Properties

Label 363.4.a.a.1.1
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -13.0000 q^{5} +12.0000 q^{6} -26.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -3.00000 q^{3} +8.00000 q^{4} -13.0000 q^{5} +12.0000 q^{6} -26.0000 q^{7} +9.00000 q^{9} +52.0000 q^{10} -24.0000 q^{12} +73.0000 q^{13} +104.000 q^{14} +39.0000 q^{15} -64.0000 q^{16} -31.0000 q^{17} -36.0000 q^{18} +108.000 q^{19} -104.000 q^{20} +78.0000 q^{21} -86.0000 q^{23} +44.0000 q^{25} -292.000 q^{26} -27.0000 q^{27} -208.000 q^{28} +207.000 q^{29} -156.000 q^{30} +208.000 q^{31} +256.000 q^{32} +124.000 q^{34} +338.000 q^{35} +72.0000 q^{36} +45.0000 q^{37} -432.000 q^{38} -219.000 q^{39} -247.000 q^{41} -312.000 q^{42} +450.000 q^{43} -117.000 q^{45} +344.000 q^{46} -500.000 q^{47} +192.000 q^{48} +333.000 q^{49} -176.000 q^{50} +93.0000 q^{51} +584.000 q^{52} -441.000 q^{53} +108.000 q^{54} -324.000 q^{57} -828.000 q^{58} +598.000 q^{59} +312.000 q^{60} -378.000 q^{61} -832.000 q^{62} -234.000 q^{63} -512.000 q^{64} -949.000 q^{65} +494.000 q^{67} -248.000 q^{68} +258.000 q^{69} -1352.00 q^{70} -594.000 q^{71} -1034.00 q^{73} -180.000 q^{74} -132.000 q^{75} +864.000 q^{76} +876.000 q^{78} -352.000 q^{79} +832.000 q^{80} +81.0000 q^{81} +988.000 q^{82} -360.000 q^{83} +624.000 q^{84} +403.000 q^{85} -1800.00 q^{86} -621.000 q^{87} -351.000 q^{89} +468.000 q^{90} -1898.00 q^{91} -688.000 q^{92} -624.000 q^{93} +2000.00 q^{94} -1404.00 q^{95} -768.000 q^{96} +1079.00 q^{97} -1332.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.00000 1.00000
\(5\) −13.0000 −1.16276 −0.581378 0.813634i \(-0.697486\pi\)
−0.581378 + 0.813634i \(0.697486\pi\)
\(6\) 12.0000 0.816497
\(7\) −26.0000 −1.40387 −0.701934 0.712242i \(-0.747680\pi\)
−0.701934 + 0.712242i \(0.747680\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 52.0000 1.64438
\(11\) 0 0
\(12\) −24.0000 −0.577350
\(13\) 73.0000 1.55743 0.778714 0.627379i \(-0.215872\pi\)
0.778714 + 0.627379i \(0.215872\pi\)
\(14\) 104.000 1.98537
\(15\) 39.0000 0.671317
\(16\) −64.0000 −1.00000
\(17\) −31.0000 −0.442271 −0.221135 0.975243i \(-0.570976\pi\)
−0.221135 + 0.975243i \(0.570976\pi\)
\(18\) −36.0000 −0.471405
\(19\) 108.000 1.30405 0.652024 0.758199i \(-0.273920\pi\)
0.652024 + 0.758199i \(0.273920\pi\)
\(20\) −104.000 −1.16276
\(21\) 78.0000 0.810524
\(22\) 0 0
\(23\) −86.0000 −0.779663 −0.389831 0.920886i \(-0.627467\pi\)
−0.389831 + 0.920886i \(0.627467\pi\)
\(24\) 0 0
\(25\) 44.0000 0.352000
\(26\) −292.000 −2.20254
\(27\) −27.0000 −0.192450
\(28\) −208.000 −1.40387
\(29\) 207.000 1.32548 0.662740 0.748849i \(-0.269393\pi\)
0.662740 + 0.748849i \(0.269393\pi\)
\(30\) −156.000 −0.949386
\(31\) 208.000 1.20509 0.602547 0.798084i \(-0.294153\pi\)
0.602547 + 0.798084i \(0.294153\pi\)
\(32\) 256.000 1.41421
\(33\) 0 0
\(34\) 124.000 0.625465
\(35\) 338.000 1.63236
\(36\) 72.0000 0.333333
\(37\) 45.0000 0.199945 0.0999724 0.994990i \(-0.468125\pi\)
0.0999724 + 0.994990i \(0.468125\pi\)
\(38\) −432.000 −1.84420
\(39\) −219.000 −0.899181
\(40\) 0 0
\(41\) −247.000 −0.940852 −0.470426 0.882440i \(-0.655900\pi\)
−0.470426 + 0.882440i \(0.655900\pi\)
\(42\) −312.000 −1.14625
\(43\) 450.000 1.59592 0.797958 0.602714i \(-0.205914\pi\)
0.797958 + 0.602714i \(0.205914\pi\)
\(44\) 0 0
\(45\) −117.000 −0.387585
\(46\) 344.000 1.10261
\(47\) −500.000 −1.55176 −0.775878 0.630883i \(-0.782693\pi\)
−0.775878 + 0.630883i \(0.782693\pi\)
\(48\) 192.000 0.577350
\(49\) 333.000 0.970845
\(50\) −176.000 −0.497803
\(51\) 93.0000 0.255345
\(52\) 584.000 1.55743
\(53\) −441.000 −1.14294 −0.571472 0.820622i \(-0.693627\pi\)
−0.571472 + 0.820622i \(0.693627\pi\)
\(54\) 108.000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −324.000 −0.752892
\(58\) −828.000 −1.87451
\(59\) 598.000 1.31954 0.659771 0.751467i \(-0.270653\pi\)
0.659771 + 0.751467i \(0.270653\pi\)
\(60\) 312.000 0.671317
\(61\) −378.000 −0.793409 −0.396704 0.917946i \(-0.629846\pi\)
−0.396704 + 0.917946i \(0.629846\pi\)
\(62\) −832.000 −1.70426
\(63\) −234.000 −0.467956
\(64\) −512.000 −1.00000
\(65\) −949.000 −1.81091
\(66\) 0 0
\(67\) 494.000 0.900772 0.450386 0.892834i \(-0.351286\pi\)
0.450386 + 0.892834i \(0.351286\pi\)
\(68\) −248.000 −0.442271
\(69\) 258.000 0.450138
\(70\) −1352.00 −2.30850
\(71\) −594.000 −0.992885 −0.496442 0.868070i \(-0.665361\pi\)
−0.496442 + 0.868070i \(0.665361\pi\)
\(72\) 0 0
\(73\) −1034.00 −1.65782 −0.828908 0.559385i \(-0.811037\pi\)
−0.828908 + 0.559385i \(0.811037\pi\)
\(74\) −180.000 −0.282765
\(75\) −132.000 −0.203227
\(76\) 864.000 1.30405
\(77\) 0 0
\(78\) 876.000 1.27163
\(79\) −352.000 −0.501305 −0.250652 0.968077i \(-0.580645\pi\)
−0.250652 + 0.968077i \(0.580645\pi\)
\(80\) 832.000 1.16276
\(81\) 81.0000 0.111111
\(82\) 988.000 1.33057
\(83\) −360.000 −0.476086 −0.238043 0.971255i \(-0.576506\pi\)
−0.238043 + 0.971255i \(0.576506\pi\)
\(84\) 624.000 0.810524
\(85\) 403.000 0.514253
\(86\) −1800.00 −2.25697
\(87\) −621.000 −0.765267
\(88\) 0 0
\(89\) −351.000 −0.418044 −0.209022 0.977911i \(-0.567028\pi\)
−0.209022 + 0.977911i \(0.567028\pi\)
\(90\) 468.000 0.548128
\(91\) −1898.00 −2.18642
\(92\) −688.000 −0.779663
\(93\) −624.000 −0.695761
\(94\) 2000.00 2.19451
\(95\) −1404.00 −1.51629
\(96\) −768.000 −0.816497
\(97\) 1079.00 1.12944 0.564721 0.825282i \(-0.308984\pi\)
0.564721 + 0.825282i \(0.308984\pi\)
\(98\) −1332.00 −1.37298
\(99\) 0 0
\(100\) 352.000 0.352000
\(101\) −86.0000 −0.0847259 −0.0423630 0.999102i \(-0.513489\pi\)
−0.0423630 + 0.999102i \(0.513489\pi\)
\(102\) −372.000 −0.361113
\(103\) −226.000 −0.216198 −0.108099 0.994140i \(-0.534476\pi\)
−0.108099 + 0.994140i \(0.534476\pi\)
\(104\) 0 0
\(105\) −1014.00 −0.942441
\(106\) 1764.00 1.61637
\(107\) 846.000 0.764354 0.382177 0.924089i \(-0.375174\pi\)
0.382177 + 0.924089i \(0.375174\pi\)
\(108\) −216.000 −0.192450
\(109\) −1079.00 −0.948160 −0.474080 0.880482i \(-0.657219\pi\)
−0.474080 + 0.880482i \(0.657219\pi\)
\(110\) 0 0
\(111\) −135.000 −0.115438
\(112\) 1664.00 1.40387
\(113\) 117.000 0.0974021 0.0487010 0.998813i \(-0.484492\pi\)
0.0487010 + 0.998813i \(0.484492\pi\)
\(114\) 1296.00 1.06475
\(115\) 1118.00 0.906557
\(116\) 1656.00 1.32548
\(117\) 657.000 0.519142
\(118\) −2392.00 −1.86611
\(119\) 806.000 0.620890
\(120\) 0 0
\(121\) 0 0
\(122\) 1512.00 1.12205
\(123\) 741.000 0.543201
\(124\) 1664.00 1.20509
\(125\) 1053.00 0.753465
\(126\) 936.000 0.661790
\(127\) −892.000 −0.623246 −0.311623 0.950206i \(-0.600873\pi\)
−0.311623 + 0.950206i \(0.600873\pi\)
\(128\) 0 0
\(129\) −1350.00 −0.921402
\(130\) 3796.00 2.56101
\(131\) 2250.00 1.50064 0.750318 0.661077i \(-0.229900\pi\)
0.750318 + 0.661077i \(0.229900\pi\)
\(132\) 0 0
\(133\) −2808.00 −1.83071
\(134\) −1976.00 −1.27388
\(135\) 351.000 0.223772
\(136\) 0 0
\(137\) −2138.00 −1.33330 −0.666648 0.745372i \(-0.732272\pi\)
−0.666648 + 0.745372i \(0.732272\pi\)
\(138\) −1032.00 −0.636592
\(139\) −350.000 −0.213573 −0.106786 0.994282i \(-0.534056\pi\)
−0.106786 + 0.994282i \(0.534056\pi\)
\(140\) 2704.00 1.63236
\(141\) 1500.00 0.895906
\(142\) 2376.00 1.40415
\(143\) 0 0
\(144\) −576.000 −0.333333
\(145\) −2691.00 −1.54121
\(146\) 4136.00 2.34451
\(147\) −999.000 −0.560518
\(148\) 360.000 0.199945
\(149\) 1543.00 0.848372 0.424186 0.905575i \(-0.360560\pi\)
0.424186 + 0.905575i \(0.360560\pi\)
\(150\) 528.000 0.287407
\(151\) 2600.00 1.40123 0.700613 0.713542i \(-0.252910\pi\)
0.700613 + 0.713542i \(0.252910\pi\)
\(152\) 0 0
\(153\) −279.000 −0.147424
\(154\) 0 0
\(155\) −2704.00 −1.40123
\(156\) −1752.00 −0.899181
\(157\) −586.000 −0.297885 −0.148942 0.988846i \(-0.547587\pi\)
−0.148942 + 0.988846i \(0.547587\pi\)
\(158\) 1408.00 0.708952
\(159\) 1323.00 0.659879
\(160\) −3328.00 −1.64438
\(161\) 2236.00 1.09454
\(162\) −324.000 −0.157135
\(163\) −1502.00 −0.721753 −0.360876 0.932614i \(-0.617522\pi\)
−0.360876 + 0.932614i \(0.617522\pi\)
\(164\) −1976.00 −0.940852
\(165\) 0 0
\(166\) 1440.00 0.673287
\(167\) −234.000 −0.108428 −0.0542140 0.998529i \(-0.517265\pi\)
−0.0542140 + 0.998529i \(0.517265\pi\)
\(168\) 0 0
\(169\) 3132.00 1.42558
\(170\) −1612.00 −0.727263
\(171\) 972.000 0.434682
\(172\) 3600.00 1.59592
\(173\) 918.000 0.403435 0.201717 0.979444i \(-0.435348\pi\)
0.201717 + 0.979444i \(0.435348\pi\)
\(174\) 2484.00 1.08225
\(175\) −1144.00 −0.494162
\(176\) 0 0
\(177\) −1794.00 −0.761838
\(178\) 1404.00 0.591204
\(179\) 216.000 0.0901933 0.0450966 0.998983i \(-0.485640\pi\)
0.0450966 + 0.998983i \(0.485640\pi\)
\(180\) −936.000 −0.387585
\(181\) −863.000 −0.354399 −0.177200 0.984175i \(-0.556704\pi\)
−0.177200 + 0.984175i \(0.556704\pi\)
\(182\) 7592.00 3.09207
\(183\) 1134.00 0.458075
\(184\) 0 0
\(185\) −585.000 −0.232487
\(186\) 2496.00 0.983955
\(187\) 0 0
\(188\) −4000.00 −1.55176
\(189\) 702.000 0.270175
\(190\) 5616.00 2.14436
\(191\) −1508.00 −0.571283 −0.285641 0.958337i \(-0.592207\pi\)
−0.285641 + 0.958337i \(0.592207\pi\)
\(192\) 1536.00 0.577350
\(193\) 2203.00 0.821634 0.410817 0.911718i \(-0.365243\pi\)
0.410817 + 0.911718i \(0.365243\pi\)
\(194\) −4316.00 −1.59727
\(195\) 2847.00 1.04553
\(196\) 2664.00 0.970845
\(197\) −949.000 −0.343215 −0.171608 0.985165i \(-0.554896\pi\)
−0.171608 + 0.985165i \(0.554896\pi\)
\(198\) 0 0
\(199\) 2322.00 0.827147 0.413573 0.910471i \(-0.364281\pi\)
0.413573 + 0.910471i \(0.364281\pi\)
\(200\) 0 0
\(201\) −1482.00 −0.520061
\(202\) 344.000 0.119821
\(203\) −5382.00 −1.86080
\(204\) 744.000 0.255345
\(205\) 3211.00 1.09398
\(206\) 904.000 0.305751
\(207\) −774.000 −0.259888
\(208\) −4672.00 −1.55743
\(209\) 0 0
\(210\) 4056.00 1.33281
\(211\) −702.000 −0.229041 −0.114521 0.993421i \(-0.536533\pi\)
−0.114521 + 0.993421i \(0.536533\pi\)
\(212\) −3528.00 −1.14294
\(213\) 1782.00 0.573242
\(214\) −3384.00 −1.08096
\(215\) −5850.00 −1.85566
\(216\) 0 0
\(217\) −5408.00 −1.69179
\(218\) 4316.00 1.34090
\(219\) 3102.00 0.957140
\(220\) 0 0
\(221\) −2263.00 −0.688805
\(222\) 540.000 0.163254
\(223\) −2240.00 −0.672652 −0.336326 0.941746i \(-0.609184\pi\)
−0.336326 + 0.941746i \(0.609184\pi\)
\(224\) −6656.00 −1.98537
\(225\) 396.000 0.117333
\(226\) −468.000 −0.137747
\(227\) −2574.00 −0.752610 −0.376305 0.926496i \(-0.622805\pi\)
−0.376305 + 0.926496i \(0.622805\pi\)
\(228\) −2592.00 −0.752892
\(229\) −3231.00 −0.932360 −0.466180 0.884690i \(-0.654370\pi\)
−0.466180 + 0.884690i \(0.654370\pi\)
\(230\) −4472.00 −1.28206
\(231\) 0 0
\(232\) 0 0
\(233\) −855.000 −0.240399 −0.120199 0.992750i \(-0.538353\pi\)
−0.120199 + 0.992750i \(0.538353\pi\)
\(234\) −2628.00 −0.734178
\(235\) 6500.00 1.80431
\(236\) 4784.00 1.31954
\(237\) 1056.00 0.289429
\(238\) −3224.00 −0.878071
\(239\) 576.000 0.155893 0.0779463 0.996958i \(-0.475164\pi\)
0.0779463 + 0.996958i \(0.475164\pi\)
\(240\) −2496.00 −0.671317
\(241\) −3770.00 −1.00766 −0.503832 0.863802i \(-0.668077\pi\)
−0.503832 + 0.863802i \(0.668077\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) −3024.00 −0.793409
\(245\) −4329.00 −1.12886
\(246\) −2964.00 −0.768202
\(247\) 7884.00 2.03096
\(248\) 0 0
\(249\) 1080.00 0.274868
\(250\) −4212.00 −1.06556
\(251\) 554.000 0.139315 0.0696577 0.997571i \(-0.477809\pi\)
0.0696577 + 0.997571i \(0.477809\pi\)
\(252\) −1872.00 −0.467956
\(253\) 0 0
\(254\) 3568.00 0.881402
\(255\) −1209.00 −0.296904
\(256\) 4096.00 1.00000
\(257\) −6799.00 −1.65023 −0.825117 0.564962i \(-0.808891\pi\)
−0.825117 + 0.564962i \(0.808891\pi\)
\(258\) 5400.00 1.30306
\(259\) −1170.00 −0.280696
\(260\) −7592.00 −1.81091
\(261\) 1863.00 0.441827
\(262\) −9000.00 −2.12222
\(263\) −716.000 −0.167872 −0.0839362 0.996471i \(-0.526749\pi\)
−0.0839362 + 0.996471i \(0.526749\pi\)
\(264\) 0 0
\(265\) 5733.00 1.32896
\(266\) 11232.0 2.58902
\(267\) 1053.00 0.241358
\(268\) 3952.00 0.900772
\(269\) 8411.00 1.90642 0.953211 0.302305i \(-0.0977560\pi\)
0.953211 + 0.302305i \(0.0977560\pi\)
\(270\) −1404.00 −0.316462
\(271\) 2366.00 0.530348 0.265174 0.964201i \(-0.414571\pi\)
0.265174 + 0.964201i \(0.414571\pi\)
\(272\) 1984.00 0.442271
\(273\) 5694.00 1.26233
\(274\) 8552.00 1.88557
\(275\) 0 0
\(276\) 2064.00 0.450138
\(277\) −5291.00 −1.14767 −0.573836 0.818970i \(-0.694545\pi\)
−0.573836 + 0.818970i \(0.694545\pi\)
\(278\) 1400.00 0.302037
\(279\) 1872.00 0.401698
\(280\) 0 0
\(281\) 2106.00 0.447094 0.223547 0.974693i \(-0.428236\pi\)
0.223547 + 0.974693i \(0.428236\pi\)
\(282\) −6000.00 −1.26700
\(283\) −6578.00 −1.38170 −0.690851 0.722997i \(-0.742764\pi\)
−0.690851 + 0.722997i \(0.742764\pi\)
\(284\) −4752.00 −0.992885
\(285\) 4212.00 0.875429
\(286\) 0 0
\(287\) 6422.00 1.32083
\(288\) 2304.00 0.471405
\(289\) −3952.00 −0.804396
\(290\) 10764.0 2.17960
\(291\) −3237.00 −0.652084
\(292\) −8272.00 −1.65782
\(293\) 7443.00 1.48404 0.742022 0.670376i \(-0.233867\pi\)
0.742022 + 0.670376i \(0.233867\pi\)
\(294\) 3996.00 0.792692
\(295\) −7774.00 −1.53430
\(296\) 0 0
\(297\) 0 0
\(298\) −6172.00 −1.19978
\(299\) −6278.00 −1.21427
\(300\) −1056.00 −0.203227
\(301\) −11700.0 −2.24045
\(302\) −10400.0 −1.98163
\(303\) 258.000 0.0489165
\(304\) −6912.00 −1.30405
\(305\) 4914.00 0.922540
\(306\) 1116.00 0.208488
\(307\) −982.000 −0.182559 −0.0912796 0.995825i \(-0.529096\pi\)
−0.0912796 + 0.995825i \(0.529096\pi\)
\(308\) 0 0
\(309\) 678.000 0.124822
\(310\) 10816.0 1.98164
\(311\) −5868.00 −1.06992 −0.534958 0.844879i \(-0.679672\pi\)
−0.534958 + 0.844879i \(0.679672\pi\)
\(312\) 0 0
\(313\) −6265.00 −1.13137 −0.565685 0.824621i \(-0.691388\pi\)
−0.565685 + 0.824621i \(0.691388\pi\)
\(314\) 2344.00 0.421273
\(315\) 3042.00 0.544118
\(316\) −2816.00 −0.501305
\(317\) −7726.00 −1.36888 −0.684441 0.729069i \(-0.739954\pi\)
−0.684441 + 0.729069i \(0.739954\pi\)
\(318\) −5292.00 −0.933210
\(319\) 0 0
\(320\) 6656.00 1.16276
\(321\) −2538.00 −0.441300
\(322\) −8944.00 −1.54792
\(323\) −3348.00 −0.576742
\(324\) 648.000 0.111111
\(325\) 3212.00 0.548214
\(326\) 6008.00 1.02071
\(327\) 3237.00 0.547421
\(328\) 0 0
\(329\) 13000.0 2.17846
\(330\) 0 0
\(331\) 1144.00 0.189970 0.0949848 0.995479i \(-0.469720\pi\)
0.0949848 + 0.995479i \(0.469720\pi\)
\(332\) −2880.00 −0.476086
\(333\) 405.000 0.0666482
\(334\) 936.000 0.153340
\(335\) −6422.00 −1.04738
\(336\) −4992.00 −0.810524
\(337\) −4849.00 −0.783804 −0.391902 0.920007i \(-0.628183\pi\)
−0.391902 + 0.920007i \(0.628183\pi\)
\(338\) −12528.0 −2.01608
\(339\) −351.000 −0.0562351
\(340\) 3224.00 0.514253
\(341\) 0 0
\(342\) −3888.00 −0.614734
\(343\) 260.000 0.0409291
\(344\) 0 0
\(345\) −3354.00 −0.523401
\(346\) −3672.00 −0.570543
\(347\) −5512.00 −0.852737 −0.426368 0.904550i \(-0.640207\pi\)
−0.426368 + 0.904550i \(0.640207\pi\)
\(348\) −4968.00 −0.765267
\(349\) 8073.00 1.23822 0.619109 0.785305i \(-0.287494\pi\)
0.619109 + 0.785305i \(0.287494\pi\)
\(350\) 4576.00 0.698850
\(351\) −1971.00 −0.299727
\(352\) 0 0
\(353\) −4203.00 −0.633720 −0.316860 0.948472i \(-0.602629\pi\)
−0.316860 + 0.948472i \(0.602629\pi\)
\(354\) 7176.00 1.07740
\(355\) 7722.00 1.15448
\(356\) −2808.00 −0.418044
\(357\) −2418.00 −0.358471
\(358\) −864.000 −0.127553
\(359\) −1930.00 −0.283737 −0.141868 0.989886i \(-0.545311\pi\)
−0.141868 + 0.989886i \(0.545311\pi\)
\(360\) 0 0
\(361\) 4805.00 0.700539
\(362\) 3452.00 0.501196
\(363\) 0 0
\(364\) −15184.0 −2.18642
\(365\) 13442.0 1.92763
\(366\) −4536.00 −0.647816
\(367\) −494.000 −0.0702632 −0.0351316 0.999383i \(-0.511185\pi\)
−0.0351316 + 0.999383i \(0.511185\pi\)
\(368\) 5504.00 0.779663
\(369\) −2223.00 −0.313617
\(370\) 2340.00 0.328786
\(371\) 11466.0 1.60454
\(372\) −4992.00 −0.695761
\(373\) −7982.00 −1.10802 −0.554011 0.832509i \(-0.686904\pi\)
−0.554011 + 0.832509i \(0.686904\pi\)
\(374\) 0 0
\(375\) −3159.00 −0.435013
\(376\) 0 0
\(377\) 15111.0 2.06434
\(378\) −2808.00 −0.382084
\(379\) −6938.00 −0.940320 −0.470160 0.882581i \(-0.655804\pi\)
−0.470160 + 0.882581i \(0.655804\pi\)
\(380\) −11232.0 −1.51629
\(381\) 2676.00 0.359831
\(382\) 6032.00 0.807916
\(383\) −8654.00 −1.15457 −0.577283 0.816544i \(-0.695887\pi\)
−0.577283 + 0.816544i \(0.695887\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8812.00 −1.16197
\(387\) 4050.00 0.531972
\(388\) 8632.00 1.12944
\(389\) 5715.00 0.744889 0.372445 0.928054i \(-0.378520\pi\)
0.372445 + 0.928054i \(0.378520\pi\)
\(390\) −11388.0 −1.47860
\(391\) 2666.00 0.344822
\(392\) 0 0
\(393\) −6750.00 −0.866393
\(394\) 3796.00 0.485380
\(395\) 4576.00 0.582895
\(396\) 0 0
\(397\) −7163.00 −0.905543 −0.452772 0.891627i \(-0.649565\pi\)
−0.452772 + 0.891627i \(0.649565\pi\)
\(398\) −9288.00 −1.16976
\(399\) 8424.00 1.05696
\(400\) −2816.00 −0.352000
\(401\) 6493.00 0.808591 0.404295 0.914628i \(-0.367517\pi\)
0.404295 + 0.914628i \(0.367517\pi\)
\(402\) 5928.00 0.735477
\(403\) 15184.0 1.87685
\(404\) −688.000 −0.0847259
\(405\) −1053.00 −0.129195
\(406\) 21528.0 2.63157
\(407\) 0 0
\(408\) 0 0
\(409\) −585.000 −0.0707247 −0.0353623 0.999375i \(-0.511259\pi\)
−0.0353623 + 0.999375i \(0.511259\pi\)
\(410\) −12844.0 −1.54712
\(411\) 6414.00 0.769779
\(412\) −1808.00 −0.216198
\(413\) −15548.0 −1.85246
\(414\) 3096.00 0.367536
\(415\) 4680.00 0.553571
\(416\) 18688.0 2.20254
\(417\) 1050.00 0.123306
\(418\) 0 0
\(419\) 7330.00 0.854639 0.427320 0.904101i \(-0.359458\pi\)
0.427320 + 0.904101i \(0.359458\pi\)
\(420\) −8112.00 −0.942441
\(421\) −16055.0 −1.85861 −0.929303 0.369319i \(-0.879591\pi\)
−0.929303 + 0.369319i \(0.879591\pi\)
\(422\) 2808.00 0.323913
\(423\) −4500.00 −0.517252
\(424\) 0 0
\(425\) −1364.00 −0.155679
\(426\) −7128.00 −0.810687
\(427\) 9828.00 1.11384
\(428\) 6768.00 0.764354
\(429\) 0 0
\(430\) 23400.0 2.62430
\(431\) −2934.00 −0.327902 −0.163951 0.986468i \(-0.552424\pi\)
−0.163951 + 0.986468i \(0.552424\pi\)
\(432\) 1728.00 0.192450
\(433\) −8549.00 −0.948819 −0.474410 0.880304i \(-0.657338\pi\)
−0.474410 + 0.880304i \(0.657338\pi\)
\(434\) 21632.0 2.39256
\(435\) 8073.00 0.889818
\(436\) −8632.00 −0.948160
\(437\) −9288.00 −1.01672
\(438\) −12408.0 −1.35360
\(439\) −14084.0 −1.53119 −0.765595 0.643323i \(-0.777555\pi\)
−0.765595 + 0.643323i \(0.777555\pi\)
\(440\) 0 0
\(441\) 2997.00 0.323615
\(442\) 9052.00 0.974117
\(443\) 13586.0 1.45709 0.728544 0.684999i \(-0.240197\pi\)
0.728544 + 0.684999i \(0.240197\pi\)
\(444\) −1080.00 −0.115438
\(445\) 4563.00 0.486083
\(446\) 8960.00 0.951274
\(447\) −4629.00 −0.489808
\(448\) 13312.0 1.40387
\(449\) −3659.00 −0.384585 −0.192293 0.981338i \(-0.561592\pi\)
−0.192293 + 0.981338i \(0.561592\pi\)
\(450\) −1584.00 −0.165934
\(451\) 0 0
\(452\) 936.000 0.0974021
\(453\) −7800.00 −0.808998
\(454\) 10296.0 1.06435
\(455\) 24674.0 2.54227
\(456\) 0 0
\(457\) 9711.00 0.994007 0.497004 0.867748i \(-0.334434\pi\)
0.497004 + 0.867748i \(0.334434\pi\)
\(458\) 12924.0 1.31856
\(459\) 837.000 0.0851151
\(460\) 8944.00 0.906557
\(461\) 6903.00 0.697407 0.348704 0.937233i \(-0.386622\pi\)
0.348704 + 0.937233i \(0.386622\pi\)
\(462\) 0 0
\(463\) 18694.0 1.87642 0.938212 0.346062i \(-0.112481\pi\)
0.938212 + 0.346062i \(0.112481\pi\)
\(464\) −13248.0 −1.32548
\(465\) 8112.00 0.809000
\(466\) 3420.00 0.339975
\(467\) 450.000 0.0445900 0.0222950 0.999751i \(-0.492903\pi\)
0.0222950 + 0.999751i \(0.492903\pi\)
\(468\) 5256.00 0.519142
\(469\) −12844.0 −1.26456
\(470\) −26000.0 −2.55168
\(471\) 1758.00 0.171984
\(472\) 0 0
\(473\) 0 0
\(474\) −4224.00 −0.409314
\(475\) 4752.00 0.459025
\(476\) 6448.00 0.620890
\(477\) −3969.00 −0.380981
\(478\) −2304.00 −0.220465
\(479\) 12298.0 1.17309 0.586545 0.809917i \(-0.300488\pi\)
0.586545 + 0.809917i \(0.300488\pi\)
\(480\) 9984.00 0.949386
\(481\) 3285.00 0.311399
\(482\) 15080.0 1.42505
\(483\) −6708.00 −0.631935
\(484\) 0 0
\(485\) −14027.0 −1.31326
\(486\) 972.000 0.0907218
\(487\) −11098.0 −1.03265 −0.516323 0.856394i \(-0.672700\pi\)
−0.516323 + 0.856394i \(0.672700\pi\)
\(488\) 0 0
\(489\) 4506.00 0.416704
\(490\) 17316.0 1.59644
\(491\) 2696.00 0.247798 0.123899 0.992295i \(-0.460460\pi\)
0.123899 + 0.992295i \(0.460460\pi\)
\(492\) 5928.00 0.543201
\(493\) −6417.00 −0.586221
\(494\) −31536.0 −2.87221
\(495\) 0 0
\(496\) −13312.0 −1.20509
\(497\) 15444.0 1.39388
\(498\) −4320.00 −0.388723
\(499\) 26.0000 0.00233250 0.00116625 0.999999i \(-0.499629\pi\)
0.00116625 + 0.999999i \(0.499629\pi\)
\(500\) 8424.00 0.753465
\(501\) 702.000 0.0626009
\(502\) −2216.00 −0.197022
\(503\) −19462.0 −1.72518 −0.862592 0.505900i \(-0.831160\pi\)
−0.862592 + 0.505900i \(0.831160\pi\)
\(504\) 0 0
\(505\) 1118.00 0.0985155
\(506\) 0 0
\(507\) −9396.00 −0.823059
\(508\) −7136.00 −0.623246
\(509\) −19782.0 −1.72264 −0.861318 0.508066i \(-0.830361\pi\)
−0.861318 + 0.508066i \(0.830361\pi\)
\(510\) 4836.00 0.419886
\(511\) 26884.0 2.32735
\(512\) −16384.0 −1.41421
\(513\) −2916.00 −0.250964
\(514\) 27196.0 2.33378
\(515\) 2938.00 0.251386
\(516\) −10800.0 −0.921402
\(517\) 0 0
\(518\) 4680.00 0.396964
\(519\) −2754.00 −0.232923
\(520\) 0 0
\(521\) −15534.0 −1.30625 −0.653126 0.757250i \(-0.726543\pi\)
−0.653126 + 0.757250i \(0.726543\pi\)
\(522\) −7452.00 −0.624838
\(523\) −376.000 −0.0314366 −0.0157183 0.999876i \(-0.505003\pi\)
−0.0157183 + 0.999876i \(0.505003\pi\)
\(524\) 18000.0 1.50064
\(525\) 3432.00 0.285304
\(526\) 2864.00 0.237407
\(527\) −6448.00 −0.532978
\(528\) 0 0
\(529\) −4771.00 −0.392126
\(530\) −22932.0 −1.87944
\(531\) 5382.00 0.439847
\(532\) −22464.0 −1.83071
\(533\) −18031.0 −1.46531
\(534\) −4212.00 −0.341332
\(535\) −10998.0 −0.888757
\(536\) 0 0
\(537\) −648.000 −0.0520731
\(538\) −33644.0 −2.69609
\(539\) 0 0
\(540\) 2808.00 0.223772
\(541\) −10322.0 −0.820291 −0.410146 0.912020i \(-0.634522\pi\)
−0.410146 + 0.912020i \(0.634522\pi\)
\(542\) −9464.00 −0.750025
\(543\) 2589.00 0.204613
\(544\) −7936.00 −0.625465
\(545\) 14027.0 1.10248
\(546\) −22776.0 −1.78521
\(547\) 13976.0 1.09245 0.546225 0.837638i \(-0.316064\pi\)
0.546225 + 0.837638i \(0.316064\pi\)
\(548\) −17104.0 −1.33330
\(549\) −3402.00 −0.264470
\(550\) 0 0
\(551\) 22356.0 1.72849
\(552\) 0 0
\(553\) 9152.00 0.703766
\(554\) 21164.0 1.62305
\(555\) 1755.00 0.134226
\(556\) −2800.00 −0.213573
\(557\) −8086.00 −0.615107 −0.307554 0.951531i \(-0.599510\pi\)
−0.307554 + 0.951531i \(0.599510\pi\)
\(558\) −7488.00 −0.568087
\(559\) 32850.0 2.48552
\(560\) −21632.0 −1.63236
\(561\) 0 0
\(562\) −8424.00 −0.632286
\(563\) 10418.0 0.779869 0.389935 0.920843i \(-0.372498\pi\)
0.389935 + 0.920843i \(0.372498\pi\)
\(564\) 12000.0 0.895906
\(565\) −1521.00 −0.113255
\(566\) 26312.0 1.95402
\(567\) −2106.00 −0.155985
\(568\) 0 0
\(569\) −20826.0 −1.53440 −0.767198 0.641410i \(-0.778350\pi\)
−0.767198 + 0.641410i \(0.778350\pi\)
\(570\) −16848.0 −1.23804
\(571\) −10018.0 −0.734221 −0.367111 0.930177i \(-0.619653\pi\)
−0.367111 + 0.930177i \(0.619653\pi\)
\(572\) 0 0
\(573\) 4524.00 0.329830
\(574\) −25688.0 −1.86794
\(575\) −3784.00 −0.274441
\(576\) −4608.00 −0.333333
\(577\) 1395.00 0.100649 0.0503246 0.998733i \(-0.483974\pi\)
0.0503246 + 0.998733i \(0.483974\pi\)
\(578\) 15808.0 1.13759
\(579\) −6609.00 −0.474371
\(580\) −21528.0 −1.54121
\(581\) 9360.00 0.668362
\(582\) 12948.0 0.922185
\(583\) 0 0
\(584\) 0 0
\(585\) −8541.00 −0.603636
\(586\) −29772.0 −2.09875
\(587\) −17320.0 −1.21784 −0.608921 0.793231i \(-0.708397\pi\)
−0.608921 + 0.793231i \(0.708397\pi\)
\(588\) −7992.00 −0.560518
\(589\) 22464.0 1.57150
\(590\) 31096.0 2.16983
\(591\) 2847.00 0.198156
\(592\) −2880.00 −0.199945
\(593\) 10309.0 0.713895 0.356948 0.934124i \(-0.383817\pi\)
0.356948 + 0.934124i \(0.383817\pi\)
\(594\) 0 0
\(595\) −10478.0 −0.721943
\(596\) 12344.0 0.848372
\(597\) −6966.00 −0.477553
\(598\) 25112.0 1.71723
\(599\) −11128.0 −0.759061 −0.379531 0.925179i \(-0.623915\pi\)
−0.379531 + 0.925179i \(0.623915\pi\)
\(600\) 0 0
\(601\) −12349.0 −0.838147 −0.419073 0.907952i \(-0.637645\pi\)
−0.419073 + 0.907952i \(0.637645\pi\)
\(602\) 46800.0 3.16848
\(603\) 4446.00 0.300257
\(604\) 20800.0 1.40123
\(605\) 0 0
\(606\) −1032.00 −0.0691784
\(607\) 2744.00 0.183485 0.0917426 0.995783i \(-0.470756\pi\)
0.0917426 + 0.995783i \(0.470756\pi\)
\(608\) 27648.0 1.84420
\(609\) 16146.0 1.07433
\(610\) −19656.0 −1.30467
\(611\) −36500.0 −2.41675
\(612\) −2232.00 −0.147424
\(613\) −19387.0 −1.27738 −0.638690 0.769464i \(-0.720523\pi\)
−0.638690 + 0.769464i \(0.720523\pi\)
\(614\) 3928.00 0.258178
\(615\) −9633.00 −0.631610
\(616\) 0 0
\(617\) 14265.0 0.930774 0.465387 0.885107i \(-0.345915\pi\)
0.465387 + 0.885107i \(0.345915\pi\)
\(618\) −2712.00 −0.176525
\(619\) 26630.0 1.72916 0.864580 0.502495i \(-0.167585\pi\)
0.864580 + 0.502495i \(0.167585\pi\)
\(620\) −21632.0 −1.40123
\(621\) 2322.00 0.150046
\(622\) 23472.0 1.51309
\(623\) 9126.00 0.586879
\(624\) 14016.0 0.899181
\(625\) −19189.0 −1.22810
\(626\) 25060.0 1.60000
\(627\) 0 0
\(628\) −4688.00 −0.297885
\(629\) −1395.00 −0.0884297
\(630\) −12168.0 −0.769500
\(631\) 12610.0 0.795557 0.397778 0.917482i \(-0.369781\pi\)
0.397778 + 0.917482i \(0.369781\pi\)
\(632\) 0 0
\(633\) 2106.00 0.132237
\(634\) 30904.0 1.93589
\(635\) 11596.0 0.724682
\(636\) 10584.0 0.659879
\(637\) 24309.0 1.51202
\(638\) 0 0
\(639\) −5346.00 −0.330962
\(640\) 0 0
\(641\) −23535.0 −1.45020 −0.725099 0.688645i \(-0.758206\pi\)
−0.725099 + 0.688645i \(0.758206\pi\)
\(642\) 10152.0 0.624093
\(643\) 728.000 0.0446493 0.0223247 0.999751i \(-0.492893\pi\)
0.0223247 + 0.999751i \(0.492893\pi\)
\(644\) 17888.0 1.09454
\(645\) 17550.0 1.07137
\(646\) 13392.0 0.815637
\(647\) 18400.0 1.11805 0.559025 0.829151i \(-0.311175\pi\)
0.559025 + 0.829151i \(0.311175\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −12848.0 −0.775292
\(651\) 16224.0 0.976757
\(652\) −12016.0 −0.721753
\(653\) 19202.0 1.15074 0.575369 0.817894i \(-0.304858\pi\)
0.575369 + 0.817894i \(0.304858\pi\)
\(654\) −12948.0 −0.774170
\(655\) −29250.0 −1.74487
\(656\) 15808.0 0.940852
\(657\) −9306.00 −0.552605
\(658\) −52000.0 −3.08081
\(659\) 4640.00 0.274277 0.137139 0.990552i \(-0.456209\pi\)
0.137139 + 0.990552i \(0.456209\pi\)
\(660\) 0 0
\(661\) −10367.0 −0.610030 −0.305015 0.952348i \(-0.598661\pi\)
−0.305015 + 0.952348i \(0.598661\pi\)
\(662\) −4576.00 −0.268658
\(663\) 6789.00 0.397682
\(664\) 0 0
\(665\) 36504.0 2.12867
\(666\) −1620.00 −0.0942548
\(667\) −17802.0 −1.03343
\(668\) −1872.00 −0.108428
\(669\) 6720.00 0.388356
\(670\) 25688.0 1.48121
\(671\) 0 0
\(672\) 19968.0 1.14625
\(673\) 18278.0 1.04690 0.523451 0.852056i \(-0.324644\pi\)
0.523451 + 0.852056i \(0.324644\pi\)
\(674\) 19396.0 1.10847
\(675\) −1188.00 −0.0677424
\(676\) 25056.0 1.42558
\(677\) −27549.0 −1.56395 −0.781975 0.623310i \(-0.785788\pi\)
−0.781975 + 0.623310i \(0.785788\pi\)
\(678\) 1404.00 0.0795285
\(679\) −28054.0 −1.58559
\(680\) 0 0
\(681\) 7722.00 0.434519
\(682\) 0 0
\(683\) −9626.00 −0.539281 −0.269640 0.962961i \(-0.586905\pi\)
−0.269640 + 0.962961i \(0.586905\pi\)
\(684\) 7776.00 0.434682
\(685\) 27794.0 1.55030
\(686\) −1040.00 −0.0578825
\(687\) 9693.00 0.538298
\(688\) −28800.0 −1.59592
\(689\) −32193.0 −1.78005
\(690\) 13416.0 0.740201
\(691\) 27170.0 1.49580 0.747898 0.663813i \(-0.231063\pi\)
0.747898 + 0.663813i \(0.231063\pi\)
\(692\) 7344.00 0.403435
\(693\) 0 0
\(694\) 22048.0 1.20595
\(695\) 4550.00 0.248333
\(696\) 0 0
\(697\) 7657.00 0.416111
\(698\) −32292.0 −1.75110
\(699\) 2565.00 0.138794
\(700\) −9152.00 −0.494162
\(701\) −31577.0 −1.70135 −0.850675 0.525691i \(-0.823807\pi\)
−0.850675 + 0.525691i \(0.823807\pi\)
\(702\) 7884.00 0.423878
\(703\) 4860.00 0.260737
\(704\) 0 0
\(705\) −19500.0 −1.04172
\(706\) 16812.0 0.896215
\(707\) 2236.00 0.118944
\(708\) −14352.0 −0.761838
\(709\) −31330.0 −1.65955 −0.829776 0.558096i \(-0.811532\pi\)
−0.829776 + 0.558096i \(0.811532\pi\)
\(710\) −30888.0 −1.63268
\(711\) −3168.00 −0.167102
\(712\) 0 0
\(713\) −17888.0 −0.939566
\(714\) 9672.00 0.506954
\(715\) 0 0
\(716\) 1728.00 0.0901933
\(717\) −1728.00 −0.0900047
\(718\) 7720.00 0.401264
\(719\) −34146.0 −1.77111 −0.885557 0.464531i \(-0.846223\pi\)
−0.885557 + 0.464531i \(0.846223\pi\)
\(720\) 7488.00 0.387585
\(721\) 5876.00 0.303514
\(722\) −19220.0 −0.990712
\(723\) 11310.0 0.581775
\(724\) −6904.00 −0.354399
\(725\) 9108.00 0.466569
\(726\) 0 0
\(727\) −8658.00 −0.441688 −0.220844 0.975309i \(-0.570881\pi\)
−0.220844 + 0.975309i \(0.570881\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −53768.0 −2.72609
\(731\) −13950.0 −0.705827
\(732\) 9072.00 0.458075
\(733\) 20349.0 1.02539 0.512693 0.858572i \(-0.328648\pi\)
0.512693 + 0.858572i \(0.328648\pi\)
\(734\) 1976.00 0.0993672
\(735\) 12987.0 0.651745
\(736\) −22016.0 −1.10261
\(737\) 0 0
\(738\) 8892.00 0.443522
\(739\) 9080.00 0.451980 0.225990 0.974130i \(-0.427438\pi\)
0.225990 + 0.974130i \(0.427438\pi\)
\(740\) −4680.00 −0.232487
\(741\) −23652.0 −1.17257
\(742\) −45864.0 −2.26916
\(743\) 24674.0 1.21831 0.609153 0.793053i \(-0.291510\pi\)
0.609153 + 0.793053i \(0.291510\pi\)
\(744\) 0 0
\(745\) −20059.0 −0.986450
\(746\) 31928.0 1.56698
\(747\) −3240.00 −0.158695
\(748\) 0 0
\(749\) −21996.0 −1.07305
\(750\) 12636.0 0.615202
\(751\) 442.000 0.0214764 0.0107382 0.999942i \(-0.496582\pi\)
0.0107382 + 0.999942i \(0.496582\pi\)
\(752\) 32000.0 1.55176
\(753\) −1662.00 −0.0804338
\(754\) −60444.0 −2.91942
\(755\) −33800.0 −1.62928
\(756\) 5616.00 0.270175
\(757\) −10999.0 −0.528092 −0.264046 0.964510i \(-0.585057\pi\)
−0.264046 + 0.964510i \(0.585057\pi\)
\(758\) 27752.0 1.32981
\(759\) 0 0
\(760\) 0 0
\(761\) 20709.0 0.986466 0.493233 0.869897i \(-0.335815\pi\)
0.493233 + 0.869897i \(0.335815\pi\)
\(762\) −10704.0 −0.508878
\(763\) 28054.0 1.33109
\(764\) −12064.0 −0.571283
\(765\) 3627.00 0.171418
\(766\) 34616.0 1.63280
\(767\) 43654.0 2.05509
\(768\) −12288.0 −0.577350
\(769\) −6985.00 −0.327549 −0.163775 0.986498i \(-0.552367\pi\)
−0.163775 + 0.986498i \(0.552367\pi\)
\(770\) 0 0
\(771\) 20397.0 0.952763
\(772\) 17624.0 0.821634
\(773\) 26442.0 1.23034 0.615170 0.788395i \(-0.289087\pi\)
0.615170 + 0.788395i \(0.289087\pi\)
\(774\) −16200.0 −0.752322
\(775\) 9152.00 0.424193
\(776\) 0 0
\(777\) 3510.00 0.162060
\(778\) −22860.0 −1.05343
\(779\) −26676.0 −1.22692
\(780\) 22776.0 1.04553
\(781\) 0 0
\(782\) −10664.0 −0.487652
\(783\) −5589.00 −0.255089
\(784\) −21312.0 −0.970845
\(785\) 7618.00 0.346367
\(786\) 27000.0 1.22526
\(787\) 13312.0 0.602950 0.301475 0.953474i \(-0.402521\pi\)
0.301475 + 0.953474i \(0.402521\pi\)
\(788\) −7592.00 −0.343215
\(789\) 2148.00 0.0969212
\(790\) −18304.0 −0.824338
\(791\) −3042.00 −0.136740
\(792\) 0 0
\(793\) −27594.0 −1.23568
\(794\) 28652.0 1.28063
\(795\) −17199.0 −0.767278
\(796\) 18576.0 0.827147
\(797\) −3614.00 −0.160620 −0.0803102 0.996770i \(-0.525591\pi\)
−0.0803102 + 0.996770i \(0.525591\pi\)
\(798\) −33696.0 −1.49477
\(799\) 15500.0 0.686296
\(800\) 11264.0 0.497803
\(801\) −3159.00 −0.139348
\(802\) −25972.0 −1.14352
\(803\) 0 0
\(804\) −11856.0 −0.520061
\(805\) −29068.0 −1.27269
\(806\) −60736.0 −2.65426
\(807\) −25233.0 −1.10067
\(808\) 0 0
\(809\) 25650.0 1.11472 0.557358 0.830272i \(-0.311815\pi\)
0.557358 + 0.830272i \(0.311815\pi\)
\(810\) 4212.00 0.182709
\(811\) 9676.00 0.418952 0.209476 0.977814i \(-0.432824\pi\)
0.209476 + 0.977814i \(0.432824\pi\)
\(812\) −43056.0 −1.86080
\(813\) −7098.00 −0.306196
\(814\) 0 0
\(815\) 19526.0 0.839222
\(816\) −5952.00 −0.255345
\(817\) 48600.0 2.08115
\(818\) 2340.00 0.100020
\(819\) −17082.0 −0.728808
\(820\) 25688.0 1.09398
\(821\) −29614.0 −1.25887 −0.629437 0.777051i \(-0.716714\pi\)
−0.629437 + 0.777051i \(0.716714\pi\)
\(822\) −25656.0 −1.08863
\(823\) −38358.0 −1.62464 −0.812318 0.583214i \(-0.801795\pi\)
−0.812318 + 0.583214i \(0.801795\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 62192.0 2.61978
\(827\) 24862.0 1.04539 0.522694 0.852520i \(-0.324927\pi\)
0.522694 + 0.852520i \(0.324927\pi\)
\(828\) −6192.00 −0.259888
\(829\) 5533.00 0.231808 0.115904 0.993260i \(-0.463023\pi\)
0.115904 + 0.993260i \(0.463023\pi\)
\(830\) −18720.0 −0.782868
\(831\) 15873.0 0.662609
\(832\) −37376.0 −1.55743
\(833\) −10323.0 −0.429377
\(834\) −4200.00 −0.174381
\(835\) 3042.00 0.126075
\(836\) 0 0
\(837\) −5616.00 −0.231920
\(838\) −29320.0 −1.20864
\(839\) −42016.0 −1.72891 −0.864454 0.502712i \(-0.832336\pi\)
−0.864454 + 0.502712i \(0.832336\pi\)
\(840\) 0 0
\(841\) 18460.0 0.756899
\(842\) 64220.0 2.62846
\(843\) −6318.00 −0.258130
\(844\) −5616.00 −0.229041
\(845\) −40716.0 −1.65760
\(846\) 18000.0 0.731504
\(847\) 0 0
\(848\) 28224.0 1.14294
\(849\) 19734.0 0.797726
\(850\) 5456.00 0.220164
\(851\) −3870.00 −0.155889
\(852\) 14256.0 0.573242
\(853\) −7579.00 −0.304220 −0.152110 0.988364i \(-0.548607\pi\)
−0.152110 + 0.988364i \(0.548607\pi\)
\(854\) −39312.0 −1.57521
\(855\) −12636.0 −0.505429
\(856\) 0 0
\(857\) 14854.0 0.592069 0.296034 0.955177i \(-0.404336\pi\)
0.296034 + 0.955177i \(0.404336\pi\)
\(858\) 0 0
\(859\) 47178.0 1.87391 0.936957 0.349444i \(-0.113629\pi\)
0.936957 + 0.349444i \(0.113629\pi\)
\(860\) −46800.0 −1.85566
\(861\) −19266.0 −0.762582
\(862\) 11736.0 0.463724
\(863\) 17442.0 0.687987 0.343993 0.938972i \(-0.388220\pi\)
0.343993 + 0.938972i \(0.388220\pi\)
\(864\) −6912.00 −0.272166
\(865\) −11934.0 −0.469096
\(866\) 34196.0 1.34183
\(867\) 11856.0 0.464419
\(868\) −43264.0 −1.69179
\(869\) 0 0
\(870\) −32292.0 −1.25839
\(871\) 36062.0 1.40289
\(872\) 0 0
\(873\) 9711.00 0.376481
\(874\) 37152.0 1.43785
\(875\) −27378.0 −1.05777
\(876\) 24816.0 0.957140
\(877\) 27261.0 1.04964 0.524822 0.851212i \(-0.324132\pi\)
0.524822 + 0.851212i \(0.324132\pi\)
\(878\) 56336.0 2.16543
\(879\) −22329.0 −0.856813
\(880\) 0 0
\(881\) 15457.0 0.591101 0.295550 0.955327i \(-0.404497\pi\)
0.295550 + 0.955327i \(0.404497\pi\)
\(882\) −11988.0 −0.457661
\(883\) −20852.0 −0.794706 −0.397353 0.917666i \(-0.630071\pi\)
−0.397353 + 0.917666i \(0.630071\pi\)
\(884\) −18104.0 −0.688805
\(885\) 23322.0 0.885831
\(886\) −54344.0 −2.06063
\(887\) 22594.0 0.855279 0.427639 0.903949i \(-0.359345\pi\)
0.427639 + 0.903949i \(0.359345\pi\)
\(888\) 0 0
\(889\) 23192.0 0.874955
\(890\) −18252.0 −0.687425
\(891\) 0 0
\(892\) −17920.0 −0.672652
\(893\) −54000.0 −2.02356
\(894\) 18516.0 0.692693
\(895\) −2808.00 −0.104873
\(896\) 0 0
\(897\) 18834.0 0.701058
\(898\) 14636.0 0.543886
\(899\) 43056.0 1.59733
\(900\) 3168.00 0.117333
\(901\) 13671.0 0.505491
\(902\) 0 0
\(903\) 35100.0 1.29353
\(904\) 0 0
\(905\) 11219.0 0.412080
\(906\) 31200.0 1.14410
\(907\) −47718.0 −1.74691 −0.873457 0.486902i \(-0.838127\pi\)
−0.873457 + 0.486902i \(0.838127\pi\)
\(908\) −20592.0 −0.752610
\(909\) −774.000 −0.0282420
\(910\) −98696.0 −3.59532
\(911\) −8982.00 −0.326660 −0.163330 0.986572i \(-0.552223\pi\)
−0.163330 + 0.986572i \(0.552223\pi\)
\(912\) 20736.0 0.752892
\(913\) 0 0
\(914\) −38844.0 −1.40574
\(915\) −14742.0 −0.532629
\(916\) −25848.0 −0.932360
\(917\) −58500.0 −2.10670
\(918\) −3348.00 −0.120371
\(919\) −25154.0 −0.902888 −0.451444 0.892300i \(-0.649091\pi\)
−0.451444 + 0.892300i \(0.649091\pi\)
\(920\) 0 0
\(921\) 2946.00 0.105401
\(922\) −27612.0 −0.986283
\(923\) −43362.0 −1.54635
\(924\) 0 0
\(925\) 1980.00 0.0703805
\(926\) −74776.0 −2.65366
\(927\) −2034.00 −0.0720662
\(928\) 52992.0 1.87451
\(929\) 27729.0 0.979288 0.489644 0.871922i \(-0.337127\pi\)
0.489644 + 0.871922i \(0.337127\pi\)
\(930\) −32448.0 −1.14410
\(931\) 35964.0 1.26603
\(932\) −6840.00 −0.240399
\(933\) 17604.0 0.617716
\(934\) −1800.00 −0.0630597
\(935\) 0 0
\(936\) 0 0
\(937\) −43525.0 −1.51750 −0.758751 0.651381i \(-0.774190\pi\)
−0.758751 + 0.651381i \(0.774190\pi\)
\(938\) 51376.0 1.78836
\(939\) 18795.0 0.653197
\(940\) 52000.0 1.80431
\(941\) 34407.0 1.19196 0.595981 0.802999i \(-0.296763\pi\)
0.595981 + 0.802999i \(0.296763\pi\)
\(942\) −7032.00 −0.243222
\(943\) 21242.0 0.733547
\(944\) −38272.0 −1.31954
\(945\) −9126.00 −0.314147
\(946\) 0 0
\(947\) 6786.00 0.232857 0.116428 0.993199i \(-0.462855\pi\)
0.116428 + 0.993199i \(0.462855\pi\)
\(948\) 8448.00 0.289429
\(949\) −75482.0 −2.58193
\(950\) −19008.0 −0.649159
\(951\) 23178.0 0.790324
\(952\) 0 0
\(953\) −13567.0 −0.461152 −0.230576 0.973054i \(-0.574061\pi\)
−0.230576 + 0.973054i \(0.574061\pi\)
\(954\) 15876.0 0.538789
\(955\) 19604.0 0.664262
\(956\) 4608.00 0.155893
\(957\) 0 0
\(958\) −49192.0 −1.65900
\(959\) 55588.0 1.87177
\(960\) −19968.0 −0.671317
\(961\) 13473.0 0.452251
\(962\) −13140.0 −0.440385
\(963\) 7614.00 0.254785
\(964\) −30160.0 −1.00766
\(965\) −28639.0 −0.955360
\(966\) 26832.0 0.893691
\(967\) 8710.00 0.289653 0.144827 0.989457i \(-0.453738\pi\)
0.144827 + 0.989457i \(0.453738\pi\)
\(968\) 0 0
\(969\) 10044.0 0.332982
\(970\) 56108.0 1.85724
\(971\) 32566.0 1.07631 0.538153 0.842847i \(-0.319122\pi\)
0.538153 + 0.842847i \(0.319122\pi\)
\(972\) −1944.00 −0.0641500
\(973\) 9100.00 0.299828
\(974\) 44392.0 1.46038
\(975\) −9636.00 −0.316512
\(976\) 24192.0 0.793409
\(977\) 22689.0 0.742974 0.371487 0.928438i \(-0.378848\pi\)
0.371487 + 0.928438i \(0.378848\pi\)
\(978\) −18024.0 −0.589309
\(979\) 0 0
\(980\) −34632.0 −1.12886
\(981\) −9711.00 −0.316053
\(982\) −10784.0 −0.350439
\(983\) 37908.0 1.22999 0.614994 0.788532i \(-0.289159\pi\)
0.614994 + 0.788532i \(0.289159\pi\)
\(984\) 0 0
\(985\) 12337.0 0.399076
\(986\) 25668.0 0.829042
\(987\) −39000.0 −1.25773
\(988\) 63072.0 2.03096
\(989\) −38700.0 −1.24428
\(990\) 0 0
\(991\) 19540.0 0.626346 0.313173 0.949696i \(-0.398608\pi\)
0.313173 + 0.949696i \(0.398608\pi\)
\(992\) 53248.0 1.70426
\(993\) −3432.00 −0.109679
\(994\) −61776.0 −1.97124
\(995\) −30186.0 −0.961769
\(996\) 8640.00 0.274868
\(997\) 26585.0 0.844489 0.422244 0.906482i \(-0.361242\pi\)
0.422244 + 0.906482i \(0.361242\pi\)
\(998\) −104.000 −0.00329866
\(999\) −1215.00 −0.0384794
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 363.4.a.a.1.1 1
3.2 odd 2 1089.4.a.j.1.1 1
11.10 odd 2 363.4.a.g.1.1 yes 1
33.32 even 2 1089.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.a.1.1 1 1.1 even 1 trivial
363.4.a.g.1.1 yes 1 11.10 odd 2
1089.4.a.b.1.1 1 33.32 even 2
1089.4.a.j.1.1 1 3.2 odd 2