Properties

Label 531.8.a.d.1.6
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(7.01814\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.01814 q^{2} -102.818 q^{4} +77.5687 q^{5} -216.829 q^{7} +1158.28 q^{8} +O(q^{10})\) \(q-5.01814 q^{2} -102.818 q^{4} +77.5687 q^{5} -216.829 q^{7} +1158.28 q^{8} -389.251 q^{10} -3376.21 q^{11} +12448.6 q^{13} +1088.08 q^{14} +7348.33 q^{16} +26362.1 q^{17} +41401.4 q^{19} -7975.48 q^{20} +16942.3 q^{22} +42118.5 q^{23} -72108.1 q^{25} -62468.7 q^{26} +22293.9 q^{28} +151717. q^{29} -239434. q^{31} -185135. q^{32} -132289. q^{34} -16819.1 q^{35} -411153. q^{37} -207758. q^{38} +89846.2 q^{40} +536752. q^{41} +583229. q^{43} +347136. q^{44} -211357. q^{46} +587465. q^{47} -776528. q^{49} +361849. q^{50} -1.27994e6 q^{52} +267710. q^{53} -261889. q^{55} -251148. q^{56} -761339. q^{58} +205379. q^{59} -498824. q^{61} +1.20151e6 q^{62} -11553.8 q^{64} +965620. q^{65} +2.87144e6 q^{67} -2.71050e6 q^{68} +84400.8 q^{70} -1.08176e6 q^{71} -4746.50 q^{73} +2.06322e6 q^{74} -4.25682e6 q^{76} +732060. q^{77} -1.41454e6 q^{79} +570000. q^{80} -2.69350e6 q^{82} -3.72243e6 q^{83} +2.04487e6 q^{85} -2.92672e6 q^{86} -3.91060e6 q^{88} -1.04746e7 q^{89} -2.69921e6 q^{91} -4.33055e6 q^{92} -2.94798e6 q^{94} +3.21145e6 q^{95} -4.51283e6 q^{97} +3.89673e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.01814 −0.443545 −0.221773 0.975098i \(-0.571184\pi\)
−0.221773 + 0.975098i \(0.571184\pi\)
\(3\) 0 0
\(4\) −102.818 −0.803268
\(5\) 77.5687 0.277518 0.138759 0.990326i \(-0.455689\pi\)
0.138759 + 0.990326i \(0.455689\pi\)
\(6\) 0 0
\(7\) −216.829 −0.238932 −0.119466 0.992838i \(-0.538118\pi\)
−0.119466 + 0.992838i \(0.538118\pi\)
\(8\) 1158.28 0.799831
\(9\) 0 0
\(10\) −389.251 −0.123092
\(11\) −3376.21 −0.764813 −0.382407 0.923994i \(-0.624905\pi\)
−0.382407 + 0.923994i \(0.624905\pi\)
\(12\) 0 0
\(13\) 12448.6 1.57151 0.785756 0.618536i \(-0.212274\pi\)
0.785756 + 0.618536i \(0.212274\pi\)
\(14\) 1088.08 0.105977
\(15\) 0 0
\(16\) 7348.33 0.448506
\(17\) 26362.1 1.30139 0.650696 0.759338i \(-0.274477\pi\)
0.650696 + 0.759338i \(0.274477\pi\)
\(18\) 0 0
\(19\) 41401.4 1.38477 0.692384 0.721529i \(-0.256560\pi\)
0.692384 + 0.721529i \(0.256560\pi\)
\(20\) −7975.48 −0.222921
\(21\) 0 0
\(22\) 16942.3 0.339229
\(23\) 42118.5 0.721815 0.360907 0.932602i \(-0.382467\pi\)
0.360907 + 0.932602i \(0.382467\pi\)
\(24\) 0 0
\(25\) −72108.1 −0.922984
\(26\) −62468.7 −0.697037
\(27\) 0 0
\(28\) 22293.9 0.191926
\(29\) 151717. 1.15516 0.577580 0.816334i \(-0.303997\pi\)
0.577580 + 0.816334i \(0.303997\pi\)
\(30\) 0 0
\(31\) −239434. −1.44351 −0.721754 0.692149i \(-0.756664\pi\)
−0.721754 + 0.692149i \(0.756664\pi\)
\(32\) −185135. −0.998764
\(33\) 0 0
\(34\) −132289. −0.577226
\(35\) −16819.1 −0.0663079
\(36\) 0 0
\(37\) −411153. −1.33444 −0.667218 0.744863i \(-0.732515\pi\)
−0.667218 + 0.744863i \(0.732515\pi\)
\(38\) −207758. −0.614208
\(39\) 0 0
\(40\) 89846.2 0.221968
\(41\) 536752. 1.21627 0.608135 0.793834i \(-0.291918\pi\)
0.608135 + 0.793834i \(0.291918\pi\)
\(42\) 0 0
\(43\) 583229. 1.11866 0.559331 0.828944i \(-0.311058\pi\)
0.559331 + 0.828944i \(0.311058\pi\)
\(44\) 347136. 0.614350
\(45\) 0 0
\(46\) −211357. −0.320158
\(47\) 587465. 0.825353 0.412677 0.910878i \(-0.364594\pi\)
0.412677 + 0.910878i \(0.364594\pi\)
\(48\) 0 0
\(49\) −776528. −0.942912
\(50\) 361849. 0.409385
\(51\) 0 0
\(52\) −1.27994e6 −1.26234
\(53\) 267710. 0.247001 0.123501 0.992344i \(-0.460588\pi\)
0.123501 + 0.992344i \(0.460588\pi\)
\(54\) 0 0
\(55\) −261889. −0.212250
\(56\) −251148. −0.191105
\(57\) 0 0
\(58\) −761339. −0.512366
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −498824. −0.281380 −0.140690 0.990054i \(-0.544932\pi\)
−0.140690 + 0.990054i \(0.544932\pi\)
\(62\) 1.20151e6 0.640261
\(63\) 0 0
\(64\) −11553.8 −0.00550929
\(65\) 965620. 0.436124
\(66\) 0 0
\(67\) 2.87144e6 1.16637 0.583187 0.812338i \(-0.301805\pi\)
0.583187 + 0.812338i \(0.301805\pi\)
\(68\) −2.71050e6 −1.04537
\(69\) 0 0
\(70\) 84400.8 0.0294106
\(71\) −1.08176e6 −0.358697 −0.179349 0.983786i \(-0.557399\pi\)
−0.179349 + 0.983786i \(0.557399\pi\)
\(72\) 0 0
\(73\) −4746.50 −0.00142805 −0.000714025 1.00000i \(-0.500227\pi\)
−0.000714025 1.00000i \(0.500227\pi\)
\(74\) 2.06322e6 0.591883
\(75\) 0 0
\(76\) −4.25682e6 −1.11234
\(77\) 732060. 0.182738
\(78\) 0 0
\(79\) −1.41454e6 −0.322791 −0.161396 0.986890i \(-0.551600\pi\)
−0.161396 + 0.986890i \(0.551600\pi\)
\(80\) 570000. 0.124469
\(81\) 0 0
\(82\) −2.69350e6 −0.539471
\(83\) −3.72243e6 −0.714583 −0.357292 0.933993i \(-0.616300\pi\)
−0.357292 + 0.933993i \(0.616300\pi\)
\(84\) 0 0
\(85\) 2.04487e6 0.361160
\(86\) −2.92672e6 −0.496178
\(87\) 0 0
\(88\) −3.91060e6 −0.611721
\(89\) −1.04746e7 −1.57497 −0.787484 0.616336i \(-0.788616\pi\)
−0.787484 + 0.616336i \(0.788616\pi\)
\(90\) 0 0
\(91\) −2.69921e6 −0.375484
\(92\) −4.33055e6 −0.579810
\(93\) 0 0
\(94\) −2.94798e6 −0.366081
\(95\) 3.21145e6 0.384299
\(96\) 0 0
\(97\) −4.51283e6 −0.502051 −0.251025 0.967980i \(-0.580768\pi\)
−0.251025 + 0.967980i \(0.580768\pi\)
\(98\) 3.89673e6 0.418224
\(99\) 0 0
\(100\) 7.41403e6 0.741403
\(101\) 1.70633e7 1.64793 0.823963 0.566644i \(-0.191758\pi\)
0.823963 + 0.566644i \(0.191758\pi\)
\(102\) 0 0
\(103\) −4.18892e6 −0.377721 −0.188861 0.982004i \(-0.560479\pi\)
−0.188861 + 0.982004i \(0.560479\pi\)
\(104\) 1.44189e7 1.25694
\(105\) 0 0
\(106\) −1.34341e6 −0.109556
\(107\) 1.20271e7 0.949116 0.474558 0.880224i \(-0.342608\pi\)
0.474558 + 0.880224i \(0.342608\pi\)
\(108\) 0 0
\(109\) −1.96024e7 −1.44983 −0.724913 0.688840i \(-0.758120\pi\)
−0.724913 + 0.688840i \(0.758120\pi\)
\(110\) 1.31419e6 0.0941424
\(111\) 0 0
\(112\) −1.59333e6 −0.107162
\(113\) 7.54916e6 0.492180 0.246090 0.969247i \(-0.420854\pi\)
0.246090 + 0.969247i \(0.420854\pi\)
\(114\) 0 0
\(115\) 3.26708e6 0.200317
\(116\) −1.55993e7 −0.927903
\(117\) 0 0
\(118\) −1.03062e6 −0.0577447
\(119\) −5.71605e6 −0.310944
\(120\) 0 0
\(121\) −8.08835e6 −0.415060
\(122\) 2.50317e6 0.124805
\(123\) 0 0
\(124\) 2.46182e7 1.15952
\(125\) −1.16534e7 −0.533663
\(126\) 0 0
\(127\) −1.36062e7 −0.589417 −0.294709 0.955587i \(-0.595223\pi\)
−0.294709 + 0.955587i \(0.595223\pi\)
\(128\) 2.37552e7 1.00121
\(129\) 0 0
\(130\) −4.84562e6 −0.193441
\(131\) 1.29419e7 0.502978 0.251489 0.967860i \(-0.419080\pi\)
0.251489 + 0.967860i \(0.419080\pi\)
\(132\) 0 0
\(133\) −8.97700e6 −0.330865
\(134\) −1.44093e7 −0.517340
\(135\) 0 0
\(136\) 3.05346e7 1.04089
\(137\) 7.65865e6 0.254466 0.127233 0.991873i \(-0.459390\pi\)
0.127233 + 0.991873i \(0.459390\pi\)
\(138\) 0 0
\(139\) 4.87660e6 0.154016 0.0770080 0.997030i \(-0.475463\pi\)
0.0770080 + 0.997030i \(0.475463\pi\)
\(140\) 1.72931e6 0.0532630
\(141\) 0 0
\(142\) 5.42844e6 0.159098
\(143\) −4.20290e7 −1.20191
\(144\) 0 0
\(145\) 1.17685e7 0.320578
\(146\) 23818.6 0.000633405 0
\(147\) 0 0
\(148\) 4.22740e7 1.07191
\(149\) −1.05059e6 −0.0260186 −0.0130093 0.999915i \(-0.504141\pi\)
−0.0130093 + 0.999915i \(0.504141\pi\)
\(150\) 0 0
\(151\) 5.81661e7 1.37484 0.687418 0.726262i \(-0.258744\pi\)
0.687418 + 0.726262i \(0.258744\pi\)
\(152\) 4.79543e7 1.10758
\(153\) 0 0
\(154\) −3.67358e6 −0.0810526
\(155\) −1.85726e7 −0.400600
\(156\) 0 0
\(157\) 4.08109e7 0.841642 0.420821 0.907144i \(-0.361742\pi\)
0.420821 + 0.907144i \(0.361742\pi\)
\(158\) 7.09838e6 0.143173
\(159\) 0 0
\(160\) −1.43607e7 −0.277175
\(161\) −9.13250e6 −0.172464
\(162\) 0 0
\(163\) −7.46918e7 −1.35088 −0.675439 0.737416i \(-0.736046\pi\)
−0.675439 + 0.737416i \(0.736046\pi\)
\(164\) −5.51879e7 −0.976990
\(165\) 0 0
\(166\) 1.86797e7 0.316950
\(167\) 5.71426e7 0.949407 0.474704 0.880146i \(-0.342555\pi\)
0.474704 + 0.880146i \(0.342555\pi\)
\(168\) 0 0
\(169\) 9.22184e7 1.46965
\(170\) −1.02615e7 −0.160191
\(171\) 0 0
\(172\) −5.99665e7 −0.898585
\(173\) −6.74021e7 −0.989720 −0.494860 0.868973i \(-0.664781\pi\)
−0.494860 + 0.868973i \(0.664781\pi\)
\(174\) 0 0
\(175\) 1.56351e7 0.220530
\(176\) −2.48095e7 −0.343024
\(177\) 0 0
\(178\) 5.25629e7 0.698569
\(179\) −5.40298e7 −0.704122 −0.352061 0.935977i \(-0.614519\pi\)
−0.352061 + 0.935977i \(0.614519\pi\)
\(180\) 0 0
\(181\) −2.98190e7 −0.373782 −0.186891 0.982381i \(-0.559841\pi\)
−0.186891 + 0.982381i \(0.559841\pi\)
\(182\) 1.35450e7 0.166544
\(183\) 0 0
\(184\) 4.87850e7 0.577330
\(185\) −3.18926e7 −0.370330
\(186\) 0 0
\(187\) −8.90040e7 −0.995322
\(188\) −6.04022e7 −0.662979
\(189\) 0 0
\(190\) −1.61155e7 −0.170454
\(191\) 1.03129e8 1.07093 0.535466 0.844557i \(-0.320136\pi\)
0.535466 + 0.844557i \(0.320136\pi\)
\(192\) 0 0
\(193\) 1.72283e8 1.72501 0.862506 0.506047i \(-0.168894\pi\)
0.862506 + 0.506047i \(0.168894\pi\)
\(194\) 2.26460e7 0.222682
\(195\) 0 0
\(196\) 7.98413e7 0.757410
\(197\) 1.27602e8 1.18912 0.594562 0.804050i \(-0.297326\pi\)
0.594562 + 0.804050i \(0.297326\pi\)
\(198\) 0 0
\(199\) −6.47395e7 −0.582349 −0.291175 0.956670i \(-0.594046\pi\)
−0.291175 + 0.956670i \(0.594046\pi\)
\(200\) −8.35213e7 −0.738231
\(201\) 0 0
\(202\) −8.56260e7 −0.730930
\(203\) −3.28967e7 −0.276004
\(204\) 0 0
\(205\) 4.16351e7 0.337537
\(206\) 2.10206e7 0.167536
\(207\) 0 0
\(208\) 9.14762e7 0.704833
\(209\) −1.39780e8 −1.05909
\(210\) 0 0
\(211\) 1.42567e8 1.04479 0.522395 0.852704i \(-0.325039\pi\)
0.522395 + 0.852704i \(0.325039\pi\)
\(212\) −2.75255e7 −0.198408
\(213\) 0 0
\(214\) −6.03539e7 −0.420976
\(215\) 4.52403e7 0.310449
\(216\) 0 0
\(217\) 5.19161e7 0.344900
\(218\) 9.83675e7 0.643064
\(219\) 0 0
\(220\) 2.69269e7 0.170493
\(221\) 3.28170e8 2.04515
\(222\) 0 0
\(223\) 2.35480e8 1.42196 0.710979 0.703213i \(-0.248252\pi\)
0.710979 + 0.703213i \(0.248252\pi\)
\(224\) 4.01425e7 0.238636
\(225\) 0 0
\(226\) −3.78828e7 −0.218304
\(227\) −3.35151e8 −1.90174 −0.950868 0.309598i \(-0.899806\pi\)
−0.950868 + 0.309598i \(0.899806\pi\)
\(228\) 0 0
\(229\) −1.68231e8 −0.925727 −0.462864 0.886430i \(-0.653178\pi\)
−0.462864 + 0.886430i \(0.653178\pi\)
\(230\) −1.63947e7 −0.0888496
\(231\) 0 0
\(232\) 1.75731e8 0.923933
\(233\) 2.76156e8 1.43024 0.715120 0.699002i \(-0.246372\pi\)
0.715120 + 0.699002i \(0.246372\pi\)
\(234\) 0 0
\(235\) 4.55689e7 0.229051
\(236\) −2.11167e7 −0.104577
\(237\) 0 0
\(238\) 2.86840e7 0.137918
\(239\) −1.64969e8 −0.781647 −0.390824 0.920466i \(-0.627810\pi\)
−0.390824 + 0.920466i \(0.627810\pi\)
\(240\) 0 0
\(241\) 2.19804e8 1.01152 0.505761 0.862674i \(-0.331212\pi\)
0.505761 + 0.862674i \(0.331212\pi\)
\(242\) 4.05885e7 0.184098
\(243\) 0 0
\(244\) 5.12882e7 0.226023
\(245\) −6.02343e7 −0.261675
\(246\) 0 0
\(247\) 5.15388e8 2.17618
\(248\) −2.77331e8 −1.15456
\(249\) 0 0
\(250\) 5.84784e7 0.236704
\(251\) 2.64722e8 1.05665 0.528327 0.849041i \(-0.322820\pi\)
0.528327 + 0.849041i \(0.322820\pi\)
\(252\) 0 0
\(253\) −1.42201e8 −0.552054
\(254\) 6.82777e7 0.261433
\(255\) 0 0
\(256\) −1.17728e8 −0.438572
\(257\) 2.39775e8 0.881125 0.440563 0.897722i \(-0.354779\pi\)
0.440563 + 0.897722i \(0.354779\pi\)
\(258\) 0 0
\(259\) 8.91498e7 0.318839
\(260\) −9.92834e7 −0.350324
\(261\) 0 0
\(262\) −6.49444e7 −0.223094
\(263\) −1.27661e8 −0.432727 −0.216364 0.976313i \(-0.569420\pi\)
−0.216364 + 0.976313i \(0.569420\pi\)
\(264\) 0 0
\(265\) 2.07659e7 0.0685474
\(266\) 4.50479e7 0.146754
\(267\) 0 0
\(268\) −2.95236e8 −0.936910
\(269\) −1.98210e8 −0.620858 −0.310429 0.950597i \(-0.600473\pi\)
−0.310429 + 0.950597i \(0.600473\pi\)
\(270\) 0 0
\(271\) 1.17159e8 0.357587 0.178794 0.983887i \(-0.442781\pi\)
0.178794 + 0.983887i \(0.442781\pi\)
\(272\) 1.93717e8 0.583683
\(273\) 0 0
\(274\) −3.84322e7 −0.112867
\(275\) 2.43452e8 0.705910
\(276\) 0 0
\(277\) 2.80002e8 0.791555 0.395778 0.918346i \(-0.370475\pi\)
0.395778 + 0.918346i \(0.370475\pi\)
\(278\) −2.44715e7 −0.0683131
\(279\) 0 0
\(280\) −1.94812e7 −0.0530351
\(281\) 6.16699e7 0.165806 0.0829032 0.996558i \(-0.473581\pi\)
0.0829032 + 0.996558i \(0.473581\pi\)
\(282\) 0 0
\(283\) −6.20410e8 −1.62715 −0.813573 0.581463i \(-0.802481\pi\)
−0.813573 + 0.581463i \(0.802481\pi\)
\(284\) 1.11225e8 0.288130
\(285\) 0 0
\(286\) 2.10908e8 0.533103
\(287\) −1.16383e8 −0.290605
\(288\) 0 0
\(289\) 2.84620e8 0.693622
\(290\) −5.90561e7 −0.142191
\(291\) 0 0
\(292\) 488027. 0.00114711
\(293\) 7.47133e8 1.73525 0.867623 0.497222i \(-0.165647\pi\)
0.867623 + 0.497222i \(0.165647\pi\)
\(294\) 0 0
\(295\) 1.59310e7 0.0361298
\(296\) −4.76230e8 −1.06732
\(297\) 0 0
\(298\) 5.27203e6 0.0115404
\(299\) 5.24316e8 1.13434
\(300\) 0 0
\(301\) −1.26461e8 −0.267284
\(302\) −2.91886e8 −0.609802
\(303\) 0 0
\(304\) 3.04231e8 0.621077
\(305\) −3.86932e7 −0.0780881
\(306\) 0 0
\(307\) 6.94608e8 1.37011 0.685056 0.728491i \(-0.259778\pi\)
0.685056 + 0.728491i \(0.259778\pi\)
\(308\) −7.52691e7 −0.146788
\(309\) 0 0
\(310\) 9.31998e7 0.177684
\(311\) −1.27745e8 −0.240815 −0.120407 0.992725i \(-0.538420\pi\)
−0.120407 + 0.992725i \(0.538420\pi\)
\(312\) 0 0
\(313\) −9.74255e8 −1.79584 −0.897920 0.440159i \(-0.854922\pi\)
−0.897920 + 0.440159i \(0.854922\pi\)
\(314\) −2.04795e8 −0.373306
\(315\) 0 0
\(316\) 1.45441e8 0.259288
\(317\) 8.99019e8 1.58512 0.792559 0.609795i \(-0.208748\pi\)
0.792559 + 0.609795i \(0.208748\pi\)
\(318\) 0 0
\(319\) −5.12230e8 −0.883482
\(320\) −896215. −0.00152893
\(321\) 0 0
\(322\) 4.58282e7 0.0764957
\(323\) 1.09143e9 1.80213
\(324\) 0 0
\(325\) −8.97643e8 −1.45048
\(326\) 3.74814e8 0.599176
\(327\) 0 0
\(328\) 6.21708e8 0.972810
\(329\) −1.27379e8 −0.197203
\(330\) 0 0
\(331\) −1.16605e9 −1.76733 −0.883666 0.468119i \(-0.844932\pi\)
−0.883666 + 0.468119i \(0.844932\pi\)
\(332\) 3.82733e8 0.574002
\(333\) 0 0
\(334\) −2.86750e8 −0.421105
\(335\) 2.22734e8 0.323690
\(336\) 0 0
\(337\) 1.23311e9 1.75509 0.877544 0.479496i \(-0.159181\pi\)
0.877544 + 0.479496i \(0.159181\pi\)
\(338\) −4.62765e8 −0.651857
\(339\) 0 0
\(340\) −2.10250e8 −0.290108
\(341\) 8.08379e8 1.10401
\(342\) 0 0
\(343\) 3.46941e8 0.464223
\(344\) 6.75541e8 0.894741
\(345\) 0 0
\(346\) 3.38233e8 0.438986
\(347\) 5.89436e8 0.757327 0.378664 0.925534i \(-0.376384\pi\)
0.378664 + 0.925534i \(0.376384\pi\)
\(348\) 0 0
\(349\) 1.15878e9 1.45919 0.729596 0.683878i \(-0.239708\pi\)
0.729596 + 0.683878i \(0.239708\pi\)
\(350\) −7.84592e7 −0.0978150
\(351\) 0 0
\(352\) 6.25054e8 0.763868
\(353\) 5.26080e8 0.636561 0.318281 0.947997i \(-0.396895\pi\)
0.318281 + 0.947997i \(0.396895\pi\)
\(354\) 0 0
\(355\) −8.39110e7 −0.0995450
\(356\) 1.07698e9 1.26512
\(357\) 0 0
\(358\) 2.71129e8 0.312310
\(359\) −1.09995e9 −1.25470 −0.627351 0.778737i \(-0.715861\pi\)
−0.627351 + 0.778737i \(0.715861\pi\)
\(360\) 0 0
\(361\) 8.20202e8 0.917583
\(362\) 1.49636e8 0.165789
\(363\) 0 0
\(364\) 2.77528e8 0.301614
\(365\) −368180. −0.000396310 0
\(366\) 0 0
\(367\) 1.21755e9 1.28575 0.642874 0.765972i \(-0.277742\pi\)
0.642874 + 0.765972i \(0.277742\pi\)
\(368\) 3.09501e8 0.323738
\(369\) 0 0
\(370\) 1.60042e8 0.164258
\(371\) −5.80473e7 −0.0590164
\(372\) 0 0
\(373\) 5.53595e8 0.552346 0.276173 0.961108i \(-0.410934\pi\)
0.276173 + 0.961108i \(0.410934\pi\)
\(374\) 4.46635e8 0.441471
\(375\) 0 0
\(376\) 6.80449e8 0.660143
\(377\) 1.88866e9 1.81535
\(378\) 0 0
\(379\) −2.47132e8 −0.233180 −0.116590 0.993180i \(-0.537196\pi\)
−0.116590 + 0.993180i \(0.537196\pi\)
\(380\) −3.30196e8 −0.308695
\(381\) 0 0
\(382\) −5.17514e8 −0.475007
\(383\) 599846. 0.000545562 0 0.000272781 1.00000i \(-0.499913\pi\)
0.000272781 1.00000i \(0.499913\pi\)
\(384\) 0 0
\(385\) 5.67850e7 0.0507132
\(386\) −8.64541e8 −0.765121
\(387\) 0 0
\(388\) 4.64001e8 0.403281
\(389\) 8.92101e8 0.768406 0.384203 0.923249i \(-0.374476\pi\)
0.384203 + 0.923249i \(0.374476\pi\)
\(390\) 0 0
\(391\) 1.11033e9 0.939364
\(392\) −8.99436e8 −0.754170
\(393\) 0 0
\(394\) −6.40327e8 −0.527430
\(395\) −1.09724e8 −0.0895805
\(396\) 0 0
\(397\) −9.58791e8 −0.769055 −0.384527 0.923114i \(-0.625636\pi\)
−0.384527 + 0.923114i \(0.625636\pi\)
\(398\) 3.24872e8 0.258298
\(399\) 0 0
\(400\) −5.29874e8 −0.413964
\(401\) −1.82830e9 −1.41593 −0.707966 0.706247i \(-0.750387\pi\)
−0.707966 + 0.706247i \(0.750387\pi\)
\(402\) 0 0
\(403\) −2.98061e9 −2.26849
\(404\) −1.75442e9 −1.32373
\(405\) 0 0
\(406\) 1.65080e8 0.122420
\(407\) 1.38814e9 1.02059
\(408\) 0 0
\(409\) −7.15047e8 −0.516776 −0.258388 0.966041i \(-0.583191\pi\)
−0.258388 + 0.966041i \(0.583191\pi\)
\(410\) −2.08931e8 −0.149713
\(411\) 0 0
\(412\) 4.30697e8 0.303411
\(413\) −4.45320e7 −0.0311062
\(414\) 0 0
\(415\) −2.88744e8 −0.198310
\(416\) −2.30466e9 −1.56957
\(417\) 0 0
\(418\) 7.01435e8 0.469754
\(419\) −2.22875e9 −1.48017 −0.740086 0.672513i \(-0.765215\pi\)
−0.740086 + 0.672513i \(0.765215\pi\)
\(420\) 0 0
\(421\) 1.49109e9 0.973906 0.486953 0.873428i \(-0.338108\pi\)
0.486953 + 0.873428i \(0.338108\pi\)
\(422\) −7.15419e8 −0.463412
\(423\) 0 0
\(424\) 3.10083e8 0.197559
\(425\) −1.90092e9 −1.20116
\(426\) 0 0
\(427\) 1.08159e8 0.0672306
\(428\) −1.23661e9 −0.762394
\(429\) 0 0
\(430\) −2.27022e8 −0.137698
\(431\) −2.66273e9 −1.60198 −0.800988 0.598680i \(-0.795692\pi\)
−0.800988 + 0.598680i \(0.795692\pi\)
\(432\) 0 0
\(433\) 7.09435e7 0.0419957 0.0209979 0.999780i \(-0.493316\pi\)
0.0209979 + 0.999780i \(0.493316\pi\)
\(434\) −2.60522e8 −0.152979
\(435\) 0 0
\(436\) 2.01548e9 1.16460
\(437\) 1.74376e9 0.999546
\(438\) 0 0
\(439\) 6.26733e7 0.0353555 0.0176777 0.999844i \(-0.494373\pi\)
0.0176777 + 0.999844i \(0.494373\pi\)
\(440\) −3.03340e8 −0.169764
\(441\) 0 0
\(442\) −1.64680e9 −0.907119
\(443\) 6.41984e8 0.350842 0.175421 0.984494i \(-0.443871\pi\)
0.175421 + 0.984494i \(0.443871\pi\)
\(444\) 0 0
\(445\) −8.12500e8 −0.437082
\(446\) −1.18167e9 −0.630703
\(447\) 0 0
\(448\) 2.50520e6 0.00131634
\(449\) 2.77507e9 1.44681 0.723406 0.690423i \(-0.242575\pi\)
0.723406 + 0.690423i \(0.242575\pi\)
\(450\) 0 0
\(451\) −1.81219e9 −0.930219
\(452\) −7.76191e8 −0.395352
\(453\) 0 0
\(454\) 1.68184e9 0.843506
\(455\) −2.09374e8 −0.104204
\(456\) 0 0
\(457\) 3.79393e9 1.85944 0.929721 0.368266i \(-0.120048\pi\)
0.929721 + 0.368266i \(0.120048\pi\)
\(458\) 8.44209e8 0.410602
\(459\) 0 0
\(460\) −3.35916e8 −0.160908
\(461\) 3.74187e9 1.77884 0.889418 0.457095i \(-0.151110\pi\)
0.889418 + 0.457095i \(0.151110\pi\)
\(462\) 0 0
\(463\) 1.68652e9 0.789691 0.394846 0.918748i \(-0.370798\pi\)
0.394846 + 0.918748i \(0.370798\pi\)
\(464\) 1.11487e9 0.518097
\(465\) 0 0
\(466\) −1.38579e9 −0.634376
\(467\) 1.90310e8 0.0864675 0.0432338 0.999065i \(-0.486234\pi\)
0.0432338 + 0.999065i \(0.486234\pi\)
\(468\) 0 0
\(469\) −6.22610e8 −0.278684
\(470\) −2.28671e8 −0.101594
\(471\) 0 0
\(472\) 2.37886e8 0.104129
\(473\) −1.96910e9 −0.855568
\(474\) 0 0
\(475\) −2.98537e9 −1.27812
\(476\) 5.87714e8 0.249771
\(477\) 0 0
\(478\) 8.27840e8 0.346696
\(479\) −1.04486e9 −0.434395 −0.217197 0.976128i \(-0.569692\pi\)
−0.217197 + 0.976128i \(0.569692\pi\)
\(480\) 0 0
\(481\) −5.11827e9 −2.09708
\(482\) −1.10301e9 −0.448656
\(483\) 0 0
\(484\) 8.31630e8 0.333404
\(485\) −3.50054e8 −0.139328
\(486\) 0 0
\(487\) −4.52457e9 −1.77511 −0.887557 0.460697i \(-0.847599\pi\)
−0.887557 + 0.460697i \(0.847599\pi\)
\(488\) −5.77778e8 −0.225056
\(489\) 0 0
\(490\) 3.02264e8 0.116065
\(491\) −1.28164e9 −0.488630 −0.244315 0.969696i \(-0.578563\pi\)
−0.244315 + 0.969696i \(0.578563\pi\)
\(492\) 0 0
\(493\) 3.99958e9 1.50332
\(494\) −2.58629e9 −0.965235
\(495\) 0 0
\(496\) −1.75944e9 −0.647423
\(497\) 2.34557e8 0.0857041
\(498\) 0 0
\(499\) −3.26234e9 −1.17538 −0.587689 0.809087i \(-0.699962\pi\)
−0.587689 + 0.809087i \(0.699962\pi\)
\(500\) 1.19818e9 0.428674
\(501\) 0 0
\(502\) −1.32841e9 −0.468674
\(503\) −5.11806e8 −0.179315 −0.0896577 0.995973i \(-0.528577\pi\)
−0.0896577 + 0.995973i \(0.528577\pi\)
\(504\) 0 0
\(505\) 1.32358e9 0.457330
\(506\) 7.13586e8 0.244861
\(507\) 0 0
\(508\) 1.39896e9 0.473460
\(509\) −5.38317e9 −1.80936 −0.904682 0.426088i \(-0.859892\pi\)
−0.904682 + 0.426088i \(0.859892\pi\)
\(510\) 0 0
\(511\) 1.02918e6 0.000341206 0
\(512\) −2.44989e9 −0.806681
\(513\) 0 0
\(514\) −1.20322e9 −0.390819
\(515\) −3.24929e8 −0.104825
\(516\) 0 0
\(517\) −1.98341e9 −0.631241
\(518\) −4.47366e8 −0.141419
\(519\) 0 0
\(520\) 1.11846e9 0.348825
\(521\) −1.92620e9 −0.596720 −0.298360 0.954453i \(-0.596440\pi\)
−0.298360 + 0.954453i \(0.596440\pi\)
\(522\) 0 0
\(523\) 3.47164e9 1.06115 0.530577 0.847637i \(-0.321975\pi\)
0.530577 + 0.847637i \(0.321975\pi\)
\(524\) −1.33067e9 −0.404026
\(525\) 0 0
\(526\) 6.40623e8 0.191934
\(527\) −6.31197e9 −1.87857
\(528\) 0 0
\(529\) −1.63086e9 −0.478984
\(530\) −1.04206e8 −0.0304039
\(531\) 0 0
\(532\) 9.23000e8 0.265773
\(533\) 6.68179e9 1.91138
\(534\) 0 0
\(535\) 9.32930e8 0.263397
\(536\) 3.32593e9 0.932902
\(537\) 0 0
\(538\) 9.94645e8 0.275379
\(539\) 2.62173e9 0.721152
\(540\) 0 0
\(541\) −1.13960e9 −0.309430 −0.154715 0.987959i \(-0.549446\pi\)
−0.154715 + 0.987959i \(0.549446\pi\)
\(542\) −5.87919e8 −0.158606
\(543\) 0 0
\(544\) −4.88053e9 −1.29978
\(545\) −1.52053e9 −0.402353
\(546\) 0 0
\(547\) −3.40030e9 −0.888303 −0.444152 0.895952i \(-0.646495\pi\)
−0.444152 + 0.895952i \(0.646495\pi\)
\(548\) −7.87449e8 −0.204405
\(549\) 0 0
\(550\) −1.22168e9 −0.313103
\(551\) 6.28131e9 1.59963
\(552\) 0 0
\(553\) 3.06714e8 0.0771250
\(554\) −1.40509e9 −0.351091
\(555\) 0 0
\(556\) −5.01404e8 −0.123716
\(557\) −1.94268e9 −0.476331 −0.238165 0.971225i \(-0.576546\pi\)
−0.238165 + 0.971225i \(0.576546\pi\)
\(558\) 0 0
\(559\) 7.26036e9 1.75799
\(560\) −1.23592e8 −0.0297395
\(561\) 0 0
\(562\) −3.09468e8 −0.0735427
\(563\) −6.16858e9 −1.45682 −0.728410 0.685142i \(-0.759740\pi\)
−0.728410 + 0.685142i \(0.759740\pi\)
\(564\) 0 0
\(565\) 5.85579e8 0.136589
\(566\) 3.11331e9 0.721713
\(567\) 0 0
\(568\) −1.25298e9 −0.286897
\(569\) 5.43476e9 1.23677 0.618383 0.785877i \(-0.287788\pi\)
0.618383 + 0.785877i \(0.287788\pi\)
\(570\) 0 0
\(571\) −5.20959e9 −1.17105 −0.585527 0.810653i \(-0.699112\pi\)
−0.585527 + 0.810653i \(0.699112\pi\)
\(572\) 4.32135e9 0.965458
\(573\) 0 0
\(574\) 5.84027e8 0.128897
\(575\) −3.03709e9 −0.666223
\(576\) 0 0
\(577\) 2.67350e9 0.579381 0.289691 0.957120i \(-0.406448\pi\)
0.289691 + 0.957120i \(0.406448\pi\)
\(578\) −1.42826e9 −0.307653
\(579\) 0 0
\(580\) −1.21002e9 −0.257510
\(581\) 8.07129e8 0.170737
\(582\) 0 0
\(583\) −9.03847e8 −0.188910
\(584\) −5.49777e6 −0.00114220
\(585\) 0 0
\(586\) −3.74922e9 −0.769660
\(587\) −4.23836e9 −0.864898 −0.432449 0.901658i \(-0.642350\pi\)
−0.432449 + 0.901658i \(0.642350\pi\)
\(588\) 0 0
\(589\) −9.91288e9 −1.99892
\(590\) −7.99440e7 −0.0160252
\(591\) 0 0
\(592\) −3.02129e9 −0.598503
\(593\) 5.59276e9 1.10137 0.550687 0.834712i \(-0.314366\pi\)
0.550687 + 0.834712i \(0.314366\pi\)
\(594\) 0 0
\(595\) −4.43387e8 −0.0862926
\(596\) 1.08020e8 0.0208999
\(597\) 0 0
\(598\) −2.63109e9 −0.503132
\(599\) 2.76072e9 0.524842 0.262421 0.964953i \(-0.415479\pi\)
0.262421 + 0.964953i \(0.415479\pi\)
\(600\) 0 0
\(601\) −3.38464e9 −0.635992 −0.317996 0.948092i \(-0.603010\pi\)
−0.317996 + 0.948092i \(0.603010\pi\)
\(602\) 6.34598e8 0.118552
\(603\) 0 0
\(604\) −5.98054e9 −1.10436
\(605\) −6.27403e8 −0.115187
\(606\) 0 0
\(607\) 3.57949e9 0.649622 0.324811 0.945779i \(-0.394699\pi\)
0.324811 + 0.945779i \(0.394699\pi\)
\(608\) −7.66483e9 −1.38306
\(609\) 0 0
\(610\) 1.94168e8 0.0346356
\(611\) 7.31310e9 1.29705
\(612\) 0 0
\(613\) 9.92114e9 1.73960 0.869801 0.493403i \(-0.164247\pi\)
0.869801 + 0.493403i \(0.164247\pi\)
\(614\) −3.48564e9 −0.607706
\(615\) 0 0
\(616\) 8.47929e8 0.146160
\(617\) 2.67237e9 0.458034 0.229017 0.973422i \(-0.426449\pi\)
0.229017 + 0.973422i \(0.426449\pi\)
\(618\) 0 0
\(619\) 5.77104e9 0.977996 0.488998 0.872285i \(-0.337363\pi\)
0.488998 + 0.872285i \(0.337363\pi\)
\(620\) 1.90960e9 0.321789
\(621\) 0 0
\(622\) 6.41044e8 0.106812
\(623\) 2.27119e9 0.376309
\(624\) 0 0
\(625\) 4.72951e9 0.774882
\(626\) 4.88895e9 0.796536
\(627\) 0 0
\(628\) −4.19610e9 −0.676064
\(629\) −1.08388e10 −1.73662
\(630\) 0 0
\(631\) −1.33017e9 −0.210768 −0.105384 0.994432i \(-0.533607\pi\)
−0.105384 + 0.994432i \(0.533607\pi\)
\(632\) −1.63844e9 −0.258178
\(633\) 0 0
\(634\) −4.51141e9 −0.703072
\(635\) −1.05541e9 −0.163574
\(636\) 0 0
\(637\) −9.66667e9 −1.48180
\(638\) 2.57044e9 0.391864
\(639\) 0 0
\(640\) 1.84266e9 0.277853
\(641\) −8.97295e8 −0.134565 −0.0672825 0.997734i \(-0.521433\pi\)
−0.0672825 + 0.997734i \(0.521433\pi\)
\(642\) 0 0
\(643\) 7.92605e9 1.17576 0.587879 0.808949i \(-0.299963\pi\)
0.587879 + 0.808949i \(0.299963\pi\)
\(644\) 9.38988e8 0.138535
\(645\) 0 0
\(646\) −5.47693e9 −0.799325
\(647\) 9.72853e9 1.41215 0.706077 0.708135i \(-0.250463\pi\)
0.706077 + 0.708135i \(0.250463\pi\)
\(648\) 0 0
\(649\) −6.93403e8 −0.0995702
\(650\) 4.50450e9 0.643354
\(651\) 0 0
\(652\) 7.67968e9 1.08512
\(653\) 4.58349e9 0.644170 0.322085 0.946711i \(-0.395616\pi\)
0.322085 + 0.946711i \(0.395616\pi\)
\(654\) 0 0
\(655\) 1.00389e9 0.139586
\(656\) 3.94423e9 0.545505
\(657\) 0 0
\(658\) 6.39208e8 0.0874684
\(659\) 1.15331e9 0.156982 0.0784908 0.996915i \(-0.474990\pi\)
0.0784908 + 0.996915i \(0.474990\pi\)
\(660\) 0 0
\(661\) −1.03408e8 −0.0139267 −0.00696334 0.999976i \(-0.502217\pi\)
−0.00696334 + 0.999976i \(0.502217\pi\)
\(662\) 5.85139e9 0.783892
\(663\) 0 0
\(664\) −4.31161e9 −0.571546
\(665\) −6.96335e8 −0.0918211
\(666\) 0 0
\(667\) 6.39011e9 0.833812
\(668\) −5.87530e9 −0.762628
\(669\) 0 0
\(670\) −1.11771e9 −0.143571
\(671\) 1.68414e9 0.215203
\(672\) 0 0
\(673\) −8.65901e9 −1.09500 −0.547502 0.836804i \(-0.684421\pi\)
−0.547502 + 0.836804i \(0.684421\pi\)
\(674\) −6.18795e9 −0.778461
\(675\) 0 0
\(676\) −9.48174e9 −1.18052
\(677\) −1.41759e10 −1.75586 −0.877930 0.478789i \(-0.841076\pi\)
−0.877930 + 0.478789i \(0.841076\pi\)
\(678\) 0 0
\(679\) 9.78511e8 0.119956
\(680\) 2.36853e9 0.288867
\(681\) 0 0
\(682\) −4.05656e9 −0.489681
\(683\) −2.10450e9 −0.252742 −0.126371 0.991983i \(-0.540333\pi\)
−0.126371 + 0.991983i \(0.540333\pi\)
\(684\) 0 0
\(685\) 5.94072e8 0.0706191
\(686\) −1.74100e9 −0.205904
\(687\) 0 0
\(688\) 4.28575e9 0.501727
\(689\) 3.33261e9 0.388166
\(690\) 0 0
\(691\) 8.19673e9 0.945078 0.472539 0.881310i \(-0.343338\pi\)
0.472539 + 0.881310i \(0.343338\pi\)
\(692\) 6.93017e9 0.795010
\(693\) 0 0
\(694\) −2.95787e9 −0.335909
\(695\) 3.78272e8 0.0427423
\(696\) 0 0
\(697\) 1.41499e10 1.58284
\(698\) −5.81493e9 −0.647218
\(699\) 0 0
\(700\) −1.60757e9 −0.177145
\(701\) 1.58794e9 0.174109 0.0870547 0.996204i \(-0.472255\pi\)
0.0870547 + 0.996204i \(0.472255\pi\)
\(702\) 0 0
\(703\) −1.70223e10 −1.84788
\(704\) 3.90082e7 0.00421358
\(705\) 0 0
\(706\) −2.63994e9 −0.282344
\(707\) −3.69981e9 −0.393741
\(708\) 0 0
\(709\) −1.39994e10 −1.47519 −0.737596 0.675242i \(-0.764039\pi\)
−0.737596 + 0.675242i \(0.764039\pi\)
\(710\) 4.21077e8 0.0441527
\(711\) 0 0
\(712\) −1.21325e10 −1.25971
\(713\) −1.00846e10 −1.04195
\(714\) 0 0
\(715\) −3.26014e9 −0.333553
\(716\) 5.55525e9 0.565598
\(717\) 0 0
\(718\) 5.51968e9 0.556517
\(719\) 3.63177e8 0.0364391 0.0182195 0.999834i \(-0.494200\pi\)
0.0182195 + 0.999834i \(0.494200\pi\)
\(720\) 0 0
\(721\) 9.08277e8 0.0902495
\(722\) −4.11589e9 −0.406990
\(723\) 0 0
\(724\) 3.06594e9 0.300247
\(725\) −1.09401e10 −1.06619
\(726\) 0 0
\(727\) −7.18128e9 −0.693157 −0.346578 0.938021i \(-0.612657\pi\)
−0.346578 + 0.938021i \(0.612657\pi\)
\(728\) −3.12643e9 −0.300324
\(729\) 0 0
\(730\) 1.84758e6 0.000175782 0
\(731\) 1.53751e10 1.45582
\(732\) 0 0
\(733\) 1.12100e10 1.05133 0.525667 0.850691i \(-0.323816\pi\)
0.525667 + 0.850691i \(0.323816\pi\)
\(734\) −6.10985e9 −0.570288
\(735\) 0 0
\(736\) −7.79760e9 −0.720922
\(737\) −9.69459e9 −0.892059
\(738\) 0 0
\(739\) −2.61736e9 −0.238566 −0.119283 0.992860i \(-0.538060\pi\)
−0.119283 + 0.992860i \(0.538060\pi\)
\(740\) 3.27914e9 0.297474
\(741\) 0 0
\(742\) 2.91289e8 0.0261765
\(743\) 2.16513e10 1.93653 0.968264 0.249930i \(-0.0804075\pi\)
0.968264 + 0.249930i \(0.0804075\pi\)
\(744\) 0 0
\(745\) −8.14933e7 −0.00722063
\(746\) −2.77802e9 −0.244991
\(747\) 0 0
\(748\) 9.15123e9 0.799510
\(749\) −2.60783e9 −0.226774
\(750\) 0 0
\(751\) −1.23735e10 −1.06599 −0.532995 0.846118i \(-0.678934\pi\)
−0.532995 + 0.846118i \(0.678934\pi\)
\(752\) 4.31689e9 0.370176
\(753\) 0 0
\(754\) −9.47759e9 −0.805190
\(755\) 4.51187e9 0.381542
\(756\) 0 0
\(757\) −1.68097e9 −0.140840 −0.0704198 0.997517i \(-0.522434\pi\)
−0.0704198 + 0.997517i \(0.522434\pi\)
\(758\) 1.24014e9 0.103426
\(759\) 0 0
\(760\) 3.71976e9 0.307374
\(761\) −1.31161e10 −1.07884 −0.539421 0.842036i \(-0.681357\pi\)
−0.539421 + 0.842036i \(0.681357\pi\)
\(762\) 0 0
\(763\) 4.25036e9 0.346409
\(764\) −1.06035e10 −0.860246
\(765\) 0 0
\(766\) −3.01011e6 −0.000241981 0
\(767\) 2.55668e9 0.204593
\(768\) 0 0
\(769\) 1.42811e10 1.13245 0.566224 0.824251i \(-0.308404\pi\)
0.566224 + 0.824251i \(0.308404\pi\)
\(770\) −2.84955e8 −0.0224936
\(771\) 0 0
\(772\) −1.77138e10 −1.38565
\(773\) 8.34783e9 0.650048 0.325024 0.945706i \(-0.394628\pi\)
0.325024 + 0.945706i \(0.394628\pi\)
\(774\) 0 0
\(775\) 1.72651e10 1.33233
\(776\) −5.22711e9 −0.401556
\(777\) 0 0
\(778\) −4.47669e9 −0.340823
\(779\) 2.22223e10 1.68425
\(780\) 0 0
\(781\) 3.65226e9 0.274336
\(782\) −5.57180e9 −0.416651
\(783\) 0 0
\(784\) −5.70618e9 −0.422902
\(785\) 3.16565e9 0.233571
\(786\) 0 0
\(787\) 1.47026e10 1.07518 0.537591 0.843206i \(-0.319334\pi\)
0.537591 + 0.843206i \(0.319334\pi\)
\(788\) −1.31199e10 −0.955184
\(789\) 0 0
\(790\) 5.50613e8 0.0397330
\(791\) −1.63687e9 −0.117597
\(792\) 0 0
\(793\) −6.20965e9 −0.442192
\(794\) 4.81135e9 0.341111
\(795\) 0 0
\(796\) 6.65640e9 0.467782
\(797\) −3.94070e9 −0.275721 −0.137860 0.990452i \(-0.544023\pi\)
−0.137860 + 0.990452i \(0.544023\pi\)
\(798\) 0 0
\(799\) 1.54868e10 1.07411
\(800\) 1.33497e10 0.921843
\(801\) 0 0
\(802\) 9.17467e9 0.628030
\(803\) 1.60252e7 0.00109219
\(804\) 0 0
\(805\) −7.08397e8 −0.0478620
\(806\) 1.49571e10 1.00618
\(807\) 0 0
\(808\) 1.97640e10 1.31806
\(809\) −2.12121e10 −1.40852 −0.704260 0.709942i \(-0.748721\pi\)
−0.704260 + 0.709942i \(0.748721\pi\)
\(810\) 0 0
\(811\) 4.13209e9 0.272018 0.136009 0.990708i \(-0.456572\pi\)
0.136009 + 0.990708i \(0.456572\pi\)
\(812\) 3.38238e9 0.221705
\(813\) 0 0
\(814\) −6.96589e9 −0.452680
\(815\) −5.79375e9 −0.374893
\(816\) 0 0
\(817\) 2.41465e10 1.54909
\(818\) 3.58821e9 0.229214
\(819\) 0 0
\(820\) −4.28085e9 −0.271133
\(821\) −4.99283e9 −0.314880 −0.157440 0.987529i \(-0.550324\pi\)
−0.157440 + 0.987529i \(0.550324\pi\)
\(822\) 0 0
\(823\) 8.56606e9 0.535650 0.267825 0.963468i \(-0.413695\pi\)
0.267825 + 0.963468i \(0.413695\pi\)
\(824\) −4.85193e9 −0.302113
\(825\) 0 0
\(826\) 2.23468e8 0.0137970
\(827\) −2.19884e10 −1.35184 −0.675920 0.736975i \(-0.736253\pi\)
−0.675920 + 0.736975i \(0.736253\pi\)
\(828\) 0 0
\(829\) −1.74819e9 −0.106573 −0.0532867 0.998579i \(-0.516970\pi\)
−0.0532867 + 0.998579i \(0.516970\pi\)
\(830\) 1.44896e9 0.0879595
\(831\) 0 0
\(832\) −1.43829e8 −0.00865792
\(833\) −2.04709e10 −1.22710
\(834\) 0 0
\(835\) 4.43248e9 0.263478
\(836\) 1.43719e10 0.850732
\(837\) 0 0
\(838\) 1.11842e10 0.656523
\(839\) −7.57544e8 −0.0442834 −0.0221417 0.999755i \(-0.507048\pi\)
−0.0221417 + 0.999755i \(0.507048\pi\)
\(840\) 0 0
\(841\) 5.76829e9 0.334396
\(842\) −7.48251e9 −0.431972
\(843\) 0 0
\(844\) −1.46584e10 −0.839246
\(845\) 7.15327e9 0.407855
\(846\) 0 0
\(847\) 1.75379e9 0.0991710
\(848\) 1.96722e9 0.110782
\(849\) 0 0
\(850\) 9.53908e9 0.532771
\(851\) −1.73172e10 −0.963215
\(852\) 0 0
\(853\) −1.72683e9 −0.0952636 −0.0476318 0.998865i \(-0.515167\pi\)
−0.0476318 + 0.998865i \(0.515167\pi\)
\(854\) −5.42759e8 −0.0298198
\(855\) 0 0
\(856\) 1.39308e10 0.759132
\(857\) 3.43055e10 1.86179 0.930895 0.365286i \(-0.119029\pi\)
0.930895 + 0.365286i \(0.119029\pi\)
\(858\) 0 0
\(859\) −1.16013e10 −0.624499 −0.312249 0.950000i \(-0.601082\pi\)
−0.312249 + 0.950000i \(0.601082\pi\)
\(860\) −4.65153e9 −0.249374
\(861\) 0 0
\(862\) 1.33619e10 0.710549
\(863\) −1.06656e10 −0.564870 −0.282435 0.959286i \(-0.591142\pi\)
−0.282435 + 0.959286i \(0.591142\pi\)
\(864\) 0 0
\(865\) −5.22830e9 −0.274665
\(866\) −3.56005e8 −0.0186270
\(867\) 0 0
\(868\) −5.33792e9 −0.277047
\(869\) 4.77580e9 0.246875
\(870\) 0 0
\(871\) 3.57453e10 1.83297
\(872\) −2.27050e10 −1.15962
\(873\) 0 0
\(874\) −8.75046e9 −0.443344
\(875\) 2.52679e9 0.127509
\(876\) 0 0
\(877\) −5.25940e9 −0.263292 −0.131646 0.991297i \(-0.542026\pi\)
−0.131646 + 0.991297i \(0.542026\pi\)
\(878\) −3.14503e8 −0.0156817
\(879\) 0 0
\(880\) −1.92444e9 −0.0951954
\(881\) 2.79482e10 1.37701 0.688507 0.725230i \(-0.258266\pi\)
0.688507 + 0.725230i \(0.258266\pi\)
\(882\) 0 0
\(883\) 3.72423e9 0.182043 0.0910214 0.995849i \(-0.470987\pi\)
0.0910214 + 0.995849i \(0.470987\pi\)
\(884\) −3.37419e10 −1.64281
\(885\) 0 0
\(886\) −3.22157e9 −0.155614
\(887\) −6.26596e9 −0.301477 −0.150739 0.988574i \(-0.548165\pi\)
−0.150739 + 0.988574i \(0.548165\pi\)
\(888\) 0 0
\(889\) 2.95021e9 0.140830
\(890\) 4.07724e9 0.193866
\(891\) 0 0
\(892\) −2.42116e10 −1.14221
\(893\) 2.43219e10 1.14292
\(894\) 0 0
\(895\) −4.19102e9 −0.195407
\(896\) −5.15081e9 −0.239220
\(897\) 0 0
\(898\) −1.39257e10 −0.641727
\(899\) −3.63263e10 −1.66748
\(900\) 0 0
\(901\) 7.05740e9 0.321446
\(902\) 9.09382e9 0.412594
\(903\) 0 0
\(904\) 8.74403e9 0.393661
\(905\) −2.31302e9 −0.103731
\(906\) 0 0
\(907\) 8.21053e9 0.365381 0.182690 0.983170i \(-0.441519\pi\)
0.182690 + 0.983170i \(0.441519\pi\)
\(908\) 3.44596e10 1.52760
\(909\) 0 0
\(910\) 1.05067e9 0.0462191
\(911\) 1.57505e10 0.690207 0.345103 0.938565i \(-0.387844\pi\)
0.345103 + 0.938565i \(0.387844\pi\)
\(912\) 0 0
\(913\) 1.25677e10 0.546523
\(914\) −1.90385e10 −0.824746
\(915\) 0 0
\(916\) 1.72973e10 0.743607
\(917\) −2.80618e9 −0.120177
\(918\) 0 0
\(919\) −6.94140e9 −0.295014 −0.147507 0.989061i \(-0.547125\pi\)
−0.147507 + 0.989061i \(0.547125\pi\)
\(920\) 3.78419e9 0.160220
\(921\) 0 0
\(922\) −1.87773e10 −0.788994
\(923\) −1.34664e10 −0.563697
\(924\) 0 0
\(925\) 2.96475e10 1.23166
\(926\) −8.46318e9 −0.350264
\(927\) 0 0
\(928\) −2.80881e10 −1.15373
\(929\) 1.64084e9 0.0671448 0.0335724 0.999436i \(-0.489312\pi\)
0.0335724 + 0.999436i \(0.489312\pi\)
\(930\) 0 0
\(931\) −3.21493e10 −1.30571
\(932\) −2.83939e10 −1.14887
\(933\) 0 0
\(934\) −9.55004e8 −0.0383523
\(935\) −6.90393e9 −0.276220
\(936\) 0 0
\(937\) 3.84565e10 1.52715 0.763575 0.645719i \(-0.223442\pi\)
0.763575 + 0.645719i \(0.223442\pi\)
\(938\) 3.12435e9 0.123609
\(939\) 0 0
\(940\) −4.68532e9 −0.183989
\(941\) −3.13483e10 −1.22645 −0.613226 0.789907i \(-0.710129\pi\)
−0.613226 + 0.789907i \(0.710129\pi\)
\(942\) 0 0
\(943\) 2.26072e10 0.877921
\(944\) 1.50919e9 0.0583905
\(945\) 0 0
\(946\) 9.88124e9 0.379483
\(947\) 4.56816e10 1.74790 0.873950 0.486016i \(-0.161550\pi\)
0.873950 + 0.486016i \(0.161550\pi\)
\(948\) 0 0
\(949\) −5.90871e7 −0.00224420
\(950\) 1.49810e10 0.566903
\(951\) 0 0
\(952\) −6.62078e9 −0.248702
\(953\) 2.57557e10 0.963937 0.481968 0.876189i \(-0.339922\pi\)
0.481968 + 0.876189i \(0.339922\pi\)
\(954\) 0 0
\(955\) 7.99955e9 0.297204
\(956\) 1.69619e10 0.627872
\(957\) 0 0
\(958\) 5.24327e9 0.192674
\(959\) −1.66061e9 −0.0608000
\(960\) 0 0
\(961\) 2.98159e10 1.08372
\(962\) 2.56842e10 0.930151
\(963\) 0 0
\(964\) −2.25998e10 −0.812522
\(965\) 1.33638e10 0.478723
\(966\) 0 0
\(967\) 2.61784e10 0.931002 0.465501 0.885047i \(-0.345874\pi\)
0.465501 + 0.885047i \(0.345874\pi\)
\(968\) −9.36857e9 −0.331978
\(969\) 0 0
\(970\) 1.75662e9 0.0617984
\(971\) 7.63541e9 0.267648 0.133824 0.991005i \(-0.457274\pi\)
0.133824 + 0.991005i \(0.457274\pi\)
\(972\) 0 0
\(973\) −1.05739e9 −0.0367993
\(974\) 2.27050e10 0.787344
\(975\) 0 0
\(976\) −3.66552e9 −0.126201
\(977\) 1.18991e10 0.408211 0.204105 0.978949i \(-0.434571\pi\)
0.204105 + 0.978949i \(0.434571\pi\)
\(978\) 0 0
\(979\) 3.53644e10 1.20456
\(980\) 6.19319e9 0.210195
\(981\) 0 0
\(982\) 6.43144e9 0.216729
\(983\) 3.88296e10 1.30384 0.651921 0.758287i \(-0.273963\pi\)
0.651921 + 0.758287i \(0.273963\pi\)
\(984\) 0 0
\(985\) 9.89796e9 0.330004
\(986\) −2.00705e10 −0.666789
\(987\) 0 0
\(988\) −5.29913e10 −1.74806
\(989\) 2.45647e10 0.807467
\(990\) 0 0
\(991\) −4.18861e10 −1.36714 −0.683569 0.729886i \(-0.739573\pi\)
−0.683569 + 0.729886i \(0.739573\pi\)
\(992\) 4.43275e10 1.44172
\(993\) 0 0
\(994\) −1.17704e9 −0.0380136
\(995\) −5.02176e9 −0.161613
\(996\) 0 0
\(997\) −2.75354e10 −0.879950 −0.439975 0.898010i \(-0.645013\pi\)
−0.439975 + 0.898010i \(0.645013\pi\)
\(998\) 1.63709e10 0.521333
\(999\) 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.d.1.6 17
3.2 odd 2 177.8.a.b.1.12 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.12 17 3.2 odd 2
531.8.a.d.1.6 17 1.1 even 1 trivial