Properties

Label 6561.2.a.d.1.3
Level $6561$
Weight $2$
Character 6561.1
Self dual yes
Analytic conductor $52.390$
Analytic rank $0$
Dimension $72$
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] = [6561,2,Mod(1,6561)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6561, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6561.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6561 = 3^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6561.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: \(52.3898487662\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: no (minimal twist has level 81)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 6561.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.54050 q^{2} +4.45416 q^{4} +3.18683 q^{5} -1.66066 q^{7} -6.23481 q^{8} -8.09616 q^{10} -1.96180 q^{11} -3.88044 q^{13} +4.21891 q^{14} +6.93125 q^{16} -4.35949 q^{17} +1.41661 q^{19} +14.1947 q^{20} +4.98397 q^{22} -2.26830 q^{23} +5.15590 q^{25} +9.85829 q^{26} -7.39684 q^{28} +9.44157 q^{29} -4.17854 q^{31} -5.13924 q^{32} +11.0753 q^{34} -5.29223 q^{35} -6.84762 q^{37} -3.59890 q^{38} -19.8693 q^{40} +3.56777 q^{41} +7.45581 q^{43} -8.73818 q^{44} +5.76263 q^{46} +6.93565 q^{47} -4.24222 q^{49} -13.0986 q^{50} -17.2841 q^{52} +2.70633 q^{53} -6.25193 q^{55} +10.3539 q^{56} -23.9864 q^{58} +3.53534 q^{59} -2.78995 q^{61} +10.6156 q^{62} -0.806236 q^{64} -12.3663 q^{65} -1.84415 q^{67} -19.4179 q^{68} +13.4449 q^{70} -2.29664 q^{71} -11.4402 q^{73} +17.3964 q^{74} +6.30981 q^{76} +3.25788 q^{77} +6.96987 q^{79} +22.0887 q^{80} -9.06394 q^{82} +0.107298 q^{83} -13.8930 q^{85} -18.9415 q^{86} +12.2315 q^{88} -5.28583 q^{89} +6.44408 q^{91} -10.1034 q^{92} -17.6201 q^{94} +4.51449 q^{95} -4.35627 q^{97} +10.7774 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 9 q^{2} + 63 q^{4} + 18 q^{5} + 27 q^{8} + 36 q^{11} + 36 q^{14} + 45 q^{16} + 36 q^{17} + 54 q^{20} + 54 q^{23} + 36 q^{25} + 45 q^{26} + 9 q^{28} + 54 q^{29} + 63 q^{32} + 72 q^{35} + 54 q^{38}+ \cdots + 81 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.54050 −1.79641 −0.898204 0.439579i \(-0.855128\pi\)
−0.898204 + 0.439579i \(0.855128\pi\)
\(3\) 0 0
\(4\) 4.45416 2.22708
\(5\) 3.18683 1.42519 0.712597 0.701573i \(-0.247519\pi\)
0.712597 + 0.701573i \(0.247519\pi\)
\(6\) 0 0
\(7\) −1.66066 −0.627669 −0.313835 0.949478i \(-0.601614\pi\)
−0.313835 + 0.949478i \(0.601614\pi\)
\(8\) −6.23481 −2.20434
\(9\) 0 0
\(10\) −8.09616 −2.56023
\(11\) −1.96180 −0.591505 −0.295753 0.955265i \(-0.595570\pi\)
−0.295753 + 0.955265i \(0.595570\pi\)
\(12\) 0 0
\(13\) −3.88044 −1.07624 −0.538121 0.842868i \(-0.680866\pi\)
−0.538121 + 0.842868i \(0.680866\pi\)
\(14\) 4.21891 1.12755
\(15\) 0 0
\(16\) 6.93125 1.73281
\(17\) −4.35949 −1.05733 −0.528666 0.848830i \(-0.677308\pi\)
−0.528666 + 0.848830i \(0.677308\pi\)
\(18\) 0 0
\(19\) 1.41661 0.324992 0.162496 0.986709i \(-0.448045\pi\)
0.162496 + 0.986709i \(0.448045\pi\)
\(20\) 14.1947 3.17403
\(21\) 0 0
\(22\) 4.98397 1.06258
\(23\) −2.26830 −0.472973 −0.236487 0.971635i \(-0.575996\pi\)
−0.236487 + 0.971635i \(0.575996\pi\)
\(24\) 0 0
\(25\) 5.15590 1.03118
\(26\) 9.85829 1.93337
\(27\) 0 0
\(28\) −7.39684 −1.39787
\(29\) 9.44157 1.75326 0.876628 0.481169i \(-0.159788\pi\)
0.876628 + 0.481169i \(0.159788\pi\)
\(30\) 0 0
\(31\) −4.17854 −0.750487 −0.375244 0.926926i \(-0.622441\pi\)
−0.375244 + 0.926926i \(0.622441\pi\)
\(32\) −5.13924 −0.908498
\(33\) 0 0
\(34\) 11.0753 1.89940
\(35\) −5.29223 −0.894551
\(36\) 0 0
\(37\) −6.84762 −1.12574 −0.562871 0.826545i \(-0.690303\pi\)
−0.562871 + 0.826545i \(0.690303\pi\)
\(38\) −3.59890 −0.583819
\(39\) 0 0
\(40\) −19.8693 −3.14161
\(41\) 3.56777 0.557192 0.278596 0.960408i \(-0.410131\pi\)
0.278596 + 0.960408i \(0.410131\pi\)
\(42\) 0 0
\(43\) 7.45581 1.13700 0.568500 0.822683i \(-0.307524\pi\)
0.568500 + 0.822683i \(0.307524\pi\)
\(44\) −8.73818 −1.31733
\(45\) 0 0
\(46\) 5.76263 0.849653
\(47\) 6.93565 1.01167 0.505835 0.862631i \(-0.331185\pi\)
0.505835 + 0.862631i \(0.331185\pi\)
\(48\) 0 0
\(49\) −4.24222 −0.606032
\(50\) −13.0986 −1.85242
\(51\) 0 0
\(52\) −17.2841 −2.39688
\(53\) 2.70633 0.371743 0.185872 0.982574i \(-0.440489\pi\)
0.185872 + 0.982574i \(0.440489\pi\)
\(54\) 0 0
\(55\) −6.25193 −0.843010
\(56\) 10.3539 1.38360
\(57\) 0 0
\(58\) −23.9864 −3.14956
\(59\) 3.53534 0.460262 0.230131 0.973160i \(-0.426085\pi\)
0.230131 + 0.973160i \(0.426085\pi\)
\(60\) 0 0
\(61\) −2.78995 −0.357217 −0.178609 0.983920i \(-0.557160\pi\)
−0.178609 + 0.983920i \(0.557160\pi\)
\(62\) 10.6156 1.34818
\(63\) 0 0
\(64\) −0.806236 −0.100779
\(65\) −12.3663 −1.53385
\(66\) 0 0
\(67\) −1.84415 −0.225299 −0.112650 0.993635i \(-0.535934\pi\)
−0.112650 + 0.993635i \(0.535934\pi\)
\(68\) −19.4179 −2.35477
\(69\) 0 0
\(70\) 13.4449 1.60698
\(71\) −2.29664 −0.272561 −0.136280 0.990670i \(-0.543515\pi\)
−0.136280 + 0.990670i \(0.543515\pi\)
\(72\) 0 0
\(73\) −11.4402 −1.33898 −0.669489 0.742822i \(-0.733487\pi\)
−0.669489 + 0.742822i \(0.733487\pi\)
\(74\) 17.3964 2.02229
\(75\) 0 0
\(76\) 6.30981 0.723785
\(77\) 3.25788 0.371270
\(78\) 0 0
\(79\) 6.96987 0.784172 0.392086 0.919929i \(-0.371754\pi\)
0.392086 + 0.919929i \(0.371754\pi\)
\(80\) 22.0887 2.46959
\(81\) 0 0
\(82\) −9.06394 −1.00094
\(83\) 0.107298 0.0117775 0.00588873 0.999983i \(-0.498126\pi\)
0.00588873 + 0.999983i \(0.498126\pi\)
\(84\) 0 0
\(85\) −13.8930 −1.50690
\(86\) −18.9415 −2.04252
\(87\) 0 0
\(88\) 12.2315 1.30388
\(89\) −5.28583 −0.560297 −0.280149 0.959957i \(-0.590384\pi\)
−0.280149 + 0.959957i \(0.590384\pi\)
\(90\) 0 0
\(91\) 6.44408 0.675524
\(92\) −10.1034 −1.05335
\(93\) 0 0
\(94\) −17.6201 −1.81737
\(95\) 4.51449 0.463177
\(96\) 0 0
\(97\) −4.35627 −0.442312 −0.221156 0.975238i \(-0.570983\pi\)
−0.221156 + 0.975238i \(0.570983\pi\)
\(98\) 10.7774 1.08868
\(99\) 0 0
\(100\) 22.9652 2.29652
\(101\) 11.2030 1.11474 0.557369 0.830265i \(-0.311811\pi\)
0.557369 + 0.830265i \(0.311811\pi\)
\(102\) 0 0
\(103\) −4.02820 −0.396910 −0.198455 0.980110i \(-0.563592\pi\)
−0.198455 + 0.980110i \(0.563592\pi\)
\(104\) 24.1939 2.37240
\(105\) 0 0
\(106\) −6.87545 −0.667803
\(107\) −16.2002 −1.56614 −0.783068 0.621936i \(-0.786346\pi\)
−0.783068 + 0.621936i \(0.786346\pi\)
\(108\) 0 0
\(109\) 12.8910 1.23473 0.617365 0.786677i \(-0.288200\pi\)
0.617365 + 0.786677i \(0.288200\pi\)
\(110\) 15.8831 1.51439
\(111\) 0 0
\(112\) −11.5104 −1.08763
\(113\) 16.1325 1.51761 0.758807 0.651315i \(-0.225782\pi\)
0.758807 + 0.651315i \(0.225782\pi\)
\(114\) 0 0
\(115\) −7.22869 −0.674079
\(116\) 42.0543 3.90464
\(117\) 0 0
\(118\) −8.98154 −0.826818
\(119\) 7.23962 0.663655
\(120\) 0 0
\(121\) −7.15134 −0.650121
\(122\) 7.08789 0.641708
\(123\) 0 0
\(124\) −18.6119 −1.67140
\(125\) 0.496823 0.0444372
\(126\) 0 0
\(127\) −3.99204 −0.354236 −0.177118 0.984190i \(-0.556677\pi\)
−0.177118 + 0.984190i \(0.556677\pi\)
\(128\) 12.3267 1.08954
\(129\) 0 0
\(130\) 31.4167 2.75543
\(131\) 9.76431 0.853111 0.426556 0.904461i \(-0.359727\pi\)
0.426556 + 0.904461i \(0.359727\pi\)
\(132\) 0 0
\(133\) −2.35250 −0.203988
\(134\) 4.68508 0.404729
\(135\) 0 0
\(136\) 27.1806 2.33072
\(137\) 7.42951 0.634745 0.317373 0.948301i \(-0.397199\pi\)
0.317373 + 0.948301i \(0.397199\pi\)
\(138\) 0 0
\(139\) 17.7537 1.50585 0.752924 0.658108i \(-0.228643\pi\)
0.752924 + 0.658108i \(0.228643\pi\)
\(140\) −23.5725 −1.99224
\(141\) 0 0
\(142\) 5.83462 0.489630
\(143\) 7.61266 0.636603
\(144\) 0 0
\(145\) 30.0887 2.49873
\(146\) 29.0640 2.40535
\(147\) 0 0
\(148\) −30.5004 −2.50712
\(149\) 14.0449 1.15061 0.575303 0.817940i \(-0.304884\pi\)
0.575303 + 0.817940i \(0.304884\pi\)
\(150\) 0 0
\(151\) 2.76578 0.225076 0.112538 0.993647i \(-0.464102\pi\)
0.112538 + 0.993647i \(0.464102\pi\)
\(152\) −8.83229 −0.716394
\(153\) 0 0
\(154\) −8.27665 −0.666952
\(155\) −13.3163 −1.06959
\(156\) 0 0
\(157\) 4.99735 0.398832 0.199416 0.979915i \(-0.436096\pi\)
0.199416 + 0.979915i \(0.436096\pi\)
\(158\) −17.7070 −1.40869
\(159\) 0 0
\(160\) −16.3779 −1.29479
\(161\) 3.76687 0.296871
\(162\) 0 0
\(163\) 12.7147 0.995894 0.497947 0.867208i \(-0.334087\pi\)
0.497947 + 0.867208i \(0.334087\pi\)
\(164\) 15.8914 1.24091
\(165\) 0 0
\(166\) −0.272591 −0.0211571
\(167\) 20.2459 1.56668 0.783338 0.621596i \(-0.213515\pi\)
0.783338 + 0.621596i \(0.213515\pi\)
\(168\) 0 0
\(169\) 2.05785 0.158296
\(170\) 35.2952 2.70701
\(171\) 0 0
\(172\) 33.2094 2.53219
\(173\) 6.79458 0.516582 0.258291 0.966067i \(-0.416841\pi\)
0.258291 + 0.966067i \(0.416841\pi\)
\(174\) 0 0
\(175\) −8.56218 −0.647240
\(176\) −13.5977 −1.02497
\(177\) 0 0
\(178\) 13.4287 1.00652
\(179\) 22.4033 1.67450 0.837252 0.546817i \(-0.184161\pi\)
0.837252 + 0.546817i \(0.184161\pi\)
\(180\) 0 0
\(181\) −22.7104 −1.68805 −0.844024 0.536306i \(-0.819819\pi\)
−0.844024 + 0.536306i \(0.819819\pi\)
\(182\) −16.3712 −1.21352
\(183\) 0 0
\(184\) 14.1424 1.04259
\(185\) −21.8222 −1.60440
\(186\) 0 0
\(187\) 8.55246 0.625418
\(188\) 30.8925 2.25307
\(189\) 0 0
\(190\) −11.4691 −0.832056
\(191\) −18.2867 −1.32318 −0.661591 0.749865i \(-0.730118\pi\)
−0.661591 + 0.749865i \(0.730118\pi\)
\(192\) 0 0
\(193\) 8.79042 0.632748 0.316374 0.948634i \(-0.397534\pi\)
0.316374 + 0.948634i \(0.397534\pi\)
\(194\) 11.0671 0.794573
\(195\) 0 0
\(196\) −18.8955 −1.34968
\(197\) −1.86015 −0.132530 −0.0662649 0.997802i \(-0.521108\pi\)
−0.0662649 + 0.997802i \(0.521108\pi\)
\(198\) 0 0
\(199\) 13.3896 0.949160 0.474580 0.880212i \(-0.342600\pi\)
0.474580 + 0.880212i \(0.342600\pi\)
\(200\) −32.1461 −2.27307
\(201\) 0 0
\(202\) −28.4612 −2.00252
\(203\) −15.6792 −1.10046
\(204\) 0 0
\(205\) 11.3699 0.794107
\(206\) 10.2336 0.713012
\(207\) 0 0
\(208\) −26.8963 −1.86492
\(209\) −2.77911 −0.192235
\(210\) 0 0
\(211\) 24.3694 1.67766 0.838831 0.544392i \(-0.183240\pi\)
0.838831 + 0.544392i \(0.183240\pi\)
\(212\) 12.0545 0.827903
\(213\) 0 0
\(214\) 41.1568 2.81342
\(215\) 23.7604 1.62045
\(216\) 0 0
\(217\) 6.93911 0.471058
\(218\) −32.7495 −2.21808
\(219\) 0 0
\(220\) −27.8471 −1.87745
\(221\) 16.9168 1.13794
\(222\) 0 0
\(223\) −28.9499 −1.93863 −0.969315 0.245821i \(-0.920943\pi\)
−0.969315 + 0.245821i \(0.920943\pi\)
\(224\) 8.53451 0.570236
\(225\) 0 0
\(226\) −40.9846 −2.72625
\(227\) −5.76469 −0.382616 −0.191308 0.981530i \(-0.561273\pi\)
−0.191308 + 0.981530i \(0.561273\pi\)
\(228\) 0 0
\(229\) −22.8731 −1.51150 −0.755748 0.654863i \(-0.772726\pi\)
−0.755748 + 0.654863i \(0.772726\pi\)
\(230\) 18.3645 1.21092
\(231\) 0 0
\(232\) −58.8665 −3.86477
\(233\) −20.4379 −1.33893 −0.669464 0.742844i \(-0.733476\pi\)
−0.669464 + 0.742844i \(0.733476\pi\)
\(234\) 0 0
\(235\) 22.1028 1.44183
\(236\) 15.7470 1.02504
\(237\) 0 0
\(238\) −18.3923 −1.19219
\(239\) 11.6116 0.751090 0.375545 0.926804i \(-0.377455\pi\)
0.375545 + 0.926804i \(0.377455\pi\)
\(240\) 0 0
\(241\) 13.2687 0.854710 0.427355 0.904084i \(-0.359446\pi\)
0.427355 + 0.904084i \(0.359446\pi\)
\(242\) 18.1680 1.16788
\(243\) 0 0
\(244\) −12.4269 −0.795552
\(245\) −13.5192 −0.863713
\(246\) 0 0
\(247\) −5.49707 −0.349770
\(248\) 26.0524 1.65433
\(249\) 0 0
\(250\) −1.26218 −0.0798274
\(251\) 7.29619 0.460532 0.230266 0.973128i \(-0.426040\pi\)
0.230266 + 0.973128i \(0.426040\pi\)
\(252\) 0 0
\(253\) 4.44996 0.279766
\(254\) 10.1418 0.636352
\(255\) 0 0
\(256\) −29.7036 −1.85648
\(257\) 13.6877 0.853818 0.426909 0.904295i \(-0.359603\pi\)
0.426909 + 0.904295i \(0.359603\pi\)
\(258\) 0 0
\(259\) 11.3715 0.706594
\(260\) −55.0816 −3.41602
\(261\) 0 0
\(262\) −24.8063 −1.53254
\(263\) 0.175949 0.0108495 0.00542473 0.999985i \(-0.498273\pi\)
0.00542473 + 0.999985i \(0.498273\pi\)
\(264\) 0 0
\(265\) 8.62463 0.529807
\(266\) 5.97654 0.366445
\(267\) 0 0
\(268\) −8.21416 −0.501760
\(269\) 8.05183 0.490929 0.245464 0.969406i \(-0.421060\pi\)
0.245464 + 0.969406i \(0.421060\pi\)
\(270\) 0 0
\(271\) 19.1488 1.16321 0.581605 0.813471i \(-0.302425\pi\)
0.581605 + 0.813471i \(0.302425\pi\)
\(272\) −30.2167 −1.83216
\(273\) 0 0
\(274\) −18.8747 −1.14026
\(275\) −10.1148 −0.609948
\(276\) 0 0
\(277\) 21.2874 1.27904 0.639518 0.768776i \(-0.279134\pi\)
0.639518 + 0.768776i \(0.279134\pi\)
\(278\) −45.1033 −2.70512
\(279\) 0 0
\(280\) 32.9961 1.97189
\(281\) −10.0072 −0.596977 −0.298489 0.954413i \(-0.596483\pi\)
−0.298489 + 0.954413i \(0.596483\pi\)
\(282\) 0 0
\(283\) 27.5972 1.64048 0.820241 0.572019i \(-0.193840\pi\)
0.820241 + 0.572019i \(0.193840\pi\)
\(284\) −10.2296 −0.607015
\(285\) 0 0
\(286\) −19.3400 −1.14360
\(287\) −5.92484 −0.349732
\(288\) 0 0
\(289\) 2.00517 0.117951
\(290\) −76.4405 −4.48874
\(291\) 0 0
\(292\) −50.9567 −2.98202
\(293\) 7.33127 0.428298 0.214149 0.976801i \(-0.431302\pi\)
0.214149 + 0.976801i \(0.431302\pi\)
\(294\) 0 0
\(295\) 11.2665 0.655963
\(296\) 42.6937 2.48152
\(297\) 0 0
\(298\) −35.6812 −2.06696
\(299\) 8.80201 0.509034
\(300\) 0 0
\(301\) −12.3815 −0.713660
\(302\) −7.02648 −0.404328
\(303\) 0 0
\(304\) 9.81887 0.563151
\(305\) −8.89112 −0.509104
\(306\) 0 0
\(307\) 5.82836 0.332642 0.166321 0.986072i \(-0.446811\pi\)
0.166321 + 0.986072i \(0.446811\pi\)
\(308\) 14.5111 0.826848
\(309\) 0 0
\(310\) 33.8301 1.92142
\(311\) −9.12122 −0.517217 −0.258608 0.965982i \(-0.583264\pi\)
−0.258608 + 0.965982i \(0.583264\pi\)
\(312\) 0 0
\(313\) 6.39310 0.361359 0.180680 0.983542i \(-0.442170\pi\)
0.180680 + 0.983542i \(0.442170\pi\)
\(314\) −12.6958 −0.716465
\(315\) 0 0
\(316\) 31.0450 1.74642
\(317\) −24.7081 −1.38775 −0.693873 0.720097i \(-0.744097\pi\)
−0.693873 + 0.720097i \(0.744097\pi\)
\(318\) 0 0
\(319\) −18.5225 −1.03706
\(320\) −2.56934 −0.143630
\(321\) 0 0
\(322\) −9.56975 −0.533301
\(323\) −6.17570 −0.343625
\(324\) 0 0
\(325\) −20.0072 −1.10980
\(326\) −32.3018 −1.78903
\(327\) 0 0
\(328\) −22.2444 −1.22824
\(329\) −11.5177 −0.634993
\(330\) 0 0
\(331\) −7.06870 −0.388531 −0.194265 0.980949i \(-0.562232\pi\)
−0.194265 + 0.980949i \(0.562232\pi\)
\(332\) 0.477922 0.0262294
\(333\) 0 0
\(334\) −51.4349 −2.81439
\(335\) −5.87701 −0.321095
\(336\) 0 0
\(337\) 0.489921 0.0266877 0.0133439 0.999911i \(-0.495752\pi\)
0.0133439 + 0.999911i \(0.495752\pi\)
\(338\) −5.22797 −0.284364
\(339\) 0 0
\(340\) −61.8816 −3.35600
\(341\) 8.19746 0.443917
\(342\) 0 0
\(343\) 18.6695 1.00806
\(344\) −46.4856 −2.50634
\(345\) 0 0
\(346\) −17.2617 −0.927993
\(347\) 18.8536 1.01211 0.506057 0.862500i \(-0.331103\pi\)
0.506057 + 0.862500i \(0.331103\pi\)
\(348\) 0 0
\(349\) 35.4610 1.89818 0.949092 0.314999i \(-0.102004\pi\)
0.949092 + 0.314999i \(0.102004\pi\)
\(350\) 21.7522 1.16271
\(351\) 0 0
\(352\) 10.0822 0.537381
\(353\) −1.97621 −0.105183 −0.0525914 0.998616i \(-0.516748\pi\)
−0.0525914 + 0.998616i \(0.516748\pi\)
\(354\) 0 0
\(355\) −7.31900 −0.388452
\(356\) −23.5440 −1.24783
\(357\) 0 0
\(358\) −56.9158 −3.00809
\(359\) −8.05736 −0.425251 −0.212626 0.977134i \(-0.568201\pi\)
−0.212626 + 0.977134i \(0.568201\pi\)
\(360\) 0 0
\(361\) −16.9932 −0.894380
\(362\) 57.6958 3.03242
\(363\) 0 0
\(364\) 28.7030 1.50445
\(365\) −36.4581 −1.90831
\(366\) 0 0
\(367\) 17.8111 0.929730 0.464865 0.885382i \(-0.346103\pi\)
0.464865 + 0.885382i \(0.346103\pi\)
\(368\) −15.7222 −0.819574
\(369\) 0 0
\(370\) 55.4395 2.88216
\(371\) −4.49429 −0.233332
\(372\) 0 0
\(373\) 2.35142 0.121752 0.0608760 0.998145i \(-0.480611\pi\)
0.0608760 + 0.998145i \(0.480611\pi\)
\(374\) −21.7276 −1.12351
\(375\) 0 0
\(376\) −43.2425 −2.23006
\(377\) −36.6375 −1.88693
\(378\) 0 0
\(379\) 11.6298 0.597382 0.298691 0.954350i \(-0.403450\pi\)
0.298691 + 0.954350i \(0.403450\pi\)
\(380\) 20.1083 1.03153
\(381\) 0 0
\(382\) 46.4575 2.37697
\(383\) −29.9411 −1.52992 −0.764958 0.644080i \(-0.777240\pi\)
−0.764958 + 0.644080i \(0.777240\pi\)
\(384\) 0 0
\(385\) 10.3823 0.529131
\(386\) −22.3321 −1.13667
\(387\) 0 0
\(388\) −19.4035 −0.985065
\(389\) 3.41170 0.172980 0.0864900 0.996253i \(-0.472435\pi\)
0.0864900 + 0.996253i \(0.472435\pi\)
\(390\) 0 0
\(391\) 9.88864 0.500090
\(392\) 26.4495 1.33590
\(393\) 0 0
\(394\) 4.72571 0.238078
\(395\) 22.2118 1.11760
\(396\) 0 0
\(397\) −2.85437 −0.143257 −0.0716285 0.997431i \(-0.522820\pi\)
−0.0716285 + 0.997431i \(0.522820\pi\)
\(398\) −34.0162 −1.70508
\(399\) 0 0
\(400\) 35.7368 1.78684
\(401\) 24.8247 1.23969 0.619843 0.784726i \(-0.287196\pi\)
0.619843 + 0.784726i \(0.287196\pi\)
\(402\) 0 0
\(403\) 16.2146 0.807705
\(404\) 49.8999 2.48261
\(405\) 0 0
\(406\) 39.8331 1.97688
\(407\) 13.4337 0.665883
\(408\) 0 0
\(409\) −14.3691 −0.710506 −0.355253 0.934770i \(-0.615605\pi\)
−0.355253 + 0.934770i \(0.615605\pi\)
\(410\) −28.8852 −1.42654
\(411\) 0 0
\(412\) −17.9422 −0.883951
\(413\) −5.87098 −0.288892
\(414\) 0 0
\(415\) 0.341940 0.0167852
\(416\) 19.9425 0.977763
\(417\) 0 0
\(418\) 7.06033 0.345332
\(419\) −18.4515 −0.901415 −0.450707 0.892672i \(-0.648828\pi\)
−0.450707 + 0.892672i \(0.648828\pi\)
\(420\) 0 0
\(421\) 17.7932 0.867188 0.433594 0.901108i \(-0.357245\pi\)
0.433594 + 0.901108i \(0.357245\pi\)
\(422\) −61.9107 −3.01377
\(423\) 0 0
\(424\) −16.8735 −0.819449
\(425\) −22.4771 −1.09030
\(426\) 0 0
\(427\) 4.63316 0.224214
\(428\) −72.1585 −3.48791
\(429\) 0 0
\(430\) −60.3634 −2.91098
\(431\) −10.8330 −0.521808 −0.260904 0.965365i \(-0.584021\pi\)
−0.260904 + 0.965365i \(0.584021\pi\)
\(432\) 0 0
\(433\) −30.1293 −1.44792 −0.723962 0.689840i \(-0.757681\pi\)
−0.723962 + 0.689840i \(0.757681\pi\)
\(434\) −17.6288 −0.846212
\(435\) 0 0
\(436\) 57.4184 2.74984
\(437\) −3.21330 −0.153713
\(438\) 0 0
\(439\) 28.0736 1.33988 0.669940 0.742415i \(-0.266320\pi\)
0.669940 + 0.742415i \(0.266320\pi\)
\(440\) 38.9796 1.85828
\(441\) 0 0
\(442\) −42.9771 −2.04421
\(443\) 14.5133 0.689546 0.344773 0.938686i \(-0.387956\pi\)
0.344773 + 0.938686i \(0.387956\pi\)
\(444\) 0 0
\(445\) −16.8451 −0.798532
\(446\) 73.5474 3.48257
\(447\) 0 0
\(448\) 1.33888 0.0632562
\(449\) 7.86431 0.371140 0.185570 0.982631i \(-0.440587\pi\)
0.185570 + 0.982631i \(0.440587\pi\)
\(450\) 0 0
\(451\) −6.99926 −0.329582
\(452\) 71.8566 3.37985
\(453\) 0 0
\(454\) 14.6452 0.687335
\(455\) 20.5362 0.962753
\(456\) 0 0
\(457\) 25.4437 1.19021 0.595103 0.803649i \(-0.297111\pi\)
0.595103 + 0.803649i \(0.297111\pi\)
\(458\) 58.1092 2.71526
\(459\) 0 0
\(460\) −32.1978 −1.50123
\(461\) −20.4311 −0.951570 −0.475785 0.879562i \(-0.657836\pi\)
−0.475785 + 0.879562i \(0.657836\pi\)
\(462\) 0 0
\(463\) 27.6385 1.28447 0.642236 0.766507i \(-0.278007\pi\)
0.642236 + 0.766507i \(0.278007\pi\)
\(464\) 65.4419 3.03806
\(465\) 0 0
\(466\) 51.9225 2.40526
\(467\) 9.16496 0.424104 0.212052 0.977258i \(-0.431985\pi\)
0.212052 + 0.977258i \(0.431985\pi\)
\(468\) 0 0
\(469\) 3.06250 0.141413
\(470\) −56.1522 −2.59011
\(471\) 0 0
\(472\) −22.0422 −1.01457
\(473\) −14.6268 −0.672542
\(474\) 0 0
\(475\) 7.30389 0.335126
\(476\) 32.2464 1.47801
\(477\) 0 0
\(478\) −29.4993 −1.34926
\(479\) 6.75559 0.308671 0.154335 0.988019i \(-0.450676\pi\)
0.154335 + 0.988019i \(0.450676\pi\)
\(480\) 0 0
\(481\) 26.5718 1.21157
\(482\) −33.7091 −1.53541
\(483\) 0 0
\(484\) −31.8532 −1.44787
\(485\) −13.8827 −0.630381
\(486\) 0 0
\(487\) −19.6769 −0.891646 −0.445823 0.895121i \(-0.647089\pi\)
−0.445823 + 0.895121i \(0.647089\pi\)
\(488\) 17.3949 0.787428
\(489\) 0 0
\(490\) 34.3457 1.55158
\(491\) −33.2134 −1.49890 −0.749449 0.662062i \(-0.769681\pi\)
−0.749449 + 0.662062i \(0.769681\pi\)
\(492\) 0 0
\(493\) −41.1605 −1.85377
\(494\) 13.9653 0.628330
\(495\) 0 0
\(496\) −28.9625 −1.30045
\(497\) 3.81393 0.171078
\(498\) 0 0
\(499\) 32.1585 1.43961 0.719805 0.694176i \(-0.244231\pi\)
0.719805 + 0.694176i \(0.244231\pi\)
\(500\) 2.21293 0.0989654
\(501\) 0 0
\(502\) −18.5360 −0.827303
\(503\) 17.2428 0.768819 0.384410 0.923163i \(-0.374405\pi\)
0.384410 + 0.923163i \(0.374405\pi\)
\(504\) 0 0
\(505\) 35.7020 1.58872
\(506\) −11.3051 −0.502574
\(507\) 0 0
\(508\) −17.7812 −0.788912
\(509\) 32.1700 1.42591 0.712954 0.701211i \(-0.247357\pi\)
0.712954 + 0.701211i \(0.247357\pi\)
\(510\) 0 0
\(511\) 18.9983 0.840436
\(512\) 50.8088 2.24545
\(513\) 0 0
\(514\) −34.7738 −1.53380
\(515\) −12.8372 −0.565674
\(516\) 0 0
\(517\) −13.6064 −0.598408
\(518\) −28.8895 −1.26933
\(519\) 0 0
\(520\) 77.1017 3.38113
\(521\) 24.6284 1.07899 0.539495 0.841989i \(-0.318615\pi\)
0.539495 + 0.841989i \(0.318615\pi\)
\(522\) 0 0
\(523\) 5.84292 0.255493 0.127747 0.991807i \(-0.459226\pi\)
0.127747 + 0.991807i \(0.459226\pi\)
\(524\) 43.4918 1.89995
\(525\) 0 0
\(526\) −0.446998 −0.0194900
\(527\) 18.2163 0.793514
\(528\) 0 0
\(529\) −17.8548 −0.776296
\(530\) −21.9109 −0.951749
\(531\) 0 0
\(532\) −10.4784 −0.454297
\(533\) −13.8445 −0.599673
\(534\) 0 0
\(535\) −51.6274 −2.23205
\(536\) 11.4980 0.496636
\(537\) 0 0
\(538\) −20.4557 −0.881908
\(539\) 8.32239 0.358471
\(540\) 0 0
\(541\) −8.50279 −0.365564 −0.182782 0.983154i \(-0.558510\pi\)
−0.182782 + 0.983154i \(0.558510\pi\)
\(542\) −48.6477 −2.08960
\(543\) 0 0
\(544\) 22.4045 0.960584
\(545\) 41.0813 1.75973
\(546\) 0 0
\(547\) −44.9602 −1.92236 −0.961179 0.275925i \(-0.911016\pi\)
−0.961179 + 0.275925i \(0.911016\pi\)
\(548\) 33.0922 1.41363
\(549\) 0 0
\(550\) 25.6968 1.09572
\(551\) 13.3750 0.569795
\(552\) 0 0
\(553\) −11.5746 −0.492201
\(554\) −54.0807 −2.29767
\(555\) 0 0
\(556\) 79.0778 3.35365
\(557\) −24.4636 −1.03656 −0.518278 0.855212i \(-0.673427\pi\)
−0.518278 + 0.855212i \(0.673427\pi\)
\(558\) 0 0
\(559\) −28.9319 −1.22369
\(560\) −36.6818 −1.55009
\(561\) 0 0
\(562\) 25.4233 1.07242
\(563\) 6.06992 0.255817 0.127908 0.991786i \(-0.459174\pi\)
0.127908 + 0.991786i \(0.459174\pi\)
\(564\) 0 0
\(565\) 51.4114 2.16290
\(566\) −70.1107 −2.94697
\(567\) 0 0
\(568\) 14.3191 0.600817
\(569\) 10.6297 0.445620 0.222810 0.974862i \(-0.428477\pi\)
0.222810 + 0.974862i \(0.428477\pi\)
\(570\) 0 0
\(571\) 27.4864 1.15027 0.575134 0.818059i \(-0.304950\pi\)
0.575134 + 0.818059i \(0.304950\pi\)
\(572\) 33.9080 1.41777
\(573\) 0 0
\(574\) 15.0521 0.628262
\(575\) −11.6951 −0.487721
\(576\) 0 0
\(577\) −5.93714 −0.247166 −0.123583 0.992334i \(-0.539439\pi\)
−0.123583 + 0.992334i \(0.539439\pi\)
\(578\) −5.09415 −0.211889
\(579\) 0 0
\(580\) 134.020 5.56488
\(581\) −0.178185 −0.00739235
\(582\) 0 0
\(583\) −5.30929 −0.219888
\(584\) 71.3278 2.95156
\(585\) 0 0
\(586\) −18.6251 −0.769397
\(587\) −11.1071 −0.458440 −0.229220 0.973375i \(-0.573617\pi\)
−0.229220 + 0.973375i \(0.573617\pi\)
\(588\) 0 0
\(589\) −5.91935 −0.243903
\(590\) −28.6227 −1.17838
\(591\) 0 0
\(592\) −47.4626 −1.95070
\(593\) 2.05072 0.0842128 0.0421064 0.999113i \(-0.486593\pi\)
0.0421064 + 0.999113i \(0.486593\pi\)
\(594\) 0 0
\(595\) 23.0714 0.945837
\(596\) 62.5584 2.56249
\(597\) 0 0
\(598\) −22.3616 −0.914432
\(599\) 2.80439 0.114584 0.0572921 0.998357i \(-0.481753\pi\)
0.0572921 + 0.998357i \(0.481753\pi\)
\(600\) 0 0
\(601\) −36.0402 −1.47011 −0.735055 0.678007i \(-0.762844\pi\)
−0.735055 + 0.678007i \(0.762844\pi\)
\(602\) 31.4554 1.28202
\(603\) 0 0
\(604\) 12.3192 0.501263
\(605\) −22.7901 −0.926550
\(606\) 0 0
\(607\) 38.5142 1.56324 0.781621 0.623754i \(-0.214393\pi\)
0.781621 + 0.623754i \(0.214393\pi\)
\(608\) −7.28029 −0.295255
\(609\) 0 0
\(610\) 22.5879 0.914558
\(611\) −26.9134 −1.08880
\(612\) 0 0
\(613\) 1.45973 0.0589580 0.0294790 0.999565i \(-0.490615\pi\)
0.0294790 + 0.999565i \(0.490615\pi\)
\(614\) −14.8070 −0.597561
\(615\) 0 0
\(616\) −20.3123 −0.818404
\(617\) −12.0248 −0.484102 −0.242051 0.970264i \(-0.577820\pi\)
−0.242051 + 0.970264i \(0.577820\pi\)
\(618\) 0 0
\(619\) 31.3697 1.26085 0.630427 0.776249i \(-0.282880\pi\)
0.630427 + 0.776249i \(0.282880\pi\)
\(620\) −59.3129 −2.38206
\(621\) 0 0
\(622\) 23.1725 0.929132
\(623\) 8.77795 0.351681
\(624\) 0 0
\(625\) −24.1962 −0.967848
\(626\) −16.2417 −0.649149
\(627\) 0 0
\(628\) 22.2590 0.888231
\(629\) 29.8522 1.19028
\(630\) 0 0
\(631\) −17.4483 −0.694605 −0.347303 0.937753i \(-0.612902\pi\)
−0.347303 + 0.937753i \(0.612902\pi\)
\(632\) −43.4559 −1.72858
\(633\) 0 0
\(634\) 62.7711 2.49296
\(635\) −12.7219 −0.504855
\(636\) 0 0
\(637\) 16.4617 0.652236
\(638\) 47.0565 1.86298
\(639\) 0 0
\(640\) 39.2832 1.55280
\(641\) 9.22820 0.364492 0.182246 0.983253i \(-0.441663\pi\)
0.182246 + 0.983253i \(0.441663\pi\)
\(642\) 0 0
\(643\) 7.30126 0.287934 0.143967 0.989583i \(-0.454014\pi\)
0.143967 + 0.989583i \(0.454014\pi\)
\(644\) 16.7782 0.661156
\(645\) 0 0
\(646\) 15.6894 0.617291
\(647\) 34.3969 1.35228 0.676142 0.736772i \(-0.263651\pi\)
0.676142 + 0.736772i \(0.263651\pi\)
\(648\) 0 0
\(649\) −6.93563 −0.272247
\(650\) 50.8283 1.99365
\(651\) 0 0
\(652\) 56.6335 2.21794
\(653\) 14.2491 0.557612 0.278806 0.960347i \(-0.410061\pi\)
0.278806 + 0.960347i \(0.410061\pi\)
\(654\) 0 0
\(655\) 31.1172 1.21585
\(656\) 24.7291 0.965509
\(657\) 0 0
\(658\) 29.2609 1.14071
\(659\) 44.0929 1.71761 0.858807 0.512300i \(-0.171206\pi\)
0.858807 + 0.512300i \(0.171206\pi\)
\(660\) 0 0
\(661\) −2.63277 −0.102403 −0.0512015 0.998688i \(-0.516305\pi\)
−0.0512015 + 0.998688i \(0.516305\pi\)
\(662\) 17.9581 0.697960
\(663\) 0 0
\(664\) −0.668982 −0.0259615
\(665\) −7.49702 −0.290722
\(666\) 0 0
\(667\) −21.4163 −0.829244
\(668\) 90.1787 3.48912
\(669\) 0 0
\(670\) 14.9306 0.576818
\(671\) 5.47334 0.211296
\(672\) 0 0
\(673\) −3.41710 −0.131720 −0.0658598 0.997829i \(-0.520979\pi\)
−0.0658598 + 0.997829i \(0.520979\pi\)
\(674\) −1.24465 −0.0479420
\(675\) 0 0
\(676\) 9.16598 0.352538
\(677\) 38.8868 1.49454 0.747271 0.664520i \(-0.231364\pi\)
0.747271 + 0.664520i \(0.231364\pi\)
\(678\) 0 0
\(679\) 7.23427 0.277626
\(680\) 86.6201 3.32173
\(681\) 0 0
\(682\) −20.8257 −0.797456
\(683\) 23.1782 0.886891 0.443445 0.896301i \(-0.353756\pi\)
0.443445 + 0.896301i \(0.353756\pi\)
\(684\) 0 0
\(685\) 23.6766 0.904636
\(686\) −47.4299 −1.81088
\(687\) 0 0
\(688\) 51.6781 1.97021
\(689\) −10.5018 −0.400086
\(690\) 0 0
\(691\) 17.3931 0.661665 0.330833 0.943689i \(-0.392670\pi\)
0.330833 + 0.943689i \(0.392670\pi\)
\(692\) 30.2642 1.15047
\(693\) 0 0
\(694\) −47.8976 −1.81817
\(695\) 56.5780 2.14613
\(696\) 0 0
\(697\) −15.5537 −0.589137
\(698\) −90.0888 −3.40991
\(699\) 0 0
\(700\) −38.1373 −1.44146
\(701\) −18.9249 −0.714785 −0.357393 0.933954i \(-0.616334\pi\)
−0.357393 + 0.933954i \(0.616334\pi\)
\(702\) 0 0
\(703\) −9.70041 −0.365858
\(704\) 1.58167 0.0596116
\(705\) 0 0
\(706\) 5.02056 0.188951
\(707\) −18.6043 −0.699687
\(708\) 0 0
\(709\) 42.7604 1.60590 0.802951 0.596045i \(-0.203262\pi\)
0.802951 + 0.596045i \(0.203262\pi\)
\(710\) 18.5940 0.697819
\(711\) 0 0
\(712\) 32.9562 1.23509
\(713\) 9.47818 0.354960
\(714\) 0 0
\(715\) 24.2603 0.907283
\(716\) 99.7882 3.72926
\(717\) 0 0
\(718\) 20.4698 0.763924
\(719\) 13.0136 0.485326 0.242663 0.970111i \(-0.421979\pi\)
0.242663 + 0.970111i \(0.421979\pi\)
\(720\) 0 0
\(721\) 6.68945 0.249128
\(722\) 43.1714 1.60667
\(723\) 0 0
\(724\) −101.156 −3.75942
\(725\) 48.6798 1.80792
\(726\) 0 0
\(727\) −50.4565 −1.87133 −0.935664 0.352891i \(-0.885199\pi\)
−0.935664 + 0.352891i \(0.885199\pi\)
\(728\) −40.1777 −1.48908
\(729\) 0 0
\(730\) 92.6220 3.42810
\(731\) −32.5035 −1.20219
\(732\) 0 0
\(733\) −15.9879 −0.590527 −0.295263 0.955416i \(-0.595407\pi\)
−0.295263 + 0.955416i \(0.595407\pi\)
\(734\) −45.2491 −1.67017
\(735\) 0 0
\(736\) 11.6573 0.429695
\(737\) 3.61786 0.133266
\(738\) 0 0
\(739\) −7.13598 −0.262501 −0.131251 0.991349i \(-0.541899\pi\)
−0.131251 + 0.991349i \(0.541899\pi\)
\(740\) −97.1998 −3.57313
\(741\) 0 0
\(742\) 11.4178 0.419159
\(743\) −33.9109 −1.24407 −0.622036 0.782989i \(-0.713694\pi\)
−0.622036 + 0.782989i \(0.713694\pi\)
\(744\) 0 0
\(745\) 44.7589 1.63984
\(746\) −5.97380 −0.218716
\(747\) 0 0
\(748\) 38.0940 1.39286
\(749\) 26.9030 0.983015
\(750\) 0 0
\(751\) −41.9333 −1.53017 −0.765085 0.643930i \(-0.777303\pi\)
−0.765085 + 0.643930i \(0.777303\pi\)
\(752\) 48.0727 1.75303
\(753\) 0 0
\(754\) 93.0777 3.38969
\(755\) 8.81408 0.320777
\(756\) 0 0
\(757\) 26.1805 0.951548 0.475774 0.879568i \(-0.342168\pi\)
0.475774 + 0.879568i \(0.342168\pi\)
\(758\) −29.5455 −1.07314
\(759\) 0 0
\(760\) −28.1470 −1.02100
\(761\) −46.8751 −1.69922 −0.849610 0.527411i \(-0.823163\pi\)
−0.849610 + 0.527411i \(0.823163\pi\)
\(762\) 0 0
\(763\) −21.4074 −0.775001
\(764\) −81.4521 −2.94683
\(765\) 0 0
\(766\) 76.0654 2.74835
\(767\) −13.7187 −0.495353
\(768\) 0 0
\(769\) −13.7103 −0.494407 −0.247203 0.968964i \(-0.579512\pi\)
−0.247203 + 0.968964i \(0.579512\pi\)
\(770\) −26.3763 −0.950536
\(771\) 0 0
\(772\) 39.1540 1.40918
\(773\) 15.2517 0.548566 0.274283 0.961649i \(-0.411559\pi\)
0.274283 + 0.961649i \(0.411559\pi\)
\(774\) 0 0
\(775\) −21.5441 −0.773887
\(776\) 27.1605 0.975006
\(777\) 0 0
\(778\) −8.66743 −0.310743
\(779\) 5.05414 0.181083
\(780\) 0 0
\(781\) 4.50555 0.161221
\(782\) −25.1221 −0.898366
\(783\) 0 0
\(784\) −29.4039 −1.05014
\(785\) 15.9257 0.568413
\(786\) 0 0
\(787\) −21.0003 −0.748580 −0.374290 0.927312i \(-0.622114\pi\)
−0.374290 + 0.927312i \(0.622114\pi\)
\(788\) −8.28539 −0.295155
\(789\) 0 0
\(790\) −56.4292 −2.00766
\(791\) −26.7905 −0.952560
\(792\) 0 0
\(793\) 10.8263 0.384452
\(794\) 7.25155 0.257348
\(795\) 0 0
\(796\) 59.6393 2.11386
\(797\) 46.1575 1.63498 0.817491 0.575941i \(-0.195364\pi\)
0.817491 + 0.575941i \(0.195364\pi\)
\(798\) 0 0
\(799\) −30.2359 −1.06967
\(800\) −26.4974 −0.936824
\(801\) 0 0
\(802\) −63.0672 −2.22698
\(803\) 22.4435 0.792013
\(804\) 0 0
\(805\) 12.0044 0.423099
\(806\) −41.1932 −1.45097
\(807\) 0 0
\(808\) −69.8485 −2.45726
\(809\) 15.0049 0.527545 0.263772 0.964585i \(-0.415033\pi\)
0.263772 + 0.964585i \(0.415033\pi\)
\(810\) 0 0
\(811\) 47.0448 1.65197 0.825983 0.563695i \(-0.190621\pi\)
0.825983 + 0.563695i \(0.190621\pi\)
\(812\) −69.8378 −2.45082
\(813\) 0 0
\(814\) −34.1283 −1.19620
\(815\) 40.5197 1.41934
\(816\) 0 0
\(817\) 10.5620 0.369516
\(818\) 36.5047 1.27636
\(819\) 0 0
\(820\) 50.6433 1.76854
\(821\) 1.52951 0.0533801 0.0266901 0.999644i \(-0.491503\pi\)
0.0266901 + 0.999644i \(0.491503\pi\)
\(822\) 0 0
\(823\) 2.16425 0.0754410 0.0377205 0.999288i \(-0.487990\pi\)
0.0377205 + 0.999288i \(0.487990\pi\)
\(824\) 25.1151 0.874924
\(825\) 0 0
\(826\) 14.9153 0.518968
\(827\) −40.8370 −1.42004 −0.710020 0.704181i \(-0.751314\pi\)
−0.710020 + 0.704181i \(0.751314\pi\)
\(828\) 0 0
\(829\) 17.1117 0.594316 0.297158 0.954828i \(-0.403961\pi\)
0.297158 + 0.954828i \(0.403961\pi\)
\(830\) −0.868701 −0.0301530
\(831\) 0 0
\(832\) 3.12855 0.108463
\(833\) 18.4939 0.640777
\(834\) 0 0
\(835\) 64.5204 2.23282
\(836\) −12.3786 −0.428122
\(837\) 0 0
\(838\) 46.8761 1.61931
\(839\) −18.3623 −0.633937 −0.316969 0.948436i \(-0.602665\pi\)
−0.316969 + 0.948436i \(0.602665\pi\)
\(840\) 0 0
\(841\) 60.1433 2.07391
\(842\) −45.2037 −1.55782
\(843\) 0 0
\(844\) 108.546 3.73629
\(845\) 6.55801 0.225602
\(846\) 0 0
\(847\) 11.8759 0.408061
\(848\) 18.7583 0.644162
\(849\) 0 0
\(850\) 57.1032 1.95862
\(851\) 15.5325 0.532446
\(852\) 0 0
\(853\) 15.5086 0.531005 0.265503 0.964110i \(-0.414462\pi\)
0.265503 + 0.964110i \(0.414462\pi\)
\(854\) −11.7706 −0.402780
\(855\) 0 0
\(856\) 101.005 3.45229
\(857\) 19.0971 0.652344 0.326172 0.945310i \(-0.394241\pi\)
0.326172 + 0.945310i \(0.394241\pi\)
\(858\) 0 0
\(859\) 2.83131 0.0966032 0.0483016 0.998833i \(-0.484619\pi\)
0.0483016 + 0.998833i \(0.484619\pi\)
\(860\) 105.833 3.60887
\(861\) 0 0
\(862\) 27.5213 0.937380
\(863\) 3.01216 0.102535 0.0512675 0.998685i \(-0.483674\pi\)
0.0512675 + 0.998685i \(0.483674\pi\)
\(864\) 0 0
\(865\) 21.6532 0.736231
\(866\) 76.5437 2.60106
\(867\) 0 0
\(868\) 30.9079 1.04908
\(869\) −13.6735 −0.463842
\(870\) 0 0
\(871\) 7.15613 0.242476
\(872\) −80.3727 −2.72176
\(873\) 0 0
\(874\) 8.16339 0.276131
\(875\) −0.825053 −0.0278919
\(876\) 0 0
\(877\) 14.5173 0.490214 0.245107 0.969496i \(-0.421177\pi\)
0.245107 + 0.969496i \(0.421177\pi\)
\(878\) −71.3212 −2.40697
\(879\) 0 0
\(880\) −43.3337 −1.46078
\(881\) 6.48384 0.218446 0.109223 0.994017i \(-0.465164\pi\)
0.109223 + 0.994017i \(0.465164\pi\)
\(882\) 0 0
\(883\) 25.3933 0.854554 0.427277 0.904121i \(-0.359473\pi\)
0.427277 + 0.904121i \(0.359473\pi\)
\(884\) 75.3500 2.53430
\(885\) 0 0
\(886\) −36.8710 −1.23871
\(887\) 0.107021 0.00359341 0.00179671 0.999998i \(-0.499428\pi\)
0.00179671 + 0.999998i \(0.499428\pi\)
\(888\) 0 0
\(889\) 6.62940 0.222343
\(890\) 42.7950 1.43449
\(891\) 0 0
\(892\) −128.948 −4.31749
\(893\) 9.82511 0.328785
\(894\) 0 0
\(895\) 71.3957 2.38650
\(896\) −20.4705 −0.683870
\(897\) 0 0
\(898\) −19.9793 −0.666718
\(899\) −39.4519 −1.31580
\(900\) 0 0
\(901\) −11.7982 −0.393056
\(902\) 17.7816 0.592064
\(903\) 0 0
\(904\) −100.583 −3.34534
\(905\) −72.3741 −2.40580
\(906\) 0 0
\(907\) 43.5972 1.44762 0.723810 0.689999i \(-0.242389\pi\)
0.723810 + 0.689999i \(0.242389\pi\)
\(908\) −25.6769 −0.852118
\(909\) 0 0
\(910\) −52.1723 −1.72950
\(911\) 55.4124 1.83590 0.917948 0.396702i \(-0.129845\pi\)
0.917948 + 0.396702i \(0.129845\pi\)
\(912\) 0 0
\(913\) −0.210497 −0.00696644
\(914\) −64.6399 −2.13810
\(915\) 0 0
\(916\) −101.880 −3.36622
\(917\) −16.2152 −0.535472
\(918\) 0 0
\(919\) −40.9441 −1.35062 −0.675311 0.737533i \(-0.735990\pi\)
−0.675311 + 0.737533i \(0.735990\pi\)
\(920\) 45.0696 1.48590
\(921\) 0 0
\(922\) 51.9053 1.70941
\(923\) 8.91198 0.293341
\(924\) 0 0
\(925\) −35.3057 −1.16084
\(926\) −70.2159 −2.30744
\(927\) 0 0
\(928\) −48.5225 −1.59283
\(929\) −32.3050 −1.05989 −0.529947 0.848031i \(-0.677788\pi\)
−0.529947 + 0.848031i \(0.677788\pi\)
\(930\) 0 0
\(931\) −6.00957 −0.196956
\(932\) −91.0336 −2.98190
\(933\) 0 0
\(934\) −23.2836 −0.761863
\(935\) 27.2552 0.891342
\(936\) 0 0
\(937\) −9.06200 −0.296043 −0.148021 0.988984i \(-0.547290\pi\)
−0.148021 + 0.988984i \(0.547290\pi\)
\(938\) −7.78031 −0.254036
\(939\) 0 0
\(940\) 98.4493 3.21106
\(941\) 23.9044 0.779263 0.389631 0.920971i \(-0.372602\pi\)
0.389631 + 0.920971i \(0.372602\pi\)
\(942\) 0 0
\(943\) −8.09278 −0.263537
\(944\) 24.5043 0.797547
\(945\) 0 0
\(946\) 37.1595 1.20816
\(947\) 8.23165 0.267493 0.133746 0.991016i \(-0.457299\pi\)
0.133746 + 0.991016i \(0.457299\pi\)
\(948\) 0 0
\(949\) 44.3932 1.44106
\(950\) −18.5556 −0.602022
\(951\) 0 0
\(952\) −45.1377 −1.46292
\(953\) −17.0755 −0.553129 −0.276564 0.960995i \(-0.589196\pi\)
−0.276564 + 0.960995i \(0.589196\pi\)
\(954\) 0 0
\(955\) −58.2768 −1.88579
\(956\) 51.7199 1.67274
\(957\) 0 0
\(958\) −17.1626 −0.554498
\(959\) −12.3379 −0.398410
\(960\) 0 0
\(961\) −13.5398 −0.436769
\(962\) −67.5058 −2.17648
\(963\) 0 0
\(964\) 59.1008 1.90351
\(965\) 28.0136 0.901790
\(966\) 0 0
\(967\) 15.3524 0.493698 0.246849 0.969054i \(-0.420605\pi\)
0.246849 + 0.969054i \(0.420605\pi\)
\(968\) 44.5873 1.43309
\(969\) 0 0
\(970\) 35.2691 1.13242
\(971\) −10.5213 −0.337645 −0.168822 0.985646i \(-0.553996\pi\)
−0.168822 + 0.985646i \(0.553996\pi\)
\(972\) 0 0
\(973\) −29.4828 −0.945174
\(974\) 49.9893 1.60176
\(975\) 0 0
\(976\) −19.3379 −0.618990
\(977\) 23.0942 0.738850 0.369425 0.929261i \(-0.379555\pi\)
0.369425 + 0.929261i \(0.379555\pi\)
\(978\) 0 0
\(979\) 10.3698 0.331419
\(980\) −60.2169 −1.92356
\(981\) 0 0
\(982\) 84.3787 2.69263
\(983\) 1.19718 0.0381843 0.0190921 0.999818i \(-0.493922\pi\)
0.0190921 + 0.999818i \(0.493922\pi\)
\(984\) 0 0
\(985\) −5.92797 −0.188881
\(986\) 104.568 3.33013
\(987\) 0 0
\(988\) −24.4849 −0.778967
\(989\) −16.9120 −0.537771
\(990\) 0 0
\(991\) −21.7076 −0.689565 −0.344783 0.938683i \(-0.612047\pi\)
−0.344783 + 0.938683i \(0.612047\pi\)
\(992\) 21.4745 0.681816
\(993\) 0 0
\(994\) −9.68930 −0.307326
\(995\) 42.6703 1.35274
\(996\) 0 0
\(997\) −34.5063 −1.09282 −0.546412 0.837516i \(-0.684007\pi\)
−0.546412 + 0.837516i \(0.684007\pi\)
\(998\) −81.6987 −2.58613
\(999\) 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 6561.2.a.d.1.3 72
3.2 odd 2 6561.2.a.c.1.70 72
81.4 even 27 729.2.g.a.136.1 144
81.7 even 27 243.2.g.a.37.8 144
81.20 odd 54 729.2.g.d.595.8 144
81.23 odd 54 81.2.g.a.43.1 144
81.31 even 27 729.2.g.b.622.8 144
81.34 even 27 729.2.g.b.109.8 144
81.47 odd 54 729.2.g.c.109.1 144
81.50 odd 54 729.2.g.c.622.1 144
81.58 even 27 243.2.g.a.46.8 144
81.61 even 27 729.2.g.a.595.1 144
81.74 odd 54 81.2.g.a.49.1 yes 144
81.77 odd 54 729.2.g.d.136.8 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.g.a.43.1 144 81.23 odd 54
81.2.g.a.49.1 yes 144 81.74 odd 54
243.2.g.a.37.8 144 81.7 even 27
243.2.g.a.46.8 144 81.58 even 27
729.2.g.a.136.1 144 81.4 even 27
729.2.g.a.595.1 144 81.61 even 27
729.2.g.b.109.8 144 81.34 even 27
729.2.g.b.622.8 144 81.31 even 27
729.2.g.c.109.1 144 81.47 odd 54
729.2.g.c.622.1 144 81.50 odd 54
729.2.g.d.136.8 144 81.77 odd 54
729.2.g.d.595.8 144 81.20 odd 54
6561.2.a.c.1.70 72 3.2 odd 2
6561.2.a.d.1.3 72 1.1 even 1 trivial