Properties

Label 3150.2.g.f.2899.2
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 350)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(1.00000i\)
Character \(\chi\) = 3150.2899
Dual form 3150.2.g.f.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} -3.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +7.00000 q^{19} -3.00000i q^{22} -2.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -3.00000 q^{34} -8.00000i q^{37} +7.00000i q^{38} +9.00000 q^{41} +8.00000i q^{43} +3.00000 q^{44} -6.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +12.0000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +7.00000i q^{67} -3.00000i q^{68} -6.00000 q^{71} +5.00000i q^{73} +8.00000 q^{74} -7.00000 q^{76} +3.00000i q^{77} -14.0000 q^{79} +9.00000i q^{82} +9.00000i q^{83} -8.00000 q^{86} +3.00000i q^{88} -15.0000 q^{89} +2.00000 q^{91} +6.00000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 6q^{11} + 2q^{14} + 2q^{16} + 14q^{19} - 4q^{26} - 12q^{29} - 8q^{31} - 6q^{34} + 18q^{41} + 6q^{44} - 2q^{49} - 2q^{56} + 24q^{59} - 20q^{61} - 2q^{64} - 12q^{71} + 16q^{74} - 14q^{76} - 28q^{79} - 16q^{86} - 30q^{89} + 4q^{91} + 12q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 3.00000i − 0.639602i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 7.00000i 1.13555i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 5.00000i 0.585206i 0.956234 + 0.292603i \(0.0945214\pi\)
−0.956234 + 0.292603i \(0.905479\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) −14.0000 −1.57512 −0.787562 0.616236i \(-0.788657\pi\)
−0.787562 + 0.616236i \(0.788657\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.00000i 0.993884i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 20.0000i 1.97066i 0.170664 + 0.985329i \(0.445409\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 10.0000i − 0.905357i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 7.00000i − 0.606977i
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 21.0000i 1.79415i 0.441877 + 0.897076i \(0.354313\pi\)
−0.441877 + 0.897076i \(0.645687\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 6.00000i − 0.503509i
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) −5.00000 −0.413803
\(147\) 0 0
\(148\) 8.00000i 0.657596i
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) − 7.00000i − 0.567775i
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) − 20.0000i − 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(158\) − 14.0000i − 1.11378i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) − 8.00000i − 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) − 15.0000i − 1.12430i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.00000i − 0.658145i
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) 5.00000i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 −1.39347
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) − 12.0000i − 0.824163i
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) − 14.0000i − 0.948200i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 3.00000i 0.194461i
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 14.0000i 0.890799i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.00000 0.429198
\(267\) 0 0
\(268\) − 7.00000i − 0.427593i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) −21.0000 −1.26866
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 7.00000i 0.419832i
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 1.00000i − 0.0594438i −0.999558 0.0297219i \(-0.990538\pi\)
0.999558 0.0297219i \(-0.00946217\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) − 9.00000i − 0.531253i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) − 5.00000i − 0.292603i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 12.0000i 0.695141i
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000i 0.399511i 0.979846 + 0.199756i \(0.0640148\pi\)
−0.979846 + 0.199756i \(0.935985\pi\)
\(308\) − 3.00000i − 0.170941i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 20.0000 1.12867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.0000i 1.16847i
\(324\) 0 0
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) 0 0
\(328\) − 9.00000i − 0.496942i
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) − 21.0000i − 1.12734i −0.826000 0.563670i \(-0.809389\pi\)
0.826000 0.563670i \(-0.190611\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 3.00000i − 0.159901i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) 3.00000i 0.158555i
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 2.00000i 0.105118i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) −17.0000 −0.873231 −0.436616 0.899648i \(-0.643823\pi\)
−0.436616 + 0.899648i \(0.643823\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.00000i 0.306987i
\(383\) 30.0000i 1.53293i 0.642287 + 0.766464i \(0.277986\pi\)
−0.642287 + 0.766464i \(0.722014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) − 14.0000i − 0.701757i
\(399\) 0 0
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 20.0000i − 0.985329i
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) − 21.0000i − 1.02714i
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 17.0000i 0.827547i
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 10.0000i 0.483934i
\(428\) − 3.00000i − 0.145010i
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 11.0000i 0.528626i 0.964437 + 0.264313i \(0.0851452\pi\)
−0.964437 + 0.264313i \(0.914855\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6.00000i − 0.285391i
\(443\) − 21.0000i − 0.997740i −0.866677 0.498870i \(-0.833748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) − 9.00000i − 0.423324i
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) − 17.0000i − 0.795226i −0.917553 0.397613i \(-0.869839\pi\)
0.917553 0.397613i \(-0.130161\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 0 0
\(472\) − 12.0000i − 0.552345i
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) − 12.0000i − 0.548867i
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) − 25.0000i − 1.13872i
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000i 0.269137i
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 15.0000i − 0.669483i
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000i 0.0887357i
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 5.00000 0.221187
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) 0 0
\(517\) 18.0000i 0.791639i
\(518\) − 8.00000i − 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) − 7.00000i − 0.306089i −0.988219 0.153044i \(-0.951092\pi\)
0.988219 0.153044i \(-0.0489077\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) − 12.0000i − 0.522728i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 7.00000i 0.303488i
\(533\) 18.0000i 0.779667i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 0 0
\(538\) − 6.00000i − 0.258678i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 2.00000i 0.0859074i
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) − 35.0000i − 1.49649i −0.663421 0.748246i \(-0.730896\pi\)
0.663421 0.748246i \(-0.269104\pi\)
\(548\) − 21.0000i − 0.897076i
\(549\) 0 0
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) 14.0000i 0.595341i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −7.00000 −0.296866
\(557\) 36.0000i 1.52537i 0.646771 + 0.762684i \(0.276119\pi\)
−0.646771 + 0.762684i \(0.723881\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000i 0.759284i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.00000 0.0420331
\(567\) 0 0
\(568\) 6.00000i 0.251754i
\(569\) −27.0000 −1.13190 −0.565949 0.824440i \(-0.691490\pi\)
−0.565949 + 0.824440i \(0.691490\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 6.00000i 0.250873i
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) 0 0
\(576\) 0 0
\(577\) 7.00000i 0.291414i 0.989328 + 0.145707i \(0.0465456\pi\)
−0.989328 + 0.145707i \(0.953454\pi\)
\(578\) 8.00000i 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 39.0000i 1.60970i 0.593477 + 0.804851i \(0.297755\pi\)
−0.593477 + 0.804851i \(0.702245\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) 0 0
\(592\) − 8.00000i − 0.328798i
\(593\) 27.0000i 1.10876i 0.832265 + 0.554379i \(0.187044\pi\)
−0.832265 + 0.554379i \(0.812956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 44.0000i − 1.78590i −0.450151 0.892952i \(-0.648630\pi\)
0.450151 0.892952i \(-0.351370\pi\)
\(608\) 7.00000i 0.283887i
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) −7.00000 −0.282497
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000i 0.721734i
\(623\) 15.0000i 0.600962i
\(624\) 0 0
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 20.0000i 0.798087i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 14.0000i 0.556890i
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 18.0000i 0.712627i
\(639\) 0 0
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −21.0000 −0.826234
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.00000i − 0.195815i
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) − 6.00000i − 0.233904i
\(659\) 9.00000 0.350590 0.175295 0.984516i \(-0.443912\pi\)
0.175295 + 0.984516i \(0.443912\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 25.0000i − 0.971653i
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) − 3.00000i − 0.114792i −0.998351 0.0573959i \(-0.981720\pi\)
0.998351 0.0573959i \(-0.0182797\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.00000i 0.304997i
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −19.0000 −0.722794 −0.361397 0.932412i \(-0.617700\pi\)
−0.361397 + 0.932412i \(0.617700\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 21.0000 0.797149
\(695\) 0 0
\(696\) 0 0
\(697\) 27.0000i 1.02270i
\(698\) − 8.00000i − 0.302804i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) − 56.0000i − 2.11208i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.0000i 0.562149i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 0 0
\(718\) − 6.00000i − 0.223918i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 20.0000 0.744839
\(722\) 30.0000i 1.11648i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 34.0000i 1.26099i 0.776193 + 0.630495i \(0.217148\pi\)
−0.776193 + 0.630495i \(0.782852\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) − 21.0000i − 0.773545i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000i 0.440534i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) 9.00000i 0.329073i
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −46.0000 −1.67856 −0.839282 0.543696i \(-0.817024\pi\)
−0.839282 + 0.543696i \(0.817024\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 34.0000i 1.23575i 0.786276 + 0.617876i \(0.212006\pi\)
−0.786276 + 0.617876i \(0.787994\pi\)
\(758\) − 17.0000i − 0.617468i
\(759\) 0 0
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 5.00000i − 0.179954i
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) − 24.0000i − 0.860442i
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) − 20.0000i − 0.710221i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 27.0000i 0.953403i
\(803\) − 15.0000i − 0.529339i
\(804\) 0 0
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) 56.0000i 1.95919i
\(818\) 25.0000i 0.874105i
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 20.0000 0.696733
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 9.00000i 0.312961i 0.987681 + 0.156480i \(0.0500148\pi\)
−0.987681 + 0.156480i \(0.949985\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 2.00000i − 0.0693375i
\(833\) − 3.00000i − 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 21.0000 0.726300
\(837\) 0 0
\(838\) − 3.00000i − 0.103633i
\(839\) −6.00000 −0.207143 −0.103572 0.994622i \(-0.533027\pi\)
−0.103572 + 0.994622i \(0.533027\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000i 0.689246i
\(843\) 0 0
\(844\) −17.0000 −0.585164
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 12.0000i 0.412082i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 15.0000i 0.512390i 0.966625 + 0.256195i \(0.0824690\pi\)
−0.966625 + 0.256195i \(0.917531\pi\)
\(858\) 0 0
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −11.0000 −0.373795
\(867\) 0 0
\(868\) − 4.00000i − 0.135769i
\(869\) 42.0000 1.42475
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 14.0000i 0.474100i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 4.00000i 0.134993i
\(879\) 0 0
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) 47.0000i 1.58168i 0.612026 + 0.790838i \(0.290355\pi\)
−0.612026 + 0.790838i \(0.709645\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 21.0000 0.705509
\(887\) 6.00000i 0.201460i 0.994914 + 0.100730i \(0.0321179\pi\)
−0.994914 + 0.100730i \(0.967882\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) − 14.0000i − 0.468755i
\(893\) − 42.0000i − 1.40548i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) − 15.0000i − 0.500556i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) − 27.0000i − 0.899002i
\(903\) 0 0
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) 0 0
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) − 27.0000i − 0.893570i
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 18.0000i − 0.592798i
\(923\) − 12.0000i − 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) − 6.00000i − 0.196960i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) − 29.0000i − 0.947389i −0.880689 0.473694i \(-0.842920\pi\)
0.880689 0.473694i \(-0.157080\pi\)
\(938\) 7.00000i 0.228558i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) − 3.00000i − 0.0972306i
\(953\) − 57.0000i − 1.84641i −0.384307 0.923206i \(-0.625559\pi\)
0.384307 0.923206i \(-0.374441\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 18.0000i 0.581554i
\(959\) 21.0000 0.678125
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 16.0000i 0.515861i
\(963\) 0 0
\(964\) 25.0000 0.805196
\(965\) 0 0
\(966\) 0 0
\(967\) 34.0000i 1.09337i 0.837340 + 0.546683i \(0.184110\pi\)
−0.837340 + 0.546683i \(0.815890\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 0 0
\(970\) 0 0
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) − 7.00000i − 0.224410i
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 3.00000i − 0.0959785i −0.998848 0.0479893i \(-0.984719\pi\)
0.998848 0.0479893i \(-0.0152813\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000i 0.382935i
\(983\) − 12.0000i − 0.382741i −0.981518 0.191370i \(-0.938707\pi\)
0.981518 0.191370i \(-0.0612931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) − 14.0000i − 0.445399i
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 0 0
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) 0 0
\(997\) − 62.0000i − 1.96356i −0.190022 0.981780i \(-0.560856\pi\)
0.190022 0.981780i \(-0.439144\pi\)
\(998\) 28.0000i 0.886325i
\(999\) 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 3150.2.g.f.2899.2 2
3.2 odd 2 350.2.c.c.99.1 2
5.2 odd 4 3150.2.a.m.1.1 1
5.3 odd 4 3150.2.a.x.1.1 1
5.4 even 2 inner 3150.2.g.f.2899.1 2
12.11 even 2 2800.2.g.i.449.1 2
15.2 even 4 350.2.a.e.1.1 yes 1
15.8 even 4 350.2.a.a.1.1 1
15.14 odd 2 350.2.c.c.99.2 2
21.20 even 2 2450.2.c.h.99.1 2
60.23 odd 4 2800.2.a.x.1.1 1
60.47 odd 4 2800.2.a.h.1.1 1
60.59 even 2 2800.2.g.i.449.2 2
105.62 odd 4 2450.2.a.x.1.1 1
105.83 odd 4 2450.2.a.m.1.1 1
105.104 even 2 2450.2.c.h.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 15.8 even 4
350.2.a.e.1.1 yes 1 15.2 even 4
350.2.c.c.99.1 2 3.2 odd 2
350.2.c.c.99.2 2 15.14 odd 2
2450.2.a.m.1.1 1 105.83 odd 4
2450.2.a.x.1.1 1 105.62 odd 4
2450.2.c.h.99.1 2 21.20 even 2
2450.2.c.h.99.2 2 105.104 even 2
2800.2.a.h.1.1 1 60.47 odd 4
2800.2.a.x.1.1 1 60.23 odd 4
2800.2.g.i.449.1 2 12.11 even 2
2800.2.g.i.449.2 2 60.59 even 2
3150.2.a.m.1.1 1 5.2 odd 4
3150.2.a.x.1.1 1 5.3 odd 4
3150.2.g.f.2899.1 2 5.4 even 2 inner
3150.2.g.f.2899.2 2 1.1 even 1 trivial