Properties

Label 2541.2.a.bg.1.3
Level 2541
Weight 2
Character 2541.1
Self dual yes
Analytic conductor 20.290
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
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 \(-0.254102\)
Character \(\chi\) = 2541.1

$q$-expansion

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

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 \(0.260117\pi\)
\(3\) 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
\(11\) 0 0
\(12\) 1.74590 0.503997
\(13\) 3.17313 0.880067 0.440034 0.897981i \(-0.354967\pi\)
0.440034 + 0.897981i \(0.354967\pi\)
\(14\) 1.93543 0.517266
\(15\) 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
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −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 \(-0.404808\pi\)
0.294617 + 0.955615i \(0.404808\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −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 \(-0.760993\pi\)
−0.731101 + 0.682269i \(0.760993\pi\)
\(42\) 1.93543 0.298644
\(43\) 10.8873 1.66029 0.830147 0.557545i \(-0.188257\pi\)
0.830147 + 0.557545i \(0.188257\pi\)
\(44\) 0 0
\(45\) 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
\(55\) 0 0
\(56\) −0.491797 −0.0657191
\(57\) 0.318669 0.0422088
\(58\) 6.14137 0.806402
\(59\) −7.04399 −0.917050 −0.458525 0.888682i \(-0.651622\pi\)
−0.458525 + 0.888682i \(0.651622\pi\)
\(60\) 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
\(63\) 1.00000 0.125988
\(64\) −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
\(69\) −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 \(-0.408846\pi\)
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 \(-0.591062\pi\)
−0.282195 + 0.959357i \(0.591062\pi\)
\(80\) −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 \(0.561964\pi\)
\(84\) 1.74590 0.190493
\(85\) −28.7170 −3.11479
\(86\) 21.0716 2.27221
\(87\) 3.17313 0.340195
\(88\) 0 0
\(89\) −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
\(99\) 0 0
\(100\) 21.9149 2.19149
\(101\) −4.88727 −0.486302 −0.243151 0.969988i \(-0.578181\pi\)
−0.243151 + 0.969988i \(0.578181\pi\)
\(102\) −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 \(-0.514714\pi\)
−0.0462104 + 0.998932i \(0.514714\pi\)
\(108\) 1.74590 0.167999
\(109\) 7.61259 0.729154 0.364577 0.931173i \(-0.381214\pi\)
0.364577 + 0.931173i \(0.381214\pi\)
\(110\) 0 0
\(111\) −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
\(121\) 0 0
\(122\) 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 \(-0.421652\pi\)
0.243659 + 0.969861i \(0.421652\pi\)
\(128\) 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 \(0.555909\pi\)
\(132\) 0 0
\(133\) 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 \(-0.375104\pi\)
0.382381 + 0.924005i \(0.375104\pi\)
\(140\) 7.31450 0.618188
\(141\) −8.06040 −0.678808
\(142\) −9.70892 −0.814754
\(143\) 0 0
\(144\) −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 \(-0.431597\pi\)
0.213244 + 0.976999i \(0.431597\pi\)
\(150\) 24.2939 1.98359
\(151\) −6.24993 −0.508612 −0.254306 0.967124i \(-0.581847\pi\)
−0.254306 + 0.967124i \(0.581847\pi\)
\(152\) −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 \(-0.757283\pi\)
−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
\(165\) 0 0
\(166\) −6.82164 −0.529462
\(167\) 11.1455 0.862468 0.431234 0.902240i \(-0.358078\pi\)
0.431234 + 0.902240i \(0.358078\pi\)
\(168\) −0.491797 −0.0379429
\(169\) −2.93126 −0.225482
\(170\) −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
\(175\) 12.5522 0.948857
\(176\) 0 0
\(177\) −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
\(187\) 0 0
\(188\) −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 \(-0.502863\pi\)
−0.00899527 + 0.999960i \(0.502863\pi\)
\(194\) −23.7089 −1.70220
\(195\) 13.2939 0.951998
\(196\) 1.74590 0.124707
\(197\) −18.4999 −1.31806 −0.659030 0.752116i \(-0.729033\pi\)
−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
\(201\) −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 0
\(210\) 8.10856 0.559544
\(211\) 4.63734 0.319248 0.159624 0.987178i \(-0.448972\pi\)
0.159624 + 0.987178i \(0.448972\pi\)
\(212\) 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
\(220\) 0 0
\(221\) −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 \(-0.504004\pi\)
−0.0125798 + 0.999921i \(0.504004\pi\)
\(228\) 0.556364 0.0368461
\(229\) 24.9424 1.64824 0.824121 0.566413i \(-0.191669\pi\)
0.824121 + 0.566413i \(0.191669\pi\)
\(230\) −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 \(0.778823\pi\)
\(234\) 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 \(-0.552503\pi\)
−0.164196 + 0.986428i \(0.552503\pi\)
\(240\) −18.6168 −1.20171
\(241\) −19.2939 −1.24283 −0.621415 0.783481i \(-0.713442\pi\)
−0.621415 + 0.783481i \(0.713442\pi\)
\(242\) 0 0
\(243\) 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
\(253\) 0 0
\(254\) 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 \(-0.528727\pi\)
−0.0901259 + 0.995930i \(0.528727\pi\)
\(264\) 0 0
\(265\) 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 \(-0.288293\pi\)
0.617136 + 0.786857i \(0.288293\pi\)
\(272\) 30.4587 1.84683
\(273\) 3.17313 0.192046
\(274\) 30.2171 1.82548
\(275\) 0 0
\(276\) −3.26634 −0.196611
\(277\) −18.0552 −1.08483 −0.542415 0.840111i \(-0.682490\pi\)
−0.542415 + 0.840111i \(0.682490\pi\)
\(278\) 17.4506 1.04662
\(279\) 9.23353 0.552797
\(280\) −2.06040 −0.123132
\(281\) −27.2939 −1.62822 −0.814110 0.580711i \(-0.802774\pi\)
−0.814110 + 0.580711i \(0.802774\pi\)
\(282\) −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
\(286\) 0 0
\(287\) −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 \(-0.459479\pi\)
0.126955 + 0.991908i \(0.459479\pi\)
\(294\) 1.93543 0.112877
\(295\) −29.5110 −1.71820
\(296\) 3.71414 0.215880
\(297\) 0 0
\(298\) 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 \(-0.298194\pi\)
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 \(-0.427167\pi\)
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
\(319\) 0 0
\(320\) −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
\(330\) 0 0
\(331\) 17.0164 0.935306 0.467653 0.883912i \(-0.345100\pi\)
0.467653 + 0.883912i \(0.345100\pi\)
\(332\) −6.15361 −0.337723
\(333\) −7.55220 −0.413858
\(334\) 21.5714 1.18034
\(335\) −11.1648 −0.609998
\(336\) −4.44364 −0.242420
\(337\) −1.52461 −0.0830508 −0.0415254 0.999137i \(-0.513222\pi\)
−0.0415254 + 0.999137i \(0.513222\pi\)
\(338\) −5.67326 −0.308585
\(339\) −7.70892 −0.418691
\(340\) −50.1369 −2.71906
\(341\) 0 0
\(342\) 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 \(-0.655169\pi\)
−0.468400 + 0.883517i \(0.655169\pi\)
\(348\) 5.53996 0.296973
\(349\) −6.85969 −0.367191 −0.183595 0.983002i \(-0.558774\pi\)
−0.183595 + 0.983002i \(0.558774\pi\)
\(350\) 24.2939 1.29856
\(351\) 3.17313 0.169369
\(352\) 0 0
\(353\) −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 \(-0.276943\pi\)
0.644795 + 0.764356i \(0.276943\pi\)
\(360\) −2.06040 −0.108593
\(361\) −18.8984 −0.994655
\(362\) −6.25827 −0.328927
\(363\) 0 0
\(364\) 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 \(-0.526222\pi\)
−0.0822845 + 0.996609i \(0.526222\pi\)
\(374\) 0 0
\(375\) 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
\(385\) 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
\(392\) −0.491797 −0.0248395
\(393\) −4.00000 −0.201773
\(394\) −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
\(399\) 0.318669 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
\(407\) 0 0
\(408\) 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
\(413\) −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 0
\(419\) −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
\(429\) 0 0
\(430\) 88.2801 4.25724
\(431\) 2.28586 0.110106 0.0550529 0.998483i \(-0.482467\pi\)
0.0550529 + 0.998483i \(0.482467\pi\)
\(432\) −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 \(-0.354222\pi\)
0.442133 + 0.896949i \(0.354222\pi\)
\(440\) 0 0
\(441\) 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
\(451\) 0 0
\(452\) −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 \(-0.401596\pi\)
0.304246 + 0.952594i \(0.401596\pi\)
\(458\) 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 \(-0.362064\pi\)
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
\(473\) 0 0
\(474\) −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 \(-0.0689307\pi\)
0.976644 + 0.214864i \(0.0689307\pi\)
\(480\) −31.9107 −1.45652
\(481\) −23.9641 −1.09267
\(482\) −37.3421 −1.70089
\(483\) −1.87086 −0.0851273
\(484\) 0 0
\(485\) −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 \(-0.753518\pi\)
−0.714879 + 0.699248i \(0.753518\pi\)
\(492\) −16.3463 −0.736946
\(493\) −21.7501 −0.979574
\(494\) 1.95707 0.0880526
\(495\) 0 0
\(496\) −41.0304 −1.84232
\(497\) −5.01641 −0.225017
\(498\) −6.82164 −0.305685
\(499\) −24.1260 −1.08003 −0.540015 0.841656i \(-0.681581\pi\)
−0.540015 + 0.841656i \(0.681581\pi\)
\(500\) 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 \(0.741773\pi\)
\(504\) −0.491797 −0.0219064
\(505\) −20.4754 −0.911143
\(506\) 0 0
\(507\) −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
\(517\) 0 0
\(518\) −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 \(-0.590420\pi\)
−0.280258 + 0.959925i \(0.590420\pi\)
\(524\) −6.98359 −0.305080
\(525\) 12.5522 0.547823
\(526\) −5.65765 −0.246685
\(527\) −63.2908 −2.75699
\(528\) 0 0
\(529\) −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
\(539\) 0 0
\(540\) 7.31450 0.314766
\(541\) 21.8625 0.939943 0.469972 0.882681i \(-0.344264\pi\)
0.469972 + 0.882681i \(0.344264\pi\)
\(542\) 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 \(-0.666901\pi\)
−0.500637 + 0.865657i \(0.666901\pi\)
\(548\) 27.2580 1.16440
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 1.01118 0.0430776
\(552\) 0.920085 0.0391614
\(553\) −5.01641 −0.213319
\(554\) −34.9446 −1.48465
\(555\) −31.6402 −1.34305
\(556\) 15.7417 0.667598
\(557\) 6.91486 0.292992 0.146496 0.989211i \(-0.453200\pi\)
0.146496 + 0.989211i \(0.453200\pi\)
\(558\) 17.8709 0.756534
\(559\) 34.5467 1.46117
\(560\) −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 \(0.580252\pi\)
\(564\) −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 \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 2.58395 0.108230
\(571\) 15.2252 0.637154 0.318577 0.947897i \(-0.396795\pi\)
0.318577 + 0.947897i \(0.396795\pi\)
\(572\) 0 0
\(573\) 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 0
\(584\) −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 \(-0.333374\pi\)
0.499890 + 0.866089i \(0.333374\pi\)
\(594\) 0 0
\(595\) −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 \(-0.814616\pi\)
−0.835145 + 0.550030i \(0.814616\pi\)
\(602\) 21.0716 0.858813
\(603\) −2.66492 −0.108524
\(604\) −10.9117 −0.443993
\(605\) 0 0
\(606\) −9.45898 −0.384245
\(607\) 14.0276 0.569362 0.284681 0.958622i \(-0.408112\pi\)
0.284681 + 0.958622i \(0.408112\pi\)
\(608\) −2.42723 −0.0984371
\(609\) 3.17313 0.128582
\(610\) 16.2171 0.656612
\(611\) −25.5767 −1.03472
\(612\) −11.9672 −0.483745
\(613\) −18.5306 −0.748442 −0.374221 0.927339i \(-0.622090\pi\)
−0.374221 + 0.927339i \(0.622090\pi\)
\(614\) 40.1760 1.62137
\(615\) −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 \(-0.709418\pi\)
−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 0
\(628\) −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
\(638\) 0 0
\(639\) −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
\(649\) 0 0
\(650\) 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 \(-0.555811\pi\)
−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
\(663\) −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
\(671\) 0 0
\(672\) −7.61676 −0.293823
\(673\) −32.2499 −1.24314 −0.621572 0.783357i \(-0.713506\pi\)
−0.621572 + 0.783357i \(0.713506\pi\)
\(674\) −2.95078 −0.113660
\(675\) 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
\(679\) −12.2499 −0.470109
\(680\) 14.1229 0.541589
\(681\) −0.379068 −0.0145259
\(682\) 0 0
\(683\) 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
\(693\) 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
\(700\) 21.9149 0.828304
\(701\) 19.2747 0.727995 0.363997 0.931400i \(-0.381412\pi\)
0.363997 + 0.931400i \(0.381412\pi\)
\(702\) 6.14137 0.231791
\(703\) −2.40665 −0.0907686
\(704\) 0 0
\(705\) −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
\(715\) 0 0
\(716\) −22.2744 −0.832434
\(717\) −5.07681 −0.189597
\(718\) 47.2908 1.76488
\(719\) 14.0276 0.523141 0.261570 0.965184i \(-0.415760\pi\)
0.261570 + 0.965184i \(0.415760\pi\)
\(720\) −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
\(726\) 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
\(729\) 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
\(735\) 4.18953 0.154533
\(736\) 14.2499 0.525259
\(737\) 0 0
\(738\) −18.1208 −0.667036
\(739\) 6.38741 0.234965 0.117482 0.993075i \(-0.462518\pi\)
0.117482 + 0.993075i \(0.462518\pi\)
\(740\) −55.2405 −2.03068
\(741\) 1.01118 0.0371466
\(742\) 0.983593 0.0361088
\(743\) 23.1096 0.847810 0.423905 0.905707i \(-0.360659\pi\)
0.423905 + 0.905707i \(0.360659\pi\)
\(744\) −4.54102 −0.166482
\(745\) 21.8105 0.799074
\(746\) −6.15149 −0.225222
\(747\) −3.52461 −0.128959
\(748\) 0 0
\(749\) −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
\(759\) 0 0
\(760\) −0.656586 −0.0238169
\(761\) 17.8625 0.647516 0.323758 0.946140i \(-0.395054\pi\)
0.323758 + 0.946140i \(0.395054\pi\)
\(762\) 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 \(-0.351910\pi\)
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 \(-0.716491\pi\)
−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
\(781\) 0 0
\(782\) 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 \(-0.412883\pi\)
0.270283 + 0.962781i \(0.412883\pi\)
\(788\) −32.2989 −1.15060
\(789\) −2.92319 −0.104068
\(790\) −40.6758 −1.44718
\(791\) −7.70892 −0.274097
\(792\) 0 0
\(793\) 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
\(803\) 0 0
\(804\) −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 \(-0.675791\pi\)
−0.524617 + 0.851338i \(0.675791\pi\)
\(810\) 8.10856 0.284906
\(811\) 16.6321 0.584032 0.292016 0.956413i \(-0.405674\pi\)
0.292016 + 0.956413i \(0.405674\pi\)
\(812\) 5.53996 0.194414
\(813\) 20.3187 0.712607
\(814\) 0 0
\(815\) −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 \(-0.677549\pi\)
−0.529309 + 0.848429i \(0.677549\pi\)
\(822\) 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
\(825\) 0 0
\(826\) −13.6332 −0.474359
\(827\) −45.4559 −1.58066 −0.790328 0.612684i \(-0.790090\pi\)
−0.790328 + 0.612684i \(0.790090\pi\)
\(828\) −3.26634 −0.113513
\(829\) −20.3463 −0.706655 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(830\) −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 0
\(837\) 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
\(847\) 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 \(0.383039\pi\)
\(854\) 3.87086 0.132458
\(855\) 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 \(0.526076\pi\)
\(858\) 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
\(861\) −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
\(869\) 0 0
\(870\) 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 \(-0.169906\pi\)
0.860893 + 0.508787i \(0.169906\pi\)
\(878\) 35.8586 1.21017
\(879\) 4.34625 0.146596
\(880\) 0 0
\(881\) 32.1895 1.08449 0.542246 0.840219i \(-0.317574\pi\)
0.542246 + 0.840219i \(0.317574\pi\)
\(882\) 1.93543 0.0651694
\(883\) 36.6154 1.23221 0.616104 0.787665i \(-0.288710\pi\)
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
\(889\) 5.49180 0.184189
\(890\) −14.1229 −0.473401
\(891\) 0 0
\(892\) −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
\(902\) 0 0
\(903\) 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
\(913\) 0 0
\(914\) 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 \(-0.345194\pi\)
0.467391 + 0.884051i \(0.345194\pi\)
\(920\) 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 \(-0.473416\pi\)
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
\(935\) 0 0
\(936\) −1.56053 −0.0510076
\(937\) −32.2088 −1.05222 −0.526108 0.850418i \(-0.676349\pi\)
−0.526108 + 0.850418i \(0.676349\pi\)
\(938\) −5.15778 −0.168407
\(939\) 8.12914 0.265284
\(940\) −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
\(945\) 4.18953 0.136286
\(946\) 0 0
\(947\) −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 \(-0.369050\pi\)
0.399884 + 0.916566i \(0.369050\pi\)
\(954\) 0.983593 0.0318450
\(955\) 87.8765 2.84362
\(956\) −8.86359 −0.286669
\(957\) 0 0
\(958\) 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 \(-0.611430\pi\)
−0.342961 + 0.939350i \(0.611430\pi\)
\(968\) 0 0
\(969\) −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
\(979\) 0 0
\(980\) 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
\(990\) 0 0
\(991\) 49.7693 1.58097 0.790487 0.612479i \(-0.209827\pi\)
0.790487 + 0.612479i \(0.209827\pi\)
\(992\) −70.3296 −2.23297
\(993\) 17.0164 0.539999
\(994\) −9.70892 −0.307948
\(995\) 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 2541.2.a.bg.1.3 3
3.2 odd 2 7623.2.a.cd.1.1 3
11.10 odd 2 231.2.a.e.1.1 3
33.32 even 2 693.2.a.l.1.3 3
44.43 even 2 3696.2.a.bo.1.3 3
55.54 odd 2 5775.2.a.bp.1.3 3
77.76 even 2 1617.2.a.t.1.1 3
231.230 odd 2 4851.2.a.bi.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.e.1.1 3 11.10 odd 2
693.2.a.l.1.3 3 33.32 even 2
1617.2.a.t.1.1 3 77.76 even 2
2541.2.a.bg.1.3 3 1.1 even 1 trivial
3696.2.a.bo.1.3 3 44.43 even 2
4851.2.a.bi.1.3 3 231.230 odd 2
5775.2.a.bp.1.3 3 55.54 odd 2
7623.2.a.cd.1.1 3 3.2 odd 2