Properties

Label 882.4.a.bd.1.1
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
Dimension $2$
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] = [882,4,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, 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("882.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(52.0396846251\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -10.4462 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -10.4462 q^{5} +8.00000 q^{8} -20.8924 q^{10} +61.1236 q^{11} -59.2311 q^{13} +16.0000 q^{16} +20.4622 q^{17} -80.3387 q^{19} -41.7849 q^{20} +122.247 q^{22} -158.247 q^{23} -15.8764 q^{25} -118.462 q^{26} -85.1236 q^{29} +243.494 q^{31} +32.0000 q^{32} +40.9244 q^{34} +290.371 q^{37} -160.677 q^{38} -83.5698 q^{40} -168.000 q^{41} +7.62934 q^{43} +244.494 q^{44} -316.494 q^{46} -169.291 q^{47} -31.7529 q^{50} -236.924 q^{52} -250.112 q^{53} -638.510 q^{55} -170.247 q^{58} -805.220 q^{59} +33.1715 q^{61} +486.988 q^{62} +64.0000 q^{64} +618.741 q^{65} -277.382 q^{67} +81.8489 q^{68} -631.506 q^{71} -768.284 q^{73} +580.741 q^{74} -321.355 q^{76} -418.730 q^{79} -167.140 q^{80} -336.000 q^{82} -761.714 q^{83} -213.753 q^{85} +15.2587 q^{86} +488.988 q^{88} -1572.10 q^{89} -632.988 q^{92} -338.581 q^{94} +839.236 q^{95} +1045.16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} - 7 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} - 7 q^{5} + 16 q^{8} - 14 q^{10} + 25 q^{11} - 49 q^{13} + 32 q^{16} - 98 q^{17} - 119 q^{19} - 28 q^{20} + 50 q^{22} - 122 q^{23} - 129 q^{25} - 98 q^{26} - 73 q^{29} + 98 q^{31} + 64 q^{32} - 196 q^{34} + 289 q^{37} - 238 q^{38} - 56 q^{40} - 336 q^{41} + 307 q^{43} + 100 q^{44} - 244 q^{46} - 672 q^{47} - 258 q^{50} - 196 q^{52} + 375 q^{53} - 763 q^{55} - 146 q^{58} - 763 q^{59} - 406 q^{61} + 196 q^{62} + 128 q^{64} + 654 q^{65} - 1041 q^{67} - 392 q^{68} - 1652 q^{71} - 189 q^{73} + 578 q^{74} - 476 q^{76} + 524 q^{79} - 112 q^{80} - 672 q^{82} - 287 q^{83} - 622 q^{85} + 614 q^{86} + 200 q^{88} - 2394 q^{89} - 488 q^{92} - 1344 q^{94} + 706 q^{95} - 63 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) −10.4462 −0.934338 −0.467169 0.884168i \(-0.654726\pi\)
−0.467169 + 0.884168i \(0.654726\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −20.8924 −0.660677
\(11\) 61.1236 1.67540 0.837702 0.546128i \(-0.183899\pi\)
0.837702 + 0.546128i \(0.183899\pi\)
\(12\) 0 0
\(13\) −59.2311 −1.26367 −0.631837 0.775102i \(-0.717699\pi\)
−0.631837 + 0.775102i \(0.717699\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 20.4622 0.291930 0.145965 0.989290i \(-0.453371\pi\)
0.145965 + 0.989290i \(0.453371\pi\)
\(18\) 0 0
\(19\) −80.3387 −0.970050 −0.485025 0.874500i \(-0.661190\pi\)
−0.485025 + 0.874500i \(0.661190\pi\)
\(20\) −41.7849 −0.467169
\(21\) 0 0
\(22\) 122.247 1.18469
\(23\) −158.247 −1.43464 −0.717322 0.696742i \(-0.754632\pi\)
−0.717322 + 0.696742i \(0.754632\pi\)
\(24\) 0 0
\(25\) −15.8764 −0.127012
\(26\) −118.462 −0.893552
\(27\) 0 0
\(28\) 0 0
\(29\) −85.1236 −0.545071 −0.272535 0.962146i \(-0.587862\pi\)
−0.272535 + 0.962146i \(0.587862\pi\)
\(30\) 0 0
\(31\) 243.494 1.41074 0.705369 0.708841i \(-0.250782\pi\)
0.705369 + 0.708841i \(0.250782\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 40.9244 0.206426
\(35\) 0 0
\(36\) 0 0
\(37\) 290.371 1.29018 0.645090 0.764107i \(-0.276820\pi\)
0.645090 + 0.764107i \(0.276820\pi\)
\(38\) −160.677 −0.685929
\(39\) 0 0
\(40\) −83.5698 −0.330339
\(41\) −168.000 −0.639932 −0.319966 0.947429i \(-0.603671\pi\)
−0.319966 + 0.947429i \(0.603671\pi\)
\(42\) 0 0
\(43\) 7.62934 0.0270573 0.0135286 0.999908i \(-0.495694\pi\)
0.0135286 + 0.999908i \(0.495694\pi\)
\(44\) 244.494 0.837702
\(45\) 0 0
\(46\) −316.494 −1.01445
\(47\) −169.291 −0.525395 −0.262698 0.964878i \(-0.584612\pi\)
−0.262698 + 0.964878i \(0.584612\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −31.7529 −0.0898107
\(51\) 0 0
\(52\) −236.924 −0.631837
\(53\) −250.112 −0.648217 −0.324109 0.946020i \(-0.605064\pi\)
−0.324109 + 0.946020i \(0.605064\pi\)
\(54\) 0 0
\(55\) −638.510 −1.56539
\(56\) 0 0
\(57\) 0 0
\(58\) −170.247 −0.385423
\(59\) −805.220 −1.77679 −0.888395 0.459079i \(-0.848179\pi\)
−0.888395 + 0.459079i \(0.848179\pi\)
\(60\) 0 0
\(61\) 33.1715 0.0696259 0.0348130 0.999394i \(-0.488916\pi\)
0.0348130 + 0.999394i \(0.488916\pi\)
\(62\) 486.988 0.997542
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 618.741 1.18070
\(66\) 0 0
\(67\) −277.382 −0.505786 −0.252893 0.967494i \(-0.581382\pi\)
−0.252893 + 0.967494i \(0.581382\pi\)
\(68\) 81.8489 0.145965
\(69\) 0 0
\(70\) 0 0
\(71\) −631.506 −1.05558 −0.527788 0.849376i \(-0.676979\pi\)
−0.527788 + 0.849376i \(0.676979\pi\)
\(72\) 0 0
\(73\) −768.284 −1.23179 −0.615896 0.787828i \(-0.711206\pi\)
−0.615896 + 0.787828i \(0.711206\pi\)
\(74\) 580.741 0.912295
\(75\) 0 0
\(76\) −321.355 −0.485025
\(77\) 0 0
\(78\) 0 0
\(79\) −418.730 −0.596339 −0.298169 0.954513i \(-0.596376\pi\)
−0.298169 + 0.954513i \(0.596376\pi\)
\(80\) −167.140 −0.233585
\(81\) 0 0
\(82\) −336.000 −0.452500
\(83\) −761.714 −1.00734 −0.503668 0.863897i \(-0.668017\pi\)
−0.503668 + 0.863897i \(0.668017\pi\)
\(84\) 0 0
\(85\) −213.753 −0.272762
\(86\) 15.2587 0.0191324
\(87\) 0 0
\(88\) 488.988 0.592345
\(89\) −1572.10 −1.87238 −0.936190 0.351494i \(-0.885674\pi\)
−0.936190 + 0.351494i \(0.885674\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −632.988 −0.717322
\(93\) 0 0
\(94\) −338.581 −0.371511
\(95\) 839.236 0.906355
\(96\) 0 0
\(97\) 1045.16 1.09402 0.547012 0.837125i \(-0.315765\pi\)
0.547012 + 0.837125i \(0.315765\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −63.5058 −0.0635058
\(101\) 975.922 0.961464 0.480732 0.876868i \(-0.340371\pi\)
0.480732 + 0.876868i \(0.340371\pi\)
\(102\) 0 0
\(103\) −643.144 −0.615251 −0.307626 0.951508i \(-0.599534\pi\)
−0.307626 + 0.951508i \(0.599534\pi\)
\(104\) −473.849 −0.446776
\(105\) 0 0
\(106\) −500.224 −0.458359
\(107\) −311.595 −0.281523 −0.140762 0.990044i \(-0.544955\pi\)
−0.140762 + 0.990044i \(0.544955\pi\)
\(108\) 0 0
\(109\) −1865.83 −1.63958 −0.819790 0.572665i \(-0.805910\pi\)
−0.819790 + 0.572665i \(0.805910\pi\)
\(110\) −1277.02 −1.10690
\(111\) 0 0
\(112\) 0 0
\(113\) −1720.49 −1.43231 −0.716153 0.697944i \(-0.754099\pi\)
−0.716153 + 0.697944i \(0.754099\pi\)
\(114\) 0 0
\(115\) 1653.08 1.34044
\(116\) −340.494 −0.272535
\(117\) 0 0
\(118\) −1610.44 −1.25638
\(119\) 0 0
\(120\) 0 0
\(121\) 2405.09 1.80698
\(122\) 66.3431 0.0492330
\(123\) 0 0
\(124\) 973.977 0.705369
\(125\) 1471.63 1.05301
\(126\) 0 0
\(127\) 142.236 0.0993808 0.0496904 0.998765i \(-0.484177\pi\)
0.0496904 + 0.998765i \(0.484177\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 1237.48 0.834880
\(131\) −2486.51 −1.65838 −0.829189 0.558969i \(-0.811197\pi\)
−0.829189 + 0.558969i \(0.811197\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −554.764 −0.357644
\(135\) 0 0
\(136\) 163.698 0.103213
\(137\) 2597.21 1.61967 0.809835 0.586658i \(-0.199557\pi\)
0.809835 + 0.586658i \(0.199557\pi\)
\(138\) 0 0
\(139\) 1600.52 0.976651 0.488325 0.872662i \(-0.337608\pi\)
0.488325 + 0.872662i \(0.337608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1263.01 −0.746405
\(143\) −3620.42 −2.11716
\(144\) 0 0
\(145\) 889.220 0.509280
\(146\) −1536.57 −0.871008
\(147\) 0 0
\(148\) 1161.48 0.645090
\(149\) 2937.71 1.61521 0.807605 0.589724i \(-0.200763\pi\)
0.807605 + 0.589724i \(0.200763\pi\)
\(150\) 0 0
\(151\) 1823.32 0.982649 0.491325 0.870977i \(-0.336513\pi\)
0.491325 + 0.870977i \(0.336513\pi\)
\(152\) −642.709 −0.342965
\(153\) 0 0
\(154\) 0 0
\(155\) −2543.59 −1.31811
\(156\) 0 0
\(157\) 633.648 0.322106 0.161053 0.986946i \(-0.448511\pi\)
0.161053 + 0.986946i \(0.448511\pi\)
\(158\) −837.460 −0.421675
\(159\) 0 0
\(160\) −334.279 −0.165169
\(161\) 0 0
\(162\) 0 0
\(163\) −1345.71 −0.646650 −0.323325 0.946288i \(-0.604801\pi\)
−0.323325 + 0.946288i \(0.604801\pi\)
\(164\) −672.000 −0.319966
\(165\) 0 0
\(166\) −1523.43 −0.712295
\(167\) 387.922 0.179750 0.0898751 0.995953i \(-0.471353\pi\)
0.0898751 + 0.995953i \(0.471353\pi\)
\(168\) 0 0
\(169\) 1311.32 0.596870
\(170\) −427.506 −0.192872
\(171\) 0 0
\(172\) 30.5174 0.0135286
\(173\) −622.660 −0.273641 −0.136821 0.990596i \(-0.543688\pi\)
−0.136821 + 0.990596i \(0.543688\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 977.977 0.418851
\(177\) 0 0
\(178\) −3144.19 −1.32397
\(179\) −449.931 −0.187874 −0.0939368 0.995578i \(-0.529945\pi\)
−0.0939368 + 0.995578i \(0.529945\pi\)
\(180\) 0 0
\(181\) 184.353 0.0757063 0.0378532 0.999283i \(-0.487948\pi\)
0.0378532 + 0.999283i \(0.487948\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1265.98 −0.507223
\(185\) −3033.28 −1.20546
\(186\) 0 0
\(187\) 1250.72 0.489101
\(188\) −677.163 −0.262698
\(189\) 0 0
\(190\) 1678.47 0.640890
\(191\) 350.965 0.132958 0.0664789 0.997788i \(-0.478823\pi\)
0.0664789 + 0.997788i \(0.478823\pi\)
\(192\) 0 0
\(193\) 1528.26 0.569982 0.284991 0.958530i \(-0.408009\pi\)
0.284991 + 0.958530i \(0.408009\pi\)
\(194\) 2090.33 0.773592
\(195\) 0 0
\(196\) 0 0
\(197\) 2874.65 1.03965 0.519823 0.854274i \(-0.325998\pi\)
0.519823 + 0.854274i \(0.325998\pi\)
\(198\) 0 0
\(199\) 4948.70 1.76284 0.881418 0.472337i \(-0.156590\pi\)
0.881418 + 0.472337i \(0.156590\pi\)
\(200\) −127.012 −0.0449054
\(201\) 0 0
\(202\) 1951.84 0.679858
\(203\) 0 0
\(204\) 0 0
\(205\) 1754.97 0.597913
\(206\) −1286.29 −0.435048
\(207\) 0 0
\(208\) −947.698 −0.315918
\(209\) −4910.58 −1.62523
\(210\) 0 0
\(211\) −5280.90 −1.72299 −0.861497 0.507762i \(-0.830473\pi\)
−0.861497 + 0.507762i \(0.830473\pi\)
\(212\) −1000.45 −0.324109
\(213\) 0 0
\(214\) −623.189 −0.199067
\(215\) −79.6978 −0.0252807
\(216\) 0 0
\(217\) 0 0
\(218\) −3731.66 −1.15936
\(219\) 0 0
\(220\) −2554.04 −0.782697
\(221\) −1212.00 −0.368905
\(222\) 0 0
\(223\) 3996.57 1.20013 0.600067 0.799950i \(-0.295141\pi\)
0.600067 + 0.799950i \(0.295141\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −3440.99 −1.01279
\(227\) 538.592 0.157478 0.0787392 0.996895i \(-0.474911\pi\)
0.0787392 + 0.996895i \(0.474911\pi\)
\(228\) 0 0
\(229\) 4171.61 1.20379 0.601895 0.798576i \(-0.294413\pi\)
0.601895 + 0.798576i \(0.294413\pi\)
\(230\) 3306.17 0.947836
\(231\) 0 0
\(232\) −680.988 −0.192712
\(233\) −1539.73 −0.432923 −0.216461 0.976291i \(-0.569452\pi\)
−0.216461 + 0.976291i \(0.569452\pi\)
\(234\) 0 0
\(235\) 1768.45 0.490897
\(236\) −3220.88 −0.888395
\(237\) 0 0
\(238\) 0 0
\(239\) −3132.67 −0.847848 −0.423924 0.905698i \(-0.639348\pi\)
−0.423924 + 0.905698i \(0.639348\pi\)
\(240\) 0 0
\(241\) 2606.69 0.696730 0.348365 0.937359i \(-0.386737\pi\)
0.348365 + 0.937359i \(0.386737\pi\)
\(242\) 4810.18 1.27773
\(243\) 0 0
\(244\) 132.686 0.0348130
\(245\) 0 0
\(246\) 0 0
\(247\) 4758.55 1.22583
\(248\) 1947.95 0.498771
\(249\) 0 0
\(250\) 2943.25 0.744591
\(251\) 3426.24 0.861603 0.430801 0.902447i \(-0.358231\pi\)
0.430801 + 0.902447i \(0.358231\pi\)
\(252\) 0 0
\(253\) −9672.63 −2.40361
\(254\) 284.471 0.0702728
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −5857.44 −1.42170 −0.710850 0.703343i \(-0.751690\pi\)
−0.710850 + 0.703343i \(0.751690\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2474.97 0.590349
\(261\) 0 0
\(262\) −4973.02 −1.17265
\(263\) 1085.93 0.254606 0.127303 0.991864i \(-0.459368\pi\)
0.127303 + 0.991864i \(0.459368\pi\)
\(264\) 0 0
\(265\) 2612.73 0.605654
\(266\) 0 0
\(267\) 0 0
\(268\) −1109.53 −0.252893
\(269\) 5579.45 1.26463 0.632315 0.774712i \(-0.282105\pi\)
0.632315 + 0.774712i \(0.282105\pi\)
\(270\) 0 0
\(271\) −560.327 −0.125599 −0.0627997 0.998026i \(-0.520003\pi\)
−0.0627997 + 0.998026i \(0.520003\pi\)
\(272\) 327.396 0.0729826
\(273\) 0 0
\(274\) 5194.42 1.14528
\(275\) −970.425 −0.212796
\(276\) 0 0
\(277\) −4508.08 −0.977849 −0.488924 0.872326i \(-0.662611\pi\)
−0.488924 + 0.872326i \(0.662611\pi\)
\(278\) 3201.04 0.690597
\(279\) 0 0
\(280\) 0 0
\(281\) 4329.75 0.919186 0.459593 0.888130i \(-0.347995\pi\)
0.459593 + 0.888130i \(0.347995\pi\)
\(282\) 0 0
\(283\) −871.051 −0.182963 −0.0914817 0.995807i \(-0.529160\pi\)
−0.0914817 + 0.995807i \(0.529160\pi\)
\(284\) −2526.02 −0.527788
\(285\) 0 0
\(286\) −7240.83 −1.49706
\(287\) 0 0
\(288\) 0 0
\(289\) −4494.30 −0.914777
\(290\) 1778.44 0.360116
\(291\) 0 0
\(292\) −3073.13 −0.615896
\(293\) −1651.91 −0.329372 −0.164686 0.986346i \(-0.552661\pi\)
−0.164686 + 0.986346i \(0.552661\pi\)
\(294\) 0 0
\(295\) 8411.50 1.66012
\(296\) 2322.97 0.456147
\(297\) 0 0
\(298\) 5875.41 1.14213
\(299\) 9373.15 1.81292
\(300\) 0 0
\(301\) 0 0
\(302\) 3646.65 0.694838
\(303\) 0 0
\(304\) −1285.42 −0.242513
\(305\) −346.517 −0.0650542
\(306\) 0 0
\(307\) 1016.42 0.188958 0.0944791 0.995527i \(-0.469881\pi\)
0.0944791 + 0.995527i \(0.469881\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5087.19 −0.932042
\(311\) 9193.05 1.67617 0.838087 0.545536i \(-0.183674\pi\)
0.838087 + 0.545536i \(0.183674\pi\)
\(312\) 0 0
\(313\) −6692.61 −1.20859 −0.604295 0.796761i \(-0.706545\pi\)
−0.604295 + 0.796761i \(0.706545\pi\)
\(314\) 1267.30 0.227763
\(315\) 0 0
\(316\) −1674.92 −0.298169
\(317\) −7052.49 −1.24955 −0.624775 0.780805i \(-0.714809\pi\)
−0.624775 + 0.780805i \(0.714809\pi\)
\(318\) 0 0
\(319\) −5203.05 −0.913213
\(320\) −668.558 −0.116792
\(321\) 0 0
\(322\) 0 0
\(323\) −1643.91 −0.283187
\(324\) 0 0
\(325\) 940.380 0.160501
\(326\) −2691.41 −0.457250
\(327\) 0 0
\(328\) −1344.00 −0.226250
\(329\) 0 0
\(330\) 0 0
\(331\) 3140.37 0.521482 0.260741 0.965409i \(-0.416033\pi\)
0.260741 + 0.965409i \(0.416033\pi\)
\(332\) −3046.86 −0.503668
\(333\) 0 0
\(334\) 775.843 0.127103
\(335\) 2897.60 0.472575
\(336\) 0 0
\(337\) 2743.87 0.443526 0.221763 0.975101i \(-0.428819\pi\)
0.221763 + 0.975101i \(0.428819\pi\)
\(338\) 2622.65 0.422051
\(339\) 0 0
\(340\) −855.012 −0.136381
\(341\) 14883.2 2.36355
\(342\) 0 0
\(343\) 0 0
\(344\) 61.0347 0.00956619
\(345\) 0 0
\(346\) −1245.32 −0.193494
\(347\) −10436.9 −1.61465 −0.807323 0.590110i \(-0.799084\pi\)
−0.807323 + 0.590110i \(0.799084\pi\)
\(348\) 0 0
\(349\) −2257.01 −0.346175 −0.173087 0.984906i \(-0.555374\pi\)
−0.173087 + 0.984906i \(0.555374\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1955.95 0.296172
\(353\) −5616.44 −0.846835 −0.423418 0.905935i \(-0.639170\pi\)
−0.423418 + 0.905935i \(0.639170\pi\)
\(354\) 0 0
\(355\) 6596.85 0.986266
\(356\) −6288.38 −0.936190
\(357\) 0 0
\(358\) −899.861 −0.132847
\(359\) 795.591 0.116963 0.0584815 0.998288i \(-0.481374\pi\)
0.0584815 + 0.998288i \(0.481374\pi\)
\(360\) 0 0
\(361\) −404.699 −0.0590026
\(362\) 368.706 0.0535325
\(363\) 0 0
\(364\) 0 0
\(365\) 8025.66 1.15091
\(366\) 0 0
\(367\) 8193.97 1.16545 0.582727 0.812668i \(-0.301986\pi\)
0.582727 + 0.812668i \(0.301986\pi\)
\(368\) −2531.95 −0.358661
\(369\) 0 0
\(370\) −6066.55 −0.852392
\(371\) 0 0
\(372\) 0 0
\(373\) 14077.4 1.95415 0.977076 0.212890i \(-0.0682876\pi\)
0.977076 + 0.212890i \(0.0682876\pi\)
\(374\) 2501.45 0.345847
\(375\) 0 0
\(376\) −1354.33 −0.185755
\(377\) 5041.96 0.688791
\(378\) 0 0
\(379\) 6221.22 0.843173 0.421587 0.906788i \(-0.361473\pi\)
0.421587 + 0.906788i \(0.361473\pi\)
\(380\) 3356.94 0.453178
\(381\) 0 0
\(382\) 701.931 0.0940154
\(383\) 10701.4 1.42772 0.713860 0.700288i \(-0.246945\pi\)
0.713860 + 0.700288i \(0.246945\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3056.52 0.403038
\(387\) 0 0
\(388\) 4180.66 0.547012
\(389\) −4221.61 −0.550242 −0.275121 0.961410i \(-0.588718\pi\)
−0.275121 + 0.961410i \(0.588718\pi\)
\(390\) 0 0
\(391\) −3238.09 −0.418816
\(392\) 0 0
\(393\) 0 0
\(394\) 5749.30 0.735141
\(395\) 4374.14 0.557182
\(396\) 0 0
\(397\) −12075.7 −1.52661 −0.763305 0.646038i \(-0.776425\pi\)
−0.763305 + 0.646038i \(0.776425\pi\)
\(398\) 9897.41 1.24651
\(399\) 0 0
\(400\) −254.023 −0.0317529
\(401\) 5506.38 0.685724 0.342862 0.939386i \(-0.388604\pi\)
0.342862 + 0.939386i \(0.388604\pi\)
\(402\) 0 0
\(403\) −14422.4 −1.78271
\(404\) 3903.69 0.480732
\(405\) 0 0
\(406\) 0 0
\(407\) 17748.5 2.16157
\(408\) 0 0
\(409\) 1428.18 0.172662 0.0863311 0.996266i \(-0.472486\pi\)
0.0863311 + 0.996266i \(0.472486\pi\)
\(410\) 3509.93 0.422788
\(411\) 0 0
\(412\) −2572.58 −0.307626
\(413\) 0 0
\(414\) 0 0
\(415\) 7957.03 0.941193
\(416\) −1895.40 −0.223388
\(417\) 0 0
\(418\) −9821.17 −1.14921
\(419\) −264.960 −0.0308929 −0.0154465 0.999881i \(-0.504917\pi\)
−0.0154465 + 0.999881i \(0.504917\pi\)
\(420\) 0 0
\(421\) −281.066 −0.0325375 −0.0162688 0.999868i \(-0.505179\pi\)
−0.0162688 + 0.999868i \(0.505179\pi\)
\(422\) −10561.8 −1.21834
\(423\) 0 0
\(424\) −2000.90 −0.229179
\(425\) −324.867 −0.0370785
\(426\) 0 0
\(427\) 0 0
\(428\) −1246.38 −0.140762
\(429\) 0 0
\(430\) −159.396 −0.0178761
\(431\) 1786.38 0.199645 0.0998223 0.995005i \(-0.468173\pi\)
0.0998223 + 0.995005i \(0.468173\pi\)
\(432\) 0 0
\(433\) 184.621 0.0204904 0.0102452 0.999948i \(-0.496739\pi\)
0.0102452 + 0.999948i \(0.496739\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7463.32 −0.819790
\(437\) 12713.4 1.39168
\(438\) 0 0
\(439\) 11274.6 1.22576 0.612881 0.790176i \(-0.290011\pi\)
0.612881 + 0.790176i \(0.290011\pi\)
\(440\) −5108.08 −0.553451
\(441\) 0 0
\(442\) −2424.00 −0.260855
\(443\) −7969.84 −0.854760 −0.427380 0.904072i \(-0.640563\pi\)
−0.427380 + 0.904072i \(0.640563\pi\)
\(444\) 0 0
\(445\) 16422.5 1.74944
\(446\) 7993.13 0.848623
\(447\) 0 0
\(448\) 0 0
\(449\) −8850.37 −0.930233 −0.465117 0.885249i \(-0.653988\pi\)
−0.465117 + 0.885249i \(0.653988\pi\)
\(450\) 0 0
\(451\) −10268.8 −1.07214
\(452\) −6881.98 −0.716153
\(453\) 0 0
\(454\) 1077.18 0.111354
\(455\) 0 0
\(456\) 0 0
\(457\) −13027.3 −1.33346 −0.666730 0.745299i \(-0.732307\pi\)
−0.666730 + 0.745299i \(0.732307\pi\)
\(458\) 8343.22 0.851207
\(459\) 0 0
\(460\) 6612.34 0.670221
\(461\) −1261.05 −0.127403 −0.0637016 0.997969i \(-0.520291\pi\)
−0.0637016 + 0.997969i \(0.520291\pi\)
\(462\) 0 0
\(463\) −4005.47 −0.402052 −0.201026 0.979586i \(-0.564427\pi\)
−0.201026 + 0.979586i \(0.564427\pi\)
\(464\) −1361.98 −0.136268
\(465\) 0 0
\(466\) −3079.46 −0.306123
\(467\) −17097.8 −1.69420 −0.847101 0.531432i \(-0.821654\pi\)
−0.847101 + 0.531432i \(0.821654\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3536.90 0.347117
\(471\) 0 0
\(472\) −6441.76 −0.628190
\(473\) 466.332 0.0453319
\(474\) 0 0
\(475\) 1275.49 0.123208
\(476\) 0 0
\(477\) 0 0
\(478\) −6265.34 −0.599519
\(479\) −11688.5 −1.11495 −0.557477 0.830193i \(-0.688230\pi\)
−0.557477 + 0.830193i \(0.688230\pi\)
\(480\) 0 0
\(481\) −17199.0 −1.63037
\(482\) 5213.39 0.492662
\(483\) 0 0
\(484\) 9620.36 0.903489
\(485\) −10918.0 −1.02219
\(486\) 0 0
\(487\) −9624.79 −0.895567 −0.447783 0.894142i \(-0.647786\pi\)
−0.447783 + 0.894142i \(0.647786\pi\)
\(488\) 265.372 0.0246165
\(489\) 0 0
\(490\) 0 0
\(491\) 6320.63 0.580949 0.290475 0.956883i \(-0.406187\pi\)
0.290475 + 0.956883i \(0.406187\pi\)
\(492\) 0 0
\(493\) −1741.82 −0.159123
\(494\) 9517.10 0.866790
\(495\) 0 0
\(496\) 3895.91 0.352684
\(497\) 0 0
\(498\) 0 0
\(499\) −15054.6 −1.35057 −0.675286 0.737556i \(-0.735980\pi\)
−0.675286 + 0.737556i \(0.735980\pi\)
\(500\) 5886.51 0.526505
\(501\) 0 0
\(502\) 6852.48 0.609245
\(503\) −16559.1 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(504\) 0 0
\(505\) −10194.7 −0.898333
\(506\) −19345.3 −1.69961
\(507\) 0 0
\(508\) 568.942 0.0496904
\(509\) 13339.0 1.16157 0.580785 0.814057i \(-0.302746\pi\)
0.580785 + 0.814057i \(0.302746\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −11714.9 −1.00529
\(515\) 6718.42 0.574853
\(516\) 0 0
\(517\) −10347.6 −0.880250
\(518\) 0 0
\(519\) 0 0
\(520\) 4949.93 0.417440
\(521\) 7182.70 0.603992 0.301996 0.953309i \(-0.402347\pi\)
0.301996 + 0.953309i \(0.402347\pi\)
\(522\) 0 0
\(523\) 9630.13 0.805155 0.402578 0.915386i \(-0.368114\pi\)
0.402578 + 0.915386i \(0.368114\pi\)
\(524\) −9946.04 −0.829189
\(525\) 0 0
\(526\) 2171.86 0.180034
\(527\) 4982.43 0.411837
\(528\) 0 0
\(529\) 12875.1 1.05820
\(530\) 5225.45 0.428262
\(531\) 0 0
\(532\) 0 0
\(533\) 9950.83 0.808664
\(534\) 0 0
\(535\) 3254.99 0.263038
\(536\) −2219.06 −0.178822
\(537\) 0 0
\(538\) 11158.9 0.894228
\(539\) 0 0
\(540\) 0 0
\(541\) 15549.5 1.23572 0.617860 0.786288i \(-0.288000\pi\)
0.617860 + 0.786288i \(0.288000\pi\)
\(542\) −1120.65 −0.0888122
\(543\) 0 0
\(544\) 654.791 0.0516065
\(545\) 19490.9 1.53192
\(546\) 0 0
\(547\) −5917.38 −0.462539 −0.231270 0.972890i \(-0.574288\pi\)
−0.231270 + 0.972890i \(0.574288\pi\)
\(548\) 10388.8 0.809835
\(549\) 0 0
\(550\) −1940.85 −0.150469
\(551\) 6838.71 0.528746
\(552\) 0 0
\(553\) 0 0
\(554\) −9016.15 −0.691444
\(555\) 0 0
\(556\) 6402.09 0.488325
\(557\) −2131.86 −0.162172 −0.0810862 0.996707i \(-0.525839\pi\)
−0.0810862 + 0.996707i \(0.525839\pi\)
\(558\) 0 0
\(559\) −451.894 −0.0341916
\(560\) 0 0
\(561\) 0 0
\(562\) 8659.51 0.649963
\(563\) 7375.91 0.552145 0.276072 0.961137i \(-0.410967\pi\)
0.276072 + 0.961137i \(0.410967\pi\)
\(564\) 0 0
\(565\) 17972.7 1.33826
\(566\) −1742.10 −0.129375
\(567\) 0 0
\(568\) −5052.05 −0.373203
\(569\) 12700.8 0.935759 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(570\) 0 0
\(571\) −7690.12 −0.563610 −0.281805 0.959472i \(-0.590933\pi\)
−0.281805 + 0.959472i \(0.590933\pi\)
\(572\) −14481.7 −1.05858
\(573\) 0 0
\(574\) 0 0
\(575\) 2512.40 0.182216
\(576\) 0 0
\(577\) 3094.87 0.223295 0.111647 0.993748i \(-0.464387\pi\)
0.111647 + 0.993748i \(0.464387\pi\)
\(578\) −8988.60 −0.646845
\(579\) 0 0
\(580\) 3556.88 0.254640
\(581\) 0 0
\(582\) 0 0
\(583\) −15287.7 −1.08603
\(584\) −6146.27 −0.435504
\(585\) 0 0
\(586\) −3303.83 −0.232901
\(587\) 9967.83 0.700880 0.350440 0.936585i \(-0.386032\pi\)
0.350440 + 0.936585i \(0.386032\pi\)
\(588\) 0 0
\(589\) −19562.0 −1.36849
\(590\) 16823.0 1.17388
\(591\) 0 0
\(592\) 4645.93 0.322545
\(593\) −1769.73 −0.122553 −0.0612766 0.998121i \(-0.519517\pi\)
−0.0612766 + 0.998121i \(0.519517\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11750.8 0.807605
\(597\) 0 0
\(598\) 18746.3 1.28193
\(599\) 6347.33 0.432963 0.216481 0.976287i \(-0.430542\pi\)
0.216481 + 0.976287i \(0.430542\pi\)
\(600\) 0 0
\(601\) 22005.7 1.49356 0.746781 0.665070i \(-0.231599\pi\)
0.746781 + 0.665070i \(0.231599\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 7293.30 0.491325
\(605\) −25124.1 −1.68833
\(606\) 0 0
\(607\) 17707.1 1.18403 0.592017 0.805925i \(-0.298332\pi\)
0.592017 + 0.805925i \(0.298332\pi\)
\(608\) −2570.84 −0.171482
\(609\) 0 0
\(610\) −693.035 −0.0460003
\(611\) 10027.3 0.663928
\(612\) 0 0
\(613\) 15846.6 1.04411 0.522055 0.852912i \(-0.325166\pi\)
0.522055 + 0.852912i \(0.325166\pi\)
\(614\) 2032.84 0.133614
\(615\) 0 0
\(616\) 0 0
\(617\) −29473.4 −1.92311 −0.961553 0.274621i \(-0.911448\pi\)
−0.961553 + 0.274621i \(0.911448\pi\)
\(618\) 0 0
\(619\) 11527.5 0.748511 0.374255 0.927326i \(-0.377898\pi\)
0.374255 + 0.927326i \(0.377898\pi\)
\(620\) −10174.4 −0.659053
\(621\) 0 0
\(622\) 18386.1 1.18523
\(623\) 0 0
\(624\) 0 0
\(625\) −13388.4 −0.856856
\(626\) −13385.2 −0.854602
\(627\) 0 0
\(628\) 2534.59 0.161053
\(629\) 5941.63 0.376643
\(630\) 0 0
\(631\) 25846.1 1.63062 0.815308 0.579027i \(-0.196568\pi\)
0.815308 + 0.579027i \(0.196568\pi\)
\(632\) −3349.84 −0.210838
\(633\) 0 0
\(634\) −14105.0 −0.883565
\(635\) −1485.82 −0.0928553
\(636\) 0 0
\(637\) 0 0
\(638\) −10406.1 −0.645739
\(639\) 0 0
\(640\) −1337.12 −0.0825846
\(641\) −5534.78 −0.341046 −0.170523 0.985354i \(-0.554546\pi\)
−0.170523 + 0.985354i \(0.554546\pi\)
\(642\) 0 0
\(643\) 1634.56 0.100250 0.0501251 0.998743i \(-0.484038\pi\)
0.0501251 + 0.998743i \(0.484038\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3287.81 −0.200244
\(647\) −7538.01 −0.458036 −0.229018 0.973422i \(-0.573552\pi\)
−0.229018 + 0.973422i \(0.573552\pi\)
\(648\) 0 0
\(649\) −49217.9 −2.97684
\(650\) 1880.76 0.113491
\(651\) 0 0
\(652\) −5382.83 −0.323325
\(653\) 13312.2 0.797773 0.398887 0.917000i \(-0.369397\pi\)
0.398887 + 0.917000i \(0.369397\pi\)
\(654\) 0 0
\(655\) 25974.6 1.54949
\(656\) −2688.00 −0.159983
\(657\) 0 0
\(658\) 0 0
\(659\) −10962.0 −0.647981 −0.323991 0.946060i \(-0.605025\pi\)
−0.323991 + 0.946060i \(0.605025\pi\)
\(660\) 0 0
\(661\) −23632.5 −1.39062 −0.695309 0.718711i \(-0.744733\pi\)
−0.695309 + 0.718711i \(0.744733\pi\)
\(662\) 6280.74 0.368743
\(663\) 0 0
\(664\) −6093.71 −0.356147
\(665\) 0 0
\(666\) 0 0
\(667\) 13470.6 0.781982
\(668\) 1551.69 0.0898751
\(669\) 0 0
\(670\) 5795.19 0.334161
\(671\) 2027.56 0.116652
\(672\) 0 0
\(673\) 23483.0 1.34503 0.672513 0.740085i \(-0.265215\pi\)
0.672513 + 0.740085i \(0.265215\pi\)
\(674\) 5487.75 0.313620
\(675\) 0 0
\(676\) 5245.30 0.298435
\(677\) −24672.5 −1.40065 −0.700325 0.713824i \(-0.746961\pi\)
−0.700325 + 0.713824i \(0.746961\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1710.02 −0.0964359
\(681\) 0 0
\(682\) 29766.5 1.67129
\(683\) −8939.46 −0.500818 −0.250409 0.968140i \(-0.580565\pi\)
−0.250409 + 0.968140i \(0.580565\pi\)
\(684\) 0 0
\(685\) −27131.1 −1.51332
\(686\) 0 0
\(687\) 0 0
\(688\) 122.069 0.00676432
\(689\) 14814.4 0.819135
\(690\) 0 0
\(691\) 15110.1 0.831858 0.415929 0.909397i \(-0.363457\pi\)
0.415929 + 0.909397i \(0.363457\pi\)
\(692\) −2490.64 −0.136821
\(693\) 0 0
\(694\) −20873.8 −1.14173
\(695\) −16719.4 −0.912523
\(696\) 0 0
\(697\) −3437.65 −0.186815
\(698\) −4514.02 −0.244783
\(699\) 0 0
\(700\) 0 0
\(701\) −18353.3 −0.988865 −0.494432 0.869216i \(-0.664624\pi\)
−0.494432 + 0.869216i \(0.664624\pi\)
\(702\) 0 0
\(703\) −23328.0 −1.25154
\(704\) 3911.91 0.209426
\(705\) 0 0
\(706\) −11232.9 −0.598803
\(707\) 0 0
\(708\) 0 0
\(709\) −10073.7 −0.533604 −0.266802 0.963751i \(-0.585967\pi\)
−0.266802 + 0.963751i \(0.585967\pi\)
\(710\) 13193.7 0.697395
\(711\) 0 0
\(712\) −12576.8 −0.661986
\(713\) −38532.3 −2.02391
\(714\) 0 0
\(715\) 37819.7 1.97815
\(716\) −1799.72 −0.0939368
\(717\) 0 0
\(718\) 1591.18 0.0827053
\(719\) −19889.3 −1.03163 −0.515817 0.856699i \(-0.672511\pi\)
−0.515817 + 0.856699i \(0.672511\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −809.397 −0.0417211
\(723\) 0 0
\(724\) 737.412 0.0378532
\(725\) 1351.46 0.0692303
\(726\) 0 0
\(727\) 13403.5 0.683780 0.341890 0.939740i \(-0.388933\pi\)
0.341890 + 0.939740i \(0.388933\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 16051.3 0.813816
\(731\) 156.113 0.00789884
\(732\) 0 0
\(733\) 13255.0 0.667919 0.333960 0.942587i \(-0.391615\pi\)
0.333960 + 0.942587i \(0.391615\pi\)
\(734\) 16387.9 0.824101
\(735\) 0 0
\(736\) −5063.91 −0.253612
\(737\) −16954.6 −0.847395
\(738\) 0 0
\(739\) 15420.4 0.767588 0.383794 0.923419i \(-0.374617\pi\)
0.383794 + 0.923419i \(0.374617\pi\)
\(740\) −12133.1 −0.602732
\(741\) 0 0
\(742\) 0 0
\(743\) −943.019 −0.0465626 −0.0232813 0.999729i \(-0.507411\pi\)
−0.0232813 + 0.999729i \(0.507411\pi\)
\(744\) 0 0
\(745\) −30687.9 −1.50915
\(746\) 28154.7 1.38179
\(747\) 0 0
\(748\) 5002.89 0.244551
\(749\) 0 0
\(750\) 0 0
\(751\) −28595.9 −1.38945 −0.694726 0.719275i \(-0.744474\pi\)
−0.694726 + 0.719275i \(0.744474\pi\)
\(752\) −2708.65 −0.131349
\(753\) 0 0
\(754\) 10083.9 0.487049
\(755\) −19046.9 −0.918127
\(756\) 0 0
\(757\) −28984.4 −1.39162 −0.695810 0.718226i \(-0.744955\pi\)
−0.695810 + 0.718226i \(0.744955\pi\)
\(758\) 12442.4 0.596213
\(759\) 0 0
\(760\) 6713.88 0.320445
\(761\) −8930.11 −0.425383 −0.212691 0.977119i \(-0.568223\pi\)
−0.212691 + 0.977119i \(0.568223\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1403.86 0.0664789
\(765\) 0 0
\(766\) 21402.8 1.00955
\(767\) 47694.0 2.24528
\(768\) 0 0
\(769\) 2373.15 0.111285 0.0556424 0.998451i \(-0.482279\pi\)
0.0556424 + 0.998451i \(0.482279\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6113.03 0.284991
\(773\) −27977.0 −1.30176 −0.650881 0.759180i \(-0.725600\pi\)
−0.650881 + 0.759180i \(0.725600\pi\)
\(774\) 0 0
\(775\) −3865.82 −0.179180
\(776\) 8361.32 0.386796
\(777\) 0 0
\(778\) −8443.23 −0.389080
\(779\) 13496.9 0.620766
\(780\) 0 0
\(781\) −38599.9 −1.76852
\(782\) −6476.17 −0.296148
\(783\) 0 0
\(784\) 0 0
\(785\) −6619.23 −0.300956
\(786\) 0 0
\(787\) 34360.9 1.55633 0.778167 0.628058i \(-0.216150\pi\)
0.778167 + 0.628058i \(0.216150\pi\)
\(788\) 11498.6 0.519823
\(789\) 0 0
\(790\) 8748.29 0.393987
\(791\) 0 0
\(792\) 0 0
\(793\) −1964.79 −0.0879844
\(794\) −24151.5 −1.07948
\(795\) 0 0
\(796\) 19794.8 0.881418
\(797\) 10399.5 0.462196 0.231098 0.972930i \(-0.425768\pi\)
0.231098 + 0.972930i \(0.425768\pi\)
\(798\) 0 0
\(799\) −3464.06 −0.153379
\(800\) −508.046 −0.0224527
\(801\) 0 0
\(802\) 11012.8 0.484880
\(803\) −46960.2 −2.06375
\(804\) 0 0
\(805\) 0 0
\(806\) −28844.9 −1.26057
\(807\) 0 0
\(808\) 7807.37 0.339929
\(809\) 17943.1 0.779783 0.389891 0.920861i \(-0.372513\pi\)
0.389891 + 0.920861i \(0.372513\pi\)
\(810\) 0 0
\(811\) −571.663 −0.0247519 −0.0123760 0.999923i \(-0.503939\pi\)
−0.0123760 + 0.999923i \(0.503939\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 35497.0 1.52846
\(815\) 14057.6 0.604190
\(816\) 0 0
\(817\) −612.931 −0.0262469
\(818\) 2856.36 0.122091
\(819\) 0 0
\(820\) 7019.86 0.298956
\(821\) −13895.7 −0.590698 −0.295349 0.955389i \(-0.595436\pi\)
−0.295349 + 0.955389i \(0.595436\pi\)
\(822\) 0 0
\(823\) −21629.3 −0.916098 −0.458049 0.888927i \(-0.651452\pi\)
−0.458049 + 0.888927i \(0.651452\pi\)
\(824\) −5145.15 −0.217524
\(825\) 0 0
\(826\) 0 0
\(827\) −4700.10 −0.197628 −0.0988140 0.995106i \(-0.531505\pi\)
−0.0988140 + 0.995106i \(0.531505\pi\)
\(828\) 0 0
\(829\) −38110.8 −1.59667 −0.798337 0.602211i \(-0.794287\pi\)
−0.798337 + 0.602211i \(0.794287\pi\)
\(830\) 15914.1 0.665524
\(831\) 0 0
\(832\) −3790.79 −0.157959
\(833\) 0 0
\(834\) 0 0
\(835\) −4052.32 −0.167948
\(836\) −19642.3 −0.812613
\(837\) 0 0
\(838\) −529.920 −0.0218446
\(839\) −20080.2 −0.826277 −0.413138 0.910668i \(-0.635567\pi\)
−0.413138 + 0.910668i \(0.635567\pi\)
\(840\) 0 0
\(841\) −17143.0 −0.702898
\(842\) −562.131 −0.0230075
\(843\) 0 0
\(844\) −21123.6 −0.861497
\(845\) −13698.4 −0.557679
\(846\) 0 0
\(847\) 0 0
\(848\) −4001.79 −0.162054
\(849\) 0 0
\(850\) −649.735 −0.0262185
\(851\) −45950.3 −1.85095
\(852\) 0 0
\(853\) −34906.8 −1.40116 −0.700578 0.713576i \(-0.747074\pi\)
−0.700578 + 0.713576i \(0.747074\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2492.76 −0.0995335
\(857\) −48046.8 −1.91511 −0.957553 0.288257i \(-0.906924\pi\)
−0.957553 + 0.288257i \(0.906924\pi\)
\(858\) 0 0
\(859\) −35206.2 −1.39840 −0.699198 0.714928i \(-0.746459\pi\)
−0.699198 + 0.714928i \(0.746459\pi\)
\(860\) −318.791 −0.0126403
\(861\) 0 0
\(862\) 3572.76 0.141170
\(863\) 35857.7 1.41438 0.707190 0.707024i \(-0.249963\pi\)
0.707190 + 0.707024i \(0.249963\pi\)
\(864\) 0 0
\(865\) 6504.44 0.255674
\(866\) 369.242 0.0144889
\(867\) 0 0
\(868\) 0 0
\(869\) −25594.3 −0.999109
\(870\) 0 0
\(871\) 16429.7 0.639148
\(872\) −14926.6 −0.579679
\(873\) 0 0
\(874\) 25426.7 0.984064
\(875\) 0 0
\(876\) 0 0
\(877\) 7366.18 0.283624 0.141812 0.989894i \(-0.454707\pi\)
0.141812 + 0.989894i \(0.454707\pi\)
\(878\) 22549.3 0.866744
\(879\) 0 0
\(880\) −10216.2 −0.391349
\(881\) −3628.00 −0.138741 −0.0693703 0.997591i \(-0.522099\pi\)
−0.0693703 + 0.997591i \(0.522099\pi\)
\(882\) 0 0
\(883\) 25123.0 0.957483 0.478741 0.877956i \(-0.341093\pi\)
0.478741 + 0.877956i \(0.341093\pi\)
\(884\) −4848.00 −0.184452
\(885\) 0 0
\(886\) −15939.7 −0.604406
\(887\) 15774.8 0.597142 0.298571 0.954387i \(-0.403490\pi\)
0.298571 + 0.954387i \(0.403490\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 32844.9 1.23704
\(891\) 0 0
\(892\) 15986.3 0.600067
\(893\) 13600.6 0.509660
\(894\) 0 0
\(895\) 4700.07 0.175538
\(896\) 0 0
\(897\) 0 0
\(898\) −17700.7 −0.657774
\(899\) −20727.1 −0.768951
\(900\) 0 0
\(901\) −5117.85 −0.189234
\(902\) −20537.5 −0.758120
\(903\) 0 0
\(904\) −13764.0 −0.506396
\(905\) −1925.79 −0.0707353
\(906\) 0 0
\(907\) −5113.98 −0.187218 −0.0936092 0.995609i \(-0.529840\pi\)
−0.0936092 + 0.995609i \(0.529840\pi\)
\(908\) 2154.37 0.0787392
\(909\) 0 0
\(910\) 0 0
\(911\) −10279.1 −0.373834 −0.186917 0.982376i \(-0.559850\pi\)
−0.186917 + 0.982376i \(0.559850\pi\)
\(912\) 0 0
\(913\) −46558.7 −1.68770
\(914\) −26054.6 −0.942899
\(915\) 0 0
\(916\) 16686.4 0.601895
\(917\) 0 0
\(918\) 0 0
\(919\) −38921.9 −1.39708 −0.698540 0.715571i \(-0.746167\pi\)
−0.698540 + 0.715571i \(0.746167\pi\)
\(920\) 13224.7 0.473918
\(921\) 0 0
\(922\) −2522.10 −0.0900877
\(923\) 37404.8 1.33390
\(924\) 0 0
\(925\) −4610.05 −0.163868
\(926\) −8010.93 −0.284293
\(927\) 0 0
\(928\) −2723.95 −0.0963558
\(929\) 14967.6 0.528603 0.264302 0.964440i \(-0.414859\pi\)
0.264302 + 0.964440i \(0.414859\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6158.92 −0.216461
\(933\) 0 0
\(934\) −34195.6 −1.19798
\(935\) −13065.3 −0.456986
\(936\) 0 0
\(937\) −22353.0 −0.779339 −0.389669 0.920955i \(-0.627411\pi\)
−0.389669 + 0.920955i \(0.627411\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 7073.79 0.245449
\(941\) 13381.5 0.463577 0.231788 0.972766i \(-0.425542\pi\)
0.231788 + 0.972766i \(0.425542\pi\)
\(942\) 0 0
\(943\) 26585.5 0.918074
\(944\) −12883.5 −0.444198
\(945\) 0 0
\(946\) 932.665 0.0320545
\(947\) 633.777 0.0217476 0.0108738 0.999941i \(-0.496539\pi\)
0.0108738 + 0.999941i \(0.496539\pi\)
\(948\) 0 0
\(949\) 45506.3 1.55658
\(950\) 2550.98 0.0871209
\(951\) 0 0
\(952\) 0 0
\(953\) 49340.0 1.67710 0.838551 0.544823i \(-0.183403\pi\)
0.838551 + 0.544823i \(0.183403\pi\)
\(954\) 0 0
\(955\) −3666.26 −0.124228
\(956\) −12530.7 −0.423924
\(957\) 0 0
\(958\) −23377.1 −0.788391
\(959\) 0 0
\(960\) 0 0
\(961\) 29498.4 0.990179
\(962\) −34398.0 −1.15284
\(963\) 0 0
\(964\) 10426.8 0.348365
\(965\) −15964.5 −0.532556
\(966\) 0 0
\(967\) −3340.71 −0.111096 −0.0555480 0.998456i \(-0.517691\pi\)
−0.0555480 + 0.998456i \(0.517691\pi\)
\(968\) 19240.7 0.638863
\(969\) 0 0
\(970\) −21836.0 −0.722797
\(971\) 4870.40 0.160967 0.0804833 0.996756i \(-0.474354\pi\)
0.0804833 + 0.996756i \(0.474354\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −19249.6 −0.633261
\(975\) 0 0
\(976\) 530.745 0.0174065
\(977\) 6841.34 0.224027 0.112013 0.993707i \(-0.464270\pi\)
0.112013 + 0.993707i \(0.464270\pi\)
\(978\) 0 0
\(979\) −96092.1 −3.13699
\(980\) 0 0
\(981\) 0 0
\(982\) 12641.3 0.410793
\(983\) −10689.9 −0.346851 −0.173425 0.984847i \(-0.555484\pi\)
−0.173425 + 0.984847i \(0.555484\pi\)
\(984\) 0 0
\(985\) −30029.2 −0.971381
\(986\) −3483.63 −0.112517
\(987\) 0 0
\(988\) 19034.2 0.612913
\(989\) −1207.32 −0.0388176
\(990\) 0 0
\(991\) 7465.69 0.239309 0.119655 0.992816i \(-0.461821\pi\)
0.119655 + 0.992816i \(0.461821\pi\)
\(992\) 7791.81 0.249385
\(993\) 0 0
\(994\) 0 0
\(995\) −51695.3 −1.64709
\(996\) 0 0
\(997\) −17354.8 −0.551286 −0.275643 0.961260i \(-0.588891\pi\)
−0.275643 + 0.961260i \(0.588891\pi\)
\(998\) −30109.1 −0.954998
\(999\) 0 0
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 882.4.a.bd.1.1 2
3.2 odd 2 882.4.a.ba.1.2 2
7.2 even 3 126.4.g.e.109.2 yes 4
7.3 odd 6 882.4.g.z.667.1 4
7.4 even 3 126.4.g.e.37.2 4
7.5 odd 6 882.4.g.z.361.1 4
7.6 odd 2 882.4.a.bh.1.2 2
21.2 odd 6 126.4.g.f.109.1 yes 4
21.5 even 6 882.4.g.bj.361.2 4
21.11 odd 6 126.4.g.f.37.1 yes 4
21.17 even 6 882.4.g.bj.667.2 4
21.20 even 2 882.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.4.g.e.37.2 4 7.4 even 3
126.4.g.e.109.2 yes 4 7.2 even 3
126.4.g.f.37.1 yes 4 21.11 odd 6
126.4.g.f.109.1 yes 4 21.2 odd 6
882.4.a.u.1.1 2 21.20 even 2
882.4.a.ba.1.2 2 3.2 odd 2
882.4.a.bd.1.1 2 1.1 even 1 trivial
882.4.a.bh.1.2 2 7.6 odd 2
882.4.g.z.361.1 4 7.5 odd 6
882.4.g.z.667.1 4 7.3 odd 6
882.4.g.bj.361.2 4 21.5 even 6
882.4.g.bj.667.2 4 21.17 even 6