Properties

Label 810.3.b.a.809.1
Level $810$
Weight $3$
Character 810.809
Analytic conductor $22.071$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,3,Mod(809,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 809.1
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 810.809
Dual form 810.3.b.a.809.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +(-4.87832 - 1.09638i) q^{5} +2.10102i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +(-4.87832 - 1.09638i) q^{5} +2.10102i q^{7} -2.82843 q^{8} +(6.89898 + 1.55051i) q^{10} -13.3636i q^{11} +23.1464i q^{13} -2.97129i q^{14} +4.00000 q^{16} +27.5057 q^{17} -15.5505 q^{19} +(-9.75663 - 2.19275i) q^{20} +18.8990i q^{22} -14.9528 q^{23} +(22.5959 + 10.6969i) q^{25} -32.7340i q^{26} +4.20204i q^{28} +3.28913i q^{29} -9.84337 q^{31} -5.65685 q^{32} -38.8990 q^{34} +(2.30351 - 10.2494i) q^{35} -33.3485i q^{37} +21.9917 q^{38} +(13.7980 + 3.10102i) q^{40} -10.8530i q^{41} -43.1918i q^{43} -26.7272i q^{44} +21.1464 q^{46} -36.0873 q^{47} +44.5857 q^{49} +(-31.9555 - 15.1278i) q^{50} +46.2929i q^{52} -53.1687 q^{53} +(-14.6515 + 65.1918i) q^{55} -5.94258i q^{56} -4.65153i q^{58} -101.252i q^{59} +0.101021 q^{61} +13.9206 q^{62} +8.00000 q^{64} +(25.3772 - 112.916i) q^{65} -55.9898i q^{67} +55.0115 q^{68} +(-3.25765 + 14.4949i) q^{70} +79.2457i q^{71} -20.8990i q^{73} +47.1619i q^{74} -31.1010 q^{76} +28.0772 q^{77} -1.48469 q^{79} +(-19.5133 - 4.38551i) q^{80} +15.3485i q^{82} -9.93160 q^{83} +(-134.182 - 30.1566i) q^{85} +61.0825i q^{86} +37.7980i q^{88} -152.974i q^{89} -48.6311 q^{91} -29.9056 q^{92} +51.0352 q^{94} +(75.8603 + 17.0492i) q^{95} -38.7878i q^{97} -63.0537 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 16 q^{10} + 32 q^{16} - 144 q^{19} + 24 q^{25} + 176 q^{31} - 272 q^{34} + 32 q^{40} + 32 q^{46} - 192 q^{49} - 176 q^{55} + 40 q^{61} + 64 q^{64} - 320 q^{70} - 288 q^{76} + 576 q^{79} - 368 q^{85} + 336 q^{91} - 160 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) −4.87832 1.09638i −0.975663 0.219275i
\(6\) 0 0
\(7\) 2.10102i 0.300146i 0.988675 + 0.150073i \(0.0479508\pi\)
−0.988675 + 0.150073i \(0.952049\pi\)
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 6.89898 + 1.55051i 0.689898 + 0.155051i
\(11\) 13.3636i 1.21487i −0.794368 0.607436i \(-0.792198\pi\)
0.794368 0.607436i \(-0.207802\pi\)
\(12\) 0 0
\(13\) 23.1464i 1.78049i 0.455478 + 0.890247i \(0.349469\pi\)
−0.455478 + 0.890247i \(0.650531\pi\)
\(14\) 2.97129i 0.212235i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 27.5057 1.61798 0.808992 0.587819i \(-0.200013\pi\)
0.808992 + 0.587819i \(0.200013\pi\)
\(18\) 0 0
\(19\) −15.5505 −0.818448 −0.409224 0.912434i \(-0.634201\pi\)
−0.409224 + 0.912434i \(0.634201\pi\)
\(20\) −9.75663 2.19275i −0.487832 0.109638i
\(21\) 0 0
\(22\) 18.8990i 0.859045i
\(23\) −14.9528 −0.650121 −0.325060 0.945693i \(-0.605385\pi\)
−0.325060 + 0.945693i \(0.605385\pi\)
\(24\) 0 0
\(25\) 22.5959 + 10.6969i 0.903837 + 0.427878i
\(26\) 32.7340i 1.25900i
\(27\) 0 0
\(28\) 4.20204i 0.150073i
\(29\) 3.28913i 0.113418i 0.998391 + 0.0567091i \(0.0180608\pi\)
−0.998391 + 0.0567091i \(0.981939\pi\)
\(30\) 0 0
\(31\) −9.84337 −0.317528 −0.158764 0.987317i \(-0.550751\pi\)
−0.158764 + 0.987317i \(0.550751\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) −38.8990 −1.14409
\(35\) 2.30351 10.2494i 0.0658145 0.292841i
\(36\) 0 0
\(37\) 33.3485i 0.901310i −0.892698 0.450655i \(-0.851190\pi\)
0.892698 0.450655i \(-0.148810\pi\)
\(38\) 21.9917 0.578730
\(39\) 0 0
\(40\) 13.7980 + 3.10102i 0.344949 + 0.0775255i
\(41\) 10.8530i 0.264707i −0.991203 0.132354i \(-0.957747\pi\)
0.991203 0.132354i \(-0.0422535\pi\)
\(42\) 0 0
\(43\) 43.1918i 1.00446i −0.864734 0.502231i \(-0.832513\pi\)
0.864734 0.502231i \(-0.167487\pi\)
\(44\) 26.7272i 0.607436i
\(45\) 0 0
\(46\) 21.1464 0.459705
\(47\) −36.0873 −0.767816 −0.383908 0.923371i \(-0.625422\pi\)
−0.383908 + 0.923371i \(0.625422\pi\)
\(48\) 0 0
\(49\) 44.5857 0.909913
\(50\) −31.9555 15.1278i −0.639109 0.302555i
\(51\) 0 0
\(52\) 46.2929i 0.890247i
\(53\) −53.1687 −1.00318 −0.501591 0.865105i \(-0.667252\pi\)
−0.501591 + 0.865105i \(0.667252\pi\)
\(54\) 0 0
\(55\) −14.6515 + 65.1918i −0.266391 + 1.18531i
\(56\) 5.94258i 0.106118i
\(57\) 0 0
\(58\) 4.65153i 0.0801988i
\(59\) 101.252i 1.71613i −0.513538 0.858067i \(-0.671665\pi\)
0.513538 0.858067i \(-0.328335\pi\)
\(60\) 0 0
\(61\) 0.101021 0.00165607 0.000828037 1.00000i \(-0.499736\pi\)
0.000828037 1.00000i \(0.499736\pi\)
\(62\) 13.9206 0.224526
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 25.3772 112.916i 0.390418 1.73716i
\(66\) 0 0
\(67\) 55.9898i 0.835669i −0.908523 0.417834i \(-0.862789\pi\)
0.908523 0.417834i \(-0.137211\pi\)
\(68\) 55.0115 0.808992
\(69\) 0 0
\(70\) −3.25765 + 14.4949i −0.0465379 + 0.207070i
\(71\) 79.2457i 1.11614i 0.829795 + 0.558069i \(0.188457\pi\)
−0.829795 + 0.558069i \(0.811543\pi\)
\(72\) 0 0
\(73\) 20.8990i 0.286287i −0.989702 0.143144i \(-0.954279\pi\)
0.989702 0.143144i \(-0.0457211\pi\)
\(74\) 47.1619i 0.637322i
\(75\) 0 0
\(76\) −31.1010 −0.409224
\(77\) 28.0772 0.364639
\(78\) 0 0
\(79\) −1.48469 −0.0187936 −0.00939679 0.999956i \(-0.502991\pi\)
−0.00939679 + 0.999956i \(0.502991\pi\)
\(80\) −19.5133 4.38551i −0.243916 0.0548188i
\(81\) 0 0
\(82\) 15.3485i 0.187176i
\(83\) −9.93160 −0.119658 −0.0598289 0.998209i \(-0.519056\pi\)
−0.0598289 + 0.998209i \(0.519056\pi\)
\(84\) 0 0
\(85\) −134.182 30.1566i −1.57861 0.354784i
\(86\) 61.0825i 0.710261i
\(87\) 0 0
\(88\) 37.7980i 0.429522i
\(89\) 152.974i 1.71881i −0.511294 0.859406i \(-0.670834\pi\)
0.511294 0.859406i \(-0.329166\pi\)
\(90\) 0 0
\(91\) −48.6311 −0.534408
\(92\) −29.9056 −0.325060
\(93\) 0 0
\(94\) 51.0352 0.542928
\(95\) 75.8603 + 17.0492i 0.798529 + 0.179465i
\(96\) 0 0
\(97\) 38.7878i 0.399874i −0.979809 0.199937i \(-0.935926\pi\)
0.979809 0.199937i \(-0.0640737\pi\)
\(98\) −63.0537 −0.643405
\(99\) 0 0
\(100\) 45.1918 + 21.3939i 0.451918 + 0.213939i
\(101\) 81.9602i 0.811487i −0.913987 0.405743i \(-0.867013\pi\)
0.913987 0.405743i \(-0.132987\pi\)
\(102\) 0 0
\(103\) 115.778i 1.12405i −0.827119 0.562027i \(-0.810022\pi\)
0.827119 0.562027i \(-0.189978\pi\)
\(104\) 65.4680i 0.629500i
\(105\) 0 0
\(106\) 75.1918 0.709357
\(107\) −119.512 −1.11693 −0.558465 0.829528i \(-0.688609\pi\)
−0.558465 + 0.829528i \(0.688609\pi\)
\(108\) 0 0
\(109\) 62.9092 0.577148 0.288574 0.957458i \(-0.406819\pi\)
0.288574 + 0.957458i \(0.406819\pi\)
\(110\) 20.7204 92.1952i 0.188367 0.838138i
\(111\) 0 0
\(112\) 8.40408i 0.0750364i
\(113\) −80.0243 −0.708179 −0.354090 0.935211i \(-0.615209\pi\)
−0.354090 + 0.935211i \(0.615209\pi\)
\(114\) 0 0
\(115\) 72.9444 + 16.3939i 0.634299 + 0.142555i
\(116\) 6.57826i 0.0567091i
\(117\) 0 0
\(118\) 143.192i 1.21349i
\(119\) 57.7901i 0.485631i
\(120\) 0 0
\(121\) −57.5857 −0.475915
\(122\) −0.142865 −0.00117102
\(123\) 0 0
\(124\) −19.6867 −0.158764
\(125\) −98.5021 76.9567i −0.788017 0.615653i
\(126\) 0 0
\(127\) 122.687i 0.966037i −0.875610 0.483019i \(-0.839540\pi\)
0.875610 0.483019i \(-0.160460\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) −35.8888 + 159.687i −0.276067 + 1.22836i
\(131\) 22.3417i 0.170547i −0.996358 0.0852736i \(-0.972824\pi\)
0.996358 0.0852736i \(-0.0271764\pi\)
\(132\) 0 0
\(133\) 32.6719i 0.245654i
\(134\) 79.1815i 0.590907i
\(135\) 0 0
\(136\) −77.7980 −0.572044
\(137\) 0.857187 0.00625684 0.00312842 0.999995i \(-0.499004\pi\)
0.00312842 + 0.999995i \(0.499004\pi\)
\(138\) 0 0
\(139\) 235.283 1.69268 0.846340 0.532642i \(-0.178801\pi\)
0.846340 + 0.532642i \(0.178801\pi\)
\(140\) 4.60702 20.4989i 0.0329073 0.146421i
\(141\) 0 0
\(142\) 112.070i 0.789228i
\(143\) 309.320 2.16307
\(144\) 0 0
\(145\) 3.60612 16.0454i 0.0248698 0.110658i
\(146\) 29.5556i 0.202436i
\(147\) 0 0
\(148\) 66.6969i 0.450655i
\(149\) 131.204i 0.880564i −0.897860 0.440282i \(-0.854878\pi\)
0.897860 0.440282i \(-0.145122\pi\)
\(150\) 0 0
\(151\) −88.4495 −0.585758 −0.292879 0.956149i \(-0.594613\pi\)
−0.292879 + 0.956149i \(0.594613\pi\)
\(152\) 43.9835 0.289365
\(153\) 0 0
\(154\) −39.7071 −0.257839
\(155\) 48.0190 + 10.7920i 0.309800 + 0.0696260i
\(156\) 0 0
\(157\) 162.697i 1.03629i −0.855294 0.518143i \(-0.826623\pi\)
0.855294 0.518143i \(-0.173377\pi\)
\(158\) 2.09967 0.0132891
\(159\) 0 0
\(160\) 27.5959 + 6.20204i 0.172474 + 0.0387628i
\(161\) 31.4161i 0.195131i
\(162\) 0 0
\(163\) 275.192i 1.68829i −0.536112 0.844147i \(-0.680108\pi\)
0.536112 0.844147i \(-0.319892\pi\)
\(164\) 21.7060i 0.132354i
\(165\) 0 0
\(166\) 14.0454 0.0846109
\(167\) −136.989 −0.820295 −0.410148 0.912019i \(-0.634523\pi\)
−0.410148 + 0.912019i \(0.634523\pi\)
\(168\) 0 0
\(169\) −366.757 −2.17016
\(170\) 189.761 + 42.6479i 1.11624 + 0.250870i
\(171\) 0 0
\(172\) 86.3837i 0.502231i
\(173\) −92.7666 −0.536223 −0.268112 0.963388i \(-0.586400\pi\)
−0.268112 + 0.963388i \(0.586400\pi\)
\(174\) 0 0
\(175\) −22.4745 + 47.4745i −0.128426 + 0.271283i
\(176\) 53.4544i 0.303718i
\(177\) 0 0
\(178\) 216.338i 1.21538i
\(179\) 103.273i 0.576944i −0.957488 0.288472i \(-0.906853\pi\)
0.957488 0.288472i \(-0.0931472\pi\)
\(180\) 0 0
\(181\) 215.050 1.18812 0.594061 0.804420i \(-0.297524\pi\)
0.594061 + 0.804420i \(0.297524\pi\)
\(182\) 68.7748 0.377883
\(183\) 0 0
\(184\) 42.2929 0.229852
\(185\) −36.5625 + 162.684i −0.197635 + 0.879375i
\(186\) 0 0
\(187\) 367.576i 1.96564i
\(188\) −72.1747 −0.383908
\(189\) 0 0
\(190\) −107.283 24.1112i −0.564646 0.126901i
\(191\) 272.693i 1.42771i 0.700293 + 0.713856i \(0.253053\pi\)
−0.700293 + 0.713856i \(0.746947\pi\)
\(192\) 0 0
\(193\) 321.621i 1.66643i 0.552949 + 0.833215i \(0.313502\pi\)
−0.552949 + 0.833215i \(0.686498\pi\)
\(194\) 54.8542i 0.282753i
\(195\) 0 0
\(196\) 89.1714 0.454956
\(197\) −37.6267 −0.190999 −0.0954993 0.995429i \(-0.530445\pi\)
−0.0954993 + 0.995429i \(0.530445\pi\)
\(198\) 0 0
\(199\) −318.520 −1.60060 −0.800301 0.599598i \(-0.795327\pi\)
−0.800301 + 0.599598i \(0.795327\pi\)
\(200\) −63.9109 30.2555i −0.319555 0.151278i
\(201\) 0 0
\(202\) 115.909i 0.573808i
\(203\) −6.91053 −0.0340420
\(204\) 0 0
\(205\) −11.8990 + 52.9444i −0.0580438 + 0.258265i
\(206\) 163.734i 0.794826i
\(207\) 0 0
\(208\) 92.5857i 0.445124i
\(209\) 207.811i 0.994310i
\(210\) 0 0
\(211\) 5.75255 0.0272633 0.0136316 0.999907i \(-0.495661\pi\)
0.0136316 + 0.999907i \(0.495661\pi\)
\(212\) −106.337 −0.501591
\(213\) 0 0
\(214\) 169.015 0.789789
\(215\) −47.3545 + 210.703i −0.220254 + 0.980016i
\(216\) 0 0
\(217\) 20.6811i 0.0953047i
\(218\) −88.9670 −0.408106
\(219\) 0 0
\(220\) −29.3031 + 130.384i −0.133196 + 0.592653i
\(221\) 636.659i 2.88081i
\(222\) 0 0
\(223\) 23.8990i 0.107170i −0.998563 0.0535852i \(-0.982935\pi\)
0.998563 0.0535852i \(-0.0170649\pi\)
\(224\) 11.8852i 0.0530588i
\(225\) 0 0
\(226\) 113.171 0.500759
\(227\) 212.931 0.938024 0.469012 0.883192i \(-0.344610\pi\)
0.469012 + 0.883192i \(0.344610\pi\)
\(228\) 0 0
\(229\) −337.474 −1.47369 −0.736844 0.676063i \(-0.763685\pi\)
−0.736844 + 0.676063i \(0.763685\pi\)
\(230\) −103.159 23.1844i −0.448517 0.100802i
\(231\) 0 0
\(232\) 9.30306i 0.0400994i
\(233\) 198.996 0.854062 0.427031 0.904237i \(-0.359560\pi\)
0.427031 + 0.904237i \(0.359560\pi\)
\(234\) 0 0
\(235\) 176.045 + 39.5653i 0.749129 + 0.168363i
\(236\) 202.504i 0.858067i
\(237\) 0 0
\(238\) 81.7276i 0.343393i
\(239\) 76.1107i 0.318455i 0.987242 + 0.159227i \(0.0509003\pi\)
−0.987242 + 0.159227i \(0.949100\pi\)
\(240\) 0 0
\(241\) 326.757 1.35584 0.677919 0.735136i \(-0.262882\pi\)
0.677919 + 0.735136i \(0.262882\pi\)
\(242\) 81.4385 0.336523
\(243\) 0 0
\(244\) 0.202041 0.000828037
\(245\) −217.503 48.8827i −0.887768 0.199521i
\(246\) 0 0
\(247\) 359.939i 1.45724i
\(248\) 27.8412 0.112263
\(249\) 0 0
\(250\) 139.303 + 108.833i 0.557212 + 0.435333i
\(251\) 289.663i 1.15404i 0.816731 + 0.577019i \(0.195784\pi\)
−0.816731 + 0.577019i \(0.804216\pi\)
\(252\) 0 0
\(253\) 199.823i 0.789814i
\(254\) 173.505i 0.683092i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 460.533 1.79196 0.895978 0.444098i \(-0.146476\pi\)
0.895978 + 0.444098i \(0.146476\pi\)
\(258\) 0 0
\(259\) 70.0658 0.270524
\(260\) 50.7544 225.831i 0.195209 0.868581i
\(261\) 0 0
\(262\) 31.5959i 0.120595i
\(263\) 211.053 0.802484 0.401242 0.915972i \(-0.368579\pi\)
0.401242 + 0.915972i \(0.368579\pi\)
\(264\) 0 0
\(265\) 259.373 + 58.2929i 0.978768 + 0.219973i
\(266\) 46.2051i 0.173703i
\(267\) 0 0
\(268\) 111.980i 0.417834i
\(269\) 187.201i 0.695915i 0.937510 + 0.347957i \(0.113125\pi\)
−0.937510 + 0.347957i \(0.886875\pi\)
\(270\) 0 0
\(271\) −199.914 −0.737689 −0.368845 0.929491i \(-0.620247\pi\)
−0.368845 + 0.929491i \(0.620247\pi\)
\(272\) 110.023 0.404496
\(273\) 0 0
\(274\) −1.21225 −0.00442426
\(275\) 142.950 301.963i 0.519817 1.09805i
\(276\) 0 0
\(277\) 107.187i 0.386958i 0.981104 + 0.193479i \(0.0619771\pi\)
−0.981104 + 0.193479i \(0.938023\pi\)
\(278\) −332.740 −1.19691
\(279\) 0 0
\(280\) −6.51531 + 28.9898i −0.0232690 + 0.103535i
\(281\) 330.786i 1.17718i 0.808433 + 0.588588i \(0.200316\pi\)
−0.808433 + 0.588588i \(0.799684\pi\)
\(282\) 0 0
\(283\) 316.576i 1.11864i 0.828951 + 0.559321i \(0.188938\pi\)
−0.828951 + 0.559321i \(0.811062\pi\)
\(284\) 158.491i 0.558069i
\(285\) 0 0
\(286\) −437.444 −1.52952
\(287\) 22.8024 0.0794508
\(288\) 0 0
\(289\) 467.565 1.61787
\(290\) −5.09983 + 22.6916i −0.0175856 + 0.0782470i
\(291\) 0 0
\(292\) 41.7980i 0.143144i
\(293\) −85.0022 −0.290110 −0.145055 0.989424i \(-0.546336\pi\)
−0.145055 + 0.989424i \(0.546336\pi\)
\(294\) 0 0
\(295\) −111.010 + 493.939i −0.376306 + 1.67437i
\(296\) 94.3237i 0.318661i
\(297\) 0 0
\(298\) 185.551i 0.622653i
\(299\) 346.104i 1.15754i
\(300\) 0 0
\(301\) 90.7469 0.301485
\(302\) 125.086 0.414194
\(303\) 0 0
\(304\) −62.2020 −0.204612
\(305\) −0.492810 0.110756i −0.00161577 0.000363136i
\(306\) 0 0
\(307\) 34.5959i 0.112690i −0.998411 0.0563451i \(-0.982055\pi\)
0.998411 0.0563451i \(-0.0179447\pi\)
\(308\) 56.1544 0.182319
\(309\) 0 0
\(310\) −67.9092 15.2622i −0.219062 0.0492330i
\(311\) 42.3478i 0.136166i −0.997680 0.0680832i \(-0.978312\pi\)
0.997680 0.0680832i \(-0.0216883\pi\)
\(312\) 0 0
\(313\) 366.924i 1.17228i 0.810209 + 0.586141i \(0.199353\pi\)
−0.810209 + 0.586141i \(0.800647\pi\)
\(314\) 230.088i 0.732765i
\(315\) 0 0
\(316\) −2.96938 −0.00939679
\(317\) 87.1307 0.274860 0.137430 0.990511i \(-0.456116\pi\)
0.137430 + 0.990511i \(0.456116\pi\)
\(318\) 0 0
\(319\) 43.9546 0.137789
\(320\) −39.0265 8.77101i −0.121958 0.0274094i
\(321\) 0 0
\(322\) 44.4291i 0.137979i
\(323\) −427.728 −1.32424
\(324\) 0 0
\(325\) −247.596 + 523.015i −0.761834 + 1.60928i
\(326\) 389.180i 1.19380i
\(327\) 0 0
\(328\) 30.6969i 0.0935882i
\(329\) 75.8202i 0.230457i
\(330\) 0 0
\(331\) −127.333 −0.384691 −0.192345 0.981327i \(-0.561609\pi\)
−0.192345 + 0.981327i \(0.561609\pi\)
\(332\) −19.8632 −0.0598289
\(333\) 0 0
\(334\) 193.732 0.580036
\(335\) −61.3859 + 273.136i −0.183241 + 0.815331i
\(336\) 0 0
\(337\) 428.595i 1.27179i −0.771774 0.635897i \(-0.780630\pi\)
0.771774 0.635897i \(-0.219370\pi\)
\(338\) 518.673 1.53454
\(339\) 0 0
\(340\) −268.363 60.3133i −0.789304 0.177392i
\(341\) 131.543i 0.385756i
\(342\) 0 0
\(343\) 196.626i 0.573252i
\(344\) 122.165i 0.355131i
\(345\) 0 0
\(346\) 131.192 0.379167
\(347\) −350.503 −1.01010 −0.505048 0.863091i \(-0.668525\pi\)
−0.505048 + 0.863091i \(0.668525\pi\)
\(348\) 0 0
\(349\) 603.170 1.72828 0.864141 0.503250i \(-0.167862\pi\)
0.864141 + 0.503250i \(0.167862\pi\)
\(350\) 31.7837 67.1391i 0.0908106 0.191826i
\(351\) 0 0
\(352\) 75.5959i 0.214761i
\(353\) 7.81426 0.0221367 0.0110684 0.999939i \(-0.496477\pi\)
0.0110684 + 0.999939i \(0.496477\pi\)
\(354\) 0 0
\(355\) 86.8832 386.586i 0.244741 1.08897i
\(356\) 305.949i 0.859406i
\(357\) 0 0
\(358\) 146.050i 0.407961i
\(359\) 159.113i 0.443211i 0.975136 + 0.221605i \(0.0711297\pi\)
−0.975136 + 0.221605i \(0.928870\pi\)
\(360\) 0 0
\(361\) −119.182 −0.330143
\(362\) −304.127 −0.840129
\(363\) 0 0
\(364\) −97.2622 −0.267204
\(365\) −22.9131 + 101.952i −0.0627757 + 0.279320i
\(366\) 0 0
\(367\) 563.292i 1.53486i −0.641136 0.767428i \(-0.721536\pi\)
0.641136 0.767428i \(-0.278464\pi\)
\(368\) −59.8111 −0.162530
\(369\) 0 0
\(370\) 51.7071 230.070i 0.139749 0.621812i
\(371\) 111.708i 0.301101i
\(372\) 0 0
\(373\) 469.312i 1.25821i 0.777320 + 0.629105i \(0.216578\pi\)
−0.777320 + 0.629105i \(0.783422\pi\)
\(374\) 519.830i 1.38992i
\(375\) 0 0
\(376\) 102.070 0.271464
\(377\) −76.1316 −0.201941
\(378\) 0 0
\(379\) −339.101 −0.894726 −0.447363 0.894353i \(-0.647637\pi\)
−0.447363 + 0.894353i \(0.647637\pi\)
\(380\) 151.721 + 34.0984i 0.399265 + 0.0897327i
\(381\) 0 0
\(382\) 385.646i 1.00954i
\(383\) −5.62150 −0.0146776 −0.00733878 0.999973i \(-0.502336\pi\)
−0.00733878 + 0.999973i \(0.502336\pi\)
\(384\) 0 0
\(385\) −136.969 30.7832i −0.355765 0.0799563i
\(386\) 454.841i 1.17834i
\(387\) 0 0
\(388\) 77.5755i 0.199937i
\(389\) 635.621i 1.63399i −0.576647 0.816993i \(-0.695639\pi\)
0.576647 0.816993i \(-0.304361\pi\)
\(390\) 0 0
\(391\) −411.287 −1.05189
\(392\) −126.107 −0.321703
\(393\) 0 0
\(394\) 53.2122 0.135056
\(395\) 7.24280 + 1.62778i 0.0183362 + 0.00412097i
\(396\) 0 0
\(397\) 626.656i 1.57848i −0.614086 0.789239i \(-0.710475\pi\)
0.614086 0.789239i \(-0.289525\pi\)
\(398\) 450.455 1.13180
\(399\) 0 0
\(400\) 90.3837 + 42.7878i 0.225959 + 0.106969i
\(401\) 459.826i 1.14670i −0.819311 0.573350i \(-0.805644\pi\)
0.819311 0.573350i \(-0.194356\pi\)
\(402\) 0 0
\(403\) 227.839i 0.565357i
\(404\) 163.920i 0.405743i
\(405\) 0 0
\(406\) 9.77296 0.0240713
\(407\) −445.655 −1.09498
\(408\) 0 0
\(409\) −59.9592 −0.146599 −0.0732997 0.997310i \(-0.523353\pi\)
−0.0732997 + 0.997310i \(0.523353\pi\)
\(410\) 16.8277 74.8747i 0.0410432 0.182621i
\(411\) 0 0
\(412\) 231.555i 0.562027i
\(413\) 212.732 0.515090
\(414\) 0 0
\(415\) 48.4495 + 10.8888i 0.116746 + 0.0262380i
\(416\) 130.936i 0.314750i
\(417\) 0 0
\(418\) 293.889i 0.703083i
\(419\) 221.438i 0.528491i 0.964455 + 0.264245i \(0.0851229\pi\)
−0.964455 + 0.264245i \(0.914877\pi\)
\(420\) 0 0
\(421\) −133.171 −0.316322 −0.158161 0.987413i \(-0.550556\pi\)
−0.158161 + 0.987413i \(0.550556\pi\)
\(422\) −8.13534 −0.0192780
\(423\) 0 0
\(424\) 150.384 0.354678
\(425\) 621.517 + 294.227i 1.46239 + 0.692299i
\(426\) 0 0
\(427\) 0.212246i 0.000497064i
\(428\) −239.023 −0.558465
\(429\) 0 0
\(430\) 66.9694 297.980i 0.155743 0.692976i
\(431\) 458.090i 1.06285i −0.847104 0.531427i \(-0.821656\pi\)
0.847104 0.531427i \(-0.178344\pi\)
\(432\) 0 0
\(433\) 240.318i 0.555007i 0.960725 + 0.277503i \(0.0895069\pi\)
−0.960725 + 0.277503i \(0.910493\pi\)
\(434\) 29.2475i 0.0673906i
\(435\) 0 0
\(436\) 125.818 0.288574
\(437\) 232.523 0.532090
\(438\) 0 0
\(439\) 440.908 1.00435 0.502173 0.864767i \(-0.332534\pi\)
0.502173 + 0.864767i \(0.332534\pi\)
\(440\) 41.4408 184.390i 0.0941836 0.419069i
\(441\) 0 0
\(442\) 900.372i 2.03704i
\(443\) −607.808 −1.37203 −0.686014 0.727588i \(-0.740641\pi\)
−0.686014 + 0.727588i \(0.740641\pi\)
\(444\) 0 0
\(445\) −167.717 + 746.257i −0.376893 + 1.67698i
\(446\) 33.7983i 0.0757809i
\(447\) 0 0
\(448\) 16.8082i 0.0375182i
\(449\) 56.1962i 0.125159i −0.998040 0.0625793i \(-0.980067\pi\)
0.998040 0.0625793i \(-0.0199326\pi\)
\(450\) 0 0
\(451\) −145.035 −0.321586
\(452\) −160.049 −0.354090
\(453\) 0 0
\(454\) −301.131 −0.663283
\(455\) 237.238 + 53.3180i 0.521402 + 0.117182i
\(456\) 0 0
\(457\) 213.975i 0.468217i 0.972211 + 0.234108i \(0.0752170\pi\)
−0.972211 + 0.234108i \(0.924783\pi\)
\(458\) 477.261 1.04205
\(459\) 0 0
\(460\) 145.889 + 32.7878i 0.317150 + 0.0712777i
\(461\) 217.749i 0.472340i −0.971712 0.236170i \(-0.924108\pi\)
0.971712 0.236170i \(-0.0758923\pi\)
\(462\) 0 0
\(463\) 141.637i 0.305911i −0.988233 0.152955i \(-0.951121\pi\)
0.988233 0.152955i \(-0.0488791\pi\)
\(464\) 13.1565i 0.0283546i
\(465\) 0 0
\(466\) −281.423 −0.603913
\(467\) 254.279 0.544495 0.272248 0.962227i \(-0.412233\pi\)
0.272248 + 0.962227i \(0.412233\pi\)
\(468\) 0 0
\(469\) 117.636 0.250822
\(470\) −248.966 55.9538i −0.529714 0.119051i
\(471\) 0 0
\(472\) 286.384i 0.606745i
\(473\) −577.198 −1.22029
\(474\) 0 0
\(475\) −351.378 166.343i −0.739743 0.350195i
\(476\) 115.580i 0.242816i
\(477\) 0 0
\(478\) 107.637i 0.225181i
\(479\) 853.370i 1.78157i −0.454430 0.890783i \(-0.650157\pi\)
0.454430 0.890783i \(-0.349843\pi\)
\(480\) 0 0
\(481\) 771.898 1.60478
\(482\) −462.104 −0.958723
\(483\) 0 0
\(484\) −115.171 −0.237957
\(485\) −42.5260 + 189.219i −0.0876824 + 0.390142i
\(486\) 0 0
\(487\) 504.261i 1.03544i −0.855549 0.517722i \(-0.826780\pi\)
0.855549 0.517722i \(-0.173220\pi\)
\(488\) −0.285729 −0.000585511
\(489\) 0 0
\(490\) 307.596 + 69.1306i 0.627747 + 0.141083i
\(491\) 439.748i 0.895618i −0.894129 0.447809i \(-0.852205\pi\)
0.894129 0.447809i \(-0.147795\pi\)
\(492\) 0 0
\(493\) 90.4699i 0.183509i
\(494\) 509.030i 1.03043i
\(495\) 0 0
\(496\) −39.3735 −0.0793820
\(497\) −166.497 −0.335004
\(498\) 0 0
\(499\) −807.614 −1.61847 −0.809233 0.587488i \(-0.800117\pi\)
−0.809233 + 0.587488i \(0.800117\pi\)
\(500\) −197.004 153.913i −0.394009 0.307827i
\(501\) 0 0
\(502\) 409.646i 0.816028i
\(503\) 526.962 1.04764 0.523819 0.851829i \(-0.324507\pi\)
0.523819 + 0.851829i \(0.324507\pi\)
\(504\) 0 0
\(505\) −89.8592 + 399.828i −0.177939 + 0.791738i
\(506\) 282.592i 0.558483i
\(507\) 0 0
\(508\) 245.373i 0.483019i
\(509\) 120.439i 0.236620i −0.992977 0.118310i \(-0.962252\pi\)
0.992977 0.118310i \(-0.0377476\pi\)
\(510\) 0 0
\(511\) 43.9092 0.0859280
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) −651.292 −1.26710
\(515\) −126.936 + 564.799i −0.246477 + 1.09670i
\(516\) 0 0
\(517\) 482.257i 0.932798i
\(518\) −99.0880 −0.191290
\(519\) 0 0
\(520\) −71.7775 + 319.373i −0.138034 + 0.614180i
\(521\) 34.7871i 0.0667699i 0.999443 + 0.0333850i \(0.0106287\pi\)
−0.999443 + 0.0333850i \(0.989371\pi\)
\(522\) 0 0
\(523\) 210.535i 0.402552i −0.979535 0.201276i \(-0.935491\pi\)
0.979535 0.201276i \(-0.0645088\pi\)
\(524\) 44.6834i 0.0852736i
\(525\) 0 0
\(526\) −298.474 −0.567442
\(527\) −270.749 −0.513755
\(528\) 0 0
\(529\) −305.414 −0.577343
\(530\) −366.809 82.4385i −0.692093 0.155544i
\(531\) 0 0
\(532\) 65.3439i 0.122827i
\(533\) 251.208 0.471310
\(534\) 0 0
\(535\) 583.015 + 131.030i 1.08975 + 0.244915i
\(536\) 158.363i 0.295453i
\(537\) 0 0
\(538\) 264.742i 0.492086i
\(539\) 595.825i 1.10543i
\(540\) 0 0
\(541\) −123.070 −0.227487 −0.113743 0.993510i \(-0.536284\pi\)
−0.113743 + 0.993510i \(0.536284\pi\)
\(542\) 282.721 0.521625
\(543\) 0 0
\(544\) −155.596 −0.286022
\(545\) −306.891 68.9721i −0.563102 0.126554i
\(546\) 0 0
\(547\) 219.627i 0.401511i −0.979641 0.200756i \(-0.935660\pi\)
0.979641 0.200756i \(-0.0643397\pi\)
\(548\) 1.71437 0.00312842
\(549\) 0 0
\(550\) −202.161 + 427.040i −0.367566 + 0.776436i
\(551\) 51.1476i 0.0928269i
\(552\) 0 0
\(553\) 3.11937i 0.00564081i
\(554\) 151.586i 0.273620i
\(555\) 0 0
\(556\) 470.565 0.846340
\(557\) 351.396 0.630873 0.315436 0.948947i \(-0.397849\pi\)
0.315436 + 0.948947i \(0.397849\pi\)
\(558\) 0 0
\(559\) 999.737 1.78844
\(560\) 9.21404 40.9978i 0.0164536 0.0732103i
\(561\) 0 0
\(562\) 467.803i 0.832389i
\(563\) 338.321 0.600926 0.300463 0.953793i \(-0.402859\pi\)
0.300463 + 0.953793i \(0.402859\pi\)
\(564\) 0 0
\(565\) 390.384 + 87.7367i 0.690945 + 0.155286i
\(566\) 447.705i 0.790999i
\(567\) 0 0
\(568\) 224.141i 0.394614i
\(569\) 75.6741i 0.132995i 0.997787 + 0.0664975i \(0.0211824\pi\)
−0.997787 + 0.0664975i \(0.978818\pi\)
\(570\) 0 0
\(571\) 503.596 0.881954 0.440977 0.897518i \(-0.354632\pi\)
0.440977 + 0.897518i \(0.354632\pi\)
\(572\) 618.639 1.08154
\(573\) 0 0
\(574\) −32.2474 −0.0561802
\(575\) −337.872 159.949i −0.587603 0.278172i
\(576\) 0 0
\(577\) 665.090i 1.15267i 0.817214 + 0.576334i \(0.195517\pi\)
−0.817214 + 0.576334i \(0.804483\pi\)
\(578\) −661.237 −1.14401
\(579\) 0 0
\(580\) 7.21225 32.0908i 0.0124349 0.0553290i
\(581\) 20.8665i 0.0359148i
\(582\) 0 0
\(583\) 710.524i 1.21874i
\(584\) 59.1112i 0.101218i
\(585\) 0 0
\(586\) 120.211 0.205139
\(587\) 460.672 0.784791 0.392396 0.919797i \(-0.371646\pi\)
0.392396 + 0.919797i \(0.371646\pi\)
\(588\) 0 0
\(589\) 153.069 0.259880
\(590\) 156.992 698.535i 0.266088 1.18396i
\(591\) 0 0
\(592\) 133.394i 0.225327i
\(593\) −585.162 −0.986782 −0.493391 0.869808i \(-0.664243\pi\)
−0.493391 + 0.869808i \(0.664243\pi\)
\(594\) 0 0
\(595\) 63.3597 281.918i 0.106487 0.473812i
\(596\) 262.408i 0.440282i
\(597\) 0 0
\(598\) 489.464i 0.818502i
\(599\) 2.79956i 0.00467373i −0.999997 0.00233686i \(-0.999256\pi\)
0.999997 0.00233686i \(-0.000743847\pi\)
\(600\) 0 0
\(601\) −498.869 −0.830066 −0.415033 0.909806i \(-0.636230\pi\)
−0.415033 + 0.909806i \(0.636230\pi\)
\(602\) −128.336 −0.213182
\(603\) 0 0
\(604\) −176.899 −0.292879
\(605\) 280.921 + 63.1356i 0.464333 + 0.104356i
\(606\) 0 0
\(607\) 30.3847i 0.0500572i 0.999687 + 0.0250286i \(0.00796768\pi\)
−0.999687 + 0.0250286i \(0.992032\pi\)
\(608\) 87.9670 0.144683
\(609\) 0 0
\(610\) 0.696938 + 0.156633i 0.00114252 + 0.000256776i
\(611\) 835.293i 1.36709i
\(612\) 0 0
\(613\) 448.449i 0.731565i 0.930700 + 0.365783i \(0.119199\pi\)
−0.930700 + 0.365783i \(0.880801\pi\)
\(614\) 48.9260i 0.0796841i
\(615\) 0 0
\(616\) −79.4143 −0.128919
\(617\) 170.733 0.276715 0.138357 0.990382i \(-0.455818\pi\)
0.138357 + 0.990382i \(0.455818\pi\)
\(618\) 0 0
\(619\) 238.247 0.384891 0.192445 0.981308i \(-0.438358\pi\)
0.192445 + 0.981308i \(0.438358\pi\)
\(620\) 96.0381 + 21.5841i 0.154900 + 0.0348130i
\(621\) 0 0
\(622\) 59.8888i 0.0962842i
\(623\) 321.402 0.515894
\(624\) 0 0
\(625\) 396.151 + 483.414i 0.633842 + 0.773463i
\(626\) 518.909i 0.828928i
\(627\) 0 0
\(628\) 325.394i 0.518143i
\(629\) 917.274i 1.45831i
\(630\) 0 0
\(631\) 1148.15 1.81957 0.909786 0.415078i \(-0.136246\pi\)
0.909786 + 0.415078i \(0.136246\pi\)
\(632\) 4.19934 0.00664453
\(633\) 0 0
\(634\) −123.221 −0.194356
\(635\) −134.511 + 598.505i −0.211828 + 0.942527i
\(636\) 0 0
\(637\) 1032.00i 1.62009i
\(638\) −62.1612 −0.0974313
\(639\) 0 0
\(640\) 55.1918 + 12.4041i 0.0862372 + 0.0193814i
\(641\) 388.533i 0.606136i 0.952969 + 0.303068i \(0.0980109\pi\)
−0.952969 + 0.303068i \(0.901989\pi\)
\(642\) 0 0
\(643\) 928.980i 1.44476i −0.691497 0.722379i \(-0.743049\pi\)
0.691497 0.722379i \(-0.256951\pi\)
\(644\) 62.8322i 0.0975655i
\(645\) 0 0
\(646\) 604.899 0.936376
\(647\) −485.464 −0.750330 −0.375165 0.926958i \(-0.622414\pi\)
−0.375165 + 0.926958i \(0.622414\pi\)
\(648\) 0 0
\(649\) −1353.09 −2.08488
\(650\) 350.154 739.655i 0.538698 1.13793i
\(651\) 0 0
\(652\) 550.384i 0.844147i
\(653\) −33.9620 −0.0520093 −0.0260046 0.999662i \(-0.508278\pi\)
−0.0260046 + 0.999662i \(0.508278\pi\)
\(654\) 0 0
\(655\) −24.4949 + 108.990i −0.0373968 + 0.166397i
\(656\) 43.4120i 0.0661769i
\(657\) 0 0
\(658\) 107.226i 0.162957i
\(659\) 67.9818i 0.103159i 0.998669 + 0.0515795i \(0.0164256\pi\)
−0.998669 + 0.0515795i \(0.983574\pi\)
\(660\) 0 0
\(661\) −883.959 −1.33731 −0.668653 0.743575i \(-0.733129\pi\)
−0.668653 + 0.743575i \(0.733129\pi\)
\(662\) 180.076 0.272017
\(663\) 0 0
\(664\) 28.0908 0.0423054
\(665\) −35.8207 + 159.384i −0.0538658 + 0.239675i
\(666\) 0 0
\(667\) 49.1816i 0.0737356i
\(668\) −273.979 −0.410148
\(669\) 0 0
\(670\) 86.8128 386.272i 0.129571 0.576526i
\(671\) 1.35000i 0.00201192i
\(672\) 0 0
\(673\) 217.448i 0.323103i 0.986864 + 0.161552i \(0.0516498\pi\)
−0.986864 + 0.161552i \(0.948350\pi\)
\(674\) 606.125i 0.899295i
\(675\) 0 0
\(676\) −733.514 −1.08508
\(677\) −926.345 −1.36831 −0.684155 0.729337i \(-0.739829\pi\)
−0.684155 + 0.729337i \(0.739829\pi\)
\(678\) 0 0
\(679\) 81.4939 0.120020
\(680\) 379.523 + 85.2958i 0.558122 + 0.125435i
\(681\) 0 0
\(682\) 186.030i 0.272771i
\(683\) −586.563 −0.858804 −0.429402 0.903113i \(-0.641276\pi\)
−0.429402 + 0.903113i \(0.641276\pi\)
\(684\) 0 0
\(685\) −4.18163 0.939800i −0.00610457 0.00137197i
\(686\) 278.070i 0.405351i
\(687\) 0 0
\(688\) 172.767i 0.251115i
\(689\) 1230.66i 1.78616i
\(690\) 0 0
\(691\) −830.161 −1.20139 −0.600696 0.799478i \(-0.705110\pi\)
−0.600696 + 0.799478i \(0.705110\pi\)
\(692\) −185.533 −0.268112
\(693\) 0 0
\(694\) 495.687 0.714246
\(695\) −1147.78 257.958i −1.65149 0.371163i
\(696\) 0 0
\(697\) 298.520i 0.428293i
\(698\) −853.012 −1.22208
\(699\) 0 0
\(700\) −44.9490 + 94.9490i −0.0642128 + 0.135641i
\(701\) 915.199i 1.30556i 0.757547 + 0.652781i \(0.226398\pi\)
−0.757547 + 0.652781i \(0.773602\pi\)
\(702\) 0 0
\(703\) 518.586i 0.737675i
\(704\) 106.909i 0.151859i
\(705\) 0 0
\(706\) −11.0510 −0.0156530
\(707\) 172.200 0.243564
\(708\) 0 0
\(709\) −18.8480 −0.0265839 −0.0132919 0.999912i \(-0.504231\pi\)
−0.0132919 + 0.999912i \(0.504231\pi\)
\(710\) −122.871 + 546.715i −0.173058 + 0.770021i
\(711\) 0 0
\(712\) 432.677i 0.607692i
\(713\) 147.186 0.206432
\(714\) 0 0
\(715\) −1508.96 339.131i −2.11043 0.474309i
\(716\) 206.546i 0.288472i
\(717\) 0 0
\(718\) 225.019i 0.313397i
\(719\) 65.5964i 0.0912328i 0.998959 + 0.0456164i \(0.0145252\pi\)
−0.998959 + 0.0456164i \(0.985475\pi\)
\(720\) 0 0
\(721\) 243.251 0.337380
\(722\) 168.548 0.233446
\(723\) 0 0
\(724\) 430.100 0.594061
\(725\) −35.1836 + 74.3209i −0.0485291 + 0.102512i
\(726\) 0 0
\(727\) 1249.52i 1.71873i −0.511364 0.859364i \(-0.670860\pi\)
0.511364 0.859364i \(-0.329140\pi\)
\(728\) 137.550 0.188942
\(729\) 0 0
\(730\) 32.4041 144.182i 0.0443892 0.197509i
\(731\) 1188.02i 1.62520i
\(732\) 0 0
\(733\) 195.991i 0.267382i −0.991023 0.133691i \(-0.957317\pi\)
0.991023 0.133691i \(-0.0426829\pi\)
\(734\) 796.615i 1.08531i
\(735\) 0 0
\(736\) 84.5857 0.114926
\(737\) −748.225 −1.01523
\(738\) 0 0
\(739\) 376.536 0.509521 0.254760 0.967004i \(-0.418003\pi\)
0.254760 + 0.967004i \(0.418003\pi\)
\(740\) −73.1249 + 325.369i −0.0988175 + 0.439687i
\(741\) 0 0
\(742\) 157.980i 0.212910i
\(743\) 1293.12 1.74041 0.870204 0.492692i \(-0.163987\pi\)
0.870204 + 0.492692i \(0.163987\pi\)
\(744\) 0 0
\(745\) −143.849 + 640.055i −0.193086 + 0.859134i
\(746\) 663.708i 0.889689i
\(747\) 0 0
\(748\) 735.151i 0.982822i
\(749\) 251.096i 0.335242i
\(750\) 0 0
\(751\) 8.50866 0.0113298 0.00566489 0.999984i \(-0.498197\pi\)
0.00566489 + 0.999984i \(0.498197\pi\)
\(752\) −144.349 −0.191954
\(753\) 0 0
\(754\) 107.666 0.142794
\(755\) 431.484 + 96.9739i 0.571503 + 0.128442i
\(756\) 0 0
\(757\) 759.469i 1.00326i 0.865082 + 0.501631i \(0.167266\pi\)
−0.865082 + 0.501631i \(0.832734\pi\)
\(758\) 479.561 0.632667
\(759\) 0 0
\(760\) −214.565 48.2225i −0.282323 0.0634506i
\(761\) 452.801i 0.595007i −0.954721 0.297504i \(-0.903846\pi\)
0.954721 0.297504i \(-0.0961540\pi\)
\(762\) 0 0
\(763\) 132.173i 0.173229i
\(764\) 545.386i 0.713856i
\(765\) 0 0
\(766\) 7.95001 0.0103786
\(767\) 2343.62 3.05557
\(768\) 0 0
\(769\) 1196.72 1.55620 0.778099 0.628142i \(-0.216184\pi\)
0.778099 + 0.628142i \(0.216184\pi\)
\(770\) 193.704 + 43.5340i 0.251564 + 0.0565376i
\(771\) 0 0
\(772\) 643.242i 0.833215i
\(773\) −287.556 −0.372000 −0.186000 0.982550i \(-0.559552\pi\)
−0.186000 + 0.982550i \(0.559552\pi\)
\(774\) 0 0
\(775\) −222.420 105.294i −0.286993 0.135863i
\(776\) 109.708i 0.141377i
\(777\) 0 0
\(778\) 898.904i 1.15540i
\(779\) 168.770i 0.216649i
\(780\) 0 0
\(781\) 1059.01 1.35596
\(782\) 581.648 0.743795
\(783\) 0 0
\(784\) 178.343 0.227478
\(785\) −178.377 + 793.687i −0.227232 + 1.01107i
\(786\) 0 0
\(787\) 341.757i 0.434253i 0.976143 + 0.217127i \(0.0696684\pi\)
−0.976143 + 0.217127i \(0.930332\pi\)
\(788\) −75.2535 −0.0954993
\(789\) 0 0
\(790\) −10.2429 2.30203i −0.0129656 0.00291396i
\(791\) 168.133i 0.212557i
\(792\) 0 0
\(793\) 2.33826i 0.00294863i
\(794\) 886.226i 1.11615i
\(795\) 0 0
\(796\) −637.040 −0.800301
\(797\) −423.701 −0.531619 −0.265810 0.964026i \(-0.585639\pi\)
−0.265810 + 0.964026i \(0.585639\pi\)
\(798\) 0 0
\(799\) −992.609 −1.24231
\(800\) −127.822 60.5110i −0.159777 0.0756388i
\(801\) 0 0
\(802\) 650.293i 0.810839i
\(803\) −279.286 −0.347803
\(804\) 0 0
\(805\) −34.4439 + 153.258i −0.0427874 + 0.190382i
\(806\) 322.213i 0.399768i
\(807\) 0 0
\(808\) 231.818i 0.286904i
\(809\) 46.4331i 0.0573957i 0.999588 + 0.0286978i \(0.00913606\pi\)
−0.999588 + 0.0286978i \(0.990864\pi\)
\(810\) 0 0
\(811\) −511.160 −0.630284 −0.315142 0.949045i \(-0.602052\pi\)
−0.315142 + 0.949045i \(0.602052\pi\)
\(812\) −13.8211 −0.0170210
\(813\) 0 0
\(814\) 630.252 0.774265
\(815\) −301.714 + 1342.47i −0.370201 + 1.64721i
\(816\) 0 0
\(817\) 671.655i 0.822099i
\(818\) 84.7951 0.103661
\(819\) 0 0
\(820\) −23.7980 + 105.889i −0.0290219 + 0.129133i
\(821\) 154.461i 0.188137i −0.995566 0.0940686i \(-0.970013\pi\)
0.995566 0.0940686i \(-0.0299873\pi\)
\(822\) 0 0
\(823\) 1038.95i 1.26239i 0.775624 + 0.631196i \(0.217435\pi\)
−0.775624 + 0.631196i \(0.782565\pi\)
\(824\) 327.468i 0.397413i
\(825\) 0 0
\(826\) −300.849 −0.364224
\(827\) 792.713 0.958540 0.479270 0.877668i \(-0.340901\pi\)
0.479270 + 0.877668i \(0.340901\pi\)
\(828\) 0 0
\(829\) 1143.31 1.37914 0.689572 0.724217i \(-0.257798\pi\)
0.689572 + 0.724217i \(0.257798\pi\)
\(830\) −68.5179 15.3991i −0.0825517 0.0185531i
\(831\) 0 0
\(832\) 185.171i 0.222562i
\(833\) 1226.36 1.47222
\(834\) 0 0
\(835\) 668.277 + 150.192i 0.800332 + 0.179870i
\(836\) 415.621i 0.497155i
\(837\) 0 0
\(838\) 313.160i 0.373700i
\(839\) 1121.25i 1.33641i 0.743976 + 0.668206i \(0.232938\pi\)
−0.743976 + 0.668206i \(0.767062\pi\)
\(840\) 0 0
\(841\) 830.182 0.987136
\(842\) 188.333 0.223673
\(843\) 0 0
\(844\) 11.5051 0.0136316
\(845\) 1789.16 + 402.104i 2.11735 + 0.475863i
\(846\) 0 0
\(847\) 120.989i 0.142844i
\(848\) −212.675 −0.250796
\(849\) 0 0
\(850\) −878.958 416.100i −1.03407 0.489529i
\(851\) 498.652i 0.585961i
\(852\) 0 0
\(853\) 1571.03i 1.84177i −0.389830 0.920887i \(-0.627466\pi\)
0.389830 0.920887i \(-0.372534\pi\)
\(854\) 0.300161i 0.000351477i
\(855\) 0 0
\(856\) 338.030 0.394894
\(857\) −1566.41 −1.82778 −0.913892 0.405957i \(-0.866938\pi\)
−0.913892 + 0.405957i \(0.866938\pi\)
\(858\) 0 0
\(859\) −54.9240 −0.0639394 −0.0319697 0.999489i \(-0.510178\pi\)
−0.0319697 + 0.999489i \(0.510178\pi\)
\(860\) −94.7090 + 421.407i −0.110127 + 0.490008i
\(861\) 0 0
\(862\) 647.837i 0.751551i
\(863\) −397.076 −0.460111 −0.230056 0.973177i \(-0.573891\pi\)
−0.230056 + 0.973177i \(0.573891\pi\)
\(864\) 0 0
\(865\) 452.545 + 101.707i 0.523173 + 0.117581i
\(866\) 339.861i 0.392449i
\(867\) 0 0
\(868\) 41.3622i 0.0476523i
\(869\) 19.8408i 0.0228318i
\(870\) 0 0
\(871\) 1295.96 1.48790
\(872\) −177.934 −0.204053
\(873\) 0 0
\(874\) −328.838 −0.376245
\(875\) 161.688 206.955i 0.184786 0.236520i
\(876\) 0 0
\(877\) 94.0250i 0.107212i −0.998562 0.0536060i \(-0.982928\pi\)
0.998562 0.0536060i \(-0.0170715\pi\)
\(878\) −623.538 −0.710180
\(879\) 0 0
\(880\) −58.6061 + 260.767i −0.0665979 + 0.296327i
\(881\) 283.024i 0.321253i −0.987015 0.160627i \(-0.948648\pi\)
0.987015 0.160627i \(-0.0513515\pi\)
\(882\) 0 0
\(883\) 1194.90i 1.35323i −0.736339 0.676613i \(-0.763447\pi\)
0.736339 0.676613i \(-0.236553\pi\)
\(884\) 1273.32i 1.44041i
\(885\) 0 0
\(886\) 859.571 0.970170
\(887\) −714.584 −0.805619 −0.402810 0.915284i \(-0.631966\pi\)
−0.402810 + 0.915284i \(0.631966\pi\)
\(888\) 0 0
\(889\) 257.767 0.289952
\(890\) 237.188 1055.37i 0.266504 1.18580i
\(891\) 0 0
\(892\) 47.7980i 0.0535852i
\(893\) 561.177 0.628417
\(894\) 0 0
\(895\) −113.226 + 503.798i −0.126510 + 0.562903i
\(896\) 23.7703i 0.0265294i
\(897\) 0 0
\(898\) 79.4735i 0.0885005i
\(899\) 32.3761i 0.0360135i
\(900\) 0 0
\(901\) −1462.44 −1.62313
\(902\) 205.111 0.227396
\(903\) 0 0
\(904\) 226.343 0.250379
\(905\) −1049.08 235.776i −1.15921 0.260526i
\(906\) 0 0
\(907\) 645.000i 0.711136i −0.934650 0.355568i \(-0.884288\pi\)
0.934650 0.355568i \(-0.115712\pi\)
\(908\) 425.863 0.469012
\(909\) 0 0
\(910\) −335.505 75.4031i −0.368687 0.0828605i
\(911\) 1002.58i 1.10052i 0.834992 + 0.550261i \(0.185472\pi\)
−0.834992 + 0.550261i \(0.814528\pi\)
\(912\) 0 0
\(913\) 132.722i 0.145369i
\(914\) 302.606i 0.331079i
\(915\) 0 0
\(916\) −674.949 −0.736844
\(917\) 46.9403 0.0511890
\(918\) 0 0
\(919\) −1583.98 −1.72359 −0.861794 0.507258i \(-0.830659\pi\)
−0.861794 + 0.507258i \(0.830659\pi\)
\(920\) −206.318 46.3689i −0.224259 0.0504010i
\(921\) 0 0
\(922\) 307.943i 0.333995i
\(923\) −1834.26 −1.98728
\(924\) 0 0
\(925\) 356.727 753.539i 0.385650 0.814637i
\(926\) 200.305i 0.216312i
\(927\) 0 0
\(928\) 18.6061i 0.0200497i
\(929\) 636.903i 0.685579i 0.939412 + 0.342790i \(0.111372\pi\)
−0.939412 + 0.342790i \(0.888628\pi\)
\(930\) 0 0
\(931\) −693.331 −0.744716
\(932\) 397.993 0.427031
\(933\) 0 0
\(934\) −359.605 −0.385016
\(935\) −403.001 + 1793.15i −0.431017 + 1.91781i
\(936\) 0 0
\(937\) 398.688i 0.425494i −0.977107 0.212747i \(-0.931759\pi\)
0.977107 0.212747i \(-0.0682410\pi\)
\(938\) −166.362 −0.177358
\(939\) 0 0
\(940\) 352.091 + 79.1306i 0.374565 + 0.0841815i
\(941\) 523.582i 0.556410i 0.960522 + 0.278205i \(0.0897395\pi\)
−0.960522 + 0.278205i \(0.910261\pi\)
\(942\) 0 0
\(943\) 162.283i 0.172092i
\(944\) 405.008i 0.429034i
\(945\) 0 0
\(946\) 816.282 0.862877
\(947\) −317.944 −0.335739 −0.167869 0.985809i \(-0.553689\pi\)
−0.167869 + 0.985809i \(0.553689\pi\)
\(948\) 0 0
\(949\) 483.737 0.509733
\(950\) 496.924 + 235.244i 0.523077 + 0.247626i
\(951\) 0 0
\(952\) 163.455i 0.171697i
\(953\) 848.148 0.889977 0.444988 0.895536i \(-0.353208\pi\)
0.444988 + 0.895536i \(0.353208\pi\)
\(954\) 0 0
\(955\) 298.974 1330.28i 0.313062 1.39297i
\(956\) 152.221i 0.159227i
\(957\) 0 0
\(958\) 1206.85i 1.25976i
\(959\) 1.80097i 0.00187797i
\(960\) 0 0
\(961\) −864.108 −0.899176
\(962\) −1091.63 −1.13475
\(963\) 0 0
\(964\) 653.514 0.677919
\(965\) 352.618 1568.97i 0.365407 1.62587i
\(966\) 0 0
\(967\) 998.725i 1.03281i 0.856345 + 0.516404i \(0.172730\pi\)
−0.856345 + 0.516404i \(0.827270\pi\)
\(968\) 162.877 0.168261
\(969\) 0 0
\(970\) 60.1408 267.596i 0.0620008 0.275872i
\(971\) 1392.89i 1.43449i 0.696820 + 0.717246i \(0.254598\pi\)
−0.696820 + 0.717246i \(0.745402\pi\)
\(972\) 0 0
\(973\) 494.334i 0.508051i
\(974\) 713.133i 0.732169i
\(975\) 0 0
\(976\) 0.404082 0.000414019
\(977\) −618.875 −0.633444 −0.316722 0.948518i \(-0.602582\pi\)
−0.316722 + 0.948518i \(0.602582\pi\)
\(978\) 0 0
\(979\) −2044.29 −2.08814
\(980\) −435.006 97.7654i −0.443884 0.0997607i
\(981\) 0 0
\(982\) 621.898i 0.633297i
\(983\) 192.813 0.196148 0.0980739 0.995179i \(-0.468732\pi\)
0.0980739 + 0.995179i \(0.468732\pi\)
\(984\) 0 0
\(985\) 183.555 + 41.2531i 0.186350 + 0.0418813i
\(986\) 127.944i 0.129760i
\(987\) 0 0
\(988\) 719.878i 0.728621i
\(989\) 645.838i 0.653021i
\(990\) 0 0
\(991\) 410.243 0.413969 0.206984 0.978344i \(-0.433635\pi\)
0.206984 + 0.978344i \(0.433635\pi\)
\(992\) 55.6825 0.0561315
\(993\) 0 0
\(994\) 235.462 0.236884
\(995\) 1553.84 + 349.218i 1.56165 + 0.350973i
\(996\) 0 0
\(997\) 453.648i 0.455013i 0.973777 + 0.227507i \(0.0730573\pi\)
−0.973777 + 0.227507i \(0.926943\pi\)
\(998\) 1142.14 1.14443
\(999\) 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 810.3.b.a.809.1 8
3.2 odd 2 inner 810.3.b.a.809.8 8
5.4 even 2 inner 810.3.b.a.809.7 8
9.2 odd 6 270.3.j.a.179.1 8
9.4 even 3 270.3.j.a.89.3 8
9.5 odd 6 90.3.j.a.29.1 8
9.7 even 3 90.3.j.a.59.4 yes 8
15.14 odd 2 inner 810.3.b.a.809.2 8
45.2 even 12 1350.3.i.c.1151.2 4
45.4 even 6 270.3.j.a.89.1 8
45.7 odd 12 450.3.i.c.401.1 4
45.13 odd 12 1350.3.i.a.251.1 4
45.14 odd 6 90.3.j.a.29.4 yes 8
45.22 odd 12 1350.3.i.c.251.2 4
45.23 even 12 450.3.i.a.101.2 4
45.29 odd 6 270.3.j.a.179.3 8
45.32 even 12 450.3.i.c.101.1 4
45.34 even 6 90.3.j.a.59.1 yes 8
45.38 even 12 1350.3.i.a.1151.1 4
45.43 odd 12 450.3.i.a.401.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.3.j.a.29.1 8 9.5 odd 6
90.3.j.a.29.4 yes 8 45.14 odd 6
90.3.j.a.59.1 yes 8 45.34 even 6
90.3.j.a.59.4 yes 8 9.7 even 3
270.3.j.a.89.1 8 45.4 even 6
270.3.j.a.89.3 8 9.4 even 3
270.3.j.a.179.1 8 9.2 odd 6
270.3.j.a.179.3 8 45.29 odd 6
450.3.i.a.101.2 4 45.23 even 12
450.3.i.a.401.2 4 45.43 odd 12
450.3.i.c.101.1 4 45.32 even 12
450.3.i.c.401.1 4 45.7 odd 12
810.3.b.a.809.1 8 1.1 even 1 trivial
810.3.b.a.809.2 8 15.14 odd 2 inner
810.3.b.a.809.7 8 5.4 even 2 inner
810.3.b.a.809.8 8 3.2 odd 2 inner
1350.3.i.a.251.1 4 45.13 odd 12
1350.3.i.a.1151.1 4 45.38 even 12
1350.3.i.c.251.2 4 45.22 odd 12
1350.3.i.c.1151.2 4 45.2 even 12