Properties

Label 1216.2.h.d.1215.1
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
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] = [1216,2,Mod(1215,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.1215");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1215.1
Root \(-0.331077 - 1.37491i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.d.1215.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.35829 q^{3} +2.56155 q^{5} -4.15286i q^{7} +2.56155 q^{9} +O(q^{10})\) \(q-2.35829 q^{3} +2.56155 q^{5} -4.15286i q^{7} +2.56155 q^{9} -2.33205i q^{11} -4.29400i q^{13} -6.04090 q^{15} -1.00000 q^{17} +(-3.68260 + 2.33205i) q^{19} +9.79366i q^{21} +1.82081i q^{23} +1.56155 q^{25} +1.03399 q^{27} -1.20565i q^{29} -1.32431 q^{31} +5.49966i q^{33} -10.6378i q^{35} +5.49966i q^{37} +10.1265i q^{39} +5.49966i q^{41} +1.30957i q^{43} +6.56155 q^{45} -6.99614i q^{47} -10.2462 q^{49} +2.35829 q^{51} -9.79366i q^{53} -5.97366i q^{55} +(8.68466 - 5.49966i) q^{57} -6.33122 q^{59} -11.6847 q^{61} -10.6378i q^{63} -10.9993i q^{65} +0.290319 q^{67} -4.29400i q^{69} +2.06798 q^{71} -0.123106 q^{73} -3.68260 q^{75} -9.68466 q^{77} -13.4061 q^{79} -10.1231 q^{81} +11.9473i q^{83} -2.56155 q^{85} +2.84329i q^{87} -14.0877i q^{89} -17.8324 q^{91} +3.12311 q^{93} +(-9.43318 + 5.97366i) q^{95} -2.41131i q^{97} -5.97366i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + 4 q^{9} - 8 q^{17} - 4 q^{25} + 36 q^{45} - 16 q^{49} + 20 q^{57} - 44 q^{61} + 32 q^{73} - 28 q^{77} - 48 q^{81} - 4 q^{85} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.35829 −1.36156 −0.680781 0.732487i \(-0.738359\pi\)
−0.680781 + 0.732487i \(0.738359\pi\)
\(4\) 0 0
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) 4.15286i 1.56963i −0.619729 0.784816i \(-0.712757\pi\)
0.619729 0.784816i \(-0.287243\pi\)
\(8\) 0 0
\(9\) 2.56155 0.853851
\(10\) 0 0
\(11\) 2.33205i 0.703139i −0.936162 0.351569i \(-0.885648\pi\)
0.936162 0.351569i \(-0.114352\pi\)
\(12\) 0 0
\(13\) 4.29400i 1.19094i −0.803377 0.595471i \(-0.796966\pi\)
0.803377 0.595471i \(-0.203034\pi\)
\(14\) 0 0
\(15\) −6.04090 −1.55975
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −3.68260 + 2.33205i −0.844847 + 0.535008i
\(20\) 0 0
\(21\) 9.79366i 2.13715i
\(22\) 0 0
\(23\) 1.82081i 0.379665i 0.981817 + 0.189832i \(0.0607944\pi\)
−0.981817 + 0.189832i \(0.939206\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) 1.03399 0.198991
\(28\) 0 0
\(29\) 1.20565i 0.223884i −0.993715 0.111942i \(-0.964293\pi\)
0.993715 0.111942i \(-0.0357071\pi\)
\(30\) 0 0
\(31\) −1.32431 −0.237853 −0.118926 0.992903i \(-0.537945\pi\)
−0.118926 + 0.992903i \(0.537945\pi\)
\(32\) 0 0
\(33\) 5.49966i 0.957367i
\(34\) 0 0
\(35\) 10.6378i 1.79811i
\(36\) 0 0
\(37\) 5.49966i 0.904138i 0.891983 + 0.452069i \(0.149314\pi\)
−0.891983 + 0.452069i \(0.850686\pi\)
\(38\) 0 0
\(39\) 10.1265i 1.62154i
\(40\) 0 0
\(41\) 5.49966i 0.858902i 0.903090 + 0.429451i \(0.141293\pi\)
−0.903090 + 0.429451i \(0.858707\pi\)
\(42\) 0 0
\(43\) 1.30957i 0.199707i 0.995002 + 0.0998536i \(0.0318375\pi\)
−0.995002 + 0.0998536i \(0.968163\pi\)
\(44\) 0 0
\(45\) 6.56155 0.978139
\(46\) 0 0
\(47\) 6.99614i 1.02049i −0.860028 0.510246i \(-0.829554\pi\)
0.860028 0.510246i \(-0.170446\pi\)
\(48\) 0 0
\(49\) −10.2462 −1.46374
\(50\) 0 0
\(51\) 2.35829 0.330227
\(52\) 0 0
\(53\) 9.79366i 1.34526i −0.739978 0.672631i \(-0.765164\pi\)
0.739978 0.672631i \(-0.234836\pi\)
\(54\) 0 0
\(55\) 5.97366i 0.805489i
\(56\) 0 0
\(57\) 8.68466 5.49966i 1.15031 0.728447i
\(58\) 0 0
\(59\) −6.33122 −0.824254 −0.412127 0.911126i \(-0.635214\pi\)
−0.412127 + 0.911126i \(0.635214\pi\)
\(60\) 0 0
\(61\) −11.6847 −1.49607 −0.748034 0.663661i \(-0.769002\pi\)
−0.748034 + 0.663661i \(0.769002\pi\)
\(62\) 0 0
\(63\) 10.6378i 1.34023i
\(64\) 0 0
\(65\) 10.9993i 1.36430i
\(66\) 0 0
\(67\) 0.290319 0.0354681 0.0177341 0.999843i \(-0.494355\pi\)
0.0177341 + 0.999843i \(0.494355\pi\)
\(68\) 0 0
\(69\) 4.29400i 0.516937i
\(70\) 0 0
\(71\) 2.06798 0.245423 0.122712 0.992442i \(-0.460841\pi\)
0.122712 + 0.992442i \(0.460841\pi\)
\(72\) 0 0
\(73\) −0.123106 −0.0144084 −0.00720421 0.999974i \(-0.502293\pi\)
−0.00720421 + 0.999974i \(0.502293\pi\)
\(74\) 0 0
\(75\) −3.68260 −0.425230
\(76\) 0 0
\(77\) −9.68466 −1.10367
\(78\) 0 0
\(79\) −13.4061 −1.50830 −0.754152 0.656700i \(-0.771952\pi\)
−0.754152 + 0.656700i \(0.771952\pi\)
\(80\) 0 0
\(81\) −10.1231 −1.12479
\(82\) 0 0
\(83\) 11.9473i 1.31139i 0.755026 + 0.655695i \(0.227624\pi\)
−0.755026 + 0.655695i \(0.772376\pi\)
\(84\) 0 0
\(85\) −2.56155 −0.277839
\(86\) 0 0
\(87\) 2.84329i 0.304832i
\(88\) 0 0
\(89\) 14.0877i 1.49329i −0.665223 0.746644i \(-0.731664\pi\)
0.665223 0.746644i \(-0.268336\pi\)
\(90\) 0 0
\(91\) −17.8324 −1.86934
\(92\) 0 0
\(93\) 3.12311 0.323851
\(94\) 0 0
\(95\) −9.43318 + 5.97366i −0.967824 + 0.612885i
\(96\) 0 0
\(97\) 2.41131i 0.244831i −0.992479 0.122416i \(-0.960936\pi\)
0.992479 0.122416i \(-0.0390641\pi\)
\(98\) 0 0
\(99\) 5.97366i 0.600376i
\(100\) 0 0
\(101\) 8.24621 0.820529 0.410264 0.911967i \(-0.365436\pi\)
0.410264 + 0.911967i \(0.365436\pi\)
\(102\) 0 0
\(103\) 12.0818 1.19045 0.595227 0.803557i \(-0.297062\pi\)
0.595227 + 0.803557i \(0.297062\pi\)
\(104\) 0 0
\(105\) 25.0870i 2.44824i
\(106\) 0 0
\(107\) 5.75058 0.555929 0.277965 0.960591i \(-0.410340\pi\)
0.277965 + 0.960591i \(0.410340\pi\)
\(108\) 0 0
\(109\) 12.2050i 1.16902i −0.811385 0.584512i \(-0.801286\pi\)
0.811385 0.584512i \(-0.198714\pi\)
\(110\) 0 0
\(111\) 12.9698i 1.23104i
\(112\) 0 0
\(113\) 5.49966i 0.517364i −0.965963 0.258682i \(-0.916712\pi\)
0.965963 0.258682i \(-0.0832882\pi\)
\(114\) 0 0
\(115\) 4.66410i 0.434929i
\(116\) 0 0
\(117\) 10.9993i 1.01689i
\(118\) 0 0
\(119\) 4.15286i 0.380692i
\(120\) 0 0
\(121\) 5.56155 0.505596
\(122\) 0 0
\(123\) 12.9698i 1.16945i
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) 6.04090 0.536043 0.268021 0.963413i \(-0.413630\pi\)
0.268021 + 0.963413i \(0.413630\pi\)
\(128\) 0 0
\(129\) 3.08835i 0.271914i
\(130\) 0 0
\(131\) 5.97366i 0.521921i −0.965349 0.260961i \(-0.915961\pi\)
0.965349 0.260961i \(-0.0840393\pi\)
\(132\) 0 0
\(133\) 9.68466 + 15.2933i 0.839766 + 1.32610i
\(134\) 0 0
\(135\) 2.64861 0.227956
\(136\) 0 0
\(137\) −12.1231 −1.03575 −0.517873 0.855457i \(-0.673276\pi\)
−0.517873 + 0.855457i \(0.673276\pi\)
\(138\) 0 0
\(139\) 18.9435i 1.60676i −0.595464 0.803382i \(-0.703032\pi\)
0.595464 0.803382i \(-0.296968\pi\)
\(140\) 0 0
\(141\) 16.4990i 1.38946i
\(142\) 0 0
\(143\) −10.0138 −0.837397
\(144\) 0 0
\(145\) 3.08835i 0.256473i
\(146\) 0 0
\(147\) 24.1636 1.99298
\(148\) 0 0
\(149\) −2.80776 −0.230021 −0.115010 0.993364i \(-0.536690\pi\)
−0.115010 + 0.993364i \(0.536690\pi\)
\(150\) 0 0
\(151\) −8.85254 −0.720409 −0.360205 0.932873i \(-0.617293\pi\)
−0.360205 + 0.932873i \(0.617293\pi\)
\(152\) 0 0
\(153\) −2.56155 −0.207089
\(154\) 0 0
\(155\) −3.39228 −0.272475
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 23.0963i 1.83166i
\(160\) 0 0
\(161\) 7.56155 0.595934
\(162\) 0 0
\(163\) 17.6339i 1.38119i 0.723240 + 0.690597i \(0.242652\pi\)
−0.723240 + 0.690597i \(0.757348\pi\)
\(164\) 0 0
\(165\) 14.0877i 1.09672i
\(166\) 0 0
\(167\) −10.7575 −0.832439 −0.416220 0.909264i \(-0.636645\pi\)
−0.416220 + 0.909264i \(0.636645\pi\)
\(168\) 0 0
\(169\) −5.43845 −0.418342
\(170\) 0 0
\(171\) −9.43318 + 5.97366i −0.721373 + 0.456817i
\(172\) 0 0
\(173\) 2.41131i 0.183328i 0.995790 + 0.0916642i \(0.0292186\pi\)
−0.995790 + 0.0916642i \(0.970781\pi\)
\(174\) 0 0
\(175\) 6.48490i 0.490213i
\(176\) 0 0
\(177\) 14.9309 1.12227
\(178\) 0 0
\(179\) 8.10887 0.606085 0.303043 0.952977i \(-0.401998\pi\)
0.303043 + 0.952977i \(0.401998\pi\)
\(180\) 0 0
\(181\) 19.5873i 1.45591i −0.685623 0.727957i \(-0.740470\pi\)
0.685623 0.727957i \(-0.259530\pi\)
\(182\) 0 0
\(183\) 27.5559 2.03699
\(184\) 0 0
\(185\) 14.0877i 1.03575i
\(186\) 0 0
\(187\) 2.33205i 0.170536i
\(188\) 0 0
\(189\) 4.29400i 0.312343i
\(190\) 0 0
\(191\) 7.79447i 0.563988i −0.959416 0.281994i \(-0.909004\pi\)
0.959416 0.281994i \(-0.0909959\pi\)
\(192\) 0 0
\(193\) 5.49966i 0.395874i 0.980215 + 0.197937i \(0.0634241\pi\)
−0.980215 + 0.197937i \(0.936576\pi\)
\(194\) 0 0
\(195\) 25.9396i 1.85757i
\(196\) 0 0
\(197\) 22.4924 1.60252 0.801259 0.598317i \(-0.204164\pi\)
0.801259 + 0.598317i \(0.204164\pi\)
\(198\) 0 0
\(199\) 5.17534i 0.366870i 0.983032 + 0.183435i \(0.0587217\pi\)
−0.983032 + 0.183435i \(0.941278\pi\)
\(200\) 0 0
\(201\) −0.684658 −0.0482921
\(202\) 0 0
\(203\) −5.00691 −0.351416
\(204\) 0 0
\(205\) 14.0877i 0.983925i
\(206\) 0 0
\(207\) 4.66410i 0.324177i
\(208\) 0 0
\(209\) 5.43845 + 8.58800i 0.376185 + 0.594045i
\(210\) 0 0
\(211\) 25.1976 1.73467 0.867336 0.497723i \(-0.165830\pi\)
0.867336 + 0.497723i \(0.165830\pi\)
\(212\) 0 0
\(213\) −4.87689 −0.334159
\(214\) 0 0
\(215\) 3.35453i 0.228777i
\(216\) 0 0
\(217\) 5.49966i 0.373341i
\(218\) 0 0
\(219\) 0.290319 0.0196180
\(220\) 0 0
\(221\) 4.29400i 0.288846i
\(222\) 0 0
\(223\) 22.2586 1.49055 0.745274 0.666758i \(-0.232319\pi\)
0.745274 + 0.666758i \(0.232319\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −24.6169 −1.63388 −0.816942 0.576720i \(-0.804332\pi\)
−0.816942 + 0.576720i \(0.804332\pi\)
\(228\) 0 0
\(229\) 6.56155 0.433600 0.216800 0.976216i \(-0.430438\pi\)
0.216800 + 0.976216i \(0.430438\pi\)
\(230\) 0 0
\(231\) 22.8393 1.50271
\(232\) 0 0
\(233\) 10.8078 0.708040 0.354020 0.935238i \(-0.384814\pi\)
0.354020 + 0.935238i \(0.384814\pi\)
\(234\) 0 0
\(235\) 17.9210i 1.16904i
\(236\) 0 0
\(237\) 31.6155 2.05365
\(238\) 0 0
\(239\) 9.83943i 0.636460i −0.948014 0.318230i \(-0.896912\pi\)
0.948014 0.318230i \(-0.103088\pi\)
\(240\) 0 0
\(241\) 22.6757i 1.46067i 0.683090 + 0.730334i \(0.260635\pi\)
−0.683090 + 0.730334i \(0.739365\pi\)
\(242\) 0 0
\(243\) 20.7713 1.33248
\(244\) 0 0
\(245\) −26.2462 −1.67681
\(246\) 0 0
\(247\) 10.0138 + 15.8131i 0.637164 + 1.00616i
\(248\) 0 0
\(249\) 28.1753i 1.78554i
\(250\) 0 0
\(251\) 21.5626i 1.36102i 0.732739 + 0.680510i \(0.238242\pi\)
−0.732739 + 0.680510i \(0.761758\pi\)
\(252\) 0 0
\(253\) 4.24621 0.266957
\(254\) 0 0
\(255\) 6.04090 0.378296
\(256\) 0 0
\(257\) 17.1760i 1.07141i −0.844405 0.535705i \(-0.820046\pi\)
0.844405 0.535705i \(-0.179954\pi\)
\(258\) 0 0
\(259\) 22.8393 1.41916
\(260\) 0 0
\(261\) 3.08835i 0.191164i
\(262\) 0 0
\(263\) 14.2794i 0.880504i −0.897874 0.440252i \(-0.854889\pi\)
0.897874 0.440252i \(-0.145111\pi\)
\(264\) 0 0
\(265\) 25.0870i 1.54108i
\(266\) 0 0
\(267\) 33.2228i 2.03321i
\(268\) 0 0
\(269\) 5.49966i 0.335320i −0.985845 0.167660i \(-0.946379\pi\)
0.985845 0.167660i \(-0.0536211\pi\)
\(270\) 0 0
\(271\) 22.0738i 1.34089i 0.741959 + 0.670445i \(0.233897\pi\)
−0.741959 + 0.670445i \(0.766103\pi\)
\(272\) 0 0
\(273\) 42.0540 2.54522
\(274\) 0 0
\(275\) 3.64162i 0.219598i
\(276\) 0 0
\(277\) −13.0540 −0.784337 −0.392169 0.919893i \(-0.628275\pi\)
−0.392169 + 0.919893i \(0.628275\pi\)
\(278\) 0 0
\(279\) −3.39228 −0.203091
\(280\) 0 0
\(281\) 28.1753i 1.68080i 0.541968 + 0.840399i \(0.317679\pi\)
−0.541968 + 0.840399i \(0.682321\pi\)
\(282\) 0 0
\(283\) 5.97366i 0.355097i −0.984112 0.177549i \(-0.943183\pi\)
0.984112 0.177549i \(-0.0568167\pi\)
\(284\) 0 0
\(285\) 22.2462 14.0877i 1.31775 0.834481i
\(286\) 0 0
\(287\) 22.8393 1.34816
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 5.68658i 0.333353i
\(292\) 0 0
\(293\) 4.29400i 0.250858i 0.992103 + 0.125429i \(0.0400308\pi\)
−0.992103 + 0.125429i \(0.959969\pi\)
\(294\) 0 0
\(295\) −16.2177 −0.944233
\(296\) 0 0
\(297\) 2.41131i 0.139918i
\(298\) 0 0
\(299\) 7.81855 0.452159
\(300\) 0 0
\(301\) 5.43845 0.313467
\(302\) 0 0
\(303\) −19.4470 −1.11720
\(304\) 0 0
\(305\) −29.9309 −1.71384
\(306\) 0 0
\(307\) 13.4061 0.765126 0.382563 0.923929i \(-0.375041\pi\)
0.382563 + 0.923929i \(0.375041\pi\)
\(308\) 0 0
\(309\) −28.4924 −1.62088
\(310\) 0 0
\(311\) 4.15286i 0.235487i −0.993044 0.117743i \(-0.962434\pi\)
0.993044 0.117743i \(-0.0375660\pi\)
\(312\) 0 0
\(313\) −0.438447 −0.0247825 −0.0123913 0.999923i \(-0.503944\pi\)
−0.0123913 + 0.999923i \(0.503944\pi\)
\(314\) 0 0
\(315\) 27.2492i 1.53532i
\(316\) 0 0
\(317\) 4.29400i 0.241175i −0.992703 0.120588i \(-0.961522\pi\)
0.992703 0.120588i \(-0.0384779\pi\)
\(318\) 0 0
\(319\) −2.81164 −0.157422
\(320\) 0 0
\(321\) −13.5616 −0.756932
\(322\) 0 0
\(323\) 3.68260 2.33205i 0.204905 0.129759i
\(324\) 0 0
\(325\) 6.70531i 0.371944i
\(326\) 0 0
\(327\) 28.7829i 1.59170i
\(328\) 0 0
\(329\) −29.0540 −1.60180
\(330\) 0 0
\(331\) 23.8733 1.31219 0.656097 0.754677i \(-0.272206\pi\)
0.656097 + 0.754677i \(0.272206\pi\)
\(332\) 0 0
\(333\) 14.0877i 0.771999i
\(334\) 0 0
\(335\) 0.743668 0.0406309
\(336\) 0 0
\(337\) 6.17669i 0.336466i −0.985747 0.168233i \(-0.946194\pi\)
0.985747 0.168233i \(-0.0538061\pi\)
\(338\) 0 0
\(339\) 12.9698i 0.704423i
\(340\) 0 0
\(341\) 3.08835i 0.167243i
\(342\) 0 0
\(343\) 13.4810i 0.727908i
\(344\) 0 0
\(345\) 10.9993i 0.592183i
\(346\) 0 0
\(347\) 29.8683i 1.60342i −0.597716 0.801708i \(-0.703925\pi\)
0.597716 0.801708i \(-0.296075\pi\)
\(348\) 0 0
\(349\) −9.05398 −0.484648 −0.242324 0.970195i \(-0.577910\pi\)
−0.242324 + 0.970195i \(0.577910\pi\)
\(350\) 0 0
\(351\) 4.43994i 0.236987i
\(352\) 0 0
\(353\) 18.6847 0.994484 0.497242 0.867612i \(-0.334346\pi\)
0.497242 + 0.867612i \(0.334346\pi\)
\(354\) 0 0
\(355\) 5.29723 0.281148
\(356\) 0 0
\(357\) 9.79366i 0.518335i
\(358\) 0 0
\(359\) 6.19782i 0.327108i −0.986534 0.163554i \(-0.947704\pi\)
0.986534 0.163554i \(-0.0522958\pi\)
\(360\) 0 0
\(361\) 8.12311 17.1760i 0.427532 0.904000i
\(362\) 0 0
\(363\) −13.1158 −0.688400
\(364\) 0 0
\(365\) −0.315342 −0.0165057
\(366\) 0 0
\(367\) 1.02248i 0.0533730i −0.999644 0.0266865i \(-0.991504\pi\)
0.999644 0.0266865i \(-0.00849559\pi\)
\(368\) 0 0
\(369\) 14.0877i 0.733374i
\(370\) 0 0
\(371\) −40.6716 −2.11157
\(372\) 0 0
\(373\) 6.70531i 0.347188i −0.984817 0.173594i \(-0.944462\pi\)
0.984817 0.173594i \(-0.0555380\pi\)
\(374\) 0 0
\(375\) 20.7713 1.07263
\(376\) 0 0
\(377\) −5.17708 −0.266633
\(378\) 0 0
\(379\) −15.7644 −0.809762 −0.404881 0.914369i \(-0.632687\pi\)
−0.404881 + 0.914369i \(0.632687\pi\)
\(380\) 0 0
\(381\) −14.2462 −0.729856
\(382\) 0 0
\(383\) −22.2586 −1.13736 −0.568682 0.822558i \(-0.692546\pi\)
−0.568682 + 0.822558i \(0.692546\pi\)
\(384\) 0 0
\(385\) −24.8078 −1.26432
\(386\) 0 0
\(387\) 3.35453i 0.170520i
\(388\) 0 0
\(389\) −14.8078 −0.750783 −0.375392 0.926866i \(-0.622492\pi\)
−0.375392 + 0.926866i \(0.622492\pi\)
\(390\) 0 0
\(391\) 1.82081i 0.0920822i
\(392\) 0 0
\(393\) 14.0877i 0.710628i
\(394\) 0 0
\(395\) −34.3404 −1.72785
\(396\) 0 0
\(397\) 3.93087 0.197285 0.0986423 0.995123i \(-0.468550\pi\)
0.0986423 + 0.995123i \(0.468550\pi\)
\(398\) 0 0
\(399\) −22.8393 36.0661i −1.14339 1.80557i
\(400\) 0 0
\(401\) 7.91096i 0.395055i 0.980297 + 0.197527i \(0.0632911\pi\)
−0.980297 + 0.197527i \(0.936709\pi\)
\(402\) 0 0
\(403\) 5.68658i 0.283269i
\(404\) 0 0
\(405\) −25.9309 −1.28852
\(406\) 0 0
\(407\) 12.8255 0.635734
\(408\) 0 0
\(409\) 27.4983i 1.35970i −0.733350 0.679851i \(-0.762044\pi\)
0.733350 0.679851i \(-0.237956\pi\)
\(410\) 0 0
\(411\) 28.5899 1.41023
\(412\) 0 0
\(413\) 26.2926i 1.29378i
\(414\) 0 0
\(415\) 30.6037i 1.50228i
\(416\) 0 0
\(417\) 44.6743i 2.18771i
\(418\) 0 0
\(419\) 39.9319i 1.95080i −0.220440 0.975401i \(-0.570749\pi\)
0.220440 0.975401i \(-0.429251\pi\)
\(420\) 0 0
\(421\) 23.2043i 1.13091i 0.824780 + 0.565454i \(0.191299\pi\)
−0.824780 + 0.565454i \(0.808701\pi\)
\(422\) 0 0
\(423\) 17.9210i 0.871348i
\(424\) 0 0
\(425\) −1.56155 −0.0757464
\(426\) 0 0
\(427\) 48.5247i 2.34827i
\(428\) 0 0
\(429\) 23.6155 1.14017
\(430\) 0 0
\(431\) −5.29723 −0.255158 −0.127579 0.991828i \(-0.540721\pi\)
−0.127579 + 0.991828i \(0.540721\pi\)
\(432\) 0 0
\(433\) 18.9103i 0.908770i −0.890805 0.454385i \(-0.849859\pi\)
0.890805 0.454385i \(-0.150141\pi\)
\(434\) 0 0
\(435\) 7.28323i 0.349204i
\(436\) 0 0
\(437\) −4.24621 6.70531i −0.203124 0.320758i
\(438\) 0 0
\(439\) 19.6100 0.935935 0.467968 0.883746i \(-0.344986\pi\)
0.467968 + 0.883746i \(0.344986\pi\)
\(440\) 0 0
\(441\) −26.2462 −1.24982
\(442\) 0 0
\(443\) 12.2344i 0.581275i 0.956833 + 0.290637i \(0.0938673\pi\)
−0.956833 + 0.290637i \(0.906133\pi\)
\(444\) 0 0
\(445\) 36.0863i 1.71065i
\(446\) 0 0
\(447\) 6.62153 0.313188
\(448\) 0 0
\(449\) 14.7647i 0.696789i 0.937348 + 0.348395i \(0.113273\pi\)
−0.937348 + 0.348395i \(0.886727\pi\)
\(450\) 0 0
\(451\) 12.8255 0.603927
\(452\) 0 0
\(453\) 20.8769 0.980882
\(454\) 0 0
\(455\) −45.6786 −2.14144
\(456\) 0 0
\(457\) 3.87689 0.181353 0.0906767 0.995880i \(-0.471097\pi\)
0.0906767 + 0.995880i \(0.471097\pi\)
\(458\) 0 0
\(459\) −1.03399 −0.0482624
\(460\) 0 0
\(461\) −31.6847 −1.47570 −0.737851 0.674964i \(-0.764159\pi\)
−0.737851 + 0.674964i \(0.764159\pi\)
\(462\) 0 0
\(463\) 16.3243i 0.758656i −0.925262 0.379328i \(-0.876155\pi\)
0.925262 0.379328i \(-0.123845\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 2.33205i 0.107914i −0.998543 0.0539572i \(-0.982817\pi\)
0.998543 0.0539572i \(-0.0171834\pi\)
\(468\) 0 0
\(469\) 1.20565i 0.0556719i
\(470\) 0 0
\(471\) 14.1498 0.651987
\(472\) 0 0
\(473\) 3.05398 0.140422
\(474\) 0 0
\(475\) −5.75058 + 3.64162i −0.263855 + 0.167089i
\(476\) 0 0
\(477\) 25.0870i 1.14865i
\(478\) 0 0
\(479\) 41.5286i 1.89749i −0.316045 0.948744i \(-0.602355\pi\)
0.316045 0.948744i \(-0.397645\pi\)
\(480\) 0 0
\(481\) 23.6155 1.07678
\(482\) 0 0
\(483\) −17.8324 −0.811401
\(484\) 0 0
\(485\) 6.17669i 0.280469i
\(486\) 0 0
\(487\) 18.8664 0.854916 0.427458 0.904035i \(-0.359409\pi\)
0.427458 + 0.904035i \(0.359409\pi\)
\(488\) 0 0
\(489\) 41.5859i 1.88058i
\(490\) 0 0
\(491\) 21.2755i 0.960151i −0.877227 0.480075i \(-0.840609\pi\)
0.877227 0.480075i \(-0.159391\pi\)
\(492\) 0 0
\(493\) 1.20565i 0.0542999i
\(494\) 0 0
\(495\) 15.3019i 0.687767i
\(496\) 0 0
\(497\) 8.58800i 0.385225i
\(498\) 0 0
\(499\) 1.30957i 0.0586243i 0.999570 + 0.0293122i \(0.00933169\pi\)
−0.999570 + 0.0293122i \(0.990668\pi\)
\(500\) 0 0
\(501\) 25.3693 1.13342
\(502\) 0 0
\(503\) 16.3873i 0.730672i 0.930876 + 0.365336i \(0.119046\pi\)
−0.930876 + 0.365336i \(0.880954\pi\)
\(504\) 0 0
\(505\) 21.1231 0.939966
\(506\) 0 0
\(507\) 12.8255 0.569599
\(508\) 0 0
\(509\) 8.58800i 0.380657i 0.981721 + 0.190328i \(0.0609552\pi\)
−0.981721 + 0.190328i \(0.939045\pi\)
\(510\) 0 0
\(511\) 0.511240i 0.0226159i
\(512\) 0 0
\(513\) −3.80776 + 2.41131i −0.168117 + 0.106462i
\(514\) 0 0
\(515\) 30.9481 1.36374
\(516\) 0 0
\(517\) −16.3153 −0.717548
\(518\) 0 0
\(519\) 5.68658i 0.249613i
\(520\) 0 0
\(521\) 22.6757i 0.993439i −0.867911 0.496719i \(-0.834538\pi\)
0.867911 0.496719i \(-0.165462\pi\)
\(522\) 0 0
\(523\) −2.19526 −0.0959922 −0.0479961 0.998848i \(-0.515284\pi\)
−0.0479961 + 0.998848i \(0.515284\pi\)
\(524\) 0 0
\(525\) 15.2933i 0.667455i
\(526\) 0 0
\(527\) 1.32431 0.0576877
\(528\) 0 0
\(529\) 19.6847 0.855855
\(530\) 0 0
\(531\) −16.2177 −0.703790
\(532\) 0 0
\(533\) 23.6155 1.02290
\(534\) 0 0
\(535\) 14.7304 0.636851
\(536\) 0 0
\(537\) −19.1231 −0.825223
\(538\) 0 0
\(539\) 23.8947i 1.02922i
\(540\) 0 0
\(541\) 32.8078 1.41052 0.705258 0.708951i \(-0.250831\pi\)
0.705258 + 0.708951i \(0.250831\pi\)
\(542\) 0 0
\(543\) 46.1927i 1.98232i
\(544\) 0 0
\(545\) 31.2637i 1.33919i
\(546\) 0 0
\(547\) −0.580639 −0.0248263 −0.0124132 0.999923i \(-0.503951\pi\)
−0.0124132 + 0.999923i \(0.503951\pi\)
\(548\) 0 0
\(549\) −29.9309 −1.27742
\(550\) 0 0
\(551\) 2.81164 + 4.43994i 0.119780 + 0.189148i
\(552\) 0 0
\(553\) 55.6736i 2.36748i
\(554\) 0 0
\(555\) 33.2228i 1.41023i
\(556\) 0 0
\(557\) −9.93087 −0.420784 −0.210392 0.977617i \(-0.567474\pi\)
−0.210392 + 0.977617i \(0.567474\pi\)
\(558\) 0 0
\(559\) 5.62329 0.237840
\(560\) 0 0
\(561\) 5.49966i 0.232196i
\(562\) 0 0
\(563\) −21.3519 −0.899877 −0.449938 0.893060i \(-0.648554\pi\)
−0.449938 + 0.893060i \(0.648554\pi\)
\(564\) 0 0
\(565\) 14.0877i 0.592672i
\(566\) 0 0
\(567\) 42.0398i 1.76551i
\(568\) 0 0
\(569\) 41.5859i 1.74337i 0.490064 + 0.871687i \(0.336973\pi\)
−0.490064 + 0.871687i \(0.663027\pi\)
\(570\) 0 0
\(571\) 15.5889i 0.652377i −0.945305 0.326188i \(-0.894236\pi\)
0.945305 0.326188i \(-0.105764\pi\)
\(572\) 0 0
\(573\) 18.3817i 0.767905i
\(574\) 0 0
\(575\) 2.84329i 0.118573i
\(576\) 0 0
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) 0 0
\(579\) 12.9698i 0.539007i
\(580\) 0 0
\(581\) 49.6155 2.05840
\(582\) 0 0
\(583\) −22.8393 −0.945906
\(584\) 0 0
\(585\) 28.1753i 1.16491i
\(586\) 0 0
\(587\) 5.97366i 0.246559i −0.992372 0.123280i \(-0.960659\pi\)
0.992372 0.123280i \(-0.0393412\pi\)
\(588\) 0 0
\(589\) 4.87689 3.08835i 0.200949 0.127253i
\(590\) 0 0
\(591\) −53.0438 −2.18193
\(592\) 0 0
\(593\) 12.7386 0.523113 0.261556 0.965188i \(-0.415764\pi\)
0.261556 + 0.965188i \(0.415764\pi\)
\(594\) 0 0
\(595\) 10.6378i 0.436106i
\(596\) 0 0
\(597\) 12.2050i 0.499516i
\(598\) 0 0
\(599\) −24.9073 −1.01768 −0.508841 0.860860i \(-0.669926\pi\)
−0.508841 + 0.860860i \(0.669926\pi\)
\(600\) 0 0
\(601\) 35.4092i 1.44437i 0.691698 + 0.722187i \(0.256863\pi\)
−0.691698 + 0.722187i \(0.743137\pi\)
\(602\) 0 0
\(603\) 0.743668 0.0302845
\(604\) 0 0
\(605\) 14.2462 0.579191
\(606\) 0 0
\(607\) 31.5288 1.27971 0.639857 0.768494i \(-0.278994\pi\)
0.639857 + 0.768494i \(0.278994\pi\)
\(608\) 0 0
\(609\) 11.8078 0.478475
\(610\) 0 0
\(611\) −30.0414 −1.21535
\(612\) 0 0
\(613\) 29.3002 1.18342 0.591712 0.806150i \(-0.298452\pi\)
0.591712 + 0.806150i \(0.298452\pi\)
\(614\) 0 0
\(615\) 33.2228i 1.33967i
\(616\) 0 0
\(617\) 41.5464 1.67259 0.836297 0.548276i \(-0.184716\pi\)
0.836297 + 0.548276i \(0.184716\pi\)
\(618\) 0 0
\(619\) 6.70906i 0.269660i −0.990869 0.134830i \(-0.956951\pi\)
0.990869 0.134830i \(-0.0430488\pi\)
\(620\) 0 0
\(621\) 1.88269i 0.0755499i
\(622\) 0 0
\(623\) −58.5040 −2.34391
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) −12.8255 20.2530i −0.512200 0.808828i
\(628\) 0 0
\(629\) 5.49966i 0.219286i
\(630\) 0 0
\(631\) 9.61528i 0.382778i 0.981514 + 0.191389i \(0.0612992\pi\)
−0.981514 + 0.191389i \(0.938701\pi\)
\(632\) 0 0
\(633\) −59.4233 −2.36186
\(634\) 0 0
\(635\) 15.4741 0.614070
\(636\) 0 0
\(637\) 43.9972i 1.74323i
\(638\) 0 0
\(639\) 5.29723 0.209555
\(640\) 0 0
\(641\) 26.8212i 1.05938i −0.848193 0.529688i \(-0.822309\pi\)
0.848193 0.529688i \(-0.177691\pi\)
\(642\) 0 0
\(643\) 14.2794i 0.563124i 0.959543 + 0.281562i \(0.0908524\pi\)
−0.959543 + 0.281562i \(0.909148\pi\)
\(644\) 0 0
\(645\) 7.91096i 0.311494i
\(646\) 0 0
\(647\) 32.1374i 1.26345i −0.775191 0.631726i \(-0.782347\pi\)
0.775191 0.631726i \(-0.217653\pi\)
\(648\) 0 0
\(649\) 14.7647i 0.579565i
\(650\) 0 0
\(651\) 12.9698i 0.508327i
\(652\) 0 0
\(653\) 25.1922 0.985848 0.492924 0.870072i \(-0.335928\pi\)
0.492924 + 0.870072i \(0.335928\pi\)
\(654\) 0 0
\(655\) 15.3019i 0.597893i
\(656\) 0 0
\(657\) −0.315342 −0.0123026
\(658\) 0 0
\(659\) −41.9960 −1.63593 −0.817965 0.575268i \(-0.804898\pi\)
−0.817965 + 0.575268i \(0.804898\pi\)
\(660\) 0 0
\(661\) 37.2919i 1.45049i 0.688492 + 0.725244i \(0.258273\pi\)
−0.688492 + 0.725244i \(0.741727\pi\)
\(662\) 0 0
\(663\) 10.1265i 0.393281i
\(664\) 0 0
\(665\) 24.8078 + 39.1746i 0.962004 + 1.51913i
\(666\) 0 0
\(667\) 2.19526 0.0850010
\(668\) 0 0
\(669\) −52.4924 −2.02947
\(670\) 0 0
\(671\) 27.2492i 1.05194i
\(672\) 0 0
\(673\) 24.4099i 0.940934i 0.882418 + 0.470467i \(0.155914\pi\)
−0.882418 + 0.470467i \(0.844086\pi\)
\(674\) 0 0
\(675\) 1.61463 0.0621470
\(676\) 0 0
\(677\) 40.3803i 1.55194i 0.630770 + 0.775970i \(0.282739\pi\)
−0.630770 + 0.775970i \(0.717261\pi\)
\(678\) 0 0
\(679\) −10.0138 −0.384295
\(680\) 0 0
\(681\) 58.0540 2.22463
\(682\) 0 0
\(683\) 31.5288 1.20642 0.603208 0.797584i \(-0.293889\pi\)
0.603208 + 0.797584i \(0.293889\pi\)
\(684\) 0 0
\(685\) −31.0540 −1.18651
\(686\) 0 0
\(687\) −15.4741 −0.590373
\(688\) 0 0
\(689\) −42.0540 −1.60213
\(690\) 0 0
\(691\) 8.01862i 0.305043i −0.988300 0.152521i \(-0.951261\pi\)
0.988300 0.152521i \(-0.0487393\pi\)
\(692\) 0 0
\(693\) −24.8078 −0.942369
\(694\) 0 0
\(695\) 48.5247i 1.84065i
\(696\) 0 0
\(697\) 5.49966i 0.208314i
\(698\) 0 0
\(699\) −25.4879 −0.964041
\(700\) 0 0
\(701\) −4.63068 −0.174898 −0.0874492 0.996169i \(-0.527872\pi\)
−0.0874492 + 0.996169i \(0.527872\pi\)
\(702\) 0 0
\(703\) −12.8255 20.2530i −0.483721 0.763858i
\(704\) 0 0
\(705\) 42.2630i 1.59172i
\(706\) 0 0
\(707\) 34.2453i 1.28793i
\(708\) 0 0
\(709\) −12.6307 −0.474355 −0.237178 0.971466i \(-0.576222\pi\)
−0.237178 + 0.971466i \(0.576222\pi\)
\(710\) 0 0
\(711\) −34.3404 −1.28787
\(712\) 0 0
\(713\) 2.41131i 0.0903042i
\(714\) 0 0
\(715\) −25.6509 −0.959290
\(716\) 0 0
\(717\) 23.2043i 0.866580i
\(718\) 0 0
\(719\) 26.4509i 0.986450i −0.869902 0.493225i \(-0.835818\pi\)
0.869902 0.493225i \(-0.164182\pi\)
\(720\) 0 0
\(721\) 50.1739i 1.86858i
\(722\) 0 0
\(723\) 53.4759i 1.98879i
\(724\) 0 0
\(725\) 1.88269i 0.0699215i
\(726\) 0 0
\(727\) 44.6589i 1.65631i −0.560500 0.828154i \(-0.689391\pi\)
0.560500 0.828154i \(-0.310609\pi\)
\(728\) 0 0
\(729\) −18.6155 −0.689464
\(730\) 0 0
\(731\) 1.30957i 0.0484361i
\(732\) 0 0
\(733\) 5.12311 0.189226 0.0946131 0.995514i \(-0.469839\pi\)
0.0946131 + 0.995514i \(0.469839\pi\)
\(734\) 0 0
\(735\) 61.8963 2.28308
\(736\) 0 0
\(737\) 0.677039i 0.0249390i
\(738\) 0 0
\(739\) 25.2042i 0.927152i 0.886057 + 0.463576i \(0.153434\pi\)
−0.886057 + 0.463576i \(0.846566\pi\)
\(740\) 0 0
\(741\) −23.6155 37.2919i −0.867538 1.36995i
\(742\) 0 0
\(743\) −11.9188 −0.437257 −0.218628 0.975808i \(-0.570158\pi\)
−0.218628 + 0.975808i \(0.570158\pi\)
\(744\) 0 0
\(745\) −7.19224 −0.263503
\(746\) 0 0
\(747\) 30.6037i 1.11973i
\(748\) 0 0
\(749\) 23.8813i 0.872604i
\(750\) 0 0
\(751\) 35.6647 1.30142 0.650712 0.759324i \(-0.274470\pi\)
0.650712 + 0.759324i \(0.274470\pi\)
\(752\) 0 0
\(753\) 50.8510i 1.85311i
\(754\) 0 0
\(755\) −22.6762 −0.825273
\(756\) 0 0
\(757\) 44.8078 1.62857 0.814283 0.580468i \(-0.197130\pi\)
0.814283 + 0.580468i \(0.197130\pi\)
\(758\) 0 0
\(759\) −10.0138 −0.363479
\(760\) 0 0
\(761\) 30.6155 1.10981 0.554906 0.831913i \(-0.312754\pi\)
0.554906 + 0.831913i \(0.312754\pi\)
\(762\) 0 0
\(763\) −50.6855 −1.83494
\(764\) 0 0
\(765\) −6.56155 −0.237233
\(766\) 0 0
\(767\) 27.1862i 0.981638i
\(768\) 0 0
\(769\) −28.2311 −1.01804 −0.509019 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(770\) 0 0
\(771\) 40.5061i 1.45879i
\(772\) 0 0
\(773\) 41.0573i 1.47673i −0.674402 0.738365i \(-0.735598\pi\)
0.674402 0.738365i \(-0.264402\pi\)
\(774\) 0 0
\(775\) −2.06798 −0.0742839
\(776\) 0 0
\(777\) −53.8617 −1.93228
\(778\) 0 0
\(779\) −12.8255 20.2530i −0.459520 0.725640i
\(780\) 0 0
\(781\) 4.82262i 0.172567i
\(782\) 0 0
\(783\) 1.24663i 0.0445510i
\(784\) 0 0
\(785\) −15.3693 −0.548554
\(786\) 0 0
\(787\) 26.3588 0.939591 0.469796 0.882775i \(-0.344328\pi\)
0.469796 + 0.882775i \(0.344328\pi\)
\(788\) 0 0
\(789\) 33.6750i 1.19886i
\(790\) 0 0
\(791\) −22.8393 −0.812071
\(792\) 0 0
\(793\) 50.1739i 1.78173i
\(794\) 0 0
\(795\) 59.1625i 2.09828i
\(796\) 0 0
\(797\) 23.2043i 0.821938i −0.911649 0.410969i \(-0.865191\pi\)
0.911649 0.410969i \(-0.134809\pi\)
\(798\) 0 0
\(799\) 6.99614i 0.247506i
\(800\) 0 0
\(801\) 36.0863i 1.27505i
\(802\) 0 0
\(803\) 0.287088i 0.0101311i
\(804\) 0 0
\(805\) 19.3693 0.682679
\(806\) 0 0
\(807\) 12.9698i 0.456559i
\(808\) 0 0
\(809\) 44.3693 1.55994 0.779971 0.625816i \(-0.215234\pi\)
0.779971 + 0.625816i \(0.215234\pi\)
\(810\) 0 0
\(811\) 29.9142 1.05043 0.525214 0.850970i \(-0.323985\pi\)
0.525214 + 0.850970i \(0.323985\pi\)
\(812\) 0 0
\(813\) 52.0566i 1.82571i
\(814\) 0 0
\(815\) 45.1702i 1.58224i
\(816\) 0 0
\(817\) −3.05398 4.82262i −0.106845 0.168722i
\(818\) 0 0
\(819\) −45.6786 −1.59614
\(820\) 0 0
\(821\) 28.4233 0.991980 0.495990 0.868328i \(-0.334805\pi\)
0.495990 + 0.868328i \(0.334805\pi\)
\(822\) 0 0
\(823\) 31.1150i 1.08460i 0.840185 + 0.542299i \(0.182446\pi\)
−0.840185 + 0.542299i \(0.817554\pi\)
\(824\) 0 0
\(825\) 8.58800i 0.298996i
\(826\) 0 0
\(827\) 49.5242 1.72212 0.861062 0.508499i \(-0.169800\pi\)
0.861062 + 0.508499i \(0.169800\pi\)
\(828\) 0 0
\(829\) 43.4686i 1.50973i −0.655881 0.754864i \(-0.727703\pi\)
0.655881 0.754864i \(-0.272297\pi\)
\(830\) 0 0
\(831\) 30.7851 1.06792
\(832\) 0 0
\(833\) 10.2462 0.355010
\(834\) 0 0
\(835\) −27.5559 −0.953610
\(836\) 0 0
\(837\) −1.36932 −0.0473305
\(838\) 0 0
\(839\) −15.8917 −0.548642 −0.274321 0.961638i \(-0.588453\pi\)
−0.274321 + 0.961638i \(0.588453\pi\)
\(840\) 0 0
\(841\) 27.5464 0.949876
\(842\) 0 0
\(843\) 66.4457i 2.28851i
\(844\) 0 0
\(845\) −13.9309 −0.479236
\(846\) 0 0
\(847\) 23.0963i 0.793599i
\(848\) 0 0
\(849\) 14.0877i 0.483487i
\(850\) 0 0
\(851\) −10.0138 −0.343269
\(852\) 0 0
\(853\) −31.3693 −1.07406 −0.537032 0.843562i \(-0.680455\pi\)
−0.537032 + 0.843562i \(0.680455\pi\)
\(854\) 0 0
\(855\) −24.1636 + 15.3019i −0.826377 + 0.523312i
\(856\) 0 0
\(857\) 39.1746i 1.33818i −0.743181 0.669090i \(-0.766684\pi\)
0.743181 0.669090i \(-0.233316\pi\)
\(858\) 0 0
\(859\) 45.9056i 1.56628i 0.621847 + 0.783139i \(0.286383\pi\)
−0.621847 + 0.783139i \(0.713617\pi\)
\(860\) 0 0
\(861\) −53.8617 −1.83560
\(862\) 0 0
\(863\) 23.4199 0.797223 0.398612 0.917120i \(-0.369492\pi\)
0.398612 + 0.917120i \(0.369492\pi\)
\(864\) 0 0
\(865\) 6.17669i 0.210014i
\(866\) 0 0
\(867\) 37.7327 1.28147
\(868\) 0 0
\(869\) 31.2637i 1.06055i
\(870\) 0 0
\(871\) 1.24663i 0.0422405i
\(872\) 0 0
\(873\) 6.17669i 0.209049i
\(874\) 0 0
\(875\) 36.5774i 1.23654i
\(876\) 0 0
\(877\) 12.8820i 0.434994i 0.976061 + 0.217497i \(0.0697893\pi\)
−0.976061 + 0.217497i \(0.930211\pi\)
\(878\) 0 0
\(879\) 10.1265i 0.341559i
\(880\) 0 0
\(881\) −10.1771 −0.342875 −0.171437 0.985195i \(-0.554841\pi\)
−0.171437 + 0.985195i \(0.554841\pi\)
\(882\) 0 0
\(883\) 48.5247i 1.63299i −0.577355 0.816493i \(-0.695915\pi\)
0.577355 0.816493i \(-0.304085\pi\)
\(884\) 0 0
\(885\) 38.2462 1.28563
\(886\) 0 0
\(887\) −23.0023 −0.772342 −0.386171 0.922427i \(-0.626203\pi\)
−0.386171 + 0.922427i \(0.626203\pi\)
\(888\) 0 0
\(889\) 25.0870i 0.841390i
\(890\) 0 0
\(891\) 23.6076i 0.790883i
\(892\) 0 0
\(893\) 16.3153 + 25.7640i 0.545972 + 0.862160i
\(894\) 0 0
\(895\) 20.7713 0.694308
\(896\) 0 0
\(897\) −18.4384 −0.615642
\(898\) 0 0
\(899\) 1.59666i 0.0532515i
\(900\) 0 0
\(901\) 9.79366i 0.326274i
\(902\) 0 0
\(903\) −12.8255 −0.426805
\(904\) 0 0
\(905\) 50.1739i 1.66784i
\(906\) 0 0
\(907\) 25.0345 0.831258 0.415629 0.909534i \(-0.363561\pi\)
0.415629 + 0.909534i \(0.363561\pi\)
\(908\) 0 0
\(909\) 21.1231 0.700609
\(910\) 0 0
\(911\) 40.3813 1.33789 0.668946 0.743311i \(-0.266746\pi\)
0.668946 + 0.743311i \(0.266746\pi\)
\(912\) 0 0
\(913\) 27.8617 0.922089
\(914\) 0 0
\(915\) 70.5858 2.33349
\(916\) 0 0
\(917\) −24.8078 −0.819225
\(918\) 0 0
\(919\) 24.1188i 0.795606i −0.917471 0.397803i \(-0.869773\pi\)
0.917471 0.397803i \(-0.130227\pi\)
\(920\) 0 0
\(921\) −31.6155 −1.04177
\(922\) 0 0
\(923\) 8.87989i 0.292285i
\(924\) 0 0
\(925\) 8.58800i 0.282372i
\(926\) 0 0
\(927\) 30.9481 1.01647
\(928\) 0 0
\(929\) −13.3153 −0.436862 −0.218431 0.975852i \(-0.570094\pi\)
−0.218431 + 0.975852i \(0.570094\pi\)
\(930\) 0 0
\(931\) 37.7327 23.8947i 1.23664 0.783116i
\(932\) 0 0
\(933\) 9.79366i 0.320630i
\(934\) 0 0
\(935\) 5.97366i 0.195360i
\(936\) 0 0
\(937\) −13.8769 −0.453338 −0.226669 0.973972i \(-0.572784\pi\)
−0.226669 + 0.973972i \(0.572784\pi\)
\(938\) 0 0
\(939\) 1.03399 0.0337429
\(940\) 0 0
\(941\) 48.2912i 1.57425i −0.616794 0.787125i \(-0.711569\pi\)
0.616794 0.787125i \(-0.288431\pi\)
\(942\) 0 0
\(943\) −10.0138 −0.326095
\(944\) 0 0
\(945\) 10.9993i 0.357808i
\(946\) 0 0
\(947\) 21.2755i 0.691361i −0.938352 0.345681i \(-0.887648\pi\)
0.938352 0.345681i \(-0.112352\pi\)
\(948\) 0 0
\(949\) 0.528616i 0.0171596i
\(950\) 0 0
\(951\) 10.1265i 0.328375i
\(952\) 0 0
\(953\) 31.2637i 1.01273i −0.862319 0.506365i \(-0.830989\pi\)
0.862319 0.506365i \(-0.169011\pi\)
\(954\) 0 0
\(955\) 19.9660i 0.646083i
\(956\) 0 0
\(957\) 6.63068 0.214340
\(958\) 0 0
\(959\) 50.3455i 1.62574i
\(960\) 0 0
\(961\) −29.2462 −0.943426
\(962\) 0 0
\(963\) 14.7304 0.474681
\(964\) 0 0
\(965\) 14.0877i 0.453498i
\(966\) 0 0
\(967\) 10.3507i 0.332855i 0.986054 + 0.166428i \(0.0532232\pi\)
−0.986054 + 0.166428i \(0.946777\pi\)
\(968\) 0 0
\(969\) −8.68466 + 5.49966i −0.278991 + 0.176674i
\(970\) 0 0
\(971\) 0.580639 0.0186336 0.00931679 0.999957i \(-0.497034\pi\)
0.00931679 + 0.999957i \(0.497034\pi\)
\(972\) 0 0
\(973\) −78.6695 −2.52203
\(974\) 0 0
\(975\) 15.8131i 0.506424i
\(976\) 0 0
\(977\) 50.8510i 1.62687i 0.581658 + 0.813433i \(0.302404\pi\)
−0.581658 + 0.813433i \(0.697596\pi\)
\(978\) 0 0
\(979\) −32.8531 −1.04999
\(980\) 0 0
\(981\) 31.2637i 0.998172i
\(982\) 0 0
\(983\) −6.78456 −0.216394 −0.108197 0.994129i \(-0.534508\pi\)
−0.108197 + 0.994129i \(0.534508\pi\)
\(984\) 0 0
\(985\) 57.6155 1.83578
\(986\) 0 0
\(987\) 68.5178 2.18095
\(988\) 0 0
\(989\) −2.38447 −0.0758218
\(990\) 0 0
\(991\) 0.417609 0.0132658 0.00663289 0.999978i \(-0.497889\pi\)
0.00663289 + 0.999978i \(0.497889\pi\)
\(992\) 0 0
\(993\) −56.3002 −1.78663
\(994\) 0 0
\(995\) 13.2569i 0.420272i
\(996\) 0 0
\(997\) −41.5464 −1.31579 −0.657894 0.753111i \(-0.728552\pi\)
−0.657894 + 0.753111i \(0.728552\pi\)
\(998\) 0 0
\(999\) 5.68658i 0.179915i
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 1216.2.h.d.1215.1 8
4.3 odd 2 inner 1216.2.h.d.1215.8 8
8.3 odd 2 76.2.d.a.75.4 yes 8
8.5 even 2 76.2.d.a.75.6 yes 8
19.18 odd 2 inner 1216.2.h.d.1215.7 8
24.5 odd 2 684.2.f.b.379.3 8
24.11 even 2 684.2.f.b.379.5 8
76.75 even 2 inner 1216.2.h.d.1215.2 8
152.37 odd 2 76.2.d.a.75.3 8
152.75 even 2 76.2.d.a.75.5 yes 8
456.227 odd 2 684.2.f.b.379.4 8
456.341 even 2 684.2.f.b.379.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.d.a.75.3 8 152.37 odd 2
76.2.d.a.75.4 yes 8 8.3 odd 2
76.2.d.a.75.5 yes 8 152.75 even 2
76.2.d.a.75.6 yes 8 8.5 even 2
684.2.f.b.379.3 8 24.5 odd 2
684.2.f.b.379.4 8 456.227 odd 2
684.2.f.b.379.5 8 24.11 even 2
684.2.f.b.379.6 8 456.341 even 2
1216.2.h.d.1215.1 8 1.1 even 1 trivial
1216.2.h.d.1215.2 8 76.75 even 2 inner
1216.2.h.d.1215.7 8 19.18 odd 2 inner
1216.2.h.d.1215.8 8 4.3 odd 2 inner