Properties

Label 5070.2.a.u.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} +1.00000 q^{45} +8.00000 q^{46} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +1.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} +12.0000 q^{86} +2.00000 q^{87} +14.0000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{93} -4.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 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
\(13\) 0 0
\(14\) −4.00000 −1.06904
\(15\) 1.00000 0.258199
\(16\) 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
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 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
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 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
\(70\) −4.00000 −0.478091
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −4.00000 −0.436436
\(85\) −2.00000 −0.216930
\(86\) 12.0000 1.29399
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 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
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −4.00000 −0.377964
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −4.00000 −0.374634
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 1.00000 0.0912871
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −4.00000 −0.356348
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 8.00000 0.681005
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) 6.00000 0.496564
\(147\) 9.00000 0.742307
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −4.00000 −0.324443
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) −32.0000 −2.52195
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) −4.00000 −0.308607
\(169\) 0 0
\(170\) −2.00000 −0.153393
\(171\) −4.00000 −0.305888
\(172\) 12.0000 0.914991
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 2.00000 0.151620
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 8.00000 0.589768
\(185\) −2.00000 −0.147043
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) −8.00000 −0.561490
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000 0.686803
\(213\) 16.0000 1.09630
\(214\) −12.0000 −0.820303
\(215\) 12.0000 0.818393
\(216\) 1.00000 0.0680414
\(217\) −32.0000 −2.17230
\(218\) −14.0000 −0.948200
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) −4.00000 −0.267261
\(225\) 1.00000 0.0666667
\(226\) −10.0000 −0.665190
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.00000 −0.519656
\(238\) 8.00000 0.518563
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 1.00000 0.0645497
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 9.00000 0.574989
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 12.0000 0.747087
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 8.00000 0.494242
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 16.0000 0.981023
\(267\) 14.0000 0.856786
\(268\) 4.00000 0.244339
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 1.00000 0.0608581
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) −20.0000 −1.19952
\(279\) 8.00000 0.478947
\(280\) −4.00000 −0.239046
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 16.0000 0.949425
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 6.00000 0.351726
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −48.0000 −2.76667
\(302\) 8.00000 0.460348
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) −10.0000 −0.572598
\(306\) −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
\(309\) −4.00000 −0.227552
\(310\) 8.00000 0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 18.0000 1.01742 0.508710 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) 18.0000 1.01580
\(315\) −4.00000 −0.225374
\(316\) −8.00000 −0.450035
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) −32.0000 −1.78329
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) −14.0000 −0.774202
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) −16.0000 −0.875481
\(335\) 4.00000 0.218543
\(336\) −4.00000 −0.218218
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −8.00000 −0.431959
\(344\) 12.0000 0.646997
\(345\) 8.00000 0.430706
\(346\) 2.00000 0.107521
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 2.00000 0.107211
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 14.0000 0.741999
\(357\) 8.00000 0.423405
\(358\) 16.0000 0.845626
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) −10.0000 −0.522708
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) −2.00000 −0.103975
\(371\) −40.0000 −2.07670
\(372\) 8.00000 0.414781
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) −4.00000 −0.205738
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) −12.0000 −0.614779
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) 12.0000 0.609994
\(388\) 6.00000 0.304604
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 9.00000 0.454569
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) −8.00000 −0.401004
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 1.00000 0.0496904
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 6.00000 0.296319
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) −4.00000 −0.195180
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) −2.00000 −0.0970143
\(426\) 16.0000 0.775203
\(427\) 40.0000 1.93574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) −32.0000 −1.53605
\(435\) 2.00000 0.0958927
\(436\) −14.0000 −0.670478
\(437\) −32.0000 −1.53077
\(438\) 6.00000 0.286691
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 14.0000 0.663664
\(446\) −12.0000 −0.568216
\(447\) −10.0000 −0.472984
\(448\) −4.00000 −0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −10.0000 −0.470360
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −14.0000 −0.654177
\(459\) −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
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 2.00000 0.0928477
\(465\) 8.00000 0.370991
\(466\) −26.0000 −1.20443
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) −4.00000 −0.183533
\(476\) 8.00000 0.366679
\(477\) 10.0000 0.457869
\(478\) −24.0000 −1.09773
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) −32.0000 −1.45605
\(484\) −11.0000 −0.500000
\(485\) 6.00000 0.272446
\(486\) 1.00000 0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −10.0000 −0.452679
\(489\) 20.0000 0.904431
\(490\) 9.00000 0.406579
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −64.0000 −2.87079
\(498\) 4.00000 0.179244
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 1.00000 0.0447214
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −4.00000 −0.178174
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −24.0000 −1.06170
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 22.0000 0.970378
\(515\) −4.00000 −0.176261
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 2.00000 0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 8.00000 0.349482
\(525\) −4.00000 −0.174574
\(526\) 24.0000 1.04645
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) 0 0
\(534\) 14.0000 0.605839
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 16.0000 0.690451
\(538\) −14.0000 −0.603583
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 32.0000 1.37452
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 8.00000 0.340503
\(553\) 32.0000 1.36078
\(554\) 18.0000 0.764747
\(555\) −2.00000 −0.0848953
\(556\) −20.0000 −0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 20.0000 0.840663
\(567\) −4.00000 −0.167984
\(568\) 16.0000 0.671345
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) −4.00000 −0.167542
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) −24.0000 −1.00174
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) 2.00000 0.0830455
\(581\) −16.0000 −0.663792
\(582\) 6.00000 0.248708
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 9.00000 0.371154
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −10.0000 −0.409616
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 1.00000 0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −48.0000 −1.95633
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) −11.0000 −0.447214
\(606\) 10.0000 0.406222
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) −4.00000 −0.162221
\(609\) −8.00000 −0.324176
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 12.0000 0.484281
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) −4.00000 −0.160904
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 8.00000 0.321288
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) −56.0000 −2.24359
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.0000 0.719425
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 4.00000 0.159490
\(630\) −4.00000 −0.159364
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −8.00000 −0.318223
\(633\) −12.0000 −0.476957
\(634\) 14.0000 0.556011
\(635\) −12.0000 −0.476205
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) 16.0000 0.632950
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) −32.0000 −1.26098
\(645\) 12.0000 0.472500
\(646\) 8.00000 0.314756
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 20.0000 0.783260
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −14.0000 −0.547443
\(655\) 8.00000 0.312586
\(656\) 6.00000 0.234261
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 16.0000 0.620453
\(666\) −2.00000 −0.0774984
\(667\) 16.0000 0.619522
\(668\) −16.0000 −0.619059
\(669\) −12.0000 −0.463947
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) −10.0000 −0.384048
\(679\) −24.0000 −0.921035
\(680\) −2.00000 −0.0766965
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) −8.00000 −0.305441
\(687\) −14.0000 −0.534133
\(688\) 12.0000 0.457496
\(689\) 0 0
\(690\) 8.00000 0.304555
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −20.0000 −0.758643
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) −26.0000 −0.983410
\(700\) −4.00000 −0.151186
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 16.0000 0.600469
\(711\) −8.00000 −0.300023
\(712\) 14.0000 0.524672
\(713\) 64.0000 2.39682
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 16.0000 0.597948
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) 1.00000 0.0372678
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) −11.0000 −0.408248
\(727\) 52.0000 1.92857 0.964287 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −24.0000 −0.887672
\(732\) −10.0000 −0.369611
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 4.00000 0.147643
\(735\) 9.00000 0.331970
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −40.0000 −1.46845
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 8.00000 0.293294
\(745\) −10.0000 −0.366372
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 1.00000 0.0365148
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) −4.00000 −0.145479
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −12.0000 −0.434714
\(763\) 56.0000 2.02734
\(764\) 8.00000 0.289430
\(765\) −2.00000 −0.0723102
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −18.0000 −0.647834
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) 12.0000 0.431331
\(775\) 8.00000 0.287368
\(776\) 6.00000 0.215387
\(777\) 8.00000 0.286998
\(778\) −14.0000 −0.501924
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) −16.0000 −0.572159
\(783\) 2.00000 0.0714742
\(784\) 9.00000 0.321429
\(785\) 18.0000 0.642448
\(786\) 8.00000 0.285351
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 6.00000 0.213741
\(789\) 24.0000 0.854423
\(790\) −8.00000 −0.284627
\(791\) 40.0000 1.42224
\(792\) 0 0
\(793\) 0 0
\(794\) −10.0000 −0.354887
\(795\) 10.0000 0.354663
\(796\) −8.00000 −0.283552
\(797\) −46.0000 −1.62940 −0.814702 0.579880i \(-0.803099\pi\)
−0.814702 + 0.579880i \(0.803099\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 14.0000 0.494666
\(802\) −26.0000 −0.918092
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) −32.0000 −1.12785
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 10.0000 0.351799
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 1.00000 0.0351364
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) −8.00000 −0.280745
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) −2.00000 −0.0700140
\(817\) −48.0000 −1.67931
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) −6.00000 −0.209274
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 8.00000 0.278019
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 4.00000 0.138842
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) −20.0000 −0.692543
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −24.0000 −0.829066
\(839\) 8.00000 0.276191 0.138095 0.990419i \(-0.455902\pi\)
0.138095 + 0.990419i \(0.455902\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) −30.0000 −1.03387
\(843\) −18.0000 −0.619953
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) 10.0000 0.343401
\(849\) 20.0000 0.686398
\(850\) −2.00000 −0.0685994
\(851\) −16.0000 −0.548473
\(852\) 16.0000 0.548151
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 40.0000 1.36877
\(855\) −4.00000 −0.136797
\(856\) −12.0000 −0.410152
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 12.0000 0.409197
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 2.00000 0.0680020
\(866\) 10.0000 0.339814
\(867\) −13.0000 −0.441503
\(868\) −32.0000 −1.08615
\(869\) 0 0
\(870\) 2.00000 0.0678064
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 6.00000 0.203069
\(874\) −32.0000 −1.08242
\(875\) −4.00000 −0.135225
\(876\) 6.00000 0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 40.0000 1.34993
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 9.00000 0.303046
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 48.0000 1.60987
\(890\) 14.0000 0.469281
\(891\) 0 0
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 16.0000 0.534821
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 16.0000 0.533630
\(900\) 1.00000 0.0333333
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) −48.0000 −1.59734
\(904\) −10.0000 −0.332595
\(905\) −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 \(0.346104\pi\)
\(908\) 12.0000 0.398234
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) −18.0000 −0.595387
\(915\) −10.0000 −0.330590
\(916\) −14.0000 −0.462573
\(917\) −32.0000 −1.05673
\(918\) −2.00000 −0.0660098
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 8.00000 0.263752
\(921\) 12.0000 0.395413
\(922\) −2.00000 −0.0658665
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) −20.0000 −0.657241
\(927\) −4.00000 −0.131377
\(928\) 2.00000 0.0656532
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 8.00000 0.262330
\(931\) −36.0000 −1.17985
\(932\) −26.0000 −0.851658
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) −16.0000 −0.522419
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 18.0000 0.586472
\(943\) 48.0000 1.56310
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 14.0000 0.453981
\(952\) 8.00000 0.259281
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 10.0000 0.323762
\(955\) 8.00000 0.258874
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 24.0000 0.775000
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −2.00000 −0.0644157
\(965\) −18.0000 −0.579441
\(966\) −32.0000 −1.02958
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −11.0000 −0.353553
\(969\) 8.00000 0.256997
\(970\) 6.00000 0.192648
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 1.00000 0.0320750
\(973\) 80.0000 2.56468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) −14.0000 −0.446986
\(982\) 0 0
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 0 0
\(989\) 96.0000 3.05262
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 8.00000 0.254000
\(993\) 20.0000 0.634681
\(994\) −64.0000 −2.02996
\(995\) −8.00000 −0.253617
\(996\) 4.00000 0.126745
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 44.0000 1.39280
\(999\) −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 5070.2.a.u.1.1 1
13.5 odd 4 5070.2.b.i.1351.1 2
13.8 odd 4 5070.2.b.i.1351.2 2
13.12 even 2 390.2.a.c.1.1 1
39.38 odd 2 1170.2.a.n.1.1 1
52.51 odd 2 3120.2.a.a.1.1 1
65.12 odd 4 1950.2.e.e.1249.1 2
65.38 odd 4 1950.2.e.e.1249.2 2
65.64 even 2 1950.2.a.n.1.1 1
156.155 even 2 9360.2.a.bc.1.1 1
195.38 even 4 5850.2.e.m.5149.1 2
195.77 even 4 5850.2.e.m.5149.2 2
195.194 odd 2 5850.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.c.1.1 1 13.12 even 2
1170.2.a.n.1.1 1 39.38 odd 2
1950.2.a.n.1.1 1 65.64 even 2
1950.2.e.e.1249.1 2 65.12 odd 4
1950.2.e.e.1249.2 2 65.38 odd 4
3120.2.a.a.1.1 1 52.51 odd 2
5070.2.a.u.1.1 1 1.1 even 1 trivial
5070.2.b.i.1351.1 2 13.5 odd 4
5070.2.b.i.1351.2 2 13.8 odd 4
5850.2.a.c.1.1 1 195.194 odd 2
5850.2.e.m.5149.1 2 195.38 even 4
5850.2.e.m.5149.2 2 195.77 even 4
9360.2.a.bc.1.1 1 156.155 even 2