Properties

Label 2800.2.g.b.449.2
Level $2800$
Weight $2$
Character 2800.449
Analytic conductor $22.358$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.b.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +5.00000 q^{11} -5.00000i q^{13} +7.00000i q^{17} -2.00000 q^{19} -3.00000 q^{21} +2.00000i q^{23} -9.00000i q^{27} -7.00000 q^{29} -4.00000 q^{31} +15.0000i q^{33} +6.00000i q^{37} +15.0000 q^{39} -12.0000 q^{41} +2.00000i q^{43} +1.00000i q^{47} -1.00000 q^{49} -21.0000 q^{51} -6.00000i q^{57} -4.00000 q^{59} +4.00000 q^{61} -6.00000i q^{63} +8.00000i q^{67} -6.00000 q^{69} +6.00000i q^{73} +5.00000i q^{77} -3.00000 q^{79} +9.00000 q^{81} +4.00000i q^{83} -21.0000i q^{87} +5.00000 q^{91} -12.0000i q^{93} -13.0000i q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{9} + O(q^{10}) \) \( 2q - 12q^{9} + 10q^{11} - 4q^{19} - 6q^{21} - 14q^{29} - 8q^{31} + 30q^{39} - 24q^{41} - 2q^{49} - 42q^{51} - 8q^{59} + 8q^{61} - 12q^{69} - 6q^{79} + 18q^{81} + 10q^{91} - 60q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 0.693375i \(-0.756123\pi\)
0.720577 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000i 1.69775i 0.528594 + 0.848875i \(0.322719\pi\)
−0.528594 + 0.848875i \(0.677281\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −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
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 15.0000i 2.61116i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 15.0000 2.40192
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000i 0.145865i 0.997337 + 0.0729325i \(0.0232358\pi\)
−0.997337 + 0.0729325i \(0.976764\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −21.0000 −2.94059
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 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 \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 21.0000i − 2.25144i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 5.00000 0.524142
\(92\) 0 0
\(93\) − 12.0000i − 1.24434i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.0000i − 1.31995i −0.751288 0.659975i \(-0.770567\pi\)
0.751288 0.659975i \(-0.229433\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 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 \(-0.221258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −18.0000 −1.70848
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 30.0000i 2.77350i
\(118\) 0 0
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) − 36.0000i − 3.24601i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) − 2.00000i − 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.00000i 0.683486i 0.939793 + 0.341743i \(0.111017\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) − 25.0000i − 2.09061i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 0 0
\(153\) − 42.0000i − 3.39550i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 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
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 0 0
\(173\) − 7.00000i − 0.532200i −0.963945 0.266100i \(-0.914265\pi\)
0.963945 0.266100i \(-0.0857352\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 35.0000i 2.55945i
\(188\) 0 0
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 0 0
\(193\) − 8.00000i − 0.575853i −0.957653 0.287926i \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) − 7.00000i − 0.491304i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.0000i − 0.834058i
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 0 0
\(223\) 19.0000i 1.27233i 0.771551 + 0.636167i \(0.219481\pi\)
−0.771551 + 0.636167i \(0.780519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.00000i 0.597351i 0.954355 + 0.298675i \(0.0965448\pi\)
−0.954355 + 0.298675i \(0.903455\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 0 0
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 9.00000i − 0.584613i
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 10.0000i 0.636285i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 10.0000i 0.628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 42.0000 2.59973
\(262\) 0 0
\(263\) 30.0000i 1.84988i 0.380114 + 0.924940i \(0.375885\pi\)
−0.380114 + 0.924940i \(0.624115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 15.0000i 0.907841i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) 25.0000i 1.48610i 0.669238 + 0.743048i \(0.266621\pi\)
−0.669238 + 0.743048i \(0.733379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 12.0000i − 0.708338i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 39.0000 2.28622
\(292\) 0 0
\(293\) 13.0000i 0.759468i 0.925096 + 0.379734i \(0.123985\pi\)
−0.925096 + 0.379734i \(0.876015\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 45.0000i − 2.61116i
\(298\) 0 0
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) − 54.0000i − 3.10222i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 17.0000i − 0.970241i −0.874447 0.485121i \(-0.838776\pi\)
0.874447 0.485121i \(-0.161224\pi\)
\(308\) 0 0
\(309\) 39.0000 2.21863
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 0 0
\(313\) − 1.00000i − 0.0565233i −0.999601 0.0282617i \(-0.991003\pi\)
0.999601 0.0282617i \(-0.00899717\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) −35.0000 −1.95962
\(320\) 0 0
\(321\) −54.0000 −3.01399
\(322\) 0 0
\(323\) − 14.0000i − 0.778981i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 15.0000i − 0.829502i
\(328\) 0 0
\(329\) −1.00000 −0.0551318
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) − 36.0000i − 1.97279i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.0000i 1.39575i 0.716218 + 0.697877i \(0.245872\pi\)
−0.716218 + 0.697877i \(0.754128\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) −45.0000 −2.40192
\(352\) 0 0
\(353\) 5.00000i 0.266123i 0.991108 + 0.133062i \(0.0424808\pi\)
−0.991108 + 0.133062i \(0.957519\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 21.0000i − 1.11144i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 42.0000i 2.20443i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 7.00000i − 0.365397i −0.983169 0.182699i \(-0.941517\pi\)
0.983169 0.182699i \(-0.0584832\pi\)
\(368\) 0 0
\(369\) 72.0000 3.74817
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.0000i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 35.0000i 1.80259i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 36.0000 1.84434
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 12.0000i − 0.609994i
\(388\) 0 0
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) −14.0000 −0.708010
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.0000i 1.15434i 0.816625 + 0.577168i \(0.195842\pi\)
−0.816625 + 0.577168i \(0.804158\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 20.0000i 0.996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.0000i 1.48704i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 0 0
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 54.0000i 2.64439i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −3.00000 −0.146211 −0.0731055 0.997324i \(-0.523291\pi\)
−0.0731055 + 0.997324i \(0.523291\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 75.0000 3.62103
\(430\) 0 0
\(431\) 25.0000 1.20421 0.602104 0.798418i \(-0.294329\pi\)
0.602104 + 0.798418i \(0.294329\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.00000i − 0.191346i
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 66.0000i 3.12169i
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) −60.0000 −2.82529
\(452\) 0 0
\(453\) 57.0000i 2.67809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) 63.0000 2.94059
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.0000i 0.509019i 0.967070 + 0.254510i \(0.0819141\pi\)
−0.967070 + 0.254510i \(0.918086\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 30.0000 1.38233
\(472\) 0 0
\(473\) 10.0000i 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.0000 1.55350 0.776750 0.629809i \(-0.216867\pi\)
0.776750 + 0.629809i \(0.216867\pi\)
\(480\) 0 0
\(481\) 30.0000 1.36788
\(482\) 0 0
\(483\) − 6.00000i − 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) 0 0
\(489\) −42.0000 −1.89931
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) − 49.0000i − 2.20685i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 −0.0447661 −0.0223831 0.999749i \(-0.507125\pi\)
−0.0223831 + 0.999749i \(0.507125\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) − 3.00000i − 0.133763i −0.997761 0.0668817i \(-0.978695\pi\)
0.997761 0.0668817i \(-0.0213050\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 36.0000i − 1.59882i
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 0 0
\(513\) 18.0000i 0.794719i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.00000i 0.219900i
\(518\) 0 0
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 28.0000i − 1.21970i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 60.0000i 2.59889i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 12.0000i − 0.517838i
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 24.0000i 1.02994i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) − 3.00000i − 0.127573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.0000i 0.677942i 0.940797 + 0.338971i \(0.110079\pi\)
−0.940797 + 0.338971i \(0.889921\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −105.000 −4.43310
\(562\) 0 0
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 39.0000i 1.62925i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 37.0000i 1.54033i 0.637845 + 0.770165i \(0.279826\pi\)
−0.637845 + 0.770165i \(0.720174\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 0 0
\(593\) 27.0000i 1.10876i 0.832265 + 0.554379i \(0.187044\pi\)
−0.832265 + 0.554379i \(0.812956\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 12.0000i − 0.491127i
\(598\) 0 0
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) − 48.0000i − 1.95471i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.00000i 0.121766i 0.998145 + 0.0608831i \(0.0193917\pi\)
−0.998145 + 0.0608831i \(0.980608\pi\)
\(608\) 0 0
\(609\) 21.0000 0.850963
\(610\) 0 0
\(611\) 5.00000 0.202278
\(612\) 0 0
\(613\) − 30.0000i − 1.21169i −0.795583 0.605844i \(-0.792835\pi\)
0.795583 0.605844i \(-0.207165\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 30.0000i − 1.19808i
\(628\) 0 0
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) − 15.0000i − 0.596196i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.00000i 0.198107i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 21.0000i − 0.828159i −0.910241 0.414080i \(-0.864104\pi\)
0.910241 0.414080i \(-0.135896\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) 38.0000i 1.48705i 0.668705 + 0.743527i \(0.266849\pi\)
−0.668705 + 0.743527i \(0.733151\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 36.0000i − 1.40449i
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 105.000i 4.07786i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 14.0000i − 0.542082i
\(668\) 0 0
\(669\) −57.0000 −2.20375
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.0000i 1.11456i 0.830324 + 0.557280i \(0.188155\pi\)
−0.830324 + 0.557280i \(0.811845\pi\)
\(678\) 0 0
\(679\) 13.0000 0.498894
\(680\) 0 0
\(681\) −27.0000 −1.03464
\(682\) 0 0
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 12.0000i − 0.457829i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 0 0
\(693\) − 30.0000i − 1.13961i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 84.0000i − 3.18173i
\(698\) 0 0
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) 35.0000 1.32193 0.660966 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) − 12.0000i − 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 18.0000i − 0.676960i
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 18.0000 0.675053
\(712\) 0 0
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.0000i 1.00833i
\(718\) 0 0
\(719\) −14.0000 −0.522112 −0.261056 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 0 0
\(721\) 13.0000 0.484145
\(722\) 0 0
\(723\) − 66.0000i − 2.45457i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 40.0000i − 1.48352i −0.670667 0.741759i \(-0.733992\pi\)
0.670667 0.741759i \(-0.266008\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −14.0000 −0.517809
\(732\) 0 0
\(733\) − 33.0000i − 1.21888i −0.792831 0.609441i \(-0.791394\pi\)
0.792831 0.609441i \(-0.208606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.0000i 1.47342i
\(738\) 0 0
\(739\) 51.0000 1.87607 0.938033 0.346547i \(-0.112646\pi\)
0.938033 + 0.346547i \(0.112646\pi\)
\(740\) 0 0
\(741\) −30.0000 −1.10208
\(742\) 0 0
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 24.0000i − 0.878114i
\(748\) 0 0
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 3.00000 0.109472 0.0547358 0.998501i \(-0.482568\pi\)
0.0547358 + 0.998501i \(0.482568\pi\)
\(752\) 0 0
\(753\) − 18.0000i − 0.655956i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000i 1.45382i 0.686730 + 0.726912i \(0.259045\pi\)
−0.686730 + 0.726912i \(0.740955\pi\)
\(758\) 0 0
\(759\) −30.0000 −1.08893
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) − 5.00000i − 0.181012i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000i 0.722158i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 54.0000 1.94476
\(772\) 0 0
\(773\) − 21.0000i − 0.755318i −0.925945 0.377659i \(-0.876729\pi\)
0.925945 0.377659i \(-0.123271\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 18.0000i − 0.645746i
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 63.0000i 2.25144i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 0 0
\(789\) −90.0000 −3.20408
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) − 20.0000i − 0.710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.0000i 1.38145i 0.723117 + 0.690725i \(0.242709\pi\)
−0.723117 + 0.690725i \(0.757291\pi\)
\(798\) 0 0
\(799\) −7.00000 −0.247642
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.0000i 1.05868i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 78.0000i − 2.74573i
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 10.0000 0.351147 0.175574 0.984466i \(-0.443822\pi\)
0.175574 + 0.984466i \(0.443822\pi\)
\(812\) 0 0
\(813\) − 36.0000i − 1.26258i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.00000i − 0.139942i
\(818\) 0 0
\(819\) −30.0000 −1.04828
\(820\) 0 0
\(821\) 1.00000 0.0349002 0.0174501 0.999848i \(-0.494445\pi\)
0.0174501 + 0.999848i \(0.494445\pi\)
\(822\) 0 0
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 6.00000 0.208138
\(832\) 0 0
\(833\) − 7.00000i − 0.242536i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 36.0000i 1.24434i
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 57.0000i 1.96318i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) −75.0000 −2.57399
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 36.0000 1.22688
\(862\) 0 0
\(863\) − 48.0000i − 1.63394i −0.576681 0.816970i \(-0.695652\pi\)
0.576681 0.816970i \(-0.304348\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 96.0000i − 3.26033i
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) 78.0000i 2.63990i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 0 0
\(879\) −39.0000 −1.31544
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 40.0000i 1.34611i 0.739594 + 0.673054i \(0.235018\pi\)
−0.739594 + 0.673054i \(0.764982\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.0000i 0.537227i 0.963248 + 0.268614i \(0.0865655\pi\)
−0.963248 + 0.268614i \(0.913434\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 45.0000 1.50756
\(892\) 0 0
\(893\) − 2.00000i − 0.0669274i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 30.0000i 1.00167i
\(898\) 0 0
\(899\) 28.0000 0.933852
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 6.00000i − 0.199667i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 18.0000i − 0.597680i −0.954303 0.298840i \(-0.903400\pi\)
0.954303 0.298840i \(-0.0965997\pi\)
\(908\) 0 0
\(909\) 108.000 3.58213
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 20.0000i 0.661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.00000i 0.198137i
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) 51.0000 1.68051
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 78.0000i 2.56186i
\(928\) 0 0
\(929\) 32.0000 1.04989 0.524943 0.851137i \(-0.324087\pi\)
0.524943 + 0.851137i \(0.324087\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 102.000i 3.33933i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 13.0000i − 0.424691i −0.977195 0.212346i \(-0.931890\pi\)
0.977195 0.212346i \(-0.0681103\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) − 24.0000i − 0.781548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.0000i 1.29983i 0.760009 + 0.649913i \(0.225195\pi\)
−0.760009 + 0.649913i \(0.774805\pi\)
\(948\) 0 0
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 105.000i − 3.39417i
\(958\) 0 0
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 108.000i − 3.48025i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.00000i 0.192947i 0.995336 + 0.0964735i \(0.0307563\pi\)
−0.995336 + 0.0964735i \(0.969244\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 0 0
\(971\) 16.0000 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) 0 0
\(973\) 18.0000i 0.577054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 26.0000i − 0.831814i −0.909407 0.415907i \(-0.863464\pi\)
0.909407 0.415907i \(-0.136536\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) 0 0
\(983\) − 3.00000i − 0.0956851i −0.998855 0.0478426i \(-0.984765\pi\)
0.998855 0.0478426i \(-0.0152346\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 3.00000i − 0.0954911i
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 36.0000i 1.14243i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.00000i 0.0950110i 0.998871 + 0.0475055i \(0.0151272\pi\)
−0.998871 + 0.0475055i \(0.984873\pi\)
\(998\) 0 0
\(999\) 54.0000 1.70848
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 2800.2.g.b.449.2 2
4.3 odd 2 1400.2.g.a.449.1 2
5.2 odd 4 560.2.a.f.1.1 1
5.3 odd 4 2800.2.a.c.1.1 1
5.4 even 2 inner 2800.2.g.b.449.1 2
15.2 even 4 5040.2.a.a.1.1 1
20.3 even 4 1400.2.a.n.1.1 1
20.7 even 4 280.2.a.a.1.1 1
20.19 odd 2 1400.2.g.a.449.2 2
35.27 even 4 3920.2.a.c.1.1 1
40.27 even 4 2240.2.a.z.1.1 1
40.37 odd 4 2240.2.a.a.1.1 1
60.47 odd 4 2520.2.a.i.1.1 1
140.27 odd 4 1960.2.a.o.1.1 1
140.47 odd 12 1960.2.q.a.361.1 2
140.67 even 12 1960.2.q.o.961.1 2
140.83 odd 4 9800.2.a.a.1.1 1
140.87 odd 12 1960.2.q.a.961.1 2
140.107 even 12 1960.2.q.o.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.a.1.1 1 20.7 even 4
560.2.a.f.1.1 1 5.2 odd 4
1400.2.a.n.1.1 1 20.3 even 4
1400.2.g.a.449.1 2 4.3 odd 2
1400.2.g.a.449.2 2 20.19 odd 2
1960.2.a.o.1.1 1 140.27 odd 4
1960.2.q.a.361.1 2 140.47 odd 12
1960.2.q.a.961.1 2 140.87 odd 12
1960.2.q.o.361.1 2 140.107 even 12
1960.2.q.o.961.1 2 140.67 even 12
2240.2.a.a.1.1 1 40.37 odd 4
2240.2.a.z.1.1 1 40.27 even 4
2520.2.a.i.1.1 1 60.47 odd 4
2800.2.a.c.1.1 1 5.3 odd 4
2800.2.g.b.449.1 2 5.4 even 2 inner
2800.2.g.b.449.2 2 1.1 even 1 trivial
3920.2.a.c.1.1 1 35.27 even 4
5040.2.a.a.1.1 1 15.2 even 4
9800.2.a.a.1.1 1 140.83 odd 4