Properties

Label 5733.2.a.bd.1.2
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $3$
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] = [5733,2,Mod(1,5733)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5733, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5733.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.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: \(45.7782354788\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.210756\) of defining polynomial
Character \(\chi\) \(=\) 5733.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.21076 q^{2} -0.534070 q^{4} -2.21076 q^{5} -3.06814 q^{8} +O(q^{10})\) \(q+1.21076 q^{2} -0.534070 q^{4} -2.21076 q^{5} -3.06814 q^{8} -2.67669 q^{10} +0.789244 q^{11} -1.00000 q^{13} -2.64663 q^{16} +1.74483 q^{17} +4.32331 q^{19} +1.18070 q^{20} +0.955582 q^{22} +1.11256 q^{23} -0.112558 q^{25} -1.21076 q^{26} +8.48965 q^{29} +5.70041 q^{31} +2.93186 q^{32} +2.11256 q^{34} +2.27890 q^{37} +5.23448 q^{38} +6.78291 q^{40} -12.1363 q^{41} +8.06814 q^{43} -0.421512 q^{44} +1.34704 q^{46} -8.74483 q^{47} -0.136281 q^{50} +0.534070 q^{52} -7.95558 q^{53} -1.74483 q^{55} +10.2789 q^{58} -10.9556 q^{59} -13.0681 q^{61} +6.90180 q^{62} +8.84302 q^{64} +2.21076 q^{65} +6.55779 q^{67} -0.931860 q^{68} -5.85738 q^{71} +8.00936 q^{73} +2.75919 q^{74} -2.30895 q^{76} -6.91116 q^{79} +5.85105 q^{80} -14.6941 q^{82} -3.14262 q^{83} -3.85738 q^{85} +9.76855 q^{86} -2.42151 q^{88} -3.39145 q^{89} -0.594184 q^{92} -10.5878 q^{94} -9.55779 q^{95} -0.0981974 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 6 q^{4} - 5 q^{5} + 6 q^{8} - 14 q^{10} + 4 q^{11} - 3 q^{13} + 4 q^{16} - 4 q^{17} + 7 q^{19} - 16 q^{20} - 8 q^{22} - q^{23} + 4 q^{25} - 2 q^{26} + 7 q^{29} - 3 q^{31} + 24 q^{32} + 2 q^{34} - 10 q^{37} - 12 q^{38} - 22 q^{40} - 6 q^{41} + 9 q^{43} + 2 q^{44} - 28 q^{46} - 17 q^{47} + 30 q^{50} - 6 q^{52} - 13 q^{53} + 4 q^{55} + 14 q^{58} - 22 q^{59} - 24 q^{61} + 18 q^{62} + 20 q^{64} + 5 q^{65} - 14 q^{67} - 18 q^{68} - 4 q^{71} + 5 q^{73} - 8 q^{74} - 8 q^{76} + q^{79} - 40 q^{80} + 20 q^{82} - 23 q^{83} + 2 q^{85} - 6 q^{86} - 4 q^{88} + 11 q^{89} - 30 q^{92} - 16 q^{94} + 5 q^{95} - 3 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21076 0.856134 0.428067 0.903747i \(-0.359195\pi\)
0.428067 + 0.903747i \(0.359195\pi\)
\(3\) 0 0
\(4\) −0.534070 −0.267035
\(5\) −2.21076 −0.988680 −0.494340 0.869269i \(-0.664590\pi\)
−0.494340 + 0.869269i \(0.664590\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −3.06814 −1.08475
\(9\) 0 0
\(10\) −2.67669 −0.846442
\(11\) 0.789244 0.237966 0.118983 0.992896i \(-0.462037\pi\)
0.118983 + 0.992896i \(0.462037\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −2.64663 −0.661657
\(17\) 1.74483 0.423182 0.211591 0.977358i \(-0.432135\pi\)
0.211591 + 0.977358i \(0.432135\pi\)
\(18\) 0 0
\(19\) 4.32331 0.991836 0.495918 0.868369i \(-0.334832\pi\)
0.495918 + 0.868369i \(0.334832\pi\)
\(20\) 1.18070 0.264012
\(21\) 0 0
\(22\) 0.955582 0.203731
\(23\) 1.11256 0.231984 0.115992 0.993250i \(-0.462995\pi\)
0.115992 + 0.993250i \(0.462995\pi\)
\(24\) 0 0
\(25\) −0.112558 −0.0225117
\(26\) −1.21076 −0.237449
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48965 1.57649 0.788244 0.615362i \(-0.210990\pi\)
0.788244 + 0.615362i \(0.210990\pi\)
\(30\) 0 0
\(31\) 5.70041 1.02382 0.511912 0.859038i \(-0.328937\pi\)
0.511912 + 0.859038i \(0.328937\pi\)
\(32\) 2.93186 0.518284
\(33\) 0 0
\(34\) 2.11256 0.362301
\(35\) 0 0
\(36\) 0 0
\(37\) 2.27890 0.374648 0.187324 0.982298i \(-0.440019\pi\)
0.187324 + 0.982298i \(0.440019\pi\)
\(38\) 5.23448 0.849144
\(39\) 0 0
\(40\) 6.78291 1.07247
\(41\) −12.1363 −1.89537 −0.947684 0.319209i \(-0.896583\pi\)
−0.947684 + 0.319209i \(0.896583\pi\)
\(42\) 0 0
\(43\) 8.06814 1.23038 0.615190 0.788379i \(-0.289079\pi\)
0.615190 + 0.788379i \(0.289079\pi\)
\(44\) −0.421512 −0.0635453
\(45\) 0 0
\(46\) 1.34704 0.198610
\(47\) −8.74483 −1.27556 −0.637782 0.770217i \(-0.720148\pi\)
−0.637782 + 0.770217i \(0.720148\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −0.136281 −0.0192730
\(51\) 0 0
\(52\) 0.534070 0.0740622
\(53\) −7.95558 −1.09278 −0.546392 0.837530i \(-0.683999\pi\)
−0.546392 + 0.837530i \(0.683999\pi\)
\(54\) 0 0
\(55\) −1.74483 −0.235272
\(56\) 0 0
\(57\) 0 0
\(58\) 10.2789 1.34969
\(59\) −10.9556 −1.42630 −0.713148 0.701014i \(-0.752731\pi\)
−0.713148 + 0.701014i \(0.752731\pi\)
\(60\) 0 0
\(61\) −13.0681 −1.67320 −0.836602 0.547811i \(-0.815461\pi\)
−0.836602 + 0.547811i \(0.815461\pi\)
\(62\) 6.90180 0.876530
\(63\) 0 0
\(64\) 8.84302 1.10538
\(65\) 2.21076 0.274211
\(66\) 0 0
\(67\) 6.55779 0.801162 0.400581 0.916261i \(-0.368808\pi\)
0.400581 + 0.916261i \(0.368808\pi\)
\(68\) −0.931860 −0.113005
\(69\) 0 0
\(70\) 0 0
\(71\) −5.85738 −0.695144 −0.347572 0.937653i \(-0.612994\pi\)
−0.347572 + 0.937653i \(0.612994\pi\)
\(72\) 0 0
\(73\) 8.00936 0.937425 0.468712 0.883351i \(-0.344718\pi\)
0.468712 + 0.883351i \(0.344718\pi\)
\(74\) 2.75919 0.320749
\(75\) 0 0
\(76\) −2.30895 −0.264855
\(77\) 0 0
\(78\) 0 0
\(79\) −6.91116 −0.777567 −0.388783 0.921329i \(-0.627105\pi\)
−0.388783 + 0.921329i \(0.627105\pi\)
\(80\) 5.85105 0.654167
\(81\) 0 0
\(82\) −14.6941 −1.62269
\(83\) −3.14262 −0.344947 −0.172473 0.985014i \(-0.555176\pi\)
−0.172473 + 0.985014i \(0.555176\pi\)
\(84\) 0 0
\(85\) −3.85738 −0.418392
\(86\) 9.76855 1.05337
\(87\) 0 0
\(88\) −2.42151 −0.258134
\(89\) −3.39145 −0.359493 −0.179747 0.983713i \(-0.557528\pi\)
−0.179747 + 0.983713i \(0.557528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.594184 −0.0619480
\(93\) 0 0
\(94\) −10.5878 −1.09205
\(95\) −9.55779 −0.980609
\(96\) 0 0
\(97\) −0.0981974 −0.00997044 −0.00498522 0.999988i \(-0.501587\pi\)
−0.00498522 + 0.999988i \(0.501587\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0.0601141 0.00601141
\(101\) −6.90180 −0.686755 −0.343378 0.939197i \(-0.611571\pi\)
−0.343378 + 0.939197i \(0.611571\pi\)
\(102\) 0 0
\(103\) −15.8223 −1.55902 −0.779510 0.626390i \(-0.784532\pi\)
−0.779510 + 0.626390i \(0.784532\pi\)
\(104\) 3.06814 0.300856
\(105\) 0 0
\(106\) −9.63227 −0.935569
\(107\) 2.02372 0.195641 0.0978203 0.995204i \(-0.468813\pi\)
0.0978203 + 0.995204i \(0.468813\pi\)
\(108\) 0 0
\(109\) −17.3470 −1.66154 −0.830772 0.556612i \(-0.812101\pi\)
−0.830772 + 0.556612i \(0.812101\pi\)
\(110\) −2.11256 −0.201425
\(111\) 0 0
\(112\) 0 0
\(113\) −10.1807 −0.957720 −0.478860 0.877891i \(-0.658950\pi\)
−0.478860 + 0.877891i \(0.658950\pi\)
\(114\) 0 0
\(115\) −2.45960 −0.229358
\(116\) −4.53407 −0.420978
\(117\) 0 0
\(118\) −13.2645 −1.22110
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3771 −0.943372
\(122\) −15.8223 −1.43249
\(123\) 0 0
\(124\) −3.04442 −0.273397
\(125\) 11.3026 1.01094
\(126\) 0 0
\(127\) 16.4452 1.45928 0.729639 0.683832i \(-0.239688\pi\)
0.729639 + 0.683832i \(0.239688\pi\)
\(128\) 4.84302 0.428067
\(129\) 0 0
\(130\) 2.67669 0.234761
\(131\) −12.3327 −1.07751 −0.538755 0.842462i \(-0.681105\pi\)
−0.538755 + 0.842462i \(0.681105\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 7.93989 0.685902
\(135\) 0 0
\(136\) −5.35337 −0.459048
\(137\) 3.34704 0.285957 0.142978 0.989726i \(-0.454332\pi\)
0.142978 + 0.989726i \(0.454332\pi\)
\(138\) 0 0
\(139\) −6.16634 −0.523022 −0.261511 0.965201i \(-0.584221\pi\)
−0.261511 + 0.965201i \(0.584221\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.09186 −0.595136
\(143\) −0.789244 −0.0659999
\(144\) 0 0
\(145\) −18.7685 −1.55864
\(146\) 9.69738 0.802561
\(147\) 0 0
\(148\) −1.21709 −0.100044
\(149\) −18.8367 −1.54316 −0.771581 0.636131i \(-0.780534\pi\)
−0.771581 + 0.636131i \(0.780534\pi\)
\(150\) 0 0
\(151\) 20.1901 1.64304 0.821522 0.570177i \(-0.193125\pi\)
0.821522 + 0.570177i \(0.193125\pi\)
\(152\) −13.2645 −1.07590
\(153\) 0 0
\(154\) 0 0
\(155\) −12.6022 −1.01223
\(156\) 0 0
\(157\) 13.8811 1.10783 0.553916 0.832572i \(-0.313133\pi\)
0.553916 + 0.832572i \(0.313133\pi\)
\(158\) −8.36773 −0.665701
\(159\) 0 0
\(160\) −6.48163 −0.512418
\(161\) 0 0
\(162\) 0 0
\(163\) 11.4897 0.899939 0.449970 0.893044i \(-0.351435\pi\)
0.449970 + 0.893044i \(0.351435\pi\)
\(164\) 6.48163 0.506130
\(165\) 0 0
\(166\) −3.80494 −0.295321
\(167\) 12.9699 1.00364 0.501822 0.864971i \(-0.332663\pi\)
0.501822 + 0.864971i \(0.332663\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.67035 −0.358200
\(171\) 0 0
\(172\) −4.30895 −0.328555
\(173\) −2.48029 −0.188573 −0.0942865 0.995545i \(-0.530057\pi\)
−0.0942865 + 0.995545i \(0.530057\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.08884 −0.157452
\(177\) 0 0
\(178\) −4.10622 −0.307774
\(179\) 8.15698 0.609681 0.304841 0.952403i \(-0.401397\pi\)
0.304841 + 0.952403i \(0.401397\pi\)
\(180\) 0 0
\(181\) 3.60855 0.268221 0.134111 0.990966i \(-0.457182\pi\)
0.134111 + 0.990966i \(0.457182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.41349 −0.251645
\(185\) −5.03808 −0.370407
\(186\) 0 0
\(187\) 1.37709 0.100703
\(188\) 4.67035 0.340620
\(189\) 0 0
\(190\) −11.5722 −0.839532
\(191\) −0.337675 −0.0244333 −0.0122167 0.999925i \(-0.503889\pi\)
−0.0122167 + 0.999925i \(0.503889\pi\)
\(192\) 0 0
\(193\) 8.42151 0.606194 0.303097 0.952960i \(-0.401979\pi\)
0.303097 + 0.952960i \(0.401979\pi\)
\(194\) −0.118893 −0.00853603
\(195\) 0 0
\(196\) 0 0
\(197\) 8.51035 0.606337 0.303169 0.952937i \(-0.401955\pi\)
0.303169 + 0.952937i \(0.401955\pi\)
\(198\) 0 0
\(199\) −14.6166 −1.03614 −0.518071 0.855338i \(-0.673350\pi\)
−0.518071 + 0.855338i \(0.673350\pi\)
\(200\) 0.345345 0.0244196
\(201\) 0 0
\(202\) −8.35640 −0.587954
\(203\) 0 0
\(204\) 0 0
\(205\) 26.8304 1.87391
\(206\) −19.1570 −1.33473
\(207\) 0 0
\(208\) 2.64663 0.183511
\(209\) 3.41215 0.236023
\(210\) 0 0
\(211\) −23.8667 −1.64305 −0.821527 0.570169i \(-0.806878\pi\)
−0.821527 + 0.570169i \(0.806878\pi\)
\(212\) 4.24884 0.291811
\(213\) 0 0
\(214\) 2.45023 0.167495
\(215\) −17.8367 −1.21645
\(216\) 0 0
\(217\) 0 0
\(218\) −21.0030 −1.42250
\(219\) 0 0
\(220\) 0.931860 0.0628260
\(221\) −1.74483 −0.117370
\(222\) 0 0
\(223\) 1.27890 0.0856412 0.0428206 0.999083i \(-0.486366\pi\)
0.0428206 + 0.999083i \(0.486366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.3263 −0.819936
\(227\) −9.57849 −0.635747 −0.317873 0.948133i \(-0.602969\pi\)
−0.317873 + 0.948133i \(0.602969\pi\)
\(228\) 0 0
\(229\) −13.5134 −0.892989 −0.446494 0.894786i \(-0.647328\pi\)
−0.446494 + 0.894786i \(0.647328\pi\)
\(230\) −2.97797 −0.196361
\(231\) 0 0
\(232\) −26.0474 −1.71010
\(233\) −28.9586 −1.89714 −0.948571 0.316565i \(-0.897470\pi\)
−0.948571 + 0.316565i \(0.897470\pi\)
\(234\) 0 0
\(235\) 19.3327 1.26112
\(236\) 5.85105 0.380871
\(237\) 0 0
\(238\) 0 0
\(239\) 24.1363 1.56125 0.780623 0.625002i \(-0.214902\pi\)
0.780623 + 0.625002i \(0.214902\pi\)
\(240\) 0 0
\(241\) −9.92552 −0.639359 −0.319680 0.947526i \(-0.603575\pi\)
−0.319680 + 0.947526i \(0.603575\pi\)
\(242\) −12.5641 −0.807653
\(243\) 0 0
\(244\) 6.97930 0.446804
\(245\) 0 0
\(246\) 0 0
\(247\) −4.32331 −0.275086
\(248\) −17.4897 −1.11059
\(249\) 0 0
\(250\) 13.6847 0.865497
\(251\) −18.4990 −1.16765 −0.583824 0.811880i \(-0.698444\pi\)
−0.583824 + 0.811880i \(0.698444\pi\)
\(252\) 0 0
\(253\) 0.878080 0.0552044
\(254\) 19.9112 1.24934
\(255\) 0 0
\(256\) −11.8223 −0.738895
\(257\) −16.6767 −1.04026 −0.520132 0.854086i \(-0.674117\pi\)
−0.520132 + 0.854086i \(0.674117\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.18070 −0.0732238
\(261\) 0 0
\(262\) −14.9319 −0.922493
\(263\) 13.4690 0.830531 0.415266 0.909700i \(-0.363689\pi\)
0.415266 + 0.909700i \(0.363689\pi\)
\(264\) 0 0
\(265\) 17.5878 1.08041
\(266\) 0 0
\(267\) 0 0
\(268\) −3.50232 −0.213938
\(269\) 30.5578 1.86314 0.931571 0.363560i \(-0.118439\pi\)
0.931571 + 0.363560i \(0.118439\pi\)
\(270\) 0 0
\(271\) 4.22512 0.256658 0.128329 0.991732i \(-0.459039\pi\)
0.128329 + 0.991732i \(0.459039\pi\)
\(272\) −4.61791 −0.280002
\(273\) 0 0
\(274\) 4.05244 0.244817
\(275\) −0.0888361 −0.00535702
\(276\) 0 0
\(277\) −3.00000 −0.180253 −0.0901263 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) −7.46593 −0.447777
\(279\) 0 0
\(280\) 0 0
\(281\) 6.27890 0.374568 0.187284 0.982306i \(-0.440032\pi\)
0.187284 + 0.982306i \(0.440032\pi\)
\(282\) 0 0
\(283\) 23.5371 1.39914 0.699568 0.714566i \(-0.253376\pi\)
0.699568 + 0.714566i \(0.253376\pi\)
\(284\) 3.12825 0.185628
\(285\) 0 0
\(286\) −0.955582 −0.0565047
\(287\) 0 0
\(288\) 0 0
\(289\) −13.9556 −0.820917
\(290\) −22.7241 −1.33441
\(291\) 0 0
\(292\) −4.27756 −0.250325
\(293\) −21.6166 −1.26285 −0.631427 0.775435i \(-0.717530\pi\)
−0.631427 + 0.775435i \(0.717530\pi\)
\(294\) 0 0
\(295\) 24.2201 1.41015
\(296\) −6.99197 −0.406400
\(297\) 0 0
\(298\) −22.8066 −1.32115
\(299\) −1.11256 −0.0643409
\(300\) 0 0
\(301\) 0 0
\(302\) 24.4452 1.40667
\(303\) 0 0
\(304\) −11.4422 −0.656256
\(305\) 28.8905 1.65426
\(306\) 0 0
\(307\) −20.3945 −1.16397 −0.581987 0.813198i \(-0.697725\pi\)
−0.581987 + 0.813198i \(0.697725\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −15.2582 −0.866608
\(311\) −12.9492 −0.734284 −0.367142 0.930165i \(-0.619664\pi\)
−0.367142 + 0.930165i \(0.619664\pi\)
\(312\) 0 0
\(313\) −33.2345 −1.87852 −0.939262 0.343201i \(-0.888489\pi\)
−0.939262 + 0.343201i \(0.888489\pi\)
\(314\) 16.8066 0.948453
\(315\) 0 0
\(316\) 3.69105 0.207638
\(317\) 4.87175 0.273624 0.136812 0.990597i \(-0.456314\pi\)
0.136812 + 0.990597i \(0.456314\pi\)
\(318\) 0 0
\(319\) 6.70041 0.375151
\(320\) −19.5498 −1.09287
\(321\) 0 0
\(322\) 0 0
\(323\) 7.54343 0.419728
\(324\) 0 0
\(325\) 0.112558 0.00624362
\(326\) 13.9112 0.770468
\(327\) 0 0
\(328\) 37.2358 2.05600
\(329\) 0 0
\(330\) 0 0
\(331\) 6.87175 0.377705 0.188853 0.982005i \(-0.439523\pi\)
0.188853 + 0.982005i \(0.439523\pi\)
\(332\) 1.67838 0.0921129
\(333\) 0 0
\(334\) 15.7034 0.859254
\(335\) −14.4977 −0.792093
\(336\) 0 0
\(337\) 11.0712 0.603085 0.301542 0.953453i \(-0.402499\pi\)
0.301542 + 0.953453i \(0.402499\pi\)
\(338\) 1.21076 0.0658564
\(339\) 0 0
\(340\) 2.06011 0.111725
\(341\) 4.49901 0.243635
\(342\) 0 0
\(343\) 0 0
\(344\) −24.7542 −1.33466
\(345\) 0 0
\(346\) −3.00303 −0.161444
\(347\) −5.62593 −0.302016 −0.151008 0.988533i \(-0.548252\pi\)
−0.151008 + 0.988533i \(0.548252\pi\)
\(348\) 0 0
\(349\) 18.4783 0.989122 0.494561 0.869143i \(-0.335329\pi\)
0.494561 + 0.869143i \(0.335329\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.31395 0.123334
\(353\) −4.67035 −0.248578 −0.124289 0.992246i \(-0.539665\pi\)
−0.124289 + 0.992246i \(0.539665\pi\)
\(354\) 0 0
\(355\) 12.9492 0.687275
\(356\) 1.81127 0.0959974
\(357\) 0 0
\(358\) 9.87611 0.521968
\(359\) 11.7211 0.618616 0.309308 0.950962i \(-0.399903\pi\)
0.309308 + 0.950962i \(0.399903\pi\)
\(360\) 0 0
\(361\) −0.308953 −0.0162607
\(362\) 4.36907 0.229633
\(363\) 0 0
\(364\) 0 0
\(365\) −17.7067 −0.926813
\(366\) 0 0
\(367\) 19.9699 1.04242 0.521211 0.853428i \(-0.325480\pi\)
0.521211 + 0.853428i \(0.325480\pi\)
\(368\) −2.94453 −0.153494
\(369\) 0 0
\(370\) −6.09989 −0.317118
\(371\) 0 0
\(372\) 0 0
\(373\) −4.42651 −0.229196 −0.114598 0.993412i \(-0.536558\pi\)
−0.114598 + 0.993412i \(0.536558\pi\)
\(374\) 1.66732 0.0862153
\(375\) 0 0
\(376\) 26.8304 1.38367
\(377\) −8.48965 −0.437239
\(378\) 0 0
\(379\) −32.5702 −1.67302 −0.836509 0.547953i \(-0.815407\pi\)
−0.836509 + 0.547953i \(0.815407\pi\)
\(380\) 5.10453 0.261857
\(381\) 0 0
\(382\) −0.408842 −0.0209182
\(383\) −14.1964 −0.725402 −0.362701 0.931906i \(-0.618145\pi\)
−0.362701 + 0.931906i \(0.618145\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.1964 0.518983
\(387\) 0 0
\(388\) 0.0524443 0.00266246
\(389\) −17.3170 −0.878006 −0.439003 0.898486i \(-0.644668\pi\)
−0.439003 + 0.898486i \(0.644668\pi\)
\(390\) 0 0
\(391\) 1.94122 0.0981718
\(392\) 0 0
\(393\) 0 0
\(394\) 10.3040 0.519106
\(395\) 15.2789 0.768765
\(396\) 0 0
\(397\) 15.7241 0.789171 0.394586 0.918859i \(-0.370888\pi\)
0.394586 + 0.918859i \(0.370888\pi\)
\(398\) −17.6971 −0.887075
\(399\) 0 0
\(400\) 0.297900 0.0148950
\(401\) −3.57849 −0.178701 −0.0893506 0.996000i \(-0.528479\pi\)
−0.0893506 + 0.996000i \(0.528479\pi\)
\(402\) 0 0
\(403\) −5.70041 −0.283958
\(404\) 3.68605 0.183388
\(405\) 0 0
\(406\) 0 0
\(407\) 1.79861 0.0891536
\(408\) 0 0
\(409\) −31.3501 −1.55016 −0.775080 0.631863i \(-0.782291\pi\)
−0.775080 + 0.631863i \(0.782291\pi\)
\(410\) 32.4850 1.60432
\(411\) 0 0
\(412\) 8.45023 0.416313
\(413\) 0 0
\(414\) 0 0
\(415\) 6.94756 0.341042
\(416\) −2.93186 −0.143746
\(417\) 0 0
\(418\) 4.13128 0.202068
\(419\) 28.7716 1.40558 0.702792 0.711396i \(-0.251937\pi\)
0.702792 + 0.711396i \(0.251937\pi\)
\(420\) 0 0
\(421\) −0.190060 −0.00926297 −0.00463148 0.999989i \(-0.501474\pi\)
−0.00463148 + 0.999989i \(0.501474\pi\)
\(422\) −28.8968 −1.40667
\(423\) 0 0
\(424\) 24.4088 1.18540
\(425\) −0.196395 −0.00952655
\(426\) 0 0
\(427\) 0 0
\(428\) −1.08081 −0.0522429
\(429\) 0 0
\(430\) −21.5959 −1.04145
\(431\) −25.6734 −1.23664 −0.618322 0.785925i \(-0.712187\pi\)
−0.618322 + 0.785925i \(0.712187\pi\)
\(432\) 0 0
\(433\) 1.82233 0.0875755 0.0437877 0.999041i \(-0.486057\pi\)
0.0437877 + 0.999041i \(0.486057\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.26454 0.443691
\(437\) 4.80994 0.230091
\(438\) 0 0
\(439\) 0.735465 0.0351018 0.0175509 0.999846i \(-0.494413\pi\)
0.0175509 + 0.999846i \(0.494413\pi\)
\(440\) 5.35337 0.255212
\(441\) 0 0
\(442\) −2.11256 −0.100484
\(443\) 3.71174 0.176350 0.0881751 0.996105i \(-0.471896\pi\)
0.0881751 + 0.996105i \(0.471896\pi\)
\(444\) 0 0
\(445\) 7.49768 0.355424
\(446\) 1.54843 0.0733203
\(447\) 0 0
\(448\) 0 0
\(449\) 5.17570 0.244256 0.122128 0.992514i \(-0.461028\pi\)
0.122128 + 0.992514i \(0.461028\pi\)
\(450\) 0 0
\(451\) −9.57849 −0.451033
\(452\) 5.43721 0.255745
\(453\) 0 0
\(454\) −11.5972 −0.544284
\(455\) 0 0
\(456\) 0 0
\(457\) −34.1299 −1.59653 −0.798266 0.602305i \(-0.794249\pi\)
−0.798266 + 0.602305i \(0.794249\pi\)
\(458\) −16.3614 −0.764518
\(459\) 0 0
\(460\) 1.31360 0.0612467
\(461\) 11.4008 0.530989 0.265494 0.964112i \(-0.414465\pi\)
0.265494 + 0.964112i \(0.414465\pi\)
\(462\) 0 0
\(463\) −30.0124 −1.39479 −0.697397 0.716685i \(-0.745659\pi\)
−0.697397 + 0.716685i \(0.745659\pi\)
\(464\) −22.4690 −1.04310
\(465\) 0 0
\(466\) −35.0618 −1.62421
\(467\) −34.1663 −1.58103 −0.790515 0.612443i \(-0.790187\pi\)
−0.790515 + 0.612443i \(0.790187\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 23.4072 1.07969
\(471\) 0 0
\(472\) 33.6133 1.54718
\(473\) 6.36773 0.292789
\(474\) 0 0
\(475\) −0.486625 −0.0223279
\(476\) 0 0
\(477\) 0 0
\(478\) 29.2231 1.33664
\(479\) 12.8761 0.588324 0.294162 0.955756i \(-0.404959\pi\)
0.294162 + 0.955756i \(0.404959\pi\)
\(480\) 0 0
\(481\) −2.27890 −0.103909
\(482\) −12.0174 −0.547377
\(483\) 0 0
\(484\) 5.54210 0.251913
\(485\) 0.217091 0.00985758
\(486\) 0 0
\(487\) −21.9399 −0.994191 −0.497096 0.867696i \(-0.665600\pi\)
−0.497096 + 0.867696i \(0.665600\pi\)
\(488\) 40.0949 1.81501
\(489\) 0 0
\(490\) 0 0
\(491\) −4.11256 −0.185597 −0.0927986 0.995685i \(-0.529581\pi\)
−0.0927986 + 0.995685i \(0.529581\pi\)
\(492\) 0 0
\(493\) 14.8130 0.667142
\(494\) −5.23448 −0.235510
\(495\) 0 0
\(496\) −15.0869 −0.677420
\(497\) 0 0
\(498\) 0 0
\(499\) 19.0331 0.852038 0.426019 0.904714i \(-0.359916\pi\)
0.426019 + 0.904714i \(0.359916\pi\)
\(500\) −6.03639 −0.269956
\(501\) 0 0
\(502\) −22.3978 −0.999662
\(503\) 35.8698 1.59935 0.799677 0.600430i \(-0.205004\pi\)
0.799677 + 0.600430i \(0.205004\pi\)
\(504\) 0 0
\(505\) 15.2582 0.678981
\(506\) 1.06314 0.0472624
\(507\) 0 0
\(508\) −8.78291 −0.389679
\(509\) −17.5465 −0.777733 −0.388867 0.921294i \(-0.627133\pi\)
−0.388867 + 0.921294i \(0.627133\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −24.0000 −1.06066
\(513\) 0 0
\(514\) −20.1914 −0.890604
\(515\) 34.9793 1.54137
\(516\) 0 0
\(517\) −6.90180 −0.303541
\(518\) 0 0
\(519\) 0 0
\(520\) −6.78291 −0.297450
\(521\) 8.84302 0.387420 0.193710 0.981059i \(-0.437948\pi\)
0.193710 + 0.981059i \(0.437948\pi\)
\(522\) 0 0
\(523\) 21.9112 0.958108 0.479054 0.877785i \(-0.340980\pi\)
0.479054 + 0.877785i \(0.340980\pi\)
\(524\) 6.58651 0.287733
\(525\) 0 0
\(526\) 16.3076 0.711046
\(527\) 9.94622 0.433264
\(528\) 0 0
\(529\) −21.7622 −0.946183
\(530\) 21.2946 0.924978
\(531\) 0 0
\(532\) 0 0
\(533\) 12.1363 0.525681
\(534\) 0 0
\(535\) −4.47396 −0.193426
\(536\) −20.1202 −0.869061
\(537\) 0 0
\(538\) 36.9980 1.59510
\(539\) 0 0
\(540\) 0 0
\(541\) 22.8654 0.983061 0.491530 0.870860i \(-0.336438\pi\)
0.491530 + 0.870860i \(0.336438\pi\)
\(542\) 5.11559 0.219733
\(543\) 0 0
\(544\) 5.11559 0.219329
\(545\) 38.3501 1.64274
\(546\) 0 0
\(547\) −35.2676 −1.50793 −0.753966 0.656913i \(-0.771862\pi\)
−0.753966 + 0.656913i \(0.771862\pi\)
\(548\) −1.78755 −0.0763605
\(549\) 0 0
\(550\) −0.107559 −0.00458632
\(551\) 36.7034 1.56362
\(552\) 0 0
\(553\) 0 0
\(554\) −3.63227 −0.154320
\(555\) 0 0
\(556\) 3.29326 0.139665
\(557\) −39.8698 −1.68934 −0.844668 0.535290i \(-0.820202\pi\)
−0.844668 + 0.535290i \(0.820202\pi\)
\(558\) 0 0
\(559\) −8.06814 −0.341246
\(560\) 0 0
\(561\) 0 0
\(562\) 7.60221 0.320680
\(563\) −25.5972 −1.07879 −0.539397 0.842052i \(-0.681348\pi\)
−0.539397 + 0.842052i \(0.681348\pi\)
\(564\) 0 0
\(565\) 22.5070 0.946878
\(566\) 28.4977 1.19785
\(567\) 0 0
\(568\) 17.9713 0.754058
\(569\) 25.0949 1.05203 0.526016 0.850475i \(-0.323685\pi\)
0.526016 + 0.850475i \(0.323685\pi\)
\(570\) 0 0
\(571\) −31.2488 −1.30772 −0.653862 0.756614i \(-0.726852\pi\)
−0.653862 + 0.756614i \(0.726852\pi\)
\(572\) 0.421512 0.0176243
\(573\) 0 0
\(574\) 0 0
\(575\) −0.125228 −0.00522236
\(576\) 0 0
\(577\) −12.8717 −0.535858 −0.267929 0.963439i \(-0.586339\pi\)
−0.267929 + 0.963439i \(0.586339\pi\)
\(578\) −16.8968 −0.702814
\(579\) 0 0
\(580\) 10.0237 0.416212
\(581\) 0 0
\(582\) 0 0
\(583\) −6.27890 −0.260045
\(584\) −24.5738 −1.01687
\(585\) 0 0
\(586\) −26.1724 −1.08117
\(587\) −9.96692 −0.411379 −0.205689 0.978617i \(-0.565944\pi\)
−0.205689 + 0.978617i \(0.565944\pi\)
\(588\) 0 0
\(589\) 24.6447 1.01547
\(590\) 29.3246 1.20728
\(591\) 0 0
\(592\) −6.03139 −0.247889
\(593\) 20.5909 0.845566 0.422783 0.906231i \(-0.361053\pi\)
0.422783 + 0.906231i \(0.361053\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0601 0.412078
\(597\) 0 0
\(598\) −1.34704 −0.0550844
\(599\) 12.5371 0.512252 0.256126 0.966643i \(-0.417554\pi\)
0.256126 + 0.966643i \(0.417554\pi\)
\(600\) 0 0
\(601\) 1.82233 0.0743343 0.0371672 0.999309i \(-0.488167\pi\)
0.0371672 + 0.999309i \(0.488167\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.7829 −0.438750
\(605\) 22.9412 0.932693
\(606\) 0 0
\(607\) −30.9780 −1.25736 −0.628678 0.777665i \(-0.716404\pi\)
−0.628678 + 0.777665i \(0.716404\pi\)
\(608\) 12.6754 0.514053
\(609\) 0 0
\(610\) 34.9793 1.41627
\(611\) 8.74483 0.353778
\(612\) 0 0
\(613\) 12.4502 0.502860 0.251430 0.967875i \(-0.419099\pi\)
0.251430 + 0.967875i \(0.419099\pi\)
\(614\) −24.6927 −0.996518
\(615\) 0 0
\(616\) 0 0
\(617\) −31.7809 −1.27945 −0.639726 0.768603i \(-0.720952\pi\)
−0.639726 + 0.768603i \(0.720952\pi\)
\(618\) 0 0
\(619\) −4.22512 −0.169822 −0.0849109 0.996389i \(-0.527061\pi\)
−0.0849109 + 0.996389i \(0.527061\pi\)
\(620\) 6.73047 0.270302
\(621\) 0 0
\(622\) −15.6784 −0.628646
\(623\) 0 0
\(624\) 0 0
\(625\) −24.4245 −0.976982
\(626\) −40.2388 −1.60827
\(627\) 0 0
\(628\) −7.41349 −0.295830
\(629\) 3.97628 0.158545
\(630\) 0 0
\(631\) 7.31198 0.291085 0.145543 0.989352i \(-0.453507\pi\)
0.145543 + 0.989352i \(0.453507\pi\)
\(632\) 21.2044 0.843467
\(633\) 0 0
\(634\) 5.89849 0.234259
\(635\) −36.3564 −1.44276
\(636\) 0 0
\(637\) 0 0
\(638\) 8.11256 0.321179
\(639\) 0 0
\(640\) −10.7067 −0.423221
\(641\) 23.0474 0.910319 0.455160 0.890410i \(-0.349582\pi\)
0.455160 + 0.890410i \(0.349582\pi\)
\(642\) 0 0
\(643\) 48.9379 1.92992 0.964961 0.262392i \(-0.0845113\pi\)
0.964961 + 0.262392i \(0.0845113\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 9.13325 0.359343
\(647\) −33.4309 −1.31430 −0.657152 0.753758i \(-0.728239\pi\)
−0.657152 + 0.753758i \(0.728239\pi\)
\(648\) 0 0
\(649\) −8.64663 −0.339410
\(650\) 0.136281 0.00534537
\(651\) 0 0
\(652\) −6.13628 −0.240315
\(653\) 10.4629 0.409445 0.204723 0.978820i \(-0.434371\pi\)
0.204723 + 0.978820i \(0.434371\pi\)
\(654\) 0 0
\(655\) 27.2645 1.06531
\(656\) 32.1202 1.25408
\(657\) 0 0
\(658\) 0 0
\(659\) 8.73849 0.340403 0.170202 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(660\) 0 0
\(661\) 4.61354 0.179446 0.0897230 0.995967i \(-0.471402\pi\)
0.0897230 + 0.995967i \(0.471402\pi\)
\(662\) 8.32001 0.323366
\(663\) 0 0
\(664\) 9.64199 0.374182
\(665\) 0 0
\(666\) 0 0
\(667\) 9.44523 0.365721
\(668\) −6.92686 −0.268008
\(669\) 0 0
\(670\) −17.5531 −0.678137
\(671\) −10.3140 −0.398166
\(672\) 0 0
\(673\) 9.83802 0.379228 0.189614 0.981859i \(-0.439276\pi\)
0.189614 + 0.981859i \(0.439276\pi\)
\(674\) 13.4045 0.516321
\(675\) 0 0
\(676\) −0.534070 −0.0205412
\(677\) 4.69541 0.180459 0.0902296 0.995921i \(-0.471240\pi\)
0.0902296 + 0.995921i \(0.471240\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 11.8350 0.453851
\(681\) 0 0
\(682\) 5.44721 0.208584
\(683\) −11.0207 −0.421695 −0.210848 0.977519i \(-0.567622\pi\)
−0.210848 + 0.977519i \(0.567622\pi\)
\(684\) 0 0
\(685\) −7.39948 −0.282720
\(686\) 0 0
\(687\) 0 0
\(688\) −21.3534 −0.814090
\(689\) 7.95558 0.303084
\(690\) 0 0
\(691\) 38.7635 1.47463 0.737317 0.675546i \(-0.236092\pi\)
0.737317 + 0.675546i \(0.236092\pi\)
\(692\) 1.32465 0.0503556
\(693\) 0 0
\(694\) −6.81163 −0.258566
\(695\) 13.6323 0.517101
\(696\) 0 0
\(697\) −21.1757 −0.802087
\(698\) 22.3727 0.846820
\(699\) 0 0
\(700\) 0 0
\(701\) 32.0681 1.21120 0.605598 0.795770i \(-0.292934\pi\)
0.605598 + 0.795770i \(0.292934\pi\)
\(702\) 0 0
\(703\) 9.85238 0.371590
\(704\) 6.97930 0.263042
\(705\) 0 0
\(706\) −5.65465 −0.212816
\(707\) 0 0
\(708\) 0 0
\(709\) 38.3451 1.44008 0.720040 0.693933i \(-0.244124\pi\)
0.720040 + 0.693933i \(0.244124\pi\)
\(710\) 15.6784 0.588399
\(711\) 0 0
\(712\) 10.4055 0.389961
\(713\) 6.34204 0.237511
\(714\) 0 0
\(715\) 1.74483 0.0652528
\(716\) −4.35640 −0.162806
\(717\) 0 0
\(718\) 14.1914 0.529618
\(719\) −8.72413 −0.325355 −0.162678 0.986679i \(-0.552013\pi\)
−0.162678 + 0.986679i \(0.552013\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.374067 −0.0139213
\(723\) 0 0
\(724\) −1.92722 −0.0716244
\(725\) −0.955582 −0.0354894
\(726\) 0 0
\(727\) −26.6754 −0.989334 −0.494667 0.869083i \(-0.664710\pi\)
−0.494667 + 0.869083i \(0.664710\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −21.4385 −0.793476
\(731\) 14.0775 0.520675
\(732\) 0 0
\(733\) −26.7211 −0.986966 −0.493483 0.869755i \(-0.664277\pi\)
−0.493483 + 0.869755i \(0.664277\pi\)
\(734\) 24.1787 0.892453
\(735\) 0 0
\(736\) 3.26187 0.120234
\(737\) 5.17570 0.190649
\(738\) 0 0
\(739\) 36.0538 1.32626 0.663130 0.748504i \(-0.269228\pi\)
0.663130 + 0.748504i \(0.269228\pi\)
\(740\) 2.69069 0.0989117
\(741\) 0 0
\(742\) 0 0
\(743\) 2.96058 0.108613 0.0543066 0.998524i \(-0.482705\pi\)
0.0543066 + 0.998524i \(0.482705\pi\)
\(744\) 0 0
\(745\) 41.6433 1.52569
\(746\) −5.35942 −0.196222
\(747\) 0 0
\(748\) −0.735465 −0.0268913
\(749\) 0 0
\(750\) 0 0
\(751\) 43.1550 1.57475 0.787374 0.616475i \(-0.211440\pi\)
0.787374 + 0.616475i \(0.211440\pi\)
\(752\) 23.1443 0.843986
\(753\) 0 0
\(754\) −10.2789 −0.374335
\(755\) −44.6353 −1.62444
\(756\) 0 0
\(757\) 18.7335 0.680880 0.340440 0.940266i \(-0.389424\pi\)
0.340440 + 0.940266i \(0.389424\pi\)
\(758\) −39.4345 −1.43233
\(759\) 0 0
\(760\) 29.3246 1.06372
\(761\) −4.22948 −0.153318 −0.0766592 0.997057i \(-0.524425\pi\)
−0.0766592 + 0.997057i \(0.524425\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.180342 0.00652456
\(765\) 0 0
\(766\) −17.1884 −0.621041
\(767\) 10.9556 0.395583
\(768\) 0 0
\(769\) 21.1299 0.761965 0.380983 0.924582i \(-0.375586\pi\)
0.380983 + 0.924582i \(0.375586\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.49768 −0.161875
\(773\) 33.0742 1.18960 0.594798 0.803875i \(-0.297232\pi\)
0.594798 + 0.803875i \(0.297232\pi\)
\(774\) 0 0
\(775\) −0.641629 −0.0230480
\(776\) 0.301284 0.0108154
\(777\) 0 0
\(778\) −20.9666 −0.751690
\(779\) −52.4690 −1.87990
\(780\) 0 0
\(781\) −4.62291 −0.165421
\(782\) 2.35034 0.0840482
\(783\) 0 0
\(784\) 0 0
\(785\) −30.6877 −1.09529
\(786\) 0 0
\(787\) −6.23948 −0.222413 −0.111207 0.993797i \(-0.535472\pi\)
−0.111207 + 0.993797i \(0.535472\pi\)
\(788\) −4.54512 −0.161913
\(789\) 0 0
\(790\) 18.4990 0.658165
\(791\) 0 0
\(792\) 0 0
\(793\) 13.0681 0.464063
\(794\) 19.0381 0.675636
\(795\) 0 0
\(796\) 7.80628 0.276686
\(797\) −32.1837 −1.14001 −0.570003 0.821643i \(-0.693058\pi\)
−0.570003 + 0.821643i \(0.693058\pi\)
\(798\) 0 0
\(799\) −15.2582 −0.539796
\(800\) −0.330006 −0.0116675
\(801\) 0 0
\(802\) −4.33268 −0.152992
\(803\) 6.32134 0.223075
\(804\) 0 0
\(805\) 0 0
\(806\) −6.90180 −0.243106
\(807\) 0 0
\(808\) 21.1757 0.744958
\(809\) −47.8016 −1.68062 −0.840308 0.542109i \(-0.817626\pi\)
−0.840308 + 0.542109i \(0.817626\pi\)
\(810\) 0 0
\(811\) 37.4957 1.31665 0.658326 0.752733i \(-0.271265\pi\)
0.658326 + 0.752733i \(0.271265\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.17767 0.0763274
\(815\) −25.4008 −0.889752
\(816\) 0 0
\(817\) 34.8811 1.22034
\(818\) −37.9573 −1.32714
\(819\) 0 0
\(820\) −14.3293 −0.500401
\(821\) −39.6447 −1.38361 −0.691804 0.722085i \(-0.743184\pi\)
−0.691804 + 0.722085i \(0.743184\pi\)
\(822\) 0 0
\(823\) −17.4659 −0.608824 −0.304412 0.952540i \(-0.598460\pi\)
−0.304412 + 0.952540i \(0.598460\pi\)
\(824\) 48.5451 1.69115
\(825\) 0 0
\(826\) 0 0
\(827\) 1.53104 0.0532396 0.0266198 0.999646i \(-0.491526\pi\)
0.0266198 + 0.999646i \(0.491526\pi\)
\(828\) 0 0
\(829\) −37.2218 −1.29277 −0.646383 0.763013i \(-0.723719\pi\)
−0.646383 + 0.763013i \(0.723719\pi\)
\(830\) 8.41179 0.291978
\(831\) 0 0
\(832\) −8.84302 −0.306577
\(833\) 0 0
\(834\) 0 0
\(835\) −28.6734 −0.992283
\(836\) −1.82233 −0.0630265
\(837\) 0 0
\(838\) 34.8354 1.20337
\(839\) 7.37209 0.254513 0.127256 0.991870i \(-0.459383\pi\)
0.127256 + 0.991870i \(0.459383\pi\)
\(840\) 0 0
\(841\) 43.0742 1.48532
\(842\) −0.230116 −0.00793034
\(843\) 0 0
\(844\) 12.7465 0.438753
\(845\) −2.21076 −0.0760523
\(846\) 0 0
\(847\) 0 0
\(848\) 21.0555 0.723048
\(849\) 0 0
\(850\) −0.237786 −0.00815600
\(851\) 2.53541 0.0869126
\(852\) 0 0
\(853\) 4.57215 0.156548 0.0782738 0.996932i \(-0.475059\pi\)
0.0782738 + 0.996932i \(0.475059\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.20906 −0.212221
\(857\) 52.0348 1.77747 0.888737 0.458417i \(-0.151584\pi\)
0.888737 + 0.458417i \(0.151584\pi\)
\(858\) 0 0
\(859\) 42.8905 1.46340 0.731702 0.681625i \(-0.238726\pi\)
0.731702 + 0.681625i \(0.238726\pi\)
\(860\) 9.52604 0.324835
\(861\) 0 0
\(862\) −31.0842 −1.05873
\(863\) 7.82233 0.266275 0.133138 0.991098i \(-0.457495\pi\)
0.133138 + 0.991098i \(0.457495\pi\)
\(864\) 0 0
\(865\) 5.48332 0.186438
\(866\) 2.20639 0.0749763
\(867\) 0 0
\(868\) 0 0
\(869\) −5.45460 −0.185034
\(870\) 0 0
\(871\) −6.55779 −0.222202
\(872\) 53.2231 1.80236
\(873\) 0 0
\(874\) 5.82366 0.196988
\(875\) 0 0
\(876\) 0 0
\(877\) −34.7191 −1.17238 −0.586191 0.810173i \(-0.699373\pi\)
−0.586191 + 0.810173i \(0.699373\pi\)
\(878\) 0.890468 0.0300518
\(879\) 0 0
\(880\) 4.61791 0.155670
\(881\) −11.4422 −0.385498 −0.192749 0.981248i \(-0.561740\pi\)
−0.192749 + 0.981248i \(0.561740\pi\)
\(882\) 0 0
\(883\) −31.0217 −1.04396 −0.521982 0.852956i \(-0.674807\pi\)
−0.521982 + 0.852956i \(0.674807\pi\)
\(884\) 0.931860 0.0313418
\(885\) 0 0
\(886\) 4.49401 0.150979
\(887\) −12.8304 −0.430801 −0.215401 0.976526i \(-0.569106\pi\)
−0.215401 + 0.976526i \(0.569106\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 9.07786 0.304291
\(891\) 0 0
\(892\) −0.683020 −0.0228692
\(893\) −37.8066 −1.26515
\(894\) 0 0
\(895\) −18.0331 −0.602780
\(896\) 0 0
\(897\) 0 0
\(898\) 6.26651 0.209116
\(899\) 48.3945 1.61405
\(900\) 0 0
\(901\) −13.8811 −0.462447
\(902\) −11.5972 −0.386145
\(903\) 0 0
\(904\) 31.2358 1.03889
\(905\) −7.97761 −0.265185
\(906\) 0 0
\(907\) 10.1620 0.337423 0.168711 0.985665i \(-0.446039\pi\)
0.168711 + 0.985665i \(0.446039\pi\)
\(908\) 5.11559 0.169767
\(909\) 0 0
\(910\) 0 0
\(911\) −34.4008 −1.13975 −0.569875 0.821731i \(-0.693008\pi\)
−0.569875 + 0.821731i \(0.693008\pi\)
\(912\) 0 0
\(913\) −2.48029 −0.0820856
\(914\) −41.3230 −1.36684
\(915\) 0 0
\(916\) 7.21709 0.238459
\(917\) 0 0
\(918\) 0 0
\(919\) 8.15500 0.269009 0.134504 0.990913i \(-0.457056\pi\)
0.134504 + 0.990913i \(0.457056\pi\)
\(920\) 7.54638 0.248797
\(921\) 0 0
\(922\) 13.8036 0.454598
\(923\) 5.85738 0.192798
\(924\) 0 0
\(925\) −0.256509 −0.00843396
\(926\) −36.3377 −1.19413
\(927\) 0 0
\(928\) 24.8905 0.817070
\(929\) −4.80994 −0.157809 −0.0789045 0.996882i \(-0.525142\pi\)
−0.0789045 + 0.996882i \(0.525142\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 15.4659 0.506603
\(933\) 0 0
\(934\) −41.3671 −1.35357
\(935\) −3.04442 −0.0995631
\(936\) 0 0
\(937\) −47.1931 −1.54173 −0.770865 0.636998i \(-0.780176\pi\)
−0.770865 + 0.636998i \(0.780176\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −10.3250 −0.336765
\(941\) −27.5702 −0.898762 −0.449381 0.893340i \(-0.648355\pi\)
−0.449381 + 0.893340i \(0.648355\pi\)
\(942\) 0 0
\(943\) −13.5023 −0.439696
\(944\) 28.9954 0.943718
\(945\) 0 0
\(946\) 7.70977 0.250666
\(947\) 27.8698 0.905646 0.452823 0.891600i \(-0.350417\pi\)
0.452823 + 0.891600i \(0.350417\pi\)
\(948\) 0 0
\(949\) −8.00936 −0.259995
\(950\) −0.589185 −0.0191157
\(951\) 0 0
\(952\) 0 0
\(953\) 27.1076 0.878100 0.439050 0.898463i \(-0.355315\pi\)
0.439050 + 0.898463i \(0.355315\pi\)
\(954\) 0 0
\(955\) 0.746518 0.0241567
\(956\) −12.8905 −0.416908
\(957\) 0 0
\(958\) 15.5898 0.503684
\(959\) 0 0
\(960\) 0 0
\(961\) 1.49465 0.0482146
\(962\) −2.75919 −0.0889598
\(963\) 0 0
\(964\) 5.30093 0.170731
\(965\) −18.6179 −0.599332
\(966\) 0 0
\(967\) −4.45657 −0.143314 −0.0716568 0.997429i \(-0.522829\pi\)
−0.0716568 + 0.997429i \(0.522829\pi\)
\(968\) 31.8384 1.02332
\(969\) 0 0
\(970\) 0.262844 0.00843940
\(971\) 18.9192 0.607146 0.303573 0.952808i \(-0.401820\pi\)
0.303573 + 0.952808i \(0.401820\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −26.5638 −0.851161
\(975\) 0 0
\(976\) 34.5865 1.10709
\(977\) −17.4359 −0.557823 −0.278911 0.960317i \(-0.589974\pi\)
−0.278911 + 0.960317i \(0.589974\pi\)
\(978\) 0 0
\(979\) −2.67669 −0.0855472
\(980\) 0 0
\(981\) 0 0
\(982\) −4.97930 −0.158896
\(983\) 15.3815 0.490592 0.245296 0.969448i \(-0.421115\pi\)
0.245296 + 0.969448i \(0.421115\pi\)
\(984\) 0 0
\(985\) −18.8143 −0.599473
\(986\) 17.9349 0.571163
\(987\) 0 0
\(988\) 2.30895 0.0734576
\(989\) 8.97628 0.285429
\(990\) 0 0
\(991\) 2.42651 0.0770807 0.0385403 0.999257i \(-0.487729\pi\)
0.0385403 + 0.999257i \(0.487729\pi\)
\(992\) 16.7128 0.530632
\(993\) 0 0
\(994\) 0 0
\(995\) 32.3137 1.02441
\(996\) 0 0
\(997\) −15.0869 −0.477806 −0.238903 0.971043i \(-0.576788\pi\)
−0.238903 + 0.971043i \(0.576788\pi\)
\(998\) 23.0444 0.729458
\(999\) 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 5733.2.a.bd.1.2 3
3.2 odd 2 637.2.a.i.1.2 yes 3
7.6 odd 2 5733.2.a.be.1.2 3
21.2 odd 6 637.2.e.k.508.2 6
21.5 even 6 637.2.e.l.508.2 6
21.11 odd 6 637.2.e.k.79.2 6
21.17 even 6 637.2.e.l.79.2 6
21.20 even 2 637.2.a.h.1.2 3
39.38 odd 2 8281.2.a.bk.1.2 3
273.272 even 2 8281.2.a.bh.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.2 3 21.20 even 2
637.2.a.i.1.2 yes 3 3.2 odd 2
637.2.e.k.79.2 6 21.11 odd 6
637.2.e.k.508.2 6 21.2 odd 6
637.2.e.l.79.2 6 21.17 even 6
637.2.e.l.508.2 6 21.5 even 6
5733.2.a.bd.1.2 3 1.1 even 1 trivial
5733.2.a.be.1.2 3 7.6 odd 2
8281.2.a.bh.1.2 3 273.272 even 2
8281.2.a.bk.1.2 3 39.38 odd 2