Properties

Label 1710.2.a.l.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -6.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -6.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} +2.00000 q^{37} -1.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -8.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} +9.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -4.00000 q^{55} -4.00000 q^{56} -6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +16.0000 q^{71} -14.0000 q^{73} +2.00000 q^{74} -1.00000 q^{76} -16.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} -10.0000 q^{82} -6.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} +24.0000 q^{91} -4.00000 q^{92} -12.0000 q^{94} +1.00000 q^{95} +14.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 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
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) −4.00000 −0.534522
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −10.0000 −1.10432
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 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
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) −14.0000 −1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −16.0000 −1.28932
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 24.0000 1.77900
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) −12.0000 −0.875190
\(189\) 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
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) −36.0000 −2.42162
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) −24.0000 −1.55569
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) −16.0000 −1.00393
\(255\) 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
\(259\) −8.00000 −0.497096
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 4.00000 0.245256
\(267\) 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
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 40.0000 2.36113
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −16.0000 −0.911685
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 16.0000 0.891645
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 23.0000 1.25104
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 24.0000 1.25794
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) −36.0000 −1.71235
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −4.00000 −0.188982
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) −24.0000 −1.12514
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) −32.0000 −1.47136
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −24.0000 −1.10004
\(477\) 0 0
\(478\) −16.0000 −0.731823
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) −9.00000 −0.406579
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −64.0000 −2.87079
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −48.0000 −2.11104
\(518\) −8.00000 −0.351500
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 60.0000 2.59889
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 40.0000 1.66957
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 32.0000 1.30422
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 72.0000 2.91281
\(612\) 0 0
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 8.00000 0.320771
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −54.0000 −2.13956
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 48.0000 1.87123
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) −32.0000 −1.22534
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −60.0000 −2.27266
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 24.0000 0.902613
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −16.0000 −0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 24.0000 0.889499
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 8.00000 0.293689
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −40.0000 −1.42224
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 22.0000 0.776847
\(803\) −56.0000 −1.97620
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 48.0000 1.69073
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) −6.00000 −0.209785
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) −24.0000 −0.795592
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) −96.0000 −3.15988
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 20.0000 0.657241
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) 8.00000 0.261768
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 48.0000 1.56726
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) 24.0000 0.779895 0.389948 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 0 0
\(949\) 84.0000 2.72676
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −48.0000 −1.53881
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) −64.0000 −2.02996
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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.l.1.1 1
3.2 odd 2 570.2.a.e.1.1 1
5.4 even 2 8550.2.a.r.1.1 1
12.11 even 2 4560.2.a.q.1.1 1
15.2 even 4 2850.2.d.a.799.1 2
15.8 even 4 2850.2.d.a.799.2 2
15.14 odd 2 2850.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.e.1.1 1 3.2 odd 2
1710.2.a.l.1.1 1 1.1 even 1 trivial
2850.2.a.v.1.1 1 15.14 odd 2
2850.2.d.a.799.1 2 15.2 even 4
2850.2.d.a.799.2 2 15.8 even 4
4560.2.a.q.1.1 1 12.11 even 2
8550.2.a.r.1.1 1 5.4 even 2