Properties

Label 128.3.f.b.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 \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 128.31
Dual form 128.3.f.b.95.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.24914 + 3.24914i) q^{3} +(0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +12.1138i q^{9} +O(q^{10})\) \(q+(3.24914 + 3.24914i) q^{3} +(0.0586332 + 0.0586332i) q^{5} +4.61555 q^{7} +12.1138i q^{9} +(5.36641 - 5.36641i) q^{11} +(-11.0552 + 11.0552i) q^{13} +0.381015i q^{15} -12.8793 q^{17} +(-2.63359 - 2.63359i) q^{19} +(14.9966 + 14.9966i) q^{21} +16.3810 q^{23} -24.9931i q^{25} +(-10.1173 + 10.1173i) q^{27} +(26.0518 - 26.0518i) q^{29} +20.2345i q^{31} +34.8724 q^{33} +(0.270624 + 0.270624i) q^{35} +(-41.2829 - 41.2829i) q^{37} -71.8398 q^{39} +3.29640i q^{41} +(0.786951 - 0.786951i) q^{43} +(-0.710272 + 0.710272i) q^{45} -79.7517i q^{47} -27.6967 q^{49} +(-41.8466 - 41.8466i) q^{51} +(-1.06207 - 1.06207i) q^{53} +0.629299 q^{55} -17.1138i q^{57} +(-32.5163 + 32.5163i) q^{59} +(-15.2897 + 15.2897i) q^{61} +55.9119i q^{63} -1.29640 q^{65} +(60.0631 + 60.0631i) q^{67} +(53.2242 + 53.2242i) q^{69} +56.3535 q^{71} -9.70663i q^{73} +(81.2062 - 81.2062i) q^{75} +(24.7689 - 24.7689i) q^{77} +84.4278i q^{79} +43.2796 q^{81} +(-26.7577 - 26.7577i) q^{83} +(-0.755154 - 0.755154i) q^{85} +169.292 q^{87} +115.555i q^{89} +(-51.0258 + 51.0258i) q^{91} +(-65.7448 + 65.7448i) q^{93} -0.308832i q^{95} -146.245 q^{97} +(65.0077 + 65.0077i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 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
\(3\) 3.24914 + 3.24914i 1.08305 + 1.08305i 0.996224 + 0.0868231i \(0.0276715\pi\)
0.0868231 + 0.996224i \(0.472329\pi\)
\(4\) 0 0
\(5\) 0.0586332 + 0.0586332i 0.0117266 + 0.0117266i 0.712946 0.701219i \(-0.247361\pi\)
−0.701219 + 0.712946i \(0.747361\pi\)
\(6\) 0 0
\(7\) 4.61555 0.659364 0.329682 0.944092i \(-0.393058\pi\)
0.329682 + 0.944092i \(0.393058\pi\)
\(8\) 0 0
\(9\) 12.1138i 1.34598i
\(10\) 0 0
\(11\) 5.36641 5.36641i 0.487855 0.487855i −0.419774 0.907629i \(-0.637891\pi\)
0.907629 + 0.419774i \(0.137891\pi\)
\(12\) 0 0
\(13\) −11.0552 + 11.0552i −0.850400 + 0.850400i −0.990182 0.139783i \(-0.955360\pi\)
0.139783 + 0.990182i \(0.455360\pi\)
\(14\) 0 0
\(15\) 0.381015i 0.0254010i
\(16\) 0 0
\(17\) −12.8793 −0.757606 −0.378803 0.925477i \(-0.623664\pi\)
−0.378803 + 0.925477i \(0.623664\pi\)
\(18\) 0 0
\(19\) −2.63359 2.63359i −0.138610 0.138610i 0.634397 0.773007i \(-0.281248\pi\)
−0.773007 + 0.634397i \(0.781248\pi\)
\(20\) 0 0
\(21\) 14.9966 + 14.9966i 0.714122 + 0.714122i
\(22\) 0 0
\(23\) 16.3810 0.712218 0.356109 0.934444i \(-0.384103\pi\)
0.356109 + 0.934444i \(0.384103\pi\)
\(24\) 0 0
\(25\) 24.9931i 0.999725i
\(26\) 0 0
\(27\) −10.1173 + 10.1173i −0.374714 + 0.374714i
\(28\) 0 0
\(29\) 26.0518 26.0518i 0.898336 0.898336i −0.0969525 0.995289i \(-0.530910\pi\)
0.995289 + 0.0969525i \(0.0309095\pi\)
\(30\) 0 0
\(31\) 20.2345i 0.652727i 0.945244 + 0.326363i \(0.105823\pi\)
−0.945244 + 0.326363i \(0.894177\pi\)
\(32\) 0 0
\(33\) 34.8724 1.05674
\(34\) 0 0
\(35\) 0.270624 + 0.270624i 0.00773212 + 0.00773212i
\(36\) 0 0
\(37\) −41.2829 41.2829i −1.11575 1.11575i −0.992358 0.123395i \(-0.960622\pi\)
−0.123395 0.992358i \(-0.539378\pi\)
\(38\) 0 0
\(39\) −71.8398 −1.84205
\(40\) 0 0
\(41\) 3.29640i 0.0804001i 0.999192 + 0.0402000i \(0.0127995\pi\)
−0.999192 + 0.0402000i \(0.987200\pi\)
\(42\) 0 0
\(43\) 0.786951 0.786951i 0.0183012 0.0183012i −0.697897 0.716198i \(-0.745881\pi\)
0.716198 + 0.697897i \(0.245881\pi\)
\(44\) 0 0
\(45\) −0.710272 + 0.710272i −0.0157838 + 0.0157838i
\(46\) 0 0
\(47\) 79.7517i 1.69685i −0.529320 0.848423i \(-0.677553\pi\)
0.529320 0.848423i \(-0.322447\pi\)
\(48\) 0 0
\(49\) −27.6967 −0.565239
\(50\) 0 0
\(51\) −41.8466 41.8466i −0.820522 0.820522i
\(52\) 0 0
\(53\) −1.06207 1.06207i −0.0200391 0.0200391i 0.697016 0.717055i \(-0.254511\pi\)
−0.717055 + 0.697016i \(0.754511\pi\)
\(54\) 0 0
\(55\) 0.629299 0.0114418
\(56\) 0 0
\(57\) 17.1138i 0.300243i
\(58\) 0 0
\(59\) −32.5163 + 32.5163i −0.551124 + 0.551124i −0.926765 0.375641i \(-0.877423\pi\)
0.375641 + 0.926765i \(0.377423\pi\)
\(60\) 0 0
\(61\) −15.2897 + 15.2897i −0.250651 + 0.250651i −0.821238 0.570586i \(-0.806716\pi\)
0.570586 + 0.821238i \(0.306716\pi\)
\(62\) 0 0
\(63\) 55.9119i 0.887491i
\(64\) 0 0
\(65\) −1.29640 −0.0199446
\(66\) 0 0
\(67\) 60.0631 + 60.0631i 0.896465 + 0.896465i 0.995122 0.0986569i \(-0.0314546\pi\)
−0.0986569 + 0.995122i \(0.531455\pi\)
\(68\) 0 0
\(69\) 53.2242 + 53.2242i 0.771365 + 0.771365i
\(70\) 0 0
\(71\) 56.3535 0.793711 0.396856 0.917881i \(-0.370101\pi\)
0.396856 + 0.917881i \(0.370101\pi\)
\(72\) 0 0
\(73\) 9.70663i 0.132968i −0.997788 0.0664838i \(-0.978822\pi\)
0.997788 0.0664838i \(-0.0211781\pi\)
\(74\) 0 0
\(75\) 81.2062 81.2062i 1.08275 1.08275i
\(76\) 0 0
\(77\) 24.7689 24.7689i 0.321674 0.321674i
\(78\) 0 0
\(79\) 84.4278i 1.06871i 0.845261 + 0.534353i \(0.179445\pi\)
−0.845261 + 0.534353i \(0.820555\pi\)
\(80\) 0 0
\(81\) 43.2796 0.534316
\(82\) 0 0
\(83\) −26.7577 26.7577i −0.322382 0.322382i 0.527298 0.849680i \(-0.323205\pi\)
−0.849680 + 0.527298i \(0.823205\pi\)
\(84\) 0 0
\(85\) −0.755154 0.755154i −0.00888416 0.00888416i
\(86\) 0 0
\(87\) 169.292 1.94588
\(88\) 0 0
\(89\) 115.555i 1.29838i 0.760628 + 0.649188i \(0.224891\pi\)
−0.760628 + 0.649188i \(0.775109\pi\)
\(90\) 0 0
\(91\) −51.0258 + 51.0258i −0.560723 + 0.560723i
\(92\) 0 0
\(93\) −65.7448 + 65.7448i −0.706934 + 0.706934i
\(94\) 0 0
\(95\) 0.308832i 0.00325086i
\(96\) 0 0
\(97\) −146.245 −1.50768 −0.753841 0.657056i \(-0.771801\pi\)
−0.753841 + 0.657056i \(0.771801\pi\)
\(98\) 0 0
\(99\) 65.0077 + 65.0077i 0.656644 + 0.656644i
\(100\) 0 0
\(101\) 53.8554 + 53.8554i 0.533222 + 0.533222i 0.921530 0.388308i \(-0.126940\pi\)
−0.388308 + 0.921530i \(0.626940\pi\)
\(102\) 0 0
\(103\) −158.184 −1.53577 −0.767885 0.640588i \(-0.778691\pi\)
−0.767885 + 0.640588i \(0.778691\pi\)
\(104\) 0 0
\(105\) 1.75859i 0.0167485i
\(106\) 0 0
\(107\) 57.6009 57.6009i 0.538327 0.538327i −0.384711 0.923037i \(-0.625699\pi\)
0.923037 + 0.384711i \(0.125699\pi\)
\(108\) 0 0
\(109\) −56.8795 + 56.8795i −0.521830 + 0.521830i −0.918124 0.396294i \(-0.870296\pi\)
0.396294 + 0.918124i \(0.370296\pi\)
\(110\) 0 0
\(111\) 268.268i 2.41683i
\(112\) 0 0
\(113\) 135.731 1.20116 0.600580 0.799565i \(-0.294936\pi\)
0.600580 + 0.799565i \(0.294936\pi\)
\(114\) 0 0
\(115\) 0.960471 + 0.960471i 0.00835192 + 0.00835192i
\(116\) 0 0
\(117\) −133.921 133.921i −1.14462 1.14462i
\(118\) 0 0
\(119\) −59.4450 −0.499538
\(120\) 0 0
\(121\) 63.4034i 0.523995i
\(122\) 0 0
\(123\) −10.7105 + 10.7105i −0.0870770 + 0.0870770i
\(124\) 0 0
\(125\) 2.93125 2.93125i 0.0234500 0.0234500i
\(126\) 0 0
\(127\) 166.552i 1.31144i 0.755006 + 0.655718i \(0.227634\pi\)
−0.755006 + 0.655718i \(0.772366\pi\)
\(128\) 0 0
\(129\) 5.11383 0.0396421
\(130\) 0 0
\(131\) −22.2547 22.2547i −0.169883 0.169883i 0.617045 0.786928i \(-0.288330\pi\)
−0.786928 + 0.617045i \(0.788330\pi\)
\(132\) 0 0
\(133\) −12.1555 12.1555i −0.0913945 0.0913945i
\(134\) 0 0
\(135\) −1.18641 −0.00878826
\(136\) 0 0
\(137\) 174.890i 1.27657i −0.769800 0.638285i \(-0.779644\pi\)
0.769800 0.638285i \(-0.220356\pi\)
\(138\) 0 0
\(139\) −99.8891 + 99.8891i −0.718627 + 0.718627i −0.968324 0.249697i \(-0.919669\pi\)
0.249697 + 0.968324i \(0.419669\pi\)
\(140\) 0 0
\(141\) 259.125 259.125i 1.83776 1.83776i
\(142\) 0 0
\(143\) 118.653i 0.829744i
\(144\) 0 0
\(145\) 3.05499 0.0210689
\(146\) 0 0
\(147\) −89.9905 89.9905i −0.612181 0.612181i
\(148\) 0 0
\(149\) 74.8860 + 74.8860i 0.502590 + 0.502590i 0.912242 0.409652i \(-0.134350\pi\)
−0.409652 + 0.912242i \(0.634350\pi\)
\(150\) 0 0
\(151\) 70.0357 0.463813 0.231906 0.972738i \(-0.425504\pi\)
0.231906 + 0.972738i \(0.425504\pi\)
\(152\) 0 0
\(153\) 156.018i 1.01972i
\(154\) 0 0
\(155\) −1.18641 + 1.18641i −0.00765429 + 0.00765429i
\(156\) 0 0
\(157\) 29.5307 29.5307i 0.188094 0.188094i −0.606778 0.794872i \(-0.707538\pi\)
0.794872 + 0.606778i \(0.207538\pi\)
\(158\) 0 0
\(159\) 6.90164i 0.0434065i
\(160\) 0 0
\(161\) 75.6074 0.469611
\(162\) 0 0
\(163\) −47.7990 47.7990i −0.293245 0.293245i 0.545116 0.838361i \(-0.316486\pi\)
−0.838361 + 0.545116i \(0.816486\pi\)
\(164\) 0 0
\(165\) 2.04468 + 2.04468i 0.0123920 + 0.0123920i
\(166\) 0 0
\(167\) −156.268 −0.935734 −0.467867 0.883799i \(-0.654977\pi\)
−0.467867 + 0.883799i \(0.654977\pi\)
\(168\) 0 0
\(169\) 75.4347i 0.446359i
\(170\) 0 0
\(171\) 31.9029 31.9029i 0.186567 0.186567i
\(172\) 0 0
\(173\) −190.103 + 190.103i −1.09886 + 1.09886i −0.104319 + 0.994544i \(0.533266\pi\)
−0.994544 + 0.104319i \(0.966734\pi\)
\(174\) 0 0
\(175\) 115.357i 0.659183i
\(176\) 0 0
\(177\) −211.300 −1.19379
\(178\) 0 0
\(179\) −54.2749 54.2749i −0.303212 0.303212i 0.539057 0.842269i \(-0.318781\pi\)
−0.842269 + 0.539057i \(0.818781\pi\)
\(180\) 0 0
\(181\) −19.7343 19.7343i −0.109029 0.109029i 0.650487 0.759517i \(-0.274565\pi\)
−0.759517 + 0.650487i \(0.774565\pi\)
\(182\) 0 0
\(183\) −99.3569 −0.542934
\(184\) 0 0
\(185\) 4.84109i 0.0261680i
\(186\) 0 0
\(187\) −69.1155 + 69.1155i −0.369602 + 0.369602i
\(188\) 0 0
\(189\) −46.6967 + 46.6967i −0.247073 + 0.247073i
\(190\) 0 0
\(191\) 166.552i 0.872002i −0.899946 0.436001i \(-0.856394\pi\)
0.899946 0.436001i \(-0.143606\pi\)
\(192\) 0 0
\(193\) 2.18257 0.0113087 0.00565434 0.999984i \(-0.498200\pi\)
0.00565434 + 0.999984i \(0.498200\pi\)
\(194\) 0 0
\(195\) −4.21219 4.21219i −0.0216010 0.0216010i
\(196\) 0 0
\(197\) −67.4310 67.4310i −0.342290 0.342290i 0.514938 0.857227i \(-0.327815\pi\)
−0.857227 + 0.514938i \(0.827815\pi\)
\(198\) 0 0
\(199\) 222.906 1.12013 0.560065 0.828449i \(-0.310776\pi\)
0.560065 + 0.828449i \(0.310776\pi\)
\(200\) 0 0
\(201\) 390.307i 1.94183i
\(202\) 0 0
\(203\) 120.243 120.243i 0.592331 0.592331i
\(204\) 0 0
\(205\) −0.193278 + 0.193278i −0.000942822 + 0.000942822i
\(206\) 0 0
\(207\) 198.437i 0.958632i
\(208\) 0 0
\(209\) −28.2659 −0.135243
\(210\) 0 0
\(211\) 147.118 + 147.118i 0.697240 + 0.697240i 0.963814 0.266574i \(-0.0858917\pi\)
−0.266574 + 0.963814i \(0.585892\pi\)
\(212\) 0 0
\(213\) 183.100 + 183.100i 0.859627 + 0.859627i
\(214\) 0 0
\(215\) 0.0922828 0.000429223
\(216\) 0 0
\(217\) 93.3934i 0.430385i
\(218\) 0 0
\(219\) 31.5382 31.5382i 0.144010 0.144010i
\(220\) 0 0
\(221\) 142.383 142.383i 0.644268 0.644268i
\(222\) 0 0
\(223\) 60.7036i 0.272213i −0.990694 0.136107i \(-0.956541\pi\)
0.990694 0.136107i \(-0.0434590\pi\)
\(224\) 0 0
\(225\) 302.762 1.34561
\(226\) 0 0
\(227\) 225.526 + 225.526i 0.993505 + 0.993505i 0.999979 0.00647371i \(-0.00206066\pi\)
−0.00647371 + 0.999979i \(0.502061\pi\)
\(228\) 0 0
\(229\) −227.796 227.796i −0.994743 0.994743i 0.00524305 0.999986i \(-0.498331\pi\)
−0.999986 + 0.00524305i \(0.998331\pi\)
\(230\) 0 0
\(231\) 160.955 0.696776
\(232\) 0 0
\(233\) 121.053i 0.519540i 0.965671 + 0.259770i \(0.0836467\pi\)
−0.965671 + 0.259770i \(0.916353\pi\)
\(234\) 0 0
\(235\) 4.67610 4.67610i 0.0198983 0.0198983i
\(236\) 0 0
\(237\) −274.318 + 274.318i −1.15746 + 1.15746i
\(238\) 0 0
\(239\) 221.393i 0.926332i 0.886271 + 0.463166i \(0.153287\pi\)
−0.886271 + 0.463166i \(0.846713\pi\)
\(240\) 0 0
\(241\) 84.2667 0.349654 0.174827 0.984599i \(-0.444063\pi\)
0.174827 + 0.984599i \(0.444063\pi\)
\(242\) 0 0
\(243\) 231.677 + 231.677i 0.953403 + 0.953403i
\(244\) 0 0
\(245\) −1.62395 1.62395i −0.00662835 0.00662835i
\(246\) 0 0
\(247\) 58.2298 0.235748
\(248\) 0 0
\(249\) 173.879i 0.698310i
\(250\) 0 0
\(251\) 176.615 176.615i 0.703646 0.703646i −0.261545 0.965191i \(-0.584232\pi\)
0.965191 + 0.261545i \(0.0842321\pi\)
\(252\) 0 0
\(253\) 87.9072 87.9072i 0.347459 0.347459i
\(254\) 0 0
\(255\) 4.90720i 0.0192439i
\(256\) 0 0
\(257\) −163.001 −0.634244 −0.317122 0.948385i \(-0.602717\pi\)
−0.317122 + 0.948385i \(0.602717\pi\)
\(258\) 0 0
\(259\) −190.543 190.543i −0.735687 0.735687i
\(260\) 0 0
\(261\) 315.587 + 315.587i 1.20914 + 1.20914i
\(262\) 0 0
\(263\) 175.001 0.665404 0.332702 0.943032i \(-0.392040\pi\)
0.332702 + 0.943032i \(0.392040\pi\)
\(264\) 0 0
\(265\) 0.124545i 0.000469982i
\(266\) 0 0
\(267\) −375.456 + 375.456i −1.40620 + 1.40620i
\(268\) 0 0
\(269\) −29.7489 + 29.7489i −0.110591 + 0.110591i −0.760237 0.649646i \(-0.774917\pi\)
0.649646 + 0.760237i \(0.274917\pi\)
\(270\) 0 0
\(271\) 275.891i 1.01805i −0.860753 0.509024i \(-0.830007\pi\)
0.860753 0.509024i \(-0.169993\pi\)
\(272\) 0 0
\(273\) −331.580 −1.21458
\(274\) 0 0
\(275\) −134.123 134.123i −0.487721 0.487721i
\(276\) 0 0
\(277\) 278.337 + 278.337i 1.00483 + 1.00483i 0.999988 + 0.00484003i \(0.00154063\pi\)
0.00484003 + 0.999988i \(0.498459\pi\)
\(278\) 0 0
\(279\) −245.118 −0.878558
\(280\) 0 0
\(281\) 202.356i 0.720128i −0.932928 0.360064i \(-0.882755\pi\)
0.932928 0.360064i \(-0.117245\pi\)
\(282\) 0 0
\(283\) 292.256 292.256i 1.03271 1.03271i 0.0332615 0.999447i \(-0.489411\pi\)
0.999447 0.0332615i \(-0.0105894\pi\)
\(284\) 0 0
\(285\) 1.00344 1.00344i 0.00352083 0.00352083i
\(286\) 0 0
\(287\) 15.2147i 0.0530129i
\(288\) 0 0
\(289\) −123.124 −0.426034
\(290\) 0 0
\(291\) −475.171 475.171i −1.63289 1.63289i
\(292\) 0 0
\(293\) −331.170 331.170i −1.13027 1.13027i −0.990132 0.140141i \(-0.955244\pi\)
−0.140141 0.990132i \(-0.544756\pi\)
\(294\) 0 0
\(295\) −3.81307 −0.0129257
\(296\) 0 0
\(297\) 108.587i 0.365612i
\(298\) 0 0
\(299\) −181.095 + 181.095i −0.605670 + 0.605670i
\(300\) 0 0
\(301\) 3.63221 3.63221i 0.0120671 0.0120671i
\(302\) 0 0
\(303\) 349.968i 1.15501i
\(304\) 0 0
\(305\) −1.79297 −0.00587859
\(306\) 0 0
\(307\) −23.7513 23.7513i −0.0773656 0.0773656i 0.667365 0.744731i \(-0.267422\pi\)
−0.744731 + 0.667365i \(0.767422\pi\)
\(308\) 0 0
\(309\) −513.963 513.963i −1.66331 1.66331i
\(310\) 0 0
\(311\) −157.757 −0.507258 −0.253629 0.967302i \(-0.581624\pi\)
−0.253629 + 0.967302i \(0.581624\pi\)
\(312\) 0 0
\(313\) 58.5936i 0.187200i −0.995610 0.0936000i \(-0.970163\pi\)
0.995610 0.0936000i \(-0.0298375\pi\)
\(314\) 0 0
\(315\) −3.27829 + 3.27829i −0.0104073 + 0.0104073i
\(316\) 0 0
\(317\) −27.0040 + 27.0040i −0.0851863 + 0.0851863i −0.748416 0.663230i \(-0.769185\pi\)
0.663230 + 0.748416i \(0.269185\pi\)
\(318\) 0 0
\(319\) 279.609i 0.876516i
\(320\) 0 0
\(321\) 374.307 1.16607
\(322\) 0 0
\(323\) 33.9188 + 33.9188i 0.105012 + 0.105012i
\(324\) 0 0
\(325\) 276.304 + 276.304i 0.850166 + 0.850166i
\(326\) 0 0
\(327\) −369.619 −1.13033
\(328\) 0 0
\(329\) 368.098i 1.11884i
\(330\) 0 0
\(331\) 182.195 182.195i 0.550437 0.550437i −0.376130 0.926567i \(-0.622745\pi\)
0.926567 + 0.376130i \(0.122745\pi\)
\(332\) 0 0
\(333\) 500.093 500.093i 1.50178 1.50178i
\(334\) 0 0
\(335\) 7.04338i 0.0210250i
\(336\) 0 0
\(337\) 510.137 1.51376 0.756881 0.653553i \(-0.226722\pi\)
0.756881 + 0.653553i \(0.226722\pi\)
\(338\) 0 0
\(339\) 441.009 + 441.009i 1.30091 + 1.30091i
\(340\) 0 0
\(341\) 108.587 + 108.587i 0.318436 + 0.318436i
\(342\) 0 0
\(343\) −353.997 −1.03206
\(344\) 0 0
\(345\) 6.24141i 0.0180910i
\(346\) 0 0
\(347\) −432.614 + 432.614i −1.24673 + 1.24673i −0.289570 + 0.957157i \(0.593512\pi\)
−0.957157 + 0.289570i \(0.906488\pi\)
\(348\) 0 0
\(349\) 148.839 148.839i 0.426472 0.426472i −0.460953 0.887425i \(-0.652492\pi\)
0.887425 + 0.460953i \(0.152492\pi\)
\(350\) 0 0
\(351\) 223.697i 0.637313i
\(352\) 0 0
\(353\) −268.587 −0.760869 −0.380434 0.924808i \(-0.624226\pi\)
−0.380434 + 0.924808i \(0.624226\pi\)
\(354\) 0 0
\(355\) 3.30418 + 3.30418i 0.00930756 + 0.00930756i
\(356\) 0 0
\(357\) −193.145 193.145i −0.541023 0.541023i
\(358\) 0 0
\(359\) 628.520 1.75075 0.875376 0.483442i \(-0.160614\pi\)
0.875376 + 0.483442i \(0.160614\pi\)
\(360\) 0 0
\(361\) 347.128i 0.961574i
\(362\) 0 0
\(363\) −206.006 + 206.006i −0.567511 + 0.567511i
\(364\) 0 0
\(365\) 0.569131 0.569131i 0.00155926 0.00155926i
\(366\) 0 0
\(367\) 396.386i 1.08007i 0.841643 + 0.540035i \(0.181589\pi\)
−0.841643 + 0.540035i \(0.818411\pi\)
\(368\) 0 0
\(369\) −39.9320 −0.108217
\(370\) 0 0
\(371\) −4.90204 4.90204i −0.0132130 0.0132130i
\(372\) 0 0
\(373\) −134.275 134.275i −0.359987 0.359987i 0.503821 0.863808i \(-0.331927\pi\)
−0.863808 + 0.503821i \(0.831927\pi\)
\(374\) 0 0
\(375\) 19.0481 0.0507950
\(376\) 0 0
\(377\) 576.015i 1.52789i
\(378\) 0 0
\(379\) 350.491 350.491i 0.924777 0.924777i −0.0725851 0.997362i \(-0.523125\pi\)
0.997362 + 0.0725851i \(0.0231249\pi\)
\(380\) 0 0
\(381\) −541.152 + 541.152i −1.42035 + 1.42035i
\(382\) 0 0
\(383\) 403.778i 1.05425i −0.849787 0.527126i \(-0.823270\pi\)
0.849787 0.527126i \(-0.176730\pi\)
\(384\) 0 0
\(385\) 2.90456 0.00754431
\(386\) 0 0
\(387\) 9.53299 + 9.53299i 0.0246330 + 0.0246330i
\(388\) 0 0
\(389\) 125.310 + 125.310i 0.322134 + 0.322134i 0.849585 0.527452i \(-0.176852\pi\)
−0.527452 + 0.849585i \(0.676852\pi\)
\(390\) 0 0
\(391\) −210.976 −0.539580
\(392\) 0 0
\(393\) 144.617i 0.367983i
\(394\) 0 0
\(395\) −4.95027 + 4.95027i −0.0125323 + 0.0125323i
\(396\) 0 0
\(397\) 69.8722 69.8722i 0.176001 0.176001i −0.613609 0.789610i \(-0.710283\pi\)
0.789610 + 0.613609i \(0.210283\pi\)
\(398\) 0 0
\(399\) 78.9897i 0.197969i
\(400\) 0 0
\(401\) 11.3010 0.0281821 0.0140911 0.999901i \(-0.495515\pi\)
0.0140911 + 0.999901i \(0.495515\pi\)
\(402\) 0 0
\(403\) −223.697 223.697i −0.555079 0.555079i
\(404\) 0 0
\(405\) 2.53762 + 2.53762i 0.00626573 + 0.00626573i
\(406\) 0 0
\(407\) −443.081 −1.08865
\(408\) 0 0
\(409\) 614.595i 1.50268i 0.659917 + 0.751339i \(0.270592\pi\)
−0.659917 + 0.751339i \(0.729408\pi\)
\(410\) 0 0
\(411\) 568.242 568.242i 1.38258 1.38258i
\(412\) 0 0
\(413\) −150.081 + 150.081i −0.363391 + 0.363391i
\(414\) 0 0
\(415\) 3.13778i 0.00756092i
\(416\) 0 0
\(417\) −649.108 −1.55661
\(418\) 0 0
\(419\) −78.7092 78.7092i −0.187850 0.187850i 0.606916 0.794766i \(-0.292406\pi\)
−0.794766 + 0.606916i \(0.792406\pi\)
\(420\) 0 0
\(421\) 374.618 + 374.618i 0.889829 + 0.889829i 0.994506 0.104678i \(-0.0333811\pi\)
−0.104678 + 0.994506i \(0.533381\pi\)
\(422\) 0 0
\(423\) 966.099 2.28392
\(424\) 0 0
\(425\) 321.894i 0.757397i
\(426\) 0 0
\(427\) −70.5705 + 70.5705i −0.165270 + 0.165270i
\(428\) 0 0
\(429\) −385.521 + 385.521i −0.898651 + 0.898651i
\(430\) 0 0
\(431\) 616.593i 1.43061i 0.698813 + 0.715305i \(0.253712\pi\)
−0.698813 + 0.715305i \(0.746288\pi\)
\(432\) 0 0
\(433\) 219.246 0.506342 0.253171 0.967422i \(-0.418526\pi\)
0.253171 + 0.967422i \(0.418526\pi\)
\(434\) 0 0
\(435\) 9.92610 + 9.92610i 0.0228186 + 0.0228186i
\(436\) 0 0
\(437\) −43.1409 43.1409i −0.0987207 0.0987207i
\(438\) 0 0
\(439\) 575.292 1.31046 0.655231 0.755429i \(-0.272571\pi\)
0.655231 + 0.755429i \(0.272571\pi\)
\(440\) 0 0
\(441\) 335.513i 0.760801i
\(442\) 0 0
\(443\) −371.895 + 371.895i −0.839492 + 0.839492i −0.988792 0.149300i \(-0.952298\pi\)
0.149300 + 0.988792i \(0.452298\pi\)
\(444\) 0 0
\(445\) −6.77538 + 6.77538i −0.0152256 + 0.0152256i
\(446\) 0 0
\(447\) 486.630i 1.08866i
\(448\) 0 0
\(449\) −498.135 −1.10943 −0.554716 0.832040i \(-0.687173\pi\)
−0.554716 + 0.832040i \(0.687173\pi\)
\(450\) 0 0
\(451\) 17.6898 + 17.6898i 0.0392236 + 0.0392236i
\(452\) 0 0
\(453\) 227.556 + 227.556i 0.502331 + 0.502331i
\(454\) 0 0
\(455\) −5.98361 −0.0131508
\(456\) 0 0
\(457\) 61.1711i 0.133854i −0.997758 0.0669268i \(-0.978681\pi\)
0.997758 0.0669268i \(-0.0213194\pi\)
\(458\) 0 0
\(459\) 130.303 130.303i 0.283885 0.283885i
\(460\) 0 0
\(461\) 443.183 443.183i 0.961352 0.961352i −0.0379287 0.999280i \(-0.512076\pi\)
0.999280 + 0.0379287i \(0.0120760\pi\)
\(462\) 0 0
\(463\) 706.883i 1.52675i −0.645958 0.763373i \(-0.723542\pi\)
0.645958 0.763373i \(-0.276458\pi\)
\(464\) 0 0
\(465\) −7.70966 −0.0165799
\(466\) 0 0
\(467\) 406.857 + 406.857i 0.871214 + 0.871214i 0.992605 0.121391i \(-0.0387355\pi\)
−0.121391 + 0.992605i \(0.538735\pi\)
\(468\) 0 0
\(469\) 277.224 + 277.224i 0.591096 + 0.591096i
\(470\) 0 0
\(471\) 191.899 0.407429
\(472\) 0 0
\(473\) 8.44620i 0.0178567i
\(474\) 0 0
\(475\) −65.8217 + 65.8217i −0.138572 + 0.138572i
\(476\) 0 0
\(477\) 12.8657 12.8657i 0.0269722 0.0269722i
\(478\) 0 0
\(479\) 133.063i 0.277793i 0.990307 + 0.138896i \(0.0443555\pi\)
−0.990307 + 0.138896i \(0.955645\pi\)
\(480\) 0 0
\(481\) 912.780 1.89767
\(482\) 0 0
\(483\) 245.659 + 245.659i 0.508611 + 0.508611i
\(484\) 0 0
\(485\) −8.57482 8.57482i −0.0176800 0.0176800i
\(486\) 0 0
\(487\) −208.075 −0.427259 −0.213629 0.976915i \(-0.568529\pi\)
−0.213629 + 0.976915i \(0.568529\pi\)
\(488\) 0 0
\(489\) 310.611i 0.635197i
\(490\) 0 0
\(491\) 98.9374 98.9374i 0.201502 0.201502i −0.599141 0.800643i \(-0.704491\pi\)
0.800643 + 0.599141i \(0.204491\pi\)
\(492\) 0 0
\(493\) −335.528 + 335.528i −0.680585 + 0.680585i
\(494\) 0 0
\(495\) 7.62322i 0.0154004i
\(496\) 0 0
\(497\) 260.102 0.523345
\(498\) 0 0
\(499\) 287.076 + 287.076i 0.575304 + 0.575304i 0.933606 0.358302i \(-0.116644\pi\)
−0.358302 + 0.933606i \(0.616644\pi\)
\(500\) 0 0
\(501\) −507.735 507.735i −1.01344 1.01344i
\(502\) 0 0
\(503\) −78.7359 −0.156533 −0.0782663 0.996932i \(-0.524938\pi\)
−0.0782663 + 0.996932i \(0.524938\pi\)
\(504\) 0 0
\(505\) 6.31543i 0.0125058i
\(506\) 0 0
\(507\) 245.098 245.098i 0.483428 0.483428i
\(508\) 0 0
\(509\) 242.477 242.477i 0.476378 0.476378i −0.427593 0.903971i \(-0.640638\pi\)
0.903971 + 0.427593i \(0.140638\pi\)
\(510\) 0 0
\(511\) 44.8014i 0.0876740i
\(512\) 0 0
\(513\) 53.2895 0.103878
\(514\) 0 0
\(515\) −9.27484 9.27484i −0.0180094 0.0180094i
\(516\) 0 0
\(517\) −427.980 427.980i −0.827815 0.827815i
\(518\) 0 0
\(519\) −1235.34 −2.38024
\(520\) 0 0
\(521\) 561.306i 1.07736i 0.842510 + 0.538681i \(0.181077\pi\)
−0.842510 + 0.538681i \(0.818923\pi\)
\(522\) 0 0
\(523\) −396.152 + 396.152i −0.757460 + 0.757460i −0.975859 0.218399i \(-0.929916\pi\)
0.218399 + 0.975859i \(0.429916\pi\)
\(524\) 0 0
\(525\) 374.811 374.811i 0.713926 0.713926i
\(526\) 0 0
\(527\) 260.607i 0.494510i
\(528\) 0 0
\(529\) −260.662 −0.492745
\(530\) 0 0
\(531\) −393.897 393.897i −0.741803 0.741803i
\(532\) 0 0
\(533\) −36.4424 36.4424i −0.0683722 0.0683722i
\(534\) 0 0
\(535\) 6.75465 0.0126255
\(536\) 0 0
\(537\) 352.694i 0.656785i
\(538\) 0 0
\(539\) −148.632 + 148.632i −0.275755 + 0.275755i
\(540\) 0 0
\(541\) 22.5728 22.5728i 0.0417242 0.0417242i −0.685937 0.727661i \(-0.740607\pi\)
0.727661 + 0.685937i \(0.240607\pi\)
\(542\) 0 0
\(543\) 128.239i 0.236168i
\(544\) 0 0
\(545\) −6.67005 −0.0122386
\(546\) 0 0
\(547\) −601.634 601.634i −1.09988 1.09988i −0.994424 0.105456i \(-0.966370\pi\)
−0.105456 0.994424i \(-0.533630\pi\)
\(548\) 0 0
\(549\) −185.217 185.217i −0.337372 0.337372i
\(550\) 0 0
\(551\) −137.219 −0.249037
\(552\) 0 0
\(553\) 389.681i 0.704666i
\(554\) 0 0
\(555\) 15.7294 15.7294i 0.0283412 0.0283412i
\(556\) 0 0
\(557\) −502.883 + 502.883i −0.902841 + 0.902841i −0.995681 0.0928399i \(-0.970406\pi\)
0.0928399 + 0.995681i \(0.470406\pi\)
\(558\) 0 0
\(559\) 17.3998i 0.0311266i
\(560\) 0 0
\(561\) −449.132 −0.800592
\(562\) 0 0
\(563\) −655.972 655.972i −1.16514 1.16514i −0.983335 0.181802i \(-0.941807\pi\)
−0.181802 0.983335i \(-0.558193\pi\)
\(564\) 0 0
\(565\) 7.95834 + 7.95834i 0.0140856 + 0.0140856i
\(566\) 0 0
\(567\) 199.759 0.352309
\(568\) 0 0
\(569\) 649.911i 1.14220i −0.820881 0.571099i \(-0.806517\pi\)
0.820881 0.571099i \(-0.193483\pi\)
\(570\) 0 0
\(571\) −269.718 + 269.718i −0.472360 + 0.472360i −0.902678 0.430317i \(-0.858402\pi\)
0.430317 + 0.902678i \(0.358402\pi\)
\(572\) 0 0
\(573\) 541.152 541.152i 0.944419 0.944419i
\(574\) 0 0
\(575\) 409.413i 0.712022i
\(576\) 0 0
\(577\) −142.675 −0.247271 −0.123635 0.992328i \(-0.539455\pi\)
−0.123635 + 0.992328i \(0.539455\pi\)
\(578\) 0 0
\(579\) 7.09149 + 7.09149i 0.0122478 + 0.0122478i
\(580\) 0 0
\(581\) −123.502 123.502i −0.212567 0.212567i
\(582\) 0 0
\(583\) −11.3990 −0.0195523
\(584\) 0 0
\(585\) 15.7044i 0.0268451i
\(586\) 0 0
\(587\) 687.876 687.876i 1.17185 1.17185i 0.190082 0.981768i \(-0.439125\pi\)
0.981768 0.190082i \(-0.0608753\pi\)
\(588\) 0 0
\(589\) 53.2895 53.2895i 0.0904746 0.0904746i
\(590\) 0 0
\(591\) 438.186i 0.741431i
\(592\) 0 0
\(593\) −58.8678 −0.0992711 −0.0496355 0.998767i \(-0.515806\pi\)
−0.0496355 + 0.998767i \(0.515806\pi\)
\(594\) 0 0
\(595\) −3.48545 3.48545i −0.00585790 0.00585790i
\(596\) 0 0
\(597\) 724.252 + 724.252i 1.21315 + 1.21315i
\(598\) 0 0
\(599\) −670.449 −1.11928 −0.559641 0.828735i \(-0.689061\pi\)
−0.559641 + 0.828735i \(0.689061\pi\)
\(600\) 0 0
\(601\) 910.721i 1.51534i −0.652636 0.757671i \(-0.726337\pi\)
0.652636 0.757671i \(-0.273663\pi\)
\(602\) 0 0
\(603\) −727.594 + 727.594i −1.20662 + 1.20662i
\(604\) 0 0
\(605\) −3.71754 + 3.71754i −0.00614469 + 0.00614469i
\(606\) 0 0
\(607\) 761.794i 1.25501i 0.778611 + 0.627507i \(0.215925\pi\)
−0.778611 + 0.627507i \(0.784075\pi\)
\(608\) 0 0
\(609\) 781.374 1.28304
\(610\) 0 0
\(611\) 881.671 + 881.671i 1.44300 + 1.44300i
\(612\) 0 0
\(613\) −273.397 273.397i −0.445999 0.445999i 0.448023 0.894022i \(-0.352128\pi\)
−0.894022 + 0.448023i \(0.852128\pi\)
\(614\) 0 0
\(615\) −1.25598 −0.00204224
\(616\) 0 0
\(617\) 1088.68i 1.76448i −0.470804 0.882238i \(-0.656036\pi\)
0.470804 0.882238i \(-0.343964\pi\)
\(618\) 0 0
\(619\) 129.299 129.299i 0.208884 0.208884i −0.594909 0.803793i \(-0.702812\pi\)
0.803793 + 0.594909i \(0.202812\pi\)
\(620\) 0 0
\(621\) −165.731 + 165.731i −0.266878 + 0.266878i
\(622\) 0 0
\(623\) 533.351i 0.856102i
\(624\) 0 0
\(625\) −624.484 −0.999175
\(626\) 0 0
\(627\) −91.8398 91.8398i −0.146475 0.146475i
\(628\) 0 0
\(629\) 531.694 + 531.694i 0.845301 + 0.845301i
\(630\) 0 0
\(631\) 455.029 0.721123 0.360562 0.932735i \(-0.382585\pi\)
0.360562 + 0.932735i \(0.382585\pi\)
\(632\) 0 0
\(633\) 956.012i 1.51029i
\(634\) 0 0
\(635\) −9.76549 + 9.76549i −0.0153787 + 0.0153787i
\(636\) 0 0
\(637\) 306.193 306.193i 0.480679 0.480679i
\(638\) 0 0
\(639\) 682.657i 1.06832i
\(640\) 0 0
\(641\) 798.626 1.24591 0.622953 0.782259i \(-0.285933\pi\)
0.622953 + 0.782259i \(0.285933\pi\)
\(642\) 0 0
\(643\) 305.718 + 305.718i 0.475455 + 0.475455i 0.903675 0.428219i \(-0.140859\pi\)
−0.428219 + 0.903675i \(0.640859\pi\)
\(644\) 0 0
\(645\) 0.299840 + 0.299840i 0.000464868 + 0.000464868i
\(646\) 0 0
\(647\) 1161.90 1.79583 0.897916 0.440167i \(-0.145081\pi\)
0.897916 + 0.440167i \(0.145081\pi\)
\(648\) 0 0
\(649\) 348.992i 0.537738i
\(650\) 0 0
\(651\) −303.448 + 303.448i −0.466127 + 0.466127i
\(652\) 0 0
\(653\) −77.5410 + 77.5410i −0.118746 + 0.118746i −0.763983 0.645237i \(-0.776759\pi\)
0.645237 + 0.763983i \(0.276759\pi\)
\(654\) 0 0
\(655\) 2.60973i 0.00398431i
\(656\) 0 0
\(657\) 117.584 0.178972
\(658\) 0 0
\(659\) −836.993 836.993i −1.27010 1.27010i −0.946037 0.324059i \(-0.894952\pi\)
−0.324059 0.946037i \(-0.605048\pi\)
\(660\) 0 0
\(661\) −121.071 121.071i −0.183164 0.183164i 0.609569 0.792733i \(-0.291342\pi\)
−0.792733 + 0.609569i \(0.791342\pi\)
\(662\) 0 0
\(663\) 925.246 1.39554
\(664\) 0 0
\(665\) 1.42543i 0.00214350i
\(666\) 0 0
\(667\) 426.754 426.754i 0.639811 0.639811i
\(668\) 0 0
\(669\) 197.235 197.235i 0.294820 0.294820i
\(670\) 0 0
\(671\) 164.102i 0.244563i
\(672\) 0 0
\(673\) −954.371 −1.41808 −0.709042 0.705166i \(-0.750872\pi\)
−0.709042 + 0.705166i \(0.750872\pi\)
\(674\) 0 0
\(675\) 252.862 + 252.862i 0.374611 + 0.374611i
\(676\) 0 0
\(677\) −245.475 245.475i −0.362593 0.362593i 0.502174 0.864767i \(-0.332534\pi\)
−0.864767 + 0.502174i \(0.832534\pi\)
\(678\) 0 0
\(679\) −675.002 −0.994112
\(680\) 0 0
\(681\) 1465.53i 2.15203i
\(682\) 0 0
\(683\) 911.271 911.271i 1.33422 1.33422i 0.432663 0.901556i \(-0.357574\pi\)
0.901556 0.432663i \(-0.142426\pi\)
\(684\) 0 0
\(685\) 10.2544 10.2544i 0.0149699 0.0149699i
\(686\) 0 0
\(687\) 1480.28i 2.15471i
\(688\) 0 0
\(689\) 23.4828 0.0340824
\(690\) 0 0
\(691\) 476.155 + 476.155i 0.689081 + 0.689081i 0.962029 0.272947i \(-0.0879985\pi\)
−0.272947 + 0.962029i \(0.587999\pi\)
\(692\) 0 0
\(693\) 300.046 + 300.046i 0.432967 + 0.432967i
\(694\) 0 0
\(695\) −11.7136 −0.0168541
\(696\) 0 0
\(697\) 42.4553i 0.0609115i
\(698\) 0 0
\(699\) −393.317 + 393.317i −0.562686 + 0.562686i
\(700\) 0 0
\(701\) −934.966 + 934.966i −1.33376 + 1.33376i −0.431782 + 0.901978i \(0.642115\pi\)
−0.901978 + 0.431782i \(0.857885\pi\)
\(702\) 0 0
\(703\) 217.444i 0.309309i
\(704\) 0 0
\(705\) 30.3866 0.0431015
\(706\) 0 0
\(707\) 248.572 + 248.572i 0.351587 + 0.351587i
\(708\) 0 0
\(709\) −5.89548 5.89548i −0.00831520 0.00831520i 0.702937 0.711252i \(-0.251872\pi\)
−0.711252 + 0.702937i \(0.751872\pi\)
\(710\) 0 0
\(711\) −1022.74 −1.43846
\(712\) 0 0
\(713\) 331.462i 0.464884i
\(714\) 0 0
\(715\) −6.95702 + 6.95702i −0.00973010 + 0.00973010i
\(716\) 0 0
\(717\) −719.338 + 719.338i −1.00326 + 1.00326i
\(718\) 0 0
\(719\) 19.5965i 0.0272552i 0.999907 + 0.0136276i \(0.00433793\pi\)
−0.999907 + 0.0136276i \(0.995662\pi\)
\(720\) 0 0
\(721\) −730.107 −1.01263
\(722\) 0 0
\(723\) 273.794 + 273.794i 0.378692 + 0.378692i
\(724\) 0 0
\(725\) −651.115 651.115i −0.898089 0.898089i
\(726\) 0 0
\(727\) 741.995 1.02063 0.510313 0.859989i \(-0.329530\pi\)
0.510313 + 0.859989i \(0.329530\pi\)
\(728\) 0 0
\(729\) 1115.99i 1.53084i
\(730\) 0 0
\(731\) −10.1354 + 10.1354i −0.0138651 + 0.0138651i
\(732\) 0 0
\(733\) −349.267 + 349.267i −0.476490 + 0.476490i −0.904007 0.427517i \(-0.859388\pi\)
0.427517 + 0.904007i \(0.359388\pi\)
\(734\) 0 0
\(735\) 10.5529i 0.0143576i
\(736\) 0 0
\(737\) 644.646 0.874690
\(738\) 0 0
\(739\) 358.932 + 358.932i 0.485700 + 0.485700i 0.906946 0.421246i \(-0.138407\pi\)
−0.421246 + 0.906946i \(0.638407\pi\)
\(740\) 0 0
\(741\) 189.197 + 189.197i 0.255326 + 0.255326i
\(742\) 0 0
\(743\) −856.214 −1.15237 −0.576187 0.817318i \(-0.695460\pi\)
−0.576187 + 0.817318i \(0.695460\pi\)
\(744\) 0 0
\(745\) 8.78160i 0.0117874i
\(746\) 0 0
\(747\) 324.139 324.139i 0.433921 0.433921i
\(748\) 0 0
\(749\) 265.860 265.860i 0.354953 0.354953i
\(750\) 0 0
\(751\) 442.218i 0.588839i −0.955676 0.294420i \(-0.904874\pi\)
0.955676 0.294420i \(-0.0951264\pi\)
\(752\) 0 0
\(753\) 1147.69 1.52416
\(754\) 0 0
\(755\) 4.10641 + 4.10641i 0.00543896 + 0.00543896i
\(756\) 0 0
\(757\) −489.198 489.198i −0.646233 0.646233i 0.305848 0.952080i \(-0.401060\pi\)
−0.952080 + 0.305848i \(0.901060\pi\)
\(758\) 0 0
\(759\) 571.246 0.752629
\(760\) 0 0
\(761\) 404.015i 0.530899i −0.964125 0.265450i \(-0.914480\pi\)
0.964125 0.265450i \(-0.0855204\pi\)
\(762\) 0 0
\(763\) −262.530 + 262.530i −0.344076 + 0.344076i
\(764\) 0 0
\(765\) 9.14780 9.14780i 0.0119579 0.0119579i
\(766\) 0 0
\(767\) 718.949i 0.937352i
\(768\) 0 0
\(769\) −387.336 −0.503688 −0.251844 0.967768i \(-0.581037\pi\)
−0.251844 + 0.967768i \(0.581037\pi\)
\(770\) 0 0
\(771\) −529.612 529.612i −0.686916 0.686916i
\(772\) 0 0
\(773\) 960.396 + 960.396i 1.24243 + 1.24243i 0.958991 + 0.283436i \(0.0914745\pi\)
0.283436 + 0.958991i \(0.408526\pi\)
\(774\) 0 0
\(775\) 505.724 0.652547
\(776\) 0 0
\(777\) 1238.20i 1.59357i
\(778\) 0 0
\(779\) 8.68138 8.68138i 0.0111443 0.0111443i
\(780\) 0 0
\(781\) 302.416 302.416i 0.387216 0.387216i
\(782\) 0 0
\(783\) 527.145i 0.673238i
\(784\) 0 0
\(785\) 3.46296 0.00441141
\(786\) 0 0
\(787\) 298.374 + 298.374i 0.379129 + 0.379129i 0.870788 0.491659i \(-0.163609\pi\)
−0.491659 + 0.870788i \(0.663609\pi\)
\(788\) 0 0
\(789\) 568.604 + 568.604i 0.720664 + 0.720664i
\(790\) 0 0
\(791\) 626.473 0.792002
\(792\) 0 0
\(793\) 338.062i 0.426308i
\(794\) 0 0
\(795\) 0.404665 0.404665i 0.000509012 0.000509012i
\(796\) 0 0
\(797\) 870.093 870.093i 1.09171 1.09171i 0.0963642 0.995346i \(-0.469279\pi\)
0.995346 0.0963642i \(-0.0307214\pi\)
\(798\) 0 0
\(799\) 1027.15i 1.28554i
\(800\) 0 0
\(801\) −1399.82 −1.74759
\(802\) 0 0
\(803\) −52.0897 52.0897i −0.0648689 0.0648689i
\(804\) 0 0
\(805\) 4.43310 + 4.43310i 0.00550695 + 0.00550695i
\(806\) 0 0
\(807\) −193.317 −0.239550
\(808\) 0 0
\(809\) 107.642i 0.133055i 0.997785 + 0.0665277i \(0.0211921\pi\)
−0.997785 + 0.0665277i \(0.978808\pi\)
\(810\) 0 0
\(811\) −829.739 + 829.739i −1.02311 + 1.02311i −0.0233795 + 0.999727i \(0.507443\pi\)
−0.999727 + 0.0233795i \(0.992557\pi\)
\(812\) 0 0
\(813\) 896.408 896.408i 1.10259 1.10259i
\(814\) 0 0
\(815\) 5.60521i 0.00687756i
\(816\) 0 0
\(817\) −4.14502 −0.00507346
\(818\) 0 0
\(819\) −618.118 618.118i −0.754722 0.754722i
\(820\) 0 0
\(821\) −506.899 506.899i −0.617416 0.617416i 0.327452 0.944868i \(-0.393810\pi\)
−0.944868 + 0.327452i \(0.893810\pi\)
\(822\) 0 0
\(823\) −927.304 −1.12674 −0.563368 0.826206i \(-0.690495\pi\)
−0.563368 + 0.826206i \(0.690495\pi\)
\(824\) 0 0
\(825\) 871.571i 1.05645i
\(826\) 0 0
\(827\) 19.4711 19.4711i 0.0235443 0.0235443i −0.695237 0.718781i \(-0.744700\pi\)
0.718781 + 0.695237i \(0.244700\pi\)
\(828\) 0 0
\(829\) −409.028 + 409.028i −0.493400 + 0.493400i −0.909376 0.415976i \(-0.863440\pi\)
0.415976 + 0.909376i \(0.363440\pi\)
\(830\) 0 0
\(831\) 1808.71i 2.17655i
\(832\) 0 0
\(833\) 356.714 0.428228
\(834\) 0 0
\(835\) −9.16246 9.16246i −0.0109730 0.0109730i
\(836\) 0 0
\(837\) −204.718 204.718i −0.244586 0.244586i
\(838\) 0 0
\(839\) 634.212 0.755914 0.377957 0.925823i \(-0.376627\pi\)
0.377957 + 0.925823i \(0.376627\pi\)
\(840\) 0 0
\(841\) 516.388i 0.614017i
\(842\) 0 0
\(843\) 657.483 657.483i 0.779933 0.779933i
\(844\) 0 0
\(845\) 4.42297 4.42297i 0.00523429 0.00523429i
\(846\) 0 0
\(847\) 292.641i 0.345503i
\(848\) 0 0
\(849\) 1899.16 2.23694
\(850\) 0 0
\(851\) −676.255 676.255i −0.794659 0.794659i
\(852\) 0 0
\(853\) −687.203 687.203i −0.805630 0.805630i 0.178339 0.983969i \(-0.442928\pi\)
−0.983969 + 0.178339i \(0.942928\pi\)
\(854\) 0 0
\(855\) 3.74114 0.00437560
\(856\) 0 0
\(857\) 995.675i 1.16181i 0.813970 + 0.580907i \(0.197302\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(858\) 0 0
\(859\) −430.241 + 430.241i −0.500863 + 0.500863i −0.911706 0.410843i \(-0.865234\pi\)
0.410843 + 0.911706i \(0.365234\pi\)
\(860\) 0 0
\(861\) −49.4347 + 49.4347i −0.0574154 + 0.0574154i
\(862\) 0 0
\(863\) 1014.03i 1.17501i −0.809222 0.587503i \(-0.800111\pi\)
0.809222 0.587503i \(-0.199889\pi\)
\(864\) 0 0
\(865\) −22.2927 −0.0257719
\(866\) 0 0
\(867\) −400.046 400.046i −0.461414 0.461414i
\(868\) 0 0
\(869\) 453.074 + 453.074i 0.521374 + 0.521374i
\(870\) 0 0
\(871\) −1328.02 −1.52471
\(872\) 0 0
\(873\) 1771.59i 2.02931i
\(874\) 0 0
\(875\) 13.5293 13.5293i 0.0154621 0.0154621i
\(876\) 0 0
\(877\) 544.315 544.315i 0.620656 0.620656i −0.325043 0.945699i \(-0.605379\pi\)
0.945699 + 0.325043i \(0.105379\pi\)
\(878\) 0 0
\(879\) 2152.03i 2.44828i
\(880\) 0 0
\(881\) −645.905 −0.733150 −0.366575 0.930388i \(-0.619470\pi\)
−0.366575 + 0.930388i \(0.619470\pi\)
\(882\) 0 0
\(883\) −586.952 586.952i −0.664725 0.664725i 0.291765 0.956490i \(-0.405757\pi\)
−0.956490 + 0.291765i \(0.905757\pi\)
\(884\) 0 0
\(885\) −12.3892 12.3892i −0.0139991 0.0139991i
\(886\) 0 0
\(887\) −1221.93 −1.37759 −0.688797 0.724955i \(-0.741861\pi\)
−0.688797 + 0.724955i \(0.741861\pi\)
\(888\) 0 0
\(889\) 768.730i 0.864713i
\(890\) 0 0
\(891\) 232.256 232.256i 0.260669 0.260669i
\(892\) 0 0
\(893\) −210.034 + 210.034i −0.235200 + 0.235200i
\(894\) 0 0
\(895\) 6.36462i 0.00711131i
\(896\) 0 0
\(897\) −1176.81 −1.31194
\(898\) 0 0
\(899\) 527.145 + 527.145i 0.586368 + 0.586368i
\(900\) 0 0
\(901\) 13.6787 + 13.6787i 0.0151817 + 0.0151817i
\(902\) 0 0
\(903\) 23.6031 0.0261386
\(904\) 0 0
\(905\) 2.31417i 0.00255710i
\(906\) 0 0
\(907\) −310.014 + 310.014i −0.341801 + 0.341801i −0.857044 0.515243i \(-0.827702\pi\)
0.515243 + 0.857044i \(0.327702\pi\)
\(908\) 0 0
\(909\) −652.395 + 652.395i −0.717707 + 0.717707i
\(910\) 0 0
\(911\) 1044.12i 1.14612i 0.819513 + 0.573060i \(0.194244\pi\)
−0.819513 + 0.573060i \(0.805756\pi\)
\(912\) 0 0
\(913\) −287.186 −0.314552
\(914\) 0 0
\(915\) −5.82561 5.82561i −0.00636679 0.00636679i
\(916\) 0 0
\(917\) −102.718 102.718i −0.112015 0.112015i
\(918\) 0 0
\(919\) 188.522 0.205138 0.102569 0.994726i \(-0.467294\pi\)
0.102569 + 0.994726i \(0.467294\pi\)
\(920\) 0 0
\(921\) 154.342i 0.167581i
\(922\) 0 0
\(923\) −622.999 + 622.999i −0.674972 + 0.674972i
\(924\) 0 0
\(925\) −1031.79 + 1031.79i −1.11545 + 1.11545i
\(926\) 0 0
\(927\) 1916.22i 2.06712i
\(928\) 0 0
\(929\) −220.366 −0.237208 −0.118604 0.992942i \(-0.537842\pi\)
−0.118604 + 0.992942i \(0.537842\pi\)
\(930\) 0 0
\(931\) 72.9419 + 72.9419i 0.0783479 + 0.0783479i
\(932\) 0 0
\(933\) −512.576 512.576i −0.549384 0.549384i
\(934\) 0 0
\(935\) −8.10493 −0.00866837
\(936\) 0 0
\(937\) 558.321i 0.595860i 0.954588 + 0.297930i \(0.0962962\pi\)
−0.954588 + 0.297930i \(0.903704\pi\)
\(938\) 0 0
\(939\) 190.379 190.379i 0.202746 0.202746i
\(940\) 0 0
\(941\) 794.760 794.760i 0.844591 0.844591i −0.144861 0.989452i \(-0.546274\pi\)
0.989452 + 0.144861i \(0.0462736\pi\)
\(942\) 0 0
\(943\) 53.9984i 0.0572624i
\(944\) 0 0
\(945\) −5.47595 −0.00579466
\(946\) 0 0
\(947\) 44.9362 + 44.9362i 0.0474511 + 0.0474511i 0.730434 0.682983i \(-0.239318\pi\)
−0.682983 + 0.730434i \(0.739318\pi\)
\(948\) 0 0
\(949\) 107.309 + 107.309i 0.113076 + 0.113076i
\(950\) 0 0
\(951\) −175.480 −0.184521
\(952\) 0 0
\(953\) 304.232i 0.319236i 0.987179 + 0.159618i \(0.0510262\pi\)
−0.987179 + 0.159618i \(0.948974\pi\)
\(954\) 0 0
\(955\) 9.76549 9.76549i 0.0102256 0.0102256i
\(956\) 0 0
\(957\) 908.488 908.488i 0.949308 0.949308i
\(958\) 0 0
\(959\) 807.213i 0.841724i
\(960\) 0 0
\(961\) 551.564 0.573948
\(962\) 0 0
\(963\) 697.768 + 697.768i 0.724577 + 0.724577i
\(964\) 0 0
\(965\) 0.127971 + 0.127971i 0.000132613 + 0.000132613i
\(966\) 0 0
\(967\) 834.409 0.862884 0.431442 0.902141i \(-0.358005\pi\)
0.431442 + 0.902141i \(0.358005\pi\)
\(968\) 0 0
\(969\) 220.414i 0.227465i
\(970\) 0 0
\(971\) −211.499 + 211.499i −0.217816 + 0.217816i −0.807577 0.589761i \(-0.799222\pi\)
0.589761 + 0.807577i \(0.299222\pi\)
\(972\) 0 0
\(973\) −461.043 + 461.043i −0.473837 + 0.473837i
\(974\) 0 0
\(975\) 1795.50i 1.84154i
\(976\) 0 0
\(977\) 891.561 0.912549 0.456275 0.889839i \(-0.349183\pi\)
0.456275 + 0.889839i \(0.349183\pi\)
\(978\) 0 0
\(979\) 620.117 + 620.117i 0.633419 + 0.633419i
\(980\) 0 0
\(981\) −689.028 689.028i −0.702374 0.702374i
\(982\) 0 0
\(983\) −181.589 −0.184730 −0.0923648 0.995725i \(-0.529443\pi\)
−0.0923648 + 0.995725i \(0.529443\pi\)
\(984\) 0 0
\(985\) 7.90739i 0.00802781i
\(986\) 0 0
\(987\) 1196.00 1196.00i 1.21175 1.21175i
\(988\) 0 0
\(989\) 12.8911 12.8911i 0.0130344 0.0130344i
\(990\) 0 0
\(991\) 1140.89i 1.15125i 0.817715 + 0.575624i \(0.195241\pi\)
−0.817715 + 0.575624i \(0.804759\pi\)
\(992\) 0 0
\(993\) 1183.95 1.19230
\(994\) 0 0
\(995\) 13.0697 + 13.0697i 0.0131354 + 0.0131354i
\(996\) 0 0
\(997\) 742.946 + 742.946i 0.745182 + 0.745182i 0.973570 0.228388i \(-0.0733455\pi\)
−0.228388 + 0.973570i \(0.573346\pi\)
\(998\) 0 0
\(999\) 835.339 0.836175
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.b.31.3 6
3.2 odd 2 1152.3.m.a.415.2 6
4.3 odd 2 128.3.f.a.31.1 6
8.3 odd 2 64.3.f.a.15.3 6
8.5 even 2 16.3.f.a.11.3 yes 6
12.11 even 2 1152.3.m.b.415.2 6
16.3 odd 4 inner 128.3.f.b.95.3 6
16.5 even 4 64.3.f.a.47.3 6
16.11 odd 4 16.3.f.a.3.3 6
16.13 even 4 128.3.f.a.95.1 6
24.5 odd 2 144.3.m.a.91.1 6
24.11 even 2 576.3.m.a.271.2 6
32.3 odd 8 1024.3.c.j.1023.1 12
32.5 even 8 1024.3.d.k.511.2 12
32.11 odd 8 1024.3.d.k.511.1 12
32.13 even 8 1024.3.c.j.1023.2 12
32.19 odd 8 1024.3.c.j.1023.12 12
32.21 even 8 1024.3.d.k.511.11 12
32.27 odd 8 1024.3.d.k.511.12 12
32.29 even 8 1024.3.c.j.1023.11 12
40.13 odd 4 400.3.k.d.299.3 6
40.29 even 2 400.3.r.c.251.1 6
40.37 odd 4 400.3.k.c.299.1 6
48.5 odd 4 576.3.m.a.559.2 6
48.11 even 4 144.3.m.a.19.1 6
48.29 odd 4 1152.3.m.b.991.2 6
48.35 even 4 1152.3.m.a.991.2 6
80.27 even 4 400.3.k.d.99.3 6
80.43 even 4 400.3.k.c.99.1 6
80.59 odd 4 400.3.r.c.51.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.3.f.a.3.3 6 16.11 odd 4
16.3.f.a.11.3 yes 6 8.5 even 2
64.3.f.a.15.3 6 8.3 odd 2
64.3.f.a.47.3 6 16.5 even 4
128.3.f.a.31.1 6 4.3 odd 2
128.3.f.a.95.1 6 16.13 even 4
128.3.f.b.31.3 6 1.1 even 1 trivial
128.3.f.b.95.3 6 16.3 odd 4 inner
144.3.m.a.19.1 6 48.11 even 4
144.3.m.a.91.1 6 24.5 odd 2
400.3.k.c.99.1 6 80.43 even 4
400.3.k.c.299.1 6 40.37 odd 4
400.3.k.d.99.3 6 80.27 even 4
400.3.k.d.299.3 6 40.13 odd 4
400.3.r.c.51.1 6 80.59 odd 4
400.3.r.c.251.1 6 40.29 even 2
576.3.m.a.271.2 6 24.11 even 2
576.3.m.a.559.2 6 48.5 odd 4
1024.3.c.j.1023.1 12 32.3 odd 8
1024.3.c.j.1023.2 12 32.13 even 8
1024.3.c.j.1023.11 12 32.29 even 8
1024.3.c.j.1023.12 12 32.19 odd 8
1024.3.d.k.511.1 12 32.11 odd 8
1024.3.d.k.511.2 12 32.5 even 8
1024.3.d.k.511.11 12 32.21 even 8
1024.3.d.k.511.12 12 32.27 odd 8
1152.3.m.a.415.2 6 3.2 odd 2
1152.3.m.a.991.2 6 48.35 even 4
1152.3.m.b.415.2 6 12.11 even 2
1152.3.m.b.991.2 6 48.29 odd 4