Properties

Label 177.4.a.c.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: \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.254436\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+0.254436 q^{2} -3.00000 q^{3} -7.93526 q^{4} -10.8225 q^{5} -0.763309 q^{6} -23.2950 q^{7} -4.05451 q^{8} +9.00000 q^{9} -2.75362 q^{10} +51.0608 q^{11} +23.8058 q^{12} +51.8661 q^{13} -5.92710 q^{14} +32.4674 q^{15} +62.4505 q^{16} -0.0421754 q^{17} +2.28993 q^{18} -85.2082 q^{19} +85.8790 q^{20} +69.8851 q^{21} +12.9917 q^{22} -8.50543 q^{23} +12.1635 q^{24} -7.87442 q^{25} +13.1966 q^{26} -27.0000 q^{27} +184.852 q^{28} -101.142 q^{29} +8.26087 q^{30} +271.338 q^{31} +48.3257 q^{32} -153.182 q^{33} -0.0107310 q^{34} +252.110 q^{35} -71.4174 q^{36} -9.54238 q^{37} -21.6800 q^{38} -155.598 q^{39} +43.8797 q^{40} -185.363 q^{41} +17.7813 q^{42} +277.532 q^{43} -405.181 q^{44} -97.4021 q^{45} -2.16409 q^{46} +309.877 q^{47} -187.351 q^{48} +199.659 q^{49} -2.00354 q^{50} +0.126526 q^{51} -411.571 q^{52} +273.504 q^{53} -6.86978 q^{54} -552.603 q^{55} +94.4499 q^{56} +255.625 q^{57} -25.7341 q^{58} -59.0000 q^{59} -257.637 q^{60} +752.893 q^{61} +69.0383 q^{62} -209.655 q^{63} -487.308 q^{64} -561.319 q^{65} -38.9751 q^{66} +751.435 q^{67} +0.334673 q^{68} +25.5163 q^{69} +64.1458 q^{70} -651.607 q^{71} -36.4906 q^{72} +842.428 q^{73} -2.42793 q^{74} +23.6233 q^{75} +676.149 q^{76} -1189.46 q^{77} -39.5898 q^{78} -368.828 q^{79} -675.868 q^{80} +81.0000 q^{81} -47.1631 q^{82} -545.888 q^{83} -554.557 q^{84} +0.456442 q^{85} +70.6143 q^{86} +303.425 q^{87} -207.026 q^{88} +634.328 q^{89} -24.7826 q^{90} -1208.22 q^{91} +67.4928 q^{92} -814.015 q^{93} +78.8440 q^{94} +922.162 q^{95} -144.977 q^{96} +316.368 q^{97} +50.8004 q^{98} +459.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + 29q^{10} - 27q^{11} - 114q^{12} + 89q^{13} - 37q^{14} + 36q^{15} + 362q^{16} + 79q^{17} + 18q^{18} + 288q^{19} + 457q^{20} - 159q^{21} + 596q^{22} + 202q^{23} - 9q^{24} + 264q^{25} + 270q^{26} - 216q^{27} + 702q^{28} - 114q^{29} - 87q^{30} + 538q^{31} + 316q^{32} + 81q^{33} + 498q^{34} - 196q^{35} + 342q^{36} + 395q^{37} + 397q^{38} - 267q^{39} + 918q^{40} - 39q^{41} + 111q^{42} + 527q^{43} + 64q^{44} - 108q^{45} - 539q^{46} + 860q^{47} - 1086q^{48} + 347q^{49} - 591q^{50} - 237q^{51} - 644q^{52} - 812q^{53} - 54q^{54} + 536q^{55} - 2218q^{56} - 864q^{57} - 1154q^{58} - 472q^{59} - 1371q^{60} - 460q^{61} - 2014q^{62} + 477q^{63} - 451q^{64} - 986q^{65} - 1788q^{66} + 1934q^{67} - 69q^{68} - 606q^{69} - 1028q^{70} - 1687q^{71} + 27q^{72} + 1980q^{73} - 2400q^{74} - 792q^{75} - 940q^{76} - 821q^{77} - 810q^{78} + 3319q^{79} - 2119q^{80} + 648q^{81} + 429q^{82} + 2057q^{83} - 2106q^{84} + 566q^{85} - 6690q^{86} + 342q^{87} + 1189q^{88} + 1668q^{89} + 261q^{90} + 2427q^{91} - 980q^{92} - 1614q^{93} + 332q^{94} + 2146q^{95} - 948q^{96} + 1956q^{97} - 2026q^{98} - 243q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.254436 0.0899568 0.0449784 0.998988i \(-0.485678\pi\)
0.0449784 + 0.998988i \(0.485678\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.93526 −0.991908
\(5\) −10.8225 −0.967990 −0.483995 0.875071i \(-0.660815\pi\)
−0.483995 + 0.875071i \(0.660815\pi\)
\(6\) −0.763309 −0.0519366
\(7\) −23.2950 −1.25781 −0.628907 0.777481i \(-0.716497\pi\)
−0.628907 + 0.777481i \(0.716497\pi\)
\(8\) −4.05451 −0.179186
\(9\) 9.00000 0.333333
\(10\) −2.75362 −0.0870773
\(11\) 51.0608 1.39958 0.699791 0.714347i \(-0.253276\pi\)
0.699791 + 0.714347i \(0.253276\pi\)
\(12\) 23.8058 0.572678
\(13\) 51.8661 1.10654 0.553272 0.833001i \(-0.313379\pi\)
0.553272 + 0.833001i \(0.313379\pi\)
\(14\) −5.92710 −0.113149
\(15\) 32.4674 0.558869
\(16\) 62.4505 0.975789
\(17\) −0.0421754 −0.000601709 0 −0.000300854 1.00000i \(-0.500096\pi\)
−0.000300854 1.00000i \(0.500096\pi\)
\(18\) 2.28993 0.0299856
\(19\) −85.2082 −1.02885 −0.514424 0.857536i \(-0.671994\pi\)
−0.514424 + 0.857536i \(0.671994\pi\)
\(20\) 85.8790 0.960157
\(21\) 69.8851 0.726199
\(22\) 12.9917 0.125902
\(23\) −8.50543 −0.0771089 −0.0385544 0.999257i \(-0.512275\pi\)
−0.0385544 + 0.999257i \(0.512275\pi\)
\(24\) 12.1635 0.103453
\(25\) −7.87442 −0.0629954
\(26\) 13.1966 0.0995411
\(27\) −27.0000 −0.192450
\(28\) 184.852 1.24764
\(29\) −101.142 −0.647639 −0.323819 0.946119i \(-0.604967\pi\)
−0.323819 + 0.946119i \(0.604967\pi\)
\(30\) 8.26087 0.0502741
\(31\) 271.338 1.57206 0.786029 0.618189i \(-0.212133\pi\)
0.786029 + 0.618189i \(0.212133\pi\)
\(32\) 48.3257 0.266964
\(33\) −153.182 −0.808049
\(34\) −0.0107310 −5.41278e−5 0
\(35\) 252.110 1.21755
\(36\) −71.4174 −0.330636
\(37\) −9.54238 −0.0423988 −0.0211994 0.999775i \(-0.506748\pi\)
−0.0211994 + 0.999775i \(0.506748\pi\)
\(38\) −21.6800 −0.0925518
\(39\) −155.598 −0.638863
\(40\) 43.8797 0.173450
\(41\) −185.363 −0.706071 −0.353035 0.935610i \(-0.614850\pi\)
−0.353035 + 0.935610i \(0.614850\pi\)
\(42\) 17.7813 0.0653265
\(43\) 277.532 0.984263 0.492131 0.870521i \(-0.336218\pi\)
0.492131 + 0.870521i \(0.336218\pi\)
\(44\) −405.181 −1.38826
\(45\) −97.4021 −0.322663
\(46\) −2.16409 −0.00693647
\(47\) 309.877 0.961708 0.480854 0.876801i \(-0.340327\pi\)
0.480854 + 0.876801i \(0.340327\pi\)
\(48\) −187.351 −0.563372
\(49\) 199.659 0.582096
\(50\) −2.00354 −0.00566686
\(51\) 0.126526 0.000347397 0
\(52\) −411.571 −1.09759
\(53\) 273.504 0.708842 0.354421 0.935086i \(-0.384678\pi\)
0.354421 + 0.935086i \(0.384678\pi\)
\(54\) −6.86978 −0.0173122
\(55\) −552.603 −1.35478
\(56\) 94.4499 0.225382
\(57\) 255.625 0.594005
\(58\) −25.7341 −0.0582595
\(59\) −59.0000 −0.130189
\(60\) −257.637 −0.554347
\(61\) 752.893 1.58030 0.790148 0.612916i \(-0.210004\pi\)
0.790148 + 0.612916i \(0.210004\pi\)
\(62\) 69.0383 0.141417
\(63\) −209.655 −0.419271
\(64\) −487.308 −0.951774
\(65\) −561.319 −1.07112
\(66\) −38.9751 −0.0726895
\(67\) 751.435 1.37018 0.685092 0.728456i \(-0.259762\pi\)
0.685092 + 0.728456i \(0.259762\pi\)
\(68\) 0.334673 0.000596840 0
\(69\) 25.5163 0.0445188
\(70\) 64.1458 0.109527
\(71\) −651.607 −1.08918 −0.544588 0.838704i \(-0.683314\pi\)
−0.544588 + 0.838704i \(0.683314\pi\)
\(72\) −36.4906 −0.0597285
\(73\) 842.428 1.35067 0.675334 0.737512i \(-0.263999\pi\)
0.675334 + 0.737512i \(0.263999\pi\)
\(74\) −2.42793 −0.00381406
\(75\) 23.6233 0.0363704
\(76\) 676.149 1.02052
\(77\) −1189.46 −1.76041
\(78\) −39.5898 −0.0574701
\(79\) −368.828 −0.525271 −0.262635 0.964895i \(-0.584592\pi\)
−0.262635 + 0.964895i \(0.584592\pi\)
\(80\) −675.868 −0.944554
\(81\) 81.0000 0.111111
\(82\) −47.1631 −0.0635158
\(83\) −545.888 −0.721916 −0.360958 0.932582i \(-0.617550\pi\)
−0.360958 + 0.932582i \(0.617550\pi\)
\(84\) −554.557 −0.720323
\(85\) 0.456442 0.000582448 0
\(86\) 70.6143 0.0885411
\(87\) 303.425 0.373914
\(88\) −207.026 −0.250785
\(89\) 634.328 0.755490 0.377745 0.925910i \(-0.376700\pi\)
0.377745 + 0.925910i \(0.376700\pi\)
\(90\) −24.7826 −0.0290258
\(91\) −1208.22 −1.39183
\(92\) 67.4928 0.0764849
\(93\) −814.015 −0.907629
\(94\) 78.8440 0.0865121
\(95\) 922.162 0.995914
\(96\) −144.977 −0.154132
\(97\) 316.368 0.331158 0.165579 0.986197i \(-0.447051\pi\)
0.165579 + 0.986197i \(0.447051\pi\)
\(98\) 50.8004 0.0523635
\(99\) 459.547 0.466528
\(100\) 62.4856 0.0624856
\(101\) −1615.25 −1.59132 −0.795662 0.605740i \(-0.792877\pi\)
−0.795662 + 0.605740i \(0.792877\pi\)
\(102\) 0.0321929 3.12507e−5 0
\(103\) 1261.07 1.20638 0.603190 0.797598i \(-0.293896\pi\)
0.603190 + 0.797598i \(0.293896\pi\)
\(104\) −210.291 −0.198277
\(105\) −756.329 −0.702954
\(106\) 69.5892 0.0637651
\(107\) 733.783 0.662967 0.331483 0.943461i \(-0.392451\pi\)
0.331483 + 0.943461i \(0.392451\pi\)
\(108\) 214.252 0.190893
\(109\) 582.848 0.512171 0.256086 0.966654i \(-0.417567\pi\)
0.256086 + 0.966654i \(0.417567\pi\)
\(110\) −140.602 −0.121872
\(111\) 28.6271 0.0244790
\(112\) −1454.79 −1.22736
\(113\) −1661.78 −1.38342 −0.691712 0.722174i \(-0.743143\pi\)
−0.691712 + 0.722174i \(0.743143\pi\)
\(114\) 65.0401 0.0534348
\(115\) 92.0496 0.0746406
\(116\) 802.585 0.642398
\(117\) 466.795 0.368848
\(118\) −15.0117 −0.0117114
\(119\) 0.982479 0.000756838 0
\(120\) −131.639 −0.100141
\(121\) 1276.20 0.958831
\(122\) 191.563 0.142158
\(123\) 556.090 0.407650
\(124\) −2153.14 −1.55934
\(125\) 1438.03 1.02897
\(126\) −53.3439 −0.0377163
\(127\) 1560.58 1.09039 0.545193 0.838310i \(-0.316456\pi\)
0.545193 + 0.838310i \(0.316456\pi\)
\(128\) −510.595 −0.352583
\(129\) −832.597 −0.568264
\(130\) −142.820 −0.0963548
\(131\) 2608.59 1.73980 0.869898 0.493232i \(-0.164185\pi\)
0.869898 + 0.493232i \(0.164185\pi\)
\(132\) 1215.54 0.801510
\(133\) 1984.93 1.29410
\(134\) 191.192 0.123257
\(135\) 292.206 0.186290
\(136\) 0.171001 0.000107818 0
\(137\) −2284.40 −1.42459 −0.712296 0.701879i \(-0.752345\pi\)
−0.712296 + 0.701879i \(0.752345\pi\)
\(138\) 6.49227 0.00400477
\(139\) −288.324 −0.175937 −0.0879687 0.996123i \(-0.528038\pi\)
−0.0879687 + 0.996123i \(0.528038\pi\)
\(140\) −2000.56 −1.20770
\(141\) −929.632 −0.555242
\(142\) −165.792 −0.0979787
\(143\) 2648.32 1.54870
\(144\) 562.054 0.325263
\(145\) 1094.60 0.626908
\(146\) 214.344 0.121502
\(147\) −598.977 −0.336073
\(148\) 75.7212 0.0420557
\(149\) 591.503 0.325220 0.162610 0.986690i \(-0.448009\pi\)
0.162610 + 0.986690i \(0.448009\pi\)
\(150\) 6.01061 0.00327176
\(151\) −2398.78 −1.29278 −0.646391 0.763007i \(-0.723722\pi\)
−0.646391 + 0.763007i \(0.723722\pi\)
\(152\) 345.477 0.184355
\(153\) −0.379579 −0.000200570 0
\(154\) −302.642 −0.158361
\(155\) −2936.55 −1.52174
\(156\) 1234.71 0.633693
\(157\) 135.478 0.0688684 0.0344342 0.999407i \(-0.489037\pi\)
0.0344342 + 0.999407i \(0.489037\pi\)
\(158\) −93.8432 −0.0472517
\(159\) −820.511 −0.409250
\(160\) −523.003 −0.258419
\(161\) 198.134 0.0969886
\(162\) 20.6093 0.00999520
\(163\) 1649.77 0.792763 0.396381 0.918086i \(-0.370266\pi\)
0.396381 + 0.918086i \(0.370266\pi\)
\(164\) 1470.91 0.700357
\(165\) 1657.81 0.782184
\(166\) −138.894 −0.0649412
\(167\) 3500.76 1.62214 0.811070 0.584949i \(-0.198886\pi\)
0.811070 + 0.584949i \(0.198886\pi\)
\(168\) −283.350 −0.130124
\(169\) 493.092 0.224439
\(170\) 0.116135 5.23951e−5 0
\(171\) −766.874 −0.342949
\(172\) −2202.29 −0.976298
\(173\) 2524.80 1.10958 0.554790 0.831991i \(-0.312799\pi\)
0.554790 + 0.831991i \(0.312799\pi\)
\(174\) 77.2023 0.0336361
\(175\) 183.435 0.0792364
\(176\) 3188.77 1.36570
\(177\) 177.000 0.0751646
\(178\) 161.396 0.0679614
\(179\) 2717.45 1.13470 0.567351 0.823476i \(-0.307968\pi\)
0.567351 + 0.823476i \(0.307968\pi\)
\(180\) 772.911 0.320052
\(181\) 2116.30 0.869077 0.434539 0.900653i \(-0.356911\pi\)
0.434539 + 0.900653i \(0.356911\pi\)
\(182\) −307.416 −0.125204
\(183\) −2258.68 −0.912384
\(184\) 34.4853 0.0138168
\(185\) 103.272 0.0410416
\(186\) −207.115 −0.0816473
\(187\) −2.15351 −0.000842141 0
\(188\) −2458.96 −0.953925
\(189\) 628.966 0.242066
\(190\) 234.631 0.0895892
\(191\) −162.049 −0.0613899 −0.0306950 0.999529i \(-0.509772\pi\)
−0.0306950 + 0.999529i \(0.509772\pi\)
\(192\) 1461.92 0.549507
\(193\) −1737.71 −0.648099 −0.324050 0.946040i \(-0.605044\pi\)
−0.324050 + 0.946040i \(0.605044\pi\)
\(194\) 80.4956 0.0297899
\(195\) 1683.96 0.618413
\(196\) −1584.35 −0.577385
\(197\) −1898.32 −0.686548 −0.343274 0.939235i \(-0.611536\pi\)
−0.343274 + 0.939235i \(0.611536\pi\)
\(198\) 116.925 0.0419673
\(199\) −925.010 −0.329509 −0.164754 0.986335i \(-0.552683\pi\)
−0.164754 + 0.986335i \(0.552683\pi\)
\(200\) 31.9269 0.0112879
\(201\) −2254.30 −0.791077
\(202\) −410.979 −0.143150
\(203\) 2356.10 0.814609
\(204\) −1.00402 −0.000344585 0
\(205\) 2006.09 0.683469
\(206\) 320.862 0.108522
\(207\) −76.5489 −0.0257030
\(208\) 3239.06 1.07975
\(209\) −4350.80 −1.43996
\(210\) −192.437 −0.0632354
\(211\) 2553.68 0.833186 0.416593 0.909093i \(-0.363224\pi\)
0.416593 + 0.909093i \(0.363224\pi\)
\(212\) −2170.32 −0.703106
\(213\) 1954.82 0.628836
\(214\) 186.701 0.0596383
\(215\) −3003.58 −0.952756
\(216\) 109.472 0.0344843
\(217\) −6320.84 −1.97736
\(218\) 148.298 0.0460733
\(219\) −2527.28 −0.779809
\(220\) 4385.05 1.34382
\(221\) −2.18748 −0.000665817 0
\(222\) 7.28378 0.00220205
\(223\) −2867.05 −0.860949 −0.430475 0.902603i \(-0.641654\pi\)
−0.430475 + 0.902603i \(0.641654\pi\)
\(224\) −1125.75 −0.335792
\(225\) −70.8698 −0.0209985
\(226\) −422.816 −0.124448
\(227\) −5544.65 −1.62120 −0.810598 0.585603i \(-0.800858\pi\)
−0.810598 + 0.585603i \(0.800858\pi\)
\(228\) −2028.45 −0.589198
\(229\) −3010.94 −0.868857 −0.434429 0.900706i \(-0.643050\pi\)
−0.434429 + 0.900706i \(0.643050\pi\)
\(230\) 23.4208 0.00671443
\(231\) 3568.39 1.01638
\(232\) 410.079 0.116048
\(233\) 132.563 0.0372724 0.0186362 0.999826i \(-0.494068\pi\)
0.0186362 + 0.999826i \(0.494068\pi\)
\(234\) 118.770 0.0331804
\(235\) −3353.63 −0.930923
\(236\) 468.180 0.129135
\(237\) 1106.48 0.303265
\(238\) 0.249978 6.80827e−5 0
\(239\) 7104.31 1.92276 0.961380 0.275224i \(-0.0887519\pi\)
0.961380 + 0.275224i \(0.0887519\pi\)
\(240\) 2027.60 0.545338
\(241\) 124.665 0.0333211 0.0166605 0.999861i \(-0.494697\pi\)
0.0166605 + 0.999861i \(0.494697\pi\)
\(242\) 324.713 0.0862534
\(243\) −243.000 −0.0641500
\(244\) −5974.40 −1.56751
\(245\) −2160.80 −0.563463
\(246\) 141.489 0.0366709
\(247\) −4419.42 −1.13846
\(248\) −1100.14 −0.281690
\(249\) 1637.67 0.416798
\(250\) 365.886 0.0925627
\(251\) −6840.81 −1.72027 −0.860135 0.510066i \(-0.829621\pi\)
−0.860135 + 0.510066i \(0.829621\pi\)
\(252\) 1663.67 0.415878
\(253\) −434.294 −0.107920
\(254\) 397.068 0.0980877
\(255\) −1.36933 −0.000336277 0
\(256\) 3768.55 0.920056
\(257\) −374.942 −0.0910049 −0.0455024 0.998964i \(-0.514489\pi\)
−0.0455024 + 0.998964i \(0.514489\pi\)
\(258\) −211.843 −0.0511192
\(259\) 222.290 0.0533298
\(260\) 4454.21 1.06246
\(261\) −910.275 −0.215880
\(262\) 663.719 0.156506
\(263\) 7174.93 1.68222 0.841112 0.540861i \(-0.181901\pi\)
0.841112 + 0.540861i \(0.181901\pi\)
\(264\) 621.079 0.144791
\(265\) −2959.98 −0.686152
\(266\) 505.038 0.116413
\(267\) −1902.98 −0.436182
\(268\) −5962.83 −1.35910
\(269\) −2997.57 −0.679425 −0.339713 0.940529i \(-0.610330\pi\)
−0.339713 + 0.940529i \(0.610330\pi\)
\(270\) 74.3479 0.0167580
\(271\) 5993.77 1.34353 0.671763 0.740766i \(-0.265537\pi\)
0.671763 + 0.740766i \(0.265537\pi\)
\(272\) −2.63388 −0.000587141 0
\(273\) 3624.67 0.803571
\(274\) −581.233 −0.128152
\(275\) −402.074 −0.0881672
\(276\) −202.478 −0.0441586
\(277\) −8759.33 −1.89999 −0.949994 0.312267i \(-0.898912\pi\)
−0.949994 + 0.312267i \(0.898912\pi\)
\(278\) −73.3600 −0.0158268
\(279\) 2442.05 0.524020
\(280\) −1022.18 −0.218168
\(281\) −4991.90 −1.05976 −0.529878 0.848074i \(-0.677762\pi\)
−0.529878 + 0.848074i \(0.677762\pi\)
\(282\) −236.532 −0.0499478
\(283\) −470.303 −0.0987866 −0.0493933 0.998779i \(-0.515729\pi\)
−0.0493933 + 0.998779i \(0.515729\pi\)
\(284\) 5170.67 1.08036
\(285\) −2766.49 −0.574991
\(286\) 673.829 0.139316
\(287\) 4318.05 0.888105
\(288\) 434.931 0.0889881
\(289\) −4913.00 −1.00000
\(290\) 278.506 0.0563946
\(291\) −949.105 −0.191194
\(292\) −6684.89 −1.33974
\(293\) 1777.56 0.354425 0.177212 0.984173i \(-0.443292\pi\)
0.177212 + 0.984173i \(0.443292\pi\)
\(294\) −152.401 −0.0302321
\(295\) 638.525 0.126022
\(296\) 38.6896 0.00759726
\(297\) −1378.64 −0.269350
\(298\) 150.500 0.0292558
\(299\) −441.143 −0.0853243
\(300\) −187.457 −0.0360761
\(301\) −6465.13 −1.23802
\(302\) −610.336 −0.116294
\(303\) 4845.76 0.918752
\(304\) −5321.29 −1.00394
\(305\) −8148.15 −1.52971
\(306\) −0.0965786 −1.80426e−5 0
\(307\) 4138.06 0.769289 0.384645 0.923065i \(-0.374324\pi\)
0.384645 + 0.923065i \(0.374324\pi\)
\(308\) 9438.70 1.74617
\(309\) −3783.21 −0.696503
\(310\) −747.164 −0.136891
\(311\) 7356.20 1.34126 0.670630 0.741792i \(-0.266024\pi\)
0.670630 + 0.741792i \(0.266024\pi\)
\(312\) 630.874 0.114475
\(313\) 6696.92 1.20937 0.604685 0.796465i \(-0.293299\pi\)
0.604685 + 0.796465i \(0.293299\pi\)
\(314\) 34.4706 0.00619518
\(315\) 2268.99 0.405850
\(316\) 2926.75 0.521020
\(317\) 7828.34 1.38701 0.693507 0.720450i \(-0.256065\pi\)
0.693507 + 0.720450i \(0.256065\pi\)
\(318\) −208.768 −0.0368148
\(319\) −5164.37 −0.906424
\(320\) 5273.87 0.921307
\(321\) −2201.35 −0.382764
\(322\) 50.4125 0.00872478
\(323\) 3.59369 0.000619066 0
\(324\) −642.756 −0.110212
\(325\) −408.415 −0.0697071
\(326\) 419.762 0.0713144
\(327\) −1748.54 −0.295702
\(328\) 751.557 0.126518
\(329\) −7218.60 −1.20965
\(330\) 421.807 0.0703627
\(331\) −791.925 −0.131505 −0.0657525 0.997836i \(-0.520945\pi\)
−0.0657525 + 0.997836i \(0.520945\pi\)
\(332\) 4331.77 0.716074
\(333\) −85.8814 −0.0141329
\(334\) 890.721 0.145922
\(335\) −8132.37 −1.32633
\(336\) 4364.36 0.708617
\(337\) 3609.06 0.583378 0.291689 0.956513i \(-0.405783\pi\)
0.291689 + 0.956513i \(0.405783\pi\)
\(338\) 125.460 0.0201898
\(339\) 4985.33 0.798720
\(340\) −3.62199 −0.000577735 0
\(341\) 13854.8 2.20023
\(342\) −195.120 −0.0308506
\(343\) 3339.14 0.525646
\(344\) −1125.26 −0.176366
\(345\) −276.149 −0.0430938
\(346\) 642.401 0.0998142
\(347\) −4501.38 −0.696389 −0.348194 0.937422i \(-0.613205\pi\)
−0.348194 + 0.937422i \(0.613205\pi\)
\(348\) −2407.76 −0.370889
\(349\) 4505.21 0.690998 0.345499 0.938419i \(-0.387710\pi\)
0.345499 + 0.938419i \(0.387710\pi\)
\(350\) 46.6725 0.00712785
\(351\) −1400.38 −0.212954
\(352\) 2467.55 0.373639
\(353\) −6156.62 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(354\) 45.0352 0.00676157
\(355\) 7051.99 1.05431
\(356\) −5033.56 −0.749376
\(357\) −2.94744 −0.000436960 0
\(358\) 691.417 0.102074
\(359\) 8811.87 1.29547 0.647734 0.761867i \(-0.275717\pi\)
0.647734 + 0.761867i \(0.275717\pi\)
\(360\) 394.918 0.0578166
\(361\) 401.436 0.0585269
\(362\) 538.462 0.0781794
\(363\) −3828.61 −0.553582
\(364\) 9587.56 1.38056
\(365\) −9117.14 −1.30743
\(366\) −574.690 −0.0820752
\(367\) 13290.6 1.89036 0.945181 0.326546i \(-0.105885\pi\)
0.945181 + 0.326546i \(0.105885\pi\)
\(368\) −531.168 −0.0752420
\(369\) −1668.27 −0.235357
\(370\) 26.2761 0.00369197
\(371\) −6371.28 −0.891591
\(372\) 6459.43 0.900284
\(373\) 10006.7 1.38908 0.694541 0.719453i \(-0.255608\pi\)
0.694541 + 0.719453i \(0.255608\pi\)
\(374\) −0.547931 −7.57563e−5 0
\(375\) −4314.08 −0.594075
\(376\) −1256.40 −0.172324
\(377\) −5245.82 −0.716641
\(378\) 160.032 0.0217755
\(379\) −3895.86 −0.528013 −0.264006 0.964521i \(-0.585044\pi\)
−0.264006 + 0.964521i \(0.585044\pi\)
\(380\) −7317.60 −0.987855
\(381\) −4681.74 −0.629535
\(382\) −41.2312 −0.00552244
\(383\) −3038.64 −0.405398 −0.202699 0.979241i \(-0.564971\pi\)
−0.202699 + 0.979241i \(0.564971\pi\)
\(384\) 1531.78 0.203564
\(385\) 12872.9 1.70406
\(386\) −442.136 −0.0583009
\(387\) 2497.79 0.328088
\(388\) −2510.47 −0.328478
\(389\) −13537.3 −1.76444 −0.882222 0.470834i \(-0.843953\pi\)
−0.882222 + 0.470834i \(0.843953\pi\)
\(390\) 428.459 0.0556305
\(391\) 0.358720 4.63971e−5 0
\(392\) −809.518 −0.104303
\(393\) −7825.76 −1.00447
\(394\) −483.002 −0.0617597
\(395\) 3991.63 0.508457
\(396\) −3646.63 −0.462752
\(397\) −3491.32 −0.441371 −0.220686 0.975345i \(-0.570829\pi\)
−0.220686 + 0.975345i \(0.570829\pi\)
\(398\) −235.356 −0.0296415
\(399\) −5954.78 −0.747148
\(400\) −491.761 −0.0614702
\(401\) 12667.7 1.57754 0.788769 0.614689i \(-0.210719\pi\)
0.788769 + 0.614689i \(0.210719\pi\)
\(402\) −573.577 −0.0711627
\(403\) 14073.3 1.73955
\(404\) 12817.5 1.57845
\(405\) −876.619 −0.107554
\(406\) 599.477 0.0732796
\(407\) −487.241 −0.0593407
\(408\) −0.513002 −6.22485e−5 0
\(409\) 10431.3 1.26111 0.630557 0.776143i \(-0.282826\pi\)
0.630557 + 0.776143i \(0.282826\pi\)
\(410\) 510.421 0.0614827
\(411\) 6853.19 0.822488
\(412\) −10006.9 −1.19662
\(413\) 1374.41 0.163753
\(414\) −19.4768 −0.00231216
\(415\) 5907.85 0.698808
\(416\) 2506.47 0.295408
\(417\) 864.971 0.101578
\(418\) −1107.00 −0.129534
\(419\) −6955.06 −0.810923 −0.405462 0.914112i \(-0.632889\pi\)
−0.405462 + 0.914112i \(0.632889\pi\)
\(420\) 6001.67 0.697265
\(421\) −2977.09 −0.344643 −0.172321 0.985041i \(-0.555127\pi\)
−0.172321 + 0.985041i \(0.555127\pi\)
\(422\) 649.748 0.0749507
\(423\) 2788.90 0.320569
\(424\) −1108.92 −0.127014
\(425\) 0.332107 3.79049e−5 0
\(426\) 497.377 0.0565681
\(427\) −17538.7 −1.98772
\(428\) −5822.76 −0.657602
\(429\) −7944.97 −0.894142
\(430\) −764.220 −0.0857069
\(431\) −9721.43 −1.08646 −0.543230 0.839584i \(-0.682799\pi\)
−0.543230 + 0.839584i \(0.682799\pi\)
\(432\) −1686.16 −0.187791
\(433\) 814.588 0.0904079 0.0452040 0.998978i \(-0.485606\pi\)
0.0452040 + 0.998978i \(0.485606\pi\)
\(434\) −1608.25 −0.177877
\(435\) −3283.80 −0.361945
\(436\) −4625.05 −0.508027
\(437\) 724.732 0.0793333
\(438\) −643.033 −0.0701491
\(439\) 15373.9 1.67143 0.835716 0.549162i \(-0.185053\pi\)
0.835716 + 0.549162i \(0.185053\pi\)
\(440\) 2240.53 0.242757
\(441\) 1796.93 0.194032
\(442\) −0.556573 −5.98947e−5 0
\(443\) 5462.95 0.585897 0.292949 0.956128i \(-0.405364\pi\)
0.292949 + 0.956128i \(0.405364\pi\)
\(444\) −227.164 −0.0242809
\(445\) −6864.99 −0.731307
\(446\) −729.480 −0.0774482
\(447\) −1774.51 −0.187766
\(448\) 11351.9 1.19715
\(449\) −9408.54 −0.988901 −0.494451 0.869206i \(-0.664631\pi\)
−0.494451 + 0.869206i \(0.664631\pi\)
\(450\) −18.0318 −0.00188895
\(451\) −9464.80 −0.988204
\(452\) 13186.6 1.37223
\(453\) 7196.34 0.746388
\(454\) −1410.76 −0.145838
\(455\) 13075.9 1.34727
\(456\) −1036.43 −0.106437
\(457\) 1254.04 0.128362 0.0641809 0.997938i \(-0.479557\pi\)
0.0641809 + 0.997938i \(0.479557\pi\)
\(458\) −766.091 −0.0781596
\(459\) 1.13874 0.000115799 0
\(460\) −730.438 −0.0740366
\(461\) −15647.0 −1.58081 −0.790407 0.612582i \(-0.790131\pi\)
−0.790407 + 0.612582i \(0.790131\pi\)
\(462\) 907.927 0.0914299
\(463\) 18387.3 1.84564 0.922821 0.385228i \(-0.125877\pi\)
0.922821 + 0.385228i \(0.125877\pi\)
\(464\) −6316.34 −0.631959
\(465\) 8809.65 0.878575
\(466\) 33.7287 0.00335290
\(467\) 13135.8 1.30161 0.650805 0.759245i \(-0.274431\pi\)
0.650805 + 0.759245i \(0.274431\pi\)
\(468\) −3704.14 −0.365863
\(469\) −17504.7 −1.72344
\(470\) −853.286 −0.0837429
\(471\) −406.435 −0.0397612
\(472\) 239.216 0.0233280
\(473\) 14171.0 1.37756
\(474\) 281.530 0.0272808
\(475\) 670.965 0.0648126
\(476\) −7.79623 −0.000750713 0
\(477\) 2461.53 0.236281
\(478\) 1807.59 0.172965
\(479\) −4918.54 −0.469173 −0.234587 0.972095i \(-0.575374\pi\)
−0.234587 + 0.972095i \(0.575374\pi\)
\(480\) 1569.01 0.149198
\(481\) −494.926 −0.0469162
\(482\) 31.7193 0.00299745
\(483\) −594.403 −0.0559964
\(484\) −10127.0 −0.951072
\(485\) −3423.88 −0.320558
\(486\) −61.8280 −0.00577073
\(487\) 12272.0 1.14189 0.570944 0.820989i \(-0.306577\pi\)
0.570944 + 0.820989i \(0.306577\pi\)
\(488\) −3052.61 −0.283166
\(489\) −4949.32 −0.457702
\(490\) −549.786 −0.0506873
\(491\) −9165.57 −0.842437 −0.421218 0.906959i \(-0.638397\pi\)
−0.421218 + 0.906959i \(0.638397\pi\)
\(492\) −4412.72 −0.404351
\(493\) 4.26569 0.000389690 0
\(494\) −1124.46 −0.102413
\(495\) −4973.43 −0.451594
\(496\) 16945.2 1.53400
\(497\) 15179.2 1.36998
\(498\) 416.681 0.0374938
\(499\) −4308.93 −0.386561 −0.193281 0.981144i \(-0.561913\pi\)
−0.193281 + 0.981144i \(0.561913\pi\)
\(500\) −11411.1 −1.02064
\(501\) −10502.3 −0.936543
\(502\) −1740.55 −0.154750
\(503\) −10004.0 −0.886795 −0.443397 0.896325i \(-0.646227\pi\)
−0.443397 + 0.896325i \(0.646227\pi\)
\(504\) 850.049 0.0751274
\(505\) 17481.0 1.54039
\(506\) −110.500 −0.00970816
\(507\) −1479.28 −0.129580
\(508\) −12383.6 −1.08156
\(509\) −11329.0 −0.986538 −0.493269 0.869877i \(-0.664198\pi\)
−0.493269 + 0.869877i \(0.664198\pi\)
\(510\) −0.348406 −3.02503e−5 0
\(511\) −19624.4 −1.69889
\(512\) 5043.61 0.435348
\(513\) 2300.62 0.198002
\(514\) −95.3989 −0.00818650
\(515\) −13647.9 −1.16776
\(516\) 6606.88 0.563666
\(517\) 15822.6 1.34599
\(518\) 56.5586 0.00479738
\(519\) −7574.41 −0.640616
\(520\) 2275.87 0.191930
\(521\) −578.251 −0.0486251 −0.0243125 0.999704i \(-0.507740\pi\)
−0.0243125 + 0.999704i \(0.507740\pi\)
\(522\) −231.607 −0.0194198
\(523\) −22559.8 −1.88618 −0.943088 0.332543i \(-0.892093\pi\)
−0.943088 + 0.332543i \(0.892093\pi\)
\(524\) −20699.8 −1.72572
\(525\) −550.305 −0.0457472
\(526\) 1825.56 0.151327
\(527\) −11.4438 −0.000945922 0
\(528\) −9566.31 −0.788486
\(529\) −12094.7 −0.994054
\(530\) −753.126 −0.0617240
\(531\) −531.000 −0.0433963
\(532\) −15750.9 −1.28363
\(533\) −9614.07 −0.781298
\(534\) −484.188 −0.0392376
\(535\) −7941.33 −0.641745
\(536\) −3046.70 −0.245517
\(537\) −8152.34 −0.655120
\(538\) −762.691 −0.0611189
\(539\) 10194.7 0.814691
\(540\) −2318.73 −0.184782
\(541\) −8649.61 −0.687386 −0.343693 0.939082i \(-0.611678\pi\)
−0.343693 + 0.939082i \(0.611678\pi\)
\(542\) 1525.03 0.120859
\(543\) −6348.89 −0.501762
\(544\) −2.03816 −0.000160635 0
\(545\) −6307.84 −0.495777
\(546\) 922.247 0.0722867
\(547\) −9720.33 −0.759801 −0.379901 0.925027i \(-0.624042\pi\)
−0.379901 + 0.925027i \(0.624042\pi\)
\(548\) 18127.3 1.41306
\(549\) 6776.04 0.526765
\(550\) −102.302 −0.00793124
\(551\) 8618.09 0.666322
\(552\) −103.456 −0.00797713
\(553\) 8591.87 0.660693
\(554\) −2228.69 −0.170917
\(555\) −309.816 −0.0236954
\(556\) 2287.92 0.174514
\(557\) 14288.6 1.08694 0.543472 0.839428i \(-0.317110\pi\)
0.543472 + 0.839428i \(0.317110\pi\)
\(558\) 621.345 0.0471391
\(559\) 14394.5 1.08913
\(560\) 15744.4 1.18807
\(561\) 6.46053 0.000486210 0
\(562\) −1270.12 −0.0953323
\(563\) −2067.53 −0.154771 −0.0773856 0.997001i \(-0.524657\pi\)
−0.0773856 + 0.997001i \(0.524657\pi\)
\(564\) 7376.87 0.550749
\(565\) 17984.5 1.33914
\(566\) −119.662 −0.00888652
\(567\) −1886.90 −0.139757
\(568\) 2641.94 0.195165
\(569\) −3196.60 −0.235516 −0.117758 0.993042i \(-0.537571\pi\)
−0.117758 + 0.993042i \(0.537571\pi\)
\(570\) −703.894 −0.0517244
\(571\) −15856.8 −1.16215 −0.581073 0.813852i \(-0.697367\pi\)
−0.581073 + 0.813852i \(0.697367\pi\)
\(572\) −21015.1 −1.53617
\(573\) 486.148 0.0354435
\(574\) 1098.67 0.0798911
\(575\) 66.9753 0.00485750
\(576\) −4385.77 −0.317258
\(577\) 18096.9 1.30569 0.652846 0.757491i \(-0.273575\pi\)
0.652846 + 0.757491i \(0.273575\pi\)
\(578\) −1250.04 −0.0899567
\(579\) 5213.13 0.374180
\(580\) −8685.94 −0.621835
\(581\) 12716.5 0.908036
\(582\) −241.487 −0.0171992
\(583\) 13965.3 0.992082
\(584\) −3415.63 −0.242020
\(585\) −5051.87 −0.357041
\(586\) 452.277 0.0318829
\(587\) 1903.98 0.133877 0.0669383 0.997757i \(-0.478677\pi\)
0.0669383 + 0.997757i \(0.478677\pi\)
\(588\) 4753.04 0.333354
\(589\) −23120.3 −1.61741
\(590\) 162.464 0.0113365
\(591\) 5694.97 0.396379
\(592\) −595.926 −0.0413723
\(593\) 16287.4 1.12790 0.563948 0.825810i \(-0.309282\pi\)
0.563948 + 0.825810i \(0.309282\pi\)
\(594\) −350.776 −0.0242298
\(595\) −10.6328 −0.000732611 0
\(596\) −4693.73 −0.322588
\(597\) 2775.03 0.190242
\(598\) −112.243 −0.00767550
\(599\) −13497.9 −0.920715 −0.460357 0.887734i \(-0.652279\pi\)
−0.460357 + 0.887734i \(0.652279\pi\)
\(600\) −95.7807 −0.00651705
\(601\) 13860.8 0.940755 0.470378 0.882465i \(-0.344118\pi\)
0.470378 + 0.882465i \(0.344118\pi\)
\(602\) −1644.96 −0.111368
\(603\) 6762.91 0.456728
\(604\) 19034.9 1.28232
\(605\) −13811.7 −0.928139
\(606\) 1232.94 0.0826479
\(607\) 13140.0 0.878641 0.439321 0.898330i \(-0.355219\pi\)
0.439321 + 0.898330i \(0.355219\pi\)
\(608\) −4117.75 −0.274666
\(609\) −7068.29 −0.470315
\(610\) −2073.18 −0.137608
\(611\) 16072.1 1.06417
\(612\) 3.01206 0.000198947 0
\(613\) −10495.5 −0.691535 −0.345768 0.938320i \(-0.612381\pi\)
−0.345768 + 0.938320i \(0.612381\pi\)
\(614\) 1052.87 0.0692028
\(615\) −6018.26 −0.394601
\(616\) 4822.69 0.315441
\(617\) 9157.76 0.597533 0.298766 0.954326i \(-0.403425\pi\)
0.298766 + 0.954326i \(0.403425\pi\)
\(618\) −962.586 −0.0626552
\(619\) 15266.3 0.991286 0.495643 0.868526i \(-0.334932\pi\)
0.495643 + 0.868526i \(0.334932\pi\)
\(620\) 23302.3 1.50942
\(621\) 229.647 0.0148396
\(622\) 1871.68 0.120655
\(623\) −14776.7 −0.950266
\(624\) −9717.19 −0.623396
\(625\) −14578.7 −0.933036
\(626\) 1703.94 0.108791
\(627\) 13052.4 0.831359
\(628\) −1075.06 −0.0683111
\(629\) 0.402454 2.55117e−5 0
\(630\) 577.312 0.0365090
\(631\) 6869.98 0.433422 0.216711 0.976236i \(-0.430467\pi\)
0.216711 + 0.976236i \(0.430467\pi\)
\(632\) 1495.42 0.0941210
\(633\) −7661.03 −0.481040
\(634\) 1991.81 0.124771
\(635\) −16889.3 −1.05548
\(636\) 6510.97 0.405938
\(637\) 10355.5 0.644114
\(638\) −1314.00 −0.0815390
\(639\) −5864.46 −0.363059
\(640\) 5525.89 0.341297
\(641\) 14205.2 0.875306 0.437653 0.899144i \(-0.355810\pi\)
0.437653 + 0.899144i \(0.355810\pi\)
\(642\) −560.102 −0.0344322
\(643\) 32050.2 1.96569 0.982843 0.184442i \(-0.0590479\pi\)
0.982843 + 0.184442i \(0.0590479\pi\)
\(644\) −1572.25 −0.0962038
\(645\) 9010.75 0.550074
\(646\) 0.914366 5.56892e−5 0
\(647\) 27275.5 1.65736 0.828679 0.559725i \(-0.189093\pi\)
0.828679 + 0.559725i \(0.189093\pi\)
\(648\) −328.415 −0.0199095
\(649\) −3012.59 −0.182210
\(650\) −103.916 −0.00627063
\(651\) 18962.5 1.14163
\(652\) −13091.4 −0.786347
\(653\) −6694.40 −0.401182 −0.200591 0.979675i \(-0.564286\pi\)
−0.200591 + 0.979675i \(0.564286\pi\)
\(654\) −444.893 −0.0266004
\(655\) −28231.3 −1.68411
\(656\) −11576.0 −0.688976
\(657\) 7581.85 0.450223
\(658\) −1836.67 −0.108816
\(659\) 1828.61 0.108092 0.0540460 0.998538i \(-0.482788\pi\)
0.0540460 + 0.998538i \(0.482788\pi\)
\(660\) −13155.2 −0.775854
\(661\) 3994.07 0.235025 0.117512 0.993071i \(-0.462508\pi\)
0.117512 + 0.993071i \(0.462508\pi\)
\(662\) −201.494 −0.0118298
\(663\) 6.56243 0.000384410 0
\(664\) 2213.31 0.129357
\(665\) −21481.8 −1.25267
\(666\) −21.8513 −0.00127135
\(667\) 860.253 0.0499387
\(668\) −27779.5 −1.60901
\(669\) 8601.14 0.497069
\(670\) −2069.17 −0.119312
\(671\) 38443.3 2.21175
\(672\) 3377.25 0.193869
\(673\) 1511.93 0.0865980 0.0432990 0.999062i \(-0.486213\pi\)
0.0432990 + 0.999062i \(0.486213\pi\)
\(674\) 918.276 0.0524788
\(675\) 212.609 0.0121235
\(676\) −3912.81 −0.222623
\(677\) 105.944 0.00601442 0.00300721 0.999995i \(-0.499043\pi\)
0.00300721 + 0.999995i \(0.499043\pi\)
\(678\) 1268.45 0.0718503
\(679\) −7369.81 −0.416535
\(680\) −1.85065 −0.000104366 0
\(681\) 16634.0 0.935998
\(682\) 3525.15 0.197925
\(683\) −5370.44 −0.300870 −0.150435 0.988620i \(-0.548067\pi\)
−0.150435 + 0.988620i \(0.548067\pi\)
\(684\) 6085.34 0.340174
\(685\) 24722.8 1.37899
\(686\) 849.597 0.0472854
\(687\) 9032.81 0.501635
\(688\) 17332.0 0.960433
\(689\) 14185.6 0.784364
\(690\) −70.2623 −0.00387658
\(691\) −11938.4 −0.657246 −0.328623 0.944461i \(-0.606584\pi\)
−0.328623 + 0.944461i \(0.606584\pi\)
\(692\) −20035.0 −1.10060
\(693\) −10705.2 −0.586805
\(694\) −1145.31 −0.0626449
\(695\) 3120.37 0.170306
\(696\) −1230.24 −0.0670001
\(697\) 7.81778 0.000424849 0
\(698\) 1146.29 0.0621599
\(699\) −397.688 −0.0215192
\(700\) −1455.60 −0.0785952
\(701\) 20482.4 1.10358 0.551791 0.833982i \(-0.313945\pi\)
0.551791 + 0.833982i \(0.313945\pi\)
\(702\) −356.309 −0.0191567
\(703\) 813.089 0.0436219
\(704\) −24882.3 −1.33209
\(705\) 10060.9 0.537469
\(706\) −1566.47 −0.0835053
\(707\) 37627.4 2.00159
\(708\) −1404.54 −0.0745564
\(709\) −9508.25 −0.503653 −0.251826 0.967772i \(-0.581031\pi\)
−0.251826 + 0.967772i \(0.581031\pi\)
\(710\) 1794.28 0.0948424
\(711\) −3319.45 −0.175090
\(712\) −2571.89 −0.135373
\(713\) −2307.85 −0.121220
\(714\) −0.749934 −3.93075e−5 0
\(715\) −28661.4 −1.49913
\(716\) −21563.7 −1.12552
\(717\) −21312.9 −1.11011
\(718\) 2242.06 0.116536
\(719\) 7683.75 0.398547 0.199274 0.979944i \(-0.436142\pi\)
0.199274 + 0.979944i \(0.436142\pi\)
\(720\) −6082.81 −0.314851
\(721\) −29376.7 −1.51740
\(722\) 102.140 0.00526489
\(723\) −373.995 −0.0192379
\(724\) −16793.4 −0.862044
\(725\) 796.432 0.0407982
\(726\) −974.138 −0.0497984
\(727\) −28207.9 −1.43903 −0.719515 0.694477i \(-0.755636\pi\)
−0.719515 + 0.694477i \(0.755636\pi\)
\(728\) 4898.75 0.249395
\(729\) 729.000 0.0370370
\(730\) −2319.73 −0.117612
\(731\) −11.7051 −0.000592239 0
\(732\) 17923.2 0.905001
\(733\) −21546.7 −1.08574 −0.542870 0.839817i \(-0.682662\pi\)
−0.542870 + 0.839817i \(0.682662\pi\)
\(734\) 3381.61 0.170051
\(735\) 6482.40 0.325316
\(736\) −411.031 −0.0205853
\(737\) 38368.9 1.91769
\(738\) −424.468 −0.0211719
\(739\) 13254.6 0.659780 0.329890 0.944019i \(-0.392988\pi\)
0.329890 + 0.944019i \(0.392988\pi\)
\(740\) −819.490 −0.0407095
\(741\) 13258.2 0.657293
\(742\) −1621.08 −0.0802046
\(743\) −13311.7 −0.657280 −0.328640 0.944455i \(-0.606590\pi\)
−0.328640 + 0.944455i \(0.606590\pi\)
\(744\) 3300.43 0.162634
\(745\) −6401.51 −0.314810
\(746\) 2546.07 0.124957
\(747\) −4913.00 −0.240639
\(748\) 17.0887 0.000835326 0
\(749\) −17093.5 −0.833889
\(750\) −1097.66 −0.0534411
\(751\) 30067.0 1.46093 0.730467 0.682948i \(-0.239302\pi\)
0.730467 + 0.682948i \(0.239302\pi\)
\(752\) 19352.0 0.938423
\(753\) 20522.4 0.993199
\(754\) −1334.73 −0.0644667
\(755\) 25960.7 1.25140
\(756\) −4991.01 −0.240108
\(757\) −40542.2 −1.94654 −0.973269 0.229666i \(-0.926236\pi\)
−0.973269 + 0.229666i \(0.926236\pi\)
\(758\) −991.247 −0.0474983
\(759\) 1302.88 0.0623078
\(760\) −3738.91 −0.178453
\(761\) −35970.2 −1.71343 −0.856713 0.515793i \(-0.827498\pi\)
−0.856713 + 0.515793i \(0.827498\pi\)
\(762\) −1191.20 −0.0566310
\(763\) −13577.5 −0.644216
\(764\) 1285.90 0.0608931
\(765\) 4.10798 0.000194149 0
\(766\) −773.140 −0.0364683
\(767\) −3060.10 −0.144060
\(768\) −11305.7 −0.531195
\(769\) 29540.6 1.38525 0.692627 0.721296i \(-0.256453\pi\)
0.692627 + 0.721296i \(0.256453\pi\)
\(770\) 3275.34 0.153292
\(771\) 1124.83 0.0525417
\(772\) 13789.2 0.642854
\(773\) 31936.1 1.48598 0.742989 0.669304i \(-0.233408\pi\)
0.742989 + 0.669304i \(0.233408\pi\)
\(774\) 635.529 0.0295137
\(775\) −2136.63 −0.0990324
\(776\) −1282.72 −0.0593388
\(777\) −666.870 −0.0307900
\(778\) −3444.38 −0.158724
\(779\) 15794.5 0.726439
\(780\) −13362.6 −0.613409
\(781\) −33271.6 −1.52439
\(782\) 0.0912714 4.17373e−6 0
\(783\) 2730.82 0.124638
\(784\) 12468.8 0.568003
\(785\) −1466.21 −0.0666640
\(786\) −1991.16 −0.0903590
\(787\) 19554.7 0.885705 0.442853 0.896594i \(-0.353966\pi\)
0.442853 + 0.896594i \(0.353966\pi\)
\(788\) 15063.7 0.680993
\(789\) −21524.8 −0.971232
\(790\) 1015.61 0.0457391
\(791\) 38711.2 1.74009
\(792\) −1863.24 −0.0835950
\(793\) 39049.6 1.74867
\(794\) −888.319 −0.0397043
\(795\) 8879.94 0.396150
\(796\) 7340.19 0.326842
\(797\) −19917.3 −0.885202 −0.442601 0.896719i \(-0.645944\pi\)
−0.442601 + 0.896719i \(0.645944\pi\)
\(798\) −1515.11 −0.0672110
\(799\) −13.0692 −0.000578668 0
\(800\) −380.537 −0.0168175
\(801\) 5708.95 0.251830
\(802\) 3223.11 0.141910
\(803\) 43015.1 1.89037
\(804\) 17888.5 0.784675
\(805\) −2144.30 −0.0938840
\(806\) 3580.75 0.156484
\(807\) 8992.72 0.392266
\(808\) 6549.06 0.285142
\(809\) −23172.3 −1.00704 −0.503520 0.863984i \(-0.667962\pi\)
−0.503520 + 0.863984i \(0.667962\pi\)
\(810\) −223.044 −0.00967525
\(811\) −9763.58 −0.422744 −0.211372 0.977406i \(-0.567793\pi\)
−0.211372 + 0.977406i \(0.567793\pi\)
\(812\) −18696.3 −0.808017
\(813\) −17981.3 −0.775685
\(814\) −123.972 −0.00533809
\(815\) −17854.6 −0.767386
\(816\) 7.90163 0.000338986 0
\(817\) −23648.0 −1.01266
\(818\) 2654.11 0.113446
\(819\) −10874.0 −0.463942
\(820\) −15918.8 −0.677939
\(821\) 24006.4 1.02050 0.510249 0.860026i \(-0.329553\pi\)
0.510249 + 0.860026i \(0.329553\pi\)
\(822\) 1743.70 0.0739884
\(823\) 36909.5 1.56329 0.781643 0.623726i \(-0.214382\pi\)
0.781643 + 0.623726i \(0.214382\pi\)
\(824\) −5113.02 −0.216166
\(825\) 1206.22 0.0509034
\(826\) 349.699 0.0147307
\(827\) −17834.8 −0.749912 −0.374956 0.927043i \(-0.622342\pi\)
−0.374956 + 0.927043i \(0.622342\pi\)
\(828\) 607.435 0.0254950
\(829\) −7164.05 −0.300142 −0.150071 0.988675i \(-0.547950\pi\)
−0.150071 + 0.988675i \(0.547950\pi\)
\(830\) 1503.17 0.0628625
\(831\) 26278.0 1.09696
\(832\) −25274.8 −1.05318
\(833\) −8.42070 −0.000350252 0
\(834\) 220.080 0.00913758
\(835\) −37886.9 −1.57021
\(836\) 34524.7 1.42830
\(837\) −7326.14 −0.302543
\(838\) −1769.62 −0.0729480
\(839\) −30792.5 −1.26708 −0.633538 0.773712i \(-0.718398\pi\)
−0.633538 + 0.773712i \(0.718398\pi\)
\(840\) 3066.54 0.125959
\(841\) −14159.4 −0.580564
\(842\) −757.480 −0.0310030
\(843\) 14975.7 0.611851
\(844\) −20264.1 −0.826444
\(845\) −5336.47 −0.217254
\(846\) 709.596 0.0288374
\(847\) −29729.2 −1.20603
\(848\) 17080.4 0.691680
\(849\) 1410.91 0.0570345
\(850\) 0.0845001 3.40980e−6 0
\(851\) 81.1620 0.00326933
\(852\) −15512.0 −0.623747
\(853\) 31914.3 1.28104 0.640519 0.767942i \(-0.278719\pi\)
0.640519 + 0.767942i \(0.278719\pi\)
\(854\) −4462.47 −0.178809
\(855\) 8299.46 0.331971
\(856\) −2975.13 −0.118794
\(857\) −48527.1 −1.93425 −0.967126 0.254296i \(-0.918156\pi\)
−0.967126 + 0.254296i \(0.918156\pi\)
\(858\) −2021.49 −0.0804341
\(859\) −5570.19 −0.221249 −0.110624 0.993862i \(-0.535285\pi\)
−0.110624 + 0.993862i \(0.535285\pi\)
\(860\) 23834.2 0.945047
\(861\) −12954.1 −0.512748
\(862\) −2473.48 −0.0977345
\(863\) −45442.4 −1.79244 −0.896221 0.443607i \(-0.853699\pi\)
−0.896221 + 0.443607i \(0.853699\pi\)
\(864\) −1304.79 −0.0513773
\(865\) −27324.6 −1.07406
\(866\) 207.261 0.00813280
\(867\) 14739.0 0.577350
\(868\) 50157.5 1.96136
\(869\) −18832.7 −0.735160
\(870\) −835.518 −0.0325594
\(871\) 38974.0 1.51617
\(872\) −2363.16 −0.0917737
\(873\) 2847.32 0.110386
\(874\) 184.398 0.00713657
\(875\) −33498.9 −1.29425
\(876\) 20054.7 0.773498
\(877\) −23890.4 −0.919865 −0.459932 0.887954i \(-0.652126\pi\)
−0.459932 + 0.887954i \(0.652126\pi\)
\(878\) 3911.69 0.150357
\(879\) −5332.69 −0.204627
\(880\) −34510.3 −1.32198
\(881\) −9587.92 −0.366657 −0.183329 0.983052i \(-0.558687\pi\)
−0.183329 + 0.983052i \(0.558687\pi\)
\(882\) 457.204 0.0174545
\(883\) −50410.3 −1.92122 −0.960612 0.277893i \(-0.910364\pi\)
−0.960612 + 0.277893i \(0.910364\pi\)
\(884\) 17.3582 0.000660429 0
\(885\) −1915.57 −0.0727586
\(886\) 1389.97 0.0527054
\(887\) 7413.01 0.280614 0.140307 0.990108i \(-0.455191\pi\)
0.140307 + 0.990108i \(0.455191\pi\)
\(888\) −116.069 −0.00438628
\(889\) −36353.8 −1.37150
\(890\) −1746.70 −0.0657860
\(891\) 4135.92 0.155509
\(892\) 22750.8 0.853982
\(893\) −26404.1 −0.989450
\(894\) −451.499 −0.0168908
\(895\) −29409.5 −1.09838
\(896\) 11894.3 0.443484
\(897\) 1323.43 0.0492620
\(898\) −2393.87 −0.0889584
\(899\) −27443.6 −1.01813
\(900\) 562.370 0.0208285
\(901\) −11.5351 −0.000426516 0
\(902\) −2408.19 −0.0888957
\(903\) 19395.4 0.714771
\(904\) 6737.69 0.247890
\(905\) −22903.5 −0.841258
\(906\) 1831.01 0.0671426
\(907\) −13964.2 −0.511216 −0.255608 0.966781i \(-0.582276\pi\)
−0.255608 + 0.966781i \(0.582276\pi\)
\(908\) 43998.3 1.60808
\(909\) −14537.3 −0.530442
\(910\) 3326.99 0.121196
\(911\) 48214.6 1.75348 0.876740 0.480965i \(-0.159714\pi\)
0.876740 + 0.480965i \(0.159714\pi\)
\(912\) 15963.9 0.579624
\(913\) −27873.5 −1.01038
\(914\) 319.072 0.0115470
\(915\) 24444.5 0.883179
\(916\) 23892.6 0.861826
\(917\) −60767.1 −2.18834
\(918\) 0.289736 1.04169e−5 0
\(919\) −41258.5 −1.48095 −0.740475 0.672084i \(-0.765399\pi\)
−0.740475 + 0.672084i \(0.765399\pi\)
\(920\) −373.216 −0.0133745
\(921\) −12414.2 −0.444149
\(922\) −3981.17 −0.142205
\(923\) −33796.3 −1.20522
\(924\) −28316.1 −1.00815
\(925\) 75.1407 0.00267093
\(926\) 4678.41 0.166028
\(927\) 11349.6 0.402126
\(928\) −4887.74 −0.172897
\(929\) 2682.75 0.0947451 0.0473726 0.998877i \(-0.484915\pi\)
0.0473726 + 0.998877i \(0.484915\pi\)
\(930\) 2241.49 0.0790338
\(931\) −17012.6 −0.598888
\(932\) −1051.92 −0.0369708
\(933\) −22068.6 −0.774376
\(934\) 3342.22 0.117089
\(935\) 23.3063 0.000815184 0
\(936\) −1892.62 −0.0660922
\(937\) 12243.8 0.426883 0.213441 0.976956i \(-0.431533\pi\)
0.213441 + 0.976956i \(0.431533\pi\)
\(938\) −4453.83 −0.155035
\(939\) −20090.8 −0.698230
\(940\) 26612.0 0.923390
\(941\) 29701.6 1.02895 0.514476 0.857505i \(-0.327986\pi\)
0.514476 + 0.857505i \(0.327986\pi\)
\(942\) −103.412 −0.00357679
\(943\) 1576.59 0.0544443
\(944\) −3684.58 −0.127037
\(945\) −6806.96 −0.234318
\(946\) 3605.62 0.123921
\(947\) 11974.6 0.410899 0.205450 0.978668i \(-0.434134\pi\)
0.205450 + 0.978668i \(0.434134\pi\)
\(948\) −8780.24 −0.300811
\(949\) 43693.5 1.49457
\(950\) 170.718 0.00583033
\(951\) −23485.0 −0.800793
\(952\) −3.98347 −0.000135614 0
\(953\) 23437.4 0.796655 0.398327 0.917243i \(-0.369591\pi\)
0.398327 + 0.917243i \(0.369591\pi\)
\(954\) 626.303 0.0212550
\(955\) 1753.77 0.0594248
\(956\) −56374.6 −1.90720
\(957\) 15493.1 0.523324
\(958\) −1251.46 −0.0422053
\(959\) 53215.1 1.79187
\(960\) −15821.6 −0.531917
\(961\) 43833.6 1.47137
\(962\) −125.927 −0.00422043
\(963\) 6604.04 0.220989
\(964\) −989.249 −0.0330514
\(965\) 18806.3 0.627353
\(966\) −151.238 −0.00503726
\(967\) 8006.86 0.266270 0.133135 0.991098i \(-0.457496\pi\)
0.133135 + 0.991098i \(0.457496\pi\)
\(968\) −5174.38 −0.171809
\(969\) −10.7811 −0.000357418 0
\(970\) −871.160 −0.0288363
\(971\) −13457.8 −0.444780 −0.222390 0.974958i \(-0.571386\pi\)
−0.222390 + 0.974958i \(0.571386\pi\)
\(972\) 1928.27 0.0636309
\(973\) 6716.51 0.221297
\(974\) 3122.45 0.102721
\(975\) 1225.25 0.0402454
\(976\) 47018.5 1.54204
\(977\) 25366.7 0.830659 0.415329 0.909671i \(-0.363666\pi\)
0.415329 + 0.909671i \(0.363666\pi\)
\(978\) −1259.29 −0.0411734
\(979\) 32389.3 1.05737
\(980\) 17146.5 0.558903
\(981\) 5245.63 0.170724
\(982\) −2332.05 −0.0757829
\(983\) 5778.72 0.187500 0.0937500 0.995596i \(-0.470115\pi\)
0.0937500 + 0.995596i \(0.470115\pi\)
\(984\) −2254.67 −0.0730450
\(985\) 20544.5 0.664572
\(986\) 1.08535 3.50552e−5 0
\(987\) 21655.8 0.698391
\(988\) 35069.2 1.12925
\(989\) −2360.53 −0.0758954
\(990\) −1265.42 −0.0406239
\(991\) 24959.9 0.800079 0.400039 0.916498i \(-0.368996\pi\)
0.400039 + 0.916498i \(0.368996\pi\)
\(992\) 13112.6 0.419684
\(993\) 2375.78 0.0759244
\(994\) 3862.14 0.123239
\(995\) 10010.9 0.318961
\(996\) −12995.3 −0.413426
\(997\) 29815.3 0.947101 0.473550 0.880767i \(-0.342972\pi\)
0.473550 + 0.880767i \(0.342972\pi\)
\(998\) −1096.35 −0.0347738
\(999\) 257.644 0.00815966
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.c.1.4 8
3.2 odd 2 531.4.a.f.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.c.1.4 8 1.1 even 1 trivial
531.4.a.f.1.5 8 3.2 odd 2