Properties

Label 128.3.f.a.31.3
Level $128$
Weight $3$
Character 128.31
Analytic conductor $3.488$
Analytic rank $0$
Dimension $6$
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] = [128,3,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.48774738381\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 31.3
Root \(1.40680 + 0.144584i\) of defining polynomial
Character \(\chi\) \(=\) 128.31
Dual form 128.3.f.a.95.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.10278 + 2.10278i) q^{3} +(4.62721 + 4.62721i) q^{5} -3.04888 q^{7} -0.156674i q^{9} +O(q^{10})\) \(q+(2.10278 + 2.10278i) q^{3} +(4.62721 + 4.62721i) q^{5} -3.04888 q^{7} -0.156674i q^{9} +(-9.15165 + 9.15165i) q^{11} +(5.78389 - 5.78389i) q^{13} +19.4600i q^{15} +17.6655 q^{17} +(-1.15165 - 1.15165i) q^{19} +(-6.41110 - 6.41110i) q^{21} +3.45998 q^{23} +17.8222i q^{25} +(19.2544 - 19.2544i) q^{27} +(-12.1950 + 12.1950i) q^{29} -38.5089i q^{31} -38.4877 q^{33} +(-14.1078 - 14.1078i) q^{35} +(0.0972356 + 0.0972356i) q^{37} +24.3244 q^{39} -51.5266i q^{41} +(-1.70172 + 1.70172i) q^{43} +(0.724965 - 0.724965i) q^{45} -24.1533i q^{47} -39.7044 q^{49} +(37.1466 + 37.1466i) q^{51} +(-27.0383 - 27.0383i) q^{53} -84.6933 q^{55} -4.84333i q^{57} +(19.5939 - 19.5939i) q^{59} +(-16.7250 + 16.7250i) q^{61} +0.477680i q^{63} +53.5266 q^{65} +(-75.8560 - 75.8560i) q^{67} +(7.27555 + 7.27555i) q^{69} +134.749 q^{71} +112.210i q^{73} +(-37.4761 + 37.4761i) q^{75} +(27.9022 - 27.9022i) q^{77} +135.915i q^{79} +79.5654 q^{81} +(74.9250 + 74.9250i) q^{83} +(81.7422 + 81.7422i) q^{85} -51.2866 q^{87} -31.4278i q^{89} +(-17.6344 + 17.6344i) q^{91} +(80.9755 - 80.9755i) q^{93} -10.6579i q^{95} +31.5456 q^{97} +(1.43383 + 1.43383i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 2 q^{5} + 4 q^{7} - 18 q^{11} + 2 q^{13} - 4 q^{17} + 30 q^{19} + 20 q^{21} - 60 q^{23} + 64 q^{27} + 18 q^{29} - 4 q^{33} - 100 q^{35} - 46 q^{37} + 196 q^{39} - 114 q^{43} - 66 q^{45} - 46 q^{49} + 156 q^{51} - 78 q^{53} - 252 q^{55} + 206 q^{59} - 30 q^{61} + 12 q^{65} - 226 q^{67} + 116 q^{69} + 260 q^{71} - 238 q^{75} + 212 q^{77} + 86 q^{81} + 318 q^{83} + 212 q^{85} - 444 q^{87} + 188 q^{91} + 32 q^{93} - 4 q^{97} - 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(3\) 2.10278 + 2.10278i 0.700925 + 0.700925i 0.964609 0.263684i \(-0.0849376\pi\)
−0.263684 + 0.964609i \(0.584938\pi\)
\(4\) 0 0
\(5\) 4.62721 + 4.62721i 0.925443 + 0.925443i 0.997407 0.0719646i \(-0.0229269\pi\)
−0.0719646 + 0.997407i \(0.522927\pi\)
\(6\) 0 0
\(7\) −3.04888 −0.435554 −0.217777 0.975999i \(-0.569881\pi\)
−0.217777 + 0.975999i \(0.569881\pi\)
\(8\) 0 0
\(9\) 0.156674i 0.0174082i
\(10\) 0 0
\(11\) −9.15165 + 9.15165i −0.831968 + 0.831968i −0.987786 0.155818i \(-0.950199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(12\) 0 0
\(13\) 5.78389 5.78389i 0.444914 0.444914i −0.448745 0.893660i \(-0.648129\pi\)
0.893660 + 0.448745i \(0.148129\pi\)
\(14\) 0 0
\(15\) 19.4600i 1.29733i
\(16\) 0 0
\(17\) 17.6655 1.03915 0.519574 0.854425i \(-0.326091\pi\)
0.519574 + 0.854425i \(0.326091\pi\)
\(18\) 0 0
\(19\) −1.15165 1.15165i −0.0606132 0.0606132i 0.676150 0.736764i \(-0.263647\pi\)
−0.736764 + 0.676150i \(0.763647\pi\)
\(20\) 0 0
\(21\) −6.41110 6.41110i −0.305290 0.305290i
\(22\) 0 0
\(23\) 3.45998 0.150434 0.0752169 0.997167i \(-0.476035\pi\)
0.0752169 + 0.997167i \(0.476035\pi\)
\(24\) 0 0
\(25\) 17.8222i 0.712888i
\(26\) 0 0
\(27\) 19.2544 19.2544i 0.713127 0.713127i
\(28\) 0 0
\(29\) −12.1950 + 12.1950i −0.420517 + 0.420517i −0.885382 0.464865i \(-0.846103\pi\)
0.464865 + 0.885382i \(0.346103\pi\)
\(30\) 0 0
\(31\) 38.5089i 1.24222i −0.783723 0.621111i \(-0.786682\pi\)
0.783723 0.621111i \(-0.213318\pi\)
\(32\) 0 0
\(33\) −38.4877 −1.16629
\(34\) 0 0
\(35\) −14.1078 14.1078i −0.403080 0.403080i
\(36\) 0 0
\(37\) 0.0972356 + 0.0972356i 0.00262799 + 0.00262799i 0.708420 0.705792i \(-0.249408\pi\)
−0.705792 + 0.708420i \(0.749408\pi\)
\(38\) 0 0
\(39\) 24.3244 0.623703
\(40\) 0 0
\(41\) 51.5266i 1.25675i −0.777912 0.628373i \(-0.783721\pi\)
0.777912 0.628373i \(-0.216279\pi\)
\(42\) 0 0
\(43\) −1.70172 + 1.70172i −0.0395749 + 0.0395749i −0.726617 0.687042i \(-0.758909\pi\)
0.687042 + 0.726617i \(0.258909\pi\)
\(44\) 0 0
\(45\) 0.724965 0.724965i 0.0161103 0.0161103i
\(46\) 0 0
\(47\) 24.1533i 0.513899i −0.966425 0.256949i \(-0.917283\pi\)
0.966425 0.256949i \(-0.0827174\pi\)
\(48\) 0 0
\(49\) −39.7044 −0.810293
\(50\) 0 0
\(51\) 37.1466 + 37.1466i 0.728365 + 0.728365i
\(52\) 0 0
\(53\) −27.0383 27.0383i −0.510157 0.510157i 0.404418 0.914574i \(-0.367474\pi\)
−0.914574 + 0.404418i \(0.867474\pi\)
\(54\) 0 0
\(55\) −84.6933 −1.53988
\(56\) 0 0
\(57\) 4.84333i 0.0849706i
\(58\) 0 0
\(59\) 19.5939 19.5939i 0.332100 0.332100i −0.521283 0.853384i \(-0.674547\pi\)
0.853384 + 0.521283i \(0.174547\pi\)
\(60\) 0 0
\(61\) −16.7250 + 16.7250i −0.274180 + 0.274180i −0.830780 0.556601i \(-0.812105\pi\)
0.556601 + 0.830780i \(0.312105\pi\)
\(62\) 0 0
\(63\) 0.477680i 0.00758222i
\(64\) 0 0
\(65\) 53.5266 0.823485
\(66\) 0 0
\(67\) −75.8560 75.8560i −1.13218 1.13218i −0.989814 0.142365i \(-0.954529\pi\)
−0.142365 0.989814i \(-0.545471\pi\)
\(68\) 0 0
\(69\) 7.27555 + 7.27555i 0.105443 + 0.105443i
\(70\) 0 0
\(71\) 134.749 1.89787 0.948935 0.315471i \(-0.102163\pi\)
0.948935 + 0.315471i \(0.102163\pi\)
\(72\) 0 0
\(73\) 112.210i 1.53712i 0.639777 + 0.768560i \(0.279026\pi\)
−0.639777 + 0.768560i \(0.720974\pi\)
\(74\) 0 0
\(75\) −37.4761 + 37.4761i −0.499681 + 0.499681i
\(76\) 0 0
\(77\) 27.9022 27.9022i 0.362367 0.362367i
\(78\) 0 0
\(79\) 135.915i 1.72045i 0.509915 + 0.860225i \(0.329677\pi\)
−0.509915 + 0.860225i \(0.670323\pi\)
\(80\) 0 0
\(81\) 79.5654 0.982289
\(82\) 0 0
\(83\) 74.9250 + 74.9250i 0.902711 + 0.902711i 0.995670 0.0929594i \(-0.0296327\pi\)
−0.0929594 + 0.995670i \(0.529633\pi\)
\(84\) 0 0
\(85\) 81.7422 + 81.7422i 0.961672 + 0.961672i
\(86\) 0 0
\(87\) −51.2866 −0.589501
\(88\) 0 0
\(89\) 31.4278i 0.353121i −0.984290 0.176561i \(-0.943503\pi\)
0.984290 0.176561i \(-0.0564971\pi\)
\(90\) 0 0
\(91\) −17.6344 + 17.6344i −0.193784 + 0.193784i
\(92\) 0 0
\(93\) 80.9755 80.9755i 0.870704 0.870704i
\(94\) 0 0
\(95\) 10.6579i 0.112188i
\(96\) 0 0
\(97\) 31.5456 0.325213 0.162606 0.986691i \(-0.448010\pi\)
0.162606 + 0.986691i \(0.448010\pi\)
\(98\) 0 0
\(99\) 1.43383 + 1.43383i 0.0144831 + 0.0144831i
\(100\) 0 0
\(101\) −27.4695 27.4695i −0.271975 0.271975i 0.557920 0.829895i \(-0.311600\pi\)
−0.829895 + 0.557920i \(0.811600\pi\)
\(102\) 0 0
\(103\) −102.882 −0.998854 −0.499427 0.866356i \(-0.666456\pi\)
−0.499427 + 0.866356i \(0.666456\pi\)
\(104\) 0 0
\(105\) 59.3311i 0.565058i
\(106\) 0 0
\(107\) −79.6605 + 79.6605i −0.744491 + 0.744491i −0.973439 0.228948i \(-0.926471\pi\)
0.228948 + 0.973439i \(0.426471\pi\)
\(108\) 0 0
\(109\) −125.408 + 125.408i −1.15053 + 1.15053i −0.164088 + 0.986446i \(0.552468\pi\)
−0.986446 + 0.164088i \(0.947532\pi\)
\(110\) 0 0
\(111\) 0.408929i 0.00368405i
\(112\) 0 0
\(113\) −96.6199 −0.855043 −0.427521 0.904005i \(-0.640613\pi\)
−0.427521 + 0.904005i \(0.640613\pi\)
\(114\) 0 0
\(115\) 16.0100 + 16.0100i 0.139218 + 0.139218i
\(116\) 0 0
\(117\) −0.906186 0.906186i −0.00774518 0.00774518i
\(118\) 0 0
\(119\) −53.8600 −0.452605
\(120\) 0 0
\(121\) 46.5054i 0.384342i
\(122\) 0 0
\(123\) 108.349 108.349i 0.880884 0.880884i
\(124\) 0 0
\(125\) 33.2132 33.2132i 0.265706 0.265706i
\(126\) 0 0
\(127\) 196.309i 1.54574i −0.634566 0.772868i \(-0.718821\pi\)
0.634566 0.772868i \(-0.281179\pi\)
\(128\) 0 0
\(129\) −7.15667 −0.0554781
\(130\) 0 0
\(131\) 17.9437 + 17.9437i 0.136975 + 0.136975i 0.772270 0.635295i \(-0.219121\pi\)
−0.635295 + 0.772270i \(0.719121\pi\)
\(132\) 0 0
\(133\) 3.51124 + 3.51124i 0.0264003 + 0.0264003i
\(134\) 0 0
\(135\) 178.189 1.31992
\(136\) 0 0
\(137\) 51.7200i 0.377518i 0.982023 + 0.188759i \(0.0604465\pi\)
−0.982023 + 0.188759i \(0.939553\pi\)
\(138\) 0 0
\(139\) −17.4640 + 17.4640i −0.125640 + 0.125640i −0.767131 0.641491i \(-0.778316\pi\)
0.641491 + 0.767131i \(0.278316\pi\)
\(140\) 0 0
\(141\) 50.7889 50.7889i 0.360205 0.360205i
\(142\) 0 0
\(143\) 105.864i 0.740309i
\(144\) 0 0
\(145\) −112.858 −0.778328
\(146\) 0 0
\(147\) −83.4893 83.4893i −0.567955 0.567955i
\(148\) 0 0
\(149\) −11.9170 11.9170i −0.0799802 0.0799802i 0.665985 0.745965i \(-0.268011\pi\)
−0.745965 + 0.665985i \(0.768011\pi\)
\(150\) 0 0
\(151\) 132.548 0.877805 0.438902 0.898535i \(-0.355367\pi\)
0.438902 + 0.898535i \(0.355367\pi\)
\(152\) 0 0
\(153\) 2.76773i 0.0180898i
\(154\) 0 0
\(155\) 178.189 178.189i 1.14960 1.14960i
\(156\) 0 0
\(157\) −106.091 + 106.091i −0.675742 + 0.675742i −0.959034 0.283292i \(-0.908573\pi\)
0.283292 + 0.959034i \(0.408573\pi\)
\(158\) 0 0
\(159\) 113.711i 0.715163i
\(160\) 0 0
\(161\) −10.5490 −0.0655220
\(162\) 0 0
\(163\) −105.577 105.577i −0.647712 0.647712i 0.304728 0.952440i \(-0.401435\pi\)
−0.952440 + 0.304728i \(0.901435\pi\)
\(164\) 0 0
\(165\) −178.091 178.091i −1.07934 1.07934i
\(166\) 0 0
\(167\) −111.591 −0.668210 −0.334105 0.942536i \(-0.608434\pi\)
−0.334105 + 0.942536i \(0.608434\pi\)
\(168\) 0 0
\(169\) 102.093i 0.604102i
\(170\) 0 0
\(171\) −0.180434 + 0.180434i −0.00105517 + 0.00105517i
\(172\) 0 0
\(173\) −14.5363 + 14.5363i −0.0840249 + 0.0840249i −0.747870 0.663845i \(-0.768923\pi\)
0.663845 + 0.747870i \(0.268923\pi\)
\(174\) 0 0
\(175\) 54.3377i 0.310501i
\(176\) 0 0
\(177\) 82.4032 0.465555
\(178\) 0 0
\(179\) −19.7371 19.7371i −0.110263 0.110263i 0.649823 0.760086i \(-0.274843\pi\)
−0.760086 + 0.649823i \(0.774843\pi\)
\(180\) 0 0
\(181\) −168.153 168.153i −0.929021 0.929021i 0.0686221 0.997643i \(-0.478140\pi\)
−0.997643 + 0.0686221i \(0.978140\pi\)
\(182\) 0 0
\(183\) −70.3377 −0.384359
\(184\) 0 0
\(185\) 0.899859i 0.00486410i
\(186\) 0 0
\(187\) −161.669 + 161.669i −0.864539 + 0.864539i
\(188\) 0 0
\(189\) −58.7044 + 58.7044i −0.310605 + 0.310605i
\(190\) 0 0
\(191\) 196.309i 1.02779i 0.857852 + 0.513897i \(0.171799\pi\)
−0.857852 + 0.513897i \(0.828201\pi\)
\(192\) 0 0
\(193\) −40.3699 −0.209170 −0.104585 0.994516i \(-0.533351\pi\)
−0.104585 + 0.994516i \(0.533351\pi\)
\(194\) 0 0
\(195\) 112.554 + 112.554i 0.577202 + 0.577202i
\(196\) 0 0
\(197\) 230.578 + 230.578i 1.17045 + 1.17045i 0.982103 + 0.188344i \(0.0603121\pi\)
0.188344 + 0.982103i \(0.439688\pi\)
\(198\) 0 0
\(199\) −61.5598 −0.309346 −0.154673 0.987966i \(-0.549432\pi\)
−0.154673 + 0.987966i \(0.549432\pi\)
\(200\) 0 0
\(201\) 319.016i 1.58715i
\(202\) 0 0
\(203\) 37.1810 37.1810i 0.183158 0.183158i
\(204\) 0 0
\(205\) 238.424 238.424i 1.16305 1.16305i
\(206\) 0 0
\(207\) 0.542089i 0.00261879i
\(208\) 0 0
\(209\) 21.0790 0.100857
\(210\) 0 0
\(211\) 151.149 + 151.149i 0.716346 + 0.716346i 0.967855 0.251509i \(-0.0809267\pi\)
−0.251509 + 0.967855i \(0.580927\pi\)
\(212\) 0 0
\(213\) 283.346 + 283.346i 1.33026 + 1.33026i
\(214\) 0 0
\(215\) −15.7485 −0.0732486
\(216\) 0 0
\(217\) 117.409i 0.541054i
\(218\) 0 0
\(219\) −235.952 + 235.952i −1.07741 + 1.07741i
\(220\) 0 0
\(221\) 102.175 102.175i 0.462332 0.462332i
\(222\) 0 0
\(223\) 115.527i 0.518056i 0.965870 + 0.259028i \(0.0834022\pi\)
−0.965870 + 0.259028i \(0.916598\pi\)
\(224\) 0 0
\(225\) 2.79228 0.0124101
\(226\) 0 0
\(227\) 25.2363 + 25.2363i 0.111173 + 0.111173i 0.760505 0.649332i \(-0.224951\pi\)
−0.649332 + 0.760505i \(0.724951\pi\)
\(228\) 0 0
\(229\) 155.318 + 155.318i 0.678244 + 0.678244i 0.959603 0.281359i \(-0.0907851\pi\)
−0.281359 + 0.959603i \(0.590785\pi\)
\(230\) 0 0
\(231\) 117.344 0.507984
\(232\) 0 0
\(233\) 119.738i 0.513899i 0.966425 + 0.256949i \(0.0827174\pi\)
−0.966425 + 0.256949i \(0.917283\pi\)
\(234\) 0 0
\(235\) 111.762 111.762i 0.475584 0.475584i
\(236\) 0 0
\(237\) −285.800 + 285.800i −1.20591 + 1.20591i
\(238\) 0 0
\(239\) 245.409i 1.02681i −0.858145 0.513407i \(-0.828383\pi\)
0.858145 0.513407i \(-0.171617\pi\)
\(240\) 0 0
\(241\) 431.216 1.78928 0.894639 0.446790i \(-0.147433\pi\)
0.894639 + 0.446790i \(0.147433\pi\)
\(242\) 0 0
\(243\) −5.98173 5.98173i −0.0246162 0.0246162i
\(244\) 0 0
\(245\) −183.721 183.721i −0.749880 0.749880i
\(246\) 0 0
\(247\) −13.3220 −0.0539354
\(248\) 0 0
\(249\) 315.101i 1.26546i
\(250\) 0 0
\(251\) −24.0171 + 24.0171i −0.0956858 + 0.0956858i −0.753329 0.657643i \(-0.771553\pi\)
0.657643 + 0.753329i \(0.271553\pi\)
\(252\) 0 0
\(253\) −31.6645 + 31.6645i −0.125156 + 0.125156i
\(254\) 0 0
\(255\) 343.771i 1.34812i
\(256\) 0 0
\(257\) −100.860 −0.392450 −0.196225 0.980559i \(-0.562868\pi\)
−0.196225 + 0.980559i \(0.562868\pi\)
\(258\) 0 0
\(259\) −0.296459 0.296459i −0.00114463 0.00114463i
\(260\) 0 0
\(261\) 1.91064 + 1.91064i 0.00732046 + 0.00732046i
\(262\) 0 0
\(263\) −216.776 −0.824242 −0.412121 0.911129i \(-0.635212\pi\)
−0.412121 + 0.911129i \(0.635212\pi\)
\(264\) 0 0
\(265\) 250.224i 0.944242i
\(266\) 0 0
\(267\) 66.0855 66.0855i 0.247511 0.247511i
\(268\) 0 0
\(269\) 256.778 256.778i 0.954567 0.954567i −0.0444453 0.999012i \(-0.514152\pi\)
0.999012 + 0.0444453i \(0.0141520\pi\)
\(270\) 0 0
\(271\) 12.8603i 0.0474551i −0.999718 0.0237275i \(-0.992447\pi\)
0.999718 0.0237275i \(-0.00755342\pi\)
\(272\) 0 0
\(273\) −74.1622 −0.271656
\(274\) 0 0
\(275\) −163.103 163.103i −0.593100 0.593100i
\(276\) 0 0
\(277\) −77.1023 77.1023i −0.278348 0.278348i 0.554102 0.832449i \(-0.313062\pi\)
−0.832449 + 0.554102i \(0.813062\pi\)
\(278\) 0 0
\(279\) −6.03334 −0.0216249
\(280\) 0 0
\(281\) 189.034i 0.672719i −0.941734 0.336360i \(-0.890804\pi\)
0.941734 0.336360i \(-0.109196\pi\)
\(282\) 0 0
\(283\) −69.4317 + 69.4317i −0.245342 + 0.245342i −0.819056 0.573714i \(-0.805502\pi\)
0.573714 + 0.819056i \(0.305502\pi\)
\(284\) 0 0
\(285\) 22.4111 22.4111i 0.0786354 0.0786354i
\(286\) 0 0
\(287\) 157.098i 0.547380i
\(288\) 0 0
\(289\) 23.0708 0.0798298
\(290\) 0 0
\(291\) 66.3333 + 66.3333i 0.227950 + 0.227950i
\(292\) 0 0
\(293\) −239.919 239.919i −0.818837 0.818837i 0.167103 0.985939i \(-0.446559\pi\)
−0.985939 + 0.167103i \(0.946559\pi\)
\(294\) 0 0
\(295\) 181.331 0.614680
\(296\) 0 0
\(297\) 352.420i 1.18660i
\(298\) 0 0
\(299\) 20.0121 20.0121i 0.0669301 0.0669301i
\(300\) 0 0
\(301\) 5.18834 5.18834i 0.0172370 0.0172370i
\(302\) 0 0
\(303\) 115.524i 0.381269i
\(304\) 0 0
\(305\) −154.780 −0.507475
\(306\) 0 0
\(307\) −231.185 231.185i −0.753046 0.753046i 0.222001 0.975046i \(-0.428741\pi\)
−0.975046 + 0.222001i \(0.928741\pi\)
\(308\) 0 0
\(309\) −216.338 216.338i −0.700122 0.700122i
\(310\) 0 0
\(311\) 513.328 1.65057 0.825287 0.564714i \(-0.191013\pi\)
0.825287 + 0.564714i \(0.191013\pi\)
\(312\) 0 0
\(313\) 345.242i 1.10301i −0.834172 0.551504i \(-0.814054\pi\)
0.834172 0.551504i \(-0.185946\pi\)
\(314\) 0 0
\(315\) −2.21033 + 2.21033i −0.00701691 + 0.00701691i
\(316\) 0 0
\(317\) −345.632 + 345.632i −1.09032 + 1.09032i −0.0948290 + 0.995494i \(0.530230\pi\)
−0.995494 + 0.0948290i \(0.969770\pi\)
\(318\) 0 0
\(319\) 223.209i 0.699713i
\(320\) 0 0
\(321\) −335.016 −1.04366
\(322\) 0 0
\(323\) −20.3445 20.3445i −0.0629861 0.0629861i
\(324\) 0 0
\(325\) 103.082 + 103.082i 0.317174 + 0.317174i
\(326\) 0 0
\(327\) −527.410 −1.61288
\(328\) 0 0
\(329\) 73.6403i 0.223831i
\(330\) 0 0
\(331\) 425.968 425.968i 1.28691 1.28691i 0.350261 0.936652i \(-0.386093\pi\)
0.936652 0.350261i \(-0.113907\pi\)
\(332\) 0 0
\(333\) 0.0152343 0.0152343i 4.57487e−5 4.57487e-5i
\(334\) 0 0
\(335\) 702.004i 2.09553i
\(336\) 0 0
\(337\) −467.297 −1.38664 −0.693319 0.720631i \(-0.743852\pi\)
−0.693319 + 0.720631i \(0.743852\pi\)
\(338\) 0 0
\(339\) −203.170 203.170i −0.599321 0.599321i
\(340\) 0 0
\(341\) 352.420 + 352.420i 1.03349 + 1.03349i
\(342\) 0 0
\(343\) 270.449 0.788480
\(344\) 0 0
\(345\) 67.3311i 0.195162i
\(346\) 0 0
\(347\) −22.0463 + 22.0463i −0.0635341 + 0.0635341i −0.738160 0.674626i \(-0.764305\pi\)
0.674626 + 0.738160i \(0.264305\pi\)
\(348\) 0 0
\(349\) 158.622 158.622i 0.454506 0.454506i −0.442341 0.896847i \(-0.645852\pi\)
0.896847 + 0.442341i \(0.145852\pi\)
\(350\) 0 0
\(351\) 222.731i 0.634561i
\(352\) 0 0
\(353\) 404.451 1.14575 0.572877 0.819642i \(-0.305827\pi\)
0.572877 + 0.819642i \(0.305827\pi\)
\(354\) 0 0
\(355\) 623.511 + 623.511i 1.75637 + 1.75637i
\(356\) 0 0
\(357\) −113.255 113.255i −0.317242 0.317242i
\(358\) 0 0
\(359\) −423.833 −1.18059 −0.590297 0.807186i \(-0.700989\pi\)
−0.590297 + 0.807186i \(0.700989\pi\)
\(360\) 0 0
\(361\) 358.347i 0.992652i
\(362\) 0 0
\(363\) 97.7905 97.7905i 0.269395 0.269395i
\(364\) 0 0
\(365\) −519.219 + 519.219i −1.42252 + 1.42252i
\(366\) 0 0
\(367\) 477.144i 1.30012i 0.759883 + 0.650059i \(0.225256\pi\)
−0.759883 + 0.650059i \(0.774744\pi\)
\(368\) 0 0
\(369\) −8.07288 −0.0218777
\(370\) 0 0
\(371\) 82.4365 + 82.4365i 0.222201 + 0.222201i
\(372\) 0 0
\(373\) −112.221 112.221i −0.300860 0.300860i 0.540490 0.841350i \(-0.318239\pi\)
−0.841350 + 0.540490i \(0.818239\pi\)
\(374\) 0 0
\(375\) 139.680 0.372479
\(376\) 0 0
\(377\) 141.069i 0.374188i
\(378\) 0 0
\(379\) 52.2069 52.2069i 0.137749 0.137749i −0.634870 0.772619i \(-0.718946\pi\)
0.772619 + 0.634870i \(0.218946\pi\)
\(380\) 0 0
\(381\) 412.793 412.793i 1.08345 1.08345i
\(382\) 0 0
\(383\) 74.8407i 0.195406i 0.995216 + 0.0977032i \(0.0311496\pi\)
−0.995216 + 0.0977032i \(0.968850\pi\)
\(384\) 0 0
\(385\) 258.219 0.670699
\(386\) 0 0
\(387\) 0.266616 + 0.266616i 0.000688930 + 0.000688930i
\(388\) 0 0
\(389\) 57.0441 + 57.0441i 0.146643 + 0.146643i 0.776617 0.629974i \(-0.216934\pi\)
−0.629974 + 0.776617i \(0.716934\pi\)
\(390\) 0 0
\(391\) 61.1223 0.156323
\(392\) 0 0
\(393\) 75.4632i 0.192018i
\(394\) 0 0
\(395\) −628.910 + 628.910i −1.59218 + 1.59218i
\(396\) 0 0
\(397\) 355.874 355.874i 0.896407 0.896407i −0.0987089 0.995116i \(-0.531471\pi\)
0.995116 + 0.0987089i \(0.0314713\pi\)
\(398\) 0 0
\(399\) 14.7667i 0.0370093i
\(400\) 0 0
\(401\) 113.892 0.284019 0.142010 0.989865i \(-0.454644\pi\)
0.142010 + 0.989865i \(0.454644\pi\)
\(402\) 0 0
\(403\) −222.731 222.731i −0.552682 0.552682i
\(404\) 0 0
\(405\) 368.166 + 368.166i 0.909052 + 0.909052i
\(406\) 0 0
\(407\) −1.77973 −0.00437281
\(408\) 0 0
\(409\) 139.909i 0.342077i −0.985264 0.171038i \(-0.945288\pi\)
0.985264 0.171038i \(-0.0547122\pi\)
\(410\) 0 0
\(411\) −108.756 + 108.756i −0.264612 + 0.264612i
\(412\) 0 0
\(413\) −59.7394 + 59.7394i −0.144648 + 0.144648i
\(414\) 0 0
\(415\) 693.388i 1.67081i
\(416\) 0 0
\(417\) −73.4456 −0.176129
\(418\) 0 0
\(419\) −370.978 370.978i −0.885389 0.885389i 0.108687 0.994076i \(-0.465335\pi\)
−0.994076 + 0.108687i \(0.965335\pi\)
\(420\) 0 0
\(421\) −465.112 465.112i −1.10478 1.10478i −0.993825 0.110955i \(-0.964609\pi\)
−0.110955 0.993825i \(-0.535391\pi\)
\(422\) 0 0
\(423\) −3.78419 −0.00894608
\(424\) 0 0
\(425\) 314.839i 0.740797i
\(426\) 0 0
\(427\) 50.9923 50.9923i 0.119420 0.119420i
\(428\) 0 0
\(429\) −222.609 + 222.609i −0.518901 + 0.518901i
\(430\) 0 0
\(431\) 409.924i 0.951099i 0.879689 + 0.475549i \(0.157751\pi\)
−0.879689 + 0.475549i \(0.842249\pi\)
\(432\) 0 0
\(433\) −20.6859 −0.0477735 −0.0238868 0.999715i \(-0.507604\pi\)
−0.0238868 + 0.999715i \(0.507604\pi\)
\(434\) 0 0
\(435\) −237.314 237.314i −0.545550 0.545550i
\(436\) 0 0
\(437\) −3.98468 3.98468i −0.00911827 0.00911827i
\(438\) 0 0
\(439\) 63.2889 0.144166 0.0720830 0.997399i \(-0.477035\pi\)
0.0720830 + 0.997399i \(0.477035\pi\)
\(440\) 0 0
\(441\) 6.22065i 0.0141058i
\(442\) 0 0
\(443\) −297.084 + 297.084i −0.670619 + 0.670619i −0.957859 0.287240i \(-0.907262\pi\)
0.287240 + 0.957859i \(0.407262\pi\)
\(444\) 0 0
\(445\) 145.423 145.423i 0.326793 0.326793i
\(446\) 0 0
\(447\) 50.1177i 0.112120i
\(448\) 0 0
\(449\) 364.701 0.812251 0.406126 0.913817i \(-0.366880\pi\)
0.406126 + 0.913817i \(0.366880\pi\)
\(450\) 0 0
\(451\) 471.553 + 471.553i 1.04557 + 1.04557i
\(452\) 0 0
\(453\) 278.720 + 278.720i 0.615275 + 0.615275i
\(454\) 0 0
\(455\) −163.196 −0.358672
\(456\) 0 0
\(457\) 640.046i 1.40054i 0.713879 + 0.700269i \(0.246936\pi\)
−0.713879 + 0.700269i \(0.753064\pi\)
\(458\) 0 0
\(459\) 340.140 340.140i 0.741045 0.741045i
\(460\) 0 0
\(461\) 239.416 239.416i 0.519341 0.519341i −0.398031 0.917372i \(-0.630306\pi\)
0.917372 + 0.398031i \(0.130306\pi\)
\(462\) 0 0
\(463\) 479.413i 1.03545i −0.855548 0.517724i \(-0.826779\pi\)
0.855548 0.517724i \(-0.173221\pi\)
\(464\) 0 0
\(465\) 749.381 1.61157
\(466\) 0 0
\(467\) 403.375 + 403.375i 0.863758 + 0.863758i 0.991772 0.128015i \(-0.0408605\pi\)
−0.128015 + 0.991772i \(0.540860\pi\)
\(468\) 0 0
\(469\) 231.276 + 231.276i 0.493125 + 0.493125i
\(470\) 0 0
\(471\) −446.173 −0.947288
\(472\) 0 0
\(473\) 31.1471i 0.0658501i
\(474\) 0 0
\(475\) 20.5250 20.5250i 0.0432104 0.0432104i
\(476\) 0 0
\(477\) −4.23621 + 4.23621i −0.00888093 + 0.00888093i
\(478\) 0 0
\(479\) 460.611i 0.961609i 0.876828 + 0.480805i \(0.159655\pi\)
−0.876828 + 0.480805i \(0.840345\pi\)
\(480\) 0 0
\(481\) 1.12480 0.00233846
\(482\) 0 0
\(483\) −22.1823 22.1823i −0.0459260 0.0459260i
\(484\) 0 0
\(485\) 145.968 + 145.968i 0.300966 + 0.300966i
\(486\) 0 0
\(487\) 575.128 1.18096 0.590481 0.807052i \(-0.298938\pi\)
0.590481 + 0.807052i \(0.298938\pi\)
\(488\) 0 0
\(489\) 444.010i 0.907995i
\(490\) 0 0
\(491\) 271.375 271.375i 0.552699 0.552699i −0.374520 0.927219i \(-0.622192\pi\)
0.927219 + 0.374520i \(0.122192\pi\)
\(492\) 0 0
\(493\) −215.431 + 215.431i −0.436979 + 0.436979i
\(494\) 0 0
\(495\) 13.2693i 0.0268066i
\(496\) 0 0
\(497\) −410.832 −0.826624
\(498\) 0 0
\(499\) 268.082 + 268.082i 0.537239 + 0.537239i 0.922717 0.385478i \(-0.125963\pi\)
−0.385478 + 0.922717i \(0.625963\pi\)
\(500\) 0 0
\(501\) −234.651 234.651i −0.468365 0.468365i
\(502\) 0 0
\(503\) 368.002 0.731615 0.365807 0.930691i \(-0.380793\pi\)
0.365807 + 0.930691i \(0.380793\pi\)
\(504\) 0 0
\(505\) 254.215i 0.503395i
\(506\) 0 0
\(507\) −214.679 + 214.679i −0.423430 + 0.423430i
\(508\) 0 0
\(509\) −297.809 + 297.809i −0.585087 + 0.585087i −0.936297 0.351210i \(-0.885770\pi\)
0.351210 + 0.936297i \(0.385770\pi\)
\(510\) 0 0
\(511\) 342.114i 0.669498i
\(512\) 0 0
\(513\) −44.3488 −0.0864498
\(514\) 0 0
\(515\) −476.057 476.057i −0.924382 0.924382i
\(516\) 0 0
\(517\) 221.042 + 221.042i 0.427548 + 0.427548i
\(518\) 0 0
\(519\) −61.1332 −0.117790
\(520\) 0 0
\(521\) 95.5605i 0.183418i −0.995786 0.0917088i \(-0.970767\pi\)
0.995786 0.0917088i \(-0.0292329\pi\)
\(522\) 0 0
\(523\) −250.389 + 250.389i −0.478756 + 0.478756i −0.904734 0.425978i \(-0.859930\pi\)
0.425978 + 0.904734i \(0.359930\pi\)
\(524\) 0 0
\(525\) 114.260 114.260i 0.217638 0.217638i
\(526\) 0 0
\(527\) 680.279i 1.29085i
\(528\) 0 0
\(529\) −517.029 −0.977370
\(530\) 0 0
\(531\) −3.06986 3.06986i −0.00578128 0.00578128i
\(532\) 0 0
\(533\) −298.024 298.024i −0.559144 0.559144i
\(534\) 0 0
\(535\) −737.212 −1.37797
\(536\) 0 0
\(537\) 83.0055i 0.154573i
\(538\) 0 0
\(539\) 363.360 363.360i 0.674138 0.674138i
\(540\) 0 0
\(541\) 81.7015 81.7015i 0.151019 0.151019i −0.627554 0.778573i \(-0.715944\pi\)
0.778573 + 0.627554i \(0.215944\pi\)
\(542\) 0 0
\(543\) 707.175i 1.30235i
\(544\) 0 0
\(545\) −1160.58 −2.12951
\(546\) 0 0
\(547\) −381.162 381.162i −0.696823 0.696823i 0.266901 0.963724i \(-0.414000\pi\)
−0.963724 + 0.266901i \(0.914000\pi\)
\(548\) 0 0
\(549\) 2.62037 + 2.62037i 0.00477299 + 0.00477299i
\(550\) 0 0
\(551\) 28.0887 0.0509777
\(552\) 0 0
\(553\) 414.389i 0.749348i
\(554\) 0 0
\(555\) −1.89220 + 1.89220i −0.00340937 + 0.00340937i
\(556\) 0 0
\(557\) 63.7634 63.7634i 0.114476 0.114476i −0.647548 0.762025i \(-0.724206\pi\)
0.762025 + 0.647548i \(0.224206\pi\)
\(558\) 0 0
\(559\) 19.6851i 0.0352149i
\(560\) 0 0
\(561\) −679.906 −1.21195
\(562\) 0 0
\(563\) 333.679 + 333.679i 0.592681 + 0.592681i 0.938355 0.345674i \(-0.112350\pi\)
−0.345674 + 0.938355i \(0.612350\pi\)
\(564\) 0 0
\(565\) −447.081 447.081i −0.791293 0.791293i
\(566\) 0 0
\(567\) −242.585 −0.427839
\(568\) 0 0
\(569\) 93.3114i 0.163992i −0.996633 0.0819960i \(-0.973871\pi\)
0.996633 0.0819960i \(-0.0261295\pi\)
\(570\) 0 0
\(571\) 196.999 196.999i 0.345007 0.345007i −0.513239 0.858246i \(-0.671555\pi\)
0.858246 + 0.513239i \(0.171555\pi\)
\(572\) 0 0
\(573\) −412.793 + 412.793i −0.720406 + 0.720406i
\(574\) 0 0
\(575\) 61.6644i 0.107242i
\(576\) 0 0
\(577\) 370.057 0.641347 0.320673 0.947190i \(-0.396091\pi\)
0.320673 + 0.947190i \(0.396091\pi\)
\(578\) 0 0
\(579\) −84.8888 84.8888i −0.146613 0.146613i
\(580\) 0 0
\(581\) −228.437 228.437i −0.393179 0.393179i
\(582\) 0 0
\(583\) 494.890 0.848869
\(584\) 0 0
\(585\) 8.38623i 0.0143354i
\(586\) 0 0
\(587\) 328.063 328.063i 0.558880 0.558880i −0.370108 0.928989i \(-0.620679\pi\)
0.928989 + 0.370108i \(0.120679\pi\)
\(588\) 0 0
\(589\) −44.3488 + 44.3488i −0.0752950 + 0.0752950i
\(590\) 0 0
\(591\) 969.708i 1.64079i
\(592\) 0 0
\(593\) 1088.78 1.83605 0.918024 0.396525i \(-0.129784\pi\)
0.918024 + 0.396525i \(0.129784\pi\)
\(594\) 0 0
\(595\) −249.222 249.222i −0.418860 0.418860i
\(596\) 0 0
\(597\) −129.446 129.446i −0.216828 0.216828i
\(598\) 0 0
\(599\) 350.354 0.584899 0.292449 0.956281i \(-0.405530\pi\)
0.292449 + 0.956281i \(0.405530\pi\)
\(600\) 0 0
\(601\) 1021.45i 1.69958i 0.527123 + 0.849789i \(0.323271\pi\)
−0.527123 + 0.849789i \(0.676729\pi\)
\(602\) 0 0
\(603\) −11.8847 + 11.8847i −0.0197093 + 0.0197093i
\(604\) 0 0
\(605\) 215.191 215.191i 0.355687 0.355687i
\(606\) 0 0
\(607\) 394.204i 0.649431i −0.945812 0.324715i \(-0.894732\pi\)
0.945812 0.324715i \(-0.105268\pi\)
\(608\) 0 0
\(609\) 156.367 0.256760
\(610\) 0 0
\(611\) −139.700 139.700i −0.228641 0.228641i
\(612\) 0 0
\(613\) −157.606 157.606i −0.257106 0.257106i 0.566770 0.823876i \(-0.308193\pi\)
−0.823876 + 0.566770i \(0.808193\pi\)
\(614\) 0 0
\(615\) 1002.71 1.63042
\(616\) 0 0
\(617\) 609.080i 0.987164i −0.869699 0.493582i \(-0.835687\pi\)
0.869699 0.493582i \(-0.164313\pi\)
\(618\) 0 0
\(619\) −497.519 + 497.519i −0.803747 + 0.803747i −0.983679 0.179932i \(-0.942412\pi\)
0.179932 + 0.983679i \(0.442412\pi\)
\(620\) 0 0
\(621\) 66.6199 66.6199i 0.107278 0.107278i
\(622\) 0 0
\(623\) 95.8194i 0.153803i
\(624\) 0 0
\(625\) 752.924 1.20468
\(626\) 0 0
\(627\) 44.3244 + 44.3244i 0.0706929 + 0.0706929i
\(628\) 0 0
\(629\) 1.71772 + 1.71772i 0.00273087 + 0.00273087i
\(630\) 0 0
\(631\) −668.065 −1.05874 −0.529370 0.848391i \(-0.677572\pi\)
−0.529370 + 0.848391i \(0.677572\pi\)
\(632\) 0 0
\(633\) 635.665i 1.00421i
\(634\) 0 0
\(635\) 908.362 908.362i 1.43049 1.43049i
\(636\) 0 0
\(637\) −229.646 + 229.646i −0.360511 + 0.360511i
\(638\) 0 0
\(639\) 21.1117i 0.0330386i
\(640\) 0 0
\(641\) −419.792 −0.654902 −0.327451 0.944868i \(-0.606190\pi\)
−0.327451 + 0.944868i \(0.606190\pi\)
\(642\) 0 0
\(643\) −138.767 138.767i −0.215813 0.215813i 0.590919 0.806731i \(-0.298765\pi\)
−0.806731 + 0.590919i \(0.798765\pi\)
\(644\) 0 0
\(645\) −33.1155 33.1155i −0.0513418 0.0513418i
\(646\) 0 0
\(647\) −647.036 −1.00006 −0.500028 0.866009i \(-0.666677\pi\)
−0.500028 + 0.866009i \(0.666677\pi\)
\(648\) 0 0
\(649\) 358.633i 0.552594i
\(650\) 0 0
\(651\) −246.884 + 246.884i −0.379238 + 0.379238i
\(652\) 0 0
\(653\) 452.293 452.293i 0.692639 0.692639i −0.270173 0.962812i \(-0.587081\pi\)
0.962812 + 0.270173i \(0.0870808\pi\)
\(654\) 0 0
\(655\) 166.059i 0.253525i
\(656\) 0 0
\(657\) 17.5804 0.0267586
\(658\) 0 0
\(659\) 382.858 + 382.858i 0.580969 + 0.580969i 0.935169 0.354201i \(-0.115247\pi\)
−0.354201 + 0.935169i \(0.615247\pi\)
\(660\) 0 0
\(661\) 841.606 + 841.606i 1.27323 + 1.27323i 0.944383 + 0.328849i \(0.106661\pi\)
0.328849 + 0.944383i \(0.393339\pi\)
\(662\) 0 0
\(663\) 429.704 0.648120
\(664\) 0 0
\(665\) 32.4945i 0.0488639i
\(666\) 0 0
\(667\) −42.1944 + 42.1944i −0.0632599 + 0.0632599i
\(668\) 0 0
\(669\) −242.926 + 242.926i −0.363119 + 0.363119i
\(670\) 0 0
\(671\) 306.122i 0.456218i
\(672\) 0 0
\(673\) −506.103 −0.752010 −0.376005 0.926618i \(-0.622703\pi\)
−0.376005 + 0.926618i \(0.622703\pi\)
\(674\) 0 0
\(675\) 343.156 + 343.156i 0.508380 + 0.508380i
\(676\) 0 0
\(677\) 430.816 + 430.816i 0.636361 + 0.636361i 0.949656 0.313295i \(-0.101433\pi\)
−0.313295 + 0.949656i \(0.601433\pi\)
\(678\) 0 0
\(679\) −96.1787 −0.141648
\(680\) 0 0
\(681\) 106.132i 0.155848i
\(682\) 0 0
\(683\) −910.083 + 910.083i −1.33248 + 1.33248i −0.429333 + 0.903146i \(0.641251\pi\)
−0.903146 + 0.429333i \(0.858749\pi\)
\(684\) 0 0
\(685\) −239.319 + 239.319i −0.349371 + 0.349371i
\(686\) 0 0
\(687\) 653.197i 0.950796i
\(688\) 0 0
\(689\) −312.773 −0.453952
\(690\) 0 0
\(691\) −601.836 601.836i −0.870964 0.870964i 0.121614 0.992577i \(-0.461193\pi\)
−0.992577 + 0.121614i \(0.961193\pi\)
\(692\) 0 0
\(693\) −4.37156 4.37156i −0.00630817 0.00630817i
\(694\) 0 0
\(695\) −161.619 −0.232545
\(696\) 0 0
\(697\) 910.244i 1.30595i
\(698\) 0 0
\(699\) −251.783 + 251.783i −0.360204 + 0.360204i
\(700\) 0 0
\(701\) 555.343 555.343i 0.792215 0.792215i −0.189639 0.981854i \(-0.560732\pi\)
0.981854 + 0.189639i \(0.0607317\pi\)
\(702\) 0 0
\(703\) 0.223963i 0.000318582i
\(704\) 0 0
\(705\) 470.022 0.666697
\(706\) 0 0
\(707\) 83.7511 + 83.7511i 0.118460 + 0.118460i
\(708\) 0 0
\(709\) 412.979 + 412.979i 0.582480 + 0.582480i 0.935584 0.353104i \(-0.114874\pi\)
−0.353104 + 0.935584i \(0.614874\pi\)
\(710\) 0 0
\(711\) 21.2945 0.0299500
\(712\) 0 0
\(713\) 133.240i 0.186872i
\(714\) 0 0
\(715\) −489.856 + 489.856i −0.685114 + 0.685114i
\(716\) 0 0
\(717\) 516.039 516.039i 0.719720 0.719720i
\(718\) 0 0
\(719\) 1173.98i 1.63279i 0.577495 + 0.816394i \(0.304030\pi\)
−0.577495 + 0.816394i \(0.695970\pi\)
\(720\) 0 0
\(721\) 313.674 0.435054
\(722\) 0 0
\(723\) 906.750 + 906.750i 1.25415 + 1.25415i
\(724\) 0 0
\(725\) −217.342 217.342i −0.299781 0.299781i
\(726\) 0 0
\(727\) −678.813 −0.933718 −0.466859 0.884332i \(-0.654614\pi\)
−0.466859 + 0.884332i \(0.654614\pi\)
\(728\) 0 0
\(729\) 741.245i 1.01680i
\(730\) 0 0
\(731\) −30.0618 + 30.0618i −0.0411242 + 0.0411242i
\(732\) 0 0
\(733\) −336.854 + 336.854i −0.459556 + 0.459556i −0.898510 0.438954i \(-0.855349\pi\)
0.438954 + 0.898510i \(0.355349\pi\)
\(734\) 0 0
\(735\) 772.646i 1.05122i
\(736\) 0 0
\(737\) 1388.42 1.88387
\(738\) 0 0
\(739\) 178.478 + 178.478i 0.241513 + 0.241513i 0.817476 0.575963i \(-0.195373\pi\)
−0.575963 + 0.817476i \(0.695373\pi\)
\(740\) 0 0
\(741\) −28.0133 28.0133i −0.0378047 0.0378047i
\(742\) 0 0
\(743\) −795.320 −1.07042 −0.535208 0.844720i \(-0.679767\pi\)
−0.535208 + 0.844720i \(0.679767\pi\)
\(744\) 0 0
\(745\) 110.285i 0.148034i
\(746\) 0 0
\(747\) 11.7388 11.7388i 0.0157146 0.0157146i
\(748\) 0 0
\(749\) 242.875 242.875i 0.324266 0.324266i
\(750\) 0 0
\(751\) 102.850i 0.136951i 0.997653 + 0.0684755i \(0.0218135\pi\)
−0.997653 + 0.0684755i \(0.978186\pi\)
\(752\) 0 0
\(753\) −101.005 −0.134137
\(754\) 0 0
\(755\) 613.330 + 613.330i 0.812358 + 0.812358i
\(756\) 0 0
\(757\) −48.6324 48.6324i −0.0642436 0.0642436i 0.674255 0.738499i \(-0.264465\pi\)
−0.738499 + 0.674255i \(0.764465\pi\)
\(758\) 0 0
\(759\) −133.167 −0.175450
\(760\) 0 0
\(761\) 947.802i 1.24547i 0.782433 + 0.622734i \(0.213978\pi\)
−0.782433 + 0.622734i \(0.786022\pi\)
\(762\) 0 0
\(763\) 382.354 382.354i 0.501119 0.501119i
\(764\) 0 0
\(765\) 12.8069 12.8069i 0.0167410 0.0167410i
\(766\) 0 0
\(767\) 226.658i 0.295512i
\(768\) 0 0
\(769\) −183.427 −0.238527 −0.119263 0.992863i \(-0.538053\pi\)
−0.119263 + 0.992863i \(0.538053\pi\)
\(770\) 0 0
\(771\) −212.085 212.085i −0.275078 0.275078i
\(772\) 0 0
\(773\) 178.338 + 178.338i 0.230710 + 0.230710i 0.812989 0.582279i \(-0.197839\pi\)
−0.582279 + 0.812989i \(0.697839\pi\)
\(774\) 0 0
\(775\) 686.312 0.885564
\(776\) 0 0
\(777\) 1.24677i 0.00160460i
\(778\) 0 0
\(779\) −59.3406 + 59.3406i −0.0761754 + 0.0761754i
\(780\) 0 0
\(781\) −1233.17 + 1233.17i −1.57897 + 1.57897i
\(782\) 0 0
\(783\) 469.615i 0.599764i
\(784\) 0 0
\(785\) −981.815 −1.25072
\(786\) 0 0
\(787\) −480.981 480.981i −0.611158 0.611158i 0.332090 0.943248i \(-0.392246\pi\)
−0.943248 + 0.332090i \(0.892246\pi\)
\(788\) 0 0
\(789\) −455.831 455.831i −0.577732 0.577732i
\(790\) 0 0
\(791\) 294.582 0.372417
\(792\) 0 0
\(793\) 193.471i 0.243973i
\(794\) 0 0
\(795\) 526.165 526.165i 0.661843 0.661843i
\(796\) 0 0
\(797\) 558.478 558.478i 0.700725 0.700725i −0.263841 0.964566i \(-0.584989\pi\)
0.964566 + 0.263841i \(0.0849894\pi\)
\(798\) 0 0
\(799\) 426.680i 0.534017i
\(800\) 0 0
\(801\) −4.92392 −0.00614722
\(802\) 0 0
\(803\) −1026.90 1026.90i −1.27884 1.27884i
\(804\) 0 0
\(805\) −48.8126 48.8126i −0.0606368 0.0606368i
\(806\) 0 0
\(807\) 1079.89 1.33816
\(808\) 0 0
\(809\) 1152.43i 1.42451i 0.701918 + 0.712257i \(0.252327\pi\)
−0.701918 + 0.712257i \(0.747673\pi\)
\(810\) 0 0
\(811\) 364.890 364.890i 0.449926 0.449926i −0.445404 0.895330i \(-0.646940\pi\)
0.895330 + 0.445404i \(0.146940\pi\)
\(812\) 0 0
\(813\) 27.0424 27.0424i 0.0332624 0.0332624i
\(814\) 0 0
\(815\) 977.055i 1.19884i
\(816\) 0 0
\(817\) 3.91958 0.00479753
\(818\) 0 0
\(819\) 2.76285 + 2.76285i 0.00337344 + 0.00337344i
\(820\) 0 0
\(821\) 618.975 + 618.975i 0.753928 + 0.753928i 0.975210 0.221282i \(-0.0710240\pi\)
−0.221282 + 0.975210i \(0.571024\pi\)
\(822\) 0 0
\(823\) −626.066 −0.760712 −0.380356 0.924840i \(-0.624199\pi\)
−0.380356 + 0.924840i \(0.624199\pi\)
\(824\) 0 0
\(825\) 685.936i 0.831438i
\(826\) 0 0
\(827\) 375.666 375.666i 0.454252 0.454252i −0.442511 0.896763i \(-0.645912\pi\)
0.896763 + 0.442511i \(0.145912\pi\)
\(828\) 0 0
\(829\) 299.648 299.648i 0.361457 0.361457i −0.502892 0.864349i \(-0.667731\pi\)
0.864349 + 0.502892i \(0.167731\pi\)
\(830\) 0 0
\(831\) 324.258i 0.390202i
\(832\) 0 0
\(833\) −701.398 −0.842015
\(834\) 0 0
\(835\) −516.356 516.356i −0.618390 0.618390i
\(836\) 0 0
\(837\) −741.466 741.466i −0.885861 0.885861i
\(838\) 0 0
\(839\) 1477.80 1.76138 0.880689 0.473694i \(-0.157080\pi\)
0.880689 + 0.473694i \(0.157080\pi\)
\(840\) 0 0
\(841\) 543.565i 0.646331i
\(842\) 0 0
\(843\) 397.496 397.496i 0.471526 0.471526i
\(844\) 0 0
\(845\) −472.407 + 472.407i −0.559062 + 0.559062i
\(846\) 0 0
\(847\) 141.789i 0.167402i
\(848\) 0 0
\(849\) −291.998 −0.343932
\(850\) 0 0
\(851\) 0.336433 + 0.336433i 0.000395338 + 0.000395338i
\(852\) 0 0
\(853\) −404.051 404.051i −0.473682 0.473682i 0.429422 0.903104i \(-0.358717\pi\)
−0.903104 + 0.429422i \(0.858717\pi\)
\(854\) 0 0
\(855\) −1.66981 −0.00195300
\(856\) 0 0
\(857\) 892.363i 1.04126i −0.853781 0.520632i \(-0.825696\pi\)
0.853781 0.520632i \(-0.174304\pi\)
\(858\) 0 0
\(859\) −378.424 + 378.424i −0.440540 + 0.440540i −0.892193 0.451654i \(-0.850834\pi\)
0.451654 + 0.892193i \(0.350834\pi\)
\(860\) 0 0
\(861\) −330.342 + 330.342i −0.383672 + 0.383672i
\(862\) 0 0
\(863\) 1457.30i 1.68865i 0.535833 + 0.844324i \(0.319998\pi\)
−0.535833 + 0.844324i \(0.680002\pi\)
\(864\) 0 0
\(865\) −134.525 −0.155520
\(866\) 0 0
\(867\) 48.5127 + 48.5127i 0.0559547 + 0.0559547i
\(868\) 0 0
\(869\) −1243.85 1243.85i −1.43136 1.43136i
\(870\) 0 0
\(871\) −877.485 −1.00745
\(872\) 0 0
\(873\) 4.94238i 0.00566138i
\(874\) 0 0
\(875\) −101.263 + 101.263i −0.115729 + 0.115729i
\(876\) 0 0
\(877\) 571.322 571.322i 0.651450 0.651450i −0.301892 0.953342i \(-0.597618\pi\)
0.953342 + 0.301892i \(0.0976181\pi\)
\(878\) 0 0
\(879\) 1008.99i 1.14789i
\(880\) 0 0
\(881\) 994.662 1.12901 0.564507 0.825428i \(-0.309066\pi\)
0.564507 + 0.825428i \(0.309066\pi\)
\(882\) 0 0
\(883\) 74.0725 + 74.0725i 0.0838873 + 0.0838873i 0.747805 0.663918i \(-0.231108\pi\)
−0.663918 + 0.747805i \(0.731108\pi\)
\(884\) 0 0
\(885\) 381.297 + 381.297i 0.430844 + 0.430844i
\(886\) 0 0
\(887\) 522.759 0.589356 0.294678 0.955597i \(-0.404788\pi\)
0.294678 + 0.955597i \(0.404788\pi\)
\(888\) 0 0
\(889\) 598.520i 0.673251i
\(890\) 0 0
\(891\) −728.155 + 728.155i −0.817233 + 0.817233i
\(892\) 0 0
\(893\) −27.8161 + 27.8161i −0.0311491 + 0.0311491i
\(894\) 0 0
\(895\) 182.656i 0.204085i
\(896\) 0 0
\(897\) 84.1619 0.0938260
\(898\) 0 0
\(899\) 469.615 + 469.615i 0.522375 + 0.522375i
\(900\) 0 0
\(901\) −477.646 477.646i −0.530129 0.530129i
\(902\) 0 0
\(903\) 21.8198 0.0241637
\(904\) 0 0
\(905\) 1556.16i 1.71951i
\(906\) 0 0
\(907\) 442.760 442.760i 0.488159 0.488159i −0.419566 0.907725i \(-0.637818\pi\)
0.907725 + 0.419566i \(0.137818\pi\)
\(908\) 0 0
\(909\) −4.30376 + 4.30376i −0.00473461 + 0.00473461i
\(910\) 0 0
\(911\) 835.738i 0.917385i −0.888595 0.458692i \(-0.848318\pi\)
0.888595 0.458692i \(-0.151682\pi\)
\(912\) 0 0
\(913\) −1371.37 −1.50205
\(914\) 0 0
\(915\) −325.467 325.467i −0.355702 0.355702i
\(916\) 0 0
\(917\) −54.7082 54.7082i −0.0596599 0.0596599i
\(918\) 0 0
\(919\) −776.423 −0.844856 −0.422428 0.906396i \(-0.638822\pi\)
−0.422428 + 0.906396i \(0.638822\pi\)
\(920\) 0 0
\(921\) 972.260i 1.05566i
\(922\) 0 0
\(923\) 779.372 779.372i 0.844390 0.844390i
\(924\) 0 0
\(925\) −1.73295 + 1.73295i −0.00187346 + 0.00187346i
\(926\) 0 0
\(927\) 16.1189i 0.0173883i
\(928\) 0 0
\(929\) −144.945 −0.156022 −0.0780112 0.996952i \(-0.524857\pi\)
−0.0780112 + 0.996952i \(0.524857\pi\)
\(930\) 0 0
\(931\) 45.7256 + 45.7256i 0.0491145 + 0.0491145i
\(932\) 0 0
\(933\) 1079.41 + 1079.41i 1.15693 + 1.15693i
\(934\) 0 0
\(935\) −1496.15 −1.60016
\(936\) 0 0
\(937\) 851.499i 0.908750i 0.890811 + 0.454375i \(0.150137\pi\)
−0.890811 + 0.454375i \(0.849863\pi\)
\(938\) 0 0
\(939\) 725.966 725.966i 0.773127 0.773127i
\(940\) 0 0
\(941\) −1251.60 + 1251.60i −1.33008 + 1.33008i −0.424778 + 0.905297i \(0.639648\pi\)
−0.905297 + 0.424778i \(0.860352\pi\)
\(942\) 0 0
\(943\) 178.281i 0.189057i
\(944\) 0 0
\(945\) −543.275 −0.574894
\(946\) 0 0
\(947\) 919.818 + 919.818i 0.971296 + 0.971296i 0.999599 0.0283032i \(-0.00901039\pi\)
−0.0283032 + 0.999599i \(0.509010\pi\)
\(948\) 0 0
\(949\) 649.009 + 649.009i 0.683887 + 0.683887i
\(950\) 0 0
\(951\) −1453.57 −1.52847
\(952\) 0 0
\(953\) 489.450i 0.513589i −0.966466 0.256794i \(-0.917334\pi\)
0.966466 0.256794i \(-0.0826663\pi\)
\(954\) 0 0
\(955\) −908.362 + 908.362i −0.951164 + 0.951164i
\(956\) 0 0
\(957\) 469.357 469.357i 0.490447 0.490447i
\(958\) 0 0
\(959\) 157.688i 0.164429i
\(960\) 0 0
\(961\) −521.932 −0.543113
\(962\) 0 0
\(963\) 12.4807 + 12.4807i 0.0129603 + 0.0129603i
\(964\) 0 0
\(965\) −186.800 186.800i −0.193575 0.193575i
\(966\) 0 0
\(967\) 1368.49 1.41519 0.707594 0.706619i \(-0.249780\pi\)
0.707594 + 0.706619i \(0.249780\pi\)
\(968\) 0 0
\(969\) 85.5599i 0.0882971i
\(970\) 0 0
\(971\) −1013.79 + 1013.79i −1.04407 + 1.04407i −0.0450900 + 0.998983i \(0.514357\pi\)
−0.998983 + 0.0450900i \(0.985643\pi\)
\(972\) 0 0
\(973\) 53.2455 53.2455i 0.0547230 0.0547230i
\(974\) 0 0
\(975\) 433.515i 0.444631i
\(976\) 0 0
\(977\) −5.19534 −0.00531765 −0.00265882 0.999996i \(-0.500846\pi\)
−0.00265882 + 0.999996i \(0.500846\pi\)
\(978\) 0 0
\(979\) 287.616 + 287.616i 0.293785 + 0.293785i
\(980\) 0 0
\(981\) 19.6482 + 19.6482i 0.0200288 + 0.0200288i
\(982\) 0 0
\(983\) 1591.90 1.61943 0.809714 0.586825i \(-0.199622\pi\)
0.809714 + 0.586825i \(0.199622\pi\)
\(984\) 0 0
\(985\) 2133.87i 2.16636i
\(986\) 0 0
\(987\) −154.849 + 154.849i −0.156888 + 0.156888i
\(988\) 0 0
\(989\) −5.88792 + 5.88792i −0.00595340 + 0.00595340i
\(990\) 0 0
\(991\) 622.896i 0.628553i −0.949331 0.314277i \(-0.898238\pi\)
0.949331 0.314277i \(-0.101762\pi\)
\(992\) 0 0
\(993\) 1791.43 1.80406
\(994\) 0 0
\(995\) −284.850 284.850i −0.286282 0.286282i
\(996\) 0 0
\(997\) 635.503 + 635.503i 0.637415 + 0.637415i 0.949917 0.312502i \(-0.101167\pi\)
−0.312502 + 0.949917i \(0.601167\pi\)
\(998\) 0 0
\(999\) 3.74443 0.00374818
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 128.3.f.a.31.3 6
3.2 odd 2 1152.3.m.b.415.1 6
4.3 odd 2 128.3.f.b.31.1 6
8.3 odd 2 16.3.f.a.11.1 yes 6
8.5 even 2 64.3.f.a.15.1 6
12.11 even 2 1152.3.m.a.415.1 6
16.3 odd 4 inner 128.3.f.a.95.3 6
16.5 even 4 16.3.f.a.3.1 6
16.11 odd 4 64.3.f.a.47.1 6
16.13 even 4 128.3.f.b.95.1 6
24.5 odd 2 576.3.m.a.271.3 6
24.11 even 2 144.3.m.a.91.3 6
32.3 odd 8 1024.3.c.j.1023.3 12
32.5 even 8 1024.3.d.k.511.4 12
32.11 odd 8 1024.3.d.k.511.3 12
32.13 even 8 1024.3.c.j.1023.4 12
32.19 odd 8 1024.3.c.j.1023.10 12
32.21 even 8 1024.3.d.k.511.9 12
32.27 odd 8 1024.3.d.k.511.10 12
32.29 even 8 1024.3.c.j.1023.9 12
40.3 even 4 400.3.k.d.299.2 6
40.19 odd 2 400.3.r.c.251.3 6
40.27 even 4 400.3.k.c.299.2 6
48.5 odd 4 144.3.m.a.19.3 6
48.11 even 4 576.3.m.a.559.3 6
48.29 odd 4 1152.3.m.a.991.1 6
48.35 even 4 1152.3.m.b.991.1 6
80.37 odd 4 400.3.k.d.99.2 6
80.53 odd 4 400.3.k.c.99.2 6
80.69 even 4 400.3.r.c.51.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.f.a.3.1 6 16.5 even 4
16.3.f.a.11.1 yes 6 8.3 odd 2
64.3.f.a.15.1 6 8.5 even 2
64.3.f.a.47.1 6 16.11 odd 4
128.3.f.a.31.3 6 1.1 even 1 trivial
128.3.f.a.95.3 6 16.3 odd 4 inner
128.3.f.b.31.1 6 4.3 odd 2
128.3.f.b.95.1 6 16.13 even 4
144.3.m.a.19.3 6 48.5 odd 4
144.3.m.a.91.3 6 24.11 even 2
400.3.k.c.99.2 6 80.53 odd 4
400.3.k.c.299.2 6 40.27 even 4
400.3.k.d.99.2 6 80.37 odd 4
400.3.k.d.299.2 6 40.3 even 4
400.3.r.c.51.3 6 80.69 even 4
400.3.r.c.251.3 6 40.19 odd 2
576.3.m.a.271.3 6 24.5 odd 2
576.3.m.a.559.3 6 48.11 even 4
1024.3.c.j.1023.3 12 32.3 odd 8
1024.3.c.j.1023.4 12 32.13 even 8
1024.3.c.j.1023.9 12 32.29 even 8
1024.3.c.j.1023.10 12 32.19 odd 8
1024.3.d.k.511.3 12 32.11 odd 8
1024.3.d.k.511.4 12 32.5 even 8
1024.3.d.k.511.9 12 32.21 even 8
1024.3.d.k.511.10 12 32.27 odd 8
1152.3.m.a.415.1 6 12.11 even 2
1152.3.m.a.991.1 6 48.29 odd 4
1152.3.m.b.415.1 6 3.2 odd 2
1152.3.m.b.991.1 6 48.35 even 4