Properties

Label 2960.2.a.q.1.2
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q+2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} -1.46410 q^{11} +1.46410 q^{13} +2.73205 q^{15} -1.46410 q^{17} +4.19615 q^{19} +3.46410 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} -8.92820 q^{29} +2.73205 q^{31} -4.00000 q^{33} +1.26795 q^{35} +1.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} +6.92820 q^{43} +4.46410 q^{45} +1.26795 q^{47} -5.39230 q^{49} -4.00000 q^{51} -6.00000 q^{53} -1.46410 q^{55} +11.4641 q^{57} -0.196152 q^{59} +8.92820 q^{61} +5.66025 q^{63} +1.46410 q^{65} -13.6603 q^{67} +21.8564 q^{69} +10.9282 q^{71} +12.9282 q^{73} +2.73205 q^{75} -1.85641 q^{77} -5.26795 q^{79} -2.46410 q^{81} +5.26795 q^{83} -1.46410 q^{85} -24.3923 q^{87} -2.00000 q^{89} +1.85641 q^{91} +7.46410 q^{93} +4.19615 q^{95} -2.00000 q^{97} -6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) −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 \(0.434920\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 0 0
\(17\) −1.46410 −0.355097 −0.177548 0.984112i \(-0.556817\pi\)
−0.177548 + 0.984112i \(0.556817\pi\)
\(18\) 0 0
\(19\) 4.19615 0.962663 0.481332 0.876539i \(-0.340153\pi\)
0.481332 + 0.876539i \(0.340153\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −8.92820 −1.65793 −0.828963 0.559304i \(-0.811069\pi\)
−0.828963 + 0.559304i \(0.811069\pi\)
\(30\) 0 0
\(31\) 2.73205 0.490691 0.245345 0.969436i \(-0.421099\pi\)
0.245345 + 0.969436i \(0.421099\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 1.26795 0.214323
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 0 0
\(45\) 4.46410 0.665469
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −1.46410 −0.197419
\(56\) 0 0
\(57\) 11.4641 1.51846
\(58\) 0 0
\(59\) −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 \(0.306335\pi\)
\(62\) 0 0
\(63\) 5.66025 0.713125
\(64\) 0 0
\(65\) 1.46410 0.181599
\(66\) 0 0
\(67\) −13.6603 −1.66887 −0.834433 0.551110i \(-0.814205\pi\)
−0.834433 + 0.551110i \(0.814205\pi\)
\(68\) 0 0
\(69\) 21.8564 2.63120
\(70\) 0 0
\(71\) 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 \(0.226876\pi\)
\(74\) 0 0
\(75\) 2.73205 0.315470
\(76\) 0 0
\(77\) −1.85641 −0.211557
\(78\) 0 0
\(79\) −5.26795 −0.592691 −0.296345 0.955081i \(-0.595768\pi\)
−0.296345 + 0.955081i \(0.595768\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) −24.3923 −2.61513
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 1.85641 0.194604
\(92\) 0 0
\(93\) 7.46410 0.773991
\(94\) 0 0
\(95\) 4.19615 0.430516
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −6.53590 −0.656883
\(100\) 0 0
\(101\) −2.53590 −0.252331 −0.126166 0.992009i \(-0.540267\pi\)
−0.126166 + 0.992009i \(0.540267\pi\)
\(102\) 0 0
\(103\) 13.4641 1.32666 0.663329 0.748328i \(-0.269143\pi\)
0.663329 + 0.748328i \(0.269143\pi\)
\(104\) 0 0
\(105\) 3.46410 0.338062
\(106\) 0 0
\(107\) −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 \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.73205 0.259315
\(112\) 0 0
\(113\) −17.4641 −1.64288 −0.821442 0.570292i \(-0.806830\pi\)
−0.821442 + 0.570292i \(0.806830\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 6.53590 0.604244
\(118\) 0 0
\(119\) −1.85641 −0.170177
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) −5.46410 −0.492681
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 13.6603 1.21215 0.606076 0.795407i \(-0.292743\pi\)
0.606076 + 0.795407i \(0.292743\pi\)
\(128\) 0 0
\(129\) 18.9282 1.66654
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 0 0
\(143\) −2.14359 −0.179256
\(144\) 0 0
\(145\) −8.92820 −0.741447
\(146\) 0 0
\(147\) −14.7321 −1.21508
\(148\) 0 0
\(149\) −16.3923 −1.34291 −0.671455 0.741045i \(-0.734330\pi\)
−0.671455 + 0.741045i \(0.734330\pi\)
\(150\) 0 0
\(151\) −8.39230 −0.682956 −0.341478 0.939890i \(-0.610927\pi\)
−0.341478 + 0.939890i \(0.610927\pi\)
\(152\) 0 0
\(153\) −6.53590 −0.528396
\(154\) 0 0
\(155\) 2.73205 0.219444
\(156\) 0 0
\(157\) 16.9282 1.35102 0.675509 0.737352i \(-0.263924\pi\)
0.675509 + 0.737352i \(0.263924\pi\)
\(158\) 0 0
\(159\) −16.3923 −1.29999
\(160\) 0 0
\(161\) 10.1436 0.799427
\(162\) 0 0
\(163\) −23.3205 −1.82660 −0.913302 0.407284i \(-0.866476\pi\)
−0.913302 + 0.407284i \(0.866476\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) 18.7321 1.43248
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 1.26795 0.0958479
\(176\) 0 0
\(177\) −0.535898 −0.0402806
\(178\) 0 0
\(179\) 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 \(0.517329\pi\)
\(182\) 0 0
\(183\) 24.3923 1.80313
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 2.14359 0.156755
\(188\) 0 0
\(189\) 5.07180 0.368919
\(190\) 0 0
\(191\) 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 \(0.640332\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) 18.7846 1.33835 0.669174 0.743106i \(-0.266648\pi\)
0.669174 + 0.743106i \(0.266648\pi\)
\(198\) 0 0
\(199\) −26.0526 −1.84682 −0.923408 0.383819i \(-0.874609\pi\)
−0.923408 + 0.383819i \(0.874609\pi\)
\(200\) 0 0
\(201\) −37.3205 −2.63239
\(202\) 0 0
\(203\) −11.3205 −0.794544
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 35.7128 2.48221
\(208\) 0 0
\(209\) −6.14359 −0.424961
\(210\) 0 0
\(211\) −9.85641 −0.678543 −0.339272 0.940688i \(-0.610181\pi\)
−0.339272 + 0.940688i \(0.610181\pi\)
\(212\) 0 0
\(213\) 29.8564 2.04573
\(214\) 0 0
\(215\) 6.92820 0.472500
\(216\) 0 0
\(217\) 3.46410 0.235159
\(218\) 0 0
\(219\) 35.3205 2.38674
\(220\) 0 0
\(221\) −2.14359 −0.144194
\(222\) 0 0
\(223\) 22.0526 1.47675 0.738374 0.674391i \(-0.235594\pi\)
0.738374 + 0.674391i \(0.235594\pi\)
\(224\) 0 0
\(225\) 4.46410 0.297607
\(226\) 0 0
\(227\) −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 \(0.324473\pi\)
\(230\) 0 0
\(231\) −5.07180 −0.333700
\(232\) 0 0
\(233\) 15.0718 0.987386 0.493693 0.869636i \(-0.335647\pi\)
0.493693 + 0.869636i \(0.335647\pi\)
\(234\) 0 0
\(235\) 1.26795 0.0827119
\(236\) 0 0
\(237\) −14.3923 −0.934881
\(238\) 0 0
\(239\) 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 \(0.592843\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 0 0
\(245\) −5.39230 −0.344502
\(246\) 0 0
\(247\) 6.14359 0.390907
\(248\) 0 0
\(249\) 14.3923 0.912075
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) −25.4641 −1.58841 −0.794204 0.607652i \(-0.792112\pi\)
−0.794204 + 0.607652i \(0.792112\pi\)
\(258\) 0 0
\(259\) 1.26795 0.0787865
\(260\) 0 0
\(261\) −39.8564 −2.46705
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −5.46410 −0.334398
\(268\) 0 0
\(269\) −0.392305 −0.0239192 −0.0119596 0.999928i \(-0.503807\pi\)
−0.0119596 + 0.999928i \(0.503807\pi\)
\(270\) 0 0
\(271\) −16.7846 −1.01959 −0.509796 0.860295i \(-0.670279\pi\)
−0.509796 + 0.860295i \(0.670279\pi\)
\(272\) 0 0
\(273\) 5.07180 0.306959
\(274\) 0 0
\(275\) −1.46410 −0.0882886
\(276\) 0 0
\(277\) −26.2487 −1.57713 −0.788566 0.614950i \(-0.789176\pi\)
−0.788566 + 0.614950i \(0.789176\pi\)
\(278\) 0 0
\(279\) 12.1962 0.730165
\(280\) 0 0
\(281\) 4.92820 0.293992 0.146996 0.989137i \(-0.453040\pi\)
0.146996 + 0.989137i \(0.453040\pi\)
\(282\) 0 0
\(283\) 4.39230 0.261095 0.130548 0.991442i \(-0.458326\pi\)
0.130548 + 0.991442i \(0.458326\pi\)
\(284\) 0 0
\(285\) 11.4641 0.679075
\(286\) 0 0
\(287\) −2.53590 −0.149689
\(288\) 0 0
\(289\) −14.8564 −0.873906
\(290\) 0 0
\(291\) −5.46410 −0.320311
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −0.196152 −0.0114204
\(296\) 0 0
\(297\) −5.85641 −0.339823
\(298\) 0 0
\(299\) 11.7128 0.677369
\(300\) 0 0
\(301\) 8.78461 0.506336
\(302\) 0 0
\(303\) −6.92820 −0.398015
\(304\) 0 0
\(305\) 8.92820 0.511227
\(306\) 0 0
\(307\) 12.5885 0.718461 0.359231 0.933249i \(-0.383039\pi\)
0.359231 + 0.933249i \(0.383039\pi\)
\(308\) 0 0
\(309\) 36.7846 2.09260
\(310\) 0 0
\(311\) −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 \(0.465239\pi\)
\(314\) 0 0
\(315\) 5.66025 0.318919
\(316\) 0 0
\(317\) −31.8564 −1.78923 −0.894617 0.446834i \(-0.852552\pi\)
−0.894617 + 0.446834i \(0.852552\pi\)
\(318\) 0 0
\(319\) 13.0718 0.731880
\(320\) 0 0
\(321\) −18.3923 −1.02656
\(322\) 0 0
\(323\) −6.14359 −0.341839
\(324\) 0 0
\(325\) 1.46410 0.0812137
\(326\) 0 0
\(327\) 5.46410 0.302166
\(328\) 0 0
\(329\) 1.60770 0.0886351
\(330\) 0 0
\(331\) 8.87564 0.487850 0.243925 0.969794i \(-0.421565\pi\)
0.243925 + 0.969794i \(0.421565\pi\)
\(332\) 0 0
\(333\) 4.46410 0.244631
\(334\) 0 0
\(335\) −13.6603 −0.746339
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) −47.7128 −2.59140
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) 21.8564 1.17671
\(346\) 0 0
\(347\) −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 \(0.327011\pi\)
\(350\) 0 0
\(351\) 5.85641 0.312592
\(352\) 0 0
\(353\) 15.8564 0.843951 0.421976 0.906607i \(-0.361337\pi\)
0.421976 + 0.906607i \(0.361337\pi\)
\(354\) 0 0
\(355\) 10.9282 0.580009
\(356\) 0 0
\(357\) −5.07180 −0.268428
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) −24.1962 −1.26997
\(364\) 0 0
\(365\) 12.9282 0.676693
\(366\) 0 0
\(367\) 21.6603 1.13066 0.565328 0.824866i \(-0.308750\pi\)
0.565328 + 0.824866i \(0.308750\pi\)
\(368\) 0 0
\(369\) −8.92820 −0.464784
\(370\) 0 0
\(371\) −7.60770 −0.394972
\(372\) 0 0
\(373\) 30.7846 1.59397 0.796983 0.604001i \(-0.206428\pi\)
0.796983 + 0.604001i \(0.206428\pi\)
\(374\) 0 0
\(375\) 2.73205 0.141082
\(376\) 0 0
\(377\) −13.0718 −0.673232
\(378\) 0 0
\(379\) −11.6077 −0.596247 −0.298124 0.954527i \(-0.596361\pi\)
−0.298124 + 0.954527i \(0.596361\pi\)
\(380\) 0 0
\(381\) 37.3205 1.91199
\(382\) 0 0
\(383\) 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
\(386\) 0 0
\(387\) 30.9282 1.57217
\(388\) 0 0
\(389\) 15.8564 0.803952 0.401976 0.915650i \(-0.368324\pi\)
0.401976 + 0.915650i \(0.368324\pi\)
\(390\) 0 0
\(391\) −11.7128 −0.592342
\(392\) 0 0
\(393\) −34.3923 −1.73486
\(394\) 0 0
\(395\) −5.26795 −0.265059
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 14.5359 0.727705
\(400\) 0 0
\(401\) −19.0718 −0.952400 −0.476200 0.879337i \(-0.657986\pi\)
−0.476200 + 0.879337i \(0.657986\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) −2.46410 −0.122442
\(406\) 0 0
\(407\) −1.46410 −0.0725728
\(408\) 0 0
\(409\) 16.9282 0.837046 0.418523 0.908206i \(-0.362548\pi\)
0.418523 + 0.908206i \(0.362548\pi\)
\(410\) 0 0
\(411\) 5.46410 0.269524
\(412\) 0 0
\(413\) −0.248711 −0.0122383
\(414\) 0 0
\(415\) 5.26795 0.258593
\(416\) 0 0
\(417\) 18.9282 0.926918
\(418\) 0 0
\(419\) 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 \(0.737508\pi\)
\(422\) 0 0
\(423\) 5.66025 0.275211
\(424\) 0 0
\(425\) −1.46410 −0.0710194
\(426\) 0 0
\(427\) 11.3205 0.547838
\(428\) 0 0
\(429\) −5.85641 −0.282750
\(430\) 0 0
\(431\) −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 \(0.804347\pi\)
\(434\) 0 0
\(435\) −24.3923 −1.16952
\(436\) 0 0
\(437\) 33.5692 1.60583
\(438\) 0 0
\(439\) 31.5167 1.50421 0.752104 0.659044i \(-0.229039\pi\)
0.752104 + 0.659044i \(0.229039\pi\)
\(440\) 0 0
\(441\) −24.0718 −1.14628
\(442\) 0 0
\(443\) 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
\(446\) 0 0
\(447\) −44.7846 −2.11824
\(448\) 0 0
\(449\) 33.7128 1.59101 0.795503 0.605950i \(-0.207207\pi\)
0.795503 + 0.605950i \(0.207207\pi\)
\(450\) 0 0
\(451\) 2.92820 0.137884
\(452\) 0 0
\(453\) −22.9282 −1.07726
\(454\) 0 0
\(455\) 1.85641 0.0870297
\(456\) 0 0
\(457\) −4.14359 −0.193829 −0.0969146 0.995293i \(-0.530897\pi\)
−0.0969146 + 0.995293i \(0.530897\pi\)
\(458\) 0 0
\(459\) −5.85641 −0.273354
\(460\) 0 0
\(461\) −26.7846 −1.24748 −0.623742 0.781630i \(-0.714388\pi\)
−0.623742 + 0.781630i \(0.714388\pi\)
\(462\) 0 0
\(463\) −5.07180 −0.235706 −0.117853 0.993031i \(-0.537601\pi\)
−0.117853 + 0.993031i \(0.537601\pi\)
\(464\) 0 0
\(465\) 7.46410 0.346139
\(466\) 0 0
\(467\) −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
\(470\) 0 0
\(471\) 46.2487 2.13103
\(472\) 0 0
\(473\) −10.1436 −0.466403
\(474\) 0 0
\(475\) 4.19615 0.192533
\(476\) 0 0
\(477\) −26.7846 −1.22638
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 27.7128 1.26098
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 16.3923 0.742806 0.371403 0.928472i \(-0.378877\pi\)
0.371403 + 0.928472i \(0.378877\pi\)
\(488\) 0 0
\(489\) −63.7128 −2.88119
\(490\) 0 0
\(491\) 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
\(494\) 0 0
\(495\) −6.53590 −0.293767
\(496\) 0 0
\(497\) 13.8564 0.621545
\(498\) 0 0
\(499\) −40.5885 −1.81699 −0.908494 0.417897i \(-0.862767\pi\)
−0.908494 + 0.417897i \(0.862767\pi\)
\(500\) 0 0
\(501\) −14.9282 −0.666943
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) −29.6603 −1.31726
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 16.3923 0.725153
\(512\) 0 0
\(513\) 16.7846 0.741059
\(514\) 0 0
\(515\) 13.4641 0.593299
\(516\) 0 0
\(517\) −1.85641 −0.0816447
\(518\) 0 0
\(519\) 27.3205 1.19924
\(520\) 0 0
\(521\) 33.4641 1.46609 0.733044 0.680181i \(-0.238099\pi\)
0.733044 + 0.680181i \(0.238099\pi\)
\(522\) 0 0
\(523\) 4.78461 0.209216 0.104608 0.994514i \(-0.466641\pi\)
0.104608 + 0.994514i \(0.466641\pi\)
\(524\) 0 0
\(525\) 3.46410 0.151186
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −0.875644 −0.0379997
\(532\) 0 0
\(533\) −2.92820 −0.126835
\(534\) 0 0
\(535\) −6.73205 −0.291052
\(536\) 0 0
\(537\) 48.2487 2.08209
\(538\) 0 0
\(539\) 7.89488 0.340057
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 39.8564 1.70103
\(550\) 0 0
\(551\) −37.4641 −1.59602
\(552\) 0 0
\(553\) −6.67949 −0.284041
\(554\) 0 0
\(555\) 2.73205 0.115969
\(556\) 0 0
\(557\) −32.1051 −1.36034 −0.680169 0.733056i \(-0.738094\pi\)
−0.680169 + 0.733056i \(0.738094\pi\)
\(558\) 0 0
\(559\) 10.1436 0.429028
\(560\) 0 0
\(561\) 5.85641 0.247258
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) −3.12436 −0.131211
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 14.3923 0.601247
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −3.60770 −0.150190 −0.0750952 0.997176i \(-0.523926\pi\)
−0.0750952 + 0.997176i \(0.523926\pi\)
\(578\) 0 0
\(579\) −32.3923 −1.34618
\(580\) 0 0
\(581\) 6.67949 0.277112
\(582\) 0 0
\(583\) 8.78461 0.363821
\(584\) 0 0
\(585\) 6.53590 0.270226
\(586\) 0 0
\(587\) −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 0
\(591\) 51.3205 2.11104
\(592\) 0 0
\(593\) 6.78461 0.278611 0.139305 0.990249i \(-0.455513\pi\)
0.139305 + 0.990249i \(0.455513\pi\)
\(594\) 0 0
\(595\) −1.85641 −0.0761052
\(596\) 0 0
\(597\) −71.1769 −2.91308
\(598\) 0 0
\(599\) −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 \(0.0514180\pi\)
\(602\) 0 0
\(603\) −60.9808 −2.48333
\(604\) 0 0
\(605\) −8.85641 −0.360064
\(606\) 0 0
\(607\) 40.7846 1.65540 0.827698 0.561174i \(-0.189650\pi\)
0.827698 + 0.561174i \(0.189650\pi\)
\(608\) 0 0
\(609\) −30.9282 −1.25327
\(610\) 0 0
\(611\) 1.85641 0.0751022
\(612\) 0 0
\(613\) 3.07180 0.124069 0.0620344 0.998074i \(-0.480241\pi\)
0.0620344 + 0.998074i \(0.480241\pi\)
\(614\) 0 0
\(615\) −5.46410 −0.220334
\(616\) 0 0
\(617\) 12.9282 0.520470 0.260235 0.965545i \(-0.416200\pi\)
0.260235 + 0.965545i \(0.416200\pi\)
\(618\) 0 0
\(619\) −9.85641 −0.396162 −0.198081 0.980186i \(-0.563471\pi\)
−0.198081 + 0.980186i \(0.563471\pi\)
\(620\) 0 0
\(621\) 32.0000 1.28412
\(622\) 0 0
\(623\) −2.53590 −0.101599
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.7846 −0.670313
\(628\) 0 0
\(629\) −1.46410 −0.0583776
\(630\) 0 0
\(631\) 20.9808 0.835231 0.417615 0.908624i \(-0.362866\pi\)
0.417615 + 0.908624i \(0.362866\pi\)
\(632\) 0 0
\(633\) −26.9282 −1.07030
\(634\) 0 0
\(635\) 13.6603 0.542091
\(636\) 0 0
\(637\) −7.89488 −0.312807
\(638\) 0 0
\(639\) 48.7846 1.92989
\(640\) 0 0
\(641\) −13.4641 −0.531800 −0.265900 0.964001i \(-0.585669\pi\)
−0.265900 + 0.964001i \(0.585669\pi\)
\(642\) 0 0
\(643\) −41.4641 −1.63518 −0.817592 0.575798i \(-0.804692\pi\)
−0.817592 + 0.575798i \(0.804692\pi\)
\(644\) 0 0
\(645\) 18.9282 0.745297
\(646\) 0 0
\(647\) −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
\(650\) 0 0
\(651\) 9.46410 0.370927
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −12.5885 −0.491872
\(656\) 0 0
\(657\) 57.7128 2.25159
\(658\) 0 0
\(659\) −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 \(0.838320\pi\)
\(662\) 0 0
\(663\) −5.85641 −0.227444
\(664\) 0 0
\(665\) 5.32051 0.206320
\(666\) 0 0
\(667\) −71.4256 −2.76561
\(668\) 0 0
\(669\) 60.2487 2.32935
\(670\) 0 0
\(671\) −13.0718 −0.504631
\(672\) 0 0
\(673\) −19.0718 −0.735164 −0.367582 0.929991i \(-0.619814\pi\)
−0.367582 + 0.929991i \(0.619814\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −31.8564 −1.22434 −0.612171 0.790726i \(-0.709703\pi\)
−0.612171 + 0.790726i \(0.709703\pi\)
\(678\) 0 0
\(679\) −2.53590 −0.0973188
\(680\) 0 0
\(681\) −9.85641 −0.377698
\(682\) 0 0
\(683\) −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
\(686\) 0 0
\(687\) 43.3205 1.65278
\(688\) 0 0
\(689\) −8.78461 −0.334667
\(690\) 0 0
\(691\) −20.3923 −0.775760 −0.387880 0.921710i \(-0.626792\pi\)
−0.387880 + 0.921710i \(0.626792\pi\)
\(692\) 0 0
\(693\) −8.28719 −0.314804
\(694\) 0 0
\(695\) 6.92820 0.262802
\(696\) 0 0
\(697\) 2.92820 0.110914
\(698\) 0 0
\(699\) 41.1769 1.55745
\(700\) 0 0
\(701\) 15.8564 0.598888 0.299444 0.954114i \(-0.403199\pi\)
0.299444 + 0.954114i \(0.403199\pi\)
\(702\) 0 0
\(703\) 4.19615 0.158261
\(704\) 0 0
\(705\) 3.46410 0.130466
\(706\) 0 0
\(707\) −3.21539 −0.120927
\(708\) 0 0
\(709\) 16.1436 0.606285 0.303143 0.952945i \(-0.401964\pi\)
0.303143 + 0.952945i \(0.401964\pi\)
\(710\) 0 0
\(711\) −23.5167 −0.881944
\(712\) 0 0
\(713\) 21.8564 0.818529
\(714\) 0 0
\(715\) −2.14359 −0.0801659
\(716\) 0 0
\(717\) 47.1769 1.76185
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) −24.3923 −0.907160
\(724\) 0 0
\(725\) −8.92820 −0.331585
\(726\) 0 0
\(727\) 32.7846 1.21591 0.607957 0.793970i \(-0.291989\pi\)
0.607957 + 0.793970i \(0.291989\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −10.1436 −0.375174
\(732\) 0 0
\(733\) 0.143594 0.00530375 0.00265187 0.999996i \(-0.499156\pi\)
0.00265187 + 0.999996i \(0.499156\pi\)
\(734\) 0 0
\(735\) −14.7321 −0.543400
\(736\) 0 0
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) 6.92820 0.254858 0.127429 0.991848i \(-0.459327\pi\)
0.127429 + 0.991848i \(0.459327\pi\)
\(740\) 0 0
\(741\) 16.7846 0.616598
\(742\) 0 0
\(743\) 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
\(746\) 0 0
\(747\) 23.5167 0.860430
\(748\) 0 0
\(749\) −8.53590 −0.311895
\(750\) 0 0
\(751\) −0.392305 −0.0143154 −0.00715770 0.999974i \(-0.502278\pi\)
−0.00715770 + 0.999974i \(0.502278\pi\)
\(752\) 0 0
\(753\) −62.1051 −2.26324
\(754\) 0 0
\(755\) −8.39230 −0.305427
\(756\) 0 0
\(757\) −53.7128 −1.95223 −0.976113 0.217265i \(-0.930286\pi\)
−0.976113 + 0.217265i \(0.930286\pi\)
\(758\) 0 0
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 2.53590 0.0918057
\(764\) 0 0
\(765\) −6.53590 −0.236306
\(766\) 0 0
\(767\) −0.287187 −0.0103697
\(768\) 0 0
\(769\) 20.9282 0.754690 0.377345 0.926073i \(-0.376837\pi\)
0.377345 + 0.926073i \(0.376837\pi\)
\(770\) 0 0
\(771\) −69.5692 −2.50547
\(772\) 0 0
\(773\) −38.7846 −1.39499 −0.697493 0.716592i \(-0.745701\pi\)
−0.697493 + 0.716592i \(0.745701\pi\)
\(774\) 0 0
\(775\) 2.73205 0.0981382
\(776\) 0 0
\(777\) 3.46410 0.124274
\(778\) 0 0
\(779\) −8.39230 −0.300686
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) −35.7128 −1.27627
\(784\) 0 0
\(785\) 16.9282 0.604193
\(786\) 0 0
\(787\) −11.8038 −0.420762 −0.210381 0.977620i \(-0.567470\pi\)
−0.210381 + 0.977620i \(0.567470\pi\)
\(788\) 0 0
\(789\) −82.1051 −2.92302
\(790\) 0 0
\(791\) −22.1436 −0.787336
\(792\) 0 0
\(793\) 13.0718 0.464193
\(794\) 0 0
\(795\) −16.3923 −0.581375
\(796\) 0 0
\(797\) 17.4641 0.618610 0.309305 0.950963i \(-0.399904\pi\)
0.309305 + 0.950963i \(0.399904\pi\)
\(798\) 0 0
\(799\) −1.85641 −0.0656749
\(800\) 0 0
\(801\) −8.92820 −0.315463
\(802\) 0 0
\(803\) −18.9282 −0.667962
\(804\) 0 0
\(805\) 10.1436 0.357515
\(806\) 0 0
\(807\) −1.07180 −0.0377290
\(808\) 0 0
\(809\) −39.5692 −1.39118 −0.695590 0.718439i \(-0.744857\pi\)
−0.695590 + 0.718439i \(0.744857\pi\)
\(810\) 0 0
\(811\) 14.1436 0.496649 0.248324 0.968677i \(-0.420120\pi\)
0.248324 + 0.968677i \(0.420120\pi\)
\(812\) 0 0
\(813\) −45.8564 −1.60825
\(814\) 0 0
\(815\) −23.3205 −0.816882
\(816\) 0 0
\(817\) 29.0718 1.01709
\(818\) 0 0
\(819\) 8.28719 0.289578
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 10.4449 0.364085 0.182043 0.983291i \(-0.441729\pi\)
0.182043 + 0.983291i \(0.441729\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −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 \(0.315305\pi\)
\(830\) 0 0
\(831\) −71.7128 −2.48769
\(832\) 0 0
\(833\) 7.89488 0.273541
\(834\) 0 0
\(835\) −5.46410 −0.189093
\(836\) 0 0
\(837\) 10.9282 0.377734
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) 13.4641 0.463728
\(844\) 0 0
\(845\) −10.8564 −0.373472
\(846\) 0 0
\(847\) −11.2295 −0.385849
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 18.7321 0.640623
\(856\) 0 0
\(857\) −4.14359 −0.141542 −0.0707712 0.997493i \(-0.522546\pi\)
−0.0707712 + 0.997493i \(0.522546\pi\)
\(858\) 0 0
\(859\) 14.4449 0.492852 0.246426 0.969162i \(-0.420744\pi\)
0.246426 + 0.969162i \(0.420744\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) −40.5885 −1.37846
\(868\) 0 0
\(869\) 7.71281 0.261639
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) −8.92820 −0.302174
\(874\) 0 0
\(875\) 1.26795 0.0428645
\(876\) 0 0
\(877\) −50.7846 −1.71487 −0.857437 0.514589i \(-0.827945\pi\)
−0.857437 + 0.514589i \(0.827945\pi\)
\(878\) 0 0
\(879\) 16.3923 0.552899
\(880\) 0 0
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 0 0
\(883\) 34.2487 1.15256 0.576280 0.817252i \(-0.304504\pi\)
0.576280 + 0.817252i \(0.304504\pi\)
\(884\) 0 0
\(885\) −0.535898 −0.0180140
\(886\) 0 0
\(887\) −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
\(890\) 0 0
\(891\) 3.60770 0.120862
\(892\) 0 0
\(893\) 5.32051 0.178044
\(894\) 0 0
\(895\) 17.6603 0.590317
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) −24.3923 −0.813529
\(900\) 0 0
\(901\) 8.78461 0.292658
\(902\) 0 0
\(903\) 24.0000 0.798670
\(904\) 0 0
\(905\) −1.46410 −0.0486684
\(906\) 0 0
\(907\) −5.75129 −0.190968 −0.0954842 0.995431i \(-0.530440\pi\)
−0.0954842 + 0.995431i \(0.530440\pi\)
\(908\) 0 0
\(909\) −11.3205 −0.375478
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) 24.3923 0.806385
\(916\) 0 0
\(917\) −15.9615 −0.527096
\(918\) 0 0
\(919\) 13.2679 0.437669 0.218835 0.975762i \(-0.429774\pi\)
0.218835 + 0.975762i \(0.429774\pi\)
\(920\) 0 0
\(921\) 34.3923 1.13326
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 60.1051 1.97411
\(928\) 0 0
\(929\) −15.8564 −0.520232 −0.260116 0.965577i \(-0.583761\pi\)
−0.260116 + 0.965577i \(0.583761\pi\)
\(930\) 0 0
\(931\) −22.6269 −0.741568
\(932\) 0 0
\(933\) −74.1051 −2.42609
\(934\) 0 0
\(935\) 2.14359 0.0701030
\(936\) 0 0
\(937\) 3.85641 0.125983 0.0629917 0.998014i \(-0.479936\pi\)
0.0629917 + 0.998014i \(0.479936\pi\)
\(938\) 0 0
\(939\) 10.5359 0.343826
\(940\) 0 0
\(941\) 28.3923 0.925563 0.462781 0.886472i \(-0.346852\pi\)
0.462781 + 0.886472i \(0.346852\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 5.07180 0.164986
\(946\) 0 0
\(947\) 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
\(950\) 0 0
\(951\) −87.0333 −2.82225
\(952\) 0 0
\(953\) −11.8564 −0.384067 −0.192033 0.981388i \(-0.561508\pi\)
−0.192033 + 0.981388i \(0.561508\pi\)
\(954\) 0 0
\(955\) 5.26795 0.170467
\(956\) 0 0
\(957\) 35.7128 1.15443
\(958\) 0 0
\(959\) 2.53590 0.0818884
\(960\) 0 0
\(961\) −23.5359 −0.759223
\(962\) 0 0
\(963\) −30.0526 −0.968430
\(964\) 0 0
\(965\) −11.8564 −0.381671
\(966\) 0 0
\(967\) 23.6077 0.759172 0.379586 0.925156i \(-0.376066\pi\)
0.379586 + 0.925156i \(0.376066\pi\)
\(968\) 0 0
\(969\) −16.7846 −0.539199
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 0 0
\(979\) 2.92820 0.0935858
\(980\) 0 0
\(981\) 8.92820 0.285056
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) 4.39230 0.139809
\(988\) 0 0
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) −5.94744 −0.188927 −0.0944633 0.995528i \(-0.530114\pi\)
−0.0944633 + 0.995528i \(0.530114\pi\)
\(992\) 0 0
\(993\) 24.2487 0.769510
\(994\) 0 0
\(995\) −26.0526 −0.825922
\(996\) 0 0
\(997\) −4.14359 −0.131229 −0.0656145 0.997845i \(-0.520901\pi\)
−0.0656145 + 0.997845i \(0.520901\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 2960.2.a.q.1.2 2
4.3 odd 2 370.2.a.e.1.1 2
12.11 even 2 3330.2.a.bd.1.2 2
20.3 even 4 1850.2.b.l.149.3 4
20.7 even 4 1850.2.b.l.149.2 4
20.19 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 4.3 odd 2
1850.2.a.x.1.2 2 20.19 odd 2
1850.2.b.l.149.2 4 20.7 even 4
1850.2.b.l.149.3 4 20.3 even 4
2960.2.a.q.1.2 2 1.1 even 1 trivial
3330.2.a.bd.1.2 2 12.11 even 2