Properties

Label 9075.2.a.m.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
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
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} +1.00000 q^{21} -1.00000 q^{23} +3.00000 q^{24} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{31} +5.00000 q^{32} -3.00000 q^{34} -1.00000 q^{36} +3.00000 q^{37} +3.00000 q^{38} -2.00000 q^{39} -3.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} -1.00000 q^{46} +1.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000 q^{57} -6.00000 q^{58} -3.00000 q^{59} +10.0000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +6.00000 q^{67} +3.00000 q^{68} +1.00000 q^{69} -7.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} +3.00000 q^{74} -3.00000 q^{76} -2.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} -6.00000 q^{83} -1.00000 q^{84} +12.0000 q^{86} +6.00000 q^{87} -14.0000 q^{89} -2.00000 q^{91} +1.00000 q^{92} -2.00000 q^{93} +1.00000 q^{94} -5.00000 q^{96} -3.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 3.00000 0.486664
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −3.00000 −0.397360
\(58\) −6.00000 −0.787839
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 3.00000 0.363803
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 3.00000 0.348743
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 1.00000 0.104257
\(93\) −2.00000 −0.207390
\(94\) 1.00000 0.103142
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 0 0
\(101\) 11.0000 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(102\) 3.00000 0.297044
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 1.00000 0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −3.00000 −0.276172
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 0 0
\(122\) 10.0000 0.905357
\(123\) 3.00000 0.270501
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) −3.00000 −0.265165
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) 9.00000 0.771744
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −7.00000 −0.587427
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 6.00000 0.494872
\(148\) −3.00000 −0.246598
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −9.00000 −0.729996
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −7.00000 −0.556890
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) −12.0000 −0.914991
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) −14.0000 −1.04934
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) −2.00000 −0.148250
\(183\) −10.0000 −0.739221
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) −1.00000 −0.0729325
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) −7.00000 −0.505181
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −3.00000 −0.215387
\(195\) 0 0
\(196\) 6.00000 0.428571
\(197\) 21.0000 1.49619 0.748094 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) 0 0
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 11.0000 0.773957
\(203\) 6.00000 0.421117
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −6.00000 −0.412082
\(213\) 7.00000 0.479632
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) −2.00000 −0.135769
\(218\) 12.0000 0.812743
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) −3.00000 −0.201347
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −22.0000 −1.46019 −0.730096 0.683345i \(-0.760525\pi\)
−0.730096 + 0.683345i \(0.760525\pi\)
\(228\) 3.00000 0.198680
\(229\) 23.0000 1.51988 0.759941 0.649992i \(-0.225228\pi\)
0.759941 + 0.649992i \(0.225228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 7.00000 0.454699
\(238\) 3.00000 0.194461
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 3.00000 0.191273
\(247\) 6.00000 0.381771
\(248\) −6.00000 −0.381000
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −19.0000 −1.19217
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −12.0000 −0.747087
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −2.00000 −0.123560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 14.0000 0.856786
\(268\) −6.00000 −0.366508
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) 3.00000 0.181902
\(273\) 2.00000 0.121046
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −4.00000 −0.239904
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 17.0000 1.01413 0.507067 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) −1.00000 −0.0595491
\(283\) −1.00000 −0.0594438 −0.0297219 0.999558i \(-0.509462\pi\)
−0.0297219 + 0.999558i \(0.509462\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) 5.00000 0.294628
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 3.00000 0.175863
\(292\) −2.00000 −0.117041
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) −15.0000 −0.868927
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 20.0000 1.15087
\(303\) −11.0000 −0.631933
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 6.00000 0.339683
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 1.00000 0.0557278
\(323\) −9.00000 −0.500773
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) −12.0000 −0.663602
\(328\) 9.00000 0.496942
\(329\) −1.00000 −0.0551318
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 6.00000 0.329293
\(333\) 3.00000 0.164399
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 3.00000 0.162221
\(343\) 13.0000 0.701934
\(344\) −36.0000 −1.94099
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) −6.00000 −0.321634
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 3.00000 0.159448
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −3.00000 −0.158777
\(358\) −15.0000 −0.792775
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) −5.00000 −0.262794
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 6.00000 0.313197 0.156599 0.987662i \(-0.449947\pi\)
0.156599 + 0.987662i \(0.449947\pi\)
\(368\) 1.00000 0.0521286
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 2.00000 0.103695
\(373\) 12.0000 0.621336 0.310668 0.950518i \(-0.399447\pi\)
0.310668 + 0.950518i \(0.399447\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) −12.0000 −0.618031
\(378\) 1.00000 0.0514344
\(379\) −14.0000 −0.719132 −0.359566 0.933120i \(-0.617075\pi\)
−0.359566 + 0.933120i \(0.617075\pi\)
\(380\) 0 0
\(381\) 19.0000 0.973399
\(382\) −3.00000 −0.153493
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 12.0000 0.609994
\(388\) 3.00000 0.152302
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 18.0000 0.909137
\(393\) 2.00000 0.100887
\(394\) 21.0000 1.05796
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −26.0000 −1.30326
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) −6.00000 −0.299253
\(403\) 4.00000 0.199254
\(404\) −11.0000 −0.547270
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) −9.00000 −0.445566
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) −8.00000 −0.394132
\(413\) 3.00000 0.147620
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −37.0000 −1.80757 −0.903784 0.427989i \(-0.859222\pi\)
−0.903784 + 0.427989i \(0.859222\pi\)
\(420\) 0 0
\(421\) −15.0000 −0.731055 −0.365528 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) −16.0000 −0.778868
\(423\) 1.00000 0.0486217
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 7.00000 0.339151
\(427\) −10.0000 −0.483934
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) −3.00000 −0.143509
\(438\) −2.00000 −0.0955637
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −6.00000 −0.285391
\(443\) −13.0000 −0.617649 −0.308824 0.951119i \(-0.599936\pi\)
−0.308824 + 0.951119i \(0.599936\pi\)
\(444\) 3.00000 0.142374
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 15.0000 0.709476
\(448\) −7.00000 −0.330719
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) −20.0000 −0.939682
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 9.00000 0.421464
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 23.0000 1.07472
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −9.00000 −0.416917
\(467\) 32.0000 1.48078 0.740392 0.672176i \(-0.234640\pi\)
0.740392 + 0.672176i \(0.234640\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 9.00000 0.414259
\(473\) 0 0
\(474\) 7.00000 0.321521
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) −22.0000 −1.00207
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −30.0000 −1.35804
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) −3.00000 −0.135250
\(493\) 18.0000 0.810679
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 7.00000 0.313993
\(498\) 6.00000 0.268866
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 20.0000 0.892644
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 19.0000 0.842989
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −11.0000 −0.486136
\(513\) −3.00000 −0.132453
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) −9.00000 −0.395056
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) −6.00000 −0.262613
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 2.00000 0.0873704
\(525\) 0 0
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 3.00000 0.130066
\(533\) −6.00000 −0.259889
\(534\) 14.0000 0.605839
\(535\) 0 0
\(536\) −18.0000 −0.777482
\(537\) 15.0000 0.647298
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −19.0000 −0.816120
\(543\) 5.00000 0.214571
\(544\) −15.0000 −0.643120
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) −16.0000 −0.683486
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) −3.00000 −0.127688
\(553\) 7.00000 0.297670
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 2.00000 0.0846668
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 17.0000 0.717102
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 1.00000 0.0421076
\(565\) 0 0
\(566\) −1.00000 −0.0420331
\(567\) −1.00000 −0.0419961
\(568\) 21.0000 0.881140
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 3.00000 0.125327
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −5.00000 −0.208153 −0.104076 0.994569i \(-0.533189\pi\)
−0.104076 + 0.994569i \(0.533189\pi\)
\(578\) −8.00000 −0.332756
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 3.00000 0.124354
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −21.0000 −0.867502
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) −6.00000 −0.247436
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) −3.00000 −0.123299
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 26.0000 1.06411
\(598\) −2.00000 −0.0817861
\(599\) 23.0000 0.939755 0.469877 0.882732i \(-0.344298\pi\)
0.469877 + 0.882732i \(0.344298\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) −12.0000 −0.489083
\(603\) 6.00000 0.244339
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −11.0000 −0.446844
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 15.0000 0.608330
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) 3.00000 0.121268
\(613\) −32.0000 −1.29247 −0.646234 0.763139i \(-0.723657\pi\)
−0.646234 + 0.763139i \(0.723657\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 8.00000 0.320771
\(623\) 14.0000 0.560898
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −19.0000 −0.759393
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −9.00000 −0.358854
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 21.0000 0.835335
\(633\) 16.0000 0.635943
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −7.00000 −0.276916
\(640\) 0 0
\(641\) 28.0000 1.10593 0.552967 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 8.00000 0.315735
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −9.00000 −0.354100
\(647\) −43.0000 −1.69050 −0.845252 0.534368i \(-0.820550\pi\)
−0.845252 + 0.534368i \(0.820550\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) −10.0000 −0.391630
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 2.00000 0.0780274
\(658\) −1.00000 −0.0389841
\(659\) 34.0000 1.32445 0.662226 0.749304i \(-0.269612\pi\)
0.662226 + 0.749304i \(0.269612\pi\)
\(660\) 0 0
\(661\) −29.0000 −1.12797 −0.563985 0.825785i \(-0.690732\pi\)
−0.563985 + 0.825785i \(0.690732\pi\)
\(662\) −16.0000 −0.621858
\(663\) 6.00000 0.233021
\(664\) 18.0000 0.698535
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) 6.00000 0.232321
\(668\) 6.00000 0.232147
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 0 0
\(672\) 5.00000 0.192879
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 18.0000 0.691286
\(679\) 3.00000 0.115129
\(680\) 0 0
\(681\) 22.0000 0.843042
\(682\) 0 0
\(683\) −31.0000 −1.18618 −0.593091 0.805135i \(-0.702093\pi\)
−0.593091 + 0.805135i \(0.702093\pi\)
\(684\) −3.00000 −0.114708
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −23.0000 −0.877505
\(688\) −12.0000 −0.457496
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) −18.0000 −0.682288
\(697\) 9.00000 0.340899
\(698\) 6.00000 0.227103
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) −23.0000 −0.868698 −0.434349 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 9.00000 0.339441
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −11.0000 −0.413698
\(708\) −3.00000 −0.112747
\(709\) −9.00000 −0.338002 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(710\) 0 0
\(711\) −7.00000 −0.262521
\(712\) 42.0000 1.57402
\(713\) −2.00000 −0.0749006
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 15.0000 0.560576
\(717\) 24.0000 0.896296
\(718\) 14.0000 0.522475
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −10.0000 −0.372161
\(723\) 22.0000 0.818189
\(724\) 5.00000 0.185824
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 10.0000 0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 6.00000 0.221464
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 0 0
\(738\) −3.00000 −0.110432
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) −6.00000 −0.220267
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) 12.0000 0.439351
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −1.00000 −0.0364662
\(753\) −20.0000 −0.728841
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −14.0000 −0.508503
\(759\) 0 0
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 19.0000 0.688297
\(763\) −12.0000 −0.434429
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −6.00000 −0.216647
\(768\) 17.0000 0.613435
\(769\) −44.0000 −1.58668 −0.793340 0.608778i \(-0.791660\pi\)
−0.793340 + 0.608778i \(0.791660\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −4.00000 −0.143963
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 9.00000 0.323081
\(777\) 3.00000 0.107624
\(778\) −12.0000 −0.430221
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 0 0
\(782\) 3.00000 0.107280
\(783\) 6.00000 0.214423
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 37.0000 1.31891 0.659454 0.751745i \(-0.270788\pi\)
0.659454 + 0.751745i \(0.270788\pi\)
\(788\) −21.0000 −0.748094
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 26.0000 0.921546
\(797\) 52.0000 1.84193 0.920967 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 3.00000 0.106199
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 6.00000 0.211604
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 6.00000 0.211210
\(808\) −33.0000 −1.16094
\(809\) 31.0000 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) 0 0
\(811\) 47.0000 1.65039 0.825197 0.564846i \(-0.191064\pi\)
0.825197 + 0.564846i \(0.191064\pi\)
\(812\) −6.00000 −0.210559
\(813\) 19.0000 0.666359
\(814\) 0 0
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 36.0000 1.25948
\(818\) −10.0000 −0.349642
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −16.0000 −0.558064
\(823\) 10.0000 0.348578 0.174289 0.984695i \(-0.444237\pi\)
0.174289 + 0.984695i \(0.444237\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) 40.0000 1.39094 0.695468 0.718557i \(-0.255197\pi\)
0.695468 + 0.718557i \(0.255197\pi\)
\(828\) 1.00000 0.0347524
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 14.0000 0.485363
\(833\) 18.0000 0.623663
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) −37.0000 −1.27814
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −15.0000 −0.516934
\(843\) −17.0000 −0.585511
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 1.00000 0.0343807
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 1.00000 0.0343199
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) −7.00000 −0.239816
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) −23.0000 −0.785665 −0.392833 0.919610i \(-0.628505\pi\)
−0.392833 + 0.919610i \(0.628505\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) 6.00000 0.204361
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 6.00000 0.203888
\(867\) 8.00000 0.271694
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) −36.0000 −1.21911
\(873\) −3.00000 −0.101535
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) −35.0000 −1.18119
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) −20.0000 −0.673817 −0.336909 0.941537i \(-0.609381\pi\)
−0.336909 + 0.941537i \(0.609381\pi\)
\(882\) −6.00000 −0.202031
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −13.0000 −0.436744
\(887\) 54.0000 1.81314 0.906571 0.422053i \(-0.138690\pi\)
0.906571 + 0.422053i \(0.138690\pi\)
\(888\) 9.00000 0.302020
\(889\) 19.0000 0.637240
\(890\) 0 0
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 3.00000 0.100391
\(894\) 15.0000 0.501675
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) 2.00000 0.0667781
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 12.0000 0.399335
\(904\) 54.0000 1.79601
\(905\) 0 0
\(906\) −20.0000 −0.664455
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 22.0000 0.730096
\(909\) 11.0000 0.364847
\(910\) 0 0
\(911\) 29.0000 0.960813 0.480406 0.877046i \(-0.340489\pi\)
0.480406 + 0.877046i \(0.340489\pi\)
\(912\) 3.00000 0.0993399
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) −23.0000 −0.759941
\(917\) 2.00000 0.0660458
\(918\) 3.00000 0.0990148
\(919\) 25.0000 0.824674 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 38.0000 1.25146
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) 8.00000 0.262754
\(928\) −30.0000 −0.984798
\(929\) 32.0000 1.04989 0.524943 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 9.00000 0.294805
\(933\) −8.00000 −0.261908
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) −6.00000 −0.195907
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) −49.0000 −1.59735 −0.798677 0.601760i \(-0.794466\pi\)
−0.798677 + 0.601760i \(0.794466\pi\)
\(942\) −18.0000 −0.586472
\(943\) 3.00000 0.0976934
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 0 0
\(947\) −37.0000 −1.20234 −0.601169 0.799122i \(-0.705298\pi\)
−0.601169 + 0.799122i \(0.705298\pi\)
\(948\) −7.00000 −0.227349
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) −9.00000 −0.291692
\(953\) −51.0000 −1.65205 −0.826026 0.563632i \(-0.809404\pi\)
−0.826026 + 0.563632i \(0.809404\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −14.0000 −0.452319
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 6.00000 0.193448
\(963\) −8.00000 −0.257796
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) −10.0000 −0.319765
\(979\) 0 0
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 30.0000 0.957338
\(983\) 1.00000 0.0318950 0.0159475 0.999873i \(-0.494924\pi\)
0.0159475 + 0.999873i \(0.494924\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 1.00000 0.0318304
\(988\) −6.00000 −0.190885
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 10.0000 0.317500
\(993\) 16.0000 0.507745
\(994\) 7.00000 0.222027
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 8.00000 0.253236
\(999\) −3.00000 −0.0949158
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 9075.2.a.m.1.1 yes 1
5.4 even 2 9075.2.a.h.1.1 yes 1
11.10 odd 2 9075.2.a.e.1.1 1
55.54 odd 2 9075.2.a.p.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.e.1.1 1 11.10 odd 2
9075.2.a.h.1.1 yes 1 5.4 even 2
9075.2.a.m.1.1 yes 1 1.1 even 1 trivial
9075.2.a.p.1.1 yes 1 55.54 odd 2