Properties

Label 1849.4.a.d.1.9
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1849,4,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1849.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 59x^{8} + 42x^{7} + 1187x^{6} - 541x^{5} - 9389x^{4} + 2180x^{3} + 22676x^{2} - 320x - 768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-3.82341\) of defining polynomial
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.82341 q^{2} -7.76517 q^{3} +6.61843 q^{4} +18.1323 q^{5} -29.6894 q^{6} +5.19796 q^{7} -5.28231 q^{8} +33.2979 q^{9} +O(q^{10})\) \(q+3.82341 q^{2} -7.76517 q^{3} +6.61843 q^{4} +18.1323 q^{5} -29.6894 q^{6} +5.19796 q^{7} -5.28231 q^{8} +33.2979 q^{9} +69.3271 q^{10} +31.5438 q^{11} -51.3932 q^{12} -71.3248 q^{13} +19.8739 q^{14} -140.800 q^{15} -73.1438 q^{16} +136.180 q^{17} +127.311 q^{18} -1.39966 q^{19} +120.007 q^{20} -40.3630 q^{21} +120.605 q^{22} -60.7163 q^{23} +41.0180 q^{24} +203.780 q^{25} -272.704 q^{26} -48.9039 q^{27} +34.4023 q^{28} +29.6008 q^{29} -538.337 q^{30} +87.2661 q^{31} -237.400 q^{32} -244.943 q^{33} +520.672 q^{34} +94.2509 q^{35} +220.379 q^{36} +255.642 q^{37} -5.35146 q^{38} +553.849 q^{39} -95.7804 q^{40} +23.1367 q^{41} -154.324 q^{42} +208.771 q^{44} +603.766 q^{45} -232.143 q^{46} -1.08229 q^{47} +567.974 q^{48} -315.981 q^{49} +779.133 q^{50} -1057.46 q^{51} -472.058 q^{52} -66.8923 q^{53} -186.979 q^{54} +571.962 q^{55} -27.4572 q^{56} +10.8686 q^{57} +113.176 q^{58} +465.968 q^{59} -931.877 q^{60} -580.345 q^{61} +333.654 q^{62} +173.081 q^{63} -322.526 q^{64} -1293.28 q^{65} -936.517 q^{66} +328.856 q^{67} +901.299 q^{68} +471.472 q^{69} +360.359 q^{70} +832.806 q^{71} -175.890 q^{72} -1049.14 q^{73} +977.423 q^{74} -1582.39 q^{75} -9.26354 q^{76} +163.963 q^{77} +2117.59 q^{78} +808.341 q^{79} -1326.27 q^{80} -519.295 q^{81} +88.4608 q^{82} +1038.31 q^{83} -267.140 q^{84} +2469.26 q^{85} -229.855 q^{87} -166.624 q^{88} +875.226 q^{89} +2308.44 q^{90} -370.744 q^{91} -401.846 q^{92} -677.636 q^{93} -4.13803 q^{94} -25.3790 q^{95} +1843.45 q^{96} +88.7982 q^{97} -1208.12 q^{98} +1050.34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 5 q^{3} + 39 q^{4} + 19 q^{5} - 15 q^{6} + 51 q^{7} - 36 q^{8} + 117 q^{9} - 27 q^{10} + 27 q^{11} + 72 q^{12} + 15 q^{13} - 96 q^{14} - 65 q^{15} + 67 q^{16} + 82 q^{17} - 247 q^{18} - 78 q^{19} + 495 q^{20} - 9 q^{21} + 190 q^{22} + 61 q^{23} - 202 q^{24} + 151 q^{25} + 21 q^{26} - 97 q^{27} + 794 q^{28} + 53 q^{29} - 627 q^{30} - 253 q^{31} - 399 q^{32} + 424 q^{33} + 231 q^{34} + 355 q^{35} + 1092 q^{36} + 129 q^{37} + 854 q^{38} + 691 q^{39} - 1345 q^{40} + 391 q^{41} + 31 q^{42} + 377 q^{44} + 944 q^{45} + 40 q^{46} - 334 q^{47} + 2401 q^{48} + 115 q^{49} - 424 q^{50} + 795 q^{51} + 564 q^{52} - 773 q^{53} + 182 q^{54} + 1242 q^{55} + 923 q^{56} + 765 q^{57} - 1328 q^{58} - 1483 q^{59} + 1075 q^{60} - 437 q^{61} - 1509 q^{62} + 2222 q^{63} - 738 q^{64} - 1063 q^{65} - 1483 q^{66} + 642 q^{67} + 1052 q^{68} + 3503 q^{69} - 85 q^{70} + 1545 q^{71} - 3834 q^{72} - 1292 q^{73} + 2232 q^{74} + 82 q^{75} + 252 q^{76} - 1448 q^{77} + 2822 q^{78} + 1405 q^{79} + 3157 q^{80} - 974 q^{81} + 3304 q^{82} - 543 q^{83} + 3652 q^{84} + 973 q^{85} + 1409 q^{87} - 2686 q^{88} + 2196 q^{89} - 742 q^{90} + 3513 q^{91} - 2629 q^{92} + 983 q^{93} + 4939 q^{94} + 149 q^{95} - 3540 q^{96} - 425 q^{97} + 213 q^{98} + 3181 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.82341 1.35178 0.675889 0.737003i \(-0.263760\pi\)
0.675889 + 0.737003i \(0.263760\pi\)
\(3\) −7.76517 −1.49441 −0.747204 0.664595i \(-0.768604\pi\)
−0.747204 + 0.664595i \(0.768604\pi\)
\(4\) 6.61843 0.827303
\(5\) 18.1323 1.62180 0.810901 0.585184i \(-0.198978\pi\)
0.810901 + 0.585184i \(0.198978\pi\)
\(6\) −29.6894 −2.02011
\(7\) 5.19796 0.280663 0.140332 0.990105i \(-0.455183\pi\)
0.140332 + 0.990105i \(0.455183\pi\)
\(8\) −5.28231 −0.233447
\(9\) 33.2979 1.23325
\(10\) 69.3271 2.19232
\(11\) 31.5438 0.864620 0.432310 0.901725i \(-0.357699\pi\)
0.432310 + 0.901725i \(0.357699\pi\)
\(12\) −51.3932 −1.23633
\(13\) −71.3248 −1.52169 −0.760844 0.648935i \(-0.775215\pi\)
−0.760844 + 0.648935i \(0.775215\pi\)
\(14\) 19.8739 0.379395
\(15\) −140.800 −2.42363
\(16\) −73.1438 −1.14287
\(17\) 136.180 1.94286 0.971428 0.237334i \(-0.0762736\pi\)
0.971428 + 0.237334i \(0.0762736\pi\)
\(18\) 127.311 1.66709
\(19\) −1.39966 −0.0169002 −0.00845010 0.999964i \(-0.502690\pi\)
−0.00845010 + 0.999964i \(0.502690\pi\)
\(20\) 120.007 1.34172
\(21\) −40.3630 −0.419425
\(22\) 120.605 1.16877
\(23\) −60.7163 −0.550444 −0.275222 0.961381i \(-0.588751\pi\)
−0.275222 + 0.961381i \(0.588751\pi\)
\(24\) 41.0180 0.348865
\(25\) 203.780 1.63024
\(26\) −272.704 −2.05698
\(27\) −48.9039 −0.348576
\(28\) 34.4023 0.232194
\(29\) 29.6008 0.189542 0.0947712 0.995499i \(-0.469788\pi\)
0.0947712 + 0.995499i \(0.469788\pi\)
\(30\) −538.337 −3.27621
\(31\) 87.2661 0.505595 0.252798 0.967519i \(-0.418649\pi\)
0.252798 + 0.967519i \(0.418649\pi\)
\(32\) −237.400 −1.31146
\(33\) −244.943 −1.29209
\(34\) 520.672 2.62631
\(35\) 94.2509 0.455180
\(36\) 220.379 1.02028
\(37\) 255.642 1.13587 0.567936 0.823073i \(-0.307742\pi\)
0.567936 + 0.823073i \(0.307742\pi\)
\(38\) −5.35146 −0.0228453
\(39\) 553.849 2.27402
\(40\) −95.7804 −0.378605
\(41\) 23.1367 0.0881302 0.0440651 0.999029i \(-0.485969\pi\)
0.0440651 + 0.999029i \(0.485969\pi\)
\(42\) −154.324 −0.566970
\(43\) 0 0
\(44\) 208.771 0.715303
\(45\) 603.766 2.00009
\(46\) −232.143 −0.744079
\(47\) −1.08229 −0.00335890 −0.00167945 0.999999i \(-0.500535\pi\)
−0.00167945 + 0.999999i \(0.500535\pi\)
\(48\) 567.974 1.70792
\(49\) −315.981 −0.921228
\(50\) 779.133 2.20372
\(51\) −1057.46 −2.90342
\(52\) −472.058 −1.25890
\(53\) −66.8923 −0.173365 −0.0866827 0.996236i \(-0.527627\pi\)
−0.0866827 + 0.996236i \(0.527627\pi\)
\(54\) −186.979 −0.471198
\(55\) 571.962 1.40224
\(56\) −27.4572 −0.0655201
\(57\) 10.8686 0.0252558
\(58\) 113.176 0.256219
\(59\) 465.968 1.02820 0.514101 0.857730i \(-0.328126\pi\)
0.514101 + 0.857730i \(0.328126\pi\)
\(60\) −931.877 −2.00508
\(61\) −580.345 −1.21812 −0.609062 0.793123i \(-0.708454\pi\)
−0.609062 + 0.793123i \(0.708454\pi\)
\(62\) 333.654 0.683452
\(63\) 173.081 0.346129
\(64\) −322.526 −0.629933
\(65\) −1293.28 −2.46788
\(66\) −936.517 −1.74663
\(67\) 328.856 0.599644 0.299822 0.953995i \(-0.403073\pi\)
0.299822 + 0.953995i \(0.403073\pi\)
\(68\) 901.299 1.60733
\(69\) 471.472 0.822588
\(70\) 360.359 0.615303
\(71\) 832.806 1.39205 0.696027 0.718016i \(-0.254949\pi\)
0.696027 + 0.718016i \(0.254949\pi\)
\(72\) −175.890 −0.287900
\(73\) −1049.14 −1.68210 −0.841048 0.540961i \(-0.818061\pi\)
−0.841048 + 0.540961i \(0.818061\pi\)
\(74\) 977.423 1.53545
\(75\) −1582.39 −2.43624
\(76\) −9.26354 −0.0139816
\(77\) 163.963 0.242667
\(78\) 2117.59 3.07397
\(79\) 808.341 1.15121 0.575605 0.817728i \(-0.304767\pi\)
0.575605 + 0.817728i \(0.304767\pi\)
\(80\) −1326.27 −1.85351
\(81\) −519.295 −0.712339
\(82\) 88.4608 0.119132
\(83\) 1038.31 1.37312 0.686560 0.727074i \(-0.259120\pi\)
0.686560 + 0.727074i \(0.259120\pi\)
\(84\) −267.140 −0.346992
\(85\) 2469.26 3.15093
\(86\) 0 0
\(87\) −229.855 −0.283254
\(88\) −166.624 −0.201843
\(89\) 875.226 1.04240 0.521201 0.853434i \(-0.325484\pi\)
0.521201 + 0.853434i \(0.325484\pi\)
\(90\) 2308.44 2.70368
\(91\) −370.744 −0.427082
\(92\) −401.846 −0.455385
\(93\) −677.636 −0.755565
\(94\) −4.13803 −0.00454048
\(95\) −25.3790 −0.0274088
\(96\) 1843.45 1.95986
\(97\) 88.7982 0.0929494 0.0464747 0.998919i \(-0.485201\pi\)
0.0464747 + 0.998919i \(0.485201\pi\)
\(98\) −1208.12 −1.24530
\(99\) 1050.34 1.06630
\(100\) 1348.70 1.34870
\(101\) 1277.36 1.25843 0.629216 0.777230i \(-0.283376\pi\)
0.629216 + 0.777230i \(0.283376\pi\)
\(102\) −4043.11 −3.92478
\(103\) −928.067 −0.887817 −0.443909 0.896072i \(-0.646409\pi\)
−0.443909 + 0.896072i \(0.646409\pi\)
\(104\) 376.760 0.355234
\(105\) −731.874 −0.680225
\(106\) −255.757 −0.234352
\(107\) −1194.14 −1.07890 −0.539448 0.842019i \(-0.681367\pi\)
−0.539448 + 0.842019i \(0.681367\pi\)
\(108\) −323.667 −0.288378
\(109\) 671.431 0.590013 0.295007 0.955495i \(-0.404678\pi\)
0.295007 + 0.955495i \(0.404678\pi\)
\(110\) 2186.84 1.89552
\(111\) −1985.10 −1.69746
\(112\) −380.199 −0.320762
\(113\) −263.783 −0.219598 −0.109799 0.993954i \(-0.535021\pi\)
−0.109799 + 0.993954i \(0.535021\pi\)
\(114\) 41.5550 0.0341402
\(115\) −1100.93 −0.892712
\(116\) 195.911 0.156809
\(117\) −2374.96 −1.87663
\(118\) 1781.59 1.38990
\(119\) 707.859 0.545289
\(120\) 743.751 0.565791
\(121\) −335.987 −0.252432
\(122\) −2218.89 −1.64663
\(123\) −179.660 −0.131702
\(124\) 577.564 0.418281
\(125\) 1428.46 1.02212
\(126\) 661.758 0.467890
\(127\) −1082.71 −0.756499 −0.378249 0.925704i \(-0.623474\pi\)
−0.378249 + 0.925704i \(0.623474\pi\)
\(128\) 666.053 0.459932
\(129\) 0 0
\(130\) −4944.74 −3.33602
\(131\) 1370.38 0.913972 0.456986 0.889474i \(-0.348929\pi\)
0.456986 + 0.889474i \(0.348929\pi\)
\(132\) −1621.14 −1.06895
\(133\) −7.27537 −0.00474327
\(134\) 1257.35 0.810586
\(135\) −886.739 −0.565321
\(136\) −719.346 −0.453555
\(137\) 1001.07 0.624283 0.312142 0.950036i \(-0.398954\pi\)
0.312142 + 0.950036i \(0.398954\pi\)
\(138\) 1802.63 1.11196
\(139\) −598.525 −0.365225 −0.182612 0.983185i \(-0.558455\pi\)
−0.182612 + 0.983185i \(0.558455\pi\)
\(140\) 623.793 0.376572
\(141\) 8.40416 0.00501956
\(142\) 3184.15 1.88175
\(143\) −2249.86 −1.31568
\(144\) −2435.53 −1.40945
\(145\) 536.730 0.307400
\(146\) −4011.30 −2.27382
\(147\) 2453.65 1.37669
\(148\) 1691.95 0.939711
\(149\) −710.011 −0.390378 −0.195189 0.980766i \(-0.562532\pi\)
−0.195189 + 0.980766i \(0.562532\pi\)
\(150\) −6050.10 −3.29326
\(151\) 1699.83 0.916093 0.458047 0.888928i \(-0.348549\pi\)
0.458047 + 0.888928i \(0.348549\pi\)
\(152\) 7.39343 0.00394531
\(153\) 4534.51 2.39603
\(154\) 626.899 0.328032
\(155\) 1582.33 0.819975
\(156\) 3665.61 1.88131
\(157\) 1361.92 0.692311 0.346155 0.938177i \(-0.387487\pi\)
0.346155 + 0.938177i \(0.387487\pi\)
\(158\) 3090.62 1.55618
\(159\) 519.430 0.259079
\(160\) −4304.61 −2.12693
\(161\) −315.601 −0.154490
\(162\) −1985.48 −0.962924
\(163\) 2881.57 1.38467 0.692336 0.721575i \(-0.256582\pi\)
0.692336 + 0.721575i \(0.256582\pi\)
\(164\) 153.128 0.0729104
\(165\) −4441.38 −2.09552
\(166\) 3969.86 1.85615
\(167\) 986.781 0.457242 0.228621 0.973515i \(-0.426578\pi\)
0.228621 + 0.973515i \(0.426578\pi\)
\(168\) 213.210 0.0979138
\(169\) 2890.23 1.31554
\(170\) 9440.98 4.25935
\(171\) −46.6056 −0.0208422
\(172\) 0 0
\(173\) 595.012 0.261491 0.130746 0.991416i \(-0.458263\pi\)
0.130746 + 0.991416i \(0.458263\pi\)
\(174\) −878.830 −0.382896
\(175\) 1059.24 0.457549
\(176\) −2307.24 −0.988150
\(177\) −3618.32 −1.53655
\(178\) 3346.34 1.40910
\(179\) 2320.31 0.968872 0.484436 0.874827i \(-0.339025\pi\)
0.484436 + 0.874827i \(0.339025\pi\)
\(180\) 3995.98 1.65468
\(181\) 1531.17 0.628792 0.314396 0.949292i \(-0.398198\pi\)
0.314396 + 0.949292i \(0.398198\pi\)
\(182\) −1417.50 −0.577320
\(183\) 4506.48 1.82037
\(184\) 320.722 0.128500
\(185\) 4635.37 1.84216
\(186\) −2590.88 −1.02136
\(187\) 4295.65 1.67983
\(188\) −7.16306 −0.00277883
\(189\) −254.200 −0.0978326
\(190\) −97.0343 −0.0370506
\(191\) 1733.51 0.656714 0.328357 0.944554i \(-0.393505\pi\)
0.328357 + 0.944554i \(0.393505\pi\)
\(192\) 2504.47 0.941377
\(193\) 1109.49 0.413798 0.206899 0.978362i \(-0.433663\pi\)
0.206899 + 0.978362i \(0.433663\pi\)
\(194\) 339.512 0.125647
\(195\) 10042.6 3.68801
\(196\) −2091.30 −0.762135
\(197\) 1579.78 0.571343 0.285672 0.958328i \(-0.407783\pi\)
0.285672 + 0.958328i \(0.407783\pi\)
\(198\) 4015.88 1.44140
\(199\) 3596.57 1.28118 0.640588 0.767885i \(-0.278691\pi\)
0.640588 + 0.767885i \(0.278691\pi\)
\(200\) −1076.43 −0.380575
\(201\) −2553.62 −0.896112
\(202\) 4883.85 1.70112
\(203\) 153.864 0.0531976
\(204\) −6998.74 −2.40201
\(205\) 419.521 0.142930
\(206\) −3548.38 −1.20013
\(207\) −2021.72 −0.678838
\(208\) 5216.97 1.73910
\(209\) −44.1506 −0.0146123
\(210\) −2798.25 −0.919513
\(211\) 3927.29 1.28135 0.640676 0.767811i \(-0.278654\pi\)
0.640676 + 0.767811i \(0.278654\pi\)
\(212\) −442.722 −0.143426
\(213\) −6466.88 −2.08030
\(214\) −4565.68 −1.45843
\(215\) 0 0
\(216\) 258.326 0.0813742
\(217\) 453.606 0.141902
\(218\) 2567.15 0.797567
\(219\) 8146.78 2.51374
\(220\) 3785.49 1.16008
\(221\) −9713.03 −2.95642
\(222\) −7589.85 −2.29458
\(223\) −249.842 −0.0750255 −0.0375127 0.999296i \(-0.511943\pi\)
−0.0375127 + 0.999296i \(0.511943\pi\)
\(224\) −1234.00 −0.368079
\(225\) 6785.44 2.01050
\(226\) −1008.55 −0.296848
\(227\) 5231.99 1.52978 0.764889 0.644163i \(-0.222794\pi\)
0.764889 + 0.644163i \(0.222794\pi\)
\(228\) 71.9330 0.0208942
\(229\) −948.401 −0.273677 −0.136839 0.990593i \(-0.543694\pi\)
−0.136839 + 0.990593i \(0.543694\pi\)
\(230\) −4209.28 −1.20675
\(231\) −1273.20 −0.362644
\(232\) −156.361 −0.0442482
\(233\) 3452.38 0.970699 0.485350 0.874320i \(-0.338692\pi\)
0.485350 + 0.874320i \(0.338692\pi\)
\(234\) −9080.45 −2.53678
\(235\) −19.6244 −0.00544747
\(236\) 3083.98 0.850635
\(237\) −6276.91 −1.72038
\(238\) 2706.43 0.737109
\(239\) −1033.96 −0.279838 −0.139919 0.990163i \(-0.544684\pi\)
−0.139919 + 0.990163i \(0.544684\pi\)
\(240\) 10298.7 2.76990
\(241\) 925.893 0.247477 0.123739 0.992315i \(-0.460512\pi\)
0.123739 + 0.992315i \(0.460512\pi\)
\(242\) −1284.62 −0.341232
\(243\) 5352.82 1.41310
\(244\) −3840.97 −1.00776
\(245\) −5729.46 −1.49405
\(246\) −686.913 −0.178032
\(247\) 99.8304 0.0257168
\(248\) −460.967 −0.118030
\(249\) −8062.62 −2.05200
\(250\) 5461.59 1.38168
\(251\) −1376.38 −0.346120 −0.173060 0.984911i \(-0.555365\pi\)
−0.173060 + 0.984911i \(0.555365\pi\)
\(252\) 1145.52 0.286354
\(253\) −1915.22 −0.475925
\(254\) −4139.65 −1.02262
\(255\) −19174.2 −4.70877
\(256\) 5126.80 1.25166
\(257\) −582.311 −0.141337 −0.0706685 0.997500i \(-0.522513\pi\)
−0.0706685 + 0.997500i \(0.522513\pi\)
\(258\) 0 0
\(259\) 1328.82 0.318798
\(260\) −8559.50 −2.04168
\(261\) 985.643 0.233754
\(262\) 5239.50 1.23549
\(263\) −130.662 −0.0306348 −0.0153174 0.999883i \(-0.504876\pi\)
−0.0153174 + 0.999883i \(0.504876\pi\)
\(264\) 1293.87 0.301636
\(265\) −1212.91 −0.281164
\(266\) −27.8167 −0.00641184
\(267\) −6796.28 −1.55777
\(268\) 2176.51 0.496088
\(269\) −7250.60 −1.64341 −0.821704 0.569914i \(-0.806976\pi\)
−0.821704 + 0.569914i \(0.806976\pi\)
\(270\) −3390.36 −0.764189
\(271\) 4219.91 0.945909 0.472955 0.881087i \(-0.343187\pi\)
0.472955 + 0.881087i \(0.343187\pi\)
\(272\) −9960.74 −2.22044
\(273\) 2878.89 0.638235
\(274\) 3827.48 0.843892
\(275\) 6428.00 1.40954
\(276\) 3120.40 0.680530
\(277\) 397.916 0.0863121 0.0431560 0.999068i \(-0.486259\pi\)
0.0431560 + 0.999068i \(0.486259\pi\)
\(278\) −2288.40 −0.493703
\(279\) 2905.77 0.623527
\(280\) −497.863 −0.106261
\(281\) −5264.19 −1.11756 −0.558781 0.829315i \(-0.688731\pi\)
−0.558781 + 0.829315i \(0.688731\pi\)
\(282\) 32.1325 0.00678533
\(283\) −4632.62 −0.973076 −0.486538 0.873659i \(-0.661740\pi\)
−0.486538 + 0.873659i \(0.661740\pi\)
\(284\) 5511.86 1.15165
\(285\) 197.072 0.0409599
\(286\) −8602.12 −1.77851
\(287\) 120.263 0.0247349
\(288\) −7904.91 −1.61737
\(289\) 13632.1 2.77469
\(290\) 2052.14 0.415537
\(291\) −689.533 −0.138904
\(292\) −6943.68 −1.39160
\(293\) −519.305 −0.103543 −0.0517716 0.998659i \(-0.516487\pi\)
−0.0517716 + 0.998659i \(0.516487\pi\)
\(294\) 9381.29 1.86098
\(295\) 8449.08 1.66754
\(296\) −1350.38 −0.265166
\(297\) −1542.62 −0.301386
\(298\) −2714.66 −0.527704
\(299\) 4330.58 0.837605
\(300\) −10472.9 −2.01551
\(301\) 0 0
\(302\) 6499.13 1.23835
\(303\) −9918.88 −1.88061
\(304\) 102.376 0.0193148
\(305\) −10523.0 −1.97555
\(306\) 17337.3 3.23891
\(307\) 1193.72 0.221920 0.110960 0.993825i \(-0.464607\pi\)
0.110960 + 0.993825i \(0.464607\pi\)
\(308\) 1085.18 0.200759
\(309\) 7206.60 1.32676
\(310\) 6049.90 1.10842
\(311\) −2588.37 −0.471938 −0.235969 0.971761i \(-0.575826\pi\)
−0.235969 + 0.971761i \(0.575826\pi\)
\(312\) −2925.60 −0.530865
\(313\) 423.736 0.0765208 0.0382604 0.999268i \(-0.487818\pi\)
0.0382604 + 0.999268i \(0.487818\pi\)
\(314\) 5207.16 0.935851
\(315\) 3138.35 0.561353
\(316\) 5349.95 0.952399
\(317\) 1368.80 0.242522 0.121261 0.992621i \(-0.461306\pi\)
0.121261 + 0.992621i \(0.461306\pi\)
\(318\) 1985.99 0.350217
\(319\) 933.723 0.163882
\(320\) −5848.13 −1.02163
\(321\) 9272.71 1.61231
\(322\) −1206.67 −0.208836
\(323\) −190.606 −0.0328347
\(324\) −3436.92 −0.589320
\(325\) −14534.6 −2.48072
\(326\) 11017.4 1.87177
\(327\) −5213.78 −0.881720
\(328\) −122.215 −0.0205738
\(329\) −5.62570 −0.000942720 0
\(330\) −16981.2 −2.83268
\(331\) −2454.58 −0.407601 −0.203801 0.979012i \(-0.565329\pi\)
−0.203801 + 0.979012i \(0.565329\pi\)
\(332\) 6871.95 1.13599
\(333\) 8512.33 1.40082
\(334\) 3772.87 0.618090
\(335\) 5962.91 0.972504
\(336\) 2952.31 0.479350
\(337\) 7409.08 1.19762 0.598810 0.800891i \(-0.295640\pi\)
0.598810 + 0.800891i \(0.295640\pi\)
\(338\) 11050.5 1.77831
\(339\) 2048.32 0.328169
\(340\) 16342.6 2.60677
\(341\) 2752.71 0.437148
\(342\) −178.192 −0.0281741
\(343\) −3425.36 −0.539218
\(344\) 0 0
\(345\) 8548.87 1.33407
\(346\) 2274.97 0.353478
\(347\) −10125.5 −1.56647 −0.783237 0.621723i \(-0.786433\pi\)
−0.783237 + 0.621723i \(0.786433\pi\)
\(348\) −1521.28 −0.234337
\(349\) 67.8059 0.0103999 0.00519995 0.999986i \(-0.498345\pi\)
0.00519995 + 0.999986i \(0.498345\pi\)
\(350\) 4049.90 0.618504
\(351\) 3488.06 0.530424
\(352\) −7488.51 −1.13392
\(353\) −6725.07 −1.01399 −0.506996 0.861948i \(-0.669244\pi\)
−0.506996 + 0.861948i \(0.669244\pi\)
\(354\) −13834.3 −2.07708
\(355\) 15100.7 2.25764
\(356\) 5792.62 0.862383
\(357\) −5496.65 −0.814883
\(358\) 8871.49 1.30970
\(359\) −9260.90 −1.36148 −0.680740 0.732525i \(-0.738342\pi\)
−0.680740 + 0.732525i \(0.738342\pi\)
\(360\) −3189.28 −0.466916
\(361\) −6857.04 −0.999714
\(362\) 5854.30 0.849987
\(363\) 2609.00 0.377236
\(364\) −2453.74 −0.353327
\(365\) −19023.4 −2.72802
\(366\) 17230.1 2.46074
\(367\) 1180.46 0.167901 0.0839504 0.996470i \(-0.473246\pi\)
0.0839504 + 0.996470i \(0.473246\pi\)
\(368\) 4441.02 0.629088
\(369\) 770.401 0.108687
\(370\) 17722.9 2.49019
\(371\) −347.704 −0.0486573
\(372\) −4484.88 −0.625082
\(373\) −6144.38 −0.852933 −0.426466 0.904503i \(-0.640242\pi\)
−0.426466 + 0.904503i \(0.640242\pi\)
\(374\) 16424.0 2.27076
\(375\) −11092.2 −1.52747
\(376\) 5.71699 0.000784126 0
\(377\) −2111.27 −0.288425
\(378\) −971.911 −0.132248
\(379\) −8452.78 −1.14562 −0.572810 0.819688i \(-0.694147\pi\)
−0.572810 + 0.819688i \(0.694147\pi\)
\(380\) −167.969 −0.0226754
\(381\) 8407.46 1.13052
\(382\) 6627.91 0.887731
\(383\) 3796.02 0.506443 0.253221 0.967408i \(-0.418510\pi\)
0.253221 + 0.967408i \(0.418510\pi\)
\(384\) −5172.02 −0.687326
\(385\) 2973.03 0.393558
\(386\) 4242.04 0.559363
\(387\) 0 0
\(388\) 587.705 0.0768974
\(389\) 12372.4 1.61261 0.806303 0.591502i \(-0.201465\pi\)
0.806303 + 0.591502i \(0.201465\pi\)
\(390\) 38396.8 4.98537
\(391\) −8268.36 −1.06943
\(392\) 1669.11 0.215058
\(393\) −10641.2 −1.36585
\(394\) 6040.14 0.772329
\(395\) 14657.1 1.86703
\(396\) 6951.61 0.882150
\(397\) −12525.5 −1.58346 −0.791731 0.610869i \(-0.790820\pi\)
−0.791731 + 0.610869i \(0.790820\pi\)
\(398\) 13751.1 1.73186
\(399\) 56.4945 0.00708837
\(400\) −14905.2 −1.86316
\(401\) 8542.42 1.06381 0.531905 0.846804i \(-0.321476\pi\)
0.531905 + 0.846804i \(0.321476\pi\)
\(402\) −9763.53 −1.21135
\(403\) −6224.24 −0.769358
\(404\) 8454.09 1.04111
\(405\) −9416.01 −1.15527
\(406\) 588.284 0.0719114
\(407\) 8063.92 0.982098
\(408\) 5585.85 0.677796
\(409\) −9148.33 −1.10601 −0.553003 0.833180i \(-0.686518\pi\)
−0.553003 + 0.833180i \(0.686518\pi\)
\(410\) 1604.00 0.193209
\(411\) −7773.45 −0.932934
\(412\) −6142.34 −0.734494
\(413\) 2422.08 0.288579
\(414\) −7729.86 −0.917638
\(415\) 18826.9 2.22693
\(416\) 16932.5 1.99564
\(417\) 4647.65 0.545794
\(418\) −168.806 −0.0197525
\(419\) 8781.89 1.02392 0.511961 0.859009i \(-0.328919\pi\)
0.511961 + 0.859009i \(0.328919\pi\)
\(420\) −4843.86 −0.562752
\(421\) −7723.93 −0.894160 −0.447080 0.894494i \(-0.647536\pi\)
−0.447080 + 0.894494i \(0.647536\pi\)
\(422\) 15015.6 1.73210
\(423\) −36.0379 −0.00414237
\(424\) 353.346 0.0404717
\(425\) 27750.8 3.16732
\(426\) −24725.5 −2.81210
\(427\) −3016.61 −0.341883
\(428\) −7903.33 −0.892575
\(429\) 17470.5 1.96617
\(430\) 0 0
\(431\) −5925.41 −0.662221 −0.331110 0.943592i \(-0.607423\pi\)
−0.331110 + 0.943592i \(0.607423\pi\)
\(432\) 3577.02 0.398378
\(433\) −3909.25 −0.433871 −0.216936 0.976186i \(-0.569606\pi\)
−0.216936 + 0.976186i \(0.569606\pi\)
\(434\) 1734.32 0.191820
\(435\) −4167.80 −0.459381
\(436\) 4443.82 0.488120
\(437\) 84.9821 0.00930262
\(438\) 31148.4 3.39801
\(439\) −4441.26 −0.482846 −0.241423 0.970420i \(-0.577614\pi\)
−0.241423 + 0.970420i \(0.577614\pi\)
\(440\) −3021.28 −0.327350
\(441\) −10521.5 −1.13611
\(442\) −37136.9 −3.99643
\(443\) −7712.27 −0.827136 −0.413568 0.910473i \(-0.635718\pi\)
−0.413568 + 0.910473i \(0.635718\pi\)
\(444\) −13138.3 −1.40431
\(445\) 15869.9 1.69057
\(446\) −955.248 −0.101418
\(447\) 5513.35 0.583384
\(448\) −1676.48 −0.176799
\(449\) 1293.48 0.135954 0.0679769 0.997687i \(-0.478346\pi\)
0.0679769 + 0.997687i \(0.478346\pi\)
\(450\) 25943.5 2.71775
\(451\) 729.818 0.0761991
\(452\) −1745.83 −0.181674
\(453\) −13199.5 −1.36902
\(454\) 20004.0 2.06792
\(455\) −6722.43 −0.692643
\(456\) −57.4113 −0.00589590
\(457\) −5561.70 −0.569290 −0.284645 0.958633i \(-0.591876\pi\)
−0.284645 + 0.958633i \(0.591876\pi\)
\(458\) −3626.12 −0.369951
\(459\) −6659.74 −0.677233
\(460\) −7286.40 −0.738543
\(461\) −15913.7 −1.60776 −0.803878 0.594795i \(-0.797233\pi\)
−0.803878 + 0.594795i \(0.797233\pi\)
\(462\) −4867.98 −0.490214
\(463\) 15646.3 1.57051 0.785256 0.619172i \(-0.212532\pi\)
0.785256 + 0.619172i \(0.212532\pi\)
\(464\) −2165.12 −0.216623
\(465\) −12287.1 −1.22538
\(466\) 13199.8 1.31217
\(467\) 9463.86 0.937762 0.468881 0.883261i \(-0.344657\pi\)
0.468881 + 0.883261i \(0.344657\pi\)
\(468\) −15718.5 −1.55254
\(469\) 1709.38 0.168298
\(470\) −75.0320 −0.00736376
\(471\) −10575.5 −1.03459
\(472\) −2461.39 −0.240031
\(473\) 0 0
\(474\) −23999.2 −2.32557
\(475\) −285.223 −0.0275514
\(476\) 4684.91 0.451119
\(477\) −2227.37 −0.213804
\(478\) −3953.24 −0.378278
\(479\) 3861.05 0.368301 0.184150 0.982898i \(-0.441047\pi\)
0.184150 + 0.982898i \(0.441047\pi\)
\(480\) 33426.0 3.17850
\(481\) −18233.6 −1.72844
\(482\) 3540.06 0.334534
\(483\) 2450.69 0.230870
\(484\) −2223.71 −0.208838
\(485\) 1610.12 0.150746
\(486\) 20466.0 1.91020
\(487\) 3513.12 0.326888 0.163444 0.986553i \(-0.447740\pi\)
0.163444 + 0.986553i \(0.447740\pi\)
\(488\) 3065.56 0.284368
\(489\) −22375.8 −2.06926
\(490\) −21906.1 −2.01962
\(491\) −19277.8 −1.77188 −0.885940 0.463799i \(-0.846486\pi\)
−0.885940 + 0.463799i \(0.846486\pi\)
\(492\) −1189.07 −0.108958
\(493\) 4031.04 0.368254
\(494\) 381.692 0.0347635
\(495\) 19045.1 1.72932
\(496\) −6382.98 −0.577831
\(497\) 4328.89 0.390699
\(498\) −30826.7 −2.77385
\(499\) −7626.32 −0.684170 −0.342085 0.939669i \(-0.611133\pi\)
−0.342085 + 0.939669i \(0.611133\pi\)
\(500\) 9454.17 0.845607
\(501\) −7662.53 −0.683306
\(502\) −5262.45 −0.467878
\(503\) 9139.85 0.810190 0.405095 0.914275i \(-0.367238\pi\)
0.405095 + 0.914275i \(0.367238\pi\)
\(504\) −914.267 −0.0808029
\(505\) 23161.4 2.04093
\(506\) −7322.68 −0.643345
\(507\) −22443.1 −1.96595
\(508\) −7165.86 −0.625854
\(509\) −13821.6 −1.20360 −0.601799 0.798647i \(-0.705549\pi\)
−0.601799 + 0.798647i \(0.705549\pi\)
\(510\) −73310.8 −6.36521
\(511\) −5453.40 −0.472103
\(512\) 14273.4 1.23203
\(513\) 68.4488 0.00589101
\(514\) −2226.41 −0.191056
\(515\) −16828.0 −1.43986
\(516\) 0 0
\(517\) −34.1396 −0.00290417
\(518\) 5080.60 0.430944
\(519\) −4620.37 −0.390774
\(520\) 6831.52 0.576119
\(521\) −10933.9 −0.919427 −0.459714 0.888067i \(-0.652048\pi\)
−0.459714 + 0.888067i \(0.652048\pi\)
\(522\) 3768.51 0.315983
\(523\) −15350.3 −1.28340 −0.641701 0.766955i \(-0.721771\pi\)
−0.641701 + 0.766955i \(0.721771\pi\)
\(524\) 9069.74 0.756132
\(525\) −8225.18 −0.683764
\(526\) −499.573 −0.0414115
\(527\) 11883.9 0.982299
\(528\) 17916.1 1.47670
\(529\) −8480.53 −0.697011
\(530\) −4637.45 −0.380072
\(531\) 15515.7 1.26803
\(532\) −48.1515 −0.00392412
\(533\) −1650.22 −0.134107
\(534\) −25984.9 −2.10576
\(535\) −21652.5 −1.74976
\(536\) −1737.12 −0.139985
\(537\) −18017.6 −1.44789
\(538\) −27722.0 −2.22152
\(539\) −9967.26 −0.796512
\(540\) −5868.82 −0.467692
\(541\) 21902.0 1.74055 0.870277 0.492562i \(-0.163940\pi\)
0.870277 + 0.492562i \(0.163940\pi\)
\(542\) 16134.4 1.27866
\(543\) −11889.8 −0.939671
\(544\) −32329.2 −2.54798
\(545\) 12174.6 0.956884
\(546\) 11007.1 0.862752
\(547\) 3326.60 0.260028 0.130014 0.991512i \(-0.458498\pi\)
0.130014 + 0.991512i \(0.458498\pi\)
\(548\) 6625.48 0.516472
\(549\) −19324.2 −1.50226
\(550\) 24576.8 1.90538
\(551\) −41.4310 −0.00320331
\(552\) −2490.46 −0.192031
\(553\) 4201.73 0.323102
\(554\) 1521.39 0.116675
\(555\) −35994.5 −2.75294
\(556\) −3961.29 −0.302152
\(557\) −16723.9 −1.27220 −0.636098 0.771608i \(-0.719453\pi\)
−0.636098 + 0.771608i \(0.719453\pi\)
\(558\) 11109.9 0.842870
\(559\) 0 0
\(560\) −6893.87 −0.520213
\(561\) −33356.4 −2.51035
\(562\) −20127.1 −1.51070
\(563\) −8139.07 −0.609273 −0.304637 0.952469i \(-0.598535\pi\)
−0.304637 + 0.952469i \(0.598535\pi\)
\(564\) 55.6223 0.00415270
\(565\) −4782.98 −0.356145
\(566\) −17712.4 −1.31538
\(567\) −2699.27 −0.199927
\(568\) −4399.14 −0.324971
\(569\) −10529.4 −0.775771 −0.387885 0.921708i \(-0.626794\pi\)
−0.387885 + 0.921708i \(0.626794\pi\)
\(570\) 753.488 0.0553686
\(571\) 21060.5 1.54353 0.771765 0.635908i \(-0.219374\pi\)
0.771765 + 0.635908i \(0.219374\pi\)
\(572\) −14890.5 −1.08847
\(573\) −13461.0 −0.981398
\(574\) 459.816 0.0334361
\(575\) −12372.8 −0.897356
\(576\) −10739.4 −0.776868
\(577\) −832.149 −0.0600395 −0.0300198 0.999549i \(-0.509557\pi\)
−0.0300198 + 0.999549i \(0.509557\pi\)
\(578\) 52120.9 3.75077
\(579\) −8615.40 −0.618383
\(580\) 3552.31 0.254313
\(581\) 5397.07 0.385384
\(582\) −2636.37 −0.187768
\(583\) −2110.04 −0.149895
\(584\) 5541.90 0.392681
\(585\) −43063.5 −3.04352
\(586\) −1985.51 −0.139967
\(587\) −21831.1 −1.53504 −0.767519 0.641027i \(-0.778509\pi\)
−0.767519 + 0.641027i \(0.778509\pi\)
\(588\) 16239.3 1.13894
\(589\) −122.143 −0.00854466
\(590\) 32304.2 2.25414
\(591\) −12267.3 −0.853820
\(592\) −18698.6 −1.29816
\(593\) 22576.3 1.56340 0.781702 0.623653i \(-0.214352\pi\)
0.781702 + 0.623653i \(0.214352\pi\)
\(594\) −5898.04 −0.407407
\(595\) 12835.1 0.884350
\(596\) −4699.15 −0.322961
\(597\) −27928.0 −1.91460
\(598\) 16557.6 1.13226
\(599\) −2926.37 −0.199613 −0.0998065 0.995007i \(-0.531822\pi\)
−0.0998065 + 0.995007i \(0.531822\pi\)
\(600\) 8358.66 0.568734
\(601\) 26815.7 1.82003 0.910013 0.414579i \(-0.136071\pi\)
0.910013 + 0.414579i \(0.136071\pi\)
\(602\) 0 0
\(603\) 10950.2 0.739513
\(604\) 11250.2 0.757887
\(605\) −6092.22 −0.409395
\(606\) −37923.9 −2.54217
\(607\) 19629.5 1.31258 0.656290 0.754508i \(-0.272125\pi\)
0.656290 + 0.754508i \(0.272125\pi\)
\(608\) 332.279 0.0221640
\(609\) −1194.78 −0.0794989
\(610\) −40233.6 −2.67051
\(611\) 77.1941 0.00511120
\(612\) 30011.3 1.98225
\(613\) −14870.5 −0.979793 −0.489896 0.871781i \(-0.662965\pi\)
−0.489896 + 0.871781i \(0.662965\pi\)
\(614\) 4564.09 0.299986
\(615\) −3257.65 −0.213595
\(616\) −866.106 −0.0566500
\(617\) 3238.39 0.211301 0.105650 0.994403i \(-0.466308\pi\)
0.105650 + 0.994403i \(0.466308\pi\)
\(618\) 27553.7 1.79349
\(619\) 20133.6 1.30733 0.653666 0.756783i \(-0.273230\pi\)
0.653666 + 0.756783i \(0.273230\pi\)
\(620\) 10472.6 0.678368
\(621\) 2969.26 0.191872
\(622\) −9896.37 −0.637955
\(623\) 4549.39 0.292564
\(624\) −40510.7 −2.59892
\(625\) 428.784 0.0274422
\(626\) 1620.12 0.103439
\(627\) 342.837 0.0218367
\(628\) 9013.75 0.572751
\(629\) 34813.4 2.20684
\(630\) 11999.2 0.758824
\(631\) −6227.34 −0.392879 −0.196439 0.980516i \(-0.562938\pi\)
−0.196439 + 0.980516i \(0.562938\pi\)
\(632\) −4269.91 −0.268747
\(633\) −30496.0 −1.91486
\(634\) 5233.47 0.327835
\(635\) −19632.1 −1.22689
\(636\) 3437.81 0.214337
\(637\) 22537.3 1.40182
\(638\) 3570.00 0.221532
\(639\) 27730.6 1.71676
\(640\) 12077.1 0.745919
\(641\) 12495.3 0.769945 0.384973 0.922928i \(-0.374211\pi\)
0.384973 + 0.922928i \(0.374211\pi\)
\(642\) 35453.3 2.17949
\(643\) 4940.90 0.303033 0.151516 0.988455i \(-0.451584\pi\)
0.151516 + 0.988455i \(0.451584\pi\)
\(644\) −2088.78 −0.127810
\(645\) 0 0
\(646\) −728.764 −0.0443852
\(647\) 6624.13 0.402506 0.201253 0.979539i \(-0.435499\pi\)
0.201253 + 0.979539i \(0.435499\pi\)
\(648\) 2743.08 0.166294
\(649\) 14698.4 0.889004
\(650\) −55571.6 −3.35338
\(651\) −3522.32 −0.212060
\(652\) 19071.4 1.14554
\(653\) −4497.38 −0.269519 −0.134760 0.990878i \(-0.543026\pi\)
−0.134760 + 0.990878i \(0.543026\pi\)
\(654\) −19934.4 −1.19189
\(655\) 24848.1 1.48228
\(656\) −1692.30 −0.100722
\(657\) −34934.2 −2.07445
\(658\) −21.5093 −0.00127435
\(659\) 26308.3 1.55512 0.777562 0.628806i \(-0.216456\pi\)
0.777562 + 0.628806i \(0.216456\pi\)
\(660\) −29395.0 −1.73363
\(661\) 15971.0 0.939788 0.469894 0.882723i \(-0.344292\pi\)
0.469894 + 0.882723i \(0.344292\pi\)
\(662\) −9384.86 −0.550986
\(663\) 75423.3 4.41810
\(664\) −5484.66 −0.320551
\(665\) −131.919 −0.00769264
\(666\) 32546.1 1.89360
\(667\) −1797.25 −0.104333
\(668\) 6530.94 0.378278
\(669\) 1940.07 0.112119
\(670\) 22798.6 1.31461
\(671\) −18306.3 −1.05321
\(672\) 9582.18 0.550061
\(673\) 3279.61 0.187845 0.0939226 0.995580i \(-0.470059\pi\)
0.0939226 + 0.995580i \(0.470059\pi\)
\(674\) 28327.9 1.61892
\(675\) −9965.63 −0.568263
\(676\) 19128.8 1.08835
\(677\) −33062.8 −1.87697 −0.938483 0.345325i \(-0.887769\pi\)
−0.938483 + 0.345325i \(0.887769\pi\)
\(678\) 7831.55 0.443612
\(679\) 461.570 0.0260875
\(680\) −13043.4 −0.735576
\(681\) −40627.3 −2.28611
\(682\) 10524.7 0.590927
\(683\) −28185.8 −1.57906 −0.789531 0.613711i \(-0.789676\pi\)
−0.789531 + 0.613711i \(0.789676\pi\)
\(684\) −308.456 −0.0172429
\(685\) 18151.6 1.01246
\(686\) −13096.5 −0.728903
\(687\) 7364.49 0.408985
\(688\) 0 0
\(689\) 4771.08 0.263808
\(690\) 32685.8 1.80337
\(691\) −20756.6 −1.14272 −0.571359 0.820700i \(-0.693584\pi\)
−0.571359 + 0.820700i \(0.693584\pi\)
\(692\) 3938.05 0.216332
\(693\) 5459.63 0.299270
\(694\) −38714.0 −2.11753
\(695\) −10852.6 −0.592322
\(696\) 1214.17 0.0661248
\(697\) 3150.75 0.171224
\(698\) 259.249 0.0140584
\(699\) −26808.3 −1.45062
\(700\) 7010.50 0.378532
\(701\) −32953.3 −1.77550 −0.887751 0.460324i \(-0.847733\pi\)
−0.887751 + 0.460324i \(0.847733\pi\)
\(702\) 13336.3 0.717016
\(703\) −357.812 −0.0191965
\(704\) −10173.7 −0.544653
\(705\) 152.387 0.00814073
\(706\) −25712.7 −1.37069
\(707\) 6639.64 0.353196
\(708\) −23947.6 −1.27120
\(709\) 10596.1 0.561276 0.280638 0.959814i \(-0.409454\pi\)
0.280638 + 0.959814i \(0.409454\pi\)
\(710\) 57736.0 3.05182
\(711\) 26916.0 1.41973
\(712\) −4623.22 −0.243346
\(713\) −5298.47 −0.278302
\(714\) −21015.9 −1.10154
\(715\) −40795.1 −2.13378
\(716\) 15356.8 0.801551
\(717\) 8028.86 0.418191
\(718\) −35408.2 −1.84042
\(719\) −27081.5 −1.40469 −0.702344 0.711838i \(-0.747863\pi\)
−0.702344 + 0.711838i \(0.747863\pi\)
\(720\) −44161.8 −2.28585
\(721\) −4824.05 −0.249178
\(722\) −26217.2 −1.35139
\(723\) −7189.72 −0.369832
\(724\) 10134.0 0.520201
\(725\) 6032.05 0.309000
\(726\) 9975.25 0.509940
\(727\) −2011.78 −0.102631 −0.0513155 0.998682i \(-0.516341\pi\)
−0.0513155 + 0.998682i \(0.516341\pi\)
\(728\) 1958.38 0.0997012
\(729\) −27544.6 −1.39941
\(730\) −72734.1 −3.68768
\(731\) 0 0
\(732\) 29825.8 1.50600
\(733\) 9401.81 0.473757 0.236878 0.971539i \(-0.423876\pi\)
0.236878 + 0.971539i \(0.423876\pi\)
\(734\) 4513.38 0.226965
\(735\) 44490.3 2.23272
\(736\) 14414.0 0.721887
\(737\) 10373.4 0.518464
\(738\) 2945.55 0.146921
\(739\) −18781.6 −0.934900 −0.467450 0.884019i \(-0.654827\pi\)
−0.467450 + 0.884019i \(0.654827\pi\)
\(740\) 30678.9 1.52402
\(741\) −775.200 −0.0384314
\(742\) −1329.41 −0.0657739
\(743\) 32099.5 1.58495 0.792474 0.609906i \(-0.208793\pi\)
0.792474 + 0.609906i \(0.208793\pi\)
\(744\) 3579.48 0.176385
\(745\) −12874.1 −0.633116
\(746\) −23492.4 −1.15298
\(747\) 34573.4 1.69340
\(748\) 28430.4 1.38973
\(749\) −6207.09 −0.302807
\(750\) −42410.2 −2.06480
\(751\) −22982.6 −1.11671 −0.558353 0.829603i \(-0.688567\pi\)
−0.558353 + 0.829603i \(0.688567\pi\)
\(752\) 79.1628 0.00383879
\(753\) 10687.8 0.517245
\(754\) −8072.25 −0.389886
\(755\) 30821.8 1.48572
\(756\) −1682.41 −0.0809372
\(757\) 4670.82 0.224259 0.112129 0.993694i \(-0.464233\pi\)
0.112129 + 0.993694i \(0.464233\pi\)
\(758\) −32318.4 −1.54862
\(759\) 14872.0 0.711226
\(760\) 134.060 0.00639851
\(761\) −14771.7 −0.703643 −0.351821 0.936067i \(-0.614438\pi\)
−0.351821 + 0.936067i \(0.614438\pi\)
\(762\) 32145.1 1.52821
\(763\) 3490.07 0.165595
\(764\) 11473.1 0.543301
\(765\) 82221.0 3.88589
\(766\) 14513.7 0.684598
\(767\) −33235.1 −1.56460
\(768\) −39810.5 −1.87049
\(769\) 15650.6 0.733910 0.366955 0.930239i \(-0.380400\pi\)
0.366955 + 0.930239i \(0.380400\pi\)
\(770\) 11367.1 0.532003
\(771\) 4521.75 0.211215
\(772\) 7343.10 0.342337
\(773\) −31462.6 −1.46395 −0.731975 0.681332i \(-0.761401\pi\)
−0.731975 + 0.681332i \(0.761401\pi\)
\(774\) 0 0
\(775\) 17783.1 0.824242
\(776\) −469.060 −0.0216988
\(777\) −10318.5 −0.476414
\(778\) 47304.6 2.17989
\(779\) −32.3834 −0.00148942
\(780\) 66465.9 3.05111
\(781\) 26269.9 1.20360
\(782\) −31613.3 −1.44564
\(783\) −1447.59 −0.0660700
\(784\) 23112.1 1.05285
\(785\) 24694.7 1.12279
\(786\) −40685.6 −1.84632
\(787\) 14116.6 0.639391 0.319696 0.947520i \(-0.396419\pi\)
0.319696 + 0.947520i \(0.396419\pi\)
\(788\) 10455.7 0.472674
\(789\) 1014.61 0.0457809
\(790\) 56040.0 2.52381
\(791\) −1371.13 −0.0616331
\(792\) −5548.23 −0.248924
\(793\) 41393.0 1.85360
\(794\) −47889.9 −2.14049
\(795\) 9418.46 0.420174
\(796\) 23803.6 1.05992
\(797\) 21531.5 0.956946 0.478473 0.878102i \(-0.341191\pi\)
0.478473 + 0.878102i \(0.341191\pi\)
\(798\) 216.001 0.00958191
\(799\) −147.386 −0.00652586
\(800\) −48377.4 −2.13800
\(801\) 29143.1 1.28555
\(802\) 32661.1 1.43804
\(803\) −33094.0 −1.45437
\(804\) −16901.0 −0.741357
\(805\) −5722.56 −0.250551
\(806\) −23797.8 −1.04000
\(807\) 56302.1 2.45592
\(808\) −6747.39 −0.293778
\(809\) −9241.88 −0.401641 −0.200820 0.979628i \(-0.564361\pi\)
−0.200820 + 0.979628i \(0.564361\pi\)
\(810\) −36001.2 −1.56167
\(811\) 25173.8 1.08998 0.544989 0.838443i \(-0.316534\pi\)
0.544989 + 0.838443i \(0.316534\pi\)
\(812\) 1018.34 0.0440106
\(813\) −32768.3 −1.41357
\(814\) 30831.6 1.32758
\(815\) 52249.4 2.24566
\(816\) 77346.9 3.31824
\(817\) 0 0
\(818\) −34977.8 −1.49507
\(819\) −12345.0 −0.526701
\(820\) 2776.57 0.118246
\(821\) −10451.1 −0.444272 −0.222136 0.975016i \(-0.571303\pi\)
−0.222136 + 0.975016i \(0.571303\pi\)
\(822\) −29721.0 −1.26112
\(823\) 4757.91 0.201519 0.100760 0.994911i \(-0.467873\pi\)
0.100760 + 0.994911i \(0.467873\pi\)
\(824\) 4902.34 0.207259
\(825\) −49914.5 −2.10642
\(826\) 9260.61 0.390094
\(827\) −15905.4 −0.668785 −0.334393 0.942434i \(-0.608531\pi\)
−0.334393 + 0.942434i \(0.608531\pi\)
\(828\) −13380.6 −0.561605
\(829\) −3943.21 −0.165203 −0.0826016 0.996583i \(-0.526323\pi\)
−0.0826016 + 0.996583i \(0.526323\pi\)
\(830\) 71982.7 3.01031
\(831\) −3089.88 −0.128985
\(832\) 23004.1 0.958562
\(833\) −43030.4 −1.78981
\(834\) 17769.8 0.737793
\(835\) 17892.6 0.741556
\(836\) −292.208 −0.0120888
\(837\) −4267.65 −0.176238
\(838\) 33576.7 1.38411
\(839\) −19902.5 −0.818962 −0.409481 0.912319i \(-0.634290\pi\)
−0.409481 + 0.912319i \(0.634290\pi\)
\(840\) 3865.99 0.158797
\(841\) −23512.8 −0.964074
\(842\) −29531.7 −1.20871
\(843\) 40877.3 1.67009
\(844\) 25992.5 1.06007
\(845\) 52406.5 2.13354
\(846\) −137.788 −0.00559957
\(847\) −1746.45 −0.0708485
\(848\) 4892.76 0.198135
\(849\) 35973.1 1.45417
\(850\) 106103. 4.28152
\(851\) −15521.6 −0.625234
\(852\) −42800.6 −1.72104
\(853\) 44071.9 1.76904 0.884521 0.466500i \(-0.154485\pi\)
0.884521 + 0.466500i \(0.154485\pi\)
\(854\) −11533.7 −0.462150
\(855\) −845.067 −0.0338020
\(856\) 6307.82 0.251866
\(857\) −17150.5 −0.683606 −0.341803 0.939772i \(-0.611038\pi\)
−0.341803 + 0.939772i \(0.611038\pi\)
\(858\) 66796.9 2.65782
\(859\) 48887.4 1.94181 0.970907 0.239457i \(-0.0769695\pi\)
0.970907 + 0.239457i \(0.0769695\pi\)
\(860\) 0 0
\(861\) −933.865 −0.0369640
\(862\) −22655.3 −0.895175
\(863\) −38864.3 −1.53297 −0.766486 0.642260i \(-0.777997\pi\)
−0.766486 + 0.642260i \(0.777997\pi\)
\(864\) 11609.8 0.457145
\(865\) 10788.9 0.424087
\(866\) −14946.6 −0.586498
\(867\) −105855. −4.14652
\(868\) 3002.16 0.117396
\(869\) 25498.2 0.995358
\(870\) −15935.2 −0.620981
\(871\) −23455.6 −0.912471
\(872\) −3546.71 −0.137737
\(873\) 2956.79 0.114630
\(874\) 324.921 0.0125751
\(875\) 7425.08 0.286873
\(876\) 53918.9 2.07962
\(877\) −12583.6 −0.484515 −0.242258 0.970212i \(-0.577888\pi\)
−0.242258 + 0.970212i \(0.577888\pi\)
\(878\) −16980.7 −0.652701
\(879\) 4032.49 0.154736
\(880\) −41835.5 −1.60258
\(881\) −27901.3 −1.06699 −0.533495 0.845803i \(-0.679122\pi\)
−0.533495 + 0.845803i \(0.679122\pi\)
\(882\) −40227.9 −1.53577
\(883\) −50179.5 −1.91243 −0.956215 0.292666i \(-0.905458\pi\)
−0.956215 + 0.292666i \(0.905458\pi\)
\(884\) −64285.0 −2.44586
\(885\) −65608.5 −2.49198
\(886\) −29487.1 −1.11810
\(887\) −9359.99 −0.354315 −0.177158 0.984182i \(-0.556690\pi\)
−0.177158 + 0.984182i \(0.556690\pi\)
\(888\) 10485.9 0.396267
\(889\) −5627.90 −0.212321
\(890\) 60676.9 2.28527
\(891\) −16380.6 −0.615903
\(892\) −1653.56 −0.0620688
\(893\) 1.51484 5.67660e−5 0
\(894\) 21079.8 0.788605
\(895\) 42072.5 1.57132
\(896\) 3462.12 0.129086
\(897\) −33627.7 −1.25172
\(898\) 4945.51 0.183779
\(899\) 2583.15 0.0958318
\(900\) 44908.9 1.66329
\(901\) −9109.41 −0.336824
\(902\) 2790.39 0.103004
\(903\) 0 0
\(904\) 1393.38 0.0512646
\(905\) 27763.7 1.01978
\(906\) −50466.9 −1.85061
\(907\) −41146.5 −1.50633 −0.753167 0.657829i \(-0.771475\pi\)
−0.753167 + 0.657829i \(0.771475\pi\)
\(908\) 34627.6 1.26559
\(909\) 42533.2 1.55197
\(910\) −25702.6 −0.936299
\(911\) 16872.6 0.613626 0.306813 0.951770i \(-0.400737\pi\)
0.306813 + 0.951770i \(0.400737\pi\)
\(912\) −794.970 −0.0288641
\(913\) 32752.1 1.18723
\(914\) −21264.6 −0.769554
\(915\) 81712.7 2.95228
\(916\) −6276.92 −0.226414
\(917\) 7123.16 0.256519
\(918\) −25462.9 −0.915469
\(919\) 47698.3 1.71210 0.856052 0.516890i \(-0.172910\pi\)
0.856052 + 0.516890i \(0.172910\pi\)
\(920\) 5815.43 0.208401
\(921\) −9269.46 −0.331639
\(922\) −60844.5 −2.17333
\(923\) −59399.7 −2.11827
\(924\) −8426.61 −0.300016
\(925\) 52094.7 1.85174
\(926\) 59822.3 2.12298
\(927\) −30902.6 −1.09490
\(928\) −7027.23 −0.248578
\(929\) 17674.8 0.624210 0.312105 0.950048i \(-0.398966\pi\)
0.312105 + 0.950048i \(0.398966\pi\)
\(930\) −46978.5 −1.65644
\(931\) 442.266 0.0155689
\(932\) 22849.3 0.803063
\(933\) 20099.1 0.705268
\(934\) 36184.2 1.26765
\(935\) 77889.9 2.72435
\(936\) 12545.3 0.438094
\(937\) 17557.9 0.612158 0.306079 0.952006i \(-0.400983\pi\)
0.306079 + 0.952006i \(0.400983\pi\)
\(938\) 6535.65 0.227502
\(939\) −3290.39 −0.114353
\(940\) −129.883 −0.00450671
\(941\) 42936.2 1.48744 0.743719 0.668492i \(-0.233060\pi\)
0.743719 + 0.668492i \(0.233060\pi\)
\(942\) −40434.5 −1.39854
\(943\) −1404.77 −0.0485108
\(944\) −34082.7 −1.17510
\(945\) −4609.24 −0.158665
\(946\) 0 0
\(947\) −4691.32 −0.160979 −0.0804897 0.996755i \(-0.525648\pi\)
−0.0804897 + 0.996755i \(0.525648\pi\)
\(948\) −41543.3 −1.42327
\(949\) 74830.0 2.55962
\(950\) −1090.52 −0.0372433
\(951\) −10629.0 −0.362426
\(952\) −3739.13 −0.127296
\(953\) −24602.9 −0.836271 −0.418135 0.908385i \(-0.637316\pi\)
−0.418135 + 0.908385i \(0.637316\pi\)
\(954\) −8516.14 −0.289015
\(955\) 31432.5 1.06506
\(956\) −6843.18 −0.231511
\(957\) −7250.51 −0.244907
\(958\) 14762.4 0.497861
\(959\) 5203.50 0.175213
\(960\) 45411.7 1.52673
\(961\) −22175.6 −0.744373
\(962\) −69714.5 −2.33647
\(963\) −39762.3 −1.33055
\(964\) 6127.96 0.204739
\(965\) 20117.7 0.671099
\(966\) 9369.99 0.312086
\(967\) 1371.38 0.0456056 0.0228028 0.999740i \(-0.492741\pi\)
0.0228028 + 0.999740i \(0.492741\pi\)
\(968\) 1774.79 0.0589296
\(969\) 1480.09 0.0490684
\(970\) 6156.12 0.203774
\(971\) 33787.3 1.11667 0.558335 0.829616i \(-0.311440\pi\)
0.558335 + 0.829616i \(0.311440\pi\)
\(972\) 35427.2 1.16906
\(973\) −3111.11 −0.102505
\(974\) 13432.1 0.441880
\(975\) 112863. 3.70720
\(976\) 42448.6 1.39216
\(977\) 9124.24 0.298782 0.149391 0.988778i \(-0.452269\pi\)
0.149391 + 0.988778i \(0.452269\pi\)
\(978\) −85551.9 −2.79719
\(979\) 27608.0 0.901282
\(980\) −37920.0 −1.23603
\(981\) 22357.2 0.727636
\(982\) −73706.7 −2.39519
\(983\) −9300.65 −0.301775 −0.150887 0.988551i \(-0.548213\pi\)
−0.150887 + 0.988551i \(0.548213\pi\)
\(984\) 949.020 0.0307456
\(985\) 28645.0 0.926605
\(986\) 15412.3 0.497797
\(987\) 43.6845 0.00140881
\(988\) 660.721 0.0212756
\(989\) 0 0
\(990\) 72817.1 2.33766
\(991\) 17367.8 0.556717 0.278358 0.960477i \(-0.410210\pi\)
0.278358 + 0.960477i \(0.410210\pi\)
\(992\) −20717.0 −0.663069
\(993\) 19060.2 0.609122
\(994\) 16551.1 0.528138
\(995\) 65214.0 2.07781
\(996\) −53361.9 −1.69763
\(997\) −36720.9 −1.16646 −0.583230 0.812307i \(-0.698211\pi\)
−0.583230 + 0.812307i \(0.698211\pi\)
\(998\) −29158.5 −0.924846
\(999\) −12501.9 −0.395938
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 1849.4.a.d.1.9 10
43.6 even 3 43.4.c.a.36.9 yes 20
43.36 even 3 43.4.c.a.6.9 20
43.42 odd 2 1849.4.a.f.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.c.a.6.9 20 43.36 even 3
43.4.c.a.36.9 yes 20 43.6 even 3
1849.4.a.d.1.9 10 1.1 even 1 trivial
1849.4.a.f.1.2 10 43.42 odd 2