Properties

Label 76.6.e.a.49.3
Level $76$
Weight $6$
Character 76.49
Analytic conductor $12.189$
Analytic rank $0$
Dimension $18$
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] = [76,6,Mod(45,76)]
 
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, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.45");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.e (of order \(3\), 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: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} - 1275481376 x^{11} + 361098181456 x^{10} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 49.3
Root \(6.29505 + 10.9033i\) of defining polynomial
Character \(\chi\) \(=\) 76.49
Dual form 76.6.e.a.45.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.79505 + 11.7694i) q^{3} +(-47.4871 + 82.2500i) q^{5} +189.860 q^{7} +(29.1546 + 50.4972i) q^{9} +O(q^{10})\) \(q+(-6.79505 + 11.7694i) q^{3} +(-47.4871 + 82.2500i) q^{5} +189.860 q^{7} +(29.1546 + 50.4972i) q^{9} -530.005 q^{11} +(-353.895 - 612.964i) q^{13} +(-645.354 - 1117.79i) q^{15} +(-764.589 + 1324.31i) q^{17} +(654.031 - 1431.20i) q^{19} +(-1290.11 + 2234.53i) q^{21} +(497.425 + 861.565i) q^{23} +(-2947.54 - 5105.29i) q^{25} -4094.82 q^{27} +(-1290.66 - 2235.49i) q^{29} -2790.81 q^{31} +(3601.41 - 6237.82i) q^{33} +(-9015.88 + 15616.0i) q^{35} +7238.14 q^{37} +9618.94 q^{39} +(2181.63 - 3778.69i) q^{41} +(-3121.70 + 5406.94i) q^{43} -5537.86 q^{45} +(12320.9 + 21340.4i) q^{47} +19239.7 q^{49} +(-10390.8 - 17997.5i) q^{51} +(-15497.6 - 26842.6i) q^{53} +(25168.4 - 43592.9i) q^{55} +(12400.2 + 17422.6i) q^{57} +(-18177.7 + 31484.8i) q^{59} +(2886.93 + 5000.30i) q^{61} +(5535.27 + 9587.38i) q^{63} +67221.7 q^{65} +(30388.4 + 52634.2i) q^{67} -13520.1 q^{69} +(14829.0 - 25684.5i) q^{71} +(-39002.3 + 67553.9i) q^{73} +80114.7 q^{75} -100627. q^{77} +(-33427.5 + 57898.2i) q^{79} +(20740.0 - 35922.7i) q^{81} -21919.5 q^{83} +(-72616.2 - 125775. i) q^{85} +35080.4 q^{87} +(-45374.8 - 78591.5i) q^{89} +(-67190.4 - 116377. i) q^{91} +(18963.7 - 32846.0i) q^{93} +(86658.4 + 121758. i) q^{95} +(-30757.3 + 53273.1i) q^{97} +(-15452.1 - 26763.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −6.79505 + 11.7694i −0.435903 + 0.755006i −0.997369 0.0724940i \(-0.976904\pi\)
0.561466 + 0.827500i \(0.310238\pi\)
\(4\) 0 0
\(5\) −47.4871 + 82.2500i −0.849474 + 1.47133i 0.0322040 + 0.999481i \(0.489747\pi\)
−0.881678 + 0.471851i \(0.843586\pi\)
\(6\) 0 0
\(7\) 189.860 1.46449 0.732247 0.681039i \(-0.238472\pi\)
0.732247 + 0.681039i \(0.238472\pi\)
\(8\) 0 0
\(9\) 29.1546 + 50.4972i 0.119978 + 0.207807i
\(10\) 0 0
\(11\) −530.005 −1.32068 −0.660341 0.750966i \(-0.729588\pi\)
−0.660341 + 0.750966i \(0.729588\pi\)
\(12\) 0 0
\(13\) −353.895 612.964i −0.580786 1.00595i −0.995386 0.0959473i \(-0.969412\pi\)
0.414600 0.910004i \(-0.363921\pi\)
\(14\) 0 0
\(15\) −645.354 1117.79i −0.740576 1.28272i
\(16\) 0 0
\(17\) −764.589 + 1324.31i −0.641661 + 1.11139i 0.343401 + 0.939189i \(0.388421\pi\)
−0.985062 + 0.172201i \(0.944912\pi\)
\(18\) 0 0
\(19\) 654.031 1431.20i 0.415637 0.909530i
\(20\) 0 0
\(21\) −1290.11 + 2234.53i −0.638377 + 1.10570i
\(22\) 0 0
\(23\) 497.425 + 861.565i 0.196068 + 0.339600i 0.947250 0.320495i \(-0.103849\pi\)
−0.751182 + 0.660095i \(0.770516\pi\)
\(24\) 0 0
\(25\) −2947.54 5105.29i −0.943213 1.63369i
\(26\) 0 0
\(27\) −4094.82 −1.08100
\(28\) 0 0
\(29\) −1290.66 2235.49i −0.284982 0.493603i 0.687623 0.726068i \(-0.258654\pi\)
−0.972605 + 0.232465i \(0.925321\pi\)
\(30\) 0 0
\(31\) −2790.81 −0.521585 −0.260793 0.965395i \(-0.583984\pi\)
−0.260793 + 0.965395i \(0.583984\pi\)
\(32\) 0 0
\(33\) 3601.41 6237.82i 0.575689 0.997122i
\(34\) 0 0
\(35\) −9015.88 + 15616.0i −1.24405 + 2.15476i
\(36\) 0 0
\(37\) 7238.14 0.869206 0.434603 0.900622i \(-0.356889\pi\)
0.434603 + 0.900622i \(0.356889\pi\)
\(38\) 0 0
\(39\) 9618.94 1.01266
\(40\) 0 0
\(41\) 2181.63 3778.69i 0.202685 0.351060i −0.746708 0.665152i \(-0.768367\pi\)
0.949393 + 0.314092i \(0.101700\pi\)
\(42\) 0 0
\(43\) −3121.70 + 5406.94i −0.257466 + 0.445944i −0.965562 0.260172i \(-0.916221\pi\)
0.708096 + 0.706116i \(0.249554\pi\)
\(44\) 0 0
\(45\) −5537.86 −0.407671
\(46\) 0 0
\(47\) 12320.9 + 21340.4i 0.813576 + 1.40915i 0.910346 + 0.413848i \(0.135815\pi\)
−0.0967705 + 0.995307i \(0.530851\pi\)
\(48\) 0 0
\(49\) 19239.7 1.14474
\(50\) 0 0
\(51\) −10390.8 17997.5i −0.559404 0.968916i
\(52\) 0 0
\(53\) −15497.6 26842.6i −0.757833 1.31261i −0.943953 0.330079i \(-0.892925\pi\)
0.186120 0.982527i \(-0.440409\pi\)
\(54\) 0 0
\(55\) 25168.4 43592.9i 1.12188 1.94316i
\(56\) 0 0
\(57\) 12400.2 + 17422.6i 0.505523 + 0.710275i
\(58\) 0 0
\(59\) −18177.7 + 31484.8i −0.679845 + 1.17753i 0.295182 + 0.955441i \(0.404620\pi\)
−0.975027 + 0.222085i \(0.928714\pi\)
\(60\) 0 0
\(61\) 2886.93 + 5000.30i 0.0993370 + 0.172057i 0.911410 0.411499i \(-0.134994\pi\)
−0.812073 + 0.583555i \(0.801661\pi\)
\(62\) 0 0
\(63\) 5535.27 + 9587.38i 0.175706 + 0.304333i
\(64\) 0 0
\(65\) 67221.7 1.97345
\(66\) 0 0
\(67\) 30388.4 + 52634.2i 0.827029 + 1.43246i 0.900359 + 0.435148i \(0.143304\pi\)
−0.0733301 + 0.997308i \(0.523363\pi\)
\(68\) 0 0
\(69\) −13520.1 −0.341867
\(70\) 0 0
\(71\) 14829.0 25684.5i 0.349112 0.604680i −0.636980 0.770880i \(-0.719817\pi\)
0.986092 + 0.166201i \(0.0531500\pi\)
\(72\) 0 0
\(73\) −39002.3 + 67553.9i −0.856609 + 1.48369i 0.0185353 + 0.999828i \(0.494100\pi\)
−0.875144 + 0.483862i \(0.839234\pi\)
\(74\) 0 0
\(75\) 80114.7 1.64460
\(76\) 0 0
\(77\) −100627. −1.93413
\(78\) 0 0
\(79\) −33427.5 + 57898.2i −0.602611 + 1.04375i 0.389814 + 0.920894i \(0.372539\pi\)
−0.992424 + 0.122858i \(0.960794\pi\)
\(80\) 0 0
\(81\) 20740.0 35922.7i 0.351233 0.608354i
\(82\) 0 0
\(83\) −21919.5 −0.349250 −0.174625 0.984635i \(-0.555871\pi\)
−0.174625 + 0.984635i \(0.555871\pi\)
\(84\) 0 0
\(85\) −72616.2 125775.i −1.09015 1.88819i
\(86\) 0 0
\(87\) 35080.4 0.496898
\(88\) 0 0
\(89\) −45374.8 78591.5i −0.607211 1.05172i −0.991698 0.128590i \(-0.958955\pi\)
0.384487 0.923131i \(-0.374378\pi\)
\(90\) 0 0
\(91\) −67190.4 116377.i −0.850558 1.47321i
\(92\) 0 0
\(93\) 18963.7 32846.0i 0.227360 0.393800i
\(94\) 0 0
\(95\) 86658.4 + 121758.i 0.985149 + 1.38416i
\(96\) 0 0
\(97\) −30757.3 + 53273.1i −0.331908 + 0.574882i −0.982886 0.184215i \(-0.941026\pi\)
0.650978 + 0.759097i \(0.274359\pi\)
\(98\) 0 0
\(99\) −15452.1 26763.7i −0.158452 0.274447i
\(100\) 0 0
\(101\) 5882.91 + 10189.5i 0.0573837 + 0.0993915i 0.893290 0.449480i \(-0.148391\pi\)
−0.835906 + 0.548872i \(0.815057\pi\)
\(102\) 0 0
\(103\) −57682.4 −0.535735 −0.267868 0.963456i \(-0.586319\pi\)
−0.267868 + 0.963456i \(0.586319\pi\)
\(104\) 0 0
\(105\) −122527. 212222.i −1.08457 1.87853i
\(106\) 0 0
\(107\) −79892.9 −0.674604 −0.337302 0.941397i \(-0.609514\pi\)
−0.337302 + 0.941397i \(0.609514\pi\)
\(108\) 0 0
\(109\) −98323.4 + 170301.i −0.792667 + 1.37294i 0.131643 + 0.991297i \(0.457975\pi\)
−0.924310 + 0.381642i \(0.875359\pi\)
\(110\) 0 0
\(111\) −49183.5 + 85188.4i −0.378889 + 0.656255i
\(112\) 0 0
\(113\) 205714. 1.51554 0.757770 0.652521i \(-0.226289\pi\)
0.757770 + 0.652521i \(0.226289\pi\)
\(114\) 0 0
\(115\) −94484.9 −0.666220
\(116\) 0 0
\(117\) 20635.3 35741.4i 0.139363 0.241383i
\(118\) 0 0
\(119\) −145165. + 251433.i −0.939709 + 1.62762i
\(120\) 0 0
\(121\) 119854. 0.744199
\(122\) 0 0
\(123\) 29648.6 + 51352.8i 0.176702 + 0.306056i
\(124\) 0 0
\(125\) 263086. 1.50599
\(126\) 0 0
\(127\) −143.620 248.756i −0.000790141 0.00136856i 0.865630 0.500684i \(-0.166918\pi\)
−0.866420 + 0.499316i \(0.833585\pi\)
\(128\) 0 0
\(129\) −42424.2 73480.9i −0.224460 0.388777i
\(130\) 0 0
\(131\) −21086.6 + 36523.1i −0.107357 + 0.185947i −0.914699 0.404137i \(-0.867572\pi\)
0.807342 + 0.590084i \(0.200905\pi\)
\(132\) 0 0
\(133\) 124174. 271728.i 0.608698 1.33200i
\(134\) 0 0
\(135\) 194451. 336799.i 0.918281 1.59051i
\(136\) 0 0
\(137\) −52318.5 90618.2i −0.238152 0.412491i 0.722032 0.691859i \(-0.243208\pi\)
−0.960184 + 0.279369i \(0.909875\pi\)
\(138\) 0 0
\(139\) 64404.7 + 111552.i 0.282736 + 0.489713i 0.972058 0.234743i \(-0.0754247\pi\)
−0.689322 + 0.724455i \(0.742091\pi\)
\(140\) 0 0
\(141\) −334885. −1.41856
\(142\) 0 0
\(143\) 187566. + 324874.i 0.767033 + 1.32854i
\(144\) 0 0
\(145\) 245159. 0.968339
\(146\) 0 0
\(147\) −130735. + 226439.i −0.498997 + 0.864288i
\(148\) 0 0
\(149\) −136506. + 236435.i −0.503716 + 0.872462i 0.496275 + 0.868165i \(0.334701\pi\)
−0.999991 + 0.00429614i \(0.998632\pi\)
\(150\) 0 0
\(151\) 68761.9 0.245418 0.122709 0.992443i \(-0.460842\pi\)
0.122709 + 0.992443i \(0.460842\pi\)
\(152\) 0 0
\(153\) −89165.0 −0.307940
\(154\) 0 0
\(155\) 132527. 229544.i 0.443073 0.767425i
\(156\) 0 0
\(157\) 275998. 478042.i 0.893628 1.54781i 0.0581337 0.998309i \(-0.481485\pi\)
0.835494 0.549500i \(-0.185182\pi\)
\(158\) 0 0
\(159\) 421227. 1.32137
\(160\) 0 0
\(161\) 94440.9 + 163576.i 0.287141 + 0.497343i
\(162\) 0 0
\(163\) −153395. −0.452211 −0.226106 0.974103i \(-0.572599\pi\)
−0.226106 + 0.974103i \(0.572599\pi\)
\(164\) 0 0
\(165\) 342041. + 592432.i 0.978065 + 1.69406i
\(166\) 0 0
\(167\) 146626. + 253963.i 0.406836 + 0.704660i 0.994533 0.104421i \(-0.0332988\pi\)
−0.587697 + 0.809081i \(0.699965\pi\)
\(168\) 0 0
\(169\) −64836.9 + 112301.i −0.174625 + 0.302459i
\(170\) 0 0
\(171\) 91339.7 8699.36i 0.238874 0.0227508i
\(172\) 0 0
\(173\) −114249. + 197885.i −0.290227 + 0.502688i −0.973863 0.227135i \(-0.927064\pi\)
0.683636 + 0.729823i \(0.260398\pi\)
\(174\) 0 0
\(175\) −559619. 969289.i −1.38133 2.39253i
\(176\) 0 0
\(177\) −247037. 427881.i −0.592693 1.02657i
\(178\) 0 0
\(179\) 247143. 0.576521 0.288261 0.957552i \(-0.406923\pi\)
0.288261 + 0.957552i \(0.406923\pi\)
\(180\) 0 0
\(181\) 353864. + 612910.i 0.802859 + 1.39059i 0.917727 + 0.397213i \(0.130022\pi\)
−0.114867 + 0.993381i \(0.536644\pi\)
\(182\) 0 0
\(183\) −78467.2 −0.173205
\(184\) 0 0
\(185\) −343718. + 595337.i −0.738368 + 1.27889i
\(186\) 0 0
\(187\) 405236. 701889.i 0.847430 1.46779i
\(188\) 0 0
\(189\) −777442. −1.58312
\(190\) 0 0
\(191\) 10502.1 0.0208301 0.0104151 0.999946i \(-0.496685\pi\)
0.0104151 + 0.999946i \(0.496685\pi\)
\(192\) 0 0
\(193\) 496794. 860472.i 0.960025 1.66281i 0.237601 0.971363i \(-0.423639\pi\)
0.722424 0.691450i \(-0.243028\pi\)
\(194\) 0 0
\(195\) −456775. + 791158.i −0.860233 + 1.48997i
\(196\) 0 0
\(197\) −185300. −0.340182 −0.170091 0.985428i \(-0.554406\pi\)
−0.170091 + 0.985428i \(0.554406\pi\)
\(198\) 0 0
\(199\) 43149.9 + 74737.8i 0.0772408 + 0.133785i 0.902059 0.431614i \(-0.142056\pi\)
−0.824818 + 0.565399i \(0.808722\pi\)
\(200\) 0 0
\(201\) −825963. −1.44202
\(202\) 0 0
\(203\) −245045. 424430.i −0.417354 0.722879i
\(204\) 0 0
\(205\) 207198. + 358878.i 0.344351 + 0.596433i
\(206\) 0 0
\(207\) −29004.4 + 50237.1i −0.0470476 + 0.0814889i
\(208\) 0 0
\(209\) −346640. + 758544.i −0.548924 + 1.20120i
\(210\) 0 0
\(211\) 151234. 261945.i 0.233853 0.405045i −0.725086 0.688659i \(-0.758200\pi\)
0.958939 + 0.283613i \(0.0915332\pi\)
\(212\) 0 0
\(213\) 201527. + 349055.i 0.304358 + 0.527163i
\(214\) 0 0
\(215\) −296481. 513519.i −0.437422 0.757636i
\(216\) 0 0
\(217\) −529861. −0.763858
\(218\) 0 0
\(219\) −530045. 918064.i −0.746796 1.29349i
\(220\) 0 0
\(221\) 1.08234e6 1.49067
\(222\) 0 0
\(223\) −339486. + 588007.i −0.457151 + 0.791809i −0.998809 0.0487901i \(-0.984463\pi\)
0.541658 + 0.840599i \(0.317797\pi\)
\(224\) 0 0
\(225\) 171868. 297685.i 0.226329 0.392013i
\(226\) 0 0
\(227\) 231581. 0.298289 0.149145 0.988815i \(-0.452348\pi\)
0.149145 + 0.988815i \(0.452348\pi\)
\(228\) 0 0
\(229\) 33377.9 0.0420601 0.0210300 0.999779i \(-0.493305\pi\)
0.0210300 + 0.999779i \(0.493305\pi\)
\(230\) 0 0
\(231\) 683762. 1.18431e6i 0.843093 1.46028i
\(232\) 0 0
\(233\) 542190. 939100.i 0.654277 1.13324i −0.327798 0.944748i \(-0.606306\pi\)
0.982075 0.188493i \(-0.0603602\pi\)
\(234\) 0 0
\(235\) −2.34033e6 −2.76445
\(236\) 0 0
\(237\) −454284. 786843.i −0.525359 0.909949i
\(238\) 0 0
\(239\) −284041. −0.321652 −0.160826 0.986983i \(-0.551416\pi\)
−0.160826 + 0.986983i \(0.551416\pi\)
\(240\) 0 0
\(241\) 306804. + 531400.i 0.340266 + 0.589357i 0.984482 0.175486i \(-0.0561497\pi\)
−0.644216 + 0.764843i \(0.722816\pi\)
\(242\) 0 0
\(243\) −215663. 373539.i −0.234293 0.405807i
\(244\) 0 0
\(245\) −913637. + 1.58246e6i −0.972430 + 1.68430i
\(246\) 0 0
\(247\) −1.10873e6 + 105598.i −1.15634 + 0.110132i
\(248\) 0 0
\(249\) 148944. 257979.i 0.152239 0.263685i
\(250\) 0 0
\(251\) 369089. + 639282.i 0.369783 + 0.640484i 0.989531 0.144317i \(-0.0460986\pi\)
−0.619748 + 0.784801i \(0.712765\pi\)
\(252\) 0 0
\(253\) −263637. 456633.i −0.258944 0.448504i
\(254\) 0 0
\(255\) 1.97372e6 1.90080
\(256\) 0 0
\(257\) 686359. + 1.18881e6i 0.648214 + 1.12274i 0.983549 + 0.180641i \(0.0578172\pi\)
−0.335335 + 0.942099i \(0.608849\pi\)
\(258\) 0 0
\(259\) 1.37423e6 1.27295
\(260\) 0 0
\(261\) 75257.3 130349.i 0.0683829 0.118443i
\(262\) 0 0
\(263\) −146329. + 253449.i −0.130449 + 0.225944i −0.923850 0.382756i \(-0.874975\pi\)
0.793401 + 0.608699i \(0.208308\pi\)
\(264\) 0 0
\(265\) 2.94373e6 2.57504
\(266\) 0 0
\(267\) 1.23330e6 1.05874
\(268\) 0 0
\(269\) −977718. + 1.69346e6i −0.823821 + 1.42690i 0.0789962 + 0.996875i \(0.474828\pi\)
−0.902817 + 0.430025i \(0.858505\pi\)
\(270\) 0 0
\(271\) −713502. + 1.23582e6i −0.590163 + 1.02219i 0.404047 + 0.914738i \(0.367603\pi\)
−0.994210 + 0.107454i \(0.965730\pi\)
\(272\) 0 0
\(273\) 1.82625e6 1.48304
\(274\) 0 0
\(275\) 1.56221e6 + 2.70583e6i 1.24568 + 2.15759i
\(276\) 0 0
\(277\) −1.20887e6 −0.946629 −0.473315 0.880893i \(-0.656943\pi\)
−0.473315 + 0.880893i \(0.656943\pi\)
\(278\) 0 0
\(279\) −81364.7 140928.i −0.0625785 0.108389i
\(280\) 0 0
\(281\) −996432. 1.72587e6i −0.752804 1.30389i −0.946459 0.322825i \(-0.895367\pi\)
0.193655 0.981070i \(-0.437966\pi\)
\(282\) 0 0
\(283\) 33435.3 57911.6i 0.0248164 0.0429832i −0.853350 0.521338i \(-0.825433\pi\)
0.878167 + 0.478354i \(0.158767\pi\)
\(284\) 0 0
\(285\) −2.02186e6 + 192566.i −1.47448 + 0.140432i
\(286\) 0 0
\(287\) 414203. 717421.i 0.296831 0.514126i
\(288\) 0 0
\(289\) −459264. 795469.i −0.323458 0.560246i
\(290\) 0 0
\(291\) −417994. 723987.i −0.289360 0.501185i
\(292\) 0 0
\(293\) 1.98637e6 1.35173 0.675866 0.737025i \(-0.263770\pi\)
0.675866 + 0.737025i \(0.263770\pi\)
\(294\) 0 0
\(295\) −1.72641e6 2.99024e6i −1.15502 2.00056i
\(296\) 0 0
\(297\) 2.17027e6 1.42766
\(298\) 0 0
\(299\) 352072. 609807.i 0.227748 0.394470i
\(300\) 0 0
\(301\) −592685. + 1.02656e6i −0.377058 + 0.653083i
\(302\) 0 0
\(303\) −159899. −0.100055
\(304\) 0 0
\(305\) −548366. −0.337537
\(306\) 0 0
\(307\) 932152. 1.61454e6i 0.564470 0.977691i −0.432629 0.901572i \(-0.642414\pi\)
0.997099 0.0761187i \(-0.0242528\pi\)
\(308\) 0 0
\(309\) 391955. 678886.i 0.233528 0.404483i
\(310\) 0 0
\(311\) 1.73570e6 1.01759 0.508796 0.860887i \(-0.330091\pi\)
0.508796 + 0.860887i \(0.330091\pi\)
\(312\) 0 0
\(313\) 26566.2 + 46014.0i 0.0153274 + 0.0265478i 0.873587 0.486667i \(-0.161788\pi\)
−0.858260 + 0.513215i \(0.828454\pi\)
\(314\) 0 0
\(315\) −1.05142e6 −0.597032
\(316\) 0 0
\(317\) −690554. 1.19607e6i −0.385966 0.668514i 0.605936 0.795513i \(-0.292799\pi\)
−0.991903 + 0.127000i \(0.959465\pi\)
\(318\) 0 0
\(319\) 684056. + 1.18482e6i 0.376370 + 0.651892i
\(320\) 0 0
\(321\) 542876. 940289.i 0.294062 0.509330i
\(322\) 0 0
\(323\) 1.39529e6 + 1.96042e6i 0.744144 + 1.04555i
\(324\) 0 0
\(325\) −2.08624e6 + 3.61347e6i −1.09561 + 1.89765i
\(326\) 0 0
\(327\) −1.33623e6 2.31441e6i −0.691051 1.19694i
\(328\) 0 0
\(329\) 2.33924e6 + 4.05169e6i 1.19148 + 2.06370i
\(330\) 0 0
\(331\) −79218.6 −0.0397427 −0.0198713 0.999803i \(-0.506326\pi\)
−0.0198713 + 0.999803i \(0.506326\pi\)
\(332\) 0 0
\(333\) 211025. + 365506.i 0.104285 + 0.180627i
\(334\) 0 0
\(335\) −5.77222e6 −2.81016
\(336\) 0 0
\(337\) 1.06653e6 1.84729e6i 0.511563 0.886053i −0.488347 0.872649i \(-0.662400\pi\)
0.999910 0.0134034i \(-0.00426657\pi\)
\(338\) 0 0
\(339\) −1.39784e6 + 2.42112e6i −0.660628 + 1.14424i
\(340\) 0 0
\(341\) 1.47914e6 0.688848
\(342\) 0 0
\(343\) 461871. 0.211976
\(344\) 0 0
\(345\) 642030. 1.11203e6i 0.290407 0.503000i
\(346\) 0 0
\(347\) 988867. 1.71277e6i 0.440874 0.763616i −0.556881 0.830592i \(-0.688002\pi\)
0.997755 + 0.0669767i \(0.0213353\pi\)
\(348\) 0 0
\(349\) −2.01158e6 −0.884043 −0.442022 0.897004i \(-0.645739\pi\)
−0.442022 + 0.897004i \(0.645739\pi\)
\(350\) 0 0
\(351\) 1.44914e6 + 2.50998e6i 0.627829 + 1.08743i
\(352\) 0 0
\(353\) −2.89132e6 −1.23498 −0.617490 0.786579i \(-0.711850\pi\)
−0.617490 + 0.786579i \(0.711850\pi\)
\(354\) 0 0
\(355\) 1.40837e6 + 2.43936e6i 0.593123 + 1.02732i
\(356\) 0 0
\(357\) −1.97280e6 3.41699e6i −0.819243 1.41897i
\(358\) 0 0
\(359\) −371558. + 643557.i −0.152156 + 0.263543i −0.932020 0.362407i \(-0.881955\pi\)
0.779864 + 0.625950i \(0.215288\pi\)
\(360\) 0 0
\(361\) −1.62059e6 1.87210e6i −0.654491 0.756070i
\(362\) 0 0
\(363\) −814414. + 1.41061e6i −0.324398 + 0.561875i
\(364\) 0 0
\(365\) −3.70420e6 6.41587e6i −1.45533 2.52071i
\(366\) 0 0
\(367\) 2.21010e6 + 3.82801e6i 0.856540 + 1.48357i 0.875209 + 0.483745i \(0.160724\pi\)
−0.0186695 + 0.999826i \(0.505943\pi\)
\(368\) 0 0
\(369\) 254418. 0.0972705
\(370\) 0 0
\(371\) −2.94236e6 5.09632e6i −1.10984 1.92230i
\(372\) 0 0
\(373\) −2.07890e6 −0.773680 −0.386840 0.922147i \(-0.626433\pi\)
−0.386840 + 0.922147i \(0.626433\pi\)
\(374\) 0 0
\(375\) −1.78768e6 + 3.09636e6i −0.656466 + 1.13703i
\(376\) 0 0
\(377\) −913517. + 1.58226e6i −0.331027 + 0.573355i
\(378\) 0 0
\(379\) 3.87208e6 1.38467 0.692335 0.721576i \(-0.256582\pi\)
0.692335 + 0.721576i \(0.256582\pi\)
\(380\) 0 0
\(381\) 3903.61 0.00137770
\(382\) 0 0
\(383\) −72504.1 + 125581.i −0.0252560 + 0.0437448i −0.878377 0.477968i \(-0.841373\pi\)
0.853121 + 0.521713i \(0.174707\pi\)
\(384\) 0 0
\(385\) 4.77846e6 8.27653e6i 1.64299 2.84575i
\(386\) 0 0
\(387\) −364047. −0.123561
\(388\) 0 0
\(389\) 670618. + 1.16154e6i 0.224699 + 0.389190i 0.956229 0.292619i \(-0.0945268\pi\)
−0.731530 + 0.681809i \(0.761193\pi\)
\(390\) 0 0
\(391\) −1.52130e6 −0.503238
\(392\) 0 0
\(393\) −286570. 496353.i −0.0935942 0.162110i
\(394\) 0 0
\(395\) −3.17475e6 5.49883e6i −1.02380 1.77328i
\(396\) 0 0
\(397\) −635788. + 1.10122e6i −0.202458 + 0.350668i −0.949320 0.314311i \(-0.898226\pi\)
0.746862 + 0.664980i \(0.231560\pi\)
\(398\) 0 0
\(399\) 2.35430e6 + 3.30786e6i 0.740336 + 1.04019i
\(400\) 0 0
\(401\) 722427. 1.25128e6i 0.224354 0.388592i −0.731772 0.681550i \(-0.761306\pi\)
0.956125 + 0.292958i \(0.0946396\pi\)
\(402\) 0 0
\(403\) 987652. + 1.71066e6i 0.302929 + 0.524689i
\(404\) 0 0
\(405\) 1.96976e6 + 3.41172e6i 0.596727 + 1.03356i
\(406\) 0 0
\(407\) −3.83625e6 −1.14794
\(408\) 0 0
\(409\) 306511. + 530893.i 0.0906020 + 0.156927i 0.907765 0.419480i \(-0.137788\pi\)
−0.817163 + 0.576407i \(0.804454\pi\)
\(410\) 0 0
\(411\) 1.42203e6 0.415244
\(412\) 0 0
\(413\) −3.45122e6 + 5.97769e6i −0.995629 + 1.72448i
\(414\) 0 0
\(415\) 1.04089e6 1.80288e6i 0.296679 0.513862i
\(416\) 0 0
\(417\) −1.75053e6 −0.492981
\(418\) 0 0
\(419\) 6.30914e6 1.75564 0.877820 0.478991i \(-0.158997\pi\)
0.877820 + 0.478991i \(0.158997\pi\)
\(420\) 0 0
\(421\) −1.28732e6 + 2.22970e6i −0.353981 + 0.613113i −0.986943 0.161070i \(-0.948506\pi\)
0.632962 + 0.774183i \(0.281839\pi\)
\(422\) 0 0
\(423\) −718421. + 1.24434e6i −0.195222 + 0.338134i
\(424\) 0 0
\(425\) 9.01463e6 2.42089
\(426\) 0 0
\(427\) 548111. + 949356.i 0.145478 + 0.251976i
\(428\) 0 0
\(429\) −5.09808e6 −1.33741
\(430\) 0 0
\(431\) −921074. 1.59535e6i −0.238837 0.413678i 0.721544 0.692369i \(-0.243433\pi\)
−0.960381 + 0.278691i \(0.910099\pi\)
\(432\) 0 0
\(433\) −2.48716e6 4.30788e6i −0.637505 1.10419i −0.985979 0.166872i \(-0.946633\pi\)
0.348474 0.937318i \(-0.386700\pi\)
\(434\) 0 0
\(435\) −1.66587e6 + 2.88537e6i −0.422102 + 0.731101i
\(436\) 0 0
\(437\) 1.55841e6 148425.i 0.390370 0.0371796i
\(438\) 0 0
\(439\) 1.97430e6 3.41959e6i 0.488936 0.846863i −0.510983 0.859591i \(-0.670718\pi\)
0.999919 + 0.0127283i \(0.00405165\pi\)
\(440\) 0 0
\(441\) 560925. + 971550.i 0.137344 + 0.237886i
\(442\) 0 0
\(443\) 3.42436e6 + 5.93117e6i 0.829031 + 1.43592i 0.898799 + 0.438360i \(0.144440\pi\)
−0.0697689 + 0.997563i \(0.522226\pi\)
\(444\) 0 0
\(445\) 8.61886e6 2.06324
\(446\) 0 0
\(447\) −1.85513e6 3.21318e6i −0.439142 0.760617i
\(448\) 0 0
\(449\) −3.05530e6 −0.715216 −0.357608 0.933872i \(-0.616408\pi\)
−0.357608 + 0.933872i \(0.616408\pi\)
\(450\) 0 0
\(451\) −1.15627e6 + 2.00272e6i −0.267682 + 0.463639i
\(452\) 0 0
\(453\) −467241. + 809285.i −0.106978 + 0.185292i
\(454\) 0 0
\(455\) 1.27627e7 2.89011
\(456\) 0 0
\(457\) −5.86875e6 −1.31448 −0.657242 0.753680i \(-0.728277\pi\)
−0.657242 + 0.753680i \(0.728277\pi\)
\(458\) 0 0
\(459\) 3.13086e6 5.42280e6i 0.693636 1.20141i
\(460\) 0 0
\(461\) 970738. 1.68137e6i 0.212740 0.368477i −0.739831 0.672793i \(-0.765095\pi\)
0.952571 + 0.304316i \(0.0984279\pi\)
\(462\) 0 0
\(463\) 2.37394e6 0.514656 0.257328 0.966324i \(-0.417158\pi\)
0.257328 + 0.966324i \(0.417158\pi\)
\(464\) 0 0
\(465\) 1.80106e6 + 3.11952e6i 0.386274 + 0.669046i
\(466\) 0 0
\(467\) −2.72608e6 −0.578425 −0.289212 0.957265i \(-0.593393\pi\)
−0.289212 + 0.957265i \(0.593393\pi\)
\(468\) 0 0
\(469\) 5.76953e6 + 9.99312e6i 1.21118 + 2.09782i
\(470\) 0 0
\(471\) 3.75084e6 + 6.49664e6i 0.779069 + 1.34939i
\(472\) 0 0
\(473\) 1.65452e6 2.86570e6i 0.340031 0.588950i
\(474\) 0 0
\(475\) −9.23449e6 + 879510.i −1.87793 + 0.178857i
\(476\) 0 0
\(477\) 903649. 1.56517e6i 0.181846 0.314967i
\(478\) 0 0
\(479\) −2.64917e6 4.58849e6i −0.527558 0.913757i −0.999484 0.0321192i \(-0.989774\pi\)
0.471926 0.881638i \(-0.343559\pi\)
\(480\) 0 0
\(481\) −2.56154e6 4.43672e6i −0.504823 0.874378i
\(482\) 0 0
\(483\) −2.56692e6 −0.500662
\(484\) 0 0
\(485\) −2.92114e6 5.05957e6i −0.563895 0.976695i
\(486\) 0 0
\(487\) 5.45937e6 1.04309 0.521543 0.853225i \(-0.325356\pi\)
0.521543 + 0.853225i \(0.325356\pi\)
\(488\) 0 0
\(489\) 1.04233e6 1.80536e6i 0.197120 0.341422i
\(490\) 0 0
\(491\) −3.95495e6 + 6.85017e6i −0.740350 + 1.28232i 0.211986 + 0.977273i \(0.432007\pi\)
−0.952336 + 0.305051i \(0.901326\pi\)
\(492\) 0 0
\(493\) 3.94730e6 0.731447
\(494\) 0 0
\(495\) 2.93509e6 0.538404
\(496\) 0 0
\(497\) 2.81542e6 4.87645e6i 0.511272 0.885550i
\(498\) 0 0
\(499\) −4.75343e6 + 8.23318e6i −0.854586 + 1.48019i 0.0224433 + 0.999748i \(0.492855\pi\)
−0.877029 + 0.480438i \(0.840478\pi\)
\(500\) 0 0
\(501\) −3.98532e6 −0.709363
\(502\) 0 0
\(503\) −3.72038e6 6.44389e6i −0.655643 1.13561i −0.981732 0.190268i \(-0.939064\pi\)
0.326090 0.945339i \(-0.394269\pi\)
\(504\) 0 0
\(505\) −1.11745e6 −0.194984
\(506\) 0 0
\(507\) −881141. 1.52618e6i −0.152239 0.263685i
\(508\) 0 0
\(509\) 513257. + 888987.i 0.0878092 + 0.152090i 0.906585 0.422024i \(-0.138680\pi\)
−0.818776 + 0.574114i \(0.805347\pi\)
\(510\) 0 0
\(511\) −7.40496e6 + 1.28258e7i −1.25450 + 2.17286i
\(512\) 0 0
\(513\) −2.67814e6 + 5.86052e6i −0.449304 + 0.983202i
\(514\) 0 0
\(515\) 2.73917e6 4.74438e6i 0.455093 0.788245i
\(516\) 0 0
\(517\) −6.53014e6 1.13105e7i −1.07447 1.86104i
\(518\) 0 0
\(519\) −1.55266e6 2.68928e6i −0.253022 0.438246i
\(520\) 0 0
\(521\) 5.83600e6 0.941936 0.470968 0.882150i \(-0.343905\pi\)
0.470968 + 0.882150i \(0.343905\pi\)
\(522\) 0 0
\(523\) 1.68243e6 + 2.91406e6i 0.268958 + 0.465848i 0.968593 0.248652i \(-0.0799875\pi\)
−0.699635 + 0.714500i \(0.746654\pi\)
\(524\) 0 0
\(525\) 1.52106e7 2.40850
\(526\) 0 0
\(527\) 2.13382e6 3.69588e6i 0.334681 0.579684i
\(528\) 0 0
\(529\) 2.72331e6 4.71691e6i 0.423114 0.732856i
\(530\) 0 0
\(531\) −2.11986e6 −0.326265
\(532\) 0 0
\(533\) −3.08827e6 −0.470866
\(534\) 0 0
\(535\) 3.79388e6 6.57119e6i 0.573058 0.992566i
\(536\) 0 0
\(537\) −1.67935e6 + 2.90872e6i −0.251307 + 0.435277i
\(538\) 0 0
\(539\) −1.01971e7 −1.51184
\(540\) 0 0
\(541\) 4.79104e6 + 8.29833e6i 0.703780 + 1.21898i 0.967130 + 0.254282i \(0.0818393\pi\)
−0.263350 + 0.964700i \(0.584827\pi\)
\(542\) 0 0
\(543\) −9.61809e6 −1.39987
\(544\) 0 0
\(545\) −9.33818e6 1.61742e7i −1.34670 2.33255i
\(546\) 0 0
\(547\) −2.53036e6 4.38271e6i −0.361588 0.626289i 0.626634 0.779314i \(-0.284432\pi\)
−0.988222 + 0.153024i \(0.951099\pi\)
\(548\) 0 0
\(549\) −168334. + 291563.i −0.0238364 + 0.0412859i
\(550\) 0 0
\(551\) −4.04357e6 + 385117.i −0.567396 + 0.0540398i
\(552\) 0 0
\(553\) −6.34654e6 + 1.09925e7i −0.882520 + 1.52857i
\(554\) 0 0
\(555\) −4.67116e6 8.09069e6i −0.643713 1.11494i
\(556\) 0 0
\(557\) −3.89885e6 6.75300e6i −0.532474 0.922271i −0.999281 0.0379124i \(-0.987929\pi\)
0.466807 0.884359i \(-0.345404\pi\)
\(558\) 0 0
\(559\) 4.41902e6 0.598131
\(560\) 0 0
\(561\) 5.50720e6 + 9.53874e6i 0.738794 + 1.27963i
\(562\) 0 0
\(563\) 4.30439e6 0.572323 0.286161 0.958181i \(-0.407621\pi\)
0.286161 + 0.958181i \(0.407621\pi\)
\(564\) 0 0
\(565\) −9.76875e6 + 1.69200e7i −1.28741 + 2.22986i
\(566\) 0 0
\(567\) 3.93768e6 6.82027e6i 0.514379 0.890930i
\(568\) 0 0
\(569\) −9.79289e6 −1.26803 −0.634016 0.773320i \(-0.718595\pi\)
−0.634016 + 0.773320i \(0.718595\pi\)
\(570\) 0 0
\(571\) −6.14671e6 −0.788955 −0.394477 0.918906i \(-0.629074\pi\)
−0.394477 + 0.918906i \(0.629074\pi\)
\(572\) 0 0
\(573\) −71362.1 + 123603.i −0.00907990 + 0.0157269i
\(574\) 0 0
\(575\) 2.93236e6 5.07899e6i 0.369869 0.640631i
\(576\) 0 0
\(577\) −704051. −0.0880369 −0.0440185 0.999031i \(-0.514016\pi\)
−0.0440185 + 0.999031i \(0.514016\pi\)
\(578\) 0 0
\(579\) 6.75148e6 + 1.16939e7i 0.836955 + 1.44965i
\(580\) 0 0
\(581\) −4.16163e6 −0.511474
\(582\) 0 0
\(583\) 8.21378e6 + 1.42267e7i 1.00086 + 1.73353i
\(584\) 0 0
\(585\) 1.95982e6 + 3.39451e6i 0.236770 + 0.410097i
\(586\) 0 0
\(587\) 4.80763e6 8.32707e6i 0.575885 0.997463i −0.420059 0.907497i \(-0.637991\pi\)
0.995945 0.0899662i \(-0.0286759\pi\)
\(588\) 0 0
\(589\) −1.82527e6 + 3.99421e6i −0.216790 + 0.474398i
\(590\) 0 0
\(591\) 1.25913e6 2.18087e6i 0.148286 0.256839i
\(592\) 0 0
\(593\) 5.57887e6 + 9.66289e6i 0.651493 + 1.12842i 0.982761 + 0.184882i \(0.0591903\pi\)
−0.331268 + 0.943537i \(0.607476\pi\)
\(594\) 0 0
\(595\) −1.37869e7 2.38796e7i −1.59652 2.76525i
\(596\) 0 0
\(597\) −1.17282e6 −0.134678
\(598\) 0 0
\(599\) 4.00898e6 + 6.94376e6i 0.456528 + 0.790729i 0.998775 0.0494902i \(-0.0157597\pi\)
−0.542247 + 0.840219i \(0.682426\pi\)
\(600\) 0 0
\(601\) 1.34115e7 1.51458 0.757290 0.653079i \(-0.226523\pi\)
0.757290 + 0.653079i \(0.226523\pi\)
\(602\) 0 0
\(603\) −1.77192e6 + 3.06905e6i −0.198450 + 0.343725i
\(604\) 0 0
\(605\) −5.69151e6 + 9.85799e6i −0.632178 + 1.09496i
\(606\) 0 0
\(607\) 3.14504e6 0.346461 0.173231 0.984881i \(-0.444579\pi\)
0.173231 + 0.984881i \(0.444579\pi\)
\(608\) 0 0
\(609\) 6.66036e6 0.727704
\(610\) 0 0
\(611\) 8.72061e6 1.51045e7i 0.945027 1.63683i
\(612\) 0 0
\(613\) 1.86161e6 3.22440e6i 0.200095 0.346575i −0.748464 0.663176i \(-0.769208\pi\)
0.948559 + 0.316601i \(0.102542\pi\)
\(614\) 0 0
\(615\) −5.63169e6 −0.600414
\(616\) 0 0
\(617\) −650699. 1.12704e6i −0.0688125 0.119187i 0.829566 0.558408i \(-0.188588\pi\)
−0.898379 + 0.439221i \(0.855254\pi\)
\(618\) 0 0
\(619\) 1.44161e7 1.51224 0.756118 0.654435i \(-0.227093\pi\)
0.756118 + 0.654435i \(0.227093\pi\)
\(620\) 0 0
\(621\) −2.03687e6 3.52795e6i −0.211950 0.367108i
\(622\) 0 0
\(623\) −8.61485e6 1.49214e7i −0.889257 1.54024i
\(624\) 0 0
\(625\) −3.28211e6 + 5.68478e6i −0.336088 + 0.582122i
\(626\) 0 0
\(627\) −6.57216e6 9.23408e6i −0.667635 0.938047i
\(628\) 0 0
\(629\) −5.53420e6 + 9.58552e6i −0.557736 + 0.966026i
\(630\) 0 0
\(631\) 1.56517e6 + 2.71095e6i 0.156490 + 0.271049i 0.933601 0.358315i \(-0.116649\pi\)
−0.777110 + 0.629364i \(0.783315\pi\)
\(632\) 0 0
\(633\) 2.05528e6 + 3.55986e6i 0.203874 + 0.353121i
\(634\) 0 0
\(635\) 27280.3 0.00268482
\(636\) 0 0
\(637\) −6.80883e6 1.17932e7i −0.664851 1.15156i
\(638\) 0 0
\(639\) 1.72933e6 0.167542
\(640\) 0 0
\(641\) −2.12325e6 + 3.67757e6i −0.204106 + 0.353522i −0.949848 0.312713i \(-0.898762\pi\)
0.745742 + 0.666235i \(0.232095\pi\)
\(642\) 0 0
\(643\) 5.08032e6 8.79938e6i 0.484578 0.839314i −0.515265 0.857031i \(-0.672306\pi\)
0.999843 + 0.0177168i \(0.00563972\pi\)
\(644\) 0 0
\(645\) 8.05840e6 0.762693
\(646\) 0 0
\(647\) 3.33300e6 0.313022 0.156511 0.987676i \(-0.449975\pi\)
0.156511 + 0.987676i \(0.449975\pi\)
\(648\) 0 0
\(649\) 9.63429e6 1.66871e7i 0.897859 1.55514i
\(650\) 0 0
\(651\) 3.60044e6 6.23614e6i 0.332968 0.576718i
\(652\) 0 0
\(653\) −3.56244e6 −0.326938 −0.163469 0.986549i \(-0.552268\pi\)
−0.163469 + 0.986549i \(0.552268\pi\)
\(654\) 0 0
\(655\) −2.00269e6 3.46875e6i −0.182394 0.315915i
\(656\) 0 0
\(657\) −4.54837e6 −0.411096
\(658\) 0 0
\(659\) 9.65732e6 + 1.67270e7i 0.866249 + 1.50039i 0.865801 + 0.500388i \(0.166809\pi\)
0.000448081 1.00000i \(0.499857\pi\)
\(660\) 0 0
\(661\) 5.55793e6 + 9.62662e6i 0.494777 + 0.856979i 0.999982 0.00602036i \(-0.00191635\pi\)
−0.505205 + 0.863000i \(0.668583\pi\)
\(662\) 0 0
\(663\) −7.35454e6 + 1.27384e7i −0.649788 + 1.12547i
\(664\) 0 0
\(665\) 1.64529e7 + 2.31169e7i 1.44274 + 2.02710i
\(666\) 0 0
\(667\) 1.28401e6 2.22398e6i 0.111752 0.193560i
\(668\) 0 0
\(669\) −4.61365e6 7.99108e6i −0.398547 0.690303i
\(670\) 0 0
\(671\) −1.53008e6 2.65018e6i −0.131193 0.227232i
\(672\) 0 0
\(673\) 4.82842e6 0.410929 0.205465 0.978665i \(-0.434129\pi\)
0.205465 + 0.978665i \(0.434129\pi\)
\(674\) 0 0
\(675\) 1.20697e7 + 2.09052e7i 1.01961 + 1.76602i
\(676\) 0 0
\(677\) −3.20527e6 −0.268777 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(678\) 0 0
\(679\) −5.83956e6 + 1.01144e7i −0.486078 + 0.841912i
\(680\) 0 0
\(681\) −1.57360e6 + 2.72556e6i −0.130025 + 0.225210i
\(682\) 0 0
\(683\) 2.29678e7 1.88395 0.941973 0.335688i \(-0.108969\pi\)
0.941973 + 0.335688i \(0.108969\pi\)
\(684\) 0 0
\(685\) 9.93780e6 0.809215
\(686\) 0 0
\(687\) −226804. + 392837.i −0.0183341 + 0.0317556i
\(688\) 0 0
\(689\) −1.09690e7 + 1.89989e7i −0.880278 + 1.52469i
\(690\) 0 0
\(691\) 7.10876e6 0.566368 0.283184 0.959066i \(-0.408609\pi\)
0.283184 + 0.959066i \(0.408609\pi\)
\(692\) 0 0
\(693\) −2.93372e6 5.08135e6i −0.232052 0.401926i
\(694\) 0 0
\(695\) −1.22336e7 −0.960707
\(696\) 0 0
\(697\) 3.33610e6 + 5.77829e6i 0.260110 + 0.450523i
\(698\) 0 0
\(699\) 7.36841e6 + 1.27625e7i 0.570402 + 0.987965i
\(700\) 0 0
\(701\) 7.73351e6 1.33948e7i 0.594403 1.02954i −0.399227 0.916852i \(-0.630722\pi\)
0.993631 0.112685i \(-0.0359451\pi\)
\(702\) 0 0
\(703\) 4.73397e6 1.03592e7i 0.361274 0.790569i
\(704\) 0 0
\(705\) 1.59027e7 2.75443e7i 1.20503 2.08717i
\(706\) 0 0
\(707\) 1.11693e6 + 1.93458e6i 0.0840381 + 0.145558i
\(708\) 0 0
\(709\) 6.30537e6 + 1.09212e7i 0.471081 + 0.815936i 0.999453 0.0330772i \(-0.0105307\pi\)
−0.528372 + 0.849013i \(0.677197\pi\)
\(710\) 0 0
\(711\) −3.89826e6 −0.289199
\(712\) 0 0
\(713\) −1.38822e6 2.40446e6i −0.102266 0.177131i
\(714\) 0 0
\(715\) −3.56278e7 −2.60630
\(716\) 0 0
\(717\) 1.93007e6 3.34298e6i 0.140209 0.242849i
\(718\) 0 0
\(719\) −1.02749e7 + 1.77967e7i −0.741234 + 1.28386i 0.210699 + 0.977551i \(0.432426\pi\)
−0.951934 + 0.306304i \(0.900907\pi\)
\(720\) 0 0
\(721\) −1.09516e7 −0.784581
\(722\) 0 0
\(723\) −8.33899e6 −0.593291
\(724\) 0 0
\(725\) −7.60855e6 + 1.31784e7i −0.537597 + 0.931146i
\(726\) 0 0
\(727\) −8.28557e6 + 1.43510e7i −0.581415 + 1.00704i 0.413897 + 0.910324i \(0.364167\pi\)
−0.995312 + 0.0967169i \(0.969166\pi\)
\(728\) 0 0
\(729\) 1.59414e7 1.11098
\(730\) 0 0
\(731\) −4.77363e6 8.26818e6i −0.330412 0.572290i
\(732\) 0 0
\(733\) 2.52568e7 1.73628 0.868139 0.496321i \(-0.165316\pi\)
0.868139 + 0.496321i \(0.165316\pi\)
\(734\) 0 0
\(735\) −1.24164e7 2.15059e7i −0.847770 1.46838i
\(736\) 0 0
\(737\) −1.61060e7 2.78964e7i −1.09224 1.89182i
\(738\) 0 0
\(739\) 6.54248e6 1.13319e7i 0.440688 0.763294i −0.557053 0.830477i \(-0.688068\pi\)
0.997741 + 0.0671833i \(0.0214012\pi\)
\(740\) 0 0
\(741\) 6.29109e6 1.37667e7i 0.420901 0.921049i
\(742\) 0 0
\(743\) 1.18603e6 2.05426e6i 0.0788177 0.136516i −0.823922 0.566703i \(-0.808219\pi\)
0.902740 + 0.430186i \(0.141552\pi\)
\(744\) 0 0
\(745\) −1.29645e7 2.24552e7i −0.855787 1.48227i
\(746\) 0 0
\(747\) −639054. 1.10687e6i −0.0419021 0.0725766i
\(748\) 0 0
\(749\) −1.51684e7 −0.987953
\(750\) 0 0
\(751\) −1.34919e7 2.33686e7i −0.872916 1.51193i −0.858966 0.512032i \(-0.828893\pi\)
−0.0139493 0.999903i \(-0.504440\pi\)
\(752\) 0 0
\(753\) −1.00319e7 −0.644758
\(754\) 0 0
\(755\) −3.26530e6 + 5.65567e6i −0.208476 + 0.361091i
\(756\) 0 0
\(757\) 1.46311e7 2.53419e7i 0.927980 1.60731i 0.141284 0.989969i \(-0.454877\pi\)
0.786697 0.617340i \(-0.211790\pi\)
\(758\) 0 0
\(759\) 7.16572e6 0.451497
\(760\) 0 0
\(761\) −2.62075e7 −1.64045 −0.820226 0.572040i \(-0.806152\pi\)
−0.820226 + 0.572040i \(0.806152\pi\)
\(762\) 0 0
\(763\) −1.86677e7 + 3.23333e7i −1.16086 + 2.01066i
\(764\) 0 0
\(765\) 4.23418e6 7.33382e6i 0.261587 0.453082i
\(766\) 0 0
\(767\) 2.57321e7 1.57938
\(768\) 0 0
\(769\) 2.54995e6 + 4.41664e6i 0.155495 + 0.269325i 0.933239 0.359256i \(-0.116969\pi\)
−0.777744 + 0.628581i \(0.783636\pi\)
\(770\) 0 0
\(771\) −1.86554e7 −1.13023
\(772\) 0 0
\(773\) −8.67472e6 1.50250e7i −0.522164 0.904414i −0.999668 0.0257842i \(-0.991792\pi\)
0.477504 0.878630i \(-0.341542\pi\)
\(774\) 0 0
\(775\) 8.22601e6 + 1.42479e7i 0.491966 + 0.852110i
\(776\) 0 0
\(777\) −9.33797e6 + 1.61738e7i −0.554881 + 0.961082i
\(778\) 0 0
\(779\) −3.98122e6 5.59373e6i −0.235057 0.330262i
\(780\) 0 0
\(781\) −7.85942e6 + 1.36129e7i −0.461066 + 0.798589i
\(782\) 0 0
\(783\) 5.28503e6 + 9.15394e6i 0.308065 + 0.533585i
\(784\) 0 0
\(785\) 2.62126e7 + 4.54016e7i 1.51823 + 2.62965i
\(786\) 0 0
\(787\) 3.89085e6 0.223928 0.111964 0.993712i \(-0.464286\pi\)
0.111964 + 0.993712i \(0.464286\pi\)
\(788\) 0 0
\(789\) −1.98862e6 3.44439e6i −0.113726 0.196979i
\(790\) 0 0
\(791\) 3.90568e7 2.21950
\(792\) 0 0
\(793\) 2.04334e6 3.53917e6i 0.115387 0.199856i
\(794\) 0 0
\(795\) −2.00028e7 + 3.46459e7i −1.12247 + 1.94417i
\(796\) 0 0
\(797\) −2.46516e6 −0.137467 −0.0687336 0.997635i \(-0.521896\pi\)
−0.0687336 + 0.997635i \(0.521896\pi\)
\(798\) 0 0
\(799\) −3.76817e7 −2.08816
\(800\) 0 0
\(801\) 2.64576e6 4.58260e6i 0.145703 0.252366i
\(802\) 0 0
\(803\) 2.06714e7 3.58039e7i 1.13131 1.95948i
\(804\) 0 0
\(805\) −1.79389e7 −0.975676
\(806\) 0 0
\(807\) −1.32873e7 2.30143e7i −0.718212 1.24398i
\(808\) 0 0
\(809\) 1.17605e7 0.631762 0.315881 0.948799i \(-0.397700\pi\)
0.315881 + 0.948799i \(0.397700\pi\)
\(810\) 0 0
\(811\) 4.70469e6 + 8.14875e6i 0.251176 + 0.435050i 0.963850 0.266446i \(-0.0858493\pi\)
−0.712674 + 0.701496i \(0.752516\pi\)
\(812\) 0 0
\(813\) −9.69656e6 1.67949e7i −0.514507 0.891153i
\(814\) 0 0
\(815\) 7.28427e6 1.26167e7i 0.384142 0.665353i
\(816\) 0 0
\(817\) 5.69674e6 + 8.00409e6i 0.298587 + 0.419524i
\(818\) 0 0
\(819\) 3.91781e6 6.78585e6i 0.204096 0.353504i
\(820\) 0 0
\(821\) 3.22161e6 + 5.57999e6i 0.166807 + 0.288918i 0.937296 0.348536i \(-0.113321\pi\)
−0.770488 + 0.637454i \(0.779988\pi\)
\(822\) 0 0
\(823\) 5.61870e6 + 9.73187e6i 0.289158 + 0.500837i 0.973609 0.228222i \(-0.0732912\pi\)
−0.684451 + 0.729059i \(0.739958\pi\)
\(824\) 0 0
\(825\) −4.24612e7 −2.17199
\(826\) 0 0
\(827\) 5.80309e6 + 1.00513e7i 0.295050 + 0.511042i 0.974997 0.222220i \(-0.0713303\pi\)
−0.679946 + 0.733262i \(0.737997\pi\)
\(828\) 0 0
\(829\) −2.34279e7 −1.18399 −0.591993 0.805943i \(-0.701659\pi\)
−0.591993 + 0.805943i \(0.701659\pi\)
\(830\) 0 0
\(831\) 8.21433e6 1.42276e7i 0.412638 0.714711i
\(832\) 0 0
\(833\) −1.47105e7 + 2.54793e7i −0.734537 + 1.27226i
\(834\) 0 0
\(835\) −2.78513e7 −1.38239
\(836\) 0 0
\(837\) 1.14278e7 0.563833
\(838\) 0 0
\(839\) −1.33046e7 + 2.30442e7i −0.652523 + 1.13020i 0.329986 + 0.943986i \(0.392956\pi\)
−0.982509 + 0.186217i \(0.940377\pi\)
\(840\) 0 0
\(841\) 6.92396e6 1.19927e7i 0.337571 0.584690i
\(842\) 0 0
\(843\) 2.70832e7 1.31260
\(844\) 0 0
\(845\) −6.15783e6 1.06657e7i −0.296678 0.513862i
\(846\) 0 0
\(847\) 2.27554e7 1.08988
\(848\) 0 0
\(849\) 454389. + 787024.i 0.0216351 + 0.0374730i
\(850\) 0 0
\(851\) 3.60043e6 + 6.23613e6i 0.170424 + 0.295183i
\(852\) 0 0
\(853\) 1.75001e7 3.03111e7i 0.823510 1.42636i −0.0795426 0.996831i \(-0.525346\pi\)
0.903053 0.429530i \(-0.141321\pi\)
\(854\) 0 0
\(855\) −3.62193e6 + 7.92580e6i −0.169443 + 0.370790i
\(856\) 0 0
\(857\) −4.77062e6 + 8.26296e6i −0.221882 + 0.384312i −0.955380 0.295381i \(-0.904553\pi\)
0.733497 + 0.679692i \(0.237887\pi\)
\(858\) 0 0
\(859\) 2.79816e6 + 4.84656e6i 0.129387 + 0.224105i 0.923439 0.383745i \(-0.125366\pi\)
−0.794052 + 0.607849i \(0.792032\pi\)
\(860\) 0 0
\(861\) 5.62906e6 + 9.74983e6i 0.258779 + 0.448218i
\(862\) 0 0
\(863\) 6.93765e6 0.317092 0.158546 0.987352i \(-0.449319\pi\)
0.158546 + 0.987352i \(0.449319\pi\)
\(864\) 0 0
\(865\) −1.08507e7 1.87940e7i −0.493081 0.854041i
\(866\) 0 0
\(867\) 1.24829e7 0.563985
\(868\) 0 0
\(869\) 1.77168e7 3.06863e7i 0.795856 1.37846i
\(870\) 0 0
\(871\) 2.15086e7 3.72540e7i 0.960653 1.66390i
\(872\) 0 0
\(873\) −3.58686e6 −0.159286
\(874\) 0 0
\(875\) 4.99494e7 2.20552
\(876\) 0 0
\(877\) −5.50749e6 + 9.53926e6i −0.241799 + 0.418809i −0.961227 0.275759i \(-0.911071\pi\)
0.719428 + 0.694567i \(0.244404\pi\)
\(878\) 0 0
\(879\) −1.34975e7 + 2.33783e7i −0.589224 + 1.02057i
\(880\) 0 0
\(881\) −3.01700e7 −1.30959 −0.654796 0.755806i \(-0.727246\pi\)
−0.654796 + 0.755806i \(0.727246\pi\)
\(882\) 0 0
\(883\) 1.52557e7 + 2.64237e7i 0.658464 + 1.14049i 0.981013 + 0.193940i \(0.0621267\pi\)
−0.322550 + 0.946552i \(0.604540\pi\)
\(884\) 0 0
\(885\) 4.69243e7 2.01391
\(886\) 0 0
\(887\) 3.89963e6 + 6.75436e6i 0.166423 + 0.288254i 0.937160 0.348900i \(-0.113445\pi\)
−0.770736 + 0.637154i \(0.780111\pi\)
\(888\) 0 0
\(889\) −27267.6 47228.8i −0.00115716 0.00200425i
\(890\) 0 0
\(891\) −1.09923e7 + 1.90392e7i −0.463867 + 0.803441i
\(892\) 0 0
\(893\) 3.86007e7 3.67641e6i 1.61982 0.154275i
\(894\) 0 0
\(895\) −1.17361e7 + 2.03275e7i −0.489740 + 0.848254i
\(896\) 0 0
\(897\) 4.78470e6 + 8.28734e6i 0.198552 + 0.343901i
\(898\) 0 0
\(899\) 3.60198e6 + 6.23882e6i 0.148642 + 0.257456i
\(900\) 0 0
\(901\) 4.73971e7 1.94509
\(902\) 0 0
\(903\) −8.05465e6 1.39511e7i −0.328721 0.569361i
\(904\) 0 0
\(905\) −6.72158e7 −2.72803
\(906\) 0 0
\(907\) −3.48982e6 + 6.04454e6i −0.140859 + 0.243975i −0.927820 0.373027i \(-0.878320\pi\)
0.786961 + 0.617002i \(0.211653\pi\)
\(908\) 0 0
\(909\) −343027. + 594141.i −0.0137695 + 0.0238495i
\(910\) 0 0
\(911\) −2.39353e7 −0.955528 −0.477764 0.878488i \(-0.658553\pi\)
−0.477764 + 0.878488i \(0.658553\pi\)
\(912\) 0 0
\(913\) 1.16175e7 0.461247
\(914\) 0 0
\(915\) 3.72618e6 6.45393e6i 0.147133 0.254842i
\(916\) 0 0
\(917\) −4.00350e6 + 6.93427e6i −0.157223 + 0.272319i
\(918\) 0 0
\(919\) −9.03520e6 −0.352898 −0.176449 0.984310i \(-0.556461\pi\)
−0.176449 + 0.984310i \(0.556461\pi\)
\(920\) 0 0
\(921\) 1.26680e7 + 2.19417e7i 0.492108 + 0.852356i
\(922\) 0 0
\(923\) −2.09916e7 −0.811037
\(924\) 0 0
\(925\) −2.13347e7 3.69528e7i −0.819846 1.42002i
\(926\) 0 0
\(927\) −1.68170e6 2.91280e6i −0.0642762 0.111330i
\(928\) 0 0
\(929\) −1.55430e6 + 2.69212e6i −0.0590874 + 0.102342i −0.894056 0.447955i \(-0.852152\pi\)
0.834969 + 0.550298i \(0.185486\pi\)
\(930\) 0 0
\(931\) 1.25834e7 2.75359e7i 0.475798 1.04118i
\(932\) 0 0
\(933\) −1.17942e7 + 2.04281e7i −0.443571 + 0.768288i
\(934\) 0 0
\(935\) 3.84869e7 + 6.66613e7i 1.43974 + 2.49370i
\(936\) 0 0
\(937\) −1.85807e7 3.21827e7i −0.691374 1.19750i −0.971388 0.237499i \(-0.923672\pi\)
0.280013 0.959996i \(-0.409661\pi\)
\(938\) 0 0
\(939\) −722074. −0.0267250
\(940\) 0 0
\(941\) 8.14651e6 + 1.41102e7i 0.299915 + 0.519467i 0.976116 0.217249i \(-0.0697085\pi\)
−0.676202 + 0.736717i \(0.736375\pi\)
\(942\) 0 0
\(943\) 4.34078e6 0.158960
\(944\) 0 0
\(945\) 3.69184e7 6.39446e7i 1.34482 2.32929i
\(946\) 0 0
\(947\) 1.90876e7 3.30607e7i 0.691634 1.19795i −0.279668 0.960097i \(-0.590224\pi\)
0.971302 0.237849i \(-0.0764424\pi\)
\(948\) 0 0
\(949\) 5.52108e7 1.99003
\(950\) 0 0
\(951\) 1.87694e7 0.672975
\(952\) 0 0
\(953\) −3.93217e6 + 6.81072e6i −0.140249 + 0.242919i −0.927590 0.373599i \(-0.878124\pi\)
0.787341 + 0.616517i \(0.211457\pi\)
\(954\) 0 0
\(955\) −498713. + 863795.i −0.0176946 + 0.0306480i
\(956\) 0 0
\(957\) −1.85928e7 −0.656243
\(958\) 0 0
\(959\) −9.93317e6 1.72047e7i −0.348772 0.604090i
\(960\) 0 0
\(961\) −2.08406e7 −0.727949
\(962\) 0 0
\(963\) −2.32924e6 4.03436e6i −0.0809373 0.140188i
\(964\) 0 0
\(965\) 4.71825e7 + 8.17225e7i 1.63103 + 2.82503i
\(966\) 0 0
\(967\) 1.96743e6 3.40769e6i 0.0676601 0.117191i −0.830211 0.557450i \(-0.811780\pi\)
0.897871 + 0.440259i \(0.145113\pi\)
\(968\) 0 0
\(969\) −3.25540e7 + 3.10050e6i −1.11377 + 0.106077i
\(970\) 0 0
\(971\) −2.28377e6 + 3.95561e6i −0.0777329 + 0.134637i −0.902271 0.431168i \(-0.858101\pi\)
0.824539 + 0.565806i \(0.191435\pi\)
\(972\) 0 0
\(973\) 1.22279e7 + 2.11793e7i 0.414065 + 0.717181i
\(974\) 0 0
\(975\) −2.83522e7 4.91075e7i −0.955158 1.65438i
\(976\) 0 0
\(977\) −1.77743e7 −0.595739 −0.297870 0.954607i \(-0.596276\pi\)
−0.297870 + 0.954607i \(0.596276\pi\)
\(978\) 0 0
\(979\) 2.40489e7 + 4.16539e7i 0.801933 + 1.38899i
\(980\) 0 0
\(981\) −1.14663e7 −0.380409
\(982\) 0 0
\(983\) 9.38981e6 1.62636e7i 0.309937 0.536826i −0.668412 0.743792i \(-0.733025\pi\)
0.978348 + 0.206966i \(0.0663588\pi\)
\(984\) 0 0
\(985\) 8.79937e6 1.52410e7i 0.288975 0.500520i
\(986\) 0 0
\(987\) −6.35811e7 −2.07747
\(988\) 0 0
\(989\) −6.21124e6 −0.201924
\(990\) 0 0
\(991\) −2.05047e7 + 3.55152e7i −0.663239 + 1.14876i 0.316521 + 0.948586i \(0.397485\pi\)
−0.979760 + 0.200178i \(0.935848\pi\)
\(992\) 0 0
\(993\) 538295. 932354.i 0.0173240 0.0300060i
\(994\) 0 0
\(995\) −8.19624e6 −0.262456
\(996\) 0 0
\(997\) −1.23888e7 2.14579e7i −0.394720 0.683676i 0.598345 0.801239i \(-0.295825\pi\)
−0.993065 + 0.117563i \(0.962492\pi\)
\(998\) 0 0
\(999\) −2.96389e7 −0.939611
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 76.6.e.a.49.3 yes 18
3.2 odd 2 684.6.k.f.505.9 18
4.3 odd 2 304.6.i.d.49.7 18
19.7 even 3 inner 76.6.e.a.45.3 18
57.26 odd 6 684.6.k.f.577.9 18
76.7 odd 6 304.6.i.d.273.7 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.e.a.45.3 18 19.7 even 3 inner
76.6.e.a.49.3 yes 18 1.1 even 1 trivial
304.6.i.d.49.7 18 4.3 odd 2
304.6.i.d.273.7 18 76.7 odd 6
684.6.k.f.505.9 18 3.2 odd 2
684.6.k.f.577.9 18 57.26 odd 6