Properties

Label 1134.2.a.p.1.1
Level $1134$
Weight $2$
Character 1134.1
Self dual yes
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1134.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.44949 q^{10} +2.00000 q^{11} +4.89898 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +2.55051 q^{19} -1.44949 q^{20} +2.00000 q^{22} -1.00000 q^{23} -2.89898 q^{25} +4.89898 q^{26} -1.00000 q^{28} -6.89898 q^{29} +6.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.44949 q^{35} +11.7980 q^{37} +2.55051 q^{38} -1.44949 q^{40} +9.79796 q^{41} +6.89898 q^{43} +2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +1.00000 q^{49} -2.89898 q^{50} +4.89898 q^{52} -10.8990 q^{53} -2.89898 q^{55} -1.00000 q^{56} -6.89898 q^{58} -2.00000 q^{59} +6.55051 q^{61} +6.00000 q^{62} +1.00000 q^{64} -7.10102 q^{65} -12.8990 q^{67} +2.00000 q^{68} +1.44949 q^{70} +0.101021 q^{71} -6.89898 q^{73} +11.7980 q^{74} +2.55051 q^{76} -2.00000 q^{77} -1.89898 q^{79} -1.44949 q^{80} +9.79796 q^{82} +2.00000 q^{83} -2.89898 q^{85} +6.89898 q^{86} +2.00000 q^{88} -16.8990 q^{89} -4.89898 q^{91} -1.00000 q^{92} +9.79796 q^{94} -3.69694 q^{95} +2.89898 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + 2q^{10} + 4q^{11} - 2q^{14} + 2q^{16} + 4q^{17} + 10q^{19} + 2q^{20} + 4q^{22} - 2q^{23} + 4q^{25} - 2q^{28} - 4q^{29} + 12q^{31} + 2q^{32} + 4q^{34} - 2q^{35} + 4q^{37} + 10q^{38} + 2q^{40} + 4q^{43} + 4q^{44} - 2q^{46} + 2q^{49} + 4q^{50} - 12q^{53} + 4q^{55} - 2q^{56} - 4q^{58} - 4q^{59} + 18q^{61} + 12q^{62} + 2q^{64} - 24q^{65} - 16q^{67} + 4q^{68} - 2q^{70} + 10q^{71} - 4q^{73} + 4q^{74} + 10q^{76} - 4q^{77} + 6q^{79} + 2q^{80} + 4q^{83} + 4q^{85} + 4q^{86} + 4q^{88} - 24q^{89} - 2q^{92} + 22q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.44949 −0.648232 −0.324116 0.946017i \(-0.605067\pi\)
−0.324116 + 0.946017i \(0.605067\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.44949 −0.458369
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 4.89898 1.35873 0.679366 0.733799i \(-0.262255\pi\)
0.679366 + 0.733799i \(0.262255\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 2.55051 0.585127 0.292564 0.956246i \(-0.405492\pi\)
0.292564 + 0.956246i \(0.405492\pi\)
\(20\) −1.44949 −0.324116
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −2.89898 −0.579796
\(26\) 4.89898 0.960769
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −6.89898 −1.28111 −0.640554 0.767913i \(-0.721295\pi\)
−0.640554 + 0.767913i \(0.721295\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.44949 0.245008
\(36\) 0 0
\(37\) 11.7980 1.93957 0.969786 0.243956i \(-0.0784453\pi\)
0.969786 + 0.243956i \(0.0784453\pi\)
\(38\) 2.55051 0.413747
\(39\) 0 0
\(40\) −1.44949 −0.229184
\(41\) 9.79796 1.53018 0.765092 0.643921i \(-0.222693\pi\)
0.765092 + 0.643921i \(0.222693\pi\)
\(42\) 0 0
\(43\) 6.89898 1.05208 0.526042 0.850458i \(-0.323675\pi\)
0.526042 + 0.850458i \(0.323675\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.89898 −0.409978
\(51\) 0 0
\(52\) 4.89898 0.679366
\(53\) −10.8990 −1.49709 −0.748545 0.663084i \(-0.769247\pi\)
−0.748545 + 0.663084i \(0.769247\pi\)
\(54\) 0 0
\(55\) −2.89898 −0.390898
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.89898 −0.905880
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 6.55051 0.838707 0.419353 0.907823i \(-0.362257\pi\)
0.419353 + 0.907823i \(0.362257\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.10102 −0.880773
\(66\) 0 0
\(67\) −12.8990 −1.57586 −0.787931 0.615764i \(-0.788847\pi\)
−0.787931 + 0.615764i \(0.788847\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 1.44949 0.173247
\(71\) 0.101021 0.0119889 0.00599446 0.999982i \(-0.498092\pi\)
0.00599446 + 0.999982i \(0.498092\pi\)
\(72\) 0 0
\(73\) −6.89898 −0.807464 −0.403732 0.914877i \(-0.632287\pi\)
−0.403732 + 0.914877i \(0.632287\pi\)
\(74\) 11.7980 1.37148
\(75\) 0 0
\(76\) 2.55051 0.292564
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −1.89898 −0.213652 −0.106826 0.994278i \(-0.534069\pi\)
−0.106826 + 0.994278i \(0.534069\pi\)
\(80\) −1.44949 −0.162058
\(81\) 0 0
\(82\) 9.79796 1.08200
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −2.89898 −0.314438
\(86\) 6.89898 0.743936
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −16.8990 −1.79129 −0.895644 0.444771i \(-0.853285\pi\)
−0.895644 + 0.444771i \(0.853285\pi\)
\(90\) 0 0
\(91\) −4.89898 −0.513553
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 9.79796 1.01058
\(95\) −3.69694 −0.379298
\(96\) 0 0
\(97\) 2.89898 0.294347 0.147173 0.989111i \(-0.452982\pi\)
0.147173 + 0.989111i \(0.452982\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −2.89898 −0.289898
\(101\) −17.2474 −1.71619 −0.858093 0.513495i \(-0.828351\pi\)
−0.858093 + 0.513495i \(0.828351\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 4.89898 0.480384
\(105\) 0 0
\(106\) −10.8990 −1.05860
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 12.6969 1.21615 0.608073 0.793881i \(-0.291943\pi\)
0.608073 + 0.793881i \(0.291943\pi\)
\(110\) −2.89898 −0.276407
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −6.10102 −0.573936 −0.286968 0.957940i \(-0.592647\pi\)
−0.286968 + 0.957940i \(0.592647\pi\)
\(114\) 0 0
\(115\) 1.44949 0.135166
\(116\) −6.89898 −0.640554
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.55051 0.593055
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 11.4495 1.02407
\(126\) 0 0
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −7.10102 −0.622801
\(131\) −8.55051 −0.747062 −0.373531 0.927618i \(-0.621853\pi\)
−0.373531 + 0.927618i \(0.621853\pi\)
\(132\) 0 0
\(133\) −2.55051 −0.221157
\(134\) −12.8990 −1.11430
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 7.79796 0.666225 0.333112 0.942887i \(-0.391901\pi\)
0.333112 + 0.942887i \(0.391901\pi\)
\(138\) 0 0
\(139\) 4.55051 0.385969 0.192985 0.981202i \(-0.438183\pi\)
0.192985 + 0.981202i \(0.438183\pi\)
\(140\) 1.44949 0.122504
\(141\) 0 0
\(142\) 0.101021 0.00847745
\(143\) 9.79796 0.819346
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) −6.89898 −0.570964
\(147\) 0 0
\(148\) 11.7980 0.969786
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 2.55051 0.206874
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) −8.69694 −0.698555
\(156\) 0 0
\(157\) −8.34847 −0.666280 −0.333140 0.942877i \(-0.608108\pi\)
−0.333140 + 0.942877i \(0.608108\pi\)
\(158\) −1.89898 −0.151075
\(159\) 0 0
\(160\) −1.44949 −0.114592
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −19.7980 −1.55070 −0.775348 0.631534i \(-0.782425\pi\)
−0.775348 + 0.631534i \(0.782425\pi\)
\(164\) 9.79796 0.765092
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −10.6969 −0.827754 −0.413877 0.910333i \(-0.635826\pi\)
−0.413877 + 0.910333i \(0.635826\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) −2.89898 −0.222342
\(171\) 0 0
\(172\) 6.89898 0.526042
\(173\) −3.10102 −0.235766 −0.117883 0.993027i \(-0.537611\pi\)
−0.117883 + 0.993027i \(0.537611\pi\)
\(174\) 0 0
\(175\) 2.89898 0.219142
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −16.8990 −1.26663
\(179\) 20.6969 1.54696 0.773481 0.633820i \(-0.218514\pi\)
0.773481 + 0.633820i \(0.218514\pi\)
\(180\) 0 0
\(181\) −10.3485 −0.769196 −0.384598 0.923084i \(-0.625660\pi\)
−0.384598 + 0.923084i \(0.625660\pi\)
\(182\) −4.89898 −0.363137
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −17.1010 −1.25729
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 9.79796 0.714590
\(189\) 0 0
\(190\) −3.69694 −0.268204
\(191\) 4.10102 0.296739 0.148370 0.988932i \(-0.452597\pi\)
0.148370 + 0.988932i \(0.452597\pi\)
\(192\) 0 0
\(193\) −17.8990 −1.28840 −0.644198 0.764858i \(-0.722809\pi\)
−0.644198 + 0.764858i \(0.722809\pi\)
\(194\) 2.89898 0.208135
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 16.6969 1.18961 0.594804 0.803871i \(-0.297230\pi\)
0.594804 + 0.803871i \(0.297230\pi\)
\(198\) 0 0
\(199\) −2.89898 −0.205503 −0.102752 0.994707i \(-0.532765\pi\)
−0.102752 + 0.994707i \(0.532765\pi\)
\(200\) −2.89898 −0.204989
\(201\) 0 0
\(202\) −17.2474 −1.21353
\(203\) 6.89898 0.484213
\(204\) 0 0
\(205\) −14.2020 −0.991914
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) 4.89898 0.339683
\(209\) 5.10102 0.352845
\(210\) 0 0
\(211\) 12.8990 0.888002 0.444001 0.896026i \(-0.353559\pi\)
0.444001 + 0.896026i \(0.353559\pi\)
\(212\) −10.8990 −0.748545
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 12.6969 0.859945
\(219\) 0 0
\(220\) −2.89898 −0.195449
\(221\) 9.79796 0.659082
\(222\) 0 0
\(223\) −11.1010 −0.743379 −0.371690 0.928357i \(-0.621221\pi\)
−0.371690 + 0.928357i \(0.621221\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.10102 −0.405834
\(227\) −5.44949 −0.361695 −0.180848 0.983511i \(-0.557884\pi\)
−0.180848 + 0.983511i \(0.557884\pi\)
\(228\) 0 0
\(229\) 1.24745 0.0824337 0.0412169 0.999150i \(-0.486877\pi\)
0.0412169 + 0.999150i \(0.486877\pi\)
\(230\) 1.44949 0.0955765
\(231\) 0 0
\(232\) −6.89898 −0.452940
\(233\) −7.00000 −0.458585 −0.229293 0.973358i \(-0.573641\pi\)
−0.229293 + 0.973358i \(0.573641\pi\)
\(234\) 0 0
\(235\) −14.2020 −0.926439
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 6.79796 0.439723 0.219862 0.975531i \(-0.429439\pi\)
0.219862 + 0.975531i \(0.429439\pi\)
\(240\) 0 0
\(241\) 0.898979 0.0579084 0.0289542 0.999581i \(-0.490782\pi\)
0.0289542 + 0.999581i \(0.490782\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 6.55051 0.419353
\(245\) −1.44949 −0.0926045
\(246\) 0 0
\(247\) 12.4949 0.795031
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 11.4495 0.724129
\(251\) 17.4495 1.10140 0.550701 0.834703i \(-0.314360\pi\)
0.550701 + 0.834703i \(0.314360\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) −3.00000 −0.188237
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.20204 0.511629 0.255815 0.966726i \(-0.417656\pi\)
0.255815 + 0.966726i \(0.417656\pi\)
\(258\) 0 0
\(259\) −11.7980 −0.733090
\(260\) −7.10102 −0.440387
\(261\) 0 0
\(262\) −8.55051 −0.528252
\(263\) 25.8990 1.59700 0.798500 0.601995i \(-0.205627\pi\)
0.798500 + 0.601995i \(0.205627\pi\)
\(264\) 0 0
\(265\) 15.7980 0.970461
\(266\) −2.55051 −0.156382
\(267\) 0 0
\(268\) −12.8990 −0.787931
\(269\) −18.3485 −1.11873 −0.559363 0.828923i \(-0.688954\pi\)
−0.559363 + 0.828923i \(0.688954\pi\)
\(270\) 0 0
\(271\) 7.10102 0.431356 0.215678 0.976465i \(-0.430804\pi\)
0.215678 + 0.976465i \(0.430804\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 7.79796 0.471092
\(275\) −5.79796 −0.349630
\(276\) 0 0
\(277\) −18.6969 −1.12339 −0.561695 0.827344i \(-0.689851\pi\)
−0.561695 + 0.827344i \(0.689851\pi\)
\(278\) 4.55051 0.272921
\(279\) 0 0
\(280\) 1.44949 0.0866236
\(281\) −19.0000 −1.13344 −0.566722 0.823909i \(-0.691789\pi\)
−0.566722 + 0.823909i \(0.691789\pi\)
\(282\) 0 0
\(283\) −25.4495 −1.51282 −0.756408 0.654101i \(-0.773047\pi\)
−0.756408 + 0.654101i \(0.773047\pi\)
\(284\) 0.101021 0.00599446
\(285\) 0 0
\(286\) 9.79796 0.579365
\(287\) −9.79796 −0.578355
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 10.0000 0.587220
\(291\) 0 0
\(292\) −6.89898 −0.403732
\(293\) 2.75255 0.160806 0.0804029 0.996762i \(-0.474379\pi\)
0.0804029 + 0.996762i \(0.474379\pi\)
\(294\) 0 0
\(295\) 2.89898 0.168785
\(296\) 11.7980 0.685742
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −4.89898 −0.283315
\(300\) 0 0
\(301\) −6.89898 −0.397651
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) 2.55051 0.146282
\(305\) −9.49490 −0.543676
\(306\) 0 0
\(307\) 25.2474 1.44095 0.720474 0.693482i \(-0.243924\pi\)
0.720474 + 0.693482i \(0.243924\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) −8.69694 −0.493953
\(311\) 30.6969 1.74066 0.870332 0.492466i \(-0.163904\pi\)
0.870332 + 0.492466i \(0.163904\pi\)
\(312\) 0 0
\(313\) −4.69694 −0.265487 −0.132743 0.991150i \(-0.542379\pi\)
−0.132743 + 0.991150i \(0.542379\pi\)
\(314\) −8.34847 −0.471131
\(315\) 0 0
\(316\) −1.89898 −0.106826
\(317\) 20.6969 1.16246 0.581228 0.813741i \(-0.302572\pi\)
0.581228 + 0.813741i \(0.302572\pi\)
\(318\) 0 0
\(319\) −13.7980 −0.772537
\(320\) −1.44949 −0.0810289
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 5.10102 0.283828
\(324\) 0 0
\(325\) −14.2020 −0.787787
\(326\) −19.7980 −1.09651
\(327\) 0 0
\(328\) 9.79796 0.541002
\(329\) −9.79796 −0.540179
\(330\) 0 0
\(331\) 4.69694 0.258167 0.129084 0.991634i \(-0.458796\pi\)
0.129084 + 0.991634i \(0.458796\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) −10.6969 −0.585310
\(335\) 18.6969 1.02152
\(336\) 0 0
\(337\) −23.3939 −1.27435 −0.637173 0.770721i \(-0.719896\pi\)
−0.637173 + 0.770721i \(0.719896\pi\)
\(338\) 11.0000 0.598321
\(339\) 0 0
\(340\) −2.89898 −0.157219
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.89898 0.371968
\(345\) 0 0
\(346\) −3.10102 −0.166712
\(347\) 19.5959 1.05196 0.525982 0.850496i \(-0.323698\pi\)
0.525982 + 0.850496i \(0.323698\pi\)
\(348\) 0 0
\(349\) 11.1010 0.594224 0.297112 0.954843i \(-0.403977\pi\)
0.297112 + 0.954843i \(0.403977\pi\)
\(350\) 2.89898 0.154957
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −0.146428 −0.00777160
\(356\) −16.8990 −0.895644
\(357\) 0 0
\(358\) 20.6969 1.09387
\(359\) −8.79796 −0.464339 −0.232169 0.972675i \(-0.574582\pi\)
−0.232169 + 0.972675i \(0.574582\pi\)
\(360\) 0 0
\(361\) −12.4949 −0.657626
\(362\) −10.3485 −0.543903
\(363\) 0 0
\(364\) −4.89898 −0.256776
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −13.7980 −0.720248 −0.360124 0.932905i \(-0.617266\pi\)
−0.360124 + 0.932905i \(0.617266\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −17.1010 −0.889040
\(371\) 10.8990 0.565847
\(372\) 0 0
\(373\) −6.89898 −0.357216 −0.178608 0.983920i \(-0.557159\pi\)
−0.178608 + 0.983920i \(0.557159\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 9.79796 0.505291
\(377\) −33.7980 −1.74068
\(378\) 0 0
\(379\) 22.4949 1.15549 0.577743 0.816219i \(-0.303934\pi\)
0.577743 + 0.816219i \(0.303934\pi\)
\(380\) −3.69694 −0.189649
\(381\) 0 0
\(382\) 4.10102 0.209826
\(383\) −2.89898 −0.148131 −0.0740655 0.997253i \(-0.523597\pi\)
−0.0740655 + 0.997253i \(0.523597\pi\)
\(384\) 0 0
\(385\) 2.89898 0.147746
\(386\) −17.8990 −0.911034
\(387\) 0 0
\(388\) 2.89898 0.147173
\(389\) −24.8990 −1.26243 −0.631214 0.775609i \(-0.717443\pi\)
−0.631214 + 0.775609i \(0.717443\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 16.6969 0.841180
\(395\) 2.75255 0.138496
\(396\) 0 0
\(397\) 38.6969 1.94214 0.971072 0.238788i \(-0.0767500\pi\)
0.971072 + 0.238788i \(0.0767500\pi\)
\(398\) −2.89898 −0.145313
\(399\) 0 0
\(400\) −2.89898 −0.144949
\(401\) −19.8990 −0.993708 −0.496854 0.867834i \(-0.665511\pi\)
−0.496854 + 0.867834i \(0.665511\pi\)
\(402\) 0 0
\(403\) 29.3939 1.46421
\(404\) −17.2474 −0.858093
\(405\) 0 0
\(406\) 6.89898 0.342391
\(407\) 23.5959 1.16961
\(408\) 0 0
\(409\) −13.7980 −0.682265 −0.341133 0.940015i \(-0.610811\pi\)
−0.341133 + 0.940015i \(0.610811\pi\)
\(410\) −14.2020 −0.701389
\(411\) 0 0
\(412\) 14.0000 0.689730
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) −2.89898 −0.142305
\(416\) 4.89898 0.240192
\(417\) 0 0
\(418\) 5.10102 0.249499
\(419\) 29.4495 1.43870 0.719351 0.694647i \(-0.244439\pi\)
0.719351 + 0.694647i \(0.244439\pi\)
\(420\) 0 0
\(421\) 22.8990 1.11603 0.558014 0.829832i \(-0.311564\pi\)
0.558014 + 0.829832i \(0.311564\pi\)
\(422\) 12.8990 0.627912
\(423\) 0 0
\(424\) −10.8990 −0.529301
\(425\) −5.79796 −0.281242
\(426\) 0 0
\(427\) −6.55051 −0.317001
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −10.0000 −0.482243
\(431\) 31.5959 1.52192 0.760961 0.648798i \(-0.224728\pi\)
0.760961 + 0.648798i \(0.224728\pi\)
\(432\) 0 0
\(433\) −7.79796 −0.374746 −0.187373 0.982289i \(-0.559997\pi\)
−0.187373 + 0.982289i \(0.559997\pi\)
\(434\) −6.00000 −0.288009
\(435\) 0 0
\(436\) 12.6969 0.608073
\(437\) −2.55051 −0.122007
\(438\) 0 0
\(439\) 2.20204 0.105098 0.0525488 0.998618i \(-0.483265\pi\)
0.0525488 + 0.998618i \(0.483265\pi\)
\(440\) −2.89898 −0.138203
\(441\) 0 0
\(442\) 9.79796 0.466041
\(443\) −14.8990 −0.707872 −0.353936 0.935270i \(-0.615157\pi\)
−0.353936 + 0.935270i \(0.615157\pi\)
\(444\) 0 0
\(445\) 24.4949 1.16117
\(446\) −11.1010 −0.525649
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 20.5959 0.971981 0.485991 0.873964i \(-0.338459\pi\)
0.485991 + 0.873964i \(0.338459\pi\)
\(450\) 0 0
\(451\) 19.5959 0.922736
\(452\) −6.10102 −0.286968
\(453\) 0 0
\(454\) −5.44949 −0.255757
\(455\) 7.10102 0.332901
\(456\) 0 0
\(457\) −17.4949 −0.818377 −0.409188 0.912450i \(-0.634188\pi\)
−0.409188 + 0.912450i \(0.634188\pi\)
\(458\) 1.24745 0.0582895
\(459\) 0 0
\(460\) 1.44949 0.0675828
\(461\) 5.65153 0.263218 0.131609 0.991302i \(-0.457986\pi\)
0.131609 + 0.991302i \(0.457986\pi\)
\(462\) 0 0
\(463\) 3.69694 0.171811 0.0859057 0.996303i \(-0.472622\pi\)
0.0859057 + 0.996303i \(0.472622\pi\)
\(464\) −6.89898 −0.320277
\(465\) 0 0
\(466\) −7.00000 −0.324269
\(467\) 10.0000 0.462745 0.231372 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 12.8990 0.595620
\(470\) −14.2020 −0.655091
\(471\) 0 0
\(472\) −2.00000 −0.0920575
\(473\) 13.7980 0.634431
\(474\) 0 0
\(475\) −7.39388 −0.339254
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 6.79796 0.310931
\(479\) 9.59592 0.438449 0.219224 0.975674i \(-0.429647\pi\)
0.219224 + 0.975674i \(0.429647\pi\)
\(480\) 0 0
\(481\) 57.7980 2.63536
\(482\) 0.898979 0.0409474
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −4.20204 −0.190805
\(486\) 0 0
\(487\) −36.3939 −1.64916 −0.824582 0.565742i \(-0.808590\pi\)
−0.824582 + 0.565742i \(0.808590\pi\)
\(488\) 6.55051 0.296528
\(489\) 0 0
\(490\) −1.44949 −0.0654813
\(491\) 15.7980 0.712952 0.356476 0.934304i \(-0.383978\pi\)
0.356476 + 0.934304i \(0.383978\pi\)
\(492\) 0 0
\(493\) −13.7980 −0.621429
\(494\) 12.4949 0.562172
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −0.101021 −0.00453139
\(498\) 0 0
\(499\) −25.3939 −1.13679 −0.568393 0.822757i \(-0.692435\pi\)
−0.568393 + 0.822757i \(0.692435\pi\)
\(500\) 11.4495 0.512037
\(501\) 0 0
\(502\) 17.4495 0.778809
\(503\) −24.4949 −1.09217 −0.546087 0.837729i \(-0.683883\pi\)
−0.546087 + 0.837729i \(0.683883\pi\)
\(504\) 0 0
\(505\) 25.0000 1.11249
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) −3.00000 −0.133103
\(509\) −7.10102 −0.314747 −0.157374 0.987539i \(-0.550303\pi\)
−0.157374 + 0.987539i \(0.550303\pi\)
\(510\) 0 0
\(511\) 6.89898 0.305193
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 8.20204 0.361777
\(515\) −20.2929 −0.894210
\(516\) 0 0
\(517\) 19.5959 0.861827
\(518\) −11.7980 −0.518373
\(519\) 0 0
\(520\) −7.10102 −0.311400
\(521\) −9.30306 −0.407575 −0.203787 0.979015i \(-0.565325\pi\)
−0.203787 + 0.979015i \(0.565325\pi\)
\(522\) 0 0
\(523\) −14.3485 −0.627415 −0.313707 0.949520i \(-0.601571\pi\)
−0.313707 + 0.949520i \(0.601571\pi\)
\(524\) −8.55051 −0.373531
\(525\) 0 0
\(526\) 25.8990 1.12925
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 15.7980 0.686219
\(531\) 0 0
\(532\) −2.55051 −0.110579
\(533\) 48.0000 2.07911
\(534\) 0 0
\(535\) 17.3939 0.752003
\(536\) −12.8990 −0.557151
\(537\) 0 0
\(538\) −18.3485 −0.791059
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −18.4949 −0.795158 −0.397579 0.917568i \(-0.630149\pi\)
−0.397579 + 0.917568i \(0.630149\pi\)
\(542\) 7.10102 0.305015
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −18.4041 −0.788344
\(546\) 0 0
\(547\) −7.59592 −0.324778 −0.162389 0.986727i \(-0.551920\pi\)
−0.162389 + 0.986727i \(0.551920\pi\)
\(548\) 7.79796 0.333112
\(549\) 0 0
\(550\) −5.79796 −0.247226
\(551\) −17.5959 −0.749611
\(552\) 0 0
\(553\) 1.89898 0.0807528
\(554\) −18.6969 −0.794357
\(555\) 0 0
\(556\) 4.55051 0.192985
\(557\) −12.8990 −0.546547 −0.273274 0.961936i \(-0.588106\pi\)
−0.273274 + 0.961936i \(0.588106\pi\)
\(558\) 0 0
\(559\) 33.7980 1.42950
\(560\) 1.44949 0.0612521
\(561\) 0 0
\(562\) −19.0000 −0.801467
\(563\) −39.9444 −1.68346 −0.841728 0.539902i \(-0.818461\pi\)
−0.841728 + 0.539902i \(0.818461\pi\)
\(564\) 0 0
\(565\) 8.84337 0.372043
\(566\) −25.4495 −1.06972
\(567\) 0 0
\(568\) 0.101021 0.00423873
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 33.7980 1.41440 0.707200 0.707013i \(-0.249958\pi\)
0.707200 + 0.707013i \(0.249958\pi\)
\(572\) 9.79796 0.409673
\(573\) 0 0
\(574\) −9.79796 −0.408959
\(575\) 2.89898 0.120896
\(576\) 0 0
\(577\) −15.5959 −0.649267 −0.324633 0.945840i \(-0.605241\pi\)
−0.324633 + 0.945840i \(0.605241\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 10.0000 0.415227
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) −21.7980 −0.902779
\(584\) −6.89898 −0.285482
\(585\) 0 0
\(586\) 2.75255 0.113707
\(587\) 16.1464 0.666434 0.333217 0.942850i \(-0.391866\pi\)
0.333217 + 0.942850i \(0.391866\pi\)
\(588\) 0 0
\(589\) 15.3031 0.630552
\(590\) 2.89898 0.119349
\(591\) 0 0
\(592\) 11.7980 0.484893
\(593\) 14.6969 0.603531 0.301765 0.953382i \(-0.402424\pi\)
0.301765 + 0.953382i \(0.402424\pi\)
\(594\) 0 0
\(595\) 2.89898 0.118847
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −4.89898 −0.200334
\(599\) −33.7980 −1.38095 −0.690474 0.723358i \(-0.742598\pi\)
−0.690474 + 0.723358i \(0.742598\pi\)
\(600\) 0 0
\(601\) 16.6969 0.681082 0.340541 0.940230i \(-0.389390\pi\)
0.340541 + 0.940230i \(0.389390\pi\)
\(602\) −6.89898 −0.281181
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) 10.1464 0.412511
\(606\) 0 0
\(607\) 20.6969 0.840063 0.420031 0.907510i \(-0.362019\pi\)
0.420031 + 0.907510i \(0.362019\pi\)
\(608\) 2.55051 0.103437
\(609\) 0 0
\(610\) −9.49490 −0.384437
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −14.6969 −0.593604 −0.296802 0.954939i \(-0.595920\pi\)
−0.296802 + 0.954939i \(0.595920\pi\)
\(614\) 25.2474 1.01890
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −15.3939 −0.619734 −0.309867 0.950780i \(-0.600285\pi\)
−0.309867 + 0.950780i \(0.600285\pi\)
\(618\) 0 0
\(619\) 30.1464 1.21169 0.605844 0.795584i \(-0.292836\pi\)
0.605844 + 0.795584i \(0.292836\pi\)
\(620\) −8.69694 −0.349277
\(621\) 0 0
\(622\) 30.6969 1.23084
\(623\) 16.8990 0.677043
\(624\) 0 0
\(625\) −2.10102 −0.0840408
\(626\) −4.69694 −0.187727
\(627\) 0 0
\(628\) −8.34847 −0.333140
\(629\) 23.5959 0.940831
\(630\) 0 0
\(631\) 27.8990 1.11064 0.555320 0.831636i \(-0.312596\pi\)
0.555320 + 0.831636i \(0.312596\pi\)
\(632\) −1.89898 −0.0755373
\(633\) 0 0
\(634\) 20.6969 0.821980
\(635\) 4.34847 0.172564
\(636\) 0 0
\(637\) 4.89898 0.194105
\(638\) −13.7980 −0.546266
\(639\) 0 0
\(640\) −1.44949 −0.0572961
\(641\) −7.49490 −0.296031 −0.148015 0.988985i \(-0.547288\pi\)
−0.148015 + 0.988985i \(0.547288\pi\)
\(642\) 0 0
\(643\) −39.3939 −1.55354 −0.776771 0.629783i \(-0.783144\pi\)
−0.776771 + 0.629783i \(0.783144\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 5.10102 0.200697
\(647\) −50.6969 −1.99310 −0.996551 0.0829807i \(-0.973556\pi\)
−0.996551 + 0.0829807i \(0.973556\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) −14.2020 −0.557050
\(651\) 0 0
\(652\) −19.7980 −0.775348
\(653\) 9.79796 0.383424 0.191712 0.981451i \(-0.438596\pi\)
0.191712 + 0.981451i \(0.438596\pi\)
\(654\) 0 0
\(655\) 12.3939 0.484269
\(656\) 9.79796 0.382546
\(657\) 0 0
\(658\) −9.79796 −0.381964
\(659\) −24.6969 −0.962056 −0.481028 0.876705i \(-0.659736\pi\)
−0.481028 + 0.876705i \(0.659736\pi\)
\(660\) 0 0
\(661\) 4.55051 0.176994 0.0884972 0.996076i \(-0.471794\pi\)
0.0884972 + 0.996076i \(0.471794\pi\)
\(662\) 4.69694 0.182552
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) 3.69694 0.143361
\(666\) 0 0
\(667\) 6.89898 0.267130
\(668\) −10.6969 −0.413877
\(669\) 0 0
\(670\) 18.6969 0.722326
\(671\) 13.1010 0.505759
\(672\) 0 0
\(673\) −8.59592 −0.331348 −0.165674 0.986181i \(-0.552980\pi\)
−0.165674 + 0.986181i \(0.552980\pi\)
\(674\) −23.3939 −0.901098
\(675\) 0 0
\(676\) 11.0000 0.423077
\(677\) 14.6969 0.564849 0.282425 0.959289i \(-0.408861\pi\)
0.282425 + 0.959289i \(0.408861\pi\)
\(678\) 0 0
\(679\) −2.89898 −0.111253
\(680\) −2.89898 −0.111171
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 51.7980 1.98199 0.990997 0.133885i \(-0.0427452\pi\)
0.990997 + 0.133885i \(0.0427452\pi\)
\(684\) 0 0
\(685\) −11.3031 −0.431868
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 6.89898 0.263021
\(689\) −53.3939 −2.03414
\(690\) 0 0
\(691\) 51.0454 1.94186 0.970929 0.239366i \(-0.0769396\pi\)
0.970929 + 0.239366i \(0.0769396\pi\)
\(692\) −3.10102 −0.117883
\(693\) 0 0
\(694\) 19.5959 0.743851
\(695\) −6.59592 −0.250197
\(696\) 0 0
\(697\) 19.5959 0.742248
\(698\) 11.1010 0.420180
\(699\) 0 0
\(700\) 2.89898 0.109571
\(701\) −7.39388 −0.279263 −0.139631 0.990204i \(-0.544592\pi\)
−0.139631 + 0.990204i \(0.544592\pi\)
\(702\) 0 0
\(703\) 30.0908 1.13490
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 17.2474 0.648657
\(708\) 0 0
\(709\) 27.5959 1.03639 0.518193 0.855264i \(-0.326605\pi\)
0.518193 + 0.855264i \(0.326605\pi\)
\(710\) −0.146428 −0.00549535
\(711\) 0 0
\(712\) −16.8990 −0.633316
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −14.2020 −0.531126
\(716\) 20.6969 0.773481
\(717\) 0 0
\(718\) −8.79796 −0.328337
\(719\) 9.79796 0.365402 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) −12.4949 −0.465012
\(723\) 0 0
\(724\) −10.3485 −0.384598
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −8.49490 −0.315058 −0.157529 0.987514i \(-0.550353\pi\)
−0.157529 + 0.987514i \(0.550353\pi\)
\(728\) −4.89898 −0.181568
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 13.7980 0.510336
\(732\) 0 0
\(733\) −17.4495 −0.644512 −0.322256 0.946653i \(-0.604441\pi\)
−0.322256 + 0.946653i \(0.604441\pi\)
\(734\) −13.7980 −0.509292
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −25.7980 −0.950280
\(738\) 0 0
\(739\) 13.5959 0.500134 0.250067 0.968229i \(-0.419547\pi\)
0.250067 + 0.968229i \(0.419547\pi\)
\(740\) −17.1010 −0.628646
\(741\) 0 0
\(742\) 10.8990 0.400114
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 8.69694 0.318631
\(746\) −6.89898 −0.252590
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 1.40408 0.0512357 0.0256178 0.999672i \(-0.491845\pi\)
0.0256178 + 0.999672i \(0.491845\pi\)
\(752\) 9.79796 0.357295
\(753\) 0 0
\(754\) −33.7980 −1.23085
\(755\) 7.24745 0.263762
\(756\) 0 0
\(757\) −35.3939 −1.28641 −0.643206 0.765693i \(-0.722396\pi\)
−0.643206 + 0.765693i \(0.722396\pi\)
\(758\) 22.4949 0.817051
\(759\) 0 0
\(760\) −3.69694 −0.134102
\(761\) 2.00000 0.0724999 0.0362500 0.999343i \(-0.488459\pi\)
0.0362500 + 0.999343i \(0.488459\pi\)
\(762\) 0 0
\(763\) −12.6969 −0.459660
\(764\) 4.10102 0.148370
\(765\) 0 0
\(766\) −2.89898 −0.104744
\(767\) −9.79796 −0.353784
\(768\) 0 0
\(769\) −34.0908 −1.22935 −0.614673 0.788782i \(-0.710712\pi\)
−0.614673 + 0.788782i \(0.710712\pi\)
\(770\) 2.89898 0.104472
\(771\) 0 0
\(772\) −17.8990 −0.644198
\(773\) 33.9444 1.22089 0.610447 0.792057i \(-0.290990\pi\)
0.610447 + 0.792057i \(0.290990\pi\)
\(774\) 0 0
\(775\) −17.3939 −0.624807
\(776\) 2.89898 0.104067
\(777\) 0 0
\(778\) −24.8990 −0.892672
\(779\) 24.9898 0.895352
\(780\) 0 0
\(781\) 0.202041 0.00722960
\(782\) −2.00000 −0.0715199
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 12.1010 0.431904
\(786\) 0 0
\(787\) −11.3939 −0.406148 −0.203074 0.979163i \(-0.565093\pi\)
−0.203074 + 0.979163i \(0.565093\pi\)
\(788\) 16.6969 0.594804
\(789\) 0 0
\(790\) 2.75255 0.0979314
\(791\) 6.10102 0.216927
\(792\) 0 0
\(793\) 32.0908 1.13958
\(794\) 38.6969 1.37330
\(795\) 0 0
\(796\) −2.89898 −0.102752
\(797\) −17.9444 −0.635623 −0.317811 0.948154i \(-0.602948\pi\)
−0.317811 + 0.948154i \(0.602948\pi\)
\(798\) 0 0
\(799\) 19.5959 0.693254
\(800\) −2.89898 −0.102494
\(801\) 0 0
\(802\) −19.8990 −0.702657
\(803\) −13.7980 −0.486919
\(804\) 0 0
\(805\) −1.44949 −0.0510878
\(806\) 29.3939 1.03536
\(807\) 0 0
\(808\) −17.2474 −0.606763
\(809\) −16.2020 −0.569633 −0.284817 0.958582i \(-0.591933\pi\)
−0.284817 + 0.958582i \(0.591933\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 6.89898 0.242107
\(813\) 0 0
\(814\) 23.5959 0.827036
\(815\) 28.6969 1.00521
\(816\) 0 0
\(817\) 17.5959 0.615603
\(818\) −13.7980 −0.482434
\(819\) 0 0
\(820\) −14.2020 −0.495957
\(821\) 0.404082 0.0141026 0.00705128 0.999975i \(-0.497755\pi\)
0.00705128 + 0.999975i \(0.497755\pi\)
\(822\) 0 0
\(823\) −13.3939 −0.466881 −0.233441 0.972371i \(-0.574998\pi\)
−0.233441 + 0.972371i \(0.574998\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 2.00000 0.0695889
\(827\) −36.4949 −1.26905 −0.634526 0.772902i \(-0.718805\pi\)
−0.634526 + 0.772902i \(0.718805\pi\)
\(828\) 0 0
\(829\) 1.30306 0.0452572 0.0226286 0.999744i \(-0.492796\pi\)
0.0226286 + 0.999744i \(0.492796\pi\)
\(830\) −2.89898 −0.100625
\(831\) 0 0
\(832\) 4.89898 0.169842
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 15.5051 0.536576
\(836\) 5.10102 0.176422
\(837\) 0 0
\(838\) 29.4495 1.01732
\(839\) −35.1010 −1.21182 −0.605911 0.795533i \(-0.707191\pi\)
−0.605911 + 0.795533i \(0.707191\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) 22.8990 0.789151
\(843\) 0 0
\(844\) 12.8990 0.444001
\(845\) −15.9444 −0.548504
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) −10.8990 −0.374272
\(849\) 0 0
\(850\) −5.79796 −0.198868
\(851\) −11.7980 −0.404429
\(852\) 0 0
\(853\) 24.8434 0.850621 0.425310 0.905048i \(-0.360165\pi\)
0.425310 + 0.905048i \(0.360165\pi\)
\(854\) −6.55051 −0.224154
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −34.8990 −1.19213 −0.596063 0.802938i \(-0.703269\pi\)
−0.596063 + 0.802938i \(0.703269\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) −10.0000 −0.340997
\(861\) 0 0
\(862\) 31.5959 1.07616
\(863\) 11.8990 0.405046 0.202523 0.979278i \(-0.435086\pi\)
0.202523 + 0.979278i \(0.435086\pi\)
\(864\) 0 0
\(865\) 4.49490 0.152831
\(866\) −7.79796 −0.264985
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) −3.79796 −0.128837
\(870\) 0 0
\(871\) −63.1918 −2.14117
\(872\) 12.6969 0.429973
\(873\) 0 0
\(874\) −2.55051 −0.0862723
\(875\) −11.4495 −0.387063
\(876\) 0 0
\(877\) 22.4949 0.759599 0.379799 0.925069i \(-0.375993\pi\)
0.379799 + 0.925069i \(0.375993\pi\)
\(878\) 2.20204 0.0743153
\(879\) 0 0
\(880\) −2.89898 −0.0977246
\(881\) 19.5959 0.660203 0.330102 0.943945i \(-0.392917\pi\)
0.330102 + 0.943945i \(0.392917\pi\)
\(882\) 0 0
\(883\) −19.7980 −0.666254 −0.333127 0.942882i \(-0.608104\pi\)
−0.333127 + 0.942882i \(0.608104\pi\)
\(884\) 9.79796 0.329541
\(885\) 0 0
\(886\) −14.8990 −0.500541
\(887\) −14.2020 −0.476858 −0.238429 0.971160i \(-0.576632\pi\)
−0.238429 + 0.971160i \(0.576632\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) 24.4949 0.821071
\(891\) 0 0
\(892\) −11.1010 −0.371690
\(893\) 24.9898 0.836252
\(894\) 0 0
\(895\) −30.0000 −1.00279
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 20.5959 0.687295
\(899\) −41.3939 −1.38056
\(900\) 0 0
\(901\) −21.7980 −0.726195
\(902\) 19.5959 0.652473
\(903\) 0 0
\(904\) −6.10102 −0.202917
\(905\) 15.0000 0.498617
\(906\) 0 0
\(907\) 2.69694 0.0895504 0.0447752 0.998997i \(-0.485743\pi\)
0.0447752 + 0.998997i \(0.485743\pi\)
\(908\) −5.44949 −0.180848
\(909\) 0 0
\(910\) 7.10102 0.235397
\(911\) 51.9898 1.72250 0.861249 0.508183i \(-0.169682\pi\)
0.861249 + 0.508183i \(0.169682\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −17.4949 −0.578680
\(915\) 0 0
\(916\) 1.24745 0.0412169
\(917\) 8.55051 0.282363
\(918\) 0 0
\(919\) −25.6969 −0.847664 −0.423832 0.905741i \(-0.639315\pi\)
−0.423832 + 0.905741i \(0.639315\pi\)
\(920\) 1.44949 0.0477883
\(921\) 0 0
\(922\) 5.65153 0.186123
\(923\) 0.494897 0.0162897
\(924\) 0 0
\(925\) −34.2020 −1.12456
\(926\) 3.69694 0.121489
\(927\) 0 0
\(928\) −6.89898 −0.226470
\(929\) 34.2929 1.12511 0.562556 0.826759i \(-0.309818\pi\)
0.562556 + 0.826759i \(0.309818\pi\)
\(930\) 0 0
\(931\) 2.55051 0.0835896
\(932\) −7.00000 −0.229293
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) −5.79796 −0.189614
\(936\) 0 0
\(937\) 45.5959 1.48955 0.744777 0.667314i \(-0.232556\pi\)
0.744777 + 0.667314i \(0.232556\pi\)
\(938\) 12.8990 0.421167
\(939\) 0 0
\(940\) −14.2020 −0.463220
\(941\) −1.44949 −0.0472520 −0.0236260 0.999721i \(-0.507521\pi\)
−0.0236260 + 0.999721i \(0.507521\pi\)
\(942\) 0 0
\(943\) −9.79796 −0.319065
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 13.7980 0.448610
\(947\) 52.4949 1.70585 0.852927 0.522029i \(-0.174825\pi\)
0.852927 + 0.522029i \(0.174825\pi\)
\(948\) 0 0
\(949\) −33.7980 −1.09713
\(950\) −7.39388 −0.239889
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) −3.39388 −0.109938 −0.0549692 0.998488i \(-0.517506\pi\)
−0.0549692 + 0.998488i \(0.517506\pi\)
\(954\) 0 0
\(955\) −5.94439 −0.192356
\(956\) 6.79796 0.219862
\(957\) 0 0
\(958\) 9.59592 0.310030
\(959\) −7.79796 −0.251809
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 57.7980 1.86348
\(963\) 0 0
\(964\) 0.898979 0.0289542
\(965\) 25.9444 0.835179
\(966\) 0 0
\(967\) 24.5959 0.790951 0.395476 0.918476i \(-0.370580\pi\)
0.395476 + 0.918476i \(0.370580\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −4.20204 −0.134919
\(971\) −0.0556128 −0.00178470 −0.000892350 1.00000i \(-0.500284\pi\)
−0.000892350 1.00000i \(0.500284\pi\)
\(972\) 0 0
\(973\) −4.55051 −0.145883
\(974\) −36.3939 −1.16614
\(975\) 0 0
\(976\) 6.55051 0.209677
\(977\) −37.5959 −1.20280 −0.601400 0.798948i \(-0.705390\pi\)
−0.601400 + 0.798948i \(0.705390\pi\)
\(978\) 0 0
\(979\) −33.7980 −1.08019
\(980\) −1.44949 −0.0463023
\(981\) 0 0
\(982\) 15.7980 0.504133
\(983\) −33.1918 −1.05866 −0.529328 0.848418i \(-0.677556\pi\)
−0.529328 + 0.848418i \(0.677556\pi\)
\(984\) 0 0
\(985\) −24.2020 −0.771141
\(986\) −13.7980 −0.439417
\(987\) 0 0
\(988\) 12.4949 0.397516
\(989\) −6.89898 −0.219375
\(990\) 0 0
\(991\) −1.79796 −0.0571140 −0.0285570 0.999592i \(-0.509091\pi\)
−0.0285570 + 0.999592i \(0.509091\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) −0.101021 −0.00320418
\(995\) 4.20204 0.133214
\(996\) 0 0
\(997\) 52.1464 1.65149 0.825747 0.564041i \(-0.190754\pi\)
0.825747 + 0.564041i \(0.190754\pi\)
\(998\) −25.3939 −0.803829
\(999\) 0 0
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 1134.2.a.p.1.1 2
3.2 odd 2 1134.2.a.i.1.2 2
4.3 odd 2 9072.2.a.bk.1.1 2
7.6 odd 2 7938.2.a.bn.1.2 2
9.2 odd 6 378.2.f.d.253.1 4
9.4 even 3 126.2.f.c.43.2 4
9.5 odd 6 378.2.f.d.127.1 4
9.7 even 3 126.2.f.c.85.1 yes 4
12.11 even 2 9072.2.a.bd.1.2 2
21.20 even 2 7938.2.a.bm.1.1 2
36.7 odd 6 1008.2.r.e.337.2 4
36.11 even 6 3024.2.r.e.1009.1 4
36.23 even 6 3024.2.r.e.2017.1 4
36.31 odd 6 1008.2.r.e.673.1 4
63.2 odd 6 2646.2.h.m.361.2 4
63.4 even 3 882.2.h.k.79.2 4
63.5 even 6 2646.2.e.k.2125.2 4
63.11 odd 6 2646.2.e.l.1549.1 4
63.13 odd 6 882.2.f.j.295.1 4
63.16 even 3 882.2.h.k.67.2 4
63.20 even 6 2646.2.f.k.1765.2 4
63.23 odd 6 2646.2.e.l.2125.1 4
63.25 even 3 882.2.e.m.373.1 4
63.31 odd 6 882.2.h.l.79.1 4
63.32 odd 6 2646.2.h.m.667.2 4
63.34 odd 6 882.2.f.j.589.2 4
63.38 even 6 2646.2.e.k.1549.2 4
63.40 odd 6 882.2.e.n.655.2 4
63.41 even 6 2646.2.f.k.883.2 4
63.47 even 6 2646.2.h.n.361.1 4
63.52 odd 6 882.2.e.n.373.2 4
63.58 even 3 882.2.e.m.655.1 4
63.59 even 6 2646.2.h.n.667.1 4
63.61 odd 6 882.2.h.l.67.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.f.c.43.2 4 9.4 even 3
126.2.f.c.85.1 yes 4 9.7 even 3
378.2.f.d.127.1 4 9.5 odd 6
378.2.f.d.253.1 4 9.2 odd 6
882.2.e.m.373.1 4 63.25 even 3
882.2.e.m.655.1 4 63.58 even 3
882.2.e.n.373.2 4 63.52 odd 6
882.2.e.n.655.2 4 63.40 odd 6
882.2.f.j.295.1 4 63.13 odd 6
882.2.f.j.589.2 4 63.34 odd 6
882.2.h.k.67.2 4 63.16 even 3
882.2.h.k.79.2 4 63.4 even 3
882.2.h.l.67.1 4 63.61 odd 6
882.2.h.l.79.1 4 63.31 odd 6
1008.2.r.e.337.2 4 36.7 odd 6
1008.2.r.e.673.1 4 36.31 odd 6
1134.2.a.i.1.2 2 3.2 odd 2
1134.2.a.p.1.1 2 1.1 even 1 trivial
2646.2.e.k.1549.2 4 63.38 even 6
2646.2.e.k.2125.2 4 63.5 even 6
2646.2.e.l.1549.1 4 63.11 odd 6
2646.2.e.l.2125.1 4 63.23 odd 6
2646.2.f.k.883.2 4 63.41 even 6
2646.2.f.k.1765.2 4 63.20 even 6
2646.2.h.m.361.2 4 63.2 odd 6
2646.2.h.m.667.2 4 63.32 odd 6
2646.2.h.n.361.1 4 63.47 even 6
2646.2.h.n.667.1 4 63.59 even 6
3024.2.r.e.1009.1 4 36.11 even 6
3024.2.r.e.2017.1 4 36.23 even 6
7938.2.a.bm.1.1 2 21.20 even 2
7938.2.a.bn.1.2 2 7.6 odd 2
9072.2.a.bd.1.2 2 12.11 even 2
9072.2.a.bk.1.1 2 4.3 odd 2