Properties

Label 9555.2.a.bq.1.3
Level $9555$
Weight $2$
Character 9555.1
Self dual yes
Analytic conductor $76.297$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9555 = 3 \cdot 5 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9555.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(76.2970591313\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 9555.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.77846 q^{2} +1.00000 q^{3} +5.71982 q^{4} +1.00000 q^{5} +2.77846 q^{6} +10.3354 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.77846 q^{2} +1.00000 q^{3} +5.71982 q^{4} +1.00000 q^{5} +2.77846 q^{6} +10.3354 q^{8} +1.00000 q^{9} +2.77846 q^{10} -2.71982 q^{11} +5.71982 q^{12} -1.00000 q^{13} +1.00000 q^{15} +17.2767 q^{16} +2.83709 q^{17} +2.77846 q^{18} +3.55691 q^{19} +5.71982 q^{20} -7.55691 q^{22} -4.83709 q^{23} +10.3354 q^{24} +1.00000 q^{25} -2.77846 q^{26} +1.00000 q^{27} +6.00000 q^{29} +2.77846 q^{30} -7.55691 q^{31} +27.3319 q^{32} -2.71982 q^{33} +7.88273 q^{34} +5.71982 q^{36} -4.27674 q^{37} +9.88273 q^{38} -1.00000 q^{39} +10.3354 q^{40} -2.83709 q^{41} +11.1138 q^{43} -15.5569 q^{44} +1.00000 q^{45} -13.4396 q^{46} +11.5569 q^{47} +17.2767 q^{48} +2.77846 q^{50} +2.83709 q^{51} -5.71982 q^{52} +1.16291 q^{53} +2.77846 q^{54} -2.71982 q^{55} +3.55691 q^{57} +16.6707 q^{58} +2.11727 q^{59} +5.71982 q^{60} -6.60256 q^{61} -20.9966 q^{62} +41.3871 q^{64} -1.00000 q^{65} -7.55691 q^{66} +1.88273 q^{67} +16.2277 q^{68} -4.83709 q^{69} -6.71982 q^{71} +10.3354 q^{72} -9.11383 q^{73} -11.8827 q^{74} +1.00000 q^{75} +20.3449 q^{76} -2.77846 q^{78} +10.2767 q^{79} +17.2767 q^{80} +1.00000 q^{81} -7.88273 q^{82} -2.11727 q^{83} +2.83709 q^{85} +30.8793 q^{86} +6.00000 q^{87} -28.1104 q^{88} -1.16291 q^{89} +2.77846 q^{90} -27.6673 q^{92} -7.55691 q^{93} +32.1104 q^{94} +3.55691 q^{95} +27.3319 q^{96} +10.8371 q^{97} -2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 8 q^{4} + 3 q^{5} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 8 q^{4} + 3 q^{5} + 6 q^{8} + 3 q^{9} + q^{11} + 8 q^{12} - 3 q^{13} + 3 q^{15} + 26 q^{16} + q^{17} - 6 q^{19} + 8 q^{20} - 6 q^{22} - 7 q^{23} + 6 q^{24} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} + 22 q^{32} + q^{33} + 22 q^{34} + 8 q^{36} + 13 q^{37} + 28 q^{38} - 3 q^{39} + 6 q^{40} - q^{41} - 30 q^{44} + 3 q^{45} - 22 q^{46} + 18 q^{47} + 26 q^{48} + q^{51} - 8 q^{52} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 8 q^{60} - 9 q^{61} - 28 q^{62} + 30 q^{64} - 3 q^{65} - 6 q^{66} + 4 q^{67} - 18 q^{68} - 7 q^{69} - 11 q^{71} + 6 q^{72} + 6 q^{73} - 34 q^{74} + 3 q^{75} - 4 q^{76} + 5 q^{79} + 26 q^{80} + 3 q^{81} - 22 q^{82} - 8 q^{83} + q^{85} + 56 q^{86} + 18 q^{87} - 16 q^{88} - 11 q^{89} + 2 q^{92} - 6 q^{93} + 28 q^{94} - 6 q^{95} + 22 q^{96} + 25 q^{97} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.77846 1.96467 0.982333 0.187142i \(-0.0599223\pi\)
0.982333 + 0.187142i \(0.0599223\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.71982 2.85991
\(5\) 1.00000 0.447214
\(6\) 2.77846 1.13430
\(7\) 0 0
\(8\) 10.3354 3.65411
\(9\) 1.00000 0.333333
\(10\) 2.77846 0.878625
\(11\) −2.71982 −0.820058 −0.410029 0.912073i \(-0.634481\pi\)
−0.410029 + 0.912073i \(0.634481\pi\)
\(12\) 5.71982 1.65117
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 17.2767 4.31918
\(17\) 2.83709 0.688095 0.344048 0.938952i \(-0.388202\pi\)
0.344048 + 0.938952i \(0.388202\pi\)
\(18\) 2.77846 0.654889
\(19\) 3.55691 0.816012 0.408006 0.912979i \(-0.366224\pi\)
0.408006 + 0.912979i \(0.366224\pi\)
\(20\) 5.71982 1.27899
\(21\) 0 0
\(22\) −7.55691 −1.61114
\(23\) −4.83709 −1.00860 −0.504302 0.863528i \(-0.668250\pi\)
−0.504302 + 0.863528i \(0.668250\pi\)
\(24\) 10.3354 2.10970
\(25\) 1.00000 0.200000
\(26\) −2.77846 −0.544900
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.77846 0.507275
\(31\) −7.55691 −1.35726 −0.678631 0.734479i \(-0.737426\pi\)
−0.678631 + 0.734479i \(0.737426\pi\)
\(32\) 27.3319 4.83165
\(33\) −2.71982 −0.473461
\(34\) 7.88273 1.35188
\(35\) 0 0
\(36\) 5.71982 0.953304
\(37\) −4.27674 −0.703091 −0.351546 0.936171i \(-0.614344\pi\)
−0.351546 + 0.936171i \(0.614344\pi\)
\(38\) 9.88273 1.60319
\(39\) −1.00000 −0.160128
\(40\) 10.3354 1.63417
\(41\) −2.83709 −0.443079 −0.221540 0.975151i \(-0.571108\pi\)
−0.221540 + 0.975151i \(0.571108\pi\)
\(42\) 0 0
\(43\) 11.1138 1.69484 0.847421 0.530921i \(-0.178154\pi\)
0.847421 + 0.530921i \(0.178154\pi\)
\(44\) −15.5569 −2.34529
\(45\) 1.00000 0.149071
\(46\) −13.4396 −1.98157
\(47\) 11.5569 1.68575 0.842875 0.538110i \(-0.180862\pi\)
0.842875 + 0.538110i \(0.180862\pi\)
\(48\) 17.2767 2.49368
\(49\) 0 0
\(50\) 2.77846 0.392933
\(51\) 2.83709 0.397272
\(52\) −5.71982 −0.793197
\(53\) 1.16291 0.159738 0.0798690 0.996805i \(-0.474550\pi\)
0.0798690 + 0.996805i \(0.474550\pi\)
\(54\) 2.77846 0.378100
\(55\) −2.71982 −0.366741
\(56\) 0 0
\(57\) 3.55691 0.471125
\(58\) 16.6707 2.18898
\(59\) 2.11727 0.275645 0.137822 0.990457i \(-0.455990\pi\)
0.137822 + 0.990457i \(0.455990\pi\)
\(60\) 5.71982 0.738426
\(61\) −6.60256 −0.845371 −0.422685 0.906276i \(-0.638912\pi\)
−0.422685 + 0.906276i \(0.638912\pi\)
\(62\) −20.9966 −2.66657
\(63\) 0 0
\(64\) 41.3871 5.17339
\(65\) −1.00000 −0.124035
\(66\) −7.55691 −0.930192
\(67\) 1.88273 0.230013 0.115006 0.993365i \(-0.463311\pi\)
0.115006 + 0.993365i \(0.463311\pi\)
\(68\) 16.2277 1.96789
\(69\) −4.83709 −0.582317
\(70\) 0 0
\(71\) −6.71982 −0.797496 −0.398748 0.917060i \(-0.630555\pi\)
−0.398748 + 0.917060i \(0.630555\pi\)
\(72\) 10.3354 1.21804
\(73\) −9.11383 −1.06669 −0.533346 0.845897i \(-0.679066\pi\)
−0.533346 + 0.845897i \(0.679066\pi\)
\(74\) −11.8827 −1.38134
\(75\) 1.00000 0.115470
\(76\) 20.3449 2.33372
\(77\) 0 0
\(78\) −2.77846 −0.314598
\(79\) 10.2767 1.15622 0.578112 0.815958i \(-0.303790\pi\)
0.578112 + 0.815958i \(0.303790\pi\)
\(80\) 17.2767 1.93160
\(81\) 1.00000 0.111111
\(82\) −7.88273 −0.870502
\(83\) −2.11727 −0.232400 −0.116200 0.993226i \(-0.537071\pi\)
−0.116200 + 0.993226i \(0.537071\pi\)
\(84\) 0 0
\(85\) 2.83709 0.307726
\(86\) 30.8793 3.32980
\(87\) 6.00000 0.643268
\(88\) −28.1104 −2.99658
\(89\) −1.16291 −0.123268 −0.0616341 0.998099i \(-0.519631\pi\)
−0.0616341 + 0.998099i \(0.519631\pi\)
\(90\) 2.77846 0.292875
\(91\) 0 0
\(92\) −27.6673 −2.88452
\(93\) −7.55691 −0.783616
\(94\) 32.1104 3.31193
\(95\) 3.55691 0.364932
\(96\) 27.3319 2.78955
\(97\) 10.8371 1.10034 0.550170 0.835053i \(-0.314563\pi\)
0.550170 + 0.835053i \(0.314563\pi\)
\(98\) 0 0
\(99\) −2.71982 −0.273353
\(100\) 5.71982 0.571982
\(101\) −7.67418 −0.763610 −0.381805 0.924243i \(-0.624697\pi\)
−0.381805 + 0.924243i \(0.624697\pi\)
\(102\) 7.88273 0.780507
\(103\) −3.76547 −0.371023 −0.185511 0.982642i \(-0.559394\pi\)
−0.185511 + 0.982642i \(0.559394\pi\)
\(104\) −10.3354 −1.01347
\(105\) 0 0
\(106\) 3.23109 0.313832
\(107\) −12.6026 −1.21834 −0.609168 0.793041i \(-0.708496\pi\)
−0.609168 + 0.793041i \(0.708496\pi\)
\(108\) 5.71982 0.550390
\(109\) 11.4396 1.09572 0.547860 0.836570i \(-0.315443\pi\)
0.547860 + 0.836570i \(0.315443\pi\)
\(110\) −7.55691 −0.720524
\(111\) −4.27674 −0.405930
\(112\) 0 0
\(113\) −13.1138 −1.23365 −0.616823 0.787102i \(-0.711580\pi\)
−0.616823 + 0.787102i \(0.711580\pi\)
\(114\) 9.88273 0.925603
\(115\) −4.83709 −0.451061
\(116\) 34.3189 3.18643
\(117\) −1.00000 −0.0924500
\(118\) 5.88273 0.541550
\(119\) 0 0
\(120\) 10.3354 0.943486
\(121\) −3.60256 −0.327505
\(122\) −18.3449 −1.66087
\(123\) −2.83709 −0.255812
\(124\) −43.2242 −3.88165
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.4396 −1.19258 −0.596288 0.802771i \(-0.703358\pi\)
−0.596288 + 0.802771i \(0.703358\pi\)
\(128\) 60.3285 5.33234
\(129\) 11.1138 0.978518
\(130\) −2.77846 −0.243687
\(131\) 9.43965 0.824746 0.412373 0.911015i \(-0.364700\pi\)
0.412373 + 0.911015i \(0.364700\pi\)
\(132\) −15.5569 −1.35406
\(133\) 0 0
\(134\) 5.23109 0.451898
\(135\) 1.00000 0.0860663
\(136\) 29.3224 2.51437
\(137\) 1.76547 0.150834 0.0754170 0.997152i \(-0.475971\pi\)
0.0754170 + 0.997152i \(0.475971\pi\)
\(138\) −13.4396 −1.14406
\(139\) 6.27674 0.532386 0.266193 0.963920i \(-0.414234\pi\)
0.266193 + 0.963920i \(0.414234\pi\)
\(140\) 0 0
\(141\) 11.5569 0.973268
\(142\) −18.6707 −1.56681
\(143\) 2.71982 0.227443
\(144\) 17.2767 1.43973
\(145\) 6.00000 0.498273
\(146\) −25.3224 −2.09570
\(147\) 0 0
\(148\) −24.4622 −2.01078
\(149\) −20.8302 −1.70648 −0.853239 0.521520i \(-0.825365\pi\)
−0.853239 + 0.521520i \(0.825365\pi\)
\(150\) 2.77846 0.226860
\(151\) 4.99656 0.406614 0.203307 0.979115i \(-0.434831\pi\)
0.203307 + 0.979115i \(0.434831\pi\)
\(152\) 36.7620 2.98179
\(153\) 2.83709 0.229365
\(154\) 0 0
\(155\) −7.55691 −0.606986
\(156\) −5.71982 −0.457952
\(157\) −8.87930 −0.708645 −0.354322 0.935123i \(-0.615288\pi\)
−0.354322 + 0.935123i \(0.615288\pi\)
\(158\) 28.5535 2.27159
\(159\) 1.16291 0.0922247
\(160\) 27.3319 2.16078
\(161\) 0 0
\(162\) 2.77846 0.218296
\(163\) −13.8337 −1.08354 −0.541768 0.840528i \(-0.682245\pi\)
−0.541768 + 0.840528i \(0.682245\pi\)
\(164\) −16.2277 −1.26717
\(165\) −2.71982 −0.211738
\(166\) −5.88273 −0.456589
\(167\) 9.88273 0.764749 0.382374 0.924007i \(-0.375106\pi\)
0.382374 + 0.924007i \(0.375106\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 7.88273 0.604578
\(171\) 3.55691 0.272004
\(172\) 63.5691 4.84710
\(173\) −13.1138 −0.997026 −0.498513 0.866882i \(-0.666120\pi\)
−0.498513 + 0.866882i \(0.666120\pi\)
\(174\) 16.6707 1.26381
\(175\) 0 0
\(176\) −46.9897 −3.54198
\(177\) 2.11727 0.159143
\(178\) −3.23109 −0.242181
\(179\) 8.55348 0.639317 0.319658 0.947533i \(-0.396432\pi\)
0.319658 + 0.947533i \(0.396432\pi\)
\(180\) 5.71982 0.426331
\(181\) −3.72326 −0.276748 −0.138374 0.990380i \(-0.544188\pi\)
−0.138374 + 0.990380i \(0.544188\pi\)
\(182\) 0 0
\(183\) −6.60256 −0.488075
\(184\) −49.9931 −3.68554
\(185\) −4.27674 −0.314432
\(186\) −20.9966 −1.53954
\(187\) −7.71639 −0.564278
\(188\) 66.1035 4.82109
\(189\) 0 0
\(190\) 9.88273 0.716969
\(191\) −4.23453 −0.306400 −0.153200 0.988195i \(-0.548958\pi\)
−0.153200 + 0.988195i \(0.548958\pi\)
\(192\) 41.3871 2.98686
\(193\) −23.3906 −1.68369 −0.841845 0.539719i \(-0.818530\pi\)
−0.841845 + 0.539719i \(0.818530\pi\)
\(194\) 30.1104 2.16180
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 14.5535 1.03689 0.518446 0.855110i \(-0.326511\pi\)
0.518446 + 0.855110i \(0.326511\pi\)
\(198\) −7.55691 −0.537047
\(199\) 15.1138 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(200\) 10.3354 0.730821
\(201\) 1.88273 0.132798
\(202\) −21.3224 −1.50024
\(203\) 0 0
\(204\) 16.2277 1.13616
\(205\) −2.83709 −0.198151
\(206\) −10.4622 −0.728935
\(207\) −4.83709 −0.336201
\(208\) −17.2767 −1.19793
\(209\) −9.67418 −0.669177
\(210\) 0 0
\(211\) −18.2277 −1.25484 −0.627422 0.778680i \(-0.715890\pi\)
−0.627422 + 0.778680i \(0.715890\pi\)
\(212\) 6.65164 0.456836
\(213\) −6.71982 −0.460435
\(214\) −35.0157 −2.39362
\(215\) 11.1138 0.757957
\(216\) 10.3354 0.703233
\(217\) 0 0
\(218\) 31.7846 2.15272
\(219\) −9.11383 −0.615855
\(220\) −15.5569 −1.04885
\(221\) −2.83709 −0.190843
\(222\) −11.8827 −0.797517
\(223\) −10.1173 −0.677502 −0.338751 0.940876i \(-0.610004\pi\)
−0.338751 + 0.940876i \(0.610004\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −36.4362 −2.42370
\(227\) −11.3224 −0.751493 −0.375746 0.926723i \(-0.622614\pi\)
−0.375746 + 0.926723i \(0.622614\pi\)
\(228\) 20.3449 1.34738
\(229\) 6.23453 0.411990 0.205995 0.978553i \(-0.433957\pi\)
0.205995 + 0.978553i \(0.433957\pi\)
\(230\) −13.4396 −0.886184
\(231\) 0 0
\(232\) 62.0122 4.07130
\(233\) 6.83709 0.447913 0.223956 0.974599i \(-0.428103\pi\)
0.223956 + 0.974599i \(0.428103\pi\)
\(234\) −2.77846 −0.181633
\(235\) 11.5569 0.753890
\(236\) 12.1104 0.788319
\(237\) 10.2767 0.667546
\(238\) 0 0
\(239\) 1.28018 0.0828077 0.0414039 0.999142i \(-0.486817\pi\)
0.0414039 + 0.999142i \(0.486817\pi\)
\(240\) 17.2767 1.11521
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −10.0096 −0.643438
\(243\) 1.00000 0.0641500
\(244\) −37.7655 −2.41769
\(245\) 0 0
\(246\) −7.88273 −0.502585
\(247\) −3.55691 −0.226321
\(248\) −78.1035 −4.95958
\(249\) −2.11727 −0.134176
\(250\) 2.77846 0.175725
\(251\) −18.2277 −1.15052 −0.575260 0.817971i \(-0.695099\pi\)
−0.575260 + 0.817971i \(0.695099\pi\)
\(252\) 0 0
\(253\) 13.1560 0.827113
\(254\) −37.3415 −2.34301
\(255\) 2.83709 0.177665
\(256\) 84.8459 5.30287
\(257\) −1.11383 −0.0694787 −0.0347394 0.999396i \(-0.511060\pi\)
−0.0347394 + 0.999396i \(0.511060\pi\)
\(258\) 30.8793 1.92246
\(259\) 0 0
\(260\) −5.71982 −0.354728
\(261\) 6.00000 0.371391
\(262\) 26.2277 1.62035
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −28.1104 −1.73007
\(265\) 1.16291 0.0714370
\(266\) 0 0
\(267\) −1.16291 −0.0711689
\(268\) 10.7689 0.657816
\(269\) −15.6742 −0.955672 −0.477836 0.878449i \(-0.658579\pi\)
−0.477836 + 0.878449i \(0.658579\pi\)
\(270\) 2.77846 0.169092
\(271\) −0.443086 −0.0269155 −0.0134578 0.999909i \(-0.504284\pi\)
−0.0134578 + 0.999909i \(0.504284\pi\)
\(272\) 49.0157 2.97201
\(273\) 0 0
\(274\) 4.90528 0.296339
\(275\) −2.71982 −0.164012
\(276\) −27.6673 −1.66538
\(277\) −4.87930 −0.293168 −0.146584 0.989198i \(-0.546828\pi\)
−0.146584 + 0.989198i \(0.546828\pi\)
\(278\) 17.4396 1.04596
\(279\) −7.55691 −0.452421
\(280\) 0 0
\(281\) −9.11383 −0.543685 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(282\) 32.1104 1.91215
\(283\) −33.3415 −1.98195 −0.990973 0.134063i \(-0.957198\pi\)
−0.990973 + 0.134063i \(0.957198\pi\)
\(284\) −38.4362 −2.28077
\(285\) 3.55691 0.210693
\(286\) 7.55691 0.446850
\(287\) 0 0
\(288\) 27.3319 1.61055
\(289\) −8.95092 −0.526525
\(290\) 16.6707 0.978940
\(291\) 10.8371 0.635281
\(292\) −52.1295 −3.05065
\(293\) 29.4328 1.71948 0.859740 0.510731i \(-0.170625\pi\)
0.859740 + 0.510731i \(0.170625\pi\)
\(294\) 0 0
\(295\) 2.11727 0.123272
\(296\) −44.2017 −2.56917
\(297\) −2.71982 −0.157820
\(298\) −57.8759 −3.35266
\(299\) 4.83709 0.279736
\(300\) 5.71982 0.330234
\(301\) 0 0
\(302\) 13.8827 0.798862
\(303\) −7.67418 −0.440870
\(304\) 61.4519 3.52451
\(305\) −6.60256 −0.378061
\(306\) 7.88273 0.450626
\(307\) 21.8337 1.24611 0.623056 0.782177i \(-0.285891\pi\)
0.623056 + 0.782177i \(0.285891\pi\)
\(308\) 0 0
\(309\) −3.76547 −0.214210
\(310\) −20.9966 −1.19252
\(311\) −25.1070 −1.42368 −0.711842 0.702339i \(-0.752139\pi\)
−0.711842 + 0.702339i \(0.752139\pi\)
\(312\) −10.3354 −0.585125
\(313\) −8.22766 −0.465055 −0.232527 0.972590i \(-0.574700\pi\)
−0.232527 + 0.972590i \(0.574700\pi\)
\(314\) −24.6707 −1.39225
\(315\) 0 0
\(316\) 58.7811 3.30670
\(317\) 27.6742 1.55434 0.777168 0.629293i \(-0.216655\pi\)
0.777168 + 0.629293i \(0.216655\pi\)
\(318\) 3.23109 0.181191
\(319\) −16.3189 −0.913685
\(320\) 41.3871 2.31361
\(321\) −12.6026 −0.703406
\(322\) 0 0
\(323\) 10.0913 0.561494
\(324\) 5.71982 0.317768
\(325\) −1.00000 −0.0554700
\(326\) −38.4362 −2.12878
\(327\) 11.4396 0.632614
\(328\) −29.3224 −1.61906
\(329\) 0 0
\(330\) −7.55691 −0.415994
\(331\) −13.2311 −0.727247 −0.363623 0.931546i \(-0.618460\pi\)
−0.363623 + 0.931546i \(0.618460\pi\)
\(332\) −12.1104 −0.664644
\(333\) −4.27674 −0.234364
\(334\) 27.4588 1.50248
\(335\) 1.88273 0.102865
\(336\) 0 0
\(337\) −4.32582 −0.235642 −0.117821 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(338\) 2.77846 0.151128
\(339\) −13.1138 −0.712245
\(340\) 16.2277 0.880068
\(341\) 20.5535 1.11303
\(342\) 9.88273 0.534397
\(343\) 0 0
\(344\) 114.866 6.19314
\(345\) −4.83709 −0.260420
\(346\) −36.4362 −1.95882
\(347\) 6.27674 0.336953 0.168476 0.985706i \(-0.446115\pi\)
0.168476 + 0.985706i \(0.446115\pi\)
\(348\) 34.3189 1.83969
\(349\) −17.6673 −0.945709 −0.472855 0.881140i \(-0.656776\pi\)
−0.472855 + 0.881140i \(0.656776\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −74.3380 −3.96223
\(353\) 13.7655 0.732662 0.366331 0.930485i \(-0.380614\pi\)
0.366331 + 0.930485i \(0.380614\pi\)
\(354\) 5.88273 0.312664
\(355\) −6.71982 −0.356651
\(356\) −6.65164 −0.352536
\(357\) 0 0
\(358\) 23.7655 1.25604
\(359\) 0.996562 0.0525965 0.0262983 0.999654i \(-0.491628\pi\)
0.0262983 + 0.999654i \(0.491628\pi\)
\(360\) 10.3354 0.544722
\(361\) −6.34836 −0.334124
\(362\) −10.3449 −0.543717
\(363\) −3.60256 −0.189085
\(364\) 0 0
\(365\) −9.11383 −0.477040
\(366\) −18.3449 −0.958905
\(367\) −14.2277 −0.742678 −0.371339 0.928497i \(-0.621101\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(368\) −83.5691 −4.35634
\(369\) −2.83709 −0.147693
\(370\) −11.8827 −0.617754
\(371\) 0 0
\(372\) −43.2242 −2.24107
\(373\) 15.6742 0.811578 0.405789 0.913967i \(-0.366997\pi\)
0.405789 + 0.913967i \(0.366997\pi\)
\(374\) −21.4396 −1.10862
\(375\) 1.00000 0.0516398
\(376\) 119.445 6.15991
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 26.2017 1.34589 0.672945 0.739693i \(-0.265029\pi\)
0.672945 + 0.739693i \(0.265029\pi\)
\(380\) 20.3449 1.04367
\(381\) −13.4396 −0.688534
\(382\) −11.7655 −0.601974
\(383\) −22.4362 −1.14644 −0.573218 0.819403i \(-0.694305\pi\)
−0.573218 + 0.819403i \(0.694305\pi\)
\(384\) 60.3285 3.07863
\(385\) 0 0
\(386\) −64.9897 −3.30789
\(387\) 11.1138 0.564948
\(388\) 61.9862 3.14687
\(389\) 31.6742 1.60594 0.802972 0.596016i \(-0.203251\pi\)
0.802972 + 0.596016i \(0.203251\pi\)
\(390\) −2.77846 −0.140693
\(391\) −13.7233 −0.694015
\(392\) 0 0
\(393\) 9.43965 0.476167
\(394\) 40.4362 2.03715
\(395\) 10.2767 0.517079
\(396\) −15.5569 −0.781764
\(397\) −17.9509 −0.900931 −0.450465 0.892794i \(-0.648742\pi\)
−0.450465 + 0.892794i \(0.648742\pi\)
\(398\) 41.9931 2.10493
\(399\) 0 0
\(400\) 17.2767 0.863837
\(401\) 13.5829 0.678297 0.339149 0.940733i \(-0.389861\pi\)
0.339149 + 0.940733i \(0.389861\pi\)
\(402\) 5.23109 0.260903
\(403\) 7.55691 0.376437
\(404\) −43.8950 −2.18386
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 11.6320 0.576576
\(408\) 29.3224 1.45167
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −7.88273 −0.389300
\(411\) 1.76547 0.0870841
\(412\) −21.5378 −1.06109
\(413\) 0 0
\(414\) −13.4396 −0.660523
\(415\) −2.11727 −0.103933
\(416\) −27.3319 −1.34006
\(417\) 6.27674 0.307373
\(418\) −26.8793 −1.31471
\(419\) −12.3189 −0.601820 −0.300910 0.953653i \(-0.597290\pi\)
−0.300910 + 0.953653i \(0.597290\pi\)
\(420\) 0 0
\(421\) 22.7880 1.11062 0.555310 0.831644i \(-0.312600\pi\)
0.555310 + 0.831644i \(0.312600\pi\)
\(422\) −50.6448 −2.46535
\(423\) 11.5569 0.561916
\(424\) 12.0191 0.583699
\(425\) 2.83709 0.137619
\(426\) −18.6707 −0.904600
\(427\) 0 0
\(428\) −72.0844 −3.48433
\(429\) 2.71982 0.131314
\(430\) 30.8793 1.48913
\(431\) −8.99656 −0.433349 −0.216675 0.976244i \(-0.569521\pi\)
−0.216675 + 0.976244i \(0.569521\pi\)
\(432\) 17.2767 0.831227
\(433\) 20.3258 0.976797 0.488398 0.872621i \(-0.337581\pi\)
0.488398 + 0.872621i \(0.337581\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 65.4328 3.13366
\(437\) −17.2051 −0.823032
\(438\) −25.3224 −1.20995
\(439\) 25.3906 1.21183 0.605913 0.795531i \(-0.292808\pi\)
0.605913 + 0.795531i \(0.292808\pi\)
\(440\) −28.1104 −1.34011
\(441\) 0 0
\(442\) −7.88273 −0.374943
\(443\) 10.9284 0.519223 0.259611 0.965713i \(-0.416406\pi\)
0.259611 + 0.965713i \(0.416406\pi\)
\(444\) −24.4622 −1.16092
\(445\) −1.16291 −0.0551272
\(446\) −28.1104 −1.33107
\(447\) −20.8302 −0.985235
\(448\) 0 0
\(449\) 2.83709 0.133891 0.0669453 0.997757i \(-0.478675\pi\)
0.0669453 + 0.997757i \(0.478675\pi\)
\(450\) 2.77846 0.130978
\(451\) 7.71639 0.363350
\(452\) −75.0088 −3.52812
\(453\) 4.99656 0.234759
\(454\) −31.4588 −1.47643
\(455\) 0 0
\(456\) 36.7620 1.72154
\(457\) −13.7164 −0.641625 −0.320813 0.947143i \(-0.603956\pi\)
−0.320813 + 0.947143i \(0.603956\pi\)
\(458\) 17.3224 0.809422
\(459\) 2.83709 0.132424
\(460\) −27.6673 −1.28999
\(461\) 19.6251 0.914032 0.457016 0.889458i \(-0.348918\pi\)
0.457016 + 0.889458i \(0.348918\pi\)
\(462\) 0 0
\(463\) 27.0388 1.25660 0.628299 0.777972i \(-0.283751\pi\)
0.628299 + 0.777972i \(0.283751\pi\)
\(464\) 103.660 4.81231
\(465\) −7.55691 −0.350444
\(466\) 18.9966 0.879999
\(467\) −28.9215 −1.33833 −0.669164 0.743115i \(-0.733348\pi\)
−0.669164 + 0.743115i \(0.733348\pi\)
\(468\) −5.71982 −0.264399
\(469\) 0 0
\(470\) 32.1104 1.48114
\(471\) −8.87930 −0.409136
\(472\) 21.8827 1.00723
\(473\) −30.2277 −1.38987
\(474\) 28.5535 1.31150
\(475\) 3.55691 0.163202
\(476\) 0 0
\(477\) 1.16291 0.0532460
\(478\) 3.55691 0.162689
\(479\) 12.1595 0.555580 0.277790 0.960642i \(-0.410398\pi\)
0.277790 + 0.960642i \(0.410398\pi\)
\(480\) 27.3319 1.24753
\(481\) 4.27674 0.195002
\(482\) 16.6707 0.759332
\(483\) 0 0
\(484\) −20.6060 −0.936636
\(485\) 10.8371 0.492087
\(486\) 2.77846 0.126033
\(487\) −0.159472 −0.00722636 −0.00361318 0.999993i \(-0.501150\pi\)
−0.00361318 + 0.999993i \(0.501150\pi\)
\(488\) −68.2399 −3.08907
\(489\) −13.8337 −0.625579
\(490\) 0 0
\(491\) −42.2277 −1.90571 −0.952854 0.303430i \(-0.901868\pi\)
−0.952854 + 0.303430i \(0.901868\pi\)
\(492\) −16.2277 −0.731599
\(493\) 17.0225 0.766657
\(494\) −9.88273 −0.444645
\(495\) −2.71982 −0.122247
\(496\) −130.559 −5.86226
\(497\) 0 0
\(498\) −5.88273 −0.263612
\(499\) 7.79145 0.348793 0.174397 0.984676i \(-0.444203\pi\)
0.174397 + 0.984676i \(0.444203\pi\)
\(500\) 5.71982 0.255798
\(501\) 9.88273 0.441528
\(502\) −50.6448 −2.26039
\(503\) −27.3484 −1.21940 −0.609702 0.792631i \(-0.708711\pi\)
−0.609702 + 0.792631i \(0.708711\pi\)
\(504\) 0 0
\(505\) −7.67418 −0.341497
\(506\) 36.5535 1.62500
\(507\) 1.00000 0.0444116
\(508\) −76.8724 −3.41066
\(509\) 33.4819 1.48406 0.742029 0.670368i \(-0.233864\pi\)
0.742029 + 0.670368i \(0.233864\pi\)
\(510\) 7.88273 0.349053
\(511\) 0 0
\(512\) 115.084 5.08603
\(513\) 3.55691 0.157042
\(514\) −3.09472 −0.136502
\(515\) −3.76547 −0.165926
\(516\) 63.5691 2.79848
\(517\) −31.4328 −1.38241
\(518\) 0 0
\(519\) −13.1138 −0.575633
\(520\) −10.3354 −0.453236
\(521\) 17.3484 0.760045 0.380023 0.924977i \(-0.375916\pi\)
0.380023 + 0.924977i \(0.375916\pi\)
\(522\) 16.6707 0.729659
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 53.9931 2.35870
\(525\) 0 0
\(526\) 22.2277 0.969172
\(527\) −21.4396 −0.933926
\(528\) −46.9897 −2.04496
\(529\) 0.397442 0.0172801
\(530\) 3.23109 0.140350
\(531\) 2.11727 0.0918815
\(532\) 0 0
\(533\) 2.83709 0.122888
\(534\) −3.23109 −0.139823
\(535\) −12.6026 −0.544856
\(536\) 19.4588 0.840490
\(537\) 8.55348 0.369110
\(538\) −43.5500 −1.87758
\(539\) 0 0
\(540\) 5.71982 0.246142
\(541\) −32.6448 −1.40351 −0.701754 0.712419i \(-0.747599\pi\)
−0.701754 + 0.712419i \(0.747599\pi\)
\(542\) −1.23109 −0.0528800
\(543\) −3.72326 −0.159780
\(544\) 77.5432 3.32464
\(545\) 11.4396 0.490021
\(546\) 0 0
\(547\) 34.2277 1.46347 0.731734 0.681590i \(-0.238711\pi\)
0.731734 + 0.681590i \(0.238711\pi\)
\(548\) 10.0982 0.431372
\(549\) −6.60256 −0.281790
\(550\) −7.55691 −0.322228
\(551\) 21.3415 0.909178
\(552\) −49.9931 −2.12785
\(553\) 0 0
\(554\) −13.5569 −0.575978
\(555\) −4.27674 −0.181537
\(556\) 35.9018 1.52258
\(557\) 6.65164 0.281839 0.140919 0.990021i \(-0.454994\pi\)
0.140919 + 0.990021i \(0.454994\pi\)
\(558\) −20.9966 −0.888855
\(559\) −11.1138 −0.470065
\(560\) 0 0
\(561\) −7.71639 −0.325786
\(562\) −25.3224 −1.06816
\(563\) −40.2699 −1.69717 −0.848586 0.529057i \(-0.822546\pi\)
−0.848586 + 0.529057i \(0.822546\pi\)
\(564\) 66.1035 2.78346
\(565\) −13.1138 −0.551703
\(566\) −92.6379 −3.89386
\(567\) 0 0
\(568\) −69.4519 −2.91414
\(569\) −13.4328 −0.563131 −0.281566 0.959542i \(-0.590854\pi\)
−0.281566 + 0.959542i \(0.590854\pi\)
\(570\) 9.88273 0.413942
\(571\) −35.7164 −1.49468 −0.747342 0.664439i \(-0.768670\pi\)
−0.747342 + 0.664439i \(0.768670\pi\)
\(572\) 15.5569 0.650467
\(573\) −4.23453 −0.176900
\(574\) 0 0
\(575\) −4.83709 −0.201721
\(576\) 41.3871 1.72446
\(577\) 13.7164 0.571021 0.285510 0.958376i \(-0.407837\pi\)
0.285510 + 0.958376i \(0.407837\pi\)
\(578\) −24.8697 −1.03444
\(579\) −23.3906 −0.972079
\(580\) 34.3189 1.42502
\(581\) 0 0
\(582\) 30.1104 1.24812
\(583\) −3.16291 −0.130994
\(584\) −94.1948 −3.89781
\(585\) −1.00000 −0.0413449
\(586\) 81.7777 3.37821
\(587\) −30.6707 −1.26592 −0.632959 0.774186i \(-0.718160\pi\)
−0.632959 + 0.774186i \(0.718160\pi\)
\(588\) 0 0
\(589\) −26.8793 −1.10754
\(590\) 5.88273 0.242188
\(591\) 14.5535 0.598650
\(592\) −73.8881 −3.03678
\(593\) 45.6673 1.87533 0.937666 0.347538i \(-0.112982\pi\)
0.937666 + 0.347538i \(0.112982\pi\)
\(594\) −7.55691 −0.310064
\(595\) 0 0
\(596\) −119.145 −4.88038
\(597\) 15.1138 0.618568
\(598\) 13.4396 0.549588
\(599\) −40.2208 −1.64338 −0.821688 0.569937i \(-0.806968\pi\)
−0.821688 + 0.569937i \(0.806968\pi\)
\(600\) 10.3354 0.421940
\(601\) −17.3974 −0.709656 −0.354828 0.934932i \(-0.615461\pi\)
−0.354828 + 0.934932i \(0.615461\pi\)
\(602\) 0 0
\(603\) 1.88273 0.0766708
\(604\) 28.5795 1.16288
\(605\) −3.60256 −0.146465
\(606\) −21.3224 −0.866163
\(607\) 14.2277 0.577483 0.288741 0.957407i \(-0.406763\pi\)
0.288741 + 0.957407i \(0.406763\pi\)
\(608\) 97.2173 3.94268
\(609\) 0 0
\(610\) −18.3449 −0.742764
\(611\) −11.5569 −0.467543
\(612\) 16.2277 0.655964
\(613\) −40.8302 −1.64912 −0.824558 0.565777i \(-0.808576\pi\)
−0.824558 + 0.565777i \(0.808576\pi\)
\(614\) 60.6639 2.44819
\(615\) −2.83709 −0.114403
\(616\) 0 0
\(617\) 5.11383 0.205875 0.102937 0.994688i \(-0.467176\pi\)
0.102937 + 0.994688i \(0.467176\pi\)
\(618\) −10.4622 −0.420851
\(619\) −11.5569 −0.464512 −0.232256 0.972655i \(-0.574611\pi\)
−0.232256 + 0.972655i \(0.574611\pi\)
\(620\) −43.2242 −1.73593
\(621\) −4.83709 −0.194106
\(622\) −69.7586 −2.79706
\(623\) 0 0
\(624\) −17.2767 −0.691623
\(625\) 1.00000 0.0400000
\(626\) −22.8602 −0.913677
\(627\) −9.67418 −0.386350
\(628\) −50.7880 −2.02666
\(629\) −12.1335 −0.483794
\(630\) 0 0
\(631\) 35.2242 1.40225 0.701127 0.713036i \(-0.252681\pi\)
0.701127 + 0.713036i \(0.252681\pi\)
\(632\) 106.214 4.22496
\(633\) −18.2277 −0.724484
\(634\) 76.8915 3.05375
\(635\) −13.4396 −0.533336
\(636\) 6.65164 0.263755
\(637\) 0 0
\(638\) −45.3415 −1.79509
\(639\) −6.71982 −0.265832
\(640\) 60.3285 2.38469
\(641\) −21.9018 −0.865071 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(642\) −35.0157 −1.38196
\(643\) −7.50783 −0.296080 −0.148040 0.988981i \(-0.547296\pi\)
−0.148040 + 0.988981i \(0.547296\pi\)
\(644\) 0 0
\(645\) 11.1138 0.437607
\(646\) 28.0382 1.10315
\(647\) 14.0422 0.552056 0.276028 0.961150i \(-0.410982\pi\)
0.276028 + 0.961150i \(0.410982\pi\)
\(648\) 10.3354 0.406012
\(649\) −5.75859 −0.226044
\(650\) −2.77846 −0.108980
\(651\) 0 0
\(652\) −79.1261 −3.09882
\(653\) 7.99312 0.312795 0.156398 0.987694i \(-0.450012\pi\)
0.156398 + 0.987694i \(0.450012\pi\)
\(654\) 31.7846 1.24288
\(655\) 9.43965 0.368838
\(656\) −49.0157 −1.91374
\(657\) −9.11383 −0.355564
\(658\) 0 0
\(659\) −25.3415 −0.987164 −0.493582 0.869699i \(-0.664313\pi\)
−0.493582 + 0.869699i \(0.664313\pi\)
\(660\) −15.5569 −0.605552
\(661\) −27.4396 −1.06728 −0.533639 0.845712i \(-0.679176\pi\)
−0.533639 + 0.845712i \(0.679176\pi\)
\(662\) −36.7620 −1.42880
\(663\) −2.83709 −0.110183
\(664\) −21.8827 −0.849215
\(665\) 0 0
\(666\) −11.8827 −0.460447
\(667\) −29.0225 −1.12376
\(668\) 56.5275 2.18711
\(669\) −10.1173 −0.391156
\(670\) 5.23109 0.202095
\(671\) 17.9578 0.693253
\(672\) 0 0
\(673\) 27.1070 1.04490 0.522448 0.852671i \(-0.325019\pi\)
0.522448 + 0.852671i \(0.325019\pi\)
\(674\) −12.0191 −0.462959
\(675\) 1.00000 0.0384900
\(676\) 5.71982 0.219993
\(677\) 36.5957 1.40649 0.703243 0.710949i \(-0.251735\pi\)
0.703243 + 0.710949i \(0.251735\pi\)
\(678\) −36.4362 −1.39932
\(679\) 0 0
\(680\) 29.3224 1.12446
\(681\) −11.3224 −0.433875
\(682\) 57.1070 2.18674
\(683\) −13.4656 −0.515248 −0.257624 0.966245i \(-0.582940\pi\)
−0.257624 + 0.966245i \(0.582940\pi\)
\(684\) 20.3449 0.777908
\(685\) 1.76547 0.0674550
\(686\) 0 0
\(687\) 6.23453 0.237862
\(688\) 192.011 7.32034
\(689\) −1.16291 −0.0443033
\(690\) −13.4396 −0.511639
\(691\) −29.5500 −1.12414 −0.562068 0.827091i \(-0.689994\pi\)
−0.562068 + 0.827091i \(0.689994\pi\)
\(692\) −75.0088 −2.85141
\(693\) 0 0
\(694\) 17.4396 0.662000
\(695\) 6.27674 0.238090
\(696\) 62.0122 2.35057
\(697\) −8.04908 −0.304881
\(698\) −49.0878 −1.85800
\(699\) 6.83709 0.258603
\(700\) 0 0
\(701\) 43.6604 1.64903 0.824516 0.565839i \(-0.191448\pi\)
0.824516 + 0.565839i \(0.191448\pi\)
\(702\) −2.77846 −0.104866
\(703\) −15.2120 −0.573731
\(704\) −112.566 −4.24248
\(705\) 11.5569 0.435259
\(706\) 38.2468 1.43944
\(707\) 0 0
\(708\) 12.1104 0.455136
\(709\) −26.7880 −1.00604 −0.503022 0.864273i \(-0.667779\pi\)
−0.503022 + 0.864273i \(0.667779\pi\)
\(710\) −18.6707 −0.700700
\(711\) 10.2767 0.385408
\(712\) −12.0191 −0.450435
\(713\) 36.5535 1.36894
\(714\) 0 0
\(715\) 2.71982 0.101716
\(716\) 48.9244 1.82839
\(717\) 1.28018 0.0478091
\(718\) 2.76891 0.103335
\(719\) 34.8793 1.30078 0.650389 0.759601i \(-0.274606\pi\)
0.650389 + 0.759601i \(0.274606\pi\)
\(720\) 17.2767 0.643866
\(721\) 0 0
\(722\) −17.6386 −0.656443
\(723\) 6.00000 0.223142
\(724\) −21.2964 −0.791475
\(725\) 6.00000 0.222834
\(726\) −10.0096 −0.371489
\(727\) −37.4396 −1.38856 −0.694280 0.719705i \(-0.744277\pi\)
−0.694280 + 0.719705i \(0.744277\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −25.3224 −0.937223
\(731\) 31.5309 1.16621
\(732\) −37.7655 −1.39585
\(733\) 47.1560 1.74175 0.870874 0.491506i \(-0.163554\pi\)
0.870874 + 0.491506i \(0.163554\pi\)
\(734\) −39.5309 −1.45911
\(735\) 0 0
\(736\) −132.207 −4.87322
\(737\) −5.12070 −0.188624
\(738\) −7.88273 −0.290167
\(739\) −31.7914 −1.16947 −0.584734 0.811225i \(-0.698801\pi\)
−0.584734 + 0.811225i \(0.698801\pi\)
\(740\) −24.4622 −0.899248
\(741\) −3.55691 −0.130667
\(742\) 0 0
\(743\) 16.6776 0.611842 0.305921 0.952057i \(-0.401036\pi\)
0.305921 + 0.952057i \(0.401036\pi\)
\(744\) −78.1035 −2.86341
\(745\) −20.8302 −0.763160
\(746\) 43.5500 1.59448
\(747\) −2.11727 −0.0774667
\(748\) −44.1364 −1.61379
\(749\) 0 0
\(750\) 2.77846 0.101455
\(751\) 16.1855 0.590616 0.295308 0.955402i \(-0.404578\pi\)
0.295308 + 0.955402i \(0.404578\pi\)
\(752\) 199.666 7.28106
\(753\) −18.2277 −0.664253
\(754\) −16.6707 −0.607113
\(755\) 4.99656 0.181844
\(756\) 0 0
\(757\) 12.3258 0.447990 0.223995 0.974590i \(-0.428090\pi\)
0.223995 + 0.974590i \(0.428090\pi\)
\(758\) 72.8002 2.64422
\(759\) 13.1560 0.477534
\(760\) 36.7620 1.33350
\(761\) 0.00687569 0.000249244 0 0.000124622 1.00000i \(-0.499960\pi\)
0.000124622 1.00000i \(0.499960\pi\)
\(762\) −37.3415 −1.35274
\(763\) 0 0
\(764\) −24.2208 −0.876277
\(765\) 2.83709 0.102575
\(766\) −62.3380 −2.25237
\(767\) −2.11727 −0.0764501
\(768\) 84.8459 3.06161
\(769\) 20.3258 0.732968 0.366484 0.930424i \(-0.380561\pi\)
0.366484 + 0.930424i \(0.380561\pi\)
\(770\) 0 0
\(771\) −1.11383 −0.0401136
\(772\) −133.790 −4.81520
\(773\) −9.90184 −0.356144 −0.178072 0.984017i \(-0.556986\pi\)
−0.178072 + 0.984017i \(0.556986\pi\)
\(774\) 30.8793 1.10993
\(775\) −7.55691 −0.271452
\(776\) 112.005 4.02076
\(777\) 0 0
\(778\) 88.0054 3.15514
\(779\) −10.0913 −0.361558
\(780\) −5.71982 −0.204803
\(781\) 18.2767 0.653993
\(782\) −38.1295 −1.36351
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −8.87930 −0.316916
\(786\) 26.2277 0.935510
\(787\) −36.3449 −1.29556 −0.647778 0.761829i \(-0.724302\pi\)
−0.647778 + 0.761829i \(0.724302\pi\)
\(788\) 83.2433 2.96542
\(789\) 8.00000 0.284808
\(790\) 28.5535 1.01589
\(791\) 0 0
\(792\) −28.1104 −0.998859
\(793\) 6.60256 0.234464
\(794\) −49.8759 −1.77003
\(795\) 1.16291 0.0412442
\(796\) 86.4484 3.06408
\(797\) −18.8371 −0.667244 −0.333622 0.942707i \(-0.608271\pi\)
−0.333622 + 0.942707i \(0.608271\pi\)
\(798\) 0 0
\(799\) 32.7880 1.15996
\(800\) 27.3319 0.966330
\(801\) −1.16291 −0.0410894
\(802\) 37.7395 1.33263
\(803\) 24.7880 0.874750
\(804\) 10.7689 0.379790
\(805\) 0 0
\(806\) 20.9966 0.739572
\(807\) −15.6742 −0.551757
\(808\) −79.3155 −2.79031
\(809\) 32.2277 1.13306 0.566532 0.824040i \(-0.308285\pi\)
0.566532 + 0.824040i \(0.308285\pi\)
\(810\) 2.77846 0.0976250
\(811\) 23.0034 0.807760 0.403880 0.914812i \(-0.367661\pi\)
0.403880 + 0.914812i \(0.367661\pi\)
\(812\) 0 0
\(813\) −0.443086 −0.0155397
\(814\) 32.3189 1.13278
\(815\) −13.8337 −0.484572
\(816\) 49.0157 1.71589
\(817\) 39.5309 1.38301
\(818\) 38.8984 1.36005
\(819\) 0 0
\(820\) −16.2277 −0.566694
\(821\) 49.9372 1.74282 0.871410 0.490556i \(-0.163206\pi\)
0.871410 + 0.490556i \(0.163206\pi\)
\(822\) 4.90528 0.171091
\(823\) 28.2345 0.984194 0.492097 0.870540i \(-0.336231\pi\)
0.492097 + 0.870540i \(0.336231\pi\)
\(824\) −38.9175 −1.35576
\(825\) −2.71982 −0.0946921
\(826\) 0 0
\(827\) −9.55004 −0.332087 −0.166044 0.986118i \(-0.553099\pi\)
−0.166044 + 0.986118i \(0.553099\pi\)
\(828\) −27.6673 −0.961505
\(829\) 37.9862 1.31932 0.659658 0.751565i \(-0.270701\pi\)
0.659658 + 0.751565i \(0.270701\pi\)
\(830\) −5.88273 −0.204193
\(831\) −4.87930 −0.169261
\(832\) −41.3871 −1.43484
\(833\) 0 0
\(834\) 17.4396 0.603886
\(835\) 9.88273 0.342006
\(836\) −55.3346 −1.91379
\(837\) −7.55691 −0.261205
\(838\) −34.2277 −1.18237
\(839\) 4.72670 0.163184 0.0815919 0.996666i \(-0.474000\pi\)
0.0815919 + 0.996666i \(0.474000\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 63.3155 2.18200
\(843\) −9.11383 −0.313897
\(844\) −104.259 −3.58874
\(845\) 1.00000 0.0344010
\(846\) 32.1104 1.10398
\(847\) 0 0
\(848\) 20.0913 0.689938
\(849\) −33.3415 −1.14428
\(850\) 7.88273 0.270376
\(851\) 20.6870 0.709140
\(852\) −38.4362 −1.31680
\(853\) 48.3611 1.65585 0.827927 0.560836i \(-0.189520\pi\)
0.827927 + 0.560836i \(0.189520\pi\)
\(854\) 0 0
\(855\) 3.55691 0.121644
\(856\) −130.252 −4.45193
\(857\) −6.83709 −0.233551 −0.116775 0.993158i \(-0.537256\pi\)
−0.116775 + 0.993158i \(0.537256\pi\)
\(858\) 7.55691 0.257989
\(859\) −12.6026 −0.429994 −0.214997 0.976615i \(-0.568974\pi\)
−0.214997 + 0.976615i \(0.568974\pi\)
\(860\) 63.5691 2.16769
\(861\) 0 0
\(862\) −24.9966 −0.851386
\(863\) 8.20855 0.279422 0.139711 0.990192i \(-0.455383\pi\)
0.139711 + 0.990192i \(0.455383\pi\)
\(864\) 27.3319 0.929851
\(865\) −13.1138 −0.445884
\(866\) 56.4744 1.91908
\(867\) −8.95092 −0.303989
\(868\) 0 0
\(869\) −27.9509 −0.948170
\(870\) 16.6707 0.565191
\(871\) −1.88273 −0.0637940
\(872\) 118.233 4.00387
\(873\) 10.8371 0.366780
\(874\) −47.8037 −1.61698
\(875\) 0 0
\(876\) −52.1295 −1.76129
\(877\) 13.5309 0.456907 0.228454 0.973555i \(-0.426633\pi\)
0.228454 + 0.973555i \(0.426633\pi\)
\(878\) 70.5466 2.38083
\(879\) 29.4328 0.992743
\(880\) −46.9897 −1.58402
\(881\) 9.34836 0.314954 0.157477 0.987523i \(-0.449664\pi\)
0.157477 + 0.987523i \(0.449664\pi\)
\(882\) 0 0
\(883\) −55.1001 −1.85427 −0.927133 0.374733i \(-0.877734\pi\)
−0.927133 + 0.374733i \(0.877734\pi\)
\(884\) −16.2277 −0.545795
\(885\) 2.11727 0.0711711
\(886\) 30.3640 1.02010
\(887\) −0.133492 −0.00448223 −0.00224112 0.999997i \(-0.500713\pi\)
−0.00224112 + 0.999997i \(0.500713\pi\)
\(888\) −44.2017 −1.48331
\(889\) 0 0
\(890\) −3.23109 −0.108307
\(891\) −2.71982 −0.0911175
\(892\) −57.8690 −1.93760
\(893\) 41.1070 1.37559
\(894\) −57.8759 −1.93566
\(895\) 8.55348 0.285911
\(896\) 0 0
\(897\) 4.83709 0.161506
\(898\) 7.88273 0.263050
\(899\) −45.3415 −1.51222
\(900\) 5.71982 0.190661
\(901\) 3.29928 0.109915
\(902\) 21.4396 0.713862
\(903\) 0 0
\(904\) −135.536 −4.50787
\(905\) −3.72326 −0.123765
\(906\) 13.8827 0.461223
\(907\) −58.5466 −1.94401 −0.972004 0.234964i \(-0.924503\pi\)
−0.972004 + 0.234964i \(0.924503\pi\)
\(908\) −64.7620 −2.14920
\(909\) −7.67418 −0.254537
\(910\) 0 0
\(911\) 50.4622 1.67189 0.835943 0.548816i \(-0.184921\pi\)
0.835943 + 0.548816i \(0.184921\pi\)
\(912\) 61.4519 2.03487
\(913\) 5.75859 0.190582
\(914\) −38.1104 −1.26058
\(915\) −6.60256 −0.218274
\(916\) 35.6604 1.17825
\(917\) 0 0
\(918\) 7.88273 0.260169
\(919\) 56.9735 1.87938 0.939691 0.342026i \(-0.111113\pi\)
0.939691 + 0.342026i \(0.111113\pi\)
\(920\) −49.9931 −1.64822
\(921\) 21.8337 0.719443
\(922\) 54.5275 1.79577
\(923\) 6.71982 0.221186
\(924\) 0 0
\(925\) −4.27674 −0.140618
\(926\) 75.1261 2.46880
\(927\) −3.76547 −0.123674
\(928\) 163.992 5.38329
\(929\) 36.5957 1.20067 0.600333 0.799750i \(-0.295035\pi\)
0.600333 + 0.799750i \(0.295035\pi\)
\(930\) −20.9966 −0.688504
\(931\) 0 0
\(932\) 39.1070 1.28099
\(933\) −25.1070 −0.821965
\(934\) −80.3572 −2.62937
\(935\) −7.71639 −0.252353
\(936\) −10.3354 −0.337822
\(937\) 47.1070 1.53892 0.769459 0.638697i \(-0.220526\pi\)
0.769459 + 0.638697i \(0.220526\pi\)
\(938\) 0 0
\(939\) −8.22766 −0.268499
\(940\) 66.1035 2.15606
\(941\) 36.3611 1.18534 0.592670 0.805446i \(-0.298074\pi\)
0.592670 + 0.805446i \(0.298074\pi\)
\(942\) −24.6707 −0.803816
\(943\) 13.7233 0.446891
\(944\) 36.5795 1.19056
\(945\) 0 0
\(946\) −83.9862 −2.73063
\(947\) 44.5795 1.44864 0.724319 0.689465i \(-0.242154\pi\)
0.724319 + 0.689465i \(0.242154\pi\)
\(948\) 58.7811 1.90912
\(949\) 9.11383 0.295847
\(950\) 9.88273 0.320638
\(951\) 27.6742 0.897397
\(952\) 0 0
\(953\) 19.8596 0.643317 0.321658 0.946856i \(-0.395760\pi\)
0.321658 + 0.946856i \(0.395760\pi\)
\(954\) 3.23109 0.104611
\(955\) −4.23453 −0.137026
\(956\) 7.32238 0.236823
\(957\) −16.3189 −0.527517
\(958\) 33.7846 1.09153
\(959\) 0 0
\(960\) 41.3871 1.33576
\(961\) 26.1070 0.842160
\(962\) 11.8827 0.383115
\(963\) −12.6026 −0.406112
\(964\) 34.3189 1.10534
\(965\) −23.3906 −0.752969
\(966\) 0 0
\(967\) 47.4068 1.52450 0.762250 0.647283i \(-0.224095\pi\)
0.762250 + 0.647283i \(0.224095\pi\)
\(968\) −37.2338 −1.19674
\(969\) 10.0913 0.324179
\(970\) 30.1104 0.966786
\(971\) 10.6448 0.341607 0.170803 0.985305i \(-0.445364\pi\)
0.170803 + 0.985305i \(0.445364\pi\)
\(972\) 5.71982 0.183463
\(973\) 0 0
\(974\) −0.443086 −0.0141974
\(975\) −1.00000 −0.0320256
\(976\) −114.071 −3.65131
\(977\) −1.21199 −0.0387750 −0.0193875 0.999812i \(-0.506172\pi\)
−0.0193875 + 0.999812i \(0.506172\pi\)
\(978\) −38.4362 −1.22905
\(979\) 3.16291 0.101087
\(980\) 0 0
\(981\) 11.4396 0.365240
\(982\) −117.328 −3.74408
\(983\) −51.8759 −1.65458 −0.827291 0.561773i \(-0.810119\pi\)
−0.827291 + 0.561773i \(0.810119\pi\)
\(984\) −29.3224 −0.934763
\(985\) 14.5535 0.463712
\(986\) 47.2964 1.50622
\(987\) 0 0
\(988\) −20.3449 −0.647258
\(989\) −53.7586 −1.70942
\(990\) −7.55691 −0.240175
\(991\) −21.6251 −0.686944 −0.343472 0.939163i \(-0.611603\pi\)
−0.343472 + 0.939163i \(0.611603\pi\)
\(992\) −206.545 −6.55781
\(993\) −13.2311 −0.419876
\(994\) 0 0
\(995\) 15.1138 0.479141
\(996\) −12.1104 −0.383732
\(997\) −23.2051 −0.734913 −0.367457 0.930041i \(-0.619771\pi\)
−0.367457 + 0.930041i \(0.619771\pi\)
\(998\) 21.6482 0.685262
\(999\) −4.27674 −0.135310
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 9555.2.a.bq.1.3 3
7.6 odd 2 195.2.a.e.1.3 3
21.20 even 2 585.2.a.n.1.1 3
28.27 even 2 3120.2.a.bj.1.3 3
35.13 even 4 975.2.c.i.274.1 6
35.27 even 4 975.2.c.i.274.6 6
35.34 odd 2 975.2.a.o.1.1 3
84.83 odd 2 9360.2.a.dd.1.3 3
91.90 odd 2 2535.2.a.bc.1.1 3
105.62 odd 4 2925.2.c.w.2224.1 6
105.83 odd 4 2925.2.c.w.2224.6 6
105.104 even 2 2925.2.a.bh.1.3 3
273.272 even 2 7605.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 7.6 odd 2
585.2.a.n.1.1 3 21.20 even 2
975.2.a.o.1.1 3 35.34 odd 2
975.2.c.i.274.1 6 35.13 even 4
975.2.c.i.274.6 6 35.27 even 4
2535.2.a.bc.1.1 3 91.90 odd 2
2925.2.a.bh.1.3 3 105.104 even 2
2925.2.c.w.2224.1 6 105.62 odd 4
2925.2.c.w.2224.6 6 105.83 odd 4
3120.2.a.bj.1.3 3 28.27 even 2
7605.2.a.bx.1.3 3 273.272 even 2
9360.2.a.dd.1.3 3 84.83 odd 2
9555.2.a.bq.1.3 3 1.1 even 1 trivial