Properties

Label 1617.4.a.be.1.9
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.06618\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.06618 q^{2} -3.00000 q^{3} -6.86326 q^{4} +3.08635 q^{5} -3.19854 q^{6} -15.8469 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.06618 q^{2} -3.00000 q^{3} -6.86326 q^{4} +3.08635 q^{5} -3.19854 q^{6} -15.8469 q^{8} +9.00000 q^{9} +3.29060 q^{10} +11.0000 q^{11} +20.5898 q^{12} -46.1874 q^{13} -9.25904 q^{15} +38.0105 q^{16} -8.33960 q^{17} +9.59561 q^{18} +117.065 q^{19} -21.1824 q^{20} +11.7280 q^{22} +22.7662 q^{23} +47.5407 q^{24} -115.474 q^{25} -49.2440 q^{26} -27.0000 q^{27} -87.7941 q^{29} -9.87179 q^{30} -94.8572 q^{31} +167.301 q^{32} -33.0000 q^{33} -8.89150 q^{34} -61.7694 q^{36} +86.0779 q^{37} +124.812 q^{38} +138.562 q^{39} -48.9090 q^{40} +223.293 q^{41} +255.054 q^{43} -75.4959 q^{44} +27.7771 q^{45} +24.2728 q^{46} +168.029 q^{47} -114.031 q^{48} -123.116 q^{50} +25.0188 q^{51} +316.996 q^{52} +326.809 q^{53} -28.7868 q^{54} +33.9498 q^{55} -351.194 q^{57} -93.6042 q^{58} +106.331 q^{59} +63.5473 q^{60} -45.3942 q^{61} -101.135 q^{62} -125.711 q^{64} -142.550 q^{65} -35.1839 q^{66} +618.307 q^{67} +57.2369 q^{68} -68.2985 q^{69} +91.0697 q^{71} -142.622 q^{72} +277.770 q^{73} +91.7744 q^{74} +346.423 q^{75} -803.447 q^{76} +147.732 q^{78} -1199.82 q^{79} +117.314 q^{80} +81.0000 q^{81} +238.070 q^{82} -917.325 q^{83} -25.7389 q^{85} +271.933 q^{86} +263.382 q^{87} -174.316 q^{88} -1619.11 q^{89} +29.6154 q^{90} -156.250 q^{92} +284.572 q^{93} +179.149 q^{94} +361.303 q^{95} -501.903 q^{96} -969.940 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.06618 0.376951 0.188475 0.982078i \(-0.439645\pi\)
0.188475 + 0.982078i \(0.439645\pi\)
\(3\) −3.00000 −0.577350
\(4\) −6.86326 −0.857908
\(5\) 3.08635 0.276051 0.138026 0.990429i \(-0.455924\pi\)
0.138026 + 0.990429i \(0.455924\pi\)
\(6\) −3.19854 −0.217633
\(7\) 0 0
\(8\) −15.8469 −0.700340
\(9\) 9.00000 0.333333
\(10\) 3.29060 0.104058
\(11\) 11.0000 0.301511
\(12\) 20.5898 0.495313
\(13\) −46.1874 −0.985390 −0.492695 0.870202i \(-0.663988\pi\)
−0.492695 + 0.870202i \(0.663988\pi\)
\(14\) 0 0
\(15\) −9.25904 −0.159378
\(16\) 38.0105 0.593914
\(17\) −8.33960 −0.118979 −0.0594897 0.998229i \(-0.518947\pi\)
−0.0594897 + 0.998229i \(0.518947\pi\)
\(18\) 9.59561 0.125650
\(19\) 117.065 1.41350 0.706750 0.707463i \(-0.250160\pi\)
0.706750 + 0.707463i \(0.250160\pi\)
\(20\) −21.1824 −0.236827
\(21\) 0 0
\(22\) 11.7280 0.113655
\(23\) 22.7662 0.206395 0.103197 0.994661i \(-0.467093\pi\)
0.103197 + 0.994661i \(0.467093\pi\)
\(24\) 47.5407 0.404342
\(25\) −115.474 −0.923796
\(26\) −49.2440 −0.371444
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −87.7941 −0.562171 −0.281086 0.959683i \(-0.590694\pi\)
−0.281086 + 0.959683i \(0.590694\pi\)
\(30\) −9.87179 −0.0600778
\(31\) −94.8572 −0.549576 −0.274788 0.961505i \(-0.588608\pi\)
−0.274788 + 0.961505i \(0.588608\pi\)
\(32\) 167.301 0.924217
\(33\) −33.0000 −0.174078
\(34\) −8.89150 −0.0448494
\(35\) 0 0
\(36\) −61.7694 −0.285969
\(37\) 86.0779 0.382463 0.191231 0.981545i \(-0.438752\pi\)
0.191231 + 0.981545i \(0.438752\pi\)
\(38\) 124.812 0.532820
\(39\) 138.562 0.568915
\(40\) −48.9090 −0.193330
\(41\) 223.293 0.850549 0.425275 0.905064i \(-0.360177\pi\)
0.425275 + 0.905064i \(0.360177\pi\)
\(42\) 0 0
\(43\) 255.054 0.904543 0.452271 0.891880i \(-0.350614\pi\)
0.452271 + 0.891880i \(0.350614\pi\)
\(44\) −75.4959 −0.258669
\(45\) 27.7771 0.0920171
\(46\) 24.2728 0.0778007
\(47\) 168.029 0.521481 0.260740 0.965409i \(-0.416033\pi\)
0.260740 + 0.965409i \(0.416033\pi\)
\(48\) −114.031 −0.342896
\(49\) 0 0
\(50\) −123.116 −0.348226
\(51\) 25.0188 0.0686928
\(52\) 316.996 0.845374
\(53\) 326.809 0.846994 0.423497 0.905897i \(-0.360802\pi\)
0.423497 + 0.905897i \(0.360802\pi\)
\(54\) −28.7868 −0.0725442
\(55\) 33.9498 0.0832326
\(56\) 0 0
\(57\) −351.194 −0.816085
\(58\) −93.6042 −0.211911
\(59\) 106.331 0.234630 0.117315 0.993095i \(-0.462571\pi\)
0.117315 + 0.993095i \(0.462571\pi\)
\(60\) 63.5473 0.136732
\(61\) −45.3942 −0.0952809 −0.0476404 0.998865i \(-0.515170\pi\)
−0.0476404 + 0.998865i \(0.515170\pi\)
\(62\) −101.135 −0.207163
\(63\) 0 0
\(64\) −125.711 −0.245530
\(65\) −142.550 −0.272018
\(66\) −35.1839 −0.0656187
\(67\) 618.307 1.12744 0.563718 0.825967i \(-0.309370\pi\)
0.563718 + 0.825967i \(0.309370\pi\)
\(68\) 57.2369 0.102073
\(69\) −68.2985 −0.119162
\(70\) 0 0
\(71\) 91.0697 0.152225 0.0761126 0.997099i \(-0.475749\pi\)
0.0761126 + 0.997099i \(0.475749\pi\)
\(72\) −142.622 −0.233447
\(73\) 277.770 0.445350 0.222675 0.974893i \(-0.428521\pi\)
0.222675 + 0.974893i \(0.428521\pi\)
\(74\) 91.7744 0.144170
\(75\) 346.423 0.533354
\(76\) −803.447 −1.21265
\(77\) 0 0
\(78\) 147.732 0.214453
\(79\) −1199.82 −1.70874 −0.854370 0.519665i \(-0.826057\pi\)
−0.854370 + 0.519665i \(0.826057\pi\)
\(80\) 117.314 0.163951
\(81\) 81.0000 0.111111
\(82\) 238.070 0.320615
\(83\) −917.325 −1.21313 −0.606564 0.795035i \(-0.707452\pi\)
−0.606564 + 0.795035i \(0.707452\pi\)
\(84\) 0 0
\(85\) −25.7389 −0.0328444
\(86\) 271.933 0.340968
\(87\) 263.382 0.324570
\(88\) −174.316 −0.211161
\(89\) −1619.11 −1.92837 −0.964187 0.265223i \(-0.914554\pi\)
−0.964187 + 0.265223i \(0.914554\pi\)
\(90\) 29.6154 0.0346859
\(91\) 0 0
\(92\) −156.250 −0.177068
\(93\) 284.572 0.317298
\(94\) 179.149 0.196573
\(95\) 361.303 0.390199
\(96\) −501.903 −0.533597
\(97\) −969.940 −1.01528 −0.507642 0.861568i \(-0.669483\pi\)
−0.507642 + 0.861568i \(0.669483\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 792.532 0.792532
\(101\) −1281.92 −1.26293 −0.631466 0.775404i \(-0.717546\pi\)
−0.631466 + 0.775404i \(0.717546\pi\)
\(102\) 26.6745 0.0258938
\(103\) 423.490 0.405124 0.202562 0.979269i \(-0.435073\pi\)
0.202562 + 0.979269i \(0.435073\pi\)
\(104\) 731.926 0.690108
\(105\) 0 0
\(106\) 348.437 0.319275
\(107\) 314.374 0.284034 0.142017 0.989864i \(-0.454641\pi\)
0.142017 + 0.989864i \(0.454641\pi\)
\(108\) 185.308 0.165104
\(109\) 1183.49 1.03998 0.519991 0.854172i \(-0.325935\pi\)
0.519991 + 0.854172i \(0.325935\pi\)
\(110\) 36.1966 0.0313746
\(111\) −258.234 −0.220815
\(112\) 0 0
\(113\) 579.892 0.482758 0.241379 0.970431i \(-0.422400\pi\)
0.241379 + 0.970431i \(0.422400\pi\)
\(114\) −374.436 −0.307624
\(115\) 70.2644 0.0569755
\(116\) 602.554 0.482291
\(117\) −415.686 −0.328463
\(118\) 113.368 0.0884439
\(119\) 0 0
\(120\) 146.727 0.111619
\(121\) 121.000 0.0909091
\(122\) −48.3983 −0.0359162
\(123\) −669.879 −0.491065
\(124\) 651.030 0.471486
\(125\) −742.188 −0.531066
\(126\) 0 0
\(127\) 681.482 0.476156 0.238078 0.971246i \(-0.423483\pi\)
0.238078 + 0.971246i \(0.423483\pi\)
\(128\) −1472.44 −1.01677
\(129\) −765.161 −0.522238
\(130\) −151.984 −0.102538
\(131\) −1196.44 −0.797966 −0.398983 0.916958i \(-0.630637\pi\)
−0.398983 + 0.916958i \(0.630637\pi\)
\(132\) 226.488 0.149343
\(133\) 0 0
\(134\) 659.226 0.424988
\(135\) −83.3314 −0.0531261
\(136\) 132.157 0.0833261
\(137\) 435.230 0.271418 0.135709 0.990749i \(-0.456669\pi\)
0.135709 + 0.990749i \(0.456669\pi\)
\(138\) −72.8184 −0.0449182
\(139\) −2222.45 −1.35616 −0.678080 0.734988i \(-0.737188\pi\)
−0.678080 + 0.734988i \(0.737188\pi\)
\(140\) 0 0
\(141\) −504.088 −0.301077
\(142\) 97.0966 0.0573814
\(143\) −508.061 −0.297106
\(144\) 342.094 0.197971
\(145\) −270.963 −0.155188
\(146\) 296.152 0.167875
\(147\) 0 0
\(148\) −590.775 −0.328118
\(149\) −2140.98 −1.17716 −0.588578 0.808441i \(-0.700312\pi\)
−0.588578 + 0.808441i \(0.700312\pi\)
\(150\) 369.349 0.201048
\(151\) 2194.82 1.18286 0.591430 0.806356i \(-0.298563\pi\)
0.591430 + 0.806356i \(0.298563\pi\)
\(152\) −1855.11 −0.989931
\(153\) −75.0564 −0.0396598
\(154\) 0 0
\(155\) −292.762 −0.151711
\(156\) −950.988 −0.488077
\(157\) −1695.71 −0.861989 −0.430994 0.902355i \(-0.641837\pi\)
−0.430994 + 0.902355i \(0.641837\pi\)
\(158\) −1279.22 −0.644111
\(159\) −980.427 −0.489012
\(160\) 516.349 0.255131
\(161\) 0 0
\(162\) 86.3604 0.0418834
\(163\) 704.214 0.338394 0.169197 0.985582i \(-0.445883\pi\)
0.169197 + 0.985582i \(0.445883\pi\)
\(164\) −1532.52 −0.729693
\(165\) −101.849 −0.0480544
\(166\) −978.033 −0.457289
\(167\) 723.607 0.335296 0.167648 0.985847i \(-0.446383\pi\)
0.167648 + 0.985847i \(0.446383\pi\)
\(168\) 0 0
\(169\) −63.7274 −0.0290065
\(170\) −27.4423 −0.0123807
\(171\) 1053.58 0.471167
\(172\) −1750.50 −0.776014
\(173\) −29.2038 −0.0128343 −0.00641713 0.999979i \(-0.502043\pi\)
−0.00641713 + 0.999979i \(0.502043\pi\)
\(174\) 280.813 0.122347
\(175\) 0 0
\(176\) 418.115 0.179072
\(177\) −318.994 −0.135464
\(178\) −1726.26 −0.726902
\(179\) −2433.91 −1.01631 −0.508153 0.861267i \(-0.669672\pi\)
−0.508153 + 0.861267i \(0.669672\pi\)
\(180\) −190.642 −0.0789422
\(181\) −4357.95 −1.78964 −0.894818 0.446432i \(-0.852695\pi\)
−0.894818 + 0.446432i \(0.852695\pi\)
\(182\) 0 0
\(183\) 136.183 0.0550104
\(184\) −360.773 −0.144546
\(185\) 265.666 0.105579
\(186\) 303.404 0.119606
\(187\) −91.7356 −0.0358737
\(188\) −1153.23 −0.447382
\(189\) 0 0
\(190\) 385.213 0.147086
\(191\) 784.335 0.297134 0.148567 0.988902i \(-0.452534\pi\)
0.148567 + 0.988902i \(0.452534\pi\)
\(192\) 377.134 0.141757
\(193\) −1417.21 −0.528567 −0.264283 0.964445i \(-0.585135\pi\)
−0.264283 + 0.964445i \(0.585135\pi\)
\(194\) −1034.13 −0.382712
\(195\) 427.651 0.157050
\(196\) 0 0
\(197\) −829.506 −0.299999 −0.150000 0.988686i \(-0.547927\pi\)
−0.150000 + 0.988686i \(0.547927\pi\)
\(198\) 105.552 0.0378850
\(199\) −1778.13 −0.633408 −0.316704 0.948524i \(-0.602576\pi\)
−0.316704 + 0.948524i \(0.602576\pi\)
\(200\) 1829.91 0.646971
\(201\) −1854.92 −0.650926
\(202\) −1366.76 −0.476063
\(203\) 0 0
\(204\) −171.711 −0.0589321
\(205\) 689.160 0.234795
\(206\) 451.516 0.152712
\(207\) 204.896 0.0687982
\(208\) −1755.60 −0.585237
\(209\) 1287.71 0.426186
\(210\) 0 0
\(211\) 3183.79 1.03877 0.519387 0.854539i \(-0.326160\pi\)
0.519387 + 0.854539i \(0.326160\pi\)
\(212\) −2242.98 −0.726643
\(213\) −273.209 −0.0878872
\(214\) 335.178 0.107067
\(215\) 787.185 0.249700
\(216\) 427.866 0.134781
\(217\) 0 0
\(218\) 1261.81 0.392022
\(219\) −833.310 −0.257123
\(220\) −233.007 −0.0714059
\(221\) 385.184 0.117241
\(222\) −275.323 −0.0832364
\(223\) −3608.27 −1.08353 −0.541766 0.840529i \(-0.682244\pi\)
−0.541766 + 0.840529i \(0.682244\pi\)
\(224\) 0 0
\(225\) −1039.27 −0.307932
\(226\) 618.269 0.181976
\(227\) −3083.62 −0.901616 −0.450808 0.892621i \(-0.648864\pi\)
−0.450808 + 0.892621i \(0.648864\pi\)
\(228\) 2410.34 0.700126
\(229\) 1173.29 0.338572 0.169286 0.985567i \(-0.445854\pi\)
0.169286 + 0.985567i \(0.445854\pi\)
\(230\) 74.9143 0.0214770
\(231\) 0 0
\(232\) 1391.26 0.393711
\(233\) −5573.88 −1.56720 −0.783599 0.621267i \(-0.786618\pi\)
−0.783599 + 0.621267i \(0.786618\pi\)
\(234\) −443.196 −0.123815
\(235\) 518.597 0.143955
\(236\) −729.780 −0.201291
\(237\) 3599.46 0.986542
\(238\) 0 0
\(239\) 3069.31 0.830698 0.415349 0.909662i \(-0.363659\pi\)
0.415349 + 0.909662i \(0.363659\pi\)
\(240\) −351.941 −0.0946570
\(241\) 836.491 0.223581 0.111791 0.993732i \(-0.464341\pi\)
0.111791 + 0.993732i \(0.464341\pi\)
\(242\) 129.008 0.0342683
\(243\) −243.000 −0.0641500
\(244\) 311.552 0.0817422
\(245\) 0 0
\(246\) −714.211 −0.185107
\(247\) −5406.92 −1.39285
\(248\) 1503.19 0.384890
\(249\) 2751.98 0.700399
\(250\) −791.305 −0.200186
\(251\) −5818.20 −1.46311 −0.731557 0.681781i \(-0.761206\pi\)
−0.731557 + 0.681781i \(0.761206\pi\)
\(252\) 0 0
\(253\) 250.428 0.0622303
\(254\) 726.582 0.179487
\(255\) 77.2167 0.0189627
\(256\) −564.193 −0.137743
\(257\) 4008.55 0.972944 0.486472 0.873696i \(-0.338284\pi\)
0.486472 + 0.873696i \(0.338284\pi\)
\(258\) −815.798 −0.196858
\(259\) 0 0
\(260\) 978.360 0.233367
\(261\) −790.147 −0.187390
\(262\) −1275.62 −0.300794
\(263\) 385.763 0.0904455 0.0452227 0.998977i \(-0.485600\pi\)
0.0452227 + 0.998977i \(0.485600\pi\)
\(264\) 522.947 0.121914
\(265\) 1008.65 0.233814
\(266\) 0 0
\(267\) 4857.33 1.11335
\(268\) −4243.61 −0.967237
\(269\) −6666.79 −1.51108 −0.755542 0.655100i \(-0.772626\pi\)
−0.755542 + 0.655100i \(0.772626\pi\)
\(270\) −88.8461 −0.0200259
\(271\) 2900.60 0.650181 0.325090 0.945683i \(-0.394605\pi\)
0.325090 + 0.945683i \(0.394605\pi\)
\(272\) −316.992 −0.0706636
\(273\) 0 0
\(274\) 464.033 0.102311
\(275\) −1270.22 −0.278535
\(276\) 468.751 0.102230
\(277\) −2776.14 −0.602174 −0.301087 0.953597i \(-0.597350\pi\)
−0.301087 + 0.953597i \(0.597350\pi\)
\(278\) −2369.53 −0.511206
\(279\) −853.715 −0.183192
\(280\) 0 0
\(281\) −991.245 −0.210437 −0.105218 0.994449i \(-0.533554\pi\)
−0.105218 + 0.994449i \(0.533554\pi\)
\(282\) −537.447 −0.113491
\(283\) −538.248 −0.113058 −0.0565292 0.998401i \(-0.518003\pi\)
−0.0565292 + 0.998401i \(0.518003\pi\)
\(284\) −625.035 −0.130595
\(285\) −1083.91 −0.225281
\(286\) −541.684 −0.111994
\(287\) 0 0
\(288\) 1505.71 0.308072
\(289\) −4843.45 −0.985844
\(290\) −288.895 −0.0584983
\(291\) 2909.82 0.586174
\(292\) −1906.41 −0.382069
\(293\) −5819.89 −1.16042 −0.580208 0.814469i \(-0.697029\pi\)
−0.580208 + 0.814469i \(0.697029\pi\)
\(294\) 0 0
\(295\) 328.175 0.0647698
\(296\) −1364.07 −0.267854
\(297\) −297.000 −0.0580259
\(298\) −2282.67 −0.443730
\(299\) −1051.51 −0.203379
\(300\) −2377.59 −0.457568
\(301\) 0 0
\(302\) 2340.07 0.445880
\(303\) 3845.77 0.729154
\(304\) 4449.69 0.839498
\(305\) −140.102 −0.0263024
\(306\) −80.0235 −0.0149498
\(307\) −4664.46 −0.867149 −0.433574 0.901118i \(-0.642748\pi\)
−0.433574 + 0.901118i \(0.642748\pi\)
\(308\) 0 0
\(309\) −1270.47 −0.233898
\(310\) −312.137 −0.0571877
\(311\) 7111.09 1.29657 0.648284 0.761398i \(-0.275487\pi\)
0.648284 + 0.761398i \(0.275487\pi\)
\(312\) −2195.78 −0.398434
\(313\) 4281.09 0.773103 0.386552 0.922268i \(-0.373666\pi\)
0.386552 + 0.922268i \(0.373666\pi\)
\(314\) −1807.93 −0.324928
\(315\) 0 0
\(316\) 8234.69 1.46594
\(317\) −9559.86 −1.69380 −0.846901 0.531750i \(-0.821535\pi\)
−0.846901 + 0.531750i \(0.821535\pi\)
\(318\) −1045.31 −0.184334
\(319\) −965.736 −0.169501
\(320\) −387.988 −0.0677788
\(321\) −943.121 −0.163987
\(322\) 0 0
\(323\) −976.274 −0.168178
\(324\) −555.924 −0.0953231
\(325\) 5333.46 0.910299
\(326\) 750.817 0.127558
\(327\) −3550.48 −0.600433
\(328\) −3538.50 −0.595674
\(329\) 0 0
\(330\) −108.590 −0.0181141
\(331\) −2546.72 −0.422902 −0.211451 0.977389i \(-0.567819\pi\)
−0.211451 + 0.977389i \(0.567819\pi\)
\(332\) 6295.85 1.04075
\(333\) 774.701 0.127488
\(334\) 771.494 0.126390
\(335\) 1908.31 0.311230
\(336\) 0 0
\(337\) −3782.63 −0.611433 −0.305716 0.952123i \(-0.598896\pi\)
−0.305716 + 0.952123i \(0.598896\pi\)
\(338\) −67.9447 −0.0109340
\(339\) −1739.68 −0.278721
\(340\) 176.653 0.0281775
\(341\) −1043.43 −0.165703
\(342\) 1123.31 0.177607
\(343\) 0 0
\(344\) −4041.81 −0.633488
\(345\) −210.793 −0.0328948
\(346\) −31.1365 −0.00483788
\(347\) 842.616 0.130357 0.0651787 0.997874i \(-0.479238\pi\)
0.0651787 + 0.997874i \(0.479238\pi\)
\(348\) −1807.66 −0.278451
\(349\) −3487.12 −0.534846 −0.267423 0.963579i \(-0.586172\pi\)
−0.267423 + 0.963579i \(0.586172\pi\)
\(350\) 0 0
\(351\) 1247.06 0.189638
\(352\) 1840.31 0.278662
\(353\) −9368.40 −1.41255 −0.706274 0.707938i \(-0.749625\pi\)
−0.706274 + 0.707938i \(0.749625\pi\)
\(354\) −340.104 −0.0510631
\(355\) 281.073 0.0420220
\(356\) 11112.4 1.65437
\(357\) 0 0
\(358\) −2594.98 −0.383098
\(359\) −7387.35 −1.08604 −0.543021 0.839719i \(-0.682720\pi\)
−0.543021 + 0.839719i \(0.682720\pi\)
\(360\) −440.181 −0.0644433
\(361\) 6845.17 0.997984
\(362\) −4646.35 −0.674605
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 857.295 0.122939
\(366\) 145.195 0.0207362
\(367\) 3930.04 0.558982 0.279491 0.960148i \(-0.409834\pi\)
0.279491 + 0.960148i \(0.409834\pi\)
\(368\) 865.354 0.122581
\(369\) 2009.64 0.283516
\(370\) 283.248 0.0397982
\(371\) 0 0
\(372\) −1953.09 −0.272212
\(373\) −1665.12 −0.231144 −0.115572 0.993299i \(-0.536870\pi\)
−0.115572 + 0.993299i \(0.536870\pi\)
\(374\) −97.8065 −0.0135226
\(375\) 2226.56 0.306611
\(376\) −2662.74 −0.365214
\(377\) 4054.98 0.553958
\(378\) 0 0
\(379\) −5479.83 −0.742692 −0.371346 0.928495i \(-0.621104\pi\)
−0.371346 + 0.928495i \(0.621104\pi\)
\(380\) −2479.72 −0.334755
\(381\) −2044.45 −0.274909
\(382\) 836.241 0.112005
\(383\) −1284.59 −0.171383 −0.0856915 0.996322i \(-0.527310\pi\)
−0.0856915 + 0.996322i \(0.527310\pi\)
\(384\) 4417.32 0.587032
\(385\) 0 0
\(386\) −1511.00 −0.199244
\(387\) 2295.48 0.301514
\(388\) 6656.96 0.871020
\(389\) 5859.56 0.763731 0.381865 0.924218i \(-0.375282\pi\)
0.381865 + 0.924218i \(0.375282\pi\)
\(390\) 455.952 0.0592001
\(391\) −189.861 −0.0245567
\(392\) 0 0
\(393\) 3589.32 0.460706
\(394\) −884.401 −0.113085
\(395\) −3703.07 −0.471700
\(396\) −679.463 −0.0862230
\(397\) 5403.18 0.683068 0.341534 0.939869i \(-0.389054\pi\)
0.341534 + 0.939869i \(0.389054\pi\)
\(398\) −1895.80 −0.238764
\(399\) 0 0
\(400\) −4389.24 −0.548655
\(401\) 9167.87 1.14170 0.570850 0.821054i \(-0.306614\pi\)
0.570850 + 0.821054i \(0.306614\pi\)
\(402\) −1977.68 −0.245367
\(403\) 4381.21 0.541547
\(404\) 8798.18 1.08348
\(405\) 249.994 0.0306724
\(406\) 0 0
\(407\) 946.857 0.115317
\(408\) −396.470 −0.0481083
\(409\) 11072.5 1.33864 0.669318 0.742976i \(-0.266586\pi\)
0.669318 + 0.742976i \(0.266586\pi\)
\(410\) 734.768 0.0885063
\(411\) −1305.69 −0.156703
\(412\) −2906.53 −0.347559
\(413\) 0 0
\(414\) 218.455 0.0259336
\(415\) −2831.19 −0.334885
\(416\) −7727.20 −0.910714
\(417\) 6667.36 0.782979
\(418\) 1372.93 0.160651
\(419\) 6832.94 0.796685 0.398343 0.917237i \(-0.369586\pi\)
0.398343 + 0.917237i \(0.369586\pi\)
\(420\) 0 0
\(421\) 10202.7 1.18111 0.590555 0.806997i \(-0.298909\pi\)
0.590555 + 0.806997i \(0.298909\pi\)
\(422\) 3394.49 0.391567
\(423\) 1512.26 0.173827
\(424\) −5178.91 −0.593184
\(425\) 963.011 0.109913
\(426\) −291.290 −0.0331292
\(427\) 0 0
\(428\) −2157.63 −0.243675
\(429\) 1524.18 0.171534
\(430\) 839.279 0.0941248
\(431\) 14692.7 1.64204 0.821021 0.570898i \(-0.193405\pi\)
0.821021 + 0.570898i \(0.193405\pi\)
\(432\) −1026.28 −0.114299
\(433\) 2952.33 0.327667 0.163834 0.986488i \(-0.447614\pi\)
0.163834 + 0.986488i \(0.447614\pi\)
\(434\) 0 0
\(435\) 812.890 0.0895979
\(436\) −8122.62 −0.892208
\(437\) 2665.12 0.291739
\(438\) −888.457 −0.0969227
\(439\) −4114.95 −0.447371 −0.223685 0.974661i \(-0.571809\pi\)
−0.223685 + 0.974661i \(0.571809\pi\)
\(440\) −537.999 −0.0582911
\(441\) 0 0
\(442\) 410.675 0.0441942
\(443\) 7194.74 0.771631 0.385815 0.922576i \(-0.373920\pi\)
0.385815 + 0.922576i \(0.373920\pi\)
\(444\) 1772.33 0.189439
\(445\) −4997.14 −0.532330
\(446\) −3847.06 −0.408438
\(447\) 6422.95 0.679631
\(448\) 0 0
\(449\) 11551.2 1.21410 0.607052 0.794662i \(-0.292352\pi\)
0.607052 + 0.794662i \(0.292352\pi\)
\(450\) −1108.05 −0.116075
\(451\) 2456.22 0.256450
\(452\) −3979.95 −0.414162
\(453\) −6584.46 −0.682925
\(454\) −3287.69 −0.339865
\(455\) 0 0
\(456\) 5565.34 0.571537
\(457\) 12720.7 1.30207 0.651036 0.759047i \(-0.274335\pi\)
0.651036 + 0.759047i \(0.274335\pi\)
\(458\) 1250.93 0.127625
\(459\) 225.169 0.0228976
\(460\) −482.243 −0.0488798
\(461\) 3351.88 0.338639 0.169319 0.985561i \(-0.445843\pi\)
0.169319 + 0.985561i \(0.445843\pi\)
\(462\) 0 0
\(463\) −19431.2 −1.95042 −0.975210 0.221282i \(-0.928976\pi\)
−0.975210 + 0.221282i \(0.928976\pi\)
\(464\) −3337.10 −0.333881
\(465\) 878.287 0.0875905
\(466\) −5942.75 −0.590757
\(467\) 1252.67 0.124126 0.0620628 0.998072i \(-0.480232\pi\)
0.0620628 + 0.998072i \(0.480232\pi\)
\(468\) 2852.96 0.281791
\(469\) 0 0
\(470\) 552.917 0.0542641
\(471\) 5087.13 0.497670
\(472\) −1685.02 −0.164321
\(473\) 2805.59 0.272730
\(474\) 3837.67 0.371878
\(475\) −13518.0 −1.30579
\(476\) 0 0
\(477\) 2941.28 0.282331
\(478\) 3272.43 0.313133
\(479\) 4849.73 0.462609 0.231305 0.972881i \(-0.425701\pi\)
0.231305 + 0.972881i \(0.425701\pi\)
\(480\) −1549.05 −0.147300
\(481\) −3975.71 −0.376875
\(482\) 891.849 0.0842792
\(483\) 0 0
\(484\) −830.455 −0.0779916
\(485\) −2993.57 −0.280270
\(486\) −259.081 −0.0241814
\(487\) −10951.7 −1.01904 −0.509518 0.860460i \(-0.670176\pi\)
−0.509518 + 0.860460i \(0.670176\pi\)
\(488\) 719.357 0.0667290
\(489\) −2112.64 −0.195372
\(490\) 0 0
\(491\) 13521.0 1.24276 0.621381 0.783508i \(-0.286572\pi\)
0.621381 + 0.783508i \(0.286572\pi\)
\(492\) 4597.56 0.421288
\(493\) 732.168 0.0668868
\(494\) −5764.74 −0.525036
\(495\) 305.548 0.0277442
\(496\) −3605.57 −0.326401
\(497\) 0 0
\(498\) 2934.10 0.264016
\(499\) 16997.5 1.52488 0.762438 0.647062i \(-0.224002\pi\)
0.762438 + 0.647062i \(0.224002\pi\)
\(500\) 5093.83 0.455606
\(501\) −2170.82 −0.193583
\(502\) −6203.24 −0.551522
\(503\) −1869.30 −0.165702 −0.0828508 0.996562i \(-0.526402\pi\)
−0.0828508 + 0.996562i \(0.526402\pi\)
\(504\) 0 0
\(505\) −3956.46 −0.348634
\(506\) 267.001 0.0234578
\(507\) 191.182 0.0167469
\(508\) −4677.19 −0.408498
\(509\) 17919.5 1.56045 0.780225 0.625499i \(-0.215104\pi\)
0.780225 + 0.625499i \(0.215104\pi\)
\(510\) 82.3268 0.00714802
\(511\) 0 0
\(512\) 11178.0 0.964847
\(513\) −3160.75 −0.272028
\(514\) 4273.83 0.366752
\(515\) 1307.04 0.111835
\(516\) 5251.50 0.448032
\(517\) 1848.32 0.157232
\(518\) 0 0
\(519\) 87.6115 0.00740986
\(520\) 2258.98 0.190505
\(521\) −8035.81 −0.675730 −0.337865 0.941195i \(-0.609705\pi\)
−0.337865 + 0.941195i \(0.609705\pi\)
\(522\) −842.438 −0.0706370
\(523\) −738.030 −0.0617052 −0.0308526 0.999524i \(-0.509822\pi\)
−0.0308526 + 0.999524i \(0.509822\pi\)
\(524\) 8211.49 0.684581
\(525\) 0 0
\(526\) 411.292 0.0340935
\(527\) 791.071 0.0653883
\(528\) −1254.35 −0.103387
\(529\) −11648.7 −0.957401
\(530\) 1075.40 0.0881364
\(531\) 956.981 0.0782099
\(532\) 0 0
\(533\) −10313.3 −0.838123
\(534\) 5178.78 0.419677
\(535\) 970.267 0.0784080
\(536\) −9798.25 −0.789589
\(537\) 7301.72 0.586765
\(538\) −7107.99 −0.569605
\(539\) 0 0
\(540\) 571.925 0.0455773
\(541\) −7344.17 −0.583642 −0.291821 0.956473i \(-0.594261\pi\)
−0.291821 + 0.956473i \(0.594261\pi\)
\(542\) 3092.56 0.245086
\(543\) 13073.9 1.03325
\(544\) −1395.22 −0.109963
\(545\) 3652.67 0.287088
\(546\) 0 0
\(547\) −9751.41 −0.762231 −0.381115 0.924528i \(-0.624460\pi\)
−0.381115 + 0.924528i \(0.624460\pi\)
\(548\) −2987.10 −0.232851
\(549\) −408.548 −0.0317603
\(550\) −1354.28 −0.104994
\(551\) −10277.6 −0.794629
\(552\) 1082.32 0.0834539
\(553\) 0 0
\(554\) −2959.86 −0.226990
\(555\) −796.999 −0.0609563
\(556\) 15253.3 1.16346
\(557\) 13932.4 1.05985 0.529924 0.848045i \(-0.322220\pi\)
0.529924 + 0.848045i \(0.322220\pi\)
\(558\) −910.213 −0.0690544
\(559\) −11780.3 −0.891327
\(560\) 0 0
\(561\) 275.207 0.0207117
\(562\) −1056.84 −0.0793244
\(563\) −10544.3 −0.789323 −0.394662 0.918827i \(-0.629138\pi\)
−0.394662 + 0.918827i \(0.629138\pi\)
\(564\) 3459.69 0.258296
\(565\) 1789.75 0.133266
\(566\) −573.869 −0.0426175
\(567\) 0 0
\(568\) −1443.17 −0.106609
\(569\) 23637.0 1.74150 0.870750 0.491727i \(-0.163634\pi\)
0.870750 + 0.491727i \(0.163634\pi\)
\(570\) −1155.64 −0.0849200
\(571\) −8709.02 −0.638286 −0.319143 0.947707i \(-0.603395\pi\)
−0.319143 + 0.947707i \(0.603395\pi\)
\(572\) 3486.96 0.254890
\(573\) −2353.01 −0.171550
\(574\) 0 0
\(575\) −2628.91 −0.190666
\(576\) −1131.40 −0.0818432
\(577\) −9187.55 −0.662882 −0.331441 0.943476i \(-0.607535\pi\)
−0.331441 + 0.943476i \(0.607535\pi\)
\(578\) −5163.98 −0.371615
\(579\) 4251.64 0.305168
\(580\) 1859.69 0.133137
\(581\) 0 0
\(582\) 3102.39 0.220959
\(583\) 3594.90 0.255378
\(584\) −4401.79 −0.311896
\(585\) −1282.95 −0.0906728
\(586\) −6205.04 −0.437420
\(587\) 10830.9 0.761563 0.380781 0.924665i \(-0.375655\pi\)
0.380781 + 0.924665i \(0.375655\pi\)
\(588\) 0 0
\(589\) −11104.4 −0.776826
\(590\) 349.893 0.0244151
\(591\) 2488.52 0.173205
\(592\) 3271.86 0.227150
\(593\) −21942.5 −1.51951 −0.759755 0.650209i \(-0.774681\pi\)
−0.759755 + 0.650209i \(0.774681\pi\)
\(594\) −316.655 −0.0218729
\(595\) 0 0
\(596\) 14694.1 1.00989
\(597\) 5334.39 0.365698
\(598\) −1121.10 −0.0766640
\(599\) −2881.94 −0.196582 −0.0982911 0.995158i \(-0.531338\pi\)
−0.0982911 + 0.995158i \(0.531338\pi\)
\(600\) −5489.73 −0.373529
\(601\) −3010.52 −0.204329 −0.102164 0.994768i \(-0.532577\pi\)
−0.102164 + 0.994768i \(0.532577\pi\)
\(602\) 0 0
\(603\) 5564.77 0.375812
\(604\) −15063.6 −1.01479
\(605\) 373.448 0.0250956
\(606\) 4100.28 0.274855
\(607\) −21364.3 −1.42859 −0.714293 0.699847i \(-0.753252\pi\)
−0.714293 + 0.699847i \(0.753252\pi\)
\(608\) 19585.1 1.30638
\(609\) 0 0
\(610\) −149.374 −0.00991472
\(611\) −7760.83 −0.513862
\(612\) 515.132 0.0340245
\(613\) −27059.3 −1.78290 −0.891449 0.453122i \(-0.850310\pi\)
−0.891449 + 0.453122i \(0.850310\pi\)
\(614\) −4973.14 −0.326873
\(615\) −2067.48 −0.135559
\(616\) 0 0
\(617\) −839.530 −0.0547783 −0.0273891 0.999625i \(-0.508719\pi\)
−0.0273891 + 0.999625i \(0.508719\pi\)
\(618\) −1354.55 −0.0881682
\(619\) −14131.5 −0.917600 −0.458800 0.888540i \(-0.651721\pi\)
−0.458800 + 0.888540i \(0.651721\pi\)
\(620\) 2009.31 0.130154
\(621\) −614.687 −0.0397207
\(622\) 7581.69 0.488743
\(623\) 0 0
\(624\) 5266.81 0.337887
\(625\) 12143.7 0.777194
\(626\) 4564.40 0.291422
\(627\) −3863.14 −0.246059
\(628\) 11638.1 0.739507
\(629\) −717.856 −0.0455052
\(630\) 0 0
\(631\) −29899.4 −1.88633 −0.943166 0.332323i \(-0.892168\pi\)
−0.943166 + 0.332323i \(0.892168\pi\)
\(632\) 19013.4 1.19670
\(633\) −9551.38 −0.599737
\(634\) −10192.5 −0.638480
\(635\) 2103.29 0.131443
\(636\) 6728.93 0.419528
\(637\) 0 0
\(638\) −1029.65 −0.0638936
\(639\) 819.627 0.0507417
\(640\) −4544.46 −0.280681
\(641\) 30077.3 1.85333 0.926663 0.375893i \(-0.122664\pi\)
0.926663 + 0.375893i \(0.122664\pi\)
\(642\) −1005.54 −0.0618151
\(643\) −3493.32 −0.214250 −0.107125 0.994246i \(-0.534165\pi\)
−0.107125 + 0.994246i \(0.534165\pi\)
\(644\) 0 0
\(645\) −2361.55 −0.144165
\(646\) −1040.88 −0.0633947
\(647\) −16531.4 −1.00451 −0.502253 0.864721i \(-0.667495\pi\)
−0.502253 + 0.864721i \(0.667495\pi\)
\(648\) −1283.60 −0.0778156
\(649\) 1169.64 0.0707435
\(650\) 5686.42 0.343138
\(651\) 0 0
\(652\) −4833.20 −0.290311
\(653\) 25437.5 1.52442 0.762209 0.647331i \(-0.224115\pi\)
0.762209 + 0.647331i \(0.224115\pi\)
\(654\) −3785.44 −0.226334
\(655\) −3692.63 −0.220279
\(656\) 8487.48 0.505153
\(657\) 2499.93 0.148450
\(658\) 0 0
\(659\) −5251.80 −0.310442 −0.155221 0.987880i \(-0.549609\pi\)
−0.155221 + 0.987880i \(0.549609\pi\)
\(660\) 699.020 0.0412262
\(661\) 17431.8 1.02575 0.512873 0.858464i \(-0.328581\pi\)
0.512873 + 0.858464i \(0.328581\pi\)
\(662\) −2715.26 −0.159413
\(663\) −1155.55 −0.0676892
\(664\) 14536.8 0.849602
\(665\) 0 0
\(666\) 825.970 0.0480566
\(667\) −1998.74 −0.116029
\(668\) −4966.31 −0.287653
\(669\) 10824.8 0.625577
\(670\) 2034.60 0.117319
\(671\) −499.336 −0.0287283
\(672\) 0 0
\(673\) −28669.4 −1.64209 −0.821044 0.570865i \(-0.806608\pi\)
−0.821044 + 0.570865i \(0.806608\pi\)
\(674\) −4032.96 −0.230480
\(675\) 3117.81 0.177785
\(676\) 437.378 0.0248849
\(677\) 13530.6 0.768129 0.384065 0.923306i \(-0.374524\pi\)
0.384065 + 0.923306i \(0.374524\pi\)
\(678\) −1854.81 −0.105064
\(679\) 0 0
\(680\) 407.882 0.0230023
\(681\) 9250.85 0.520549
\(682\) −1112.48 −0.0624621
\(683\) −23637.4 −1.32425 −0.662123 0.749396i \(-0.730344\pi\)
−0.662123 + 0.749396i \(0.730344\pi\)
\(684\) −7231.02 −0.404218
\(685\) 1343.27 0.0749252
\(686\) 0 0
\(687\) −3519.86 −0.195475
\(688\) 9694.72 0.537221
\(689\) −15094.5 −0.834619
\(690\) −224.743 −0.0123997
\(691\) 30049.8 1.65434 0.827169 0.561953i \(-0.189950\pi\)
0.827169 + 0.561953i \(0.189950\pi\)
\(692\) 200.434 0.0110106
\(693\) 0 0
\(694\) 898.379 0.0491383
\(695\) −6859.27 −0.374370
\(696\) −4173.79 −0.227309
\(697\) −1862.18 −0.101198
\(698\) −3717.89 −0.201611
\(699\) 16721.6 0.904822
\(700\) 0 0
\(701\) 1348.42 0.0726520 0.0363260 0.999340i \(-0.488435\pi\)
0.0363260 + 0.999340i \(0.488435\pi\)
\(702\) 1329.59 0.0714844
\(703\) 10076.7 0.540611
\(704\) −1382.82 −0.0740300
\(705\) −1555.79 −0.0831127
\(706\) −9988.39 −0.532462
\(707\) 0 0
\(708\) 2189.34 0.116215
\(709\) 19139.3 1.01381 0.506904 0.862002i \(-0.330790\pi\)
0.506904 + 0.862002i \(0.330790\pi\)
\(710\) 299.674 0.0158402
\(711\) −10798.4 −0.569580
\(712\) 25657.9 1.35052
\(713\) −2159.54 −0.113430
\(714\) 0 0
\(715\) −1568.05 −0.0820166
\(716\) 16704.5 0.871897
\(717\) −9207.92 −0.479604
\(718\) −7876.23 −0.409385
\(719\) 22559.3 1.17012 0.585062 0.810988i \(-0.301070\pi\)
0.585062 + 0.810988i \(0.301070\pi\)
\(720\) 1055.82 0.0546503
\(721\) 0 0
\(722\) 7298.17 0.376191
\(723\) −2509.47 −0.129085
\(724\) 29909.8 1.53534
\(725\) 10138.0 0.519331
\(726\) −387.023 −0.0197848
\(727\) −12489.4 −0.637148 −0.318574 0.947898i \(-0.603204\pi\)
−0.318574 + 0.947898i \(0.603204\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 914.029 0.0463421
\(731\) −2127.05 −0.107622
\(732\) −934.657 −0.0471939
\(733\) 18260.9 0.920164 0.460082 0.887876i \(-0.347820\pi\)
0.460082 + 0.887876i \(0.347820\pi\)
\(734\) 4190.12 0.210709
\(735\) 0 0
\(736\) 3808.81 0.190753
\(737\) 6801.38 0.339935
\(738\) 2142.63 0.106872
\(739\) −14478.9 −0.720725 −0.360363 0.932812i \(-0.617347\pi\)
−0.360363 + 0.932812i \(0.617347\pi\)
\(740\) −1823.34 −0.0905774
\(741\) 16220.7 0.804162
\(742\) 0 0
\(743\) −33004.7 −1.62964 −0.814822 0.579711i \(-0.803165\pi\)
−0.814822 + 0.579711i \(0.803165\pi\)
\(744\) −4509.58 −0.222217
\(745\) −6607.82 −0.324955
\(746\) −1775.32 −0.0871299
\(747\) −8255.93 −0.404376
\(748\) 629.606 0.0307763
\(749\) 0 0
\(750\) 2373.91 0.115577
\(751\) −38073.8 −1.84998 −0.924989 0.379994i \(-0.875926\pi\)
−0.924989 + 0.379994i \(0.875926\pi\)
\(752\) 6386.88 0.309715
\(753\) 17454.6 0.844729
\(754\) 4323.33 0.208815
\(755\) 6773.98 0.326530
\(756\) 0 0
\(757\) −5203.06 −0.249813 −0.124907 0.992169i \(-0.539863\pi\)
−0.124907 + 0.992169i \(0.539863\pi\)
\(758\) −5842.48 −0.279958
\(759\) −751.284 −0.0359287
\(760\) −5725.53 −0.273272
\(761\) 3531.41 0.168218 0.0841089 0.996457i \(-0.473196\pi\)
0.0841089 + 0.996457i \(0.473196\pi\)
\(762\) −2179.75 −0.103627
\(763\) 0 0
\(764\) −5383.10 −0.254913
\(765\) −231.650 −0.0109481
\(766\) −1369.61 −0.0646030
\(767\) −4911.16 −0.231202
\(768\) 1692.58 0.0795257
\(769\) 14098.0 0.661101 0.330550 0.943788i \(-0.392766\pi\)
0.330550 + 0.943788i \(0.392766\pi\)
\(770\) 0 0
\(771\) −12025.7 −0.561729
\(772\) 9726.72 0.453461
\(773\) −12595.0 −0.586040 −0.293020 0.956106i \(-0.594660\pi\)
−0.293020 + 0.956106i \(0.594660\pi\)
\(774\) 2447.40 0.113656
\(775\) 10953.6 0.507696
\(776\) 15370.5 0.711044
\(777\) 0 0
\(778\) 6247.33 0.287889
\(779\) 26139.8 1.20225
\(780\) −2935.08 −0.134734
\(781\) 1001.77 0.0458976
\(782\) −202.426 −0.00925668
\(783\) 2370.44 0.108190
\(784\) 0 0
\(785\) −5233.55 −0.237953
\(786\) 3826.86 0.173663
\(787\) 10385.8 0.470413 0.235207 0.971945i \(-0.424423\pi\)
0.235207 + 0.971945i \(0.424423\pi\)
\(788\) 5693.12 0.257372
\(789\) −1157.29 −0.0522187
\(790\) −3948.13 −0.177808
\(791\) 0 0
\(792\) −1568.84 −0.0703868
\(793\) 2096.64 0.0938888
\(794\) 5760.76 0.257483
\(795\) −3025.94 −0.134992
\(796\) 12203.8 0.543406
\(797\) −22720.1 −1.00977 −0.504886 0.863186i \(-0.668465\pi\)
−0.504886 + 0.863186i \(0.668465\pi\)
\(798\) 0 0
\(799\) −1401.30 −0.0620455
\(800\) −19319.0 −0.853787
\(801\) −14572.0 −0.642791
\(802\) 9774.59 0.430365
\(803\) 3055.47 0.134278
\(804\) 12730.8 0.558435
\(805\) 0 0
\(806\) 4671.15 0.204137
\(807\) 20000.4 0.872425
\(808\) 20314.5 0.884482
\(809\) −13976.0 −0.607381 −0.303691 0.952771i \(-0.598219\pi\)
−0.303691 + 0.952771i \(0.598219\pi\)
\(810\) 266.538 0.0115620
\(811\) −6555.19 −0.283827 −0.141914 0.989879i \(-0.545326\pi\)
−0.141914 + 0.989879i \(0.545326\pi\)
\(812\) 0 0
\(813\) −8701.80 −0.375382
\(814\) 1009.52 0.0434688
\(815\) 2173.45 0.0934142
\(816\) 950.977 0.0407976
\(817\) 29857.8 1.27857
\(818\) 11805.3 0.504600
\(819\) 0 0
\(820\) −4729.89 −0.201433
\(821\) 2818.18 0.119799 0.0598997 0.998204i \(-0.480922\pi\)
0.0598997 + 0.998204i \(0.480922\pi\)
\(822\) −1392.10 −0.0590694
\(823\) −1833.38 −0.0776521 −0.0388261 0.999246i \(-0.512362\pi\)
−0.0388261 + 0.999246i \(0.512362\pi\)
\(824\) −6711.00 −0.283724
\(825\) 3810.66 0.160812
\(826\) 0 0
\(827\) −41648.1 −1.75120 −0.875602 0.483034i \(-0.839535\pi\)
−0.875602 + 0.483034i \(0.839535\pi\)
\(828\) −1406.25 −0.0590225
\(829\) −7384.14 −0.309363 −0.154681 0.987964i \(-0.549435\pi\)
−0.154681 + 0.987964i \(0.549435\pi\)
\(830\) −3018.55 −0.126235
\(831\) 8328.43 0.347666
\(832\) 5806.27 0.241942
\(833\) 0 0
\(834\) 7108.60 0.295145
\(835\) 2233.30 0.0925589
\(836\) −8837.91 −0.365629
\(837\) 2561.15 0.105766
\(838\) 7285.14 0.300311
\(839\) 15513.2 0.638348 0.319174 0.947696i \(-0.396595\pi\)
0.319174 + 0.947696i \(0.396595\pi\)
\(840\) 0 0
\(841\) −16681.2 −0.683964
\(842\) 10877.9 0.445221
\(843\) 2973.74 0.121496
\(844\) −21851.2 −0.891173
\(845\) −196.685 −0.00800729
\(846\) 1612.34 0.0655242
\(847\) 0 0
\(848\) 12422.2 0.503042
\(849\) 1614.74 0.0652743
\(850\) 1026.74 0.0414317
\(851\) 1959.67 0.0789383
\(852\) 1875.11 0.0753992
\(853\) 13573.5 0.544841 0.272420 0.962178i \(-0.412176\pi\)
0.272420 + 0.962178i \(0.412176\pi\)
\(854\) 0 0
\(855\) 3251.72 0.130066
\(856\) −4981.84 −0.198921
\(857\) 29984.1 1.19514 0.597572 0.801815i \(-0.296132\pi\)
0.597572 + 0.801815i \(0.296132\pi\)
\(858\) 1625.05 0.0646601
\(859\) −20975.7 −0.833157 −0.416578 0.909100i \(-0.636771\pi\)
−0.416578 + 0.909100i \(0.636771\pi\)
\(860\) −5402.66 −0.214220
\(861\) 0 0
\(862\) 15665.0 0.618970
\(863\) 6495.44 0.256208 0.128104 0.991761i \(-0.459111\pi\)
0.128104 + 0.991761i \(0.459111\pi\)
\(864\) −4517.13 −0.177866
\(865\) −90.1332 −0.00354291
\(866\) 3147.71 0.123514
\(867\) 14530.4 0.569177
\(868\) 0 0
\(869\) −13198.0 −0.515205
\(870\) 866.686 0.0337740
\(871\) −28558.0 −1.11096
\(872\) −18754.7 −0.728341
\(873\) −8729.46 −0.338428
\(874\) 2841.49 0.109971
\(875\) 0 0
\(876\) 5719.23 0.220588
\(877\) 8633.30 0.332413 0.166206 0.986091i \(-0.446848\pi\)
0.166206 + 0.986091i \(0.446848\pi\)
\(878\) −4387.27 −0.168637
\(879\) 17459.7 0.669966
\(880\) 1290.45 0.0494330
\(881\) 43842.5 1.67661 0.838303 0.545204i \(-0.183548\pi\)
0.838303 + 0.545204i \(0.183548\pi\)
\(882\) 0 0
\(883\) −22525.9 −0.858503 −0.429251 0.903185i \(-0.641223\pi\)
−0.429251 + 0.903185i \(0.641223\pi\)
\(884\) −2643.62 −0.100582
\(885\) −984.526 −0.0373949
\(886\) 7670.88 0.290867
\(887\) −23652.5 −0.895348 −0.447674 0.894197i \(-0.647747\pi\)
−0.447674 + 0.894197i \(0.647747\pi\)
\(888\) 4092.20 0.154646
\(889\) 0 0
\(890\) −5327.84 −0.200662
\(891\) 891.000 0.0335013
\(892\) 24764.5 0.929570
\(893\) 19670.3 0.737113
\(894\) 6848.01 0.256188
\(895\) −7511.88 −0.280553
\(896\) 0 0
\(897\) 3154.53 0.117421
\(898\) 12315.6 0.457658
\(899\) 8327.91 0.308956
\(900\) 7132.78 0.264177
\(901\) −2725.46 −0.100775
\(902\) 2618.77 0.0966692
\(903\) 0 0
\(904\) −9189.49 −0.338095
\(905\) −13450.2 −0.494031
\(906\) −7020.21 −0.257429
\(907\) 16767.1 0.613829 0.306914 0.951737i \(-0.400704\pi\)
0.306914 + 0.951737i \(0.400704\pi\)
\(908\) 21163.7 0.773504
\(909\) −11537.3 −0.420977
\(910\) 0 0
\(911\) 13684.8 0.497692 0.248846 0.968543i \(-0.419949\pi\)
0.248846 + 0.968543i \(0.419949\pi\)
\(912\) −13349.1 −0.484684
\(913\) −10090.6 −0.365772
\(914\) 13562.5 0.490818
\(915\) 420.307 0.0151857
\(916\) −8052.58 −0.290464
\(917\) 0 0
\(918\) 240.071 0.00863127
\(919\) 20488.7 0.735431 0.367715 0.929938i \(-0.380140\pi\)
0.367715 + 0.929938i \(0.380140\pi\)
\(920\) −1113.47 −0.0399022
\(921\) 13993.4 0.500649
\(922\) 3573.70 0.127650
\(923\) −4206.27 −0.150001
\(924\) 0 0
\(925\) −9939.80 −0.353317
\(926\) −20717.1 −0.735213
\(927\) 3811.41 0.135041
\(928\) −14688.1 −0.519568
\(929\) 2367.39 0.0836076 0.0418038 0.999126i \(-0.486690\pi\)
0.0418038 + 0.999126i \(0.486690\pi\)
\(930\) 936.411 0.0330173
\(931\) 0 0
\(932\) 38255.0 1.34451
\(933\) −21333.3 −0.748574
\(934\) 1335.57 0.0467893
\(935\) −283.128 −0.00990297
\(936\) 6587.33 0.230036
\(937\) −12226.8 −0.426290 −0.213145 0.977021i \(-0.568371\pi\)
−0.213145 + 0.977021i \(0.568371\pi\)
\(938\) 0 0
\(939\) −12843.3 −0.446351
\(940\) −3559.27 −0.123501
\(941\) −8964.97 −0.310574 −0.155287 0.987869i \(-0.549630\pi\)
−0.155287 + 0.987869i \(0.549630\pi\)
\(942\) 5423.78 0.187597
\(943\) 5083.53 0.175549
\(944\) 4041.70 0.139350
\(945\) 0 0
\(946\) 2991.26 0.102806
\(947\) 19439.1 0.667040 0.333520 0.942743i \(-0.391764\pi\)
0.333520 + 0.942743i \(0.391764\pi\)
\(948\) −24704.1 −0.846362
\(949\) −12829.5 −0.438843
\(950\) −14412.6 −0.492217
\(951\) 28679.6 0.977917
\(952\) 0 0
\(953\) 9050.67 0.307639 0.153819 0.988099i \(-0.450843\pi\)
0.153819 + 0.988099i \(0.450843\pi\)
\(954\) 3135.93 0.106425
\(955\) 2420.73 0.0820241
\(956\) −21065.5 −0.712663
\(957\) 2897.21 0.0978614
\(958\) 5170.68 0.174381
\(959\) 0 0
\(960\) 1163.97 0.0391321
\(961\) −20793.1 −0.697966
\(962\) −4238.82 −0.142063
\(963\) 2829.36 0.0946780
\(964\) −5741.06 −0.191812
\(965\) −4374.02 −0.145912
\(966\) 0 0
\(967\) −18590.8 −0.618243 −0.309122 0.951023i \(-0.600035\pi\)
−0.309122 + 0.951023i \(0.600035\pi\)
\(968\) −1917.47 −0.0636673
\(969\) 2928.82 0.0970973
\(970\) −3191.68 −0.105648
\(971\) −12511.7 −0.413511 −0.206756 0.978393i \(-0.566291\pi\)
−0.206756 + 0.978393i \(0.566291\pi\)
\(972\) 1667.77 0.0550348
\(973\) 0 0
\(974\) −11676.5 −0.384126
\(975\) −16000.4 −0.525561
\(976\) −1725.46 −0.0565886
\(977\) 42667.0 1.39717 0.698587 0.715525i \(-0.253812\pi\)
0.698587 + 0.715525i \(0.253812\pi\)
\(978\) −2252.45 −0.0736457
\(979\) −17810.2 −0.581427
\(980\) 0 0
\(981\) 10651.4 0.346660
\(982\) 14415.9 0.468461
\(983\) 23289.4 0.755663 0.377832 0.925874i \(-0.376670\pi\)
0.377832 + 0.925874i \(0.376670\pi\)
\(984\) 10615.5 0.343913
\(985\) −2560.14 −0.0828152
\(986\) 780.622 0.0252130
\(987\) 0 0
\(988\) 37109.1 1.19494
\(989\) 5806.60 0.186693
\(990\) 325.769 0.0104582
\(991\) −36273.1 −1.16272 −0.581358 0.813648i \(-0.697478\pi\)
−0.581358 + 0.813648i \(0.697478\pi\)
\(992\) −15869.7 −0.507927
\(993\) 7640.17 0.244162
\(994\) 0 0
\(995\) −5487.93 −0.174853
\(996\) −18887.5 −0.600878
\(997\) 35428.0 1.12539 0.562697 0.826664i \(-0.309764\pi\)
0.562697 + 0.826664i \(0.309764\pi\)
\(998\) 18122.4 0.574803
\(999\) −2324.10 −0.0736050
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 1617.4.a.be.1.9 16
7.6 odd 2 1617.4.a.bf.1.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.9 16 1.1 even 1 trivial
1617.4.a.bf.1.9 yes 16 7.6 odd 2