Properties

Label 33.2.a.a.1.1
Level 33
Weight 2
Character 33.1
Self dual yes
Analytic conductor 0.264
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.263506326670\)
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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} +8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{42} -1.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{55} -12.0000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -8.00000 q^{70} -3.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{77} +2.00000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} +4.00000 q^{85} +6.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -8.00000 q^{91} -8.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} +2.00000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) 1.00000 0.213201
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000 0.883883
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) −8.00000 −1.35225
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 6.00000 0.948683
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −4.00000 −0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.00000 −0.298142
\(46\) 8.00000 1.17954
\(47\) 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
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −2.00000 −0.258199
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) −1.00000 −0.123091
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 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
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −2.00000 −0.210819
\(91\) −8.00000 −0.838628
\(92\) −8.00000 −0.834058
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 9.00000 0.909137
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) 8.00000 0.780720
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −16.0000 −1.49201
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) −8.00000 −0.733359
\(120\) −6.00000 −0.547723
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −8.00000 −0.681005
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 8.00000 0.676123
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) −1.00000 −0.0833333
\(145\) 12.0000 0.996546
\(146\) −14.0000 −1.15865
\(147\) −9.00000 −0.742307
\(148\) −6.00000 −0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 4.00000 0.322329
\(155\) 16.0000 1.28515
\(156\) −2.00000 −0.160128
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −4.00000 −0.318223
\(159\) −6.00000 −0.475831
\(160\) −10.0000 −0.790569
\(161\) 32.0000 2.52195
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) 2.00000 0.155700
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 12.0000 0.925820
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) −4.00000 −0.302372
\(176\) −1.00000 −0.0753778
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −8.00000 −0.592999
\(183\) −6.00000 −0.443533
\(184\) −24.0000 −1.76930
\(185\) −12.0000 −0.882258
\(186\) 8.00000 0.586588
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) −9.00000 −0.642857
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 1.00000 0.0710669
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 3.00000 0.212132
\(201\) 4.00000 0.282138
\(202\) 2.00000 0.140720
\(203\) −24.0000 −1.68447
\(204\) −2.00000 −0.140028
\(205\) 4.00000 0.279372
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) −32.0000 −2.17230
\(218\) −2.00000 −0.135457
\(219\) 14.0000 0.946032
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) −6.00000 −0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 20.0000 1.33631
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −16.0000 −1.05501
\(231\) −4.00000 −0.263181
\(232\) 18.0000 1.18176
\(233\) 30.0000 1.96537 0.982683 0.185296i \(-0.0593245\pi\)
0.982683 + 0.185296i \(0.0593245\pi\)
\(234\) −2.00000 −0.130744
\(235\) −16.0000 −1.04372
\(236\) 4.00000 0.260378
\(237\) 4.00000 0.259828
\(238\) −8.00000 −0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −2.00000 −0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −18.0000 −1.14998
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 24.0000 1.52400
\(249\) −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
\(253\) 8.00000 0.502956
\(254\) −4.00000 −0.250982
\(255\) −4.00000 −0.250490
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 3.00000 0.184637
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 2.00000 0.121716
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 2.00000 0.121268
\(273\) 8.00000 0.484182
\(274\) 2.00000 0.120824
\(275\) −1.00000 −0.0603023
\(276\) 8.00000 0.481543
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) −8.00000 −0.479808
\(279\) −8.00000 −0.478947
\(280\) 24.0000 1.43427
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −8.00000 −0.476393
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) −8.00000 −0.472225
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 12.0000 0.704664
\(291\) −2.00000 −0.117242
\(292\) 14.0000 0.819288
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −9.00000 −0.524891
\(295\) 8.00000 0.465778
\(296\) −18.0000 −1.04623
\(297\) −1.00000 −0.0580259
\(298\) −22.0000 −1.27443
\(299\) −16.0000 −0.925304
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) −2.00000 −0.114332
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) −4.00000 −0.227921
\(309\) −8.00000 −0.455104
\(310\) 16.0000 0.908739
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −6.00000 −0.339683
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 14.0000 0.790066
\(315\) −8.00000 −0.450749
\(316\) 4.00000 0.225018
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −6.00000 −0.336463
\(319\) −6.00000 −0.335936
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 32.0000 1.78329
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) 6.00000 0.331295
\(329\) 32.0000 1.76422
\(330\) 2.00000 0.110096
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −12.0000 −0.658586
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 4.00000 0.218218
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) −4.00000 −0.216930
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) −6.00000 −0.322562
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −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
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 8.00000 0.423405
\(358\) 12.0000 0.634220
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 6.00000 0.316228
\(361\) −19.0000 −1.00000
\(362\) 22.0000 1.15629
\(363\) −1.00000 −0.0524864
\(364\) 8.00000 0.419314
\(365\) 28.0000 1.46559
\(366\) −6.00000 −0.313625
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −8.00000 −0.417029
\(369\) −2.00000 −0.104116
\(370\) −12.0000 −0.623850
\(371\) 24.0000 1.24602
\(372\) −8.00000 −0.414781
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) −2.00000 −0.103418
\(375\) −12.0000 −0.619677
\(376\) −24.0000 −1.23771
\(377\) 12.0000 0.618031
\(378\) −4.00000 −0.205738
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 3.00000 0.153093
\(385\) −8.00000 −0.407718
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) −4.00000 −0.202548
\(391\) −16.0000 −0.809155
\(392\) −27.0000 −1.36371
\(393\) 12.0000 0.605320
\(394\) −14.0000 −0.705310
\(395\) 8.00000 0.402524
\(396\) −1.00000 −0.0502519
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 4.00000 0.199502
\(403\) 16.0000 0.797017
\(404\) −2.00000 −0.0995037
\(405\) −2.00000 −0.0993808
\(406\) −24.0000 −1.19110
\(407\) 6.00000 0.297409
\(408\) −6.00000 −0.297044
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 4.00000 0.197546
\(411\) −2.00000 −0.0986527
\(412\) −8.00000 −0.394132
\(413\) −16.0000 −0.787309
\(414\) 8.00000 0.393179
\(415\) −24.0000 −1.17811
\(416\) −10.0000 −0.490290
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −8.00000 −0.390360
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) −18.0000 −0.874157
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 12.0000 0.580042
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −32.0000 −1.53605
\(435\) −12.0000 −0.575356
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 14.0000 0.668946
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 6.00000 0.286039
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 6.00000 0.284747
\(445\) 12.0000 0.568855
\(446\) 16.0000 0.757622
\(447\) 22.0000 1.04056
\(448\) 28.0000 1.32288
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 6.00000 0.282216
\(453\) −20.0000 −0.939682
\(454\) 12.0000 0.563188
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 6.00000 0.280362
\(459\) 2.00000 0.0933520
\(460\) 16.0000 0.746004
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) −4.00000 −0.186097
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) −16.0000 −0.741982
\(466\) 30.0000 1.38972
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) −16.0000 −0.738025
\(471\) −14.0000 −0.645086
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 10.0000 0.456435
\(481\) −12.0000 −0.547153
\(482\) 10.0000 0.455488
\(483\) −32.0000 −1.45605
\(484\) −1.00000 −0.0454545
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −18.0000 −0.814822
\(489\) −4.00000 −0.180886
\(490\) −18.0000 −0.813157
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −32.0000 −1.42681 −0.713405 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) −12.0000 −0.534522
\(505\) −4.00000 −0.177998
\(506\) 8.00000 0.355643
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −4.00000 −0.177123
\(511\) −56.0000 −2.47729
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 24.0000 1.05450
\(519\) 6.00000 0.263371
\(520\) −12.0000 −0.526235
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 4.00000 0.174574
\(526\) −16.0000 −0.697633
\(527\) 16.0000 0.696971
\(528\) 1.00000 0.0435194
\(529\) 41.0000 1.78261
\(530\) −12.0000 −0.521247
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 6.00000 0.259645
\(535\) 24.0000 1.03761
\(536\) 12.0000 0.518321
\(537\) −12.0000 −0.517838
\(538\) −2.00000 −0.0862261
\(539\) 9.00000 0.387657
\(540\) −2.00000 −0.0860663
\(541\) 46.0000 1.97769 0.988847 0.148933i \(-0.0475840\pi\)
0.988847 + 0.148933i \(0.0475840\pi\)
\(542\) 20.0000 0.859074
\(543\) −22.0000 −0.944110
\(544\) −10.0000 −0.428746
\(545\) 4.00000 0.171341
\(546\) 8.00000 0.342368
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 6.00000 0.256074
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 24.0000 1.02151
\(553\) −16.0000 −0.680389
\(554\) −26.0000 −1.10463
\(555\) 12.0000 0.509372
\(556\) 8.00000 0.339276
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 8.00000 0.338062
\(561\) 2.00000 0.0844401
\(562\) −18.0000 −0.759284
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 8.00000 0.336861
\(565\) 12.0000 0.504844
\(566\) 16.0000 0.672530
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 2.00000 0.0836242
\(573\) −8.00000 −0.334205
\(574\) −8.00000 −0.333914
\(575\) −8.00000 −0.333623
\(576\) 7.00000 0.291667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.0000 0.581820
\(580\) −12.0000 −0.498273
\(581\) 48.0000 1.99138
\(582\) −2.00000 −0.0829027
\(583\) 6.00000 0.248495
\(584\) 42.0000 1.73797
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 8.00000 0.329355
\(591\) 14.0000 0.575883
\(592\) −6.00000 −0.246598
\(593\) 38.0000 1.56047 0.780236 0.625485i \(-0.215099\pi\)
0.780236 + 0.625485i \(0.215099\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 16.0000 0.655936
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) −3.00000 −0.122474
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −20.0000 −0.813788
\(605\) −2.00000 −0.0813116
\(606\) −2.00000 −0.0812444
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) −12.0000 −0.485866
\(611\) −16.0000 −0.647291
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 32.0000 1.29141
\(615\) −4.00000 −0.161296
\(616\) −12.0000 −0.483494
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) −16.0000 −0.642575
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) −24.0000 −0.961540
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) −12.0000 −0.478471
\(630\) −8.00000 −0.318728
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) 8.00000 0.317470
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 6.00000 0.237171
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000 0.473602
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −32.0000 −1.26098
\(645\) 0 0
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) −3.00000 −0.117851
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 32.0000 1.25418
\(652\) −4.00000 −0.156652
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 2.00000 0.0782062
\(655\) 24.0000 0.937758
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 32.0000 1.24749
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −20.0000 −0.777322
\(663\) −4.00000 −0.155347
\(664\) −36.0000 −1.39707
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −48.0000 −1.85857
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 8.00000 0.309067
\(671\) 6.00000 0.231627
\(672\) −20.0000 −0.771517
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 6.00000 0.230429
\(679\) 8.00000 0.307012
\(680\) −12.0000 −0.460179
\(681\) −12.0000 −0.459841
\(682\) −8.00000 −0.306336
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 8.00000 0.305441
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 16.0000 0.609110
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 6.00000 0.228086
\(693\) 4.00000 0.151947
\(694\) 4.00000 0.151838
\(695\) 16.0000 0.606915
\(696\) −18.0000 −0.682288
\(697\) 4.00000 0.151511
\(698\) 6.00000 0.227103
\(699\) −30.0000 −1.13470
\(700\) 4.00000 0.151186
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 7.00000 0.263822
\(705\) 16.0000 0.602595
\(706\) 18.0000 0.677439
\(707\) 8.00000 0.300871
\(708\) −4.00000 −0.150329
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) −64.0000 −2.39682
\(714\) 8.00000 0.299392
\(715\) 4.00000 0.149592
\(716\) −12.0000 −0.448461
\(717\) −24.0000 −0.896296
\(718\) −8.00000 −0.298557
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 2.00000 0.0745356
\(721\) 32.0000 1.19174
\(722\) −19.0000 −0.707107
\(723\) −10.0000 −0.371904
\(724\) −22.0000 −0.817624
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 24.0000 0.889499
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 0 0
\(732\) 6.00000 0.221766
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) −32.0000 −1.18114
\(735\) 18.0000 0.663940
\(736\) 40.0000 1.47442
\(737\) −4.00000 −0.147342
\(738\) −2.00000 −0.0736210
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) −24.0000 −0.879883
\(745\) 44.0000 1.61204
\(746\) −2.00000 −0.0732252
\(747\) 12.0000 0.439057
\(748\) 2.00000 0.0731272
\(749\) −48.0000 −1.75388
\(750\) −12.0000 −0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −8.00000 −0.291730
\(753\) −4.00000 −0.145768
\(754\) 12.0000 0.437014
\(755\) −40.0000 −1.45575
\(756\) 4.00000 0.145479
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 28.0000 1.01701
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000 0.144905
\(763\) −8.00000 −0.289619
\(764\) −8.00000 −0.289430
\(765\) 4.00000 0.144620
\(766\) −16.0000 −0.578103
\(767\) 8.00000 0.288863
\(768\) 17.0000 0.613435
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −8.00000 −0.288300
\(771\) 14.0000 0.504198
\(772\) 14.0000 0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −6.00000 −0.215387
\(777\) −24.0000 −0.860995
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 6.00000 0.214423
\(784\) −9.00000 −0.321429
\(785\) −28.0000 −0.999363
\(786\) 12.0000 0.428026
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 14.0000 0.498729
\(789\) 16.0000 0.569615
\(790\) 8.00000 0.284627
\(791\) −24.0000 −0.853342
\(792\) −3.00000 −0.106600
\(793\) −12.0000 −0.426132
\(794\) −2.00000 −0.0709773
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) −5.00000 −0.176777
\(801\) −6.00000 −0.212000
\(802\) 26.0000 0.918092
\(803\) −14.0000 −0.494049
\(804\) −4.00000 −0.141069
\(805\) −64.0000 −2.25570
\(806\) 16.0000 0.563576
\(807\) 2.00000 0.0704033
\(808\) −6.00000 −0.211079
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −56.0000 −1.96643 −0.983213 0.182462i \(-0.941593\pi\)
−0.983213 + 0.182462i \(0.941593\pi\)
\(812\) 24.0000 0.842235
\(813\) −20.0000 −0.701431
\(814\) 6.00000 0.210300
\(815\) −8.00000 −0.280228
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) −8.00000 −0.279543
\(820\) −4.00000 −0.139686
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −24.0000 −0.836080
\(825\) 1.00000 0.0348155
\(826\) −16.0000 −0.556711
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) −8.00000 −0.278019
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −24.0000 −0.833052
\(831\) 26.0000 0.901930
\(832\) −14.0000 −0.485363
\(833\) −18.0000 −0.623663
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −4.00000 −0.138178
\(839\) −56.0000 −1.93333 −0.966667 0.256036i \(-0.917584\pi\)
−0.966667 + 0.256036i \(0.917584\pi\)
\(840\) −24.0000 −0.828079
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) 18.0000 0.619219
\(846\) 8.00000 0.275046
\(847\) 4.00000 0.137442
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 2.00000 0.0685994
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 2.00000 0.0682789
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) −24.0000 −0.817443
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −5.00000 −0.170103
\(865\) 12.0000 0.408012
\(866\) 34.0000 1.15537
\(867\) 13.0000 0.441503
\(868\) 32.0000 1.08615
\(869\) −4.00000 −0.135691
\(870\) −12.0000 −0.406838
\(871\) 8.00000 0.271070
\(872\) 6.00000 0.203186
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) −14.0000 −0.473016
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −20.0000 −0.674967
\(879\) 6.00000 0.202375
\(880\) 2.00000 0.0674200
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 9.00000 0.303046
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) −8.00000 −0.268917
\(886\) 28.0000 0.940678
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 18.0000 0.604040
\(889\) −16.0000 −0.536623
\(890\) 12.0000 0.402241
\(891\) 1.00000 0.0335013
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) −24.0000 −0.802232
\(896\) −12.0000 −0.400892
\(897\) 16.0000 0.534224
\(898\) 2.00000 0.0667409
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) −44.0000 −1.46261
\(906\) −20.0000 −0.664455
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −12.0000 −0.398234
\(909\) 2.00000 0.0663358
\(910\) 16.0000 0.530395
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 18.0000 0.595387
\(915\) 12.0000 0.396708
\(916\) −6.00000 −0.198246
\(917\) −48.0000 −1.58510
\(918\) 2.00000 0.0660098
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 48.0000 1.58251
\(921\) −32.0000 −1.05444
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) −6.00000 −0.197279
\(926\) 16.0000 0.525793
\(927\) 8.00000 0.262754
\(928\) −30.0000 −0.984798
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) −16.0000 −0.524661
\(931\) 0 0
\(932\) −30.0000 −0.982683
\(933\) 24.0000 0.785725
\(934\) −12.0000 −0.392652
\(935\) 4.00000 0.130814
\(936\) 6.00000 0.196116
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −16.0000 −0.522419
\(939\) 22.0000 0.717943
\(940\) 16.0000 0.521862
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) −14.0000 −0.456145
\(943\) −16.0000 −0.521032
\(944\) 4.00000 0.130189
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −4.00000 −0.129914
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 24.0000 0.777844
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 6.00000 0.194257
\(955\) −16.0000 −0.517748
\(956\) −24.0000 −0.776215
\(957\) 6.00000 0.193952
\(958\) 8.00000 0.258468
\(959\) 8.00000 0.258333
\(960\) 14.0000 0.451848
\(961\) 33.0000 1.06452
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 28.0000 0.901352
\(966\) −32.0000 −1.02958
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 1.00000 0.0320750
\(973\) −32.0000 −1.02587
\(974\) −16.0000 −0.512673
\(975\) −2.00000 −0.0640513
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −4.00000 −0.127906
\(979\) −6.00000 −0.191761
\(980\) 18.0000 0.574989
\(981\) −2.00000 −0.0638551
\(982\) 4.00000 0.127645
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 28.0000 0.892154
\(986\) 12.0000 0.382158
\(987\) −32.0000 −1.01857
\(988\) 0 0
\(989\) 0 0
\(990\) −2.00000 −0.0635642
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −40.0000 −1.27000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −4.00000 −0.126618
\(999\) −6.00000 −0.189832
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 33.2.a.a.1.1 1
3.2 odd 2 99.2.a.b.1.1 1
4.3 odd 2 528.2.a.g.1.1 1
5.2 odd 4 825.2.c.a.199.2 2
5.3 odd 4 825.2.c.a.199.1 2
5.4 even 2 825.2.a.a.1.1 1
7.6 odd 2 1617.2.a.j.1.1 1
8.3 odd 2 2112.2.a.j.1.1 1
8.5 even 2 2112.2.a.bb.1.1 1
9.2 odd 6 891.2.e.g.595.1 2
9.4 even 3 891.2.e.e.298.1 2
9.5 odd 6 891.2.e.g.298.1 2
9.7 even 3 891.2.e.e.595.1 2
11.2 odd 10 363.2.e.g.202.1 4
11.3 even 5 363.2.e.e.130.1 4
11.4 even 5 363.2.e.e.148.1 4
11.5 even 5 363.2.e.e.124.1 4
11.6 odd 10 363.2.e.g.124.1 4
11.7 odd 10 363.2.e.g.148.1 4
11.8 odd 10 363.2.e.g.130.1 4
11.9 even 5 363.2.e.e.202.1 4
11.10 odd 2 363.2.a.b.1.1 1
12.11 even 2 1584.2.a.o.1.1 1
13.12 even 2 5577.2.a.a.1.1 1
15.2 even 4 2475.2.c.d.199.1 2
15.8 even 4 2475.2.c.d.199.2 2
15.14 odd 2 2475.2.a.g.1.1 1
17.16 even 2 9537.2.a.m.1.1 1
21.20 even 2 4851.2.a.b.1.1 1
24.5 odd 2 6336.2.a.x.1.1 1
24.11 even 2 6336.2.a.n.1.1 1
33.32 even 2 1089.2.a.j.1.1 1
44.43 even 2 5808.2.a.t.1.1 1
55.54 odd 2 9075.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 1 1.1 even 1 trivial
99.2.a.b.1.1 1 3.2 odd 2
363.2.a.b.1.1 1 11.10 odd 2
363.2.e.e.124.1 4 11.5 even 5
363.2.e.e.130.1 4 11.3 even 5
363.2.e.e.148.1 4 11.4 even 5
363.2.e.e.202.1 4 11.9 even 5
363.2.e.g.124.1 4 11.6 odd 10
363.2.e.g.130.1 4 11.8 odd 10
363.2.e.g.148.1 4 11.7 odd 10
363.2.e.g.202.1 4 11.2 odd 10
528.2.a.g.1.1 1 4.3 odd 2
825.2.a.a.1.1 1 5.4 even 2
825.2.c.a.199.1 2 5.3 odd 4
825.2.c.a.199.2 2 5.2 odd 4
891.2.e.e.298.1 2 9.4 even 3
891.2.e.e.595.1 2 9.7 even 3
891.2.e.g.298.1 2 9.5 odd 6
891.2.e.g.595.1 2 9.2 odd 6
1089.2.a.j.1.1 1 33.32 even 2
1584.2.a.o.1.1 1 12.11 even 2
1617.2.a.j.1.1 1 7.6 odd 2
2112.2.a.j.1.1 1 8.3 odd 2
2112.2.a.bb.1.1 1 8.5 even 2
2475.2.a.g.1.1 1 15.14 odd 2
2475.2.c.d.199.1 2 15.2 even 4
2475.2.c.d.199.2 2 15.8 even 4
4851.2.a.b.1.1 1 21.20 even 2
5577.2.a.a.1.1 1 13.12 even 2
5808.2.a.t.1.1 1 44.43 even 2
6336.2.a.n.1.1 1 24.11 even 2
6336.2.a.x.1.1 1 24.5 odd 2
9075.2.a.q.1.1 1 55.54 odd 2
9537.2.a.m.1.1 1 17.16 even 2