Properties

Label 531.4.a.f.1.6
Level $531$
Weight $4$
Character 531.1
Self dual yes
Analytic conductor $31.330$
Analytic rank $0$
Dimension $8$
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] = [531,4,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(31.3300142130\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.67303\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.67303 q^{2} -5.20096 q^{4} +6.76323 q^{5} -19.0526 q^{7} -22.0856 q^{8} +O(q^{10})\) \(q+1.67303 q^{2} -5.20096 q^{4} +6.76323 q^{5} -19.0526 q^{7} -22.0856 q^{8} +11.3151 q^{10} +65.4248 q^{11} -46.8971 q^{13} -31.8757 q^{14} +4.65770 q^{16} -48.0195 q^{17} +147.256 q^{19} -35.1753 q^{20} +109.458 q^{22} -33.1371 q^{23} -79.2587 q^{25} -78.4604 q^{26} +99.0919 q^{28} +73.6497 q^{29} +142.529 q^{31} +184.478 q^{32} -80.3382 q^{34} -128.857 q^{35} +397.714 q^{37} +246.363 q^{38} -149.370 q^{40} +100.625 q^{41} +388.024 q^{43} -340.272 q^{44} -55.4395 q^{46} +138.226 q^{47} +20.0023 q^{49} -132.602 q^{50} +243.910 q^{52} +439.471 q^{53} +442.483 q^{55} +420.789 q^{56} +123.218 q^{58} +59.0000 q^{59} -602.884 q^{61} +238.455 q^{62} +271.375 q^{64} -317.176 q^{65} -154.728 q^{67} +249.748 q^{68} -215.582 q^{70} -552.436 q^{71} -107.785 q^{73} +665.389 q^{74} -765.871 q^{76} -1246.51 q^{77} +989.162 q^{79} +31.5011 q^{80} +168.349 q^{82} +730.585 q^{83} -324.767 q^{85} +649.177 q^{86} -1444.95 q^{88} +1375.89 q^{89} +893.513 q^{91} +172.345 q^{92} +231.257 q^{94} +995.923 q^{95} -268.232 q^{97} +33.4645 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.67303 0.591506 0.295753 0.955264i \(-0.404429\pi\)
0.295753 + 0.955264i \(0.404429\pi\)
\(3\) 0 0
\(4\) −5.20096 −0.650120
\(5\) 6.76323 0.604922 0.302461 0.953162i \(-0.402192\pi\)
0.302461 + 0.953162i \(0.402192\pi\)
\(6\) 0 0
\(7\) −19.0526 −1.02874 −0.514372 0.857567i \(-0.671975\pi\)
−0.514372 + 0.857567i \(0.671975\pi\)
\(8\) −22.0856 −0.976057
\(9\) 0 0
\(10\) 11.3151 0.357815
\(11\) 65.4248 1.79330 0.896651 0.442737i \(-0.145993\pi\)
0.896651 + 0.442737i \(0.145993\pi\)
\(12\) 0 0
\(13\) −46.8971 −1.00053 −0.500266 0.865872i \(-0.666764\pi\)
−0.500266 + 0.865872i \(0.666764\pi\)
\(14\) −31.8757 −0.608509
\(15\) 0 0
\(16\) 4.65770 0.0727766
\(17\) −48.0195 −0.685085 −0.342542 0.939502i \(-0.611288\pi\)
−0.342542 + 0.939502i \(0.611288\pi\)
\(18\) 0 0
\(19\) 147.256 1.77804 0.889020 0.457869i \(-0.151387\pi\)
0.889020 + 0.457869i \(0.151387\pi\)
\(20\) −35.1753 −0.393272
\(21\) 0 0
\(22\) 109.458 1.06075
\(23\) −33.1371 −0.300416 −0.150208 0.988654i \(-0.547994\pi\)
−0.150208 + 0.988654i \(0.547994\pi\)
\(24\) 0 0
\(25\) −79.2587 −0.634070
\(26\) −78.4604 −0.591821
\(27\) 0 0
\(28\) 99.0919 0.668808
\(29\) 73.6497 0.471600 0.235800 0.971802i \(-0.424229\pi\)
0.235800 + 0.971802i \(0.424229\pi\)
\(30\) 0 0
\(31\) 142.529 0.825771 0.412886 0.910783i \(-0.364521\pi\)
0.412886 + 0.910783i \(0.364521\pi\)
\(32\) 184.478 1.01910
\(33\) 0 0
\(34\) −80.3382 −0.405232
\(35\) −128.857 −0.622310
\(36\) 0 0
\(37\) 397.714 1.76713 0.883565 0.468308i \(-0.155136\pi\)
0.883565 + 0.468308i \(0.155136\pi\)
\(38\) 246.363 1.05172
\(39\) 0 0
\(40\) −149.370 −0.590438
\(41\) 100.625 0.383293 0.191646 0.981464i \(-0.438617\pi\)
0.191646 + 0.981464i \(0.438617\pi\)
\(42\) 0 0
\(43\) 388.024 1.37612 0.688060 0.725654i \(-0.258463\pi\)
0.688060 + 0.725654i \(0.258463\pi\)
\(44\) −340.272 −1.16586
\(45\) 0 0
\(46\) −55.4395 −0.177698
\(47\) 138.226 0.428987 0.214493 0.976725i \(-0.431190\pi\)
0.214493 + 0.976725i \(0.431190\pi\)
\(48\) 0 0
\(49\) 20.0023 0.0583158
\(50\) −132.602 −0.375056
\(51\) 0 0
\(52\) 243.910 0.650466
\(53\) 439.471 1.13898 0.569491 0.821998i \(-0.307141\pi\)
0.569491 + 0.821998i \(0.307141\pi\)
\(54\) 0 0
\(55\) 442.483 1.08481
\(56\) 420.789 1.00411
\(57\) 0 0
\(58\) 123.218 0.278954
\(59\) 59.0000 0.130189
\(60\) 0 0
\(61\) −602.884 −1.26543 −0.632716 0.774384i \(-0.718060\pi\)
−0.632716 + 0.774384i \(0.718060\pi\)
\(62\) 238.455 0.488449
\(63\) 0 0
\(64\) 271.375 0.530030
\(65\) −317.176 −0.605244
\(66\) 0 0
\(67\) −154.728 −0.282134 −0.141067 0.990000i \(-0.545053\pi\)
−0.141067 + 0.990000i \(0.545053\pi\)
\(68\) 249.748 0.445388
\(69\) 0 0
\(70\) −215.582 −0.368100
\(71\) −552.436 −0.923410 −0.461705 0.887033i \(-0.652762\pi\)
−0.461705 + 0.887033i \(0.652762\pi\)
\(72\) 0 0
\(73\) −107.785 −0.172812 −0.0864062 0.996260i \(-0.527538\pi\)
−0.0864062 + 0.996260i \(0.527538\pi\)
\(74\) 665.389 1.04527
\(75\) 0 0
\(76\) −765.871 −1.15594
\(77\) −1246.51 −1.84485
\(78\) 0 0
\(79\) 989.162 1.40873 0.704363 0.709840i \(-0.251233\pi\)
0.704363 + 0.709840i \(0.251233\pi\)
\(80\) 31.5011 0.0440241
\(81\) 0 0
\(82\) 168.349 0.226720
\(83\) 730.585 0.966170 0.483085 0.875573i \(-0.339516\pi\)
0.483085 + 0.875573i \(0.339516\pi\)
\(84\) 0 0
\(85\) −324.767 −0.414423
\(86\) 649.177 0.813983
\(87\) 0 0
\(88\) −1444.95 −1.75036
\(89\) 1375.89 1.63869 0.819346 0.573300i \(-0.194337\pi\)
0.819346 + 0.573300i \(0.194337\pi\)
\(90\) 0 0
\(91\) 893.513 1.02929
\(92\) 172.345 0.195306
\(93\) 0 0
\(94\) 231.257 0.253748
\(95\) 995.923 1.07557
\(96\) 0 0
\(97\) −268.232 −0.280772 −0.140386 0.990097i \(-0.544834\pi\)
−0.140386 + 0.990097i \(0.544834\pi\)
\(98\) 33.4645 0.0344942
\(99\) 0 0
\(100\) 412.222 0.412222
\(101\) −198.697 −0.195753 −0.0978766 0.995199i \(-0.531205\pi\)
−0.0978766 + 0.995199i \(0.531205\pi\)
\(102\) 0 0
\(103\) 1107.72 1.05968 0.529838 0.848099i \(-0.322253\pi\)
0.529838 + 0.848099i \(0.322253\pi\)
\(104\) 1035.75 0.976576
\(105\) 0 0
\(106\) 735.249 0.673715
\(107\) 562.412 0.508134 0.254067 0.967187i \(-0.418232\pi\)
0.254067 + 0.967187i \(0.418232\pi\)
\(108\) 0 0
\(109\) 1069.38 0.939711 0.469855 0.882743i \(-0.344306\pi\)
0.469855 + 0.882743i \(0.344306\pi\)
\(110\) 740.289 0.641671
\(111\) 0 0
\(112\) −88.7414 −0.0748685
\(113\) −1601.88 −1.33356 −0.666781 0.745254i \(-0.732328\pi\)
−0.666781 + 0.745254i \(0.732328\pi\)
\(114\) 0 0
\(115\) −224.114 −0.181728
\(116\) −383.049 −0.306597
\(117\) 0 0
\(118\) 98.7089 0.0770076
\(119\) 914.898 0.704778
\(120\) 0 0
\(121\) 2949.41 2.21593
\(122\) −1008.64 −0.748511
\(123\) 0 0
\(124\) −741.286 −0.536850
\(125\) −1381.45 −0.988484
\(126\) 0 0
\(127\) −251.661 −0.175837 −0.0879184 0.996128i \(-0.528021\pi\)
−0.0879184 + 0.996128i \(0.528021\pi\)
\(128\) −1021.80 −0.705588
\(129\) 0 0
\(130\) −530.646 −0.358005
\(131\) −2491.36 −1.66161 −0.830806 0.556562i \(-0.812120\pi\)
−0.830806 + 0.556562i \(0.812120\pi\)
\(132\) 0 0
\(133\) −2805.60 −1.82915
\(134\) −258.864 −0.166884
\(135\) 0 0
\(136\) 1060.54 0.668682
\(137\) −2395.27 −1.49374 −0.746869 0.664971i \(-0.768444\pi\)
−0.746869 + 0.664971i \(0.768444\pi\)
\(138\) 0 0
\(139\) −1720.23 −1.04970 −0.524850 0.851195i \(-0.675879\pi\)
−0.524850 + 0.851195i \(0.675879\pi\)
\(140\) 670.182 0.404576
\(141\) 0 0
\(142\) −924.244 −0.546203
\(143\) −3068.24 −1.79426
\(144\) 0 0
\(145\) 498.110 0.285281
\(146\) −180.328 −0.102220
\(147\) 0 0
\(148\) −2068.50 −1.14885
\(149\) 3053.69 1.67898 0.839490 0.543375i \(-0.182854\pi\)
0.839490 + 0.543375i \(0.182854\pi\)
\(150\) 0 0
\(151\) 1862.79 1.00392 0.501960 0.864891i \(-0.332612\pi\)
0.501960 + 0.864891i \(0.332612\pi\)
\(152\) −3252.23 −1.73547
\(153\) 0 0
\(154\) −2085.46 −1.09124
\(155\) 963.954 0.499527
\(156\) 0 0
\(157\) 2654.55 1.34940 0.674701 0.738091i \(-0.264272\pi\)
0.674701 + 0.738091i \(0.264272\pi\)
\(158\) 1654.90 0.833271
\(159\) 0 0
\(160\) 1247.66 0.616478
\(161\) 631.349 0.309051
\(162\) 0 0
\(163\) −1355.54 −0.651375 −0.325688 0.945477i \(-0.605596\pi\)
−0.325688 + 0.945477i \(0.605596\pi\)
\(164\) −523.348 −0.249186
\(165\) 0 0
\(166\) 1222.29 0.571496
\(167\) −1406.91 −0.651915 −0.325958 0.945384i \(-0.605687\pi\)
−0.325958 + 0.945384i \(0.605687\pi\)
\(168\) 0 0
\(169\) 2.33930 0.00106477
\(170\) −543.346 −0.245134
\(171\) 0 0
\(172\) −2018.10 −0.894643
\(173\) −3037.13 −1.33473 −0.667366 0.744730i \(-0.732578\pi\)
−0.667366 + 0.744730i \(0.732578\pi\)
\(174\) 0 0
\(175\) 1510.09 0.652296
\(176\) 304.729 0.130510
\(177\) 0 0
\(178\) 2301.90 0.969296
\(179\) 1561.28 0.651932 0.325966 0.945382i \(-0.394311\pi\)
0.325966 + 0.945382i \(0.394311\pi\)
\(180\) 0 0
\(181\) −1573.85 −0.646316 −0.323158 0.946345i \(-0.604745\pi\)
−0.323158 + 0.946345i \(0.604745\pi\)
\(182\) 1494.88 0.608833
\(183\) 0 0
\(184\) 731.854 0.293223
\(185\) 2689.83 1.06898
\(186\) 0 0
\(187\) −3141.67 −1.22856
\(188\) −718.910 −0.278893
\(189\) 0 0
\(190\) 1666.21 0.636209
\(191\) −820.690 −0.310906 −0.155453 0.987843i \(-0.549684\pi\)
−0.155453 + 0.987843i \(0.549684\pi\)
\(192\) 0 0
\(193\) 4227.89 1.57684 0.788419 0.615138i \(-0.210900\pi\)
0.788419 + 0.615138i \(0.210900\pi\)
\(194\) −448.761 −0.166078
\(195\) 0 0
\(196\) −104.031 −0.0379123
\(197\) 319.247 0.115459 0.0577295 0.998332i \(-0.481614\pi\)
0.0577295 + 0.998332i \(0.481614\pi\)
\(198\) 0 0
\(199\) −265.391 −0.0945381 −0.0472691 0.998882i \(-0.515052\pi\)
−0.0472691 + 0.998882i \(0.515052\pi\)
\(200\) 1750.48 0.618888
\(201\) 0 0
\(202\) −332.426 −0.115789
\(203\) −1403.22 −0.485156
\(204\) 0 0
\(205\) 680.551 0.231862
\(206\) 1853.25 0.626805
\(207\) 0 0
\(208\) −218.433 −0.0728153
\(209\) 9634.17 3.18856
\(210\) 0 0
\(211\) −3536.65 −1.15390 −0.576950 0.816779i \(-0.695757\pi\)
−0.576950 + 0.816779i \(0.695757\pi\)
\(212\) −2285.67 −0.740475
\(213\) 0 0
\(214\) 940.933 0.300565
\(215\) 2624.30 0.832445
\(216\) 0 0
\(217\) −2715.54 −0.849508
\(218\) 1789.12 0.555845
\(219\) 0 0
\(220\) −2301.34 −0.705255
\(221\) 2251.98 0.685450
\(222\) 0 0
\(223\) 4045.80 1.21492 0.607460 0.794350i \(-0.292189\pi\)
0.607460 + 0.794350i \(0.292189\pi\)
\(224\) −3514.78 −1.04840
\(225\) 0 0
\(226\) −2680.00 −0.788811
\(227\) −4609.18 −1.34767 −0.673837 0.738880i \(-0.735355\pi\)
−0.673837 + 0.738880i \(0.735355\pi\)
\(228\) 0 0
\(229\) −1842.43 −0.531663 −0.265832 0.964019i \(-0.585647\pi\)
−0.265832 + 0.964019i \(0.585647\pi\)
\(230\) −374.950 −0.107493
\(231\) 0 0
\(232\) −1626.60 −0.460308
\(233\) 1938.03 0.544913 0.272456 0.962168i \(-0.412164\pi\)
0.272456 + 0.962168i \(0.412164\pi\)
\(234\) 0 0
\(235\) 934.857 0.259503
\(236\) −306.857 −0.0846384
\(237\) 0 0
\(238\) 1530.65 0.416880
\(239\) −4453.09 −1.20522 −0.602608 0.798038i \(-0.705872\pi\)
−0.602608 + 0.798038i \(0.705872\pi\)
\(240\) 0 0
\(241\) −1268.11 −0.338948 −0.169474 0.985535i \(-0.554207\pi\)
−0.169474 + 0.985535i \(0.554207\pi\)
\(242\) 4934.46 1.31074
\(243\) 0 0
\(244\) 3135.57 0.822683
\(245\) 135.280 0.0352765
\(246\) 0 0
\(247\) −6905.86 −1.77899
\(248\) −3147.84 −0.805999
\(249\) 0 0
\(250\) −2311.21 −0.584695
\(251\) −3848.45 −0.967776 −0.483888 0.875130i \(-0.660776\pi\)
−0.483888 + 0.875130i \(0.660776\pi\)
\(252\) 0 0
\(253\) −2167.99 −0.538737
\(254\) −421.037 −0.104009
\(255\) 0 0
\(256\) −3880.51 −0.947390
\(257\) −3723.47 −0.903749 −0.451874 0.892082i \(-0.649244\pi\)
−0.451874 + 0.892082i \(0.649244\pi\)
\(258\) 0 0
\(259\) −7577.50 −1.81793
\(260\) 1649.62 0.393481
\(261\) 0 0
\(262\) −4168.13 −0.982854
\(263\) 1776.07 0.416414 0.208207 0.978085i \(-0.433237\pi\)
0.208207 + 0.978085i \(0.433237\pi\)
\(264\) 0 0
\(265\) 2972.24 0.688994
\(266\) −4693.87 −1.08195
\(267\) 0 0
\(268\) 804.733 0.183421
\(269\) 6356.42 1.44073 0.720367 0.693593i \(-0.243973\pi\)
0.720367 + 0.693593i \(0.243973\pi\)
\(270\) 0 0
\(271\) 3734.20 0.837036 0.418518 0.908209i \(-0.362550\pi\)
0.418518 + 0.908209i \(0.362550\pi\)
\(272\) −223.661 −0.0498581
\(273\) 0 0
\(274\) −4007.37 −0.883556
\(275\) −5185.49 −1.13708
\(276\) 0 0
\(277\) 2987.88 0.648102 0.324051 0.946040i \(-0.394955\pi\)
0.324051 + 0.946040i \(0.394955\pi\)
\(278\) −2878.01 −0.620904
\(279\) 0 0
\(280\) 2845.89 0.607410
\(281\) 8120.56 1.72396 0.861978 0.506945i \(-0.169225\pi\)
0.861978 + 0.506945i \(0.169225\pi\)
\(282\) 0 0
\(283\) 5876.00 1.23425 0.617123 0.786866i \(-0.288298\pi\)
0.617123 + 0.786866i \(0.288298\pi\)
\(284\) 2873.20 0.600328
\(285\) 0 0
\(286\) −5133.26 −1.06131
\(287\) −1917.17 −0.394311
\(288\) 0 0
\(289\) −2607.13 −0.530659
\(290\) 833.354 0.168746
\(291\) 0 0
\(292\) 560.587 0.112349
\(293\) 1680.29 0.335029 0.167515 0.985870i \(-0.446426\pi\)
0.167515 + 0.985870i \(0.446426\pi\)
\(294\) 0 0
\(295\) 399.031 0.0787541
\(296\) −8783.77 −1.72482
\(297\) 0 0
\(298\) 5108.92 0.993127
\(299\) 1554.03 0.300576
\(300\) 0 0
\(301\) −7392.88 −1.41568
\(302\) 3116.51 0.593825
\(303\) 0 0
\(304\) 685.872 0.129400
\(305\) −4077.44 −0.765487
\(306\) 0 0
\(307\) −10436.9 −1.94028 −0.970138 0.242552i \(-0.922015\pi\)
−0.970138 + 0.242552i \(0.922015\pi\)
\(308\) 6483.07 1.19937
\(309\) 0 0
\(310\) 1612.73 0.295473
\(311\) 46.2544 0.00843359 0.00421680 0.999991i \(-0.498658\pi\)
0.00421680 + 0.999991i \(0.498658\pi\)
\(312\) 0 0
\(313\) −1814.13 −0.327607 −0.163803 0.986493i \(-0.552376\pi\)
−0.163803 + 0.986493i \(0.552376\pi\)
\(314\) 4441.15 0.798180
\(315\) 0 0
\(316\) −5144.59 −0.915842
\(317\) 2234.67 0.395935 0.197968 0.980209i \(-0.436566\pi\)
0.197968 + 0.980209i \(0.436566\pi\)
\(318\) 0 0
\(319\) 4818.52 0.845722
\(320\) 1835.37 0.320627
\(321\) 0 0
\(322\) 1056.27 0.182806
\(323\) −7071.14 −1.21811
\(324\) 0 0
\(325\) 3717.01 0.634407
\(326\) −2267.87 −0.385293
\(327\) 0 0
\(328\) −2222.37 −0.374116
\(329\) −2633.57 −0.441318
\(330\) 0 0
\(331\) 8239.89 1.36829 0.684147 0.729344i \(-0.260175\pi\)
0.684147 + 0.729344i \(0.260175\pi\)
\(332\) −3799.74 −0.628127
\(333\) 0 0
\(334\) −2353.80 −0.385612
\(335\) −1046.46 −0.170669
\(336\) 0 0
\(337\) −12009.1 −1.94118 −0.970590 0.240738i \(-0.922610\pi\)
−0.970590 + 0.240738i \(0.922610\pi\)
\(338\) 3.91372 0.000629818 0
\(339\) 0 0
\(340\) 1689.10 0.269425
\(341\) 9324.91 1.48086
\(342\) 0 0
\(343\) 6153.95 0.968753
\(344\) −8569.76 −1.34317
\(345\) 0 0
\(346\) −5081.21 −0.789502
\(347\) −2372.02 −0.366965 −0.183483 0.983023i \(-0.558737\pi\)
−0.183483 + 0.983023i \(0.558737\pi\)
\(348\) 0 0
\(349\) 9595.08 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(350\) 2526.42 0.385837
\(351\) 0 0
\(352\) 12069.4 1.82756
\(353\) 1102.51 0.166234 0.0831168 0.996540i \(-0.473513\pi\)
0.0831168 + 0.996540i \(0.473513\pi\)
\(354\) 0 0
\(355\) −3736.25 −0.558591
\(356\) −7155.93 −1.06535
\(357\) 0 0
\(358\) 2612.08 0.385622
\(359\) 1670.88 0.245643 0.122821 0.992429i \(-0.460806\pi\)
0.122821 + 0.992429i \(0.460806\pi\)
\(360\) 0 0
\(361\) 14825.2 2.16142
\(362\) −2633.10 −0.382300
\(363\) 0 0
\(364\) −4647.13 −0.669164
\(365\) −728.976 −0.104538
\(366\) 0 0
\(367\) 6862.08 0.976015 0.488008 0.872839i \(-0.337724\pi\)
0.488008 + 0.872839i \(0.337724\pi\)
\(368\) −154.343 −0.0218632
\(369\) 0 0
\(370\) 4500.18 0.632306
\(371\) −8373.08 −1.17172
\(372\) 0 0
\(373\) −6460.37 −0.896796 −0.448398 0.893834i \(-0.648005\pi\)
−0.448398 + 0.893834i \(0.648005\pi\)
\(374\) −5256.11 −0.726704
\(375\) 0 0
\(376\) −3052.82 −0.418716
\(377\) −3453.96 −0.471851
\(378\) 0 0
\(379\) −1758.86 −0.238382 −0.119191 0.992871i \(-0.538030\pi\)
−0.119191 + 0.992871i \(0.538030\pi\)
\(380\) −5179.76 −0.699253
\(381\) 0 0
\(382\) −1373.04 −0.183903
\(383\) −5111.74 −0.681978 −0.340989 0.940067i \(-0.610762\pi\)
−0.340989 + 0.940067i \(0.610762\pi\)
\(384\) 0 0
\(385\) −8430.46 −1.11599
\(386\) 7073.39 0.932710
\(387\) 0 0
\(388\) 1395.07 0.182535
\(389\) 7458.10 0.972084 0.486042 0.873936i \(-0.338440\pi\)
0.486042 + 0.873936i \(0.338440\pi\)
\(390\) 0 0
\(391\) 1591.23 0.205810
\(392\) −441.764 −0.0569195
\(393\) 0 0
\(394\) 534.111 0.0682947
\(395\) 6689.93 0.852169
\(396\) 0 0
\(397\) 12359.2 1.56245 0.781223 0.624252i \(-0.214596\pi\)
0.781223 + 0.624252i \(0.214596\pi\)
\(398\) −444.008 −0.0559199
\(399\) 0 0
\(400\) −369.163 −0.0461454
\(401\) 4262.28 0.530793 0.265396 0.964139i \(-0.414497\pi\)
0.265396 + 0.964139i \(0.414497\pi\)
\(402\) 0 0
\(403\) −6684.18 −0.826211
\(404\) 1033.41 0.127263
\(405\) 0 0
\(406\) −2347.63 −0.286973
\(407\) 26020.4 3.16900
\(408\) 0 0
\(409\) −3546.96 −0.428816 −0.214408 0.976744i \(-0.568782\pi\)
−0.214408 + 0.976744i \(0.568782\pi\)
\(410\) 1138.58 0.137148
\(411\) 0 0
\(412\) −5761.20 −0.688917
\(413\) −1124.10 −0.133931
\(414\) 0 0
\(415\) 4941.11 0.584457
\(416\) −8651.47 −1.01965
\(417\) 0 0
\(418\) 16118.3 1.88605
\(419\) 7458.96 0.869676 0.434838 0.900509i \(-0.356806\pi\)
0.434838 + 0.900509i \(0.356806\pi\)
\(420\) 0 0
\(421\) 8742.87 1.01212 0.506059 0.862499i \(-0.331102\pi\)
0.506059 + 0.862499i \(0.331102\pi\)
\(422\) −5916.93 −0.682540
\(423\) 0 0
\(424\) −9706.00 −1.11171
\(425\) 3805.97 0.434392
\(426\) 0 0
\(427\) 11486.5 1.30181
\(428\) −2925.08 −0.330348
\(429\) 0 0
\(430\) 4390.53 0.492396
\(431\) 784.232 0.0876452 0.0438226 0.999039i \(-0.486046\pi\)
0.0438226 + 0.999039i \(0.486046\pi\)
\(432\) 0 0
\(433\) 13723.6 1.52313 0.761564 0.648090i \(-0.224432\pi\)
0.761564 + 0.648090i \(0.224432\pi\)
\(434\) −4543.19 −0.502489
\(435\) 0 0
\(436\) −5561.83 −0.610925
\(437\) −4879.62 −0.534151
\(438\) 0 0
\(439\) 17585.6 1.91188 0.955939 0.293564i \(-0.0948415\pi\)
0.955939 + 0.293564i \(0.0948415\pi\)
\(440\) −9772.52 −1.05883
\(441\) 0 0
\(442\) 3767.63 0.405448
\(443\) −12123.5 −1.30023 −0.650117 0.759834i \(-0.725280\pi\)
−0.650117 + 0.759834i \(0.725280\pi\)
\(444\) 0 0
\(445\) 9305.43 0.991280
\(446\) 6768.76 0.718633
\(447\) 0 0
\(448\) −5170.41 −0.545266
\(449\) −12374.8 −1.30068 −0.650338 0.759645i \(-0.725373\pi\)
−0.650338 + 0.759645i \(0.725373\pi\)
\(450\) 0 0
\(451\) 6583.38 0.687360
\(452\) 8331.34 0.866976
\(453\) 0 0
\(454\) −7711.31 −0.797158
\(455\) 6043.03 0.622641
\(456\) 0 0
\(457\) 4878.50 0.499358 0.249679 0.968329i \(-0.419675\pi\)
0.249679 + 0.968329i \(0.419675\pi\)
\(458\) −3082.44 −0.314482
\(459\) 0 0
\(460\) 1165.61 0.118145
\(461\) −3931.03 −0.397150 −0.198575 0.980086i \(-0.563631\pi\)
−0.198575 + 0.980086i \(0.563631\pi\)
\(462\) 0 0
\(463\) −342.424 −0.0343710 −0.0171855 0.999852i \(-0.505471\pi\)
−0.0171855 + 0.999852i \(0.505471\pi\)
\(464\) 343.038 0.0343214
\(465\) 0 0
\(466\) 3242.39 0.322319
\(467\) −7929.38 −0.785713 −0.392857 0.919600i \(-0.628513\pi\)
−0.392857 + 0.919600i \(0.628513\pi\)
\(468\) 0 0
\(469\) 2947.97 0.290244
\(470\) 1564.05 0.153498
\(471\) 0 0
\(472\) −1303.05 −0.127072
\(473\) 25386.4 2.46780
\(474\) 0 0
\(475\) −11671.3 −1.12740
\(476\) −4758.35 −0.458190
\(477\) 0 0
\(478\) −7450.16 −0.712892
\(479\) 15158.1 1.44591 0.722957 0.690893i \(-0.242783\pi\)
0.722957 + 0.690893i \(0.242783\pi\)
\(480\) 0 0
\(481\) −18651.7 −1.76807
\(482\) −2121.60 −0.200490
\(483\) 0 0
\(484\) −15339.8 −1.44062
\(485\) −1814.12 −0.169845
\(486\) 0 0
\(487\) −3551.47 −0.330457 −0.165228 0.986255i \(-0.552836\pi\)
−0.165228 + 0.986255i \(0.552836\pi\)
\(488\) 13315.1 1.23513
\(489\) 0 0
\(490\) 226.328 0.0208663
\(491\) −1681.06 −0.154511 −0.0772556 0.997011i \(-0.524616\pi\)
−0.0772556 + 0.997011i \(0.524616\pi\)
\(492\) 0 0
\(493\) −3536.62 −0.323086
\(494\) −11553.7 −1.05228
\(495\) 0 0
\(496\) 663.856 0.0600968
\(497\) 10525.4 0.949953
\(498\) 0 0
\(499\) −5475.45 −0.491212 −0.245606 0.969370i \(-0.578987\pi\)
−0.245606 + 0.969370i \(0.578987\pi\)
\(500\) 7184.86 0.642634
\(501\) 0 0
\(502\) −6438.58 −0.572446
\(503\) −11336.5 −1.00491 −0.502453 0.864604i \(-0.667569\pi\)
−0.502453 + 0.864604i \(0.667569\pi\)
\(504\) 0 0
\(505\) −1343.83 −0.118415
\(506\) −3627.12 −0.318666
\(507\) 0 0
\(508\) 1308.88 0.114315
\(509\) 15867.2 1.38173 0.690866 0.722983i \(-0.257229\pi\)
0.690866 + 0.722983i \(0.257229\pi\)
\(510\) 0 0
\(511\) 2053.59 0.177780
\(512\) 1682.19 0.145201
\(513\) 0 0
\(514\) −6229.48 −0.534573
\(515\) 7491.75 0.641021
\(516\) 0 0
\(517\) 9043.44 0.769303
\(518\) −12677.4 −1.07531
\(519\) 0 0
\(520\) 7005.03 0.590752
\(521\) −831.612 −0.0699301 −0.0349650 0.999389i \(-0.511132\pi\)
−0.0349650 + 0.999389i \(0.511132\pi\)
\(522\) 0 0
\(523\) −2839.22 −0.237381 −0.118690 0.992931i \(-0.537870\pi\)
−0.118690 + 0.992931i \(0.537870\pi\)
\(524\) 12957.5 1.08025
\(525\) 0 0
\(526\) 2971.42 0.246312
\(527\) −6844.16 −0.565723
\(528\) 0 0
\(529\) −11068.9 −0.909750
\(530\) 4972.66 0.407545
\(531\) 0 0
\(532\) 14591.8 1.18917
\(533\) −4719.03 −0.383497
\(534\) 0 0
\(535\) 3803.72 0.307381
\(536\) 3417.26 0.275379
\(537\) 0 0
\(538\) 10634.5 0.852204
\(539\) 1308.65 0.104578
\(540\) 0 0
\(541\) −2903.19 −0.230717 −0.115359 0.993324i \(-0.536802\pi\)
−0.115359 + 0.993324i \(0.536802\pi\)
\(542\) 6247.44 0.495112
\(543\) 0 0
\(544\) −8858.53 −0.698173
\(545\) 7232.49 0.568451
\(546\) 0 0
\(547\) −5807.96 −0.453986 −0.226993 0.973896i \(-0.572889\pi\)
−0.226993 + 0.973896i \(0.572889\pi\)
\(548\) 12457.7 0.971109
\(549\) 0 0
\(550\) −8675.49 −0.672589
\(551\) 10845.3 0.838524
\(552\) 0 0
\(553\) −18846.1 −1.44922
\(554\) 4998.82 0.383356
\(555\) 0 0
\(556\) 8946.87 0.682431
\(557\) −20165.3 −1.53399 −0.766993 0.641656i \(-0.778248\pi\)
−0.766993 + 0.641656i \(0.778248\pi\)
\(558\) 0 0
\(559\) −18197.2 −1.37685
\(560\) −600.178 −0.0452896
\(561\) 0 0
\(562\) 13586.0 1.01973
\(563\) 5801.26 0.434270 0.217135 0.976142i \(-0.430329\pi\)
0.217135 + 0.976142i \(0.430329\pi\)
\(564\) 0 0
\(565\) −10833.9 −0.806701
\(566\) 9830.74 0.730065
\(567\) 0 0
\(568\) 12200.9 0.901301
\(569\) 24062.5 1.77285 0.886425 0.462873i \(-0.153181\pi\)
0.886425 + 0.462873i \(0.153181\pi\)
\(570\) 0 0
\(571\) 9913.83 0.726586 0.363293 0.931675i \(-0.381652\pi\)
0.363293 + 0.931675i \(0.381652\pi\)
\(572\) 15957.8 1.16648
\(573\) 0 0
\(574\) −3207.49 −0.233237
\(575\) 2626.41 0.190485
\(576\) 0 0
\(577\) 18980.4 1.36944 0.684718 0.728808i \(-0.259925\pi\)
0.684718 + 0.728808i \(0.259925\pi\)
\(578\) −4361.81 −0.313888
\(579\) 0 0
\(580\) −2590.65 −0.185467
\(581\) −13919.6 −0.993942
\(582\) 0 0
\(583\) 28752.3 2.04254
\(584\) 2380.51 0.168675
\(585\) 0 0
\(586\) 2811.18 0.198172
\(587\) −2678.77 −0.188356 −0.0941778 0.995555i \(-0.530022\pi\)
−0.0941778 + 0.995555i \(0.530022\pi\)
\(588\) 0 0
\(589\) 20988.1 1.46825
\(590\) 667.591 0.0465835
\(591\) 0 0
\(592\) 1852.43 0.128606
\(593\) −15769.4 −1.09202 −0.546012 0.837777i \(-0.683855\pi\)
−0.546012 + 0.837777i \(0.683855\pi\)
\(594\) 0 0
\(595\) 6187.66 0.426335
\(596\) −15882.1 −1.09154
\(597\) 0 0
\(598\) 2599.95 0.177792
\(599\) 20847.2 1.42202 0.711012 0.703180i \(-0.248237\pi\)
0.711012 + 0.703180i \(0.248237\pi\)
\(600\) 0 0
\(601\) −7083.71 −0.480783 −0.240392 0.970676i \(-0.577276\pi\)
−0.240392 + 0.970676i \(0.577276\pi\)
\(602\) −12368.5 −0.837381
\(603\) 0 0
\(604\) −9688.31 −0.652669
\(605\) 19947.5 1.34047
\(606\) 0 0
\(607\) 9394.28 0.628175 0.314087 0.949394i \(-0.398301\pi\)
0.314087 + 0.949394i \(0.398301\pi\)
\(608\) 27165.4 1.81201
\(609\) 0 0
\(610\) −6821.69 −0.452790
\(611\) −6482.42 −0.429215
\(612\) 0 0
\(613\) −8222.66 −0.541778 −0.270889 0.962611i \(-0.587318\pi\)
−0.270889 + 0.962611i \(0.587318\pi\)
\(614\) −17461.3 −1.14769
\(615\) 0 0
\(616\) 27530.1 1.80068
\(617\) −23878.4 −1.55803 −0.779017 0.627003i \(-0.784281\pi\)
−0.779017 + 0.627003i \(0.784281\pi\)
\(618\) 0 0
\(619\) −5763.65 −0.374250 −0.187125 0.982336i \(-0.559917\pi\)
−0.187125 + 0.982336i \(0.559917\pi\)
\(620\) −5013.49 −0.324752
\(621\) 0 0
\(622\) 77.3851 0.00498852
\(623\) −26214.2 −1.68580
\(624\) 0 0
\(625\) 564.287 0.0361143
\(626\) −3035.11 −0.193782
\(627\) 0 0
\(628\) −13806.2 −0.877274
\(629\) −19098.0 −1.21063
\(630\) 0 0
\(631\) −1351.42 −0.0852601 −0.0426301 0.999091i \(-0.513574\pi\)
−0.0426301 + 0.999091i \(0.513574\pi\)
\(632\) −21846.3 −1.37500
\(633\) 0 0
\(634\) 3738.67 0.234198
\(635\) −1702.04 −0.106368
\(636\) 0 0
\(637\) −938.051 −0.0583468
\(638\) 8061.54 0.500250
\(639\) 0 0
\(640\) −6910.67 −0.426826
\(641\) 3444.20 0.212227 0.106114 0.994354i \(-0.466159\pi\)
0.106114 + 0.994354i \(0.466159\pi\)
\(642\) 0 0
\(643\) −4572.20 −0.280420 −0.140210 0.990122i \(-0.544778\pi\)
−0.140210 + 0.990122i \(0.544778\pi\)
\(644\) −3283.62 −0.200920
\(645\) 0 0
\(646\) −11830.3 −0.720519
\(647\) 2938.89 0.178578 0.0892889 0.996006i \(-0.471541\pi\)
0.0892889 + 0.996006i \(0.471541\pi\)
\(648\) 0 0
\(649\) 3860.06 0.233468
\(650\) 6218.67 0.375256
\(651\) 0 0
\(652\) 7050.12 0.423472
\(653\) −15550.7 −0.931924 −0.465962 0.884805i \(-0.654292\pi\)
−0.465962 + 0.884805i \(0.654292\pi\)
\(654\) 0 0
\(655\) −16849.6 −1.00515
\(656\) 468.682 0.0278947
\(657\) 0 0
\(658\) −4406.06 −0.261042
\(659\) 10697.7 0.632356 0.316178 0.948700i \(-0.397600\pi\)
0.316178 + 0.948700i \(0.397600\pi\)
\(660\) 0 0
\(661\) −4979.38 −0.293004 −0.146502 0.989210i \(-0.546801\pi\)
−0.146502 + 0.989210i \(0.546801\pi\)
\(662\) 13785.6 0.809355
\(663\) 0 0
\(664\) −16135.4 −0.943037
\(665\) −18974.9 −1.10649
\(666\) 0 0
\(667\) −2440.54 −0.141676
\(668\) 7317.27 0.423823
\(669\) 0 0
\(670\) −1750.76 −0.100952
\(671\) −39443.6 −2.26930
\(672\) 0 0
\(673\) 26662.2 1.52712 0.763559 0.645738i \(-0.223450\pi\)
0.763559 + 0.645738i \(0.223450\pi\)
\(674\) −20091.6 −1.14822
\(675\) 0 0
\(676\) −12.1666 −0.000692228 0
\(677\) 27190.8 1.54362 0.771808 0.635856i \(-0.219353\pi\)
0.771808 + 0.635856i \(0.219353\pi\)
\(678\) 0 0
\(679\) 5110.53 0.288842
\(680\) 7172.69 0.404500
\(681\) 0 0
\(682\) 15600.9 0.875937
\(683\) −4166.70 −0.233433 −0.116716 0.993165i \(-0.537237\pi\)
−0.116716 + 0.993165i \(0.537237\pi\)
\(684\) 0 0
\(685\) −16199.8 −0.903594
\(686\) 10295.8 0.573023
\(687\) 0 0
\(688\) 1807.30 0.100149
\(689\) −20609.9 −1.13959
\(690\) 0 0
\(691\) 25963.4 1.42937 0.714684 0.699447i \(-0.246570\pi\)
0.714684 + 0.699447i \(0.246570\pi\)
\(692\) 15796.0 0.867736
\(693\) 0 0
\(694\) −3968.48 −0.217062
\(695\) −11634.3 −0.634986
\(696\) 0 0
\(697\) −4831.97 −0.262588
\(698\) 16052.9 0.870502
\(699\) 0 0
\(700\) −7853.90 −0.424071
\(701\) 31569.0 1.70092 0.850459 0.526041i \(-0.176324\pi\)
0.850459 + 0.526041i \(0.176324\pi\)
\(702\) 0 0
\(703\) 58565.6 3.14203
\(704\) 17754.7 0.950504
\(705\) 0 0
\(706\) 1844.53 0.0983283
\(707\) 3785.70 0.201380
\(708\) 0 0
\(709\) 10296.1 0.545383 0.272691 0.962102i \(-0.412086\pi\)
0.272691 + 0.962102i \(0.412086\pi\)
\(710\) −6250.87 −0.330410
\(711\) 0 0
\(712\) −30387.3 −1.59946
\(713\) −4722.99 −0.248075
\(714\) 0 0
\(715\) −20751.2 −1.08538
\(716\) −8120.18 −0.423834
\(717\) 0 0
\(718\) 2795.44 0.145299
\(719\) −14401.2 −0.746976 −0.373488 0.927635i \(-0.621838\pi\)
−0.373488 + 0.927635i \(0.621838\pi\)
\(720\) 0 0
\(721\) −21104.9 −1.09014
\(722\) 24803.0 1.27850
\(723\) 0 0
\(724\) 8185.53 0.420183
\(725\) −5837.38 −0.299027
\(726\) 0 0
\(727\) −1687.62 −0.0860940 −0.0430470 0.999073i \(-0.513707\pi\)
−0.0430470 + 0.999073i \(0.513707\pi\)
\(728\) −19733.8 −1.00465
\(729\) 0 0
\(730\) −1219.60 −0.0618349
\(731\) −18632.7 −0.942759
\(732\) 0 0
\(733\) −28217.6 −1.42189 −0.710943 0.703250i \(-0.751732\pi\)
−0.710943 + 0.703250i \(0.751732\pi\)
\(734\) 11480.5 0.577319
\(735\) 0 0
\(736\) −6113.05 −0.306155
\(737\) −10123.0 −0.505952
\(738\) 0 0
\(739\) −21535.4 −1.07198 −0.535989 0.844225i \(-0.680061\pi\)
−0.535989 + 0.844225i \(0.680061\pi\)
\(740\) −13989.7 −0.694963
\(741\) 0 0
\(742\) −14008.4 −0.693080
\(743\) −7391.04 −0.364941 −0.182470 0.983211i \(-0.558409\pi\)
−0.182470 + 0.983211i \(0.558409\pi\)
\(744\) 0 0
\(745\) 20652.8 1.01565
\(746\) −10808.4 −0.530461
\(747\) 0 0
\(748\) 16339.7 0.798715
\(749\) −10715.4 −0.522741
\(750\) 0 0
\(751\) 4661.31 0.226489 0.113245 0.993567i \(-0.463876\pi\)
0.113245 + 0.993567i \(0.463876\pi\)
\(752\) 643.817 0.0312202
\(753\) 0 0
\(754\) −5778.58 −0.279103
\(755\) 12598.5 0.607293
\(756\) 0 0
\(757\) −20667.4 −0.992297 −0.496149 0.868238i \(-0.665253\pi\)
−0.496149 + 0.868238i \(0.665253\pi\)
\(758\) −2942.63 −0.141004
\(759\) 0 0
\(760\) −21995.6 −1.04982
\(761\) −35282.4 −1.68066 −0.840332 0.542071i \(-0.817640\pi\)
−0.840332 + 0.542071i \(0.817640\pi\)
\(762\) 0 0
\(763\) −20374.6 −0.966722
\(764\) 4268.38 0.202126
\(765\) 0 0
\(766\) −8552.11 −0.403395
\(767\) −2766.93 −0.130258
\(768\) 0 0
\(769\) −3764.58 −0.176533 −0.0882667 0.996097i \(-0.528133\pi\)
−0.0882667 + 0.996097i \(0.528133\pi\)
\(770\) −14104.4 −0.660115
\(771\) 0 0
\(772\) −21989.1 −1.02513
\(773\) −13143.1 −0.611547 −0.305773 0.952104i \(-0.598915\pi\)
−0.305773 + 0.952104i \(0.598915\pi\)
\(774\) 0 0
\(775\) −11296.6 −0.523596
\(776\) 5924.08 0.274049
\(777\) 0 0
\(778\) 12477.6 0.574994
\(779\) 14817.6 0.681510
\(780\) 0 0
\(781\) −36143.0 −1.65595
\(782\) 2662.18 0.121738
\(783\) 0 0
\(784\) 93.1648 0.00424402
\(785\) 17953.3 0.816283
\(786\) 0 0
\(787\) −34738.5 −1.57344 −0.786718 0.617312i \(-0.788222\pi\)
−0.786718 + 0.617312i \(0.788222\pi\)
\(788\) −1660.39 −0.0750622
\(789\) 0 0
\(790\) 11192.5 0.504064
\(791\) 30520.1 1.37190
\(792\) 0 0
\(793\) 28273.5 1.26611
\(794\) 20677.4 0.924196
\(795\) 0 0
\(796\) 1380.29 0.0614612
\(797\) 4155.16 0.184672 0.0923359 0.995728i \(-0.470567\pi\)
0.0923359 + 0.995728i \(0.470567\pi\)
\(798\) 0 0
\(799\) −6637.56 −0.293892
\(800\) −14621.5 −0.646183
\(801\) 0 0
\(802\) 7130.93 0.313967
\(803\) −7051.83 −0.309905
\(804\) 0 0
\(805\) 4269.96 0.186952
\(806\) −11182.9 −0.488709
\(807\) 0 0
\(808\) 4388.35 0.191066
\(809\) −25096.7 −1.09067 −0.545335 0.838218i \(-0.683598\pi\)
−0.545335 + 0.838218i \(0.683598\pi\)
\(810\) 0 0
\(811\) −15819.9 −0.684969 −0.342485 0.939523i \(-0.611268\pi\)
−0.342485 + 0.939523i \(0.611268\pi\)
\(812\) 7298.09 0.315410
\(813\) 0 0
\(814\) 43533.0 1.87448
\(815\) −9167.84 −0.394031
\(816\) 0 0
\(817\) 57138.7 2.44679
\(818\) −5934.18 −0.253647
\(819\) 0 0
\(820\) −3539.52 −0.150738
\(821\) −4697.39 −0.199684 −0.0998418 0.995003i \(-0.531834\pi\)
−0.0998418 + 0.995003i \(0.531834\pi\)
\(822\) 0 0
\(823\) −26318.7 −1.11472 −0.557359 0.830272i \(-0.688185\pi\)
−0.557359 + 0.830272i \(0.688185\pi\)
\(824\) −24464.7 −1.03430
\(825\) 0 0
\(826\) −1880.66 −0.0792211
\(827\) −24232.9 −1.01894 −0.509469 0.860489i \(-0.670158\pi\)
−0.509469 + 0.860489i \(0.670158\pi\)
\(828\) 0 0
\(829\) −6139.03 −0.257198 −0.128599 0.991697i \(-0.541048\pi\)
−0.128599 + 0.991697i \(0.541048\pi\)
\(830\) 8266.64 0.345710
\(831\) 0 0
\(832\) −12726.7 −0.530312
\(833\) −960.502 −0.0399513
\(834\) 0 0
\(835\) −9515.24 −0.394357
\(836\) −50107.0 −2.07295
\(837\) 0 0
\(838\) 12479.1 0.514419
\(839\) −19605.1 −0.806727 −0.403364 0.915040i \(-0.632159\pi\)
−0.403364 + 0.915040i \(0.632159\pi\)
\(840\) 0 0
\(841\) −18964.7 −0.777593
\(842\) 14627.1 0.598674
\(843\) 0 0
\(844\) 18394.0 0.750174
\(845\) 15.8212 0.000644102 0
\(846\) 0 0
\(847\) −56194.0 −2.27963
\(848\) 2046.92 0.0828911
\(849\) 0 0
\(850\) 6367.51 0.256945
\(851\) −13179.1 −0.530874
\(852\) 0 0
\(853\) −33730.2 −1.35393 −0.676963 0.736017i \(-0.736704\pi\)
−0.676963 + 0.736017i \(0.736704\pi\)
\(854\) 19217.3 0.770027
\(855\) 0 0
\(856\) −12421.2 −0.495968
\(857\) 28613.3 1.14050 0.570251 0.821470i \(-0.306846\pi\)
0.570251 + 0.821470i \(0.306846\pi\)
\(858\) 0 0
\(859\) −3049.90 −0.121142 −0.0605712 0.998164i \(-0.519292\pi\)
−0.0605712 + 0.998164i \(0.519292\pi\)
\(860\) −13648.9 −0.541189
\(861\) 0 0
\(862\) 1312.04 0.0518427
\(863\) 26717.9 1.05387 0.526934 0.849906i \(-0.323341\pi\)
0.526934 + 0.849906i \(0.323341\pi\)
\(864\) 0 0
\(865\) −20540.8 −0.807408
\(866\) 22960.0 0.900940
\(867\) 0 0
\(868\) 14123.4 0.552282
\(869\) 64715.8 2.52627
\(870\) 0 0
\(871\) 7256.28 0.282284
\(872\) −23618.0 −0.917211
\(873\) 0 0
\(874\) −8163.77 −0.315954
\(875\) 26320.2 1.01690
\(876\) 0 0
\(877\) −10007.0 −0.385303 −0.192652 0.981267i \(-0.561709\pi\)
−0.192652 + 0.981267i \(0.561709\pi\)
\(878\) 29421.3 1.13089
\(879\) 0 0
\(880\) 2060.95 0.0789486
\(881\) −19155.2 −0.732527 −0.366263 0.930511i \(-0.619363\pi\)
−0.366263 + 0.930511i \(0.619363\pi\)
\(882\) 0 0
\(883\) −37716.3 −1.43743 −0.718716 0.695303i \(-0.755270\pi\)
−0.718716 + 0.695303i \(0.755270\pi\)
\(884\) −11712.4 −0.445625
\(885\) 0 0
\(886\) −20283.0 −0.769096
\(887\) −44807.5 −1.69616 −0.848078 0.529872i \(-0.822240\pi\)
−0.848078 + 0.529872i \(0.822240\pi\)
\(888\) 0 0
\(889\) 4794.80 0.180891
\(890\) 15568.3 0.586348
\(891\) 0 0
\(892\) −21042.1 −0.789844
\(893\) 20354.6 0.762756
\(894\) 0 0
\(895\) 10559.3 0.394368
\(896\) 19468.0 0.725870
\(897\) 0 0
\(898\) −20703.5 −0.769358
\(899\) 10497.2 0.389434
\(900\) 0 0
\(901\) −21103.2 −0.780299
\(902\) 11014.2 0.406578
\(903\) 0 0
\(904\) 35378.6 1.30163
\(905\) −10644.3 −0.390971
\(906\) 0 0
\(907\) 548.725 0.0200883 0.0100442 0.999950i \(-0.496803\pi\)
0.0100442 + 0.999950i \(0.496803\pi\)
\(908\) 23972.2 0.876151
\(909\) 0 0
\(910\) 10110.2 0.368296
\(911\) −23571.3 −0.857246 −0.428623 0.903483i \(-0.641001\pi\)
−0.428623 + 0.903483i \(0.641001\pi\)
\(912\) 0 0
\(913\) 47798.4 1.73264
\(914\) 8161.89 0.295373
\(915\) 0 0
\(916\) 9582.38 0.345645
\(917\) 47467.0 1.70938
\(918\) 0 0
\(919\) 24363.5 0.874513 0.437256 0.899337i \(-0.355950\pi\)
0.437256 + 0.899337i \(0.355950\pi\)
\(920\) 4949.70 0.177377
\(921\) 0 0
\(922\) −6576.73 −0.234917
\(923\) 25907.7 0.923902
\(924\) 0 0
\(925\) −31522.3 −1.12048
\(926\) −572.886 −0.0203307
\(927\) 0 0
\(928\) 13586.7 0.480610
\(929\) −7503.02 −0.264980 −0.132490 0.991184i \(-0.542297\pi\)
−0.132490 + 0.991184i \(0.542297\pi\)
\(930\) 0 0
\(931\) 2945.45 0.103688
\(932\) −10079.6 −0.354259
\(933\) 0 0
\(934\) −13266.1 −0.464754
\(935\) −21247.8 −0.743185
\(936\) 0 0
\(937\) 33994.7 1.18523 0.592613 0.805487i \(-0.298096\pi\)
0.592613 + 0.805487i \(0.298096\pi\)
\(938\) 4932.05 0.171681
\(939\) 0 0
\(940\) −4862.15 −0.168708
\(941\) −35769.7 −1.23917 −0.619585 0.784929i \(-0.712699\pi\)
−0.619585 + 0.784929i \(0.712699\pi\)
\(942\) 0 0
\(943\) −3334.43 −0.115147
\(944\) 274.804 0.00947470
\(945\) 0 0
\(946\) 42472.3 1.45972
\(947\) −15352.1 −0.526798 −0.263399 0.964687i \(-0.584844\pi\)
−0.263399 + 0.964687i \(0.584844\pi\)
\(948\) 0 0
\(949\) 5054.81 0.172904
\(950\) −19526.4 −0.666865
\(951\) 0 0
\(952\) −20206.1 −0.687903
\(953\) 38129.6 1.29605 0.648027 0.761617i \(-0.275594\pi\)
0.648027 + 0.761617i \(0.275594\pi\)
\(954\) 0 0
\(955\) −5550.52 −0.188074
\(956\) 23160.3 0.783535
\(957\) 0 0
\(958\) 25360.0 0.855267
\(959\) 45636.3 1.53668
\(960\) 0 0
\(961\) −9476.58 −0.318102
\(962\) −31204.8 −1.04583
\(963\) 0 0
\(964\) 6595.41 0.220357
\(965\) 28594.2 0.953864
\(966\) 0 0
\(967\) −32504.3 −1.08094 −0.540469 0.841364i \(-0.681753\pi\)
−0.540469 + 0.841364i \(0.681753\pi\)
\(968\) −65139.6 −2.16288
\(969\) 0 0
\(970\) −3035.08 −0.100464
\(971\) 20184.6 0.667100 0.333550 0.942732i \(-0.391753\pi\)
0.333550 + 0.942732i \(0.391753\pi\)
\(972\) 0 0
\(973\) 32775.0 1.07987
\(974\) −5941.72 −0.195467
\(975\) 0 0
\(976\) −2808.05 −0.0920938
\(977\) −34251.1 −1.12159 −0.560794 0.827956i \(-0.689504\pi\)
−0.560794 + 0.827956i \(0.689504\pi\)
\(978\) 0 0
\(979\) 90017.0 2.93867
\(980\) −703.587 −0.0229340
\(981\) 0 0
\(982\) −2812.46 −0.0913944
\(983\) −26633.5 −0.864169 −0.432084 0.901833i \(-0.642222\pi\)
−0.432084 + 0.901833i \(0.642222\pi\)
\(984\) 0 0
\(985\) 2159.14 0.0698436
\(986\) −5916.89 −0.191108
\(987\) 0 0
\(988\) 35917.1 1.15655
\(989\) −12858.0 −0.413408
\(990\) 0 0
\(991\) −28729.1 −0.920900 −0.460450 0.887686i \(-0.652312\pi\)
−0.460450 + 0.887686i \(0.652312\pi\)
\(992\) 26293.3 0.841547
\(993\) 0 0
\(994\) 17609.3 0.561903
\(995\) −1794.90 −0.0571882
\(996\) 0 0
\(997\) −8221.82 −0.261171 −0.130586 0.991437i \(-0.541686\pi\)
−0.130586 + 0.991437i \(0.541686\pi\)
\(998\) −9160.61 −0.290555
\(999\) 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 531.4.a.f.1.6 8
3.2 odd 2 177.4.a.c.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.c.1.3 8 3.2 odd 2
531.4.a.f.1.6 8 1.1 even 1 trivial