Properties

Label 145.2.a.a.1.1
Level 145
Weight 2
Character 145.1
Self dual yes
Analytic conductor 1.158
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 145 = 5 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 145.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.15783082931\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 145.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{22} +2.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -1.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} +3.00000 q^{36} +10.0000 q^{37} +2.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +8.00000 q^{43} +6.00000 q^{44} +3.00000 q^{45} -2.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} +6.00000 q^{55} -6.00000 q^{56} +1.00000 q^{58} -8.00000 q^{59} -6.00000 q^{61} -2.00000 q^{62} +6.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -12.0000 q^{71} -9.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} +2.00000 q^{76} +12.0000 q^{77} -10.0000 q^{79} +1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} -14.0000 q^{83} +2.00000 q^{85} -8.00000 q^{86} -18.0000 q^{88} +18.0000 q^{89} -3.00000 q^{90} -4.00000 q^{91} -2.00000 q^{92} +12.0000 q^{94} +2.00000 q^{95} +2.00000 q^{97} +3.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 3.00000 0.707107
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 3.00000 0.500000
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 6.00000 0.904534
\(45\) 3.00000 0.447214
\(46\) −2.00000 −0.294884
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −2.00000 −0.254000
\(63\) 6.00000 0.755929
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −9.00000 −1.06066
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) −18.0000 −1.91881
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) −3.00000 −0.316228
\(91\) −4.00000 −0.419314
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) 18.0000 1.80907
\(100\) −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
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −6.00000 −0.572078
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 1.00000 0.0928477
\(117\) −6.00000 −0.554700
\(118\) 8.00000 0.736460
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) −6.00000 −0.534522
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) −12.0000 −1.00349
\(144\) 3.00000 0.250000
\(145\) 1.00000 0.0830455
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) −12.0000 −0.966988
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −4.00000 −0.315244
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 6.00000 0.458831
\(172\) −8.00000 −0.609994
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −3.00000 −0.223607
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −18.0000 −1.27920
\(199\) −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
\(205\) −2.00000 −0.139686
\(206\) 6.00000 0.418040
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) −6.00000 −0.404520
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 10.0000 0.668153
\(225\) −3.00000 −0.200000
\(226\) −2.00000 −0.133038
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 6.00000 0.392232
\(235\) 12.0000 0.782794
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −25.0000 −1.60706
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) −6.00000 −0.377964
\(253\) −12.0000 −0.754434
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 2.00000 0.124035
\(261\) 3.00000 0.185695
\(262\) −14.0000 −0.864923
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 6.00000 0.358569
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −4.00000 −0.236113
\(288\) 15.0000 0.883883
\(289\) −13.0000 −0.764706
\(290\) −1.00000 −0.0587220
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 4.00000 0.230174
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 6.00000 0.343559
\(306\) −6.00000 −0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) 2.00000 0.113592
\(311\) −22.0000 −1.24751 −0.623753 0.781622i \(-0.714393\pi\)
−0.623753 + 0.781622i \(0.714393\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −22.0000 −1.24153
\(315\) −6.00000 −0.338062
\(316\) 10.0000 0.562544
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 4.00000 0.222566
\(324\) −9.00000 −0.500000
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) 14.0000 0.768350
\(333\) −30.0000 −1.64399
\(334\) −18.0000 −0.984916
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) −12.0000 −0.649836
\(342\) −6.00000 −0.324443
\(343\) 20.0000 1.07990
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 30.0000 1.59901
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 22.0000 1.16112 0.580558 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) 9.00000 0.474342
\(361\) −15.0000 −0.789474
\(362\) −6.00000 −0.315353
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) −2.00000 −0.104257
\(369\) −6.00000 −0.312348
\(370\) 10.0000 0.519875
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 22.0000 1.12562
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 10.0000 0.508987
\(387\) −24.0000 −1.21999
\(388\) −2.00000 −0.101535
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 10.0000 0.503155
\(396\) −18.0000 −0.904534
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −10.0000 −0.497519
\(405\) −9.00000 −0.447214
\(406\) −2.00000 −0.0992583
\(407\) −60.0000 −2.97409
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) 6.00000 0.295599
\(413\) 16.0000 0.787309
\(414\) 6.00000 0.294884
\(415\) 14.0000 0.687233
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.0000 −0.681509
\(423\) 36.0000 1.75038
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 12.0000 0.580721
\(428\) −6.00000 −0.290021
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 18.0000 0.858116
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 3.00000 0.141421
\(451\) −12.0000 −0.565058
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −22.0000 −1.03251
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 6.00000 0.277350
\(469\) −4.00000 −0.184703
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) −24.0000 −1.10469
\(473\) −48.0000 −2.20704
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) −4.00000 −0.183340
\(477\) 18.0000 0.824163
\(478\) 12.0000 0.548867
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) −25.0000 −1.13636
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) −18.0000 −0.814822
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) 4.00000 0.179969
\(495\) −18.0000 −0.809040
\(496\) −2.00000 −0.0898027
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 6.00000 0.267793
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 18.0000 0.801784
\(505\) −10.0000 −0.444994
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 30.0000 1.32324
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 72.0000 3.16656
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −3.00000 −0.131306
\(523\) 42.0000 1.83653 0.918266 0.395964i \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −6.00000 −0.260623
\(531\) 24.0000 1.04151
\(532\) −4.00000 −0.173422
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) −26.0000 −1.12094
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −6.00000 −0.256307
\(549\) 18.0000 0.768221
\(550\) 6.00000 0.255841
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) 0 0
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 6.00000 0.254000
\(559\) 16.0000 0.676728
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −22.0000 −0.924729
\(567\) −18.0000 −0.755929
\(568\) −36.0000 −1.51053
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 2.00000 0.0834058
\(576\) −21.0000 −0.875000
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −1.00000 −0.0415227
\(581\) 28.0000 1.16164
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) −18.0000 −0.744845
\(585\) 6.00000 0.248069
\(586\) 2.00000 0.0826192
\(587\) −18.0000 −0.742940 −0.371470 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 16.0000 0.652111
\(603\) −6.00000 −0.244339
\(604\) 4.00000 0.162758
\(605\) −25.0000 −1.01639
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 10.0000 0.405554
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −24.0000 −0.970936
\(612\) −6.00000 −0.242536
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 22.0000 0.882120
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) −20.0000 −0.797452
\(630\) 6.00000 0.239046
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) −6.00000 −0.237542
\(639\) 36.0000 1.42414
\(640\) −3.00000 −0.118585
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 27.0000 1.06066
\(649\) 48.0000 1.88416
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) −14.0000 −0.547025
\(656\) −2.00000 −0.0780869
\(657\) 18.0000 0.702247
\(658\) −24.0000 −0.935617
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 18.0000 0.699590
\(663\) 0 0
\(664\) −42.0000 −1.62992
\(665\) −4.00000 −0.155113
\(666\) 30.0000 1.16248
\(667\) −2.00000 −0.0774403
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) −6.00000 −0.229416
\(685\) −6.00000 −0.229248
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) −36.0000 −1.36753
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) −34.0000 −1.28692
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) −42.0000 −1.58293
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −12.0000 −0.450352
\(711\) 30.0000 1.12509
\(712\) 54.0000 2.02374
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −22.0000 −0.821033
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −3.00000 −0.111803
\(721\) 12.0000 0.446903
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −12.0000 −0.444750
\(729\) −27.0000 −1.00000
\(730\) −6.00000 −0.222070
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) −10.0000 −0.368605
\(737\) −12.0000 −0.442026
\(738\) 6.00000 0.220863
\(739\) 46.0000 1.69214 0.846069 0.533074i \(-0.178963\pi\)
0.846069 + 0.533074i \(0.178963\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) −18.0000 −0.659027
\(747\) 42.0000 1.53670
\(748\) −12.0000 −0.438763
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 2.00000 0.0728357
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −6.00000 −0.217930
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) 22.0000 0.795932
\(765\) −6.00000 −0.216930
\(766\) −14.0000 −0.505841
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 12.0000 0.432450
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 24.0000 0.862662
\(775\) 2.00000 0.0718421
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 72.0000 2.57636
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) −4.00000 −0.142224
\(792\) 54.0000 1.91881
\(793\) −12.0000 −0.426132
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −5.00000 −0.176777
\(801\) −54.0000 −1.90800
\(802\) 14.0000 0.494357
\(803\) 36.0000 1.27041
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 30.0000 1.05540
\(809\) −14.0000 −0.492214 −0.246107 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 9.00000 0.316228
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 60.0000 2.10300
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 22.0000 0.769212
\(819\) 12.0000 0.419314
\(820\) 2.00000 0.0698430
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −18.0000 −0.627060
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 6.00000 0.208514
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) −14.0000 −0.485947
\(831\) 0 0
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −2.00000 −0.0689246
\(843\) 0 0
\(844\) −14.0000 −0.481900
\(845\) 9.00000 0.309609
\(846\) −36.0000 −1.23771
\(847\) −50.0000 −1.71802
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −12.0000 −0.410632
\(855\) −6.00000 −0.205196
\(856\) 18.0000 0.615227
\(857\) 58.0000 1.98124 0.990621 0.136637i \(-0.0436295\pi\)
0.990621 + 0.136637i \(0.0436295\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −42.0000 −1.42970 −0.714848 0.699280i \(-0.753504\pi\)
−0.714848 + 0.699280i \(0.753504\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 60.0000 2.03536
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) −42.0000 −1.42230
\(873\) −6.00000 −0.203069
\(874\) 4.00000 0.135302
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −6.00000 −0.202260
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −9.00000 −0.303046
\(883\) −50.0000 −1.68263 −0.841317 0.540542i \(-0.818219\pi\)
−0.841317 + 0.540542i \(0.818219\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 18.0000 0.603361
\(891\) −54.0000 −1.80907
\(892\) 14.0000 0.468755
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −2.00000 −0.0667037
\(900\) 3.00000 0.100000
\(901\) 12.0000 0.399778
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −22.0000 −0.730096
\(909\) −30.0000 −0.995037
\(910\) −4.00000 −0.132599
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 0 0
\(913\) 84.0000 2.77999
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) −28.0000 −0.924641
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −22.0000 −0.722965
\(927\) 18.0000 0.591198
\(928\) 5.00000 0.164133
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) −12.0000 −0.392442
\(936\) −18.0000 −0.588348
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −12.0000 −0.391397
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −18.0000 −0.582772
\(955\) 22.0000 0.711903
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 14.0000 0.452319
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −20.0000 −0.644826
\(963\) −18.0000 −0.580042
\(964\) 26.0000 0.837404
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 75.0000 2.41059
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −108.000 −3.45169
\(980\) −3.00000 −0.0958315
\(981\) 42.0000 1.34096
\(982\) 2.00000 0.0638226
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 16.0000 0.508770
\(990\) 18.0000 0.572078
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) −10.0000 −0.317500
\(993\) 0 0
\(994\) −24.0000 −0.761234
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 12.0000 0.379853
\(999\) 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 145.2.a.a.1.1 1
3.2 odd 2 1305.2.a.f.1.1 1
4.3 odd 2 2320.2.a.e.1.1 1
5.2 odd 4 725.2.b.a.349.1 2
5.3 odd 4 725.2.b.a.349.2 2
5.4 even 2 725.2.a.a.1.1 1
7.6 odd 2 7105.2.a.b.1.1 1
8.3 odd 2 9280.2.a.o.1.1 1
8.5 even 2 9280.2.a.l.1.1 1
15.14 odd 2 6525.2.a.d.1.1 1
29.28 even 2 4205.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.a.1.1 1 1.1 even 1 trivial
725.2.a.a.1.1 1 5.4 even 2
725.2.b.a.349.1 2 5.2 odd 4
725.2.b.a.349.2 2 5.3 odd 4
1305.2.a.f.1.1 1 3.2 odd 2
2320.2.a.e.1.1 1 4.3 odd 2
4205.2.a.a.1.1 1 29.28 even 2
6525.2.a.d.1.1 1 15.14 odd 2
7105.2.a.b.1.1 1 7.6 odd 2
9280.2.a.l.1.1 1 8.5 even 2
9280.2.a.o.1.1 1 8.3 odd 2