Properties

Label 1600.2.n.p.1343.2
Level $1600$
Weight $2$
Character 1600.1343
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1343,1600)]
 
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, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1343.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1343
Dual form 1600.2.n.p.1407.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.224745 - 0.224745i) q^{3} +(-2.00000 - 2.00000i) q^{7} +2.89898i q^{9} +O(q^{10})\) \(q+(0.224745 - 0.224745i) q^{3} +(-2.00000 - 2.00000i) q^{7} +2.89898i q^{9} +1.44949i q^{11} +(-0.449490 - 0.449490i) q^{13} +(3.22474 - 3.22474i) q^{17} -7.44949 q^{19} -0.898979 q^{21} +(4.44949 - 4.44949i) q^{23} +(1.32577 + 1.32577i) q^{27} -6.89898i q^{29} -6.89898i q^{31} +(0.325765 + 0.325765i) q^{33} +(-6.00000 + 6.00000i) q^{37} -0.202041 q^{39} +5.00000 q^{41} +(6.89898 - 6.89898i) q^{43} +(-7.34847 - 7.34847i) q^{47} +1.00000i q^{49} -1.44949i q^{51} +(-1.55051 - 1.55051i) q^{53} +(-1.67423 + 1.67423i) q^{57} -11.7980 q^{59} -8.89898 q^{61} +(5.79796 - 5.79796i) q^{63} +(-1.77526 - 1.77526i) q^{67} -2.00000i q^{69} +2.00000i q^{71} +(1.67423 + 1.67423i) q^{73} +(2.89898 - 2.89898i) q^{77} -8.89898 q^{79} -8.10102 q^{81} +(2.67423 - 2.67423i) q^{83} +(-1.55051 - 1.55051i) q^{87} +2.10102i q^{89} +1.79796i q^{91} +(-1.55051 - 1.55051i) q^{93} +(4.89898 - 4.89898i) q^{97} -4.20204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 8 q^{7} + 8 q^{13} + 8 q^{17} - 20 q^{19} + 16 q^{21} + 8 q^{23} + 20 q^{27} + 16 q^{33} - 24 q^{37} - 40 q^{39} + 20 q^{41} + 8 q^{43} - 16 q^{53} + 8 q^{57} - 8 q^{59} - 16 q^{61} - 16 q^{63} - 12 q^{67} - 8 q^{73} - 8 q^{77} - 16 q^{79} - 52 q^{81} - 4 q^{83} - 16 q^{87} - 16 q^{93} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.224745 0.224745i 0.129757 0.129757i −0.639246 0.769002i \(-0.720753\pi\)
0.769002 + 0.639246i \(0.220753\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 2.00000i −0.755929 0.755929i 0.219650 0.975579i \(-0.429509\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) 0 0
\(9\) 2.89898i 0.966326i
\(10\) 0 0
\(11\) 1.44949i 0.437038i 0.975833 + 0.218519i \(0.0701225\pi\)
−0.975833 + 0.218519i \(0.929878\pi\)
\(12\) 0 0
\(13\) −0.449490 0.449490i −0.124666 0.124666i 0.642021 0.766687i \(-0.278096\pi\)
−0.766687 + 0.642021i \(0.778096\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.22474 3.22474i 0.782116 0.782116i −0.198072 0.980187i \(-0.563468\pi\)
0.980187 + 0.198072i \(0.0634680\pi\)
\(18\) 0 0
\(19\) −7.44949 −1.70903 −0.854515 0.519427i \(-0.826146\pi\)
−0.854515 + 0.519427i \(0.826146\pi\)
\(20\) 0 0
\(21\) −0.898979 −0.196173
\(22\) 0 0
\(23\) 4.44949 4.44949i 0.927783 0.927783i −0.0697797 0.997562i \(-0.522230\pi\)
0.997562 + 0.0697797i \(0.0222296\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.32577 + 1.32577i 0.255144 + 0.255144i
\(28\) 0 0
\(29\) 6.89898i 1.28111i −0.767913 0.640554i \(-0.778705\pi\)
0.767913 0.640554i \(-0.221295\pi\)
\(30\) 0 0
\(31\) 6.89898i 1.23909i −0.784960 0.619547i \(-0.787316\pi\)
0.784960 0.619547i \(-0.212684\pi\)
\(32\) 0 0
\(33\) 0.325765 + 0.325765i 0.0567085 + 0.0567085i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 + 6.00000i −0.986394 + 0.986394i −0.999909 0.0135147i \(-0.995698\pi\)
0.0135147 + 0.999909i \(0.495698\pi\)
\(38\) 0 0
\(39\) −0.202041 −0.0323525
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 6.89898 6.89898i 1.05208 1.05208i 0.0535176 0.998567i \(-0.482957\pi\)
0.998567 0.0535176i \(-0.0170433\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.34847 7.34847i −1.07188 1.07188i −0.997208 0.0746766i \(-0.976208\pi\)
−0.0746766 0.997208i \(-0.523792\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 1.44949i 0.202969i
\(52\) 0 0
\(53\) −1.55051 1.55051i −0.212979 0.212979i 0.592553 0.805532i \(-0.298120\pi\)
−0.805532 + 0.592553i \(0.798120\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.67423 + 1.67423i −0.221758 + 0.221758i
\(58\) 0 0
\(59\) −11.7980 −1.53596 −0.767982 0.640472i \(-0.778739\pi\)
−0.767982 + 0.640472i \(0.778739\pi\)
\(60\) 0 0
\(61\) −8.89898 −1.13940 −0.569699 0.821854i \(-0.692940\pi\)
−0.569699 + 0.821854i \(0.692940\pi\)
\(62\) 0 0
\(63\) 5.79796 5.79796i 0.730474 0.730474i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.77526 1.77526i −0.216882 0.216882i 0.590301 0.807183i \(-0.299009\pi\)
−0.807183 + 0.590301i \(0.799009\pi\)
\(68\) 0 0
\(69\) 2.00000i 0.240772i
\(70\) 0 0
\(71\) 2.00000i 0.237356i 0.992933 + 0.118678i \(0.0378657\pi\)
−0.992933 + 0.118678i \(0.962134\pi\)
\(72\) 0 0
\(73\) 1.67423 + 1.67423i 0.195954 + 0.195954i 0.798263 0.602309i \(-0.205752\pi\)
−0.602309 + 0.798263i \(0.705752\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.89898 2.89898i 0.330369 0.330369i
\(78\) 0 0
\(79\) −8.89898 −1.00121 −0.500607 0.865675i \(-0.666890\pi\)
−0.500607 + 0.865675i \(0.666890\pi\)
\(80\) 0 0
\(81\) −8.10102 −0.900113
\(82\) 0 0
\(83\) 2.67423 2.67423i 0.293535 0.293535i −0.544940 0.838475i \(-0.683447\pi\)
0.838475 + 0.544940i \(0.183447\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.55051 1.55051i −0.166232 0.166232i
\(88\) 0 0
\(89\) 2.10102i 0.222708i 0.993781 + 0.111354i \(0.0355187\pi\)
−0.993781 + 0.111354i \(0.964481\pi\)
\(90\) 0 0
\(91\) 1.79796i 0.188477i
\(92\) 0 0
\(93\) −1.55051 1.55051i −0.160780 0.160780i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.89898 4.89898i 0.497416 0.497416i −0.413217 0.910633i \(-0.635595\pi\)
0.910633 + 0.413217i \(0.135595\pi\)
\(98\) 0 0
\(99\) −4.20204 −0.422321
\(100\) 0 0
\(101\) 14.6969 1.46240 0.731200 0.682163i \(-0.238961\pi\)
0.731200 + 0.682163i \(0.238961\pi\)
\(102\) 0 0
\(103\) 2.44949 2.44949i 0.241355 0.241355i −0.576055 0.817411i \(-0.695409\pi\)
0.817411 + 0.576055i \(0.195409\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.5732 11.5732i −1.11882 1.11882i −0.991914 0.126911i \(-0.959494\pi\)
−0.126911 0.991914i \(-0.540506\pi\)
\(108\) 0 0
\(109\) 12.8990i 1.23550i 0.786375 + 0.617749i \(0.211955\pi\)
−0.786375 + 0.617749i \(0.788045\pi\)
\(110\) 0 0
\(111\) 2.69694i 0.255982i
\(112\) 0 0
\(113\) −4.32577 4.32577i −0.406934 0.406934i 0.473734 0.880668i \(-0.342906\pi\)
−0.880668 + 0.473734i \(0.842906\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.30306 1.30306i 0.120468 0.120468i
\(118\) 0 0
\(119\) −12.8990 −1.18245
\(120\) 0 0
\(121\) 8.89898 0.808998
\(122\) 0 0
\(123\) 1.12372 1.12372i 0.101323 0.101323i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.55051 + 3.55051i 0.315057 + 0.315057i 0.846865 0.531808i \(-0.178487\pi\)
−0.531808 + 0.846865i \(0.678487\pi\)
\(128\) 0 0
\(129\) 3.10102i 0.273030i
\(130\) 0 0
\(131\) 3.79796i 0.331829i −0.986140 0.165915i \(-0.946942\pi\)
0.986140 0.165915i \(-0.0530576\pi\)
\(132\) 0 0
\(133\) 14.8990 + 14.8990i 1.29191 + 1.29191i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.1237 + 10.1237i −0.864928 + 0.864928i −0.991906 0.126977i \(-0.959472\pi\)
0.126977 + 0.991906i \(0.459472\pi\)
\(138\) 0 0
\(139\) 15.2474 1.29327 0.646636 0.762799i \(-0.276175\pi\)
0.646636 + 0.762799i \(0.276175\pi\)
\(140\) 0 0
\(141\) −3.30306 −0.278168
\(142\) 0 0
\(143\) 0.651531 0.651531i 0.0544837 0.0544837i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.224745 + 0.224745i 0.0185366 + 0.0185366i
\(148\) 0 0
\(149\) 9.10102i 0.745585i 0.927915 + 0.372792i \(0.121600\pi\)
−0.927915 + 0.372792i \(0.878400\pi\)
\(150\) 0 0
\(151\) 6.69694i 0.544989i −0.962157 0.272495i \(-0.912151\pi\)
0.962157 0.272495i \(-0.0878487\pi\)
\(152\) 0 0
\(153\) 9.34847 + 9.34847i 0.755779 + 0.755779i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.2474 16.2474i 1.29669 1.29669i 0.366120 0.930568i \(-0.380686\pi\)
0.930568 0.366120i \(-0.119314\pi\)
\(158\) 0 0
\(159\) −0.696938 −0.0552708
\(160\) 0 0
\(161\) −17.7980 −1.40268
\(162\) 0 0
\(163\) 10.2247 10.2247i 0.800864 0.800864i −0.182367 0.983231i \(-0.558376\pi\)
0.983231 + 0.182367i \(0.0583758\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.55051 + 5.55051i 0.429511 + 0.429511i 0.888462 0.458950i \(-0.151774\pi\)
−0.458950 + 0.888462i \(0.651774\pi\)
\(168\) 0 0
\(169\) 12.5959i 0.968917i
\(170\) 0 0
\(171\) 21.5959i 1.65148i
\(172\) 0 0
\(173\) −13.7980 13.7980i −1.04904 1.04904i −0.998734 0.0503055i \(-0.983980\pi\)
−0.0503055 0.998734i \(-0.516020\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.65153 + 2.65153i −0.199301 + 0.199301i
\(178\) 0 0
\(179\) 9.24745 0.691187 0.345593 0.938384i \(-0.387678\pi\)
0.345593 + 0.938384i \(0.387678\pi\)
\(180\) 0 0
\(181\) 4.20204 0.312335 0.156168 0.987731i \(-0.450086\pi\)
0.156168 + 0.987731i \(0.450086\pi\)
\(182\) 0 0
\(183\) −2.00000 + 2.00000i −0.147844 + 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.67423 + 4.67423i 0.341814 + 0.341814i
\(188\) 0 0
\(189\) 5.30306i 0.385741i
\(190\) 0 0
\(191\) 22.8990i 1.65691i 0.560054 + 0.828456i \(0.310780\pi\)
−0.560054 + 0.828456i \(0.689220\pi\)
\(192\) 0 0
\(193\) −16.1237 16.1237i −1.16061 1.16061i −0.984342 0.176269i \(-0.943597\pi\)
−0.176269 0.984342i \(-0.556403\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.44949 + 6.44949i −0.459507 + 0.459507i −0.898494 0.438987i \(-0.855338\pi\)
0.438987 + 0.898494i \(0.355338\pi\)
\(198\) 0 0
\(199\) −9.79796 −0.694559 −0.347279 0.937762i \(-0.612894\pi\)
−0.347279 + 0.937762i \(0.612894\pi\)
\(200\) 0 0
\(201\) −0.797959 −0.0562837
\(202\) 0 0
\(203\) −13.7980 + 13.7980i −0.968427 + 0.968427i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.8990 + 12.8990i 0.896541 + 0.896541i
\(208\) 0 0
\(209\) 10.7980i 0.746910i
\(210\) 0 0
\(211\) 11.2474i 0.774306i 0.922015 + 0.387153i \(0.126542\pi\)
−0.922015 + 0.387153i \(0.873458\pi\)
\(212\) 0 0
\(213\) 0.449490 + 0.449490i 0.0307985 + 0.0307985i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −13.7980 + 13.7980i −0.936666 + 0.936666i
\(218\) 0 0
\(219\) 0.752551 0.0508527
\(220\) 0 0
\(221\) −2.89898 −0.195006
\(222\) 0 0
\(223\) −7.10102 + 7.10102i −0.475520 + 0.475520i −0.903695 0.428176i \(-0.859156\pi\)
0.428176 + 0.903695i \(0.359156\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.0000 10.0000i −0.663723 0.663723i 0.292532 0.956256i \(-0.405502\pi\)
−0.956256 + 0.292532i \(0.905502\pi\)
\(228\) 0 0
\(229\) 5.79796i 0.383140i 0.981479 + 0.191570i \(0.0613579\pi\)
−0.981479 + 0.191570i \(0.938642\pi\)
\(230\) 0 0
\(231\) 1.30306i 0.0857352i
\(232\) 0 0
\(233\) 7.10102 + 7.10102i 0.465203 + 0.465203i 0.900356 0.435153i \(-0.143306\pi\)
−0.435153 + 0.900356i \(0.643306\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 + 2.00000i −0.129914 + 0.129914i
\(238\) 0 0
\(239\) 13.5959 0.879447 0.439723 0.898133i \(-0.355076\pi\)
0.439723 + 0.898133i \(0.355076\pi\)
\(240\) 0 0
\(241\) −3.89898 −0.251155 −0.125578 0.992084i \(-0.540078\pi\)
−0.125578 + 0.992084i \(0.540078\pi\)
\(242\) 0 0
\(243\) −5.79796 + 5.79796i −0.371939 + 0.371939i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.34847 + 3.34847i 0.213058 + 0.213058i
\(248\) 0 0
\(249\) 1.20204i 0.0761763i
\(250\) 0 0
\(251\) 12.5505i 0.792181i 0.918211 + 0.396091i \(0.129633\pi\)
−0.918211 + 0.396091i \(0.870367\pi\)
\(252\) 0 0
\(253\) 6.44949 + 6.44949i 0.405476 + 0.405476i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.6969 10.6969i 0.667257 0.667257i −0.289823 0.957080i \(-0.593597\pi\)
0.957080 + 0.289823i \(0.0935966\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 20.0000 1.23797
\(262\) 0 0
\(263\) −18.2474 + 18.2474i −1.12519 + 1.12519i −0.134237 + 0.990949i \(0.542858\pi\)
−0.990949 + 0.134237i \(0.957142\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.472194 + 0.472194i 0.0288978 + 0.0288978i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 17.1010i 1.03881i −0.854527 0.519407i \(-0.826153\pi\)
0.854527 0.519407i \(-0.173847\pi\)
\(272\) 0 0
\(273\) 0.404082 + 0.404082i 0.0244562 + 0.0244562i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −12.4495 + 12.4495i −0.748017 + 0.748017i −0.974107 0.226089i \(-0.927406\pi\)
0.226089 + 0.974107i \(0.427406\pi\)
\(278\) 0 0
\(279\) 20.0000 1.19737
\(280\) 0 0
\(281\) 12.2020 0.727913 0.363956 0.931416i \(-0.381426\pi\)
0.363956 + 0.931416i \(0.381426\pi\)
\(282\) 0 0
\(283\) −1.57321 + 1.57321i −0.0935179 + 0.0935179i −0.752318 0.658800i \(-0.771064\pi\)
0.658800 + 0.752318i \(0.271064\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 10.0000i −0.590281 0.590281i
\(288\) 0 0
\(289\) 3.79796i 0.223409i
\(290\) 0 0
\(291\) 2.20204i 0.129086i
\(292\) 0 0
\(293\) 2.89898 + 2.89898i 0.169360 + 0.169360i 0.786698 0.617338i \(-0.211789\pi\)
−0.617338 + 0.786698i \(0.711789\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.92168 + 1.92168i −0.111507 + 0.111507i
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −27.5959 −1.59060
\(302\) 0 0
\(303\) 3.30306 3.30306i 0.189756 0.189756i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.5732 + 17.5732i 1.00296 + 1.00296i 0.999996 + 0.00296069i \(0.000942417\pi\)
0.00296069 + 0.999996i \(0.499058\pi\)
\(308\) 0 0
\(309\) 1.10102i 0.0626349i
\(310\) 0 0
\(311\) 24.8990i 1.41189i 0.708266 + 0.705946i \(0.249478\pi\)
−0.708266 + 0.705946i \(0.750522\pi\)
\(312\) 0 0
\(313\) −7.10102 7.10102i −0.401373 0.401373i 0.477343 0.878717i \(-0.341600\pi\)
−0.878717 + 0.477343i \(0.841600\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.6969 22.6969i 1.27479 1.27479i 0.331241 0.943546i \(-0.392533\pi\)
0.943546 0.331241i \(-0.107467\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) −5.20204 −0.290350
\(322\) 0 0
\(323\) −24.0227 + 24.0227i −1.33666 + 1.33666i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.89898 + 2.89898i 0.160314 + 0.160314i
\(328\) 0 0
\(329\) 29.3939i 1.62054i
\(330\) 0 0
\(331\) 4.34847i 0.239013i 0.992833 + 0.119507i \(0.0381313\pi\)
−0.992833 + 0.119507i \(0.961869\pi\)
\(332\) 0 0
\(333\) −17.3939 17.3939i −0.953179 0.953179i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.5732 24.5732i 1.33859 1.33859i 0.441160 0.897429i \(-0.354567\pi\)
0.897429 0.441160i \(-0.145433\pi\)
\(338\) 0 0
\(339\) −1.94439 −0.105605
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) −12.0000 + 12.0000i −0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.3258 + 13.3258i 0.715365 + 0.715365i 0.967652 0.252288i \(-0.0811829\pi\)
−0.252288 + 0.967652i \(0.581183\pi\)
\(348\) 0 0
\(349\) 18.6969i 1.00082i −0.865787 0.500412i \(-0.833182\pi\)
0.865787 0.500412i \(-0.166818\pi\)
\(350\) 0 0
\(351\) 1.19184i 0.0636155i
\(352\) 0 0
\(353\) 16.8990 + 16.8990i 0.899442 + 0.899442i 0.995387 0.0959447i \(-0.0305872\pi\)
−0.0959447 + 0.995387i \(0.530587\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.89898 + 2.89898i −0.153430 + 0.153430i
\(358\) 0 0
\(359\) 28.8990 1.52523 0.762615 0.646853i \(-0.223915\pi\)
0.762615 + 0.646853i \(0.223915\pi\)
\(360\) 0 0
\(361\) 36.4949 1.92078
\(362\) 0 0
\(363\) 2.00000 2.00000i 0.104973 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.6969 + 10.6969i 0.558376 + 0.558376i 0.928845 0.370469i \(-0.120803\pi\)
−0.370469 + 0.928845i \(0.620803\pi\)
\(368\) 0 0
\(369\) 14.4949i 0.754574i
\(370\) 0 0
\(371\) 6.20204i 0.321994i
\(372\) 0 0
\(373\) −2.44949 2.44949i −0.126830 0.126830i 0.640843 0.767672i \(-0.278585\pi\)
−0.767672 + 0.640843i \(0.778585\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.10102 + 3.10102i −0.159711 + 0.159711i
\(378\) 0 0
\(379\) 11.0454 0.567364 0.283682 0.958918i \(-0.408444\pi\)
0.283682 + 0.958918i \(0.408444\pi\)
\(380\) 0 0
\(381\) 1.59592 0.0817614
\(382\) 0 0
\(383\) −7.79796 + 7.79796i −0.398457 + 0.398457i −0.877689 0.479231i \(-0.840915\pi\)
0.479231 + 0.877689i \(0.340915\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.0000 + 20.0000i 1.01666 + 1.01666i
\(388\) 0 0
\(389\) 7.79796i 0.395372i −0.980265 0.197686i \(-0.936657\pi\)
0.980265 0.197686i \(-0.0633427\pi\)
\(390\) 0 0
\(391\) 28.6969i 1.45127i
\(392\) 0 0
\(393\) −0.853572 0.853572i −0.0430570 0.0430570i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −13.7980 + 13.7980i −0.692500 + 0.692500i −0.962781 0.270282i \(-0.912883\pi\)
0.270282 + 0.962781i \(0.412883\pi\)
\(398\) 0 0
\(399\) 6.69694 0.335266
\(400\) 0 0
\(401\) −13.8990 −0.694082 −0.347041 0.937850i \(-0.612814\pi\)
−0.347041 + 0.937850i \(0.612814\pi\)
\(402\) 0 0
\(403\) −3.10102 + 3.10102i −0.154473 + 0.154473i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.69694 8.69694i −0.431091 0.431091i
\(408\) 0 0
\(409\) 8.79796i 0.435031i 0.976057 + 0.217516i \(0.0697953\pi\)
−0.976057 + 0.217516i \(0.930205\pi\)
\(410\) 0 0
\(411\) 4.55051i 0.224460i
\(412\) 0 0
\(413\) 23.5959 + 23.5959i 1.16108 + 1.16108i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.42679 3.42679i 0.167810 0.167810i
\(418\) 0 0
\(419\) −12.1464 −0.593392 −0.296696 0.954972i \(-0.595885\pi\)
−0.296696 + 0.954972i \(0.595885\pi\)
\(420\) 0 0
\(421\) −11.5959 −0.565150 −0.282575 0.959245i \(-0.591189\pi\)
−0.282575 + 0.959245i \(0.591189\pi\)
\(422\) 0 0
\(423\) 21.3031 21.3031i 1.03579 1.03579i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 17.7980 + 17.7980i 0.861304 + 0.861304i
\(428\) 0 0
\(429\) 0.292856i 0.0141392i
\(430\) 0 0
\(431\) 3.10102i 0.149371i −0.997207 0.0746855i \(-0.976205\pi\)
0.997207 0.0746855i \(-0.0237953\pi\)
\(432\) 0 0
\(433\) −13.6742 13.6742i −0.657142 0.657142i 0.297561 0.954703i \(-0.403827\pi\)
−0.954703 + 0.297561i \(0.903827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −33.1464 + 33.1464i −1.58561 + 1.58561i
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) −2.89898 −0.138047
\(442\) 0 0
\(443\) −14.9217 + 14.9217i −0.708951 + 0.708951i −0.966315 0.257364i \(-0.917146\pi\)
0.257364 + 0.966315i \(0.417146\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.04541 + 2.04541i 0.0967445 + 0.0967445i
\(448\) 0 0
\(449\) 3.89898i 0.184004i 0.995759 + 0.0920021i \(0.0293267\pi\)
−0.995759 + 0.0920021i \(0.970673\pi\)
\(450\) 0 0
\(451\) 7.24745i 0.341269i
\(452\) 0 0
\(453\) −1.50510 1.50510i −0.0707159 0.0707159i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.977296 + 0.977296i −0.0457160 + 0.0457160i −0.729595 0.683879i \(-0.760292\pi\)
0.683879 + 0.729595i \(0.260292\pi\)
\(458\) 0 0
\(459\) 8.55051 0.399104
\(460\) 0 0
\(461\) −21.7980 −1.01523 −0.507616 0.861583i \(-0.669473\pi\)
−0.507616 + 0.861583i \(0.669473\pi\)
\(462\) 0 0
\(463\) −26.6969 + 26.6969i −1.24071 + 1.24071i −0.281006 + 0.959706i \(0.590668\pi\)
−0.959706 + 0.281006i \(0.909332\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.4949 22.4949i −1.04094 1.04094i −0.999125 0.0418143i \(-0.986686\pi\)
−0.0418143 0.999125i \(-0.513314\pi\)
\(468\) 0 0
\(469\) 7.10102i 0.327895i
\(470\) 0 0
\(471\) 7.30306i 0.336507i
\(472\) 0 0
\(473\) 10.0000 + 10.0000i 0.459800 + 0.459800i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.49490 4.49490i 0.205807 0.205807i
\(478\) 0 0
\(479\) 6.89898 0.315222 0.157611 0.987501i \(-0.449621\pi\)
0.157611 + 0.987501i \(0.449621\pi\)
\(480\) 0 0
\(481\) 5.39388 0.245940
\(482\) 0 0
\(483\) −4.00000 + 4.00000i −0.182006 + 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −7.55051 7.55051i −0.342146 0.342146i 0.515027 0.857174i \(-0.327782\pi\)
−0.857174 + 0.515027i \(0.827782\pi\)
\(488\) 0 0
\(489\) 4.59592i 0.207835i
\(490\) 0 0
\(491\) 37.5959i 1.69668i 0.529452 + 0.848340i \(0.322398\pi\)
−0.529452 + 0.848340i \(0.677602\pi\)
\(492\) 0 0
\(493\) −22.2474 22.2474i −1.00197 1.00197i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.00000 4.00000i 0.179425 0.179425i
\(498\) 0 0
\(499\) −11.7980 −0.528149 −0.264075 0.964502i \(-0.585067\pi\)
−0.264075 + 0.964502i \(0.585067\pi\)
\(500\) 0 0
\(501\) 2.49490 0.111464
\(502\) 0 0
\(503\) 5.79796 5.79796i 0.258518 0.258518i −0.565933 0.824451i \(-0.691484\pi\)
0.824451 + 0.565933i \(0.191484\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.83087 2.83087i −0.125723 0.125723i
\(508\) 0 0
\(509\) 39.7980i 1.76401i −0.471237 0.882007i \(-0.656192\pi\)
0.471237 0.882007i \(-0.343808\pi\)
\(510\) 0 0
\(511\) 6.69694i 0.296255i
\(512\) 0 0
\(513\) −9.87628 9.87628i −0.436048 0.436048i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 10.6515 10.6515i 0.468454 0.468454i
\(518\) 0 0
\(519\) −6.20204 −0.272239
\(520\) 0 0
\(521\) −19.0000 −0.832405 −0.416203 0.909272i \(-0.636639\pi\)
−0.416203 + 0.909272i \(0.636639\pi\)
\(522\) 0 0
\(523\) 5.77526 5.77526i 0.252534 0.252534i −0.569475 0.822009i \(-0.692853\pi\)
0.822009 + 0.569475i \(0.192853\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22.2474 22.2474i −0.969114 0.969114i
\(528\) 0 0
\(529\) 16.5959i 0.721562i
\(530\) 0 0
\(531\) 34.2020i 1.48424i
\(532\) 0 0
\(533\) −2.24745 2.24745i −0.0973478 0.0973478i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.07832 2.07832i 0.0896860 0.0896860i
\(538\) 0 0
\(539\) −1.44949 −0.0624339
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0.944387 0.944387i 0.0405275 0.0405275i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 8.67423 + 8.67423i 0.370883 + 0.370883i 0.867799 0.496915i \(-0.165534\pi\)
−0.496915 + 0.867799i \(0.665534\pi\)
\(548\) 0 0
\(549\) 25.7980i 1.10103i
\(550\) 0 0
\(551\) 51.3939i 2.18945i
\(552\) 0 0
\(553\) 17.7980 + 17.7980i 0.756846 + 0.756846i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.55051 3.55051i 0.150440 0.150440i −0.627875 0.778315i \(-0.716075\pi\)
0.778315 + 0.627875i \(0.216075\pi\)
\(558\) 0 0
\(559\) −6.20204 −0.262318
\(560\) 0 0
\(561\) 2.10102 0.0887052
\(562\) 0 0
\(563\) 1.10102 1.10102i 0.0464025 0.0464025i −0.683525 0.729927i \(-0.739554\pi\)
0.729927 + 0.683525i \(0.239554\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 16.2020 + 16.2020i 0.680422 + 0.680422i
\(568\) 0 0
\(569\) 5.40408i 0.226551i 0.993564 + 0.113275i \(0.0361343\pi\)
−0.993564 + 0.113275i \(0.963866\pi\)
\(570\) 0 0
\(571\) 14.0000i 0.585882i −0.956131 0.292941i \(-0.905366\pi\)
0.956131 0.292941i \(-0.0946339\pi\)
\(572\) 0 0
\(573\) 5.14643 + 5.14643i 0.214995 + 0.214995i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −6.77526 + 6.77526i −0.282058 + 0.282058i −0.833929 0.551872i \(-0.813914\pi\)
0.551872 + 0.833929i \(0.313914\pi\)
\(578\) 0 0
\(579\) −7.24745 −0.301194
\(580\) 0 0
\(581\) −10.6969 −0.443784
\(582\) 0 0
\(583\) 2.24745 2.24745i 0.0930798 0.0930798i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.57321 1.57321i −0.0649335 0.0649335i 0.673894 0.738828i \(-0.264620\pi\)
−0.738828 + 0.673894i \(0.764620\pi\)
\(588\) 0 0
\(589\) 51.3939i 2.11765i
\(590\) 0 0
\(591\) 2.89898i 0.119248i
\(592\) 0 0
\(593\) 2.77526 + 2.77526i 0.113966 + 0.113966i 0.761790 0.647824i \(-0.224321\pi\)
−0.647824 + 0.761790i \(0.724321\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.20204 + 2.20204i −0.0901235 + 0.0901235i
\(598\) 0 0
\(599\) −30.6969 −1.25424 −0.627121 0.778921i \(-0.715767\pi\)
−0.627121 + 0.778921i \(0.715767\pi\)
\(600\) 0 0
\(601\) 15.6969 0.640291 0.320146 0.947368i \(-0.396268\pi\)
0.320146 + 0.947368i \(0.396268\pi\)
\(602\) 0 0
\(603\) 5.14643 5.14643i 0.209579 0.209579i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13.7980 + 13.7980i 0.560042 + 0.560042i 0.929319 0.369277i \(-0.120395\pi\)
−0.369277 + 0.929319i \(0.620395\pi\)
\(608\) 0 0
\(609\) 6.20204i 0.251319i
\(610\) 0 0
\(611\) 6.60612i 0.267255i
\(612\) 0 0
\(613\) 29.1464 + 29.1464i 1.17721 + 1.17721i 0.980451 + 0.196762i \(0.0630426\pi\)
0.196762 + 0.980451i \(0.436957\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −15.5959 + 15.5959i −0.627868 + 0.627868i −0.947531 0.319663i \(-0.896430\pi\)
0.319663 + 0.947531i \(0.396430\pi\)
\(618\) 0 0
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 0 0
\(621\) 11.7980 0.473436
\(622\) 0 0
\(623\) 4.20204 4.20204i 0.168351 0.168351i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.42679 2.42679i −0.0969165 0.0969165i
\(628\) 0 0
\(629\) 38.6969i 1.54295i
\(630\) 0 0
\(631\) 24.6969i 0.983170i −0.870830 0.491585i \(-0.836418\pi\)
0.870830 0.491585i \(-0.163582\pi\)
\(632\) 0 0
\(633\) 2.52781 + 2.52781i 0.100471 + 0.100471i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.449490 0.449490i 0.0178094 0.0178094i
\(638\) 0 0
\(639\) −5.79796 −0.229364
\(640\) 0 0
\(641\) −3.79796 −0.150010 −0.0750052 0.997183i \(-0.523897\pi\)
−0.0750052 + 0.997183i \(0.523897\pi\)
\(642\) 0 0
\(643\) 28.6969 28.6969i 1.13170 1.13170i 0.141802 0.989895i \(-0.454710\pi\)
0.989895 0.141802i \(-0.0452895\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11.5959 11.5959i −0.455883 0.455883i 0.441419 0.897301i \(-0.354475\pi\)
−0.897301 + 0.441419i \(0.854475\pi\)
\(648\) 0 0
\(649\) 17.1010i 0.671274i
\(650\) 0 0
\(651\) 6.20204i 0.243077i
\(652\) 0 0
\(653\) −26.0000 26.0000i −1.01746 1.01746i −0.999845 0.0176138i \(-0.994393\pi\)
−0.0176138 0.999845i \(-0.505607\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.85357 + 4.85357i −0.189356 + 0.189356i
\(658\) 0 0
\(659\) 30.1464 1.17434 0.587169 0.809464i \(-0.300242\pi\)
0.587169 + 0.809464i \(0.300242\pi\)
\(660\) 0 0
\(661\) 28.2929 1.10046 0.550232 0.835012i \(-0.314539\pi\)
0.550232 + 0.835012i \(0.314539\pi\)
\(662\) 0 0
\(663\) −0.651531 + 0.651531i −0.0253034 + 0.0253034i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.6969 30.6969i −1.18859 1.18859i
\(668\) 0 0
\(669\) 3.19184i 0.123404i
\(670\) 0 0
\(671\) 12.8990i 0.497960i
\(672\) 0 0
\(673\) −5.79796 5.79796i −0.223495 0.223495i 0.586474 0.809968i \(-0.300516\pi\)
−0.809968 + 0.586474i \(0.800516\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.0000 28.0000i 1.07613 1.07613i 0.0792746 0.996853i \(-0.474740\pi\)
0.996853 0.0792746i \(-0.0252604\pi\)
\(678\) 0 0
\(679\) −19.5959 −0.752022
\(680\) 0 0
\(681\) −4.49490 −0.172245
\(682\) 0 0
\(683\) 9.77526 9.77526i 0.374040 0.374040i −0.494906 0.868946i \(-0.664798\pi\)
0.868946 + 0.494906i \(0.164798\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.30306 + 1.30306i 0.0497149 + 0.0497149i
\(688\) 0 0
\(689\) 1.39388i 0.0531025i
\(690\) 0 0
\(691\) 13.9444i 0.530469i −0.964184 0.265235i \(-0.914551\pi\)
0.964184 0.265235i \(-0.0854495\pi\)
\(692\) 0 0
\(693\) 8.40408 + 8.40408i 0.319245 + 0.319245i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.1237 16.1237i 0.610730 0.610730i
\(698\) 0 0
\(699\) 3.19184 0.120726
\(700\) 0 0
\(701\) 22.8990 0.864883 0.432441 0.901662i \(-0.357652\pi\)
0.432441 + 0.901662i \(0.357652\pi\)
\(702\) 0 0
\(703\) 44.6969 44.6969i 1.68578 1.68578i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.3939 29.3939i −1.10547 1.10547i
\(708\) 0 0
\(709\) 12.0000i 0.450669i −0.974281 0.225335i \(-0.927652\pi\)
0.974281 0.225335i \(-0.0723476\pi\)
\(710\) 0 0
\(711\) 25.7980i 0.967499i
\(712\) 0 0
\(713\) −30.6969 30.6969i −1.14961 1.14961i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.05561 3.05561i 0.114114 0.114114i
\(718\) 0 0
\(719\) −40.8990 −1.52527 −0.762637 0.646826i \(-0.776096\pi\)
−0.762637 + 0.646826i \(0.776096\pi\)
\(720\) 0 0
\(721\) −9.79796 −0.364895
\(722\) 0 0
\(723\) −0.876276 + 0.876276i −0.0325890 + 0.0325890i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.1464 + 17.1464i 0.635926 + 0.635926i 0.949548 0.313622i \(-0.101542\pi\)
−0.313622 + 0.949548i \(0.601542\pi\)
\(728\) 0 0
\(729\) 21.6969i 0.803590i
\(730\) 0 0
\(731\) 44.4949i 1.64570i
\(732\) 0 0
\(733\) 20.4495 + 20.4495i 0.755319 + 0.755319i 0.975467 0.220147i \(-0.0706539\pi\)
−0.220147 + 0.975467i \(0.570654\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.57321 2.57321i 0.0947856 0.0947856i
\(738\) 0 0
\(739\) 5.59592 0.205849 0.102925 0.994689i \(-0.467180\pi\)
0.102925 + 0.994689i \(0.467180\pi\)
\(740\) 0 0
\(741\) 1.50510 0.0552913
\(742\) 0 0
\(743\) −9.34847 + 9.34847i −0.342962 + 0.342962i −0.857480 0.514518i \(-0.827971\pi\)
0.514518 + 0.857480i \(0.327971\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7.75255 + 7.75255i 0.283651 + 0.283651i
\(748\) 0 0
\(749\) 46.2929i 1.69150i
\(750\) 0 0
\(751\) 37.1918i 1.35715i 0.734531 + 0.678575i \(0.237402\pi\)
−0.734531 + 0.678575i \(0.762598\pi\)
\(752\) 0 0
\(753\) 2.82066 + 2.82066i 0.102791 + 0.102791i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.6969 14.6969i 0.534169 0.534169i −0.387641 0.921810i \(-0.626710\pi\)
0.921810 + 0.387641i \(0.126710\pi\)
\(758\) 0 0
\(759\) 2.89898 0.105226
\(760\) 0 0
\(761\) −17.0000 −0.616250 −0.308125 0.951346i \(-0.599701\pi\)
−0.308125 + 0.951346i \(0.599701\pi\)
\(762\) 0 0
\(763\) 25.7980 25.7980i 0.933949 0.933949i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.30306 + 5.30306i 0.191482 + 0.191482i
\(768\) 0 0
\(769\) 3.69694i 0.133315i 0.997776 + 0.0666575i \(0.0212335\pi\)
−0.997776 + 0.0666575i \(0.978767\pi\)
\(770\) 0 0
\(771\) 4.80816i 0.173162i
\(772\) 0 0
\(773\) −24.9444 24.9444i −0.897187 0.897187i 0.0979992 0.995186i \(-0.468756\pi\)
−0.995186 + 0.0979992i \(0.968756\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.39388 5.39388i 0.193504 0.193504i
\(778\) 0 0
\(779\) −37.2474 −1.33453
\(780\) 0 0
\(781\) −2.89898 −0.103734
\(782\) 0 0
\(783\) 9.14643 9.14643i 0.326867 0.326867i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.89898 + 2.89898i 0.103337 + 0.103337i 0.756885 0.653548i \(-0.226720\pi\)
−0.653548 + 0.756885i \(0.726720\pi\)
\(788\) 0 0
\(789\) 8.20204i 0.292000i
\(790\) 0 0
\(791\) 17.3031i 0.615226i
\(792\) 0 0
\(793\) 4.00000 + 4.00000i 0.142044 + 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.34847 3.34847i 0.118609 0.118609i −0.645311 0.763920i \(-0.723272\pi\)
0.763920 + 0.645311i \(0.223272\pi\)
\(798\) 0 0
\(799\) −47.3939 −1.67667
\(800\) 0 0
\(801\) −6.09082 −0.215208
\(802\) 0 0
\(803\) −2.42679 + 2.42679i −0.0856394 + 0.0856394i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.7980i 0.696059i −0.937484 0.348030i \(-0.886851\pi\)
0.937484 0.348030i \(-0.113149\pi\)
\(810\) 0 0
\(811\) 26.0000i 0.912983i −0.889728 0.456492i \(-0.849106\pi\)
0.889728 0.456492i \(-0.150894\pi\)
\(812\) 0 0
\(813\) −3.84337 3.84337i −0.134793 0.134793i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −51.3939 + 51.3939i −1.79804 + 1.79804i
\(818\) 0 0
\(819\) −5.21225 −0.182131
\(820\) 0 0
\(821\) 3.79796 0.132550 0.0662748 0.997801i \(-0.478889\pi\)
0.0662748 + 0.997801i \(0.478889\pi\)
\(822\) 0 0
\(823\) 18.6969 18.6969i 0.651734 0.651734i −0.301676 0.953410i \(-0.597546\pi\)
0.953410 + 0.301676i \(0.0975461\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −34.9217 34.9217i −1.21435 1.21435i −0.969583 0.244763i \(-0.921290\pi\)
−0.244763 0.969583i \(-0.578710\pi\)
\(828\) 0 0
\(829\) 46.8990i 1.62887i −0.580256 0.814434i \(-0.697047\pi\)
0.580256 0.814434i \(-0.302953\pi\)
\(830\) 0 0
\(831\) 5.59592i 0.194120i
\(832\) 0 0
\(833\) 3.22474 + 3.22474i 0.111731 + 0.111731i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.14643 9.14643i 0.316147 0.316147i
\(838\) 0 0
\(839\) −39.1918 −1.35305 −0.676526 0.736419i \(-0.736515\pi\)
−0.676526 + 0.736419i \(0.736515\pi\)
\(840\) 0 0
\(841\) −18.5959 −0.641239
\(842\) 0 0
\(843\) 2.74235 2.74235i 0.0944514 0.0944514i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −17.7980 17.7980i −0.611545 0.611545i
\(848\) 0 0
\(849\) 0.707144i 0.0242691i
\(850\) 0 0
\(851\) 53.3939i 1.83032i
\(852\) 0 0
\(853\) −19.3485 19.3485i −0.662479 0.662479i 0.293485 0.955964i \(-0.405185\pi\)
−0.955964 + 0.293485i \(0.905185\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.5732 + 24.5732i −0.839405 + 0.839405i −0.988781 0.149376i \(-0.952274\pi\)
0.149376 + 0.988781i \(0.452274\pi\)
\(858\) 0 0
\(859\) 27.6515 0.943458 0.471729 0.881744i \(-0.343630\pi\)
0.471729 + 0.881744i \(0.343630\pi\)
\(860\) 0 0
\(861\) −4.49490 −0.153186
\(862\) 0 0
\(863\) −10.4495 + 10.4495i −0.355705 + 0.355705i −0.862227 0.506522i \(-0.830931\pi\)
0.506522 + 0.862227i \(0.330931\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.853572 0.853572i −0.0289888 0.0289888i
\(868\) 0 0
\(869\) 12.8990i 0.437568i
\(870\) 0 0
\(871\) 1.59592i 0.0540756i
\(872\) 0 0
\(873\) 14.2020 + 14.2020i 0.480666 + 0.480666i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18.6969 + 18.6969i −0.631351 + 0.631351i −0.948407 0.317056i \(-0.897306\pi\)
0.317056 + 0.948407i \(0.397306\pi\)
\(878\) 0 0
\(879\) 1.30306 0.0439512
\(880\) 0 0
\(881\) 35.7980 1.20606 0.603032 0.797717i \(-0.293959\pi\)
0.603032 + 0.797717i \(0.293959\pi\)
\(882\) 0 0
\(883\) 19.8207 19.8207i 0.667018 0.667018i −0.290006 0.957025i \(-0.593657\pi\)
0.957025 + 0.290006i \(0.0936574\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40.0000 40.0000i −1.34307 1.34307i −0.892988 0.450081i \(-0.851395\pi\)
−0.450081 0.892988i \(-0.648605\pi\)
\(888\) 0 0
\(889\) 14.2020i 0.476321i
\(890\) 0 0
\(891\) 11.7423i 0.393383i
\(892\) 0 0
\(893\) 54.7423 + 54.7423i 1.83188 + 1.83188i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.898979 + 0.898979i −0.0300161 + 0.0300161i
\(898\) 0 0
\(899\) −47.5959 −1.58741
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) −6.20204 + 6.20204i −0.206391 + 0.206391i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.10102 1.10102i −0.0365588 0.0365588i 0.688591 0.725150i \(-0.258230\pi\)
−0.725150 + 0.688591i \(0.758230\pi\)
\(908\) 0 0
\(909\) 42.6061i 1.41316i
\(910\) 0 0
\(911\) 10.2020i 0.338009i 0.985615 + 0.169004i \(0.0540552\pi\)
−0.985615 + 0.169004i \(0.945945\pi\)
\(912\) 0 0
\(913\) 3.87628 + 3.87628i 0.128286 + 0.128286i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.59592 + 7.59592i −0.250839 + 0.250839i
\(918\) 0 0
\(919\) −27.3031 −0.900645 −0.450322 0.892866i \(-0.648691\pi\)
−0.450322 + 0.892866i \(0.648691\pi\)
\(920\) 0 0
\(921\) 7.89898 0.260280
\(922\) 0 0
\(923\) 0.898979 0.898979i 0.0295903 0.0295903i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.10102 + 7.10102i 0.233228 + 0.233228i
\(928\) 0 0
\(929\) 45.5959i 1.49595i −0.663725 0.747977i \(-0.731025\pi\)
0.663725 0.747977i \(-0.268975\pi\)
\(930\) 0 0
\(931\) 7.44949i 0.244147i
\(932\) 0 0
\(933\) 5.59592 + 5.59592i 0.183202 + 0.183202i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.6742 21.6742i 0.708066 0.708066i −0.258062 0.966128i \(-0.583084\pi\)
0.966128 + 0.258062i \(0.0830839\pi\)
\(938\) 0 0
\(939\) −3.19184 −0.104162
\(940\) 0 0
\(941\) 18.4949 0.602916 0.301458 0.953479i \(-0.402527\pi\)
0.301458 + 0.953479i \(0.402527\pi\)
\(942\) 0 0
\(943\) 22.2474 22.2474i 0.724477 0.724477i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.1010 17.1010i −0.555708 0.555708i 0.372374 0.928083i \(-0.378544\pi\)
−0.928083 + 0.372374i \(0.878544\pi\)
\(948\) 0 0
\(949\) 1.50510i 0.0488577i
\(950\) 0 0
\(951\) 10.2020i 0.330824i
\(952\) 0 0
\(953\) −30.1237 30.1237i −0.975803 0.975803i 0.0239109 0.999714i \(-0.492388\pi\)
−0.999714 + 0.0239109i \(0.992388\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.24745 2.24745i 0.0726497 0.0726497i
\(958\) 0 0
\(959\) 40.4949 1.30765
\(960\) 0 0
\(961\) −16.5959 −0.535352
\(962\) 0 0
\(963\) 33.5505 33.5505i 1.08115 1.08115i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.10102 + 7.10102i 0.228353 + 0.228353i 0.812005 0.583651i \(-0.198376\pi\)
−0.583651 + 0.812005i \(0.698376\pi\)
\(968\) 0 0
\(969\) 10.7980i 0.346880i
\(970\) 0 0
\(971\) 2.14643i 0.0688822i 0.999407 + 0.0344411i \(0.0109651\pi\)
−0.999407 + 0.0344411i \(0.989035\pi\)
\(972\) 0 0
\(973\) −30.4949 30.4949i −0.977622 0.977622i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.3258 26.3258i 0.842236 0.842236i −0.146913 0.989149i \(-0.546934\pi\)
0.989149 + 0.146913i \(0.0469339\pi\)
\(978\) 0 0
\(979\) −3.04541 −0.0973317
\(980\) 0 0
\(981\) −37.3939 −1.19389
\(982\) 0 0
\(983\) 6.89898 6.89898i 0.220043 0.220043i −0.588473 0.808517i \(-0.700271\pi\)
0.808517 + 0.588473i \(0.200271\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.60612 + 6.60612i 0.210275 + 0.210275i
\(988\) 0 0
\(989\) 61.3939i 1.95221i
\(990\) 0 0
\(991\) 23.1010i 0.733828i −0.930255 0.366914i \(-0.880414\pi\)
0.930255 0.366914i \(-0.119586\pi\)
\(992\) 0 0
\(993\) 0.977296 + 0.977296i 0.0310136 + 0.0310136i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.79796 1.79796i 0.0569419 0.0569419i −0.678062 0.735004i \(-0.737180\pi\)
0.735004 + 0.678062i \(0.237180\pi\)
\(998\) 0 0
\(999\) −15.9092 −0.503344
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.n.p.1343.2 4
4.3 odd 2 1600.2.n.t.1343.1 4
5.2 odd 4 1600.2.n.t.1407.1 4
5.3 odd 4 1600.2.n.o.1407.2 4
5.4 even 2 1600.2.n.s.1343.1 4
8.3 odd 2 800.2.n.k.543.2 4
8.5 even 2 800.2.n.m.543.1 yes 4
20.3 even 4 1600.2.n.s.1407.1 4
20.7 even 4 inner 1600.2.n.p.1407.2 4
20.19 odd 2 1600.2.n.o.1343.2 4
40.3 even 4 800.2.n.l.607.2 yes 4
40.13 odd 4 800.2.n.n.607.1 yes 4
40.19 odd 2 800.2.n.n.543.1 yes 4
40.27 even 4 800.2.n.m.607.1 yes 4
40.29 even 2 800.2.n.l.543.2 yes 4
40.37 odd 4 800.2.n.k.607.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.n.k.543.2 4 8.3 odd 2
800.2.n.k.607.2 yes 4 40.37 odd 4
800.2.n.l.543.2 yes 4 40.29 even 2
800.2.n.l.607.2 yes 4 40.3 even 4
800.2.n.m.543.1 yes 4 8.5 even 2
800.2.n.m.607.1 yes 4 40.27 even 4
800.2.n.n.543.1 yes 4 40.19 odd 2
800.2.n.n.607.1 yes 4 40.13 odd 4
1600.2.n.o.1343.2 4 20.19 odd 2
1600.2.n.o.1407.2 4 5.3 odd 4
1600.2.n.p.1343.2 4 1.1 even 1 trivial
1600.2.n.p.1407.2 4 20.7 even 4 inner
1600.2.n.s.1343.1 4 5.4 even 2
1600.2.n.s.1407.1 4 20.3 even 4
1600.2.n.t.1343.1 4 4.3 odd 2
1600.2.n.t.1407.1 4 5.2 odd 4