Properties

Label 1617.2.a.t.1.1
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
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 1.1
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.93543 q^{2} -1.00000 q^{3} +1.74590 q^{4} -4.18953 q^{5} +1.93543 q^{6} +0.491797 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.93543 q^{2} -1.00000 q^{3} +1.74590 q^{4} -4.18953 q^{5} +1.93543 q^{6} +0.491797 q^{8} +1.00000 q^{9} +8.10856 q^{10} -1.00000 q^{11} -1.74590 q^{12} +3.17313 q^{13} +4.18953 q^{15} -4.44364 q^{16} -6.85446 q^{17} -1.93543 q^{18} +0.318669 q^{19} -7.31450 q^{20} +1.93543 q^{22} -1.87086 q^{23} -0.491797 q^{24} +12.5522 q^{25} -6.14137 q^{26} -1.00000 q^{27} -3.17313 q^{29} -8.10856 q^{30} -9.23353 q^{31} +7.61676 q^{32} +1.00000 q^{33} +13.2663 q^{34} +1.74590 q^{36} -7.55220 q^{37} -0.616763 q^{38} -3.17313 q^{39} -2.06040 q^{40} -9.36266 q^{41} -10.8873 q^{43} -1.74590 q^{44} -4.18953 q^{45} +3.62093 q^{46} +8.06040 q^{47} +4.44364 q^{48} -24.2939 q^{50} +6.85446 q^{51} +5.53996 q^{52} +0.508203 q^{53} +1.93543 q^{54} +4.18953 q^{55} -0.318669 q^{57} +6.14137 q^{58} +7.04399 q^{59} +7.31450 q^{60} +2.00000 q^{61} +17.8709 q^{62} -5.85446 q^{64} -13.2939 q^{65} -1.93543 q^{66} -2.66492 q^{67} -11.9672 q^{68} +1.87086 q^{69} -5.01641 q^{71} +0.491797 q^{72} +4.82687 q^{73} +14.6168 q^{74} -12.5522 q^{75} +0.556364 q^{76} +6.14137 q^{78} +5.01641 q^{79} +18.6168 q^{80} +1.00000 q^{81} +18.1208 q^{82} -3.52461 q^{83} +28.7170 q^{85} +21.0716 q^{86} +3.17313 q^{87} -0.491797 q^{88} +1.74173 q^{89} +8.10856 q^{90} -3.26634 q^{92} +9.23353 q^{93} -15.6004 q^{94} -1.33508 q^{95} -7.61676 q^{96} +12.2499 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + 11 q^{10} - 3 q^{11} - 6 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 3 q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{24} + 15 q^{25} + q^{26} - 3 q^{27} - 4 q^{29} - 11 q^{30} + 2 q^{31} + 8 q^{32} + 3 q^{33} + 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 18 q^{40} - 14 q^{41} - 14 q^{43} - 6 q^{44} - 4 q^{45} + 28 q^{46} + 4 q^{48} - 19 q^{50} + 8 q^{51} + 29 q^{52} - 2 q^{54} + 4 q^{55} - 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} - 5 q^{64} + 14 q^{65} + 2 q^{66} - 4 q^{67} - 42 q^{68} - 10 q^{69} - 12 q^{71} + 3 q^{72} + 20 q^{73} + 29 q^{74} - 15 q^{75} + 11 q^{76} - q^{78} + 12 q^{79} + 41 q^{80} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{85} + 24 q^{86} + 4 q^{87} - 3 q^{88} - 26 q^{89} + 11 q^{90} + 26 q^{92} - 2 q^{93} - 35 q^{94} - 8 q^{95} - 8 q^{96} + 4 q^{97} - 3 q^{99} + O(q^{100}) \)

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 \(-0.739883\pi\)
−0.684279 + 0.729221i \(0.739883\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.74590 0.872949
\(5\) −4.18953 −1.87362 −0.936808 0.349843i \(-0.886235\pi\)
−0.936808 + 0.349843i \(0.886235\pi\)
\(6\) 1.93543 0.790137
\(7\) 0 0
\(8\) 0.491797 0.173876
\(9\) 1.00000 0.333333
\(10\) 8.10856 2.56415
\(11\) −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 \(0.354967\pi\)
\(14\) 0 0
\(15\) 4.18953 1.08173
\(16\) −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 \(-0.488362\pi\)
0.0365539 + 0.999332i \(0.488362\pi\)
\(20\) −7.31450 −1.63557
\(21\) 0 0
\(22\) 1.93543 0.412636
\(23\) −1.87086 −0.390102 −0.195051 0.980793i \(-0.562487\pi\)
−0.195051 + 0.980793i \(0.562487\pi\)
\(24\) −0.491797 −0.100388
\(25\) 12.5522 2.51044
\(26\) −6.14137 −1.20442
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.17313 −0.589235 −0.294617 0.955615i \(-0.595192\pi\)
−0.294617 + 0.955615i \(0.595192\pi\)
\(30\) −8.10856 −1.48041
\(31\) −9.23353 −1.65839 −0.829195 0.558959i \(-0.811201\pi\)
−0.829195 + 0.558959i \(0.811201\pi\)
\(32\) 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 \(-0.713187\pi\)
−0.620787 + 0.783980i \(0.713187\pi\)
\(38\) −0.616763 −0.100052
\(39\) −3.17313 −0.508107
\(40\) −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
\(45\) −4.18953 −0.624539
\(46\) 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
\(49\) 0 0
\(50\) −24.2939 −3.43568
\(51\) 6.85446 0.959816
\(52\) 5.53996 0.768254
\(53\) 0.508203 0.0698071 0.0349036 0.999391i \(-0.488888\pi\)
0.0349036 + 0.999391i \(0.488888\pi\)
\(54\) 1.93543 0.263379
\(55\) 4.18953 0.564917
\(56\) 0 0
\(57\) −0.318669 −0.0422088
\(58\) 6.14137 0.806402
\(59\) 7.04399 0.917050 0.458525 0.888682i \(-0.348378\pi\)
0.458525 + 0.888682i \(0.348378\pi\)
\(60\) 7.31450 0.944298
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 17.8709 2.26960
\(63\) 0 0
\(64\) −5.85446 −0.731807
\(65\) −13.2939 −1.64891
\(66\) −1.93543 −0.238235
\(67\) −2.66492 −0.325572 −0.162786 0.986661i \(-0.552048\pi\)
−0.162786 + 0.986661i \(0.552048\pi\)
\(68\) −11.9672 −1.45123
\(69\) 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
\(75\) −12.5522 −1.44940
\(76\) 0.556364 0.0638194
\(77\) 0 0
\(78\) 6.14137 0.695374
\(79\) 5.01641 0.564390 0.282195 0.959357i \(-0.408938\pi\)
0.282195 + 0.959357i \(0.408938\pi\)
\(80\) 18.6168 2.08142
\(81\) 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 \(0.561964\pi\)
\(84\) 0 0
\(85\) 28.7170 3.11479
\(86\) 21.0716 2.27221
\(87\) 3.17313 0.340195
\(88\) −0.491797 −0.0524257
\(89\) 1.74173 0.184623 0.0923115 0.995730i \(-0.470574\pi\)
0.0923115 + 0.995730i \(0.470574\pi\)
\(90\) 8.10856 0.854717
\(91\) 0 0
\(92\) −3.26634 −0.340539
\(93\) 9.23353 0.957472
\(94\) −15.6004 −1.60905
\(95\) −1.33508 −0.136976
\(96\) −7.61676 −0.777383
\(97\) 12.2499 1.24379 0.621896 0.783100i \(-0.286363\pi\)
0.621896 + 0.783100i \(0.286363\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 21.9149 2.19149
\(101\) −4.88727 −0.486302 −0.243151 0.969988i \(-0.578181\pi\)
−0.243151 + 0.969988i \(0.578181\pi\)
\(102\) −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 \(-0.485286\pi\)
0.0462104 + 0.998932i \(0.485286\pi\)
\(108\) −1.74590 −0.167999
\(109\) −7.61259 −0.729154 −0.364577 0.931173i \(-0.618786\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(110\) −8.10856 −0.773121
\(111\) 7.55220 0.716823
\(112\) 0 0
\(113\) −7.70892 −0.725194 −0.362597 0.931946i \(-0.618110\pi\)
−0.362597 + 0.931946i \(0.618110\pi\)
\(114\) 0.616763 0.0577651
\(115\) 7.83805 0.730902
\(116\) −5.53996 −0.514372
\(117\) 3.17313 0.293356
\(118\) −13.6332 −1.25504
\(119\) 0 0
\(120\) 2.06040 0.188088
\(121\) 1.00000 0.0909091
\(122\) −3.87086 −0.350452
\(123\) 9.36266 0.844203
\(124\) −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
\(129\) 10.8873 0.958571
\(130\) 25.7295 2.25663
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 1.74590 0.151961
\(133\) 0 0
\(134\) 5.15778 0.445564
\(135\) 4.18953 0.360578
\(136\) −3.37100 −0.289061
\(137\) 15.6126 1.33387 0.666937 0.745114i \(-0.267605\pi\)
0.666937 + 0.745114i \(0.267605\pi\)
\(138\) −3.62093 −0.308234
\(139\) 9.01641 0.764762 0.382381 0.924005i \(-0.375104\pi\)
0.382381 + 0.924005i \(0.375104\pi\)
\(140\) 0 0
\(141\) −8.06040 −0.678808
\(142\) 9.70892 0.814754
\(143\) −3.17313 −0.265350
\(144\) −4.44364 −0.370303
\(145\) 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 \(-0.568403\pi\)
−0.213244 + 0.976999i \(0.568403\pi\)
\(150\) 24.2939 1.98359
\(151\) 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
\(154\) 0 0
\(155\) 38.6842 3.10719
\(156\) −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
\(160\) −31.9107 −2.52276
\(161\) 0 0
\(162\) −1.93543 −0.152062
\(163\) −2.66492 −0.208733 −0.104366 0.994539i \(-0.533281\pi\)
−0.104366 + 0.994539i \(0.533281\pi\)
\(164\) −16.3463 −1.27643
\(165\) −4.18953 −0.326155
\(166\) 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
\(171\) 0.318669 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 \(0.658201\pi\)
\(180\) −7.31450 −0.545191
\(181\) 3.23353 0.240346 0.120173 0.992753i \(-0.461655\pi\)
0.120173 + 0.992753i \(0.461655\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −0.920085 −0.0678296
\(185\) 31.6402 2.32623
\(186\) −17.8709 −1.31036
\(187\) 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 \(-0.225758\pi\)
0.758858 + 0.651256i \(0.225758\pi\)
\(192\) 5.85446 0.422509
\(193\) 0.249933 0.0179905 0.00899527 0.999960i \(-0.497137\pi\)
0.00899527 + 0.999960i \(0.497137\pi\)
\(194\) −23.7089 −1.70220
\(195\) 13.2939 0.951998
\(196\) 0 0
\(197\) 18.4999 1.31806 0.659030 0.752116i \(-0.270967\pi\)
0.659030 + 0.752116i \(0.270967\pi\)
\(198\) 1.93543 0.137545
\(199\) −9.87086 −0.699727 −0.349864 0.936801i \(-0.613772\pi\)
−0.349864 + 0.936801i \(0.613772\pi\)
\(200\) 6.17313 0.436506
\(201\) 2.66492 0.187969
\(202\) 9.45898 0.665532
\(203\) 0 0
\(204\) 11.9672 0.837871
\(205\) 39.2252 2.73961
\(206\) −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
\(215\) 45.6126 3.11075
\(216\) −0.491797 −0.0334625
\(217\) 0 0
\(218\) 14.7337 0.997889
\(219\) −4.82687 −0.326170
\(220\) 7.31450 0.493144
\(221\) −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 \(0.289447\pi\)
\(224\) 0 0
\(225\) 12.5522 0.836813
\(226\) 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 \(-0.808331\pi\)
−0.824121 + 0.566413i \(0.808331\pi\)
\(230\) −15.1700 −1.00028
\(231\) 0 0
\(232\) −1.56053 −0.102454
\(233\) 23.4506 1.53630 0.768151 0.640268i \(-0.221177\pi\)
0.768151 + 0.640268i \(0.221177\pi\)
\(234\) −6.14137 −0.401474
\(235\) −33.7693 −2.20287
\(236\) 12.2981 0.800538
\(237\) −5.01641 −0.325851
\(238\) 0 0
\(239\) 5.07681 0.328391 0.164196 0.986428i \(-0.447497\pi\)
0.164196 + 0.986428i \(0.447497\pi\)
\(240\) −18.6168 −1.20171
\(241\) −19.2939 −1.24283 −0.621415 0.783481i \(-0.713442\pi\)
−0.621415 + 0.783481i \(0.713442\pi\)
\(242\) −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
\(250\) 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
\(255\) −28.7170 −1.79833
\(256\) 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
\(259\) 0 0
\(260\) −23.2098 −1.43941
\(261\) −3.17313 −0.196412
\(262\) 7.74173 0.478286
\(263\) 2.92319 0.180252 0.0901259 0.995930i \(-0.471273\pi\)
0.0901259 + 0.995930i \(0.471273\pi\)
\(264\) 0.491797 0.0302680
\(265\) −2.12914 −0.130792
\(266\) 0 0
\(267\) −1.74173 −0.106592
\(268\) −4.65269 −0.284208
\(269\) −11.9672 −0.729652 −0.364826 0.931076i \(-0.618872\pi\)
−0.364826 + 0.931076i \(0.618872\pi\)
\(270\) −8.10856 −0.493471
\(271\) 20.3187 1.23427 0.617136 0.786857i \(-0.288293\pi\)
0.617136 + 0.786857i \(0.288293\pi\)
\(272\) 30.4587 1.84683
\(273\) 0 0
\(274\) −30.2171 −1.82548
\(275\) −12.5522 −0.756926
\(276\) 3.26634 0.196611
\(277\) 18.0552 1.08483 0.542415 0.840111i \(-0.317510\pi\)
0.542415 + 0.840111i \(0.317510\pi\)
\(278\) −17.4506 −1.04662
\(279\) −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
\(285\) 1.33508 0.0790831
\(286\) 6.14137 0.363147
\(287\) 0 0
\(288\) 7.61676 0.448822
\(289\) 29.9836 1.76374
\(290\) −25.7295 −1.51089
\(291\) −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 \(0.459479\pi\)
\(294\) 0 0
\(295\) −29.5110 −1.71820
\(296\) −3.71414 −0.215880
\(297\) 1.00000 0.0580259
\(298\) 10.0757 0.583672
\(299\) −5.93649 −0.343316
\(300\) −21.9149 −1.26525
\(301\) 0 0
\(302\) −12.0963 −0.696065
\(303\) 4.88727 0.280766
\(304\) −1.41605 −0.0812161
\(305\) −8.37907 −0.479784
\(306\) 13.2663 0.758386
\(307\) 20.7581 1.18473 0.592365 0.805670i \(-0.298194\pi\)
0.592365 + 0.805670i \(0.298194\pi\)
\(308\) 0 0
\(309\) −0.637339 −0.0362569
\(310\) −74.8706 −4.25236
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −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 \(-0.689745\pi\)
−0.561422 + 0.827530i \(0.689745\pi\)
\(318\) 0.983593 0.0551572
\(319\) 3.17313 0.177661
\(320\) 24.5275 1.37113
\(321\) −0.956008 −0.0533592
\(322\) 0 0
\(323\) −2.18431 −0.121538
\(324\) 1.74590 0.0969944
\(325\) 39.8297 2.20935
\(326\) 5.15778 0.285663
\(327\) 7.61259 0.420977
\(328\) −4.60453 −0.254242
\(329\) 0 0
\(330\) 8.10856 0.446362
\(331\) 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
\(335\) 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
\(339\) 7.70892 0.418691
\(340\) 50.1369 2.71906
\(341\) 9.23353 0.500023
\(342\) −0.616763 −0.0333507
\(343\) 0 0
\(344\) −5.35432 −0.288686
\(345\) −7.83805 −0.421986
\(346\) −48.0880 −2.58523
\(347\) 17.4506 0.936800 0.468400 0.883517i \(-0.344831\pi\)
0.468400 + 0.883517i \(0.344831\pi\)
\(348\) 5.53996 0.296973
\(349\) −6.85969 −0.367191 −0.183595 0.983002i \(-0.558774\pi\)
−0.183595 + 0.983002i \(0.558774\pi\)
\(350\) 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
\(355\) 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 \(-0.723057\pi\)
−0.644795 + 0.764356i \(0.723057\pi\)
\(360\) −2.06040 −0.108593
\(361\) −18.8984 −0.994655
\(362\) −6.25827 −0.328927
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −20.2223 −1.05849
\(366\) 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
\(370\) −61.2374 −3.18358
\(371\) 0 0
\(372\) 16.1208 0.835824
\(373\) 3.17836 0.164569 0.0822845 0.996609i \(-0.473778\pi\)
0.0822845 + 0.996609i \(0.473778\pi\)
\(374\) −13.2663 −0.685986
\(375\) 31.6402 1.63389
\(376\) 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
\(381\) 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 \(0.470474\pi\)
\(390\) −25.7295 −1.30286
\(391\) 12.8238 0.648526
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) −35.8052 −1.80384
\(395\) −21.0164 −1.05745
\(396\) −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 0
\(400\) −55.7774 −2.78887
\(401\) −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
\(405\) −4.18953 −0.208180
\(406\) 0 0
\(407\) 7.55220 0.374348
\(408\) 3.37100 0.166889
\(409\) −5.30749 −0.262439 −0.131219 0.991353i \(-0.541889\pi\)
−0.131219 + 0.991353i \(0.541889\pi\)
\(410\) −75.9177 −3.74931
\(411\) −15.6126 −0.770112
\(412\) 1.11273 0.0548202
\(413\) 0 0
\(414\) 3.62093 0.177959
\(415\) 14.7665 0.724858
\(416\) 24.1690 1.18498
\(417\) −9.01641 −0.441535
\(418\) 0.616763 0.0301669
\(419\) 11.4231 0.558053 0.279026 0.960283i \(-0.409988\pi\)
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
\(423\) 8.06040 0.391910
\(424\) 0.249933 0.0121378
\(425\) −86.0385 −4.17348
\(426\) −9.70892 −0.470399
\(427\) 0 0
\(428\) 1.66909 0.0806786
\(429\) 3.17313 0.153200
\(430\) −88.2801 −4.25724
\(431\) −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 \(0.774123\pi\)
\(434\) 0 0
\(435\) −13.2939 −0.637395
\(436\) −13.2908 −0.636515
\(437\) −0.596187 −0.0285195
\(438\) 9.34209 0.446382
\(439\) 18.5275 0.884267 0.442133 0.896949i \(-0.354222\pi\)
0.442133 + 0.896949i \(0.354222\pi\)
\(440\) 2.06040 0.0982257
\(441\) 0 0
\(442\) 42.0958 2.00229
\(443\) 28.4342 1.35095 0.675476 0.737382i \(-0.263938\pi\)
0.675476 + 0.737382i \(0.263938\pi\)
\(444\) 13.1854 0.625750
\(445\) −7.29703 −0.345913
\(446\) −35.5079 −1.68135
\(447\) 5.20594 0.246233
\(448\) 0 0
\(449\) −5.68656 −0.268365 −0.134183 0.990957i \(-0.542841\pi\)
−0.134183 + 0.990957i \(0.542841\pi\)
\(450\) −24.2939 −1.14523
\(451\) 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 \(-0.598404\pi\)
−0.304246 + 0.952594i \(0.598404\pi\)
\(458\) 48.2744 2.25571
\(459\) 6.85446 0.319939
\(460\) 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
\(465\) −38.6842 −1.79394
\(466\) −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
\(469\) 0 0
\(470\) 65.3582 3.01475
\(471\) −18.1208 −0.834962
\(472\) 3.46421 0.159453
\(473\) 10.8873 0.500597
\(474\) 9.70892 0.445945
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0.508203 0.0232690
\(478\) −9.82581 −0.449422
\(479\) 42.7498 1.95329 0.976644 0.214864i \(-0.0689307\pi\)
0.976644 + 0.214864i \(0.0689307\pi\)
\(480\) 31.9107 1.45652
\(481\) −23.9641 −1.09267
\(482\) 37.3421 1.70089
\(483\) 0 0
\(484\) 1.74590 0.0793590
\(485\) −51.3215 −2.33039
\(486\) 1.93543 0.0877930
\(487\) 26.7909 1.21401 0.607007 0.794697i \(-0.292370\pi\)
0.607007 + 0.794697i \(0.292370\pi\)
\(488\) 0.983593 0.0445252
\(489\) 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
\(495\) 4.18953 0.188306
\(496\) 41.0304 1.84232
\(497\) 0 0
\(498\) −6.82164 −0.305685
\(499\) −24.1260 −1.08003 −0.540015 0.841656i \(-0.681581\pi\)
−0.540015 + 0.841656i \(0.681581\pi\)
\(500\) −55.2405 −2.47043
\(501\) −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 0
\(505\) 20.4754 0.911143
\(506\) −3.62093 −0.160970
\(507\) 2.93126 0.130182
\(508\) −9.58812 −0.425404
\(509\) −28.2088 −1.25033 −0.625166 0.780492i \(-0.714969\pi\)
−0.625166 + 0.780492i \(0.714969\pi\)
\(510\) 55.5798 2.46111
\(511\) 0 0
\(512\) −29.4754 −1.30264
\(513\) −0.318669 −0.0140696
\(514\) 28.8667 1.27326
\(515\) −2.67015 −0.117661
\(516\) 19.0081 0.836784
\(517\) −8.06040 −0.354496
\(518\) 0 0
\(519\) −24.8461 −1.09062
\(520\) −6.53791 −0.286706
\(521\) 33.7610 1.47910 0.739548 0.673104i \(-0.235039\pi\)
0.739548 + 0.673104i \(0.235039\pi\)
\(522\) 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
\(530\) 4.12080 0.178996
\(531\) 7.04399 0.305683
\(532\) 0 0
\(533\) −29.7089 −1.28684
\(534\) 3.37100 0.145877
\(535\) −4.00523 −0.173161
\(536\) −1.31060 −0.0566093
\(537\) 12.7581 0.550554
\(538\) 23.1617 0.998571
\(539\) 0 0
\(540\) 7.31450 0.314766
\(541\) −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
\(545\) 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
\(549\) 2.00000 0.0853579
\(550\) 24.2939 1.03590
\(551\) −1.01118 −0.0430776
\(552\) 0.920085 0.0391614
\(553\) 0 0
\(554\) −34.9446 −1.48465
\(555\) −31.6402 −1.34305
\(556\) 15.7417 0.667598
\(557\) −6.91486 −0.292992 −0.146496 0.989211i \(-0.546800\pi\)
−0.146496 + 0.989211i \(0.546800\pi\)
\(558\) 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
\(565\) 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 \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −2.58395 −0.108230
\(571\) −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
\(574\) 0 0
\(575\) −23.4835 −0.979328
\(576\) −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
\(580\) 23.2098 0.963736
\(581\) 0 0
\(582\) 23.7089 0.982766
\(583\) −0.508203 −0.0210476
\(584\) 2.37384 0.0982302
\(585\) −13.2939 −0.549636
\(586\) −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
\(591\) −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 \(0.358741\pi\)
\(600\) −6.17313 −0.252017
\(601\) −40.9477 −1.67029 −0.835145 0.550030i \(-0.814616\pi\)
−0.835145 + 0.550030i \(0.814616\pi\)
\(602\) 0 0
\(603\) −2.66492 −0.108524
\(604\) 10.9117 0.443993
\(605\) −4.18953 −0.170329
\(606\) −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
\(609\) 0 0
\(610\) 16.2171 0.656612
\(611\) 25.5767 1.03472
\(612\) −11.9672 −0.483745
\(613\) 18.5306 0.748442 0.374221 0.927339i \(-0.377910\pi\)
0.374221 + 0.927339i \(0.377910\pi\)
\(614\) −40.1760 −1.62137
\(615\) −39.2252 −1.58171
\(616\) 0 0
\(617\) 2.43424 0.0979987 0.0489994 0.998799i \(-0.484397\pi\)
0.0489994 + 0.998799i \(0.484397\pi\)
\(618\) 1.23353 0.0496197
\(619\) 30.4259 1.22292 0.611460 0.791275i \(-0.290582\pi\)
0.611460 + 0.791275i \(0.290582\pi\)
\(620\) 67.5386 2.71242
\(621\) 1.87086 0.0750752
\(622\) 15.4835 0.620830
\(623\) 0 0
\(624\) 14.1002 0.564461
\(625\) 69.7966 2.79187
\(626\) 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
\(635\) 23.0081 0.913047
\(636\) −0.887271 −0.0351826
\(637\) 0 0
\(638\) −6.14137 −0.243139
\(639\) −5.01641 −0.198446
\(640\) 16.3502 0.646297
\(641\) 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 \(0.467856\pi\)
\(644\) 0 0
\(645\) −45.6126 −1.79599
\(646\) 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
\(650\) −77.0877 −3.02363
\(651\) 0 0
\(652\) −4.65269 −0.182213
\(653\) −47.8953 −1.87429 −0.937145 0.348941i \(-0.886541\pi\)
−0.937145 + 0.348941i \(0.886541\pi\)
\(654\) −14.7337 −0.576132
\(655\) 16.7581 0.654795
\(656\) 41.6043 1.62437
\(657\) 4.82687 0.188314
\(658\) 0 0
\(659\) 8.95601 0.348877 0.174438 0.984668i \(-0.444189\pi\)
0.174438 + 0.984668i \(0.444189\pi\)
\(660\) −7.31450 −0.284717
\(661\) 10.7993 0.420044 0.210022 0.977697i \(-0.432647\pi\)
0.210022 + 0.977697i \(0.432647\pi\)
\(662\) −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
\(670\) −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
\(675\) −12.5522 −0.483134
\(676\) −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
\(679\) 0 0
\(680\) 14.1229 0.541589
\(681\) 0.379068 0.0145259
\(682\) −17.8709 −0.684311
\(683\) 22.3156 0.853881 0.426941 0.904280i \(-0.359591\pi\)
0.426941 + 0.904280i \(0.359591\pi\)
\(684\) 0.556364 0.0212731
\(685\) −65.4095 −2.49917
\(686\) 0 0
\(687\) 24.9424 0.951614
\(688\) 48.3791 1.84443
\(689\) 1.61259 0.0614349
\(690\) 15.1700 0.577513
\(691\) −25.1372 −0.956264 −0.478132 0.878288i \(-0.658686\pi\)
−0.478132 + 0.878288i \(0.658686\pi\)
\(692\) 43.3788 1.64901
\(693\) 0 0
\(694\) −33.7745 −1.28206
\(695\) −37.7745 −1.43287
\(696\) 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
\(705\) 33.7693 1.27183
\(706\) −61.8008 −2.32590
\(707\) 0 0
\(708\) −12.2981 −0.462191
\(709\) 9.76098 0.366581 0.183291 0.983059i \(-0.441325\pi\)
0.183291 + 0.983059i \(0.441325\pi\)
\(710\) −40.6758 −1.52654
\(711\) 5.01641 0.188130
\(712\) 0.856577 0.0321016
\(713\) 17.2747 0.646942
\(714\) 0 0
\(715\) 13.2939 0.497165
\(716\) −22.2744 −0.832434
\(717\) −5.07681 −0.189597
\(718\) 47.2908 1.76488
\(719\) −14.0276 −0.523141 −0.261570 0.965184i \(-0.584240\pi\)
−0.261570 + 0.965184i \(0.584240\pi\)
\(720\) 18.6168 0.693806
\(721\) 0 0
\(722\) 36.5767 1.36124
\(723\) 19.2939 0.717549
\(724\) 5.64541 0.209810
\(725\) −39.8297 −1.47924
\(726\) 1.93543 0.0718306
\(727\) 34.3051 1.27231 0.636153 0.771563i \(-0.280525\pi\)
0.636153 + 0.771563i \(0.280525\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 39.1390 1.44860
\(731\) 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 \(0.537482\pi\)
\(740\) 55.2405 2.03068
\(741\) −1.01118 −0.0371466
\(742\) 0 0
\(743\) −23.1096 −0.847810 −0.423905 0.905707i \(-0.639341\pi\)
−0.423905 + 0.905707i \(0.639341\pi\)
\(744\) 4.54102 0.166482
\(745\) 21.8105 0.799074
\(746\) −6.15149 −0.225222
\(747\) −3.52461 −0.128959
\(748\) 11.9672 0.437564
\(749\) 0 0
\(750\) −61.2374 −2.23607
\(751\) 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
\(755\) −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
\(759\) −1.87086 −0.0679081
\(760\) −0.656586 −0.0238169
\(761\) 17.8625 0.647516 0.323758 0.946140i \(-0.395054\pi\)
0.323758 + 0.946140i \(0.395054\pi\)
\(762\) −10.6290 −0.385048
\(763\) 0 0
\(764\) 36.6207 1.32489
\(765\) 28.7170 1.03826
\(766\) −47.9177 −1.73134
\(767\) 22.3515 0.807065
\(768\) −19.2622 −0.695064
\(769\) 24.8820 0.897269 0.448635 0.893715i \(-0.351910\pi\)
0.448635 + 0.893715i \(0.351910\pi\)
\(770\) 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
\(775\) −115.901 −4.16329
\(776\) 6.02448 0.216266
\(777\) 0 0
\(778\) −7.07158 −0.253528
\(779\) −2.98359 −0.106898
\(780\) 23.2098 0.831046
\(781\) 5.01641 0.179501
\(782\) −24.8195 −0.887544
\(783\) 3.17313 0.113398
\(784\) 0 0
\(785\) −75.9177 −2.70962
\(786\) −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
\(790\) 40.6758 1.44718
\(791\) 0 0
\(792\) −0.491797 −0.0174752
\(793\) 6.34625 0.225362
\(794\) 8.83210 0.313440
\(795\) 2.12914 0.0755126
\(796\) −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
\(801\) 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 \(0.324209\pi\)
\(810\) 8.10856 0.284906
\(811\) 16.6321 0.584032 0.292016 0.956413i \(-0.405674\pi\)
0.292016 + 0.956413i \(0.405674\pi\)
\(812\) 0 0
\(813\) −20.3187 −0.712607
\(814\) −14.6168 −0.512317
\(815\) 11.1648 0.391086
\(816\) −30.4587 −1.06627
\(817\) −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 \(-0.322451\pi\)
0.529309 + 0.848429i \(0.322451\pi\)
\(822\) 30.2171 1.05394
\(823\) −18.4067 −0.641616 −0.320808 0.947144i \(-0.603954\pi\)
−0.320808 + 0.947144i \(0.603954\pi\)
\(824\) 0.313441 0.0109192
\(825\) 12.5522 0.437011
\(826\) 0 0
\(827\) 45.4559 1.58066 0.790328 0.612684i \(-0.209910\pi\)
0.790328 + 0.612684i \(0.209910\pi\)
\(828\) −3.26634 −0.113513
\(829\) 20.3463 0.706655 0.353327 0.935500i \(-0.385050\pi\)
0.353327 + 0.935500i \(0.385050\pi\)
\(830\) −28.5795 −0.992009
\(831\) −18.0552 −0.626327
\(832\) −18.5769 −0.644040
\(833\) 0 0
\(834\) 17.4506 0.604266
\(835\) −46.6946 −1.61593
\(836\) −0.556364 −0.0192423
\(837\) 9.23353 0.319157
\(838\) −22.1086 −0.763728
\(839\) −16.1812 −0.558637 −0.279318 0.960199i \(-0.590109\pi\)
−0.279318 + 0.960199i \(0.590109\pi\)
\(840\) 0 0
\(841\) −18.9313 −0.652802
\(842\) 53.0593 1.82855
\(843\) −27.2939 −0.940053
\(844\) −8.09632 −0.278687
\(845\) 12.2806 0.422466
\(846\) −15.6004 −0.536351
\(847\) 0 0
\(848\) −2.25827 −0.0775493
\(849\) 29.8901 1.02583
\(850\) 166.522 5.71165
\(851\) 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 \(0.383039\pi\)
\(854\) 0 0
\(855\) −1.33508 −0.0456586
\(856\) 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 \(-0.445611\pi\)
0.170038 + 0.985438i \(0.445611\pi\)
\(860\) 79.6350 2.71553
\(861\) 0 0
\(862\) 4.42412 0.150686
\(863\) 25.5470 0.869629 0.434814 0.900520i \(-0.356814\pi\)
0.434814 + 0.900520i \(0.356814\pi\)
\(864\) −7.61676 −0.259128
\(865\) −104.094 −3.53929
\(866\) 61.1044 2.07641
\(867\) −29.9836 −1.01830
\(868\) 0 0
\(869\) −5.01641 −0.170170
\(870\) 25.7295 0.872311
\(871\) −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 \(-0.830094\pi\)
−0.860893 + 0.508787i \(0.830094\pi\)
\(878\) −35.8586 −1.21017
\(879\) −4.34625 −0.146596
\(880\) −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
\(885\) 29.5110 0.992003
\(886\) −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
\(889\) 0 0
\(890\) 14.1229 0.473401
\(891\) −1.00000 −0.0335013
\(892\) 32.0307 1.07247
\(893\) 2.56860 0.0859550
\(894\) −10.0757 −0.336983
\(895\) 53.4506 1.78666
\(896\) 0 0
\(897\) 5.93649 0.198214
\(898\) 11.0060 0.367273
\(899\) 29.2992 0.977181
\(900\) 21.9149 0.730495
\(901\) −3.48346 −0.116051
\(902\) −18.1208 −0.603357
\(903\) 0 0
\(904\) −3.79122 −0.126094
\(905\) −13.5470 −0.450316
\(906\) 12.0963 0.401873
\(907\) 22.9013 0.760425 0.380212 0.924899i \(-0.375851\pi\)
0.380212 + 0.924899i \(0.375851\pi\)
\(908\) −0.661814 −0.0219631
\(909\) −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
\(915\) 8.37907 0.277003
\(916\) −43.5470 −1.43883
\(917\) 0 0
\(918\) −13.2663 −0.437854
\(919\) −28.3379 −0.934782 −0.467391 0.884051i \(-0.654806\pi\)
−0.467391 + 0.884051i \(0.654806\pi\)
\(920\) 3.85473 0.127087
\(921\) −20.7581 −0.684004
\(922\) −12.3463 −0.406602
\(923\) −15.9177 −0.523937
\(924\) 0 0
\(925\) −94.7966 −3.11689
\(926\) −65.9851 −2.16841
\(927\) 0.637339 0.0209329
\(928\) −24.1690 −0.793385
\(929\) −5.08514 −0.166838 −0.0834191 0.996515i \(-0.526584\pi\)
−0.0834191 + 0.996515i \(0.526584\pi\)
\(930\) 74.8706 2.45510
\(931\) 0 0
\(932\) 40.9424 1.34111
\(933\) 8.00000 0.261908
\(934\) 35.1250 1.14932
\(935\) −28.7170 −0.939146
\(936\) 1.56053 0.0510076
\(937\) −32.2088 −1.05222 −0.526108 0.850418i \(-0.676349\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(938\) 0 0
\(939\) 8.12914 0.265284
\(940\) −58.9578 −1.92299
\(941\) −32.7805 −1.06861 −0.534307 0.845291i \(-0.679427\pi\)
−0.534307 + 0.845291i \(0.679427\pi\)
\(942\) 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 \(-0.646648\pi\)
−0.444584 + 0.895737i \(0.646648\pi\)
\(948\) −8.75814 −0.284451
\(949\) 15.3163 0.497188
\(950\) −7.74173 −0.251175
\(951\) 19.9917 0.648274
\(952\) 0 0
\(953\) −24.6894 −0.799768 −0.399884 0.916566i \(-0.630950\pi\)
−0.399884 + 0.916566i \(0.630950\pi\)
\(954\) −0.983593 −0.0318450
\(955\) −87.8765 −2.84362
\(956\) 8.86359 0.286669
\(957\) −3.17313 −0.102573
\(958\) −82.7393 −2.67319
\(959\) 0 0
\(960\) −24.5275 −0.791620
\(961\) 54.2580 1.75026
\(962\) 46.3808 1.49538
\(963\) 0.956008 0.0308069
\(964\) −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
\(969\) 2.18431 0.0701700
\(970\) 99.3293 3.18927
\(971\) −28.4946 −0.914436 −0.457218 0.889355i \(-0.651154\pi\)
−0.457218 + 0.889355i \(0.651154\pi\)
\(972\) −1.74590 −0.0559997
\(973\) 0 0
\(974\) −51.8521 −1.66145
\(975\) −39.8297 −1.27557
\(976\) −8.88727 −0.284475
\(977\) −19.8157 −0.633960 −0.316980 0.948432i \(-0.602669\pi\)
−0.316980 + 0.948432i \(0.602669\pi\)
\(978\) −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
\(985\) −77.5058 −2.46954
\(986\) −42.0958 −1.34060
\(987\) 0 0
\(988\) 1.76541 0.0561653
\(989\) 20.3686 0.647684
\(990\) −8.10856 −0.257707
\(991\) 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
\(994\) 0 0
\(995\) 41.3543 1.31102
\(996\) 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 1617.2.a.t.1.1 3
3.2 odd 2 4851.2.a.bi.1.3 3
7.6 odd 2 231.2.a.e.1.1 3
21.20 even 2 693.2.a.l.1.3 3
28.27 even 2 3696.2.a.bo.1.3 3
35.34 odd 2 5775.2.a.bp.1.3 3
77.76 even 2 2541.2.a.bg.1.3 3
231.230 odd 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 7.6 odd 2
693.2.a.l.1.3 3 21.20 even 2
1617.2.a.t.1.1 3 1.1 even 1 trivial
2541.2.a.bg.1.3 3 77.76 even 2
3696.2.a.bo.1.3 3 28.27 even 2
4851.2.a.bi.1.3 3 3.2 odd 2
5775.2.a.bp.1.3 3 35.34 odd 2
7623.2.a.cd.1.1 3 231.230 odd 2