Properties

Label 8001.2.a.q.1.7
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $15$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + 726 x^{3} + 145 x^{2} - 83 x - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.601235\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.601235 q^{2} -1.63852 q^{4} -4.40463 q^{5} -1.00000 q^{7} +2.18760 q^{8} +O(q^{10})\) \(q-0.601235 q^{2} -1.63852 q^{4} -4.40463 q^{5} -1.00000 q^{7} +2.18760 q^{8} +2.64822 q^{10} -4.30434 q^{11} -1.91150 q^{13} +0.601235 q^{14} +1.96177 q^{16} -7.97603 q^{17} +2.00095 q^{19} +7.21707 q^{20} +2.58792 q^{22} -2.11447 q^{23} +14.4008 q^{25} +1.14926 q^{26} +1.63852 q^{28} +4.21397 q^{29} -2.87794 q^{31} -5.55469 q^{32} +4.79547 q^{34} +4.40463 q^{35} -2.54141 q^{37} -1.20304 q^{38} -9.63559 q^{40} +6.34847 q^{41} +10.2731 q^{43} +7.05274 q^{44} +1.27129 q^{46} -10.7858 q^{47} +1.00000 q^{49} -8.65827 q^{50} +3.13202 q^{52} +4.18365 q^{53} +18.9591 q^{55} -2.18760 q^{56} -2.53358 q^{58} +14.3788 q^{59} -8.44301 q^{61} +1.73032 q^{62} -0.583865 q^{64} +8.41946 q^{65} -7.52093 q^{67} +13.0689 q^{68} -2.64822 q^{70} -10.3166 q^{71} -3.45838 q^{73} +1.52799 q^{74} -3.27860 q^{76} +4.30434 q^{77} +14.3417 q^{79} -8.64088 q^{80} -3.81692 q^{82} +5.58186 q^{83} +35.1315 q^{85} -6.17654 q^{86} -9.41619 q^{88} +7.39046 q^{89} +1.91150 q^{91} +3.46459 q^{92} +6.48482 q^{94} -8.81347 q^{95} +13.7121 q^{97} -0.601235 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.601235 −0.425137 −0.212569 0.977146i \(-0.568183\pi\)
−0.212569 + 0.977146i \(0.568183\pi\)
\(3\) 0 0
\(4\) −1.63852 −0.819258
\(5\) −4.40463 −1.96981 −0.984906 0.173088i \(-0.944625\pi\)
−0.984906 + 0.173088i \(0.944625\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.18760 0.773435
\(9\) 0 0
\(10\) 2.64822 0.837441
\(11\) −4.30434 −1.29781 −0.648904 0.760870i \(-0.724772\pi\)
−0.648904 + 0.760870i \(0.724772\pi\)
\(12\) 0 0
\(13\) −1.91150 −0.530154 −0.265077 0.964227i \(-0.585397\pi\)
−0.265077 + 0.964227i \(0.585397\pi\)
\(14\) 0.601235 0.160687
\(15\) 0 0
\(16\) 1.96177 0.490442
\(17\) −7.97603 −1.93447 −0.967235 0.253881i \(-0.918293\pi\)
−0.967235 + 0.253881i \(0.918293\pi\)
\(18\) 0 0
\(19\) 2.00095 0.459050 0.229525 0.973303i \(-0.426283\pi\)
0.229525 + 0.973303i \(0.426283\pi\)
\(20\) 7.21707 1.61379
\(21\) 0 0
\(22\) 2.58792 0.551747
\(23\) −2.11447 −0.440897 −0.220448 0.975399i \(-0.570752\pi\)
−0.220448 + 0.975399i \(0.570752\pi\)
\(24\) 0 0
\(25\) 14.4008 2.88016
\(26\) 1.14926 0.225388
\(27\) 0 0
\(28\) 1.63852 0.309651
\(29\) 4.21397 0.782514 0.391257 0.920281i \(-0.372040\pi\)
0.391257 + 0.920281i \(0.372040\pi\)
\(30\) 0 0
\(31\) −2.87794 −0.516894 −0.258447 0.966025i \(-0.583211\pi\)
−0.258447 + 0.966025i \(0.583211\pi\)
\(32\) −5.55469 −0.981940
\(33\) 0 0
\(34\) 4.79547 0.822416
\(35\) 4.40463 0.744519
\(36\) 0 0
\(37\) −2.54141 −0.417805 −0.208903 0.977936i \(-0.566989\pi\)
−0.208903 + 0.977936i \(0.566989\pi\)
\(38\) −1.20304 −0.195159
\(39\) 0 0
\(40\) −9.63559 −1.52352
\(41\) 6.34847 0.991464 0.495732 0.868475i \(-0.334900\pi\)
0.495732 + 0.868475i \(0.334900\pi\)
\(42\) 0 0
\(43\) 10.2731 1.56663 0.783316 0.621624i \(-0.213527\pi\)
0.783316 + 0.621624i \(0.213527\pi\)
\(44\) 7.05274 1.06324
\(45\) 0 0
\(46\) 1.27129 0.187442
\(47\) −10.7858 −1.57328 −0.786638 0.617415i \(-0.788180\pi\)
−0.786638 + 0.617415i \(0.788180\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.65827 −1.22446
\(51\) 0 0
\(52\) 3.13202 0.434333
\(53\) 4.18365 0.574668 0.287334 0.957830i \(-0.407231\pi\)
0.287334 + 0.957830i \(0.407231\pi\)
\(54\) 0 0
\(55\) 18.9591 2.55644
\(56\) −2.18760 −0.292331
\(57\) 0 0
\(58\) −2.53358 −0.332676
\(59\) 14.3788 1.87196 0.935982 0.352049i \(-0.114515\pi\)
0.935982 + 0.352049i \(0.114515\pi\)
\(60\) 0 0
\(61\) −8.44301 −1.08102 −0.540509 0.841338i \(-0.681768\pi\)
−0.540509 + 0.841338i \(0.681768\pi\)
\(62\) 1.73032 0.219751
\(63\) 0 0
\(64\) −0.583865 −0.0729831
\(65\) 8.41946 1.04430
\(66\) 0 0
\(67\) −7.52093 −0.918828 −0.459414 0.888222i \(-0.651941\pi\)
−0.459414 + 0.888222i \(0.651941\pi\)
\(68\) 13.0689 1.58483
\(69\) 0 0
\(70\) −2.64822 −0.316523
\(71\) −10.3166 −1.22436 −0.612179 0.790720i \(-0.709707\pi\)
−0.612179 + 0.790720i \(0.709707\pi\)
\(72\) 0 0
\(73\) −3.45838 −0.404773 −0.202387 0.979306i \(-0.564870\pi\)
−0.202387 + 0.979306i \(0.564870\pi\)
\(74\) 1.52799 0.177625
\(75\) 0 0
\(76\) −3.27860 −0.376081
\(77\) 4.30434 0.490525
\(78\) 0 0
\(79\) 14.3417 1.61356 0.806782 0.590850i \(-0.201207\pi\)
0.806782 + 0.590850i \(0.201207\pi\)
\(80\) −8.64088 −0.966080
\(81\) 0 0
\(82\) −3.81692 −0.421508
\(83\) 5.58186 0.612688 0.306344 0.951921i \(-0.400894\pi\)
0.306344 + 0.951921i \(0.400894\pi\)
\(84\) 0 0
\(85\) 35.1315 3.81054
\(86\) −6.17654 −0.666034
\(87\) 0 0
\(88\) −9.41619 −1.00377
\(89\) 7.39046 0.783387 0.391693 0.920096i \(-0.371889\pi\)
0.391693 + 0.920096i \(0.371889\pi\)
\(90\) 0 0
\(91\) 1.91150 0.200380
\(92\) 3.46459 0.361208
\(93\) 0 0
\(94\) 6.48482 0.668858
\(95\) −8.81347 −0.904243
\(96\) 0 0
\(97\) 13.7121 1.39226 0.696128 0.717917i \(-0.254904\pi\)
0.696128 + 0.717917i \(0.254904\pi\)
\(98\) −0.601235 −0.0607339
\(99\) 0 0
\(100\) −23.5960 −2.35960
\(101\) −1.24756 −0.124137 −0.0620683 0.998072i \(-0.519770\pi\)
−0.0620683 + 0.998072i \(0.519770\pi\)
\(102\) 0 0
\(103\) 5.12235 0.504720 0.252360 0.967633i \(-0.418793\pi\)
0.252360 + 0.967633i \(0.418793\pi\)
\(104\) −4.18160 −0.410040
\(105\) 0 0
\(106\) −2.51536 −0.244313
\(107\) 16.0637 1.55293 0.776467 0.630157i \(-0.217010\pi\)
0.776467 + 0.630157i \(0.217010\pi\)
\(108\) 0 0
\(109\) −8.30523 −0.795496 −0.397748 0.917495i \(-0.630208\pi\)
−0.397748 + 0.917495i \(0.630208\pi\)
\(110\) −11.3988 −1.08684
\(111\) 0 0
\(112\) −1.96177 −0.185370
\(113\) 17.1150 1.61004 0.805020 0.593247i \(-0.202154\pi\)
0.805020 + 0.593247i \(0.202154\pi\)
\(114\) 0 0
\(115\) 9.31345 0.868484
\(116\) −6.90466 −0.641081
\(117\) 0 0
\(118\) −8.64505 −0.795841
\(119\) 7.97603 0.731161
\(120\) 0 0
\(121\) 7.52736 0.684306
\(122\) 5.07624 0.459581
\(123\) 0 0
\(124\) 4.71556 0.423470
\(125\) −41.4071 −3.70357
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 11.4604 1.01297
\(129\) 0 0
\(130\) −5.06207 −0.443973
\(131\) −4.01168 −0.350502 −0.175251 0.984524i \(-0.556074\pi\)
−0.175251 + 0.984524i \(0.556074\pi\)
\(132\) 0 0
\(133\) −2.00095 −0.173505
\(134\) 4.52185 0.390628
\(135\) 0 0
\(136\) −17.4484 −1.49619
\(137\) 6.88474 0.588203 0.294102 0.955774i \(-0.404980\pi\)
0.294102 + 0.955774i \(0.404980\pi\)
\(138\) 0 0
\(139\) −6.93504 −0.588222 −0.294111 0.955771i \(-0.595024\pi\)
−0.294111 + 0.955771i \(0.595024\pi\)
\(140\) −7.21707 −0.609954
\(141\) 0 0
\(142\) 6.20271 0.520520
\(143\) 8.22775 0.688039
\(144\) 0 0
\(145\) −18.5610 −1.54141
\(146\) 2.07930 0.172084
\(147\) 0 0
\(148\) 4.16414 0.342291
\(149\) 14.3035 1.17179 0.585896 0.810386i \(-0.300743\pi\)
0.585896 + 0.810386i \(0.300743\pi\)
\(150\) 0 0
\(151\) 3.87716 0.315519 0.157760 0.987478i \(-0.449573\pi\)
0.157760 + 0.987478i \(0.449573\pi\)
\(152\) 4.37729 0.355045
\(153\) 0 0
\(154\) −2.58792 −0.208541
\(155\) 12.6763 1.01818
\(156\) 0 0
\(157\) −20.8139 −1.66113 −0.830565 0.556921i \(-0.811983\pi\)
−0.830565 + 0.556921i \(0.811983\pi\)
\(158\) −8.62271 −0.685986
\(159\) 0 0
\(160\) 24.4664 1.93424
\(161\) 2.11447 0.166643
\(162\) 0 0
\(163\) −24.2929 −1.90277 −0.951383 0.308011i \(-0.900337\pi\)
−0.951383 + 0.308011i \(0.900337\pi\)
\(164\) −10.4021 −0.812265
\(165\) 0 0
\(166\) −3.35601 −0.260477
\(167\) −4.02158 −0.311200 −0.155600 0.987820i \(-0.549731\pi\)
−0.155600 + 0.987820i \(0.549731\pi\)
\(168\) 0 0
\(169\) −9.34617 −0.718936
\(170\) −21.1223 −1.62000
\(171\) 0 0
\(172\) −16.8326 −1.28348
\(173\) 5.91524 0.449727 0.224864 0.974390i \(-0.427806\pi\)
0.224864 + 0.974390i \(0.427806\pi\)
\(174\) 0 0
\(175\) −14.4008 −1.08860
\(176\) −8.44413 −0.636500
\(177\) 0 0
\(178\) −4.44340 −0.333047
\(179\) 1.75316 0.131038 0.0655188 0.997851i \(-0.479130\pi\)
0.0655188 + 0.997851i \(0.479130\pi\)
\(180\) 0 0
\(181\) 1.44921 0.107719 0.0538593 0.998549i \(-0.482848\pi\)
0.0538593 + 0.998549i \(0.482848\pi\)
\(182\) −1.14926 −0.0851888
\(183\) 0 0
\(184\) −4.62561 −0.341005
\(185\) 11.1940 0.822998
\(186\) 0 0
\(187\) 34.3316 2.51057
\(188\) 17.6728 1.28892
\(189\) 0 0
\(190\) 5.29897 0.384428
\(191\) −2.95252 −0.213637 −0.106818 0.994279i \(-0.534066\pi\)
−0.106818 + 0.994279i \(0.534066\pi\)
\(192\) 0 0
\(193\) 19.5764 1.40914 0.704570 0.709634i \(-0.251140\pi\)
0.704570 + 0.709634i \(0.251140\pi\)
\(194\) −8.24422 −0.591900
\(195\) 0 0
\(196\) −1.63852 −0.117037
\(197\) −9.65828 −0.688124 −0.344062 0.938947i \(-0.611803\pi\)
−0.344062 + 0.938947i \(0.611803\pi\)
\(198\) 0 0
\(199\) −17.5931 −1.24714 −0.623570 0.781768i \(-0.714318\pi\)
−0.623570 + 0.781768i \(0.714318\pi\)
\(200\) 31.5033 2.22762
\(201\) 0 0
\(202\) 0.750075 0.0527751
\(203\) −4.21397 −0.295763
\(204\) 0 0
\(205\) −27.9627 −1.95300
\(206\) −3.07973 −0.214575
\(207\) 0 0
\(208\) −3.74992 −0.260010
\(209\) −8.61279 −0.595759
\(210\) 0 0
\(211\) −3.30191 −0.227313 −0.113656 0.993520i \(-0.536256\pi\)
−0.113656 + 0.993520i \(0.536256\pi\)
\(212\) −6.85498 −0.470802
\(213\) 0 0
\(214\) −9.65805 −0.660210
\(215\) −45.2492 −3.08597
\(216\) 0 0
\(217\) 2.87794 0.195368
\(218\) 4.99339 0.338195
\(219\) 0 0
\(220\) −31.0647 −2.09438
\(221\) 15.2462 1.02557
\(222\) 0 0
\(223\) 16.9302 1.13373 0.566866 0.823810i \(-0.308156\pi\)
0.566866 + 0.823810i \(0.308156\pi\)
\(224\) 5.55469 0.371138
\(225\) 0 0
\(226\) −10.2901 −0.684488
\(227\) −6.21508 −0.412509 −0.206255 0.978498i \(-0.566128\pi\)
−0.206255 + 0.978498i \(0.566128\pi\)
\(228\) 0 0
\(229\) −1.17160 −0.0774215 −0.0387108 0.999250i \(-0.512325\pi\)
−0.0387108 + 0.999250i \(0.512325\pi\)
\(230\) −5.59957 −0.369225
\(231\) 0 0
\(232\) 9.21849 0.605224
\(233\) 23.8516 1.56257 0.781286 0.624173i \(-0.214564\pi\)
0.781286 + 0.624173i \(0.214564\pi\)
\(234\) 0 0
\(235\) 47.5077 3.09906
\(236\) −23.5599 −1.53362
\(237\) 0 0
\(238\) −4.79547 −0.310844
\(239\) −14.9800 −0.968977 −0.484488 0.874798i \(-0.660994\pi\)
−0.484488 + 0.874798i \(0.660994\pi\)
\(240\) 0 0
\(241\) −10.1108 −0.651291 −0.325646 0.945492i \(-0.605582\pi\)
−0.325646 + 0.945492i \(0.605582\pi\)
\(242\) −4.52571 −0.290924
\(243\) 0 0
\(244\) 13.8340 0.885632
\(245\) −4.40463 −0.281402
\(246\) 0 0
\(247\) −3.82482 −0.243368
\(248\) −6.29580 −0.399784
\(249\) 0 0
\(250\) 24.8954 1.57452
\(251\) −2.23745 −0.141227 −0.0706135 0.997504i \(-0.522496\pi\)
−0.0706135 + 0.997504i \(0.522496\pi\)
\(252\) 0 0
\(253\) 9.10138 0.572199
\(254\) −0.601235 −0.0377248
\(255\) 0 0
\(256\) −5.72268 −0.357667
\(257\) −9.85947 −0.615017 −0.307508 0.951545i \(-0.599495\pi\)
−0.307508 + 0.951545i \(0.599495\pi\)
\(258\) 0 0
\(259\) 2.54141 0.157916
\(260\) −13.7954 −0.855555
\(261\) 0 0
\(262\) 2.41196 0.149012
\(263\) 8.92520 0.550352 0.275176 0.961394i \(-0.411264\pi\)
0.275176 + 0.961394i \(0.411264\pi\)
\(264\) 0 0
\(265\) −18.4274 −1.13199
\(266\) 1.20304 0.0737633
\(267\) 0 0
\(268\) 12.3232 0.752758
\(269\) −0.300991 −0.0183518 −0.00917588 0.999958i \(-0.502921\pi\)
−0.00917588 + 0.999958i \(0.502921\pi\)
\(270\) 0 0
\(271\) −4.19940 −0.255095 −0.127548 0.991832i \(-0.540711\pi\)
−0.127548 + 0.991832i \(0.540711\pi\)
\(272\) −15.6471 −0.948746
\(273\) 0 0
\(274\) −4.13935 −0.250067
\(275\) −61.9860 −3.73790
\(276\) 0 0
\(277\) 15.0056 0.901600 0.450800 0.892625i \(-0.351139\pi\)
0.450800 + 0.892625i \(0.351139\pi\)
\(278\) 4.16959 0.250075
\(279\) 0 0
\(280\) 9.63559 0.575837
\(281\) 5.79057 0.345437 0.172718 0.984971i \(-0.444745\pi\)
0.172718 + 0.984971i \(0.444745\pi\)
\(282\) 0 0
\(283\) 23.2225 1.38043 0.690217 0.723603i \(-0.257515\pi\)
0.690217 + 0.723603i \(0.257515\pi\)
\(284\) 16.9039 1.00306
\(285\) 0 0
\(286\) −4.94681 −0.292511
\(287\) −6.34847 −0.374738
\(288\) 0 0
\(289\) 46.6170 2.74218
\(290\) 11.1595 0.655309
\(291\) 0 0
\(292\) 5.66662 0.331614
\(293\) 20.2897 1.18534 0.592668 0.805447i \(-0.298075\pi\)
0.592668 + 0.805447i \(0.298075\pi\)
\(294\) 0 0
\(295\) −63.3335 −3.68742
\(296\) −5.55960 −0.323145
\(297\) 0 0
\(298\) −8.59979 −0.498173
\(299\) 4.04180 0.233743
\(300\) 0 0
\(301\) −10.2731 −0.592131
\(302\) −2.33109 −0.134139
\(303\) 0 0
\(304\) 3.92541 0.225138
\(305\) 37.1884 2.12940
\(306\) 0 0
\(307\) 16.9061 0.964883 0.482442 0.875928i \(-0.339750\pi\)
0.482442 + 0.875928i \(0.339750\pi\)
\(308\) −7.05274 −0.401867
\(309\) 0 0
\(310\) −7.62143 −0.432868
\(311\) −10.4239 −0.591085 −0.295543 0.955330i \(-0.595500\pi\)
−0.295543 + 0.955330i \(0.595500\pi\)
\(312\) 0 0
\(313\) −21.5137 −1.21603 −0.608013 0.793927i \(-0.708033\pi\)
−0.608013 + 0.793927i \(0.708033\pi\)
\(314\) 12.5140 0.706209
\(315\) 0 0
\(316\) −23.4991 −1.32193
\(317\) 4.79749 0.269454 0.134727 0.990883i \(-0.456984\pi\)
0.134727 + 0.990883i \(0.456984\pi\)
\(318\) 0 0
\(319\) −18.1384 −1.01555
\(320\) 2.57171 0.143763
\(321\) 0 0
\(322\) −1.27129 −0.0708463
\(323\) −15.9597 −0.888020
\(324\) 0 0
\(325\) −27.5271 −1.52693
\(326\) 14.6057 0.808937
\(327\) 0 0
\(328\) 13.8879 0.766833
\(329\) 10.7858 0.594642
\(330\) 0 0
\(331\) −20.8566 −1.14638 −0.573190 0.819422i \(-0.694294\pi\)
−0.573190 + 0.819422i \(0.694294\pi\)
\(332\) −9.14596 −0.501950
\(333\) 0 0
\(334\) 2.41792 0.132303
\(335\) 33.1270 1.80992
\(336\) 0 0
\(337\) 0.154395 0.00841045 0.00420522 0.999991i \(-0.498661\pi\)
0.00420522 + 0.999991i \(0.498661\pi\)
\(338\) 5.61924 0.305647
\(339\) 0 0
\(340\) −57.5635 −3.12182
\(341\) 12.3877 0.670829
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 22.4735 1.21169
\(345\) 0 0
\(346\) −3.55645 −0.191196
\(347\) 17.7217 0.951350 0.475675 0.879621i \(-0.342204\pi\)
0.475675 + 0.879621i \(0.342204\pi\)
\(348\) 0 0
\(349\) 12.2962 0.658202 0.329101 0.944295i \(-0.393254\pi\)
0.329101 + 0.944295i \(0.393254\pi\)
\(350\) 8.65827 0.462804
\(351\) 0 0
\(352\) 23.9093 1.27437
\(353\) −16.1440 −0.859258 −0.429629 0.903005i \(-0.641356\pi\)
−0.429629 + 0.903005i \(0.641356\pi\)
\(354\) 0 0
\(355\) 45.4409 2.41175
\(356\) −12.1094 −0.641796
\(357\) 0 0
\(358\) −1.05406 −0.0557089
\(359\) −10.9875 −0.579900 −0.289950 0.957042i \(-0.593639\pi\)
−0.289950 + 0.957042i \(0.593639\pi\)
\(360\) 0 0
\(361\) −14.9962 −0.789273
\(362\) −0.871314 −0.0457952
\(363\) 0 0
\(364\) −3.13202 −0.164163
\(365\) 15.2329 0.797327
\(366\) 0 0
\(367\) 15.7238 0.820778 0.410389 0.911911i \(-0.365393\pi\)
0.410389 + 0.911911i \(0.365393\pi\)
\(368\) −4.14809 −0.216234
\(369\) 0 0
\(370\) −6.73022 −0.349887
\(371\) −4.18365 −0.217204
\(372\) 0 0
\(373\) 3.75838 0.194602 0.0973008 0.995255i \(-0.468979\pi\)
0.0973008 + 0.995255i \(0.468979\pi\)
\(374\) −20.6413 −1.06734
\(375\) 0 0
\(376\) −23.5951 −1.21683
\(377\) −8.05500 −0.414853
\(378\) 0 0
\(379\) 9.46136 0.485997 0.242999 0.970027i \(-0.421869\pi\)
0.242999 + 0.970027i \(0.421869\pi\)
\(380\) 14.4410 0.740809
\(381\) 0 0
\(382\) 1.77516 0.0908249
\(383\) 10.3557 0.529150 0.264575 0.964365i \(-0.414768\pi\)
0.264575 + 0.964365i \(0.414768\pi\)
\(384\) 0 0
\(385\) −18.9591 −0.966243
\(386\) −11.7700 −0.599078
\(387\) 0 0
\(388\) −22.4676 −1.14062
\(389\) −17.0981 −0.866907 −0.433453 0.901176i \(-0.642705\pi\)
−0.433453 + 0.901176i \(0.642705\pi\)
\(390\) 0 0
\(391\) 16.8650 0.852902
\(392\) 2.18760 0.110491
\(393\) 0 0
\(394\) 5.80689 0.292547
\(395\) −63.1698 −3.17842
\(396\) 0 0
\(397\) −22.1102 −1.10968 −0.554840 0.831957i \(-0.687220\pi\)
−0.554840 + 0.831957i \(0.687220\pi\)
\(398\) 10.5776 0.530205
\(399\) 0 0
\(400\) 28.2511 1.41255
\(401\) −38.1546 −1.90535 −0.952675 0.303990i \(-0.901681\pi\)
−0.952675 + 0.303990i \(0.901681\pi\)
\(402\) 0 0
\(403\) 5.50119 0.274034
\(404\) 2.04414 0.101700
\(405\) 0 0
\(406\) 2.53358 0.125740
\(407\) 10.9391 0.542231
\(408\) 0 0
\(409\) 9.50413 0.469949 0.234974 0.972002i \(-0.424499\pi\)
0.234974 + 0.972002i \(0.424499\pi\)
\(410\) 16.8121 0.830293
\(411\) 0 0
\(412\) −8.39305 −0.413496
\(413\) −14.3788 −0.707536
\(414\) 0 0
\(415\) −24.5860 −1.20688
\(416\) 10.6178 0.520580
\(417\) 0 0
\(418\) 5.17831 0.253279
\(419\) −5.66006 −0.276512 −0.138256 0.990397i \(-0.544150\pi\)
−0.138256 + 0.990397i \(0.544150\pi\)
\(420\) 0 0
\(421\) −10.0019 −0.487462 −0.243731 0.969843i \(-0.578371\pi\)
−0.243731 + 0.969843i \(0.578371\pi\)
\(422\) 1.98522 0.0966391
\(423\) 0 0
\(424\) 9.15216 0.444468
\(425\) −114.861 −5.57159
\(426\) 0 0
\(427\) 8.44301 0.408586
\(428\) −26.3206 −1.27225
\(429\) 0 0
\(430\) 27.2054 1.31196
\(431\) −2.95645 −0.142407 −0.0712036 0.997462i \(-0.522684\pi\)
−0.0712036 + 0.997462i \(0.522684\pi\)
\(432\) 0 0
\(433\) −38.2222 −1.83684 −0.918420 0.395607i \(-0.870534\pi\)
−0.918420 + 0.395607i \(0.870534\pi\)
\(434\) −1.73032 −0.0830580
\(435\) 0 0
\(436\) 13.6083 0.651717
\(437\) −4.23095 −0.202394
\(438\) 0 0
\(439\) 3.57924 0.170828 0.0854140 0.996346i \(-0.472779\pi\)
0.0854140 + 0.996346i \(0.472779\pi\)
\(440\) 41.4749 1.97724
\(441\) 0 0
\(442\) −9.16653 −0.436007
\(443\) 1.21879 0.0579064 0.0289532 0.999581i \(-0.490783\pi\)
0.0289532 + 0.999581i \(0.490783\pi\)
\(444\) 0 0
\(445\) −32.5523 −1.54313
\(446\) −10.1790 −0.481992
\(447\) 0 0
\(448\) 0.583865 0.0275850
\(449\) −36.5812 −1.72637 −0.863186 0.504885i \(-0.831535\pi\)
−0.863186 + 0.504885i \(0.831535\pi\)
\(450\) 0 0
\(451\) −27.3260 −1.28673
\(452\) −28.0432 −1.31904
\(453\) 0 0
\(454\) 3.73672 0.175373
\(455\) −8.41946 −0.394710
\(456\) 0 0
\(457\) −0.708080 −0.0331226 −0.0165613 0.999863i \(-0.505272\pi\)
−0.0165613 + 0.999863i \(0.505272\pi\)
\(458\) 0.704407 0.0329148
\(459\) 0 0
\(460\) −15.2602 −0.711512
\(461\) 7.44049 0.346538 0.173269 0.984875i \(-0.444567\pi\)
0.173269 + 0.984875i \(0.444567\pi\)
\(462\) 0 0
\(463\) 22.9967 1.06875 0.534374 0.845248i \(-0.320548\pi\)
0.534374 + 0.845248i \(0.320548\pi\)
\(464\) 8.26683 0.383778
\(465\) 0 0
\(466\) −14.3404 −0.664308
\(467\) 23.3688 1.08138 0.540691 0.841222i \(-0.318163\pi\)
0.540691 + 0.841222i \(0.318163\pi\)
\(468\) 0 0
\(469\) 7.52093 0.347284
\(470\) −28.5633 −1.31753
\(471\) 0 0
\(472\) 31.4552 1.44784
\(473\) −44.2189 −2.03319
\(474\) 0 0
\(475\) 28.8154 1.32214
\(476\) −13.0689 −0.599010
\(477\) 0 0
\(478\) 9.00651 0.411948
\(479\) −10.9284 −0.499333 −0.249667 0.968332i \(-0.580321\pi\)
−0.249667 + 0.968332i \(0.580321\pi\)
\(480\) 0 0
\(481\) 4.85790 0.221501
\(482\) 6.07894 0.276888
\(483\) 0 0
\(484\) −12.3337 −0.560623
\(485\) −60.3970 −2.74248
\(486\) 0 0
\(487\) −0.259244 −0.0117475 −0.00587374 0.999983i \(-0.501870\pi\)
−0.00587374 + 0.999983i \(0.501870\pi\)
\(488\) −18.4700 −0.836096
\(489\) 0 0
\(490\) 2.64822 0.119634
\(491\) −13.2585 −0.598346 −0.299173 0.954199i \(-0.596711\pi\)
−0.299173 + 0.954199i \(0.596711\pi\)
\(492\) 0 0
\(493\) −33.6107 −1.51375
\(494\) 2.29962 0.103465
\(495\) 0 0
\(496\) −5.64586 −0.253507
\(497\) 10.3166 0.462763
\(498\) 0 0
\(499\) −32.5419 −1.45677 −0.728387 0.685166i \(-0.759730\pi\)
−0.728387 + 0.685166i \(0.759730\pi\)
\(500\) 67.8463 3.03418
\(501\) 0 0
\(502\) 1.34524 0.0600408
\(503\) −39.1034 −1.74354 −0.871768 0.489920i \(-0.837026\pi\)
−0.871768 + 0.489920i \(0.837026\pi\)
\(504\) 0 0
\(505\) 5.49503 0.244526
\(506\) −5.47207 −0.243263
\(507\) 0 0
\(508\) −1.63852 −0.0726974
\(509\) −16.7722 −0.743414 −0.371707 0.928350i \(-0.621227\pi\)
−0.371707 + 0.928350i \(0.621227\pi\)
\(510\) 0 0
\(511\) 3.45838 0.152990
\(512\) −19.4802 −0.860910
\(513\) 0 0
\(514\) 5.92786 0.261467
\(515\) −22.5621 −0.994203
\(516\) 0 0
\(517\) 46.4259 2.04181
\(518\) −1.52799 −0.0671358
\(519\) 0 0
\(520\) 18.4184 0.807701
\(521\) 7.87075 0.344824 0.172412 0.985025i \(-0.444844\pi\)
0.172412 + 0.985025i \(0.444844\pi\)
\(522\) 0 0
\(523\) 15.7546 0.688899 0.344450 0.938805i \(-0.388066\pi\)
0.344450 + 0.938805i \(0.388066\pi\)
\(524\) 6.57321 0.287152
\(525\) 0 0
\(526\) −5.36614 −0.233975
\(527\) 22.9546 0.999916
\(528\) 0 0
\(529\) −18.5290 −0.805610
\(530\) 11.0792 0.481251
\(531\) 0 0
\(532\) 3.27860 0.142145
\(533\) −12.1351 −0.525629
\(534\) 0 0
\(535\) −70.7547 −3.05899
\(536\) −16.4528 −0.710653
\(537\) 0 0
\(538\) 0.180966 0.00780202
\(539\) −4.30434 −0.185401
\(540\) 0 0
\(541\) 24.5723 1.05645 0.528223 0.849106i \(-0.322858\pi\)
0.528223 + 0.849106i \(0.322858\pi\)
\(542\) 2.52482 0.108451
\(543\) 0 0
\(544\) 44.3044 1.89953
\(545\) 36.5815 1.56698
\(546\) 0 0
\(547\) 18.1043 0.774083 0.387041 0.922062i \(-0.373497\pi\)
0.387041 + 0.922062i \(0.373497\pi\)
\(548\) −11.2808 −0.481890
\(549\) 0 0
\(550\) 37.2682 1.58912
\(551\) 8.43196 0.359213
\(552\) 0 0
\(553\) −14.3417 −0.609870
\(554\) −9.02190 −0.383304
\(555\) 0 0
\(556\) 11.3632 0.481906
\(557\) −0.484330 −0.0205217 −0.0102609 0.999947i \(-0.503266\pi\)
−0.0102609 + 0.999947i \(0.503266\pi\)
\(558\) 0 0
\(559\) −19.6370 −0.830557
\(560\) 8.64088 0.365144
\(561\) 0 0
\(562\) −3.48149 −0.146858
\(563\) −25.0910 −1.05746 −0.528731 0.848790i \(-0.677332\pi\)
−0.528731 + 0.848790i \(0.677332\pi\)
\(564\) 0 0
\(565\) −75.3852 −3.17148
\(566\) −13.9622 −0.586874
\(567\) 0 0
\(568\) −22.5687 −0.946960
\(569\) 31.4579 1.31878 0.659392 0.751799i \(-0.270814\pi\)
0.659392 + 0.751799i \(0.270814\pi\)
\(570\) 0 0
\(571\) 20.3592 0.852007 0.426004 0.904721i \(-0.359921\pi\)
0.426004 + 0.904721i \(0.359921\pi\)
\(572\) −13.4813 −0.563681
\(573\) 0 0
\(574\) 3.81692 0.159315
\(575\) −30.4500 −1.26985
\(576\) 0 0
\(577\) −1.95608 −0.0814326 −0.0407163 0.999171i \(-0.512964\pi\)
−0.0407163 + 0.999171i \(0.512964\pi\)
\(578\) −28.0278 −1.16580
\(579\) 0 0
\(580\) 30.4125 1.26281
\(581\) −5.58186 −0.231574
\(582\) 0 0
\(583\) −18.0079 −0.745809
\(584\) −7.56557 −0.313065
\(585\) 0 0
\(586\) −12.1989 −0.503930
\(587\) 17.7291 0.731760 0.365880 0.930662i \(-0.380768\pi\)
0.365880 + 0.930662i \(0.380768\pi\)
\(588\) 0 0
\(589\) −5.75863 −0.237280
\(590\) 38.0783 1.56766
\(591\) 0 0
\(592\) −4.98566 −0.204909
\(593\) 42.1260 1.72991 0.864954 0.501852i \(-0.167348\pi\)
0.864954 + 0.501852i \(0.167348\pi\)
\(594\) 0 0
\(595\) −35.1315 −1.44025
\(596\) −23.4366 −0.960000
\(597\) 0 0
\(598\) −2.43007 −0.0993730
\(599\) 8.55477 0.349538 0.174769 0.984609i \(-0.444082\pi\)
0.174769 + 0.984609i \(0.444082\pi\)
\(600\) 0 0
\(601\) 5.49228 0.224035 0.112017 0.993706i \(-0.464269\pi\)
0.112017 + 0.993706i \(0.464269\pi\)
\(602\) 6.17654 0.251737
\(603\) 0 0
\(604\) −6.35280 −0.258492
\(605\) −33.1553 −1.34795
\(606\) 0 0
\(607\) −13.0024 −0.527749 −0.263875 0.964557i \(-0.585001\pi\)
−0.263875 + 0.964557i \(0.585001\pi\)
\(608\) −11.1147 −0.450760
\(609\) 0 0
\(610\) −22.3590 −0.905288
\(611\) 20.6171 0.834079
\(612\) 0 0
\(613\) −9.24230 −0.373293 −0.186646 0.982427i \(-0.559762\pi\)
−0.186646 + 0.982427i \(0.559762\pi\)
\(614\) −10.1645 −0.410208
\(615\) 0 0
\(616\) 9.41619 0.379389
\(617\) −33.9563 −1.36703 −0.683515 0.729936i \(-0.739550\pi\)
−0.683515 + 0.729936i \(0.739550\pi\)
\(618\) 0 0
\(619\) −22.8026 −0.916515 −0.458258 0.888819i \(-0.651526\pi\)
−0.458258 + 0.888819i \(0.651526\pi\)
\(620\) −20.7703 −0.834156
\(621\) 0 0
\(622\) 6.26721 0.251292
\(623\) −7.39046 −0.296092
\(624\) 0 0
\(625\) 110.379 4.41517
\(626\) 12.9348 0.516978
\(627\) 0 0
\(628\) 34.1039 1.36090
\(629\) 20.2704 0.808232
\(630\) 0 0
\(631\) 2.92274 0.116352 0.0581762 0.998306i \(-0.481471\pi\)
0.0581762 + 0.998306i \(0.481471\pi\)
\(632\) 31.3739 1.24799
\(633\) 0 0
\(634\) −2.88442 −0.114555
\(635\) −4.40463 −0.174793
\(636\) 0 0
\(637\) −1.91150 −0.0757363
\(638\) 10.9054 0.431750
\(639\) 0 0
\(640\) −50.4790 −1.99536
\(641\) 47.5449 1.87791 0.938955 0.344040i \(-0.111795\pi\)
0.938955 + 0.344040i \(0.111795\pi\)
\(642\) 0 0
\(643\) −4.78341 −0.188639 −0.0943197 0.995542i \(-0.530068\pi\)
−0.0943197 + 0.995542i \(0.530068\pi\)
\(644\) −3.46459 −0.136524
\(645\) 0 0
\(646\) 9.59551 0.377530
\(647\) −27.4077 −1.07751 −0.538753 0.842464i \(-0.681104\pi\)
−0.538753 + 0.842464i \(0.681104\pi\)
\(648\) 0 0
\(649\) −61.8914 −2.42945
\(650\) 16.5503 0.649155
\(651\) 0 0
\(652\) 39.8043 1.55886
\(653\) 1.65336 0.0647009 0.0323504 0.999477i \(-0.489701\pi\)
0.0323504 + 0.999477i \(0.489701\pi\)
\(654\) 0 0
\(655\) 17.6700 0.690424
\(656\) 12.4542 0.486256
\(657\) 0 0
\(658\) −6.48482 −0.252805
\(659\) −21.2478 −0.827697 −0.413848 0.910346i \(-0.635816\pi\)
−0.413848 + 0.910346i \(0.635816\pi\)
\(660\) 0 0
\(661\) 33.1581 1.28970 0.644850 0.764309i \(-0.276920\pi\)
0.644850 + 0.764309i \(0.276920\pi\)
\(662\) 12.5397 0.487369
\(663\) 0 0
\(664\) 12.2109 0.473874
\(665\) 8.81347 0.341772
\(666\) 0 0
\(667\) −8.91029 −0.345008
\(668\) 6.58943 0.254953
\(669\) 0 0
\(670\) −19.9171 −0.769464
\(671\) 36.3416 1.40295
\(672\) 0 0
\(673\) 11.3989 0.439394 0.219697 0.975568i \(-0.429493\pi\)
0.219697 + 0.975568i \(0.429493\pi\)
\(674\) −0.0928278 −0.00357559
\(675\) 0 0
\(676\) 15.3139 0.588994
\(677\) 18.8081 0.722852 0.361426 0.932401i \(-0.382290\pi\)
0.361426 + 0.932401i \(0.382290\pi\)
\(678\) 0 0
\(679\) −13.7121 −0.526224
\(680\) 76.8538 2.94721
\(681\) 0 0
\(682\) −7.44789 −0.285195
\(683\) 22.0996 0.845619 0.422810 0.906219i \(-0.361044\pi\)
0.422810 + 0.906219i \(0.361044\pi\)
\(684\) 0 0
\(685\) −30.3248 −1.15865
\(686\) 0.601235 0.0229553
\(687\) 0 0
\(688\) 20.1534 0.768343
\(689\) −7.99704 −0.304663
\(690\) 0 0
\(691\) −10.0595 −0.382682 −0.191341 0.981524i \(-0.561284\pi\)
−0.191341 + 0.981524i \(0.561284\pi\)
\(692\) −9.69222 −0.368443
\(693\) 0 0
\(694\) −10.6549 −0.404454
\(695\) 30.5463 1.15869
\(696\) 0 0
\(697\) −50.6356 −1.91796
\(698\) −7.39292 −0.279826
\(699\) 0 0
\(700\) 23.5960 0.891844
\(701\) −4.48043 −0.169224 −0.0846118 0.996414i \(-0.526965\pi\)
−0.0846118 + 0.996414i \(0.526965\pi\)
\(702\) 0 0
\(703\) −5.08525 −0.191794
\(704\) 2.51316 0.0947181
\(705\) 0 0
\(706\) 9.70633 0.365303
\(707\) 1.24756 0.0469192
\(708\) 0 0
\(709\) 16.8448 0.632618 0.316309 0.948656i \(-0.397556\pi\)
0.316309 + 0.948656i \(0.397556\pi\)
\(710\) −27.3207 −1.02533
\(711\) 0 0
\(712\) 16.1674 0.605898
\(713\) 6.08531 0.227897
\(714\) 0 0
\(715\) −36.2402 −1.35531
\(716\) −2.87259 −0.107354
\(717\) 0 0
\(718\) 6.60609 0.246537
\(719\) −12.2620 −0.457295 −0.228647 0.973509i \(-0.573430\pi\)
−0.228647 + 0.973509i \(0.573430\pi\)
\(720\) 0 0
\(721\) −5.12235 −0.190766
\(722\) 9.01623 0.335549
\(723\) 0 0
\(724\) −2.37455 −0.0882494
\(725\) 60.6845 2.25377
\(726\) 0 0
\(727\) 50.2692 1.86438 0.932190 0.361969i \(-0.117895\pi\)
0.932190 + 0.361969i \(0.117895\pi\)
\(728\) 4.18160 0.154980
\(729\) 0 0
\(730\) −9.15856 −0.338973
\(731\) −81.9385 −3.03060
\(732\) 0 0
\(733\) 22.1666 0.818742 0.409371 0.912368i \(-0.365748\pi\)
0.409371 + 0.912368i \(0.365748\pi\)
\(734\) −9.45372 −0.348943
\(735\) 0 0
\(736\) 11.7452 0.432934
\(737\) 32.3727 1.19246
\(738\) 0 0
\(739\) −13.9133 −0.511808 −0.255904 0.966702i \(-0.582373\pi\)
−0.255904 + 0.966702i \(0.582373\pi\)
\(740\) −18.3415 −0.674248
\(741\) 0 0
\(742\) 2.51536 0.0923416
\(743\) 42.1511 1.54637 0.773187 0.634178i \(-0.218661\pi\)
0.773187 + 0.634178i \(0.218661\pi\)
\(744\) 0 0
\(745\) −63.0019 −2.30821
\(746\) −2.25967 −0.0827324
\(747\) 0 0
\(748\) −56.2528 −2.05681
\(749\) −16.0637 −0.586954
\(750\) 0 0
\(751\) −7.52694 −0.274662 −0.137331 0.990525i \(-0.543852\pi\)
−0.137331 + 0.990525i \(0.543852\pi\)
\(752\) −21.1593 −0.771601
\(753\) 0 0
\(754\) 4.84294 0.176370
\(755\) −17.0775 −0.621513
\(756\) 0 0
\(757\) −25.7235 −0.934936 −0.467468 0.884010i \(-0.654834\pi\)
−0.467468 + 0.884010i \(0.654834\pi\)
\(758\) −5.68850 −0.206616
\(759\) 0 0
\(760\) −19.2804 −0.699373
\(761\) 10.1165 0.366723 0.183361 0.983046i \(-0.441302\pi\)
0.183361 + 0.983046i \(0.441302\pi\)
\(762\) 0 0
\(763\) 8.30523 0.300669
\(764\) 4.83775 0.175023
\(765\) 0 0
\(766\) −6.22619 −0.224961
\(767\) −27.4851 −0.992430
\(768\) 0 0
\(769\) −6.87069 −0.247763 −0.123882 0.992297i \(-0.539534\pi\)
−0.123882 + 0.992297i \(0.539534\pi\)
\(770\) 11.3988 0.410786
\(771\) 0 0
\(772\) −32.0763 −1.15445
\(773\) −17.8361 −0.641520 −0.320760 0.947160i \(-0.603938\pi\)
−0.320760 + 0.947160i \(0.603938\pi\)
\(774\) 0 0
\(775\) −41.4447 −1.48874
\(776\) 29.9967 1.07682
\(777\) 0 0
\(778\) 10.2800 0.368554
\(779\) 12.7030 0.455132
\(780\) 0 0
\(781\) 44.4062 1.58898
\(782\) −10.1398 −0.362600
\(783\) 0 0
\(784\) 1.96177 0.0700632
\(785\) 91.6777 3.27212
\(786\) 0 0
\(787\) −15.3891 −0.548562 −0.274281 0.961650i \(-0.588440\pi\)
−0.274281 + 0.961650i \(0.588440\pi\)
\(788\) 15.8252 0.563751
\(789\) 0 0
\(790\) 37.9799 1.35126
\(791\) −17.1150 −0.608538
\(792\) 0 0
\(793\) 16.1388 0.573106
\(794\) 13.2934 0.471766
\(795\) 0 0
\(796\) 28.8265 1.02173
\(797\) 39.1163 1.38557 0.692786 0.721143i \(-0.256383\pi\)
0.692786 + 0.721143i \(0.256383\pi\)
\(798\) 0 0
\(799\) 86.0281 3.04346
\(800\) −79.9920 −2.82815
\(801\) 0 0
\(802\) 22.9399 0.810035
\(803\) 14.8861 0.525318
\(804\) 0 0
\(805\) −9.31345 −0.328256
\(806\) −3.30751 −0.116502
\(807\) 0 0
\(808\) −2.72916 −0.0960115
\(809\) 6.83182 0.240194 0.120097 0.992762i \(-0.461679\pi\)
0.120097 + 0.992762i \(0.461679\pi\)
\(810\) 0 0
\(811\) 36.2145 1.27166 0.635832 0.771828i \(-0.280657\pi\)
0.635832 + 0.771828i \(0.280657\pi\)
\(812\) 6.90466 0.242306
\(813\) 0 0
\(814\) −6.57697 −0.230523
\(815\) 107.001 3.74809
\(816\) 0 0
\(817\) 20.5560 0.719163
\(818\) −5.71421 −0.199793
\(819\) 0 0
\(820\) 45.8173 1.60001
\(821\) −14.8622 −0.518693 −0.259346 0.965784i \(-0.583507\pi\)
−0.259346 + 0.965784i \(0.583507\pi\)
\(822\) 0 0
\(823\) 15.4230 0.537611 0.268805 0.963195i \(-0.413371\pi\)
0.268805 + 0.963195i \(0.413371\pi\)
\(824\) 11.2057 0.390368
\(825\) 0 0
\(826\) 8.64505 0.300800
\(827\) 35.9214 1.24911 0.624554 0.780981i \(-0.285281\pi\)
0.624554 + 0.780981i \(0.285281\pi\)
\(828\) 0 0
\(829\) 31.5985 1.09746 0.548730 0.836000i \(-0.315111\pi\)
0.548730 + 0.836000i \(0.315111\pi\)
\(830\) 14.7820 0.513090
\(831\) 0 0
\(832\) 1.11606 0.0386923
\(833\) −7.97603 −0.276353
\(834\) 0 0
\(835\) 17.7136 0.613005
\(836\) 14.1122 0.488081
\(837\) 0 0
\(838\) 3.40302 0.117556
\(839\) 39.5472 1.36532 0.682660 0.730736i \(-0.260823\pi\)
0.682660 + 0.730736i \(0.260823\pi\)
\(840\) 0 0
\(841\) −11.2425 −0.387671
\(842\) 6.01349 0.207238
\(843\) 0 0
\(844\) 5.41023 0.186228
\(845\) 41.1665 1.41617
\(846\) 0 0
\(847\) −7.52736 −0.258643
\(848\) 8.20735 0.281842
\(849\) 0 0
\(850\) 69.0586 2.36869
\(851\) 5.37373 0.184209
\(852\) 0 0
\(853\) −5.00383 −0.171328 −0.0856639 0.996324i \(-0.527301\pi\)
−0.0856639 + 0.996324i \(0.527301\pi\)
\(854\) −5.07624 −0.173705
\(855\) 0 0
\(856\) 35.1410 1.20109
\(857\) −41.4504 −1.41592 −0.707959 0.706254i \(-0.750384\pi\)
−0.707959 + 0.706254i \(0.750384\pi\)
\(858\) 0 0
\(859\) 27.5845 0.941171 0.470585 0.882354i \(-0.344043\pi\)
0.470585 + 0.882354i \(0.344043\pi\)
\(860\) 74.1416 2.52821
\(861\) 0 0
\(862\) 1.77752 0.0605426
\(863\) −39.6613 −1.35009 −0.675043 0.737778i \(-0.735875\pi\)
−0.675043 + 0.737778i \(0.735875\pi\)
\(864\) 0 0
\(865\) −26.0545 −0.885879
\(866\) 22.9805 0.780909
\(867\) 0 0
\(868\) −4.71556 −0.160057
\(869\) −61.7314 −2.09410
\(870\) 0 0
\(871\) 14.3763 0.487121
\(872\) −18.1685 −0.615264
\(873\) 0 0
\(874\) 2.54379 0.0860451
\(875\) 41.4071 1.39982
\(876\) 0 0
\(877\) −4.96526 −0.167665 −0.0838324 0.996480i \(-0.526716\pi\)
−0.0838324 + 0.996480i \(0.526716\pi\)
\(878\) −2.15197 −0.0726253
\(879\) 0 0
\(880\) 37.1933 1.25379
\(881\) −32.2888 −1.08784 −0.543919 0.839138i \(-0.683060\pi\)
−0.543919 + 0.839138i \(0.683060\pi\)
\(882\) 0 0
\(883\) −19.3735 −0.651969 −0.325984 0.945375i \(-0.605696\pi\)
−0.325984 + 0.945375i \(0.605696\pi\)
\(884\) −24.9811 −0.840205
\(885\) 0 0
\(886\) −0.732779 −0.0246182
\(887\) 2.87426 0.0965082 0.0482541 0.998835i \(-0.484634\pi\)
0.0482541 + 0.998835i \(0.484634\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 19.5716 0.656040
\(891\) 0 0
\(892\) −27.7405 −0.928819
\(893\) −21.5820 −0.722213
\(894\) 0 0
\(895\) −7.72204 −0.258119
\(896\) −11.4604 −0.382866
\(897\) 0 0
\(898\) 21.9939 0.733946
\(899\) −12.1276 −0.404477
\(900\) 0 0
\(901\) −33.3689 −1.11168
\(902\) 16.4293 0.547037
\(903\) 0 0
\(904\) 37.4408 1.24526
\(905\) −6.38323 −0.212186
\(906\) 0 0
\(907\) −31.5100 −1.04627 −0.523137 0.852249i \(-0.675238\pi\)
−0.523137 + 0.852249i \(0.675238\pi\)
\(908\) 10.1835 0.337952
\(909\) 0 0
\(910\) 5.06207 0.167806
\(911\) −8.79160 −0.291279 −0.145639 0.989338i \(-0.546524\pi\)
−0.145639 + 0.989338i \(0.546524\pi\)
\(912\) 0 0
\(913\) −24.0262 −0.795152
\(914\) 0.425722 0.0140816
\(915\) 0 0
\(916\) 1.91969 0.0634282
\(917\) 4.01168 0.132477
\(918\) 0 0
\(919\) 11.9465 0.394079 0.197040 0.980396i \(-0.436867\pi\)
0.197040 + 0.980396i \(0.436867\pi\)
\(920\) 20.3741 0.671715
\(921\) 0 0
\(922\) −4.47348 −0.147326
\(923\) 19.7202 0.649098
\(924\) 0 0
\(925\) −36.5984 −1.20335
\(926\) −13.8264 −0.454364
\(927\) 0 0
\(928\) −23.4073 −0.768382
\(929\) −14.7987 −0.485530 −0.242765 0.970085i \(-0.578054\pi\)
−0.242765 + 0.970085i \(0.578054\pi\)
\(930\) 0 0
\(931\) 2.00095 0.0655786
\(932\) −39.0813 −1.28015
\(933\) 0 0
\(934\) −14.0502 −0.459736
\(935\) −151.218 −4.94536
\(936\) 0 0
\(937\) 30.4352 0.994274 0.497137 0.867672i \(-0.334385\pi\)
0.497137 + 0.867672i \(0.334385\pi\)
\(938\) −4.52185 −0.147644
\(939\) 0 0
\(940\) −77.8421 −2.53893
\(941\) −57.3813 −1.87058 −0.935288 0.353887i \(-0.884860\pi\)
−0.935288 + 0.353887i \(0.884860\pi\)
\(942\) 0 0
\(943\) −13.4236 −0.437133
\(944\) 28.2079 0.918090
\(945\) 0 0
\(946\) 26.5860 0.864384
\(947\) 28.7119 0.933012 0.466506 0.884518i \(-0.345513\pi\)
0.466506 + 0.884518i \(0.345513\pi\)
\(948\) 0 0
\(949\) 6.61069 0.214592
\(950\) −17.3248 −0.562091
\(951\) 0 0
\(952\) 17.4484 0.565505
\(953\) 33.7610 1.09363 0.546814 0.837254i \(-0.315841\pi\)
0.546814 + 0.837254i \(0.315841\pi\)
\(954\) 0 0
\(955\) 13.0048 0.420824
\(956\) 24.5450 0.793842
\(957\) 0 0
\(958\) 6.57056 0.212285
\(959\) −6.88474 −0.222320
\(960\) 0 0
\(961\) −22.7174 −0.732821
\(962\) −2.92074 −0.0941685
\(963\) 0 0
\(964\) 16.5666 0.533576
\(965\) −86.2269 −2.77574
\(966\) 0 0
\(967\) 28.9226 0.930089 0.465045 0.885287i \(-0.346038\pi\)
0.465045 + 0.885287i \(0.346038\pi\)
\(968\) 16.4669 0.529266
\(969\) 0 0
\(970\) 36.3128 1.16593
\(971\) −0.773763 −0.0248312 −0.0124156 0.999923i \(-0.503952\pi\)
−0.0124156 + 0.999923i \(0.503952\pi\)
\(972\) 0 0
\(973\) 6.93504 0.222327
\(974\) 0.155867 0.00499430
\(975\) 0 0
\(976\) −16.5633 −0.530177
\(977\) 18.8567 0.603278 0.301639 0.953422i \(-0.402466\pi\)
0.301639 + 0.953422i \(0.402466\pi\)
\(978\) 0 0
\(979\) −31.8111 −1.01669
\(980\) 7.21707 0.230541
\(981\) 0 0
\(982\) 7.97145 0.254379
\(983\) −4.28493 −0.136668 −0.0683340 0.997663i \(-0.521768\pi\)
−0.0683340 + 0.997663i \(0.521768\pi\)
\(984\) 0 0
\(985\) 42.5412 1.35547
\(986\) 20.2079 0.643552
\(987\) 0 0
\(988\) 6.26703 0.199381
\(989\) −21.7221 −0.690723
\(990\) 0 0
\(991\) 1.52397 0.0484105 0.0242052 0.999707i \(-0.492294\pi\)
0.0242052 + 0.999707i \(0.492294\pi\)
\(992\) 15.9861 0.507559
\(993\) 0 0
\(994\) −6.20271 −0.196738
\(995\) 77.4910 2.45663
\(996\) 0 0
\(997\) −35.2873 −1.11756 −0.558780 0.829316i \(-0.688730\pi\)
−0.558780 + 0.829316i \(0.688730\pi\)
\(998\) 19.5653 0.619329
\(999\) 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 8001.2.a.q.1.7 15
3.2 odd 2 889.2.a.b.1.9 15
21.20 even 2 6223.2.a.j.1.9 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.b.1.9 15 3.2 odd 2
6223.2.a.j.1.9 15 21.20 even 2
8001.2.a.q.1.7 15 1.1 even 1 trivial