Properties

Label 2100.2.k.f.1849.1
Level 2100
Weight 2
Character 2100.1849
Analytic conductor 16.769
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1849
Dual form 2100.2.k.f.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -1.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} +4.00000i q^{13} -2.00000i q^{17} +4.00000 q^{19} -1.00000 q^{21} +7.00000i q^{23} +1.00000i q^{27} +9.00000 q^{29} -2.00000 q^{31} -1.00000i q^{33} -1.00000i q^{37} +4.00000 q^{39} +8.00000 q^{41} -9.00000i q^{43} -4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -4.00000i q^{57} -4.00000 q^{59} +4.00000 q^{61} +1.00000i q^{63} -9.00000i q^{67} +7.00000 q^{69} +5.00000 q^{71} +10.0000i q^{73} -1.00000i q^{77} +15.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{87} -8.00000 q^{89} +4.00000 q^{91} +2.00000i q^{93} -10.0000i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 2q^{11} + 8q^{19} - 2q^{21} + 18q^{29} - 4q^{31} + 8q^{39} + 16q^{41} - 2q^{49} - 4q^{51} - 8q^{59} + 8q^{61} + 14q^{69} + 10q^{71} + 30q^{79} + 2q^{81} - 16q^{89} + 8q^{91} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 7.00000i 1.45960i 0.683660 + 0.729800i \(0.260387\pi\)
−0.683660 + 0.729800i \(0.739613\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) − 1.00000i − 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) − 9.00000i − 1.37249i −0.727372 0.686244i \(-0.759258\pi\)
0.727372 0.686244i \(-0.240742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 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
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.00000i − 1.09952i −0.835321 0.549762i \(-0.814718\pi\)
0.835321 0.549762i \(-0.185282\pi\)
\(68\) 0 0
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 5.00000 0.593391 0.296695 0.954972i \(-0.404115\pi\)
0.296695 + 0.954972i \(0.404115\pi\)
\(72\) 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
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.00000i − 0.964901i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 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
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) 13.0000i 1.22294i 0.791269 + 0.611469i \(0.209421\pi\)
−0.791269 + 0.611469i \(0.790579\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.00000i − 0.369800i
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 9.00000i − 0.798621i −0.916816 0.399310i \(-0.869250\pi\)
0.916816 0.399310i \(-0.130750\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 11.0000 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 20.0000i − 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 7.00000 0.551677
\(162\) 0 0
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 8.00000i 0.608229i 0.952636 + 0.304114i \(0.0983605\pi\)
−0.952636 + 0.304114i \(0.901639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) − 4.00000i − 0.295689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.00000i − 0.146254i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) − 17.0000i − 1.22369i −0.790979 0.611843i \(-0.790428\pi\)
0.790979 0.611843i \(-0.209572\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000i 1.06871i 0.845262 + 0.534353i \(0.179445\pi\)
−0.845262 + 0.534353i \(0.820555\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) 0 0
\(203\) − 9.00000i − 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 7.00000i − 0.486534i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) − 5.00000i − 0.342594i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 19.0000i 1.24473i 0.782727 + 0.622366i \(0.213828\pi\)
−0.782727 + 0.622366i \(0.786172\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 15.0000i − 0.974355i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 7.00000i 0.440086i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 26.0000i − 1.62184i −0.585160 0.810918i \(-0.698968\pi\)
0.585160 0.810918i \(-0.301032\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) − 15.0000i − 0.924940i −0.886635 0.462470i \(-0.846963\pi\)
0.886635 0.462470i \(-0.153037\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) − 4.00000i − 0.242091i
\(274\) 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
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 19.0000 1.13344 0.566722 0.823909i \(-0.308211\pi\)
0.566722 + 0.823909i \(0.308211\pi\)
\(282\) 0 0
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.00000i − 0.472225i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000i 0.0580259i
\(298\) 0 0
\(299\) −28.0000 −1.61928
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 0 0
\(303\) − 6.00000i − 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.0000i 1.74113i 0.492050 + 0.870567i \(0.336248\pi\)
−0.492050 + 0.870567i \(0.663752\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000i 0.0553001i
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 0 0
\(333\) 1.00000i 0.0547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 0 0
\(339\) 13.0000 0.706063
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.0000i 1.44944i 0.689046 + 0.724718i \(0.258030\pi\)
−0.689046 + 0.724718i \(0.741970\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 0 0
\(359\) 35.0000 1.84723 0.923615 0.383322i \(-0.125220\pi\)
0.923615 + 0.383322i \(0.125220\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 11.0000i 0.569558i 0.958593 + 0.284779i \(0.0919203\pi\)
−0.958593 + 0.284779i \(0.908080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) −9.00000 −0.461084
\(382\) 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
\(387\) 9.00000i 0.457496i
\(388\) 0 0
\(389\) −27.0000 −1.36895 −0.684477 0.729034i \(-0.739969\pi\)
−0.684477 + 0.729034i \(0.739969\pi\)
\(390\) 0 0
\(391\) 14.0000 0.708010
\(392\) 0 0
\(393\) − 18.0000i − 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −33.0000 −1.64794 −0.823971 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) 0 0
\(403\) − 8.00000i − 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.00000i − 0.0495682i
\(408\) 0 0
\(409\) −40.0000 −1.97787 −0.988936 0.148340i \(-0.952607\pi\)
−0.988936 + 0.148340i \(0.952607\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 10.0000i − 0.489702i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.00000i − 0.193574i
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.0000i 1.33942i
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3.00000i − 0.141895i
\(448\) 0 0
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) − 11.0000i − 0.516825i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000i 0.233890i 0.993138 + 0.116945i \(0.0373101\pi\)
−0.993138 + 0.116945i \(0.962690\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.0000i 1.75843i 0.476425 + 0.879215i \(0.341932\pi\)
−0.476425 + 0.879215i \(0.658068\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) − 9.00000i − 0.413820i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) − 7.00000i − 0.318511i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 3.00000i − 0.135943i −0.997687 0.0679715i \(-0.978347\pi\)
0.997687 0.0679715i \(-0.0216527\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.00000i − 0.224281i
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) 0 0
\(503\) − 14.0000i − 0.624229i −0.950044 0.312115i \(-0.898963\pi\)
0.950044 0.312115i \(-0.101037\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000i 0.133235i
\(508\) 0 0
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 4.00000i − 0.175920i
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 40.0000i − 1.74908i −0.484955 0.874539i \(-0.661164\pi\)
0.484955 0.874539i \(-0.338836\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) −26.0000 −1.13043
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 32.0000i 1.38607i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 10.0000i 0.429141i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 35.0000i 1.49649i 0.663421 + 0.748246i \(0.269104\pi\)
−0.663421 + 0.748246i \(0.730896\pi\)
\(548\) 0 0
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) − 15.0000i − 0.637865i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.00000i − 0.0423714i −0.999776 0.0211857i \(-0.993256\pi\)
0.999776 0.0211857i \(-0.00674412\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) 0 0
\(563\) − 30.0000i − 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 45.0000 1.88650 0.943249 0.332086i \(-0.107752\pi\)
0.943249 + 0.332086i \(0.107752\pi\)
\(570\) 0 0
\(571\) −23.0000 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(572\) 0 0
\(573\) − 20.0000i − 0.835512i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.00000i − 0.333044i −0.986038 0.166522i \(-0.946746\pi\)
0.986038 0.166522i \(-0.0532537\pi\)
\(578\) 0 0
\(579\) −17.0000 −0.706496
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.0000i 1.23823i 0.785299 + 0.619116i \(0.212509\pi\)
−0.785299 + 0.619116i \(0.787491\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 15.0000 0.617018
\(592\) 0 0
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) −35.0000 −1.43006 −0.715031 0.699093i \(-0.753587\pi\)
−0.715031 + 0.699093i \(0.753587\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 9.00000i 0.366508i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 29.0000i 1.17130i 0.810564 + 0.585649i \(0.199160\pi\)
−0.810564 + 0.585649i \(0.800840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 31.0000i − 1.24801i −0.781419 0.624007i \(-0.785504\pi\)
0.781419 0.624007i \(-0.214496\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) −7.00000 −0.280900
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4.00000i − 0.159745i
\(628\) 0 0
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −27.0000 −1.07485 −0.537427 0.843311i \(-0.680603\pi\)
−0.537427 + 0.843311i \(0.680603\pi\)
\(632\) 0 0
\(633\) 4.00000i 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.00000i − 0.158486i
\(638\) 0 0
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 46.0000i − 1.80845i −0.427060 0.904223i \(-0.640451\pi\)
0.427060 0.904223i \(-0.359549\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 26.0000i 1.01746i 0.860927 + 0.508729i \(0.169885\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) − 8.00000i − 0.310694i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 63.0000i 2.43937i
\(668\) 0 0
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 2.00000i 0.0770943i 0.999257 + 0.0385472i \(0.0122730\pi\)
−0.999257 + 0.0385472i \(0.987727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 19.0000i − 0.727015i −0.931591 0.363507i \(-0.881579\pi\)
0.931591 0.363507i \(-0.118421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 22.0000i 0.839352i
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 1.00000i 0.0379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) 0 0
\(699\) 19.0000 0.718646
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −15.0000 −0.562544
\(712\) 0 0
\(713\) − 14.0000i − 0.524304i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 16.0000i − 0.597531i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 30.0000i − 1.11264i −0.830969 0.556319i \(-0.812213\pi\)
0.830969 0.556319i \(-0.187787\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) − 36.0000i − 1.32969i −0.746981 0.664845i \(-0.768498\pi\)
0.746981 0.664845i \(-0.231502\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9.00000i − 0.331519i
\(738\) 0 0
\(739\) 9.00000 0.331070 0.165535 0.986204i \(-0.447065\pi\)
0.165535 + 0.986204i \(0.447065\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 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
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 2.00000i 0.0728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.00000i 0.254419i 0.991876 + 0.127210i \(0.0406021\pi\)
−0.991876 + 0.127210i \(0.959398\pi\)
\(758\) 0 0
\(759\) 7.00000 0.254084
\(760\) 0 0
\(761\) −4.00000 −0.145000 −0.0724999 0.997368i \(-0.523098\pi\)
−0.0724999 + 0.997368i \(0.523098\pi\)
\(762\) 0 0
\(763\) 1.00000i 0.0362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 16.0000i − 0.577727i
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −26.0000 −0.936367
\(772\) 0 0
\(773\) − 26.0000i − 0.935155i −0.883952 0.467578i \(-0.845127\pi\)
0.883952 0.467578i \(-0.154873\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.00000i 0.0358748i
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 5.00000 0.178914
\(782\) 0 0
\(783\) 9.00000i 0.321634i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 26.0000i − 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) 0 0
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) 13.0000 0.462227
\(792\) 0 0
\(793\) 16.0000i 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 54.0000i 1.91278i 0.292096 + 0.956389i \(0.405647\pi\)
−0.292096 + 0.956389i \(0.594353\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 8.00000 0.282666
\(802\) 0 0
\(803\) 10.0000i 0.352892i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000i 0.633630i
\(808\) 0 0
\(809\) 23.0000 0.808637 0.404318 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 36.0000i − 1.25948i
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) − 11.0000i − 0.383436i −0.981450 0.191718i \(-0.938594\pi\)
0.981450 0.191718i \(-0.0614059\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 33.0000i − 1.14752i −0.819023 0.573761i \(-0.805484\pi\)
0.819023 0.573761i \(-0.194516\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) 2.00000i 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 2.00000i − 0.0691301i
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) − 19.0000i − 0.654395i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) 0 0
\(853\) 28.0000i 0.958702i 0.877623 + 0.479351i \(0.159128\pi\)
−0.877623 + 0.479351i \(0.840872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.00000i 0.136637i 0.997664 + 0.0683187i \(0.0217635\pi\)
−0.997664 + 0.0683187i \(0.978237\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) 33.0000i 1.12333i 0.827364 + 0.561667i \(0.189840\pi\)
−0.827364 + 0.561667i \(0.810160\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 0 0
\(873\) 10.0000i 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) 1.00000i 0.0336527i 0.999858 + 0.0168263i \(0.00535624\pi\)
−0.999858 + 0.0168263i \(0.994644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 52.0000i − 1.74599i −0.487730 0.872995i \(-0.662175\pi\)
0.487730 0.872995i \(-0.337825\pi\)
\(888\) 0 0
\(889\) −9.00000 −0.301850
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 28.0000i 0.934893i
\(898\) 0 0
\(899\) −18.0000 −0.600334
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 9.00000i 0.299501i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 24.0000i − 0.796907i −0.917189 0.398453i \(-0.869547\pi\)
0.917189 0.398453i \(-0.130453\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −9.00000 −0.298183 −0.149092 0.988823i \(-0.547635\pi\)
−0.149092 + 0.988823i \(0.547635\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 18.0000i − 0.594412i
\(918\) 0 0
\(919\) 59.0000 1.94623 0.973115 0.230319i \(-0.0739769\pi\)
0.973115 + 0.230319i \(0.0739769\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) 0 0
\(923\) 20.0000i 0.658308i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00000i 0.131377i
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) − 2.00000i − 0.0654771i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 36.0000i − 1.17607i −0.808836 0.588034i \(-0.799902\pi\)
0.808836 0.588034i \(-0.200098\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 0 0
\(943\) 56.0000i 1.82361i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) 31.0000 1.00524
\(952\) 0 0
\(953\) 21.0000i 0.680257i 0.940379 + 0.340128i \(0.110471\pi\)
−0.940379 + 0.340128i \(0.889529\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 9.00000i − 0.290929i
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) − 10.0000i − 0.320585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.0000i 1.05576i 0.849318 + 0.527882i \(0.177014\pi\)
−0.849318 + 0.527882i \(0.822986\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 1.00000 0.0319275
\(982\) 0 0
\(983\) − 58.0000i − 1.84991i −0.380073 0.924956i \(-0.624101\pi\)
0.380073 0.924956i \(-0.375899\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.00000i 0.127321i
\(988\) 0 0
\(989\) 63.0000 2.00328
\(990\) 0 0
\(991\) 37.0000 1.17534 0.587672 0.809099i \(-0.300045\pi\)
0.587672 + 0.809099i \(0.300045\pi\)
\(992\) 0 0
\(993\) 23.0000i 0.729883i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28.0000i 0.886769i 0.896332 + 0.443384i \(0.146222\pi\)
−0.896332 + 0.443384i \(0.853778\pi\)
\(998\) 0 0
\(999\) 1.00000 0.0316386
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 2100.2.k.f.1849.1 2
3.2 odd 2 6300.2.k.h.6049.1 2
5.2 odd 4 2100.2.a.g.1.1 1
5.3 odd 4 2100.2.a.m.1.1 yes 1
5.4 even 2 inner 2100.2.k.f.1849.2 2
15.2 even 4 6300.2.a.x.1.1 1
15.8 even 4 6300.2.a.f.1.1 1
15.14 odd 2 6300.2.k.h.6049.2 2
20.3 even 4 8400.2.a.x.1.1 1
20.7 even 4 8400.2.a.bv.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.a.g.1.1 1 5.2 odd 4
2100.2.a.m.1.1 yes 1 5.3 odd 4
2100.2.k.f.1849.1 2 1.1 even 1 trivial
2100.2.k.f.1849.2 2 5.4 even 2 inner
6300.2.a.f.1.1 1 15.8 even 4
6300.2.a.x.1.1 1 15.2 even 4
6300.2.k.h.6049.1 2 3.2 odd 2
6300.2.k.h.6049.2 2 15.14 odd 2
8400.2.a.x.1.1 1 20.3 even 4
8400.2.a.bv.1.1 1 20.7 even 4