Properties

Label 315.2.a.a.1.1
Level 315
Weight 2
Character 315.1
Self dual yes
Analytic conductor 2.515
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.51528766367\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 315.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} +1.00000 q^{20} -8.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} +8.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} +8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{52} -10.0000 q^{53} +3.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +6.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} +12.0000 q^{71} -2.00000 q^{73} +2.00000 q^{74} +8.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} -6.00000 q^{91} +8.00000 q^{92} +8.00000 q^{94} +8.00000 q^{95} -18.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 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
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 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
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −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 \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −18.0000 −1.76505
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −24.0000 −1.94666
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 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 \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) −24.0000 −1.76930
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 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
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 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 \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 48.0000 3.05417
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 4.00000 0.204658
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 0 0
\(388\) 18.0000 0.913812
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 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
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −24.0000 −1.19553
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 30.0000 1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −30.0000 −1.45693
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −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 \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 64.0000 3.06154
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) 7.00000 0.330719
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 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
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −4.00000 −0.182956
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −6.00000 −0.271607
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) −48.0000 −2.15962
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 18.0000 0.789352
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 18.0000 0.771035
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 0 0
\(551\) −16.0000 −0.681623
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −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 \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) 36.0000 1.51053
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) −48.0000 −1.96287
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 18.0000 0.734235 0.367118 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 24.0000 0.954669
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 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
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) 0 0
\(679\) −18.0000 −0.690777
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 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
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −36.0000 −1.34351
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) −18.0000 −0.667124
\(729\) 0 0
\(730\) −2.00000 −0.0740233
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 10.0000 0.367112
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 54.0000 1.95750 0.978749 0.205061i \(-0.0657392\pi\)
0.978749 + 0.205061i \(0.0657392\pi\)
\(762\) 0 0
\(763\) −18.0000 −0.651644
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −32.0000 −1.15621
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) −54.0000 −1.93849
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 24.0000 0.845364
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) −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 \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −42.0000 −1.45609
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 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
\(861\) 0 0
\(862\) 28.0000 0.953684
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) −54.0000 −1.82867
\(873\) 0 0
\(874\) −64.0000 −2.16483
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −58.0000 −1.95852 −0.979260 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) −28.0000 −0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) 64.0000 2.14168
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) −6.00000 −0.198898
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 24.0000 0.791257
\(921\) 0 0
\(922\) −2.00000 −0.0658665
\(923\) −72.0000 −2.36991
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 24.0000 0.788689
\(927\) 0 0
\(928\) −10.0000 −0.328266
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) 4.00000 0.129437
\(956\) −4.00000 −0.129369
\(957\) 0 0
\(958\) 32.0000 1.03387
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) 6.00000 0.193247
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) −18.0000 −0.577945
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −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
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −20.0000 −0.633089
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.2.a.a.1.1 1
3.2 odd 2 105.2.a.a.1.1 1
4.3 odd 2 5040.2.a.d.1.1 1
5.2 odd 4 1575.2.d.b.1324.1 2
5.3 odd 4 1575.2.d.b.1324.2 2
5.4 even 2 1575.2.a.h.1.1 1
7.6 odd 2 2205.2.a.b.1.1 1
12.11 even 2 1680.2.a.f.1.1 1
15.2 even 4 525.2.d.b.274.2 2
15.8 even 4 525.2.d.b.274.1 2
15.14 odd 2 525.2.a.a.1.1 1
21.2 odd 6 735.2.i.a.361.1 2
21.5 even 6 735.2.i.b.361.1 2
21.11 odd 6 735.2.i.a.226.1 2
21.17 even 6 735.2.i.b.226.1 2
21.20 even 2 735.2.a.f.1.1 1
24.5 odd 2 6720.2.a.p.1.1 1
24.11 even 2 6720.2.a.bk.1.1 1
60.59 even 2 8400.2.a.co.1.1 1
105.104 even 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 3.2 odd 2
315.2.a.a.1.1 1 1.1 even 1 trivial
525.2.a.a.1.1 1 15.14 odd 2
525.2.d.b.274.1 2 15.8 even 4
525.2.d.b.274.2 2 15.2 even 4
735.2.a.f.1.1 1 21.20 even 2
735.2.i.a.226.1 2 21.11 odd 6
735.2.i.a.361.1 2 21.2 odd 6
735.2.i.b.226.1 2 21.17 even 6
735.2.i.b.361.1 2 21.5 even 6
1575.2.a.h.1.1 1 5.4 even 2
1575.2.d.b.1324.1 2 5.2 odd 4
1575.2.d.b.1324.2 2 5.3 odd 4
1680.2.a.f.1.1 1 12.11 even 2
2205.2.a.b.1.1 1 7.6 odd 2
3675.2.a.f.1.1 1 105.104 even 2
5040.2.a.d.1.1 1 4.3 odd 2
6720.2.a.p.1.1 1 24.5 odd 2
6720.2.a.bk.1.1 1 24.11 even 2
8400.2.a.co.1.1 1 60.59 even 2