Properties

Label 435.4.a.i.1.2
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $0$
Dimension $7$
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] = [435,4,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(25.6658308525\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.57720\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.57720 q^{2} -3.00000 q^{3} +4.79637 q^{4} +5.00000 q^{5} +10.7316 q^{6} +15.0281 q^{7} +11.4600 q^{8} +9.00000 q^{9} -17.8860 q^{10} +70.8260 q^{11} -14.3891 q^{12} +62.2117 q^{13} -53.7587 q^{14} -15.0000 q^{15} -79.3658 q^{16} -67.1848 q^{17} -32.1948 q^{18} +118.988 q^{19} +23.9818 q^{20} -45.0844 q^{21} -253.359 q^{22} +72.5383 q^{23} -34.3801 q^{24} +25.0000 q^{25} -222.544 q^{26} -27.0000 q^{27} +72.0804 q^{28} +29.0000 q^{29} +53.6580 q^{30} -180.227 q^{31} +192.227 q^{32} -212.478 q^{33} +240.334 q^{34} +75.1407 q^{35} +43.1673 q^{36} -47.8439 q^{37} -425.646 q^{38} -186.635 q^{39} +57.3002 q^{40} +371.956 q^{41} +161.276 q^{42} -409.069 q^{43} +339.707 q^{44} +45.0000 q^{45} -259.484 q^{46} +125.891 q^{47} +238.097 q^{48} -117.155 q^{49} -89.4300 q^{50} +201.555 q^{51} +298.390 q^{52} -215.764 q^{53} +96.5844 q^{54} +354.130 q^{55} +172.223 q^{56} -356.965 q^{57} -103.739 q^{58} +356.919 q^{59} -71.9455 q^{60} -466.115 q^{61} +644.707 q^{62} +135.253 q^{63} -52.7086 q^{64} +311.059 q^{65} +760.076 q^{66} -578.359 q^{67} -322.243 q^{68} -217.615 q^{69} -268.793 q^{70} +870.924 q^{71} +103.140 q^{72} +411.904 q^{73} +171.147 q^{74} -75.0000 q^{75} +570.712 q^{76} +1064.38 q^{77} +667.631 q^{78} +1120.23 q^{79} -396.829 q^{80} +81.0000 q^{81} -1330.56 q^{82} +1079.28 q^{83} -216.241 q^{84} -335.924 q^{85} +1463.32 q^{86} -87.0000 q^{87} +811.668 q^{88} +388.993 q^{89} -160.974 q^{90} +934.926 q^{91} +347.920 q^{92} +540.680 q^{93} -450.339 q^{94} +594.942 q^{95} -576.681 q^{96} -1528.82 q^{97} +419.088 q^{98} +637.434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9} - 10 q^{10} + 76 q^{11} - 66 q^{12} + 30 q^{13} + 89 q^{14} - 105 q^{15} + 138 q^{16} - 140 q^{17} - 18 q^{18} + 90 q^{19}+ \cdots + 684 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.57720 −1.26473 −0.632366 0.774670i \(-0.717916\pi\)
−0.632366 + 0.774670i \(0.717916\pi\)
\(3\) −3.00000 −0.577350
\(4\) 4.79637 0.599546
\(5\) 5.00000 0.447214
\(6\) 10.7316 0.730193
\(7\) 15.0281 0.811443 0.405721 0.913997i \(-0.367020\pi\)
0.405721 + 0.913997i \(0.367020\pi\)
\(8\) 11.4600 0.506467
\(9\) 9.00000 0.333333
\(10\) −17.8860 −0.565605
\(11\) 70.8260 1.94135 0.970674 0.240399i \(-0.0772781\pi\)
0.970674 + 0.240399i \(0.0772781\pi\)
\(12\) −14.3891 −0.346148
\(13\) 62.2117 1.32726 0.663632 0.748059i \(-0.269014\pi\)
0.663632 + 0.748059i \(0.269014\pi\)
\(14\) −53.7587 −1.02626
\(15\) −15.0000 −0.258199
\(16\) −79.3658 −1.24009
\(17\) −67.1848 −0.958513 −0.479256 0.877675i \(-0.659094\pi\)
−0.479256 + 0.877675i \(0.659094\pi\)
\(18\) −32.1948 −0.421577
\(19\) 118.988 1.43673 0.718364 0.695667i \(-0.244891\pi\)
0.718364 + 0.695667i \(0.244891\pi\)
\(20\) 23.9818 0.268125
\(21\) −45.0844 −0.468487
\(22\) −253.359 −2.45528
\(23\) 72.5383 0.657621 0.328810 0.944396i \(-0.393352\pi\)
0.328810 + 0.944396i \(0.393352\pi\)
\(24\) −34.3801 −0.292409
\(25\) 25.0000 0.200000
\(26\) −222.544 −1.67863
\(27\) −27.0000 −0.192450
\(28\) 72.0804 0.486497
\(29\) 29.0000 0.185695
\(30\) 53.6580 0.326552
\(31\) −180.227 −1.04418 −0.522092 0.852889i \(-0.674848\pi\)
−0.522092 + 0.852889i \(0.674848\pi\)
\(32\) 192.227 1.06191
\(33\) −212.478 −1.12084
\(34\) 240.334 1.21226
\(35\) 75.1407 0.362888
\(36\) 43.1673 0.199849
\(37\) −47.8439 −0.212581 −0.106290 0.994335i \(-0.533897\pi\)
−0.106290 + 0.994335i \(0.533897\pi\)
\(38\) −425.646 −1.81708
\(39\) −186.635 −0.766296
\(40\) 57.3002 0.226499
\(41\) 371.956 1.41682 0.708411 0.705800i \(-0.249412\pi\)
0.708411 + 0.705800i \(0.249412\pi\)
\(42\) 161.276 0.592510
\(43\) −409.069 −1.45075 −0.725377 0.688352i \(-0.758334\pi\)
−0.725377 + 0.688352i \(0.758334\pi\)
\(44\) 339.707 1.16393
\(45\) 45.0000 0.149071
\(46\) −259.484 −0.831714
\(47\) 125.891 0.390705 0.195353 0.980733i \(-0.437415\pi\)
0.195353 + 0.980733i \(0.437415\pi\)
\(48\) 238.097 0.715967
\(49\) −117.155 −0.341560
\(50\) −89.4300 −0.252946
\(51\) 201.555 0.553398
\(52\) 298.390 0.795755
\(53\) −215.764 −0.559198 −0.279599 0.960117i \(-0.590202\pi\)
−0.279599 + 0.960117i \(0.590202\pi\)
\(54\) 96.5844 0.243398
\(55\) 354.130 0.868197
\(56\) 172.223 0.410969
\(57\) −356.965 −0.829495
\(58\) −103.739 −0.234855
\(59\) 356.919 0.787575 0.393788 0.919201i \(-0.371165\pi\)
0.393788 + 0.919201i \(0.371165\pi\)
\(60\) −71.9455 −0.154802
\(61\) −466.115 −0.978358 −0.489179 0.872183i \(-0.662704\pi\)
−0.489179 + 0.872183i \(0.662704\pi\)
\(62\) 644.707 1.32061
\(63\) 135.253 0.270481
\(64\) −52.7086 −0.102946
\(65\) 311.059 0.593570
\(66\) 760.076 1.41756
\(67\) −578.359 −1.05459 −0.527297 0.849681i \(-0.676794\pi\)
−0.527297 + 0.849681i \(0.676794\pi\)
\(68\) −322.243 −0.574672
\(69\) −217.615 −0.379678
\(70\) −268.793 −0.458956
\(71\) 870.924 1.45577 0.727885 0.685699i \(-0.240503\pi\)
0.727885 + 0.685699i \(0.240503\pi\)
\(72\) 103.140 0.168822
\(73\) 411.904 0.660407 0.330204 0.943910i \(-0.392883\pi\)
0.330204 + 0.943910i \(0.392883\pi\)
\(74\) 171.147 0.268857
\(75\) −75.0000 −0.115470
\(76\) 570.712 0.861384
\(77\) 1064.38 1.57529
\(78\) 667.631 0.969159
\(79\) 1120.23 1.59538 0.797692 0.603065i \(-0.206054\pi\)
0.797692 + 0.603065i \(0.206054\pi\)
\(80\) −396.829 −0.554585
\(81\) 81.0000 0.111111
\(82\) −1330.56 −1.79190
\(83\) 1079.28 1.42730 0.713652 0.700501i \(-0.247040\pi\)
0.713652 + 0.700501i \(0.247040\pi\)
\(84\) −216.241 −0.280879
\(85\) −335.924 −0.428660
\(86\) 1463.32 1.83481
\(87\) −87.0000 −0.107211
\(88\) 811.668 0.983229
\(89\) 388.993 0.463294 0.231647 0.972800i \(-0.425589\pi\)
0.231647 + 0.972800i \(0.425589\pi\)
\(90\) −160.974 −0.188535
\(91\) 934.926 1.07700
\(92\) 347.920 0.394274
\(93\) 540.680 0.602859
\(94\) −450.339 −0.494137
\(95\) 594.942 0.642524
\(96\) −576.681 −0.613097
\(97\) −1528.82 −1.60029 −0.800147 0.599803i \(-0.795245\pi\)
−0.800147 + 0.599803i \(0.795245\pi\)
\(98\) 419.088 0.431982
\(99\) 637.434 0.647116
\(100\) 119.909 0.119909
\(101\) −1425.50 −1.40439 −0.702193 0.711987i \(-0.747796\pi\)
−0.702193 + 0.711987i \(0.747796\pi\)
\(102\) −721.001 −0.699899
\(103\) −1595.68 −1.52648 −0.763240 0.646116i \(-0.776392\pi\)
−0.763240 + 0.646116i \(0.776392\pi\)
\(104\) 712.949 0.672215
\(105\) −225.422 −0.209514
\(106\) 771.832 0.707236
\(107\) −155.104 −0.140135 −0.0700675 0.997542i \(-0.522321\pi\)
−0.0700675 + 0.997542i \(0.522321\pi\)
\(108\) −129.502 −0.115383
\(109\) 1212.23 1.06524 0.532618 0.846356i \(-0.321208\pi\)
0.532618 + 0.846356i \(0.321208\pi\)
\(110\) −1266.79 −1.09804
\(111\) 143.532 0.122733
\(112\) −1192.72 −1.00626
\(113\) −1152.01 −0.959043 −0.479521 0.877530i \(-0.659190\pi\)
−0.479521 + 0.877530i \(0.659190\pi\)
\(114\) 1276.94 1.04909
\(115\) 362.691 0.294097
\(116\) 139.095 0.111333
\(117\) 559.905 0.442421
\(118\) −1276.77 −0.996072
\(119\) −1009.66 −0.777778
\(120\) −171.901 −0.130769
\(121\) 3685.32 2.76883
\(122\) 1667.39 1.23736
\(123\) −1115.87 −0.818003
\(124\) −864.434 −0.626036
\(125\) 125.000 0.0894427
\(126\) −483.828 −0.342086
\(127\) 398.743 0.278604 0.139302 0.990250i \(-0.455514\pi\)
0.139302 + 0.990250i \(0.455514\pi\)
\(128\) −1349.27 −0.931715
\(129\) 1227.21 0.837593
\(130\) −1112.72 −0.750707
\(131\) −385.241 −0.256937 −0.128468 0.991714i \(-0.541006\pi\)
−0.128468 + 0.991714i \(0.541006\pi\)
\(132\) −1019.12 −0.671994
\(133\) 1788.17 1.16582
\(134\) 2068.90 1.33378
\(135\) −135.000 −0.0860663
\(136\) −769.941 −0.485455
\(137\) −351.877 −0.219437 −0.109719 0.993963i \(-0.534995\pi\)
−0.109719 + 0.993963i \(0.534995\pi\)
\(138\) 778.452 0.480190
\(139\) 1138.48 0.694711 0.347355 0.937734i \(-0.387080\pi\)
0.347355 + 0.937734i \(0.387080\pi\)
\(140\) 360.402 0.217568
\(141\) −377.674 −0.225574
\(142\) −3115.47 −1.84116
\(143\) 4406.21 2.57668
\(144\) −714.292 −0.413364
\(145\) 145.000 0.0830455
\(146\) −1473.46 −0.835238
\(147\) 351.466 0.197200
\(148\) −229.477 −0.127452
\(149\) 1200.59 0.660107 0.330053 0.943962i \(-0.392933\pi\)
0.330053 + 0.943962i \(0.392933\pi\)
\(150\) 268.290 0.146039
\(151\) −1539.35 −0.829605 −0.414802 0.909911i \(-0.636149\pi\)
−0.414802 + 0.909911i \(0.636149\pi\)
\(152\) 1363.61 0.727655
\(153\) −604.664 −0.319504
\(154\) −3807.51 −1.99232
\(155\) −901.134 −0.466973
\(156\) −895.171 −0.459430
\(157\) 223.134 0.113427 0.0567136 0.998390i \(-0.481938\pi\)
0.0567136 + 0.998390i \(0.481938\pi\)
\(158\) −4007.28 −2.01773
\(159\) 647.293 0.322853
\(160\) 961.136 0.474903
\(161\) 1090.11 0.533622
\(162\) −289.753 −0.140526
\(163\) −3114.71 −1.49671 −0.748353 0.663301i \(-0.769155\pi\)
−0.748353 + 0.663301i \(0.769155\pi\)
\(164\) 1784.04 0.849450
\(165\) −1062.39 −0.501254
\(166\) −3860.79 −1.80516
\(167\) −1720.78 −0.797351 −0.398676 0.917092i \(-0.630530\pi\)
−0.398676 + 0.917092i \(0.630530\pi\)
\(168\) −516.669 −0.237273
\(169\) 1673.30 0.761629
\(170\) 1201.67 0.542140
\(171\) 1070.90 0.478909
\(172\) −1962.04 −0.869793
\(173\) 2438.70 1.07174 0.535870 0.844301i \(-0.319984\pi\)
0.535870 + 0.844301i \(0.319984\pi\)
\(174\) 311.216 0.135593
\(175\) 375.703 0.162289
\(176\) −5621.16 −2.40745
\(177\) −1070.76 −0.454707
\(178\) −1391.50 −0.585942
\(179\) −2248.00 −0.938676 −0.469338 0.883018i \(-0.655507\pi\)
−0.469338 + 0.883018i \(0.655507\pi\)
\(180\) 215.837 0.0893750
\(181\) −2357.13 −0.967977 −0.483989 0.875074i \(-0.660812\pi\)
−0.483989 + 0.875074i \(0.660812\pi\)
\(182\) −3344.42 −1.36211
\(183\) 1398.34 0.564855
\(184\) 831.292 0.333063
\(185\) −239.219 −0.0950689
\(186\) −1934.12 −0.762455
\(187\) −4758.43 −1.86081
\(188\) 603.821 0.234246
\(189\) −405.760 −0.156162
\(190\) −2128.23 −0.812621
\(191\) 1922.56 0.728331 0.364166 0.931334i \(-0.381354\pi\)
0.364166 + 0.931334i \(0.381354\pi\)
\(192\) 158.126 0.0594362
\(193\) 1840.68 0.686504 0.343252 0.939243i \(-0.388472\pi\)
0.343252 + 0.939243i \(0.388472\pi\)
\(194\) 5468.91 2.02394
\(195\) −933.176 −0.342698
\(196\) −561.919 −0.204781
\(197\) −2712.79 −0.981109 −0.490555 0.871410i \(-0.663206\pi\)
−0.490555 + 0.871410i \(0.663206\pi\)
\(198\) −2280.23 −0.818428
\(199\) 224.631 0.0800185 0.0400093 0.999199i \(-0.487261\pi\)
0.0400093 + 0.999199i \(0.487261\pi\)
\(200\) 286.501 0.101293
\(201\) 1735.08 0.608870
\(202\) 5099.31 1.77617
\(203\) 435.816 0.150681
\(204\) 966.729 0.331787
\(205\) 1859.78 0.633622
\(206\) 5708.08 1.93059
\(207\) 652.845 0.219207
\(208\) −4937.48 −1.64593
\(209\) 8427.48 2.78919
\(210\) 806.380 0.264979
\(211\) −4160.10 −1.35731 −0.678656 0.734456i \(-0.737437\pi\)
−0.678656 + 0.734456i \(0.737437\pi\)
\(212\) −1034.88 −0.335265
\(213\) −2612.77 −0.840489
\(214\) 554.837 0.177233
\(215\) −2045.34 −0.648797
\(216\) −309.421 −0.0974696
\(217\) −2708.47 −0.847295
\(218\) −4336.40 −1.34724
\(219\) −1235.71 −0.381286
\(220\) 1698.54 0.520524
\(221\) −4179.68 −1.27220
\(222\) −513.441 −0.155225
\(223\) −1746.41 −0.524432 −0.262216 0.965009i \(-0.584453\pi\)
−0.262216 + 0.965009i \(0.584453\pi\)
\(224\) 2888.81 0.861683
\(225\) 225.000 0.0666667
\(226\) 4120.97 1.21293
\(227\) 129.508 0.0378666 0.0189333 0.999821i \(-0.493973\pi\)
0.0189333 + 0.999821i \(0.493973\pi\)
\(228\) −1712.14 −0.497320
\(229\) −682.166 −0.196851 −0.0984253 0.995144i \(-0.531381\pi\)
−0.0984253 + 0.995144i \(0.531381\pi\)
\(230\) −1297.42 −0.371954
\(231\) −3193.15 −0.909496
\(232\) 332.341 0.0940486
\(233\) 1731.11 0.486732 0.243366 0.969935i \(-0.421748\pi\)
0.243366 + 0.969935i \(0.421748\pi\)
\(234\) −2002.89 −0.559544
\(235\) 629.457 0.174729
\(236\) 1711.92 0.472188
\(237\) −3360.68 −0.921095
\(238\) 3611.77 0.983681
\(239\) 3308.38 0.895402 0.447701 0.894183i \(-0.352243\pi\)
0.447701 + 0.894183i \(0.352243\pi\)
\(240\) 1190.49 0.320190
\(241\) 3586.39 0.958588 0.479294 0.877654i \(-0.340893\pi\)
0.479294 + 0.877654i \(0.340893\pi\)
\(242\) −13183.1 −3.50183
\(243\) −243.000 −0.0641500
\(244\) −2235.66 −0.586571
\(245\) −585.776 −0.152750
\(246\) 3991.68 1.03455
\(247\) 7402.48 1.90692
\(248\) −2065.41 −0.528844
\(249\) −3237.83 −0.824054
\(250\) −447.150 −0.113121
\(251\) −2956.22 −0.743406 −0.371703 0.928352i \(-0.621226\pi\)
−0.371703 + 0.928352i \(0.621226\pi\)
\(252\) 648.724 0.162166
\(253\) 5137.59 1.27667
\(254\) −1426.39 −0.352360
\(255\) 1007.77 0.247487
\(256\) 5248.27 1.28132
\(257\) 2258.12 0.548084 0.274042 0.961718i \(-0.411639\pi\)
0.274042 + 0.961718i \(0.411639\pi\)
\(258\) −4389.96 −1.05933
\(259\) −719.004 −0.172497
\(260\) 1491.95 0.355873
\(261\) 261.000 0.0618984
\(262\) 1378.09 0.324956
\(263\) 2908.67 0.681963 0.340981 0.940070i \(-0.389241\pi\)
0.340981 + 0.940070i \(0.389241\pi\)
\(264\) −2435.01 −0.567667
\(265\) −1078.82 −0.250081
\(266\) −6396.66 −1.47445
\(267\) −1166.98 −0.267483
\(268\) −2774.02 −0.632277
\(269\) −297.910 −0.0675238 −0.0337619 0.999430i \(-0.510749\pi\)
−0.0337619 + 0.999430i \(0.510749\pi\)
\(270\) 482.922 0.108851
\(271\) 6993.04 1.56752 0.783758 0.621067i \(-0.213300\pi\)
0.783758 + 0.621067i \(0.213300\pi\)
\(272\) 5332.18 1.18864
\(273\) −2804.78 −0.621805
\(274\) 1258.73 0.277529
\(275\) 1770.65 0.388270
\(276\) −1043.76 −0.227634
\(277\) 5701.84 1.23679 0.618394 0.785868i \(-0.287784\pi\)
0.618394 + 0.785868i \(0.287784\pi\)
\(278\) −4072.58 −0.878622
\(279\) −1622.04 −0.348061
\(280\) 861.115 0.183791
\(281\) −1216.34 −0.258223 −0.129112 0.991630i \(-0.541213\pi\)
−0.129112 + 0.991630i \(0.541213\pi\)
\(282\) 1351.02 0.285290
\(283\) 4096.17 0.860396 0.430198 0.902735i \(-0.358444\pi\)
0.430198 + 0.902735i \(0.358444\pi\)
\(284\) 4177.27 0.872801
\(285\) −1784.83 −0.370962
\(286\) −15761.9 −3.25881
\(287\) 5589.80 1.14967
\(288\) 1730.04 0.353972
\(289\) −399.197 −0.0812532
\(290\) −518.694 −0.105030
\(291\) 4586.47 0.923931
\(292\) 1975.64 0.395944
\(293\) 5854.29 1.16727 0.583637 0.812015i \(-0.301629\pi\)
0.583637 + 0.812015i \(0.301629\pi\)
\(294\) −1257.26 −0.249405
\(295\) 1784.60 0.352214
\(296\) −548.292 −0.107665
\(297\) −1912.30 −0.373613
\(298\) −4294.74 −0.834858
\(299\) 4512.73 0.872836
\(300\) −359.728 −0.0692296
\(301\) −6147.54 −1.17720
\(302\) 5506.56 1.04923
\(303\) 4276.51 0.810822
\(304\) −9443.62 −1.78167
\(305\) −2330.57 −0.437535
\(306\) 2163.00 0.404087
\(307\) −2748.38 −0.510940 −0.255470 0.966817i \(-0.582230\pi\)
−0.255470 + 0.966817i \(0.582230\pi\)
\(308\) 5105.17 0.944461
\(309\) 4787.05 0.881313
\(310\) 3223.54 0.590595
\(311\) −10289.0 −1.87600 −0.938000 0.346634i \(-0.887325\pi\)
−0.938000 + 0.346634i \(0.887325\pi\)
\(312\) −2138.85 −0.388104
\(313\) 8540.93 1.54237 0.771185 0.636611i \(-0.219664\pi\)
0.771185 + 0.636611i \(0.219664\pi\)
\(314\) −798.196 −0.143455
\(315\) 676.266 0.120963
\(316\) 5373.02 0.956506
\(317\) −131.153 −0.0232375 −0.0116187 0.999933i \(-0.503698\pi\)
−0.0116187 + 0.999933i \(0.503698\pi\)
\(318\) −2315.50 −0.408323
\(319\) 2053.95 0.360499
\(320\) −263.543 −0.0460391
\(321\) 465.311 0.0809070
\(322\) −3899.56 −0.674888
\(323\) −7994.22 −1.37712
\(324\) 388.506 0.0666162
\(325\) 1555.29 0.265453
\(326\) 11142.0 1.89293
\(327\) −3636.70 −0.615014
\(328\) 4262.63 0.717574
\(329\) 1891.91 0.317035
\(330\) 3800.38 0.633952
\(331\) −7274.79 −1.20803 −0.604016 0.796972i \(-0.706434\pi\)
−0.604016 + 0.796972i \(0.706434\pi\)
\(332\) 5176.61 0.855734
\(333\) −430.595 −0.0708602
\(334\) 6155.56 1.00844
\(335\) −2891.79 −0.471628
\(336\) 3578.16 0.580966
\(337\) 4663.56 0.753830 0.376915 0.926248i \(-0.376985\pi\)
0.376915 + 0.926248i \(0.376985\pi\)
\(338\) −5985.72 −0.963256
\(339\) 3456.03 0.553704
\(340\) −1611.22 −0.257001
\(341\) −12764.7 −2.02712
\(342\) −3830.81 −0.605692
\(343\) −6915.27 −1.08860
\(344\) −4687.94 −0.734759
\(345\) −1088.07 −0.169797
\(346\) −8723.72 −1.35546
\(347\) 6050.81 0.936093 0.468046 0.883704i \(-0.344958\pi\)
0.468046 + 0.883704i \(0.344958\pi\)
\(348\) −417.284 −0.0642781
\(349\) 12824.0 1.96691 0.983455 0.181152i \(-0.0579825\pi\)
0.983455 + 0.181152i \(0.0579825\pi\)
\(350\) −1343.97 −0.205251
\(351\) −1679.72 −0.255432
\(352\) 13614.7 2.06155
\(353\) 12514.5 1.88691 0.943457 0.331494i \(-0.107553\pi\)
0.943457 + 0.331494i \(0.107553\pi\)
\(354\) 3830.32 0.575082
\(355\) 4354.62 0.651040
\(356\) 1865.75 0.277766
\(357\) 3028.99 0.449051
\(358\) 8041.53 1.18717
\(359\) 3394.70 0.499068 0.249534 0.968366i \(-0.419723\pi\)
0.249534 + 0.968366i \(0.419723\pi\)
\(360\) 515.702 0.0754996
\(361\) 7299.26 1.06419
\(362\) 8431.92 1.22423
\(363\) −11056.0 −1.59859
\(364\) 4484.25 0.645710
\(365\) 2059.52 0.295343
\(366\) −5002.16 −0.714391
\(367\) −9991.93 −1.42118 −0.710592 0.703604i \(-0.751573\pi\)
−0.710592 + 0.703604i \(0.751573\pi\)
\(368\) −5757.06 −0.815509
\(369\) 3347.60 0.472274
\(370\) 855.735 0.120237
\(371\) −3242.54 −0.453758
\(372\) 2593.30 0.361442
\(373\) 7219.40 1.00216 0.501081 0.865400i \(-0.332936\pi\)
0.501081 + 0.865400i \(0.332936\pi\)
\(374\) 17021.9 2.35342
\(375\) −375.000 −0.0516398
\(376\) 1442.72 0.197879
\(377\) 1804.14 0.246467
\(378\) 1451.48 0.197503
\(379\) 10797.8 1.46345 0.731724 0.681601i \(-0.238716\pi\)
0.731724 + 0.681601i \(0.238716\pi\)
\(380\) 2853.56 0.385223
\(381\) −1196.23 −0.160852
\(382\) −6877.37 −0.921144
\(383\) −3396.75 −0.453174 −0.226587 0.973991i \(-0.572757\pi\)
−0.226587 + 0.973991i \(0.572757\pi\)
\(384\) 4047.80 0.537926
\(385\) 5321.91 0.704493
\(386\) −6584.49 −0.868243
\(387\) −3681.62 −0.483585
\(388\) −7332.80 −0.959450
\(389\) −4927.07 −0.642192 −0.321096 0.947047i \(-0.604051\pi\)
−0.321096 + 0.947047i \(0.604051\pi\)
\(390\) 3338.16 0.433421
\(391\) −4873.47 −0.630338
\(392\) −1342.60 −0.172989
\(393\) 1155.72 0.148342
\(394\) 9704.21 1.24084
\(395\) 5601.13 0.713477
\(396\) 3057.37 0.387976
\(397\) −7777.55 −0.983234 −0.491617 0.870812i \(-0.663594\pi\)
−0.491617 + 0.870812i \(0.663594\pi\)
\(398\) −803.551 −0.101202
\(399\) −5364.52 −0.673088
\(400\) −1984.14 −0.248018
\(401\) 6058.22 0.754446 0.377223 0.926122i \(-0.376879\pi\)
0.377223 + 0.926122i \(0.376879\pi\)
\(402\) −6206.71 −0.770057
\(403\) −11212.2 −1.38591
\(404\) −6837.24 −0.841994
\(405\) 405.000 0.0496904
\(406\) −1559.00 −0.190571
\(407\) −3388.59 −0.412693
\(408\) 2309.82 0.280278
\(409\) −15078.2 −1.82291 −0.911454 0.411403i \(-0.865039\pi\)
−0.911454 + 0.411403i \(0.865039\pi\)
\(410\) −6652.80 −0.801362
\(411\) 1055.63 0.126692
\(412\) −7653.48 −0.915194
\(413\) 5363.83 0.639073
\(414\) −2335.36 −0.277238
\(415\) 5396.39 0.638309
\(416\) 11958.8 1.40944
\(417\) −3415.45 −0.401091
\(418\) −30146.8 −3.52758
\(419\) 5698.22 0.664383 0.332192 0.943212i \(-0.392212\pi\)
0.332192 + 0.943212i \(0.392212\pi\)
\(420\) −1081.21 −0.125613
\(421\) 846.838 0.0980341 0.0490171 0.998798i \(-0.484391\pi\)
0.0490171 + 0.998798i \(0.484391\pi\)
\(422\) 14881.5 1.71664
\(423\) 1133.02 0.130235
\(424\) −2472.67 −0.283215
\(425\) −1679.62 −0.191703
\(426\) 9346.41 1.06299
\(427\) −7004.83 −0.793882
\(428\) −743.934 −0.0840173
\(429\) −13218.6 −1.48765
\(430\) 7316.61 0.820554
\(431\) −7518.27 −0.840237 −0.420119 0.907469i \(-0.638012\pi\)
−0.420119 + 0.907469i \(0.638012\pi\)
\(432\) 2142.88 0.238656
\(433\) 12686.0 1.40797 0.703987 0.710213i \(-0.251401\pi\)
0.703987 + 0.710213i \(0.251401\pi\)
\(434\) 9688.75 1.07160
\(435\) −435.000 −0.0479463
\(436\) 5814.31 0.638658
\(437\) 8631.22 0.944822
\(438\) 4420.39 0.482225
\(439\) 10530.4 1.14485 0.572424 0.819958i \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(440\) 4058.34 0.439713
\(441\) −1054.40 −0.113853
\(442\) 14951.6 1.60899
\(443\) −5199.17 −0.557608 −0.278804 0.960348i \(-0.589938\pi\)
−0.278804 + 0.960348i \(0.589938\pi\)
\(444\) 688.430 0.0735843
\(445\) 1944.96 0.207191
\(446\) 6247.27 0.663266
\(447\) −3601.76 −0.381113
\(448\) −792.112 −0.0835352
\(449\) −11836.7 −1.24412 −0.622058 0.782971i \(-0.713703\pi\)
−0.622058 + 0.782971i \(0.713703\pi\)
\(450\) −804.870 −0.0843154
\(451\) 26344.1 2.75055
\(452\) −5525.46 −0.574990
\(453\) 4618.04 0.478973
\(454\) −463.274 −0.0478911
\(455\) 4674.63 0.481648
\(456\) −4090.84 −0.420112
\(457\) 404.451 0.0413992 0.0206996 0.999786i \(-0.493411\pi\)
0.0206996 + 0.999786i \(0.493411\pi\)
\(458\) 2440.24 0.248963
\(459\) 1813.99 0.184466
\(460\) 1739.60 0.176325
\(461\) 7072.59 0.714541 0.357270 0.934001i \(-0.383707\pi\)
0.357270 + 0.934001i \(0.383707\pi\)
\(462\) 11422.5 1.15027
\(463\) −6500.22 −0.652464 −0.326232 0.945290i \(-0.605779\pi\)
−0.326232 + 0.945290i \(0.605779\pi\)
\(464\) −2301.61 −0.230279
\(465\) 2703.40 0.269607
\(466\) −6192.51 −0.615585
\(467\) −8850.61 −0.876997 −0.438498 0.898732i \(-0.644489\pi\)
−0.438498 + 0.898732i \(0.644489\pi\)
\(468\) 2685.51 0.265252
\(469\) −8691.65 −0.855742
\(470\) −2251.69 −0.220985
\(471\) −669.403 −0.0654872
\(472\) 4090.31 0.398881
\(473\) −28972.7 −2.81642
\(474\) 12021.8 1.16494
\(475\) 2974.71 0.287346
\(476\) −4842.71 −0.466314
\(477\) −1941.88 −0.186399
\(478\) −11834.7 −1.13244
\(479\) −8582.91 −0.818712 −0.409356 0.912375i \(-0.634247\pi\)
−0.409356 + 0.912375i \(0.634247\pi\)
\(480\) −2883.41 −0.274185
\(481\) −2976.45 −0.282150
\(482\) −12829.2 −1.21236
\(483\) −3270.34 −0.308087
\(484\) 17676.1 1.66004
\(485\) −7644.12 −0.715674
\(486\) 869.260 0.0811326
\(487\) −2823.62 −0.262732 −0.131366 0.991334i \(-0.541936\pi\)
−0.131366 + 0.991334i \(0.541936\pi\)
\(488\) −5341.69 −0.495506
\(489\) 9344.14 0.864124
\(490\) 2095.44 0.193188
\(491\) −13116.2 −1.20555 −0.602776 0.797911i \(-0.705939\pi\)
−0.602776 + 0.797911i \(0.705939\pi\)
\(492\) −5352.11 −0.490430
\(493\) −1948.36 −0.177991
\(494\) −26480.2 −2.41174
\(495\) 3187.17 0.289399
\(496\) 14303.8 1.29488
\(497\) 13088.4 1.18127
\(498\) 11582.4 1.04221
\(499\) −16495.4 −1.47983 −0.739916 0.672699i \(-0.765135\pi\)
−0.739916 + 0.672699i \(0.765135\pi\)
\(500\) 599.546 0.0536250
\(501\) 5162.33 0.460351
\(502\) 10575.0 0.940209
\(503\) −16677.3 −1.47834 −0.739169 0.673520i \(-0.764781\pi\)
−0.739169 + 0.673520i \(0.764781\pi\)
\(504\) 1550.01 0.136990
\(505\) −7127.52 −0.628060
\(506\) −18378.2 −1.61465
\(507\) −5019.89 −0.439726
\(508\) 1912.52 0.167036
\(509\) 9386.35 0.817373 0.408686 0.912675i \(-0.365987\pi\)
0.408686 + 0.912675i \(0.365987\pi\)
\(510\) −3605.01 −0.313005
\(511\) 6190.15 0.535883
\(512\) −7979.98 −0.688806
\(513\) −3212.69 −0.276498
\(514\) −8077.75 −0.693180
\(515\) −7978.42 −0.682662
\(516\) 5886.13 0.502175
\(517\) 8916.37 0.758495
\(518\) 2572.02 0.218162
\(519\) −7316.10 −0.618769
\(520\) 3564.74 0.300624
\(521\) −16530.2 −1.39002 −0.695009 0.719001i \(-0.744600\pi\)
−0.695009 + 0.719001i \(0.744600\pi\)
\(522\) −933.649 −0.0782849
\(523\) −18295.9 −1.52968 −0.764842 0.644217i \(-0.777183\pi\)
−0.764842 + 0.644217i \(0.777183\pi\)
\(524\) −1847.76 −0.154045
\(525\) −1127.11 −0.0936974
\(526\) −10404.9 −0.862500
\(527\) 12108.5 1.00086
\(528\) 16863.5 1.38994
\(529\) −6905.20 −0.567535
\(530\) 3859.16 0.316285
\(531\) 3212.27 0.262525
\(532\) 8576.74 0.698964
\(533\) 23140.0 1.88050
\(534\) 4174.51 0.338294
\(535\) −775.518 −0.0626703
\(536\) −6628.01 −0.534117
\(537\) 6743.99 0.541945
\(538\) 1065.68 0.0853994
\(539\) −8297.63 −0.663088
\(540\) −647.510 −0.0516007
\(541\) −949.695 −0.0754724 −0.0377362 0.999288i \(-0.512015\pi\)
−0.0377362 + 0.999288i \(0.512015\pi\)
\(542\) −25015.5 −1.98249
\(543\) 7071.38 0.558862
\(544\) −12914.7 −1.01786
\(545\) 6061.16 0.476388
\(546\) 10033.3 0.786417
\(547\) 18017.4 1.40835 0.704177 0.710024i \(-0.251316\pi\)
0.704177 + 0.710024i \(0.251316\pi\)
\(548\) −1687.73 −0.131563
\(549\) −4195.03 −0.326119
\(550\) −6333.97 −0.491057
\(551\) 3450.67 0.266794
\(552\) −2493.87 −0.192294
\(553\) 16834.9 1.29456
\(554\) −20396.6 −1.56420
\(555\) 717.658 0.0548881
\(556\) 5460.58 0.416511
\(557\) −11387.6 −0.866263 −0.433131 0.901331i \(-0.642591\pi\)
−0.433131 + 0.901331i \(0.642591\pi\)
\(558\) 5802.37 0.440204
\(559\) −25448.9 −1.92553
\(560\) −5963.60 −0.450014
\(561\) 14275.3 1.07434
\(562\) 4351.09 0.326583
\(563\) −277.822 −0.0207971 −0.0103986 0.999946i \(-0.503310\pi\)
−0.0103986 + 0.999946i \(0.503310\pi\)
\(564\) −1811.46 −0.135242
\(565\) −5760.04 −0.428897
\(566\) −14652.8 −1.08817
\(567\) 1217.28 0.0901603
\(568\) 9980.82 0.737299
\(569\) 15475.5 1.14019 0.570093 0.821580i \(-0.306907\pi\)
0.570093 + 0.821580i \(0.306907\pi\)
\(570\) 6384.69 0.469167
\(571\) −3245.82 −0.237887 −0.118944 0.992901i \(-0.537951\pi\)
−0.118944 + 0.992901i \(0.537951\pi\)
\(572\) 21133.8 1.54484
\(573\) −5767.67 −0.420502
\(574\) −19995.8 −1.45402
\(575\) 1813.46 0.131524
\(576\) −474.377 −0.0343155
\(577\) −23087.9 −1.66579 −0.832895 0.553430i \(-0.813318\pi\)
−0.832895 + 0.553430i \(0.813318\pi\)
\(578\) 1428.01 0.102764
\(579\) −5522.05 −0.396353
\(580\) 695.473 0.0497896
\(581\) 16219.5 1.15818
\(582\) −16406.7 −1.16852
\(583\) −15281.7 −1.08560
\(584\) 4720.44 0.334474
\(585\) 2799.53 0.197857
\(586\) −20942.0 −1.47629
\(587\) 13526.1 0.951075 0.475537 0.879696i \(-0.342254\pi\)
0.475537 + 0.879696i \(0.342254\pi\)
\(588\) 1685.76 0.118230
\(589\) −21444.9 −1.50021
\(590\) −6383.86 −0.445457
\(591\) 8138.38 0.566444
\(592\) 3797.17 0.263619
\(593\) 15964.6 1.10554 0.552770 0.833334i \(-0.313571\pi\)
0.552770 + 0.833334i \(0.313571\pi\)
\(594\) 6840.69 0.472520
\(595\) −5048.31 −0.347833
\(596\) 5758.46 0.395764
\(597\) −673.894 −0.0461987
\(598\) −16142.9 −1.10390
\(599\) 22.9584 0.00156603 0.000783017 1.00000i \(-0.499751\pi\)
0.000783017 1.00000i \(0.499751\pi\)
\(600\) −859.503 −0.0584818
\(601\) −7471.32 −0.507090 −0.253545 0.967324i \(-0.581597\pi\)
−0.253545 + 0.967324i \(0.581597\pi\)
\(602\) 21991.0 1.48885
\(603\) −5205.23 −0.351531
\(604\) −7383.28 −0.497386
\(605\) 18426.6 1.23826
\(606\) −15297.9 −1.02547
\(607\) 23036.9 1.54042 0.770212 0.637788i \(-0.220151\pi\)
0.770212 + 0.637788i \(0.220151\pi\)
\(608\) 22872.8 1.52568
\(609\) −1307.45 −0.0869958
\(610\) 8336.93 0.553365
\(611\) 7831.91 0.518569
\(612\) −2900.19 −0.191557
\(613\) −20040.0 −1.32040 −0.660202 0.751088i \(-0.729530\pi\)
−0.660202 + 0.751088i \(0.729530\pi\)
\(614\) 9831.52 0.646202
\(615\) −5579.34 −0.365822
\(616\) 12197.9 0.797834
\(617\) −27464.4 −1.79202 −0.896010 0.444035i \(-0.853547\pi\)
−0.896010 + 0.444035i \(0.853547\pi\)
\(618\) −17124.2 −1.11462
\(619\) 3585.29 0.232803 0.116402 0.993202i \(-0.462864\pi\)
0.116402 + 0.993202i \(0.462864\pi\)
\(620\) −4322.17 −0.279972
\(621\) −1958.53 −0.126559
\(622\) 36805.9 2.37264
\(623\) 5845.83 0.375936
\(624\) 14812.4 0.950276
\(625\) 625.000 0.0400000
\(626\) −30552.6 −1.95069
\(627\) −25282.4 −1.61034
\(628\) 1070.23 0.0680048
\(629\) 3214.38 0.203761
\(630\) −2419.14 −0.152985
\(631\) −27135.3 −1.71195 −0.855973 0.517020i \(-0.827041\pi\)
−0.855973 + 0.517020i \(0.827041\pi\)
\(632\) 12837.8 0.808009
\(633\) 12480.3 0.783645
\(634\) 469.160 0.0293892
\(635\) 1993.72 0.124596
\(636\) 3104.65 0.193565
\(637\) −7288.43 −0.453341
\(638\) −7347.40 −0.455935
\(639\) 7838.32 0.485257
\(640\) −6746.34 −0.416676
\(641\) 23443.5 1.44456 0.722278 0.691603i \(-0.243095\pi\)
0.722278 + 0.691603i \(0.243095\pi\)
\(642\) −1664.51 −0.102326
\(643\) −9339.29 −0.572792 −0.286396 0.958111i \(-0.592457\pi\)
−0.286396 + 0.958111i \(0.592457\pi\)
\(644\) 5228.59 0.319931
\(645\) 6136.03 0.374583
\(646\) 28596.9 1.74169
\(647\) 1103.56 0.0670562 0.0335281 0.999438i \(-0.489326\pi\)
0.0335281 + 0.999438i \(0.489326\pi\)
\(648\) 928.263 0.0562741
\(649\) 25279.2 1.52896
\(650\) −5563.60 −0.335726
\(651\) 8125.41 0.489186
\(652\) −14939.3 −0.897344
\(653\) −3127.51 −0.187425 −0.0937127 0.995599i \(-0.529874\pi\)
−0.0937127 + 0.995599i \(0.529874\pi\)
\(654\) 13009.2 0.777828
\(655\) −1926.21 −0.114906
\(656\) −29520.6 −1.75699
\(657\) 3707.14 0.220136
\(658\) −6767.75 −0.400964
\(659\) 16004.5 0.946050 0.473025 0.881049i \(-0.343162\pi\)
0.473025 + 0.881049i \(0.343162\pi\)
\(660\) −5095.61 −0.300525
\(661\) 10807.3 0.635939 0.317970 0.948101i \(-0.396999\pi\)
0.317970 + 0.948101i \(0.396999\pi\)
\(662\) 26023.4 1.52784
\(663\) 12539.1 0.734504
\(664\) 12368.6 0.722882
\(665\) 8940.87 0.521372
\(666\) 1540.32 0.0896191
\(667\) 2103.61 0.122117
\(668\) −8253.47 −0.478049
\(669\) 5239.24 0.302781
\(670\) 10344.5 0.596483
\(671\) −33013.0 −1.89933
\(672\) −8666.44 −0.497493
\(673\) 23835.6 1.36522 0.682612 0.730781i \(-0.260844\pi\)
0.682612 + 0.730781i \(0.260844\pi\)
\(674\) −16682.5 −0.953392
\(675\) −675.000 −0.0384900
\(676\) 8025.75 0.456631
\(677\) 5081.10 0.288453 0.144227 0.989545i \(-0.453931\pi\)
0.144227 + 0.989545i \(0.453931\pi\)
\(678\) −12362.9 −0.700286
\(679\) −22975.4 −1.29855
\(680\) −3849.70 −0.217102
\(681\) −388.523 −0.0218623
\(682\) 45662.0 2.56377
\(683\) 7815.19 0.437833 0.218917 0.975744i \(-0.429748\pi\)
0.218917 + 0.975744i \(0.429748\pi\)
\(684\) 5136.41 0.287128
\(685\) −1759.38 −0.0981352
\(686\) 24737.3 1.37679
\(687\) 2046.50 0.113652
\(688\) 32466.1 1.79907
\(689\) −13423.1 −0.742204
\(690\) 3892.26 0.214748
\(691\) 8815.96 0.485347 0.242674 0.970108i \(-0.421976\pi\)
0.242674 + 0.970108i \(0.421976\pi\)
\(692\) 11696.9 0.642557
\(693\) 9579.44 0.525098
\(694\) −21644.9 −1.18391
\(695\) 5692.41 0.310684
\(696\) −997.023 −0.0542990
\(697\) −24989.8 −1.35804
\(698\) −45873.9 −2.48761
\(699\) −5193.32 −0.281015
\(700\) 1802.01 0.0972994
\(701\) −4100.40 −0.220927 −0.110464 0.993880i \(-0.535234\pi\)
−0.110464 + 0.993880i \(0.535234\pi\)
\(702\) 6008.68 0.323053
\(703\) −5692.87 −0.305420
\(704\) −3733.14 −0.199855
\(705\) −1888.37 −0.100880
\(706\) −44767.0 −2.38644
\(707\) −21422.7 −1.13958
\(708\) −5135.75 −0.272618
\(709\) −17803.3 −0.943042 −0.471521 0.881855i \(-0.656295\pi\)
−0.471521 + 0.881855i \(0.656295\pi\)
\(710\) −15577.4 −0.823391
\(711\) 10082.0 0.531795
\(712\) 4457.87 0.234643
\(713\) −13073.3 −0.686677
\(714\) −10835.3 −0.567928
\(715\) 22031.0 1.15233
\(716\) −10782.2 −0.562780
\(717\) −9925.13 −0.516961
\(718\) −12143.5 −0.631187
\(719\) −4213.37 −0.218543 −0.109271 0.994012i \(-0.534852\pi\)
−0.109271 + 0.994012i \(0.534852\pi\)
\(720\) −3571.46 −0.184862
\(721\) −23980.1 −1.23865
\(722\) −26110.9 −1.34591
\(723\) −10759.2 −0.553441
\(724\) −11305.6 −0.580347
\(725\) 725.000 0.0371391
\(726\) 39549.4 2.02178
\(727\) 29559.3 1.50797 0.753984 0.656893i \(-0.228130\pi\)
0.753984 + 0.656893i \(0.228130\pi\)
\(728\) 10714.3 0.545464
\(729\) 729.000 0.0370370
\(730\) −7367.32 −0.373530
\(731\) 27483.2 1.39057
\(732\) 6706.97 0.338657
\(733\) 23312.9 1.17474 0.587369 0.809319i \(-0.300164\pi\)
0.587369 + 0.809319i \(0.300164\pi\)
\(734\) 35743.1 1.79742
\(735\) 1757.33 0.0881905
\(736\) 13943.8 0.698337
\(737\) −40962.8 −2.04733
\(738\) −11975.0 −0.597300
\(739\) 27985.2 1.39303 0.696517 0.717540i \(-0.254732\pi\)
0.696517 + 0.717540i \(0.254732\pi\)
\(740\) −1147.38 −0.0569982
\(741\) −22207.4 −1.10096
\(742\) 11599.2 0.573881
\(743\) −2542.48 −0.125538 −0.0627690 0.998028i \(-0.519993\pi\)
−0.0627690 + 0.998028i \(0.519993\pi\)
\(744\) 6196.22 0.305328
\(745\) 6002.94 0.295209
\(746\) −25825.3 −1.26747
\(747\) 9713.50 0.475768
\(748\) −22823.2 −1.11564
\(749\) −2330.92 −0.113712
\(750\) 1341.45 0.0653105
\(751\) 33600.2 1.63261 0.816305 0.577622i \(-0.196019\pi\)
0.816305 + 0.577622i \(0.196019\pi\)
\(752\) −9991.46 −0.484510
\(753\) 8868.65 0.429206
\(754\) −6453.77 −0.311714
\(755\) −7696.74 −0.371011
\(756\) −1946.17 −0.0936264
\(757\) −34022.4 −1.63351 −0.816754 0.576986i \(-0.804229\pi\)
−0.816754 + 0.576986i \(0.804229\pi\)
\(758\) −38626.0 −1.85087
\(759\) −15412.8 −0.737086
\(760\) 6818.06 0.325417
\(761\) 15032.8 0.716084 0.358042 0.933705i \(-0.383444\pi\)
0.358042 + 0.933705i \(0.383444\pi\)
\(762\) 4279.16 0.203435
\(763\) 18217.6 0.864378
\(764\) 9221.28 0.436668
\(765\) −3023.32 −0.142887
\(766\) 12150.8 0.573144
\(767\) 22204.6 1.04532
\(768\) −15744.8 −0.739768
\(769\) −24042.9 −1.12745 −0.563725 0.825963i \(-0.690632\pi\)
−0.563725 + 0.825963i \(0.690632\pi\)
\(770\) −19037.5 −0.890994
\(771\) −6774.36 −0.316437
\(772\) 8828.59 0.411590
\(773\) 17441.1 0.811530 0.405765 0.913977i \(-0.367005\pi\)
0.405765 + 0.913977i \(0.367005\pi\)
\(774\) 13169.9 0.611605
\(775\) −4505.67 −0.208837
\(776\) −17520.4 −0.810496
\(777\) 2157.01 0.0995912
\(778\) 17625.1 0.812200
\(779\) 44258.5 2.03559
\(780\) −4475.85 −0.205463
\(781\) 61684.0 2.82616
\(782\) 17433.4 0.797208
\(783\) −783.000 −0.0357371
\(784\) 9298.12 0.423566
\(785\) 1115.67 0.0507262
\(786\) −4134.26 −0.187613
\(787\) −38211.9 −1.73076 −0.865379 0.501119i \(-0.832922\pi\)
−0.865379 + 0.501119i \(0.832922\pi\)
\(788\) −13011.6 −0.588220
\(789\) −8726.00 −0.393731
\(790\) −20036.4 −0.902357
\(791\) −17312.5 −0.778208
\(792\) 7305.02 0.327743
\(793\) −28997.8 −1.29854
\(794\) 27821.8 1.24353
\(795\) 3236.47 0.144384
\(796\) 1077.41 0.0479748
\(797\) 18149.9 0.806655 0.403327 0.915056i \(-0.367854\pi\)
0.403327 + 0.915056i \(0.367854\pi\)
\(798\) 19190.0 0.851276
\(799\) −8457.99 −0.374496
\(800\) 4805.68 0.212383
\(801\) 3500.93 0.154431
\(802\) −21671.5 −0.954172
\(803\) 29173.5 1.28208
\(804\) 8322.06 0.365045
\(805\) 5450.57 0.238643
\(806\) 40108.3 1.75280
\(807\) 893.730 0.0389849
\(808\) −16336.3 −0.711275
\(809\) −339.522 −0.0147552 −0.00737759 0.999973i \(-0.502348\pi\)
−0.00737759 + 0.999973i \(0.502348\pi\)
\(810\) −1448.77 −0.0628450
\(811\) 26295.4 1.13854 0.569270 0.822151i \(-0.307226\pi\)
0.569270 + 0.822151i \(0.307226\pi\)
\(812\) 2090.33 0.0903403
\(813\) −20979.1 −0.905005
\(814\) 12121.7 0.521946
\(815\) −15573.6 −0.669347
\(816\) −15996.5 −0.686263
\(817\) −48674.5 −2.08434
\(818\) 53937.8 2.30549
\(819\) 8414.33 0.359000
\(820\) 8920.18 0.379886
\(821\) −46289.6 −1.96775 −0.983873 0.178867i \(-0.942757\pi\)
−0.983873 + 0.178867i \(0.942757\pi\)
\(822\) −3776.20 −0.160231
\(823\) −28099.3 −1.19013 −0.595067 0.803676i \(-0.702874\pi\)
−0.595067 + 0.803676i \(0.702874\pi\)
\(824\) −18286.6 −0.773111
\(825\) −5311.95 −0.224168
\(826\) −19187.5 −0.808255
\(827\) −38732.3 −1.62860 −0.814301 0.580443i \(-0.802879\pi\)
−0.814301 + 0.580443i \(0.802879\pi\)
\(828\) 3131.28 0.131425
\(829\) −9334.33 −0.391067 −0.195534 0.980697i \(-0.562644\pi\)
−0.195534 + 0.980697i \(0.562644\pi\)
\(830\) −19304.0 −0.807290
\(831\) −17105.5 −0.714060
\(832\) −3279.09 −0.136637
\(833\) 7871.06 0.327390
\(834\) 12217.7 0.507273
\(835\) −8603.88 −0.356586
\(836\) 40421.3 1.67225
\(837\) 4866.12 0.200953
\(838\) −20383.7 −0.840266
\(839\) −40742.1 −1.67649 −0.838245 0.545294i \(-0.816418\pi\)
−0.838245 + 0.545294i \(0.816418\pi\)
\(840\) −2583.34 −0.106112
\(841\) 841.000 0.0344828
\(842\) −3029.31 −0.123987
\(843\) 3649.02 0.149085
\(844\) −19953.4 −0.813771
\(845\) 8366.49 0.340611
\(846\) −4053.05 −0.164712
\(847\) 55383.5 2.24675
\(848\) 17124.3 0.693457
\(849\) −12288.5 −0.496750
\(850\) 6008.34 0.242452
\(851\) −3470.51 −0.139797
\(852\) −12531.8 −0.503912
\(853\) −29088.9 −1.16762 −0.583812 0.811889i \(-0.698440\pi\)
−0.583812 + 0.811889i \(0.698440\pi\)
\(854\) 25057.7 1.00405
\(855\) 5354.48 0.214175
\(856\) −1777.49 −0.0709737
\(857\) 31564.0 1.25812 0.629059 0.777357i \(-0.283440\pi\)
0.629059 + 0.777357i \(0.283440\pi\)
\(858\) 47285.6 1.88147
\(859\) 21654.5 0.860120 0.430060 0.902800i \(-0.358492\pi\)
0.430060 + 0.902800i \(0.358492\pi\)
\(860\) −9810.22 −0.388983
\(861\) −16769.4 −0.663763
\(862\) 26894.4 1.06267
\(863\) 2304.51 0.0908996 0.0454498 0.998967i \(-0.485528\pi\)
0.0454498 + 0.998967i \(0.485528\pi\)
\(864\) −5190.13 −0.204366
\(865\) 12193.5 0.479297
\(866\) −45380.5 −1.78071
\(867\) 1197.59 0.0469116
\(868\) −12990.8 −0.507992
\(869\) 79341.1 3.09720
\(870\) 1556.08 0.0606392
\(871\) −35980.7 −1.39972
\(872\) 13892.2 0.539507
\(873\) −13759.4 −0.533432
\(874\) −30875.6 −1.19495
\(875\) 1878.52 0.0725777
\(876\) −5926.93 −0.228599
\(877\) 42141.0 1.62258 0.811289 0.584645i \(-0.198766\pi\)
0.811289 + 0.584645i \(0.198766\pi\)
\(878\) −37669.4 −1.44793
\(879\) −17562.9 −0.673926
\(880\) −28105.8 −1.07664
\(881\) 3482.39 0.133172 0.0665861 0.997781i \(-0.478789\pi\)
0.0665861 + 0.997781i \(0.478789\pi\)
\(882\) 3771.79 0.143994
\(883\) 28559.6 1.08846 0.544229 0.838937i \(-0.316822\pi\)
0.544229 + 0.838937i \(0.316822\pi\)
\(884\) −20047.3 −0.762742
\(885\) −5353.79 −0.203351
\(886\) 18598.5 0.705224
\(887\) −42485.5 −1.60826 −0.804128 0.594456i \(-0.797367\pi\)
−0.804128 + 0.594456i \(0.797367\pi\)
\(888\) 1644.88 0.0621604
\(889\) 5992.37 0.226072
\(890\) −6957.52 −0.262041
\(891\) 5736.90 0.215705
\(892\) −8376.43 −0.314421
\(893\) 14979.6 0.561337
\(894\) 12884.2 0.482005
\(895\) −11240.0 −0.419789
\(896\) −20277.0 −0.756034
\(897\) −13538.2 −0.503932
\(898\) 42342.3 1.57347
\(899\) −5226.58 −0.193900
\(900\) 1079.18 0.0399697
\(901\) 14496.1 0.535999
\(902\) −94238.2 −3.47870
\(903\) 18442.6 0.679659
\(904\) −13202.1 −0.485724
\(905\) −11785.6 −0.432893
\(906\) −16519.7 −0.605772
\(907\) −22359.4 −0.818559 −0.409280 0.912409i \(-0.634220\pi\)
−0.409280 + 0.912409i \(0.634220\pi\)
\(908\) 621.166 0.0227028
\(909\) −12829.5 −0.468128
\(910\) −16722.1 −0.609156
\(911\) −14833.6 −0.539471 −0.269735 0.962934i \(-0.586936\pi\)
−0.269735 + 0.962934i \(0.586936\pi\)
\(912\) 28330.9 1.02865
\(913\) 76440.9 2.77089
\(914\) −1446.80 −0.0523588
\(915\) 6991.72 0.252611
\(916\) −3271.92 −0.118021
\(917\) −5789.46 −0.208489
\(918\) −6489.01 −0.233300
\(919\) −41161.8 −1.47748 −0.738740 0.673991i \(-0.764579\pi\)
−0.738740 + 0.673991i \(0.764579\pi\)
\(920\) 4156.46 0.148950
\(921\) 8245.15 0.294991
\(922\) −25300.1 −0.903702
\(923\) 54181.7 1.93219
\(924\) −15315.5 −0.545285
\(925\) −1196.10 −0.0425161
\(926\) 23252.6 0.825192
\(927\) −14361.2 −0.508826
\(928\) 5574.59 0.197193
\(929\) 29321.8 1.03554 0.517771 0.855519i \(-0.326762\pi\)
0.517771 + 0.855519i \(0.326762\pi\)
\(930\) −9670.61 −0.340980
\(931\) −13940.1 −0.490729
\(932\) 8303.02 0.291818
\(933\) 30867.0 1.08311
\(934\) 31660.4 1.10917
\(935\) −23792.2 −0.832178
\(936\) 6416.54 0.224072
\(937\) −42922.5 −1.49650 −0.748249 0.663418i \(-0.769105\pi\)
−0.748249 + 0.663418i \(0.769105\pi\)
\(938\) 31091.8 1.08228
\(939\) −25622.8 −0.890488
\(940\) 3019.10 0.104758
\(941\) −13239.3 −0.458650 −0.229325 0.973350i \(-0.573652\pi\)
−0.229325 + 0.973350i \(0.573652\pi\)
\(942\) 2394.59 0.0828237
\(943\) 26981.0 0.931732
\(944\) −28327.2 −0.976665
\(945\) −2028.80 −0.0698379
\(946\) 103641. 3.56201
\(947\) −19266.8 −0.661126 −0.330563 0.943784i \(-0.607239\pi\)
−0.330563 + 0.943784i \(0.607239\pi\)
\(948\) −16119.1 −0.552239
\(949\) 25625.3 0.876534
\(950\) −10641.1 −0.363415
\(951\) 393.459 0.0134162
\(952\) −11570.8 −0.393919
\(953\) −14816.2 −0.503615 −0.251807 0.967777i \(-0.581025\pi\)
−0.251807 + 0.967777i \(0.581025\pi\)
\(954\) 6946.49 0.235745
\(955\) 9612.78 0.325720
\(956\) 15868.2 0.536835
\(957\) −6161.86 −0.208134
\(958\) 30702.8 1.03545
\(959\) −5288.05 −0.178061
\(960\) 790.629 0.0265807
\(961\) 2690.68 0.0903185
\(962\) 10647.4 0.356845
\(963\) −1395.93 −0.0467117
\(964\) 17201.6 0.574717
\(965\) 9203.41 0.307014
\(966\) 11698.7 0.389647
\(967\) 1342.65 0.0446502 0.0223251 0.999751i \(-0.492893\pi\)
0.0223251 + 0.999751i \(0.492893\pi\)
\(968\) 42233.9 1.40232
\(969\) 23982.7 0.795082
\(970\) 27344.6 0.905135
\(971\) −11211.1 −0.370526 −0.185263 0.982689i \(-0.559314\pi\)
−0.185263 + 0.982689i \(0.559314\pi\)
\(972\) −1165.52 −0.0384609
\(973\) 17109.3 0.563718
\(974\) 10100.7 0.332285
\(975\) −4665.88 −0.153259
\(976\) 36993.6 1.21325
\(977\) 30870.2 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(978\) −33425.9 −1.09288
\(979\) 27550.8 0.899414
\(980\) −2809.60 −0.0915809
\(981\) 10910.1 0.355079
\(982\) 46919.3 1.52470
\(983\) 28549.4 0.926332 0.463166 0.886272i \(-0.346713\pi\)
0.463166 + 0.886272i \(0.346713\pi\)
\(984\) −12787.9 −0.414292
\(985\) −13564.0 −0.438765
\(986\) 6969.68 0.225111
\(987\) −5675.73 −0.183040
\(988\) 35505.0 1.14328
\(989\) −29673.1 −0.954046
\(990\) −11401.1 −0.366012
\(991\) −20806.0 −0.666927 −0.333464 0.942763i \(-0.608217\pi\)
−0.333464 + 0.942763i \(0.608217\pi\)
\(992\) −34644.5 −1.10883
\(993\) 21824.4 0.697458
\(994\) −46819.7 −1.49399
\(995\) 1123.16 0.0357854
\(996\) −15529.8 −0.494058
\(997\) 57999.9 1.84240 0.921201 0.389088i \(-0.127210\pi\)
0.921201 + 0.389088i \(0.127210\pi\)
\(998\) 59007.4 1.87159
\(999\) 1291.78 0.0409112
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 435.4.a.i.1.2 7
3.2 odd 2 1305.4.a.n.1.6 7
5.4 even 2 2175.4.a.n.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.i.1.2 7 1.1 even 1 trivial
1305.4.a.n.1.6 7 3.2 odd 2
2175.4.a.n.1.6 7 5.4 even 2