Properties

Label 2013.4.a.d.1.4
Level $2013$
Weight $4$
Character 2013.1
Self dual yes
Analytic conductor $118.771$
Analytic rank $1$
Dimension $37$
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] = [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: \(37\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
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)
 
\(f(q)\) \(=\) \(q-4.94279 q^{2} -3.00000 q^{3} +16.4312 q^{4} +0.994625 q^{5} +14.8284 q^{6} +23.9386 q^{7} -41.6735 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.94279 q^{2} -3.00000 q^{3} +16.4312 q^{4} +0.994625 q^{5} +14.8284 q^{6} +23.9386 q^{7} -41.6735 q^{8} +9.00000 q^{9} -4.91622 q^{10} +11.0000 q^{11} -49.2935 q^{12} -47.9281 q^{13} -118.323 q^{14} -2.98388 q^{15} +74.5342 q^{16} +89.5307 q^{17} -44.4851 q^{18} +94.1328 q^{19} +16.3429 q^{20} -71.8157 q^{21} -54.3707 q^{22} -65.6083 q^{23} +125.021 q^{24} -124.011 q^{25} +236.899 q^{26} -27.0000 q^{27} +393.339 q^{28} +230.134 q^{29} +14.7487 q^{30} -116.788 q^{31} -35.0184 q^{32} -33.0000 q^{33} -442.532 q^{34} +23.8099 q^{35} +147.881 q^{36} +63.6647 q^{37} -465.279 q^{38} +143.784 q^{39} -41.4495 q^{40} +211.619 q^{41} +354.970 q^{42} -340.632 q^{43} +180.743 q^{44} +8.95163 q^{45} +324.288 q^{46} -379.587 q^{47} -223.602 q^{48} +230.055 q^{49} +612.959 q^{50} -268.592 q^{51} -787.515 q^{52} +169.264 q^{53} +133.455 q^{54} +10.9409 q^{55} -997.604 q^{56} -282.398 q^{57} -1137.50 q^{58} -198.872 q^{59} -49.0286 q^{60} -61.0000 q^{61} +577.258 q^{62} +215.447 q^{63} -423.185 q^{64} -47.6705 q^{65} +163.112 q^{66} -847.117 q^{67} +1471.10 q^{68} +196.825 q^{69} -117.687 q^{70} -1070.85 q^{71} -375.062 q^{72} -1053.15 q^{73} -314.681 q^{74} +372.032 q^{75} +1546.71 q^{76} +263.324 q^{77} -710.696 q^{78} -197.798 q^{79} +74.1335 q^{80} +81.0000 q^{81} -1045.99 q^{82} +20.8936 q^{83} -1180.02 q^{84} +89.0495 q^{85} +1683.67 q^{86} -690.402 q^{87} -458.409 q^{88} +317.430 q^{89} -44.2460 q^{90} -1147.33 q^{91} -1078.02 q^{92} +350.363 q^{93} +1876.22 q^{94} +93.6268 q^{95} +105.055 q^{96} +76.5033 q^{97} -1137.11 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 37 q - 4 q^{2} - 111 q^{3} + 158 q^{4} - 15 q^{5} + 12 q^{6} - 77 q^{7} - 69 q^{8} + 333 q^{9} - 45 q^{10} + 407 q^{11} - 474 q^{12} - 169 q^{13} + 102 q^{14} + 45 q^{15} + 598 q^{16} - 338 q^{17} - 36 q^{18} - 235 q^{19} - 550 q^{20} + 231 q^{21} - 44 q^{22} - 53 q^{23} + 207 q^{24} + 750 q^{25} - 75 q^{26} - 999 q^{27} - 1378 q^{28} - 30 q^{29} + 135 q^{30} - 506 q^{31} - 841 q^{32} - 1221 q^{33} - 316 q^{34} - 822 q^{35} + 1422 q^{36} - 830 q^{37} - 371 q^{38} + 507 q^{39} - 613 q^{40} + 16 q^{41} - 306 q^{42} - 1137 q^{43} + 1738 q^{44} - 135 q^{45} - 659 q^{46} - 489 q^{47} - 1794 q^{48} + 2214 q^{49} + 1066 q^{50} + 1014 q^{51} - 2342 q^{52} + 731 q^{53} + 108 q^{54} - 165 q^{55} + 3051 q^{56} + 705 q^{57} - 611 q^{58} - 425 q^{59} + 1650 q^{60} - 2257 q^{61} + 453 q^{62} - 693 q^{63} + 4919 q^{64} + 1346 q^{65} + 132 q^{66} - 1907 q^{67} - 3236 q^{68} + 159 q^{69} - 1050 q^{70} - 561 q^{71} - 621 q^{72} - 2397 q^{73} - 1840 q^{74} - 2250 q^{75} - 3868 q^{76} - 847 q^{77} + 225 q^{78} + 393 q^{79} - 4031 q^{80} + 2997 q^{81} - 1946 q^{82} - 4191 q^{83} + 4134 q^{84} - 2667 q^{85} + 2405 q^{86} + 90 q^{87} - 759 q^{88} + 1437 q^{89} - 405 q^{90} - 5192 q^{91} - 737 q^{92} + 1518 q^{93} - 1960 q^{94} + 1356 q^{95} + 2523 q^{96} - 2368 q^{97} - 3014 q^{98} + 3663 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.94279 −1.74754 −0.873770 0.486339i \(-0.838332\pi\)
−0.873770 + 0.486339i \(0.838332\pi\)
\(3\) −3.00000 −0.577350
\(4\) 16.4312 2.05390
\(5\) 0.994625 0.0889620 0.0444810 0.999010i \(-0.485837\pi\)
0.0444810 + 0.999010i \(0.485837\pi\)
\(6\) 14.8284 1.00894
\(7\) 23.9386 1.29256 0.646280 0.763100i \(-0.276324\pi\)
0.646280 + 0.763100i \(0.276324\pi\)
\(8\) −41.6735 −1.84173
\(9\) 9.00000 0.333333
\(10\) −4.91622 −0.155465
\(11\) 11.0000 0.301511
\(12\) −49.2935 −1.18582
\(13\) −47.9281 −1.02253 −0.511264 0.859424i \(-0.670823\pi\)
−0.511264 + 0.859424i \(0.670823\pi\)
\(14\) −118.323 −2.25880
\(15\) −2.98388 −0.0513622
\(16\) 74.5342 1.16460
\(17\) 89.5307 1.27732 0.638658 0.769490i \(-0.279490\pi\)
0.638658 + 0.769490i \(0.279490\pi\)
\(18\) −44.4851 −0.582513
\(19\) 94.1328 1.13661 0.568304 0.822819i \(-0.307600\pi\)
0.568304 + 0.822819i \(0.307600\pi\)
\(20\) 16.3429 0.182719
\(21\) −71.8157 −0.746260
\(22\) −54.3707 −0.526903
\(23\) −65.6083 −0.594795 −0.297397 0.954754i \(-0.596119\pi\)
−0.297397 + 0.954754i \(0.596119\pi\)
\(24\) 125.021 1.06332
\(25\) −124.011 −0.992086
\(26\) 236.899 1.78691
\(27\) −27.0000 −0.192450
\(28\) 393.339 2.65479
\(29\) 230.134 1.47361 0.736807 0.676104i \(-0.236333\pi\)
0.736807 + 0.676104i \(0.236333\pi\)
\(30\) 14.7487 0.0897575
\(31\) −116.788 −0.676636 −0.338318 0.941032i \(-0.609858\pi\)
−0.338318 + 0.941032i \(0.609858\pi\)
\(32\) −35.0184 −0.193451
\(33\) −33.0000 −0.174078
\(34\) −442.532 −2.23216
\(35\) 23.8099 0.114989
\(36\) 147.881 0.684632
\(37\) 63.6647 0.282876 0.141438 0.989947i \(-0.454827\pi\)
0.141438 + 0.989947i \(0.454827\pi\)
\(38\) −465.279 −1.98627
\(39\) 143.784 0.590357
\(40\) −41.4495 −0.163844
\(41\) 211.619 0.806081 0.403040 0.915182i \(-0.367953\pi\)
0.403040 + 0.915182i \(0.367953\pi\)
\(42\) 354.970 1.30412
\(43\) −340.632 −1.20804 −0.604021 0.796968i \(-0.706436\pi\)
−0.604021 + 0.796968i \(0.706436\pi\)
\(44\) 180.743 0.619273
\(45\) 8.95163 0.0296540
\(46\) 324.288 1.03943
\(47\) −379.587 −1.17805 −0.589026 0.808114i \(-0.700489\pi\)
−0.589026 + 0.808114i \(0.700489\pi\)
\(48\) −223.602 −0.672380
\(49\) 230.055 0.670713
\(50\) 612.959 1.73371
\(51\) −268.592 −0.737459
\(52\) −787.515 −2.10017
\(53\) 169.264 0.438682 0.219341 0.975648i \(-0.429609\pi\)
0.219341 + 0.975648i \(0.429609\pi\)
\(54\) 133.455 0.336314
\(55\) 10.9409 0.0268230
\(56\) −997.604 −2.38054
\(57\) −282.398 −0.656220
\(58\) −1137.50 −2.57520
\(59\) −198.872 −0.438829 −0.219415 0.975632i \(-0.570415\pi\)
−0.219415 + 0.975632i \(0.570415\pi\)
\(60\) −49.0286 −0.105493
\(61\) −61.0000 −0.128037
\(62\) 577.258 1.18245
\(63\) 215.447 0.430854
\(64\) −423.185 −0.826533
\(65\) −47.6705 −0.0909661
\(66\) 163.112 0.304208
\(67\) −847.117 −1.54465 −0.772327 0.635226i \(-0.780907\pi\)
−0.772327 + 0.635226i \(0.780907\pi\)
\(68\) 1471.10 2.62348
\(69\) 196.825 0.343405
\(70\) −117.687 −0.200947
\(71\) −1070.85 −1.78995 −0.894973 0.446121i \(-0.852805\pi\)
−0.894973 + 0.446121i \(0.852805\pi\)
\(72\) −375.062 −0.613909
\(73\) −1053.15 −1.68852 −0.844258 0.535937i \(-0.819958\pi\)
−0.844258 + 0.535937i \(0.819958\pi\)
\(74\) −314.681 −0.494337
\(75\) 372.032 0.572781
\(76\) 1546.71 2.33447
\(77\) 263.324 0.389722
\(78\) −710.696 −1.03167
\(79\) −197.798 −0.281696 −0.140848 0.990031i \(-0.544983\pi\)
−0.140848 + 0.990031i \(0.544983\pi\)
\(80\) 74.1335 0.103605
\(81\) 81.0000 0.111111
\(82\) −1045.99 −1.40866
\(83\) 20.8936 0.0276310 0.0138155 0.999905i \(-0.495602\pi\)
0.0138155 + 0.999905i \(0.495602\pi\)
\(84\) −1180.02 −1.53274
\(85\) 89.0495 0.113633
\(86\) 1683.67 2.11110
\(87\) −690.402 −0.850791
\(88\) −458.409 −0.555302
\(89\) 317.430 0.378061 0.189031 0.981971i \(-0.439465\pi\)
0.189031 + 0.981971i \(0.439465\pi\)
\(90\) −44.2460 −0.0518215
\(91\) −1147.33 −1.32168
\(92\) −1078.02 −1.22165
\(93\) 350.363 0.390656
\(94\) 1876.22 2.05869
\(95\) 93.6268 0.101115
\(96\) 105.055 0.111689
\(97\) 76.5033 0.0800798 0.0400399 0.999198i \(-0.487252\pi\)
0.0400399 + 0.999198i \(0.487252\pi\)
\(98\) −1137.11 −1.17210
\(99\) 99.0000 0.100504
\(100\) −2037.64 −2.03764
\(101\) 492.052 0.484763 0.242381 0.970181i \(-0.422071\pi\)
0.242381 + 0.970181i \(0.422071\pi\)
\(102\) 1327.59 1.28874
\(103\) 948.726 0.907580 0.453790 0.891109i \(-0.350072\pi\)
0.453790 + 0.891109i \(0.350072\pi\)
\(104\) 1997.33 1.88322
\(105\) −71.4297 −0.0663888
\(106\) −836.635 −0.766615
\(107\) −521.992 −0.471616 −0.235808 0.971800i \(-0.575774\pi\)
−0.235808 + 0.971800i \(0.575774\pi\)
\(108\) −443.642 −0.395273
\(109\) −1836.07 −1.61343 −0.806716 0.590940i \(-0.798757\pi\)
−0.806716 + 0.590940i \(0.798757\pi\)
\(110\) −54.0785 −0.0468743
\(111\) −190.994 −0.163319
\(112\) 1784.24 1.50531
\(113\) −1054.84 −0.878150 −0.439075 0.898450i \(-0.644694\pi\)
−0.439075 + 0.898450i \(0.644694\pi\)
\(114\) 1395.84 1.14677
\(115\) −65.2556 −0.0529141
\(116\) 3781.37 3.02665
\(117\) −431.353 −0.340843
\(118\) 982.982 0.766871
\(119\) 2143.24 1.65101
\(120\) 124.349 0.0945952
\(121\) 121.000 0.0909091
\(122\) 301.510 0.223750
\(123\) −634.857 −0.465391
\(124\) −1918.96 −1.38974
\(125\) −247.672 −0.177220
\(126\) −1064.91 −0.752934
\(127\) −598.999 −0.418524 −0.209262 0.977860i \(-0.567106\pi\)
−0.209262 + 0.977860i \(0.567106\pi\)
\(128\) 2371.86 1.63785
\(129\) 1021.90 0.697464
\(130\) 235.625 0.158967
\(131\) 1004.02 0.669630 0.334815 0.942284i \(-0.391326\pi\)
0.334815 + 0.942284i \(0.391326\pi\)
\(132\) −542.229 −0.357538
\(133\) 2253.40 1.46913
\(134\) 4187.12 2.69934
\(135\) −26.8549 −0.0171207
\(136\) −3731.06 −2.35247
\(137\) 1872.65 1.16782 0.583909 0.811819i \(-0.301522\pi\)
0.583909 + 0.811819i \(0.301522\pi\)
\(138\) −972.864 −0.600114
\(139\) −298.707 −0.182273 −0.0911367 0.995838i \(-0.529050\pi\)
−0.0911367 + 0.995838i \(0.529050\pi\)
\(140\) 391.224 0.236175
\(141\) 1138.76 0.680149
\(142\) 5292.97 3.12800
\(143\) −527.209 −0.308304
\(144\) 670.807 0.388199
\(145\) 228.897 0.131096
\(146\) 5205.49 2.95075
\(147\) −690.164 −0.387236
\(148\) 1046.09 0.580998
\(149\) −2009.97 −1.10512 −0.552561 0.833473i \(-0.686349\pi\)
−0.552561 + 0.833473i \(0.686349\pi\)
\(150\) −1838.88 −1.00096
\(151\) 218.136 0.117561 0.0587803 0.998271i \(-0.481279\pi\)
0.0587803 + 0.998271i \(0.481279\pi\)
\(152\) −3922.85 −2.09332
\(153\) 805.776 0.425772
\(154\) −1301.56 −0.681054
\(155\) −116.160 −0.0601949
\(156\) 2362.55 1.21253
\(157\) 26.0045 0.0132190 0.00660951 0.999978i \(-0.497896\pi\)
0.00660951 + 0.999978i \(0.497896\pi\)
\(158\) 977.672 0.492275
\(159\) −507.791 −0.253273
\(160\) −34.8302 −0.0172098
\(161\) −1570.57 −0.768808
\(162\) −400.366 −0.194171
\(163\) −1566.24 −0.752624 −0.376312 0.926493i \(-0.622808\pi\)
−0.376312 + 0.926493i \(0.622808\pi\)
\(164\) 3477.15 1.65561
\(165\) −32.8226 −0.0154863
\(166\) −103.273 −0.0482863
\(167\) 1390.28 0.644210 0.322105 0.946704i \(-0.395610\pi\)
0.322105 + 0.946704i \(0.395610\pi\)
\(168\) 2992.81 1.37441
\(169\) 100.104 0.0455642
\(170\) −440.153 −0.198578
\(171\) 847.195 0.378869
\(172\) −5596.98 −2.48120
\(173\) 276.137 0.121355 0.0606773 0.998157i \(-0.480674\pi\)
0.0606773 + 0.998157i \(0.480674\pi\)
\(174\) 3412.51 1.48679
\(175\) −2968.64 −1.28233
\(176\) 819.876 0.351139
\(177\) 596.616 0.253358
\(178\) −1568.99 −0.660678
\(179\) −2853.94 −1.19170 −0.595849 0.803097i \(-0.703184\pi\)
−0.595849 + 0.803097i \(0.703184\pi\)
\(180\) 147.086 0.0609062
\(181\) −493.314 −0.202584 −0.101292 0.994857i \(-0.532298\pi\)
−0.101292 + 0.994857i \(0.532298\pi\)
\(182\) 5671.01 2.30969
\(183\) 183.000 0.0739221
\(184\) 2734.13 1.09545
\(185\) 63.3225 0.0251652
\(186\) −1731.77 −0.682687
\(187\) 984.838 0.385126
\(188\) −6237.06 −2.41960
\(189\) −646.341 −0.248753
\(190\) −462.778 −0.176702
\(191\) 2816.11 1.06684 0.533419 0.845851i \(-0.320907\pi\)
0.533419 + 0.845851i \(0.320907\pi\)
\(192\) 1269.55 0.477199
\(193\) 4246.41 1.58375 0.791875 0.610684i \(-0.209105\pi\)
0.791875 + 0.610684i \(0.209105\pi\)
\(194\) −378.140 −0.139943
\(195\) 143.012 0.0525193
\(196\) 3780.07 1.37758
\(197\) −4707.06 −1.70236 −0.851178 0.524877i \(-0.824111\pi\)
−0.851178 + 0.524877i \(0.824111\pi\)
\(198\) −489.336 −0.175634
\(199\) 2628.86 0.936457 0.468228 0.883608i \(-0.344892\pi\)
0.468228 + 0.883608i \(0.344892\pi\)
\(200\) 5167.97 1.82715
\(201\) 2541.35 0.891806
\(202\) −2432.11 −0.847142
\(203\) 5509.07 1.90473
\(204\) −4413.29 −1.51467
\(205\) 210.481 0.0717105
\(206\) −4689.35 −1.58603
\(207\) −590.475 −0.198265
\(208\) −3572.28 −1.19083
\(209\) 1035.46 0.342700
\(210\) 353.062 0.116017
\(211\) 4801.43 1.56656 0.783279 0.621670i \(-0.213546\pi\)
0.783279 + 0.621670i \(0.213546\pi\)
\(212\) 2781.20 0.901008
\(213\) 3212.54 1.03343
\(214\) 2580.10 0.824167
\(215\) −338.801 −0.107470
\(216\) 1125.19 0.354441
\(217\) −2795.73 −0.874593
\(218\) 9075.33 2.81954
\(219\) 3159.44 0.974865
\(220\) 179.771 0.0550918
\(221\) −4291.04 −1.30609
\(222\) 944.044 0.285406
\(223\) 842.574 0.253018 0.126509 0.991965i \(-0.459623\pi\)
0.126509 + 0.991965i \(0.459623\pi\)
\(224\) −838.289 −0.250047
\(225\) −1116.10 −0.330695
\(226\) 5213.85 1.53460
\(227\) 2288.30 0.669073 0.334536 0.942383i \(-0.391420\pi\)
0.334536 + 0.942383i \(0.391420\pi\)
\(228\) −4640.14 −1.34781
\(229\) 2626.72 0.757984 0.378992 0.925400i \(-0.376271\pi\)
0.378992 + 0.925400i \(0.376271\pi\)
\(230\) 322.545 0.0924695
\(231\) −789.972 −0.225006
\(232\) −9590.49 −2.71399
\(233\) −969.360 −0.272553 −0.136277 0.990671i \(-0.543514\pi\)
−0.136277 + 0.990671i \(0.543514\pi\)
\(234\) 2132.09 0.595636
\(235\) −377.547 −0.104802
\(236\) −3267.70 −0.901310
\(237\) 593.393 0.162637
\(238\) −10593.6 −2.88521
\(239\) −3010.38 −0.814750 −0.407375 0.913261i \(-0.633556\pi\)
−0.407375 + 0.913261i \(0.633556\pi\)
\(240\) −222.401 −0.0598162
\(241\) −1925.35 −0.514618 −0.257309 0.966329i \(-0.582836\pi\)
−0.257309 + 0.966329i \(0.582836\pi\)
\(242\) −598.078 −0.158867
\(243\) −243.000 −0.0641500
\(244\) −1002.30 −0.262975
\(245\) 228.818 0.0596679
\(246\) 3137.96 0.813290
\(247\) −4511.61 −1.16221
\(248\) 4866.96 1.24618
\(249\) −62.6808 −0.0159528
\(250\) 1224.19 0.309699
\(251\) 5763.38 1.44933 0.724664 0.689102i \(-0.241995\pi\)
0.724664 + 0.689102i \(0.241995\pi\)
\(252\) 3540.05 0.884929
\(253\) −721.691 −0.179337
\(254\) 2960.73 0.731388
\(255\) −267.148 −0.0656058
\(256\) −8338.13 −2.03568
\(257\) 2199.45 0.533843 0.266922 0.963718i \(-0.413993\pi\)
0.266922 + 0.963718i \(0.413993\pi\)
\(258\) −5051.01 −1.21885
\(259\) 1524.04 0.365634
\(260\) −783.282 −0.186835
\(261\) 2071.20 0.491204
\(262\) −4962.65 −1.17021
\(263\) −5575.32 −1.30718 −0.653592 0.756847i \(-0.726739\pi\)
−0.653592 + 0.756847i \(0.726739\pi\)
\(264\) 1375.23 0.320604
\(265\) 168.354 0.0390260
\(266\) −11138.1 −2.56737
\(267\) −952.289 −0.218274
\(268\) −13919.1 −3.17256
\(269\) −3312.34 −0.750770 −0.375385 0.926869i \(-0.622490\pi\)
−0.375385 + 0.926869i \(0.622490\pi\)
\(270\) 132.738 0.0299192
\(271\) 897.640 0.201209 0.100605 0.994926i \(-0.467922\pi\)
0.100605 + 0.994926i \(0.467922\pi\)
\(272\) 6673.10 1.48756
\(273\) 3441.99 0.763072
\(274\) −9256.10 −2.04081
\(275\) −1364.12 −0.299125
\(276\) 3234.06 0.705318
\(277\) −5615.99 −1.21817 −0.609083 0.793107i \(-0.708462\pi\)
−0.609083 + 0.793107i \(0.708462\pi\)
\(278\) 1476.45 0.318530
\(279\) −1051.09 −0.225545
\(280\) −992.242 −0.211778
\(281\) −3111.46 −0.660548 −0.330274 0.943885i \(-0.607141\pi\)
−0.330274 + 0.943885i \(0.607141\pi\)
\(282\) −5628.66 −1.18859
\(283\) 3424.08 0.719225 0.359612 0.933102i \(-0.382909\pi\)
0.359612 + 0.933102i \(0.382909\pi\)
\(284\) −17595.3 −3.67636
\(285\) −280.880 −0.0583787
\(286\) 2605.89 0.538773
\(287\) 5065.85 1.04191
\(288\) −315.165 −0.0644837
\(289\) 3102.75 0.631539
\(290\) −1131.39 −0.229095
\(291\) −229.510 −0.0462341
\(292\) −17304.5 −3.46804
\(293\) 2733.03 0.544932 0.272466 0.962165i \(-0.412161\pi\)
0.272466 + 0.962165i \(0.412161\pi\)
\(294\) 3411.33 0.676711
\(295\) −197.803 −0.0390391
\(296\) −2653.13 −0.520981
\(297\) −297.000 −0.0580259
\(298\) 9934.86 1.93124
\(299\) 3144.48 0.608194
\(300\) 6112.93 1.17643
\(301\) −8154.23 −1.56147
\(302\) −1078.20 −0.205442
\(303\) −1476.16 −0.279878
\(304\) 7016.11 1.32369
\(305\) −60.6721 −0.0113904
\(306\) −3982.78 −0.744054
\(307\) 2391.95 0.444677 0.222338 0.974970i \(-0.428631\pi\)
0.222338 + 0.974970i \(0.428631\pi\)
\(308\) 4326.73 0.800448
\(309\) −2846.18 −0.523991
\(310\) 574.155 0.105193
\(311\) 9993.31 1.82209 0.911043 0.412312i \(-0.135279\pi\)
0.911043 + 0.412312i \(0.135279\pi\)
\(312\) −5992.00 −1.08728
\(313\) −1001.54 −0.180864 −0.0904320 0.995903i \(-0.528825\pi\)
−0.0904320 + 0.995903i \(0.528825\pi\)
\(314\) −128.535 −0.0231008
\(315\) 214.289 0.0383296
\(316\) −3250.05 −0.578574
\(317\) −1030.24 −0.182537 −0.0912684 0.995826i \(-0.529092\pi\)
−0.0912684 + 0.995826i \(0.529092\pi\)
\(318\) 2509.90 0.442605
\(319\) 2531.47 0.444311
\(320\) −420.910 −0.0735300
\(321\) 1565.98 0.272287
\(322\) 7762.99 1.34352
\(323\) 8427.77 1.45181
\(324\) 1330.93 0.228211
\(325\) 5943.60 1.01444
\(326\) 7741.61 1.31524
\(327\) 5508.22 0.931515
\(328\) −8818.91 −1.48458
\(329\) −9086.77 −1.52270
\(330\) 162.235 0.0270629
\(331\) −7925.99 −1.31617 −0.658085 0.752944i \(-0.728633\pi\)
−0.658085 + 0.752944i \(0.728633\pi\)
\(332\) 343.307 0.0567512
\(333\) 572.982 0.0942920
\(334\) −6871.86 −1.12578
\(335\) −842.563 −0.137415
\(336\) −5352.72 −0.869092
\(337\) 3732.81 0.603380 0.301690 0.953406i \(-0.402449\pi\)
0.301690 + 0.953406i \(0.402449\pi\)
\(338\) −494.795 −0.0796252
\(339\) 3164.52 0.507000
\(340\) 1463.19 0.233390
\(341\) −1284.67 −0.204013
\(342\) −4187.51 −0.662089
\(343\) −2703.75 −0.425623
\(344\) 14195.3 2.22489
\(345\) 195.767 0.0305500
\(346\) −1364.89 −0.212072
\(347\) −1248.79 −0.193195 −0.0965973 0.995324i \(-0.530796\pi\)
−0.0965973 + 0.995324i \(0.530796\pi\)
\(348\) −11344.1 −1.74744
\(349\) 834.492 0.127992 0.0639962 0.997950i \(-0.479615\pi\)
0.0639962 + 0.997950i \(0.479615\pi\)
\(350\) 14673.4 2.24093
\(351\) 1294.06 0.196786
\(352\) −385.202 −0.0583277
\(353\) −5827.55 −0.878667 −0.439333 0.898324i \(-0.644785\pi\)
−0.439333 + 0.898324i \(0.644785\pi\)
\(354\) −2948.95 −0.442753
\(355\) −1065.09 −0.159237
\(356\) 5215.74 0.776499
\(357\) −6429.71 −0.953211
\(358\) 14106.5 2.08254
\(359\) −232.438 −0.0341717 −0.0170858 0.999854i \(-0.505439\pi\)
−0.0170858 + 0.999854i \(0.505439\pi\)
\(360\) −373.046 −0.0546146
\(361\) 2001.98 0.291876
\(362\) 2438.35 0.354024
\(363\) −363.000 −0.0524864
\(364\) −18852.0 −2.71459
\(365\) −1047.49 −0.150214
\(366\) −904.531 −0.129182
\(367\) −10954.4 −1.55808 −0.779039 0.626975i \(-0.784293\pi\)
−0.779039 + 0.626975i \(0.784293\pi\)
\(368\) −4890.06 −0.692695
\(369\) 1904.57 0.268694
\(370\) −312.990 −0.0439772
\(371\) 4051.93 0.567023
\(372\) 5756.88 0.802367
\(373\) −5422.99 −0.752793 −0.376397 0.926459i \(-0.622837\pi\)
−0.376397 + 0.926459i \(0.622837\pi\)
\(374\) −4867.85 −0.673022
\(375\) 743.017 0.102318
\(376\) 15818.7 2.16965
\(377\) −11029.9 −1.50681
\(378\) 3194.73 0.434707
\(379\) −6148.13 −0.833267 −0.416634 0.909074i \(-0.636790\pi\)
−0.416634 + 0.909074i \(0.636790\pi\)
\(380\) 1538.40 0.207679
\(381\) 1797.00 0.241635
\(382\) −13919.4 −1.86434
\(383\) 12509.4 1.66893 0.834467 0.551058i \(-0.185776\pi\)
0.834467 + 0.551058i \(0.185776\pi\)
\(384\) −7115.58 −0.945613
\(385\) 261.909 0.0346704
\(386\) −20989.1 −2.76767
\(387\) −3065.69 −0.402681
\(388\) 1257.04 0.164476
\(389\) 4748.98 0.618979 0.309490 0.950903i \(-0.399842\pi\)
0.309490 + 0.950903i \(0.399842\pi\)
\(390\) −706.876 −0.0917796
\(391\) −5873.96 −0.759741
\(392\) −9587.19 −1.23527
\(393\) −3012.06 −0.386611
\(394\) 23266.0 2.97494
\(395\) −196.734 −0.0250602
\(396\) 1626.69 0.206424
\(397\) −8079.84 −1.02145 −0.510725 0.859744i \(-0.670623\pi\)
−0.510725 + 0.859744i \(0.670623\pi\)
\(398\) −12993.9 −1.63650
\(399\) −6760.21 −0.848205
\(400\) −9243.03 −1.15538
\(401\) 2771.81 0.345181 0.172590 0.984994i \(-0.444786\pi\)
0.172590 + 0.984994i \(0.444786\pi\)
\(402\) −12561.4 −1.55847
\(403\) 5597.42 0.691879
\(404\) 8085.00 0.995653
\(405\) 80.5646 0.00988466
\(406\) −27230.2 −3.32860
\(407\) 700.312 0.0852903
\(408\) 11193.2 1.35820
\(409\) −5765.16 −0.696990 −0.348495 0.937311i \(-0.613307\pi\)
−0.348495 + 0.937311i \(0.613307\pi\)
\(410\) −1040.37 −0.125317
\(411\) −5617.94 −0.674240
\(412\) 15588.7 1.86408
\(413\) −4760.71 −0.567213
\(414\) 2918.59 0.346476
\(415\) 20.7813 0.00245811
\(416\) 1678.36 0.197809
\(417\) 896.121 0.105236
\(418\) −5118.06 −0.598882
\(419\) −5313.26 −0.619499 −0.309749 0.950818i \(-0.600245\pi\)
−0.309749 + 0.950818i \(0.600245\pi\)
\(420\) −1173.67 −0.136356
\(421\) 11151.1 1.29091 0.645453 0.763800i \(-0.276668\pi\)
0.645453 + 0.763800i \(0.276668\pi\)
\(422\) −23732.4 −2.73762
\(423\) −3416.28 −0.392684
\(424\) −7053.82 −0.807933
\(425\) −11102.8 −1.26721
\(426\) −15878.9 −1.80595
\(427\) −1460.25 −0.165495
\(428\) −8576.94 −0.968650
\(429\) 1581.63 0.177999
\(430\) 1674.62 0.187808
\(431\) 1509.61 0.168713 0.0843567 0.996436i \(-0.473116\pi\)
0.0843567 + 0.996436i \(0.473116\pi\)
\(432\) −2012.42 −0.224127
\(433\) −208.125 −0.0230989 −0.0115495 0.999933i \(-0.503676\pi\)
−0.0115495 + 0.999933i \(0.503676\pi\)
\(434\) 13818.7 1.52839
\(435\) −686.691 −0.0756880
\(436\) −30168.9 −3.31382
\(437\) −6175.89 −0.676048
\(438\) −15616.5 −1.70362
\(439\) −15002.1 −1.63101 −0.815503 0.578753i \(-0.803540\pi\)
−0.815503 + 0.578753i \(0.803540\pi\)
\(440\) −455.945 −0.0494007
\(441\) 2070.49 0.223571
\(442\) 21209.7 2.28245
\(443\) −15932.9 −1.70879 −0.854397 0.519620i \(-0.826073\pi\)
−0.854397 + 0.519620i \(0.826073\pi\)
\(444\) −3138.26 −0.335440
\(445\) 315.723 0.0336331
\(446\) −4164.67 −0.442158
\(447\) 6029.91 0.638042
\(448\) −10130.4 −1.06834
\(449\) −3370.19 −0.354230 −0.177115 0.984190i \(-0.556676\pi\)
−0.177115 + 0.984190i \(0.556676\pi\)
\(450\) 5516.63 0.577903
\(451\) 2327.81 0.243043
\(452\) −17332.3 −1.80363
\(453\) −654.407 −0.0678736
\(454\) −11310.6 −1.16923
\(455\) −1141.16 −0.117579
\(456\) 11768.5 1.20858
\(457\) −18661.6 −1.91018 −0.955089 0.296319i \(-0.904241\pi\)
−0.955089 + 0.296319i \(0.904241\pi\)
\(458\) −12983.3 −1.32461
\(459\) −2417.33 −0.245820
\(460\) −1072.23 −0.108680
\(461\) 9582.81 0.968147 0.484074 0.875027i \(-0.339157\pi\)
0.484074 + 0.875027i \(0.339157\pi\)
\(462\) 3904.67 0.393207
\(463\) 4729.90 0.474767 0.237383 0.971416i \(-0.423710\pi\)
0.237383 + 0.971416i \(0.423710\pi\)
\(464\) 17152.8 1.71616
\(465\) 348.480 0.0347535
\(466\) 4791.34 0.476297
\(467\) −9491.13 −0.940465 −0.470233 0.882542i \(-0.655830\pi\)
−0.470233 + 0.882542i \(0.655830\pi\)
\(468\) −7087.64 −0.700056
\(469\) −20278.7 −1.99656
\(470\) 1866.13 0.183145
\(471\) −78.0136 −0.00763201
\(472\) 8287.70 0.808204
\(473\) −3746.95 −0.364239
\(474\) −2933.02 −0.284215
\(475\) −11673.5 −1.12761
\(476\) 35215.9 3.39100
\(477\) 1523.37 0.146227
\(478\) 14879.7 1.42381
\(479\) −11089.0 −1.05776 −0.528880 0.848696i \(-0.677388\pi\)
−0.528880 + 0.848696i \(0.677388\pi\)
\(480\) 104.490 0.00993607
\(481\) −3051.33 −0.289249
\(482\) 9516.62 0.899316
\(483\) 4711.70 0.443872
\(484\) 1988.17 0.186718
\(485\) 76.0921 0.00712405
\(486\) 1201.10 0.112105
\(487\) −12873.4 −1.19784 −0.598920 0.800809i \(-0.704403\pi\)
−0.598920 + 0.800809i \(0.704403\pi\)
\(488\) 2542.09 0.235809
\(489\) 4698.73 0.434527
\(490\) −1131.00 −0.104272
\(491\) −5566.31 −0.511617 −0.255808 0.966728i \(-0.582342\pi\)
−0.255808 + 0.966728i \(0.582342\pi\)
\(492\) −10431.4 −0.955865
\(493\) 20604.1 1.88227
\(494\) 22299.9 2.03101
\(495\) 98.4679 0.00894101
\(496\) −8704.68 −0.788007
\(497\) −25634.5 −2.31361
\(498\) 309.818 0.0278781
\(499\) 4195.12 0.376352 0.188176 0.982135i \(-0.439743\pi\)
0.188176 + 0.982135i \(0.439743\pi\)
\(500\) −4069.55 −0.363991
\(501\) −4170.84 −0.371935
\(502\) −28487.2 −2.53276
\(503\) −5672.18 −0.502803 −0.251401 0.967883i \(-0.580891\pi\)
−0.251401 + 0.967883i \(0.580891\pi\)
\(504\) −8978.44 −0.793515
\(505\) 489.408 0.0431254
\(506\) 3567.17 0.313399
\(507\) −300.313 −0.0263065
\(508\) −9842.25 −0.859605
\(509\) −3458.25 −0.301147 −0.150574 0.988599i \(-0.548112\pi\)
−0.150574 + 0.988599i \(0.548112\pi\)
\(510\) 1320.46 0.114649
\(511\) −25210.8 −2.18251
\(512\) 22238.7 1.91958
\(513\) −2541.58 −0.218740
\(514\) −10871.4 −0.932913
\(515\) 943.626 0.0807401
\(516\) 16790.9 1.43252
\(517\) −4175.46 −0.355196
\(518\) −7533.02 −0.638961
\(519\) −828.412 −0.0700641
\(520\) 1986.60 0.167535
\(521\) −1565.88 −0.131674 −0.0658372 0.997830i \(-0.520972\pi\)
−0.0658372 + 0.997830i \(0.520972\pi\)
\(522\) −10237.5 −0.858400
\(523\) 16391.4 1.37045 0.685225 0.728331i \(-0.259704\pi\)
0.685225 + 0.728331i \(0.259704\pi\)
\(524\) 16497.2 1.37535
\(525\) 8905.91 0.740354
\(526\) 27557.7 2.28436
\(527\) −10456.1 −0.864278
\(528\) −2459.63 −0.202730
\(529\) −7862.55 −0.646219
\(530\) −832.138 −0.0681996
\(531\) −1789.85 −0.146276
\(532\) 37026.1 3.01745
\(533\) −10142.5 −0.824241
\(534\) 4706.96 0.381442
\(535\) −519.186 −0.0419558
\(536\) 35302.3 2.84483
\(537\) 8561.83 0.688027
\(538\) 16372.2 1.31200
\(539\) 2530.60 0.202228
\(540\) −441.257 −0.0351642
\(541\) 13779.7 1.09507 0.547537 0.836781i \(-0.315565\pi\)
0.547537 + 0.836781i \(0.315565\pi\)
\(542\) −4436.85 −0.351622
\(543\) 1479.94 0.116962
\(544\) −3135.22 −0.247098
\(545\) −1826.21 −0.143534
\(546\) −17013.0 −1.33350
\(547\) 17157.8 1.34116 0.670580 0.741837i \(-0.266045\pi\)
0.670580 + 0.741837i \(0.266045\pi\)
\(548\) 30769.8 2.39858
\(549\) −549.000 −0.0426790
\(550\) 6742.55 0.522733
\(551\) 21663.1 1.67492
\(552\) −8202.39 −0.632458
\(553\) −4734.99 −0.364109
\(554\) 27758.6 2.12879
\(555\) −189.968 −0.0145291
\(556\) −4908.11 −0.374371
\(557\) 11266.1 0.857018 0.428509 0.903538i \(-0.359039\pi\)
0.428509 + 0.903538i \(0.359039\pi\)
\(558\) 5195.32 0.394149
\(559\) 16325.8 1.23526
\(560\) 1774.65 0.133915
\(561\) −2954.51 −0.222352
\(562\) 15379.3 1.15433
\(563\) −11396.4 −0.853112 −0.426556 0.904461i \(-0.640273\pi\)
−0.426556 + 0.904461i \(0.640273\pi\)
\(564\) 18711.2 1.39696
\(565\) −1049.17 −0.0781220
\(566\) −16924.5 −1.25687
\(567\) 1939.02 0.143618
\(568\) 44626.0 3.29659
\(569\) 3493.84 0.257415 0.128708 0.991683i \(-0.458917\pi\)
0.128708 + 0.991683i \(0.458917\pi\)
\(570\) 1388.33 0.102019
\(571\) 19828.2 1.45321 0.726606 0.687054i \(-0.241096\pi\)
0.726606 + 0.687054i \(0.241096\pi\)
\(572\) −8662.67 −0.633224
\(573\) −8448.32 −0.615940
\(574\) −25039.4 −1.82078
\(575\) 8136.13 0.590087
\(576\) −3808.66 −0.275511
\(577\) −892.360 −0.0643838 −0.0321919 0.999482i \(-0.510249\pi\)
−0.0321919 + 0.999482i \(0.510249\pi\)
\(578\) −15336.2 −1.10364
\(579\) −12739.2 −0.914378
\(580\) 3761.05 0.269257
\(581\) 500.163 0.0357147
\(582\) 1134.42 0.0807959
\(583\) 1861.90 0.132268
\(584\) 43888.4 3.10979
\(585\) −429.035 −0.0303220
\(586\) −13508.8 −0.952291
\(587\) −22118.1 −1.55522 −0.777608 0.628750i \(-0.783567\pi\)
−0.777608 + 0.628750i \(0.783567\pi\)
\(588\) −11340.2 −0.795344
\(589\) −10993.6 −0.769069
\(590\) 977.699 0.0682224
\(591\) 14121.2 0.982856
\(592\) 4745.19 0.329436
\(593\) −9554.16 −0.661623 −0.330811 0.943697i \(-0.607322\pi\)
−0.330811 + 0.943697i \(0.607322\pi\)
\(594\) 1468.01 0.101403
\(595\) 2131.72 0.146877
\(596\) −33026.2 −2.26981
\(597\) −7886.58 −0.540663
\(598\) −15542.5 −1.06284
\(599\) 3768.21 0.257037 0.128518 0.991707i \(-0.458978\pi\)
0.128518 + 0.991707i \(0.458978\pi\)
\(600\) −15503.9 −1.05491
\(601\) 8296.73 0.563112 0.281556 0.959545i \(-0.409149\pi\)
0.281556 + 0.959545i \(0.409149\pi\)
\(602\) 40304.7 2.72873
\(603\) −7624.05 −0.514884
\(604\) 3584.23 0.241457
\(605\) 120.350 0.00808745
\(606\) 7296.33 0.489098
\(607\) 5812.47 0.388667 0.194333 0.980936i \(-0.437746\pi\)
0.194333 + 0.980936i \(0.437746\pi\)
\(608\) −3296.38 −0.219878
\(609\) −16527.2 −1.09970
\(610\) 299.890 0.0199052
\(611\) 18192.9 1.20459
\(612\) 13239.9 0.874493
\(613\) 4868.11 0.320752 0.160376 0.987056i \(-0.448729\pi\)
0.160376 + 0.987056i \(0.448729\pi\)
\(614\) −11822.9 −0.777090
\(615\) −631.444 −0.0414021
\(616\) −10973.6 −0.717761
\(617\) 631.980 0.0412359 0.0206180 0.999787i \(-0.493437\pi\)
0.0206180 + 0.999787i \(0.493437\pi\)
\(618\) 14068.1 0.915696
\(619\) −20297.7 −1.31798 −0.658992 0.752150i \(-0.729017\pi\)
−0.658992 + 0.752150i \(0.729017\pi\)
\(620\) −1908.65 −0.123634
\(621\) 1771.42 0.114468
\(622\) −49394.8 −3.18417
\(623\) 7598.81 0.488667
\(624\) 10716.8 0.687527
\(625\) 15255.0 0.976320
\(626\) 4950.41 0.316067
\(627\) −3106.38 −0.197858
\(628\) 427.285 0.0271505
\(629\) 5699.95 0.361322
\(630\) −1059.19 −0.0669825
\(631\) 13604.1 0.858276 0.429138 0.903239i \(-0.358817\pi\)
0.429138 + 0.903239i \(0.358817\pi\)
\(632\) 8242.93 0.518807
\(633\) −14404.3 −0.904453
\(634\) 5092.27 0.318990
\(635\) −595.779 −0.0372327
\(636\) −8343.61 −0.520197
\(637\) −11026.1 −0.685823
\(638\) −12512.5 −0.776452
\(639\) −9637.62 −0.596648
\(640\) 2359.11 0.145706
\(641\) 30896.0 1.90378 0.951888 0.306446i \(-0.0991399\pi\)
0.951888 + 0.306446i \(0.0991399\pi\)
\(642\) −7740.29 −0.475833
\(643\) −24107.8 −1.47857 −0.739284 0.673394i \(-0.764836\pi\)
−0.739284 + 0.673394i \(0.764836\pi\)
\(644\) −25806.3 −1.57905
\(645\) 1016.40 0.0620478
\(646\) −41656.7 −2.53709
\(647\) 7172.65 0.435836 0.217918 0.975967i \(-0.430073\pi\)
0.217918 + 0.975967i \(0.430073\pi\)
\(648\) −3375.56 −0.204636
\(649\) −2187.59 −0.132312
\(650\) −29378.0 −1.77277
\(651\) 8387.20 0.504946
\(652\) −25735.2 −1.54581
\(653\) 1741.09 0.104340 0.0521701 0.998638i \(-0.483386\pi\)
0.0521701 + 0.998638i \(0.483386\pi\)
\(654\) −27226.0 −1.62786
\(655\) 998.622 0.0595716
\(656\) 15772.8 0.938759
\(657\) −9478.33 −0.562839
\(658\) 44914.0 2.66099
\(659\) 18651.6 1.10252 0.551261 0.834333i \(-0.314147\pi\)
0.551261 + 0.834333i \(0.314147\pi\)
\(660\) −539.314 −0.0318072
\(661\) 18880.3 1.11098 0.555491 0.831522i \(-0.312530\pi\)
0.555491 + 0.831522i \(0.312530\pi\)
\(662\) 39176.5 2.30006
\(663\) 12873.1 0.754073
\(664\) −870.711 −0.0508887
\(665\) 2241.29 0.130697
\(666\) −2832.13 −0.164779
\(667\) −15098.7 −0.876497
\(668\) 22843.9 1.32314
\(669\) −2527.72 −0.146080
\(670\) 4164.61 0.240139
\(671\) −671.000 −0.0386046
\(672\) 2514.87 0.144365
\(673\) −2703.36 −0.154839 −0.0774197 0.996999i \(-0.524668\pi\)
−0.0774197 + 0.996999i \(0.524668\pi\)
\(674\) −18450.5 −1.05443
\(675\) 3348.29 0.190927
\(676\) 1644.83 0.0935841
\(677\) −8428.11 −0.478462 −0.239231 0.970963i \(-0.576895\pi\)
−0.239231 + 0.970963i \(0.576895\pi\)
\(678\) −15641.6 −0.886003
\(679\) 1831.38 0.103508
\(680\) −3711.01 −0.209280
\(681\) −6864.89 −0.386289
\(682\) 6349.83 0.356522
\(683\) 27381.4 1.53400 0.766999 0.641648i \(-0.221749\pi\)
0.766999 + 0.641648i \(0.221749\pi\)
\(684\) 13920.4 0.778158
\(685\) 1862.58 0.103891
\(686\) 13364.1 0.743794
\(687\) −7880.15 −0.437622
\(688\) −25388.7 −1.40688
\(689\) −8112.49 −0.448565
\(690\) −967.635 −0.0533873
\(691\) 15987.3 0.880152 0.440076 0.897961i \(-0.354952\pi\)
0.440076 + 0.897961i \(0.354952\pi\)
\(692\) 4537.26 0.249250
\(693\) 2369.92 0.129907
\(694\) 6172.50 0.337615
\(695\) −297.101 −0.0162154
\(696\) 28771.5 1.56693
\(697\) 18946.4 1.02962
\(698\) −4124.72 −0.223672
\(699\) 2908.08 0.157359
\(700\) −48778.2 −2.63378
\(701\) 7109.81 0.383073 0.191536 0.981486i \(-0.438653\pi\)
0.191536 + 0.981486i \(0.438653\pi\)
\(702\) −6396.26 −0.343891
\(703\) 5992.93 0.321519
\(704\) −4655.03 −0.249209
\(705\) 1132.64 0.0605074
\(706\) 28804.4 1.53551
\(707\) 11779.0 0.626585
\(708\) 9803.10 0.520371
\(709\) 12182.2 0.645291 0.322645 0.946520i \(-0.395428\pi\)
0.322645 + 0.946520i \(0.395428\pi\)
\(710\) 5264.52 0.278273
\(711\) −1780.18 −0.0938986
\(712\) −13228.4 −0.696286
\(713\) 7662.25 0.402459
\(714\) 31780.7 1.66577
\(715\) −524.376 −0.0274273
\(716\) −46893.7 −2.44762
\(717\) 9031.13 0.470396
\(718\) 1148.89 0.0597164
\(719\) −26753.0 −1.38765 −0.693824 0.720144i \(-0.744075\pi\)
−0.693824 + 0.720144i \(0.744075\pi\)
\(720\) 667.202 0.0345349
\(721\) 22711.1 1.17310
\(722\) −9895.35 −0.510065
\(723\) 5776.06 0.297115
\(724\) −8105.74 −0.416087
\(725\) −28539.1 −1.46195
\(726\) 1794.23 0.0917221
\(727\) −11951.9 −0.609725 −0.304863 0.952396i \(-0.598611\pi\)
−0.304863 + 0.952396i \(0.598611\pi\)
\(728\) 47813.3 2.43417
\(729\) 729.000 0.0370370
\(730\) 5177.51 0.262505
\(731\) −30497.0 −1.54305
\(732\) 3006.91 0.151828
\(733\) 20469.5 1.03146 0.515729 0.856751i \(-0.327521\pi\)
0.515729 + 0.856751i \(0.327521\pi\)
\(734\) 54145.3 2.72280
\(735\) −686.454 −0.0344493
\(736\) 2297.50 0.115064
\(737\) −9318.28 −0.465730
\(738\) −9413.89 −0.469553
\(739\) −17968.0 −0.894402 −0.447201 0.894433i \(-0.647579\pi\)
−0.447201 + 0.894433i \(0.647579\pi\)
\(740\) 1040.46 0.0516867
\(741\) 13534.8 0.671004
\(742\) −20027.8 −0.990896
\(743\) 29478.7 1.45554 0.727771 0.685820i \(-0.240556\pi\)
0.727771 + 0.685820i \(0.240556\pi\)
\(744\) −14600.9 −0.719482
\(745\) −1999.17 −0.0983138
\(746\) 26804.7 1.31554
\(747\) 188.042 0.00921033
\(748\) 16182.0 0.791008
\(749\) −12495.7 −0.609592
\(750\) −3672.58 −0.178805
\(751\) −38472.7 −1.86936 −0.934680 0.355491i \(-0.884314\pi\)
−0.934680 + 0.355491i \(0.884314\pi\)
\(752\) −28292.2 −1.37196
\(753\) −17290.1 −0.836770
\(754\) 54518.4 2.63321
\(755\) 216.963 0.0104584
\(756\) −10620.1 −0.510914
\(757\) −27092.8 −1.30080 −0.650399 0.759592i \(-0.725398\pi\)
−0.650399 + 0.759592i \(0.725398\pi\)
\(758\) 30388.9 1.45617
\(759\) 2165.07 0.103540
\(760\) −3901.76 −0.186226
\(761\) 9383.97 0.447002 0.223501 0.974704i \(-0.428251\pi\)
0.223501 + 0.974704i \(0.428251\pi\)
\(762\) −8882.18 −0.422267
\(763\) −43953.0 −2.08546
\(764\) 46271.9 2.19118
\(765\) 801.445 0.0378775
\(766\) −61831.5 −2.91653
\(767\) 9531.55 0.448715
\(768\) 25014.4 1.17530
\(769\) 28687.2 1.34524 0.672619 0.739989i \(-0.265169\pi\)
0.672619 + 0.739989i \(0.265169\pi\)
\(770\) −1294.56 −0.0605879
\(771\) −6598.34 −0.308215
\(772\) 69773.6 3.25286
\(773\) 27568.6 1.28276 0.641381 0.767223i \(-0.278362\pi\)
0.641381 + 0.767223i \(0.278362\pi\)
\(774\) 15153.0 0.703701
\(775\) 14482.9 0.671281
\(776\) −3188.16 −0.147485
\(777\) −4572.12 −0.211099
\(778\) −23473.2 −1.08169
\(779\) 19920.3 0.916197
\(780\) 2349.85 0.107869
\(781\) −11779.3 −0.539689
\(782\) 29033.7 1.32768
\(783\) −6213.61 −0.283597
\(784\) 17146.9 0.781110
\(785\) 25.8647 0.00117599
\(786\) 14888.0 0.675618
\(787\) −35460.2 −1.60612 −0.803062 0.595896i \(-0.796797\pi\)
−0.803062 + 0.595896i \(0.796797\pi\)
\(788\) −77342.6 −3.49646
\(789\) 16726.0 0.754703
\(790\) 972.417 0.0437937
\(791\) −25251.3 −1.13506
\(792\) −4125.68 −0.185101
\(793\) 2923.62 0.130921
\(794\) 39937.0 1.78502
\(795\) −505.062 −0.0225317
\(796\) 43195.2 1.92339
\(797\) −21603.2 −0.960131 −0.480066 0.877233i \(-0.659387\pi\)
−0.480066 + 0.877233i \(0.659387\pi\)
\(798\) 33414.3 1.48227
\(799\) −33984.7 −1.50475
\(800\) 4342.65 0.191920
\(801\) 2856.87 0.126020
\(802\) −13700.5 −0.603218
\(803\) −11584.6 −0.509107
\(804\) 41757.4 1.83168
\(805\) −1562.13 −0.0683947
\(806\) −27666.9 −1.20909
\(807\) 9937.03 0.433457
\(808\) −20505.6 −0.892801
\(809\) 26195.0 1.13840 0.569200 0.822199i \(-0.307253\pi\)
0.569200 + 0.822199i \(0.307253\pi\)
\(810\) −398.214 −0.0172738
\(811\) −34492.8 −1.49347 −0.746735 0.665121i \(-0.768380\pi\)
−0.746735 + 0.665121i \(0.768380\pi\)
\(812\) 90520.5 3.91213
\(813\) −2692.92 −0.116168
\(814\) −3461.49 −0.149048
\(815\) −1557.82 −0.0669549
\(816\) −20019.3 −0.858842
\(817\) −32064.6 −1.37307
\(818\) 28496.0 1.21802
\(819\) −10326.0 −0.440560
\(820\) 3458.46 0.147286
\(821\) −34296.1 −1.45791 −0.728954 0.684563i \(-0.759993\pi\)
−0.728954 + 0.684563i \(0.759993\pi\)
\(822\) 27768.3 1.17826
\(823\) −27005.1 −1.14379 −0.571894 0.820328i \(-0.693791\pi\)
−0.571894 + 0.820328i \(0.693791\pi\)
\(824\) −39536.8 −1.67151
\(825\) 4092.35 0.172700
\(826\) 23531.2 0.991228
\(827\) −18435.1 −0.775153 −0.387576 0.921838i \(-0.626688\pi\)
−0.387576 + 0.921838i \(0.626688\pi\)
\(828\) −9702.19 −0.407216
\(829\) 27.5036 0.00115228 0.000576139 1.00000i \(-0.499817\pi\)
0.000576139 1.00000i \(0.499817\pi\)
\(830\) −102.718 −0.00429564
\(831\) 16848.0 0.703308
\(832\) 20282.4 0.845153
\(833\) 20597.0 0.856713
\(834\) −4429.34 −0.183903
\(835\) 1382.81 0.0573102
\(836\) 17013.8 0.703870
\(837\) 3153.27 0.130219
\(838\) 26262.3 1.08260
\(839\) 41533.5 1.70905 0.854526 0.519409i \(-0.173848\pi\)
0.854526 + 0.519409i \(0.173848\pi\)
\(840\) 2976.73 0.122270
\(841\) 28572.6 1.17154
\(842\) −55117.6 −2.25591
\(843\) 9334.37 0.381368
\(844\) 78893.1 3.21755
\(845\) 99.5664 0.00405348
\(846\) 16886.0 0.686231
\(847\) 2896.57 0.117506
\(848\) 12615.9 0.510888
\(849\) −10272.2 −0.415245
\(850\) 54878.7 2.21450
\(851\) −4176.93 −0.168253
\(852\) 52785.8 2.12255
\(853\) −1143.31 −0.0458922 −0.0229461 0.999737i \(-0.507305\pi\)
−0.0229461 + 0.999737i \(0.507305\pi\)
\(854\) 7217.72 0.289210
\(855\) 842.641 0.0337049
\(856\) 21753.2 0.868587
\(857\) 9107.41 0.363014 0.181507 0.983390i \(-0.441902\pi\)
0.181507 + 0.983390i \(0.441902\pi\)
\(858\) −7817.66 −0.311061
\(859\) −28420.4 −1.12886 −0.564430 0.825481i \(-0.690904\pi\)
−0.564430 + 0.825481i \(0.690904\pi\)
\(860\) −5566.90 −0.220732
\(861\) −15197.6 −0.601546
\(862\) −7461.70 −0.294833
\(863\) −31.5674 −0.00124515 −0.000622577 1.00000i \(-0.500198\pi\)
−0.000622577 1.00000i \(0.500198\pi\)
\(864\) 945.496 0.0372297
\(865\) 274.653 0.0107959
\(866\) 1028.72 0.0403663
\(867\) −9308.25 −0.364619
\(868\) −45937.2 −1.79632
\(869\) −2175.77 −0.0849345
\(870\) 3394.17 0.132268
\(871\) 40600.7 1.57945
\(872\) 76515.7 2.97150
\(873\) 688.530 0.0266933
\(874\) 30526.1 1.18142
\(875\) −5928.92 −0.229067
\(876\) 51913.4 2.00227
\(877\) −23337.9 −0.898592 −0.449296 0.893383i \(-0.648325\pi\)
−0.449296 + 0.893383i \(0.648325\pi\)
\(878\) 74152.3 2.85025
\(879\) −8199.08 −0.314617
\(880\) 815.469 0.0312380
\(881\) −8155.91 −0.311895 −0.155948 0.987765i \(-0.549843\pi\)
−0.155948 + 0.987765i \(0.549843\pi\)
\(882\) −10234.0 −0.390699
\(883\) −19097.3 −0.727831 −0.363916 0.931432i \(-0.618560\pi\)
−0.363916 + 0.931432i \(0.618560\pi\)
\(884\) −70506.8 −2.68258
\(885\) 593.409 0.0225392
\(886\) 78753.1 2.98619
\(887\) −40907.0 −1.54850 −0.774252 0.632877i \(-0.781874\pi\)
−0.774252 + 0.632877i \(0.781874\pi\)
\(888\) 7959.40 0.300788
\(889\) −14339.2 −0.540968
\(890\) −1560.55 −0.0587752
\(891\) 891.000 0.0335013
\(892\) 13844.5 0.519672
\(893\) −35731.6 −1.33898
\(894\) −29804.6 −1.11500
\(895\) −2838.60 −0.106016
\(896\) 56778.9 2.11702
\(897\) −9433.45 −0.351141
\(898\) 16658.2 0.619031
\(899\) −26876.8 −0.997100
\(900\) −18338.8 −0.679214
\(901\) 15154.3 0.560336
\(902\) −11505.9 −0.424727
\(903\) 24462.7 0.901514
\(904\) 43958.9 1.61731
\(905\) −490.663 −0.0180223
\(906\) 3234.60 0.118612
\(907\) −7.84024 −0.000287024 0 −0.000143512 1.00000i \(-0.500046\pi\)
−0.000143512 1.00000i \(0.500046\pi\)
\(908\) 37599.4 1.37421
\(909\) 4428.47 0.161588
\(910\) 5640.53 0.205474
\(911\) 24894.6 0.905373 0.452687 0.891670i \(-0.350466\pi\)
0.452687 + 0.891670i \(0.350466\pi\)
\(912\) −21048.3 −0.764232
\(913\) 229.830 0.00833105
\(914\) 92240.2 3.33811
\(915\) 182.016 0.00657626
\(916\) 43160.0 1.55682
\(917\) 24034.8 0.865537
\(918\) 11948.4 0.429580
\(919\) −27657.3 −0.992742 −0.496371 0.868110i \(-0.665334\pi\)
−0.496371 + 0.868110i \(0.665334\pi\)
\(920\) 2719.43 0.0974533
\(921\) −7175.85 −0.256734
\(922\) −47365.8 −1.69188
\(923\) 51323.7 1.83027
\(924\) −12980.2 −0.462139
\(925\) −7895.11 −0.280637
\(926\) −23378.9 −0.829674
\(927\) 8538.53 0.302527
\(928\) −8058.91 −0.285072
\(929\) −27859.9 −0.983910 −0.491955 0.870621i \(-0.663717\pi\)
−0.491955 + 0.870621i \(0.663717\pi\)
\(930\) −1722.46 −0.0607332
\(931\) 21655.7 0.762337
\(932\) −15927.7 −0.559796
\(933\) −29979.9 −1.05198
\(934\) 46912.7 1.64350
\(935\) 979.544 0.0342615
\(936\) 17976.0 0.627740
\(937\) −19777.0 −0.689528 −0.344764 0.938689i \(-0.612041\pi\)
−0.344764 + 0.938689i \(0.612041\pi\)
\(938\) 100234. 3.48907
\(939\) 3004.62 0.104422
\(940\) −6203.54 −0.215252
\(941\) −23726.0 −0.821938 −0.410969 0.911649i \(-0.634810\pi\)
−0.410969 + 0.911649i \(0.634810\pi\)
\(942\) 385.605 0.0133372
\(943\) −13884.0 −0.479452
\(944\) −14822.7 −0.511059
\(945\) −642.867 −0.0221296
\(946\) 18520.4 0.636522
\(947\) −19624.9 −0.673414 −0.336707 0.941609i \(-0.609313\pi\)
−0.336707 + 0.941609i \(0.609313\pi\)
\(948\) 9750.14 0.334040
\(949\) 50475.4 1.72656
\(950\) 57699.5 1.97055
\(951\) 3090.73 0.105388
\(952\) −89316.2 −3.04071
\(953\) −1209.15 −0.0410999 −0.0205500 0.999789i \(-0.506542\pi\)
−0.0205500 + 0.999789i \(0.506542\pi\)
\(954\) −7529.71 −0.255538
\(955\) 2800.97 0.0949081
\(956\) −49464.0 −1.67341
\(957\) −7594.42 −0.256523
\(958\) 54810.4 1.84848
\(959\) 44828.5 1.50948
\(960\) 1262.73 0.0424525
\(961\) −16151.6 −0.542164
\(962\) 15082.1 0.505474
\(963\) −4697.93 −0.157205
\(964\) −31635.8 −1.05697
\(965\) 4223.59 0.140893
\(966\) −23289.0 −0.775683
\(967\) −56991.1 −1.89525 −0.947627 0.319378i \(-0.896526\pi\)
−0.947627 + 0.319378i \(0.896526\pi\)
\(968\) −5042.50 −0.167430
\(969\) −25283.3 −0.838202
\(970\) −376.107 −0.0124496
\(971\) −33215.1 −1.09776 −0.548879 0.835902i \(-0.684945\pi\)
−0.548879 + 0.835902i \(0.684945\pi\)
\(972\) −3992.78 −0.131758
\(973\) −7150.61 −0.235599
\(974\) 63630.3 2.09327
\(975\) −17830.8 −0.585685
\(976\) −4546.58 −0.149111
\(977\) −16932.1 −0.554459 −0.277229 0.960804i \(-0.589416\pi\)
−0.277229 + 0.960804i \(0.589416\pi\)
\(978\) −23224.8 −0.759354
\(979\) 3491.73 0.113990
\(980\) 3759.75 0.122552
\(981\) −16524.7 −0.537810
\(982\) 27513.1 0.894071
\(983\) 25288.2 0.820518 0.410259 0.911969i \(-0.365438\pi\)
0.410259 + 0.911969i \(0.365438\pi\)
\(984\) 26456.7 0.857123
\(985\) −4681.76 −0.151445
\(986\) −101842. −3.28934
\(987\) 27260.3 0.879134
\(988\) −74131.0 −2.38707
\(989\) 22348.3 0.718537
\(990\) −486.706 −0.0156248
\(991\) −22868.3 −0.733033 −0.366517 0.930411i \(-0.619450\pi\)
−0.366517 + 0.930411i \(0.619450\pi\)
\(992\) 4089.72 0.130896
\(993\) 23778.0 0.759891
\(994\) 126706. 4.04313
\(995\) 2614.73 0.0833090
\(996\) −1029.92 −0.0327653
\(997\) 41127.5 1.30644 0.653219 0.757169i \(-0.273418\pi\)
0.653219 + 0.757169i \(0.273418\pi\)
\(998\) −20735.6 −0.657690
\(999\) −1718.95 −0.0544395
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.d.1.4 37
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.4.a.d.1.4 37 1.1 even 1 trivial