Properties

Label 1617.4.a.be.1.3
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 1.3
Root \(-4.31445\) 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)
 
\(f(q)\) \(=\) \(q-4.31445 q^{2} -3.00000 q^{3} +10.6145 q^{4} -14.6492 q^{5} +12.9433 q^{6} -11.2800 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.31445 q^{2} -3.00000 q^{3} +10.6145 q^{4} -14.6492 q^{5} +12.9433 q^{6} -11.2800 q^{8} +9.00000 q^{9} +63.2031 q^{10} +11.0000 q^{11} -31.8434 q^{12} -24.6988 q^{13} +43.9475 q^{15} -36.2488 q^{16} +61.9469 q^{17} -38.8300 q^{18} -16.6166 q^{19} -155.493 q^{20} -47.4589 q^{22} -50.6775 q^{23} +33.8400 q^{24} +89.5985 q^{25} +106.562 q^{26} -27.0000 q^{27} -100.342 q^{29} -189.609 q^{30} -113.542 q^{31} +246.634 q^{32} -33.0000 q^{33} -267.267 q^{34} +95.5302 q^{36} -1.84796 q^{37} +71.6914 q^{38} +74.0963 q^{39} +165.243 q^{40} +62.0499 q^{41} -302.869 q^{43} +116.759 q^{44} -131.843 q^{45} +218.645 q^{46} -41.4573 q^{47} +108.746 q^{48} -386.568 q^{50} -185.841 q^{51} -262.164 q^{52} +678.402 q^{53} +116.490 q^{54} -161.141 q^{55} +49.8498 q^{57} +432.921 q^{58} -144.369 q^{59} +466.480 q^{60} -228.420 q^{61} +489.871 q^{62} -774.097 q^{64} +361.817 q^{65} +142.377 q^{66} +482.768 q^{67} +657.534 q^{68} +152.032 q^{69} +559.130 q^{71} -101.520 q^{72} -218.369 q^{73} +7.97293 q^{74} -268.796 q^{75} -176.376 q^{76} -319.685 q^{78} +1175.69 q^{79} +531.015 q^{80} +81.0000 q^{81} -267.711 q^{82} -187.873 q^{83} -907.471 q^{85} +1306.71 q^{86} +301.027 q^{87} -124.080 q^{88} +34.7130 q^{89} +568.828 q^{90} -537.915 q^{92} +340.626 q^{93} +178.865 q^{94} +243.419 q^{95} -739.901 q^{96} -712.441 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.31445 −1.52539 −0.762694 0.646759i \(-0.776124\pi\)
−0.762694 + 0.646759i \(0.776124\pi\)
\(3\) −3.00000 −0.577350
\(4\) 10.6145 1.32681
\(5\) −14.6492 −1.31026 −0.655131 0.755515i \(-0.727387\pi\)
−0.655131 + 0.755515i \(0.727387\pi\)
\(6\) 12.9433 0.880683
\(7\) 0 0
\(8\) −11.2800 −0.498510
\(9\) 9.00000 0.333333
\(10\) 63.2031 1.99866
\(11\) 11.0000 0.301511
\(12\) −31.8434 −0.766033
\(13\) −24.6988 −0.526939 −0.263469 0.964668i \(-0.584867\pi\)
−0.263469 + 0.964668i \(0.584867\pi\)
\(14\) 0 0
\(15\) 43.9475 0.756480
\(16\) −36.2488 −0.566387
\(17\) 61.9469 0.883784 0.441892 0.897068i \(-0.354307\pi\)
0.441892 + 0.897068i \(0.354307\pi\)
\(18\) −38.8300 −0.508463
\(19\) −16.6166 −0.200637 −0.100319 0.994955i \(-0.531986\pi\)
−0.100319 + 0.994955i \(0.531986\pi\)
\(20\) −155.493 −1.73847
\(21\) 0 0
\(22\) −47.4589 −0.459922
\(23\) −50.6775 −0.459434 −0.229717 0.973257i \(-0.573780\pi\)
−0.229717 + 0.973257i \(0.573780\pi\)
\(24\) 33.8400 0.287815
\(25\) 89.5985 0.716788
\(26\) 106.562 0.803786
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −100.342 −0.642520 −0.321260 0.946991i \(-0.604106\pi\)
−0.321260 + 0.946991i \(0.604106\pi\)
\(30\) −189.609 −1.15393
\(31\) −113.542 −0.657830 −0.328915 0.944359i \(-0.606683\pi\)
−0.328915 + 0.944359i \(0.606683\pi\)
\(32\) 246.634 1.36247
\(33\) −33.0000 −0.174078
\(34\) −267.267 −1.34811
\(35\) 0 0
\(36\) 95.5302 0.442270
\(37\) −1.84796 −0.00821088 −0.00410544 0.999992i \(-0.501307\pi\)
−0.00410544 + 0.999992i \(0.501307\pi\)
\(38\) 71.6914 0.306050
\(39\) 74.0963 0.304228
\(40\) 165.243 0.653179
\(41\) 62.0499 0.236355 0.118178 0.992992i \(-0.462295\pi\)
0.118178 + 0.992992i \(0.462295\pi\)
\(42\) 0 0
\(43\) −302.869 −1.07412 −0.537059 0.843545i \(-0.680465\pi\)
−0.537059 + 0.843545i \(0.680465\pi\)
\(44\) 116.759 0.400048
\(45\) −131.843 −0.436754
\(46\) 218.645 0.700815
\(47\) −41.4573 −0.128663 −0.0643316 0.997929i \(-0.520492\pi\)
−0.0643316 + 0.997929i \(0.520492\pi\)
\(48\) 108.746 0.327004
\(49\) 0 0
\(50\) −386.568 −1.09338
\(51\) −185.841 −0.510253
\(52\) −262.164 −0.699147
\(53\) 678.402 1.75822 0.879110 0.476619i \(-0.158138\pi\)
0.879110 + 0.476619i \(0.158138\pi\)
\(54\) 116.490 0.293561
\(55\) −161.141 −0.395059
\(56\) 0 0
\(57\) 49.8498 0.115838
\(58\) 432.921 0.980092
\(59\) −144.369 −0.318563 −0.159281 0.987233i \(-0.550918\pi\)
−0.159281 + 0.987233i \(0.550918\pi\)
\(60\) 466.480 1.00370
\(61\) −228.420 −0.479446 −0.239723 0.970841i \(-0.577057\pi\)
−0.239723 + 0.970841i \(0.577057\pi\)
\(62\) 489.871 1.00345
\(63\) 0 0
\(64\) −774.097 −1.51191
\(65\) 361.817 0.690428
\(66\) 142.377 0.265536
\(67\) 482.768 0.880291 0.440146 0.897926i \(-0.354927\pi\)
0.440146 + 0.897926i \(0.354927\pi\)
\(68\) 657.534 1.17261
\(69\) 152.032 0.265254
\(70\) 0 0
\(71\) 559.130 0.934599 0.467299 0.884099i \(-0.345227\pi\)
0.467299 + 0.884099i \(0.345227\pi\)
\(72\) −101.520 −0.166170
\(73\) −218.369 −0.350112 −0.175056 0.984558i \(-0.556011\pi\)
−0.175056 + 0.984558i \(0.556011\pi\)
\(74\) 7.97293 0.0125248
\(75\) −268.796 −0.413838
\(76\) −176.376 −0.266207
\(77\) 0 0
\(78\) −319.685 −0.464066
\(79\) 1175.69 1.67438 0.837189 0.546914i \(-0.184197\pi\)
0.837189 + 0.546914i \(0.184197\pi\)
\(80\) 531.015 0.742116
\(81\) 81.0000 0.111111
\(82\) −267.711 −0.360534
\(83\) −187.873 −0.248455 −0.124227 0.992254i \(-0.539645\pi\)
−0.124227 + 0.992254i \(0.539645\pi\)
\(84\) 0 0
\(85\) −907.471 −1.15799
\(86\) 1306.71 1.63845
\(87\) 301.027 0.370959
\(88\) −124.080 −0.150306
\(89\) 34.7130 0.0413435 0.0206718 0.999786i \(-0.493420\pi\)
0.0206718 + 0.999786i \(0.493420\pi\)
\(90\) 568.828 0.666220
\(91\) 0 0
\(92\) −537.915 −0.609581
\(93\) 340.626 0.379798
\(94\) 178.865 0.196261
\(95\) 243.419 0.262887
\(96\) −739.901 −0.786623
\(97\) −712.441 −0.745747 −0.372873 0.927882i \(-0.621627\pi\)
−0.372873 + 0.927882i \(0.621627\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 951.041 0.951041
\(101\) 89.9994 0.0886661 0.0443331 0.999017i \(-0.485884\pi\)
0.0443331 + 0.999017i \(0.485884\pi\)
\(102\) 801.800 0.778334
\(103\) −1109.04 −1.06094 −0.530469 0.847705i \(-0.677984\pi\)
−0.530469 + 0.847705i \(0.677984\pi\)
\(104\) 278.602 0.262684
\(105\) 0 0
\(106\) −2926.93 −2.68197
\(107\) 950.441 0.858716 0.429358 0.903134i \(-0.358740\pi\)
0.429358 + 0.903134i \(0.358740\pi\)
\(108\) −286.591 −0.255344
\(109\) 1574.34 1.38343 0.691716 0.722169i \(-0.256855\pi\)
0.691716 + 0.722169i \(0.256855\pi\)
\(110\) 695.235 0.602618
\(111\) 5.54388 0.00474056
\(112\) 0 0
\(113\) 1593.80 1.32683 0.663417 0.748250i \(-0.269105\pi\)
0.663417 + 0.748250i \(0.269105\pi\)
\(114\) −215.074 −0.176698
\(115\) 742.384 0.601979
\(116\) −1065.08 −0.852501
\(117\) −222.289 −0.175646
\(118\) 622.872 0.485932
\(119\) 0 0
\(120\) −495.728 −0.377113
\(121\) 121.000 0.0909091
\(122\) 985.508 0.731342
\(123\) −186.150 −0.136460
\(124\) −1205.19 −0.872815
\(125\) 518.603 0.371082
\(126\) 0 0
\(127\) −2056.40 −1.43682 −0.718409 0.695621i \(-0.755130\pi\)
−0.718409 + 0.695621i \(0.755130\pi\)
\(128\) 1366.74 0.943777
\(129\) 908.607 0.620142
\(130\) −1561.04 −1.05317
\(131\) −95.2553 −0.0635305 −0.0317652 0.999495i \(-0.510113\pi\)
−0.0317652 + 0.999495i \(0.510113\pi\)
\(132\) −350.278 −0.230968
\(133\) 0 0
\(134\) −2082.88 −1.34279
\(135\) 395.528 0.252160
\(136\) −698.761 −0.440575
\(137\) 1468.93 0.916052 0.458026 0.888939i \(-0.348557\pi\)
0.458026 + 0.888939i \(0.348557\pi\)
\(138\) −655.936 −0.404616
\(139\) 1864.27 1.13759 0.568795 0.822479i \(-0.307410\pi\)
0.568795 + 0.822479i \(0.307410\pi\)
\(140\) 0 0
\(141\) 124.372 0.0742837
\(142\) −2412.34 −1.42563
\(143\) −271.686 −0.158878
\(144\) −326.239 −0.188796
\(145\) 1469.93 0.841870
\(146\) 942.142 0.534056
\(147\) 0 0
\(148\) −19.6151 −0.0108943
\(149\) 485.799 0.267102 0.133551 0.991042i \(-0.457362\pi\)
0.133551 + 0.991042i \(0.457362\pi\)
\(150\) 1159.70 0.631263
\(151\) −2444.09 −1.31720 −0.658600 0.752494i \(-0.728851\pi\)
−0.658600 + 0.752494i \(0.728851\pi\)
\(152\) 187.435 0.100020
\(153\) 557.522 0.294595
\(154\) 0 0
\(155\) 1663.30 0.861930
\(156\) 786.493 0.403653
\(157\) 3473.99 1.76595 0.882977 0.469417i \(-0.155536\pi\)
0.882977 + 0.469417i \(0.155536\pi\)
\(158\) −5072.47 −2.55408
\(159\) −2035.21 −1.01511
\(160\) −3612.98 −1.78519
\(161\) 0 0
\(162\) −349.470 −0.169488
\(163\) −1383.85 −0.664977 −0.332488 0.943107i \(-0.607888\pi\)
−0.332488 + 0.943107i \(0.607888\pi\)
\(164\) 658.627 0.313598
\(165\) 483.423 0.228087
\(166\) 810.569 0.378990
\(167\) 2609.95 1.20936 0.604682 0.796467i \(-0.293300\pi\)
0.604682 + 0.796467i \(0.293300\pi\)
\(168\) 0 0
\(169\) −1586.97 −0.722335
\(170\) 3915.24 1.76638
\(171\) −149.549 −0.0668791
\(172\) −3214.79 −1.42515
\(173\) −1836.78 −0.807213 −0.403607 0.914933i \(-0.632244\pi\)
−0.403607 + 0.914933i \(0.632244\pi\)
\(174\) −1298.76 −0.565857
\(175\) 0 0
\(176\) −398.737 −0.170772
\(177\) 433.106 0.183922
\(178\) −149.768 −0.0630649
\(179\) 3474.43 1.45079 0.725394 0.688334i \(-0.241657\pi\)
0.725394 + 0.688334i \(0.241657\pi\)
\(180\) −1399.44 −0.579489
\(181\) 1930.00 0.792574 0.396287 0.918127i \(-0.370298\pi\)
0.396287 + 0.918127i \(0.370298\pi\)
\(182\) 0 0
\(183\) 685.261 0.276808
\(184\) 571.642 0.229033
\(185\) 27.0711 0.0107584
\(186\) −1469.61 −0.579340
\(187\) 681.416 0.266471
\(188\) −440.047 −0.170711
\(189\) 0 0
\(190\) −1050.22 −0.401005
\(191\) 4739.40 1.79545 0.897726 0.440555i \(-0.145218\pi\)
0.897726 + 0.440555i \(0.145218\pi\)
\(192\) 2322.29 0.872901
\(193\) 616.631 0.229980 0.114990 0.993367i \(-0.463316\pi\)
0.114990 + 0.993367i \(0.463316\pi\)
\(194\) 3073.79 1.13755
\(195\) −1085.45 −0.398619
\(196\) 0 0
\(197\) 266.547 0.0963996 0.0481998 0.998838i \(-0.484652\pi\)
0.0481998 + 0.998838i \(0.484652\pi\)
\(198\) −427.130 −0.153307
\(199\) −1763.43 −0.628173 −0.314086 0.949394i \(-0.601698\pi\)
−0.314086 + 0.949394i \(0.601698\pi\)
\(200\) −1010.67 −0.357326
\(201\) −1448.30 −0.508236
\(202\) −388.298 −0.135250
\(203\) 0 0
\(204\) −1972.60 −0.677008
\(205\) −908.981 −0.309688
\(206\) 4784.88 1.61834
\(207\) −456.097 −0.153145
\(208\) 895.300 0.298451
\(209\) −182.783 −0.0604944
\(210\) 0 0
\(211\) −1939.41 −0.632769 −0.316384 0.948631i \(-0.602469\pi\)
−0.316384 + 0.948631i \(0.602469\pi\)
\(212\) 7200.87 2.33282
\(213\) −1677.39 −0.539591
\(214\) −4100.63 −1.30988
\(215\) 4436.78 1.40738
\(216\) 304.560 0.0959383
\(217\) 0 0
\(218\) −6792.40 −2.11027
\(219\) 655.107 0.202137
\(220\) −1710.43 −0.524168
\(221\) −1530.01 −0.465700
\(222\) −23.9188 −0.00723119
\(223\) −1505.23 −0.452007 −0.226004 0.974126i \(-0.572566\pi\)
−0.226004 + 0.974126i \(0.572566\pi\)
\(224\) 0 0
\(225\) 806.387 0.238929
\(226\) −6876.38 −2.02394
\(227\) −5793.53 −1.69397 −0.846983 0.531620i \(-0.821583\pi\)
−0.846983 + 0.531620i \(0.821583\pi\)
\(228\) 529.129 0.153695
\(229\) 5867.31 1.69311 0.846557 0.532299i \(-0.178672\pi\)
0.846557 + 0.532299i \(0.178672\pi\)
\(230\) −3202.98 −0.918252
\(231\) 0 0
\(232\) 1131.86 0.320303
\(233\) 1959.31 0.550894 0.275447 0.961316i \(-0.411174\pi\)
0.275447 + 0.961316i \(0.411174\pi\)
\(234\) 959.054 0.267929
\(235\) 607.316 0.168583
\(236\) −1532.40 −0.422672
\(237\) −3527.08 −0.966703
\(238\) 0 0
\(239\) 407.911 0.110400 0.0552000 0.998475i \(-0.482420\pi\)
0.0552000 + 0.998475i \(0.482420\pi\)
\(240\) −1593.05 −0.428461
\(241\) −3637.51 −0.972251 −0.486125 0.873889i \(-0.661590\pi\)
−0.486125 + 0.873889i \(0.661590\pi\)
\(242\) −522.048 −0.138672
\(243\) −243.000 −0.0641500
\(244\) −2424.56 −0.636134
\(245\) 0 0
\(246\) 803.134 0.208154
\(247\) 410.409 0.105724
\(248\) 1280.75 0.327935
\(249\) 563.619 0.143445
\(250\) −2237.48 −0.566044
\(251\) 603.356 0.151727 0.0758635 0.997118i \(-0.475829\pi\)
0.0758635 + 0.997118i \(0.475829\pi\)
\(252\) 0 0
\(253\) −557.452 −0.138525
\(254\) 8872.23 2.19171
\(255\) 2722.41 0.668566
\(256\) 296.068 0.0722823
\(257\) −3594.38 −0.872418 −0.436209 0.899845i \(-0.643679\pi\)
−0.436209 + 0.899845i \(0.643679\pi\)
\(258\) −3920.14 −0.945958
\(259\) 0 0
\(260\) 3840.49 0.916066
\(261\) −903.080 −0.214173
\(262\) 410.974 0.0969087
\(263\) −3805.24 −0.892173 −0.446086 0.894990i \(-0.647183\pi\)
−0.446086 + 0.894990i \(0.647183\pi\)
\(264\) 372.240 0.0867795
\(265\) −9938.03 −2.30373
\(266\) 0 0
\(267\) −104.139 −0.0238697
\(268\) 5124.33 1.16798
\(269\) −6927.58 −1.57019 −0.785097 0.619373i \(-0.787387\pi\)
−0.785097 + 0.619373i \(0.787387\pi\)
\(270\) −1706.48 −0.384642
\(271\) 853.216 0.191251 0.0956257 0.995417i \(-0.469515\pi\)
0.0956257 + 0.995417i \(0.469515\pi\)
\(272\) −2245.50 −0.500564
\(273\) 0 0
\(274\) −6337.62 −1.39734
\(275\) 985.584 0.216120
\(276\) 1613.74 0.351942
\(277\) 5412.54 1.17404 0.587018 0.809574i \(-0.300302\pi\)
0.587018 + 0.809574i \(0.300302\pi\)
\(278\) −8043.29 −1.73527
\(279\) −1021.88 −0.219277
\(280\) 0 0
\(281\) −659.873 −0.140088 −0.0700440 0.997544i \(-0.522314\pi\)
−0.0700440 + 0.997544i \(0.522314\pi\)
\(282\) −536.596 −0.113312
\(283\) 5841.18 1.22693 0.613467 0.789721i \(-0.289774\pi\)
0.613467 + 0.789721i \(0.289774\pi\)
\(284\) 5934.87 1.24003
\(285\) −730.258 −0.151778
\(286\) 1172.18 0.242351
\(287\) 0 0
\(288\) 2219.70 0.454157
\(289\) −1075.58 −0.218925
\(290\) −6341.94 −1.28418
\(291\) 2137.32 0.430557
\(292\) −2317.87 −0.464531
\(293\) −3035.44 −0.605230 −0.302615 0.953113i \(-0.597860\pi\)
−0.302615 + 0.953113i \(0.597860\pi\)
\(294\) 0 0
\(295\) 2114.88 0.417401
\(296\) 20.8450 0.00409321
\(297\) −297.000 −0.0580259
\(298\) −2095.95 −0.407434
\(299\) 1251.67 0.242094
\(300\) −2853.12 −0.549084
\(301\) 0 0
\(302\) 10544.9 2.00924
\(303\) −269.998 −0.0511914
\(304\) 602.331 0.113638
\(305\) 3346.17 0.628201
\(306\) −2405.40 −0.449371
\(307\) 4354.54 0.809534 0.404767 0.914420i \(-0.367353\pi\)
0.404767 + 0.914420i \(0.367353\pi\)
\(308\) 0 0
\(309\) 3327.11 0.612532
\(310\) −7176.21 −1.31478
\(311\) −3364.84 −0.613513 −0.306756 0.951788i \(-0.599244\pi\)
−0.306756 + 0.951788i \(0.599244\pi\)
\(312\) −835.806 −0.151661
\(313\) −7739.96 −1.39773 −0.698863 0.715255i \(-0.746310\pi\)
−0.698863 + 0.715255i \(0.746310\pi\)
\(314\) −14988.4 −2.69376
\(315\) 0 0
\(316\) 12479.4 2.22158
\(317\) −10892.4 −1.92990 −0.964949 0.262438i \(-0.915474\pi\)
−0.964949 + 0.262438i \(0.915474\pi\)
\(318\) 8780.79 1.54843
\(319\) −1103.76 −0.193727
\(320\) 11339.9 1.98100
\(321\) −2851.32 −0.495780
\(322\) 0 0
\(323\) −1029.35 −0.177320
\(324\) 859.772 0.147423
\(325\) −2212.97 −0.377704
\(326\) 5970.53 1.01435
\(327\) −4723.01 −0.798725
\(328\) −699.923 −0.117826
\(329\) 0 0
\(330\) −2085.70 −0.347922
\(331\) 4314.69 0.716486 0.358243 0.933628i \(-0.383376\pi\)
0.358243 + 0.933628i \(0.383376\pi\)
\(332\) −1994.17 −0.329652
\(333\) −16.6316 −0.00273696
\(334\) −11260.5 −1.84475
\(335\) −7072.16 −1.15341
\(336\) 0 0
\(337\) −5671.76 −0.916797 −0.458398 0.888747i \(-0.651577\pi\)
−0.458398 + 0.888747i \(0.651577\pi\)
\(338\) 6846.91 1.10184
\(339\) −4781.41 −0.766048
\(340\) −9632.33 −1.53643
\(341\) −1248.96 −0.198343
\(342\) 645.223 0.102017
\(343\) 0 0
\(344\) 3416.36 0.535459
\(345\) −2227.15 −0.347553
\(346\) 7924.70 1.23131
\(347\) −3069.37 −0.474848 −0.237424 0.971406i \(-0.576303\pi\)
−0.237424 + 0.971406i \(0.576303\pi\)
\(348\) 3195.24 0.492192
\(349\) −81.8995 −0.0125615 −0.00628077 0.999980i \(-0.501999\pi\)
−0.00628077 + 0.999980i \(0.501999\pi\)
\(350\) 0 0
\(351\) 666.867 0.101409
\(352\) 2712.97 0.410800
\(353\) −11726.5 −1.76809 −0.884047 0.467399i \(-0.845191\pi\)
−0.884047 + 0.467399i \(0.845191\pi\)
\(354\) −1868.61 −0.280553
\(355\) −8190.80 −1.22457
\(356\) 368.460 0.0548550
\(357\) 0 0
\(358\) −14990.3 −2.21302
\(359\) −9275.28 −1.36359 −0.681797 0.731541i \(-0.738801\pi\)
−0.681797 + 0.731541i \(0.738801\pi\)
\(360\) 1487.18 0.217726
\(361\) −6582.89 −0.959745
\(362\) −8326.90 −1.20898
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) 3198.93 0.458739
\(366\) −2956.52 −0.422240
\(367\) −3892.49 −0.553641 −0.276820 0.960922i \(-0.589281\pi\)
−0.276820 + 0.960922i \(0.589281\pi\)
\(368\) 1837.00 0.260218
\(369\) 558.449 0.0787851
\(370\) −116.797 −0.0164108
\(371\) 0 0
\(372\) 3615.56 0.503920
\(373\) −11469.4 −1.59213 −0.796063 0.605213i \(-0.793088\pi\)
−0.796063 + 0.605213i \(0.793088\pi\)
\(374\) −2939.93 −0.406472
\(375\) −1555.81 −0.214244
\(376\) 467.638 0.0641399
\(377\) 2478.33 0.338569
\(378\) 0 0
\(379\) 6852.41 0.928719 0.464360 0.885647i \(-0.346284\pi\)
0.464360 + 0.885647i \(0.346284\pi\)
\(380\) 2583.77 0.348801
\(381\) 6169.20 0.829548
\(382\) −20447.9 −2.73876
\(383\) −6017.45 −0.802813 −0.401406 0.915900i \(-0.631478\pi\)
−0.401406 + 0.915900i \(0.631478\pi\)
\(384\) −4100.21 −0.544890
\(385\) 0 0
\(386\) −2660.42 −0.350808
\(387\) −2725.82 −0.358039
\(388\) −7562.18 −0.989463
\(389\) 9876.82 1.28734 0.643669 0.765304i \(-0.277411\pi\)
0.643669 + 0.765304i \(0.277411\pi\)
\(390\) 4683.12 0.608049
\(391\) −3139.31 −0.406041
\(392\) 0 0
\(393\) 285.766 0.0366794
\(394\) −1150.01 −0.147047
\(395\) −17223.0 −2.19388
\(396\) 1050.83 0.133349
\(397\) −7729.30 −0.977135 −0.488567 0.872526i \(-0.662480\pi\)
−0.488567 + 0.872526i \(0.662480\pi\)
\(398\) 7608.24 0.958207
\(399\) 0 0
\(400\) −3247.84 −0.405980
\(401\) −5461.32 −0.680113 −0.340056 0.940405i \(-0.610446\pi\)
−0.340056 + 0.940405i \(0.610446\pi\)
\(402\) 6248.63 0.775258
\(403\) 2804.35 0.346636
\(404\) 955.296 0.117643
\(405\) −1186.58 −0.145585
\(406\) 0 0
\(407\) −20.3276 −0.00247567
\(408\) 2096.28 0.254366
\(409\) −184.816 −0.0223436 −0.0111718 0.999938i \(-0.503556\pi\)
−0.0111718 + 0.999938i \(0.503556\pi\)
\(410\) 3921.75 0.472394
\(411\) −4406.79 −0.528883
\(412\) −11771.8 −1.40766
\(413\) 0 0
\(414\) 1967.81 0.233605
\(415\) 2752.19 0.325541
\(416\) −6091.54 −0.717939
\(417\) −5592.80 −0.656788
\(418\) 788.606 0.0922774
\(419\) 5175.78 0.603469 0.301735 0.953392i \(-0.402434\pi\)
0.301735 + 0.953392i \(0.402434\pi\)
\(420\) 0 0
\(421\) 1782.44 0.206345 0.103172 0.994664i \(-0.467101\pi\)
0.103172 + 0.994664i \(0.467101\pi\)
\(422\) 8367.46 0.965218
\(423\) −373.116 −0.0428877
\(424\) −7652.37 −0.876490
\(425\) 5550.35 0.633486
\(426\) 7237.01 0.823086
\(427\) 0 0
\(428\) 10088.4 1.13935
\(429\) 815.059 0.0917283
\(430\) −19142.3 −2.14680
\(431\) 11768.2 1.31521 0.657603 0.753365i \(-0.271571\pi\)
0.657603 + 0.753365i \(0.271571\pi\)
\(432\) 978.717 0.109001
\(433\) −3809.95 −0.422851 −0.211426 0.977394i \(-0.567811\pi\)
−0.211426 + 0.977394i \(0.567811\pi\)
\(434\) 0 0
\(435\) −4409.79 −0.486054
\(436\) 16710.8 1.83555
\(437\) 842.087 0.0921796
\(438\) −2826.43 −0.308338
\(439\) −16810.6 −1.82762 −0.913809 0.406144i \(-0.866873\pi\)
−0.913809 + 0.406144i \(0.866873\pi\)
\(440\) 1817.67 0.196941
\(441\) 0 0
\(442\) 6601.16 0.710374
\(443\) −2512.40 −0.269453 −0.134726 0.990883i \(-0.543016\pi\)
−0.134726 + 0.990883i \(0.543016\pi\)
\(444\) 58.8453 0.00628981
\(445\) −508.518 −0.0541709
\(446\) 6494.23 0.689486
\(447\) −1457.40 −0.154211
\(448\) 0 0
\(449\) 3925.62 0.412609 0.206305 0.978488i \(-0.433856\pi\)
0.206305 + 0.978488i \(0.433856\pi\)
\(450\) −3479.11 −0.364460
\(451\) 682.549 0.0712638
\(452\) 16917.4 1.76046
\(453\) 7332.26 0.760485
\(454\) 24995.9 2.58395
\(455\) 0 0
\(456\) −562.305 −0.0577464
\(457\) 6761.59 0.692109 0.346055 0.938214i \(-0.387521\pi\)
0.346055 + 0.938214i \(0.387521\pi\)
\(458\) −25314.2 −2.58265
\(459\) −1672.57 −0.170084
\(460\) 7880.01 0.798712
\(461\) 11942.7 1.20656 0.603282 0.797528i \(-0.293859\pi\)
0.603282 + 0.797528i \(0.293859\pi\)
\(462\) 0 0
\(463\) 2063.81 0.207156 0.103578 0.994621i \(-0.466971\pi\)
0.103578 + 0.994621i \(0.466971\pi\)
\(464\) 3637.28 0.363915
\(465\) −4989.89 −0.497636
\(466\) −8453.32 −0.840328
\(467\) −9098.55 −0.901565 −0.450782 0.892634i \(-0.648855\pi\)
−0.450782 + 0.892634i \(0.648855\pi\)
\(468\) −2359.48 −0.233049
\(469\) 0 0
\(470\) −2620.23 −0.257154
\(471\) −10422.0 −1.01957
\(472\) 1628.48 0.158807
\(473\) −3331.56 −0.323859
\(474\) 15217.4 1.47460
\(475\) −1488.82 −0.143814
\(476\) 0 0
\(477\) 6105.62 0.586073
\(478\) −1759.91 −0.168403
\(479\) 1993.08 0.190117 0.0950586 0.995472i \(-0.469696\pi\)
0.0950586 + 0.995472i \(0.469696\pi\)
\(480\) 10838.9 1.03068
\(481\) 45.6423 0.00432663
\(482\) 15693.8 1.48306
\(483\) 0 0
\(484\) 1284.35 0.120619
\(485\) 10436.7 0.977124
\(486\) 1048.41 0.0978537
\(487\) 8513.77 0.792189 0.396094 0.918210i \(-0.370365\pi\)
0.396094 + 0.918210i \(0.370365\pi\)
\(488\) 2576.58 0.239009
\(489\) 4151.54 0.383924
\(490\) 0 0
\(491\) 12336.6 1.13390 0.566950 0.823752i \(-0.308123\pi\)
0.566950 + 0.823752i \(0.308123\pi\)
\(492\) −1975.88 −0.181056
\(493\) −6215.89 −0.567849
\(494\) −1770.69 −0.161269
\(495\) −1450.27 −0.131686
\(496\) 4115.76 0.372587
\(497\) 0 0
\(498\) −2431.71 −0.218810
\(499\) 11809.5 1.05945 0.529726 0.848169i \(-0.322295\pi\)
0.529726 + 0.848169i \(0.322295\pi\)
\(500\) 5504.69 0.492355
\(501\) −7829.84 −0.698226
\(502\) −2603.15 −0.231442
\(503\) 7808.72 0.692194 0.346097 0.938199i \(-0.387507\pi\)
0.346097 + 0.938199i \(0.387507\pi\)
\(504\) 0 0
\(505\) −1318.42 −0.116176
\(506\) 2405.10 0.211304
\(507\) 4760.91 0.417041
\(508\) −21827.6 −1.90638
\(509\) 14748.0 1.28427 0.642137 0.766590i \(-0.278048\pi\)
0.642137 + 0.766590i \(0.278048\pi\)
\(510\) −11745.7 −1.01982
\(511\) 0 0
\(512\) −12211.3 −1.05404
\(513\) 448.648 0.0386127
\(514\) 15507.8 1.33078
\(515\) 16246.5 1.39011
\(516\) 9644.38 0.822810
\(517\) −456.030 −0.0387934
\(518\) 0 0
\(519\) 5510.34 0.466045
\(520\) −4081.29 −0.344185
\(521\) −1652.60 −0.138967 −0.0694835 0.997583i \(-0.522135\pi\)
−0.0694835 + 0.997583i \(0.522135\pi\)
\(522\) 3896.29 0.326697
\(523\) −17241.3 −1.44151 −0.720754 0.693191i \(-0.756204\pi\)
−0.720754 + 0.693191i \(0.756204\pi\)
\(524\) −1011.08 −0.0842928
\(525\) 0 0
\(526\) 16417.5 1.36091
\(527\) −7033.57 −0.581380
\(528\) 1196.21 0.0985954
\(529\) −9598.79 −0.788920
\(530\) 42877.1 3.51408
\(531\) −1299.32 −0.106188
\(532\) 0 0
\(533\) −1532.56 −0.124545
\(534\) 449.303 0.0364106
\(535\) −13923.2 −1.12514
\(536\) −5445.62 −0.438834
\(537\) −10423.3 −0.837613
\(538\) 29888.7 2.39515
\(539\) 0 0
\(540\) 4198.32 0.334568
\(541\) −18725.5 −1.48812 −0.744061 0.668112i \(-0.767103\pi\)
−0.744061 + 0.668112i \(0.767103\pi\)
\(542\) −3681.16 −0.291733
\(543\) −5790.01 −0.457593
\(544\) 15278.2 1.20413
\(545\) −23062.8 −1.81266
\(546\) 0 0
\(547\) 5629.03 0.440000 0.220000 0.975500i \(-0.429394\pi\)
0.220000 + 0.975500i \(0.429394\pi\)
\(548\) 15591.9 1.21543
\(549\) −2055.78 −0.159815
\(550\) −4252.25 −0.329666
\(551\) 1667.35 0.128913
\(552\) −1714.93 −0.132232
\(553\) 0 0
\(554\) −23352.1 −1.79086
\(555\) −81.2133 −0.00621137
\(556\) 19788.2 1.50937
\(557\) 1374.50 0.104559 0.0522797 0.998632i \(-0.483351\pi\)
0.0522797 + 0.998632i \(0.483351\pi\)
\(558\) 4408.84 0.334482
\(559\) 7480.49 0.565994
\(560\) 0 0
\(561\) −2044.25 −0.153847
\(562\) 2846.99 0.213689
\(563\) 16202.9 1.21292 0.606458 0.795116i \(-0.292590\pi\)
0.606458 + 0.795116i \(0.292590\pi\)
\(564\) 1320.14 0.0985603
\(565\) −23347.9 −1.73850
\(566\) −25201.5 −1.87155
\(567\) 0 0
\(568\) −6306.98 −0.465907
\(569\) −1798.25 −0.132490 −0.0662448 0.997803i \(-0.521102\pi\)
−0.0662448 + 0.997803i \(0.521102\pi\)
\(570\) 3150.66 0.231521
\(571\) −3775.13 −0.276680 −0.138340 0.990385i \(-0.544177\pi\)
−0.138340 + 0.990385i \(0.544177\pi\)
\(572\) −2883.81 −0.210801
\(573\) −14218.2 −1.03660
\(574\) 0 0
\(575\) −4540.63 −0.329317
\(576\) −6966.88 −0.503970
\(577\) 15695.5 1.13243 0.566216 0.824257i \(-0.308407\pi\)
0.566216 + 0.824257i \(0.308407\pi\)
\(578\) 4640.54 0.333946
\(579\) −1849.89 −0.132779
\(580\) 15602.5 1.11700
\(581\) 0 0
\(582\) −9221.37 −0.656767
\(583\) 7462.42 0.530123
\(584\) 2463.20 0.174534
\(585\) 3256.35 0.230143
\(586\) 13096.2 0.923210
\(587\) −9710.67 −0.682798 −0.341399 0.939918i \(-0.610901\pi\)
−0.341399 + 0.939918i \(0.610901\pi\)
\(588\) 0 0
\(589\) 1886.68 0.131985
\(590\) −9124.56 −0.636699
\(591\) −799.642 −0.0556563
\(592\) 66.9863 0.00465054
\(593\) 18925.8 1.31061 0.655304 0.755365i \(-0.272540\pi\)
0.655304 + 0.755365i \(0.272540\pi\)
\(594\) 1281.39 0.0885120
\(595\) 0 0
\(596\) 5156.49 0.354393
\(597\) 5290.30 0.362676
\(598\) −5400.27 −0.369287
\(599\) −9502.00 −0.648149 −0.324075 0.946032i \(-0.605053\pi\)
−0.324075 + 0.946032i \(0.605053\pi\)
\(600\) 3032.01 0.206302
\(601\) 25373.8 1.72216 0.861082 0.508467i \(-0.169788\pi\)
0.861082 + 0.508467i \(0.169788\pi\)
\(602\) 0 0
\(603\) 4344.91 0.293430
\(604\) −25942.7 −1.74767
\(605\) −1772.55 −0.119115
\(606\) 1164.89 0.0780867
\(607\) −11459.1 −0.766247 −0.383123 0.923697i \(-0.625152\pi\)
−0.383123 + 0.923697i \(0.625152\pi\)
\(608\) −4098.21 −0.273362
\(609\) 0 0
\(610\) −14436.9 −0.958250
\(611\) 1023.94 0.0677976
\(612\) 5917.80 0.390871
\(613\) −27701.7 −1.82522 −0.912611 0.408829i \(-0.865937\pi\)
−0.912611 + 0.408829i \(0.865937\pi\)
\(614\) −18787.4 −1.23485
\(615\) 2726.94 0.178798
\(616\) 0 0
\(617\) −19793.9 −1.29153 −0.645764 0.763537i \(-0.723461\pi\)
−0.645764 + 0.763537i \(0.723461\pi\)
\(618\) −14354.6 −0.934350
\(619\) 20932.3 1.35919 0.679597 0.733586i \(-0.262155\pi\)
0.679597 + 0.733586i \(0.262155\pi\)
\(620\) 17655.0 1.14362
\(621\) 1368.29 0.0884181
\(622\) 14517.4 0.935845
\(623\) 0 0
\(624\) −2685.90 −0.172311
\(625\) −18796.9 −1.20300
\(626\) 33393.7 2.13207
\(627\) 548.348 0.0349265
\(628\) 36874.6 2.34308
\(629\) −114.475 −0.00725665
\(630\) 0 0
\(631\) −17090.6 −1.07823 −0.539117 0.842231i \(-0.681242\pi\)
−0.539117 + 0.842231i \(0.681242\pi\)
\(632\) −13261.8 −0.834694
\(633\) 5818.22 0.365329
\(634\) 46994.7 2.94384
\(635\) 30124.6 1.88261
\(636\) −21602.6 −1.34685
\(637\) 0 0
\(638\) 4762.14 0.295509
\(639\) 5032.17 0.311533
\(640\) −20021.6 −1.23660
\(641\) 24542.3 1.51226 0.756132 0.654419i \(-0.227087\pi\)
0.756132 + 0.654419i \(0.227087\pi\)
\(642\) 12301.9 0.756257
\(643\) 14247.9 0.873846 0.436923 0.899499i \(-0.356068\pi\)
0.436923 + 0.899499i \(0.356068\pi\)
\(644\) 0 0
\(645\) −13310.3 −0.812549
\(646\) 4441.06 0.270482
\(647\) 17073.8 1.03747 0.518733 0.854936i \(-0.326404\pi\)
0.518733 + 0.854936i \(0.326404\pi\)
\(648\) −913.680 −0.0553900
\(649\) −1588.06 −0.0960503
\(650\) 9547.76 0.576144
\(651\) 0 0
\(652\) −14688.8 −0.882297
\(653\) −9800.30 −0.587313 −0.293657 0.955911i \(-0.594872\pi\)
−0.293657 + 0.955911i \(0.594872\pi\)
\(654\) 20377.2 1.21837
\(655\) 1395.41 0.0832416
\(656\) −2249.24 −0.133869
\(657\) −1965.32 −0.116704
\(658\) 0 0
\(659\) 7476.99 0.441976 0.220988 0.975277i \(-0.429072\pi\)
0.220988 + 0.975277i \(0.429072\pi\)
\(660\) 5131.28 0.302628
\(661\) 8361.23 0.492004 0.246002 0.969269i \(-0.420883\pi\)
0.246002 + 0.969269i \(0.420883\pi\)
\(662\) −18615.5 −1.09292
\(663\) 4590.04 0.268872
\(664\) 2119.21 0.123857
\(665\) 0 0
\(666\) 71.7564 0.00417493
\(667\) 5085.09 0.295196
\(668\) 27703.2 1.60459
\(669\) 4515.69 0.260966
\(670\) 30512.5 1.75940
\(671\) −2512.62 −0.144558
\(672\) 0 0
\(673\) 30985.6 1.77475 0.887375 0.461049i \(-0.152527\pi\)
0.887375 + 0.461049i \(0.152527\pi\)
\(674\) 24470.5 1.39847
\(675\) −2419.16 −0.137946
\(676\) −16844.9 −0.958401
\(677\) −15795.6 −0.896711 −0.448356 0.893855i \(-0.647990\pi\)
−0.448356 + 0.893855i \(0.647990\pi\)
\(678\) 20629.1 1.16852
\(679\) 0 0
\(680\) 10236.3 0.577269
\(681\) 17380.6 0.978011
\(682\) 5388.58 0.302550
\(683\) −14968.4 −0.838582 −0.419291 0.907852i \(-0.637721\pi\)
−0.419291 + 0.907852i \(0.637721\pi\)
\(684\) −1587.39 −0.0887357
\(685\) −21518.6 −1.20027
\(686\) 0 0
\(687\) −17601.9 −0.977519
\(688\) 10978.6 0.608367
\(689\) −16755.7 −0.926474
\(690\) 9608.93 0.530153
\(691\) −27858.1 −1.53368 −0.766840 0.641838i \(-0.778172\pi\)
−0.766840 + 0.641838i \(0.778172\pi\)
\(692\) −19496.5 −1.07102
\(693\) 0 0
\(694\) 13242.6 0.724327
\(695\) −27310.0 −1.49054
\(696\) −3395.58 −0.184927
\(697\) 3843.80 0.208887
\(698\) 353.351 0.0191612
\(699\) −5877.92 −0.318059
\(700\) 0 0
\(701\) −1192.90 −0.0642729 −0.0321365 0.999483i \(-0.510231\pi\)
−0.0321365 + 0.999483i \(0.510231\pi\)
\(702\) −2877.16 −0.154689
\(703\) 30.7068 0.00164741
\(704\) −8515.07 −0.455858
\(705\) −1821.95 −0.0973312
\(706\) 50593.2 2.69703
\(707\) 0 0
\(708\) 4597.19 0.244030
\(709\) −13888.3 −0.735664 −0.367832 0.929892i \(-0.619900\pi\)
−0.367832 + 0.929892i \(0.619900\pi\)
\(710\) 35338.8 1.86794
\(711\) 10581.2 0.558126
\(712\) −391.563 −0.0206102
\(713\) 5754.02 0.302230
\(714\) 0 0
\(715\) 3979.98 0.208172
\(716\) 36879.2 1.92492
\(717\) −1223.73 −0.0637394
\(718\) 40017.7 2.08001
\(719\) −31737.7 −1.64620 −0.823098 0.567899i \(-0.807756\pi\)
−0.823098 + 0.567899i \(0.807756\pi\)
\(720\) 4779.14 0.247372
\(721\) 0 0
\(722\) 28401.5 1.46398
\(723\) 10912.5 0.561329
\(724\) 20486.0 1.05159
\(725\) −8990.51 −0.460551
\(726\) 1566.14 0.0800621
\(727\) 14730.4 0.751472 0.375736 0.926727i \(-0.377390\pi\)
0.375736 + 0.926727i \(0.377390\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −13801.6 −0.699754
\(731\) −18761.8 −0.949288
\(732\) 7273.68 0.367272
\(733\) 6008.51 0.302769 0.151384 0.988475i \(-0.451627\pi\)
0.151384 + 0.988475i \(0.451627\pi\)
\(734\) 16793.9 0.844517
\(735\) 0 0
\(736\) −12498.8 −0.625966
\(737\) 5310.45 0.265418
\(738\) −2409.40 −0.120178
\(739\) 8723.11 0.434215 0.217108 0.976148i \(-0.430338\pi\)
0.217108 + 0.976148i \(0.430338\pi\)
\(740\) 287.345 0.0142744
\(741\) −1231.23 −0.0610395
\(742\) 0 0
\(743\) 28162.9 1.39057 0.695286 0.718733i \(-0.255278\pi\)
0.695286 + 0.718733i \(0.255278\pi\)
\(744\) −3842.26 −0.189333
\(745\) −7116.55 −0.349974
\(746\) 49484.2 2.42861
\(747\) −1690.86 −0.0828183
\(748\) 7232.87 0.353556
\(749\) 0 0
\(750\) 6712.45 0.326806
\(751\) −29044.8 −1.41126 −0.705632 0.708578i \(-0.749337\pi\)
−0.705632 + 0.708578i \(0.749337\pi\)
\(752\) 1502.78 0.0728732
\(753\) −1810.07 −0.0875996
\(754\) −10692.6 −0.516449
\(755\) 35803.9 1.72588
\(756\) 0 0
\(757\) 13181.3 0.632871 0.316435 0.948614i \(-0.397514\pi\)
0.316435 + 0.948614i \(0.397514\pi\)
\(758\) −29564.4 −1.41666
\(759\) 1672.36 0.0799772
\(760\) −2745.77 −0.131052
\(761\) 32209.1 1.53427 0.767134 0.641487i \(-0.221682\pi\)
0.767134 + 0.641487i \(0.221682\pi\)
\(762\) −26616.7 −1.26538
\(763\) 0 0
\(764\) 50306.2 2.38222
\(765\) −8167.24 −0.385996
\(766\) 25962.0 1.22460
\(767\) 3565.73 0.167863
\(768\) −888.205 −0.0417322
\(769\) −11714.3 −0.549320 −0.274660 0.961541i \(-0.588565\pi\)
−0.274660 + 0.961541i \(0.588565\pi\)
\(770\) 0 0
\(771\) 10783.1 0.503691
\(772\) 6545.21 0.305139
\(773\) −22457.9 −1.04496 −0.522481 0.852651i \(-0.674993\pi\)
−0.522481 + 0.852651i \(0.674993\pi\)
\(774\) 11760.4 0.546149
\(775\) −10173.2 −0.471525
\(776\) 8036.33 0.371762
\(777\) 0 0
\(778\) −42613.0 −1.96369
\(779\) −1031.06 −0.0474217
\(780\) −11521.5 −0.528891
\(781\) 6150.43 0.281792
\(782\) 13544.4 0.619370
\(783\) 2709.24 0.123653
\(784\) 0 0
\(785\) −50891.1 −2.31386
\(786\) −1232.92 −0.0559502
\(787\) −12982.2 −0.588010 −0.294005 0.955804i \(-0.594988\pi\)
−0.294005 + 0.955804i \(0.594988\pi\)
\(788\) 2829.26 0.127904
\(789\) 11415.7 0.515096
\(790\) 74307.5 3.34651
\(791\) 0 0
\(792\) −1116.72 −0.0501022
\(793\) 5641.70 0.252639
\(794\) 33347.7 1.49051
\(795\) 29814.1 1.33006
\(796\) −18717.9 −0.833465
\(797\) −18858.7 −0.838154 −0.419077 0.907951i \(-0.637646\pi\)
−0.419077 + 0.907951i \(0.637646\pi\)
\(798\) 0 0
\(799\) −2568.15 −0.113710
\(800\) 22098.0 0.976603
\(801\) 312.417 0.0137812
\(802\) 23562.6 1.03744
\(803\) −2402.06 −0.105563
\(804\) −15373.0 −0.674332
\(805\) 0 0
\(806\) −12099.2 −0.528755
\(807\) 20782.7 0.906552
\(808\) −1015.19 −0.0442009
\(809\) 21479.6 0.933478 0.466739 0.884395i \(-0.345429\pi\)
0.466739 + 0.884395i \(0.345429\pi\)
\(810\) 5119.45 0.222073
\(811\) 4450.75 0.192709 0.0963545 0.995347i \(-0.469282\pi\)
0.0963545 + 0.995347i \(0.469282\pi\)
\(812\) 0 0
\(813\) −2559.65 −0.110419
\(814\) 87.7022 0.00377636
\(815\) 20272.2 0.871294
\(816\) 6736.50 0.289001
\(817\) 5032.65 0.215508
\(818\) 797.377 0.0340827
\(819\) 0 0
\(820\) −9648.35 −0.410896
\(821\) −32636.1 −1.38734 −0.693670 0.720293i \(-0.744007\pi\)
−0.693670 + 0.720293i \(0.744007\pi\)
\(822\) 19012.9 0.806752
\(823\) 13553.8 0.574065 0.287033 0.957921i \(-0.407331\pi\)
0.287033 + 0.957921i \(0.407331\pi\)
\(824\) 12509.9 0.528888
\(825\) −2956.75 −0.124777
\(826\) 0 0
\(827\) −14489.8 −0.609263 −0.304631 0.952470i \(-0.598533\pi\)
−0.304631 + 0.952470i \(0.598533\pi\)
\(828\) −4841.23 −0.203194
\(829\) −26927.9 −1.12816 −0.564079 0.825721i \(-0.690769\pi\)
−0.564079 + 0.825721i \(0.690769\pi\)
\(830\) −11874.2 −0.496576
\(831\) −16237.6 −0.677830
\(832\) 19119.2 0.796684
\(833\) 0 0
\(834\) 24129.9 1.00186
\(835\) −38233.6 −1.58458
\(836\) −1940.14 −0.0802645
\(837\) 3065.63 0.126599
\(838\) −22330.6 −0.920524
\(839\) −34465.9 −1.41823 −0.709114 0.705094i \(-0.750905\pi\)
−0.709114 + 0.705094i \(0.750905\pi\)
\(840\) 0 0
\(841\) −14320.4 −0.587168
\(842\) −7690.27 −0.314755
\(843\) 1979.62 0.0808799
\(844\) −20585.8 −0.839563
\(845\) 23247.8 0.946449
\(846\) 1609.79 0.0654204
\(847\) 0 0
\(848\) −24591.2 −0.995833
\(849\) −17523.5 −0.708370
\(850\) −23946.7 −0.966312
\(851\) 93.6500 0.00377236
\(852\) −17804.6 −0.715934
\(853\) −951.037 −0.0381745 −0.0190873 0.999818i \(-0.506076\pi\)
−0.0190873 + 0.999818i \(0.506076\pi\)
\(854\) 0 0
\(855\) 2190.78 0.0876292
\(856\) −10721.0 −0.428079
\(857\) 4628.31 0.184481 0.0922404 0.995737i \(-0.470597\pi\)
0.0922404 + 0.995737i \(0.470597\pi\)
\(858\) −3516.53 −0.139921
\(859\) 30205.9 1.19978 0.599891 0.800082i \(-0.295211\pi\)
0.599891 + 0.800082i \(0.295211\pi\)
\(860\) 47094.1 1.86732
\(861\) 0 0
\(862\) −50773.2 −2.00620
\(863\) −30606.9 −1.20727 −0.603634 0.797262i \(-0.706281\pi\)
−0.603634 + 0.797262i \(0.706281\pi\)
\(864\) −6659.11 −0.262208
\(865\) 26907.3 1.05766
\(866\) 16437.8 0.645012
\(867\) 3226.74 0.126397
\(868\) 0 0
\(869\) 12932.6 0.504844
\(870\) 19025.8 0.741421
\(871\) −11923.8 −0.463860
\(872\) −17758.5 −0.689655
\(873\) −6411.97 −0.248582
\(874\) −3633.14 −0.140610
\(875\) 0 0
\(876\) 6953.62 0.268197
\(877\) 20525.2 0.790291 0.395146 0.918619i \(-0.370694\pi\)
0.395146 + 0.918619i \(0.370694\pi\)
\(878\) 72528.3 2.78783
\(879\) 9106.32 0.349429
\(880\) 5841.17 0.223756
\(881\) 34579.5 1.32237 0.661187 0.750221i \(-0.270053\pi\)
0.661187 + 0.750221i \(0.270053\pi\)
\(882\) 0 0
\(883\) 11174.0 0.425861 0.212930 0.977067i \(-0.431699\pi\)
0.212930 + 0.977067i \(0.431699\pi\)
\(884\) −16240.3 −0.617895
\(885\) −6344.65 −0.240987
\(886\) 10839.6 0.411020
\(887\) −28892.2 −1.09369 −0.546847 0.837233i \(-0.684172\pi\)
−0.546847 + 0.837233i \(0.684172\pi\)
\(888\) −62.5349 −0.00236322
\(889\) 0 0
\(890\) 2193.97 0.0826316
\(891\) 891.000 0.0335013
\(892\) −15977.2 −0.599727
\(893\) 688.879 0.0258146
\(894\) 6287.86 0.235232
\(895\) −50897.6 −1.90091
\(896\) 0 0
\(897\) −3755.01 −0.139773
\(898\) −16936.9 −0.629389
\(899\) 11393.1 0.422669
\(900\) 8559.37 0.317014
\(901\) 42024.9 1.55389
\(902\) −2944.82 −0.108705
\(903\) 0 0
\(904\) −17978.1 −0.661440
\(905\) −28273.0 −1.03848
\(906\) −31634.7 −1.16004
\(907\) −13080.3 −0.478859 −0.239429 0.970914i \(-0.576960\pi\)
−0.239429 + 0.970914i \(0.576960\pi\)
\(908\) −61495.3 −2.24757
\(909\) 809.995 0.0295554
\(910\) 0 0
\(911\) −13848.9 −0.503659 −0.251829 0.967772i \(-0.581032\pi\)
−0.251829 + 0.967772i \(0.581032\pi\)
\(912\) −1806.99 −0.0656092
\(913\) −2066.60 −0.0749119
\(914\) −29172.6 −1.05574
\(915\) −10038.5 −0.362692
\(916\) 62278.4 2.24644
\(917\) 0 0
\(918\) 7216.20 0.259445
\(919\) 41613.6 1.49370 0.746848 0.664995i \(-0.231566\pi\)
0.746848 + 0.664995i \(0.231566\pi\)
\(920\) −8374.08 −0.300093
\(921\) −13063.6 −0.467384
\(922\) −51526.1 −1.84048
\(923\) −13809.8 −0.492476
\(924\) 0 0
\(925\) −165.574 −0.00588546
\(926\) −8904.21 −0.315994
\(927\) −9981.32 −0.353646
\(928\) −24747.8 −0.875415
\(929\) 13583.7 0.479728 0.239864 0.970806i \(-0.422897\pi\)
0.239864 + 0.970806i \(0.422897\pi\)
\(930\) 21528.6 0.759088
\(931\) 0 0
\(932\) 20797.0 0.730931
\(933\) 10094.5 0.354212
\(934\) 39255.2 1.37524
\(935\) −9982.19 −0.349147
\(936\) 2507.42 0.0875614
\(937\) −47931.8 −1.67114 −0.835572 0.549381i \(-0.814864\pi\)
−0.835572 + 0.549381i \(0.814864\pi\)
\(938\) 0 0
\(939\) 23219.9 0.806978
\(940\) 6446.33 0.223677
\(941\) 30613.0 1.06053 0.530263 0.847833i \(-0.322093\pi\)
0.530263 + 0.847833i \(0.322093\pi\)
\(942\) 44965.1 1.55525
\(943\) −3144.54 −0.108590
\(944\) 5233.19 0.180430
\(945\) 0 0
\(946\) 14373.8 0.494010
\(947\) −6345.46 −0.217740 −0.108870 0.994056i \(-0.534723\pi\)
−0.108870 + 0.994056i \(0.534723\pi\)
\(948\) −37438.1 −1.28263
\(949\) 5393.45 0.184488
\(950\) 6423.45 0.219373
\(951\) 32677.2 1.11423
\(952\) 0 0
\(953\) −2674.07 −0.0908936 −0.0454468 0.998967i \(-0.514471\pi\)
−0.0454468 + 0.998967i \(0.514471\pi\)
\(954\) −26342.4 −0.893989
\(955\) −69428.4 −2.35251
\(956\) 4329.76 0.146480
\(957\) 3311.29 0.111848
\(958\) −8599.04 −0.290003
\(959\) 0 0
\(960\) −34019.7 −1.14373
\(961\) −16899.2 −0.567259
\(962\) −196.921 −0.00659980
\(963\) 8553.97 0.286239
\(964\) −38610.2 −1.28999
\(965\) −9033.14 −0.301334
\(966\) 0 0
\(967\) −20187.8 −0.671351 −0.335676 0.941978i \(-0.608965\pi\)
−0.335676 + 0.941978i \(0.608965\pi\)
\(968\) −1364.88 −0.0453191
\(969\) 3088.04 0.102376
\(970\) −45028.5 −1.49049
\(971\) 8015.21 0.264902 0.132451 0.991190i \(-0.457715\pi\)
0.132451 + 0.991190i \(0.457715\pi\)
\(972\) −2579.32 −0.0851148
\(973\) 0 0
\(974\) −36732.2 −1.20840
\(975\) 6638.92 0.218067
\(976\) 8279.96 0.271552
\(977\) −10285.8 −0.336819 −0.168410 0.985717i \(-0.553863\pi\)
−0.168410 + 0.985717i \(0.553863\pi\)
\(978\) −17911.6 −0.585634
\(979\) 381.843 0.0124655
\(980\) 0 0
\(981\) 14169.0 0.461144
\(982\) −53225.8 −1.72964
\(983\) −40809.2 −1.32412 −0.662061 0.749450i \(-0.730318\pi\)
−0.662061 + 0.749450i \(0.730318\pi\)
\(984\) 2099.77 0.0680266
\(985\) −3904.70 −0.126309
\(986\) 26818.1 0.866190
\(987\) 0 0
\(988\) 4356.28 0.140275
\(989\) 15348.6 0.493486
\(990\) 6257.11 0.200873
\(991\) −61319.6 −1.96557 −0.982786 0.184749i \(-0.940853\pi\)
−0.982786 + 0.184749i \(0.940853\pi\)
\(992\) −28003.3 −0.896274
\(993\) −12944.1 −0.413663
\(994\) 0 0
\(995\) 25832.8 0.823071
\(996\) 5982.52 0.190325
\(997\) −22436.4 −0.712705 −0.356353 0.934352i \(-0.615980\pi\)
−0.356353 + 0.934352i \(0.615980\pi\)
\(998\) −50951.6 −1.61608
\(999\) 49.8949 0.00158019
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.1.3 16
7.6 odd 2 1617.4.a.bf.1.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.3 16 1.1 even 1 trivial
1617.4.a.bf.1.3 yes 16 7.6 odd 2