Properties

Label 78.2.a.a.1.1
Level $78$
Weight $2$
Character 78.1
Self dual yes
Analytic conductor $0.623$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 78.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +8.00000 q^{38} -1.00000 q^{39} -2.00000 q^{40} -10.0000 q^{41} +4.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -8.00000 q^{55} -4.00000 q^{56} +8.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -8.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -16.0000 q^{77} +1.00000 q^{78} +8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} +4.00000 q^{93} -8.00000 q^{94} -16.0000 q^{95} +1.00000 q^{96} +10.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −4.00000 −1.06904
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −2.00000 −0.342997
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 8.00000 1.29777
\(39\) −1.00000 −0.160128
\(40\) −2.00000 −0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 1.00000 0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.00000 −1.07872
\(56\) −4.00000 −0.534522
\(57\) 8.00000 1.05963
\(58\) −6.00000 −0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −4.00000 −0.492366
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) −16.0000 −1.82337
\(78\) 1.00000 0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) −16.0000 −1.64157
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −9.00000 −0.909137
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 2.00000 0.198030
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −8.00000 −0.780720
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 8.00000 0.733359
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) −4.00000 −0.356348
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000 0.348155
\(133\) −32.0000 −2.77475
\(134\) 16.0000 1.38219
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 8.00000 0.676123
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) −2.00000 −0.165521
\(147\) −9.00000 −0.742307
\(148\) −2.00000 −0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 8.00000 0.648886
\(153\) 2.00000 0.161690
\(154\) 16.0000 1.28932
\(155\) −8.00000 −0.642575
\(156\) −1.00000 −0.0800641
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −10.0000 −0.780869
\(165\) 8.00000 0.622799
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 4.00000 0.308607
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) −8.00000 −0.611775
\(172\) 4.00000 0.304997
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 6.00000 0.454859
\(175\) −4.00000 −0.302372
\(176\) −4.00000 −0.301511
\(177\) −4.00000 −0.300658
\(178\) −14.0000 −1.04934
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) −4.00000 −0.293294
\(187\) −8.00000 −0.585018
\(188\) 8.00000 0.583460
\(189\) −4.00000 −0.290957
\(190\) 16.0000 1.16076
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −10.0000 −0.717958
\(195\) −2.00000 −0.143223
\(196\) 9.00000 0.642857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 16.0000 1.12855
\(202\) 2.00000 0.140720
\(203\) 24.0000 1.68447
\(204\) −2.00000 −0.140028
\(205\) −20.0000 −1.39686
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) 32.0000 2.21349
\(210\) 8.00000 0.552052
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −10.0000 −0.686803
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) −2.00000 −0.135147
\(220\) −8.00000 −0.539360
\(221\) 2.00000 0.134535
\(222\) −2.00000 −0.134231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 8.00000 0.529813
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 16.0000 1.04372
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 18.0000 1.14998
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) −12.0000 −0.760469
\(250\) 12.0000 0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 4.00000 0.249029
\(259\) −8.00000 −0.497096
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) −4.00000 −0.247121
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) −20.0000 −1.22859
\(266\) 32.0000 1.96205
\(267\) −14.0000 −0.856786
\(268\) −16.0000 −0.977356
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 2.00000 0.121716
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 2.00000 0.121268
\(273\) −4.00000 −0.242091
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −12.0000 −0.719712
\(279\) −4.00000 −0.239474
\(280\) −8.00000 −0.478091
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 8.00000 0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) 16.0000 0.947758
\(286\) 4.00000 0.236525
\(287\) −40.0000 −2.36113
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −12.0000 −0.704664
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 9.00000 0.524891
\(295\) 8.00000 0.465778
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) −12.0000 −0.690522
\(303\) 2.00000 0.114897
\(304\) −8.00000 −0.458831
\(305\) −4.00000 −0.229039
\(306\) −2.00000 −0.114332
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −16.0000 −0.911685
\(309\) −16.0000 −0.910208
\(310\) 8.00000 0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 1.00000 0.0566139
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −14.0000 −0.790066
\(315\) 8.00000 0.450749
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −10.0000 −0.560772
\(319\) −24.0000 −1.34374
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 16.0000 0.886158
\(327\) 2.00000 0.110600
\(328\) 10.0000 0.552158
\(329\) 32.0000 1.76422
\(330\) −8.00000 −0.440386
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −32.0000 −1.74835
\(336\) −4.00000 −0.218218
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.00000 0.325875
\(340\) 4.00000 0.216930
\(341\) 16.0000 0.866449
\(342\) 8.00000 0.432590
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 10.0000 0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 4.00000 0.213809
\(351\) −1.00000 −0.0533761
\(352\) 4.00000 0.213201
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 4.00000 0.212598
\(355\) −16.0000 −0.849192
\(356\) 14.0000 0.741999
\(357\) −8.00000 −0.423405
\(358\) 12.0000 0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) −5.00000 −0.262432
\(364\) 4.00000 0.209657
\(365\) 4.00000 0.209370
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 4.00000 0.207950
\(371\) −40.0000 −2.07670
\(372\) 4.00000 0.207390
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 8.00000 0.413670
\(375\) 12.0000 0.619677
\(376\) −8.00000 −0.412568
\(377\) 6.00000 0.309016
\(378\) 4.00000 0.205738
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) −16.0000 −0.820783
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) −32.0000 −1.63087
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) −4.00000 −0.201773
\(394\) −18.0000 −0.906827
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 8.00000 0.401004
\(399\) 32.0000 1.60200
\(400\) −1.00000 −0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −16.0000 −0.798007
\(403\) −4.00000 −0.199254
\(404\) −2.00000 −0.0995037
\(405\) 2.00000 0.0993808
\(406\) −24.0000 −1.19110
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 20.0000 0.987730
\(411\) 10.0000 0.493264
\(412\) 16.0000 0.788263
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) 24.0000 1.17811
\(416\) −1.00000 −0.0490290
\(417\) −12.0000 −0.587643
\(418\) −32.0000 −1.56517
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −8.00000 −0.390360
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −12.0000 −0.584151
\(423\) 8.00000 0.388973
\(424\) 10.0000 0.485643
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) −8.00000 −0.387147
\(428\) 12.0000 0.580042
\(429\) 4.00000 0.193122
\(430\) −8.00000 −0.385794
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 16.0000 0.768025
\(435\) −12.0000 −0.575356
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 8.00000 0.381385
\(441\) 9.00000 0.428571
\(442\) −2.00000 −0.0951303
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) 28.0000 1.32733
\(446\) 4.00000 0.189405
\(447\) 6.00000 0.283790
\(448\) 4.00000 0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −6.00000 −0.282216
\(453\) −12.0000 −0.563809
\(454\) −20.0000 −0.938647
\(455\) 8.00000 0.375046
\(456\) −8.00000 −0.374634
\(457\) −30.0000 −1.40334 −0.701670 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) −22.0000 −1.02799
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −16.0000 −0.744387
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −18.0000 −0.833834
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 1.00000 0.0462250
\(469\) −64.0000 −2.95525
\(470\) −16.0000 −0.738025
\(471\) −14.0000 −0.645086
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) 8.00000 0.367452
\(475\) 8.00000 0.367065
\(476\) 8.00000 0.366679
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 2.00000 0.0912871
\(481\) −2.00000 −0.0911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 2.00000 0.0905357
\(489\) 16.0000 0.723545
\(490\) −18.0000 −0.813157
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 10.0000 0.450835
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) −8.00000 −0.359573
\(496\) −4.00000 −0.179605
\(497\) −32.0000 −1.43540
\(498\) 12.0000 0.537733
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −4.00000 −0.178174
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 4.00000 0.177123
\(511\) 8.00000 0.353899
\(512\) −1.00000 −0.0441942
\(513\) 8.00000 0.353209
\(514\) 6.00000 0.264649
\(515\) 32.0000 1.41009
\(516\) −4.00000 −0.176090
\(517\) −32.0000 −1.40736
\(518\) 8.00000 0.351500
\(519\) 10.0000 0.438951
\(520\) −2.00000 −0.0877058
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 4.00000 0.174741
\(525\) 4.00000 0.174574
\(526\) −8.00000 −0.348817
\(527\) −8.00000 −0.348485
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 20.0000 0.868744
\(531\) 4.00000 0.173585
\(532\) −32.0000 −1.38738
\(533\) −10.0000 −0.433148
\(534\) 14.0000 0.605839
\(535\) 24.0000 1.03761
\(536\) 16.0000 0.691095
\(537\) 12.0000 0.517838
\(538\) 26.0000 1.12094
\(539\) −36.0000 −1.55063
\(540\) −2.00000 −0.0860663
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 4.00000 0.171815
\(543\) 10.0000 0.429141
\(544\) −2.00000 −0.0857493
\(545\) −4.00000 −0.171341
\(546\) 4.00000 0.171184
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) −22.0000 −0.934690
\(555\) 4.00000 0.169791
\(556\) 12.0000 0.508913
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) 4.00000 0.169182
\(560\) 8.00000 0.338062
\(561\) 8.00000 0.337760
\(562\) 26.0000 1.09674
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −8.00000 −0.336861
\(565\) −12.0000 −0.504844
\(566\) 4.00000 0.168133
\(567\) 4.00000 0.167984
\(568\) 8.00000 0.335673
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −16.0000 −0.670166
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −4.00000 −0.167248
\(573\) 8.00000 0.334205
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 13.0000 0.540729
\(579\) 14.0000 0.581820
\(580\) 12.0000 0.498273
\(581\) 48.0000 1.99138
\(582\) 10.0000 0.414513
\(583\) 40.0000 1.65663
\(584\) −2.00000 −0.0827606
\(585\) 2.00000 0.0826898
\(586\) −26.0000 −1.07405
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) −9.00000 −0.371154
\(589\) 32.0000 1.31854
\(590\) −8.00000 −0.329355
\(591\) −18.0000 −0.740421
\(592\) −2.00000 −0.0821995
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) −4.00000 −0.164122
\(595\) 16.0000 0.655936
\(596\) −6.00000 −0.245770
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −16.0000 −0.652111
\(603\) −16.0000 −0.651570
\(604\) 12.0000 0.488273
\(605\) 10.0000 0.406558
\(606\) −2.00000 −0.0812444
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 8.00000 0.324443
\(609\) −24.0000 −0.972529
\(610\) 4.00000 0.161955
\(611\) 8.00000 0.323645
\(612\) 2.00000 0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 8.00000 0.322854
\(615\) 20.0000 0.806478
\(616\) 16.0000 0.644658
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 16.0000 0.643614
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 56.0000 2.24359
\(624\) −1.00000 −0.0400320
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) −32.0000 −1.27796
\(628\) 14.0000 0.558661
\(629\) −4.00000 −0.159490
\(630\) −8.00000 −0.318728
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) −8.00000 −0.318223
\(633\) −12.0000 −0.476957
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 9.00000 0.356593
\(638\) 24.0000 0.950169
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 12.0000 0.473602
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 16.0000 0.629512
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 1.00000 0.0392232
\(651\) 16.0000 0.627089
\(652\) −16.0000 −0.626608
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 8.00000 0.312586
\(656\) −10.0000 −0.390434
\(657\) 2.00000 0.0780274
\(658\) −32.0000 −1.24749
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 8.00000 0.311400
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −8.00000 −0.310929
\(663\) −2.00000 −0.0776736
\(664\) −12.0000 −0.465690
\(665\) −64.0000 −2.48181
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 32.0000 1.23627
\(671\) 8.00000 0.308837
\(672\) 4.00000 0.154303
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) −6.00000 −0.230429
\(679\) 40.0000 1.53506
\(680\) −4.00000 −0.153393
\(681\) −20.0000 −0.766402
\(682\) −16.0000 −0.612672
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −8.00000 −0.305888
\(685\) −20.0000 −0.764161
\(686\) −8.00000 −0.305441
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) −10.0000 −0.380143
\(693\) −16.0000 −0.607790
\(694\) 12.0000 0.455514
\(695\) 24.0000 0.910372
\(696\) 6.00000 0.227429
\(697\) −20.0000 −0.757554
\(698\) −6.00000 −0.227103
\(699\) −18.0000 −0.680823
\(700\) −4.00000 −0.151186
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 1.00000 0.0377426
\(703\) 16.0000 0.603451
\(704\) −4.00000 −0.150756
\(705\) −16.0000 −0.602595
\(706\) −14.0000 −0.526897
\(707\) −8.00000 −0.300871
\(708\) −4.00000 −0.150329
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 16.0000 0.600469
\(711\) 8.00000 0.300023
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) −8.00000 −0.299183
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 2.00000 0.0745356
\(721\) 64.0000 2.38348
\(722\) −45.0000 −1.67473
\(723\) −10.0000 −0.371904
\(724\) −10.0000 −0.371647
\(725\) −6.00000 −0.222834
\(726\) 5.00000 0.185567
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) −16.0000 −0.590571
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) 64.0000 2.35747
\(738\) 10.0000 0.368105
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) −4.00000 −0.147043
\(741\) 8.00000 0.293887
\(742\) 40.0000 1.46845
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −4.00000 −0.146647
\(745\) −12.0000 −0.439646
\(746\) −6.00000 −0.219676
\(747\) 12.0000 0.439057
\(748\) −8.00000 −0.292509
\(749\) 48.0000 1.75388
\(750\) −12.0000 −0.438178
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 8.00000 0.291730
\(753\) −4.00000 −0.145768
\(754\) −6.00000 −0.218507
\(755\) 24.0000 0.873449
\(756\) −4.00000 −0.145479
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 16.0000 0.580381
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) −8.00000 −0.289430
\(765\) 4.00000 0.144620
\(766\) 24.0000 0.867155
\(767\) 4.00000 0.144432
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 32.0000 1.15320
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) −10.0000 −0.358979
\(777\) 8.00000 0.286998
\(778\) 26.0000 0.932145
\(779\) 80.0000 2.86630
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 28.0000 0.999363
\(786\) 4.00000 0.142675
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 18.0000 0.641223
\(789\) −8.00000 −0.284808
\(790\) −16.0000 −0.569254
\(791\) −24.0000 −0.853342
\(792\) 4.00000 0.142134
\(793\) −2.00000 −0.0710221
\(794\) −6.00000 −0.212932
\(795\) 20.0000 0.709327
\(796\) −8.00000 −0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −32.0000 −1.13279
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) 14.0000 0.494666
\(802\) −6.00000 −0.211867
\(803\) −8.00000 −0.282314
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 26.0000 0.915243
\(808\) 2.00000 0.0703598
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 24.0000 0.842235
\(813\) 4.00000 0.140286
\(814\) −8.00000 −0.280400
\(815\) −32.0000 −1.12091
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) −2.00000 −0.0699284
\(819\) 4.00000 0.139771
\(820\) −20.0000 −0.698430
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −10.0000 −0.348790
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −16.0000 −0.557386
\(825\) −4.00000 −0.139262
\(826\) −16.0000 −0.556711
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −24.0000 −0.833052
\(831\) −22.0000 −0.763172
\(832\) 1.00000 0.0346688
\(833\) 18.0000 0.623663
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 32.0000 1.10674
\(837\) 4.00000 0.138260
\(838\) −4.00000 −0.138178
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 8.00000 0.276026
\(841\) 7.00000 0.241379
\(842\) −22.0000 −0.758170
\(843\) 26.0000 0.895488
\(844\) 12.0000 0.413057
\(845\) 2.00000 0.0688021
\(846\) −8.00000 −0.275046
\(847\) 20.0000 0.687208
\(848\) −10.0000 −0.343401
\(849\) 4.00000 0.137280
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 8.00000 0.273754
\(855\) −16.0000 −0.547188
\(856\) −12.0000 −0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −4.00000 −0.136558
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 8.00000 0.272798
\(861\) 40.0000 1.36320
\(862\) 8.00000 0.272481
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 1.00000 0.0340207
\(865\) −20.0000 −0.680020
\(866\) 30.0000 1.01944
\(867\) 13.0000 0.441503
\(868\) −16.0000 −0.543075
\(869\) −32.0000 −1.08553
\(870\) 12.0000 0.406838
\(871\) −16.0000 −0.542139
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) −2.00000 −0.0675737
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −16.0000 −0.539974
\(879\) −26.0000 −0.876958
\(880\) −8.00000 −0.269680
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) −9.00000 −0.303046
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 2.00000 0.0672673
\(885\) −8.00000 −0.268917
\(886\) 4.00000 0.134383
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) −28.0000 −0.938562
\(891\) −4.00000 −0.134005
\(892\) −4.00000 −0.133930
\(893\) −64.0000 −2.14168
\(894\) −6.00000 −0.200670
\(895\) −24.0000 −0.802232
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −24.0000 −0.800445
\(900\) −1.00000 −0.0333333
\(901\) −20.0000 −0.666297
\(902\) −40.0000 −1.33185
\(903\) −16.0000 −0.532447
\(904\) 6.00000 0.199557
\(905\) −20.0000 −0.664822
\(906\) 12.0000 0.398673
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 20.0000 0.663723
\(909\) −2.00000 −0.0663358
\(910\) −8.00000 −0.265197
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 8.00000 0.264906
\(913\) −48.0000 −1.58857
\(914\) 30.0000 0.992312
\(915\) 4.00000 0.132236
\(916\) 22.0000 0.726900
\(917\) 16.0000 0.528367
\(918\) 2.00000 0.0660098
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 6.00000 0.197599
\(923\) −8.00000 −0.263323
\(924\) 16.0000 0.526361
\(925\) 2.00000 0.0657596
\(926\) 20.0000 0.657241
\(927\) 16.0000 0.525509
\(928\) −6.00000 −0.196960
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) −8.00000 −0.262330
\(931\) −72.0000 −2.35970
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 4.00000 0.130884
\(935\) −16.0000 −0.523256
\(936\) −1.00000 −0.0326860
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 64.0000 2.08967
\(939\) 6.00000 0.195803
\(940\) 16.0000 0.521862
\(941\) −46.0000 −1.49956 −0.749779 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) 14.0000 0.456145
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) −8.00000 −0.260240
\(946\) 16.0000 0.520205
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −8.00000 −0.259828
\(949\) 2.00000 0.0649227
\(950\) −8.00000 −0.259554
\(951\) 6.00000 0.194563
\(952\) −8.00000 −0.259281
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 10.0000 0.323762
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 24.0000 0.775810
\(958\) 16.0000 0.516937
\(959\) −40.0000 −1.29167
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) 2.00000 0.0644826
\(963\) 12.0000 0.386695
\(964\) 10.0000 0.322078
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −5.00000 −0.160706
\(969\) 16.0000 0.513994
\(970\) −20.0000 −0.642161
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.0000 1.53881
\(974\) −4.00000 −0.128168
\(975\) 1.00000 0.0320256
\(976\) −2.00000 −0.0640184
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) −16.0000 −0.511624
\(979\) −56.0000 −1.78977
\(980\) 18.0000 0.574989
\(981\) −2.00000 −0.0638551
\(982\) −36.0000 −1.14881
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −10.0000 −0.318788
\(985\) 36.0000 1.14706
\(986\) −12.0000 −0.382158
\(987\) −32.0000 −1.01857
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 4.00000 0.127000
\(993\) −8.00000 −0.253872
\(994\) 32.0000 1.01498
\(995\) −16.0000 −0.507234
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 78.2.a.a.1.1 1
3.2 odd 2 234.2.a.c.1.1 1
4.3 odd 2 624.2.a.h.1.1 1
5.2 odd 4 1950.2.e.i.1249.1 2
5.3 odd 4 1950.2.e.i.1249.2 2
5.4 even 2 1950.2.a.w.1.1 1
7.6 odd 2 3822.2.a.j.1.1 1
8.3 odd 2 2496.2.a.b.1.1 1
8.5 even 2 2496.2.a.t.1.1 1
9.2 odd 6 2106.2.e.j.1405.1 2
9.4 even 3 2106.2.e.q.703.1 2
9.5 odd 6 2106.2.e.j.703.1 2
9.7 even 3 2106.2.e.q.1405.1 2
11.10 odd 2 9438.2.a.t.1.1 1
12.11 even 2 1872.2.a.c.1.1 1
13.2 odd 12 1014.2.i.d.823.1 4
13.3 even 3 1014.2.e.f.529.1 2
13.4 even 6 1014.2.e.c.991.1 2
13.5 odd 4 1014.2.b.b.337.2 2
13.6 odd 12 1014.2.i.d.361.2 4
13.7 odd 12 1014.2.i.d.361.1 4
13.8 odd 4 1014.2.b.b.337.1 2
13.9 even 3 1014.2.e.f.991.1 2
13.10 even 6 1014.2.e.c.529.1 2
13.11 odd 12 1014.2.i.d.823.2 4
13.12 even 2 1014.2.a.d.1.1 1
15.2 even 4 5850.2.e.bb.5149.2 2
15.8 even 4 5850.2.e.bb.5149.1 2
15.14 odd 2 5850.2.a.d.1.1 1
24.5 odd 2 7488.2.a.bz.1.1 1
24.11 even 2 7488.2.a.bk.1.1 1
39.5 even 4 3042.2.b.g.1351.1 2
39.8 even 4 3042.2.b.g.1351.2 2
39.38 odd 2 3042.2.a.f.1.1 1
52.51 odd 2 8112.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.a.a.1.1 1 1.1 even 1 trivial
234.2.a.c.1.1 1 3.2 odd 2
624.2.a.h.1.1 1 4.3 odd 2
1014.2.a.d.1.1 1 13.12 even 2
1014.2.b.b.337.1 2 13.8 odd 4
1014.2.b.b.337.2 2 13.5 odd 4
1014.2.e.c.529.1 2 13.10 even 6
1014.2.e.c.991.1 2 13.4 even 6
1014.2.e.f.529.1 2 13.3 even 3
1014.2.e.f.991.1 2 13.9 even 3
1014.2.i.d.361.1 4 13.7 odd 12
1014.2.i.d.361.2 4 13.6 odd 12
1014.2.i.d.823.1 4 13.2 odd 12
1014.2.i.d.823.2 4 13.11 odd 12
1872.2.a.c.1.1 1 12.11 even 2
1950.2.a.w.1.1 1 5.4 even 2
1950.2.e.i.1249.1 2 5.2 odd 4
1950.2.e.i.1249.2 2 5.3 odd 4
2106.2.e.j.703.1 2 9.5 odd 6
2106.2.e.j.1405.1 2 9.2 odd 6
2106.2.e.q.703.1 2 9.4 even 3
2106.2.e.q.1405.1 2 9.7 even 3
2496.2.a.b.1.1 1 8.3 odd 2
2496.2.a.t.1.1 1 8.5 even 2
3042.2.a.f.1.1 1 39.38 odd 2
3042.2.b.g.1351.1 2 39.5 even 4
3042.2.b.g.1351.2 2 39.8 even 4
3822.2.a.j.1.1 1 7.6 odd 2
5850.2.a.d.1.1 1 15.14 odd 2
5850.2.e.bb.5149.1 2 15.8 even 4
5850.2.e.bb.5149.2 2 15.2 even 4
7488.2.a.bk.1.1 1 24.11 even 2
7488.2.a.bz.1.1 1 24.5 odd 2
8112.2.a.v.1.1 1 52.51 odd 2
9438.2.a.t.1.1 1 11.10 odd 2