Properties

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

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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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.a.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{13} -3.00000i q^{17} -5.00000 q^{19} +3.00000 q^{21} +2.00000i q^{23} +9.00000i q^{27} +5.00000 q^{29} -5.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -18.0000 q^{39} -2.00000 q^{41} -12.0000i q^{43} -2.00000i q^{47} +6.00000 q^{49} -9.00000 q^{51} -13.0000i q^{53} +15.0000i q^{57} +2.00000 q^{59} +1.00000 q^{61} -6.00000i q^{63} +16.0000i q^{67} +6.00000 q^{69} -15.0000 q^{71} +10.0000i q^{73} +1.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} -15.0000i q^{87} -9.00000 q^{89} +6.00000 q^{91} +15.0000i q^{93} +16.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 2 q^{11} - 10 q^{19} + 6 q^{21} + 10 q^{29} - 10 q^{31} - 36 q^{39} - 4 q^{41} + 12 q^{49} - 18 q^{51} + 4 q^{59} + 2 q^{61} + 12 q^{69} - 30 q^{71} + 4 q^{79} + 18 q^{81} - 18 q^{89} + 12 q^{91} - 12 q^{99}+O(q^{100}) \) 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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000i 0.164399i 0.996616 + 0.0821995i \(0.0261945\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 0 0
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) 0 0
\(53\) − 13.0000i − 1.78569i −0.450367 0.892844i \(-0.648707\pi\)
0.450367 0.892844i \(-0.351293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15.0000i 1.98680i
\(58\) 0 0
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) − 6.00000i − 0.755929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 16.0000i 1.95471i 0.211604 + 0.977356i \(0.432131\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 15.0000i − 1.60817i
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 15.0000i 1.55543i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) − 16.0000i − 1.50515i −0.658505 0.752577i \(-0.728811\pi\)
0.658505 0.752577i \(-0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 36.0000i 3.32820i
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 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
\(129\) −36.0000 −3.16962
\(130\) 0 0
\(131\) −7.00000 −0.611593 −0.305796 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) − 5.00000i − 0.433555i
\(134\) 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
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 18.0000i − 1.48461i
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −18.0000 −1.46482 −0.732410 0.680864i \(-0.761604\pi\)
−0.732410 + 0.680864i \(0.761604\pi\)
\(152\) 0 0
\(153\) 18.0000i 1.45521i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 7.00000i − 0.558661i −0.960195 0.279330i \(-0.909888\pi\)
0.960195 0.279330i \(-0.0901125\pi\)
\(158\) 0 0
\(159\) −39.0000 −3.09290
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 1.00000i 0.0783260i 0.999233 + 0.0391630i \(0.0124692\pi\)
−0.999233 + 0.0391630i \(0.987531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000i 0.0773823i 0.999251 + 0.0386912i \(0.0123189\pi\)
−0.999251 + 0.0386912i \(0.987681\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 30.0000 2.29416
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 6.00000i − 0.450988i
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) − 3.00000i − 0.221766i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 3.00000i − 0.219382i
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 7.00000i 0.503871i 0.967744 + 0.251936i \(0.0810671\pi\)
−0.967744 + 0.251936i \(0.918933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.0000i 1.70993i 0.518686 + 0.854965i \(0.326421\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 48.0000 3.38566
\(202\) 0 0
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.0000i − 0.834058i
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) 0 0
\(213\) 45.0000i 3.08335i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 5.00000i − 0.339422i
\(218\) 0 0
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) − 26.0000i − 1.74109i −0.492090 0.870544i \(-0.663767\pi\)
0.492090 0.870544i \(-0.336233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 11.0000i 0.720634i 0.932830 + 0.360317i \(0.117331\pi\)
−0.932830 + 0.360317i \(0.882669\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 6.00000i − 0.389742i
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 30.0000i 1.90885i
\(248\) 0 0
\(249\) 42.0000 2.66164
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −30.0000 −1.85695
\(262\) 0 0
\(263\) 13.0000i 0.801614i 0.916162 + 0.400807i \(0.131270\pi\)
−0.916162 + 0.400807i \(0.868730\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 27.0000i 1.65237i
\(268\) 0 0
\(269\) −20.0000 −1.21942 −0.609711 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) − 18.0000i − 1.08941i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 16.0000i − 0.961347i −0.876900 0.480673i \(-0.840392\pi\)
0.876900 0.480673i \(-0.159608\pi\)
\(278\) 0 0
\(279\) 30.0000 1.79605
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 10.0000i − 0.594438i −0.954809 0.297219i \(-0.903941\pi\)
0.954809 0.297219i \(-0.0960592\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.00000i − 0.118056i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 48.0000 2.81381
\(292\) 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
\(297\) 9.00000i 0.522233i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 30.0000i 1.72345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.0000i − 1.25561i −0.778372 0.627803i \(-0.783954\pi\)
0.778372 0.627803i \(-0.216046\pi\)
\(308\) 0 0
\(309\) 48.0000 2.73062
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) − 30.0000i − 1.69570i −0.530236 0.847850i \(-0.677897\pi\)
0.530236 0.847850i \(-0.322103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000i 0.168497i 0.996445 + 0.0842484i \(0.0268489\pi\)
−0.996445 + 0.0842484i \(0.973151\pi\)
\(318\) 0 0
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 15.0000i 0.834622i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 30.0000i 1.65900i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 29.0000i − 1.57973i −0.613280 0.789865i \(-0.710150\pi\)
0.613280 0.789865i \(-0.289850\pi\)
\(338\) 0 0
\(339\) −48.0000 −2.60700
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 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
\(356\) 0 0
\(357\) − 9.00000i − 0.476331i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) − 3.00000i − 0.157459i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 12.0000i − 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 13.0000 0.674926
\(372\) 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
\(376\) 0 0
\(377\) − 30.0000i − 1.54508i
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) 14.0000i 0.715367i 0.933843 + 0.357683i \(0.116433\pi\)
−0.933843 + 0.357683i \(0.883567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 72.0000i 3.65997i
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 21.0000i 1.05931i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 0 0
\(399\) −15.0000 −0.750939
\(400\) 0 0
\(401\) −21.0000 −1.04869 −0.524345 0.851506i \(-0.675690\pi\)
−0.524345 + 0.851506i \(0.675690\pi\)
\(402\) 0 0
\(403\) 30.0000i 1.49441i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000i 0.0495682i
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 0 0
\(413\) 2.00000i 0.0984136i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 0 0
\(423\) 12.0000i 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.00000i 0.0483934i
\(428\) 0 0
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 10.0000i − 0.478365i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 45.0000i − 2.12843i
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 0 0
\(453\) 54.0000i 2.53714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 33.0000i − 1.54367i −0.635820 0.771837i \(-0.719338\pi\)
0.635820 0.771837i \(-0.280662\pi\)
\(458\) 0 0
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 5.00000 0.232873 0.116437 0.993198i \(-0.462853\pi\)
0.116437 + 0.993198i \(0.462853\pi\)
\(462\) 0 0
\(463\) − 6.00000i − 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −21.0000 −0.967629
\(472\) 0 0
\(473\) − 12.0000i − 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 78.0000i 3.57137i
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 12.0000i − 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 3.00000 0.135665
\(490\) 0 0
\(491\) 13.0000 0.586682 0.293341 0.956008i \(-0.405233\pi\)
0.293341 + 0.956008i \(0.405233\pi\)
\(492\) 0 0
\(493\) − 15.0000i − 0.675566i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 15.0000i − 0.672842i
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 69.0000i 3.06440i
\(508\) 0 0
\(509\) −8.00000 −0.354594 −0.177297 0.984157i \(-0.556735\pi\)
−0.177297 + 0.984157i \(0.556735\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) − 45.0000i − 1.98680i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.00000i − 0.0879599i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.0000i 0.653410i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 12.0000i − 0.517838i
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 15.0000 0.644900 0.322450 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) 0 0
\(543\) − 66.0000i − 2.83233i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −25.0000 −1.06504
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) −72.0000 −3.04528
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) 36.0000i 1.50392i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.0000i 0.582828i 0.956597 + 0.291414i \(0.0941257\pi\)
−0.956597 + 0.291414i \(0.905874\pi\)
\(578\) 0 0
\(579\) 21.0000 0.872730
\(580\) 0 0
\(581\) −14.0000 −0.580818
\(582\) 0 0
\(583\) − 13.0000i − 0.538405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 39.0000i − 1.60970i −0.593477 0.804851i \(-0.702245\pi\)
0.593477 0.804851i \(-0.297755\pi\)
\(588\) 0 0
\(589\) 25.0000 1.03011
\(590\) 0 0
\(591\) 72.0000 2.96168
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3.00000i 0.122782i
\(598\) 0 0
\(599\) −27.0000 −1.10319 −0.551595 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) − 96.0000i − 3.90942i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 21.0000i − 0.852364i −0.904638 0.426182i \(-0.859858\pi\)
0.904638 0.426182i \(-0.140142\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12.0000i − 0.483102i −0.970388 0.241551i \(-0.922344\pi\)
0.970388 0.241551i \(-0.0776561\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) − 9.00000i − 0.360577i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 15.0000i 0.599042i
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) − 3.00000i − 0.119239i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 36.0000i − 1.42637i
\(638\) 0 0
\(639\) 90.0000 3.56034
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) − 7.00000i − 0.276053i −0.990429 0.138027i \(-0.955924\pi\)
0.990429 0.138027i \(-0.0440759\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 6.00000i − 0.235884i −0.993020 0.117942i \(-0.962370\pi\)
0.993020 0.117942i \(-0.0376297\pi\)
\(648\) 0 0
\(649\) 2.00000 0.0785069
\(650\) 0 0
\(651\) −15.0000 −0.587896
\(652\) 0 0
\(653\) − 19.0000i − 0.743527i −0.928327 0.371764i \(-0.878753\pi\)
0.928327 0.371764i \(-0.121247\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 60.0000i − 2.34082i
\(658\) 0 0
\(659\) 49.0000 1.90877 0.954384 0.298580i \(-0.0965131\pi\)
0.954384 + 0.298580i \(0.0965131\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 54.0000i 2.09719i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.0000i 0.387202i
\(668\) 0 0
\(669\) −78.0000 −3.01565
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) − 1.00000i − 0.0385472i −0.999814 0.0192736i \(-0.993865\pi\)
0.999814 0.0192736i \(-0.00613535\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) 0 0
\(683\) 19.0000i 0.727015i 0.931591 + 0.363507i \(0.118421\pi\)
−0.931591 + 0.363507i \(0.881579\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 18.0000i 0.686743i
\(688\) 0 0
\(689\) −78.0000 −2.97156
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.00000i 0.227266i
\(698\) 0 0
\(699\) 33.0000 1.24817
\(700\) 0 0
\(701\) −31.0000 −1.17085 −0.585427 0.810725i \(-0.699073\pi\)
−0.585427 + 0.810725i \(0.699073\pi\)
\(702\) 0 0
\(703\) − 5.00000i − 0.188579i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 10.0000i − 0.376089i
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) − 10.0000i − 0.374503i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000i 0.224074i
\(718\) 0 0
\(719\) 1.00000 0.0372937 0.0186469 0.999826i \(-0.494064\pi\)
0.0186469 + 0.999826i \(0.494064\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) − 54.0000i − 2.00828i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 26.0000i − 0.964287i −0.876092 0.482143i \(-0.839858\pi\)
0.876092 0.482143i \(-0.160142\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) − 32.0000i − 1.18195i −0.806691 0.590973i \(-0.798744\pi\)
0.806691 0.590973i \(-0.201256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 0 0
\(741\) 90.0000 3.30623
\(742\) 0 0
\(743\) − 33.0000i − 1.21065i −0.795977 0.605326i \(-0.793043\pi\)
0.795977 0.605326i \(-0.206957\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 84.0000i − 3.07340i
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 7.00000 0.255434 0.127717 0.991811i \(-0.459235\pi\)
0.127717 + 0.991811i \(0.459235\pi\)
\(752\) 0 0
\(753\) − 30.0000i − 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) − 10.0000i − 0.362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 12.0000i − 0.433295i
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) 66.0000 2.37693
\(772\) 0 0
\(773\) − 9.00000i − 0.323708i −0.986815 0.161854i \(-0.948253\pi\)
0.986815 0.161854i \(-0.0517473\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.00000i 0.107624i
\(778\) 0 0
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) −15.0000 −0.536742
\(782\) 0 0
\(783\) 45.0000i 1.60817i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 48.0000i − 1.71102i −0.517790 0.855508i \(-0.673245\pi\)
0.517790 0.855508i \(-0.326755\pi\)
\(788\) 0 0
\(789\) 39.0000 1.38844
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) − 6.00000i − 0.213066i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34.0000i − 1.20434i −0.798367 0.602171i \(-0.794303\pi\)
0.798367 0.602171i \(-0.205697\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 54.0000 1.90800
\(802\) 0 0
\(803\) 10.0000i 0.352892i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 60.0000i 2.11210i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −1.00000 −0.0351147 −0.0175574 0.999846i \(-0.505589\pi\)
−0.0175574 + 0.999846i \(0.505589\pi\)
\(812\) 0 0
\(813\) 72.0000i 2.52515i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 60.0000i 2.09913i
\(818\) 0 0
\(819\) −36.0000 −1.25794
\(820\) 0 0
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) − 44.0000i − 1.53374i −0.641800 0.766872i \(-0.721812\pi\)
0.641800 0.766872i \(-0.278188\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 52.0000i − 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) −48.0000 −1.66510
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 45.0000i − 1.55543i
\(838\) 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
\(843\) − 54.0000i − 1.85986i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 22.0000i 0.753266i 0.926363 + 0.376633i \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0000i 0.717346i 0.933463 + 0.358673i \(0.116771\pi\)
−0.933463 + 0.358673i \(0.883229\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 46.0000i 1.56586i 0.622111 + 0.782929i \(0.286275\pi\)
−0.622111 + 0.782929i \(0.713725\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 24.0000i − 0.815083i
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 96.0000 3.25284
\(872\) 0 0
\(873\) − 96.0000i − 3.24911i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) 0 0
\(883\) − 5.00000i − 0.168263i −0.996455 0.0841317i \(-0.973188\pi\)
0.996455 0.0841317i \(-0.0268116\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 40.0000i − 1.34307i −0.740973 0.671534i \(-0.765636\pi\)
0.740973 0.671534i \(-0.234364\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 36.0000i − 1.20201i
\(898\) 0 0
\(899\) −25.0000 −0.833797
\(900\) 0 0
\(901\) −39.0000 −1.29928
\(902\) 0 0
\(903\) − 36.0000i − 1.19800i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 29.0000i − 0.962929i −0.876466 0.481465i \(-0.840105\pi\)
0.876466 0.481465i \(-0.159895\pi\)
\(908\) 0 0
\(909\) 60.0000 1.99007
\(910\) 0 0
\(911\) −3.00000 −0.0993944 −0.0496972 0.998764i \(-0.515826\pi\)
−0.0496972 + 0.998764i \(0.515826\pi\)
\(912\) 0 0
\(913\) 14.0000i 0.463332i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 7.00000i − 0.231160i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −66.0000 −2.17477
\(922\) 0 0
\(923\) 90.0000i 2.96239i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 96.0000i − 3.15305i
\(928\) 0 0
\(929\) −59.0000 −1.93573 −0.967864 0.251476i \(-0.919084\pi\)
−0.967864 + 0.251476i \(0.919084\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 0 0
\(933\) 63.0000i 2.06253i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) −90.0000 −2.93704
\(940\) 0 0
\(941\) −41.0000 −1.33656 −0.668281 0.743909i \(-0.732970\pi\)
−0.668281 + 0.743909i \(0.732970\pi\)
\(942\) 0 0
\(943\) − 4.00000i − 0.130258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41.0000i 1.33232i 0.745808 + 0.666160i \(0.232063\pi\)
−0.745808 + 0.666160i \(0.767937\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) 29.0000i 0.939402i 0.882826 + 0.469701i \(0.155638\pi\)
−0.882826 + 0.469701i \(0.844362\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 15.0000i − 0.484881i
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) 24.0000i 0.773389i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 55.0000i 1.76868i 0.466843 + 0.884340i \(0.345391\pi\)
−0.466843 + 0.884340i \(0.654609\pi\)
\(968\) 0 0
\(969\) 45.0000 1.44561
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) − 4.00000i − 0.128234i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.00000i 0.255943i 0.991778 + 0.127971i \(0.0408466\pi\)
−0.991778 + 0.127971i \(0.959153\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) 14.0000i 0.446531i 0.974758 + 0.223265i \(0.0716716\pi\)
−0.974758 + 0.223265i \(0.928328\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.00000i − 0.190982i
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) − 60.0000i − 1.90404i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 8.00000i − 0.253363i −0.991943 0.126681i \(-0.959567\pi\)
0.991943 0.126681i \(-0.0404325\pi\)
\(998\) 0 0
\(999\) −9.00000 −0.284747
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.a.4049.1 2
4.3 odd 2 2200.2.b.b.1849.2 2
5.2 odd 4 880.2.a.a.1.1 1
5.3 odd 4 4400.2.a.be.1.1 1
5.4 even 2 inner 4400.2.b.a.4049.2 2
15.2 even 4 7920.2.a.e.1.1 1
20.3 even 4 2200.2.a.a.1.1 1
20.7 even 4 440.2.a.d.1.1 1
20.19 odd 2 2200.2.b.b.1849.1 2
40.27 even 4 3520.2.a.a.1.1 1
40.37 odd 4 3520.2.a.bh.1.1 1
55.32 even 4 9680.2.a.a.1.1 1
60.47 odd 4 3960.2.a.f.1.1 1
220.87 odd 4 4840.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.d.1.1 1 20.7 even 4
880.2.a.a.1.1 1 5.2 odd 4
2200.2.a.a.1.1 1 20.3 even 4
2200.2.b.b.1849.1 2 20.19 odd 2
2200.2.b.b.1849.2 2 4.3 odd 2
3520.2.a.a.1.1 1 40.27 even 4
3520.2.a.bh.1.1 1 40.37 odd 4
3960.2.a.f.1.1 1 60.47 odd 4
4400.2.a.be.1.1 1 5.3 odd 4
4400.2.b.a.4049.1 2 1.1 even 1 trivial
4400.2.b.a.4049.2 2 5.4 even 2 inner
4840.2.a.i.1.1 1 220.87 odd 4
7920.2.a.e.1.1 1 15.2 even 4
9680.2.a.a.1.1 1 55.32 even 4