Properties

Label 2450.2.c.w.99.1
Level $2450$
Weight $2$
Character 2450.99
Analytic conductor $19.563$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 490)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2450.99
Dual form 2450.2.c.w.99.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.585786i q^{3} -1.00000 q^{4} +0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.585786i q^{3} -1.00000 q^{4} +0.585786 q^{6} +1.00000i q^{8} +2.65685 q^{9} +4.82843 q^{11} -0.585786i q^{12} +0.828427i q^{13} +1.00000 q^{16} -5.41421i q^{17} -2.65685i q^{18} +3.41421 q^{19} -4.82843i q^{22} +6.82843i q^{23} -0.585786 q^{24} +0.828427 q^{26} +3.31371i q^{27} -0.828427 q^{29} -2.82843 q^{31} -1.00000i q^{32} +2.82843i q^{33} -5.41421 q^{34} -2.65685 q^{36} +3.65685i q^{37} -3.41421i q^{38} -0.485281 q^{39} -11.0711 q^{41} +3.17157i q^{43} -4.82843 q^{44} +6.82843 q^{46} -10.8284i q^{47} +0.585786i q^{48} +3.17157 q^{51} -0.828427i q^{52} +10.4853i q^{53} +3.31371 q^{54} +2.00000i q^{57} +0.828427i q^{58} +11.4142 q^{59} +13.3137 q^{61} +2.82843i q^{62} -1.00000 q^{64} +2.82843 q^{66} -9.65685i q^{67} +5.41421i q^{68} -4.00000 q^{69} +12.4853 q^{71} +2.65685i q^{72} -6.58579i q^{73} +3.65685 q^{74} -3.41421 q^{76} +0.485281i q^{78} +1.17157 q^{79} +6.02944 q^{81} +11.0711i q^{82} +6.24264i q^{83} +3.17157 q^{86} -0.485281i q^{87} +4.82843i q^{88} -12.7279 q^{89} -6.82843i q^{92} -1.65685i q^{93} -10.8284 q^{94} +0.585786 q^{96} +16.2426i q^{97} +12.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 8q^{6} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 8q^{6} - 12q^{9} + 8q^{11} + 4q^{16} + 8q^{19} - 8q^{24} - 8q^{26} + 8q^{29} - 16q^{34} + 12q^{36} + 32q^{39} - 16q^{41} - 8q^{44} + 16q^{46} + 24q^{51} - 32q^{54} + 40q^{59} + 8q^{61} - 4q^{64} - 16q^{69} + 16q^{71} - 8q^{74} - 8q^{76} + 16q^{79} + 92q^{81} + 24q^{86} - 32q^{94} + 8q^{96} + 40q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.585786i 0.338204i 0.985599 + 0.169102i \(0.0540867\pi\)
−0.985599 + 0.169102i \(0.945913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.585786 0.239146
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) 2.65685 0.885618
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) − 0.585786i − 0.169102i
\(13\) 0.828427i 0.229764i 0.993379 + 0.114882i \(0.0366490\pi\)
−0.993379 + 0.114882i \(0.963351\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.41421i − 1.31314i −0.754265 0.656570i \(-0.772007\pi\)
0.754265 0.656570i \(-0.227993\pi\)
\(18\) − 2.65685i − 0.626227i
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.82843i − 1.02942i
\(23\) 6.82843i 1.42383i 0.702268 + 0.711913i \(0.252171\pi\)
−0.702268 + 0.711913i \(0.747829\pi\)
\(24\) −0.585786 −0.119573
\(25\) 0 0
\(26\) 0.828427 0.162468
\(27\) 3.31371i 0.637723i
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.82843i 0.492366i
\(34\) −5.41421 −0.928530
\(35\) 0 0
\(36\) −2.65685 −0.442809
\(37\) 3.65685i 0.601183i 0.953753 + 0.300592i \(0.0971841\pi\)
−0.953753 + 0.300592i \(0.902816\pi\)
\(38\) − 3.41421i − 0.553859i
\(39\) −0.485281 −0.0777072
\(40\) 0 0
\(41\) −11.0711 −1.72901 −0.864505 0.502624i \(-0.832368\pi\)
−0.864505 + 0.502624i \(0.832368\pi\)
\(42\) 0 0
\(43\) 3.17157i 0.483660i 0.970319 + 0.241830i \(0.0777477\pi\)
−0.970319 + 0.241830i \(0.922252\pi\)
\(44\) −4.82843 −0.727913
\(45\) 0 0
\(46\) 6.82843 1.00680
\(47\) − 10.8284i − 1.57949i −0.613436 0.789744i \(-0.710213\pi\)
0.613436 0.789744i \(-0.289787\pi\)
\(48\) 0.585786i 0.0845510i
\(49\) 0 0
\(50\) 0 0
\(51\) 3.17157 0.444109
\(52\) − 0.828427i − 0.114882i
\(53\) 10.4853i 1.44026i 0.693837 + 0.720132i \(0.255919\pi\)
−0.693837 + 0.720132i \(0.744081\pi\)
\(54\) 3.31371 0.450939
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000i 0.264906i
\(58\) 0.828427i 0.108778i
\(59\) 11.4142 1.48600 0.743002 0.669289i \(-0.233401\pi\)
0.743002 + 0.669289i \(0.233401\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 2.82843i 0.359211i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.82843 0.348155
\(67\) − 9.65685i − 1.17977i −0.807486 0.589886i \(-0.799173\pi\)
0.807486 0.589886i \(-0.200827\pi\)
\(68\) 5.41421i 0.656570i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 2.65685i 0.313113i
\(73\) − 6.58579i − 0.770808i −0.922748 0.385404i \(-0.874062\pi\)
0.922748 0.385404i \(-0.125938\pi\)
\(74\) 3.65685 0.425101
\(75\) 0 0
\(76\) −3.41421 −0.391637
\(77\) 0 0
\(78\) 0.485281i 0.0549473i
\(79\) 1.17157 0.131812 0.0659061 0.997826i \(-0.479006\pi\)
0.0659061 + 0.997826i \(0.479006\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 11.0711i 1.22259i
\(83\) 6.24264i 0.685219i 0.939478 + 0.342609i \(0.111311\pi\)
−0.939478 + 0.342609i \(0.888689\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.17157 0.341999
\(87\) − 0.485281i − 0.0520276i
\(88\) 4.82843i 0.514712i
\(89\) −12.7279 −1.34916 −0.674579 0.738203i \(-0.735675\pi\)
−0.674579 + 0.738203i \(0.735675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.82843i − 0.711913i
\(93\) − 1.65685i − 0.171808i
\(94\) −10.8284 −1.11687
\(95\) 0 0
\(96\) 0.585786 0.0597866
\(97\) 16.2426i 1.64919i 0.565723 + 0.824595i \(0.308597\pi\)
−0.565723 + 0.824595i \(0.691403\pi\)
\(98\) 0 0
\(99\) 12.8284 1.28931
\(100\) 0 0
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) − 3.17157i − 0.314033i
\(103\) − 9.17157i − 0.903702i −0.892093 0.451851i \(-0.850764\pi\)
0.892093 0.451851i \(-0.149236\pi\)
\(104\) −0.828427 −0.0812340
\(105\) 0 0
\(106\) 10.4853 1.01842
\(107\) − 1.65685i − 0.160174i −0.996788 0.0800871i \(-0.974480\pi\)
0.996788 0.0800871i \(-0.0255198\pi\)
\(108\) − 3.31371i − 0.318862i
\(109\) 14.4853 1.38744 0.693719 0.720246i \(-0.255971\pi\)
0.693719 + 0.720246i \(0.255971\pi\)
\(110\) 0 0
\(111\) −2.14214 −0.203323
\(112\) 0 0
\(113\) − 7.31371i − 0.688016i −0.938967 0.344008i \(-0.888215\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) 0.828427 0.0769175
\(117\) 2.20101i 0.203483i
\(118\) − 11.4142i − 1.05076i
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) − 13.3137i − 1.20537i
\(123\) − 6.48528i − 0.584758i
\(124\) 2.82843 0.254000
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.82843i − 0.250982i −0.992095 0.125491i \(-0.959949\pi\)
0.992095 0.125491i \(-0.0400507\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −1.85786 −0.163576
\(130\) 0 0
\(131\) −2.24264 −0.195940 −0.0979702 0.995189i \(-0.531235\pi\)
−0.0979702 + 0.995189i \(0.531235\pi\)
\(132\) − 2.82843i − 0.246183i
\(133\) 0 0
\(134\) −9.65685 −0.834225
\(135\) 0 0
\(136\) 5.41421 0.464265
\(137\) − 16.0000i − 1.36697i −0.729964 0.683486i \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) 4.00000i 0.340503i
\(139\) 0.100505 0.00852473 0.00426236 0.999991i \(-0.498643\pi\)
0.00426236 + 0.999991i \(0.498643\pi\)
\(140\) 0 0
\(141\) 6.34315 0.534189
\(142\) − 12.4853i − 1.04774i
\(143\) 4.00000i 0.334497i
\(144\) 2.65685 0.221405
\(145\) 0 0
\(146\) −6.58579 −0.545044
\(147\) 0 0
\(148\) − 3.65685i − 0.300592i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 3.41421i 0.276929i
\(153\) − 14.3848i − 1.16294i
\(154\) 0 0
\(155\) 0 0
\(156\) 0.485281 0.0388536
\(157\) 10.4853i 0.836817i 0.908259 + 0.418408i \(0.137412\pi\)
−0.908259 + 0.418408i \(0.862588\pi\)
\(158\) − 1.17157i − 0.0932053i
\(159\) −6.14214 −0.487103
\(160\) 0 0
\(161\) 0 0
\(162\) − 6.02944i − 0.473717i
\(163\) − 8.14214i − 0.637741i −0.947798 0.318871i \(-0.896696\pi\)
0.947798 0.318871i \(-0.103304\pi\)
\(164\) 11.0711 0.864505
\(165\) 0 0
\(166\) 6.24264 0.484523
\(167\) 23.7990i 1.84162i 0.390010 + 0.920811i \(0.372471\pi\)
−0.390010 + 0.920811i \(0.627529\pi\)
\(168\) 0 0
\(169\) 12.3137 0.947208
\(170\) 0 0
\(171\) 9.07107 0.693682
\(172\) − 3.17157i − 0.241830i
\(173\) − 3.17157i − 0.241130i −0.992705 0.120565i \(-0.961529\pi\)
0.992705 0.120565i \(-0.0384707\pi\)
\(174\) −0.485281 −0.0367891
\(175\) 0 0
\(176\) 4.82843 0.363956
\(177\) 6.68629i 0.502572i
\(178\) 12.7279i 0.953998i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −14.4853 −1.07668 −0.538341 0.842727i \(-0.680949\pi\)
−0.538341 + 0.842727i \(0.680949\pi\)
\(182\) 0 0
\(183\) 7.79899i 0.576518i
\(184\) −6.82843 −0.503398
\(185\) 0 0
\(186\) −1.65685 −0.121486
\(187\) − 26.1421i − 1.91170i
\(188\) 10.8284i 0.789744i
\(189\) 0 0
\(190\) 0 0
\(191\) −18.1421 −1.31272 −0.656359 0.754448i \(-0.727904\pi\)
−0.656359 + 0.754448i \(0.727904\pi\)
\(192\) − 0.585786i − 0.0422755i
\(193\) 5.65685i 0.407189i 0.979055 + 0.203595i \(0.0652625\pi\)
−0.979055 + 0.203595i \(0.934738\pi\)
\(194\) 16.2426 1.16615
\(195\) 0 0
\(196\) 0 0
\(197\) 13.7990i 0.983137i 0.870839 + 0.491569i \(0.163576\pi\)
−0.870839 + 0.491569i \(0.836424\pi\)
\(198\) − 12.8284i − 0.911677i
\(199\) −0.485281 −0.0344007 −0.0172003 0.999852i \(-0.505475\pi\)
−0.0172003 + 0.999852i \(0.505475\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) − 9.31371i − 0.655310i
\(203\) 0 0
\(204\) −3.17157 −0.222055
\(205\) 0 0
\(206\) −9.17157 −0.639014
\(207\) 18.1421i 1.26097i
\(208\) 0.828427i 0.0574411i
\(209\) 16.4853 1.14031
\(210\) 0 0
\(211\) −26.6274 −1.83311 −0.916553 0.399912i \(-0.869041\pi\)
−0.916553 + 0.399912i \(0.869041\pi\)
\(212\) − 10.4853i − 0.720132i
\(213\) 7.31371i 0.501127i
\(214\) −1.65685 −0.113260
\(215\) 0 0
\(216\) −3.31371 −0.225469
\(217\) 0 0
\(218\) − 14.4853i − 0.981067i
\(219\) 3.85786 0.260690
\(220\) 0 0
\(221\) 4.48528 0.301713
\(222\) 2.14214i 0.143771i
\(223\) 15.3137i 1.02548i 0.858543 + 0.512741i \(0.171370\pi\)
−0.858543 + 0.512741i \(0.828630\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.31371 −0.486501
\(227\) − 9.75736i − 0.647619i −0.946122 0.323809i \(-0.895036\pi\)
0.946122 0.323809i \(-0.104964\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 12.1421 0.802375 0.401187 0.915996i \(-0.368598\pi\)
0.401187 + 0.915996i \(0.368598\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 0.828427i − 0.0543889i
\(233\) − 0.686292i − 0.0449605i −0.999747 0.0224802i \(-0.992844\pi\)
0.999747 0.0224802i \(-0.00715628\pi\)
\(234\) 2.20101 0.143885
\(235\) 0 0
\(236\) −11.4142 −0.743002
\(237\) 0.686292i 0.0445794i
\(238\) 0 0
\(239\) 9.65685 0.624650 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(240\) 0 0
\(241\) 10.5858 0.681890 0.340945 0.940083i \(-0.389253\pi\)
0.340945 + 0.940083i \(0.389253\pi\)
\(242\) − 12.3137i − 0.791555i
\(243\) 13.4731i 0.864299i
\(244\) −13.3137 −0.852323
\(245\) 0 0
\(246\) −6.48528 −0.413486
\(247\) 2.82843i 0.179969i
\(248\) − 2.82843i − 0.179605i
\(249\) −3.65685 −0.231744
\(250\) 0 0
\(251\) −3.41421 −0.215503 −0.107752 0.994178i \(-0.534365\pi\)
−0.107752 + 0.994178i \(0.534365\pi\)
\(252\) 0 0
\(253\) 32.9706i 2.07284i
\(254\) −2.82843 −0.177471
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 9.89949i − 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 1.85786i 0.115666i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.20101 −0.136239
\(262\) 2.24264i 0.138551i
\(263\) − 28.0000i − 1.72655i −0.504730 0.863277i \(-0.668408\pi\)
0.504730 0.863277i \(-0.331592\pi\)
\(264\) −2.82843 −0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) − 7.45584i − 0.456290i
\(268\) 9.65685i 0.589886i
\(269\) 1.51472 0.0923540 0.0461770 0.998933i \(-0.485296\pi\)
0.0461770 + 0.998933i \(0.485296\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) − 5.41421i − 0.328285i
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 20.1421i 1.21022i 0.796140 + 0.605112i \(0.206872\pi\)
−0.796140 + 0.605112i \(0.793128\pi\)
\(278\) − 0.100505i − 0.00602789i
\(279\) −7.51472 −0.449894
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) − 6.34315i − 0.377729i
\(283\) 6.24264i 0.371086i 0.982636 + 0.185543i \(0.0594045\pi\)
−0.982636 + 0.185543i \(0.940596\pi\)
\(284\) −12.4853 −0.740865
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) − 2.65685i − 0.156557i
\(289\) −12.3137 −0.724336
\(290\) 0 0
\(291\) −9.51472 −0.557763
\(292\) 6.58579i 0.385404i
\(293\) − 19.6569i − 1.14837i −0.818727 0.574183i \(-0.805320\pi\)
0.818727 0.574183i \(-0.194680\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.65685 −0.212550
\(297\) 16.0000i 0.928414i
\(298\) − 6.00000i − 0.347571i
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) − 11.3137i − 0.651031i
\(303\) 5.45584i 0.313430i
\(304\) 3.41421 0.195819
\(305\) 0 0
\(306\) −14.3848 −0.822323
\(307\) − 29.0711i − 1.65917i −0.558378 0.829587i \(-0.688576\pi\)
0.558378 0.829587i \(-0.311424\pi\)
\(308\) 0 0
\(309\) 5.37258 0.305636
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) − 0.485281i − 0.0274736i
\(313\) − 22.3848i − 1.26526i −0.774453 0.632631i \(-0.781975\pi\)
0.774453 0.632631i \(-0.218025\pi\)
\(314\) 10.4853 0.591719
\(315\) 0 0
\(316\) −1.17157 −0.0659061
\(317\) − 6.48528i − 0.364250i −0.983275 0.182125i \(-0.941702\pi\)
0.983275 0.182125i \(-0.0582975\pi\)
\(318\) 6.14214i 0.344434i
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 0.970563 0.0541715
\(322\) 0 0
\(323\) − 18.4853i − 1.02855i
\(324\) −6.02944 −0.334969
\(325\) 0 0
\(326\) −8.14214 −0.450951
\(327\) 8.48528i 0.469237i
\(328\) − 11.0711i − 0.611297i
\(329\) 0 0
\(330\) 0 0
\(331\) −5.79899 −0.318741 −0.159371 0.987219i \(-0.550946\pi\)
−0.159371 + 0.987219i \(0.550946\pi\)
\(332\) − 6.24264i − 0.342609i
\(333\) 9.71573i 0.532419i
\(334\) 23.7990 1.30222
\(335\) 0 0
\(336\) 0 0
\(337\) 6.00000i 0.326841i 0.986557 + 0.163420i \(0.0522527\pi\)
−0.986557 + 0.163420i \(0.947747\pi\)
\(338\) − 12.3137i − 0.669777i
\(339\) 4.28427 0.232690
\(340\) 0 0
\(341\) −13.6569 −0.739560
\(342\) − 9.07107i − 0.490507i
\(343\) 0 0
\(344\) −3.17157 −0.171000
\(345\) 0 0
\(346\) −3.17157 −0.170505
\(347\) − 8.82843i − 0.473935i −0.971518 0.236967i \(-0.923847\pi\)
0.971518 0.236967i \(-0.0761535\pi\)
\(348\) 0.485281i 0.0260138i
\(349\) −14.4853 −0.775379 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(350\) 0 0
\(351\) −2.74517 −0.146526
\(352\) − 4.82843i − 0.257356i
\(353\) 34.3848i 1.83012i 0.403321 + 0.915058i \(0.367856\pi\)
−0.403321 + 0.915058i \(0.632144\pi\)
\(354\) 6.68629 0.355372
\(355\) 0 0
\(356\) 12.7279 0.674579
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −7.34315 −0.386481
\(362\) 14.4853i 0.761329i
\(363\) 7.21320i 0.378595i
\(364\) 0 0
\(365\) 0 0
\(366\) 7.79899 0.407660
\(367\) 8.97056i 0.468260i 0.972205 + 0.234130i \(0.0752241\pi\)
−0.972205 + 0.234130i \(0.924776\pi\)
\(368\) 6.82843i 0.355956i
\(369\) −29.4142 −1.53124
\(370\) 0 0
\(371\) 0 0
\(372\) 1.65685i 0.0859039i
\(373\) 13.5147i 0.699766i 0.936793 + 0.349883i \(0.113779\pi\)
−0.936793 + 0.349883i \(0.886221\pi\)
\(374\) −26.1421 −1.35178
\(375\) 0 0
\(376\) 10.8284 0.558433
\(377\) − 0.686292i − 0.0353458i
\(378\) 0 0
\(379\) −17.5147 −0.899671 −0.449835 0.893112i \(-0.648517\pi\)
−0.449835 + 0.893112i \(0.648517\pi\)
\(380\) 0 0
\(381\) 1.65685 0.0848832
\(382\) 18.1421i 0.928232i
\(383\) − 15.5147i − 0.792765i −0.918085 0.396383i \(-0.870265\pi\)
0.918085 0.396383i \(-0.129735\pi\)
\(384\) −0.585786 −0.0298933
\(385\) 0 0
\(386\) 5.65685 0.287926
\(387\) 8.42641i 0.428338i
\(388\) − 16.2426i − 0.824595i
\(389\) 0.142136 0.00720656 0.00360328 0.999994i \(-0.498853\pi\)
0.00360328 + 0.999994i \(0.498853\pi\)
\(390\) 0 0
\(391\) 36.9706 1.86968
\(392\) 0 0
\(393\) − 1.31371i − 0.0662678i
\(394\) 13.7990 0.695183
\(395\) 0 0
\(396\) −12.8284 −0.644653
\(397\) 5.79899i 0.291043i 0.989355 + 0.145521i \(0.0464860\pi\)
−0.989355 + 0.145521i \(0.953514\pi\)
\(398\) 0.485281i 0.0243250i
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) − 5.65685i − 0.282138i
\(403\) − 2.34315i − 0.116720i
\(404\) −9.31371 −0.463374
\(405\) 0 0
\(406\) 0 0
\(407\) 17.6569i 0.875218i
\(408\) 3.17157i 0.157016i
\(409\) −13.4142 −0.663290 −0.331645 0.943404i \(-0.607604\pi\)
−0.331645 + 0.943404i \(0.607604\pi\)
\(410\) 0 0
\(411\) 9.37258 0.462315
\(412\) 9.17157i 0.451851i
\(413\) 0 0
\(414\) 18.1421 0.891637
\(415\) 0 0
\(416\) 0.828427 0.0406170
\(417\) 0.0588745i 0.00288310i
\(418\) − 16.4853i − 0.806321i
\(419\) 32.8701 1.60581 0.802904 0.596109i \(-0.203287\pi\)
0.802904 + 0.596109i \(0.203287\pi\)
\(420\) 0 0
\(421\) −5.31371 −0.258974 −0.129487 0.991581i \(-0.541333\pi\)
−0.129487 + 0.991581i \(0.541333\pi\)
\(422\) 26.6274i 1.29620i
\(423\) − 28.7696i − 1.39882i
\(424\) −10.4853 −0.509210
\(425\) 0 0
\(426\) 7.31371 0.354350
\(427\) 0 0
\(428\) 1.65685i 0.0800871i
\(429\) −2.34315 −0.113128
\(430\) 0 0
\(431\) −33.6569 −1.62119 −0.810597 0.585605i \(-0.800857\pi\)
−0.810597 + 0.585605i \(0.800857\pi\)
\(432\) 3.31371i 0.159431i
\(433\) 13.4142i 0.644646i 0.946630 + 0.322323i \(0.104464\pi\)
−0.946630 + 0.322323i \(0.895536\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.4853 −0.693719
\(437\) 23.3137i 1.11525i
\(438\) − 3.85786i − 0.184336i
\(439\) 8.97056 0.428142 0.214071 0.976818i \(-0.431328\pi\)
0.214071 + 0.976818i \(0.431328\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 4.48528i − 0.213343i
\(443\) − 36.9706i − 1.75652i −0.478179 0.878262i \(-0.658703\pi\)
0.478179 0.878262i \(-0.341297\pi\)
\(444\) 2.14214 0.101661
\(445\) 0 0
\(446\) 15.3137 0.725125
\(447\) 3.51472i 0.166240i
\(448\) 0 0
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) −53.4558 −2.51714
\(452\) 7.31371i 0.344008i
\(453\) 6.62742i 0.311383i
\(454\) −9.75736 −0.457936
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 10.3431i 0.483832i 0.970297 + 0.241916i \(0.0777758\pi\)
−0.970297 + 0.241916i \(0.922224\pi\)
\(458\) − 12.1421i − 0.567365i
\(459\) 17.9411 0.837420
\(460\) 0 0
\(461\) 7.17157 0.334013 0.167007 0.985956i \(-0.446590\pi\)
0.167007 + 0.985956i \(0.446590\pi\)
\(462\) 0 0
\(463\) − 16.9706i − 0.788689i −0.918963 0.394344i \(-0.870972\pi\)
0.918963 0.394344i \(-0.129028\pi\)
\(464\) −0.828427 −0.0384588
\(465\) 0 0
\(466\) −0.686292 −0.0317918
\(467\) − 3.89949i − 0.180447i −0.995922 0.0902236i \(-0.971242\pi\)
0.995922 0.0902236i \(-0.0287582\pi\)
\(468\) − 2.20101i − 0.101742i
\(469\) 0 0
\(470\) 0 0
\(471\) −6.14214 −0.283015
\(472\) 11.4142i 0.525382i
\(473\) 15.3137i 0.704125i
\(474\) 0.686292 0.0315224
\(475\) 0 0
\(476\) 0 0
\(477\) 27.8579i 1.27552i
\(478\) − 9.65685i − 0.441694i
\(479\) −22.8284 −1.04306 −0.521529 0.853234i \(-0.674638\pi\)
−0.521529 + 0.853234i \(0.674638\pi\)
\(480\) 0 0
\(481\) −3.02944 −0.138130
\(482\) − 10.5858i − 0.482169i
\(483\) 0 0
\(484\) −12.3137 −0.559714
\(485\) 0 0
\(486\) 13.4731 0.611152
\(487\) − 7.79899i − 0.353406i −0.984264 0.176703i \(-0.943457\pi\)
0.984264 0.176703i \(-0.0565432\pi\)
\(488\) 13.3137i 0.602683i
\(489\) 4.76955 0.215687
\(490\) 0 0
\(491\) −24.2843 −1.09593 −0.547967 0.836500i \(-0.684598\pi\)
−0.547967 + 0.836500i \(0.684598\pi\)
\(492\) 6.48528i 0.292379i
\(493\) 4.48528i 0.202007i
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) −2.82843 −0.127000
\(497\) 0 0
\(498\) 3.65685i 0.163868i
\(499\) −41.6569 −1.86482 −0.932408 0.361406i \(-0.882297\pi\)
−0.932408 + 0.361406i \(0.882297\pi\)
\(500\) 0 0
\(501\) −13.9411 −0.622844
\(502\) 3.41421i 0.152384i
\(503\) 6.34315i 0.282827i 0.989951 + 0.141413i \(0.0451647\pi\)
−0.989951 + 0.141413i \(0.954835\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 32.9706 1.46572
\(507\) 7.21320i 0.320350i
\(508\) 2.82843i 0.125491i
\(509\) −33.7990 −1.49811 −0.749057 0.662506i \(-0.769493\pi\)
−0.749057 + 0.662506i \(0.769493\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 11.3137i 0.499512i
\(514\) −9.89949 −0.436648
\(515\) 0 0
\(516\) 1.85786 0.0817879
\(517\) − 52.2843i − 2.29946i
\(518\) 0 0
\(519\) 1.85786 0.0815512
\(520\) 0 0
\(521\) −4.92893 −0.215940 −0.107970 0.994154i \(-0.534435\pi\)
−0.107970 + 0.994154i \(0.534435\pi\)
\(522\) 2.20101i 0.0963356i
\(523\) 4.10051i 0.179303i 0.995973 + 0.0896513i \(0.0285753\pi\)
−0.995973 + 0.0896513i \(0.971425\pi\)
\(524\) 2.24264 0.0979702
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 15.3137i 0.667076i
\(528\) 2.82843i 0.123091i
\(529\) −23.6274 −1.02728
\(530\) 0 0
\(531\) 30.3259 1.31603
\(532\) 0 0
\(533\) − 9.17157i − 0.397265i
\(534\) −7.45584 −0.322646
\(535\) 0 0
\(536\) 9.65685 0.417113
\(537\) − 2.34315i − 0.101114i
\(538\) − 1.51472i − 0.0653042i
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 12.0000i 0.515444i
\(543\) − 8.48528i − 0.364138i
\(544\) −5.41421 −0.232132
\(545\) 0 0
\(546\) 0 0
\(547\) 6.48528i 0.277291i 0.990342 + 0.138645i \(0.0442748\pi\)
−0.990342 + 0.138645i \(0.955725\pi\)
\(548\) 16.0000i 0.683486i
\(549\) 35.3726 1.50967
\(550\) 0 0
\(551\) −2.82843 −0.120495
\(552\) − 4.00000i − 0.170251i
\(553\) 0 0
\(554\) 20.1421 0.855757
\(555\) 0 0
\(556\) −0.100505 −0.00426236
\(557\) 20.8284i 0.882529i 0.897377 + 0.441264i \(0.145470\pi\)
−0.897377 + 0.441264i \(0.854530\pi\)
\(558\) 7.51472i 0.318123i
\(559\) −2.62742 −0.111128
\(560\) 0 0
\(561\) 15.3137 0.646545
\(562\) − 8.00000i − 0.337460i
\(563\) − 39.4142i − 1.66111i −0.556936 0.830556i \(-0.688023\pi\)
0.556936 0.830556i \(-0.311977\pi\)
\(564\) −6.34315 −0.267095
\(565\) 0 0
\(566\) 6.24264 0.262398
\(567\) 0 0
\(568\) 12.4853i 0.523871i
\(569\) 6.68629 0.280304 0.140152 0.990130i \(-0.455241\pi\)
0.140152 + 0.990130i \(0.455241\pi\)
\(570\) 0 0
\(571\) −41.7990 −1.74923 −0.874617 0.484815i \(-0.838887\pi\)
−0.874617 + 0.484815i \(0.838887\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) − 10.6274i − 0.443967i
\(574\) 0 0
\(575\) 0 0
\(576\) −2.65685 −0.110702
\(577\) − 25.8995i − 1.07821i −0.842239 0.539105i \(-0.818763\pi\)
0.842239 0.539105i \(-0.181237\pi\)
\(578\) 12.3137i 0.512183i
\(579\) −3.31371 −0.137713
\(580\) 0 0
\(581\) 0 0
\(582\) 9.51472i 0.394398i
\(583\) 50.6274i 2.09677i
\(584\) 6.58579 0.272522
\(585\) 0 0
\(586\) −19.6569 −0.812017
\(587\) − 2.92893i − 0.120890i −0.998172 0.0604450i \(-0.980748\pi\)
0.998172 0.0604450i \(-0.0192520\pi\)
\(588\) 0 0
\(589\) −9.65685 −0.397904
\(590\) 0 0
\(591\) −8.08326 −0.332501
\(592\) 3.65685i 0.150296i
\(593\) 28.7279i 1.17971i 0.807508 + 0.589857i \(0.200816\pi\)
−0.807508 + 0.589857i \(0.799184\pi\)
\(594\) 16.0000 0.656488
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) − 0.284271i − 0.0116344i
\(598\) 5.65685i 0.231326i
\(599\) 5.17157 0.211305 0.105652 0.994403i \(-0.466307\pi\)
0.105652 + 0.994403i \(0.466307\pi\)
\(600\) 0 0
\(601\) −9.41421 −0.384014 −0.192007 0.981394i \(-0.561500\pi\)
−0.192007 + 0.981394i \(0.561500\pi\)
\(602\) 0 0
\(603\) − 25.6569i − 1.04483i
\(604\) −11.3137 −0.460348
\(605\) 0 0
\(606\) 5.45584 0.221629
\(607\) 40.2843i 1.63509i 0.575866 + 0.817544i \(0.304665\pi\)
−0.575866 + 0.817544i \(0.695335\pi\)
\(608\) − 3.41421i − 0.138465i
\(609\) 0 0
\(610\) 0 0
\(611\) 8.97056 0.362910
\(612\) 14.3848i 0.581470i
\(613\) 23.6569i 0.955491i 0.878498 + 0.477746i \(0.158546\pi\)
−0.878498 + 0.477746i \(0.841454\pi\)
\(614\) −29.0711 −1.17321
\(615\) 0 0
\(616\) 0 0
\(617\) − 10.6863i − 0.430214i −0.976590 0.215107i \(-0.930990\pi\)
0.976590 0.215107i \(-0.0690100\pi\)
\(618\) − 5.37258i − 0.216117i
\(619\) 14.9289 0.600044 0.300022 0.953932i \(-0.403006\pi\)
0.300022 + 0.953932i \(0.403006\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 4.00000i 0.160385i
\(623\) 0 0
\(624\) −0.485281 −0.0194268
\(625\) 0 0
\(626\) −22.3848 −0.894676
\(627\) 9.65685i 0.385658i
\(628\) − 10.4853i − 0.418408i
\(629\) 19.7990 0.789437
\(630\) 0 0
\(631\) 4.48528 0.178556 0.0892781 0.996007i \(-0.471544\pi\)
0.0892781 + 0.996007i \(0.471544\pi\)
\(632\) 1.17157i 0.0466027i
\(633\) − 15.5980i − 0.619964i
\(634\) −6.48528 −0.257563
\(635\) 0 0
\(636\) 6.14214 0.243552
\(637\) 0 0
\(638\) 4.00000i 0.158362i
\(639\) 33.1716 1.31225
\(640\) 0 0
\(641\) 20.6274 0.814734 0.407367 0.913265i \(-0.366447\pi\)
0.407367 + 0.913265i \(0.366447\pi\)
\(642\) − 0.970563i − 0.0383051i
\(643\) 47.2132i 1.86191i 0.365138 + 0.930953i \(0.381022\pi\)
−0.365138 + 0.930953i \(0.618978\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −18.4853 −0.727294
\(647\) − 39.1127i − 1.53768i −0.639442 0.768839i \(-0.720835\pi\)
0.639442 0.768839i \(-0.279165\pi\)
\(648\) 6.02944i 0.236859i
\(649\) 55.1127 2.16336
\(650\) 0 0
\(651\) 0 0
\(652\) 8.14214i 0.318871i
\(653\) − 15.6569i − 0.612700i −0.951919 0.306350i \(-0.900892\pi\)
0.951919 0.306350i \(-0.0991078\pi\)
\(654\) 8.48528 0.331801
\(655\) 0 0
\(656\) −11.0711 −0.432253
\(657\) − 17.4975i − 0.682642i
\(658\) 0 0
\(659\) 32.8284 1.27881 0.639407 0.768868i \(-0.279180\pi\)
0.639407 + 0.768868i \(0.279180\pi\)
\(660\) 0 0
\(661\) −18.2843 −0.711176 −0.355588 0.934643i \(-0.615719\pi\)
−0.355588 + 0.934643i \(0.615719\pi\)
\(662\) 5.79899i 0.225384i
\(663\) 2.62742i 0.102040i
\(664\) −6.24264 −0.242261
\(665\) 0 0
\(666\) 9.71573 0.376477
\(667\) − 5.65685i − 0.219034i
\(668\) − 23.7990i − 0.920811i
\(669\) −8.97056 −0.346822
\(670\) 0 0
\(671\) 64.2843 2.48167
\(672\) 0 0
\(673\) 48.0000i 1.85026i 0.379646 + 0.925132i \(0.376046\pi\)
−0.379646 + 0.925132i \(0.623954\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −12.3137 −0.473604
\(677\) 11.4558i 0.440284i 0.975468 + 0.220142i \(0.0706521\pi\)
−0.975468 + 0.220142i \(0.929348\pi\)
\(678\) − 4.28427i − 0.164536i
\(679\) 0 0
\(680\) 0 0
\(681\) 5.71573 0.219027
\(682\) 13.6569i 0.522948i
\(683\) − 22.3431i − 0.854937i −0.904030 0.427468i \(-0.859406\pi\)
0.904030 0.427468i \(-0.140594\pi\)
\(684\) −9.07107 −0.346841
\(685\) 0 0
\(686\) 0 0
\(687\) 7.11270i 0.271366i
\(688\) 3.17157i 0.120915i
\(689\) −8.68629 −0.330921
\(690\) 0 0
\(691\) −10.2426 −0.389648 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(692\) 3.17157i 0.120565i
\(693\) 0 0
\(694\) −8.82843 −0.335123
\(695\) 0 0
\(696\) 0.485281 0.0183945
\(697\) 59.9411i 2.27043i
\(698\) 14.4853i 0.548276i
\(699\) 0.402020 0.0152058
\(700\) 0 0
\(701\) −14.4853 −0.547102 −0.273551 0.961858i \(-0.588198\pi\)
−0.273551 + 0.961858i \(0.588198\pi\)
\(702\) 2.74517i 0.103610i
\(703\) 12.4853i 0.470891i
\(704\) −4.82843 −0.181978
\(705\) 0 0
\(706\) 34.3848 1.29409
\(707\) 0 0
\(708\) − 6.68629i − 0.251286i
\(709\) 17.1127 0.642681 0.321340 0.946964i \(-0.395867\pi\)
0.321340 + 0.946964i \(0.395867\pi\)
\(710\) 0 0
\(711\) 3.11270 0.116735
\(712\) − 12.7279i − 0.476999i
\(713\) − 19.3137i − 0.723304i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 5.65685i 0.211259i
\(718\) − 28.2843i − 1.05556i
\(719\) −9.45584 −0.352643 −0.176322 0.984333i \(-0.556420\pi\)
−0.176322 + 0.984333i \(0.556420\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.34315i 0.273284i
\(723\) 6.20101i 0.230618i
\(724\) 14.4853 0.538341
\(725\) 0 0
\(726\) 7.21320 0.267707
\(727\) 20.4853i 0.759757i 0.925036 + 0.379879i \(0.124034\pi\)
−0.925036 + 0.379879i \(0.875966\pi\)
\(728\) 0 0
\(729\) 10.1960 0.377628
\(730\) 0 0
\(731\) 17.1716 0.635114
\(732\) − 7.79899i − 0.288259i
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 8.97056 0.331110
\(735\) 0 0
\(736\) 6.82843 0.251699
\(737\) − 46.6274i − 1.71754i
\(738\) 29.4142i 1.08275i
\(739\) −8.82843 −0.324759 −0.162379 0.986728i \(-0.551917\pi\)
−0.162379 + 0.986728i \(0.551917\pi\)
\(740\) 0 0
\(741\) −1.65685 −0.0608661
\(742\) 0 0
\(743\) − 12.2010i − 0.447612i −0.974634 0.223806i \(-0.928152\pi\)
0.974634 0.223806i \(-0.0718481\pi\)
\(744\) 1.65685 0.0607432
\(745\) 0 0
\(746\) 13.5147 0.494809
\(747\) 16.5858i 0.606842i
\(748\) 26.1421i 0.955851i
\(749\) 0 0
\(750\) 0 0
\(751\) −16.6863 −0.608891 −0.304446 0.952530i \(-0.598471\pi\)
−0.304446 + 0.952530i \(0.598471\pi\)
\(752\) − 10.8284i − 0.394872i
\(753\) − 2.00000i − 0.0728841i
\(754\) −0.686292 −0.0249933
\(755\) 0 0
\(756\) 0 0
\(757\) − 7.65685i − 0.278293i −0.990272 0.139147i \(-0.955564\pi\)
0.990272 0.139147i \(-0.0444359\pi\)
\(758\) 17.5147i 0.636163i
\(759\) −19.3137 −0.701043
\(760\) 0 0
\(761\) 14.3848 0.521448 0.260724 0.965413i \(-0.416039\pi\)
0.260724 + 0.965413i \(0.416039\pi\)
\(762\) − 1.65685i − 0.0600215i
\(763\) 0 0
\(764\) 18.1421 0.656359
\(765\) 0 0
\(766\) −15.5147 −0.560570
\(767\) 9.45584i 0.341431i
\(768\) 0.585786i 0.0211377i
\(769\) −11.5563 −0.416733 −0.208366 0.978051i \(-0.566815\pi\)
−0.208366 + 0.978051i \(0.566815\pi\)
\(770\) 0 0
\(771\) 5.79899 0.208846
\(772\) − 5.65685i − 0.203595i
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 8.42641 0.302881
\(775\) 0 0
\(776\) −16.2426 −0.583077
\(777\) 0 0
\(778\) − 0.142136i − 0.00509581i
\(779\) −37.7990 −1.35429
\(780\) 0 0
\(781\) 60.2843 2.15714
\(782\) − 36.9706i − 1.32206i
\(783\) − 2.74517i − 0.0981042i
\(784\) 0 0
\(785\) 0 0
\(786\) −1.31371 −0.0468584
\(787\) 26.7279i 0.952748i 0.879243 + 0.476374i \(0.158049\pi\)
−0.879243 + 0.476374i \(0.841951\pi\)
\(788\) − 13.7990i − 0.491569i
\(789\) 16.4020 0.583927
\(790\) 0 0
\(791\) 0 0
\(792\) 12.8284i 0.455838i
\(793\) 11.0294i 0.391667i
\(794\) 5.79899 0.205798
\(795\) 0 0
\(796\) 0.485281 0.0172003
\(797\) 2.20101i 0.0779638i 0.999240 + 0.0389819i \(0.0124115\pi\)
−0.999240 + 0.0389819i \(0.987589\pi\)
\(798\) 0 0
\(799\) −58.6274 −2.07409
\(800\) 0 0
\(801\) −33.8162 −1.19484
\(802\) 6.00000i 0.211867i
\(803\) − 31.7990i − 1.12216i
\(804\) −5.65685 −0.199502
\(805\) 0 0
\(806\) −2.34315 −0.0825338
\(807\) 0.887302i 0.0312345i
\(808\) 9.31371i 0.327655i
\(809\) −36.9706 −1.29982 −0.649908 0.760013i \(-0.725193\pi\)
−0.649908 + 0.760013i \(0.725193\pi\)
\(810\) 0 0
\(811\) −35.4142 −1.24356 −0.621781 0.783191i \(-0.713590\pi\)
−0.621781 + 0.783191i \(0.713590\pi\)
\(812\) 0 0
\(813\) − 7.02944i − 0.246533i
\(814\) 17.6569 0.618872
\(815\) 0 0
\(816\) 3.17157 0.111027
\(817\) 10.8284i 0.378839i
\(818\) 13.4142i 0.469017i
\(819\) 0 0
\(820\) 0 0
\(821\) −5.31371 −0.185450 −0.0927249 0.995692i \(-0.529558\pi\)
−0.0927249 + 0.995692i \(0.529558\pi\)
\(822\) − 9.37258i − 0.326906i
\(823\) 36.2843i 1.26479i 0.774646 + 0.632395i \(0.217928\pi\)
−0.774646 + 0.632395i \(0.782072\pi\)
\(824\) 9.17157 0.319507
\(825\) 0 0
\(826\) 0 0
\(827\) − 50.6274i − 1.76049i −0.474522 0.880244i \(-0.657379\pi\)
0.474522 0.880244i \(-0.342621\pi\)
\(828\) − 18.1421i − 0.630483i
\(829\) −38.9706 −1.35350 −0.676752 0.736211i \(-0.736613\pi\)
−0.676752 + 0.736211i \(0.736613\pi\)
\(830\) 0 0
\(831\) −11.7990 −0.409302
\(832\) − 0.828427i − 0.0287205i
\(833\) 0 0
\(834\) 0.0588745 0.00203866
\(835\) 0 0
\(836\) −16.4853 −0.570155
\(837\) − 9.37258i − 0.323964i
\(838\) − 32.8701i − 1.13548i
\(839\) −13.8579 −0.478427 −0.239213 0.970967i \(-0.576890\pi\)
−0.239213 + 0.970967i \(0.576890\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 5.31371i 0.183122i
\(843\) 4.68629i 0.161404i
\(844\) 26.6274 0.916553
\(845\) 0 0
\(846\) −28.7696 −0.989118
\(847\) 0 0
\(848\) 10.4853i 0.360066i
\(849\) −3.65685 −0.125503
\(850\) 0 0
\(851\) −24.9706 −0.855980
\(852\) − 7.31371i − 0.250564i
\(853\) − 48.8284i − 1.67185i −0.548841 0.835927i \(-0.684931\pi\)
0.548841 0.835927i \(-0.315069\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.65685 0.0566301
\(857\) 19.0711i 0.651455i 0.945464 + 0.325728i \(0.105609\pi\)
−0.945464 + 0.325728i \(0.894391\pi\)
\(858\) 2.34315i 0.0799937i
\(859\) 35.2132 1.20146 0.600729 0.799452i \(-0.294877\pi\)
0.600729 + 0.799452i \(0.294877\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 33.6569i 1.14636i
\(863\) − 28.9706i − 0.986169i −0.869981 0.493085i \(-0.835869\pi\)
0.869981 0.493085i \(-0.164131\pi\)
\(864\) 3.31371 0.112735
\(865\) 0 0
\(866\) 13.4142 0.455834
\(867\) − 7.21320i − 0.244973i
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 14.4853i 0.490534i
\(873\) 43.1543i 1.46055i
\(874\) 23.3137 0.788598
\(875\) 0 0
\(876\) −3.85786 −0.130345
\(877\) 26.2843i 0.887557i 0.896137 + 0.443778i \(0.146362\pi\)
−0.896137 + 0.443778i \(0.853638\pi\)
\(878\) − 8.97056i − 0.302742i
\(879\) 11.5147 0.388382
\(880\) 0 0
\(881\) −34.3848 −1.15845 −0.579226 0.815167i \(-0.696645\pi\)
−0.579226 + 0.815167i \(0.696645\pi\)
\(882\) 0 0
\(883\) 30.3431i 1.02113i 0.859840 + 0.510564i \(0.170563\pi\)
−0.859840 + 0.510564i \(0.829437\pi\)
\(884\) −4.48528 −0.150856
\(885\) 0 0
\(886\) −36.9706 −1.24205
\(887\) − 7.11270i − 0.238821i −0.992845 0.119411i \(-0.961900\pi\)
0.992845 0.119411i \(-0.0381005\pi\)
\(888\) − 2.14214i − 0.0718854i
\(889\) 0 0
\(890\) 0 0
\(891\) 29.1127 0.975312
\(892\) − 15.3137i − 0.512741i
\(893\) − 36.9706i − 1.23717i
\(894\) 3.51472 0.117550
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.31371i − 0.110642i
\(898\) 28.6274i 0.955309i
\(899\) 2.34315 0.0781483
\(900\) 0 0
\(901\) 56.7696 1.89127
\(902\) 53.4558i 1.77988i
\(903\) 0 0
\(904\) 7.31371 0.243250
\(905\) 0 0
\(906\) 6.62742 0.220181
\(907\) − 56.2843i − 1.86889i −0.356109 0.934444i \(-0.615897\pi\)
0.356109 0.934444i \(-0.384103\pi\)
\(908\) 9.75736i 0.323809i
\(909\) 24.7452 0.820745
\(910\) 0 0
\(911\) −20.2843 −0.672048 −0.336024 0.941853i \(-0.609082\pi\)
−0.336024 + 0.941853i \(0.609082\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) 30.1421i 0.997559i
\(914\) 10.3431 0.342121
\(915\) 0 0
\(916\) −12.1421 −0.401187
\(917\) 0 0
\(918\) − 17.9411i − 0.592145i
\(919\) −32.4853 −1.07159 −0.535795 0.844348i \(-0.679988\pi\)
−0.535795 + 0.844348i \(0.679988\pi\)
\(920\) 0 0
\(921\) 17.0294 0.561139
\(922\) − 7.17157i − 0.236183i
\(923\) 10.3431i 0.340449i
\(924\) 0 0
\(925\) 0 0
\(926\) −16.9706 −0.557687
\(927\) − 24.3675i − 0.800335i
\(928\) 0.828427i 0.0271945i
\(929\) 25.2132 0.827218 0.413609 0.910455i \(-0.364268\pi\)
0.413609 + 0.910455i \(0.364268\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.686292i 0.0224802i
\(933\) − 2.34315i − 0.0767111i
\(934\) −3.89949 −0.127595
\(935\) 0 0
\(936\) −2.20101 −0.0719423
\(937\) − 11.7574i − 0.384096i −0.981386 0.192048i \(-0.938487\pi\)
0.981386 0.192048i \(-0.0615130\pi\)
\(938\) 0 0
\(939\) 13.1127 0.427917
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 6.14214i 0.200122i
\(943\) − 75.5980i − 2.46181i
\(944\) 11.4142 0.371501
\(945\) 0 0
\(946\) 15.3137 0.497892
\(947\) 0.828427i 0.0269203i 0.999909 + 0.0134601i \(0.00428462\pi\)
−0.999909 + 0.0134601i \(0.995715\pi\)
\(948\) − 0.686292i − 0.0222897i
\(949\) 5.45584 0.177104
\(950\) 0 0
\(951\) 3.79899 0.123191
\(952\) 0 0
\(953\) − 11.6569i − 0.377603i −0.982015 0.188801i \(-0.939540\pi\)
0.982015 0.188801i \(-0.0604602\pi\)
\(954\) 27.8579 0.901932
\(955\) 0 0
\(956\) −9.65685 −0.312325
\(957\) − 2.34315i − 0.0757431i
\(958\) 22.8284i 0.737553i
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 3.02944i 0.0976730i
\(963\) − 4.40202i − 0.141853i
\(964\) −10.5858 −0.340945
\(965\) 0 0
\(966\) 0 0
\(967\) 13.4558i 0.432711i 0.976315 + 0.216355i \(0.0694170\pi\)
−0.976315 + 0.216355i \(0.930583\pi\)
\(968\) 12.3137i 0.395778i
\(969\) 10.8284 0.347859
\(970\) 0 0
\(971\) −37.3553 −1.19879 −0.599395 0.800453i \(-0.704592\pi\)
−0.599395 + 0.800453i \(0.704592\pi\)
\(972\) − 13.4731i − 0.432150i
\(973\) 0 0
\(974\) −7.79899 −0.249896
\(975\) 0 0
\(976\) 13.3137 0.426161
\(977\) − 35.3137i − 1.12979i −0.825164 0.564893i \(-0.808918\pi\)
0.825164 0.564893i \(-0.191082\pi\)
\(978\) − 4.76955i − 0.152513i
\(979\) −61.4558 −1.96414
\(980\) 0 0
\(981\) 38.4853 1.22874
\(982\) 24.2843i 0.774942i
\(983\) − 51.7990i − 1.65213i −0.563574 0.826066i \(-0.690574\pi\)
0.563574 0.826066i \(-0.309426\pi\)
\(984\) 6.48528 0.206743
\(985\) 0 0
\(986\) 4.48528 0.142840
\(987\) 0 0
\(988\) − 2.82843i − 0.0899843i
\(989\) −21.6569 −0.688648
\(990\) 0 0
\(991\) −28.7696 −0.913895 −0.456947 0.889494i \(-0.651057\pi\)
−0.456947 + 0.889494i \(0.651057\pi\)
\(992\) 2.82843i 0.0898027i
\(993\) − 3.39697i − 0.107800i
\(994\) 0 0
\(995\) 0 0
\(996\) 3.65685 0.115872
\(997\) − 38.2843i − 1.21248i −0.795284 0.606238i \(-0.792678\pi\)
0.795284 0.606238i \(-0.207322\pi\)
\(998\) 41.6569i 1.31862i
\(999\) −12.1177 −0.383389
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 2450.2.c.w.99.1 4
5.2 odd 4 2450.2.a.bs.1.1 2
5.3 odd 4 490.2.a.l.1.2 2
5.4 even 2 inner 2450.2.c.w.99.4 4
7.6 odd 2 2450.2.c.t.99.2 4
15.8 even 4 4410.2.a.by.1.1 2
20.3 even 4 3920.2.a.ca.1.1 2
35.3 even 12 490.2.e.i.471.2 4
35.13 even 4 490.2.a.m.1.1 yes 2
35.18 odd 12 490.2.e.j.471.1 4
35.23 odd 12 490.2.e.j.361.1 4
35.27 even 4 2450.2.a.bn.1.2 2
35.33 even 12 490.2.e.i.361.2 4
35.34 odd 2 2450.2.c.t.99.3 4
105.83 odd 4 4410.2.a.bt.1.1 2
140.83 odd 4 3920.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.2 2 5.3 odd 4
490.2.a.m.1.1 yes 2 35.13 even 4
490.2.e.i.361.2 4 35.33 even 12
490.2.e.i.471.2 4 35.3 even 12
490.2.e.j.361.1 4 35.23 odd 12
490.2.e.j.471.1 4 35.18 odd 12
2450.2.a.bn.1.2 2 35.27 even 4
2450.2.a.bs.1.1 2 5.2 odd 4
2450.2.c.t.99.2 4 7.6 odd 2
2450.2.c.t.99.3 4 35.34 odd 2
2450.2.c.w.99.1 4 1.1 even 1 trivial
2450.2.c.w.99.4 4 5.4 even 2 inner
3920.2.a.bm.1.2 2 140.83 odd 4
3920.2.a.ca.1.1 2 20.3 even 4
4410.2.a.bt.1.1 2 105.83 odd 4
4410.2.a.by.1.1 2 15.8 even 4