Properties

Label 1600.2.o.k.543.1
Level $1600$
Weight $2$
Character 1600.543
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 543.1
Root \(0.323042 - 0.323042i\) of defining polynomial
Character \(\chi\) \(=\) 1600.543
Dual form 1600.2.o.k.607.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.87083 - 1.87083i) q^{3} +(-2.44949 - 2.44949i) q^{7} +4.00000i q^{9} +O(q^{10})\) \(q+(-1.87083 - 1.87083i) q^{3} +(-2.44949 - 2.44949i) q^{7} +4.00000i q^{9} -4.58258 q^{11} +(-3.74166 + 3.74166i) q^{13} +(1.22474 - 1.22474i) q^{17} +4.58258i q^{19} +9.16515i q^{21} +(4.89898 - 4.89898i) q^{23} +(1.87083 - 1.87083i) q^{27} +9.16515 q^{29} +4.00000i q^{31} +(8.57321 + 8.57321i) q^{33} +(-3.74166 - 3.74166i) q^{37} +14.0000 q^{39} -9.00000 q^{41} +(-3.74166 - 3.74166i) q^{43} +(2.44949 + 2.44949i) q^{47} +5.00000i q^{49} -4.58258 q^{51} +(8.57321 - 8.57321i) q^{57} -9.16515i q^{61} +(9.79796 - 9.79796i) q^{63} +(-1.87083 + 1.87083i) q^{67} -18.3303 q^{69} -6.00000i q^{71} +(6.12372 + 6.12372i) q^{73} +(11.2250 + 11.2250i) q^{77} +10.0000 q^{79} +5.00000 q^{81} +(5.61249 + 5.61249i) q^{83} +(-17.1464 - 17.1464i) q^{87} +15.0000i q^{89} +18.3303 q^{91} +(7.48331 - 7.48331i) q^{93} +(4.89898 - 4.89898i) q^{97} -18.3303i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 112 q^{39} - 72 q^{41} + 80 q^{79} + 40 q^{81}+O(q^{100}) \) 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
\(3\) −1.87083 1.87083i −1.08012 1.08012i −0.996497 0.0836263i \(-0.973350\pi\)
−0.0836263 0.996497i \(-0.526650\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 2.44949i −0.925820 0.925820i 0.0716124 0.997433i \(-0.477186\pi\)
−0.997433 + 0.0716124i \(0.977186\pi\)
\(8\) 0 0
\(9\) 4.00000i 1.33333i
\(10\) 0 0
\(11\) −4.58258 −1.38170 −0.690849 0.722999i \(-0.742763\pi\)
−0.690849 + 0.722999i \(0.742763\pi\)
\(12\) 0 0
\(13\) −3.74166 + 3.74166i −1.03775 + 1.03775i −0.0384901 + 0.999259i \(0.512255\pi\)
−0.999259 + 0.0384901i \(0.987745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.22474 1.22474i 0.297044 0.297044i −0.542811 0.839855i \(-0.682640\pi\)
0.839855 + 0.542811i \(0.182640\pi\)
\(18\) 0 0
\(19\) 4.58258i 1.05131i 0.850696 + 0.525657i \(0.176181\pi\)
−0.850696 + 0.525657i \(0.823819\pi\)
\(20\) 0 0
\(21\) 9.16515i 2.00000i
\(22\) 0 0
\(23\) 4.89898 4.89898i 1.02151 1.02151i 0.0217443 0.999764i \(-0.493078\pi\)
0.999764 0.0217443i \(-0.00692196\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.87083 1.87083i 0.360041 0.360041i
\(28\) 0 0
\(29\) 9.16515 1.70193 0.850963 0.525226i \(-0.176019\pi\)
0.850963 + 0.525226i \(0.176019\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 8.57321 + 8.57321i 1.49241 + 1.49241i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.74166 3.74166i −0.615125 0.615125i 0.329152 0.944277i \(-0.393237\pi\)
−0.944277 + 0.329152i \(0.893237\pi\)
\(38\) 0 0
\(39\) 14.0000 2.24179
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −3.74166 3.74166i −0.570597 0.570597i 0.361698 0.932295i \(-0.382197\pi\)
−0.932295 + 0.361698i \(0.882197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.44949 + 2.44949i 0.357295 + 0.357295i 0.862815 0.505520i \(-0.168699\pi\)
−0.505520 + 0.862815i \(0.668699\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) −4.58258 −0.641689
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.57321 8.57321i 1.13555 1.13555i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 9.16515i 1.17348i −0.809776 0.586739i \(-0.800412\pi\)
0.809776 0.586739i \(-0.199588\pi\)
\(62\) 0 0
\(63\) 9.79796 9.79796i 1.23443 1.23443i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.87083 + 1.87083i −0.228558 + 0.228558i −0.812090 0.583532i \(-0.801670\pi\)
0.583532 + 0.812090i \(0.301670\pi\)
\(68\) 0 0
\(69\) −18.3303 −2.20671
\(70\) 0 0
\(71\) 6.00000i 0.712069i −0.934473 0.356034i \(-0.884129\pi\)
0.934473 0.356034i \(-0.115871\pi\)
\(72\) 0 0
\(73\) 6.12372 + 6.12372i 0.716728 + 0.716728i 0.967934 0.251206i \(-0.0808271\pi\)
−0.251206 + 0.967934i \(0.580827\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.2250 + 11.2250i 1.27920 + 1.27920i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 5.61249 + 5.61249i 0.616050 + 0.616050i 0.944516 0.328466i \(-0.106531\pi\)
−0.328466 + 0.944516i \(0.606531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −17.1464 17.1464i −1.83829 1.83829i
\(88\) 0 0
\(89\) 15.0000i 1.59000i 0.606612 + 0.794998i \(0.292528\pi\)
−0.606612 + 0.794998i \(0.707472\pi\)
\(90\) 0 0
\(91\) 18.3303 1.92154
\(92\) 0 0
\(93\) 7.48331 7.48331i 0.775984 0.775984i
\(94\) 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
\(99\) 18.3303i 1.84226i
\(100\) 0 0
\(101\) 9.16515i 0.911967i −0.889988 0.455983i \(-0.849288\pi\)
0.889988 0.455983i \(-0.150712\pi\)
\(102\) 0 0
\(103\) −2.44949 + 2.44949i −0.241355 + 0.241355i −0.817411 0.576055i \(-0.804591\pi\)
0.576055 + 0.817411i \(0.304591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.61249 + 5.61249i −0.542580 + 0.542580i −0.924284 0.381705i \(-0.875337\pi\)
0.381705 + 0.924284i \(0.375337\pi\)
\(108\) 0 0
\(109\) 9.16515 0.877862 0.438931 0.898521i \(-0.355357\pi\)
0.438931 + 0.898521i \(0.355357\pi\)
\(110\) 0 0
\(111\) 14.0000i 1.32882i
\(112\) 0 0
\(113\) 1.22474 + 1.22474i 0.115214 + 0.115214i 0.762363 0.647149i \(-0.224039\pi\)
−0.647149 + 0.762363i \(0.724039\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −14.9666 14.9666i −1.38367 1.38367i
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 10.0000 0.909091
\(122\) 0 0
\(123\) 16.8375 + 16.8375i 1.51818 + 1.51818i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 14.0000i 1.23263i
\(130\) 0 0
\(131\) 18.3303 1.60153 0.800763 0.598981i \(-0.204428\pi\)
0.800763 + 0.598981i \(0.204428\pi\)
\(132\) 0 0
\(133\) 11.2250 11.2250i 0.973329 0.973329i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.0227 + 11.0227i −0.941733 + 0.941733i −0.998394 0.0566604i \(-0.981955\pi\)
0.0566604 + 0.998394i \(0.481955\pi\)
\(138\) 0 0
\(139\) 4.58258i 0.388689i 0.980933 + 0.194344i \(0.0622580\pi\)
−0.980933 + 0.194344i \(0.937742\pi\)
\(140\) 0 0
\(141\) 9.16515i 0.771845i
\(142\) 0 0
\(143\) 17.1464 17.1464i 1.43386 1.43386i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.35414 9.35414i 0.771517 0.771517i
\(148\) 0 0
\(149\) −9.16515 −0.750838 −0.375419 0.926855i \(-0.622501\pi\)
−0.375419 + 0.926855i \(0.622501\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) 0 0
\(153\) 4.89898 + 4.89898i 0.396059 + 0.396059i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.9666 + 14.9666i 1.19447 + 1.19447i 0.975799 + 0.218668i \(0.0701711\pi\)
0.218668 + 0.975799i \(0.429829\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 1.87083 + 1.87083i 0.146535 + 0.146535i 0.776568 0.630033i \(-0.216959\pi\)
−0.630033 + 0.776568i \(0.716959\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.34847 + 7.34847i 0.568642 + 0.568642i 0.931748 0.363106i \(-0.118284\pi\)
−0.363106 + 0.931748i \(0.618284\pi\)
\(168\) 0 0
\(169\) 15.0000i 1.15385i
\(170\) 0 0
\(171\) −18.3303 −1.40175
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7477i 1.02755i 0.857924 + 0.513777i \(0.171754\pi\)
−0.857924 + 0.513777i \(0.828246\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −17.1464 + 17.1464i −1.26750 + 1.26750i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.61249 + 5.61249i −0.410426 + 0.410426i
\(188\) 0 0
\(189\) −9.16515 −0.666667
\(190\) 0 0
\(191\) 24.0000i 1.73658i 0.496058 + 0.868290i \(0.334780\pi\)
−0.496058 + 0.868290i \(0.665220\pi\)
\(192\) 0 0
\(193\) −6.12372 6.12372i −0.440795 0.440795i 0.451484 0.892279i \(-0.350895\pi\)
−0.892279 + 0.451484i \(0.850895\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) 0 0
\(203\) −22.4499 22.4499i −1.57568 1.57568i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 19.5959 + 19.5959i 1.36201 + 1.36201i
\(208\) 0 0
\(209\) 21.0000i 1.45260i
\(210\) 0 0
\(211\) −13.7477 −0.946433 −0.473216 0.880946i \(-0.656907\pi\)
−0.473216 + 0.880946i \(0.656907\pi\)
\(212\) 0 0
\(213\) −11.2250 + 11.2250i −0.769122 + 0.769122i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.79796 9.79796i 0.665129 0.665129i
\(218\) 0 0
\(219\) 22.9129i 1.54831i
\(220\) 0 0
\(221\) 9.16515i 0.616515i
\(222\) 0 0
\(223\) −4.89898 + 4.89898i −0.328060 + 0.328060i −0.851848 0.523788i \(-0.824518\pi\)
0.523788 + 0.851848i \(0.324518\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.2250 + 11.2250i −0.745028 + 0.745028i −0.973541 0.228513i \(-0.926614\pi\)
0.228513 + 0.973541i \(0.426614\pi\)
\(228\) 0 0
\(229\) 18.3303 1.21130 0.605650 0.795731i \(-0.292913\pi\)
0.605650 + 0.795731i \(0.292913\pi\)
\(230\) 0 0
\(231\) 42.0000i 2.76340i
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −18.7083 18.7083i −1.21523 1.21523i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) −14.9666 14.9666i −0.960110 0.960110i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −17.1464 17.1464i −1.09100 1.09100i
\(248\) 0 0
\(249\) 21.0000i 1.33082i
\(250\) 0 0
\(251\) 4.58258 0.289250 0.144625 0.989487i \(-0.453802\pi\)
0.144625 + 0.989487i \(0.453802\pi\)
\(252\) 0 0
\(253\) −22.4499 + 22.4499i −1.41142 + 1.41142i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.89898 + 4.89898i −0.305590 + 0.305590i −0.843196 0.537606i \(-0.819329\pi\)
0.537606 + 0.843196i \(0.319329\pi\)
\(258\) 0 0
\(259\) 18.3303i 1.13899i
\(260\) 0 0
\(261\) 36.6606i 2.26923i
\(262\) 0 0
\(263\) −14.6969 + 14.6969i −0.906252 + 0.906252i −0.995967 0.0897154i \(-0.971404\pi\)
0.0897154 + 0.995967i \(0.471404\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 28.0624 28.0624i 1.71739 1.71739i
\(268\) 0 0
\(269\) 18.3303 1.11762 0.558809 0.829296i \(-0.311258\pi\)
0.558809 + 0.829296i \(0.311258\pi\)
\(270\) 0 0
\(271\) 8.00000i 0.485965i −0.970031 0.242983i \(-0.921874\pi\)
0.970031 0.242983i \(-0.0781258\pi\)
\(272\) 0 0
\(273\) −34.2929 34.2929i −2.07550 2.07550i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.7083 + 18.7083i 1.12407 + 1.12407i 0.991123 + 0.132949i \(0.0424447\pi\)
0.132949 + 0.991123i \(0.457555\pi\)
\(278\) 0 0
\(279\) −16.0000 −0.957895
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 9.35414 + 9.35414i 0.556046 + 0.556046i 0.928179 0.372133i \(-0.121374\pi\)
−0.372133 + 0.928179i \(0.621374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 22.0454 + 22.0454i 1.30130 + 1.30130i
\(288\) 0 0
\(289\) 14.0000i 0.823529i
\(290\) 0 0
\(291\) −18.3303 −1.07454
\(292\) 0 0
\(293\) −11.2250 + 11.2250i −0.655770 + 0.655770i −0.954376 0.298606i \(-0.903478\pi\)
0.298606 + 0.954376i \(0.403478\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.57321 + 8.57321i −0.497468 + 0.497468i
\(298\) 0 0
\(299\) 36.6606i 2.12014i
\(300\) 0 0
\(301\) 18.3303i 1.05654i
\(302\) 0 0
\(303\) −17.1464 + 17.1464i −0.985037 + 0.985037i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.35414 + 9.35414i −0.533869 + 0.533869i −0.921722 0.387852i \(-0.873217\pi\)
0.387852 + 0.921722i \(0.373217\pi\)
\(308\) 0 0
\(309\) 9.16515 0.521387
\(310\) 0 0
\(311\) 18.0000i 1.02069i 0.859971 + 0.510343i \(0.170482\pi\)
−0.859971 + 0.510343i \(0.829518\pi\)
\(312\) 0 0
\(313\) −14.6969 14.6969i −0.830720 0.830720i 0.156895 0.987615i \(-0.449852\pi\)
−0.987615 + 0.156895i \(0.949852\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.4499 22.4499i −1.26091 1.26091i −0.950650 0.310264i \(-0.899583\pi\)
−0.310264 0.950650i \(-0.600417\pi\)
\(318\) 0 0
\(319\) −42.0000 −2.35155
\(320\) 0 0
\(321\) 21.0000 1.17211
\(322\) 0 0
\(323\) 5.61249 + 5.61249i 0.312287 + 0.312287i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −17.1464 17.1464i −0.948200 0.948200i
\(328\) 0 0
\(329\) 12.0000i 0.661581i
\(330\) 0 0
\(331\) −32.0780 −1.76317 −0.881584 0.472027i \(-0.843522\pi\)
−0.881584 + 0.472027i \(0.843522\pi\)
\(332\) 0 0
\(333\) 14.9666 14.9666i 0.820166 0.820166i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −18.3712 + 18.3712i −1.00074 + 1.00074i −0.000741840 1.00000i \(0.500236\pi\)
−1.00000 0.000741840i \(0.999764\pi\)
\(338\) 0 0
\(339\) 4.58258i 0.248891i
\(340\) 0 0
\(341\) 18.3303i 0.992642i
\(342\) 0 0
\(343\) −4.89898 + 4.89898i −0.264520 + 0.264520i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.61249 + 5.61249i −0.301294 + 0.301294i −0.841520 0.540226i \(-0.818339\pi\)
0.540226 + 0.841520i \(0.318339\pi\)
\(348\) 0 0
\(349\) −9.16515 −0.490599 −0.245300 0.969447i \(-0.578886\pi\)
−0.245300 + 0.969447i \(0.578886\pi\)
\(350\) 0 0
\(351\) 14.0000i 0.747265i
\(352\) 0 0
\(353\) 24.4949 + 24.4949i 1.30373 + 1.30373i 0.925856 + 0.377875i \(0.123345\pi\)
0.377875 + 0.925856i \(0.376655\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 11.2250 + 11.2250i 0.594089 + 0.594089i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −2.00000 −0.105263
\(362\) 0 0
\(363\) −18.7083 18.7083i −0.981930 0.981930i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.6969 + 14.6969i 0.767174 + 0.767174i 0.977608 0.210434i \(-0.0674877\pi\)
−0.210434 + 0.977608i \(0.567488\pi\)
\(368\) 0 0
\(369\) 36.0000i 1.87409i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.9666 14.9666i 0.774943 0.774943i −0.204023 0.978966i \(-0.565402\pi\)
0.978966 + 0.204023i \(0.0654018\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −34.2929 + 34.2929i −1.76617 + 1.76617i
\(378\) 0 0
\(379\) 13.7477i 0.706173i −0.935591 0.353087i \(-0.885132\pi\)
0.935591 0.353087i \(-0.114868\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.44949 2.44949i 0.125163 0.125163i −0.641750 0.766914i \(-0.721792\pi\)
0.766914 + 0.641750i \(0.221792\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 14.9666 14.9666i 0.760797 0.760797i
\(388\) 0 0
\(389\) 18.3303 0.929383 0.464692 0.885473i \(-0.346165\pi\)
0.464692 + 0.885473i \(0.346165\pi\)
\(390\) 0 0
\(391\) 12.0000i 0.606866i
\(392\) 0 0
\(393\) −34.2929 34.2929i −1.72985 1.72985i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.48331 7.48331i −0.375577 0.375577i 0.493927 0.869504i \(-0.335561\pi\)
−0.869504 + 0.493927i \(0.835561\pi\)
\(398\) 0 0
\(399\) −42.0000 −2.10263
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) −14.9666 14.9666i −0.745541 0.745541i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.1464 + 17.1464i 0.849917 + 0.849917i
\(408\) 0 0
\(409\) 17.0000i 0.840596i 0.907386 + 0.420298i \(0.138074\pi\)
−0.907386 + 0.420298i \(0.861926\pi\)
\(410\) 0 0
\(411\) 41.2432 2.03438
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.57321 8.57321i 0.419832 0.419832i
\(418\) 0 0
\(419\) 22.9129i 1.11937i −0.828706 0.559684i \(-0.810923\pi\)
0.828706 0.559684i \(-0.189077\pi\)
\(420\) 0 0
\(421\) 36.6606i 1.78673i −0.449333 0.893364i \(-0.648338\pi\)
0.449333 0.893364i \(-0.351662\pi\)
\(422\) 0 0
\(423\) −9.79796 + 9.79796i −0.476393 + 0.476393i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −22.4499 + 22.4499i −1.08643 + 1.08643i
\(428\) 0 0
\(429\) −64.1561 −3.09748
\(430\) 0 0
\(431\) 30.0000i 1.44505i 0.691345 + 0.722525i \(0.257018\pi\)
−0.691345 + 0.722525i \(0.742982\pi\)
\(432\) 0 0
\(433\) −13.4722 13.4722i −0.647432 0.647432i 0.304939 0.952372i \(-0.401364\pi\)
−0.952372 + 0.304939i \(0.901364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.4499 + 22.4499i 1.07393 + 1.07393i
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) −20.0000 −0.952381
\(442\) 0 0
\(443\) 28.0624 + 28.0624i 1.33329 + 1.33329i 0.902412 + 0.430874i \(0.141795\pi\)
0.430874 + 0.902412i \(0.358205\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 17.1464 + 17.1464i 0.810998 + 0.810998i
\(448\) 0 0
\(449\) 27.0000i 1.27421i −0.770778 0.637104i \(-0.780132\pi\)
0.770778 0.637104i \(-0.219868\pi\)
\(450\) 0 0
\(451\) 41.2432 1.94207
\(452\) 0 0
\(453\) −18.7083 + 18.7083i −0.878992 + 0.878992i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.7196 25.7196i 1.20311 1.20311i 0.229900 0.973214i \(-0.426160\pi\)
0.973214 0.229900i \(-0.0738399\pi\)
\(458\) 0 0
\(459\) 4.58258i 0.213896i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −9.79796 + 9.79796i −0.455350 + 0.455350i −0.897126 0.441776i \(-0.854349\pi\)
0.441776 + 0.897126i \(0.354349\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.2250 11.2250i 0.519430 0.519430i −0.397969 0.917399i \(-0.630285\pi\)
0.917399 + 0.397969i \(0.130285\pi\)
\(468\) 0 0
\(469\) 9.16515 0.423207
\(470\) 0 0
\(471\) 56.0000i 2.58034i
\(472\) 0 0
\(473\) 17.1464 + 17.1464i 0.788394 + 0.788394i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 28.0000 1.27669
\(482\) 0 0
\(483\) 44.8999 + 44.8999i 2.04302 + 2.04302i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.6969 14.6969i −0.665982 0.665982i 0.290802 0.956783i \(-0.406078\pi\)
−0.956783 + 0.290802i \(0.906078\pi\)
\(488\) 0 0
\(489\) 7.00000i 0.316551i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 11.2250 11.2250i 0.505547 0.505547i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.6969 + 14.6969i −0.659248 + 0.659248i
\(498\) 0 0
\(499\) 18.3303i 0.820577i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(500\) 0 0
\(501\) 27.4955i 1.22841i
\(502\) 0 0
\(503\) −9.79796 + 9.79796i −0.436869 + 0.436869i −0.890957 0.454088i \(-0.849965\pi\)
0.454088 + 0.890957i \(0.349965\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −28.0624 + 28.0624i −1.24630 + 1.24630i
\(508\) 0 0
\(509\) −18.3303 −0.812476 −0.406238 0.913767i \(-0.633160\pi\)
−0.406238 + 0.913767i \(0.633160\pi\)
\(510\) 0 0
\(511\) 30.0000i 1.32712i
\(512\) 0 0
\(513\) 8.57321 + 8.57321i 0.378517 + 0.378517i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −11.2250 11.2250i −0.493674 0.493674i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 9.35414 + 9.35414i 0.409028 + 0.409028i 0.881400 0.472371i \(-0.156602\pi\)
−0.472371 + 0.881400i \(0.656602\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.89898 + 4.89898i 0.213403 + 0.213403i
\(528\) 0 0
\(529\) 25.0000i 1.08696i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 33.6749 33.6749i 1.45862 1.45862i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 25.7196 25.7196i 1.10988 1.10988i
\(538\) 0 0
\(539\) 22.9129i 0.986928i
\(540\) 0 0
\(541\) 18.3303i 0.788081i 0.919093 + 0.394041i \(0.128923\pi\)
−0.919093 + 0.394041i \(0.871077\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0958 13.0958i 0.559936 0.559936i −0.369353 0.929289i \(-0.620421\pi\)
0.929289 + 0.369353i \(0.120421\pi\)
\(548\) 0 0
\(549\) 36.6606 1.56464
\(550\) 0 0
\(551\) 42.0000i 1.78926i
\(552\) 0 0
\(553\) −24.4949 24.4949i −1.04163 1.04163i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.2250 + 11.2250i 0.475617 + 0.475617i 0.903727 0.428110i \(-0.140820\pi\)
−0.428110 + 0.903727i \(0.640820\pi\)
\(558\) 0 0
\(559\) 28.0000 1.18427
\(560\) 0 0
\(561\) 21.0000 0.886621
\(562\) 0 0
\(563\) −11.2250 11.2250i −0.473076 0.473076i 0.429832 0.902909i \(-0.358573\pi\)
−0.902909 + 0.429832i \(0.858573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −12.2474 12.2474i −0.514344 0.514344i
\(568\) 0 0
\(569\) 3.00000i 0.125767i 0.998021 + 0.0628833i \(0.0200296\pi\)
−0.998021 + 0.0628833i \(0.979970\pi\)
\(570\) 0 0
\(571\) 36.6606 1.53420 0.767099 0.641528i \(-0.221699\pi\)
0.767099 + 0.641528i \(0.221699\pi\)
\(572\) 0 0
\(573\) 44.8999 44.8999i 1.87572 1.87572i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.1691 28.1691i 1.17270 1.17270i 0.191132 0.981564i \(-0.438784\pi\)
0.981564 0.191132i \(-0.0612158\pi\)
\(578\) 0 0
\(579\) 22.9129i 0.952227i
\(580\) 0 0
\(581\) 27.4955i 1.14070i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0624 28.0624i 1.15826 1.15826i 0.173411 0.984850i \(-0.444521\pi\)
0.984850 0.173411i \(-0.0554789\pi\)
\(588\) 0 0
\(589\) −18.3303 −0.755287
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.12372 + 6.12372i 0.251471 + 0.251471i 0.821574 0.570102i \(-0.193096\pi\)
−0.570102 + 0.821574i \(0.693096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.48331 7.48331i −0.306272 0.306272i
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) −7.48331 7.48331i −0.304744 0.304744i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −19.5959 19.5959i −0.795374 0.795374i 0.186988 0.982362i \(-0.440127\pi\)
−0.982362 + 0.186988i \(0.940127\pi\)
\(608\) 0 0
\(609\) 84.0000i 3.40385i
\(610\) 0 0
\(611\) −18.3303 −0.741565
\(612\) 0 0
\(613\) −7.48331 + 7.48331i −0.302248 + 0.302248i −0.841893 0.539645i \(-0.818559\pi\)
0.539645 + 0.841893i \(0.318559\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.4949 + 24.4949i −0.986127 + 0.986127i −0.999905 0.0137776i \(-0.995614\pi\)
0.0137776 + 0.999905i \(0.495614\pi\)
\(618\) 0 0
\(619\) 36.6606i 1.47351i 0.676158 + 0.736757i \(0.263644\pi\)
−0.676158 + 0.736757i \(0.736356\pi\)
\(620\) 0 0
\(621\) 18.3303i 0.735570i
\(622\) 0 0
\(623\) 36.7423 36.7423i 1.47205 1.47205i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −39.2874 + 39.2874i −1.56899 + 1.56899i
\(628\) 0 0
\(629\) −9.16515 −0.365439
\(630\) 0 0
\(631\) 20.0000i 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 0 0
\(633\) 25.7196 + 25.7196i 1.02226 + 1.02226i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.7083 18.7083i −0.741249 0.741249i
\(638\) 0 0
\(639\) 24.0000 0.949425
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −3.74166 3.74166i −0.147557 0.147557i 0.629469 0.777026i \(-0.283272\pi\)
−0.777026 + 0.629469i \(0.783272\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.6969 + 14.6969i 0.577796 + 0.577796i 0.934296 0.356499i \(-0.116030\pi\)
−0.356499 + 0.934296i \(0.616030\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −36.6606 −1.43684
\(652\) 0 0
\(653\) −33.6749 + 33.6749i −1.31780 + 1.31780i −0.402288 + 0.915513i \(0.631785\pi\)
−0.915513 + 0.402288i \(0.868215\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −24.4949 + 24.4949i −0.955637 + 0.955637i
\(658\) 0 0
\(659\) 13.7477i 0.535535i −0.963484 0.267768i \(-0.913714\pi\)
0.963484 0.267768i \(-0.0862860\pi\)
\(660\) 0 0
\(661\) 9.16515i 0.356483i −0.983987 0.178242i \(-0.942959\pi\)
0.983987 0.178242i \(-0.0570408\pi\)
\(662\) 0 0
\(663\) 17.1464 17.1464i 0.665912 0.665912i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 44.8999 44.8999i 1.73853 1.73853i
\(668\) 0 0
\(669\) 18.3303 0.708690
\(670\) 0 0
\(671\) 42.0000i 1.62139i
\(672\) 0 0
\(673\) 24.4949 + 24.4949i 0.944209 + 0.944209i 0.998524 0.0543149i \(-0.0172975\pi\)
−0.0543149 + 0.998524i \(0.517297\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.4499 + 22.4499i 0.862821 + 0.862821i 0.991665 0.128843i \(-0.0411265\pi\)
−0.128843 + 0.991665i \(0.541126\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) −5.61249 5.61249i −0.214756 0.214756i 0.591528 0.806284i \(-0.298525\pi\)
−0.806284 + 0.591528i \(0.798525\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −34.2929 34.2929i −1.30835 1.30835i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −13.7477 −0.522988 −0.261494 0.965205i \(-0.584215\pi\)
−0.261494 + 0.965205i \(0.584215\pi\)
\(692\) 0 0
\(693\) −44.8999 + 44.8999i −1.70561 + 1.70561i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −11.0227 + 11.0227i −0.417515 + 0.417515i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.8258i 1.73081i 0.501069 + 0.865407i \(0.332940\pi\)
−0.501069 + 0.865407i \(0.667060\pi\)
\(702\) 0 0
\(703\) 17.1464 17.1464i 0.646690 0.646690i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −22.4499 + 22.4499i −0.844317 + 0.844317i
\(708\) 0 0
\(709\) −36.6606 −1.37682 −0.688409 0.725323i \(-0.741691\pi\)
−0.688409 + 0.725323i \(0.741691\pi\)
\(710\) 0 0
\(711\) 40.0000i 1.50012i
\(712\) 0 0
\(713\) 19.5959 + 19.5959i 0.733873 + 0.733873i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.2250 + 11.2250i 0.419204 + 0.419204i
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) −13.0958 13.0958i −0.487038 0.487038i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.2474 + 12.2474i 0.454233 + 0.454233i 0.896757 0.442524i \(-0.145917\pi\)
−0.442524 + 0.896757i \(0.645917\pi\)
\(728\) 0 0
\(729\) 41.0000i 1.51852i
\(730\) 0 0
\(731\) −9.16515 −0.338985
\(732\) 0 0
\(733\) −18.7083 + 18.7083i −0.691006 + 0.691006i −0.962453 0.271447i \(-0.912498\pi\)
0.271447 + 0.962453i \(0.412498\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.57321 8.57321i 0.315798 0.315798i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 64.1561i 2.35683i
\(742\) 0 0
\(743\) 9.79796 9.79796i 0.359452 0.359452i −0.504159 0.863611i \(-0.668197\pi\)
0.863611 + 0.504159i \(0.168197\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −22.4499 + 22.4499i −0.821401 + 0.821401i
\(748\) 0 0
\(749\) 27.4955 1.00466
\(750\) 0 0
\(751\) 22.0000i 0.802791i 0.915905 + 0.401396i \(0.131475\pi\)
−0.915905 + 0.401396i \(0.868525\pi\)
\(752\) 0 0
\(753\) −8.57321 8.57321i −0.312425 0.312425i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.48331 7.48331i −0.271986 0.271986i 0.557913 0.829899i \(-0.311602\pi\)
−0.829899 + 0.557913i \(0.811602\pi\)
\(758\) 0 0
\(759\) 84.0000 3.04901
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −22.4499 22.4499i −0.812743 0.812743i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000i 1.26213i 0.775729 + 0.631066i \(0.217382\pi\)
−0.775729 + 0.631066i \(0.782618\pi\)
\(770\) 0 0
\(771\) 18.3303 0.660150
\(772\) 0 0
\(773\) 11.2250 11.2250i 0.403734 0.403734i −0.475813 0.879547i \(-0.657846\pi\)
0.879547 + 0.475813i \(0.157846\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 34.2929 34.2929i 1.23025 1.23025i
\(778\) 0 0
\(779\) 41.2432i 1.47769i
\(780\) 0 0
\(781\) 27.4955i 0.983865i
\(782\) 0 0
\(783\) 17.1464 17.1464i 0.612763 0.612763i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 18.7083 18.7083i 0.666878 0.666878i −0.290114 0.956992i \(-0.593693\pi\)
0.956992 + 0.290114i \(0.0936931\pi\)
\(788\) 0 0
\(789\) 54.9909 1.95773
\(790\) 0 0
\(791\) 6.00000i 0.213335i
\(792\) 0 0
\(793\) 34.2929 + 34.2929i 1.21778 + 1.21778i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.4499 22.4499i −0.795218 0.795218i 0.187119 0.982337i \(-0.440085\pi\)
−0.982337 + 0.187119i \(0.940085\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) −60.0000 −2.12000
\(802\) 0 0
\(803\) −28.0624 28.0624i −0.990302 0.990302i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −34.2929 34.2929i −1.20717 1.20717i
\(808\) 0 0
\(809\) 18.0000i 0.632846i 0.948618 + 0.316423i \(0.102482\pi\)
−0.948618 + 0.316423i \(0.897518\pi\)
\(810\) 0 0
\(811\) −36.6606 −1.28733 −0.643664 0.765308i \(-0.722587\pi\)
−0.643664 + 0.765308i \(0.722587\pi\)
\(812\) 0 0
\(813\) −14.9666 + 14.9666i −0.524903 + 0.524903i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 17.1464 17.1464i 0.599878 0.599878i
\(818\) 0 0
\(819\) 73.3212i 2.56205i
\(820\) 0 0
\(821\) 54.9909i 1.91920i 0.281375 + 0.959598i \(0.409210\pi\)
−0.281375 + 0.959598i \(0.590790\pi\)
\(822\) 0 0
\(823\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.61249 + 5.61249i −0.195165 + 0.195165i −0.797924 0.602758i \(-0.794068\pi\)
0.602758 + 0.797924i \(0.294068\pi\)
\(828\) 0 0
\(829\) 45.8258 1.59159 0.795797 0.605563i \(-0.207052\pi\)
0.795797 + 0.605563i \(0.207052\pi\)
\(830\) 0 0
\(831\) 70.0000i 2.42827i
\(832\) 0 0
\(833\) 6.12372 + 6.12372i 0.212174 + 0.212174i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 7.48331 + 7.48331i 0.258661 + 0.258661i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 55.0000 1.89655
\(842\) 0 0
\(843\) −11.2250 11.2250i −0.386609 0.386609i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −24.4949 24.4949i −0.841655 0.841655i
\(848\) 0 0
\(849\) 35.0000i 1.20120i
\(850\) 0 0
\(851\) −36.6606 −1.25671
\(852\) 0 0
\(853\) −7.48331 + 7.48331i −0.256224 + 0.256224i −0.823516 0.567293i \(-0.807991\pi\)
0.567293 + 0.823516i \(0.307991\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.0227 11.0227i 0.376528 0.376528i −0.493320 0.869848i \(-0.664217\pi\)
0.869848 + 0.493320i \(0.164217\pi\)
\(858\) 0 0
\(859\) 4.58258i 0.156355i 0.996939 + 0.0781777i \(0.0249102\pi\)
−0.996939 + 0.0781777i \(0.975090\pi\)
\(860\) 0 0
\(861\) 82.4864i 2.81113i
\(862\) 0 0
\(863\) 12.2474 12.2474i 0.416908 0.416908i −0.467229 0.884137i \(-0.654747\pi\)
0.884137 + 0.467229i \(0.154747\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 26.1916 26.1916i 0.889513 0.889513i
\(868\) 0 0
\(869\) −45.8258 −1.55453
\(870\) 0 0
\(871\) 14.0000i 0.474372i
\(872\) 0 0
\(873\) 19.5959 + 19.5959i 0.663221 + 0.663221i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.9666 14.9666i −0.505387 0.505387i 0.407720 0.913107i \(-0.366324\pi\)
−0.913107 + 0.407720i \(0.866324\pi\)
\(878\) 0 0
\(879\) 42.0000 1.41662
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 20.5791 + 20.5791i 0.692542 + 0.692542i 0.962791 0.270248i \(-0.0871058\pi\)
−0.270248 + 0.962791i \(0.587106\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.1918 + 39.1918i 1.31593 + 1.31593i 0.916965 + 0.398968i \(0.130632\pi\)
0.398968 + 0.916965i \(0.369368\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −22.9129 −0.767610
\(892\) 0 0
\(893\) −11.2250 + 11.2250i −0.375629 + 0.375629i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 68.5857 68.5857i 2.29001 2.29001i
\(898\) 0 0
\(899\) 36.6606i 1.22270i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 34.2929 34.2929i 1.14119 1.14119i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.74166 3.74166i 0.124240 0.124240i −0.642253 0.766493i \(-0.722000\pi\)
0.766493 + 0.642253i \(0.222000\pi\)
\(908\) 0 0
\(909\) 36.6606 1.21596
\(910\) 0 0
\(911\) 36.0000i 1.19273i −0.802712 0.596367i \(-0.796610\pi\)
0.802712 0.596367i \(-0.203390\pi\)
\(912\) 0 0
\(913\) −25.7196 25.7196i −0.851196 0.851196i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.8999 44.8999i −1.48272 1.48272i
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) 35.0000 1.15329
\(922\) 0 0
\(923\) 22.4499 + 22.4499i 0.738949 + 0.738949i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.79796 9.79796i −0.321807 0.321807i
\(928\) 0 0
\(929\) 42.0000i 1.37798i 0.724773 + 0.688988i \(0.241945\pi\)
−0.724773 + 0.688988i \(0.758055\pi\)
\(930\) 0 0
\(931\) −22.9129 −0.750939
\(932\) 0 0
\(933\) 33.6749 33.6749i 1.10247 1.10247i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.22474 1.22474i 0.0400107 0.0400107i −0.686818 0.726829i \(-0.740993\pi\)
0.726829 + 0.686818i \(0.240993\pi\)
\(938\) 0 0
\(939\) 54.9909i 1.79456i
\(940\) 0 0
\(941\) 9.16515i 0.298775i −0.988779 0.149388i \(-0.952270\pi\)
0.988779 0.149388i \(-0.0477302\pi\)
\(942\) 0 0
\(943\) −44.0908 + 44.0908i −1.43579 + 1.43579i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.6749 33.6749i 1.09429 1.09429i 0.0992225 0.995065i \(-0.468364\pi\)
0.995065 0.0992225i \(-0.0316355\pi\)
\(948\) 0 0
\(949\) −45.8258 −1.48757
\(950\) 0 0
\(951\) 84.0000i 2.72389i
\(952\) 0 0
\(953\) 8.57321 + 8.57321i 0.277714 + 0.277714i 0.832196 0.554482i \(-0.187084\pi\)
−0.554482 + 0.832196i \(0.687084\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 78.5748 + 78.5748i 2.53996 + 2.53996i
\(958\) 0 0
\(959\) 54.0000 1.74375
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −22.4499 22.4499i −0.723439 0.723439i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.4949 24.4949i −0.787703 0.787703i 0.193414 0.981117i \(-0.438044\pi\)
−0.981117 + 0.193414i \(0.938044\pi\)
\(968\) 0 0
\(969\) 21.0000i 0.674617i
\(970\) 0 0
\(971\) 22.9129 0.735309 0.367655 0.929962i \(-0.380161\pi\)
0.367655 + 0.929962i \(0.380161\pi\)
\(972\) 0 0
\(973\) 11.2250 11.2250i 0.359856 0.359856i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −28.1691 + 28.1691i −0.901210 + 0.901210i −0.995541 0.0943306i \(-0.969929\pi\)
0.0943306 + 0.995541i \(0.469929\pi\)
\(978\) 0 0
\(979\) 68.7386i 2.19690i
\(980\) 0 0
\(981\) 36.6606i 1.17048i
\(982\) 0 0
\(983\) 31.8434 31.8434i 1.01565 1.01565i 0.0157700 0.999876i \(-0.494980\pi\)
0.999876 0.0157700i \(-0.00501996\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −22.4499 + 22.4499i −0.714590 + 0.714590i
\(988\) 0 0
\(989\) −36.6606 −1.16574
\(990\) 0 0
\(991\) 2.00000i 0.0635321i 0.999495 + 0.0317660i \(0.0101131\pi\)
−0.999495 + 0.0317660i \(0.989887\pi\)
\(992\) 0 0
\(993\) 60.0125 + 60.0125i 1.90444 + 1.90444i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.48331 7.48331i −0.236999 0.236999i 0.578607 0.815606i \(-0.303596\pi\)
−0.815606 + 0.578607i \(0.803596\pi\)
\(998\) 0 0
\(999\) −14.0000 −0.442940
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.k.543.1 yes 8
4.3 odd 2 1600.2.o.f.543.4 yes 8
5.2 odd 4 1600.2.o.f.607.2 yes 8
5.3 odd 4 1600.2.o.f.607.3 yes 8
5.4 even 2 inner 1600.2.o.k.543.4 yes 8
8.3 odd 2 1600.2.o.f.543.2 yes 8
8.5 even 2 inner 1600.2.o.k.543.3 yes 8
20.3 even 4 inner 1600.2.o.k.607.2 yes 8
20.7 even 4 inner 1600.2.o.k.607.3 yes 8
20.19 odd 2 1600.2.o.f.543.1 8
40.3 even 4 inner 1600.2.o.k.607.4 yes 8
40.13 odd 4 1600.2.o.f.607.1 yes 8
40.19 odd 2 1600.2.o.f.543.3 yes 8
40.27 even 4 inner 1600.2.o.k.607.1 yes 8
40.29 even 2 inner 1600.2.o.k.543.2 yes 8
40.37 odd 4 1600.2.o.f.607.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.o.f.543.1 8 20.19 odd 2
1600.2.o.f.543.2 yes 8 8.3 odd 2
1600.2.o.f.543.3 yes 8 40.19 odd 2
1600.2.o.f.543.4 yes 8 4.3 odd 2
1600.2.o.f.607.1 yes 8 40.13 odd 4
1600.2.o.f.607.2 yes 8 5.2 odd 4
1600.2.o.f.607.3 yes 8 5.3 odd 4
1600.2.o.f.607.4 yes 8 40.37 odd 4
1600.2.o.k.543.1 yes 8 1.1 even 1 trivial
1600.2.o.k.543.2 yes 8 40.29 even 2 inner
1600.2.o.k.543.3 yes 8 8.5 even 2 inner
1600.2.o.k.543.4 yes 8 5.4 even 2 inner
1600.2.o.k.607.1 yes 8 40.27 even 4 inner
1600.2.o.k.607.2 yes 8 20.3 even 4 inner
1600.2.o.k.607.3 yes 8 20.7 even 4 inner
1600.2.o.k.607.4 yes 8 40.3 even 4 inner