Properties

Label 1710.2.a.q.1.1
Level $1710$
Weight $2$
Character 1710.1
Self dual yes
Analytic conductor $13.654$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1710.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -4.00000 q^{11} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -2.00000 q^{28} -6.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} +10.0000 q^{37} +1.00000 q^{38} +1.00000 q^{40} -4.00000 q^{41} +12.0000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -4.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -6.00000 q^{58} -10.0000 q^{59} +2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +12.0000 q^{67} -4.00000 q^{68} -2.00000 q^{70} -8.00000 q^{71} -2.00000 q^{73} +10.0000 q^{74} +1.00000 q^{76} +8.00000 q^{77} +10.0000 q^{79} +1.00000 q^{80} -4.00000 q^{82} -2.00000 q^{83} -4.00000 q^{85} +12.0000 q^{86} -4.00000 q^{88} +8.00000 q^{89} +12.0000 q^{91} -4.00000 q^{92} -4.00000 q^{94} +1.00000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 12.0000 0.889499
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) 16.0000 1.17004
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) 12.0000 0.733017
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) −6.00000 −0.340777
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −2.00000 −0.109764
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −22.0000 −1.10834
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) −24.0000 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 36.0000 1.79329
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) −40.0000 −1.98273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) −2.00000 −0.0981761
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −38.0000 −1.81364 −0.906821 0.421517i \(-0.861498\pi\)
−0.906821 + 0.421517i \(0.861498\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 16.0000 0.711287
\(507\) 0 0
\(508\) −20.0000 −0.887357
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) −20.0000 −0.878750
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −20.0000 −0.850487
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) −72.0000 −3.04528
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −4.00000 −0.168730
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −22.0000 −0.908037 −0.454019 0.890992i \(-0.650010\pi\)
−0.454019 + 0.890992i \(0.650010\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −24.0000 −0.978167
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) −16.0000 −0.641542
\(623\) −16.0000 −0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 12.0000 0.478852
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −20.0000 −0.793676
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 24.0000 0.950169
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 20.0000 0.789953 0.394976 0.918691i \(-0.370753\pi\)
0.394976 + 0.918691i \(0.370753\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 40.0000 1.57014
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −22.0000 −0.856998 −0.428499 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 24.0000 0.919007
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 12.0000 0.457496
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 8.00000 0.299813
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 12.0000 0.444750
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) −20.0000 −0.734223
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 16.0000 0.585018
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) 32.0000 1.15848
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 60.0000 2.16647
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) 10.0000 0.355784
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 24.0000 0.847469
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 36.0000 1.26805
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) −2.00000 −0.0694210
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) −40.0000 −1.37118
\(852\) 0 0
\(853\) −36.0000 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) 40.0000 1.36241
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 30.0000 1.01944
\(867\) 0 0
\(868\) 12.0000 0.407307
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) −16.0000 −0.541828
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −38.0000 −1.28244
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 8.00000 0.268161
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) −4.00000 −0.133855
\(894\) 0 0
\(895\) 2.00000 0.0668526
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) 12.0000 0.397796
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −18.0000 −0.591517
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 22.0000 0.719862
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −60.0000 −1.93448
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) −6.00000 −0.190500
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
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 1710.2.a.q.1.1 1
3.2 odd 2 570.2.a.a.1.1 1
5.4 even 2 8550.2.a.n.1.1 1
12.11 even 2 4560.2.a.t.1.1 1
15.2 even 4 2850.2.d.r.799.1 2
15.8 even 4 2850.2.d.r.799.2 2
15.14 odd 2 2850.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.a.1.1 1 3.2 odd 2
1710.2.a.q.1.1 1 1.1 even 1 trivial
2850.2.a.bb.1.1 1 15.14 odd 2
2850.2.d.r.799.1 2 15.2 even 4
2850.2.d.r.799.2 2 15.8 even 4
4560.2.a.t.1.1 1 12.11 even 2
8550.2.a.n.1.1 1 5.4 even 2