Properties

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

$q$-expansion

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

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 \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 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 \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 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
\(19\) −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 \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 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 \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 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 \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.00000i − 0.145865i −0.997337 0.0729325i \(-0.976764\pi\)
0.997337 0.0729325i \(-0.0232358\pi\)
\(48\) 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 \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 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 \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 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 \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 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 \(0.229433\pi\)
−0.751288 + 0.659975i \(0.770567\pi\)
\(98\) 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 \(0.221258\pi\)
−0.767988 + 0.640464i \(0.778742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 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 \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 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 \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) 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 \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 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 \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 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 \(-0.815282\pi\)
0.836293 0.548282i \(-0.184718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 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 \(0.0857352\pi\)
−0.963945 + 0.266100i \(0.914265\pi\)
\(174\) 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 \(0.0929658\pi\)
−0.957653 + 0.287926i \(0.907034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 8.00000i − 0.569976i −0.958531 0.284988i \(-0.908010\pi\)
0.958531 0.284988i \(-0.0919897\pi\)
\(198\) 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 \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.00000i − 0.597351i −0.954355 0.298675i \(-0.903455\pi\)
0.954355 0.298675i \(-0.0965448\pi\)
\(228\) 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 \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 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 \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 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 \(-0.624115\pi\)
0.380114 0.924940i \(-0.375885\pi\)
\(264\) 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 \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 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 \(-0.733379\pi\)
0.669238 0.743048i \(-0.266621\pi\)
\(284\) 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 \(-0.876015\pi\)
0.925096 0.379734i \(-0.123985\pi\)
\(294\) 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 \(0.161224\pi\)
−0.874447 + 0.485121i \(0.838776\pi\)
\(308\) 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 \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 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 \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) 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 \(-0.754128\pi\)
0.716218 0.697877i \(-0.245872\pi\)
\(348\) 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 \(-0.957519\pi\)
0.991108 0.133062i \(-0.0424808\pi\)
\(354\) 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 \(0.0584832\pi\)
−0.983169 + 0.182699i \(0.941517\pi\)
\(368\) 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 \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) 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 \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 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 \(-0.804158\pi\)
0.816625 0.577168i \(-0.195842\pi\)
\(398\) 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 \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 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 \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 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 \(-0.940087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 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 \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 11.0000i − 0.509019i −0.967070 0.254510i \(-0.918086\pi\)
0.967070 0.254510i \(-0.0819141\pi\)
\(468\) 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 \(-0.833900\pi\)
0.866914 0.498458i \(-0.166100\pi\)
\(488\) 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 \(0.0213050\pi\)
−0.997761 + 0.0668817i \(0.978695\pi\)
\(504\) 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 \(-0.588041\pi\)
0.273075 0.961993i \(-0.411959\pi\)
\(524\) 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 \(0.204275\pi\)
−0.801050 + 0.598597i \(0.795725\pi\)
\(548\) 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 \(-0.889921\pi\)
0.940797 0.338971i \(-0.110079\pi\)
\(558\) 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 \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 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 \(-0.720174\pi\)
0.637845 0.770165i \(-0.279826\pi\)
\(578\) 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 \(-0.812956\pi\)
0.832265 0.554379i \(-0.187044\pi\)
\(594\) 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 \(-0.980608\pi\)
0.998145 0.0608831i \(-0.0193917\pi\)
\(608\) 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 \(0.207165\pi\)
−0.795583 + 0.605844i \(0.792835\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 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 \(0.135896\pi\)
−0.910241 + 0.414080i \(0.864104\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 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 \(-0.733151\pi\)
0.668705 0.743527i \(-0.266849\pi\)
\(654\) 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 \(-0.811845\pi\)
0.830324 0.557280i \(-0.188155\pi\)
\(678\) 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 \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 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 \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 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 \(0.208606\pi\)
−0.792831 + 0.609441i \(0.791394\pi\)
\(734\) 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 \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 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 \(-0.740955\pi\)
0.686730 0.726912i \(-0.259045\pi\)
\(758\) 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 \(0.123271\pi\)
−0.925945 + 0.377659i \(0.876729\pi\)
\(774\) 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 \(-0.902014\pi\)
0.952993 0.302992i \(-0.0979856\pi\)
\(788\) 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 \(-0.757291\pi\)
0.723117 0.690725i \(-0.242709\pi\)
\(798\) 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 \(0.162277\pi\)
−0.872838 + 0.488009i \(0.837723\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.00000i − 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 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 \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.00000i 0.204956i 0.994735 + 0.102478i \(0.0326771\pi\)
−0.994735 + 0.102478i \(0.967323\pi\)
\(858\) 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 \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 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 \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 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 \(-0.764982\pi\)
0.739594 0.673054i \(-0.235018\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 16.0000i − 0.537227i −0.963248 0.268614i \(-0.913434\pi\)
0.963248 0.268614i \(-0.0865655\pi\)
\(888\) 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 \(0.0965997\pi\)
−0.954303 + 0.298840i \(0.903400\pi\)
\(908\) 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 \(0.0681103\pi\)
−0.977195 + 0.212346i \(0.931890\pi\)
\(938\) 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 \(-0.774805\pi\)
0.760009 0.649913i \(-0.225195\pi\)
\(948\) 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 \(-0.969244\pi\)
0.995336 0.0964735i \(-0.0307563\pi\)
\(968\) 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 \(0.136536\pi\)
−0.909407 + 0.415907i \(0.863464\pi\)
\(978\) 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 \(0.0152346\pi\)
−0.998855 + 0.0478426i \(0.984765\pi\)
\(984\) 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 \(-0.984873\pi\)
0.998871 0.0475055i \(-0.0151272\pi\)
\(998\) 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.449.1 2
4.3 odd 2 1400.2.g.a.449.2 2
5.2 odd 4 2800.2.a.c.1.1 1
5.3 odd 4 560.2.a.f.1.1 1
5.4 even 2 inner 2800.2.g.b.449.2 2
15.8 even 4 5040.2.a.a.1.1 1
20.3 even 4 280.2.a.a.1.1 1
20.7 even 4 1400.2.a.n.1.1 1
20.19 odd 2 1400.2.g.a.449.1 2
35.13 even 4 3920.2.a.c.1.1 1
40.3 even 4 2240.2.a.z.1.1 1
40.13 odd 4 2240.2.a.a.1.1 1
60.23 odd 4 2520.2.a.i.1.1 1
140.3 odd 12 1960.2.q.a.961.1 2
140.23 even 12 1960.2.q.o.361.1 2
140.27 odd 4 9800.2.a.a.1.1 1
140.83 odd 4 1960.2.a.o.1.1 1
140.103 odd 12 1960.2.q.a.361.1 2
140.123 even 12 1960.2.q.o.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.a.1.1 1 20.3 even 4
560.2.a.f.1.1 1 5.3 odd 4
1400.2.a.n.1.1 1 20.7 even 4
1400.2.g.a.449.1 2 20.19 odd 2
1400.2.g.a.449.2 2 4.3 odd 2
1960.2.a.o.1.1 1 140.83 odd 4
1960.2.q.a.361.1 2 140.103 odd 12
1960.2.q.a.961.1 2 140.3 odd 12
1960.2.q.o.361.1 2 140.23 even 12
1960.2.q.o.961.1 2 140.123 even 12
2240.2.a.a.1.1 1 40.13 odd 4
2240.2.a.z.1.1 1 40.3 even 4
2520.2.a.i.1.1 1 60.23 odd 4
2800.2.a.c.1.1 1 5.2 odd 4
2800.2.g.b.449.1 2 1.1 even 1 trivial
2800.2.g.b.449.2 2 5.4 even 2 inner
3920.2.a.c.1.1 1 35.13 even 4
5040.2.a.a.1.1 1 15.8 even 4
9800.2.a.a.1.1 1 140.27 odd 4