Properties

Label 338.4.a.b.1.1
Level $338$
Weight $4$
Character 338.1
Self dual yes
Analytic conductor $19.943$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -17.0000 q^{5} +2.00000 q^{6} +35.0000 q^{7} -8.00000 q^{8} -26.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} -17.0000 q^{5} +2.00000 q^{6} +35.0000 q^{7} -8.00000 q^{8} -26.0000 q^{9} +34.0000 q^{10} -2.00000 q^{11} -4.00000 q^{12} -70.0000 q^{14} +17.0000 q^{15} +16.0000 q^{16} -19.0000 q^{17} +52.0000 q^{18} -94.0000 q^{19} -68.0000 q^{20} -35.0000 q^{21} +4.00000 q^{22} -72.0000 q^{23} +8.00000 q^{24} +164.000 q^{25} +53.0000 q^{27} +140.000 q^{28} +246.000 q^{29} -34.0000 q^{30} +100.000 q^{31} -32.0000 q^{32} +2.00000 q^{33} +38.0000 q^{34} -595.000 q^{35} -104.000 q^{36} +11.0000 q^{37} +188.000 q^{38} +136.000 q^{40} +280.000 q^{41} +70.0000 q^{42} +241.000 q^{43} -8.00000 q^{44} +442.000 q^{45} +144.000 q^{46} -137.000 q^{47} -16.0000 q^{48} +882.000 q^{49} -328.000 q^{50} +19.0000 q^{51} -232.000 q^{53} -106.000 q^{54} +34.0000 q^{55} -280.000 q^{56} +94.0000 q^{57} -492.000 q^{58} +386.000 q^{59} +68.0000 q^{60} +64.0000 q^{61} -200.000 q^{62} -910.000 q^{63} +64.0000 q^{64} -4.00000 q^{66} +670.000 q^{67} -76.0000 q^{68} +72.0000 q^{69} +1190.00 q^{70} -55.0000 q^{71} +208.000 q^{72} +838.000 q^{73} -22.0000 q^{74} -164.000 q^{75} -376.000 q^{76} -70.0000 q^{77} +1016.00 q^{79} -272.000 q^{80} +649.000 q^{81} -560.000 q^{82} -420.000 q^{83} -140.000 q^{84} +323.000 q^{85} -482.000 q^{86} -246.000 q^{87} +16.0000 q^{88} +934.000 q^{89} -884.000 q^{90} -288.000 q^{92} -100.000 q^{93} +274.000 q^{94} +1598.00 q^{95} +32.0000 q^{96} +1154.00 q^{97} -1764.00 q^{98} +52.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −1.00000 −0.192450 −0.0962250 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 4.00000 0.500000
\(5\) −17.0000 −1.52053 −0.760263 0.649615i \(-0.774930\pi\)
−0.760263 + 0.649615i \(0.774930\pi\)
\(6\) 2.00000 0.136083
\(7\) 35.0000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) −8.00000 −0.353553
\(9\) −26.0000 −0.962963
\(10\) 34.0000 1.07517
\(11\) −2.00000 −0.0548202 −0.0274101 0.999624i \(-0.508726\pi\)
−0.0274101 + 0.999624i \(0.508726\pi\)
\(12\) −4.00000 −0.0962250
\(13\) 0 0
\(14\) −70.0000 −1.33631
\(15\) 17.0000 0.292625
\(16\) 16.0000 0.250000
\(17\) −19.0000 −0.271069 −0.135535 0.990773i \(-0.543275\pi\)
−0.135535 + 0.990773i \(0.543275\pi\)
\(18\) 52.0000 0.680918
\(19\) −94.0000 −1.13500 −0.567502 0.823372i \(-0.692090\pi\)
−0.567502 + 0.823372i \(0.692090\pi\)
\(20\) −68.0000 −0.760263
\(21\) −35.0000 −0.363696
\(22\) 4.00000 0.0387638
\(23\) −72.0000 −0.652741 −0.326370 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 8.00000 0.0680414
\(25\) 164.000 1.31200
\(26\) 0 0
\(27\) 53.0000 0.377772
\(28\) 140.000 0.944911
\(29\) 246.000 1.57521 0.787604 0.616181i \(-0.211321\pi\)
0.787604 + 0.616181i \(0.211321\pi\)
\(30\) −34.0000 −0.206917
\(31\) 100.000 0.579372 0.289686 0.957122i \(-0.406449\pi\)
0.289686 + 0.957122i \(0.406449\pi\)
\(32\) −32.0000 −0.176777
\(33\) 2.00000 0.0105502
\(34\) 38.0000 0.191675
\(35\) −595.000 −2.87352
\(36\) −104.000 −0.481481
\(37\) 11.0000 0.0488754 0.0244377 0.999701i \(-0.492220\pi\)
0.0244377 + 0.999701i \(0.492220\pi\)
\(38\) 188.000 0.802569
\(39\) 0 0
\(40\) 136.000 0.537587
\(41\) 280.000 1.06655 0.533276 0.845941i \(-0.320961\pi\)
0.533276 + 0.845941i \(0.320961\pi\)
\(42\) 70.0000 0.257172
\(43\) 241.000 0.854701 0.427351 0.904086i \(-0.359447\pi\)
0.427351 + 0.904086i \(0.359447\pi\)
\(44\) −8.00000 −0.0274101
\(45\) 442.000 1.46421
\(46\) 144.000 0.461557
\(47\) −137.000 −0.425181 −0.212590 0.977141i \(-0.568190\pi\)
−0.212590 + 0.977141i \(0.568190\pi\)
\(48\) −16.0000 −0.0481125
\(49\) 882.000 2.57143
\(50\) −328.000 −0.927724
\(51\) 19.0000 0.0521673
\(52\) 0 0
\(53\) −232.000 −0.601276 −0.300638 0.953738i \(-0.597200\pi\)
−0.300638 + 0.953738i \(0.597200\pi\)
\(54\) −106.000 −0.267125
\(55\) 34.0000 0.0833556
\(56\) −280.000 −0.668153
\(57\) 94.0000 0.218432
\(58\) −492.000 −1.11384
\(59\) 386.000 0.851744 0.425872 0.904783i \(-0.359967\pi\)
0.425872 + 0.904783i \(0.359967\pi\)
\(60\) 68.0000 0.146313
\(61\) 64.0000 0.134334 0.0671669 0.997742i \(-0.478604\pi\)
0.0671669 + 0.997742i \(0.478604\pi\)
\(62\) −200.000 −0.409678
\(63\) −910.000 −1.81983
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.00746009
\(67\) 670.000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −76.0000 −0.135535
\(69\) 72.0000 0.125620
\(70\) 1190.00 2.03189
\(71\) −55.0000 −0.0919338 −0.0459669 0.998943i \(-0.514637\pi\)
−0.0459669 + 0.998943i \(0.514637\pi\)
\(72\) 208.000 0.340459
\(73\) 838.000 1.34357 0.671784 0.740747i \(-0.265528\pi\)
0.671784 + 0.740747i \(0.265528\pi\)
\(74\) −22.0000 −0.0345601
\(75\) −164.000 −0.252495
\(76\) −376.000 −0.567502
\(77\) −70.0000 −0.103601
\(78\) 0 0
\(79\) 1016.00 1.44695 0.723474 0.690351i \(-0.242544\pi\)
0.723474 + 0.690351i \(0.242544\pi\)
\(80\) −272.000 −0.380132
\(81\) 649.000 0.890261
\(82\) −560.000 −0.754167
\(83\) −420.000 −0.555434 −0.277717 0.960663i \(-0.589578\pi\)
−0.277717 + 0.960663i \(0.589578\pi\)
\(84\) −140.000 −0.181848
\(85\) 323.000 0.412168
\(86\) −482.000 −0.604365
\(87\) −246.000 −0.303149
\(88\) 16.0000 0.0193819
\(89\) 934.000 1.11240 0.556201 0.831048i \(-0.312258\pi\)
0.556201 + 0.831048i \(0.312258\pi\)
\(90\) −884.000 −1.03535
\(91\) 0 0
\(92\) −288.000 −0.326370
\(93\) −100.000 −0.111500
\(94\) 274.000 0.300648
\(95\) 1598.00 1.72580
\(96\) 32.0000 0.0340207
\(97\) 1154.00 1.20795 0.603974 0.797004i \(-0.293583\pi\)
0.603974 + 0.797004i \(0.293583\pi\)
\(98\) −1764.00 −1.81827
\(99\) 52.0000 0.0527899
\(100\) 656.000 0.656000
\(101\) 292.000 0.287674 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(102\) −38.0000 −0.0368878
\(103\) 52.0000 0.0497448 0.0248724 0.999691i \(-0.492082\pi\)
0.0248724 + 0.999691i \(0.492082\pi\)
\(104\) 0 0
\(105\) 595.000 0.553010
\(106\) 464.000 0.425167
\(107\) 12.0000 0.0108419 0.00542095 0.999985i \(-0.498274\pi\)
0.00542095 + 0.999985i \(0.498274\pi\)
\(108\) 212.000 0.188886
\(109\) −1213.00 −1.06591 −0.532956 0.846143i \(-0.678919\pi\)
−0.532956 + 0.846143i \(0.678919\pi\)
\(110\) −68.0000 −0.0589413
\(111\) −11.0000 −0.00940607
\(112\) 560.000 0.472456
\(113\) −1158.00 −0.964031 −0.482015 0.876163i \(-0.660095\pi\)
−0.482015 + 0.876163i \(0.660095\pi\)
\(114\) −188.000 −0.154455
\(115\) 1224.00 0.992509
\(116\) 984.000 0.787604
\(117\) 0 0
\(118\) −772.000 −0.602274
\(119\) −665.000 −0.512273
\(120\) −136.000 −0.103459
\(121\) −1327.00 −0.996995
\(122\) −128.000 −0.0949883
\(123\) −280.000 −0.205258
\(124\) 400.000 0.289686
\(125\) −663.000 −0.474404
\(126\) 1820.00 1.28681
\(127\) −540.000 −0.377301 −0.188651 0.982044i \(-0.560411\pi\)
−0.188651 + 0.982044i \(0.560411\pi\)
\(128\) −128.000 −0.0883883
\(129\) −241.000 −0.164487
\(130\) 0 0
\(131\) −1571.00 −1.04778 −0.523889 0.851787i \(-0.675519\pi\)
−0.523889 + 0.851787i \(0.675519\pi\)
\(132\) 8.00000 0.00527508
\(133\) −3290.00 −2.14496
\(134\) −1340.00 −0.863868
\(135\) −901.000 −0.574413
\(136\) 152.000 0.0958374
\(137\) 568.000 0.354215 0.177108 0.984191i \(-0.443326\pi\)
0.177108 + 0.984191i \(0.443326\pi\)
\(138\) −144.000 −0.0888268
\(139\) 1845.00 1.12583 0.562917 0.826514i \(-0.309679\pi\)
0.562917 + 0.826514i \(0.309679\pi\)
\(140\) −2380.00 −1.43676
\(141\) 137.000 0.0818261
\(142\) 110.000 0.0650070
\(143\) 0 0
\(144\) −416.000 −0.240741
\(145\) −4182.00 −2.39515
\(146\) −1676.00 −0.950046
\(147\) −882.000 −0.494872
\(148\) 44.0000 0.0244377
\(149\) −2850.00 −1.56699 −0.783494 0.621400i \(-0.786564\pi\)
−0.783494 + 0.621400i \(0.786564\pi\)
\(150\) 328.000 0.178541
\(151\) −2763.00 −1.48907 −0.744536 0.667583i \(-0.767329\pi\)
−0.744536 + 0.667583i \(0.767329\pi\)
\(152\) 752.000 0.401285
\(153\) 494.000 0.261030
\(154\) 140.000 0.0732566
\(155\) −1700.00 −0.880950
\(156\) 0 0
\(157\) −466.000 −0.236884 −0.118442 0.992961i \(-0.537790\pi\)
−0.118442 + 0.992961i \(0.537790\pi\)
\(158\) −2032.00 −1.02315
\(159\) 232.000 0.115716
\(160\) 544.000 0.268794
\(161\) −2520.00 −1.23356
\(162\) −1298.00 −0.629509
\(163\) −2552.00 −1.22631 −0.613154 0.789964i \(-0.710099\pi\)
−0.613154 + 0.789964i \(0.710099\pi\)
\(164\) 1120.00 0.533276
\(165\) −34.0000 −0.0160418
\(166\) 840.000 0.392751
\(167\) 3768.00 1.74597 0.872984 0.487749i \(-0.162182\pi\)
0.872984 + 0.487749i \(0.162182\pi\)
\(168\) 280.000 0.128586
\(169\) 0 0
\(170\) −646.000 −0.291447
\(171\) 2444.00 1.09297
\(172\) 964.000 0.427351
\(173\) 3296.00 1.44850 0.724249 0.689538i \(-0.242187\pi\)
0.724249 + 0.689538i \(0.242187\pi\)
\(174\) 492.000 0.214359
\(175\) 5740.00 2.47945
\(176\) −32.0000 −0.0137051
\(177\) −386.000 −0.163918
\(178\) −1868.00 −0.786587
\(179\) 45.0000 0.0187903 0.00939513 0.999956i \(-0.497009\pi\)
0.00939513 + 0.999956i \(0.497009\pi\)
\(180\) 1768.00 0.732105
\(181\) −364.000 −0.149480 −0.0747401 0.997203i \(-0.523813\pi\)
−0.0747401 + 0.997203i \(0.523813\pi\)
\(182\) 0 0
\(183\) −64.0000 −0.0258525
\(184\) 576.000 0.230779
\(185\) −187.000 −0.0743163
\(186\) 200.000 0.0788425
\(187\) 38.0000 0.0148601
\(188\) −548.000 −0.212590
\(189\) 1855.00 0.713923
\(190\) −3196.00 −1.22033
\(191\) 2482.00 0.940268 0.470134 0.882595i \(-0.344206\pi\)
0.470134 + 0.882595i \(0.344206\pi\)
\(192\) −64.0000 −0.0240563
\(193\) 1220.00 0.455013 0.227507 0.973777i \(-0.426943\pi\)
0.227507 + 0.973777i \(0.426943\pi\)
\(194\) −2308.00 −0.854148
\(195\) 0 0
\(196\) 3528.00 1.28571
\(197\) 1593.00 0.576125 0.288062 0.957612i \(-0.406989\pi\)
0.288062 + 0.957612i \(0.406989\pi\)
\(198\) −104.000 −0.0373281
\(199\) 1606.00 0.572092 0.286046 0.958216i \(-0.407659\pi\)
0.286046 + 0.958216i \(0.407659\pi\)
\(200\) −1312.00 −0.463862
\(201\) −670.000 −0.235115
\(202\) −584.000 −0.203416
\(203\) 8610.00 2.97686
\(204\) 76.0000 0.0260836
\(205\) −4760.00 −1.62172
\(206\) −104.000 −0.0351749
\(207\) 1872.00 0.628565
\(208\) 0 0
\(209\) 188.000 0.0622212
\(210\) −1190.00 −0.391037
\(211\) 2469.00 0.805559 0.402780 0.915297i \(-0.368044\pi\)
0.402780 + 0.915297i \(0.368044\pi\)
\(212\) −928.000 −0.300638
\(213\) 55.0000 0.0176927
\(214\) −24.0000 −0.00766638
\(215\) −4097.00 −1.29960
\(216\) −424.000 −0.133563
\(217\) 3500.00 1.09491
\(218\) 2426.00 0.753713
\(219\) −838.000 −0.258570
\(220\) 136.000 0.0416778
\(221\) 0 0
\(222\) 22.0000 0.00665110
\(223\) −1943.00 −0.583466 −0.291733 0.956500i \(-0.594232\pi\)
−0.291733 + 0.956500i \(0.594232\pi\)
\(224\) −1120.00 −0.334077
\(225\) −4264.00 −1.26341
\(226\) 2316.00 0.681673
\(227\) −3032.00 −0.886524 −0.443262 0.896392i \(-0.646179\pi\)
−0.443262 + 0.896392i \(0.646179\pi\)
\(228\) 376.000 0.109216
\(229\) −5311.00 −1.53258 −0.766290 0.642495i \(-0.777899\pi\)
−0.766290 + 0.642495i \(0.777899\pi\)
\(230\) −2448.00 −0.701810
\(231\) 70.0000 0.0199379
\(232\) −1968.00 −0.556920
\(233\) −507.000 −0.142552 −0.0712761 0.997457i \(-0.522707\pi\)
−0.0712761 + 0.997457i \(0.522707\pi\)
\(234\) 0 0
\(235\) 2329.00 0.646499
\(236\) 1544.00 0.425872
\(237\) −1016.00 −0.278465
\(238\) 1330.00 0.362231
\(239\) 6795.00 1.83905 0.919523 0.393036i \(-0.128575\pi\)
0.919523 + 0.393036i \(0.128575\pi\)
\(240\) 272.000 0.0731564
\(241\) 4442.00 1.18728 0.593640 0.804731i \(-0.297690\pi\)
0.593640 + 0.804731i \(0.297690\pi\)
\(242\) 2654.00 0.704982
\(243\) −2080.00 −0.549103
\(244\) 256.000 0.0671669
\(245\) −14994.0 −3.90992
\(246\) 560.000 0.145139
\(247\) 0 0
\(248\) −800.000 −0.204839
\(249\) 420.000 0.106893
\(250\) 1326.00 0.335454
\(251\) −5024.00 −1.26339 −0.631697 0.775215i \(-0.717641\pi\)
−0.631697 + 0.775215i \(0.717641\pi\)
\(252\) −3640.00 −0.909914
\(253\) 144.000 0.0357834
\(254\) 1080.00 0.266792
\(255\) −323.000 −0.0793217
\(256\) 256.000 0.0625000
\(257\) 7065.00 1.71480 0.857398 0.514654i \(-0.172080\pi\)
0.857398 + 0.514654i \(0.172080\pi\)
\(258\) 482.000 0.116310
\(259\) 385.000 0.0923658
\(260\) 0 0
\(261\) −6396.00 −1.51687
\(262\) 3142.00 0.740891
\(263\) 1364.00 0.319802 0.159901 0.987133i \(-0.448883\pi\)
0.159901 + 0.987133i \(0.448883\pi\)
\(264\) −16.0000 −0.00373005
\(265\) 3944.00 0.914257
\(266\) 6580.00 1.51671
\(267\) −934.000 −0.214082
\(268\) 2680.00 0.610847
\(269\) −4304.00 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 1802.00 0.406171
\(271\) 1519.00 0.340490 0.170245 0.985402i \(-0.445544\pi\)
0.170245 + 0.985402i \(0.445544\pi\)
\(272\) −304.000 −0.0677673
\(273\) 0 0
\(274\) −1136.00 −0.250468
\(275\) −328.000 −0.0719242
\(276\) 288.000 0.0628100
\(277\) 1996.00 0.432953 0.216477 0.976288i \(-0.430544\pi\)
0.216477 + 0.976288i \(0.430544\pi\)
\(278\) −3690.00 −0.796085
\(279\) −2600.00 −0.557914
\(280\) 4760.00 1.01594
\(281\) 2918.00 0.619478 0.309739 0.950822i \(-0.399758\pi\)
0.309739 + 0.950822i \(0.399758\pi\)
\(282\) −274.000 −0.0578598
\(283\) 812.000 0.170560 0.0852798 0.996357i \(-0.472822\pi\)
0.0852798 + 0.996357i \(0.472822\pi\)
\(284\) −220.000 −0.0459669
\(285\) −1598.00 −0.332131
\(286\) 0 0
\(287\) 9800.00 2.01559
\(288\) 832.000 0.170229
\(289\) −4552.00 −0.926521
\(290\) 8364.00 1.69362
\(291\) −1154.00 −0.232470
\(292\) 3352.00 0.671784
\(293\) 1855.00 0.369864 0.184932 0.982751i \(-0.440793\pi\)
0.184932 + 0.982751i \(0.440793\pi\)
\(294\) 1764.00 0.349927
\(295\) −6562.00 −1.29510
\(296\) −88.0000 −0.0172801
\(297\) −106.000 −0.0207096
\(298\) 5700.00 1.10803
\(299\) 0 0
\(300\) −656.000 −0.126247
\(301\) 8435.00 1.61523
\(302\) 5526.00 1.05293
\(303\) −292.000 −0.0553629
\(304\) −1504.00 −0.283751
\(305\) −1088.00 −0.204258
\(306\) −988.000 −0.184576
\(307\) −5350.00 −0.994595 −0.497297 0.867580i \(-0.665674\pi\)
−0.497297 + 0.867580i \(0.665674\pi\)
\(308\) −280.000 −0.0518003
\(309\) −52.0000 −0.00957339
\(310\) 3400.00 0.622926
\(311\) 2262.00 0.412432 0.206216 0.978507i \(-0.433885\pi\)
0.206216 + 0.978507i \(0.433885\pi\)
\(312\) 0 0
\(313\) −1857.00 −0.335348 −0.167674 0.985843i \(-0.553626\pi\)
−0.167674 + 0.985843i \(0.553626\pi\)
\(314\) 932.000 0.167503
\(315\) 15470.0 2.76710
\(316\) 4064.00 0.723474
\(317\) 3870.00 0.685681 0.342840 0.939394i \(-0.388611\pi\)
0.342840 + 0.939394i \(0.388611\pi\)
\(318\) −464.000 −0.0818234
\(319\) −492.000 −0.0863533
\(320\) −1088.00 −0.190066
\(321\) −12.0000 −0.00208653
\(322\) 5040.00 0.872262
\(323\) 1786.00 0.307665
\(324\) 2596.00 0.445130
\(325\) 0 0
\(326\) 5104.00 0.867130
\(327\) 1213.00 0.205135
\(328\) −2240.00 −0.377083
\(329\) −4795.00 −0.803516
\(330\) 68.0000 0.0113433
\(331\) −10520.0 −1.74692 −0.873461 0.486893i \(-0.838130\pi\)
−0.873461 + 0.486893i \(0.838130\pi\)
\(332\) −1680.00 −0.277717
\(333\) −286.000 −0.0470652
\(334\) −7536.00 −1.23459
\(335\) −11390.0 −1.85762
\(336\) −560.000 −0.0909241
\(337\) 7839.00 1.26711 0.633557 0.773696i \(-0.281594\pi\)
0.633557 + 0.773696i \(0.281594\pi\)
\(338\) 0 0
\(339\) 1158.00 0.185528
\(340\) 1292.00 0.206084
\(341\) −200.000 −0.0317613
\(342\) −4888.00 −0.772844
\(343\) 18865.0 2.96972
\(344\) −1928.00 −0.302183
\(345\) −1224.00 −0.191009
\(346\) −6592.00 −1.02424
\(347\) −1275.00 −0.197250 −0.0986248 0.995125i \(-0.531444\pi\)
−0.0986248 + 0.995125i \(0.531444\pi\)
\(348\) −984.000 −0.151575
\(349\) 375.000 0.0575166 0.0287583 0.999586i \(-0.490845\pi\)
0.0287583 + 0.999586i \(0.490845\pi\)
\(350\) −11480.0 −1.75323
\(351\) 0 0
\(352\) 64.0000 0.00969094
\(353\) −1592.00 −0.240039 −0.120019 0.992772i \(-0.538296\pi\)
−0.120019 + 0.992772i \(0.538296\pi\)
\(354\) 772.000 0.115908
\(355\) 935.000 0.139788
\(356\) 3736.00 0.556201
\(357\) 665.000 0.0985869
\(358\) −90.0000 −0.0132867
\(359\) 2424.00 0.356362 0.178181 0.983998i \(-0.442979\pi\)
0.178181 + 0.983998i \(0.442979\pi\)
\(360\) −3536.00 −0.517677
\(361\) 1977.00 0.288234
\(362\) 728.000 0.105698
\(363\) 1327.00 0.191872
\(364\) 0 0
\(365\) −14246.0 −2.04293
\(366\) 128.000 0.0182805
\(367\) −7970.00 −1.13360 −0.566799 0.823856i \(-0.691818\pi\)
−0.566799 + 0.823856i \(0.691818\pi\)
\(368\) −1152.00 −0.163185
\(369\) −7280.00 −1.02705
\(370\) 374.000 0.0525496
\(371\) −8120.00 −1.13631
\(372\) −400.000 −0.0557501
\(373\) 12492.0 1.73408 0.867039 0.498240i \(-0.166020\pi\)
0.867039 + 0.498240i \(0.166020\pi\)
\(374\) −76.0000 −0.0105077
\(375\) 663.000 0.0912991
\(376\) 1096.00 0.150324
\(377\) 0 0
\(378\) −3710.00 −0.504820
\(379\) 4284.00 0.580618 0.290309 0.956933i \(-0.406242\pi\)
0.290309 + 0.956933i \(0.406242\pi\)
\(380\) 6392.00 0.862902
\(381\) 540.000 0.0726116
\(382\) −4964.00 −0.664870
\(383\) 2337.00 0.311789 0.155894 0.987774i \(-0.450174\pi\)
0.155894 + 0.987774i \(0.450174\pi\)
\(384\) 128.000 0.0170103
\(385\) 1190.00 0.157527
\(386\) −2440.00 −0.321743
\(387\) −6266.00 −0.823046
\(388\) 4616.00 0.603974
\(389\) 3562.00 0.464269 0.232134 0.972684i \(-0.425429\pi\)
0.232134 + 0.972684i \(0.425429\pi\)
\(390\) 0 0
\(391\) 1368.00 0.176938
\(392\) −7056.00 −0.909137
\(393\) 1571.00 0.201645
\(394\) −3186.00 −0.407382
\(395\) −17272.0 −2.20012
\(396\) 208.000 0.0263949
\(397\) 5274.00 0.666737 0.333368 0.942797i \(-0.391815\pi\)
0.333368 + 0.942797i \(0.391815\pi\)
\(398\) −3212.00 −0.404530
\(399\) 3290.00 0.412797
\(400\) 2624.00 0.328000
\(401\) −4812.00 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 1340.00 0.166252
\(403\) 0 0
\(404\) 1168.00 0.143837
\(405\) −11033.0 −1.35366
\(406\) −17220.0 −2.10496
\(407\) −22.0000 −0.00267936
\(408\) −152.000 −0.0184439
\(409\) 9448.00 1.14223 0.571117 0.820869i \(-0.306510\pi\)
0.571117 + 0.820869i \(0.306510\pi\)
\(410\) 9520.00 1.14673
\(411\) −568.000 −0.0681688
\(412\) 208.000 0.0248724
\(413\) 13510.0 1.60965
\(414\) −3744.00 −0.444463
\(415\) 7140.00 0.844551
\(416\) 0 0
\(417\) −1845.00 −0.216667
\(418\) −376.000 −0.0439970
\(419\) −521.000 −0.0607459 −0.0303729 0.999539i \(-0.509669\pi\)
−0.0303729 + 0.999539i \(0.509669\pi\)
\(420\) 2380.00 0.276505
\(421\) −7141.00 −0.826677 −0.413339 0.910577i \(-0.635637\pi\)
−0.413339 + 0.910577i \(0.635637\pi\)
\(422\) −4938.00 −0.569616
\(423\) 3562.00 0.409433
\(424\) 1856.00 0.212583
\(425\) −3116.00 −0.355643
\(426\) −110.000 −0.0125106
\(427\) 2240.00 0.253867
\(428\) 48.0000 0.00542095
\(429\) 0 0
\(430\) 8194.00 0.918953
\(431\) 6299.00 0.703973 0.351986 0.936005i \(-0.385506\pi\)
0.351986 + 0.936005i \(0.385506\pi\)
\(432\) 848.000 0.0944431
\(433\) 3231.00 0.358596 0.179298 0.983795i \(-0.442617\pi\)
0.179298 + 0.983795i \(0.442617\pi\)
\(434\) −7000.00 −0.774218
\(435\) 4182.00 0.460946
\(436\) −4852.00 −0.532956
\(437\) 6768.00 0.740863
\(438\) 1676.00 0.182836
\(439\) −410.000 −0.0445746 −0.0222873 0.999752i \(-0.507095\pi\)
−0.0222873 + 0.999752i \(0.507095\pi\)
\(440\) −272.000 −0.0294707
\(441\) −22932.0 −2.47619
\(442\) 0 0
\(443\) −9397.00 −1.00782 −0.503911 0.863756i \(-0.668106\pi\)
−0.503911 + 0.863756i \(0.668106\pi\)
\(444\) −44.0000 −0.00470304
\(445\) −15878.0 −1.69144
\(446\) 3886.00 0.412573
\(447\) 2850.00 0.301567
\(448\) 2240.00 0.236228
\(449\) −654.000 −0.0687398 −0.0343699 0.999409i \(-0.510942\pi\)
−0.0343699 + 0.999409i \(0.510942\pi\)
\(450\) 8528.00 0.893364
\(451\) −560.000 −0.0584687
\(452\) −4632.00 −0.482015
\(453\) 2763.00 0.286572
\(454\) 6064.00 0.626867
\(455\) 0 0
\(456\) −752.000 −0.0772273
\(457\) −14982.0 −1.53354 −0.766771 0.641921i \(-0.778138\pi\)
−0.766771 + 0.641921i \(0.778138\pi\)
\(458\) 10622.0 1.08370
\(459\) −1007.00 −0.102402
\(460\) 4896.00 0.496255
\(461\) −10845.0 −1.09567 −0.547833 0.836588i \(-0.684547\pi\)
−0.547833 + 0.836588i \(0.684547\pi\)
\(462\) −140.000 −0.0140982
\(463\) −13072.0 −1.31211 −0.656055 0.754713i \(-0.727776\pi\)
−0.656055 + 0.754713i \(0.727776\pi\)
\(464\) 3936.00 0.393802
\(465\) 1700.00 0.169539
\(466\) 1014.00 0.100800
\(467\) −15340.0 −1.52002 −0.760011 0.649910i \(-0.774807\pi\)
−0.760011 + 0.649910i \(0.774807\pi\)
\(468\) 0 0
\(469\) 23450.0 2.30879
\(470\) −4658.00 −0.457144
\(471\) 466.000 0.0455884
\(472\) −3088.00 −0.301137
\(473\) −482.000 −0.0468549
\(474\) 2032.00 0.196905
\(475\) −15416.0 −1.48913
\(476\) −2660.00 −0.256136
\(477\) 6032.00 0.579007
\(478\) −13590.0 −1.30040
\(479\) 19615.0 1.87105 0.935524 0.353262i \(-0.114928\pi\)
0.935524 + 0.353262i \(0.114928\pi\)
\(480\) −544.000 −0.0517294
\(481\) 0 0
\(482\) −8884.00 −0.839533
\(483\) 2520.00 0.237400
\(484\) −5308.00 −0.498497
\(485\) −19618.0 −1.83672
\(486\) 4160.00 0.388275
\(487\) 10904.0 1.01459 0.507297 0.861771i \(-0.330645\pi\)
0.507297 + 0.861771i \(0.330645\pi\)
\(488\) −512.000 −0.0474942
\(489\) 2552.00 0.236003
\(490\) 29988.0 2.76473
\(491\) −18519.0 −1.70214 −0.851070 0.525052i \(-0.824046\pi\)
−0.851070 + 0.525052i \(0.824046\pi\)
\(492\) −1120.00 −0.102629
\(493\) −4674.00 −0.426991
\(494\) 0 0
\(495\) −884.000 −0.0802684
\(496\) 1600.00 0.144843
\(497\) −1925.00 −0.173739
\(498\) −840.000 −0.0755849
\(499\) 14400.0 1.29185 0.645924 0.763401i \(-0.276472\pi\)
0.645924 + 0.763401i \(0.276472\pi\)
\(500\) −2652.00 −0.237202
\(501\) −3768.00 −0.336012
\(502\) 10048.0 0.893355
\(503\) 20422.0 1.81028 0.905141 0.425111i \(-0.139765\pi\)
0.905141 + 0.425111i \(0.139765\pi\)
\(504\) 7280.00 0.643407
\(505\) −4964.00 −0.437416
\(506\) −288.000 −0.0253027
\(507\) 0 0
\(508\) −2160.00 −0.188651
\(509\) −15606.0 −1.35899 −0.679493 0.733682i \(-0.737800\pi\)
−0.679493 + 0.733682i \(0.737800\pi\)
\(510\) 646.000 0.0560889
\(511\) 29330.0 2.53911
\(512\) −512.000 −0.0441942
\(513\) −4982.00 −0.428773
\(514\) −14130.0 −1.21254
\(515\) −884.000 −0.0756382
\(516\) −964.000 −0.0822437
\(517\) 274.000 0.0233085
\(518\) −770.000 −0.0653125
\(519\) −3296.00 −0.278764
\(520\) 0 0
\(521\) 1783.00 0.149932 0.0749661 0.997186i \(-0.476115\pi\)
0.0749661 + 0.997186i \(0.476115\pi\)
\(522\) 12792.0 1.07259
\(523\) 11140.0 0.931392 0.465696 0.884945i \(-0.345804\pi\)
0.465696 + 0.884945i \(0.345804\pi\)
\(524\) −6284.00 −0.523889
\(525\) −5740.00 −0.477170
\(526\) −2728.00 −0.226134
\(527\) −1900.00 −0.157050
\(528\) 32.0000 0.00263754
\(529\) −6983.00 −0.573929
\(530\) −7888.00 −0.646477
\(531\) −10036.0 −0.820198
\(532\) −13160.0 −1.07248
\(533\) 0 0
\(534\) 1868.00 0.151379
\(535\) −204.000 −0.0164854
\(536\) −5360.00 −0.431934
\(537\) −45.0000 −0.00361619
\(538\) 8608.00 0.689809
\(539\) −1764.00 −0.140966
\(540\) −3604.00 −0.287206
\(541\) −839.000 −0.0666755 −0.0333377 0.999444i \(-0.510614\pi\)
−0.0333377 + 0.999444i \(0.510614\pi\)
\(542\) −3038.00 −0.240762
\(543\) 364.000 0.0287675
\(544\) 608.000 0.0479187
\(545\) 20621.0 1.62075
\(546\) 0 0
\(547\) 17279.0 1.35063 0.675317 0.737528i \(-0.264007\pi\)
0.675317 + 0.737528i \(0.264007\pi\)
\(548\) 2272.00 0.177108
\(549\) −1664.00 −0.129358
\(550\) 656.000 0.0508581
\(551\) −23124.0 −1.78787
\(552\) −576.000 −0.0444134
\(553\) 35560.0 2.73448
\(554\) −3992.00 −0.306144
\(555\) 187.000 0.0143022
\(556\) 7380.00 0.562917
\(557\) 20577.0 1.56531 0.782653 0.622458i \(-0.213866\pi\)
0.782653 + 0.622458i \(0.213866\pi\)
\(558\) 5200.00 0.394505
\(559\) 0 0
\(560\) −9520.00 −0.718381
\(561\) −38.0000 −0.00285982
\(562\) −5836.00 −0.438037
\(563\) 11785.0 0.882200 0.441100 0.897458i \(-0.354588\pi\)
0.441100 + 0.897458i \(0.354588\pi\)
\(564\) 548.000 0.0409131
\(565\) 19686.0 1.46583
\(566\) −1624.00 −0.120604
\(567\) 22715.0 1.68243
\(568\) 440.000 0.0325035
\(569\) 10887.0 0.802121 0.401060 0.916052i \(-0.368642\pi\)
0.401060 + 0.916052i \(0.368642\pi\)
\(570\) 3196.00 0.234852
\(571\) −11453.0 −0.839393 −0.419696 0.907665i \(-0.637863\pi\)
−0.419696 + 0.907665i \(0.637863\pi\)
\(572\) 0 0
\(573\) −2482.00 −0.180955
\(574\) −19600.0 −1.42524
\(575\) −11808.0 −0.856396
\(576\) −1664.00 −0.120370
\(577\) −14382.0 −1.03766 −0.518831 0.854877i \(-0.673632\pi\)
−0.518831 + 0.854877i \(0.673632\pi\)
\(578\) 9104.00 0.655150
\(579\) −1220.00 −0.0875673
\(580\) −16728.0 −1.19757
\(581\) −14700.0 −1.04967
\(582\) 2308.00 0.164381
\(583\) 464.000 0.0329621
\(584\) −6704.00 −0.475023
\(585\) 0 0
\(586\) −3710.00 −0.261534
\(587\) 10160.0 0.714392 0.357196 0.934029i \(-0.383733\pi\)
0.357196 + 0.934029i \(0.383733\pi\)
\(588\) −3528.00 −0.247436
\(589\) −9400.00 −0.657590
\(590\) 13124.0 0.915774
\(591\) −1593.00 −0.110875
\(592\) 176.000 0.0122188
\(593\) −16842.0 −1.16630 −0.583152 0.812363i \(-0.698181\pi\)
−0.583152 + 0.812363i \(0.698181\pi\)
\(594\) 212.000 0.0146439
\(595\) 11305.0 0.778924
\(596\) −11400.0 −0.783494
\(597\) −1606.00 −0.110099
\(598\) 0 0
\(599\) −13830.0 −0.943370 −0.471685 0.881767i \(-0.656354\pi\)
−0.471685 + 0.881767i \(0.656354\pi\)
\(600\) 1312.00 0.0892703
\(601\) −15835.0 −1.07475 −0.537374 0.843344i \(-0.680583\pi\)
−0.537374 + 0.843344i \(0.680583\pi\)
\(602\) −16870.0 −1.14214
\(603\) −17420.0 −1.17645
\(604\) −11052.0 −0.744536
\(605\) 22559.0 1.51596
\(606\) 584.000 0.0391475
\(607\) −22894.0 −1.53087 −0.765436 0.643513i \(-0.777476\pi\)
−0.765436 + 0.643513i \(0.777476\pi\)
\(608\) 3008.00 0.200642
\(609\) −8610.00 −0.572898
\(610\) 2176.00 0.144432
\(611\) 0 0
\(612\) 1976.00 0.130515
\(613\) −9886.00 −0.651373 −0.325687 0.945478i \(-0.605595\pi\)
−0.325687 + 0.945478i \(0.605595\pi\)
\(614\) 10700.0 0.703285
\(615\) 4760.00 0.312100
\(616\) 560.000 0.0366283
\(617\) −8856.00 −0.577843 −0.288922 0.957353i \(-0.593297\pi\)
−0.288922 + 0.957353i \(0.593297\pi\)
\(618\) 104.000 0.00676941
\(619\) −3764.00 −0.244407 −0.122204 0.992505i \(-0.538996\pi\)
−0.122204 + 0.992505i \(0.538996\pi\)
\(620\) −6800.00 −0.440475
\(621\) −3816.00 −0.246587
\(622\) −4524.00 −0.291633
\(623\) 32690.0 2.10224
\(624\) 0 0
\(625\) −9229.00 −0.590656
\(626\) 3714.00 0.237127
\(627\) −188.000 −0.0119745
\(628\) −1864.00 −0.118442
\(629\) −209.000 −0.0132486
\(630\) −30940.0 −1.95663
\(631\) −17775.0 −1.12141 −0.560706 0.828015i \(-0.689470\pi\)
−0.560706 + 0.828015i \(0.689470\pi\)
\(632\) −8128.00 −0.511574
\(633\) −2469.00 −0.155030
\(634\) −7740.00 −0.484850
\(635\) 9180.00 0.573696
\(636\) 928.000 0.0578579
\(637\) 0 0
\(638\) 984.000 0.0610610
\(639\) 1430.00 0.0885288
\(640\) 2176.00 0.134397
\(641\) 13358.0 0.823103 0.411552 0.911386i \(-0.364987\pi\)
0.411552 + 0.911386i \(0.364987\pi\)
\(642\) 24.0000 0.00147540
\(643\) 26062.0 1.59842 0.799211 0.601051i \(-0.205251\pi\)
0.799211 + 0.601051i \(0.205251\pi\)
\(644\) −10080.0 −0.616782
\(645\) 4097.00 0.250107
\(646\) −3572.00 −0.217552
\(647\) 20598.0 1.25161 0.625804 0.779980i \(-0.284771\pi\)
0.625804 + 0.779980i \(0.284771\pi\)
\(648\) −5192.00 −0.314755
\(649\) −772.000 −0.0466928
\(650\) 0 0
\(651\) −3500.00 −0.210716
\(652\) −10208.0 −0.613154
\(653\) 27104.0 1.62429 0.812145 0.583456i \(-0.198300\pi\)
0.812145 + 0.583456i \(0.198300\pi\)
\(654\) −2426.00 −0.145052
\(655\) 26707.0 1.59317
\(656\) 4480.00 0.266638
\(657\) −21788.0 −1.29381
\(658\) 9590.00 0.568172
\(659\) 7500.00 0.443336 0.221668 0.975122i \(-0.428850\pi\)
0.221668 + 0.975122i \(0.428850\pi\)
\(660\) −136.000 −0.00802090
\(661\) −10402.0 −0.612089 −0.306045 0.952017i \(-0.599006\pi\)
−0.306045 + 0.952017i \(0.599006\pi\)
\(662\) 21040.0 1.23526
\(663\) 0 0
\(664\) 3360.00 0.196375
\(665\) 55930.0 3.26146
\(666\) 572.000 0.0332801
\(667\) −17712.0 −1.02820
\(668\) 15072.0 0.872984
\(669\) 1943.00 0.112288
\(670\) 22780.0 1.31353
\(671\) −128.000 −0.00736421
\(672\) 1120.00 0.0642931
\(673\) −19715.0 −1.12921 −0.564604 0.825362i \(-0.690971\pi\)
−0.564604 + 0.825362i \(0.690971\pi\)
\(674\) −15678.0 −0.895985
\(675\) 8692.00 0.495637
\(676\) 0 0
\(677\) −1168.00 −0.0663071 −0.0331535 0.999450i \(-0.510555\pi\)
−0.0331535 + 0.999450i \(0.510555\pi\)
\(678\) −2316.00 −0.131188
\(679\) 40390.0 2.28281
\(680\) −2584.00 −0.145723
\(681\) 3032.00 0.170612
\(682\) 400.000 0.0224586
\(683\) −3800.00 −0.212889 −0.106444 0.994319i \(-0.533947\pi\)
−0.106444 + 0.994319i \(0.533947\pi\)
\(684\) 9776.00 0.546483
\(685\) −9656.00 −0.538594
\(686\) −37730.0 −2.09991
\(687\) 5311.00 0.294945
\(688\) 3856.00 0.213675
\(689\) 0 0
\(690\) 2448.00 0.135063
\(691\) 17336.0 0.954403 0.477202 0.878794i \(-0.341651\pi\)
0.477202 + 0.878794i \(0.341651\pi\)
\(692\) 13184.0 0.724249
\(693\) 1820.00 0.0997635
\(694\) 2550.00 0.139476
\(695\) −31365.0 −1.71186
\(696\) 1968.00 0.107179
\(697\) −5320.00 −0.289110
\(698\) −750.000 −0.0406704
\(699\) 507.000 0.0274342
\(700\) 22960.0 1.23972
\(701\) −10088.0 −0.543536 −0.271768 0.962363i \(-0.587608\pi\)
−0.271768 + 0.962363i \(0.587608\pi\)
\(702\) 0 0
\(703\) −1034.00 −0.0554738
\(704\) −128.000 −0.00685253
\(705\) −2329.00 −0.124419
\(706\) 3184.00 0.169733
\(707\) 10220.0 0.543653
\(708\) −1544.00 −0.0819591
\(709\) −11730.0 −0.621339 −0.310670 0.950518i \(-0.600553\pi\)
−0.310670 + 0.950518i \(0.600553\pi\)
\(710\) −1870.00 −0.0988449
\(711\) −26416.0 −1.39336
\(712\) −7472.00 −0.393294
\(713\) −7200.00 −0.378180
\(714\) −1330.00 −0.0697115
\(715\) 0 0
\(716\) 180.000 0.00939513
\(717\) −6795.00 −0.353925
\(718\) −4848.00 −0.251986
\(719\) −2190.00 −0.113593 −0.0567964 0.998386i \(-0.518089\pi\)
−0.0567964 + 0.998386i \(0.518089\pi\)
\(720\) 7072.00 0.366053
\(721\) 1820.00 0.0940088
\(722\) −3954.00 −0.203813
\(723\) −4442.00 −0.228492
\(724\) −1456.00 −0.0747401
\(725\) 40344.0 2.06667
\(726\) −2654.00 −0.135674
\(727\) −2470.00 −0.126007 −0.0630036 0.998013i \(-0.520068\pi\)
−0.0630036 + 0.998013i \(0.520068\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 28492.0 1.44457
\(731\) −4579.00 −0.231683
\(732\) −256.000 −0.0129263
\(733\) 8693.00 0.438040 0.219020 0.975720i \(-0.429714\pi\)
0.219020 + 0.975720i \(0.429714\pi\)
\(734\) 15940.0 0.801575
\(735\) 14994.0 0.752465
\(736\) 2304.00 0.115389
\(737\) −1340.00 −0.0669736
\(738\) 14560.0 0.726234
\(739\) 14636.0 0.728544 0.364272 0.931293i \(-0.381318\pi\)
0.364272 + 0.931293i \(0.381318\pi\)
\(740\) −748.000 −0.0371581
\(741\) 0 0
\(742\) 16240.0 0.803489
\(743\) −27501.0 −1.35789 −0.678946 0.734188i \(-0.737563\pi\)
−0.678946 + 0.734188i \(0.737563\pi\)
\(744\) 800.000 0.0394213
\(745\) 48450.0 2.38265
\(746\) −24984.0 −1.22618
\(747\) 10920.0 0.534862
\(748\) 152.000 0.00743004
\(749\) 420.000 0.0204893
\(750\) −1326.00 −0.0645582
\(751\) 11888.0 0.577629 0.288814 0.957385i \(-0.406739\pi\)
0.288814 + 0.957385i \(0.406739\pi\)
\(752\) −2192.00 −0.106295
\(753\) 5024.00 0.243140
\(754\) 0 0
\(755\) 46971.0 2.26417
\(756\) 7420.00 0.356961
\(757\) 10776.0 0.517385 0.258692 0.965960i \(-0.416708\pi\)
0.258692 + 0.965960i \(0.416708\pi\)
\(758\) −8568.00 −0.410559
\(759\) −144.000 −0.00688652
\(760\) −12784.0 −0.610164
\(761\) 21758.0 1.03643 0.518217 0.855249i \(-0.326596\pi\)
0.518217 + 0.855249i \(0.326596\pi\)
\(762\) −1080.00 −0.0513442
\(763\) −42455.0 −2.01438
\(764\) 9928.00 0.470134
\(765\) −8398.00 −0.396902
\(766\) −4674.00 −0.220468
\(767\) 0 0
\(768\) −256.000 −0.0120281
\(769\) 8080.00 0.378898 0.189449 0.981891i \(-0.439330\pi\)
0.189449 + 0.981891i \(0.439330\pi\)
\(770\) −2380.00 −0.111389
\(771\) −7065.00 −0.330013
\(772\) 4880.00 0.227507
\(773\) −3981.00 −0.185235 −0.0926175 0.995702i \(-0.529523\pi\)
−0.0926175 + 0.995702i \(0.529523\pi\)
\(774\) 12532.0 0.581981
\(775\) 16400.0 0.760136
\(776\) −9232.00 −0.427074
\(777\) −385.000 −0.0177758
\(778\) −7124.00 −0.328288
\(779\) −26320.0 −1.21054
\(780\) 0 0
\(781\) 110.000 0.00503983
\(782\) −2736.00 −0.125114
\(783\) 13038.0 0.595070
\(784\) 14112.0 0.642857
\(785\) 7922.00 0.360189
\(786\) −3142.00 −0.142585
\(787\) 26048.0 1.17981 0.589905 0.807472i \(-0.299165\pi\)
0.589905 + 0.807472i \(0.299165\pi\)
\(788\) 6372.00 0.288062
\(789\) −1364.00 −0.0615459
\(790\) 34544.0 1.55572
\(791\) −40530.0 −1.82185
\(792\) −416.000 −0.0186640
\(793\) 0 0
\(794\) −10548.0 −0.471454
\(795\) −3944.00 −0.175949
\(796\) 6424.00 0.286046
\(797\) −12434.0 −0.552616 −0.276308 0.961069i \(-0.589111\pi\)
−0.276308 + 0.961069i \(0.589111\pi\)
\(798\) −6580.00 −0.291892
\(799\) 2603.00 0.115253
\(800\) −5248.00 −0.231931
\(801\) −24284.0 −1.07120
\(802\) 9624.00 0.423735
\(803\) −1676.00 −0.0736547
\(804\) −2680.00 −0.117558
\(805\) 42840.0 1.87567
\(806\) 0 0
\(807\) 4304.00 0.187742
\(808\) −2336.00 −0.101708
\(809\) −8113.00 −0.352581 −0.176290 0.984338i \(-0.556410\pi\)
−0.176290 + 0.984338i \(0.556410\pi\)
\(810\) 22066.0 0.957185
\(811\) −732.000 −0.0316942 −0.0158471 0.999874i \(-0.505044\pi\)
−0.0158471 + 0.999874i \(0.505044\pi\)
\(812\) 34440.0 1.48843
\(813\) −1519.00 −0.0655273
\(814\) 44.0000 0.00189459
\(815\) 43384.0 1.86463
\(816\) 304.000 0.0130418
\(817\) −22654.0 −0.970090
\(818\) −18896.0 −0.807681
\(819\) 0 0
\(820\) −19040.0 −0.810861
\(821\) 32559.0 1.38406 0.692032 0.721867i \(-0.256716\pi\)
0.692032 + 0.721867i \(0.256716\pi\)
\(822\) 1136.00 0.0482026
\(823\) −43582.0 −1.84590 −0.922948 0.384924i \(-0.874228\pi\)
−0.922948 + 0.384924i \(0.874228\pi\)
\(824\) −416.000 −0.0175874
\(825\) 328.000 0.0138418
\(826\) −27020.0 −1.13819
\(827\) 23042.0 0.968862 0.484431 0.874829i \(-0.339027\pi\)
0.484431 + 0.874829i \(0.339027\pi\)
\(828\) 7488.00 0.314283
\(829\) −20690.0 −0.866820 −0.433410 0.901197i \(-0.642690\pi\)
−0.433410 + 0.901197i \(0.642690\pi\)
\(830\) −14280.0 −0.597188
\(831\) −1996.00 −0.0833219
\(832\) 0 0
\(833\) −16758.0 −0.697035
\(834\) 3690.00 0.153207
\(835\) −64056.0 −2.65479
\(836\) 752.000 0.0311106
\(837\) 5300.00 0.218871
\(838\) 1042.00 0.0429538
\(839\) 43000.0 1.76940 0.884699 0.466163i \(-0.154364\pi\)
0.884699 + 0.466163i \(0.154364\pi\)
\(840\) −4760.00 −0.195519
\(841\) 36127.0 1.48128
\(842\) 14282.0 0.584549
\(843\) −2918.00 −0.119219
\(844\) 9876.00 0.402780
\(845\) 0 0
\(846\) −7124.00 −0.289513
\(847\) −46445.0 −1.88414
\(848\) −3712.00 −0.150319
\(849\) −812.000 −0.0328242
\(850\) 6232.00 0.251477
\(851\) −792.000 −0.0319029
\(852\) 220.000 0.00884633
\(853\) 33065.0 1.32723 0.663613 0.748076i \(-0.269022\pi\)
0.663613 + 0.748076i \(0.269022\pi\)
\(854\) −4480.00 −0.179511
\(855\) −41548.0 −1.66188
\(856\) −96.0000 −0.00383319
\(857\) −28938.0 −1.15345 −0.576723 0.816940i \(-0.695669\pi\)
−0.576723 + 0.816940i \(0.695669\pi\)
\(858\) 0 0
\(859\) 12508.0 0.496819 0.248409 0.968655i \(-0.420092\pi\)
0.248409 + 0.968655i \(0.420092\pi\)
\(860\) −16388.0 −0.649798
\(861\) −9800.00 −0.387901
\(862\) −12598.0 −0.497784
\(863\) 19535.0 0.770544 0.385272 0.922803i \(-0.374108\pi\)
0.385272 + 0.922803i \(0.374108\pi\)
\(864\) −1696.00 −0.0667814
\(865\) −56032.0 −2.20248
\(866\) −6462.00 −0.253565
\(867\) 4552.00 0.178309
\(868\) 14000.0 0.547455
\(869\) −2032.00 −0.0793221
\(870\) −8364.00 −0.325938
\(871\) 0 0
\(872\) 9704.00 0.376857
\(873\) −30004.0 −1.16321
\(874\) −13536.0 −0.523870
\(875\) −23205.0 −0.896540
\(876\) −3352.00 −0.129285
\(877\) −18775.0 −0.722904 −0.361452 0.932391i \(-0.617719\pi\)
−0.361452 + 0.932391i \(0.617719\pi\)
\(878\) 820.000 0.0315190
\(879\) −1855.00 −0.0711804
\(880\) 544.000 0.0208389
\(881\) −43531.0 −1.66470 −0.832348 0.554254i \(-0.813004\pi\)
−0.832348 + 0.554254i \(0.813004\pi\)
\(882\) 45864.0 1.75093
\(883\) 6955.00 0.265067 0.132534 0.991179i \(-0.457689\pi\)
0.132534 + 0.991179i \(0.457689\pi\)
\(884\) 0 0
\(885\) 6562.00 0.249242
\(886\) 18794.0 0.712637
\(887\) 40728.0 1.54173 0.770864 0.637000i \(-0.219825\pi\)
0.770864 + 0.637000i \(0.219825\pi\)
\(888\) 88.0000 0.00332555
\(889\) −18900.0 −0.713032
\(890\) 31756.0 1.19603
\(891\) −1298.00 −0.0488043
\(892\) −7772.00 −0.291733
\(893\) 12878.0 0.482582
\(894\) −5700.00 −0.213240
\(895\) −765.000 −0.0285711
\(896\) −4480.00 −0.167038
\(897\) 0 0
\(898\) 1308.00 0.0486064
\(899\) 24600.0 0.912632
\(900\) −17056.0 −0.631704
\(901\) 4408.00 0.162988
\(902\) 1120.00 0.0413436
\(903\) −8435.00 −0.310852
\(904\) 9264.00 0.340836
\(905\) 6188.00 0.227288
\(906\) −5526.00 −0.202637
\(907\) 23653.0 0.865915 0.432958 0.901414i \(-0.357470\pi\)
0.432958 + 0.901414i \(0.357470\pi\)
\(908\) −12128.0 −0.443262
\(909\) −7592.00 −0.277020
\(910\) 0 0
\(911\) −206.000 −0.00749186 −0.00374593 0.999993i \(-0.501192\pi\)
−0.00374593 + 0.999993i \(0.501192\pi\)
\(912\) 1504.00 0.0546079
\(913\) 840.000 0.0304490
\(914\) 29964.0 1.08438
\(915\) 1088.00 0.0393095
\(916\) −21244.0 −0.766290
\(917\) −54985.0 −1.98011
\(918\) 2014.00 0.0724095
\(919\) 28352.0 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(920\) −9792.00 −0.350905
\(921\) 5350.00 0.191410
\(922\) 21690.0 0.774753
\(923\) 0 0
\(924\) 280.000 0.00996897
\(925\) 1804.00 0.0641245
\(926\) 26144.0 0.927803
\(927\) −1352.00 −0.0479024
\(928\) −7872.00 −0.278460
\(929\) 17612.0 0.621992 0.310996 0.950411i \(-0.399337\pi\)
0.310996 + 0.950411i \(0.399337\pi\)
\(930\) −3400.00 −0.119882
\(931\) −82908.0 −2.91858
\(932\) −2028.00 −0.0712761
\(933\) −2262.00 −0.0793725
\(934\) 30680.0 1.07482
\(935\) −646.000 −0.0225951
\(936\) 0 0
\(937\) 12014.0 0.418869 0.209435 0.977823i \(-0.432838\pi\)
0.209435 + 0.977823i \(0.432838\pi\)
\(938\) −46900.0 −1.63256
\(939\) 1857.00 0.0645377
\(940\) 9316.00 0.323249
\(941\) 26737.0 0.926250 0.463125 0.886293i \(-0.346728\pi\)
0.463125 + 0.886293i \(0.346728\pi\)
\(942\) −932.000 −0.0322359
\(943\) −20160.0 −0.696182
\(944\) 6176.00 0.212936
\(945\) −31535.0 −1.08554
\(946\) 964.000 0.0331314
\(947\) 35566.0 1.22042 0.610211 0.792239i \(-0.291085\pi\)
0.610211 + 0.792239i \(0.291085\pi\)
\(948\) −4064.00 −0.139233
\(949\) 0 0
\(950\) 30832.0 1.05297
\(951\) −3870.00 −0.131959
\(952\) 5320.00 0.181116
\(953\) 12279.0 0.417372 0.208686 0.977983i \(-0.433081\pi\)
0.208686 + 0.977983i \(0.433081\pi\)
\(954\) −12064.0 −0.409420
\(955\) −42194.0 −1.42970
\(956\) 27180.0 0.919523
\(957\) 492.000 0.0166187
\(958\) −39230.0 −1.32303
\(959\) 19880.0 0.669404
\(960\) 1088.00 0.0365782
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) −312.000 −0.0104404
\(964\) 17768.0 0.593640
\(965\) −20740.0 −0.691859
\(966\) −5040.00 −0.167867
\(967\) −51523.0 −1.71341 −0.856705 0.515806i \(-0.827492\pi\)
−0.856705 + 0.515806i \(0.827492\pi\)
\(968\) 10616.0 0.352491
\(969\) −1786.00 −0.0592101
\(970\) 39236.0 1.29875
\(971\) 41027.0 1.35594 0.677971 0.735089i \(-0.262860\pi\)
0.677971 + 0.735089i \(0.262860\pi\)
\(972\) −8320.00 −0.274552
\(973\) 64575.0 2.12763
\(974\) −21808.0 −0.717426
\(975\) 0 0
\(976\) 1024.00 0.0335834
\(977\) −23562.0 −0.771561 −0.385781 0.922591i \(-0.626068\pi\)
−0.385781 + 0.922591i \(0.626068\pi\)
\(978\) −5104.00 −0.166879
\(979\) −1868.00 −0.0609822
\(980\) −59976.0 −1.95496
\(981\) 31538.0 1.02643
\(982\) 37038.0 1.20359
\(983\) 41307.0 1.34027 0.670137 0.742238i \(-0.266235\pi\)
0.670137 + 0.742238i \(0.266235\pi\)
\(984\) 2240.00 0.0725697
\(985\) −27081.0 −0.876013
\(986\) 9348.00 0.301928
\(987\) 4795.00 0.154637
\(988\) 0 0
\(989\) −17352.0 −0.557898
\(990\) 1768.00 0.0567583
\(991\) −26586.0 −0.852202 −0.426101 0.904676i \(-0.640113\pi\)
−0.426101 + 0.904676i \(0.640113\pi\)
\(992\) −3200.00 −0.102419
\(993\) 10520.0 0.336195
\(994\) 3850.00 0.122852
\(995\) −27302.0 −0.869881
\(996\) 1680.00 0.0534466
\(997\) 39298.0 1.24833 0.624163 0.781295i \(-0.285440\pi\)
0.624163 + 0.781295i \(0.285440\pi\)
\(998\) −28800.0 −0.913475
\(999\) 583.000 0.0184638
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 338.4.a.b.1.1 1
13.2 odd 12 338.4.e.c.147.1 4
13.3 even 3 338.4.c.g.191.1 2
13.4 even 6 338.4.c.c.315.1 2
13.5 odd 4 338.4.b.b.337.2 2
13.6 odd 12 338.4.e.c.23.2 4
13.7 odd 12 338.4.e.c.23.1 4
13.8 odd 4 338.4.b.b.337.1 2
13.9 even 3 338.4.c.g.315.1 2
13.10 even 6 338.4.c.c.191.1 2
13.11 odd 12 338.4.e.c.147.2 4
13.12 even 2 26.4.a.b.1.1 1
39.38 odd 2 234.4.a.a.1.1 1
52.51 odd 2 208.4.a.e.1.1 1
65.12 odd 4 650.4.b.d.599.2 2
65.38 odd 4 650.4.b.d.599.1 2
65.64 even 2 650.4.a.c.1.1 1
91.90 odd 2 1274.4.a.f.1.1 1
104.51 odd 2 832.4.a.g.1.1 1
104.77 even 2 832.4.a.j.1.1 1
156.155 even 2 1872.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.b.1.1 1 13.12 even 2
208.4.a.e.1.1 1 52.51 odd 2
234.4.a.a.1.1 1 39.38 odd 2
338.4.a.b.1.1 1 1.1 even 1 trivial
338.4.b.b.337.1 2 13.8 odd 4
338.4.b.b.337.2 2 13.5 odd 4
338.4.c.c.191.1 2 13.10 even 6
338.4.c.c.315.1 2 13.4 even 6
338.4.c.g.191.1 2 13.3 even 3
338.4.c.g.315.1 2 13.9 even 3
338.4.e.c.23.1 4 13.7 odd 12
338.4.e.c.23.2 4 13.6 odd 12
338.4.e.c.147.1 4 13.2 odd 12
338.4.e.c.147.2 4 13.11 odd 12
650.4.a.c.1.1 1 65.64 even 2
650.4.b.d.599.1 2 65.38 odd 4
650.4.b.d.599.2 2 65.12 odd 4
832.4.a.g.1.1 1 104.51 odd 2
832.4.a.j.1.1 1 104.77 even 2
1274.4.a.f.1.1 1 91.90 odd 2
1872.4.a.b.1.1 1 156.155 even 2