Properties

Label 1815.4.a.bb.1.4
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,4,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 44x^{4} + 495x^{2} - 1200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.82790\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} +5.48370 q^{6} -15.0916 q^{7} -23.1390 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.82790 q^{2} +3.00000 q^{3} -4.65878 q^{4} +5.00000 q^{5} +5.48370 q^{6} -15.0916 q^{7} -23.1390 q^{8} +9.00000 q^{9} +9.13951 q^{10} -13.9763 q^{12} -11.4358 q^{13} -27.5859 q^{14} +15.0000 q^{15} -5.02557 q^{16} +11.9707 q^{17} +16.4511 q^{18} -91.0843 q^{19} -23.2939 q^{20} -45.2747 q^{21} +103.172 q^{23} -69.4170 q^{24} +25.0000 q^{25} -20.9034 q^{26} +27.0000 q^{27} +70.3082 q^{28} -99.3326 q^{29} +27.4185 q^{30} -39.9053 q^{31} +175.926 q^{32} +21.8813 q^{34} -75.4578 q^{35} -41.9290 q^{36} +149.172 q^{37} -166.493 q^{38} -34.3073 q^{39} -115.695 q^{40} -135.891 q^{41} -82.7576 q^{42} +304.928 q^{43} +45.0000 q^{45} +188.588 q^{46} +113.073 q^{47} -15.0767 q^{48} -115.245 q^{49} +45.6975 q^{50} +35.9122 q^{51} +53.2766 q^{52} +552.888 q^{53} +49.3533 q^{54} +349.203 q^{56} -273.253 q^{57} -181.570 q^{58} -251.961 q^{59} -69.8817 q^{60} -211.282 q^{61} -72.9430 q^{62} -135.824 q^{63} +361.779 q^{64} -57.1788 q^{65} +947.582 q^{67} -55.7690 q^{68} +309.515 q^{69} -137.929 q^{70} -175.376 q^{71} -208.251 q^{72} +768.287 q^{73} +272.671 q^{74} +75.0000 q^{75} +424.342 q^{76} -62.7103 q^{78} -710.879 q^{79} -25.1279 q^{80} +81.0000 q^{81} -248.395 q^{82} +902.213 q^{83} +210.925 q^{84} +59.8537 q^{85} +557.377 q^{86} -297.998 q^{87} -482.884 q^{89} +82.2556 q^{90} +172.583 q^{91} -480.654 q^{92} -119.716 q^{93} +206.687 q^{94} -455.422 q^{95} +527.777 q^{96} +724.303 q^{97} -210.656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82790 0.646261 0.323130 0.946354i \(-0.395265\pi\)
0.323130 + 0.946354i \(0.395265\pi\)
\(3\) 3.00000 0.577350
\(4\) −4.65878 −0.582347
\(5\) 5.00000 0.447214
\(6\) 5.48370 0.373119
\(7\) −15.0916 −0.814867 −0.407434 0.913235i \(-0.633576\pi\)
−0.407434 + 0.913235i \(0.633576\pi\)
\(8\) −23.1390 −1.02261
\(9\) 9.00000 0.333333
\(10\) 9.13951 0.289017
\(11\) 0 0
\(12\) −13.9763 −0.336218
\(13\) −11.4358 −0.243977 −0.121989 0.992531i \(-0.538927\pi\)
−0.121989 + 0.992531i \(0.538927\pi\)
\(14\) −27.5859 −0.526617
\(15\) 15.0000 0.258199
\(16\) −5.02557 −0.0785246
\(17\) 11.9707 0.170784 0.0853920 0.996347i \(-0.472786\pi\)
0.0853920 + 0.996347i \(0.472786\pi\)
\(18\) 16.4511 0.215420
\(19\) −91.0843 −1.09980 −0.549899 0.835231i \(-0.685334\pi\)
−0.549899 + 0.835231i \(0.685334\pi\)
\(20\) −23.2939 −0.260434
\(21\) −45.2747 −0.470464
\(22\) 0 0
\(23\) 103.172 0.935339 0.467669 0.883903i \(-0.345094\pi\)
0.467669 + 0.883903i \(0.345094\pi\)
\(24\) −69.4170 −0.590403
\(25\) 25.0000 0.200000
\(26\) −20.9034 −0.157673
\(27\) 27.0000 0.192450
\(28\) 70.3082 0.474536
\(29\) −99.3326 −0.636055 −0.318028 0.948081i \(-0.603020\pi\)
−0.318028 + 0.948081i \(0.603020\pi\)
\(30\) 27.4185 0.166864
\(31\) −39.9053 −0.231200 −0.115600 0.993296i \(-0.536879\pi\)
−0.115600 + 0.993296i \(0.536879\pi\)
\(32\) 175.926 0.971861
\(33\) 0 0
\(34\) 21.8813 0.110371
\(35\) −75.4578 −0.364420
\(36\) −41.9290 −0.194116
\(37\) 149.172 0.662802 0.331401 0.943490i \(-0.392479\pi\)
0.331401 + 0.943490i \(0.392479\pi\)
\(38\) −166.493 −0.710757
\(39\) −34.3073 −0.140860
\(40\) −115.695 −0.457325
\(41\) −135.891 −0.517623 −0.258812 0.965928i \(-0.583331\pi\)
−0.258812 + 0.965928i \(0.583331\pi\)
\(42\) −82.7576 −0.304042
\(43\) 304.928 1.08142 0.540710 0.841209i \(-0.318156\pi\)
0.540710 + 0.841209i \(0.318156\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 188.588 0.604473
\(47\) 113.073 0.350924 0.175462 0.984486i \(-0.443858\pi\)
0.175462 + 0.984486i \(0.443858\pi\)
\(48\) −15.0767 −0.0453362
\(49\) −115.245 −0.335991
\(50\) 45.6975 0.129252
\(51\) 35.9122 0.0986022
\(52\) 53.2766 0.142080
\(53\) 552.888 1.43292 0.716462 0.697626i \(-0.245760\pi\)
0.716462 + 0.697626i \(0.245760\pi\)
\(54\) 49.3533 0.124373
\(55\) 0 0
\(56\) 349.203 0.833290
\(57\) −273.253 −0.634969
\(58\) −181.570 −0.411057
\(59\) −251.961 −0.555975 −0.277988 0.960585i \(-0.589667\pi\)
−0.277988 + 0.960585i \(0.589667\pi\)
\(60\) −69.8817 −0.150361
\(61\) −211.282 −0.443473 −0.221737 0.975107i \(-0.571172\pi\)
−0.221737 + 0.975107i \(0.571172\pi\)
\(62\) −72.9430 −0.149416
\(63\) −135.824 −0.271622
\(64\) 361.779 0.706600
\(65\) −57.1788 −0.109110
\(66\) 0 0
\(67\) 947.582 1.72784 0.863922 0.503625i \(-0.168001\pi\)
0.863922 + 0.503625i \(0.168001\pi\)
\(68\) −55.7690 −0.0994556
\(69\) 309.515 0.540018
\(70\) −137.929 −0.235510
\(71\) −175.376 −0.293146 −0.146573 0.989200i \(-0.546824\pi\)
−0.146573 + 0.989200i \(0.546824\pi\)
\(72\) −208.251 −0.340870
\(73\) 768.287 1.23180 0.615898 0.787826i \(-0.288793\pi\)
0.615898 + 0.787826i \(0.288793\pi\)
\(74\) 272.671 0.428343
\(75\) 75.0000 0.115470
\(76\) 424.342 0.640465
\(77\) 0 0
\(78\) −62.7103 −0.0910326
\(79\) −710.879 −1.01241 −0.506204 0.862414i \(-0.668952\pi\)
−0.506204 + 0.862414i \(0.668952\pi\)
\(80\) −25.1279 −0.0351173
\(81\) 81.0000 0.111111
\(82\) −248.395 −0.334519
\(83\) 902.213 1.19314 0.596571 0.802561i \(-0.296530\pi\)
0.596571 + 0.802561i \(0.296530\pi\)
\(84\) 210.925 0.273973
\(85\) 59.8537 0.0763770
\(86\) 557.377 0.698879
\(87\) −297.998 −0.367227
\(88\) 0 0
\(89\) −482.884 −0.575119 −0.287560 0.957763i \(-0.592844\pi\)
−0.287560 + 0.957763i \(0.592844\pi\)
\(90\) 82.2556 0.0963388
\(91\) 172.583 0.198809
\(92\) −480.654 −0.544692
\(93\) −119.716 −0.133484
\(94\) 206.687 0.226788
\(95\) −455.422 −0.491845
\(96\) 527.777 0.561104
\(97\) 724.303 0.758163 0.379082 0.925363i \(-0.376240\pi\)
0.379082 + 0.925363i \(0.376240\pi\)
\(98\) −210.656 −0.217138
\(99\) 0 0
\(100\) −116.469 −0.116469
\(101\) 73.5266 0.0724374 0.0362187 0.999344i \(-0.488469\pi\)
0.0362187 + 0.999344i \(0.488469\pi\)
\(102\) 65.6439 0.0637227
\(103\) −156.047 −0.149279 −0.0746395 0.997211i \(-0.523781\pi\)
−0.0746395 + 0.997211i \(0.523781\pi\)
\(104\) 264.612 0.249493
\(105\) −226.373 −0.210398
\(106\) 1010.62 0.926042
\(107\) 1998.62 1.80573 0.902867 0.429920i \(-0.141458\pi\)
0.902867 + 0.429920i \(0.141458\pi\)
\(108\) −125.787 −0.112073
\(109\) 642.090 0.564230 0.282115 0.959381i \(-0.408964\pi\)
0.282115 + 0.959381i \(0.408964\pi\)
\(110\) 0 0
\(111\) 447.515 0.382669
\(112\) 75.8437 0.0639871
\(113\) 227.521 0.189411 0.0947054 0.995505i \(-0.469809\pi\)
0.0947054 + 0.995505i \(0.469809\pi\)
\(114\) −499.479 −0.410355
\(115\) 515.859 0.418296
\(116\) 462.768 0.370405
\(117\) −102.922 −0.0813258
\(118\) −460.560 −0.359305
\(119\) −180.657 −0.139166
\(120\) −347.085 −0.264036
\(121\) 0 0
\(122\) −386.202 −0.286599
\(123\) −407.672 −0.298850
\(124\) 185.910 0.134639
\(125\) 125.000 0.0894427
\(126\) −248.273 −0.175539
\(127\) −1189.73 −0.831274 −0.415637 0.909531i \(-0.636441\pi\)
−0.415637 + 0.909531i \(0.636441\pi\)
\(128\) −746.109 −0.515213
\(129\) 914.783 0.624358
\(130\) −104.517 −0.0705135
\(131\) 1401.48 0.934713 0.467357 0.884069i \(-0.345206\pi\)
0.467357 + 0.884069i \(0.345206\pi\)
\(132\) 0 0
\(133\) 1374.60 0.896190
\(134\) 1732.09 1.11664
\(135\) 135.000 0.0860663
\(136\) −276.991 −0.174645
\(137\) 826.015 0.515119 0.257559 0.966262i \(-0.417082\pi\)
0.257559 + 0.966262i \(0.417082\pi\)
\(138\) 565.763 0.348992
\(139\) 1920.31 1.17179 0.585895 0.810387i \(-0.300743\pi\)
0.585895 + 0.810387i \(0.300743\pi\)
\(140\) 351.541 0.212219
\(141\) 339.220 0.202606
\(142\) −320.571 −0.189448
\(143\) 0 0
\(144\) −45.2302 −0.0261749
\(145\) −496.663 −0.284453
\(146\) 1404.35 0.796062
\(147\) −345.735 −0.193985
\(148\) −694.958 −0.385981
\(149\) 1057.30 0.581325 0.290662 0.956826i \(-0.406124\pi\)
0.290662 + 0.956826i \(0.406124\pi\)
\(150\) 137.093 0.0746238
\(151\) 2243.75 1.20923 0.604614 0.796519i \(-0.293327\pi\)
0.604614 + 0.796519i \(0.293327\pi\)
\(152\) 2107.60 1.12466
\(153\) 107.737 0.0569280
\(154\) 0 0
\(155\) −199.527 −0.103396
\(156\) 159.830 0.0820297
\(157\) 382.672 0.194526 0.0972628 0.995259i \(-0.468991\pi\)
0.0972628 + 0.995259i \(0.468991\pi\)
\(158\) −1299.42 −0.654279
\(159\) 1658.66 0.827299
\(160\) 879.628 0.434630
\(161\) −1557.02 −0.762177
\(162\) 148.060 0.0718067
\(163\) 2513.97 1.20803 0.604017 0.796971i \(-0.293566\pi\)
0.604017 + 0.796971i \(0.293566\pi\)
\(164\) 633.084 0.301436
\(165\) 0 0
\(166\) 1649.16 0.771080
\(167\) −4280.74 −1.98356 −0.991778 0.127971i \(-0.959154\pi\)
−0.991778 + 0.127971i \(0.959154\pi\)
\(168\) 1047.61 0.481100
\(169\) −2066.22 −0.940475
\(170\) 109.407 0.0493594
\(171\) −819.759 −0.366600
\(172\) −1420.59 −0.629761
\(173\) −2214.27 −0.973110 −0.486555 0.873650i \(-0.661747\pi\)
−0.486555 + 0.873650i \(0.661747\pi\)
\(174\) −544.711 −0.237324
\(175\) −377.289 −0.162973
\(176\) 0 0
\(177\) −755.883 −0.320992
\(178\) −882.664 −0.371677
\(179\) 1570.64 0.655837 0.327918 0.944706i \(-0.393653\pi\)
0.327918 + 0.944706i \(0.393653\pi\)
\(180\) −209.645 −0.0868112
\(181\) 2867.18 1.17743 0.588717 0.808339i \(-0.299633\pi\)
0.588717 + 0.808339i \(0.299633\pi\)
\(182\) 315.465 0.128483
\(183\) −633.845 −0.256039
\(184\) −2387.29 −0.956486
\(185\) 745.859 0.296414
\(186\) −218.829 −0.0862652
\(187\) 0 0
\(188\) −526.783 −0.204360
\(189\) −407.472 −0.156821
\(190\) −832.466 −0.317860
\(191\) −2081.99 −0.788732 −0.394366 0.918953i \(-0.629036\pi\)
−0.394366 + 0.918953i \(0.629036\pi\)
\(192\) 1085.34 0.407956
\(193\) 2362.68 0.881190 0.440595 0.897706i \(-0.354768\pi\)
0.440595 + 0.897706i \(0.354768\pi\)
\(194\) 1323.95 0.489971
\(195\) −171.536 −0.0629947
\(196\) 536.901 0.195664
\(197\) −232.619 −0.0841292 −0.0420646 0.999115i \(-0.513394\pi\)
−0.0420646 + 0.999115i \(0.513394\pi\)
\(198\) 0 0
\(199\) 563.211 0.200628 0.100314 0.994956i \(-0.468015\pi\)
0.100314 + 0.994956i \(0.468015\pi\)
\(200\) −578.475 −0.204522
\(201\) 2842.75 0.997572
\(202\) 134.399 0.0468134
\(203\) 1499.08 0.518301
\(204\) −167.307 −0.0574207
\(205\) −679.453 −0.231488
\(206\) −285.238 −0.0964731
\(207\) 928.546 0.311780
\(208\) 57.4712 0.0191582
\(209\) 0 0
\(210\) −413.788 −0.135972
\(211\) −48.3870 −0.0157872 −0.00789360 0.999969i \(-0.502513\pi\)
−0.00789360 + 0.999969i \(0.502513\pi\)
\(212\) −2575.78 −0.834459
\(213\) −526.129 −0.169248
\(214\) 3653.27 1.16697
\(215\) 1524.64 0.483625
\(216\) −624.753 −0.196801
\(217\) 602.233 0.188398
\(218\) 1173.68 0.364639
\(219\) 2304.86 0.711178
\(220\) 0 0
\(221\) −136.894 −0.0416675
\(222\) 818.014 0.247304
\(223\) 2363.27 0.709669 0.354834 0.934929i \(-0.384537\pi\)
0.354834 + 0.934929i \(0.384537\pi\)
\(224\) −2654.99 −0.791938
\(225\) 225.000 0.0666667
\(226\) 415.887 0.122409
\(227\) −1124.95 −0.328924 −0.164462 0.986383i \(-0.552589\pi\)
−0.164462 + 0.986383i \(0.552589\pi\)
\(228\) 1273.02 0.369772
\(229\) 1931.86 0.557471 0.278735 0.960368i \(-0.410085\pi\)
0.278735 + 0.960368i \(0.410085\pi\)
\(230\) 942.939 0.270328
\(231\) 0 0
\(232\) 2298.46 0.650436
\(233\) −4282.36 −1.20406 −0.602031 0.798473i \(-0.705642\pi\)
−0.602031 + 0.798473i \(0.705642\pi\)
\(234\) −188.131 −0.0525577
\(235\) 565.366 0.156938
\(236\) 1173.83 0.323770
\(237\) −2132.64 −0.584514
\(238\) −330.223 −0.0899377
\(239\) −4549.61 −1.23134 −0.615669 0.788005i \(-0.711114\pi\)
−0.615669 + 0.788005i \(0.711114\pi\)
\(240\) −75.3836 −0.0202750
\(241\) 5704.24 1.52466 0.762328 0.647191i \(-0.224056\pi\)
0.762328 + 0.647191i \(0.224056\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 984.315 0.258255
\(245\) −576.225 −0.150260
\(246\) −745.184 −0.193135
\(247\) 1041.62 0.268326
\(248\) 923.369 0.236427
\(249\) 2706.64 0.688860
\(250\) 228.488 0.0578033
\(251\) 2352.11 0.591489 0.295745 0.955267i \(-0.404432\pi\)
0.295745 + 0.955267i \(0.404432\pi\)
\(252\) 632.774 0.158179
\(253\) 0 0
\(254\) −2174.72 −0.537220
\(255\) 179.561 0.0440963
\(256\) −4258.05 −1.03956
\(257\) −1735.01 −0.421115 −0.210558 0.977581i \(-0.567528\pi\)
−0.210558 + 0.977581i \(0.567528\pi\)
\(258\) 1672.13 0.403498
\(259\) −2251.23 −0.540096
\(260\) 266.383 0.0635399
\(261\) −893.993 −0.212018
\(262\) 2561.76 0.604069
\(263\) −3674.36 −0.861485 −0.430742 0.902475i \(-0.641748\pi\)
−0.430742 + 0.902475i \(0.641748\pi\)
\(264\) 0 0
\(265\) 2764.44 0.640823
\(266\) 2512.64 0.579172
\(267\) −1448.65 −0.332045
\(268\) −4414.58 −1.00621
\(269\) 6014.78 1.36330 0.681649 0.731679i \(-0.261263\pi\)
0.681649 + 0.731679i \(0.261263\pi\)
\(270\) 246.767 0.0556213
\(271\) −3038.66 −0.681128 −0.340564 0.940221i \(-0.610618\pi\)
−0.340564 + 0.940221i \(0.610618\pi\)
\(272\) −60.1598 −0.0134107
\(273\) 517.750 0.114783
\(274\) 1509.87 0.332901
\(275\) 0 0
\(276\) −1441.96 −0.314478
\(277\) 5284.84 1.14634 0.573168 0.819438i \(-0.305714\pi\)
0.573168 + 0.819438i \(0.305714\pi\)
\(278\) 3510.14 0.757282
\(279\) −359.148 −0.0770668
\(280\) 1746.02 0.372659
\(281\) −6666.85 −1.41534 −0.707671 0.706542i \(-0.750254\pi\)
−0.707671 + 0.706542i \(0.750254\pi\)
\(282\) 620.060 0.130936
\(283\) 2563.96 0.538557 0.269279 0.963062i \(-0.413215\pi\)
0.269279 + 0.963062i \(0.413215\pi\)
\(284\) 817.039 0.170713
\(285\) −1366.26 −0.283967
\(286\) 0 0
\(287\) 2050.80 0.421794
\(288\) 1583.33 0.323954
\(289\) −4769.70 −0.970833
\(290\) −907.851 −0.183830
\(291\) 2172.91 0.437726
\(292\) −3579.28 −0.717333
\(293\) 5016.44 1.00022 0.500109 0.865963i \(-0.333293\pi\)
0.500109 + 0.865963i \(0.333293\pi\)
\(294\) −631.969 −0.125365
\(295\) −1259.80 −0.248640
\(296\) −3451.68 −0.677787
\(297\) 0 0
\(298\) 1932.64 0.375687
\(299\) −1179.85 −0.228202
\(300\) −349.408 −0.0672437
\(301\) −4601.83 −0.881213
\(302\) 4101.35 0.781476
\(303\) 220.580 0.0418217
\(304\) 457.751 0.0863612
\(305\) −1056.41 −0.198327
\(306\) 196.932 0.0367903
\(307\) −7224.25 −1.34303 −0.671514 0.740991i \(-0.734356\pi\)
−0.671514 + 0.740991i \(0.734356\pi\)
\(308\) 0 0
\(309\) −468.140 −0.0861862
\(310\) −364.715 −0.0668207
\(311\) 5271.00 0.961065 0.480532 0.876977i \(-0.340443\pi\)
0.480532 + 0.876977i \(0.340443\pi\)
\(312\) 793.835 0.144045
\(313\) −752.960 −0.135974 −0.0679870 0.997686i \(-0.521658\pi\)
−0.0679870 + 0.997686i \(0.521658\pi\)
\(314\) 699.486 0.125714
\(315\) −679.120 −0.121473
\(316\) 3311.83 0.589572
\(317\) 7792.04 1.38058 0.690291 0.723531i \(-0.257482\pi\)
0.690291 + 0.723531i \(0.257482\pi\)
\(318\) 3031.87 0.534651
\(319\) 0 0
\(320\) 1808.90 0.316001
\(321\) 5995.85 1.04254
\(322\) −2846.08 −0.492565
\(323\) −1090.35 −0.187828
\(324\) −377.361 −0.0647052
\(325\) −285.894 −0.0487955
\(326\) 4595.30 0.780705
\(327\) 1926.27 0.325758
\(328\) 3144.37 0.529326
\(329\) −1706.45 −0.285957
\(330\) 0 0
\(331\) 8292.19 1.37698 0.688489 0.725247i \(-0.258274\pi\)
0.688489 + 0.725247i \(0.258274\pi\)
\(332\) −4203.21 −0.694823
\(333\) 1342.55 0.220934
\(334\) −7824.77 −1.28189
\(335\) 4737.91 0.772716
\(336\) 227.531 0.0369430
\(337\) 925.955 0.149674 0.0748368 0.997196i \(-0.476156\pi\)
0.0748368 + 0.997196i \(0.476156\pi\)
\(338\) −3776.85 −0.607792
\(339\) 682.564 0.109356
\(340\) −278.845 −0.0444779
\(341\) 0 0
\(342\) −1498.44 −0.236919
\(343\) 6915.63 1.08866
\(344\) −7055.72 −1.10587
\(345\) 1547.58 0.241503
\(346\) −4047.47 −0.628883
\(347\) 2727.83 0.422010 0.211005 0.977485i \(-0.432326\pi\)
0.211005 + 0.977485i \(0.432326\pi\)
\(348\) 1388.31 0.213853
\(349\) 8846.92 1.35692 0.678459 0.734638i \(-0.262648\pi\)
0.678459 + 0.734638i \(0.262648\pi\)
\(350\) −689.647 −0.105323
\(351\) −308.765 −0.0469535
\(352\) 0 0
\(353\) −167.372 −0.0252360 −0.0126180 0.999920i \(-0.504017\pi\)
−0.0126180 + 0.999920i \(0.504017\pi\)
\(354\) −1381.68 −0.207445
\(355\) −876.882 −0.131099
\(356\) 2249.65 0.334919
\(357\) −541.971 −0.0803477
\(358\) 2870.97 0.423841
\(359\) −10259.3 −1.50826 −0.754130 0.656725i \(-0.771941\pi\)
−0.754130 + 0.656725i \(0.771941\pi\)
\(360\) −1041.25 −0.152442
\(361\) 1437.35 0.209557
\(362\) 5240.91 0.760929
\(363\) 0 0
\(364\) −804.027 −0.115776
\(365\) 3841.43 0.550876
\(366\) −1158.61 −0.165468
\(367\) 89.0428 0.0126648 0.00633242 0.999980i \(-0.497984\pi\)
0.00633242 + 0.999980i \(0.497984\pi\)
\(368\) −518.497 −0.0734471
\(369\) −1223.02 −0.172541
\(370\) 1363.36 0.191561
\(371\) −8343.94 −1.16764
\(372\) 557.730 0.0777338
\(373\) −9532.06 −1.32319 −0.661597 0.749860i \(-0.730121\pi\)
−0.661597 + 0.749860i \(0.730121\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) −2616.40 −0.358858
\(377\) 1135.94 0.155183
\(378\) −744.818 −0.101347
\(379\) −8887.37 −1.20452 −0.602261 0.798300i \(-0.705733\pi\)
−0.602261 + 0.798300i \(0.705733\pi\)
\(380\) 2121.71 0.286424
\(381\) −3569.20 −0.479936
\(382\) −3805.68 −0.509726
\(383\) 2061.68 0.275057 0.137529 0.990498i \(-0.456084\pi\)
0.137529 + 0.990498i \(0.456084\pi\)
\(384\) −2238.33 −0.297459
\(385\) 0 0
\(386\) 4318.75 0.569478
\(387\) 2744.35 0.360473
\(388\) −3374.37 −0.441514
\(389\) −7778.76 −1.01388 −0.506940 0.861982i \(-0.669223\pi\)
−0.506940 + 0.861982i \(0.669223\pi\)
\(390\) −313.551 −0.0407110
\(391\) 1235.04 0.159741
\(392\) 2666.65 0.343588
\(393\) 4204.43 0.539657
\(394\) −425.205 −0.0543694
\(395\) −3554.40 −0.452762
\(396\) 0 0
\(397\) 9988.30 1.26272 0.631358 0.775491i \(-0.282498\pi\)
0.631358 + 0.775491i \(0.282498\pi\)
\(398\) 1029.49 0.129658
\(399\) 4123.81 0.517415
\(400\) −125.639 −0.0157049
\(401\) 6331.91 0.788530 0.394265 0.918997i \(-0.370999\pi\)
0.394265 + 0.918997i \(0.370999\pi\)
\(402\) 5196.26 0.644691
\(403\) 456.347 0.0564077
\(404\) −342.544 −0.0421837
\(405\) 405.000 0.0496904
\(406\) 2740.18 0.334957
\(407\) 0 0
\(408\) −830.972 −0.100831
\(409\) 530.445 0.0641291 0.0320646 0.999486i \(-0.489792\pi\)
0.0320646 + 0.999486i \(0.489792\pi\)
\(410\) −1241.97 −0.149602
\(411\) 2478.05 0.297404
\(412\) 726.987 0.0869322
\(413\) 3802.48 0.453046
\(414\) 1697.29 0.201491
\(415\) 4511.06 0.533589
\(416\) −2011.84 −0.237112
\(417\) 5760.94 0.676533
\(418\) 0 0
\(419\) 575.390 0.0670875 0.0335437 0.999437i \(-0.489321\pi\)
0.0335437 + 0.999437i \(0.489321\pi\)
\(420\) 1054.62 0.122525
\(421\) 13829.7 1.60100 0.800499 0.599334i \(-0.204568\pi\)
0.800499 + 0.599334i \(0.204568\pi\)
\(422\) −88.4466 −0.0102026
\(423\) 1017.66 0.116975
\(424\) −12793.3 −1.46532
\(425\) 299.268 0.0341568
\(426\) −961.712 −0.109378
\(427\) 3188.57 0.361372
\(428\) −9311.11 −1.05156
\(429\) 0 0
\(430\) 2786.89 0.312548
\(431\) −4758.89 −0.531850 −0.265925 0.963994i \(-0.585677\pi\)
−0.265925 + 0.963994i \(0.585677\pi\)
\(432\) −135.690 −0.0151121
\(433\) −4516.02 −0.501215 −0.250608 0.968089i \(-0.580630\pi\)
−0.250608 + 0.968089i \(0.580630\pi\)
\(434\) 1100.82 0.121754
\(435\) −1489.99 −0.164229
\(436\) −2991.35 −0.328578
\(437\) −9397.33 −1.02868
\(438\) 4213.06 0.459606
\(439\) −13916.2 −1.51295 −0.756474 0.654024i \(-0.773080\pi\)
−0.756474 + 0.654024i \(0.773080\pi\)
\(440\) 0 0
\(441\) −1037.20 −0.111997
\(442\) −250.229 −0.0269280
\(443\) 10294.2 1.10405 0.552023 0.833829i \(-0.313856\pi\)
0.552023 + 0.833829i \(0.313856\pi\)
\(444\) −2084.87 −0.222846
\(445\) −2414.42 −0.257201
\(446\) 4319.82 0.458631
\(447\) 3171.90 0.335628
\(448\) −5459.81 −0.575786
\(449\) −4191.42 −0.440546 −0.220273 0.975438i \(-0.570695\pi\)
−0.220273 + 0.975438i \(0.570695\pi\)
\(450\) 411.278 0.0430840
\(451\) 0 0
\(452\) −1059.97 −0.110303
\(453\) 6731.24 0.698148
\(454\) −2056.30 −0.212571
\(455\) 862.916 0.0889102
\(456\) 6322.80 0.649325
\(457\) −8294.55 −0.849021 −0.424511 0.905423i \(-0.639554\pi\)
−0.424511 + 0.905423i \(0.639554\pi\)
\(458\) 3531.24 0.360271
\(459\) 323.210 0.0328674
\(460\) −2403.27 −0.243594
\(461\) −1778.74 −0.179706 −0.0898529 0.995955i \(-0.528640\pi\)
−0.0898529 + 0.995955i \(0.528640\pi\)
\(462\) 0 0
\(463\) 4654.76 0.467225 0.233612 0.972330i \(-0.424945\pi\)
0.233612 + 0.972330i \(0.424945\pi\)
\(464\) 499.203 0.0499460
\(465\) −598.580 −0.0596957
\(466\) −7827.72 −0.778138
\(467\) −3886.35 −0.385093 −0.192547 0.981288i \(-0.561675\pi\)
−0.192547 + 0.981288i \(0.561675\pi\)
\(468\) 479.490 0.0473599
\(469\) −14300.5 −1.40796
\(470\) 1033.43 0.101423
\(471\) 1148.02 0.112309
\(472\) 5830.12 0.568545
\(473\) 0 0
\(474\) −3898.25 −0.377748
\(475\) −2277.11 −0.219960
\(476\) 841.641 0.0810431
\(477\) 4975.99 0.477641
\(478\) −8316.23 −0.795765
\(479\) 11614.3 1.10787 0.553936 0.832559i \(-0.313125\pi\)
0.553936 + 0.832559i \(0.313125\pi\)
\(480\) 2638.89 0.250934
\(481\) −1705.89 −0.161709
\(482\) 10426.8 0.985326
\(483\) −4671.07 −0.440043
\(484\) 0 0
\(485\) 3621.52 0.339061
\(486\) 444.180 0.0414576
\(487\) −17965.5 −1.67165 −0.835826 0.548995i \(-0.815011\pi\)
−0.835826 + 0.548995i \(0.815011\pi\)
\(488\) 4888.85 0.453499
\(489\) 7541.92 0.697459
\(490\) −1053.28 −0.0971070
\(491\) −15494.4 −1.42414 −0.712072 0.702107i \(-0.752243\pi\)
−0.712072 + 0.702107i \(0.752243\pi\)
\(492\) 1899.25 0.174034
\(493\) −1189.08 −0.108628
\(494\) 1903.97 0.173409
\(495\) 0 0
\(496\) 200.547 0.0181549
\(497\) 2646.70 0.238875
\(498\) 4947.47 0.445183
\(499\) −15279.1 −1.37072 −0.685358 0.728207i \(-0.740354\pi\)
−0.685358 + 0.728207i \(0.740354\pi\)
\(500\) −582.347 −0.0520867
\(501\) −12842.2 −1.14521
\(502\) 4299.42 0.382256
\(503\) 10613.4 0.940810 0.470405 0.882451i \(-0.344108\pi\)
0.470405 + 0.882451i \(0.344108\pi\)
\(504\) 3142.83 0.277763
\(505\) 367.633 0.0323950
\(506\) 0 0
\(507\) −6198.67 −0.542983
\(508\) 5542.71 0.484090
\(509\) 15287.8 1.33127 0.665637 0.746276i \(-0.268160\pi\)
0.665637 + 0.746276i \(0.268160\pi\)
\(510\) 328.220 0.0284977
\(511\) −11594.6 −1.00375
\(512\) −1814.42 −0.156615
\(513\) −2459.28 −0.211656
\(514\) −3171.42 −0.272150
\(515\) −780.233 −0.0667596
\(516\) −4261.77 −0.363593
\(517\) 0 0
\(518\) −4115.03 −0.349043
\(519\) −6642.82 −0.561825
\(520\) 1323.06 0.111577
\(521\) 9592.74 0.806652 0.403326 0.915056i \(-0.367854\pi\)
0.403326 + 0.915056i \(0.367854\pi\)
\(522\) −1634.13 −0.137019
\(523\) 4245.93 0.354994 0.177497 0.984121i \(-0.443200\pi\)
0.177497 + 0.984121i \(0.443200\pi\)
\(524\) −6529.16 −0.544328
\(525\) −1131.87 −0.0940928
\(526\) −6716.36 −0.556744
\(527\) −477.696 −0.0394853
\(528\) 0 0
\(529\) −1522.59 −0.125141
\(530\) 5053.12 0.414139
\(531\) −2267.65 −0.185325
\(532\) −6403.97 −0.521894
\(533\) 1554.01 0.126288
\(534\) −2647.99 −0.214588
\(535\) 9993.08 0.807549
\(536\) −21926.1 −1.76691
\(537\) 4711.91 0.378648
\(538\) 10994.4 0.881046
\(539\) 0 0
\(540\) −628.935 −0.0501205
\(541\) −14877.6 −1.18232 −0.591161 0.806554i \(-0.701330\pi\)
−0.591161 + 0.806554i \(0.701330\pi\)
\(542\) −5554.38 −0.440186
\(543\) 8601.53 0.679792
\(544\) 2105.96 0.165978
\(545\) 3210.45 0.252331
\(546\) 946.395 0.0741795
\(547\) −22641.9 −1.76983 −0.884915 0.465753i \(-0.845784\pi\)
−0.884915 + 0.465753i \(0.845784\pi\)
\(548\) −3848.22 −0.299978
\(549\) −1901.54 −0.147824
\(550\) 0 0
\(551\) 9047.64 0.699533
\(552\) −7161.87 −0.552227
\(553\) 10728.3 0.824977
\(554\) 9660.16 0.740832
\(555\) 2237.58 0.171135
\(556\) −8946.31 −0.682389
\(557\) −9738.83 −0.740840 −0.370420 0.928864i \(-0.620786\pi\)
−0.370420 + 0.928864i \(0.620786\pi\)
\(558\) −656.487 −0.0498052
\(559\) −3487.08 −0.263842
\(560\) 379.219 0.0286159
\(561\) 0 0
\(562\) −12186.3 −0.914680
\(563\) −8888.54 −0.665377 −0.332688 0.943037i \(-0.607956\pi\)
−0.332688 + 0.943037i \(0.607956\pi\)
\(564\) −1580.35 −0.117987
\(565\) 1137.61 0.0847071
\(566\) 4686.67 0.348048
\(567\) −1222.42 −0.0905408
\(568\) 4058.03 0.299773
\(569\) 12443.0 0.916764 0.458382 0.888755i \(-0.348429\pi\)
0.458382 + 0.888755i \(0.348429\pi\)
\(570\) −2497.40 −0.183517
\(571\) 17508.4 1.28320 0.641598 0.767041i \(-0.278272\pi\)
0.641598 + 0.767041i \(0.278272\pi\)
\(572\) 0 0
\(573\) −6245.98 −0.455375
\(574\) 3748.66 0.272589
\(575\) 2579.29 0.187068
\(576\) 3256.01 0.235533
\(577\) −26750.4 −1.93004 −0.965021 0.262172i \(-0.915561\pi\)
−0.965021 + 0.262172i \(0.915561\pi\)
\(578\) −8718.54 −0.627411
\(579\) 7088.05 0.508755
\(580\) 2313.84 0.165650
\(581\) −13615.8 −0.972252
\(582\) 3971.86 0.282885
\(583\) 0 0
\(584\) −17777.4 −1.25965
\(585\) −514.609 −0.0363700
\(586\) 9169.56 0.646401
\(587\) 16631.8 1.16945 0.584727 0.811230i \(-0.301202\pi\)
0.584727 + 0.811230i \(0.301202\pi\)
\(588\) 1610.70 0.112966
\(589\) 3634.75 0.254274
\(590\) −2302.80 −0.160686
\(591\) −697.858 −0.0485720
\(592\) −749.673 −0.0520463
\(593\) −4248.11 −0.294181 −0.147090 0.989123i \(-0.546991\pi\)
−0.147090 + 0.989123i \(0.546991\pi\)
\(594\) 0 0
\(595\) −903.285 −0.0622371
\(596\) −4925.73 −0.338533
\(597\) 1689.63 0.115833
\(598\) −2156.64 −0.147478
\(599\) −22152.5 −1.51106 −0.755531 0.655113i \(-0.772621\pi\)
−0.755531 + 0.655113i \(0.772621\pi\)
\(600\) −1735.42 −0.118081
\(601\) 1870.46 0.126951 0.0634756 0.997983i \(-0.479782\pi\)
0.0634756 + 0.997983i \(0.479782\pi\)
\(602\) −8411.69 −0.569493
\(603\) 8528.24 0.575948
\(604\) −10453.1 −0.704190
\(605\) 0 0
\(606\) 403.198 0.0270277
\(607\) 17092.6 1.14294 0.571472 0.820622i \(-0.306373\pi\)
0.571472 + 0.820622i \(0.306373\pi\)
\(608\) −16024.1 −1.06885
\(609\) 4497.25 0.299241
\(610\) −1931.01 −0.128171
\(611\) −1293.08 −0.0856176
\(612\) −501.921 −0.0331519
\(613\) 29572.2 1.94846 0.974232 0.225550i \(-0.0724179\pi\)
0.974232 + 0.225550i \(0.0724179\pi\)
\(614\) −13205.2 −0.867947
\(615\) −2038.36 −0.133650
\(616\) 0 0
\(617\) −10496.0 −0.684854 −0.342427 0.939544i \(-0.611249\pi\)
−0.342427 + 0.939544i \(0.611249\pi\)
\(618\) −855.713 −0.0556988
\(619\) −2943.78 −0.191148 −0.0955740 0.995422i \(-0.530469\pi\)
−0.0955740 + 0.995422i \(0.530469\pi\)
\(620\) 929.550 0.0602123
\(621\) 2785.64 0.180006
\(622\) 9634.87 0.621098
\(623\) 7287.47 0.468646
\(624\) 172.414 0.0110610
\(625\) 625.000 0.0400000
\(626\) −1376.34 −0.0878746
\(627\) 0 0
\(628\) −1782.78 −0.113281
\(629\) 1785.69 0.113196
\(630\) −1241.36 −0.0785034
\(631\) 15591.4 0.983653 0.491827 0.870693i \(-0.336329\pi\)
0.491827 + 0.870693i \(0.336329\pi\)
\(632\) 16449.0 1.03530
\(633\) −145.161 −0.00911474
\(634\) 14243.1 0.892216
\(635\) −5948.67 −0.371757
\(636\) −7727.34 −0.481775
\(637\) 1317.91 0.0819743
\(638\) 0 0
\(639\) −1578.39 −0.0977152
\(640\) −3730.54 −0.230410
\(641\) 22162.7 1.36564 0.682820 0.730586i \(-0.260753\pi\)
0.682820 + 0.730586i \(0.260753\pi\)
\(642\) 10959.8 0.673753
\(643\) −24199.0 −1.48416 −0.742081 0.670310i \(-0.766161\pi\)
−0.742081 + 0.670310i \(0.766161\pi\)
\(644\) 7253.82 0.443852
\(645\) 4573.91 0.279221
\(646\) −1993.04 −0.121386
\(647\) 23519.5 1.42913 0.714564 0.699570i \(-0.246625\pi\)
0.714564 + 0.699570i \(0.246625\pi\)
\(648\) −1874.26 −0.113623
\(649\) 0 0
\(650\) −522.586 −0.0315346
\(651\) 1806.70 0.108771
\(652\) −11712.0 −0.703496
\(653\) 10747.9 0.644099 0.322050 0.946723i \(-0.395628\pi\)
0.322050 + 0.946723i \(0.395628\pi\)
\(654\) 3521.03 0.210525
\(655\) 7007.38 0.418017
\(656\) 682.928 0.0406461
\(657\) 6914.58 0.410599
\(658\) −3119.22 −0.184802
\(659\) 29529.6 1.74554 0.872769 0.488133i \(-0.162322\pi\)
0.872769 + 0.488133i \(0.162322\pi\)
\(660\) 0 0
\(661\) 33178.7 1.95235 0.976175 0.216984i \(-0.0696218\pi\)
0.976175 + 0.216984i \(0.0696218\pi\)
\(662\) 15157.3 0.889887
\(663\) −410.683 −0.0240567
\(664\) −20876.3 −1.22012
\(665\) 6873.02 0.400788
\(666\) 2454.04 0.142781
\(667\) −10248.3 −0.594927
\(668\) 19943.0 1.15512
\(669\) 7089.80 0.409727
\(670\) 8660.43 0.499376
\(671\) 0 0
\(672\) −7964.98 −0.457226
\(673\) 18591.2 1.06484 0.532419 0.846481i \(-0.321283\pi\)
0.532419 + 0.846481i \(0.321283\pi\)
\(674\) 1692.55 0.0967281
\(675\) 675.000 0.0384900
\(676\) 9626.08 0.547683
\(677\) −5323.51 −0.302214 −0.151107 0.988517i \(-0.548284\pi\)
−0.151107 + 0.988517i \(0.548284\pi\)
\(678\) 1247.66 0.0706727
\(679\) −10930.9 −0.617802
\(680\) −1384.95 −0.0781037
\(681\) −3374.86 −0.189904
\(682\) 0 0
\(683\) −19258.0 −1.07890 −0.539450 0.842018i \(-0.681368\pi\)
−0.539450 + 0.842018i \(0.681368\pi\)
\(684\) 3819.07 0.213488
\(685\) 4130.08 0.230368
\(686\) 12641.1 0.703555
\(687\) 5795.57 0.321856
\(688\) −1532.44 −0.0849180
\(689\) −6322.69 −0.349601
\(690\) 2828.82 0.156074
\(691\) −25193.1 −1.38696 −0.693481 0.720475i \(-0.743924\pi\)
−0.693481 + 0.720475i \(0.743924\pi\)
\(692\) 10315.8 0.566688
\(693\) 0 0
\(694\) 4986.20 0.272729
\(695\) 9601.56 0.524040
\(696\) 6895.37 0.375529
\(697\) −1626.71 −0.0884018
\(698\) 16171.3 0.876923
\(699\) −12847.1 −0.695166
\(700\) 1757.70 0.0949071
\(701\) 16986.7 0.915233 0.457617 0.889150i \(-0.348703\pi\)
0.457617 + 0.889150i \(0.348703\pi\)
\(702\) −564.392 −0.0303442
\(703\) −13587.2 −0.728949
\(704\) 0 0
\(705\) 1696.10 0.0906082
\(706\) −305.940 −0.0163090
\(707\) −1109.63 −0.0590268
\(708\) 3521.49 0.186929
\(709\) −27674.9 −1.46594 −0.732972 0.680259i \(-0.761867\pi\)
−0.732972 + 0.680259i \(0.761867\pi\)
\(710\) −1602.85 −0.0847239
\(711\) −6397.91 −0.337469
\(712\) 11173.4 0.588122
\(713\) −4117.10 −0.216251
\(714\) −990.669 −0.0519256
\(715\) 0 0
\(716\) −7317.24 −0.381925
\(717\) −13648.8 −0.710913
\(718\) −18753.0 −0.974729
\(719\) 34365.9 1.78252 0.891260 0.453494i \(-0.149822\pi\)
0.891260 + 0.453494i \(0.149822\pi\)
\(720\) −226.151 −0.0117058
\(721\) 2354.99 0.121643
\(722\) 2627.33 0.135428
\(723\) 17112.7 0.880261
\(724\) −13357.5 −0.685675
\(725\) −2483.31 −0.127211
\(726\) 0 0
\(727\) −10929.0 −0.557543 −0.278772 0.960357i \(-0.589927\pi\)
−0.278772 + 0.960357i \(0.589927\pi\)
\(728\) −3993.40 −0.203304
\(729\) 729.000 0.0370370
\(730\) 7021.76 0.356010
\(731\) 3650.21 0.184689
\(732\) 2952.94 0.149104
\(733\) 34482.0 1.73755 0.868774 0.495209i \(-0.164908\pi\)
0.868774 + 0.495209i \(0.164908\pi\)
\(734\) 162.761 0.00818478
\(735\) −1728.67 −0.0867526
\(736\) 18150.6 0.909020
\(737\) 0 0
\(738\) −2235.55 −0.111506
\(739\) −20970.7 −1.04387 −0.521935 0.852985i \(-0.674790\pi\)
−0.521935 + 0.852985i \(0.674790\pi\)
\(740\) −3474.79 −0.172616
\(741\) 3124.85 0.154918
\(742\) −15251.9 −0.754602
\(743\) 23789.1 1.17461 0.587306 0.809365i \(-0.300188\pi\)
0.587306 + 0.809365i \(0.300188\pi\)
\(744\) 2770.11 0.136501
\(745\) 5286.50 0.259976
\(746\) −17423.7 −0.855128
\(747\) 8119.92 0.397714
\(748\) 0 0
\(749\) −30162.2 −1.47143
\(750\) 685.463 0.0333728
\(751\) −16312.5 −0.792610 −0.396305 0.918119i \(-0.629708\pi\)
−0.396305 + 0.918119i \(0.629708\pi\)
\(752\) −568.258 −0.0275562
\(753\) 7056.33 0.341496
\(754\) 2076.39 0.100289
\(755\) 11218.7 0.540783
\(756\) 1898.32 0.0913244
\(757\) −13835.0 −0.664258 −0.332129 0.943234i \(-0.607767\pi\)
−0.332129 + 0.943234i \(0.607767\pi\)
\(758\) −16245.2 −0.778435
\(759\) 0 0
\(760\) 10538.0 0.502965
\(761\) −27153.6 −1.29345 −0.646727 0.762722i \(-0.723863\pi\)
−0.646727 + 0.762722i \(0.723863\pi\)
\(762\) −6524.15 −0.310164
\(763\) −9690.13 −0.459772
\(764\) 9699.55 0.459316
\(765\) 538.683 0.0254590
\(766\) 3768.55 0.177759
\(767\) 2881.36 0.135645
\(768\) −12774.1 −0.600192
\(769\) 4108.08 0.192641 0.0963207 0.995350i \(-0.469293\pi\)
0.0963207 + 0.995350i \(0.469293\pi\)
\(770\) 0 0
\(771\) −5205.02 −0.243131
\(772\) −11007.2 −0.513158
\(773\) −8280.36 −0.385283 −0.192642 0.981269i \(-0.561705\pi\)
−0.192642 + 0.981269i \(0.561705\pi\)
\(774\) 5016.40 0.232960
\(775\) −997.633 −0.0462401
\(776\) −16759.6 −0.775304
\(777\) −6753.70 −0.311825
\(778\) −14218.8 −0.655230
\(779\) 12377.5 0.569281
\(780\) 799.149 0.0366848
\(781\) 0 0
\(782\) 2257.53 0.103234
\(783\) −2681.98 −0.122409
\(784\) 579.172 0.0263836
\(785\) 1913.36 0.0869945
\(786\) 7685.28 0.348759
\(787\) −5701.06 −0.258222 −0.129111 0.991630i \(-0.541212\pi\)
−0.129111 + 0.991630i \(0.541212\pi\)
\(788\) 1083.72 0.0489924
\(789\) −11023.1 −0.497378
\(790\) −6497.08 −0.292602
\(791\) −3433.65 −0.154345
\(792\) 0 0
\(793\) 2416.17 0.108197
\(794\) 18257.6 0.816044
\(795\) 8293.32 0.369979
\(796\) −2623.87 −0.116835
\(797\) −12921.0 −0.574262 −0.287131 0.957891i \(-0.592702\pi\)
−0.287131 + 0.957891i \(0.592702\pi\)
\(798\) 7537.92 0.334385
\(799\) 1353.57 0.0599322
\(800\) 4398.14 0.194372
\(801\) −4345.96 −0.191706
\(802\) 11574.1 0.509596
\(803\) 0 0
\(804\) −13243.7 −0.580933
\(805\) −7785.11 −0.340856
\(806\) 834.158 0.0364540
\(807\) 18044.3 0.787101
\(808\) −1701.33 −0.0740751
\(809\) 19951.1 0.867048 0.433524 0.901142i \(-0.357270\pi\)
0.433524 + 0.901142i \(0.357270\pi\)
\(810\) 740.300 0.0321129
\(811\) −4604.36 −0.199360 −0.0996800 0.995020i \(-0.531782\pi\)
−0.0996800 + 0.995020i \(0.531782\pi\)
\(812\) −6983.90 −0.301831
\(813\) −9115.99 −0.393249
\(814\) 0 0
\(815\) 12569.9 0.540250
\(816\) −180.479 −0.00774270
\(817\) −27774.1 −1.18934
\(818\) 969.601 0.0414441
\(819\) 1553.25 0.0662698
\(820\) 3165.42 0.134806
\(821\) −6932.84 −0.294711 −0.147356 0.989084i \(-0.547076\pi\)
−0.147356 + 0.989084i \(0.547076\pi\)
\(822\) 4529.62 0.192200
\(823\) −23247.6 −0.984642 −0.492321 0.870414i \(-0.663851\pi\)
−0.492321 + 0.870414i \(0.663851\pi\)
\(824\) 3610.76 0.152654
\(825\) 0 0
\(826\) 6950.56 0.292786
\(827\) −30227.9 −1.27101 −0.635505 0.772097i \(-0.719208\pi\)
−0.635505 + 0.772097i \(0.719208\pi\)
\(828\) −4325.89 −0.181564
\(829\) 22612.6 0.947366 0.473683 0.880695i \(-0.342924\pi\)
0.473683 + 0.880695i \(0.342924\pi\)
\(830\) 8245.78 0.344838
\(831\) 15854.5 0.661838
\(832\) −4137.22 −0.172395
\(833\) −1379.57 −0.0573819
\(834\) 10530.4 0.437217
\(835\) −21403.7 −0.887073
\(836\) 0 0
\(837\) −1077.44 −0.0444945
\(838\) 1051.76 0.0433560
\(839\) −36758.8 −1.51258 −0.756290 0.654236i \(-0.772990\pi\)
−0.756290 + 0.654236i \(0.772990\pi\)
\(840\) 5238.05 0.215155
\(841\) −14522.0 −0.595434
\(842\) 25279.4 1.03466
\(843\) −20000.6 −0.817148
\(844\) 225.424 0.00919363
\(845\) −10331.1 −0.420593
\(846\) 1860.18 0.0755961
\(847\) 0 0
\(848\) −2778.58 −0.112520
\(849\) 7691.88 0.310936
\(850\) 547.033 0.0220742
\(851\) 15390.3 0.619945
\(852\) 2451.12 0.0985609
\(853\) 16399.4 0.658269 0.329135 0.944283i \(-0.393243\pi\)
0.329135 + 0.944283i \(0.393243\pi\)
\(854\) 5828.39 0.233540
\(855\) −4098.79 −0.163948
\(856\) −46246.0 −1.84656
\(857\) −9966.88 −0.397272 −0.198636 0.980073i \(-0.563651\pi\)
−0.198636 + 0.980073i \(0.563651\pi\)
\(858\) 0 0
\(859\) 1074.79 0.0426907 0.0213453 0.999772i \(-0.493205\pi\)
0.0213453 + 0.999772i \(0.493205\pi\)
\(860\) −7102.95 −0.281638
\(861\) 6152.40 0.243523
\(862\) −8698.77 −0.343714
\(863\) −23860.5 −0.941159 −0.470580 0.882357i \(-0.655955\pi\)
−0.470580 + 0.882357i \(0.655955\pi\)
\(864\) 4749.99 0.187035
\(865\) −11071.4 −0.435188
\(866\) −8254.84 −0.323916
\(867\) −14309.1 −0.560511
\(868\) −2805.67 −0.109713
\(869\) 0 0
\(870\) −2723.55 −0.106135
\(871\) −10836.3 −0.421555
\(872\) −14857.3 −0.576986
\(873\) 6518.73 0.252721
\(874\) −17177.4 −0.664798
\(875\) −1886.44 −0.0728839
\(876\) −10737.8 −0.414153
\(877\) 11183.2 0.430591 0.215295 0.976549i \(-0.430929\pi\)
0.215295 + 0.976549i \(0.430929\pi\)
\(878\) −25437.4 −0.977759
\(879\) 15049.3 0.577476
\(880\) 0 0
\(881\) 20779.5 0.794642 0.397321 0.917680i \(-0.369940\pi\)
0.397321 + 0.917680i \(0.369940\pi\)
\(882\) −1895.91 −0.0723793
\(883\) −6724.99 −0.256301 −0.128150 0.991755i \(-0.540904\pi\)
−0.128150 + 0.991755i \(0.540904\pi\)
\(884\) 637.760 0.0242649
\(885\) −3779.41 −0.143552
\(886\) 18816.8 0.713501
\(887\) −14957.5 −0.566205 −0.283102 0.959090i \(-0.591364\pi\)
−0.283102 + 0.959090i \(0.591364\pi\)
\(888\) −10355.1 −0.391321
\(889\) 17954.9 0.677378
\(890\) −4413.32 −0.166219
\(891\) 0 0
\(892\) −11009.9 −0.413274
\(893\) −10299.2 −0.385946
\(894\) 5797.92 0.216903
\(895\) 7853.18 0.293299
\(896\) 11259.9 0.419831
\(897\) −3539.54 −0.131752
\(898\) −7661.50 −0.284708
\(899\) 3963.90 0.147056
\(900\) −1048.22 −0.0388231
\(901\) 6618.47 0.244721
\(902\) 0 0
\(903\) −13805.5 −0.508769
\(904\) −5264.62 −0.193693
\(905\) 14335.9 0.526564
\(906\) 12304.0 0.451186
\(907\) −52411.3 −1.91873 −0.959366 0.282164i \(-0.908948\pi\)
−0.959366 + 0.282164i \(0.908948\pi\)
\(908\) 5240.91 0.191548
\(909\) 661.740 0.0241458
\(910\) 1577.33 0.0574592
\(911\) −19174.4 −0.697339 −0.348670 0.937246i \(-0.613366\pi\)
−0.348670 + 0.937246i \(0.613366\pi\)
\(912\) 1373.25 0.0498607
\(913\) 0 0
\(914\) −15161.6 −0.548689
\(915\) −3169.23 −0.114504
\(916\) −9000.09 −0.324641
\(917\) −21150.4 −0.761667
\(918\) 590.795 0.0212409
\(919\) −18396.3 −0.660325 −0.330162 0.943924i \(-0.607104\pi\)
−0.330162 + 0.943924i \(0.607104\pi\)
\(920\) −11936.5 −0.427753
\(921\) −21672.8 −0.775398
\(922\) −3251.37 −0.116137
\(923\) 2005.56 0.0715209
\(924\) 0 0
\(925\) 3729.29 0.132560
\(926\) 8508.44 0.301949
\(927\) −1404.42 −0.0497597
\(928\) −17475.2 −0.618157
\(929\) −17244.1 −0.608998 −0.304499 0.952513i \(-0.598489\pi\)
−0.304499 + 0.952513i \(0.598489\pi\)
\(930\) −1094.14 −0.0385790
\(931\) 10497.0 0.369523
\(932\) 19950.5 0.701182
\(933\) 15813.0 0.554871
\(934\) −7103.86 −0.248871
\(935\) 0 0
\(936\) 2381.51 0.0831645
\(937\) −8269.68 −0.288323 −0.144162 0.989554i \(-0.546048\pi\)
−0.144162 + 0.989554i \(0.546048\pi\)
\(938\) −26139.9 −0.909912
\(939\) −2258.88 −0.0785046
\(940\) −2633.92 −0.0913924
\(941\) −55133.8 −1.91000 −0.955000 0.296606i \(-0.904145\pi\)
−0.955000 + 0.296606i \(0.904145\pi\)
\(942\) 2098.46 0.0725812
\(943\) −14020.1 −0.484153
\(944\) 1266.25 0.0436577
\(945\) −2037.36 −0.0701326
\(946\) 0 0
\(947\) −49363.9 −1.69389 −0.846944 0.531682i \(-0.821560\pi\)
−0.846944 + 0.531682i \(0.821560\pi\)
\(948\) 9935.48 0.340390
\(949\) −8785.94 −0.300531
\(950\) −4162.33 −0.142151
\(951\) 23376.1 0.797080
\(952\) 4180.22 0.142313
\(953\) 44934.5 1.52736 0.763678 0.645597i \(-0.223391\pi\)
0.763678 + 0.645597i \(0.223391\pi\)
\(954\) 9095.62 0.308681
\(955\) −10410.0 −0.352732
\(956\) 21195.6 0.717066
\(957\) 0 0
\(958\) 21229.8 0.715975
\(959\) −12465.9 −0.419753
\(960\) 5426.69 0.182443
\(961\) −28198.6 −0.946546
\(962\) −3118.20 −0.104506
\(963\) 17987.5 0.601911
\(964\) −26574.8 −0.887879
\(965\) 11813.4 0.394080
\(966\) −8538.25 −0.284383
\(967\) 258.493 0.00859625 0.00429813 0.999991i \(-0.498632\pi\)
0.00429813 + 0.999991i \(0.498632\pi\)
\(968\) 0 0
\(969\) −3271.04 −0.108443
\(970\) 6619.77 0.219122
\(971\) 47477.2 1.56912 0.784561 0.620052i \(-0.212889\pi\)
0.784561 + 0.620052i \(0.212889\pi\)
\(972\) −1132.08 −0.0373576
\(973\) −28980.5 −0.954853
\(974\) −32839.2 −1.08032
\(975\) −857.681 −0.0281721
\(976\) 1061.81 0.0348235
\(977\) 27864.9 0.912463 0.456231 0.889861i \(-0.349199\pi\)
0.456231 + 0.889861i \(0.349199\pi\)
\(978\) 13785.9 0.450740
\(979\) 0 0
\(980\) 2684.50 0.0875034
\(981\) 5778.81 0.188077
\(982\) −28322.3 −0.920368
\(983\) 8177.20 0.265323 0.132661 0.991161i \(-0.457648\pi\)
0.132661 + 0.991161i \(0.457648\pi\)
\(984\) 9433.12 0.305606
\(985\) −1163.10 −0.0376237
\(986\) −2173.53 −0.0702021
\(987\) −5119.35 −0.165097
\(988\) −4852.66 −0.156259
\(989\) 31459.9 1.01149
\(990\) 0 0
\(991\) 51787.1 1.66001 0.830005 0.557756i \(-0.188337\pi\)
0.830005 + 0.557756i \(0.188337\pi\)
\(992\) −7020.37 −0.224695
\(993\) 24876.6 0.794999
\(994\) 4837.91 0.154375
\(995\) 2816.05 0.0897235
\(996\) −12609.6 −0.401156
\(997\) −22176.1 −0.704439 −0.352219 0.935917i \(-0.614573\pi\)
−0.352219 + 0.935917i \(0.614573\pi\)
\(998\) −27928.7 −0.885839
\(999\) 4027.64 0.127556
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 1815.4.a.bb.1.4 yes 6
11.10 odd 2 inner 1815.4.a.bb.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.bb.1.3 6 11.10 odd 2 inner
1815.4.a.bb.1.4 yes 6 1.1 even 1 trivial