Properties

Label 76.3.h.a.69.3
Level $76$
Weight $3$
Character 76.69
Analytic conductor $2.071$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,3,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.h (of order \(6\), degree \(2\), 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: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 917x^{4} - 1582x^{3} + 4294x^{2} - 3528x + 4971 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 69.3
Root \(0.500000 - 1.77696i\) of defining polynomial
Character \(\chi\) \(=\) 76.69
Dual form 76.3.h.a.65.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.28889 - 1.32149i) q^{3} +(-4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +(-1.00732 + 1.74474i) q^{9} +O(q^{10})\) \(q+(2.28889 - 1.32149i) q^{3} +(-4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +(-1.00732 + 1.74474i) q^{9} -4.61883 q^{11} +(19.8169 + 11.4413i) q^{13} +(-22.0500 - 12.7306i) q^{15} +(3.02053 + 5.23171i) q^{17} +(-2.21998 - 18.8699i) q^{19} +(17.3782 - 10.0333i) q^{21} +(-11.8651 + 20.5510i) q^{23} +(-33.9020 + 58.7200i) q^{25} +29.1115i q^{27} +(10.1836 + 5.87948i) q^{29} -4.22078i q^{31} +(-10.5720 + 6.10375i) q^{33} +(-36.5708 - 63.3424i) q^{35} -11.8554i q^{37} +60.4782 q^{39} +(-7.64514 + 4.41392i) q^{41} +(-11.5058 - 19.9286i) q^{43} +19.4081 q^{45} +(33.0941 - 57.3207i) q^{47} +8.64494 q^{49} +(13.8273 + 7.98320i) q^{51} +(40.7377 + 23.5199i) q^{53} +(22.2477 + 38.5342i) q^{55} +(-30.0176 - 40.2573i) q^{57} +(-7.52242 + 4.34307i) q^{59} +(-3.57907 + 6.19914i) q^{61} +(-7.64803 + 13.2468i) q^{63} -220.439i q^{65} +(-77.3224 - 44.6421i) q^{67} +62.7186i q^{69} +(-67.9840 + 39.2506i) q^{71} +(43.1234 + 74.6919i) q^{73} +179.205i q^{75} -35.0682 q^{77} +(57.9637 - 33.4653i) q^{79} +(29.4047 + 50.9304i) q^{81} +5.30752 q^{83} +(29.0982 - 50.3996i) q^{85} +31.0787 q^{87} +(-99.6601 - 57.5388i) q^{89} +(150.458 + 86.8671i) q^{91} +(-5.57772 - 9.66090i) q^{93} +(-146.735 + 109.412i) q^{95} +(-49.2806 + 28.4521i) q^{97} +(4.65266 - 8.05865i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} - q^{5} - 12 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} - q^{5} - 12 q^{7} + 16 q^{9} - 10 q^{11} + 9 q^{13} + 33 q^{15} + 23 q^{17} - 33 q^{19} - 31 q^{23} - 73 q^{25} - 105 q^{29} - 111 q^{33} - 68 q^{35} + 234 q^{39} + 18 q^{41} - 41 q^{43} + 200 q^{45} + 107 q^{47} + 312 q^{49} - 9 q^{51} + 39 q^{53} + 70 q^{55} - 381 q^{57} + 348 q^{59} - 45 q^{61} - 358 q^{63} - 432 q^{67} - 243 q^{71} + 16 q^{73} + 544 q^{77} + 75 q^{79} - 68 q^{81} - 82 q^{83} + 109 q^{85} + 414 q^{87} - 213 q^{89} + 222 q^{91} + 288 q^{93} - 385 q^{95} + 144 q^{97} - 388 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 2.28889 1.32149i 0.762963 0.440497i −0.0673955 0.997726i \(-0.521469\pi\)
0.830359 + 0.557229i \(0.188136\pi\)
\(4\) 0 0
\(5\) −4.81674 8.34284i −0.963348 1.66857i −0.713989 0.700157i \(-0.753113\pi\)
−0.249360 0.968411i \(-0.580220\pi\)
\(6\) 0 0
\(7\) 7.59243 1.08463 0.542316 0.840174i \(-0.317547\pi\)
0.542316 + 0.840174i \(0.317547\pi\)
\(8\) 0 0
\(9\) −1.00732 + 1.74474i −0.111925 + 0.193860i
\(10\) 0 0
\(11\) −4.61883 −0.419894 −0.209947 0.977713i \(-0.567329\pi\)
−0.209947 + 0.977713i \(0.567329\pi\)
\(12\) 0 0
\(13\) 19.8169 + 11.4413i 1.52438 + 0.880099i 0.999583 + 0.0288692i \(0.00919064\pi\)
0.524793 + 0.851230i \(0.324143\pi\)
\(14\) 0 0
\(15\) −22.0500 12.7306i −1.47000 0.848704i
\(16\) 0 0
\(17\) 3.02053 + 5.23171i 0.177678 + 0.307748i 0.941085 0.338171i \(-0.109808\pi\)
−0.763407 + 0.645918i \(0.776475\pi\)
\(18\) 0 0
\(19\) −2.21998 18.8699i −0.116841 0.993151i
\(20\) 0 0
\(21\) 17.3782 10.0333i 0.827534 0.477777i
\(22\) 0 0
\(23\) −11.8651 + 20.5510i −0.515875 + 0.893521i 0.483955 + 0.875093i \(0.339200\pi\)
−0.999830 + 0.0184287i \(0.994134\pi\)
\(24\) 0 0
\(25\) −33.9020 + 58.7200i −1.35608 + 2.34880i
\(26\) 0 0
\(27\) 29.1115i 1.07820i
\(28\) 0 0
\(29\) 10.1836 + 5.87948i 0.351157 + 0.202741i 0.665195 0.746670i \(-0.268348\pi\)
−0.314038 + 0.949411i \(0.601682\pi\)
\(30\) 0 0
\(31\) 4.22078i 0.136154i −0.997680 0.0680771i \(-0.978314\pi\)
0.997680 0.0680771i \(-0.0216864\pi\)
\(32\) 0 0
\(33\) −10.5720 + 6.10375i −0.320364 + 0.184962i
\(34\) 0 0
\(35\) −36.5708 63.3424i −1.04488 1.80978i
\(36\) 0 0
\(37\) 11.8554i 0.320417i −0.987083 0.160209i \(-0.948783\pi\)
0.987083 0.160209i \(-0.0512167\pi\)
\(38\) 0 0
\(39\) 60.4782 1.55072
\(40\) 0 0
\(41\) −7.64514 + 4.41392i −0.186467 + 0.107657i −0.590327 0.807164i \(-0.701001\pi\)
0.403861 + 0.914821i \(0.367668\pi\)
\(42\) 0 0
\(43\) −11.5058 19.9286i −0.267576 0.463456i 0.700659 0.713496i \(-0.252889\pi\)
−0.968235 + 0.250041i \(0.919556\pi\)
\(44\) 0 0
\(45\) 19.4081 0.431291
\(46\) 0 0
\(47\) 33.0941 57.3207i 0.704130 1.21959i −0.262874 0.964830i \(-0.584670\pi\)
0.967004 0.254759i \(-0.0819962\pi\)
\(48\) 0 0
\(49\) 8.64494 0.176427
\(50\) 0 0
\(51\) 13.8273 + 7.98320i 0.271124 + 0.156533i
\(52\) 0 0
\(53\) 40.7377 + 23.5199i 0.768635 + 0.443772i 0.832388 0.554194i \(-0.186973\pi\)
−0.0637523 + 0.997966i \(0.520307\pi\)
\(54\) 0 0
\(55\) 22.2477 + 38.5342i 0.404504 + 0.700622i
\(56\) 0 0
\(57\) −30.0176 40.2573i −0.526625 0.706269i
\(58\) 0 0
\(59\) −7.52242 + 4.34307i −0.127499 + 0.0736114i −0.562393 0.826870i \(-0.690119\pi\)
0.434894 + 0.900482i \(0.356786\pi\)
\(60\) 0 0
\(61\) −3.57907 + 6.19914i −0.0586733 + 0.101625i −0.893870 0.448326i \(-0.852020\pi\)
0.835197 + 0.549951i \(0.185354\pi\)
\(62\) 0 0
\(63\) −7.64803 + 13.2468i −0.121397 + 0.210266i
\(64\) 0 0
\(65\) 220.439i 3.39137i
\(66\) 0 0
\(67\) −77.3224 44.6421i −1.15407 0.666300i −0.204191 0.978931i \(-0.565456\pi\)
−0.949875 + 0.312631i \(0.898790\pi\)
\(68\) 0 0
\(69\) 62.7186i 0.908965i
\(70\) 0 0
\(71\) −67.9840 + 39.2506i −0.957521 + 0.552825i −0.895409 0.445244i \(-0.853117\pi\)
−0.0621119 + 0.998069i \(0.519784\pi\)
\(72\) 0 0
\(73\) 43.1234 + 74.6919i 0.590731 + 1.02318i 0.994134 + 0.108154i \(0.0344939\pi\)
−0.403403 + 0.915022i \(0.632173\pi\)
\(74\) 0 0
\(75\) 179.205i 2.38940i
\(76\) 0 0
\(77\) −35.0682 −0.455431
\(78\) 0 0
\(79\) 57.9637 33.4653i 0.733717 0.423612i −0.0860632 0.996290i \(-0.527429\pi\)
0.819781 + 0.572678i \(0.194095\pi\)
\(80\) 0 0
\(81\) 29.4047 + 50.9304i 0.363021 + 0.628770i
\(82\) 0 0
\(83\) 5.30752 0.0639460 0.0319730 0.999489i \(-0.489821\pi\)
0.0319730 + 0.999489i \(0.489821\pi\)
\(84\) 0 0
\(85\) 29.0982 50.3996i 0.342332 0.592936i
\(86\) 0 0
\(87\) 31.0787 0.357227
\(88\) 0 0
\(89\) −99.6601 57.5388i −1.11978 0.646503i −0.178432 0.983952i \(-0.557103\pi\)
−0.941344 + 0.337449i \(0.890436\pi\)
\(90\) 0 0
\(91\) 150.458 + 86.8671i 1.65339 + 0.954584i
\(92\) 0 0
\(93\) −5.57772 9.66090i −0.0599755 0.103881i
\(94\) 0 0
\(95\) −146.735 + 109.412i −1.54458 + 1.15171i
\(96\) 0 0
\(97\) −49.2806 + 28.4521i −0.508047 + 0.293321i −0.732031 0.681272i \(-0.761427\pi\)
0.223983 + 0.974593i \(0.428094\pi\)
\(98\) 0 0
\(99\) 4.65266 8.05865i 0.0469966 0.0814005i
\(100\) 0 0
\(101\) −17.2533 + 29.8836i −0.170825 + 0.295877i −0.938708 0.344712i \(-0.887977\pi\)
0.767884 + 0.640589i \(0.221310\pi\)
\(102\) 0 0
\(103\) 50.3577i 0.488910i 0.969661 + 0.244455i \(0.0786090\pi\)
−0.969661 + 0.244455i \(0.921391\pi\)
\(104\) 0 0
\(105\) −167.413 96.6558i −1.59441 0.920532i
\(106\) 0 0
\(107\) 184.188i 1.72138i −0.509126 0.860692i \(-0.670031\pi\)
0.509126 0.860692i \(-0.329969\pi\)
\(108\) 0 0
\(109\) −17.7493 + 10.2476i −0.162838 + 0.0940145i −0.579204 0.815182i \(-0.696637\pi\)
0.416366 + 0.909197i \(0.363303\pi\)
\(110\) 0 0
\(111\) −15.6669 27.1358i −0.141143 0.244467i
\(112\) 0 0
\(113\) 138.686i 1.22731i 0.789576 + 0.613653i \(0.210301\pi\)
−0.789576 + 0.613653i \(0.789699\pi\)
\(114\) 0 0
\(115\) 228.605 1.98787
\(116\) 0 0
\(117\) −39.9241 + 23.0502i −0.341231 + 0.197010i
\(118\) 0 0
\(119\) 22.9331 + 39.7214i 0.192715 + 0.333793i
\(120\) 0 0
\(121\) −99.6664 −0.823689
\(122\) 0 0
\(123\) −11.6659 + 20.2060i −0.0948449 + 0.164276i
\(124\) 0 0
\(125\) 412.351 3.29881
\(126\) 0 0
\(127\) −151.220 87.3070i −1.19071 0.687457i −0.232242 0.972658i \(-0.574606\pi\)
−0.958468 + 0.285201i \(0.907940\pi\)
\(128\) 0 0
\(129\) −52.6709 30.4096i −0.408302 0.235733i
\(130\) 0 0
\(131\) 11.2446 + 19.4763i 0.0858370 + 0.148674i 0.905748 0.423818i \(-0.139310\pi\)
−0.819911 + 0.572492i \(0.805977\pi\)
\(132\) 0 0
\(133\) −16.8550 143.268i −0.126730 1.07720i
\(134\) 0 0
\(135\) 242.873 140.223i 1.79906 1.03869i
\(136\) 0 0
\(137\) 38.3769 66.4708i 0.280124 0.485188i −0.691291 0.722576i \(-0.742958\pi\)
0.971415 + 0.237388i \(0.0762912\pi\)
\(138\) 0 0
\(139\) −84.7624 + 146.813i −0.609801 + 1.05621i 0.381472 + 0.924381i \(0.375417\pi\)
−0.991273 + 0.131826i \(0.957916\pi\)
\(140\) 0 0
\(141\) 174.934i 1.24067i
\(142\) 0 0
\(143\) −91.5309 52.8454i −0.640077 0.369548i
\(144\) 0 0
\(145\) 113.280i 0.781240i
\(146\) 0 0
\(147\) 19.7873 11.4242i 0.134608 0.0777157i
\(148\) 0 0
\(149\) 97.6855 + 169.196i 0.655607 + 1.13555i 0.981741 + 0.190222i \(0.0609207\pi\)
−0.326134 + 0.945324i \(0.605746\pi\)
\(150\) 0 0
\(151\) 162.149i 1.07383i −0.843635 0.536917i \(-0.819589\pi\)
0.843635 0.536917i \(-0.180411\pi\)
\(152\) 0 0
\(153\) −12.1706 −0.0795464
\(154\) 0 0
\(155\) −35.2133 + 20.3304i −0.227183 + 0.131164i
\(156\) 0 0
\(157\) −65.6679 113.740i −0.418267 0.724460i 0.577498 0.816392i \(-0.304029\pi\)
−0.995765 + 0.0919322i \(0.970696\pi\)
\(158\) 0 0
\(159\) 124.325 0.781920
\(160\) 0 0
\(161\) −90.0851 + 156.032i −0.559535 + 0.969142i
\(162\) 0 0
\(163\) −99.1996 −0.608586 −0.304293 0.952578i \(-0.598420\pi\)
−0.304293 + 0.952578i \(0.598420\pi\)
\(164\) 0 0
\(165\) 101.845 + 58.8003i 0.617244 + 0.356366i
\(166\) 0 0
\(167\) −183.454 105.917i −1.09853 0.634235i −0.162693 0.986677i \(-0.552018\pi\)
−0.935834 + 0.352442i \(0.885351\pi\)
\(168\) 0 0
\(169\) 177.306 + 307.103i 1.04915 + 1.81718i
\(170\) 0 0
\(171\) 35.1592 + 15.1348i 0.205609 + 0.0885075i
\(172\) 0 0
\(173\) 147.823 85.3459i 0.854471 0.493329i −0.00768594 0.999970i \(-0.502447\pi\)
0.862157 + 0.506641i \(0.169113\pi\)
\(174\) 0 0
\(175\) −257.398 + 445.827i −1.47085 + 2.54758i
\(176\) 0 0
\(177\) −11.4787 + 19.8816i −0.0648512 + 0.112326i
\(178\) 0 0
\(179\) 192.935i 1.07785i 0.842354 + 0.538925i \(0.181170\pi\)
−0.842354 + 0.538925i \(0.818830\pi\)
\(180\) 0 0
\(181\) −263.715 152.256i −1.45699 0.841194i −0.458129 0.888886i \(-0.651480\pi\)
−0.998862 + 0.0476914i \(0.984814\pi\)
\(182\) 0 0
\(183\) 18.9188i 0.103382i
\(184\) 0 0
\(185\) −98.9081 + 57.1046i −0.534638 + 0.308674i
\(186\) 0 0
\(187\) −13.9513 24.1644i −0.0746060 0.129221i
\(188\) 0 0
\(189\) 221.027i 1.16946i
\(190\) 0 0
\(191\) 201.179 1.05329 0.526647 0.850084i \(-0.323449\pi\)
0.526647 + 0.850084i \(0.323449\pi\)
\(192\) 0 0
\(193\) 33.0963 19.1081i 0.171483 0.0990059i −0.411802 0.911273i \(-0.635100\pi\)
0.583285 + 0.812268i \(0.301767\pi\)
\(194\) 0 0
\(195\) −291.308 504.560i −1.49389 2.58749i
\(196\) 0 0
\(197\) 100.500 0.510151 0.255076 0.966921i \(-0.417900\pi\)
0.255076 + 0.966921i \(0.417900\pi\)
\(198\) 0 0
\(199\) 49.2159 85.2444i 0.247316 0.428364i −0.715464 0.698649i \(-0.753785\pi\)
0.962780 + 0.270285i \(0.0871181\pi\)
\(200\) 0 0
\(201\) −235.977 −1.17401
\(202\) 0 0
\(203\) 77.3179 + 44.6395i 0.380876 + 0.219899i
\(204\) 0 0
\(205\) 73.6493 + 42.5215i 0.359265 + 0.207422i
\(206\) 0 0
\(207\) −23.9040 41.4030i −0.115478 0.200015i
\(208\) 0 0
\(209\) 10.2537 + 87.1568i 0.0490609 + 0.417018i
\(210\) 0 0
\(211\) 279.837 161.564i 1.32624 0.765707i 0.341527 0.939872i \(-0.389056\pi\)
0.984716 + 0.174165i \(0.0557226\pi\)
\(212\) 0 0
\(213\) −103.739 + 179.680i −0.487036 + 0.843570i
\(214\) 0 0
\(215\) −110.841 + 191.982i −0.515538 + 0.892938i
\(216\) 0 0
\(217\) 32.0460i 0.147677i
\(218\) 0 0
\(219\) 197.409 + 113.974i 0.901412 + 0.520430i
\(220\) 0 0
\(221\) 138.235i 0.625497i
\(222\) 0 0
\(223\) 256.629 148.165i 1.15080 0.664415i 0.201718 0.979444i \(-0.435347\pi\)
0.949082 + 0.315028i \(0.102014\pi\)
\(224\) 0 0
\(225\) −68.3006 118.300i −0.303558 0.525778i
\(226\) 0 0
\(227\) 214.461i 0.944762i 0.881394 + 0.472381i \(0.156605\pi\)
−0.881394 + 0.472381i \(0.843395\pi\)
\(228\) 0 0
\(229\) 118.681 0.518256 0.259128 0.965843i \(-0.416565\pi\)
0.259128 + 0.965843i \(0.416565\pi\)
\(230\) 0 0
\(231\) −80.2671 + 46.3423i −0.347477 + 0.200616i
\(232\) 0 0
\(233\) 91.9447 + 159.253i 0.394612 + 0.683489i 0.993052 0.117679i \(-0.0375455\pi\)
−0.598439 + 0.801168i \(0.704212\pi\)
\(234\) 0 0
\(235\) −637.623 −2.71329
\(236\) 0 0
\(237\) 88.4483 153.197i 0.373200 0.646401i
\(238\) 0 0
\(239\) −21.2428 −0.0888822 −0.0444411 0.999012i \(-0.514151\pi\)
−0.0444411 + 0.999012i \(0.514151\pi\)
\(240\) 0 0
\(241\) 68.4562 + 39.5232i 0.284051 + 0.163997i 0.635256 0.772302i \(-0.280895\pi\)
−0.351205 + 0.936299i \(0.614228\pi\)
\(242\) 0 0
\(243\) −92.2937 53.2858i −0.379809 0.219283i
\(244\) 0 0
\(245\) −41.6404 72.1234i −0.169961 0.294381i
\(246\) 0 0
\(247\) 171.902 399.341i 0.695961 1.61677i
\(248\) 0 0
\(249\) 12.1483 7.01383i 0.0487884 0.0281680i
\(250\) 0 0
\(251\) 180.962 313.436i 0.720965 1.24875i −0.239649 0.970860i \(-0.577032\pi\)
0.960614 0.277888i \(-0.0896344\pi\)
\(252\) 0 0
\(253\) 54.8030 94.9216i 0.216613 0.375184i
\(254\) 0 0
\(255\) 153.812i 0.603184i
\(256\) 0 0
\(257\) 111.378 + 64.3044i 0.433379 + 0.250212i 0.700785 0.713372i \(-0.252833\pi\)
−0.267406 + 0.963584i \(0.586166\pi\)
\(258\) 0 0
\(259\) 90.0116i 0.347535i
\(260\) 0 0
\(261\) −20.5163 + 11.8451i −0.0786065 + 0.0453835i
\(262\) 0 0
\(263\) −115.623 200.266i −0.439633 0.761466i 0.558028 0.829822i \(-0.311558\pi\)
−0.997661 + 0.0683558i \(0.978225\pi\)
\(264\) 0 0
\(265\) 453.157i 1.71003i
\(266\) 0 0
\(267\) −304.148 −1.13913
\(268\) 0 0
\(269\) −211.625 + 122.182i −0.786710 + 0.454207i −0.838803 0.544435i \(-0.816744\pi\)
0.0520931 + 0.998642i \(0.483411\pi\)
\(270\) 0 0
\(271\) 221.354 + 383.397i 0.816805 + 1.41475i 0.908025 + 0.418917i \(0.137590\pi\)
−0.0912198 + 0.995831i \(0.529077\pi\)
\(272\) 0 0
\(273\) 459.176 1.68197
\(274\) 0 0
\(275\) 156.588 271.218i 0.569410 0.986247i
\(276\) 0 0
\(277\) 89.7022 0.323835 0.161917 0.986804i \(-0.448232\pi\)
0.161917 + 0.986804i \(0.448232\pi\)
\(278\) 0 0
\(279\) 7.36415 + 4.25170i 0.0263948 + 0.0152391i
\(280\) 0 0
\(281\) 368.458 + 212.729i 1.31124 + 0.757044i 0.982301 0.187308i \(-0.0599762\pi\)
0.328937 + 0.944352i \(0.393310\pi\)
\(282\) 0 0
\(283\) 74.4153 + 128.891i 0.262952 + 0.455446i 0.967025 0.254682i \(-0.0819707\pi\)
−0.704073 + 0.710127i \(0.748637\pi\)
\(284\) 0 0
\(285\) −191.273 + 444.342i −0.671135 + 1.55909i
\(286\) 0 0
\(287\) −58.0452 + 33.5124i −0.202248 + 0.116768i
\(288\) 0 0
\(289\) 126.253 218.676i 0.436861 0.756665i
\(290\) 0 0
\(291\) −75.1985 + 130.248i −0.258414 + 0.447586i
\(292\) 0 0
\(293\) 42.5442i 0.145202i −0.997361 0.0726011i \(-0.976870\pi\)
0.997361 0.0726011i \(-0.0231300\pi\)
\(294\) 0 0
\(295\) 72.4671 + 41.8389i 0.245651 + 0.141827i
\(296\) 0 0
\(297\) 134.461i 0.452732i
\(298\) 0 0
\(299\) −470.260 + 271.505i −1.57277 + 0.908042i
\(300\) 0 0
\(301\) −87.3568 151.306i −0.290222 0.502679i
\(302\) 0 0
\(303\) 91.2004i 0.300991i
\(304\) 0 0
\(305\) 68.9579 0.226091
\(306\) 0 0
\(307\) 249.511 144.055i 0.812740 0.469235i −0.0351668 0.999381i \(-0.511196\pi\)
0.847906 + 0.530146i \(0.177863\pi\)
\(308\) 0 0
\(309\) 66.5473 + 115.263i 0.215363 + 0.373020i
\(310\) 0 0
\(311\) 269.870 0.867750 0.433875 0.900973i \(-0.357146\pi\)
0.433875 + 0.900973i \(0.357146\pi\)
\(312\) 0 0
\(313\) −88.8271 + 153.853i −0.283793 + 0.491543i −0.972316 0.233671i \(-0.924926\pi\)
0.688523 + 0.725214i \(0.258259\pi\)
\(314\) 0 0
\(315\) 147.354 0.467792
\(316\) 0 0
\(317\) −505.362 291.771i −1.59420 0.920413i −0.992575 0.121635i \(-0.961186\pi\)
−0.601627 0.798777i \(-0.705480\pi\)
\(318\) 0 0
\(319\) −47.0362 27.1563i −0.147449 0.0851296i
\(320\) 0 0
\(321\) −243.403 421.586i −0.758264 1.31335i
\(322\) 0 0
\(323\) 92.0161 68.6112i 0.284880 0.212419i
\(324\) 0 0
\(325\) −1343.66 + 775.765i −4.13435 + 2.38697i
\(326\) 0 0
\(327\) −27.0842 + 46.9112i −0.0828262 + 0.143459i
\(328\) 0 0
\(329\) 251.265 435.203i 0.763722 1.32281i
\(330\) 0 0
\(331\) 32.6891i 0.0987586i 0.998780 + 0.0493793i \(0.0157243\pi\)
−0.998780 + 0.0493793i \(0.984276\pi\)
\(332\) 0 0
\(333\) 20.6846 + 11.9423i 0.0621160 + 0.0358627i
\(334\) 0 0
\(335\) 860.118i 2.56752i
\(336\) 0 0
\(337\) −255.755 + 147.660i −0.758918 + 0.438161i −0.828907 0.559386i \(-0.811037\pi\)
0.0699892 + 0.997548i \(0.477704\pi\)
\(338\) 0 0
\(339\) 183.272 + 317.436i 0.540624 + 0.936389i
\(340\) 0 0
\(341\) 19.4951i 0.0571704i
\(342\) 0 0
\(343\) −306.393 −0.893274
\(344\) 0 0
\(345\) 523.251 302.099i 1.51667 0.875650i
\(346\) 0 0
\(347\) 85.7263 + 148.482i 0.247050 + 0.427903i 0.962706 0.270550i \(-0.0872055\pi\)
−0.715656 + 0.698453i \(0.753872\pi\)
\(348\) 0 0
\(349\) 127.644 0.365743 0.182872 0.983137i \(-0.441461\pi\)
0.182872 + 0.983137i \(0.441461\pi\)
\(350\) 0 0
\(351\) −333.073 + 576.900i −0.948926 + 1.64359i
\(352\) 0 0
\(353\) −184.494 −0.522647 −0.261323 0.965251i \(-0.584159\pi\)
−0.261323 + 0.965251i \(0.584159\pi\)
\(354\) 0 0
\(355\) 654.923 + 378.120i 1.84485 + 1.06513i
\(356\) 0 0
\(357\) 104.983 + 60.6119i 0.294070 + 0.169781i
\(358\) 0 0
\(359\) −131.720 228.145i −0.366907 0.635502i 0.622173 0.782880i \(-0.286250\pi\)
−0.989080 + 0.147378i \(0.952917\pi\)
\(360\) 0 0
\(361\) −351.143 + 83.7815i −0.972696 + 0.232082i
\(362\) 0 0
\(363\) −228.125 + 131.708i −0.628444 + 0.362832i
\(364\) 0 0
\(365\) 415.428 719.543i 1.13816 1.97135i
\(366\) 0 0
\(367\) 60.5658 104.903i 0.165029 0.285839i −0.771636 0.636064i \(-0.780561\pi\)
0.936666 + 0.350225i \(0.113895\pi\)
\(368\) 0 0
\(369\) 17.7850i 0.0481979i
\(370\) 0 0
\(371\) 309.298 + 178.573i 0.833687 + 0.481329i
\(372\) 0 0
\(373\) 335.187i 0.898625i −0.893375 0.449312i \(-0.851669\pi\)
0.893375 0.449312i \(-0.148331\pi\)
\(374\) 0 0
\(375\) 943.827 544.919i 2.51687 1.45312i
\(376\) 0 0
\(377\) 134.538 + 233.026i 0.356864 + 0.618106i
\(378\) 0 0
\(379\) 315.980i 0.833721i −0.908970 0.416861i \(-0.863130\pi\)
0.908970 0.416861i \(-0.136870\pi\)
\(380\) 0 0
\(381\) −461.502 −1.21129
\(382\) 0 0
\(383\) −408.291 + 235.727i −1.06603 + 0.615475i −0.927095 0.374825i \(-0.877703\pi\)
−0.138939 + 0.990301i \(0.544369\pi\)
\(384\) 0 0
\(385\) 168.914 + 292.568i 0.438738 + 0.759917i
\(386\) 0 0
\(387\) 46.3602 0.119794
\(388\) 0 0
\(389\) 235.508 407.912i 0.605420 1.04862i −0.386566 0.922262i \(-0.626339\pi\)
0.991985 0.126355i \(-0.0403280\pi\)
\(390\) 0 0
\(391\) −143.356 −0.366639
\(392\) 0 0
\(393\) 51.4755 + 29.7194i 0.130981 + 0.0756219i
\(394\) 0 0
\(395\) −558.392 322.388i −1.41365 0.816172i
\(396\) 0 0
\(397\) −110.236 190.934i −0.277672 0.480942i 0.693134 0.720809i \(-0.256230\pi\)
−0.970806 + 0.239867i \(0.922896\pi\)
\(398\) 0 0
\(399\) −227.907 305.651i −0.571195 0.766042i
\(400\) 0 0
\(401\) 79.1787 45.7138i 0.197453 0.114000i −0.398014 0.917379i \(-0.630300\pi\)
0.595467 + 0.803380i \(0.296967\pi\)
\(402\) 0 0
\(403\) 48.2912 83.6428i 0.119829 0.207550i
\(404\) 0 0
\(405\) 283.269 490.637i 0.699431 1.21145i
\(406\) 0 0
\(407\) 54.7583i 0.134541i
\(408\) 0 0
\(409\) 291.494 + 168.294i 0.712699 + 0.411477i 0.812060 0.583575i \(-0.198346\pi\)
−0.0993606 + 0.995051i \(0.531680\pi\)
\(410\) 0 0
\(411\) 202.859i 0.493574i
\(412\) 0 0
\(413\) −57.1134 + 32.9745i −0.138289 + 0.0798413i
\(414\) 0 0
\(415\) −25.5649 44.2798i −0.0616023 0.106698i
\(416\) 0 0
\(417\) 448.051i 1.07446i
\(418\) 0 0
\(419\) −213.808 −0.510281 −0.255141 0.966904i \(-0.582122\pi\)
−0.255141 + 0.966904i \(0.582122\pi\)
\(420\) 0 0
\(421\) 524.575 302.863i 1.24602 0.719391i 0.275708 0.961242i \(-0.411088\pi\)
0.970313 + 0.241851i \(0.0777545\pi\)
\(422\) 0 0
\(423\) 66.6730 + 115.481i 0.157619 + 0.273005i
\(424\) 0 0
\(425\) −409.608 −0.963783
\(426\) 0 0
\(427\) −27.1738 + 47.0665i −0.0636390 + 0.110226i
\(428\) 0 0
\(429\) −279.339 −0.651140
\(430\) 0 0
\(431\) 138.813 + 80.1437i 0.322072 + 0.185948i 0.652316 0.757947i \(-0.273798\pi\)
−0.330244 + 0.943896i \(0.607131\pi\)
\(432\) 0 0
\(433\) 377.175 + 217.762i 0.871073 + 0.502914i 0.867705 0.497080i \(-0.165594\pi\)
0.00336832 + 0.999994i \(0.498928\pi\)
\(434\) 0 0
\(435\) −149.698 259.285i −0.344134 0.596057i
\(436\) 0 0
\(437\) 414.135 + 178.270i 0.947677 + 0.407941i
\(438\) 0 0
\(439\) 320.878 185.259i 0.730930 0.422003i −0.0878324 0.996135i \(-0.527994\pi\)
0.818762 + 0.574133i \(0.194661\pi\)
\(440\) 0 0
\(441\) −8.70826 + 15.0831i −0.0197466 + 0.0342021i
\(442\) 0 0
\(443\) −144.384 + 250.081i −0.325924 + 0.564517i −0.981699 0.190439i \(-0.939009\pi\)
0.655775 + 0.754957i \(0.272342\pi\)
\(444\) 0 0
\(445\) 1108.60i 2.49123i
\(446\) 0 0
\(447\) 447.183 + 258.181i 1.00041 + 0.577586i
\(448\) 0 0
\(449\) 76.8398i 0.171135i −0.996332 0.0855677i \(-0.972730\pi\)
0.996332 0.0855677i \(-0.0272704\pi\)
\(450\) 0 0
\(451\) 35.3116 20.3872i 0.0782963 0.0452044i
\(452\) 0 0
\(453\) −214.278 371.141i −0.473021 0.819296i
\(454\) 0 0
\(455\) 1673.67i 3.67839i
\(456\) 0 0
\(457\) 97.5752 0.213513 0.106756 0.994285i \(-0.465954\pi\)
0.106756 + 0.994285i \(0.465954\pi\)
\(458\) 0 0
\(459\) −152.303 + 87.9321i −0.331815 + 0.191573i
\(460\) 0 0
\(461\) −156.003 270.204i −0.338401 0.586127i 0.645732 0.763565i \(-0.276553\pi\)
−0.984132 + 0.177438i \(0.943219\pi\)
\(462\) 0 0
\(463\) 114.000 0.246220 0.123110 0.992393i \(-0.460713\pi\)
0.123110 + 0.992393i \(0.460713\pi\)
\(464\) 0 0
\(465\) −53.7329 + 93.0681i −0.115555 + 0.200147i
\(466\) 0 0
\(467\) −651.304 −1.39466 −0.697328 0.716752i \(-0.745628\pi\)
−0.697328 + 0.716752i \(0.745628\pi\)
\(468\) 0 0
\(469\) −587.065 338.942i −1.25174 0.722691i
\(470\) 0 0
\(471\) −300.613 173.559i −0.638245 0.368491i
\(472\) 0 0
\(473\) 53.1433 + 92.0469i 0.112354 + 0.194602i
\(474\) 0 0
\(475\) 1183.30 + 509.369i 2.49116 + 1.07236i
\(476\) 0 0
\(477\) −82.0721 + 47.3843i −0.172059 + 0.0993382i
\(478\) 0 0
\(479\) −216.179 + 374.433i −0.451313 + 0.781697i −0.998468 0.0553341i \(-0.982378\pi\)
0.547155 + 0.837031i \(0.315711\pi\)
\(480\) 0 0
\(481\) 135.642 234.938i 0.281999 0.488437i
\(482\) 0 0
\(483\) 476.186i 0.985893i
\(484\) 0 0
\(485\) 474.743 + 274.093i 0.978853 + 0.565141i
\(486\) 0 0
\(487\) 100.949i 0.207287i 0.994615 + 0.103643i \(0.0330501\pi\)
−0.994615 + 0.103643i \(0.966950\pi\)
\(488\) 0 0
\(489\) −227.057 + 131.091i −0.464329 + 0.268080i
\(490\) 0 0
\(491\) −112.182 194.304i −0.228476 0.395732i 0.728881 0.684641i \(-0.240041\pi\)
−0.957357 + 0.288909i \(0.906708\pi\)
\(492\) 0 0
\(493\) 71.0365i 0.144090i
\(494\) 0 0
\(495\) −89.6427 −0.181096
\(496\) 0 0
\(497\) −516.164 + 298.007i −1.03856 + 0.599612i
\(498\) 0 0
\(499\) −332.278 575.523i −0.665888 1.15335i −0.979044 0.203651i \(-0.934719\pi\)
0.313155 0.949702i \(-0.398614\pi\)
\(500\) 0 0
\(501\) −559.874 −1.11751
\(502\) 0 0
\(503\) −180.202 + 312.119i −0.358255 + 0.620515i −0.987669 0.156554i \(-0.949961\pi\)
0.629415 + 0.777070i \(0.283295\pi\)
\(504\) 0 0
\(505\) 332.419 0.658255
\(506\) 0 0
\(507\) 811.668 + 468.617i 1.60092 + 0.924294i
\(508\) 0 0
\(509\) −670.407 387.060i −1.31711 0.760432i −0.333845 0.942628i \(-0.608346\pi\)
−0.983262 + 0.182196i \(0.941679\pi\)
\(510\) 0 0
\(511\) 327.411 + 567.092i 0.640726 + 1.10977i
\(512\) 0 0
\(513\) 549.330 64.6270i 1.07082 0.125979i
\(514\) 0 0
\(515\) 420.126 242.560i 0.815780 0.470991i
\(516\) 0 0
\(517\) −152.856 + 264.755i −0.295660 + 0.512098i
\(518\) 0 0
\(519\) 225.568 390.695i 0.434620 0.752784i
\(520\) 0 0
\(521\) 196.045i 0.376285i 0.982142 + 0.188143i \(0.0602468\pi\)
−0.982142 + 0.188143i \(0.939753\pi\)
\(522\) 0 0
\(523\) −285.968 165.104i −0.546784 0.315686i 0.201040 0.979583i \(-0.435568\pi\)
−0.747824 + 0.663897i \(0.768901\pi\)
\(524\) 0 0
\(525\) 1360.60i 2.59162i
\(526\) 0 0
\(527\) 22.0819 12.7490i 0.0419011 0.0241916i
\(528\) 0 0
\(529\) −17.0622 29.5526i −0.0322537 0.0558651i
\(530\) 0 0
\(531\) 17.4995i 0.0329558i
\(532\) 0 0
\(533\) −202.004 −0.378994
\(534\) 0 0
\(535\) −1536.65 + 887.186i −2.87225 + 1.65829i
\(536\) 0 0
\(537\) 254.962 + 441.607i 0.474790 + 0.822360i
\(538\) 0 0
\(539\) −39.9296 −0.0740808
\(540\) 0 0
\(541\) −328.577 + 569.111i −0.607350 + 1.05196i 0.384325 + 0.923198i \(0.374434\pi\)
−0.991675 + 0.128764i \(0.958899\pi\)
\(542\) 0 0
\(543\) −804.820 −1.48217
\(544\) 0 0
\(545\) 170.988 + 98.7199i 0.313739 + 0.181137i
\(546\) 0 0
\(547\) 51.3126 + 29.6253i 0.0938072 + 0.0541596i 0.546170 0.837675i \(-0.316085\pi\)
−0.452363 + 0.891834i \(0.649419\pi\)
\(548\) 0 0
\(549\) −7.21057 12.4891i −0.0131340 0.0227488i
\(550\) 0 0
\(551\) 88.3377 205.215i 0.160322 0.372440i
\(552\) 0 0
\(553\) 440.085 254.083i 0.795814 0.459463i
\(554\) 0 0
\(555\) −150.926 + 261.412i −0.271940 + 0.471013i
\(556\) 0 0
\(557\) 120.377 208.499i 0.216117 0.374325i −0.737501 0.675346i \(-0.763994\pi\)
0.953617 + 0.301021i \(0.0973275\pi\)
\(558\) 0 0
\(559\) 526.564i 0.941974i
\(560\) 0 0
\(561\) −63.8660 36.8731i −0.113843 0.0657274i
\(562\) 0 0
\(563\) 919.141i 1.63258i −0.577645 0.816288i \(-0.696028\pi\)
0.577645 0.816288i \(-0.303972\pi\)
\(564\) 0 0
\(565\) 1157.03 668.012i 2.04784 1.18232i
\(566\) 0 0
\(567\) 223.253 + 386.685i 0.393744 + 0.681985i
\(568\) 0 0
\(569\) 726.404i 1.27663i −0.769774 0.638316i \(-0.779631\pi\)
0.769774 0.638316i \(-0.220369\pi\)
\(570\) 0 0
\(571\) −924.876 −1.61975 −0.809874 0.586604i \(-0.800464\pi\)
−0.809874 + 0.586604i \(0.800464\pi\)
\(572\) 0 0
\(573\) 460.477 265.857i 0.803625 0.463973i
\(574\) 0 0
\(575\) −804.503 1393.44i −1.39913 2.42337i
\(576\) 0 0
\(577\) 161.154 0.279296 0.139648 0.990201i \(-0.455403\pi\)
0.139648 + 0.990201i \(0.455403\pi\)
\(578\) 0 0
\(579\) 50.5024 87.4728i 0.0872236 0.151076i
\(580\) 0 0
\(581\) 40.2969 0.0693579
\(582\) 0 0
\(583\) −188.161 108.635i −0.322745 0.186337i
\(584\) 0 0
\(585\) 384.608 + 222.053i 0.657449 + 0.379578i
\(586\) 0 0
\(587\) 294.997 + 510.950i 0.502550 + 0.870442i 0.999996 + 0.00294722i \(0.000938130\pi\)
−0.497445 + 0.867495i \(0.665729\pi\)
\(588\) 0 0
\(589\) −79.6456 + 9.37006i −0.135222 + 0.0159084i
\(590\) 0 0
\(591\) 230.033 132.810i 0.389226 0.224720i
\(592\) 0 0
\(593\) −9.99383 + 17.3098i −0.0168530 + 0.0291903i −0.874329 0.485334i \(-0.838698\pi\)
0.857476 + 0.514524i \(0.172031\pi\)
\(594\) 0 0
\(595\) 220.926 382.655i 0.371304 0.643118i
\(596\) 0 0
\(597\) 260.153i 0.435768i
\(598\) 0 0
\(599\) 931.550 + 537.831i 1.55518 + 0.897881i 0.997707 + 0.0676825i \(0.0215605\pi\)
0.557468 + 0.830198i \(0.311773\pi\)
\(600\) 0 0
\(601\) 14.5736i 0.0242489i 0.999926 + 0.0121244i \(0.00385942\pi\)
−0.999926 + 0.0121244i \(0.996141\pi\)
\(602\) 0 0
\(603\) 155.777 89.9382i 0.258337 0.149151i
\(604\) 0 0
\(605\) 480.067 + 831.501i 0.793499 + 1.37438i
\(606\) 0 0
\(607\) 437.145i 0.720173i 0.932919 + 0.360086i \(0.117253\pi\)
−0.932919 + 0.360086i \(0.882747\pi\)
\(608\) 0 0
\(609\) 235.963 0.387460
\(610\) 0 0
\(611\) 1311.65 757.279i 2.14672 1.23941i
\(612\) 0 0
\(613\) 534.944 + 926.551i 0.872666 + 1.51150i 0.859228 + 0.511592i \(0.170944\pi\)
0.0134375 + 0.999910i \(0.495723\pi\)
\(614\) 0 0
\(615\) 224.767 0.365475
\(616\) 0 0
\(617\) −99.8317 + 172.914i −0.161802 + 0.280249i −0.935515 0.353287i \(-0.885064\pi\)
0.773713 + 0.633536i \(0.218397\pi\)
\(618\) 0 0
\(619\) −432.305 −0.698393 −0.349196 0.937050i \(-0.613545\pi\)
−0.349196 + 0.937050i \(0.613545\pi\)
\(620\) 0 0
\(621\) −598.270 345.412i −0.963399 0.556218i
\(622\) 0 0
\(623\) −756.662 436.859i −1.21455 0.701218i
\(624\) 0 0
\(625\) −1138.64 1972.18i −1.82182 3.15549i
\(626\) 0 0
\(627\) 138.647 + 185.942i 0.221127 + 0.296558i
\(628\) 0 0
\(629\) 62.0242 35.8097i 0.0986077 0.0569312i
\(630\) 0 0
\(631\) 102.220 177.051i 0.161997 0.280587i −0.773588 0.633689i \(-0.781540\pi\)
0.935585 + 0.353102i \(0.114873\pi\)
\(632\) 0 0
\(633\) 427.011 739.605i 0.674583 1.16841i
\(634\) 0 0
\(635\) 1682.14i 2.64904i
\(636\) 0 0
\(637\) 171.316 + 98.9093i 0.268942 + 0.155274i
\(638\) 0 0
\(639\) 158.152i 0.247500i
\(640\) 0 0
\(641\) −34.0248 + 19.6442i −0.0530808 + 0.0306462i −0.526306 0.850296i \(-0.676423\pi\)
0.473225 + 0.880942i \(0.343090\pi\)
\(642\) 0 0
\(643\) −383.822 664.799i −0.596923 1.03390i −0.993272 0.115801i \(-0.963056\pi\)
0.396349 0.918100i \(-0.370277\pi\)
\(644\) 0 0
\(645\) 585.900i 0.908372i
\(646\) 0 0
\(647\) 963.932 1.48985 0.744924 0.667149i \(-0.232486\pi\)
0.744924 + 0.667149i \(0.232486\pi\)
\(648\) 0 0
\(649\) 34.7448 20.0599i 0.0535359 0.0309090i
\(650\) 0 0
\(651\) −42.3485 73.3497i −0.0650514 0.112672i
\(652\) 0 0
\(653\) −824.035 −1.26192 −0.630961 0.775815i \(-0.717339\pi\)
−0.630961 + 0.775815i \(0.717339\pi\)
\(654\) 0 0
\(655\) 108.325 187.625i 0.165382 0.286450i
\(656\) 0 0
\(657\) −173.757 −0.264470
\(658\) 0 0
\(659\) −144.310 83.3177i −0.218984 0.126430i 0.386496 0.922291i \(-0.373685\pi\)
−0.605480 + 0.795861i \(0.707019\pi\)
\(660\) 0 0
\(661\) 273.938 + 158.158i 0.414430 + 0.239271i 0.692692 0.721234i \(-0.256425\pi\)
−0.278261 + 0.960505i \(0.589758\pi\)
\(662\) 0 0
\(663\) 182.676 + 316.404i 0.275530 + 0.477231i
\(664\) 0 0
\(665\) −1114.08 + 830.704i −1.67530 + 1.24918i
\(666\) 0 0
\(667\) −241.658 + 139.522i −0.362306 + 0.209178i
\(668\) 0 0
\(669\) 391.596 678.265i 0.585346 1.01385i
\(670\) 0 0
\(671\) 16.5311 28.6328i 0.0246366 0.0426718i
\(672\) 0 0
\(673\) 582.475i 0.865491i −0.901516 0.432745i \(-0.857545\pi\)
0.901516 0.432745i \(-0.142455\pi\)
\(674\) 0 0
\(675\) −1709.43 986.938i −2.53248 1.46213i
\(676\) 0 0
\(677\) 956.276i 1.41252i −0.707953 0.706260i \(-0.750381\pi\)
0.707953 0.706260i \(-0.249619\pi\)
\(678\) 0 0
\(679\) −374.159 + 216.021i −0.551044 + 0.318146i
\(680\) 0 0
\(681\) 283.408 + 490.878i 0.416165 + 0.720819i
\(682\) 0 0
\(683\) 240.373i 0.351937i −0.984396 0.175969i \(-0.943694\pi\)
0.984396 0.175969i \(-0.0563057\pi\)
\(684\) 0 0
\(685\) −739.407 −1.07943
\(686\) 0 0
\(687\) 271.647 156.835i 0.395410 0.228290i
\(688\) 0 0
\(689\) 538.196 + 932.183i 0.781126 + 1.35295i
\(690\) 0 0
\(691\) 868.890 1.25744 0.628719 0.777632i \(-0.283579\pi\)
0.628719 + 0.777632i \(0.283579\pi\)
\(692\) 0 0
\(693\) 35.3250 61.1847i 0.0509740 0.0882896i
\(694\) 0 0
\(695\) 1633.11 2.34980
\(696\) 0 0
\(697\) −46.1847 26.6648i −0.0662622 0.0382565i
\(698\) 0 0
\(699\) 420.902 + 243.008i 0.602149 + 0.347651i
\(700\) 0 0
\(701\) 467.106 + 809.052i 0.666343 + 1.15414i 0.978919 + 0.204247i \(0.0654746\pi\)
−0.312576 + 0.949893i \(0.601192\pi\)
\(702\) 0 0
\(703\) −223.711 + 26.3189i −0.318223 + 0.0374379i
\(704\) 0 0
\(705\) −1459.45 + 842.613i −2.07014 + 1.19520i
\(706\) 0 0
\(707\) −130.994 + 226.889i −0.185282 + 0.320918i
\(708\) 0 0
\(709\) −514.968 + 891.950i −0.726330 + 1.25804i 0.232095 + 0.972693i \(0.425442\pi\)
−0.958424 + 0.285347i \(0.907891\pi\)
\(710\) 0 0
\(711\) 134.842i 0.189651i
\(712\) 0 0
\(713\) 86.7413 + 50.0801i 0.121657 + 0.0702386i
\(714\) 0 0
\(715\) 1018.17i 1.42402i
\(716\) 0 0
\(717\) −48.6225 + 28.0722i −0.0678138 + 0.0391523i
\(718\) 0 0
\(719\) 352.663 + 610.831i 0.490491 + 0.849556i 0.999940 0.0109450i \(-0.00348397\pi\)
−0.509449 + 0.860501i \(0.670151\pi\)
\(720\) 0 0
\(721\) 382.337i 0.530288i
\(722\) 0 0
\(723\) 208.918 0.288960
\(724\) 0 0
\(725\) −690.486 + 398.652i −0.952394 + 0.549865i
\(726\) 0 0
\(727\) 426.231 + 738.253i 0.586287 + 1.01548i 0.994714 + 0.102688i \(0.0327442\pi\)
−0.408427 + 0.912791i \(0.633922\pi\)
\(728\) 0 0
\(729\) −810.951 −1.11242
\(730\) 0 0
\(731\) 69.5070 120.390i 0.0950849 0.164692i
\(732\) 0 0
\(733\) 1327.83 1.81150 0.905751 0.423809i \(-0.139307\pi\)
0.905751 + 0.423809i \(0.139307\pi\)
\(734\) 0 0
\(735\) −190.621 110.055i −0.259348 0.149735i
\(736\) 0 0
\(737\) 357.139 + 206.195i 0.484585 + 0.279776i
\(738\) 0 0
\(739\) −197.639 342.320i −0.267441 0.463221i 0.700760 0.713398i \(-0.252845\pi\)
−0.968200 + 0.250177i \(0.919511\pi\)
\(740\) 0 0
\(741\) −134.261 1141.22i −0.181188 1.54010i
\(742\) 0 0
\(743\) −916.709 + 529.262i −1.23379 + 0.712332i −0.967819 0.251648i \(-0.919027\pi\)
−0.265976 + 0.963980i \(0.585694\pi\)
\(744\) 0 0
\(745\) 941.052 1629.95i 1.26316 2.18785i
\(746\) 0 0
\(747\) −5.34639 + 9.26022i −0.00715715 + 0.0123965i
\(748\) 0 0
\(749\) 1398.43i 1.86707i
\(750\) 0 0
\(751\) 207.032 + 119.530i 0.275674 + 0.159161i 0.631464 0.775406i \(-0.282455\pi\)
−0.355789 + 0.934566i \(0.615788\pi\)
\(752\) 0 0
\(753\) 956.559i 1.27033i
\(754\) 0 0
\(755\) −1352.78 + 781.030i −1.79177 + 1.03448i
\(756\) 0 0
\(757\) 92.7004 + 160.562i 0.122458 + 0.212103i 0.920736 0.390185i \(-0.127589\pi\)
−0.798279 + 0.602288i \(0.794256\pi\)
\(758\) 0 0
\(759\) 289.687i 0.381669i
\(760\) 0 0
\(761\) −712.211 −0.935888 −0.467944 0.883758i \(-0.655005\pi\)
−0.467944 + 0.883758i \(0.655005\pi\)
\(762\) 0 0
\(763\) −134.760 + 77.8040i −0.176619 + 0.101971i
\(764\) 0 0
\(765\) 58.6226 + 101.537i 0.0766309 + 0.132729i
\(766\) 0 0
\(767\) −198.761 −0.259141
\(768\) 0 0
\(769\) −395.236 + 684.568i −0.513961 + 0.890206i 0.485908 + 0.874010i \(0.338489\pi\)
−0.999869 + 0.0161960i \(0.994844\pi\)
\(770\) 0 0
\(771\) 339.911 0.440870
\(772\) 0 0
\(773\) 334.502 + 193.125i 0.432732 + 0.249838i 0.700510 0.713643i \(-0.252956\pi\)
−0.267778 + 0.963481i \(0.586289\pi\)
\(774\) 0 0
\(775\) 247.844 + 143.093i 0.319799 + 0.184636i
\(776\) 0 0
\(777\) −118.949 206.027i −0.153088 0.265156i
\(778\) 0 0
\(779\) 100.262 + 134.464i 0.128706 + 0.172611i
\(780\) 0 0
\(781\) 314.007 181.292i 0.402057 0.232128i
\(782\) 0 0
\(783\) −171.161 + 296.459i −0.218596 + 0.378619i
\(784\) 0 0
\(785\) −632.611 + 1095.71i −0.805874 + 1.39581i
\(786\) 0 0
\(787\) 606.206i 0.770274i −0.922859 0.385137i \(-0.874154\pi\)
0.922859 0.385137i \(-0.125846\pi\)
\(788\) 0 0
\(789\) −529.298 305.590i −0.670847 0.387314i
\(790\) 0 0
\(791\) 1052.96i 1.33118i
\(792\) 0 0
\(793\) −141.852 + 81.8984i −0.178880 + 0.103277i
\(794\) 0 0
\(795\) −598.843 1037.23i −0.753262 1.30469i
\(796\) 0 0
\(797\) 332.769i 0.417526i −0.977966 0.208763i \(-0.933056\pi\)
0.977966 0.208763i \(-0.0669438\pi\)
\(798\) 0 0
\(799\) 399.847 0.500434
\(800\) 0 0
\(801\) 200.780 115.920i 0.250662 0.144720i
\(802\) 0 0
\(803\) −199.180 344.989i −0.248044 0.429626i
\(804\) 0 0
\(805\) 1735.67 2.15611
\(806\) 0 0
\(807\) −322.924 + 559.321i −0.400154 + 0.693087i
\(808\) 0 0
\(809\) −603.251 −0.745675 −0.372838 0.927897i \(-0.621615\pi\)
−0.372838 + 0.927897i \(0.621615\pi\)
\(810\) 0 0
\(811\) −905.988 523.073i −1.11713 0.644972i −0.176460 0.984308i \(-0.556465\pi\)
−0.940665 + 0.339335i \(0.889798\pi\)
\(812\) 0 0
\(813\) 1013.31 + 585.035i 1.24638 + 0.719600i
\(814\) 0 0
\(815\) 477.819 + 827.606i 0.586281 + 1.01547i
\(816\) 0 0
\(817\) −350.507 + 261.354i −0.429017 + 0.319894i
\(818\) 0 0
\(819\) −303.121 + 175.007i −0.370111 + 0.213683i
\(820\) 0 0
\(821\) −166.613 + 288.583i −0.202940 + 0.351502i −0.949474 0.313845i \(-0.898383\pi\)
0.746535 + 0.665346i \(0.231716\pi\)
\(822\) 0 0
\(823\) 208.418 360.990i 0.253241 0.438627i −0.711175 0.703015i \(-0.751837\pi\)
0.964416 + 0.264388i \(0.0851700\pi\)
\(824\) 0 0
\(825\) 827.717i 1.00329i
\(826\) 0 0
\(827\) 734.915 + 424.303i 0.888651 + 0.513063i 0.873501 0.486822i \(-0.161844\pi\)
0.0151501 + 0.999885i \(0.495177\pi\)
\(828\) 0 0
\(829\) 958.797i 1.15657i −0.815835 0.578285i \(-0.803722\pi\)
0.815835 0.578285i \(-0.196278\pi\)
\(830\) 0 0
\(831\) 205.318 118.541i 0.247074 0.142648i
\(832\) 0 0
\(833\) 26.1123 + 45.2278i 0.0313473 + 0.0542951i
\(834\) 0 0
\(835\) 2040.70i 2.44396i
\(836\) 0 0
\(837\) 122.873 0.146802
\(838\) 0 0
\(839\) 931.117 537.581i 1.10979 0.640740i 0.171018 0.985268i \(-0.445294\pi\)
0.938776 + 0.344528i \(0.111961\pi\)
\(840\) 0 0
\(841\) −351.363 608.579i −0.417792 0.723638i
\(842\) 0 0
\(843\) 1124.48 1.33390
\(844\) 0 0
\(845\) 1708.08 2958.47i 2.02139 3.50115i
\(846\) 0 0
\(847\) −756.710 −0.893400
\(848\) 0 0
\(849\) 340.657 + 196.678i 0.401245 + 0.231659i
\(850\) 0 0
\(851\) 243.641 + 140.666i 0.286300 + 0.165295i
\(852\) 0 0
\(853\) −83.4052 144.462i −0.0977787 0.169358i 0.812986 0.582283i \(-0.197840\pi\)
−0.910765 + 0.412925i \(0.864507\pi\)
\(854\) 0 0
\(855\) −43.0856 366.228i −0.0503925 0.428337i
\(856\) 0 0
\(857\) 581.227 335.571i 0.678211 0.391565i −0.120970 0.992656i \(-0.538600\pi\)
0.799181 + 0.601091i \(0.205267\pi\)
\(858\) 0 0
\(859\) −396.070 + 686.014i −0.461083 + 0.798619i −0.999015 0.0443690i \(-0.985872\pi\)
0.537932 + 0.842988i \(0.319206\pi\)
\(860\) 0 0
\(861\) −88.5726 + 153.412i −0.102872 + 0.178179i
\(862\) 0 0
\(863\) 570.060i 0.660556i −0.943884 0.330278i \(-0.892857\pi\)
0.943884 0.330278i \(-0.107143\pi\)
\(864\) 0 0
\(865\) −1424.05 822.178i −1.64631 0.950495i
\(866\) 0 0
\(867\) 667.368i 0.769744i
\(868\) 0 0
\(869\) −267.725 + 154.571i −0.308084 + 0.177872i
\(870\) 0 0
\(871\) −1021.53 1769.34i −1.17282 2.03138i
\(872\) 0 0
\(873\) 114.642i 0.131320i
\(874\) 0 0
\(875\) 3130.75 3.57800
\(876\) 0 0
\(877\) 255.489 147.506i 0.291321 0.168194i −0.347216 0.937785i \(-0.612873\pi\)
0.638537 + 0.769591i \(0.279540\pi\)
\(878\) 0 0
\(879\) −56.2218 97.3790i −0.0639611 0.110784i
\(880\) 0 0
\(881\) −976.298 −1.10817 −0.554085 0.832460i \(-0.686932\pi\)
−0.554085 + 0.832460i \(0.686932\pi\)
\(882\) 0 0
\(883\) 147.883 256.141i 0.167478 0.290081i −0.770054 0.637978i \(-0.779771\pi\)
0.937533 + 0.347897i \(0.113104\pi\)
\(884\) 0 0
\(885\) 221.159 0.249897
\(886\) 0 0
\(887\) 793.272 + 457.996i 0.894331 + 0.516342i 0.875357 0.483477i \(-0.160626\pi\)
0.0189746 + 0.999820i \(0.493960\pi\)
\(888\) 0 0
\(889\) −1148.13 662.872i −1.29148 0.745638i
\(890\) 0 0
\(891\) −135.815 235.239i −0.152430 0.264017i
\(892\) 0 0
\(893\) −1155.10 497.231i −1.29351 0.556809i
\(894\) 0 0
\(895\) 1609.63 929.319i 1.79847 1.03835i
\(896\) 0 0
\(897\) −717.582 + 1242.89i −0.799979 + 1.38560i
\(898\) 0 0
\(899\) 24.8160 42.9826i 0.0276040 0.0478116i
\(900\) 0 0
\(901\) 284.170i 0.315394i
\(902\) 0 0
\(903\) −399.900 230.882i −0.442857 0.255684i
\(904\) 0 0
\(905\) 2933.51i 3.24145i
\(906\) 0 0
\(907\) −407.014 + 234.989i −0.448747 + 0.259084i −0.707301 0.706913i \(-0.750087\pi\)
0.258554 + 0.965997i \(0.416754\pi\)
\(908\) 0 0
\(909\) −34.7593 60.2050i −0.0382391 0.0662321i
\(910\) 0 0
\(911\) 440.064i 0.483056i 0.970394 + 0.241528i \(0.0776486\pi\)
−0.970394 + 0.241528i \(0.922351\pi\)
\(912\) 0 0
\(913\) −24.5145 −0.0268505
\(914\) 0 0
\(915\) 157.837 91.1272i 0.172499 0.0995925i
\(916\) 0 0
\(917\) 85.3741 + 147.872i 0.0931016 + 0.161257i
\(918\) 0 0
\(919\) −609.475 −0.663194 −0.331597 0.943421i \(-0.607587\pi\)
−0.331597 + 0.943421i \(0.607587\pi\)
\(920\) 0 0
\(921\) 380.735 659.453i 0.413394 0.716019i
\(922\) 0 0
\(923\) −1796.31 −1.94616
\(924\) 0 0
\(925\) 696.151 + 401.923i 0.752596 + 0.434512i
\(926\) 0 0
\(927\) −87.8610 50.7265i −0.0947799 0.0547212i
\(928\) 0 0
\(929\) 663.351 + 1148.96i 0.714048 + 1.23677i 0.963326 + 0.268335i \(0.0864736\pi\)
−0.249277 + 0.968432i \(0.580193\pi\)
\(930\) 0 0
\(931\) −19.1916 163.129i −0.0206140 0.175219i
\(932\) 0 0
\(933\) 617.703 356.631i 0.662061 0.382241i
\(934\) 0 0
\(935\) −134.400 + 232.787i −0.143743 + 0.248970i
\(936\) 0 0
\(937\) −911.033 + 1577.96i −0.972288 + 1.68405i −0.283678 + 0.958920i \(0.591555\pi\)
−0.688610 + 0.725132i \(0.741779\pi\)
\(938\) 0 0
\(939\) 469.537i 0.500039i
\(940\) 0 0
\(941\) 961.497 + 555.121i 1.02178 + 0.589926i 0.914620 0.404315i \(-0.132490\pi\)
0.107163 + 0.994242i \(0.465823\pi\)
\(942\) 0 0
\(943\) 209.487i 0.222149i
\(944\) 0 0
\(945\) 1843.99 1064.63i 1.95132 1.12659i
\(946\) 0 0
\(947\) 431.526 + 747.425i 0.455677 + 0.789255i 0.998727 0.0504450i \(-0.0160639\pi\)
−0.543050 + 0.839700i \(0.682731\pi\)
\(948\) 0 0
\(949\) 1973.55i 2.07961i
\(950\) 0 0
\(951\) −1542.29 −1.62176
\(952\) 0 0
\(953\) −1494.57 + 862.890i −1.56828 + 0.905446i −0.571909 + 0.820317i \(0.693797\pi\)
−0.996370 + 0.0851297i \(0.972870\pi\)
\(954\) 0 0
\(955\) −969.029 1678.41i −1.01469 1.75749i
\(956\) 0 0
\(957\) −143.547 −0.149997
\(958\) 0 0
\(959\) 291.374 504.675i 0.303831 0.526251i
\(960\) 0 0
\(961\) 943.185 0.981462
\(962\) 0 0
\(963\) 321.360 + 185.537i 0.333707 + 0.192666i
\(964\) 0 0
\(965\) −318.832 184.078i −0.330396 0.190754i
\(966\) 0 0
\(967\) −767.279 1328.97i −0.793463 1.37432i −0.923811 0.382850i \(-0.874943\pi\)
0.130348 0.991468i \(-0.458391\pi\)
\(968\) 0 0
\(969\) 119.946 278.642i 0.123783 0.287556i
\(970\) 0 0
\(971\) 1048.94 605.609i 1.08027 0.623696i 0.149303 0.988792i \(-0.452297\pi\)
0.930970 + 0.365096i \(0.118964\pi\)
\(972\) 0 0
\(973\) −643.552 + 1114.67i −0.661410 + 1.14560i
\(974\) 0 0
\(975\) −2050.33 + 3551.28i −2.10291 + 3.64234i
\(976\) 0 0
\(977\) 918.806i 0.940436i −0.882550 0.470218i \(-0.844175\pi\)
0.882550 0.470218i \(-0.155825\pi\)
\(978\) 0 0
\(979\) 460.313 + 265.762i 0.470187 + 0.271463i
\(980\) 0 0
\(981\) 41.2905i 0.0420903i
\(982\) 0 0
\(983\) −542.556 + 313.245i −0.551939 + 0.318662i −0.749904 0.661547i \(-0.769900\pi\)
0.197965 + 0.980209i \(0.436567\pi\)
\(984\) 0 0
\(985\) −484.081 838.453i −0.491453 0.851222i
\(986\) 0 0
\(987\) 1328.18i 1.34567i
\(988\) 0 0
\(989\) 546.070 0.552143
\(990\) 0 0
\(991\) −216.951 + 125.257i −0.218921 + 0.126394i −0.605451 0.795883i \(-0.707007\pi\)
0.386530 + 0.922277i \(0.373674\pi\)
\(992\) 0 0
\(993\) 43.1983 + 74.8217i 0.0435028 + 0.0753491i
\(994\) 0 0
\(995\) −948.241 −0.953006
\(996\) 0 0
\(997\) −643.120 + 1113.92i −0.645055 + 1.11727i 0.339234 + 0.940702i \(0.389832\pi\)
−0.984289 + 0.176566i \(0.943501\pi\)
\(998\) 0 0
\(999\) 345.130 0.345475
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 76.3.h.a.69.3 yes 8
3.2 odd 2 684.3.y.h.145.4 8
4.3 odd 2 304.3.r.c.145.2 8
19.7 even 3 1444.3.c.b.721.6 8
19.8 odd 6 inner 76.3.h.a.65.3 8
19.12 odd 6 1444.3.c.b.721.3 8
57.8 even 6 684.3.y.h.217.4 8
76.27 even 6 304.3.r.c.65.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.h.a.65.3 8 19.8 odd 6 inner
76.3.h.a.69.3 yes 8 1.1 even 1 trivial
304.3.r.c.65.2 8 76.27 even 6
304.3.r.c.145.2 8 4.3 odd 2
684.3.y.h.145.4 8 3.2 odd 2
684.3.y.h.217.4 8 57.8 even 6
1444.3.c.b.721.3 8 19.12 odd 6
1444.3.c.b.721.6 8 19.7 even 3