Properties

Label 1764.2.bm.b.1685.2
Level $1764$
Weight $2$
Character 1764.1685
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1685,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1685");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.bm (of order \(6\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1685.2
Root \(-1.69483 - 0.357142i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1685
Dual form 1764.2.bm.b.1697.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.69483 - 0.357142i) q^{3} -2.42488 q^{5} +(2.74490 + 1.21059i) q^{9} +O(q^{10})\) \(q+(-1.69483 - 0.357142i) q^{3} -2.42488 q^{5} +(2.74490 + 1.21059i) q^{9} -2.42118i q^{11} +(-4.73574 + 2.73418i) q^{13} +(4.10976 + 0.866025i) q^{15} +(-1.29034 - 2.23494i) q^{17} +(0.348755 + 0.201354i) q^{19} +3.54372i q^{23} +0.880040 q^{25} +(-4.21979 - 3.03206i) q^{27} +(-6.31784 - 3.64761i) q^{29} +(3.63732 + 2.10001i) q^{31} +(-0.864703 + 4.10349i) q^{33} +(1.59680 - 2.76574i) q^{37} +(9.00276 - 2.94264i) q^{39} +(4.03924 + 6.99618i) q^{41} +(-4.22573 + 7.31918i) q^{43} +(-6.65605 - 2.93553i) q^{45} +(-2.25769 - 3.91043i) q^{47} +(1.38872 + 4.24868i) q^{51} +(12.1493 - 7.01442i) q^{53} +5.87106i q^{55} +(-0.519169 - 0.465816i) q^{57} +(-0.0779043 + 0.134934i) q^{59} +(10.2288 - 5.90561i) q^{61} +(11.4836 - 6.63005i) q^{65} +(2.53682 - 4.39390i) q^{67} +(1.26561 - 6.00600i) q^{69} +8.73987i q^{71} +(-7.62339 + 4.40137i) q^{73} +(-1.49152 - 0.314299i) q^{75} +(5.66575 + 9.81337i) q^{79} +(6.06895 + 6.64589i) q^{81} +(7.50937 - 13.0066i) q^{83} +(3.12893 + 5.41946i) q^{85} +(9.40495 + 8.43843i) q^{87} +(7.83339 - 13.5678i) q^{89} +(-5.41464 - 4.85819i) q^{93} +(-0.845690 - 0.488259i) q^{95} +(4.97713 + 2.87355i) q^{97} +(2.93105 - 6.64589i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{9} - 12 q^{15} + 16 q^{25} - 12 q^{29} - 2 q^{37} + 18 q^{39} + 4 q^{43} + 6 q^{51} - 36 q^{53} - 42 q^{57} + 24 q^{65} + 14 q^{67} + 20 q^{79} + 54 q^{81} + 6 q^{85} + 30 q^{93} + 60 q^{95} + 90 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.69483 0.357142i −0.978511 0.206196i
\(4\) 0 0
\(5\) −2.42488 −1.08444 −0.542220 0.840237i \(-0.682416\pi\)
−0.542220 + 0.840237i \(0.682416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.74490 + 1.21059i 0.914967 + 0.403530i
\(10\) 0 0
\(11\) 2.42118i 0.730013i −0.931005 0.365006i \(-0.881067\pi\)
0.931005 0.365006i \(-0.118933\pi\)
\(12\) 0 0
\(13\) −4.73574 + 2.73418i −1.31346 + 0.758325i −0.982667 0.185380i \(-0.940648\pi\)
−0.330790 + 0.943704i \(0.607315\pi\)
\(14\) 0 0
\(15\) 4.10976 + 0.866025i 1.06114 + 0.223607i
\(16\) 0 0
\(17\) −1.29034 2.23494i −0.312954 0.542053i 0.666046 0.745911i \(-0.267985\pi\)
−0.979001 + 0.203858i \(0.934652\pi\)
\(18\) 0 0
\(19\) 0.348755 + 0.201354i 0.0800100 + 0.0461938i 0.539471 0.842004i \(-0.318624\pi\)
−0.459461 + 0.888198i \(0.651957\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.54372i 0.738916i 0.929247 + 0.369458i \(0.120457\pi\)
−0.929247 + 0.369458i \(0.879543\pi\)
\(24\) 0 0
\(25\) 0.880040 0.176008
\(26\) 0 0
\(27\) −4.21979 3.03206i −0.812098 0.583520i
\(28\) 0 0
\(29\) −6.31784 3.64761i −1.17319 0.677343i −0.218763 0.975778i \(-0.570202\pi\)
−0.954430 + 0.298435i \(0.903535\pi\)
\(30\) 0 0
\(31\) 3.63732 + 2.10001i 0.653282 + 0.377172i 0.789712 0.613477i \(-0.210230\pi\)
−0.136431 + 0.990650i \(0.543563\pi\)
\(32\) 0 0
\(33\) −0.864703 + 4.10349i −0.150526 + 0.714325i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.59680 2.76574i 0.262513 0.454685i −0.704396 0.709807i \(-0.748782\pi\)
0.966909 + 0.255122i \(0.0821155\pi\)
\(38\) 0 0
\(39\) 9.00276 2.94264i 1.44159 0.471199i
\(40\) 0 0
\(41\) 4.03924 + 6.99618i 0.630824 + 1.09262i 0.987384 + 0.158346i \(0.0506163\pi\)
−0.356560 + 0.934273i \(0.616050\pi\)
\(42\) 0 0
\(43\) −4.22573 + 7.31918i −0.644418 + 1.11616i 0.340018 + 0.940419i \(0.389567\pi\)
−0.984436 + 0.175745i \(0.943766\pi\)
\(44\) 0 0
\(45\) −6.65605 2.93553i −0.992225 0.437603i
\(46\) 0 0
\(47\) −2.25769 3.91043i −0.329317 0.570395i 0.653059 0.757307i \(-0.273485\pi\)
−0.982377 + 0.186912i \(0.940152\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.38872 + 4.24868i 0.194460 + 0.594934i
\(52\) 0 0
\(53\) 12.1493 7.01442i 1.66884 0.963505i 0.700575 0.713579i \(-0.252927\pi\)
0.968265 0.249926i \(-0.0804065\pi\)
\(54\) 0 0
\(55\) 5.87106i 0.791654i
\(56\) 0 0
\(57\) −0.519169 0.465816i −0.0687657 0.0616989i
\(58\) 0 0
\(59\) −0.0779043 + 0.134934i −0.0101423 + 0.0175669i −0.871052 0.491191i \(-0.836562\pi\)
0.860910 + 0.508758i \(0.169895\pi\)
\(60\) 0 0
\(61\) 10.2288 5.90561i 1.30967 0.756136i 0.327626 0.944808i \(-0.393752\pi\)
0.982040 + 0.188672i \(0.0604182\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.4836 6.63005i 1.42436 0.822357i
\(66\) 0 0
\(67\) 2.53682 4.39390i 0.309922 0.536801i −0.668423 0.743781i \(-0.733030\pi\)
0.978345 + 0.206981i \(0.0663637\pi\)
\(68\) 0 0
\(69\) 1.26561 6.00600i 0.152361 0.723037i
\(70\) 0 0
\(71\) 8.73987i 1.03723i 0.855007 + 0.518616i \(0.173552\pi\)
−0.855007 + 0.518616i \(0.826448\pi\)
\(72\) 0 0
\(73\) −7.62339 + 4.40137i −0.892251 + 0.515141i −0.874678 0.484704i \(-0.838927\pi\)
−0.0175727 + 0.999846i \(0.505594\pi\)
\(74\) 0 0
\(75\) −1.49152 0.314299i −0.172226 0.0362921i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.66575 + 9.81337i 0.637447 + 1.10409i 0.985991 + 0.166798i \(0.0533427\pi\)
−0.348544 + 0.937292i \(0.613324\pi\)
\(80\) 0 0
\(81\) 6.06895 + 6.64589i 0.674328 + 0.738432i
\(82\) 0 0
\(83\) 7.50937 13.0066i 0.824260 1.42766i −0.0782227 0.996936i \(-0.524925\pi\)
0.902483 0.430725i \(-0.141742\pi\)
\(84\) 0 0
\(85\) 3.12893 + 5.41946i 0.339380 + 0.587823i
\(86\) 0 0
\(87\) 9.40495 + 8.43843i 1.00832 + 0.904695i
\(88\) 0 0
\(89\) 7.83339 13.5678i 0.830338 1.43819i −0.0674328 0.997724i \(-0.521481\pi\)
0.897771 0.440463i \(-0.145186\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.41464 4.85819i −0.561472 0.503771i
\(94\) 0 0
\(95\) −0.845690 0.488259i −0.0867660 0.0500944i
\(96\) 0 0
\(97\) 4.97713 + 2.87355i 0.505351 + 0.291765i 0.730921 0.682462i \(-0.239091\pi\)
−0.225569 + 0.974227i \(0.572424\pi\)
\(98\) 0 0
\(99\) 2.93105 6.64589i 0.294582 0.667937i
\(100\) 0 0
\(101\) 9.67675 0.962873 0.481436 0.876481i \(-0.340115\pi\)
0.481436 + 0.876481i \(0.340115\pi\)
\(102\) 0 0
\(103\) 19.2676i 1.89850i 0.314526 + 0.949249i \(0.398155\pi\)
−0.314526 + 0.949249i \(0.601845\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.944715 + 0.545431i 0.0913290 + 0.0527288i 0.544969 0.838456i \(-0.316541\pi\)
−0.453640 + 0.891185i \(0.649875\pi\)
\(108\) 0 0
\(109\) −1.15678 2.00360i −0.110800 0.191910i 0.805293 0.592877i \(-0.202008\pi\)
−0.916093 + 0.400966i \(0.868674\pi\)
\(110\) 0 0
\(111\) −3.69407 + 4.11718i −0.350626 + 0.390785i
\(112\) 0 0
\(113\) 13.8868 8.01754i 1.30636 0.754227i 0.324872 0.945758i \(-0.394679\pi\)
0.981487 + 0.191531i \(0.0613453\pi\)
\(114\) 0 0
\(115\) 8.59309i 0.801309i
\(116\) 0 0
\(117\) −16.3091 + 1.77202i −1.50778 + 0.163823i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.13790 0.467082
\(122\) 0 0
\(123\) −4.34721 13.2999i −0.391974 1.19921i
\(124\) 0 0
\(125\) 9.99041 0.893569
\(126\) 0 0
\(127\) −3.06425 −0.271909 −0.135954 0.990715i \(-0.543410\pi\)
−0.135954 + 0.990715i \(0.543410\pi\)
\(128\) 0 0
\(129\) 9.77588 10.8956i 0.860718 0.959303i
\(130\) 0 0
\(131\) 11.4630 1.00153 0.500765 0.865584i \(-0.333052\pi\)
0.500765 + 0.865584i \(0.333052\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 10.2325 + 7.35238i 0.880671 + 0.632792i
\(136\) 0 0
\(137\) 1.97456i 0.168698i −0.996436 0.0843488i \(-0.973119\pi\)
0.996436 0.0843488i \(-0.0268810\pi\)
\(138\) 0 0
\(139\) 5.37804 3.10501i 0.456159 0.263364i −0.254269 0.967134i \(-0.581835\pi\)
0.710428 + 0.703770i \(0.248501\pi\)
\(140\) 0 0
\(141\) 2.42982 + 7.43383i 0.204628 + 0.626041i
\(142\) 0 0
\(143\) 6.61993 + 11.4661i 0.553587 + 0.958840i
\(144\) 0 0
\(145\) 15.3200 + 8.84500i 1.27226 + 0.734537i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.1692i 0.996943i 0.866906 + 0.498472i \(0.166105\pi\)
−0.866906 + 0.498472i \(0.833895\pi\)
\(150\) 0 0
\(151\) 10.6357 0.865519 0.432759 0.901509i \(-0.357540\pi\)
0.432759 + 0.901509i \(0.357540\pi\)
\(152\) 0 0
\(153\) −0.836270 7.69677i −0.0676084 0.622247i
\(154\) 0 0
\(155\) −8.82006 5.09226i −0.708444 0.409021i
\(156\) 0 0
\(157\) −3.06154 1.76758i −0.244337 0.141068i 0.372831 0.927899i \(-0.378387\pi\)
−0.617169 + 0.786831i \(0.711720\pi\)
\(158\) 0 0
\(159\) −23.0962 + 7.54922i −1.83165 + 0.598692i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.16575 + 5.48324i −0.247961 + 0.429481i −0.962960 0.269645i \(-0.913094\pi\)
0.714999 + 0.699125i \(0.246427\pi\)
\(164\) 0 0
\(165\) 2.09680 9.95046i 0.163236 0.774642i
\(166\) 0 0
\(167\) −8.39779 14.5454i −0.649840 1.12556i −0.983161 0.182743i \(-0.941502\pi\)
0.333320 0.942814i \(-0.391831\pi\)
\(168\) 0 0
\(169\) 8.45146 14.6384i 0.650112 1.12603i
\(170\) 0 0
\(171\) 0.713542 + 0.974896i 0.0545659 + 0.0745522i
\(172\) 0 0
\(173\) −8.30850 14.3907i −0.631684 1.09411i −0.987207 0.159441i \(-0.949031\pi\)
0.355524 0.934667i \(-0.384303\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.180225 0.200868i 0.0135466 0.0150981i
\(178\) 0 0
\(179\) −12.6082 + 7.27937i −0.942384 + 0.544086i −0.890707 0.454578i \(-0.849790\pi\)
−0.0516773 + 0.998664i \(0.516457\pi\)
\(180\) 0 0
\(181\) 4.02355i 0.299068i 0.988757 + 0.149534i \(0.0477774\pi\)
−0.988757 + 0.149534i \(0.952223\pi\)
\(182\) 0 0
\(183\) −19.4452 + 6.35587i −1.43743 + 0.469839i
\(184\) 0 0
\(185\) −3.87205 + 6.70659i −0.284679 + 0.493078i
\(186\) 0 0
\(187\) −5.41119 + 3.12415i −0.395705 + 0.228461i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.28998 + 4.20887i −0.527485 + 0.304543i −0.739992 0.672616i \(-0.765171\pi\)
0.212507 + 0.977160i \(0.431837\pi\)
\(192\) 0 0
\(193\) 4.31784 7.47871i 0.310805 0.538330i −0.667732 0.744402i \(-0.732735\pi\)
0.978537 + 0.206072i \(0.0660681\pi\)
\(194\) 0 0
\(195\) −21.8306 + 7.13555i −1.56332 + 0.510987i
\(196\) 0 0
\(197\) 23.3303i 1.66221i 0.556112 + 0.831107i \(0.312292\pi\)
−0.556112 + 0.831107i \(0.687708\pi\)
\(198\) 0 0
\(199\) −12.2441 + 7.06913i −0.867960 + 0.501117i −0.866670 0.498882i \(-0.833744\pi\)
−0.00129041 + 0.999999i \(0.500411\pi\)
\(200\) 0 0
\(201\) −5.86873 + 6.54092i −0.413948 + 0.461361i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.79468 16.9649i −0.684090 1.18488i
\(206\) 0 0
\(207\) −4.28998 + 9.72715i −0.298175 + 0.676083i
\(208\) 0 0
\(209\) 0.487514 0.844399i 0.0337221 0.0584083i
\(210\) 0 0
\(211\) −6.75786 11.7050i −0.465230 0.805802i 0.533982 0.845496i \(-0.320695\pi\)
−0.999212 + 0.0396938i \(0.987362\pi\)
\(212\) 0 0
\(213\) 3.12137 14.8126i 0.213873 1.01494i
\(214\) 0 0
\(215\) 10.2469 17.7481i 0.698832 1.21041i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.4923 4.73694i 0.979297 0.320093i
\(220\) 0 0
\(221\) 12.2215 + 7.05606i 0.822104 + 0.474642i
\(222\) 0 0
\(223\) 13.4054 + 7.73961i 0.897692 + 0.518283i 0.876451 0.481492i \(-0.159905\pi\)
0.0212411 + 0.999774i \(0.493238\pi\)
\(224\) 0 0
\(225\) 2.41562 + 1.06537i 0.161041 + 0.0710245i
\(226\) 0 0
\(227\) −15.0167 −0.996694 −0.498347 0.866978i \(-0.666059\pi\)
−0.498347 + 0.866978i \(0.666059\pi\)
\(228\) 0 0
\(229\) 19.5935i 1.29478i 0.762160 + 0.647389i \(0.224139\pi\)
−0.762160 + 0.647389i \(0.775861\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.9283 9.77356i −1.10901 0.640287i −0.170437 0.985369i \(-0.554518\pi\)
−0.938573 + 0.345081i \(0.887851\pi\)
\(234\) 0 0
\(235\) 5.47462 + 9.48232i 0.357125 + 0.618558i
\(236\) 0 0
\(237\) −6.09772 18.6555i −0.396090 1.21180i
\(238\) 0 0
\(239\) 7.36210 4.25051i 0.476215 0.274943i −0.242623 0.970121i \(-0.578008\pi\)
0.718838 + 0.695178i \(0.244674\pi\)
\(240\) 0 0
\(241\) 8.33094i 0.536643i −0.963329 0.268321i \(-0.913531\pi\)
0.963329 0.268321i \(-0.0864689\pi\)
\(242\) 0 0
\(243\) −7.91231 13.4311i −0.507575 0.861607i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.20215 −0.140120
\(248\) 0 0
\(249\) −17.3723 + 19.3621i −1.10093 + 1.22702i
\(250\) 0 0
\(251\) 13.5763 0.856928 0.428464 0.903559i \(-0.359055\pi\)
0.428464 + 0.903559i \(0.359055\pi\)
\(252\) 0 0
\(253\) 8.57997 0.539418
\(254\) 0 0
\(255\) −3.36749 10.3025i −0.210880 0.645170i
\(256\) 0 0
\(257\) −5.98060 −0.373059 −0.186530 0.982449i \(-0.559724\pi\)
−0.186530 + 0.982449i \(0.559724\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −12.9261 17.6606i −0.800104 1.09316i
\(262\) 0 0
\(263\) 3.36602i 0.207558i 0.994600 + 0.103779i \(0.0330934\pi\)
−0.994600 + 0.103779i \(0.966907\pi\)
\(264\) 0 0
\(265\) −29.4607 + 17.0091i −1.80976 + 1.04486i
\(266\) 0 0
\(267\) −18.1219 + 20.1975i −1.10904 + 1.23607i
\(268\) 0 0
\(269\) 6.14112 + 10.6367i 0.374431 + 0.648533i 0.990242 0.139361i \(-0.0445048\pi\)
−0.615811 + 0.787894i \(0.711171\pi\)
\(270\) 0 0
\(271\) 25.4823 + 14.7122i 1.54794 + 0.893703i 0.998299 + 0.0583012i \(0.0185684\pi\)
0.549640 + 0.835402i \(0.314765\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.13073i 0.128488i
\(276\) 0 0
\(277\) −9.20215 −0.552904 −0.276452 0.961028i \(-0.589159\pi\)
−0.276452 + 0.961028i \(0.589159\pi\)
\(278\) 0 0
\(279\) 7.44183 + 10.1676i 0.445531 + 0.608719i
\(280\) 0 0
\(281\) 1.05254 + 0.607682i 0.0627891 + 0.0362513i 0.531066 0.847331i \(-0.321792\pi\)
−0.468277 + 0.883582i \(0.655125\pi\)
\(282\) 0 0
\(283\) 10.5776 + 6.10696i 0.628771 + 0.363021i 0.780276 0.625435i \(-0.215079\pi\)
−0.151505 + 0.988457i \(0.548412\pi\)
\(284\) 0 0
\(285\) 1.25892 + 1.12955i 0.0745722 + 0.0669086i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.17002 8.95475i 0.304119 0.526750i
\(290\) 0 0
\(291\) −7.40913 6.64772i −0.434331 0.389696i
\(292\) 0 0
\(293\) 3.15082 + 5.45739i 0.184073 + 0.318824i 0.943264 0.332044i \(-0.107738\pi\)
−0.759191 + 0.650868i \(0.774405\pi\)
\(294\) 0 0
\(295\) 0.188909 0.327199i 0.0109987 0.0190503i
\(296\) 0 0
\(297\) −7.34116 + 10.2169i −0.425977 + 0.592842i
\(298\) 0 0
\(299\) −9.68915 16.7821i −0.560338 0.970534i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −16.4004 3.45597i −0.942181 0.198540i
\(304\) 0 0
\(305\) −24.8036 + 14.3204i −1.42025 + 0.819983i
\(306\) 0 0
\(307\) 28.7264i 1.63950i 0.572721 + 0.819750i \(0.305888\pi\)
−0.572721 + 0.819750i \(0.694112\pi\)
\(308\) 0 0
\(309\) 6.88128 32.6554i 0.391462 1.85770i
\(310\) 0 0
\(311\) 8.86623 15.3568i 0.502758 0.870802i −0.497237 0.867615i \(-0.665652\pi\)
0.999995 0.00318766i \(-0.00101467\pi\)
\(312\) 0 0
\(313\) 23.4526 13.5404i 1.32562 0.765348i 0.341003 0.940062i \(-0.389233\pi\)
0.984619 + 0.174714i \(0.0559001\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.65389 2.68692i 0.261388 0.150913i −0.363579 0.931563i \(-0.618445\pi\)
0.624968 + 0.780651i \(0.285112\pi\)
\(318\) 0 0
\(319\) −8.83150 + 15.2966i −0.494469 + 0.856446i
\(320\) 0 0
\(321\) −1.40633 1.26181i −0.0784940 0.0704274i
\(322\) 0 0
\(323\) 1.03926i 0.0578262i
\(324\) 0 0
\(325\) −4.16764 + 2.40619i −0.231179 + 0.133471i
\(326\) 0 0
\(327\) 1.24498 + 3.80890i 0.0688474 + 0.210633i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.72104 + 6.44502i 0.204527 + 0.354250i 0.949982 0.312305i \(-0.101101\pi\)
−0.745455 + 0.666556i \(0.767768\pi\)
\(332\) 0 0
\(333\) 7.73124 5.65861i 0.423669 0.310090i
\(334\) 0 0
\(335\) −6.15149 + 10.6547i −0.336092 + 0.582128i
\(336\) 0 0
\(337\) −5.31784 9.21076i −0.289681 0.501742i 0.684053 0.729433i \(-0.260216\pi\)
−0.973734 + 0.227690i \(0.926883\pi\)
\(338\) 0 0
\(339\) −26.3991 + 8.62882i −1.43380 + 0.468653i
\(340\) 0 0
\(341\) 5.08449 8.80660i 0.275341 0.476904i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.06895 + 14.5638i −0.165227 + 0.784090i
\(346\) 0 0
\(347\) 1.63842 + 0.945944i 0.0879552 + 0.0507809i 0.543332 0.839518i \(-0.317162\pi\)
−0.455377 + 0.890299i \(0.650496\pi\)
\(348\) 0 0
\(349\) −16.1105 9.30140i −0.862375 0.497892i 0.00243201 0.999997i \(-0.499226\pi\)
−0.864807 + 0.502105i \(0.832559\pi\)
\(350\) 0 0
\(351\) 28.2740 + 2.82139i 1.50915 + 0.150595i
\(352\) 0 0
\(353\) 3.89010 0.207049 0.103525 0.994627i \(-0.466988\pi\)
0.103525 + 0.994627i \(0.466988\pi\)
\(354\) 0 0
\(355\) 21.1931i 1.12481i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.46992 + 4.31276i 0.394248 + 0.227619i 0.683999 0.729483i \(-0.260239\pi\)
−0.289751 + 0.957102i \(0.593573\pi\)
\(360\) 0 0
\(361\) −9.41891 16.3140i −0.495732 0.858633i
\(362\) 0 0
\(363\) −8.70786 1.83496i −0.457044 0.0963103i
\(364\) 0 0
\(365\) 18.4858 10.6728i 0.967592 0.558639i
\(366\) 0 0
\(367\) 18.3252i 0.956568i 0.878205 + 0.478284i \(0.158741\pi\)
−0.878205 + 0.478284i \(0.841259\pi\)
\(368\) 0 0
\(369\) 2.61783 + 24.0937i 0.136279 + 1.25427i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.69583 −0.398475 −0.199237 0.979951i \(-0.563846\pi\)
−0.199237 + 0.979951i \(0.563846\pi\)
\(374\) 0 0
\(375\) −16.9320 3.56799i −0.874367 0.184250i
\(376\) 0 0
\(377\) 39.8928 2.05458
\(378\) 0 0
\(379\) −7.52510 −0.386539 −0.193269 0.981146i \(-0.561909\pi\)
−0.193269 + 0.981146i \(0.561909\pi\)
\(380\) 0 0
\(381\) 5.19339 + 1.09437i 0.266065 + 0.0560664i
\(382\) 0 0
\(383\) 3.40934 0.174209 0.0871047 0.996199i \(-0.472239\pi\)
0.0871047 + 0.996199i \(0.472239\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.4597 + 14.9748i −1.04003 + 0.761211i
\(388\) 0 0
\(389\) 9.62356i 0.487934i −0.969784 0.243967i \(-0.921551\pi\)
0.969784 0.243967i \(-0.0784489\pi\)
\(390\) 0 0
\(391\) 7.92000 4.57261i 0.400532 0.231247i
\(392\) 0 0
\(393\) −19.4279 4.09392i −0.980007 0.206511i
\(394\) 0 0
\(395\) −13.7388 23.7962i −0.691272 1.19732i
\(396\) 0 0
\(397\) −20.3349 11.7404i −1.02058 0.589232i −0.106308 0.994333i \(-0.533903\pi\)
−0.914272 + 0.405101i \(0.867236\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 25.6365i 1.28023i −0.768281 0.640113i \(-0.778887\pi\)
0.768281 0.640113i \(-0.221113\pi\)
\(402\) 0 0
\(403\) −22.9672 −1.14408
\(404\) 0 0
\(405\) −14.7165 16.1155i −0.731267 0.800785i
\(406\) 0 0
\(407\) −6.69635 3.86614i −0.331926 0.191637i
\(408\) 0 0
\(409\) −13.3646 7.71603i −0.660835 0.381533i 0.131760 0.991282i \(-0.457937\pi\)
−0.792595 + 0.609748i \(0.791270\pi\)
\(410\) 0 0
\(411\) −0.705196 + 3.34654i −0.0347848 + 0.165073i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −18.2093 + 31.5395i −0.893860 + 1.54821i
\(416\) 0 0
\(417\) −10.2238 + 3.34175i −0.500661 + 0.163646i
\(418\) 0 0
\(419\) 7.10643 + 12.3087i 0.347172 + 0.601319i 0.985746 0.168241i \(-0.0538088\pi\)
−0.638574 + 0.769560i \(0.720475\pi\)
\(420\) 0 0
\(421\) −0.849964 + 1.47218i −0.0414247 + 0.0717497i −0.885994 0.463696i \(-0.846523\pi\)
0.844570 + 0.535446i \(0.179856\pi\)
\(422\) 0 0
\(423\) −1.46320 13.4669i −0.0711433 0.654781i
\(424\) 0 0
\(425\) −1.13555 1.96684i −0.0550825 0.0954057i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −7.12465 21.7973i −0.343982 1.05238i
\(430\) 0 0
\(431\) −28.2868 + 16.3314i −1.36253 + 0.786656i −0.989960 0.141349i \(-0.954856\pi\)
−0.372568 + 0.928005i \(0.621523\pi\)
\(432\) 0 0
\(433\) 13.6919i 0.657992i 0.944331 + 0.328996i \(0.106710\pi\)
−0.944331 + 0.328996i \(0.893290\pi\)
\(434\) 0 0
\(435\) −22.8059 20.4622i −1.09346 0.981087i
\(436\) 0 0
\(437\) −0.713542 + 1.23589i −0.0341333 + 0.0591207i
\(438\) 0 0
\(439\) 7.64139 4.41176i 0.364704 0.210562i −0.306438 0.951890i \(-0.599137\pi\)
0.671142 + 0.741329i \(0.265804\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.3562 + 12.3300i −1.01466 + 0.585816i −0.912554 0.408957i \(-0.865893\pi\)
−0.102109 + 0.994773i \(0.532559\pi\)
\(444\) 0 0
\(445\) −18.9950 + 32.9004i −0.900451 + 1.55963i
\(446\) 0 0
\(447\) 4.34614 20.6248i 0.205566 0.975520i
\(448\) 0 0
\(449\) 4.61306i 0.217704i −0.994058 0.108852i \(-0.965283\pi\)
0.994058 0.108852i \(-0.0347174\pi\)
\(450\) 0 0
\(451\) 16.9390 9.77973i 0.797626 0.460509i
\(452\) 0 0
\(453\) −18.0257 3.79844i −0.846920 0.178466i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.85935 6.68460i −0.180533 0.312692i 0.761529 0.648131i \(-0.224449\pi\)
−0.942062 + 0.335438i \(0.891116\pi\)
\(458\) 0 0
\(459\) −1.33150 + 13.3434i −0.0621491 + 0.622816i
\(460\) 0 0
\(461\) 9.28621 16.0842i 0.432502 0.749115i −0.564586 0.825374i \(-0.690964\pi\)
0.997088 + 0.0762589i \(0.0242975\pi\)
\(462\) 0 0
\(463\) 17.7046 + 30.6653i 0.822804 + 1.42514i 0.903586 + 0.428406i \(0.140925\pi\)
−0.0807828 + 0.996732i \(0.525742\pi\)
\(464\) 0 0
\(465\) 13.1298 + 11.7805i 0.608882 + 0.546309i
\(466\) 0 0
\(467\) −7.92166 + 13.7207i −0.366571 + 0.634919i −0.989027 0.147736i \(-0.952802\pi\)
0.622456 + 0.782655i \(0.286135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.55751 + 4.08915i 0.209999 + 0.188418i
\(472\) 0 0
\(473\) 17.7210 + 10.2312i 0.814814 + 0.470433i
\(474\) 0 0
\(475\) 0.306919 + 0.177200i 0.0140824 + 0.00813048i
\(476\) 0 0
\(477\) 41.8403 4.54603i 1.91574 0.208149i
\(478\) 0 0
\(479\) 26.5025 1.21093 0.605465 0.795872i \(-0.292987\pi\)
0.605465 + 0.795872i \(0.292987\pi\)
\(480\) 0 0
\(481\) 17.4638i 0.796279i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.0689 6.96801i −0.548023 0.316401i
\(486\) 0 0
\(487\) 3.50469 + 6.07031i 0.158813 + 0.275072i 0.934441 0.356118i \(-0.115900\pi\)
−0.775628 + 0.631190i \(0.782567\pi\)
\(488\) 0 0
\(489\) 7.32370 8.16254i 0.331189 0.369123i
\(490\) 0 0
\(491\) −16.4508 + 9.49785i −0.742413 + 0.428632i −0.822946 0.568120i \(-0.807671\pi\)
0.0805333 + 0.996752i \(0.474338\pi\)
\(492\) 0 0
\(493\) 18.8267i 0.847910i
\(494\) 0 0
\(495\) −7.10745 + 16.1155i −0.319456 + 0.724337i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.25786 0.0563094 0.0281547 0.999604i \(-0.491037\pi\)
0.0281547 + 0.999604i \(0.491037\pi\)
\(500\) 0 0
\(501\) 9.03806 + 27.6512i 0.403791 + 1.23536i
\(502\) 0 0
\(503\) −37.9507 −1.69214 −0.846070 0.533072i \(-0.821037\pi\)
−0.846070 + 0.533072i \(0.821037\pi\)
\(504\) 0 0
\(505\) −23.4650 −1.04418
\(506\) 0 0
\(507\) −19.5518 + 21.7912i −0.868324 + 0.967780i
\(508\) 0 0
\(509\) −5.80942 −0.257498 −0.128749 0.991677i \(-0.541096\pi\)
−0.128749 + 0.991677i \(0.541096\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −0.861156 1.90712i −0.0380210 0.0842014i
\(514\) 0 0
\(515\) 46.7217i 2.05881i
\(516\) 0 0
\(517\) −9.46784 + 5.46626i −0.416395 + 0.240406i
\(518\) 0 0
\(519\) 8.94197 + 27.3572i 0.392509 + 1.20085i
\(520\) 0 0
\(521\) 6.84995 + 11.8645i 0.300102 + 0.519791i 0.976159 0.217058i \(-0.0696461\pi\)
−0.676057 + 0.736849i \(0.736313\pi\)
\(522\) 0 0
\(523\) 21.2073 + 12.2440i 0.927330 + 0.535394i 0.885966 0.463750i \(-0.153496\pi\)
0.0413640 + 0.999144i \(0.486830\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.8389i 0.472151i
\(528\) 0 0
\(529\) 10.4421 0.454003
\(530\) 0 0
\(531\) −0.377189 + 0.276071i −0.0163686 + 0.0119805i
\(532\) 0 0
\(533\) −38.2576 22.0880i −1.65712 0.956739i
\(534\) 0 0
\(535\) −2.29082 1.32261i −0.0990408 0.0571812i
\(536\) 0 0
\(537\) 23.9686 7.83437i 1.03432 0.338078i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.5804 32.1822i 0.798833 1.38362i −0.121543 0.992586i \(-0.538784\pi\)
0.920376 0.391034i \(-0.127882\pi\)
\(542\) 0 0
\(543\) 1.43698 6.81924i 0.0616666 0.292641i
\(544\) 0 0
\(545\) 2.80506 + 4.85850i 0.120155 + 0.208115i
\(546\) 0 0
\(547\) −3.31826 + 5.74739i −0.141878 + 0.245741i −0.928204 0.372072i \(-0.878648\pi\)
0.786326 + 0.617812i \(0.211981\pi\)
\(548\) 0 0
\(549\) 35.2263 3.82741i 1.50342 0.163350i
\(550\) 0 0
\(551\) −1.46892 2.54424i −0.0625781 0.108388i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.95767 9.98366i 0.380232 0.423783i
\(556\) 0 0
\(557\) −21.3205 + 12.3094i −0.903378 + 0.521565i −0.878295 0.478120i \(-0.841318\pi\)
−0.0250832 + 0.999685i \(0.507985\pi\)
\(558\) 0 0
\(559\) 46.2156i 1.95471i
\(560\) 0 0
\(561\) 10.2868 3.36235i 0.434310 0.141958i
\(562\) 0 0
\(563\) 6.28555 10.8869i 0.264904 0.458828i −0.702634 0.711551i \(-0.747993\pi\)
0.967539 + 0.252724i \(0.0813263\pi\)
\(564\) 0 0
\(565\) −33.6738 + 19.4416i −1.41667 + 0.817913i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.8872 7.44043i 0.540260 0.311919i −0.204924 0.978778i \(-0.565695\pi\)
0.745184 + 0.666859i \(0.232362\pi\)
\(570\) 0 0
\(571\) 8.45573 14.6458i 0.353861 0.612906i −0.633061 0.774102i \(-0.718202\pi\)
0.986922 + 0.161196i \(0.0515351\pi\)
\(572\) 0 0
\(573\) 13.8585 4.52977i 0.578945 0.189234i
\(574\) 0 0
\(575\) 3.11861i 0.130055i
\(576\) 0 0
\(577\) 18.1011 10.4507i 0.753558 0.435067i −0.0734203 0.997301i \(-0.523391\pi\)
0.826978 + 0.562234i \(0.190058\pi\)
\(578\) 0 0
\(579\) −9.98896 + 11.1331i −0.415127 + 0.462675i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −16.9832 29.4157i −0.703371 1.21827i
\(584\) 0 0
\(585\) 39.5476 4.29692i 1.63509 0.177656i
\(586\) 0 0
\(587\) 14.8542 25.7283i 0.613100 1.06192i −0.377615 0.925963i \(-0.623256\pi\)
0.990715 0.135957i \(-0.0434109\pi\)
\(588\) 0 0
\(589\) 0.845690 + 1.46478i 0.0348461 + 0.0603551i
\(590\) 0 0
\(591\) 8.33222 39.5409i 0.342742 1.62649i
\(592\) 0 0
\(593\) −5.89603 + 10.2122i −0.242121 + 0.419365i −0.961318 0.275440i \(-0.911176\pi\)
0.719197 + 0.694806i \(0.244510\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.2763 7.60810i 0.952637 0.311379i
\(598\) 0 0
\(599\) 27.7991 + 16.0498i 1.13584 + 0.655778i 0.945397 0.325921i \(-0.105674\pi\)
0.190443 + 0.981698i \(0.439008\pi\)
\(600\) 0 0
\(601\) 16.1636 + 9.33208i 0.659329 + 0.380664i 0.792021 0.610494i \(-0.209029\pi\)
−0.132692 + 0.991157i \(0.542362\pi\)
\(602\) 0 0
\(603\) 12.2825 8.98978i 0.500183 0.366092i
\(604\) 0 0
\(605\) −12.4588 −0.506522
\(606\) 0 0
\(607\) 13.8684i 0.562900i −0.959576 0.281450i \(-0.909185\pi\)
0.959576 0.281450i \(-0.0908154\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.3836 + 12.3458i 0.865089 + 0.499459i
\(612\) 0 0
\(613\) 10.4510 + 18.1017i 0.422114 + 0.731122i 0.996146 0.0877104i \(-0.0279550\pi\)
−0.574032 + 0.818833i \(0.694622\pi\)
\(614\) 0 0
\(615\) 10.5415 + 32.2507i 0.425072 + 1.30047i
\(616\) 0 0
\(617\) 15.2904 8.82792i 0.615569 0.355399i −0.159573 0.987186i \(-0.551012\pi\)
0.775142 + 0.631787i \(0.217678\pi\)
\(618\) 0 0
\(619\) 5.52857i 0.222212i 0.993809 + 0.111106i \(0.0354393\pi\)
−0.993809 + 0.111106i \(0.964561\pi\)
\(620\) 0 0
\(621\) 10.7448 14.9537i 0.431173 0.600073i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −28.6257 −1.14503
\(626\) 0 0
\(627\) −1.12782 + 1.25700i −0.0450409 + 0.0501998i
\(628\) 0 0
\(629\) −8.24169 −0.328618
\(630\) 0 0
\(631\) 24.3544 0.969533 0.484766 0.874644i \(-0.338905\pi\)
0.484766 + 0.874644i \(0.338905\pi\)
\(632\) 0 0
\(633\) 7.27310 + 22.2514i 0.289080 + 0.884415i
\(634\) 0 0
\(635\) 7.43045 0.294868
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −10.5804 + 23.9901i −0.418554 + 0.949032i
\(640\) 0 0
\(641\) 29.2340i 1.15467i −0.816506 0.577337i \(-0.804092\pi\)
0.816506 0.577337i \(-0.195908\pi\)
\(642\) 0 0
\(643\) 34.6535 20.0072i 1.36660 0.789008i 0.376109 0.926575i \(-0.377262\pi\)
0.990492 + 0.137567i \(0.0439284\pi\)
\(644\) 0 0
\(645\) −23.7053 + 26.4205i −0.933396 + 1.04031i
\(646\) 0 0
\(647\) −3.78276 6.55194i −0.148716 0.257583i 0.782037 0.623232i \(-0.214181\pi\)
−0.930753 + 0.365648i \(0.880847\pi\)
\(648\) 0 0
\(649\) 0.326700 + 0.188620i 0.0128241 + 0.00740399i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.1877i 1.53353i 0.641927 + 0.766766i \(0.278135\pi\)
−0.641927 + 0.766766i \(0.721865\pi\)
\(654\) 0 0
\(655\) −27.7965 −1.08610
\(656\) 0 0
\(657\) −26.2537 + 2.85252i −1.02425 + 0.111287i
\(658\) 0 0
\(659\) −22.8594 13.1979i −0.890474 0.514115i −0.0163765 0.999866i \(-0.505213\pi\)
−0.874098 + 0.485751i \(0.838546\pi\)
\(660\) 0 0
\(661\) −27.3501 15.7906i −1.06380 0.614184i −0.137317 0.990527i \(-0.543848\pi\)
−0.926480 + 0.376343i \(0.877181\pi\)
\(662\) 0 0
\(663\) −18.1933 16.3236i −0.706568 0.633957i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.9261 22.3886i 0.500500 0.866891i
\(668\) 0 0
\(669\) −19.9557 17.9049i −0.771533 0.692245i
\(670\) 0 0
\(671\) −14.2985 24.7658i −0.551989 0.956073i
\(672\) 0 0
\(673\) −7.21676 + 12.4998i −0.278186 + 0.481832i −0.970934 0.239348i \(-0.923066\pi\)
0.692748 + 0.721180i \(0.256400\pi\)
\(674\) 0 0
\(675\) −3.71358 2.66834i −0.142936 0.102704i
\(676\) 0 0
\(677\) 17.2099 + 29.8084i 0.661430 + 1.14563i 0.980240 + 0.197812i \(0.0633837\pi\)
−0.318809 + 0.947819i \(0.603283\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 25.4508 + 5.36309i 0.975275 + 0.205514i
\(682\) 0 0
\(683\) 16.4694 9.50861i 0.630184 0.363837i −0.150639 0.988589i \(-0.548133\pi\)
0.780823 + 0.624752i \(0.214800\pi\)
\(684\) 0 0
\(685\) 4.78806i 0.182942i
\(686\) 0 0
\(687\) 6.99767 33.2077i 0.266978 1.26695i
\(688\) 0 0
\(689\) −38.3574 + 66.4369i −1.46130 + 2.53104i
\(690\) 0 0
\(691\) −16.7795 + 9.68764i −0.638321 + 0.368535i −0.783968 0.620802i \(-0.786807\pi\)
0.145646 + 0.989337i \(0.453474\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −13.0411 + 7.52928i −0.494677 + 0.285602i
\(696\) 0 0
\(697\) 10.4240 18.0549i 0.394838 0.683880i
\(698\) 0 0
\(699\) 25.2001 + 22.6103i 0.953154 + 0.855201i
\(700\) 0 0
\(701\) 23.0297i 0.869819i 0.900474 + 0.434910i \(0.143220\pi\)
−0.900474 + 0.434910i \(0.856780\pi\)
\(702\) 0 0
\(703\) 1.11379 0.643045i 0.0420073 0.0242529i
\(704\) 0 0
\(705\) −5.89202 18.0261i −0.221906 0.678903i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 20.4119 + 35.3544i 0.766585 + 1.32776i 0.939405 + 0.342810i \(0.111379\pi\)
−0.172820 + 0.984953i \(0.555288\pi\)
\(710\) 0 0
\(711\) 3.67196 + 33.7956i 0.137709 + 1.26743i
\(712\) 0 0
\(713\) −7.44183 + 12.8896i −0.278699 + 0.482720i
\(714\) 0 0
\(715\) −16.0525 27.8038i −0.600331 1.03980i
\(716\) 0 0
\(717\) −13.9955 + 4.57458i −0.522673 + 0.170841i
\(718\) 0 0
\(719\) 21.1578 36.6464i 0.789054 1.36668i −0.137494 0.990503i \(-0.543905\pi\)
0.926547 0.376178i \(-0.122762\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.97532 + 14.1195i −0.110654 + 0.525111i
\(724\) 0 0
\(725\) −5.55995 3.21004i −0.206491 0.119218i
\(726\) 0 0
\(727\) −4.58754 2.64862i −0.170142 0.0982318i 0.412511 0.910953i \(-0.364652\pi\)
−0.582653 + 0.812721i \(0.697985\pi\)
\(728\) 0 0
\(729\) 8.61321 + 25.5893i 0.319008 + 0.947752i
\(730\) 0 0
\(731\) 21.8106 0.806694
\(732\) 0 0
\(733\) 18.6499i 0.688852i −0.938814 0.344426i \(-0.888074\pi\)
0.938814 0.344426i \(-0.111926\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.6384 6.14210i −0.391871 0.226247i
\(738\) 0 0
\(739\) −20.5918 35.6661i −0.757483 1.31200i −0.944131 0.329572i \(-0.893096\pi\)
0.186648 0.982427i \(-0.440238\pi\)
\(740\) 0 0
\(741\) 3.73227 + 0.786480i 0.137108 + 0.0288921i
\(742\) 0 0
\(743\) 20.5831 11.8837i 0.755122 0.435970i −0.0724196 0.997374i \(-0.523072\pi\)
0.827542 + 0.561404i \(0.189739\pi\)
\(744\) 0 0
\(745\) 29.5089i 1.08112i
\(746\) 0 0
\(747\) 36.3581 26.6111i 1.33027 0.973649i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.1230 0.661318 0.330659 0.943750i \(-0.392729\pi\)
0.330659 + 0.943750i \(0.392729\pi\)
\(752\) 0 0
\(753\) −23.0095 4.84866i −0.838513 0.176695i
\(754\) 0 0
\(755\) −25.7902 −0.938603
\(756\) 0 0
\(757\) 37.1059 1.34864 0.674319 0.738440i \(-0.264437\pi\)
0.674319 + 0.738440i \(0.264437\pi\)
\(758\) 0 0
\(759\) −14.5416 3.06426i −0.527826 0.111226i
\(760\) 0 0
\(761\) 11.8107 0.428139 0.214070 0.976818i \(-0.431328\pi\)
0.214070 + 0.976818i \(0.431328\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.02785 + 18.6637i 0.0733172 + 0.674789i
\(766\) 0 0
\(767\) 0.852017i 0.0307646i
\(768\) 0 0
\(769\) 3.14015 1.81297i 0.113237 0.0653773i −0.442312 0.896861i \(-0.645842\pi\)
0.555549 + 0.831484i \(0.312508\pi\)
\(770\) 0 0
\(771\) 10.1361 + 2.13592i 0.365043 + 0.0769233i
\(772\) 0 0
\(773\) 6.05553 + 10.4885i 0.217802 + 0.377245i 0.954136 0.299374i \(-0.0967778\pi\)
−0.736333 + 0.676619i \(0.763445\pi\)
\(774\) 0 0
\(775\) 3.20099 + 1.84809i 0.114983 + 0.0663854i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.25327i 0.116561i
\(780\) 0 0
\(781\) 21.1608 0.757192
\(782\) 0 0
\(783\) 15.6002 + 34.5482i 0.557505 + 1.23465i
\(784\) 0 0
\(785\) 7.42386 + 4.28617i 0.264969 + 0.152980i
\(786\) 0 0
\(787\) 17.5995 + 10.1611i 0.627354 + 0.362203i 0.779727 0.626120i \(-0.215358\pi\)
−0.152372 + 0.988323i \(0.548691\pi\)
\(788\) 0 0
\(789\) 1.20215 5.70483i 0.0427975 0.203097i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −32.2940 + 55.9348i −1.14679 + 1.98630i
\(794\) 0 0
\(795\) 56.0055 18.3060i 1.98631 0.649245i
\(796\) 0 0
\(797\) 16.8973 + 29.2669i 0.598532 + 1.03669i 0.993038 + 0.117795i \(0.0375824\pi\)
−0.394506 + 0.918893i \(0.629084\pi\)
\(798\) 0 0
\(799\) −5.82639 + 10.0916i −0.206123 + 0.357015i
\(800\) 0 0
\(801\) 37.9269 27.7593i 1.34008 0.980827i
\(802\) 0 0
\(803\) 10.6565 + 18.4576i 0.376060 + 0.651354i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.60934 20.2207i −0.232660 0.711803i
\(808\) 0 0
\(809\) 4.19308 2.42087i 0.147421 0.0851134i −0.424475 0.905440i \(-0.639541\pi\)
0.571896 + 0.820326i \(0.306208\pi\)
\(810\) 0 0
\(811\) 0.493486i 0.0173286i 0.999962 + 0.00866432i \(0.00275797\pi\)
−0.999962 + 0.00866432i \(0.997242\pi\)
\(812\) 0 0
\(813\) −37.9338 34.0355i −1.33040 1.19368i
\(814\) 0 0
\(815\) 7.67656 13.2962i 0.268898 0.465745i
\(816\) 0 0
\(817\) −2.94749 + 1.70174i −0.103120 + 0.0595362i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.6931 15.4113i 0.931595 0.537857i 0.0442792 0.999019i \(-0.485901\pi\)
0.887316 + 0.461163i \(0.152568\pi\)
\(822\) 0 0
\(823\) 7.94249 13.7568i 0.276858 0.479532i −0.693744 0.720221i \(-0.744040\pi\)
0.970602 + 0.240690i \(0.0773736\pi\)
\(824\) 0 0
\(825\) −0.760974 + 3.61123i −0.0264937 + 0.125727i
\(826\) 0 0
\(827\) 35.6756i 1.24056i 0.784379 + 0.620282i \(0.212982\pi\)
−0.784379 + 0.620282i \(0.787018\pi\)
\(828\) 0 0
\(829\) −2.87997 + 1.66275i −0.100026 + 0.0577498i −0.549178 0.835705i \(-0.685059\pi\)
0.449153 + 0.893455i \(0.351726\pi\)
\(830\) 0 0
\(831\) 15.5961 + 3.28647i 0.541022 + 0.114006i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 20.3636 + 35.2708i 0.704712 + 1.22060i
\(836\) 0 0
\(837\) −8.98136 19.8902i −0.310441 0.687504i
\(838\) 0 0
\(839\) −20.9689 + 36.3192i −0.723926 + 1.25388i 0.235488 + 0.971877i \(0.424331\pi\)
−0.959414 + 0.282000i \(0.909002\pi\)
\(840\) 0 0
\(841\) 12.1100 + 20.9752i 0.417588 + 0.723283i
\(842\) 0 0
\(843\) −1.56684 1.40582i −0.0539649 0.0484191i
\(844\) 0 0
\(845\) −20.4938 + 35.4963i −0.705007 + 1.22111i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −15.7461 14.1280i −0.540406 0.484870i
\(850\) 0 0
\(851\) 9.80100 + 5.65861i 0.335974 + 0.193975i
\(852\) 0 0
\(853\) −12.5330 7.23594i −0.429122 0.247754i 0.269851 0.962902i \(-0.413026\pi\)
−0.698973 + 0.715149i \(0.746359\pi\)
\(854\) 0 0
\(855\) −1.73025 2.36401i −0.0591734 0.0808473i
\(856\) 0 0
\(857\) −47.4657 −1.62140 −0.810699 0.585463i \(-0.800913\pi\)
−0.810699 + 0.585463i \(0.800913\pi\)
\(858\) 0 0
\(859\) 6.00809i 0.204993i −0.994733 0.102497i \(-0.967317\pi\)
0.994733 0.102497i \(-0.0326831\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.1064 + 15.6499i 0.922714 + 0.532729i 0.884500 0.466540i \(-0.154500\pi\)
0.0382141 + 0.999270i \(0.487833\pi\)
\(864\) 0 0
\(865\) 20.1471 + 34.8958i 0.685022 + 1.18649i
\(866\) 0 0
\(867\) −11.9604 + 13.3303i −0.406197 + 0.452722i
\(868\) 0 0
\(869\) 23.7599 13.7178i 0.806000 0.465344i
\(870\) 0 0
\(871\) 27.7445i 0.940086i
\(872\) 0 0
\(873\) 10.1830 + 13.9129i 0.344644 + 0.470879i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.3764 1.39718 0.698591 0.715521i \(-0.253811\pi\)
0.698591 + 0.715521i \(0.253811\pi\)
\(878\) 0 0
\(879\) −3.39105 10.3746i −0.114377 0.349928i
\(880\) 0 0
\(881\) −39.7350 −1.33870 −0.669352 0.742945i \(-0.733428\pi\)
−0.669352 + 0.742945i \(0.733428\pi\)
\(882\) 0 0
\(883\) −48.1324 −1.61978 −0.809892 0.586579i \(-0.800474\pi\)
−0.809892 + 0.586579i \(0.800474\pi\)
\(884\) 0 0
\(885\) −0.437024 + 0.487080i −0.0146904 + 0.0163730i
\(886\) 0 0
\(887\) −28.1122 −0.943916 −0.471958 0.881621i \(-0.656453\pi\)
−0.471958 + 0.881621i \(0.656453\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 16.0909 14.6940i 0.539065 0.492268i
\(892\) 0 0
\(893\) 1.81838i 0.0608497i
\(894\) 0 0
\(895\) 30.5735 17.6516i 1.02196 0.590028i
\(896\) 0 0
\(897\) 10.4279 + 31.9032i 0.348177 + 1.06522i
\(898\) 0 0
\(899\) −15.3200 26.5350i −0.510950 0.884992i
\(900\) 0 0
\(901\) −31.3537 18.1020i −1.04454 0.603066i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.75663i 0.324321i
\(906\) 0 0
\(907\) 59.4859 1.97520 0.987599 0.156997i \(-0.0501812\pi\)
0.987599 + 0.156997i \(0.0501812\pi\)
\(908\) 0 0
\(909\) 26.5617 + 11.7146i 0.880996 + 0.388548i
\(910\) 0 0
\(911\) 25.4502 + 14.6937i 0.843204 + 0.486824i 0.858352 0.513061i \(-0.171489\pi\)
−0.0151480 + 0.999885i \(0.504822\pi\)
\(912\) 0 0
\(913\) −31.4913 18.1815i −1.04221 0.601721i
\(914\) 0 0
\(915\) 47.1524 15.4122i 1.55881 0.509512i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 20.7994 36.0256i 0.686108 1.18837i −0.286979 0.957937i \(-0.592651\pi\)
0.973087 0.230437i \(-0.0740156\pi\)
\(920\) 0 0
\(921\) 10.2594 48.6863i 0.338058 1.60427i
\(922\) 0 0
\(923\) −23.8964 41.3897i −0.786558 1.36236i
\(924\) 0 0
\(925\) 1.40525 2.43396i 0.0462043 0.0800282i
\(926\) 0 0
\(927\) −23.3252 + 52.8878i −0.766100 + 1.73706i
\(928\) 0 0
\(929\) −1.21614 2.10641i −0.0399002 0.0691092i 0.845386 0.534156i \(-0.179371\pi\)
−0.885286 + 0.465047i \(0.846037\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −20.5113 + 22.8606i −0.671510 + 0.748423i
\(934\) 0 0
\(935\) 13.1215 7.57569i 0.429118 0.247752i
\(936\) 0 0
\(937\) 7.29837i 0.238427i 0.992869 + 0.119214i \(0.0380374\pi\)
−0.992869 + 0.119214i \(0.961963\pi\)
\(938\) 0 0
\(939\) −44.5841 + 14.5728i −1.45495 + 0.475564i
\(940\) 0 0
\(941\) 21.2367 36.7831i 0.692298 1.19910i −0.278785 0.960354i \(-0.589932\pi\)
0.971083 0.238742i \(-0.0767350\pi\)
\(942\) 0 0
\(943\) −24.7925 + 14.3139i −0.807354 + 0.466126i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.3119 27.3155i 1.53743 0.887636i 0.538443 0.842662i \(-0.319013\pi\)
0.998988 0.0449739i \(-0.0143205\pi\)
\(948\) 0 0
\(949\) 24.0683 41.6874i 0.781288 1.35323i
\(950\) 0 0
\(951\) −8.84717 + 2.89178i −0.286889 + 0.0937725i
\(952\) 0 0
\(953\) 17.1877i 0.556764i 0.960470 + 0.278382i \(0.0897981\pi\)
−0.960470 + 0.278382i \(0.910202\pi\)
\(954\) 0 0
\(955\) 17.6773 10.2060i 0.572025 0.330259i
\(956\) 0 0
\(957\) 20.4310 22.7711i 0.660439 0.736084i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −6.67994 11.5700i −0.215482 0.373226i
\(962\) 0 0
\(963\) 1.93285 + 2.64082i 0.0622854 + 0.0850991i
\(964\) 0 0
\(965\) −10.4702 + 18.1350i −0.337049 + 0.583786i
\(966\) 0 0
\(967\) −11.1546 19.3203i −0.358706 0.621298i 0.629039 0.777374i \(-0.283449\pi\)
−0.987745 + 0.156076i \(0.950115\pi\)
\(968\) 0 0
\(969\) −0.371164 + 1.76138i −0.0119235 + 0.0565836i
\(970\) 0 0
\(971\) −27.1241 + 46.9803i −0.870453 + 1.50767i −0.00892375 + 0.999960i \(0.502841\pi\)
−0.861529 + 0.507708i \(0.830493\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7.92279 2.58964i 0.253732 0.0829349i
\(976\) 0 0
\(977\) 35.6936 + 20.6077i 1.14194 + 0.659299i 0.946910 0.321498i \(-0.104186\pi\)
0.195029 + 0.980797i \(0.437520\pi\)
\(978\) 0 0
\(979\) −32.8501 18.9660i −1.04989 0.606157i
\(980\) 0 0
\(981\) −0.749708 6.90008i −0.0239363 0.220303i
\(982\) 0 0
\(983\) −18.4394 −0.588125 −0.294062 0.955786i \(-0.595007\pi\)
−0.294062 + 0.955786i \(0.595007\pi\)
\(984\) 0 0
\(985\) 56.5731i 1.80257i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −25.9371 14.9748i −0.824752 0.476171i
\(990\) 0 0
\(991\) 1.86116 + 3.22362i 0.0591216 + 0.102402i 0.894071 0.447925i \(-0.147837\pi\)
−0.834950 + 0.550326i \(0.814503\pi\)
\(992\) 0 0
\(993\) −4.00474 12.2522i −0.127087 0.388810i
\(994\) 0 0
\(995\) 29.6904 17.1418i 0.941250 0.543431i
\(996\) 0 0
\(997\) 0.475330i 0.0150539i 0.999972 + 0.00752693i \(0.00239592\pi\)
−0.999972 + 0.00752693i \(0.997604\pi\)
\(998\) 0 0
\(999\) −15.1241 + 6.82924i −0.478504 + 0.216068i
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 1764.2.bm.b.1685.2 16
3.2 odd 2 5292.2.bm.b.4625.7 16
7.2 even 3 252.2.x.a.209.7 yes 16
7.3 odd 6 1764.2.w.a.1109.4 16
7.4 even 3 1764.2.w.a.1109.5 16
7.5 odd 6 252.2.x.a.209.2 yes 16
7.6 odd 2 inner 1764.2.bm.b.1685.7 16
9.4 even 3 5292.2.w.a.1097.7 16
9.5 odd 6 1764.2.w.a.509.4 16
21.2 odd 6 756.2.x.a.629.2 16
21.5 even 6 756.2.x.a.629.7 16
21.11 odd 6 5292.2.w.a.521.2 16
21.17 even 6 5292.2.w.a.521.7 16
21.20 even 2 5292.2.bm.b.4625.2 16
28.19 even 6 1008.2.cc.c.209.7 16
28.23 odd 6 1008.2.cc.c.209.2 16
63.2 odd 6 2268.2.f.b.1133.14 16
63.4 even 3 5292.2.bm.b.2285.2 16
63.5 even 6 252.2.x.a.41.7 yes 16
63.13 odd 6 5292.2.w.a.1097.2 16
63.16 even 3 2268.2.f.b.1133.4 16
63.23 odd 6 252.2.x.a.41.2 16
63.31 odd 6 5292.2.bm.b.2285.7 16
63.32 odd 6 inner 1764.2.bm.b.1697.7 16
63.40 odd 6 756.2.x.a.125.2 16
63.41 even 6 1764.2.w.a.509.5 16
63.47 even 6 2268.2.f.b.1133.3 16
63.58 even 3 756.2.x.a.125.7 16
63.59 even 6 inner 1764.2.bm.b.1697.2 16
63.61 odd 6 2268.2.f.b.1133.13 16
84.23 even 6 3024.2.cc.c.2897.2 16
84.47 odd 6 3024.2.cc.c.2897.7 16
252.23 even 6 1008.2.cc.c.545.7 16
252.103 even 6 3024.2.cc.c.881.2 16
252.131 odd 6 1008.2.cc.c.545.2 16
252.247 odd 6 3024.2.cc.c.881.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.x.a.41.2 16 63.23 odd 6
252.2.x.a.41.7 yes 16 63.5 even 6
252.2.x.a.209.2 yes 16 7.5 odd 6
252.2.x.a.209.7 yes 16 7.2 even 3
756.2.x.a.125.2 16 63.40 odd 6
756.2.x.a.125.7 16 63.58 even 3
756.2.x.a.629.2 16 21.2 odd 6
756.2.x.a.629.7 16 21.5 even 6
1008.2.cc.c.209.2 16 28.23 odd 6
1008.2.cc.c.209.7 16 28.19 even 6
1008.2.cc.c.545.2 16 252.131 odd 6
1008.2.cc.c.545.7 16 252.23 even 6
1764.2.w.a.509.4 16 9.5 odd 6
1764.2.w.a.509.5 16 63.41 even 6
1764.2.w.a.1109.4 16 7.3 odd 6
1764.2.w.a.1109.5 16 7.4 even 3
1764.2.bm.b.1685.2 16 1.1 even 1 trivial
1764.2.bm.b.1685.7 16 7.6 odd 2 inner
1764.2.bm.b.1697.2 16 63.59 even 6 inner
1764.2.bm.b.1697.7 16 63.32 odd 6 inner
2268.2.f.b.1133.3 16 63.47 even 6
2268.2.f.b.1133.4 16 63.16 even 3
2268.2.f.b.1133.13 16 63.61 odd 6
2268.2.f.b.1133.14 16 63.2 odd 6
3024.2.cc.c.881.2 16 252.103 even 6
3024.2.cc.c.881.7 16 252.247 odd 6
3024.2.cc.c.2897.2 16 84.23 even 6
3024.2.cc.c.2897.7 16 84.47 odd 6
5292.2.w.a.521.2 16 21.11 odd 6
5292.2.w.a.521.7 16 21.17 even 6
5292.2.w.a.1097.2 16 63.13 odd 6
5292.2.w.a.1097.7 16 9.4 even 3
5292.2.bm.b.2285.2 16 63.4 even 3
5292.2.bm.b.2285.7 16 63.31 odd 6
5292.2.bm.b.4625.2 16 21.20 even 2
5292.2.bm.b.4625.7 16 3.2 odd 2