Properties

Label 208.4.i.e.81.1
Level $208$
Weight $4$
Character 208.81
Analytic conductor $12.272$
Analytic rank $0$
Dimension $4$
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] = [208,4,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), 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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(-0.780776 - 1.35234i\) of defining polynomial
Character \(\chi\) \(=\) 208.81
Dual form 208.4.i.e.113.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.84233 - 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 - 4.70983i) q^{7} +(6.71165 - 11.6249i) q^{9} +O(q^{10})\) \(q+(-1.84233 - 3.19101i) q^{3} -17.8078 q^{5} +(2.71922 - 4.70983i) q^{7} +(6.71165 - 11.6249i) q^{9} +(-11.2116 - 19.4191i) q^{11} +(21.9730 + 41.4027i) q^{13} +(32.8078 + 56.8247i) q^{15} +(-33.9924 + 58.8766i) q^{17} +(-40.4039 + 69.9816i) q^{19} -20.0388 q^{21} +(70.2656 + 121.704i) q^{23} +192.116 q^{25} -148.946 q^{27} +(53.3466 + 92.3990i) q^{29} +276.155 q^{31} +(-41.3111 + 71.5529i) q^{33} +(-48.4233 + 83.8716i) q^{35} +(2.14584 + 3.71670i) q^{37} +(91.6349 - 146.394i) q^{39} +(-113.884 - 197.254i) q^{41} +(13.7647 - 23.8411i) q^{43} +(-119.519 + 207.014i) q^{45} -318.617 q^{47} +(156.712 + 271.433i) q^{49} +250.501 q^{51} -67.6562 q^{53} +(199.654 + 345.811i) q^{55} +297.749 q^{57} +(-145.557 + 252.113i) q^{59} +(-331.655 + 574.444i) q^{61} +(-36.5009 - 63.2215i) q^{63} +(-391.290 - 737.290i) q^{65} +(-212.551 - 368.149i) q^{67} +(258.905 - 448.436i) q^{69} +(-76.4815 + 132.470i) q^{71} +117.268 q^{73} +(-353.942 - 613.045i) q^{75} -121.948 q^{77} -202.462 q^{79} +(93.1932 + 161.415i) q^{81} -336.155 q^{83} +(605.329 - 1048.46i) q^{85} +(196.564 - 340.459i) q^{87} +(-359.097 - 621.974i) q^{89} +(254.750 + 9.09407i) q^{91} +(-508.769 - 881.214i) q^{93} +(719.503 - 1246.22i) q^{95} +(-379.684 + 657.632i) q^{97} -300.994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9} + 17 q^{11} + 125 q^{13} + 90 q^{15} - 70 q^{17} - 141 q^{19} + 126 q^{21} + 145 q^{23} + 150 q^{25} - 670 q^{27} - 34 q^{29} + 280 q^{31} - 425 q^{33} - 70 q^{35} + 190 q^{37} + 181 q^{39} - 538 q^{41} + 455 q^{43} - 375 q^{45} - 120 q^{47} + 565 q^{49} + 466 q^{51} + 1090 q^{53} + 510 q^{55} - 450 q^{57} - 809 q^{59} - 502 q^{61} + 390 q^{63} - 555 q^{65} - 475 q^{67} + 479 q^{69} + 127 q^{71} + 1170 q^{73} - 1725 q^{75} + 510 q^{77} - 480 q^{79} - 122 q^{81} - 520 q^{83} + 1205 q^{85} + 1615 q^{87} - 921 q^{89} + 1287 q^{91} - 2200 q^{93} + 1270 q^{95} + 415 q^{97} - 4420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −1.84233 3.19101i −0.354556 0.614110i 0.632486 0.774572i \(-0.282035\pi\)
−0.987042 + 0.160462i \(0.948701\pi\)
\(4\) 0 0
\(5\) −17.8078 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 2.71922 4.70983i 0.146824 0.254307i −0.783228 0.621735i \(-0.786428\pi\)
0.930052 + 0.367428i \(0.119761\pi\)
\(8\) 0 0
\(9\) 6.71165 11.6249i 0.248579 0.430552i
\(10\) 0 0
\(11\) −11.2116 19.4191i −0.307313 0.532281i 0.670461 0.741945i \(-0.266096\pi\)
−0.977774 + 0.209664i \(0.932763\pi\)
\(12\) 0 0
\(13\) 21.9730 + 41.4027i 0.468786 + 0.883312i
\(14\) 0 0
\(15\) 32.8078 + 56.8247i 0.564729 + 0.978139i
\(16\) 0 0
\(17\) −33.9924 + 58.8766i −0.484963 + 0.839981i −0.999851 0.0172769i \(-0.994500\pi\)
0.514888 + 0.857258i \(0.327834\pi\)
\(18\) 0 0
\(19\) −40.4039 + 69.9816i −0.487857 + 0.844993i −0.999902 0.0139650i \(-0.995555\pi\)
0.512045 + 0.858958i \(0.328888\pi\)
\(20\) 0 0
\(21\) −20.0388 −0.208230
\(22\) 0 0
\(23\) 70.2656 + 121.704i 0.637017 + 1.10335i 0.986084 + 0.166248i \(0.0531653\pi\)
−0.349067 + 0.937098i \(0.613501\pi\)
\(24\) 0 0
\(25\) 192.116 1.53693
\(26\) 0 0
\(27\) −148.946 −1.06165
\(28\) 0 0
\(29\) 53.3466 + 92.3990i 0.341594 + 0.591657i 0.984729 0.174095i \(-0.0557001\pi\)
−0.643135 + 0.765753i \(0.722367\pi\)
\(30\) 0 0
\(31\) 276.155 1.59997 0.799983 0.600023i \(-0.204842\pi\)
0.799983 + 0.600023i \(0.204842\pi\)
\(32\) 0 0
\(33\) −41.3111 + 71.5529i −0.217919 + 0.377447i
\(34\) 0 0
\(35\) −48.4233 + 83.8716i −0.233858 + 0.405054i
\(36\) 0 0
\(37\) 2.14584 + 3.71670i 0.00953442 + 0.0165141i 0.870753 0.491720i \(-0.163632\pi\)
−0.861219 + 0.508234i \(0.830298\pi\)
\(38\) 0 0
\(39\) 91.6349 146.394i 0.376239 0.601070i
\(40\) 0 0
\(41\) −113.884 197.254i −0.433799 0.751362i 0.563398 0.826186i \(-0.309494\pi\)
−0.997197 + 0.0748237i \(0.976161\pi\)
\(42\) 0 0
\(43\) 13.7647 23.8411i 0.0488162 0.0845521i −0.840585 0.541680i \(-0.817788\pi\)
0.889401 + 0.457128i \(0.151122\pi\)
\(44\) 0 0
\(45\) −119.519 + 207.014i −0.395931 + 0.685773i
\(46\) 0 0
\(47\) −318.617 −0.988832 −0.494416 0.869225i \(-0.664618\pi\)
−0.494416 + 0.869225i \(0.664618\pi\)
\(48\) 0 0
\(49\) 156.712 + 271.433i 0.456885 + 0.791348i
\(50\) 0 0
\(51\) 250.501 0.687787
\(52\) 0 0
\(53\) −67.6562 −0.175345 −0.0876726 0.996149i \(-0.527943\pi\)
−0.0876726 + 0.996149i \(0.527943\pi\)
\(54\) 0 0
\(55\) 199.654 + 345.811i 0.489480 + 0.847804i
\(56\) 0 0
\(57\) 297.749 0.691892
\(58\) 0 0
\(59\) −145.557 + 252.113i −0.321186 + 0.556310i −0.980733 0.195353i \(-0.937415\pi\)
0.659547 + 0.751663i \(0.270748\pi\)
\(60\) 0 0
\(61\) −331.655 + 574.444i −0.696133 + 1.20574i 0.273664 + 0.961825i \(0.411764\pi\)
−0.969797 + 0.243912i \(0.921569\pi\)
\(62\) 0 0
\(63\) −36.5009 63.2215i −0.0729950 0.126431i
\(64\) 0 0
\(65\) −391.290 737.290i −0.746670 1.40692i
\(66\) 0 0
\(67\) −212.551 368.149i −0.387570 0.671291i 0.604552 0.796566i \(-0.293352\pi\)
−0.992122 + 0.125275i \(0.960019\pi\)
\(68\) 0 0
\(69\) 258.905 448.436i 0.451717 0.782397i
\(70\) 0 0
\(71\) −76.4815 + 132.470i −0.127841 + 0.221427i −0.922840 0.385184i \(-0.874138\pi\)
0.794999 + 0.606611i \(0.207471\pi\)
\(72\) 0 0
\(73\) 117.268 0.188016 0.0940081 0.995571i \(-0.470032\pi\)
0.0940081 + 0.995571i \(0.470032\pi\)
\(74\) 0 0
\(75\) −353.942 613.045i −0.544929 0.943845i
\(76\) 0 0
\(77\) −121.948 −0.180484
\(78\) 0 0
\(79\) −202.462 −0.288339 −0.144169 0.989553i \(-0.546051\pi\)
−0.144169 + 0.989553i \(0.546051\pi\)
\(80\) 0 0
\(81\) 93.1932 + 161.415i 0.127837 + 0.221420i
\(82\) 0 0
\(83\) −336.155 −0.444552 −0.222276 0.974984i \(-0.571349\pi\)
−0.222276 + 0.974984i \(0.571349\pi\)
\(84\) 0 0
\(85\) 605.329 1048.46i 0.772437 1.33790i
\(86\) 0 0
\(87\) 196.564 340.459i 0.242228 0.419552i
\(88\) 0 0
\(89\) −359.097 621.974i −0.427688 0.740777i 0.568979 0.822352i \(-0.307338\pi\)
−0.996667 + 0.0815748i \(0.974005\pi\)
\(90\) 0 0
\(91\) 254.750 + 9.09407i 0.293462 + 0.0104760i
\(92\) 0 0
\(93\) −508.769 881.214i −0.567278 0.982555i
\(94\) 0 0
\(95\) 719.503 1246.22i 0.777047 1.34588i
\(96\) 0 0
\(97\) −379.684 + 657.632i −0.397434 + 0.688376i −0.993409 0.114628i \(-0.963433\pi\)
0.595975 + 0.803003i \(0.296766\pi\)
\(98\) 0 0
\(99\) −300.994 −0.305566
\(100\) 0 0
\(101\) 174.348 + 301.980i 0.171766 + 0.297507i 0.939037 0.343816i \(-0.111720\pi\)
−0.767272 + 0.641322i \(0.778386\pi\)
\(102\) 0 0
\(103\) 580.303 0.555136 0.277568 0.960706i \(-0.410472\pi\)
0.277568 + 0.960706i \(0.410472\pi\)
\(104\) 0 0
\(105\) 356.847 0.331663
\(106\) 0 0
\(107\) 285.747 + 494.928i 0.258170 + 0.447163i 0.965752 0.259468i \(-0.0835473\pi\)
−0.707582 + 0.706631i \(0.750214\pi\)
\(108\) 0 0
\(109\) 176.004 0.154661 0.0773307 0.997005i \(-0.475360\pi\)
0.0773307 + 0.997005i \(0.475360\pi\)
\(110\) 0 0
\(111\) 7.90668 13.6948i 0.00676098 0.0117104i
\(112\) 0 0
\(113\) −632.441 + 1095.42i −0.526505 + 0.911933i 0.473018 + 0.881053i \(0.343165\pi\)
−0.999523 + 0.0308807i \(0.990169\pi\)
\(114\) 0 0
\(115\) −1251.27 2167.27i −1.01462 1.75738i
\(116\) 0 0
\(117\) 628.778 + 22.4462i 0.496842 + 0.0177363i
\(118\) 0 0
\(119\) 184.866 + 320.197i 0.142409 + 0.246659i
\(120\) 0 0
\(121\) 414.098 717.239i 0.311118 0.538872i
\(122\) 0 0
\(123\) −419.625 + 726.812i −0.307613 + 0.532801i
\(124\) 0 0
\(125\) −1195.19 −0.855211
\(126\) 0 0
\(127\) 1302.05 + 2255.22i 0.909752 + 1.57574i 0.814408 + 0.580293i \(0.197062\pi\)
0.0953448 + 0.995444i \(0.469605\pi\)
\(128\) 0 0
\(129\) −101.436 −0.0692323
\(130\) 0 0
\(131\) −2131.70 −1.42174 −0.710870 0.703324i \(-0.751698\pi\)
−0.710870 + 0.703324i \(0.751698\pi\)
\(132\) 0 0
\(133\) 219.734 + 380.591i 0.143259 + 0.248131i
\(134\) 0 0
\(135\) 2652.40 1.69098
\(136\) 0 0
\(137\) −343.992 + 595.812i −0.214520 + 0.371560i −0.953124 0.302580i \(-0.902152\pi\)
0.738604 + 0.674140i \(0.235485\pi\)
\(138\) 0 0
\(139\) −339.790 + 588.534i −0.207343 + 0.359128i −0.950877 0.309570i \(-0.899815\pi\)
0.743534 + 0.668698i \(0.233148\pi\)
\(140\) 0 0
\(141\) 586.998 + 1016.71i 0.350597 + 0.607252i
\(142\) 0 0
\(143\) 557.652 890.890i 0.326106 0.520979i
\(144\) 0 0
\(145\) −949.983 1645.42i −0.544082 0.942377i
\(146\) 0 0
\(147\) 577.429 1000.14i 0.323983 0.561155i
\(148\) 0 0
\(149\) 987.731 1710.80i 0.543074 0.940632i −0.455651 0.890159i \(-0.650593\pi\)
0.998725 0.0504739i \(-0.0160732\pi\)
\(150\) 0 0
\(151\) −1803.24 −0.971824 −0.485912 0.874008i \(-0.661513\pi\)
−0.485912 + 0.874008i \(0.661513\pi\)
\(152\) 0 0
\(153\) 456.290 + 790.318i 0.241104 + 0.417604i
\(154\) 0 0
\(155\) −4917.71 −2.54839
\(156\) 0 0
\(157\) −397.168 −0.201894 −0.100947 0.994892i \(-0.532187\pi\)
−0.100947 + 0.994892i \(0.532187\pi\)
\(158\) 0 0
\(159\) 124.645 + 215.892i 0.0621698 + 0.107681i
\(160\) 0 0
\(161\) 764.272 0.374118
\(162\) 0 0
\(163\) 470.696 815.270i 0.226183 0.391760i −0.730491 0.682922i \(-0.760709\pi\)
0.956674 + 0.291162i \(0.0940420\pi\)
\(164\) 0 0
\(165\) 735.658 1274.20i 0.347096 0.601189i
\(166\) 0 0
\(167\) 1840.22 + 3187.35i 0.852696 + 1.47691i 0.878766 + 0.477252i \(0.158367\pi\)
−0.0260704 + 0.999660i \(0.508299\pi\)
\(168\) 0 0
\(169\) −1231.37 + 1819.49i −0.560479 + 0.828168i
\(170\) 0 0
\(171\) 542.353 + 939.383i 0.242543 + 0.420096i
\(172\) 0 0
\(173\) −711.387 + 1232.16i −0.312634 + 0.541499i −0.978932 0.204187i \(-0.934545\pi\)
0.666297 + 0.745686i \(0.267878\pi\)
\(174\) 0 0
\(175\) 522.408 904.837i 0.225659 0.390853i
\(176\) 0 0
\(177\) 1072.66 0.455514
\(178\) 0 0
\(179\) 583.946 + 1011.42i 0.243833 + 0.422331i 0.961803 0.273743i \(-0.0882617\pi\)
−0.717970 + 0.696074i \(0.754928\pi\)
\(180\) 0 0
\(181\) −1133.96 −0.465673 −0.232836 0.972516i \(-0.574801\pi\)
−0.232836 + 0.972516i \(0.574801\pi\)
\(182\) 0 0
\(183\) 2444.07 0.987274
\(184\) 0 0
\(185\) −38.2126 66.1861i −0.0151862 0.0263032i
\(186\) 0 0
\(187\) 1524.44 0.596141
\(188\) 0 0
\(189\) −405.018 + 701.511i −0.155877 + 0.269986i
\(190\) 0 0
\(191\) 1341.06 2322.78i 0.508040 0.879952i −0.491916 0.870642i \(-0.663703\pi\)
0.999957 0.00930919i \(-0.00296325\pi\)
\(192\) 0 0
\(193\) 985.333 + 1706.65i 0.367491 + 0.636514i 0.989173 0.146757i \(-0.0468834\pi\)
−0.621681 + 0.783270i \(0.713550\pi\)
\(194\) 0 0
\(195\) −1631.81 + 2606.94i −0.599265 + 0.957369i
\(196\) 0 0
\(197\) 2008.02 + 3478.00i 0.726222 + 1.25785i 0.958469 + 0.285197i \(0.0920590\pi\)
−0.232247 + 0.972657i \(0.574608\pi\)
\(198\) 0 0
\(199\) −2113.03 + 3659.87i −0.752707 + 1.30373i 0.193800 + 0.981041i \(0.437919\pi\)
−0.946506 + 0.322685i \(0.895415\pi\)
\(200\) 0 0
\(201\) −783.177 + 1356.50i −0.274831 + 0.476021i
\(202\) 0 0
\(203\) 580.245 0.200617
\(204\) 0 0
\(205\) 2028.03 + 3512.65i 0.690944 + 1.19675i
\(206\) 0 0
\(207\) 1886.39 0.633398
\(208\) 0 0
\(209\) 1811.98 0.599699
\(210\) 0 0
\(211\) 682.334 + 1181.84i 0.222625 + 0.385597i 0.955604 0.294654i \(-0.0952043\pi\)
−0.732980 + 0.680251i \(0.761871\pi\)
\(212\) 0 0
\(213\) 563.617 0.181307
\(214\) 0 0
\(215\) −245.118 + 424.557i −0.0777532 + 0.134672i
\(216\) 0 0
\(217\) 750.928 1300.65i 0.234914 0.406883i
\(218\) 0 0
\(219\) −216.046 374.203i −0.0666624 0.115463i
\(220\) 0 0
\(221\) −3184.57 113.683i −0.969309 0.0346025i
\(222\) 0 0
\(223\) −529.734 917.527i −0.159075 0.275525i 0.775461 0.631396i \(-0.217518\pi\)
−0.934535 + 0.355871i \(0.884184\pi\)
\(224\) 0 0
\(225\) 1289.42 2233.34i 0.382050 0.661729i
\(226\) 0 0
\(227\) 1732.10 3000.08i 0.506446 0.877190i −0.493526 0.869731i \(-0.664292\pi\)
0.999972 0.00745930i \(-0.00237439\pi\)
\(228\) 0 0
\(229\) −2324.64 −0.670815 −0.335407 0.942073i \(-0.608874\pi\)
−0.335407 + 0.942073i \(0.608874\pi\)
\(230\) 0 0
\(231\) 224.668 + 389.137i 0.0639917 + 0.110837i
\(232\) 0 0
\(233\) −3731.01 −1.04904 −0.524521 0.851398i \(-0.675755\pi\)
−0.524521 + 0.851398i \(0.675755\pi\)
\(234\) 0 0
\(235\) 5673.86 1.57499
\(236\) 0 0
\(237\) 373.002 + 646.058i 0.102232 + 0.177072i
\(238\) 0 0
\(239\) −6044.47 −1.63592 −0.817958 0.575278i \(-0.804894\pi\)
−0.817958 + 0.575278i \(0.804894\pi\)
\(240\) 0 0
\(241\) 2586.98 4480.78i 0.691461 1.19765i −0.279898 0.960030i \(-0.590301\pi\)
0.971359 0.237616i \(-0.0763659\pi\)
\(242\) 0 0
\(243\) −1667.39 + 2888.00i −0.440176 + 0.762408i
\(244\) 0 0
\(245\) −2790.68 4833.61i −0.727715 1.26044i
\(246\) 0 0
\(247\) −3785.22 135.125i −0.975093 0.0348090i
\(248\) 0 0
\(249\) 619.309 + 1072.67i 0.157619 + 0.273004i
\(250\) 0 0
\(251\) 2810.37 4867.70i 0.706728 1.22409i −0.259337 0.965787i \(-0.583504\pi\)
0.966064 0.258301i \(-0.0831628\pi\)
\(252\) 0 0
\(253\) 1575.59 2729.00i 0.391527 0.678144i
\(254\) 0 0
\(255\) −4460.86 −1.09549
\(256\) 0 0
\(257\) 837.070 + 1449.85i 0.203171 + 0.351903i 0.949549 0.313620i \(-0.101542\pi\)
−0.746377 + 0.665523i \(0.768209\pi\)
\(258\) 0 0
\(259\) 23.3401 0.00559954
\(260\) 0 0
\(261\) 1432.17 0.339653
\(262\) 0 0
\(263\) −3154.59 5463.91i −0.739622 1.28106i −0.952666 0.304020i \(-0.901671\pi\)
0.213044 0.977043i \(-0.431662\pi\)
\(264\) 0 0
\(265\) 1204.81 0.279285
\(266\) 0 0
\(267\) −1323.15 + 2291.76i −0.303279 + 0.525294i
\(268\) 0 0
\(269\) 1241.37 2150.11i 0.281366 0.487340i −0.690356 0.723470i \(-0.742546\pi\)
0.971721 + 0.236131i \(0.0758793\pi\)
\(270\) 0 0
\(271\) 1417.86 + 2455.81i 0.317819 + 0.550478i 0.980033 0.198837i \(-0.0637163\pi\)
−0.662214 + 0.749315i \(0.730383\pi\)
\(272\) 0 0
\(273\) −440.313 829.662i −0.0976153 0.183932i
\(274\) 0 0
\(275\) −2153.94 3730.74i −0.472318 0.818080i
\(276\) 0 0
\(277\) 1918.76 3323.38i 0.416198 0.720876i −0.579355 0.815075i \(-0.696696\pi\)
0.995553 + 0.0941989i \(0.0300290\pi\)
\(278\) 0 0
\(279\) 1853.46 3210.28i 0.397719 0.688869i
\(280\) 0 0
\(281\) −9122.13 −1.93659 −0.968293 0.249819i \(-0.919629\pi\)
−0.968293 + 0.249819i \(0.919629\pi\)
\(282\) 0 0
\(283\) 1063.92 + 1842.77i 0.223476 + 0.387072i 0.955861 0.293819i \(-0.0949262\pi\)
−0.732385 + 0.680891i \(0.761593\pi\)
\(284\) 0 0
\(285\) −5302.24 −1.10203
\(286\) 0 0
\(287\) −1238.71 −0.254769
\(288\) 0 0
\(289\) 145.530 + 252.066i 0.0296215 + 0.0513059i
\(290\) 0 0
\(291\) 2798.01 0.563651
\(292\) 0 0
\(293\) −4137.38 + 7166.16i −0.824944 + 1.42884i 0.0770183 + 0.997030i \(0.475460\pi\)
−0.901962 + 0.431815i \(0.857873\pi\)
\(294\) 0 0
\(295\) 2592.05 4489.56i 0.511576 0.886076i
\(296\) 0 0
\(297\) 1669.93 + 2892.40i 0.326260 + 0.565099i
\(298\) 0 0
\(299\) −3494.92 + 5583.38i −0.675974 + 1.07992i
\(300\) 0 0
\(301\) −74.8585 129.659i −0.0143348 0.0248286i
\(302\) 0 0
\(303\) 642.414 1112.69i 0.121801 0.210966i
\(304\) 0 0
\(305\) 5906.04 10229.6i 1.10878 1.92047i
\(306\) 0 0
\(307\) 3610.49 0.671211 0.335605 0.942003i \(-0.391059\pi\)
0.335605 + 0.942003i \(0.391059\pi\)
\(308\) 0 0
\(309\) −1069.11 1851.75i −0.196827 0.340914i
\(310\) 0 0
\(311\) −3331.06 −0.607354 −0.303677 0.952775i \(-0.598214\pi\)
−0.303677 + 0.952775i \(0.598214\pi\)
\(312\) 0 0
\(313\) −358.125 −0.0646724 −0.0323362 0.999477i \(-0.510295\pi\)
−0.0323362 + 0.999477i \(0.510295\pi\)
\(314\) 0 0
\(315\) 650.000 + 1125.83i 0.116265 + 0.201376i
\(316\) 0 0
\(317\) 3047.46 0.539944 0.269972 0.962868i \(-0.412986\pi\)
0.269972 + 0.962868i \(0.412986\pi\)
\(318\) 0 0
\(319\) 1196.21 2071.89i 0.209952 0.363647i
\(320\) 0 0
\(321\) 1052.88 1823.64i 0.183072 0.317089i
\(322\) 0 0
\(323\) −2746.85 4757.69i −0.473185 0.819581i
\(324\) 0 0
\(325\) 4221.38 + 7954.15i 0.720492 + 1.35759i
\(326\) 0 0
\(327\) −324.257 561.629i −0.0548362 0.0949791i
\(328\) 0 0
\(329\) −866.392 + 1500.63i −0.145185 + 0.251467i
\(330\) 0 0
\(331\) 3847.39 6663.87i 0.638887 1.10658i −0.346791 0.937942i \(-0.612729\pi\)
0.985677 0.168641i \(-0.0539380\pi\)
\(332\) 0 0
\(333\) 57.6084 0.00948025
\(334\) 0 0
\(335\) 3785.05 + 6555.90i 0.617312 + 1.06922i
\(336\) 0 0
\(337\) 4712.21 0.761693 0.380846 0.924638i \(-0.375633\pi\)
0.380846 + 0.924638i \(0.375633\pi\)
\(338\) 0 0
\(339\) 4660.66 0.746703
\(340\) 0 0
\(341\) −3096.16 5362.70i −0.491690 0.851632i
\(342\) 0 0
\(343\) 3569.92 0.561976
\(344\) 0 0
\(345\) −4610.52 + 7985.65i −0.719484 + 1.24618i
\(346\) 0 0
\(347\) −2630.99 + 4557.01i −0.407029 + 0.704995i −0.994555 0.104210i \(-0.966768\pi\)
0.587526 + 0.809205i \(0.300102\pi\)
\(348\) 0 0
\(349\) −25.1672 43.5909i −0.00386009 0.00668587i 0.864089 0.503339i \(-0.167895\pi\)
−0.867949 + 0.496653i \(0.834562\pi\)
\(350\) 0 0
\(351\) −3272.79 6166.77i −0.497689 0.937772i
\(352\) 0 0
\(353\) −4528.82 7844.14i −0.682846 1.18272i −0.974109 0.226081i \(-0.927409\pi\)
0.291263 0.956643i \(-0.405925\pi\)
\(354\) 0 0
\(355\) 1361.96 2358.99i 0.203621 0.352683i
\(356\) 0 0
\(357\) 681.168 1179.82i 0.100984 0.174909i
\(358\) 0 0
\(359\) −7177.86 −1.05525 −0.527623 0.849479i \(-0.676917\pi\)
−0.527623 + 0.849479i \(0.676917\pi\)
\(360\) 0 0
\(361\) 164.553 + 285.014i 0.0239908 + 0.0415532i
\(362\) 0 0
\(363\) −3051.62 −0.441235
\(364\) 0 0
\(365\) −2088.28 −0.299467
\(366\) 0 0
\(367\) 2002.07 + 3467.69i 0.284761 + 0.493221i 0.972551 0.232689i \(-0.0747524\pi\)
−0.687790 + 0.725910i \(0.741419\pi\)
\(368\) 0 0
\(369\) −3057.41 −0.431334
\(370\) 0 0
\(371\) −183.972 + 318.649i −0.0257449 + 0.0445915i
\(372\) 0 0
\(373\) 5007.09 8672.53i 0.695060 1.20388i −0.275101 0.961415i \(-0.588711\pi\)
0.970161 0.242464i \(-0.0779555\pi\)
\(374\) 0 0
\(375\) 2201.94 + 3813.87i 0.303221 + 0.525194i
\(376\) 0 0
\(377\) −2653.39 + 4238.98i −0.362484 + 0.579094i
\(378\) 0 0
\(379\) −4084.56 7074.66i −0.553587 0.958842i −0.998012 0.0630252i \(-0.979925\pi\)
0.444425 0.895816i \(-0.353408\pi\)
\(380\) 0 0
\(381\) 4797.62 8309.73i 0.645117 1.11738i
\(382\) 0 0
\(383\) −3655.12 + 6330.86i −0.487645 + 0.844626i −0.999899 0.0142079i \(-0.995477\pi\)
0.512254 + 0.858834i \(0.328811\pi\)
\(384\) 0 0
\(385\) 2171.62 0.287470
\(386\) 0 0
\(387\) −184.767 320.027i −0.0242694 0.0420358i
\(388\) 0 0
\(389\) 8785.47 1.14509 0.572546 0.819872i \(-0.305956\pi\)
0.572546 + 0.819872i \(0.305956\pi\)
\(390\) 0 0
\(391\) −9553.99 −1.23572
\(392\) 0 0
\(393\) 3927.30 + 6802.29i 0.504087 + 0.873104i
\(394\) 0 0
\(395\) 3605.40 0.459259
\(396\) 0 0
\(397\) −5633.40 + 9757.33i −0.712171 + 1.23352i 0.251869 + 0.967761i \(0.418955\pi\)
−0.964040 + 0.265756i \(0.914379\pi\)
\(398\) 0 0
\(399\) 809.646 1402.35i 0.101586 0.175953i
\(400\) 0 0
\(401\) −788.117 1365.06i −0.0981464 0.169995i 0.812771 0.582583i \(-0.197958\pi\)
−0.910917 + 0.412589i \(0.864625\pi\)
\(402\) 0 0
\(403\) 6067.96 + 11433.6i 0.750042 + 1.41327i
\(404\) 0 0
\(405\) −1659.56 2874.45i −0.203616 0.352672i
\(406\) 0 0
\(407\) 48.1168 83.3407i 0.00586010 0.0101500i
\(408\) 0 0
\(409\) −3377.89 + 5850.68i −0.408377 + 0.707329i −0.994708 0.102742i \(-0.967238\pi\)
0.586331 + 0.810071i \(0.300572\pi\)
\(410\) 0 0
\(411\) 2534.99 0.304238
\(412\) 0 0
\(413\) 791.606 + 1371.10i 0.0943157 + 0.163360i
\(414\) 0 0
\(415\) 5986.17 0.708072
\(416\) 0 0
\(417\) 2504.02 0.294059
\(418\) 0 0
\(419\) 5378.09 + 9315.13i 0.627057 + 1.08610i 0.988139 + 0.153561i \(0.0490742\pi\)
−0.361082 + 0.932534i \(0.617592\pi\)
\(420\) 0 0
\(421\) 7886.03 0.912925 0.456463 0.889743i \(-0.349116\pi\)
0.456463 + 0.889743i \(0.349116\pi\)
\(422\) 0 0
\(423\) −2138.45 + 3703.90i −0.245803 + 0.425744i
\(424\) 0 0
\(425\) −6530.50 + 11311.2i −0.745355 + 1.29099i
\(426\) 0 0
\(427\) 1803.69 + 3124.08i 0.204418 + 0.354063i
\(428\) 0 0
\(429\) −3870.22 138.159i −0.435561 0.0155487i
\(430\) 0 0
\(431\) −7042.31 12197.6i −0.787044 1.36320i −0.927770 0.373152i \(-0.878277\pi\)
0.140726 0.990049i \(-0.455056\pi\)
\(432\) 0 0
\(433\) −932.072 + 1614.40i −0.103447 + 0.179175i −0.913103 0.407730i \(-0.866321\pi\)
0.809656 + 0.586905i \(0.199654\pi\)
\(434\) 0 0
\(435\) −3500.36 + 6062.81i −0.385815 + 0.668252i
\(436\) 0 0
\(437\) −11356.0 −1.24309
\(438\) 0 0
\(439\) 3077.24 + 5329.94i 0.334553 + 0.579463i 0.983399 0.181457i \(-0.0580813\pi\)
−0.648846 + 0.760920i \(0.724748\pi\)
\(440\) 0 0
\(441\) 4207.17 0.454289
\(442\) 0 0
\(443\) 14539.3 1.55933 0.779663 0.626200i \(-0.215391\pi\)
0.779663 + 0.626200i \(0.215391\pi\)
\(444\) 0 0
\(445\) 6394.72 + 11076.0i 0.681210 + 1.17989i
\(446\) 0 0
\(447\) −7278.90 −0.770202
\(448\) 0 0
\(449\) 3521.93 6100.17i 0.370179 0.641169i −0.619414 0.785065i \(-0.712630\pi\)
0.989593 + 0.143896i \(0.0459630\pi\)
\(450\) 0 0
\(451\) −2553.66 + 4423.08i −0.266624 + 0.461806i
\(452\) 0 0
\(453\) 3322.16 + 5754.15i 0.344567 + 0.596807i
\(454\) 0 0
\(455\) −4536.52 161.945i −0.467418 0.0166859i
\(456\) 0 0
\(457\) −7049.43 12210.0i −0.721572 1.24980i −0.960370 0.278730i \(-0.910087\pi\)
0.238798 0.971069i \(-0.423247\pi\)
\(458\) 0 0
\(459\) 5063.04 8769.44i 0.514863 0.891770i
\(460\) 0 0
\(461\) −7224.85 + 12513.8i −0.729924 + 1.26426i 0.226991 + 0.973897i \(0.427111\pi\)
−0.956915 + 0.290368i \(0.906222\pi\)
\(462\) 0 0
\(463\) −15806.5 −1.58659 −0.793293 0.608840i \(-0.791635\pi\)
−0.793293 + 0.608840i \(0.791635\pi\)
\(464\) 0 0
\(465\) 9060.04 + 15692.4i 0.903547 + 1.56499i
\(466\) 0 0
\(467\) 15071.3 1.49340 0.746699 0.665162i \(-0.231638\pi\)
0.746699 + 0.665162i \(0.231638\pi\)
\(468\) 0 0
\(469\) −2311.89 −0.227619
\(470\) 0 0
\(471\) 731.713 + 1267.36i 0.0715830 + 0.123985i
\(472\) 0 0
\(473\) −617.299 −0.0600073
\(474\) 0 0
\(475\) −7762.25 + 13444.6i −0.749803 + 1.29870i
\(476\) 0 0
\(477\) −454.085 + 786.498i −0.0435872 + 0.0754953i
\(478\) 0 0
\(479\) −196.272 339.954i −0.0187222 0.0324277i 0.856513 0.516126i \(-0.172626\pi\)
−0.875235 + 0.483698i \(0.839293\pi\)
\(480\) 0 0
\(481\) −106.731 + 170.511i −0.0101175 + 0.0161634i
\(482\) 0 0
\(483\) −1408.04 2438.80i −0.132646 0.229750i
\(484\) 0 0
\(485\) 6761.33 11711.0i 0.633023 1.09643i
\(486\) 0 0
\(487\) 4748.94 8225.41i 0.441879 0.765357i −0.555950 0.831216i \(-0.687645\pi\)
0.997829 + 0.0658588i \(0.0209787\pi\)
\(488\) 0 0
\(489\) −3468.71 −0.320778
\(490\) 0 0
\(491\) −946.912 1640.10i −0.0870337 0.150747i 0.819222 0.573476i \(-0.194405\pi\)
−0.906256 + 0.422729i \(0.861072\pi\)
\(492\) 0 0
\(493\) −7253.52 −0.662641
\(494\) 0 0
\(495\) 5360.04 0.486699
\(496\) 0 0
\(497\) 415.941 + 720.430i 0.0375402 + 0.0650216i
\(498\) 0 0
\(499\) 13370.1 1.19945 0.599727 0.800205i \(-0.295276\pi\)
0.599727 + 0.800205i \(0.295276\pi\)
\(500\) 0 0
\(501\) 6780.57 11744.3i 0.604658 1.04730i
\(502\) 0 0
\(503\) 2777.36 4810.52i 0.246195 0.426423i −0.716272 0.697822i \(-0.754153\pi\)
0.962467 + 0.271399i \(0.0874862\pi\)
\(504\) 0 0
\(505\) −3104.76 5377.60i −0.273584 0.473861i
\(506\) 0 0
\(507\) 8074.59 + 577.231i 0.707308 + 0.0505635i
\(508\) 0 0
\(509\) −1098.78 1903.13i −0.0956824 0.165727i 0.814211 0.580569i \(-0.197170\pi\)
−0.909893 + 0.414843i \(0.863837\pi\)
\(510\) 0 0
\(511\) 318.878 552.313i 0.0276053 0.0478139i
\(512\) 0 0
\(513\) 6018.00 10423.5i 0.517936 0.897091i
\(514\) 0 0
\(515\) −10333.9 −0.884206
\(516\) 0 0
\(517\) 3572.23 + 6187.28i 0.303881 + 0.526337i
\(518\) 0 0
\(519\) 5242.44 0.443386
\(520\) 0 0
\(521\) 17005.2 1.42997 0.714983 0.699142i \(-0.246435\pi\)
0.714983 + 0.699142i \(0.246435\pi\)
\(522\) 0 0
\(523\) −7243.11 12545.4i −0.605581 1.04890i −0.991959 0.126557i \(-0.959607\pi\)
0.386378 0.922341i \(-0.373726\pi\)
\(524\) 0 0
\(525\) −3849.79 −0.320035
\(526\) 0 0
\(527\) −9387.19 + 16259.1i −0.775925 + 1.34394i
\(528\) 0 0
\(529\) −3791.02 + 6566.23i −0.311582 + 0.539675i
\(530\) 0 0
\(531\) 1953.86 + 3384.18i 0.159680 + 0.276574i
\(532\) 0 0
\(533\) 5664.46 9049.39i 0.460328 0.735408i
\(534\) 0 0
\(535\) −5088.51 8813.56i −0.411206 0.712230i
\(536\) 0 0
\(537\) 2151.64 3726.75i 0.172905 0.299481i
\(538\) 0 0
\(539\) 3513.99 6086.41i 0.280813 0.486383i
\(540\) 0 0
\(541\) −15266.7 −1.21325 −0.606623 0.794990i \(-0.707476\pi\)
−0.606623 + 0.794990i \(0.707476\pi\)
\(542\) 0 0
\(543\) 2089.13 + 3618.48i 0.165107 + 0.285974i
\(544\) 0 0
\(545\) −3134.23 −0.246341
\(546\) 0 0
\(547\) −15260.5 −1.19286 −0.596430 0.802665i \(-0.703414\pi\)
−0.596430 + 0.802665i \(0.703414\pi\)
\(548\) 0 0
\(549\) 4451.91 + 7710.93i 0.346089 + 0.599443i
\(550\) 0 0
\(551\) −8621.64 −0.666595
\(552\) 0 0
\(553\) −550.540 + 953.563i −0.0423351 + 0.0733266i
\(554\) 0 0
\(555\) −140.800 + 243.873i −0.0107687 + 0.0186520i
\(556\) 0 0
\(557\) 5221.05 + 9043.12i 0.397169 + 0.687916i 0.993375 0.114915i \(-0.0366595\pi\)
−0.596207 + 0.802831i \(0.703326\pi\)
\(558\) 0 0
\(559\) 1289.54 + 46.0341i 0.0975702 + 0.00348307i
\(560\) 0 0
\(561\) −2808.53 4864.51i −0.211366 0.366096i
\(562\) 0 0
\(563\) 3572.63 6187.98i 0.267440 0.463219i −0.700760 0.713397i \(-0.747156\pi\)
0.968200 + 0.250178i \(0.0804891\pi\)
\(564\) 0 0
\(565\) 11262.4 19507.0i 0.838604 1.45250i
\(566\) 0 0
\(567\) 1013.65 0.0750783
\(568\) 0 0
\(569\) −2219.43 3844.17i −0.163521 0.283226i 0.772608 0.634883i \(-0.218952\pi\)
−0.936129 + 0.351657i \(0.885618\pi\)
\(570\) 0 0
\(571\) −10117.3 −0.741497 −0.370748 0.928733i \(-0.620899\pi\)
−0.370748 + 0.928733i \(0.620899\pi\)
\(572\) 0 0
\(573\) −9882.70 −0.720516
\(574\) 0 0
\(575\) 13499.2 + 23381.3i 0.979052 + 1.69577i
\(576\) 0 0
\(577\) 3105.60 0.224069 0.112035 0.993704i \(-0.464263\pi\)
0.112035 + 0.993704i \(0.464263\pi\)
\(578\) 0 0
\(579\) 3630.62 6288.41i 0.260593 0.451360i
\(580\) 0 0
\(581\) −914.081 + 1583.24i −0.0652711 + 0.113053i
\(582\) 0 0
\(583\) 758.538 + 1313.83i 0.0538858 + 0.0933329i
\(584\) 0 0
\(585\) −11197.1 399.716i −0.791358 0.0282500i
\(586\) 0 0
\(587\) 9831.16 + 17028.1i 0.691270 + 1.19731i 0.971422 + 0.237359i \(0.0762818\pi\)
−0.280152 + 0.959956i \(0.590385\pi\)
\(588\) 0 0
\(589\) −11157.7 + 19325.8i −0.780555 + 1.35196i
\(590\) 0 0
\(591\) 7398.88 12815.2i 0.514974 0.891960i
\(592\) 0 0
\(593\) 6395.51 0.442888 0.221444 0.975173i \(-0.428923\pi\)
0.221444 + 0.975173i \(0.428923\pi\)
\(594\) 0 0
\(595\) −3292.05 5702.00i −0.226825 0.392872i
\(596\) 0 0
\(597\) 15571.6 1.06751
\(598\) 0 0
\(599\) −8878.48 −0.605618 −0.302809 0.953051i \(-0.597924\pi\)
−0.302809 + 0.953051i \(0.597924\pi\)
\(600\) 0 0
\(601\) −9550.29 16541.6i −0.648194 1.12270i −0.983554 0.180615i \(-0.942191\pi\)
0.335360 0.942090i \(-0.391142\pi\)
\(602\) 0 0
\(603\) −5706.26 −0.385368
\(604\) 0 0
\(605\) −7374.16 + 12772.4i −0.495541 + 0.858302i
\(606\) 0 0
\(607\) 8297.88 14372.4i 0.554861 0.961047i −0.443053 0.896495i \(-0.646105\pi\)
0.997914 0.0645522i \(-0.0205619\pi\)
\(608\) 0 0
\(609\) −1069.00 1851.57i −0.0711300 0.123201i
\(610\) 0 0
\(611\) −7000.98 13191.6i −0.463551 0.873447i
\(612\) 0 0
\(613\) −8234.58 14262.7i −0.542564 0.939748i −0.998756 0.0498668i \(-0.984120\pi\)
0.456192 0.889881i \(-0.349213\pi\)
\(614\) 0 0
\(615\) 7472.59 12942.9i 0.489958 0.848631i
\(616\) 0 0
\(617\) −5057.99 + 8760.69i −0.330027 + 0.571624i −0.982517 0.186174i \(-0.940391\pi\)
0.652489 + 0.757798i \(0.273725\pi\)
\(618\) 0 0
\(619\) −18854.8 −1.22430 −0.612148 0.790743i \(-0.709694\pi\)
−0.612148 + 0.790743i \(0.709694\pi\)
\(620\) 0 0
\(621\) −10465.8 18127.3i −0.676292 1.17137i
\(622\) 0 0
\(623\) −3905.86 −0.251180
\(624\) 0 0
\(625\) −2730.82 −0.174773
\(626\) 0 0
\(627\) −3338.26 5782.03i −0.212627 0.368281i
\(628\) 0 0
\(629\) −291.769 −0.0184954
\(630\) 0 0
\(631\) 9473.12 16407.9i 0.597653 1.03517i −0.395514 0.918460i \(-0.629433\pi\)
0.993167 0.116705i \(-0.0372333\pi\)
\(632\) 0 0
\(633\) 2514.17 4354.66i 0.157866 0.273432i
\(634\) 0 0
\(635\) −23186.7 40160.5i −1.44903 2.50979i
\(636\) 0 0
\(637\) −7794.62 + 12452.5i −0.484826 + 0.774545i
\(638\) 0 0
\(639\) 1026.63 + 1778.18i 0.0635571 + 0.110084i
\(640\) 0 0
\(641\) 11793.5 20426.9i 0.726698 1.25868i −0.231573 0.972818i \(-0.574387\pi\)
0.958271 0.285861i \(-0.0922795\pi\)
\(642\) 0 0
\(643\) −13576.5 + 23515.2i −0.832669 + 1.44222i 0.0632461 + 0.997998i \(0.479855\pi\)
−0.895915 + 0.444226i \(0.853479\pi\)
\(644\) 0 0
\(645\) 1806.35 0.110272
\(646\) 0 0
\(647\) 3428.36 + 5938.09i 0.208319 + 0.360820i 0.951185 0.308620i \(-0.0998673\pi\)
−0.742866 + 0.669440i \(0.766534\pi\)
\(648\) 0 0
\(649\) 6527.75 0.394817
\(650\) 0 0
\(651\) −5533.83 −0.333161
\(652\) 0 0
\(653\) 4036.95 + 6992.20i 0.241926 + 0.419029i 0.961263 0.275633i \(-0.0888874\pi\)
−0.719337 + 0.694662i \(0.755554\pi\)
\(654\) 0 0
\(655\) 37960.9 2.26451
\(656\) 0 0
\(657\) 787.061 1363.23i 0.0467370 0.0809508i
\(658\) 0 0
\(659\) 2652.86 4594.89i 0.156815 0.271611i −0.776904 0.629620i \(-0.783211\pi\)
0.933718 + 0.358008i \(0.116544\pi\)
\(660\) 0 0
\(661\) −12924.2 22385.3i −0.760502 1.31723i −0.942592 0.333946i \(-0.891620\pi\)
0.182091 0.983282i \(-0.441714\pi\)
\(662\) 0 0
\(663\) 5504.26 + 10371.4i 0.322425 + 0.607531i
\(664\) 0 0
\(665\) −3912.98 6777.48i −0.228179 0.395217i
\(666\) 0 0
\(667\) −7496.86 + 12984.9i −0.435202 + 0.753792i
\(668\) 0 0
\(669\) −1951.89 + 3380.77i −0.112802 + 0.195379i
\(670\) 0 0
\(671\) 14873.6 0.855722
\(672\) 0 0
\(673\) 7264.55 + 12582.6i 0.416089 + 0.720687i 0.995542 0.0943186i \(-0.0300673\pi\)
−0.579453 + 0.815005i \(0.696734\pi\)
\(674\) 0 0
\(675\) −28615.0 −1.63169
\(676\) 0 0
\(677\) 12058.1 0.684535 0.342267 0.939603i \(-0.388805\pi\)
0.342267 + 0.939603i \(0.388805\pi\)
\(678\) 0 0
\(679\) 2064.89 + 3576.50i 0.116706 + 0.202140i
\(680\) 0 0
\(681\) −12764.4 −0.718255
\(682\) 0 0
\(683\) 15014.4 26005.7i 0.841156 1.45693i −0.0477615 0.998859i \(-0.515209\pi\)
0.888918 0.458067i \(-0.151458\pi\)
\(684\) 0 0
\(685\) 6125.74 10610.1i 0.341682 0.591811i
\(686\) 0 0
\(687\) 4282.75 + 7417.94i 0.237842 + 0.411954i
\(688\) 0 0
\(689\) −1486.61 2801.15i −0.0821994 0.154884i
\(690\) 0 0
\(691\) 224.848 + 389.448i 0.0123786 + 0.0214404i 0.872148 0.489241i \(-0.162726\pi\)
−0.859770 + 0.510682i \(0.829393\pi\)
\(692\) 0 0
\(693\) −818.471 + 1417.63i −0.0448646 + 0.0777077i
\(694\) 0 0
\(695\) 6050.90 10480.5i 0.330250 0.572010i
\(696\) 0 0
\(697\) 15484.8 0.841506
\(698\) 0 0
\(699\) 6873.75 + 11905.7i 0.371944 + 0.644227i
\(700\) 0 0
\(701\) −26986.0 −1.45399 −0.726994 0.686644i \(-0.759083\pi\)
−0.726994 + 0.686644i \(0.759083\pi\)
\(702\) 0 0
\(703\) −346.801 −0.0186057
\(704\) 0 0
\(705\) −10453.1 18105.3i −0.558422 0.967215i
\(706\) 0 0
\(707\) 1896.37 0.100877
\(708\) 0 0
\(709\) 4549.44 7879.85i 0.240984 0.417396i −0.720011 0.693963i \(-0.755863\pi\)
0.960995 + 0.276566i \(0.0891965\pi\)
\(710\) 0 0
\(711\) −1358.85 + 2353.60i −0.0716751 + 0.124145i
\(712\) 0 0
\(713\) 19404.2 + 33609.1i 1.01921 + 1.76532i
\(714\) 0 0
\(715\) −9930.53 + 15864.8i −0.519414 + 0.829802i
\(716\) 0 0
\(717\) 11135.9 + 19287.9i 0.580025 + 1.00463i
\(718\) 0 0
\(719\) −3146.78 + 5450.38i −0.163220 + 0.282705i −0.936022 0.351942i \(-0.885521\pi\)
0.772802 + 0.634647i \(0.218855\pi\)
\(720\) 0 0
\(721\) 1577.97 2733.13i 0.0815074 0.141175i
\(722\) 0 0
\(723\) −19064.3 −0.980648
\(724\) 0 0
\(725\) 10248.8 + 17751.4i 0.525006 + 0.909337i
\(726\) 0 0
\(727\) −18070.7 −0.921878 −0.460939 0.887432i \(-0.652487\pi\)
−0.460939 + 0.887432i \(0.652487\pi\)
\(728\) 0 0
\(729\) 17319.9 0.879944
\(730\) 0 0
\(731\) 935.790 + 1620.84i 0.0473481 + 0.0820093i
\(732\) 0 0
\(733\) 34771.5 1.75214 0.876068 0.482188i \(-0.160158\pi\)
0.876068 + 0.482188i \(0.160158\pi\)
\(734\) 0 0
\(735\) −10282.7 + 17810.2i −0.516032 + 0.893794i
\(736\) 0 0
\(737\) −4766.09 + 8255.10i −0.238210 + 0.412592i
\(738\) 0 0
\(739\) 11815.7 + 20465.4i 0.588158 + 1.01872i 0.994474 + 0.104986i \(0.0334799\pi\)
−0.406316 + 0.913733i \(0.633187\pi\)
\(740\) 0 0
\(741\) 6542.44 + 12327.6i 0.324349 + 0.611156i
\(742\) 0 0
\(743\) 16251.4 + 28148.3i 0.802431 + 1.38985i 0.918012 + 0.396553i \(0.129794\pi\)
−0.115581 + 0.993298i \(0.536873\pi\)
\(744\) 0 0
\(745\) −17589.3 + 30465.5i −0.864995 + 1.49822i
\(746\) 0 0
\(747\) −2256.16 + 3907.78i −0.110507 + 0.191403i
\(748\) 0 0
\(749\) 3108.04 0.151622
\(750\) 0 0
\(751\) 1010.43 + 1750.12i 0.0490960 + 0.0850368i 0.889529 0.456879i \(-0.151033\pi\)
−0.840433 + 0.541915i \(0.817699\pi\)
\(752\) 0 0
\(753\) −20710.5 −1.00230
\(754\) 0 0
\(755\) 32111.6 1.54790
\(756\) 0 0
\(757\) −6284.11 10884.4i −0.301717 0.522589i 0.674808 0.737993i \(-0.264226\pi\)
−0.976525 + 0.215404i \(0.930893\pi\)
\(758\) 0 0
\(759\) −11611.0 −0.555273
\(760\) 0 0
\(761\) 4352.40 7538.59i 0.207325 0.359098i −0.743546 0.668685i \(-0.766857\pi\)
0.950871 + 0.309587i \(0.100191\pi\)
\(762\) 0 0
\(763\) 478.593 828.948i 0.0227081 0.0393315i
\(764\) 0 0
\(765\) −8125.51 14073.8i −0.384024 0.665149i
\(766\) 0 0
\(767\) −13636.5 486.797i −0.641962 0.0229168i
\(768\) 0 0
\(769\) 10957.9 + 18979.7i 0.513853 + 0.890020i 0.999871 + 0.0160706i \(0.00511566\pi\)
−0.486018 + 0.873949i \(0.661551\pi\)
\(770\) 0 0
\(771\) 3084.32 5342.19i 0.144071 0.249539i
\(772\) 0 0
\(773\) 11538.8 19985.7i 0.536896 0.929930i −0.462173 0.886790i \(-0.652930\pi\)
0.999069 0.0431408i \(-0.0137364\pi\)
\(774\) 0 0
\(775\) 53054.0 2.45904
\(776\) 0 0
\(777\) −43.0001 74.4783i −0.00198535 0.00343873i
\(778\) 0 0
\(779\) 18405.5 0.846528
\(780\) 0 0
\(781\) 3429.94 0.157148
\(782\) 0 0
\(783\) −7945.76 13762.5i −0.362654 0.628136i
\(784\) 0 0
\(785\) 7072.67 0.321572
\(786\) 0 0
\(787\) −8261.21 + 14308.8i −0.374181 + 0.648100i −0.990204 0.139628i \(-0.955409\pi\)
0.616023 + 0.787728i \(0.288743\pi\)
\(788\) 0 0
\(789\) −11623.6 + 20132.7i −0.524475 + 0.908418i
\(790\) 0 0
\(791\) 3439.50 + 5957.39i 0.154607 + 0.267788i
\(792\) 0 0
\(793\) −31071.0 1109.18i −1.39138 0.0496696i
\(794\) 0 0
\(795\) −2219.65 3844.55i −0.0990224 0.171512i
\(796\) 0 0
\(797\) 5859.68 10149.3i 0.260427 0.451073i −0.705928 0.708283i \(-0.749470\pi\)
0.966356 + 0.257210i \(0.0828032\pi\)
\(798\) 0 0
\(799\) 10830.6 18759.1i 0.479547 0.830600i
\(800\) 0 0
\(801\) −9640.53 −0.425258
\(802\) 0 0
\(803\) −1314.77 2277.24i −0.0577797 0.100077i
\(804\) 0 0
\(805\) −13610.0 −0.595886
\(806\) 0 0
\(807\) −9148.01 −0.399040
\(808\) 0 0
\(809\) 12048.0 + 20867.8i 0.523592 + 0.906888i 0.999623 + 0.0274594i \(0.00874171\pi\)
−0.476031 + 0.879429i \(0.657925\pi\)
\(810\) 0 0
\(811\) −16622.6 −0.719729 −0.359864 0.933005i \(-0.617177\pi\)
−0.359864 + 0.933005i \(0.617177\pi\)
\(812\) 0 0
\(813\) 5224.33 9048.81i 0.225369 0.390351i
\(814\) 0 0
\(815\) −8382.05 + 14518.1i −0.360258 + 0.623985i
\(816\) 0 0
\(817\) 1112.29 + 1926.55i 0.0476306 + 0.0824987i
\(818\) 0 0
\(819\) 1815.51 2900.40i 0.0774590 0.123746i
\(820\) 0 0
\(821\) 19002.8 + 32913.8i 0.807797 + 1.39915i 0.914387 + 0.404842i \(0.132674\pi\)
−0.106590 + 0.994303i \(0.533993\pi\)
\(822\) 0 0
\(823\) −7929.75 + 13734.7i −0.335861 + 0.581728i −0.983650 0.180092i \(-0.942361\pi\)
0.647789 + 0.761820i \(0.275694\pi\)
\(824\) 0 0
\(825\) −7936.54 + 13746.5i −0.334927 + 0.580111i
\(826\) 0 0
\(827\) −12201.0 −0.513023 −0.256512 0.966541i \(-0.582573\pi\)
−0.256512 + 0.966541i \(0.582573\pi\)
\(828\) 0 0
\(829\) 2715.71 + 4703.74i 0.113776 + 0.197066i 0.917290 0.398220i \(-0.130372\pi\)
−0.803514 + 0.595286i \(0.797039\pi\)
\(830\) 0 0
\(831\) −14139.9 −0.590263
\(832\) 0 0
\(833\) −21308.0 −0.886290
\(834\) 0 0
\(835\) −32770.1 56759.5i −1.35815 2.35239i
\(836\) 0 0
\(837\) −41132.2 −1.69861
\(838\) 0 0
\(839\) −3980.45 + 6894.34i −0.163791 + 0.283694i −0.936225 0.351401i \(-0.885706\pi\)
0.772434 + 0.635095i \(0.219039\pi\)
\(840\) 0 0
\(841\) 6502.78 11263.2i 0.266628 0.461813i
\(842\) 0 0
\(843\) 16806.0 + 29108.8i 0.686629 + 1.18928i
\(844\) 0 0
\(845\) 21928.0 32401.0i 0.892718 1.31909i
\(846\) 0 0
\(847\) −2252.05 3900.67i −0.0913593 0.158239i
\(848\) 0 0
\(849\) 3920.20 6789.98i 0.158470 0.274478i
\(850\) 0 0
\(851\) −301.557 + 522.313i −0.0121472 + 0.0210395i
\(852\) 0 0
\(853\) 13576.7 0.544969 0.272485 0.962160i \(-0.412155\pi\)
0.272485 + 0.962160i \(0.412155\pi\)
\(854\) 0 0
\(855\) −9658.10 16728.3i −0.386316 0.669118i
\(856\) 0 0
\(857\) −31223.9 −1.24456 −0.622281 0.782794i \(-0.713794\pi\)
−0.622281 + 0.782794i \(0.713794\pi\)
\(858\) 0 0
\(859\) 11815.8 0.469323 0.234661 0.972077i \(-0.424602\pi\)
0.234661 + 0.972077i \(0.424602\pi\)
\(860\) 0 0
\(861\) 2282.11 + 3952.73i 0.0903300 + 0.156456i
\(862\) 0 0
\(863\) 1790.84 0.0706384 0.0353192 0.999376i \(-0.488755\pi\)
0.0353192 + 0.999376i \(0.488755\pi\)
\(864\) 0 0
\(865\) 12668.2 21942.0i 0.497956 0.862485i
\(866\) 0 0
\(867\) 536.230 928.777i 0.0210050 0.0363817i
\(868\) 0 0
\(869\) 2269.93 + 3931.64i 0.0886102 + 0.153477i
\(870\) 0 0
\(871\) 10572.0 16889.5i 0.411272 0.657037i
\(872\) 0 0
\(873\) 5096.61 + 8827.59i 0.197588 + 0.342232i
\(874\) 0 0
\(875\) −3250.00 + 5629.17i −0.125566 + 0.217486i
\(876\) 0 0
\(877\) −21771.2 + 37708.9i −0.838270 + 1.45193i 0.0530701 + 0.998591i \(0.483099\pi\)
−0.891340 + 0.453335i \(0.850234\pi\)
\(878\) 0 0
\(879\) 30489.7 1.16996
\(880\) 0 0
\(881\) 510.020 + 883.380i 0.0195040 + 0.0337819i 0.875613 0.483014i \(-0.160458\pi\)
−0.856109 + 0.516796i \(0.827125\pi\)
\(882\) 0 0
\(883\) 34781.9 1.32560 0.662800 0.748797i \(-0.269368\pi\)
0.662800 + 0.748797i \(0.269368\pi\)
\(884\) 0 0
\(885\) −19101.6 −0.725531
\(886\) 0 0
\(887\) 24892.5 + 43115.2i 0.942288 + 1.63209i 0.761091 + 0.648645i \(0.224664\pi\)
0.181197 + 0.983447i \(0.442003\pi\)
\(888\) 0 0
\(889\) 14162.3 0.534295
\(890\) 0 0
\(891\) 2089.70 3619.46i 0.0785718 0.136090i
\(892\) 0 0
\(893\) 12873.4 22297.3i 0.482409 0.835557i
\(894\) 0 0
\(895\) −10398.8 18011.2i −0.388371 0.672679i
\(896\) 0 0
\(897\) 24255.4 + 865.872i 0.902859 + 0.0322304i
\(898\) 0 0
\(899\) 14731.9 + 25516.5i 0.546538 + 0.946632i
\(900\) 0 0
\(901\) 2299.80 3983.37i 0.0850360 0.147287i
\(902\) 0 0
\(903\) −275.828 + 477.748i −0.0101650 + 0.0176063i
\(904\) 0 0
\(905\) 20193.3 0.741712
\(906\) 0 0
\(907\) 8694.93 + 15060.1i 0.318314 + 0.551335i 0.980136 0.198325i \(-0.0635502\pi\)
−0.661823 + 0.749660i \(0.730217\pi\)
\(908\) 0 0
\(909\) 4680.66 0.170790
\(910\) 0 0
\(911\) −20419.5 −0.742621 −0.371311 0.928509i \(-0.621091\pi\)
−0.371311 + 0.928509i \(0.621091\pi\)
\(912\) 0 0
\(913\) 3768.85 + 6527.85i 0.136616 + 0.236627i
\(914\) 0 0
\(915\) −43523.5 −1.57250
\(916\) 0 0
\(917\) −5796.58 + 10040.0i −0.208746 + 0.361558i
\(918\) 0 0
\(919\) −16615.9 + 28779.6i −0.596417 + 1.03303i 0.396928 + 0.917850i \(0.370076\pi\)
−0.993345 + 0.115175i \(0.963257\pi\)
\(920\) 0 0
\(921\) −6651.71 11521.1i −0.237982 0.412197i
\(922\) 0 0
\(923\) −7165.15 255.782i −0.255519 0.00912153i
\(924\) 0 0
\(925\) 412.251 + 714.039i 0.0146538 + 0.0253810i
\(926\) 0 0
\(927\) 3894.79 6745.97i 0.137995 0.239015i
\(928\) 0 0
\(929\) 12611.4 21843.6i 0.445390 0.771438i −0.552689 0.833387i \(-0.686398\pi\)
0.998079 + 0.0619492i \(0.0197317\pi\)
\(930\) 0 0
\(931\) −25327.0 −0.891579
\(932\) 0 0
\(933\) 6136.91 + 10629.4i 0.215341 + 0.372982i
\(934\) 0 0
\(935\) −27146.9 −0.949519
\(936\) 0 0
\(937\) −26979.4 −0.940639 −0.470319 0.882496i \(-0.655861\pi\)
−0.470319 + 0.882496i \(0.655861\pi\)
\(938\) 0 0
\(939\) 659.785 + 1142.78i 0.0229300 + 0.0397159i
\(940\) 0 0
\(941\) 7641.67 0.264730 0.132365 0.991201i \(-0.457743\pi\)
0.132365 + 0.991201i \(0.457743\pi\)
\(942\) 0 0
\(943\) 16004.3 27720.3i 0.552675 0.957261i
\(944\) 0 0
\(945\) 7212.46 12492.3i 0.248276 0.430027i
\(946\) 0 0
\(947\) −1434.66 2484.90i −0.0492293 0.0852677i 0.840361 0.542028i \(-0.182343\pi\)
−0.889590 + 0.456760i \(0.849010\pi\)
\(948\) 0 0
\(949\) 2576.73 + 4855.22i 0.0881393 + 0.166077i
\(950\) 0 0
\(951\) −5614.42 9724.46i −0.191441 0.331585i
\(952\) 0 0
\(953\) −6156.79 + 10663.9i −0.209274 + 0.362473i −0.951486 0.307692i \(-0.900443\pi\)
0.742212 + 0.670165i \(0.233777\pi\)
\(954\) 0 0
\(955\) −23881.3 + 41363.6i −0.809194 + 1.40156i
\(956\) 0 0
\(957\) −8815.22 −0.297759
\(958\) 0 0
\(959\) 1870.78 + 3240.29i 0.0629935 + 0.109108i
\(960\) 0 0
\(961\) 46470.7 1.55989
\(962\) 0 0
\(963\) 7671.32 0.256703
\(964\) 0 0
\(965\) −17546.6 30391.6i −0.585331 1.01382i
\(966\) 0 0
\(967\) 17838.0 0.593207 0.296603 0.955001i \(-0.404146\pi\)
0.296603 + 0.955001i \(0.404146\pi\)
\(968\) 0 0
\(969\) −10121.2 + 17530.5i −0.335542 + 0.581176i
\(970\) 0 0
\(971\) 20762.7 35962.0i 0.686206 1.18854i −0.286851 0.957975i \(-0.592608\pi\)
0.973056 0.230568i \(-0.0740584\pi\)
\(972\) 0 0
\(973\) 1847.93 + 3200.71i 0.0608859 + 0.105457i
\(974\) 0 0
\(975\) 17604.6 28124.6i 0.578254 0.923803i
\(976\) 0 0
\(977\) 15827.2 + 27413.5i 0.518277 + 0.897682i 0.999775 + 0.0212344i \(0.00675962\pi\)
−0.481498 + 0.876447i \(0.659907\pi\)
\(978\) 0 0
\(979\) −8052.14 + 13946.7i −0.262868 + 0.455300i
\(980\) 0 0
\(981\) 1181.27 2046.03i 0.0384457 0.0665899i
\(982\) 0 0
\(983\) 39913.2 1.29505 0.647525 0.762045i \(-0.275804\pi\)
0.647525 + 0.762045i \(0.275804\pi\)
\(984\) 0 0
\(985\) −35758.4 61935.4i −1.15671 2.00348i
\(986\) 0 0
\(987\) 6384.72 0.205905
\(988\) 0 0
\(989\) 3868.74 0.124387
\(990\) 0 0
\(991\) −1350.47 2339.08i −0.0432887 0.0749781i 0.843569 0.537020i \(-0.180450\pi\)
−0.886858 + 0.462042i \(0.847117\pi\)
\(992\) 0 0
\(993\) −28352.6 −0.906085
\(994\) 0 0
\(995\) 37628.3 65174.2i 1.19889 2.07654i
\(996\) 0 0
\(997\) 4864.54 8425.63i 0.154525 0.267645i −0.778361 0.627817i \(-0.783949\pi\)
0.932886 + 0.360172i \(0.117282\pi\)
\(998\) 0 0
\(999\) −319.614 553.588i −0.0101223 0.0175323i
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 208.4.i.e.81.1 4
4.3 odd 2 13.4.c.b.3.1 4
12.11 even 2 117.4.g.d.55.2 4
13.9 even 3 inner 208.4.i.e.113.1 4
52.3 odd 6 169.4.a.f.1.2 2
52.7 even 12 169.4.e.g.147.3 8
52.11 even 12 169.4.b.e.168.2 4
52.15 even 12 169.4.b.e.168.3 4
52.19 even 12 169.4.e.g.147.2 8
52.23 odd 6 169.4.a.j.1.1 2
52.31 even 4 169.4.e.g.23.3 8
52.35 odd 6 13.4.c.b.9.1 yes 4
52.43 odd 6 169.4.c.f.22.2 4
52.47 even 4 169.4.e.g.23.2 8
52.51 odd 2 169.4.c.f.146.2 4
156.23 even 6 1521.4.a.l.1.2 2
156.35 even 6 117.4.g.d.100.2 4
156.107 even 6 1521.4.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 4.3 odd 2
13.4.c.b.9.1 yes 4 52.35 odd 6
117.4.g.d.55.2 4 12.11 even 2
117.4.g.d.100.2 4 156.35 even 6
169.4.a.f.1.2 2 52.3 odd 6
169.4.a.j.1.1 2 52.23 odd 6
169.4.b.e.168.2 4 52.11 even 12
169.4.b.e.168.3 4 52.15 even 12
169.4.c.f.22.2 4 52.43 odd 6
169.4.c.f.146.2 4 52.51 odd 2
169.4.e.g.23.2 8 52.47 even 4
169.4.e.g.23.3 8 52.31 even 4
169.4.e.g.147.2 8 52.19 even 12
169.4.e.g.147.3 8 52.7 even 12
208.4.i.e.81.1 4 1.1 even 1 trivial
208.4.i.e.113.1 4 13.9 even 3 inner
1521.4.a.l.1.2 2 156.23 even 6
1521.4.a.t.1.1 2 156.107 even 6