Properties

Label 4400.2.b.i.4049.1
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.i.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -3.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} -7.00000i q^{17} +5.00000 q^{19} -3.00000 q^{21} -6.00000i q^{23} -5.00000i q^{27} -5.00000 q^{29} +3.00000 q^{31} +1.00000i q^{33} +3.00000i q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} +2.00000i q^{47} -2.00000 q^{49} -7.00000 q^{51} +1.00000i q^{53} -5.00000i q^{57} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{63} -8.00000i q^{67} -6.00000 q^{69} -7.00000 q^{71} -14.0000i q^{73} +3.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +5.00000i q^{87} +15.0000 q^{89} +18.0000 q^{91} -3.00000i q^{93} -12.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 2 q^{11} + 10 q^{19} - 6 q^{21} - 10 q^{29} + 6 q^{31} + 12 q^{39} + 4 q^{41} - 4 q^{49} - 14 q^{51} - 20 q^{59} + 14 q^{61} - 12 q^{69} - 14 q^{71} + 20 q^{79} + 2 q^{81} + 30 q^{89} + 36 q^{91} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 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
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 5.00000i − 0.662266i
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) − 6.00000i − 0.755929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −7.00000 −0.830747 −0.415374 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.00000i 0.536056i
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 0 0
\(93\) − 3.00000i − 0.311086i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.0000i − 1.21842i −0.793011 0.609208i \(-0.791488\pi\)
0.793011 0.609208i \(-0.208512\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) 16.0000i 1.50515i 0.658505 + 0.752577i \(0.271189\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000i 1.10940i
\(118\) 0 0
\(119\) −21.0000 −1.92507
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −17.0000 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) 0 0
\(133\) − 15.0000i − 1.30066i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) − 14.0000i − 1.13183i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.00000i 0.239426i 0.992809 + 0.119713i \(0.0381975\pi\)
−0.992809 + 0.119713i \(0.961803\pi\)
\(158\) 0 0
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) −18.0000 −1.41860
\(162\) 0 0
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 3.00000i − 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 10.0000 0.764719
\(172\) 0 0
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) − 7.00000i − 0.517455i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7.00000i 0.511891i
\(188\) 0 0
\(189\) −15.0000 −1.09109
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) −25.0000 −1.77220 −0.886102 0.463491i \(-0.846597\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 15.0000i 1.05279i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.0000i − 0.834058i
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 0 0
\(213\) 7.00000i 0.479632i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.00000i − 0.610960i
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 42.0000 2.82523
\(222\) 0 0
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.00000i 0.132745i 0.997795 + 0.0663723i \(0.0211425\pi\)
−0.997795 + 0.0663723i \(0.978857\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) − 9.00000i − 0.589610i −0.955557 0.294805i \(-0.904745\pi\)
0.955557 0.294805i \(-0.0952546\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 10.0000i − 0.649570i
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) − 16.0000i − 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.0000i 1.90885i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) 9.00000 0.559233
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 9.00000i 0.554964i 0.960731 + 0.277482i \(0.0894999\pi\)
−0.960731 + 0.277482i \(0.910500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 15.0000i − 0.917985i
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) − 18.0000i − 1.08941i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.0000i − 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.00000i − 0.354169i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) − 2.00000i − 0.114897i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 7.00000i − 0.393159i −0.980488 0.196580i \(-0.937017\pi\)
0.980488 0.196580i \(-0.0629834\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) − 35.0000i − 1.94745i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10.0000i − 0.553001i
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 17.0000i − 0.926049i −0.886345 0.463025i \(-0.846764\pi\)
0.886345 0.463025i \(-0.153236\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) − 15.0000i − 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.0000i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 30.0000 1.60128
\(352\) 0 0
\(353\) − 34.0000i − 1.80964i −0.425797 0.904819i \(-0.640006\pi\)
0.425797 0.904819i \(-0.359994\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 21.0000i 1.11144i
\(358\) 0 0
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) − 1.00000i − 0.0524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 30.0000i − 1.54508i
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 34.0000i 1.73732i 0.495410 + 0.868659i \(0.335018\pi\)
−0.495410 + 0.868659i \(0.664982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −42.0000 −2.12403
\(392\) 0 0
\(393\) 17.0000i 0.857537i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) −15.0000 −0.750939
\(400\) 0 0
\(401\) −13.0000 −0.649189 −0.324595 0.945853i \(-0.605228\pi\)
−0.324595 + 0.945853i \(0.605228\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 30.0000i 1.47620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 21.0000i − 1.01626i
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 30.0000i − 1.43509i
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 15.0000i 0.709476i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) 2.00000i 0.0939682i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000i 0.140334i 0.997535 + 0.0701670i \(0.0223532\pi\)
−0.997535 + 0.0701670i \(0.977647\pi\)
\(458\) 0 0
\(459\) −35.0000 −1.63366
\(460\) 0 0
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) 0 0
\(463\) 34.0000i 1.58011i 0.613033 + 0.790057i \(0.289949\pi\)
−0.613033 + 0.790057i \(0.710051\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 23.0000i − 1.06431i −0.846646 0.532157i \(-0.821382\pi\)
0.846646 0.532157i \(-0.178618\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −18.0000 −0.820729
\(482\) 0 0
\(483\) 18.0000i 0.819028i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 0 0
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) 3.00000 0.135388 0.0676941 0.997706i \(-0.478436\pi\)
0.0676941 + 0.997706i \(0.478436\pi\)
\(492\) 0 0
\(493\) 35.0000i 1.57632i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.0000i 0.941979i
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −3.00000 −0.134030
\(502\) 0 0
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −42.0000 −1.85797
\(512\) 0 0
\(513\) − 25.0000i − 1.10378i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.00000i − 0.0879599i
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 21.0000i − 0.914774i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) −20.0000 −0.867926
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −23.0000 −0.988847 −0.494424 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) 0 0
\(543\) − 2.00000i − 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) −25.0000 −1.06504
\(552\) 0 0
\(553\) − 30.0000i − 1.27573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 7.00000 0.295540
\(562\) 0 0
\(563\) − 6.00000i − 0.252870i −0.991975 0.126435i \(-0.959647\pi\)
0.991975 0.126435i \(-0.0403535\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 3.00000i − 0.125988i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −27.0000 −1.12991 −0.564957 0.825120i \(-0.691107\pi\)
−0.564957 + 0.825120i \(0.691107\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 38.0000i 1.58196i 0.611842 + 0.790980i \(0.290429\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −18.0000 −0.746766
\(582\) 0 0
\(583\) − 1.00000i − 0.0414158i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.0000i 1.11441i 0.830375 + 0.557205i \(0.188126\pi\)
−0.830375 + 0.557205i \(0.811874\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.0000i 1.02318i
\(598\) 0 0
\(599\) 45.0000 1.83865 0.919325 0.393499i \(-0.128735\pi\)
0.919325 + 0.393499i \(0.128735\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) − 16.0000i − 0.651570i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 47.0000i 1.90767i 0.300329 + 0.953836i \(0.402903\pi\)
−0.300329 + 0.953836i \(0.597097\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000i 0.322068i 0.986949 + 0.161034i \(0.0514829\pi\)
−0.986949 + 0.161034i \(0.948517\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −30.0000 −1.20386
\(622\) 0 0
\(623\) − 45.0000i − 1.80289i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 5.00000i 0.199681i
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 33.0000 1.31371 0.656855 0.754017i \(-0.271887\pi\)
0.656855 + 0.754017i \(0.271887\pi\)
\(632\) 0 0
\(633\) − 23.0000i − 0.914168i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 12.0000i − 0.475457i
\(638\) 0 0
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 19.0000i 0.749287i 0.927169 + 0.374643i \(0.122235\pi\)
−0.927169 + 0.374643i \(0.877765\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000i 1.65119i 0.564263 + 0.825595i \(0.309160\pi\)
−0.564263 + 0.825595i \(0.690840\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 0 0
\(653\) 31.0000i 1.21312i 0.795036 + 0.606562i \(0.207452\pi\)
−0.795036 + 0.606562i \(0.792548\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 28.0000i − 1.09238i
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) − 42.0000i − 1.63114i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0000i 1.16160i
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −7.00000 −0.270232
\(672\) 0 0
\(673\) − 29.0000i − 1.11787i −0.829212 0.558934i \(-0.811211\pi\)
0.829212 0.558934i \(-0.188789\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.0000i 1.07613i 0.842904 + 0.538064i \(0.180844\pi\)
−0.842904 + 0.538064i \(0.819156\pi\)
\(678\) 0 0
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) − 31.0000i − 1.18618i −0.805135 0.593091i \(-0.797907\pi\)
0.805135 0.593091i \(-0.202093\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 38.0000 1.44559 0.722794 0.691063i \(-0.242858\pi\)
0.722794 + 0.691063i \(0.242858\pi\)
\(692\) 0 0
\(693\) 6.00000i 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 14.0000i − 0.530288i
\(698\) 0 0
\(699\) −9.00000 −0.340411
\(700\) 0 0
\(701\) 7.00000 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(702\) 0 0
\(703\) 15.0000i 0.565736i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) − 18.0000i − 0.674105i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 10.0000i − 0.373457i
\(718\) 0 0
\(719\) 25.0000 0.932343 0.466171 0.884694i \(-0.345633\pi\)
0.466171 + 0.884694i \(0.345633\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 18.0000i 0.669427i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 22.0000i 0.815935i 0.912996 + 0.407967i \(0.133762\pi\)
−0.912996 + 0.407967i \(0.866238\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) 0 0
\(733\) − 24.0000i − 0.886460i −0.896408 0.443230i \(-0.853832\pi\)
0.896408 0.443230i \(-0.146168\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 30.0000 1.10208
\(742\) 0 0
\(743\) − 21.0000i − 0.770415i −0.922830 0.385208i \(-0.874130\pi\)
0.922830 0.385208i \(-0.125870\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12.0000i − 0.439057i
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −17.0000 −0.620339 −0.310169 0.950681i \(-0.600386\pi\)
−0.310169 + 0.950681i \(0.600386\pi\)
\(752\) 0 0
\(753\) 2.00000i 0.0728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) − 30.0000i − 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 60.0000i − 2.16647i
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) − 19.0000i − 0.683383i −0.939812 0.341691i \(-0.889000\pi\)
0.939812 0.341691i \(-0.111000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.00000i − 0.322873i
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 7.00000 0.250480
\(782\) 0 0
\(783\) 25.0000i 0.893427i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 48.0000 1.70668
\(792\) 0 0
\(793\) 42.0000i 1.49146i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 14.0000i 0.494049i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 20.0000i − 0.704033i
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 20.0000i 0.699711i
\(818\) 0 0
\(819\) 36.0000 1.25794
\(820\) 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 \(-0.0899574\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 0 0
\(833\) 14.0000i 0.485071i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 15.0000i − 0.518476i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 3.00000i − 0.103081i
\(848\) 0 0
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) 18.0000 0.617032
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 7.00000i − 0.239115i −0.992827 0.119558i \(-0.961852\pi\)
0.992827 0.119558i \(-0.0381477\pi\)
\(858\) 0 0
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 32.0000i 1.08678i
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) − 24.0000i − 0.812277i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 9.00000i 0.302874i 0.988467 + 0.151437i \(0.0483901\pi\)
−0.988467 + 0.151437i \(0.951610\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 36.0000i − 1.20201i
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 7.00000 0.233204
\(902\) 0 0
\(903\) − 12.0000i − 0.399335i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 57.0000i 1.89265i 0.323211 + 0.946327i \(0.395238\pi\)
−0.323211 + 0.946327i \(0.604762\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) −27.0000 −0.894550 −0.447275 0.894397i \(-0.647605\pi\)
−0.447275 + 0.894397i \(0.647605\pi\)
\(912\) 0 0
\(913\) 6.00000i 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 51.0000i 1.68417i
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 0 0
\(923\) − 42.0000i − 1.38245i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −35.0000 −1.14831 −0.574156 0.818746i \(-0.694670\pi\)
−0.574156 + 0.818746i \(0.694670\pi\)
\(930\) 0 0
\(931\) −10.0000 −0.327737
\(932\) 0 0
\(933\) − 3.00000i − 0.0982156i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 17.0000 0.554184 0.277092 0.960843i \(-0.410629\pi\)
0.277092 + 0.960843i \(0.410629\pi\)
\(942\) 0 0
\(943\) − 12.0000i − 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0000i 0.877382i 0.898638 + 0.438691i \(0.144558\pi\)
−0.898638 + 0.438691i \(0.855442\pi\)
\(948\) 0 0
\(949\) 84.0000 2.72676
\(950\) 0 0
\(951\) −7.00000 −0.226991
\(952\) 0 0
\(953\) − 39.0000i − 1.26333i −0.775240 0.631667i \(-0.782371\pi\)
0.775240 0.631667i \(-0.217629\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5.00000i − 0.161627i
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) − 16.0000i − 0.515593i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 27.0000i 0.868261i 0.900850 + 0.434131i \(0.142944\pi\)
−0.900850 + 0.434131i \(0.857056\pi\)
\(968\) 0 0
\(969\) −35.0000 −1.12436
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) 60.0000i 1.92351i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) 54.0000i 1.72233i 0.508323 + 0.861166i \(0.330265\pi\)
−0.508323 + 0.861166i \(0.669735\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.00000i − 0.190982i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) − 28.0000i − 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 32.0000i − 1.01345i −0.862108 0.506725i \(-0.830856\pi\)
0.862108 0.506725i \(-0.169144\pi\)
\(998\) 0 0
\(999\) 15.0000 0.474579
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 4400.2.b.i.4049.1 2
4.3 odd 2 550.2.b.a.199.2 2
5.2 odd 4 4400.2.a.l.1.1 1
5.3 odd 4 880.2.a.i.1.1 1
5.4 even 2 inner 4400.2.b.i.4049.2 2
12.11 even 2 4950.2.c.m.199.1 2
15.8 even 4 7920.2.a.d.1.1 1
20.3 even 4 110.2.a.b.1.1 1
20.7 even 4 550.2.a.f.1.1 1
20.19 odd 2 550.2.b.a.199.1 2
40.3 even 4 3520.2.a.y.1.1 1
40.13 odd 4 3520.2.a.h.1.1 1
55.43 even 4 9680.2.a.x.1.1 1
60.23 odd 4 990.2.a.d.1.1 1
60.47 odd 4 4950.2.a.bc.1.1 1
60.59 even 2 4950.2.c.m.199.2 2
140.83 odd 4 5390.2.a.bf.1.1 1
220.43 odd 4 1210.2.a.b.1.1 1
220.87 odd 4 6050.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.b.1.1 1 20.3 even 4
550.2.a.f.1.1 1 20.7 even 4
550.2.b.a.199.1 2 20.19 odd 2
550.2.b.a.199.2 2 4.3 odd 2
880.2.a.i.1.1 1 5.3 odd 4
990.2.a.d.1.1 1 60.23 odd 4
1210.2.a.b.1.1 1 220.43 odd 4
3520.2.a.h.1.1 1 40.13 odd 4
3520.2.a.y.1.1 1 40.3 even 4
4400.2.a.l.1.1 1 5.2 odd 4
4400.2.b.i.4049.1 2 1.1 even 1 trivial
4400.2.b.i.4049.2 2 5.4 even 2 inner
4950.2.a.bc.1.1 1 60.47 odd 4
4950.2.c.m.199.1 2 12.11 even 2
4950.2.c.m.199.2 2 60.59 even 2
5390.2.a.bf.1.1 1 140.83 odd 4
6050.2.a.bj.1.1 1 220.87 odd 4
7920.2.a.d.1.1 1 15.8 even 4
9680.2.a.x.1.1 1 55.43 even 4