930199
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 12.0000 0.652714
\(339\) 51.0000 2.76994
\(340\) −8.00000 −0.433861
\(341\) 4.00000 0.216612
\(342\) 36.0000 1.94666
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) −24.0000 −1.29212
\(346\) −5.00000 −0.268802
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 3.00000 0.160817
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 1.00000 0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 27.0000 1.43503
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −13.0000 −0.687071
\(359\) −31.0000 −1.63612 −0.818059 0.575135i \(-0.804950\pi\)
−0.818059 + 0.575135i \(0.804950\pi\)
\(360\) −24.0000 −1.26491
\(361\) 17.0000 0.894737
\(362\) −22.0000 −1.15629
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) −15.0000 −0.784063
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) −2.00000 −0.104257
\(369\) 12.0000 0.624695
\(370\) 8.00000 0.415900
\(371\) 0 0
\(372\) −12.0000 −0.622171
\(373\) −7.00000 −0.362446 −0.181223 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) −2.00000 −0.103418
\(375\) 72.0000 3.71806
\(376\) 2.00000 0.103142
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) −24.0000 −1.23117
\(381\) 15.0000 0.768473
\(382\) −14.0000 −0.716302
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 24.0000 1.21999
\(388\) 5.00000 0.253837
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −12.0000 −0.607644
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 54.0000 2.72394
\(394\) 3.00000 0.151138
\(395\) −60.0000 −3.01893
\(396\) −6.00000 −0.301511
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 27.0000 1.34664
\(403\) −4.00000 −0.199254
\(404\) 15.0000 0.746278
\(405\) 36.0000 1.78885
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 6.00000 0.297044
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) −8.00000 −0.395092
\(411\) −27.0000 −1.33181
\(412\) −12.0000 −0.591198
\(413\) 0 0
\(414\) 12.0000 0.589768
\(415\) 24.0000 1.17811
\(416\) −1.00000 −0.0490290
\(417\) −24.0000 −1.17529
\(418\) −6.00000 −0.293470
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 14.0000 0.681509
\(423\) −12.0000 −0.583460
\(424\) 12.0000 0.582772
\(425\) −22.0000 −1.06716
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) −3.00000 −0.144841
\(430\) −16.0000 −0.771589
\(431\) 1.00000 0.0481683 0.0240842 0.999710i \(-0.492333\pi\)
0.0240842 + 0.999710i \(0.492333\pi\)
\(432\) 9.00000 0.433013
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 10.0000 0.478913
\(437\) 12.0000 0.574038
\(438\) −6.00000 −0.286691
\(439\) 5.00000 0.238637 0.119318 0.992856i \(-0.461929\pi\)
0.119318 + 0.992856i \(0.461929\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) 2.00000 0.0951303
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −6.00000 −0.284747
\(445\) −24.0000 −1.13771
\(446\) −26.0000 −1.23114
\(447\) −30.0000 −1.41895
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −66.0000 −3.11127
\(451\) −2.00000 −0.0941763
\(452\) 17.0000 0.799613
\(453\) 27.0000 1.26857
\(454\) 10.0000 0.469323
\(455\) 0 0
\(456\) 18.0000 0.842927
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −16.0000 −0.747631
\(459\) −18.0000 −0.840168
\(460\) −8.00000 −0.373002
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 1.00000 0.0464238
\(465\) −48.0000 −2.22595
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 8.00000 0.369012
\(471\) −12.0000 −0.552931
\(472\) 9.00000 0.414259
\(473\) −4.00000 −0.183920
\(474\) 45.0000 2.06692
\(475\) −66.0000 −3.02829
\(476\) 0 0
\(477\) −72.0000 −3.29665
\(478\) −19.0000 −0.869040
\(479\) 37.0000 1.69057 0.845287 0.534313i \(-0.179430\pi\)
0.845287 + 0.534313i \(0.179430\pi\)
\(480\) −12.0000 −0.547723
\(481\) −2.00000 −0.0911922
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 20.0000 0.908153
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −5.00000 −0.226339
\(489\) 39.0000 1.76364
\(490\) 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
\(493\) −2.00000 −0.0900755
\(494\) 6.00000 0.269953
\(495\) −24.0000 −1.07872
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −18.0000 −0.806599
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 24.0000 1.07331
\(501\) 51.0000 2.27851
\(502\) 12.0000 0.535586
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 60.0000 2.66996
\(506\) −2.00000 −0.0889108
\(507\) −36.0000 −1.59882
\(508\) 5.00000 0.221839
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 24.0000 1.06274
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −54.0000 −2.38416
\(514\) −21.0000 −0.926270
\(515\) −48.0000 −2.11513
\(516\) 12.0000 0.528271
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 15.0000 0.658427
\(520\) −4.00000 −0.175412
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −3.00000 −0.130806
\(527\) 8.00000 0.348485
\(528\) −3.00000 −0.130558
\(529\) −19.0000 −0.826087
\(530\) 48.0000 2.08499
\(531\) −54.0000 −2.34340
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 18.0000 0.778936
\(535\) 32.0000 1.38348
\(536\) 9.00000 0.388741
\(537\) 39.0000 1.68297
\(538\) 12.0000 0.517357
\(539\) 0 0
\(540\) 36.0000 1.54919
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 25.0000 1.07384
\(543\) 66.0000 2.83233
\(544\) 2.00000 0.0857493
\(545\) 40.0000 1.71341
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −9.00000 −0.384461
\(549\) 30.0000 1.28037
\(550\) 11.0000 0.469042
\(551\) −6.00000 −0.255609
\(552\) 6.00000 0.255377
\(553\) 0 0
\(554\) 3.00000 0.127458
\(555\) −24.0000 −1.01874
\(556\) −8.00000 −0.339276
\(557\) 38.0000 1.61011 0.805056 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 24.0000 1.01600
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 10.0000 0.421825
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) −6.00000 −0.252646
\(565\) 68.0000 2.86078
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 72.0000 3.01575
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 42.0000 1.75458
\(574\) 0 0
\(575\) −22.0000 −0.917463
\(576\) 6.00000 0.250000
\(577\) 7.00000 0.291414 0.145707 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 13.0000 0.540729
\(579\) −66.0000 −2.74287
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −15.0000 −0.621770
\(583\) 12.0000 0.496989
\(584\) −2.00000 −0.0827606
\(585\) 24.0000 0.992278
\(586\) −6.00000 −0.247858
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 36.0000 1.48210
\(591\) −9.00000 −0.370211
\(592\) −2.00000 −0.0821995
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 9.00000 0.369274
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −30.0000 −1.22782
\(598\) 2.00000 0.0817861
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −33.0000 −1.34722
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −54.0000 −2.19905
\(604\) 9.00000 0.366205
\(605\) 4.00000 0.162623
\(606\) −45.0000 −1.82800
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) −2.00000 −0.0809113
\(612\) −12.0000 −0.485071
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 32.0000 1.29141
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 36.0000 1.44813
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −16.0000 −0.642575
\(621\) −18.0000 −0.722315
\(622\) −28.0000 −1.12270
\(623\) 0 0
\(624\) 3.00000 0.120096
\(625\) 41.0000 1.64000
\(626\) 1.00000 0.0399680
\(627\) 18.0000 0.718851
\(628\) −4.00000 −0.159617
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 15.0000 0.596668
\(633\) −42.0000 −1.66935
\(634\) 12.0000 0.476581
\(635\) 20.0000 0.793676
\(636\) −36.0000 −1.42749
\(637\) 0 0
\(638\) 1.00000 0.0395904
\(639\) 24.0000 0.949425
\(640\) −4.00000 −0.158114
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) −24.0000 −0.947204
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 0 0
\(645\) 48.0000 1.89000
\(646\) −12.0000 −0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −9.00000 −0.353553
\(649\) 9.00000 0.353281
\(650\) −11.0000 −0.431455
\(651\) 0 0
\(652\) 13.0000 0.509119
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) −30.0000 −1.17309
\(655\) 72.0000 2.81327
\(656\) 2.00000 0.0780869
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) −12.0000 −0.467099
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) −7.00000 −0.272063
\(663\) −6.00000 −0.233021
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) −2.00000 −0.0774403
\(668\) 17.0000 0.657750
\(669\) 78.0000 3.01565
\(670\) 36.0000 1.39080
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 30.0000 1.15556
\(675\) 99.0000 3.81051
\(676\) −12.0000 −0.461538
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −51.0000 −1.95864
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) −30.0000 −1.14960
\(682\) −4.00000 −0.153168
\(683\) −33.0000 −1.26271 −0.631355 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) −36.0000 −1.37649
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 48.0000 1.83131
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 24.0000 0.913664
\(691\) −15.0000 −0.570627 −0.285313 0.958434i \(-0.592098\pi\)
−0.285313 + 0.958434i \(0.592098\pi\)
\(692\) 5.00000 0.190071
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) −32.0000 −1.21383
\(696\) −3.00000 −0.113715
\(697\) −4.00000 −0.151511
\(698\) 2.00000 0.0757011
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) −9.00000 −0.339683
\(703\) 12.0000 0.452589
\(704\) −1.00000 −0.0376889
\(705\) −24.0000 −0.903892
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −27.0000 −1.01472
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −16.0000 −0.600469
\(711\) −90.0000 −3.37526
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 13.0000 0.485833
\(717\) 57.0000 2.12870
\(718\) 31.0000 1.15691
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 24.0000 0.894427
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −90.0000 −3.34714
\(724\) 22.0000 0.817624
\(725\) 11.0000 0.408530
\(726\) −3.00000 −0.111340
\(727\) 34.0000 1.26099 0.630495 0.776193i \(-0.282852\pi\)
0.630495 + 0.776193i \(0.282852\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −8.00000 −0.296093
\(731\) −8.00000 −0.295891
\(732\) 15.0000 0.554416
\(733\) 1.00000 0.0369358 0.0184679 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 9.00000 0.331519
\(738\) −12.0000 −0.441726
\(739\) 18.0000 0.662141 0.331070 0.943606i \(-0.392590\pi\)
0.331070 + 0.943606i \(0.392590\pi\)
\(740\) −8.00000 −0.294086
\(741\) −18.0000 −0.661247
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 12.0000 0.439941
\(745\) −40.0000 −1.46549
\(746\) 7.00000 0.256288
\(747\) 36.0000 1.31717
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) −72.0000 −2.62907
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) −2.00000 −0.0729325
\(753\) −36.0000 −1.31191
\(754\) −1.00000 −0.0364179
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 29.0000 1.05333
\(759\) 6.00000 0.217786
\(760\) 24.0000 0.870572
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) −15.0000 −0.543393
\(763\) 0 0
\(764\) 14.0000 0.506502
\(765\) −48.0000 −1.73544
\(766\) 8.00000 0.289052
\(767\) −9.00000 −0.324971
\(768\) 3.00000 0.108253
\(769\) 38.0000 1.37032 0.685158 0.728395i \(-0.259733\pi\)
0.685158 + 0.728395i \(0.259733\pi\)
\(770\) 0 0
\(771\) 63.0000 2.26889
\(772\) −22.0000 −0.791797
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) −24.0000 −0.862662
\(775\) −44.0000 −1.58053
\(776\) −5.00000 −0.179490
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 12.0000 0.429669
\(781\) −4.00000 −0.143131
\(782\) −4.00000 −0.143040
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −16.0000 −0.571064
\(786\) −54.0000 −1.92612
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −3.00000 −0.106871
\(789\) 9.00000 0.320408
\(790\) 60.0000 2.13470
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 5.00000 0.177555
\(794\) −30.0000 −1.06466
\(795\) −144.000 −5.10715
\(796\) −10.0000 −0.354441
\(797\) −38.0000 −1.34603 −0.673015 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) −11.0000 −0.388909
\(801\) −36.0000 −1.27200
\(802\) −15.0000 −0.529668
\(803\) −2.00000 −0.0705785
\(804\) −27.0000 −0.952217
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −36.0000 −1.26726
\(808\) −15.0000 −0.527698
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) −36.0000 −1.26491
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −75.0000 −2.63036
\(814\) −2.00000 −0.0701000
\(815\) 52.0000 1.82148
\(816\) −6.00000 −0.210042
\(817\) −24.0000 −0.839654
\(818\) −32.0000 −1.11885
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) 27.0000 0.941733
\(823\) 38.0000 1.32460 0.662298 0.749240i \(-0.269581\pi\)
0.662298 + 0.749240i \(0.269581\pi\)
\(824\) 12.0000 0.418040
\(825\) −33.0000 −1.14891
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) −12.0000 −0.417029
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) −24.0000 −0.833052
\(831\) −9.00000 −0.312207
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 24.0000 0.831052
\(835\) 68.0000 2.35324
\(836\) 6.00000 0.207514
\(837\) −36.0000 −1.24434
\(838\) −20.0000 −0.690889
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 20.0000 0.689246
\(843\) −30.0000 −1.03325
\(844\) −14.0000 −0.481900
\(845\) −48.0000 −1.65125
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 18.0000 0.617758
\(850\) 22.0000 0.754594
\(851\) 4.00000 0.137118
\(852\) 12.0000 0.411113
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) −144.000 −4.92470
\(856\) −8.00000 −0.273434
\(857\) 4.00000 0.136637 0.0683187 0.997664i \(-0.478237\pi\)
0.0683187 + 0.997664i \(0.478237\pi\)
\(858\) 3.00000 0.102418
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) −1.00000 −0.0340601
\(863\) 10.0000 0.340404 0.170202 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(864\) −9.00000 −0.306186
\(865\) 20.0000 0.680020
\(866\) −10.0000 −0.339814
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) −12.0000 −0.406838
\(871\) −9.00000 −0.304953
\(872\) −10.0000 −0.338643
\(873\) 30.0000 1.01535
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 25.0000 0.844190 0.422095 0.906552i \(-0.361295\pi\)
0.422095 + 0.906552i \(0.361295\pi\)
\(878\) −5.00000 −0.168742
\(879\) 18.0000 0.607125
\(880\) −4.00000 −0.134840
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −108.000 −3.63038
\(886\) 12.0000 0.403148
\(887\) 17.0000 0.570804 0.285402 0.958408i \(-0.407873\pi\)
0.285402 + 0.958408i \(0.407873\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 24.0000 0.804482
\(891\) −9.00000 −0.301511
\(892\) 26.0000 0.870544
\(893\) 12.0000 0.401565
\(894\) 30.0000 1.00335
\(895\) 52.0000 1.73817
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −30.0000 −1.00111
\(899\) −4.00000 −0.133407
\(900\) 66.0000 2.20000
\(901\) 24.0000 0.799556
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) −17.0000 −0.565412
\(905\) 88.0000 2.92522
\(906\) −27.0000 −0.897015
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −10.0000 −0.331862
\(909\) 90.0000 2.98511
\(910\) 0 0
\(911\) −54.0000 −1.78910 −0.894550 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) −18.0000 −0.596040
\(913\) −6.00000 −0.198571
\(914\) −2.00000 −0.0661541
\(915\) 60.0000 1.98354
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 18.0000 0.594089
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 8.00000 0.263752
\(921\) −96.0000 −3.16331
\(922\) −3.00000 −0.0987997
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 14.0000 0.460069
\(927\) −72.0000 −2.36479
\(928\) −1.00000 −0.0328266
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 48.0000 1.57398
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 84.0000 2.75004
\(934\) −12.0000 −0.392652
\(935\) 8.00000 0.261628
\(936\) −6.00000 −0.196116
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) 0 0
\(939\) −3.00000 −0.0979013
\(940\) −8.00000 −0.260931
\(941\) −55.0000 −1.79295 −0.896474 0.443096i \(-0.853880\pi\)
−0.896474 + 0.443096i \(0.853880\pi\)
\(942\) 12.0000 0.390981
\(943\) −4.00000 −0.130258
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −16.0000 −0.519930 −0.259965 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) −45.0000 −1.46153
\(949\) 2.00000 0.0649227
\(950\) 66.0000 2.14132
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) 56.0000 1.81402 0.907009 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 72.0000 2.33109
\(955\) 56.0000 1.81212
\(956\) 19.0000 0.614504
\(957\) −3.00000 −0.0969762
\(958\) −37.0000 −1.19542
\(959\) 0 0
\(960\) 12.0000 0.387298
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 48.0000 1.54678
\(964\) −30.0000 −0.966235
\(965\) −88.0000 −2.83282
\(966\) 0 0
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 36.0000 1.15649
\(970\) −20.0000 −0.642161
\(971\) 13.0000 0.417190 0.208595 0.978002i \(-0.433111\pi\)
0.208595 + 0.978002i \(0.433111\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 4.00000 0.128168
\(975\) 33.0000 1.05685
\(976\) 5.00000 0.160046
\(977\) 14.0000 0.447900 0.223950 0.974601i \(-0.428105\pi\)
0.223950 + 0.974601i \(0.428105\pi\)
\(978\) −39.0000 −1.24708
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) −18.0000 −0.574403
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) −6.00000 −0.191273
\(985\) −12.0000 −0.382352
\(986\) 2.00000 0.0636930
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −8.00000 −0.254385
\(990\) 24.0000 0.762770
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 4.00000 0.127000
\(993\) 21.0000 0.666415
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) 18.0000 0.570352
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 16.0000 0.506471
\(999\) −18.0000 −0.569495
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 1078.2.a.f.1.1 1
3.2 odd 2 9702.2.a.bb.1.1 1
4.3 odd 2 8624.2.a.d.1.1 1
7.2 even 3 1078.2.e.g.67.1 2
7.3 odd 6 154.2.e.d.23.1 2
7.4 even 3 1078.2.e.g.177.1 2
7.5 odd 6 154.2.e.d.67.1 yes 2
7.6 odd 2 1078.2.a.a.1.1 1
21.5 even 6 1386.2.k.a.991.1 2
21.17 even 6 1386.2.k.a.793.1 2
21.20 even 2 9702.2.a.cg.1.1 1
28.3 even 6 1232.2.q.a.177.1 2
28.19 even 6 1232.2.q.a.529.1 2
28.27 even 2 8624.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.d.23.1 2 7.3 odd 6
154.2.e.d.67.1 yes 2 7.5 odd 6
1078.2.a.a.1.1 1 7.6 odd 2
1078.2.a.f.1.1 1 1.1 even 1 trivial
1078.2.e.g.67.1 2 7.2 even 3
1078.2.e.g.177.1 2 7.4 even 3
1232.2.q.a.177.1 2 28.3 even 6
1232.2.q.a.529.1 2 28.19 even 6
1386.2.k.a.793.1 2 21.17 even 6
1386.2.k.a.991.1 2 21.5 even 6
8624.2.a.d.1.1 1 4.3 odd 2
8624.2.a.bd.1.1 1 28.27 even 2
9702.2.a.bb.1.1 1 3.2 odd 2
9702.2.a.cg.1.1 1 21.20 even 2