Properties

Label 177.4.a.d.1.4
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.03574\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.0357401 q^{2} +3.00000 q^{3} -7.99872 q^{4} -7.80970 q^{5} -0.107220 q^{6} +4.90513 q^{7} +0.571796 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.0357401 q^{2} +3.00000 q^{3} -7.99872 q^{4} -7.80970 q^{5} -0.107220 q^{6} +4.90513 q^{7} +0.571796 q^{8} +9.00000 q^{9} +0.279120 q^{10} +37.9474 q^{11} -23.9962 q^{12} +27.2075 q^{13} -0.175310 q^{14} -23.4291 q^{15} +63.9693 q^{16} +63.3515 q^{17} -0.321661 q^{18} +97.9685 q^{19} +62.4676 q^{20} +14.7154 q^{21} -1.35625 q^{22} +9.58224 q^{23} +1.71539 q^{24} -64.0086 q^{25} -0.972399 q^{26} +27.0000 q^{27} -39.2348 q^{28} +65.2932 q^{29} +0.837359 q^{30} -165.898 q^{31} -6.86064 q^{32} +113.842 q^{33} -2.26419 q^{34} -38.3076 q^{35} -71.9885 q^{36} +215.965 q^{37} -3.50141 q^{38} +81.6224 q^{39} -4.46556 q^{40} +61.9926 q^{41} -0.525930 q^{42} +77.4402 q^{43} -303.531 q^{44} -70.2873 q^{45} -0.342470 q^{46} +189.151 q^{47} +191.908 q^{48} -318.940 q^{49} +2.28768 q^{50} +190.055 q^{51} -217.625 q^{52} +558.039 q^{53} -0.964984 q^{54} -296.358 q^{55} +2.80474 q^{56} +293.906 q^{57} -2.33359 q^{58} +59.0000 q^{59} +187.403 q^{60} +451.056 q^{61} +5.92922 q^{62} +44.1462 q^{63} -511.510 q^{64} -212.482 q^{65} -4.06874 q^{66} -730.287 q^{67} -506.731 q^{68} +28.7467 q^{69} +1.36912 q^{70} +1040.27 q^{71} +5.14617 q^{72} -529.757 q^{73} -7.71862 q^{74} -192.026 q^{75} -783.623 q^{76} +186.137 q^{77} -2.91720 q^{78} -1007.53 q^{79} -499.581 q^{80} +81.0000 q^{81} -2.21562 q^{82} -393.059 q^{83} -117.704 q^{84} -494.756 q^{85} -2.76772 q^{86} +195.880 q^{87} +21.6982 q^{88} -50.5608 q^{89} +2.51208 q^{90} +133.456 q^{91} -76.6457 q^{92} -497.694 q^{93} -6.76028 q^{94} -765.105 q^{95} -20.5819 q^{96} -1043.06 q^{97} +11.3989 q^{98} +341.527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + 21q^{10} + 67q^{11} + 102q^{12} + 33q^{13} + 79q^{14} + 126q^{15} - 30q^{16} + 139q^{17} + 54q^{18} + 64q^{19} + 117q^{20} + 159q^{21} - 84q^{22} + 226q^{23} + 153q^{24} + 96q^{25} + 24q^{26} + 216q^{27} + 34q^{28} + 456q^{29} + 63q^{30} + 124q^{31} + 174q^{32} + 201q^{33} - 114q^{34} + 556q^{35} + 306q^{36} + 127q^{37} + 237q^{38} + 99q^{39} - 188q^{40} + 425q^{41} + 237q^{42} - 115q^{43} + 510q^{44} + 378q^{45} - 711q^{46} + 420q^{47} - 90q^{48} + 171q^{49} - 137q^{50} + 417q^{51} - 922q^{52} + 98q^{53} + 162q^{54} - 616q^{55} - 412q^{56} + 192q^{57} - 1548q^{58} + 472q^{59} + 351q^{60} - 1254q^{61} - 766q^{62} + 477q^{63} - 2019q^{64} - 734q^{65} - 252q^{66} - 1010q^{67} - 503q^{68} + 678q^{69} - 2956q^{70} - 17q^{71} + 459q^{72} - 1180q^{73} - 1228q^{74} + 288q^{75} - 2008q^{76} + 441q^{77} + 72q^{78} - 873q^{79} - 865q^{80} + 648q^{81} - 3645q^{82} + 759q^{83} + 102q^{84} - 850q^{85} - 1226q^{86} + 1368q^{87} - 3047q^{88} + 988q^{89} + 189q^{90} - 2111q^{91} - 1062q^{92} + 372q^{93} - 2240q^{94} + 1822q^{95} + 522q^{96} - 668q^{97} - 1368q^{98} + 603q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0357401 −0.0126360 −0.00631802 0.999980i \(-0.502011\pi\)
−0.00631802 + 0.999980i \(0.502011\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.99872 −0.999840
\(5\) −7.80970 −0.698521 −0.349260 0.937026i \(-0.613567\pi\)
−0.349260 + 0.937026i \(0.613567\pi\)
\(6\) −0.107220 −0.00729542
\(7\) 4.90513 0.264852 0.132426 0.991193i \(-0.457723\pi\)
0.132426 + 0.991193i \(0.457723\pi\)
\(8\) 0.571796 0.0252701
\(9\) 9.00000 0.333333
\(10\) 0.279120 0.00882654
\(11\) 37.9474 1.04014 0.520071 0.854123i \(-0.325905\pi\)
0.520071 + 0.854123i \(0.325905\pi\)
\(12\) −23.9962 −0.577258
\(13\) 27.2075 0.580461 0.290231 0.956957i \(-0.406268\pi\)
0.290231 + 0.956957i \(0.406268\pi\)
\(14\) −0.175310 −0.00334668
\(15\) −23.4291 −0.403291
\(16\) 63.9693 0.999521
\(17\) 63.3515 0.903824 0.451912 0.892063i \(-0.350742\pi\)
0.451912 + 0.892063i \(0.350742\pi\)
\(18\) −0.321661 −0.00421202
\(19\) 97.9685 1.18292 0.591461 0.806334i \(-0.298551\pi\)
0.591461 + 0.806334i \(0.298551\pi\)
\(20\) 62.4676 0.698409
\(21\) 14.7154 0.152912
\(22\) −1.35625 −0.0131433
\(23\) 9.58224 0.0868711 0.0434355 0.999056i \(-0.486170\pi\)
0.0434355 + 0.999056i \(0.486170\pi\)
\(24\) 1.71539 0.0145897
\(25\) −64.0086 −0.512069
\(26\) −0.972399 −0.00733473
\(27\) 27.0000 0.192450
\(28\) −39.2348 −0.264810
\(29\) 65.2932 0.418091 0.209046 0.977906i \(-0.432964\pi\)
0.209046 + 0.977906i \(0.432964\pi\)
\(30\) 0.837359 0.00509600
\(31\) −165.898 −0.961167 −0.480583 0.876949i \(-0.659575\pi\)
−0.480583 + 0.876949i \(0.659575\pi\)
\(32\) −6.86064 −0.0379001
\(33\) 113.842 0.600527
\(34\) −2.26419 −0.0114208
\(35\) −38.3076 −0.185005
\(36\) −71.9885 −0.333280
\(37\) 215.965 0.959580 0.479790 0.877383i \(-0.340713\pi\)
0.479790 + 0.877383i \(0.340713\pi\)
\(38\) −3.50141 −0.0149475
\(39\) 81.6224 0.335129
\(40\) −4.46556 −0.0176517
\(41\) 61.9926 0.236137 0.118068 0.993005i \(-0.462330\pi\)
0.118068 + 0.993005i \(0.462330\pi\)
\(42\) −0.525930 −0.00193221
\(43\) 77.4402 0.274640 0.137320 0.990527i \(-0.456151\pi\)
0.137320 + 0.990527i \(0.456151\pi\)
\(44\) −303.531 −1.03998
\(45\) −70.2873 −0.232840
\(46\) −0.342470 −0.00109771
\(47\) 189.151 0.587032 0.293516 0.955954i \(-0.405175\pi\)
0.293516 + 0.955954i \(0.405175\pi\)
\(48\) 191.908 0.577074
\(49\) −318.940 −0.929853
\(50\) 2.28768 0.00647053
\(51\) 190.055 0.521823
\(52\) −217.625 −0.580368
\(53\) 558.039 1.44627 0.723137 0.690704i \(-0.242699\pi\)
0.723137 + 0.690704i \(0.242699\pi\)
\(54\) −0.964984 −0.00243181
\(55\) −296.358 −0.726561
\(56\) 2.80474 0.00669283
\(57\) 293.906 0.682960
\(58\) −2.33359 −0.00528302
\(59\) 59.0000 0.130189
\(60\) 187.403 0.403227
\(61\) 451.056 0.946752 0.473376 0.880861i \(-0.343035\pi\)
0.473376 + 0.880861i \(0.343035\pi\)
\(62\) 5.92922 0.0121453
\(63\) 44.1462 0.0882841
\(64\) −511.510 −0.999042
\(65\) −212.482 −0.405464
\(66\) −4.06874 −0.00758828
\(67\) −730.287 −1.33162 −0.665811 0.746120i \(-0.731914\pi\)
−0.665811 + 0.746120i \(0.731914\pi\)
\(68\) −506.731 −0.903679
\(69\) 28.7467 0.0501550
\(70\) 1.36912 0.00233773
\(71\) 1040.27 1.73883 0.869414 0.494084i \(-0.164497\pi\)
0.869414 + 0.494084i \(0.164497\pi\)
\(72\) 5.14617 0.00842336
\(73\) −529.757 −0.849361 −0.424680 0.905343i \(-0.639613\pi\)
−0.424680 + 0.905343i \(0.639613\pi\)
\(74\) −7.71862 −0.0121253
\(75\) −192.026 −0.295643
\(76\) −783.623 −1.18273
\(77\) 186.137 0.275484
\(78\) −2.91720 −0.00423471
\(79\) −1007.53 −1.43488 −0.717440 0.696620i \(-0.754686\pi\)
−0.717440 + 0.696620i \(0.754686\pi\)
\(80\) −499.581 −0.698186
\(81\) 81.0000 0.111111
\(82\) −2.21562 −0.00298384
\(83\) −393.059 −0.519805 −0.259903 0.965635i \(-0.583690\pi\)
−0.259903 + 0.965635i \(0.583690\pi\)
\(84\) −117.704 −0.152888
\(85\) −494.756 −0.631339
\(86\) −2.76772 −0.00347036
\(87\) 195.880 0.241385
\(88\) 21.6982 0.0262845
\(89\) −50.5608 −0.0602183 −0.0301092 0.999547i \(-0.509585\pi\)
−0.0301092 + 0.999547i \(0.509585\pi\)
\(90\) 2.51208 0.00294218
\(91\) 133.456 0.153736
\(92\) −76.6457 −0.0868572
\(93\) −497.694 −0.554930
\(94\) −6.76028 −0.00741776
\(95\) −765.105 −0.826296
\(96\) −20.5819 −0.0218816
\(97\) −1043.06 −1.09182 −0.545912 0.837843i \(-0.683817\pi\)
−0.545912 + 0.837843i \(0.683817\pi\)
\(98\) 11.3989 0.0117497
\(99\) 341.527 0.346714
\(100\) 511.987 0.511987
\(101\) 1481.74 1.45978 0.729892 0.683563i \(-0.239570\pi\)
0.729892 + 0.683563i \(0.239570\pi\)
\(102\) −6.79258 −0.00659378
\(103\) 61.1482 0.0584962 0.0292481 0.999572i \(-0.490689\pi\)
0.0292481 + 0.999572i \(0.490689\pi\)
\(104\) 15.5571 0.0146683
\(105\) −114.923 −0.106813
\(106\) −19.9444 −0.0182752
\(107\) −1057.95 −0.955847 −0.477923 0.878401i \(-0.658610\pi\)
−0.477923 + 0.878401i \(0.658610\pi\)
\(108\) −215.966 −0.192419
\(109\) 773.766 0.679939 0.339970 0.940436i \(-0.389583\pi\)
0.339970 + 0.940436i \(0.389583\pi\)
\(110\) 10.5919 0.00918086
\(111\) 647.895 0.554014
\(112\) 313.778 0.264725
\(113\) 170.203 0.141694 0.0708469 0.997487i \(-0.477430\pi\)
0.0708469 + 0.997487i \(0.477430\pi\)
\(114\) −10.5042 −0.00862992
\(115\) −74.8344 −0.0606812
\(116\) −522.262 −0.418024
\(117\) 244.867 0.193487
\(118\) −2.10867 −0.00164507
\(119\) 310.748 0.239380
\(120\) −13.3967 −0.0101912
\(121\) 109.005 0.0818971
\(122\) −16.1208 −0.0119632
\(123\) 185.978 0.136334
\(124\) 1326.97 0.961013
\(125\) 1476.10 1.05621
\(126\) −1.57779 −0.00111556
\(127\) 148.411 0.103695 0.0518477 0.998655i \(-0.483489\pi\)
0.0518477 + 0.998655i \(0.483489\pi\)
\(128\) 73.1666 0.0505240
\(129\) 232.321 0.158563
\(130\) 7.59414 0.00512346
\(131\) −514.797 −0.343344 −0.171672 0.985154i \(-0.554917\pi\)
−0.171672 + 0.985154i \(0.554917\pi\)
\(132\) −910.592 −0.600431
\(133\) 480.549 0.313300
\(134\) 26.1005 0.0168264
\(135\) −210.862 −0.134430
\(136\) 36.2242 0.0228397
\(137\) 2145.47 1.33795 0.668976 0.743284i \(-0.266733\pi\)
0.668976 + 0.743284i \(0.266733\pi\)
\(138\) −1.02741 −0.000633761 0
\(139\) 951.765 0.580775 0.290387 0.956909i \(-0.406216\pi\)
0.290387 + 0.956909i \(0.406216\pi\)
\(140\) 306.412 0.184975
\(141\) 567.452 0.338923
\(142\) −37.1792 −0.0219719
\(143\) 1032.45 0.603762
\(144\) 575.724 0.333174
\(145\) −509.920 −0.292045
\(146\) 18.9336 0.0107326
\(147\) −956.819 −0.536851
\(148\) −1727.45 −0.959426
\(149\) −2363.38 −1.29944 −0.649718 0.760175i \(-0.725113\pi\)
−0.649718 + 0.760175i \(0.725113\pi\)
\(150\) 6.86303 0.00373576
\(151\) 2047.04 1.10322 0.551609 0.834103i \(-0.314014\pi\)
0.551609 + 0.834103i \(0.314014\pi\)
\(152\) 56.0181 0.0298925
\(153\) 570.164 0.301275
\(154\) −6.65256 −0.00348103
\(155\) 1295.61 0.671395
\(156\) −652.875 −0.335076
\(157\) 1779.17 0.904416 0.452208 0.891912i \(-0.350636\pi\)
0.452208 + 0.891912i \(0.350636\pi\)
\(158\) 36.0091 0.0181312
\(159\) 1674.12 0.835007
\(160\) 53.5796 0.0264740
\(161\) 47.0021 0.0230080
\(162\) −2.89495 −0.00140401
\(163\) 389.908 0.187362 0.0936808 0.995602i \(-0.470137\pi\)
0.0936808 + 0.995602i \(0.470137\pi\)
\(164\) −495.861 −0.236099
\(165\) −889.073 −0.419480
\(166\) 14.0480 0.00656828
\(167\) −1317.81 −0.610632 −0.305316 0.952251i \(-0.598762\pi\)
−0.305316 + 0.952251i \(0.598762\pi\)
\(168\) 8.41421 0.00386411
\(169\) −1456.75 −0.663065
\(170\) 17.6827 0.00797763
\(171\) 881.717 0.394307
\(172\) −619.423 −0.274596
\(173\) −821.841 −0.361176 −0.180588 0.983559i \(-0.557800\pi\)
−0.180588 + 0.983559i \(0.557800\pi\)
\(174\) −7.00076 −0.00305015
\(175\) −313.971 −0.135623
\(176\) 2427.47 1.03964
\(177\) 177.000 0.0751646
\(178\) 1.80705 0.000760921 0
\(179\) −592.023 −0.247206 −0.123603 0.992332i \(-0.539445\pi\)
−0.123603 + 0.992332i \(0.539445\pi\)
\(180\) 562.208 0.232803
\(181\) −812.424 −0.333630 −0.166815 0.985988i \(-0.553348\pi\)
−0.166815 + 0.985988i \(0.553348\pi\)
\(182\) −4.76974 −0.00194262
\(183\) 1353.17 0.546607
\(184\) 5.47909 0.00219524
\(185\) −1686.62 −0.670286
\(186\) 17.7877 0.00701212
\(187\) 2404.03 0.940106
\(188\) −1512.96 −0.586938
\(189\) 132.439 0.0509708
\(190\) 27.3449 0.0104411
\(191\) 2821.53 1.06890 0.534448 0.845202i \(-0.320520\pi\)
0.534448 + 0.845202i \(0.320520\pi\)
\(192\) −1534.53 −0.576797
\(193\) −2254.59 −0.840875 −0.420437 0.907322i \(-0.638123\pi\)
−0.420437 + 0.907322i \(0.638123\pi\)
\(194\) 37.2792 0.0137963
\(195\) −637.446 −0.234095
\(196\) 2551.11 0.929705
\(197\) −2019.03 −0.730203 −0.365102 0.930968i \(-0.618966\pi\)
−0.365102 + 0.930968i \(0.618966\pi\)
\(198\) −12.2062 −0.00438110
\(199\) 1504.81 0.536047 0.268024 0.963412i \(-0.413630\pi\)
0.268024 + 0.963412i \(0.413630\pi\)
\(200\) −36.5999 −0.0129400
\(201\) −2190.86 −0.768813
\(202\) −52.9574 −0.0184459
\(203\) 320.272 0.110732
\(204\) −1520.19 −0.521740
\(205\) −484.143 −0.164946
\(206\) −2.18544 −0.000739161 0
\(207\) 86.2401 0.0289570
\(208\) 1740.44 0.580183
\(209\) 3717.65 1.23041
\(210\) 4.10736 0.00134969
\(211\) 1789.09 0.583724 0.291862 0.956460i \(-0.405725\pi\)
0.291862 + 0.956460i \(0.405725\pi\)
\(212\) −4463.60 −1.44604
\(213\) 3120.80 1.00391
\(214\) 37.8112 0.0120781
\(215\) −604.784 −0.191842
\(216\) 15.4385 0.00486323
\(217\) −813.752 −0.254567
\(218\) −27.6545 −0.00859174
\(219\) −1589.27 −0.490379
\(220\) 2370.48 0.726445
\(221\) 1723.63 0.524634
\(222\) −23.1559 −0.00700054
\(223\) −1853.90 −0.556710 −0.278355 0.960478i \(-0.589789\pi\)
−0.278355 + 0.960478i \(0.589789\pi\)
\(224\) −33.6524 −0.0100379
\(225\) −576.078 −0.170690
\(226\) −6.08309 −0.00179045
\(227\) 2113.71 0.618026 0.309013 0.951058i \(-0.400001\pi\)
0.309013 + 0.951058i \(0.400001\pi\)
\(228\) −2350.87 −0.682851
\(229\) −993.951 −0.286822 −0.143411 0.989663i \(-0.545807\pi\)
−0.143411 + 0.989663i \(0.545807\pi\)
\(230\) 2.67459 0.000766771 0
\(231\) 558.411 0.159051
\(232\) 37.3344 0.0105652
\(233\) 1982.82 0.557505 0.278753 0.960363i \(-0.410079\pi\)
0.278753 + 0.960363i \(0.410079\pi\)
\(234\) −8.75159 −0.00244491
\(235\) −1477.21 −0.410054
\(236\) −471.925 −0.130168
\(237\) −3022.58 −0.828429
\(238\) −11.1062 −0.00302481
\(239\) −4186.15 −1.13297 −0.566485 0.824072i \(-0.691697\pi\)
−0.566485 + 0.824072i \(0.691697\pi\)
\(240\) −1498.74 −0.403098
\(241\) 455.312 0.121698 0.0608490 0.998147i \(-0.480619\pi\)
0.0608490 + 0.998147i \(0.480619\pi\)
\(242\) −3.89586 −0.00103486
\(243\) 243.000 0.0641500
\(244\) −3607.87 −0.946600
\(245\) 2490.82 0.649522
\(246\) −6.64687 −0.00172272
\(247\) 2665.48 0.686640
\(248\) −94.8599 −0.0242887
\(249\) −1179.18 −0.300110
\(250\) −52.7560 −0.0133463
\(251\) −7160.56 −1.80068 −0.900340 0.435188i \(-0.856682\pi\)
−0.900340 + 0.435188i \(0.856682\pi\)
\(252\) −353.113 −0.0882700
\(253\) 363.621 0.0903583
\(254\) −5.30421 −0.00131030
\(255\) −1484.27 −0.364504
\(256\) 4089.46 0.998404
\(257\) −813.094 −0.197352 −0.0986759 0.995120i \(-0.531461\pi\)
−0.0986759 + 0.995120i \(0.531461\pi\)
\(258\) −8.30317 −0.00200361
\(259\) 1059.34 0.254147
\(260\) 1699.59 0.405399
\(261\) 587.639 0.139364
\(262\) 18.3989 0.00433851
\(263\) −3135.36 −0.735112 −0.367556 0.930001i \(-0.619805\pi\)
−0.367556 + 0.930001i \(0.619805\pi\)
\(264\) 65.0946 0.0151754
\(265\) −4358.12 −1.01025
\(266\) −17.1749 −0.00395887
\(267\) −151.682 −0.0347671
\(268\) 5841.36 1.33141
\(269\) −4459.37 −1.01075 −0.505377 0.862899i \(-0.668646\pi\)
−0.505377 + 0.862899i \(0.668646\pi\)
\(270\) 7.53623 0.00169867
\(271\) 5557.17 1.24566 0.622830 0.782357i \(-0.285983\pi\)
0.622830 + 0.782357i \(0.285983\pi\)
\(272\) 4052.56 0.903391
\(273\) 400.369 0.0887598
\(274\) −76.6792 −0.0169064
\(275\) −2428.96 −0.532625
\(276\) −229.937 −0.0501470
\(277\) −3618.46 −0.784881 −0.392440 0.919777i \(-0.628369\pi\)
−0.392440 + 0.919777i \(0.628369\pi\)
\(278\) −34.0162 −0.00733869
\(279\) −1493.08 −0.320389
\(280\) −21.9041 −0.00467508
\(281\) −4971.01 −1.05532 −0.527661 0.849455i \(-0.676931\pi\)
−0.527661 + 0.849455i \(0.676931\pi\)
\(282\) −20.2808 −0.00428264
\(283\) −4836.09 −1.01582 −0.507908 0.861412i \(-0.669581\pi\)
−0.507908 + 0.861412i \(0.669581\pi\)
\(284\) −8320.80 −1.73855
\(285\) −2295.31 −0.477062
\(286\) −36.9000 −0.00762917
\(287\) 304.082 0.0625414
\(288\) −61.7458 −0.0126334
\(289\) −899.584 −0.183103
\(290\) 18.2246 0.00369030
\(291\) −3129.19 −0.630365
\(292\) 4237.38 0.849225
\(293\) −6061.44 −1.20858 −0.604288 0.796766i \(-0.706543\pi\)
−0.604288 + 0.796766i \(0.706543\pi\)
\(294\) 34.1968 0.00678367
\(295\) −460.772 −0.0909396
\(296\) 123.488 0.0242486
\(297\) 1024.58 0.200176
\(298\) 84.4677 0.0164197
\(299\) 260.708 0.0504253
\(300\) 1535.96 0.295596
\(301\) 379.854 0.0727390
\(302\) −73.1615 −0.0139403
\(303\) 4445.21 0.842807
\(304\) 6266.98 1.18236
\(305\) −3522.61 −0.661326
\(306\) −20.3777 −0.00380692
\(307\) −3807.74 −0.707880 −0.353940 0.935268i \(-0.615158\pi\)
−0.353940 + 0.935268i \(0.615158\pi\)
\(308\) −1488.86 −0.275440
\(309\) 183.445 0.0337728
\(310\) −46.3054 −0.00848377
\(311\) 7633.34 1.39179 0.695896 0.718143i \(-0.255008\pi\)
0.695896 + 0.718143i \(0.255008\pi\)
\(312\) 46.6714 0.00846874
\(313\) −5921.30 −1.06930 −0.534651 0.845073i \(-0.679557\pi\)
−0.534651 + 0.845073i \(0.679557\pi\)
\(314\) −63.5878 −0.0114282
\(315\) −344.768 −0.0616682
\(316\) 8058.92 1.43465
\(317\) −1748.08 −0.309722 −0.154861 0.987936i \(-0.549493\pi\)
−0.154861 + 0.987936i \(0.549493\pi\)
\(318\) −59.8332 −0.0105512
\(319\) 2477.71 0.434875
\(320\) 3994.74 0.697852
\(321\) −3173.84 −0.551858
\(322\) −1.67986 −0.000290730 0
\(323\) 6206.46 1.06915
\(324\) −647.897 −0.111093
\(325\) −1741.51 −0.297236
\(326\) −13.9354 −0.00236751
\(327\) 2321.30 0.392563
\(328\) 35.4471 0.00596720
\(329\) 927.810 0.155477
\(330\) 31.7756 0.00530057
\(331\) 10853.1 1.80223 0.901116 0.433579i \(-0.142749\pi\)
0.901116 + 0.433579i \(0.142749\pi\)
\(332\) 3143.97 0.519722
\(333\) 1943.69 0.319860
\(334\) 47.0989 0.00771597
\(335\) 5703.32 0.930166
\(336\) 941.334 0.152839
\(337\) 8975.80 1.45087 0.725435 0.688291i \(-0.241639\pi\)
0.725435 + 0.688291i \(0.241639\pi\)
\(338\) 52.0646 0.00837852
\(339\) 510.610 0.0818069
\(340\) 3957.42 0.631239
\(341\) −6295.40 −0.999750
\(342\) −31.5127 −0.00498249
\(343\) −3246.90 −0.511126
\(344\) 44.2800 0.00694017
\(345\) −224.503 −0.0350343
\(346\) 29.3727 0.00456384
\(347\) 6240.09 0.965377 0.482688 0.875792i \(-0.339660\pi\)
0.482688 + 0.875792i \(0.339660\pi\)
\(348\) −1566.79 −0.241347
\(349\) −12565.4 −1.92724 −0.963621 0.267271i \(-0.913878\pi\)
−0.963621 + 0.267271i \(0.913878\pi\)
\(350\) 11.2214 0.00171373
\(351\) 734.602 0.111710
\(352\) −260.344 −0.0394215
\(353\) 4825.08 0.727516 0.363758 0.931494i \(-0.381494\pi\)
0.363758 + 0.931494i \(0.381494\pi\)
\(354\) −6.32600 −0.000949783 0
\(355\) −8124.16 −1.21461
\(356\) 404.422 0.0602087
\(357\) 932.243 0.138206
\(358\) 21.1590 0.00312371
\(359\) −2919.83 −0.429256 −0.214628 0.976696i \(-0.568854\pi\)
−0.214628 + 0.976696i \(0.568854\pi\)
\(360\) −40.1900 −0.00588389
\(361\) 2738.83 0.399305
\(362\) 29.0362 0.00421576
\(363\) 327.015 0.0472833
\(364\) −1067.48 −0.153712
\(365\) 4137.24 0.593296
\(366\) −48.3624 −0.00690695
\(367\) −9268.74 −1.31832 −0.659161 0.752002i \(-0.729088\pi\)
−0.659161 + 0.752002i \(0.729088\pi\)
\(368\) 612.969 0.0868295
\(369\) 557.933 0.0787123
\(370\) 60.2801 0.00846977
\(371\) 2737.26 0.383049
\(372\) 3980.92 0.554841
\(373\) 6510.93 0.903816 0.451908 0.892065i \(-0.350744\pi\)
0.451908 + 0.892065i \(0.350744\pi\)
\(374\) −85.9202 −0.0118792
\(375\) 4428.30 0.609804
\(376\) 108.156 0.0148343
\(377\) 1776.46 0.242686
\(378\) −4.73337 −0.000644070 0
\(379\) −2068.45 −0.280341 −0.140171 0.990127i \(-0.544765\pi\)
−0.140171 + 0.990127i \(0.544765\pi\)
\(380\) 6119.86 0.826164
\(381\) 445.232 0.0598685
\(382\) −100.842 −0.0135066
\(383\) 12258.5 1.63545 0.817727 0.575606i \(-0.195234\pi\)
0.817727 + 0.575606i \(0.195234\pi\)
\(384\) 219.500 0.0291700
\(385\) −1453.67 −0.192431
\(386\) 80.5793 0.0106253
\(387\) 696.962 0.0915466
\(388\) 8343.16 1.09165
\(389\) −4389.85 −0.572171 −0.286085 0.958204i \(-0.592354\pi\)
−0.286085 + 0.958204i \(0.592354\pi\)
\(390\) 22.7824 0.00295803
\(391\) 607.049 0.0785161
\(392\) −182.369 −0.0234975
\(393\) −1544.39 −0.198230
\(394\) 72.1604 0.00922688
\(395\) 7868.48 1.00229
\(396\) −2731.78 −0.346659
\(397\) 4517.53 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(398\) −53.7822 −0.00677352
\(399\) 1441.65 0.180884
\(400\) −4094.59 −0.511824
\(401\) −4353.84 −0.542195 −0.271098 0.962552i \(-0.587387\pi\)
−0.271098 + 0.962552i \(0.587387\pi\)
\(402\) 78.3016 0.00971475
\(403\) −4513.66 −0.557920
\(404\) −11852.0 −1.45955
\(405\) −632.586 −0.0776134
\(406\) −11.4466 −0.00139922
\(407\) 8195.31 0.998100
\(408\) 108.673 0.0131865
\(409\) −12382.7 −1.49702 −0.748512 0.663121i \(-0.769231\pi\)
−0.748512 + 0.663121i \(0.769231\pi\)
\(410\) 17.3033 0.00208427
\(411\) 6436.40 0.772467
\(412\) −489.107 −0.0584869
\(413\) 289.403 0.0344808
\(414\) −3.08223 −0.000365902 0
\(415\) 3069.67 0.363095
\(416\) −186.661 −0.0219995
\(417\) 2855.30 0.335310
\(418\) −132.869 −0.0155475
\(419\) 2273.01 0.265020 0.132510 0.991182i \(-0.457696\pi\)
0.132510 + 0.991182i \(0.457696\pi\)
\(420\) 919.236 0.106795
\(421\) −11870.2 −1.37415 −0.687076 0.726586i \(-0.741106\pi\)
−0.687076 + 0.726586i \(0.741106\pi\)
\(422\) −63.9422 −0.00737597
\(423\) 1702.36 0.195677
\(424\) 319.085 0.0365475
\(425\) −4055.04 −0.462820
\(426\) −111.538 −0.0126855
\(427\) 2212.49 0.250749
\(428\) 8462.23 0.955694
\(429\) 3097.36 0.348582
\(430\) 21.6151 0.00242412
\(431\) −15025.7 −1.67926 −0.839629 0.543160i \(-0.817228\pi\)
−0.839629 + 0.543160i \(0.817228\pi\)
\(432\) 1727.17 0.192358
\(433\) −16251.4 −1.80368 −0.901838 0.432074i \(-0.857782\pi\)
−0.901838 + 0.432074i \(0.857782\pi\)
\(434\) 29.0836 0.00321672
\(435\) −1529.76 −0.168612
\(436\) −6189.14 −0.679831
\(437\) 938.758 0.102762
\(438\) 56.8007 0.00619645
\(439\) 2884.92 0.313644 0.156822 0.987627i \(-0.449875\pi\)
0.156822 + 0.987627i \(0.449875\pi\)
\(440\) −169.456 −0.0183603
\(441\) −2870.46 −0.309951
\(442\) −61.6029 −0.00662930
\(443\) 9495.38 1.01837 0.509187 0.860656i \(-0.329946\pi\)
0.509187 + 0.860656i \(0.329946\pi\)
\(444\) −5182.34 −0.553925
\(445\) 394.864 0.0420637
\(446\) 66.2586 0.00703461
\(447\) −7090.15 −0.750230
\(448\) −2509.02 −0.264599
\(449\) 8540.14 0.897626 0.448813 0.893626i \(-0.351847\pi\)
0.448813 + 0.893626i \(0.351847\pi\)
\(450\) 20.5891 0.00215684
\(451\) 2352.46 0.245616
\(452\) −1361.41 −0.141671
\(453\) 6141.12 0.636943
\(454\) −75.5443 −0.00780940
\(455\) −1042.25 −0.107388
\(456\) 168.054 0.0172585
\(457\) −7167.88 −0.733696 −0.366848 0.930281i \(-0.619563\pi\)
−0.366848 + 0.930281i \(0.619563\pi\)
\(458\) 35.5239 0.00362429
\(459\) 1710.49 0.173941
\(460\) 598.579 0.0606715
\(461\) −5091.59 −0.514401 −0.257200 0.966358i \(-0.582800\pi\)
−0.257200 + 0.966358i \(0.582800\pi\)
\(462\) −19.9577 −0.00200977
\(463\) 9870.55 0.990764 0.495382 0.868675i \(-0.335028\pi\)
0.495382 + 0.868675i \(0.335028\pi\)
\(464\) 4176.76 0.417891
\(465\) 3886.84 0.387630
\(466\) −70.8662 −0.00704466
\(467\) −16600.2 −1.64489 −0.822446 0.568844i \(-0.807391\pi\)
−0.822446 + 0.568844i \(0.807391\pi\)
\(468\) −1958.62 −0.193456
\(469\) −3582.15 −0.352683
\(470\) 52.7957 0.00518146
\(471\) 5337.52 0.522165
\(472\) 33.7360 0.00328988
\(473\) 2938.65 0.285665
\(474\) 108.027 0.0104681
\(475\) −6270.83 −0.605738
\(476\) −2485.58 −0.239341
\(477\) 5022.35 0.482092
\(478\) 149.614 0.0143163
\(479\) 11476.4 1.09472 0.547360 0.836897i \(-0.315633\pi\)
0.547360 + 0.836897i \(0.315633\pi\)
\(480\) 160.739 0.0152848
\(481\) 5875.86 0.556999
\(482\) −16.2729 −0.00153778
\(483\) 141.006 0.0132837
\(484\) −871.901 −0.0818840
\(485\) 8146.00 0.762661
\(486\) −8.68485 −0.000810603 0
\(487\) 15808.1 1.47091 0.735455 0.677574i \(-0.236969\pi\)
0.735455 + 0.677574i \(0.236969\pi\)
\(488\) 257.912 0.0239245
\(489\) 1169.72 0.108173
\(490\) −89.0223 −0.00820739
\(491\) 279.729 0.0257108 0.0128554 0.999917i \(-0.495908\pi\)
0.0128554 + 0.999917i \(0.495908\pi\)
\(492\) −1487.58 −0.136312
\(493\) 4136.42 0.377881
\(494\) −95.2645 −0.00867642
\(495\) −2667.22 −0.242187
\(496\) −10612.4 −0.960706
\(497\) 5102.64 0.460533
\(498\) 42.1440 0.00379220
\(499\) −11541.3 −1.03539 −0.517695 0.855565i \(-0.673210\pi\)
−0.517695 + 0.855565i \(0.673210\pi\)
\(500\) −11806.9 −1.05604
\(501\) −3953.44 −0.352549
\(502\) 255.919 0.0227535
\(503\) −6614.84 −0.586364 −0.293182 0.956057i \(-0.594714\pi\)
−0.293182 + 0.956057i \(0.594714\pi\)
\(504\) 25.2426 0.00223094
\(505\) −11571.9 −1.01969
\(506\) −12.9959 −0.00114177
\(507\) −4370.26 −0.382821
\(508\) −1187.09 −0.103679
\(509\) 4802.47 0.418204 0.209102 0.977894i \(-0.432946\pi\)
0.209102 + 0.977894i \(0.432946\pi\)
\(510\) 53.0480 0.00460589
\(511\) −2598.53 −0.224955
\(512\) −731.491 −0.0631399
\(513\) 2645.15 0.227653
\(514\) 29.0601 0.00249375
\(515\) −477.549 −0.0408608
\(516\) −1858.27 −0.158538
\(517\) 7177.78 0.610597
\(518\) −37.8609 −0.00321141
\(519\) −2465.52 −0.208525
\(520\) −121.497 −0.0102461
\(521\) −13950.7 −1.17311 −0.586557 0.809908i \(-0.699517\pi\)
−0.586557 + 0.809908i \(0.699517\pi\)
\(522\) −21.0023 −0.00176101
\(523\) −4290.15 −0.358690 −0.179345 0.983786i \(-0.557398\pi\)
−0.179345 + 0.983786i \(0.557398\pi\)
\(524\) 4117.72 0.343289
\(525\) −941.912 −0.0783017
\(526\) 112.058 0.00928891
\(527\) −10509.9 −0.868725
\(528\) 7282.41 0.600239
\(529\) −12075.2 −0.992453
\(530\) 155.760 0.0127656
\(531\) 531.000 0.0433963
\(532\) −3843.77 −0.313250
\(533\) 1686.66 0.137068
\(534\) 5.42115 0.000439318 0
\(535\) 8262.25 0.667679
\(536\) −417.575 −0.0336502
\(537\) −1776.07 −0.142724
\(538\) 159.378 0.0127719
\(539\) −12102.9 −0.967180
\(540\) 1686.63 0.134409
\(541\) −11817.9 −0.939173 −0.469586 0.882887i \(-0.655597\pi\)
−0.469586 + 0.882887i \(0.655597\pi\)
\(542\) −198.614 −0.0157402
\(543\) −2437.27 −0.192621
\(544\) −434.632 −0.0342550
\(545\) −6042.88 −0.474952
\(546\) −14.3092 −0.00112157
\(547\) 9698.77 0.758116 0.379058 0.925373i \(-0.376248\pi\)
0.379058 + 0.925373i \(0.376248\pi\)
\(548\) −17161.0 −1.33774
\(549\) 4059.51 0.315584
\(550\) 86.8114 0.00673027
\(551\) 6396.68 0.494569
\(552\) 16.4373 0.00126742
\(553\) −4942.05 −0.380031
\(554\) 129.324 0.00991779
\(555\) −5059.87 −0.386990
\(556\) −7612.90 −0.580682
\(557\) −12145.8 −0.923939 −0.461970 0.886896i \(-0.652857\pi\)
−0.461970 + 0.886896i \(0.652857\pi\)
\(558\) 53.3630 0.00404845
\(559\) 2106.95 0.159418
\(560\) −2450.51 −0.184916
\(561\) 7212.08 0.542770
\(562\) 177.664 0.0133351
\(563\) 18085.9 1.35387 0.676935 0.736043i \(-0.263308\pi\)
0.676935 + 0.736043i \(0.263308\pi\)
\(564\) −4538.89 −0.338869
\(565\) −1329.24 −0.0989760
\(566\) 172.842 0.0128359
\(567\) 397.316 0.0294280
\(568\) 594.820 0.0439403
\(569\) 8313.29 0.612498 0.306249 0.951952i \(-0.400926\pi\)
0.306249 + 0.951952i \(0.400926\pi\)
\(570\) 82.0348 0.00602818
\(571\) −10880.3 −0.797418 −0.398709 0.917078i \(-0.630542\pi\)
−0.398709 + 0.917078i \(0.630542\pi\)
\(572\) −8258.30 −0.603666
\(573\) 8464.60 0.617127
\(574\) −10.8679 −0.000790276 0
\(575\) −613.346 −0.0444840
\(576\) −4603.59 −0.333014
\(577\) 680.956 0.0491310 0.0245655 0.999698i \(-0.492180\pi\)
0.0245655 + 0.999698i \(0.492180\pi\)
\(578\) 32.1512 0.00231369
\(579\) −6763.76 −0.485479
\(580\) 4078.71 0.291999
\(581\) −1928.01 −0.137672
\(582\) 111.837 0.00796532
\(583\) 21176.1 1.50433
\(584\) −302.913 −0.0214634
\(585\) −1912.34 −0.135155
\(586\) 216.637 0.0152716
\(587\) −2603.48 −0.183062 −0.0915309 0.995802i \(-0.529176\pi\)
−0.0915309 + 0.995802i \(0.529176\pi\)
\(588\) 7653.33 0.536765
\(589\) −16252.8 −1.13699
\(590\) 16.4681 0.00114912
\(591\) −6057.09 −0.421583
\(592\) 13815.1 0.959120
\(593\) 25521.8 1.76737 0.883687 0.468078i \(-0.155054\pi\)
0.883687 + 0.468078i \(0.155054\pi\)
\(594\) −36.6186 −0.00252943
\(595\) −2426.84 −0.167212
\(596\) 18904.1 1.29923
\(597\) 4514.44 0.309487
\(598\) −9.31775 −0.000637176 0
\(599\) −2552.55 −0.174114 −0.0870570 0.996203i \(-0.527746\pi\)
−0.0870570 + 0.996203i \(0.527746\pi\)
\(600\) −109.800 −0.00747092
\(601\) 18890.3 1.28211 0.641057 0.767493i \(-0.278496\pi\)
0.641057 + 0.767493i \(0.278496\pi\)
\(602\) −13.5760 −0.000919133 0
\(603\) −6572.58 −0.443874
\(604\) −16373.7 −1.10304
\(605\) −851.297 −0.0572068
\(606\) −158.872 −0.0106497
\(607\) 8916.27 0.596211 0.298106 0.954533i \(-0.403645\pi\)
0.298106 + 0.954533i \(0.403645\pi\)
\(608\) −672.127 −0.0448328
\(609\) 960.815 0.0639314
\(610\) 125.899 0.00835654
\(611\) 5146.31 0.340749
\(612\) −4560.58 −0.301226
\(613\) 7796.34 0.513689 0.256844 0.966453i \(-0.417317\pi\)
0.256844 + 0.966453i \(0.417317\pi\)
\(614\) 136.089 0.00894481
\(615\) −1452.43 −0.0952319
\(616\) 106.432 0.00696150
\(617\) −27806.2 −1.81432 −0.907161 0.420784i \(-0.861755\pi\)
−0.907161 + 0.420784i \(0.861755\pi\)
\(618\) −6.55633 −0.000426755 0
\(619\) 11643.6 0.756050 0.378025 0.925795i \(-0.376603\pi\)
0.378025 + 0.925795i \(0.376603\pi\)
\(620\) −10363.3 −0.671287
\(621\) 258.720 0.0167183
\(622\) −272.817 −0.0175867
\(623\) −248.007 −0.0159490
\(624\) 5221.33 0.334969
\(625\) −3526.82 −0.225716
\(626\) 211.628 0.0135118
\(627\) 11153.0 0.710376
\(628\) −14231.1 −0.904272
\(629\) 13681.7 0.867291
\(630\) 12.3221 0.000779243 0
\(631\) −2815.28 −0.177614 −0.0888070 0.996049i \(-0.528305\pi\)
−0.0888070 + 0.996049i \(0.528305\pi\)
\(632\) −576.100 −0.0362595
\(633\) 5367.26 0.337013
\(634\) 62.4765 0.00391366
\(635\) −1159.04 −0.0724333
\(636\) −13390.8 −0.834874
\(637\) −8677.54 −0.539744
\(638\) −88.5536 −0.00549509
\(639\) 9362.39 0.579609
\(640\) −571.409 −0.0352921
\(641\) 19548.3 1.20454 0.602271 0.798292i \(-0.294263\pi\)
0.602271 + 0.798292i \(0.294263\pi\)
\(642\) 113.434 0.00697331
\(643\) 20675.9 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(644\) −375.957 −0.0230043
\(645\) −1814.35 −0.110760
\(646\) −221.820 −0.0135099
\(647\) 19267.9 1.17078 0.585392 0.810750i \(-0.300941\pi\)
0.585392 + 0.810750i \(0.300941\pi\)
\(648\) 46.3155 0.00280779
\(649\) 2238.90 0.135415
\(650\) 62.2419 0.00375589
\(651\) −2441.25 −0.146974
\(652\) −3118.76 −0.187332
\(653\) 17386.3 1.04193 0.520963 0.853579i \(-0.325573\pi\)
0.520963 + 0.853579i \(0.325573\pi\)
\(654\) −82.9635 −0.00496045
\(655\) 4020.41 0.239833
\(656\) 3965.62 0.236024
\(657\) −4767.81 −0.283120
\(658\) −33.1600 −0.00196461
\(659\) −15986.9 −0.945007 −0.472504 0.881329i \(-0.656650\pi\)
−0.472504 + 0.881329i \(0.656650\pi\)
\(660\) 7111.45 0.419413
\(661\) 14387.2 0.846592 0.423296 0.905991i \(-0.360873\pi\)
0.423296 + 0.905991i \(0.360873\pi\)
\(662\) −387.890 −0.0227731
\(663\) 5170.90 0.302898
\(664\) −224.750 −0.0131355
\(665\) −3752.94 −0.218846
\(666\) −69.4676 −0.00404176
\(667\) 625.655 0.0363200
\(668\) 10540.8 0.610535
\(669\) −5561.70 −0.321417
\(670\) −203.837 −0.0117536
\(671\) 17116.4 0.984757
\(672\) −100.957 −0.00579539
\(673\) −21373.2 −1.22418 −0.612092 0.790787i \(-0.709672\pi\)
−0.612092 + 0.790787i \(0.709672\pi\)
\(674\) −320.796 −0.0183333
\(675\) −1728.23 −0.0985477
\(676\) 11652.2 0.662959
\(677\) 13150.7 0.746564 0.373282 0.927718i \(-0.378232\pi\)
0.373282 + 0.927718i \(0.378232\pi\)
\(678\) −18.2493 −0.00103372
\(679\) −5116.36 −0.289172
\(680\) −282.900 −0.0159540
\(681\) 6341.13 0.356817
\(682\) 224.998 0.0126329
\(683\) 14825.3 0.830563 0.415282 0.909693i \(-0.363683\pi\)
0.415282 + 0.909693i \(0.363683\pi\)
\(684\) −7052.61 −0.394244
\(685\) −16755.4 −0.934587
\(686\) 116.045 0.00645861
\(687\) −2981.85 −0.165596
\(688\) 4953.80 0.274508
\(689\) 15182.8 0.839506
\(690\) 8.02377 0.000442695 0
\(691\) −7056.59 −0.388488 −0.194244 0.980953i \(-0.562225\pi\)
−0.194244 + 0.980953i \(0.562225\pi\)
\(692\) 6573.68 0.361118
\(693\) 1675.23 0.0918280
\(694\) −223.022 −0.0121985
\(695\) −7433.00 −0.405683
\(696\) 112.003 0.00609982
\(697\) 3927.32 0.213426
\(698\) 449.087 0.0243527
\(699\) 5948.46 0.321876
\(700\) 2511.36 0.135601
\(701\) −15864.1 −0.854751 −0.427376 0.904074i \(-0.640562\pi\)
−0.427376 + 0.904074i \(0.640562\pi\)
\(702\) −26.2548 −0.00141157
\(703\) 21157.8 1.13511
\(704\) −19410.5 −1.03915
\(705\) −4431.63 −0.236745
\(706\) −172.449 −0.00919292
\(707\) 7268.11 0.386627
\(708\) −1415.77 −0.0751526
\(709\) −9641.46 −0.510709 −0.255354 0.966848i \(-0.582192\pi\)
−0.255354 + 0.966848i \(0.582192\pi\)
\(710\) 290.359 0.0153478
\(711\) −9067.74 −0.478294
\(712\) −28.9105 −0.00152172
\(713\) −1589.67 −0.0834976
\(714\) −33.3185 −0.00174638
\(715\) −8063.14 −0.421741
\(716\) 4735.43 0.247167
\(717\) −12558.5 −0.654121
\(718\) 104.355 0.00542409
\(719\) −31356.7 −1.62644 −0.813218 0.581959i \(-0.802286\pi\)
−0.813218 + 0.581959i \(0.802286\pi\)
\(720\) −4496.23 −0.232729
\(721\) 299.940 0.0154929
\(722\) −97.8863 −0.00504564
\(723\) 1365.94 0.0702623
\(724\) 6498.36 0.333577
\(725\) −4179.33 −0.214092
\(726\) −11.6876 −0.000597474 0
\(727\) 25527.1 1.30227 0.651133 0.758964i \(-0.274294\pi\)
0.651133 + 0.758964i \(0.274294\pi\)
\(728\) 76.3098 0.00388493
\(729\) 729.000 0.0370370
\(730\) −147.865 −0.00749691
\(731\) 4905.95 0.248226
\(732\) −10823.6 −0.546520
\(733\) 32735.8 1.64956 0.824779 0.565456i \(-0.191300\pi\)
0.824779 + 0.565456i \(0.191300\pi\)
\(734\) 331.266 0.0166584
\(735\) 7472.47 0.375002
\(736\) −65.7403 −0.00329242
\(737\) −27712.5 −1.38508
\(738\) −19.9406 −0.000994612 0
\(739\) 26530.4 1.32062 0.660308 0.750995i \(-0.270426\pi\)
0.660308 + 0.750995i \(0.270426\pi\)
\(740\) 13490.8 0.670179
\(741\) 7996.43 0.396432
\(742\) −97.8299 −0.00484023
\(743\) −25269.1 −1.24769 −0.623845 0.781548i \(-0.714430\pi\)
−0.623845 + 0.781548i \(0.714430\pi\)
\(744\) −284.580 −0.0140231
\(745\) 18457.3 0.907683
\(746\) −232.702 −0.0114207
\(747\) −3537.53 −0.173268
\(748\) −19229.1 −0.939956
\(749\) −5189.37 −0.253158
\(750\) −158.268 −0.00770551
\(751\) 34954.2 1.69840 0.849199 0.528072i \(-0.177085\pi\)
0.849199 + 0.528072i \(0.177085\pi\)
\(752\) 12099.9 0.586750
\(753\) −21481.7 −1.03962
\(754\) −63.4910 −0.00306659
\(755\) −15986.8 −0.770620
\(756\) −1059.34 −0.0509627
\(757\) 38062.0 1.82746 0.913730 0.406322i \(-0.133189\pi\)
0.913730 + 0.406322i \(0.133189\pi\)
\(758\) 73.9268 0.00354241
\(759\) 1090.86 0.0521684
\(760\) −437.484 −0.0208806
\(761\) −6796.71 −0.323759 −0.161879 0.986811i \(-0.551756\pi\)
−0.161879 + 0.986811i \(0.551756\pi\)
\(762\) −15.9126 −0.000756501 0
\(763\) 3795.43 0.180083
\(764\) −22568.7 −1.06872
\(765\) −4452.81 −0.210446
\(766\) −438.120 −0.0206657
\(767\) 1605.24 0.0755696
\(768\) 12268.4 0.576429
\(769\) −34493.8 −1.61753 −0.808763 0.588135i \(-0.799862\pi\)
−0.808763 + 0.588135i \(0.799862\pi\)
\(770\) 51.9545 0.00243157
\(771\) −2439.28 −0.113941
\(772\) 18033.8 0.840740
\(773\) 17764.5 0.826576 0.413288 0.910600i \(-0.364380\pi\)
0.413288 + 0.910600i \(0.364380\pi\)
\(774\) −24.9095 −0.00115679
\(775\) 10618.9 0.492184
\(776\) −596.419 −0.0275905
\(777\) 3178.01 0.146732
\(778\) 156.894 0.00722997
\(779\) 6073.32 0.279332
\(780\) 5098.76 0.234057
\(781\) 39475.4 1.80863
\(782\) −21.6960 −0.000992133 0
\(783\) 1762.92 0.0804617
\(784\) −20402.4 −0.929408
\(785\) −13894.8 −0.631753
\(786\) 55.1967 0.00250484
\(787\) 15912.1 0.720717 0.360359 0.932814i \(-0.382654\pi\)
0.360359 + 0.932814i \(0.382654\pi\)
\(788\) 16149.7 0.730086
\(789\) −9406.08 −0.424417
\(790\) −281.220 −0.0126650
\(791\) 834.870 0.0375279
\(792\) 195.284 0.00876150
\(793\) 12272.1 0.549552
\(794\) −161.457 −0.00721650
\(795\) −13074.4 −0.583270
\(796\) −12036.6 −0.535961
\(797\) −11535.5 −0.512685 −0.256343 0.966586i \(-0.582517\pi\)
−0.256343 + 0.966586i \(0.582517\pi\)
\(798\) −51.5246 −0.00228565
\(799\) 11983.0 0.530573
\(800\) 439.140 0.0194074
\(801\) −455.047 −0.0200728
\(802\) 155.607 0.00685120
\(803\) −20102.9 −0.883456
\(804\) 17524.1 0.768690
\(805\) −367.072 −0.0160716
\(806\) 161.319 0.00704990
\(807\) −13378.1 −0.583559
\(808\) 847.251 0.0368888
\(809\) −43528.4 −1.89169 −0.945846 0.324616i \(-0.894765\pi\)
−0.945846 + 0.324616i \(0.894765\pi\)
\(810\) 22.6087 0.000980726 0
\(811\) −7310.19 −0.316517 −0.158259 0.987398i \(-0.550588\pi\)
−0.158259 + 0.987398i \(0.550588\pi\)
\(812\) −2561.77 −0.110715
\(813\) 16671.5 0.719182
\(814\) −292.902 −0.0126120
\(815\) −3045.06 −0.130876
\(816\) 12157.7 0.521573
\(817\) 7586.70 0.324878
\(818\) 442.558 0.0189165
\(819\) 1201.11 0.0512455
\(820\) 3872.53 0.164920
\(821\) 14986.7 0.637076 0.318538 0.947910i \(-0.396808\pi\)
0.318538 + 0.947910i \(0.396808\pi\)
\(822\) −230.038 −0.00976093
\(823\) −25042.3 −1.06066 −0.530328 0.847792i \(-0.677931\pi\)
−0.530328 + 0.847792i \(0.677931\pi\)
\(824\) 34.9643 0.00147820
\(825\) −7286.88 −0.307511
\(826\) −10.3433 −0.000435701 0
\(827\) −31526.4 −1.32561 −0.662806 0.748791i \(-0.730635\pi\)
−0.662806 + 0.748791i \(0.730635\pi\)
\(828\) −689.811 −0.0289524
\(829\) −27719.8 −1.16134 −0.580668 0.814141i \(-0.697208\pi\)
−0.580668 + 0.814141i \(0.697208\pi\)
\(830\) −109.711 −0.00458808
\(831\) −10855.4 −0.453151
\(832\) −13916.9 −0.579905
\(833\) −20205.3 −0.840423
\(834\) −102.049 −0.00423700
\(835\) 10291.7 0.426539
\(836\) −29736.5 −1.23021
\(837\) −4479.25 −0.184977
\(838\) −81.2375 −0.00334881
\(839\) 29512.0 1.21438 0.607192 0.794555i \(-0.292296\pi\)
0.607192 + 0.794555i \(0.292296\pi\)
\(840\) −65.7124 −0.00269916
\(841\) −20125.8 −0.825200
\(842\) 424.242 0.0173638
\(843\) −14913.0 −0.609290
\(844\) −14310.4 −0.583631
\(845\) 11376.8 0.463165
\(846\) −60.8425 −0.00247259
\(847\) 534.684 0.0216906
\(848\) 35697.4 1.44558
\(849\) −14508.3 −0.586481
\(850\) 144.928 0.00584821
\(851\) 2069.43 0.0833597
\(852\) −24962.4 −1.00375
\(853\) −26933.2 −1.08109 −0.540547 0.841314i \(-0.681783\pi\)
−0.540547 + 0.841314i \(0.681783\pi\)
\(854\) −79.0747 −0.00316848
\(855\) −6885.94 −0.275432
\(856\) −604.931 −0.0241543
\(857\) 26634.2 1.06162 0.530809 0.847491i \(-0.321888\pi\)
0.530809 + 0.847491i \(0.321888\pi\)
\(858\) −110.700 −0.00440470
\(859\) −6411.56 −0.254668 −0.127334 0.991860i \(-0.540642\pi\)
−0.127334 + 0.991860i \(0.540642\pi\)
\(860\) 4837.50 0.191811
\(861\) 912.245 0.0361083
\(862\) 537.019 0.0212192
\(863\) −20915.9 −0.825011 −0.412505 0.910955i \(-0.635346\pi\)
−0.412505 + 0.910955i \(0.635346\pi\)
\(864\) −185.237 −0.00729387
\(865\) 6418.33 0.252289
\(866\) 580.827 0.0227913
\(867\) −2698.75 −0.105714
\(868\) 6508.97 0.254526
\(869\) −38233.0 −1.49248
\(870\) 54.6739 0.00213059
\(871\) −19869.3 −0.772955
\(872\) 442.437 0.0171821
\(873\) −9387.56 −0.363941
\(874\) −33.5513 −0.00129850
\(875\) 7240.47 0.279740
\(876\) 12712.1 0.490300
\(877\) −14800.0 −0.569852 −0.284926 0.958549i \(-0.591969\pi\)
−0.284926 + 0.958549i \(0.591969\pi\)
\(878\) −103.107 −0.00396322
\(879\) −18184.3 −0.697772
\(880\) −18957.8 −0.726213
\(881\) 25932.2 0.991690 0.495845 0.868411i \(-0.334858\pi\)
0.495845 + 0.868411i \(0.334858\pi\)
\(882\) 102.591 0.00391656
\(883\) 20302.7 0.773772 0.386886 0.922128i \(-0.373551\pi\)
0.386886 + 0.922128i \(0.373551\pi\)
\(884\) −13786.9 −0.524551
\(885\) −1382.32 −0.0525040
\(886\) −339.366 −0.0128682
\(887\) −39704.8 −1.50299 −0.751497 0.659736i \(-0.770668\pi\)
−0.751497 + 0.659736i \(0.770668\pi\)
\(888\) 370.464 0.0140000
\(889\) 727.973 0.0274639
\(890\) −14.1125 −0.000531519 0
\(891\) 3073.74 0.115571
\(892\) 14828.8 0.556621
\(893\) 18530.8 0.694413
\(894\) 253.403 0.00947994
\(895\) 4623.52 0.172678
\(896\) 358.892 0.0133814
\(897\) 782.125 0.0291130
\(898\) −305.226 −0.0113424
\(899\) −10832.0 −0.401855
\(900\) 4607.88 0.170662
\(901\) 35352.6 1.30718
\(902\) −84.0771 −0.00310362
\(903\) 1139.56 0.0419959
\(904\) 97.3217 0.00358061
\(905\) 6344.79 0.233047
\(906\) −219.485 −0.00804844
\(907\) −23723.8 −0.868507 −0.434254 0.900791i \(-0.642988\pi\)
−0.434254 + 0.900791i \(0.642988\pi\)
\(908\) −16907.0 −0.617927
\(909\) 13335.6 0.486595
\(910\) 37.2503 0.00135696
\(911\) −23515.4 −0.855212 −0.427606 0.903965i \(-0.640643\pi\)
−0.427606 + 0.903965i \(0.640643\pi\)
\(912\) 18800.9 0.682633
\(913\) −14915.6 −0.540672
\(914\) 256.181 0.00927102
\(915\) −10567.8 −0.381816
\(916\) 7950.34 0.286776
\(917\) −2525.15 −0.0909353
\(918\) −61.1332 −0.00219793
\(919\) 4610.25 0.165482 0.0827411 0.996571i \(-0.473633\pi\)
0.0827411 + 0.996571i \(0.473633\pi\)
\(920\) −42.7900 −0.00153342
\(921\) −11423.2 −0.408695
\(922\) 181.974 0.00649999
\(923\) 28303.0 1.00932
\(924\) −4466.57 −0.159025
\(925\) −13823.6 −0.491371
\(926\) −352.775 −0.0125193
\(927\) 550.334 0.0194987
\(928\) −447.954 −0.0158457
\(929\) 34994.2 1.23587 0.617935 0.786229i \(-0.287969\pi\)
0.617935 + 0.786229i \(0.287969\pi\)
\(930\) −138.916 −0.00489811
\(931\) −31246.1 −1.09994
\(932\) −15860.0 −0.557416
\(933\) 22900.0 0.803551
\(934\) 593.292 0.0207849
\(935\) −18774.7 −0.656683
\(936\) 140.014 0.00488943
\(937\) 50967.1 1.77697 0.888485 0.458905i \(-0.151758\pi\)
0.888485 + 0.458905i \(0.151758\pi\)
\(938\) 128.027 0.00445652
\(939\) −17763.9 −0.617362
\(940\) 11815.8 0.409988
\(941\) −34297.5 −1.18817 −0.594084 0.804403i \(-0.702485\pi\)
−0.594084 + 0.804403i \(0.702485\pi\)
\(942\) −190.764 −0.00659810
\(943\) 594.027 0.0205135
\(944\) 3774.19 0.130127
\(945\) −1034.31 −0.0356042
\(946\) −105.028 −0.00360967
\(947\) 15651.4 0.537066 0.268533 0.963270i \(-0.413461\pi\)
0.268533 + 0.963270i \(0.413461\pi\)
\(948\) 24176.8 0.828297
\(949\) −14413.3 −0.493021
\(950\) 224.120 0.00765413
\(951\) −5244.23 −0.178818
\(952\) 177.684 0.00604914
\(953\) 44875.1 1.52534 0.762669 0.646789i \(-0.223889\pi\)
0.762669 + 0.646789i \(0.223889\pi\)
\(954\) −179.500 −0.00609173
\(955\) −22035.3 −0.746645
\(956\) 33483.9 1.13279
\(957\) 7433.12 0.251075
\(958\) −410.169 −0.0138329
\(959\) 10523.8 0.354360
\(960\) 11984.2 0.402905
\(961\) −2268.85 −0.0761589
\(962\) −210.004 −0.00703826
\(963\) −9521.53 −0.318616
\(964\) −3641.91 −0.121679
\(965\) 17607.7 0.587368
\(966\) −5.03959 −0.000167853 0
\(967\) −53906.6 −1.79268 −0.896338 0.443371i \(-0.853783\pi\)
−0.896338 + 0.443371i \(0.853783\pi\)
\(968\) 62.3287 0.00206955
\(969\) 18619.4 0.617276
\(970\) −291.139 −0.00963702
\(971\) −15575.1 −0.514755 −0.257378 0.966311i \(-0.582858\pi\)
−0.257378 + 0.966311i \(0.582858\pi\)
\(972\) −1943.69 −0.0641398
\(973\) 4668.53 0.153819
\(974\) −564.983 −0.0185865
\(975\) −5224.54 −0.171609
\(976\) 28853.8 0.946298
\(977\) −20898.7 −0.684348 −0.342174 0.939637i \(-0.611163\pi\)
−0.342174 + 0.939637i \(0.611163\pi\)
\(978\) −41.8061 −0.00136688
\(979\) −1918.65 −0.0626357
\(980\) −19923.4 −0.649418
\(981\) 6963.90 0.226646
\(982\) −9.99756 −0.000324883 0
\(983\) 54689.4 1.77449 0.887243 0.461303i \(-0.152618\pi\)
0.887243 + 0.461303i \(0.152618\pi\)
\(984\) 106.341 0.00344516
\(985\) 15768.0 0.510062
\(986\) −147.836 −0.00477492
\(987\) 2783.43 0.0897645
\(988\) −21320.4 −0.686531
\(989\) 742.050 0.0238583
\(990\) 95.3268 0.00306029
\(991\) −14912.4 −0.478011 −0.239005 0.971018i \(-0.576821\pi\)
−0.239005 + 0.971018i \(0.576821\pi\)
\(992\) 1138.17 0.0364283
\(993\) 32559.2 1.04052
\(994\) −182.369 −0.00581931
\(995\) −11752.1 −0.374440
\(996\) 9431.91 0.300062
\(997\) 20492.3 0.650951 0.325475 0.945550i \(-0.394476\pi\)
0.325475 + 0.945550i \(0.394476\pi\)
\(998\) 412.488 0.0130832
\(999\) 5831.06 0.184671
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 177.4.a.d.1.4 8
3.2 odd 2 531.4.a.e.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.d.1.4 8 1.1 even 1 trivial
531.4.a.e.1.5 8 3.2 odd 2