Properties

Label 2450.2.c.v.99.4
Level $2450$
Weight $2$
Character 2450.99
Analytic conductor $19.563$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.41421i q^{3} -1.00000 q^{4} -1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.41421i q^{3} -1.00000 q^{4} -1.41421 q^{6} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{11} -1.41421i q^{12} +1.00000 q^{16} -1.41421i q^{17} +1.00000i q^{18} +7.07107 q^{19} -2.00000i q^{22} +4.00000i q^{23} +1.41421 q^{24} +5.65685i q^{27} -2.00000 q^{29} +8.48528 q^{31} +1.00000i q^{32} -2.82843i q^{33} +1.41421 q^{34} -1.00000 q^{36} +10.0000i q^{37} +7.07107i q^{38} -9.89949 q^{41} -2.00000i q^{43} +2.00000 q^{44} -4.00000 q^{46} +2.82843i q^{47} +1.41421i q^{48} +2.00000 q^{51} +2.00000i q^{53} -5.65685 q^{54} +10.0000i q^{57} -2.00000i q^{58} +1.41421 q^{59} +2.82843 q^{61} +8.48528i q^{62} -1.00000 q^{64} +2.82843 q^{66} +12.0000i q^{67} +1.41421i q^{68} -5.65685 q^{69} -12.0000 q^{71} -1.00000i q^{72} +1.41421i q^{73} -10.0000 q^{74} -7.07107 q^{76} +4.00000 q^{79} -5.00000 q^{81} -9.89949i q^{82} -9.89949i q^{83} +2.00000 q^{86} -2.82843i q^{87} +2.00000i q^{88} +7.07107 q^{89} -4.00000i q^{92} +12.0000i q^{93} -2.82843 q^{94} -1.41421 q^{96} +9.89949i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{9} - 8q^{11} + 4q^{16} - 8q^{29} - 4q^{36} + 8q^{44} - 16q^{46} + 8q^{51} - 4q^{64} - 48q^{71} - 40q^{74} + 16q^{79} - 20q^{81} + 8q^{86} - 8q^{99} + O(q^{100}) \)

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
\(3\) 1.41421i 0.816497i 0.912871 + 0.408248i \(0.133860\pi\)
−0.912871 + 0.408248i \(0.866140\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) − 1.41421i − 0.408248i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.41421i − 0.342997i −0.985184 0.171499i \(-0.945139\pi\)
0.985184 0.171499i \(-0.0548609\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 7.07107 1.62221 0.811107 0.584898i \(-0.198865\pi\)
0.811107 + 0.584898i \(0.198865\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.00000i − 0.426401i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 1.41421 0.288675
\(25\) 0 0
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 2.82843i − 0.492366i
\(34\) 1.41421 0.242536
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 7.07107i 1.14708i
\(39\) 0 0
\(40\) 0 0
\(41\) −9.89949 −1.54604 −0.773021 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 1.41421i 0.204124i
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −5.65685 −0.769800
\(55\) 0 0
\(56\) 0 0
\(57\) 10.0000i 1.32453i
\(58\) − 2.00000i − 0.262613i
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 8.48528i 1.07763i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.82843 0.348155
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 1.41421i 0.171499i
\(69\) −5.65685 −0.681005
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 1.41421i 0.165521i 0.996569 + 0.0827606i \(0.0263737\pi\)
−0.996569 + 0.0827606i \(0.973626\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −7.07107 −0.811107
\(77\) 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
\(82\) − 9.89949i − 1.09322i
\(83\) − 9.89949i − 1.08661i −0.839535 0.543305i \(-0.817173\pi\)
0.839535 0.543305i \(-0.182827\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) − 2.82843i − 0.303239i
\(88\) 2.00000i 0.213201i
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) 12.0000i 1.24434i
\(94\) −2.82843 −0.291730
\(95\) 0 0
\(96\) −1.41421 −0.144338
\(97\) 9.89949i 1.00514i 0.864536 + 0.502571i \(0.167612\pi\)
−0.864536 + 0.502571i \(0.832388\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 8.48528 0.844317 0.422159 0.906522i \(-0.361273\pi\)
0.422159 + 0.906522i \(0.361273\pi\)
\(102\) 2.00000i 0.198030i
\(103\) − 2.82843i − 0.278693i −0.990244 0.139347i \(-0.955500\pi\)
0.990244 0.139347i \(-0.0445002\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) − 5.65685i − 0.544331i
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −14.1421 −1.34231
\(112\) 0 0
\(113\) 12.0000i 1.12887i 0.825479 + 0.564433i \(0.190905\pi\)
−0.825479 + 0.564433i \(0.809095\pi\)
\(114\) −10.0000 −0.936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 1.41421i 0.130189i
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.82843i 0.256074i
\(123\) − 14.0000i − 1.26234i
\(124\) −8.48528 −0.762001
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 2.82843 0.249029
\(130\) 0 0
\(131\) 12.7279 1.11204 0.556022 0.831168i \(-0.312327\pi\)
0.556022 + 0.831168i \(0.312327\pi\)
\(132\) 2.82843i 0.246183i
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 5.65685i − 0.481543i
\(139\) −9.89949 −0.839664 −0.419832 0.907602i \(-0.637911\pi\)
−0.419832 + 0.907602i \(0.637911\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) − 12.0000i − 1.00702i
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −1.41421 −0.117041
\(147\) 0 0
\(148\) − 10.0000i − 0.821995i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 7.07107i − 0.573539i
\(153\) − 1.41421i − 0.114332i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 11.3137i − 0.902932i −0.892288 0.451466i \(-0.850901\pi\)
0.892288 0.451466i \(-0.149099\pi\)
\(158\) 4.00000i 0.318223i
\(159\) −2.82843 −0.224309
\(160\) 0 0
\(161\) 0 0
\(162\) − 5.00000i − 0.392837i
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 9.89949 0.773021
\(165\) 0 0
\(166\) 9.89949 0.768350
\(167\) − 19.7990i − 1.53209i −0.642786 0.766046i \(-0.722221\pi\)
0.642786 0.766046i \(-0.277779\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 7.07107 0.540738
\(172\) 2.00000i 0.152499i
\(173\) 16.9706i 1.29025i 0.764078 + 0.645124i \(0.223194\pi\)
−0.764078 + 0.645124i \(0.776806\pi\)
\(174\) 2.82843 0.214423
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 2.00000i 0.150329i
\(178\) 7.07107i 0.529999i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) 2.82843i 0.206835i
\(188\) − 2.82843i − 0.206284i
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) − 1.41421i − 0.102062i
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) −9.89949 −0.710742
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) −8.48528 −0.601506 −0.300753 0.953702i \(-0.597238\pi\)
−0.300753 + 0.953702i \(0.597238\pi\)
\(200\) 0 0
\(201\) −16.9706 −1.19701
\(202\) 8.48528i 0.597022i
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 2.82843 0.197066
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) −14.1421 −0.978232
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) − 16.9706i − 1.16280i
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 5.65685 0.384900
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) − 14.1421i − 0.949158i
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) − 21.2132i − 1.40797i −0.710215 0.703985i \(-0.751402\pi\)
0.710215 0.703985i \(-0.248598\pi\)
\(228\) − 10.0000i − 0.662266i
\(229\) 16.9706 1.12145 0.560723 0.828003i \(-0.310523\pi\)
0.560723 + 0.828003i \(0.310523\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) − 24.0000i − 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.41421 −0.0920575
\(237\) 5.65685i 0.367452i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −21.2132 −1.36646 −0.683231 0.730202i \(-0.739426\pi\)
−0.683231 + 0.730202i \(0.739426\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 9.89949i 0.635053i
\(244\) −2.82843 −0.181071
\(245\) 0 0
\(246\) 14.0000 0.892607
\(247\) 0 0
\(248\) − 8.48528i − 0.538816i
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 9.89949 0.624851 0.312425 0.949942i \(-0.398859\pi\)
0.312425 + 0.949942i \(0.398859\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.7279i 0.793946i 0.917830 + 0.396973i \(0.129939\pi\)
−0.917830 + 0.396973i \(0.870061\pi\)
\(258\) 2.82843i 0.176090i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 12.7279i 0.786334i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) −2.82843 −0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) − 12.0000i − 0.733017i
\(269\) 11.3137 0.689809 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(270\) 0 0
\(271\) 22.6274 1.37452 0.687259 0.726413i \(-0.258814\pi\)
0.687259 + 0.726413i \(0.258814\pi\)
\(272\) − 1.41421i − 0.0857493i
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 5.65685 0.340503
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 9.89949i − 0.593732i
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) 1.41421i 0.0840663i 0.999116 + 0.0420331i \(0.0133835\pi\)
−0.999116 + 0.0420331i \(0.986616\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) − 1.41421i − 0.0827606i
\(293\) 19.7990i 1.15667i 0.815800 + 0.578335i \(0.196297\pi\)
−0.815800 + 0.578335i \(0.803703\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) − 11.3137i − 0.656488i
\(298\) − 10.0000i − 0.579284i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 16.0000i − 0.920697i
\(303\) 12.0000i 0.689382i
\(304\) 7.07107 0.405554
\(305\) 0 0
\(306\) 1.41421 0.0808452
\(307\) − 9.89949i − 0.564994i −0.959268 0.282497i \(-0.908837\pi\)
0.959268 0.282497i \(-0.0911627\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) − 12.7279i − 0.719425i −0.933063 0.359712i \(-0.882875\pi\)
0.933063 0.359712i \(-0.117125\pi\)
\(314\) 11.3137 0.638470
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 10.0000i 0.561656i 0.959758 + 0.280828i \(0.0906090\pi\)
−0.959758 + 0.280828i \(0.909391\pi\)
\(318\) − 2.82843i − 0.158610i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 5.65685 0.315735
\(322\) 0 0
\(323\) − 10.0000i − 0.556415i
\(324\) 5.00000 0.277778
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) 2.82843i 0.156412i
\(328\) 9.89949i 0.546608i
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 9.89949i 0.543305i
\(333\) 10.0000i 0.547997i
\(334\) 19.7990 1.08335
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 13.0000i 0.707107i
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) −16.9706 −0.919007
\(342\) 7.07107i 0.382360i
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −16.9706 −0.912343
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 2.82843i 0.151620i
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 2.00000i − 0.106600i
\(353\) 1.41421i 0.0752710i 0.999292 + 0.0376355i \(0.0119826\pi\)
−0.999292 + 0.0376355i \(0.988017\pi\)
\(354\) −2.00000 −0.106299
\(355\) 0 0
\(356\) −7.07107 −0.374766
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) − 9.89949i − 0.519589i
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 28.2843i 1.47643i 0.674567 + 0.738213i \(0.264330\pi\)
−0.674567 + 0.738213i \(0.735670\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −9.89949 −0.515347
\(370\) 0 0
\(371\) 0 0
\(372\) − 12.0000i − 0.622171i
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) −2.82843 −0.146254
\(375\) 0 0
\(376\) 2.82843 0.145865
\(377\) 0 0
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −22.6274 −1.15924
\(382\) − 4.00000i − 0.204658i
\(383\) 36.7696i 1.87884i 0.342773 + 0.939418i \(0.388634\pi\)
−0.342773 + 0.939418i \(0.611366\pi\)
\(384\) 1.41421 0.0721688
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) − 2.00000i − 0.101666i
\(388\) − 9.89949i − 0.502571i
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 22.6274i 1.13564i 0.823154 + 0.567819i \(0.192213\pi\)
−0.823154 + 0.567819i \(0.807787\pi\)
\(398\) − 8.48528i − 0.425329i
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) − 16.9706i − 0.846415i
\(403\) 0 0
\(404\) −8.48528 −0.422159
\(405\) 0 0
\(406\) 0 0
\(407\) − 20.0000i − 0.991363i
\(408\) − 2.00000i − 0.0990148i
\(409\) −38.1838 −1.88807 −0.944033 0.329851i \(-0.893001\pi\)
−0.944033 + 0.329851i \(0.893001\pi\)
\(410\) 0 0
\(411\) −16.9706 −0.837096
\(412\) 2.82843i 0.139347i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) − 14.0000i − 0.685583i
\(418\) − 14.1421i − 0.691714i
\(419\) 9.89949 0.483622 0.241811 0.970323i \(-0.422259\pi\)
0.241811 + 0.970323i \(0.422259\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) − 12.0000i − 0.584151i
\(423\) 2.82843i 0.137523i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 16.9706 0.822226
\(427\) 0 0
\(428\) 4.00000i 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 5.65685i 0.272166i
\(433\) 29.6985i 1.42722i 0.700544 + 0.713609i \(0.252941\pi\)
−0.700544 + 0.713609i \(0.747059\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 28.2843i 1.35302i
\(438\) − 2.00000i − 0.0955637i
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 14.1421 0.671156
\(445\) 0 0
\(446\) 0 0
\(447\) − 14.1421i − 0.668900i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 19.7990 0.932298
\(452\) − 12.0000i − 0.564433i
\(453\) − 22.6274i − 1.06313i
\(454\) 21.2132 0.995585
\(455\) 0 0
\(456\) 10.0000 0.468293
\(457\) 24.0000i 1.12267i 0.827588 + 0.561336i \(0.189713\pi\)
−0.827588 + 0.561336i \(0.810287\pi\)
\(458\) 16.9706i 0.792982i
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) 39.5980 1.84426 0.922131 0.386878i \(-0.126447\pi\)
0.922131 + 0.386878i \(0.126447\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 32.5269i 1.50517i 0.658497 + 0.752583i \(0.271192\pi\)
−0.658497 + 0.752583i \(0.728808\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) − 1.41421i − 0.0650945i
\(473\) 4.00000i 0.183920i
\(474\) −5.65685 −0.259828
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 12.0000i 0.548867i
\(479\) 31.1127 1.42158 0.710788 0.703407i \(-0.248339\pi\)
0.710788 + 0.703407i \(0.248339\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 21.2132i − 0.966235i
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −9.89949 −0.449050
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) − 2.82843i − 0.128037i
\(489\) 14.1421 0.639529
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 14.0000i 0.631169i
\(493\) 2.82843i 0.127386i
\(494\) 0 0
\(495\) 0 0
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 14.0000i 0.627355i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 28.0000 1.25095
\(502\) 9.89949i 0.441836i
\(503\) − 39.5980i − 1.76559i −0.469762 0.882793i \(-0.655660\pi\)
0.469762 0.882793i \(-0.344340\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 18.3848i 0.816497i
\(508\) − 16.0000i − 0.709885i
\(509\) −22.6274 −1.00294 −0.501471 0.865174i \(-0.667208\pi\)
−0.501471 + 0.865174i \(0.667208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 40.0000i 1.76604i
\(514\) −12.7279 −0.561405
\(515\) 0 0
\(516\) −2.82843 −0.124515
\(517\) − 5.65685i − 0.248788i
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −1.41421 −0.0619578 −0.0309789 0.999520i \(-0.509862\pi\)
−0.0309789 + 0.999520i \(0.509862\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 12.7279i − 0.556553i −0.960501 0.278277i \(-0.910237\pi\)
0.960501 0.278277i \(-0.0897632\pi\)
\(524\) −12.7279 −0.556022
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) − 12.0000i − 0.522728i
\(528\) − 2.82843i − 0.123091i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 1.41421 0.0613716
\(532\) 0 0
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 16.9706i − 0.732334i
\(538\) 11.3137i 0.487769i
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 22.6274i 0.971931i
\(543\) 0 0
\(544\) 1.41421 0.0606339
\(545\) 0 0
\(546\) 0 0
\(547\) − 26.0000i − 1.11168i −0.831289 0.555840i \(-0.812397\pi\)
0.831289 0.555840i \(-0.187603\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 2.82843 0.120714
\(550\) 0 0
\(551\) −14.1421 −0.602475
\(552\) 5.65685i 0.240772i
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 9.89949 0.419832
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 8.48528i 0.359211i
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 16.0000i 0.674919i
\(563\) 1.41421i 0.0596020i 0.999556 + 0.0298010i \(0.00948736\pi\)
−0.999556 + 0.0298010i \(0.990513\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −1.41421 −0.0594438
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) − 5.65685i − 0.236318i
\(574\) 0 0
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) − 21.2132i − 0.883117i −0.897232 0.441559i \(-0.854426\pi\)
0.897232 0.441559i \(-0.145574\pi\)
\(578\) 15.0000i 0.623918i
\(579\) −22.6274 −0.940363
\(580\) 0 0
\(581\) 0 0
\(582\) − 14.0000i − 0.580319i
\(583\) − 4.00000i − 0.165663i
\(584\) 1.41421 0.0585206
\(585\) 0 0
\(586\) −19.7990 −0.817889
\(587\) − 29.6985i − 1.22579i −0.790165 0.612894i \(-0.790005\pi\)
0.790165 0.612894i \(-0.209995\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) 0 0
\(591\) −2.82843 −0.116346
\(592\) 10.0000i 0.410997i
\(593\) 7.07107i 0.290374i 0.989404 + 0.145187i \(0.0463784\pi\)
−0.989404 + 0.145187i \(0.953622\pi\)
\(594\) 11.3137 0.464207
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) − 12.0000i − 0.491127i
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 29.6985 1.21143 0.605713 0.795683i \(-0.292888\pi\)
0.605713 + 0.795683i \(0.292888\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) − 16.9706i − 0.688814i −0.938820 0.344407i \(-0.888080\pi\)
0.938820 0.344407i \(-0.111920\pi\)
\(608\) 7.07107i 0.286770i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.41421i 0.0571662i
\(613\) 30.0000i 1.21169i 0.795583 + 0.605844i \(0.207165\pi\)
−0.795583 + 0.605844i \(0.792835\pi\)
\(614\) 9.89949 0.399511
\(615\) 0 0
\(616\) 0 0
\(617\) − 26.0000i − 1.04672i −0.852111 0.523360i \(-0.824678\pi\)
0.852111 0.523360i \(-0.175322\pi\)
\(618\) 4.00000i 0.160904i
\(619\) −18.3848 −0.738947 −0.369473 0.929241i \(-0.620462\pi\)
−0.369473 + 0.929241i \(0.620462\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) − 11.3137i − 0.453638i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 12.7279 0.508710
\(627\) − 20.0000i − 0.798723i
\(628\) 11.3137i 0.451466i
\(629\) 14.1421 0.563884
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) − 16.9706i − 0.674519i
\(634\) −10.0000 −0.397151
\(635\) 0 0
\(636\) 2.82843 0.112154
\(637\) 0 0
\(638\) 4.00000i 0.158362i
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 5.65685i 0.223258i
\(643\) − 9.89949i − 0.390398i −0.980764 0.195199i \(-0.937465\pi\)
0.980764 0.195199i \(-0.0625353\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) 8.48528i 0.333591i 0.985992 + 0.166795i \(0.0533419\pi\)
−0.985992 + 0.166795i \(0.946658\pi\)
\(648\) 5.00000i 0.196419i
\(649\) −2.82843 −0.111025
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000i 0.391630i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) −2.82843 −0.110600
\(655\) 0 0
\(656\) −9.89949 −0.386510
\(657\) 1.41421i 0.0551737i
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) 10.0000i 0.388661i
\(663\) 0 0
\(664\) −9.89949 −0.384175
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) − 8.00000i − 0.309761i
\(668\) 19.7990i 0.766046i
\(669\) 0 0
\(670\) 0 0
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) 12.0000i 0.462566i 0.972887 + 0.231283i \(0.0742923\pi\)
−0.972887 + 0.231283i \(0.925708\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) − 16.9706i − 0.652232i −0.945330 0.326116i \(-0.894260\pi\)
0.945330 0.326116i \(-0.105740\pi\)
\(678\) − 16.9706i − 0.651751i
\(679\) 0 0
\(680\) 0 0
\(681\) 30.0000 1.14960
\(682\) − 16.9706i − 0.649836i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) −7.07107 −0.270369
\(685\) 0 0
\(686\) 0 0
\(687\) 24.0000i 0.915657i
\(688\) − 2.00000i − 0.0762493i
\(689\) 0 0
\(690\) 0 0
\(691\) 12.7279 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(692\) − 16.9706i − 0.645124i
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) −2.82843 −0.107211
\(697\) 14.0000i 0.530288i
\(698\) 0 0
\(699\) 33.9411 1.28377
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 70.7107i 2.66690i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) − 2.00000i − 0.0751646i
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) − 7.07107i − 0.264999i
\(713\) 33.9411i 1.27111i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 16.9706i 0.633777i
\(718\) 32.0000i 1.19423i
\(719\) −2.82843 −0.105483 −0.0527413 0.998608i \(-0.516796\pi\)
−0.0527413 + 0.998608i \(0.516796\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 31.0000i 1.15370i
\(723\) − 30.0000i − 1.11571i
\(724\) 0 0
\(725\) 0 0
\(726\) 9.89949 0.367405
\(727\) − 19.7990i − 0.734304i −0.930161 0.367152i \(-0.880333\pi\)
0.930161 0.367152i \(-0.119667\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) −2.82843 −0.104613
\(732\) − 4.00000i − 0.147844i
\(733\) − 42.4264i − 1.56706i −0.621357 0.783528i \(-0.713418\pi\)
0.621357 0.783528i \(-0.286582\pi\)
\(734\) −28.2843 −1.04399
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) − 24.0000i − 0.884051i
\(738\) − 9.89949i − 0.364405i
\(739\) 30.0000 1.10357 0.551784 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) − 9.89949i − 0.362204i
\(748\) − 2.82843i − 0.103418i
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 2.82843i 0.103142i
\(753\) 14.0000i 0.510188i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 26.0000i 0.944363i
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) −7.07107 −0.256326 −0.128163 0.991753i \(-0.540908\pi\)
−0.128163 + 0.991753i \(0.540908\pi\)
\(762\) − 22.6274i − 0.819705i
\(763\) 0 0
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) −36.7696 −1.32854
\(767\) 0 0
\(768\) 1.41421i 0.0510310i
\(769\) −29.6985 −1.07095 −0.535477 0.844550i \(-0.679868\pi\)
−0.535477 + 0.844550i \(0.679868\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) − 16.0000i − 0.575853i
\(773\) − 48.0833i − 1.72943i −0.502259 0.864717i \(-0.667498\pi\)
0.502259 0.864717i \(-0.332502\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 9.89949 0.355371
\(777\) 0 0
\(778\) − 26.0000i − 0.932145i
\(779\) −70.0000 −2.50801
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 5.65685i 0.202289i
\(783\) − 11.3137i − 0.404319i
\(784\) 0 0
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) − 1.41421i − 0.0504113i −0.999682 0.0252056i \(-0.991976\pi\)
0.999682 0.0252056i \(-0.00802405\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) 16.9706 0.604168
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000i 0.0710669i
\(793\) 0 0
\(794\) −22.6274 −0.803017
\(795\) 0 0
\(796\) 8.48528 0.300753
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) 7.07107 0.249844
\(802\) − 18.0000i − 0.635602i
\(803\) − 2.82843i − 0.0998130i
\(804\) 16.9706 0.598506
\(805\) 0 0
\(806\) 0 0
\(807\) 16.0000i 0.563227i
\(808\) − 8.48528i − 0.298511i
\(809\) 16.0000 0.562530 0.281265 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) 32.0000i 1.12229i
\(814\) 20.0000 0.701000
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) − 14.1421i − 0.494771i
\(818\) − 38.1838i − 1.33506i
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) − 16.9706i − 0.591916i
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) −2.82843 −0.0985329
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 31.1127 1.08059 0.540294 0.841476i \(-0.318313\pi\)
0.540294 + 0.841476i \(0.318313\pi\)
\(830\) 0 0
\(831\) 2.82843 0.0981170
\(832\) 0 0
\(833\) 0 0
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) 14.1421 0.489116
\(837\) 48.0000i 1.65912i
\(838\) 9.89949i 0.341972i
\(839\) −19.7990 −0.683537 −0.341769 0.939784i \(-0.611026\pi\)
−0.341769 + 0.939784i \(0.611026\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 30.0000i 1.03387i
\(843\) 22.6274i 0.779330i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −2.82843 −0.0972433
\(847\) 0 0
\(848\) 2.00000i 0.0686803i
\(849\) −2.00000 −0.0686398
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) 16.9706i 0.581402i
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 18.3848i 0.628012i 0.949421 + 0.314006i \(0.101671\pi\)
−0.949421 + 0.314006i \(0.898329\pi\)
\(858\) 0 0
\(859\) 26.8701 0.916795 0.458397 0.888747i \(-0.348424\pi\)
0.458397 + 0.888747i \(0.348424\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000i 0.408722i
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) −5.65685 −0.192450
\(865\) 0 0
\(866\) −29.6985 −1.00920
\(867\) 21.2132i 0.720438i
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) − 2.00000i − 0.0677285i
\(873\) 9.89949i 0.335047i
\(874\) −28.2843 −0.956730
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) 16.9706i 0.572729i
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) 29.6985 1.00057 0.500284 0.865862i \(-0.333229\pi\)
0.500284 + 0.865862i \(0.333229\pi\)
\(882\) 0 0
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 36.7696i − 1.23460i −0.786728 0.617300i \(-0.788226\pi\)
0.786728 0.617300i \(-0.211774\pi\)
\(888\) 14.1421i 0.474579i
\(889\) 0 0
\(890\) 0 0
\(891\) 10.0000 0.335013
\(892\) 0 0
\(893\) 20.0000i 0.669274i
\(894\) 14.1421 0.472984
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 30.0000i − 1.00111i
\(899\) −16.9706 −0.566000
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) 19.7990i 0.659234i
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) 22.6274 0.751746
\(907\) − 44.0000i − 1.46100i −0.682915 0.730498i \(-0.739288\pi\)
0.682915 0.730498i \(-0.260712\pi\)
\(908\) 21.2132i 0.703985i
\(909\) 8.48528 0.281439
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 10.0000i 0.331133i
\(913\) 19.7990i 0.655251i
\(914\) −24.0000 −0.793849
\(915\) 0 0
\(916\) −16.9706 −0.560723
\(917\) 0 0
\(918\) 8.00000i 0.264039i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 39.5980i 1.30409i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) − 2.82843i − 0.0928977i
\(928\) − 2.00000i − 0.0656532i
\(929\) −32.5269 −1.06717 −0.533587 0.845745i \(-0.679156\pi\)
−0.533587 + 0.845745i \(0.679156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0000i 0.786146i
\(933\) − 16.0000i − 0.523816i
\(934\) −32.5269 −1.06431
\(935\) 0 0
\(936\) 0 0
\(937\) 9.89949i 0.323402i 0.986840 + 0.161701i \(0.0516981\pi\)
−0.986840 + 0.161701i \(0.948302\pi\)
\(938\) 0 0
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) −31.1127 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(942\) 16.0000i 0.521308i
\(943\) − 39.5980i − 1.28949i
\(944\) 1.41421 0.0460287
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) − 5.65685i − 0.183726i
\(949\) 0 0
\(950\) 0 0
\(951\) −14.1421 −0.458590
\(952\) 0 0
\(953\) 26.0000i 0.842223i 0.907009 + 0.421111i \(0.138360\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 5.65685i 0.182860i
\(958\) 31.1127i 1.00521i
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 21.2132 0.683231
\(965\) 0 0
\(966\) 0 0
\(967\) − 12.0000i − 0.385894i −0.981209 0.192947i \(-0.938195\pi\)
0.981209 0.192947i \(-0.0618045\pi\)
\(968\) 7.00000i 0.224989i
\(969\) 14.1421 0.454311
\(970\) 0 0
\(971\) 32.5269 1.04384 0.521919 0.852995i \(-0.325216\pi\)
0.521919 + 0.852995i \(0.325216\pi\)
\(972\) − 9.89949i − 0.317526i
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 2.82843 0.0905357
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 14.1421i 0.452216i
\(979\) −14.1421 −0.451985
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) − 12.0000i − 0.382935i
\(983\) − 48.0833i − 1.53362i −0.641875 0.766809i \(-0.721843\pi\)
0.641875 0.766809i \(-0.278157\pi\)
\(984\) −14.0000 −0.446304
\(985\) 0 0
\(986\) −2.82843 −0.0900755
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.48528i 0.269408i
\(993\) 14.1421i 0.448787i
\(994\) 0 0
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) − 31.1127i − 0.985349i −0.870214 0.492675i \(-0.836019\pi\)
0.870214 0.492675i \(-0.163981\pi\)
\(998\) 4.00000i 0.126618i
\(999\) −56.5685 −1.78975
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.v.99.4 4
5.2 odd 4 2450.2.a.bj.1.2 2
5.3 odd 4 98.2.a.b.1.1 2
5.4 even 2 inner 2450.2.c.v.99.1 4
7.6 odd 2 inner 2450.2.c.v.99.3 4
15.8 even 4 882.2.a.n.1.1 2
20.3 even 4 784.2.a.l.1.2 2
35.3 even 12 98.2.c.c.79.1 4
35.13 even 4 98.2.a.b.1.2 yes 2
35.18 odd 12 98.2.c.c.79.2 4
35.23 odd 12 98.2.c.c.67.2 4
35.27 even 4 2450.2.a.bj.1.1 2
35.33 even 12 98.2.c.c.67.1 4
35.34 odd 2 inner 2450.2.c.v.99.2 4
40.3 even 4 3136.2.a.bm.1.1 2
40.13 odd 4 3136.2.a.bn.1.2 2
60.23 odd 4 7056.2.a.cl.1.1 2
105.23 even 12 882.2.g.l.361.2 4
105.38 odd 12 882.2.g.l.667.1 4
105.53 even 12 882.2.g.l.667.2 4
105.68 odd 12 882.2.g.l.361.1 4
105.83 odd 4 882.2.a.n.1.2 2
140.3 odd 12 784.2.i.m.177.2 4
140.23 even 12 784.2.i.m.753.1 4
140.83 odd 4 784.2.a.l.1.1 2
140.103 odd 12 784.2.i.m.753.2 4
140.123 even 12 784.2.i.m.177.1 4
280.13 even 4 3136.2.a.bn.1.1 2
280.83 odd 4 3136.2.a.bm.1.2 2
420.83 even 4 7056.2.a.cl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 5.3 odd 4
98.2.a.b.1.2 yes 2 35.13 even 4
98.2.c.c.67.1 4 35.33 even 12
98.2.c.c.67.2 4 35.23 odd 12
98.2.c.c.79.1 4 35.3 even 12
98.2.c.c.79.2 4 35.18 odd 12
784.2.a.l.1.1 2 140.83 odd 4
784.2.a.l.1.2 2 20.3 even 4
784.2.i.m.177.1 4 140.123 even 12
784.2.i.m.177.2 4 140.3 odd 12
784.2.i.m.753.1 4 140.23 even 12
784.2.i.m.753.2 4 140.103 odd 12
882.2.a.n.1.1 2 15.8 even 4
882.2.a.n.1.2 2 105.83 odd 4
882.2.g.l.361.1 4 105.68 odd 12
882.2.g.l.361.2 4 105.23 even 12
882.2.g.l.667.1 4 105.38 odd 12
882.2.g.l.667.2 4 105.53 even 12
2450.2.a.bj.1.1 2 35.27 even 4
2450.2.a.bj.1.2 2 5.2 odd 4
2450.2.c.v.99.1 4 5.4 even 2 inner
2450.2.c.v.99.2 4 35.34 odd 2 inner
2450.2.c.v.99.3 4 7.6 odd 2 inner
2450.2.c.v.99.4 4 1.1 even 1 trivial
3136.2.a.bm.1.1 2 40.3 even 4
3136.2.a.bm.1.2 2 280.83 odd 4
3136.2.a.bn.1.1 2 280.13 even 4
3136.2.a.bn.1.2 2 40.13 odd 4
7056.2.a.cl.1.1 2 60.23 odd 4
7056.2.a.cl.1.2 2 420.83 even 4