Properties

Label 63.2.a.b.1.2
Level $63$
Weight $2$
Character 63.1
Self dual yes
Analytic conductor $0.503$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 63.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{7} -1.73205 q^{8} -6.00000 q^{10} +3.46410 q^{11} +2.00000 q^{13} +1.73205 q^{14} -5.00000 q^{16} +3.46410 q^{17} -4.00000 q^{19} -3.46410 q^{20} +6.00000 q^{22} -3.46410 q^{23} +7.00000 q^{25} +3.46410 q^{26} +1.00000 q^{28} -4.00000 q^{31} -5.19615 q^{32} +6.00000 q^{34} -3.46410 q^{35} +2.00000 q^{37} -6.92820 q^{38} +6.00000 q^{40} +10.3923 q^{41} -4.00000 q^{43} +3.46410 q^{44} -6.00000 q^{46} +6.92820 q^{47} +1.00000 q^{49} +12.1244 q^{50} +2.00000 q^{52} -6.92820 q^{53} -12.0000 q^{55} -1.73205 q^{56} -6.92820 q^{59} -10.0000 q^{61} -6.92820 q^{62} +1.00000 q^{64} -6.92820 q^{65} -4.00000 q^{67} +3.46410 q^{68} -6.00000 q^{70} -10.3923 q^{71} +14.0000 q^{73} +3.46410 q^{74} -4.00000 q^{76} +3.46410 q^{77} +8.00000 q^{79} +17.3205 q^{80} +18.0000 q^{82} -12.0000 q^{85} -6.92820 q^{86} -6.00000 q^{88} -3.46410 q^{89} +2.00000 q^{91} -3.46410 q^{92} +12.0000 q^{94} +13.8564 q^{95} +14.0000 q^{97} +1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{7} - 12q^{10} + 4q^{13} - 10q^{16} - 8q^{19} + 12q^{22} + 14q^{25} + 2q^{28} - 8q^{31} + 12q^{34} + 4q^{37} + 12q^{40} - 8q^{43} - 12q^{46} + 2q^{49} + 4q^{52} - 24q^{55} - 20q^{61} + 2q^{64} - 8q^{67} - 12q^{70} + 28q^{73} - 8q^{76} + 16q^{79} + 36q^{82} - 24q^{85} - 12q^{88} + 4q^{91} + 24q^{94} + 28q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −6.00000 −1.89737
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.73205 0.462910
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −3.46410 −0.774597
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −6.92820 −1.12390
\(39\) 0 0
\(40\) 6.00000 0.948683
\(41\) 10.3923 1.62301 0.811503 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 12.1244 1.71464
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −6.92820 −0.951662 −0.475831 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) −1.73205 −0.231455
\(57\) 0 0
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.92820 −0.879883
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 17.3205 1.93649
\(81\) 0 0
\(82\) 18.0000 1.98777
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −3.46410 −0.361158
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 13.8564 1.42164
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 1.73205 0.174964
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 17.3205 1.67444 0.837218 0.546869i \(-0.184180\pi\)
0.837218 + 0.546869i \(0.184180\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −20.7846 −1.98173
\(111\) 0 0
\(112\) −5.00000 −0.472456
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 0 0
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −17.3205 −1.56813
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) −13.8564 −1.21064 −0.605320 0.795982i \(-0.706955\pi\)
−0.605320 + 0.795982i \(0.706955\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −6.92820 −0.598506
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −6.92820 −0.591916 −0.295958 0.955201i \(-0.595639\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −3.46410 −0.292770
\(141\) 0 0
\(142\) −18.0000 −1.51053
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) 0 0
\(146\) 24.2487 2.00684
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.92820 0.567581 0.283790 0.958886i \(-0.408408\pi\)
0.283790 + 0.958886i \(0.408408\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 6.92820 0.561951
\(153\) 0 0
\(154\) 6.00000 0.483494
\(155\) 13.8564 1.11297
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 13.8564 1.10236
\(159\) 0 0
\(160\) 18.0000 1.42302
\(161\) −3.46410 −0.273009
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 10.3923 0.811503
\(165\) 0 0
\(166\) 0 0
\(167\) −20.7846 −1.60836 −0.804181 0.594385i \(-0.797396\pi\)
−0.804181 + 0.594385i \(0.797396\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −20.7846 −1.59411
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 17.3205 1.31685 0.658427 0.752645i \(-0.271222\pi\)
0.658427 + 0.752645i \(0.271222\pi\)
\(174\) 0 0
\(175\) 7.00000 0.529150
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −17.3205 −1.29460 −0.647298 0.762237i \(-0.724101\pi\)
−0.647298 + 0.762237i \(0.724101\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) −24.2487 −1.75458 −0.877288 0.479965i \(-0.840649\pi\)
−0.877288 + 0.479965i \(0.840649\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 24.2487 1.74096
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 20.7846 1.48084 0.740421 0.672143i \(-0.234626\pi\)
0.740421 + 0.672143i \(0.234626\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −12.1244 −0.857321
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) −36.0000 −2.51435
\(206\) −6.92820 −0.482711
\(207\) 0 0
\(208\) −10.0000 −0.693375
\(209\) −13.8564 −0.958468
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −6.92820 −0.475831
\(213\) 0 0
\(214\) 30.0000 2.05076
\(215\) 13.8564 0.944999
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 3.46410 0.234619
\(219\) 0 0
\(220\) −12.0000 −0.809040
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −5.19615 −0.347183
\(225\) 0 0
\(226\) 0 0
\(227\) −6.92820 −0.459841 −0.229920 0.973209i \(-0.573847\pi\)
−0.229920 + 0.973209i \(0.573847\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 20.7846 1.37050
\(231\) 0 0
\(232\) 0 0
\(233\) 6.92820 0.453882 0.226941 0.973909i \(-0.427128\pi\)
0.226941 + 0.973909i \(0.427128\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.73205 0.111340
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −3.46410 −0.221313
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 6.92820 0.439941
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 20.7846 1.31191 0.655956 0.754799i \(-0.272265\pi\)
0.655956 + 0.754799i \(0.272265\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 13.8564 0.869428
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −3.46410 −0.216085 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −6.92820 −0.429669
\(261\) 0 0
\(262\) −24.0000 −1.48272
\(263\) −17.3205 −1.06803 −0.534014 0.845476i \(-0.679317\pi\)
−0.534014 + 0.845476i \(0.679317\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −17.3205 −1.05605 −0.528025 0.849229i \(-0.677067\pi\)
−0.528025 + 0.849229i \(0.677067\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −17.3205 −1.05021
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 24.2487 1.46225
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −27.7128 −1.66210
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) −20.7846 −1.23991 −0.619953 0.784639i \(-0.712848\pi\)
−0.619953 + 0.784639i \(0.712848\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −10.3923 −0.616670
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 10.3923 0.613438
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) −10.3923 −0.607125 −0.303562 0.952812i \(-0.598176\pi\)
−0.303562 + 0.952812i \(0.598176\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) −3.46410 −0.201347
\(297\) 0 0
\(298\) 12.0000 0.695141
\(299\) −6.92820 −0.400668
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 13.8564 0.797347
\(303\) 0 0
\(304\) 20.0000 1.14708
\(305\) 34.6410 1.98354
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 3.46410 0.197386
\(309\) 0 0
\(310\) 24.0000 1.36311
\(311\) 34.6410 1.96431 0.982156 0.188069i \(-0.0602227\pi\)
0.982156 + 0.188069i \(0.0602227\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −17.3205 −0.977453
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.92820 0.389127 0.194563 0.980890i \(-0.437671\pi\)
0.194563 + 0.980890i \(0.437671\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.46410 −0.193649
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) −13.8564 −0.770991
\(324\) 0 0
\(325\) 14.0000 0.776580
\(326\) 34.6410 1.91859
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −36.0000 −1.96983
\(335\) 13.8564 0.757056
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −15.5885 −0.847900
\(339\) 0 0
\(340\) −12.0000 −0.650791
\(341\) −13.8564 −0.750366
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 6.92820 0.373544
\(345\) 0 0
\(346\) 30.0000 1.61281
\(347\) −17.3205 −0.929814 −0.464907 0.885360i \(-0.653912\pi\)
−0.464907 + 0.885360i \(0.653912\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 12.1244 0.648074
\(351\) 0 0
\(352\) −18.0000 −0.959403
\(353\) 3.46410 0.184376 0.0921878 0.995742i \(-0.470614\pi\)
0.0921878 + 0.995742i \(0.470614\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) −3.46410 −0.183597
\(357\) 0 0
\(358\) −30.0000 −1.58555
\(359\) −24.2487 −1.27980 −0.639899 0.768459i \(-0.721024\pi\)
−0.639899 + 0.768459i \(0.721024\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 3.46410 0.182069
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −48.4974 −2.53847
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 17.3205 0.902894
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) −6.92820 −0.359694
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 20.7846 1.07475
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 13.8564 0.710819
\(381\) 0 0
\(382\) −42.0000 −2.14891
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 24.2487 1.23423
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 13.8564 0.702548 0.351274 0.936273i \(-0.385749\pi\)
0.351274 + 0.936273i \(0.385749\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) −1.73205 −0.0874818
\(393\) 0 0
\(394\) 36.0000 1.81365
\(395\) −27.7128 −1.39438
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) −27.7128 −1.38912
\(399\) 0 0
\(400\) −35.0000 −1.75000
\(401\) 6.92820 0.345978 0.172989 0.984924i \(-0.444657\pi\)
0.172989 + 0.984924i \(0.444657\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 3.46410 0.172345
\(405\) 0 0
\(406\) 0 0
\(407\) 6.92820 0.343418
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −62.3538 −3.07944
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) −10.3923 −0.509525
\(417\) 0 0
\(418\) −24.0000 −1.17388
\(419\) 20.7846 1.01539 0.507697 0.861536i \(-0.330497\pi\)
0.507697 + 0.861536i \(0.330497\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 34.6410 1.68630
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 24.2487 1.17624
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 17.3205 0.837218
\(429\) 0 0
\(430\) 24.0000 1.15738
\(431\) 3.46410 0.166860 0.0834300 0.996514i \(-0.473413\pi\)
0.0834300 + 0.996514i \(0.473413\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −6.92820 −0.332564
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 13.8564 0.662842
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 20.7846 0.990867
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) −3.46410 −0.164584 −0.0822922 0.996608i \(-0.526224\pi\)
−0.0822922 + 0.996608i \(0.526224\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 13.8564 0.656120
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 41.5692 1.96177 0.980886 0.194581i \(-0.0623348\pi\)
0.980886 + 0.194581i \(0.0623348\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −38.1051 −1.78054
\(459\) 0 0
\(460\) 12.0000 0.559503
\(461\) 31.1769 1.45205 0.726027 0.687666i \(-0.241365\pi\)
0.726027 + 0.687666i \(0.241365\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 6.92820 0.320599 0.160300 0.987068i \(-0.448754\pi\)
0.160300 + 0.987068i \(0.448754\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −41.5692 −1.91745
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −13.8564 −0.637118
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) 3.46410 0.158777
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) −6.92820 −0.316558 −0.158279 0.987394i \(-0.550594\pi\)
−0.158279 + 0.987394i \(0.550594\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −17.3205 −0.788928
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −48.4974 −2.20215
\(486\) 0 0
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 17.3205 0.784063
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) 10.3923 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −13.8564 −0.623429
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) −10.3923 −0.466159
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −6.92820 −0.309839
\(501\) 0 0
\(502\) 36.0000 1.60676
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) −20.7846 −0.923989
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −3.46410 −0.153544 −0.0767718 0.997049i \(-0.524461\pi\)
−0.0767718 + 0.997049i \(0.524461\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 13.8564 0.610586
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 3.46410 0.152204
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) 3.46410 0.151765 0.0758825 0.997117i \(-0.475823\pi\)
0.0758825 + 0.997117i \(0.475823\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −13.8564 −0.605320
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) −13.8564 −0.603595
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 41.5692 1.80565
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 20.7846 0.900281
\(534\) 0 0
\(535\) −60.0000 −2.59403
\(536\) 6.92820 0.299253
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 3.46410 0.149209
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 34.6410 1.48796
\(543\) 0 0
\(544\) −18.0000 −0.771744
\(545\) −6.92820 −0.296772
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −6.92820 −0.295958
\(549\) 0 0
\(550\) 42.0000 1.79089
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −17.3205 −0.735878
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) −6.92820 −0.293557 −0.146779 0.989169i \(-0.546891\pi\)
−0.146779 + 0.989169i \(0.546891\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 17.3205 0.731925
\(561\) 0 0
\(562\) −36.0000 −1.51857
\(563\) 34.6410 1.45994 0.729972 0.683477i \(-0.239533\pi\)
0.729972 + 0.683477i \(0.239533\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.92820 −0.291214
\(567\) 0 0
\(568\) 18.0000 0.755263
\(569\) 6.92820 0.290445 0.145223 0.989399i \(-0.453610\pi\)
0.145223 + 0.989399i \(0.453610\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 6.92820 0.289683
\(573\) 0 0
\(574\) 18.0000 0.751305
\(575\) −24.2487 −1.01124
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −8.66025 −0.360219
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) −24.2487 −1.00342
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 41.5692 1.71138
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −24.2487 −0.995775 −0.497888 0.867242i \(-0.665891\pi\)
−0.497888 + 0.867242i \(0.665891\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 6.92820 0.283790
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) 45.0333 1.84001 0.920006 0.391905i \(-0.128184\pi\)
0.920006 + 0.391905i \(0.128184\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −6.92820 −0.282372
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) −3.46410 −0.140836
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 20.7846 0.842927
\(609\) 0 0
\(610\) 60.0000 2.42933
\(611\) 13.8564 0.560570
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −48.4974 −1.95720
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −20.7846 −0.836757 −0.418378 0.908273i \(-0.637401\pi\)
−0.418378 + 0.908273i \(0.637401\pi\)
\(618\) 0 0
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 13.8564 0.556487
\(621\) 0 0
\(622\) 60.0000 2.40578
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 3.46410 0.138453
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 6.92820 0.276246
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −13.8564 −0.551178
\(633\) 0 0
\(634\) 12.0000 0.476581
\(635\) −27.7128 −1.09975
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) −42.0000 −1.66020
\(641\) −48.4974 −1.91553 −0.957767 0.287547i \(-0.907160\pi\)
−0.957767 + 0.287547i \(0.907160\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) −3.46410 −0.136505
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −6.92820 −0.272376 −0.136188 0.990683i \(-0.543485\pi\)
−0.136188 + 0.990683i \(0.543485\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 24.2487 0.951113
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 27.7128 1.08449 0.542243 0.840222i \(-0.317575\pi\)
0.542243 + 0.840222i \(0.317575\pi\)
\(654\) 0 0
\(655\) 48.0000 1.87552
\(656\) −51.9615 −2.02876
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) −10.3923 −0.404827 −0.202413 0.979300i \(-0.564878\pi\)
−0.202413 + 0.979300i \(0.564878\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 34.6410 1.34636
\(663\) 0 0
\(664\) 0 0
\(665\) 13.8564 0.537328
\(666\) 0 0
\(667\) 0 0
\(668\) −20.7846 −0.804181
\(669\) 0 0
\(670\) 24.0000 0.927201
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 24.2487 0.934025
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −24.2487 −0.931954 −0.465977 0.884797i \(-0.654297\pi\)
−0.465977 + 0.884797i \(0.654297\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 20.7846 0.797053
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) 24.2487 0.927851 0.463926 0.885874i \(-0.346441\pi\)
0.463926 + 0.885874i \(0.346441\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 1.73205 0.0661300
\(687\) 0 0
\(688\) 20.0000 0.762493
\(689\) −13.8564 −0.527887
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 17.3205 0.658427
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 55.4256 2.10241
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 24.2487 0.917827
\(699\) 0 0
\(700\) 7.00000 0.264575
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 3.46410 0.130281
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 62.3538 2.34010
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 13.8564 0.518927
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) −17.3205 −0.647298
\(717\) 0 0
\(718\) −42.0000 −1.56743
\(719\) 27.7128 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −5.19615 −0.193381
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) −4.00000 −0.148352 −0.0741759 0.997245i \(-0.523633\pi\)
−0.0741759 + 0.997245i \(0.523633\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) −84.0000 −3.10898
\(731\) −13.8564 −0.512498
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −27.7128 −1.02290
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) −13.8564 −0.510407
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −6.92820 −0.254686
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −10.3923 −0.381257 −0.190628 0.981662i \(-0.561053\pi\)
−0.190628 + 0.981662i \(0.561053\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) −17.3205 −0.634149
\(747\) 0 0
\(748\) 12.0000 0.438763
\(749\) 17.3205 0.632878
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −34.6410 −1.26323
\(753\) 0 0
\(754\) 0 0
\(755\) −27.7128 −1.00857
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −48.4974 −1.76151
\(759\) 0 0
\(760\) −24.0000 −0.870572
\(761\) 38.1051 1.38131 0.690655 0.723185i \(-0.257322\pi\)
0.690655 + 0.723185i \(0.257322\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −24.2487 −0.877288
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) −20.7846 −0.749025
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 45.0333 1.61974 0.809868 0.586612i \(-0.199539\pi\)
0.809868 + 0.586612i \(0.199539\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) −24.2487 −0.870478
\(777\) 0 0
\(778\) 24.0000 0.860442
\(779\) −41.5692 −1.48937
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) −20.7846 −0.743256
\(783\) 0 0
\(784\) −5.00000 −0.178571
\(785\) 34.6410 1.23639
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 20.7846 0.740421
\(789\) 0 0
\(790\) −48.0000 −1.70776
\(791\) 0 0
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 65.8179 2.33579
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 31.1769 1.10434 0.552171 0.833731i \(-0.313799\pi\)
0.552171 + 0.833731i \(0.313799\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) −36.3731 −1.28598
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 48.4974 1.71144
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) −13.8564 −0.488071
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −27.7128 −0.974331 −0.487165 0.873310i \(-0.661969\pi\)
−0.487165 + 0.873310i \(0.661969\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −69.2820 −2.42684
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 24.2487 0.847836
\(819\) 0 0
\(820\) −36.0000 −1.25717
\(821\) 6.92820 0.241796 0.120898 0.992665i \(-0.461423\pi\)
0.120898 + 0.992665i \(0.461423\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 6.92820 0.241355
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −10.3923 −0.361376 −0.180688 0.983540i \(-0.557832\pi\)
−0.180688 + 0.983540i \(0.557832\pi\)
\(828\) 0 0
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) 72.0000 2.49166
\(836\) −13.8564 −0.479234
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) 20.7846 0.717564 0.358782 0.933421i \(-0.383192\pi\)
0.358782 + 0.933421i \(0.383192\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −17.3205 −0.596904
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) 31.1769 1.07252
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 34.6410 1.18958
\(849\) 0 0
\(850\) 42.0000 1.44059
\(851\) −6.92820 −0.237496
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −17.3205 −0.592696
\(855\) 0 0
\(856\) −30.0000 −1.02538
\(857\) −17.3205 −0.591657 −0.295829 0.955241i \(-0.595596\pi\)
−0.295829 + 0.955241i \(0.595596\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 13.8564 0.472500
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 38.1051 1.29711 0.648557 0.761166i \(-0.275373\pi\)
0.648557 + 0.761166i \(0.275373\pi\)
\(864\) 0 0
\(865\) −60.0000 −2.04006
\(866\) 45.0333 1.53029
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 27.7128 0.940093
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −3.46410 −0.117309
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) −6.92820 −0.234216
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −27.7128 −0.935262
\(879\) 0 0
\(880\) 60.0000 2.02260
\(881\) −51.9615 −1.75063 −0.875314 0.483555i \(-0.839345\pi\)
−0.875314 + 0.483555i \(0.839345\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 6.92820 0.233021
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 6.92820 0.232626 0.116313 0.993213i \(-0.462892\pi\)
0.116313 + 0.993213i \(0.462892\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 20.7846 0.696702
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) −27.7128 −0.927374
\(894\) 0 0
\(895\) 60.0000 2.00558
\(896\) 12.1244 0.405046
\(897\) 0 0
\(898\) 72.0000 2.40267
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 62.3538 2.07616
\(903\) 0 0
\(904\) 0 0
\(905\) −6.92820 −0.230301
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −6.92820 −0.229920
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 31.1769 1.03294 0.516469 0.856306i \(-0.327246\pi\)
0.516469 + 0.856306i \(0.327246\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.3205 −0.572911
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −13.8564 −0.457579
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −20.7846 −0.685248
\(921\) 0 0
\(922\) 54.0000 1.77840
\(923\) −20.7846 −0.684134
\(924\) 0 0
\(925\) 14.0000 0.460317
\(926\) 55.4256 1.82140
\(927\) 0 0
\(928\) 0 0
\(929\) −45.0333 −1.47750 −0.738748 0.673982i \(-0.764582\pi\)
−0.738748 + 0.673982i \(0.764582\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 6.92820 0.226941
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −41.5692 −1.35946
\(936\) 0 0
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) −6.92820 −0.226214
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) −17.3205 −0.564632 −0.282316 0.959321i \(-0.591103\pi\)
−0.282316 + 0.959321i \(0.591103\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) −24.2487 −0.787977 −0.393989 0.919115i \(-0.628905\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) −48.4974 −1.57346
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −41.5692 −1.34656 −0.673280 0.739388i \(-0.735115\pi\)
−0.673280 + 0.739388i \(0.735115\pi\)
\(954\) 0 0
\(955\) 84.0000 2.71818
\(956\) 10.3923 0.336111
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) −6.92820 −0.223723
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.92820 0.223374
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −48.4974 −1.56119
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) −84.0000 −2.69708
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) −69.2820 −2.21994
\(975\) 0 0
\(976\) 50.0000 1.60046
\(977\) 34.6410 1.10826 0.554132 0.832429i \(-0.313050\pi\)
0.554132 + 0.832429i \(0.313050\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) −3.46410 −0.110657
\(981\) 0 0
\(982\) 18.0000 0.574403
\(983\) 13.8564 0.441951 0.220975 0.975279i \(-0.429076\pi\)
0.220975 + 0.975279i \(0.429076\pi\)
\(984\) 0 0
\(985\) −72.0000 −2.29411
\(986\) 0 0
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 13.8564 0.440608
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 20.7846 0.659912
\(993\) 0 0
\(994\) −18.0000 −0.570925
\(995\) 55.4256 1.75711
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −6.92820 −0.219308
\(999\) 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 63.2.a.b.1.2 yes 2
3.2 odd 2 inner 63.2.a.b.1.1 2
4.3 odd 2 1008.2.a.n.1.1 2
5.2 odd 4 1575.2.d.i.1324.4 4
5.3 odd 4 1575.2.d.i.1324.1 4
5.4 even 2 1575.2.a.q.1.1 2
7.2 even 3 441.2.e.j.361.1 4
7.3 odd 6 441.2.e.i.226.1 4
7.4 even 3 441.2.e.j.226.1 4
7.5 odd 6 441.2.e.i.361.1 4
7.6 odd 2 441.2.a.g.1.2 2
8.3 odd 2 4032.2.a.bq.1.2 2
8.5 even 2 4032.2.a.bt.1.2 2
9.2 odd 6 567.2.f.j.190.2 4
9.4 even 3 567.2.f.j.379.1 4
9.5 odd 6 567.2.f.j.379.2 4
9.7 even 3 567.2.f.j.190.1 4
11.10 odd 2 7623.2.a.bi.1.1 2
12.11 even 2 1008.2.a.n.1.2 2
15.2 even 4 1575.2.d.i.1324.2 4
15.8 even 4 1575.2.d.i.1324.3 4
15.14 odd 2 1575.2.a.q.1.2 2
21.2 odd 6 441.2.e.j.361.2 4
21.5 even 6 441.2.e.i.361.2 4
21.11 odd 6 441.2.e.j.226.2 4
21.17 even 6 441.2.e.i.226.2 4
21.20 even 2 441.2.a.g.1.1 2
24.5 odd 2 4032.2.a.bt.1.1 2
24.11 even 2 4032.2.a.bq.1.1 2
28.27 even 2 7056.2.a.cm.1.2 2
33.32 even 2 7623.2.a.bi.1.2 2
84.83 odd 2 7056.2.a.cm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 2 3.2 odd 2 inner
63.2.a.b.1.2 yes 2 1.1 even 1 trivial
441.2.a.g.1.1 2 21.20 even 2
441.2.a.g.1.2 2 7.6 odd 2
441.2.e.i.226.1 4 7.3 odd 6
441.2.e.i.226.2 4 21.17 even 6
441.2.e.i.361.1 4 7.5 odd 6
441.2.e.i.361.2 4 21.5 even 6
441.2.e.j.226.1 4 7.4 even 3
441.2.e.j.226.2 4 21.11 odd 6
441.2.e.j.361.1 4 7.2 even 3
441.2.e.j.361.2 4 21.2 odd 6
567.2.f.j.190.1 4 9.7 even 3
567.2.f.j.190.2 4 9.2 odd 6
567.2.f.j.379.1 4 9.4 even 3
567.2.f.j.379.2 4 9.5 odd 6
1008.2.a.n.1.1 2 4.3 odd 2
1008.2.a.n.1.2 2 12.11 even 2
1575.2.a.q.1.1 2 5.4 even 2
1575.2.a.q.1.2 2 15.14 odd 2
1575.2.d.i.1324.1 4 5.3 odd 4
1575.2.d.i.1324.2 4 15.2 even 4
1575.2.d.i.1324.3 4 15.8 even 4
1575.2.d.i.1324.4 4 5.2 odd 4
4032.2.a.bq.1.1 2 24.11 even 2
4032.2.a.bq.1.2 2 8.3 odd 2
4032.2.a.bt.1.1 2 24.5 odd 2
4032.2.a.bt.1.2 2 8.5 even 2
7056.2.a.cm.1.1 2 84.83 odd 2
7056.2.a.cm.1.2 2 28.27 even 2
7623.2.a.bi.1.1 2 11.10 odd 2
7623.2.a.bi.1.2 2 33.32 even 2