Properties

Label 2450.2.a.bj.1.2
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $1$
Dimension $2$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
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 98)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.41421 0.408248
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.07107 −1.62221 −0.811107 0.584898i \(-0.801135\pi\)
−0.811107 + 0.584898i \(0.801135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.41421 −0.288675
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.82843 −0.492366
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 7.07107 1.14708
\(39\) 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
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 1.41421 0.204124
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 5.65685 0.769800
\(55\) 0 0
\(56\) 0 0
\(57\) −10.0000 −1.32453
\(58\) −2.00000 −0.262613
\(59\) −1.41421 −0.184115 −0.0920575 0.995754i \(-0.529344\pi\)
−0.0920575 + 0.995754i \(0.529344\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.82843 0.348155
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 1.41421 0.171499
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 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 \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 9.89949 1.09322
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 2.82843 0.303239
\(88\) 2.00000 0.213201
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 12.0000 1.24434
\(94\) 2.82843 0.291730
\(95\) 0 0
\(96\) −1.41421 −0.144338
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 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 \(0.361273\pi\)
\(102\) −2.00000 −0.198030
\(103\) −2.82843 −0.278693 −0.139347 0.990244i \(-0.544500\pi\)
−0.139347 + 0.990244i \(0.544500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −5.65685 −0.544331
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −14.1421 −1.34231
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 10.0000 0.936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 1.41421 0.130189
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.82843 −0.256074
\(123\) −14.0000 −1.26234
\(124\) 8.48528 0.762001
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.82843 −0.249029
\(130\) 0 0
\(131\) 12.7279 1.11204 0.556022 0.831168i \(-0.312327\pi\)
0.556022 + 0.831168i \(0.312327\pi\)
\(132\) −2.82843 −0.246183
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −5.65685 −0.481543
\(139\) 9.89949 0.839664 0.419832 0.907602i \(-0.362089\pi\)
0.419832 + 0.907602i \(0.362089\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −1.41421 −0.117041
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 7.07107 0.573539
\(153\) −1.41421 −0.114332
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 11.3137 0.902932 0.451466 0.892288i \(-0.350901\pi\)
0.451466 + 0.892288i \(0.350901\pi\)
\(158\) 4.00000 0.318223
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) 0 0
\(162\) 5.00000 0.392837
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −9.89949 −0.773021
\(165\) 0 0
\(166\) 9.89949 0.768350
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 7.07107 0.540738
\(172\) −2.00000 −0.152499
\(173\) 16.9706 1.29025 0.645124 0.764078i \(-0.276806\pi\)
0.645124 + 0.764078i \(0.276806\pi\)
\(174\) −2.82843 −0.214423
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −2.00000 −0.150329
\(178\) 7.07107 0.529999
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) −2.82843 −0.206835
\(188\) −2.82843 −0.206284
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.41421 0.102062
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 9.89949 0.710742
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −2.00000 −0.142134
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) −16.9706 −1.19701
\(202\) −8.48528 −0.597022
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 2.82843 0.197066
\(207\) −4.00000 −0.278019
\(208\) 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 \(0.635539\pi\)
\(212\) 2.00000 0.137361
\(213\) −16.9706 −1.16280
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 5.65685 0.384900
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 14.1421 0.949158
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) −10.0000 −0.662266
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.41421 −0.0920575
\(237\) −5.65685 −0.367452
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −21.2132 −1.36646 −0.683231 0.730202i \(-0.739426\pi\)
−0.683231 + 0.730202i \(0.739426\pi\)
\(242\) 7.00000 0.449977
\(243\) 9.89949 0.635053
\(244\) 2.82843 0.181071
\(245\) 0 0
\(246\) 14.0000 0.892607
\(247\) 0 0
\(248\) −8.48528 −0.538816
\(249\) −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
\(253\) −8.00000 −0.502956
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.7279 −0.793946 −0.396973 0.917830i \(-0.629939\pi\)
−0.396973 + 0.917830i \(0.629939\pi\)
\(258\) 2.82843 0.176090
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −12.7279 −0.786334
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 2.82843 0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) −11.3137 −0.689809 −0.344904 0.938638i \(-0.612089\pi\)
−0.344904 + 0.938638i \(0.612089\pi\)
\(270\) 0 0
\(271\) 22.6274 1.37452 0.687259 0.726413i \(-0.258814\pi\)
0.687259 + 0.726413i \(0.258814\pi\)
\(272\) 1.41421 0.0857493
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 5.65685 0.340503
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −9.89949 −0.593732
\(279\) −8.48528 −0.508001
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 4.00000 0.238197
\(283\) 1.41421 0.0840663 0.0420331 0.999116i \(-0.486616\pi\)
0.0420331 + 0.999116i \(0.486616\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 1.41421 0.0827606
\(293\) 19.7990 1.15667 0.578335 0.815800i \(-0.303703\pi\)
0.578335 + 0.815800i \(0.303703\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 11.3137 0.656488
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 12.0000 0.689382
\(304\) −7.07107 −0.405554
\(305\) 0 0
\(306\) 1.41421 0.0808452
\(307\) 9.89949 0.564994 0.282497 0.959268i \(-0.408837\pi\)
0.282497 + 0.959268i \(0.408837\pi\)
\(308\) 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
\(313\) −12.7279 −0.719425 −0.359712 0.933063i \(-0.617125\pi\)
−0.359712 + 0.933063i \(0.617125\pi\)
\(314\) −11.3137 −0.638470
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) −2.82843 −0.158610
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 5.65685 0.315735
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) −5.00000 −0.277778
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) −2.82843 −0.156412
\(328\) 9.89949 0.546608
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −9.89949 −0.543305
\(333\) 10.0000 0.547997
\(334\) −19.7990 −1.08335
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 13.0000 0.707107
\(339\) 16.9706 0.921714
\(340\) 0 0
\(341\) −16.9706 −0.919007
\(342\) −7.07107 −0.382360
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −16.9706 −0.912343
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 2.82843 0.151620
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 2.00000 0.106299
\(355\) 0 0
\(356\) −7.07107 −0.374766
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) −28.2843 −1.47643 −0.738213 0.674567i \(-0.764330\pi\)
−0.738213 + 0.674567i \(0.764330\pi\)
\(368\) 4.00000 0.208514
\(369\) 9.89949 0.515347
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 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 \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) −22.6274 −1.15924
\(382\) 4.00000 0.204658
\(383\) 36.7696 1.87884 0.939418 0.342773i \(-0.111366\pi\)
0.939418 + 0.342773i \(0.111366\pi\)
\(384\) −1.41421 −0.0721688
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) 2.00000 0.101666
\(388\) −9.89949 −0.502571
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −22.6274 −1.13564 −0.567819 0.823154i \(-0.692213\pi\)
−0.567819 + 0.823154i \(0.692213\pi\)
\(398\) −8.48528 −0.425329
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 16.9706 0.846415
\(403\) 0 0
\(404\) 8.48528 0.422159
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) −2.00000 −0.0990148
\(409\) 38.1838 1.88807 0.944033 0.329851i \(-0.106999\pi\)
0.944033 + 0.329851i \(0.106999\pi\)
\(410\) 0 0
\(411\) −16.9706 −0.837096
\(412\) −2.82843 −0.139347
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) −14.1421 −0.691714
\(419\) −9.89949 −0.483622 −0.241811 0.970323i \(-0.577741\pi\)
−0.241811 + 0.970323i \(0.577741\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 12.0000 0.584151
\(423\) 2.82843 0.137523
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 16.9706 0.822226
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −5.65685 −0.272166
\(433\) 29.6985 1.42722 0.713609 0.700544i \(-0.247059\pi\)
0.713609 + 0.700544i \(0.247059\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) −28.2843 −1.35302
\(438\) −2.00000 −0.0955637
\(439\) −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
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −14.1421 −0.671156
\(445\) 0 0
\(446\) 0 0
\(447\) 14.1421 0.668900
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 19.7990 0.932298
\(452\) 12.0000 0.564433
\(453\) −22.6274 −1.06313
\(454\) −21.2132 −0.995585
\(455\) 0 0
\(456\) 10.0000 0.468293
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 16.9706 0.792982
\(459\) −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
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −32.5269 −1.50517 −0.752583 0.658497i \(-0.771192\pi\)
−0.752583 + 0.658497i \(0.771192\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 1.41421 0.0650945
\(473\) 4.00000 0.183920
\(474\) 5.65685 0.259828
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 12.0000 0.548867
\(479\) −31.1127 −1.42158 −0.710788 0.703407i \(-0.751661\pi\)
−0.710788 + 0.703407i \(0.751661\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 21.2132 0.966235
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −9.89949 −0.449050
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −2.82843 −0.128037
\(489\) −14.1421 −0.639529
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −14.0000 −0.631169
\(493\) 2.82843 0.127386
\(494\) 0 0
\(495\) 0 0
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 14.0000 0.627355
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 28.0000 1.25095
\(502\) −9.89949 −0.441836
\(503\) −39.5980 −1.76559 −0.882793 0.469762i \(-0.844340\pi\)
−0.882793 + 0.469762i \(0.844340\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) −18.3848 −0.816497
\(508\) −16.0000 −0.709885
\(509\) 22.6274 1.00294 0.501471 0.865174i \(-0.332792\pi\)
0.501471 + 0.865174i \(0.332792\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 40.0000 1.76604
\(514\) 12.7279 0.561405
\(515\) 0 0
\(516\) −2.82843 −0.124515
\(517\) 5.65685 0.248788
\(518\) 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 \(0.509862\pi\)
\(522\) 2.00000 0.0875376
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 12.7279 0.556022
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 12.0000 0.522728
\(528\) −2.82843 −0.123091
\(529\) −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
\(537\) 16.9706 0.732334
\(538\) 11.3137 0.487769
\(539\) 0 0
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −22.6274 −0.971931
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) 0 0
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −12.0000 −0.512615
\(549\) −2.82843 −0.120714
\(550\) 0 0
\(551\) −14.1421 −0.602475
\(552\) −5.65685 −0.240772
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 9.89949 0.419832
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 8.48528 0.359211
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −16.0000 −0.674919
\(563\) 1.41421 0.0596020 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −1.41421 −0.0594438
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) −5.65685 −0.236318
\(574\) 0 0
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 21.2132 0.883117 0.441559 0.897232i \(-0.354426\pi\)
0.441559 + 0.897232i \(0.354426\pi\)
\(578\) 15.0000 0.623918
\(579\) 22.6274 0.940363
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000 0.580319
\(583\) −4.00000 −0.165663
\(584\) −1.41421 −0.0585206
\(585\) 0 0
\(586\) −19.7990 −0.817889
\(587\) 29.6985 1.22579 0.612894 0.790165i \(-0.290005\pi\)
0.612894 + 0.790165i \(0.290005\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) 0 0
\(591\) −2.82843 −0.116346
\(592\) −10.0000 −0.410997
\(593\) 7.07107 0.290374 0.145187 0.989404i \(-0.453622\pi\)
0.145187 + 0.989404i \(0.453622\pi\)
\(594\) −11.3137 −0.464207
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 29.6985 1.21143 0.605713 0.795683i \(-0.292888\pi\)
0.605713 + 0.795683i \(0.292888\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −12.0000 −0.487467
\(607\) 16.9706 0.688814 0.344407 0.938820i \(-0.388080\pi\)
0.344407 + 0.938820i \(0.388080\pi\)
\(608\) 7.07107 0.286770
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.41421 −0.0571662
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) −9.89949 −0.399511
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 4.00000 0.160904
\(619\) 18.3848 0.738947 0.369473 0.929241i \(-0.379538\pi\)
0.369473 + 0.929241i \(0.379538\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 11.3137 0.453638
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 12.7279 0.508710
\(627\) 20.0000 0.798723
\(628\) 11.3137 0.451466
\(629\) −14.1421 −0.563884
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 4.00000 0.159111
\(633\) −16.9706 −0.674519
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 2.82843 0.112154
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −5.65685 −0.223258
\(643\) −9.89949 −0.390398 −0.195199 0.980764i \(-0.562535\pi\)
−0.195199 + 0.980764i \(0.562535\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) −8.48528 −0.333591 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(648\) 5.00000 0.196419
\(649\) 2.82843 0.111025
\(650\) 0 0
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 2.82843 0.110600
\(655\) 0 0
\(656\) −9.89949 −0.386510
\(657\) −1.41421 −0.0551737
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 9.89949 0.384175
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 8.00000 0.309761
\(668\) 19.7990 0.766046
\(669\) 0 0
\(670\) 0 0
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) −16.9706 −0.651751
\(679\) 0 0
\(680\) 0 0
\(681\) 30.0000 1.14960
\(682\) 16.9706 0.649836
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 7.07107 0.270369
\(685\) 0 0
\(686\) 0 0
\(687\) −24.0000 −0.915657
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) 0 0
\(691\) 12.7279 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(692\) 16.9706 0.645124
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 0 0
\(696\) −2.82843 −0.107211
\(697\) −14.0000 −0.530288
\(698\) 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
\(703\) 70.7107 2.66690
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) −2.00000 −0.0751646
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 7.07107 0.264999
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −16.9706 −0.633777
\(718\) 32.0000 1.19423
\(719\) 2.82843 0.105483 0.0527413 0.998608i \(-0.483204\pi\)
0.0527413 + 0.998608i \(0.483204\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −31.0000 −1.15370
\(723\) −30.0000 −1.11571
\(724\) 0 0
\(725\) 0 0
\(726\) 9.89949 0.367405
\(727\) 19.7990 0.734304 0.367152 0.930161i \(-0.380333\pi\)
0.367152 + 0.930161i \(0.380333\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −2.82843 −0.104613
\(732\) 4.00000 0.147844
\(733\) −42.4264 −1.56706 −0.783528 0.621357i \(-0.786582\pi\)
−0.783528 + 0.621357i \(0.786582\pi\)
\(734\) 28.2843 1.04399
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 24.0000 0.884051
\(738\) −9.89949 −0.364405
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 9.89949 0.362204
\(748\) −2.82843 −0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −2.82843 −0.103142
\(753\) 14.0000 0.510188
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 26.0000 0.944363
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) −7.07107 −0.256326 −0.128163 0.991753i \(-0.540908\pi\)
−0.128163 + 0.991753i \(0.540908\pi\)
\(762\) 22.6274 0.819705
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −36.7696 −1.32854
\(767\) 0 0
\(768\) 1.41421 0.0510310
\(769\) 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
\(772\) 16.0000 0.575853
\(773\) −48.0833 −1.72943 −0.864717 0.502259i \(-0.832502\pi\)
−0.864717 + 0.502259i \(0.832502\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) 9.89949 0.355371
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 70.0000 2.50801
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −5.65685 −0.202289
\(783\) −11.3137 −0.404319
\(784\) 0 0
\(785\) 0 0
\(786\) −18.0000 −0.642039
\(787\) 1.41421 0.0504113 0.0252056 0.999682i \(-0.491976\pi\)
0.0252056 + 0.999682i \(0.491976\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −16.9706 −0.604168
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) 22.6274 0.803017
\(795\) 0 0
\(796\) 8.48528 0.300753
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 7.07107 0.249844
\(802\) 18.0000 0.635602
\(803\) −2.82843 −0.0998130
\(804\) −16.9706 −0.598506
\(805\) 0 0
\(806\) 0 0
\(807\) −16.0000 −0.563227
\(808\) −8.48528 −0.298511
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) −29.6985 −1.04285 −0.521427 0.853296i \(-0.674600\pi\)
−0.521427 + 0.853296i \(0.674600\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 14.1421 0.494771
\(818\) −38.1838 −1.33506
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 16.9706 0.591916
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 2.82843 0.0985329
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) −31.1127 −1.08059 −0.540294 0.841476i \(-0.681687\pi\)
−0.540294 + 0.841476i \(0.681687\pi\)
\(830\) 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
\(837\) −48.0000 −1.65912
\(838\) 9.89949 0.341972
\(839\) 19.7990 0.683537 0.341769 0.939784i \(-0.388974\pi\)
0.341769 + 0.939784i \(0.388974\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −30.0000 −1.03387
\(843\) 22.6274 0.779330
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) −2.82843 −0.0972433
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) −16.9706 −0.581402
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −18.3848 −0.628012 −0.314006 0.949421i \(-0.601671\pi\)
−0.314006 + 0.949421i \(0.601671\pi\)
\(858\) 0 0
\(859\) −26.8701 −0.916795 −0.458397 0.888747i \(-0.651576\pi\)
−0.458397 + 0.888747i \(0.651576\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 5.65685 0.192450
\(865\) 0 0
\(866\) −29.6985 −1.00920
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 9.89949 0.335047
\(874\) 28.2843 0.956730
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 16.9706 0.572729
\(879\) 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
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 36.7696 1.23460 0.617300 0.786728i \(-0.288226\pi\)
0.617300 + 0.786728i \(0.288226\pi\)
\(888\) 14.1421 0.474579
\(889\) 0 0
\(890\) 0 0
\(891\) 10.0000 0.335013
\(892\) 0 0
\(893\) 20.0000 0.669274
\(894\) −14.1421 −0.472984
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 16.9706 0.566000
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) −19.7990 −0.659234
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 22.6274 0.751746
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 21.2132 0.703985
\(909\) −8.48528 −0.281439
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −10.0000 −0.331133
\(913\) 19.7990 0.655251
\(914\) 24.0000 0.793849
\(915\) 0 0
\(916\) −16.9706 −0.560723
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) −39.5980 −1.30409
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 2.82843 0.0928977
\(928\) −2.00000 −0.0656532
\(929\) 32.5269 1.06717 0.533587 0.845745i \(-0.320844\pi\)
0.533587 + 0.845745i \(0.320844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −24.0000 −0.786146
\(933\) −16.0000 −0.523816
\(934\) 32.5269 1.06431
\(935\) 0 0
\(936\) 0 0
\(937\) −9.89949 −0.323402 −0.161701 0.986840i \(-0.551698\pi\)
−0.161701 + 0.986840i \(0.551698\pi\)
\(938\) 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 \(0.669291\pi\)
\(942\) −16.0000 −0.521308
\(943\) −39.5980 −1.28949
\(944\) −1.41421 −0.0460287
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 18.0000 0.584921 0.292461 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) −5.65685 −0.183726
\(949\) 0 0
\(950\) 0 0
\(951\) −14.1421 −0.458590
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −5.65685 −0.182860
\(958\) 31.1127 1.00521
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) −21.2132 −0.683231
\(965\) 0 0
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 7.00000 0.224989
\(969\) −14.1421 −0.454311
\(970\) 0 0
\(971\) 32.5269 1.04384 0.521919 0.852995i \(-0.325216\pi\)
0.521919 + 0.852995i \(0.325216\pi\)
\(972\) 9.89949 0.317526
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 2.82843 0.0905357
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 14.1421 0.452216
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −48.0833 −1.53362 −0.766809 0.641875i \(-0.778157\pi\)
−0.766809 + 0.641875i \(0.778157\pi\)
\(984\) 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 \(0.581789\pi\)
\(992\) −8.48528 −0.269408
\(993\) 14.1421 0.448787
\(994\) 0 0
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) 31.1127 0.985349 0.492675 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(998\) 4.00000 0.126618
\(999\) 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.a.bj.1.2 2
5.2 odd 4 2450.2.c.v.99.1 4
5.3 odd 4 2450.2.c.v.99.4 4
5.4 even 2 98.2.a.b.1.1 2
7.6 odd 2 inner 2450.2.a.bj.1.1 2
15.14 odd 2 882.2.a.n.1.1 2
20.19 odd 2 784.2.a.l.1.2 2
35.4 even 6 98.2.c.c.79.2 4
35.9 even 6 98.2.c.c.67.2 4
35.13 even 4 2450.2.c.v.99.3 4
35.19 odd 6 98.2.c.c.67.1 4
35.24 odd 6 98.2.c.c.79.1 4
35.27 even 4 2450.2.c.v.99.2 4
35.34 odd 2 98.2.a.b.1.2 yes 2
40.19 odd 2 3136.2.a.bm.1.1 2
40.29 even 2 3136.2.a.bn.1.2 2
60.59 even 2 7056.2.a.cl.1.1 2
105.44 odd 6 882.2.g.l.361.2 4
105.59 even 6 882.2.g.l.667.1 4
105.74 odd 6 882.2.g.l.667.2 4
105.89 even 6 882.2.g.l.361.1 4
105.104 even 2 882.2.a.n.1.2 2
140.19 even 6 784.2.i.m.753.2 4
140.39 odd 6 784.2.i.m.177.1 4
140.59 even 6 784.2.i.m.177.2 4
140.79 odd 6 784.2.i.m.753.1 4
140.139 even 2 784.2.a.l.1.1 2
280.69 odd 2 3136.2.a.bn.1.1 2
280.139 even 2 3136.2.a.bm.1.2 2
420.419 odd 2 7056.2.a.cl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 5.4 even 2
98.2.a.b.1.2 yes 2 35.34 odd 2
98.2.c.c.67.1 4 35.19 odd 6
98.2.c.c.67.2 4 35.9 even 6
98.2.c.c.79.1 4 35.24 odd 6
98.2.c.c.79.2 4 35.4 even 6
784.2.a.l.1.1 2 140.139 even 2
784.2.a.l.1.2 2 20.19 odd 2
784.2.i.m.177.1 4 140.39 odd 6
784.2.i.m.177.2 4 140.59 even 6
784.2.i.m.753.1 4 140.79 odd 6
784.2.i.m.753.2 4 140.19 even 6
882.2.a.n.1.1 2 15.14 odd 2
882.2.a.n.1.2 2 105.104 even 2
882.2.g.l.361.1 4 105.89 even 6
882.2.g.l.361.2 4 105.44 odd 6
882.2.g.l.667.1 4 105.59 even 6
882.2.g.l.667.2 4 105.74 odd 6
2450.2.a.bj.1.1 2 7.6 odd 2 inner
2450.2.a.bj.1.2 2 1.1 even 1 trivial
2450.2.c.v.99.1 4 5.2 odd 4
2450.2.c.v.99.2 4 35.27 even 4
2450.2.c.v.99.3 4 35.13 even 4
2450.2.c.v.99.4 4 5.3 odd 4
3136.2.a.bm.1.1 2 40.19 odd 2
3136.2.a.bm.1.2 2 280.139 even 2
3136.2.a.bn.1.1 2 280.69 odd 2
3136.2.a.bn.1.2 2 40.29 even 2
7056.2.a.cl.1.1 2 60.59 even 2
7056.2.a.cl.1.2 2 420.419 odd 2