Properties

Label 3330.2.a.bd.1.2
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.26795 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.46410 q^{11} +1.46410 q^{13} -1.26795 q^{14} +1.00000 q^{16} +1.46410 q^{17} -4.19615 q^{19} -1.00000 q^{20} -1.46410 q^{22} +8.00000 q^{23} +1.00000 q^{25} +1.46410 q^{26} -1.26795 q^{28} +8.92820 q^{29} -2.73205 q^{31} +1.00000 q^{32} +1.46410 q^{34} +1.26795 q^{35} +1.00000 q^{37} -4.19615 q^{38} -1.00000 q^{40} +2.00000 q^{41} -6.92820 q^{43} -1.46410 q^{44} +8.00000 q^{46} +1.26795 q^{47} -5.39230 q^{49} +1.00000 q^{50} +1.46410 q^{52} +6.00000 q^{53} +1.46410 q^{55} -1.26795 q^{56} +8.92820 q^{58} -0.196152 q^{59} +8.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} -1.46410 q^{65} +13.6603 q^{67} +1.46410 q^{68} +1.26795 q^{70} +10.9282 q^{71} +12.9282 q^{73} +1.00000 q^{74} -4.19615 q^{76} +1.85641 q^{77} +5.26795 q^{79} -1.00000 q^{80} +2.00000 q^{82} +5.26795 q^{83} -1.46410 q^{85} -6.92820 q^{86} -1.46410 q^{88} +2.00000 q^{89} -1.85641 q^{91} +8.00000 q^{92} +1.26795 q^{94} +4.19615 q^{95} -2.00000 q^{97} -5.39230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 6 q^{7} + 2 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} - 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{19} - 2 q^{20} + 4 q^{22} + 16 q^{23} + 2 q^{25} - 4 q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{35} + 2 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 4 q^{55} - 6 q^{56} + 4 q^{58} + 10 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 4 q^{68} + 6 q^{70} + 8 q^{71} + 12 q^{73} + 2 q^{74} + 2 q^{76} - 24 q^{77} + 14 q^{79} - 2 q^{80} + 4 q^{82} + 14 q^{83} + 4 q^{85} + 4 q^{88} + 4 q^{89} + 24 q^{91} + 16 q^{92} + 6 q^{94} - 2 q^{95} - 4 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.26795 −0.479240 −0.239620 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −1.46410 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 0 0
\(13\) 1.46410 0.406069 0.203034 0.979172i \(-0.434920\pi\)
0.203034 + 0.979172i \(0.434920\pi\)
\(14\) −1.26795 −0.338874
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.46410 0.355097 0.177548 0.984112i \(-0.443183\pi\)
0.177548 + 0.984112i \(0.443183\pi\)
\(18\) 0 0
\(19\) −4.19615 −0.962663 −0.481332 0.876539i \(-0.659847\pi\)
−0.481332 + 0.876539i \(0.659847\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.46410 −0.312148
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.46410 0.287134
\(27\) 0 0
\(28\) −1.26795 −0.239620
\(29\) 8.92820 1.65793 0.828963 0.559304i \(-0.188931\pi\)
0.828963 + 0.559304i \(0.188931\pi\)
\(30\) 0 0
\(31\) −2.73205 −0.490691 −0.245345 0.969436i \(-0.578901\pi\)
−0.245345 + 0.969436i \(0.578901\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.46410 0.251091
\(35\) 1.26795 0.214323
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −4.19615 −0.680706
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) −1.46410 −0.220722
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 1.26795 0.184949 0.0924747 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.46410 0.203034
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 1.46410 0.197419
\(56\) −1.26795 −0.169437
\(57\) 0 0
\(58\) 8.92820 1.17233
\(59\) −0.196152 −0.0255369 −0.0127684 0.999918i \(-0.504064\pi\)
−0.0127684 + 0.999918i \(0.504064\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) −2.73205 −0.346971
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.46410 −0.181599
\(66\) 0 0
\(67\) 13.6603 1.66887 0.834433 0.551110i \(-0.185795\pi\)
0.834433 + 0.551110i \(0.185795\pi\)
\(68\) 1.46410 0.177548
\(69\) 0 0
\(70\) 1.26795 0.151549
\(71\) 10.9282 1.29694 0.648470 0.761241i \(-0.275409\pi\)
0.648470 + 0.761241i \(0.275409\pi\)
\(72\) 0 0
\(73\) 12.9282 1.51313 0.756566 0.653917i \(-0.226876\pi\)
0.756566 + 0.653917i \(0.226876\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.19615 −0.481332
\(77\) 1.85641 0.211557
\(78\) 0 0
\(79\) 5.26795 0.592691 0.296345 0.955081i \(-0.404232\pi\)
0.296345 + 0.955081i \(0.404232\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 5.26795 0.578233 0.289116 0.957294i \(-0.406639\pi\)
0.289116 + 0.957294i \(0.406639\pi\)
\(84\) 0 0
\(85\) −1.46410 −0.158804
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) −1.46410 −0.156074
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −1.85641 −0.194604
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 1.26795 0.130779
\(95\) 4.19615 0.430516
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −5.39230 −0.544705
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.53590 0.252331 0.126166 0.992009i \(-0.459733\pi\)
0.126166 + 0.992009i \(0.459733\pi\)
\(102\) 0 0
\(103\) −13.4641 −1.32666 −0.663329 0.748328i \(-0.730857\pi\)
−0.663329 + 0.748328i \(0.730857\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −6.73205 −0.650812 −0.325406 0.945574i \(-0.605501\pi\)
−0.325406 + 0.945574i \(0.605501\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.46410 0.139597
\(111\) 0 0
\(112\) −1.26795 −0.119810
\(113\) 17.4641 1.64288 0.821442 0.570292i \(-0.193170\pi\)
0.821442 + 0.570292i \(0.193170\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 8.92820 0.828963
\(117\) 0 0
\(118\) −0.196152 −0.0180573
\(119\) −1.85641 −0.170177
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 8.92820 0.808322
\(123\) 0 0
\(124\) −2.73205 −0.245345
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.6603 −1.21215 −0.606076 0.795407i \(-0.707257\pi\)
−0.606076 + 0.795407i \(0.707257\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.46410 −0.128410
\(131\) −12.5885 −1.09986 −0.549929 0.835211i \(-0.685345\pi\)
−0.549929 + 0.835211i \(0.685345\pi\)
\(132\) 0 0
\(133\) 5.32051 0.461347
\(134\) 13.6603 1.18007
\(135\) 0 0
\(136\) 1.46410 0.125546
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) 1.26795 0.107161
\(141\) 0 0
\(142\) 10.9282 0.917074
\(143\) −2.14359 −0.179256
\(144\) 0 0
\(145\) −8.92820 −0.741447
\(146\) 12.9282 1.06995
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 16.3923 1.34291 0.671455 0.741045i \(-0.265670\pi\)
0.671455 + 0.741045i \(0.265670\pi\)
\(150\) 0 0
\(151\) 8.39230 0.682956 0.341478 0.939890i \(-0.389073\pi\)
0.341478 + 0.939890i \(0.389073\pi\)
\(152\) −4.19615 −0.340353
\(153\) 0 0
\(154\) 1.85641 0.149593
\(155\) 2.73205 0.219444
\(156\) 0 0
\(157\) 16.9282 1.35102 0.675509 0.737352i \(-0.263924\pi\)
0.675509 + 0.737352i \(0.263924\pi\)
\(158\) 5.26795 0.419096
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −10.1436 −0.799427
\(162\) 0 0
\(163\) 23.3205 1.82660 0.913302 0.407284i \(-0.133524\pi\)
0.913302 + 0.407284i \(0.133524\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 5.26795 0.408872
\(167\) −5.46410 −0.422825 −0.211412 0.977397i \(-0.567806\pi\)
−0.211412 + 0.977397i \(0.567806\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) −1.46410 −0.112291
\(171\) 0 0
\(172\) −6.92820 −0.528271
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −1.26795 −0.0958479
\(176\) −1.46410 −0.110361
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 17.6603 1.31999 0.659995 0.751270i \(-0.270559\pi\)
0.659995 + 0.751270i \(0.270559\pi\)
\(180\) 0 0
\(181\) −1.46410 −0.108826 −0.0544129 0.998519i \(-0.517329\pi\)
−0.0544129 + 0.998519i \(0.517329\pi\)
\(182\) −1.85641 −0.137606
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −2.14359 −0.156755
\(188\) 1.26795 0.0924747
\(189\) 0 0
\(190\) 4.19615 0.304421
\(191\) 5.26795 0.381175 0.190588 0.981670i \(-0.438961\pi\)
0.190588 + 0.981670i \(0.438961\pi\)
\(192\) 0 0
\(193\) −11.8564 −0.853443 −0.426721 0.904383i \(-0.640332\pi\)
−0.426721 + 0.904383i \(0.640332\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −5.39230 −0.385165
\(197\) −18.7846 −1.33835 −0.669174 0.743106i \(-0.733352\pi\)
−0.669174 + 0.743106i \(0.733352\pi\)
\(198\) 0 0
\(199\) 26.0526 1.84682 0.923408 0.383819i \(-0.125391\pi\)
0.923408 + 0.383819i \(0.125391\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 2.53590 0.178425
\(203\) −11.3205 −0.794544
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −13.4641 −0.938088
\(207\) 0 0
\(208\) 1.46410 0.101517
\(209\) 6.14359 0.424961
\(210\) 0 0
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −6.73205 −0.460194
\(215\) 6.92820 0.472500
\(216\) 0 0
\(217\) 3.46410 0.235159
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 1.46410 0.0987097
\(221\) 2.14359 0.144194
\(222\) 0 0
\(223\) −22.0526 −1.47675 −0.738374 0.674391i \(-0.764406\pi\)
−0.738374 + 0.674391i \(0.764406\pi\)
\(224\) −1.26795 −0.0847184
\(225\) 0 0
\(226\) 17.4641 1.16169
\(227\) −3.60770 −0.239451 −0.119726 0.992807i \(-0.538201\pi\)
−0.119726 + 0.992807i \(0.538201\pi\)
\(228\) 0 0
\(229\) 15.8564 1.04782 0.523910 0.851773i \(-0.324473\pi\)
0.523910 + 0.851773i \(0.324473\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 8.92820 0.586165
\(233\) −15.0718 −0.987386 −0.493693 0.869636i \(-0.664353\pi\)
−0.493693 + 0.869636i \(0.664353\pi\)
\(234\) 0 0
\(235\) −1.26795 −0.0827119
\(236\) −0.196152 −0.0127684
\(237\) 0 0
\(238\) −1.85641 −0.120333
\(239\) 17.2679 1.11697 0.558485 0.829514i \(-0.311383\pi\)
0.558485 + 0.829514i \(0.311383\pi\)
\(240\) 0 0
\(241\) −8.92820 −0.575116 −0.287558 0.957763i \(-0.592843\pi\)
−0.287558 + 0.957763i \(0.592843\pi\)
\(242\) −8.85641 −0.569311
\(243\) 0 0
\(244\) 8.92820 0.571570
\(245\) 5.39230 0.344502
\(246\) 0 0
\(247\) −6.14359 −0.390907
\(248\) −2.73205 −0.173485
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −22.7321 −1.43483 −0.717417 0.696644i \(-0.754676\pi\)
−0.717417 + 0.696644i \(0.754676\pi\)
\(252\) 0 0
\(253\) −11.7128 −0.736378
\(254\) −13.6603 −0.857121
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.4641 1.58841 0.794204 0.607652i \(-0.207888\pi\)
0.794204 + 0.607652i \(0.207888\pi\)
\(258\) 0 0
\(259\) −1.26795 −0.0787865
\(260\) −1.46410 −0.0907997
\(261\) 0 0
\(262\) −12.5885 −0.777717
\(263\) −30.0526 −1.85312 −0.926560 0.376147i \(-0.877249\pi\)
−0.926560 + 0.376147i \(0.877249\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 5.32051 0.326221
\(267\) 0 0
\(268\) 13.6603 0.834433
\(269\) 0.392305 0.0239192 0.0119596 0.999928i \(-0.496193\pi\)
0.0119596 + 0.999928i \(0.496193\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) 1.46410 0.0887742
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −1.46410 −0.0882886
\(276\) 0 0
\(277\) −26.2487 −1.57713 −0.788566 0.614950i \(-0.789176\pi\)
−0.788566 + 0.614950i \(0.789176\pi\)
\(278\) −6.92820 −0.415526
\(279\) 0 0
\(280\) 1.26795 0.0757745
\(281\) −4.92820 −0.293992 −0.146996 0.989137i \(-0.546960\pi\)
−0.146996 + 0.989137i \(0.546960\pi\)
\(282\) 0 0
\(283\) −4.39230 −0.261095 −0.130548 0.991442i \(-0.541674\pi\)
−0.130548 + 0.991442i \(0.541674\pi\)
\(284\) 10.9282 0.648470
\(285\) 0 0
\(286\) −2.14359 −0.126753
\(287\) −2.53590 −0.149689
\(288\) 0 0
\(289\) −14.8564 −0.873906
\(290\) −8.92820 −0.524282
\(291\) 0 0
\(292\) 12.9282 0.756566
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0.196152 0.0114204
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 16.3923 0.949581
\(299\) 11.7128 0.677369
\(300\) 0 0
\(301\) 8.78461 0.506336
\(302\) 8.39230 0.482923
\(303\) 0 0
\(304\) −4.19615 −0.240666
\(305\) −8.92820 −0.511227
\(306\) 0 0
\(307\) −12.5885 −0.718461 −0.359231 0.933249i \(-0.616961\pi\)
−0.359231 + 0.933249i \(0.616961\pi\)
\(308\) 1.85641 0.105779
\(309\) 0 0
\(310\) 2.73205 0.155170
\(311\) −27.1244 −1.53808 −0.769041 0.639200i \(-0.779266\pi\)
−0.769041 + 0.639200i \(0.779266\pi\)
\(312\) 0 0
\(313\) 3.85641 0.217977 0.108988 0.994043i \(-0.465239\pi\)
0.108988 + 0.994043i \(0.465239\pi\)
\(314\) 16.9282 0.955314
\(315\) 0 0
\(316\) 5.26795 0.296345
\(317\) 31.8564 1.78923 0.894617 0.446834i \(-0.147448\pi\)
0.894617 + 0.446834i \(0.147448\pi\)
\(318\) 0 0
\(319\) −13.0718 −0.731880
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −10.1436 −0.565280
\(323\) −6.14359 −0.341839
\(324\) 0 0
\(325\) 1.46410 0.0812137
\(326\) 23.3205 1.29160
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −1.60770 −0.0886351
\(330\) 0 0
\(331\) −8.87564 −0.487850 −0.243925 0.969794i \(-0.578435\pi\)
−0.243925 + 0.969794i \(0.578435\pi\)
\(332\) 5.26795 0.289116
\(333\) 0 0
\(334\) −5.46410 −0.298982
\(335\) −13.6603 −0.746339
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −10.8564 −0.590511
\(339\) 0 0
\(340\) −1.46410 −0.0794021
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 15.7128 0.848412
\(344\) −6.92820 −0.373544
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −17.0718 −0.916462 −0.458231 0.888833i \(-0.651517\pi\)
−0.458231 + 0.888833i \(0.651517\pi\)
\(348\) 0 0
\(349\) 19.3205 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) −1.26795 −0.0677747
\(351\) 0 0
\(352\) −1.46410 −0.0780369
\(353\) −15.8564 −0.843951 −0.421976 0.906607i \(-0.638663\pi\)
−0.421976 + 0.906607i \(0.638663\pi\)
\(354\) 0 0
\(355\) −10.9282 −0.580009
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 17.6603 0.933373
\(359\) −8.39230 −0.442929 −0.221464 0.975168i \(-0.571084\pi\)
−0.221464 + 0.975168i \(0.571084\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) −1.46410 −0.0769515
\(363\) 0 0
\(364\) −1.85641 −0.0973021
\(365\) −12.9282 −0.676693
\(366\) 0 0
\(367\) −21.6603 −1.13066 −0.565328 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −7.60770 −0.394972
\(372\) 0 0
\(373\) 30.7846 1.59397 0.796983 0.604001i \(-0.206428\pi\)
0.796983 + 0.604001i \(0.206428\pi\)
\(374\) −2.14359 −0.110843
\(375\) 0 0
\(376\) 1.26795 0.0653895
\(377\) 13.0718 0.673232
\(378\) 0 0
\(379\) 11.6077 0.596247 0.298124 0.954527i \(-0.403639\pi\)
0.298124 + 0.954527i \(0.403639\pi\)
\(380\) 4.19615 0.215258
\(381\) 0 0
\(382\) 5.26795 0.269532
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) −1.85641 −0.0946112
\(386\) −11.8564 −0.603475
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) −15.8564 −0.803952 −0.401976 0.915650i \(-0.631676\pi\)
−0.401976 + 0.915650i \(0.631676\pi\)
\(390\) 0 0
\(391\) 11.7128 0.592342
\(392\) −5.39230 −0.272353
\(393\) 0 0
\(394\) −18.7846 −0.946355
\(395\) −5.26795 −0.265059
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 26.0526 1.30590
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 19.0718 0.952400 0.476200 0.879337i \(-0.342014\pi\)
0.476200 + 0.879337i \(0.342014\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 2.53590 0.126166
\(405\) 0 0
\(406\) −11.3205 −0.561827
\(407\) −1.46410 −0.0725728
\(408\) 0 0
\(409\) 16.9282 0.837046 0.418523 0.908206i \(-0.362548\pi\)
0.418523 + 0.908206i \(0.362548\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −13.4641 −0.663329
\(413\) 0.248711 0.0122383
\(414\) 0 0
\(415\) −5.26795 −0.258593
\(416\) 1.46410 0.0717835
\(417\) 0 0
\(418\) 6.14359 0.300493
\(419\) 34.2487 1.67316 0.836580 0.547846i \(-0.184552\pi\)
0.836580 + 0.547846i \(0.184552\pi\)
\(420\) 0 0
\(421\) −27.8564 −1.35764 −0.678819 0.734306i \(-0.737508\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(422\) 9.85641 0.479802
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 1.46410 0.0710194
\(426\) 0 0
\(427\) −11.3205 −0.547838
\(428\) −6.73205 −0.325406
\(429\) 0 0
\(430\) 6.92820 0.334108
\(431\) −8.19615 −0.394795 −0.197397 0.980324i \(-0.563249\pi\)
−0.197397 + 0.980324i \(0.563249\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 3.46410 0.166282
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −33.5692 −1.60583
\(438\) 0 0
\(439\) −31.5167 −1.50421 −0.752104 0.659044i \(-0.770961\pi\)
−0.752104 + 0.659044i \(0.770961\pi\)
\(440\) 1.46410 0.0697983
\(441\) 0 0
\(442\) 2.14359 0.101960
\(443\) 39.1244 1.85885 0.929427 0.369006i \(-0.120302\pi\)
0.929427 + 0.369006i \(0.120302\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) −22.0526 −1.04422
\(447\) 0 0
\(448\) −1.26795 −0.0599050
\(449\) −33.7128 −1.59101 −0.795503 0.605950i \(-0.792793\pi\)
−0.795503 + 0.605950i \(0.792793\pi\)
\(450\) 0 0
\(451\) −2.92820 −0.137884
\(452\) 17.4641 0.821442
\(453\) 0 0
\(454\) −3.60770 −0.169318
\(455\) 1.85641 0.0870297
\(456\) 0 0
\(457\) −4.14359 −0.193829 −0.0969146 0.995293i \(-0.530897\pi\)
−0.0969146 + 0.995293i \(0.530897\pi\)
\(458\) 15.8564 0.740921
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 26.7846 1.24748 0.623742 0.781630i \(-0.285612\pi\)
0.623742 + 0.781630i \(0.285612\pi\)
\(462\) 0 0
\(463\) 5.07180 0.235706 0.117853 0.993031i \(-0.462399\pi\)
0.117853 + 0.993031i \(0.462399\pi\)
\(464\) 8.92820 0.414481
\(465\) 0 0
\(466\) −15.0718 −0.698188
\(467\) −37.1769 −1.72034 −0.860171 0.510005i \(-0.829643\pi\)
−0.860171 + 0.510005i \(0.829643\pi\)
\(468\) 0 0
\(469\) −17.3205 −0.799787
\(470\) −1.26795 −0.0584861
\(471\) 0 0
\(472\) −0.196152 −0.00902865
\(473\) 10.1436 0.466403
\(474\) 0 0
\(475\) −4.19615 −0.192533
\(476\) −1.85641 −0.0850883
\(477\) 0 0
\(478\) 17.2679 0.789818
\(479\) 34.0526 1.55590 0.777951 0.628325i \(-0.216259\pi\)
0.777951 + 0.628325i \(0.216259\pi\)
\(480\) 0 0
\(481\) 1.46410 0.0667573
\(482\) −8.92820 −0.406669
\(483\) 0 0
\(484\) −8.85641 −0.402564
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −16.3923 −0.742806 −0.371403 0.928472i \(-0.621123\pi\)
−0.371403 + 0.928472i \(0.621123\pi\)
\(488\) 8.92820 0.404161
\(489\) 0 0
\(490\) 5.39230 0.243600
\(491\) 1.07180 0.0483695 0.0241848 0.999708i \(-0.492301\pi\)
0.0241848 + 0.999708i \(0.492301\pi\)
\(492\) 0 0
\(493\) 13.0718 0.588724
\(494\) −6.14359 −0.276413
\(495\) 0 0
\(496\) −2.73205 −0.122673
\(497\) −13.8564 −0.621545
\(498\) 0 0
\(499\) 40.5885 1.81699 0.908494 0.417897i \(-0.137233\pi\)
0.908494 + 0.417897i \(0.137233\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −22.7321 −1.01458
\(503\) 5.07180 0.226140 0.113070 0.993587i \(-0.463932\pi\)
0.113070 + 0.993587i \(0.463932\pi\)
\(504\) 0 0
\(505\) −2.53590 −0.112846
\(506\) −11.7128 −0.520698
\(507\) 0 0
\(508\) −13.6603 −0.606076
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −16.3923 −0.725153
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 25.4641 1.12317
\(515\) 13.4641 0.593299
\(516\) 0 0
\(517\) −1.85641 −0.0816447
\(518\) −1.26795 −0.0557105
\(519\) 0 0
\(520\) −1.46410 −0.0642051
\(521\) −33.4641 −1.46609 −0.733044 0.680181i \(-0.761901\pi\)
−0.733044 + 0.680181i \(0.761901\pi\)
\(522\) 0 0
\(523\) −4.78461 −0.209216 −0.104608 0.994514i \(-0.533359\pi\)
−0.104608 + 0.994514i \(0.533359\pi\)
\(524\) −12.5885 −0.549929
\(525\) 0 0
\(526\) −30.0526 −1.31035
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 5.32051 0.230673
\(533\) 2.92820 0.126835
\(534\) 0 0
\(535\) 6.73205 0.291052
\(536\) 13.6603 0.590033
\(537\) 0 0
\(538\) 0.392305 0.0169135
\(539\) 7.89488 0.340057
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 16.7846 0.720961
\(543\) 0 0
\(544\) 1.46410 0.0627728
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) −1.46410 −0.0624295
\(551\) −37.4641 −1.59602
\(552\) 0 0
\(553\) −6.67949 −0.284041
\(554\) −26.2487 −1.11520
\(555\) 0 0
\(556\) −6.92820 −0.293821
\(557\) 32.1051 1.36034 0.680169 0.733056i \(-0.261906\pi\)
0.680169 + 0.733056i \(0.261906\pi\)
\(558\) 0 0
\(559\) −10.1436 −0.429028
\(560\) 1.26795 0.0535806
\(561\) 0 0
\(562\) −4.92820 −0.207884
\(563\) −17.0718 −0.719490 −0.359745 0.933051i \(-0.617136\pi\)
−0.359745 + 0.933051i \(0.617136\pi\)
\(564\) 0 0
\(565\) −17.4641 −0.734720
\(566\) −4.39230 −0.184622
\(567\) 0 0
\(568\) 10.9282 0.458537
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −2.14359 −0.0896281
\(573\) 0 0
\(574\) −2.53590 −0.105846
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −3.60770 −0.150190 −0.0750952 0.997176i \(-0.523926\pi\)
−0.0750952 + 0.997176i \(0.523926\pi\)
\(578\) −14.8564 −0.617945
\(579\) 0 0
\(580\) −8.92820 −0.370723
\(581\) −6.67949 −0.277112
\(582\) 0 0
\(583\) −8.78461 −0.363821
\(584\) 12.9282 0.534973
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −3.21539 −0.132713 −0.0663567 0.997796i \(-0.521138\pi\)
−0.0663567 + 0.997796i \(0.521138\pi\)
\(588\) 0 0
\(589\) 11.4641 0.472370
\(590\) 0.196152 0.00807547
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −6.78461 −0.278611 −0.139305 0.990249i \(-0.544487\pi\)
−0.139305 + 0.990249i \(0.544487\pi\)
\(594\) 0 0
\(595\) 1.85641 0.0761052
\(596\) 16.3923 0.671455
\(597\) 0 0
\(598\) 11.7128 0.478973
\(599\) −2.53590 −0.103614 −0.0518070 0.998657i \(-0.516498\pi\)
−0.0518070 + 0.998657i \(0.516498\pi\)
\(600\) 0 0
\(601\) 48.3923 1.97396 0.986982 0.160833i \(-0.0514180\pi\)
0.986982 + 0.160833i \(0.0514180\pi\)
\(602\) 8.78461 0.358034
\(603\) 0 0
\(604\) 8.39230 0.341478
\(605\) 8.85641 0.360064
\(606\) 0 0
\(607\) −40.7846 −1.65540 −0.827698 0.561174i \(-0.810350\pi\)
−0.827698 + 0.561174i \(0.810350\pi\)
\(608\) −4.19615 −0.170176
\(609\) 0 0
\(610\) −8.92820 −0.361492
\(611\) 1.85641 0.0751022
\(612\) 0 0
\(613\) 3.07180 0.124069 0.0620344 0.998074i \(-0.480241\pi\)
0.0620344 + 0.998074i \(0.480241\pi\)
\(614\) −12.5885 −0.508029
\(615\) 0 0
\(616\) 1.85641 0.0747967
\(617\) −12.9282 −0.520470 −0.260235 0.965545i \(-0.583800\pi\)
−0.260235 + 0.965545i \(0.583800\pi\)
\(618\) 0 0
\(619\) 9.85641 0.396162 0.198081 0.980186i \(-0.436529\pi\)
0.198081 + 0.980186i \(0.436529\pi\)
\(620\) 2.73205 0.109722
\(621\) 0 0
\(622\) −27.1244 −1.08759
\(623\) −2.53590 −0.101599
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.85641 0.154133
\(627\) 0 0
\(628\) 16.9282 0.675509
\(629\) 1.46410 0.0583776
\(630\) 0 0
\(631\) −20.9808 −0.835231 −0.417615 0.908624i \(-0.637134\pi\)
−0.417615 + 0.908624i \(0.637134\pi\)
\(632\) 5.26795 0.209548
\(633\) 0 0
\(634\) 31.8564 1.26518
\(635\) 13.6603 0.542091
\(636\) 0 0
\(637\) −7.89488 −0.312807
\(638\) −13.0718 −0.517517
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 13.4641 0.531800 0.265900 0.964001i \(-0.414331\pi\)
0.265900 + 0.964001i \(0.414331\pi\)
\(642\) 0 0
\(643\) 41.4641 1.63518 0.817592 0.575798i \(-0.195308\pi\)
0.817592 + 0.575798i \(0.195308\pi\)
\(644\) −10.1436 −0.399714
\(645\) 0 0
\(646\) −6.14359 −0.241716
\(647\) −19.7128 −0.774991 −0.387495 0.921872i \(-0.626660\pi\)
−0.387495 + 0.921872i \(0.626660\pi\)
\(648\) 0 0
\(649\) 0.287187 0.0112731
\(650\) 1.46410 0.0574268
\(651\) 0 0
\(652\) 23.3205 0.913302
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 12.5885 0.491872
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −1.60770 −0.0626745
\(659\) −6.92820 −0.269884 −0.134942 0.990853i \(-0.543085\pi\)
−0.134942 + 0.990853i \(0.543085\pi\)
\(660\) 0 0
\(661\) −44.9282 −1.74750 −0.873752 0.486371i \(-0.838320\pi\)
−0.873752 + 0.486371i \(0.838320\pi\)
\(662\) −8.87564 −0.344962
\(663\) 0 0
\(664\) 5.26795 0.204436
\(665\) −5.32051 −0.206320
\(666\) 0 0
\(667\) 71.4256 2.76561
\(668\) −5.46410 −0.211412
\(669\) 0 0
\(670\) −13.6603 −0.527742
\(671\) −13.0718 −0.504631
\(672\) 0 0
\(673\) −19.0718 −0.735164 −0.367582 0.929991i \(-0.619814\pi\)
−0.367582 + 0.929991i \(0.619814\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 31.8564 1.22434 0.612171 0.790726i \(-0.290297\pi\)
0.612171 + 0.790726i \(0.290297\pi\)
\(678\) 0 0
\(679\) 2.53590 0.0973188
\(680\) −1.46410 −0.0561457
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −36.7846 −1.40752 −0.703762 0.710436i \(-0.748498\pi\)
−0.703762 + 0.710436i \(0.748498\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 15.7128 0.599918
\(687\) 0 0
\(688\) −6.92820 −0.264135
\(689\) 8.78461 0.334667
\(690\) 0 0
\(691\) 20.3923 0.775760 0.387880 0.921710i \(-0.373208\pi\)
0.387880 + 0.921710i \(0.373208\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −17.0718 −0.648037
\(695\) 6.92820 0.262802
\(696\) 0 0
\(697\) 2.92820 0.110914
\(698\) 19.3205 0.731292
\(699\) 0 0
\(700\) −1.26795 −0.0479240
\(701\) −15.8564 −0.598888 −0.299444 0.954114i \(-0.596801\pi\)
−0.299444 + 0.954114i \(0.596801\pi\)
\(702\) 0 0
\(703\) −4.19615 −0.158261
\(704\) −1.46410 −0.0551804
\(705\) 0 0
\(706\) −15.8564 −0.596764
\(707\) −3.21539 −0.120927
\(708\) 0 0
\(709\) 16.1436 0.606285 0.303143 0.952945i \(-0.401964\pi\)
0.303143 + 0.952945i \(0.401964\pi\)
\(710\) −10.9282 −0.410128
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) −21.8564 −0.818529
\(714\) 0 0
\(715\) 2.14359 0.0801659
\(716\) 17.6603 0.659995
\(717\) 0 0
\(718\) −8.39230 −0.313198
\(719\) −8.39230 −0.312980 −0.156490 0.987680i \(-0.550018\pi\)
−0.156490 + 0.987680i \(0.550018\pi\)
\(720\) 0 0
\(721\) 17.0718 0.635787
\(722\) −1.39230 −0.0518162
\(723\) 0 0
\(724\) −1.46410 −0.0544129
\(725\) 8.92820 0.331585
\(726\) 0 0
\(727\) −32.7846 −1.21591 −0.607957 0.793970i \(-0.708011\pi\)
−0.607957 + 0.793970i \(0.708011\pi\)
\(728\) −1.85641 −0.0688030
\(729\) 0 0
\(730\) −12.9282 −0.478494
\(731\) −10.1436 −0.375174
\(732\) 0 0
\(733\) 0.143594 0.00530375 0.00265187 0.999996i \(-0.499156\pi\)
0.00265187 + 0.999996i \(0.499156\pi\)
\(734\) −21.6603 −0.799495
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) −6.92820 −0.254858 −0.127429 0.991848i \(-0.540673\pi\)
−0.127429 + 0.991848i \(0.540673\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −7.60770 −0.279287
\(743\) 7.12436 0.261367 0.130684 0.991424i \(-0.458283\pi\)
0.130684 + 0.991424i \(0.458283\pi\)
\(744\) 0 0
\(745\) −16.3923 −0.600568
\(746\) 30.7846 1.12710
\(747\) 0 0
\(748\) −2.14359 −0.0783775
\(749\) 8.53590 0.311895
\(750\) 0 0
\(751\) 0.392305 0.0143154 0.00715770 0.999974i \(-0.497722\pi\)
0.00715770 + 0.999974i \(0.497722\pi\)
\(752\) 1.26795 0.0462373
\(753\) 0 0
\(754\) 13.0718 0.476047
\(755\) −8.39230 −0.305427
\(756\) 0 0
\(757\) −53.7128 −1.95223 −0.976113 0.217265i \(-0.930286\pi\)
−0.976113 + 0.217265i \(0.930286\pi\)
\(758\) 11.6077 0.421610
\(759\) 0 0
\(760\) 4.19615 0.152210
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) −2.53590 −0.0918057
\(764\) 5.26795 0.190588
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −0.287187 −0.0103697
\(768\) 0 0
\(769\) 20.9282 0.754690 0.377345 0.926073i \(-0.376837\pi\)
0.377345 + 0.926073i \(0.376837\pi\)
\(770\) −1.85641 −0.0669002
\(771\) 0 0
\(772\) −11.8564 −0.426721
\(773\) 38.7846 1.39499 0.697493 0.716592i \(-0.254299\pi\)
0.697493 + 0.716592i \(0.254299\pi\)
\(774\) 0 0
\(775\) −2.73205 −0.0981382
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) −15.8564 −0.568480
\(779\) −8.39230 −0.300686
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 11.7128 0.418849
\(783\) 0 0
\(784\) −5.39230 −0.192582
\(785\) −16.9282 −0.604193
\(786\) 0 0
\(787\) 11.8038 0.420762 0.210381 0.977620i \(-0.432530\pi\)
0.210381 + 0.977620i \(0.432530\pi\)
\(788\) −18.7846 −0.669174
\(789\) 0 0
\(790\) −5.26795 −0.187425
\(791\) −22.1436 −0.787336
\(792\) 0 0
\(793\) 13.0718 0.464193
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 26.0526 0.923408
\(797\) −17.4641 −0.618610 −0.309305 0.950963i \(-0.600096\pi\)
−0.309305 + 0.950963i \(0.600096\pi\)
\(798\) 0 0
\(799\) 1.85641 0.0656749
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 19.0718 0.673449
\(803\) −18.9282 −0.667962
\(804\) 0 0
\(805\) 10.1436 0.357515
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 2.53590 0.0892126
\(809\) 39.5692 1.39118 0.695590 0.718439i \(-0.255143\pi\)
0.695590 + 0.718439i \(0.255143\pi\)
\(810\) 0 0
\(811\) −14.1436 −0.496649 −0.248324 0.968677i \(-0.579880\pi\)
−0.248324 + 0.968677i \(0.579880\pi\)
\(812\) −11.3205 −0.397272
\(813\) 0 0
\(814\) −1.46410 −0.0513167
\(815\) −23.3205 −0.816882
\(816\) 0 0
\(817\) 29.0718 1.01709
\(818\) 16.9282 0.591881
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) −10.4449 −0.364085 −0.182043 0.983291i \(-0.558271\pi\)
−0.182043 + 0.983291i \(0.558271\pi\)
\(824\) −13.4641 −0.469044
\(825\) 0 0
\(826\) 0.248711 0.00865377
\(827\) −4.39230 −0.152735 −0.0763677 0.997080i \(-0.524332\pi\)
−0.0763677 + 0.997080i \(0.524332\pi\)
\(828\) 0 0
\(829\) 31.5692 1.09644 0.548222 0.836333i \(-0.315305\pi\)
0.548222 + 0.836333i \(0.315305\pi\)
\(830\) −5.26795 −0.182853
\(831\) 0 0
\(832\) 1.46410 0.0507586
\(833\) −7.89488 −0.273541
\(834\) 0 0
\(835\) 5.46410 0.189093
\(836\) 6.14359 0.212481
\(837\) 0 0
\(838\) 34.2487 1.18310
\(839\) −32.7846 −1.13185 −0.565925 0.824457i \(-0.691481\pi\)
−0.565925 + 0.824457i \(0.691481\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) −27.8564 −0.959995
\(843\) 0 0
\(844\) 9.85641 0.339272
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) 11.2295 0.385849
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 1.46410 0.0502183
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −11.3205 −0.387380
\(855\) 0 0
\(856\) −6.73205 −0.230097
\(857\) 4.14359 0.141542 0.0707712 0.997493i \(-0.477454\pi\)
0.0707712 + 0.997493i \(0.477454\pi\)
\(858\) 0 0
\(859\) −14.4449 −0.492852 −0.246426 0.969162i \(-0.579256\pi\)
−0.246426 + 0.969162i \(0.579256\pi\)
\(860\) 6.92820 0.236250
\(861\) 0 0
\(862\) −8.19615 −0.279162
\(863\) −38.4449 −1.30868 −0.654339 0.756201i \(-0.727053\pi\)
−0.654339 + 0.756201i \(0.727053\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 3.46410 0.117579
\(869\) −7.71281 −0.261639
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −33.5692 −1.13550
\(875\) 1.26795 0.0428645
\(876\) 0 0
\(877\) −50.7846 −1.71487 −0.857437 0.514589i \(-0.827945\pi\)
−0.857437 + 0.514589i \(0.827945\pi\)
\(878\) −31.5167 −1.06364
\(879\) 0 0
\(880\) 1.46410 0.0493549
\(881\) −47.3205 −1.59427 −0.797134 0.603802i \(-0.793652\pi\)
−0.797134 + 0.603802i \(0.793652\pi\)
\(882\) 0 0
\(883\) −34.2487 −1.15256 −0.576280 0.817252i \(-0.695496\pi\)
−0.576280 + 0.817252i \(0.695496\pi\)
\(884\) 2.14359 0.0720969
\(885\) 0 0
\(886\) 39.1244 1.31441
\(887\) −20.1962 −0.678120 −0.339060 0.940765i \(-0.610109\pi\)
−0.339060 + 0.940765i \(0.610109\pi\)
\(888\) 0 0
\(889\) 17.3205 0.580911
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) −22.0526 −0.738374
\(893\) −5.32051 −0.178044
\(894\) 0 0
\(895\) −17.6603 −0.590317
\(896\) −1.26795 −0.0423592
\(897\) 0 0
\(898\) −33.7128 −1.12501
\(899\) −24.3923 −0.813529
\(900\) 0 0
\(901\) 8.78461 0.292658
\(902\) −2.92820 −0.0974985
\(903\) 0 0
\(904\) 17.4641 0.580847
\(905\) 1.46410 0.0486684
\(906\) 0 0
\(907\) 5.75129 0.190968 0.0954842 0.995431i \(-0.469560\pi\)
0.0954842 + 0.995431i \(0.469560\pi\)
\(908\) −3.60770 −0.119726
\(909\) 0 0
\(910\) 1.85641 0.0615393
\(911\) 27.9090 0.924665 0.462333 0.886707i \(-0.347013\pi\)
0.462333 + 0.886707i \(0.347013\pi\)
\(912\) 0 0
\(913\) −7.71281 −0.255257
\(914\) −4.14359 −0.137058
\(915\) 0 0
\(916\) 15.8564 0.523910
\(917\) 15.9615 0.527096
\(918\) 0 0
\(919\) −13.2679 −0.437669 −0.218835 0.975762i \(-0.570226\pi\)
−0.218835 + 0.975762i \(0.570226\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) 26.7846 0.882104
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 5.07180 0.166670
\(927\) 0 0
\(928\) 8.92820 0.293083
\(929\) 15.8564 0.520232 0.260116 0.965577i \(-0.416239\pi\)
0.260116 + 0.965577i \(0.416239\pi\)
\(930\) 0 0
\(931\) 22.6269 0.741568
\(932\) −15.0718 −0.493693
\(933\) 0 0
\(934\) −37.1769 −1.21647
\(935\) 2.14359 0.0701030
\(936\) 0 0
\(937\) 3.85641 0.125983 0.0629917 0.998014i \(-0.479936\pi\)
0.0629917 + 0.998014i \(0.479936\pi\)
\(938\) −17.3205 −0.565535
\(939\) 0 0
\(940\) −1.26795 −0.0413559
\(941\) −28.3923 −0.925563 −0.462781 0.886472i \(-0.653148\pi\)
−0.462781 + 0.886472i \(0.653148\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −0.196152 −0.00638422
\(945\) 0 0
\(946\) 10.1436 0.329797
\(947\) 45.1769 1.46805 0.734026 0.679121i \(-0.237639\pi\)
0.734026 + 0.679121i \(0.237639\pi\)
\(948\) 0 0
\(949\) 18.9282 0.614435
\(950\) −4.19615 −0.136141
\(951\) 0 0
\(952\) −1.85641 −0.0601665
\(953\) 11.8564 0.384067 0.192033 0.981388i \(-0.438492\pi\)
0.192033 + 0.981388i \(0.438492\pi\)
\(954\) 0 0
\(955\) −5.26795 −0.170467
\(956\) 17.2679 0.558485
\(957\) 0 0
\(958\) 34.0526 1.10019
\(959\) 2.53590 0.0818884
\(960\) 0 0
\(961\) −23.5359 −0.759223
\(962\) 1.46410 0.0472045
\(963\) 0 0
\(964\) −8.92820 −0.287558
\(965\) 11.8564 0.381671
\(966\) 0 0
\(967\) −23.6077 −0.759172 −0.379586 0.925156i \(-0.623934\pi\)
−0.379586 + 0.925156i \(0.623934\pi\)
\(968\) −8.85641 −0.284656
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −14.9282 −0.479069 −0.239534 0.970888i \(-0.576995\pi\)
−0.239534 + 0.970888i \(0.576995\pi\)
\(972\) 0 0
\(973\) 8.78461 0.281622
\(974\) −16.3923 −0.525243
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 0 0
\(979\) −2.92820 −0.0935858
\(980\) 5.39230 0.172251
\(981\) 0 0
\(982\) 1.07180 0.0342024
\(983\) 38.4449 1.22620 0.613100 0.790005i \(-0.289922\pi\)
0.613100 + 0.790005i \(0.289922\pi\)
\(984\) 0 0
\(985\) 18.7846 0.598527
\(986\) 13.0718 0.416291
\(987\) 0 0
\(988\) −6.14359 −0.195454
\(989\) −55.4256 −1.76243
\(990\) 0 0
\(991\) 5.94744 0.188927 0.0944633 0.995528i \(-0.469886\pi\)
0.0944633 + 0.995528i \(0.469886\pi\)
\(992\) −2.73205 −0.0867427
\(993\) 0 0
\(994\) −13.8564 −0.439499
\(995\) −26.0526 −0.825922
\(996\) 0 0
\(997\) −4.14359 −0.131229 −0.0656145 0.997845i \(-0.520901\pi\)
−0.0656145 + 0.997845i \(0.520901\pi\)
\(998\) 40.5885 1.28481
\(999\) 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 3330.2.a.bd.1.2 2
3.2 odd 2 370.2.a.e.1.1 2
12.11 even 2 2960.2.a.q.1.2 2
15.2 even 4 1850.2.b.l.149.2 4
15.8 even 4 1850.2.b.l.149.3 4
15.14 odd 2 1850.2.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 3.2 odd 2
1850.2.a.x.1.2 2 15.14 odd 2
1850.2.b.l.149.2 4 15.2 even 4
1850.2.b.l.149.3 4 15.8 even 4
2960.2.a.q.1.2 2 12.11 even 2
3330.2.a.bd.1.2 2 1.1 even 1 trivial