Properties

Label 5070.2.a.w.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 $1$

Related objects

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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 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} +4.00000 q^{84} +6.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -18.0000 q^{89} +1.00000 q^{90} -8.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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 \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 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
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.00000 0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 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
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) 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 \(0.421429\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −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
\(76\) 4.00000 0.458831
\(77\) 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
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 4.00000 0.436436
\(85\) 6.00000 0.650791
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 6.00000 0.594089
\(103\) −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
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 4.00000 0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 24.0000 2.20008
\(120\) 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
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −2.00000 −0.165521
\(147\) 9.00000 0.742307
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 4.00000 0.308607
\(169\) 0 0
\(170\) 6.00000 0.460179
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −6.00000 −0.454859
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −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
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) −2.00000 −0.143592
\(195\) 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 \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 18.0000 1.26648
\(203\) −24.0000 −1.68447
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) −32.0000 −2.17230
\(218\) 10.0000 0.677285
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) −18.0000 −1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 4.00000 0.264906
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 24.0000 1.55569
\(239\) −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
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 16.0000 0.981023
\(267\) −18.0000 −1.10158
\(268\) 4.00000 0.244339
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 1.00000 0.0608581
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) −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
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) 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
\(298\) 6.00000 0.347571
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −16.0000 −0.922225
\(302\) −8.00000 −0.460348
\(303\) 18.0000 1.03407
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) 6.00000 0.342997
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 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
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 2.00000 0.112867
\(315\) 4.00000 0.225374
\(316\) 8.00000 0.450035
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 10.0000 0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 4.00000 0.218218
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 24.0000 1.27021
\(358\) 24.0000 1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) −10.0000 −0.522708
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) −2.00000 −0.103975
\(371\) −24.0000 −1.24602
\(372\) −8.00000 −0.414781
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 4.00000 0.205196
\(381\) 20.0000 1.02463
\(382\) −24.0000 −1.22795
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −4.00000 −0.203331
\(388\) −2.00000 −0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 8.00000 0.401004
\(399\) 16.0000 0.801002
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 1.00000 0.0496904
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 6.00000 0.296319
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 4.00000 0.195180
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −40.0000 −1.93574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −32.0000 −1.53605
\(435\) −6.00000 −0.287678
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −18.0000 −0.853282
\(446\) −20.0000 −0.947027
\(447\) 6.00000 0.283790
\(448\) 4.00000 0.188982
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) −8.00000 −0.375873
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 10.0000 0.467269
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −6.00000 −0.278543
\(465\) −8.00000 −0.370991
\(466\) −18.0000 −0.833834
\(467\) −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
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 4.00000 0.183533
\(476\) 24.0000 1.10004
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) 9.00000 0.406579
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 6.00000 0.270501
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −24.0000 −1.07117
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 4.00000 0.178174
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 6.00000 0.265684
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −18.0000 −0.793946
\(515\) −4.00000 −0.176261
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 24.0000 1.03568
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 16.0000 0.687259
\(543\) 14.0000 0.600798
\(544\) 6.00000 0.257248
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 2.00000 0.0849719
\(555\) −2.00000 −0.0848953
\(556\) −4.00000 −0.169638
\(557\) −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
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) −28.0000 −1.17693
\(567\) 4.00000 0.167984
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 4.00000 0.167542
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) 22.0000 0.914289
\(580\) −6.00000 −0.249136
\(581\) −48.0000 −1.99138
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 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
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 6.00000 0.245770
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −16.0000 −0.652111
\(603\) 4.00000 0.162893
\(604\) −8.00000 −0.325515
\(605\) −11.0000 −0.447214
\(606\) 18.0000 0.731200
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 4.00000 0.162221
\(609\) −24.0000 −0.972529
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −20.0000 −0.807134
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −4.00000 −0.160904
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) −72.0000 −2.88462
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −12.0000 −0.478471
\(630\) 4.00000 0.159364
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 8.00000 0.318223
\(633\) 20.0000 0.794929
\(634\) −18.0000 −0.714871
\(635\) 20.0000 0.793676
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 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
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 24.0000 0.944267
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 4.00000 0.156652
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 16.0000 0.620453
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) −20.0000 −0.773245
\(670\) 4.00000 0.154533
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −18.0000 −0.691286
\(679\) −8.00000 −0.307012
\(680\) 6.00000 0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 4.00000 0.152944
\(685\) −6.00000 −0.229248
\(686\) 8.00000 0.305441
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −4.00000 −0.151729
\(696\) −6.00000 −0.227429
\(697\) 36.0000 1.36360
\(698\) 10.0000 0.378506
\(699\) −18.0000 −0.680823
\(700\) 4.00000 0.151186
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 72.0000 2.70784
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) −24.0000 −0.896296
\(718\) 24.0000 0.895672
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) 14.0000 0.520306
\(725\) −6.00000 −0.222834
\(726\) −11.0000 −0.408248
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) −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 \(0.366821\pi\)
\(734\) −28.0000 −1.03350
\(735\) 9.00000 0.331970
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 6.00000 0.219823
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 1.00000 0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −24.0000 −0.874609
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 4.00000 0.145479
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 20.0000 0.724524
\(763\) 40.0000 1.44810
\(764\) −24.0000 −0.868290
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 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
\(771\) −18.0000 −0.648254
\(772\) 22.0000 0.791797
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −4.00000 −0.143777
\(775\) −8.00000 −0.287368
\(776\) −2.00000 −0.0717958
\(777\) −8.00000 −0.286998
\(778\) −6.00000 −0.215110
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 9.00000 0.321429
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −72.0000 −2.56003
\(792\) 0 0
\(793\) 0 0
\(794\) 22.0000 0.780751
\(795\) −6.00000 −0.212798
\(796\) 8.00000 0.283552
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 16.0000 0.566394
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −18.0000 −0.635999
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 18.0000 0.633238
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 1.00000 0.0351364
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −24.0000 −0.842235
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 6.00000 0.210042
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −6.00000 −0.209274
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −12.0000 −0.416526
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −18.0000 −0.619953
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −44.0000 −1.51186
\(848\) −6.00000 −0.206041
\(849\) −28.0000 −0.960958
\(850\) 6.00000 0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −40.0000 −1.36877
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −4.00000 −0.136399
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 18.0000 0.612018
\(866\) 26.0000 0.883516
\(867\) 19.0000 0.645274
\(868\) −32.0000 −1.08615
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) 0 0
\(872\) 10.0000 0.338643
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 4.00000 0.135225
\(876\) −2.00000 −0.0675737
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 8.00000 0.269987
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 9.00000 0.303046
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 80.0000 2.68311
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −20.0000 −0.669650
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 24.0000 0.802232
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −16.0000 −0.532447
\(904\) −18.0000 −0.598671
\(905\) 14.0000 0.465376
\(906\) −8.00000 −0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) −10.0000 −0.330590
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 4.00000 0.131448
\(927\) −4.00000 −0.131377
\(928\) −6.00000 −0.196960
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) −8.00000 −0.262330
\(931\) 36.0000 1.17985
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000 0.522419
\(939\) 2.00000 0.0652675
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) −18.0000 −0.583690
\(952\) 24.0000 0.777844
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −6.00000 −0.194257
\(955\) −24.0000 −0.776622
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −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
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −11.0000 −0.353553
\(969\) 24.0000 0.770991
\(970\) −2.00000 −0.0642161
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 4.00000 0.127906
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) 10.0000 0.319275
\(982\) 24.0000 0.765871
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −8.00000 −0.254000
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) −12.0000 −0.380235
\(997\) 26.0000 0.823428 0.411714 0.911313i \(-0.364930\pi\)
0.411714 + 0.911313i \(0.364930\pi\)
\(998\) 4.00000 0.126618
\(999\) −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.w.1.1 1
13.5 odd 4 5070.2.b.k.1351.1 2
13.8 odd 4 5070.2.b.k.1351.2 2
13.12 even 2 30.2.a.a.1.1 1
39.38 odd 2 90.2.a.c.1.1 1
52.51 odd 2 240.2.a.b.1.1 1
65.12 odd 4 150.2.c.a.49.1 2
65.38 odd 4 150.2.c.a.49.2 2
65.64 even 2 150.2.a.b.1.1 1
91.12 odd 6 1470.2.i.q.361.1 2
91.25 even 6 1470.2.i.o.961.1 2
91.38 odd 6 1470.2.i.q.961.1 2
91.51 even 6 1470.2.i.o.361.1 2
91.90 odd 2 1470.2.a.d.1.1 1
104.51 odd 2 960.2.a.p.1.1 1
104.77 even 2 960.2.a.e.1.1 1
117.25 even 6 810.2.e.l.271.1 2
117.38 odd 6 810.2.e.b.271.1 2
117.77 odd 6 810.2.e.b.541.1 2
117.103 even 6 810.2.e.l.541.1 2
143.142 odd 2 3630.2.a.w.1.1 1
156.155 even 2 720.2.a.j.1.1 1
195.38 even 4 450.2.c.b.199.1 2
195.77 even 4 450.2.c.b.199.2 2
195.194 odd 2 450.2.a.d.1.1 1
208.51 odd 4 3840.2.k.f.1921.1 2
208.77 even 4 3840.2.k.y.1921.2 2
208.155 odd 4 3840.2.k.f.1921.2 2
208.181 even 4 3840.2.k.y.1921.1 2
221.220 even 2 8670.2.a.g.1.1 1
260.103 even 4 1200.2.f.e.49.1 2
260.207 even 4 1200.2.f.e.49.2 2
260.259 odd 2 1200.2.a.k.1.1 1
273.272 even 2 4410.2.a.z.1.1 1
312.77 odd 2 2880.2.a.a.1.1 1
312.155 even 2 2880.2.a.q.1.1 1
455.454 odd 2 7350.2.a.ct.1.1 1
520.77 odd 4 4800.2.f.p.3649.2 2
520.259 odd 2 4800.2.a.d.1.1 1
520.363 even 4 4800.2.f.w.3649.2 2
520.389 even 2 4800.2.a.cq.1.1 1
520.467 even 4 4800.2.f.w.3649.1 2
520.493 odd 4 4800.2.f.p.3649.1 2
780.467 odd 4 3600.2.f.i.2449.2 2
780.623 odd 4 3600.2.f.i.2449.1 2
780.779 even 2 3600.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 13.12 even 2
90.2.a.c.1.1 1 39.38 odd 2
150.2.a.b.1.1 1 65.64 even 2
150.2.c.a.49.1 2 65.12 odd 4
150.2.c.a.49.2 2 65.38 odd 4
240.2.a.b.1.1 1 52.51 odd 2
450.2.a.d.1.1 1 195.194 odd 2
450.2.c.b.199.1 2 195.38 even 4
450.2.c.b.199.2 2 195.77 even 4
720.2.a.j.1.1 1 156.155 even 2
810.2.e.b.271.1 2 117.38 odd 6
810.2.e.b.541.1 2 117.77 odd 6
810.2.e.l.271.1 2 117.25 even 6
810.2.e.l.541.1 2 117.103 even 6
960.2.a.e.1.1 1 104.77 even 2
960.2.a.p.1.1 1 104.51 odd 2
1200.2.a.k.1.1 1 260.259 odd 2
1200.2.f.e.49.1 2 260.103 even 4
1200.2.f.e.49.2 2 260.207 even 4
1470.2.a.d.1.1 1 91.90 odd 2
1470.2.i.o.361.1 2 91.51 even 6
1470.2.i.o.961.1 2 91.25 even 6
1470.2.i.q.361.1 2 91.12 odd 6
1470.2.i.q.961.1 2 91.38 odd 6
2880.2.a.a.1.1 1 312.77 odd 2
2880.2.a.q.1.1 1 312.155 even 2
3600.2.a.f.1.1 1 780.779 even 2
3600.2.f.i.2449.1 2 780.623 odd 4
3600.2.f.i.2449.2 2 780.467 odd 4
3630.2.a.w.1.1 1 143.142 odd 2
3840.2.k.f.1921.1 2 208.51 odd 4
3840.2.k.f.1921.2 2 208.155 odd 4
3840.2.k.y.1921.1 2 208.181 even 4
3840.2.k.y.1921.2 2 208.77 even 4
4410.2.a.z.1.1 1 273.272 even 2
4800.2.a.d.1.1 1 520.259 odd 2
4800.2.a.cq.1.1 1 520.389 even 2
4800.2.f.p.3649.1 2 520.493 odd 4
4800.2.f.p.3649.2 2 520.77 odd 4
4800.2.f.w.3649.1 2 520.467 even 4
4800.2.f.w.3649.2 2 520.363 even 4
5070.2.a.w.1.1 1 1.1 even 1 trivial
5070.2.b.k.1351.1 2 13.5 odd 4
5070.2.b.k.1351.2 2 13.8 odd 4
7350.2.a.ct.1.1 1 455.454 odd 2
8670.2.a.g.1.1 1 221.220 even 2