Properties

Label 231.2.a.e.1.1
Level 231
Weight 2
Character 231.1
Self dual yes
Analytic conductor 1.845
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.254102\)
Character \(\chi\) = 231.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.93543 q^{2} +1.00000 q^{3} +1.74590 q^{4} +4.18953 q^{5} -1.93543 q^{6} -1.00000 q^{7} +0.491797 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.93543 q^{2} +1.00000 q^{3} +1.74590 q^{4} +4.18953 q^{5} -1.93543 q^{6} -1.00000 q^{7} +0.491797 q^{8} +1.00000 q^{9} -8.10856 q^{10} -1.00000 q^{11} +1.74590 q^{12} -3.17313 q^{13} +1.93543 q^{14} +4.18953 q^{15} -4.44364 q^{16} +6.85446 q^{17} -1.93543 q^{18} -0.318669 q^{19} +7.31450 q^{20} -1.00000 q^{21} +1.93543 q^{22} -1.87086 q^{23} +0.491797 q^{24} +12.5522 q^{25} +6.14137 q^{26} +1.00000 q^{27} -1.74590 q^{28} -3.17313 q^{29} -8.10856 q^{30} +9.23353 q^{31} +7.61676 q^{32} -1.00000 q^{33} -13.2663 q^{34} -4.18953 q^{35} +1.74590 q^{36} -7.55220 q^{37} +0.616763 q^{38} -3.17313 q^{39} +2.06040 q^{40} +9.36266 q^{41} +1.93543 q^{42} -10.8873 q^{43} -1.74590 q^{44} +4.18953 q^{45} +3.62093 q^{46} -8.06040 q^{47} -4.44364 q^{48} +1.00000 q^{49} -24.2939 q^{50} +6.85446 q^{51} -5.53996 q^{52} +0.508203 q^{53} -1.93543 q^{54} -4.18953 q^{55} -0.491797 q^{56} -0.318669 q^{57} +6.14137 q^{58} -7.04399 q^{59} +7.31450 q^{60} -2.00000 q^{61} -17.8709 q^{62} -1.00000 q^{63} -5.85446 q^{64} -13.2939 q^{65} +1.93543 q^{66} -2.66492 q^{67} +11.9672 q^{68} -1.87086 q^{69} +8.10856 q^{70} -5.01641 q^{71} +0.491797 q^{72} -4.82687 q^{73} +14.6168 q^{74} +12.5522 q^{75} -0.556364 q^{76} +1.00000 q^{77} +6.14137 q^{78} +5.01641 q^{79} -18.6168 q^{80} +1.00000 q^{81} -18.1208 q^{82} +3.52461 q^{83} -1.74590 q^{84} +28.7170 q^{85} +21.0716 q^{86} -3.17313 q^{87} -0.491797 q^{88} -1.74173 q^{89} -8.10856 q^{90} +3.17313 q^{91} -3.26634 q^{92} +9.23353 q^{93} +15.6004 q^{94} -1.33508 q^{95} +7.61676 q^{96} -12.2499 q^{97} -1.93543 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{2} + 3q^{3} + 6q^{4} + 4q^{5} + 2q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 2q^{2} + 3q^{3} + 6q^{4} + 4q^{5} + 2q^{6} - 3q^{7} + 3q^{8} + 3q^{9} - 11q^{10} - 3q^{11} + 6q^{12} - 4q^{13} - 2q^{14} + 4q^{15} - 4q^{16} + 8q^{17} + 2q^{18} - 8q^{19} - 3q^{20} - 3q^{21} - 2q^{22} + 10q^{23} + 3q^{24} + 15q^{25} - q^{26} + 3q^{27} - 6q^{28} - 4q^{29} - 11q^{30} - 2q^{31} + 8q^{32} - 3q^{33} - 4q^{34} - 4q^{35} + 6q^{36} - 13q^{38} - 4q^{39} - 18q^{40} + 14q^{41} - 2q^{42} - 14q^{43} - 6q^{44} + 4q^{45} + 28q^{46} - 4q^{48} + 3q^{49} - 19q^{50} + 8q^{51} - 29q^{52} + 2q^{54} - 4q^{55} - 3q^{56} - 8q^{57} - q^{58} - 3q^{60} - 6q^{61} - 38q^{62} - 3q^{63} - 5q^{64} + 14q^{65} - 2q^{66} - 4q^{67} + 42q^{68} + 10q^{69} + 11q^{70} - 12q^{71} + 3q^{72} - 20q^{73} + 29q^{74} + 15q^{75} - 11q^{76} + 3q^{77} - q^{78} + 12q^{79} - 41q^{80} + 3q^{81} - 6q^{82} + 6q^{83} - 6q^{84} - 6q^{85} + 24q^{86} - 4q^{87} - 3q^{88} + 26q^{89} - 11q^{90} + 4q^{91} + 26q^{92} - 2q^{93} + 35q^{94} - 8q^{95} + 8q^{96} - 4q^{97} + 2q^{98} - 3q^{99} + O(q^{100}) \)

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 \(-0.113765\pi\)
0.936808 + 0.349843i \(0.113765\pi\)
\(6\) −1.93543 −0.790137
\(7\) −1.00000 −0.377964
\(8\) 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 \(-0.645033\pi\)
−0.440034 + 0.897981i \(0.645033\pi\)
\(14\) 1.93543 0.517266
\(15\) 4.18953 1.08173
\(16\) −4.44364 −1.11091
\(17\) 6.85446 1.66245 0.831225 0.555936i \(-0.187640\pi\)
0.831225 + 0.555936i \(0.187640\pi\)
\(18\) −1.93543 −0.456186
\(19\) −0.318669 −0.0731078 −0.0365539 0.999332i \(-0.511638\pi\)
−0.0365539 + 0.999332i \(0.511638\pi\)
\(20\) 7.31450 1.63557
\(21\) −1.00000 −0.218218
\(22\) 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
\(28\) −1.74590 −0.329944
\(29\) −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 \(-0.188799\pi\)
0.829195 + 0.558959i \(0.188799\pi\)
\(32\) 7.61676 1.34647
\(33\) −1.00000 −0.174078
\(34\) −13.2663 −2.27516
\(35\) −4.18953 −0.708161
\(36\) 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 \(-0.239007\pi\)
0.731101 + 0.682269i \(0.239007\pi\)
\(42\) 1.93543 0.298644
\(43\) −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 \(-0.700031\pi\)
−0.587865 + 0.808959i \(0.700031\pi\)
\(48\) −4.44364 −0.641384
\(49\) 1.00000 0.142857
\(50\) −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
\(56\) −0.491797 −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 \(0.651622\pi\)
\(60\) 7.31450 0.944298
\(61\) −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
\(65\) −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
\(70\) 8.10856 0.969158
\(71\) −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 \(-0.591154\pi\)
−0.282471 + 0.959276i \(0.591154\pi\)
\(74\) 14.6168 1.69916
\(75\) 12.5522 1.44940
\(76\) −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 \(0.408938\pi\)
\(80\) −18.6168 −2.08142
\(81\) 1.00000 0.111111
\(82\) −18.1208 −2.00111
\(83\) 3.52461 0.386876 0.193438 0.981112i \(-0.438036\pi\)
0.193438 + 0.981112i \(0.438036\pi\)
\(84\) −1.74590 −0.190493
\(85\) 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 \(-0.529426\pi\)
−0.0923115 + 0.995730i \(0.529426\pi\)
\(90\) −8.10856 −0.854717
\(91\) 3.17313 0.332634
\(92\) −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 \(-0.713637\pi\)
−0.621896 + 0.783100i \(0.713637\pi\)
\(98\) −1.93543 −0.195508
\(99\) −1.00000 −0.100504
\(100\) 21.9149 2.19149
\(101\) 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 \(-0.509996\pi\)
−0.0313994 + 0.999507i \(0.509996\pi\)
\(104\) −1.56053 −0.153023
\(105\) −4.18953 −0.408857
\(106\) −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
\(112\) 4.44364 0.419884
\(113\) −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
\(119\) −6.85446 −0.628347
\(120\) 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
\(126\) 1.93543 0.172422
\(127\) −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 \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.74590 −0.151961
\(133\) 0.318669 0.0276321
\(134\) 5.15778 0.445564
\(135\) 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 \(-0.624896\pi\)
−0.382381 + 0.924005i \(0.624896\pi\)
\(140\) −7.31450 −0.618188
\(141\) −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
\(147\) 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 \(0.568403\pi\)
\(150\) −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
\(154\) −1.93543 −0.155962
\(155\) 38.6842 3.10719
\(156\) −5.53996 −0.443552
\(157\) −18.1208 −1.44620 −0.723099 0.690745i \(-0.757283\pi\)
−0.723099 + 0.690745i \(0.757283\pi\)
\(158\) −9.70892 −0.772400
\(159\) 0.508203 0.0403031
\(160\) 31.9107 2.52276
\(161\) 1.87086 0.147445
\(162\) −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 \(-0.641922\pi\)
−0.431234 + 0.902240i \(0.641922\pi\)
\(168\) −0.491797 −0.0379429
\(169\) −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 \(-0.893461\pi\)
−0.944508 + 0.328489i \(0.893461\pi\)
\(174\) 6.14137 0.465576
\(175\) −12.5522 −0.948857
\(176\) 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 \(-0.538345\pi\)
−0.120173 + 0.992753i \(0.538345\pi\)
\(182\) −6.14137 −0.455229
\(183\) −2.00000 −0.147844
\(184\) −0.920085 −0.0678296
\(185\) −31.6402 −2.32623
\(186\) −17.8709 −1.31036
\(187\) −6.85446 −0.501248
\(188\) −14.0726 −1.02635
\(189\) −1.00000 −0.0727393
\(190\) 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
\(196\) 1.74590 0.124707
\(197\) 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 \(-0.386228\pi\)
0.349864 + 0.936801i \(0.386228\pi\)
\(200\) 6.17313 0.436506
\(201\) −2.66492 −0.187969
\(202\) −9.45898 −0.665532
\(203\) 3.17313 0.222710
\(204\) 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
\(210\) 8.10856 0.559544
\(211\) −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
\(217\) −9.23353 −0.626813
\(218\) 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 \(-0.710553\pi\)
−0.614278 + 0.789090i \(0.710553\pi\)
\(224\) −7.61676 −0.508916
\(225\) 12.5522 0.836813
\(226\) 14.9201 0.992469
\(227\) 0.379068 0.0251596 0.0125798 0.999921i \(-0.495996\pi\)
0.0125798 + 0.999921i \(0.495996\pi\)
\(228\) −0.556364 −0.0368461
\(229\) 24.9424 1.64824 0.824121 0.566413i \(-0.191669\pi\)
0.824121 + 0.566413i \(0.191669\pi\)
\(230\) 15.1700 1.00028
\(231\) 1.00000 0.0657952
\(232\) −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
\(238\) 13.2663 0.859929
\(239\) 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 \(-0.286558\pi\)
0.621415 + 0.783481i \(0.286558\pi\)
\(242\) −1.93543 −0.124414
\(243\) 1.00000 0.0641500
\(244\) −3.49180 −0.223539
\(245\) 4.18953 0.267660
\(246\) −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 \(-0.229481\pi\)
0.751188 + 0.660088i \(0.229481\pi\)
\(252\) −1.74590 −0.109981
\(253\) 1.87086 0.117620
\(254\) 10.6290 0.666923
\(255\) 28.7170 1.79833
\(256\) 19.2622 1.20389
\(257\) 14.9149 0.930363 0.465182 0.885215i \(-0.345989\pi\)
0.465182 + 0.885215i \(0.345989\pi\)
\(258\) 21.0716 1.31186
\(259\) 7.55220 0.469271
\(260\) −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
\(266\) −0.616763 −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 \(0.381128\pi\)
\(270\) −8.10856 −0.493471
\(271\) −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
\(275\) −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
\(280\) −2.06040 −0.123132
\(281\) 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 \(-0.151823\pi\)
0.888391 + 0.459087i \(0.151823\pi\)
\(284\) −8.75814 −0.519700
\(285\) −1.33508 −0.0790831
\(286\) −6.14137 −0.363147
\(287\) −9.36266 −0.552660
\(288\) 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 \(-0.540521\pi\)
−0.126955 + 0.991908i \(0.540521\pi\)
\(294\) −1.93543 −0.112877
\(295\) −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
\(301\) 10.8873 0.627532
\(302\) −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 \(-0.701806\pi\)
−0.592365 + 0.805670i \(0.701806\pi\)
\(308\) 1.74590 0.0994818
\(309\) −0.637339 −0.0362569
\(310\) −74.8706 −4.25236
\(311\) 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 \(-0.426211\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(314\) 35.0716 1.97920
\(315\) −4.18953 −0.236054
\(316\) 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
\(322\) −3.62093 −0.201787
\(323\) −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
\(329\) 8.06040 0.444384
\(330\) 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
\(336\) 4.44364 0.242420
\(337\) 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
\(343\) −1.00000 −0.0539949
\(344\) −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 \(-0.441226\pi\)
0.183595 + 0.983002i \(0.441226\pi\)
\(350\) 24.2939 1.29856
\(351\) −3.17313 −0.169369
\(352\) −7.61676 −0.405975
\(353\) −31.9313 −1.69953 −0.849765 0.527162i \(-0.823256\pi\)
−0.849765 + 0.527162i \(0.823256\pi\)
\(354\) 13.6332 0.724595
\(355\) −21.0164 −1.11544
\(356\) −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 \(0.723057\pi\)
\(360\) 2.06040 0.108593
\(361\) −18.8984 −0.994655
\(362\) 6.25827 0.328927
\(363\) 1.00000 0.0524864
\(364\) 5.53996 0.290373
\(365\) −20.2223 −1.05849
\(366\) 3.87086 0.202333
\(367\) 17.0716 0.891129 0.445565 0.895250i \(-0.353003\pi\)
0.445565 + 0.895250i \(0.353003\pi\)
\(368\) 8.31344 0.433368
\(369\) 9.36266 0.487401
\(370\) 61.2374 3.18358
\(371\) −0.508203 −0.0263846
\(372\) 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
\(378\) 1.93543 0.0995479
\(379\) 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 \(-0.717988\pi\)
−0.632541 + 0.774527i \(0.717988\pi\)
\(384\) −3.90262 −0.199155
\(385\) 4.18953 0.213518
\(386\) −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
\(392\) 0.491797 0.0248395
\(393\) 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 \(-0.463469\pi\)
0.114515 + 0.993422i \(0.463469\pi\)
\(398\) −19.1044 −0.957617
\(399\) 0.318669 0.0159534
\(400\) −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
\(406\) −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 \(0.458111\pi\)
\(410\) −75.9177 −3.74931
\(411\) 15.6126 0.770112
\(412\) −1.11273 −0.0548202
\(413\) 7.04399 0.346612
\(414\) 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 \(-0.590012\pi\)
−0.279026 + 0.960283i \(0.590012\pi\)
\(420\) −7.31450 −0.356911
\(421\) −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
\(427\) 2.00000 0.0967868
\(428\) 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 \(-0.225877\pi\)
0.758613 + 0.651541i \(0.225877\pi\)
\(434\) 17.8709 0.857829
\(435\) −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 \(-0.645778\pi\)
−0.442133 + 0.896949i \(0.645778\pi\)
\(440\) −2.06040 −0.0982257
\(441\) 1.00000 0.0476190
\(442\) 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
\(448\) 5.85446 0.276597
\(449\) −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
\(455\) 13.2939 0.623229
\(456\) −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 \(-0.547461\pi\)
−0.148551 + 0.988905i \(0.547461\pi\)
\(462\) −1.93543 −0.0900445
\(463\) 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 \(-0.362064\pi\)
0.419903 + 0.907569i \(0.362064\pi\)
\(468\) −5.53996 −0.256085
\(469\) 2.66492 0.123055
\(470\) 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
\(476\) −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 \(0.931069\pi\)
\(480\) 31.9107 1.45652
\(481\) 23.9641 1.09267
\(482\) −37.3421 −1.70089
\(483\) 1.87086 0.0851273
\(484\) 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
\(490\) −8.10856 −0.366307
\(491\) 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
\(497\) 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 \(0.681581\pi\)
\(500\) 55.2405 2.47043
\(501\) −11.1455 −0.497946
\(502\) −46.0674 −2.05609
\(503\) 30.8873 1.37720 0.688598 0.725144i \(-0.258227\pi\)
0.688598 + 0.725144i \(0.258227\pi\)
\(504\) −0.491797 −0.0219064
\(505\) 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 \(-0.285031\pi\)
0.625166 + 0.780492i \(0.285031\pi\)
\(510\) −55.5798 −2.46111
\(511\) 4.82687 0.213528
\(512\) −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
\(518\) −14.6168 −0.642224
\(519\) −24.8461 −1.09062
\(520\) −6.53791 −0.286706
\(521\) −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 \(-0.409580\pi\)
0.280258 + 0.959925i \(0.409580\pi\)
\(524\) 6.98359 0.305080
\(525\) −12.5522 −0.547823
\(526\) −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
\(532\) 0.556364 0.0241215
\(533\) −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
\(539\) −1.00000 −0.0430730
\(540\) 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
\(546\) −6.14137 −0.262827
\(547\) 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
\(553\) −5.01641 −0.213319
\(554\) −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
\(560\) 18.6168 0.786702
\(561\) −6.85446 −0.289395
\(562\) −52.8255 −2.22831
\(563\) 11.8381 0.498914 0.249457 0.968386i \(-0.419748\pi\)
0.249457 + 0.968386i \(0.419748\pi\)
\(564\) −14.0726 −0.592565
\(565\) −32.2968 −1.35874
\(566\) −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 \(0.432780\pi\)
\(570\) 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
\(574\) 18.1208 0.756347
\(575\) −23.4835 −0.979328
\(576\) −5.85446 −0.243936
\(577\) 17.3215 0.721104 0.360552 0.932739i \(-0.382588\pi\)
0.360552 + 0.932739i \(0.382588\pi\)
\(578\) −58.0312 −2.41378
\(579\) 0.249933 0.0103868
\(580\) −23.2098 −0.963736
\(581\) −3.52461 −0.146225
\(582\) 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 \(-0.304958\pi\)
0.575113 + 0.818074i \(0.304958\pi\)
\(588\) 1.74590 0.0719996
\(589\) −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 \(-0.666626\pi\)
−0.499890 + 0.866089i \(0.666626\pi\)
\(594\) 1.93543 0.0794118
\(595\) −28.7170 −1.17728
\(596\) −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 \(-0.185384\pi\)
0.835145 + 0.550030i \(0.185384\pi\)
\(602\) −21.0716 −0.858813
\(603\) −2.66492 −0.108524
\(604\) 10.9117 0.443993
\(605\) 4.18953 0.170329
\(606\) −9.45898 −0.384245
\(607\) −14.0276 −0.569362 −0.284681 0.958622i \(-0.591888\pi\)
−0.284681 + 0.958622i \(0.591888\pi\)
\(608\) −2.42723 −0.0984371
\(609\) 3.17313 0.128582
\(610\) 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
\(616\) 0.491797 0.0198151
\(617\) 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 \(-0.709418\pi\)
−0.611460 + 0.791275i \(0.709418\pi\)
\(620\) 67.5386 2.71242
\(621\) −1.87086 −0.0750752
\(622\) −15.4835 −0.620830
\(623\) 1.74173 0.0697809
\(624\) 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
\(630\) 8.10856 0.323053
\(631\) 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
\(637\) −3.17313 −0.125724
\(638\) −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 \(-0.532144\pi\)
−0.100813 + 0.994905i \(0.532144\pi\)
\(644\) 3.26634 0.128712
\(645\) −45.6126 −1.79599
\(646\) 4.22758 0.166332
\(647\) 25.7693 1.01310 0.506548 0.862212i \(-0.330921\pi\)
0.506548 + 0.862212i \(0.330921\pi\)
\(648\) 0.491797 0.0193196
\(649\) 7.04399 0.276501
\(650\) 77.0877 3.02363
\(651\) −9.23353 −0.361890
\(652\) −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
\(658\) −15.6004 −0.608165
\(659\) 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 \(-0.567353\pi\)
−0.210022 + 0.977697i \(0.567353\pi\)
\(662\) −32.9341 −1.28002
\(663\) −21.7501 −0.844703
\(664\) 1.73339 0.0672686
\(665\) 1.33508 0.0517720
\(666\) 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
\(672\) −7.61676 −0.293823
\(673\) 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 \(-0.678736\pi\)
−0.532470 + 0.846449i \(0.678736\pi\)
\(678\) 14.9201 0.573002
\(679\) 12.2499 0.470109
\(680\) 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
\(686\) 1.93543 0.0738951
\(687\) 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 \(-0.341314\pi\)
0.478132 + 0.878288i \(0.341314\pi\)
\(692\) −43.3788 −1.64901
\(693\) 1.00000 0.0379869
\(694\) −33.7745 −1.28206
\(695\) −37.7745 −1.43287
\(696\) −1.56053 −0.0591519
\(697\) 64.1760 2.43084
\(698\) −13.2765 −0.502521
\(699\) 23.4506 0.886985
\(700\) −21.9149 −0.828304
\(701\) −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
\(707\) −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 \(0.441325\pi\)
\(710\) 40.6758 1.52654
\(711\) 5.01641 0.188130
\(712\) −0.856577 −0.0321016
\(713\) −17.2747 −0.646942
\(714\) 13.2663 0.496480
\(715\) 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 \(-0.415760\pi\)
0.261570 + 0.965184i \(0.415760\pi\)
\(720\) −18.6168 −0.693806
\(721\) 0.637339 0.0237357
\(722\) 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 \(-0.719475\pi\)
−0.636153 + 0.771563i \(0.719475\pi\)
\(728\) 1.56053 0.0578372
\(729\) 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 \(-0.563666\pi\)
−0.198680 + 0.980064i \(0.563666\pi\)
\(734\) −33.0409 −1.21956
\(735\) 4.18953 0.154533
\(736\) −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
\(742\) 0.983593 0.0361088
\(743\) −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
\(749\) −0.956008 −0.0349318
\(750\) −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
\(756\) −1.74590 −0.0634977
\(757\) −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 \(-0.604946\pi\)
−0.323758 + 0.946140i \(0.604946\pi\)
\(762\) 10.6290 0.385048
\(763\) 7.61259 0.275594
\(764\) 36.6207 1.32489
\(765\) 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 \(-0.648090\pi\)
−0.448635 + 0.893715i \(0.648090\pi\)
\(770\) −8.10856 −0.292212
\(771\) 14.9149 0.537145
\(772\) 0.436357 0.0157048
\(773\) −34.9700 −1.25778 −0.628892 0.777492i \(-0.716491\pi\)
−0.628892 + 0.777492i \(0.716491\pi\)
\(774\) 21.0716 0.757402
\(775\) 115.901 4.16329
\(776\) −6.02448 −0.216266
\(777\) 7.55220 0.270933
\(778\) −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
\(784\) −4.44364 −0.158701
\(785\) −75.9177 −2.70962
\(786\) −7.74173 −0.276138
\(787\) −15.1648 −0.540566 −0.270283 0.962781i \(-0.587117\pi\)
−0.270283 + 0.962781i \(0.587117\pi\)
\(788\) 32.2989 1.15060
\(789\) 2.92319 0.104068
\(790\) −40.6758 −1.44718
\(791\) 7.70892 0.274097
\(792\) −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 \(-0.453353\pi\)
0.146021 + 0.989281i \(0.453353\pi\)
\(798\) −0.616763 −0.0218332
\(799\) −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
\(805\) 7.83805 0.276255
\(806\) 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 \(-0.594326\pi\)
−0.292016 + 0.956413i \(0.594326\pi\)
\(812\) 5.53996 0.194414
\(813\) −20.3187 −0.712607
\(814\) −14.6168 −0.512317
\(815\) −11.1648 −0.391086
\(816\) −30.4587 −1.06627
\(817\) 3.46944 0.121380
\(818\) −10.2723 −0.359162
\(819\) 3.17313 0.110878
\(820\) 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
\(826\) −13.6332 −0.474359
\(827\) 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 \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(830\) −28.5795 −0.992009
\(831\) 18.0552 0.626327
\(832\) 18.5769 0.644040
\(833\) 6.85446 0.237493
\(834\) 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 \(-0.409891\pi\)
0.279318 + 0.960199i \(0.409891\pi\)
\(840\) −2.06040 −0.0710905
\(841\) −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
\(847\) −1.00000 −0.0343604
\(848\) −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 \(-0.616961\pi\)
−0.359232 + 0.933248i \(0.616961\pi\)
\(854\) −3.87086 −0.132458
\(855\) −1.33508 −0.0456586
\(856\) 0.470162 0.0160698
\(857\) 4.79095 0.163656 0.0818279 0.996646i \(-0.473924\pi\)
0.0818279 + 0.996646i \(0.473924\pi\)
\(858\) −6.14137 −0.209663
\(859\) −9.96719 −0.340076 −0.170038 0.985438i \(-0.554389\pi\)
−0.170038 + 0.985438i \(0.554389\pi\)
\(860\) −79.6350 −2.71553
\(861\) −9.36266 −0.319079
\(862\) 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
\(868\) −16.1208 −0.547176
\(869\) −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
\(875\) −31.6402 −1.06963
\(876\) −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 \(-0.317574\pi\)
0.542246 + 0.840219i \(0.317574\pi\)
\(882\) −1.93543 −0.0651694
\(883\) 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 \(-0.800103\pi\)
−0.809206 + 0.587525i \(0.800103\pi\)
\(888\) −3.71414 −0.124639
\(889\) 5.49180 0.184189
\(890\) 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
\(896\) 3.90262 0.130377
\(897\) 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
\(903\) 10.8873 0.362306
\(904\) −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
\(910\) −25.7295 −0.852924
\(911\) −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
\(917\) −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 \(0.654806\pi\)
\(920\) −3.85473 −0.127087
\(921\) −20.7581 −0.684004
\(922\) 12.3463 0.406602
\(923\) 15.9177 0.523937
\(924\) 1.74590 0.0574358
\(925\) −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 \(-0.473416\pi\)
0.0834191 + 0.996515i \(0.473416\pi\)
\(930\) −74.8706 −2.45510
\(931\) −0.318669 −0.0104440
\(932\) 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 \(-0.323651\pi\)
0.526108 + 0.850418i \(0.323651\pi\)
\(938\) −5.15778 −0.168407
\(939\) 8.12914 0.265284
\(940\) −58.9578 −1.92299
\(941\) 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
\(945\) −4.18953 −0.136286
\(946\) −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
\(952\) −3.37100 −0.109255
\(953\) −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
\(959\) −15.6126 −0.504157
\(960\) −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
\(966\) −3.62093 −0.116502
\(967\) 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 \(-0.348846\pi\)
0.457218 + 0.889355i \(0.348846\pi\)
\(972\) 1.74590 0.0559997
\(973\) 9.01641 0.289053
\(974\) −51.8521 −1.66145
\(975\) −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
\(980\) 7.31450 0.233653
\(981\) −7.61259 −0.243051
\(982\) −61.3171 −1.95671
\(983\) −53.7745 −1.71514 −0.857571 0.514366i \(-0.828027\pi\)
−0.857571 + 0.514366i \(0.828027\pi\)
\(984\) 4.60453 0.146787
\(985\) 77.5058 2.46954
\(986\) 42.0958 1.34060
\(987\) 8.06040 0.256565
\(988\) 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
\(994\) −9.70892 −0.307948
\(995\) 41.3543 1.31102
\(996\) 6.15361 0.194985
\(997\) 0.659696 0.0208928 0.0104464 0.999945i \(-0.496675\pi\)
0.0104464 + 0.999945i \(0.496675\pi\)
\(998\) 46.6943 1.47808
\(999\) −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 231.2.a.e.1.1 3
3.2 odd 2 693.2.a.l.1.3 3
4.3 odd 2 3696.2.a.bo.1.3 3
5.4 even 2 5775.2.a.bp.1.3 3
7.6 odd 2 1617.2.a.t.1.1 3
11.10 odd 2 2541.2.a.bg.1.3 3
21.20 even 2 4851.2.a.bi.1.3 3
33.32 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 1.1 even 1 trivial
693.2.a.l.1.3 3 3.2 odd 2
1617.2.a.t.1.1 3 7.6 odd 2
2541.2.a.bg.1.3 3 11.10 odd 2
3696.2.a.bo.1.3 3 4.3 odd 2
4851.2.a.bi.1.3 3 21.20 even 2
5775.2.a.bp.1.3 3 5.4 even 2
7623.2.a.cd.1.1 3 33.32 even 2