Properties

Label 1600.2.o.e.543.4
Level $1600$
Weight $2$
Character 1600.543
Analytic conductor $12.776$
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] = [1600,2,Mod(543,1600)]
 
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, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.543");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 543.4
Root \(-1.09445 - 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 1600.543
Dual form 1600.2.o.e.607.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.18890 + 2.18890i) q^{3} +(1.79129 + 1.79129i) q^{7} +6.58258i q^{9} +O(q^{10})\) \(q+(2.18890 + 2.18890i) q^{3} +(1.79129 + 1.79129i) q^{7} +6.58258i q^{9} +0.913701 q^{11} +(1.73205 - 1.73205i) q^{13} +(-3.00000 + 3.00000i) q^{17} +3.46410i q^{19} +7.84190i q^{21} +(3.79129 - 3.79129i) q^{23} +(-7.84190 + 7.84190i) q^{27} -5.29150 q^{29} -7.58258i q^{31} +(2.00000 + 2.00000i) q^{33} +(5.19615 + 5.19615i) q^{37} +7.58258 q^{39} +1.58258 q^{41} +(0.361500 + 0.361500i) q^{43} +(-3.79129 - 3.79129i) q^{47} -0.582576i q^{49} -13.1334 q^{51} +(-2.64575 + 2.64575i) q^{53} +(-7.58258 + 7.58258i) q^{57} +5.29150i q^{59} -6.20520i q^{61} +(-11.7913 + 11.7913i) q^{63} +(-0.361500 + 0.361500i) q^{67} +16.5975 q^{69} -4.41742i q^{71} +(8.58258 + 8.58258i) q^{73} +(1.63670 + 1.63670i) q^{77} -12.0000 q^{79} -14.5826 q^{81} +(3.10260 + 3.10260i) q^{83} +(-11.5826 - 11.5826i) q^{87} -15.1652i q^{89} +6.20520 q^{91} +(16.5975 - 16.5975i) q^{93} +(8.58258 - 8.58258i) q^{97} +6.01450i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 24 q^{17} + 12 q^{23} + 16 q^{33} + 24 q^{39} - 24 q^{41} - 12 q^{47} - 24 q^{57} - 76 q^{63} + 32 q^{73} - 96 q^{79} - 80 q^{81} - 56 q^{87} + 32 q^{97}+O(q^{100}) \) 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
\(3\) 2.18890 + 2.18890i 1.26376 + 1.26376i 0.949255 + 0.314508i \(0.101839\pi\)
0.314508 + 0.949255i \(0.398161\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.79129 + 1.79129i 0.677043 + 0.677043i 0.959330 0.282287i \(-0.0910930\pi\)
−0.282287 + 0.959330i \(0.591093\pi\)
\(8\) 0 0
\(9\) 6.58258i 2.19419i
\(10\) 0 0
\(11\) 0.913701 0.275491 0.137746 0.990468i \(-0.456014\pi\)
0.137746 + 0.990468i \(0.456014\pi\)
\(12\) 0 0
\(13\) 1.73205 1.73205i 0.480384 0.480384i −0.424870 0.905254i \(-0.639680\pi\)
0.905254 + 0.424870i \(0.139680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 + 3.00000i −0.727607 + 0.727607i −0.970143 0.242536i \(-0.922021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 7.84190i 1.71124i
\(22\) 0 0
\(23\) 3.79129 3.79129i 0.790538 0.790538i −0.191043 0.981582i \(-0.561187\pi\)
0.981582 + 0.191043i \(0.0611871\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −7.84190 + 7.84190i −1.50918 + 1.50918i
\(28\) 0 0
\(29\) −5.29150 −0.982607 −0.491304 0.870988i \(-0.663479\pi\)
−0.491304 + 0.870988i \(0.663479\pi\)
\(30\) 0 0
\(31\) 7.58258i 1.36187i −0.732343 0.680935i \(-0.761573\pi\)
0.732343 0.680935i \(-0.238427\pi\)
\(32\) 0 0
\(33\) 2.00000 + 2.00000i 0.348155 + 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.19615 + 5.19615i 0.854242 + 0.854242i 0.990652 0.136410i \(-0.0435565\pi\)
−0.136410 + 0.990652i \(0.543557\pi\)
\(38\) 0 0
\(39\) 7.58258 1.21418
\(40\) 0 0
\(41\) 1.58258 0.247157 0.123578 0.992335i \(-0.460563\pi\)
0.123578 + 0.992335i \(0.460563\pi\)
\(42\) 0 0
\(43\) 0.361500 + 0.361500i 0.0551282 + 0.0551282i 0.734133 0.679005i \(-0.237589\pi\)
−0.679005 + 0.734133i \(0.737589\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.79129 3.79129i −0.553016 0.553016i 0.374294 0.927310i \(-0.377885\pi\)
−0.927310 + 0.374294i \(0.877885\pi\)
\(48\) 0 0
\(49\) 0.582576i 0.0832251i
\(50\) 0 0
\(51\) −13.1334 −1.83904
\(52\) 0 0
\(53\) −2.64575 + 2.64575i −0.363422 + 0.363422i −0.865071 0.501649i \(-0.832727\pi\)
0.501649 + 0.865071i \(0.332727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.58258 + 7.58258i −1.00434 + 1.00434i
\(58\) 0 0
\(59\) 5.29150i 0.688895i 0.938806 + 0.344447i \(0.111934\pi\)
−0.938806 + 0.344447i \(0.888066\pi\)
\(60\) 0 0
\(61\) 6.20520i 0.794495i −0.917712 0.397247i \(-0.869965\pi\)
0.917712 0.397247i \(-0.130035\pi\)
\(62\) 0 0
\(63\) −11.7913 + 11.7913i −1.48556 + 1.48556i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.361500 + 0.361500i −0.0441643 + 0.0441643i −0.728844 0.684680i \(-0.759942\pi\)
0.684680 + 0.728844i \(0.259942\pi\)
\(68\) 0 0
\(69\) 16.5975 1.99811
\(70\) 0 0
\(71\) 4.41742i 0.524252i −0.965034 0.262126i \(-0.915576\pi\)
0.965034 0.262126i \(-0.0844236\pi\)
\(72\) 0 0
\(73\) 8.58258 + 8.58258i 1.00451 + 1.00451i 0.999990 + 0.00452474i \(0.00144028\pi\)
0.00452474 + 0.999990i \(0.498560\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.63670 + 1.63670i 0.186519 + 0.186519i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) −14.5826 −1.62029
\(82\) 0 0
\(83\) 3.10260 + 3.10260i 0.340555 + 0.340555i 0.856576 0.516021i \(-0.172587\pi\)
−0.516021 + 0.856576i \(0.672587\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.5826 11.5826i −1.24178 1.24178i
\(88\) 0 0
\(89\) 15.1652i 1.60750i −0.594965 0.803751i \(-0.702834\pi\)
0.594965 0.803751i \(-0.297166\pi\)
\(90\) 0 0
\(91\) 6.20520 0.650482
\(92\) 0 0
\(93\) 16.5975 16.5975i 1.72108 1.72108i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.58258 8.58258i 0.871429 0.871429i −0.121200 0.992628i \(-0.538674\pi\)
0.992628 + 0.121200i \(0.0386741\pi\)
\(98\) 0 0
\(99\) 6.01450i 0.604480i
\(100\) 0 0
\(101\) 17.5112i 1.74243i 0.490901 + 0.871215i \(0.336668\pi\)
−0.490901 + 0.871215i \(0.663332\pi\)
\(102\) 0 0
\(103\) −3.37386 + 3.37386i −0.332437 + 0.332437i −0.853511 0.521075i \(-0.825531\pi\)
0.521075 + 0.853511i \(0.325531\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.361500 + 0.361500i −0.0349475 + 0.0349475i −0.724365 0.689417i \(-0.757867\pi\)
0.689417 + 0.724365i \(0.257867\pi\)
\(108\) 0 0
\(109\) 9.66930 0.926151 0.463076 0.886319i \(-0.346746\pi\)
0.463076 + 0.886319i \(0.346746\pi\)
\(110\) 0 0
\(111\) 22.7477i 2.15912i
\(112\) 0 0
\(113\) −4.58258 4.58258i −0.431092 0.431092i 0.457907 0.889000i \(-0.348599\pi\)
−0.889000 + 0.457907i \(0.848599\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11.4014 + 11.4014i 1.05406 + 1.05406i
\(118\) 0 0
\(119\) −10.7477 −0.985243
\(120\) 0 0
\(121\) −10.1652 −0.924105
\(122\) 0 0
\(123\) 3.46410 + 3.46410i 0.312348 + 0.312348i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.79129 + 5.79129i 0.513894 + 0.513894i 0.915717 0.401823i \(-0.131623\pi\)
−0.401823 + 0.915717i \(0.631623\pi\)
\(128\) 0 0
\(129\) 1.58258i 0.139338i
\(130\) 0 0
\(131\) −6.01450 −0.525490 −0.262745 0.964865i \(-0.584628\pi\)
−0.262745 + 0.964865i \(0.584628\pi\)
\(132\) 0 0
\(133\) −6.20520 + 6.20520i −0.538059 + 0.538059i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.16515 6.16515i 0.526724 0.526724i −0.392870 0.919594i \(-0.628518\pi\)
0.919594 + 0.392870i \(0.128518\pi\)
\(138\) 0 0
\(139\) 22.8027i 1.93410i −0.254585 0.967050i \(-0.581939\pi\)
0.254585 0.967050i \(-0.418061\pi\)
\(140\) 0 0
\(141\) 16.5975i 1.39776i
\(142\) 0 0
\(143\) 1.58258 1.58258i 0.132342 0.132342i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.27520 1.27520i 0.105177 0.105177i
\(148\) 0 0
\(149\) −0.913701 −0.0748533 −0.0374266 0.999299i \(-0.511916\pi\)
−0.0374266 + 0.999299i \(0.511916\pi\)
\(150\) 0 0
\(151\) 14.7477i 1.20015i −0.799943 0.600077i \(-0.795137\pi\)
0.799943 0.600077i \(-0.204863\pi\)
\(152\) 0 0
\(153\) −19.7477 19.7477i −1.59651 1.59651i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.47315 + 4.47315i 0.356996 + 0.356996i 0.862705 0.505708i \(-0.168769\pi\)
−0.505708 + 0.862705i \(0.668769\pi\)
\(158\) 0 0
\(159\) −11.5826 −0.918558
\(160\) 0 0
\(161\) 13.5826 1.07046
\(162\) 0 0
\(163\) 0.361500 + 0.361500i 0.0283149 + 0.0283149i 0.721122 0.692808i \(-0.243626\pi\)
−0.692808 + 0.721122i \(0.743626\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.7913 15.7913i −1.22197 1.22197i −0.966932 0.255035i \(-0.917913\pi\)
−0.255035 0.966932i \(-0.582087\pi\)
\(168\) 0 0
\(169\) 7.00000i 0.538462i
\(170\) 0 0
\(171\) −22.8027 −1.74377
\(172\) 0 0
\(173\) −8.66025 + 8.66025i −0.658427 + 0.658427i −0.955008 0.296581i \(-0.904154\pi\)
0.296581 + 0.955008i \(0.404154\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −11.5826 + 11.5826i −0.870600 + 0.870600i
\(178\) 0 0
\(179\) 19.1479i 1.43118i −0.698520 0.715591i \(-0.746157\pi\)
0.698520 0.715591i \(-0.253843\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 13.5826 13.5826i 1.00405 1.00405i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.74110 + 2.74110i −0.200449 + 0.200449i
\(188\) 0 0
\(189\) −28.0942 −2.04355
\(190\) 0 0
\(191\) 4.41742i 0.319634i 0.987147 + 0.159817i \(0.0510903\pi\)
−0.987147 + 0.159817i \(0.948910\pi\)
\(192\) 0 0
\(193\) 4.16515 + 4.16515i 0.299814 + 0.299814i 0.840941 0.541127i \(-0.182002\pi\)
−0.541127 + 0.840941i \(0.682002\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.64575 2.64575i −0.188502 0.188502i 0.606546 0.795048i \(-0.292554\pi\)
−0.795048 + 0.606546i \(0.792554\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −1.58258 −0.111626
\(202\) 0 0
\(203\) −9.47860 9.47860i −0.665268 0.665268i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 24.9564 + 24.9564i 1.73459 + 1.73459i
\(208\) 0 0
\(209\) 3.16515i 0.218938i
\(210\) 0 0
\(211\) 23.5257 1.61958 0.809788 0.586722i \(-0.199582\pi\)
0.809788 + 0.586722i \(0.199582\pi\)
\(212\) 0 0
\(213\) 9.66930 9.66930i 0.662530 0.662530i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 13.5826 13.5826i 0.922045 0.922045i
\(218\) 0 0
\(219\) 37.5728i 2.53894i
\(220\) 0 0
\(221\) 10.3923i 0.699062i
\(222\) 0 0
\(223\) 1.37386 1.37386i 0.0920007 0.0920007i −0.659609 0.751609i \(-0.729278\pi\)
0.751609 + 0.659609i \(0.229278\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.8582 + 11.8582i −0.787057 + 0.787057i −0.981011 0.193954i \(-0.937869\pi\)
0.193954 + 0.981011i \(0.437869\pi\)
\(228\) 0 0
\(229\) −3.46410 −0.228914 −0.114457 0.993428i \(-0.536513\pi\)
−0.114457 + 0.993428i \(0.536513\pi\)
\(230\) 0 0
\(231\) 7.16515i 0.471432i
\(232\) 0 0
\(233\) 12.1652 + 12.1652i 0.796966 + 0.796966i 0.982616 0.185650i \(-0.0594392\pi\)
−0.185650 + 0.982616i \(0.559439\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −26.2668 26.2668i −1.70621 1.70621i
\(238\) 0 0
\(239\) 15.1652 0.980952 0.490476 0.871455i \(-0.336823\pi\)
0.490476 + 0.871455i \(0.336823\pi\)
\(240\) 0 0
\(241\) 16.7477 1.07882 0.539408 0.842045i \(-0.318648\pi\)
0.539408 + 0.842045i \(0.318648\pi\)
\(242\) 0 0
\(243\) −8.39410 8.39410i −0.538482 0.538482i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 + 6.00000i 0.381771 + 0.381771i
\(248\) 0 0
\(249\) 13.5826i 0.860761i
\(250\) 0 0
\(251\) 12.9427 0.816936 0.408468 0.912773i \(-0.366063\pi\)
0.408468 + 0.912773i \(0.366063\pi\)
\(252\) 0 0
\(253\) 3.46410 3.46410i 0.217786 0.217786i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.74773 + 7.74773i −0.483290 + 0.483290i −0.906181 0.422891i \(-0.861015\pi\)
0.422891 + 0.906181i \(0.361015\pi\)
\(258\) 0 0
\(259\) 18.6156i 1.15672i
\(260\) 0 0
\(261\) 34.8317i 2.15603i
\(262\) 0 0
\(263\) 14.2087 14.2087i 0.876147 0.876147i −0.116987 0.993133i \(-0.537324\pi\)
0.993133 + 0.116987i \(0.0373235\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 33.1950 33.1950i 2.03150 2.03150i
\(268\) 0 0
\(269\) 13.3241 0.812385 0.406193 0.913788i \(-0.366856\pi\)
0.406193 + 0.913788i \(0.366856\pi\)
\(270\) 0 0
\(271\) 18.7477i 1.13884i −0.822046 0.569422i \(-0.807167\pi\)
0.822046 0.569422i \(-0.192833\pi\)
\(272\) 0 0
\(273\) 13.5826 + 13.5826i 0.822055 + 0.822055i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −21.7937 21.7937i −1.30945 1.30945i −0.921808 0.387646i \(-0.873288\pi\)
−0.387646 0.921808i \(-0.626712\pi\)
\(278\) 0 0
\(279\) 49.9129 2.98821
\(280\) 0 0
\(281\) 16.7477 0.999086 0.499543 0.866289i \(-0.333501\pi\)
0.499543 + 0.866289i \(0.333501\pi\)
\(282\) 0 0
\(283\) 22.4412 + 22.4412i 1.33399 + 1.33399i 0.901765 + 0.432226i \(0.142272\pi\)
0.432226 + 0.901765i \(0.357728\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.83485 + 2.83485i 0.167336 + 0.167336i
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 0 0
\(291\) 37.5728 2.20256
\(292\) 0 0
\(293\) −18.1389 + 18.1389i −1.05968 + 1.05968i −0.0615814 + 0.998102i \(0.519614\pi\)
−0.998102 + 0.0615814i \(0.980386\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.16515 + 7.16515i −0.415764 + 0.415764i
\(298\) 0 0
\(299\) 13.1334i 0.759525i
\(300\) 0 0
\(301\) 1.29510i 0.0746484i
\(302\) 0 0
\(303\) −38.3303 + 38.3303i −2.20202 + 2.20202i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.56670 6.56670i 0.374782 0.374782i −0.494434 0.869215i \(-0.664625\pi\)
0.869215 + 0.494434i \(0.164625\pi\)
\(308\) 0 0
\(309\) −14.7701 −0.840242
\(310\) 0 0
\(311\) 19.5826i 1.11043i −0.831708 0.555213i \(-0.812637\pi\)
0.831708 0.555213i \(-0.187363\pi\)
\(312\) 0 0
\(313\) 6.58258 + 6.58258i 0.372069 + 0.372069i 0.868230 0.496161i \(-0.165258\pi\)
−0.496161 + 0.868230i \(0.665258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.1425 + 14.1425i 0.794320 + 0.794320i 0.982193 0.187874i \(-0.0601596\pi\)
−0.187874 + 0.982193i \(0.560160\pi\)
\(318\) 0 0
\(319\) −4.83485 −0.270700
\(320\) 0 0
\(321\) −1.58258 −0.0883308
\(322\) 0 0
\(323\) −10.3923 10.3923i −0.578243 0.578243i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 21.1652 + 21.1652i 1.17044 + 1.17044i
\(328\) 0 0
\(329\) 13.5826i 0.748832i
\(330\) 0 0
\(331\) −9.66930 −0.531473 −0.265737 0.964046i \(-0.585615\pi\)
−0.265737 + 0.964046i \(0.585615\pi\)
\(332\) 0 0
\(333\) −34.2041 + 34.2041i −1.87437 + 1.87437i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 + 1.00000i −0.0544735 + 0.0544735i −0.733819 0.679345i \(-0.762264\pi\)
0.679345 + 0.733819i \(0.262264\pi\)
\(338\) 0 0
\(339\) 20.0616i 1.08960i
\(340\) 0 0
\(341\) 6.92820i 0.375183i
\(342\) 0 0
\(343\) 13.5826 13.5826i 0.733390 0.733390i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.27520 + 1.27520i −0.0684564 + 0.0684564i −0.740506 0.672050i \(-0.765414\pi\)
0.672050 + 0.740506i \(0.265414\pi\)
\(348\) 0 0
\(349\) 10.3923 0.556287 0.278144 0.960539i \(-0.410281\pi\)
0.278144 + 0.960539i \(0.410281\pi\)
\(350\) 0 0
\(351\) 27.1652i 1.44997i
\(352\) 0 0
\(353\) −24.1652 24.1652i −1.28618 1.28618i −0.937088 0.349093i \(-0.886490\pi\)
−0.349093 0.937088i \(-0.613510\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −23.5257 23.5257i −1.24511 1.24511i
\(358\) 0 0
\(359\) 27.1652 1.43372 0.716861 0.697216i \(-0.245578\pi\)
0.716861 + 0.697216i \(0.245578\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −22.2505 22.2505i −1.16785 1.16785i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −16.5390 16.5390i −0.863330 0.863330i 0.128394 0.991723i \(-0.459018\pi\)
−0.991723 + 0.128394i \(0.959018\pi\)
\(368\) 0 0
\(369\) 10.4174i 0.542309i
\(370\) 0 0
\(371\) −9.47860 −0.492105
\(372\) 0 0
\(373\) 5.19615 5.19615i 0.269047 0.269047i −0.559669 0.828716i \(-0.689072\pi\)
0.828716 + 0.559669i \(0.189072\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.16515 + 9.16515i −0.472029 + 0.472029i
\(378\) 0 0
\(379\) 10.3923i 0.533817i 0.963722 + 0.266908i \(0.0860021\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(380\) 0 0
\(381\) 25.3531i 1.29888i
\(382\) 0 0
\(383\) 17.3739 17.3739i 0.887763 0.887763i −0.106545 0.994308i \(-0.533979\pi\)
0.994308 + 0.106545i \(0.0339788\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.37960 + 2.37960i −0.120962 + 0.120962i
\(388\) 0 0
\(389\) −35.5547 −1.80270 −0.901348 0.433096i \(-0.857421\pi\)
−0.901348 + 0.433096i \(0.857421\pi\)
\(390\) 0 0
\(391\) 22.7477i 1.15040i
\(392\) 0 0
\(393\) −13.1652 13.1652i −0.664094 0.664094i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.93725 + 7.93725i 0.398359 + 0.398359i 0.877654 0.479295i \(-0.159107\pi\)
−0.479295 + 0.877654i \(0.659107\pi\)
\(398\) 0 0
\(399\) −27.1652 −1.35996
\(400\) 0 0
\(401\) −24.3303 −1.21500 −0.607499 0.794321i \(-0.707827\pi\)
−0.607499 + 0.794321i \(0.707827\pi\)
\(402\) 0 0
\(403\) −13.1334 13.1334i −0.654222 0.654222i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.74773 + 4.74773i 0.235336 + 0.235336i
\(408\) 0 0
\(409\) 16.4174i 0.811789i −0.913920 0.405895i \(-0.866960\pi\)
0.913920 0.405895i \(-0.133040\pi\)
\(410\) 0 0
\(411\) 26.9898 1.33131
\(412\) 0 0
\(413\) −9.47860 + 9.47860i −0.466412 + 0.466412i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 49.9129 49.9129i 2.44424 2.44424i
\(418\) 0 0
\(419\) 20.9753i 1.02471i −0.858773 0.512355i \(-0.828773\pi\)
0.858773 0.512355i \(-0.171227\pi\)
\(420\) 0 0
\(421\) 13.1334i 0.640083i −0.947404 0.320042i \(-0.896303\pi\)
0.947404 0.320042i \(-0.103697\pi\)
\(422\) 0 0
\(423\) 24.9564 24.9564i 1.21342 1.21342i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.1153 11.1153i 0.537907 0.537907i
\(428\) 0 0
\(429\) 6.92820 0.334497
\(430\) 0 0
\(431\) 10.7477i 0.517700i −0.965918 0.258850i \(-0.916656\pi\)
0.965918 0.258850i \(-0.0833435\pi\)
\(432\) 0 0
\(433\) 5.41742 + 5.41742i 0.260345 + 0.260345i 0.825194 0.564849i \(-0.191066\pi\)
−0.564849 + 0.825194i \(0.691066\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.1334 + 13.1334i 0.628256 + 0.628256i
\(438\) 0 0
\(439\) −25.4955 −1.21683 −0.608416 0.793618i \(-0.708195\pi\)
−0.608416 + 0.793618i \(0.708195\pi\)
\(440\) 0 0
\(441\) 3.83485 0.182612
\(442\) 0 0
\(443\) −14.4086 14.4086i −0.684574 0.684574i 0.276454 0.961027i \(-0.410841\pi\)
−0.961027 + 0.276454i \(0.910841\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.00000 2.00000i −0.0945968 0.0945968i
\(448\) 0 0
\(449\) 19.5826i 0.924159i −0.886839 0.462079i \(-0.847103\pi\)
0.886839 0.462079i \(-0.152897\pi\)
\(450\) 0 0
\(451\) 1.44600 0.0680895
\(452\) 0 0
\(453\) 32.2813 32.2813i 1.51671 1.51671i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.5826 + 12.5826i −0.588588 + 0.588588i −0.937249 0.348661i \(-0.886636\pi\)
0.348661 + 0.937249i \(0.386636\pi\)
\(458\) 0 0
\(459\) 47.0514i 2.19617i
\(460\) 0 0
\(461\) 3.65480i 0.170221i −0.996372 0.0851106i \(-0.972876\pi\)
0.996372 0.0851106i \(-0.0271243\pi\)
\(462\) 0 0
\(463\) −12.2087 + 12.2087i −0.567387 + 0.567387i −0.931396 0.364009i \(-0.881408\pi\)
0.364009 + 0.931396i \(0.381408\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.2360 16.2360i 0.751313 0.751313i −0.223411 0.974724i \(-0.571719\pi\)
0.974724 + 0.223411i \(0.0717193\pi\)
\(468\) 0 0
\(469\) −1.29510 −0.0598022
\(470\) 0 0
\(471\) 19.5826i 0.902317i
\(472\) 0 0
\(473\) 0.330303 + 0.330303i 0.0151873 + 0.0151873i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −17.4159 17.4159i −0.797417 0.797417i
\(478\) 0 0
\(479\) −33.4955 −1.53045 −0.765223 0.643765i \(-0.777371\pi\)
−0.765223 + 0.643765i \(0.777371\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 29.7309 + 29.7309i 1.35280 + 1.35280i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −13.3739 13.3739i −0.606028 0.606028i 0.335878 0.941906i \(-0.390967\pi\)
−0.941906 + 0.335878i \(0.890967\pi\)
\(488\) 0 0
\(489\) 1.58258i 0.0715665i
\(490\) 0 0
\(491\) −27.1805 −1.22664 −0.613320 0.789835i \(-0.710166\pi\)
−0.613320 + 0.789835i \(0.710166\pi\)
\(492\) 0 0
\(493\) 15.8745 15.8745i 0.714952 0.714952i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.91288 7.91288i 0.354941 0.354941i
\(498\) 0 0
\(499\) 24.2487i 1.08552i 0.839887 + 0.542761i \(0.182621\pi\)
−0.839887 + 0.542761i \(0.817379\pi\)
\(500\) 0 0
\(501\) 69.1311i 3.08855i
\(502\) 0 0
\(503\) −14.5390 + 14.5390i −0.648263 + 0.648263i −0.952573 0.304310i \(-0.901574\pi\)
0.304310 + 0.952573i \(0.401574\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −15.3223 + 15.3223i −0.680488 + 0.680488i
\(508\) 0 0
\(509\) 3.84550 0.170449 0.0852244 0.996362i \(-0.472839\pi\)
0.0852244 + 0.996362i \(0.472839\pi\)
\(510\) 0 0
\(511\) 30.7477i 1.36020i
\(512\) 0 0
\(513\) −27.1652 27.1652i −1.19937 1.19937i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −3.46410 3.46410i −0.152351 0.152351i
\(518\) 0 0
\(519\) −37.9129 −1.66419
\(520\) 0 0
\(521\) −14.8348 −0.649927 −0.324963 0.945727i \(-0.605352\pi\)
−0.324963 + 0.945727i \(0.605352\pi\)
\(522\) 0 0
\(523\) 23.8872 + 23.8872i 1.04451 + 1.04451i 0.998962 + 0.0455529i \(0.0145050\pi\)
0.0455529 + 0.998962i \(0.485495\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.7477 + 22.7477i 0.990907 + 0.990907i
\(528\) 0 0
\(529\) 5.74773i 0.249901i
\(530\) 0 0
\(531\) −34.8317 −1.51157
\(532\) 0 0
\(533\) 2.74110 2.74110i 0.118730 0.118730i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 41.9129 41.9129i 1.80867 1.80867i
\(538\) 0 0
\(539\) 0.532300i 0.0229278i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −15.1652 + 15.1652i −0.650799 + 0.650799i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 23.1642 23.1642i 0.990430 0.990430i −0.00952449 0.999955i \(-0.503032\pi\)
0.999955 + 0.00952449i \(0.00303179\pi\)
\(548\) 0 0
\(549\) 40.8462 1.74327
\(550\) 0 0
\(551\) 18.3303i 0.780897i
\(552\) 0 0
\(553\) −21.4955 21.4955i −0.914080 0.914080i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.8800 + 20.8800i 0.884712 + 0.884712i 0.994009 0.109297i \(-0.0348599\pi\)
−0.109297 + 0.994009i \(0.534860\pi\)
\(558\) 0 0
\(559\) 1.25227 0.0529655
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) −11.6675 11.6675i −0.491727 0.491727i 0.417123 0.908850i \(-0.363038\pi\)
−0.908850 + 0.417123i \(0.863038\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −26.1216 26.1216i −1.09700 1.09700i
\(568\) 0 0
\(569\) 22.7477i 0.953634i 0.879002 + 0.476817i \(0.158210\pi\)
−0.879002 + 0.476817i \(0.841790\pi\)
\(570\) 0 0
\(571\) 8.22330 0.344135 0.172067 0.985085i \(-0.444955\pi\)
0.172067 + 0.985085i \(0.444955\pi\)
\(572\) 0 0
\(573\) −9.66930 + 9.66930i −0.403941 + 0.403941i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15.7477 + 15.7477i −0.655586 + 0.655586i −0.954333 0.298746i \(-0.903432\pi\)
0.298746 + 0.954333i \(0.403432\pi\)
\(578\) 0 0
\(579\) 18.2342i 0.757788i
\(580\) 0 0
\(581\) 11.1153i 0.461141i
\(582\) 0 0
\(583\) −2.41742 + 2.41742i −0.100119 + 0.100119i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.54860 + 4.54860i −0.187741 + 0.187741i −0.794719 0.606978i \(-0.792382\pi\)
0.606978 + 0.794719i \(0.292382\pi\)
\(588\) 0 0
\(589\) 26.2668 1.08231
\(590\) 0 0
\(591\) 11.5826i 0.476444i
\(592\) 0 0
\(593\) 18.1652 + 18.1652i 0.745953 + 0.745953i 0.973717 0.227763i \(-0.0731412\pi\)
−0.227763 + 0.973717i \(0.573141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.75560 + 8.75560i 0.358343 + 0.358343i
\(598\) 0 0
\(599\) −5.66970 −0.231658 −0.115829 0.993269i \(-0.536952\pi\)
−0.115829 + 0.993269i \(0.536952\pi\)
\(600\) 0 0
\(601\) 24.7477 1.00948 0.504740 0.863271i \(-0.331588\pi\)
0.504740 + 0.863271i \(0.331588\pi\)
\(602\) 0 0
\(603\) −2.37960 2.37960i −0.0969049 0.0969049i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.37386 + 7.37386i 0.299296 + 0.299296i 0.840738 0.541442i \(-0.182121\pi\)
−0.541442 + 0.840738i \(0.682121\pi\)
\(608\) 0 0
\(609\) 41.4955i 1.68148i
\(610\) 0 0
\(611\) −13.1334 −0.531321
\(612\) 0 0
\(613\) 8.66025 8.66025i 0.349784 0.349784i −0.510245 0.860029i \(-0.670445\pi\)
0.860029 + 0.510245i \(0.170445\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.00000 + 3.00000i −0.120775 + 0.120775i −0.764911 0.644136i \(-0.777217\pi\)
0.644136 + 0.764911i \(0.277217\pi\)
\(618\) 0 0
\(619\) 2.01810i 0.0811143i −0.999177 0.0405572i \(-0.987087\pi\)
0.999177 0.0405572i \(-0.0129133\pi\)
\(620\) 0 0
\(621\) 59.4618i 2.38612i
\(622\) 0 0
\(623\) 27.1652 27.1652i 1.08835 1.08835i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.92820 + 6.92820i −0.276686 + 0.276686i
\(628\) 0 0
\(629\) −31.1769 −1.24310
\(630\) 0 0
\(631\) 17.0780i 0.679866i 0.940450 + 0.339933i \(0.110404\pi\)
−0.940450 + 0.339933i \(0.889596\pi\)
\(632\) 0 0
\(633\) 51.4955 + 51.4955i 2.04676 + 2.04676i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00905 1.00905i −0.0399800 0.0399800i
\(638\) 0 0
\(639\) 29.0780 1.15031
\(640\) 0 0
\(641\) −19.9129 −0.786511 −0.393256 0.919429i \(-0.628651\pi\)
−0.393256 + 0.919429i \(0.628651\pi\)
\(642\) 0 0
\(643\) −10.7538 10.7538i −0.424089 0.424089i 0.462520 0.886609i \(-0.346945\pi\)
−0.886609 + 0.462520i \(0.846945\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.5390 32.5390i −1.27924 1.27924i −0.941093 0.338148i \(-0.890200\pi\)
−0.338148 0.941093i \(-0.609800\pi\)
\(648\) 0 0
\(649\) 4.83485i 0.189784i
\(650\) 0 0
\(651\) 59.4618 2.33049
\(652\) 0 0
\(653\) −4.47315 + 4.47315i −0.175048 + 0.175048i −0.789193 0.614145i \(-0.789501\pi\)
0.614145 + 0.789193i \(0.289501\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −56.4955 + 56.4955i −2.20410 + 2.20410i
\(658\) 0 0
\(659\) 38.4865i 1.49922i −0.661879 0.749611i \(-0.730241\pi\)
0.661879 0.749611i \(-0.269759\pi\)
\(660\) 0 0
\(661\) 37.9542i 1.47625i 0.674665 + 0.738124i \(0.264288\pi\)
−0.674665 + 0.738124i \(0.735712\pi\)
\(662\) 0 0
\(663\) −22.7477 + 22.7477i −0.883449 + 0.883449i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.0616 + 20.0616i −0.776789 + 0.776789i
\(668\) 0 0
\(669\) 6.01450 0.232534
\(670\) 0 0
\(671\) 5.66970i 0.218876i
\(672\) 0 0
\(673\) 9.41742 + 9.41742i 0.363015 + 0.363015i 0.864922 0.501907i \(-0.167368\pi\)
−0.501907 + 0.864922i \(0.667368\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.45505 + 2.45505i 0.0943553 + 0.0943553i 0.752709 0.658354i \(-0.228747\pi\)
−0.658354 + 0.752709i \(0.728747\pi\)
\(678\) 0 0
\(679\) 30.7477 1.17999
\(680\) 0 0
\(681\) −51.9129 −1.98931
\(682\) 0 0
\(683\) −32.8335 32.8335i −1.25634 1.25634i −0.952828 0.303512i \(-0.901841\pi\)
−0.303512 0.952828i \(-0.598159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −7.58258 7.58258i −0.289293 0.289293i
\(688\) 0 0
\(689\) 9.16515i 0.349164i
\(690\) 0 0
\(691\) −15.1515 −0.576391 −0.288195 0.957572i \(-0.593055\pi\)
−0.288195 + 0.957572i \(0.593055\pi\)
\(692\) 0 0
\(693\) −10.7737 + 10.7737i −0.409259 + 0.409259i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.74773 + 4.74773i −0.179833 + 0.179833i
\(698\) 0 0
\(699\) 53.2566i 2.01435i
\(700\) 0 0
\(701\) 25.1624i 0.950371i −0.879886 0.475186i \(-0.842381\pi\)
0.879886 0.475186i \(-0.157619\pi\)
\(702\) 0 0
\(703\) −18.0000 + 18.0000i −0.678883 + 0.678883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.3676 + 31.3676i −1.17970 + 1.17970i
\(708\) 0 0
\(709\) −36.6591 −1.37676 −0.688381 0.725349i \(-0.741678\pi\)
−0.688381 + 0.725349i \(0.741678\pi\)
\(710\) 0 0
\(711\) 78.9909i 2.96239i
\(712\) 0 0
\(713\) −28.7477 28.7477i −1.07661 1.07661i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33.1950 + 33.1950i 1.23969 + 1.23969i
\(718\) 0 0
\(719\) 51.8258 1.93277 0.966387 0.257091i \(-0.0827639\pi\)
0.966387 + 0.257091i \(0.0827639\pi\)
\(720\) 0 0
\(721\) −12.0871 −0.450148
\(722\) 0 0
\(723\) 36.6591 + 36.6591i 1.36337 + 1.36337i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 16.9564 + 16.9564i 0.628880 + 0.628880i 0.947786 0.318907i \(-0.103316\pi\)
−0.318907 + 0.947786i \(0.603316\pi\)
\(728\) 0 0
\(729\) 7.00000i 0.259259i
\(730\) 0 0
\(731\) −2.16900 −0.0802234
\(732\) 0 0
\(733\) −14.1425 + 14.1425i −0.522364 + 0.522364i −0.918285 0.395921i \(-0.870425\pi\)
0.395921 + 0.918285i \(0.370425\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.330303 + 0.330303i −0.0121669 + 0.0121669i
\(738\) 0 0
\(739\) 29.7309i 1.09367i 0.837241 + 0.546835i \(0.184167\pi\)
−0.837241 + 0.546835i \(0.815833\pi\)
\(740\) 0 0
\(741\) 26.2668i 0.964935i
\(742\) 0 0
\(743\) −21.7913 + 21.7913i −0.799445 + 0.799445i −0.983008 0.183563i \(-0.941237\pi\)
0.183563 + 0.983008i \(0.441237\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −20.4231 + 20.4231i −0.747243 + 0.747243i
\(748\) 0 0
\(749\) −1.29510 −0.0473220
\(750\) 0 0
\(751\) 41.0780i 1.49896i 0.662028 + 0.749479i \(0.269696\pi\)
−0.662028 + 0.749479i \(0.730304\pi\)
\(752\) 0 0
\(753\) 28.3303 + 28.3303i 1.03241 + 1.03241i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −27.2759 27.2759i −0.991358 0.991358i 0.00860486 0.999963i \(-0.497261\pi\)
−0.999963 + 0.00860486i \(0.997261\pi\)
\(758\) 0 0
\(759\) 15.1652 0.550460
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 17.3205 + 17.3205i 0.627044 + 0.627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.16515 + 9.16515i 0.330934 + 0.330934i
\(768\) 0 0
\(769\) 17.4955i 0.630902i 0.948942 + 0.315451i \(0.102156\pi\)
−0.948942 + 0.315451i \(0.897844\pi\)
\(770\) 0 0
\(771\) −33.9180 −1.22153
\(772\) 0 0
\(773\) −31.4630 + 31.4630i −1.13164 + 1.13164i −0.141740 + 0.989904i \(0.545270\pi\)
−0.989904 + 0.141740i \(0.954730\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −40.7477 + 40.7477i −1.46182 + 1.46182i
\(778\) 0 0
\(779\) 5.48220i 0.196420i
\(780\) 0 0
\(781\) 4.03620i 0.144427i
\(782\) 0 0
\(783\) 41.4955 41.4955i 1.48293 1.48293i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.9771 18.9771i 0.676461 0.676461i −0.282737 0.959198i \(-0.591242\pi\)
0.959198 + 0.282737i \(0.0912423\pi\)
\(788\) 0 0
\(789\) 62.2029 2.21448
\(790\) 0 0
\(791\) 16.4174i 0.583736i
\(792\) 0 0
\(793\) −10.7477 10.7477i −0.381663 0.381663i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.17805 + 3.17805i 0.112572 + 0.112572i 0.761149 0.648577i \(-0.224635\pi\)
−0.648577 + 0.761149i \(0.724635\pi\)
\(798\) 0 0
\(799\) 22.7477 0.804757
\(800\) 0 0
\(801\) 99.8258 3.52717
\(802\) 0 0
\(803\) 7.84190 + 7.84190i 0.276735 + 0.276735i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 29.1652 + 29.1652i 1.02666 + 1.02666i
\(808\) 0 0
\(809\) 6.33030i 0.222562i −0.993789 0.111281i \(-0.964505\pi\)
0.993789 0.111281i \(-0.0354953\pi\)
\(810\) 0 0
\(811\) −49.7925 −1.74845 −0.874226 0.485519i \(-0.838631\pi\)
−0.874226 + 0.485519i \(0.838631\pi\)
\(812\) 0 0
\(813\) 41.0369 41.0369i 1.43923 1.43923i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.25227 + 1.25227i −0.0438115 + 0.0438115i
\(818\) 0 0
\(819\) 40.8462i 1.42728i
\(820\) 0 0
\(821\) 17.8528i 0.623067i 0.950235 + 0.311534i \(0.100843\pi\)
−0.950235 + 0.311534i \(0.899157\pi\)
\(822\) 0 0
\(823\) 7.79129 7.79129i 0.271587 0.271587i −0.558152 0.829739i \(-0.688489\pi\)
0.829739 + 0.558152i \(0.188489\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.6428 + 32.6428i −1.13510 + 1.13510i −0.145786 + 0.989316i \(0.546571\pi\)
−0.989316 + 0.145786i \(0.953429\pi\)
\(828\) 0 0
\(829\) 23.5257 0.817082 0.408541 0.912740i \(-0.366038\pi\)
0.408541 + 0.912740i \(0.366038\pi\)
\(830\) 0 0
\(831\) 95.4083i 3.30968i
\(832\) 0 0
\(833\) 1.74773 + 1.74773i 0.0605552 + 0.0605552i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 59.4618 + 59.4618i 2.05530 + 2.05530i
\(838\) 0 0
\(839\) −6.33030 −0.218546 −0.109273 0.994012i \(-0.534852\pi\)
−0.109273 + 0.994012i \(0.534852\pi\)
\(840\) 0 0
\(841\) −1.00000 −0.0344828
\(842\) 0 0
\(843\) 36.6591 + 36.6591i 1.26261 + 1.26261i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −18.2087 18.2087i −0.625659 0.625659i
\(848\) 0 0
\(849\) 98.2432i 3.37170i
\(850\) 0 0
\(851\) 39.4002 1.35062
\(852\) 0 0
\(853\) 26.5529 26.5529i 0.909153 0.909153i −0.0870511 0.996204i \(-0.527744\pi\)
0.996204 + 0.0870511i \(0.0277443\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.66970 + 2.66970i −0.0911951 + 0.0911951i −0.751233 0.660038i \(-0.770540\pi\)
0.660038 + 0.751233i \(0.270540\pi\)
\(858\) 0 0
\(859\) 29.7309i 1.01441i −0.861827 0.507203i \(-0.830679\pi\)
0.861827 0.507203i \(-0.169321\pi\)
\(860\) 0 0
\(861\) 12.4104i 0.422946i
\(862\) 0 0
\(863\) −17.7042 + 17.7042i −0.602657 + 0.602657i −0.941017 0.338360i \(-0.890128\pi\)
0.338360 + 0.941017i \(0.390128\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.18890 2.18890i 0.0743390 0.0743390i
\(868\) 0 0
\(869\) −10.9644 −0.371942
\(870\) 0 0
\(871\) 1.25227i 0.0424316i
\(872\) 0 0
\(873\) 56.4955 + 56.4955i 1.91208 + 1.91208i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −35.6501 35.6501i −1.20382 1.20382i −0.972996 0.230821i \(-0.925859\pi\)
−0.230821 0.972996i \(-0.574141\pi\)
\(878\) 0 0
\(879\) −79.4083 −2.67838
\(880\) 0 0
\(881\) −46.4174 −1.56384 −0.781921 0.623377i \(-0.785760\pi\)
−0.781921 + 0.623377i \(0.785760\pi\)
\(882\) 0 0
\(883\) 14.2179 + 14.2179i 0.478471 + 0.478471i 0.904642 0.426172i \(-0.140138\pi\)
−0.426172 + 0.904642i \(0.640138\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.956439 + 0.956439i 0.0321141 + 0.0321141i 0.722981 0.690867i \(-0.242771\pi\)
−0.690867 + 0.722981i \(0.742771\pi\)
\(888\) 0 0
\(889\) 20.7477i 0.695856i
\(890\) 0 0
\(891\) −13.3241 −0.446374
\(892\) 0 0
\(893\) 13.1334 13.1334i 0.439493 0.439493i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 28.7477 28.7477i 0.959859 0.959859i
\(898\) 0 0
\(899\) 40.1232i 1.33818i
\(900\) 0 0
\(901\) 15.8745i 0.528857i
\(902\) 0 0
\(903\) −2.83485 + 2.83485i −0.0943379 + 0.0943379i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −12.7719 + 12.7719i −0.424084 + 0.424084i −0.886607 0.462523i \(-0.846944\pi\)
0.462523 + 0.886607i \(0.346944\pi\)
\(908\) 0 0
\(909\) −115.269 −3.82323
\(910\) 0 0
\(911\) 40.4174i 1.33909i 0.742772 + 0.669545i \(0.233511\pi\)
−0.742772 + 0.669545i \(0.766489\pi\)
\(912\) 0 0
\(913\) 2.83485 + 2.83485i 0.0938198 + 0.0938198i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.7737 10.7737i −0.355779 0.355779i
\(918\) 0 0
\(919\) −5.66970 −0.187026 −0.0935130 0.995618i \(-0.529810\pi\)
−0.0935130 + 0.995618i \(0.529810\pi\)
\(920\) 0 0
\(921\) 28.7477 0.947270
\(922\) 0 0
\(923\) −7.65120 7.65120i −0.251842 0.251842i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −22.2087 22.2087i −0.729430 0.729430i
\(928\) 0 0
\(929\) 25.9129i 0.850174i 0.905153 + 0.425087i \(0.139756\pi\)
−0.905153 + 0.425087i \(0.860244\pi\)
\(930\) 0 0
\(931\) 2.01810 0.0661406
\(932\) 0 0
\(933\) 42.8643 42.8643i 1.40331 1.40331i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −15.8348 + 15.8348i −0.517302 + 0.517302i −0.916754 0.399452i \(-0.869200\pi\)
0.399452 + 0.916754i \(0.369200\pi\)
\(938\) 0 0
\(939\) 28.8172i 0.940414i
\(940\) 0 0
\(941\) 3.27340i 0.106710i 0.998576 + 0.0533549i \(0.0169915\pi\)
−0.998576 + 0.0533549i \(0.983009\pi\)
\(942\) 0 0
\(943\) 6.00000 6.00000i 0.195387 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.2831 + 30.2831i −0.984069 + 0.984069i −0.999875 0.0158061i \(-0.994969\pi\)
0.0158061 + 0.999875i \(0.494969\pi\)
\(948\) 0 0
\(949\) 29.7309 0.965106
\(950\) 0 0
\(951\) 61.9129i 2.00766i
\(952\) 0 0
\(953\) −34.9129 34.9129i −1.13094 1.13094i −0.990021 0.140918i \(-0.954995\pi\)
−0.140918 0.990021i \(-0.545005\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −10.5830 10.5830i −0.342100 0.342100i
\(958\) 0 0
\(959\) 22.0871 0.713230
\(960\) 0 0
\(961\) −26.4955 −0.854692
\(962\) 0 0
\(963\) −2.37960 2.37960i −0.0766816 0.0766816i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.20871 + 4.20871i 0.135343 + 0.135343i 0.771533 0.636190i \(-0.219490\pi\)
−0.636190 + 0.771533i \(0.719490\pi\)
\(968\) 0 0
\(969\) 45.4955i 1.46152i
\(970\) 0 0
\(971\) −0.532300 −0.0170823 −0.00854116 0.999964i \(-0.502719\pi\)
−0.00854116 + 0.999964i \(0.502719\pi\)
\(972\) 0 0
\(973\) 40.8462 40.8462i 1.30947 1.30947i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.4174 + 19.4174i −0.621218 + 0.621218i −0.945843 0.324625i \(-0.894762\pi\)
0.324625 + 0.945843i \(0.394762\pi\)
\(978\) 0 0
\(979\) 13.8564i 0.442853i
\(980\) 0 0
\(981\) 63.6489i 2.03215i
\(982\) 0 0
\(983\) 23.7042 23.7042i 0.756045 0.756045i −0.219555 0.975600i \(-0.570460\pi\)
0.975600 + 0.219555i \(0.0704604\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 29.7309 29.7309i 0.946345 0.946345i
\(988\) 0 0
\(989\) 2.74110 0.0871620
\(990\) 0 0
\(991\) 30.7477i 0.976734i −0.872638 0.488367i \(-0.837593\pi\)
0.872638 0.488367i \(-0.162407\pi\)
\(992\) 0 0
\(993\) −21.1652 21.1652i −0.671656 0.671656i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.1425 + 14.1425i 0.447896 + 0.447896i 0.894655 0.446758i \(-0.147422\pi\)
−0.446758 + 0.894655i \(0.647422\pi\)
\(998\) 0 0
\(999\) −81.4955 −2.57840
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.o.e.543.4 8
4.3 odd 2 1600.2.o.l.543.1 8
5.2 odd 4 1600.2.o.l.607.4 8
5.3 odd 4 320.2.o.e.287.1 yes 8
5.4 even 2 320.2.o.f.223.1 yes 8
8.3 odd 2 1600.2.o.l.543.4 8
8.5 even 2 inner 1600.2.o.e.543.1 8
20.3 even 4 320.2.o.f.287.4 yes 8
20.7 even 4 inner 1600.2.o.e.607.1 8
20.19 odd 2 320.2.o.e.223.4 yes 8
40.3 even 4 320.2.o.f.287.1 yes 8
40.13 odd 4 320.2.o.e.287.4 yes 8
40.19 odd 2 320.2.o.e.223.1 8
40.27 even 4 inner 1600.2.o.e.607.4 8
40.29 even 2 320.2.o.f.223.4 yes 8
40.37 odd 4 1600.2.o.l.607.1 8
80.3 even 4 1280.2.n.n.767.4 8
80.13 odd 4 1280.2.n.p.767.1 8
80.19 odd 4 1280.2.n.p.1023.1 8
80.29 even 4 1280.2.n.n.1023.4 8
80.43 even 4 1280.2.n.n.767.1 8
80.53 odd 4 1280.2.n.p.767.4 8
80.59 odd 4 1280.2.n.p.1023.4 8
80.69 even 4 1280.2.n.n.1023.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.o.e.223.1 8 40.19 odd 2
320.2.o.e.223.4 yes 8 20.19 odd 2
320.2.o.e.287.1 yes 8 5.3 odd 4
320.2.o.e.287.4 yes 8 40.13 odd 4
320.2.o.f.223.1 yes 8 5.4 even 2
320.2.o.f.223.4 yes 8 40.29 even 2
320.2.o.f.287.1 yes 8 40.3 even 4
320.2.o.f.287.4 yes 8 20.3 even 4
1280.2.n.n.767.1 8 80.43 even 4
1280.2.n.n.767.4 8 80.3 even 4
1280.2.n.n.1023.1 8 80.69 even 4
1280.2.n.n.1023.4 8 80.29 even 4
1280.2.n.p.767.1 8 80.13 odd 4
1280.2.n.p.767.4 8 80.53 odd 4
1280.2.n.p.1023.1 8 80.19 odd 4
1280.2.n.p.1023.4 8 80.59 odd 4
1600.2.o.e.543.1 8 8.5 even 2 inner
1600.2.o.e.543.4 8 1.1 even 1 trivial
1600.2.o.e.607.1 8 20.7 even 4 inner
1600.2.o.e.607.4 8 40.27 even 4 inner
1600.2.o.l.543.1 8 4.3 odd 2
1600.2.o.l.543.4 8 8.3 odd 2
1600.2.o.l.607.1 8 40.37 odd 4
1600.2.o.l.607.4 8 5.2 odd 4