Properties

Label 3150.2.g.k.2899.1
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: yes
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.k.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +3.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} -3.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} -2.00000i q^{37} +2.00000i q^{38} -3.00000 q^{41} -7.00000i q^{43} +3.00000 q^{46} -6.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +9.00000i q^{53} +1.00000 q^{56} +3.00000i q^{58} -3.00000 q^{59} -1.00000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -8.00000i q^{67} +3.00000i q^{68} -4.00000i q^{73} -2.00000 q^{74} +2.00000 q^{76} -8.00000 q^{79} +3.00000i q^{82} +15.0000i q^{83} -7.00000 q^{86} -6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -6.00000 q^{94} -8.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{14} + 2q^{16} - 4q^{19} - 2q^{26} - 6q^{29} - 2q^{31} - 6q^{34} - 6q^{41} + 6q^{46} - 2q^{49} + 2q^{56} - 6q^{59} - 2q^{61} - 2q^{64} - 4q^{74} + 4q^{76} - 16q^{79} - 14q^{86} - 12q^{89} - 2q^{91} - 12q^{94} + O(q^{100}) \)

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
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) − 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
\(52\) 1.00000i 0.138675i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 3.00000i 0.393919i
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000i 0.331295i
\(83\) 15.0000i 1.64646i 0.567705 + 0.823232i \(0.307831\pi\)
−0.567705 + 0.823232i \(0.692169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) − 3.00000i − 0.312772i
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) − 13.0000i − 1.28093i −0.767988 0.640464i \(-0.778742\pi\)
0.767988 0.640464i \(-0.221258\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 3.00000i 0.276172i
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 1.00000i 0.0905357i
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 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
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 2.00000i 0.173422i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) 2.00000i 0.164399i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) − 2.00000i − 0.162221i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) − 1.00000i − 0.0783260i −0.999233 0.0391630i \(-0.987531\pi\)
0.999233 0.0391630i \(-0.0124692\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 15.0000 1.16423
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 7.00000i 0.533745i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.00000i 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 18.0000i 1.26648i
\(203\) 3.00000i 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) −13.0000 −0.905753
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) −7.00000 −0.481900 −0.240950 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) − 9.00000i − 0.618123i
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) 1.00000i 0.0678844i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) − 13.0000i − 0.870544i −0.900299 0.435272i \(-0.856652\pi\)
0.900299 0.435272i \(-0.143348\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) − 21.0000i − 1.39382i −0.717159 0.696909i \(-0.754558\pi\)
0.717159 0.696909i \(-0.245442\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 0 0
\(238\) 3.00000i 0.194461i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) − 1.00000i − 0.0635001i
\(249\) 0 0
\(250\) 0 0
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 3.00000i − 0.187135i −0.995613 0.0935674i \(-0.970173\pi\)
0.995613 0.0935674i \(-0.0298271\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 15.0000i 0.924940i 0.886635 + 0.462470i \(0.153037\pi\)
−0.886635 + 0.462470i \(0.846963\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000i 0.177084i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000i 0.234082i
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 15.0000i 0.868927i
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) 10.0000i 0.575435i
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) 0 0
\(307\) − 14.0000i − 0.799022i −0.916728 0.399511i \(-0.869180\pi\)
0.916728 0.399511i \(-0.130820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) − 28.0000i − 1.58265i −0.611393 0.791327i \(-0.709391\pi\)
0.611393 0.791327i \(-0.290609\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 21.0000i − 1.17948i −0.807594 0.589739i \(-0.799231\pi\)
0.807594 0.589739i \(-0.200769\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) − 3.00000i − 0.167183i
\(323\) 6.00000i 0.333849i
\(324\) 0 0
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 0 0
\(328\) − 3.00000i − 0.165647i
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) − 15.0000i − 0.823232i
\(333\) 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 \(0.884795\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 0 0
\(349\) 31.0000 1.65939 0.829696 0.558216i \(-0.188514\pi\)
0.829696 + 0.558216i \(0.188514\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 6.00000i 0.317110i
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) − 2.00000i − 0.105118i
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) − 17.0000i − 0.887393i −0.896177 0.443696i \(-0.853667\pi\)
0.896177 0.443696i \(-0.146333\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 3.00000i 0.154508i
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.00000i 0.460480i
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 8.00000i 0.406138i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) 3.00000 0.151138
\(395\) 0 0
\(396\) 0 0
\(397\) 31.0000i 1.55585i 0.628360 + 0.777923i \(0.283727\pi\)
−0.628360 + 0.777923i \(0.716273\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 1.00000i 0.0498135i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 13.0000i 0.640464i
\(413\) 3.00000i 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 7.00000i 0.340755i
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 1.00000i 0.0483934i
\(428\) 6.00000i 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) 39.0000 1.87856 0.939282 0.343146i \(-0.111493\pi\)
0.939282 + 0.343146i \(0.111493\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) − 6.00000i − 0.287019i
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.00000i 0.142695i
\(443\) − 18.0000i − 0.855206i −0.903967 0.427603i \(-0.859358\pi\)
0.903967 0.427603i \(-0.140642\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −13.0000 −0.615568
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 12.0000i − 0.564433i
\(453\) 0 0
\(454\) −21.0000 −0.985579
\(455\) 0 0
\(456\) 0 0
\(457\) 1.00000i 0.0467780i 0.999726 + 0.0233890i \(0.00744563\pi\)
−0.999726 + 0.0233890i \(0.992554\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) − 28.0000i − 1.30127i −0.759390 0.650635i \(-0.774503\pi\)
0.759390 0.650635i \(-0.225497\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) − 3.00000i − 0.138086i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 0 0
\(478\) 24.0000i 1.09773i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 22.0000i 1.00207i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 0 0
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) − 1.00000i − 0.0452679i
\(489\) 0 0
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 9.00000i 0.405340i
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) −23.0000 −1.02962 −0.514811 0.857304i \(-0.672138\pi\)
−0.514811 + 0.857304i \(0.672138\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.00000i − 0.133897i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000i 0.0887357i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 2.00000i 0.0878750i
\(519\) 0 0
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 0 0
\(523\) − 34.0000i − 1.48672i −0.668894 0.743358i \(-0.733232\pi\)
0.668894 0.743358i \(-0.266768\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 15.0000 0.654031
\(527\) 3.00000i 0.130682i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) − 2.00000i − 0.0867110i
\(533\) 3.00000i 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) − 18.0000i − 0.776035i
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0000i 0.555840i 0.960604 + 0.277920i \(0.0896450\pi\)
−0.960604 + 0.277920i \(0.910355\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −7.00000 −0.296068
\(560\) 0 0
\(561\) 0 0
\(562\) − 24.0000i − 1.01238i
\(563\) − 45.0000i − 1.89652i −0.317489 0.948262i \(-0.602840\pi\)
0.317489 0.948262i \(-0.397160\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 36.0000 1.50920 0.754599 0.656186i \(-0.227831\pi\)
0.754599 + 0.656186i \(0.227831\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) 15.0000 0.622305
\(582\) 0 0
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 9.00000i 0.371470i 0.982600 + 0.185735i \(0.0594666\pi\)
−0.982600 + 0.185735i \(0.940533\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.00000i − 0.0821995i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 0 0
\(598\) − 3.00000i − 0.122679i
\(599\) −15.0000 −0.612883 −0.306442 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 7.00000i 0.285299i
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) − 8.00000i − 0.324710i −0.986732 0.162355i \(-0.948091\pi\)
0.986732 0.162355i \(-0.0519090\pi\)
\(608\) 2.00000i 0.0811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) −14.0000 −0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.0000i 1.20289i
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) 2.00000i 0.0798087i
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 0 0
\(634\) −21.0000 −0.834017
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000i 0.0391630i
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) 6.00000i 0.233904i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 25.0000i 0.971653i
\(663\) 0 0
\(664\) −15.0000 −0.582113
\(665\) 0 0
\(666\) 0 0
\(667\) − 9.00000i − 0.348481i
\(668\) − 12.0000i − 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 25.0000i − 0.963679i −0.876259 0.481840i \(-0.839969\pi\)
0.876259 0.481840i \(-0.160031\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 6.00000i − 0.229584i −0.993390 0.114792i \(-0.963380\pi\)
0.993390 0.114792i \(-0.0366201\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) − 7.00000i − 0.266872i
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 9.00000i 0.340899i
\(698\) − 31.0000i − 1.17337i
\(699\) 0 0
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 0 0
\(703\) 4.00000i 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 18.0000i 0.676960i
\(708\) 0 0
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 6.00000i − 0.224860i
\(713\) − 3.00000i − 0.112351i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 0 0
\(718\) − 9.00000i − 0.335877i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 15.0000i 0.558242i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 35.0000i − 1.29808i −0.760755 0.649039i \(-0.775171\pi\)
0.760755 0.649039i \(-0.224829\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −21.0000 −0.776713
\(732\) 0 0
\(733\) 5.00000i 0.184679i 0.995728 + 0.0923396i \(0.0294345\pi\)
−0.995728 + 0.0923396i \(0.970565\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) 0 0
\(738\) 0 0
\(739\) −29.0000 −1.06678 −0.533391 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 9.00000i − 0.330400i
\(743\) − 15.0000i − 0.550297i −0.961402 0.275148i \(-0.911273\pi\)
0.961402 0.275148i \(-0.0887270\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) 16.0000i 0.581530i 0.956795 + 0.290765i \(0.0939098\pi\)
−0.956795 + 0.290765i \(0.906090\pi\)
\(758\) 5.00000i 0.181608i
\(759\) 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
\(763\) 2.00000i 0.0724049i
\(764\) 9.00000 0.325609
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 3.00000i 0.108324i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 14.0000i − 0.503871i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) − 9.00000i − 0.321839i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 50.0000i − 1.78231i −0.453701 0.891154i \(-0.649897\pi\)
0.453701 0.891154i \(-0.350103\pi\)
\(788\) − 3.00000i − 0.106871i
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 1.00000i 0.0355110i
\(794\) 31.0000 1.10015
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 36.0000i 1.27519i 0.770374 + 0.637593i \(0.220070\pi\)
−0.770374 + 0.637593i \(0.779930\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 24.0000i 0.847469i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 1.00000 0.0352235
\(807\) 0 0
\(808\) − 18.0000i − 0.633238i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) − 3.00000i − 0.105279i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 14.0000i 0.489798i
\(818\) − 28.0000i − 0.978997i
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) 30.0000i 1.04320i 0.853189 + 0.521601i \(0.174665\pi\)
−0.853189 + 0.521601i \(0.825335\pi\)
\(828\) 0 0
\(829\) −5.00000 −0.173657 −0.0868286 0.996223i \(-0.527673\pi\)
−0.0868286 + 0.996223i \(0.527673\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) 3.00000i 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 15.0000i − 0.518166i
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 16.0000i 0.551396i
\(843\) 0 0
\(844\) 7.00000 0.240950
\(845\) 0 0
\(846\) 0 0
\(847\) 11.0000i 0.377964i
\(848\) 9.00000i 0.309061i
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) 17.0000i 0.582069i 0.956713 + 0.291034i \(0.0939994\pi\)
−0.956713 + 0.291034i \(0.906001\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 39.0000i − 1.32835i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) − 1.00000i − 0.0339422i
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 2.00000i − 0.0677285i
\(873\) 0 0
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) 0 0
\(877\) 40.0000i 1.35070i 0.737496 + 0.675352i \(0.236008\pi\)
−0.737496 + 0.675352i \(0.763992\pi\)
\(878\) − 13.0000i − 0.438729i
\(879\) 0 0
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) − 25.0000i − 0.841317i −0.907219 0.420658i \(-0.861799\pi\)
0.907219 0.420658i \(-0.138201\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −18.0000 −0.604722
\(887\) 18.0000i 0.604381i 0.953248 + 0.302190i \(0.0977178\pi\)
−0.953248 + 0.302190i \(0.902282\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0000i 0.435272i
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) − 6.00000i − 0.200223i
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) 27.0000 0.899500
\(902\) 0 0
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) − 35.0000i − 1.16216i −0.813848 0.581078i \(-0.802631\pi\)
0.813848 0.581078i \(-0.197369\pi\)
\(908\) 21.0000i 0.696909i
\(909\) 0 0
\(910\) 0 0
\(911\) 57.0000 1.88849 0.944247 0.329238i \(-0.106792\pi\)
0.944247 + 0.329238i \(0.106792\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.00000 0.0330771
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 30.0000i − 0.987997i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −28.0000 −0.920137
\(927\) 0 0
\(928\) 3.00000i 0.0984798i
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 0 0
\(943\) − 9.00000i − 0.293080i
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) 0 0
\(947\) − 30.0000i − 0.974869i −0.873160 0.487435i \(-0.837933\pi\)
0.873160 0.487435i \(-0.162067\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) − 3.00000i − 0.0972306i
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) − 30.0000i − 0.969256i
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 2.00000i 0.0644826i
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 22.0000i 0.707472i 0.935345 + 0.353736i \(0.115089\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) − 4.00000i − 0.128234i
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) − 30.0000i − 0.957338i
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 0 0
\(988\) − 2.00000i − 0.0636285i
\(989\) 21.0000 0.667761
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) 23.0000i 0.728052i
\(999\) 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.k.2899.1 2
3.2 odd 2 3150.2.g.n.2899.2 2
5.2 odd 4 3150.2.a.bm.1.1 yes 1
5.3 odd 4 3150.2.a.h.1.1 1
5.4 even 2 inner 3150.2.g.k.2899.2 2
15.2 even 4 3150.2.a.o.1.1 yes 1
15.8 even 4 3150.2.a.bc.1.1 yes 1
15.14 odd 2 3150.2.g.n.2899.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.a.h.1.1 1 5.3 odd 4
3150.2.a.o.1.1 yes 1 15.2 even 4
3150.2.a.bc.1.1 yes 1 15.8 even 4
3150.2.a.bm.1.1 yes 1 5.2 odd 4
3150.2.g.k.2899.1 2 1.1 even 1 trivial
3150.2.g.k.2899.2 2 5.4 even 2 inner
3150.2.g.n.2899.1 2 15.14 odd 2
3150.2.g.n.2899.2 2 3.2 odd 2