Properties

Label 6762.2.a.cf.1.3
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3132.1
Defining polynomial: \(x^{3} - 15 x - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.05932\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.05932 q^{11} +1.00000 q^{12} +6.26870 q^{13} -1.00000 q^{15} +1.00000 q^{16} -3.84994 q^{17} -1.00000 q^{18} -3.20938 q^{19} -1.00000 q^{20} -4.05932 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -6.26870 q^{26} +1.00000 q^{27} -1.26870 q^{29} +1.00000 q^{30} -6.47808 q^{31} -1.00000 q^{32} +4.05932 q^{33} +3.84994 q^{34} +1.00000 q^{36} -10.9093 q^{37} +3.20938 q^{38} +6.26870 q^{39} +1.00000 q^{40} +4.79062 q^{41} -8.11864 q^{43} +4.05932 q^{44} -1.00000 q^{45} +1.00000 q^{46} -5.47808 q^{47} +1.00000 q^{48} +4.00000 q^{50} -3.84994 q^{51} +6.26870 q^{52} -9.11864 q^{53} -1.00000 q^{54} -4.05932 q^{55} -3.20938 q^{57} +1.26870 q^{58} +6.84994 q^{59} -1.00000 q^{60} -10.0000 q^{61} +6.47808 q^{62} +1.00000 q^{64} -6.26870 q^{65} -4.05932 q^{66} -13.4781 q^{67} -3.84994 q^{68} -1.00000 q^{69} +12.3873 q^{71} -1.00000 q^{72} -9.05932 q^{73} +10.9093 q^{74} -4.00000 q^{75} -3.20938 q^{76} -6.26870 q^{78} +9.38734 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.79062 q^{82} -0.0593201 q^{83} +3.84994 q^{85} +8.11864 q^{86} -1.26870 q^{87} -4.05932 q^{88} -10.9093 q^{89} +1.00000 q^{90} -1.00000 q^{92} -6.47808 q^{93} +5.47808 q^{94} +3.20938 q^{95} -1.00000 q^{96} +15.8061 q^{97} +4.05932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} - 6q^{19} - 3q^{20} - 3q^{23} - 3q^{24} - 12q^{25} - 3q^{26} + 3q^{27} + 12q^{29} + 3q^{30} - 3q^{32} + 3q^{34} + 3q^{36} - 12q^{37} + 6q^{38} + 3q^{39} + 3q^{40} + 18q^{41} - 3q^{45} + 3q^{46} + 3q^{47} + 3q^{48} + 12q^{50} - 3q^{51} + 3q^{52} - 3q^{53} - 3q^{54} - 6q^{57} - 12q^{58} + 12q^{59} - 3q^{60} - 30q^{61} + 3q^{64} - 3q^{65} - 21q^{67} - 3q^{68} - 3q^{69} - 3q^{71} - 3q^{72} - 15q^{73} + 12q^{74} - 12q^{75} - 6q^{76} - 3q^{78} - 12q^{79} - 3q^{80} + 3q^{81} - 18q^{82} + 12q^{83} + 3q^{85} + 12q^{87} - 12q^{89} + 3q^{90} - 3q^{92} - 3q^{94} + 6q^{95} - 3q^{96} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.05932 1.22393 0.611966 0.790884i \(-0.290379\pi\)
0.611966 + 0.790884i \(0.290379\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.26870 1.73862 0.869312 0.494263i \(-0.164562\pi\)
0.869312 + 0.494263i \(0.164562\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.84994 −0.933748 −0.466874 0.884324i \(-0.654620\pi\)
−0.466874 + 0.884324i \(0.654620\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.20938 −0.736282 −0.368141 0.929770i \(-0.620006\pi\)
−0.368141 + 0.929770i \(0.620006\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −4.05932 −0.865450
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −6.26870 −1.22939
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.26870 −0.235592 −0.117796 0.993038i \(-0.537583\pi\)
−0.117796 + 0.993038i \(0.537583\pi\)
\(30\) 1.00000 0.182574
\(31\) −6.47808 −1.16350 −0.581749 0.813369i \(-0.697631\pi\)
−0.581749 + 0.813369i \(0.697631\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.05932 0.706637
\(34\) 3.84994 0.660259
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.9093 −1.79347 −0.896736 0.442567i \(-0.854068\pi\)
−0.896736 + 0.442567i \(0.854068\pi\)
\(38\) 3.20938 0.520630
\(39\) 6.26870 1.00380
\(40\) 1.00000 0.158114
\(41\) 4.79062 0.748169 0.374085 0.927395i \(-0.377957\pi\)
0.374085 + 0.927395i \(0.377957\pi\)
\(42\) 0 0
\(43\) −8.11864 −1.23808 −0.619041 0.785359i \(-0.712478\pi\)
−0.619041 + 0.785359i \(0.712478\pi\)
\(44\) 4.05932 0.611966
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) −5.47808 −0.799060 −0.399530 0.916720i \(-0.630827\pi\)
−0.399530 + 0.916720i \(0.630827\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −3.84994 −0.539099
\(52\) 6.26870 0.869312
\(53\) −9.11864 −1.25254 −0.626271 0.779606i \(-0.715420\pi\)
−0.626271 + 0.779606i \(0.715420\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.05932 −0.547359
\(56\) 0 0
\(57\) −3.20938 −0.425093
\(58\) 1.26870 0.166588
\(59\) 6.84994 0.891786 0.445893 0.895086i \(-0.352886\pi\)
0.445893 + 0.895086i \(0.352886\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 6.47808 0.822717
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.26870 −0.777537
\(66\) −4.05932 −0.499668
\(67\) −13.4781 −1.64661 −0.823305 0.567600i \(-0.807872\pi\)
−0.823305 + 0.567600i \(0.807872\pi\)
\(68\) −3.84994 −0.466874
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.3873 1.47011 0.735053 0.678009i \(-0.237157\pi\)
0.735053 + 0.678009i \(0.237157\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.05932 −1.06031 −0.530157 0.847900i \(-0.677867\pi\)
−0.530157 + 0.847900i \(0.677867\pi\)
\(74\) 10.9093 1.26818
\(75\) −4.00000 −0.461880
\(76\) −3.20938 −0.368141
\(77\) 0 0
\(78\) −6.26870 −0.709791
\(79\) 9.38734 1.05616 0.528079 0.849195i \(-0.322912\pi\)
0.528079 + 0.849195i \(0.322912\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −4.79062 −0.529035
\(83\) −0.0593201 −0.00651123 −0.00325562 0.999995i \(-0.501036\pi\)
−0.00325562 + 0.999995i \(0.501036\pi\)
\(84\) 0 0
\(85\) 3.84994 0.417585
\(86\) 8.11864 0.875455
\(87\) −1.26870 −0.136019
\(88\) −4.05932 −0.432725
\(89\) −10.9093 −1.15638 −0.578190 0.815902i \(-0.696241\pi\)
−0.578190 + 0.815902i \(0.696241\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −6.47808 −0.671746
\(94\) 5.47808 0.565021
\(95\) 3.20938 0.329275
\(96\) −1.00000 −0.102062
\(97\) 15.8061 1.60487 0.802433 0.596742i \(-0.203538\pi\)
0.802433 + 0.596742i \(0.203538\pi\)
\(98\) 0 0
\(99\) 4.05932 0.407977
\(100\) −4.00000 −0.400000
\(101\) 14.1186 1.40486 0.702429 0.711754i \(-0.252099\pi\)
0.702429 + 0.711754i \(0.252099\pi\)
\(102\) 3.84994 0.381201
\(103\) −0.521920 −0.0514263 −0.0257132 0.999669i \(-0.508186\pi\)
−0.0257132 + 0.999669i \(0.508186\pi\)
\(104\) −6.26870 −0.614697
\(105\) 0 0
\(106\) 9.11864 0.885681
\(107\) −14.0593 −1.35917 −0.679583 0.733599i \(-0.737839\pi\)
−0.679583 + 0.733599i \(0.737839\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.1186 −1.16076 −0.580378 0.814347i \(-0.697095\pi\)
−0.580378 + 0.814347i \(0.697095\pi\)
\(110\) 4.05932 0.387041
\(111\) −10.9093 −1.03546
\(112\) 0 0
\(113\) −5.43118 −0.510922 −0.255461 0.966819i \(-0.582227\pi\)
−0.255461 + 0.966819i \(0.582227\pi\)
\(114\) 3.20938 0.300586
\(115\) 1.00000 0.0932505
\(116\) −1.26870 −0.117796
\(117\) 6.26870 0.579542
\(118\) −6.84994 −0.630588
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 5.47808 0.498007
\(122\) 10.0000 0.905357
\(123\) 4.79062 0.431956
\(124\) −6.47808 −0.581749
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −16.0593 −1.42503 −0.712517 0.701655i \(-0.752445\pi\)
−0.712517 + 0.701655i \(0.752445\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.11864 −0.714806
\(130\) 6.26870 0.549801
\(131\) 18.8654 1.64828 0.824140 0.566387i \(-0.191659\pi\)
0.824140 + 0.566387i \(0.191659\pi\)
\(132\) 4.05932 0.353318
\(133\) 0 0
\(134\) 13.4781 1.16433
\(135\) −1.00000 −0.0860663
\(136\) 3.84994 0.330130
\(137\) 15.5967 1.33252 0.666259 0.745721i \(-0.267895\pi\)
0.666259 + 0.745721i \(0.267895\pi\)
\(138\) 1.00000 0.0851257
\(139\) −6.83752 −0.579951 −0.289975 0.957034i \(-0.593647\pi\)
−0.289975 + 0.957034i \(0.593647\pi\)
\(140\) 0 0
\(141\) −5.47808 −0.461338
\(142\) −12.3873 −1.03952
\(143\) 25.4467 2.12796
\(144\) 1.00000 0.0833333
\(145\) 1.26870 0.105360
\(146\) 9.05932 0.749755
\(147\) 0 0
\(148\) −10.9093 −0.896736
\(149\) 3.17796 0.260349 0.130174 0.991491i \(-0.458446\pi\)
0.130174 + 0.991491i \(0.458446\pi\)
\(150\) 4.00000 0.326599
\(151\) 10.0593 0.818616 0.409308 0.912396i \(-0.365770\pi\)
0.409308 + 0.912396i \(0.365770\pi\)
\(152\) 3.20938 0.260315
\(153\) −3.84994 −0.311249
\(154\) 0 0
\(155\) 6.47808 0.520332
\(156\) 6.26870 0.501898
\(157\) −2.07174 −0.165343 −0.0826714 0.996577i \(-0.526345\pi\)
−0.0826714 + 0.996577i \(0.526345\pi\)
\(158\) −9.38734 −0.746817
\(159\) −9.11864 −0.723155
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.07174 0.318923 0.159462 0.987204i \(-0.449024\pi\)
0.159462 + 0.987204i \(0.449024\pi\)
\(164\) 4.79062 0.374085
\(165\) −4.05932 −0.316018
\(166\) 0.0593201 0.00460414
\(167\) −18.0155 −1.39408 −0.697040 0.717032i \(-0.745500\pi\)
−0.697040 + 0.717032i \(0.745500\pi\)
\(168\) 0 0
\(169\) 26.2966 2.02282
\(170\) −3.84994 −0.295277
\(171\) −3.20938 −0.245427
\(172\) −8.11864 −0.619041
\(173\) 14.5374 1.10526 0.552629 0.833427i \(-0.313625\pi\)
0.552629 + 0.833427i \(0.313625\pi\)
\(174\) 1.26870 0.0961799
\(175\) 0 0
\(176\) 4.05932 0.305983
\(177\) 6.84994 0.514873
\(178\) 10.9093 0.817684
\(179\) −12.7592 −0.953667 −0.476834 0.878994i \(-0.658216\pi\)
−0.476834 + 0.878994i \(0.658216\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 12.5374 0.931898 0.465949 0.884812i \(-0.345713\pi\)
0.465949 + 0.884812i \(0.345713\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 1.00000 0.0737210
\(185\) 10.9093 0.802065
\(186\) 6.47808 0.474996
\(187\) −15.6281 −1.14284
\(188\) −5.47808 −0.399530
\(189\) 0 0
\(190\) −3.20938 −0.232833
\(191\) 4.07174 0.294621 0.147310 0.989090i \(-0.452938\pi\)
0.147310 + 0.989090i \(0.452938\pi\)
\(192\) 1.00000 0.0721688
\(193\) 23.7747 1.71134 0.855669 0.517523i \(-0.173146\pi\)
0.855669 + 0.517523i \(0.173146\pi\)
\(194\) −15.8061 −1.13481
\(195\) −6.26870 −0.448911
\(196\) 0 0
\(197\) 0.418760 0.0298354 0.0149177 0.999889i \(-0.495251\pi\)
0.0149177 + 0.999889i \(0.495251\pi\)
\(198\) −4.05932 −0.288483
\(199\) 5.69988 0.404054 0.202027 0.979380i \(-0.435247\pi\)
0.202027 + 0.979380i \(0.435247\pi\)
\(200\) 4.00000 0.282843
\(201\) −13.4781 −0.950670
\(202\) −14.1186 −0.993384
\(203\) 0 0
\(204\) −3.84994 −0.269550
\(205\) −4.79062 −0.334591
\(206\) 0.521920 0.0363639
\(207\) −1.00000 −0.0695048
\(208\) 6.26870 0.434656
\(209\) −13.0279 −0.901159
\(210\) 0 0
\(211\) −4.41876 −0.304200 −0.152100 0.988365i \(-0.548604\pi\)
−0.152100 + 0.988365i \(0.548604\pi\)
\(212\) −9.11864 −0.626271
\(213\) 12.3873 0.848767
\(214\) 14.0593 0.961075
\(215\) 8.11864 0.553687
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 12.1186 0.820778
\(219\) −9.05932 −0.612172
\(220\) −4.05932 −0.273679
\(221\) −24.1341 −1.62344
\(222\) 10.9093 0.732182
\(223\) −11.7592 −0.787454 −0.393727 0.919227i \(-0.628815\pi\)
−0.393727 + 0.919227i \(0.628815\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 5.43118 0.361277
\(227\) −13.7592 −0.913230 −0.456615 0.889664i \(-0.650938\pi\)
−0.456615 + 0.889664i \(0.650938\pi\)
\(228\) −3.20938 −0.212546
\(229\) −21.6281 −1.42923 −0.714614 0.699519i \(-0.753398\pi\)
−0.714614 + 0.699519i \(0.753398\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 1.26870 0.0832942
\(233\) −26.6091 −1.74322 −0.871611 0.490198i \(-0.836925\pi\)
−0.871611 + 0.490198i \(0.836925\pi\)
\(234\) −6.26870 −0.409798
\(235\) 5.47808 0.357351
\(236\) 6.84994 0.445893
\(237\) 9.38734 0.609773
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −28.2249 −1.81812 −0.909062 0.416662i \(-0.863200\pi\)
−0.909062 + 0.416662i \(0.863200\pi\)
\(242\) −5.47808 −0.352144
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −4.79062 −0.305439
\(247\) −20.1186 −1.28012
\(248\) 6.47808 0.411358
\(249\) −0.0593201 −0.00375926
\(250\) −9.00000 −0.569210
\(251\) 8.29660 0.523677 0.261838 0.965112i \(-0.415671\pi\)
0.261838 + 0.965112i \(0.415671\pi\)
\(252\) 0 0
\(253\) −4.05932 −0.255207
\(254\) 16.0593 1.00765
\(255\) 3.84994 0.241093
\(256\) 1.00000 0.0625000
\(257\) −19.0279 −1.18693 −0.593464 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(258\) 8.11864 0.505444
\(259\) 0 0
\(260\) −6.26870 −0.388768
\(261\) −1.26870 −0.0785306
\(262\) −18.8654 −1.16551
\(263\) 0.118640 0.00731567 0.00365784 0.999993i \(-0.498836\pi\)
0.00365784 + 0.999993i \(0.498836\pi\)
\(264\) −4.05932 −0.249834
\(265\) 9.11864 0.560154
\(266\) 0 0
\(267\) −10.9093 −0.667636
\(268\) −13.4781 −0.823305
\(269\) −2.96858 −0.180998 −0.0904988 0.995897i \(-0.528846\pi\)
−0.0904988 + 0.995897i \(0.528846\pi\)
\(270\) 1.00000 0.0608581
\(271\) 12.8968 0.783427 0.391714 0.920087i \(-0.371882\pi\)
0.391714 + 0.920087i \(0.371882\pi\)
\(272\) −3.84994 −0.233437
\(273\) 0 0
\(274\) −15.5967 −0.942232
\(275\) −16.2373 −0.979145
\(276\) −1.00000 −0.0601929
\(277\) 11.1780 0.671619 0.335809 0.941930i \(-0.390990\pi\)
0.335809 + 0.941930i \(0.390990\pi\)
\(278\) 6.83752 0.410087
\(279\) −6.47808 −0.387833
\(280\) 0 0
\(281\) −31.4621 −1.87687 −0.938437 0.345450i \(-0.887726\pi\)
−0.938437 + 0.345450i \(0.887726\pi\)
\(282\) 5.47808 0.326215
\(283\) 10.0155 0.595358 0.297679 0.954666i \(-0.403787\pi\)
0.297679 + 0.954666i \(0.403787\pi\)
\(284\) 12.3873 0.735053
\(285\) 3.20938 0.190107
\(286\) −25.4467 −1.50469
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −2.17796 −0.128115
\(290\) −1.26870 −0.0745006
\(291\) 15.8061 0.926570
\(292\) −9.05932 −0.530157
\(293\) −3.83752 −0.224190 −0.112095 0.993697i \(-0.535756\pi\)
−0.112095 + 0.993697i \(0.535756\pi\)
\(294\) 0 0
\(295\) −6.84994 −0.398819
\(296\) 10.9093 0.634088
\(297\) 4.05932 0.235546
\(298\) −3.17796 −0.184094
\(299\) −6.26870 −0.362528
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −10.0593 −0.578849
\(303\) 14.1186 0.811095
\(304\) −3.20938 −0.184071
\(305\) 10.0000 0.572598
\(306\) 3.84994 0.220086
\(307\) −13.9841 −0.798112 −0.399056 0.916926i \(-0.630662\pi\)
−0.399056 + 0.916926i \(0.630662\pi\)
\(308\) 0 0
\(309\) −0.521920 −0.0296910
\(310\) −6.47808 −0.367930
\(311\) 26.3873 1.49629 0.748144 0.663536i \(-0.230945\pi\)
0.748144 + 0.663536i \(0.230945\pi\)
\(312\) −6.26870 −0.354895
\(313\) −9.15006 −0.517192 −0.258596 0.965986i \(-0.583260\pi\)
−0.258596 + 0.965986i \(0.583260\pi\)
\(314\) 2.07174 0.116915
\(315\) 0 0
\(316\) 9.38734 0.528079
\(317\) 20.2249 1.13594 0.567971 0.823049i \(-0.307729\pi\)
0.567971 + 0.823049i \(0.307729\pi\)
\(318\) 9.11864 0.511348
\(319\) −5.15006 −0.288348
\(320\) −1.00000 −0.0559017
\(321\) −14.0593 −0.784715
\(322\) 0 0
\(323\) 12.3559 0.687502
\(324\) 1.00000 0.0555556
\(325\) −25.0748 −1.39090
\(326\) −4.07174 −0.225513
\(327\) −12.1186 −0.670162
\(328\) −4.79062 −0.264518
\(329\) 0 0
\(330\) 4.05932 0.223458
\(331\) −20.6091 −1.13278 −0.566390 0.824137i \(-0.691661\pi\)
−0.566390 + 0.824137i \(0.691661\pi\)
\(332\) −0.0593201 −0.00325562
\(333\) −10.9093 −0.597824
\(334\) 18.0155 0.985763
\(335\) 13.4781 0.736386
\(336\) 0 0
\(337\) −26.2249 −1.42856 −0.714280 0.699860i \(-0.753245\pi\)
−0.714280 + 0.699860i \(0.753245\pi\)
\(338\) −26.2966 −1.43035
\(339\) −5.43118 −0.294981
\(340\) 3.84994 0.208792
\(341\) −26.2966 −1.42404
\(342\) 3.20938 0.173543
\(343\) 0 0
\(344\) 8.11864 0.437728
\(345\) 1.00000 0.0538382
\(346\) −14.5374 −0.781535
\(347\) 25.5967 1.37410 0.687052 0.726608i \(-0.258904\pi\)
0.687052 + 0.726608i \(0.258904\pi\)
\(348\) −1.26870 −0.0680095
\(349\) −1.61266 −0.0863237 −0.0431618 0.999068i \(-0.513743\pi\)
−0.0431618 + 0.999068i \(0.513743\pi\)
\(350\) 0 0
\(351\) 6.26870 0.334598
\(352\) −4.05932 −0.216362
\(353\) 26.6091 1.41626 0.708131 0.706081i \(-0.249539\pi\)
0.708131 + 0.706081i \(0.249539\pi\)
\(354\) −6.84994 −0.364070
\(355\) −12.3873 −0.657452
\(356\) −10.9093 −0.578190
\(357\) 0 0
\(358\) 12.7592 0.674345
\(359\) −18.8375 −0.994206 −0.497103 0.867691i \(-0.665603\pi\)
−0.497103 + 0.867691i \(0.665603\pi\)
\(360\) 1.00000 0.0527046
\(361\) −8.69988 −0.457888
\(362\) −12.5374 −0.658951
\(363\) 5.47808 0.287525
\(364\) 0 0
\(365\) 9.05932 0.474186
\(366\) 10.0000 0.522708
\(367\) −0.746780 −0.0389816 −0.0194908 0.999810i \(-0.506205\pi\)
−0.0194908 + 0.999810i \(0.506205\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 4.79062 0.249390
\(370\) −10.9093 −0.567145
\(371\) 0 0
\(372\) −6.47808 −0.335873
\(373\) 18.9562 0.981513 0.490756 0.871297i \(-0.336721\pi\)
0.490756 + 0.871297i \(0.336721\pi\)
\(374\) 15.6281 0.808112
\(375\) 9.00000 0.464758
\(376\) 5.47808 0.282510
\(377\) −7.95310 −0.409606
\(378\) 0 0
\(379\) 11.7154 0.601778 0.300889 0.953659i \(-0.402717\pi\)
0.300889 + 0.953659i \(0.402717\pi\)
\(380\) 3.20938 0.164638
\(381\) −16.0593 −0.822744
\(382\) −4.07174 −0.208328
\(383\) 26.1655 1.33700 0.668498 0.743714i \(-0.266937\pi\)
0.668498 + 0.743714i \(0.266937\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −23.7747 −1.21010
\(387\) −8.11864 −0.412694
\(388\) 15.8061 0.802433
\(389\) −27.7937 −1.40920 −0.704598 0.709607i \(-0.748873\pi\)
−0.704598 + 0.709607i \(0.748873\pi\)
\(390\) 6.26870 0.317428
\(391\) 3.84994 0.194700
\(392\) 0 0
\(393\) 18.8654 0.951635
\(394\) −0.418760 −0.0210968
\(395\) −9.38734 −0.472328
\(396\) 4.05932 0.203989
\(397\) −3.84994 −0.193223 −0.0966115 0.995322i \(-0.530800\pi\)
−0.0966115 + 0.995322i \(0.530800\pi\)
\(398\) −5.69988 −0.285709
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 7.17796 0.358450 0.179225 0.983808i \(-0.442641\pi\)
0.179225 + 0.983808i \(0.442641\pi\)
\(402\) 13.4781 0.672226
\(403\) −40.6091 −2.02289
\(404\) 14.1186 0.702429
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −44.2842 −2.19509
\(408\) 3.84994 0.190600
\(409\) 16.3749 0.809688 0.404844 0.914386i \(-0.367326\pi\)
0.404844 + 0.914386i \(0.367326\pi\)
\(410\) 4.79062 0.236592
\(411\) 15.5967 0.769329
\(412\) −0.521920 −0.0257132
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0.0593201 0.00291191
\(416\) −6.26870 −0.307348
\(417\) −6.83752 −0.334835
\(418\) 13.0279 0.637215
\(419\) 25.9372 1.26711 0.633557 0.773696i \(-0.281594\pi\)
0.633557 + 0.773696i \(0.281594\pi\)
\(420\) 0 0
\(421\) 3.13764 0.152919 0.0764596 0.997073i \(-0.475638\pi\)
0.0764596 + 0.997073i \(0.475638\pi\)
\(422\) 4.41876 0.215102
\(423\) −5.47808 −0.266353
\(424\) 9.11864 0.442840
\(425\) 15.3998 0.746998
\(426\) −12.3873 −0.600169
\(427\) 0 0
\(428\) −14.0593 −0.679583
\(429\) 25.4467 1.22858
\(430\) −8.11864 −0.391516
\(431\) −17.0748 −0.822464 −0.411232 0.911531i \(-0.634901\pi\)
−0.411232 + 0.911531i \(0.634901\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.4746 1.65674 0.828371 0.560180i \(-0.189268\pi\)
0.828371 + 0.560180i \(0.189268\pi\)
\(434\) 0 0
\(435\) 1.26870 0.0608295
\(436\) −12.1186 −0.580378
\(437\) 3.20938 0.153525
\(438\) 9.05932 0.432871
\(439\) −2.59672 −0.123935 −0.0619673 0.998078i \(-0.519737\pi\)
−0.0619673 + 0.998078i \(0.519737\pi\)
\(440\) 4.05932 0.193520
\(441\) 0 0
\(442\) 24.1341 1.14794
\(443\) 4.02790 0.191371 0.0956857 0.995412i \(-0.469496\pi\)
0.0956857 + 0.995412i \(0.469496\pi\)
\(444\) −10.9093 −0.517731
\(445\) 10.9093 0.517149
\(446\) 11.7592 0.556814
\(447\) 3.17796 0.150312
\(448\) 0 0
\(449\) −22.1655 −1.04606 −0.523028 0.852315i \(-0.675198\pi\)
−0.523028 + 0.852315i \(0.675198\pi\)
\(450\) 4.00000 0.188562
\(451\) 19.4467 0.915707
\(452\) −5.43118 −0.255461
\(453\) 10.0593 0.472628
\(454\) 13.7592 0.645751
\(455\) 0 0
\(456\) 3.20938 0.150293
\(457\) 32.7623 1.53255 0.766277 0.642510i \(-0.222107\pi\)
0.766277 + 0.642510i \(0.222107\pi\)
\(458\) 21.6281 1.01062
\(459\) −3.84994 −0.179700
\(460\) 1.00000 0.0466252
\(461\) 10.2373 0.476798 0.238399 0.971167i \(-0.423377\pi\)
0.238399 + 0.971167i \(0.423377\pi\)
\(462\) 0 0
\(463\) −0.118640 −0.00551368 −0.00275684 0.999996i \(-0.500878\pi\)
−0.00275684 + 0.999996i \(0.500878\pi\)
\(464\) −1.26870 −0.0588979
\(465\) 6.47808 0.300414
\(466\) 26.6091 1.23264
\(467\) 8.11864 0.375686 0.187843 0.982199i \(-0.439850\pi\)
0.187843 + 0.982199i \(0.439850\pi\)
\(468\) 6.26870 0.289771
\(469\) 0 0
\(470\) −5.47808 −0.252685
\(471\) −2.07174 −0.0954608
\(472\) −6.84994 −0.315294
\(473\) −32.9562 −1.51533
\(474\) −9.38734 −0.431175
\(475\) 12.8375 0.589026
\(476\) 0 0
\(477\) −9.11864 −0.417514
\(478\) 16.0000 0.731823
\(479\) 4.55334 0.208047 0.104024 0.994575i \(-0.466828\pi\)
0.104024 + 0.994575i \(0.466828\pi\)
\(480\) 1.00000 0.0456435
\(481\) −68.3869 −3.11817
\(482\) 28.2249 1.28561
\(483\) 0 0
\(484\) 5.47808 0.249004
\(485\) −15.8061 −0.717718
\(486\) −1.00000 −0.0453609
\(487\) −21.3156 −0.965902 −0.482951 0.875647i \(-0.660435\pi\)
−0.482951 + 0.875647i \(0.660435\pi\)
\(488\) 10.0000 0.452679
\(489\) 4.07174 0.184130
\(490\) 0 0
\(491\) 2.09074 0.0943538 0.0471769 0.998887i \(-0.484978\pi\)
0.0471769 + 0.998887i \(0.484978\pi\)
\(492\) 4.79062 0.215978
\(493\) 4.88442 0.219983
\(494\) 20.1186 0.905180
\(495\) −4.05932 −0.182453
\(496\) −6.47808 −0.290874
\(497\) 0 0
\(498\) 0.0593201 0.00265820
\(499\) −7.13764 −0.319525 −0.159762 0.987156i \(-0.551073\pi\)
−0.159762 + 0.987156i \(0.551073\pi\)
\(500\) 9.00000 0.402492
\(501\) −18.0155 −0.804872
\(502\) −8.29660 −0.370295
\(503\) −11.9372 −0.532252 −0.266126 0.963938i \(-0.585744\pi\)
−0.266126 + 0.963938i \(0.585744\pi\)
\(504\) 0 0
\(505\) −14.1186 −0.628271
\(506\) 4.05932 0.180459
\(507\) 26.2966 1.16787
\(508\) −16.0593 −0.712517
\(509\) −1.56882 −0.0695367 −0.0347684 0.999395i \(-0.511069\pi\)
−0.0347684 + 0.999395i \(0.511069\pi\)
\(510\) −3.84994 −0.170478
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −3.20938 −0.141698
\(514\) 19.0279 0.839285
\(515\) 0.521920 0.0229985
\(516\) −8.11864 −0.357403
\(517\) −22.2373 −0.977994
\(518\) 0 0
\(519\) 14.5374 0.638121
\(520\) 6.26870 0.274901
\(521\) −6.75920 −0.296126 −0.148063 0.988978i \(-0.547304\pi\)
−0.148063 + 0.988978i \(0.547304\pi\)
\(522\) 1.26870 0.0555295
\(523\) −7.71536 −0.337369 −0.168685 0.985670i \(-0.553952\pi\)
−0.168685 + 0.985670i \(0.553952\pi\)
\(524\) 18.8654 0.824140
\(525\) 0 0
\(526\) −0.118640 −0.00517296
\(527\) 24.9402 1.08641
\(528\) 4.05932 0.176659
\(529\) 1.00000 0.0434783
\(530\) −9.11864 −0.396088
\(531\) 6.84994 0.297262
\(532\) 0 0
\(533\) 30.0310 1.30079
\(534\) 10.9093 0.472090
\(535\) 14.0593 0.607837
\(536\) 13.4781 0.582164
\(537\) −12.7592 −0.550600
\(538\) 2.96858 0.127985
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −1.84994 −0.0795351 −0.0397676 0.999209i \(-0.512662\pi\)
−0.0397676 + 0.999209i \(0.512662\pi\)
\(542\) −12.8968 −0.553967
\(543\) 12.5374 0.538031
\(544\) 3.84994 0.165065
\(545\) 12.1186 0.519106
\(546\) 0 0
\(547\) −37.6839 −1.61125 −0.805624 0.592427i \(-0.798170\pi\)
−0.805624 + 0.592427i \(0.798170\pi\)
\(548\) 15.5967 0.666259
\(549\) −10.0000 −0.426790
\(550\) 16.2373 0.692360
\(551\) 4.07174 0.173462
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −11.1780 −0.474906
\(555\) 10.9093 0.463072
\(556\) −6.83752 −0.289975
\(557\) 21.9965 0.932021 0.466010 0.884779i \(-0.345691\pi\)
0.466010 + 0.884779i \(0.345691\pi\)
\(558\) 6.47808 0.274239
\(559\) −50.8933 −2.15256
\(560\) 0 0
\(561\) −15.6281 −0.659821
\(562\) 31.4621 1.32715
\(563\) 35.8778 1.51207 0.756035 0.654531i \(-0.227134\pi\)
0.756035 + 0.654531i \(0.227134\pi\)
\(564\) −5.47808 −0.230669
\(565\) 5.43118 0.228491
\(566\) −10.0155 −0.420982
\(567\) 0 0
\(568\) −12.3873 −0.519761
\(569\) −21.0593 −0.882853 −0.441426 0.897298i \(-0.645527\pi\)
−0.441426 + 0.897298i \(0.645527\pi\)
\(570\) −3.20938 −0.134426
\(571\) −34.0624 −1.42547 −0.712733 0.701435i \(-0.752543\pi\)
−0.712733 + 0.701435i \(0.752543\pi\)
\(572\) 25.4467 1.06398
\(573\) 4.07174 0.170099
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −34.7154 −1.44522 −0.722610 0.691256i \(-0.757058\pi\)
−0.722610 + 0.691256i \(0.757058\pi\)
\(578\) 2.17796 0.0905912
\(579\) 23.7747 0.988042
\(580\) 1.26870 0.0526799
\(581\) 0 0
\(582\) −15.8061 −0.655184
\(583\) −37.0155 −1.53302
\(584\) 9.05932 0.374877
\(585\) −6.26870 −0.259179
\(586\) 3.83752 0.158526
\(587\) 41.5843 1.71637 0.858184 0.513342i \(-0.171593\pi\)
0.858184 + 0.513342i \(0.171593\pi\)
\(588\) 0 0
\(589\) 20.7906 0.856663
\(590\) 6.84994 0.282008
\(591\) 0.418760 0.0172255
\(592\) −10.9093 −0.448368
\(593\) 25.2811 1.03817 0.519086 0.854722i \(-0.326273\pi\)
0.519086 + 0.854722i \(0.326273\pi\)
\(594\) −4.05932 −0.166556
\(595\) 0 0
\(596\) 3.17796 0.130174
\(597\) 5.69988 0.233281
\(598\) 6.26870 0.256346
\(599\) 21.5250 0.879487 0.439743 0.898123i \(-0.355069\pi\)
0.439743 + 0.898123i \(0.355069\pi\)
\(600\) 4.00000 0.163299
\(601\) −4.17796 −0.170423 −0.0852113 0.996363i \(-0.527157\pi\)
−0.0852113 + 0.996363i \(0.527157\pi\)
\(602\) 0 0
\(603\) −13.4781 −0.548870
\(604\) 10.0593 0.409308
\(605\) −5.47808 −0.222716
\(606\) −14.1186 −0.573531
\(607\) 21.2528 0.862623 0.431311 0.902203i \(-0.358051\pi\)
0.431311 + 0.902203i \(0.358051\pi\)
\(608\) 3.20938 0.130158
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −34.3404 −1.38927
\(612\) −3.84994 −0.155625
\(613\) 2.35592 0.0951547 0.0475774 0.998868i \(-0.484850\pi\)
0.0475774 + 0.998868i \(0.484850\pi\)
\(614\) 13.9841 0.564351
\(615\) −4.79062 −0.193176
\(616\) 0 0
\(617\) −12.8778 −0.518442 −0.259221 0.965818i \(-0.583466\pi\)
−0.259221 + 0.965818i \(0.583466\pi\)
\(618\) 0.521920 0.0209947
\(619\) 11.9437 0.480059 0.240030 0.970766i \(-0.422843\pi\)
0.240030 + 0.970766i \(0.422843\pi\)
\(620\) 6.47808 0.260166
\(621\) −1.00000 −0.0401286
\(622\) −26.3873 −1.05804
\(623\) 0 0
\(624\) 6.26870 0.250949
\(625\) 11.0000 0.440000
\(626\) 9.15006 0.365710
\(627\) −13.0279 −0.520284
\(628\) −2.07174 −0.0826714
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 10.9841 0.437269 0.218634 0.975807i \(-0.429840\pi\)
0.218634 + 0.975807i \(0.429840\pi\)
\(632\) −9.38734 −0.373408
\(633\) −4.41876 −0.175630
\(634\) −20.2249 −0.803232
\(635\) 16.0593 0.637295
\(636\) −9.11864 −0.361578
\(637\) 0 0
\(638\) 5.15006 0.203893
\(639\) 12.3873 0.490036
\(640\) 1.00000 0.0395285
\(641\) 27.4621 1.08469 0.542345 0.840156i \(-0.317537\pi\)
0.542345 + 0.840156i \(0.317537\pi\)
\(642\) 14.0593 0.554877
\(643\) 3.32802 0.131244 0.0656222 0.997845i \(-0.479097\pi\)
0.0656222 + 0.997845i \(0.479097\pi\)
\(644\) 0 0
\(645\) 8.11864 0.319671
\(646\) −12.3559 −0.486137
\(647\) −31.8495 −1.25213 −0.626066 0.779770i \(-0.715336\pi\)
−0.626066 + 0.779770i \(0.715336\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 27.8061 1.09148
\(650\) 25.0748 0.983515
\(651\) 0 0
\(652\) 4.07174 0.159462
\(653\) −2.28770 −0.0895246 −0.0447623 0.998998i \(-0.514253\pi\)
−0.0447623 + 0.998998i \(0.514253\pi\)
\(654\) 12.1186 0.473876
\(655\) −18.8654 −0.737133
\(656\) 4.79062 0.187042
\(657\) −9.05932 −0.353438
\(658\) 0 0
\(659\) −8.18148 −0.318705 −0.159353 0.987222i \(-0.550941\pi\)
−0.159353 + 0.987222i \(0.550941\pi\)
\(660\) −4.05932 −0.158009
\(661\) 26.6809 1.03777 0.518883 0.854845i \(-0.326348\pi\)
0.518883 + 0.854845i \(0.326348\pi\)
\(662\) 20.6091 0.800997
\(663\) −24.1341 −0.937292
\(664\) 0.0593201 0.00230207
\(665\) 0 0
\(666\) 10.9093 0.422725
\(667\) 1.26870 0.0491243
\(668\) −18.0155 −0.697040
\(669\) −11.7592 −0.454637
\(670\) −13.4781 −0.520704
\(671\) −40.5932 −1.56708
\(672\) 0 0
\(673\) −48.6525 −1.87542 −0.937708 0.347423i \(-0.887057\pi\)
−0.937708 + 0.347423i \(0.887057\pi\)
\(674\) 26.2249 1.01014
\(675\) −4.00000 −0.153960
\(676\) 26.2966 1.01141
\(677\) −1.23728 −0.0475526 −0.0237763 0.999717i \(-0.507569\pi\)
−0.0237763 + 0.999717i \(0.507569\pi\)
\(678\) 5.43118 0.208583
\(679\) 0 0
\(680\) −3.84994 −0.147638
\(681\) −13.7592 −0.527254
\(682\) 26.2966 1.00695
\(683\) 5.34702 0.204598 0.102299 0.994754i \(-0.467380\pi\)
0.102299 + 0.994754i \(0.467380\pi\)
\(684\) −3.20938 −0.122714
\(685\) −15.5967 −0.595920
\(686\) 0 0
\(687\) −21.6281 −0.825165
\(688\) −8.11864 −0.309520
\(689\) −57.1620 −2.17770
\(690\) −1.00000 −0.0380693
\(691\) −3.53434 −0.134453 −0.0672263 0.997738i \(-0.521415\pi\)
−0.0672263 + 0.997738i \(0.521415\pi\)
\(692\) 14.5374 0.552629
\(693\) 0 0
\(694\) −25.5967 −0.971638
\(695\) 6.83752 0.259362
\(696\) 1.26870 0.0480900
\(697\) −18.4436 −0.698601
\(698\) 1.61266 0.0610401
\(699\) −26.6091 −1.00645
\(700\) 0 0
\(701\) −32.6999 −1.23506 −0.617529 0.786548i \(-0.711866\pi\)
−0.617529 + 0.786548i \(0.711866\pi\)
\(702\) −6.26870 −0.236597
\(703\) 35.0120 1.32050
\(704\) 4.05932 0.152991
\(705\) 5.47808 0.206316
\(706\) −26.6091 −1.00145
\(707\) 0 0
\(708\) 6.84994 0.257437
\(709\) 42.2213 1.58566 0.792828 0.609446i \(-0.208608\pi\)
0.792828 + 0.609446i \(0.208608\pi\)
\(710\) 12.3873 0.464889
\(711\) 9.38734 0.352053
\(712\) 10.9093 0.408842
\(713\) 6.47808 0.242606
\(714\) 0 0
\(715\) −25.4467 −0.951651
\(716\) −12.7592 −0.476834
\(717\) −16.0000 −0.597531
\(718\) 18.8375 0.703010
\(719\) −23.2966 −0.868817 −0.434408 0.900716i \(-0.643042\pi\)
−0.434408 + 0.900716i \(0.643042\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 8.69988 0.323776
\(723\) −28.2249 −1.04969
\(724\) 12.5374 0.465949
\(725\) 5.07480 0.188473
\(726\) −5.47808 −0.203311
\(727\) 32.5498 1.20721 0.603603 0.797285i \(-0.293731\pi\)
0.603603 + 0.797285i \(0.293731\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −9.05932 −0.335300
\(731\) 31.2563 1.15606
\(732\) −10.0000 −0.369611
\(733\) −8.76578 −0.323771 −0.161886 0.986810i \(-0.551758\pi\)
−0.161886 + 0.986810i \(0.551758\pi\)
\(734\) 0.746780 0.0275642
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −54.7118 −2.01534
\(738\) −4.79062 −0.176345
\(739\) −25.0996 −0.923305 −0.461653 0.887061i \(-0.652743\pi\)
−0.461653 + 0.887061i \(0.652743\pi\)
\(740\) 10.9093 0.401032
\(741\) −20.1186 −0.739077
\(742\) 0 0
\(743\) 18.3090 0.671693 0.335846 0.941917i \(-0.390978\pi\)
0.335846 + 0.941917i \(0.390978\pi\)
\(744\) 6.47808 0.237498
\(745\) −3.17796 −0.116431
\(746\) −18.9562 −0.694034
\(747\) −0.0593201 −0.00217041
\(748\) −15.6281 −0.571421
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) −28.0434 −1.02332 −0.511659 0.859189i \(-0.670969\pi\)
−0.511659 + 0.859189i \(0.670969\pi\)
\(752\) −5.47808 −0.199765
\(753\) 8.29660 0.302345
\(754\) 7.95310 0.289635
\(755\) −10.0593 −0.366096
\(756\) 0 0
\(757\) 21.7468 0.790400 0.395200 0.918595i \(-0.370675\pi\)
0.395200 + 0.918595i \(0.370675\pi\)
\(758\) −11.7154 −0.425521
\(759\) −4.05932 −0.147344
\(760\) −3.20938 −0.116416
\(761\) 39.9212 1.44714 0.723572 0.690249i \(-0.242499\pi\)
0.723572 + 0.690249i \(0.242499\pi\)
\(762\) 16.0593 0.581768
\(763\) 0 0
\(764\) 4.07174 0.147310
\(765\) 3.84994 0.139195
\(766\) −26.1655 −0.945399
\(767\) 42.9402 1.55048
\(768\) 1.00000 0.0360844
\(769\) 29.3245 1.05747 0.528734 0.848787i \(-0.322667\pi\)
0.528734 + 0.848787i \(0.322667\pi\)
\(770\) 0 0
\(771\) −19.0279 −0.685273
\(772\) 23.7747 0.855669
\(773\) −5.35944 −0.192766 −0.0963828 0.995344i \(-0.530727\pi\)
−0.0963828 + 0.995344i \(0.530727\pi\)
\(774\) 8.11864 0.291818
\(775\) 25.9123 0.930798
\(776\) −15.8061 −0.567406
\(777\) 0 0
\(778\) 27.7937 0.996452
\(779\) −15.3749 −0.550864
\(780\) −6.26870 −0.224455
\(781\) 50.2842 1.79931
\(782\) −3.84994 −0.137674
\(783\) −1.26870 −0.0453396
\(784\) 0 0
\(785\) 2.07174 0.0739436
\(786\) −18.8654 −0.672907
\(787\) −27.8968 −0.994415 −0.497207 0.867632i \(-0.665641\pi\)
−0.497207 + 0.867632i \(0.665641\pi\)
\(788\) 0.418760 0.0149177
\(789\) 0.118640 0.00422371
\(790\) 9.38734 0.333987
\(791\) 0 0
\(792\) −4.05932 −0.144242
\(793\) −62.6870 −2.22608
\(794\) 3.84994 0.136629
\(795\) 9.11864 0.323405
\(796\) 5.69988 0.202027
\(797\) −13.6560 −0.483722 −0.241861 0.970311i \(-0.577758\pi\)
−0.241861 + 0.970311i \(0.577758\pi\)
\(798\) 0 0
\(799\) 21.0903 0.746120
\(800\) 4.00000 0.141421
\(801\) −10.9093 −0.385460
\(802\) −7.17796 −0.253463
\(803\) −36.7747 −1.29775
\(804\) −13.4781 −0.475335
\(805\) 0 0
\(806\) 40.6091 1.43040
\(807\) −2.96858 −0.104499
\(808\) −14.1186 −0.496692
\(809\) −16.7278 −0.588117 −0.294059 0.955787i \(-0.595006\pi\)
−0.294059 + 0.955787i \(0.595006\pi\)
\(810\) 1.00000 0.0351364
\(811\) 17.3998 0.610988 0.305494 0.952194i \(-0.401178\pi\)
0.305494 + 0.952194i \(0.401178\pi\)
\(812\) 0 0
\(813\) 12.8968 0.452312
\(814\) 44.2842 1.55216
\(815\) −4.07174 −0.142627
\(816\) −3.84994 −0.134775
\(817\) 26.0558 0.911577
\(818\) −16.3749 −0.572536
\(819\) 0 0
\(820\) −4.79062 −0.167296
\(821\) −28.7313 −1.00273 −0.501365 0.865236i \(-0.667168\pi\)
−0.501365 + 0.865236i \(0.667168\pi\)
\(822\) −15.5967 −0.543998
\(823\) −32.0310 −1.11653 −0.558265 0.829663i \(-0.688533\pi\)
−0.558265 + 0.829663i \(0.688533\pi\)
\(824\) 0.521920 0.0181819
\(825\) −16.2373 −0.565310
\(826\) 0 0
\(827\) 26.3214 0.915286 0.457643 0.889136i \(-0.348694\pi\)
0.457643 + 0.889136i \(0.348694\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 21.5339 0.747903 0.373951 0.927448i \(-0.378003\pi\)
0.373951 + 0.927448i \(0.378003\pi\)
\(830\) −0.0593201 −0.00205903
\(831\) 11.1780 0.387759
\(832\) 6.26870 0.217328
\(833\) 0 0
\(834\) 6.83752 0.236764
\(835\) 18.0155 0.623451
\(836\) −13.0279 −0.450579
\(837\) −6.47808 −0.223915
\(838\) −25.9372 −0.895984
\(839\) −6.25322 −0.215885 −0.107943 0.994157i \(-0.534426\pi\)
−0.107943 + 0.994157i \(0.534426\pi\)
\(840\) 0 0
\(841\) −27.3904 −0.944497
\(842\) −3.13764 −0.108130
\(843\) −31.4621 −1.08361
\(844\) −4.41876 −0.152100
\(845\) −26.2966 −0.904631
\(846\) 5.47808 0.188340
\(847\) 0 0
\(848\) −9.11864 −0.313135
\(849\) 10.0155 0.343730
\(850\) −15.3998 −0.528207
\(851\) 10.9093 0.373965
\(852\) 12.3873 0.424383
\(853\) −4.65604 −0.159420 −0.0797099 0.996818i \(-0.525399\pi\)
−0.0797099 + 0.996818i \(0.525399\pi\)
\(854\) 0 0
\(855\) 3.20938 0.109758
\(856\) 14.0593 0.480538
\(857\) 45.6839 1.56053 0.780267 0.625447i \(-0.215083\pi\)
0.780267 + 0.625447i \(0.215083\pi\)
\(858\) −25.4467 −0.868735
\(859\) −42.0089 −1.43333 −0.716663 0.697420i \(-0.754331\pi\)
−0.716663 + 0.697420i \(0.754331\pi\)
\(860\) 8.11864 0.276843
\(861\) 0 0
\(862\) 17.0748 0.581570
\(863\) −50.2059 −1.70903 −0.854514 0.519429i \(-0.826145\pi\)
−0.854514 + 0.519429i \(0.826145\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.5374 −0.494286
\(866\) −34.4746 −1.17149
\(867\) −2.17796 −0.0739674
\(868\) 0 0
\(869\) 38.1062 1.29266
\(870\) −1.26870 −0.0430130
\(871\) −84.4900 −2.86284
\(872\) 12.1186 0.410389
\(873\) 15.8061 0.534955
\(874\) −3.20938 −0.108559
\(875\) 0 0
\(876\) −9.05932 −0.306086
\(877\) −8.45018 −0.285342 −0.142671 0.989770i \(-0.545569\pi\)
−0.142671 + 0.989770i \(0.545569\pi\)
\(878\) 2.59672 0.0876351
\(879\) −3.83752 −0.129436
\(880\) −4.05932 −0.136840
\(881\) −1.17796 −0.0396865 −0.0198432 0.999803i \(-0.506317\pi\)
−0.0198432 + 0.999803i \(0.506317\pi\)
\(882\) 0 0
\(883\) 27.5653 0.927646 0.463823 0.885928i \(-0.346477\pi\)
0.463823 + 0.885928i \(0.346477\pi\)
\(884\) −24.1341 −0.811718
\(885\) −6.84994 −0.230258
\(886\) −4.02790 −0.135320
\(887\) −48.2373 −1.61965 −0.809825 0.586672i \(-0.800438\pi\)
−0.809825 + 0.586672i \(0.800438\pi\)
\(888\) 10.9093 0.366091
\(889\) 0 0
\(890\) −10.9093 −0.365679
\(891\) 4.05932 0.135992
\(892\) −11.7592 −0.393727
\(893\) 17.5812 0.588334
\(894\) −3.17796 −0.106287
\(895\) 12.7592 0.426493
\(896\) 0 0
\(897\) −6.26870 −0.209306
\(898\) 22.1655 0.739674
\(899\) 8.21874 0.274110
\(900\) −4.00000 −0.133333
\(901\) 35.1062 1.16956
\(902\) −19.4467 −0.647503
\(903\) 0 0
\(904\) 5.43118 0.180638
\(905\) −12.5374 −0.416757
\(906\) −10.0593 −0.334199
\(907\) −24.2687 −0.805829 −0.402914 0.915238i \(-0.632003\pi\)
−0.402914 + 0.915238i \(0.632003\pi\)
\(908\) −13.7592 −0.456615
\(909\) 14.1186 0.468286
\(910\) 0 0
\(911\) −6.68088 −0.221347 −0.110674 0.993857i \(-0.535301\pi\)
−0.110674 + 0.993857i \(0.535301\pi\)
\(912\) −3.20938 −0.106273
\(913\) −0.240799 −0.00796930
\(914\) −32.7623 −1.08368
\(915\) 10.0000 0.330590
\(916\) −21.6281 −0.714614
\(917\) 0 0
\(918\) 3.84994 0.127067
\(919\) 4.28464 0.141337 0.0706686 0.997500i \(-0.477487\pi\)
0.0706686 + 0.997500i \(0.477487\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −13.9841 −0.460790
\(922\) −10.2373 −0.337147
\(923\) 77.6525 2.55596
\(924\) 0 0
\(925\) 43.6370 1.43478
\(926\) 0.118640 0.00389876
\(927\) −0.521920 −0.0171421
\(928\) 1.26870 0.0416471
\(929\) 3.18454 0.104481 0.0522407 0.998635i \(-0.483364\pi\)
0.0522407 + 0.998635i \(0.483364\pi\)
\(930\) −6.47808 −0.212425
\(931\) 0 0
\(932\) −26.6091 −0.871611
\(933\) 26.3873 0.863883
\(934\) −8.11864 −0.265650
\(935\) 15.6281 0.511095
\(936\) −6.26870 −0.204899
\(937\) −25.8619 −0.844871 −0.422436 0.906393i \(-0.638825\pi\)
−0.422436 + 0.906393i \(0.638825\pi\)
\(938\) 0 0
\(939\) −9.15006 −0.298601
\(940\) 5.47808 0.178675
\(941\) −3.70340 −0.120727 −0.0603637 0.998176i \(-0.519226\pi\)
−0.0603637 + 0.998176i \(0.519226\pi\)
\(942\) 2.07174 0.0675010
\(943\) −4.79062 −0.156004
\(944\) 6.84994 0.222947
\(945\) 0 0
\(946\) 32.9562 1.07150
\(947\) −50.0155 −1.62528 −0.812642 0.582763i \(-0.801972\pi\)
−0.812642 + 0.582763i \(0.801972\pi\)
\(948\) 9.38734 0.304887
\(949\) −56.7902 −1.84349
\(950\) −12.8375 −0.416504
\(951\) 20.2249 0.655836
\(952\) 0 0
\(953\) 24.4188 0.791001 0.395501 0.918466i \(-0.370571\pi\)
0.395501 + 0.918466i \(0.370571\pi\)
\(954\) 9.11864 0.295227
\(955\) −4.07174 −0.131758
\(956\) −16.0000 −0.517477
\(957\) −5.15006 −0.166478
\(958\) −4.55334 −0.147112
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 10.9655 0.353726
\(962\) 68.3869 2.20488
\(963\) −14.0593 −0.453055
\(964\) −28.2249 −0.909062
\(965\) −23.7747 −0.765334
\(966\) 0 0
\(967\) −18.4152 −0.592194 −0.296097 0.955158i \(-0.595685\pi\)
−0.296097 + 0.955158i \(0.595685\pi\)
\(968\) −5.47808 −0.176072
\(969\) 12.3559 0.396929
\(970\) 15.8061 0.507503
\(971\) 17.9407 0.575744 0.287872 0.957669i \(-0.407052\pi\)
0.287872 + 0.957669i \(0.407052\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 21.3156 0.682996
\(975\) −25.0748 −0.803036
\(976\) −10.0000 −0.320092
\(977\) 28.4342 0.909692 0.454846 0.890570i \(-0.349694\pi\)
0.454846 + 0.890570i \(0.349694\pi\)
\(978\) −4.07174 −0.130200
\(979\) −44.2842 −1.41533
\(980\) 0 0
\(981\) −12.1186 −0.386918
\(982\) −2.09074 −0.0667182
\(983\) −4.60914 −0.147009 −0.0735044 0.997295i \(-0.523418\pi\)
−0.0735044 + 0.997295i \(0.523418\pi\)
\(984\) −4.79062 −0.152719
\(985\) −0.418760 −0.0133428
\(986\) −4.88442 −0.155552
\(987\) 0 0
\(988\) −20.1186 −0.640059
\(989\) 8.11864 0.258158
\(990\) 4.05932 0.129014
\(991\) −41.8778 −1.33029 −0.665147 0.746713i \(-0.731631\pi\)
−0.665147 + 0.746713i \(0.731631\pi\)
\(992\) 6.47808 0.205679
\(993\) −20.6091 −0.654011
\(994\) 0 0
\(995\) −5.69988 −0.180698
\(996\) −0.0593201 −0.00187963
\(997\) −33.3280 −1.05551 −0.527754 0.849397i \(-0.676966\pi\)
−0.527754 + 0.849397i \(0.676966\pi\)
\(998\) 7.13764 0.225938
\(999\) −10.9093 −0.345154
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 6762.2.a.cf.1.3 3
7.3 odd 6 966.2.i.k.415.2 yes 6
7.5 odd 6 966.2.i.k.277.2 6
7.6 odd 2 6762.2.a.ce.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.k.277.2 6 7.5 odd 6
966.2.i.k.415.2 yes 6 7.3 odd 6
6762.2.a.ce.1.3 3 7.6 odd 2
6762.2.a.cf.1.3 3 1.1 even 1 trivial