Properties

Label 1001.2.a.c.1.1
Level 1001
Weight 2
Character 1001.1
Self dual yes
Analytic conductor 7.993
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.99302524233\)
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 \(\chi\) = 1001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -4.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +4.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} -2.00000 q^{21} -5.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +2.00000 q^{28} -5.00000 q^{29} -9.00000 q^{31} -2.00000 q^{33} +1.00000 q^{35} -2.00000 q^{36} +10.0000 q^{37} -2.00000 q^{39} -2.00000 q^{41} -13.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -7.00000 q^{47} +8.00000 q^{48} +1.00000 q^{49} +4.00000 q^{51} +2.00000 q^{52} -1.00000 q^{53} +1.00000 q^{55} -2.00000 q^{57} +8.00000 q^{59} +4.00000 q^{60} -2.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} -4.00000 q^{68} -10.0000 q^{69} +6.00000 q^{71} +5.00000 q^{73} -8.00000 q^{75} +2.00000 q^{76} +1.00000 q^{77} +11.0000 q^{79} -4.00000 q^{80} -11.0000 q^{81} -3.00000 q^{83} +4.00000 q^{84} -2.00000 q^{85} -10.0000 q^{87} +5.00000 q^{89} +1.00000 q^{91} +10.0000 q^{92} -18.0000 q^{93} +1.00000 q^{95} +5.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −4.00000 −1.15470
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 4.00000 1.00000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 2.00000 0.377964
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −13.0000 −1.98248 −0.991241 0.132068i \(-0.957838\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 8.00000 1.15470
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 2.00000 0.277350
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 4.00000 0.516398
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −4.00000 −0.485071
\(69\) −10.0000 −1.20386
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 2.00000 0.229416
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −4.00000 −0.447214
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 4.00000 0.436436
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) 5.00000 0.529999 0.264999 0.964249i \(-0.414628\pi\)
0.264999 + 0.964249i \(0.414628\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 10.0000 1.04257
\(93\) −18.0000 −1.86651
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 8.00000 0.800000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 8.00000 0.769800
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) −4.00000 −0.377964
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 5.00000 0.466252
\(116\) 10.0000 0.928477
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 18.0000 1.61645
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −26.0000 −2.28917
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000 0.348155
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −2.00000 −0.169031
\(141\) −14.0000 −1.17901
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 4.00000 0.333333
\(145\) 5.00000 0.415227
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) −20.0000 −1.64399
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 4.00000 0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 4.00000 0.312348
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) −17.0000 −1.31550 −0.657750 0.753237i \(-0.728492\pi\)
−0.657750 + 0.753237i \(0.728492\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 26.0000 1.98248
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) 16.0000 1.20263
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 2.00000 0.149071
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 14.0000 1.02105
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −16.0000 −1.15470
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) −2.00000 −0.142857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) −8.00000 −0.560112
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −5.00000 −0.347524
\(208\) −4.00000 −0.277350
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 3.00000 0.206529 0.103264 0.994654i \(-0.467071\pi\)
0.103264 + 0.994654i \(0.467071\pi\)
\(212\) 2.00000 0.137361
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 13.0000 0.886593
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) −2.00000 −0.134840
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 4.00000 0.264906
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 11.0000 0.720634 0.360317 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) −16.0000 −1.04151
\(237\) 22.0000 1.42905
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −8.00000 −0.516398
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 2.00000 0.125988
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 16.0000 1.00000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) −2.00000 −0.124035
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) −27.0000 −1.66489 −0.832446 0.554107i \(-0.813060\pi\)
−0.832446 + 0.554107i \(0.813060\pi\)
\(264\) 0 0
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 8.00000 0.485071
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 20.0000 1.20386
\(277\) −31.0000 −1.86261 −0.931305 0.364241i \(-0.881328\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −12.0000 −0.712069
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −10.0000 −0.585206
\(293\) −1.00000 −0.0584206 −0.0292103 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 16.0000 0.923760
\(301\) 13.0000 0.749308
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −3.00000 −0.171219 −0.0856095 0.996329i \(-0.527284\pi\)
−0.0856095 + 0.996329i \(0.527284\pi\)
\(308\) −2.00000 −0.113961
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 12.0000 0.678280 0.339140 0.940736i \(-0.389864\pi\)
0.339140 + 0.940736i \(0.389864\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) −22.0000 −1.23760
\(317\) −16.0000 −0.898650 −0.449325 0.893368i \(-0.648335\pi\)
−0.449325 + 0.893368i \(0.648335\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 8.00000 0.447214
\(321\) 40.0000 2.23258
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 22.0000 1.22222
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) 6.00000 0.329293
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) −8.00000 −0.436436
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 4.00000 0.216930
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 10.0000 0.538382
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 20.0000 1.07211
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) −10.0000 −0.529999
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) −2.00000 −0.104828
\(365\) −5.00000 −0.261712
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) −20.0000 −1.04257
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 36.0000 1.86651
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 18.0000 0.929516
\(376\) 0 0
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −2.00000 −0.102598
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −13.0000 −0.660827
\(388\) −10.0000 −0.507673
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) −24.0000 −1.21064
\(394\) 0 0
\(395\) −11.0000 −0.553470
\(396\) 2.00000 0.100504
\(397\) 33.0000 1.65622 0.828111 0.560564i \(-0.189416\pi\)
0.828111 + 0.560564i \(0.189416\pi\)
\(398\) 0 0
\(399\) 2.00000 0.100125
\(400\) −16.0000 −0.800000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 9.00000 0.448322
\(404\) 24.0000 1.19404
\(405\) 11.0000 0.546594
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 9.00000 0.445021 0.222511 0.974930i \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(410\) 0 0
\(411\) 44.0000 2.17036
\(412\) −16.0000 −0.788263
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −4.00000 −0.195180
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −7.00000 −0.340352
\(424\) 0 0
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −40.0000 −1.93347
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −16.0000 −0.769800
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 10.0000 0.479463
\(436\) −8.00000 −0.383131
\(437\) 5.00000 0.239182
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 31.0000 1.47285 0.736427 0.676517i \(-0.236511\pi\)
0.736427 + 0.676517i \(0.236511\pi\)
\(444\) −40.0000 −1.89832
\(445\) −5.00000 −0.237023
\(446\) 0 0
\(447\) 16.0000 0.756774
\(448\) 8.00000 0.377964
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −2.00000 −0.0940721
\(453\) −28.0000 −1.31555
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) −10.0000 −0.466252
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −20.0000 −0.928477
\(465\) 18.0000 0.834730
\(466\) 0 0
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 2.00000 0.0924500
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 13.0000 0.597741
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 4.00000 0.183340
\(477\) −1.00000 −0.0457869
\(478\) 0 0
\(479\) −33.0000 −1.50781 −0.753904 0.656984i \(-0.771832\pi\)
−0.753904 + 0.656984i \(0.771832\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) 0 0
\(483\) 10.0000 0.455016
\(484\) −2.00000 −0.0909091
\(485\) −5.00000 −0.227038
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) −32.0000 −1.44709
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 8.00000 0.360668
\(493\) −10.0000 −0.450377
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) −36.0000 −1.61645
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −26.0000 −1.16392 −0.581960 0.813217i \(-0.697714\pi\)
−0.581960 + 0.813217i \(0.697714\pi\)
\(500\) −18.0000 −0.804984
\(501\) −34.0000 −1.51901
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 2.00000 0.0888231
\(508\) 16.0000 0.709885
\(509\) 39.0000 1.72864 0.864322 0.502938i \(-0.167748\pi\)
0.864322 + 0.502938i \(0.167748\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 52.0000 2.28917
\(517\) 7.00000 0.307860
\(518\) 0 0
\(519\) −28.0000 −1.22906
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 0 0
\(523\) −38.0000 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) 24.0000 1.04844
\(525\) 8.00000 0.349149
\(526\) 0 0
\(527\) −18.0000 −0.784092
\(528\) −8.00000 −0.348155
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) −2.00000 −0.0867110
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) −8.00000 −0.344265
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) −40.0000 −1.71656
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) −44.0000 −1.87959
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 5.00000 0.213007
\(552\) 0 0
\(553\) −11.0000 −0.467768
\(554\) 0 0
\(555\) −20.0000 −0.848953
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 13.0000 0.549841
\(560\) 4.00000 0.169031
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 28.0000 1.17901
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 7.00000 0.293455 0.146728 0.989177i \(-0.453126\pi\)
0.146728 + 0.989177i \(0.453126\pi\)
\(570\) 0 0
\(571\) −5.00000 −0.209243 −0.104622 0.994512i \(-0.533363\pi\)
−0.104622 + 0.994512i \(0.533363\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) −8.00000 −0.333333
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) −10.0000 −0.415227
\(581\) 3.00000 0.124461
\(582\) 0 0
\(583\) 1.00000 0.0414158
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) −4.00000 −0.164957
\(589\) 9.00000 0.370839
\(590\) 0 0
\(591\) −44.0000 −1.80992
\(592\) 40.0000 1.64399
\(593\) 15.0000 0.615976 0.307988 0.951390i \(-0.400344\pi\)
0.307988 + 0.951390i \(0.400344\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −16.0000 −0.655386
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −33.0000 −1.34834 −0.674172 0.738575i \(-0.735499\pi\)
−0.674172 + 0.738575i \(0.735499\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 28.0000 1.13930
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) −4.00000 −0.161690
\(613\) −44.0000 −1.77714 −0.888572 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) 0 0
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −18.0000 −0.722897
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) −5.00000 −0.200321
\(624\) −8.00000 −0.320256
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 2.00000 0.0798723
\(628\) −4.00000 −0.159617
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 0 0
\(633\) 6.00000 0.238479
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 4.00000 0.158610
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −10.0000 −0.394055
\(645\) 26.0000 1.02375
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 32.0000 1.25322
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) −8.00000 −0.312348
\(657\) 5.00000 0.195069
\(658\) 0 0
\(659\) −47.0000 −1.83086 −0.915430 0.402477i \(-0.868149\pi\)
−0.915430 + 0.402477i \(0.868149\pi\)
\(660\) −4.00000 −0.155700
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) 25.0000 0.968004
\(668\) 34.0000 1.31550
\(669\) −38.0000 −1.46916
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 49.0000 1.88881 0.944406 0.328783i \(-0.106638\pi\)
0.944406 + 0.328783i \(0.106638\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) −2.00000 −0.0769231
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) −5.00000 −0.191882
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 2.00000 0.0764719
\(685\) −22.0000 −0.840577
\(686\) 0 0
\(687\) 36.0000 1.37349
\(688\) −52.0000 −1.98248
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) −15.0000 −0.570627 −0.285313 0.958434i \(-0.592098\pi\)
−0.285313 + 0.958434i \(0.592098\pi\)
\(692\) 28.0000 1.06440
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) −8.00000 −0.302372
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 8.00000 0.301511
\(705\) 14.0000 0.527271
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) −32.0000 −1.20263
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 11.0000 0.412532
\(712\) 0 0
\(713\) 45.0000 1.68526
\(714\) 0 0
\(715\) −1.00000 −0.0373979
\(716\) 18.0000 0.672692
\(717\) 48.0000 1.79259
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) −4.00000 −0.149071
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −14.0000 −0.520666
\(724\) 40.0000 1.48659
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −26.0000 −0.961645
\(732\) 8.00000 0.295689
\(733\) 41.0000 1.51437 0.757185 0.653201i \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 20.0000 0.735215
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) −3.00000 −0.109764
\(748\) 4.00000 0.146254
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 27.0000 0.985244 0.492622 0.870243i \(-0.336039\pi\)
0.492622 + 0.870243i \(0.336039\pi\)
\(752\) −28.0000 −1.02105
\(753\) 48.0000 1.74922
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) −8.00000 −0.290957
\(757\) −27.0000 −0.981332 −0.490666 0.871348i \(-0.663246\pi\)
−0.490666 + 0.871348i \(0.663246\pi\)
\(758\) 0 0
\(759\) 10.0000 0.362977
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 32.0000 1.15470
\(769\) −45.0000 −1.62274 −0.811371 0.584532i \(-0.801278\pi\)
−0.811371 + 0.584532i \(0.801278\pi\)
\(770\) 0 0
\(771\) 48.0000 1.72868
\(772\) 12.0000 0.431889
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) 0 0
\(777\) −20.0000 −0.717496
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) −4.00000 −0.143223
\(781\) −6.00000 −0.214697
\(782\) 0 0
\(783\) 20.0000 0.714742
\(784\) 4.00000 0.142857
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 3.00000 0.106938 0.0534692 0.998569i \(-0.482972\pi\)
0.0534692 + 0.998569i \(0.482972\pi\)
\(788\) 44.0000 1.56744
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 2.00000 0.0709327
\(796\) −4.00000 −0.141776
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) −14.0000 −0.495284
\(800\) 0 0
\(801\) 5.00000 0.176666
\(802\) 0 0
\(803\) −5.00000 −0.176446
\(804\) −8.00000 −0.282138
\(805\) −5.00000 −0.176227
\(806\) 0 0
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) −15.0000 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) −10.0000 −0.350931
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 16.0000 0.560112
\(817\) 13.0000 0.454812
\(818\) 0 0
\(819\) 1.00000 0.0349428
\(820\) −4.00000 −0.139686
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 10.0000 0.347524
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) −62.0000 −2.15076
\(832\) 8.00000 0.277350
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 17.0000 0.588309
\(836\) −2.00000 −0.0691714
\(837\) 36.0000 1.24434
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −48.0000 −1.65321
\(844\) −6.00000 −0.206529
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) 0 0
\(851\) −50.0000 −1.71398
\(852\) −24.0000 −0.822226
\(853\) 31.0000 1.06142 0.530710 0.847554i \(-0.321925\pi\)
0.530710 + 0.847554i \(0.321925\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) −26.0000 −0.886593
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) −26.0000 −0.883006
\(868\) −18.0000 −0.610960
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) 0 0
\(873\) 5.00000 0.169224
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) −20.0000 −0.675737
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) −2.00000 −0.0674583
\(880\) 4.00000 0.134840
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 4.00000 0.134535
\(885\) −16.0000 −0.537834
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 38.0000 1.27233
\(893\) 7.00000 0.234246
\(894\) 0 0
\(895\) 9.00000 0.300837
\(896\) 0 0
\(897\) 10.0000 0.333890
\(898\) 0 0
\(899\) 45.0000 1.50083
\(900\) 8.00000 0.266667
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) 26.0000 0.865226
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −5.00000 −0.166022 −0.0830111 0.996549i \(-0.526454\pi\)
−0.0830111 + 0.996549i \(0.526454\pi\)
\(908\) −48.0000 −1.59294
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 25.0000 0.828287 0.414143 0.910212i \(-0.364081\pi\)
0.414143 + 0.910212i \(0.364081\pi\)
\(912\) −8.00000 −0.264906
\(913\) 3.00000 0.0992855
\(914\) 0 0
\(915\) 4.00000 0.132236
\(916\) −36.0000 −1.18947
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) −4.00000 −0.131590
\(925\) −40.0000 −1.31519
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −22.0000 −0.720634
\(933\) 16.0000 0.523816
\(934\) 0 0
\(935\) 2.00000 0.0654070
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 0 0
\(939\) 24.0000 0.783210
\(940\) −14.0000 −0.456630
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 0 0
\(943\) 10.0000 0.325645
\(944\) 32.0000 1.04151
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) −44.0000 −1.42905
\(949\) −5.00000 −0.162307
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −48.0000 −1.55243
\(957\) 10.0000 0.323254
\(958\) 0 0
\(959\) −22.0000 −0.710417
\(960\) 16.0000 0.516398
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 14.0000 0.450910
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) −44.0000 −1.41494 −0.707472 0.706741i \(-0.750165\pi\)
−0.707472 + 0.706741i \(0.750165\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 14.0000 0.449281 0.224641 0.974442i \(-0.427879\pi\)
0.224641 + 0.974442i \(0.427879\pi\)
\(972\) 20.0000 0.641500
\(973\) 0 0
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) −8.00000 −0.256074
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) −5.00000 −0.159801
\(980\) 2.00000 0.0638877
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) −37.0000 −1.18012 −0.590058 0.807361i \(-0.700895\pi\)
−0.590058 + 0.807361i \(0.700895\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 14.0000 0.445625
\(988\) −2.00000 −0.0636285
\(989\) 65.0000 2.06688
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −2.00000 −0.0634043
\(996\) 12.0000 0.380235
\(997\) 48.0000 1.52018 0.760088 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 1001.2.a.c.1.1 1
3.2 odd 2 9009.2.a.h.1.1 1
7.6 odd 2 7007.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1001.2.a.c.1.1 1 1.1 even 1 trivial
7007.2.a.c.1.1 1 7.6 odd 2
9009.2.a.h.1.1 1 3.2 odd 2