Properties

Label 256.4.g.a.33.3
Level $256$
Weight $4$
Character 256.33
Analytic conductor $15.104$
Analytic rank $0$
Dimension $44$
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] = [256,4,Mod(33,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 256.g (of order \(8\), degree \(4\), 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: \(15.1044889615\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 33.3
Character \(\chi\) \(=\) 256.33
Dual form 256.4.g.a.225.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.90169 + 4.59109i) q^{3} +(-0.188811 + 0.0782080i) q^{5} +(11.4103 - 11.4103i) q^{7} +(1.63019 + 1.63019i) q^{9} +O(q^{10})\) \(q+(-1.90169 + 4.59109i) q^{3} +(-0.188811 + 0.0782080i) q^{5} +(11.4103 - 11.4103i) q^{7} +(1.63019 + 1.63019i) q^{9} +(18.6604 + 45.0502i) q^{11} +(-18.9369 - 7.84392i) q^{13} -1.01558i q^{15} -85.7028i q^{17} +(110.749 + 45.8739i) q^{19} +(30.6868 + 74.0844i) q^{21} +(74.2331 + 74.2331i) q^{23} +(-88.3588 + 88.3588i) q^{25} +(-134.544 + 55.7299i) q^{27} +(-64.4881 + 155.688i) q^{29} -36.6720 q^{31} -242.316 q^{33} +(-1.26201 + 3.04676i) q^{35} +(-313.271 + 129.761i) q^{37} +(72.0243 - 72.0243i) q^{39} +(196.689 + 196.689i) q^{41} +(-20.8770 - 50.4016i) q^{43} +(-0.435291 - 0.180303i) q^{45} +508.601i q^{47} +82.6118i q^{49} +(393.469 + 162.980i) q^{51} +(-73.8289 - 178.239i) q^{53} +(-7.04657 - 7.04657i) q^{55} +(-421.223 + 421.223i) q^{57} +(40.9489 - 16.9616i) q^{59} +(324.831 - 784.211i) q^{61} +37.2017 q^{63} +4.18895 q^{65} +(-49.4427 + 119.365i) q^{67} +(-481.979 + 199.642i) q^{69} +(362.308 - 362.308i) q^{71} +(239.192 + 239.192i) q^{73} +(-237.632 - 573.695i) q^{75} +(726.954 + 301.114i) q^{77} +1018.36i q^{79} -661.438i q^{81} +(231.267 + 95.7940i) q^{83} +(6.70265 + 16.1816i) q^{85} +(-592.141 - 592.141i) q^{87} +(1103.08 - 1103.08i) q^{89} +(-305.576 + 126.574i) q^{91} +(69.7388 - 168.364i) q^{93} -24.4984 q^{95} -74.0528 q^{97} +(-43.0203 + 103.860i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 4 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q - 4 q^{3} + 4 q^{5} + 4 q^{7} - 4 q^{9} - 4 q^{11} + 4 q^{13} - 4 q^{19} + 4 q^{21} - 324 q^{23} - 4 q^{25} - 268 q^{27} + 4 q^{29} + 752 q^{31} - 8 q^{33} - 460 q^{35} + 4 q^{37} - 596 q^{39} - 4 q^{41} + 804 q^{43} - 104 q^{45} - 1384 q^{51} - 748 q^{53} + 292 q^{55} - 4 q^{57} + 1372 q^{59} + 1828 q^{61} - 2512 q^{63} - 8 q^{65} + 2036 q^{67} + 1060 q^{69} - 220 q^{71} - 4 q^{73} - 1712 q^{75} - 1900 q^{77} + 2436 q^{83} - 496 q^{85} + 1292 q^{87} - 4 q^{89} - 3604 q^{91} + 112 q^{93} + 6088 q^{95} - 8 q^{97} - 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\)

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
\(3\) −1.90169 + 4.59109i −0.365981 + 0.883556i 0.628419 + 0.777875i \(0.283702\pi\)
−0.994400 + 0.105681i \(0.966298\pi\)
\(4\) 0 0
\(5\) −0.188811 + 0.0782080i −0.0168878 + 0.00699514i −0.391111 0.920343i \(-0.627909\pi\)
0.374224 + 0.927339i \(0.377909\pi\)
\(6\) 0 0
\(7\) 11.4103 11.4103i 0.616096 0.616096i −0.328432 0.944528i \(-0.606520\pi\)
0.944528 + 0.328432i \(0.106520\pi\)
\(8\) 0 0
\(9\) 1.63019 + 1.63019i 0.0603773 + 0.0603773i
\(10\) 0 0
\(11\) 18.6604 + 45.0502i 0.511483 + 1.23483i 0.943020 + 0.332735i \(0.107971\pi\)
−0.431537 + 0.902095i \(0.642029\pi\)
\(12\) 0 0
\(13\) −18.9369 7.84392i −0.404011 0.167347i 0.171418 0.985198i \(-0.445165\pi\)
−0.575430 + 0.817851i \(0.695165\pi\)
\(14\) 0 0
\(15\) 1.01558i 0.0174814i
\(16\) 0 0
\(17\) 85.7028i 1.22270i −0.791359 0.611352i \(-0.790626\pi\)
0.791359 0.611352i \(-0.209374\pi\)
\(18\) 0 0
\(19\) 110.749 + 45.8739i 1.33725 + 0.553905i 0.932713 0.360619i \(-0.117434\pi\)
0.404532 + 0.914524i \(0.367434\pi\)
\(20\) 0 0
\(21\) 30.6868 + 74.0844i 0.318876 + 0.769835i
\(22\) 0 0
\(23\) 74.2331 + 74.2331i 0.672985 + 0.672985i 0.958403 0.285418i \(-0.0921324\pi\)
−0.285418 + 0.958403i \(0.592132\pi\)
\(24\) 0 0
\(25\) −88.3588 + 88.3588i −0.706871 + 0.706871i
\(26\) 0 0
\(27\) −134.544 + 55.7299i −0.959000 + 0.397231i
\(28\) 0 0
\(29\) −64.4881 + 155.688i −0.412936 + 0.996915i 0.571410 + 0.820665i \(0.306397\pi\)
−0.984345 + 0.176250i \(0.943603\pi\)
\(30\) 0 0
\(31\) −36.6720 −0.212467 −0.106234 0.994341i \(-0.533879\pi\)
−0.106234 + 0.994341i \(0.533879\pi\)
\(32\) 0 0
\(33\) −242.316 −1.27824
\(34\) 0 0
\(35\) −1.26201 + 3.04676i −0.00609481 + 0.0147142i
\(36\) 0 0
\(37\) −313.271 + 129.761i −1.39193 + 0.576557i −0.947644 0.319328i \(-0.896543\pi\)
−0.444287 + 0.895884i \(0.646543\pi\)
\(38\) 0 0
\(39\) 72.0243 72.0243i 0.295721 0.295721i
\(40\) 0 0
\(41\) 196.689 + 196.689i 0.749211 + 0.749211i 0.974331 0.225120i \(-0.0722773\pi\)
−0.225120 + 0.974331i \(0.572277\pi\)
\(42\) 0 0
\(43\) −20.8770 50.4016i −0.0740400 0.178748i 0.882526 0.470263i \(-0.155841\pi\)
−0.956566 + 0.291514i \(0.905841\pi\)
\(44\) 0 0
\(45\) −0.435291 0.180303i −0.00144198 0.000597290i
\(46\) 0 0
\(47\) 508.601i 1.57845i 0.614106 + 0.789224i \(0.289517\pi\)
−0.614106 + 0.789224i \(0.710483\pi\)
\(48\) 0 0
\(49\) 82.6118i 0.240851i
\(50\) 0 0
\(51\) 393.469 + 162.980i 1.08033 + 0.447487i
\(52\) 0 0
\(53\) −73.8289 178.239i −0.191343 0.461943i 0.798871 0.601503i \(-0.205431\pi\)
−0.990214 + 0.139560i \(0.955431\pi\)
\(54\) 0 0
\(55\) −7.04657 7.04657i −0.0172756 0.0172756i
\(56\) 0 0
\(57\) −421.223 + 421.223i −0.978813 + 0.978813i
\(58\) 0 0
\(59\) 40.9489 16.9616i 0.0903576 0.0374273i −0.337047 0.941488i \(-0.609428\pi\)
0.427404 + 0.904061i \(0.359428\pi\)
\(60\) 0 0
\(61\) 324.831 784.211i 0.681809 1.64603i −0.0788549 0.996886i \(-0.525126\pi\)
0.760664 0.649146i \(-0.224874\pi\)
\(62\) 0 0
\(63\) 37.2017 0.0743964
\(64\) 0 0
\(65\) 4.18895 0.00799346
\(66\) 0 0
\(67\) −49.4427 + 119.365i −0.0901550 + 0.217654i −0.962525 0.271192i \(-0.912582\pi\)
0.872370 + 0.488846i \(0.162582\pi\)
\(68\) 0 0
\(69\) −481.979 + 199.642i −0.840920 + 0.348321i
\(70\) 0 0
\(71\) 362.308 362.308i 0.605606 0.605606i −0.336189 0.941795i \(-0.609138\pi\)
0.941795 + 0.336189i \(0.109138\pi\)
\(72\) 0 0
\(73\) 239.192 + 239.192i 0.383498 + 0.383498i 0.872361 0.488863i \(-0.162588\pi\)
−0.488863 + 0.872361i \(0.662588\pi\)
\(74\) 0 0
\(75\) −237.632 573.695i −0.365859 0.883261i
\(76\) 0 0
\(77\) 726.954 + 301.114i 1.07590 + 0.445651i
\(78\) 0 0
\(79\) 1018.36i 1.45030i 0.688589 + 0.725152i \(0.258231\pi\)
−0.688589 + 0.725152i \(0.741769\pi\)
\(80\) 0 0
\(81\) 661.438i 0.907323i
\(82\) 0 0
\(83\) 231.267 + 95.7940i 0.305842 + 0.126684i 0.530326 0.847794i \(-0.322070\pi\)
−0.224484 + 0.974478i \(0.572070\pi\)
\(84\) 0 0
\(85\) 6.70265 + 16.1816i 0.00855299 + 0.0206487i
\(86\) 0 0
\(87\) −592.141 592.141i −0.729704 0.729704i
\(88\) 0 0
\(89\) 1103.08 1103.08i 1.31378 1.31378i 0.395174 0.918606i \(-0.370684\pi\)
0.918606 0.395174i \(-0.129316\pi\)
\(90\) 0 0
\(91\) −305.576 + 126.574i −0.352012 + 0.145808i
\(92\) 0 0
\(93\) 69.7388 168.364i 0.0777589 0.187727i
\(94\) 0 0
\(95\) −24.4984 −0.0264577
\(96\) 0 0
\(97\) −74.0528 −0.0775147 −0.0387573 0.999249i \(-0.512340\pi\)
−0.0387573 + 0.999249i \(0.512340\pi\)
\(98\) 0 0
\(99\) −43.0203 + 103.860i −0.0436737 + 0.105438i
\(100\) 0 0
\(101\) −564.281 + 233.733i −0.555922 + 0.230270i −0.642914 0.765939i \(-0.722275\pi\)
0.0869918 + 0.996209i \(0.472275\pi\)
\(102\) 0 0
\(103\) 504.784 504.784i 0.482892 0.482892i −0.423162 0.906054i \(-0.639080\pi\)
0.906054 + 0.423162i \(0.139080\pi\)
\(104\) 0 0
\(105\) −11.5880 11.5880i −0.0107702 0.0107702i
\(106\) 0 0
\(107\) −399.644 964.826i −0.361075 0.871713i −0.995143 0.0984356i \(-0.968616\pi\)
0.634068 0.773277i \(-0.281384\pi\)
\(108\) 0 0
\(109\) 1362.55 + 564.385i 1.19732 + 0.495948i 0.890133 0.455700i \(-0.150611\pi\)
0.307191 + 0.951648i \(0.400611\pi\)
\(110\) 0 0
\(111\) 1685.02i 1.44086i
\(112\) 0 0
\(113\) 1171.99i 0.975681i −0.872933 0.487840i \(-0.837785\pi\)
0.872933 0.487840i \(-0.162215\pi\)
\(114\) 0 0
\(115\) −19.8216 8.21039i −0.0160728 0.00665759i
\(116\) 0 0
\(117\) −18.0836 43.6577i −0.0142892 0.0344971i
\(118\) 0 0
\(119\) −977.891 977.891i −0.753304 0.753304i
\(120\) 0 0
\(121\) −740.147 + 740.147i −0.556084 + 0.556084i
\(122\) 0 0
\(123\) −1277.06 + 528.976i −0.936168 + 0.387773i
\(124\) 0 0
\(125\) 19.5487 47.1948i 0.0139879 0.0337699i
\(126\) 0 0
\(127\) −2033.56 −1.42086 −0.710432 0.703766i \(-0.751500\pi\)
−0.710432 + 0.703766i \(0.751500\pi\)
\(128\) 0 0
\(129\) 271.100 0.185031
\(130\) 0 0
\(131\) 268.953 649.310i 0.179378 0.433057i −0.808458 0.588553i \(-0.799698\pi\)
0.987836 + 0.155496i \(0.0496977\pi\)
\(132\) 0 0
\(133\) 1787.11 740.247i 1.16513 0.482613i
\(134\) 0 0
\(135\) 21.0448 21.0448i 0.0134167 0.0134167i
\(136\) 0 0
\(137\) 1077.41 + 1077.41i 0.671894 + 0.671894i 0.958152 0.286259i \(-0.0924116\pi\)
−0.286259 + 0.958152i \(0.592412\pi\)
\(138\) 0 0
\(139\) 285.862 + 690.132i 0.174435 + 0.421124i 0.986783 0.162050i \(-0.0518106\pi\)
−0.812347 + 0.583174i \(0.801811\pi\)
\(140\) 0 0
\(141\) −2335.03 967.202i −1.39465 0.577682i
\(142\) 0 0
\(143\) 999.480i 0.584481i
\(144\) 0 0
\(145\) 34.4391i 0.0197242i
\(146\) 0 0
\(147\) −379.278 157.102i −0.212805 0.0881468i
\(148\) 0 0
\(149\) 634.226 + 1531.16i 0.348710 + 0.841860i 0.996773 + 0.0802736i \(0.0255794\pi\)
−0.648063 + 0.761587i \(0.724421\pi\)
\(150\) 0 0
\(151\) 805.847 + 805.847i 0.434297 + 0.434297i 0.890087 0.455790i \(-0.150643\pi\)
−0.455790 + 0.890087i \(0.650643\pi\)
\(152\) 0 0
\(153\) 139.712 139.712i 0.0738236 0.0738236i
\(154\) 0 0
\(155\) 6.92407 2.86804i 0.00358809 0.00148624i
\(156\) 0 0
\(157\) 308.064 743.733i 0.156600 0.378066i −0.826034 0.563620i \(-0.809408\pi\)
0.982634 + 0.185554i \(0.0594081\pi\)
\(158\) 0 0
\(159\) 958.710 0.478180
\(160\) 0 0
\(161\) 1694.04 0.829248
\(162\) 0 0
\(163\) 396.959 958.344i 0.190750 0.460511i −0.799352 0.600863i \(-0.794824\pi\)
0.990102 + 0.140353i \(0.0448236\pi\)
\(164\) 0 0
\(165\) 45.7519 18.9510i 0.0215865 0.00894143i
\(166\) 0 0
\(167\) −307.489 + 307.489i −0.142480 + 0.142480i −0.774749 0.632269i \(-0.782124\pi\)
0.632269 + 0.774749i \(0.282124\pi\)
\(168\) 0 0
\(169\) −1256.43 1256.43i −0.571887 0.571887i
\(170\) 0 0
\(171\) 105.759 + 255.325i 0.0472960 + 0.114183i
\(172\) 0 0
\(173\) −328.204 135.946i −0.144236 0.0597446i 0.309398 0.950933i \(-0.399873\pi\)
−0.453634 + 0.891188i \(0.649873\pi\)
\(174\) 0 0
\(175\) 2016.39i 0.871001i
\(176\) 0 0
\(177\) 220.256i 0.0935337i
\(178\) 0 0
\(179\) −1270.86 526.407i −0.530662 0.219807i 0.101231 0.994863i \(-0.467722\pi\)
−0.631893 + 0.775056i \(0.717722\pi\)
\(180\) 0 0
\(181\) −1449.19 3498.66i −0.595125 1.43676i −0.878496 0.477749i \(-0.841453\pi\)
0.283371 0.959010i \(-0.408547\pi\)
\(182\) 0 0
\(183\) 2982.66 + 2982.66i 1.20483 + 1.20483i
\(184\) 0 0
\(185\) 49.0007 49.0007i 0.0194735 0.0194735i
\(186\) 0 0
\(187\) 3860.92 1599.25i 1.50983 0.625393i
\(188\) 0 0
\(189\) −899.289 + 2171.08i −0.346104 + 0.835569i
\(190\) 0 0
\(191\) 2124.08 0.804677 0.402338 0.915491i \(-0.368197\pi\)
0.402338 + 0.915491i \(0.368197\pi\)
\(192\) 0 0
\(193\) −2384.50 −0.889328 −0.444664 0.895697i \(-0.646677\pi\)
−0.444664 + 0.895697i \(0.646677\pi\)
\(194\) 0 0
\(195\) −7.96609 + 19.2319i −0.00292546 + 0.00706268i
\(196\) 0 0
\(197\) 3010.53 1247.00i 1.08879 0.450990i 0.235204 0.971946i \(-0.424424\pi\)
0.853584 + 0.520956i \(0.174424\pi\)
\(198\) 0 0
\(199\) −2542.97 + 2542.97i −0.905862 + 0.905862i −0.995935 0.0900734i \(-0.971290\pi\)
0.0900734 + 0.995935i \(0.471290\pi\)
\(200\) 0 0
\(201\) −453.992 453.992i −0.159314 0.159314i
\(202\) 0 0
\(203\) 1040.61 + 2512.27i 0.359787 + 0.868604i
\(204\) 0 0
\(205\) −52.5197 21.7544i −0.0178933 0.00741167i
\(206\) 0 0
\(207\) 242.028i 0.0812661i
\(208\) 0 0
\(209\) 5845.30i 1.93458i
\(210\) 0 0
\(211\) 411.497 + 170.448i 0.134259 + 0.0556118i 0.448801 0.893632i \(-0.351851\pi\)
−0.314543 + 0.949243i \(0.601851\pi\)
\(212\) 0 0
\(213\) 974.390 + 2352.39i 0.313447 + 0.756727i
\(214\) 0 0
\(215\) 7.88363 + 7.88363i 0.00250074 + 0.00250074i
\(216\) 0 0
\(217\) −418.437 + 418.437i −0.130900 + 0.130900i
\(218\) 0 0
\(219\) −1553.02 + 643.284i −0.479195 + 0.198489i
\(220\) 0 0
\(221\) −672.246 + 1622.94i −0.204616 + 0.493987i
\(222\) 0 0
\(223\) 1847.81 0.554880 0.277440 0.960743i \(-0.410514\pi\)
0.277440 + 0.960743i \(0.410514\pi\)
\(224\) 0 0
\(225\) −288.083 −0.0853579
\(226\) 0 0
\(227\) −904.304 + 2183.18i −0.264409 + 0.638339i −0.999202 0.0399527i \(-0.987279\pi\)
0.734793 + 0.678291i \(0.237279\pi\)
\(228\) 0 0
\(229\) −4319.52 + 1789.20i −1.24647 + 0.516305i −0.905731 0.423852i \(-0.860678\pi\)
−0.340740 + 0.940158i \(0.610678\pi\)
\(230\) 0 0
\(231\) −2764.89 + 2764.89i −0.787516 + 0.787516i
\(232\) 0 0
\(233\) −3523.28 3523.28i −0.990634 0.990634i 0.00932269 0.999957i \(-0.497032\pi\)
−0.999957 + 0.00932269i \(0.997032\pi\)
\(234\) 0 0
\(235\) −39.7767 96.0294i −0.0110415 0.0266564i
\(236\) 0 0
\(237\) −4675.37 1936.60i −1.28143 0.530784i
\(238\) 0 0
\(239\) 6222.79i 1.68418i −0.539337 0.842090i \(-0.681325\pi\)
0.539337 0.842090i \(-0.318675\pi\)
\(240\) 0 0
\(241\) 4271.08i 1.14160i 0.821091 + 0.570798i \(0.193366\pi\)
−0.821091 + 0.570798i \(0.806634\pi\)
\(242\) 0 0
\(243\) −595.962 246.856i −0.157329 0.0651679i
\(244\) 0 0
\(245\) −6.46091 15.5980i −0.00168478 0.00406743i
\(246\) 0 0
\(247\) −1737.42 1737.42i −0.447568 0.447568i
\(248\) 0 0
\(249\) −879.598 + 879.598i −0.223864 + 0.223864i
\(250\) 0 0
\(251\) 4861.21 2013.58i 1.22246 0.506358i 0.324267 0.945966i \(-0.394883\pi\)
0.898190 + 0.439608i \(0.144883\pi\)
\(252\) 0 0
\(253\) −1958.99 + 4729.43i −0.486802 + 1.17524i
\(254\) 0 0
\(255\) −87.0377 −0.0213746
\(256\) 0 0
\(257\) 2043.19 0.495917 0.247958 0.968771i \(-0.420240\pi\)
0.247958 + 0.968771i \(0.420240\pi\)
\(258\) 0 0
\(259\) −2093.90 + 5055.12i −0.502349 + 1.21278i
\(260\) 0 0
\(261\) −358.928 + 148.673i −0.0851230 + 0.0352591i
\(262\) 0 0
\(263\) 3376.45 3376.45i 0.791638 0.791638i −0.190123 0.981760i \(-0.560889\pi\)
0.981760 + 0.190123i \(0.0608886\pi\)
\(264\) 0 0
\(265\) 27.8794 + 27.8794i 0.00646271 + 0.00646271i
\(266\) 0 0
\(267\) 2966.63 + 7162.07i 0.679980 + 1.64162i
\(268\) 0 0
\(269\) 3758.60 + 1556.86i 0.851917 + 0.352876i 0.765541 0.643387i \(-0.222471\pi\)
0.0863762 + 0.996263i \(0.472471\pi\)
\(270\) 0 0
\(271\) 4182.58i 0.937540i 0.883320 + 0.468770i \(0.155303\pi\)
−0.883320 + 0.468770i \(0.844697\pi\)
\(272\) 0 0
\(273\) 1643.63i 0.364385i
\(274\) 0 0
\(275\) −5629.39 2331.77i −1.23442 0.511313i
\(276\) 0 0
\(277\) 26.8731 + 64.8775i 0.00582906 + 0.0140726i 0.926767 0.375636i \(-0.122576\pi\)
−0.920938 + 0.389709i \(0.872576\pi\)
\(278\) 0 0
\(279\) −59.7822 59.7822i −0.0128282 0.0128282i
\(280\) 0 0
\(281\) 3756.73 3756.73i 0.797536 0.797536i −0.185171 0.982706i \(-0.559284\pi\)
0.982706 + 0.185171i \(0.0592837\pi\)
\(282\) 0 0
\(283\) 47.8634 19.8257i 0.0100536 0.00416436i −0.377651 0.925948i \(-0.623268\pi\)
0.387705 + 0.921784i \(0.373268\pi\)
\(284\) 0 0
\(285\) 46.5885 112.474i 0.00968303 0.0233769i
\(286\) 0 0
\(287\) 4488.55 0.923173
\(288\) 0 0
\(289\) −2431.97 −0.495007
\(290\) 0 0
\(291\) 140.826 339.983i 0.0283689 0.0684886i
\(292\) 0 0
\(293\) 3659.53 1515.83i 0.729666 0.302238i 0.0132515 0.999912i \(-0.495782\pi\)
0.716415 + 0.697674i \(0.245782\pi\)
\(294\) 0 0
\(295\) −6.40507 + 6.40507i −0.00126413 + 0.00126413i
\(296\) 0 0
\(297\) −5021.28 5021.28i −0.981025 0.981025i
\(298\) 0 0
\(299\) −823.465 1988.02i −0.159272 0.384516i
\(300\) 0 0
\(301\) −813.309 336.883i −0.155742 0.0645104i
\(302\) 0 0
\(303\) 3035.16i 0.575463i
\(304\) 0 0
\(305\) 173.472i 0.0325671i
\(306\) 0 0
\(307\) 1166.44 + 483.155i 0.216847 + 0.0898211i 0.488463 0.872585i \(-0.337558\pi\)
−0.271615 + 0.962406i \(0.587558\pi\)
\(308\) 0 0
\(309\) 1357.57 + 3277.46i 0.249933 + 0.603391i
\(310\) 0 0
\(311\) −811.491 811.491i −0.147960 0.147960i 0.629246 0.777206i \(-0.283364\pi\)
−0.777206 + 0.629246i \(0.783364\pi\)
\(312\) 0 0
\(313\) −748.416 + 748.416i −0.135153 + 0.135153i −0.771447 0.636294i \(-0.780467\pi\)
0.636294 + 0.771447i \(0.280467\pi\)
\(314\) 0 0
\(315\) −7.02409 + 2.90947i −0.00125639 + 0.000520414i
\(316\) 0 0
\(317\) 1647.93 3978.45i 0.291977 0.704896i −0.708022 0.706191i \(-0.750412\pi\)
0.999999 + 0.00129494i \(0.000412192\pi\)
\(318\) 0 0
\(319\) −8217.14 −1.44223
\(320\) 0 0
\(321\) 5189.61 0.902354
\(322\) 0 0
\(323\) 3931.52 9491.54i 0.677263 1.63506i
\(324\) 0 0
\(325\) 2366.32 980.162i 0.403876 0.167291i
\(326\) 0 0
\(327\) −5182.29 + 5182.29i −0.876395 + 0.876395i
\(328\) 0 0
\(329\) 5803.27 + 5803.27i 0.972476 + 0.972476i
\(330\) 0 0
\(331\) −2379.90 5745.58i −0.395199 0.954095i −0.988788 0.149327i \(-0.952289\pi\)
0.593589 0.804769i \(-0.297711\pi\)
\(332\) 0 0
\(333\) −722.225 299.156i −0.118852 0.0492301i
\(334\) 0 0
\(335\) 26.4043i 0.00430633i
\(336\) 0 0
\(337\) 10731.5i 1.73466i −0.497730 0.867332i \(-0.665833\pi\)
0.497730 0.867332i \(-0.334167\pi\)
\(338\) 0 0
\(339\) 5380.73 + 2228.77i 0.862069 + 0.357081i
\(340\) 0 0
\(341\) −684.313 1652.08i −0.108673 0.262361i
\(342\) 0 0
\(343\) 4856.34 + 4856.34i 0.764484 + 0.764484i
\(344\) 0 0
\(345\) 75.3893 75.3893i 0.0117647 0.0117647i
\(346\) 0 0
\(347\) −5414.63 + 2242.81i −0.837673 + 0.346976i −0.759936 0.649998i \(-0.774770\pi\)
−0.0777376 + 0.996974i \(0.524770\pi\)
\(348\) 0 0
\(349\) 443.942 1071.77i 0.0680907 0.164385i −0.886171 0.463359i \(-0.846644\pi\)
0.954261 + 0.298973i \(0.0966441\pi\)
\(350\) 0 0
\(351\) 2984.99 0.453922
\(352\) 0 0
\(353\) 812.561 0.122516 0.0612582 0.998122i \(-0.480489\pi\)
0.0612582 + 0.998122i \(0.480489\pi\)
\(354\) 0 0
\(355\) −40.0723 + 96.7430i −0.00599103 + 0.0144636i
\(356\) 0 0
\(357\) 6349.24 2629.94i 0.941281 0.389891i
\(358\) 0 0
\(359\) 3733.47 3733.47i 0.548871 0.548871i −0.377243 0.926114i \(-0.623128\pi\)
0.926114 + 0.377243i \(0.123128\pi\)
\(360\) 0 0
\(361\) 5310.98 + 5310.98i 0.774308 + 0.774308i
\(362\) 0 0
\(363\) −1990.55 4805.62i −0.287815 0.694847i
\(364\) 0 0
\(365\) −63.8689 26.4554i −0.00915904 0.00379380i
\(366\) 0 0
\(367\) 6820.92i 0.970161i −0.874469 0.485081i \(-0.838790\pi\)
0.874469 0.485081i \(-0.161210\pi\)
\(368\) 0 0
\(369\) 641.280i 0.0904707i
\(370\) 0 0
\(371\) −2876.16 1191.34i −0.402487 0.166716i
\(372\) 0 0
\(373\) −3875.45 9356.15i −0.537970 1.29878i −0.926137 0.377186i \(-0.876892\pi\)
0.388167 0.921589i \(-0.373108\pi\)
\(374\) 0 0
\(375\) 179.500 + 179.500i 0.0247183 + 0.0247183i
\(376\) 0 0
\(377\) 2442.41 2442.41i 0.333661 0.333661i
\(378\) 0 0
\(379\) −4336.19 + 1796.11i −0.587692 + 0.243430i −0.656657 0.754189i \(-0.728030\pi\)
0.0689653 + 0.997619i \(0.478030\pi\)
\(380\) 0 0
\(381\) 3867.22 9336.29i 0.520009 1.25541i
\(382\) 0 0
\(383\) −8452.79 −1.12772 −0.563861 0.825870i \(-0.690685\pi\)
−0.563861 + 0.825870i \(0.690685\pi\)
\(384\) 0 0
\(385\) −160.806 −0.0212869
\(386\) 0 0
\(387\) 48.1306 116.198i 0.00632201 0.0152627i
\(388\) 0 0
\(389\) 468.925 194.235i 0.0611194 0.0253165i −0.351914 0.936032i \(-0.614469\pi\)
0.413034 + 0.910716i \(0.364469\pi\)
\(390\) 0 0
\(391\) 6361.98 6361.98i 0.822862 0.822862i
\(392\) 0 0
\(393\) 2469.58 + 2469.58i 0.316981 + 0.316981i
\(394\) 0 0
\(395\) −79.6437 192.277i −0.0101451 0.0244924i
\(396\) 0 0
\(397\) 3306.01 + 1369.40i 0.417945 + 0.173118i 0.581738 0.813376i \(-0.302373\pi\)
−0.163793 + 0.986495i \(0.552373\pi\)
\(398\) 0 0
\(399\) 9612.53i 1.20609i
\(400\) 0 0
\(401\) 3887.84i 0.484164i 0.970256 + 0.242082i \(0.0778303\pi\)
−0.970256 + 0.242082i \(0.922170\pi\)
\(402\) 0 0
\(403\) 694.453 + 287.652i 0.0858392 + 0.0355557i
\(404\) 0 0
\(405\) 51.7298 + 124.887i 0.00634685 + 0.0153227i
\(406\) 0 0
\(407\) −11691.5 11691.5i −1.42390 1.42390i
\(408\) 0 0
\(409\) −7188.15 + 7188.15i −0.869025 + 0.869025i −0.992365 0.123339i \(-0.960640\pi\)
0.123339 + 0.992365i \(0.460640\pi\)
\(410\) 0 0
\(411\) −6995.40 + 2897.59i −0.839556 + 0.347756i
\(412\) 0 0
\(413\) 273.702 660.775i 0.0326101 0.0787278i
\(414\) 0 0
\(415\) −51.1576 −0.00605115
\(416\) 0 0
\(417\) −3712.08 −0.435927
\(418\) 0 0
\(419\) 4190.33 10116.4i 0.488571 1.17951i −0.466869 0.884327i \(-0.654618\pi\)
0.955439 0.295187i \(-0.0953821\pi\)
\(420\) 0 0
\(421\) −14568.6 + 6034.51i −1.68653 + 0.698584i −0.999604 0.0281419i \(-0.991041\pi\)
−0.686927 + 0.726726i \(0.741041\pi\)
\(422\) 0 0
\(423\) −829.114 + 829.114i −0.0953024 + 0.0953024i
\(424\) 0 0
\(425\) 7572.60 + 7572.60i 0.864294 + 0.864294i
\(426\) 0 0
\(427\) −5241.65 12654.5i −0.594054 1.43417i
\(428\) 0 0
\(429\) 4588.71 + 1900.70i 0.516422 + 0.213909i
\(430\) 0 0
\(431\) 844.040i 0.0943295i 0.998887 + 0.0471647i \(0.0150186\pi\)
−0.998887 + 0.0471647i \(0.984981\pi\)
\(432\) 0 0
\(433\) 11447.7i 1.27053i −0.772293 0.635266i \(-0.780890\pi\)
0.772293 0.635266i \(-0.219110\pi\)
\(434\) 0 0
\(435\) 158.113 + 65.4925i 0.0174274 + 0.00721868i
\(436\) 0 0
\(437\) 4815.91 + 11626.6i 0.527177 + 1.27272i
\(438\) 0 0
\(439\) −2040.34 2040.34i −0.221823 0.221823i 0.587443 0.809266i \(-0.300135\pi\)
−0.809266 + 0.587443i \(0.800135\pi\)
\(440\) 0 0
\(441\) −134.673 + 134.673i −0.0145419 + 0.0145419i
\(442\) 0 0
\(443\) 12477.8 5168.47i 1.33823 0.554315i 0.405241 0.914210i \(-0.367187\pi\)
0.932993 + 0.359895i \(0.117187\pi\)
\(444\) 0 0
\(445\) −122.004 + 294.544i −0.0129967 + 0.0313769i
\(446\) 0 0
\(447\) −8235.78 −0.871452
\(448\) 0 0
\(449\) −3601.74 −0.378567 −0.189283 0.981922i \(-0.560616\pi\)
−0.189283 + 0.981922i \(0.560616\pi\)
\(450\) 0 0
\(451\) −5190.58 + 12531.2i −0.541940 + 1.30836i
\(452\) 0 0
\(453\) −5232.19 + 2167.25i −0.542671 + 0.224782i
\(454\) 0 0
\(455\) 47.7970 47.7970i 0.00492474 0.00492474i
\(456\) 0 0
\(457\) −10018.4 10018.4i −1.02547 1.02547i −0.999667 0.0258078i \(-0.991784\pi\)
−0.0258078 0.999667i \(-0.508216\pi\)
\(458\) 0 0
\(459\) 4776.21 + 11530.8i 0.485696 + 1.17257i
\(460\) 0 0
\(461\) 1495.66 + 619.524i 0.151106 + 0.0625902i 0.456955 0.889490i \(-0.348940\pi\)
−0.305849 + 0.952080i \(0.598940\pi\)
\(462\) 0 0
\(463\) 14517.8i 1.45724i 0.684921 + 0.728618i \(0.259837\pi\)
−0.684921 + 0.728618i \(0.740163\pi\)
\(464\) 0 0
\(465\) 37.2432i 0.00371422i
\(466\) 0 0
\(467\) 5649.95 + 2340.29i 0.559847 + 0.231896i 0.644619 0.764504i \(-0.277016\pi\)
−0.0847717 + 0.996400i \(0.527016\pi\)
\(468\) 0 0
\(469\) 797.835 + 1926.14i 0.0785513 + 0.189640i
\(470\) 0 0
\(471\) 2828.70 + 2828.70i 0.276730 + 0.276730i
\(472\) 0 0
\(473\) 1881.03 1881.03i 0.182854 0.182854i
\(474\) 0 0
\(475\) −13839.1 + 5732.32i −1.33680 + 0.553720i
\(476\) 0 0
\(477\) 170.207 410.917i 0.0163381 0.0394436i
\(478\) 0 0
\(479\) 10244.1 0.977172 0.488586 0.872516i \(-0.337513\pi\)
0.488586 + 0.872516i \(0.337513\pi\)
\(480\) 0 0
\(481\) 6950.22 0.658841
\(482\) 0 0
\(483\) −3221.54 + 7777.48i −0.303489 + 0.732687i
\(484\) 0 0
\(485\) 13.9820 5.79153i 0.00130905 0.000542226i
\(486\) 0 0
\(487\) 12510.0 12510.0i 1.16403 1.16403i 0.180446 0.983585i \(-0.442246\pi\)
0.983585 0.180446i \(-0.0577542\pi\)
\(488\) 0 0
\(489\) 3644.95 + 3644.95i 0.337076 + 0.337076i
\(490\) 0 0
\(491\) −4847.78 11703.6i −0.445575 1.07571i −0.973962 0.226710i \(-0.927203\pi\)
0.528387 0.849004i \(-0.322797\pi\)
\(492\) 0 0
\(493\) 13342.9 + 5526.81i 1.21893 + 0.504898i
\(494\) 0 0
\(495\) 22.9744i 0.00208611i
\(496\) 0 0
\(497\) 8268.05i 0.746223i
\(498\) 0 0
\(499\) 18011.3 + 7460.51i 1.61582 + 0.669295i 0.993538 0.113498i \(-0.0362055\pi\)
0.622282 + 0.782793i \(0.286206\pi\)
\(500\) 0 0
\(501\) −826.961 1996.46i −0.0737443 0.178034i
\(502\) 0 0
\(503\) 3687.97 + 3687.97i 0.326915 + 0.326915i 0.851412 0.524497i \(-0.175747\pi\)
−0.524497 + 0.851412i \(0.675747\pi\)
\(504\) 0 0
\(505\) 88.2627 88.2627i 0.00777750 0.00777750i
\(506\) 0 0
\(507\) 8157.76 3379.06i 0.714594 0.295994i
\(508\) 0 0
\(509\) −1071.88 + 2587.76i −0.0933407 + 0.225344i −0.963654 0.267155i \(-0.913917\pi\)
0.870313 + 0.492499i \(0.163917\pi\)
\(510\) 0 0
\(511\) 5458.49 0.472543
\(512\) 0 0
\(513\) −17457.2 −1.50245
\(514\) 0 0
\(515\) −55.8306 + 134.787i −0.00477707 + 0.0115329i
\(516\) 0 0
\(517\) −22912.5 + 9490.68i −1.94911 + 0.807350i
\(518\) 0 0
\(519\) 1248.29 1248.29i 0.105575 0.105575i
\(520\) 0 0
\(521\) −8038.89 8038.89i −0.675988 0.675988i 0.283102 0.959090i \(-0.408637\pi\)
−0.959090 + 0.283102i \(0.908637\pi\)
\(522\) 0 0
\(523\) 7860.76 + 18977.6i 0.657222 + 1.58667i 0.802076 + 0.597222i \(0.203729\pi\)
−0.144854 + 0.989453i \(0.546271\pi\)
\(524\) 0 0
\(525\) −9257.45 3834.56i −0.769578 0.318770i
\(526\) 0 0
\(527\) 3142.89i 0.259785i
\(528\) 0 0
\(529\) 1145.90i 0.0941813i
\(530\) 0 0
\(531\) 94.4050 + 39.1038i 0.00771531 + 0.00319579i
\(532\) 0 0
\(533\) −2181.87 5267.49i −0.177312 0.428068i
\(534\) 0 0
\(535\) 150.914 + 150.914i 0.0121955 + 0.0121955i
\(536\) 0 0
\(537\) 4833.57 4833.57i 0.388424 0.388424i
\(538\) 0 0
\(539\) −3721.67 + 1541.57i −0.297410 + 0.123191i
\(540\) 0 0
\(541\) 1316.22 3177.63i 0.104600 0.252527i −0.862911 0.505356i \(-0.831361\pi\)
0.967511 + 0.252829i \(0.0813612\pi\)
\(542\) 0 0
\(543\) 18818.6 1.48726
\(544\) 0 0
\(545\) −301.403 −0.0236893
\(546\) 0 0
\(547\) −6431.02 + 15525.9i −0.502689 + 1.21360i 0.445325 + 0.895369i \(0.353088\pi\)
−0.948014 + 0.318229i \(0.896912\pi\)
\(548\) 0 0
\(549\) 1807.95 748.876i 0.140549 0.0582172i
\(550\) 0 0
\(551\) −14284.0 + 14284.0i −1.10439 + 1.10439i
\(552\) 0 0
\(553\) 11619.7 + 11619.7i 0.893527 + 0.893527i
\(554\) 0 0
\(555\) 131.782 + 318.151i 0.0100790 + 0.0243329i
\(556\) 0 0
\(557\) 5335.11 + 2209.88i 0.405846 + 0.168107i 0.576261 0.817266i \(-0.304511\pi\)
−0.170416 + 0.985372i \(0.554511\pi\)
\(558\) 0 0
\(559\) 1118.21i 0.0846068i
\(560\) 0 0
\(561\) 20767.1i 1.56290i
\(562\) 0 0
\(563\) 5228.46 + 2165.70i 0.391391 + 0.162120i 0.569696 0.821856i \(-0.307061\pi\)
−0.178304 + 0.983975i \(0.557061\pi\)
\(564\) 0 0
\(565\) 91.6594 + 221.285i 0.00682502 + 0.0164771i
\(566\) 0 0
\(567\) −7547.18 7547.18i −0.558998 0.558998i
\(568\) 0 0
\(569\) −4183.25 + 4183.25i −0.308209 + 0.308209i −0.844215 0.536005i \(-0.819933\pi\)
0.536005 + 0.844215i \(0.319933\pi\)
\(570\) 0 0
\(571\) 7441.94 3082.55i 0.545421 0.225921i −0.0929211 0.995673i \(-0.529620\pi\)
0.638342 + 0.769753i \(0.279620\pi\)
\(572\) 0 0
\(573\) −4039.35 + 9751.87i −0.294496 + 0.710977i
\(574\) 0 0
\(575\) −13118.3 −0.951427
\(576\) 0 0
\(577\) 2828.16 0.204052 0.102026 0.994782i \(-0.467468\pi\)
0.102026 + 0.994782i \(0.467468\pi\)
\(578\) 0 0
\(579\) 4534.59 10947.5i 0.325477 0.785771i
\(580\) 0 0
\(581\) 3731.85 1545.78i 0.266477 0.110379i
\(582\) 0 0
\(583\) 6652.00 6652.00i 0.472552 0.472552i
\(584\) 0 0
\(585\) 6.82877 + 6.82877i 0.000482624 + 0.000482624i
\(586\) 0 0
\(587\) 1835.52 + 4431.34i 0.129063 + 0.311586i 0.975181 0.221410i \(-0.0710659\pi\)
−0.846118 + 0.532996i \(0.821066\pi\)
\(588\) 0 0
\(589\) −4061.40 1682.29i −0.284121 0.117687i
\(590\) 0 0
\(591\) 16193.0i 1.12706i
\(592\) 0 0
\(593\) 13545.0i 0.937989i 0.883201 + 0.468995i \(0.155384\pi\)
−0.883201 + 0.468995i \(0.844616\pi\)
\(594\) 0 0
\(595\) 261.116 + 108.158i 0.0179911 + 0.00745215i
\(596\) 0 0
\(597\) −6839.07 16511.0i −0.468852 1.13191i
\(598\) 0 0
\(599\) 12379.3 + 12379.3i 0.844414 + 0.844414i 0.989429 0.145015i \(-0.0463231\pi\)
−0.145015 + 0.989429i \(0.546323\pi\)
\(600\) 0 0
\(601\) −17513.1 + 17513.1i −1.18864 + 1.18864i −0.211202 + 0.977442i \(0.567738\pi\)
−0.977442 + 0.211202i \(0.932262\pi\)
\(602\) 0 0
\(603\) −275.189 + 113.987i −0.0185846 + 0.00769801i
\(604\) 0 0
\(605\) 81.8624 197.633i 0.00550112 0.0132809i
\(606\) 0 0
\(607\) −21984.1 −1.47003 −0.735013 0.678053i \(-0.762824\pi\)
−0.735013 + 0.678053i \(0.762824\pi\)
\(608\) 0 0
\(609\) −13513.0 −0.899135
\(610\) 0 0
\(611\) 3989.42 9631.31i 0.264148 0.637711i
\(612\) 0 0
\(613\) 13582.2 5625.93i 0.894910 0.370684i 0.112649 0.993635i \(-0.464066\pi\)
0.782261 + 0.622951i \(0.214066\pi\)
\(614\) 0 0
\(615\) 199.753 199.753i 0.0130972 0.0130972i
\(616\) 0 0
\(617\) −4045.14 4045.14i −0.263941 0.263941i 0.562712 0.826653i \(-0.309758\pi\)
−0.826653 + 0.562712i \(0.809758\pi\)
\(618\) 0 0
\(619\) −4901.51 11833.3i −0.318268 0.768368i −0.999346 0.0361568i \(-0.988488\pi\)
0.681078 0.732211i \(-0.261512\pi\)
\(620\) 0 0
\(621\) −14124.6 5850.61i −0.912723 0.378062i
\(622\) 0 0
\(623\) 25172.9i 1.61883i
\(624\) 0 0
\(625\) 15609.3i 0.998998i
\(626\) 0 0
\(627\) −26836.3 11116.0i −1.70931 0.708021i
\(628\) 0 0
\(629\) 11120.9 + 26848.2i 0.704959 + 1.70192i
\(630\) 0 0
\(631\) 14073.2 + 14073.2i 0.887870 + 0.887870i 0.994318 0.106448i \(-0.0339478\pi\)
−0.106448 + 0.994318i \(0.533948\pi\)
\(632\) 0 0
\(633\) −1565.08 + 1565.08i −0.0982723 + 0.0982723i
\(634\) 0 0
\(635\) 383.959 159.041i 0.0239952 0.00993914i
\(636\) 0 0
\(637\) 648.000 1564.41i 0.0403057 0.0973065i
\(638\) 0 0
\(639\) 1181.26 0.0731297
\(640\) 0 0
\(641\) 14401.9 0.887430 0.443715 0.896168i \(-0.353660\pi\)
0.443715 + 0.896168i \(0.353660\pi\)
\(642\) 0 0
\(643\) 3465.77 8367.12i 0.212561 0.513168i −0.781254 0.624213i \(-0.785420\pi\)
0.993815 + 0.111045i \(0.0354198\pi\)
\(644\) 0 0
\(645\) −51.1867 + 21.2022i −0.00312477 + 0.00129432i
\(646\) 0 0
\(647\) −17607.8 + 17607.8i −1.06991 + 1.06991i −0.0725490 + 0.997365i \(0.523113\pi\)
−0.997365 + 0.0725490i \(0.976887\pi\)
\(648\) 0 0
\(649\) 1528.25 + 1528.25i 0.0924328 + 0.0924328i
\(650\) 0 0
\(651\) −1125.34 2716.82i −0.0677507 0.163565i
\(652\) 0 0
\(653\) −25243.9 10456.4i −1.51282 0.626629i −0.536680 0.843786i \(-0.680322\pi\)
−0.976137 + 0.217157i \(0.930322\pi\)
\(654\) 0 0
\(655\) 143.631i 0.00856814i
\(656\) 0 0
\(657\) 779.856i 0.0463091i
\(658\) 0 0
\(659\) −2555.01 1058.32i −0.151030 0.0625588i 0.305888 0.952068i \(-0.401047\pi\)
−0.456918 + 0.889509i \(0.651047\pi\)
\(660\) 0 0
\(661\) 1008.92 + 2435.76i 0.0593685 + 0.143328i 0.950780 0.309866i \(-0.100284\pi\)
−0.891412 + 0.453195i \(0.850284\pi\)
\(662\) 0 0
\(663\) −6172.68 6172.68i −0.361579 0.361579i
\(664\) 0 0
\(665\) −279.533 + 279.533i −0.0163005 + 0.0163005i
\(666\) 0 0
\(667\) −16344.3 + 6770.05i −0.948809 + 0.393009i
\(668\) 0 0
\(669\) −3513.96 + 8483.45i −0.203076 + 0.490268i
\(670\) 0 0
\(671\) 41390.3 2.38130
\(672\) 0 0
\(673\) 23085.3 1.32225 0.661125 0.750276i \(-0.270079\pi\)
0.661125 + 0.750276i \(0.270079\pi\)
\(674\) 0 0
\(675\) 6963.91 16812.4i 0.397098 0.958679i
\(676\) 0 0
\(677\) 29890.9 12381.2i 1.69690 0.702879i 0.697001 0.717070i \(-0.254517\pi\)
0.999899 + 0.0141909i \(0.00451726\pi\)
\(678\) 0 0
\(679\) −844.962 + 844.962i −0.0477565 + 0.0477565i
\(680\) 0 0
\(681\) −8303.48 8303.48i −0.467240 0.467240i
\(682\) 0 0
\(683\) 4968.96 + 11996.1i 0.278378 + 0.672063i 0.999791 0.0204414i \(-0.00650715\pi\)
−0.721413 + 0.692505i \(0.756507\pi\)
\(684\) 0 0
\(685\) −287.689 119.165i −0.0160468 0.00664679i
\(686\) 0 0
\(687\) 23233.8i 1.29029i
\(688\) 0 0
\(689\) 3954.39i 0.218651i
\(690\) 0 0
\(691\) −7791.33 3227.27i −0.428938 0.177672i 0.157760 0.987477i \(-0.449573\pi\)
−0.586698 + 0.809805i \(0.699573\pi\)
\(692\) 0 0
\(693\) 694.198 + 1675.94i 0.0380525 + 0.0918670i
\(694\) 0 0
\(695\) −107.948 107.948i −0.00589165 0.00589165i
\(696\) 0 0
\(697\) 16856.8 16856.8i 0.916064 0.916064i
\(698\) 0 0
\(699\) 22875.9 9475.51i 1.23783 0.512728i
\(700\) 0 0
\(701\) 3402.06 8213.30i 0.183301 0.442528i −0.805342 0.592810i \(-0.798018\pi\)
0.988643 + 0.150283i \(0.0480184\pi\)
\(702\) 0 0
\(703\) −40647.3 −2.18071
\(704\) 0 0
\(705\) 516.523 0.0275934
\(706\) 0 0
\(707\) −3771.64 + 9105.55i −0.200633 + 0.484370i
\(708\) 0 0
\(709\) 6190.43 2564.16i 0.327908 0.135824i −0.212656 0.977127i \(-0.568211\pi\)
0.540563 + 0.841303i \(0.318211\pi\)
\(710\) 0 0
\(711\) −1660.11 + 1660.11i −0.0875655 + 0.0875655i
\(712\) 0 0
\(713\) −2722.27 2722.27i −0.142987 0.142987i
\(714\) 0 0
\(715\) 78.1674 + 188.713i 0.00408852 + 0.00987057i
\(716\) 0 0
\(717\) 28569.4 + 11833.8i 1.48807 + 0.616378i
\(718\) 0 0
\(719\) 7000.41i 0.363103i 0.983381 + 0.181552i \(0.0581120\pi\)
−0.983381 + 0.181552i \(0.941888\pi\)
\(720\) 0 0
\(721\) 11519.4i 0.595016i
\(722\) 0 0
\(723\) −19608.9 8122.29i −1.00866 0.417802i
\(724\) 0 0
\(725\) −8058.32 19454.5i −0.412798 0.996582i
\(726\) 0 0
\(727\) 4176.48 + 4176.48i 0.213063 + 0.213063i 0.805567 0.592504i \(-0.201861\pi\)
−0.592504 + 0.805567i \(0.701861\pi\)
\(728\) 0 0
\(729\) 14894.8 14894.8i 0.756733 0.756733i
\(730\) 0 0
\(731\) −4319.56 + 1789.22i −0.218556 + 0.0905291i
\(732\) 0 0
\(733\) −8727.96 + 21071.2i −0.439801 + 1.06177i 0.536216 + 0.844081i \(0.319853\pi\)
−0.976017 + 0.217694i \(0.930147\pi\)
\(734\) 0 0
\(735\) 83.8986 0.00421040
\(736\) 0 0
\(737\) −6300.04 −0.314878
\(738\) 0 0
\(739\) −1771.01 + 4275.60i −0.0881566 + 0.212829i −0.961809 0.273722i \(-0.911745\pi\)
0.873652 + 0.486551i \(0.161745\pi\)
\(740\) 0 0
\(741\) 11280.7 4672.61i 0.559253 0.231650i
\(742\) 0 0
\(743\) −331.818 + 331.818i −0.0163839 + 0.0163839i −0.715251 0.698867i \(-0.753688\pi\)
0.698867 + 0.715251i \(0.253688\pi\)
\(744\) 0 0
\(745\) −239.497 239.497i −0.0117779 0.0117779i
\(746\) 0 0
\(747\) 220.846 + 533.171i 0.0108171 + 0.0261147i
\(748\) 0 0
\(749\) −15569.0 6448.88i −0.759516 0.314602i
\(750\) 0 0
\(751\) 35952.8i 1.74692i −0.486897 0.873459i \(-0.661871\pi\)
0.486897 0.873459i \(-0.338129\pi\)
\(752\) 0 0
\(753\) 26147.5i 1.26543i
\(754\) 0 0
\(755\) −215.177 89.1290i −0.0103723 0.00429634i
\(756\) 0 0
\(757\) −8248.62 19913.9i −0.396038 0.956121i −0.988596 0.150593i \(-0.951882\pi\)
0.592557 0.805528i \(-0.298118\pi\)
\(758\) 0 0
\(759\) −17987.8 17987.8i −0.860234 0.860234i
\(760\) 0 0
\(761\) 20845.8 20845.8i 0.992983 0.992983i −0.00699229 0.999976i \(-0.502226\pi\)
0.999976 + 0.00699229i \(0.00222573\pi\)
\(762\) 0 0
\(763\) 21986.8 9107.23i 1.04322 0.432115i
\(764\) 0 0
\(765\) −15.4525 + 37.3056i −0.000730309 + 0.00176312i
\(766\) 0 0
\(767\) −908.491 −0.0427689
\(768\) 0 0
\(769\) 14131.4 0.662669 0.331335 0.943513i \(-0.392501\pi\)
0.331335 + 0.943513i \(0.392501\pi\)
\(770\) 0 0
\(771\) −3885.52 + 9380.46i −0.181496 + 0.438170i
\(772\) 0 0
\(773\) 15774.0 6533.82i 0.733962 0.304017i 0.0157831 0.999875i \(-0.494976\pi\)
0.718179 + 0.695858i \(0.244976\pi\)
\(774\) 0 0
\(775\) 3240.29 3240.29i 0.150187 0.150187i
\(776\) 0 0
\(777\) −19226.6 19226.6i −0.887708 0.887708i
\(778\) 0 0
\(779\) 12760.3 + 30806.1i 0.586888 + 1.41687i
\(780\) 0 0
\(781\) 23082.8 + 9561.22i 1.05758 + 0.438063i
\(782\) 0 0
\(783\) 24540.8i 1.12007i
\(784\) 0 0
\(785\) 164.518i 0.00748013i
\(786\) 0 0
\(787\) 9893.89 + 4098.18i 0.448131 + 0.185622i 0.595324 0.803486i \(-0.297024\pi\)
−0.147193 + 0.989108i \(0.547024\pi\)
\(788\) 0 0
\(789\) 9080.62 + 21922.6i 0.409732 + 0.989181i
\(790\) 0 0
\(791\) −13372.8 13372.8i −0.601113 0.601113i
\(792\) 0 0
\(793\) −12302.6 + 12302.6i −0.550917 + 0.550917i
\(794\) 0 0
\(795\) −181.015 + 74.9788i −0.00807539 + 0.00334494i
\(796\) 0 0
\(797\) 10234.0 24707.0i 0.454838 1.09807i −0.515623 0.856815i \(-0.672440\pi\)
0.970461 0.241259i \(-0.0775605\pi\)
\(798\) 0 0
\(799\) 43588.5 1.92998
\(800\) 0 0
\(801\) 3596.46 0.158645
\(802\) 0 0
\(803\) −6312.23 + 15239.1i −0.277402 + 0.669707i
\(804\) 0 0
\(805\) −319.853 + 132.487i −0.0140041 + 0.00580070i
\(806\) 0 0
\(807\) −14295.4 + 14295.4i −0.623571 + 0.623571i
\(808\) 0 0
\(809\) −4366.67 4366.67i −0.189770 0.189770i 0.605827 0.795597i \(-0.292843\pi\)
−0.795597 + 0.605827i \(0.792843\pi\)
\(810\) 0 0
\(811\) −16742.8 40420.6i −0.724930 1.75014i −0.658792 0.752325i \(-0.728932\pi\)
−0.0661382 0.997810i \(-0.521068\pi\)
\(812\) 0 0
\(813\) −19202.6 7953.98i −0.828370 0.343122i
\(814\) 0 0
\(815\) 211.991i 0.00911132i
\(816\) 0 0
\(817\) 6539.67i 0.280042i
\(818\) 0 0
\(819\) −704.485 291.807i −0.0300570 0.0124500i
\(820\) 0 0
\(821\) 2730.75 + 6592.63i 0.116083 + 0.280249i 0.971233 0.238132i \(-0.0765351\pi\)
−0.855150 + 0.518381i \(0.826535\pi\)
\(822\) 0 0
\(823\) 7832.94 + 7832.94i 0.331761 + 0.331761i 0.853255 0.521494i \(-0.174625\pi\)
−0.521494 + 0.853255i \(0.674625\pi\)
\(824\) 0 0
\(825\) 21410.7 21410.7i 0.903547 0.903547i
\(826\) 0 0
\(827\) −18116.8 + 7504.23i −0.761769 + 0.315535i −0.729533 0.683945i \(-0.760263\pi\)
−0.0322359 + 0.999480i \(0.510263\pi\)
\(828\) 0 0
\(829\) 3888.12 9386.75i 0.162895 0.393264i −0.821265 0.570547i \(-0.806731\pi\)
0.984160 + 0.177284i \(0.0567311\pi\)
\(830\) 0 0
\(831\) −348.963 −0.0145673
\(832\) 0 0
\(833\) 7080.06 0.294489
\(834\) 0 0
\(835\) 34.0092 82.1054i 0.00140950 0.00340284i
\(836\) 0 0
\(837\) 4933.99 2043.73i 0.203756 0.0843985i
\(838\) 0 0
\(839\) 4919.87 4919.87i 0.202447 0.202447i −0.598601 0.801047i \(-0.704276\pi\)
0.801047 + 0.598601i \(0.204276\pi\)
\(840\) 0 0
\(841\) −2834.41 2834.41i −0.116217 0.116217i
\(842\) 0 0
\(843\) 10103.3 + 24391.6i 0.412785 + 0.996551i
\(844\) 0 0
\(845\) 335.492 + 138.965i 0.0136583 + 0.00565746i
\(846\) 0 0
\(847\) 16890.5i 0.685202i
\(848\) 0 0
\(849\) 257.447i 0.0104070i
\(850\) 0 0
\(851\) −32887.6 13622.5i −1.32476 0.548735i
\(852\) 0 0
\(853\) 2809.03 + 6781.60i 0.112754 + 0.272213i 0.970176 0.242401i \(-0.0779351\pi\)
−0.857422 + 0.514614i \(0.827935\pi\)
\(854\) 0 0
\(855\) −39.9370 39.9370i −0.00159745 0.00159745i
\(856\) 0 0
\(857\) −7846.03 + 7846.03i −0.312737 + 0.312737i −0.845969 0.533232i \(-0.820977\pi\)
0.533232 + 0.845969i \(0.320977\pi\)
\(858\) 0 0
\(859\) 10278.4 4257.47i 0.408261 0.169107i −0.169095 0.985600i \(-0.554085\pi\)
0.577356 + 0.816493i \(0.304085\pi\)
\(860\) 0 0
\(861\) −8535.84 + 20607.3i −0.337864 + 0.815675i
\(862\) 0 0
\(863\) −28549.2 −1.12610 −0.563050 0.826423i \(-0.690372\pi\)
−0.563050 + 0.826423i \(0.690372\pi\)
\(864\) 0 0
\(865\) 72.6005 0.00285375
\(866\) 0 0
\(867\) 4624.86 11165.4i 0.181163 0.437366i
\(868\) 0 0
\(869\) −45877.1 + 19002.9i −1.79088 + 0.741807i
\(870\) 0 0
\(871\) 1872.58 1872.58i 0.0728473 0.0728473i
\(872\) 0 0
\(873\) −120.720 120.720i −0.00468013 0.00468013i
\(874\) 0 0
\(875\) −315.449 761.562i −0.0121876 0.0294234i
\(876\) 0 0
\(877\) 8149.22 + 3375.52i 0.313774 + 0.129969i 0.534012 0.845477i \(-0.320684\pi\)
−0.220239 + 0.975446i \(0.570684\pi\)
\(878\) 0 0
\(879\) 19683.9i 0.755315i
\(880\) 0 0
\(881\) 22699.0i 0.868046i 0.900902 + 0.434023i \(0.142906\pi\)
−0.900902 + 0.434023i \(0.857094\pi\)
\(882\) 0 0
\(883\) −3173.59 1314.54i −0.120951 0.0500995i 0.321387 0.946948i \(-0.395851\pi\)
−0.442338 + 0.896848i \(0.645851\pi\)
\(884\) 0 0
\(885\) −17.2258 41.5868i −0.000654282 0.00157958i
\(886\) 0 0
\(887\) −389.896 389.896i −0.0147592 0.0147592i 0.699689 0.714448i \(-0.253322\pi\)
−0.714448 + 0.699689i \(0.753322\pi\)
\(888\) 0 0
\(889\) −23203.5 + 23203.5i −0.875389 + 0.875389i
\(890\) 0 0
\(891\) 29797.9 12342.7i 1.12039 0.464081i
\(892\) 0 0
\(893\) −23331.5 + 56327.2i −0.874310 + 2.11077i
\(894\) 0 0
\(895\) 281.121 0.0104993
\(896\) 0 0
\(897\) 10693.2 0.398032
\(898\) 0 0
\(899\) 2364.91 5709.39i 0.0877353 0.211812i
\(900\) 0 0
\(901\) −15275.6 + 6327.34i −0.564820 + 0.233956i
\(902\) 0 0
\(903\) 3093.33 3093.33i 0.113997 0.113997i
\(904\) 0 0
\(905\) 547.247 + 547.247i 0.0201007 + 0.0201007i
\(906\) 0 0
\(907\) −14791.8 35710.6i −0.541516 1.30733i −0.923653 0.383229i \(-0.874812\pi\)
0.382138 0.924105i \(-0.375188\pi\)
\(908\) 0 0
\(909\) −1300.91 538.856i −0.0474682 0.0196620i
\(910\) 0 0
\(911\) 26874.8i 0.977388i 0.872455 + 0.488694i \(0.162527\pi\)
−0.872455 + 0.488694i \(0.837473\pi\)
\(912\) 0 0
\(913\) 12206.2i 0.442459i
\(914\) 0 0
\(915\) −796.426 329.891i −0.0287749 0.0119190i
\(916\) 0 0
\(917\) −4339.97 10477.6i −0.156291 0.377319i
\(918\) 0 0
\(919\) −9495.37 9495.37i −0.340831 0.340831i 0.515849 0.856680i \(-0.327477\pi\)
−0.856680 + 0.515849i \(0.827477\pi\)
\(920\) 0 0
\(921\) −4436.42 + 4436.42i −0.158724 + 0.158724i
\(922\) 0 0
\(923\) −9702.89 + 4019.07i −0.346018 + 0.143325i
\(924\) 0 0
\(925\) 16214.7 39145.8i 0.576364 1.39147i
\(926\) 0 0
\(927\) 1645.79 0.0583114
\(928\) 0 0
\(929\) −51635.3 −1.82357 −0.911786 0.410665i \(-0.865297\pi\)
−0.911786 + 0.410665i \(0.865297\pi\)
\(930\) 0 0
\(931\) −3789.73 + 9149.21i −0.133409 + 0.322077i
\(932\) 0 0
\(933\) 5268.84 2182.42i 0.184881 0.0765803i
\(934\) 0 0
\(935\) −603.911 + 603.911i −0.0211230 + 0.0211230i
\(936\) 0 0
\(937\) 3704.24 + 3704.24i 0.129149 + 0.129149i 0.768726 0.639578i \(-0.220891\pi\)
−0.639578 + 0.768726i \(0.720891\pi\)
\(938\) 0 0
\(939\) −2012.79 4859.30i −0.0699520 0.168879i
\(940\) 0 0
\(941\) −6682.02 2767.78i −0.231485 0.0958843i 0.263926 0.964543i \(-0.414983\pi\)
−0.495411 + 0.868659i \(0.664983\pi\)
\(942\) 0 0
\(943\) 29201.7i 1.00842i
\(944\) 0 0
\(945\) 480.254i 0.0165319i
\(946\) 0 0
\(947\) 494.002 + 204.622i 0.0169513 + 0.00702147i 0.391143 0.920330i \(-0.372080\pi\)
−0.374192 + 0.927351i \(0.622080\pi\)
\(948\) 0 0
\(949\) −2653.35 6405.76i −0.0907603 0.219115i
\(950\) 0 0
\(951\) 15131.6 + 15131.6i 0.515957 + 0.515957i
\(952\) 0 0
\(953\) −28698.0 + 28698.0i −0.975465 + 0.975465i −0.999706 0.0242408i \(-0.992283\pi\)
0.0242408 + 0.999706i \(0.492283\pi\)
\(954\) 0 0
\(955\) −401.050 + 166.120i −0.0135892 + 0.00562883i
\(956\) 0 0
\(957\) 15626.5 37725.6i 0.527829 1.27429i
\(958\) 0 0
\(959\) 24587.1 0.827903
\(960\) 0 0
\(961\) −28446.2 −0.954858
\(962\) 0 0
\(963\) 921.352 2224.34i 0.0308309 0.0744324i
\(964\) 0 0
\(965\) 450.220 186.487i 0.0150188 0.00622098i
\(966\) 0 0
\(967\) −22689.5 + 22689.5i −0.754544 + 0.754544i −0.975324 0.220780i \(-0.929140\pi\)
0.220780 + 0.975324i \(0.429140\pi\)
\(968\) 0 0
\(969\) 36100.0 + 36100.0i 1.19680 + 1.19680i
\(970\) 0 0
\(971\) 18313.2 + 44211.9i 0.605250 + 1.46120i 0.868111 + 0.496369i \(0.165334\pi\)
−0.262861 + 0.964834i \(0.584666\pi\)
\(972\) 0 0
\(973\) 11136.4 + 4612.83i 0.366922 + 0.151984i
\(974\) 0 0
\(975\) 12728.0i 0.418073i
\(976\) 0 0
\(977\) 3076.94i 0.100757i 0.998730 + 0.0503787i \(0.0160428\pi\)
−0.998730 + 0.0503787i \(0.983957\pi\)
\(978\) 0 0
\(979\) 70277.9 + 29110.1i 2.29427 + 0.950319i
\(980\) 0 0
\(981\) 1301.15 + 3141.26i 0.0423472 + 0.102235i
\(982\) 0 0
\(983\) 40340.6 + 40340.6i 1.30892 + 1.30892i 0.922197 + 0.386722i \(0.126393\pi\)
0.386722 + 0.922197i \(0.373607\pi\)
\(984\) 0 0
\(985\) −470.895 + 470.895i −0.0152324 + 0.0152324i
\(986\) 0 0
\(987\) −37679.4 + 15607.3i −1.21514 + 0.503329i
\(988\) 0 0
\(989\) 2191.70 5291.24i 0.0704672 0.170123i
\(990\) 0 0
\(991\) −19825.3 −0.635492 −0.317746 0.948176i \(-0.602926\pi\)
−0.317746 + 0.948176i \(0.602926\pi\)
\(992\) 0 0
\(993\) 30904.3 0.987632
\(994\) 0 0
\(995\) 281.260 679.022i 0.00896135 0.0216346i
\(996\) 0 0
\(997\) 12624.9 5229.39i 0.401036 0.166115i −0.173043 0.984914i \(-0.555360\pi\)
0.574079 + 0.818800i \(0.305360\pi\)
\(998\) 0 0
\(999\) 34917.2 34917.2i 1.10584 1.10584i
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 256.4.g.a.33.3 44
4.3 odd 2 256.4.g.b.33.9 44
8.3 odd 2 32.4.g.a.13.6 yes 44
8.5 even 2 128.4.g.a.17.9 44
32.5 even 8 inner 256.4.g.a.225.3 44
32.11 odd 8 32.4.g.a.5.6 44
32.21 even 8 128.4.g.a.113.9 44
32.27 odd 8 256.4.g.b.225.9 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.5.6 44 32.11 odd 8
32.4.g.a.13.6 yes 44 8.3 odd 2
128.4.g.a.17.9 44 8.5 even 2
128.4.g.a.113.9 44 32.21 even 8
256.4.g.a.33.3 44 1.1 even 1 trivial
256.4.g.a.225.3 44 32.5 even 8 inner
256.4.g.b.33.9 44 4.3 odd 2
256.4.g.b.225.9 44 32.27 odd 8