Properties

Label 693.2.a.l.1.3
Level $693$
Weight $2$
Character 693.1
Self dual yes
Analytic conductor $5.534$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.53363286007\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 693.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} -1.00000 q^{7} -0.491797 q^{8} +O(q^{10})\) \(q+1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} -1.00000 q^{7} -0.491797 q^{8} -8.10856 q^{10} +1.00000 q^{11} -3.17313 q^{13} -1.93543 q^{14} -4.44364 q^{16} -6.85446 q^{17} -0.318669 q^{19} -7.31450 q^{20} +1.93543 q^{22} +1.87086 q^{23} +12.5522 q^{25} -6.14137 q^{26} -1.74590 q^{28} +3.17313 q^{29} +9.23353 q^{31} -7.61676 q^{32} -13.2663 q^{34} +4.18953 q^{35} -7.55220 q^{37} -0.616763 q^{38} +2.06040 q^{40} -9.36266 q^{41} -10.8873 q^{43} +1.74590 q^{44} +3.62093 q^{46} +8.06040 q^{47} +1.00000 q^{49} +24.2939 q^{50} -5.53996 q^{52} -0.508203 q^{53} -4.18953 q^{55} +0.491797 q^{56} +6.14137 q^{58} +7.04399 q^{59} -2.00000 q^{61} +17.8709 q^{62} -5.85446 q^{64} +13.2939 q^{65} -2.66492 q^{67} -11.9672 q^{68} +8.10856 q^{70} +5.01641 q^{71} -4.82687 q^{73} -14.6168 q^{74} -0.556364 q^{76} -1.00000 q^{77} +5.01641 q^{79} +18.6168 q^{80} -18.1208 q^{82} -3.52461 q^{83} +28.7170 q^{85} -21.0716 q^{86} -0.491797 q^{88} +1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} +15.6004 q^{94} +1.33508 q^{95} -12.2499 q^{97} +1.93543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} + O(q^{10}) \) \( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29} - 2 q^{31} - 8 q^{32} - 4 q^{34} + 4 q^{35} + 13 q^{38} - 18 q^{40} - 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 3 q^{49} + 19 q^{50} - 29 q^{52} - 4 q^{55} + 3 q^{56} - q^{58} - 6 q^{61} + 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} - 42 q^{68} + 11 q^{70} + 12 q^{71} - 20 q^{73} - 29 q^{74} - 11 q^{76} - 3 q^{77} + 12 q^{79} + 41 q^{80} - 6 q^{82} - 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} - 26 q^{89} + 4 q^{91} - 26 q^{92} + 35 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98} + O(q^{100}) \)

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 \(0.260117\pi\)
\(3\) 0 0
\(4\) 1.74590 0.872949
\(5\) −4.18953 −1.87362 −0.936808 0.349843i \(-0.886235\pi\)
−0.936808 + 0.349843i \(0.886235\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.491797 −0.173876
\(9\) 0 0
\(10\) −8.10856 −2.56415
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.17313 −0.880067 −0.440034 0.897981i \(-0.645033\pi\)
−0.440034 + 0.897981i \(0.645033\pi\)
\(14\) −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 \(0.812360\pi\)
\(18\) 0 0
\(19\) −0.318669 −0.0731078 −0.0365539 0.999332i \(-0.511638\pi\)
−0.0365539 + 0.999332i \(0.511638\pi\)
\(20\) −7.31450 −1.63557
\(21\) 0 0
\(22\) 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 0
\(25\) 12.5522 2.51044
\(26\) −6.14137 −1.20442
\(27\) 0 0
\(28\) −1.74590 −0.329944
\(29\) 3.17313 0.589235 0.294617 0.955615i \(-0.404808\pi\)
0.294617 + 0.955615i \(0.404808\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −13.2663 −2.27516
\(35\) 4.18953 0.708161
\(36\) 0 0
\(37\) −7.55220 −1.24157 −0.620787 0.783980i \(-0.713187\pi\)
−0.620787 + 0.783980i \(0.713187\pi\)
\(38\) −0.616763 −0.100052
\(39\) 0 0
\(40\) 2.06040 0.325778
\(41\) −9.36266 −1.46220 −0.731101 0.682269i \(-0.760993\pi\)
−0.731101 + 0.682269i \(0.760993\pi\)
\(42\) 0 0
\(43\) −10.8873 −1.66029 −0.830147 0.557545i \(-0.811743\pi\)
−0.830147 + 0.557545i \(0.811743\pi\)
\(44\) 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 \(0.299969\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 24.2939 3.43568
\(51\) 0 0
\(52\) −5.53996 −0.768254
\(53\) −0.508203 −0.0698071 −0.0349036 0.999391i \(-0.511112\pi\)
−0.0349036 + 0.999391i \(0.511112\pi\)
\(54\) 0 0
\(55\) −4.18953 −0.564917
\(56\) 0.491797 0.0657191
\(57\) 0 0
\(58\) 6.14137 0.806402
\(59\) 7.04399 0.917050 0.458525 0.888682i \(-0.348378\pi\)
0.458525 + 0.888682i \(0.348378\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −5.85446 −0.731807
\(65\) 13.2939 1.64891
\(66\) 0 0
\(67\) −2.66492 −0.325572 −0.162786 0.986661i \(-0.552048\pi\)
−0.162786 + 0.986661i \(0.552048\pi\)
\(68\) −11.9672 −1.45123
\(69\) 0 0
\(70\) 8.10856 0.969158
\(71\) 5.01641 0.595338 0.297669 0.954669i \(-0.403791\pi\)
0.297669 + 0.954669i \(0.403791\pi\)
\(72\) 0 0
\(73\) −4.82687 −0.564943 −0.282471 0.959276i \(-0.591154\pi\)
−0.282471 + 0.959276i \(0.591154\pi\)
\(74\) −14.6168 −1.69916
\(75\) 0 0
\(76\) −0.556364 −0.0638194
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 5.01641 0.564390 0.282195 0.959357i \(-0.408938\pi\)
0.282195 + 0.959357i \(0.408938\pi\)
\(80\) 18.6168 2.08142
\(81\) 0 0
\(82\) −18.1208 −2.00111
\(83\) −3.52461 −0.386876 −0.193438 0.981112i \(-0.561964\pi\)
−0.193438 + 0.981112i \(0.561964\pi\)
\(84\) 0 0
\(85\) 28.7170 3.11479
\(86\) −21.0716 −2.27221
\(87\) 0 0
\(88\) −0.491797 −0.0524257
\(89\) 1.74173 0.184623 0.0923115 0.995730i \(-0.470574\pi\)
0.0923115 + 0.995730i \(0.470574\pi\)
\(90\) 0 0
\(91\) 3.17313 0.332634
\(92\) 3.26634 0.340539
\(93\) 0 0
\(94\) 15.6004 1.60905
\(95\) 1.33508 0.136976
\(96\) 0 0
\(97\) −12.2499 −1.24379 −0.621896 0.783100i \(-0.713637\pi\)
−0.621896 + 0.783100i \(0.713637\pi\)
\(98\) 1.93543 0.195508
\(99\) 0 0
\(100\) 21.9149 2.19149
\(101\) −4.88727 −0.486302 −0.243151 0.969988i \(-0.578181\pi\)
−0.243151 + 0.969988i \(0.578181\pi\)
\(102\) 0 0
\(103\) −0.637339 −0.0627988 −0.0313994 0.999507i \(-0.509996\pi\)
−0.0313994 + 0.999507i \(0.509996\pi\)
\(104\) 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 \(0.514714\pi\)
\(108\) 0 0
\(109\) −7.61259 −0.729154 −0.364577 0.931173i \(-0.618786\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(110\) −8.10856 −0.773121
\(111\) 0 0
\(112\) 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 0
\(115\) −7.83805 −0.730902
\(116\) 5.53996 0.514372
\(117\) 0 0
\(118\) 13.6332 1.25504
\(119\) 6.85446 0.628347
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.87086 −0.350452
\(123\) 0 0
\(124\) 16.1208 1.44769
\(125\) −31.6402 −2.82998
\(126\) 0 0
\(127\) −5.49180 −0.487318 −0.243659 0.969861i \(-0.578348\pi\)
−0.243659 + 0.969861i \(0.578348\pi\)
\(128\) 3.90262 0.344946
\(129\) 0 0
\(130\) 25.7295 2.25663
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 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 \(0.732395\pi\)
\(138\) 0 0
\(139\) −9.01641 −0.764762 −0.382381 0.924005i \(-0.624896\pi\)
−0.382381 + 0.924005i \(0.624896\pi\)
\(140\) 7.31450 0.618188
\(141\) 0 0
\(142\) 9.70892 0.814754
\(143\) −3.17313 −0.265350
\(144\) 0 0
\(145\) −13.2939 −1.10400
\(146\) −9.34209 −0.773157
\(147\) 0 0
\(148\) −13.1854 −1.08383
\(149\) 5.20594 0.426487 0.213244 0.976999i \(-0.431597\pi\)
0.213244 + 0.976999i \(0.431597\pi\)
\(150\) 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
\(153\) 0 0
\(154\) −1.93543 −0.155962
\(155\) −38.6842 −3.10719
\(156\) 0 0
\(157\) −18.1208 −1.44620 −0.723099 0.690745i \(-0.757283\pi\)
−0.723099 + 0.690745i \(0.757283\pi\)
\(158\) 9.70892 0.772400
\(159\) 0 0
\(160\) 31.9107 2.52276
\(161\) −1.87086 −0.147445
\(162\) 0 0
\(163\) −2.66492 −0.208733 −0.104366 0.994539i \(-0.533281\pi\)
−0.104366 + 0.994539i \(0.533281\pi\)
\(164\) −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 0
\(169\) −2.93126 −0.225482
\(170\) 55.5798 4.26277
\(171\) 0 0
\(172\) −19.0081 −1.44935
\(173\) 24.8461 1.88902 0.944508 0.328489i \(-0.106539\pi\)
0.944508 + 0.328489i \(0.106539\pi\)
\(174\) 0 0
\(175\) −12.5522 −0.948857
\(176\) −4.44364 −0.334952
\(177\) 0 0
\(178\) 3.37100 0.252667
\(179\) 12.7581 0.953588 0.476794 0.879015i \(-0.341799\pi\)
0.476794 + 0.879015i \(0.341799\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −0.920085 −0.0678296
\(185\) 31.6402 2.32623
\(186\) 0 0
\(187\) −6.85446 −0.501248
\(188\) 14.0726 1.02635
\(189\) 0 0
\(190\) 2.58395 0.187459
\(191\) −20.9753 −1.51772 −0.758858 0.651256i \(-0.774242\pi\)
−0.758858 + 0.651256i \(0.774242\pi\)
\(192\) 0 0
\(193\) 0.249933 0.0179905 0.00899527 0.999960i \(-0.497137\pi\)
0.00899527 + 0.999960i \(0.497137\pi\)
\(194\) −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 \(0.729033\pi\)
\(198\) 0 0
\(199\) 9.87086 0.699727 0.349864 0.936801i \(-0.386228\pi\)
0.349864 + 0.936801i \(0.386228\pi\)
\(200\) −6.17313 −0.436506
\(201\) 0 0
\(202\) −9.45898 −0.665532
\(203\) −3.17313 −0.222710
\(204\) 0 0
\(205\) 39.2252 2.73961
\(206\) −1.23353 −0.0859438
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) −1.85029 −0.126483
\(215\) 45.6126 3.11075
\(216\) 0 0
\(217\) −9.23353 −0.626813
\(218\) −14.7337 −0.997889
\(219\) 0 0
\(220\) −7.31450 −0.493144
\(221\) 21.7501 1.46307
\(222\) 0 0
\(223\) −18.3463 −1.22856 −0.614278 0.789090i \(-0.710553\pi\)
−0.614278 + 0.789090i \(0.710553\pi\)
\(224\) 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 0
\(229\) 24.9424 1.64824 0.824121 0.566413i \(-0.191669\pi\)
0.824121 + 0.566413i \(0.191669\pi\)
\(230\) −15.1700 −1.00028
\(231\) 0 0
\(232\) −1.56053 −0.102454
\(233\) −23.4506 −1.53630 −0.768151 0.640268i \(-0.778823\pi\)
−0.768151 + 0.640268i \(0.778823\pi\)
\(234\) 0 0
\(235\) −33.7693 −2.20287
\(236\) 12.2981 0.800538
\(237\) 0 0
\(238\) 13.2663 0.859929
\(239\) −5.07681 −0.328391 −0.164196 0.986428i \(-0.552503\pi\)
−0.164196 + 0.986428i \(0.552503\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −3.49180 −0.223539
\(245\) −4.18953 −0.267660
\(246\) 0 0
\(247\) 1.01118 0.0643397
\(248\) −4.54102 −0.288355
\(249\) 0 0
\(250\) −61.2374 −3.87299
\(251\) −23.8021 −1.50238 −0.751188 0.660088i \(-0.770519\pi\)
−0.751188 + 0.660088i \(0.770519\pi\)
\(252\) 0 0
\(253\) 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 \(0.654011\pi\)
\(258\) 0 0
\(259\) 7.55220 0.469271
\(260\) 23.2098 1.43941
\(261\) 0 0
\(262\) −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 0
\(265\) 2.12914 0.130792
\(266\) 0.616763 0.0378162
\(267\) 0 0
\(268\) −4.65269 −0.284208
\(269\) −11.9672 −0.729652 −0.364826 0.931076i \(-0.618872\pi\)
−0.364826 + 0.931076i \(0.618872\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −30.2171 −1.82548
\(275\) 12.5522 0.756926
\(276\) 0 0
\(277\) 18.0552 1.08483 0.542415 0.840111i \(-0.317510\pi\)
0.542415 + 0.840111i \(0.317510\pi\)
\(278\) −17.4506 −1.04662
\(279\) 0 0
\(280\) −2.06040 −0.123132
\(281\) −27.2939 −1.62822 −0.814110 0.580711i \(-0.802774\pi\)
−0.814110 + 0.580711i \(0.802774\pi\)
\(282\) 0 0
\(283\) 29.8901 1.77678 0.888391 0.459087i \(-0.151823\pi\)
0.888391 + 0.459087i \(0.151823\pi\)
\(284\) 8.75814 0.519700
\(285\) 0 0
\(286\) −6.14137 −0.363147
\(287\) 9.36266 0.552660
\(288\) 0 0
\(289\) 29.9836 1.76374
\(290\) −25.7295 −1.51089
\(291\) 0 0
\(292\) −8.42723 −0.493166
\(293\) 4.34625 0.253911 0.126955 0.991908i \(-0.459479\pi\)
0.126955 + 0.991908i \(0.459479\pi\)
\(294\) 0 0
\(295\) −29.5110 −1.71820
\(296\) 3.71414 0.215880
\(297\) 0 0
\(298\) 10.0757 0.583672
\(299\) −5.93649 −0.343316
\(300\) 0 0
\(301\) 10.8873 0.627532
\(302\) 12.0963 0.696065
\(303\) 0 0
\(304\) 1.41605 0.0812161
\(305\) 8.37907 0.479784
\(306\) 0 0
\(307\) −20.7581 −1.18473 −0.592365 0.805670i \(-0.701806\pi\)
−0.592365 + 0.805670i \(0.701806\pi\)
\(308\) −1.74590 −0.0994818
\(309\) 0 0
\(310\) −74.8706 −4.25236
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 8.12914 0.459486 0.229743 0.973251i \(-0.426211\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(314\) −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 0
\(319\) 3.17313 0.177661
\(320\) 24.5275 1.37113
\(321\) 0 0
\(322\) −3.62093 −0.201787
\(323\) 2.18431 0.121538
\(324\) 0 0
\(325\) −39.8297 −2.20935
\(326\) −5.15778 −0.285663
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 21.5714 1.18034
\(335\) 11.1648 0.609998
\(336\) 0 0
\(337\) 1.52461 0.0830508 0.0415254 0.999137i \(-0.486778\pi\)
0.0415254 + 0.999137i \(0.486778\pi\)
\(338\) −5.67326 −0.308585
\(339\) 0 0
\(340\) 50.1369 2.71906
\(341\) 9.23353 0.500023
\(342\) 0 0
\(343\) −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 \(0.655169\pi\)
\(348\) 0 0
\(349\) 6.85969 0.367191 0.183595 0.983002i \(-0.441226\pi\)
0.183595 + 0.983002i \(0.441226\pi\)
\(350\) −24.2939 −1.29856
\(351\) 0 0
\(352\) −7.61676 −0.405975
\(353\) 31.9313 1.69953 0.849765 0.527162i \(-0.176744\pi\)
0.849765 + 0.527162i \(0.176744\pi\)
\(354\) 0 0
\(355\) −21.0164 −1.11544
\(356\) 3.04088 0.161166
\(357\) 0 0
\(358\) 24.6925 1.30504
\(359\) 24.4342 1.28959 0.644795 0.764356i \(-0.276943\pi\)
0.644795 + 0.764356i \(0.276943\pi\)
\(360\) 0 0
\(361\) −18.8984 −0.994655
\(362\) −6.25827 −0.328927
\(363\) 0 0
\(364\) 5.53996 0.290373
\(365\) 20.2223 1.05849
\(366\) 0 0
\(367\) 17.0716 0.891129 0.445565 0.895250i \(-0.353003\pi\)
0.445565 + 0.895250i \(0.353003\pi\)
\(368\) −8.31344 −0.433368
\(369\) 0 0
\(370\) 61.2374 3.18358
\(371\) 0.508203 0.0263846
\(372\) 0 0
\(373\) 3.17836 0.164569 0.0822845 0.996609i \(-0.473778\pi\)
0.0822845 + 0.996609i \(0.473778\pi\)
\(374\) −13.2663 −0.685986
\(375\) 0 0
\(376\) −3.96408 −0.204432
\(377\) −10.0687 −0.518566
\(378\) 0 0
\(379\) 3.93960 0.202364 0.101182 0.994868i \(-0.467738\pi\)
0.101182 + 0.994868i \(0.467738\pi\)
\(380\) 2.33091 0.119573
\(381\) 0 0
\(382\) −40.5962 −2.07708
\(383\) 24.7581 1.26508 0.632541 0.774527i \(-0.282012\pi\)
0.632541 + 0.774527i \(0.282012\pi\)
\(384\) 0 0
\(385\) 4.18953 0.213518
\(386\) 0.483728 0.0246211
\(387\) 0 0
\(388\) −21.3871 −1.08577
\(389\) −3.65375 −0.185252 −0.0926261 0.995701i \(-0.529526\pi\)
−0.0926261 + 0.995701i \(0.529526\pi\)
\(390\) 0 0
\(391\) −12.8238 −0.648526
\(392\) −0.491797 −0.0248395
\(393\) 0 0
\(394\) −35.8052 −1.80384
\(395\) −21.0164 −1.05745
\(396\) 0 0
\(397\) 4.56337 0.229029 0.114515 0.993422i \(-0.463469\pi\)
0.114515 + 0.993422i \(0.463469\pi\)
\(398\) 19.1044 0.957617
\(399\) 0 0
\(400\) −55.7774 −2.78887
\(401\) 7.23353 0.361225 0.180613 0.983554i \(-0.442192\pi\)
0.180613 + 0.983554i \(0.442192\pi\)
\(402\) 0 0
\(403\) −29.2992 −1.45949
\(404\) −8.53268 −0.424517
\(405\) 0 0
\(406\) −6.14137 −0.304791
\(407\) −7.55220 −0.374348
\(408\) 0 0
\(409\) 5.30749 0.262439 0.131219 0.991353i \(-0.458111\pi\)
0.131219 + 0.991353i \(0.458111\pi\)
\(410\) 75.9177 3.74931
\(411\) 0 0
\(412\) −1.11273 −0.0548202
\(413\) −7.04399 −0.346612
\(414\) 0 0
\(415\) 14.7665 0.724858
\(416\) 24.1690 1.18498
\(417\) 0 0
\(418\) −0.616763 −0.0301669
\(419\) 11.4231 0.558053 0.279026 0.960283i \(-0.409988\pi\)
0.279026 + 0.960283i \(0.409988\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 0.249933 0.0121378
\(425\) −86.0385 −4.17348
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −1.66909 −0.0806786
\(429\) 0 0
\(430\) 88.2801 4.25724
\(431\) 2.28586 0.110106 0.0550529 0.998483i \(-0.482467\pi\)
0.0550529 + 0.998483i \(0.482467\pi\)
\(432\) 0 0
\(433\) 31.5714 1.51723 0.758613 0.651541i \(-0.225877\pi\)
0.758613 + 0.651541i \(0.225877\pi\)
\(434\) −17.8709 −0.857829
\(435\) 0 0
\(436\) −13.2908 −0.636515
\(437\) −0.596187 −0.0285195
\(438\) 0 0
\(439\) −18.5275 −0.884267 −0.442133 0.896949i \(-0.645778\pi\)
−0.442133 + 0.896949i \(0.645778\pi\)
\(440\) 2.06040 0.0982257
\(441\) 0 0
\(442\) 42.0958 2.00229
\(443\) −28.4342 −1.35095 −0.675476 0.737382i \(-0.736062\pi\)
−0.675476 + 0.737382i \(0.736062\pi\)
\(444\) 0 0
\(445\) −7.29703 −0.345913
\(446\) −35.5079 −1.68135
\(447\) 0 0
\(448\) 5.85446 0.276597
\(449\) 5.68656 0.268365 0.134183 0.990957i \(-0.457159\pi\)
0.134183 + 0.990957i \(0.457159\pi\)
\(450\) 0 0
\(451\) −9.36266 −0.440871
\(452\) 13.4590 0.633057
\(453\) 0 0
\(454\) −0.733661 −0.0344324
\(455\) −13.2939 −0.623229
\(456\) 0 0
\(457\) −13.0081 −0.608492 −0.304246 0.952594i \(-0.598404\pi\)
−0.304246 + 0.952594i \(0.598404\pi\)
\(458\) 48.2744 2.25571
\(459\) 0 0
\(460\) −13.6844 −0.638040
\(461\) 6.37907 0.297103 0.148551 0.988905i \(-0.452539\pi\)
0.148551 + 0.988905i \(0.452539\pi\)
\(462\) 0 0
\(463\) 34.0932 1.58445 0.792223 0.610232i \(-0.208924\pi\)
0.792223 + 0.610232i \(0.208924\pi\)
\(464\) −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 \(0.637936\pi\)
\(468\) 0 0
\(469\) 2.66492 0.123055
\(470\) −65.3582 −3.01475
\(471\) 0 0
\(472\) −3.46421 −0.159453
\(473\) −10.8873 −0.500597
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 11.9672 0.548515
\(477\) 0 0
\(478\) −9.82581 −0.449422
\(479\) 42.7498 1.95329 0.976644 0.214864i \(-0.0689307\pi\)
0.976644 + 0.214864i \(0.0689307\pi\)
\(480\) 0 0
\(481\) 23.9641 1.09267
\(482\) 37.3421 1.70089
\(483\) 0 0
\(484\) 1.74590 0.0793590
\(485\) 51.3215 2.33039
\(486\) 0 0
\(487\) 26.7909 1.21401 0.607007 0.794697i \(-0.292370\pi\)
0.607007 + 0.794697i \(0.292370\pi\)
\(488\) 0.983593 0.0445252
\(489\) 0 0
\(490\) −8.10856 −0.366307
\(491\) −31.6813 −1.42976 −0.714879 0.699248i \(-0.753518\pi\)
−0.714879 + 0.699248i \(0.753518\pi\)
\(492\) 0 0
\(493\) −21.7501 −0.979574
\(494\) 1.95707 0.0880526
\(495\) 0 0
\(496\) −41.0304 −1.84232
\(497\) −5.01641 −0.225017
\(498\) 0 0
\(499\) −24.1260 −1.08003 −0.540015 0.841656i \(-0.681581\pi\)
−0.540015 + 0.841656i \(0.681581\pi\)
\(500\) −55.2405 −2.47043
\(501\) 0 0
\(502\) −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 0
\(505\) 20.4754 0.911143
\(506\) 3.62093 0.160970
\(507\) 0 0
\(508\) −9.58812 −0.425404
\(509\) −28.2088 −1.25033 −0.625166 0.780492i \(-0.714969\pi\)
−0.625166 + 0.780492i \(0.714969\pi\)
\(510\) 0 0
\(511\) 4.82687 0.213528
\(512\) 29.4754 1.30264
\(513\) 0 0
\(514\) −28.8667 −1.27326
\(515\) 2.67015 0.117661
\(516\) 0 0
\(517\) 8.06040 0.354496
\(518\) 14.6168 0.642224
\(519\) 0 0
\(520\) −6.53791 −0.286706
\(521\) 33.7610 1.47910 0.739548 0.673104i \(-0.235039\pi\)
0.739548 + 0.673104i \(0.235039\pi\)
\(522\) 0 0
\(523\) 12.8185 0.560515 0.280258 0.959925i \(-0.409580\pi\)
0.280258 + 0.959925i \(0.409580\pi\)
\(524\) −6.98359 −0.305080
\(525\) 0 0
\(526\) −5.65765 −0.246685
\(527\) −63.2908 −2.75699
\(528\) 0 0
\(529\) −19.4999 −0.847820
\(530\) 4.12080 0.178996
\(531\) 0 0
\(532\) 0.556364 0.0241215
\(533\) 29.7089 1.28684
\(534\) 0 0
\(535\) 4.00523 0.173161
\(536\) 1.31060 0.0566093
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) 52.2088 2.23843
\(545\) 31.8932 1.36616
\(546\) 0 0
\(547\) 23.4178 1.00127 0.500637 0.865657i \(-0.333099\pi\)
0.500637 + 0.865657i \(0.333099\pi\)
\(548\) −27.2580 −1.16440
\(549\) 0 0
\(550\) 24.2939 1.03590
\(551\) −1.01118 −0.0430776
\(552\) 0 0
\(553\) −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 \(0.453200\pi\)
\(558\) 0 0
\(559\) 34.5467 1.46117
\(560\) −18.6168 −0.786702
\(561\) 0 0
\(562\) −52.8255 −2.22831
\(563\) −11.8381 −0.498914 −0.249457 0.968386i \(-0.580252\pi\)
−0.249457 + 0.968386i \(0.580252\pi\)
\(564\) 0 0
\(565\) −32.2968 −1.35874
\(566\) 57.8503 2.43163
\(567\) 0 0
\(568\) −2.46705 −0.103515
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 18.1208 0.756347
\(575\) 23.4835 0.979328
\(576\) 0 0
\(577\) 17.3215 0.721104 0.360552 0.932739i \(-0.382588\pi\)
0.360552 + 0.932739i \(0.382588\pi\)
\(578\) 58.0312 2.41378
\(579\) 0 0
\(580\) −23.2098 −0.963736
\(581\) 3.52461 0.146225
\(582\) 0 0
\(583\) −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 \(0.695042\pi\)
\(588\) 0 0
\(589\) −2.94244 −0.121241
\(590\) −57.1166 −2.35145
\(591\) 0 0
\(592\) 33.5592 1.37927
\(593\) 24.3463 0.999781 0.499890 0.866089i \(-0.333374\pi\)
0.499890 + 0.866089i \(0.333374\pi\)
\(594\) 0 0
\(595\) −28.7170 −1.17728
\(596\) 9.08904 0.372302
\(597\) 0 0
\(598\) −11.4897 −0.469848
\(599\) −21.0164 −0.858707 −0.429354 0.903136i \(-0.641259\pi\)
−0.429354 + 0.903136i \(0.641259\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 10.9117 0.443993
\(605\) −4.18953 −0.170329
\(606\) 0 0
\(607\) −14.0276 −0.569362 −0.284681 0.958622i \(-0.591888\pi\)
−0.284681 + 0.958622i \(0.591888\pi\)
\(608\) 2.42723 0.0984371
\(609\) 0 0
\(610\) 16.2171 0.656612
\(611\) −25.5767 −1.03472
\(612\) 0 0
\(613\) 18.5306 0.748442 0.374221 0.927339i \(-0.377910\pi\)
0.374221 + 0.927339i \(0.377910\pi\)
\(614\) −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 \(0.515603\pi\)
\(618\) 0 0
\(619\) −30.4259 −1.22292 −0.611460 0.791275i \(-0.709418\pi\)
−0.611460 + 0.791275i \(0.709418\pi\)
\(620\) −67.5386 −2.71242
\(621\) 0 0
\(622\) −15.4835 −0.620830
\(623\) −1.74173 −0.0697809
\(624\) 0 0
\(625\) 69.7966 2.79187
\(626\) 15.7334 0.628833
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 38.6925 1.53668
\(635\) 23.0081 0.913047
\(636\) 0 0
\(637\) −3.17313 −0.125724
\(638\) 6.14137 0.243139
\(639\) 0 0
\(640\) −16.3502 −0.646297
\(641\) −8.56337 −0.338233 −0.169116 0.985596i \(-0.554091\pi\)
−0.169116 + 0.985596i \(0.554091\pi\)
\(642\) 0 0
\(643\) −5.11273 −0.201626 −0.100813 0.994905i \(-0.532144\pi\)
−0.100813 + 0.994905i \(0.532144\pi\)
\(644\) −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 0
\(649\) 7.04399 0.276501
\(650\) −77.0877 −3.02363
\(651\) 0 0
\(652\) −4.65269 −0.182213
\(653\) 47.8953 1.87429 0.937145 0.348941i \(-0.113459\pi\)
0.937145 + 0.348941i \(0.113459\pi\)
\(654\) 0 0
\(655\) 16.7581 0.654795
\(656\) 41.6043 1.62437
\(657\) 0 0
\(658\) −15.6004 −0.608165
\(659\) −8.95601 −0.348877 −0.174438 0.984668i \(-0.555811\pi\)
−0.174438 + 0.984668i \(0.555811\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 1.73339 0.0672686
\(665\) −1.33508 −0.0517720
\(666\) 0 0
\(667\) 5.93649 0.229862
\(668\) 19.4590 0.752891
\(669\) 0 0
\(670\) 21.6087 0.834817
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 32.2499 1.24314 0.621572 0.783357i \(-0.286494\pi\)
0.621572 + 0.783357i \(0.286494\pi\)
\(674\) 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 \(0.321264\pi\)
\(678\) 0 0
\(679\) 12.2499 0.470109
\(680\) −14.1229 −0.541589
\(681\) 0 0
\(682\) 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 0
\(685\) 65.4095 2.49917
\(686\) −1.93543 −0.0738951
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) −33.7745 −1.28206
\(695\) 37.7745 1.43287
\(696\) 0 0
\(697\) 64.1760 2.43084
\(698\) 13.2765 0.502521
\(699\) 0 0
\(700\) −21.9149 −0.828304
\(701\) 19.2747 0.727995 0.363997 0.931400i \(-0.381412\pi\)
0.363997 + 0.931400i \(0.381412\pi\)
\(702\) 0 0
\(703\) 2.40665 0.0907686
\(704\) −5.85446 −0.220648
\(705\) 0 0
\(706\) 61.8008 2.32590
\(707\) 4.88727 0.183805
\(708\) 0 0
\(709\) 9.76098 0.366581 0.183291 0.983059i \(-0.441325\pi\)
0.183291 + 0.983059i \(0.441325\pi\)
\(710\) −40.6758 −1.52654
\(711\) 0 0
\(712\) −0.856577 −0.0321016
\(713\) 17.2747 0.646942
\(714\) 0 0
\(715\) 13.2939 0.497165
\(716\) 22.2744 0.832434
\(717\) 0 0
\(718\) 47.2908 1.76488
\(719\) −14.0276 −0.523141 −0.261570 0.965184i \(-0.584240\pi\)
−0.261570 + 0.965184i \(0.584240\pi\)
\(720\) 0 0
\(721\) 0.637339 0.0237357
\(722\) −36.5767 −1.36124
\(723\) 0 0
\(724\) −5.64541 −0.209810
\(725\) 39.8297 1.47924
\(726\) 0 0
\(727\) −34.3051 −1.27231 −0.636153 0.771563i \(-0.719475\pi\)
−0.636153 + 0.771563i \(0.719475\pi\)
\(728\) −1.56053 −0.0578372
\(729\) 0 0
\(730\) 39.1390 1.44860
\(731\) 74.6263 2.76016
\(732\) 0 0
\(733\) −10.7581 −0.397361 −0.198680 0.980064i \(-0.563666\pi\)
−0.198680 + 0.980064i \(0.563666\pi\)
\(734\) 33.0409 1.21956
\(735\) 0 0
\(736\) −14.2499 −0.525259
\(737\) −2.66492 −0.0981637
\(738\) 0 0
\(739\) −6.38741 −0.234965 −0.117482 0.993075i \(-0.537482\pi\)
−0.117482 + 0.993075i \(0.537482\pi\)
\(740\) 55.2405 2.03068
\(741\) 0 0
\(742\) 0.983593 0.0361088
\(743\) 23.1096 0.847810 0.423905 0.905707i \(-0.360659\pi\)
0.423905 + 0.905707i \(0.360659\pi\)
\(744\) 0 0
\(745\) −21.8105 −0.799074
\(746\) 6.15149 0.225222
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) −19.4874 −0.709688
\(755\) −26.1843 −0.952944
\(756\) 0 0
\(757\) −48.3103 −1.75587 −0.877934 0.478781i \(-0.841079\pi\)
−0.877934 + 0.478781i \(0.841079\pi\)
\(758\) 7.62483 0.276946
\(759\) 0 0
\(760\) −0.656586 −0.0238169
\(761\) 17.8625 0.647516 0.323758 0.946140i \(-0.395054\pi\)
0.323758 + 0.946140i \(0.395054\pi\)
\(762\) 0 0
\(763\) 7.61259 0.275594
\(764\) −36.6207 −1.32489
\(765\) 0 0
\(766\) 47.9177 1.73134
\(767\) −22.3515 −0.807065
\(768\) 0 0
\(769\) −24.8820 −0.897269 −0.448635 0.893715i \(-0.648090\pi\)
−0.448635 + 0.893715i \(0.648090\pi\)
\(770\) 8.10856 0.292212
\(771\) 0 0
\(772\) 0.436357 0.0157048
\(773\) 34.9700 1.25778 0.628892 0.777492i \(-0.283509\pi\)
0.628892 + 0.777492i \(0.283509\pi\)
\(774\) 0 0
\(775\) 115.901 4.16329
\(776\) 6.02448 0.216266
\(777\) 0 0
\(778\) −7.07158 −0.253528
\(779\) 2.98359 0.106898
\(780\) 0 0
\(781\) 5.01641 0.179501
\(782\) −24.8195 −0.887544
\(783\) 0 0
\(784\) −4.44364 −0.158701
\(785\) 75.9177 2.70962
\(786\) 0 0
\(787\) −15.1648 −0.540566 −0.270283 0.962781i \(-0.587117\pi\)
−0.270283 + 0.962781i \(0.587117\pi\)
\(788\) −32.2989 −1.15060
\(789\) 0 0
\(790\) −40.6758 −1.44718
\(791\) −7.70892 −0.274097
\(792\) 0 0
\(793\) 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 0
\(799\) −55.2497 −1.95459
\(800\) −95.6071 −3.38022
\(801\) 0 0
\(802\) 14.0000 0.494357
\(803\) −4.82687 −0.170337
\(804\) 0 0
\(805\) 7.83805 0.276255
\(806\) −56.7065 −1.99740
\(807\) 0 0
\(808\) 2.40354 0.0845564
\(809\) −29.8433 −1.04923 −0.524617 0.851338i \(-0.675791\pi\)
−0.524617 + 0.851338i \(0.675791\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −14.6168 −0.512317
\(815\) 11.1648 0.391086
\(816\) 0 0
\(817\) 3.46944 0.121380
\(818\) 10.2723 0.359162
\(819\) 0 0
\(820\) 68.4832 2.39154
\(821\) −30.3327 −1.05862 −0.529309 0.848429i \(-0.677549\pi\)
−0.529309 + 0.848429i \(0.677549\pi\)
\(822\) 0 0
\(823\) −18.4067 −0.641616 −0.320808 0.947144i \(-0.603954\pi\)
−0.320808 + 0.947144i \(0.603954\pi\)
\(824\) 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 \(0.790090\pi\)
\(828\) 0 0
\(829\) −20.3463 −0.706655 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(830\) 28.5795 0.992009
\(831\) 0 0
\(832\) 18.5769 0.644040
\(833\) −6.85446 −0.237493
\(834\) 0 0
\(835\) −46.6946 −1.61593
\(836\) −0.556364 −0.0192423
\(837\) 0 0
\(838\) 22.1086 0.763728
\(839\) −16.1812 −0.558637 −0.279318 0.960199i \(-0.590109\pi\)
−0.279318 + 0.960199i \(0.590109\pi\)
\(840\) 0 0
\(841\) −18.9313 −0.652802
\(842\) −53.0593 −1.82855
\(843\) 0 0
\(844\) −8.09632 −0.278687
\(845\) 12.2806 0.422466
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 2.25827 0.0775493
\(849\) 0 0
\(850\) −166.522 −5.71165
\(851\) −14.1291 −0.484341
\(852\) 0 0
\(853\) −20.9836 −0.718465 −0.359232 0.933248i \(-0.616961\pi\)
−0.359232 + 0.933248i \(0.616961\pi\)
\(854\) 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 \(0.526076\pi\)
\(858\) 0 0
\(859\) −9.96719 −0.340076 −0.170038 0.985438i \(-0.554389\pi\)
−0.170038 + 0.985438i \(0.554389\pi\)
\(860\) 79.6350 2.71553
\(861\) 0 0
\(862\) 4.42412 0.150686
\(863\) −25.5470 −0.869629 −0.434814 0.900520i \(-0.643186\pi\)
−0.434814 + 0.900520i \(0.643186\pi\)
\(864\) 0 0
\(865\) −104.094 −3.53929
\(866\) 61.1044 2.07641
\(867\) 0 0
\(868\) −16.1208 −0.547176
\(869\) 5.01641 0.170170
\(870\) 0 0
\(871\) 8.45614 0.286525
\(872\) 3.74385 0.126783
\(873\) 0 0
\(874\) −1.15388 −0.0390306
\(875\) 31.6402 1.06963
\(876\) 0 0
\(877\) −50.9893 −1.72179 −0.860893 0.508787i \(-0.830094\pi\)
−0.860893 + 0.508787i \(0.830094\pi\)
\(878\) −35.8586 −1.21017
\(879\) 0 0
\(880\) 18.6168 0.627571
\(881\) −32.1895 −1.08449 −0.542246 0.840219i \(-0.682426\pi\)
−0.542246 + 0.840219i \(0.682426\pi\)
\(882\) 0 0
\(883\) 36.6154 1.23221 0.616104 0.787665i \(-0.288710\pi\)
0.616104 + 0.787665i \(0.288710\pi\)
\(884\) 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 \(0.199897\pi\)
\(888\) 0 0
\(889\) 5.49180 0.184189
\(890\) −14.1229 −0.473401
\(891\) 0 0
\(892\) −32.0307 −1.07247
\(893\) −2.56860 −0.0859550
\(894\) 0 0
\(895\) −53.4506 −1.78666
\(896\) −3.90262 −0.130377
\(897\) 0 0
\(898\) 11.0060 0.367273
\(899\) 29.2992 0.977181
\(900\) 0 0
\(901\) 3.48346 0.116051
\(902\) −18.1208 −0.603357
\(903\) 0 0
\(904\) −3.79122 −0.126094
\(905\) 13.5470 0.450316
\(906\) 0 0
\(907\) 22.9013 0.760425 0.380212 0.924899i \(-0.375851\pi\)
0.380212 + 0.924899i \(0.375851\pi\)
\(908\) −0.661814 −0.0219631
\(909\) 0 0
\(910\) −25.7295 −0.852924
\(911\) 36.0552 1.19456 0.597281 0.802032i \(-0.296248\pi\)
0.597281 + 0.802032i \(0.296248\pi\)
\(912\) 0 0
\(913\) −3.52461 −0.116648
\(914\) −25.1762 −0.832756
\(915\) 0 0
\(916\) 43.5470 1.43883
\(917\) 4.00000 0.132092
\(918\) 0 0
\(919\) −28.3379 −0.934782 −0.467391 0.884051i \(-0.654806\pi\)
−0.467391 + 0.884051i \(0.654806\pi\)
\(920\) 3.85473 0.127087
\(921\) 0 0
\(922\) 12.3463 0.406602
\(923\) −15.9177 −0.523937
\(924\) 0 0
\(925\) −94.7966 −3.11689
\(926\) 65.9851 2.16841
\(927\) 0 0
\(928\) −24.1690 −0.793385
\(929\) −5.08514 −0.166838 −0.0834191 0.996515i \(-0.526584\pi\)
−0.0834191 + 0.996515i \(0.526584\pi\)
\(930\) 0 0
\(931\) −0.318669 −0.0104440
\(932\) −40.9424 −1.34111
\(933\) 0 0
\(934\) −35.1250 −1.14932
\(935\) 28.7170 0.939146
\(936\) 0 0
\(937\) 32.2088 1.05222 0.526108 0.850418i \(-0.323651\pi\)
0.526108 + 0.850418i \(0.323651\pi\)
\(938\) 5.15778 0.168407
\(939\) 0 0
\(940\) −58.9578 −1.92299
\(941\) −32.7805 −1.06861 −0.534307 0.845291i \(-0.679427\pi\)
−0.534307 + 0.845291i \(0.679427\pi\)
\(942\) 0 0
\(943\) −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 \(0.353352\pi\)
\(948\) 0 0
\(949\) 15.3163 0.497188
\(950\) −7.74173 −0.251175
\(951\) 0 0
\(952\) −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 0
\(955\) 87.8765 2.84362
\(956\) −8.86359 −0.286669
\(957\) 0 0
\(958\) 82.7393 2.67319
\(959\) 15.6126 0.504157
\(960\) 0 0
\(961\) 54.2580 1.75026
\(962\) 46.3808 1.49538
\(963\) 0 0
\(964\) 33.6852 1.08493
\(965\) −1.04710 −0.0337074
\(966\) 0 0
\(967\) 21.3298 0.685922 0.342961 0.939350i \(-0.388570\pi\)
0.342961 + 0.939350i \(0.388570\pi\)
\(968\) −0.491797 −0.0158069
\(969\) 0 0
\(970\) 99.3293 3.18927
\(971\) −28.4946 −0.914436 −0.457218 0.889355i \(-0.651154\pi\)
−0.457218 + 0.889355i \(0.651154\pi\)
\(972\) 0 0
\(973\) 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 \(0.397331\pi\)
\(978\) 0 0
\(979\) 1.74173 0.0556659
\(980\) −7.31450 −0.233653
\(981\) 0 0
\(982\) −61.3171 −1.95671
\(983\) 53.7745 1.71514 0.857571 0.514366i \(-0.171973\pi\)
0.857571 + 0.514366i \(0.171973\pi\)
\(984\) 0 0
\(985\) 77.5058 2.46954
\(986\) −42.0958 −1.34060
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −9.70892 −0.307948
\(995\) −41.3543 −1.31102
\(996\) 0 0
\(997\) 0.659696 0.0208928 0.0104464 0.999945i \(-0.496675\pi\)
0.0104464 + 0.999945i \(0.496675\pi\)
\(998\) −46.6943 −1.47808
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 693.2.a.l.1.3 3
3.2 odd 2 231.2.a.e.1.1 3
7.6 odd 2 4851.2.a.bi.1.3 3
11.10 odd 2 7623.2.a.cd.1.1 3
12.11 even 2 3696.2.a.bo.1.3 3
15.14 odd 2 5775.2.a.bp.1.3 3
21.20 even 2 1617.2.a.t.1.1 3
33.32 even 2 2541.2.a.bg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 3.2 odd 2
693.2.a.l.1.3 3 1.1 even 1 trivial
1617.2.a.t.1.1 3 21.20 even 2
2541.2.a.bg.1.3 3 33.32 even 2
3696.2.a.bo.1.3 3 12.11 even 2
4851.2.a.bi.1.3 3 7.6 odd 2
5775.2.a.bp.1.3 3 15.14 odd 2
7623.2.a.cd.1.1 3 11.10 odd 2