Properties

Label 1521.4.a.bh.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
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] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.39012\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.390115 q^{2} -7.84781 q^{4} -7.52136 q^{5} -19.5446 q^{7} +6.18247 q^{8} +O(q^{10})\) \(q-0.390115 q^{2} -7.84781 q^{4} -7.52136 q^{5} -19.5446 q^{7} +6.18247 q^{8} +2.93420 q^{10} +45.8243 q^{11} +7.62463 q^{14} +60.3706 q^{16} -86.5200 q^{17} -148.737 q^{19} +59.0262 q^{20} -17.8767 q^{22} +91.5351 q^{23} -68.4292 q^{25} +153.382 q^{28} -258.901 q^{29} -31.2317 q^{31} -73.0113 q^{32} +33.7528 q^{34} +147.002 q^{35} -148.436 q^{37} +58.0245 q^{38} -46.5006 q^{40} +95.9135 q^{41} -80.9930 q^{43} -359.620 q^{44} -35.7093 q^{46} +94.3777 q^{47} +38.9898 q^{49} +26.6953 q^{50} -493.555 q^{53} -344.661 q^{55} -120.834 q^{56} +101.001 q^{58} -575.686 q^{59} -40.2865 q^{61} +12.1840 q^{62} -454.482 q^{64} -601.141 q^{67} +678.992 q^{68} -57.3476 q^{70} -518.800 q^{71} +1055.21 q^{73} +57.9072 q^{74} +1167.26 q^{76} -895.615 q^{77} -320.840 q^{79} -454.069 q^{80} -37.4173 q^{82} -32.4841 q^{83} +650.748 q^{85} +31.5966 q^{86} +283.307 q^{88} +450.795 q^{89} -718.350 q^{92} -36.8182 q^{94} +1118.70 q^{95} -231.743 q^{97} -15.2105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{2} + 37 q^{4} + 30 q^{5} - 38 q^{7} + 60 q^{8} - 147 q^{10} + 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} - 161 q^{19} + 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} - 344 q^{28} - 280 q^{29} - 706 q^{31} + 680 q^{32} - 216 q^{34} - 20 q^{35} - 298 q^{37} + 739 q^{38} + 13 q^{40} + 1201 q^{41} - 533 q^{43} + 355 q^{44} + 840 q^{46} + 956 q^{47} + 403 q^{49} - 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} + 2877 q^{58} + 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} + 931 q^{67} + 1536 q^{68} + 4854 q^{70} + 2046 q^{71} + 45 q^{73} + 1990 q^{74} + 3608 q^{76} + 718 q^{77} + 412 q^{79} - 787 q^{80} + 2757 q^{82} + 3709 q^{83} + 2106 q^{85} + 125 q^{86} - 636 q^{88} + 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} + 1087 q^{97} - 282 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.390115 −0.137927 −0.0689633 0.997619i \(-0.521969\pi\)
−0.0689633 + 0.997619i \(0.521969\pi\)
\(3\) 0 0
\(4\) −7.84781 −0.980976
\(5\) −7.52136 −0.672731 −0.336365 0.941732i \(-0.609198\pi\)
−0.336365 + 0.941732i \(0.609198\pi\)
\(6\) 0 0
\(7\) −19.5446 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(8\) 6.18247 0.273229
\(9\) 0 0
\(10\) 2.93420 0.0927875
\(11\) 45.8243 1.25605 0.628024 0.778194i \(-0.283864\pi\)
0.628024 + 0.778194i \(0.283864\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 7.62463 0.145555
\(15\) 0 0
\(16\) 60.3706 0.943291
\(17\) −86.5200 −1.23436 −0.617182 0.786821i \(-0.711726\pi\)
−0.617182 + 0.786821i \(0.711726\pi\)
\(18\) 0 0
\(19\) −148.737 −1.79593 −0.897963 0.440072i \(-0.854953\pi\)
−0.897963 + 0.440072i \(0.854953\pi\)
\(20\) 59.0262 0.659933
\(21\) 0 0
\(22\) −17.8767 −0.173242
\(23\) 91.5351 0.829843 0.414922 0.909857i \(-0.363809\pi\)
0.414922 + 0.909857i \(0.363809\pi\)
\(24\) 0 0
\(25\) −68.4292 −0.547433
\(26\) 0 0
\(27\) 0 0
\(28\) 153.382 1.03523
\(29\) −258.901 −1.65782 −0.828909 0.559383i \(-0.811038\pi\)
−0.828909 + 0.559383i \(0.811038\pi\)
\(30\) 0 0
\(31\) −31.2317 −0.180948 −0.0904739 0.995899i \(-0.528838\pi\)
−0.0904739 + 0.995899i \(0.528838\pi\)
\(32\) −73.0113 −0.403334
\(33\) 0 0
\(34\) 33.7528 0.170252
\(35\) 147.002 0.709938
\(36\) 0 0
\(37\) −148.436 −0.659533 −0.329767 0.944062i \(-0.606970\pi\)
−0.329767 + 0.944062i \(0.606970\pi\)
\(38\) 58.0245 0.247706
\(39\) 0 0
\(40\) −46.5006 −0.183810
\(41\) 95.9135 0.365346 0.182673 0.983174i \(-0.441525\pi\)
0.182673 + 0.983174i \(0.441525\pi\)
\(42\) 0 0
\(43\) −80.9930 −0.287240 −0.143620 0.989633i \(-0.545874\pi\)
−0.143620 + 0.989633i \(0.545874\pi\)
\(44\) −359.620 −1.23215
\(45\) 0 0
\(46\) −35.7093 −0.114457
\(47\) 94.3777 0.292902 0.146451 0.989218i \(-0.453215\pi\)
0.146451 + 0.989218i \(0.453215\pi\)
\(48\) 0 0
\(49\) 38.9898 0.113673
\(50\) 26.6953 0.0755056
\(51\) 0 0
\(52\) 0 0
\(53\) −493.555 −1.27915 −0.639575 0.768729i \(-0.720890\pi\)
−0.639575 + 0.768729i \(0.720890\pi\)
\(54\) 0 0
\(55\) −344.661 −0.844983
\(56\) −120.834 −0.288341
\(57\) 0 0
\(58\) 101.001 0.228657
\(59\) −575.686 −1.27030 −0.635152 0.772387i \(-0.719062\pi\)
−0.635152 + 0.772387i \(0.719062\pi\)
\(60\) 0 0
\(61\) −40.2865 −0.0845599 −0.0422800 0.999106i \(-0.513462\pi\)
−0.0422800 + 0.999106i \(0.513462\pi\)
\(62\) 12.1840 0.0249575
\(63\) 0 0
\(64\) −454.482 −0.887660
\(65\) 0 0
\(66\) 0 0
\(67\) −601.141 −1.09613 −0.548067 0.836434i \(-0.684636\pi\)
−0.548067 + 0.836434i \(0.684636\pi\)
\(68\) 678.992 1.21088
\(69\) 0 0
\(70\) −57.3476 −0.0979193
\(71\) −518.800 −0.867187 −0.433593 0.901109i \(-0.642755\pi\)
−0.433593 + 0.901109i \(0.642755\pi\)
\(72\) 0 0
\(73\) 1055.21 1.69182 0.845908 0.533328i \(-0.179059\pi\)
0.845908 + 0.533328i \(0.179059\pi\)
\(74\) 57.9072 0.0909672
\(75\) 0 0
\(76\) 1167.26 1.76176
\(77\) −895.615 −1.32552
\(78\) 0 0
\(79\) −320.840 −0.456928 −0.228464 0.973552i \(-0.573370\pi\)
−0.228464 + 0.973552i \(0.573370\pi\)
\(80\) −454.069 −0.634581
\(81\) 0 0
\(82\) −37.4173 −0.0503909
\(83\) −32.4841 −0.0429590 −0.0214795 0.999769i \(-0.506838\pi\)
−0.0214795 + 0.999769i \(0.506838\pi\)
\(84\) 0 0
\(85\) 650.748 0.830394
\(86\) 31.5966 0.0396180
\(87\) 0 0
\(88\) 283.307 0.343189
\(89\) 450.795 0.536901 0.268450 0.963293i \(-0.413488\pi\)
0.268450 + 0.963293i \(0.413488\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −718.350 −0.814057
\(93\) 0 0
\(94\) −36.8182 −0.0403990
\(95\) 1118.70 1.20817
\(96\) 0 0
\(97\) −231.743 −0.242576 −0.121288 0.992617i \(-0.538702\pi\)
−0.121288 + 0.992617i \(0.538702\pi\)
\(98\) −15.2105 −0.0156785
\(99\) 0 0
\(100\) 537.019 0.537019
\(101\) −570.125 −0.561679 −0.280840 0.959755i \(-0.590613\pi\)
−0.280840 + 0.959755i \(0.590613\pi\)
\(102\) 0 0
\(103\) 969.551 0.927502 0.463751 0.885965i \(-0.346503\pi\)
0.463751 + 0.885965i \(0.346503\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 192.543 0.176429
\(107\) 343.156 0.310039 0.155019 0.987911i \(-0.450456\pi\)
0.155019 + 0.987911i \(0.450456\pi\)
\(108\) 0 0
\(109\) 83.1640 0.0730795 0.0365398 0.999332i \(-0.488366\pi\)
0.0365398 + 0.999332i \(0.488366\pi\)
\(110\) 134.457 0.116546
\(111\) 0 0
\(112\) −1179.92 −0.995461
\(113\) 2116.18 1.76171 0.880856 0.473384i \(-0.156968\pi\)
0.880856 + 0.473384i \(0.156968\pi\)
\(114\) 0 0
\(115\) −688.469 −0.558261
\(116\) 2031.81 1.62628
\(117\) 0 0
\(118\) 224.584 0.175209
\(119\) 1691.00 1.30263
\(120\) 0 0
\(121\) 768.863 0.577658
\(122\) 15.7164 0.0116631
\(123\) 0 0
\(124\) 245.101 0.177506
\(125\) 1454.85 1.04101
\(126\) 0 0
\(127\) −1176.69 −0.822161 −0.411081 0.911599i \(-0.634848\pi\)
−0.411081 + 0.911599i \(0.634848\pi\)
\(128\) 761.391 0.525766
\(129\) 0 0
\(130\) 0 0
\(131\) −775.336 −0.517110 −0.258555 0.965996i \(-0.583246\pi\)
−0.258555 + 0.965996i \(0.583246\pi\)
\(132\) 0 0
\(133\) 2907.00 1.89525
\(134\) 234.514 0.151186
\(135\) 0 0
\(136\) −534.907 −0.337264
\(137\) 2542.62 1.58563 0.792814 0.609464i \(-0.208615\pi\)
0.792814 + 0.609464i \(0.208615\pi\)
\(138\) 0 0
\(139\) −286.317 −0.174713 −0.0873564 0.996177i \(-0.527842\pi\)
−0.0873564 + 0.996177i \(0.527842\pi\)
\(140\) −1153.64 −0.696432
\(141\) 0 0
\(142\) 202.392 0.119608
\(143\) 0 0
\(144\) 0 0
\(145\) 1947.29 1.11527
\(146\) −411.652 −0.233346
\(147\) 0 0
\(148\) 1164.90 0.646987
\(149\) 2354.13 1.29435 0.647173 0.762343i \(-0.275951\pi\)
0.647173 + 0.762343i \(0.275951\pi\)
\(150\) 0 0
\(151\) 165.158 0.0890089 0.0445045 0.999009i \(-0.485829\pi\)
0.0445045 + 0.999009i \(0.485829\pi\)
\(152\) −919.562 −0.490699
\(153\) 0 0
\(154\) 349.393 0.182824
\(155\) 234.905 0.121729
\(156\) 0 0
\(157\) −3095.72 −1.57367 −0.786833 0.617166i \(-0.788281\pi\)
−0.786833 + 0.617166i \(0.788281\pi\)
\(158\) 125.165 0.0630226
\(159\) 0 0
\(160\) 549.144 0.271335
\(161\) −1789.01 −0.875740
\(162\) 0 0
\(163\) 299.310 0.143827 0.0719134 0.997411i \(-0.477089\pi\)
0.0719134 + 0.997411i \(0.477089\pi\)
\(164\) −752.711 −0.358395
\(165\) 0 0
\(166\) 12.6726 0.00592519
\(167\) 3005.17 1.39250 0.696249 0.717801i \(-0.254851\pi\)
0.696249 + 0.717801i \(0.254851\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −253.867 −0.114533
\(171\) 0 0
\(172\) 635.618 0.281776
\(173\) −455.853 −0.200334 −0.100167 0.994971i \(-0.531938\pi\)
−0.100167 + 0.994971i \(0.531938\pi\)
\(174\) 0 0
\(175\) 1337.42 0.577710
\(176\) 2766.44 1.18482
\(177\) 0 0
\(178\) −175.862 −0.0740529
\(179\) −1364.19 −0.569635 −0.284818 0.958582i \(-0.591933\pi\)
−0.284818 + 0.958582i \(0.591933\pi\)
\(180\) 0 0
\(181\) −2026.11 −0.832041 −0.416021 0.909355i \(-0.636576\pi\)
−0.416021 + 0.909355i \(0.636576\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 565.914 0.226738
\(185\) 1116.44 0.443688
\(186\) 0 0
\(187\) −3964.71 −1.55042
\(188\) −740.658 −0.287330
\(189\) 0 0
\(190\) −436.423 −0.166639
\(191\) −2161.06 −0.818686 −0.409343 0.912380i \(-0.634242\pi\)
−0.409343 + 0.912380i \(0.634242\pi\)
\(192\) 0 0
\(193\) 1207.88 0.450491 0.225246 0.974302i \(-0.427682\pi\)
0.225246 + 0.974302i \(0.427682\pi\)
\(194\) 90.4063 0.0334577
\(195\) 0 0
\(196\) −305.985 −0.111510
\(197\) 4926.76 1.78181 0.890906 0.454187i \(-0.150070\pi\)
0.890906 + 0.454187i \(0.150070\pi\)
\(198\) 0 0
\(199\) 1009.45 0.359589 0.179795 0.983704i \(-0.442457\pi\)
0.179795 + 0.983704i \(0.442457\pi\)
\(200\) −423.061 −0.149575
\(201\) 0 0
\(202\) 222.415 0.0774705
\(203\) 5060.11 1.74951
\(204\) 0 0
\(205\) −721.400 −0.245779
\(206\) −378.237 −0.127927
\(207\) 0 0
\(208\) 0 0
\(209\) −6815.76 −2.25577
\(210\) 0 0
\(211\) −4911.32 −1.60241 −0.801206 0.598388i \(-0.795808\pi\)
−0.801206 + 0.598388i \(0.795808\pi\)
\(212\) 3873.32 1.25482
\(213\) 0 0
\(214\) −133.871 −0.0427626
\(215\) 609.178 0.193235
\(216\) 0 0
\(217\) 610.410 0.190956
\(218\) −32.4435 −0.0100796
\(219\) 0 0
\(220\) 2704.83 0.828908
\(221\) 0 0
\(222\) 0 0
\(223\) 1364.58 0.409771 0.204885 0.978786i \(-0.434318\pi\)
0.204885 + 0.978786i \(0.434318\pi\)
\(224\) 1426.97 0.425641
\(225\) 0 0
\(226\) −825.554 −0.242987
\(227\) −4169.88 −1.21923 −0.609614 0.792699i \(-0.708675\pi\)
−0.609614 + 0.792699i \(0.708675\pi\)
\(228\) 0 0
\(229\) 3506.89 1.01197 0.505987 0.862541i \(-0.331128\pi\)
0.505987 + 0.862541i \(0.331128\pi\)
\(230\) 268.582 0.0769991
\(231\) 0 0
\(232\) −1600.65 −0.452965
\(233\) 570.253 0.160337 0.0801684 0.996781i \(-0.474454\pi\)
0.0801684 + 0.996781i \(0.474454\pi\)
\(234\) 0 0
\(235\) −709.848 −0.197044
\(236\) 4517.87 1.24614
\(237\) 0 0
\(238\) −659.683 −0.179668
\(239\) −231.056 −0.0625347 −0.0312674 0.999511i \(-0.509954\pi\)
−0.0312674 + 0.999511i \(0.509954\pi\)
\(240\) 0 0
\(241\) 3088.50 0.825508 0.412754 0.910842i \(-0.364567\pi\)
0.412754 + 0.910842i \(0.364567\pi\)
\(242\) −299.945 −0.0796744
\(243\) 0 0
\(244\) 316.161 0.0829513
\(245\) −293.256 −0.0764713
\(246\) 0 0
\(247\) 0 0
\(248\) −193.089 −0.0494403
\(249\) 0 0
\(250\) −567.559 −0.143582
\(251\) 2445.64 0.615009 0.307504 0.951547i \(-0.400506\pi\)
0.307504 + 0.951547i \(0.400506\pi\)
\(252\) 0 0
\(253\) 4194.53 1.04232
\(254\) 459.045 0.113398
\(255\) 0 0
\(256\) 3338.83 0.815143
\(257\) −3273.99 −0.794654 −0.397327 0.917677i \(-0.630062\pi\)
−0.397327 + 0.917677i \(0.630062\pi\)
\(258\) 0 0
\(259\) 2901.12 0.696010
\(260\) 0 0
\(261\) 0 0
\(262\) 302.471 0.0713233
\(263\) 4631.27 1.08584 0.542921 0.839784i \(-0.317318\pi\)
0.542921 + 0.839784i \(0.317318\pi\)
\(264\) 0 0
\(265\) 3712.20 0.860524
\(266\) −1134.06 −0.261406
\(267\) 0 0
\(268\) 4717.64 1.07528
\(269\) 2838.51 0.643373 0.321686 0.946846i \(-0.395750\pi\)
0.321686 + 0.946846i \(0.395750\pi\)
\(270\) 0 0
\(271\) 7052.31 1.58080 0.790401 0.612589i \(-0.209872\pi\)
0.790401 + 0.612589i \(0.209872\pi\)
\(272\) −5223.26 −1.16436
\(273\) 0 0
\(274\) −991.917 −0.218700
\(275\) −3135.72 −0.687603
\(276\) 0 0
\(277\) −1938.31 −0.420439 −0.210220 0.977654i \(-0.567418\pi\)
−0.210220 + 0.977654i \(0.567418\pi\)
\(278\) 111.697 0.0240975
\(279\) 0 0
\(280\) 908.834 0.193976
\(281\) 3290.74 0.698609 0.349305 0.937009i \(-0.386418\pi\)
0.349305 + 0.937009i \(0.386418\pi\)
\(282\) 0 0
\(283\) −7918.08 −1.66318 −0.831592 0.555388i \(-0.812570\pi\)
−0.831592 + 0.555388i \(0.812570\pi\)
\(284\) 4071.45 0.850690
\(285\) 0 0
\(286\) 0 0
\(287\) −1874.59 −0.385552
\(288\) 0 0
\(289\) 2572.71 0.523653
\(290\) −759.667 −0.153825
\(291\) 0 0
\(292\) −8281.06 −1.65963
\(293\) −5675.43 −1.13161 −0.565806 0.824539i \(-0.691435\pi\)
−0.565806 + 0.824539i \(0.691435\pi\)
\(294\) 0 0
\(295\) 4329.94 0.854572
\(296\) −917.702 −0.180204
\(297\) 0 0
\(298\) −918.381 −0.178525
\(299\) 0 0
\(300\) 0 0
\(301\) 1582.97 0.303126
\(302\) −64.4306 −0.0122767
\(303\) 0 0
\(304\) −8979.33 −1.69408
\(305\) 303.009 0.0568861
\(306\) 0 0
\(307\) −4338.86 −0.806618 −0.403309 0.915064i \(-0.632140\pi\)
−0.403309 + 0.915064i \(0.632140\pi\)
\(308\) 7028.62 1.30030
\(309\) 0 0
\(310\) −91.6401 −0.0167897
\(311\) 5234.75 0.954454 0.477227 0.878780i \(-0.341642\pi\)
0.477227 + 0.878780i \(0.341642\pi\)
\(312\) 0 0
\(313\) 2167.86 0.391484 0.195742 0.980655i \(-0.437288\pi\)
0.195742 + 0.980655i \(0.437288\pi\)
\(314\) 1207.69 0.217050
\(315\) 0 0
\(316\) 2517.89 0.448236
\(317\) −4863.71 −0.861744 −0.430872 0.902413i \(-0.641794\pi\)
−0.430872 + 0.902413i \(0.641794\pi\)
\(318\) 0 0
\(319\) −11864.0 −2.08230
\(320\) 3418.32 0.597156
\(321\) 0 0
\(322\) 697.922 0.120788
\(323\) 12868.7 2.21682
\(324\) 0 0
\(325\) 0 0
\(326\) −116.765 −0.0198375
\(327\) 0 0
\(328\) 592.983 0.0998231
\(329\) −1844.57 −0.309102
\(330\) 0 0
\(331\) −2685.26 −0.445908 −0.222954 0.974829i \(-0.571570\pi\)
−0.222954 + 0.974829i \(0.571570\pi\)
\(332\) 254.929 0.0421417
\(333\) 0 0
\(334\) −1172.36 −0.192062
\(335\) 4521.39 0.737404
\(336\) 0 0
\(337\) −6518.36 −1.05364 −0.526821 0.849976i \(-0.676616\pi\)
−0.526821 + 0.849976i \(0.676616\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −5106.95 −0.814597
\(341\) −1431.17 −0.227279
\(342\) 0 0
\(343\) 5941.75 0.935347
\(344\) −500.737 −0.0784824
\(345\) 0 0
\(346\) 177.835 0.0276314
\(347\) −75.7963 −0.0117261 −0.00586305 0.999983i \(-0.501866\pi\)
−0.00586305 + 0.999983i \(0.501866\pi\)
\(348\) 0 0
\(349\) −3682.09 −0.564750 −0.282375 0.959304i \(-0.591122\pi\)
−0.282375 + 0.959304i \(0.591122\pi\)
\(350\) −521.747 −0.0796816
\(351\) 0 0
\(352\) −3345.69 −0.506607
\(353\) −10031.5 −1.51254 −0.756268 0.654261i \(-0.772980\pi\)
−0.756268 + 0.654261i \(0.772980\pi\)
\(354\) 0 0
\(355\) 3902.08 0.583383
\(356\) −3537.75 −0.526687
\(357\) 0 0
\(358\) 532.193 0.0785678
\(359\) 6869.76 1.00995 0.504975 0.863134i \(-0.331502\pi\)
0.504975 + 0.863134i \(0.331502\pi\)
\(360\) 0 0
\(361\) 15263.7 2.22535
\(362\) 790.416 0.114761
\(363\) 0 0
\(364\) 0 0
\(365\) −7936.59 −1.13814
\(366\) 0 0
\(367\) 8883.29 1.26350 0.631749 0.775173i \(-0.282337\pi\)
0.631749 + 0.775173i \(0.282337\pi\)
\(368\) 5526.03 0.782784
\(369\) 0 0
\(370\) −435.541 −0.0611964
\(371\) 9646.31 1.34990
\(372\) 0 0
\(373\) −5454.50 −0.757166 −0.378583 0.925567i \(-0.623589\pi\)
−0.378583 + 0.925567i \(0.623589\pi\)
\(374\) 1546.70 0.213844
\(375\) 0 0
\(376\) 583.487 0.0800294
\(377\) 0 0
\(378\) 0 0
\(379\) 6642.15 0.900222 0.450111 0.892973i \(-0.351384\pi\)
0.450111 + 0.892973i \(0.351384\pi\)
\(380\) −8779.37 −1.18519
\(381\) 0 0
\(382\) 843.064 0.112919
\(383\) −1262.33 −0.168413 −0.0842066 0.996448i \(-0.526836\pi\)
−0.0842066 + 0.996448i \(0.526836\pi\)
\(384\) 0 0
\(385\) 6736.24 0.891716
\(386\) −471.211 −0.0621347
\(387\) 0 0
\(388\) 1818.67 0.237962
\(389\) 2793.42 0.364093 0.182046 0.983290i \(-0.441728\pi\)
0.182046 + 0.983290i \(0.441728\pi\)
\(390\) 0 0
\(391\) −7919.62 −1.02433
\(392\) 241.053 0.0310588
\(393\) 0 0
\(394\) −1922.00 −0.245759
\(395\) 2413.15 0.307390
\(396\) 0 0
\(397\) 6848.91 0.865836 0.432918 0.901433i \(-0.357484\pi\)
0.432918 + 0.901433i \(0.357484\pi\)
\(398\) −393.803 −0.0495969
\(399\) 0 0
\(400\) −4131.11 −0.516389
\(401\) −11024.5 −1.37291 −0.686456 0.727171i \(-0.740835\pi\)
−0.686456 + 0.727171i \(0.740835\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4474.24 0.550994
\(405\) 0 0
\(406\) −1974.03 −0.241304
\(407\) −6801.97 −0.828406
\(408\) 0 0
\(409\) 3530.56 0.426833 0.213417 0.976961i \(-0.431541\pi\)
0.213417 + 0.976961i \(0.431541\pi\)
\(410\) 281.429 0.0338995
\(411\) 0 0
\(412\) −7608.86 −0.909858
\(413\) 11251.5 1.34056
\(414\) 0 0
\(415\) 244.325 0.0288998
\(416\) 0 0
\(417\) 0 0
\(418\) 2658.93 0.311131
\(419\) 4179.22 0.487275 0.243637 0.969866i \(-0.421659\pi\)
0.243637 + 0.969866i \(0.421659\pi\)
\(420\) 0 0
\(421\) −6209.31 −0.718820 −0.359410 0.933180i \(-0.617022\pi\)
−0.359410 + 0.933180i \(0.617022\pi\)
\(422\) 1915.98 0.221015
\(423\) 0 0
\(424\) −3051.39 −0.349501
\(425\) 5920.49 0.675732
\(426\) 0 0
\(427\) 787.382 0.0892367
\(428\) −2693.03 −0.304141
\(429\) 0 0
\(430\) −237.650 −0.0266523
\(431\) 11880.2 1.32773 0.663864 0.747854i \(-0.268916\pi\)
0.663864 + 0.747854i \(0.268916\pi\)
\(432\) 0 0
\(433\) 8725.71 0.968432 0.484216 0.874949i \(-0.339105\pi\)
0.484216 + 0.874949i \(0.339105\pi\)
\(434\) −238.130 −0.0263378
\(435\) 0 0
\(436\) −652.655 −0.0716893
\(437\) −13614.7 −1.49034
\(438\) 0 0
\(439\) −1200.37 −0.130503 −0.0652514 0.997869i \(-0.520785\pi\)
−0.0652514 + 0.997869i \(0.520785\pi\)
\(440\) −2130.86 −0.230874
\(441\) 0 0
\(442\) 0 0
\(443\) −2258.86 −0.242261 −0.121130 0.992637i \(-0.538652\pi\)
−0.121130 + 0.992637i \(0.538652\pi\)
\(444\) 0 0
\(445\) −3390.59 −0.361190
\(446\) −532.343 −0.0565183
\(447\) 0 0
\(448\) 8882.65 0.936754
\(449\) −14662.4 −1.54112 −0.770561 0.637367i \(-0.780024\pi\)
−0.770561 + 0.637367i \(0.780024\pi\)
\(450\) 0 0
\(451\) 4395.16 0.458892
\(452\) −16607.4 −1.72820
\(453\) 0 0
\(454\) 1626.73 0.168164
\(455\) 0 0
\(456\) 0 0
\(457\) −9334.94 −0.955514 −0.477757 0.878492i \(-0.658550\pi\)
−0.477757 + 0.878492i \(0.658550\pi\)
\(458\) −1368.09 −0.139578
\(459\) 0 0
\(460\) 5402.97 0.547641
\(461\) 10736.9 1.08474 0.542370 0.840140i \(-0.317527\pi\)
0.542370 + 0.840140i \(0.317527\pi\)
\(462\) 0 0
\(463\) −10650.0 −1.06900 −0.534501 0.845168i \(-0.679501\pi\)
−0.534501 + 0.845168i \(0.679501\pi\)
\(464\) −15630.0 −1.56380
\(465\) 0 0
\(466\) −222.464 −0.0221147
\(467\) 2638.11 0.261407 0.130703 0.991422i \(-0.458276\pi\)
0.130703 + 0.991422i \(0.458276\pi\)
\(468\) 0 0
\(469\) 11749.0 1.15676
\(470\) 276.923 0.0271776
\(471\) 0 0
\(472\) −3559.16 −0.347084
\(473\) −3711.45 −0.360787
\(474\) 0 0
\(475\) 10177.9 0.983149
\(476\) −13270.6 −1.27785
\(477\) 0 0
\(478\) 90.1386 0.00862520
\(479\) −3225.31 −0.307658 −0.153829 0.988097i \(-0.549160\pi\)
−0.153829 + 0.988097i \(0.549160\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1204.87 −0.113860
\(483\) 0 0
\(484\) −6033.89 −0.566669
\(485\) 1743.02 0.163189
\(486\) 0 0
\(487\) 1496.39 0.139236 0.0696181 0.997574i \(-0.477822\pi\)
0.0696181 + 0.997574i \(0.477822\pi\)
\(488\) −249.070 −0.0231043
\(489\) 0 0
\(490\) 114.404 0.0105474
\(491\) 518.408 0.0476485 0.0238243 0.999716i \(-0.492416\pi\)
0.0238243 + 0.999716i \(0.492416\pi\)
\(492\) 0 0
\(493\) 22400.1 2.04635
\(494\) 0 0
\(495\) 0 0
\(496\) −1885.48 −0.170686
\(497\) 10139.7 0.915148
\(498\) 0 0
\(499\) 2405.01 0.215757 0.107879 0.994164i \(-0.465594\pi\)
0.107879 + 0.994164i \(0.465594\pi\)
\(500\) −11417.4 −1.02120
\(501\) 0 0
\(502\) −954.080 −0.0848260
\(503\) −2413.76 −0.213965 −0.106983 0.994261i \(-0.534119\pi\)
−0.106983 + 0.994261i \(0.534119\pi\)
\(504\) 0 0
\(505\) 4288.12 0.377859
\(506\) −1636.35 −0.143764
\(507\) 0 0
\(508\) 9234.45 0.806520
\(509\) 15678.3 1.36528 0.682641 0.730754i \(-0.260831\pi\)
0.682641 + 0.730754i \(0.260831\pi\)
\(510\) 0 0
\(511\) −20623.6 −1.78539
\(512\) −7393.65 −0.638196
\(513\) 0 0
\(514\) 1277.24 0.109604
\(515\) −7292.34 −0.623959
\(516\) 0 0
\(517\) 4324.79 0.367899
\(518\) −1131.77 −0.0959983
\(519\) 0 0
\(520\) 0 0
\(521\) 11691.5 0.983135 0.491568 0.870839i \(-0.336424\pi\)
0.491568 + 0.870839i \(0.336424\pi\)
\(522\) 0 0
\(523\) −3878.95 −0.324311 −0.162156 0.986765i \(-0.551845\pi\)
−0.162156 + 0.986765i \(0.551845\pi\)
\(524\) 6084.69 0.507273
\(525\) 0 0
\(526\) −1806.73 −0.149767
\(527\) 2702.17 0.223355
\(528\) 0 0
\(529\) −3788.32 −0.311360
\(530\) −1448.19 −0.118689
\(531\) 0 0
\(532\) −22813.6 −1.85920
\(533\) 0 0
\(534\) 0 0
\(535\) −2581.00 −0.208573
\(536\) −3716.54 −0.299496
\(537\) 0 0
\(538\) −1107.35 −0.0887382
\(539\) 1786.68 0.142779
\(540\) 0 0
\(541\) 16353.0 1.29958 0.649788 0.760115i \(-0.274858\pi\)
0.649788 + 0.760115i \(0.274858\pi\)
\(542\) −2751.22 −0.218035
\(543\) 0 0
\(544\) 6316.93 0.497861
\(545\) −625.506 −0.0491628
\(546\) 0 0
\(547\) 2748.67 0.214853 0.107426 0.994213i \(-0.465739\pi\)
0.107426 + 0.994213i \(0.465739\pi\)
\(548\) −19954.0 −1.55546
\(549\) 0 0
\(550\) 1223.29 0.0948387
\(551\) 38508.1 2.97732
\(552\) 0 0
\(553\) 6270.68 0.482200
\(554\) 756.163 0.0579897
\(555\) 0 0
\(556\) 2246.96 0.171389
\(557\) −16765.6 −1.27537 −0.637686 0.770297i \(-0.720108\pi\)
−0.637686 + 0.770297i \(0.720108\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 8874.58 0.669677
\(561\) 0 0
\(562\) −1283.77 −0.0963568
\(563\) −18492.4 −1.38430 −0.692151 0.721752i \(-0.743337\pi\)
−0.692151 + 0.721752i \(0.743337\pi\)
\(564\) 0 0
\(565\) −15916.5 −1.18516
\(566\) 3088.96 0.229397
\(567\) 0 0
\(568\) −3207.47 −0.236941
\(569\) −1563.28 −0.115178 −0.0575888 0.998340i \(-0.518341\pi\)
−0.0575888 + 0.998340i \(0.518341\pi\)
\(570\) 0 0
\(571\) 9165.98 0.671776 0.335888 0.941902i \(-0.390964\pi\)
0.335888 + 0.941902i \(0.390964\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 731.305 0.0531778
\(575\) −6263.67 −0.454284
\(576\) 0 0
\(577\) 18762.5 1.35372 0.676858 0.736114i \(-0.263341\pi\)
0.676858 + 0.736114i \(0.263341\pi\)
\(578\) −1003.65 −0.0722257
\(579\) 0 0
\(580\) −15281.9 −1.09405
\(581\) 634.888 0.0453349
\(582\) 0 0
\(583\) −22616.8 −1.60667
\(584\) 6523.79 0.462254
\(585\) 0 0
\(586\) 2214.07 0.156079
\(587\) −12646.3 −0.889212 −0.444606 0.895726i \(-0.646656\pi\)
−0.444606 + 0.895726i \(0.646656\pi\)
\(588\) 0 0
\(589\) 4645.31 0.324969
\(590\) −1689.18 −0.117868
\(591\) 0 0
\(592\) −8961.17 −0.622132
\(593\) 9662.74 0.669142 0.334571 0.942371i \(-0.391409\pi\)
0.334571 + 0.942371i \(0.391409\pi\)
\(594\) 0 0
\(595\) −12718.6 −0.876321
\(596\) −18474.7 −1.26972
\(597\) 0 0
\(598\) 0 0
\(599\) 26968.7 1.83959 0.919794 0.392402i \(-0.128356\pi\)
0.919794 + 0.392402i \(0.128356\pi\)
\(600\) 0 0
\(601\) 11280.1 0.765600 0.382800 0.923831i \(-0.374960\pi\)
0.382800 + 0.923831i \(0.374960\pi\)
\(602\) −617.542 −0.0418092
\(603\) 0 0
\(604\) −1296.13 −0.0873156
\(605\) −5782.89 −0.388608
\(606\) 0 0
\(607\) 12052.6 0.805931 0.402965 0.915215i \(-0.367979\pi\)
0.402965 + 0.915215i \(0.367979\pi\)
\(608\) 10859.5 0.724358
\(609\) 0 0
\(610\) −118.208 −0.00784610
\(611\) 0 0
\(612\) 0 0
\(613\) 11259.7 0.741883 0.370941 0.928656i \(-0.379035\pi\)
0.370941 + 0.928656i \(0.379035\pi\)
\(614\) 1692.65 0.111254
\(615\) 0 0
\(616\) −5537.12 −0.362170
\(617\) 24058.2 1.56976 0.784882 0.619645i \(-0.212723\pi\)
0.784882 + 0.619645i \(0.212723\pi\)
\(618\) 0 0
\(619\) −2793.41 −0.181384 −0.0906919 0.995879i \(-0.528908\pi\)
−0.0906919 + 0.995879i \(0.528908\pi\)
\(620\) −1843.49 −0.119413
\(621\) 0 0
\(622\) −2042.15 −0.131645
\(623\) −8810.59 −0.566595
\(624\) 0 0
\(625\) −2388.81 −0.152884
\(626\) −845.714 −0.0539960
\(627\) 0 0
\(628\) 24294.6 1.54373
\(629\) 12842.7 0.814104
\(630\) 0 0
\(631\) −24850.0 −1.56777 −0.783883 0.620908i \(-0.786764\pi\)
−0.783883 + 0.620908i \(0.786764\pi\)
\(632\) −1983.59 −0.124846
\(633\) 0 0
\(634\) 1897.41 0.118857
\(635\) 8850.32 0.553093
\(636\) 0 0
\(637\) 0 0
\(638\) 4628.31 0.287205
\(639\) 0 0
\(640\) −5726.69 −0.353699
\(641\) 7794.71 0.480300 0.240150 0.970736i \(-0.422803\pi\)
0.240150 + 0.970736i \(0.422803\pi\)
\(642\) 0 0
\(643\) 27709.9 1.69949 0.849744 0.527195i \(-0.176756\pi\)
0.849744 + 0.527195i \(0.176756\pi\)
\(644\) 14039.8 0.859080
\(645\) 0 0
\(646\) −5020.28 −0.305759
\(647\) −11150.9 −0.677571 −0.338786 0.940864i \(-0.610016\pi\)
−0.338786 + 0.940864i \(0.610016\pi\)
\(648\) 0 0
\(649\) −26380.4 −1.59556
\(650\) 0 0
\(651\) 0 0
\(652\) −2348.93 −0.141091
\(653\) −29137.5 −1.74615 −0.873077 0.487583i \(-0.837879\pi\)
−0.873077 + 0.487583i \(0.837879\pi\)
\(654\) 0 0
\(655\) 5831.58 0.347876
\(656\) 5790.36 0.344627
\(657\) 0 0
\(658\) 719.595 0.0426333
\(659\) 5300.12 0.313298 0.156649 0.987654i \(-0.449931\pi\)
0.156649 + 0.987654i \(0.449931\pi\)
\(660\) 0 0
\(661\) −27412.5 −1.61305 −0.806523 0.591203i \(-0.798653\pi\)
−0.806523 + 0.591203i \(0.798653\pi\)
\(662\) 1047.56 0.0615025
\(663\) 0 0
\(664\) −200.832 −0.0117377
\(665\) −21864.6 −1.27499
\(666\) 0 0
\(667\) −23698.6 −1.37573
\(668\) −23584.0 −1.36601
\(669\) 0 0
\(670\) −1763.87 −0.101708
\(671\) −1846.10 −0.106211
\(672\) 0 0
\(673\) −21282.1 −1.21896 −0.609482 0.792800i \(-0.708623\pi\)
−0.609482 + 0.792800i \(0.708623\pi\)
\(674\) 2542.91 0.145325
\(675\) 0 0
\(676\) 0 0
\(677\) 13544.2 0.768904 0.384452 0.923145i \(-0.374390\pi\)
0.384452 + 0.923145i \(0.374390\pi\)
\(678\) 0 0
\(679\) 4529.31 0.255992
\(680\) 4023.23 0.226888
\(681\) 0 0
\(682\) 558.322 0.0313479
\(683\) −10350.1 −0.579849 −0.289924 0.957050i \(-0.593630\pi\)
−0.289924 + 0.957050i \(0.593630\pi\)
\(684\) 0 0
\(685\) −19124.0 −1.06670
\(686\) −2317.97 −0.129009
\(687\) 0 0
\(688\) −4889.60 −0.270951
\(689\) 0 0
\(690\) 0 0
\(691\) −25714.8 −1.41569 −0.707843 0.706370i \(-0.750331\pi\)
−0.707843 + 0.706370i \(0.750331\pi\)
\(692\) 3577.45 0.196523
\(693\) 0 0
\(694\) 29.5693 0.00161734
\(695\) 2153.49 0.117535
\(696\) 0 0
\(697\) −8298.43 −0.450969
\(698\) 1436.44 0.0778940
\(699\) 0 0
\(700\) −10495.8 −0.566720
\(701\) −7431.30 −0.400394 −0.200197 0.979756i \(-0.564158\pi\)
−0.200197 + 0.979756i \(0.564158\pi\)
\(702\) 0 0
\(703\) 22077.9 1.18447
\(704\) −20826.3 −1.11494
\(705\) 0 0
\(706\) 3913.46 0.208619
\(707\) 11142.8 0.592744
\(708\) 0 0
\(709\) −19986.8 −1.05870 −0.529351 0.848403i \(-0.677565\pi\)
−0.529351 + 0.848403i \(0.677565\pi\)
\(710\) −1522.26 −0.0804641
\(711\) 0 0
\(712\) 2787.03 0.146697
\(713\) −2858.80 −0.150158
\(714\) 0 0
\(715\) 0 0
\(716\) 10705.9 0.558798
\(717\) 0 0
\(718\) −2680.00 −0.139299
\(719\) −36101.9 −1.87256 −0.936281 0.351251i \(-0.885756\pi\)
−0.936281 + 0.351251i \(0.885756\pi\)
\(720\) 0 0
\(721\) −18949.5 −0.978800
\(722\) −5954.59 −0.306935
\(723\) 0 0
\(724\) 15900.5 0.816213
\(725\) 17716.4 0.907545
\(726\) 0 0
\(727\) −1751.90 −0.0893735 −0.0446868 0.999001i \(-0.514229\pi\)
−0.0446868 + 0.999001i \(0.514229\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 3096.18 0.156979
\(731\) 7007.52 0.354559
\(732\) 0 0
\(733\) −20031.3 −1.00938 −0.504688 0.863302i \(-0.668393\pi\)
−0.504688 + 0.863302i \(0.668393\pi\)
\(734\) −3465.51 −0.174270
\(735\) 0 0
\(736\) −6683.10 −0.334704
\(737\) −27546.8 −1.37680
\(738\) 0 0
\(739\) 18632.9 0.927502 0.463751 0.885966i \(-0.346503\pi\)
0.463751 + 0.885966i \(0.346503\pi\)
\(740\) −8761.62 −0.435248
\(741\) 0 0
\(742\) −3763.17 −0.186187
\(743\) −4907.69 −0.242323 −0.121161 0.992633i \(-0.538662\pi\)
−0.121161 + 0.992633i \(0.538662\pi\)
\(744\) 0 0
\(745\) −17706.2 −0.870747
\(746\) 2127.88 0.104433
\(747\) 0 0
\(748\) 31114.3 1.52093
\(749\) −6706.84 −0.327186
\(750\) 0 0
\(751\) 31156.9 1.51389 0.756945 0.653479i \(-0.226691\pi\)
0.756945 + 0.653479i \(0.226691\pi\)
\(752\) 5697.64 0.276292
\(753\) 0 0
\(754\) 0 0
\(755\) −1242.21 −0.0598790
\(756\) 0 0
\(757\) −11047.0 −0.530396 −0.265198 0.964194i \(-0.585437\pi\)
−0.265198 + 0.964194i \(0.585437\pi\)
\(758\) −2591.20 −0.124165
\(759\) 0 0
\(760\) 6916.35 0.330109
\(761\) −27606.1 −1.31501 −0.657503 0.753452i \(-0.728387\pi\)
−0.657503 + 0.753452i \(0.728387\pi\)
\(762\) 0 0
\(763\) −1625.40 −0.0771213
\(764\) 16959.6 0.803112
\(765\) 0 0
\(766\) 492.456 0.0232287
\(767\) 0 0
\(768\) 0 0
\(769\) −3694.72 −0.173257 −0.0866287 0.996241i \(-0.527609\pi\)
−0.0866287 + 0.996241i \(0.527609\pi\)
\(770\) −2627.91 −0.122991
\(771\) 0 0
\(772\) −9479.18 −0.441921
\(773\) −19808.0 −0.921662 −0.460831 0.887488i \(-0.652449\pi\)
−0.460831 + 0.887488i \(0.652449\pi\)
\(774\) 0 0
\(775\) 2137.16 0.0990569
\(776\) −1432.74 −0.0662789
\(777\) 0 0
\(778\) −1089.76 −0.0502181
\(779\) −14265.9 −0.656133
\(780\) 0 0
\(781\) −23773.6 −1.08923
\(782\) 3089.56 0.141282
\(783\) 0 0
\(784\) 2353.84 0.107227
\(785\) 23284.0 1.05865
\(786\) 0 0
\(787\) −4116.79 −0.186464 −0.0932322 0.995644i \(-0.529720\pi\)
−0.0932322 + 0.995644i \(0.529720\pi\)
\(788\) −38664.3 −1.74792
\(789\) 0 0
\(790\) −941.408 −0.0423972
\(791\) −41359.8 −1.85915
\(792\) 0 0
\(793\) 0 0
\(794\) −2671.86 −0.119422
\(795\) 0 0
\(796\) −7922.00 −0.352748
\(797\) −25359.3 −1.12707 −0.563533 0.826094i \(-0.690558\pi\)
−0.563533 + 0.826094i \(0.690558\pi\)
\(798\) 0 0
\(799\) −8165.55 −0.361548
\(800\) 4996.10 0.220799
\(801\) 0 0
\(802\) 4300.83 0.189361
\(803\) 48354.1 2.12500
\(804\) 0 0
\(805\) 13455.8 0.589137
\(806\) 0 0
\(807\) 0 0
\(808\) −3524.78 −0.153467
\(809\) 5558.73 0.241576 0.120788 0.992678i \(-0.461458\pi\)
0.120788 + 0.992678i \(0.461458\pi\)
\(810\) 0 0
\(811\) 15021.4 0.650399 0.325199 0.945646i \(-0.394569\pi\)
0.325199 + 0.945646i \(0.394569\pi\)
\(812\) −39710.8 −1.71623
\(813\) 0 0
\(814\) 2653.55 0.114259
\(815\) −2251.22 −0.0967567
\(816\) 0 0
\(817\) 12046.6 0.515862
\(818\) −1377.32 −0.0588717
\(819\) 0 0
\(820\) 5661.41 0.241104
\(821\) 15901.6 0.675966 0.337983 0.941152i \(-0.390255\pi\)
0.337983 + 0.941152i \(0.390255\pi\)
\(822\) 0 0
\(823\) −40279.3 −1.70601 −0.853006 0.521901i \(-0.825223\pi\)
−0.853006 + 0.521901i \(0.825223\pi\)
\(824\) 5994.23 0.253421
\(825\) 0 0
\(826\) −4389.39 −0.184899
\(827\) 5251.09 0.220796 0.110398 0.993887i \(-0.464787\pi\)
0.110398 + 0.993887i \(0.464787\pi\)
\(828\) 0 0
\(829\) 33964.4 1.42296 0.711479 0.702708i \(-0.248026\pi\)
0.711479 + 0.702708i \(0.248026\pi\)
\(830\) −95.3148 −0.00398606
\(831\) 0 0
\(832\) 0 0
\(833\) −3373.40 −0.140314
\(834\) 0 0
\(835\) −22603.0 −0.936776
\(836\) 53488.8 2.21286
\(837\) 0 0
\(838\) −1630.38 −0.0672081
\(839\) −15485.0 −0.637188 −0.318594 0.947891i \(-0.603211\pi\)
−0.318594 + 0.947891i \(0.603211\pi\)
\(840\) 0 0
\(841\) 42640.8 1.74836
\(842\) 2422.35 0.0991444
\(843\) 0 0
\(844\) 38543.1 1.57193
\(845\) 0 0
\(846\) 0 0
\(847\) −15027.1 −0.609606
\(848\) −29796.2 −1.20661
\(849\) 0 0
\(850\) −2309.67 −0.0932014
\(851\) −13587.1 −0.547309
\(852\) 0 0
\(853\) −20057.8 −0.805118 −0.402559 0.915394i \(-0.631879\pi\)
−0.402559 + 0.915394i \(0.631879\pi\)
\(854\) −307.170 −0.0123081
\(855\) 0 0
\(856\) 2121.55 0.0847117
\(857\) 8066.23 0.321514 0.160757 0.986994i \(-0.448606\pi\)
0.160757 + 0.986994i \(0.448606\pi\)
\(858\) 0 0
\(859\) 39719.0 1.57764 0.788821 0.614623i \(-0.210692\pi\)
0.788821 + 0.614623i \(0.210692\pi\)
\(860\) −4780.71 −0.189559
\(861\) 0 0
\(862\) −4634.66 −0.183129
\(863\) −24473.8 −0.965351 −0.482676 0.875799i \(-0.660335\pi\)
−0.482676 + 0.875799i \(0.660335\pi\)
\(864\) 0 0
\(865\) 3428.63 0.134771
\(866\) −3404.03 −0.133572
\(867\) 0 0
\(868\) −4790.38 −0.187323
\(869\) −14702.3 −0.573924
\(870\) 0 0
\(871\) 0 0
\(872\) 514.159 0.0199675
\(873\) 0 0
\(874\) 5311.28 0.205557
\(875\) −28434.4 −1.09858
\(876\) 0 0
\(877\) 25326.6 0.975162 0.487581 0.873078i \(-0.337879\pi\)
0.487581 + 0.873078i \(0.337879\pi\)
\(878\) 468.284 0.0179998
\(879\) 0 0
\(880\) −20807.4 −0.797064
\(881\) −1327.44 −0.0507634 −0.0253817 0.999678i \(-0.508080\pi\)
−0.0253817 + 0.999678i \(0.508080\pi\)
\(882\) 0 0
\(883\) −2112.05 −0.0804941 −0.0402470 0.999190i \(-0.512814\pi\)
−0.0402470 + 0.999190i \(0.512814\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 881.214 0.0334142
\(887\) 40935.4 1.54958 0.774790 0.632219i \(-0.217856\pi\)
0.774790 + 0.632219i \(0.217856\pi\)
\(888\) 0 0
\(889\) 22997.9 0.867632
\(890\) 1322.72 0.0498177
\(891\) 0 0
\(892\) −10708.9 −0.401975
\(893\) −14037.4 −0.526030
\(894\) 0 0
\(895\) 10260.6 0.383211
\(896\) −14881.0 −0.554845
\(897\) 0 0
\(898\) 5720.04 0.212562
\(899\) 8085.93 0.299979
\(900\) 0 0
\(901\) 42702.3 1.57894
\(902\) −1714.62 −0.0632934
\(903\) 0 0
\(904\) 13083.2 0.481351
\(905\) 15239.1 0.559740
\(906\) 0 0
\(907\) 153.968 0.00563663 0.00281831 0.999996i \(-0.499103\pi\)
0.00281831 + 0.999996i \(0.499103\pi\)
\(908\) 32724.4 1.19603
\(909\) 0 0
\(910\) 0 0
\(911\) 733.607 0.0266800 0.0133400 0.999911i \(-0.495754\pi\)
0.0133400 + 0.999911i \(0.495754\pi\)
\(912\) 0 0
\(913\) −1488.56 −0.0539586
\(914\) 3641.70 0.131791
\(915\) 0 0
\(916\) −27521.4 −0.992722
\(917\) 15153.6 0.545710
\(918\) 0 0
\(919\) −51106.4 −1.83443 −0.917216 0.398390i \(-0.869569\pi\)
−0.917216 + 0.398390i \(0.869569\pi\)
\(920\) −4256.44 −0.152533
\(921\) 0 0
\(922\) −4188.61 −0.149615
\(923\) 0 0
\(924\) 0 0
\(925\) 10157.4 0.361051
\(926\) 4154.73 0.147444
\(927\) 0 0
\(928\) 18902.7 0.668655
\(929\) 30645.2 1.08228 0.541139 0.840933i \(-0.317993\pi\)
0.541139 + 0.840933i \(0.317993\pi\)
\(930\) 0 0
\(931\) −5799.22 −0.204148
\(932\) −4475.23 −0.157287
\(933\) 0 0
\(934\) −1029.17 −0.0360550
\(935\) 29820.0 1.04302
\(936\) 0 0
\(937\) −24422.2 −0.851482 −0.425741 0.904845i \(-0.639987\pi\)
−0.425741 + 0.904845i \(0.639987\pi\)
\(938\) −4583.48 −0.159548
\(939\) 0 0
\(940\) 5570.75 0.193296
\(941\) 16475.9 0.570774 0.285387 0.958412i \(-0.407878\pi\)
0.285387 + 0.958412i \(0.407878\pi\)
\(942\) 0 0
\(943\) 8779.46 0.303180
\(944\) −34754.5 −1.19827
\(945\) 0 0
\(946\) 1447.89 0.0497622
\(947\) 10122.1 0.347332 0.173666 0.984805i \(-0.444439\pi\)
0.173666 + 0.984805i \(0.444439\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −3970.57 −0.135602
\(951\) 0 0
\(952\) 10454.5 0.355917
\(953\) −38097.2 −1.29495 −0.647475 0.762086i \(-0.724175\pi\)
−0.647475 + 0.762086i \(0.724175\pi\)
\(954\) 0 0
\(955\) 16254.1 0.550756
\(956\) 1813.29 0.0613451
\(957\) 0 0
\(958\) 1258.24 0.0424342
\(959\) −49694.5 −1.67332
\(960\) 0 0
\(961\) −28815.6 −0.967258
\(962\) 0 0
\(963\) 0 0
\(964\) −24237.9 −0.809804
\(965\) −9084.87 −0.303059
\(966\) 0 0
\(967\) −44515.5 −1.48037 −0.740187 0.672401i \(-0.765263\pi\)
−0.740187 + 0.672401i \(0.765263\pi\)
\(968\) 4753.47 0.157833
\(969\) 0 0
\(970\) −679.978 −0.0225080
\(971\) 25444.3 0.840934 0.420467 0.907308i \(-0.361866\pi\)
0.420467 + 0.907308i \(0.361866\pi\)
\(972\) 0 0
\(973\) 5595.94 0.184376
\(974\) −583.765 −0.0192044
\(975\) 0 0
\(976\) −2432.12 −0.0797646
\(977\) −39467.9 −1.29242 −0.646208 0.763161i \(-0.723646\pi\)
−0.646208 + 0.763161i \(0.723646\pi\)
\(978\) 0 0
\(979\) 20657.4 0.674374
\(980\) 2301.42 0.0750165
\(981\) 0 0
\(982\) −202.239 −0.00657200
\(983\) 14970.4 0.485740 0.242870 0.970059i \(-0.421911\pi\)
0.242870 + 0.970059i \(0.421911\pi\)
\(984\) 0 0
\(985\) −37055.9 −1.19868
\(986\) −8738.63 −0.282246
\(987\) 0 0
\(988\) 0 0
\(989\) −7413.71 −0.238364
\(990\) 0 0
\(991\) 58422.4 1.87270 0.936352 0.351063i \(-0.114180\pi\)
0.936352 + 0.351063i \(0.114180\pi\)
\(992\) 2280.27 0.0729825
\(993\) 0 0
\(994\) −3955.66 −0.126223
\(995\) −7592.46 −0.241907
\(996\) 0 0
\(997\) 20131.6 0.639493 0.319746 0.947503i \(-0.396402\pi\)
0.319746 + 0.947503i \(0.396402\pi\)
\(998\) −938.230 −0.0297587
\(999\) 0 0
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 1521.4.a.bh.1.4 9
3.2 odd 2 169.4.a.k.1.6 9
13.12 even 2 1521.4.a.bg.1.6 9
39.2 even 12 169.4.e.h.147.11 36
39.5 even 4 169.4.b.g.168.8 18
39.8 even 4 169.4.b.g.168.11 18
39.11 even 12 169.4.e.h.147.8 36
39.17 odd 6 169.4.c.k.146.6 18
39.20 even 12 169.4.e.h.23.11 36
39.23 odd 6 169.4.c.k.22.6 18
39.29 odd 6 169.4.c.l.22.4 18
39.32 even 12 169.4.e.h.23.8 36
39.35 odd 6 169.4.c.l.146.4 18
39.38 odd 2 169.4.a.l.1.4 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.4.a.k.1.6 9 3.2 odd 2
169.4.a.l.1.4 yes 9 39.38 odd 2
169.4.b.g.168.8 18 39.5 even 4
169.4.b.g.168.11 18 39.8 even 4
169.4.c.k.22.6 18 39.23 odd 6
169.4.c.k.146.6 18 39.17 odd 6
169.4.c.l.22.4 18 39.29 odd 6
169.4.c.l.146.4 18 39.35 odd 6
169.4.e.h.23.8 36 39.32 even 12
169.4.e.h.23.11 36 39.20 even 12
169.4.e.h.147.8 36 39.11 even 12
169.4.e.h.147.11 36 39.2 even 12
1521.4.a.bg.1.6 9 13.12 even 2
1521.4.a.bh.1.4 9 1.1 even 1 trivial