Properties

Label 5775.2.a.bp.1.3
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 5775.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.93543 q^{2} -1.00000 q^{3} +1.74590 q^{4} -1.93543 q^{6} +1.00000 q^{7} -0.491797 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.93543 q^{2} -1.00000 q^{3} +1.74590 q^{4} -1.93543 q^{6} +1.00000 q^{7} -0.491797 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.74590 q^{12} +3.17313 q^{13} +1.93543 q^{14} -4.44364 q^{16} -6.85446 q^{17} +1.93543 q^{18} -0.318669 q^{19} -1.00000 q^{21} -1.93543 q^{22} +1.87086 q^{23} +0.491797 q^{24} +6.14137 q^{26} -1.00000 q^{27} +1.74590 q^{28} -3.17313 q^{29} +9.23353 q^{31} -7.61676 q^{32} +1.00000 q^{33} -13.2663 q^{34} +1.74590 q^{36} +7.55220 q^{37} -0.616763 q^{38} -3.17313 q^{39} +9.36266 q^{41} -1.93543 q^{42} +10.8873 q^{43} -1.74590 q^{44} +3.62093 q^{46} +8.06040 q^{47} +4.44364 q^{48} +1.00000 q^{49} +6.85446 q^{51} +5.53996 q^{52} -0.508203 q^{53} -1.93543 q^{54} -0.491797 q^{56} +0.318669 q^{57} -6.14137 q^{58} -7.04399 q^{59} -2.00000 q^{61} +17.8709 q^{62} +1.00000 q^{63} -5.85446 q^{64} +1.93543 q^{66} +2.66492 q^{67} -11.9672 q^{68} -1.87086 q^{69} -5.01641 q^{71} -0.491797 q^{72} +4.82687 q^{73} +14.6168 q^{74} -0.556364 q^{76} -1.00000 q^{77} -6.14137 q^{78} +5.01641 q^{79} +1.00000 q^{81} +18.1208 q^{82} -3.52461 q^{83} -1.74590 q^{84} +21.0716 q^{86} +3.17313 q^{87} +0.491797 q^{88} -1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} -9.23353 q^{93} +15.6004 q^{94} +7.61676 q^{96} +12.2499 q^{97} +1.93543 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - 2 q^{2} - 3 q^{3} + 6 q^{4} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} + 4 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 3 q^{21} + 2 q^{22} - 10 q^{23} + 3 q^{24} - q^{26} - 3 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} - 8 q^{32} + 3 q^{33} - 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 14 q^{41} + 2 q^{42} + 14 q^{43} - 6 q^{44} + 28 q^{46} + 4 q^{48} + 3 q^{49} + 8 q^{51} + 29 q^{52} + 2 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} - 6 q^{61} + 38 q^{62} + 3 q^{63} - 5 q^{64} - 2 q^{66} + 4 q^{67} - 42 q^{68} + 10 q^{69} - 12 q^{71} - 3 q^{72} + 20 q^{73} + 29 q^{74} - 11 q^{76} - 3 q^{77} + q^{78} + 12 q^{79} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{84} + 24 q^{86} + 4 q^{87} + 3 q^{88} + 26 q^{89} + 4 q^{91} - 26 q^{92} + 2 q^{93} + 35 q^{94} + 8 q^{96} + 4 q^{97} - 2 q^{98} - 3 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.93543 1.36856 0.684279 0.729221i \(-0.260117\pi\)
0.684279 + 0.729221i \(0.260117\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.74590 0.872949
\(5\) 0 0
\(6\) −1.93543 −0.790137
\(7\) 1.00000 0.377964
\(8\) −0.491797 −0.173876
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.74590 −0.503997
\(13\) 3.17313 0.880067 0.440034 0.897981i \(-0.354967\pi\)
0.440034 + 0.897981i \(0.354967\pi\)
\(14\) 1.93543 0.517266
\(15\) 0 0
\(16\) −4.44364 −1.11091
\(17\) −6.85446 −1.66245 −0.831225 0.555936i \(-0.812360\pi\)
−0.831225 + 0.555936i \(0.812360\pi\)
\(18\) 1.93543 0.456186
\(19\) −0.318669 −0.0731078 −0.0365539 0.999332i \(-0.511638\pi\)
−0.0365539 + 0.999332i \(0.511638\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −1.93543 −0.412636
\(23\) 1.87086 0.390102 0.195051 0.980793i \(-0.437513\pi\)
0.195051 + 0.980793i \(0.437513\pi\)
\(24\) 0.491797 0.100388
\(25\) 0 0
\(26\) 6.14137 1.20442
\(27\) −1.00000 −0.192450
\(28\) 1.74590 0.329944
\(29\) −3.17313 −0.589235 −0.294617 0.955615i \(-0.595192\pi\)
−0.294617 + 0.955615i \(0.595192\pi\)
\(30\) 0 0
\(31\) 9.23353 1.65839 0.829195 0.558959i \(-0.188799\pi\)
0.829195 + 0.558959i \(0.188799\pi\)
\(32\) −7.61676 −1.34647
\(33\) 1.00000 0.174078
\(34\) −13.2663 −2.27516
\(35\) 0 0
\(36\) 1.74590 0.290983
\(37\) 7.55220 1.24157 0.620787 0.783980i \(-0.286813\pi\)
0.620787 + 0.783980i \(0.286813\pi\)
\(38\) −0.616763 −0.100052
\(39\) −3.17313 −0.508107
\(40\) 0 0
\(41\) 9.36266 1.46220 0.731101 0.682269i \(-0.239007\pi\)
0.731101 + 0.682269i \(0.239007\pi\)
\(42\) −1.93543 −0.298644
\(43\) 10.8873 1.66029 0.830147 0.557545i \(-0.188257\pi\)
0.830147 + 0.557545i \(0.188257\pi\)
\(44\) −1.74590 −0.263204
\(45\) 0 0
\(46\) 3.62093 0.533877
\(47\) 8.06040 1.17573 0.587865 0.808959i \(-0.299969\pi\)
0.587865 + 0.808959i \(0.299969\pi\)
\(48\) 4.44364 0.641384
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.85446 0.959816
\(52\) 5.53996 0.768254
\(53\) −0.508203 −0.0698071 −0.0349036 0.999391i \(-0.511112\pi\)
−0.0349036 + 0.999391i \(0.511112\pi\)
\(54\) −1.93543 −0.263379
\(55\) 0 0
\(56\) −0.491797 −0.0657191
\(57\) 0.318669 0.0422088
\(58\) −6.14137 −0.806402
\(59\) −7.04399 −0.917050 −0.458525 0.888682i \(-0.651622\pi\)
−0.458525 + 0.888682i \(0.651622\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 17.8709 2.26960
\(63\) 1.00000 0.125988
\(64\) −5.85446 −0.731807
\(65\) 0 0
\(66\) 1.93543 0.238235
\(67\) 2.66492 0.325572 0.162786 0.986661i \(-0.447952\pi\)
0.162786 + 0.986661i \(0.447952\pi\)
\(68\) −11.9672 −1.45123
\(69\) −1.87086 −0.225226
\(70\) 0 0
\(71\) −5.01641 −0.595338 −0.297669 0.954669i \(-0.596209\pi\)
−0.297669 + 0.954669i \(0.596209\pi\)
\(72\) −0.491797 −0.0579588
\(73\) 4.82687 0.564943 0.282471 0.959276i \(-0.408846\pi\)
0.282471 + 0.959276i \(0.408846\pi\)
\(74\) 14.6168 1.69916
\(75\) 0 0
\(76\) −0.556364 −0.0638194
\(77\) −1.00000 −0.113961
\(78\) −6.14137 −0.695374
\(79\) 5.01641 0.564390 0.282195 0.959357i \(-0.408938\pi\)
0.282195 + 0.959357i \(0.408938\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 18.1208 2.00111
\(83\) −3.52461 −0.386876 −0.193438 0.981112i \(-0.561964\pi\)
−0.193438 + 0.981112i \(0.561964\pi\)
\(84\) −1.74590 −0.190493
\(85\) 0 0
\(86\) 21.0716 2.27221
\(87\) 3.17313 0.340195
\(88\) 0.491797 0.0524257
\(89\) −1.74173 −0.184623 −0.0923115 0.995730i \(-0.529426\pi\)
−0.0923115 + 0.995730i \(0.529426\pi\)
\(90\) 0 0
\(91\) 3.17313 0.332634
\(92\) 3.26634 0.340539
\(93\) −9.23353 −0.957472
\(94\) 15.6004 1.60905
\(95\) 0 0
\(96\) 7.61676 0.777383
\(97\) 12.2499 1.24379 0.621896 0.783100i \(-0.286363\pi\)
0.621896 + 0.783100i \(0.286363\pi\)
\(98\) 1.93543 0.195508
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 4.88727 0.486302 0.243151 0.969988i \(-0.421819\pi\)
0.243151 + 0.969988i \(0.421819\pi\)
\(102\) 13.2663 1.31356
\(103\) 0.637339 0.0627988 0.0313994 0.999507i \(-0.490004\pi\)
0.0313994 + 0.999507i \(0.490004\pi\)
\(104\) −1.56053 −0.153023
\(105\) 0 0
\(106\) −0.983593 −0.0955350
\(107\) −0.956008 −0.0924208 −0.0462104 0.998932i \(-0.514714\pi\)
−0.0462104 + 0.998932i \(0.514714\pi\)
\(108\) −1.74590 −0.167999
\(109\) −7.61259 −0.729154 −0.364577 0.931173i \(-0.618786\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(110\) 0 0
\(111\) −7.55220 −0.716823
\(112\) −4.44364 −0.419884
\(113\) 7.70892 0.725194 0.362597 0.931946i \(-0.381890\pi\)
0.362597 + 0.931946i \(0.381890\pi\)
\(114\) 0.616763 0.0577651
\(115\) 0 0
\(116\) −5.53996 −0.514372
\(117\) 3.17313 0.293356
\(118\) −13.6332 −1.25504
\(119\) −6.85446 −0.628347
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.87086 −0.350452
\(123\) −9.36266 −0.844203
\(124\) 16.1208 1.44769
\(125\) 0 0
\(126\) 1.93543 0.172422
\(127\) 5.49180 0.487318 0.243659 0.969861i \(-0.421652\pi\)
0.243659 + 0.969861i \(0.421652\pi\)
\(128\) 3.90262 0.344946
\(129\) −10.8873 −0.958571
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 1.74590 0.151961
\(133\) −0.318669 −0.0276321
\(134\) 5.15778 0.445564
\(135\) 0 0
\(136\) 3.37100 0.289061
\(137\) −15.6126 −1.33387 −0.666937 0.745114i \(-0.732395\pi\)
−0.666937 + 0.745114i \(0.732395\pi\)
\(138\) −3.62093 −0.308234
\(139\) −9.01641 −0.764762 −0.382381 0.924005i \(-0.624896\pi\)
−0.382381 + 0.924005i \(0.624896\pi\)
\(140\) 0 0
\(141\) −8.06040 −0.678808
\(142\) −9.70892 −0.814754
\(143\) −3.17313 −0.265350
\(144\) −4.44364 −0.370303
\(145\) 0 0
\(146\) 9.34209 0.773157
\(147\) −1.00000 −0.0824786
\(148\) 13.1854 1.08383
\(149\) −5.20594 −0.426487 −0.213244 0.976999i \(-0.568403\pi\)
−0.213244 + 0.976999i \(0.568403\pi\)
\(150\) 0 0
\(151\) 6.24993 0.508612 0.254306 0.967124i \(-0.418153\pi\)
0.254306 + 0.967124i \(0.418153\pi\)
\(152\) 0.156721 0.0127117
\(153\) −6.85446 −0.554150
\(154\) −1.93543 −0.155962
\(155\) 0 0
\(156\) −5.53996 −0.443552
\(157\) 18.1208 1.44620 0.723099 0.690745i \(-0.242717\pi\)
0.723099 + 0.690745i \(0.242717\pi\)
\(158\) 9.70892 0.772400
\(159\) 0.508203 0.0403031
\(160\) 0 0
\(161\) 1.87086 0.147445
\(162\) 1.93543 0.152062
\(163\) 2.66492 0.208733 0.104366 0.994539i \(-0.466719\pi\)
0.104366 + 0.994539i \(0.466719\pi\)
\(164\) 16.3463 1.27643
\(165\) 0 0
\(166\) −6.82164 −0.529462
\(167\) 11.1455 0.862468 0.431234 0.902240i \(-0.358078\pi\)
0.431234 + 0.902240i \(0.358078\pi\)
\(168\) 0.491797 0.0379429
\(169\) −2.93126 −0.225482
\(170\) 0 0
\(171\) −0.318669 −0.0243693
\(172\) 19.0081 1.44935
\(173\) 24.8461 1.88902 0.944508 0.328489i \(-0.106539\pi\)
0.944508 + 0.328489i \(0.106539\pi\)
\(174\) 6.14137 0.465576
\(175\) 0 0
\(176\) 4.44364 0.334952
\(177\) 7.04399 0.529459
\(178\) −3.37100 −0.252667
\(179\) −12.7581 −0.953588 −0.476794 0.879015i \(-0.658201\pi\)
−0.476794 + 0.879015i \(0.658201\pi\)
\(180\) 0 0
\(181\) −3.23353 −0.240346 −0.120173 0.992753i \(-0.538345\pi\)
−0.120173 + 0.992753i \(0.538345\pi\)
\(182\) 6.14137 0.455229
\(183\) 2.00000 0.147844
\(184\) −0.920085 −0.0678296
\(185\) 0 0
\(186\) −17.8709 −1.31036
\(187\) 6.85446 0.501248
\(188\) 14.0726 1.02635
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 20.9753 1.51772 0.758858 0.651256i \(-0.225758\pi\)
0.758858 + 0.651256i \(0.225758\pi\)
\(192\) 5.85446 0.422509
\(193\) −0.249933 −0.0179905 −0.00899527 0.999960i \(-0.502863\pi\)
−0.00899527 + 0.999960i \(0.502863\pi\)
\(194\) 23.7089 1.70220
\(195\) 0 0
\(196\) 1.74590 0.124707
\(197\) −18.4999 −1.31806 −0.659030 0.752116i \(-0.729033\pi\)
−0.659030 + 0.752116i \(0.729033\pi\)
\(198\) −1.93543 −0.137545
\(199\) 9.87086 0.699727 0.349864 0.936801i \(-0.386228\pi\)
0.349864 + 0.936801i \(0.386228\pi\)
\(200\) 0 0
\(201\) −2.66492 −0.187969
\(202\) 9.45898 0.665532
\(203\) −3.17313 −0.222710
\(204\) 11.9672 0.837871
\(205\) 0 0
\(206\) 1.23353 0.0859438
\(207\) 1.87086 0.130034
\(208\) −14.1002 −0.977674
\(209\) 0.318669 0.0220428
\(210\) 0 0
\(211\) −4.63734 −0.319248 −0.159624 0.987178i \(-0.551028\pi\)
−0.159624 + 0.987178i \(0.551028\pi\)
\(212\) −0.887271 −0.0609381
\(213\) 5.01641 0.343719
\(214\) −1.85029 −0.126483
\(215\) 0 0
\(216\) 0.491797 0.0334625
\(217\) 9.23353 0.626813
\(218\) −14.7337 −0.997889
\(219\) −4.82687 −0.326170
\(220\) 0 0
\(221\) −21.7501 −1.46307
\(222\) −14.6168 −0.981013
\(223\) 18.3463 1.22856 0.614278 0.789090i \(-0.289447\pi\)
0.614278 + 0.789090i \(0.289447\pi\)
\(224\) −7.61676 −0.508916
\(225\) 0 0
\(226\) 14.9201 0.992469
\(227\) −0.379068 −0.0251596 −0.0125798 0.999921i \(-0.504004\pi\)
−0.0125798 + 0.999921i \(0.504004\pi\)
\(228\) 0.556364 0.0368461
\(229\) 24.9424 1.64824 0.824121 0.566413i \(-0.191669\pi\)
0.824121 + 0.566413i \(0.191669\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 1.56053 0.102454
\(233\) −23.4506 −1.53630 −0.768151 0.640268i \(-0.778823\pi\)
−0.768151 + 0.640268i \(0.778823\pi\)
\(234\) 6.14137 0.401474
\(235\) 0 0
\(236\) −12.2981 −0.800538
\(237\) −5.01641 −0.325851
\(238\) −13.2663 −0.859929
\(239\) 5.07681 0.328391 0.164196 0.986428i \(-0.447497\pi\)
0.164196 + 0.986428i \(0.447497\pi\)
\(240\) 0 0
\(241\) 19.2939 1.24283 0.621415 0.783481i \(-0.286558\pi\)
0.621415 + 0.783481i \(0.286558\pi\)
\(242\) 1.93543 0.124414
\(243\) −1.00000 −0.0641500
\(244\) −3.49180 −0.223539
\(245\) 0 0
\(246\) −18.1208 −1.15534
\(247\) −1.01118 −0.0643397
\(248\) −4.54102 −0.288355
\(249\) 3.52461 0.223363
\(250\) 0 0
\(251\) 23.8021 1.50238 0.751188 0.660088i \(-0.229481\pi\)
0.751188 + 0.660088i \(0.229481\pi\)
\(252\) 1.74590 0.109981
\(253\) −1.87086 −0.117620
\(254\) 10.6290 0.666923
\(255\) 0 0
\(256\) 19.2622 1.20389
\(257\) −14.9149 −0.930363 −0.465182 0.885215i \(-0.654011\pi\)
−0.465182 + 0.885215i \(0.654011\pi\)
\(258\) −21.0716 −1.31186
\(259\) 7.55220 0.469271
\(260\) 0 0
\(261\) −3.17313 −0.196412
\(262\) 7.74173 0.478286
\(263\) −2.92319 −0.180252 −0.0901259 0.995930i \(-0.528727\pi\)
−0.0901259 + 0.995930i \(0.528727\pi\)
\(264\) −0.491797 −0.0302680
\(265\) 0 0
\(266\) −0.616763 −0.0378162
\(267\) 1.74173 0.106592
\(268\) 4.65269 0.284208
\(269\) 11.9672 0.729652 0.364826 0.931076i \(-0.381128\pi\)
0.364826 + 0.931076i \(0.381128\pi\)
\(270\) 0 0
\(271\) −20.3187 −1.23427 −0.617136 0.786857i \(-0.711707\pi\)
−0.617136 + 0.786857i \(0.711707\pi\)
\(272\) 30.4587 1.84683
\(273\) −3.17313 −0.192046
\(274\) −30.2171 −1.82548
\(275\) 0 0
\(276\) −3.26634 −0.196611
\(277\) −18.0552 −1.08483 −0.542415 0.840111i \(-0.682490\pi\)
−0.542415 + 0.840111i \(0.682490\pi\)
\(278\) −17.4506 −1.04662
\(279\) 9.23353 0.552797
\(280\) 0 0
\(281\) 27.2939 1.62822 0.814110 0.580711i \(-0.197226\pi\)
0.814110 + 0.580711i \(0.197226\pi\)
\(282\) −15.6004 −0.928988
\(283\) −29.8901 −1.77678 −0.888391 0.459087i \(-0.848177\pi\)
−0.888391 + 0.459087i \(0.848177\pi\)
\(284\) −8.75814 −0.519700
\(285\) 0 0
\(286\) −6.14137 −0.363147
\(287\) 9.36266 0.552660
\(288\) −7.61676 −0.448822
\(289\) 29.9836 1.76374
\(290\) 0 0
\(291\) −12.2499 −0.718104
\(292\) 8.42723 0.493166
\(293\) 4.34625 0.253911 0.126955 0.991908i \(-0.459479\pi\)
0.126955 + 0.991908i \(0.459479\pi\)
\(294\) −1.93543 −0.112877
\(295\) 0 0
\(296\) −3.71414 −0.215880
\(297\) 1.00000 0.0580259
\(298\) −10.0757 −0.583672
\(299\) 5.93649 0.343316
\(300\) 0 0
\(301\) 10.8873 0.627532
\(302\) 12.0963 0.696065
\(303\) −4.88727 −0.280766
\(304\) 1.41605 0.0812161
\(305\) 0 0
\(306\) −13.2663 −0.758386
\(307\) 20.7581 1.18473 0.592365 0.805670i \(-0.298194\pi\)
0.592365 + 0.805670i \(0.298194\pi\)
\(308\) −1.74590 −0.0994818
\(309\) −0.637339 −0.0362569
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 1.56053 0.0883478
\(313\) −8.12914 −0.459486 −0.229743 0.973251i \(-0.573789\pi\)
−0.229743 + 0.973251i \(0.573789\pi\)
\(314\) 35.0716 1.97920
\(315\) 0 0
\(316\) 8.75814 0.492684
\(317\) 19.9917 1.12284 0.561422 0.827530i \(-0.310255\pi\)
0.561422 + 0.827530i \(0.310255\pi\)
\(318\) 0.983593 0.0551572
\(319\) 3.17313 0.177661
\(320\) 0 0
\(321\) 0.956008 0.0533592
\(322\) 3.62093 0.201787
\(323\) 2.18431 0.121538
\(324\) 1.74590 0.0969944
\(325\) 0 0
\(326\) 5.15778 0.285663
\(327\) 7.61259 0.420977
\(328\) −4.60453 −0.254242
\(329\) 8.06040 0.444384
\(330\) 0 0
\(331\) 17.0164 0.935306 0.467653 0.883912i \(-0.345100\pi\)
0.467653 + 0.883912i \(0.345100\pi\)
\(332\) −6.15361 −0.337723
\(333\) 7.55220 0.413858
\(334\) 21.5714 1.18034
\(335\) 0 0
\(336\) 4.44364 0.242420
\(337\) −1.52461 −0.0830508 −0.0415254 0.999137i \(-0.513222\pi\)
−0.0415254 + 0.999137i \(0.513222\pi\)
\(338\) −5.67326 −0.308585
\(339\) −7.70892 −0.418691
\(340\) 0 0
\(341\) −9.23353 −0.500023
\(342\) −0.616763 −0.0333507
\(343\) 1.00000 0.0539949
\(344\) −5.35432 −0.288686
\(345\) 0 0
\(346\) 48.0880 2.58523
\(347\) −17.4506 −0.936800 −0.468400 0.883517i \(-0.655169\pi\)
−0.468400 + 0.883517i \(0.655169\pi\)
\(348\) 5.53996 0.296973
\(349\) 6.85969 0.367191 0.183595 0.983002i \(-0.441226\pi\)
0.183595 + 0.983002i \(0.441226\pi\)
\(350\) 0 0
\(351\) −3.17313 −0.169369
\(352\) 7.61676 0.405975
\(353\) 31.9313 1.69953 0.849765 0.527162i \(-0.176744\pi\)
0.849765 + 0.527162i \(0.176744\pi\)
\(354\) 13.6332 0.724595
\(355\) 0 0
\(356\) −3.04088 −0.161166
\(357\) 6.85446 0.362776
\(358\) −24.6925 −1.30504
\(359\) −24.4342 −1.28959 −0.644795 0.764356i \(-0.723057\pi\)
−0.644795 + 0.764356i \(0.723057\pi\)
\(360\) 0 0
\(361\) −18.8984 −0.994655
\(362\) −6.25827 −0.328927
\(363\) −1.00000 −0.0524864
\(364\) 5.53996 0.290373
\(365\) 0 0
\(366\) 3.87086 0.202333
\(367\) −17.0716 −0.891129 −0.445565 0.895250i \(-0.646997\pi\)
−0.445565 + 0.895250i \(0.646997\pi\)
\(368\) −8.31344 −0.433368
\(369\) 9.36266 0.487401
\(370\) 0 0
\(371\) −0.508203 −0.0263846
\(372\) −16.1208 −0.835824
\(373\) −3.17836 −0.164569 −0.0822845 0.996609i \(-0.526222\pi\)
−0.0822845 + 0.996609i \(0.526222\pi\)
\(374\) 13.2663 0.685986
\(375\) 0 0
\(376\) −3.96408 −0.204432
\(377\) −10.0687 −0.518566
\(378\) −1.93543 −0.0995479
\(379\) 3.93960 0.202364 0.101182 0.994868i \(-0.467738\pi\)
0.101182 + 0.994868i \(0.467738\pi\)
\(380\) 0 0
\(381\) −5.49180 −0.281353
\(382\) 40.5962 2.07708
\(383\) 24.7581 1.26508 0.632541 0.774527i \(-0.282012\pi\)
0.632541 + 0.774527i \(0.282012\pi\)
\(384\) −3.90262 −0.199155
\(385\) 0 0
\(386\) −0.483728 −0.0246211
\(387\) 10.8873 0.553431
\(388\) 21.3871 1.08577
\(389\) 3.65375 0.185252 0.0926261 0.995701i \(-0.470474\pi\)
0.0926261 + 0.995701i \(0.470474\pi\)
\(390\) 0 0
\(391\) −12.8238 −0.648526
\(392\) −0.491797 −0.0248395
\(393\) −4.00000 −0.201773
\(394\) −35.8052 −1.80384
\(395\) 0 0
\(396\) −1.74590 −0.0877347
\(397\) −4.56337 −0.229029 −0.114515 0.993422i \(-0.536531\pi\)
−0.114515 + 0.993422i \(0.536531\pi\)
\(398\) 19.1044 0.957617
\(399\) 0.318669 0.0159534
\(400\) 0 0
\(401\) −7.23353 −0.361225 −0.180613 0.983554i \(-0.557808\pi\)
−0.180613 + 0.983554i \(0.557808\pi\)
\(402\) −5.15778 −0.257247
\(403\) 29.2992 1.45949
\(404\) 8.53268 0.424517
\(405\) 0 0
\(406\) −6.14137 −0.304791
\(407\) −7.55220 −0.374348
\(408\) −3.37100 −0.166889
\(409\) 5.30749 0.262439 0.131219 0.991353i \(-0.458111\pi\)
0.131219 + 0.991353i \(0.458111\pi\)
\(410\) 0 0
\(411\) 15.6126 0.770112
\(412\) 1.11273 0.0548202
\(413\) −7.04399 −0.346612
\(414\) 3.62093 0.177959
\(415\) 0 0
\(416\) −24.1690 −1.18498
\(417\) 9.01641 0.441535
\(418\) 0.616763 0.0301669
\(419\) −11.4231 −0.558053 −0.279026 0.960283i \(-0.590012\pi\)
−0.279026 + 0.960283i \(0.590012\pi\)
\(420\) 0 0
\(421\) −27.4147 −1.33611 −0.668056 0.744111i \(-0.732873\pi\)
−0.668056 + 0.744111i \(0.732873\pi\)
\(422\) −8.97526 −0.436909
\(423\) 8.06040 0.391910
\(424\) 0.249933 0.0121378
\(425\) 0 0
\(426\) 9.70892 0.470399
\(427\) −2.00000 −0.0967868
\(428\) −1.66909 −0.0806786
\(429\) 3.17313 0.153200
\(430\) 0 0
\(431\) −2.28586 −0.110106 −0.0550529 0.998483i \(-0.517533\pi\)
−0.0550529 + 0.998483i \(0.517533\pi\)
\(432\) 4.44364 0.213795
\(433\) −31.5714 −1.51723 −0.758613 0.651541i \(-0.774123\pi\)
−0.758613 + 0.651541i \(0.774123\pi\)
\(434\) 17.8709 0.857829
\(435\) 0 0
\(436\) −13.2908 −0.636515
\(437\) −0.596187 −0.0285195
\(438\) −9.34209 −0.446382
\(439\) −18.5275 −0.884267 −0.442133 0.896949i \(-0.645778\pi\)
−0.442133 + 0.896949i \(0.645778\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −42.0958 −2.00229
\(443\) −28.4342 −1.35095 −0.675476 0.737382i \(-0.736062\pi\)
−0.675476 + 0.737382i \(0.736062\pi\)
\(444\) −13.1854 −0.625750
\(445\) 0 0
\(446\) 35.5079 1.68135
\(447\) 5.20594 0.246233
\(448\) −5.85446 −0.276597
\(449\) −5.68656 −0.268365 −0.134183 0.990957i \(-0.542841\pi\)
−0.134183 + 0.990957i \(0.542841\pi\)
\(450\) 0 0
\(451\) −9.36266 −0.440871
\(452\) 13.4590 0.633057
\(453\) −6.24993 −0.293647
\(454\) −0.733661 −0.0344324
\(455\) 0 0
\(456\) −0.156721 −0.00733911
\(457\) 13.0081 0.608492 0.304246 0.952594i \(-0.401596\pi\)
0.304246 + 0.952594i \(0.401596\pi\)
\(458\) 48.2744 2.25571
\(459\) 6.85446 0.319939
\(460\) 0 0
\(461\) −6.37907 −0.297103 −0.148551 0.988905i \(-0.547461\pi\)
−0.148551 + 0.988905i \(0.547461\pi\)
\(462\) 1.93543 0.0900445
\(463\) −34.0932 −1.58445 −0.792223 0.610232i \(-0.791076\pi\)
−0.792223 + 0.610232i \(0.791076\pi\)
\(464\) 14.1002 0.654586
\(465\) 0 0
\(466\) −45.3871 −2.10252
\(467\) −18.1484 −0.839807 −0.419903 0.907569i \(-0.637936\pi\)
−0.419903 + 0.907569i \(0.637936\pi\)
\(468\) 5.53996 0.256085
\(469\) 2.66492 0.123055
\(470\) 0 0
\(471\) −18.1208 −0.834962
\(472\) 3.46421 0.159453
\(473\) −10.8873 −0.500597
\(474\) −9.70892 −0.445945
\(475\) 0 0
\(476\) −11.9672 −0.548515
\(477\) −0.508203 −0.0232690
\(478\) 9.82581 0.449422
\(479\) −42.7498 −1.95329 −0.976644 0.214864i \(-0.931069\pi\)
−0.976644 + 0.214864i \(0.931069\pi\)
\(480\) 0 0
\(481\) 23.9641 1.09267
\(482\) 37.3421 1.70089
\(483\) −1.87086 −0.0851273
\(484\) 1.74590 0.0793590
\(485\) 0 0
\(486\) −1.93543 −0.0877930
\(487\) −26.7909 −1.21401 −0.607007 0.794697i \(-0.707630\pi\)
−0.607007 + 0.794697i \(0.707630\pi\)
\(488\) 0.983593 0.0445252
\(489\) −2.66492 −0.120512
\(490\) 0 0
\(491\) 31.6813 1.42976 0.714879 0.699248i \(-0.246482\pi\)
0.714879 + 0.699248i \(0.246482\pi\)
\(492\) −16.3463 −0.736946
\(493\) 21.7501 0.979574
\(494\) −1.95707 −0.0880526
\(495\) 0 0
\(496\) −41.0304 −1.84232
\(497\) −5.01641 −0.225017
\(498\) 6.82164 0.305685
\(499\) −24.1260 −1.08003 −0.540015 0.841656i \(-0.681581\pi\)
−0.540015 + 0.841656i \(0.681581\pi\)
\(500\) 0 0
\(501\) −11.1455 −0.497946
\(502\) 46.0674 2.05609
\(503\) −30.8873 −1.37720 −0.688598 0.725144i \(-0.741773\pi\)
−0.688598 + 0.725144i \(0.741773\pi\)
\(504\) −0.491797 −0.0219064
\(505\) 0 0
\(506\) −3.62093 −0.160970
\(507\) 2.93126 0.130182
\(508\) 9.58812 0.425404
\(509\) 28.2088 1.25033 0.625166 0.780492i \(-0.285031\pi\)
0.625166 + 0.780492i \(0.285031\pi\)
\(510\) 0 0
\(511\) 4.82687 0.213528
\(512\) 29.4754 1.30264
\(513\) 0.318669 0.0140696
\(514\) −28.8667 −1.27326
\(515\) 0 0
\(516\) −19.0081 −0.836784
\(517\) −8.06040 −0.354496
\(518\) 14.6168 0.642224
\(519\) −24.8461 −1.09062
\(520\) 0 0
\(521\) −33.7610 −1.47910 −0.739548 0.673104i \(-0.764961\pi\)
−0.739548 + 0.673104i \(0.764961\pi\)
\(522\) −6.14137 −0.268801
\(523\) −12.8185 −0.560515 −0.280258 0.959925i \(-0.590420\pi\)
−0.280258 + 0.959925i \(0.590420\pi\)
\(524\) 6.98359 0.305080
\(525\) 0 0
\(526\) −5.65765 −0.246685
\(527\) −63.2908 −2.75699
\(528\) −4.44364 −0.193384
\(529\) −19.4999 −0.847820
\(530\) 0 0
\(531\) −7.04399 −0.305683
\(532\) −0.556364 −0.0241215
\(533\) 29.7089 1.28684
\(534\) 3.37100 0.145877
\(535\) 0 0
\(536\) −1.31060 −0.0566093
\(537\) 12.7581 0.550554
\(538\) 23.1617 0.998571
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −21.8625 −0.939943 −0.469972 0.882681i \(-0.655736\pi\)
−0.469972 + 0.882681i \(0.655736\pi\)
\(542\) −39.3254 −1.68917
\(543\) 3.23353 0.138764
\(544\) 52.2088 2.23843
\(545\) 0 0
\(546\) −6.14137 −0.262827
\(547\) −23.4178 −1.00127 −0.500637 0.865657i \(-0.666901\pi\)
−0.500637 + 0.865657i \(0.666901\pi\)
\(548\) −27.2580 −1.16440
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 1.01118 0.0430776
\(552\) 0.920085 0.0391614
\(553\) 5.01641 0.213319
\(554\) −34.9446 −1.48465
\(555\) 0 0
\(556\) −15.7417 −0.667598
\(557\) 6.91486 0.292992 0.146496 0.989211i \(-0.453200\pi\)
0.146496 + 0.989211i \(0.453200\pi\)
\(558\) 17.8709 0.756534
\(559\) 34.5467 1.46117
\(560\) 0 0
\(561\) −6.85446 −0.289395
\(562\) 52.8255 2.22831
\(563\) −11.8381 −0.498914 −0.249457 0.968386i \(-0.580252\pi\)
−0.249457 + 0.968386i \(0.580252\pi\)
\(564\) −14.0726 −0.592565
\(565\) 0 0
\(566\) −57.8503 −2.43163
\(567\) 1.00000 0.0419961
\(568\) 2.46705 0.103515
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −15.2252 −0.637154 −0.318577 0.947897i \(-0.603205\pi\)
−0.318577 + 0.947897i \(0.603205\pi\)
\(572\) −5.53996 −0.231637
\(573\) −20.9753 −0.876254
\(574\) 18.1208 0.756347
\(575\) 0 0
\(576\) −5.85446 −0.243936
\(577\) −17.3215 −0.721104 −0.360552 0.932739i \(-0.617412\pi\)
−0.360552 + 0.932739i \(0.617412\pi\)
\(578\) 58.0312 2.41378
\(579\) 0.249933 0.0103868
\(580\) 0 0
\(581\) −3.52461 −0.146225
\(582\) −23.7089 −0.982766
\(583\) 0.508203 0.0210476
\(584\) −2.37384 −0.0982302
\(585\) 0 0
\(586\) 8.41188 0.347492
\(587\) −27.8678 −1.15023 −0.575113 0.818074i \(-0.695042\pi\)
−0.575113 + 0.818074i \(0.695042\pi\)
\(588\) −1.74590 −0.0719996
\(589\) −2.94244 −0.121241
\(590\) 0 0
\(591\) 18.4999 0.760983
\(592\) −33.5592 −1.37927
\(593\) 24.3463 0.999781 0.499890 0.866089i \(-0.333374\pi\)
0.499890 + 0.866089i \(0.333374\pi\)
\(594\) 1.93543 0.0794118
\(595\) 0 0
\(596\) −9.08904 −0.372302
\(597\) −9.87086 −0.403988
\(598\) 11.4897 0.469848
\(599\) 21.0164 0.858707 0.429354 0.903136i \(-0.358741\pi\)
0.429354 + 0.903136i \(0.358741\pi\)
\(600\) 0 0
\(601\) 40.9477 1.67029 0.835145 0.550030i \(-0.185384\pi\)
0.835145 + 0.550030i \(0.185384\pi\)
\(602\) 21.0716 0.858813
\(603\) 2.66492 0.108524
\(604\) 10.9117 0.443993
\(605\) 0 0
\(606\) −9.45898 −0.384245
\(607\) 14.0276 0.569362 0.284681 0.958622i \(-0.408112\pi\)
0.284681 + 0.958622i \(0.408112\pi\)
\(608\) 2.42723 0.0984371
\(609\) 3.17313 0.128582
\(610\) 0 0
\(611\) 25.5767 1.03472
\(612\) −11.9672 −0.483745
\(613\) −18.5306 −0.748442 −0.374221 0.927339i \(-0.622090\pi\)
−0.374221 + 0.927339i \(0.622090\pi\)
\(614\) 40.1760 1.62137
\(615\) 0 0
\(616\) 0.491797 0.0198151
\(617\) −2.43424 −0.0979987 −0.0489994 0.998799i \(-0.515603\pi\)
−0.0489994 + 0.998799i \(0.515603\pi\)
\(618\) −1.23353 −0.0496197
\(619\) −30.4259 −1.22292 −0.611460 0.791275i \(-0.709418\pi\)
−0.611460 + 0.791275i \(0.709418\pi\)
\(620\) 0 0
\(621\) −1.87086 −0.0750752
\(622\) 15.4835 0.620830
\(623\) −1.74173 −0.0697809
\(624\) 14.1002 0.564461
\(625\) 0 0
\(626\) −15.7334 −0.628833
\(627\) −0.318669 −0.0127264
\(628\) 31.6371 1.26246
\(629\) −51.7662 −2.06405
\(630\) 0 0
\(631\) 34.9836 1.39267 0.696337 0.717715i \(-0.254812\pi\)
0.696337 + 0.717715i \(0.254812\pi\)
\(632\) −2.46705 −0.0981341
\(633\) 4.63734 0.184318
\(634\) 38.6925 1.53668
\(635\) 0 0
\(636\) 0.887271 0.0351826
\(637\) 3.17313 0.125724
\(638\) 6.14137 0.243139
\(639\) −5.01641 −0.198446
\(640\) 0 0
\(641\) 8.56337 0.338233 0.169116 0.985596i \(-0.445909\pi\)
0.169116 + 0.985596i \(0.445909\pi\)
\(642\) 1.85029 0.0730251
\(643\) 5.11273 0.201626 0.100813 0.994905i \(-0.467856\pi\)
0.100813 + 0.994905i \(0.467856\pi\)
\(644\) 3.26634 0.128712
\(645\) 0 0
\(646\) 4.22758 0.166332
\(647\) −25.7693 −1.01310 −0.506548 0.862212i \(-0.669079\pi\)
−0.506548 + 0.862212i \(0.669079\pi\)
\(648\) −0.491797 −0.0193196
\(649\) 7.04399 0.276501
\(650\) 0 0
\(651\) −9.23353 −0.361890
\(652\) 4.65269 0.182213
\(653\) 47.8953 1.87429 0.937145 0.348941i \(-0.113459\pi\)
0.937145 + 0.348941i \(0.113459\pi\)
\(654\) 14.7337 0.576132
\(655\) 0 0
\(656\) −41.6043 −1.62437
\(657\) 4.82687 0.188314
\(658\) 15.6004 0.608165
\(659\) 8.95601 0.348877 0.174438 0.984668i \(-0.444189\pi\)
0.174438 + 0.984668i \(0.444189\pi\)
\(660\) 0 0
\(661\) −10.7993 −0.420044 −0.210022 0.977697i \(-0.567353\pi\)
−0.210022 + 0.977697i \(0.567353\pi\)
\(662\) 32.9341 1.28002
\(663\) 21.7501 0.844703
\(664\) 1.73339 0.0672686
\(665\) 0 0
\(666\) 14.6168 0.566388
\(667\) −5.93649 −0.229862
\(668\) 19.4590 0.752891
\(669\) −18.3463 −0.709307
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 7.61676 0.293823
\(673\) −32.2499 −1.24314 −0.621572 0.783357i \(-0.713506\pi\)
−0.621572 + 0.783357i \(0.713506\pi\)
\(674\) −2.95078 −0.113660
\(675\) 0 0
\(676\) −5.11769 −0.196834
\(677\) 27.7089 1.06494 0.532470 0.846449i \(-0.321264\pi\)
0.532470 + 0.846449i \(0.321264\pi\)
\(678\) −14.9201 −0.573002
\(679\) 12.2499 0.470109
\(680\) 0 0
\(681\) 0.379068 0.0145259
\(682\) −17.8709 −0.684311
\(683\) −22.3156 −0.853881 −0.426941 0.904280i \(-0.640409\pi\)
−0.426941 + 0.904280i \(0.640409\pi\)
\(684\) −0.556364 −0.0212731
\(685\) 0 0
\(686\) 1.93543 0.0738951
\(687\) −24.9424 −0.951614
\(688\) −48.3791 −1.84443
\(689\) −1.61259 −0.0614349
\(690\) 0 0
\(691\) 25.1372 0.956264 0.478132 0.878288i \(-0.341314\pi\)
0.478132 + 0.878288i \(0.341314\pi\)
\(692\) 43.3788 1.64901
\(693\) −1.00000 −0.0379869
\(694\) −33.7745 −1.28206
\(695\) 0 0
\(696\) −1.56053 −0.0591519
\(697\) −64.1760 −2.43084
\(698\) 13.2765 0.502521
\(699\) 23.4506 0.886985
\(700\) 0 0
\(701\) −19.2747 −0.727995 −0.363997 0.931400i \(-0.618588\pi\)
−0.363997 + 0.931400i \(0.618588\pi\)
\(702\) −6.14137 −0.231791
\(703\) −2.40665 −0.0907686
\(704\) 5.85446 0.220648
\(705\) 0 0
\(706\) 61.8008 2.32590
\(707\) 4.88727 0.183805
\(708\) 12.2981 0.462191
\(709\) 9.76098 0.366581 0.183291 0.983059i \(-0.441325\pi\)
0.183291 + 0.983059i \(0.441325\pi\)
\(710\) 0 0
\(711\) 5.01641 0.188130
\(712\) 0.856577 0.0321016
\(713\) 17.2747 0.646942
\(714\) 13.2663 0.496480
\(715\) 0 0
\(716\) −22.2744 −0.832434
\(717\) −5.07681 −0.189597
\(718\) −47.2908 −1.76488
\(719\) 14.0276 0.523141 0.261570 0.965184i \(-0.415760\pi\)
0.261570 + 0.965184i \(0.415760\pi\)
\(720\) 0 0
\(721\) 0.637339 0.0237357
\(722\) −36.5767 −1.36124
\(723\) −19.2939 −0.717549
\(724\) −5.64541 −0.209810
\(725\) 0 0
\(726\) −1.93543 −0.0718306
\(727\) 34.3051 1.27231 0.636153 0.771563i \(-0.280525\pi\)
0.636153 + 0.771563i \(0.280525\pi\)
\(728\) −1.56053 −0.0578372
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −74.6263 −2.76016
\(732\) 3.49180 0.129061
\(733\) 10.7581 0.397361 0.198680 0.980064i \(-0.436334\pi\)
0.198680 + 0.980064i \(0.436334\pi\)
\(734\) −33.0409 −1.21956
\(735\) 0 0
\(736\) −14.2499 −0.525259
\(737\) −2.66492 −0.0981637
\(738\) 18.1208 0.667036
\(739\) −6.38741 −0.234965 −0.117482 0.993075i \(-0.537482\pi\)
−0.117482 + 0.993075i \(0.537482\pi\)
\(740\) 0 0
\(741\) 1.01118 0.0371466
\(742\) −0.983593 −0.0361088
\(743\) 23.1096 0.847810 0.423905 0.905707i \(-0.360659\pi\)
0.423905 + 0.905707i \(0.360659\pi\)
\(744\) 4.54102 0.166482
\(745\) 0 0
\(746\) −6.15149 −0.225222
\(747\) −3.52461 −0.128959
\(748\) 11.9672 0.437564
\(749\) −0.956008 −0.0349318
\(750\) 0 0
\(751\) 31.8678 1.16287 0.581435 0.813593i \(-0.302491\pi\)
0.581435 + 0.813593i \(0.302491\pi\)
\(752\) −35.8175 −1.30613
\(753\) −23.8021 −0.867398
\(754\) −19.4874 −0.709688
\(755\) 0 0
\(756\) −1.74590 −0.0634977
\(757\) 48.3103 1.75587 0.877934 0.478781i \(-0.158921\pi\)
0.877934 + 0.478781i \(0.158921\pi\)
\(758\) 7.62483 0.276946
\(759\) 1.87086 0.0679081
\(760\) 0 0
\(761\) −17.8625 −0.647516 −0.323758 0.946140i \(-0.604946\pi\)
−0.323758 + 0.946140i \(0.604946\pi\)
\(762\) −10.6290 −0.385048
\(763\) −7.61259 −0.275594
\(764\) 36.6207 1.32489
\(765\) 0 0
\(766\) 47.9177 1.73134
\(767\) −22.3515 −0.807065
\(768\) −19.2622 −0.695064
\(769\) −24.8820 −0.897269 −0.448635 0.893715i \(-0.648090\pi\)
−0.448635 + 0.893715i \(0.648090\pi\)
\(770\) 0 0
\(771\) 14.9149 0.537145
\(772\) −0.436357 −0.0157048
\(773\) 34.9700 1.25778 0.628892 0.777492i \(-0.283509\pi\)
0.628892 + 0.777492i \(0.283509\pi\)
\(774\) 21.0716 0.757402
\(775\) 0 0
\(776\) −6.02448 −0.216266
\(777\) −7.55220 −0.270933
\(778\) 7.07158 0.253528
\(779\) −2.98359 −0.106898
\(780\) 0 0
\(781\) 5.01641 0.179501
\(782\) −24.8195 −0.887544
\(783\) 3.17313 0.113398
\(784\) −4.44364 −0.158701
\(785\) 0 0
\(786\) −7.74173 −0.276138
\(787\) 15.1648 0.540566 0.270283 0.962781i \(-0.412883\pi\)
0.270283 + 0.962781i \(0.412883\pi\)
\(788\) −32.2989 −1.15060
\(789\) 2.92319 0.104068
\(790\) 0 0
\(791\) 7.70892 0.274097
\(792\) 0.491797 0.0174752
\(793\) −6.34625 −0.225362
\(794\) −8.83210 −0.313440
\(795\) 0 0
\(796\) 17.2335 0.610826
\(797\) −8.24470 −0.292042 −0.146021 0.989281i \(-0.546647\pi\)
−0.146021 + 0.989281i \(0.546647\pi\)
\(798\) 0.616763 0.0218332
\(799\) −55.2497 −1.95459
\(800\) 0 0
\(801\) −1.74173 −0.0615410
\(802\) −14.0000 −0.494357
\(803\) −4.82687 −0.170337
\(804\) −4.65269 −0.164088
\(805\) 0 0
\(806\) 56.7065 1.99740
\(807\) −11.9672 −0.421265
\(808\) −2.40354 −0.0845564
\(809\) 29.8433 1.04923 0.524617 0.851338i \(-0.324209\pi\)
0.524617 + 0.851338i \(0.324209\pi\)
\(810\) 0 0
\(811\) −16.6321 −0.584032 −0.292016 0.956413i \(-0.594326\pi\)
−0.292016 + 0.956413i \(0.594326\pi\)
\(812\) −5.53996 −0.194414
\(813\) 20.3187 0.712607
\(814\) −14.6168 −0.512317
\(815\) 0 0
\(816\) −30.4587 −1.06627
\(817\) −3.46944 −0.121380
\(818\) 10.2723 0.359162
\(819\) 3.17313 0.110878
\(820\) 0 0
\(821\) 30.3327 1.05862 0.529309 0.848429i \(-0.322451\pi\)
0.529309 + 0.848429i \(0.322451\pi\)
\(822\) 30.2171 1.05394
\(823\) 18.4067 0.641616 0.320808 0.947144i \(-0.396046\pi\)
0.320808 + 0.947144i \(0.396046\pi\)
\(824\) −0.313441 −0.0109192
\(825\) 0 0
\(826\) −13.6332 −0.474359
\(827\) −45.4559 −1.58066 −0.790328 0.612684i \(-0.790090\pi\)
−0.790328 + 0.612684i \(0.790090\pi\)
\(828\) 3.26634 0.113513
\(829\) −20.3463 −0.706655 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(830\) 0 0
\(831\) 18.0552 0.626327
\(832\) −18.5769 −0.644040
\(833\) −6.85446 −0.237493
\(834\) 17.4506 0.604266
\(835\) 0 0
\(836\) 0.556364 0.0192423
\(837\) −9.23353 −0.319157
\(838\) −22.1086 −0.763728
\(839\) 16.1812 0.558637 0.279318 0.960199i \(-0.409891\pi\)
0.279318 + 0.960199i \(0.409891\pi\)
\(840\) 0 0
\(841\) −18.9313 −0.652802
\(842\) −53.0593 −1.82855
\(843\) −27.2939 −0.940053
\(844\) −8.09632 −0.278687
\(845\) 0 0
\(846\) 15.6004 0.536351
\(847\) 1.00000 0.0343604
\(848\) 2.25827 0.0775493
\(849\) 29.8901 1.02583
\(850\) 0 0
\(851\) 14.1291 0.484341
\(852\) 8.75814 0.300049
\(853\) 20.9836 0.718465 0.359232 0.933248i \(-0.383039\pi\)
0.359232 + 0.933248i \(0.383039\pi\)
\(854\) −3.87086 −0.132458
\(855\) 0 0
\(856\) 0.470162 0.0160698
\(857\) −4.79095 −0.163656 −0.0818279 0.996646i \(-0.526076\pi\)
−0.0818279 + 0.996646i \(0.526076\pi\)
\(858\) 6.14137 0.209663
\(859\) −9.96719 −0.340076 −0.170038 0.985438i \(-0.554389\pi\)
−0.170038 + 0.985438i \(0.554389\pi\)
\(860\) 0 0
\(861\) −9.36266 −0.319079
\(862\) −4.42412 −0.150686
\(863\) −25.5470 −0.869629 −0.434814 0.900520i \(-0.643186\pi\)
−0.434814 + 0.900520i \(0.643186\pi\)
\(864\) 7.61676 0.259128
\(865\) 0 0
\(866\) −61.1044 −2.07641
\(867\) −29.9836 −1.01830
\(868\) 16.1208 0.547176
\(869\) −5.01641 −0.170170
\(870\) 0 0
\(871\) 8.45614 0.286525
\(872\) 3.74385 0.126783
\(873\) 12.2499 0.414597
\(874\) −1.15388 −0.0390306
\(875\) 0 0
\(876\) −8.42723 −0.284730
\(877\) 50.9893 1.72179 0.860893 0.508787i \(-0.169906\pi\)
0.860893 + 0.508787i \(0.169906\pi\)
\(878\) −35.8586 −1.21017
\(879\) −4.34625 −0.146596
\(880\) 0 0
\(881\) 32.1895 1.08449 0.542246 0.840219i \(-0.317574\pi\)
0.542246 + 0.840219i \(0.317574\pi\)
\(882\) 1.93543 0.0651694
\(883\) −36.6154 −1.23221 −0.616104 0.787665i \(-0.711290\pi\)
−0.616104 + 0.787665i \(0.711290\pi\)
\(884\) −37.9734 −1.27718
\(885\) 0 0
\(886\) −55.0325 −1.84885
\(887\) 48.2004 1.61841 0.809206 0.587525i \(-0.199897\pi\)
0.809206 + 0.587525i \(0.199897\pi\)
\(888\) 3.71414 0.124639
\(889\) 5.49180 0.184189
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 32.0307 1.07247
\(893\) −2.56860 −0.0859550
\(894\) 10.0757 0.336983
\(895\) 0 0
\(896\) 3.90262 0.130377
\(897\) −5.93649 −0.198214
\(898\) −11.0060 −0.367273
\(899\) −29.2992 −0.977181
\(900\) 0 0
\(901\) 3.48346 0.116051
\(902\) −18.1208 −0.603357
\(903\) −10.8873 −0.362306
\(904\) −3.79122 −0.126094
\(905\) 0 0
\(906\) −12.0963 −0.401873
\(907\) −22.9013 −0.760425 −0.380212 0.924899i \(-0.624149\pi\)
−0.380212 + 0.924899i \(0.624149\pi\)
\(908\) −0.661814 −0.0219631
\(909\) 4.88727 0.162101
\(910\) 0 0
\(911\) −36.0552 −1.19456 −0.597281 0.802032i \(-0.703752\pi\)
−0.597281 + 0.802032i \(0.703752\pi\)
\(912\) −1.41605 −0.0468901
\(913\) 3.52461 0.116648
\(914\) 25.1762 0.832756
\(915\) 0 0
\(916\) 43.5470 1.43883
\(917\) 4.00000 0.132092
\(918\) 13.2663 0.437854
\(919\) −28.3379 −0.934782 −0.467391 0.884051i \(-0.654806\pi\)
−0.467391 + 0.884051i \(0.654806\pi\)
\(920\) 0 0
\(921\) −20.7581 −0.684004
\(922\) −12.3463 −0.406602
\(923\) −15.9177 −0.523937
\(924\) 1.74590 0.0574358
\(925\) 0 0
\(926\) −65.9851 −2.16841
\(927\) 0.637339 0.0209329
\(928\) 24.1690 0.793385
\(929\) 5.08514 0.166838 0.0834191 0.996515i \(-0.473416\pi\)
0.0834191 + 0.996515i \(0.473416\pi\)
\(930\) 0 0
\(931\) −0.318669 −0.0104440
\(932\) −40.9424 −1.34111
\(933\) −8.00000 −0.261908
\(934\) −35.1250 −1.14932
\(935\) 0 0
\(936\) −1.56053 −0.0510076
\(937\) −32.2088 −1.05222 −0.526108 0.850418i \(-0.676349\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(938\) 5.15778 0.168407
\(939\) 8.12914 0.265284
\(940\) 0 0
\(941\) 32.7805 1.06861 0.534307 0.845291i \(-0.320573\pi\)
0.534307 + 0.845291i \(0.320573\pi\)
\(942\) −35.0716 −1.14269
\(943\) 17.5163 0.570408
\(944\) 31.3009 1.01876
\(945\) 0 0
\(946\) −21.0716 −0.685096
\(947\) 27.3627 0.889167 0.444584 0.895737i \(-0.353352\pi\)
0.444584 + 0.895737i \(0.353352\pi\)
\(948\) −8.75814 −0.284451
\(949\) 15.3163 0.497188
\(950\) 0 0
\(951\) −19.9917 −0.648274
\(952\) 3.37100 0.109255
\(953\) 24.6894 0.799768 0.399884 0.916566i \(-0.369050\pi\)
0.399884 + 0.916566i \(0.369050\pi\)
\(954\) −0.983593 −0.0318450
\(955\) 0 0
\(956\) 8.86359 0.286669
\(957\) −3.17313 −0.102573
\(958\) −82.7393 −2.67319
\(959\) −15.6126 −0.504157
\(960\) 0 0
\(961\) 54.2580 1.75026
\(962\) 46.3808 1.49538
\(963\) −0.956008 −0.0308069
\(964\) 33.6852 1.08493
\(965\) 0 0
\(966\) −3.62093 −0.116502
\(967\) −21.3298 −0.685922 −0.342961 0.939350i \(-0.611430\pi\)
−0.342961 + 0.939350i \(0.611430\pi\)
\(968\) −0.491797 −0.0158069
\(969\) −2.18431 −0.0701700
\(970\) 0 0
\(971\) 28.4946 0.914436 0.457218 0.889355i \(-0.348846\pi\)
0.457218 + 0.889355i \(0.348846\pi\)
\(972\) −1.74590 −0.0559997
\(973\) −9.01641 −0.289053
\(974\) −51.8521 −1.66145
\(975\) 0 0
\(976\) 8.88727 0.284475
\(977\) 19.8157 0.633960 0.316980 0.948432i \(-0.397331\pi\)
0.316980 + 0.948432i \(0.397331\pi\)
\(978\) −5.15778 −0.164928
\(979\) 1.74173 0.0556659
\(980\) 0 0
\(981\) −7.61259 −0.243051
\(982\) 61.3171 1.95671
\(983\) 53.7745 1.71514 0.857571 0.514366i \(-0.171973\pi\)
0.857571 + 0.514366i \(0.171973\pi\)
\(984\) 4.60453 0.146787
\(985\) 0 0
\(986\) 42.0958 1.34060
\(987\) −8.06040 −0.256565
\(988\) −1.76541 −0.0561653
\(989\) 20.3686 0.647684
\(990\) 0 0
\(991\) 49.7693 1.58097 0.790487 0.612479i \(-0.209827\pi\)
0.790487 + 0.612479i \(0.209827\pi\)
\(992\) −70.3296 −2.23297
\(993\) −17.0164 −0.539999
\(994\) −9.70892 −0.307948
\(995\) 0 0
\(996\) 6.15361 0.194985
\(997\) −0.659696 −0.0208928 −0.0104464 0.999945i \(-0.503325\pi\)
−0.0104464 + 0.999945i \(0.503325\pi\)
\(998\) −46.6943 −1.47808
\(999\) −7.55220 −0.238941
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 5775.2.a.bp.1.3 3
5.4 even 2 231.2.a.e.1.1 3
15.14 odd 2 693.2.a.l.1.3 3
20.19 odd 2 3696.2.a.bo.1.3 3
35.34 odd 2 1617.2.a.t.1.1 3
55.54 odd 2 2541.2.a.bg.1.3 3
105.104 even 2 4851.2.a.bi.1.3 3
165.164 even 2 7623.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 5.4 even 2
693.2.a.l.1.3 3 15.14 odd 2
1617.2.a.t.1.1 3 35.34 odd 2
2541.2.a.bg.1.3 3 55.54 odd 2
3696.2.a.bo.1.3 3 20.19 odd 2
4851.2.a.bi.1.3 3 105.104 even 2
5775.2.a.bp.1.3 3 1.1 even 1 trivial
7623.2.a.cd.1.1 3 165.164 even 2