Properties

Label 81.5.b.a.80.3
Level $81$
Weight $5$
Character 81.80
Analytic conductor $8.373$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,5,Mod(80,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.80");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.37296700979\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 80.3
Root \(-0.102534 + 0.177594i\) of defining polynomial
Character \(\chi\) \(=\) 81.80
Dual form 81.5.b.a.80.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.355188i q^{2} +15.8738 q^{4} -34.7338i q^{5} -31.2109 q^{7} -11.3212i q^{8} -12.3370 q^{10} -57.7314i q^{11} -73.2956 q^{13} +11.0857i q^{14} +249.960 q^{16} -386.985i q^{17} +115.791 q^{19} -551.359i q^{20} -20.5055 q^{22} -548.312i q^{23} -581.437 q^{25} +26.0337i q^{26} -495.436 q^{28} +785.291i q^{29} +544.734 q^{31} -269.922i q^{32} -137.452 q^{34} +1084.07i q^{35} +898.827 q^{37} -41.1275i q^{38} -393.228 q^{40} +2588.85i q^{41} +2000.11 q^{43} -916.419i q^{44} -194.753 q^{46} +811.345i q^{47} -1426.88 q^{49} +206.519i q^{50} -1163.48 q^{52} +2221.00i q^{53} -2005.23 q^{55} +353.344i q^{56} +278.926 q^{58} +1512.26i q^{59} -1902.56 q^{61} -193.483i q^{62} +3903.49 q^{64} +2545.83i q^{65} +4507.09 q^{67} -6142.93i q^{68} +385.049 q^{70} +3993.54i q^{71} -3436.70 q^{73} -319.252i q^{74} +1838.05 q^{76} +1801.85i q^{77} +1202.78 q^{79} -8682.07i q^{80} +919.529 q^{82} -9256.34i q^{83} -13441.4 q^{85} -710.413i q^{86} -653.588 q^{88} -8929.99i q^{89} +2287.62 q^{91} -8703.81i q^{92} +288.180 q^{94} -4021.86i q^{95} +6670.29 q^{97} +506.811i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.355188i − 0.0887969i −0.999014 0.0443984i \(-0.985863\pi\)
0.999014 0.0443984i \(-0.0141371\pi\)
\(3\) 0 0
\(4\) 15.8738 0.992115
\(5\) − 34.7338i − 1.38935i −0.719323 0.694676i \(-0.755548\pi\)
0.719323 0.694676i \(-0.244452\pi\)
\(6\) 0 0
\(7\) −31.2109 −0.636956 −0.318478 0.947930i \(-0.603172\pi\)
−0.318478 + 0.947930i \(0.603172\pi\)
\(8\) − 11.3212i − 0.176894i
\(9\) 0 0
\(10\) −12.3370 −0.123370
\(11\) − 57.7314i − 0.477119i −0.971128 0.238559i \(-0.923325\pi\)
0.971128 0.238559i \(-0.0766752\pi\)
\(12\) 0 0
\(13\) −73.2956 −0.433702 −0.216851 0.976205i \(-0.569578\pi\)
−0.216851 + 0.976205i \(0.569578\pi\)
\(14\) 11.0857i 0.0565597i
\(15\) 0 0
\(16\) 249.960 0.976408
\(17\) − 386.985i − 1.33905i −0.742791 0.669524i \(-0.766498\pi\)
0.742791 0.669524i \(-0.233502\pi\)
\(18\) 0 0
\(19\) 115.791 0.320750 0.160375 0.987056i \(-0.448730\pi\)
0.160375 + 0.987056i \(0.448730\pi\)
\(20\) − 551.359i − 1.37840i
\(21\) 0 0
\(22\) −20.5055 −0.0423667
\(23\) − 548.312i − 1.03651i −0.855227 0.518253i \(-0.826583\pi\)
0.855227 0.518253i \(-0.173417\pi\)
\(24\) 0 0
\(25\) −581.437 −0.930299
\(26\) 26.0337i 0.0385113i
\(27\) 0 0
\(28\) −495.436 −0.631934
\(29\) 785.291i 0.933758i 0.884321 + 0.466879i \(0.154622\pi\)
−0.884321 + 0.466879i \(0.845378\pi\)
\(30\) 0 0
\(31\) 544.734 0.566840 0.283420 0.958996i \(-0.408531\pi\)
0.283420 + 0.958996i \(0.408531\pi\)
\(32\) − 269.922i − 0.263596i
\(33\) 0 0
\(34\) −137.452 −0.118903
\(35\) 1084.07i 0.884957i
\(36\) 0 0
\(37\) 898.827 0.656557 0.328279 0.944581i \(-0.393532\pi\)
0.328279 + 0.944581i \(0.393532\pi\)
\(38\) − 41.1275i − 0.0284816i
\(39\) 0 0
\(40\) −393.228 −0.245768
\(41\) 2588.85i 1.54007i 0.638003 + 0.770034i \(0.279761\pi\)
−0.638003 + 0.770034i \(0.720239\pi\)
\(42\) 0 0
\(43\) 2000.11 1.08172 0.540862 0.841111i \(-0.318098\pi\)
0.540862 + 0.841111i \(0.318098\pi\)
\(44\) − 916.419i − 0.473357i
\(45\) 0 0
\(46\) −194.753 −0.0920385
\(47\) 811.345i 0.367291i 0.982993 + 0.183645i \(0.0587898\pi\)
−0.982993 + 0.183645i \(0.941210\pi\)
\(48\) 0 0
\(49\) −1426.88 −0.594287
\(50\) 206.519i 0.0826077i
\(51\) 0 0
\(52\) −1163.48 −0.430282
\(53\) 2221.00i 0.790672i 0.918537 + 0.395336i \(0.129372\pi\)
−0.918537 + 0.395336i \(0.870628\pi\)
\(54\) 0 0
\(55\) −2005.23 −0.662886
\(56\) 353.344i 0.112674i
\(57\) 0 0
\(58\) 278.926 0.0829148
\(59\) 1512.26i 0.434432i 0.976124 + 0.217216i \(0.0696976\pi\)
−0.976124 + 0.217216i \(0.930302\pi\)
\(60\) 0 0
\(61\) −1902.56 −0.511304 −0.255652 0.966769i \(-0.582290\pi\)
−0.255652 + 0.966769i \(0.582290\pi\)
\(62\) − 193.483i − 0.0503337i
\(63\) 0 0
\(64\) 3903.49 0.953001
\(65\) 2545.83i 0.602564i
\(66\) 0 0
\(67\) 4507.09 1.00403 0.502015 0.864859i \(-0.332592\pi\)
0.502015 + 0.864859i \(0.332592\pi\)
\(68\) − 6142.93i − 1.32849i
\(69\) 0 0
\(70\) 385.049 0.0785814
\(71\) 3993.54i 0.792213i 0.918205 + 0.396106i \(0.129639\pi\)
−0.918205 + 0.396106i \(0.870361\pi\)
\(72\) 0 0
\(73\) −3436.70 −0.644905 −0.322452 0.946586i \(-0.604507\pi\)
−0.322452 + 0.946586i \(0.604507\pi\)
\(74\) − 319.252i − 0.0583002i
\(75\) 0 0
\(76\) 1838.05 0.318221
\(77\) 1801.85i 0.303904i
\(78\) 0 0
\(79\) 1202.78 0.192722 0.0963608 0.995346i \(-0.469280\pi\)
0.0963608 + 0.995346i \(0.469280\pi\)
\(80\) − 8682.07i − 1.35657i
\(81\) 0 0
\(82\) 919.529 0.136753
\(83\) − 9256.34i − 1.34364i −0.740714 0.671820i \(-0.765513\pi\)
0.740714 0.671820i \(-0.234487\pi\)
\(84\) 0 0
\(85\) −13441.4 −1.86041
\(86\) − 710.413i − 0.0960537i
\(87\) 0 0
\(88\) −653.588 −0.0843993
\(89\) − 8929.99i − 1.12738i −0.825986 0.563691i \(-0.809381\pi\)
0.825986 0.563691i \(-0.190619\pi\)
\(90\) 0 0
\(91\) 2287.62 0.276249
\(92\) − 8703.81i − 1.02833i
\(93\) 0 0
\(94\) 288.180 0.0326143
\(95\) − 4021.86i − 0.445635i
\(96\) 0 0
\(97\) 6670.29 0.708926 0.354463 0.935070i \(-0.384664\pi\)
0.354463 + 0.935070i \(0.384664\pi\)
\(98\) 506.811i 0.0527708i
\(99\) 0 0
\(100\) −9229.64 −0.922964
\(101\) 9150.06i 0.896977i 0.893789 + 0.448489i \(0.148038\pi\)
−0.893789 + 0.448489i \(0.851962\pi\)
\(102\) 0 0
\(103\) 15312.4 1.44334 0.721670 0.692237i \(-0.243375\pi\)
0.721670 + 0.692237i \(0.243375\pi\)
\(104\) 829.793i 0.0767190i
\(105\) 0 0
\(106\) 788.870 0.0702092
\(107\) 6099.28i 0.532735i 0.963872 + 0.266367i \(0.0858234\pi\)
−0.963872 + 0.266367i \(0.914177\pi\)
\(108\) 0 0
\(109\) 15169.5 1.27679 0.638393 0.769710i \(-0.279599\pi\)
0.638393 + 0.769710i \(0.279599\pi\)
\(110\) 712.233i 0.0588622i
\(111\) 0 0
\(112\) −7801.48 −0.621929
\(113\) − 1373.06i − 0.107531i −0.998554 0.0537655i \(-0.982878\pi\)
0.998554 0.0537655i \(-0.0171224\pi\)
\(114\) 0 0
\(115\) −19045.0 −1.44007
\(116\) 12465.6i 0.926396i
\(117\) 0 0
\(118\) 537.135 0.0385762
\(119\) 12078.1i 0.852914i
\(120\) 0 0
\(121\) 11308.1 0.772358
\(122\) 675.767i 0.0454022i
\(123\) 0 0
\(124\) 8647.01 0.562371
\(125\) − 1513.10i − 0.0968386i
\(126\) 0 0
\(127\) −19152.4 −1.18745 −0.593726 0.804667i \(-0.702344\pi\)
−0.593726 + 0.804667i \(0.702344\pi\)
\(128\) − 5705.22i − 0.348219i
\(129\) 0 0
\(130\) 904.248 0.0535058
\(131\) − 3020.00i − 0.175980i −0.996121 0.0879902i \(-0.971956\pi\)
0.996121 0.0879902i \(-0.0280444\pi\)
\(132\) 0 0
\(133\) −3613.93 −0.204304
\(134\) − 1600.86i − 0.0891548i
\(135\) 0 0
\(136\) −4381.13 −0.236869
\(137\) 10147.9i 0.540671i 0.962766 + 0.270336i \(0.0871346\pi\)
−0.962766 + 0.270336i \(0.912865\pi\)
\(138\) 0 0
\(139\) −35126.4 −1.81804 −0.909021 0.416751i \(-0.863169\pi\)
−0.909021 + 0.416751i \(0.863169\pi\)
\(140\) 17208.4i 0.877979i
\(141\) 0 0
\(142\) 1418.46 0.0703460
\(143\) 4231.45i 0.206927i
\(144\) 0 0
\(145\) 27276.1 1.29732
\(146\) 1220.67i 0.0572656i
\(147\) 0 0
\(148\) 14267.8 0.651380
\(149\) 37382.0i 1.68380i 0.539636 + 0.841899i \(0.318562\pi\)
−0.539636 + 0.841899i \(0.681438\pi\)
\(150\) 0 0
\(151\) 33270.0 1.45915 0.729573 0.683903i \(-0.239719\pi\)
0.729573 + 0.683903i \(0.239719\pi\)
\(152\) − 1310.89i − 0.0567387i
\(153\) 0 0
\(154\) 639.993 0.0269857
\(155\) − 18920.7i − 0.787541i
\(156\) 0 0
\(157\) 8080.33 0.327816 0.163908 0.986476i \(-0.447590\pi\)
0.163908 + 0.986476i \(0.447590\pi\)
\(158\) − 427.211i − 0.0171131i
\(159\) 0 0
\(160\) −9375.41 −0.366227
\(161\) 17113.3i 0.660209i
\(162\) 0 0
\(163\) −25427.1 −0.957022 −0.478511 0.878081i \(-0.658823\pi\)
−0.478511 + 0.878081i \(0.658823\pi\)
\(164\) 41095.0i 1.52792i
\(165\) 0 0
\(166\) −3287.74 −0.119311
\(167\) − 35052.8i − 1.25687i −0.777863 0.628434i \(-0.783696\pi\)
0.777863 0.628434i \(-0.216304\pi\)
\(168\) 0 0
\(169\) −23188.8 −0.811903
\(170\) 4774.24i 0.165198i
\(171\) 0 0
\(172\) 31749.4 1.07319
\(173\) − 14700.5i − 0.491177i −0.969374 0.245589i \(-0.921019\pi\)
0.969374 0.245589i \(-0.0789813\pi\)
\(174\) 0 0
\(175\) 18147.2 0.592560
\(176\) − 14430.6i − 0.465862i
\(177\) 0 0
\(178\) −3171.82 −0.100108
\(179\) − 37052.5i − 1.15641i −0.815892 0.578205i \(-0.803753\pi\)
0.815892 0.578205i \(-0.196247\pi\)
\(180\) 0 0
\(181\) −39664.7 −1.21073 −0.605365 0.795948i \(-0.706973\pi\)
−0.605365 + 0.795948i \(0.706973\pi\)
\(182\) − 812.533i − 0.0245300i
\(183\) 0 0
\(184\) −6207.54 −0.183351
\(185\) − 31219.7i − 0.912189i
\(186\) 0 0
\(187\) −22341.2 −0.638885
\(188\) 12879.2i 0.364395i
\(189\) 0 0
\(190\) −1428.51 −0.0395710
\(191\) − 57637.0i − 1.57992i −0.613160 0.789959i \(-0.710102\pi\)
0.613160 0.789959i \(-0.289898\pi\)
\(192\) 0 0
\(193\) 4179.63 0.112208 0.0561039 0.998425i \(-0.482132\pi\)
0.0561039 + 0.998425i \(0.482132\pi\)
\(194\) − 2369.20i − 0.0629504i
\(195\) 0 0
\(196\) −22650.1 −0.589601
\(197\) − 22191.5i − 0.571812i −0.958258 0.285906i \(-0.907705\pi\)
0.958258 0.285906i \(-0.0922945\pi\)
\(198\) 0 0
\(199\) 50608.7 1.27797 0.638983 0.769221i \(-0.279355\pi\)
0.638983 + 0.769221i \(0.279355\pi\)
\(200\) 6582.56i 0.164564i
\(201\) 0 0
\(202\) 3249.99 0.0796488
\(203\) − 24509.6i − 0.594763i
\(204\) 0 0
\(205\) 89920.7 2.13970
\(206\) − 5438.77i − 0.128164i
\(207\) 0 0
\(208\) −18321.0 −0.423469
\(209\) − 6684.77i − 0.153036i
\(210\) 0 0
\(211\) −8918.05 −0.200311 −0.100156 0.994972i \(-0.531934\pi\)
−0.100156 + 0.994972i \(0.531934\pi\)
\(212\) 35255.7i 0.784437i
\(213\) 0 0
\(214\) 2166.39 0.0473052
\(215\) − 69471.4i − 1.50290i
\(216\) 0 0
\(217\) −17001.6 −0.361052
\(218\) − 5388.02i − 0.113375i
\(219\) 0 0
\(220\) −31830.7 −0.657659
\(221\) 28364.3i 0.580747i
\(222\) 0 0
\(223\) −8497.53 −0.170877 −0.0854384 0.996343i \(-0.527229\pi\)
−0.0854384 + 0.996343i \(0.527229\pi\)
\(224\) 8424.49i 0.167899i
\(225\) 0 0
\(226\) −487.695 −0.00954842
\(227\) − 19102.8i − 0.370719i −0.982671 0.185360i \(-0.940655\pi\)
0.982671 0.185360i \(-0.0593450\pi\)
\(228\) 0 0
\(229\) −10723.0 −0.204478 −0.102239 0.994760i \(-0.532601\pi\)
−0.102239 + 0.994760i \(0.532601\pi\)
\(230\) 6764.53i 0.127874i
\(231\) 0 0
\(232\) 8890.43 0.165176
\(233\) 102200.i 1.88251i 0.337694 + 0.941256i \(0.390353\pi\)
−0.337694 + 0.941256i \(0.609647\pi\)
\(234\) 0 0
\(235\) 28181.1 0.510296
\(236\) 24005.3i 0.431006i
\(237\) 0 0
\(238\) 4290.00 0.0757362
\(239\) 15520.2i 0.271707i 0.990729 + 0.135854i \(0.0433777\pi\)
−0.990729 + 0.135854i \(0.956622\pi\)
\(240\) 0 0
\(241\) −71957.8 −1.23892 −0.619461 0.785028i \(-0.712649\pi\)
−0.619461 + 0.785028i \(0.712649\pi\)
\(242\) − 4016.49i − 0.0685830i
\(243\) 0 0
\(244\) −30201.0 −0.507272
\(245\) 49561.1i 0.825674i
\(246\) 0 0
\(247\) −8486.96 −0.139110
\(248\) − 6167.03i − 0.100270i
\(249\) 0 0
\(250\) −537.435 −0.00859897
\(251\) − 41487.8i − 0.658526i −0.944238 0.329263i \(-0.893200\pi\)
0.944238 0.329263i \(-0.106800\pi\)
\(252\) 0 0
\(253\) −31654.8 −0.494537
\(254\) 6802.70i 0.105442i
\(255\) 0 0
\(256\) 60429.5 0.922080
\(257\) − 72529.3i − 1.09811i −0.835785 0.549057i \(-0.814987\pi\)
0.835785 0.549057i \(-0.185013\pi\)
\(258\) 0 0
\(259\) −28053.1 −0.418198
\(260\) 40412.2i 0.597813i
\(261\) 0 0
\(262\) −1072.67 −0.0156265
\(263\) 85890.2i 1.24174i 0.783912 + 0.620872i \(0.213221\pi\)
−0.783912 + 0.620872i \(0.786779\pi\)
\(264\) 0 0
\(265\) 77143.7 1.09852
\(266\) 1283.62i 0.0181416i
\(267\) 0 0
\(268\) 71544.9 0.996114
\(269\) 88967.6i 1.22950i 0.788724 + 0.614748i \(0.210742\pi\)
−0.788724 + 0.614748i \(0.789258\pi\)
\(270\) 0 0
\(271\) 96541.6 1.31455 0.657273 0.753652i \(-0.271710\pi\)
0.657273 + 0.753652i \(0.271710\pi\)
\(272\) − 96730.8i − 1.30746i
\(273\) 0 0
\(274\) 3604.39 0.0480099
\(275\) 33567.2i 0.443863i
\(276\) 0 0
\(277\) −23541.6 −0.306815 −0.153407 0.988163i \(-0.549025\pi\)
−0.153407 + 0.988163i \(0.549025\pi\)
\(278\) 12476.5i 0.161436i
\(279\) 0 0
\(280\) 12273.0 0.156543
\(281\) 58409.9i 0.739731i 0.929085 + 0.369865i \(0.120596\pi\)
−0.929085 + 0.369865i \(0.879404\pi\)
\(282\) 0 0
\(283\) −76117.3 −0.950408 −0.475204 0.879876i \(-0.657626\pi\)
−0.475204 + 0.879876i \(0.657626\pi\)
\(284\) 63392.9i 0.785966i
\(285\) 0 0
\(286\) 1502.96 0.0183745
\(287\) − 80800.3i − 0.980956i
\(288\) 0 0
\(289\) −66236.1 −0.793047
\(290\) − 9688.14i − 0.115198i
\(291\) 0 0
\(292\) −54553.6 −0.639820
\(293\) 139911.i 1.62973i 0.579649 + 0.814866i \(0.303190\pi\)
−0.579649 + 0.814866i \(0.696810\pi\)
\(294\) 0 0
\(295\) 52526.4 0.603579
\(296\) − 10175.8i − 0.116141i
\(297\) 0 0
\(298\) 13277.6 0.149516
\(299\) 40188.8i 0.449534i
\(300\) 0 0
\(301\) −62425.1 −0.689011
\(302\) − 11817.1i − 0.129568i
\(303\) 0 0
\(304\) 28943.1 0.313183
\(305\) 66083.2i 0.710381i
\(306\) 0 0
\(307\) −81796.8 −0.867880 −0.433940 0.900942i \(-0.642877\pi\)
−0.433940 + 0.900942i \(0.642877\pi\)
\(308\) 28602.2i 0.301508i
\(309\) 0 0
\(310\) −6720.39 −0.0699312
\(311\) − 1702.71i − 0.0176043i −0.999961 0.00880216i \(-0.997198\pi\)
0.999961 0.00880216i \(-0.00280185\pi\)
\(312\) 0 0
\(313\) −69960.1 −0.714105 −0.357052 0.934084i \(-0.616218\pi\)
−0.357052 + 0.934084i \(0.616218\pi\)
\(314\) − 2870.03i − 0.0291090i
\(315\) 0 0
\(316\) 19092.7 0.191202
\(317\) − 116629.i − 1.16062i −0.814397 0.580309i \(-0.802932\pi\)
0.814397 0.580309i \(-0.197068\pi\)
\(318\) 0 0
\(319\) 45335.9 0.445514
\(320\) − 135583.i − 1.32405i
\(321\) 0 0
\(322\) 6078.42 0.0586245
\(323\) − 44809.3i − 0.429500i
\(324\) 0 0
\(325\) 42616.8 0.403472
\(326\) 9031.40i 0.0849806i
\(327\) 0 0
\(328\) 29308.9 0.272428
\(329\) − 25322.8i − 0.233948i
\(330\) 0 0
\(331\) 100836. 0.920366 0.460183 0.887824i \(-0.347784\pi\)
0.460183 + 0.887824i \(0.347784\pi\)
\(332\) − 146934.i − 1.33305i
\(333\) 0 0
\(334\) −12450.3 −0.111606
\(335\) − 156549.i − 1.39495i
\(336\) 0 0
\(337\) 40094.5 0.353041 0.176520 0.984297i \(-0.443516\pi\)
0.176520 + 0.984297i \(0.443516\pi\)
\(338\) 8236.36i 0.0720945i
\(339\) 0 0
\(340\) −213367. −1.84574
\(341\) − 31448.2i − 0.270450i
\(342\) 0 0
\(343\) 119471. 1.01549
\(344\) − 22643.6i − 0.191350i
\(345\) 0 0
\(346\) −5221.42 −0.0436150
\(347\) − 226362.i − 1.87994i −0.341254 0.939971i \(-0.610852\pi\)
0.341254 0.939971i \(-0.389148\pi\)
\(348\) 0 0
\(349\) 79598.3 0.653511 0.326755 0.945109i \(-0.394045\pi\)
0.326755 + 0.945109i \(0.394045\pi\)
\(350\) − 6445.64i − 0.0526175i
\(351\) 0 0
\(352\) −15583.0 −0.125766
\(353\) 139919.i 1.12286i 0.827523 + 0.561431i \(0.189749\pi\)
−0.827523 + 0.561431i \(0.810251\pi\)
\(354\) 0 0
\(355\) 138711. 1.10066
\(356\) − 141753.i − 1.11849i
\(357\) 0 0
\(358\) −13160.6 −0.102686
\(359\) 211616.i 1.64195i 0.570963 + 0.820976i \(0.306570\pi\)
−0.570963 + 0.820976i \(0.693430\pi\)
\(360\) 0 0
\(361\) −116913. −0.897119
\(362\) 14088.4i 0.107509i
\(363\) 0 0
\(364\) 36313.3 0.274071
\(365\) 119370.i 0.896000i
\(366\) 0 0
\(367\) −31993.6 −0.237537 −0.118769 0.992922i \(-0.537895\pi\)
−0.118769 + 0.992922i \(0.537895\pi\)
\(368\) − 137056.i − 1.01205i
\(369\) 0 0
\(370\) −11088.8 −0.0809995
\(371\) − 69319.2i − 0.503623i
\(372\) 0 0
\(373\) −123148. −0.885133 −0.442567 0.896736i \(-0.645932\pi\)
−0.442567 + 0.896736i \(0.645932\pi\)
\(374\) 7935.30i 0.0567310i
\(375\) 0 0
\(376\) 9185.40 0.0649714
\(377\) − 57558.3i − 0.404972i
\(378\) 0 0
\(379\) −116524. −0.811218 −0.405609 0.914047i \(-0.632941\pi\)
−0.405609 + 0.914047i \(0.632941\pi\)
\(380\) − 63842.4i − 0.442122i
\(381\) 0 0
\(382\) −20471.9 −0.140292
\(383\) 181112.i 1.23467i 0.786701 + 0.617334i \(0.211787\pi\)
−0.786701 + 0.617334i \(0.788213\pi\)
\(384\) 0 0
\(385\) 62585.0 0.422229
\(386\) − 1484.55i − 0.00996371i
\(387\) 0 0
\(388\) 105883. 0.703336
\(389\) 50827.4i 0.335892i 0.985796 + 0.167946i \(0.0537134\pi\)
−0.985796 + 0.167946i \(0.946287\pi\)
\(390\) 0 0
\(391\) −212188. −1.38793
\(392\) 16154.0i 0.105126i
\(393\) 0 0
\(394\) −7882.13 −0.0507751
\(395\) − 41777.0i − 0.267758i
\(396\) 0 0
\(397\) 228710. 1.45112 0.725561 0.688158i \(-0.241580\pi\)
0.725561 + 0.688158i \(0.241580\pi\)
\(398\) − 17975.6i − 0.113479i
\(399\) 0 0
\(400\) −145336. −0.908351
\(401\) 188930.i 1.17493i 0.809250 + 0.587465i \(0.199874\pi\)
−0.809250 + 0.587465i \(0.800126\pi\)
\(402\) 0 0
\(403\) −39926.5 −0.245840
\(404\) 145247.i 0.889904i
\(405\) 0 0
\(406\) −8705.50 −0.0528131
\(407\) − 51890.5i − 0.313256i
\(408\) 0 0
\(409\) 277427. 1.65845 0.829223 0.558918i \(-0.188783\pi\)
0.829223 + 0.558918i \(0.188783\pi\)
\(410\) − 31938.7i − 0.189998i
\(411\) 0 0
\(412\) 243067. 1.43196
\(413\) − 47198.8i − 0.276714i
\(414\) 0 0
\(415\) −321508. −1.86679
\(416\) 19784.1i 0.114322i
\(417\) 0 0
\(418\) −2374.35 −0.0135891
\(419\) − 102422.i − 0.583397i −0.956510 0.291699i \(-0.905780\pi\)
0.956510 0.291699i \(-0.0942205\pi\)
\(420\) 0 0
\(421\) 47135.8 0.265942 0.132971 0.991120i \(-0.457548\pi\)
0.132971 + 0.991120i \(0.457548\pi\)
\(422\) 3167.58i 0.0177870i
\(423\) 0 0
\(424\) 25144.3 0.139865
\(425\) 225007.i 1.24571i
\(426\) 0 0
\(427\) 59380.6 0.325678
\(428\) 96819.0i 0.528534i
\(429\) 0 0
\(430\) −24675.4 −0.133452
\(431\) 45556.1i 0.245240i 0.992454 + 0.122620i \(0.0391297\pi\)
−0.992454 + 0.122620i \(0.960870\pi\)
\(432\) 0 0
\(433\) 209599. 1.11793 0.558965 0.829192i \(-0.311199\pi\)
0.558965 + 0.829192i \(0.311199\pi\)
\(434\) 6038.76i 0.0320603i
\(435\) 0 0
\(436\) 240798. 1.26672
\(437\) − 63489.5i − 0.332460i
\(438\) 0 0
\(439\) −183684. −0.953110 −0.476555 0.879145i \(-0.658115\pi\)
−0.476555 + 0.879145i \(0.658115\pi\)
\(440\) 22701.6i 0.117260i
\(441\) 0 0
\(442\) 10074.6 0.0515685
\(443\) 115943.i 0.590795i 0.955374 + 0.295398i \(0.0954521\pi\)
−0.955374 + 0.295398i \(0.904548\pi\)
\(444\) 0 0
\(445\) −310172. −1.56633
\(446\) 3018.22i 0.0151733i
\(447\) 0 0
\(448\) −121831. −0.607020
\(449\) 328940.i 1.63164i 0.578305 + 0.815820i \(0.303714\pi\)
−0.578305 + 0.815820i \(0.696286\pi\)
\(450\) 0 0
\(451\) 149458. 0.734795
\(452\) − 21795.8i − 0.106683i
\(453\) 0 0
\(454\) −6785.07 −0.0329187
\(455\) − 79457.6i − 0.383807i
\(456\) 0 0
\(457\) −212737. −1.01862 −0.509308 0.860584i \(-0.670099\pi\)
−0.509308 + 0.860584i \(0.670099\pi\)
\(458\) 3808.68i 0.0181570i
\(459\) 0 0
\(460\) −302317. −1.42872
\(461\) 48448.1i 0.227968i 0.993483 + 0.113984i \(0.0363613\pi\)
−0.993483 + 0.113984i \(0.963639\pi\)
\(462\) 0 0
\(463\) 169101. 0.788832 0.394416 0.918932i \(-0.370947\pi\)
0.394416 + 0.918932i \(0.370947\pi\)
\(464\) 196292.i 0.911729i
\(465\) 0 0
\(466\) 36300.1 0.167161
\(467\) − 54646.4i − 0.250569i −0.992121 0.125285i \(-0.960016\pi\)
0.992121 0.125285i \(-0.0399844\pi\)
\(468\) 0 0
\(469\) −140670. −0.639524
\(470\) − 10009.6i − 0.0453127i
\(471\) 0 0
\(472\) 17120.6 0.0768482
\(473\) − 115469.i − 0.516111i
\(474\) 0 0
\(475\) −67325.1 −0.298394
\(476\) 191726.i 0.846189i
\(477\) 0 0
\(478\) 5512.58 0.0241268
\(479\) 138577.i 0.603978i 0.953311 + 0.301989i \(0.0976506\pi\)
−0.953311 + 0.301989i \(0.902349\pi\)
\(480\) 0 0
\(481\) −65880.0 −0.284750
\(482\) 25558.5i 0.110012i
\(483\) 0 0
\(484\) 179503. 0.766268
\(485\) − 231684.i − 0.984948i
\(486\) 0 0
\(487\) 23464.1 0.0989340 0.0494670 0.998776i \(-0.484248\pi\)
0.0494670 + 0.998776i \(0.484248\pi\)
\(488\) 21539.3i 0.0904464i
\(489\) 0 0
\(490\) 17603.5 0.0733172
\(491\) 74326.7i 0.308306i 0.988047 + 0.154153i \(0.0492649\pi\)
−0.988047 + 0.154153i \(0.950735\pi\)
\(492\) 0 0
\(493\) 303895. 1.25035
\(494\) 3014.46i 0.0123525i
\(495\) 0 0
\(496\) 136162. 0.553467
\(497\) − 124642.i − 0.504605i
\(498\) 0 0
\(499\) −18972.0 −0.0761925 −0.0380963 0.999274i \(-0.512129\pi\)
−0.0380963 + 0.999274i \(0.512129\pi\)
\(500\) − 24018.8i − 0.0960750i
\(501\) 0 0
\(502\) −14735.9 −0.0584751
\(503\) − 117856.i − 0.465818i −0.972498 0.232909i \(-0.925175\pi\)
0.972498 0.232909i \(-0.0748245\pi\)
\(504\) 0 0
\(505\) 317816. 1.24622
\(506\) 11243.4i 0.0439133i
\(507\) 0 0
\(508\) −304022. −1.17809
\(509\) 183460.i 0.708120i 0.935223 + 0.354060i \(0.115199\pi\)
−0.935223 + 0.354060i \(0.884801\pi\)
\(510\) 0 0
\(511\) 107262. 0.410776
\(512\) − 112747.i − 0.430097i
\(513\) 0 0
\(514\) −25761.5 −0.0975091
\(515\) − 531858.i − 2.00531i
\(516\) 0 0
\(517\) 46840.1 0.175241
\(518\) 9964.13i 0.0371347i
\(519\) 0 0
\(520\) 28821.9 0.106590
\(521\) − 409498.i − 1.50861i −0.656526 0.754303i \(-0.727975\pi\)
0.656526 0.754303i \(-0.272025\pi\)
\(522\) 0 0
\(523\) −211852. −0.774513 −0.387256 0.921972i \(-0.626577\pi\)
−0.387256 + 0.921972i \(0.626577\pi\)
\(524\) − 47939.0i − 0.174593i
\(525\) 0 0
\(526\) 30507.1 0.110263
\(527\) − 210804.i − 0.759026i
\(528\) 0 0
\(529\) −20804.7 −0.0743448
\(530\) − 27400.5i − 0.0975453i
\(531\) 0 0
\(532\) −57367.0 −0.202693
\(533\) − 189751.i − 0.667930i
\(534\) 0 0
\(535\) 211851. 0.740156
\(536\) − 51025.7i − 0.177607i
\(537\) 0 0
\(538\) 31600.2 0.109175
\(539\) 82375.9i 0.283545i
\(540\) 0 0
\(541\) 44016.5 0.150391 0.0751954 0.997169i \(-0.476042\pi\)
0.0751954 + 0.997169i \(0.476042\pi\)
\(542\) − 34290.4i − 0.116728i
\(543\) 0 0
\(544\) −104456. −0.352967
\(545\) − 526894.i − 1.77391i
\(546\) 0 0
\(547\) −42859.1 −0.143241 −0.0716207 0.997432i \(-0.522817\pi\)
−0.0716207 + 0.997432i \(0.522817\pi\)
\(548\) 161085.i 0.536408i
\(549\) 0 0
\(550\) 11922.6 0.0394137
\(551\) 90929.5i 0.299503i
\(552\) 0 0
\(553\) −37539.6 −0.122755
\(554\) 8361.69i 0.0272442i
\(555\) 0 0
\(556\) −557591. −1.80371
\(557\) − 311007.i − 1.00244i −0.865319 0.501222i \(-0.832884\pi\)
0.865319 0.501222i \(-0.167116\pi\)
\(558\) 0 0
\(559\) −146599. −0.469145
\(560\) 270975.i 0.864078i
\(561\) 0 0
\(562\) 20746.5 0.0656858
\(563\) − 285768.i − 0.901563i −0.892634 0.450782i \(-0.851145\pi\)
0.892634 0.450782i \(-0.148855\pi\)
\(564\) 0 0
\(565\) −47691.7 −0.149398
\(566\) 27035.9i 0.0843933i
\(567\) 0 0
\(568\) 45211.7 0.140137
\(569\) 136683.i 0.422171i 0.977468 + 0.211086i \(0.0676999\pi\)
−0.977468 + 0.211086i \(0.932300\pi\)
\(570\) 0 0
\(571\) −28510.0 −0.0874431 −0.0437215 0.999044i \(-0.513921\pi\)
−0.0437215 + 0.999044i \(0.513921\pi\)
\(572\) 67169.4i 0.205296i
\(573\) 0 0
\(574\) −28699.3 −0.0871058
\(575\) 318809.i 0.964261i
\(576\) 0 0
\(577\) −293742. −0.882297 −0.441149 0.897434i \(-0.645429\pi\)
−0.441149 + 0.897434i \(0.645429\pi\)
\(578\) 23526.2i 0.0704201i
\(579\) 0 0
\(580\) 432977. 1.28709
\(581\) 288898.i 0.855840i
\(582\) 0 0
\(583\) 128221. 0.377244
\(584\) 38907.5i 0.114080i
\(585\) 0 0
\(586\) 49694.6 0.144715
\(587\) − 386055.i − 1.12040i −0.828357 0.560200i \(-0.810724\pi\)
0.828357 0.560200i \(-0.189276\pi\)
\(588\) 0 0
\(589\) 63075.2 0.181814
\(590\) − 18656.7i − 0.0535959i
\(591\) 0 0
\(592\) 224671. 0.641067
\(593\) 305581.i 0.868995i 0.900673 + 0.434497i \(0.143074\pi\)
−0.900673 + 0.434497i \(0.856926\pi\)
\(594\) 0 0
\(595\) 419519. 1.18500
\(596\) 593396.i 1.67052i
\(597\) 0 0
\(598\) 14274.6 0.0399172
\(599\) 264738.i 0.737840i 0.929461 + 0.368920i \(0.120272\pi\)
−0.929461 + 0.368920i \(0.879728\pi\)
\(600\) 0 0
\(601\) −132203. −0.366009 −0.183005 0.983112i \(-0.558582\pi\)
−0.183005 + 0.983112i \(0.558582\pi\)
\(602\) 22172.6i 0.0611820i
\(603\) 0 0
\(604\) 528123. 1.44764
\(605\) − 392773.i − 1.07308i
\(606\) 0 0
\(607\) −254146. −0.689772 −0.344886 0.938645i \(-0.612082\pi\)
−0.344886 + 0.938645i \(0.612082\pi\)
\(608\) − 31254.5i − 0.0845484i
\(609\) 0 0
\(610\) 23471.9 0.0630797
\(611\) − 59468.0i − 0.159295i
\(612\) 0 0
\(613\) −492878. −1.31165 −0.655826 0.754912i \(-0.727680\pi\)
−0.655826 + 0.754912i \(0.727680\pi\)
\(614\) 29053.2i 0.0770650i
\(615\) 0 0
\(616\) 20399.0 0.0537587
\(617\) − 121278.i − 0.318574i −0.987232 0.159287i \(-0.949081\pi\)
0.987232 0.159287i \(-0.0509195\pi\)
\(618\) 0 0
\(619\) 307364. 0.802180 0.401090 0.916039i \(-0.368631\pi\)
0.401090 + 0.916039i \(0.368631\pi\)
\(620\) − 300344.i − 0.781331i
\(621\) 0 0
\(622\) −604.780 −0.00156321
\(623\) 278713.i 0.718092i
\(624\) 0 0
\(625\) −415954. −1.06484
\(626\) 24849.0i 0.0634103i
\(627\) 0 0
\(628\) 128266. 0.325231
\(629\) − 347832.i − 0.879161i
\(630\) 0 0
\(631\) −254196. −0.638425 −0.319212 0.947683i \(-0.603418\pi\)
−0.319212 + 0.947683i \(0.603418\pi\)
\(632\) − 13616.9i − 0.0340912i
\(633\) 0 0
\(634\) −41425.3 −0.103059
\(635\) 665236.i 1.64979i
\(636\) 0 0
\(637\) 104584. 0.257743
\(638\) − 16102.8i − 0.0395602i
\(639\) 0 0
\(640\) −198164. −0.483799
\(641\) − 365987.i − 0.890736i −0.895348 0.445368i \(-0.853073\pi\)
0.895348 0.445368i \(-0.146927\pi\)
\(642\) 0 0
\(643\) 230783. 0.558189 0.279094 0.960264i \(-0.409966\pi\)
0.279094 + 0.960264i \(0.409966\pi\)
\(644\) 271653.i 0.655003i
\(645\) 0 0
\(646\) −15915.7 −0.0381383
\(647\) 278596.i 0.665529i 0.943010 + 0.332764i \(0.107981\pi\)
−0.943010 + 0.332764i \(0.892019\pi\)
\(648\) 0 0
\(649\) 87304.7 0.207276
\(650\) − 15136.9i − 0.0358271i
\(651\) 0 0
\(652\) −403626. −0.949476
\(653\) − 78506.8i − 0.184111i −0.995754 0.0920557i \(-0.970656\pi\)
0.995754 0.0920557i \(-0.0293438\pi\)
\(654\) 0 0
\(655\) −104896. −0.244499
\(656\) 647111.i 1.50373i
\(657\) 0 0
\(658\) −8994.34 −0.0207739
\(659\) 857273.i 1.97401i 0.160703 + 0.987003i \(0.448624\pi\)
−0.160703 + 0.987003i \(0.551376\pi\)
\(660\) 0 0
\(661\) 22344.0 0.0511396 0.0255698 0.999673i \(-0.491860\pi\)
0.0255698 + 0.999673i \(0.491860\pi\)
\(662\) − 35815.8i − 0.0817257i
\(663\) 0 0
\(664\) −104793. −0.237681
\(665\) 125526.i 0.283850i
\(666\) 0 0
\(667\) 430584. 0.967846
\(668\) − 556423.i − 1.24696i
\(669\) 0 0
\(670\) −55604.1 −0.123867
\(671\) 109838.i 0.243953i
\(672\) 0 0
\(673\) −663478. −1.46486 −0.732430 0.680842i \(-0.761614\pi\)
−0.732430 + 0.680842i \(0.761614\pi\)
\(674\) − 14241.1i − 0.0313489i
\(675\) 0 0
\(676\) −368095. −0.805501
\(677\) 228121.i 0.497722i 0.968539 + 0.248861i \(0.0800563\pi\)
−0.968539 + 0.248861i \(0.919944\pi\)
\(678\) 0 0
\(679\) −208185. −0.451555
\(680\) 152173.i 0.329094i
\(681\) 0 0
\(682\) −11170.0 −0.0240151
\(683\) 53606.6i 0.114915i 0.998348 + 0.0574575i \(0.0182994\pi\)
−0.998348 + 0.0574575i \(0.981701\pi\)
\(684\) 0 0
\(685\) 352474. 0.751182
\(686\) − 42434.8i − 0.0901724i
\(687\) 0 0
\(688\) 499948. 1.05620
\(689\) − 162789.i − 0.342915i
\(690\) 0 0
\(691\) 757326. 1.58609 0.793043 0.609166i \(-0.208496\pi\)
0.793043 + 0.609166i \(0.208496\pi\)
\(692\) − 233353.i − 0.487305i
\(693\) 0 0
\(694\) −80400.9 −0.166933
\(695\) 1.22007e6i 2.52590i
\(696\) 0 0
\(697\) 1.00185e6 2.06222
\(698\) − 28272.3i − 0.0580297i
\(699\) 0 0
\(700\) 288065. 0.587888
\(701\) 506359.i 1.03044i 0.857058 + 0.515220i \(0.172290\pi\)
−0.857058 + 0.515220i \(0.827710\pi\)
\(702\) 0 0
\(703\) 104076. 0.210591
\(704\) − 225354.i − 0.454695i
\(705\) 0 0
\(706\) 49697.4 0.0997067
\(707\) − 285581.i − 0.571335i
\(708\) 0 0
\(709\) −651243. −1.29554 −0.647770 0.761836i \(-0.724298\pi\)
−0.647770 + 0.761836i \(0.724298\pi\)
\(710\) − 49268.4i − 0.0977354i
\(711\) 0 0
\(712\) −101098. −0.199427
\(713\) − 298684.i − 0.587533i
\(714\) 0 0
\(715\) 146974. 0.287495
\(716\) − 588166.i − 1.14729i
\(717\) 0 0
\(718\) 75163.5 0.145800
\(719\) − 69724.6i − 0.134874i −0.997724 0.0674370i \(-0.978518\pi\)
0.997724 0.0674370i \(-0.0214822\pi\)
\(720\) 0 0
\(721\) −477913. −0.919345
\(722\) 41526.2i 0.0796614i
\(723\) 0 0
\(724\) −629631. −1.20118
\(725\) − 456597.i − 0.868675i
\(726\) 0 0
\(727\) −747728. −1.41473 −0.707367 0.706846i \(-0.750117\pi\)
−0.707367 + 0.706846i \(0.750117\pi\)
\(728\) − 25898.6i − 0.0488667i
\(729\) 0 0
\(730\) 42398.6 0.0795620
\(731\) − 774011.i − 1.44848i
\(732\) 0 0
\(733\) 65738.5 0.122352 0.0611761 0.998127i \(-0.480515\pi\)
0.0611761 + 0.998127i \(0.480515\pi\)
\(734\) 11363.7i 0.0210925i
\(735\) 0 0
\(736\) −148001. −0.273218
\(737\) − 260201.i − 0.479042i
\(738\) 0 0
\(739\) 977921. 1.79067 0.895334 0.445395i \(-0.146937\pi\)
0.895334 + 0.445395i \(0.146937\pi\)
\(740\) − 495576.i − 0.904996i
\(741\) 0 0
\(742\) −24621.3 −0.0447202
\(743\) 504874.i 0.914545i 0.889327 + 0.457272i \(0.151174\pi\)
−0.889327 + 0.457272i \(0.848826\pi\)
\(744\) 0 0
\(745\) 1.29842e6 2.33939
\(746\) 43740.5i 0.0785971i
\(747\) 0 0
\(748\) −354640. −0.633847
\(749\) − 190364.i − 0.339329i
\(750\) 0 0
\(751\) −532791. −0.944663 −0.472332 0.881421i \(-0.656588\pi\)
−0.472332 + 0.881421i \(0.656588\pi\)
\(752\) 202804.i 0.358626i
\(753\) 0 0
\(754\) −20444.0 −0.0359603
\(755\) − 1.15559e6i − 2.02727i
\(756\) 0 0
\(757\) −293571. −0.512297 −0.256148 0.966637i \(-0.582454\pi\)
−0.256148 + 0.966637i \(0.582454\pi\)
\(758\) 41388.0i 0.0720337i
\(759\) 0 0
\(760\) −45532.2 −0.0788301
\(761\) − 25691.2i − 0.0443624i −0.999754 0.0221812i \(-0.992939\pi\)
0.999754 0.0221812i \(-0.00706108\pi\)
\(762\) 0 0
\(763\) −473453. −0.813257
\(764\) − 914920.i − 1.56746i
\(765\) 0 0
\(766\) 64328.9 0.109635
\(767\) − 110842.i − 0.188414i
\(768\) 0 0
\(769\) −647144. −1.09433 −0.547165 0.837025i \(-0.684293\pi\)
−0.547165 + 0.837025i \(0.684293\pi\)
\(770\) − 22229.4i − 0.0374927i
\(771\) 0 0
\(772\) 66346.8 0.111323
\(773\) 610238.i 1.02127i 0.859798 + 0.510634i \(0.170589\pi\)
−0.859798 + 0.510634i \(0.829411\pi\)
\(774\) 0 0
\(775\) −316728. −0.527331
\(776\) − 75515.6i − 0.125405i
\(777\) 0 0
\(778\) 18053.3 0.0298261
\(779\) 299766.i 0.493977i
\(780\) 0 0
\(781\) 230553. 0.377980
\(782\) 75366.6i 0.123244i
\(783\) 0 0
\(784\) −356664. −0.580266
\(785\) − 280661.i − 0.455451i
\(786\) 0 0
\(787\) −580321. −0.936955 −0.468477 0.883475i \(-0.655197\pi\)
−0.468477 + 0.883475i \(0.655197\pi\)
\(788\) − 352264.i − 0.567303i
\(789\) 0 0
\(790\) −14838.7 −0.0237761
\(791\) 42854.5i 0.0684926i
\(792\) 0 0
\(793\) 139449. 0.221753
\(794\) − 81234.9i − 0.128855i
\(795\) 0 0
\(796\) 803355. 1.26789
\(797\) − 1.05714e6i − 1.66424i −0.554598 0.832118i \(-0.687128\pi\)
0.554598 0.832118i \(-0.312872\pi\)
\(798\) 0 0
\(799\) 313978. 0.491820
\(800\) 156943.i 0.245223i
\(801\) 0 0
\(802\) 67105.5 0.104330
\(803\) 198405.i 0.307696i
\(804\) 0 0
\(805\) 594409. 0.917263
\(806\) 14181.4i 0.0218298i
\(807\) 0 0
\(808\) 103590. 0.158670
\(809\) − 355969.i − 0.543895i −0.962312 0.271947i \(-0.912332\pi\)
0.962312 0.271947i \(-0.0876677\pi\)
\(810\) 0 0
\(811\) −136417. −0.207409 −0.103704 0.994608i \(-0.533070\pi\)
−0.103704 + 0.994608i \(0.533070\pi\)
\(812\) − 389061.i − 0.590074i
\(813\) 0 0
\(814\) −18430.9 −0.0278161
\(815\) 883181.i 1.32964i
\(816\) 0 0
\(817\) 231594. 0.346964
\(818\) − 98538.4i − 0.147265i
\(819\) 0 0
\(820\) 1.42739e6 2.12282
\(821\) 455979.i 0.676485i 0.941059 + 0.338243i \(0.109832\pi\)
−0.941059 + 0.338243i \(0.890168\pi\)
\(822\) 0 0
\(823\) 503590. 0.743494 0.371747 0.928334i \(-0.378759\pi\)
0.371747 + 0.928334i \(0.378759\pi\)
\(824\) − 173355.i − 0.255318i
\(825\) 0 0
\(826\) −16764.4 −0.0245713
\(827\) 551861.i 0.806898i 0.915002 + 0.403449i \(0.132189\pi\)
−0.915002 + 0.403449i \(0.867811\pi\)
\(828\) 0 0
\(829\) −182308. −0.265275 −0.132638 0.991165i \(-0.542345\pi\)
−0.132638 + 0.991165i \(0.542345\pi\)
\(830\) 114196.i 0.165765i
\(831\) 0 0
\(832\) −286109. −0.413318
\(833\) 552182.i 0.795778i
\(834\) 0 0
\(835\) −1.21752e6 −1.74623
\(836\) − 106113.i − 0.151829i
\(837\) 0 0
\(838\) −36379.0 −0.0518039
\(839\) 415684.i 0.590526i 0.955416 + 0.295263i \(0.0954073\pi\)
−0.955416 + 0.295263i \(0.904593\pi\)
\(840\) 0 0
\(841\) 90599.4 0.128095
\(842\) − 16742.1i − 0.0236148i
\(843\) 0 0
\(844\) −141564. −0.198732
\(845\) 805434.i 1.12802i
\(846\) 0 0
\(847\) −352935. −0.491958
\(848\) 555161.i 0.772018i
\(849\) 0 0
\(850\) 79919.8 0.110616
\(851\) − 492837.i − 0.680525i
\(852\) 0 0
\(853\) 150271. 0.206527 0.103263 0.994654i \(-0.467072\pi\)
0.103263 + 0.994654i \(0.467072\pi\)
\(854\) − 21091.3i − 0.0289192i
\(855\) 0 0
\(856\) 69051.1 0.0942374
\(857\) − 144768.i − 0.197111i −0.995132 0.0985554i \(-0.968578\pi\)
0.995132 0.0985554i \(-0.0314222\pi\)
\(858\) 0 0
\(859\) 333897. 0.452508 0.226254 0.974068i \(-0.427352\pi\)
0.226254 + 0.974068i \(0.427352\pi\)
\(860\) − 1.10278e6i − 1.49105i
\(861\) 0 0
\(862\) 16180.9 0.0217766
\(863\) 1.13280e6i 1.52101i 0.649330 + 0.760507i \(0.275049\pi\)
−0.649330 + 0.760507i \(0.724951\pi\)
\(864\) 0 0
\(865\) −510603. −0.682418
\(866\) − 74447.1i − 0.0992686i
\(867\) 0 0
\(868\) −269881. −0.358206
\(869\) − 69437.9i − 0.0919511i
\(870\) 0 0
\(871\) −330350. −0.435450
\(872\) − 171737.i − 0.225855i
\(873\) 0 0
\(874\) −22550.7 −0.0295214
\(875\) 47225.2i 0.0616820i
\(876\) 0 0
\(877\) 496541. 0.645589 0.322795 0.946469i \(-0.395378\pi\)
0.322795 + 0.946469i \(0.395378\pi\)
\(878\) 65242.4i 0.0846332i
\(879\) 0 0
\(880\) −501228. −0.647247
\(881\) − 1.21533e6i − 1.56583i −0.622131 0.782913i \(-0.713733\pi\)
0.622131 0.782913i \(-0.286267\pi\)
\(882\) 0 0
\(883\) −999070. −1.28137 −0.640685 0.767804i \(-0.721349\pi\)
−0.640685 + 0.767804i \(0.721349\pi\)
\(884\) 450250.i 0.576168i
\(885\) 0 0
\(886\) 41181.5 0.0524608
\(887\) 1.08316e6i 1.37672i 0.725367 + 0.688362i \(0.241670\pi\)
−0.725367 + 0.688362i \(0.758330\pi\)
\(888\) 0 0
\(889\) 597763. 0.756355
\(890\) 110169.i 0.139085i
\(891\) 0 0
\(892\) −134888. −0.169529
\(893\) 93946.4i 0.117809i
\(894\) 0 0
\(895\) −1.28698e6 −1.60666
\(896\) 178065.i 0.221800i
\(897\) 0 0
\(898\) 116836. 0.144885
\(899\) 427774.i 0.529292i
\(900\) 0 0
\(901\) 859491. 1.05875
\(902\) − 53085.6i − 0.0652475i
\(903\) 0 0
\(904\) −15544.7 −0.0190216
\(905\) 1.37771e6i 1.68213i
\(906\) 0 0
\(907\) 1.27685e6 1.55212 0.776061 0.630658i \(-0.217215\pi\)
0.776061 + 0.630658i \(0.217215\pi\)
\(908\) − 303235.i − 0.367796i
\(909\) 0 0
\(910\) −28222.4 −0.0340809
\(911\) − 273486.i − 0.329532i −0.986333 0.164766i \(-0.947313\pi\)
0.986333 0.164766i \(-0.0526869\pi\)
\(912\) 0 0
\(913\) −534381. −0.641076
\(914\) 75561.6i 0.0904500i
\(915\) 0 0
\(916\) −170215. −0.202865
\(917\) 94256.7i 0.112092i
\(918\) 0 0
\(919\) 1.15882e6 1.37210 0.686051 0.727553i \(-0.259343\pi\)
0.686051 + 0.727553i \(0.259343\pi\)
\(920\) 215612.i 0.254740i
\(921\) 0 0
\(922\) 17208.1 0.0202429
\(923\) − 292709.i − 0.343584i
\(924\) 0 0
\(925\) −522611. −0.610795
\(926\) − 60062.6i − 0.0700459i
\(927\) 0 0
\(928\) 211967. 0.246135
\(929\) − 852706.i − 0.988025i −0.869455 0.494012i \(-0.835530\pi\)
0.869455 0.494012i \(-0.164470\pi\)
\(930\) 0 0
\(931\) −165220. −0.190618
\(932\) 1.62230e6i 1.86767i
\(933\) 0 0
\(934\) −19409.7 −0.0222498
\(935\) 775993.i 0.887636i
\(936\) 0 0
\(937\) 629989. 0.717552 0.358776 0.933424i \(-0.383194\pi\)
0.358776 + 0.933424i \(0.383194\pi\)
\(938\) 49964.3i 0.0567877i
\(939\) 0 0
\(940\) 447343. 0.506273
\(941\) − 423252.i − 0.477991i −0.971021 0.238996i \(-0.923182\pi\)
0.971021 0.238996i \(-0.0768182\pi\)
\(942\) 0 0
\(943\) 1.41950e6 1.59629
\(944\) 378004.i 0.424182i
\(945\) 0 0
\(946\) −41013.1 −0.0458290
\(947\) 1.39444e6i 1.55489i 0.628952 + 0.777444i \(0.283484\pi\)
−0.628952 + 0.777444i \(0.716516\pi\)
\(948\) 0 0
\(949\) 251895. 0.279696
\(950\) 23913.1i 0.0264965i
\(951\) 0 0
\(952\) 136739. 0.150875
\(953\) 24001.0i 0.0264267i 0.999913 + 0.0132134i \(0.00420607\pi\)
−0.999913 + 0.0132134i \(0.995794\pi\)
\(954\) 0 0
\(955\) −2.00195e6 −2.19506
\(956\) 246365.i 0.269565i
\(957\) 0 0
\(958\) 49220.9 0.0536314
\(959\) − 316723.i − 0.344384i
\(960\) 0 0
\(961\) −626786. −0.678692
\(962\) 23399.8i 0.0252849i
\(963\) 0 0
\(964\) −1.14225e6 −1.22915
\(965\) − 145174.i − 0.155896i
\(966\) 0 0
\(967\) 1.04406e6 1.11654 0.558268 0.829660i \(-0.311466\pi\)
0.558268 + 0.829660i \(0.311466\pi\)
\(968\) − 128021.i − 0.136625i
\(969\) 0 0
\(970\) −82291.4 −0.0874603
\(971\) − 605213.i − 0.641904i −0.947096 0.320952i \(-0.895997\pi\)
0.947096 0.320952i \(-0.104003\pi\)
\(972\) 0 0
\(973\) 1.09632e6 1.15801
\(974\) − 8334.15i − 0.00878503i
\(975\) 0 0
\(976\) −475565. −0.499241
\(977\) 48105.1i 0.0503967i 0.999682 + 0.0251983i \(0.00802173\pi\)
−0.999682 + 0.0251983i \(0.991978\pi\)
\(978\) 0 0
\(979\) −515540. −0.537895
\(980\) 786724.i 0.819163i
\(981\) 0 0
\(982\) 26399.9 0.0273766
\(983\) 123883.i 0.128205i 0.997943 + 0.0641024i \(0.0204184\pi\)
−0.997943 + 0.0641024i \(0.979582\pi\)
\(984\) 0 0
\(985\) −770794. −0.794448
\(986\) − 107940.i − 0.111027i
\(987\) 0 0
\(988\) −134721. −0.138013
\(989\) − 1.09668e6i − 1.12121i
\(990\) 0 0
\(991\) 851418. 0.866953 0.433477 0.901165i \(-0.357287\pi\)
0.433477 + 0.901165i \(0.357287\pi\)
\(992\) − 147036.i − 0.149417i
\(993\) 0 0
\(994\) −44271.3 −0.0448073
\(995\) − 1.75783e6i − 1.77555i
\(996\) 0 0
\(997\) −1.60305e6 −1.61271 −0.806355 0.591432i \(-0.798563\pi\)
−0.806355 + 0.591432i \(0.798563\pi\)
\(998\) 6738.62i 0.00676566i
\(999\) 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 81.5.b.a.80.3 6
3.2 odd 2 inner 81.5.b.a.80.4 6
4.3 odd 2 1296.5.e.c.161.1 6
9.2 odd 6 9.5.d.a.5.2 yes 6
9.4 even 3 9.5.d.a.2.2 6
9.5 odd 6 27.5.d.a.8.2 6
9.7 even 3 27.5.d.a.17.2 6
12.11 even 2 1296.5.e.c.161.6 6
36.7 odd 6 432.5.q.a.17.3 6
36.11 even 6 144.5.q.a.113.1 6
36.23 even 6 432.5.q.a.305.3 6
36.31 odd 6 144.5.q.a.65.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.5.d.a.2.2 6 9.4 even 3
9.5.d.a.5.2 yes 6 9.2 odd 6
27.5.d.a.8.2 6 9.5 odd 6
27.5.d.a.17.2 6 9.7 even 3
81.5.b.a.80.3 6 1.1 even 1 trivial
81.5.b.a.80.4 6 3.2 odd 2 inner
144.5.q.a.65.1 6 36.31 odd 6
144.5.q.a.113.1 6 36.11 even 6
432.5.q.a.17.3 6 36.7 odd 6
432.5.q.a.305.3 6 36.23 even 6
1296.5.e.c.161.1 6 4.3 odd 2
1296.5.e.c.161.6 6 12.11 even 2