Properties

Label 289.4.a.g.1.5
Level $289$
Weight $4$
Character 289.1
Self dual yes
Analytic conductor $17.052$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.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: \(17.0515519917\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.47083\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.70547 q^{2} -4.44379 q^{3} -5.09138 q^{4} +6.80983 q^{5} +7.57875 q^{6} -13.8760 q^{7} +22.3269 q^{8} -7.25269 q^{9} +O(q^{10})\) \(q-1.70547 q^{2} -4.44379 q^{3} -5.09138 q^{4} +6.80983 q^{5} +7.57875 q^{6} -13.8760 q^{7} +22.3269 q^{8} -7.25269 q^{9} -11.6140 q^{10} +30.8361 q^{11} +22.6250 q^{12} +66.0130 q^{13} +23.6651 q^{14} -30.2615 q^{15} +2.65318 q^{16} +12.3692 q^{18} -79.7150 q^{19} -34.6714 q^{20} +61.6622 q^{21} -52.5899 q^{22} +28.8986 q^{23} -99.2163 q^{24} -78.6262 q^{25} -112.583 q^{26} +152.212 q^{27} +70.6481 q^{28} +266.201 q^{29} +51.6100 q^{30} -35.7906 q^{31} -183.140 q^{32} -137.029 q^{33} -94.4935 q^{35} +36.9262 q^{36} -357.875 q^{37} +135.951 q^{38} -293.348 q^{39} +152.043 q^{40} +32.7452 q^{41} -105.163 q^{42} -516.157 q^{43} -156.998 q^{44} -49.3896 q^{45} -49.2857 q^{46} -210.602 q^{47} -11.7902 q^{48} -150.456 q^{49} +134.094 q^{50} -336.097 q^{52} +87.2324 q^{53} -259.593 q^{54} +209.989 q^{55} -309.809 q^{56} +354.237 q^{57} -453.997 q^{58} -310.728 q^{59} +154.073 q^{60} -365.247 q^{61} +61.0397 q^{62} +100.639 q^{63} +291.115 q^{64} +449.537 q^{65} +233.699 q^{66} -660.131 q^{67} -128.420 q^{69} +161.156 q^{70} +398.030 q^{71} -161.930 q^{72} +643.443 q^{73} +610.344 q^{74} +349.399 q^{75} +405.859 q^{76} -427.882 q^{77} +500.296 q^{78} -384.511 q^{79} +18.0677 q^{80} -480.576 q^{81} -55.8460 q^{82} -153.488 q^{83} -313.946 q^{84} +880.289 q^{86} -1182.94 q^{87} +688.475 q^{88} -599.053 q^{89} +84.2324 q^{90} -915.998 q^{91} -147.134 q^{92} +159.046 q^{93} +359.175 q^{94} -542.846 q^{95} +813.838 q^{96} +44.5693 q^{97} +256.598 q^{98} -223.645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.70547 −0.602974 −0.301487 0.953470i \(-0.597483\pi\)
−0.301487 + 0.953470i \(0.597483\pi\)
\(3\) −4.44379 −0.855209 −0.427604 0.903966i \(-0.640642\pi\)
−0.427604 + 0.903966i \(0.640642\pi\)
\(4\) −5.09138 −0.636422
\(5\) 6.80983 0.609090 0.304545 0.952498i \(-0.401496\pi\)
0.304545 + 0.952498i \(0.401496\pi\)
\(6\) 7.57875 0.515668
\(7\) −13.8760 −0.749235 −0.374618 0.927179i \(-0.622226\pi\)
−0.374618 + 0.927179i \(0.622226\pi\)
\(8\) 22.3269 0.986720
\(9\) −7.25269 −0.268618
\(10\) −11.6140 −0.367265
\(11\) 30.8361 0.845221 0.422610 0.906311i \(-0.361114\pi\)
0.422610 + 0.906311i \(0.361114\pi\)
\(12\) 22.6250 0.544274
\(13\) 66.0130 1.40836 0.704181 0.710021i \(-0.251314\pi\)
0.704181 + 0.710021i \(0.251314\pi\)
\(14\) 23.6651 0.451769
\(15\) −30.2615 −0.520899
\(16\) 2.65318 0.0414559
\(17\) 0 0
\(18\) 12.3692 0.161970
\(19\) −79.7150 −0.962520 −0.481260 0.876578i \(-0.659821\pi\)
−0.481260 + 0.876578i \(0.659821\pi\)
\(20\) −34.6714 −0.387639
\(21\) 61.6622 0.640752
\(22\) −52.5899 −0.509646
\(23\) 28.8986 0.261990 0.130995 0.991383i \(-0.458183\pi\)
0.130995 + 0.991383i \(0.458183\pi\)
\(24\) −99.2163 −0.843851
\(25\) −78.6262 −0.629009
\(26\) −112.583 −0.849205
\(27\) 152.212 1.08493
\(28\) 70.6481 0.476830
\(29\) 266.201 1.70456 0.852281 0.523084i \(-0.175219\pi\)
0.852281 + 0.523084i \(0.175219\pi\)
\(30\) 51.6100 0.314089
\(31\) −35.7906 −0.207361 −0.103680 0.994611i \(-0.533062\pi\)
−0.103680 + 0.994611i \(0.533062\pi\)
\(32\) −183.140 −1.01172
\(33\) −137.029 −0.722840
\(34\) 0 0
\(35\) −94.4935 −0.456352
\(36\) 36.9262 0.170955
\(37\) −357.875 −1.59011 −0.795057 0.606535i \(-0.792559\pi\)
−0.795057 + 0.606535i \(0.792559\pi\)
\(38\) 135.951 0.580374
\(39\) −293.348 −1.20444
\(40\) 152.043 0.601001
\(41\) 32.7452 0.124730 0.0623652 0.998053i \(-0.480136\pi\)
0.0623652 + 0.998053i \(0.480136\pi\)
\(42\) −105.163 −0.386357
\(43\) −516.157 −1.83054 −0.915270 0.402841i \(-0.868023\pi\)
−0.915270 + 0.402841i \(0.868023\pi\)
\(44\) −156.998 −0.537917
\(45\) −49.3896 −0.163613
\(46\) −49.2857 −0.157973
\(47\) −210.602 −0.653606 −0.326803 0.945093i \(-0.605971\pi\)
−0.326803 + 0.945093i \(0.605971\pi\)
\(48\) −11.7902 −0.0354534
\(49\) −150.456 −0.438647
\(50\) 134.094 0.379276
\(51\) 0 0
\(52\) −336.097 −0.896313
\(53\) 87.2324 0.226081 0.113040 0.993590i \(-0.463941\pi\)
0.113040 + 0.993590i \(0.463941\pi\)
\(54\) −259.593 −0.654186
\(55\) 209.989 0.514815
\(56\) −309.809 −0.739285
\(57\) 354.237 0.823155
\(58\) −453.997 −1.02781
\(59\) −310.728 −0.685651 −0.342825 0.939399i \(-0.611384\pi\)
−0.342825 + 0.939399i \(0.611384\pi\)
\(60\) 154.073 0.331512
\(61\) −365.247 −0.766641 −0.383321 0.923615i \(-0.625220\pi\)
−0.383321 + 0.923615i \(0.625220\pi\)
\(62\) 61.0397 0.125033
\(63\) 100.639 0.201258
\(64\) 291.115 0.568583
\(65\) 449.537 0.857819
\(66\) 233.699 0.435854
\(67\) −660.131 −1.20370 −0.601850 0.798609i \(-0.705569\pi\)
−0.601850 + 0.798609i \(0.705569\pi\)
\(68\) 0 0
\(69\) −128.420 −0.224056
\(70\) 161.156 0.275168
\(71\) 398.030 0.665316 0.332658 0.943047i \(-0.392054\pi\)
0.332658 + 0.943047i \(0.392054\pi\)
\(72\) −161.930 −0.265051
\(73\) 643.443 1.03163 0.515817 0.856699i \(-0.327489\pi\)
0.515817 + 0.856699i \(0.327489\pi\)
\(74\) 610.344 0.958797
\(75\) 349.399 0.537934
\(76\) 405.859 0.612569
\(77\) −427.882 −0.633269
\(78\) 500.296 0.726248
\(79\) −384.511 −0.547605 −0.273803 0.961786i \(-0.588282\pi\)
−0.273803 + 0.961786i \(0.588282\pi\)
\(80\) 18.0677 0.0252504
\(81\) −480.576 −0.659226
\(82\) −55.8460 −0.0752092
\(83\) −153.488 −0.202983 −0.101491 0.994836i \(-0.532361\pi\)
−0.101491 + 0.994836i \(0.532361\pi\)
\(84\) −313.946 −0.407789
\(85\) 0 0
\(86\) 880.289 1.10377
\(87\) −1182.94 −1.45776
\(88\) 688.475 0.833996
\(89\) −599.053 −0.713477 −0.356738 0.934204i \(-0.616111\pi\)
−0.356738 + 0.934204i \(0.616111\pi\)
\(90\) 84.2324 0.0986542
\(91\) −915.998 −1.05519
\(92\) −147.134 −0.166737
\(93\) 159.046 0.177336
\(94\) 359.175 0.394107
\(95\) −542.846 −0.586261
\(96\) 813.838 0.865229
\(97\) 44.5693 0.0466529 0.0233265 0.999728i \(-0.492574\pi\)
0.0233265 + 0.999728i \(0.492574\pi\)
\(98\) 256.598 0.264492
\(99\) −223.645 −0.227042
\(100\) 400.316 0.400316
\(101\) 1478.33 1.45643 0.728215 0.685348i \(-0.240350\pi\)
0.728215 + 0.685348i \(0.240350\pi\)
\(102\) 0 0
\(103\) 618.133 0.591325 0.295662 0.955293i \(-0.404460\pi\)
0.295662 + 0.955293i \(0.404460\pi\)
\(104\) 1473.87 1.38966
\(105\) 419.909 0.390276
\(106\) −148.772 −0.136321
\(107\) 138.979 0.125567 0.0627833 0.998027i \(-0.480002\pi\)
0.0627833 + 0.998027i \(0.480002\pi\)
\(108\) −774.969 −0.690476
\(109\) 823.466 0.723612 0.361806 0.932253i \(-0.382160\pi\)
0.361806 + 0.932253i \(0.382160\pi\)
\(110\) −358.129 −0.310420
\(111\) 1590.32 1.35988
\(112\) −36.8156 −0.0310602
\(113\) −602.474 −0.501558 −0.250779 0.968044i \(-0.580687\pi\)
−0.250779 + 0.968044i \(0.580687\pi\)
\(114\) −604.140 −0.496341
\(115\) 196.795 0.159576
\(116\) −1355.33 −1.08482
\(117\) −478.772 −0.378312
\(118\) 529.937 0.413430
\(119\) 0 0
\(120\) −675.646 −0.513982
\(121\) −380.136 −0.285602
\(122\) 622.917 0.462265
\(123\) −145.513 −0.106671
\(124\) 182.223 0.131969
\(125\) −1386.66 −0.992213
\(126\) −171.636 −0.121354
\(127\) 1506.98 1.05294 0.526468 0.850195i \(-0.323516\pi\)
0.526468 + 0.850195i \(0.323516\pi\)
\(128\) 968.636 0.668876
\(129\) 2293.70 1.56549
\(130\) −766.671 −0.517243
\(131\) −2783.58 −1.85651 −0.928254 0.371946i \(-0.878691\pi\)
−0.928254 + 0.371946i \(0.878691\pi\)
\(132\) 697.667 0.460032
\(133\) 1106.13 0.721154
\(134\) 1125.83 0.725799
\(135\) 1036.54 0.660822
\(136\) 0 0
\(137\) −629.783 −0.392744 −0.196372 0.980529i \(-0.562916\pi\)
−0.196372 + 0.980529i \(0.562916\pi\)
\(138\) 219.015 0.135100
\(139\) −2654.10 −1.61955 −0.809775 0.586740i \(-0.800411\pi\)
−0.809775 + 0.586740i \(0.800411\pi\)
\(140\) 481.102 0.290432
\(141\) 935.872 0.558969
\(142\) −678.827 −0.401168
\(143\) 2035.58 1.19038
\(144\) −19.2427 −0.0111358
\(145\) 1812.78 1.03823
\(146\) −1097.37 −0.622048
\(147\) 668.595 0.375134
\(148\) 1822.08 1.01198
\(149\) −2216.30 −1.21857 −0.609283 0.792953i \(-0.708543\pi\)
−0.609283 + 0.792953i \(0.708543\pi\)
\(150\) −595.888 −0.324360
\(151\) 2133.15 1.14963 0.574813 0.818285i \(-0.305075\pi\)
0.574813 + 0.818285i \(0.305075\pi\)
\(152\) −1779.79 −0.949737
\(153\) 0 0
\(154\) 729.740 0.381845
\(155\) −243.728 −0.126301
\(156\) 1493.55 0.766534
\(157\) −345.107 −0.175430 −0.0877152 0.996146i \(-0.527957\pi\)
−0.0877152 + 0.996146i \(0.527957\pi\)
\(158\) 655.770 0.330192
\(159\) −387.643 −0.193346
\(160\) −1247.16 −0.616227
\(161\) −400.998 −0.196292
\(162\) 819.606 0.397496
\(163\) −678.526 −0.326050 −0.163025 0.986622i \(-0.552125\pi\)
−0.163025 + 0.986622i \(0.552125\pi\)
\(164\) −166.718 −0.0793812
\(165\) −933.146 −0.440275
\(166\) 261.770 0.122393
\(167\) −799.790 −0.370596 −0.185298 0.982682i \(-0.559325\pi\)
−0.185298 + 0.982682i \(0.559325\pi\)
\(168\) 1376.73 0.632243
\(169\) 2160.71 0.983483
\(170\) 0 0
\(171\) 578.148 0.258550
\(172\) 2627.95 1.16500
\(173\) 1885.48 0.828615 0.414307 0.910137i \(-0.364024\pi\)
0.414307 + 0.910137i \(0.364024\pi\)
\(174\) 2017.47 0.878989
\(175\) 1091.02 0.471276
\(176\) 81.8136 0.0350394
\(177\) 1380.81 0.586374
\(178\) 1021.66 0.430208
\(179\) −3588.86 −1.49857 −0.749286 0.662247i \(-0.769603\pi\)
−0.749286 + 0.662247i \(0.769603\pi\)
\(180\) 251.461 0.104127
\(181\) −461.347 −0.189457 −0.0947284 0.995503i \(-0.530198\pi\)
−0.0947284 + 0.995503i \(0.530198\pi\)
\(182\) 1562.20 0.636255
\(183\) 1623.08 0.655638
\(184\) 645.218 0.258511
\(185\) −2437.07 −0.968522
\(186\) −271.248 −0.106929
\(187\) 0 0
\(188\) 1072.26 0.415969
\(189\) −2112.10 −0.812870
\(190\) 925.806 0.353500
\(191\) −4519.91 −1.71230 −0.856149 0.516728i \(-0.827150\pi\)
−0.856149 + 0.516728i \(0.827150\pi\)
\(192\) −1293.65 −0.486257
\(193\) 2928.92 1.09238 0.546188 0.837663i \(-0.316078\pi\)
0.546188 + 0.837663i \(0.316078\pi\)
\(194\) −76.0116 −0.0281305
\(195\) −1997.65 −0.733614
\(196\) 766.027 0.279165
\(197\) −1926.13 −0.696606 −0.348303 0.937382i \(-0.613242\pi\)
−0.348303 + 0.937382i \(0.613242\pi\)
\(198\) 381.419 0.136900
\(199\) −3086.61 −1.09952 −0.549759 0.835323i \(-0.685280\pi\)
−0.549759 + 0.835323i \(0.685280\pi\)
\(200\) −1755.48 −0.620656
\(201\) 2933.49 1.02941
\(202\) −2521.25 −0.878190
\(203\) −3693.81 −1.27712
\(204\) 0 0
\(205\) 222.990 0.0759721
\(206\) −1054.21 −0.356553
\(207\) −209.593 −0.0703754
\(208\) 175.144 0.0583849
\(209\) −2458.10 −0.813541
\(210\) −716.142 −0.235326
\(211\) −3602.40 −1.17535 −0.587676 0.809097i \(-0.699957\pi\)
−0.587676 + 0.809097i \(0.699957\pi\)
\(212\) −444.133 −0.143883
\(213\) −1768.76 −0.568984
\(214\) −237.024 −0.0757133
\(215\) −3514.94 −1.11496
\(216\) 3398.42 1.07053
\(217\) 496.631 0.155362
\(218\) −1404.39 −0.436319
\(219\) −2859.33 −0.882262
\(220\) −1069.13 −0.327640
\(221\) 0 0
\(222\) −2712.24 −0.819972
\(223\) −1577.43 −0.473688 −0.236844 0.971548i \(-0.576113\pi\)
−0.236844 + 0.971548i \(0.576113\pi\)
\(224\) 2541.26 0.758014
\(225\) 570.252 0.168963
\(226\) 1027.50 0.302426
\(227\) 4292.90 1.25520 0.627599 0.778537i \(-0.284038\pi\)
0.627599 + 0.778537i \(0.284038\pi\)
\(228\) −1803.55 −0.523874
\(229\) 580.893 0.167627 0.0838133 0.996481i \(-0.473290\pi\)
0.0838133 + 0.996481i \(0.473290\pi\)
\(230\) −335.627 −0.0962200
\(231\) 1901.42 0.541577
\(232\) 5943.45 1.68193
\(233\) 671.778 0.188883 0.0944413 0.995530i \(-0.469894\pi\)
0.0944413 + 0.995530i \(0.469894\pi\)
\(234\) 816.530 0.228112
\(235\) −1434.16 −0.398105
\(236\) 1582.04 0.436363
\(237\) 1708.69 0.468317
\(238\) 0 0
\(239\) −273.635 −0.0740584 −0.0370292 0.999314i \(-0.511789\pi\)
−0.0370292 + 0.999314i \(0.511789\pi\)
\(240\) −80.2891 −0.0215943
\(241\) −6486.56 −1.73376 −0.866879 0.498518i \(-0.833878\pi\)
−0.866879 + 0.498518i \(0.833878\pi\)
\(242\) 648.310 0.172211
\(243\) −1974.14 −0.521158
\(244\) 1859.61 0.487908
\(245\) −1024.58 −0.267175
\(246\) 248.168 0.0643195
\(247\) −5262.22 −1.35558
\(248\) −799.093 −0.204607
\(249\) 682.071 0.173592
\(250\) 2364.90 0.598279
\(251\) −2527.58 −0.635615 −0.317808 0.948155i \(-0.602947\pi\)
−0.317808 + 0.948155i \(0.602947\pi\)
\(252\) −512.389 −0.128085
\(253\) 891.120 0.221440
\(254\) −2570.11 −0.634893
\(255\) 0 0
\(256\) −3980.89 −0.971898
\(257\) 1702.44 0.413211 0.206605 0.978424i \(-0.433758\pi\)
0.206605 + 0.978424i \(0.433758\pi\)
\(258\) −3911.82 −0.943952
\(259\) 4965.88 1.19137
\(260\) −2288.76 −0.545935
\(261\) −1930.67 −0.457877
\(262\) 4747.31 1.11943
\(263\) −4274.40 −1.00217 −0.501085 0.865398i \(-0.667066\pi\)
−0.501085 + 0.865398i \(0.667066\pi\)
\(264\) −3059.44 −0.713241
\(265\) 594.038 0.137704
\(266\) −1886.46 −0.434837
\(267\) 2662.07 0.610172
\(268\) 3360.98 0.766061
\(269\) 7996.34 1.81244 0.906218 0.422810i \(-0.138956\pi\)
0.906218 + 0.422810i \(0.138956\pi\)
\(270\) −1767.78 −0.398458
\(271\) 474.268 0.106309 0.0531545 0.998586i \(-0.483072\pi\)
0.0531545 + 0.998586i \(0.483072\pi\)
\(272\) 0 0
\(273\) 4070.51 0.902411
\(274\) 1074.07 0.236815
\(275\) −2424.52 −0.531652
\(276\) 653.833 0.142595
\(277\) −2826.25 −0.613043 −0.306521 0.951864i \(-0.599165\pi\)
−0.306521 + 0.951864i \(0.599165\pi\)
\(278\) 4526.48 0.976547
\(279\) 259.578 0.0557008
\(280\) −2109.75 −0.450291
\(281\) −2498.15 −0.530346 −0.265173 0.964201i \(-0.585429\pi\)
−0.265173 + 0.964201i \(0.585429\pi\)
\(282\) −1596.10 −0.337044
\(283\) 2417.07 0.507704 0.253852 0.967243i \(-0.418302\pi\)
0.253852 + 0.967243i \(0.418302\pi\)
\(284\) −2026.52 −0.423422
\(285\) 2412.29 0.501375
\(286\) −3471.62 −0.717766
\(287\) −454.374 −0.0934524
\(288\) 1328.26 0.271766
\(289\) 0 0
\(290\) −3091.65 −0.626027
\(291\) −198.057 −0.0398980
\(292\) −3276.01 −0.656555
\(293\) 6652.17 1.32636 0.663181 0.748459i \(-0.269206\pi\)
0.663181 + 0.748459i \(0.269206\pi\)
\(294\) −1140.27 −0.226196
\(295\) −2116.01 −0.417623
\(296\) −7990.24 −1.56900
\(297\) 4693.62 0.917008
\(298\) 3779.83 0.734764
\(299\) 1907.68 0.368977
\(300\) −1778.92 −0.342353
\(301\) 7162.21 1.37150
\(302\) −3638.02 −0.693194
\(303\) −6569.40 −1.24555
\(304\) −211.498 −0.0399021
\(305\) −2487.27 −0.466954
\(306\) 0 0
\(307\) −2511.18 −0.466842 −0.233421 0.972376i \(-0.574992\pi\)
−0.233421 + 0.972376i \(0.574992\pi\)
\(308\) 2178.51 0.403027
\(309\) −2746.85 −0.505706
\(310\) 415.670 0.0761563
\(311\) 3585.10 0.653674 0.326837 0.945081i \(-0.394017\pi\)
0.326837 + 0.945081i \(0.394017\pi\)
\(312\) −6549.56 −1.18845
\(313\) −2612.32 −0.471747 −0.235874 0.971784i \(-0.575795\pi\)
−0.235874 + 0.971784i \(0.575795\pi\)
\(314\) 588.570 0.105780
\(315\) 685.332 0.122584
\(316\) 1957.69 0.348508
\(317\) −3903.80 −0.691669 −0.345834 0.938296i \(-0.612404\pi\)
−0.345834 + 0.938296i \(0.612404\pi\)
\(318\) 661.112 0.116583
\(319\) 8208.60 1.44073
\(320\) 1982.44 0.346318
\(321\) −617.594 −0.107386
\(322\) 683.890 0.118359
\(323\) 0 0
\(324\) 2446.79 0.419546
\(325\) −5190.35 −0.885873
\(326\) 1157.20 0.196600
\(327\) −3659.31 −0.618839
\(328\) 731.101 0.123074
\(329\) 2922.32 0.489704
\(330\) 1591.45 0.265474
\(331\) −6236.03 −1.03554 −0.517769 0.855520i \(-0.673237\pi\)
−0.517769 + 0.855520i \(0.673237\pi\)
\(332\) 781.468 0.129183
\(333\) 2595.55 0.427134
\(334\) 1364.02 0.223460
\(335\) −4495.38 −0.733161
\(336\) 163.601 0.0265630
\(337\) 3842.08 0.621043 0.310521 0.950566i \(-0.399496\pi\)
0.310521 + 0.950566i \(0.399496\pi\)
\(338\) −3685.02 −0.593014
\(339\) 2677.27 0.428936
\(340\) 0 0
\(341\) −1103.64 −0.175265
\(342\) −986.014 −0.155899
\(343\) 6847.21 1.07788
\(344\) −11524.2 −1.80623
\(345\) −874.516 −0.136471
\(346\) −3215.62 −0.499633
\(347\) 8301.86 1.28434 0.642172 0.766561i \(-0.278034\pi\)
0.642172 + 0.766561i \(0.278034\pi\)
\(348\) 6022.81 0.927748
\(349\) 4929.17 0.756024 0.378012 0.925801i \(-0.376608\pi\)
0.378012 + 0.925801i \(0.376608\pi\)
\(350\) −1860.70 −0.284167
\(351\) 10048.0 1.52798
\(352\) −5647.33 −0.855124
\(353\) −9607.54 −1.44861 −0.724303 0.689482i \(-0.757838\pi\)
−0.724303 + 0.689482i \(0.757838\pi\)
\(354\) −2354.93 −0.353568
\(355\) 2710.52 0.405237
\(356\) 3050.00 0.454073
\(357\) 0 0
\(358\) 6120.69 0.903600
\(359\) −1962.76 −0.288553 −0.144277 0.989537i \(-0.546085\pi\)
−0.144277 + 0.989537i \(0.546085\pi\)
\(360\) −1102.72 −0.161440
\(361\) −504.522 −0.0735561
\(362\) 786.813 0.114238
\(363\) 1689.25 0.244249
\(364\) 4663.69 0.671549
\(365\) 4381.74 0.628358
\(366\) −2768.12 −0.395333
\(367\) 3458.29 0.491883 0.245941 0.969285i \(-0.420903\pi\)
0.245941 + 0.969285i \(0.420903\pi\)
\(368\) 76.6732 0.0108610
\(369\) −237.491 −0.0335049
\(370\) 4156.34 0.583994
\(371\) −1210.44 −0.169388
\(372\) −809.763 −0.112861
\(373\) 2661.22 0.369418 0.184709 0.982793i \(-0.440866\pi\)
0.184709 + 0.982793i \(0.440866\pi\)
\(374\) 0 0
\(375\) 6162.03 0.848549
\(376\) −4702.10 −0.644926
\(377\) 17572.7 2.40064
\(378\) 3602.11 0.490140
\(379\) 5583.03 0.756677 0.378339 0.925667i \(-0.376495\pi\)
0.378339 + 0.925667i \(0.376495\pi\)
\(380\) 2763.83 0.373110
\(381\) −6696.71 −0.900480
\(382\) 7708.56 1.03247
\(383\) 7832.13 1.04492 0.522459 0.852665i \(-0.325015\pi\)
0.522459 + 0.852665i \(0.325015\pi\)
\(384\) −4304.42 −0.572029
\(385\) −2913.81 −0.385718
\(386\) −4995.18 −0.658674
\(387\) 3743.53 0.491717
\(388\) −226.919 −0.0296910
\(389\) 7432.95 0.968807 0.484403 0.874845i \(-0.339037\pi\)
0.484403 + 0.874845i \(0.339037\pi\)
\(390\) 3406.93 0.442350
\(391\) 0 0
\(392\) −3359.22 −0.432821
\(393\) 12369.7 1.58770
\(394\) 3284.96 0.420035
\(395\) −2618.45 −0.333541
\(396\) 1138.66 0.144494
\(397\) −9540.26 −1.20608 −0.603038 0.797713i \(-0.706043\pi\)
−0.603038 + 0.797713i \(0.706043\pi\)
\(398\) 5264.12 0.662981
\(399\) −4915.40 −0.616737
\(400\) −208.609 −0.0260761
\(401\) −8663.30 −1.07886 −0.539432 0.842029i \(-0.681361\pi\)
−0.539432 + 0.842029i \(0.681361\pi\)
\(402\) −5002.97 −0.620710
\(403\) −2362.64 −0.292039
\(404\) −7526.75 −0.926905
\(405\) −3272.64 −0.401528
\(406\) 6299.68 0.770069
\(407\) −11035.4 −1.34400
\(408\) 0 0
\(409\) 5279.17 0.638235 0.319117 0.947715i \(-0.396614\pi\)
0.319117 + 0.947715i \(0.396614\pi\)
\(410\) −380.302 −0.0458092
\(411\) 2798.62 0.335878
\(412\) −3147.15 −0.376332
\(413\) 4311.68 0.513714
\(414\) 357.454 0.0424346
\(415\) −1045.23 −0.123635
\(416\) −12089.6 −1.42486
\(417\) 11794.3 1.38505
\(418\) 4192.21 0.490544
\(419\) −9408.24 −1.09695 −0.548476 0.836166i \(-0.684792\pi\)
−0.548476 + 0.836166i \(0.684792\pi\)
\(420\) −2137.92 −0.248380
\(421\) −8191.97 −0.948343 −0.474171 0.880433i \(-0.657252\pi\)
−0.474171 + 0.880433i \(0.657252\pi\)
\(422\) 6143.77 0.708706
\(423\) 1527.43 0.175570
\(424\) 1947.63 0.223079
\(425\) 0 0
\(426\) 3016.57 0.343083
\(427\) 5068.18 0.574395
\(428\) −707.595 −0.0799133
\(429\) −9045.70 −1.01802
\(430\) 5994.62 0.672294
\(431\) −1950.29 −0.217964 −0.108982 0.994044i \(-0.534759\pi\)
−0.108982 + 0.994044i \(0.534759\pi\)
\(432\) 403.845 0.0449769
\(433\) 11152.3 1.23775 0.618876 0.785489i \(-0.287588\pi\)
0.618876 + 0.785489i \(0.287588\pi\)
\(434\) −846.988 −0.0936791
\(435\) −8055.64 −0.887905
\(436\) −4192.58 −0.460523
\(437\) −2303.65 −0.252171
\(438\) 4876.49 0.531981
\(439\) −470.158 −0.0511149 −0.0255574 0.999673i \(-0.508136\pi\)
−0.0255574 + 0.999673i \(0.508136\pi\)
\(440\) 4688.40 0.507979
\(441\) 1091.21 0.117829
\(442\) 0 0
\(443\) −13132.9 −1.40849 −0.704246 0.709956i \(-0.748715\pi\)
−0.704246 + 0.709956i \(0.748715\pi\)
\(444\) −8096.93 −0.865457
\(445\) −4079.45 −0.434572
\(446\) 2690.25 0.285621
\(447\) 9848.78 1.04213
\(448\) −4039.51 −0.426002
\(449\) 10433.0 1.09658 0.548291 0.836288i \(-0.315279\pi\)
0.548291 + 0.836288i \(0.315279\pi\)
\(450\) −972.546 −0.101881
\(451\) 1009.73 0.105425
\(452\) 3067.43 0.319203
\(453\) −9479.28 −0.983169
\(454\) −7321.40 −0.756851
\(455\) −6237.79 −0.642708
\(456\) 7909.02 0.812224
\(457\) −253.385 −0.0259362 −0.0129681 0.999916i \(-0.504128\pi\)
−0.0129681 + 0.999916i \(0.504128\pi\)
\(458\) −990.694 −0.101074
\(459\) 0 0
\(460\) −1001.96 −0.101558
\(461\) 15560.2 1.57204 0.786019 0.618202i \(-0.212139\pi\)
0.786019 + 0.618202i \(0.212139\pi\)
\(462\) −3242.81 −0.326557
\(463\) 10727.9 1.07682 0.538409 0.842684i \(-0.319026\pi\)
0.538409 + 0.842684i \(0.319026\pi\)
\(464\) 706.279 0.0706641
\(465\) 1083.08 0.108014
\(466\) −1145.70 −0.113891
\(467\) −1960.48 −0.194262 −0.0971308 0.995272i \(-0.530967\pi\)
−0.0971308 + 0.995272i \(0.530967\pi\)
\(468\) 2437.61 0.240766
\(469\) 9160.00 0.901854
\(470\) 2445.92 0.240047
\(471\) 1533.59 0.150030
\(472\) −6937.61 −0.676545
\(473\) −15916.3 −1.54721
\(474\) −2914.11 −0.282383
\(475\) 6267.68 0.605434
\(476\) 0 0
\(477\) −632.670 −0.0607295
\(478\) 466.675 0.0446553
\(479\) 4294.19 0.409617 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(480\) 5542.10 0.527002
\(481\) −23624.4 −2.23946
\(482\) 11062.6 1.04541
\(483\) 1781.95 0.167871
\(484\) 1935.42 0.181764
\(485\) 303.510 0.0284158
\(486\) 3366.84 0.314244
\(487\) 18013.5 1.67612 0.838060 0.545578i \(-0.183690\pi\)
0.838060 + 0.545578i \(0.183690\pi\)
\(488\) −8154.85 −0.756460
\(489\) 3015.23 0.278841
\(490\) 1747.39 0.161100
\(491\) 13733.2 1.26226 0.631129 0.775678i \(-0.282592\pi\)
0.631129 + 0.775678i \(0.282592\pi\)
\(492\) 740.862 0.0678875
\(493\) 0 0
\(494\) 8974.55 0.817377
\(495\) −1522.98 −0.138289
\(496\) −94.9587 −0.00859632
\(497\) −5523.07 −0.498478
\(498\) −1163.25 −0.104672
\(499\) −7747.71 −0.695060 −0.347530 0.937669i \(-0.612980\pi\)
−0.347530 + 0.937669i \(0.612980\pi\)
\(500\) 7060.01 0.631467
\(501\) 3554.10 0.316937
\(502\) 4310.71 0.383259
\(503\) −19203.7 −1.70229 −0.851143 0.524934i \(-0.824090\pi\)
−0.851143 + 0.524934i \(0.824090\pi\)
\(504\) 2246.95 0.198586
\(505\) 10067.2 0.887098
\(506\) −1519.78 −0.133522
\(507\) −9601.76 −0.841083
\(508\) −7672.61 −0.670112
\(509\) 151.325 0.0131775 0.00658876 0.999978i \(-0.497903\pi\)
0.00658876 + 0.999978i \(0.497903\pi\)
\(510\) 0 0
\(511\) −8928.43 −0.772936
\(512\) −959.802 −0.0828470
\(513\) −12133.6 −1.04427
\(514\) −2903.45 −0.249155
\(515\) 4209.38 0.360170
\(516\) −11678.1 −0.996315
\(517\) −6494.14 −0.552441
\(518\) −8469.15 −0.718365
\(519\) −8378.68 −0.708638
\(520\) 10036.8 0.846427
\(521\) 8548.62 0.718852 0.359426 0.933174i \(-0.382973\pi\)
0.359426 + 0.933174i \(0.382973\pi\)
\(522\) 3292.70 0.276088
\(523\) 17874.0 1.49441 0.747205 0.664593i \(-0.231395\pi\)
0.747205 + 0.664593i \(0.231395\pi\)
\(524\) 14172.3 1.18152
\(525\) −4848.26 −0.403039
\(526\) 7289.85 0.604283
\(527\) 0 0
\(528\) −363.563 −0.0299660
\(529\) −11331.9 −0.931361
\(530\) −1013.11 −0.0830317
\(531\) 2253.62 0.184178
\(532\) −5631.71 −0.458958
\(533\) 2161.61 0.175666
\(534\) −4540.07 −0.367918
\(535\) 946.425 0.0764813
\(536\) −14738.7 −1.18771
\(537\) 15948.2 1.28159
\(538\) −13637.5 −1.09285
\(539\) −4639.47 −0.370753
\(540\) −5277.41 −0.420562
\(541\) −2122.24 −0.168655 −0.0843273 0.996438i \(-0.526874\pi\)
−0.0843273 + 0.996438i \(0.526874\pi\)
\(542\) −808.849 −0.0641015
\(543\) 2050.13 0.162025
\(544\) 0 0
\(545\) 5607.67 0.440745
\(546\) −6942.12 −0.544130
\(547\) −284.066 −0.0222044 −0.0111022 0.999938i \(-0.503534\pi\)
−0.0111022 + 0.999938i \(0.503534\pi\)
\(548\) 3206.46 0.249951
\(549\) 2649.03 0.205934
\(550\) 4134.95 0.320572
\(551\) −21220.2 −1.64067
\(552\) −2867.21 −0.221081
\(553\) 5335.48 0.410285
\(554\) 4820.08 0.369649
\(555\) 10829.8 0.828289
\(556\) 13513.0 1.03072
\(557\) 9915.36 0.754268 0.377134 0.926159i \(-0.376910\pi\)
0.377134 + 0.926159i \(0.376910\pi\)
\(558\) −442.702 −0.0335862
\(559\) −34073.1 −2.57806
\(560\) −250.708 −0.0189185
\(561\) 0 0
\(562\) 4260.52 0.319785
\(563\) −5048.52 −0.377922 −0.188961 0.981985i \(-0.560512\pi\)
−0.188961 + 0.981985i \(0.560512\pi\)
\(564\) −4764.88 −0.355741
\(565\) −4102.75 −0.305494
\(566\) −4122.24 −0.306132
\(567\) 6668.48 0.493915
\(568\) 8886.78 0.656481
\(569\) 10964.5 0.807828 0.403914 0.914797i \(-0.367650\pi\)
0.403914 + 0.914797i \(0.367650\pi\)
\(570\) −4114.09 −0.302316
\(571\) −6757.57 −0.495263 −0.247632 0.968854i \(-0.579652\pi\)
−0.247632 + 0.968854i \(0.579652\pi\)
\(572\) −10363.9 −0.757582
\(573\) 20085.5 1.46437
\(574\) 774.920 0.0563494
\(575\) −2272.19 −0.164794
\(576\) −2111.36 −0.152732
\(577\) −18050.9 −1.30237 −0.651187 0.758917i \(-0.725729\pi\)
−0.651187 + 0.758917i \(0.725729\pi\)
\(578\) 0 0
\(579\) −13015.5 −0.934209
\(580\) −9229.57 −0.660754
\(581\) 2129.81 0.152082
\(582\) 337.780 0.0240574
\(583\) 2689.90 0.191088
\(584\) 14366.1 1.01793
\(585\) −3260.36 −0.230426
\(586\) −11345.1 −0.799762
\(587\) −7447.06 −0.523634 −0.261817 0.965118i \(-0.584322\pi\)
−0.261817 + 0.965118i \(0.584322\pi\)
\(588\) −3404.07 −0.238744
\(589\) 2853.04 0.199589
\(590\) 3608.78 0.251816
\(591\) 8559.34 0.595743
\(592\) −949.505 −0.0659196
\(593\) −1229.32 −0.0851298 −0.0425649 0.999094i \(-0.513553\pi\)
−0.0425649 + 0.999094i \(0.513553\pi\)
\(594\) −8004.82 −0.552932
\(595\) 0 0
\(596\) 11284.0 0.775523
\(597\) 13716.3 0.940317
\(598\) −3253.49 −0.222484
\(599\) 10655.6 0.726840 0.363420 0.931625i \(-0.381609\pi\)
0.363420 + 0.931625i \(0.381609\pi\)
\(600\) 7800.99 0.530790
\(601\) −2134.49 −0.144871 −0.0724356 0.997373i \(-0.523077\pi\)
−0.0724356 + 0.997373i \(0.523077\pi\)
\(602\) −12214.9 −0.826982
\(603\) 4787.73 0.323336
\(604\) −10860.7 −0.731647
\(605\) −2588.67 −0.173957
\(606\) 11203.9 0.751036
\(607\) 12364.9 0.826812 0.413406 0.910547i \(-0.364339\pi\)
0.413406 + 0.910547i \(0.364339\pi\)
\(608\) 14599.0 0.973797
\(609\) 16414.5 1.09220
\(610\) 4241.96 0.281561
\(611\) −13902.5 −0.920513
\(612\) 0 0
\(613\) 4888.87 0.322120 0.161060 0.986945i \(-0.448509\pi\)
0.161060 + 0.986945i \(0.448509\pi\)
\(614\) 4282.73 0.281493
\(615\) −990.920 −0.0649720
\(616\) −9553.30 −0.624859
\(617\) −24750.0 −1.61491 −0.807453 0.589932i \(-0.799155\pi\)
−0.807453 + 0.589932i \(0.799155\pi\)
\(618\) 4684.67 0.304927
\(619\) −10794.8 −0.700937 −0.350469 0.936574i \(-0.613978\pi\)
−0.350469 + 0.936574i \(0.613978\pi\)
\(620\) 1240.91 0.0803809
\(621\) 4398.72 0.284242
\(622\) −6114.28 −0.394148
\(623\) 8312.47 0.534562
\(624\) −778.304 −0.0499313
\(625\) 385.346 0.0246621
\(626\) 4455.22 0.284451
\(627\) 10923.3 0.695748
\(628\) 1757.07 0.111648
\(629\) 0 0
\(630\) −1168.81 −0.0739152
\(631\) −7579.90 −0.478211 −0.239106 0.970994i \(-0.576854\pi\)
−0.239106 + 0.970994i \(0.576854\pi\)
\(632\) −8584.94 −0.540333
\(633\) 16008.3 1.00517
\(634\) 6657.80 0.417058
\(635\) 10262.3 0.641333
\(636\) 1973.64 0.123050
\(637\) −9932.03 −0.617773
\(638\) −13999.5 −0.868723
\(639\) −2886.79 −0.178716
\(640\) 6596.25 0.407406
\(641\) 3181.53 0.196042 0.0980208 0.995184i \(-0.468749\pi\)
0.0980208 + 0.995184i \(0.468749\pi\)
\(642\) 1053.29 0.0647507
\(643\) −10552.8 −0.647219 −0.323609 0.946191i \(-0.604896\pi\)
−0.323609 + 0.946191i \(0.604896\pi\)
\(644\) 2041.63 0.124925
\(645\) 15619.7 0.953526
\(646\) 0 0
\(647\) −10952.0 −0.665483 −0.332742 0.943018i \(-0.607974\pi\)
−0.332742 + 0.943018i \(0.607974\pi\)
\(648\) −10729.8 −0.650471
\(649\) −9581.64 −0.579526
\(650\) 8851.97 0.534158
\(651\) −2206.93 −0.132867
\(652\) 3454.63 0.207506
\(653\) 6526.76 0.391136 0.195568 0.980690i \(-0.437345\pi\)
0.195568 + 0.980690i \(0.437345\pi\)
\(654\) 6240.84 0.373144
\(655\) −18955.7 −1.13078
\(656\) 86.8789 0.00517081
\(657\) −4666.69 −0.277116
\(658\) −4983.92 −0.295279
\(659\) −25717.9 −1.52022 −0.760111 0.649793i \(-0.774855\pi\)
−0.760111 + 0.649793i \(0.774855\pi\)
\(660\) 4751.00 0.280201
\(661\) 11808.9 0.694878 0.347439 0.937703i \(-0.387051\pi\)
0.347439 + 0.937703i \(0.387051\pi\)
\(662\) 10635.3 0.624403
\(663\) 0 0
\(664\) −3426.93 −0.200287
\(665\) 7532.54 0.439247
\(666\) −4426.64 −0.257550
\(667\) 7692.85 0.446579
\(668\) 4072.03 0.235856
\(669\) 7009.76 0.405102
\(670\) 7666.73 0.442077
\(671\) −11262.8 −0.647981
\(672\) −11292.8 −0.648260
\(673\) 6282.85 0.359860 0.179930 0.983679i \(-0.442413\pi\)
0.179930 + 0.983679i \(0.442413\pi\)
\(674\) −6552.54 −0.374472
\(675\) −11967.8 −0.682433
\(676\) −11001.0 −0.625910
\(677\) 18502.8 1.05040 0.525200 0.850979i \(-0.323991\pi\)
0.525200 + 0.850979i \(0.323991\pi\)
\(678\) −4566.00 −0.258638
\(679\) −618.446 −0.0349540
\(680\) 0 0
\(681\) −19076.8 −1.07346
\(682\) 1882.22 0.105680
\(683\) −137.518 −0.00770424 −0.00385212 0.999993i \(-0.501226\pi\)
−0.00385212 + 0.999993i \(0.501226\pi\)
\(684\) −2943.57 −0.164547
\(685\) −4288.71 −0.239217
\(686\) −11677.7 −0.649936
\(687\) −2581.37 −0.143356
\(688\) −1369.46 −0.0758867
\(689\) 5758.47 0.318404
\(690\) 1491.46 0.0822882
\(691\) −303.235 −0.0166940 −0.00834702 0.999965i \(-0.502657\pi\)
−0.00834702 + 0.999965i \(0.502657\pi\)
\(692\) −9599.69 −0.527349
\(693\) 3103.30 0.170108
\(694\) −14158.6 −0.774426
\(695\) −18074.0 −0.986452
\(696\) −26411.5 −1.43840
\(697\) 0 0
\(698\) −8406.54 −0.455863
\(699\) −2985.24 −0.161534
\(700\) −5554.79 −0.299931
\(701\) −23434.3 −1.26263 −0.631313 0.775528i \(-0.717484\pi\)
−0.631313 + 0.775528i \(0.717484\pi\)
\(702\) −17136.5 −0.921331
\(703\) 28528.0 1.53052
\(704\) 8976.83 0.480578
\(705\) 6373.13 0.340463
\(706\) 16385.4 0.873472
\(707\) −20513.4 −1.09121
\(708\) −7030.24 −0.373182
\(709\) −52.1860 −0.00276430 −0.00138215 0.999999i \(-0.500440\pi\)
−0.00138215 + 0.999999i \(0.500440\pi\)
\(710\) −4622.70 −0.244348
\(711\) 2788.74 0.147097
\(712\) −13375.0 −0.704002
\(713\) −1034.30 −0.0543265
\(714\) 0 0
\(715\) 13862.0 0.725046
\(716\) 18272.3 0.953725
\(717\) 1215.98 0.0633354
\(718\) 3347.43 0.173990
\(719\) −2244.80 −0.116435 −0.0582175 0.998304i \(-0.518542\pi\)
−0.0582175 + 0.998304i \(0.518542\pi\)
\(720\) −131.039 −0.00678271
\(721\) −8577.23 −0.443041
\(722\) 860.445 0.0443524
\(723\) 28824.9 1.48272
\(724\) 2348.89 0.120575
\(725\) −20930.4 −1.07219
\(726\) −2880.96 −0.147276
\(727\) −2971.90 −0.151612 −0.0758059 0.997123i \(-0.524153\pi\)
−0.0758059 + 0.997123i \(0.524153\pi\)
\(728\) −20451.4 −1.04118
\(729\) 21748.2 1.10492
\(730\) −7472.91 −0.378883
\(731\) 0 0
\(732\) −8263.73 −0.417263
\(733\) 31704.8 1.59760 0.798802 0.601594i \(-0.205468\pi\)
0.798802 + 0.601594i \(0.205468\pi\)
\(734\) −5898.00 −0.296593
\(735\) 4553.02 0.228491
\(736\) −5292.50 −0.265060
\(737\) −20355.9 −1.01739
\(738\) 405.034 0.0202026
\(739\) −36985.6 −1.84105 −0.920526 0.390681i \(-0.872240\pi\)
−0.920526 + 0.390681i \(0.872240\pi\)
\(740\) 12408.0 0.616389
\(741\) 23384.2 1.15930
\(742\) 2064.36 0.102136
\(743\) 10601.9 0.523481 0.261741 0.965138i \(-0.415704\pi\)
0.261741 + 0.965138i \(0.415704\pi\)
\(744\) 3551.01 0.174981
\(745\) −15092.6 −0.742217
\(746\) −4538.63 −0.222750
\(747\) 1113.21 0.0545248
\(748\) 0 0
\(749\) −1928.48 −0.0940789
\(750\) −10509.1 −0.511653
\(751\) 29161.5 1.41694 0.708468 0.705743i \(-0.249387\pi\)
0.708468 + 0.705743i \(0.249387\pi\)
\(752\) −558.765 −0.0270958
\(753\) 11232.0 0.543583
\(754\) −29969.7 −1.44752
\(755\) 14526.4 0.700225
\(756\) 10753.5 0.517329
\(757\) −34447.9 −1.65394 −0.826969 0.562248i \(-0.809937\pi\)
−0.826969 + 0.562248i \(0.809937\pi\)
\(758\) −9521.67 −0.456257
\(759\) −3959.96 −0.189377
\(760\) −12120.1 −0.578476
\(761\) −14414.5 −0.686632 −0.343316 0.939220i \(-0.611550\pi\)
−0.343316 + 0.939220i \(0.611550\pi\)
\(762\) 11421.0 0.542966
\(763\) −11426.4 −0.542156
\(764\) 23012.6 1.08975
\(765\) 0 0
\(766\) −13357.5 −0.630058
\(767\) −20512.1 −0.965644
\(768\) 17690.3 0.831175
\(769\) −7049.33 −0.330566 −0.165283 0.986246i \(-0.552854\pi\)
−0.165283 + 0.986246i \(0.552854\pi\)
\(770\) 4969.40 0.232578
\(771\) −7565.28 −0.353381
\(772\) −14912.3 −0.695212
\(773\) 26768.3 1.24552 0.622760 0.782413i \(-0.286011\pi\)
0.622760 + 0.782413i \(0.286011\pi\)
\(774\) −6384.47 −0.296492
\(775\) 2814.08 0.130432
\(776\) 995.097 0.0460334
\(777\) −22067.3 −1.01887
\(778\) −12676.7 −0.584165
\(779\) −2610.29 −0.120055
\(780\) 10170.8 0.466889
\(781\) 12273.7 0.562339
\(782\) 0 0
\(783\) 40519.0 1.84934
\(784\) −399.186 −0.0181845
\(785\) −2350.12 −0.106853
\(786\) −21096.1 −0.957343
\(787\) 26085.5 1.18151 0.590755 0.806851i \(-0.298830\pi\)
0.590755 + 0.806851i \(0.298830\pi\)
\(788\) 9806.68 0.443336
\(789\) 18994.6 0.857065
\(790\) 4465.69 0.201116
\(791\) 8359.95 0.375785
\(792\) −4993.30 −0.224027
\(793\) −24111.1 −1.07971
\(794\) 16270.6 0.727232
\(795\) −2639.78 −0.117765
\(796\) 15715.1 0.699758
\(797\) 18108.6 0.804818 0.402409 0.915460i \(-0.368173\pi\)
0.402409 + 0.915460i \(0.368173\pi\)
\(798\) 8383.06 0.371876
\(799\) 0 0
\(800\) 14399.6 0.636379
\(801\) 4344.75 0.191653
\(802\) 14775.0 0.650527
\(803\) 19841.2 0.871958
\(804\) −14935.5 −0.655142
\(805\) −2730.73 −0.119560
\(806\) 4029.41 0.176092
\(807\) −35534.1 −1.55001
\(808\) 33006.6 1.43709
\(809\) −7304.62 −0.317450 −0.158725 0.987323i \(-0.550738\pi\)
−0.158725 + 0.987323i \(0.550738\pi\)
\(810\) 5581.38 0.242111
\(811\) −19511.7 −0.844817 −0.422409 0.906405i \(-0.638815\pi\)
−0.422409 + 0.906405i \(0.638815\pi\)
\(812\) 18806.6 0.812786
\(813\) −2107.55 −0.0909163
\(814\) 18820.6 0.810395
\(815\) −4620.65 −0.198594
\(816\) 0 0
\(817\) 41145.5 1.76193
\(818\) −9003.45 −0.384839
\(819\) 6643.45 0.283444
\(820\) −1135.32 −0.0483503
\(821\) 29364.3 1.24826 0.624131 0.781320i \(-0.285453\pi\)
0.624131 + 0.781320i \(0.285453\pi\)
\(822\) −4772.96 −0.202526
\(823\) 28112.4 1.19069 0.595345 0.803470i \(-0.297015\pi\)
0.595345 + 0.803470i \(0.297015\pi\)
\(824\) 13801.0 0.583472
\(825\) 10774.1 0.454673
\(826\) −7353.43 −0.309756
\(827\) 39360.5 1.65502 0.827509 0.561453i \(-0.189757\pi\)
0.827509 + 0.561453i \(0.189757\pi\)
\(828\) 1067.12 0.0447885
\(829\) 15525.3 0.650443 0.325221 0.945638i \(-0.394561\pi\)
0.325221 + 0.945638i \(0.394561\pi\)
\(830\) 1782.61 0.0745485
\(831\) 12559.3 0.524280
\(832\) 19217.3 0.800771
\(833\) 0 0
\(834\) −20114.7 −0.835151
\(835\) −5446.44 −0.225727
\(836\) 12515.1 0.517756
\(837\) −5447.75 −0.224972
\(838\) 16045.5 0.661433
\(839\) −30007.9 −1.23479 −0.617394 0.786654i \(-0.711811\pi\)
−0.617394 + 0.786654i \(0.711811\pi\)
\(840\) 9375.29 0.385093
\(841\) 46474.0 1.90553
\(842\) 13971.1 0.571826
\(843\) 11101.3 0.453556
\(844\) 18341.2 0.748020
\(845\) 14714.1 0.599029
\(846\) −2604.99 −0.105864
\(847\) 5274.78 0.213983
\(848\) 231.443 0.00937239
\(849\) −10741.0 −0.434193
\(850\) 0 0
\(851\) −10342.1 −0.416595
\(852\) 9005.44 0.362114
\(853\) 4351.43 0.174666 0.0873331 0.996179i \(-0.472166\pi\)
0.0873331 + 0.996179i \(0.472166\pi\)
\(854\) −8643.62 −0.346345
\(855\) 3937.09 0.157480
\(856\) 3102.98 0.123899
\(857\) −25861.5 −1.03082 −0.515410 0.856944i \(-0.672360\pi\)
−0.515410 + 0.856944i \(0.672360\pi\)
\(858\) 15427.2 0.613840
\(859\) 19084.0 0.758017 0.379009 0.925393i \(-0.376265\pi\)
0.379009 + 0.925393i \(0.376265\pi\)
\(860\) 17895.9 0.709588
\(861\) 2019.14 0.0799213
\(862\) 3326.16 0.131426
\(863\) 24790.2 0.977831 0.488916 0.872331i \(-0.337393\pi\)
0.488916 + 0.872331i \(0.337393\pi\)
\(864\) −27876.1 −1.09765
\(865\) 12839.8 0.504701
\(866\) −19019.9 −0.746332
\(867\) 0 0
\(868\) −2528.54 −0.0988757
\(869\) −11856.8 −0.462847
\(870\) 13738.6 0.535383
\(871\) −43577.2 −1.69524
\(872\) 18385.5 0.714003
\(873\) −323.248 −0.0125318
\(874\) 3928.81 0.152053
\(875\) 19241.3 0.743401
\(876\) 14557.9 0.561491
\(877\) 14394.0 0.554219 0.277109 0.960838i \(-0.410624\pi\)
0.277109 + 0.960838i \(0.410624\pi\)
\(878\) 801.839 0.0308209
\(879\) −29560.9 −1.13432
\(880\) 557.137 0.0213421
\(881\) 15328.6 0.586190 0.293095 0.956083i \(-0.405315\pi\)
0.293095 + 0.956083i \(0.405315\pi\)
\(882\) −1861.02 −0.0710475
\(883\) 8385.90 0.319602 0.159801 0.987149i \(-0.448915\pi\)
0.159801 + 0.987149i \(0.448915\pi\)
\(884\) 0 0
\(885\) 9403.11 0.357155
\(886\) 22397.7 0.849283
\(887\) −9215.90 −0.348861 −0.174430 0.984669i \(-0.555808\pi\)
−0.174430 + 0.984669i \(0.555808\pi\)
\(888\) 35507.0 1.34182
\(889\) −20910.9 −0.788897
\(890\) 6957.37 0.262035
\(891\) −14819.1 −0.557191
\(892\) 8031.28 0.301465
\(893\) 16788.1 0.629108
\(894\) −16796.8 −0.628376
\(895\) −24439.6 −0.912765
\(896\) −13440.8 −0.501146
\(897\) −8477.36 −0.315553
\(898\) −17793.2 −0.661211
\(899\) −9527.49 −0.353459
\(900\) −2903.37 −0.107532
\(901\) 0 0
\(902\) −1722.07 −0.0635684
\(903\) −31827.4 −1.17292
\(904\) −13451.4 −0.494897
\(905\) −3141.70 −0.115396
\(906\) 16166.6 0.592825
\(907\) 7468.98 0.273433 0.136716 0.990610i \(-0.456345\pi\)
0.136716 + 0.990610i \(0.456345\pi\)
\(908\) −21856.8 −0.798836
\(909\) −10721.9 −0.391224
\(910\) 10638.4 0.387536
\(911\) 30698.3 1.11644 0.558222 0.829692i \(-0.311484\pi\)
0.558222 + 0.829692i \(0.311484\pi\)
\(912\) 939.854 0.0341246
\(913\) −4732.98 −0.171565
\(914\) 432.140 0.0156389
\(915\) 11052.9 0.399343
\(916\) −2957.55 −0.106681
\(917\) 38625.1 1.39096
\(918\) 0 0
\(919\) −24550.8 −0.881237 −0.440619 0.897694i \(-0.645241\pi\)
−0.440619 + 0.897694i \(0.645241\pi\)
\(920\) 4393.82 0.157457
\(921\) 11159.2 0.399247
\(922\) −26537.4 −0.947898
\(923\) 26275.1 0.937006
\(924\) −9680.86 −0.344672
\(925\) 28138.3 1.00020
\(926\) −18296.0 −0.649293
\(927\) −4483.13 −0.158841
\(928\) −48752.1 −1.72453
\(929\) −41655.0 −1.47110 −0.735552 0.677468i \(-0.763077\pi\)
−0.735552 + 0.677468i \(0.763077\pi\)
\(930\) −1847.15 −0.0651296
\(931\) 11993.6 0.422206
\(932\) −3420.28 −0.120209
\(933\) −15931.5 −0.559027
\(934\) 3343.54 0.117135
\(935\) 0 0
\(936\) −10689.5 −0.373288
\(937\) 83.7368 0.00291949 0.00145975 0.999999i \(-0.499535\pi\)
0.00145975 + 0.999999i \(0.499535\pi\)
\(938\) −15622.1 −0.543795
\(939\) 11608.6 0.403442
\(940\) 7301.88 0.253363
\(941\) −49294.1 −1.70770 −0.853848 0.520523i \(-0.825737\pi\)
−0.853848 + 0.520523i \(0.825737\pi\)
\(942\) −2615.48 −0.0904639
\(943\) 946.293 0.0326782
\(944\) −824.418 −0.0284243
\(945\) −14383.0 −0.495111
\(946\) 27144.7 0.932927
\(947\) −9445.27 −0.324108 −0.162054 0.986782i \(-0.551812\pi\)
−0.162054 + 0.986782i \(0.551812\pi\)
\(948\) −8699.57 −0.298047
\(949\) 42475.5 1.45291
\(950\) −10689.3 −0.365061
\(951\) 17347.7 0.591521
\(952\) 0 0
\(953\) 41693.4 1.41719 0.708595 0.705616i \(-0.249329\pi\)
0.708595 + 0.705616i \(0.249329\pi\)
\(954\) 1079.00 0.0366183
\(955\) −30779.8 −1.04294
\(956\) 1393.18 0.0471324
\(957\) −36477.3 −1.23213
\(958\) −7323.60 −0.246988
\(959\) 8738.88 0.294258
\(960\) −8809.56 −0.296174
\(961\) −28510.0 −0.957002
\(962\) 40290.6 1.35033
\(963\) −1007.97 −0.0337295
\(964\) 33025.5 1.10340
\(965\) 19945.5 0.665355
\(966\) −3039.06 −0.101222
\(967\) 32380.0 1.07681 0.538403 0.842688i \(-0.319028\pi\)
0.538403 + 0.842688i \(0.319028\pi\)
\(968\) −8487.28 −0.281809
\(969\) 0 0
\(970\) −517.626 −0.0171340
\(971\) 5401.25 0.178511 0.0892556 0.996009i \(-0.471551\pi\)
0.0892556 + 0.996009i \(0.471551\pi\)
\(972\) 10051.1 0.331676
\(973\) 36828.3 1.21342
\(974\) −30721.5 −1.01066
\(975\) 23064.8 0.757606
\(976\) −969.066 −0.0317818
\(977\) −25086.2 −0.821473 −0.410737 0.911754i \(-0.634728\pi\)
−0.410737 + 0.911754i \(0.634728\pi\)
\(978\) −5142.37 −0.168134
\(979\) −18472.4 −0.603045
\(980\) 5216.52 0.170036
\(981\) −5972.35 −0.194375
\(982\) −23421.4 −0.761108
\(983\) −59631.6 −1.93485 −0.967423 0.253166i \(-0.918528\pi\)
−0.967423 + 0.253166i \(0.918528\pi\)
\(984\) −3248.86 −0.105254
\(985\) −13116.7 −0.424296
\(986\) 0 0
\(987\) −12986.2 −0.418799
\(988\) 26792.0 0.862719
\(989\) −14916.2 −0.479584
\(990\) 2597.40 0.0833846
\(991\) −40063.8 −1.28423 −0.642114 0.766609i \(-0.721942\pi\)
−0.642114 + 0.766609i \(0.721942\pi\)
\(992\) 6554.70 0.209790
\(993\) 27711.6 0.885601
\(994\) 9419.43 0.300569
\(995\) −21019.3 −0.669706
\(996\) −3472.68 −0.110478
\(997\) −10818.4 −0.343652 −0.171826 0.985127i \(-0.554967\pi\)
−0.171826 + 0.985127i \(0.554967\pi\)
\(998\) 13213.5 0.419103
\(999\) −54472.8 −1.72517
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 289.4.a.g.1.5 12
17.4 even 4 289.4.b.e.288.8 12
17.11 odd 16 17.4.d.a.2.2 12
17.13 even 4 289.4.b.e.288.7 12
17.14 odd 16 17.4.d.a.9.2 yes 12
17.16 even 2 inner 289.4.a.g.1.6 12
51.11 even 16 153.4.l.a.19.2 12
51.14 even 16 153.4.l.a.145.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.2.2 12 17.11 odd 16
17.4.d.a.9.2 yes 12 17.14 odd 16
153.4.l.a.19.2 12 51.11 even 16
153.4.l.a.145.2 12 51.14 even 16
289.4.a.g.1.5 12 1.1 even 1 trivial
289.4.a.g.1.6 12 17.16 even 2 inner
289.4.b.e.288.7 12 17.13 even 4
289.4.b.e.288.8 12 17.4 even 4