Properties

Label 1815.4.a.o.1.1
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.70156 q^{2} +3.00000 q^{3} -0.701562 q^{4} +5.00000 q^{5} -8.10469 q^{6} +10.1047 q^{7} +23.5078 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.70156 q^{2} +3.00000 q^{3} -0.701562 q^{4} +5.00000 q^{5} -8.10469 q^{6} +10.1047 q^{7} +23.5078 q^{8} +9.00000 q^{9} -13.5078 q^{10} -2.10469 q^{12} +26.7016 q^{13} -27.2984 q^{14} +15.0000 q^{15} -57.8953 q^{16} -36.1938 q^{17} -24.3141 q^{18} -66.3141 q^{19} -3.50781 q^{20} +30.3141 q^{21} +70.7484 q^{23} +70.5234 q^{24} +25.0000 q^{25} -72.1359 q^{26} +27.0000 q^{27} -7.08907 q^{28} -41.0891 q^{29} -40.5234 q^{30} -24.2672 q^{31} -31.6547 q^{32} +97.7797 q^{34} +50.5234 q^{35} -6.31406 q^{36} -367.423 q^{37} +179.152 q^{38} +80.1047 q^{39} +117.539 q^{40} -10.2094 q^{41} -81.8953 q^{42} -79.8844 q^{43} +45.0000 q^{45} -191.131 q^{46} -99.3985 q^{47} -173.686 q^{48} -240.895 q^{49} -67.5391 q^{50} -108.581 q^{51} -18.7328 q^{52} -460.298 q^{53} -72.9422 q^{54} +237.539 q^{56} -198.942 q^{57} +111.005 q^{58} -753.475 q^{59} -10.5234 q^{60} +424.183 q^{61} +65.5593 q^{62} +90.9422 q^{63} +548.680 q^{64} +133.508 q^{65} -1049.23 q^{67} +25.3922 q^{68} +212.245 q^{69} -136.492 q^{70} +263.695 q^{71} +211.570 q^{72} +117.550 q^{73} +992.617 q^{74} +75.0000 q^{75} +46.5234 q^{76} -216.408 q^{78} +707.408 q^{79} -289.477 q^{80} +81.0000 q^{81} +27.5813 q^{82} +535.758 q^{83} -21.2672 q^{84} -180.969 q^{85} +215.813 q^{86} -123.267 q^{87} -823.287 q^{89} -121.570 q^{90} +269.811 q^{91} -49.6344 q^{92} -72.8016 q^{93} +268.531 q^{94} -331.570 q^{95} -94.9641 q^{96} +128.481 q^{97} +650.794 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 6q^{3} + 5q^{4} + 10q^{5} + 3q^{6} + q^{7} + 15q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + q^{2} + 6q^{3} + 5q^{4} + 10q^{5} + 3q^{6} + q^{7} + 15q^{8} + 18q^{9} + 5q^{10} + 15q^{12} + 47q^{13} - 61q^{14} + 30q^{15} - 135q^{16} - 98q^{17} + 9q^{18} - 75q^{19} + 25q^{20} + 3q^{21} - 57q^{23} + 45q^{24} + 50q^{25} + 3q^{26} + 54q^{27} - 59q^{28} - 127q^{29} + 15q^{30} - 183q^{31} - 249q^{32} - 131q^{34} + 5q^{35} + 45q^{36} - 229q^{37} + 147q^{38} + 141q^{39} + 75q^{40} + 18q^{41} - 183q^{42} + 186q^{43} + 90q^{45} - 664q^{46} - 615q^{47} - 405q^{48} - 501q^{49} + 25q^{50} - 294q^{51} + 97q^{52} - 927q^{53} + 27q^{54} + 315q^{56} - 225q^{57} - 207q^{58} - 380q^{59} + 75q^{60} + 509q^{61} - 522q^{62} + 9q^{63} + 361q^{64} + 235q^{65} - 1138q^{67} - 327q^{68} - 171q^{69} - 305q^{70} - 273q^{71} + 135q^{72} + 440q^{73} + 1505q^{74} + 150q^{75} - 3q^{76} + 9q^{78} + 973q^{79} - 675q^{80} + 162q^{81} + 132q^{82} + 15q^{83} - 177q^{84} - 490q^{85} + 1200q^{86} - 381q^{87} - 1288q^{89} + 45q^{90} + 85q^{91} - 778q^{92} - 549q^{93} - 1640q^{94} - 375q^{95} - 747q^{96} - 76q^{97} - 312q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70156 −0.955146 −0.477573 0.878592i \(-0.658483\pi\)
−0.477573 + 0.878592i \(0.658483\pi\)
\(3\) 3.00000 0.577350
\(4\) −0.701562 −0.0876953
\(5\) 5.00000 0.447214
\(6\) −8.10469 −0.551454
\(7\) 10.1047 0.545602 0.272801 0.962071i \(-0.412050\pi\)
0.272801 + 0.962071i \(0.412050\pi\)
\(8\) 23.5078 1.03891
\(9\) 9.00000 0.333333
\(10\) −13.5078 −0.427154
\(11\) 0 0
\(12\) −2.10469 −0.0506309
\(13\) 26.7016 0.569668 0.284834 0.958577i \(-0.408062\pi\)
0.284834 + 0.958577i \(0.408062\pi\)
\(14\) −27.2984 −0.521130
\(15\) 15.0000 0.258199
\(16\) −57.8953 −0.904614
\(17\) −36.1938 −0.516369 −0.258185 0.966096i \(-0.583124\pi\)
−0.258185 + 0.966096i \(0.583124\pi\)
\(18\) −24.3141 −0.318382
\(19\) −66.3141 −0.800710 −0.400355 0.916360i \(-0.631113\pi\)
−0.400355 + 0.916360i \(0.631113\pi\)
\(20\) −3.50781 −0.0392185
\(21\) 30.3141 0.315003
\(22\) 0 0
\(23\) 70.7484 0.641394 0.320697 0.947182i \(-0.396083\pi\)
0.320697 + 0.947182i \(0.396083\pi\)
\(24\) 70.5234 0.599814
\(25\) 25.0000 0.200000
\(26\) −72.1359 −0.544116
\(27\) 27.0000 0.192450
\(28\) −7.08907 −0.0478467
\(29\) −41.0891 −0.263105 −0.131553 0.991309i \(-0.541996\pi\)
−0.131553 + 0.991309i \(0.541996\pi\)
\(30\) −40.5234 −0.246618
\(31\) −24.2672 −0.140597 −0.0702987 0.997526i \(-0.522395\pi\)
−0.0702987 + 0.997526i \(0.522395\pi\)
\(32\) −31.6547 −0.174869
\(33\) 0 0
\(34\) 97.7797 0.493208
\(35\) 50.5234 0.244001
\(36\) −6.31406 −0.0292318
\(37\) −367.423 −1.63254 −0.816271 0.577669i \(-0.803962\pi\)
−0.816271 + 0.577669i \(0.803962\pi\)
\(38\) 179.152 0.764795
\(39\) 80.1047 0.328898
\(40\) 117.539 0.464614
\(41\) −10.2094 −0.0388887 −0.0194443 0.999811i \(-0.506190\pi\)
−0.0194443 + 0.999811i \(0.506190\pi\)
\(42\) −81.8953 −0.300874
\(43\) −79.8844 −0.283308 −0.141654 0.989916i \(-0.545242\pi\)
−0.141654 + 0.989916i \(0.545242\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) −191.131 −0.612625
\(47\) −99.3985 −0.308484 −0.154242 0.988033i \(-0.549294\pi\)
−0.154242 + 0.988033i \(0.549294\pi\)
\(48\) −173.686 −0.522279
\(49\) −240.895 −0.702319
\(50\) −67.5391 −0.191029
\(51\) −108.581 −0.298126
\(52\) −18.7328 −0.0499572
\(53\) −460.298 −1.19296 −0.596480 0.802628i \(-0.703434\pi\)
−0.596480 + 0.802628i \(0.703434\pi\)
\(54\) −72.9422 −0.183818
\(55\) 0 0
\(56\) 237.539 0.566830
\(57\) −198.942 −0.462290
\(58\) 111.005 0.251304
\(59\) −753.475 −1.66261 −0.831306 0.555815i \(-0.812406\pi\)
−0.831306 + 0.555815i \(0.812406\pi\)
\(60\) −10.5234 −0.0226428
\(61\) 424.183 0.890345 0.445172 0.895445i \(-0.353142\pi\)
0.445172 + 0.895445i \(0.353142\pi\)
\(62\) 65.5593 0.134291
\(63\) 90.9422 0.181867
\(64\) 548.680 1.07164
\(65\) 133.508 0.254763
\(66\) 0 0
\(67\) −1049.23 −1.91320 −0.956600 0.291405i \(-0.905877\pi\)
−0.956600 + 0.291405i \(0.905877\pi\)
\(68\) 25.3922 0.0452831
\(69\) 212.245 0.370309
\(70\) −136.492 −0.233056
\(71\) 263.695 0.440773 0.220386 0.975413i \(-0.429268\pi\)
0.220386 + 0.975413i \(0.429268\pi\)
\(72\) 211.570 0.346303
\(73\) 117.550 0.188468 0.0942342 0.995550i \(-0.469960\pi\)
0.0942342 + 0.995550i \(0.469960\pi\)
\(74\) 992.617 1.55932
\(75\) 75.0000 0.115470
\(76\) 46.5234 0.0702185
\(77\) 0 0
\(78\) −216.408 −0.314146
\(79\) 707.408 1.00746 0.503732 0.863860i \(-0.331960\pi\)
0.503732 + 0.863860i \(0.331960\pi\)
\(80\) −289.477 −0.404556
\(81\) 81.0000 0.111111
\(82\) 27.5813 0.0371444
\(83\) 535.758 0.708519 0.354259 0.935147i \(-0.384733\pi\)
0.354259 + 0.935147i \(0.384733\pi\)
\(84\) −21.2672 −0.0276243
\(85\) −180.969 −0.230927
\(86\) 215.813 0.270601
\(87\) −123.267 −0.151904
\(88\) 0 0
\(89\) −823.287 −0.980543 −0.490271 0.871570i \(-0.663102\pi\)
−0.490271 + 0.871570i \(0.663102\pi\)
\(90\) −121.570 −0.142385
\(91\) 269.811 0.310812
\(92\) −49.6344 −0.0562472
\(93\) −72.8016 −0.0811739
\(94\) 268.531 0.294648
\(95\) −331.570 −0.358088
\(96\) −94.9641 −0.100961
\(97\) 128.481 0.134488 0.0672438 0.997737i \(-0.478579\pi\)
0.0672438 + 0.997737i \(0.478579\pi\)
\(98\) 650.794 0.670817
\(99\) 0 0
\(100\) −17.5391 −0.0175391
\(101\) 225.748 0.222404 0.111202 0.993798i \(-0.464530\pi\)
0.111202 + 0.993798i \(0.464530\pi\)
\(102\) 293.339 0.284754
\(103\) −979.798 −0.937305 −0.468652 0.883383i \(-0.655260\pi\)
−0.468652 + 0.883383i \(0.655260\pi\)
\(104\) 627.695 0.591833
\(105\) 151.570 0.140874
\(106\) 1243.52 1.13945
\(107\) −185.961 −0.168014 −0.0840071 0.996465i \(-0.526772\pi\)
−0.0840071 + 0.996465i \(0.526772\pi\)
\(108\) −18.9422 −0.0168770
\(109\) 5.66566 0.00497864 0.00248932 0.999997i \(-0.499208\pi\)
0.00248932 + 0.999997i \(0.499208\pi\)
\(110\) 0 0
\(111\) −1102.27 −0.942548
\(112\) −585.014 −0.493559
\(113\) 415.311 0.345745 0.172872 0.984944i \(-0.444695\pi\)
0.172872 + 0.984944i \(0.444695\pi\)
\(114\) 537.455 0.441555
\(115\) 353.742 0.286840
\(116\) 28.8265 0.0230731
\(117\) 240.314 0.189889
\(118\) 2035.56 1.58804
\(119\) −365.727 −0.281732
\(120\) 352.617 0.268245
\(121\) 0 0
\(122\) −1145.96 −0.850410
\(123\) −30.6281 −0.0224524
\(124\) 17.0249 0.0123297
\(125\) 125.000 0.0894427
\(126\) −245.686 −0.173710
\(127\) 113.159 0.0790650 0.0395325 0.999218i \(-0.487413\pi\)
0.0395325 + 0.999218i \(0.487413\pi\)
\(128\) −1229.05 −0.848704
\(129\) −239.653 −0.163568
\(130\) −360.680 −0.243336
\(131\) 943.106 0.629004 0.314502 0.949257i \(-0.398162\pi\)
0.314502 + 0.949257i \(0.398162\pi\)
\(132\) 0 0
\(133\) −670.083 −0.436869
\(134\) 2834.57 1.82739
\(135\) 135.000 0.0860663
\(136\) −850.836 −0.536460
\(137\) 2767.42 1.72582 0.862909 0.505359i \(-0.168640\pi\)
0.862909 + 0.505359i \(0.168640\pi\)
\(138\) −573.394 −0.353699
\(139\) −363.369 −0.221731 −0.110865 0.993835i \(-0.535362\pi\)
−0.110865 + 0.993835i \(0.535362\pi\)
\(140\) −35.4453 −0.0213977
\(141\) −298.195 −0.178103
\(142\) −712.389 −0.421003
\(143\) 0 0
\(144\) −521.058 −0.301538
\(145\) −205.445 −0.117664
\(146\) −317.569 −0.180015
\(147\) −722.686 −0.405484
\(148\) 257.770 0.143166
\(149\) −161.100 −0.0885760 −0.0442880 0.999019i \(-0.514102\pi\)
−0.0442880 + 0.999019i \(0.514102\pi\)
\(150\) −202.617 −0.110291
\(151\) 176.217 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(152\) −1558.90 −0.831864
\(153\) −325.744 −0.172123
\(154\) 0 0
\(155\) −121.336 −0.0628770
\(156\) −56.1984 −0.0288428
\(157\) −137.895 −0.0700971 −0.0350485 0.999386i \(-0.511159\pi\)
−0.0350485 + 0.999386i \(0.511159\pi\)
\(158\) −1911.11 −0.962275
\(159\) −1380.90 −0.688755
\(160\) −158.273 −0.0782039
\(161\) 714.891 0.349946
\(162\) −218.827 −0.106127
\(163\) −952.656 −0.457778 −0.228889 0.973453i \(-0.573509\pi\)
−0.228889 + 0.973453i \(0.573509\pi\)
\(164\) 7.16251 0.00341035
\(165\) 0 0
\(166\) −1447.38 −0.676739
\(167\) −2100.45 −0.973282 −0.486641 0.873602i \(-0.661778\pi\)
−0.486641 + 0.873602i \(0.661778\pi\)
\(168\) 712.617 0.327260
\(169\) −1484.03 −0.675479
\(170\) 488.898 0.220569
\(171\) −596.827 −0.266903
\(172\) 56.0438 0.0248448
\(173\) −1040.22 −0.457147 −0.228573 0.973527i \(-0.573406\pi\)
−0.228573 + 0.973527i \(0.573406\pi\)
\(174\) 333.014 0.145090
\(175\) 252.617 0.109120
\(176\) 0 0
\(177\) −2260.42 −0.959909
\(178\) 2224.16 0.936562
\(179\) 1570.93 0.655959 0.327980 0.944685i \(-0.393632\pi\)
0.327980 + 0.944685i \(0.393632\pi\)
\(180\) −31.5703 −0.0130728
\(181\) 2883.25 1.18403 0.592017 0.805925i \(-0.298332\pi\)
0.592017 + 0.805925i \(0.298332\pi\)
\(182\) −728.911 −0.296871
\(183\) 1272.55 0.514041
\(184\) 1663.14 0.666350
\(185\) −1837.12 −0.730095
\(186\) 196.678 0.0775330
\(187\) 0 0
\(188\) 69.7342 0.0270526
\(189\) 272.827 0.105001
\(190\) 895.758 0.342027
\(191\) −4176.94 −1.58237 −0.791186 0.611575i \(-0.790536\pi\)
−0.791186 + 0.611575i \(0.790536\pi\)
\(192\) 1646.04 0.618712
\(193\) −1346.96 −0.502363 −0.251181 0.967940i \(-0.580819\pi\)
−0.251181 + 0.967940i \(0.580819\pi\)
\(194\) −347.100 −0.128455
\(195\) 400.523 0.147088
\(196\) 169.003 0.0615900
\(197\) −3488.00 −1.26147 −0.630736 0.775998i \(-0.717247\pi\)
−0.630736 + 0.775998i \(0.717247\pi\)
\(198\) 0 0
\(199\) −1969.15 −0.701454 −0.350727 0.936478i \(-0.614066\pi\)
−0.350727 + 0.936478i \(0.614066\pi\)
\(200\) 587.695 0.207782
\(201\) −3147.70 −1.10459
\(202\) −609.873 −0.212428
\(203\) −415.192 −0.143551
\(204\) 76.1765 0.0261442
\(205\) −51.0469 −0.0173915
\(206\) 2646.99 0.895263
\(207\) 636.736 0.213798
\(208\) −1545.90 −0.515330
\(209\) 0 0
\(210\) −409.477 −0.134555
\(211\) 3829.56 1.24947 0.624734 0.780837i \(-0.285207\pi\)
0.624734 + 0.780837i \(0.285207\pi\)
\(212\) 322.928 0.104617
\(213\) 791.086 0.254480
\(214\) 502.385 0.160478
\(215\) −399.422 −0.126699
\(216\) 634.711 0.199938
\(217\) −245.212 −0.0767101
\(218\) −15.3061 −0.00475533
\(219\) 352.650 0.108812
\(220\) 0 0
\(221\) −966.430 −0.294159
\(222\) 2977.85 0.900272
\(223\) 3783.58 1.13618 0.568088 0.822968i \(-0.307683\pi\)
0.568088 + 0.822968i \(0.307683\pi\)
\(224\) −319.861 −0.0954089
\(225\) 225.000 0.0666667
\(226\) −1121.99 −0.330237
\(227\) −3305.66 −0.966539 −0.483270 0.875472i \(-0.660551\pi\)
−0.483270 + 0.875472i \(0.660551\pi\)
\(228\) 139.570 0.0405407
\(229\) 1881.69 0.542994 0.271497 0.962439i \(-0.412481\pi\)
0.271497 + 0.962439i \(0.412481\pi\)
\(230\) −955.656 −0.273974
\(231\) 0 0
\(232\) −965.914 −0.273342
\(233\) 6167.54 1.73412 0.867058 0.498207i \(-0.166008\pi\)
0.867058 + 0.498207i \(0.166008\pi\)
\(234\) −649.223 −0.181372
\(235\) −496.992 −0.137958
\(236\) 528.609 0.145803
\(237\) 2122.22 0.581659
\(238\) 988.033 0.269095
\(239\) −2574.48 −0.696776 −0.348388 0.937350i \(-0.613271\pi\)
−0.348388 + 0.937350i \(0.613271\pi\)
\(240\) −868.430 −0.233570
\(241\) 1125.12 0.300728 0.150364 0.988631i \(-0.451955\pi\)
0.150364 + 0.988631i \(0.451955\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −297.591 −0.0780790
\(245\) −1204.48 −0.314086
\(246\) 82.7438 0.0214453
\(247\) −1770.69 −0.456139
\(248\) −570.469 −0.146068
\(249\) 1607.27 0.409063
\(250\) −337.695 −0.0854309
\(251\) 558.309 0.140399 0.0701995 0.997533i \(-0.477636\pi\)
0.0701995 + 0.997533i \(0.477636\pi\)
\(252\) −63.8016 −0.0159489
\(253\) 0 0
\(254\) −305.707 −0.0755187
\(255\) −542.906 −0.133326
\(256\) −1069.07 −0.261003
\(257\) −166.553 −0.0404252 −0.0202126 0.999796i \(-0.506434\pi\)
−0.0202126 + 0.999796i \(0.506434\pi\)
\(258\) 647.438 0.156231
\(259\) −3712.70 −0.890718
\(260\) −93.6640 −0.0223415
\(261\) −369.802 −0.0877017
\(262\) −2547.86 −0.600791
\(263\) −5678.97 −1.33148 −0.665742 0.746182i \(-0.731885\pi\)
−0.665742 + 0.746182i \(0.731885\pi\)
\(264\) 0 0
\(265\) −2301.49 −0.533508
\(266\) 1810.27 0.417274
\(267\) −2469.86 −0.566117
\(268\) 736.103 0.167779
\(269\) 2056.03 0.466015 0.233008 0.972475i \(-0.425143\pi\)
0.233008 + 0.972475i \(0.425143\pi\)
\(270\) −364.711 −0.0822059
\(271\) 6349.58 1.42328 0.711641 0.702544i \(-0.247952\pi\)
0.711641 + 0.702544i \(0.247952\pi\)
\(272\) 2095.45 0.467115
\(273\) 809.433 0.179447
\(274\) −7476.37 −1.64841
\(275\) 0 0
\(276\) −148.903 −0.0324744
\(277\) −5318.77 −1.15370 −0.576848 0.816851i \(-0.695718\pi\)
−0.576848 + 0.816851i \(0.695718\pi\)
\(278\) 981.664 0.211785
\(279\) −218.405 −0.0468658
\(280\) 1187.70 0.253494
\(281\) −8731.32 −1.85362 −0.926810 0.375532i \(-0.877460\pi\)
−0.926810 + 0.375532i \(0.877460\pi\)
\(282\) 805.593 0.170115
\(283\) 8899.43 1.86931 0.934657 0.355550i \(-0.115706\pi\)
0.934657 + 0.355550i \(0.115706\pi\)
\(284\) −184.999 −0.0386537
\(285\) −994.711 −0.206742
\(286\) 0 0
\(287\) −103.163 −0.0212177
\(288\) −284.892 −0.0582897
\(289\) −3603.01 −0.733363
\(290\) 555.023 0.112387
\(291\) 385.444 0.0776464
\(292\) −82.4686 −0.0165278
\(293\) −8071.93 −1.60944 −0.804722 0.593652i \(-0.797686\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(294\) 1952.38 0.387296
\(295\) −3767.37 −0.743542
\(296\) −8637.32 −1.69606
\(297\) 0 0
\(298\) 435.222 0.0846031
\(299\) 1889.09 0.365382
\(300\) −52.6172 −0.0101262
\(301\) −807.206 −0.154573
\(302\) −476.062 −0.0907096
\(303\) 677.245 0.128405
\(304\) 3839.27 0.724334
\(305\) 2120.91 0.398174
\(306\) 880.017 0.164403
\(307\) −2698.56 −0.501677 −0.250839 0.968029i \(-0.580706\pi\)
−0.250839 + 0.968029i \(0.580706\pi\)
\(308\) 0 0
\(309\) −2939.39 −0.541153
\(310\) 327.797 0.0600568
\(311\) −2955.76 −0.538926 −0.269463 0.963011i \(-0.586846\pi\)
−0.269463 + 0.963011i \(0.586846\pi\)
\(312\) 1883.09 0.341695
\(313\) −2124.40 −0.383636 −0.191818 0.981431i \(-0.561438\pi\)
−0.191818 + 0.981431i \(0.561438\pi\)
\(314\) 372.533 0.0669530
\(315\) 454.711 0.0813335
\(316\) −496.291 −0.0883498
\(317\) −2404.95 −0.426105 −0.213053 0.977041i \(-0.568341\pi\)
−0.213053 + 0.977041i \(0.568341\pi\)
\(318\) 3730.57 0.657862
\(319\) 0 0
\(320\) 2743.40 0.479252
\(321\) −557.882 −0.0970030
\(322\) −1931.32 −0.334250
\(323\) 2400.15 0.413462
\(324\) −56.8265 −0.00974392
\(325\) 667.539 0.113934
\(326\) 2573.66 0.437245
\(327\) 16.9970 0.00287442
\(328\) −240.000 −0.0404018
\(329\) −1004.39 −0.168310
\(330\) 0 0
\(331\) −6375.50 −1.05870 −0.529349 0.848404i \(-0.677564\pi\)
−0.529349 + 0.848404i \(0.677564\pi\)
\(332\) −375.867 −0.0621337
\(333\) −3306.81 −0.544181
\(334\) 5674.51 0.929627
\(335\) −5246.17 −0.855609
\(336\) −1755.04 −0.284957
\(337\) 8482.64 1.37115 0.685577 0.728000i \(-0.259550\pi\)
0.685577 + 0.728000i \(0.259550\pi\)
\(338\) 4009.19 0.645181
\(339\) 1245.93 0.199616
\(340\) 126.961 0.0202512
\(341\) 0 0
\(342\) 1612.36 0.254932
\(343\) −5900.08 −0.928788
\(344\) −1877.91 −0.294331
\(345\) 1061.23 0.165607
\(346\) 2810.22 0.436642
\(347\) −1062.72 −0.164409 −0.0822044 0.996615i \(-0.526196\pi\)
−0.0822044 + 0.996615i \(0.526196\pi\)
\(348\) 86.4796 0.0133212
\(349\) 5284.62 0.810542 0.405271 0.914197i \(-0.367177\pi\)
0.405271 + 0.914197i \(0.367177\pi\)
\(350\) −682.461 −0.104226
\(351\) 720.942 0.109633
\(352\) 0 0
\(353\) −12074.0 −1.82049 −0.910244 0.414072i \(-0.864106\pi\)
−0.910244 + 0.414072i \(0.864106\pi\)
\(354\) 6106.68 0.916854
\(355\) 1318.48 0.197120
\(356\) 577.587 0.0859889
\(357\) −1097.18 −0.162658
\(358\) −4243.96 −0.626537
\(359\) −6002.96 −0.882519 −0.441259 0.897380i \(-0.645468\pi\)
−0.441259 + 0.897380i \(0.645468\pi\)
\(360\) 1057.85 0.154871
\(361\) −2461.45 −0.358864
\(362\) −7789.28 −1.13093
\(363\) 0 0
\(364\) −189.289 −0.0272567
\(365\) 587.750 0.0842856
\(366\) −3437.87 −0.490984
\(367\) 2684.43 0.381815 0.190908 0.981608i \(-0.438857\pi\)
0.190908 + 0.981608i \(0.438857\pi\)
\(368\) −4096.00 −0.580214
\(369\) −91.8844 −0.0129629
\(370\) 4963.09 0.697347
\(371\) −4651.17 −0.650881
\(372\) 51.0748 0.00711857
\(373\) 9694.41 1.34573 0.672866 0.739765i \(-0.265063\pi\)
0.672866 + 0.739765i \(0.265063\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) −2336.64 −0.320487
\(377\) −1097.14 −0.149882
\(378\) −737.058 −0.100291
\(379\) 8409.86 1.13980 0.569902 0.821713i \(-0.306981\pi\)
0.569902 + 0.821713i \(0.306981\pi\)
\(380\) 232.617 0.0314027
\(381\) 339.478 0.0456482
\(382\) 11284.3 1.51140
\(383\) −13673.0 −1.82417 −0.912085 0.410001i \(-0.865528\pi\)
−0.912085 + 0.410001i \(0.865528\pi\)
\(384\) −3687.16 −0.489999
\(385\) 0 0
\(386\) 3638.89 0.479830
\(387\) −718.959 −0.0944360
\(388\) −90.1376 −0.0117939
\(389\) −6628.44 −0.863946 −0.431973 0.901886i \(-0.642182\pi\)
−0.431973 + 0.901886i \(0.642182\pi\)
\(390\) −1082.04 −0.140490
\(391\) −2560.65 −0.331196
\(392\) −5662.92 −0.729645
\(393\) 2829.32 0.363156
\(394\) 9423.06 1.20489
\(395\) 3537.04 0.450551
\(396\) 0 0
\(397\) 1787.98 0.226036 0.113018 0.993593i \(-0.463948\pi\)
0.113018 + 0.993593i \(0.463948\pi\)
\(398\) 5319.78 0.669992
\(399\) −2010.25 −0.252226
\(400\) −1447.38 −0.180923
\(401\) −5846.25 −0.728049 −0.364025 0.931389i \(-0.618598\pi\)
−0.364025 + 0.931389i \(0.618598\pi\)
\(402\) 8503.72 1.05504
\(403\) −647.972 −0.0800938
\(404\) −158.377 −0.0195038
\(405\) 405.000 0.0496904
\(406\) 1121.67 0.137112
\(407\) 0 0
\(408\) −2552.51 −0.309725
\(409\) −12683.5 −1.53339 −0.766695 0.642011i \(-0.778100\pi\)
−0.766695 + 0.642011i \(0.778100\pi\)
\(410\) 137.906 0.0166115
\(411\) 8302.27 0.996401
\(412\) 687.389 0.0821972
\(413\) −7613.63 −0.907124
\(414\) −1720.18 −0.204208
\(415\) 2678.79 0.316859
\(416\) −845.230 −0.0996173
\(417\) −1090.11 −0.128016
\(418\) 0 0
\(419\) 6841.28 0.797657 0.398829 0.917025i \(-0.369417\pi\)
0.398829 + 0.917025i \(0.369417\pi\)
\(420\) −106.336 −0.0123540
\(421\) −1807.76 −0.209275 −0.104637 0.994510i \(-0.533368\pi\)
−0.104637 + 0.994510i \(0.533368\pi\)
\(422\) −10345.8 −1.19343
\(423\) −894.586 −0.102828
\(424\) −10820.6 −1.23938
\(425\) −904.844 −0.103274
\(426\) −2137.17 −0.243066
\(427\) 4286.23 0.485774
\(428\) 130.463 0.0147340
\(429\) 0 0
\(430\) 1079.06 0.121016
\(431\) −1777.64 −0.198667 −0.0993337 0.995054i \(-0.531671\pi\)
−0.0993337 + 0.995054i \(0.531671\pi\)
\(432\) −1563.17 −0.174093
\(433\) 12979.4 1.44053 0.720264 0.693700i \(-0.244021\pi\)
0.720264 + 0.693700i \(0.244021\pi\)
\(434\) 662.457 0.0732694
\(435\) −616.336 −0.0679334
\(436\) −3.97481 −0.000436603 0
\(437\) −4691.62 −0.513571
\(438\) −952.706 −0.103932
\(439\) 4471.38 0.486121 0.243061 0.970011i \(-0.421849\pi\)
0.243061 + 0.970011i \(0.421849\pi\)
\(440\) 0 0
\(441\) −2168.06 −0.234106
\(442\) 2610.87 0.280965
\(443\) 5115.99 0.548686 0.274343 0.961632i \(-0.411540\pi\)
0.274343 + 0.961632i \(0.411540\pi\)
\(444\) 773.311 0.0826570
\(445\) −4116.44 −0.438512
\(446\) −10221.6 −1.08522
\(447\) −483.300 −0.0511394
\(448\) 5544.24 0.584689
\(449\) −5834.81 −0.613278 −0.306639 0.951826i \(-0.599204\pi\)
−0.306639 + 0.951826i \(0.599204\pi\)
\(450\) −607.851 −0.0636764
\(451\) 0 0
\(452\) −291.366 −0.0303202
\(453\) 528.652 0.0548305
\(454\) 8930.45 0.923186
\(455\) 1349.05 0.138999
\(456\) −4676.70 −0.480277
\(457\) −15429.8 −1.57938 −0.789691 0.613504i \(-0.789759\pi\)
−0.789691 + 0.613504i \(0.789759\pi\)
\(458\) −5083.51 −0.518639
\(459\) −977.231 −0.0993753
\(460\) −248.172 −0.0251545
\(461\) 1187.92 0.120015 0.0600074 0.998198i \(-0.480888\pi\)
0.0600074 + 0.998198i \(0.480888\pi\)
\(462\) 0 0
\(463\) 3781.30 0.379550 0.189775 0.981828i \(-0.439224\pi\)
0.189775 + 0.981828i \(0.439224\pi\)
\(464\) 2378.86 0.238009
\(465\) −364.008 −0.0363021
\(466\) −16662.0 −1.65634
\(467\) 590.287 0.0584908 0.0292454 0.999572i \(-0.490690\pi\)
0.0292454 + 0.999572i \(0.490690\pi\)
\(468\) −168.595 −0.0166524
\(469\) −10602.2 −1.04385
\(470\) 1342.66 0.131770
\(471\) −413.686 −0.0404706
\(472\) −17712.5 −1.72730
\(473\) 0 0
\(474\) −5733.32 −0.555570
\(475\) −1657.85 −0.160142
\(476\) 256.580 0.0247066
\(477\) −4142.69 −0.397653
\(478\) 6955.12 0.665523
\(479\) −17061.5 −1.62747 −0.813735 0.581236i \(-0.802569\pi\)
−0.813735 + 0.581236i \(0.802569\pi\)
\(480\) −474.820 −0.0451510
\(481\) −9810.78 −0.930006
\(482\) −3039.59 −0.287239
\(483\) 2144.67 0.202041
\(484\) 0 0
\(485\) 642.406 0.0601447
\(486\) −656.480 −0.0612727
\(487\) −12480.6 −1.16129 −0.580646 0.814156i \(-0.697200\pi\)
−0.580646 + 0.814156i \(0.697200\pi\)
\(488\) 9971.61 0.924987
\(489\) −2857.97 −0.264298
\(490\) 3253.97 0.299999
\(491\) −12257.3 −1.12661 −0.563303 0.826250i \(-0.690470\pi\)
−0.563303 + 0.826250i \(0.690470\pi\)
\(492\) 21.4875 0.00196897
\(493\) 1487.17 0.135859
\(494\) 4783.63 0.435679
\(495\) 0 0
\(496\) 1404.96 0.127186
\(497\) 2664.56 0.240486
\(498\) −4342.15 −0.390716
\(499\) 2873.79 0.257813 0.128907 0.991657i \(-0.458853\pi\)
0.128907 + 0.991657i \(0.458853\pi\)
\(500\) −87.6953 −0.00784370
\(501\) −6301.36 −0.561925
\(502\) −1508.31 −0.134102
\(503\) 1150.72 0.102004 0.0510020 0.998699i \(-0.483759\pi\)
0.0510020 + 0.998699i \(0.483759\pi\)
\(504\) 2137.85 0.188943
\(505\) 1128.74 0.0994621
\(506\) 0 0
\(507\) −4452.08 −0.389988
\(508\) −79.3882 −0.00693363
\(509\) 1002.22 0.0872745 0.0436372 0.999047i \(-0.486105\pi\)
0.0436372 + 0.999047i \(0.486105\pi\)
\(510\) 1466.70 0.127346
\(511\) 1187.81 0.102829
\(512\) 12720.6 1.09800
\(513\) −1790.48 −0.154097
\(514\) 449.953 0.0386120
\(515\) −4898.99 −0.419175
\(516\) 168.132 0.0143441
\(517\) 0 0
\(518\) 10030.1 0.850766
\(519\) −3120.66 −0.263934
\(520\) 3138.48 0.264676
\(521\) −14545.1 −1.22310 −0.611549 0.791207i \(-0.709453\pi\)
−0.611549 + 0.791207i \(0.709453\pi\)
\(522\) 999.042 0.0837680
\(523\) 8685.21 0.726152 0.363076 0.931759i \(-0.381726\pi\)
0.363076 + 0.931759i \(0.381726\pi\)
\(524\) −661.647 −0.0551607
\(525\) 757.851 0.0630007
\(526\) 15342.1 1.27176
\(527\) 878.321 0.0726001
\(528\) 0 0
\(529\) −7161.66 −0.588613
\(530\) 6217.62 0.509578
\(531\) −6781.27 −0.554204
\(532\) 470.105 0.0383113
\(533\) −272.606 −0.0221536
\(534\) 6672.49 0.540724
\(535\) −929.804 −0.0751382
\(536\) −24665.2 −1.98764
\(537\) 4712.78 0.378718
\(538\) −5554.48 −0.445113
\(539\) 0 0
\(540\) −94.7109 −0.00754761
\(541\) −21517.2 −1.70998 −0.854988 0.518648i \(-0.826436\pi\)
−0.854988 + 0.518648i \(0.826436\pi\)
\(542\) −17153.8 −1.35944
\(543\) 8649.75 0.683603
\(544\) 1145.70 0.0902970
\(545\) 28.3283 0.00222651
\(546\) −2186.73 −0.171398
\(547\) 12053.4 0.942170 0.471085 0.882088i \(-0.343862\pi\)
0.471085 + 0.882088i \(0.343862\pi\)
\(548\) −1941.52 −0.151346
\(549\) 3817.65 0.296782
\(550\) 0 0
\(551\) 2724.78 0.210671
\(552\) 4989.42 0.384717
\(553\) 7148.13 0.549674
\(554\) 14369.0 1.10195
\(555\) −5511.35 −0.421520
\(556\) 254.926 0.0194447
\(557\) −4454.51 −0.338857 −0.169429 0.985542i \(-0.554192\pi\)
−0.169429 + 0.985542i \(0.554192\pi\)
\(558\) 590.034 0.0447637
\(559\) −2133.04 −0.161392
\(560\) −2925.07 −0.220726
\(561\) 0 0
\(562\) 23588.2 1.77048
\(563\) −13628.5 −1.02020 −0.510102 0.860114i \(-0.670392\pi\)
−0.510102 + 0.860114i \(0.670392\pi\)
\(564\) 209.203 0.0156188
\(565\) 2076.55 0.154622
\(566\) −24042.4 −1.78547
\(567\) 818.480 0.0606224
\(568\) 6198.90 0.457923
\(569\) 2064.07 0.152075 0.0760373 0.997105i \(-0.475773\pi\)
0.0760373 + 0.997105i \(0.475773\pi\)
\(570\) 2687.27 0.197469
\(571\) −666.014 −0.0488123 −0.0244061 0.999702i \(-0.507769\pi\)
−0.0244061 + 0.999702i \(0.507769\pi\)
\(572\) 0 0
\(573\) −12530.8 −0.913583
\(574\) 278.700 0.0202660
\(575\) 1768.71 0.128279
\(576\) 4938.12 0.357213
\(577\) 2115.23 0.152614 0.0763068 0.997084i \(-0.475687\pi\)
0.0763068 + 0.997084i \(0.475687\pi\)
\(578\) 9733.76 0.700469
\(579\) −4040.87 −0.290039
\(580\) 144.133 0.0103186
\(581\) 5413.66 0.386569
\(582\) −1041.30 −0.0741637
\(583\) 0 0
\(584\) 2763.34 0.195801
\(585\) 1201.57 0.0849211
\(586\) 21806.8 1.53725
\(587\) −16525.0 −1.16194 −0.580969 0.813926i \(-0.697326\pi\)
−0.580969 + 0.813926i \(0.697326\pi\)
\(588\) 507.009 0.0355590
\(589\) 1609.26 0.112578
\(590\) 10177.8 0.710192
\(591\) −10464.0 −0.728311
\(592\) 21272.1 1.47682
\(593\) 26736.2 1.85147 0.925736 0.378171i \(-0.123447\pi\)
0.925736 + 0.378171i \(0.123447\pi\)
\(594\) 0 0
\(595\) −1828.63 −0.125994
\(596\) 113.022 0.00776770
\(597\) −5907.45 −0.404985
\(598\) −5103.50 −0.348993
\(599\) 15220.7 1.03823 0.519117 0.854703i \(-0.326261\pi\)
0.519117 + 0.854703i \(0.326261\pi\)
\(600\) 1763.09 0.119963
\(601\) 24398.7 1.65598 0.827989 0.560744i \(-0.189485\pi\)
0.827989 + 0.560744i \(0.189485\pi\)
\(602\) 2180.72 0.147640
\(603\) −9443.11 −0.637733
\(604\) −123.627 −0.00832836
\(605\) 0 0
\(606\) −1829.62 −0.122646
\(607\) −23619.4 −1.57938 −0.789688 0.613509i \(-0.789758\pi\)
−0.789688 + 0.613509i \(0.789758\pi\)
\(608\) 2099.15 0.140019
\(609\) −1245.58 −0.0828790
\(610\) −5729.78 −0.380315
\(611\) −2654.09 −0.175733
\(612\) 228.529 0.0150944
\(613\) 23250.9 1.53197 0.765984 0.642859i \(-0.222252\pi\)
0.765984 + 0.642859i \(0.222252\pi\)
\(614\) 7290.33 0.479175
\(615\) −153.141 −0.0100410
\(616\) 0 0
\(617\) 15687.9 1.02362 0.511809 0.859099i \(-0.328976\pi\)
0.511809 + 0.859099i \(0.328976\pi\)
\(618\) 7940.96 0.516881
\(619\) 15447.8 1.00307 0.501534 0.865138i \(-0.332769\pi\)
0.501534 + 0.865138i \(0.332769\pi\)
\(620\) 85.1247 0.00551402
\(621\) 1910.21 0.123436
\(622\) 7985.18 0.514753
\(623\) −8319.06 −0.534986
\(624\) −4637.69 −0.297526
\(625\) 625.000 0.0400000
\(626\) 5739.19 0.366428
\(627\) 0 0
\(628\) 96.7421 0.00614718
\(629\) 13298.4 0.842994
\(630\) −1228.43 −0.0776854
\(631\) −26681.2 −1.68330 −0.841648 0.540026i \(-0.818414\pi\)
−0.841648 + 0.540026i \(0.818414\pi\)
\(632\) 16629.6 1.04666
\(633\) 11488.7 0.721381
\(634\) 6497.12 0.406993
\(635\) 565.796 0.0353590
\(636\) 968.784 0.0604006
\(637\) −6432.28 −0.400088
\(638\) 0 0
\(639\) 2373.26 0.146924
\(640\) −6145.27 −0.379552
\(641\) −15244.1 −0.939321 −0.469661 0.882847i \(-0.655624\pi\)
−0.469661 + 0.882847i \(0.655624\pi\)
\(642\) 1507.15 0.0926521
\(643\) −24128.9 −1.47986 −0.739929 0.672684i \(-0.765141\pi\)
−0.739929 + 0.672684i \(0.765141\pi\)
\(644\) −501.540 −0.0306886
\(645\) −1198.27 −0.0731498
\(646\) −6484.17 −0.394917
\(647\) 5415.41 0.329060 0.164530 0.986372i \(-0.447389\pi\)
0.164530 + 0.986372i \(0.447389\pi\)
\(648\) 1904.13 0.115434
\(649\) 0 0
\(650\) −1803.40 −0.108823
\(651\) −735.637 −0.0442886
\(652\) 668.347 0.0401449
\(653\) 11335.3 0.679303 0.339651 0.940551i \(-0.389691\pi\)
0.339651 + 0.940551i \(0.389691\pi\)
\(654\) −45.9184 −0.00274549
\(655\) 4715.53 0.281299
\(656\) 591.075 0.0351793
\(657\) 1057.95 0.0628228
\(658\) 2713.42 0.160760
\(659\) 25441.6 1.50389 0.751945 0.659226i \(-0.229116\pi\)
0.751945 + 0.659226i \(0.229116\pi\)
\(660\) 0 0
\(661\) 312.429 0.0183844 0.00919219 0.999958i \(-0.497074\pi\)
0.00919219 + 0.999958i \(0.497074\pi\)
\(662\) 17223.8 1.01121
\(663\) −2899.29 −0.169833
\(664\) 12594.5 0.736086
\(665\) −3350.41 −0.195374
\(666\) 8933.55 0.519772
\(667\) −2906.99 −0.168754
\(668\) 1473.60 0.0853522
\(669\) 11350.7 0.655972
\(670\) 14172.9 0.817232
\(671\) 0 0
\(672\) −959.582 −0.0550844
\(673\) 13815.1 0.791280 0.395640 0.918406i \(-0.370523\pi\)
0.395640 + 0.918406i \(0.370523\pi\)
\(674\) −22916.4 −1.30965
\(675\) 675.000 0.0384900
\(676\) 1041.14 0.0592363
\(677\) 23931.4 1.35858 0.679290 0.733870i \(-0.262288\pi\)
0.679290 + 0.733870i \(0.262288\pi\)
\(678\) −3365.96 −0.190662
\(679\) 1298.26 0.0733766
\(680\) −4254.18 −0.239912
\(681\) −9916.98 −0.558032
\(682\) 0 0
\(683\) −6301.98 −0.353058 −0.176529 0.984295i \(-0.556487\pi\)
−0.176529 + 0.984295i \(0.556487\pi\)
\(684\) 418.711 0.0234062
\(685\) 13837.1 0.771809
\(686\) 15939.4 0.887129
\(687\) 5645.08 0.313498
\(688\) 4624.93 0.256285
\(689\) −12290.7 −0.679591
\(690\) −2866.97 −0.158179
\(691\) 8554.84 0.470972 0.235486 0.971878i \(-0.424332\pi\)
0.235486 + 0.971878i \(0.424332\pi\)
\(692\) 729.778 0.0400896
\(693\) 0 0
\(694\) 2871.01 0.157034
\(695\) −1816.84 −0.0991609
\(696\) −2897.74 −0.157814
\(697\) 369.515 0.0200809
\(698\) −14276.7 −0.774187
\(699\) 18502.6 1.00119
\(700\) −177.227 −0.00956934
\(701\) −29410.3 −1.58461 −0.792306 0.610124i \(-0.791120\pi\)
−0.792306 + 0.610124i \(0.791120\pi\)
\(702\) −1947.67 −0.104715
\(703\) 24365.3 1.30719
\(704\) 0 0
\(705\) −1490.98 −0.0796503
\(706\) 32618.6 1.73883
\(707\) 2281.12 0.121344
\(708\) 1585.83 0.0841795
\(709\) 15207.4 0.805537 0.402768 0.915302i \(-0.368048\pi\)
0.402768 + 0.915302i \(0.368048\pi\)
\(710\) −3561.95 −0.188278
\(711\) 6366.67 0.335821
\(712\) −19353.7 −1.01869
\(713\) −1716.87 −0.0901783
\(714\) 2964.10 0.155362
\(715\) 0 0
\(716\) −1102.10 −0.0575245
\(717\) −7723.45 −0.402284
\(718\) 16217.4 0.842935
\(719\) −13773.8 −0.714431 −0.357215 0.934022i \(-0.616274\pi\)
−0.357215 + 0.934022i \(0.616274\pi\)
\(720\) −2605.29 −0.134852
\(721\) −9900.55 −0.511395
\(722\) 6649.75 0.342767
\(723\) 3375.37 0.173625
\(724\) −2022.78 −0.103834
\(725\) −1027.23 −0.0526210
\(726\) 0 0
\(727\) 33429.1 1.70539 0.852694 0.522410i \(-0.174967\pi\)
0.852694 + 0.522410i \(0.174967\pi\)
\(728\) 6342.66 0.322905
\(729\) 729.000 0.0370370
\(730\) −1587.84 −0.0805051
\(731\) 2891.31 0.146292
\(732\) −892.772 −0.0450790
\(733\) 7774.73 0.391768 0.195884 0.980627i \(-0.437242\pi\)
0.195884 + 0.980627i \(0.437242\pi\)
\(734\) −7252.16 −0.364690
\(735\) −3613.43 −0.181338
\(736\) −2239.52 −0.112160
\(737\) 0 0
\(738\) 248.231 0.0123815
\(739\) −37474.1 −1.86537 −0.932684 0.360693i \(-0.882540\pi\)
−0.932684 + 0.360693i \(0.882540\pi\)
\(740\) 1288.85 0.0640259
\(741\) −5312.07 −0.263352
\(742\) 12565.4 0.621687
\(743\) 30510.7 1.50650 0.753250 0.657735i \(-0.228485\pi\)
0.753250 + 0.657735i \(0.228485\pi\)
\(744\) −1711.41 −0.0843322
\(745\) −805.500 −0.0396124
\(746\) −26190.1 −1.28537
\(747\) 4821.82 0.236173
\(748\) 0 0
\(749\) −1879.08 −0.0916688
\(750\) −1013.09 −0.0493236
\(751\) 34091.1 1.65646 0.828229 0.560389i \(-0.189348\pi\)
0.828229 + 0.560389i \(0.189348\pi\)
\(752\) 5754.71 0.279059
\(753\) 1674.93 0.0810594
\(754\) 2964.00 0.143160
\(755\) 881.087 0.0424716
\(756\) −191.405 −0.00920810
\(757\) 901.287 0.0432733 0.0216366 0.999766i \(-0.493112\pi\)
0.0216366 + 0.999766i \(0.493112\pi\)
\(758\) −22719.8 −1.08868
\(759\) 0 0
\(760\) −7794.49 −0.372021
\(761\) −5030.20 −0.239612 −0.119806 0.992797i \(-0.538227\pi\)
−0.119806 + 0.992797i \(0.538227\pi\)
\(762\) −917.120 −0.0436007
\(763\) 57.2497 0.00271635
\(764\) 2930.39 0.138767
\(765\) −1628.72 −0.0769758
\(766\) 36938.4 1.74235
\(767\) −20119.0 −0.947136
\(768\) −3207.21 −0.150690
\(769\) 13447.0 0.630574 0.315287 0.948996i \(-0.397899\pi\)
0.315287 + 0.948996i \(0.397899\pi\)
\(770\) 0 0
\(771\) −499.658 −0.0233395
\(772\) 944.974 0.0440548
\(773\) 35885.7 1.66975 0.834876 0.550438i \(-0.185539\pi\)
0.834876 + 0.550438i \(0.185539\pi\)
\(774\) 1942.31 0.0902003
\(775\) −606.680 −0.0281195
\(776\) 3020.31 0.139720
\(777\) −11138.1 −0.514256
\(778\) 17907.1 0.825195
\(779\) 677.025 0.0311386
\(780\) −280.992 −0.0128989
\(781\) 0 0
\(782\) 6917.76 0.316341
\(783\) −1109.40 −0.0506346
\(784\) 13946.7 0.635328
\(785\) −689.477 −0.0313484
\(786\) −7643.58 −0.346867
\(787\) 29894.2 1.35402 0.677010 0.735974i \(-0.263275\pi\)
0.677010 + 0.735974i \(0.263275\pi\)
\(788\) 2447.05 0.110625
\(789\) −17036.9 −0.768732
\(790\) −9555.53 −0.430342
\(791\) 4196.59 0.188639
\(792\) 0 0
\(793\) 11326.3 0.507201
\(794\) −4830.34 −0.215897
\(795\) −6904.48 −0.308021
\(796\) 1381.48 0.0615142
\(797\) 14033.8 0.623716 0.311858 0.950129i \(-0.399049\pi\)
0.311858 + 0.950129i \(0.399049\pi\)
\(798\) 5430.81 0.240913
\(799\) 3597.60 0.159292
\(800\) −791.367 −0.0349738
\(801\) −7409.59 −0.326848
\(802\) 15794.0 0.695394
\(803\) 0 0
\(804\) 2208.31 0.0968670
\(805\) 3574.45 0.156501
\(806\) 1750.54 0.0765013
\(807\) 6168.08 0.269054
\(808\) 5306.85 0.231057
\(809\) 45008.7 1.95602 0.978011 0.208552i \(-0.0668751\pi\)
0.978011 + 0.208552i \(0.0668751\pi\)
\(810\) −1094.13 −0.0474616
\(811\) −26921.1 −1.16563 −0.582816 0.812604i \(-0.698049\pi\)
−0.582816 + 0.812604i \(0.698049\pi\)
\(812\) 291.283 0.0125887
\(813\) 19048.7 0.821732
\(814\) 0 0
\(815\) −4763.28 −0.204724
\(816\) 6286.35 0.269689
\(817\) 5297.46 0.226848
\(818\) 34265.2 1.46461
\(819\) 2428.30 0.103604
\(820\) 35.8125 0.00152516
\(821\) 11197.0 0.475976 0.237988 0.971268i \(-0.423512\pi\)
0.237988 + 0.971268i \(0.423512\pi\)
\(822\) −22429.1 −0.951709
\(823\) 1247.54 0.0528392 0.0264196 0.999651i \(-0.491589\pi\)
0.0264196 + 0.999651i \(0.491589\pi\)
\(824\) −23032.9 −0.973774
\(825\) 0 0
\(826\) 20568.7 0.866436
\(827\) −11594.1 −0.487504 −0.243752 0.969838i \(-0.578378\pi\)
−0.243752 + 0.969838i \(0.578378\pi\)
\(828\) −446.710 −0.0187491
\(829\) 1658.14 0.0694689 0.0347344 0.999397i \(-0.488941\pi\)
0.0347344 + 0.999397i \(0.488941\pi\)
\(830\) −7236.91 −0.302647
\(831\) −15956.3 −0.666087
\(832\) 14650.6 0.610479
\(833\) 8718.91 0.362656
\(834\) 2944.99 0.122274
\(835\) −10502.3 −0.435265
\(836\) 0 0
\(837\) −655.214 −0.0270580
\(838\) −18482.1 −0.761880
\(839\) −15906.9 −0.654549 −0.327275 0.944929i \(-0.606130\pi\)
−0.327275 + 0.944929i \(0.606130\pi\)
\(840\) 3563.09 0.146355
\(841\) −22700.7 −0.930776
\(842\) 4883.77 0.199888
\(843\) −26194.0 −1.07019
\(844\) −2686.68 −0.109573
\(845\) −7420.13 −0.302083
\(846\) 2416.78 0.0982159
\(847\) 0 0
\(848\) 26649.1 1.07917
\(849\) 26698.3 1.07925
\(850\) 2444.49 0.0986416
\(851\) −25994.6 −1.04710
\(852\) −554.996 −0.0223167
\(853\) 9055.06 0.363469 0.181735 0.983348i \(-0.441829\pi\)
0.181735 + 0.983348i \(0.441829\pi\)
\(854\) −11579.5 −0.463985
\(855\) −2984.13 −0.119363
\(856\) −4371.53 −0.174551
\(857\) 8173.52 0.325790 0.162895 0.986643i \(-0.447917\pi\)
0.162895 + 0.986643i \(0.447917\pi\)
\(858\) 0 0
\(859\) 41286.0 1.63988 0.819941 0.572447i \(-0.194006\pi\)
0.819941 + 0.572447i \(0.194006\pi\)
\(860\) 280.219 0.0111109
\(861\) −309.488 −0.0122501
\(862\) 4802.39 0.189757
\(863\) 18537.1 0.731182 0.365591 0.930776i \(-0.380867\pi\)
0.365591 + 0.930776i \(0.380867\pi\)
\(864\) −854.677 −0.0336536
\(865\) −5201.09 −0.204442
\(866\) −35064.6 −1.37591
\(867\) −10809.0 −0.423407
\(868\) 172.032 0.00672712
\(869\) 0 0
\(870\) 1665.07 0.0648864
\(871\) −28016.2 −1.08989
\(872\) 133.187 0.00517235
\(873\) 1156.33 0.0448292
\(874\) 12674.7 0.490535
\(875\) 1263.09 0.0488001
\(876\) −247.406 −0.00954232
\(877\) 22220.2 0.855558 0.427779 0.903883i \(-0.359296\pi\)
0.427779 + 0.903883i \(0.359296\pi\)
\(878\) −12079.7 −0.464317
\(879\) −24215.8 −0.929213
\(880\) 0 0
\(881\) −50496.5 −1.93107 −0.965534 0.260275i \(-0.916187\pi\)
−0.965534 + 0.260275i \(0.916187\pi\)
\(882\) 5857.14 0.223606
\(883\) 22313.1 0.850391 0.425196 0.905102i \(-0.360205\pi\)
0.425196 + 0.905102i \(0.360205\pi\)
\(884\) 678.010 0.0257963
\(885\) −11302.1 −0.429284
\(886\) −13821.2 −0.524076
\(887\) −41393.3 −1.56691 −0.783456 0.621447i \(-0.786545\pi\)
−0.783456 + 0.621447i \(0.786545\pi\)
\(888\) −25912.0 −0.979221
\(889\) 1143.44 0.0431380
\(890\) 11120.8 0.418843
\(891\) 0 0
\(892\) −2654.42 −0.0996373
\(893\) 6591.52 0.247006
\(894\) 1305.67 0.0488456
\(895\) 7854.64 0.293354
\(896\) −12419.2 −0.463054
\(897\) 5667.28 0.210953
\(898\) 15763.1 0.585770
\(899\) 997.116 0.0369919
\(900\) −157.851 −0.00584635
\(901\) 16659.9 0.616007
\(902\) 0 0
\(903\) −2421.62 −0.0892430
\(904\) 9763.05 0.359197
\(905\) 14416.2 0.529516
\(906\) −1428.19 −0.0523712
\(907\) 47821.8 1.75071 0.875356 0.483479i \(-0.160627\pi\)
0.875356 + 0.483479i \(0.160627\pi\)
\(908\) 2319.13 0.0847609
\(909\) 2031.74 0.0741347
\(910\) −3644.55 −0.132765
\(911\) 32560.8 1.18418 0.592091 0.805871i \(-0.298303\pi\)
0.592091 + 0.805871i \(0.298303\pi\)
\(912\) 11517.8 0.418194
\(913\) 0 0
\(914\) 41684.7 1.50854
\(915\) 6362.74 0.229886
\(916\) −1320.12 −0.0476180
\(917\) 9529.79 0.343186
\(918\) 2640.05 0.0949179
\(919\) 13638.7 0.489554 0.244777 0.969579i \(-0.421285\pi\)
0.244777 + 0.969579i \(0.421285\pi\)
\(920\) 8315.70 0.298001
\(921\) −8095.68 −0.289643
\(922\) −3209.23 −0.114632
\(923\) 7041.08 0.251094
\(924\) 0 0
\(925\) −9185.59 −0.326508
\(926\) −10215.4 −0.362526
\(927\) −8818.18 −0.312435
\(928\) 1300.66 0.0460090
\(929\) −17716.2 −0.625672 −0.312836 0.949807i \(-0.601279\pi\)
−0.312836 + 0.949807i \(0.601279\pi\)
\(930\) 983.390 0.0346738
\(931\) 15974.7 0.562354
\(932\) −4326.91 −0.152074
\(933\) −8867.29 −0.311149
\(934\) −1594.70 −0.0558673
\(935\) 0 0
\(936\) 5649.26 0.197278
\(937\) −46320.5 −1.61497 −0.807483 0.589890i \(-0.799171\pi\)
−0.807483 + 0.589890i \(0.799171\pi\)
\(938\) 28642.5 0.997025
\(939\) −6373.19 −0.221492
\(940\) 348.671 0.0120983
\(941\) −7828.97 −0.271219 −0.135609 0.990762i \(-0.543299\pi\)
−0.135609 + 0.990762i \(0.543299\pi\)
\(942\) 1117.60 0.0386553
\(943\) −722.297 −0.0249430
\(944\) 43622.7 1.50402
\(945\) 1364.13 0.0469579
\(946\) 0 0
\(947\) −23815.7 −0.817219 −0.408610 0.912709i \(-0.633986\pi\)
−0.408610 + 0.912709i \(0.633986\pi\)
\(948\) −1488.87 −0.0510088
\(949\) 3138.77 0.107364
\(950\) 4478.79 0.152959
\(951\) −7214.85 −0.246012
\(952\) −8597.43 −0.292694
\(953\) 2187.90 0.0743685 0.0371842 0.999308i \(-0.488161\pi\)
0.0371842 + 0.999308i \(0.488161\pi\)
\(954\) 11191.7 0.379817
\(955\) −20884.7 −0.707658
\(956\) 1806.16 0.0611039
\(957\) 0 0
\(958\) 46092.6 1.55447
\(959\) 27964.0 0.941609
\(960\) 8230.19 0.276696
\(961\) −29202.1 −0.980232
\(962\) 26504.4 0.888292
\(963\) −1673.65 −0.0560047
\(964\) −789.343 −0.0263724
\(965\) −6734.78 −0.224664
\(966\) −5793.96 −0.192979
\(967\) −14139.2 −0.470201 −0.235101 0.971971i \(-0.575542\pi\)
−0.235101 + 0.971971i \(0.575542\pi\)
\(968\) 0 0
\(969\) 7200.46 0.238712
\(970\) −1735.50 −0.0574470
\(971\) −6358.97 −0.210164 −0.105082 0.994464i \(-0.533510\pi\)
−0.105082 + 0.994464i \(0.533510\pi\)
\(972\) −170.480 −0.00562565
\(973\) −3671.73 −0.120977
\(974\) 33717.1 1.10920
\(975\) 2002.62 0.0657796
\(976\) −24558.2 −0.805419
\(977\) 46153.9 1.51136 0.755678 0.654944i \(-0.227308\pi\)
0.755678 + 0.654944i \(0.227308\pi\)
\(978\) 7720.98 0.252443
\(979\) 0 0
\(980\) 845.015 0.0275439
\(981\) 50.9909 0.00165955
\(982\) 33113.8 1.07607
\(983\) 20863.0 0.676935 0.338467 0.940978i \(-0.390091\pi\)
0.338467 + 0.940978i \(0.390091\pi\)
\(984\) −720.000 −0.0233260
\(985\) −17440.0 −0.564147
\(986\) −4017.68 −0.129766
\(987\) −3013.17 −0.0971735
\(988\) 1242.25 0.0400012
\(989\) −5651.69 −0.181712
\(990\) 0 0
\(991\) 21394.4 0.685788 0.342894 0.939374i \(-0.388593\pi\)
0.342894 + 0.939374i \(0.388593\pi\)
\(992\) 768.171 0.0245861
\(993\) −19126.5 −0.611240
\(994\) −7198.47 −0.229700
\(995\) −9845.76 −0.313700
\(996\) −1127.60 −0.0358729
\(997\) −46133.4 −1.46546 −0.732728 0.680521i \(-0.761753\pi\)
−0.732728 + 0.680521i \(0.761753\pi\)
\(998\) −7763.73 −0.246249
\(999\) −9920.43 −0.314183
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 1815.4.a.o.1.1 yes 2
11.10 odd 2 1815.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.i.1.2 2 11.10 odd 2
1815.4.a.o.1.1 yes 2 1.1 even 1 trivial