Properties

Label 1792.2.b.k.897.4
Level $1792$
Weight $2$
Character 1792.897
Analytic conductor $14.309$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(897,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.897");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 897.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1792.897
Dual form 1792.2.b.k.897.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.23607i q^{3} -1.23607i q^{5} -1.00000 q^{7} -7.47214 q^{9} +O(q^{10})\) \(q+3.23607i q^{3} -1.23607i q^{5} -1.00000 q^{7} -7.47214 q^{9} +2.47214i q^{11} -5.23607i q^{13} +4.00000 q^{15} -4.47214 q^{17} -3.23607i q^{19} -3.23607i q^{21} -4.00000 q^{23} +3.47214 q^{25} -14.4721i q^{27} -4.47214i q^{29} +6.47214 q^{31} -8.00000 q^{33} +1.23607i q^{35} +4.47214i q^{37} +16.9443 q^{39} -0.472136 q^{41} -2.47214i q^{43} +9.23607i q^{45} +1.52786 q^{47} +1.00000 q^{49} -14.4721i q^{51} -10.0000i q^{53} +3.05573 q^{55} +10.4721 q^{57} -4.76393i q^{59} -6.76393i q^{61} +7.47214 q^{63} -6.47214 q^{65} -4.00000i q^{67} -12.9443i q^{69} -12.9443 q^{71} -14.9443 q^{73} +11.2361i q^{75} -2.47214i q^{77} -4.94427 q^{79} +24.4164 q^{81} +4.76393i q^{83} +5.52786i q^{85} +14.4721 q^{87} +6.00000 q^{89} +5.23607i q^{91} +20.9443i q^{93} -4.00000 q^{95} +3.52786 q^{97} -18.4721i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 12 q^{9} + 16 q^{15} - 16 q^{23} - 4 q^{25} + 8 q^{31} - 32 q^{33} + 32 q^{39} + 16 q^{41} + 24 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} + 12 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{73} + 16 q^{79} + 44 q^{81} + 40 q^{87} + 24 q^{89} - 16 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.23607i 1.86834i 0.356822 + 0.934172i \(0.383860\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) − 1.23607i − 0.552786i −0.961045 0.276393i \(-0.910861\pi\)
0.961045 0.276393i \(-0.0891392\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) 2.47214i 0.745377i 0.927957 + 0.372689i \(0.121564\pi\)
−0.927957 + 0.372689i \(0.878436\pi\)
\(12\) 0 0
\(13\) − 5.23607i − 1.45222i −0.687576 0.726112i \(-0.741325\pi\)
0.687576 0.726112i \(-0.258675\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) − 3.23607i − 0.742405i −0.928552 0.371202i \(-0.878946\pi\)
0.928552 0.371202i \(-0.121054\pi\)
\(20\) 0 0
\(21\) − 3.23607i − 0.706168i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 3.47214 0.694427
\(26\) 0 0
\(27\) − 14.4721i − 2.78516i
\(28\) 0 0
\(29\) − 4.47214i − 0.830455i −0.909718 0.415227i \(-0.863702\pi\)
0.909718 0.415227i \(-0.136298\pi\)
\(30\) 0 0
\(31\) 6.47214 1.16243 0.581215 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 0 0
\(33\) −8.00000 −1.39262
\(34\) 0 0
\(35\) 1.23607i 0.208934i
\(36\) 0 0
\(37\) 4.47214i 0.735215i 0.929981 + 0.367607i \(0.119823\pi\)
−0.929981 + 0.367607i \(0.880177\pi\)
\(38\) 0 0
\(39\) 16.9443 2.71325
\(40\) 0 0
\(41\) −0.472136 −0.0737352 −0.0368676 0.999320i \(-0.511738\pi\)
−0.0368676 + 0.999320i \(0.511738\pi\)
\(42\) 0 0
\(43\) − 2.47214i − 0.376997i −0.982073 0.188499i \(-0.939638\pi\)
0.982073 0.188499i \(-0.0603621\pi\)
\(44\) 0 0
\(45\) 9.23607i 1.37683i
\(46\) 0 0
\(47\) 1.52786 0.222862 0.111431 0.993772i \(-0.464457\pi\)
0.111431 + 0.993772i \(0.464457\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 14.4721i − 2.02650i
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 3.05573 0.412034
\(56\) 0 0
\(57\) 10.4721 1.38707
\(58\) 0 0
\(59\) − 4.76393i − 0.620211i −0.950702 0.310106i \(-0.899636\pi\)
0.950702 0.310106i \(-0.100364\pi\)
\(60\) 0 0
\(61\) − 6.76393i − 0.866033i −0.901386 0.433016i \(-0.857449\pi\)
0.901386 0.433016i \(-0.142551\pi\)
\(62\) 0 0
\(63\) 7.47214 0.941401
\(64\) 0 0
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) − 12.9443i − 1.55831i
\(70\) 0 0
\(71\) −12.9443 −1.53620 −0.768101 0.640328i \(-0.778798\pi\)
−0.768101 + 0.640328i \(0.778798\pi\)
\(72\) 0 0
\(73\) −14.9443 −1.74909 −0.874547 0.484940i \(-0.838841\pi\)
−0.874547 + 0.484940i \(0.838841\pi\)
\(74\) 0 0
\(75\) 11.2361i 1.29743i
\(76\) 0 0
\(77\) − 2.47214i − 0.281726i
\(78\) 0 0
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 4.76393i 0.522909i 0.965216 + 0.261455i \(0.0842022\pi\)
−0.965216 + 0.261455i \(0.915798\pi\)
\(84\) 0 0
\(85\) 5.52786i 0.599581i
\(86\) 0 0
\(87\) 14.4721 1.55158
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 5.23607i 0.548889i
\(92\) 0 0
\(93\) 20.9443i 2.17182i
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 3.52786 0.358200 0.179100 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 0 0
\(99\) − 18.4721i − 1.85652i
\(100\) 0 0
\(101\) 11.7082i 1.16501i 0.812827 + 0.582505i \(0.197927\pi\)
−0.812827 + 0.582505i \(0.802073\pi\)
\(102\) 0 0
\(103\) 14.4721 1.42598 0.712991 0.701173i \(-0.247340\pi\)
0.712991 + 0.701173i \(0.247340\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) − 8.94427i − 0.864675i −0.901712 0.432338i \(-0.857689\pi\)
0.901712 0.432338i \(-0.142311\pi\)
\(108\) 0 0
\(109\) 0.472136i 0.0452224i 0.999744 + 0.0226112i \(0.00719799\pi\)
−0.999744 + 0.0226112i \(0.992802\pi\)
\(110\) 0 0
\(111\) −14.4721 −1.37363
\(112\) 0 0
\(113\) −3.52786 −0.331874 −0.165937 0.986136i \(-0.553065\pi\)
−0.165937 + 0.986136i \(0.553065\pi\)
\(114\) 0 0
\(115\) 4.94427i 0.461056i
\(116\) 0 0
\(117\) 39.1246i 3.61707i
\(118\) 0 0
\(119\) 4.47214 0.409960
\(120\) 0 0
\(121\) 4.88854 0.444413
\(122\) 0 0
\(123\) − 1.52786i − 0.137763i
\(124\) 0 0
\(125\) − 10.4721i − 0.936656i
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 1.70820i 0.149246i 0.997212 + 0.0746232i \(0.0237754\pi\)
−0.997212 + 0.0746232i \(0.976225\pi\)
\(132\) 0 0
\(133\) 3.23607i 0.280603i
\(134\) 0 0
\(135\) −17.8885 −1.53960
\(136\) 0 0
\(137\) 2.94427 0.251546 0.125773 0.992059i \(-0.459859\pi\)
0.125773 + 0.992059i \(0.459859\pi\)
\(138\) 0 0
\(139\) − 3.23607i − 0.274480i −0.990538 0.137240i \(-0.956177\pi\)
0.990538 0.137240i \(-0.0438231\pi\)
\(140\) 0 0
\(141\) 4.94427i 0.416383i
\(142\) 0 0
\(143\) 12.9443 1.08245
\(144\) 0 0
\(145\) −5.52786 −0.459064
\(146\) 0 0
\(147\) 3.23607i 0.266906i
\(148\) 0 0
\(149\) − 14.9443i − 1.22428i −0.790748 0.612141i \(-0.790308\pi\)
0.790748 0.612141i \(-0.209692\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 0 0
\(153\) 33.4164 2.70156
\(154\) 0 0
\(155\) − 8.00000i − 0.642575i
\(156\) 0 0
\(157\) − 5.23607i − 0.417884i −0.977928 0.208942i \(-0.932998\pi\)
0.977928 0.208942i \(-0.0670019\pi\)
\(158\) 0 0
\(159\) 32.3607 2.56637
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) − 23.4164i − 1.83411i −0.398755 0.917057i \(-0.630558\pi\)
0.398755 0.917057i \(-0.369442\pi\)
\(164\) 0 0
\(165\) 9.88854i 0.769822i
\(166\) 0 0
\(167\) 3.41641 0.264370 0.132185 0.991225i \(-0.457801\pi\)
0.132185 + 0.991225i \(0.457801\pi\)
\(168\) 0 0
\(169\) −14.4164 −1.10895
\(170\) 0 0
\(171\) 24.1803i 1.84912i
\(172\) 0 0
\(173\) 7.70820i 0.586044i 0.956106 + 0.293022i \(0.0946609\pi\)
−0.956106 + 0.293022i \(0.905339\pi\)
\(174\) 0 0
\(175\) −3.47214 −0.262469
\(176\) 0 0
\(177\) 15.4164 1.15877
\(178\) 0 0
\(179\) 24.9443i 1.86442i 0.361915 + 0.932211i \(0.382123\pi\)
−0.361915 + 0.932211i \(0.617877\pi\)
\(180\) 0 0
\(181\) 10.1803i 0.756699i 0.925663 + 0.378349i \(0.123508\pi\)
−0.925663 + 0.378349i \(0.876492\pi\)
\(182\) 0 0
\(183\) 21.8885 1.61805
\(184\) 0 0
\(185\) 5.52786 0.406417
\(186\) 0 0
\(187\) − 11.0557i − 0.808475i
\(188\) 0 0
\(189\) 14.4721i 1.05269i
\(190\) 0 0
\(191\) −12.9443 −0.936615 −0.468307 0.883566i \(-0.655136\pi\)
−0.468307 + 0.883566i \(0.655136\pi\)
\(192\) 0 0
\(193\) −8.47214 −0.609838 −0.304919 0.952378i \(-0.598629\pi\)
−0.304919 + 0.952378i \(0.598629\pi\)
\(194\) 0 0
\(195\) − 20.9443i − 1.49985i
\(196\) 0 0
\(197\) − 6.94427i − 0.494759i −0.968919 0.247379i \(-0.920431\pi\)
0.968919 0.247379i \(-0.0795694\pi\)
\(198\) 0 0
\(199\) 11.4164 0.809288 0.404644 0.914474i \(-0.367395\pi\)
0.404644 + 0.914474i \(0.367395\pi\)
\(200\) 0 0
\(201\) 12.9443 0.913019
\(202\) 0 0
\(203\) 4.47214i 0.313882i
\(204\) 0 0
\(205\) 0.583592i 0.0407598i
\(206\) 0 0
\(207\) 29.8885 2.07740
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) − 41.8885i − 2.87016i
\(214\) 0 0
\(215\) −3.05573 −0.208399
\(216\) 0 0
\(217\) −6.47214 −0.439357
\(218\) 0 0
\(219\) − 48.3607i − 3.26791i
\(220\) 0 0
\(221\) 23.4164i 1.57516i
\(222\) 0 0
\(223\) −4.94427 −0.331093 −0.165546 0.986202i \(-0.552939\pi\)
−0.165546 + 0.986202i \(0.552939\pi\)
\(224\) 0 0
\(225\) −25.9443 −1.72962
\(226\) 0 0
\(227\) 12.7639i 0.847172i 0.905856 + 0.423586i \(0.139229\pi\)
−0.905856 + 0.423586i \(0.860771\pi\)
\(228\) 0 0
\(229\) − 25.5967i − 1.69148i −0.533595 0.845740i \(-0.679159\pi\)
0.533595 0.845740i \(-0.320841\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −19.8885 −1.30294 −0.651471 0.758674i \(-0.725848\pi\)
−0.651471 + 0.758674i \(0.725848\pi\)
\(234\) 0 0
\(235\) − 1.88854i − 0.123195i
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) 21.8885 1.41585 0.707926 0.706287i \(-0.249631\pi\)
0.707926 + 0.706287i \(0.249631\pi\)
\(240\) 0 0
\(241\) 3.52786 0.227250 0.113625 0.993524i \(-0.463754\pi\)
0.113625 + 0.993524i \(0.463754\pi\)
\(242\) 0 0
\(243\) 35.5967i 2.28353i
\(244\) 0 0
\(245\) − 1.23607i − 0.0789695i
\(246\) 0 0
\(247\) −16.9443 −1.07814
\(248\) 0 0
\(249\) −15.4164 −0.976975
\(250\) 0 0
\(251\) 17.7082i 1.11773i 0.829258 + 0.558866i \(0.188763\pi\)
−0.829258 + 0.558866i \(0.811237\pi\)
\(252\) 0 0
\(253\) − 9.88854i − 0.621687i
\(254\) 0 0
\(255\) −17.8885 −1.12022
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) − 4.47214i − 0.277885i
\(260\) 0 0
\(261\) 33.4164i 2.06842i
\(262\) 0 0
\(263\) −28.9443 −1.78478 −0.892390 0.451265i \(-0.850973\pi\)
−0.892390 + 0.451265i \(0.850973\pi\)
\(264\) 0 0
\(265\) −12.3607 −0.759311
\(266\) 0 0
\(267\) 19.4164i 1.18826i
\(268\) 0 0
\(269\) − 18.1803i − 1.10847i −0.832359 0.554237i \(-0.813010\pi\)
0.832359 0.554237i \(-0.186990\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) −16.9443 −1.02551
\(274\) 0 0
\(275\) 8.58359i 0.517610i
\(276\) 0 0
\(277\) − 27.8885i − 1.67566i −0.545931 0.837830i \(-0.683824\pi\)
0.545931 0.837830i \(-0.316176\pi\)
\(278\) 0 0
\(279\) −48.3607 −2.89528
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 16.1803i 0.961821i 0.876770 + 0.480911i \(0.159694\pi\)
−0.876770 + 0.480911i \(0.840306\pi\)
\(284\) 0 0
\(285\) − 12.9443i − 0.766752i
\(286\) 0 0
\(287\) 0.472136 0.0278693
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 11.4164i 0.669242i
\(292\) 0 0
\(293\) − 17.2361i − 1.00694i −0.864012 0.503471i \(-0.832056\pi\)
0.864012 0.503471i \(-0.167944\pi\)
\(294\) 0 0
\(295\) −5.88854 −0.342844
\(296\) 0 0
\(297\) 35.7771 2.07600
\(298\) 0 0
\(299\) 20.9443i 1.21124i
\(300\) 0 0
\(301\) 2.47214i 0.142492i
\(302\) 0 0
\(303\) −37.8885 −2.17664
\(304\) 0 0
\(305\) −8.36068 −0.478731
\(306\) 0 0
\(307\) − 24.1803i − 1.38004i −0.723788 0.690022i \(-0.757601\pi\)
0.723788 0.690022i \(-0.242399\pi\)
\(308\) 0 0
\(309\) 46.8328i 2.66423i
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −0.472136 −0.0266867 −0.0133434 0.999911i \(-0.504247\pi\)
−0.0133434 + 0.999911i \(0.504247\pi\)
\(314\) 0 0
\(315\) − 9.23607i − 0.520393i
\(316\) 0 0
\(317\) − 26.9443i − 1.51334i −0.653796 0.756671i \(-0.726825\pi\)
0.653796 0.756671i \(-0.273175\pi\)
\(318\) 0 0
\(319\) 11.0557 0.619002
\(320\) 0 0
\(321\) 28.9443 1.61551
\(322\) 0 0
\(323\) 14.4721i 0.805251i
\(324\) 0 0
\(325\) − 18.1803i − 1.00846i
\(326\) 0 0
\(327\) −1.52786 −0.0844911
\(328\) 0 0
\(329\) −1.52786 −0.0842339
\(330\) 0 0
\(331\) 13.5279i 0.743559i 0.928321 + 0.371779i \(0.121252\pi\)
−0.928321 + 0.371779i \(0.878748\pi\)
\(332\) 0 0
\(333\) − 33.4164i − 1.83121i
\(334\) 0 0
\(335\) −4.94427 −0.270134
\(336\) 0 0
\(337\) −34.3607 −1.87175 −0.935873 0.352338i \(-0.885387\pi\)
−0.935873 + 0.352338i \(0.885387\pi\)
\(338\) 0 0
\(339\) − 11.4164i − 0.620054i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) − 2.47214i − 0.132711i −0.997796 0.0663556i \(-0.978863\pi\)
0.997796 0.0663556i \(-0.0211372\pi\)
\(348\) 0 0
\(349\) 4.65248i 0.249041i 0.992217 + 0.124521i \(0.0397393\pi\)
−0.992217 + 0.124521i \(0.960261\pi\)
\(350\) 0 0
\(351\) −75.7771 −4.04468
\(352\) 0 0
\(353\) 19.8885 1.05856 0.529280 0.848447i \(-0.322462\pi\)
0.529280 + 0.848447i \(0.322462\pi\)
\(354\) 0 0
\(355\) 16.0000i 0.849192i
\(356\) 0 0
\(357\) 14.4721i 0.765947i
\(358\) 0 0
\(359\) −0.944272 −0.0498368 −0.0249184 0.999689i \(-0.507933\pi\)
−0.0249184 + 0.999689i \(0.507933\pi\)
\(360\) 0 0
\(361\) 8.52786 0.448835
\(362\) 0 0
\(363\) 15.8197i 0.830317i
\(364\) 0 0
\(365\) 18.4721i 0.966876i
\(366\) 0 0
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) 0 0
\(369\) 3.52786 0.183653
\(370\) 0 0
\(371\) 10.0000i 0.519174i
\(372\) 0 0
\(373\) − 14.9443i − 0.773785i −0.922125 0.386893i \(-0.873548\pi\)
0.922125 0.386893i \(-0.126452\pi\)
\(374\) 0 0
\(375\) 33.8885 1.75000
\(376\) 0 0
\(377\) −23.4164 −1.20601
\(378\) 0 0
\(379\) − 31.4164i − 1.61375i −0.590721 0.806876i \(-0.701156\pi\)
0.590721 0.806876i \(-0.298844\pi\)
\(380\) 0 0
\(381\) − 28.9443i − 1.48286i
\(382\) 0 0
\(383\) 11.4164 0.583351 0.291676 0.956517i \(-0.405787\pi\)
0.291676 + 0.956517i \(0.405787\pi\)
\(384\) 0 0
\(385\) −3.05573 −0.155734
\(386\) 0 0
\(387\) 18.4721i 0.938991i
\(388\) 0 0
\(389\) 4.47214i 0.226746i 0.993552 + 0.113373i \(0.0361656\pi\)
−0.993552 + 0.113373i \(0.963834\pi\)
\(390\) 0 0
\(391\) 17.8885 0.904663
\(392\) 0 0
\(393\) −5.52786 −0.278844
\(394\) 0 0
\(395\) 6.11146i 0.307501i
\(396\) 0 0
\(397\) 10.7639i 0.540226i 0.962829 + 0.270113i \(0.0870611\pi\)
−0.962829 + 0.270113i \(0.912939\pi\)
\(398\) 0 0
\(399\) −10.4721 −0.524263
\(400\) 0 0
\(401\) −32.4721 −1.62158 −0.810791 0.585336i \(-0.800962\pi\)
−0.810791 + 0.585336i \(0.800962\pi\)
\(402\) 0 0
\(403\) − 33.8885i − 1.68811i
\(404\) 0 0
\(405\) − 30.1803i − 1.49967i
\(406\) 0 0
\(407\) −11.0557 −0.548012
\(408\) 0 0
\(409\) −5.41641 −0.267824 −0.133912 0.990993i \(-0.542754\pi\)
−0.133912 + 0.990993i \(0.542754\pi\)
\(410\) 0 0
\(411\) 9.52786i 0.469975i
\(412\) 0 0
\(413\) 4.76393i 0.234418i
\(414\) 0 0
\(415\) 5.88854 0.289057
\(416\) 0 0
\(417\) 10.4721 0.512823
\(418\) 0 0
\(419\) 0.180340i 0.00881018i 0.999990 + 0.00440509i \(0.00140219\pi\)
−0.999990 + 0.00440509i \(0.998598\pi\)
\(420\) 0 0
\(421\) − 2.00000i − 0.0974740i −0.998812 0.0487370i \(-0.984480\pi\)
0.998812 0.0487370i \(-0.0155196\pi\)
\(422\) 0 0
\(423\) −11.4164 −0.555085
\(424\) 0 0
\(425\) −15.5279 −0.753212
\(426\) 0 0
\(427\) 6.76393i 0.327330i
\(428\) 0 0
\(429\) 41.8885i 2.02240i
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) −17.4164 −0.836979 −0.418490 0.908222i \(-0.637440\pi\)
−0.418490 + 0.908222i \(0.637440\pi\)
\(434\) 0 0
\(435\) − 17.8885i − 0.857690i
\(436\) 0 0
\(437\) 12.9443i 0.619208i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −7.47214 −0.355816
\(442\) 0 0
\(443\) 21.8885i 1.03996i 0.854180 + 0.519978i \(0.174060\pi\)
−0.854180 + 0.519978i \(0.825940\pi\)
\(444\) 0 0
\(445\) − 7.41641i − 0.351571i
\(446\) 0 0
\(447\) 48.3607 2.28738
\(448\) 0 0
\(449\) 27.8885 1.31614 0.658071 0.752956i \(-0.271373\pi\)
0.658071 + 0.752956i \(0.271373\pi\)
\(450\) 0 0
\(451\) − 1.16718i − 0.0549606i
\(452\) 0 0
\(453\) − 28.9443i − 1.35992i
\(454\) 0 0
\(455\) 6.47214 0.303418
\(456\) 0 0
\(457\) 16.4721 0.770534 0.385267 0.922805i \(-0.374109\pi\)
0.385267 + 0.922805i \(0.374109\pi\)
\(458\) 0 0
\(459\) 64.7214i 3.02093i
\(460\) 0 0
\(461\) − 8.29180i − 0.386187i −0.981180 0.193094i \(-0.938148\pi\)
0.981180 0.193094i \(-0.0618521\pi\)
\(462\) 0 0
\(463\) 35.7771 1.66270 0.831351 0.555748i \(-0.187568\pi\)
0.831351 + 0.555748i \(0.187568\pi\)
\(464\) 0 0
\(465\) 25.8885 1.20055
\(466\) 0 0
\(467\) − 26.0689i − 1.20632i −0.797619 0.603162i \(-0.793907\pi\)
0.797619 0.603162i \(-0.206093\pi\)
\(468\) 0 0
\(469\) 4.00000i 0.184703i
\(470\) 0 0
\(471\) 16.9443 0.780751
\(472\) 0 0
\(473\) 6.11146 0.281005
\(474\) 0 0
\(475\) − 11.2361i − 0.515546i
\(476\) 0 0
\(477\) 74.7214i 3.42126i
\(478\) 0 0
\(479\) −35.4164 −1.61822 −0.809108 0.587659i \(-0.800050\pi\)
−0.809108 + 0.587659i \(0.800050\pi\)
\(480\) 0 0
\(481\) 23.4164 1.06770
\(482\) 0 0
\(483\) 12.9443i 0.588985i
\(484\) 0 0
\(485\) − 4.36068i − 0.198008i
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 75.7771 3.42676
\(490\) 0 0
\(491\) 2.11146i 0.0952887i 0.998864 + 0.0476443i \(0.0151714\pi\)
−0.998864 + 0.0476443i \(0.984829\pi\)
\(492\) 0 0
\(493\) 20.0000i 0.900755i
\(494\) 0 0
\(495\) −22.8328 −1.02626
\(496\) 0 0
\(497\) 12.9443 0.580630
\(498\) 0 0
\(499\) 13.8885i 0.621737i 0.950453 + 0.310868i \(0.100620\pi\)
−0.950453 + 0.310868i \(0.899380\pi\)
\(500\) 0 0
\(501\) 11.0557i 0.493934i
\(502\) 0 0
\(503\) −12.9443 −0.577157 −0.288578 0.957456i \(-0.593183\pi\)
−0.288578 + 0.957456i \(0.593183\pi\)
\(504\) 0 0
\(505\) 14.4721 0.644002
\(506\) 0 0
\(507\) − 46.6525i − 2.07191i
\(508\) 0 0
\(509\) 0.875388i 0.0388009i 0.999812 + 0.0194004i \(0.00617574\pi\)
−0.999812 + 0.0194004i \(0.993824\pi\)
\(510\) 0 0
\(511\) 14.9443 0.661096
\(512\) 0 0
\(513\) −46.8328 −2.06772
\(514\) 0 0
\(515\) − 17.8885i − 0.788263i
\(516\) 0 0
\(517\) 3.77709i 0.166116i
\(518\) 0 0
\(519\) −24.9443 −1.09493
\(520\) 0 0
\(521\) 33.4164 1.46400 0.732000 0.681305i \(-0.238587\pi\)
0.732000 + 0.681305i \(0.238587\pi\)
\(522\) 0 0
\(523\) − 17.7082i − 0.774326i −0.922011 0.387163i \(-0.873455\pi\)
0.922011 0.387163i \(-0.126545\pi\)
\(524\) 0 0
\(525\) − 11.2361i − 0.490382i
\(526\) 0 0
\(527\) −28.9443 −1.26083
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 35.5967i 1.54477i
\(532\) 0 0
\(533\) 2.47214i 0.107080i
\(534\) 0 0
\(535\) −11.0557 −0.477981
\(536\) 0 0
\(537\) −80.7214 −3.48338
\(538\) 0 0
\(539\) 2.47214i 0.106482i
\(540\) 0 0
\(541\) 22.9443i 0.986451i 0.869901 + 0.493226i \(0.164182\pi\)
−0.869901 + 0.493226i \(0.835818\pi\)
\(542\) 0 0
\(543\) −32.9443 −1.41377
\(544\) 0 0
\(545\) 0.583592 0.0249983
\(546\) 0 0
\(547\) 31.4164i 1.34327i 0.740883 + 0.671634i \(0.234407\pi\)
−0.740883 + 0.671634i \(0.765593\pi\)
\(548\) 0 0
\(549\) 50.5410i 2.15704i
\(550\) 0 0
\(551\) −14.4721 −0.616534
\(552\) 0 0
\(553\) 4.94427 0.210252
\(554\) 0 0
\(555\) 17.8885i 0.759326i
\(556\) 0 0
\(557\) − 26.9443i − 1.14167i −0.821066 0.570833i \(-0.806620\pi\)
0.821066 0.570833i \(-0.193380\pi\)
\(558\) 0 0
\(559\) −12.9443 −0.547484
\(560\) 0 0
\(561\) 35.7771 1.51051
\(562\) 0 0
\(563\) 40.1803i 1.69340i 0.532071 + 0.846700i \(0.321414\pi\)
−0.532071 + 0.846700i \(0.678586\pi\)
\(564\) 0 0
\(565\) 4.36068i 0.183455i
\(566\) 0 0
\(567\) −24.4164 −1.02539
\(568\) 0 0
\(569\) 18.3607 0.769720 0.384860 0.922975i \(-0.374250\pi\)
0.384860 + 0.922975i \(0.374250\pi\)
\(570\) 0 0
\(571\) − 5.52786i − 0.231334i −0.993288 0.115667i \(-0.963099\pi\)
0.993288 0.115667i \(-0.0369005\pi\)
\(572\) 0 0
\(573\) − 41.8885i − 1.74992i
\(574\) 0 0
\(575\) −13.8885 −0.579192
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) − 27.4164i − 1.13939i
\(580\) 0 0
\(581\) − 4.76393i − 0.197641i
\(582\) 0 0
\(583\) 24.7214 1.02385
\(584\) 0 0
\(585\) 48.3607 1.99947
\(586\) 0 0
\(587\) 9.70820i 0.400700i 0.979724 + 0.200350i \(0.0642080\pi\)
−0.979724 + 0.200350i \(0.935792\pi\)
\(588\) 0 0
\(589\) − 20.9443i − 0.862994i
\(590\) 0 0
\(591\) 22.4721 0.924380
\(592\) 0 0
\(593\) −20.8328 −0.855501 −0.427751 0.903897i \(-0.640694\pi\)
−0.427751 + 0.903897i \(0.640694\pi\)
\(594\) 0 0
\(595\) − 5.52786i − 0.226620i
\(596\) 0 0
\(597\) 36.9443i 1.51203i
\(598\) 0 0
\(599\) 17.8885 0.730906 0.365453 0.930830i \(-0.380914\pi\)
0.365453 + 0.930830i \(0.380914\pi\)
\(600\) 0 0
\(601\) 41.7771 1.70412 0.852061 0.523442i \(-0.175352\pi\)
0.852061 + 0.523442i \(0.175352\pi\)
\(602\) 0 0
\(603\) 29.8885i 1.21716i
\(604\) 0 0
\(605\) − 6.04257i − 0.245666i
\(606\) 0 0
\(607\) −25.8885 −1.05078 −0.525392 0.850860i \(-0.676081\pi\)
−0.525392 + 0.850860i \(0.676081\pi\)
\(608\) 0 0
\(609\) −14.4721 −0.586441
\(610\) 0 0
\(611\) − 8.00000i − 0.323645i
\(612\) 0 0
\(613\) 2.58359i 0.104350i 0.998638 + 0.0521752i \(0.0166154\pi\)
−0.998638 + 0.0521752i \(0.983385\pi\)
\(614\) 0 0
\(615\) −1.88854 −0.0761534
\(616\) 0 0
\(617\) 10.3607 0.417105 0.208553 0.978011i \(-0.433125\pi\)
0.208553 + 0.978011i \(0.433125\pi\)
\(618\) 0 0
\(619\) 10.0689i 0.404703i 0.979313 + 0.202351i \(0.0648583\pi\)
−0.979313 + 0.202351i \(0.935142\pi\)
\(620\) 0 0
\(621\) 57.8885i 2.32299i
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 25.8885i 1.03389i
\(628\) 0 0
\(629\) − 20.0000i − 0.797452i
\(630\) 0 0
\(631\) 27.0557 1.07707 0.538536 0.842603i \(-0.318978\pi\)
0.538536 + 0.842603i \(0.318978\pi\)
\(632\) 0 0
\(633\) −38.8328 −1.54347
\(634\) 0 0
\(635\) 11.0557i 0.438733i
\(636\) 0 0
\(637\) − 5.23607i − 0.207461i
\(638\) 0 0
\(639\) 96.7214 3.82624
\(640\) 0 0
\(641\) 41.4164 1.63585 0.817925 0.575325i \(-0.195124\pi\)
0.817925 + 0.575325i \(0.195124\pi\)
\(642\) 0 0
\(643\) 30.2918i 1.19459i 0.802021 + 0.597296i \(0.203758\pi\)
−0.802021 + 0.597296i \(0.796242\pi\)
\(644\) 0 0
\(645\) − 9.88854i − 0.389361i
\(646\) 0 0
\(647\) −24.3607 −0.957717 −0.478859 0.877892i \(-0.658949\pi\)
−0.478859 + 0.877892i \(0.658949\pi\)
\(648\) 0 0
\(649\) 11.7771 0.462291
\(650\) 0 0
\(651\) − 20.9443i − 0.820871i
\(652\) 0 0
\(653\) − 17.4164i − 0.681557i −0.940144 0.340778i \(-0.889309\pi\)
0.940144 0.340778i \(-0.110691\pi\)
\(654\) 0 0
\(655\) 2.11146 0.0825014
\(656\) 0 0
\(657\) 111.666 4.35649
\(658\) 0 0
\(659\) 25.3050i 0.985741i 0.870103 + 0.492870i \(0.164052\pi\)
−0.870103 + 0.492870i \(0.835948\pi\)
\(660\) 0 0
\(661\) 8.65248i 0.336542i 0.985741 + 0.168271i \(0.0538184\pi\)
−0.985741 + 0.168271i \(0.946182\pi\)
\(662\) 0 0
\(663\) −75.7771 −2.94294
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 17.8885i 0.692647i
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) 16.7214 0.645521
\(672\) 0 0
\(673\) 14.9443 0.576059 0.288030 0.957621i \(-0.407000\pi\)
0.288030 + 0.957621i \(0.407000\pi\)
\(674\) 0 0
\(675\) − 50.2492i − 1.93409i
\(676\) 0 0
\(677\) 42.1803i 1.62112i 0.585654 + 0.810561i \(0.300838\pi\)
−0.585654 + 0.810561i \(0.699162\pi\)
\(678\) 0 0
\(679\) −3.52786 −0.135387
\(680\) 0 0
\(681\) −41.3050 −1.58281
\(682\) 0 0
\(683\) − 47.7771i − 1.82814i −0.405557 0.914070i \(-0.632922\pi\)
0.405557 0.914070i \(-0.367078\pi\)
\(684\) 0 0
\(685\) − 3.63932i − 0.139051i
\(686\) 0 0
\(687\) 82.8328 3.16027
\(688\) 0 0
\(689\) −52.3607 −1.99478
\(690\) 0 0
\(691\) − 8.18034i − 0.311195i −0.987821 0.155597i \(-0.950270\pi\)
0.987821 0.155597i \(-0.0497302\pi\)
\(692\) 0 0
\(693\) 18.4721i 0.701698i
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 2.11146 0.0799771
\(698\) 0 0
\(699\) − 64.3607i − 2.43434i
\(700\) 0 0
\(701\) 14.5836i 0.550815i 0.961328 + 0.275407i \(0.0888127\pi\)
−0.961328 + 0.275407i \(0.911187\pi\)
\(702\) 0 0
\(703\) 14.4721 0.545827
\(704\) 0 0
\(705\) 6.11146 0.230171
\(706\) 0 0
\(707\) − 11.7082i − 0.440332i
\(708\) 0 0
\(709\) 46.3607i 1.74111i 0.492069 + 0.870556i \(0.336241\pi\)
−0.492069 + 0.870556i \(0.663759\pi\)
\(710\) 0 0
\(711\) 36.9443 1.38552
\(712\) 0 0
\(713\) −25.8885 −0.969534
\(714\) 0 0
\(715\) − 16.0000i − 0.598366i
\(716\) 0 0
\(717\) 70.8328i 2.64530i
\(718\) 0 0
\(719\) −47.1935 −1.76002 −0.880010 0.474955i \(-0.842464\pi\)
−0.880010 + 0.474955i \(0.842464\pi\)
\(720\) 0 0
\(721\) −14.4721 −0.538971
\(722\) 0 0
\(723\) 11.4164i 0.424581i
\(724\) 0 0
\(725\) − 15.5279i − 0.576690i
\(726\) 0 0
\(727\) 32.3607 1.20019 0.600096 0.799928i \(-0.295129\pi\)
0.600096 + 0.799928i \(0.295129\pi\)
\(728\) 0 0
\(729\) −41.9443 −1.55349
\(730\) 0 0
\(731\) 11.0557i 0.408911i
\(732\) 0 0
\(733\) 9.23607i 0.341142i 0.985345 + 0.170571i \(0.0545612\pi\)
−0.985345 + 0.170571i \(0.945439\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) 9.88854 0.364249
\(738\) 0 0
\(739\) − 23.4164i − 0.861386i −0.902498 0.430693i \(-0.858269\pi\)
0.902498 0.430693i \(-0.141731\pi\)
\(740\) 0 0
\(741\) − 54.8328i − 2.01433i
\(742\) 0 0
\(743\) −7.05573 −0.258850 −0.129425 0.991589i \(-0.541313\pi\)
−0.129425 + 0.991589i \(0.541313\pi\)
\(744\) 0 0
\(745\) −18.4721 −0.676767
\(746\) 0 0
\(747\) − 35.5967i − 1.30242i
\(748\) 0 0
\(749\) 8.94427i 0.326817i
\(750\) 0 0
\(751\) 36.0000 1.31366 0.656829 0.754039i \(-0.271897\pi\)
0.656829 + 0.754039i \(0.271897\pi\)
\(752\) 0 0
\(753\) −57.3050 −2.08831
\(754\) 0 0
\(755\) 11.0557i 0.402359i
\(756\) 0 0
\(757\) − 23.3050i − 0.847033i −0.905888 0.423516i \(-0.860796\pi\)
0.905888 0.423516i \(-0.139204\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 12.4721 0.452115 0.226057 0.974114i \(-0.427416\pi\)
0.226057 + 0.974114i \(0.427416\pi\)
\(762\) 0 0
\(763\) − 0.472136i − 0.0170925i
\(764\) 0 0
\(765\) − 41.3050i − 1.49338i
\(766\) 0 0
\(767\) −24.9443 −0.900685
\(768\) 0 0
\(769\) 26.3607 0.950590 0.475295 0.879826i \(-0.342341\pi\)
0.475295 + 0.879826i \(0.342341\pi\)
\(770\) 0 0
\(771\) − 45.3050i − 1.63162i
\(772\) 0 0
\(773\) 17.8197i 0.640929i 0.947261 + 0.320464i \(0.103839\pi\)
−0.947261 + 0.320464i \(0.896161\pi\)
\(774\) 0 0
\(775\) 22.4721 0.807223
\(776\) 0 0
\(777\) 14.4721 0.519185
\(778\) 0 0
\(779\) 1.52786i 0.0547414i
\(780\) 0 0
\(781\) − 32.0000i − 1.14505i
\(782\) 0 0
\(783\) −64.7214 −2.31295
\(784\) 0 0
\(785\) −6.47214 −0.231000
\(786\) 0 0
\(787\) 25.7082i 0.916399i 0.888850 + 0.458199i \(0.151505\pi\)
−0.888850 + 0.458199i \(0.848495\pi\)
\(788\) 0 0
\(789\) − 93.6656i − 3.33458i
\(790\) 0 0
\(791\) 3.52786 0.125436
\(792\) 0 0
\(793\) −35.4164 −1.25767
\(794\) 0 0
\(795\) − 40.0000i − 1.41865i
\(796\) 0 0
\(797\) 29.8197i 1.05627i 0.849161 + 0.528133i \(0.177108\pi\)
−0.849161 + 0.528133i \(0.822892\pi\)
\(798\) 0 0
\(799\) −6.83282 −0.241728
\(800\) 0 0
\(801\) −44.8328 −1.58409
\(802\) 0 0
\(803\) − 36.9443i − 1.30374i
\(804\) 0 0
\(805\) − 4.94427i − 0.174263i
\(806\) 0 0
\(807\) 58.8328 2.07101
\(808\) 0 0
\(809\) −9.41641 −0.331063 −0.165532 0.986204i \(-0.552934\pi\)
−0.165532 + 0.986204i \(0.552934\pi\)
\(810\) 0 0
\(811\) 23.0132i 0.808101i 0.914737 + 0.404051i \(0.132398\pi\)
−0.914737 + 0.404051i \(0.867602\pi\)
\(812\) 0 0
\(813\) − 77.6656i − 2.72385i
\(814\) 0 0
\(815\) −28.9443 −1.01387
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) − 39.1246i − 1.36712i
\(820\) 0 0
\(821\) 39.8885i 1.39212i 0.717984 + 0.696060i \(0.245065\pi\)
−0.717984 + 0.696060i \(0.754935\pi\)
\(822\) 0 0
\(823\) −51.7771 −1.80484 −0.902418 0.430862i \(-0.858210\pi\)
−0.902418 + 0.430862i \(0.858210\pi\)
\(824\) 0 0
\(825\) −27.7771 −0.967074
\(826\) 0 0
\(827\) 32.9443i 1.14558i 0.819701 + 0.572792i \(0.194140\pi\)
−0.819701 + 0.572792i \(0.805860\pi\)
\(828\) 0 0
\(829\) − 6.76393i − 0.234921i −0.993078 0.117461i \(-0.962525\pi\)
0.993078 0.117461i \(-0.0374754\pi\)
\(830\) 0 0
\(831\) 90.2492 3.13071
\(832\) 0 0
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) − 4.22291i − 0.146140i
\(836\) 0 0
\(837\) − 93.6656i − 3.23756i
\(838\) 0 0
\(839\) 30.4721 1.05201 0.526007 0.850480i \(-0.323688\pi\)
0.526007 + 0.850480i \(0.323688\pi\)
\(840\) 0 0
\(841\) 9.00000 0.310345
\(842\) 0 0
\(843\) − 84.1378i − 2.89786i
\(844\) 0 0
\(845\) 17.8197i 0.613015i
\(846\) 0 0
\(847\) −4.88854 −0.167972
\(848\) 0 0
\(849\) −52.3607 −1.79701
\(850\) 0 0
\(851\) − 17.8885i − 0.613211i
\(852\) 0 0
\(853\) 0.291796i 0.00999091i 0.999988 + 0.00499545i \(0.00159011\pi\)
−0.999988 + 0.00499545i \(0.998410\pi\)
\(854\) 0 0
\(855\) 29.8885 1.02217
\(856\) 0 0
\(857\) 36.4721 1.24586 0.622932 0.782276i \(-0.285941\pi\)
0.622932 + 0.782276i \(0.285941\pi\)
\(858\) 0 0
\(859\) − 56.1803i − 1.91685i −0.285347 0.958424i \(-0.592109\pi\)
0.285347 0.958424i \(-0.407891\pi\)
\(860\) 0 0
\(861\) 1.52786i 0.0520695i
\(862\) 0 0
\(863\) −3.05573 −0.104018 −0.0520091 0.998647i \(-0.516562\pi\)
−0.0520091 + 0.998647i \(0.516562\pi\)
\(864\) 0 0
\(865\) 9.52786 0.323957
\(866\) 0 0
\(867\) 9.70820i 0.329708i
\(868\) 0 0
\(869\) − 12.2229i − 0.414634i
\(870\) 0 0
\(871\) −20.9443 −0.709670
\(872\) 0 0
\(873\) −26.3607 −0.892174
\(874\) 0 0
\(875\) 10.4721i 0.354023i
\(876\) 0 0
\(877\) 13.4164i 0.453040i 0.974007 + 0.226520i \(0.0727348\pi\)
−0.974007 + 0.226520i \(0.927265\pi\)
\(878\) 0 0
\(879\) 55.7771 1.88131
\(880\) 0 0
\(881\) −28.8328 −0.971402 −0.485701 0.874125i \(-0.661436\pi\)
−0.485701 + 0.874125i \(0.661436\pi\)
\(882\) 0 0
\(883\) − 40.9443i − 1.37788i −0.724816 0.688942i \(-0.758075\pi\)
0.724816 0.688942i \(-0.241925\pi\)
\(884\) 0 0
\(885\) − 19.0557i − 0.640551i
\(886\) 0 0
\(887\) −29.3050 −0.983964 −0.491982 0.870605i \(-0.663727\pi\)
−0.491982 + 0.870605i \(0.663727\pi\)
\(888\) 0 0
\(889\) 8.94427 0.299981
\(890\) 0 0
\(891\) 60.3607i 2.02216i
\(892\) 0 0
\(893\) − 4.94427i − 0.165454i
\(894\) 0 0
\(895\) 30.8328 1.03063
\(896\) 0 0
\(897\) −67.7771 −2.26301
\(898\) 0 0
\(899\) − 28.9443i − 0.965346i
\(900\) 0 0
\(901\) 44.7214i 1.48988i
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 12.5836 0.418293
\(906\) 0 0
\(907\) − 16.9443i − 0.562625i −0.959616 0.281313i \(-0.909230\pi\)
0.959616 0.281313i \(-0.0907698\pi\)
\(908\) 0 0
\(909\) − 87.4853i − 2.90170i
\(910\) 0 0
\(911\) 18.8328 0.623959 0.311980 0.950089i \(-0.399008\pi\)
0.311980 + 0.950089i \(0.399008\pi\)
\(912\) 0 0
\(913\) −11.7771 −0.389765
\(914\) 0 0
\(915\) − 27.0557i − 0.894435i
\(916\) 0 0
\(917\) − 1.70820i − 0.0564099i
\(918\) 0 0
\(919\) −35.7771 −1.18018 −0.590089 0.807338i \(-0.700907\pi\)
−0.590089 + 0.807338i \(0.700907\pi\)
\(920\) 0 0
\(921\) 78.2492 2.57840
\(922\) 0 0
\(923\) 67.7771i 2.23091i
\(924\) 0 0
\(925\) 15.5279i 0.510553i
\(926\) 0 0
\(927\) −108.138 −3.55171
\(928\) 0 0
\(929\) 15.3050 0.502139 0.251070 0.967969i \(-0.419218\pi\)
0.251070 + 0.967969i \(0.419218\pi\)
\(930\) 0 0
\(931\) − 3.23607i − 0.106058i
\(932\) 0 0
\(933\) − 25.8885i − 0.847553i
\(934\) 0 0
\(935\) −13.6656 −0.446914
\(936\) 0 0
\(937\) 26.9443 0.880231 0.440115 0.897941i \(-0.354937\pi\)
0.440115 + 0.897941i \(0.354937\pi\)
\(938\) 0 0
\(939\) − 1.52786i − 0.0498600i
\(940\) 0 0
\(941\) − 13.5967i − 0.443241i −0.975133 0.221621i \(-0.928865\pi\)
0.975133 0.221621i \(-0.0711347\pi\)
\(942\) 0 0
\(943\) 1.88854 0.0614994
\(944\) 0 0
\(945\) 17.8885 0.581914
\(946\) 0 0
\(947\) 31.4164i 1.02090i 0.859909 + 0.510448i \(0.170520\pi\)
−0.859909 + 0.510448i \(0.829480\pi\)
\(948\) 0 0
\(949\) 78.2492i 2.54008i
\(950\) 0 0
\(951\) 87.1935 2.82744
\(952\) 0 0
\(953\) −16.1115 −0.521901 −0.260951 0.965352i \(-0.584036\pi\)
−0.260951 + 0.965352i \(0.584036\pi\)
\(954\) 0 0
\(955\) 16.0000i 0.517748i
\(956\) 0 0
\(957\) 35.7771i 1.15651i
\(958\) 0 0
\(959\) −2.94427 −0.0950755
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) 66.8328i 2.15366i
\(964\) 0 0
\(965\) 10.4721i 0.337110i
\(966\) 0 0
\(967\) 5.88854 0.189363 0.0946814 0.995508i \(-0.469817\pi\)
0.0946814 + 0.995508i \(0.469817\pi\)
\(968\) 0 0
\(969\) −46.8328 −1.50449
\(970\) 0 0
\(971\) 27.2361i 0.874047i 0.899450 + 0.437024i \(0.143967\pi\)
−0.899450 + 0.437024i \(0.856033\pi\)
\(972\) 0 0
\(973\) 3.23607i 0.103744i
\(974\) 0 0
\(975\) 58.8328 1.88416
\(976\) 0 0
\(977\) 40.8328 1.30636 0.653179 0.757204i \(-0.273435\pi\)
0.653179 + 0.757204i \(0.273435\pi\)
\(978\) 0 0
\(979\) 14.8328i 0.474059i
\(980\) 0 0
\(981\) − 3.52786i − 0.112636i
\(982\) 0 0
\(983\) 14.4721 0.461589 0.230795 0.973002i \(-0.425867\pi\)
0.230795 + 0.973002i \(0.425867\pi\)
\(984\) 0 0
\(985\) −8.58359 −0.273496
\(986\) 0 0
\(987\) − 4.94427i − 0.157378i
\(988\) 0 0
\(989\) 9.88854i 0.314437i
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) −43.7771 −1.38922
\(994\) 0 0
\(995\) − 14.1115i − 0.447363i
\(996\) 0 0
\(997\) − 40.0689i − 1.26899i −0.772925 0.634497i \(-0.781207\pi\)
0.772925 0.634497i \(-0.218793\pi\)
\(998\) 0 0
\(999\) 64.7214 2.04769
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 1792.2.b.k.897.4 4
4.3 odd 2 1792.2.b.m.897.1 4
8.3 odd 2 1792.2.b.m.897.4 4
8.5 even 2 inner 1792.2.b.k.897.1 4
16.3 odd 4 224.2.a.d.1.2 yes 2
16.5 even 4 448.2.a.j.1.2 2
16.11 odd 4 448.2.a.i.1.1 2
16.13 even 4 224.2.a.c.1.1 2
48.5 odd 4 4032.2.a.bw.1.1 2
48.11 even 4 4032.2.a.bv.1.1 2
48.29 odd 4 2016.2.a.r.1.2 2
48.35 even 4 2016.2.a.o.1.2 2
80.19 odd 4 5600.2.a.z.1.1 2
80.29 even 4 5600.2.a.bk.1.2 2
112.3 even 12 1568.2.i.w.961.2 4
112.13 odd 4 1568.2.a.v.1.2 2
112.19 even 12 1568.2.i.w.1537.2 4
112.27 even 4 3136.2.a.by.1.2 2
112.45 odd 12 1568.2.i.n.961.1 4
112.51 odd 12 1568.2.i.m.1537.1 4
112.61 odd 12 1568.2.i.n.1537.1 4
112.67 odd 12 1568.2.i.m.961.1 4
112.69 odd 4 3136.2.a.bf.1.1 2
112.83 even 4 1568.2.a.k.1.1 2
112.93 even 12 1568.2.i.v.1537.2 4
112.109 even 12 1568.2.i.v.961.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.1 2 16.13 even 4
224.2.a.d.1.2 yes 2 16.3 odd 4
448.2.a.i.1.1 2 16.11 odd 4
448.2.a.j.1.2 2 16.5 even 4
1568.2.a.k.1.1 2 112.83 even 4
1568.2.a.v.1.2 2 112.13 odd 4
1568.2.i.m.961.1 4 112.67 odd 12
1568.2.i.m.1537.1 4 112.51 odd 12
1568.2.i.n.961.1 4 112.45 odd 12
1568.2.i.n.1537.1 4 112.61 odd 12
1568.2.i.v.961.2 4 112.109 even 12
1568.2.i.v.1537.2 4 112.93 even 12
1568.2.i.w.961.2 4 112.3 even 12
1568.2.i.w.1537.2 4 112.19 even 12
1792.2.b.k.897.1 4 8.5 even 2 inner
1792.2.b.k.897.4 4 1.1 even 1 trivial
1792.2.b.m.897.1 4 4.3 odd 2
1792.2.b.m.897.4 4 8.3 odd 2
2016.2.a.o.1.2 2 48.35 even 4
2016.2.a.r.1.2 2 48.29 odd 4
3136.2.a.bf.1.1 2 112.69 odd 4
3136.2.a.by.1.2 2 112.27 even 4
4032.2.a.bv.1.1 2 48.11 even 4
4032.2.a.bw.1.1 2 48.5 odd 4
5600.2.a.z.1.1 2 80.19 odd 4
5600.2.a.bk.1.2 2 80.29 even 4