Properties

Label 81.5.b.a.80.5
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 80.5
Root \(-1.28901 - 2.23263i\) of defining polynomial
Character \(\chi\) \(=\) 81.80
Dual form 81.5.b.a.80.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.46526i q^{2} -3.93857 q^{4} -16.0226i q^{5} +72.4837 q^{7} +53.8574i q^{8} +71.5451 q^{10} +96.1576i q^{11} +153.906 q^{13} +323.659i q^{14} -303.505 q^{16} +72.7905i q^{17} -190.660 q^{19} +63.1062i q^{20} -429.369 q^{22} +14.4552i q^{23} +368.276 q^{25} +687.231i q^{26} -285.482 q^{28} +716.262i q^{29} -302.568 q^{31} -493.509i q^{32} -325.029 q^{34} -1161.38i q^{35} +826.277 q^{37} -851.348i q^{38} +862.936 q^{40} -556.119i q^{41} +892.680 q^{43} -378.723i q^{44} -64.5464 q^{46} -3955.43i q^{47} +2852.89 q^{49} +1644.45i q^{50} -606.170 q^{52} -1966.96i q^{53} +1540.69 q^{55} +3903.79i q^{56} -3198.30 q^{58} -5414.94i q^{59} -1712.42 q^{61} -1351.04i q^{62} -2652.43 q^{64} -2465.97i q^{65} -4634.49 q^{67} -286.691i q^{68} +5185.86 q^{70} +6697.12i q^{71} -4823.86 q^{73} +3689.54i q^{74} +750.929 q^{76} +6969.86i q^{77} -5728.79 q^{79} +4862.94i q^{80} +2483.22 q^{82} +2832.23i q^{83} +1166.29 q^{85} +3986.05i q^{86} -5178.80 q^{88} -14277.7i q^{89} +11155.7 q^{91} -56.9330i q^{92} +17662.0 q^{94} +3054.87i q^{95} +7165.31 q^{97} +12738.9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 30 q^{4} - 24 q^{7} - 36 q^{10} + 12 q^{13} - 30 q^{16} - 258 q^{19} + 738 q^{22} + 546 q^{25} + 1308 q^{28} - 2580 q^{31} - 1026 q^{34} + 12 q^{37} + 2628 q^{40} + 570 q^{43} - 5760 q^{46} + 3726 q^{49}+ \cdots + 57918 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.46526i 1.11632i 0.829735 + 0.558158i \(0.188492\pi\)
−0.829735 + 0.558158i \(0.811508\pi\)
\(3\) 0 0
\(4\) −3.93857 −0.246161
\(5\) − 16.0226i − 0.640904i −0.947265 0.320452i \(-0.896165\pi\)
0.947265 0.320452i \(-0.103835\pi\)
\(6\) 0 0
\(7\) 72.4837 1.47926 0.739630 0.673014i \(-0.235001\pi\)
0.739630 + 0.673014i \(0.235001\pi\)
\(8\) 53.8574i 0.841523i
\(9\) 0 0
\(10\) 71.5451 0.715451
\(11\) 96.1576i 0.794691i 0.917669 + 0.397345i \(0.130068\pi\)
−0.917669 + 0.397345i \(0.869932\pi\)
\(12\) 0 0
\(13\) 153.906 0.910686 0.455343 0.890316i \(-0.349517\pi\)
0.455343 + 0.890316i \(0.349517\pi\)
\(14\) 323.659i 1.65132i
\(15\) 0 0
\(16\) −303.505 −1.18557
\(17\) 72.7905i 0.251870i 0.992038 + 0.125935i \(0.0401931\pi\)
−0.992038 + 0.125935i \(0.959807\pi\)
\(18\) 0 0
\(19\) −190.660 −0.528145 −0.264072 0.964503i \(-0.585066\pi\)
−0.264072 + 0.964503i \(0.585066\pi\)
\(20\) 63.1062i 0.157765i
\(21\) 0 0
\(22\) −429.369 −0.887126
\(23\) 14.4552i 0.0273256i 0.999907 + 0.0136628i \(0.00434914\pi\)
−0.999907 + 0.0136628i \(0.995651\pi\)
\(24\) 0 0
\(25\) 368.276 0.589242
\(26\) 687.231i 1.01661i
\(27\) 0 0
\(28\) −285.482 −0.364135
\(29\) 716.262i 0.851679i 0.904799 + 0.425839i \(0.140021\pi\)
−0.904799 + 0.425839i \(0.859979\pi\)
\(30\) 0 0
\(31\) −302.568 −0.314847 −0.157423 0.987531i \(-0.550319\pi\)
−0.157423 + 0.987531i \(0.550319\pi\)
\(32\) − 493.509i − 0.481943i
\(33\) 0 0
\(34\) −325.029 −0.281167
\(35\) − 1161.38i − 0.948063i
\(36\) 0 0
\(37\) 826.277 0.603562 0.301781 0.953377i \(-0.402419\pi\)
0.301781 + 0.953377i \(0.402419\pi\)
\(38\) − 851.348i − 0.589576i
\(39\) 0 0
\(40\) 862.936 0.539335
\(41\) − 556.119i − 0.330826i −0.986224 0.165413i \(-0.947104\pi\)
0.986224 0.165413i \(-0.0528958\pi\)
\(42\) 0 0
\(43\) 892.680 0.482791 0.241395 0.970427i \(-0.422395\pi\)
0.241395 + 0.970427i \(0.422395\pi\)
\(44\) − 378.723i − 0.195622i
\(45\) 0 0
\(46\) −64.5464 −0.0305040
\(47\) − 3955.43i − 1.79060i −0.445465 0.895299i \(-0.646962\pi\)
0.445465 0.895299i \(-0.353038\pi\)
\(48\) 0 0
\(49\) 2852.89 1.18821
\(50\) 1644.45i 0.657780i
\(51\) 0 0
\(52\) −606.170 −0.224175
\(53\) − 1966.96i − 0.700234i −0.936706 0.350117i \(-0.886142\pi\)
0.936706 0.350117i \(-0.113858\pi\)
\(54\) 0 0
\(55\) 1540.69 0.509320
\(56\) 3903.79i 1.24483i
\(57\) 0 0
\(58\) −3198.30 −0.950742
\(59\) − 5414.94i − 1.55557i −0.628530 0.777785i \(-0.716343\pi\)
0.628530 0.777785i \(-0.283657\pi\)
\(60\) 0 0
\(61\) −1712.42 −0.460204 −0.230102 0.973166i \(-0.573906\pi\)
−0.230102 + 0.973166i \(0.573906\pi\)
\(62\) − 1351.04i − 0.351468i
\(63\) 0 0
\(64\) −2652.43 −0.647565
\(65\) − 2465.97i − 0.583663i
\(66\) 0 0
\(67\) −4634.49 −1.03241 −0.516205 0.856465i \(-0.672656\pi\)
−0.516205 + 0.856465i \(0.672656\pi\)
\(68\) − 286.691i − 0.0620006i
\(69\) 0 0
\(70\) 5185.86 1.05834
\(71\) 6697.12i 1.32853i 0.747497 + 0.664265i \(0.231256\pi\)
−0.747497 + 0.664265i \(0.768744\pi\)
\(72\) 0 0
\(73\) −4823.86 −0.905208 −0.452604 0.891712i \(-0.649505\pi\)
−0.452604 + 0.891712i \(0.649505\pi\)
\(74\) 3689.54i 0.673766i
\(75\) 0 0
\(76\) 750.929 0.130008
\(77\) 6969.86i 1.17555i
\(78\) 0 0
\(79\) −5728.79 −0.917929 −0.458964 0.888455i \(-0.651779\pi\)
−0.458964 + 0.888455i \(0.651779\pi\)
\(80\) 4862.94i 0.759834i
\(81\) 0 0
\(82\) 2483.22 0.369307
\(83\) 2832.23i 0.411124i 0.978644 + 0.205562i \(0.0659022\pi\)
−0.978644 + 0.205562i \(0.934098\pi\)
\(84\) 0 0
\(85\) 1166.29 0.161425
\(86\) 3986.05i 0.538947i
\(87\) 0 0
\(88\) −5178.80 −0.668750
\(89\) − 14277.7i − 1.80251i −0.433290 0.901255i \(-0.642647\pi\)
0.433290 0.901255i \(-0.357353\pi\)
\(90\) 0 0
\(91\) 11155.7 1.34714
\(92\) − 56.9330i − 0.00672649i
\(93\) 0 0
\(94\) 17662.0 1.99887
\(95\) 3054.87i 0.338490i
\(96\) 0 0
\(97\) 7165.31 0.761538 0.380769 0.924670i \(-0.375659\pi\)
0.380769 + 0.924670i \(0.375659\pi\)
\(98\) 12738.9i 1.32641i
\(99\) 0 0
\(100\) −1450.48 −0.145048
\(101\) − 1959.20i − 0.192060i −0.995378 0.0960298i \(-0.969386\pi\)
0.995378 0.0960298i \(-0.0306144\pi\)
\(102\) 0 0
\(103\) −5155.48 −0.485953 −0.242976 0.970032i \(-0.578124\pi\)
−0.242976 + 0.970032i \(0.578124\pi\)
\(104\) 8288.98i 0.766363i
\(105\) 0 0
\(106\) 8782.98 0.781682
\(107\) 9117.08i 0.796321i 0.917316 + 0.398161i \(0.130351\pi\)
−0.917316 + 0.398161i \(0.869649\pi\)
\(108\) 0 0
\(109\) −16161.1 −1.36024 −0.680122 0.733099i \(-0.738073\pi\)
−0.680122 + 0.733099i \(0.738073\pi\)
\(110\) 6879.60i 0.568562i
\(111\) 0 0
\(112\) −21999.2 −1.75376
\(113\) − 21099.2i − 1.65238i −0.563393 0.826189i \(-0.690504\pi\)
0.563393 0.826189i \(-0.309496\pi\)
\(114\) 0 0
\(115\) 231.611 0.0175131
\(116\) − 2821.05i − 0.209650i
\(117\) 0 0
\(118\) 24179.1 1.73651
\(119\) 5276.13i 0.372582i
\(120\) 0 0
\(121\) 5394.72 0.368467
\(122\) − 7646.41i − 0.513733i
\(123\) 0 0
\(124\) 1191.68 0.0775029
\(125\) − 15914.9i − 1.01855i
\(126\) 0 0
\(127\) −20660.9 −1.28098 −0.640489 0.767968i \(-0.721268\pi\)
−0.640489 + 0.767968i \(0.721268\pi\)
\(128\) − 19739.9i − 1.20483i
\(129\) 0 0
\(130\) 11011.2 0.651552
\(131\) − 3201.45i − 0.186554i −0.995640 0.0932768i \(-0.970266\pi\)
0.995640 0.0932768i \(-0.0297342\pi\)
\(132\) 0 0
\(133\) −13819.8 −0.781263
\(134\) − 20694.2i − 1.15250i
\(135\) 0 0
\(136\) −3920.31 −0.211955
\(137\) − 3871.92i − 0.206293i −0.994666 0.103147i \(-0.967109\pi\)
0.994666 0.103147i \(-0.0328911\pi\)
\(138\) 0 0
\(139\) −11679.2 −0.604484 −0.302242 0.953231i \(-0.597735\pi\)
−0.302242 + 0.953231i \(0.597735\pi\)
\(140\) 4574.17i 0.233376i
\(141\) 0 0
\(142\) −29904.4 −1.48306
\(143\) 14799.2i 0.723714i
\(144\) 0 0
\(145\) 11476.4 0.545844
\(146\) − 21539.8i − 1.01050i
\(147\) 0 0
\(148\) −3254.35 −0.148573
\(149\) 15091.8i 0.679778i 0.940466 + 0.339889i \(0.110390\pi\)
−0.940466 + 0.339889i \(0.889610\pi\)
\(150\) 0 0
\(151\) 30255.3 1.32693 0.663465 0.748207i \(-0.269085\pi\)
0.663465 + 0.748207i \(0.269085\pi\)
\(152\) − 10268.5i − 0.444446i
\(153\) 0 0
\(154\) −31122.2 −1.31229
\(155\) 4847.92i 0.201787i
\(156\) 0 0
\(157\) 20622.7 0.836656 0.418328 0.908296i \(-0.362616\pi\)
0.418328 + 0.908296i \(0.362616\pi\)
\(158\) − 25580.6i − 1.02470i
\(159\) 0 0
\(160\) −7907.31 −0.308879
\(161\) 1047.77i 0.0404216i
\(162\) 0 0
\(163\) 39790.7 1.49764 0.748818 0.662776i \(-0.230622\pi\)
0.748818 + 0.662776i \(0.230622\pi\)
\(164\) 2190.31i 0.0814364i
\(165\) 0 0
\(166\) −12646.7 −0.458944
\(167\) 27450.7i 0.984283i 0.870515 + 0.492141i \(0.163786\pi\)
−0.870515 + 0.492141i \(0.836214\pi\)
\(168\) 0 0
\(169\) −4873.95 −0.170651
\(170\) 5207.81i 0.180201i
\(171\) 0 0
\(172\) −3515.89 −0.118844
\(173\) 10079.7i 0.336787i 0.985720 + 0.168393i \(0.0538578\pi\)
−0.985720 + 0.168393i \(0.946142\pi\)
\(174\) 0 0
\(175\) 26694.0 0.871642
\(176\) − 29184.3i − 0.942158i
\(177\) 0 0
\(178\) 63753.6 2.01217
\(179\) 1215.45i 0.0379343i 0.999820 + 0.0189672i \(0.00603779\pi\)
−0.999820 + 0.0189672i \(0.993962\pi\)
\(180\) 0 0
\(181\) 28359.9 0.865661 0.432831 0.901475i \(-0.357515\pi\)
0.432831 + 0.901475i \(0.357515\pi\)
\(182\) 49813.0i 1.50383i
\(183\) 0 0
\(184\) −778.522 −0.0229951
\(185\) − 13239.1i − 0.386826i
\(186\) 0 0
\(187\) −6999.36 −0.200159
\(188\) 15578.8i 0.440775i
\(189\) 0 0
\(190\) −13640.8 −0.377862
\(191\) − 9751.84i − 0.267313i −0.991028 0.133656i \(-0.957328\pi\)
0.991028 0.133656i \(-0.0426719\pi\)
\(192\) 0 0
\(193\) −53402.8 −1.43367 −0.716836 0.697242i \(-0.754410\pi\)
−0.716836 + 0.697242i \(0.754410\pi\)
\(194\) 31995.0i 0.850117i
\(195\) 0 0
\(196\) −11236.3 −0.292490
\(197\) 68537.7i 1.76603i 0.469349 + 0.883013i \(0.344489\pi\)
−0.469349 + 0.883013i \(0.655511\pi\)
\(198\) 0 0
\(199\) −8237.42 −0.208010 −0.104005 0.994577i \(-0.533166\pi\)
−0.104005 + 0.994577i \(0.533166\pi\)
\(200\) 19834.4i 0.495860i
\(201\) 0 0
\(202\) 8748.34 0.214399
\(203\) 51917.3i 1.25985i
\(204\) 0 0
\(205\) −8910.47 −0.212028
\(206\) − 23020.6i − 0.542477i
\(207\) 0 0
\(208\) −46711.2 −1.07968
\(209\) − 18333.4i − 0.419712i
\(210\) 0 0
\(211\) 40787.4 0.916138 0.458069 0.888917i \(-0.348541\pi\)
0.458069 + 0.888917i \(0.348541\pi\)
\(212\) 7747.00i 0.172370i
\(213\) 0 0
\(214\) −40710.2 −0.888946
\(215\) − 14303.1i − 0.309423i
\(216\) 0 0
\(217\) −21931.2 −0.465740
\(218\) − 72163.4i − 1.51846i
\(219\) 0 0
\(220\) −6068.13 −0.125375
\(221\) 11202.9i 0.229375i
\(222\) 0 0
\(223\) −59919.6 −1.20492 −0.602461 0.798148i \(-0.705813\pi\)
−0.602461 + 0.798148i \(0.705813\pi\)
\(224\) − 35771.4i − 0.712918i
\(225\) 0 0
\(226\) 94213.5 1.84458
\(227\) 98810.4i 1.91757i 0.284134 + 0.958785i \(0.408294\pi\)
−0.284134 + 0.958785i \(0.591706\pi\)
\(228\) 0 0
\(229\) −5317.94 −0.101408 −0.0507040 0.998714i \(-0.516147\pi\)
−0.0507040 + 0.998714i \(0.516147\pi\)
\(230\) 1034.20i 0.0195501i
\(231\) 0 0
\(232\) −38576.0 −0.716707
\(233\) − 18467.8i − 0.340176i −0.985429 0.170088i \(-0.945595\pi\)
0.985429 0.170088i \(-0.0544052\pi\)
\(234\) 0 0
\(235\) −63376.3 −1.14760
\(236\) 21327.1i 0.382920i
\(237\) 0 0
\(238\) −23559.3 −0.415919
\(239\) − 25846.2i − 0.452482i −0.974071 0.226241i \(-0.927356\pi\)
0.974071 0.226241i \(-0.0726437\pi\)
\(240\) 0 0
\(241\) 83072.3 1.43028 0.715142 0.698979i \(-0.246362\pi\)
0.715142 + 0.698979i \(0.246362\pi\)
\(242\) 24088.9i 0.411325i
\(243\) 0 0
\(244\) 6744.49 0.113284
\(245\) − 45710.7i − 0.761527i
\(246\) 0 0
\(247\) −29343.8 −0.480974
\(248\) − 16295.5i − 0.264951i
\(249\) 0 0
\(250\) 71064.1 1.13703
\(251\) 49051.6i 0.778585i 0.921114 + 0.389292i \(0.127280\pi\)
−0.921114 + 0.389292i \(0.872720\pi\)
\(252\) 0 0
\(253\) −1389.98 −0.0217154
\(254\) − 92256.3i − 1.42998i
\(255\) 0 0
\(256\) 45705.2 0.697405
\(257\) − 97105.7i − 1.47021i −0.677955 0.735104i \(-0.737133\pi\)
0.677955 0.735104i \(-0.262867\pi\)
\(258\) 0 0
\(259\) 59891.6 0.892825
\(260\) 9712.41i 0.143675i
\(261\) 0 0
\(262\) 14295.3 0.208253
\(263\) 41478.0i 0.599662i 0.953992 + 0.299831i \(0.0969303\pi\)
−0.953992 + 0.299831i \(0.903070\pi\)
\(264\) 0 0
\(265\) −31515.8 −0.448783
\(266\) − 61708.9i − 0.872136i
\(267\) 0 0
\(268\) 18253.3 0.254139
\(269\) − 115737.i − 1.59944i −0.600370 0.799722i \(-0.704980\pi\)
0.600370 0.799722i \(-0.295020\pi\)
\(270\) 0 0
\(271\) −5077.71 −0.0691400 −0.0345700 0.999402i \(-0.511006\pi\)
−0.0345700 + 0.999402i \(0.511006\pi\)
\(272\) − 22092.3i − 0.298609i
\(273\) 0 0
\(274\) 17289.1 0.230289
\(275\) 35412.5i 0.468265i
\(276\) 0 0
\(277\) −42385.8 −0.552409 −0.276205 0.961099i \(-0.589077\pi\)
−0.276205 + 0.961099i \(0.589077\pi\)
\(278\) − 52150.9i − 0.674795i
\(279\) 0 0
\(280\) 62548.8 0.797817
\(281\) − 41456.6i − 0.525027i −0.964928 0.262513i \(-0.915449\pi\)
0.964928 0.262513i \(-0.0845514\pi\)
\(282\) 0 0
\(283\) 38388.0 0.479316 0.239658 0.970857i \(-0.422965\pi\)
0.239658 + 0.970857i \(0.422965\pi\)
\(284\) − 26377.1i − 0.327032i
\(285\) 0 0
\(286\) −66082.4 −0.807893
\(287\) − 40309.6i − 0.489378i
\(288\) 0 0
\(289\) 78222.5 0.936561
\(290\) 51245.0i 0.609335i
\(291\) 0 0
\(292\) 18999.1 0.222827
\(293\) 12978.2i 0.151175i 0.997139 + 0.0755876i \(0.0240833\pi\)
−0.997139 + 0.0755876i \(0.975917\pi\)
\(294\) 0 0
\(295\) −86761.4 −0.996971
\(296\) 44501.2i 0.507911i
\(297\) 0 0
\(298\) −67388.7 −0.758847
\(299\) 2224.75i 0.0248850i
\(300\) 0 0
\(301\) 64704.8 0.714173
\(302\) 135098.i 1.48127i
\(303\) 0 0
\(304\) 57866.3 0.626150
\(305\) 27437.4i 0.294947i
\(306\) 0 0
\(307\) −126105. −1.33799 −0.668997 0.743265i \(-0.733276\pi\)
−0.668997 + 0.743265i \(0.733276\pi\)
\(308\) − 27451.3i − 0.289375i
\(309\) 0 0
\(310\) −21647.2 −0.225258
\(311\) 99803.5i 1.03187i 0.856628 + 0.515935i \(0.172555\pi\)
−0.856628 + 0.515935i \(0.827445\pi\)
\(312\) 0 0
\(313\) −2225.35 −0.0227148 −0.0113574 0.999936i \(-0.503615\pi\)
−0.0113574 + 0.999936i \(0.503615\pi\)
\(314\) 92085.9i 0.933972i
\(315\) 0 0
\(316\) 22563.3 0.225958
\(317\) − 168069.i − 1.67251i −0.548342 0.836254i \(-0.684741\pi\)
0.548342 0.836254i \(-0.315259\pi\)
\(318\) 0 0
\(319\) −68874.0 −0.676821
\(320\) 42498.8i 0.415027i
\(321\) 0 0
\(322\) −4678.56 −0.0451233
\(323\) − 13878.3i − 0.133024i
\(324\) 0 0
\(325\) 56679.9 0.536615
\(326\) 177676.i 1.67183i
\(327\) 0 0
\(328\) 29951.1 0.278398
\(329\) − 286704.i − 2.64876i
\(330\) 0 0
\(331\) 6758.86 0.0616904 0.0308452 0.999524i \(-0.490180\pi\)
0.0308452 + 0.999524i \(0.490180\pi\)
\(332\) − 11154.9i − 0.101202i
\(333\) 0 0
\(334\) −122574. −1.09877
\(335\) 74256.6i 0.661676i
\(336\) 0 0
\(337\) −222093. −1.95557 −0.977787 0.209600i \(-0.932784\pi\)
−0.977787 + 0.209600i \(0.932784\pi\)
\(338\) − 21763.5i − 0.190500i
\(339\) 0 0
\(340\) −4593.53 −0.0397364
\(341\) − 29094.2i − 0.250206i
\(342\) 0 0
\(343\) 32754.4 0.278408
\(344\) 48077.5i 0.406280i
\(345\) 0 0
\(346\) −45008.4 −0.375960
\(347\) 162548.i 1.34996i 0.737835 + 0.674981i \(0.235848\pi\)
−0.737835 + 0.674981i \(0.764152\pi\)
\(348\) 0 0
\(349\) −72696.9 −0.596850 −0.298425 0.954433i \(-0.596461\pi\)
−0.298425 + 0.954433i \(0.596461\pi\)
\(350\) 119196.i 0.973027i
\(351\) 0 0
\(352\) 47454.7 0.382995
\(353\) 128629.i 1.03226i 0.856509 + 0.516131i \(0.172628\pi\)
−0.856509 + 0.516131i \(0.827372\pi\)
\(354\) 0 0
\(355\) 107305. 0.851461
\(356\) 56233.6i 0.443707i
\(357\) 0 0
\(358\) −5427.32 −0.0423467
\(359\) 95302.8i 0.739463i 0.929139 + 0.369732i \(0.120550\pi\)
−0.929139 + 0.369732i \(0.879450\pi\)
\(360\) 0 0
\(361\) −93969.7 −0.721063
\(362\) 126635.i 0.966351i
\(363\) 0 0
\(364\) −43937.4 −0.331613
\(365\) 77290.7i 0.580152i
\(366\) 0 0
\(367\) 104146. 0.773233 0.386616 0.922241i \(-0.373644\pi\)
0.386616 + 0.922241i \(0.373644\pi\)
\(368\) − 4387.23i − 0.0323963i
\(369\) 0 0
\(370\) 59116.1 0.431820
\(371\) − 142572.i − 1.03583i
\(372\) 0 0
\(373\) 245831. 1.76693 0.883463 0.468501i \(-0.155206\pi\)
0.883463 + 0.468501i \(0.155206\pi\)
\(374\) − 31254.0i − 0.223441i
\(375\) 0 0
\(376\) 213029. 1.50683
\(377\) 110237.i 0.775612i
\(378\) 0 0
\(379\) 200830. 1.39814 0.699070 0.715053i \(-0.253597\pi\)
0.699070 + 0.715053i \(0.253597\pi\)
\(380\) − 12031.8i − 0.0833230i
\(381\) 0 0
\(382\) 43544.5 0.298406
\(383\) − 135338.i − 0.922615i −0.887240 0.461308i \(-0.847380\pi\)
0.887240 0.461308i \(-0.152620\pi\)
\(384\) 0 0
\(385\) 111675. 0.753417
\(386\) − 238458.i − 1.60043i
\(387\) 0 0
\(388\) −28221.1 −0.187461
\(389\) − 202177.i − 1.33608i −0.744126 0.668039i \(-0.767134\pi\)
0.744126 0.668039i \(-0.232866\pi\)
\(390\) 0 0
\(391\) −1052.20 −0.00688251
\(392\) 153649.i 0.999904i
\(393\) 0 0
\(394\) −306039. −1.97144
\(395\) 91790.2i 0.588304i
\(396\) 0 0
\(397\) 102384. 0.649608 0.324804 0.945781i \(-0.394702\pi\)
0.324804 + 0.945781i \(0.394702\pi\)
\(398\) − 36782.2i − 0.232205i
\(399\) 0 0
\(400\) −111774. −0.698585
\(401\) 90097.2i 0.560302i 0.959956 + 0.280151i \(0.0903846\pi\)
−0.959956 + 0.280151i \(0.909615\pi\)
\(402\) 0 0
\(403\) −46567.0 −0.286727
\(404\) 7716.45i 0.0472775i
\(405\) 0 0
\(406\) −231824. −1.40639
\(407\) 79452.8i 0.479645i
\(408\) 0 0
\(409\) 149722. 0.895033 0.447517 0.894276i \(-0.352309\pi\)
0.447517 + 0.894276i \(0.352309\pi\)
\(410\) − 39787.6i − 0.236690i
\(411\) 0 0
\(412\) 20305.2 0.119623
\(413\) − 392495.i − 2.30109i
\(414\) 0 0
\(415\) 45379.7 0.263491
\(416\) − 75954.1i − 0.438899i
\(417\) 0 0
\(418\) 81863.6 0.468531
\(419\) − 126351.i − 0.719697i −0.933011 0.359848i \(-0.882828\pi\)
0.933011 0.359848i \(-0.117172\pi\)
\(420\) 0 0
\(421\) −215289. −1.21467 −0.607335 0.794446i \(-0.707762\pi\)
−0.607335 + 0.794446i \(0.707762\pi\)
\(422\) 182126.i 1.02270i
\(423\) 0 0
\(424\) 105935. 0.589263
\(425\) 26807.0i 0.148413i
\(426\) 0 0
\(427\) −124123. −0.680762
\(428\) − 35908.3i − 0.196023i
\(429\) 0 0
\(430\) 63866.9 0.345413
\(431\) − 121972.i − 0.656607i −0.944572 0.328304i \(-0.893523\pi\)
0.944572 0.328304i \(-0.106477\pi\)
\(432\) 0 0
\(433\) −152419. −0.812951 −0.406475 0.913662i \(-0.633242\pi\)
−0.406475 + 0.913662i \(0.633242\pi\)
\(434\) − 97928.7i − 0.519913i
\(435\) 0 0
\(436\) 63651.5 0.334839
\(437\) − 2756.04i − 0.0144319i
\(438\) 0 0
\(439\) 231128. 1.19929 0.599643 0.800267i \(-0.295309\pi\)
0.599643 + 0.800267i \(0.295309\pi\)
\(440\) 82977.9i 0.428605i
\(441\) 0 0
\(442\) −50023.9 −0.256055
\(443\) − 20185.7i − 0.102858i −0.998677 0.0514289i \(-0.983622\pi\)
0.998677 0.0514289i \(-0.0163775\pi\)
\(444\) 0 0
\(445\) −228766. −1.15524
\(446\) − 267557.i − 1.34507i
\(447\) 0 0
\(448\) −192258. −0.957917
\(449\) − 92962.8i − 0.461123i −0.973058 0.230561i \(-0.925944\pi\)
0.973058 0.230561i \(-0.0740562\pi\)
\(450\) 0 0
\(451\) 53475.0 0.262905
\(452\) 83100.7i 0.406750i
\(453\) 0 0
\(454\) −441215. −2.14061
\(455\) − 178743.i − 0.863388i
\(456\) 0 0
\(457\) 220887. 1.05764 0.528819 0.848735i \(-0.322635\pi\)
0.528819 + 0.848735i \(0.322635\pi\)
\(458\) − 23746.0i − 0.113203i
\(459\) 0 0
\(460\) −912.215 −0.00431103
\(461\) 106195.i 0.499690i 0.968286 + 0.249845i \(0.0803797\pi\)
−0.968286 + 0.249845i \(0.919620\pi\)
\(462\) 0 0
\(463\) −68179.9 −0.318049 −0.159025 0.987275i \(-0.550835\pi\)
−0.159025 + 0.987275i \(0.550835\pi\)
\(464\) − 217389.i − 1.00972i
\(465\) 0 0
\(466\) 82463.6 0.379744
\(467\) 224350.i 1.02871i 0.857578 + 0.514353i \(0.171968\pi\)
−0.857578 + 0.514353i \(0.828032\pi\)
\(468\) 0 0
\(469\) −335925. −1.52720
\(470\) − 282992.i − 1.28109i
\(471\) 0 0
\(472\) 291635. 1.30905
\(473\) 85838.0i 0.383669i
\(474\) 0 0
\(475\) −70215.6 −0.311205
\(476\) − 20780.4i − 0.0917149i
\(477\) 0 0
\(478\) 115410. 0.505113
\(479\) − 109169.i − 0.475802i −0.971289 0.237901i \(-0.923541\pi\)
0.971289 0.237901i \(-0.0764594\pi\)
\(480\) 0 0
\(481\) 127169. 0.549656
\(482\) 370940.i 1.59665i
\(483\) 0 0
\(484\) −21247.5 −0.0907021
\(485\) − 114807.i − 0.488073i
\(486\) 0 0
\(487\) 106309. 0.448240 0.224120 0.974562i \(-0.428049\pi\)
0.224120 + 0.974562i \(0.428049\pi\)
\(488\) − 92226.6i − 0.387272i
\(489\) 0 0
\(490\) 204110. 0.850105
\(491\) 74446.6i 0.308803i 0.988008 + 0.154402i \(0.0493450\pi\)
−0.988008 + 0.154402i \(0.950655\pi\)
\(492\) 0 0
\(493\) −52137.1 −0.214513
\(494\) − 131028.i − 0.536919i
\(495\) 0 0
\(496\) 91830.8 0.373271
\(497\) 485432.i 1.96524i
\(498\) 0 0
\(499\) −111207. −0.446612 −0.223306 0.974748i \(-0.571685\pi\)
−0.223306 + 0.974748i \(0.571685\pi\)
\(500\) 62681.8i 0.250727i
\(501\) 0 0
\(502\) −219028. −0.869146
\(503\) 115897.i 0.458077i 0.973417 + 0.229038i \(0.0735581\pi\)
−0.973417 + 0.229038i \(0.926442\pi\)
\(504\) 0 0
\(505\) −31391.5 −0.123092
\(506\) − 6206.63i − 0.0242412i
\(507\) 0 0
\(508\) 81374.4 0.315326
\(509\) 196644.i 0.759007i 0.925190 + 0.379503i \(0.123905\pi\)
−0.925190 + 0.379503i \(0.876095\pi\)
\(510\) 0 0
\(511\) −349651. −1.33904
\(512\) − 111753.i − 0.426305i
\(513\) 0 0
\(514\) 433603. 1.64122
\(515\) 82604.1i 0.311449i
\(516\) 0 0
\(517\) 380345. 1.42297
\(518\) 267432.i 0.996675i
\(519\) 0 0
\(520\) 132811. 0.491165
\(521\) 299306.i 1.10266i 0.834289 + 0.551328i \(0.185879\pi\)
−0.834289 + 0.551328i \(0.814121\pi\)
\(522\) 0 0
\(523\) −162892. −0.595520 −0.297760 0.954641i \(-0.596240\pi\)
−0.297760 + 0.954641i \(0.596240\pi\)
\(524\) 12609.1i 0.0459222i
\(525\) 0 0
\(526\) −185210. −0.669412
\(527\) − 22024.1i − 0.0793006i
\(528\) 0 0
\(529\) 279632. 0.999253
\(530\) − 140726.i − 0.500983i
\(531\) 0 0
\(532\) 54430.1 0.192316
\(533\) − 85590.0i − 0.301279i
\(534\) 0 0
\(535\) 146079. 0.510365
\(536\) − 249602.i − 0.868796i
\(537\) 0 0
\(538\) 516798. 1.78548
\(539\) 274327.i 0.944257i
\(540\) 0 0
\(541\) −62918.7 −0.214974 −0.107487 0.994207i \(-0.534280\pi\)
−0.107487 + 0.994207i \(0.534280\pi\)
\(542\) − 22673.3i − 0.0771820i
\(543\) 0 0
\(544\) 35922.8 0.121387
\(545\) 258942.i 0.871786i
\(546\) 0 0
\(547\) −14807.1 −0.0494876 −0.0247438 0.999694i \(-0.507877\pi\)
−0.0247438 + 0.999694i \(0.507877\pi\)
\(548\) 15249.8i 0.0507813i
\(549\) 0 0
\(550\) −158126. −0.522732
\(551\) − 136563.i − 0.449810i
\(552\) 0 0
\(553\) −415244. −1.35785
\(554\) − 189264.i − 0.616663i
\(555\) 0 0
\(556\) 45999.5 0.148800
\(557\) 254392.i 0.819962i 0.912094 + 0.409981i \(0.134465\pi\)
−0.912094 + 0.409981i \(0.865535\pi\)
\(558\) 0 0
\(559\) 137389. 0.439671
\(560\) 352484.i 1.12399i
\(561\) 0 0
\(562\) 185115. 0.586096
\(563\) − 93307.5i − 0.294374i −0.989109 0.147187i \(-0.952978\pi\)
0.989109 0.147187i \(-0.0470220\pi\)
\(564\) 0 0
\(565\) −338064. −1.05902
\(566\) 171412.i 0.535068i
\(567\) 0 0
\(568\) −360690. −1.11799
\(569\) − 158024.i − 0.488089i −0.969764 0.244045i \(-0.921526\pi\)
0.969764 0.244045i \(-0.0784744\pi\)
\(570\) 0 0
\(571\) −438375. −1.34454 −0.672270 0.740306i \(-0.734681\pi\)
−0.672270 + 0.740306i \(0.734681\pi\)
\(572\) − 58287.8i − 0.178150i
\(573\) 0 0
\(574\) 179993. 0.546300
\(575\) 5323.52i 0.0161014i
\(576\) 0 0
\(577\) 214770. 0.645094 0.322547 0.946554i \(-0.395461\pi\)
0.322547 + 0.946554i \(0.395461\pi\)
\(578\) 349284.i 1.04550i
\(579\) 0 0
\(580\) −45200.5 −0.134365
\(581\) 205291.i 0.608158i
\(582\) 0 0
\(583\) 189138. 0.556469
\(584\) − 259801.i − 0.761753i
\(585\) 0 0
\(586\) −57951.3 −0.168759
\(587\) 656412.i 1.90502i 0.304502 + 0.952512i \(0.401510\pi\)
−0.304502 + 0.952512i \(0.598490\pi\)
\(588\) 0 0
\(589\) 57687.6 0.166285
\(590\) − 387413.i − 1.11293i
\(591\) 0 0
\(592\) −250779. −0.715563
\(593\) − 118369.i − 0.336612i −0.985735 0.168306i \(-0.946170\pi\)
0.985735 0.168306i \(-0.0538296\pi\)
\(594\) 0 0
\(595\) 84537.3 0.238789
\(596\) − 59440.0i − 0.167335i
\(597\) 0 0
\(598\) −9934.08 −0.0277796
\(599\) 354139.i 0.987006i 0.869744 + 0.493503i \(0.164284\pi\)
−0.869744 + 0.493503i \(0.835716\pi\)
\(600\) 0 0
\(601\) −595639. −1.64905 −0.824526 0.565824i \(-0.808558\pi\)
−0.824526 + 0.565824i \(0.808558\pi\)
\(602\) 288924.i 0.797242i
\(603\) 0 0
\(604\) −119163. −0.326638
\(605\) − 86437.5i − 0.236152i
\(606\) 0 0
\(607\) 9721.62 0.0263853 0.0131926 0.999913i \(-0.495801\pi\)
0.0131926 + 0.999913i \(0.495801\pi\)
\(608\) 94092.6i 0.254536i
\(609\) 0 0
\(610\) −122515. −0.329254
\(611\) − 608765.i − 1.63067i
\(612\) 0 0
\(613\) 194450. 0.517473 0.258736 0.965948i \(-0.416694\pi\)
0.258736 + 0.965948i \(0.416694\pi\)
\(614\) − 563090.i − 1.49362i
\(615\) 0 0
\(616\) −375379. −0.989255
\(617\) − 487639.i − 1.28094i −0.767984 0.640469i \(-0.778740\pi\)
0.767984 0.640469i \(-0.221260\pi\)
\(618\) 0 0
\(619\) 17256.5 0.0450371 0.0225185 0.999746i \(-0.492832\pi\)
0.0225185 + 0.999746i \(0.492832\pi\)
\(620\) − 19093.9i − 0.0496719i
\(621\) 0 0
\(622\) −445649. −1.15189
\(623\) − 1.03490e6i − 2.66638i
\(624\) 0 0
\(625\) −24825.0 −0.0635520
\(626\) − 9936.78i − 0.0253569i
\(627\) 0 0
\(628\) −81224.1 −0.205952
\(629\) 60145.1i 0.152019i
\(630\) 0 0
\(631\) −429836. −1.07955 −0.539777 0.841808i \(-0.681491\pi\)
−0.539777 + 0.841808i \(0.681491\pi\)
\(632\) − 308538.i − 0.772458i
\(633\) 0 0
\(634\) 750471. 1.86705
\(635\) 331041.i 0.820984i
\(636\) 0 0
\(637\) 439076. 1.08208
\(638\) − 307540.i − 0.755546i
\(639\) 0 0
\(640\) −316285. −0.772181
\(641\) − 32308.0i − 0.0786310i −0.999227 0.0393155i \(-0.987482\pi\)
0.999227 0.0393155i \(-0.0125177\pi\)
\(642\) 0 0
\(643\) 277048. 0.670091 0.335045 0.942202i \(-0.391248\pi\)
0.335045 + 0.942202i \(0.391248\pi\)
\(644\) − 4126.71i − 0.00995022i
\(645\) 0 0
\(646\) 61970.1 0.148497
\(647\) − 125452.i − 0.299689i −0.988710 0.149844i \(-0.952123\pi\)
0.988710 0.149844i \(-0.0478773\pi\)
\(648\) 0 0
\(649\) 520687. 1.23620
\(650\) 253091.i 0.599031i
\(651\) 0 0
\(652\) −156718. −0.368659
\(653\) − 283982.i − 0.665985i −0.942930 0.332992i \(-0.891942\pi\)
0.942930 0.332992i \(-0.108058\pi\)
\(654\) 0 0
\(655\) −51295.5 −0.119563
\(656\) 168785.i 0.392216i
\(657\) 0 0
\(658\) 1.28021e6 2.95685
\(659\) 359272.i 0.827280i 0.910440 + 0.413640i \(0.135743\pi\)
−0.910440 + 0.413640i \(0.864257\pi\)
\(660\) 0 0
\(661\) 197804. 0.452723 0.226362 0.974043i \(-0.427317\pi\)
0.226362 + 0.974043i \(0.427317\pi\)
\(662\) 30180.1i 0.0688659i
\(663\) 0 0
\(664\) −152537. −0.345970
\(665\) 221429.i 0.500715i
\(666\) 0 0
\(667\) −10353.7 −0.0232726
\(668\) − 108116.i − 0.242292i
\(669\) 0 0
\(670\) −331575. −0.738639
\(671\) − 164662.i − 0.365720i
\(672\) 0 0
\(673\) 238072. 0.525628 0.262814 0.964847i \(-0.415349\pi\)
0.262814 + 0.964847i \(0.415349\pi\)
\(674\) − 991702.i − 2.18304i
\(675\) 0 0
\(676\) 19196.4 0.0420075
\(677\) 250150.i 0.545787i 0.962044 + 0.272894i \(0.0879807\pi\)
−0.962044 + 0.272894i \(0.912019\pi\)
\(678\) 0 0
\(679\) 519368. 1.12651
\(680\) 62813.6i 0.135843i
\(681\) 0 0
\(682\) 129913. 0.279309
\(683\) 267358.i 0.573129i 0.958061 + 0.286564i \(0.0925132\pi\)
−0.958061 + 0.286564i \(0.907487\pi\)
\(684\) 0 0
\(685\) −62038.2 −0.132214
\(686\) 146257.i 0.310791i
\(687\) 0 0
\(688\) −270933. −0.572380
\(689\) − 302727.i − 0.637694i
\(690\) 0 0
\(691\) 336686. 0.705129 0.352565 0.935787i \(-0.385310\pi\)
0.352565 + 0.935787i \(0.385310\pi\)
\(692\) − 39699.5i − 0.0829036i
\(693\) 0 0
\(694\) −725817. −1.50698
\(695\) 187132.i 0.387417i
\(696\) 0 0
\(697\) 40480.2 0.0833253
\(698\) − 324611.i − 0.666273i
\(699\) 0 0
\(700\) −105136. −0.214564
\(701\) 635795.i 1.29384i 0.762557 + 0.646921i \(0.223944\pi\)
−0.762557 + 0.646921i \(0.776056\pi\)
\(702\) 0 0
\(703\) −157538. −0.318768
\(704\) − 255051.i − 0.514614i
\(705\) 0 0
\(706\) −574363. −1.15233
\(707\) − 142010.i − 0.284106i
\(708\) 0 0
\(709\) 431026. 0.857455 0.428728 0.903434i \(-0.358962\pi\)
0.428728 + 0.903434i \(0.358962\pi\)
\(710\) 479147.i 0.950499i
\(711\) 0 0
\(712\) 768959. 1.51685
\(713\) − 4373.69i − 0.00860337i
\(714\) 0 0
\(715\) 237122. 0.463831
\(716\) − 4787.15i − 0.00933793i
\(717\) 0 0
\(718\) −425552. −0.825475
\(719\) − 798370.i − 1.54435i −0.635409 0.772176i \(-0.719168\pi\)
0.635409 0.772176i \(-0.280832\pi\)
\(720\) 0 0
\(721\) −373688. −0.718850
\(722\) − 419599.i − 0.804934i
\(723\) 0 0
\(724\) −111698. −0.213092
\(725\) 263782.i 0.501845i
\(726\) 0 0
\(727\) 317113. 0.599992 0.299996 0.953940i \(-0.403015\pi\)
0.299996 + 0.953940i \(0.403015\pi\)
\(728\) 600816.i 1.13365i
\(729\) 0 0
\(730\) −345123. −0.647632
\(731\) 64978.7i 0.121601i
\(732\) 0 0
\(733\) −977843. −1.81996 −0.909979 0.414654i \(-0.863902\pi\)
−0.909979 + 0.414654i \(0.863902\pi\)
\(734\) 465039.i 0.863172i
\(735\) 0 0
\(736\) 7133.80 0.0131694
\(737\) − 445641.i − 0.820446i
\(738\) 0 0
\(739\) −392565. −0.718825 −0.359412 0.933179i \(-0.617023\pi\)
−0.359412 + 0.933179i \(0.617023\pi\)
\(740\) 52143.2i 0.0952213i
\(741\) 0 0
\(742\) 636623. 1.15631
\(743\) − 1.03860e6i − 1.88135i −0.339314 0.940673i \(-0.610195\pi\)
0.339314 0.940673i \(-0.389805\pi\)
\(744\) 0 0
\(745\) 241809. 0.435673
\(746\) 1.09770e6i 1.97245i
\(747\) 0 0
\(748\) 27567.5 0.0492713
\(749\) 660840.i 1.17797i
\(750\) 0 0
\(751\) 319565. 0.566604 0.283302 0.959031i \(-0.408570\pi\)
0.283302 + 0.959031i \(0.408570\pi\)
\(752\) 1.20049e6i 2.12287i
\(753\) 0 0
\(754\) −492237. −0.865828
\(755\) − 484769.i − 0.850435i
\(756\) 0 0
\(757\) 500321. 0.873085 0.436543 0.899684i \(-0.356203\pi\)
0.436543 + 0.899684i \(0.356203\pi\)
\(758\) 896760.i 1.56077i
\(759\) 0 0
\(760\) −164528. −0.284847
\(761\) 904875.i 1.56250i 0.624220 + 0.781249i \(0.285417\pi\)
−0.624220 + 0.781249i \(0.714583\pi\)
\(762\) 0 0
\(763\) −1.17141e6 −2.01215
\(764\) 38408.3i 0.0658019i
\(765\) 0 0
\(766\) 604318. 1.02993
\(767\) − 833392.i − 1.41664i
\(768\) 0 0
\(769\) 143377. 0.242453 0.121227 0.992625i \(-0.461317\pi\)
0.121227 + 0.992625i \(0.461317\pi\)
\(770\) 498659.i 0.841051i
\(771\) 0 0
\(772\) 210331. 0.352914
\(773\) 710063.i 1.18833i 0.804342 + 0.594166i \(0.202518\pi\)
−0.804342 + 0.594166i \(0.797482\pi\)
\(774\) 0 0
\(775\) −111428. −0.185521
\(776\) 385905.i 0.640851i
\(777\) 0 0
\(778\) 902772. 1.49148
\(779\) 106030.i 0.174724i
\(780\) 0 0
\(781\) −643979. −1.05577
\(782\) − 4698.37i − 0.00768305i
\(783\) 0 0
\(784\) −865865. −1.40870
\(785\) − 330430.i − 0.536216i
\(786\) 0 0
\(787\) −601711. −0.971490 −0.485745 0.874101i \(-0.661452\pi\)
−0.485745 + 0.874101i \(0.661452\pi\)
\(788\) − 269941.i − 0.434726i
\(789\) 0 0
\(790\) −409867. −0.656733
\(791\) − 1.52935e6i − 2.44429i
\(792\) 0 0
\(793\) −263552. −0.419102
\(794\) 457172.i 0.725168i
\(795\) 0 0
\(796\) 32443.7 0.0512040
\(797\) 429811.i 0.676645i 0.941030 + 0.338322i \(0.109859\pi\)
−0.941030 + 0.338322i \(0.890141\pi\)
\(798\) 0 0
\(799\) 287918. 0.450999
\(800\) − 181748.i − 0.283981i
\(801\) 0 0
\(802\) −402308. −0.625474
\(803\) − 463850.i − 0.719361i
\(804\) 0 0
\(805\) 16788.0 0.0259064
\(806\) − 207934.i − 0.320077i
\(807\) 0 0
\(808\) 105517. 0.161622
\(809\) 629639.i 0.962042i 0.876709 + 0.481021i \(0.159734\pi\)
−0.876709 + 0.481021i \(0.840266\pi\)
\(810\) 0 0
\(811\) 577760. 0.878428 0.439214 0.898383i \(-0.355257\pi\)
0.439214 + 0.898383i \(0.355257\pi\)
\(812\) − 204480.i − 0.310126i
\(813\) 0 0
\(814\) −354778. −0.535436
\(815\) − 637550.i − 0.959841i
\(816\) 0 0
\(817\) −170199. −0.254984
\(818\) 668548.i 0.999140i
\(819\) 0 0
\(820\) 35094.5 0.0521929
\(821\) 44719.6i 0.0663455i 0.999450 + 0.0331727i \(0.0105612\pi\)
−0.999450 + 0.0331727i \(0.989439\pi\)
\(822\) 0 0
\(823\) −292578. −0.431958 −0.215979 0.976398i \(-0.569294\pi\)
−0.215979 + 0.976398i \(0.569294\pi\)
\(824\) − 277661.i − 0.408940i
\(825\) 0 0
\(826\) 1.75259e6 2.56874
\(827\) 806491.i 1.17920i 0.807694 + 0.589601i \(0.200715\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(828\) 0 0
\(829\) 204904. 0.298154 0.149077 0.988826i \(-0.452370\pi\)
0.149077 + 0.988826i \(0.452370\pi\)
\(830\) 202632.i 0.294139i
\(831\) 0 0
\(832\) −408224. −0.589729
\(833\) 207663.i 0.299274i
\(834\) 0 0
\(835\) 439831. 0.630831
\(836\) 72207.5i 0.103317i
\(837\) 0 0
\(838\) 564189. 0.803409
\(839\) 771105.i 1.09544i 0.836661 + 0.547721i \(0.184505\pi\)
−0.836661 + 0.547721i \(0.815495\pi\)
\(840\) 0 0
\(841\) 194250. 0.274643
\(842\) − 961324.i − 1.35596i
\(843\) 0 0
\(844\) −160644. −0.225517
\(845\) 78093.4i 0.109371i
\(846\) 0 0
\(847\) 391030. 0.545058
\(848\) 596981.i 0.830173i
\(849\) 0 0
\(850\) −119700. −0.165675
\(851\) 11944.0i 0.0164927i
\(852\) 0 0
\(853\) 1.08237e6 1.48757 0.743784 0.668420i \(-0.233029\pi\)
0.743784 + 0.668420i \(0.233029\pi\)
\(854\) − 554240.i − 0.759945i
\(855\) 0 0
\(856\) −491023. −0.670122
\(857\) − 533345.i − 0.726183i −0.931753 0.363092i \(-0.881721\pi\)
0.931753 0.363092i \(-0.118279\pi\)
\(858\) 0 0
\(859\) 1.13105e6 1.53283 0.766417 0.642343i \(-0.222038\pi\)
0.766417 + 0.642343i \(0.222038\pi\)
\(860\) 56333.6i 0.0761677i
\(861\) 0 0
\(862\) 544637. 0.732981
\(863\) − 110025.i − 0.147730i −0.997268 0.0738649i \(-0.976467\pi\)
0.997268 0.0738649i \(-0.0235334\pi\)
\(864\) 0 0
\(865\) 161503. 0.215848
\(866\) − 680592.i − 0.907510i
\(867\) 0 0
\(868\) 86377.7 0.114647
\(869\) − 550867.i − 0.729470i
\(870\) 0 0
\(871\) −713275. −0.940202
\(872\) − 870394.i − 1.14468i
\(873\) 0 0
\(874\) 12306.4 0.0161105
\(875\) − 1.15357e6i − 1.50670i
\(876\) 0 0
\(877\) −408753. −0.531450 −0.265725 0.964049i \(-0.585611\pi\)
−0.265725 + 0.964049i \(0.585611\pi\)
\(878\) 1.03205e6i 1.33878i
\(879\) 0 0
\(880\) −467608. −0.603833
\(881\) 235809.i 0.303815i 0.988395 + 0.151907i \(0.0485416\pi\)
−0.988395 + 0.151907i \(0.951458\pi\)
\(882\) 0 0
\(883\) −273719. −0.351061 −0.175531 0.984474i \(-0.556164\pi\)
−0.175531 + 0.984474i \(0.556164\pi\)
\(884\) − 44123.4i − 0.0564631i
\(885\) 0 0
\(886\) 90134.6 0.114822
\(887\) − 371896.i − 0.472687i −0.971670 0.236344i \(-0.924051\pi\)
0.971670 0.236344i \(-0.0759491\pi\)
\(888\) 0 0
\(889\) −1.49758e6 −1.89490
\(890\) − 1.02150e6i − 1.28961i
\(891\) 0 0
\(892\) 235997. 0.296604
\(893\) 754144.i 0.945695i
\(894\) 0 0
\(895\) 19474.7 0.0243122
\(896\) − 1.43082e6i − 1.78226i
\(897\) 0 0
\(898\) 415103. 0.514758
\(899\) − 216718.i − 0.268148i
\(900\) 0 0
\(901\) 143176. 0.176368
\(902\) 238780.i 0.293484i
\(903\) 0 0
\(904\) 1.13635e6 1.39051
\(905\) − 454400.i − 0.554806i
\(906\) 0 0
\(907\) −143931. −0.174960 −0.0874801 0.996166i \(-0.527881\pi\)
−0.0874801 + 0.996166i \(0.527881\pi\)
\(908\) − 389172.i − 0.472030i
\(909\) 0 0
\(910\) 798134. 0.963814
\(911\) − 891661.i − 1.07439i −0.843457 0.537196i \(-0.819483\pi\)
0.843457 0.537196i \(-0.180517\pi\)
\(912\) 0 0
\(913\) −272340. −0.326716
\(914\) 986317.i 1.18066i
\(915\) 0 0
\(916\) 20945.1 0.0249627
\(917\) − 232053.i − 0.275961i
\(918\) 0 0
\(919\) −582156. −0.689300 −0.344650 0.938731i \(-0.612002\pi\)
−0.344650 + 0.938731i \(0.612002\pi\)
\(920\) 12474.0i 0.0147377i
\(921\) 0 0
\(922\) −474187. −0.557812
\(923\) 1.03073e6i 1.20987i
\(924\) 0 0
\(925\) 304298. 0.355644
\(926\) − 304441.i − 0.355043i
\(927\) 0 0
\(928\) 353482. 0.410460
\(929\) − 771371.i − 0.893782i −0.894588 0.446891i \(-0.852531\pi\)
0.894588 0.446891i \(-0.147469\pi\)
\(930\) 0 0
\(931\) −543932. −0.627546
\(932\) 72736.8i 0.0837380i
\(933\) 0 0
\(934\) −1.00178e6 −1.14836
\(935\) 112148.i 0.128283i
\(936\) 0 0
\(937\) −1.46097e6 −1.66403 −0.832015 0.554753i \(-0.812813\pi\)
−0.832015 + 0.554753i \(0.812813\pi\)
\(938\) − 1.49999e6i − 1.70484i
\(939\) 0 0
\(940\) 249612. 0.282494
\(941\) − 169346.i − 0.191248i −0.995418 0.0956240i \(-0.969515\pi\)
0.995418 0.0956240i \(-0.0304846\pi\)
\(942\) 0 0
\(943\) 8038.83 0.00904002
\(944\) 1.64346e6i 1.84423i
\(945\) 0 0
\(946\) −383289. −0.428296
\(947\) 593888.i 0.662224i 0.943591 + 0.331112i \(0.107424\pi\)
−0.943591 + 0.331112i \(0.892576\pi\)
\(948\) 0 0
\(949\) −742420. −0.824361
\(950\) − 313531.i − 0.347403i
\(951\) 0 0
\(952\) −284159. −0.313536
\(953\) − 1.25477e6i − 1.38159i −0.723050 0.690796i \(-0.757260\pi\)
0.723050 0.690796i \(-0.242740\pi\)
\(954\) 0 0
\(955\) −156250. −0.171322
\(956\) 101797.i 0.111383i
\(957\) 0 0
\(958\) 487466. 0.531145
\(959\) − 280651.i − 0.305161i
\(960\) 0 0
\(961\) −831974. −0.900872
\(962\) 567843.i 0.613590i
\(963\) 0 0
\(964\) −327186. −0.352080
\(965\) 855652.i 0.918846i
\(966\) 0 0
\(967\) −1.05786e6 −1.13129 −0.565645 0.824649i \(-0.691373\pi\)
−0.565645 + 0.824649i \(0.691373\pi\)
\(968\) 290546.i 0.310073i
\(969\) 0 0
\(970\) 512643. 0.544843
\(971\) 775745.i 0.822774i 0.911461 + 0.411387i \(0.134956\pi\)
−0.911461 + 0.411387i \(0.865044\pi\)
\(972\) 0 0
\(973\) −846555. −0.894189
\(974\) 474696.i 0.500377i
\(975\) 0 0
\(976\) 519728. 0.545603
\(977\) 496516.i 0.520168i 0.965586 + 0.260084i \(0.0837503\pi\)
−0.965586 + 0.260084i \(0.916250\pi\)
\(978\) 0 0
\(979\) 1.37291e6 1.43244
\(980\) 180035.i 0.187458i
\(981\) 0 0
\(982\) −332424. −0.344722
\(983\) 300964.i 0.311464i 0.987799 + 0.155732i \(0.0497736\pi\)
−0.987799 + 0.155732i \(0.950226\pi\)
\(984\) 0 0
\(985\) 1.09815e6 1.13185
\(986\) − 232806.i − 0.239464i
\(987\) 0 0
\(988\) 115572. 0.118397
\(989\) 12903.9i 0.0131925i
\(990\) 0 0
\(991\) 1.70094e6 1.73197 0.865987 0.500067i \(-0.166691\pi\)
0.865987 + 0.500067i \(0.166691\pi\)
\(992\) 149320.i 0.151738i
\(993\) 0 0
\(994\) −2.16758e6 −2.19383
\(995\) 131985.i 0.133315i
\(996\) 0 0
\(997\) −625837. −0.629609 −0.314805 0.949156i \(-0.601939\pi\)
−0.314805 + 0.949156i \(0.601939\pi\)
\(998\) − 496568.i − 0.498560i
\(999\) 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.80.5 6
3.2 odd 2 inner 81.5.b.a.80.2 6
4.3 odd 2 1296.5.e.c.161.2 6
9.2 odd 6 27.5.d.a.17.1 6
9.4 even 3 27.5.d.a.8.1 6
9.5 odd 6 9.5.d.a.2.3 6
9.7 even 3 9.5.d.a.5.3 yes 6
12.11 even 2 1296.5.e.c.161.5 6
36.7 odd 6 144.5.q.a.113.3 6
36.11 even 6 432.5.q.a.17.1 6
36.23 even 6 144.5.q.a.65.3 6
36.31 odd 6 432.5.q.a.305.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.5.d.a.2.3 6 9.5 odd 6
9.5.d.a.5.3 yes 6 9.7 even 3
27.5.d.a.8.1 6 9.4 even 3
27.5.d.a.17.1 6 9.2 odd 6
81.5.b.a.80.2 6 3.2 odd 2 inner
81.5.b.a.80.5 6 1.1 even 1 trivial
144.5.q.a.65.3 6 36.23 even 6
144.5.q.a.113.3 6 36.7 odd 6
432.5.q.a.17.1 6 36.11 even 6
432.5.q.a.305.1 6 36.31 odd 6
1296.5.e.c.161.2 6 4.3 odd 2
1296.5.e.c.161.5 6 12.11 even 2