Properties

Label 3150.2.g.s.2899.1
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000 q^{19} -4.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -8.00000i q^{37} +6.00000i q^{38} -10.0000 q^{41} -2.00000i q^{43} -4.00000 q^{44} +10.0000i q^{47} -1.00000 q^{49} -6.00000i q^{52} -14.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000i q^{67} -4.00000i q^{68} +2.00000 q^{71} -10.0000i q^{73} -8.00000 q^{74} +6.00000 q^{76} +4.00000i q^{77} -16.0000 q^{79} +10.0000i q^{82} +8.00000i q^{83} -2.00000 q^{86} +4.00000i q^{88} +2.00000 q^{89} -6.00000 q^{91} +10.0000 q^{94} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 8q^{11} + 2q^{14} + 2q^{16} - 12q^{19} + 12q^{26} - 12q^{29} - 8q^{31} + 8q^{34} - 20q^{41} - 8q^{44} - 2q^{49} - 2q^{56} - 8q^{59} - 16q^{61} - 2q^{64} + 4q^{71} - 16q^{74} + 12q^{76} - 32q^{79} - 4q^{86} + 4q^{89} - 12q^{91} + 20q^{94} + O(q^{100}) \)

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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 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
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 6.00000i 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 6.00000i − 0.832050i
\(53\) − 14.0000i − 1.92305i −0.274721 0.961524i \(-0.588586\pi\)
0.274721 0.961524i \(-0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.00000i − 0.733017i −0.930415 0.366508i \(-0.880553\pi\)
0.930415 0.366508i \(-0.119447\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.0000i 1.10432i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 4.00000i 0.426401i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 8.00000i 0.724286i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.00000i − 0.167836i
\(143\) 24.0000i 2.00698i
\(144\) 0 0
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 8.00000i 0.657596i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 6.00000i − 0.486664i
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 6.00000i 0.464294i 0.972681 + 0.232147i \(0.0745750\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000i 0.152499i
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) − 2.00000i − 0.149906i
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.0000i 1.17004i
\(188\) − 10.0000i − 0.729325i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 6.00000i − 0.422159i
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 6.00000i 0.416025i
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 14.0000i 0.961524i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) − 18.0000i − 1.21911i
\(219\) 0 0
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 4.00000i 0.259281i
\(239\) −22.0000 −1.42306 −0.711531 0.702655i \(-0.751998\pi\)
−0.711531 + 0.702655i \(0.751998\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) − 36.0000i − 2.29063i
\(248\) − 4.00000i − 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.0000i 0.998053i 0.866587 + 0.499026i \(0.166309\pi\)
−0.866587 + 0.499026i \(0.833691\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 4.00000i 0.247121i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 6.00000i 0.366508i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 12.0000i 0.721010i 0.932757 + 0.360505i \(0.117396\pi\)
−0.932757 + 0.360505i \(0.882604\pi\)
\(278\) 10.0000i 0.599760i
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) − 10.0000i − 0.590281i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000i 0.585206i
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 16.0000i 0.920697i
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) − 10.0000i − 0.552158i
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) 0 0
\(334\) 6.00000 0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 4.00000i − 0.213201i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 24.0000i 1.26844i
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 8.00000i 0.420471i
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.0000 0.726844
\(372\) 0 0
\(373\) 32.0000i 1.65690i 0.560065 + 0.828449i \(0.310776\pi\)
−0.560065 + 0.828449i \(0.689224\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) − 36.0000i − 1.85409i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 18.0000i − 0.920960i
\(383\) 30.0000i 1.53293i 0.642287 + 0.766464i \(0.277986\pi\)
−0.642287 + 0.766464i \(0.722014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 2.00000i 0.101535i
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 0 0
\(403\) − 24.0000i − 1.19553i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) − 32.0000i − 1.58618i
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 24.0000i 1.17388i
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) − 12.0000i − 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) − 22.0000i − 1.05725i −0.848855 0.528626i \(-0.822707\pi\)
0.848855 0.528626i \(-0.177293\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.0000i 1.14156i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) − 14.0000i − 0.658505i
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000i 0.654892i 0.944870 + 0.327446i \(0.106188\pi\)
−0.944870 + 0.327446i \(0.893812\pi\)
\(458\) 4.00000i 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 32.0000i − 1.48078i −0.672176 0.740392i \(-0.734640\pi\)
0.672176 0.740392i \(-0.265360\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 0 0
\(472\) − 4.00000i − 0.184115i
\(473\) − 8.00000i − 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 22.0000i 1.00626i
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) − 10.0000i − 0.455488i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) − 24.0000i − 1.08091i
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 2.00000i 0.0897123i
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 38.0000 1.68432 0.842160 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 16.0000 0.705730
\(515\) 0 0
\(516\) 0 0
\(517\) 40.0000i 1.75920i
\(518\) − 8.00000i − 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) − 16.0000i − 0.696971i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000i 0.260133i
\(533\) − 60.0000i − 2.59889i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 12.0000i 0.515444i
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) − 30.0000i − 1.28271i −0.767245 0.641354i \(-0.778373\pi\)
0.767245 0.641354i \(-0.221627\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 0 0
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) − 16.0000i − 0.680389i
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) − 8.00000i − 0.337460i
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 2.00000i 0.0839181i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) − 24.0000i − 1.00349i
\(573\) 0 0
\(574\) −10.0000 −0.417392
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) − 56.0000i − 2.31928i
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) − 8.00000i − 0.328798i
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) − 2.00000i − 0.0815139i
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 6.00000i 0.243332i
\(609\) 0 0
\(610\) 0 0
\(611\) −60.0000 −2.42734
\(612\) 0 0
\(613\) 48.0000i 1.93870i 0.245680 + 0.969351i \(0.420989\pi\)
−0.245680 + 0.969351i \(0.579011\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 10.0000i 0.402585i 0.979531 + 0.201292i \(0.0645141\pi\)
−0.979531 + 0.201292i \(0.935486\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 12.0000i − 0.481156i
\(623\) 2.00000i 0.0801283i
\(624\) 0 0
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) 24.0000i 0.950169i
\(639\) 0 0
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 2.00000i 0.0786281i 0.999227 + 0.0393141i \(0.0125173\pi\)
−0.999227 + 0.0393141i \(0.987483\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000i 0.391630i
\(653\) 38.0000i 1.48705i 0.668705 + 0.743527i \(0.266849\pi\)
−0.668705 + 0.743527i \(0.733151\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 10.0000i 0.389841i
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) − 16.0000i − 0.621858i
\(663\) 0 0
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 6.00000i − 0.232147i
\(669\) 0 0
\(670\) 0 0
\(671\) −32.0000 −1.23535
\(672\) 0 0
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 16.0000i 0.612672i
\(683\) − 52.0000i − 1.98972i −0.101237 0.994862i \(-0.532280\pi\)
0.101237 0.994862i \(-0.467720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) − 2.00000i − 0.0762493i
\(689\) 84.0000 3.20015
\(690\) 0 0
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) − 2.00000i − 0.0760286i
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) − 40.0000i − 1.51511i
\(698\) 28.0000i 1.05982i
\(699\) 0 0
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 48.0000i 1.81035i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.00000i 0.0749532i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) − 34.0000i − 1.26887i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) − 17.0000i − 0.632674i
\(723\) 0 0
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) − 6.00000i − 0.222375i
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 14.0000i − 0.513956i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) − 16.0000i − 0.585018i
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 10.0000i 0.364662i
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) − 40.0000i − 1.45382i −0.686730 0.726912i \(-0.740955\pi\)
0.686730 0.726912i \(-0.259045\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 18.0000i 0.651644i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000i 0.359908i
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) − 22.0000i − 0.788738i
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) − 48.0000i − 1.70453i
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) 0 0
\(802\) − 16.0000i − 0.564980i
\(803\) − 40.0000i − 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 6.00000i 0.210559i
\(813\) 0 0
\(814\) −32.0000 −1.12160
\(815\) 0 0
\(816\) 0 0
\(817\) 12.0000i 0.419827i
\(818\) 18.0000i 0.629355i
\(819\) 0 0
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) 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
\(832\) − 6.00000i − 0.208013i
\(833\) − 4.00000i − 0.138592i
\(834\) 0 0
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 36.0000i 1.24360i
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 14.0000i − 0.482472i
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000i 0.171802i
\(848\) − 14.0000i − 0.480762i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 48.0000i − 1.63965i −0.572615 0.819824i \(-0.694071\pi\)
0.572615 0.819824i \(-0.305929\pi\)
\(858\) 0 0
\(859\) −30.0000 −1.02359 −0.511793 0.859109i \(-0.671019\pi\)
−0.511793 + 0.859109i \(0.671019\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30.0000i 1.02180i
\(863\) − 44.0000i − 1.49778i −0.662696 0.748889i \(-0.730588\pi\)
0.662696 0.748889i \(-0.269412\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 18.0000i 0.609557i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.0000i 0.945493i 0.881199 + 0.472746i \(0.156737\pi\)
−0.881199 + 0.472746i \(0.843263\pi\)
\(878\) 28.0000i 0.944954i
\(879\) 0 0
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) 6.00000i 0.201916i 0.994891 + 0.100958i \(0.0321908\pi\)
−0.994891 + 0.100958i \(0.967809\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) − 2.00000i − 0.0671534i −0.999436 0.0335767i \(-0.989310\pi\)
0.999436 0.0335767i \(-0.0106898\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) − 60.0000i − 2.00782i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 12.0000i 0.400445i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 56.0000 1.86563
\(902\) 40.0000i 1.33185i
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 10.0000i 0.332045i 0.986122 + 0.166022i \(0.0530924\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) 20.0000i 0.663723i
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 32.0000i 1.05905i
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) − 4.00000i − 0.132092i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.00000i 0.0658665i
\(923\) 12.0000i 0.394985i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) 6.00000i 0.196960i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) − 6.00000i − 0.196537i
\(933\) 0 0
\(934\) −32.0000 −1.04707
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) − 6.00000i − 0.195907i
\(939\) 0 0
\(940\) 0 0
\(941\) −58.0000 −1.89075 −0.945373 0.325991i \(-0.894302\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) 0 0
\(952\) − 4.00000i − 0.129641i
\(953\) 30.0000i 0.971795i 0.874016 + 0.485898i \(0.161507\pi\)
−0.874016 + 0.485898i \(0.838493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 22.0000 0.711531
\(957\) 0 0
\(958\) − 36.0000i − 1.16311i
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 48.0000i − 1.54758i
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) − 10.0000i − 0.320585i
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 0 0
\(982\) 24.0000i 0.765871i
\(983\) 38.0000i 1.21201i 0.795460 + 0.606006i \(0.207229\pi\)
−0.795460 + 0.606006i \(0.792771\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 36.0000i 1.14531i
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 0 0
\(994\) 2.00000 0.0634361
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.0000i − 0.443384i −0.975117 0.221692i \(-0.928842\pi\)
0.975117 0.221692i \(-0.0711580\pi\)
\(998\) − 24.0000i − 0.759707i
\(999\) 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.s.2899.1 2
3.2 odd 2 3150.2.g.b.2899.2 2
5.2 odd 4 630.2.a.g.1.1 yes 1
5.3 odd 4 3150.2.a.s.1.1 1
5.4 even 2 inner 3150.2.g.s.2899.2 2
15.2 even 4 630.2.a.e.1.1 1
15.8 even 4 3150.2.a.bh.1.1 1
15.14 odd 2 3150.2.g.b.2899.1 2
20.7 even 4 5040.2.a.n.1.1 1
35.27 even 4 4410.2.a.bl.1.1 1
60.47 odd 4 5040.2.a.bp.1.1 1
105.62 odd 4 4410.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.a.e.1.1 1 15.2 even 4
630.2.a.g.1.1 yes 1 5.2 odd 4
3150.2.a.s.1.1 1 5.3 odd 4
3150.2.a.bh.1.1 1 15.8 even 4
3150.2.g.b.2899.1 2 15.14 odd 2
3150.2.g.b.2899.2 2 3.2 odd 2
3150.2.g.s.2899.1 2 1.1 even 1 trivial
3150.2.g.s.2899.2 2 5.4 even 2 inner
4410.2.a.a.1.1 1 105.62 odd 4
4410.2.a.bl.1.1 1 35.27 even 4
5040.2.a.n.1.1 1 20.7 even 4
5040.2.a.bp.1.1 1 60.47 odd 4