Properties

Label 1617.4.a.be.1.1
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 1.1
Root \(-5.15488\) 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)
 
\(f(q)\) \(=\) \(q-5.15488 q^{2} -3.00000 q^{3} +18.5728 q^{4} +4.10022 q^{5} +15.4646 q^{6} -54.5014 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.15488 q^{2} -3.00000 q^{3} +18.5728 q^{4} +4.10022 q^{5} +15.4646 q^{6} -54.5014 q^{8} +9.00000 q^{9} -21.1361 q^{10} +11.0000 q^{11} -55.7184 q^{12} +20.8231 q^{13} -12.3007 q^{15} +132.366 q^{16} -44.4422 q^{17} -46.3939 q^{18} -35.8954 q^{19} +76.1525 q^{20} -56.7037 q^{22} +176.700 q^{23} +163.504 q^{24} -108.188 q^{25} -107.341 q^{26} -27.0000 q^{27} +111.286 q^{29} +63.4084 q^{30} +2.06128 q^{31} -246.320 q^{32} -33.0000 q^{33} +229.094 q^{34} +167.155 q^{36} -121.486 q^{37} +185.036 q^{38} -62.4694 q^{39} -223.468 q^{40} +234.931 q^{41} -69.4132 q^{43} +204.301 q^{44} +36.9020 q^{45} -910.869 q^{46} -401.595 q^{47} -397.098 q^{48} +557.697 q^{50} +133.327 q^{51} +386.744 q^{52} -199.418 q^{53} +139.182 q^{54} +45.1024 q^{55} +107.686 q^{57} -573.668 q^{58} -212.553 q^{59} -228.457 q^{60} +151.192 q^{61} -10.6256 q^{62} +210.820 q^{64} +85.3794 q^{65} +170.111 q^{66} -808.228 q^{67} -825.416 q^{68} -530.101 q^{69} +453.654 q^{71} -490.513 q^{72} -522.350 q^{73} +626.245 q^{74} +324.565 q^{75} -666.677 q^{76} +322.022 q^{78} -93.3975 q^{79} +542.730 q^{80} +81.0000 q^{81} -1211.04 q^{82} +666.204 q^{83} -182.223 q^{85} +357.817 q^{86} -333.859 q^{87} -599.516 q^{88} -1288.54 q^{89} -190.225 q^{90} +3281.82 q^{92} -6.18383 q^{93} +2070.17 q^{94} -147.179 q^{95} +738.959 q^{96} -844.386 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.15488 −1.82253 −0.911263 0.411826i \(-0.864891\pi\)
−0.911263 + 0.411826i \(0.864891\pi\)
\(3\) −3.00000 −0.577350
\(4\) 18.5728 2.32160
\(5\) 4.10022 0.366735 0.183367 0.983044i \(-0.441300\pi\)
0.183367 + 0.983044i \(0.441300\pi\)
\(6\) 15.4646 1.05224
\(7\) 0 0
\(8\) −54.5014 −2.40865
\(9\) 9.00000 0.333333
\(10\) −21.1361 −0.668383
\(11\) 11.0000 0.301511
\(12\) −55.7184 −1.34038
\(13\) 20.8231 0.444254 0.222127 0.975018i \(-0.428700\pi\)
0.222127 + 0.975018i \(0.428700\pi\)
\(14\) 0 0
\(15\) −12.3007 −0.211734
\(16\) 132.366 2.06822
\(17\) −44.4422 −0.634049 −0.317024 0.948417i \(-0.602684\pi\)
−0.317024 + 0.948417i \(0.602684\pi\)
\(18\) −46.3939 −0.607508
\(19\) −35.8954 −0.433419 −0.216709 0.976236i \(-0.569532\pi\)
−0.216709 + 0.976236i \(0.569532\pi\)
\(20\) 76.1525 0.851410
\(21\) 0 0
\(22\) −56.7037 −0.549512
\(23\) 176.700 1.60194 0.800969 0.598706i \(-0.204318\pi\)
0.800969 + 0.598706i \(0.204318\pi\)
\(24\) 163.504 1.39063
\(25\) −108.188 −0.865506
\(26\) −107.341 −0.809664
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 111.286 0.712599 0.356299 0.934372i \(-0.384038\pi\)
0.356299 + 0.934372i \(0.384038\pi\)
\(30\) 63.4084 0.385891
\(31\) 2.06128 0.0119425 0.00597123 0.999982i \(-0.498099\pi\)
0.00597123 + 0.999982i \(0.498099\pi\)
\(32\) −246.320 −1.36074
\(33\) −33.0000 −0.174078
\(34\) 229.094 1.15557
\(35\) 0 0
\(36\) 167.155 0.773866
\(37\) −121.486 −0.539788 −0.269894 0.962890i \(-0.586989\pi\)
−0.269894 + 0.962890i \(0.586989\pi\)
\(38\) 185.036 0.789917
\(39\) −62.4694 −0.256490
\(40\) −223.468 −0.883334
\(41\) 234.931 0.894881 0.447441 0.894314i \(-0.352336\pi\)
0.447441 + 0.894314i \(0.352336\pi\)
\(42\) 0 0
\(43\) −69.4132 −0.246172 −0.123086 0.992396i \(-0.539279\pi\)
−0.123086 + 0.992396i \(0.539279\pi\)
\(44\) 204.301 0.699988
\(45\) 36.9020 0.122245
\(46\) −910.869 −2.91957
\(47\) −401.595 −1.24635 −0.623176 0.782081i \(-0.714158\pi\)
−0.623176 + 0.782081i \(0.714158\pi\)
\(48\) −397.098 −1.19409
\(49\) 0 0
\(50\) 557.697 1.57741
\(51\) 133.327 0.366068
\(52\) 386.744 1.03138
\(53\) −199.418 −0.516834 −0.258417 0.966033i \(-0.583201\pi\)
−0.258417 + 0.966033i \(0.583201\pi\)
\(54\) 139.182 0.350745
\(55\) 45.1024 0.110575
\(56\) 0 0
\(57\) 107.686 0.250235
\(58\) −573.668 −1.29873
\(59\) −212.553 −0.469017 −0.234508 0.972114i \(-0.575348\pi\)
−0.234508 + 0.972114i \(0.575348\pi\)
\(60\) −228.457 −0.491562
\(61\) 151.192 0.317346 0.158673 0.987331i \(-0.449278\pi\)
0.158673 + 0.987331i \(0.449278\pi\)
\(62\) −10.6256 −0.0217654
\(63\) 0 0
\(64\) 210.820 0.411758
\(65\) 85.3794 0.162923
\(66\) 170.111 0.317261
\(67\) −808.228 −1.47374 −0.736871 0.676033i \(-0.763698\pi\)
−0.736871 + 0.676033i \(0.763698\pi\)
\(68\) −825.416 −1.47201
\(69\) −530.101 −0.924880
\(70\) 0 0
\(71\) 453.654 0.758294 0.379147 0.925336i \(-0.376217\pi\)
0.379147 + 0.925336i \(0.376217\pi\)
\(72\) −490.513 −0.802882
\(73\) −522.350 −0.837486 −0.418743 0.908105i \(-0.637529\pi\)
−0.418743 + 0.908105i \(0.637529\pi\)
\(74\) 626.245 0.983777
\(75\) 324.565 0.499700
\(76\) −666.677 −1.00622
\(77\) 0 0
\(78\) 322.022 0.467460
\(79\) −93.3975 −0.133013 −0.0665065 0.997786i \(-0.521185\pi\)
−0.0665065 + 0.997786i \(0.521185\pi\)
\(80\) 542.730 0.758488
\(81\) 81.0000 0.111111
\(82\) −1211.04 −1.63094
\(83\) 666.204 0.881029 0.440514 0.897746i \(-0.354796\pi\)
0.440514 + 0.897746i \(0.354796\pi\)
\(84\) 0 0
\(85\) −182.223 −0.232528
\(86\) 357.817 0.448655
\(87\) −333.859 −0.411419
\(88\) −599.516 −0.726234
\(89\) −1288.54 −1.53466 −0.767330 0.641253i \(-0.778415\pi\)
−0.767330 + 0.641253i \(0.778415\pi\)
\(90\) −190.225 −0.222794
\(91\) 0 0
\(92\) 3281.82 3.71906
\(93\) −6.18383 −0.00689498
\(94\) 2070.17 2.27151
\(95\) −147.179 −0.158950
\(96\) 738.959 0.785622
\(97\) −844.386 −0.883860 −0.441930 0.897050i \(-0.645706\pi\)
−0.441930 + 0.897050i \(0.645706\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −2009.36 −2.00936
\(101\) 493.849 0.486533 0.243267 0.969959i \(-0.421781\pi\)
0.243267 + 0.969959i \(0.421781\pi\)
\(102\) −687.283 −0.667169
\(103\) −942.052 −0.901196 −0.450598 0.892727i \(-0.648789\pi\)
−0.450598 + 0.892727i \(0.648789\pi\)
\(104\) −1134.89 −1.07005
\(105\) 0 0
\(106\) 1027.98 0.941943
\(107\) −1064.86 −0.962092 −0.481046 0.876695i \(-0.659743\pi\)
−0.481046 + 0.876695i \(0.659743\pi\)
\(108\) −501.465 −0.446792
\(109\) 714.701 0.628036 0.314018 0.949417i \(-0.398325\pi\)
0.314018 + 0.949417i \(0.398325\pi\)
\(110\) −232.497 −0.201525
\(111\) 364.458 0.311647
\(112\) 0 0
\(113\) 2149.38 1.78935 0.894675 0.446719i \(-0.147407\pi\)
0.894675 + 0.446719i \(0.147407\pi\)
\(114\) −555.109 −0.456059
\(115\) 724.510 0.587486
\(116\) 2066.90 1.65437
\(117\) 187.408 0.148085
\(118\) 1095.68 0.854795
\(119\) 0 0
\(120\) 670.403 0.509993
\(121\) 121.000 0.0909091
\(122\) −779.375 −0.578371
\(123\) −704.794 −0.516660
\(124\) 38.2836 0.0277256
\(125\) −956.122 −0.684146
\(126\) 0 0
\(127\) 251.745 0.175895 0.0879477 0.996125i \(-0.471969\pi\)
0.0879477 + 0.996125i \(0.471969\pi\)
\(128\) 883.807 0.610298
\(129\) 208.239 0.142128
\(130\) −440.121 −0.296932
\(131\) 982.348 0.655176 0.327588 0.944821i \(-0.393764\pi\)
0.327588 + 0.944821i \(0.393764\pi\)
\(132\) −612.902 −0.404138
\(133\) 0 0
\(134\) 4166.32 2.68593
\(135\) −110.706 −0.0705781
\(136\) 2422.17 1.52720
\(137\) 1683.45 1.04983 0.524916 0.851154i \(-0.324097\pi\)
0.524916 + 0.851154i \(0.324097\pi\)
\(138\) 2732.61 1.68562
\(139\) 2377.02 1.45048 0.725238 0.688499i \(-0.241730\pi\)
0.725238 + 0.688499i \(0.241730\pi\)
\(140\) 0 0
\(141\) 1204.78 0.719582
\(142\) −2338.53 −1.38201
\(143\) 229.055 0.133948
\(144\) 1191.29 0.689407
\(145\) 456.299 0.261335
\(146\) 2692.65 1.52634
\(147\) 0 0
\(148\) −2256.33 −1.25317
\(149\) −1285.07 −0.706556 −0.353278 0.935518i \(-0.614933\pi\)
−0.353278 + 0.935518i \(0.614933\pi\)
\(150\) −1673.09 −0.910716
\(151\) −395.537 −0.213168 −0.106584 0.994304i \(-0.533991\pi\)
−0.106584 + 0.994304i \(0.533991\pi\)
\(152\) 1956.35 1.04395
\(153\) −399.980 −0.211350
\(154\) 0 0
\(155\) 8.45168 0.00437971
\(156\) −1160.23 −0.595467
\(157\) 641.407 0.326050 0.163025 0.986622i \(-0.447875\pi\)
0.163025 + 0.986622i \(0.447875\pi\)
\(158\) 481.453 0.242420
\(159\) 598.255 0.298394
\(160\) −1009.96 −0.499029
\(161\) 0 0
\(162\) −417.545 −0.202503
\(163\) 1378.98 0.662638 0.331319 0.943519i \(-0.392506\pi\)
0.331319 + 0.943519i \(0.392506\pi\)
\(164\) 4363.33 2.07755
\(165\) −135.307 −0.0638403
\(166\) −3434.20 −1.60570
\(167\) 2428.71 1.12539 0.562693 0.826666i \(-0.309765\pi\)
0.562693 + 0.826666i \(0.309765\pi\)
\(168\) 0 0
\(169\) −1763.40 −0.802638
\(170\) 939.337 0.423787
\(171\) −323.058 −0.144473
\(172\) −1289.20 −0.571513
\(173\) 1923.61 0.845372 0.422686 0.906276i \(-0.361087\pi\)
0.422686 + 0.906276i \(0.361087\pi\)
\(174\) 1721.00 0.749822
\(175\) 0 0
\(176\) 1456.03 0.623592
\(177\) 637.658 0.270787
\(178\) 6642.25 2.79695
\(179\) −2234.57 −0.933072 −0.466536 0.884502i \(-0.654498\pi\)
−0.466536 + 0.884502i \(0.654498\pi\)
\(180\) 685.372 0.283803
\(181\) 3135.20 1.28750 0.643750 0.765236i \(-0.277378\pi\)
0.643750 + 0.765236i \(0.277378\pi\)
\(182\) 0 0
\(183\) −453.575 −0.183220
\(184\) −9630.43 −3.85850
\(185\) −498.118 −0.197959
\(186\) 31.8769 0.0125663
\(187\) −488.865 −0.191173
\(188\) −7458.73 −2.89353
\(189\) 0 0
\(190\) 758.689 0.289690
\(191\) 2558.75 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(192\) −632.460 −0.237728
\(193\) 2116.47 0.789361 0.394681 0.918818i \(-0.370855\pi\)
0.394681 + 0.918818i \(0.370855\pi\)
\(194\) 4352.71 1.61086
\(195\) −256.138 −0.0940638
\(196\) 0 0
\(197\) −5202.85 −1.88166 −0.940832 0.338872i \(-0.889954\pi\)
−0.940832 + 0.338872i \(0.889954\pi\)
\(198\) −510.333 −0.183171
\(199\) −3204.04 −1.14135 −0.570674 0.821177i \(-0.693318\pi\)
−0.570674 + 0.821177i \(0.693318\pi\)
\(200\) 5896.41 2.08470
\(201\) 2424.68 0.850866
\(202\) −2545.73 −0.886719
\(203\) 0 0
\(204\) 2476.25 0.849863
\(205\) 963.270 0.328184
\(206\) 4856.17 1.64245
\(207\) 1590.30 0.533979
\(208\) 2756.28 0.918815
\(209\) −394.849 −0.130681
\(210\) 0 0
\(211\) 1413.85 0.461294 0.230647 0.973037i \(-0.425916\pi\)
0.230647 + 0.973037i \(0.425916\pi\)
\(212\) −3703.75 −1.19988
\(213\) −1360.96 −0.437801
\(214\) 5489.22 1.75344
\(215\) −284.609 −0.0902799
\(216\) 1471.54 0.463544
\(217\) 0 0
\(218\) −3684.20 −1.14461
\(219\) 1567.05 0.483523
\(220\) 837.677 0.256710
\(221\) −925.427 −0.281679
\(222\) −1878.73 −0.567984
\(223\) 2421.91 0.727279 0.363639 0.931540i \(-0.381534\pi\)
0.363639 + 0.931540i \(0.381534\pi\)
\(224\) 0 0
\(225\) −973.694 −0.288502
\(226\) −11079.8 −3.26113
\(227\) −1752.69 −0.512468 −0.256234 0.966615i \(-0.582482\pi\)
−0.256234 + 0.966615i \(0.582482\pi\)
\(228\) 2000.03 0.580944
\(229\) −2799.01 −0.807702 −0.403851 0.914825i \(-0.632329\pi\)
−0.403851 + 0.914825i \(0.632329\pi\)
\(230\) −3734.76 −1.07071
\(231\) 0 0
\(232\) −6065.27 −1.71640
\(233\) 3673.99 1.03301 0.516504 0.856285i \(-0.327233\pi\)
0.516504 + 0.856285i \(0.327233\pi\)
\(234\) −966.067 −0.269888
\(235\) −1646.63 −0.457081
\(236\) −3947.69 −1.08887
\(237\) 280.192 0.0767951
\(238\) 0 0
\(239\) 824.498 0.223148 0.111574 0.993756i \(-0.464411\pi\)
0.111574 + 0.993756i \(0.464411\pi\)
\(240\) −1628.19 −0.437913
\(241\) −6573.79 −1.75707 −0.878537 0.477674i \(-0.841480\pi\)
−0.878537 + 0.477674i \(0.841480\pi\)
\(242\) −623.740 −0.165684
\(243\) −243.000 −0.0641500
\(244\) 2808.05 0.736750
\(245\) 0 0
\(246\) 3633.13 0.941626
\(247\) −747.454 −0.192548
\(248\) −112.342 −0.0287651
\(249\) −1998.61 −0.508662
\(250\) 4928.70 1.24687
\(251\) −4372.58 −1.09958 −0.549791 0.835302i \(-0.685293\pi\)
−0.549791 + 0.835302i \(0.685293\pi\)
\(252\) 0 0
\(253\) 1943.70 0.483003
\(254\) −1297.71 −0.320574
\(255\) 546.669 0.134250
\(256\) −6242.48 −1.52404
\(257\) −4933.11 −1.19735 −0.598675 0.800992i \(-0.704306\pi\)
−0.598675 + 0.800992i \(0.704306\pi\)
\(258\) −1073.45 −0.259031
\(259\) 0 0
\(260\) 1585.73 0.378242
\(261\) 1001.58 0.237533
\(262\) −5063.88 −1.19408
\(263\) 572.985 0.134341 0.0671706 0.997742i \(-0.478603\pi\)
0.0671706 + 0.997742i \(0.478603\pi\)
\(264\) 1798.55 0.419291
\(265\) −817.658 −0.189541
\(266\) 0 0
\(267\) 3865.61 0.886036
\(268\) −15011.0 −3.42144
\(269\) 3302.45 0.748527 0.374263 0.927322i \(-0.377896\pi\)
0.374263 + 0.927322i \(0.377896\pi\)
\(270\) 570.675 0.128630
\(271\) −4618.64 −1.03529 −0.517643 0.855597i \(-0.673190\pi\)
−0.517643 + 0.855597i \(0.673190\pi\)
\(272\) −5882.65 −1.31135
\(273\) 0 0
\(274\) −8677.99 −1.91335
\(275\) −1190.07 −0.260960
\(276\) −9845.46 −2.14720
\(277\) −6194.19 −1.34358 −0.671792 0.740740i \(-0.734475\pi\)
−0.671792 + 0.740740i \(0.734475\pi\)
\(278\) −12253.2 −2.64353
\(279\) 18.5515 0.00398082
\(280\) 0 0
\(281\) 8222.64 1.74563 0.872815 0.488052i \(-0.162292\pi\)
0.872815 + 0.488052i \(0.162292\pi\)
\(282\) −6210.51 −1.31146
\(283\) −4423.62 −0.929176 −0.464588 0.885527i \(-0.653798\pi\)
−0.464588 + 0.885527i \(0.653798\pi\)
\(284\) 8425.63 1.76045
\(285\) 441.536 0.0917697
\(286\) −1180.75 −0.244123
\(287\) 0 0
\(288\) −2216.88 −0.453579
\(289\) −2937.89 −0.597982
\(290\) −2352.16 −0.476289
\(291\) 2533.16 0.510297
\(292\) −9701.50 −1.94431
\(293\) −8597.95 −1.71433 −0.857164 0.515044i \(-0.827775\pi\)
−0.857164 + 0.515044i \(0.827775\pi\)
\(294\) 0 0
\(295\) −871.512 −0.172005
\(296\) 6621.15 1.30016
\(297\) −297.000 −0.0580259
\(298\) 6624.37 1.28772
\(299\) 3679.46 0.711667
\(300\) 6028.07 1.16010
\(301\) 0 0
\(302\) 2038.95 0.388504
\(303\) −1481.55 −0.280900
\(304\) −4751.33 −0.896406
\(305\) 619.919 0.116382
\(306\) 2061.85 0.385190
\(307\) −4268.73 −0.793582 −0.396791 0.917909i \(-0.629876\pi\)
−0.396791 + 0.917909i \(0.629876\pi\)
\(308\) 0 0
\(309\) 2826.16 0.520306
\(310\) −43.5674 −0.00798213
\(311\) 8173.52 1.49028 0.745141 0.666907i \(-0.232382\pi\)
0.745141 + 0.666907i \(0.232382\pi\)
\(312\) 3404.67 0.617794
\(313\) −4288.23 −0.774393 −0.387196 0.921997i \(-0.626556\pi\)
−0.387196 + 0.921997i \(0.626556\pi\)
\(314\) −3306.38 −0.594234
\(315\) 0 0
\(316\) −1734.65 −0.308803
\(317\) 4750.02 0.841602 0.420801 0.907153i \(-0.361749\pi\)
0.420801 + 0.907153i \(0.361749\pi\)
\(318\) −3083.93 −0.543831
\(319\) 1224.15 0.214857
\(320\) 864.407 0.151006
\(321\) 3194.58 0.555464
\(322\) 0 0
\(323\) 1595.27 0.274809
\(324\) 1504.40 0.257955
\(325\) −2252.82 −0.384504
\(326\) −7108.48 −1.20768
\(327\) −2144.10 −0.362597
\(328\) −12804.1 −2.15545
\(329\) 0 0
\(330\) 697.492 0.116351
\(331\) 3668.55 0.609189 0.304594 0.952482i \(-0.401479\pi\)
0.304594 + 0.952482i \(0.401479\pi\)
\(332\) 12373.3 2.04539
\(333\) −1093.37 −0.179929
\(334\) −12519.7 −2.05105
\(335\) −3313.91 −0.540473
\(336\) 0 0
\(337\) −5900.30 −0.953739 −0.476870 0.878974i \(-0.658229\pi\)
−0.476870 + 0.878974i \(0.658229\pi\)
\(338\) 9090.10 1.46283
\(339\) −6448.13 −1.03308
\(340\) −3384.39 −0.539836
\(341\) 22.6740 0.00360078
\(342\) 1665.33 0.263306
\(343\) 0 0
\(344\) 3783.12 0.592942
\(345\) −2173.53 −0.339185
\(346\) −9915.98 −1.54071
\(347\) −6781.16 −1.04908 −0.524541 0.851385i \(-0.675763\pi\)
−0.524541 + 0.851385i \(0.675763\pi\)
\(348\) −6200.70 −0.955150
\(349\) −4474.05 −0.686219 −0.343109 0.939295i \(-0.611480\pi\)
−0.343109 + 0.939295i \(0.611480\pi\)
\(350\) 0 0
\(351\) −562.225 −0.0854967
\(352\) −2709.52 −0.410278
\(353\) −12106.0 −1.82532 −0.912661 0.408718i \(-0.865976\pi\)
−0.912661 + 0.408718i \(0.865976\pi\)
\(354\) −3287.05 −0.493516
\(355\) 1860.08 0.278093
\(356\) −23931.7 −3.56286
\(357\) 0 0
\(358\) 11519.0 1.70055
\(359\) 7720.76 1.13506 0.567530 0.823353i \(-0.307899\pi\)
0.567530 + 0.823353i \(0.307899\pi\)
\(360\) −2011.21 −0.294445
\(361\) −5570.52 −0.812148
\(362\) −16161.6 −2.34650
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −2141.75 −0.307135
\(366\) 2338.13 0.333923
\(367\) −8058.61 −1.14620 −0.573101 0.819485i \(-0.694260\pi\)
−0.573101 + 0.819485i \(0.694260\pi\)
\(368\) 23389.1 3.31316
\(369\) 2114.38 0.298294
\(370\) 2567.74 0.360785
\(371\) 0 0
\(372\) −114.851 −0.0160074
\(373\) 10589.7 1.47001 0.735004 0.678063i \(-0.237180\pi\)
0.735004 + 0.678063i \(0.237180\pi\)
\(374\) 2520.04 0.348417
\(375\) 2868.37 0.394992
\(376\) 21887.5 3.00202
\(377\) 2317.33 0.316575
\(378\) 0 0
\(379\) −2361.53 −0.320063 −0.160032 0.987112i \(-0.551160\pi\)
−0.160032 + 0.987112i \(0.551160\pi\)
\(380\) −2733.52 −0.369017
\(381\) −755.234 −0.101553
\(382\) −13190.0 −1.76665
\(383\) −10762.6 −1.43588 −0.717941 0.696104i \(-0.754915\pi\)
−0.717941 + 0.696104i \(0.754915\pi\)
\(384\) −2651.42 −0.352356
\(385\) 0 0
\(386\) −10910.1 −1.43863
\(387\) −624.718 −0.0820574
\(388\) −15682.6 −2.05197
\(389\) 6387.85 0.832588 0.416294 0.909230i \(-0.363329\pi\)
0.416294 + 0.909230i \(0.363329\pi\)
\(390\) 1320.36 0.171434
\(391\) −7852.96 −1.01571
\(392\) 0 0
\(393\) −2947.04 −0.378266
\(394\) 26820.1 3.42938
\(395\) −382.950 −0.0487805
\(396\) 1838.71 0.233329
\(397\) 7154.04 0.904411 0.452205 0.891914i \(-0.350637\pi\)
0.452205 + 0.891914i \(0.350637\pi\)
\(398\) 16516.4 2.08014
\(399\) 0 0
\(400\) −14320.4 −1.79006
\(401\) 9428.12 1.17411 0.587055 0.809547i \(-0.300287\pi\)
0.587055 + 0.809547i \(0.300287\pi\)
\(402\) −12499.0 −1.55072
\(403\) 42.9222 0.00530548
\(404\) 9172.16 1.12953
\(405\) 332.118 0.0407483
\(406\) 0 0
\(407\) −1336.34 −0.162752
\(408\) −7266.50 −0.881729
\(409\) 7133.14 0.862374 0.431187 0.902263i \(-0.358095\pi\)
0.431187 + 0.902263i \(0.358095\pi\)
\(410\) −4965.54 −0.598124
\(411\) −5050.36 −0.606121
\(412\) −17496.5 −2.09221
\(413\) 0 0
\(414\) −8197.82 −0.973191
\(415\) 2731.58 0.323104
\(416\) −5129.15 −0.604513
\(417\) −7131.05 −0.837432
\(418\) 2035.40 0.238169
\(419\) −2702.73 −0.315124 −0.157562 0.987509i \(-0.550363\pi\)
−0.157562 + 0.987509i \(0.550363\pi\)
\(420\) 0 0
\(421\) 14187.6 1.64242 0.821212 0.570623i \(-0.193298\pi\)
0.821212 + 0.570623i \(0.193298\pi\)
\(422\) −7288.20 −0.840721
\(423\) −3614.35 −0.415451
\(424\) 10868.6 1.24487
\(425\) 4808.13 0.548773
\(426\) 7015.60 0.797904
\(427\) 0 0
\(428\) −19777.4 −2.23359
\(429\) −687.164 −0.0773347
\(430\) 1467.13 0.164537
\(431\) −13215.8 −1.47699 −0.738497 0.674257i \(-0.764464\pi\)
−0.738497 + 0.674257i \(0.764464\pi\)
\(432\) −3573.88 −0.398029
\(433\) −10474.5 −1.16253 −0.581264 0.813715i \(-0.697442\pi\)
−0.581264 + 0.813715i \(0.697442\pi\)
\(434\) 0 0
\(435\) −1368.90 −0.150882
\(436\) 13274.0 1.45805
\(437\) −6342.73 −0.694310
\(438\) −8077.96 −0.881232
\(439\) 12254.2 1.33226 0.666128 0.745838i \(-0.267951\pi\)
0.666128 + 0.745838i \(0.267951\pi\)
\(440\) −2458.15 −0.266335
\(441\) 0 0
\(442\) 4770.47 0.513366
\(443\) 328.965 0.0352813 0.0176406 0.999844i \(-0.494385\pi\)
0.0176406 + 0.999844i \(0.494385\pi\)
\(444\) 6768.99 0.723518
\(445\) −5283.28 −0.562813
\(446\) −12484.7 −1.32548
\(447\) 3855.21 0.407931
\(448\) 0 0
\(449\) −12984.7 −1.36478 −0.682391 0.730988i \(-0.739060\pi\)
−0.682391 + 0.730988i \(0.739060\pi\)
\(450\) 5019.28 0.525802
\(451\) 2584.25 0.269817
\(452\) 39919.9 4.15415
\(453\) 1186.61 0.123073
\(454\) 9034.92 0.933986
\(455\) 0 0
\(456\) −5869.05 −0.602726
\(457\) 14340.0 1.46783 0.733913 0.679243i \(-0.237692\pi\)
0.733913 + 0.679243i \(0.237692\pi\)
\(458\) 14428.6 1.47206
\(459\) 1199.94 0.122023
\(460\) 13456.2 1.36391
\(461\) 18118.0 1.83045 0.915227 0.402938i \(-0.132011\pi\)
0.915227 + 0.402938i \(0.132011\pi\)
\(462\) 0 0
\(463\) −10593.6 −1.06334 −0.531671 0.846951i \(-0.678436\pi\)
−0.531671 + 0.846951i \(0.678436\pi\)
\(464\) 14730.5 1.47381
\(465\) −25.3550 −0.00252863
\(466\) −18939.0 −1.88268
\(467\) −13704.4 −1.35796 −0.678978 0.734159i \(-0.737577\pi\)
−0.678978 + 0.734159i \(0.737577\pi\)
\(468\) 3480.69 0.343793
\(469\) 0 0
\(470\) 8488.15 0.833041
\(471\) −1924.22 −0.188245
\(472\) 11584.4 1.12969
\(473\) −763.545 −0.0742237
\(474\) −1444.36 −0.139961
\(475\) 3883.45 0.375127
\(476\) 0 0
\(477\) −1794.76 −0.172278
\(478\) −4250.19 −0.406692
\(479\) −7822.36 −0.746164 −0.373082 0.927798i \(-0.621699\pi\)
−0.373082 + 0.927798i \(0.621699\pi\)
\(480\) 3029.89 0.288115
\(481\) −2529.72 −0.239803
\(482\) 33887.1 3.20231
\(483\) 0 0
\(484\) 2247.31 0.211054
\(485\) −3462.17 −0.324142
\(486\) 1252.64 0.116915
\(487\) −4053.25 −0.377146 −0.188573 0.982059i \(-0.560386\pi\)
−0.188573 + 0.982059i \(0.560386\pi\)
\(488\) −8240.17 −0.764375
\(489\) −4136.94 −0.382574
\(490\) 0 0
\(491\) 3840.20 0.352965 0.176482 0.984304i \(-0.443528\pi\)
0.176482 + 0.984304i \(0.443528\pi\)
\(492\) −13090.0 −1.19948
\(493\) −4945.82 −0.451822
\(494\) 3853.04 0.350924
\(495\) 405.922 0.0368582
\(496\) 272.843 0.0246996
\(497\) 0 0
\(498\) 10302.6 0.927050
\(499\) −21207.5 −1.90256 −0.951279 0.308330i \(-0.900230\pi\)
−0.951279 + 0.308330i \(0.900230\pi\)
\(500\) −17757.9 −1.58831
\(501\) −7286.14 −0.649742
\(502\) 22540.1 2.00402
\(503\) −14084.9 −1.24854 −0.624270 0.781209i \(-0.714603\pi\)
−0.624270 + 0.781209i \(0.714603\pi\)
\(504\) 0 0
\(505\) 2024.89 0.178429
\(506\) −10019.6 −0.880284
\(507\) 5290.19 0.463404
\(508\) 4675.60 0.408359
\(509\) −15961.5 −1.38994 −0.694971 0.719038i \(-0.744583\pi\)
−0.694971 + 0.719038i \(0.744583\pi\)
\(510\) −2818.01 −0.244674
\(511\) 0 0
\(512\) 25108.8 2.16731
\(513\) 969.175 0.0834115
\(514\) 25429.6 2.18220
\(515\) −3862.62 −0.330500
\(516\) 3867.59 0.329963
\(517\) −4417.54 −0.375790
\(518\) 0 0
\(519\) −5770.83 −0.488076
\(520\) −4653.30 −0.392425
\(521\) 11583.9 0.974085 0.487043 0.873378i \(-0.338076\pi\)
0.487043 + 0.873378i \(0.338076\pi\)
\(522\) −5163.01 −0.432910
\(523\) −16086.9 −1.34499 −0.672497 0.740100i \(-0.734778\pi\)
−0.672497 + 0.740100i \(0.734778\pi\)
\(524\) 18244.9 1.52106
\(525\) 0 0
\(526\) −2953.67 −0.244840
\(527\) −91.6077 −0.00757210
\(528\) −4368.08 −0.360031
\(529\) 19056.0 1.56621
\(530\) 4214.93 0.345443
\(531\) −1912.97 −0.156339
\(532\) 0 0
\(533\) 4892.01 0.397555
\(534\) −19926.8 −1.61482
\(535\) −4366.15 −0.352832
\(536\) 44049.6 3.54972
\(537\) 6703.72 0.538709
\(538\) −17023.7 −1.36421
\(539\) 0 0
\(540\) −2056.12 −0.163854
\(541\) 9041.80 0.718553 0.359277 0.933231i \(-0.383023\pi\)
0.359277 + 0.933231i \(0.383023\pi\)
\(542\) 23808.5 1.88683
\(543\) −9405.60 −0.743338
\(544\) 10947.0 0.862774
\(545\) 2930.43 0.230323
\(546\) 0 0
\(547\) 9298.69 0.726843 0.363422 0.931625i \(-0.381608\pi\)
0.363422 + 0.931625i \(0.381608\pi\)
\(548\) 31266.4 2.43729
\(549\) 1360.73 0.105782
\(550\) 6134.67 0.475606
\(551\) −3994.67 −0.308854
\(552\) 28891.3 2.22771
\(553\) 0 0
\(554\) 31930.3 2.44871
\(555\) 1494.36 0.114292
\(556\) 44147.8 3.36742
\(557\) 4872.37 0.370644 0.185322 0.982678i \(-0.440667\pi\)
0.185322 + 0.982678i \(0.440667\pi\)
\(558\) −95.6306 −0.00725514
\(559\) −1445.40 −0.109363
\(560\) 0 0
\(561\) 1466.59 0.110374
\(562\) −42386.7 −3.18145
\(563\) −3636.78 −0.272242 −0.136121 0.990692i \(-0.543464\pi\)
−0.136121 + 0.990692i \(0.543464\pi\)
\(564\) 22376.2 1.67058
\(565\) 8812.92 0.656216
\(566\) 22803.2 1.69345
\(567\) 0 0
\(568\) −24724.8 −1.82646
\(569\) −11853.3 −0.873313 −0.436657 0.899628i \(-0.643837\pi\)
−0.436657 + 0.899628i \(0.643837\pi\)
\(570\) −2276.07 −0.167253
\(571\) −15122.7 −1.10834 −0.554172 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(572\) 4254.18 0.310972
\(573\) −7676.24 −0.559650
\(574\) 0 0
\(575\) −19116.9 −1.38649
\(576\) 1897.38 0.137253
\(577\) −21220.6 −1.53107 −0.765534 0.643395i \(-0.777525\pi\)
−0.765534 + 0.643395i \(0.777525\pi\)
\(578\) 15144.5 1.08984
\(579\) −6349.40 −0.455738
\(580\) 8474.74 0.606714
\(581\) 0 0
\(582\) −13058.1 −0.930028
\(583\) −2193.60 −0.155831
\(584\) 28468.8 2.01721
\(585\) 768.415 0.0543078
\(586\) 44321.4 3.12440
\(587\) −20307.3 −1.42789 −0.713946 0.700201i \(-0.753094\pi\)
−0.713946 + 0.700201i \(0.753094\pi\)
\(588\) 0 0
\(589\) −73.9902 −0.00517608
\(590\) 4492.54 0.313483
\(591\) 15608.6 1.08638
\(592\) −16080.6 −1.11640
\(593\) −19863.3 −1.37553 −0.687764 0.725934i \(-0.741408\pi\)
−0.687764 + 0.725934i \(0.741408\pi\)
\(594\) 1531.00 0.105754
\(595\) 0 0
\(596\) −23867.3 −1.64034
\(597\) 9612.11 0.658957
\(598\) −18967.2 −1.29703
\(599\) 9641.40 0.657658 0.328829 0.944389i \(-0.393346\pi\)
0.328829 + 0.944389i \(0.393346\pi\)
\(600\) −17689.2 −1.20360
\(601\) 24.4025 0.00165623 0.000828117 1.00000i \(-0.499736\pi\)
0.000828117 1.00000i \(0.499736\pi\)
\(602\) 0 0
\(603\) −7274.05 −0.491248
\(604\) −7346.23 −0.494891
\(605\) 496.126 0.0333395
\(606\) 7637.20 0.511947
\(607\) 10722.6 0.716995 0.358498 0.933531i \(-0.383289\pi\)
0.358498 + 0.933531i \(0.383289\pi\)
\(608\) 8841.73 0.589769
\(609\) 0 0
\(610\) −3195.61 −0.212109
\(611\) −8362.46 −0.553697
\(612\) −7428.75 −0.490669
\(613\) 2259.85 0.148898 0.0744490 0.997225i \(-0.476280\pi\)
0.0744490 + 0.997225i \(0.476280\pi\)
\(614\) 22004.8 1.44632
\(615\) −2889.81 −0.189477
\(616\) 0 0
\(617\) −10857.4 −0.708434 −0.354217 0.935163i \(-0.615253\pi\)
−0.354217 + 0.935163i \(0.615253\pi\)
\(618\) −14568.5 −0.948270
\(619\) −21603.8 −1.40279 −0.701397 0.712771i \(-0.747440\pi\)
−0.701397 + 0.712771i \(0.747440\pi\)
\(620\) 156.971 0.0101679
\(621\) −4770.91 −0.308293
\(622\) −42133.5 −2.71608
\(623\) 0 0
\(624\) −8268.83 −0.530478
\(625\) 9603.22 0.614606
\(626\) 22105.3 1.41135
\(627\) 1184.55 0.0754486
\(628\) 11912.7 0.756957
\(629\) 5399.10 0.342252
\(630\) 0 0
\(631\) 17836.8 1.12531 0.562657 0.826691i \(-0.309779\pi\)
0.562657 + 0.826691i \(0.309779\pi\)
\(632\) 5090.30 0.320381
\(633\) −4241.54 −0.266328
\(634\) −24485.8 −1.53384
\(635\) 1032.21 0.0645070
\(636\) 11111.3 0.692751
\(637\) 0 0
\(638\) −6310.35 −0.391582
\(639\) 4082.89 0.252765
\(640\) 3623.80 0.223818
\(641\) −7761.90 −0.478278 −0.239139 0.970985i \(-0.576865\pi\)
−0.239139 + 0.970985i \(0.576865\pi\)
\(642\) −16467.7 −1.01235
\(643\) −6432.71 −0.394528 −0.197264 0.980350i \(-0.563206\pi\)
−0.197264 + 0.980350i \(0.563206\pi\)
\(644\) 0 0
\(645\) 853.827 0.0521231
\(646\) −8223.43 −0.500846
\(647\) −3424.50 −0.208085 −0.104043 0.994573i \(-0.533178\pi\)
−0.104043 + 0.994573i \(0.533178\pi\)
\(648\) −4414.62 −0.267627
\(649\) −2338.08 −0.141414
\(650\) 11613.0 0.700769
\(651\) 0 0
\(652\) 25611.5 1.53838
\(653\) −8157.08 −0.488838 −0.244419 0.969670i \(-0.578597\pi\)
−0.244419 + 0.969670i \(0.578597\pi\)
\(654\) 11052.6 0.660842
\(655\) 4027.84 0.240276
\(656\) 31097.0 1.85081
\(657\) −4701.15 −0.279162
\(658\) 0 0
\(659\) 589.850 0.0348669 0.0174335 0.999848i \(-0.494450\pi\)
0.0174335 + 0.999848i \(0.494450\pi\)
\(660\) −2513.03 −0.148212
\(661\) 9917.28 0.583567 0.291783 0.956484i \(-0.405751\pi\)
0.291783 + 0.956484i \(0.405751\pi\)
\(662\) −18910.9 −1.11026
\(663\) 2776.28 0.162627
\(664\) −36309.1 −2.12209
\(665\) 0 0
\(666\) 5636.20 0.327926
\(667\) 19664.4 1.14154
\(668\) 45108.0 2.61270
\(669\) −7265.74 −0.419895
\(670\) 17082.8 0.985025
\(671\) 1663.11 0.0956835
\(672\) 0 0
\(673\) −17692.0 −1.01334 −0.506670 0.862140i \(-0.669124\pi\)
−0.506670 + 0.862140i \(0.669124\pi\)
\(674\) 30415.4 1.73821
\(675\) 2921.08 0.166567
\(676\) −32751.2 −1.86340
\(677\) −25180.5 −1.42949 −0.714746 0.699384i \(-0.753458\pi\)
−0.714746 + 0.699384i \(0.753458\pi\)
\(678\) 33239.4 1.88282
\(679\) 0 0
\(680\) 9931.41 0.560077
\(681\) 5258.08 0.295874
\(682\) −116.882 −0.00656252
\(683\) −1415.08 −0.0792778 −0.0396389 0.999214i \(-0.512621\pi\)
−0.0396389 + 0.999214i \(0.512621\pi\)
\(684\) −6000.09 −0.335408
\(685\) 6902.52 0.385010
\(686\) 0 0
\(687\) 8397.03 0.466327
\(688\) −9187.95 −0.509138
\(689\) −4152.51 −0.229605
\(690\) 11204.3 0.618174
\(691\) 8234.05 0.453311 0.226656 0.973975i \(-0.427221\pi\)
0.226656 + 0.973975i \(0.427221\pi\)
\(692\) 35726.8 1.96261
\(693\) 0 0
\(694\) 34956.1 1.91198
\(695\) 9746.29 0.531939
\(696\) 18195.8 0.990963
\(697\) −10440.9 −0.567398
\(698\) 23063.2 1.25065
\(699\) −11022.0 −0.596408
\(700\) 0 0
\(701\) 869.294 0.0468371 0.0234185 0.999726i \(-0.492545\pi\)
0.0234185 + 0.999726i \(0.492545\pi\)
\(702\) 2898.20 0.155820
\(703\) 4360.78 0.233954
\(704\) 2319.02 0.124150
\(705\) 4939.88 0.263896
\(706\) 62405.1 3.32669
\(707\) 0 0
\(708\) 11843.1 0.628658
\(709\) −14134.3 −0.748696 −0.374348 0.927288i \(-0.622133\pi\)
−0.374348 + 0.927288i \(0.622133\pi\)
\(710\) −9588.50 −0.506831
\(711\) −840.577 −0.0443377
\(712\) 70227.1 3.69645
\(713\) 364.228 0.0191311
\(714\) 0 0
\(715\) 939.174 0.0491232
\(716\) −41502.3 −2.16622
\(717\) −2473.49 −0.128834
\(718\) −39799.6 −2.06867
\(719\) −28596.7 −1.48328 −0.741638 0.670800i \(-0.765951\pi\)
−0.741638 + 0.670800i \(0.765951\pi\)
\(720\) 4884.57 0.252829
\(721\) 0 0
\(722\) 28715.4 1.48016
\(723\) 19721.4 1.01445
\(724\) 58229.4 2.98906
\(725\) −12039.9 −0.616759
\(726\) 1871.22 0.0956578
\(727\) 7952.80 0.405713 0.202856 0.979209i \(-0.434978\pi\)
0.202856 + 0.979209i \(0.434978\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 11040.5 0.559761
\(731\) 3084.88 0.156085
\(732\) −8424.15 −0.425363
\(733\) −11887.3 −0.599002 −0.299501 0.954096i \(-0.596820\pi\)
−0.299501 + 0.954096i \(0.596820\pi\)
\(734\) 41541.2 2.08898
\(735\) 0 0
\(736\) −43524.8 −2.17982
\(737\) −8890.51 −0.444350
\(738\) −10899.4 −0.543648
\(739\) 1657.84 0.0825231 0.0412615 0.999148i \(-0.486862\pi\)
0.0412615 + 0.999148i \(0.486862\pi\)
\(740\) −9251.45 −0.459581
\(741\) 2242.36 0.111168
\(742\) 0 0
\(743\) −13830.2 −0.682879 −0.341440 0.939904i \(-0.610914\pi\)
−0.341440 + 0.939904i \(0.610914\pi\)
\(744\) 337.027 0.0166076
\(745\) −5269.06 −0.259119
\(746\) −54588.5 −2.67913
\(747\) 5995.83 0.293676
\(748\) −9079.58 −0.443827
\(749\) 0 0
\(750\) −14786.1 −0.719882
\(751\) 10765.6 0.523095 0.261547 0.965191i \(-0.415767\pi\)
0.261547 + 0.965191i \(0.415767\pi\)
\(752\) −53157.5 −2.57773
\(753\) 13117.8 0.634844
\(754\) −11945.6 −0.576966
\(755\) −1621.79 −0.0781761
\(756\) 0 0
\(757\) 32858.7 1.57764 0.788819 0.614626i \(-0.210693\pi\)
0.788819 + 0.614626i \(0.210693\pi\)
\(758\) 12173.4 0.583323
\(759\) −5831.11 −0.278862
\(760\) 8021.46 0.382854
\(761\) 19534.7 0.930526 0.465263 0.885172i \(-0.345960\pi\)
0.465263 + 0.885172i \(0.345960\pi\)
\(762\) 3893.14 0.185083
\(763\) 0 0
\(764\) 47523.1 2.25042
\(765\) −1640.01 −0.0775092
\(766\) 55479.9 2.61693
\(767\) −4426.01 −0.208362
\(768\) 18727.4 0.879906
\(769\) 20557.6 0.964014 0.482007 0.876167i \(-0.339908\pi\)
0.482007 + 0.876167i \(0.339908\pi\)
\(770\) 0 0
\(771\) 14799.3 0.691290
\(772\) 39308.7 1.83258
\(773\) 18633.1 0.866994 0.433497 0.901155i \(-0.357280\pi\)
0.433497 + 0.901155i \(0.357280\pi\)
\(774\) 3220.35 0.149552
\(775\) −223.006 −0.0103363
\(776\) 46020.2 2.12890
\(777\) 0 0
\(778\) −32928.6 −1.51741
\(779\) −8432.95 −0.387859
\(780\) −4757.20 −0.218378
\(781\) 4990.20 0.228634
\(782\) 40481.1 1.85115
\(783\) −3004.73 −0.137140
\(784\) 0 0
\(785\) 2629.91 0.119574
\(786\) 15191.7 0.689400
\(787\) 11340.5 0.513655 0.256827 0.966457i \(-0.417323\pi\)
0.256827 + 0.966457i \(0.417323\pi\)
\(788\) −96631.5 −4.36847
\(789\) −1718.95 −0.0775619
\(790\) 1974.06 0.0889037
\(791\) 0 0
\(792\) −5395.64 −0.242078
\(793\) 3148.29 0.140982
\(794\) −36878.2 −1.64831
\(795\) 2452.97 0.109431
\(796\) −59507.9 −2.64975
\(797\) 30268.1 1.34523 0.672617 0.739991i \(-0.265170\pi\)
0.672617 + 0.739991i \(0.265170\pi\)
\(798\) 0 0
\(799\) 17847.8 0.790248
\(800\) 26648.9 1.17773
\(801\) −11596.8 −0.511553
\(802\) −48600.8 −2.13984
\(803\) −5745.85 −0.252511
\(804\) 45033.1 1.97537
\(805\) 0 0
\(806\) −221.259 −0.00966937
\(807\) −9907.34 −0.432162
\(808\) −26915.5 −1.17189
\(809\) 33341.8 1.44899 0.724496 0.689280i \(-0.242073\pi\)
0.724496 + 0.689280i \(0.242073\pi\)
\(810\) −1712.03 −0.0742648
\(811\) 19598.4 0.848572 0.424286 0.905528i \(-0.360525\pi\)
0.424286 + 0.905528i \(0.360525\pi\)
\(812\) 0 0
\(813\) 13855.9 0.597722
\(814\) 6888.69 0.296620
\(815\) 5654.12 0.243012
\(816\) 17647.9 0.757110
\(817\) 2491.61 0.106696
\(818\) −36770.5 −1.57170
\(819\) 0 0
\(820\) 17890.6 0.761911
\(821\) 25874.0 1.09989 0.549945 0.835201i \(-0.314649\pi\)
0.549945 + 0.835201i \(0.314649\pi\)
\(822\) 26034.0 1.10467
\(823\) 6010.03 0.254552 0.127276 0.991867i \(-0.459377\pi\)
0.127276 + 0.991867i \(0.459377\pi\)
\(824\) 51343.2 2.17066
\(825\) 3570.21 0.150665
\(826\) 0 0
\(827\) −22679.5 −0.953619 −0.476810 0.879007i \(-0.658207\pi\)
−0.476810 + 0.879007i \(0.658207\pi\)
\(828\) 29536.4 1.23969
\(829\) −45120.9 −1.89037 −0.945183 0.326541i \(-0.894117\pi\)
−0.945183 + 0.326541i \(0.894117\pi\)
\(830\) −14081.0 −0.588865
\(831\) 18582.6 0.775718
\(832\) 4389.93 0.182925
\(833\) 0 0
\(834\) 36759.7 1.52624
\(835\) 9958.26 0.412718
\(836\) −7333.44 −0.303388
\(837\) −55.6544 −0.00229833
\(838\) 13932.3 0.574322
\(839\) −27012.2 −1.11152 −0.555760 0.831343i \(-0.687573\pi\)
−0.555760 + 0.831343i \(0.687573\pi\)
\(840\) 0 0
\(841\) −12004.3 −0.492203
\(842\) −73135.3 −2.99336
\(843\) −24667.9 −1.00784
\(844\) 26259.0 1.07094
\(845\) −7230.31 −0.294355
\(846\) 18631.5 0.757170
\(847\) 0 0
\(848\) −26396.2 −1.06893
\(849\) 13270.9 0.536460
\(850\) −24785.3 −1.00015
\(851\) −21466.6 −0.864707
\(852\) −25276.9 −1.01640
\(853\) −34662.5 −1.39135 −0.695675 0.718357i \(-0.744895\pi\)
−0.695675 + 0.718357i \(0.744895\pi\)
\(854\) 0 0
\(855\) −1324.61 −0.0529832
\(856\) 58036.3 2.31734
\(857\) 2799.64 0.111592 0.0557958 0.998442i \(-0.482230\pi\)
0.0557958 + 0.998442i \(0.482230\pi\)
\(858\) 3542.25 0.140944
\(859\) −39111.2 −1.55350 −0.776750 0.629809i \(-0.783133\pi\)
−0.776750 + 0.629809i \(0.783133\pi\)
\(860\) −5285.98 −0.209594
\(861\) 0 0
\(862\) 68126.0 2.69186
\(863\) 16191.6 0.638667 0.319333 0.947642i \(-0.396541\pi\)
0.319333 + 0.947642i \(0.396541\pi\)
\(864\) 6650.63 0.261874
\(865\) 7887.22 0.310027
\(866\) 53995.0 2.11874
\(867\) 8813.66 0.345245
\(868\) 0 0
\(869\) −1027.37 −0.0401050
\(870\) 7056.49 0.274986
\(871\) −16829.8 −0.654716
\(872\) −38952.2 −1.51272
\(873\) −7599.47 −0.294620
\(874\) 32696.0 1.26540
\(875\) 0 0
\(876\) 29104.5 1.12255
\(877\) −20172.6 −0.776715 −0.388358 0.921509i \(-0.626957\pi\)
−0.388358 + 0.921509i \(0.626957\pi\)
\(878\) −63168.8 −2.42807
\(879\) 25793.9 0.989767
\(880\) 5970.03 0.228693
\(881\) −15801.5 −0.604277 −0.302138 0.953264i \(-0.597700\pi\)
−0.302138 + 0.953264i \(0.597700\pi\)
\(882\) 0 0
\(883\) −29892.5 −1.13926 −0.569628 0.821902i \(-0.692913\pi\)
−0.569628 + 0.821902i \(0.692913\pi\)
\(884\) −17187.8 −0.653945
\(885\) 2614.53 0.0993069
\(886\) −1695.78 −0.0643010
\(887\) −27083.3 −1.02522 −0.512608 0.858623i \(-0.671321\pi\)
−0.512608 + 0.858623i \(0.671321\pi\)
\(888\) −19863.5 −0.750647
\(889\) 0 0
\(890\) 27234.7 1.02574
\(891\) 891.000 0.0335013
\(892\) 44981.6 1.68845
\(893\) 14415.4 0.540193
\(894\) −19873.1 −0.743464
\(895\) −9162.24 −0.342190
\(896\) 0 0
\(897\) −11038.4 −0.410881
\(898\) 66934.7 2.48735
\(899\) 229.392 0.00851018
\(900\) −18084.2 −0.669786
\(901\) 8862.59 0.327698
\(902\) −13321.5 −0.491748
\(903\) 0 0
\(904\) −117144. −4.30991
\(905\) 12855.0 0.472171
\(906\) −6116.84 −0.224303
\(907\) −27867.6 −1.02021 −0.510104 0.860113i \(-0.670393\pi\)
−0.510104 + 0.860113i \(0.670393\pi\)
\(908\) −32552.4 −1.18975
\(909\) 4444.64 0.162178
\(910\) 0 0
\(911\) −22233.0 −0.808575 −0.404288 0.914632i \(-0.632481\pi\)
−0.404288 + 0.914632i \(0.632481\pi\)
\(912\) 14254.0 0.517540
\(913\) 7328.24 0.265640
\(914\) −73920.9 −2.67515
\(915\) −1859.76 −0.0671931
\(916\) −51985.4 −1.87516
\(917\) 0 0
\(918\) −6185.55 −0.222390
\(919\) 15831.1 0.568247 0.284124 0.958788i \(-0.408297\pi\)
0.284124 + 0.958788i \(0.408297\pi\)
\(920\) −39486.8 −1.41505
\(921\) 12806.2 0.458174
\(922\) −93396.1 −3.33605
\(923\) 9446.51 0.336875
\(924\) 0 0
\(925\) 13143.3 0.467190
\(926\) 54608.9 1.93797
\(927\) −8478.47 −0.300399
\(928\) −27412.0 −0.969660
\(929\) 10402.2 0.367369 0.183684 0.982985i \(-0.441198\pi\)
0.183684 + 0.982985i \(0.441198\pi\)
\(930\) 130.702 0.00460849
\(931\) 0 0
\(932\) 68236.2 2.39823
\(933\) −24520.6 −0.860415
\(934\) 70644.7 2.47491
\(935\) −2004.45 −0.0701097
\(936\) −10214.0 −0.356683
\(937\) −42248.2 −1.47299 −0.736493 0.676445i \(-0.763520\pi\)
−0.736493 + 0.676445i \(0.763520\pi\)
\(938\) 0 0
\(939\) 12864.7 0.447096
\(940\) −30582.4 −1.06116
\(941\) −32247.6 −1.11715 −0.558576 0.829453i \(-0.688652\pi\)
−0.558576 + 0.829453i \(0.688652\pi\)
\(942\) 9919.13 0.343081
\(943\) 41512.5 1.43354
\(944\) −28134.7 −0.970030
\(945\) 0 0
\(946\) 3935.98 0.135275
\(947\) −46963.8 −1.61153 −0.805764 0.592237i \(-0.798245\pi\)
−0.805764 + 0.592237i \(0.798245\pi\)
\(948\) 5203.95 0.178287
\(949\) −10877.0 −0.372056
\(950\) −20018.7 −0.683678
\(951\) −14250.1 −0.485899
\(952\) 0 0
\(953\) −29188.1 −0.992126 −0.496063 0.868286i \(-0.665222\pi\)
−0.496063 + 0.868286i \(0.665222\pi\)
\(954\) 9251.79 0.313981
\(955\) 10491.4 0.355492
\(956\) 15313.2 0.518059
\(957\) −3672.45 −0.124048
\(958\) 40323.3 1.35990
\(959\) 0 0
\(960\) −2593.22 −0.0871832
\(961\) −29786.8 −0.999857
\(962\) 13040.4 0.437047
\(963\) −9583.73 −0.320697
\(964\) −122094. −4.07922
\(965\) 8677.98 0.289486
\(966\) 0 0
\(967\) −2456.33 −0.0816860 −0.0408430 0.999166i \(-0.513004\pi\)
−0.0408430 + 0.999166i \(0.513004\pi\)
\(968\) −6594.67 −0.218968
\(969\) −4785.81 −0.158661
\(970\) 17847.0 0.590757
\(971\) −36668.5 −1.21189 −0.605947 0.795505i \(-0.707206\pi\)
−0.605947 + 0.795505i \(0.707206\pi\)
\(972\) −4513.19 −0.148931
\(973\) 0 0
\(974\) 20894.0 0.687359
\(975\) 6758.46 0.221994
\(976\) 20012.7 0.656342
\(977\) −14293.9 −0.468068 −0.234034 0.972228i \(-0.575193\pi\)
−0.234034 + 0.972228i \(0.575193\pi\)
\(978\) 21325.4 0.697252
\(979\) −14173.9 −0.462717
\(980\) 0 0
\(981\) 6432.31 0.209345
\(982\) −19795.8 −0.643287
\(983\) −13150.4 −0.426686 −0.213343 0.976977i \(-0.568435\pi\)
−0.213343 + 0.976977i \(0.568435\pi\)
\(984\) 38412.3 1.24445
\(985\) −21332.8 −0.690072
\(986\) 25495.1 0.823458
\(987\) 0 0
\(988\) −13882.3 −0.447019
\(989\) −12265.3 −0.394353
\(990\) −2092.48 −0.0671750
\(991\) 7110.30 0.227917 0.113959 0.993485i \(-0.463647\pi\)
0.113959 + 0.993485i \(0.463647\pi\)
\(992\) −507.733 −0.0162505
\(993\) −11005.6 −0.351715
\(994\) 0 0
\(995\) −13137.3 −0.418572
\(996\) −37119.8 −1.18091
\(997\) 3722.83 0.118258 0.0591290 0.998250i \(-0.481168\pi\)
0.0591290 + 0.998250i \(0.481168\pi\)
\(998\) 109322. 3.46746
\(999\) 3280.12 0.103882
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.1.1 16
7.6 odd 2 1617.4.a.bf.1.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.1 16 1.1 even 1 trivial
1617.4.a.bf.1.1 yes 16 7.6 odd 2