Properties

Label 531.8.a.d.1.9
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 1.9
Root \(2.39686\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +O(q^{10})\) \(q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +98.1157 q^{10} +2477.11 q^{11} +9417.61 q^{13} +259.118 q^{14} +16323.5 q^{16} -12577.2 q^{17} -41390.0 q^{19} +31606.9 q^{20} -983.055 q^{22} +59781.4 q^{23} -17000.9 q^{25} -3737.43 q^{26} +83472.1 q^{28} -212559. q^{29} -7793.74 q^{31} -19474.2 q^{32} +4991.34 q^{34} +161426. q^{35} -447343. q^{37} +16425.8 q^{38} -25102.2 q^{40} -14016.1 q^{41} -896369. q^{43} -316680. q^{44} -23724.6 q^{46} -21326.5 q^{47} -397227. q^{49} +6746.89 q^{50} -1.20397e6 q^{52} -910896. q^{53} -612424. q^{55} -66293.5 q^{56} +84355.0 q^{58} +205379. q^{59} +810282. q^{61} +3092.99 q^{62} -2.08169e6 q^{64} -2.32834e6 q^{65} -3.83166e6 q^{67} +1.60791e6 q^{68} -64062.6 q^{70} +3.35674e6 q^{71} +461542. q^{73} +177530. q^{74} +5.29140e6 q^{76} -1.61738e6 q^{77} -1.37060e6 q^{79} -4.03572e6 q^{80} +5562.36 q^{82} +7.63422e6 q^{83} +3.10951e6 q^{85} +355729. q^{86} +251507. q^{88} +5.26057e6 q^{89} -6.14903e6 q^{91} -7.64260e6 q^{92} +8463.51 q^{94} +1.02330e7 q^{95} -9.23173e6 q^{97} +157641. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.396855 −0.0350774 −0.0175387 0.999846i \(-0.505583\pi\)
−0.0175387 + 0.999846i \(0.505583\pi\)
\(3\) 0 0
\(4\) −127.843 −0.998770
\(5\) −247.233 −0.884527 −0.442264 0.896885i \(-0.645824\pi\)
−0.442264 + 0.896885i \(0.645824\pi\)
\(6\) 0 0
\(7\) −652.929 −0.719487 −0.359743 0.933051i \(-0.617136\pi\)
−0.359743 + 0.933051i \(0.617136\pi\)
\(8\) 101.532 0.0701116
\(9\) 0 0
\(10\) 98.1157 0.0310269
\(11\) 2477.11 0.561140 0.280570 0.959834i \(-0.409476\pi\)
0.280570 + 0.959834i \(0.409476\pi\)
\(12\) 0 0
\(13\) 9417.61 1.18888 0.594442 0.804139i \(-0.297373\pi\)
0.594442 + 0.804139i \(0.297373\pi\)
\(14\) 259.118 0.0252377
\(15\) 0 0
\(16\) 16323.5 0.996310
\(17\) −12577.2 −0.620889 −0.310444 0.950592i \(-0.600478\pi\)
−0.310444 + 0.950592i \(0.600478\pi\)
\(18\) 0 0
\(19\) −41390.0 −1.38439 −0.692194 0.721711i \(-0.743356\pi\)
−0.692194 + 0.721711i \(0.743356\pi\)
\(20\) 31606.9 0.883439
\(21\) 0 0
\(22\) −983.055 −0.0196833
\(23\) 59781.4 1.02452 0.512258 0.858832i \(-0.328809\pi\)
0.512258 + 0.858832i \(0.328809\pi\)
\(24\) 0 0
\(25\) −17000.9 −0.217611
\(26\) −3737.43 −0.0417029
\(27\) 0 0
\(28\) 83472.1 0.718602
\(29\) −212559. −1.61840 −0.809199 0.587534i \(-0.800099\pi\)
−0.809199 + 0.587534i \(0.800099\pi\)
\(30\) 0 0
\(31\) −7793.74 −0.0469872 −0.0234936 0.999724i \(-0.507479\pi\)
−0.0234936 + 0.999724i \(0.507479\pi\)
\(32\) −19474.2 −0.105060
\(33\) 0 0
\(34\) 4991.34 0.0217792
\(35\) 161426. 0.636406
\(36\) 0 0
\(37\) −447343. −1.45189 −0.725946 0.687751i \(-0.758598\pi\)
−0.725946 + 0.687751i \(0.758598\pi\)
\(38\) 16425.8 0.0485607
\(39\) 0 0
\(40\) −25102.2 −0.0620156
\(41\) −14016.1 −0.0317602 −0.0158801 0.999874i \(-0.505055\pi\)
−0.0158801 + 0.999874i \(0.505055\pi\)
\(42\) 0 0
\(43\) −896369. −1.71928 −0.859641 0.510899i \(-0.829313\pi\)
−0.859641 + 0.510899i \(0.829313\pi\)
\(44\) −316680. −0.560450
\(45\) 0 0
\(46\) −23724.6 −0.0359373
\(47\) −21326.5 −0.0299624 −0.0149812 0.999888i \(-0.504769\pi\)
−0.0149812 + 0.999888i \(0.504769\pi\)
\(48\) 0 0
\(49\) −397227. −0.482339
\(50\) 6746.89 0.00763324
\(51\) 0 0
\(52\) −1.20397e6 −1.18742
\(53\) −910896. −0.840433 −0.420217 0.907424i \(-0.638046\pi\)
−0.420217 + 0.907424i \(0.638046\pi\)
\(54\) 0 0
\(55\) −612424. −0.496344
\(56\) −66293.5 −0.0504444
\(57\) 0 0
\(58\) 84355.0 0.0567692
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 810282. 0.457069 0.228535 0.973536i \(-0.426607\pi\)
0.228535 + 0.973536i \(0.426607\pi\)
\(62\) 3092.99 0.00164819
\(63\) 0 0
\(64\) −2.08169e6 −0.992625
\(65\) −2.32834e6 −1.05160
\(66\) 0 0
\(67\) −3.83166e6 −1.55641 −0.778207 0.628007i \(-0.783871\pi\)
−0.778207 + 0.628007i \(0.783871\pi\)
\(68\) 1.60791e6 0.620125
\(69\) 0 0
\(70\) −64062.6 −0.0223234
\(71\) 3.35674e6 1.11305 0.556524 0.830832i \(-0.312135\pi\)
0.556524 + 0.830832i \(0.312135\pi\)
\(72\) 0 0
\(73\) 461542. 0.138861 0.0694306 0.997587i \(-0.477882\pi\)
0.0694306 + 0.997587i \(0.477882\pi\)
\(74\) 177530. 0.0509286
\(75\) 0 0
\(76\) 5.29140e6 1.38269
\(77\) −1.61738e6 −0.403733
\(78\) 0 0
\(79\) −1.37060e6 −0.312762 −0.156381 0.987697i \(-0.549983\pi\)
−0.156381 + 0.987697i \(0.549983\pi\)
\(80\) −4.03572e6 −0.881264
\(81\) 0 0
\(82\) 5562.36 0.00111407
\(83\) 7.63422e6 1.46552 0.732759 0.680488i \(-0.238232\pi\)
0.732759 + 0.680488i \(0.238232\pi\)
\(84\) 0 0
\(85\) 3.10951e6 0.549193
\(86\) 355729. 0.0603079
\(87\) 0 0
\(88\) 251507. 0.0393424
\(89\) 5.26057e6 0.790984 0.395492 0.918469i \(-0.370574\pi\)
0.395492 + 0.918469i \(0.370574\pi\)
\(90\) 0 0
\(91\) −6.14903e6 −0.855386
\(92\) −7.64260e6 −1.02326
\(93\) 0 0
\(94\) 8463.51 0.00105100
\(95\) 1.02330e7 1.22453
\(96\) 0 0
\(97\) −9.23173e6 −1.02703 −0.513514 0.858081i \(-0.671657\pi\)
−0.513514 + 0.858081i \(0.671657\pi\)
\(98\) 157641. 0.0169192
\(99\) 0 0
\(100\) 2.17344e6 0.217344
\(101\) −4.85554e6 −0.468935 −0.234468 0.972124i \(-0.575335\pi\)
−0.234468 + 0.972124i \(0.575335\pi\)
\(102\) 0 0
\(103\) −4.34436e6 −0.391738 −0.195869 0.980630i \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(104\) 956193. 0.0833545
\(105\) 0 0
\(106\) 361494. 0.0294802
\(107\) 6.60510e6 0.521238 0.260619 0.965442i \(-0.416073\pi\)
0.260619 + 0.965442i \(0.416073\pi\)
\(108\) 0 0
\(109\) −2.74784e6 −0.203235 −0.101617 0.994824i \(-0.532402\pi\)
−0.101617 + 0.994824i \(0.532402\pi\)
\(110\) 243044. 0.0174104
\(111\) 0 0
\(112\) −1.06581e7 −0.716832
\(113\) 3.08697e6 0.201260 0.100630 0.994924i \(-0.467914\pi\)
0.100630 + 0.994924i \(0.467914\pi\)
\(114\) 0 0
\(115\) −1.47799e7 −0.906212
\(116\) 2.71740e7 1.61641
\(117\) 0 0
\(118\) −81505.7 −0.00456669
\(119\) 8.21204e6 0.446721
\(120\) 0 0
\(121\) −1.33511e7 −0.685122
\(122\) −321565. −0.0160328
\(123\) 0 0
\(124\) 996371. 0.0469294
\(125\) 2.35182e7 1.07701
\(126\) 0 0
\(127\) 1.02662e7 0.444731 0.222365 0.974963i \(-0.428622\pi\)
0.222365 + 0.974963i \(0.428622\pi\)
\(128\) 3.31883e6 0.139878
\(129\) 0 0
\(130\) 924016. 0.0368874
\(131\) 3.04995e7 1.18534 0.592671 0.805445i \(-0.298073\pi\)
0.592671 + 0.805445i \(0.298073\pi\)
\(132\) 0 0
\(133\) 2.70247e7 0.996049
\(134\) 1.52062e6 0.0545950
\(135\) 0 0
\(136\) −1.27700e6 −0.0435315
\(137\) −2.18105e7 −0.724676 −0.362338 0.932047i \(-0.618021\pi\)
−0.362338 + 0.932047i \(0.618021\pi\)
\(138\) 0 0
\(139\) 5.59222e7 1.76617 0.883085 0.469213i \(-0.155462\pi\)
0.883085 + 0.469213i \(0.155462\pi\)
\(140\) −2.06370e7 −0.635623
\(141\) 0 0
\(142\) −1.33214e6 −0.0390428
\(143\) 2.33285e7 0.667130
\(144\) 0 0
\(145\) 5.25515e7 1.43152
\(146\) −183165. −0.00487089
\(147\) 0 0
\(148\) 5.71894e7 1.45011
\(149\) −4.25096e7 −1.05277 −0.526387 0.850245i \(-0.676454\pi\)
−0.526387 + 0.850245i \(0.676454\pi\)
\(150\) 0 0
\(151\) −2.69136e7 −0.636140 −0.318070 0.948067i \(-0.603035\pi\)
−0.318070 + 0.948067i \(0.603035\pi\)
\(152\) −4.20243e6 −0.0970617
\(153\) 0 0
\(154\) 641865. 0.0141619
\(155\) 1.92687e6 0.0415615
\(156\) 0 0
\(157\) −6.09453e7 −1.25687 −0.628436 0.777861i \(-0.716305\pi\)
−0.628436 + 0.777861i \(0.716305\pi\)
\(158\) 543928. 0.0109709
\(159\) 0 0
\(160\) 4.81467e6 0.0929281
\(161\) −3.90330e7 −0.737126
\(162\) 0 0
\(163\) 2.43192e7 0.439839 0.219919 0.975518i \(-0.429421\pi\)
0.219919 + 0.975518i \(0.429421\pi\)
\(164\) 1.79185e6 0.0317212
\(165\) 0 0
\(166\) −3.02968e6 −0.0514066
\(167\) −2.18099e7 −0.362366 −0.181183 0.983449i \(-0.557993\pi\)
−0.181183 + 0.983449i \(0.557993\pi\)
\(168\) 0 0
\(169\) 2.59429e7 0.413443
\(170\) −1.23402e6 −0.0192643
\(171\) 0 0
\(172\) 1.14594e8 1.71717
\(173\) −7.57905e7 −1.11289 −0.556446 0.830884i \(-0.687836\pi\)
−0.556446 + 0.830884i \(0.687836\pi\)
\(174\) 0 0
\(175\) 1.11004e7 0.156568
\(176\) 4.04353e7 0.559070
\(177\) 0 0
\(178\) −2.08769e6 −0.0277457
\(179\) 1.14237e6 0.0148874 0.00744372 0.999972i \(-0.497631\pi\)
0.00744372 + 0.999972i \(0.497631\pi\)
\(180\) 0 0
\(181\) 1.35346e8 1.69656 0.848280 0.529548i \(-0.177638\pi\)
0.848280 + 0.529548i \(0.177638\pi\)
\(182\) 2.44028e6 0.0300047
\(183\) 0 0
\(184\) 6.06975e6 0.0718304
\(185\) 1.10598e8 1.28424
\(186\) 0 0
\(187\) −3.11552e7 −0.348406
\(188\) 2.72643e6 0.0299255
\(189\) 0 0
\(190\) −4.06101e6 −0.0429533
\(191\) −1.11443e8 −1.15727 −0.578637 0.815585i \(-0.696415\pi\)
−0.578637 + 0.815585i \(0.696415\pi\)
\(192\) 0 0
\(193\) −4.58821e7 −0.459402 −0.229701 0.973261i \(-0.573775\pi\)
−0.229701 + 0.973261i \(0.573775\pi\)
\(194\) 3.66366e6 0.0360254
\(195\) 0 0
\(196\) 5.07824e7 0.481745
\(197\) −1.22654e8 −1.14301 −0.571504 0.820600i \(-0.693640\pi\)
−0.571504 + 0.820600i \(0.693640\pi\)
\(198\) 0 0
\(199\) −1.02506e7 −0.0922072 −0.0461036 0.998937i \(-0.514680\pi\)
−0.0461036 + 0.998937i \(0.514680\pi\)
\(200\) −1.72614e6 −0.0152571
\(201\) 0 0
\(202\) 1.92695e6 0.0164490
\(203\) 1.38786e8 1.16442
\(204\) 0 0
\(205\) 3.46524e6 0.0280928
\(206\) 1.72408e6 0.0137411
\(207\) 0 0
\(208\) 1.53729e8 1.18450
\(209\) −1.02528e8 −0.776836
\(210\) 0 0
\(211\) −1.63127e8 −1.19547 −0.597734 0.801694i \(-0.703932\pi\)
−0.597734 + 0.801694i \(0.703932\pi\)
\(212\) 1.16451e8 0.839399
\(213\) 0 0
\(214\) −2.62127e6 −0.0182837
\(215\) 2.21612e8 1.52075
\(216\) 0 0
\(217\) 5.08876e6 0.0338067
\(218\) 1.09049e6 0.00712895
\(219\) 0 0
\(220\) 7.82938e7 0.495733
\(221\) −1.18448e8 −0.738164
\(222\) 0 0
\(223\) 2.16056e8 1.30466 0.652332 0.757933i \(-0.273791\pi\)
0.652332 + 0.757933i \(0.273791\pi\)
\(224\) 1.27153e7 0.0755890
\(225\) 0 0
\(226\) −1.22508e6 −0.00705968
\(227\) 9.46539e7 0.537091 0.268546 0.963267i \(-0.413457\pi\)
0.268546 + 0.963267i \(0.413457\pi\)
\(228\) 0 0
\(229\) 5.62090e7 0.309301 0.154651 0.987969i \(-0.450575\pi\)
0.154651 + 0.987969i \(0.450575\pi\)
\(230\) 5.86549e6 0.0317876
\(231\) 0 0
\(232\) −2.15816e7 −0.113469
\(233\) −3.12602e8 −1.61900 −0.809500 0.587120i \(-0.800261\pi\)
−0.809500 + 0.587120i \(0.800261\pi\)
\(234\) 0 0
\(235\) 5.27260e6 0.0265025
\(236\) −2.62562e7 −0.130029
\(237\) 0 0
\(238\) −3.25899e6 −0.0156698
\(239\) −3.24955e8 −1.53968 −0.769840 0.638237i \(-0.779664\pi\)
−0.769840 + 0.638237i \(0.779664\pi\)
\(240\) 0 0
\(241\) 2.55841e8 1.17737 0.588683 0.808364i \(-0.299647\pi\)
0.588683 + 0.808364i \(0.299647\pi\)
\(242\) 5.29845e6 0.0240323
\(243\) 0 0
\(244\) −1.03589e8 −0.456507
\(245\) 9.82075e7 0.426642
\(246\) 0 0
\(247\) −3.89795e8 −1.64588
\(248\) −791317. −0.00329435
\(249\) 0 0
\(250\) −9.33334e6 −0.0377787
\(251\) −1.47199e8 −0.587552 −0.293776 0.955874i \(-0.594912\pi\)
−0.293776 + 0.955874i \(0.594912\pi\)
\(252\) 0 0
\(253\) 1.48085e8 0.574897
\(254\) −4.07420e6 −0.0156000
\(255\) 0 0
\(256\) 2.65139e8 0.987718
\(257\) 3.26518e8 1.19989 0.599944 0.800042i \(-0.295190\pi\)
0.599944 + 0.800042i \(0.295190\pi\)
\(258\) 0 0
\(259\) 2.92083e8 1.04462
\(260\) 2.97661e8 1.05031
\(261\) 0 0
\(262\) −1.21039e7 −0.0415787
\(263\) 1.12281e8 0.380592 0.190296 0.981727i \(-0.439055\pi\)
0.190296 + 0.981727i \(0.439055\pi\)
\(264\) 0 0
\(265\) 2.25203e8 0.743386
\(266\) −1.07249e7 −0.0349388
\(267\) 0 0
\(268\) 4.89849e8 1.55450
\(269\) 5.12589e7 0.160560 0.0802798 0.996772i \(-0.474419\pi\)
0.0802798 + 0.996772i \(0.474419\pi\)
\(270\) 0 0
\(271\) −5.28326e8 −1.61254 −0.806269 0.591549i \(-0.798516\pi\)
−0.806269 + 0.591549i \(0.798516\pi\)
\(272\) −2.05305e8 −0.618598
\(273\) 0 0
\(274\) 8.65561e6 0.0254197
\(275\) −4.21131e7 −0.122110
\(276\) 0 0
\(277\) −4.48700e7 −0.126846 −0.0634230 0.997987i \(-0.520202\pi\)
−0.0634230 + 0.997987i \(0.520202\pi\)
\(278\) −2.21930e7 −0.0619526
\(279\) 0 0
\(280\) 1.63899e7 0.0446194
\(281\) 5.27134e8 1.41726 0.708629 0.705581i \(-0.249314\pi\)
0.708629 + 0.705581i \(0.249314\pi\)
\(282\) 0 0
\(283\) 3.08524e8 0.809163 0.404582 0.914502i \(-0.367417\pi\)
0.404582 + 0.914502i \(0.367417\pi\)
\(284\) −4.29134e8 −1.11168
\(285\) 0 0
\(286\) −9.25804e6 −0.0234012
\(287\) 9.15152e6 0.0228511
\(288\) 0 0
\(289\) −2.52152e8 −0.614497
\(290\) −2.08553e7 −0.0502139
\(291\) 0 0
\(292\) −5.90047e7 −0.138690
\(293\) 8.22037e8 1.90921 0.954607 0.297869i \(-0.0962760\pi\)
0.954607 + 0.297869i \(0.0962760\pi\)
\(294\) 0 0
\(295\) −5.07764e7 −0.115156
\(296\) −4.54198e7 −0.101795
\(297\) 0 0
\(298\) 1.68702e7 0.0369285
\(299\) 5.62998e8 1.21803
\(300\) 0 0
\(301\) 5.85265e8 1.23700
\(302\) 1.06808e7 0.0223141
\(303\) 0 0
\(304\) −6.75632e8 −1.37928
\(305\) −2.00328e8 −0.404290
\(306\) 0 0
\(307\) 1.21225e8 0.239116 0.119558 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(308\) 2.06770e8 0.403236
\(309\) 0 0
\(310\) −764688. −0.00145787
\(311\) 4.99117e8 0.940894 0.470447 0.882428i \(-0.344093\pi\)
0.470447 + 0.882428i \(0.344093\pi\)
\(312\) 0 0
\(313\) 6.71603e8 1.23796 0.618981 0.785406i \(-0.287546\pi\)
0.618981 + 0.785406i \(0.287546\pi\)
\(314\) 2.41865e7 0.0440878
\(315\) 0 0
\(316\) 1.75220e8 0.312377
\(317\) 7.55861e8 1.33271 0.666353 0.745636i \(-0.267854\pi\)
0.666353 + 0.745636i \(0.267854\pi\)
\(318\) 0 0
\(319\) −5.26531e8 −0.908149
\(320\) 5.14661e8 0.878004
\(321\) 0 0
\(322\) 1.54905e7 0.0258564
\(323\) 5.20572e8 0.859552
\(324\) 0 0
\(325\) −1.60108e8 −0.258714
\(326\) −9.65122e6 −0.0154284
\(327\) 0 0
\(328\) −1.42309e6 −0.00222676
\(329\) 1.39247e7 0.0215575
\(330\) 0 0
\(331\) 7.06940e7 0.107148 0.0535740 0.998564i \(-0.482939\pi\)
0.0535740 + 0.998564i \(0.482939\pi\)
\(332\) −9.75978e8 −1.46372
\(333\) 0 0
\(334\) 8.65539e6 0.0127108
\(335\) 9.47313e8 1.37669
\(336\) 0 0
\(337\) −6.96987e8 −0.992020 −0.496010 0.868317i \(-0.665202\pi\)
−0.496010 + 0.868317i \(0.665202\pi\)
\(338\) −1.02956e7 −0.0145025
\(339\) 0 0
\(340\) −3.97527e8 −0.548518
\(341\) −1.93060e7 −0.0263664
\(342\) 0 0
\(343\) 7.97076e8 1.06652
\(344\) −9.10105e7 −0.120542
\(345\) 0 0
\(346\) 3.00778e7 0.0390374
\(347\) 3.13443e8 0.402721 0.201361 0.979517i \(-0.435464\pi\)
0.201361 + 0.979517i \(0.435464\pi\)
\(348\) 0 0
\(349\) 6.69924e8 0.843599 0.421800 0.906689i \(-0.361399\pi\)
0.421800 + 0.906689i \(0.361399\pi\)
\(350\) −4.40524e6 −0.00549201
\(351\) 0 0
\(352\) −4.82399e7 −0.0589531
\(353\) 1.16673e9 1.41175 0.705875 0.708336i \(-0.250554\pi\)
0.705875 + 0.708336i \(0.250554\pi\)
\(354\) 0 0
\(355\) −8.29897e8 −0.984521
\(356\) −6.72525e8 −0.790011
\(357\) 0 0
\(358\) −453354. −0.000522212 0
\(359\) −1.70302e9 −1.94263 −0.971316 0.237794i \(-0.923576\pi\)
−0.971316 + 0.237794i \(0.923576\pi\)
\(360\) 0 0
\(361\) 8.19262e8 0.916532
\(362\) −5.37126e7 −0.0595109
\(363\) 0 0
\(364\) 7.86108e8 0.854333
\(365\) −1.14108e8 −0.122827
\(366\) 0 0
\(367\) 9.54527e8 1.00799 0.503996 0.863706i \(-0.331863\pi\)
0.503996 + 0.863706i \(0.331863\pi\)
\(368\) 9.75845e8 1.02074
\(369\) 0 0
\(370\) −4.38914e7 −0.0450477
\(371\) 5.94751e8 0.604681
\(372\) 0 0
\(373\) 1.42139e9 1.41819 0.709093 0.705115i \(-0.249105\pi\)
0.709093 + 0.705115i \(0.249105\pi\)
\(374\) 1.23641e7 0.0122212
\(375\) 0 0
\(376\) −2.16533e6 −0.00210071
\(377\) −2.00179e9 −1.92409
\(378\) 0 0
\(379\) 2.98909e8 0.282035 0.141017 0.990007i \(-0.454963\pi\)
0.141017 + 0.990007i \(0.454963\pi\)
\(380\) −1.30821e9 −1.22302
\(381\) 0 0
\(382\) 4.42268e7 0.0405941
\(383\) 2.63974e8 0.240086 0.120043 0.992769i \(-0.461697\pi\)
0.120043 + 0.992769i \(0.461697\pi\)
\(384\) 0 0
\(385\) 3.99869e8 0.357113
\(386\) 1.82086e7 0.0161146
\(387\) 0 0
\(388\) 1.18021e9 1.02576
\(389\) 1.71619e9 1.47823 0.739116 0.673578i \(-0.235243\pi\)
0.739116 + 0.673578i \(0.235243\pi\)
\(390\) 0 0
\(391\) −7.51885e8 −0.636111
\(392\) −4.03314e7 −0.0338175
\(393\) 0 0
\(394\) 4.86758e7 0.0400937
\(395\) 3.38856e8 0.276647
\(396\) 0 0
\(397\) 5.74825e8 0.461072 0.230536 0.973064i \(-0.425952\pi\)
0.230536 + 0.973064i \(0.425952\pi\)
\(398\) 4.06802e6 0.00323439
\(399\) 0 0
\(400\) −2.77515e8 −0.216808
\(401\) 1.14785e9 0.888953 0.444476 0.895791i \(-0.353390\pi\)
0.444476 + 0.895791i \(0.353390\pi\)
\(402\) 0 0
\(403\) −7.33984e7 −0.0558623
\(404\) 6.20745e8 0.468358
\(405\) 0 0
\(406\) −5.50778e7 −0.0408447
\(407\) −1.10812e9 −0.814715
\(408\) 0 0
\(409\) −4.45907e6 −0.00322265 −0.00161132 0.999999i \(-0.500513\pi\)
−0.00161132 + 0.999999i \(0.500513\pi\)
\(410\) −1.37520e6 −0.000985422 0
\(411\) 0 0
\(412\) 5.55394e8 0.391256
\(413\) −1.34098e8 −0.0936692
\(414\) 0 0
\(415\) −1.88743e9 −1.29629
\(416\) −1.83401e8 −0.124904
\(417\) 0 0
\(418\) 4.06887e7 0.0272494
\(419\) −2.72506e9 −1.80978 −0.904892 0.425642i \(-0.860048\pi\)
−0.904892 + 0.425642i \(0.860048\pi\)
\(420\) 0 0
\(421\) −1.98552e9 −1.29684 −0.648420 0.761283i \(-0.724570\pi\)
−0.648420 + 0.761283i \(0.724570\pi\)
\(422\) 6.47379e7 0.0419339
\(423\) 0 0
\(424\) −9.24855e7 −0.0589241
\(425\) 2.13824e8 0.135112
\(426\) 0 0
\(427\) −5.29057e8 −0.328855
\(428\) −8.44413e8 −0.520597
\(429\) 0 0
\(430\) −8.79478e7 −0.0533440
\(431\) 8.94337e8 0.538060 0.269030 0.963132i \(-0.413297\pi\)
0.269030 + 0.963132i \(0.413297\pi\)
\(432\) 0 0
\(433\) −2.81913e9 −1.66881 −0.834405 0.551152i \(-0.814188\pi\)
−0.834405 + 0.551152i \(0.814188\pi\)
\(434\) −2.01950e6 −0.00118585
\(435\) 0 0
\(436\) 3.51290e8 0.202985
\(437\) −2.47435e9 −1.41833
\(438\) 0 0
\(439\) 1.50706e9 0.850166 0.425083 0.905154i \(-0.360245\pi\)
0.425083 + 0.905154i \(0.360245\pi\)
\(440\) −6.21809e7 −0.0347995
\(441\) 0 0
\(442\) 4.70065e7 0.0258929
\(443\) −1.30737e9 −0.714471 −0.357235 0.934014i \(-0.616281\pi\)
−0.357235 + 0.934014i \(0.616281\pi\)
\(444\) 0 0
\(445\) −1.30059e9 −0.699647
\(446\) −8.57428e7 −0.0457642
\(447\) 0 0
\(448\) 1.35919e9 0.714181
\(449\) 2.59024e9 1.35045 0.675223 0.737613i \(-0.264047\pi\)
0.675223 + 0.737613i \(0.264047\pi\)
\(450\) 0 0
\(451\) −3.47195e7 −0.0178219
\(452\) −3.94646e8 −0.201012
\(453\) 0 0
\(454\) −3.75639e7 −0.0188397
\(455\) 1.52024e9 0.756612
\(456\) 0 0
\(457\) −3.00834e9 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(458\) −2.23068e7 −0.0108495
\(459\) 0 0
\(460\) 1.88950e9 0.905097
\(461\) −1.29563e9 −0.615926 −0.307963 0.951398i \(-0.599647\pi\)
−0.307963 + 0.951398i \(0.599647\pi\)
\(462\) 0 0
\(463\) 7.89028e7 0.0369453 0.0184726 0.999829i \(-0.494120\pi\)
0.0184726 + 0.999829i \(0.494120\pi\)
\(464\) −3.46971e9 −1.61243
\(465\) 0 0
\(466\) 1.24058e8 0.0567903
\(467\) 1.97263e9 0.896264 0.448132 0.893967i \(-0.352089\pi\)
0.448132 + 0.893967i \(0.352089\pi\)
\(468\) 0 0
\(469\) 2.50180e9 1.11982
\(470\) −2.09246e6 −0.000929639 0
\(471\) 0 0
\(472\) 2.08526e7 0.00912775
\(473\) −2.22041e9 −0.964758
\(474\) 0 0
\(475\) 7.03667e8 0.301259
\(476\) −1.04985e9 −0.446172
\(477\) 0 0
\(478\) 1.28960e8 0.0540080
\(479\) −8.20556e7 −0.0341141 −0.0170571 0.999855i \(-0.505430\pi\)
−0.0170571 + 0.999855i \(0.505430\pi\)
\(480\) 0 0
\(481\) −4.21290e9 −1.72613
\(482\) −1.01532e8 −0.0412989
\(483\) 0 0
\(484\) 1.70684e9 0.684279
\(485\) 2.28239e9 0.908434
\(486\) 0 0
\(487\) 2.81731e9 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(488\) 8.22699e7 0.0320458
\(489\) 0 0
\(490\) −3.89742e7 −0.0149655
\(491\) 4.34251e9 1.65560 0.827800 0.561023i \(-0.189592\pi\)
0.827800 + 0.561023i \(0.189592\pi\)
\(492\) 0 0
\(493\) 2.67340e9 1.00485
\(494\) 1.54692e8 0.0577330
\(495\) 0 0
\(496\) −1.27221e8 −0.0468138
\(497\) −2.19171e9 −0.800823
\(498\) 0 0
\(499\) −2.81881e9 −1.01558 −0.507789 0.861481i \(-0.669537\pi\)
−0.507789 + 0.861481i \(0.669537\pi\)
\(500\) −3.00663e9 −1.07569
\(501\) 0 0
\(502\) 5.84167e7 0.0206098
\(503\) −3.80145e9 −1.33187 −0.665933 0.746011i \(-0.731967\pi\)
−0.665933 + 0.746011i \(0.731967\pi\)
\(504\) 0 0
\(505\) 1.20045e9 0.414786
\(506\) −5.87684e7 −0.0201659
\(507\) 0 0
\(508\) −1.31246e9 −0.444184
\(509\) −1.38927e9 −0.466956 −0.233478 0.972362i \(-0.575011\pi\)
−0.233478 + 0.972362i \(0.575011\pi\)
\(510\) 0 0
\(511\) −3.01354e8 −0.0999089
\(512\) −5.30032e8 −0.174525
\(513\) 0 0
\(514\) −1.29580e8 −0.0420889
\(515\) 1.07407e9 0.346503
\(516\) 0 0
\(517\) −5.28280e7 −0.0168131
\(518\) −1.15915e8 −0.0366425
\(519\) 0 0
\(520\) −2.36402e8 −0.0737293
\(521\) 2.67044e9 0.827277 0.413639 0.910441i \(-0.364258\pi\)
0.413639 + 0.910441i \(0.364258\pi\)
\(522\) 0 0
\(523\) 2.60473e8 0.0796173 0.0398086 0.999207i \(-0.487325\pi\)
0.0398086 + 0.999207i \(0.487325\pi\)
\(524\) −3.89914e9 −1.18388
\(525\) 0 0
\(526\) −4.45592e7 −0.0133502
\(527\) 9.80236e7 0.0291738
\(528\) 0 0
\(529\) 1.68991e8 0.0496327
\(530\) −8.93732e7 −0.0260760
\(531\) 0 0
\(532\) −3.45491e9 −0.994824
\(533\) −1.31998e8 −0.0377592
\(534\) 0 0
\(535\) −1.63300e9 −0.461049
\(536\) −3.89038e8 −0.109123
\(537\) 0 0
\(538\) −2.03424e7 −0.00563201
\(539\) −9.83975e8 −0.270660
\(540\) 0 0
\(541\) −9.32970e7 −0.0253324 −0.0126662 0.999920i \(-0.504032\pi\)
−0.0126662 + 0.999920i \(0.504032\pi\)
\(542\) 2.09669e8 0.0565636
\(543\) 0 0
\(544\) 2.44932e8 0.0652303
\(545\) 6.79356e8 0.179767
\(546\) 0 0
\(547\) 6.23723e9 1.62943 0.814716 0.579861i \(-0.196893\pi\)
0.814716 + 0.579861i \(0.196893\pi\)
\(548\) 2.78831e9 0.723784
\(549\) 0 0
\(550\) 1.67128e7 0.00428332
\(551\) 8.79780e9 2.24049
\(552\) 0 0
\(553\) 8.94902e8 0.225028
\(554\) 1.78069e7 0.00444943
\(555\) 0 0
\(556\) −7.14923e9 −1.76400
\(557\) −1.77646e9 −0.435574 −0.217787 0.975996i \(-0.569884\pi\)
−0.217787 + 0.975996i \(0.569884\pi\)
\(558\) 0 0
\(559\) −8.44166e9 −2.04403
\(560\) 2.63504e9 0.634058
\(561\) 0 0
\(562\) −2.09196e8 −0.0497137
\(563\) −2.36725e9 −0.559067 −0.279534 0.960136i \(-0.590180\pi\)
−0.279534 + 0.960136i \(0.590180\pi\)
\(564\) 0 0
\(565\) −7.63200e8 −0.178020
\(566\) −1.22439e8 −0.0283833
\(567\) 0 0
\(568\) 3.40818e8 0.0780375
\(569\) 4.20570e9 0.957073 0.478536 0.878068i \(-0.341167\pi\)
0.478536 + 0.878068i \(0.341167\pi\)
\(570\) 0 0
\(571\) 5.59664e9 1.25806 0.629029 0.777382i \(-0.283453\pi\)
0.629029 + 0.777382i \(0.283453\pi\)
\(572\) −2.98237e9 −0.666309
\(573\) 0 0
\(574\) −3.63183e6 −0.000801556 0
\(575\) −1.01634e9 −0.222946
\(576\) 0 0
\(577\) −3.81070e9 −0.825828 −0.412914 0.910770i \(-0.635489\pi\)
−0.412914 + 0.910770i \(0.635489\pi\)
\(578\) 1.00068e8 0.0215549
\(579\) 0 0
\(580\) −6.71831e9 −1.42976
\(581\) −4.98460e9 −1.05442
\(582\) 0 0
\(583\) −2.25639e9 −0.471601
\(584\) 4.68615e7 0.00973579
\(585\) 0 0
\(586\) −3.26230e8 −0.0669702
\(587\) −1.29560e9 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(588\) 0 0
\(589\) 3.22583e8 0.0650486
\(590\) 2.01509e7 0.00403936
\(591\) 0 0
\(592\) −7.30222e9 −1.44654
\(593\) −3.30125e9 −0.650111 −0.325055 0.945695i \(-0.605383\pi\)
−0.325055 + 0.945695i \(0.605383\pi\)
\(594\) 0 0
\(595\) −2.03029e9 −0.395137
\(596\) 5.43453e9 1.05148
\(597\) 0 0
\(598\) −2.23429e8 −0.0427253
\(599\) −3.03218e9 −0.576448 −0.288224 0.957563i \(-0.593065\pi\)
−0.288224 + 0.957563i \(0.593065\pi\)
\(600\) 0 0
\(601\) 3.17671e9 0.596922 0.298461 0.954422i \(-0.403527\pi\)
0.298461 + 0.954422i \(0.403527\pi\)
\(602\) −2.32266e8 −0.0433908
\(603\) 0 0
\(604\) 3.44071e9 0.635358
\(605\) 3.30083e9 0.606009
\(606\) 0 0
\(607\) 2.80510e9 0.509083 0.254541 0.967062i \(-0.418075\pi\)
0.254541 + 0.967062i \(0.418075\pi\)
\(608\) 8.06039e8 0.145443
\(609\) 0 0
\(610\) 7.95014e7 0.0141814
\(611\) −2.00844e8 −0.0356218
\(612\) 0 0
\(613\) −9.45694e9 −1.65821 −0.829104 0.559094i \(-0.811149\pi\)
−0.829104 + 0.559094i \(0.811149\pi\)
\(614\) −4.81088e7 −0.00838755
\(615\) 0 0
\(616\) −1.64216e8 −0.0283064
\(617\) −5.14249e9 −0.881405 −0.440702 0.897653i \(-0.645271\pi\)
−0.440702 + 0.897653i \(0.645271\pi\)
\(618\) 0 0
\(619\) 7.42513e9 1.25831 0.629153 0.777281i \(-0.283402\pi\)
0.629153 + 0.777281i \(0.283402\pi\)
\(620\) −2.46336e8 −0.0415103
\(621\) 0 0
\(622\) −1.98077e8 −0.0330041
\(623\) −3.43478e9 −0.569103
\(624\) 0 0
\(625\) −4.48629e9 −0.735034
\(626\) −2.66529e8 −0.0434245
\(627\) 0 0
\(628\) 7.79140e9 1.25533
\(629\) 5.62634e9 0.901464
\(630\) 0 0
\(631\) 6.01569e8 0.0953197 0.0476598 0.998864i \(-0.484824\pi\)
0.0476598 + 0.998864i \(0.484824\pi\)
\(632\) −1.39160e8 −0.0219283
\(633\) 0 0
\(634\) −2.99967e8 −0.0467479
\(635\) −2.53815e9 −0.393377
\(636\) 0 0
\(637\) −3.74093e9 −0.573444
\(638\) 2.08957e8 0.0318555
\(639\) 0 0
\(640\) −8.20524e8 −0.123726
\(641\) −3.94333e9 −0.591371 −0.295685 0.955285i \(-0.595548\pi\)
−0.295685 + 0.955285i \(0.595548\pi\)
\(642\) 0 0
\(643\) 3.67937e9 0.545802 0.272901 0.962042i \(-0.412017\pi\)
0.272901 + 0.962042i \(0.412017\pi\)
\(644\) 4.99008e9 0.736219
\(645\) 0 0
\(646\) −2.06592e8 −0.0301508
\(647\) −9.80484e9 −1.42323 −0.711616 0.702569i \(-0.752036\pi\)
−0.711616 + 0.702569i \(0.752036\pi\)
\(648\) 0 0
\(649\) 5.08747e8 0.0730542
\(650\) 6.35396e7 0.00907503
\(651\) 0 0
\(652\) −3.10903e9 −0.439297
\(653\) 4.40696e9 0.619360 0.309680 0.950841i \(-0.399778\pi\)
0.309680 + 0.950841i \(0.399778\pi\)
\(654\) 0 0
\(655\) −7.54049e9 −1.04847
\(656\) −2.28792e8 −0.0316430
\(657\) 0 0
\(658\) −5.52607e6 −0.000756182 0
\(659\) 1.35935e10 1.85025 0.925127 0.379657i \(-0.123958\pi\)
0.925127 + 0.379657i \(0.123958\pi\)
\(660\) 0 0
\(661\) −5.30552e9 −0.714534 −0.357267 0.934002i \(-0.616291\pi\)
−0.357267 + 0.934002i \(0.616291\pi\)
\(662\) −2.80553e7 −0.00375847
\(663\) 0 0
\(664\) 7.75121e8 0.102750
\(665\) −6.68141e9 −0.881033
\(666\) 0 0
\(667\) −1.27070e10 −1.65807
\(668\) 2.78824e9 0.361920
\(669\) 0 0
\(670\) −3.75946e8 −0.0482907
\(671\) 2.00716e9 0.256480
\(672\) 0 0
\(673\) 3.77543e9 0.477434 0.238717 0.971089i \(-0.423273\pi\)
0.238717 + 0.971089i \(0.423273\pi\)
\(674\) 2.76603e8 0.0347975
\(675\) 0 0
\(676\) −3.31661e9 −0.412934
\(677\) 7.29031e9 0.902995 0.451498 0.892272i \(-0.350890\pi\)
0.451498 + 0.892272i \(0.350890\pi\)
\(678\) 0 0
\(679\) 6.02767e9 0.738933
\(680\) 3.15716e8 0.0385048
\(681\) 0 0
\(682\) 7.66167e6 0.000924865 0
\(683\) −7.10884e9 −0.853742 −0.426871 0.904313i \(-0.640384\pi\)
−0.426871 + 0.904313i \(0.640384\pi\)
\(684\) 0 0
\(685\) 5.39227e9 0.640996
\(686\) −3.16324e8 −0.0374108
\(687\) 0 0
\(688\) −1.46319e10 −1.71294
\(689\) −8.57847e9 −0.999177
\(690\) 0 0
\(691\) −1.26683e10 −1.46065 −0.730324 0.683100i \(-0.760631\pi\)
−0.730324 + 0.683100i \(0.760631\pi\)
\(692\) 9.68924e9 1.11152
\(693\) 0 0
\(694\) −1.24391e8 −0.0141264
\(695\) −1.38258e10 −1.56223
\(696\) 0 0
\(697\) 1.76284e8 0.0197196
\(698\) −2.65863e8 −0.0295913
\(699\) 0 0
\(700\) −1.41910e9 −0.156376
\(701\) 1.78862e10 1.96113 0.980564 0.196201i \(-0.0628606\pi\)
0.980564 + 0.196201i \(0.0628606\pi\)
\(702\) 0 0
\(703\) 1.85155e10 2.00998
\(704\) −5.15657e9 −0.557002
\(705\) 0 0
\(706\) −4.63022e8 −0.0495205
\(707\) 3.17033e9 0.337393
\(708\) 0 0
\(709\) −1.24346e10 −1.31030 −0.655150 0.755499i \(-0.727395\pi\)
−0.655150 + 0.755499i \(0.727395\pi\)
\(710\) 3.29349e8 0.0345344
\(711\) 0 0
\(712\) 5.34119e8 0.0554572
\(713\) −4.65920e8 −0.0481392
\(714\) 0 0
\(715\) −5.76757e9 −0.590095
\(716\) −1.46043e8 −0.0148691
\(717\) 0 0
\(718\) 6.75854e8 0.0681424
\(719\) −2.69952e9 −0.270854 −0.135427 0.990787i \(-0.543241\pi\)
−0.135427 + 0.990787i \(0.543241\pi\)
\(720\) 0 0
\(721\) 2.83656e9 0.281850
\(722\) −3.25128e8 −0.0321495
\(723\) 0 0
\(724\) −1.73029e10 −1.69447
\(725\) 3.61368e9 0.352182
\(726\) 0 0
\(727\) −4.43778e9 −0.428346 −0.214173 0.976796i \(-0.568706\pi\)
−0.214173 + 0.976796i \(0.568706\pi\)
\(728\) −6.24327e8 −0.0599725
\(729\) 0 0
\(730\) 4.52845e7 0.00430844
\(731\) 1.12738e10 1.06748
\(732\) 0 0
\(733\) 8.21884e9 0.770808 0.385404 0.922748i \(-0.374062\pi\)
0.385404 + 0.922748i \(0.374062\pi\)
\(734\) −3.78809e8 −0.0353577
\(735\) 0 0
\(736\) −1.16420e9 −0.107635
\(737\) −9.49146e9 −0.873367
\(738\) 0 0
\(739\) −8.71653e9 −0.794490 −0.397245 0.917713i \(-0.630034\pi\)
−0.397245 + 0.917713i \(0.630034\pi\)
\(740\) −1.41391e10 −1.28266
\(741\) 0 0
\(742\) −2.36030e8 −0.0212106
\(743\) 1.01228e9 0.0905396 0.0452698 0.998975i \(-0.485585\pi\)
0.0452698 + 0.998975i \(0.485585\pi\)
\(744\) 0 0
\(745\) 1.05098e10 0.931207
\(746\) −5.64087e8 −0.0497462
\(747\) 0 0
\(748\) 3.98296e9 0.347977
\(749\) −4.31266e9 −0.375024
\(750\) 0 0
\(751\) 1.76039e10 1.51659 0.758295 0.651911i \(-0.226032\pi\)
0.758295 + 0.651911i \(0.226032\pi\)
\(752\) −3.48123e8 −0.0298518
\(753\) 0 0
\(754\) 7.94422e8 0.0674919
\(755\) 6.65393e9 0.562684
\(756\) 0 0
\(757\) 1.48620e10 1.24520 0.622602 0.782539i \(-0.286076\pi\)
0.622602 + 0.782539i \(0.286076\pi\)
\(758\) −1.18624e8 −0.00989304
\(759\) 0 0
\(760\) 1.03898e9 0.0858537
\(761\) −1.01144e10 −0.831939 −0.415970 0.909378i \(-0.636558\pi\)
−0.415970 + 0.909378i \(0.636558\pi\)
\(762\) 0 0
\(763\) 1.79414e9 0.146225
\(764\) 1.42472e10 1.15585
\(765\) 0 0
\(766\) −1.04760e8 −0.00842157
\(767\) 1.93418e9 0.154779
\(768\) 0 0
\(769\) −1.01233e10 −0.802748 −0.401374 0.915914i \(-0.631467\pi\)
−0.401374 + 0.915914i \(0.631467\pi\)
\(770\) −1.58690e8 −0.0125266
\(771\) 0 0
\(772\) 5.86568e9 0.458836
\(773\) −1.87926e10 −1.46339 −0.731693 0.681635i \(-0.761269\pi\)
−0.731693 + 0.681635i \(0.761269\pi\)
\(774\) 0 0
\(775\) 1.32500e8 0.0102250
\(776\) −9.37320e8 −0.0720066
\(777\) 0 0
\(778\) −6.81081e8 −0.0518525
\(779\) 5.80127e8 0.0439685
\(780\) 0 0
\(781\) 8.31503e9 0.624576
\(782\) 2.98389e8 0.0223131
\(783\) 0 0
\(784\) −6.48415e9 −0.480559
\(785\) 1.50677e10 1.11174
\(786\) 0 0
\(787\) 1.04611e10 0.765009 0.382505 0.923954i \(-0.375062\pi\)
0.382505 + 0.923954i \(0.375062\pi\)
\(788\) 1.56804e10 1.14160
\(789\) 0 0
\(790\) −1.34477e8 −0.00970405
\(791\) −2.01557e9 −0.144804
\(792\) 0 0
\(793\) 7.63092e9 0.543402
\(794\) −2.28122e8 −0.0161732
\(795\) 0 0
\(796\) 1.31047e9 0.0920937
\(797\) −2.15776e10 −1.50973 −0.754865 0.655880i \(-0.772298\pi\)
−0.754865 + 0.655880i \(0.772298\pi\)
\(798\) 0 0
\(799\) 2.68228e8 0.0186033
\(800\) 3.31079e8 0.0228622
\(801\) 0 0
\(802\) −4.55529e8 −0.0311821
\(803\) 1.14329e9 0.0779206
\(804\) 0 0
\(805\) 9.65025e9 0.652008
\(806\) 2.91285e7 0.00195950
\(807\) 0 0
\(808\) −4.92995e8 −0.0328778
\(809\) −1.89775e10 −1.26014 −0.630070 0.776538i \(-0.716974\pi\)
−0.630070 + 0.776538i \(0.716974\pi\)
\(810\) 0 0
\(811\) −4.70722e9 −0.309879 −0.154939 0.987924i \(-0.549518\pi\)
−0.154939 + 0.987924i \(0.549518\pi\)
\(812\) −1.77427e10 −1.16298
\(813\) 0 0
\(814\) 4.39763e8 0.0285781
\(815\) −6.01252e9 −0.389049
\(816\) 0 0
\(817\) 3.71007e10 2.38015
\(818\) 1.76961e6 0.000113042 0
\(819\) 0 0
\(820\) −4.43005e8 −0.0280582
\(821\) 2.74939e10 1.73395 0.866973 0.498355i \(-0.166062\pi\)
0.866973 + 0.498355i \(0.166062\pi\)
\(822\) 0 0
\(823\) −1.47691e9 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(824\) −4.41093e8 −0.0274654
\(825\) 0 0
\(826\) 5.32175e7 0.00328567
\(827\) 3.11067e10 1.91243 0.956213 0.292671i \(-0.0945439\pi\)
0.956213 + 0.292671i \(0.0945439\pi\)
\(828\) 0 0
\(829\) 2.06946e9 0.126158 0.0630792 0.998009i \(-0.479908\pi\)
0.0630792 + 0.998009i \(0.479908\pi\)
\(830\) 7.49037e8 0.0454705
\(831\) 0 0
\(832\) −1.96045e10 −1.18012
\(833\) 4.99601e9 0.299479
\(834\) 0 0
\(835\) 5.39214e9 0.320522
\(836\) 1.31074e10 0.775880
\(837\) 0 0
\(838\) 1.08145e9 0.0634825
\(839\) −2.02258e9 −0.118233 −0.0591166 0.998251i \(-0.518828\pi\)
−0.0591166 + 0.998251i \(0.518828\pi\)
\(840\) 0 0
\(841\) 2.79312e10 1.61921
\(842\) 7.87963e8 0.0454898
\(843\) 0 0
\(844\) 2.08546e10 1.19400
\(845\) −6.41395e9 −0.365702
\(846\) 0 0
\(847\) 8.71731e9 0.492936
\(848\) −1.48691e10 −0.837332
\(849\) 0 0
\(850\) −8.48572e7 −0.00473939
\(851\) −2.67428e10 −1.48749
\(852\) 0 0
\(853\) −3.22240e10 −1.77770 −0.888849 0.458201i \(-0.848494\pi\)
−0.888849 + 0.458201i \(0.848494\pi\)
\(854\) 2.09959e8 0.0115354
\(855\) 0 0
\(856\) 6.70632e8 0.0365448
\(857\) −6.94237e9 −0.376769 −0.188384 0.982095i \(-0.560325\pi\)
−0.188384 + 0.982095i \(0.560325\pi\)
\(858\) 0 0
\(859\) −6.91744e9 −0.372365 −0.186183 0.982515i \(-0.559612\pi\)
−0.186183 + 0.982515i \(0.559612\pi\)
\(860\) −2.83314e10 −1.51888
\(861\) 0 0
\(862\) −3.54922e8 −0.0188737
\(863\) 2.83705e10 1.50255 0.751275 0.659990i \(-0.229439\pi\)
0.751275 + 0.659990i \(0.229439\pi\)
\(864\) 0 0
\(865\) 1.87379e10 0.984384
\(866\) 1.11878e9 0.0585375
\(867\) 0 0
\(868\) −6.50559e8 −0.0337651
\(869\) −3.39512e9 −0.175504
\(870\) 0 0
\(871\) −3.60851e10 −1.85040
\(872\) −2.78995e8 −0.0142491
\(873\) 0 0
\(874\) 9.81960e8 0.0497512
\(875\) −1.53557e10 −0.774895
\(876\) 0 0
\(877\) −3.27420e9 −0.163910 −0.0819552 0.996636i \(-0.526116\pi\)
−0.0819552 + 0.996636i \(0.526116\pi\)
\(878\) −5.98083e8 −0.0298216
\(879\) 0 0
\(880\) −9.99693e9 −0.494512
\(881\) −1.44579e10 −0.712346 −0.356173 0.934420i \(-0.615919\pi\)
−0.356173 + 0.934420i \(0.615919\pi\)
\(882\) 0 0
\(883\) 8.75453e9 0.427928 0.213964 0.976842i \(-0.431363\pi\)
0.213964 + 0.976842i \(0.431363\pi\)
\(884\) 1.51426e10 0.737256
\(885\) 0 0
\(886\) 5.18835e8 0.0250618
\(887\) −1.54047e10 −0.741175 −0.370588 0.928798i \(-0.620844\pi\)
−0.370588 + 0.928798i \(0.620844\pi\)
\(888\) 0 0
\(889\) −6.70311e9 −0.319978
\(890\) 5.16144e8 0.0245418
\(891\) 0 0
\(892\) −2.76211e10 −1.30306
\(893\) 8.82702e8 0.0414796
\(894\) 0 0
\(895\) −2.82431e8 −0.0131684
\(896\) −2.16696e9 −0.100641
\(897\) 0 0
\(898\) −1.02795e9 −0.0473701
\(899\) 1.65662e9 0.0760441
\(900\) 0 0
\(901\) 1.14566e10 0.521816
\(902\) 1.37786e7 0.000625147 0
\(903\) 0 0
\(904\) 3.13427e8 0.0141107
\(905\) −3.34619e10 −1.50065
\(906\) 0 0
\(907\) 4.05126e10 1.80287 0.901435 0.432915i \(-0.142515\pi\)
0.901435 + 0.432915i \(0.142515\pi\)
\(908\) −1.21008e10 −0.536430
\(909\) 0 0
\(910\) −6.03317e8 −0.0265400
\(911\) −9.95690e9 −0.436325 −0.218162 0.975912i \(-0.570006\pi\)
−0.218162 + 0.975912i \(0.570006\pi\)
\(912\) 0 0
\(913\) 1.89108e10 0.822361
\(914\) 1.19387e9 0.0517186
\(915\) 0 0
\(916\) −7.18590e9 −0.308921
\(917\) −1.99140e10 −0.852838
\(918\) 0 0
\(919\) 1.67638e10 0.712472 0.356236 0.934396i \(-0.384060\pi\)
0.356236 + 0.934396i \(0.384060\pi\)
\(920\) −1.50064e9 −0.0635360
\(921\) 0 0
\(922\) 5.14179e8 0.0216051
\(923\) 3.16125e10 1.32328
\(924\) 0 0
\(925\) 7.60523e9 0.315948
\(926\) −3.13130e7 −0.00129594
\(927\) 0 0
\(928\) 4.13941e9 0.170028
\(929\) 2.97001e10 1.21536 0.607678 0.794184i \(-0.292101\pi\)
0.607678 + 0.794184i \(0.292101\pi\)
\(930\) 0 0
\(931\) 1.64412e10 0.667744
\(932\) 3.99639e10 1.61701
\(933\) 0 0
\(934\) −7.82847e8 −0.0314386
\(935\) 7.70260e9 0.308174
\(936\) 0 0
\(937\) 3.54569e10 1.40803 0.704015 0.710185i \(-0.251389\pi\)
0.704015 + 0.710185i \(0.251389\pi\)
\(938\) −9.92854e8 −0.0392804
\(939\) 0 0
\(940\) −6.74063e8 −0.0264699
\(941\) −1.78640e10 −0.698900 −0.349450 0.936955i \(-0.613631\pi\)
−0.349450 + 0.936955i \(0.613631\pi\)
\(942\) 0 0
\(943\) −8.37902e8 −0.0325389
\(944\) 3.35251e9 0.129709
\(945\) 0 0
\(946\) 8.81180e8 0.0338412
\(947\) 4.19426e10 1.60483 0.802417 0.596764i \(-0.203547\pi\)
0.802417 + 0.596764i \(0.203547\pi\)
\(948\) 0 0
\(949\) 4.34662e9 0.165090
\(950\) −2.79254e8 −0.0105674
\(951\) 0 0
\(952\) 8.33789e8 0.0313204
\(953\) 2.78513e10 1.04236 0.521182 0.853445i \(-0.325491\pi\)
0.521182 + 0.853445i \(0.325491\pi\)
\(954\) 0 0
\(955\) 2.75524e10 1.02364
\(956\) 4.15431e10 1.53779
\(957\) 0 0
\(958\) 3.25642e7 0.00119663
\(959\) 1.42407e10 0.521395
\(960\) 0 0
\(961\) −2.74519e10 −0.997792
\(962\) 1.67191e9 0.0605482
\(963\) 0 0
\(964\) −3.27074e10 −1.17592
\(965\) 1.13436e10 0.406353
\(966\) 0 0
\(967\) −1.00043e10 −0.355789 −0.177895 0.984050i \(-0.556929\pi\)
−0.177895 + 0.984050i \(0.556929\pi\)
\(968\) −1.35557e9 −0.0480350
\(969\) 0 0
\(970\) −9.05778e8 −0.0318655
\(971\) 2.19879e10 0.770755 0.385377 0.922759i \(-0.374071\pi\)
0.385377 + 0.922759i \(0.374071\pi\)
\(972\) 0 0
\(973\) −3.65132e10 −1.27074
\(974\) −1.11807e9 −0.0387714
\(975\) 0 0
\(976\) 1.32267e10 0.455383
\(977\) 4.29793e10 1.47444 0.737222 0.675651i \(-0.236137\pi\)
0.737222 + 0.675651i \(0.236137\pi\)
\(978\) 0 0
\(979\) 1.30310e10 0.443853
\(980\) −1.25551e10 −0.426117
\(981\) 0 0
\(982\) −1.72335e9 −0.0580741
\(983\) 2.32107e10 0.779383 0.389691 0.920946i \(-0.372582\pi\)
0.389691 + 0.920946i \(0.372582\pi\)
\(984\) 0 0
\(985\) 3.03240e10 1.01102
\(986\) −1.06095e9 −0.0352474
\(987\) 0 0
\(988\) 4.98324e10 1.64385
\(989\) −5.35862e10 −1.76143
\(990\) 0 0
\(991\) −1.22560e10 −0.400030 −0.200015 0.979793i \(-0.564099\pi\)
−0.200015 + 0.979793i \(0.564099\pi\)
\(992\) 1.51777e8 0.00493646
\(993\) 0 0
\(994\) 8.69793e8 0.0280908
\(995\) 2.53429e9 0.0815598
\(996\) 0 0
\(997\) −1.51059e10 −0.482740 −0.241370 0.970433i \(-0.577597\pi\)
−0.241370 + 0.970433i \(0.577597\pi\)
\(998\) 1.11866e9 0.0356238
\(999\) 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.1.9 17
3.2 odd 2 177.8.a.b.1.9 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.9 17 3.2 odd 2
531.8.a.d.1.9 17 1.1 even 1 trivial