Properties

Label 531.8.a.e.1.12
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,8,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

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: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} + \cdots + 51\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-6.66616\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.66616 q^{2} -95.8947 q^{4} +15.0142 q^{5} +499.350 q^{7} -1268.62 q^{8} +O(q^{10})\) \(q+5.66616 q^{2} -95.8947 q^{4} +15.0142 q^{5} +499.350 q^{7} -1268.62 q^{8} +85.0729 q^{10} -5251.00 q^{11} +114.657 q^{13} +2829.39 q^{14} +5086.30 q^{16} +16772.2 q^{17} +5407.16 q^{19} -1439.78 q^{20} -29753.0 q^{22} +74684.5 q^{23} -77899.6 q^{25} +649.662 q^{26} -47885.0 q^{28} -58710.0 q^{29} +194742. q^{31} +191203. q^{32} +95033.9 q^{34} +7497.35 q^{35} +319137. q^{37} +30637.8 q^{38} -19047.4 q^{40} -172353. q^{41} +17400.1 q^{43} +503543. q^{44} +423174. q^{46} -701890. q^{47} -574193. q^{49} -441391. q^{50} -10995.0 q^{52} -467762. q^{53} -78839.7 q^{55} -633486. q^{56} -332660. q^{58} -205379. q^{59} -878795. q^{61} +1.10344e6 q^{62} +432342. q^{64} +1721.48 q^{65} -301315. q^{67} -1.60836e6 q^{68} +42481.1 q^{70} -722433. q^{71} +6.53656e6 q^{73} +1.80828e6 q^{74} -518518. q^{76} -2.62209e6 q^{77} +3.78720e6 q^{79} +76366.8 q^{80} -976580. q^{82} +190031. q^{83} +251821. q^{85} +98591.5 q^{86} +6.66154e6 q^{88} -2.70581e6 q^{89} +57253.7 q^{91} -7.16185e6 q^{92} -3.97702e6 q^{94} +81184.2 q^{95} +1.47369e7 q^{97} -3.25347e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 24 q^{2} + 1358 q^{4} - 678 q^{5} + 3081 q^{7} - 4107 q^{8} + 3609 q^{10} - 15070 q^{11} + 13662 q^{13} - 20861 q^{14} + 60482 q^{16} - 71919 q^{17} + 56231 q^{19} - 143053 q^{20} + 274198 q^{22} - 150029 q^{23} + 399672 q^{25} - 182846 q^{26} + 434150 q^{28} - 591285 q^{29} + 426733 q^{31} - 1205630 q^{32} + 403548 q^{34} - 912879 q^{35} + 7703 q^{37} + 417859 q^{38} + 618020 q^{40} - 770959 q^{41} + 793050 q^{43} - 2591274 q^{44} - 4068019 q^{46} - 1410373 q^{47} + 1637427 q^{49} - 1021549 q^{50} - 3749190 q^{52} - 1037934 q^{53} + 331974 q^{55} + 391748 q^{56} + 653724 q^{58} - 3696822 q^{59} - 1374623 q^{61} - 5251718 q^{62} + 5077197 q^{64} - 3257170 q^{65} - 2436904 q^{67} - 14119909 q^{68} + 5185580 q^{70} - 14289172 q^{71} + 5482515 q^{73} - 14934154 q^{74} + 3822912 q^{76} - 23157109 q^{77} + 19786414 q^{79} - 31978143 q^{80} + 9749509 q^{82} - 30227337 q^{83} + 9946981 q^{85} - 44295864 q^{86} + 39970897 q^{88} - 31061677 q^{89} + 26377785 q^{91} - 4719698 q^{92} + 44488296 q^{94} - 15534599 q^{95} + 12084118 q^{97} - 42274744 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.66616 0.500822 0.250411 0.968140i \(-0.419434\pi\)
0.250411 + 0.968140i \(0.419434\pi\)
\(3\) 0 0
\(4\) −95.8947 −0.749177
\(5\) 15.0142 0.0537165 0.0268582 0.999639i \(-0.491450\pi\)
0.0268582 + 0.999639i \(0.491450\pi\)
\(6\) 0 0
\(7\) 499.350 0.550252 0.275126 0.961408i \(-0.411280\pi\)
0.275126 + 0.961408i \(0.411280\pi\)
\(8\) −1268.62 −0.876027
\(9\) 0 0
\(10\) 85.0729 0.0269024
\(11\) −5251.00 −1.18951 −0.594755 0.803907i \(-0.702751\pi\)
−0.594755 + 0.803907i \(0.702751\pi\)
\(12\) 0 0
\(13\) 114.657 0.0144743 0.00723714 0.999974i \(-0.497696\pi\)
0.00723714 + 0.999974i \(0.497696\pi\)
\(14\) 2829.39 0.275579
\(15\) 0 0
\(16\) 5086.30 0.310443
\(17\) 16772.2 0.827977 0.413989 0.910282i \(-0.364135\pi\)
0.413989 + 0.910282i \(0.364135\pi\)
\(18\) 0 0
\(19\) 5407.16 0.180855 0.0904277 0.995903i \(-0.471177\pi\)
0.0904277 + 0.995903i \(0.471177\pi\)
\(20\) −1439.78 −0.0402432
\(21\) 0 0
\(22\) −29753.0 −0.595733
\(23\) 74684.5 1.27992 0.639961 0.768408i \(-0.278951\pi\)
0.639961 + 0.768408i \(0.278951\pi\)
\(24\) 0 0
\(25\) −77899.6 −0.997115
\(26\) 649.662 0.00724905
\(27\) 0 0
\(28\) −47885.0 −0.412236
\(29\) −58710.0 −0.447012 −0.223506 0.974703i \(-0.571750\pi\)
−0.223506 + 0.974703i \(0.571750\pi\)
\(30\) 0 0
\(31\) 194742. 1.17407 0.587034 0.809562i \(-0.300296\pi\)
0.587034 + 0.809562i \(0.300296\pi\)
\(32\) 191203. 1.03150
\(33\) 0 0
\(34\) 95033.9 0.414670
\(35\) 7497.35 0.0295576
\(36\) 0 0
\(37\) 319137. 1.03579 0.517895 0.855444i \(-0.326716\pi\)
0.517895 + 0.855444i \(0.326716\pi\)
\(38\) 30637.8 0.0905764
\(39\) 0 0
\(40\) −19047.4 −0.0470571
\(41\) −172353. −0.390549 −0.195275 0.980749i \(-0.562560\pi\)
−0.195275 + 0.980749i \(0.562560\pi\)
\(42\) 0 0
\(43\) 17400.1 0.0333742 0.0166871 0.999861i \(-0.494688\pi\)
0.0166871 + 0.999861i \(0.494688\pi\)
\(44\) 503543. 0.891153
\(45\) 0 0
\(46\) 423174. 0.641013
\(47\) −701890. −0.986113 −0.493057 0.869997i \(-0.664120\pi\)
−0.493057 + 0.869997i \(0.664120\pi\)
\(48\) 0 0
\(49\) −574193. −0.697223
\(50\) −441391. −0.499377
\(51\) 0 0
\(52\) −10995.0 −0.0108438
\(53\) −467762. −0.431578 −0.215789 0.976440i \(-0.569232\pi\)
−0.215789 + 0.976440i \(0.569232\pi\)
\(54\) 0 0
\(55\) −78839.7 −0.0638963
\(56\) −633486. −0.482036
\(57\) 0 0
\(58\) −332660. −0.223874
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −878795. −0.495716 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(62\) 1.10344e6 0.587999
\(63\) 0 0
\(64\) 432342. 0.206157
\(65\) 1721.48 0.000777508 0
\(66\) 0 0
\(67\) −301315. −0.122394 −0.0611968 0.998126i \(-0.519492\pi\)
−0.0611968 + 0.998126i \(0.519492\pi\)
\(68\) −1.60836e6 −0.620302
\(69\) 0 0
\(70\) 42481.1 0.0148031
\(71\) −722433. −0.239548 −0.119774 0.992801i \(-0.538217\pi\)
−0.119774 + 0.992801i \(0.538217\pi\)
\(72\) 0 0
\(73\) 6.53656e6 1.96661 0.983307 0.181956i \(-0.0582429\pi\)
0.983307 + 0.181956i \(0.0582429\pi\)
\(74\) 1.80828e6 0.518746
\(75\) 0 0
\(76\) −518518. −0.135493
\(77\) −2.62209e6 −0.654530
\(78\) 0 0
\(79\) 3.78720e6 0.864217 0.432109 0.901822i \(-0.357770\pi\)
0.432109 + 0.901822i \(0.357770\pi\)
\(80\) 76366.8 0.0166759
\(81\) 0 0
\(82\) −976580. −0.195596
\(83\) 190031. 0.0364797 0.0182398 0.999834i \(-0.494194\pi\)
0.0182398 + 0.999834i \(0.494194\pi\)
\(84\) 0 0
\(85\) 251821. 0.0444760
\(86\) 98591.5 0.0167146
\(87\) 0 0
\(88\) 6.66154e6 1.04204
\(89\) −2.70581e6 −0.406848 −0.203424 0.979091i \(-0.565207\pi\)
−0.203424 + 0.979091i \(0.565207\pi\)
\(90\) 0 0
\(91\) 57253.7 0.00796451
\(92\) −7.16185e6 −0.958888
\(93\) 0 0
\(94\) −3.97702e6 −0.493867
\(95\) 81184.2 0.00971492
\(96\) 0 0
\(97\) 1.47369e7 1.63948 0.819738 0.572739i \(-0.194119\pi\)
0.819738 + 0.572739i \(0.194119\pi\)
\(98\) −3.25347e6 −0.349185
\(99\) 0 0
\(100\) 7.47015e6 0.747015
\(101\) −3.16363e6 −0.305535 −0.152767 0.988262i \(-0.548819\pi\)
−0.152767 + 0.988262i \(0.548819\pi\)
\(102\) 0 0
\(103\) −8.88176e6 −0.800883 −0.400441 0.916322i \(-0.631143\pi\)
−0.400441 + 0.916322i \(0.631143\pi\)
\(104\) −145456. −0.0126799
\(105\) 0 0
\(106\) −2.65041e6 −0.216144
\(107\) −1.68081e7 −1.32641 −0.663203 0.748440i \(-0.730803\pi\)
−0.663203 + 0.748440i \(0.730803\pi\)
\(108\) 0 0
\(109\) 6.20279e6 0.458769 0.229385 0.973336i \(-0.426329\pi\)
0.229385 + 0.973336i \(0.426329\pi\)
\(110\) −446718. −0.0320007
\(111\) 0 0
\(112\) 2.53984e6 0.170822
\(113\) −1.00745e7 −0.656821 −0.328410 0.944535i \(-0.606513\pi\)
−0.328410 + 0.944535i \(0.606513\pi\)
\(114\) 0 0
\(115\) 1.12133e6 0.0687529
\(116\) 5.62998e6 0.334891
\(117\) 0 0
\(118\) −1.16371e6 −0.0652015
\(119\) 8.37519e6 0.455596
\(120\) 0 0
\(121\) 8.08587e6 0.414933
\(122\) −4.97939e6 −0.248266
\(123\) 0 0
\(124\) −1.86747e7 −0.879585
\(125\) −2.34259e6 −0.107278
\(126\) 0 0
\(127\) 8.08324e6 0.350165 0.175082 0.984554i \(-0.443981\pi\)
0.175082 + 0.984554i \(0.443981\pi\)
\(128\) −2.20243e7 −0.928256
\(129\) 0 0
\(130\) 9754.17 0.000389393 0
\(131\) −4.08601e7 −1.58800 −0.793999 0.607919i \(-0.792005\pi\)
−0.793999 + 0.607919i \(0.792005\pi\)
\(132\) 0 0
\(133\) 2.70006e6 0.0995161
\(134\) −1.70730e6 −0.0612974
\(135\) 0 0
\(136\) −2.12776e7 −0.725331
\(137\) −4.34315e7 −1.44306 −0.721528 0.692385i \(-0.756560\pi\)
−0.721528 + 0.692385i \(0.756560\pi\)
\(138\) 0 0
\(139\) −3.00038e7 −0.947598 −0.473799 0.880633i \(-0.657118\pi\)
−0.473799 + 0.880633i \(0.657118\pi\)
\(140\) −718955. −0.0221439
\(141\) 0 0
\(142\) −4.09342e6 −0.119971
\(143\) −602062. −0.0172173
\(144\) 0 0
\(145\) −881485. −0.0240119
\(146\) 3.70372e7 0.984924
\(147\) 0 0
\(148\) −3.06036e7 −0.775990
\(149\) 2.00795e7 0.497279 0.248640 0.968596i \(-0.420017\pi\)
0.248640 + 0.968596i \(0.420017\pi\)
\(150\) 0 0
\(151\) −4.42468e7 −1.04583 −0.522917 0.852384i \(-0.675156\pi\)
−0.522917 + 0.852384i \(0.675156\pi\)
\(152\) −6.85964e6 −0.158434
\(153\) 0 0
\(154\) −1.48572e7 −0.327803
\(155\) 2.92390e6 0.0630668
\(156\) 0 0
\(157\) −5.91607e7 −1.22007 −0.610035 0.792375i \(-0.708844\pi\)
−0.610035 + 0.792375i \(0.708844\pi\)
\(158\) 2.14588e7 0.432819
\(159\) 0 0
\(160\) 2.87077e6 0.0554088
\(161\) 3.72937e7 0.704279
\(162\) 0 0
\(163\) −3.45882e7 −0.625564 −0.312782 0.949825i \(-0.601261\pi\)
−0.312782 + 0.949825i \(0.601261\pi\)
\(164\) 1.65277e7 0.292590
\(165\) 0 0
\(166\) 1.07674e6 0.0182698
\(167\) 1.07115e8 1.77969 0.889844 0.456265i \(-0.150813\pi\)
0.889844 + 0.456265i \(0.150813\pi\)
\(168\) 0 0
\(169\) −6.27354e7 −0.999790
\(170\) 1.42686e6 0.0222746
\(171\) 0 0
\(172\) −1.66857e6 −0.0250032
\(173\) −2.97790e7 −0.437269 −0.218634 0.975807i \(-0.570160\pi\)
−0.218634 + 0.975807i \(0.570160\pi\)
\(174\) 0 0
\(175\) −3.88991e7 −0.548664
\(176\) −2.67082e7 −0.369275
\(177\) 0 0
\(178\) −1.53315e7 −0.203758
\(179\) −1.22880e8 −1.60139 −0.800693 0.599075i \(-0.795535\pi\)
−0.800693 + 0.599075i \(0.795535\pi\)
\(180\) 0 0
\(181\) 7.94319e7 0.995680 0.497840 0.867269i \(-0.334127\pi\)
0.497840 + 0.867269i \(0.334127\pi\)
\(182\) 324409. 0.00398880
\(183\) 0 0
\(184\) −9.47465e7 −1.12125
\(185\) 4.79160e6 0.0556390
\(186\) 0 0
\(187\) −8.80708e7 −0.984887
\(188\) 6.73075e7 0.738773
\(189\) 0 0
\(190\) 460003. 0.00486545
\(191\) −1.37986e8 −1.43291 −0.716453 0.697635i \(-0.754236\pi\)
−0.716453 + 0.697635i \(0.754236\pi\)
\(192\) 0 0
\(193\) 871407. 0.00872510 0.00436255 0.999990i \(-0.498611\pi\)
0.00436255 + 0.999990i \(0.498611\pi\)
\(194\) 8.35016e7 0.821086
\(195\) 0 0
\(196\) 5.50620e7 0.522343
\(197\) −1.71832e8 −1.60130 −0.800648 0.599135i \(-0.795511\pi\)
−0.800648 + 0.599135i \(0.795511\pi\)
\(198\) 0 0
\(199\) −9.52117e7 −0.856455 −0.428227 0.903671i \(-0.640862\pi\)
−0.428227 + 0.903671i \(0.640862\pi\)
\(200\) 9.88251e7 0.873499
\(201\) 0 0
\(202\) −1.79256e7 −0.153019
\(203\) −2.93168e7 −0.245969
\(204\) 0 0
\(205\) −2.58775e6 −0.0209789
\(206\) −5.03255e7 −0.401100
\(207\) 0 0
\(208\) 583178. 0.00449344
\(209\) −2.83930e7 −0.215129
\(210\) 0 0
\(211\) 1.10926e8 0.812917 0.406458 0.913669i \(-0.366764\pi\)
0.406458 + 0.913669i \(0.366764\pi\)
\(212\) 4.48559e7 0.323328
\(213\) 0 0
\(214\) −9.52375e7 −0.664293
\(215\) 261248. 0.00179275
\(216\) 0 0
\(217\) 9.72443e7 0.646033
\(218\) 3.51460e7 0.229762
\(219\) 0 0
\(220\) 7.56031e6 0.0478696
\(221\) 1.92304e6 0.0119844
\(222\) 0 0
\(223\) −1.48718e8 −0.898042 −0.449021 0.893521i \(-0.648227\pi\)
−0.449021 + 0.893521i \(0.648227\pi\)
\(224\) 9.54774e7 0.567587
\(225\) 0 0
\(226\) −5.70834e7 −0.328951
\(227\) −4.45860e7 −0.252993 −0.126496 0.991967i \(-0.540373\pi\)
−0.126496 + 0.991967i \(0.540373\pi\)
\(228\) 0 0
\(229\) −9.60064e7 −0.528294 −0.264147 0.964482i \(-0.585090\pi\)
−0.264147 + 0.964482i \(0.585090\pi\)
\(230\) 6.35363e6 0.0344330
\(231\) 0 0
\(232\) 7.44809e7 0.391595
\(233\) 1.49924e8 0.776470 0.388235 0.921560i \(-0.373085\pi\)
0.388235 + 0.921560i \(0.373085\pi\)
\(234\) 0 0
\(235\) −1.05383e7 −0.0529705
\(236\) 1.96947e7 0.0975345
\(237\) 0 0
\(238\) 4.74551e7 0.228173
\(239\) 4.99588e7 0.236712 0.118356 0.992971i \(-0.462238\pi\)
0.118356 + 0.992971i \(0.462238\pi\)
\(240\) 0 0
\(241\) 3.94054e6 0.0181341 0.00906705 0.999959i \(-0.497114\pi\)
0.00906705 + 0.999959i \(0.497114\pi\)
\(242\) 4.58158e7 0.207808
\(243\) 0 0
\(244\) 8.42717e7 0.371379
\(245\) −8.62106e6 −0.0374524
\(246\) 0 0
\(247\) 619966. 0.00261775
\(248\) −2.47054e8 −1.02851
\(249\) 0 0
\(250\) −1.32735e7 −0.0537272
\(251\) 6.89865e7 0.275363 0.137682 0.990477i \(-0.456035\pi\)
0.137682 + 0.990477i \(0.456035\pi\)
\(252\) 0 0
\(253\) −3.92169e8 −1.52248
\(254\) 4.58009e7 0.175370
\(255\) 0 0
\(256\) −1.80133e8 −0.671048
\(257\) −6.74827e7 −0.247986 −0.123993 0.992283i \(-0.539570\pi\)
−0.123993 + 0.992283i \(0.539570\pi\)
\(258\) 0 0
\(259\) 1.59361e8 0.569945
\(260\) −165081. −0.000582491 0
\(261\) 0 0
\(262\) −2.31520e8 −0.795305
\(263\) −7.91658e7 −0.268344 −0.134172 0.990958i \(-0.542838\pi\)
−0.134172 + 0.990958i \(0.542838\pi\)
\(264\) 0 0
\(265\) −7.02308e6 −0.0231829
\(266\) 1.52990e7 0.0498399
\(267\) 0 0
\(268\) 2.88945e7 0.0916944
\(269\) −2.24443e8 −0.703030 −0.351515 0.936182i \(-0.614333\pi\)
−0.351515 + 0.936182i \(0.614333\pi\)
\(270\) 0 0
\(271\) 5.85302e7 0.178644 0.0893218 0.996003i \(-0.471530\pi\)
0.0893218 + 0.996003i \(0.471530\pi\)
\(272\) 8.53084e7 0.257040
\(273\) 0 0
\(274\) −2.46090e8 −0.722715
\(275\) 4.09051e8 1.18608
\(276\) 0 0
\(277\) 3.20396e8 0.905749 0.452874 0.891574i \(-0.350399\pi\)
0.452874 + 0.891574i \(0.350399\pi\)
\(278\) −1.70006e8 −0.474578
\(279\) 0 0
\(280\) −9.51130e6 −0.0258933
\(281\) 2.95974e8 0.795758 0.397879 0.917438i \(-0.369746\pi\)
0.397879 + 0.917438i \(0.369746\pi\)
\(282\) 0 0
\(283\) −2.63828e8 −0.691940 −0.345970 0.938246i \(-0.612450\pi\)
−0.345970 + 0.938246i \(0.612450\pi\)
\(284\) 6.92775e7 0.179464
\(285\) 0 0
\(286\) −3.41138e6 −0.00862281
\(287\) −8.60645e7 −0.214901
\(288\) 0 0
\(289\) −1.29032e8 −0.314453
\(290\) −4.99463e6 −0.0120257
\(291\) 0 0
\(292\) −6.26821e8 −1.47334
\(293\) −3.79473e8 −0.881342 −0.440671 0.897669i \(-0.645259\pi\)
−0.440671 + 0.897669i \(0.645259\pi\)
\(294\) 0 0
\(295\) −3.08360e6 −0.00699329
\(296\) −4.04865e8 −0.907379
\(297\) 0 0
\(298\) 1.13773e8 0.249048
\(299\) 8.56307e6 0.0185260
\(300\) 0 0
\(301\) 8.68872e6 0.0183642
\(302\) −2.50709e8 −0.523777
\(303\) 0 0
\(304\) 2.75024e7 0.0561453
\(305\) −1.31944e7 −0.0266281
\(306\) 0 0
\(307\) 4.67759e8 0.922652 0.461326 0.887231i \(-0.347374\pi\)
0.461326 + 0.887231i \(0.347374\pi\)
\(308\) 2.51444e8 0.490359
\(309\) 0 0
\(310\) 1.65673e7 0.0315853
\(311\) 3.46804e7 0.0653767 0.0326883 0.999466i \(-0.489593\pi\)
0.0326883 + 0.999466i \(0.489593\pi\)
\(312\) 0 0
\(313\) 3.26866e8 0.602511 0.301256 0.953543i \(-0.402594\pi\)
0.301256 + 0.953543i \(0.402594\pi\)
\(314\) −3.35214e8 −0.611038
\(315\) 0 0
\(316\) −3.63172e8 −0.647452
\(317\) −4.00162e8 −0.705552 −0.352776 0.935708i \(-0.614762\pi\)
−0.352776 + 0.935708i \(0.614762\pi\)
\(318\) 0 0
\(319\) 3.08287e8 0.531725
\(320\) 6.49128e6 0.0110740
\(321\) 0 0
\(322\) 2.11312e8 0.352719
\(323\) 9.06899e7 0.149744
\(324\) 0 0
\(325\) −8.93170e6 −0.0144325
\(326\) −1.95982e8 −0.313296
\(327\) 0 0
\(328\) 2.18651e8 0.342132
\(329\) −3.50489e8 −0.542611
\(330\) 0 0
\(331\) −2.08461e8 −0.315957 −0.157978 0.987443i \(-0.550498\pi\)
−0.157978 + 0.987443i \(0.550498\pi\)
\(332\) −1.82229e7 −0.0273297
\(333\) 0 0
\(334\) 6.06932e8 0.891308
\(335\) −4.52400e6 −0.00657455
\(336\) 0 0
\(337\) −1.17533e9 −1.67284 −0.836422 0.548086i \(-0.815357\pi\)
−0.836422 + 0.548086i \(0.815357\pi\)
\(338\) −3.55468e8 −0.500717
\(339\) 0 0
\(340\) −2.41483e7 −0.0333204
\(341\) −1.02259e9 −1.39656
\(342\) 0 0
\(343\) −6.97959e8 −0.933900
\(344\) −2.20741e7 −0.0292367
\(345\) 0 0
\(346\) −1.68732e8 −0.218994
\(347\) −3.03478e8 −0.389919 −0.194960 0.980811i \(-0.562458\pi\)
−0.194960 + 0.980811i \(0.562458\pi\)
\(348\) 0 0
\(349\) 7.65904e8 0.964463 0.482232 0.876044i \(-0.339826\pi\)
0.482232 + 0.876044i \(0.339826\pi\)
\(350\) −2.20409e8 −0.274783
\(351\) 0 0
\(352\) −1.00401e9 −1.22698
\(353\) −1.07514e9 −1.30092 −0.650462 0.759539i \(-0.725425\pi\)
−0.650462 + 0.759539i \(0.725425\pi\)
\(354\) 0 0
\(355\) −1.08468e7 −0.0128677
\(356\) 2.59473e8 0.304801
\(357\) 0 0
\(358\) −6.96258e8 −0.802010
\(359\) 8.01759e8 0.914562 0.457281 0.889322i \(-0.348823\pi\)
0.457281 + 0.889322i \(0.348823\pi\)
\(360\) 0 0
\(361\) −8.64634e8 −0.967291
\(362\) 4.50074e8 0.498659
\(363\) 0 0
\(364\) −5.49033e6 −0.00596683
\(365\) 9.81413e7 0.105640
\(366\) 0 0
\(367\) 2.39324e8 0.252729 0.126364 0.991984i \(-0.459669\pi\)
0.126364 + 0.991984i \(0.459669\pi\)
\(368\) 3.79868e8 0.397343
\(369\) 0 0
\(370\) 2.71499e7 0.0278652
\(371\) −2.33577e8 −0.237477
\(372\) 0 0
\(373\) 1.44976e9 1.44649 0.723245 0.690592i \(-0.242650\pi\)
0.723245 + 0.690592i \(0.242650\pi\)
\(374\) −4.99023e8 −0.493253
\(375\) 0 0
\(376\) 8.90434e8 0.863862
\(377\) −6.73149e6 −0.00647018
\(378\) 0 0
\(379\) 5.83143e8 0.550221 0.275111 0.961413i \(-0.411285\pi\)
0.275111 + 0.961413i \(0.411285\pi\)
\(380\) −7.78513e6 −0.00727819
\(381\) 0 0
\(382\) −7.81850e8 −0.717631
\(383\) 2.12967e9 1.93694 0.968472 0.249121i \(-0.0801417\pi\)
0.968472 + 0.249121i \(0.0801417\pi\)
\(384\) 0 0
\(385\) −3.93686e7 −0.0351591
\(386\) 4.93753e6 0.00436973
\(387\) 0 0
\(388\) −1.41319e9 −1.22826
\(389\) −1.02632e9 −0.884017 −0.442008 0.897011i \(-0.645734\pi\)
−0.442008 + 0.897011i \(0.645734\pi\)
\(390\) 0 0
\(391\) 1.25262e9 1.05975
\(392\) 7.28434e8 0.610786
\(393\) 0 0
\(394\) −9.73626e8 −0.801965
\(395\) 5.68618e7 0.0464227
\(396\) 0 0
\(397\) 8.48731e8 0.680775 0.340387 0.940285i \(-0.389442\pi\)
0.340387 + 0.940285i \(0.389442\pi\)
\(398\) −5.39484e8 −0.428932
\(399\) 0 0
\(400\) −3.96221e8 −0.309547
\(401\) 1.05558e9 0.817498 0.408749 0.912647i \(-0.365965\pi\)
0.408749 + 0.912647i \(0.365965\pi\)
\(402\) 0 0
\(403\) 2.23284e7 0.0169938
\(404\) 3.03375e8 0.228900
\(405\) 0 0
\(406\) −1.66114e8 −0.123187
\(407\) −1.67579e9 −1.23208
\(408\) 0 0
\(409\) 3.62710e7 0.0262137 0.0131068 0.999914i \(-0.495828\pi\)
0.0131068 + 0.999914i \(0.495828\pi\)
\(410\) −1.46626e7 −0.0105067
\(411\) 0 0
\(412\) 8.51714e8 0.600003
\(413\) −1.02556e8 −0.0716367
\(414\) 0 0
\(415\) 2.85316e6 0.00195956
\(416\) 2.19227e7 0.0149303
\(417\) 0 0
\(418\) −1.60879e8 −0.107742
\(419\) 2.92602e8 0.194325 0.0971623 0.995269i \(-0.469023\pi\)
0.0971623 + 0.995269i \(0.469023\pi\)
\(420\) 0 0
\(421\) −1.22412e9 −0.799534 −0.399767 0.916617i \(-0.630909\pi\)
−0.399767 + 0.916617i \(0.630909\pi\)
\(422\) 6.28526e8 0.407127
\(423\) 0 0
\(424\) 5.93413e8 0.378074
\(425\) −1.30655e9 −0.825588
\(426\) 0 0
\(427\) −4.38826e8 −0.272769
\(428\) 1.61181e9 0.993712
\(429\) 0 0
\(430\) 1.48027e6 0.000897848 0
\(431\) 6.26512e8 0.376929 0.188464 0.982080i \(-0.439649\pi\)
0.188464 + 0.982080i \(0.439649\pi\)
\(432\) 0 0
\(433\) −2.79784e9 −1.65621 −0.828106 0.560571i \(-0.810582\pi\)
−0.828106 + 0.560571i \(0.810582\pi\)
\(434\) 5.51001e8 0.323548
\(435\) 0 0
\(436\) −5.94814e8 −0.343699
\(437\) 4.03831e8 0.231481
\(438\) 0 0
\(439\) −6.09498e8 −0.343832 −0.171916 0.985112i \(-0.554996\pi\)
−0.171916 + 0.985112i \(0.554996\pi\)
\(440\) 1.00018e8 0.0559749
\(441\) 0 0
\(442\) 1.08963e7 0.00600205
\(443\) 1.80771e9 0.987905 0.493953 0.869489i \(-0.335552\pi\)
0.493953 + 0.869489i \(0.335552\pi\)
\(444\) 0 0
\(445\) −4.06256e7 −0.0218544
\(446\) −8.42660e8 −0.449760
\(447\) 0 0
\(448\) 2.15890e8 0.113438
\(449\) −6.32760e8 −0.329896 −0.164948 0.986302i \(-0.552746\pi\)
−0.164948 + 0.986302i \(0.552746\pi\)
\(450\) 0 0
\(451\) 9.05027e8 0.464562
\(452\) 9.66086e8 0.492075
\(453\) 0 0
\(454\) −2.52631e8 −0.126704
\(455\) 859620. 0.000427825 0
\(456\) 0 0
\(457\) −3.19962e8 −0.156816 −0.0784082 0.996921i \(-0.524984\pi\)
−0.0784082 + 0.996921i \(0.524984\pi\)
\(458\) −5.43987e8 −0.264582
\(459\) 0 0
\(460\) −1.07530e8 −0.0515081
\(461\) 9.55318e8 0.454145 0.227073 0.973878i \(-0.427085\pi\)
0.227073 + 0.973878i \(0.427085\pi\)
\(462\) 0 0
\(463\) 3.55134e9 1.66287 0.831437 0.555619i \(-0.187519\pi\)
0.831437 + 0.555619i \(0.187519\pi\)
\(464\) −2.98617e8 −0.138772
\(465\) 0 0
\(466\) 8.49491e8 0.388873
\(467\) −4.08842e9 −1.85758 −0.928788 0.370613i \(-0.879148\pi\)
−0.928788 + 0.370613i \(0.879148\pi\)
\(468\) 0 0
\(469\) −1.50461e8 −0.0673473
\(470\) −5.97119e7 −0.0265288
\(471\) 0 0
\(472\) 2.60548e8 0.114049
\(473\) −9.13679e7 −0.0396990
\(474\) 0 0
\(475\) −4.21215e8 −0.180334
\(476\) −8.03136e8 −0.341322
\(477\) 0 0
\(478\) 2.83075e8 0.118550
\(479\) −2.21317e9 −0.920112 −0.460056 0.887890i \(-0.652171\pi\)
−0.460056 + 0.887890i \(0.652171\pi\)
\(480\) 0 0
\(481\) 3.65912e7 0.0149923
\(482\) 2.23277e7 0.00908196
\(483\) 0 0
\(484\) −7.75392e8 −0.310858
\(485\) 2.21263e8 0.0880669
\(486\) 0 0
\(487\) −4.68349e9 −1.83746 −0.918731 0.394883i \(-0.870785\pi\)
−0.918731 + 0.394883i \(0.870785\pi\)
\(488\) 1.11486e9 0.434261
\(489\) 0 0
\(490\) −4.88483e7 −0.0187570
\(491\) 1.79902e9 0.685885 0.342942 0.939356i \(-0.388576\pi\)
0.342942 + 0.939356i \(0.388576\pi\)
\(492\) 0 0
\(493\) −9.84696e8 −0.370116
\(494\) 3.51283e6 0.00131103
\(495\) 0 0
\(496\) 9.90515e8 0.364481
\(497\) −3.60747e8 −0.131812
\(498\) 0 0
\(499\) −8.51942e8 −0.306943 −0.153472 0.988153i \(-0.549045\pi\)
−0.153472 + 0.988153i \(0.549045\pi\)
\(500\) 2.24642e8 0.0803702
\(501\) 0 0
\(502\) 3.90888e8 0.137908
\(503\) 3.29333e9 1.15385 0.576923 0.816798i \(-0.304253\pi\)
0.576923 + 0.816798i \(0.304253\pi\)
\(504\) 0 0
\(505\) −4.74994e7 −0.0164123
\(506\) −2.22209e9 −0.762491
\(507\) 0 0
\(508\) −7.75139e8 −0.262335
\(509\) 1.92293e9 0.646325 0.323162 0.946344i \(-0.395254\pi\)
0.323162 + 0.946344i \(0.395254\pi\)
\(510\) 0 0
\(511\) 3.26403e9 1.08213
\(512\) 1.79845e9 0.592180
\(513\) 0 0
\(514\) −3.82367e8 −0.124197
\(515\) −1.33353e8 −0.0430206
\(516\) 0 0
\(517\) 3.68563e9 1.17299
\(518\) 9.02965e8 0.285441
\(519\) 0 0
\(520\) −2.18391e6 −0.000681118 0
\(521\) −5.43560e9 −1.68390 −0.841949 0.539558i \(-0.818591\pi\)
−0.841949 + 0.539558i \(0.818591\pi\)
\(522\) 0 0
\(523\) −3.26406e9 −0.997705 −0.498853 0.866687i \(-0.666245\pi\)
−0.498853 + 0.866687i \(0.666245\pi\)
\(524\) 3.91827e9 1.18969
\(525\) 0 0
\(526\) −4.48566e8 −0.134393
\(527\) 3.26625e9 0.972102
\(528\) 0 0
\(529\) 2.17296e9 0.638199
\(530\) −3.97939e7 −0.0116105
\(531\) 0 0
\(532\) −2.58922e8 −0.0745551
\(533\) −1.97614e7 −0.00565292
\(534\) 0 0
\(535\) −2.52361e8 −0.0712498
\(536\) 3.82255e8 0.107220
\(537\) 0 0
\(538\) −1.27173e9 −0.352093
\(539\) 3.01509e9 0.829353
\(540\) 0 0
\(541\) 6.87940e9 1.86793 0.933964 0.357367i \(-0.116325\pi\)
0.933964 + 0.357367i \(0.116325\pi\)
\(542\) 3.31641e8 0.0894687
\(543\) 0 0
\(544\) 3.20690e9 0.854062
\(545\) 9.31300e7 0.0246435
\(546\) 0 0
\(547\) −3.21914e9 −0.840977 −0.420489 0.907298i \(-0.638141\pi\)
−0.420489 + 0.907298i \(0.638141\pi\)
\(548\) 4.16485e9 1.08110
\(549\) 0 0
\(550\) 2.31775e9 0.594014
\(551\) −3.17454e8 −0.0808446
\(552\) 0 0
\(553\) 1.89114e9 0.475537
\(554\) 1.81541e9 0.453619
\(555\) 0 0
\(556\) 2.87720e9 0.709918
\(557\) 2.80717e9 0.688295 0.344148 0.938916i \(-0.388168\pi\)
0.344148 + 0.938916i \(0.388168\pi\)
\(558\) 0 0
\(559\) 1.99503e6 0.000483068 0
\(560\) 3.81338e7 0.00917596
\(561\) 0 0
\(562\) 1.67703e9 0.398533
\(563\) −6.17910e8 −0.145930 −0.0729652 0.997334i \(-0.523246\pi\)
−0.0729652 + 0.997334i \(0.523246\pi\)
\(564\) 0 0
\(565\) −1.51260e8 −0.0352821
\(566\) −1.49489e9 −0.346539
\(567\) 0 0
\(568\) 9.16495e8 0.209851
\(569\) 4.23484e9 0.963705 0.481852 0.876252i \(-0.339964\pi\)
0.481852 + 0.876252i \(0.339964\pi\)
\(570\) 0 0
\(571\) 6.61606e8 0.148721 0.0743607 0.997231i \(-0.476308\pi\)
0.0743607 + 0.997231i \(0.476308\pi\)
\(572\) 5.77345e7 0.0128988
\(573\) 0 0
\(574\) −4.87655e8 −0.107627
\(575\) −5.81789e9 −1.27623
\(576\) 0 0
\(577\) 9.05424e8 0.196217 0.0981085 0.995176i \(-0.468721\pi\)
0.0981085 + 0.995176i \(0.468721\pi\)
\(578\) −7.31118e8 −0.157485
\(579\) 0 0
\(580\) 8.45297e7 0.0179892
\(581\) 9.48919e7 0.0200730
\(582\) 0 0
\(583\) 2.45622e9 0.513366
\(584\) −8.29242e9 −1.72281
\(585\) 0 0
\(586\) −2.15016e9 −0.441396
\(587\) −5.48391e9 −1.11907 −0.559534 0.828807i \(-0.689020\pi\)
−0.559534 + 0.828807i \(0.689020\pi\)
\(588\) 0 0
\(589\) 1.05300e9 0.212336
\(590\) −1.74722e7 −0.00350240
\(591\) 0 0
\(592\) 1.62323e9 0.321554
\(593\) 6.47258e9 1.27464 0.637318 0.770601i \(-0.280044\pi\)
0.637318 + 0.770601i \(0.280044\pi\)
\(594\) 0 0
\(595\) 1.25747e8 0.0244730
\(596\) −1.92551e9 −0.372550
\(597\) 0 0
\(598\) 4.85197e7 0.00927821
\(599\) −2.50605e8 −0.0476426 −0.0238213 0.999716i \(-0.507583\pi\)
−0.0238213 + 0.999716i \(0.507583\pi\)
\(600\) 0 0
\(601\) −4.78670e9 −0.899447 −0.449724 0.893168i \(-0.648477\pi\)
−0.449724 + 0.893168i \(0.648477\pi\)
\(602\) 4.92317e7 0.00919723
\(603\) 0 0
\(604\) 4.24303e9 0.783514
\(605\) 1.21403e8 0.0222888
\(606\) 0 0
\(607\) −2.08136e9 −0.377735 −0.188868 0.982003i \(-0.560482\pi\)
−0.188868 + 0.982003i \(0.560482\pi\)
\(608\) 1.03387e9 0.186553
\(609\) 0 0
\(610\) −7.47616e7 −0.0133360
\(611\) −8.04763e7 −0.0142733
\(612\) 0 0
\(613\) 9.25255e9 1.62237 0.811185 0.584790i \(-0.198823\pi\)
0.811185 + 0.584790i \(0.198823\pi\)
\(614\) 2.65040e9 0.462085
\(615\) 0 0
\(616\) 3.32644e9 0.573386
\(617\) −4.36405e9 −0.747982 −0.373991 0.927432i \(-0.622011\pi\)
−0.373991 + 0.927432i \(0.622011\pi\)
\(618\) 0 0
\(619\) −5.18190e9 −0.878156 −0.439078 0.898449i \(-0.644695\pi\)
−0.439078 + 0.898449i \(0.644695\pi\)
\(620\) −2.80386e8 −0.0472482
\(621\) 0 0
\(622\) 1.96505e8 0.0327421
\(623\) −1.35114e9 −0.223869
\(624\) 0 0
\(625\) 6.05073e9 0.991352
\(626\) 1.85208e9 0.301751
\(627\) 0 0
\(628\) 5.67319e9 0.914048
\(629\) 5.35263e9 0.857610
\(630\) 0 0
\(631\) 5.15467e9 0.816767 0.408383 0.912810i \(-0.366093\pi\)
0.408383 + 0.912810i \(0.366093\pi\)
\(632\) −4.80452e9 −0.757077
\(633\) 0 0
\(634\) −2.26738e9 −0.353356
\(635\) 1.21363e8 0.0188096
\(636\) 0 0
\(637\) −6.58350e7 −0.0100918
\(638\) 1.74680e9 0.266300
\(639\) 0 0
\(640\) −3.30678e8 −0.0498626
\(641\) 4.91786e9 0.737518 0.368759 0.929525i \(-0.379783\pi\)
0.368759 + 0.929525i \(0.379783\pi\)
\(642\) 0 0
\(643\) −1.42616e9 −0.211558 −0.105779 0.994390i \(-0.533734\pi\)
−0.105779 + 0.994390i \(0.533734\pi\)
\(644\) −3.57627e9 −0.527630
\(645\) 0 0
\(646\) 5.13863e8 0.0749952
\(647\) −6.26814e9 −0.909858 −0.454929 0.890528i \(-0.650335\pi\)
−0.454929 + 0.890528i \(0.650335\pi\)
\(648\) 0 0
\(649\) 1.07845e9 0.154861
\(650\) −5.06084e7 −0.00722813
\(651\) 0 0
\(652\) 3.31683e9 0.468658
\(653\) −1.00126e10 −1.40719 −0.703593 0.710603i \(-0.748422\pi\)
−0.703593 + 0.710603i \(0.748422\pi\)
\(654\) 0 0
\(655\) −6.13483e8 −0.0853017
\(656\) −8.76640e8 −0.121243
\(657\) 0 0
\(658\) −1.98592e9 −0.271752
\(659\) −3.69501e9 −0.502941 −0.251470 0.967865i \(-0.580914\pi\)
−0.251470 + 0.967865i \(0.580914\pi\)
\(660\) 0 0
\(661\) 2.49829e9 0.336464 0.168232 0.985747i \(-0.446194\pi\)
0.168232 + 0.985747i \(0.446194\pi\)
\(662\) −1.18117e9 −0.158238
\(663\) 0 0
\(664\) −2.41077e8 −0.0319572
\(665\) 4.05393e7 0.00534565
\(666\) 0 0
\(667\) −4.38473e9 −0.572140
\(668\) −1.02718e10 −1.33330
\(669\) 0 0
\(670\) −2.56337e7 −0.00329268
\(671\) 4.61455e9 0.589659
\(672\) 0 0
\(673\) 3.99102e9 0.504698 0.252349 0.967636i \(-0.418797\pi\)
0.252349 + 0.967636i \(0.418797\pi\)
\(674\) −6.65961e9 −0.837798
\(675\) 0 0
\(676\) 6.01599e9 0.749020
\(677\) 3.69552e9 0.457737 0.228868 0.973457i \(-0.426497\pi\)
0.228868 + 0.973457i \(0.426497\pi\)
\(678\) 0 0
\(679\) 7.35887e9 0.902125
\(680\) −3.19466e8 −0.0389622
\(681\) 0 0
\(682\) −5.79416e9 −0.699431
\(683\) 6.33943e9 0.761338 0.380669 0.924711i \(-0.375694\pi\)
0.380669 + 0.924711i \(0.375694\pi\)
\(684\) 0 0
\(685\) −6.52091e8 −0.0775159
\(686\) −3.95475e9 −0.467718
\(687\) 0 0
\(688\) 8.85020e7 0.0103608
\(689\) −5.36320e7 −0.00624679
\(690\) 0 0
\(691\) 9.02991e9 1.04114 0.520572 0.853818i \(-0.325719\pi\)
0.520572 + 0.853818i \(0.325719\pi\)
\(692\) 2.85564e9 0.327592
\(693\) 0 0
\(694\) −1.71956e9 −0.195280
\(695\) −4.50483e8 −0.0509016
\(696\) 0 0
\(697\) −2.89074e9 −0.323366
\(698\) 4.33974e9 0.483025
\(699\) 0 0
\(700\) 3.73022e9 0.411047
\(701\) 4.09079e9 0.448533 0.224267 0.974528i \(-0.428001\pi\)
0.224267 + 0.974528i \(0.428001\pi\)
\(702\) 0 0
\(703\) 1.72563e9 0.187328
\(704\) −2.27023e9 −0.245226
\(705\) 0 0
\(706\) −6.09189e9 −0.651532
\(707\) −1.57976e9 −0.168121
\(708\) 0 0
\(709\) −1.70129e10 −1.79273 −0.896366 0.443315i \(-0.853802\pi\)
−0.896366 + 0.443315i \(0.853802\pi\)
\(710\) −6.14595e7 −0.00644443
\(711\) 0 0
\(712\) 3.43265e9 0.356410
\(713\) 1.45442e10 1.50271
\(714\) 0 0
\(715\) −9.03949e6 −0.000924853 0
\(716\) 1.17835e10 1.19972
\(717\) 0 0
\(718\) 4.54289e9 0.458033
\(719\) 1.05938e10 1.06292 0.531460 0.847083i \(-0.321643\pi\)
0.531460 + 0.847083i \(0.321643\pi\)
\(720\) 0 0
\(721\) −4.43511e9 −0.440687
\(722\) −4.89915e9 −0.484441
\(723\) 0 0
\(724\) −7.61709e9 −0.745941
\(725\) 4.57349e9 0.445722
\(726\) 0 0
\(727\) 6.96354e9 0.672140 0.336070 0.941837i \(-0.390902\pi\)
0.336070 + 0.941837i \(0.390902\pi\)
\(728\) −7.26334e7 −0.00697712
\(729\) 0 0
\(730\) 5.56084e8 0.0529067
\(731\) 2.91837e8 0.0276331
\(732\) 0 0
\(733\) −2.65861e9 −0.249340 −0.124670 0.992198i \(-0.539787\pi\)
−0.124670 + 0.992198i \(0.539787\pi\)
\(734\) 1.35605e9 0.126572
\(735\) 0 0
\(736\) 1.42799e10 1.32024
\(737\) 1.58220e9 0.145588
\(738\) 0 0
\(739\) −4.87522e8 −0.0444364 −0.0222182 0.999753i \(-0.507073\pi\)
−0.0222182 + 0.999753i \(0.507073\pi\)
\(740\) −4.59488e8 −0.0416834
\(741\) 0 0
\(742\) −1.32348e9 −0.118934
\(743\) 1.88059e10 1.68202 0.841012 0.541016i \(-0.181960\pi\)
0.841012 + 0.541016i \(0.181960\pi\)
\(744\) 0 0
\(745\) 3.01477e8 0.0267121
\(746\) 8.21457e9 0.724434
\(747\) 0 0
\(748\) 8.44552e9 0.737855
\(749\) −8.39314e9 −0.729857
\(750\) 0 0
\(751\) 2.25057e10 1.93889 0.969444 0.245315i \(-0.0788912\pi\)
0.969444 + 0.245315i \(0.0788912\pi\)
\(752\) −3.57003e9 −0.306132
\(753\) 0 0
\(754\) −3.81417e7 −0.00324041
\(755\) −6.64331e8 −0.0561785
\(756\) 0 0
\(757\) 9.35131e8 0.0783496 0.0391748 0.999232i \(-0.487527\pi\)
0.0391748 + 0.999232i \(0.487527\pi\)
\(758\) 3.30418e9 0.275563
\(759\) 0 0
\(760\) −1.02992e8 −0.00851053
\(761\) 8.03138e9 0.660608 0.330304 0.943875i \(-0.392849\pi\)
0.330304 + 0.943875i \(0.392849\pi\)
\(762\) 0 0
\(763\) 3.09736e9 0.252439
\(764\) 1.32321e10 1.07350
\(765\) 0 0
\(766\) 1.20671e10 0.970065
\(767\) −2.35481e7 −0.00188439
\(768\) 0 0
\(769\) −2.58797e9 −0.205219 −0.102610 0.994722i \(-0.532719\pi\)
−0.102610 + 0.994722i \(0.532719\pi\)
\(770\) −2.23069e8 −0.0176084
\(771\) 0 0
\(772\) −8.35633e7 −0.00653665
\(773\) 1.46523e10 1.14098 0.570489 0.821306i \(-0.306754\pi\)
0.570489 + 0.821306i \(0.306754\pi\)
\(774\) 0 0
\(775\) −1.51703e10 −1.17068
\(776\) −1.86956e10 −1.43622
\(777\) 0 0
\(778\) −5.81531e9 −0.442735
\(779\) −9.31941e8 −0.0706329
\(780\) 0 0
\(781\) 3.79350e9 0.284945
\(782\) 7.09756e9 0.530744
\(783\) 0 0
\(784\) −2.92052e9 −0.216448
\(785\) −8.88251e8 −0.0655378
\(786\) 0 0
\(787\) −7.14096e9 −0.522210 −0.261105 0.965310i \(-0.584087\pi\)
−0.261105 + 0.965310i \(0.584087\pi\)
\(788\) 1.64777e10 1.19965
\(789\) 0 0
\(790\) 3.22188e8 0.0232495
\(791\) −5.03068e9 −0.361417
\(792\) 0 0
\(793\) −1.00760e8 −0.00717514
\(794\) 4.80904e9 0.340947
\(795\) 0 0
\(796\) 9.13029e9 0.641636
\(797\) −1.01959e10 −0.713379 −0.356689 0.934223i \(-0.616095\pi\)
−0.356689 + 0.934223i \(0.616095\pi\)
\(798\) 0 0
\(799\) −1.17722e10 −0.816480
\(800\) −1.48947e10 −1.02853
\(801\) 0 0
\(802\) 5.98110e9 0.409421
\(803\) −3.43235e10 −2.33931
\(804\) 0 0
\(805\) 5.59936e8 0.0378314
\(806\) 1.26516e8 0.00851087
\(807\) 0 0
\(808\) 4.01345e9 0.267657
\(809\) 1.75531e10 1.16556 0.582780 0.812630i \(-0.301965\pi\)
0.582780 + 0.812630i \(0.301965\pi\)
\(810\) 0 0
\(811\) −2.83520e10 −1.86643 −0.933214 0.359321i \(-0.883008\pi\)
−0.933214 + 0.359321i \(0.883008\pi\)
\(812\) 2.81133e9 0.184275
\(813\) 0 0
\(814\) −9.49529e9 −0.617054
\(815\) −5.19315e8 −0.0336031
\(816\) 0 0
\(817\) 9.40849e7 0.00603591
\(818\) 2.05517e8 0.0131284
\(819\) 0 0
\(820\) 2.48151e8 0.0157169
\(821\) 2.80476e10 1.76886 0.884432 0.466669i \(-0.154546\pi\)
0.884432 + 0.466669i \(0.154546\pi\)
\(822\) 0 0
\(823\) −2.23504e10 −1.39761 −0.698806 0.715311i \(-0.746285\pi\)
−0.698806 + 0.715311i \(0.746285\pi\)
\(824\) 1.12676e10 0.701595
\(825\) 0 0
\(826\) −5.81098e8 −0.0358773
\(827\) −1.26693e10 −0.778900 −0.389450 0.921048i \(-0.627335\pi\)
−0.389450 + 0.921048i \(0.627335\pi\)
\(828\) 0 0
\(829\) −2.22306e10 −1.35522 −0.677610 0.735421i \(-0.736984\pi\)
−0.677610 + 0.735421i \(0.736984\pi\)
\(830\) 1.61665e7 0.000981391 0
\(831\) 0 0
\(832\) 4.95709e7 0.00298397
\(833\) −9.63047e9 −0.577285
\(834\) 0 0
\(835\) 1.60825e9 0.0955986
\(836\) 2.72274e9 0.161170
\(837\) 0 0
\(838\) 1.65793e9 0.0973221
\(839\) 2.27124e10 1.32769 0.663845 0.747870i \(-0.268923\pi\)
0.663845 + 0.747870i \(0.268923\pi\)
\(840\) 0 0
\(841\) −1.38030e10 −0.800180
\(842\) −6.93606e9 −0.400425
\(843\) 0 0
\(844\) −1.06372e10 −0.609019
\(845\) −9.41922e8 −0.0537052
\(846\) 0 0
\(847\) 4.03768e9 0.228318
\(848\) −2.37918e9 −0.133981
\(849\) 0 0
\(850\) −7.40310e9 −0.413473
\(851\) 2.38346e10 1.32573
\(852\) 0 0
\(853\) −2.92944e10 −1.61608 −0.808041 0.589127i \(-0.799472\pi\)
−0.808041 + 0.589127i \(0.799472\pi\)
\(854\) −2.48646e9 −0.136609
\(855\) 0 0
\(856\) 2.13232e10 1.16197
\(857\) 2.01612e9 0.109417 0.0547084 0.998502i \(-0.482577\pi\)
0.0547084 + 0.998502i \(0.482577\pi\)
\(858\) 0 0
\(859\) −1.62001e10 −0.872052 −0.436026 0.899934i \(-0.643614\pi\)
−0.436026 + 0.899934i \(0.643614\pi\)
\(860\) −2.50523e7 −0.00134309
\(861\) 0 0
\(862\) 3.54992e9 0.188774
\(863\) −4.82303e9 −0.255436 −0.127718 0.991811i \(-0.540765\pi\)
−0.127718 + 0.991811i \(0.540765\pi\)
\(864\) 0 0
\(865\) −4.47108e8 −0.0234885
\(866\) −1.58530e10 −0.829468
\(867\) 0 0
\(868\) −9.32521e9 −0.483993
\(869\) −1.98866e10 −1.02799
\(870\) 0 0
\(871\) −3.45477e7 −0.00177156
\(872\) −7.86900e9 −0.401894
\(873\) 0 0
\(874\) 2.28817e9 0.115931
\(875\) −1.16977e9 −0.0590299
\(876\) 0 0
\(877\) 4.98280e9 0.249445 0.124723 0.992192i \(-0.460196\pi\)
0.124723 + 0.992192i \(0.460196\pi\)
\(878\) −3.45351e9 −0.172199
\(879\) 0 0
\(880\) −4.01003e8 −0.0198362
\(881\) −1.27067e10 −0.626061 −0.313031 0.949743i \(-0.601344\pi\)
−0.313031 + 0.949743i \(0.601344\pi\)
\(882\) 0 0
\(883\) −1.01601e9 −0.0496633 −0.0248316 0.999692i \(-0.507905\pi\)
−0.0248316 + 0.999692i \(0.507905\pi\)
\(884\) −1.84409e8 −0.00897843
\(885\) 0 0
\(886\) 1.02428e10 0.494765
\(887\) −1.10442e10 −0.531376 −0.265688 0.964059i \(-0.585599\pi\)
−0.265688 + 0.964059i \(0.585599\pi\)
\(888\) 0 0
\(889\) 4.03636e9 0.192679
\(890\) −2.30191e8 −0.0109452
\(891\) 0 0
\(892\) 1.42613e10 0.672793
\(893\) −3.79523e9 −0.178344
\(894\) 0 0
\(895\) −1.84495e9 −0.0860208
\(896\) −1.09978e10 −0.510775
\(897\) 0 0
\(898\) −3.58532e9 −0.165219
\(899\) −1.14333e10 −0.524823
\(900\) 0 0
\(901\) −7.84539e9 −0.357337
\(902\) 5.12803e9 0.232663
\(903\) 0 0
\(904\) 1.27807e10 0.575393
\(905\) 1.19261e9 0.0534844
\(906\) 0 0
\(907\) −2.36818e10 −1.05388 −0.526938 0.849904i \(-0.676660\pi\)
−0.526938 + 0.849904i \(0.676660\pi\)
\(908\) 4.27556e9 0.189536
\(909\) 0 0
\(910\) 4.87074e6 0.000214265 0
\(911\) −1.78985e10 −0.784338 −0.392169 0.919893i \(-0.628275\pi\)
−0.392169 + 0.919893i \(0.628275\pi\)
\(912\) 0 0
\(913\) −9.97853e8 −0.0433929
\(914\) −1.81295e9 −0.0785371
\(915\) 0 0
\(916\) 9.20650e9 0.395786
\(917\) −2.04035e10 −0.873799
\(918\) 0 0
\(919\) −5.16659e9 −0.219584 −0.109792 0.993955i \(-0.535018\pi\)
−0.109792 + 0.993955i \(0.535018\pi\)
\(920\) −1.42254e9 −0.0602294
\(921\) 0 0
\(922\) 5.41298e9 0.227446
\(923\) −8.28317e7 −0.00346729
\(924\) 0 0
\(925\) −2.48607e10 −1.03280
\(926\) 2.01225e10 0.832804
\(927\) 0 0
\(928\) −1.12256e10 −0.461095
\(929\) −2.17904e10 −0.891681 −0.445840 0.895112i \(-0.647095\pi\)
−0.445840 + 0.895112i \(0.647095\pi\)
\(930\) 0 0
\(931\) −3.10475e9 −0.126096
\(932\) −1.43769e10 −0.581713
\(933\) 0 0
\(934\) −2.31656e10 −0.930315
\(935\) −1.32231e9 −0.0529047
\(936\) 0 0
\(937\) −2.40380e10 −0.954575 −0.477287 0.878747i \(-0.658380\pi\)
−0.477287 + 0.878747i \(0.658380\pi\)
\(938\) −8.52538e8 −0.0337290
\(939\) 0 0
\(940\) 1.01057e9 0.0396843
\(941\) −4.36782e10 −1.70884 −0.854419 0.519585i \(-0.826087\pi\)
−0.854419 + 0.519585i \(0.826087\pi\)
\(942\) 0 0
\(943\) −1.28721e10 −0.499872
\(944\) −1.04462e9 −0.0404163
\(945\) 0 0
\(946\) −5.17705e8 −0.0198821
\(947\) −1.94958e10 −0.745961 −0.372980 0.927839i \(-0.621664\pi\)
−0.372980 + 0.927839i \(0.621664\pi\)
\(948\) 0 0
\(949\) 7.49459e8 0.0284653
\(950\) −2.38667e9 −0.0903151
\(951\) 0 0
\(952\) −1.06250e10 −0.399115
\(953\) −1.31470e10 −0.492040 −0.246020 0.969265i \(-0.579123\pi\)
−0.246020 + 0.969265i \(0.579123\pi\)
\(954\) 0 0
\(955\) −2.07175e9 −0.0769707
\(956\) −4.79079e9 −0.177339
\(957\) 0 0
\(958\) −1.25402e10 −0.460813
\(959\) −2.16875e10 −0.794045
\(960\) 0 0
\(961\) 1.04117e10 0.378435
\(962\) 2.07331e8 0.00750848
\(963\) 0 0
\(964\) −3.77877e8 −0.0135857
\(965\) 1.30835e7 0.000468682 0
\(966\) 0 0
\(967\) 4.78218e10 1.70072 0.850361 0.526200i \(-0.176384\pi\)
0.850361 + 0.526200i \(0.176384\pi\)
\(968\) −1.02579e10 −0.363493
\(969\) 0 0
\(970\) 1.25371e9 0.0441059
\(971\) −2.03563e10 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(972\) 0 0
\(973\) −1.49824e10 −0.521418
\(974\) −2.65374e10 −0.920242
\(975\) 0 0
\(976\) −4.46982e9 −0.153892
\(977\) 4.11493e10 1.41166 0.705832 0.708379i \(-0.250573\pi\)
0.705832 + 0.708379i \(0.250573\pi\)
\(978\) 0 0
\(979\) 1.42082e10 0.483949
\(980\) 8.26713e8 0.0280584
\(981\) 0 0
\(982\) 1.01935e10 0.343506
\(983\) 2.78989e10 0.936805 0.468402 0.883515i \(-0.344830\pi\)
0.468402 + 0.883515i \(0.344830\pi\)
\(984\) 0 0
\(985\) −2.57992e9 −0.0860160
\(986\) −5.57944e9 −0.185362
\(987\) 0 0
\(988\) −5.94514e7 −0.00196116
\(989\) 1.29952e9 0.0427164
\(990\) 0 0
\(991\) −5.57475e10 −1.81957 −0.909783 0.415084i \(-0.863752\pi\)
−0.909783 + 0.415084i \(0.863752\pi\)
\(992\) 3.72353e10 1.21106
\(993\) 0 0
\(994\) −2.04405e9 −0.0660144
\(995\) −1.42953e9 −0.0460058
\(996\) 0 0
\(997\) 1.05869e10 0.338326 0.169163 0.985588i \(-0.445894\pi\)
0.169163 + 0.985588i \(0.445894\pi\)
\(998\) −4.82724e9 −0.153724
\(999\) 0 0
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 531.8.a.e.1.12 18
3.2 odd 2 177.8.a.d.1.7 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.7 18 3.2 odd 2
531.8.a.e.1.12 18 1.1 even 1 trivial