Properties

Label 2394.2.a.f.1.1
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2394.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -4.00000 q^{10} +6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{19} +4.00000 q^{20} -6.00000 q^{22} -2.00000 q^{23} +11.0000 q^{25} +4.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -4.00000 q^{35} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} +6.00000 q^{44} +2.00000 q^{46} +8.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} -4.00000 q^{52} +14.0000 q^{53} +24.0000 q^{55} +1.00000 q^{56} +2.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -16.0000 q^{65} +10.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +4.00000 q^{71} -14.0000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -6.00000 q^{77} +2.00000 q^{79} +4.00000 q^{80} +6.00000 q^{82} +16.0000 q^{85} -6.00000 q^{88} -6.00000 q^{89} +4.00000 q^{91} -2.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{98} +O(q^{100})\)

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
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −4.00000 −1.26491
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 4.00000 0.894427
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −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
\(40\) −4.00000 −0.632456
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 24.0000 3.23616
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 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
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 16.0000 1.73544
\(86\) 0 0
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −24.0000 −2.28831
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 16.0000 1.40329
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −16.0000 −1.22714
\(171\) 0 0
\(172\) 0 0
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −11.0000 −0.831522
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 14.0000 1.01300 0.506502 0.862239i \(-0.330938\pi\)
0.506502 + 0.862239i \(0.330938\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −11.0000 −0.777817
\(201\) 0 0
\(202\) 4.00000 0.281439
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 24.0000 1.61808
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 32.0000 2.08745
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 22.0000 1.42306 0.711531 0.702655i \(-0.248002\pi\)
0.711531 + 0.702655i \(0.248002\pi\)
\(240\) 0 0
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) −25.0000 −1.60706
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −16.0000 −0.992278
\(261\) 0 0
\(262\) −20.0000 −1.23560
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) 56.0000 3.44005
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 66.0000 3.97995
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 14.0000 0.805609
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −40.0000 −2.29039
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −16.0000 −0.908739
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 4.00000 0.223607
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −44.0000 −2.44068
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 40.0000 2.18543
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 16.0000 0.867722
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 11.0000 0.587975
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −16.0000 −0.840941
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −56.0000 −2.93117
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) −14.0000 −0.726844
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −24.0000 −1.24101
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −14.0000 −0.716302
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) −26.0000 −1.32337
\(387\) 0 0
\(388\) 0 0
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 24.0000 1.18528
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) −14.0000 −0.679900
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −24.0000 −1.14416
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) −32.0000 −1.47605
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) −11.0000 −0.504715
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) −22.0000 −1.00626
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) −4.00000 −0.180702
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 48.0000 2.11104
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 16.0000 0.701646
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −2.00000 −0.0872041
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −56.0000 −2.43248
\(531\) 0 0
\(532\) 1.00000 0.0433555
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 10.0000 0.427569 0.213785 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −66.0000 −2.81425
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) −72.0000 −3.02906
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −22.0000 −0.917463
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −8.00000 −0.332182
\(581\) 0 0
\(582\) 0 0
\(583\) 84.0000 3.47892
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −16.0000 −0.657041 −0.328521 0.944497i \(-0.606550\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 100.000 4.06558
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 40.0000 1.61955
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 14.0000 0.559553
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 12.0000 0.475085
\(639\) 0 0
\(640\) −4.00000 −0.158114
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 44.0000 1.72582
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 80.0000 3.12586
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) −40.0000 −1.54533
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −16.0000 −0.613572
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) 72.0000 2.75098
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 0 0
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) 80.0000 3.03457
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 30.0000 1.13552
\(699\) 0 0
\(700\) −11.0000 −0.415761
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −16.0000 −0.600469
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −96.0000 −3.59020
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −34.0000 −1.26887
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 16.0000 0.594635
\(725\) −22.0000 −0.817059
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) 56.0000 2.07265
\(731\) 0 0
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 60.0000 2.21013
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 14.0000 0.513956
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 0 0
\(745\) 72.0000 2.63788
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) −56.0000 −2.03805
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 34.0000 1.23494
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −16.0000 −0.580000 −0.290000 0.957027i \(-0.593655\pi\)
−0.290000 + 0.957027i \(0.593655\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 14.0000 0.506502
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 24.0000 0.864900
\(771\) 0 0
\(772\) 26.0000 0.935760
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 44.0000 1.58053
\(776\) 0 0
\(777\) 0 0
\(778\) 34.0000 1.21896
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −88.0000 −3.14085
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) −11.0000 −0.388909
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) −84.0000 −2.96430
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 4.00000 0.140720
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) −96.0000 −3.36273
\(816\) 0 0
\(817\) 0 0
\(818\) −24.0000 −0.839140
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) −44.0000 −1.52818 −0.764092 0.645108i \(-0.776812\pi\)
−0.764092 + 0.645108i \(0.776812\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 10.0000 0.344214
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) −44.0000 −1.50919
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) −2.00000 −0.0676510
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −54.0000 −1.82345 −0.911725 0.410801i \(-0.865249\pi\)
−0.911725 + 0.410801i \(0.865249\pi\)
\(878\) −20.0000 −0.674967
\(879\) 0 0
\(880\) 24.0000 0.809040
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 24.0000 0.804482
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 56.0000 1.86563
\(902\) 36.0000 1.19867
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 64.0000 2.12743
\(906\) 0 0
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 20.0000 0.663723
\(909\) 0 0
\(910\) −16.0000 −0.530395
\(911\) −52.0000 −1.72284 −0.861418 0.507896i \(-0.830423\pi\)
−0.861418 + 0.507896i \(0.830423\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) 36.0000 1.18560
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) 0 0
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 52.0000 1.70606 0.853032 0.521858i \(-0.174761\pi\)
0.853032 + 0.521858i \(0.174761\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 96.0000 3.13954
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 10.0000 0.326512
\(939\) 0 0
\(940\) 32.0000 1.04372
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) 11.0000 0.356887
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) 56.0000 1.81212
\(956\) 22.0000 0.711531
\(957\) 0 0
\(958\) 4.00000 0.129234
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) 104.000 3.34788
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 4.00000 0.127775
\(981\) 0 0
\(982\) 30.0000 0.957338
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 0 0
\(985\) −88.0000 −2.80391
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 0 0
\(990\) 0 0
\(991\) −30.0000 −0.952981 −0.476491 0.879180i \(-0.658091\pi\)
−0.476491 + 0.879180i \(0.658091\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 4.00000 0.126872
\(995\) 32.0000 1.01447
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 2394.2.a.f.1.1 1
3.2 odd 2 798.2.a.h.1.1 1
12.11 even 2 6384.2.a.c.1.1 1
21.20 even 2 5586.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.a.h.1.1 1 3.2 odd 2
2394.2.a.f.1.1 1 1.1 even 1 trivial
5586.2.a.x.1.1 1 21.20 even 2
6384.2.a.c.1.1 1 12.11 even 2