Properties

Label 2013.4.a.b.1.5
Level $2013$
Weight $4$
Character 2013.1
Self dual yes
Analytic conductor $118.771$
Analytic rank $1$
Dimension $36$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,4,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(118.770844842\)
Analytic rank: \(1\)
Dimension: \(36\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.58366 q^{2} -3.00000 q^{3} +13.0099 q^{4} -16.2221 q^{5} +13.7510 q^{6} +3.97810 q^{7} -22.9636 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.58366 q^{2} -3.00000 q^{3} +13.0099 q^{4} -16.2221 q^{5} +13.7510 q^{6} +3.97810 q^{7} -22.9636 q^{8} +9.00000 q^{9} +74.3567 q^{10} -11.0000 q^{11} -39.0297 q^{12} -79.7218 q^{13} -18.2342 q^{14} +48.6664 q^{15} +1.17823 q^{16} -16.3106 q^{17} -41.2529 q^{18} -57.2836 q^{19} -211.048 q^{20} -11.9343 q^{21} +50.4202 q^{22} +129.684 q^{23} +68.8909 q^{24} +138.158 q^{25} +365.417 q^{26} -27.0000 q^{27} +51.7546 q^{28} +211.000 q^{29} -223.070 q^{30} -129.512 q^{31} +178.308 q^{32} +33.0000 q^{33} +74.7622 q^{34} -64.5332 q^{35} +117.089 q^{36} -7.92892 q^{37} +262.568 q^{38} +239.166 q^{39} +372.519 q^{40} -367.734 q^{41} +54.7027 q^{42} +196.158 q^{43} -143.109 q^{44} -145.999 q^{45} -594.428 q^{46} -209.255 q^{47} -3.53470 q^{48} -327.175 q^{49} -633.267 q^{50} +48.9318 q^{51} -1037.17 q^{52} -220.297 q^{53} +123.759 q^{54} +178.443 q^{55} -91.3515 q^{56} +171.851 q^{57} -967.150 q^{58} +637.881 q^{59} +633.145 q^{60} +61.0000 q^{61} +593.638 q^{62} +35.8029 q^{63} -826.731 q^{64} +1293.26 q^{65} -151.261 q^{66} -653.829 q^{67} -212.199 q^{68} -389.053 q^{69} +295.798 q^{70} +129.870 q^{71} -206.673 q^{72} -658.004 q^{73} +36.3434 q^{74} -414.473 q^{75} -745.254 q^{76} -43.7590 q^{77} -1096.25 q^{78} +1058.66 q^{79} -19.1135 q^{80} +81.0000 q^{81} +1685.57 q^{82} +797.979 q^{83} -155.264 q^{84} +264.593 q^{85} -899.121 q^{86} -632.999 q^{87} +252.600 q^{88} +651.829 q^{89} +669.210 q^{90} -317.141 q^{91} +1687.18 q^{92} +388.536 q^{93} +959.152 q^{94} +929.263 q^{95} -534.925 q^{96} +403.693 q^{97} +1499.66 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 2 q^{2} - 108 q^{3} + 118 q^{4} - 5 q^{5} - 6 q^{6} - 63 q^{7} + 3 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 2 q^{2} - 108 q^{3} + 118 q^{4} - 5 q^{5} - 6 q^{6} - 63 q^{7} + 3 q^{8} + 324 q^{9} - 45 q^{10} - 396 q^{11} - 354 q^{12} - 13 q^{13} + 82 q^{14} + 15 q^{15} + 262 q^{16} + 204 q^{17} + 18 q^{18} - 431 q^{19} + 354 q^{20} + 189 q^{21} - 22 q^{22} - 179 q^{23} - 9 q^{24} + 711 q^{25} + 331 q^{26} - 972 q^{27} - 296 q^{28} + 478 q^{29} + 135 q^{30} - 574 q^{31} - 149 q^{32} + 1188 q^{33} + 276 q^{34} - 194 q^{35} + 1062 q^{36} - 12 q^{37} + 325 q^{38} + 39 q^{39} - 185 q^{40} + 900 q^{41} - 246 q^{42} - 1053 q^{43} - 1298 q^{44} - 45 q^{45} - 407 q^{46} - 653 q^{47} - 786 q^{48} + 753 q^{49} - 1520 q^{50} - 612 q^{51} + 60 q^{52} + 735 q^{53} - 54 q^{54} + 55 q^{55} - 809 q^{56} + 1293 q^{57} - 1399 q^{58} - 1127 q^{59} - 1062 q^{60} + 2196 q^{61} - 1795 q^{62} - 567 q^{63} - 2133 q^{64} + 1886 q^{65} + 66 q^{66} - 989 q^{67} + 10 q^{68} + 537 q^{69} - 2130 q^{70} + 61 q^{71} + 27 q^{72} - 1471 q^{73} - 122 q^{74} - 2133 q^{75} - 4064 q^{76} + 693 q^{77} - 993 q^{78} - 1853 q^{79} + 2197 q^{80} + 2916 q^{81} - 2566 q^{82} - 3523 q^{83} + 888 q^{84} - 449 q^{85} - 771 q^{86} - 1434 q^{87} - 33 q^{88} + 2209 q^{89} - 405 q^{90} - 1668 q^{91} - 1999 q^{92} + 1722 q^{93} - 2844 q^{94} + 1220 q^{95} + 447 q^{96} - 3622 q^{97} + 3846 q^{98} - 3564 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.58366 −1.62057 −0.810283 0.586038i \(-0.800687\pi\)
−0.810283 + 0.586038i \(0.800687\pi\)
\(3\) −3.00000 −0.577350
\(4\) 13.0099 1.62624
\(5\) −16.2221 −1.45095 −0.725476 0.688248i \(-0.758380\pi\)
−0.725476 + 0.688248i \(0.758380\pi\)
\(6\) 13.7510 0.935635
\(7\) 3.97810 0.214797 0.107398 0.994216i \(-0.465748\pi\)
0.107398 + 0.994216i \(0.465748\pi\)
\(8\) −22.9636 −1.01486
\(9\) 9.00000 0.333333
\(10\) 74.3567 2.35136
\(11\) −11.0000 −0.301511
\(12\) −39.0297 −0.938908
\(13\) −79.7218 −1.70084 −0.850418 0.526108i \(-0.823651\pi\)
−0.850418 + 0.526108i \(0.823651\pi\)
\(14\) −18.2342 −0.348093
\(15\) 48.6664 0.837707
\(16\) 1.17823 0.0184099
\(17\) −16.3106 −0.232700 −0.116350 0.993208i \(-0.537119\pi\)
−0.116350 + 0.993208i \(0.537119\pi\)
\(18\) −41.2529 −0.540189
\(19\) −57.2836 −0.691672 −0.345836 0.938295i \(-0.612405\pi\)
−0.345836 + 0.938295i \(0.612405\pi\)
\(20\) −211.048 −2.35959
\(21\) −11.9343 −0.124013
\(22\) 50.4202 0.488619
\(23\) 129.684 1.17570 0.587849 0.808971i \(-0.299975\pi\)
0.587849 + 0.808971i \(0.299975\pi\)
\(24\) 68.8909 0.585929
\(25\) 138.158 1.10526
\(26\) 365.417 2.75632
\(27\) −27.0000 −0.192450
\(28\) 51.7546 0.349311
\(29\) 211.000 1.35109 0.675546 0.737318i \(-0.263908\pi\)
0.675546 + 0.737318i \(0.263908\pi\)
\(30\) −223.070 −1.35756
\(31\) −129.512 −0.750356 −0.375178 0.926953i \(-0.622418\pi\)
−0.375178 + 0.926953i \(0.622418\pi\)
\(32\) 178.308 0.985025
\(33\) 33.0000 0.174078
\(34\) 74.7622 0.377106
\(35\) −64.5332 −0.311660
\(36\) 117.089 0.542079
\(37\) −7.92892 −0.0352299 −0.0176149 0.999845i \(-0.505607\pi\)
−0.0176149 + 0.999845i \(0.505607\pi\)
\(38\) 262.568 1.12090
\(39\) 239.166 0.981978
\(40\) 372.519 1.47251
\(41\) −367.734 −1.40074 −0.700371 0.713779i \(-0.746982\pi\)
−0.700371 + 0.713779i \(0.746982\pi\)
\(42\) 54.7027 0.200971
\(43\) 196.158 0.695670 0.347835 0.937556i \(-0.386917\pi\)
0.347835 + 0.937556i \(0.386917\pi\)
\(44\) −143.109 −0.490329
\(45\) −145.999 −0.483651
\(46\) −594.428 −1.90530
\(47\) −209.255 −0.649425 −0.324712 0.945813i \(-0.605267\pi\)
−0.324712 + 0.945813i \(0.605267\pi\)
\(48\) −3.53470 −0.0106290
\(49\) −327.175 −0.953862
\(50\) −633.267 −1.79115
\(51\) 48.9318 0.134350
\(52\) −1037.17 −2.76596
\(53\) −220.297 −0.570947 −0.285473 0.958387i \(-0.592151\pi\)
−0.285473 + 0.958387i \(0.592151\pi\)
\(54\) 123.759 0.311878
\(55\) 178.443 0.437478
\(56\) −91.3515 −0.217989
\(57\) 171.851 0.399337
\(58\) −967.150 −2.18953
\(59\) 637.881 1.40754 0.703772 0.710426i \(-0.251498\pi\)
0.703772 + 0.710426i \(0.251498\pi\)
\(60\) 633.145 1.36231
\(61\) 61.0000 0.128037
\(62\) 593.638 1.21600
\(63\) 35.8029 0.0715990
\(64\) −826.731 −1.61471
\(65\) 1293.26 2.46783
\(66\) −151.261 −0.282104
\(67\) −653.829 −1.19221 −0.596104 0.802907i \(-0.703285\pi\)
−0.596104 + 0.802907i \(0.703285\pi\)
\(68\) −212.199 −0.378426
\(69\) −389.053 −0.678789
\(70\) 295.798 0.505066
\(71\) 129.870 0.217081 0.108540 0.994092i \(-0.465382\pi\)
0.108540 + 0.994092i \(0.465382\pi\)
\(72\) −206.673 −0.338286
\(73\) −658.004 −1.05498 −0.527490 0.849561i \(-0.676867\pi\)
−0.527490 + 0.849561i \(0.676867\pi\)
\(74\) 36.3434 0.0570924
\(75\) −414.473 −0.638123
\(76\) −745.254 −1.12482
\(77\) −43.7590 −0.0647637
\(78\) −1096.25 −1.59136
\(79\) 1058.66 1.50771 0.753853 0.657043i \(-0.228193\pi\)
0.753853 + 0.657043i \(0.228193\pi\)
\(80\) −19.1135 −0.0267119
\(81\) 81.0000 0.111111
\(82\) 1685.57 2.27000
\(83\) 797.979 1.05530 0.527648 0.849463i \(-0.323074\pi\)
0.527648 + 0.849463i \(0.323074\pi\)
\(84\) −155.264 −0.201675
\(85\) 264.593 0.337637
\(86\) −899.121 −1.12738
\(87\) −632.999 −0.780053
\(88\) 252.600 0.305992
\(89\) 651.829 0.776334 0.388167 0.921589i \(-0.373108\pi\)
0.388167 + 0.921589i \(0.373108\pi\)
\(90\) 669.210 0.783788
\(91\) −317.141 −0.365334
\(92\) 1687.18 1.91196
\(93\) 388.536 0.433218
\(94\) 959.152 1.05244
\(95\) 929.263 1.00358
\(96\) −534.925 −0.568704
\(97\) 403.693 0.422565 0.211283 0.977425i \(-0.432236\pi\)
0.211283 + 0.977425i \(0.432236\pi\)
\(98\) 1499.66 1.54580
\(99\) −99.0000 −0.100504
\(100\) 1797.42 1.79742
\(101\) −509.163 −0.501620 −0.250810 0.968036i \(-0.580697\pi\)
−0.250810 + 0.968036i \(0.580697\pi\)
\(102\) −224.287 −0.217722
\(103\) 652.898 0.624582 0.312291 0.949986i \(-0.398904\pi\)
0.312291 + 0.949986i \(0.398904\pi\)
\(104\) 1830.70 1.72611
\(105\) 193.600 0.179937
\(106\) 1009.77 0.925257
\(107\) 248.959 0.224932 0.112466 0.993656i \(-0.464125\pi\)
0.112466 + 0.993656i \(0.464125\pi\)
\(108\) −351.267 −0.312969
\(109\) 236.771 0.208060 0.104030 0.994574i \(-0.466826\pi\)
0.104030 + 0.994574i \(0.466826\pi\)
\(110\) −817.923 −0.708963
\(111\) 23.7868 0.0203400
\(112\) 4.68713 0.00395439
\(113\) 1650.74 1.37423 0.687116 0.726548i \(-0.258876\pi\)
0.687116 + 0.726548i \(0.258876\pi\)
\(114\) −787.705 −0.647152
\(115\) −2103.76 −1.70588
\(116\) 2745.08 2.19720
\(117\) −717.497 −0.566945
\(118\) −2923.83 −2.28102
\(119\) −64.8852 −0.0499833
\(120\) −1117.56 −0.850155
\(121\) 121.000 0.0909091
\(122\) −279.603 −0.207492
\(123\) 1103.20 0.808718
\(124\) −1684.94 −1.22026
\(125\) −213.444 −0.152728
\(126\) −164.108 −0.116031
\(127\) −478.717 −0.334483 −0.167241 0.985916i \(-0.553486\pi\)
−0.167241 + 0.985916i \(0.553486\pi\)
\(128\) 2362.98 1.63172
\(129\) −588.474 −0.401646
\(130\) −5927.85 −3.99928
\(131\) −50.5922 −0.0337425 −0.0168712 0.999858i \(-0.505371\pi\)
−0.0168712 + 0.999858i \(0.505371\pi\)
\(132\) 429.327 0.283092
\(133\) −227.880 −0.148569
\(134\) 2996.93 1.93205
\(135\) 437.998 0.279236
\(136\) 374.551 0.236158
\(137\) 933.261 0.581999 0.291000 0.956723i \(-0.406012\pi\)
0.291000 + 0.956723i \(0.406012\pi\)
\(138\) 1783.28 1.10002
\(139\) 86.8769 0.0530130 0.0265065 0.999649i \(-0.491562\pi\)
0.0265065 + 0.999649i \(0.491562\pi\)
\(140\) −839.570 −0.506833
\(141\) 627.765 0.374945
\(142\) −595.279 −0.351794
\(143\) 876.940 0.512821
\(144\) 10.6041 0.00613664
\(145\) −3422.87 −1.96037
\(146\) 3016.06 1.70967
\(147\) 981.524 0.550713
\(148\) −103.154 −0.0572922
\(149\) 1653.80 0.909293 0.454647 0.890672i \(-0.349766\pi\)
0.454647 + 0.890672i \(0.349766\pi\)
\(150\) 1899.80 1.03412
\(151\) 1556.69 0.838950 0.419475 0.907767i \(-0.362214\pi\)
0.419475 + 0.907767i \(0.362214\pi\)
\(152\) 1315.44 0.701950
\(153\) −146.796 −0.0775668
\(154\) 200.576 0.104954
\(155\) 2100.96 1.08873
\(156\) 3111.52 1.59693
\(157\) −357.842 −0.181904 −0.0909518 0.995855i \(-0.528991\pi\)
−0.0909518 + 0.995855i \(0.528991\pi\)
\(158\) −4852.54 −2.44334
\(159\) 660.892 0.329636
\(160\) −2892.54 −1.42922
\(161\) 515.896 0.252536
\(162\) −371.276 −0.180063
\(163\) 1101.66 0.529377 0.264688 0.964334i \(-0.414731\pi\)
0.264688 + 0.964334i \(0.414731\pi\)
\(164\) −4784.18 −2.27794
\(165\) −535.330 −0.252578
\(166\) −3657.66 −1.71018
\(167\) 2499.39 1.15813 0.579067 0.815280i \(-0.303417\pi\)
0.579067 + 0.815280i \(0.303417\pi\)
\(168\) 274.055 0.125856
\(169\) 4158.57 1.89284
\(170\) −1212.80 −0.547163
\(171\) −515.553 −0.230557
\(172\) 2552.00 1.13133
\(173\) −2049.10 −0.900523 −0.450261 0.892897i \(-0.648669\pi\)
−0.450261 + 0.892897i \(0.648669\pi\)
\(174\) 2901.45 1.26413
\(175\) 549.604 0.237407
\(176\) −12.9606 −0.00555080
\(177\) −1913.64 −0.812646
\(178\) −2987.76 −1.25810
\(179\) 1188.44 0.496246 0.248123 0.968728i \(-0.420186\pi\)
0.248123 + 0.968728i \(0.420186\pi\)
\(180\) −1899.43 −0.786530
\(181\) −1331.90 −0.546959 −0.273479 0.961878i \(-0.588175\pi\)
−0.273479 + 0.961878i \(0.588175\pi\)
\(182\) 1453.67 0.592049
\(183\) −183.000 −0.0739221
\(184\) −2978.02 −1.19317
\(185\) 128.624 0.0511169
\(186\) −1780.91 −0.702059
\(187\) 179.417 0.0701618
\(188\) −2722.38 −1.05612
\(189\) −107.409 −0.0413377
\(190\) −4259.42 −1.62637
\(191\) −99.9724 −0.0378730 −0.0189365 0.999821i \(-0.506028\pi\)
−0.0189365 + 0.999821i \(0.506028\pi\)
\(192\) 2480.19 0.932252
\(193\) 407.993 0.152166 0.0760828 0.997102i \(-0.475759\pi\)
0.0760828 + 0.997102i \(0.475759\pi\)
\(194\) −1850.39 −0.684796
\(195\) −3879.77 −1.42480
\(196\) −4256.51 −1.55121
\(197\) −1123.81 −0.406439 −0.203219 0.979133i \(-0.565140\pi\)
−0.203219 + 0.979133i \(0.565140\pi\)
\(198\) 453.782 0.162873
\(199\) 3456.17 1.23116 0.615582 0.788073i \(-0.288921\pi\)
0.615582 + 0.788073i \(0.288921\pi\)
\(200\) −3172.60 −1.12168
\(201\) 1961.49 0.688322
\(202\) 2333.83 0.812908
\(203\) 839.377 0.290210
\(204\) 636.598 0.218484
\(205\) 5965.43 2.03241
\(206\) −2992.66 −1.01218
\(207\) 1167.16 0.391899
\(208\) −93.9310 −0.0313122
\(209\) 630.120 0.208547
\(210\) −887.394 −0.291600
\(211\) 1623.45 0.529683 0.264841 0.964292i \(-0.414680\pi\)
0.264841 + 0.964292i \(0.414680\pi\)
\(212\) −2866.05 −0.928495
\(213\) −389.610 −0.125332
\(214\) −1141.14 −0.364518
\(215\) −3182.10 −1.00938
\(216\) 620.018 0.195310
\(217\) −515.211 −0.161174
\(218\) −1085.28 −0.337176
\(219\) 1974.01 0.609093
\(220\) 2321.53 0.711444
\(221\) 1300.31 0.395785
\(222\) −109.030 −0.0329623
\(223\) −1227.51 −0.368611 −0.184306 0.982869i \(-0.559004\pi\)
−0.184306 + 0.982869i \(0.559004\pi\)
\(224\) 709.328 0.211580
\(225\) 1243.42 0.368420
\(226\) −7566.41 −2.22704
\(227\) −657.451 −0.192232 −0.0961158 0.995370i \(-0.530642\pi\)
−0.0961158 + 0.995370i \(0.530642\pi\)
\(228\) 2235.76 0.649417
\(229\) −601.476 −0.173566 −0.0867831 0.996227i \(-0.527659\pi\)
−0.0867831 + 0.996227i \(0.527659\pi\)
\(230\) 9642.89 2.76449
\(231\) 131.277 0.0373913
\(232\) −4845.32 −1.37117
\(233\) 4374.82 1.23006 0.615030 0.788504i \(-0.289144\pi\)
0.615030 + 0.788504i \(0.289144\pi\)
\(234\) 3288.76 0.918772
\(235\) 3394.56 0.942284
\(236\) 8298.77 2.28900
\(237\) −3175.99 −0.870475
\(238\) 297.411 0.0810013
\(239\) 3599.93 0.974310 0.487155 0.873316i \(-0.338035\pi\)
0.487155 + 0.873316i \(0.338035\pi\)
\(240\) 57.3404 0.0154221
\(241\) 4012.26 1.07242 0.536209 0.844085i \(-0.319856\pi\)
0.536209 + 0.844085i \(0.319856\pi\)
\(242\) −554.622 −0.147324
\(243\) −243.000 −0.0641500
\(244\) 793.604 0.208218
\(245\) 5307.47 1.38401
\(246\) −5056.70 −1.31058
\(247\) 4566.76 1.17642
\(248\) 2974.07 0.761505
\(249\) −2393.94 −0.609275
\(250\) 978.355 0.247506
\(251\) 726.525 0.182701 0.0913503 0.995819i \(-0.470882\pi\)
0.0913503 + 0.995819i \(0.470882\pi\)
\(252\) 465.791 0.116437
\(253\) −1426.53 −0.354486
\(254\) 2194.28 0.542052
\(255\) −793.779 −0.194935
\(256\) −4217.24 −1.02960
\(257\) −5727.50 −1.39016 −0.695081 0.718932i \(-0.744631\pi\)
−0.695081 + 0.718932i \(0.744631\pi\)
\(258\) 2697.36 0.650893
\(259\) −31.5420 −0.00756727
\(260\) 16825.2 4.01328
\(261\) 1899.00 0.450364
\(262\) 231.897 0.0546819
\(263\) −3815.81 −0.894651 −0.447325 0.894371i \(-0.647623\pi\)
−0.447325 + 0.894371i \(0.647623\pi\)
\(264\) −757.800 −0.176664
\(265\) 3573.69 0.828416
\(266\) 1044.52 0.240766
\(267\) −1955.49 −0.448217
\(268\) −8506.25 −1.93881
\(269\) 1031.45 0.233786 0.116893 0.993144i \(-0.462706\pi\)
0.116893 + 0.993144i \(0.462706\pi\)
\(270\) −2007.63 −0.452520
\(271\) −1355.75 −0.303896 −0.151948 0.988389i \(-0.548555\pi\)
−0.151948 + 0.988389i \(0.548555\pi\)
\(272\) −19.2177 −0.00428399
\(273\) 951.423 0.210926
\(274\) −4277.75 −0.943169
\(275\) −1519.73 −0.333249
\(276\) −5061.54 −1.10387
\(277\) 793.175 0.172048 0.0860239 0.996293i \(-0.472584\pi\)
0.0860239 + 0.996293i \(0.472584\pi\)
\(278\) −398.214 −0.0859110
\(279\) −1165.61 −0.250119
\(280\) 1481.92 0.316291
\(281\) −2136.81 −0.453634 −0.226817 0.973937i \(-0.572832\pi\)
−0.226817 + 0.973937i \(0.572832\pi\)
\(282\) −2877.46 −0.607624
\(283\) 727.204 0.152748 0.0763742 0.997079i \(-0.475666\pi\)
0.0763742 + 0.997079i \(0.475666\pi\)
\(284\) 1689.60 0.353025
\(285\) −2787.79 −0.579419
\(286\) −4019.59 −0.831061
\(287\) −1462.88 −0.300875
\(288\) 1604.78 0.328342
\(289\) −4646.96 −0.945851
\(290\) 15689.2 3.17691
\(291\) −1211.08 −0.243968
\(292\) −8560.57 −1.71565
\(293\) −7465.12 −1.48845 −0.744227 0.667926i \(-0.767182\pi\)
−0.744227 + 0.667926i \(0.767182\pi\)
\(294\) −4498.97 −0.892467
\(295\) −10347.8 −2.04228
\(296\) 182.077 0.0357534
\(297\) 297.000 0.0580259
\(298\) −7580.46 −1.47357
\(299\) −10338.7 −1.99967
\(300\) −5392.25 −1.03774
\(301\) 780.336 0.149428
\(302\) −7135.32 −1.35957
\(303\) 1527.49 0.289610
\(304\) −67.4936 −0.0127336
\(305\) −989.550 −0.185775
\(306\) 672.860 0.125702
\(307\) −7738.58 −1.43864 −0.719322 0.694677i \(-0.755547\pi\)
−0.719322 + 0.694677i \(0.755547\pi\)
\(308\) −569.301 −0.105321
\(309\) −1958.69 −0.360603
\(310\) −9630.08 −1.76436
\(311\) 6943.90 1.26609 0.633043 0.774117i \(-0.281806\pi\)
0.633043 + 0.774117i \(0.281806\pi\)
\(312\) −5492.11 −0.996569
\(313\) 7632.13 1.37825 0.689127 0.724641i \(-0.257994\pi\)
0.689127 + 0.724641i \(0.257994\pi\)
\(314\) 1640.22 0.294787
\(315\) −580.799 −0.103887
\(316\) 13773.1 2.45189
\(317\) −4942.99 −0.875791 −0.437896 0.899026i \(-0.644276\pi\)
−0.437896 + 0.899026i \(0.644276\pi\)
\(318\) −3029.30 −0.534197
\(319\) −2321.00 −0.407369
\(320\) 13411.3 2.34286
\(321\) −746.876 −0.129865
\(322\) −2364.69 −0.409252
\(323\) 934.331 0.160952
\(324\) 1053.80 0.180693
\(325\) −11014.2 −1.87987
\(326\) −5049.61 −0.857890
\(327\) −710.314 −0.120124
\(328\) 8444.51 1.42156
\(329\) −832.436 −0.139494
\(330\) 2453.77 0.409320
\(331\) 6861.55 1.13941 0.569706 0.821849i \(-0.307057\pi\)
0.569706 + 0.821849i \(0.307057\pi\)
\(332\) 10381.6 1.71616
\(333\) −71.3603 −0.0117433
\(334\) −11456.3 −1.87683
\(335\) 10606.5 1.72984
\(336\) −14.0614 −0.00228307
\(337\) −2231.10 −0.360641 −0.180320 0.983608i \(-0.557713\pi\)
−0.180320 + 0.983608i \(0.557713\pi\)
\(338\) −19061.5 −3.06747
\(339\) −4952.21 −0.793413
\(340\) 3442.33 0.549078
\(341\) 1424.63 0.226241
\(342\) 2363.12 0.373634
\(343\) −2666.02 −0.419684
\(344\) −4504.50 −0.706007
\(345\) 6311.27 0.984890
\(346\) 9392.38 1.45936
\(347\) −2911.74 −0.450463 −0.225231 0.974305i \(-0.572314\pi\)
−0.225231 + 0.974305i \(0.572314\pi\)
\(348\) −8235.25 −1.26855
\(349\) −6886.68 −1.05626 −0.528131 0.849163i \(-0.677107\pi\)
−0.528131 + 0.849163i \(0.677107\pi\)
\(350\) −2519.20 −0.384733
\(351\) 2152.49 0.327326
\(352\) −1961.39 −0.296996
\(353\) 6385.52 0.962796 0.481398 0.876502i \(-0.340129\pi\)
0.481398 + 0.876502i \(0.340129\pi\)
\(354\) 8771.49 1.31695
\(355\) −2106.77 −0.314974
\(356\) 8480.22 1.26250
\(357\) 194.656 0.0288579
\(358\) −5447.40 −0.804201
\(359\) −5381.57 −0.791166 −0.395583 0.918430i \(-0.629457\pi\)
−0.395583 + 0.918430i \(0.629457\pi\)
\(360\) 3352.67 0.490837
\(361\) −3577.58 −0.521590
\(362\) 6104.98 0.886383
\(363\) −363.000 −0.0524864
\(364\) −4125.97 −0.594120
\(365\) 10674.2 1.53073
\(366\) 838.809 0.119796
\(367\) 8673.70 1.23369 0.616844 0.787086i \(-0.288411\pi\)
0.616844 + 0.787086i \(0.288411\pi\)
\(368\) 152.798 0.0216445
\(369\) −3309.61 −0.466914
\(370\) −589.568 −0.0828383
\(371\) −876.364 −0.122638
\(372\) 5054.81 0.704515
\(373\) 7643.56 1.06104 0.530521 0.847672i \(-0.321996\pi\)
0.530521 + 0.847672i \(0.321996\pi\)
\(374\) −822.385 −0.113702
\(375\) 640.333 0.0881777
\(376\) 4805.25 0.659074
\(377\) −16821.3 −2.29798
\(378\) 492.324 0.0669905
\(379\) 4039.97 0.547544 0.273772 0.961795i \(-0.411729\pi\)
0.273772 + 0.961795i \(0.411729\pi\)
\(380\) 12089.6 1.63206
\(381\) 1436.15 0.193114
\(382\) 458.239 0.0613758
\(383\) −10864.5 −1.44948 −0.724741 0.689021i \(-0.758041\pi\)
−0.724741 + 0.689021i \(0.758041\pi\)
\(384\) −7088.94 −0.942073
\(385\) 709.865 0.0939690
\(386\) −1870.10 −0.246595
\(387\) 1765.42 0.231890
\(388\) 5252.01 0.687192
\(389\) 2682.34 0.349614 0.174807 0.984603i \(-0.444070\pi\)
0.174807 + 0.984603i \(0.444070\pi\)
\(390\) 17783.5 2.30899
\(391\) −2115.23 −0.273585
\(392\) 7513.12 0.968036
\(393\) 151.777 0.0194812
\(394\) 5151.18 0.658661
\(395\) −17173.8 −2.18761
\(396\) −1287.98 −0.163443
\(397\) −9484.81 −1.19907 −0.599533 0.800350i \(-0.704647\pi\)
−0.599533 + 0.800350i \(0.704647\pi\)
\(398\) −15841.9 −1.99518
\(399\) 683.639 0.0857764
\(400\) 162.782 0.0203478
\(401\) −8668.93 −1.07957 −0.539783 0.841805i \(-0.681494\pi\)
−0.539783 + 0.841805i \(0.681494\pi\)
\(402\) −8990.78 −1.11547
\(403\) 10324.9 1.27623
\(404\) −6624.16 −0.815753
\(405\) −1313.99 −0.161217
\(406\) −3847.41 −0.470305
\(407\) 87.2181 0.0106222
\(408\) −1123.65 −0.136346
\(409\) 11225.7 1.35715 0.678576 0.734530i \(-0.262598\pi\)
0.678576 + 0.734530i \(0.262598\pi\)
\(410\) −27343.5 −3.29365
\(411\) −2799.78 −0.336017
\(412\) 8494.14 1.01572
\(413\) 2537.55 0.302336
\(414\) −5349.85 −0.635099
\(415\) −12944.9 −1.53118
\(416\) −14215.1 −1.67536
\(417\) −260.631 −0.0306070
\(418\) −2888.25 −0.337964
\(419\) 14523.3 1.69334 0.846671 0.532117i \(-0.178603\pi\)
0.846671 + 0.532117i \(0.178603\pi\)
\(420\) 2518.71 0.292620
\(421\) −16735.5 −1.93739 −0.968694 0.248259i \(-0.920142\pi\)
−0.968694 + 0.248259i \(0.920142\pi\)
\(422\) −7441.34 −0.858386
\(423\) −1883.29 −0.216475
\(424\) 5058.83 0.579430
\(425\) −2253.44 −0.257194
\(426\) 1785.84 0.203108
\(427\) 242.664 0.0275019
\(428\) 3238.93 0.365793
\(429\) −2630.82 −0.296077
\(430\) 14585.7 1.63577
\(431\) 17367.9 1.94102 0.970512 0.241055i \(-0.0774933\pi\)
0.970512 + 0.241055i \(0.0774933\pi\)
\(432\) −31.8123 −0.00354299
\(433\) −15888.1 −1.76336 −0.881679 0.471850i \(-0.843586\pi\)
−0.881679 + 0.471850i \(0.843586\pi\)
\(434\) 2361.55 0.261193
\(435\) 10268.6 1.13182
\(436\) 3080.37 0.338355
\(437\) −7428.79 −0.813197
\(438\) −9048.19 −0.987076
\(439\) −8211.06 −0.892693 −0.446347 0.894860i \(-0.647275\pi\)
−0.446347 + 0.894860i \(0.647275\pi\)
\(440\) −4097.71 −0.443979
\(441\) −2944.57 −0.317954
\(442\) −5960.18 −0.641396
\(443\) −7167.93 −0.768756 −0.384378 0.923176i \(-0.625584\pi\)
−0.384378 + 0.923176i \(0.625584\pi\)
\(444\) 309.463 0.0330776
\(445\) −10574.1 −1.12642
\(446\) 5626.49 0.597359
\(447\) −4961.40 −0.524981
\(448\) −3288.81 −0.346834
\(449\) −12378.3 −1.30104 −0.650522 0.759487i \(-0.725450\pi\)
−0.650522 + 0.759487i \(0.725450\pi\)
\(450\) −5699.40 −0.597050
\(451\) 4045.07 0.422339
\(452\) 21475.9 2.23483
\(453\) −4670.06 −0.484368
\(454\) 3013.53 0.311524
\(455\) 5144.70 0.530082
\(456\) −3946.32 −0.405271
\(457\) −6377.38 −0.652782 −0.326391 0.945235i \(-0.605833\pi\)
−0.326391 + 0.945235i \(0.605833\pi\)
\(458\) 2756.96 0.281276
\(459\) 440.387 0.0447832
\(460\) −27369.6 −2.77417
\(461\) −4951.11 −0.500209 −0.250105 0.968219i \(-0.580465\pi\)
−0.250105 + 0.968219i \(0.580465\pi\)
\(462\) −601.729 −0.0605952
\(463\) 12712.0 1.27597 0.637987 0.770047i \(-0.279767\pi\)
0.637987 + 0.770047i \(0.279767\pi\)
\(464\) 248.607 0.0248735
\(465\) −6302.88 −0.628579
\(466\) −20052.7 −1.99339
\(467\) 4980.38 0.493500 0.246750 0.969079i \(-0.420637\pi\)
0.246750 + 0.969079i \(0.420637\pi\)
\(468\) −9334.56 −0.921987
\(469\) −2600.99 −0.256083
\(470\) −15559.5 −1.52703
\(471\) 1073.52 0.105022
\(472\) −14648.1 −1.42846
\(473\) −2157.74 −0.209753
\(474\) 14557.6 1.41066
\(475\) −7914.17 −0.764478
\(476\) −844.149 −0.0812847
\(477\) −1982.68 −0.190316
\(478\) −16500.8 −1.57893
\(479\) 1408.45 0.134350 0.0671752 0.997741i \(-0.478601\pi\)
0.0671752 + 0.997741i \(0.478601\pi\)
\(480\) 8677.63 0.825162
\(481\) 632.108 0.0599202
\(482\) −18390.8 −1.73792
\(483\) −1547.69 −0.145802
\(484\) 1574.20 0.147840
\(485\) −6548.77 −0.613122
\(486\) 1113.83 0.103959
\(487\) −17539.5 −1.63202 −0.816008 0.578041i \(-0.803817\pi\)
−0.816008 + 0.578041i \(0.803817\pi\)
\(488\) −1400.78 −0.129939
\(489\) −3304.97 −0.305636
\(490\) −24327.6 −2.24288
\(491\) −17628.3 −1.62027 −0.810137 0.586240i \(-0.800608\pi\)
−0.810137 + 0.586240i \(0.800608\pi\)
\(492\) 14352.5 1.31517
\(493\) −3441.53 −0.314399
\(494\) −20932.4 −1.90647
\(495\) 1605.99 0.145826
\(496\) −152.595 −0.0138140
\(497\) 516.635 0.0466283
\(498\) 10973.0 0.987371
\(499\) 8332.95 0.747563 0.373782 0.927517i \(-0.378061\pi\)
0.373782 + 0.927517i \(0.378061\pi\)
\(500\) −2776.89 −0.248372
\(501\) −7498.16 −0.668649
\(502\) −3330.14 −0.296078
\(503\) 10603.3 0.939919 0.469959 0.882688i \(-0.344269\pi\)
0.469959 + 0.882688i \(0.344269\pi\)
\(504\) −822.164 −0.0726629
\(505\) 8259.71 0.727826
\(506\) 6538.71 0.574468
\(507\) −12475.7 −1.09283
\(508\) −6228.06 −0.543948
\(509\) 856.192 0.0745580 0.0372790 0.999305i \(-0.488131\pi\)
0.0372790 + 0.999305i \(0.488131\pi\)
\(510\) 3638.41 0.315905
\(511\) −2617.60 −0.226607
\(512\) 426.542 0.0368177
\(513\) 1546.66 0.133112
\(514\) 26252.9 2.25285
\(515\) −10591.4 −0.906238
\(516\) −7655.99 −0.653171
\(517\) 2301.80 0.195809
\(518\) 144.578 0.0122633
\(519\) 6147.31 0.519917
\(520\) −29697.9 −2.50450
\(521\) −4899.62 −0.412008 −0.206004 0.978551i \(-0.566046\pi\)
−0.206004 + 0.978551i \(0.566046\pi\)
\(522\) −8704.35 −0.729845
\(523\) −2008.61 −0.167936 −0.0839681 0.996468i \(-0.526759\pi\)
−0.0839681 + 0.996468i \(0.526759\pi\)
\(524\) −658.199 −0.0548732
\(525\) −1648.81 −0.137067
\(526\) 17490.4 1.44984
\(527\) 2112.42 0.174608
\(528\) 38.8817 0.00320475
\(529\) 4651.01 0.382264
\(530\) −16380.6 −1.34250
\(531\) 5740.93 0.469181
\(532\) −2964.69 −0.241609
\(533\) 29316.4 2.38243
\(534\) 8963.28 0.726365
\(535\) −4038.64 −0.326366
\(536\) 15014.3 1.20992
\(537\) −3565.32 −0.286508
\(538\) −4727.81 −0.378867
\(539\) 3598.92 0.287600
\(540\) 5698.30 0.454104
\(541\) −8895.89 −0.706958 −0.353479 0.935442i \(-0.615001\pi\)
−0.353479 + 0.935442i \(0.615001\pi\)
\(542\) 6214.27 0.492483
\(543\) 3995.71 0.315787
\(544\) −2908.32 −0.229215
\(545\) −3840.94 −0.301885
\(546\) −4361.00 −0.341819
\(547\) 1736.07 0.135702 0.0678512 0.997695i \(-0.478386\pi\)
0.0678512 + 0.997695i \(0.478386\pi\)
\(548\) 12141.6 0.946469
\(549\) 549.000 0.0426790
\(550\) 6965.93 0.540052
\(551\) −12086.8 −0.934512
\(552\) 8934.07 0.688875
\(553\) 4211.46 0.323851
\(554\) −3635.64 −0.278815
\(555\) −385.872 −0.0295123
\(556\) 1130.26 0.0862116
\(557\) −6202.01 −0.471791 −0.235896 0.971778i \(-0.575802\pi\)
−0.235896 + 0.971778i \(0.575802\pi\)
\(558\) 5342.74 0.405334
\(559\) −15638.1 −1.18322
\(560\) −76.0352 −0.00573763
\(561\) −538.250 −0.0405079
\(562\) 9794.38 0.735144
\(563\) 11792.8 0.882785 0.441393 0.897314i \(-0.354485\pi\)
0.441393 + 0.897314i \(0.354485\pi\)
\(564\) 8167.15 0.609750
\(565\) −26778.5 −1.99394
\(566\) −3333.25 −0.247539
\(567\) 322.226 0.0238663
\(568\) −2982.29 −0.220306
\(569\) −720.606 −0.0530920 −0.0265460 0.999648i \(-0.508451\pi\)
−0.0265460 + 0.999648i \(0.508451\pi\)
\(570\) 12778.3 0.938987
\(571\) 6777.60 0.496731 0.248366 0.968666i \(-0.420107\pi\)
0.248366 + 0.968666i \(0.420107\pi\)
\(572\) 11408.9 0.833969
\(573\) 299.917 0.0218660
\(574\) 6705.34 0.487588
\(575\) 17916.9 1.29945
\(576\) −7440.57 −0.538236
\(577\) −4736.39 −0.341730 −0.170865 0.985294i \(-0.554656\pi\)
−0.170865 + 0.985294i \(0.554656\pi\)
\(578\) 21300.1 1.53281
\(579\) −1223.98 −0.0878528
\(580\) −44531.1 −3.18802
\(581\) 3174.43 0.226674
\(582\) 5551.17 0.395367
\(583\) 2423.27 0.172147
\(584\) 15110.2 1.07066
\(585\) 11639.3 0.822610
\(586\) 34217.6 2.41214
\(587\) −17899.6 −1.25860 −0.629299 0.777163i \(-0.716658\pi\)
−0.629299 + 0.777163i \(0.716658\pi\)
\(588\) 12769.5 0.895589
\(589\) 7418.92 0.519000
\(590\) 47430.7 3.30965
\(591\) 3371.44 0.234658
\(592\) −9.34212 −0.000648579 0
\(593\) 8710.21 0.603179 0.301590 0.953438i \(-0.402483\pi\)
0.301590 + 0.953438i \(0.402483\pi\)
\(594\) −1361.35 −0.0940348
\(595\) 1052.58 0.0725234
\(596\) 21515.8 1.47873
\(597\) −10368.5 −0.710812
\(598\) 47388.9 3.24060
\(599\) 8887.52 0.606234 0.303117 0.952953i \(-0.401973\pi\)
0.303117 + 0.952953i \(0.401973\pi\)
\(600\) 9517.80 0.647604
\(601\) −193.942 −0.0131632 −0.00658158 0.999978i \(-0.502095\pi\)
−0.00658158 + 0.999978i \(0.502095\pi\)
\(602\) −3576.79 −0.242158
\(603\) −5884.46 −0.397403
\(604\) 20252.3 1.36433
\(605\) −1962.88 −0.131905
\(606\) −7001.48 −0.469333
\(607\) −2042.05 −0.136547 −0.0682736 0.997667i \(-0.521749\pi\)
−0.0682736 + 0.997667i \(0.521749\pi\)
\(608\) −10214.2 −0.681314
\(609\) −2518.13 −0.167553
\(610\) 4535.76 0.301061
\(611\) 16682.2 1.10456
\(612\) −1909.79 −0.126142
\(613\) −19534.6 −1.28710 −0.643552 0.765402i \(-0.722540\pi\)
−0.643552 + 0.765402i \(0.722540\pi\)
\(614\) 35471.0 2.33142
\(615\) −17896.3 −1.17341
\(616\) 1004.87 0.0657260
\(617\) −25909.6 −1.69057 −0.845286 0.534315i \(-0.820570\pi\)
−0.845286 + 0.534315i \(0.820570\pi\)
\(618\) 8977.98 0.584381
\(619\) 5678.56 0.368724 0.184362 0.982858i \(-0.440978\pi\)
0.184362 + 0.982858i \(0.440978\pi\)
\(620\) 27333.3 1.77053
\(621\) −3501.48 −0.226263
\(622\) −31828.4 −2.05178
\(623\) 2593.04 0.166754
\(624\) 281.793 0.0180781
\(625\) −13807.2 −0.883659
\(626\) −34983.1 −2.23355
\(627\) −1890.36 −0.120405
\(628\) −4655.48 −0.295818
\(629\) 129.325 0.00819801
\(630\) 2662.18 0.168355
\(631\) 12122.5 0.764800 0.382400 0.923997i \(-0.375098\pi\)
0.382400 + 0.923997i \(0.375098\pi\)
\(632\) −24310.7 −1.53011
\(633\) −4870.36 −0.305812
\(634\) 22657.0 1.41928
\(635\) 7765.82 0.485318
\(636\) 8598.14 0.536067
\(637\) 26083.0 1.62236
\(638\) 10638.6 0.660170
\(639\) 1168.83 0.0723602
\(640\) −38332.6 −2.36754
\(641\) 26350.5 1.62369 0.811843 0.583876i \(-0.198465\pi\)
0.811843 + 0.583876i \(0.198465\pi\)
\(642\) 3423.42 0.210454
\(643\) 11612.0 0.712182 0.356091 0.934451i \(-0.384109\pi\)
0.356091 + 0.934451i \(0.384109\pi\)
\(644\) 6711.76 0.410684
\(645\) 9546.31 0.582768
\(646\) −4282.65 −0.260834
\(647\) −12052.1 −0.732327 −0.366164 0.930551i \(-0.619329\pi\)
−0.366164 + 0.930551i \(0.619329\pi\)
\(648\) −1860.05 −0.112762
\(649\) −7016.70 −0.424390
\(650\) 50485.2 3.04645
\(651\) 1545.63 0.0930539
\(652\) 14332.4 0.860892
\(653\) −27660.9 −1.65766 −0.828831 0.559499i \(-0.810994\pi\)
−0.828831 + 0.559499i \(0.810994\pi\)
\(654\) 3255.83 0.194668
\(655\) 820.713 0.0489587
\(656\) −433.277 −0.0257875
\(657\) −5922.04 −0.351660
\(658\) 3815.60 0.226060
\(659\) 33255.7 1.96580 0.982898 0.184149i \(-0.0589530\pi\)
0.982898 + 0.184149i \(0.0589530\pi\)
\(660\) −6964.59 −0.410752
\(661\) 8330.75 0.490210 0.245105 0.969497i \(-0.421178\pi\)
0.245105 + 0.969497i \(0.421178\pi\)
\(662\) −31451.0 −1.84649
\(663\) −3900.94 −0.228506
\(664\) −18324.5 −1.07098
\(665\) 3696.70 0.215567
\(666\) 327.091 0.0190308
\(667\) 27363.3 1.58848
\(668\) 32516.8 1.88340
\(669\) 3682.54 0.212818
\(670\) −48616.6 −2.80332
\(671\) −671.000 −0.0386046
\(672\) −2127.98 −0.122156
\(673\) −7169.40 −0.410639 −0.205320 0.978695i \(-0.565823\pi\)
−0.205320 + 0.978695i \(0.565823\pi\)
\(674\) 10226.6 0.584443
\(675\) −3730.26 −0.212708
\(676\) 54102.6 3.07821
\(677\) 17876.2 1.01483 0.507415 0.861702i \(-0.330601\pi\)
0.507415 + 0.861702i \(0.330601\pi\)
\(678\) 22699.2 1.28578
\(679\) 1605.93 0.0907658
\(680\) −6076.02 −0.342654
\(681\) 1972.35 0.110985
\(682\) −6530.02 −0.366638
\(683\) −25975.2 −1.45522 −0.727608 0.685993i \(-0.759368\pi\)
−0.727608 + 0.685993i \(0.759368\pi\)
\(684\) −6707.29 −0.374941
\(685\) −15139.5 −0.844453
\(686\) 12220.1 0.680125
\(687\) 1804.43 0.100208
\(688\) 231.120 0.0128072
\(689\) 17562.5 0.971086
\(690\) −28928.7 −1.59608
\(691\) −11074.9 −0.609710 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(692\) −26658.6 −1.46446
\(693\) −393.831 −0.0215879
\(694\) 13346.4 0.730005
\(695\) −1409.33 −0.0769192
\(696\) 14536.0 0.791644
\(697\) 5997.97 0.325953
\(698\) 31566.2 1.71174
\(699\) −13124.5 −0.710175
\(700\) 7150.29 0.386079
\(701\) 13647.6 0.735324 0.367662 0.929960i \(-0.380158\pi\)
0.367662 + 0.929960i \(0.380158\pi\)
\(702\) −9866.27 −0.530454
\(703\) 454.197 0.0243675
\(704\) 9094.04 0.486853
\(705\) −10183.7 −0.544028
\(706\) −29269.0 −1.56028
\(707\) −2025.50 −0.107746
\(708\) −24896.3 −1.32155
\(709\) −12631.7 −0.669101 −0.334551 0.942378i \(-0.608585\pi\)
−0.334551 + 0.942378i \(0.608585\pi\)
\(710\) 9656.70 0.510436
\(711\) 9527.96 0.502569
\(712\) −14968.4 −0.787869
\(713\) −16795.7 −0.882191
\(714\) −892.234 −0.0467661
\(715\) −14225.8 −0.744079
\(716\) 15461.5 0.807014
\(717\) −10799.8 −0.562518
\(718\) 24667.3 1.28214
\(719\) 19598.3 1.01654 0.508270 0.861198i \(-0.330285\pi\)
0.508270 + 0.861198i \(0.330285\pi\)
\(720\) −172.021 −0.00890396
\(721\) 2597.29 0.134158
\(722\) 16398.4 0.845271
\(723\) −12036.8 −0.619161
\(724\) −17327.9 −0.889484
\(725\) 29151.2 1.49331
\(726\) 1663.87 0.0850577
\(727\) 3377.93 0.172325 0.0861627 0.996281i \(-0.472540\pi\)
0.0861627 + 0.996281i \(0.472540\pi\)
\(728\) 7282.71 0.370763
\(729\) 729.000 0.0370370
\(730\) −48927.0 −2.48064
\(731\) −3199.46 −0.161883
\(732\) −2380.81 −0.120215
\(733\) −25011.2 −1.26031 −0.630157 0.776467i \(-0.717010\pi\)
−0.630157 + 0.776467i \(0.717010\pi\)
\(734\) −39757.2 −1.99927
\(735\) −15922.4 −0.799057
\(736\) 23123.8 1.15809
\(737\) 7192.12 0.359464
\(738\) 15170.1 0.756665
\(739\) 10710.5 0.533143 0.266572 0.963815i \(-0.414109\pi\)
0.266572 + 0.963815i \(0.414109\pi\)
\(740\) 1673.38 0.0831281
\(741\) −13700.3 −0.679207
\(742\) 4016.95 0.198742
\(743\) 11349.4 0.560388 0.280194 0.959943i \(-0.409601\pi\)
0.280194 + 0.959943i \(0.409601\pi\)
\(744\) −8922.20 −0.439655
\(745\) −26828.2 −1.31934
\(746\) −35035.5 −1.71949
\(747\) 7181.81 0.351765
\(748\) 2334.19 0.114100
\(749\) 990.381 0.0483148
\(750\) −2935.07 −0.142898
\(751\) 4149.89 0.201640 0.100820 0.994905i \(-0.467853\pi\)
0.100820 + 0.994905i \(0.467853\pi\)
\(752\) −246.551 −0.0119558
\(753\) −2179.57 −0.105482
\(754\) 77103.0 3.72404
\(755\) −25252.8 −1.21728
\(756\) −1397.37 −0.0672249
\(757\) 1194.12 0.0573327 0.0286664 0.999589i \(-0.490874\pi\)
0.0286664 + 0.999589i \(0.490874\pi\)
\(758\) −18517.8 −0.887332
\(759\) 4279.58 0.204663
\(760\) −21339.3 −1.01850
\(761\) −4267.65 −0.203288 −0.101644 0.994821i \(-0.532410\pi\)
−0.101644 + 0.994821i \(0.532410\pi\)
\(762\) −6582.83 −0.312954
\(763\) 941.899 0.0446907
\(764\) −1300.63 −0.0615905
\(765\) 2381.34 0.112546
\(766\) 49799.3 2.34898
\(767\) −50853.1 −2.39400
\(768\) 12651.7 0.594440
\(769\) 12460.3 0.584304 0.292152 0.956372i \(-0.405629\pi\)
0.292152 + 0.956372i \(0.405629\pi\)
\(770\) −3253.78 −0.152283
\(771\) 17182.5 0.802610
\(772\) 5307.94 0.247457
\(773\) 6865.85 0.319466 0.159733 0.987160i \(-0.448937\pi\)
0.159733 + 0.987160i \(0.448937\pi\)
\(774\) −8092.09 −0.375793
\(775\) −17893.1 −0.829339
\(776\) −9270.27 −0.428844
\(777\) 94.6260 0.00436897
\(778\) −12294.9 −0.566573
\(779\) 21065.1 0.968854
\(780\) −50475.5 −2.31707
\(781\) −1428.57 −0.0654523
\(782\) 9695.49 0.443363
\(783\) −5696.99 −0.260018
\(784\) −385.489 −0.0175605
\(785\) 5804.95 0.263933
\(786\) −695.692 −0.0315706
\(787\) −25953.2 −1.17552 −0.587758 0.809037i \(-0.699989\pi\)
−0.587758 + 0.809037i \(0.699989\pi\)
\(788\) −14620.7 −0.660966
\(789\) 11447.4 0.516527
\(790\) 78718.6 3.54517
\(791\) 6566.79 0.295181
\(792\) 2273.40 0.101997
\(793\) −4863.03 −0.217770
\(794\) 43475.1 1.94317
\(795\) −10721.1 −0.478286
\(796\) 44964.4 2.00216
\(797\) −36630.5 −1.62800 −0.814001 0.580863i \(-0.802715\pi\)
−0.814001 + 0.580863i \(0.802715\pi\)
\(798\) −3133.57 −0.139006
\(799\) 3413.07 0.151121
\(800\) 24634.7 1.08871
\(801\) 5866.46 0.258778
\(802\) 39735.4 1.74951
\(803\) 7238.05 0.318089
\(804\) 25518.7 1.11937
\(805\) −8368.94 −0.366418
\(806\) −47325.9 −2.06822
\(807\) −3094.35 −0.134977
\(808\) 11692.2 0.509073
\(809\) −20199.6 −0.877849 −0.438925 0.898524i \(-0.644641\pi\)
−0.438925 + 0.898524i \(0.644641\pi\)
\(810\) 6022.89 0.261263
\(811\) 27715.6 1.20003 0.600017 0.799987i \(-0.295161\pi\)
0.600017 + 0.799987i \(0.295161\pi\)
\(812\) 10920.2 0.471951
\(813\) 4067.24 0.175454
\(814\) −399.778 −0.0172140
\(815\) −17871.2 −0.768100
\(816\) 57.6532 0.00247336
\(817\) −11236.7 −0.481176
\(818\) −51454.7 −2.19935
\(819\) −2854.27 −0.121778
\(820\) 77609.6 3.30518
\(821\) −19695.3 −0.837236 −0.418618 0.908162i \(-0.637485\pi\)
−0.418618 + 0.908162i \(0.637485\pi\)
\(822\) 12833.2 0.544539
\(823\) 2782.00 0.117830 0.0589152 0.998263i \(-0.481236\pi\)
0.0589152 + 0.998263i \(0.481236\pi\)
\(824\) −14992.9 −0.633863
\(825\) 4559.20 0.192401
\(826\) −11631.3 −0.489956
\(827\) 25074.1 1.05431 0.527153 0.849771i \(-0.323260\pi\)
0.527153 + 0.849771i \(0.323260\pi\)
\(828\) 15184.6 0.637321
\(829\) −8669.64 −0.363219 −0.181610 0.983371i \(-0.558131\pi\)
−0.181610 + 0.983371i \(0.558131\pi\)
\(830\) 59335.0 2.48138
\(831\) −2379.52 −0.0993318
\(832\) 65908.5 2.74635
\(833\) 5336.42 0.221964
\(834\) 1194.64 0.0496008
\(835\) −40545.4 −1.68040
\(836\) 8197.80 0.339147
\(837\) 3496.82 0.144406
\(838\) −66569.9 −2.74417
\(839\) −38859.9 −1.59904 −0.799518 0.600642i \(-0.794912\pi\)
−0.799518 + 0.600642i \(0.794912\pi\)
\(840\) −4445.75 −0.182611
\(841\) 20131.9 0.825449
\(842\) 76709.9 3.13967
\(843\) 6410.42 0.261906
\(844\) 21120.9 0.861389
\(845\) −67460.9 −2.74642
\(846\) 8632.37 0.350812
\(847\) 481.350 0.0195270
\(848\) −259.562 −0.0105111
\(849\) −2181.61 −0.0881893
\(850\) 10329.0 0.416801
\(851\) −1028.26 −0.0414197
\(852\) −5068.79 −0.203819
\(853\) −6284.73 −0.252268 −0.126134 0.992013i \(-0.540257\pi\)
−0.126134 + 0.992013i \(0.540257\pi\)
\(854\) −1112.29 −0.0445687
\(855\) 8363.37 0.334528
\(856\) −5717.00 −0.228275
\(857\) 8942.69 0.356449 0.178224 0.983990i \(-0.442965\pi\)
0.178224 + 0.983990i \(0.442965\pi\)
\(858\) 12058.8 0.479813
\(859\) −11753.8 −0.466863 −0.233431 0.972373i \(-0.574995\pi\)
−0.233431 + 0.972373i \(0.574995\pi\)
\(860\) −41398.8 −1.64150
\(861\) 4388.64 0.173710
\(862\) −79608.3 −3.14556
\(863\) 43212.4 1.70448 0.852240 0.523150i \(-0.175243\pi\)
0.852240 + 0.523150i \(0.175243\pi\)
\(864\) −4814.33 −0.189568
\(865\) 33240.8 1.30662
\(866\) 72825.6 2.85764
\(867\) 13940.9 0.546087
\(868\) −6702.84 −0.262107
\(869\) −11645.3 −0.454591
\(870\) −47067.7 −1.83419
\(871\) 52124.5 2.02775
\(872\) −5437.13 −0.211152
\(873\) 3633.24 0.140855
\(874\) 34051.0 1.31784
\(875\) −849.102 −0.0328056
\(876\) 25681.7 0.990530
\(877\) −28589.3 −1.10079 −0.550395 0.834904i \(-0.685523\pi\)
−0.550395 + 0.834904i \(0.685523\pi\)
\(878\) 37636.6 1.44667
\(879\) 22395.4 0.859360
\(880\) 210.248 0.00805394
\(881\) 30596.4 1.17006 0.585028 0.811013i \(-0.301083\pi\)
0.585028 + 0.811013i \(0.301083\pi\)
\(882\) 13496.9 0.515266
\(883\) −4764.07 −0.181567 −0.0907836 0.995871i \(-0.528937\pi\)
−0.0907836 + 0.995871i \(0.528937\pi\)
\(884\) 16916.9 0.643640
\(885\) 31043.4 1.17911
\(886\) 32855.3 1.24582
\(887\) 21955.5 0.831110 0.415555 0.909568i \(-0.363587\pi\)
0.415555 + 0.909568i \(0.363587\pi\)
\(888\) −546.230 −0.0206422
\(889\) −1904.38 −0.0718459
\(890\) 48467.8 1.82544
\(891\) −891.000 −0.0335013
\(892\) −15969.8 −0.599449
\(893\) 11986.9 0.449189
\(894\) 22741.4 0.850766
\(895\) −19279.0 −0.720030
\(896\) 9400.16 0.350488
\(897\) 31016.0 1.15451
\(898\) 56738.0 2.10843
\(899\) −27327.0 −1.01380
\(900\) 16176.7 0.599139
\(901\) 3593.18 0.132859
\(902\) −18541.2 −0.684429
\(903\) −2341.01 −0.0862722
\(904\) −37906.9 −1.39465
\(905\) 21606.3 0.793610
\(906\) 21406.0 0.784951
\(907\) −7846.51 −0.287254 −0.143627 0.989632i \(-0.545877\pi\)
−0.143627 + 0.989632i \(0.545877\pi\)
\(908\) −8553.37 −0.312614
\(909\) −4582.47 −0.167207
\(910\) −23581.6 −0.859034
\(911\) −26447.4 −0.961844 −0.480922 0.876763i \(-0.659698\pi\)
−0.480922 + 0.876763i \(0.659698\pi\)
\(912\) 202.481 0.00735176
\(913\) −8777.76 −0.318184
\(914\) 29231.7 1.05788
\(915\) 2968.65 0.107257
\(916\) −7825.14 −0.282260
\(917\) −201.261 −0.00724778
\(918\) −2018.58 −0.0725742
\(919\) 33796.3 1.21310 0.606549 0.795046i \(-0.292553\pi\)
0.606549 + 0.795046i \(0.292553\pi\)
\(920\) 48309.9 1.73123
\(921\) 23215.7 0.830602
\(922\) 22694.2 0.810622
\(923\) −10353.5 −0.369219
\(924\) 1707.90 0.0608072
\(925\) −1095.44 −0.0389382
\(926\) −58267.4 −2.06780
\(927\) 5876.08 0.208194
\(928\) 37623.0 1.33086
\(929\) −45691.8 −1.61367 −0.806834 0.590778i \(-0.798821\pi\)
−0.806834 + 0.590778i \(0.798821\pi\)
\(930\) 28890.2 1.01865
\(931\) 18741.8 0.659760
\(932\) 56915.9 2.00037
\(933\) −20831.7 −0.730975
\(934\) −22828.4 −0.799751
\(935\) −2910.52 −0.101801
\(936\) 16476.3 0.575369
\(937\) 28028.5 0.977216 0.488608 0.872503i \(-0.337505\pi\)
0.488608 + 0.872503i \(0.337505\pi\)
\(938\) 11922.1 0.414999
\(939\) −22896.4 −0.795735
\(940\) 44162.9 1.53238
\(941\) 21039.8 0.728881 0.364441 0.931227i \(-0.381260\pi\)
0.364441 + 0.931227i \(0.381260\pi\)
\(942\) −4920.67 −0.170195
\(943\) −47689.3 −1.64685
\(944\) 751.574 0.0259128
\(945\) 1742.40 0.0599790
\(946\) 9890.33 0.339918
\(947\) −54566.5 −1.87241 −0.936205 0.351455i \(-0.885687\pi\)
−0.936205 + 0.351455i \(0.885687\pi\)
\(948\) −41319.3 −1.41560
\(949\) 52457.3 1.79435
\(950\) 36275.8 1.23889
\(951\) 14829.0 0.505638
\(952\) 1490.00 0.0507260
\(953\) 20333.0 0.691135 0.345568 0.938394i \(-0.387686\pi\)
0.345568 + 0.938394i \(0.387686\pi\)
\(954\) 9087.90 0.308419
\(955\) 1621.77 0.0549519
\(956\) 46834.7 1.58446
\(957\) 6962.99 0.235195
\(958\) −6455.86 −0.217724
\(959\) 3712.60 0.125012
\(960\) −40234.0 −1.35265
\(961\) −13017.7 −0.436966
\(962\) −2897.36 −0.0971048
\(963\) 2240.63 0.0749774
\(964\) 52199.1 1.74401
\(965\) −6618.51 −0.220785
\(966\) 7094.07 0.236282
\(967\) −56345.3 −1.87378 −0.936889 0.349627i \(-0.886308\pi\)
−0.936889 + 0.349627i \(0.886308\pi\)
\(968\) −2778.60 −0.0922599
\(969\) −2802.99 −0.0929258
\(970\) 30017.3 0.993605
\(971\) −58451.1 −1.93181 −0.965903 0.258904i \(-0.916639\pi\)
−0.965903 + 0.258904i \(0.916639\pi\)
\(972\) −3161.40 −0.104323
\(973\) 345.604 0.0113870
\(974\) 80395.1 2.64479
\(975\) 33042.5 1.08534
\(976\) 71.8723 0.00235715
\(977\) 43020.4 1.40875 0.704373 0.709830i \(-0.251228\pi\)
0.704373 + 0.709830i \(0.251228\pi\)
\(978\) 15148.8 0.495303
\(979\) −7170.12 −0.234073
\(980\) 69049.7 2.25073
\(981\) 2130.94 0.0693534
\(982\) 80802.1 2.62576
\(983\) 5497.83 0.178386 0.0891930 0.996014i \(-0.471571\pi\)
0.0891930 + 0.996014i \(0.471571\pi\)
\(984\) −25333.5 −0.820735
\(985\) 18230.7 0.589723
\(986\) 15774.8 0.509505
\(987\) 2497.31 0.0805371
\(988\) 59413.0 1.91314
\(989\) 25438.6 0.817898
\(990\) −7361.31 −0.236321
\(991\) −22617.8 −0.725003 −0.362502 0.931983i \(-0.618077\pi\)
−0.362502 + 0.931983i \(0.618077\pi\)
\(992\) −23093.1 −0.739119
\(993\) −20584.7 −0.657839
\(994\) −2368.08 −0.0755642
\(995\) −56066.5 −1.78636
\(996\) −31144.9 −0.990826
\(997\) 23209.6 0.737266 0.368633 0.929575i \(-0.379826\pi\)
0.368633 + 0.929575i \(0.379826\pi\)
\(998\) −38195.4 −1.21148
\(999\) 214.081 0.00678000
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 2013.4.a.b.1.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.4.a.b.1.5 36 1.1 even 1 trivial