Properties

Label 6762.2.a.bz.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +4.47214 q^{13} +2.00000 q^{15} +1.00000 q^{16} -4.47214 q^{17} +1.00000 q^{18} -2.47214 q^{19} -2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +4.47214 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +2.47214 q^{31} +1.00000 q^{32} -4.47214 q^{34} +1.00000 q^{36} +6.94427 q^{37} -2.47214 q^{38} -4.47214 q^{39} -2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -2.00000 q^{45} -1.00000 q^{46} +2.47214 q^{47} -1.00000 q^{48} -1.00000 q^{50} +4.47214 q^{51} +4.47214 q^{52} -10.9443 q^{53} -1.00000 q^{54} +2.47214 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} -8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +1.00000 q^{69} +4.94427 q^{71} +1.00000 q^{72} -14.9443 q^{73} +6.94427 q^{74} +1.00000 q^{75} -2.47214 q^{76} -4.47214 q^{78} -4.94427 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} -4.94427 q^{86} +2.00000 q^{87} -9.41641 q^{89} -2.00000 q^{90} -1.00000 q^{92} -2.47214 q^{93} +2.47214 q^{94} +4.94427 q^{95} -1.00000 q^{96} +0.472136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} + 2q^{8} + 2q^{9} - 4q^{10} - 2q^{12} + 4q^{15} + 2q^{16} + 2q^{18} + 4q^{19} - 4q^{20} - 2q^{23} - 2q^{24} - 2q^{25} - 2q^{27} - 4q^{29} + 4q^{30} - 4q^{31} + 2q^{32} + 2q^{36} - 4q^{37} + 4q^{38} - 4q^{40} + 12q^{41} + 8q^{43} - 4q^{45} - 2q^{46} - 4q^{47} - 2q^{48} - 2q^{50} - 4q^{53} - 2q^{54} - 4q^{57} - 4q^{58} - 8q^{59} + 4q^{60} + 12q^{61} - 4q^{62} + 2q^{64} - 8q^{67} + 2q^{69} - 8q^{71} + 2q^{72} - 12q^{73} - 4q^{74} + 2q^{75} + 4q^{76} + 8q^{79} - 4q^{80} + 2q^{81} + 12q^{82} + 4q^{83} + 8q^{86} + 4q^{87} + 8q^{89} - 4q^{90} - 2q^{92} + 4q^{93} - 4q^{94} - 8q^{95} - 2q^{96} - 8q^{97} + O(q^{100}) \)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.47214 0.877058
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −4.47214 −0.766965
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) −2.47214 −0.401033
\(39\) −4.47214 −0.716115
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.94427 −0.753994 −0.376997 0.926214i \(-0.623043\pi\)
−0.376997 + 0.926214i \(0.623043\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) 2.47214 0.360598 0.180299 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 4.47214 0.626224
\(52\) 4.47214 0.620174
\(53\) −10.9443 −1.50331 −0.751656 0.659556i \(-0.770744\pi\)
−0.751656 + 0.659556i \(0.770744\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 2.47214 0.327442
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 2.47214 0.313962
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.94427 −1.10940
\(66\) 0 0
\(67\) 4.94427 0.604039 0.302019 0.953302i \(-0.402339\pi\)
0.302019 + 0.953302i \(0.402339\pi\)
\(68\) −4.47214 −0.542326
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.9443 −1.74909 −0.874547 0.484940i \(-0.838841\pi\)
−0.874547 + 0.484940i \(0.838841\pi\)
\(74\) 6.94427 0.807255
\(75\) 1.00000 0.115470
\(76\) −2.47214 −0.283573
\(77\) 0 0
\(78\) −4.47214 −0.506370
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −2.47214 −0.271352 −0.135676 0.990753i \(-0.543321\pi\)
−0.135676 + 0.990753i \(0.543321\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) −4.94427 −0.533155
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −9.41641 −0.998137 −0.499069 0.866562i \(-0.666324\pi\)
−0.499069 + 0.866562i \(0.666324\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −2.47214 −0.256349
\(94\) 2.47214 0.254981
\(95\) 4.94427 0.507272
\(96\) −1.00000 −0.102062
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 9.41641 0.936968 0.468484 0.883472i \(-0.344800\pi\)
0.468484 + 0.883472i \(0.344800\pi\)
\(102\) 4.47214 0.442807
\(103\) 4.94427 0.487174 0.243587 0.969879i \(-0.421676\pi\)
0.243587 + 0.969879i \(0.421676\pi\)
\(104\) 4.47214 0.438529
\(105\) 0 0
\(106\) −10.9443 −1.06300
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) −6.94427 −0.659121
\(112\) 0 0
\(113\) −18.9443 −1.78213 −0.891064 0.453878i \(-0.850040\pi\)
−0.891064 + 0.453878i \(0.850040\pi\)
\(114\) 2.47214 0.231537
\(115\) 2.00000 0.186501
\(116\) −2.00000 −0.185695
\(117\) 4.47214 0.413449
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) 6.00000 0.543214
\(123\) −6.00000 −0.541002
\(124\) 2.47214 0.222004
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 20.9443 1.85850 0.929252 0.369447i \(-0.120453\pi\)
0.929252 + 0.369447i \(0.120453\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.94427 0.435319
\(130\) −8.94427 −0.784465
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.94427 0.427120
\(135\) 2.00000 0.172133
\(136\) −4.47214 −0.383482
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 1.00000 0.0851257
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) 4.94427 0.414914
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) −14.9443 −1.23680
\(147\) 0 0
\(148\) 6.94427 0.570816
\(149\) −1.05573 −0.0864886 −0.0432443 0.999065i \(-0.513769\pi\)
−0.0432443 + 0.999065i \(0.513769\pi\)
\(150\) 1.00000 0.0816497
\(151\) 4.94427 0.402359 0.201180 0.979554i \(-0.435523\pi\)
0.201180 + 0.979554i \(0.435523\pi\)
\(152\) −2.47214 −0.200517
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) −4.94427 −0.397133
\(156\) −4.47214 −0.358057
\(157\) 2.94427 0.234978 0.117489 0.993074i \(-0.462515\pi\)
0.117489 + 0.993074i \(0.462515\pi\)
\(158\) −4.94427 −0.393345
\(159\) 10.9443 0.867937
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.94427 −0.700569 −0.350285 0.936643i \(-0.613915\pi\)
−0.350285 + 0.936643i \(0.613915\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −2.47214 −0.191875
\(167\) 15.4164 1.19296 0.596479 0.802629i \(-0.296566\pi\)
0.596479 + 0.802629i \(0.296566\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 8.94427 0.685994
\(171\) −2.47214 −0.189049
\(172\) −4.94427 −0.376997
\(173\) −0.472136 −0.0358958 −0.0179479 0.999839i \(-0.505713\pi\)
−0.0179479 + 0.999839i \(0.505713\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) −9.41641 −0.705790
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) −1.00000 −0.0737210
\(185\) −13.8885 −1.02111
\(186\) −2.47214 −0.181266
\(187\) 0 0
\(188\) 2.47214 0.180299
\(189\) 0 0
\(190\) 4.94427 0.358695
\(191\) −3.05573 −0.221105 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.8885 0.855756 0.427878 0.903836i \(-0.359261\pi\)
0.427878 + 0.903836i \(0.359261\pi\)
\(194\) 0.472136 0.0338974
\(195\) 8.94427 0.640513
\(196\) 0 0
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) −3.05573 −0.216615 −0.108307 0.994117i \(-0.534543\pi\)
−0.108307 + 0.994117i \(0.534543\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.94427 −0.348742
\(202\) 9.41641 0.662536
\(203\) 0 0
\(204\) 4.47214 0.313112
\(205\) −12.0000 −0.838116
\(206\) 4.94427 0.344484
\(207\) −1.00000 −0.0695048
\(208\) 4.47214 0.310087
\(209\) 0 0
\(210\) 0 0
\(211\) −0.944272 −0.0650064 −0.0325032 0.999472i \(-0.510348\pi\)
−0.0325032 + 0.999472i \(0.510348\pi\)
\(212\) −10.9443 −0.751656
\(213\) −4.94427 −0.338776
\(214\) −8.00000 −0.546869
\(215\) 9.88854 0.674393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −2.94427 −0.199411
\(219\) 14.9443 1.00984
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) −6.94427 −0.466069
\(223\) −18.4721 −1.23699 −0.618493 0.785790i \(-0.712256\pi\)
−0.618493 + 0.785790i \(0.712256\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −18.9443 −1.26015
\(227\) −5.52786 −0.366897 −0.183449 0.983029i \(-0.558726\pi\)
−0.183449 + 0.983029i \(0.558726\pi\)
\(228\) 2.47214 0.163721
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.47214 0.292353
\(235\) −4.94427 −0.322529
\(236\) −4.00000 −0.260378
\(237\) 4.94427 0.321165
\(238\) 0 0
\(239\) 20.9443 1.35477 0.677386 0.735628i \(-0.263113\pi\)
0.677386 + 0.735628i \(0.263113\pi\)
\(240\) 2.00000 0.129099
\(241\) −20.4721 −1.31873 −0.659363 0.751825i \(-0.729174\pi\)
−0.659363 + 0.751825i \(0.729174\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −11.0557 −0.703459
\(248\) 2.47214 0.156981
\(249\) 2.47214 0.156665
\(250\) 12.0000 0.758947
\(251\) 5.52786 0.348916 0.174458 0.984665i \(-0.444183\pi\)
0.174458 + 0.984665i \(0.444183\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 20.9443 1.31416
\(255\) −8.94427 −0.560112
\(256\) 1.00000 0.0625000
\(257\) 2.94427 0.183659 0.0918293 0.995775i \(-0.470729\pi\)
0.0918293 + 0.995775i \(0.470729\pi\)
\(258\) 4.94427 0.307817
\(259\) 0 0
\(260\) −8.94427 −0.554700
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 21.8885 1.34460
\(266\) 0 0
\(267\) 9.41641 0.576275
\(268\) 4.94427 0.302019
\(269\) −16.4721 −1.00432 −0.502162 0.864774i \(-0.667462\pi\)
−0.502162 + 0.864774i \(0.667462\pi\)
\(270\) 2.00000 0.121716
\(271\) −15.4164 −0.936480 −0.468240 0.883601i \(-0.655112\pi\)
−0.468240 + 0.883601i \(0.655112\pi\)
\(272\) −4.47214 −0.271163
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −11.8885 −0.714313 −0.357157 0.934044i \(-0.616254\pi\)
−0.357157 + 0.934044i \(0.616254\pi\)
\(278\) −8.94427 −0.536442
\(279\) 2.47214 0.148003
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −2.47214 −0.147214
\(283\) −18.4721 −1.09805 −0.549027 0.835804i \(-0.685002\pi\)
−0.549027 + 0.835804i \(0.685002\pi\)
\(284\) 4.94427 0.293389
\(285\) −4.94427 −0.292873
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 3.00000 0.176471
\(290\) 4.00000 0.234888
\(291\) −0.472136 −0.0276771
\(292\) −14.9443 −0.874547
\(293\) −27.8885 −1.62927 −0.814633 0.579977i \(-0.803062\pi\)
−0.814633 + 0.579977i \(0.803062\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 6.94427 0.403628
\(297\) 0 0
\(298\) −1.05573 −0.0611567
\(299\) −4.47214 −0.258630
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 4.94427 0.284511
\(303\) −9.41641 −0.540958
\(304\) −2.47214 −0.141787
\(305\) −12.0000 −0.687118
\(306\) −4.47214 −0.255655
\(307\) −24.9443 −1.42364 −0.711822 0.702360i \(-0.752130\pi\)
−0.711822 + 0.702360i \(0.752130\pi\)
\(308\) 0 0
\(309\) −4.94427 −0.281270
\(310\) −4.94427 −0.280816
\(311\) 2.47214 0.140182 0.0700910 0.997541i \(-0.477671\pi\)
0.0700910 + 0.997541i \(0.477671\pi\)
\(312\) −4.47214 −0.253185
\(313\) 13.4164 0.758340 0.379170 0.925327i \(-0.376210\pi\)
0.379170 + 0.925327i \(0.376210\pi\)
\(314\) 2.94427 0.166155
\(315\) 0 0
\(316\) −4.94427 −0.278137
\(317\) −13.0557 −0.733283 −0.366641 0.930362i \(-0.619492\pi\)
−0.366641 + 0.930362i \(0.619492\pi\)
\(318\) 10.9443 0.613724
\(319\) 0 0
\(320\) −2.00000 −0.111803
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 11.0557 0.615157
\(324\) 1.00000 0.0555556
\(325\) −4.47214 −0.248069
\(326\) −8.94427 −0.495377
\(327\) 2.94427 0.162819
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −13.8885 −0.763383 −0.381692 0.924290i \(-0.624658\pi\)
−0.381692 + 0.924290i \(0.624658\pi\)
\(332\) −2.47214 −0.135676
\(333\) 6.94427 0.380544
\(334\) 15.4164 0.843548
\(335\) −9.88854 −0.540269
\(336\) 0 0
\(337\) −2.94427 −0.160385 −0.0801924 0.996779i \(-0.525553\pi\)
−0.0801924 + 0.996779i \(0.525553\pi\)
\(338\) 7.00000 0.380750
\(339\) 18.9443 1.02891
\(340\) 8.94427 0.485071
\(341\) 0 0
\(342\) −2.47214 −0.133678
\(343\) 0 0
\(344\) −4.94427 −0.266577
\(345\) −2.00000 −0.107676
\(346\) −0.472136 −0.0253822
\(347\) −0.944272 −0.0506912 −0.0253456 0.999679i \(-0.508069\pi\)
−0.0253456 + 0.999679i \(0.508069\pi\)
\(348\) 2.00000 0.107211
\(349\) −24.4721 −1.30996 −0.654982 0.755645i \(-0.727324\pi\)
−0.654982 + 0.755645i \(0.727324\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 4.00000 0.212598
\(355\) −9.88854 −0.524829
\(356\) −9.41641 −0.499069
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −30.8328 −1.62729 −0.813647 0.581359i \(-0.802521\pi\)
−0.813647 + 0.581359i \(0.802521\pi\)
\(360\) −2.00000 −0.105409
\(361\) −12.8885 −0.678344
\(362\) −2.00000 −0.105118
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 29.8885 1.56444
\(366\) −6.00000 −0.313625
\(367\) −30.8328 −1.60946 −0.804730 0.593641i \(-0.797690\pi\)
−0.804730 + 0.593641i \(0.797690\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 6.00000 0.312348
\(370\) −13.8885 −0.722031
\(371\) 0 0
\(372\) −2.47214 −0.128174
\(373\) 6.94427 0.359561 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 2.47214 0.127491
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 4.94427 0.253636
\(381\) −20.9443 −1.07301
\(382\) −3.05573 −0.156345
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 11.8885 0.605111
\(387\) −4.94427 −0.251331
\(388\) 0.472136 0.0239691
\(389\) −18.9443 −0.960513 −0.480256 0.877128i \(-0.659456\pi\)
−0.480256 + 0.877128i \(0.659456\pi\)
\(390\) 8.94427 0.452911
\(391\) 4.47214 0.226166
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) −14.9443 −0.752882
\(395\) 9.88854 0.497547
\(396\) 0 0
\(397\) 36.4721 1.83048 0.915242 0.402905i \(-0.131999\pi\)
0.915242 + 0.402905i \(0.131999\pi\)
\(398\) −3.05573 −0.153170
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) −4.94427 −0.246598
\(403\) 11.0557 0.550725
\(404\) 9.41641 0.468484
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 4.47214 0.221404
\(409\) 15.8885 0.785638 0.392819 0.919616i \(-0.371500\pi\)
0.392819 + 0.919616i \(0.371500\pi\)
\(410\) −12.0000 −0.592638
\(411\) 14.0000 0.690569
\(412\) 4.94427 0.243587
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 4.94427 0.242705
\(416\) 4.47214 0.219265
\(417\) 8.94427 0.438003
\(418\) 0 0
\(419\) 7.41641 0.362315 0.181158 0.983454i \(-0.442016\pi\)
0.181158 + 0.983454i \(0.442016\pi\)
\(420\) 0 0
\(421\) 32.8328 1.60017 0.800087 0.599884i \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(422\) −0.944272 −0.0459664
\(423\) 2.47214 0.120199
\(424\) −10.9443 −0.531501
\(425\) 4.47214 0.216930
\(426\) −4.94427 −0.239551
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) 9.88854 0.476868
\(431\) 17.8885 0.861661 0.430830 0.902433i \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −25.4164 −1.22143 −0.610717 0.791849i \(-0.709119\pi\)
−0.610717 + 0.791849i \(0.709119\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) −2.94427 −0.141005
\(437\) 2.47214 0.118258
\(438\) 14.9443 0.714065
\(439\) −28.3607 −1.35358 −0.676791 0.736175i \(-0.736630\pi\)
−0.676791 + 0.736175i \(0.736630\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −20.0000 −0.951303
\(443\) 34.8328 1.65496 0.827479 0.561497i \(-0.189775\pi\)
0.827479 + 0.561497i \(0.189775\pi\)
\(444\) −6.94427 −0.329561
\(445\) 18.8328 0.892761
\(446\) −18.4721 −0.874681
\(447\) 1.05573 0.0499342
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −18.9443 −0.891064
\(453\) −4.94427 −0.232302
\(454\) −5.52786 −0.259436
\(455\) 0 0
\(456\) 2.47214 0.115768
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 14.0000 0.654177
\(459\) 4.47214 0.208741
\(460\) 2.00000 0.0932505
\(461\) −6.58359 −0.306628 −0.153314 0.988177i \(-0.548995\pi\)
−0.153314 + 0.988177i \(0.548995\pi\)
\(462\) 0 0
\(463\) 6.11146 0.284023 0.142012 0.989865i \(-0.454643\pi\)
0.142012 + 0.989865i \(0.454643\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 4.94427 0.229285
\(466\) −6.00000 −0.277945
\(467\) −33.3050 −1.54117 −0.770585 0.637338i \(-0.780036\pi\)
−0.770585 + 0.637338i \(0.780036\pi\)
\(468\) 4.47214 0.206725
\(469\) 0 0
\(470\) −4.94427 −0.228062
\(471\) −2.94427 −0.135665
\(472\) −4.00000 −0.184115
\(473\) 0 0
\(474\) 4.94427 0.227098
\(475\) 2.47214 0.113429
\(476\) 0 0
\(477\) −10.9443 −0.501104
\(478\) 20.9443 0.957969
\(479\) 4.94427 0.225910 0.112955 0.993600i \(-0.463968\pi\)
0.112955 + 0.993600i \(0.463968\pi\)
\(480\) 2.00000 0.0912871
\(481\) 31.0557 1.41602
\(482\) −20.4721 −0.932480
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −0.944272 −0.0428772
\(486\) −1.00000 −0.0453609
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 6.00000 0.271607
\(489\) 8.94427 0.404474
\(490\) 0 0
\(491\) 40.9443 1.84779 0.923895 0.382647i \(-0.124987\pi\)
0.923895 + 0.382647i \(0.124987\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.94427 0.402830
\(494\) −11.0557 −0.497421
\(495\) 0 0
\(496\) 2.47214 0.111002
\(497\) 0 0
\(498\) 2.47214 0.110779
\(499\) 32.9443 1.47479 0.737394 0.675463i \(-0.236056\pi\)
0.737394 + 0.675463i \(0.236056\pi\)
\(500\) 12.0000 0.536656
\(501\) −15.4164 −0.688754
\(502\) 5.52786 0.246721
\(503\) 9.88854 0.440908 0.220454 0.975397i \(-0.429246\pi\)
0.220454 + 0.975397i \(0.429246\pi\)
\(504\) 0 0
\(505\) −18.8328 −0.838049
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 20.9443 0.929252
\(509\) −6.58359 −0.291813 −0.145906 0.989298i \(-0.546610\pi\)
−0.145906 + 0.989298i \(0.546610\pi\)
\(510\) −8.94427 −0.396059
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.47214 0.109147
\(514\) 2.94427 0.129866
\(515\) −9.88854 −0.435741
\(516\) 4.94427 0.217659
\(517\) 0 0
\(518\) 0 0
\(519\) 0.472136 0.0207245
\(520\) −8.94427 −0.392232
\(521\) −33.4164 −1.46400 −0.732000 0.681305i \(-0.761413\pi\)
−0.732000 + 0.681305i \(0.761413\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 25.3050 1.10651 0.553254 0.833013i \(-0.313386\pi\)
0.553254 + 0.833013i \(0.313386\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −11.0557 −0.481595
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 21.8885 0.950778
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 26.8328 1.16226
\(534\) 9.41641 0.407488
\(535\) 16.0000 0.691740
\(536\) 4.94427 0.213560
\(537\) 20.0000 0.863064
\(538\) −16.4721 −0.710164
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −15.4164 −0.662191
\(543\) 2.00000 0.0858282
\(544\) −4.47214 −0.191741
\(545\) 5.88854 0.252238
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 4.94427 0.210633
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −11.8885 −0.505096
\(555\) 13.8885 0.589536
\(556\) −8.94427 −0.379322
\(557\) −17.0557 −0.722674 −0.361337 0.932435i \(-0.617680\pi\)
−0.361337 + 0.932435i \(0.617680\pi\)
\(558\) 2.47214 0.104654
\(559\) −22.1115 −0.935215
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −26.4721 −1.11567 −0.557834 0.829953i \(-0.688367\pi\)
−0.557834 + 0.829953i \(0.688367\pi\)
\(564\) −2.47214 −0.104096
\(565\) 37.8885 1.59398
\(566\) −18.4721 −0.776442
\(567\) 0 0
\(568\) 4.94427 0.207457
\(569\) −7.88854 −0.330705 −0.165352 0.986235i \(-0.552876\pi\)
−0.165352 + 0.986235i \(0.552876\pi\)
\(570\) −4.94427 −0.207093
\(571\) −12.9443 −0.541701 −0.270850 0.962621i \(-0.587305\pi\)
−0.270850 + 0.962621i \(0.587305\pi\)
\(572\) 0 0
\(573\) 3.05573 0.127655
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.94427 0.122572 0.0612858 0.998120i \(-0.480480\pi\)
0.0612858 + 0.998120i \(0.480480\pi\)
\(578\) 3.00000 0.124784
\(579\) −11.8885 −0.494071
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) −0.472136 −0.0195707
\(583\) 0 0
\(584\) −14.9443 −0.618398
\(585\) −8.94427 −0.369800
\(586\) −27.8885 −1.15207
\(587\) 0.944272 0.0389743 0.0194871 0.999810i \(-0.493797\pi\)
0.0194871 + 0.999810i \(0.493797\pi\)
\(588\) 0 0
\(589\) −6.11146 −0.251818
\(590\) 8.00000 0.329355
\(591\) 14.9443 0.614725
\(592\) 6.94427 0.285408
\(593\) 28.8328 1.18402 0.592011 0.805930i \(-0.298334\pi\)
0.592011 + 0.805930i \(0.298334\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.05573 −0.0432443
\(597\) 3.05573 0.125063
\(598\) −4.47214 −0.182879
\(599\) 33.8885 1.38465 0.692324 0.721587i \(-0.256587\pi\)
0.692324 + 0.721587i \(0.256587\pi\)
\(600\) 1.00000 0.0408248
\(601\) −40.8328 −1.66561 −0.832803 0.553570i \(-0.813265\pi\)
−0.832803 + 0.553570i \(0.813265\pi\)
\(602\) 0 0
\(603\) 4.94427 0.201346
\(604\) 4.94427 0.201180
\(605\) 22.0000 0.894427
\(606\) −9.41641 −0.382515
\(607\) 28.3607 1.15112 0.575562 0.817758i \(-0.304783\pi\)
0.575562 + 0.817758i \(0.304783\pi\)
\(608\) −2.47214 −0.100258
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 11.0557 0.447267
\(612\) −4.47214 −0.180775
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) −24.9443 −1.00667
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −4.94427 −0.198888
\(619\) 34.4721 1.38555 0.692776 0.721153i \(-0.256387\pi\)
0.692776 + 0.721153i \(0.256387\pi\)
\(620\) −4.94427 −0.198567
\(621\) 1.00000 0.0401286
\(622\) 2.47214 0.0991236
\(623\) 0 0
\(624\) −4.47214 −0.179029
\(625\) −19.0000 −0.760000
\(626\) 13.4164 0.536228
\(627\) 0 0
\(628\) 2.94427 0.117489
\(629\) −31.0557 −1.23827
\(630\) 0 0
\(631\) −11.0557 −0.440122 −0.220061 0.975486i \(-0.570626\pi\)
−0.220061 + 0.975486i \(0.570626\pi\)
\(632\) −4.94427 −0.196673
\(633\) 0.944272 0.0375314
\(634\) −13.0557 −0.518509
\(635\) −41.8885 −1.66230
\(636\) 10.9443 0.433969
\(637\) 0 0
\(638\) 0 0
\(639\) 4.94427 0.195592
\(640\) −2.00000 −0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 8.00000 0.315735
\(643\) 10.4721 0.412981 0.206490 0.978449i \(-0.433796\pi\)
0.206490 + 0.978449i \(0.433796\pi\)
\(644\) 0 0
\(645\) −9.88854 −0.389361
\(646\) 11.0557 0.434982
\(647\) −12.3607 −0.485948 −0.242974 0.970033i \(-0.578123\pi\)
−0.242974 + 0.970033i \(0.578123\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −4.47214 −0.175412
\(651\) 0 0
\(652\) −8.94427 −0.350285
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 2.94427 0.115130
\(655\) 24.0000 0.937758
\(656\) 6.00000 0.234261
\(657\) −14.9443 −0.583032
\(658\) 0 0
\(659\) 46.8328 1.82435 0.912174 0.409804i \(-0.134403\pi\)
0.912174 + 0.409804i \(0.134403\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) −13.8885 −0.539794
\(663\) 20.0000 0.776736
\(664\) −2.47214 −0.0959375
\(665\) 0 0
\(666\) 6.94427 0.269085
\(667\) 2.00000 0.0774403
\(668\) 15.4164 0.596479
\(669\) 18.4721 0.714174
\(670\) −9.88854 −0.382028
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −2.94427 −0.113409
\(675\) 1.00000 0.0384900
\(676\) 7.00000 0.269231
\(677\) 4.11146 0.158016 0.0790080 0.996874i \(-0.474825\pi\)
0.0790080 + 0.996874i \(0.474825\pi\)
\(678\) 18.9443 0.727550
\(679\) 0 0
\(680\) 8.94427 0.342997
\(681\) 5.52786 0.211828
\(682\) 0 0
\(683\) 16.9443 0.648355 0.324177 0.945996i \(-0.394913\pi\)
0.324177 + 0.945996i \(0.394913\pi\)
\(684\) −2.47214 −0.0945245
\(685\) 28.0000 1.06983
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −4.94427 −0.188499
\(689\) −48.9443 −1.86463
\(690\) −2.00000 −0.0761387
\(691\) 15.0557 0.572747 0.286373 0.958118i \(-0.407550\pi\)
0.286373 + 0.958118i \(0.407550\pi\)
\(692\) −0.472136 −0.0179479
\(693\) 0 0
\(694\) −0.944272 −0.0358441
\(695\) 17.8885 0.678551
\(696\) 2.00000 0.0758098
\(697\) −26.8328 −1.01637
\(698\) −24.4721 −0.926284
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −20.8328 −0.786845 −0.393422 0.919358i \(-0.628709\pi\)
−0.393422 + 0.919358i \(0.628709\pi\)
\(702\) −4.47214 −0.168790
\(703\) −17.1672 −0.647473
\(704\) 0 0
\(705\) 4.94427 0.186212
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) 6.94427 0.260798 0.130399 0.991462i \(-0.458374\pi\)
0.130399 + 0.991462i \(0.458374\pi\)
\(710\) −9.88854 −0.371110
\(711\) −4.94427 −0.185425
\(712\) −9.41641 −0.352895
\(713\) −2.47214 −0.0925822
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −20.9443 −0.782178
\(718\) −30.8328 −1.15067
\(719\) 5.52786 0.206155 0.103077 0.994673i \(-0.467131\pi\)
0.103077 + 0.994673i \(0.467131\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −12.8885 −0.479662
\(723\) 20.4721 0.761367
\(724\) −2.00000 −0.0743294
\(725\) 2.00000 0.0742781
\(726\) 11.0000 0.408248
\(727\) −3.05573 −0.113331 −0.0566653 0.998393i \(-0.518047\pi\)
−0.0566653 + 0.998393i \(0.518047\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 29.8885 1.10622
\(731\) 22.1115 0.817822
\(732\) −6.00000 −0.221766
\(733\) 50.9443 1.88167 0.940835 0.338866i \(-0.110043\pi\)
0.940835 + 0.338866i \(0.110043\pi\)
\(734\) −30.8328 −1.13806
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) −13.8885 −0.510553
\(741\) 11.0557 0.406142
\(742\) 0 0
\(743\) −22.8328 −0.837655 −0.418827 0.908066i \(-0.637559\pi\)
−0.418827 + 0.908066i \(0.637559\pi\)
\(744\) −2.47214 −0.0906329
\(745\) 2.11146 0.0773578
\(746\) 6.94427 0.254248
\(747\) −2.47214 −0.0904507
\(748\) 0 0
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 17.8885 0.652762 0.326381 0.945238i \(-0.394171\pi\)
0.326381 + 0.945238i \(0.394171\pi\)
\(752\) 2.47214 0.0901495
\(753\) −5.52786 −0.201447
\(754\) −8.94427 −0.325731
\(755\) −9.88854 −0.359881
\(756\) 0 0
\(757\) −42.9443 −1.56084 −0.780418 0.625258i \(-0.784994\pi\)
−0.780418 + 0.625258i \(0.784994\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) 4.94427 0.179348
\(761\) −21.0557 −0.763270 −0.381635 0.924313i \(-0.624639\pi\)
−0.381635 + 0.924313i \(0.624639\pi\)
\(762\) −20.9443 −0.758731
\(763\) 0 0
\(764\) −3.05573 −0.110552
\(765\) 8.94427 0.323381
\(766\) −16.0000 −0.578103
\(767\) −17.8885 −0.645918
\(768\) −1.00000 −0.0360844
\(769\) −30.3607 −1.09483 −0.547417 0.836860i \(-0.684389\pi\)
−0.547417 + 0.836860i \(0.684389\pi\)
\(770\) 0 0
\(771\) −2.94427 −0.106035
\(772\) 11.8885 0.427878
\(773\) −5.05573 −0.181842 −0.0909210 0.995858i \(-0.528981\pi\)
−0.0909210 + 0.995858i \(0.528981\pi\)
\(774\) −4.94427 −0.177718
\(775\) −2.47214 −0.0888017
\(776\) 0.472136 0.0169487
\(777\) 0 0
\(778\) −18.9443 −0.679185
\(779\) −14.8328 −0.531441
\(780\) 8.94427 0.320256
\(781\) 0 0
\(782\) 4.47214 0.159923
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −5.88854 −0.210171
\(786\) 12.0000 0.428026
\(787\) 0.583592 0.0208028 0.0104014 0.999946i \(-0.496689\pi\)
0.0104014 + 0.999946i \(0.496689\pi\)
\(788\) −14.9443 −0.532368
\(789\) 24.0000 0.854423
\(790\) 9.88854 0.351819
\(791\) 0 0
\(792\) 0 0
\(793\) 26.8328 0.952861
\(794\) 36.4721 1.29435
\(795\) −21.8885 −0.776307
\(796\) −3.05573 −0.108307
\(797\) 28.8328 1.02131 0.510655 0.859785i \(-0.329403\pi\)
0.510655 + 0.859785i \(0.329403\pi\)
\(798\) 0 0
\(799\) −11.0557 −0.391124
\(800\) −1.00000 −0.0353553
\(801\) −9.41641 −0.332712
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) −4.94427 −0.174371
\(805\) 0 0
\(806\) 11.0557 0.389421
\(807\) 16.4721 0.579847
\(808\) 9.41641 0.331268
\(809\) 35.8885 1.26177 0.630887 0.775875i \(-0.282691\pi\)
0.630887 + 0.775875i \(0.282691\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 37.8885 1.33045 0.665223 0.746644i \(-0.268336\pi\)
0.665223 + 0.746644i \(0.268336\pi\)
\(812\) 0 0
\(813\) 15.4164 0.540677
\(814\) 0 0
\(815\) 17.8885 0.626608
\(816\) 4.47214 0.156556
\(817\) 12.2229 0.427626
\(818\) 15.8885 0.555530
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −16.8328 −0.587469 −0.293735 0.955887i \(-0.594898\pi\)
−0.293735 + 0.955887i \(0.594898\pi\)
\(822\) 14.0000 0.488306
\(823\) 20.9443 0.730071 0.365036 0.930994i \(-0.381057\pi\)
0.365036 + 0.930994i \(0.381057\pi\)
\(824\) 4.94427 0.172242
\(825\) 0 0
\(826\) 0 0
\(827\) 33.8885 1.17842 0.589210 0.807980i \(-0.299439\pi\)
0.589210 + 0.807980i \(0.299439\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 4.94427 0.171618
\(831\) 11.8885 0.412409
\(832\) 4.47214 0.155043
\(833\) 0 0
\(834\) 8.94427 0.309715
\(835\) −30.8328 −1.06701
\(836\) 0 0
\(837\) −2.47214 −0.0854495
\(838\) 7.41641 0.256196
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 32.8328 1.13149
\(843\) −10.0000 −0.344418
\(844\) −0.944272 −0.0325032
\(845\) −14.0000 −0.481615
\(846\) 2.47214 0.0849938
\(847\) 0 0
\(848\) −10.9443 −0.375828
\(849\) 18.4721 0.633962
\(850\) 4.47214 0.153393
\(851\) −6.94427 −0.238047
\(852\) −4.94427 −0.169388
\(853\) 6.36068 0.217786 0.108893 0.994054i \(-0.465269\pi\)
0.108893 + 0.994054i \(0.465269\pi\)
\(854\) 0 0
\(855\) 4.94427 0.169091
\(856\) −8.00000 −0.273434
\(857\) 47.8885 1.63584 0.817921 0.575331i \(-0.195127\pi\)
0.817921 + 0.575331i \(0.195127\pi\)
\(858\) 0 0
\(859\) −32.9443 −1.12404 −0.562022 0.827122i \(-0.689976\pi\)
−0.562022 + 0.827122i \(0.689976\pi\)
\(860\) 9.88854 0.337197
\(861\) 0 0
\(862\) 17.8885 0.609286
\(863\) 20.9443 0.712951 0.356476 0.934305i \(-0.383978\pi\)
0.356476 + 0.934305i \(0.383978\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0.944272 0.0321062
\(866\) −25.4164 −0.863685
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) 22.1115 0.749218
\(872\) −2.94427 −0.0997056
\(873\) 0.472136 0.0159794
\(874\) 2.47214 0.0836212
\(875\) 0 0
\(876\) 14.9443 0.504920
\(877\) −34.7214 −1.17246 −0.586229 0.810146i \(-0.699388\pi\)
−0.586229 + 0.810146i \(0.699388\pi\)
\(878\) −28.3607 −0.957127
\(879\) 27.8885 0.940657
\(880\) 0 0
\(881\) −22.3607 −0.753350 −0.376675 0.926345i \(-0.622933\pi\)
−0.376675 + 0.926345i \(0.622933\pi\)
\(882\) 0 0
\(883\) 2.83282 0.0953318 0.0476659 0.998863i \(-0.484822\pi\)
0.0476659 + 0.998863i \(0.484822\pi\)
\(884\) −20.0000 −0.672673
\(885\) −8.00000 −0.268917
\(886\) 34.8328 1.17023
\(887\) 5.52786 0.185608 0.0928038 0.995684i \(-0.470417\pi\)
0.0928038 + 0.995684i \(0.470417\pi\)
\(888\) −6.94427 −0.233035
\(889\) 0 0
\(890\) 18.8328 0.631277
\(891\) 0 0
\(892\) −18.4721 −0.618493
\(893\) −6.11146 −0.204512
\(894\) 1.05573 0.0353088
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) 4.47214 0.149320
\(898\) 18.0000 0.600668
\(899\) −4.94427 −0.164901
\(900\) −1.00000 −0.0333333
\(901\) 48.9443 1.63057
\(902\) 0 0
\(903\) 0 0
\(904\) −18.9443 −0.630077
\(905\) 4.00000 0.132964
\(906\) −4.94427 −0.164262
\(907\) −41.8885 −1.39089 −0.695443 0.718581i \(-0.744792\pi\)
−0.695443 + 0.718581i \(0.744792\pi\)
\(908\) −5.52786 −0.183449
\(909\) 9.41641 0.312323
\(910\) 0 0
\(911\) −48.7214 −1.61421 −0.807105 0.590407i \(-0.798967\pi\)
−0.807105 + 0.590407i \(0.798967\pi\)
\(912\) 2.47214 0.0818606
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) 12.0000 0.396708
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 4.47214 0.147602
\(919\) 35.0557 1.15638 0.578191 0.815902i \(-0.303759\pi\)
0.578191 + 0.815902i \(0.303759\pi\)
\(920\) 2.00000 0.0659380
\(921\) 24.9443 0.821942
\(922\) −6.58359 −0.216819
\(923\) 22.1115 0.727807
\(924\) 0 0
\(925\) −6.94427 −0.228326
\(926\) 6.11146 0.200835
\(927\) 4.94427 0.162391
\(928\) −2.00000 −0.0656532
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 4.94427 0.162129
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −2.47214 −0.0809341
\(934\) −33.3050 −1.08977
\(935\) 0 0
\(936\) 4.47214 0.146176
\(937\) −40.2492 −1.31488 −0.657442 0.753505i \(-0.728362\pi\)
−0.657442 + 0.753505i \(0.728362\pi\)
\(938\) 0 0
\(939\) −13.4164 −0.437828
\(940\) −4.94427 −0.161264
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −2.94427 −0.0959296
\(943\) −6.00000 −0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) 0.944272 0.0306847 0.0153424 0.999882i \(-0.495116\pi\)
0.0153424 + 0.999882i \(0.495116\pi\)
\(948\) 4.94427 0.160582
\(949\) −66.8328 −2.16949
\(950\) 2.47214 0.0802067
\(951\) 13.0557 0.423361
\(952\) 0 0
\(953\) −9.05573 −0.293344 −0.146672 0.989185i \(-0.546856\pi\)
−0.146672 + 0.989185i \(0.546856\pi\)
\(954\) −10.9443 −0.354334
\(955\) 6.11146 0.197762
\(956\) 20.9443 0.677386
\(957\) 0 0
\(958\) 4.94427 0.159742
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −24.8885 −0.802856
\(962\) 31.0557 1.00128
\(963\) −8.00000 −0.257796
\(964\) −20.4721 −0.659363
\(965\) −23.7771 −0.765412
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −11.0000 −0.353553
\(969\) −11.0557 −0.355161
\(970\) −0.944272 −0.0303187
\(971\) 23.4164 0.751468 0.375734 0.926727i \(-0.377391\pi\)
0.375734 + 0.926727i \(0.377391\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 24.0000 0.769010
\(975\) 4.47214 0.143223
\(976\) 6.00000 0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 8.94427 0.286006
\(979\) 0 0
\(980\) 0 0
\(981\) −2.94427 −0.0940034
\(982\) 40.9443 1.30658
\(983\) 22.1115 0.705246 0.352623 0.935765i \(-0.385290\pi\)
0.352623 + 0.935765i \(0.385290\pi\)
\(984\) −6.00000 −0.191273
\(985\) 29.8885 0.952328
\(986\) 8.94427 0.284844
\(987\) 0 0
\(988\) −11.0557 −0.351730
\(989\) 4.94427 0.157219
\(990\) 0 0
\(991\) 60.9443 1.93596 0.967979 0.251030i \(-0.0807693\pi\)
0.967979 + 0.251030i \(0.0807693\pi\)
\(992\) 2.47214 0.0784904
\(993\) 13.8885 0.440740
\(994\) 0 0
\(995\) 6.11146 0.193746
\(996\) 2.47214 0.0783326
\(997\) 46.3607 1.46826 0.734129 0.679010i \(-0.237591\pi\)
0.734129 + 0.679010i \(0.237591\pi\)
\(998\) 32.9443 1.04283
\(999\) −6.94427 −0.219707
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.bz.1.2 2
7.6 odd 2 966.2.a.p.1.1 2
21.20 even 2 2898.2.a.v.1.1 2
28.27 even 2 7728.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.1 2 7.6 odd 2
2898.2.a.v.1.1 2 21.20 even 2
6762.2.a.bz.1.2 2 1.1 even 1 trivial
7728.2.a.bd.1.1 2 28.27 even 2