Properties

Label 784.2.a.k.1.2
Level $784$
Weight $2$
Character 784.1
Self dual yes
Analytic conductor $6.260$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.26027151847\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} -2.82843 q^{5} -1.00000 q^{9} -6.00000 q^{11} +5.65685 q^{13} -4.00000 q^{15} -1.41421 q^{17} -4.24264 q^{19} -4.00000 q^{23} +3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} +2.82843 q^{31} -8.48528 q^{33} +2.00000 q^{37} +8.00000 q^{39} +1.41421 q^{41} -10.0000 q^{43} +2.82843 q^{45} -2.82843 q^{47} -2.00000 q^{51} -2.00000 q^{53} +16.9706 q^{55} -6.00000 q^{57} +1.41421 q^{59} +8.48528 q^{61} -16.0000 q^{65} -4.00000 q^{67} -5.65685 q^{69} +12.0000 q^{71} +9.89949 q^{73} +4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} +1.41421 q^{83} +4.00000 q^{85} -8.48528 q^{87} +4.24264 q^{89} +4.00000 q^{93} +12.0000 q^{95} -12.7279 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 12q^{11} - 8q^{15} - 8q^{23} + 6q^{25} - 12q^{29} + 4q^{37} + 16q^{39} - 20q^{43} - 4q^{51} - 4q^{53} - 12q^{57} - 32q^{65} - 8q^{67} + 24q^{71} + 8q^{79} - 10q^{81} + 8q^{85} + 8q^{93} + 24q^{95} + 12q^{99} + O(q^{100}) \)

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
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) −8.48528 −1.47710
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 1.41421 0.220863 0.110432 0.993884i \(-0.464777\pi\)
0.110432 + 0.993884i \(0.464777\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 16.9706 2.28831
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 8.48528 1.08643 0.543214 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −5.65685 −0.681005
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 9.89949 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(74\) 0 0
\(75\) 4.24264 0.489898
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 1.41421 0.155230 0.0776151 0.996983i \(-0.475269\pi\)
0.0776151 + 0.996983i \(0.475269\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −8.48528 −0.909718
\(88\) 0 0
\(89\) 4.24264 0.449719 0.224860 0.974391i \(-0.427808\pi\)
0.224860 + 0.974391i \(0.427808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) −12.7279 −1.29232 −0.646162 0.763200i \(-0.723627\pi\)
−0.646162 + 0.763200i \(0.723627\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 2.82843 0.281439 0.140720 0.990050i \(-0.455058\pi\)
0.140720 + 0.990050i \(0.455058\pi\)
\(102\) 0 0
\(103\) −14.1421 −1.39347 −0.696733 0.717331i \(-0.745364\pi\)
−0.696733 + 0.717331i \(0.745364\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 2.82843 0.268462
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) 11.3137 1.05501
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −14.1421 −1.24515
\(130\) 0 0
\(131\) 21.2132 1.85341 0.926703 0.375794i \(-0.122630\pi\)
0.926703 + 0.375794i \(0.122630\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) −21.2132 −1.79928 −0.899640 0.436632i \(-0.856171\pi\)
−0.899640 + 0.436632i \(0.856171\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) −33.9411 −2.83830
\(144\) 0 0
\(145\) 16.9706 1.40933
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 1.41421 0.114332
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −16.9706 −1.35440 −0.677199 0.735800i \(-0.736806\pi\)
−0.677199 + 0.735800i \(0.736806\pi\)
\(158\) 0 0
\(159\) −2.82843 −0.224309
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 24.0000 1.86840
\(166\) 0 0
\(167\) −2.82843 −0.218870 −0.109435 0.993994i \(-0.534904\pi\)
−0.109435 + 0.993994i \(0.534904\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 4.24264 0.324443
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −16.9706 −1.26141 −0.630706 0.776022i \(-0.717235\pi\)
−0.630706 + 0.776022i \(0.717235\pi\)
\(182\) 0 0
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) −5.65685 −0.415900
\(186\) 0 0
\(187\) 8.48528 0.620505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) −22.6274 −1.62038
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 25.4558 1.80452 0.902258 0.431196i \(-0.141908\pi\)
0.902258 + 0.431196i \(0.141908\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 25.4558 1.76082
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 16.9706 1.16280
\(214\) 0 0
\(215\) 28.2843 1.92897
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 11.3137 0.757622 0.378811 0.925474i \(-0.376333\pi\)
0.378811 + 0.925474i \(0.376333\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 9.89949 0.657053 0.328526 0.944495i \(-0.393448\pi\)
0.328526 + 0.944495i \(0.393448\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 1.41421 0.0910975 0.0455488 0.998962i \(-0.485496\pi\)
0.0455488 + 0.998962i \(0.485496\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 1.41421 0.0892644 0.0446322 0.999003i \(-0.485788\pi\)
0.0446322 + 0.999003i \(0.485788\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) 12.7279 0.793946 0.396973 0.917830i \(-0.370061\pi\)
0.396973 + 0.917830i \(0.370061\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 5.65685 0.347498
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −28.2843 −1.72452 −0.862261 0.506464i \(-0.830952\pi\)
−0.862261 + 0.506464i \(0.830952\pi\)
\(270\) 0 0
\(271\) −11.3137 −0.687259 −0.343629 0.939105i \(-0.611656\pi\)
−0.343629 + 0.939105i \(0.611656\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.0000 −1.08544
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) −21.2132 −1.26099 −0.630497 0.776192i \(-0.717149\pi\)
−0.630497 + 0.776192i \(0.717149\pi\)
\(284\) 0 0
\(285\) 16.9706 1.00525
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) −2.82843 −0.165238 −0.0826192 0.996581i \(-0.526329\pi\)
−0.0826192 + 0.996581i \(0.526329\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 33.9411 1.96946
\(298\) 0 0
\(299\) −22.6274 −1.30858
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) −1.41421 −0.0807134 −0.0403567 0.999185i \(-0.512849\pi\)
−0.0403567 + 0.999185i \(0.512849\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) −21.2132 −1.19904 −0.599521 0.800359i \(-0.704642\pi\)
−0.599521 + 0.800359i \(0.704642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) −5.65685 −0.315735
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 16.9706 0.941357
\(326\) 0 0
\(327\) −14.1421 −0.782062
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) −16.9706 −0.919007
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) −16.9706 −0.908413 −0.454207 0.890896i \(-0.650077\pi\)
−0.454207 + 0.890896i \(0.650077\pi\)
\(350\) 0 0
\(351\) −32.0000 −1.70803
\(352\) 0 0
\(353\) −24.0416 −1.27961 −0.639803 0.768539i \(-0.720984\pi\)
−0.639803 + 0.768539i \(0.720984\pi\)
\(354\) 0 0
\(355\) −33.9411 −1.80141
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 35.3553 1.85567
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) 5.65685 0.295285 0.147643 0.989041i \(-0.452831\pi\)
0.147643 + 0.989041i \(0.452831\pi\)
\(368\) 0 0
\(369\) −1.41421 −0.0736210
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 8.00000 0.413118
\(376\) 0 0
\(377\) −33.9411 −1.74806
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 11.3137 0.579619
\(382\) 0 0
\(383\) 14.1421 0.722629 0.361315 0.932444i \(-0.382328\pi\)
0.361315 + 0.932444i \(0.382328\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 30.0000 1.51330
\(394\) 0 0
\(395\) −11.3137 −0.569254
\(396\) 0 0
\(397\) −28.2843 −1.41955 −0.709773 0.704430i \(-0.751203\pi\)
−0.709773 + 0.704430i \(0.751203\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 14.1421 0.702728
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) 4.24264 0.209785 0.104893 0.994484i \(-0.466550\pi\)
0.104893 + 0.994484i \(0.466550\pi\)
\(410\) 0 0
\(411\) −5.65685 −0.279032
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −30.0000 −1.46911
\(418\) 0 0
\(419\) 32.5269 1.58904 0.794522 0.607236i \(-0.207722\pi\)
0.794522 + 0.607236i \(0.207722\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) −4.24264 −0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −48.0000 −2.31746
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 0 0
\(433\) 15.5563 0.747590 0.373795 0.927511i \(-0.378056\pi\)
0.373795 + 0.927511i \(0.378056\pi\)
\(434\) 0 0
\(435\) 24.0000 1.15071
\(436\) 0 0
\(437\) 16.9706 0.811812
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) −8.48528 −0.401340
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −8.48528 −0.399556
\(452\) 0 0
\(453\) −11.3137 −0.531564
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 0 0
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 22.6274 1.05386 0.526932 0.849907i \(-0.323342\pi\)
0.526932 + 0.849907i \(0.323342\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −11.3137 −0.524661
\(466\) 0 0
\(467\) −21.2132 −0.981630 −0.490815 0.871264i \(-0.663301\pi\)
−0.490815 + 0.871264i \(0.663301\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 0 0
\(473\) 60.0000 2.75880
\(474\) 0 0
\(475\) −12.7279 −0.583997
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 0 0
\(479\) 19.7990 0.904639 0.452319 0.891856i \(-0.350597\pi\)
0.452319 + 0.891856i \(0.350597\pi\)
\(480\) 0 0
\(481\) 11.3137 0.515861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 36.0000 1.63468
\(486\) 0 0
\(487\) −36.0000 −1.63132 −0.815658 0.578535i \(-0.803625\pi\)
−0.815658 + 0.578535i \(0.803625\pi\)
\(488\) 0 0
\(489\) −2.82843 −0.127906
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 0 0
\(495\) −16.9706 −0.762770
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −4.00000 −0.178707
\(502\) 0 0
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 26.8701 1.19334
\(508\) 0 0
\(509\) 39.5980 1.75515 0.877575 0.479440i \(-0.159160\pi\)
0.877575 + 0.479440i \(0.159160\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) 40.0000 1.76261
\(516\) 0 0
\(517\) 16.9706 0.746364
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 32.5269 1.42503 0.712515 0.701657i \(-0.247556\pi\)
0.712515 + 0.701657i \(0.247556\pi\)
\(522\) 0 0
\(523\) 32.5269 1.42230 0.711151 0.703039i \(-0.248174\pi\)
0.711151 + 0.703039i \(0.248174\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −1.41421 −0.0613716
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 0 0
\(537\) −28.2843 −1.22056
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) −24.0000 −1.02994
\(544\) 0 0
\(545\) 28.2843 1.21157
\(546\) 0 0
\(547\) −14.0000 −0.598597 −0.299298 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) 0 0
\(549\) −8.48528 −0.362143
\(550\) 0 0
\(551\) 25.4558 1.08446
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −56.5685 −2.39259
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) −21.2132 −0.894030 −0.447015 0.894526i \(-0.647513\pi\)
−0.447015 + 0.894526i \(0.647513\pi\)
\(564\) 0 0
\(565\) −11.3137 −0.475971
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −21.2132 −0.883117 −0.441559 0.897232i \(-0.645574\pi\)
−0.441559 + 0.897232i \(0.645574\pi\)
\(578\) 0 0
\(579\) 22.6274 0.940363
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) 0 0
\(585\) 16.0000 0.661519
\(586\) 0 0
\(587\) −4.24264 −0.175113 −0.0875563 0.996160i \(-0.527906\pi\)
−0.0875563 + 0.996160i \(0.527906\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 25.4558 1.04711
\(592\) 0 0
\(593\) −7.07107 −0.290374 −0.145187 0.989404i \(-0.546378\pi\)
−0.145187 + 0.989404i \(0.546378\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 36.0000 1.47338
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −26.8701 −1.09605 −0.548026 0.836461i \(-0.684621\pi\)
−0.548026 + 0.836461i \(0.684621\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −70.7107 −2.87480
\(606\) 0 0
\(607\) 39.5980 1.60723 0.803616 0.595148i \(-0.202907\pi\)
0.803616 + 0.595148i \(0.202907\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) 0 0
\(615\) −5.65685 −0.228106
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 4.24264 0.170526 0.0852631 0.996358i \(-0.472827\pi\)
0.0852631 + 0.996358i \(0.472827\pi\)
\(620\) 0 0
\(621\) 22.6274 0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 36.0000 1.43770
\(628\) 0 0
\(629\) −2.82843 −0.112777
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) −16.9706 −0.674519
\(634\) 0 0
\(635\) −22.6274 −0.897942
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) 35.3553 1.39428 0.697139 0.716936i \(-0.254456\pi\)
0.697139 + 0.716936i \(0.254456\pi\)
\(644\) 0 0
\(645\) 40.0000 1.57500
\(646\) 0 0
\(647\) −42.4264 −1.66795 −0.833977 0.551799i \(-0.813942\pi\)
−0.833977 + 0.551799i \(0.813942\pi\)
\(648\) 0 0
\(649\) −8.48528 −0.333076
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) −60.0000 −2.34439
\(656\) 0 0
\(657\) −9.89949 −0.386216
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 36.7696 1.43017 0.715085 0.699038i \(-0.246388\pi\)
0.715085 + 0.699038i \(0.246388\pi\)
\(662\) 0 0
\(663\) −11.3137 −0.439388
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −50.9117 −1.96542
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 0 0
\(675\) −16.9706 −0.653197
\(676\) 0 0
\(677\) 33.9411 1.30446 0.652232 0.758020i \(-0.273833\pi\)
0.652232 + 0.758020i \(0.273833\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 11.3137 0.432275
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −11.3137 −0.431018
\(690\) 0 0
\(691\) 43.8406 1.66778 0.833888 0.551934i \(-0.186110\pi\)
0.833888 + 0.551934i \(0.186110\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 60.0000 2.27593
\(696\) 0 0
\(697\) −2.00000 −0.0757554
\(698\) 0 0
\(699\) 11.3137 0.427924
\(700\) 0 0
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) −8.48528 −0.320028
\(704\) 0 0
\(705\) 11.3137 0.426099
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −11.3137 −0.423702
\(714\) 0 0
\(715\) 96.0000 3.59020
\(716\) 0 0
\(717\) 5.65685 0.211259
\(718\) 0 0
\(719\) −2.82843 −0.105483 −0.0527413 0.998608i \(-0.516796\pi\)
−0.0527413 + 0.998608i \(0.516796\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) −25.4558 −0.944105 −0.472052 0.881570i \(-0.656487\pi\)
−0.472052 + 0.881570i \(0.656487\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 14.1421 0.523066
\(732\) 0 0
\(733\) −8.48528 −0.313411 −0.156706 0.987645i \(-0.550087\pi\)
−0.156706 + 0.987645i \(0.550087\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 6.00000 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) 0 0
\(741\) −33.9411 −1.24686
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 16.9706 0.621753
\(746\) 0 0
\(747\) −1.41421 −0.0517434
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 22.6274 0.823496
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 33.9411 1.23198
\(760\) 0 0
\(761\) 26.8701 0.974039 0.487019 0.873391i \(-0.338084\pi\)
0.487019 + 0.873391i \(0.338084\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 19.7990 0.712120 0.356060 0.934463i \(-0.384120\pi\)
0.356060 + 0.934463i \(0.384120\pi\)
\(774\) 0 0
\(775\) 8.48528 0.304800
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) 0 0
\(783\) 33.9411 1.21296
\(784\) 0 0
\(785\) 48.0000 1.71319
\(786\) 0 0
\(787\) −32.5269 −1.15946 −0.579730 0.814809i \(-0.696842\pi\)
−0.579730 + 0.814809i \(0.696842\pi\)
\(788\) 0 0
\(789\) 5.65685 0.201389
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) −39.5980 −1.40263 −0.701316 0.712850i \(-0.747404\pi\)
−0.701316 + 0.712850i \(0.747404\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) −4.24264 −0.149906
\(802\) 0 0
\(803\) −59.3970 −2.09607
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −40.0000 −1.40807
\(808\) 0 0
\(809\) −32.0000 −1.12506 −0.562530 0.826777i \(-0.690172\pi\)
−0.562530 + 0.826777i \(0.690172\pi\)
\(810\) 0 0
\(811\) 7.07107 0.248299 0.124149 0.992264i \(-0.460380\pi\)
0.124149 + 0.992264i \(0.460380\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 5.65685 0.198151
\(816\) 0 0
\(817\) 42.4264 1.48431
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 56.0000 1.95204 0.976019 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) 0 0
\(825\) −25.4558 −0.886259
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) −25.4558 −0.884118 −0.442059 0.896986i \(-0.645752\pi\)
−0.442059 + 0.896986i \(0.645752\pi\)
\(830\) 0 0
\(831\) 19.7990 0.686819
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −31.1127 −1.07413 −0.537065 0.843541i \(-0.680467\pi\)
−0.537065 + 0.843541i \(0.680467\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −45.2548 −1.55866
\(844\) 0 0
\(845\) −53.7401 −1.84872
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 16.9706 0.581061 0.290531 0.956866i \(-0.406168\pi\)
0.290531 + 0.956866i \(0.406168\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 7.07107 0.241543 0.120772 0.992680i \(-0.461463\pi\)
0.120772 + 0.992680i \(0.461463\pi\)
\(858\) 0 0
\(859\) 15.5563 0.530776 0.265388 0.964142i \(-0.414500\pi\)
0.265388 + 0.964142i \(0.414500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −22.6274 −0.766701
\(872\) 0 0
\(873\) 12.7279 0.430775
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) −4.00000 −0.134917
\(880\) 0 0
\(881\) −38.1838 −1.28644 −0.643222 0.765680i \(-0.722403\pi\)
−0.643222 + 0.765680i \(0.722403\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) −5.65685 −0.190153
\(886\) 0 0
\(887\) 36.7696 1.23460 0.617300 0.786728i \(-0.288226\pi\)
0.617300 + 0.786728i \(0.288226\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 30.0000 1.00504
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 56.5685 1.89088
\(896\) 0 0
\(897\) −32.0000 −1.06845
\(898\) 0 0
\(899\) −16.9706 −0.566000
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) −2.82843 −0.0938130
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −8.48528 −0.280822
\(914\) 0 0
\(915\) −33.9411 −1.12206
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 0 0
\(923\) 67.8823 2.23437
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) 14.1421 0.464489
\(928\) 0 0
\(929\) 32.5269 1.06717 0.533587 0.845745i \(-0.320844\pi\)
0.533587 + 0.845745i \(0.320844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) 21.2132 0.693005 0.346503 0.938049i \(-0.387369\pi\)
0.346503 + 0.938049i \(0.387369\pi\)
\(938\) 0 0
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) 31.1127 1.01424 0.507122 0.861874i \(-0.330709\pi\)
0.507122 + 0.861874i \(0.330709\pi\)
\(942\) 0 0
\(943\) −5.65685 −0.184213
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0000 1.36482 0.682408 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) 0 0
\(951\) −31.1127 −1.00890
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 0 0
\(957\) 50.9117 1.64574
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −45.2548 −1.45680
\(966\) 0 0
\(967\) −60.0000 −1.92947 −0.964735 0.263223i \(-0.915214\pi\)
−0.964735 + 0.263223i \(0.915214\pi\)
\(968\) 0 0
\(969\) 8.48528 0.272587
\(970\) 0 0
\(971\) −32.5269 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 24.0000 0.768615
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −25.4558 −0.813572
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 53.7401 1.71404 0.857022 0.515280i \(-0.172312\pi\)
0.857022 + 0.515280i \(0.172312\pi\)
\(984\) 0 0
\(985\) −50.9117 −1.62218
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 19.7990 0.628302
\(994\) 0 0
\(995\) −72.0000 −2.28255
\(996\) 0 0
\(997\) 8.48528 0.268732 0.134366 0.990932i \(-0.457100\pi\)
0.134366 + 0.990932i \(0.457100\pi\)
\(998\) 0 0
\(999\) −11.3137 −0.357950
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 784.2.a.k.1.2 2
3.2 odd 2 7056.2.a.ct.1.2 2
4.3 odd 2 392.2.a.g.1.1 2
7.2 even 3 784.2.i.n.753.1 4
7.3 odd 6 784.2.i.n.177.2 4
7.4 even 3 784.2.i.n.177.1 4
7.5 odd 6 784.2.i.n.753.2 4
7.6 odd 2 inner 784.2.a.k.1.1 2
8.3 odd 2 3136.2.a.bk.1.2 2
8.5 even 2 3136.2.a.bp.1.1 2
12.11 even 2 3528.2.a.be.1.2 2
20.19 odd 2 9800.2.a.bv.1.2 2
21.20 even 2 7056.2.a.ct.1.1 2
28.3 even 6 392.2.i.h.177.1 4
28.11 odd 6 392.2.i.h.177.2 4
28.19 even 6 392.2.i.h.361.1 4
28.23 odd 6 392.2.i.h.361.2 4
28.27 even 2 392.2.a.g.1.2 yes 2
56.13 odd 2 3136.2.a.bp.1.2 2
56.27 even 2 3136.2.a.bk.1.1 2
84.11 even 6 3528.2.s.bj.3313.1 4
84.23 even 6 3528.2.s.bj.361.1 4
84.47 odd 6 3528.2.s.bj.361.2 4
84.59 odd 6 3528.2.s.bj.3313.2 4
84.83 odd 2 3528.2.a.be.1.1 2
140.139 even 2 9800.2.a.bv.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.2.a.g.1.1 2 4.3 odd 2
392.2.a.g.1.2 yes 2 28.27 even 2
392.2.i.h.177.1 4 28.3 even 6
392.2.i.h.177.2 4 28.11 odd 6
392.2.i.h.361.1 4 28.19 even 6
392.2.i.h.361.2 4 28.23 odd 6
784.2.a.k.1.1 2 7.6 odd 2 inner
784.2.a.k.1.2 2 1.1 even 1 trivial
784.2.i.n.177.1 4 7.4 even 3
784.2.i.n.177.2 4 7.3 odd 6
784.2.i.n.753.1 4 7.2 even 3
784.2.i.n.753.2 4 7.5 odd 6
3136.2.a.bk.1.1 2 56.27 even 2
3136.2.a.bk.1.2 2 8.3 odd 2
3136.2.a.bp.1.1 2 8.5 even 2
3136.2.a.bp.1.2 2 56.13 odd 2
3528.2.a.be.1.1 2 84.83 odd 2
3528.2.a.be.1.2 2 12.11 even 2
3528.2.s.bj.361.1 4 84.23 even 6
3528.2.s.bj.361.2 4 84.47 odd 6
3528.2.s.bj.3313.1 4 84.11 even 6
3528.2.s.bj.3313.2 4 84.59 odd 6
7056.2.a.ct.1.1 2 21.20 even 2
7056.2.a.ct.1.2 2 3.2 odd 2
9800.2.a.bv.1.1 2 140.139 even 2
9800.2.a.bv.1.2 2 20.19 odd 2