Properties

Label 105.2.a.a.1.1
Level $105$
Weight $2$
Character 105.1
Self dual yes
Analytic conductor $0.838$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.838429221223\)
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 \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} +8.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -8.00000 q^{38} -6.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{45} +8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -3.00000 q^{56} -8.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} +1.00000 q^{70} -12.0000 q^{71} -3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -6.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} +1.00000 q^{90} -6.00000 q^{91} -8.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -8.00000 q^{95} +5.00000 q^{96} -18.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) −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 \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 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
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) −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
\(39\) −6.00000 −0.960769
\(40\) −3.00000 −0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 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
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) 6.00000 0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −8.00000 −1.05963
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) 8.00000 0.963087
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −3.00000 −0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.00000 −0.628971
\(92\) −8.00000 −0.834058
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) −8.00000 −0.820783
\(96\) 5.00000 0.510310
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 18.0000 1.76505
\(105\) 1.00000 0.0975900
\(106\) 10.0000 0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −8.00000 −0.749269
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) 4.00000 0.368230
\(119\) 2.00000 0.183340
\(120\) −3.00000 −0.273861
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.00000 0.352180
\(130\) −6.00000 −0.526235
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) −6.00000 −0.514496
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 8.00000 0.681005
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 8.00000 0.673722
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −2.00000 −0.166091
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 24.0000 1.94666
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 6.00000 0.480384
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 8.00000 0.636446
\(159\) 10.0000 0.793052
\(160\) 5.00000 0.395285
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −3.00000 −0.231455
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) −8.00000 −0.611775
\(172\) −4.00000 −0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −2.00000 −0.151620
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −6.00000 −0.444750
\(183\) −2.00000 −0.147844
\(184\) −24.0000 −1.76930
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) −8.00000 −0.580381
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 7.00000 0.505181
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) −18.0000 −1.29232
\(195\) −6.00000 −0.429669
\(196\) −1.00000 −0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −3.00000 −0.212132
\(201\) 4.00000 0.282138
\(202\) −10.0000 −0.703598
\(203\) −2.00000 −0.140372
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −10.0000 −0.686803
\(213\) −12.0000 −0.822226
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) −3.00000 −0.204124
\(217\) 4.00000 0.271538
\(218\) −18.0000 −1.21911
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −2.00000 −0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 5.00000 0.334077
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 8.00000 0.529813
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −6.00000 −0.392232
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 2.00000 0.129641
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 1.00000 0.0638877
\(246\) −6.00000 −0.382546
\(247\) 48.0000 3.05417
\(248\) −12.0000 −0.762001
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 2.00000 0.125245
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 4.00000 0.249029
\(259\) −2.00000 −0.124274
\(260\) 6.00000 0.372104
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) −8.00000 −0.490511
\(267\) −6.00000 −0.367194
\(268\) −4.00000 −0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) −6.00000 −0.363137
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) −3.00000 −0.179284
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 8.00000 0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.0000 0.712069
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) −18.0000 −1.05518
\(292\) 2.00000 0.117041
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 1.00000 0.0583212
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) −48.0000 −2.77591
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) 8.00000 0.458831
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 4.00000 0.227185
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 18.0000 1.01905
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −14.0000 −0.790066
\(315\) 1.00000 0.0563436
\(316\) −8.00000 −0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) −12.0000 −0.669775
\(322\) 8.00000 0.445823
\(323\) −16.0000 −0.890264
\(324\) −1.00000 −0.0555556
\(325\) −6.00000 −0.332820
\(326\) 12.0000 0.664619
\(327\) −18.0000 −0.995402
\(328\) 18.0000 0.993884
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) 8.00000 0.437741
\(335\) 4.00000 0.218543
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 23.0000 1.25104
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 1.00000 0.0539949
\(344\) −12.0000 −0.646997
\(345\) 8.00000 0.430706
\(346\) 6.00000 0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 2.00000 0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 1.00000 0.0534522
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) −12.0000 −0.636894
\(356\) 6.00000 0.317999
\(357\) 2.00000 0.105851
\(358\) −24.0000 −1.26844
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) −3.00000 −0.158114
\(361\) 45.0000 2.36842
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 6.00000 0.314485
\(365\) −2.00000 −0.104685
\(366\) −2.00000 −0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −8.00000 −0.417029
\(369\) −6.00000 −0.312348
\(370\) −2.00000 −0.103975
\(371\) 10.0000 0.519174
\(372\) −4.00000 −0.207390
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −24.0000 −1.23771
\(377\) 12.0000 0.618031
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 8.00000 0.410391
\(381\) 8.00000 0.409852
\(382\) 4.00000 0.204658
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 4.00000 0.203331
\(388\) 18.0000 0.913812
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −6.00000 −0.303822
\(391\) 16.0000 0.809155
\(392\) −3.00000 −0.151523
\(393\) 20.0000 1.00887
\(394\) 18.0000 0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −4.00000 −0.200502
\(399\) −8.00000 −0.400501
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 4.00000 0.199502
\(403\) −24.0000 −1.19553
\(404\) 10.0000 0.497519
\(405\) 1.00000 0.0496904
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) −10.0000 −0.493264
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) 8.00000 0.393179
\(415\) −4.00000 −0.196352
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) −30.0000 −1.45693
\(425\) 2.00000 0.0970143
\(426\) −12.0000 −0.581402
\(427\) −2.00000 −0.0967868
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 4.00000 0.192006
\(435\) −2.00000 −0.0958927
\(436\) 18.0000 0.862044
\(437\) −64.0000 −3.06154
\(438\) −2.00000 −0.0955637
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) −6.00000 −0.284427
\(446\) −24.0000 −1.13643
\(447\) 14.0000 0.662177
\(448\) 7.00000 0.330719
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) −4.00000 −0.187729
\(455\) −6.00000 −0.281284
\(456\) 24.0000 1.12390
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 22.0000 1.02799
\(459\) 2.00000 0.0933520
\(460\) −8.00000 −0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) 4.00000 0.185496
\(466\) −18.0000 −0.833834
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 6.00000 0.277350
\(469\) 4.00000 0.184703
\(470\) 8.00000 0.369012
\(471\) −14.0000 −0.645086
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) −8.00000 −0.367065
\(476\) −2.00000 −0.0916698
\(477\) 10.0000 0.457869
\(478\) −4.00000 −0.182956
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 5.00000 0.228218
\(481\) 12.0000 0.547153
\(482\) −6.00000 −0.273293
\(483\) 8.00000 0.364013
\(484\) 11.0000 0.500000
\(485\) −18.0000 −0.817338
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 6.00000 0.271607
\(489\) 12.0000 0.542659
\(490\) 1.00000 0.0451754
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) 48.0000 2.15962
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −12.0000 −0.538274
\(498\) −4.00000 −0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.00000 0.357414
\(502\) −12.0000 −0.535586
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) −3.00000 −0.133631
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) −8.00000 −0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 2.00000 0.0885615
\(511\) −2.00000 −0.0884748
\(512\) −11.0000 −0.486136
\(513\) −8.00000 −0.353209
\(514\) −6.00000 −0.264649
\(515\) 8.00000 0.352522
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 6.00000 0.263371
\(520\) 18.0000 0.789352
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −20.0000 −0.873704
\(525\) 1.00000 0.0436436
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 10.0000 0.434372
\(531\) 4.00000 0.173585
\(532\) 8.00000 0.346844
\(533\) 36.0000 1.55933
\(534\) −6.00000 −0.259645
\(535\) −12.0000 −0.518805
\(536\) −12.0000 −0.518321
\(537\) −24.0000 −1.03568
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 12.0000 0.515444
\(543\) −2.00000 −0.0858282
\(544\) 10.0000 0.428746
\(545\) −18.0000 −0.771035
\(546\) −6.00000 −0.256776
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) −24.0000 −1.02151
\(553\) 8.00000 0.340195
\(554\) 14.0000 0.594803
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 4.00000 0.169334
\(559\) −24.0000 −1.01509
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 36.0000 1.51053
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) −8.00000 −0.335083
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) −6.00000 −0.250435
\(575\) 8.00000 0.333623
\(576\) 7.00000 0.291667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −13.0000 −0.540729
\(579\) 18.0000 0.748054
\(580\) 2.00000 0.0830455
\(581\) −4.00000 −0.165948
\(582\) −18.0000 −0.746124
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) −6.00000 −0.248069
\(586\) 14.0000 0.578335
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) 4.00000 0.164677
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −14.0000 −0.573462
\(597\) −4.00000 −0.163709
\(598\) −48.0000 −1.96287
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −3.00000 −0.122474
\(601\) 18.0000 0.734235 0.367118 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −11.0000 −0.447214
\(606\) −10.0000 −0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −40.0000 −1.62221
\(609\) −2.00000 −0.0810441
\(610\) −2.00000 −0.0809776
\(611\) −48.0000 −1.94187
\(612\) −2.00000 −0.0808452
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 12.0000 0.484281
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 8.00000 0.321807
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −4.00000 −0.160644
\(621\) 8.00000 0.321029
\(622\) 24.0000 0.962312
\(623\) −6.00000 −0.240385
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −4.00000 −0.159490
\(630\) 1.00000 0.0398410
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −24.0000 −0.954669
\(633\) −20.0000 −0.794929
\(634\) 2.00000 0.0794301
\(635\) 8.00000 0.317470
\(636\) −10.0000 −0.396526
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −3.00000 −0.118585
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) −8.00000 −0.315244
\(645\) 4.00000 0.157500
\(646\) −16.0000 −0.629512
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 4.00000 0.156772
\(652\) −12.0000 −0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −18.0000 −0.703856
\(655\) 20.0000 0.781465
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 8.00000 0.311872
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −12.0000 −0.466393
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) −8.00000 −0.310227
\(666\) −2.00000 −0.0774984
\(667\) −16.0000 −0.619522
\(668\) −8.00000 −0.309529
\(669\) −24.0000 −0.927894
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 5.00000 0.192879
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −14.0000 −0.539260
\(675\) 1.00000 0.0384900
\(676\) −23.0000 −0.884615
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 6.00000 0.230429
\(679\) −18.0000 −0.690777
\(680\) −6.00000 −0.230089
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 8.00000 0.305888
\(685\) −10.0000 −0.382080
\(686\) 1.00000 0.0381802
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) −60.0000 −2.28582
\(690\) 8.00000 0.304555
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −12.0000 −0.454532
\(698\) 14.0000 0.529908
\(699\) −18.0000 −0.680823
\(700\) −1.00000 −0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −6.00000 −0.226455
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) −10.0000 −0.376089
\(708\) −4.00000 −0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −12.0000 −0.450352
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) 32.0000 1.19841
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) −4.00000 −0.149383
\(718\) −36.0000 −1.34351
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 45.0000 1.67473
\(723\) −6.00000 −0.223142
\(724\) 2.00000 0.0743294
\(725\) −2.00000 −0.0742781
\(726\) −11.0000 −0.408248
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 18.0000 0.667124
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 8.00000 0.295891
\(732\) 2.00000 0.0739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −8.00000 −0.295285
\(735\) 1.00000 0.0368856
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 2.00000 0.0735215
\(741\) 48.0000 1.76332
\(742\) 10.0000 0.367112
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −12.0000 −0.439941
\(745\) 14.0000 0.512920
\(746\) −10.0000 −0.366126
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) 8.00000 0.291150
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 8.00000 0.289809
\(763\) −18.0000 −0.651644
\(764\) −4.00000 −0.144715
\(765\) 2.00000 0.0723102
\(766\) −32.0000 −1.15621
\(767\) −24.0000 −0.866590
\(768\) −17.0000 −0.613435
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −18.0000 −0.647834
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) 54.0000 1.93849
\(777\) −2.00000 −0.0717496
\(778\) 30.0000 1.07555
\(779\) 48.0000 1.71978
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) −2.00000 −0.0714742
\(784\) −1.00000 −0.0357143
\(785\) −14.0000 −0.499681
\(786\) 20.0000 0.713376
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −18.0000 −0.641223
\(789\) 16.0000 0.569615
\(790\) 8.00000 0.284627
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −22.0000 −0.780751
\(795\) 10.0000 0.354663
\(796\) 4.00000 0.141776
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) −8.00000 −0.283197
\(799\) 16.0000 0.566039
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 8.00000 0.281963
\(806\) −24.0000 −0.845364
\(807\) 14.0000 0.492823
\(808\) 30.0000 1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 2.00000 0.0701862
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) −2.00000 −0.0700140
\(817\) −32.0000 −1.11954
\(818\) −22.0000 −0.769212
\(819\) −6.00000 −0.209657
\(820\) 6.00000 0.209529
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −10.0000 −0.348790
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −8.00000 −0.278019
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) −4.00000 −0.138842
\(831\) 14.0000 0.485655
\(832\) −42.0000 −1.45609
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −3.00000 −0.103510
\(841\) −25.0000 −0.862069
\(842\) −26.0000 −0.896019
\(843\) 18.0000 0.619953
\(844\) 20.0000 0.688428
\(845\) 23.0000 0.791224
\(846\) 8.00000 0.275046
\(847\) −11.0000 −0.377964
\(848\) −10.0000 −0.343401
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) −16.0000 −0.548473
\(852\) 12.0000 0.411113
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −8.00000 −0.273594
\(856\) 36.0000 1.23045
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −4.00000 −0.136399
\(861\) −6.00000 −0.204479
\(862\) 28.0000 0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) −2.00000 −0.0679628
\(867\) −13.0000 −0.441503
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) −24.0000 −0.813209
\(872\) 54.0000 1.82867
\(873\) −18.0000 −0.609208
\(874\) −64.0000 −2.16483
\(875\) 1.00000 0.0338062
\(876\) 2.00000 0.0675737
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 28.0000 0.944954
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 1.00000 0.0336718
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 12.0000 0.403604
\(885\) 4.00000 0.134459
\(886\) 12.0000 0.403148
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 6.00000 0.201347
\(889\) 8.00000 0.268311
\(890\) −6.00000 −0.201120
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) −64.0000 −2.14168
\(894\) 14.0000 0.468230
\(895\) −24.0000 −0.802232
\(896\) −3.00000 −0.100223
\(897\) −48.0000 −1.60267
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) −18.0000 −0.598671
\(905\) −2.00000 −0.0664822
\(906\) 8.00000 0.265782
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 4.00000 0.132745
\(909\) −10.0000 −0.331679
\(910\) −6.00000 −0.198898
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) −2.00000 −0.0661180
\(916\) −22.0000 −0.726900
\(917\) 20.0000 0.660458
\(918\) 2.00000 0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −24.0000 −0.791257
\(921\) 12.0000 0.395413
\(922\) −2.00000 −0.0658665
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −24.0000 −0.788689
\(927\) 8.00000 0.262754
\(928\) −10.0000 −0.328266
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 4.00000 0.131165
\(931\) −8.00000 −0.262189
\(932\) 18.0000 0.589610
\(933\) 24.0000 0.785725
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 4.00000 0.130605
\(939\) −10.0000 −0.326338
\(940\) −8.00000 −0.260931
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −14.0000 −0.456145
\(943\) −48.0000 −1.56310
\(944\) −4.00000 −0.130189
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) −8.00000 −0.259828
\(949\) 12.0000 0.389536
\(950\) −8.00000 −0.259554
\(951\) 2.00000 0.0648544
\(952\) −6.00000 −0.194461
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 10.0000 0.323762
\(955\) 4.00000 0.129437
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) −10.0000 −0.322917
\(960\) 7.00000 0.225924
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) −12.0000 −0.386695
\(964\) 6.00000 0.193247
\(965\) 18.0000 0.579441
\(966\) 8.00000 0.257396
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 33.0000 1.06066
\(969\) −16.0000 −0.513994
\(970\) −18.0000 −0.577945
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −6.00000 −0.192154
\(976\) 2.00000 0.0640184
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 18.0000 0.573819
\(985\) 18.0000 0.573528
\(986\) −4.00000 −0.127386
\(987\) 8.00000 0.254643
\(988\) −48.0000 −1.52708
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 20.0000 0.635001
\(993\) −12.0000 −0.380808
\(994\) −12.0000 −0.380617
\(995\) −4.00000 −0.126809
\(996\) 4.00000 0.126745
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 20.0000 0.633089
\(999\) −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 105.2.a.a.1.1 1
3.2 odd 2 315.2.a.a.1.1 1
4.3 odd 2 1680.2.a.f.1.1 1
5.2 odd 4 525.2.d.b.274.2 2
5.3 odd 4 525.2.d.b.274.1 2
5.4 even 2 525.2.a.a.1.1 1
7.2 even 3 735.2.i.a.361.1 2
7.3 odd 6 735.2.i.b.226.1 2
7.4 even 3 735.2.i.a.226.1 2
7.5 odd 6 735.2.i.b.361.1 2
7.6 odd 2 735.2.a.f.1.1 1
8.3 odd 2 6720.2.a.bk.1.1 1
8.5 even 2 6720.2.a.p.1.1 1
12.11 even 2 5040.2.a.d.1.1 1
15.2 even 4 1575.2.d.b.1324.1 2
15.8 even 4 1575.2.d.b.1324.2 2
15.14 odd 2 1575.2.a.h.1.1 1
20.19 odd 2 8400.2.a.co.1.1 1
21.20 even 2 2205.2.a.b.1.1 1
35.34 odd 2 3675.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.a.1.1 1 1.1 even 1 trivial
315.2.a.a.1.1 1 3.2 odd 2
525.2.a.a.1.1 1 5.4 even 2
525.2.d.b.274.1 2 5.3 odd 4
525.2.d.b.274.2 2 5.2 odd 4
735.2.a.f.1.1 1 7.6 odd 2
735.2.i.a.226.1 2 7.4 even 3
735.2.i.a.361.1 2 7.2 even 3
735.2.i.b.226.1 2 7.3 odd 6
735.2.i.b.361.1 2 7.5 odd 6
1575.2.a.h.1.1 1 15.14 odd 2
1575.2.d.b.1324.1 2 15.2 even 4
1575.2.d.b.1324.2 2 15.8 even 4
1680.2.a.f.1.1 1 4.3 odd 2
2205.2.a.b.1.1 1 21.20 even 2
3675.2.a.f.1.1 1 35.34 odd 2
5040.2.a.d.1.1 1 12.11 even 2
6720.2.a.p.1.1 1 8.5 even 2
6720.2.a.bk.1.1 1 8.3 odd 2
8400.2.a.co.1.1 1 20.19 odd 2