Properties

Label 1600.2.s.e.207.4
Level $1600$
Weight $2$
Character 1600.207
Analytic conductor $12.776$
Analytic rank $0$
Dimension $24$
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] = [1600,2,Mod(207,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.207");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.s (of order \(4\), degree \(2\), not 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 207.4
Character \(\chi\) \(=\) 1600.207
Dual form 1600.2.s.e.943.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.86755 q^{3} +(0.719989 + 0.719989i) q^{7} +0.487737 q^{9} +O(q^{10})\) \(q-1.86755 q^{3} +(0.719989 + 0.719989i) q^{7} +0.487737 q^{9} +(0.805654 - 0.805654i) q^{11} +5.90473i q^{13} +(-5.17145 - 5.17145i) q^{17} +(1.16370 - 1.16370i) q^{19} +(-1.34461 - 1.34461i) q^{21} +(-2.30177 + 2.30177i) q^{23} +4.69177 q^{27} +(3.71953 + 3.71953i) q^{29} -9.82775i q^{31} +(-1.50460 + 1.50460i) q^{33} -1.71983i q^{37} -11.0274i q^{39} +3.93637i q^{41} +8.82362i q^{43} +(-3.21130 + 3.21130i) q^{47} -5.96323i q^{49} +(9.65793 + 9.65793i) q^{51} -8.60748 q^{53} +(-2.17326 + 2.17326i) q^{57} +(-5.24522 - 5.24522i) q^{59} +(1.59176 - 1.59176i) q^{61} +(0.351165 + 0.351165i) q^{63} -9.29532i q^{67} +(4.29867 - 4.29867i) q^{69} -9.33581 q^{71} +(-8.57821 - 8.57821i) q^{73} +1.16012 q^{77} +1.70231 q^{79} -10.2253 q^{81} -13.8974 q^{83} +(-6.94640 - 6.94640i) q^{87} -4.48540 q^{89} +(-4.25134 + 4.25134i) q^{91} +18.3538i q^{93} +(-4.46476 - 4.46476i) q^{97} +(0.392947 - 0.392947i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 40 q^{9} + 20 q^{11} - 12 q^{19} - 8 q^{29} - 20 q^{51} + 8 q^{59} - 48 q^{61} + 64 q^{69} + 16 q^{71} - 104 q^{79} + 48 q^{81} - 96 q^{89} - 64 q^{91} + 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.86755 −1.07823 −0.539115 0.842232i \(-0.681241\pi\)
−0.539115 + 0.842232i \(0.681241\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.719989 + 0.719989i 0.272130 + 0.272130i 0.829957 0.557827i \(-0.188365\pi\)
−0.557827 + 0.829957i \(0.688365\pi\)
\(8\) 0 0
\(9\) 0.487737 0.162579
\(10\) 0 0
\(11\) 0.805654 0.805654i 0.242914 0.242914i −0.575141 0.818055i \(-0.695053\pi\)
0.818055 + 0.575141i \(0.195053\pi\)
\(12\) 0 0
\(13\) 5.90473i 1.63768i 0.574023 + 0.818839i \(0.305382\pi\)
−0.574023 + 0.818839i \(0.694618\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.17145 5.17145i −1.25426 1.25426i −0.953792 0.300469i \(-0.902857\pi\)
−0.300469 0.953792i \(-0.597143\pi\)
\(18\) 0 0
\(19\) 1.16370 1.16370i 0.266971 0.266971i −0.560908 0.827878i \(-0.689548\pi\)
0.827878 + 0.560908i \(0.189548\pi\)
\(20\) 0 0
\(21\) −1.34461 1.34461i −0.293419 0.293419i
\(22\) 0 0
\(23\) −2.30177 + 2.30177i −0.479953 + 0.479953i −0.905116 0.425164i \(-0.860217\pi\)
0.425164 + 0.905116i \(0.360217\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.69177 0.902932
\(28\) 0 0
\(29\) 3.71953 + 3.71953i 0.690699 + 0.690699i 0.962386 0.271687i \(-0.0875814\pi\)
−0.271687 + 0.962386i \(0.587581\pi\)
\(30\) 0 0
\(31\) 9.82775i 1.76512i −0.470204 0.882558i \(-0.655820\pi\)
0.470204 0.882558i \(-0.344180\pi\)
\(32\) 0 0
\(33\) −1.50460 + 1.50460i −0.261917 + 0.261917i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.71983i 0.282739i −0.989957 0.141369i \(-0.954850\pi\)
0.989957 0.141369i \(-0.0451505\pi\)
\(38\) 0 0
\(39\) 11.0274i 1.76579i
\(40\) 0 0
\(41\) 3.93637i 0.614758i 0.951587 + 0.307379i \(0.0994519\pi\)
−0.951587 + 0.307379i \(0.900548\pi\)
\(42\) 0 0
\(43\) 8.82362i 1.34559i 0.739829 + 0.672794i \(0.234906\pi\)
−0.739829 + 0.672794i \(0.765094\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.21130 + 3.21130i −0.468417 + 0.468417i −0.901401 0.432985i \(-0.857460\pi\)
0.432985 + 0.901401i \(0.357460\pi\)
\(48\) 0 0
\(49\) 5.96323i 0.851890i
\(50\) 0 0
\(51\) 9.65793 + 9.65793i 1.35238 + 1.35238i
\(52\) 0 0
\(53\) −8.60748 −1.18233 −0.591164 0.806551i \(-0.701332\pi\)
−0.591164 + 0.806551i \(0.701332\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.17326 + 2.17326i −0.287856 + 0.287856i
\(58\) 0 0
\(59\) −5.24522 5.24522i −0.682870 0.682870i 0.277776 0.960646i \(-0.410403\pi\)
−0.960646 + 0.277776i \(0.910403\pi\)
\(60\) 0 0
\(61\) 1.59176 1.59176i 0.203804 0.203804i −0.597824 0.801628i \(-0.703968\pi\)
0.801628 + 0.597824i \(0.203968\pi\)
\(62\) 0 0
\(63\) 0.351165 + 0.351165i 0.0442426 + 0.0442426i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.29532i 1.13560i −0.823165 0.567802i \(-0.807794\pi\)
0.823165 0.567802i \(-0.192206\pi\)
\(68\) 0 0
\(69\) 4.29867 4.29867i 0.517499 0.517499i
\(70\) 0 0
\(71\) −9.33581 −1.10796 −0.553979 0.832531i \(-0.686891\pi\)
−0.553979 + 0.832531i \(0.686891\pi\)
\(72\) 0 0
\(73\) −8.57821 8.57821i −1.00400 1.00400i −0.999992 0.00401200i \(-0.998723\pi\)
−0.00401200 0.999992i \(-0.501277\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.16012 0.132208
\(78\) 0 0
\(79\) 1.70231 0.191525 0.0957625 0.995404i \(-0.469471\pi\)
0.0957625 + 0.995404i \(0.469471\pi\)
\(80\) 0 0
\(81\) −10.2253 −1.13615
\(82\) 0 0
\(83\) −13.8974 −1.52544 −0.762718 0.646732i \(-0.776135\pi\)
−0.762718 + 0.646732i \(0.776135\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.94640 6.94640i −0.744732 0.744732i
\(88\) 0 0
\(89\) −4.48540 −0.475452 −0.237726 0.971332i \(-0.576402\pi\)
−0.237726 + 0.971332i \(0.576402\pi\)
\(90\) 0 0
\(91\) −4.25134 + 4.25134i −0.445662 + 0.445662i
\(92\) 0 0
\(93\) 18.3538i 1.90320i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.46476 4.46476i −0.453327 0.453327i 0.443130 0.896457i \(-0.353868\pi\)
−0.896457 + 0.443130i \(0.853868\pi\)
\(98\) 0 0
\(99\) 0.392947 0.392947i 0.0394927 0.0394927i
\(100\) 0 0
\(101\) −9.04273 9.04273i −0.899785 0.899785i 0.0956319 0.995417i \(-0.469513\pi\)
−0.995417 + 0.0956319i \(0.969513\pi\)
\(102\) 0 0
\(103\) 8.89360 8.89360i 0.876312 0.876312i −0.116839 0.993151i \(-0.537276\pi\)
0.993151 + 0.116839i \(0.0372760\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.29532 −0.898612 −0.449306 0.893378i \(-0.648329\pi\)
−0.449306 + 0.893378i \(0.648329\pi\)
\(108\) 0 0
\(109\) −4.10635 4.10635i −0.393317 0.393317i 0.482551 0.875868i \(-0.339710\pi\)
−0.875868 + 0.482551i \(0.839710\pi\)
\(110\) 0 0
\(111\) 3.21187i 0.304857i
\(112\) 0 0
\(113\) 7.51147 7.51147i 0.706619 0.706619i −0.259203 0.965823i \(-0.583460\pi\)
0.965823 + 0.259203i \(0.0834600\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.87996i 0.266252i
\(118\) 0 0
\(119\) 7.44677i 0.682644i
\(120\) 0 0
\(121\) 9.70184i 0.881986i
\(122\) 0 0
\(123\) 7.35136i 0.662850i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.66639 7.66639i 0.680282 0.680282i −0.279782 0.960064i \(-0.590262\pi\)
0.960064 + 0.279782i \(0.0902621\pi\)
\(128\) 0 0
\(129\) 16.4785i 1.45085i
\(130\) 0 0
\(131\) 1.70610 + 1.70610i 0.149062 + 0.149062i 0.777699 0.628637i \(-0.216387\pi\)
−0.628637 + 0.777699i \(0.716387\pi\)
\(132\) 0 0
\(133\) 1.67570 0.145302
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.69414 + 1.69414i −0.144740 + 0.144740i −0.775764 0.631024i \(-0.782635\pi\)
0.631024 + 0.775764i \(0.282635\pi\)
\(138\) 0 0
\(139\) 2.38206 + 2.38206i 0.202044 + 0.202044i 0.800875 0.598831i \(-0.204368\pi\)
−0.598831 + 0.800875i \(0.704368\pi\)
\(140\) 0 0
\(141\) 5.99726 5.99726i 0.505061 0.505061i
\(142\) 0 0
\(143\) 4.75717 + 4.75717i 0.397815 + 0.397815i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.1366i 0.918533i
\(148\) 0 0
\(149\) 2.49691 2.49691i 0.204555 0.204555i −0.597393 0.801948i \(-0.703797\pi\)
0.801948 + 0.597393i \(0.203797\pi\)
\(150\) 0 0
\(151\) −16.5505 −1.34686 −0.673431 0.739250i \(-0.735180\pi\)
−0.673431 + 0.739250i \(0.735180\pi\)
\(152\) 0 0
\(153\) −2.52231 2.52231i −0.203916 0.203916i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.07087 0.404699 0.202350 0.979313i \(-0.435142\pi\)
0.202350 + 0.979313i \(0.435142\pi\)
\(158\) 0 0
\(159\) 16.0749 1.27482
\(160\) 0 0
\(161\) −3.31450 −0.261219
\(162\) 0 0
\(163\) −15.3065 −1.19890 −0.599450 0.800412i \(-0.704614\pi\)
−0.599450 + 0.800412i \(0.704614\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.78800 4.78800i −0.370506 0.370506i 0.497155 0.867662i \(-0.334378\pi\)
−0.867662 + 0.497155i \(0.834378\pi\)
\(168\) 0 0
\(169\) −21.8659 −1.68199
\(170\) 0 0
\(171\) 0.567579 0.567579i 0.0434038 0.0434038i
\(172\) 0 0
\(173\) 9.32156i 0.708705i −0.935112 0.354353i \(-0.884701\pi\)
0.935112 0.354353i \(-0.115299\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.79570 + 9.79570i 0.736290 + 0.736290i
\(178\) 0 0
\(179\) −15.6273 + 15.6273i −1.16804 + 1.16804i −0.185369 + 0.982669i \(0.559348\pi\)
−0.982669 + 0.185369i \(0.940652\pi\)
\(180\) 0 0
\(181\) 17.0056 + 17.0056i 1.26401 + 1.26401i 0.949131 + 0.314882i \(0.101965\pi\)
0.314882 + 0.949131i \(0.398035\pi\)
\(182\) 0 0
\(183\) −2.97268 + 2.97268i −0.219747 + 0.219747i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.33280 −0.609355
\(188\) 0 0
\(189\) 3.37802 + 3.37802i 0.245715 + 0.245715i
\(190\) 0 0
\(191\) 3.88531i 0.281131i −0.990071 0.140566i \(-0.955108\pi\)
0.990071 0.140566i \(-0.0448921\pi\)
\(192\) 0 0
\(193\) −3.68299 + 3.68299i −0.265108 + 0.265108i −0.827125 0.562018i \(-0.810025\pi\)
0.562018 + 0.827125i \(0.310025\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.2901i 0.875633i −0.899064 0.437816i \(-0.855752\pi\)
0.899064 0.437816i \(-0.144248\pi\)
\(198\) 0 0
\(199\) 12.6548i 0.897075i 0.893764 + 0.448537i \(0.148055\pi\)
−0.893764 + 0.448537i \(0.851945\pi\)
\(200\) 0 0
\(201\) 17.3595i 1.22444i
\(202\) 0 0
\(203\) 5.35604i 0.375920i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.12266 + 1.12266i −0.0780302 + 0.0780302i
\(208\) 0 0
\(209\) 1.87508i 0.129702i
\(210\) 0 0
\(211\) −2.57346 2.57346i −0.177164 0.177164i 0.612954 0.790118i \(-0.289981\pi\)
−0.790118 + 0.612954i \(0.789981\pi\)
\(212\) 0 0
\(213\) 17.4351 1.19463
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.07587 7.07587i 0.480341 0.480341i
\(218\) 0 0
\(219\) 16.0202 + 16.0202i 1.08255 + 1.08255i
\(220\) 0 0
\(221\) 30.5360 30.5360i 2.05407 2.05407i
\(222\) 0 0
\(223\) 9.55375 + 9.55375i 0.639766 + 0.639766i 0.950498 0.310731i \(-0.100574\pi\)
−0.310731 + 0.950498i \(0.600574\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.8481i 0.852759i 0.904544 + 0.426380i \(0.140211\pi\)
−0.904544 + 0.426380i \(0.859789\pi\)
\(228\) 0 0
\(229\) 1.96090 1.96090i 0.129580 0.129580i −0.639342 0.768922i \(-0.720793\pi\)
0.768922 + 0.639342i \(0.220793\pi\)
\(230\) 0 0
\(231\) −2.16659 −0.142551
\(232\) 0 0
\(233\) −7.39089 7.39089i −0.484193 0.484193i 0.422275 0.906468i \(-0.361232\pi\)
−0.906468 + 0.422275i \(0.861232\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −3.17915 −0.206508
\(238\) 0 0
\(239\) −10.7765 −0.697074 −0.348537 0.937295i \(-0.613321\pi\)
−0.348537 + 0.937295i \(0.613321\pi\)
\(240\) 0 0
\(241\) −14.5670 −0.938344 −0.469172 0.883107i \(-0.655448\pi\)
−0.469172 + 0.883107i \(0.655448\pi\)
\(242\) 0 0
\(243\) 5.02097 0.322095
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.87133 + 6.87133i 0.437212 + 0.437212i
\(248\) 0 0
\(249\) 25.9540 1.64477
\(250\) 0 0
\(251\) 12.1001 12.1001i 0.763750 0.763750i −0.213248 0.976998i \(-0.568404\pi\)
0.976998 + 0.213248i \(0.0684043\pi\)
\(252\) 0 0
\(253\) 3.70887i 0.233174i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.6083 + 17.6083i 1.09838 + 1.09838i 0.994601 + 0.103775i \(0.0330922\pi\)
0.103775 + 0.994601i \(0.466908\pi\)
\(258\) 0 0
\(259\) 1.23826 1.23826i 0.0769417 0.0769417i
\(260\) 0 0
\(261\) 1.81415 + 1.81415i 0.112293 + 0.112293i
\(262\) 0 0
\(263\) 12.1083 12.1083i 0.746630 0.746630i −0.227214 0.973845i \(-0.572962\pi\)
0.973845 + 0.227214i \(0.0729618\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.37671 0.512646
\(268\) 0 0
\(269\) 6.20149 + 6.20149i 0.378112 + 0.378112i 0.870421 0.492309i \(-0.163847\pi\)
−0.492309 + 0.870421i \(0.663847\pi\)
\(270\) 0 0
\(271\) 5.11166i 0.310511i −0.987874 0.155256i \(-0.950380\pi\)
0.987874 0.155256i \(-0.0496201\pi\)
\(272\) 0 0
\(273\) 7.93959 7.93959i 0.480526 0.480526i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.65720i 0.279824i 0.990164 + 0.139912i \(0.0446820\pi\)
−0.990164 + 0.139912i \(0.955318\pi\)
\(278\) 0 0
\(279\) 4.79335i 0.286971i
\(280\) 0 0
\(281\) 26.7402i 1.59519i −0.603194 0.797595i \(-0.706105\pi\)
0.603194 0.797595i \(-0.293895\pi\)
\(282\) 0 0
\(283\) 2.12034i 0.126041i −0.998012 0.0630205i \(-0.979927\pi\)
0.998012 0.0630205i \(-0.0200733\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.83414 + 2.83414i −0.167294 + 0.167294i
\(288\) 0 0
\(289\) 36.4877i 2.14634i
\(290\) 0 0
\(291\) 8.33815 + 8.33815i 0.488791 + 0.488791i
\(292\) 0 0
\(293\) 15.7244 0.918632 0.459316 0.888273i \(-0.348095\pi\)
0.459316 + 0.888273i \(0.348095\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.77995 3.77995i 0.219335 0.219335i
\(298\) 0 0
\(299\) −13.5914 13.5914i −0.786008 0.786008i
\(300\) 0 0
\(301\) −6.35291 + 6.35291i −0.366175 + 0.366175i
\(302\) 0 0
\(303\) 16.8877 + 16.8877i 0.970175 + 0.970175i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.32655i 0.304002i 0.988380 + 0.152001i \(0.0485717\pi\)
−0.988380 + 0.152001i \(0.951428\pi\)
\(308\) 0 0
\(309\) −16.6092 + 16.6092i −0.944866 + 0.944866i
\(310\) 0 0
\(311\) 15.8269 0.897459 0.448730 0.893668i \(-0.351877\pi\)
0.448730 + 0.893668i \(0.351877\pi\)
\(312\) 0 0
\(313\) 14.4637 + 14.4637i 0.817539 + 0.817539i 0.985751 0.168212i \(-0.0537993\pi\)
−0.168212 + 0.985751i \(0.553799\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.38384 −0.302387 −0.151193 0.988504i \(-0.548312\pi\)
−0.151193 + 0.988504i \(0.548312\pi\)
\(318\) 0 0
\(319\) 5.99331 0.335561
\(320\) 0 0
\(321\) 17.3595 0.968910
\(322\) 0 0
\(323\) −12.0360 −0.669702
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 7.66882 + 7.66882i 0.424086 + 0.424086i
\(328\) 0 0
\(329\) −4.62420 −0.254941
\(330\) 0 0
\(331\) 5.04895 5.04895i 0.277516 0.277516i −0.554601 0.832116i \(-0.687129\pi\)
0.832116 + 0.554601i \(0.187129\pi\)
\(332\) 0 0
\(333\) 0.838825i 0.0459673i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.10233 8.10233i −0.441362 0.441362i 0.451107 0.892470i \(-0.351029\pi\)
−0.892470 + 0.451107i \(0.851029\pi\)
\(338\) 0 0
\(339\) −14.0280 + 14.0280i −0.761898 + 0.761898i
\(340\) 0 0
\(341\) −7.91777 7.91777i −0.428771 0.428771i
\(342\) 0 0
\(343\) 9.33338 9.33338i 0.503955 0.503955i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.87019 0.422494 0.211247 0.977433i \(-0.432248\pi\)
0.211247 + 0.977433i \(0.432248\pi\)
\(348\) 0 0
\(349\) 18.7492 + 18.7492i 1.00362 + 1.00362i 0.999993 + 0.00363019i \(0.00115553\pi\)
0.00363019 + 0.999993i \(0.498844\pi\)
\(350\) 0 0
\(351\) 27.7037i 1.47871i
\(352\) 0 0
\(353\) 0.0830593 0.0830593i 0.00442080 0.00442080i −0.704893 0.709314i \(-0.749005\pi\)
0.709314 + 0.704893i \(0.249005\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 13.9072i 0.736047i
\(358\) 0 0
\(359\) 35.9409i 1.89689i 0.316941 + 0.948445i \(0.397344\pi\)
−0.316941 + 0.948445i \(0.602656\pi\)
\(360\) 0 0
\(361\) 16.2916i 0.857453i
\(362\) 0 0
\(363\) 18.1187i 0.950983i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 23.6922 23.6922i 1.23672 1.23672i 0.275389 0.961333i \(-0.411193\pi\)
0.961333 0.275389i \(-0.0888066\pi\)
\(368\) 0 0
\(369\) 1.91991i 0.0999467i
\(370\) 0 0
\(371\) −6.19729 6.19729i −0.321747 0.321747i
\(372\) 0 0
\(373\) −27.8655 −1.44282 −0.721410 0.692508i \(-0.756506\pi\)
−0.721410 + 0.692508i \(0.756506\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −21.9628 + 21.9628i −1.13114 + 1.13114i
\(378\) 0 0
\(379\) 18.4005 + 18.4005i 0.945168 + 0.945168i 0.998573 0.0534045i \(-0.0170073\pi\)
−0.0534045 + 0.998573i \(0.517007\pi\)
\(380\) 0 0
\(381\) −14.3173 + 14.3173i −0.733500 + 0.733500i
\(382\) 0 0
\(383\) −27.3966 27.3966i −1.39990 1.39990i −0.800298 0.599603i \(-0.795325\pi\)
−0.599603 0.800298i \(-0.704675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.30360i 0.218764i
\(388\) 0 0
\(389\) −9.05190 + 9.05190i −0.458950 + 0.458950i −0.898311 0.439361i \(-0.855205\pi\)
0.439361 + 0.898311i \(0.355205\pi\)
\(390\) 0 0
\(391\) 23.8070 1.20397
\(392\) 0 0
\(393\) −3.18622 3.18622i −0.160724 0.160724i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.6052 0.582448 0.291224 0.956655i \(-0.405937\pi\)
0.291224 + 0.956655i \(0.405937\pi\)
\(398\) 0 0
\(399\) −3.12945 −0.156668
\(400\) 0 0
\(401\) −9.94759 −0.496759 −0.248379 0.968663i \(-0.579898\pi\)
−0.248379 + 0.968663i \(0.579898\pi\)
\(402\) 0 0
\(403\) 58.0302 2.89069
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.38559 1.38559i −0.0686811 0.0686811i
\(408\) 0 0
\(409\) −24.7129 −1.22198 −0.610988 0.791640i \(-0.709228\pi\)
−0.610988 + 0.791640i \(0.709228\pi\)
\(410\) 0 0
\(411\) 3.16389 3.16389i 0.156063 0.156063i
\(412\) 0 0
\(413\) 7.55300i 0.371659i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.44861 4.44861i −0.217849 0.217849i
\(418\) 0 0
\(419\) 12.9537 12.9537i 0.632829 0.632829i −0.315947 0.948777i \(-0.602322\pi\)
0.948777 + 0.315947i \(0.102322\pi\)
\(420\) 0 0
\(421\) −8.99009 8.99009i −0.438150 0.438150i 0.453239 0.891389i \(-0.350268\pi\)
−0.891389 + 0.453239i \(0.850268\pi\)
\(422\) 0 0
\(423\) −1.56627 + 1.56627i −0.0761547 + 0.0761547i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.29210 0.110922
\(428\) 0 0
\(429\) −8.88425 8.88425i −0.428936 0.428936i
\(430\) 0 0
\(431\) 20.8177i 1.00275i −0.865230 0.501376i \(-0.832828\pi\)
0.865230 0.501376i \(-0.167172\pi\)
\(432\) 0 0
\(433\) 7.70002 7.70002i 0.370039 0.370039i −0.497452 0.867491i \(-0.665731\pi\)
0.867491 + 0.497452i \(0.165731\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.35714i 0.256267i
\(438\) 0 0
\(439\) 38.4535i 1.83529i −0.397407 0.917643i \(-0.630090\pi\)
0.397407 0.917643i \(-0.369910\pi\)
\(440\) 0 0
\(441\) 2.90849i 0.138499i
\(442\) 0 0
\(443\) 17.1242i 0.813595i −0.913518 0.406797i \(-0.866645\pi\)
0.913518 0.406797i \(-0.133355\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.66310 + 4.66310i −0.220557 + 0.220557i
\(448\) 0 0
\(449\) 12.1296i 0.572431i 0.958165 + 0.286215i \(0.0923973\pi\)
−0.958165 + 0.286215i \(0.907603\pi\)
\(450\) 0 0
\(451\) 3.17135 + 3.17135i 0.149333 + 0.149333i
\(452\) 0 0
\(453\) 30.9089 1.45223
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.96532 + 3.96532i −0.185490 + 0.185490i −0.793743 0.608253i \(-0.791871\pi\)
0.608253 + 0.793743i \(0.291871\pi\)
\(458\) 0 0
\(459\) −24.2633 24.2633i −1.13251 1.13251i
\(460\) 0 0
\(461\) −25.6620 + 25.6620i −1.19520 + 1.19520i −0.219614 + 0.975587i \(0.570480\pi\)
−0.975587 + 0.219614i \(0.929520\pi\)
\(462\) 0 0
\(463\) 3.98789 + 3.98789i 0.185333 + 0.185333i 0.793675 0.608342i \(-0.208165\pi\)
−0.608342 + 0.793675i \(0.708165\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.8576i 0.502429i −0.967931 0.251215i \(-0.919170\pi\)
0.967931 0.251215i \(-0.0808300\pi\)
\(468\) 0 0
\(469\) 6.69253 6.69253i 0.309032 0.309032i
\(470\) 0 0
\(471\) −9.47009 −0.436359
\(472\) 0 0
\(473\) 7.10878 + 7.10878i 0.326862 + 0.326862i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.19819 −0.192222
\(478\) 0 0
\(479\) −0.144583 −0.00660618 −0.00330309 0.999995i \(-0.501051\pi\)
−0.00330309 + 0.999995i \(0.501051\pi\)
\(480\) 0 0
\(481\) 10.1551 0.463035
\(482\) 0 0
\(483\) 6.18999 0.281654
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 24.9905 + 24.9905i 1.13243 + 1.13243i 0.989772 + 0.142655i \(0.0455639\pi\)
0.142655 + 0.989772i \(0.454436\pi\)
\(488\) 0 0
\(489\) 28.5857 1.29269
\(490\) 0 0
\(491\) −16.8603 + 16.8603i −0.760893 + 0.760893i −0.976484 0.215591i \(-0.930832\pi\)
0.215591 + 0.976484i \(0.430832\pi\)
\(492\) 0 0
\(493\) 38.4707i 1.73263i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.72168 6.72168i −0.301509 0.301509i
\(498\) 0 0
\(499\) 12.2949 12.2949i 0.550397 0.550397i −0.376159 0.926555i \(-0.622755\pi\)
0.926555 + 0.376159i \(0.122755\pi\)
\(500\) 0 0
\(501\) 8.94182 + 8.94182i 0.399491 + 0.399491i
\(502\) 0 0
\(503\) 2.55961 2.55961i 0.114128 0.114128i −0.647737 0.761864i \(-0.724284\pi\)
0.761864 + 0.647737i \(0.224284\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 40.8356 1.81357
\(508\) 0 0
\(509\) −6.59987 6.59987i −0.292534 0.292534i 0.545547 0.838080i \(-0.316322\pi\)
−0.838080 + 0.545547i \(0.816322\pi\)
\(510\) 0 0
\(511\) 12.3524i 0.546440i
\(512\) 0 0
\(513\) 5.45981 5.45981i 0.241056 0.241056i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.17440i 0.227570i
\(518\) 0 0
\(519\) 17.4085i 0.764147i
\(520\) 0 0
\(521\) 13.9510i 0.611204i −0.952159 0.305602i \(-0.901142\pi\)
0.952159 0.305602i \(-0.0988577\pi\)
\(522\) 0 0
\(523\) 24.6076i 1.07601i 0.842941 + 0.538007i \(0.180822\pi\)
−0.842941 + 0.538007i \(0.819178\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −50.8237 + 50.8237i −2.21391 + 2.21391i
\(528\) 0 0
\(529\) 12.4037i 0.539291i
\(530\) 0 0
\(531\) −2.55829 2.55829i −0.111020 0.111020i
\(532\) 0 0
\(533\) −23.2432 −1.00678
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 29.1847 29.1847i 1.25941 1.25941i
\(538\) 0 0
\(539\) −4.80430 4.80430i −0.206936 0.206936i
\(540\) 0 0
\(541\) −14.4785 + 14.4785i −0.622481 + 0.622481i −0.946165 0.323684i \(-0.895078\pi\)
0.323684 + 0.946165i \(0.395078\pi\)
\(542\) 0 0
\(543\) −31.7587 31.7587i −1.36290 1.36290i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 27.5523i 1.17805i −0.808114 0.589026i \(-0.799512\pi\)
0.808114 0.589026i \(-0.200488\pi\)
\(548\) 0 0
\(549\) 0.776359 0.776359i 0.0331342 0.0331342i
\(550\) 0 0
\(551\) 8.65682 0.368793
\(552\) 0 0
\(553\) 1.22565 + 1.22565i 0.0521197 + 0.0521197i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −23.5208 −0.996608 −0.498304 0.867002i \(-0.666044\pi\)
−0.498304 + 0.867002i \(0.666044\pi\)
\(558\) 0 0
\(559\) −52.1011 −2.20364
\(560\) 0 0
\(561\) 15.5619 0.657024
\(562\) 0 0
\(563\) 11.3970 0.480327 0.240163 0.970732i \(-0.422799\pi\)
0.240163 + 0.970732i \(0.422799\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.36212 7.36212i −0.309180 0.309180i
\(568\) 0 0
\(569\) 21.2444 0.890613 0.445307 0.895378i \(-0.353095\pi\)
0.445307 + 0.895378i \(0.353095\pi\)
\(570\) 0 0
\(571\) −23.0980 + 23.0980i −0.966622 + 0.966622i −0.999461 0.0328390i \(-0.989545\pi\)
0.0328390 + 0.999461i \(0.489545\pi\)
\(572\) 0 0
\(573\) 7.25601i 0.303124i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.54721 8.54721i −0.355825 0.355825i 0.506446 0.862271i \(-0.330959\pi\)
−0.862271 + 0.506446i \(0.830959\pi\)
\(578\) 0 0
\(579\) 6.87817 6.87817i 0.285847 0.285847i
\(580\) 0 0
\(581\) −10.0060 10.0060i −0.415117 0.415117i
\(582\) 0 0
\(583\) −6.93466 + 6.93466i −0.287204 + 0.287204i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.9980 −1.19687 −0.598437 0.801170i \(-0.704211\pi\)
−0.598437 + 0.801170i \(0.704211\pi\)
\(588\) 0 0
\(589\) −11.4365 11.4365i −0.471234 0.471234i
\(590\) 0 0
\(591\) 22.9523i 0.944133i
\(592\) 0 0
\(593\) 1.73827 1.73827i 0.0713822 0.0713822i −0.670514 0.741897i \(-0.733927\pi\)
0.741897 + 0.670514i \(0.233927\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.6334i 0.967252i
\(598\) 0 0
\(599\) 23.3429i 0.953764i 0.878967 + 0.476882i \(0.158233\pi\)
−0.878967 + 0.476882i \(0.841767\pi\)
\(600\) 0 0
\(601\) 14.2850i 0.582697i −0.956617 0.291348i \(-0.905896\pi\)
0.956617 0.291348i \(-0.0941039\pi\)
\(602\) 0 0
\(603\) 4.53367i 0.184625i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.84325 4.84325i 0.196581 0.196581i −0.601951 0.798533i \(-0.705610\pi\)
0.798533 + 0.601951i \(0.205610\pi\)
\(608\) 0 0
\(609\) 10.0027i 0.405328i
\(610\) 0 0
\(611\) −18.9619 18.9619i −0.767116 0.767116i
\(612\) 0 0
\(613\) −6.37122 −0.257331 −0.128666 0.991688i \(-0.541069\pi\)
−0.128666 + 0.991688i \(0.541069\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.7251 + 13.7251i −0.552553 + 0.552553i −0.927177 0.374624i \(-0.877772\pi\)
0.374624 + 0.927177i \(0.377772\pi\)
\(618\) 0 0
\(619\) 6.92352 + 6.92352i 0.278280 + 0.278280i 0.832422 0.554142i \(-0.186954\pi\)
−0.554142 + 0.832422i \(0.686954\pi\)
\(620\) 0 0
\(621\) −10.7994 + 10.7994i −0.433365 + 0.433365i
\(622\) 0 0
\(623\) −3.22944 3.22944i −0.129385 0.129385i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.50180i 0.139848i
\(628\) 0 0
\(629\) −8.89402 + 8.89402i −0.354628 + 0.354628i
\(630\) 0 0
\(631\) −0.299394 −0.0119187 −0.00595935 0.999982i \(-0.501897\pi\)
−0.00595935 + 0.999982i \(0.501897\pi\)
\(632\) 0 0
\(633\) 4.80606 + 4.80606i 0.191024 + 0.191024i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 35.2113 1.39512
\(638\) 0 0
\(639\) −4.55342 −0.180130
\(640\) 0 0
\(641\) 45.7708 1.80784 0.903920 0.427702i \(-0.140677\pi\)
0.903920 + 0.427702i \(0.140677\pi\)
\(642\) 0 0
\(643\) −9.26732 −0.365467 −0.182734 0.983162i \(-0.558495\pi\)
−0.182734 + 0.983162i \(0.558495\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.284672 0.284672i −0.0111916 0.0111916i 0.701489 0.712680i \(-0.252519\pi\)
−0.712680 + 0.701489i \(0.752519\pi\)
\(648\) 0 0
\(649\) −8.45167 −0.331757
\(650\) 0 0
\(651\) −13.2145 + 13.2145i −0.517918 + 0.517918i
\(652\) 0 0
\(653\) 11.1970i 0.438173i −0.975705 0.219087i \(-0.929692\pi\)
0.975705 0.219087i \(-0.0703078\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.18391 4.18391i −0.163230 0.163230i
\(658\) 0 0
\(659\) 5.66498 5.66498i 0.220676 0.220676i −0.588107 0.808783i \(-0.700127\pi\)
0.808783 + 0.588107i \(0.200127\pi\)
\(660\) 0 0
\(661\) 23.3785 + 23.3785i 0.909320 + 0.909320i 0.996217 0.0868975i \(-0.0276953\pi\)
−0.0868975 + 0.996217i \(0.527695\pi\)
\(662\) 0 0
\(663\) −57.0275 + 57.0275i −2.21476 + 2.21476i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −17.1230 −0.663006
\(668\) 0 0
\(669\) −17.8421 17.8421i −0.689815 0.689815i
\(670\) 0 0
\(671\) 2.56481i 0.0990135i
\(672\) 0 0
\(673\) −26.6324 + 26.6324i −1.02660 + 1.02660i −0.0269656 + 0.999636i \(0.508584\pi\)
−0.999636 + 0.0269656i \(0.991416\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.5539i 0.520918i 0.965485 + 0.260459i \(0.0838739\pi\)
−0.965485 + 0.260459i \(0.916126\pi\)
\(678\) 0 0
\(679\) 6.42915i 0.246728i
\(680\) 0 0
\(681\) 23.9945i 0.919470i
\(682\) 0 0
\(683\) 15.3467i 0.587225i 0.955925 + 0.293613i \(0.0948575\pi\)
−0.955925 + 0.293613i \(0.905142\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −3.66207 + 3.66207i −0.139717 + 0.139717i
\(688\) 0 0
\(689\) 50.8249i 1.93627i
\(690\) 0 0
\(691\) −10.6170 10.6170i −0.403891 0.403891i 0.475710 0.879602i \(-0.342191\pi\)
−0.879602 + 0.475710i \(0.842191\pi\)
\(692\) 0 0
\(693\) 0.565835 0.0214943
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.3567 20.3567i 0.771067 0.771067i
\(698\) 0 0
\(699\) 13.8028 + 13.8028i 0.522072 + 0.522072i
\(700\) 0 0
\(701\) −5.96737 + 5.96737i −0.225384 + 0.225384i −0.810761 0.585377i \(-0.800947\pi\)
0.585377 + 0.810761i \(0.300947\pi\)
\(702\) 0 0
\(703\) −2.00137 2.00137i −0.0754829 0.0754829i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0213i 0.489717i
\(708\) 0 0
\(709\) 35.3379 35.3379i 1.32714 1.32714i 0.419287 0.907854i \(-0.362280\pi\)
0.907854 0.419287i \(-0.137720\pi\)
\(710\) 0 0
\(711\) 0.830280 0.0311379
\(712\) 0 0
\(713\) 22.6212 + 22.6212i 0.847172 + 0.847172i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.1256 0.751606
\(718\) 0 0
\(719\) 37.8418 1.41126 0.705631 0.708580i \(-0.250664\pi\)
0.705631 + 0.708580i \(0.250664\pi\)
\(720\) 0 0
\(721\) 12.8066 0.476942
\(722\) 0 0
\(723\) 27.2046 1.01175
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −32.3722 32.3722i −1.20062 1.20062i −0.973978 0.226641i \(-0.927226\pi\)
−0.226641 0.973978i \(-0.572774\pi\)
\(728\) 0 0
\(729\) 21.2991 0.788855
\(730\) 0 0
\(731\) 45.6309 45.6309i 1.68772 1.68772i
\(732\) 0 0
\(733\) 1.96701i 0.0726532i −0.999340 0.0363266i \(-0.988434\pi\)
0.999340 0.0363266i \(-0.0115657\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.48881 7.48881i −0.275854 0.275854i
\(738\) 0 0
\(739\) 14.3605 14.3605i 0.528261 0.528261i −0.391793 0.920054i \(-0.628145\pi\)
0.920054 + 0.391793i \(0.128145\pi\)
\(740\) 0 0
\(741\) −12.8325 12.8325i −0.471415 0.471415i
\(742\) 0 0
\(743\) −2.28846 + 2.28846i −0.0839556 + 0.0839556i −0.747837 0.663882i \(-0.768908\pi\)
0.663882 + 0.747837i \(0.268908\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6.77826 −0.248004
\(748\) 0 0
\(749\) −6.69253 6.69253i −0.244540 0.244540i
\(750\) 0 0
\(751\) 23.7064i 0.865058i 0.901620 + 0.432529i \(0.142379\pi\)
−0.901620 + 0.432529i \(0.857621\pi\)
\(752\) 0 0
\(753\) −22.5975 + 22.5975i −0.823497 + 0.823497i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.5469i 1.11025i 0.831768 + 0.555124i \(0.187329\pi\)
−0.831768 + 0.555124i \(0.812671\pi\)
\(758\) 0 0
\(759\) 6.92649i 0.251416i
\(760\) 0 0
\(761\) 23.6988i 0.859080i −0.903048 0.429540i \(-0.858676\pi\)
0.903048 0.429540i \(-0.141324\pi\)
\(762\) 0 0
\(763\) 5.91306i 0.214067i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.9716 30.9716i 1.11832 1.11832i
\(768\) 0 0
\(769\) 2.76629i 0.0997548i 0.998755 + 0.0498774i \(0.0158831\pi\)
−0.998755 + 0.0498774i \(0.984117\pi\)
\(770\) 0 0
\(771\) −32.8844 32.8844i −1.18430 1.18430i
\(772\) 0 0
\(773\) −36.0726 −1.29744 −0.648720 0.761027i \(-0.724696\pi\)
−0.648720 + 0.761027i \(0.724696\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.31251 + 2.31251i −0.0829608 + 0.0829608i
\(778\) 0 0
\(779\) 4.58075 + 4.58075i 0.164122 + 0.164122i
\(780\) 0 0
\(781\) −7.52144 + 7.52144i −0.269138 + 0.269138i
\(782\) 0 0
\(783\) 17.4512 + 17.4512i 0.623654 + 0.623654i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.1024i 0.431402i 0.976459 + 0.215701i \(0.0692037\pi\)
−0.976459 + 0.215701i \(0.930796\pi\)
\(788\) 0 0
\(789\) −22.6129 + 22.6129i −0.805039 + 0.805039i
\(790\) 0 0
\(791\) 10.8163 0.384585
\(792\) 0 0
\(793\) 9.39890 + 9.39890i 0.333765 + 0.333765i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.6614 −0.838130 −0.419065 0.907956i \(-0.637642\pi\)
−0.419065 + 0.907956i \(0.637642\pi\)
\(798\) 0 0
\(799\) 33.2142 1.17503
\(800\) 0 0
\(801\) −2.18770 −0.0772984
\(802\) 0 0
\(803\) −13.8221 −0.487773
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11.5816 11.5816i −0.407691 0.407691i
\(808\) 0 0
\(809\) −23.5574 −0.828235 −0.414117 0.910223i \(-0.635910\pi\)
−0.414117 + 0.910223i \(0.635910\pi\)
\(810\) 0 0
\(811\) −14.0698 + 14.0698i −0.494056 + 0.494056i −0.909581 0.415526i \(-0.863598\pi\)
0.415526 + 0.909581i \(0.363598\pi\)
\(812\) 0 0
\(813\) 9.54627i 0.334802i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 10.2680 + 10.2680i 0.359233 + 0.359233i
\(818\) 0 0
\(819\) −2.07354 + 2.07354i −0.0724552 + 0.0724552i
\(820\) 0 0
\(821\) −28.1332 28.1332i −0.981855 0.981855i 0.0179834 0.999838i \(-0.494275\pi\)
−0.999838 + 0.0179834i \(0.994275\pi\)
\(822\) 0 0
\(823\) −27.6068 + 27.6068i −0.962313 + 0.962313i −0.999315 0.0370020i \(-0.988219\pi\)
0.0370020 + 0.999315i \(0.488219\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.79320 0.236223 0.118111 0.993000i \(-0.462316\pi\)
0.118111 + 0.993000i \(0.462316\pi\)
\(828\) 0 0
\(829\) −25.7474 25.7474i −0.894245 0.894245i 0.100675 0.994919i \(-0.467900\pi\)
−0.994919 + 0.100675i \(0.967900\pi\)
\(830\) 0 0
\(831\) 8.69755i 0.301715i
\(832\) 0 0
\(833\) −30.8385 + 30.8385i −1.06849 + 1.06849i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 46.1096i 1.59378i
\(838\) 0 0
\(839\) 11.4280i 0.394538i 0.980349 + 0.197269i \(0.0632073\pi\)
−0.980349 + 0.197269i \(0.936793\pi\)
\(840\) 0 0
\(841\) 1.33022i 0.0458696i
\(842\) 0 0
\(843\) 49.9387i 1.71998i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.98522 + 6.98522i −0.240015 + 0.240015i
\(848\) 0 0
\(849\) 3.95983i 0.135901i
\(850\) 0 0
\(851\) 3.95866 + 3.95866i 0.135701 + 0.135701i
\(852\) 0 0
\(853\) 9.67475 0.331257 0.165629 0.986188i \(-0.447035\pi\)
0.165629 + 0.986188i \(0.447035\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −26.5831 + 26.5831i −0.908061 + 0.908061i −0.996116 0.0880544i \(-0.971935\pi\)
0.0880544 + 0.996116i \(0.471935\pi\)
\(858\) 0 0
\(859\) 1.58572 + 1.58572i 0.0541040 + 0.0541040i 0.733641 0.679537i \(-0.237819\pi\)
−0.679537 + 0.733641i \(0.737819\pi\)
\(860\) 0 0
\(861\) 5.29290 5.29290i 0.180382 0.180382i
\(862\) 0 0
\(863\) −9.59115 9.59115i −0.326486 0.326486i 0.524762 0.851249i \(-0.324154\pi\)
−0.851249 + 0.524762i \(0.824154\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 68.1426i 2.31424i
\(868\) 0 0
\(869\) 1.37147 1.37147i 0.0465241 0.0465241i
\(870\) 0 0
\(871\) 54.8864 1.85975
\(872\) 0 0
\(873\) −2.17763 2.17763i −0.0737015 0.0737015i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32.7116 −1.10459 −0.552295 0.833649i \(-0.686248\pi\)
−0.552295 + 0.833649i \(0.686248\pi\)
\(878\) 0 0
\(879\) −29.3661 −0.990496
\(880\) 0 0
\(881\) 9.42337 0.317481 0.158741 0.987320i \(-0.449257\pi\)
0.158741 + 0.987320i \(0.449257\pi\)
\(882\) 0 0
\(883\) −13.1729 −0.443304 −0.221652 0.975126i \(-0.571145\pi\)
−0.221652 + 0.975126i \(0.571145\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.04550 + 7.04550i 0.236565 + 0.236565i 0.815426 0.578861i \(-0.196503\pi\)
−0.578861 + 0.815426i \(0.696503\pi\)
\(888\) 0 0
\(889\) 11.0394 0.370251
\(890\) 0 0
\(891\) −8.23808 + 8.23808i −0.275986 + 0.275986i
\(892\) 0 0
\(893\) 7.47397i 0.250107i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 25.3825 + 25.3825i 0.847497 + 0.847497i
\(898\) 0 0
\(899\) 36.5546 36.5546i 1.21916 1.21916i
\(900\) 0 0
\(901\) 44.5131 + 44.5131i 1.48295 + 1.48295i
\(902\) 0 0
\(903\) 11.8644 11.8644i 0.394821 0.394821i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.4995 −1.51079 −0.755394 0.655271i \(-0.772554\pi\)
−0.755394 + 0.655271i \(0.772554\pi\)
\(908\) 0 0
\(909\) −4.41047 4.41047i −0.146286 0.146286i
\(910\) 0 0
\(911\) 41.2904i 1.36801i 0.729477 + 0.684005i \(0.239763\pi\)
−0.729477 + 0.684005i \(0.760237\pi\)
\(912\) 0 0
\(913\) −11.1965 + 11.1965i −0.370549 + 0.370549i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.45674i 0.0811288i
\(918\) 0 0
\(919\) 20.1715i 0.665397i 0.943033 + 0.332698i \(0.107959\pi\)
−0.943033 + 0.332698i \(0.892041\pi\)
\(920\) 0 0
\(921\) 9.94759i 0.327784i
\(922\) 0 0
\(923\) 55.1255i 1.81448i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.33774 4.33774i 0.142470 0.142470i
\(928\) 0 0
\(929\) 8.18969i 0.268695i −0.990934 0.134348i \(-0.957106\pi\)
0.990934 0.134348i \(-0.0428939\pi\)
\(930\) 0 0
\(931\) −6.93940 6.93940i −0.227430 0.227430i
\(932\) 0 0
\(933\) −29.5574 −0.967667
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.0511 33.0511i 1.07973 1.07973i 0.0831985 0.996533i \(-0.473486\pi\)
0.996533 0.0831985i \(-0.0265136\pi\)
\(938\) 0 0
\(939\) −27.0117 27.0117i −0.881495 0.881495i
\(940\) 0 0
\(941\) 26.6371 26.6371i 0.868345 0.868345i −0.123944 0.992289i \(-0.539554\pi\)
0.992289 + 0.123944i \(0.0395544\pi\)
\(942\) 0 0
\(943\) −9.06063 9.06063i −0.295055 0.295055i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.5777i 1.22111i 0.791973 + 0.610556i \(0.209054\pi\)
−0.791973 + 0.610556i \(0.790946\pi\)
\(948\) 0 0
\(949\) 50.6521 50.6521i 1.64424 1.64424i
\(950\) 0 0
\(951\) 10.0546 0.326042
\(952\) 0 0
\(953\) −7.96284 7.96284i −0.257942 0.257942i 0.566275 0.824217i \(-0.308384\pi\)
−0.824217 + 0.566275i \(0.808384\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −11.1928 −0.361812
\(958\) 0 0
\(959\) −2.43952 −0.0787763
\(960\) 0 0
\(961\) −65.5846 −2.11563
\(962\) 0 0
\(963\) −4.53367 −0.146095
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.6144 + 16.6144i 0.534282 + 0.534282i 0.921844 0.387562i \(-0.126683\pi\)
−0.387562 + 0.921844i \(0.626683\pi\)
\(968\) 0 0
\(969\) 22.4778 0.722092
\(970\) 0 0
\(971\) 17.9269 17.9269i 0.575301 0.575301i −0.358304 0.933605i \(-0.616645\pi\)
0.933605 + 0.358304i \(0.116645\pi\)
\(972\) 0 0
\(973\) 3.43011i 0.109964i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.0721 + 35.0721i 1.12206 + 1.12206i 0.991432 + 0.130625i \(0.0416983\pi\)
0.130625 + 0.991432i \(0.458302\pi\)
\(978\) 0 0
\(979\) −3.61368 + 3.61368i −0.115494 + 0.115494i
\(980\) 0 0
\(981\) −2.00282 2.00282i −0.0639451 0.0639451i
\(982\) 0 0
\(983\) −24.9265 + 24.9265i −0.795033 + 0.795033i −0.982308 0.187274i \(-0.940035\pi\)
0.187274 + 0.982308i \(0.440035\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.63592 0.274884
\(988\) 0 0
\(989\) −20.3100 20.3100i −0.645819 0.645819i
\(990\) 0 0
\(991\) 10.7686i 0.342076i 0.985264 + 0.171038i \(0.0547121\pi\)
−0.985264 + 0.171038i \(0.945288\pi\)
\(992\) 0 0
\(993\) −9.42916 + 9.42916i −0.299226 + 0.299226i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.3858i 0.328922i 0.986384 + 0.164461i \(0.0525885\pi\)
−0.986384 + 0.164461i \(0.947412\pi\)
\(998\) 0 0
\(999\) 8.06906i 0.255294i
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 1600.2.s.e.207.4 24
4.3 odd 2 400.2.s.e.107.11 yes 24
5.2 odd 4 1600.2.j.e.143.9 24
5.3 odd 4 1600.2.j.e.143.4 24
5.4 even 2 inner 1600.2.s.e.207.9 24
16.3 odd 4 1600.2.j.e.1007.9 24
16.13 even 4 400.2.j.e.307.5 yes 24
20.3 even 4 400.2.j.e.43.5 24
20.7 even 4 400.2.j.e.43.8 yes 24
20.19 odd 2 400.2.s.e.107.2 yes 24
80.3 even 4 inner 1600.2.s.e.943.4 24
80.13 odd 4 400.2.s.e.243.11 yes 24
80.19 odd 4 1600.2.j.e.1007.4 24
80.29 even 4 400.2.j.e.307.8 yes 24
80.67 even 4 inner 1600.2.s.e.943.9 24
80.77 odd 4 400.2.s.e.243.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.e.43.5 24 20.3 even 4
400.2.j.e.43.8 yes 24 20.7 even 4
400.2.j.e.307.5 yes 24 16.13 even 4
400.2.j.e.307.8 yes 24 80.29 even 4
400.2.s.e.107.2 yes 24 20.19 odd 2
400.2.s.e.107.11 yes 24 4.3 odd 2
400.2.s.e.243.2 yes 24 80.77 odd 4
400.2.s.e.243.11 yes 24 80.13 odd 4
1600.2.j.e.143.4 24 5.3 odd 4
1600.2.j.e.143.9 24 5.2 odd 4
1600.2.j.e.1007.4 24 80.19 odd 4
1600.2.j.e.1007.9 24 16.3 odd 4
1600.2.s.e.207.4 24 1.1 even 1 trivial
1600.2.s.e.207.9 24 5.4 even 2 inner
1600.2.s.e.943.4 24 80.3 even 4 inner
1600.2.s.e.943.9 24 80.67 even 4 inner