Properties

Label 3150.2.g.v.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.v.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +3.00000 q^{19} +5.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} -8.00000i q^{37} +3.00000i q^{38} -11.0000 q^{41} -8.00000i q^{43} -5.00000 q^{44} +2.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -4.00000i q^{53} +1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -9.00000i q^{67} +1.00000i q^{68} +10.0000 q^{71} -7.00000i q^{73} +8.00000 q^{74} -3.00000 q^{76} +5.00000i q^{77} +2.00000 q^{79} -11.0000i q^{82} -11.0000i q^{83} +8.00000 q^{86} -5.00000i q^{88} -11.0000 q^{89} +6.00000 q^{91} -2.00000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 10q^{11} - 2q^{14} + 2q^{16} + 6q^{19} + 12q^{26} - 12q^{29} - 8q^{31} + 2q^{34} - 22q^{41} - 10q^{44} - 2q^{49} + 2q^{56} + 8q^{59} - 4q^{61} - 2q^{64} + 20q^{71} + 16q^{74} - 6q^{76} + 4q^{79} + 16q^{86} - 22q^{89} + 12q^{91} - 4q^{94} + O(q^{100}) \)

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
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.00000i − 0.242536i −0.992620 0.121268i \(-0.961304\pi\)
0.992620 0.121268i \(-0.0386960\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.00000i 1.06600i
\(23\) 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
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 3.00000i 0.486664i
\(39\) 0 0
\(40\) 0 0
\(41\) −11.0000 −1.71791 −0.858956 0.512050i \(-0.828886\pi\)
−0.858956 + 0.512050i \(0.828886\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000i 0.832050i
\(53\) − 4.00000i − 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) 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 \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.00000i − 1.09952i −0.835321 0.549762i \(-0.814718\pi\)
0.835321 0.549762i \(-0.185282\pi\)
\(68\) 1.00000i 0.121268i
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 11.0000i − 1.21475i
\(83\) − 11.0000i − 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) − 5.00000i − 0.533002i
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 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
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 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
\(113\) 1.00000i 0.0940721i 0.998893 + 0.0470360i \(0.0149776\pi\)
−0.998893 + 0.0470360i \(0.985022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) − 2.00000i − 0.181071i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 9.00000 0.777482
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.0000i 0.839181i
\(143\) − 30.0000i − 2.50873i
\(144\) 0 0
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) 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 \(0.394462\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) 0 0
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 2.00000i 0.159111i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 19.0000i − 1.48819i −0.668071 0.744097i \(-0.732880\pi\)
0.668071 0.744097i \(-0.267120\pi\)
\(164\) 11.0000 0.858956
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000i 0.609994i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) − 11.0000i − 0.824485i
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.00000i − 0.365636i
\(188\) − 2.00000i − 0.145865i
\(189\) 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
\(193\) − 19.0000i − 1.36765i −0.729646 0.683825i \(-0.760315\pi\)
0.729646 0.683825i \(-0.239685\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) − 6.00000i − 0.416025i
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 4.00000i 0.274721i
\(213\) 0 0
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 18.0000i 1.21911i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) − 28.0000i − 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 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
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 1.00000i 0.0648204i
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 14.0000i 0.899954i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 18.0000i − 1.14531i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) − 8.00000i − 0.494242i
\(263\) 10.0000i 0.616626i 0.951285 + 0.308313i \(0.0997645\pi\)
−0.951285 + 0.308313i \(0.900236\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) 9.00000i 0.549762i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) − 1.00000i − 0.0606339i
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000i 1.80253i 0.433273 + 0.901263i \(0.357359\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(278\) 11.0000i 0.659736i
\(279\) 0 0
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) − 13.0000i − 0.772770i −0.922338 0.386385i \(-0.873724\pi\)
0.922338 0.386385i \(-0.126276\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) 30.0000 1.77394
\(287\) − 11.0000i − 0.649309i
\(288\) 0 0
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 7.00000i 0.409644i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 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
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 0 0
\(307\) − 13.0000i − 0.741949i −0.928643 0.370975i \(-0.879024\pi\)
0.928643 0.370975i \(-0.120976\pi\)
\(308\) − 5.00000i − 0.284901i
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 4.00000i 0.224662i 0.993671 + 0.112331i \(0.0358318\pi\)
−0.993671 + 0.112331i \(0.964168\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 19.0000 1.05231
\(327\) 0 0
\(328\) 11.0000i 0.607373i
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 11.0000i 0.603703i
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) 0 0
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 19.0000i 1.01997i 0.860182 + 0.509987i \(0.170350\pi\)
−0.860182 + 0.509987i \(0.829650\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 11.0000 0.582999
\(357\) 0 0
\(358\) 3.00000i 0.158555i
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 10.0000i 0.525588i
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) −9.00000 −0.462299 −0.231149 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.00000i 0.306987i
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.0000 0.967075
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) 8.00000 0.405616 0.202808 0.979219i \(-0.434993\pi\)
0.202808 + 0.979219i \(0.434993\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0 0
\(400\) 0 0
\(401\) −37.0000 −1.84769 −0.923846 0.382765i \(-0.874972\pi\)
−0.923846 + 0.382765i \(0.874972\pi\)
\(402\) 0 0
\(403\) 24.0000i 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) − 40.0000i − 1.98273i
\(408\) 0 0
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 15.0000i 0.733674i
\(419\) −39.0000 −1.90527 −0.952637 0.304109i \(-0.901641\pi\)
−0.952637 + 0.304109i \(0.901641\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 1.00000i 0.0486792i
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) − 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
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) 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
\(442\) − 6.00000i − 0.285391i
\(443\) − 37.0000i − 1.75792i −0.476893 0.878962i \(-0.658237\pi\)
0.476893 0.878962i \(-0.341763\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −22.0000 −1.04173
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) −55.0000 −2.58985
\(452\) − 1.00000i − 0.0470360i
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) − 25.0000i − 1.16945i −0.811231 0.584725i \(-0.801202\pi\)
0.811231 0.584725i \(-0.198798\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 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
\(467\) − 4.00000i − 0.185098i −0.995708 0.0925490i \(-0.970499\pi\)
0.995708 0.0925490i \(-0.0295015\pi\)
\(468\) 0 0
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) − 4.00000i − 0.184115i
\(473\) − 40.0000i − 1.83920i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) 4.00000i 0.182956i
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) − 5.00000i − 0.227744i
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 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
\(493\) 6.00000i 0.270226i
\(494\) 18.0000 0.809858
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 10.0000i 0.448561i
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 27.0000i − 1.20507i
\(503\) 30.0000i 1.33763i 0.743427 + 0.668817i \(0.233199\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 14.0000i − 0.621150i
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 0 0
\(517\) 10.0000i 0.439799i
\(518\) 8.00000i 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 0 0
\(523\) 13.0000i 0.568450i 0.958758 + 0.284225i \(0.0917363\pi\)
−0.958758 + 0.284225i \(0.908264\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) − 3.00000i − 0.130066i
\(533\) 66.0000i 2.85878i
\(534\) 0 0
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) 0 0
\(538\) 18.0000i 0.776035i
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) − 6.00000i − 0.257722i
\(543\) 0 0
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) − 27.0000i − 1.15444i −0.816590 0.577218i \(-0.804138\pi\)
0.816590 0.577218i \(-0.195862\pi\)
\(548\) 3.00000i 0.128154i
\(549\) 0 0
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) −11.0000 −0.466504
\(557\) 4.00000i 0.169485i 0.996403 + 0.0847427i \(0.0270068\pi\)
−0.996403 + 0.0847427i \(0.972993\pi\)
\(558\) 0 0
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) − 14.0000i − 0.590554i
\(563\) − 20.0000i − 0.842900i −0.906852 0.421450i \(-0.861521\pi\)
0.906852 0.421450i \(-0.138479\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 13.0000 0.546431
\(567\) 0 0
\(568\) − 10.0000i − 0.419591i
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 30.0000i 1.25436i
\(573\) 0 0
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 0 0
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) 16.0000i 0.665512i
\(579\) 0 0
\(580\) 0 0
\(581\) 11.0000 0.456357
\(582\) 0 0
\(583\) − 20.0000i − 0.828315i
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) − 13.0000i − 0.536567i −0.963340 0.268284i \(-0.913544\pi\)
0.963340 0.268284i \(-0.0864565\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) − 8.00000i − 0.328798i
\(593\) 39.0000i 1.60154i 0.598973 + 0.800769i \(0.295576\pi\)
−0.598973 + 0.800769i \(0.704424\pi\)
\(594\) 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 \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 3.00000i 0.121666i
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) 13.0000 0.524637
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 14.0000i 0.563619i 0.959470 + 0.281809i \(0.0909346\pi\)
−0.959470 + 0.281809i \(0.909065\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 6.00000i − 0.240578i
\(623\) − 11.0000i − 0.440706i
\(624\) 0 0
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 4.00000i 0.159617i
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 2.00000i − 0.0795557i
\(633\) 0 0
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) − 30.0000i − 1.18771i
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 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
\(646\) 3.00000 0.118033
\(647\) − 8.00000i − 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 19.0000i 0.744097i
\(653\) 28.0000i 1.09572i 0.836569 + 0.547862i \(0.184558\pi\)
−0.836569 + 0.547862i \(0.815442\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −11.0000 −0.429478
\(657\) 0 0
\(658\) − 2.00000i − 0.0779681i
\(659\) 1.00000 0.0389545 0.0194772 0.999810i \(-0.493800\pi\)
0.0194772 + 0.999810i \(0.493800\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) − 17.0000i − 0.660724i
\(663\) 0 0
\(664\) −11.0000 −0.426883
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) −29.0000 −1.11704
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 48.0000i − 1.84479i −0.386248 0.922395i \(-0.626229\pi\)
0.386248 0.922395i \(-0.373771\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) − 20.0000i − 0.765840i
\(683\) 13.0000i 0.497431i 0.968577 + 0.248716i \(0.0800084\pi\)
−0.968577 + 0.248716i \(0.919992\pi\)
\(684\) 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
\(691\) 49.0000 1.86405 0.932024 0.362397i \(-0.118041\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) 2.00000i 0.0760286i
\(693\) 0 0
\(694\) −19.0000 −0.721230
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0000i 0.416655i
\(698\) 8.00000i 0.302804i
\(699\) 0 0
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) 0 0
\(703\) − 24.0000i − 0.905177i
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 11.0000i 0.412242i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) 0 0
\(718\) 26.0000i 0.970311i
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) − 10.0000i − 0.372161i
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) − 6.00000i − 0.222528i −0.993791 0.111264i \(-0.964510\pi\)
0.993791 0.111264i \(-0.0354899\pi\)
\(728\) − 6.00000i − 0.222375i
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) − 45.0000i − 1.65760i
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000i 0.146845i
\(743\) 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
\(748\) 5.00000i 0.182818i
\(749\) −3.00000 −0.109618
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) 2.00000i 0.0729325i
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 9.00000i − 0.326895i
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 18.0000i 0.651644i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 19.0000i 0.683825i
\(773\) − 36.0000i − 1.29483i −0.762138 0.647415i \(-0.775850\pi\)
0.762138 0.647415i \(-0.224150\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 8.00000i 0.286814i
\(779\) −33.0000 −1.18235
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) 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
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) 2.00000 0.0707549
\(800\) 0 0
\(801\) 0 0
\(802\) − 37.0000i − 1.30652i
\(803\) − 35.0000i − 1.23512i
\(804\) 0 0
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 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
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 6.00000i 0.210559i
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) − 24.0000i − 0.839654i
\(818\) 21.0000i 0.734248i
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 10.0000i 0.348578i 0.984695 + 0.174289i \(0.0557627\pi\)
−0.984695 + 0.174289i \(0.944237\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) 41.0000i 1.42571i 0.701312 + 0.712855i \(0.252598\pi\)
−0.701312 + 0.712855i \(0.747402\pi\)
\(828\) 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
\(833\) 1.00000i 0.0346479i
\(834\) 0 0
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 0 0
\(838\) − 39.0000i − 1.34723i
\(839\) 2.00000 0.0690477 0.0345238 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000i 0.689246i
\(843\) 0 0
\(844\) −1.00000 −0.0344214
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) − 4.00000i − 0.137361i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 34.0000i − 1.16414i −0.813139 0.582069i \(-0.802243\pi\)
0.813139 0.582069i \(-0.197757\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 3.00000i 0.102478i 0.998686 + 0.0512390i \(0.0163170\pi\)
−0.998686 + 0.0512390i \(0.983683\pi\)
\(858\) 0 0
\(859\) 51.0000 1.74010 0.870049 0.492966i \(-0.164087\pi\)
0.870049 + 0.492966i \(0.164087\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.00000 0.0339814
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −54.0000 −1.82972
\(872\) − 18.0000i − 0.609557i
\(873\) 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 \(-0.181682\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) 0 0
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) 15.0000i 0.504790i 0.967624 + 0.252395i \(0.0812183\pi\)
−0.967624 + 0.252395i \(0.918782\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 37.0000 1.24304
\(887\) − 34.0000i − 1.14161i −0.821086 0.570804i \(-0.806632\pi\)
0.821086 0.570804i \(-0.193368\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) − 22.0000i − 0.736614i
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 33.0000i 1.10122i
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) − 55.0000i − 1.83130i
\(903\) 0 0
\(904\) 1.00000 0.0332595
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) 28.0000i 0.929213i
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) − 55.0000i − 1.82023i
\(914\) 25.0000 0.826927
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) − 8.00000i − 0.264183i
\(918\) 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
\(922\) 38.0000i 1.25146i
\(923\) − 60.0000i − 1.97492i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) − 6.00000i − 0.196960i
\(929\) 46.0000 1.50921 0.754606 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 9.00000i 0.293860i
\(939\) 0 0
\(940\) 0 0
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 40.0000 1.30051
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) 0 0
\(952\) − 1.00000i − 0.0324102i
\(953\) − 9.00000i − 0.291539i −0.989319 0.145769i \(-0.953434\pi\)
0.989319 0.145769i \(-0.0465657\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 0 0
\(958\) − 6.00000i − 0.193851i
\(959\) 3.00000 0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 48.0000i − 1.54758i
\(963\) 0 0
\(964\) 5.00000 0.161039
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) − 14.0000i − 0.449977i
\(969\) 0 0
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) 11.0000i 0.352644i
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 21.0000i 0.671850i 0.941889 + 0.335925i \(0.109049\pi\)
−0.941889 + 0.335925i \(0.890951\pi\)
\(978\) 0 0
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000i 0.382935i
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 18.0000i 0.572656i
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 0 0
\(994\) −10.0000 −0.317181
\(995\) 0 0
\(996\) 0 0
\(997\) 58.0000i 1.83688i 0.395562 + 0.918439i \(0.370550\pi\)
−0.395562 + 0.918439i \(0.629450\pi\)
\(998\) − 36.0000i − 1.13956i
\(999\) 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.v.2899.2 2
3.2 odd 2 350.2.c.a.99.1 2
5.2 odd 4 3150.2.a.j.1.1 1
5.3 odd 4 3150.2.a.bq.1.1 1
5.4 even 2 inner 3150.2.g.v.2899.1 2
12.11 even 2 2800.2.g.a.449.2 2
15.2 even 4 350.2.a.d.1.1 yes 1
15.8 even 4 350.2.a.c.1.1 1
15.14 odd 2 350.2.c.a.99.2 2
21.20 even 2 2450.2.c.r.99.1 2
60.23 odd 4 2800.2.a.b.1.1 1
60.47 odd 4 2800.2.a.bg.1.1 1
60.59 even 2 2800.2.g.a.449.1 2
105.62 odd 4 2450.2.a.bg.1.1 1
105.83 odd 4 2450.2.a.a.1.1 1
105.104 even 2 2450.2.c.r.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.c.1.1 1 15.8 even 4
350.2.a.d.1.1 yes 1 15.2 even 4
350.2.c.a.99.1 2 3.2 odd 2
350.2.c.a.99.2 2 15.14 odd 2
2450.2.a.a.1.1 1 105.83 odd 4
2450.2.a.bg.1.1 1 105.62 odd 4
2450.2.c.r.99.1 2 21.20 even 2
2450.2.c.r.99.2 2 105.104 even 2
2800.2.a.b.1.1 1 60.23 odd 4
2800.2.a.bg.1.1 1 60.47 odd 4
2800.2.g.a.449.1 2 60.59 even 2
2800.2.g.a.449.2 2 12.11 even 2
3150.2.a.j.1.1 1 5.2 odd 4
3150.2.a.bq.1.1 1 5.3 odd 4
3150.2.g.v.2899.1 2 5.4 even 2 inner
3150.2.g.v.2899.2 2 1.1 even 1 trivial