Properties

Label 531.8.a.d.1.3
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 1.3
Root \(17.5255\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-15.5255 q^{2} +113.041 q^{4} +141.845 q^{5} +105.285 q^{7} +232.240 q^{8} +O(q^{10})\) \(q-15.5255 q^{2} +113.041 q^{4} +141.845 q^{5} +105.285 q^{7} +232.240 q^{8} -2202.21 q^{10} +555.209 q^{11} -11289.1 q^{13} -1634.61 q^{14} -18074.9 q^{16} +3139.99 q^{17} -21947.5 q^{19} +16034.3 q^{20} -8619.91 q^{22} -28575.9 q^{23} -58005.1 q^{25} +175269. q^{26} +11901.6 q^{28} -160286. q^{29} -163594. q^{31} +250896. q^{32} -48750.0 q^{34} +14934.2 q^{35} -131107. q^{37} +340746. q^{38} +32942.1 q^{40} +277081. q^{41} +621893. q^{43} +62761.6 q^{44} +443655. q^{46} +1.39412e6 q^{47} -812458. q^{49} +900558. q^{50} -1.27614e6 q^{52} -1.49504e6 q^{53} +78753.5 q^{55} +24451.5 q^{56} +2.48851e6 q^{58} +205379. q^{59} -2.41889e6 q^{61} +2.53988e6 q^{62} -1.58169e6 q^{64} -1.60130e6 q^{65} +2.34039e6 q^{67} +354949. q^{68} -231861. q^{70} -52758.6 q^{71} +1.12536e6 q^{73} +2.03550e6 q^{74} -2.48098e6 q^{76} +58455.5 q^{77} +3.33432e6 q^{79} -2.56384e6 q^{80} -4.30183e6 q^{82} +5.69160e6 q^{83} +445391. q^{85} -9.65520e6 q^{86} +128942. q^{88} +5.02101e6 q^{89} -1.18858e6 q^{91} -3.23026e6 q^{92} -2.16444e7 q^{94} -3.11314e6 q^{95} +9.04920e6 q^{97} +1.26138e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.5255 −1.37227 −0.686137 0.727472i \(-0.740695\pi\)
−0.686137 + 0.727472i \(0.740695\pi\)
\(3\) 0 0
\(4\) 113.041 0.883136
\(5\) 141.845 0.507479 0.253740 0.967273i \(-0.418339\pi\)
0.253740 + 0.967273i \(0.418339\pi\)
\(6\) 0 0
\(7\) 105.285 0.116018 0.0580090 0.998316i \(-0.481525\pi\)
0.0580090 + 0.998316i \(0.481525\pi\)
\(8\) 232.240 0.160370
\(9\) 0 0
\(10\) −2202.21 −0.696401
\(11\) 555.209 0.125772 0.0628858 0.998021i \(-0.479970\pi\)
0.0628858 + 0.998021i \(0.479970\pi\)
\(12\) 0 0
\(13\) −11289.1 −1.42514 −0.712572 0.701599i \(-0.752470\pi\)
−0.712572 + 0.701599i \(0.752470\pi\)
\(14\) −1634.61 −0.159208
\(15\) 0 0
\(16\) −18074.9 −1.10321
\(17\) 3139.99 0.155009 0.0775046 0.996992i \(-0.475305\pi\)
0.0775046 + 0.996992i \(0.475305\pi\)
\(18\) 0 0
\(19\) −21947.5 −0.734088 −0.367044 0.930204i \(-0.619630\pi\)
−0.367044 + 0.930204i \(0.619630\pi\)
\(20\) 16034.3 0.448173
\(21\) 0 0
\(22\) −8619.91 −0.172593
\(23\) −28575.9 −0.489725 −0.244863 0.969558i \(-0.578743\pi\)
−0.244863 + 0.969558i \(0.578743\pi\)
\(24\) 0 0
\(25\) −58005.1 −0.742465
\(26\) 175269. 1.95569
\(27\) 0 0
\(28\) 11901.6 0.102460
\(29\) −160286. −1.22040 −0.610199 0.792248i \(-0.708910\pi\)
−0.610199 + 0.792248i \(0.708910\pi\)
\(30\) 0 0
\(31\) −163594. −0.986284 −0.493142 0.869949i \(-0.664152\pi\)
−0.493142 + 0.869949i \(0.664152\pi\)
\(32\) 250896. 1.35353
\(33\) 0 0
\(34\) −48750.0 −0.212715
\(35\) 14934.2 0.0588767
\(36\) 0 0
\(37\) −131107. −0.425519 −0.212759 0.977105i \(-0.568245\pi\)
−0.212759 + 0.977105i \(0.568245\pi\)
\(38\) 340746. 1.00737
\(39\) 0 0
\(40\) 32942.1 0.0813843
\(41\) 277081. 0.627862 0.313931 0.949446i \(-0.398354\pi\)
0.313931 + 0.949446i \(0.398354\pi\)
\(42\) 0 0
\(43\) 621893. 1.19282 0.596411 0.802679i \(-0.296593\pi\)
0.596411 + 0.802679i \(0.296593\pi\)
\(44\) 62761.6 0.111073
\(45\) 0 0
\(46\) 443655. 0.672037
\(47\) 1.39412e6 1.95865 0.979325 0.202292i \(-0.0648390\pi\)
0.979325 + 0.202292i \(0.0648390\pi\)
\(48\) 0 0
\(49\) −812458. −0.986540
\(50\) 900558. 1.01887
\(51\) 0 0
\(52\) −1.27614e6 −1.25860
\(53\) −1.49504e6 −1.37939 −0.689696 0.724099i \(-0.742256\pi\)
−0.689696 + 0.724099i \(0.742256\pi\)
\(54\) 0 0
\(55\) 78753.5 0.0638265
\(56\) 24451.5 0.0186058
\(57\) 0 0
\(58\) 2.48851e6 1.67472
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −2.41889e6 −1.36446 −0.682230 0.731137i \(-0.738990\pi\)
−0.682230 + 0.731137i \(0.738990\pi\)
\(62\) 2.53988e6 1.35345
\(63\) 0 0
\(64\) −1.58169e6 −0.754210
\(65\) −1.60130e6 −0.723231
\(66\) 0 0
\(67\) 2.34039e6 0.950664 0.475332 0.879807i \(-0.342328\pi\)
0.475332 + 0.879807i \(0.342328\pi\)
\(68\) 354949. 0.136894
\(69\) 0 0
\(70\) −231861. −0.0807950
\(71\) −52758.6 −0.0174940 −0.00874700 0.999962i \(-0.502784\pi\)
−0.00874700 + 0.999962i \(0.502784\pi\)
\(72\) 0 0
\(73\) 1.12536e6 0.338581 0.169290 0.985566i \(-0.445852\pi\)
0.169290 + 0.985566i \(0.445852\pi\)
\(74\) 2.03550e6 0.583928
\(75\) 0 0
\(76\) −2.48098e6 −0.648299
\(77\) 58455.5 0.0145918
\(78\) 0 0
\(79\) 3.33432e6 0.760873 0.380436 0.924807i \(-0.375774\pi\)
0.380436 + 0.924807i \(0.375774\pi\)
\(80\) −2.56384e6 −0.559855
\(81\) 0 0
\(82\) −4.30183e6 −0.861598
\(83\) 5.69160e6 1.09260 0.546300 0.837590i \(-0.316036\pi\)
0.546300 + 0.837590i \(0.316036\pi\)
\(84\) 0 0
\(85\) 445391. 0.0786639
\(86\) −9.65520e6 −1.63688
\(87\) 0 0
\(88\) 128942. 0.0201700
\(89\) 5.02101e6 0.754963 0.377482 0.926017i \(-0.376790\pi\)
0.377482 + 0.926017i \(0.376790\pi\)
\(90\) 0 0
\(91\) −1.18858e6 −0.165342
\(92\) −3.23026e6 −0.432494
\(93\) 0 0
\(94\) −2.16444e7 −2.68781
\(95\) −3.11314e6 −0.372534
\(96\) 0 0
\(97\) 9.04920e6 1.00672 0.503361 0.864076i \(-0.332097\pi\)
0.503361 + 0.864076i \(0.332097\pi\)
\(98\) 1.26138e7 1.35380
\(99\) 0 0
\(100\) −6.55697e6 −0.655697
\(101\) −1.74915e6 −0.168928 −0.0844639 0.996427i \(-0.526918\pi\)
−0.0844639 + 0.996427i \(0.526918\pi\)
\(102\) 0 0
\(103\) 1.68825e7 1.52232 0.761160 0.648564i \(-0.224630\pi\)
0.761160 + 0.648564i \(0.224630\pi\)
\(104\) −2.62179e6 −0.228550
\(105\) 0 0
\(106\) 2.32113e7 1.89290
\(107\) −7.69410e6 −0.607176 −0.303588 0.952803i \(-0.598185\pi\)
−0.303588 + 0.952803i \(0.598185\pi\)
\(108\) 0 0
\(109\) −2.31641e7 −1.71326 −0.856629 0.515932i \(-0.827446\pi\)
−0.856629 + 0.515932i \(0.827446\pi\)
\(110\) −1.22269e6 −0.0875874
\(111\) 0 0
\(112\) −1.90303e6 −0.127992
\(113\) −2.01725e7 −1.31518 −0.657590 0.753376i \(-0.728424\pi\)
−0.657590 + 0.753376i \(0.728424\pi\)
\(114\) 0 0
\(115\) −4.05334e6 −0.248525
\(116\) −1.81189e7 −1.07778
\(117\) 0 0
\(118\) −3.18861e6 −0.178655
\(119\) 330595. 0.0179838
\(120\) 0 0
\(121\) −1.91789e7 −0.984182
\(122\) 3.75544e7 1.87241
\(123\) 0 0
\(124\) −1.84929e7 −0.871023
\(125\) −1.93093e7 −0.884265
\(126\) 0 0
\(127\) 1.39057e6 0.0602395 0.0301197 0.999546i \(-0.490411\pi\)
0.0301197 + 0.999546i \(0.490411\pi\)
\(128\) −7.55808e6 −0.318549
\(129\) 0 0
\(130\) 2.48611e7 0.992471
\(131\) 4.16332e6 0.161804 0.0809022 0.996722i \(-0.474220\pi\)
0.0809022 + 0.996722i \(0.474220\pi\)
\(132\) 0 0
\(133\) −2.31075e6 −0.0851673
\(134\) −3.63358e7 −1.30457
\(135\) 0 0
\(136\) 729232. 0.0248588
\(137\) −3.71579e7 −1.23461 −0.617304 0.786724i \(-0.711775\pi\)
−0.617304 + 0.786724i \(0.711775\pi\)
\(138\) 0 0
\(139\) −738784. −0.0233327 −0.0116664 0.999932i \(-0.503714\pi\)
−0.0116664 + 0.999932i \(0.503714\pi\)
\(140\) 1.68818e6 0.0519961
\(141\) 0 0
\(142\) 819104. 0.0240066
\(143\) −6.26783e6 −0.179243
\(144\) 0 0
\(145\) −2.27357e7 −0.619327
\(146\) −1.74718e7 −0.464626
\(147\) 0 0
\(148\) −1.48205e7 −0.375791
\(149\) 7.16242e7 1.77381 0.886906 0.461949i \(-0.152850\pi\)
0.886906 + 0.461949i \(0.152850\pi\)
\(150\) 0 0
\(151\) −3.49956e7 −0.827168 −0.413584 0.910466i \(-0.635723\pi\)
−0.413584 + 0.910466i \(0.635723\pi\)
\(152\) −5.09710e6 −0.117725
\(153\) 0 0
\(154\) −907551. −0.0200239
\(155\) −2.32050e7 −0.500519
\(156\) 0 0
\(157\) −3.84270e7 −0.792478 −0.396239 0.918147i \(-0.629685\pi\)
−0.396239 + 0.918147i \(0.629685\pi\)
\(158\) −5.17669e7 −1.04413
\(159\) 0 0
\(160\) 3.55883e7 0.686890
\(161\) −3.00863e6 −0.0568169
\(162\) 0 0
\(163\) −4.69770e7 −0.849627 −0.424813 0.905281i \(-0.639660\pi\)
−0.424813 + 0.905281i \(0.639660\pi\)
\(164\) 3.13217e7 0.554487
\(165\) 0 0
\(166\) −8.83650e7 −1.49935
\(167\) −5.23525e7 −0.869820 −0.434910 0.900474i \(-0.643220\pi\)
−0.434910 + 0.900474i \(0.643220\pi\)
\(168\) 0 0
\(169\) 6.46959e7 1.03103
\(170\) −6.91493e6 −0.107948
\(171\) 0 0
\(172\) 7.02996e7 1.05342
\(173\) 1.05438e8 1.54823 0.774115 0.633046i \(-0.218195\pi\)
0.774115 + 0.633046i \(0.218195\pi\)
\(174\) 0 0
\(175\) −6.10709e6 −0.0861392
\(176\) −1.00354e7 −0.138752
\(177\) 0 0
\(178\) −7.79537e7 −1.03602
\(179\) 7.74219e7 1.00897 0.504485 0.863421i \(-0.331682\pi\)
0.504485 + 0.863421i \(0.331682\pi\)
\(180\) 0 0
\(181\) 1.44384e8 1.80986 0.904930 0.425561i \(-0.139923\pi\)
0.904930 + 0.425561i \(0.139923\pi\)
\(182\) 1.84533e7 0.226895
\(183\) 0 0
\(184\) −6.63647e6 −0.0785371
\(185\) −1.85968e7 −0.215942
\(186\) 0 0
\(187\) 1.74335e6 0.0194957
\(188\) 1.57593e8 1.72975
\(189\) 0 0
\(190\) 4.83331e7 0.511219
\(191\) −1.83193e8 −1.90236 −0.951178 0.308642i \(-0.900125\pi\)
−0.951178 + 0.308642i \(0.900125\pi\)
\(192\) 0 0
\(193\) 9.07393e7 0.908542 0.454271 0.890864i \(-0.349900\pi\)
0.454271 + 0.890864i \(0.349900\pi\)
\(194\) −1.40493e8 −1.38150
\(195\) 0 0
\(196\) −9.18414e7 −0.871249
\(197\) −9.27275e7 −0.864125 −0.432063 0.901844i \(-0.642214\pi\)
−0.432063 + 0.901844i \(0.642214\pi\)
\(198\) 0 0
\(199\) 7.94761e7 0.714909 0.357455 0.933931i \(-0.383645\pi\)
0.357455 + 0.933931i \(0.383645\pi\)
\(200\) −1.34711e7 −0.119069
\(201\) 0 0
\(202\) 2.71564e7 0.231815
\(203\) −1.68757e7 −0.141588
\(204\) 0 0
\(205\) 3.93026e7 0.318627
\(206\) −2.62109e8 −2.08904
\(207\) 0 0
\(208\) 2.04050e8 1.57223
\(209\) −1.21855e7 −0.0923273
\(210\) 0 0
\(211\) 1.26710e8 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(212\) −1.69002e8 −1.21819
\(213\) 0 0
\(214\) 1.19455e8 0.833212
\(215\) 8.82123e7 0.605333
\(216\) 0 0
\(217\) −1.72241e7 −0.114427
\(218\) 3.59635e8 2.35106
\(219\) 0 0
\(220\) 8.90241e6 0.0563674
\(221\) −3.54478e7 −0.220910
\(222\) 0 0
\(223\) 1.08372e8 0.654410 0.327205 0.944953i \(-0.393893\pi\)
0.327205 + 0.944953i \(0.393893\pi\)
\(224\) 2.64157e7 0.157034
\(225\) 0 0
\(226\) 3.13188e8 1.80479
\(227\) −1.82593e8 −1.03608 −0.518040 0.855356i \(-0.673338\pi\)
−0.518040 + 0.855356i \(0.673338\pi\)
\(228\) 0 0
\(229\) 1.61240e8 0.887256 0.443628 0.896211i \(-0.353691\pi\)
0.443628 + 0.896211i \(0.353691\pi\)
\(230\) 6.29302e7 0.341045
\(231\) 0 0
\(232\) −3.72247e7 −0.195715
\(233\) 1.36205e8 0.705421 0.352711 0.935732i \(-0.385260\pi\)
0.352711 + 0.935732i \(0.385260\pi\)
\(234\) 0 0
\(235\) 1.97748e8 0.993975
\(236\) 2.32163e7 0.114974
\(237\) 0 0
\(238\) −5.13266e6 −0.0246788
\(239\) 1.32804e8 0.629241 0.314620 0.949218i \(-0.398123\pi\)
0.314620 + 0.949218i \(0.398123\pi\)
\(240\) 0 0
\(241\) −1.82912e8 −0.841749 −0.420874 0.907119i \(-0.638277\pi\)
−0.420874 + 0.907119i \(0.638277\pi\)
\(242\) 2.97762e8 1.35057
\(243\) 0 0
\(244\) −2.73434e8 −1.20500
\(245\) −1.15243e8 −0.500649
\(246\) 0 0
\(247\) 2.47768e8 1.04618
\(248\) −3.79931e7 −0.158170
\(249\) 0 0
\(250\) 2.99787e8 1.21345
\(251\) 3.69029e8 1.47300 0.736500 0.676438i \(-0.236477\pi\)
0.736500 + 0.676438i \(0.236477\pi\)
\(252\) 0 0
\(253\) −1.58656e7 −0.0615935
\(254\) −2.15894e7 −0.0826651
\(255\) 0 0
\(256\) 3.19800e8 1.19135
\(257\) 2.11987e8 0.779010 0.389505 0.921024i \(-0.372646\pi\)
0.389505 + 0.921024i \(0.372646\pi\)
\(258\) 0 0
\(259\) −1.38036e7 −0.0493678
\(260\) −1.81014e8 −0.638711
\(261\) 0 0
\(262\) −6.46377e7 −0.222040
\(263\) 1.26702e8 0.429476 0.214738 0.976672i \(-0.431110\pi\)
0.214738 + 0.976672i \(0.431110\pi\)
\(264\) 0 0
\(265\) −2.12064e8 −0.700013
\(266\) 3.58756e7 0.116873
\(267\) 0 0
\(268\) 2.64561e8 0.839565
\(269\) 2.67421e8 0.837650 0.418825 0.908067i \(-0.362442\pi\)
0.418825 + 0.908067i \(0.362442\pi\)
\(270\) 0 0
\(271\) −2.12216e7 −0.0647718 −0.0323859 0.999475i \(-0.510311\pi\)
−0.0323859 + 0.999475i \(0.510311\pi\)
\(272\) −5.67552e7 −0.171007
\(273\) 0 0
\(274\) 5.76895e8 1.69422
\(275\) −3.22050e7 −0.0933809
\(276\) 0 0
\(277\) 2.91983e8 0.825425 0.412713 0.910861i \(-0.364581\pi\)
0.412713 + 0.910861i \(0.364581\pi\)
\(278\) 1.14700e7 0.0320189
\(279\) 0 0
\(280\) 3.46832e6 0.00944204
\(281\) −2.59836e8 −0.698597 −0.349298 0.937012i \(-0.613580\pi\)
−0.349298 + 0.937012i \(0.613580\pi\)
\(282\) 0 0
\(283\) 3.61529e8 0.948180 0.474090 0.880476i \(-0.342777\pi\)
0.474090 + 0.880476i \(0.342777\pi\)
\(284\) −5.96391e6 −0.0154496
\(285\) 0 0
\(286\) 9.73112e7 0.245970
\(287\) 2.91726e7 0.0728432
\(288\) 0 0
\(289\) −4.00479e8 −0.975972
\(290\) 3.52983e8 0.849886
\(291\) 0 0
\(292\) 1.27213e8 0.299013
\(293\) 6.52155e8 1.51466 0.757328 0.653034i \(-0.226504\pi\)
0.757328 + 0.653034i \(0.226504\pi\)
\(294\) 0 0
\(295\) 2.91319e7 0.0660682
\(296\) −3.04482e7 −0.0682404
\(297\) 0 0
\(298\) −1.11200e9 −2.43416
\(299\) 3.22597e8 0.697929
\(300\) 0 0
\(301\) 6.54763e7 0.138389
\(302\) 5.43324e8 1.13510
\(303\) 0 0
\(304\) 3.96700e8 0.809851
\(305\) −3.43106e8 −0.692436
\(306\) 0 0
\(307\) −5.34821e8 −1.05493 −0.527466 0.849576i \(-0.676858\pi\)
−0.527466 + 0.849576i \(0.676858\pi\)
\(308\) 6.60789e6 0.0128865
\(309\) 0 0
\(310\) 3.60269e8 0.686849
\(311\) 8.03720e8 1.51511 0.757554 0.652773i \(-0.226394\pi\)
0.757554 + 0.652773i \(0.226394\pi\)
\(312\) 0 0
\(313\) −3.67380e8 −0.677190 −0.338595 0.940932i \(-0.609952\pi\)
−0.338595 + 0.940932i \(0.609952\pi\)
\(314\) 5.96598e8 1.08750
\(315\) 0 0
\(316\) 3.76916e8 0.671954
\(317\) 6.78407e8 1.19614 0.598071 0.801443i \(-0.295934\pi\)
0.598071 + 0.801443i \(0.295934\pi\)
\(318\) 0 0
\(319\) −8.89920e7 −0.153491
\(320\) −2.24355e8 −0.382746
\(321\) 0 0
\(322\) 4.67105e7 0.0779684
\(323\) −6.89150e7 −0.113790
\(324\) 0 0
\(325\) 6.54827e8 1.05812
\(326\) 7.29341e8 1.16592
\(327\) 0 0
\(328\) 6.43494e7 0.100690
\(329\) 1.46780e8 0.227239
\(330\) 0 0
\(331\) −1.80151e8 −0.273048 −0.136524 0.990637i \(-0.543593\pi\)
−0.136524 + 0.990637i \(0.543593\pi\)
\(332\) 6.43386e8 0.964914
\(333\) 0 0
\(334\) 8.12799e8 1.19363
\(335\) 3.31973e8 0.482442
\(336\) 0 0
\(337\) −6.10617e8 −0.869089 −0.434544 0.900650i \(-0.643091\pi\)
−0.434544 + 0.900650i \(0.643091\pi\)
\(338\) −1.00444e9 −1.41486
\(339\) 0 0
\(340\) 5.03477e7 0.0694709
\(341\) −9.08290e7 −0.124046
\(342\) 0 0
\(343\) −1.72247e8 −0.230474
\(344\) 1.44429e8 0.191293
\(345\) 0 0
\(346\) −1.63698e9 −2.12459
\(347\) 1.97837e8 0.254188 0.127094 0.991891i \(-0.459435\pi\)
0.127094 + 0.991891i \(0.459435\pi\)
\(348\) 0 0
\(349\) −9.01885e8 −1.13570 −0.567848 0.823133i \(-0.692224\pi\)
−0.567848 + 0.823133i \(0.692224\pi\)
\(350\) 9.48156e7 0.118207
\(351\) 0 0
\(352\) 1.39300e8 0.170236
\(353\) 8.58268e8 1.03851 0.519256 0.854619i \(-0.326209\pi\)
0.519256 + 0.854619i \(0.326209\pi\)
\(354\) 0 0
\(355\) −7.48353e6 −0.00887784
\(356\) 5.67582e8 0.666735
\(357\) 0 0
\(358\) −1.20201e9 −1.38458
\(359\) 4.75915e8 0.542874 0.271437 0.962456i \(-0.412501\pi\)
0.271437 + 0.962456i \(0.412501\pi\)
\(360\) 0 0
\(361\) −4.12178e8 −0.461115
\(362\) −2.24164e9 −2.48362
\(363\) 0 0
\(364\) −1.34359e8 −0.146020
\(365\) 1.59627e8 0.171823
\(366\) 0 0
\(367\) 7.17854e8 0.758062 0.379031 0.925384i \(-0.376257\pi\)
0.379031 + 0.925384i \(0.376257\pi\)
\(368\) 5.16508e8 0.540268
\(369\) 0 0
\(370\) 2.88725e8 0.296332
\(371\) −1.57406e8 −0.160034
\(372\) 0 0
\(373\) −5.85967e8 −0.584645 −0.292322 0.956320i \(-0.594428\pi\)
−0.292322 + 0.956320i \(0.594428\pi\)
\(374\) −2.70664e7 −0.0267535
\(375\) 0 0
\(376\) 3.23770e8 0.314108
\(377\) 1.80948e9 1.73924
\(378\) 0 0
\(379\) 1.74459e8 0.164610 0.0823050 0.996607i \(-0.473772\pi\)
0.0823050 + 0.996607i \(0.473772\pi\)
\(380\) −3.51914e8 −0.328998
\(381\) 0 0
\(382\) 2.84416e9 2.61055
\(383\) 6.24930e8 0.568376 0.284188 0.958769i \(-0.408276\pi\)
0.284188 + 0.958769i \(0.408276\pi\)
\(384\) 0 0
\(385\) 8.29160e6 0.00740501
\(386\) −1.40877e9 −1.24677
\(387\) 0 0
\(388\) 1.02293e9 0.889071
\(389\) 2.05004e8 0.176579 0.0882896 0.996095i \(-0.471860\pi\)
0.0882896 + 0.996095i \(0.471860\pi\)
\(390\) 0 0
\(391\) −8.97281e7 −0.0759119
\(392\) −1.88685e8 −0.158211
\(393\) 0 0
\(394\) 1.43964e9 1.18582
\(395\) 4.72955e8 0.386127
\(396\) 0 0
\(397\) −1.96710e9 −1.57783 −0.788915 0.614502i \(-0.789357\pi\)
−0.788915 + 0.614502i \(0.789357\pi\)
\(398\) −1.23391e9 −0.981051
\(399\) 0 0
\(400\) 1.04844e9 0.819092
\(401\) 2.09175e9 1.61996 0.809979 0.586458i \(-0.199478\pi\)
0.809979 + 0.586458i \(0.199478\pi\)
\(402\) 0 0
\(403\) 1.84684e9 1.40560
\(404\) −1.97726e8 −0.149186
\(405\) 0 0
\(406\) 2.62004e8 0.194298
\(407\) −7.27917e7 −0.0535182
\(408\) 0 0
\(409\) 3.66808e8 0.265099 0.132549 0.991176i \(-0.457684\pi\)
0.132549 + 0.991176i \(0.457684\pi\)
\(410\) −6.10192e8 −0.437243
\(411\) 0 0
\(412\) 1.90842e9 1.34441
\(413\) 2.16234e7 0.0151042
\(414\) 0 0
\(415\) 8.07324e8 0.554472
\(416\) −2.83240e9 −1.92898
\(417\) 0 0
\(418\) 1.89186e8 0.126698
\(419\) −1.27304e9 −0.845463 −0.422731 0.906255i \(-0.638929\pi\)
−0.422731 + 0.906255i \(0.638929\pi\)
\(420\) 0 0
\(421\) −6.98288e8 −0.456087 −0.228043 0.973651i \(-0.573233\pi\)
−0.228043 + 0.973651i \(0.573233\pi\)
\(422\) −1.96724e9 −1.27428
\(423\) 0 0
\(424\) −3.47209e8 −0.221213
\(425\) −1.82135e8 −0.115089
\(426\) 0 0
\(427\) −2.54674e8 −0.158302
\(428\) −8.69752e8 −0.536219
\(429\) 0 0
\(430\) −1.36954e9 −0.830683
\(431\) −2.63872e8 −0.158753 −0.0793765 0.996845i \(-0.525293\pi\)
−0.0793765 + 0.996845i \(0.525293\pi\)
\(432\) 0 0
\(433\) 2.13875e9 1.26606 0.633028 0.774128i \(-0.281812\pi\)
0.633028 + 0.774128i \(0.281812\pi\)
\(434\) 2.67413e8 0.157025
\(435\) 0 0
\(436\) −2.61850e9 −1.51304
\(437\) 6.27170e8 0.359501
\(438\) 0 0
\(439\) 2.80384e9 1.58171 0.790856 0.612003i \(-0.209636\pi\)
0.790856 + 0.612003i \(0.209636\pi\)
\(440\) 1.82897e7 0.0102358
\(441\) 0 0
\(442\) 5.50345e8 0.303149
\(443\) −2.51161e9 −1.37259 −0.686293 0.727326i \(-0.740763\pi\)
−0.686293 + 0.727326i \(0.740763\pi\)
\(444\) 0 0
\(445\) 7.12204e8 0.383128
\(446\) −1.68253e9 −0.898030
\(447\) 0 0
\(448\) −1.66529e8 −0.0875019
\(449\) −2.93296e9 −1.52913 −0.764564 0.644548i \(-0.777045\pi\)
−0.764564 + 0.644548i \(0.777045\pi\)
\(450\) 0 0
\(451\) 1.53838e8 0.0789671
\(452\) −2.28033e9 −1.16148
\(453\) 0 0
\(454\) 2.83485e9 1.42179
\(455\) −1.68594e8 −0.0839078
\(456\) 0 0
\(457\) −3.30374e9 −1.61920 −0.809598 0.586984i \(-0.800315\pi\)
−0.809598 + 0.586984i \(0.800315\pi\)
\(458\) −2.50333e9 −1.21756
\(459\) 0 0
\(460\) −4.58195e8 −0.219482
\(461\) −6.28637e7 −0.0298845 −0.0149423 0.999888i \(-0.504756\pi\)
−0.0149423 + 0.999888i \(0.504756\pi\)
\(462\) 0 0
\(463\) −2.07437e9 −0.971300 −0.485650 0.874153i \(-0.661417\pi\)
−0.485650 + 0.874153i \(0.661417\pi\)
\(464\) 2.89715e9 1.34635
\(465\) 0 0
\(466\) −2.11466e9 −0.968031
\(467\) 1.86804e9 0.848743 0.424372 0.905488i \(-0.360495\pi\)
0.424372 + 0.905488i \(0.360495\pi\)
\(468\) 0 0
\(469\) 2.46409e8 0.110294
\(470\) −3.07014e9 −1.36401
\(471\) 0 0
\(472\) 4.76973e7 0.0208784
\(473\) 3.45281e8 0.150023
\(474\) 0 0
\(475\) 1.27307e9 0.545034
\(476\) 3.73710e7 0.0158822
\(477\) 0 0
\(478\) −2.06184e9 −0.863491
\(479\) −7.39665e8 −0.307511 −0.153755 0.988109i \(-0.549137\pi\)
−0.153755 + 0.988109i \(0.549137\pi\)
\(480\) 0 0
\(481\) 1.48008e9 0.606426
\(482\) 2.83980e9 1.15511
\(483\) 0 0
\(484\) −2.16801e9 −0.869166
\(485\) 1.28358e9 0.510890
\(486\) 0 0
\(487\) −3.16893e9 −1.24326 −0.621629 0.783312i \(-0.713529\pi\)
−0.621629 + 0.783312i \(0.713529\pi\)
\(488\) −5.61763e8 −0.218818
\(489\) 0 0
\(490\) 1.78920e9 0.687027
\(491\) 1.78015e9 0.678692 0.339346 0.940662i \(-0.389794\pi\)
0.339346 + 0.940662i \(0.389794\pi\)
\(492\) 0 0
\(493\) −5.03295e8 −0.189173
\(494\) −3.84673e9 −1.43565
\(495\) 0 0
\(496\) 2.95696e9 1.08808
\(497\) −5.55471e6 −0.00202962
\(498\) 0 0
\(499\) −1.30086e9 −0.468684 −0.234342 0.972154i \(-0.575294\pi\)
−0.234342 + 0.972154i \(0.575294\pi\)
\(500\) −2.18275e9 −0.780926
\(501\) 0 0
\(502\) −5.72937e9 −2.02136
\(503\) 1.92822e9 0.675569 0.337784 0.941224i \(-0.390323\pi\)
0.337784 + 0.941224i \(0.390323\pi\)
\(504\) 0 0
\(505\) −2.48107e8 −0.0857274
\(506\) 2.46322e8 0.0845232
\(507\) 0 0
\(508\) 1.57192e8 0.0531996
\(509\) 3.91015e9 1.31426 0.657129 0.753778i \(-0.271771\pi\)
0.657129 + 0.753778i \(0.271771\pi\)
\(510\) 0 0
\(511\) 1.18484e8 0.0392815
\(512\) −3.99762e9 −1.31631
\(513\) 0 0
\(514\) −3.29120e9 −1.06902
\(515\) 2.39469e9 0.772546
\(516\) 0 0
\(517\) 7.74027e8 0.246343
\(518\) 2.14308e8 0.0677462
\(519\) 0 0
\(520\) −3.71887e8 −0.115984
\(521\) −3.14164e9 −0.973251 −0.486626 0.873611i \(-0.661772\pi\)
−0.486626 + 0.873611i \(0.661772\pi\)
\(522\) 0 0
\(523\) −3.32735e8 −0.101705 −0.0508525 0.998706i \(-0.516194\pi\)
−0.0508525 + 0.998706i \(0.516194\pi\)
\(524\) 4.70627e8 0.142895
\(525\) 0 0
\(526\) −1.96712e9 −0.589359
\(527\) −5.13684e8 −0.152883
\(528\) 0 0
\(529\) −2.58824e9 −0.760169
\(530\) 3.29240e9 0.960610
\(531\) 0 0
\(532\) −2.61211e8 −0.0752143
\(533\) −3.12801e9 −0.894793
\(534\) 0 0
\(535\) −1.09137e9 −0.308129
\(536\) 5.43533e8 0.152458
\(537\) 0 0
\(538\) −4.15185e9 −1.14949
\(539\) −4.51084e8 −0.124079
\(540\) 0 0
\(541\) 3.91262e9 1.06237 0.531187 0.847255i \(-0.321746\pi\)
0.531187 + 0.847255i \(0.321746\pi\)
\(542\) 3.29476e8 0.0888847
\(543\) 0 0
\(544\) 7.87811e8 0.209810
\(545\) −3.28571e9 −0.869443
\(546\) 0 0
\(547\) −2.61916e9 −0.684237 −0.342118 0.939657i \(-0.611144\pi\)
−0.342118 + 0.939657i \(0.611144\pi\)
\(548\) −4.20038e9 −1.09033
\(549\) 0 0
\(550\) 4.99998e8 0.128144
\(551\) 3.51787e9 0.895879
\(552\) 0 0
\(553\) 3.51055e8 0.0882749
\(554\) −4.53318e9 −1.13271
\(555\) 0 0
\(556\) −8.35131e7 −0.0206060
\(557\) 4.99537e9 1.22483 0.612413 0.790538i \(-0.290199\pi\)
0.612413 + 0.790538i \(0.290199\pi\)
\(558\) 0 0
\(559\) −7.02063e9 −1.69994
\(560\) −2.69935e8 −0.0649532
\(561\) 0 0
\(562\) 4.03408e9 0.958666
\(563\) −1.65832e9 −0.391643 −0.195821 0.980640i \(-0.562737\pi\)
−0.195821 + 0.980640i \(0.562737\pi\)
\(564\) 0 0
\(565\) −2.86137e9 −0.667427
\(566\) −5.61292e9 −1.30116
\(567\) 0 0
\(568\) −1.22527e7 −0.00280551
\(569\) −6.06474e9 −1.38013 −0.690064 0.723748i \(-0.742418\pi\)
−0.690064 + 0.723748i \(0.742418\pi\)
\(570\) 0 0
\(571\) 7.56971e9 1.70158 0.850791 0.525504i \(-0.176123\pi\)
0.850791 + 0.525504i \(0.176123\pi\)
\(572\) −7.08524e8 −0.158295
\(573\) 0 0
\(574\) −4.52920e8 −0.0999608
\(575\) 1.65755e9 0.363604
\(576\) 0 0
\(577\) 4.65513e9 1.00883 0.504413 0.863462i \(-0.331709\pi\)
0.504413 + 0.863462i \(0.331709\pi\)
\(578\) 6.21764e9 1.33930
\(579\) 0 0
\(580\) −2.57007e9 −0.546949
\(581\) 5.99243e8 0.126761
\(582\) 0 0
\(583\) −8.30062e8 −0.173488
\(584\) 2.61354e8 0.0542981
\(585\) 0 0
\(586\) −1.01250e10 −2.07852
\(587\) 4.59010e9 0.936675 0.468337 0.883550i \(-0.344853\pi\)
0.468337 + 0.883550i \(0.344853\pi\)
\(588\) 0 0
\(589\) 3.59049e9 0.724019
\(590\) −4.52288e8 −0.0906636
\(591\) 0 0
\(592\) 2.36975e9 0.469435
\(593\) 3.99474e9 0.786679 0.393339 0.919393i \(-0.371320\pi\)
0.393339 + 0.919393i \(0.371320\pi\)
\(594\) 0 0
\(595\) 4.68932e7 0.00912643
\(596\) 8.09650e9 1.56652
\(597\) 0 0
\(598\) −5.00848e9 −0.957750
\(599\) 7.94405e9 1.51025 0.755124 0.655582i \(-0.227577\pi\)
0.755124 + 0.655582i \(0.227577\pi\)
\(600\) 0 0
\(601\) 2.34327e9 0.440313 0.220157 0.975465i \(-0.429343\pi\)
0.220157 + 0.975465i \(0.429343\pi\)
\(602\) −1.01655e9 −0.189907
\(603\) 0 0
\(604\) −3.95595e9 −0.730502
\(605\) −2.72043e9 −0.499452
\(606\) 0 0
\(607\) 4.14140e9 0.751600 0.375800 0.926701i \(-0.377368\pi\)
0.375800 + 0.926701i \(0.377368\pi\)
\(608\) −5.50654e9 −0.993611
\(609\) 0 0
\(610\) 5.32690e9 0.950211
\(611\) −1.57384e10 −2.79136
\(612\) 0 0
\(613\) −8.67426e9 −1.52097 −0.760485 0.649355i \(-0.775039\pi\)
−0.760485 + 0.649355i \(0.775039\pi\)
\(614\) 8.30337e9 1.44766
\(615\) 0 0
\(616\) 1.35757e7 0.00234008
\(617\) −6.63163e9 −1.13664 −0.568319 0.822808i \(-0.692406\pi\)
−0.568319 + 0.822808i \(0.692406\pi\)
\(618\) 0 0
\(619\) 1.16526e9 0.197473 0.0987364 0.995114i \(-0.468520\pi\)
0.0987364 + 0.995114i \(0.468520\pi\)
\(620\) −2.62312e9 −0.442026
\(621\) 0 0
\(622\) −1.24782e10 −2.07914
\(623\) 5.28639e8 0.0875893
\(624\) 0 0
\(625\) 1.79272e9 0.293719
\(626\) 5.70376e9 0.929290
\(627\) 0 0
\(628\) −4.34384e9 −0.699866
\(629\) −4.11674e8 −0.0659593
\(630\) 0 0
\(631\) −5.61888e9 −0.890322 −0.445161 0.895450i \(-0.646854\pi\)
−0.445161 + 0.895450i \(0.646854\pi\)
\(632\) 7.74362e8 0.122021
\(633\) 0 0
\(634\) −1.05326e10 −1.64144
\(635\) 1.97246e8 0.0305703
\(636\) 0 0
\(637\) 9.17194e9 1.40596
\(638\) 1.38165e9 0.210632
\(639\) 0 0
\(640\) −1.07207e9 −0.161657
\(641\) −9.83028e8 −0.147422 −0.0737111 0.997280i \(-0.523484\pi\)
−0.0737111 + 0.997280i \(0.523484\pi\)
\(642\) 0 0
\(643\) 7.69653e9 1.14171 0.570856 0.821050i \(-0.306612\pi\)
0.570856 + 0.821050i \(0.306612\pi\)
\(644\) −3.40099e8 −0.0501771
\(645\) 0 0
\(646\) 1.06994e9 0.156151
\(647\) 4.42661e9 0.642549 0.321275 0.946986i \(-0.395889\pi\)
0.321275 + 0.946986i \(0.395889\pi\)
\(648\) 0 0
\(649\) 1.14028e8 0.0163741
\(650\) −1.01665e10 −1.45203
\(651\) 0 0
\(652\) −5.31034e9 −0.750336
\(653\) 3.99185e7 0.00561020 0.00280510 0.999996i \(-0.499107\pi\)
0.00280510 + 0.999996i \(0.499107\pi\)
\(654\) 0 0
\(655\) 5.90545e8 0.0821124
\(656\) −5.00823e9 −0.692661
\(657\) 0 0
\(658\) −2.27884e9 −0.311834
\(659\) −2.06680e9 −0.281319 −0.140660 0.990058i \(-0.544922\pi\)
−0.140660 + 0.990058i \(0.544922\pi\)
\(660\) 0 0
\(661\) −5.72369e8 −0.0770853 −0.0385426 0.999257i \(-0.512272\pi\)
−0.0385426 + 0.999257i \(0.512272\pi\)
\(662\) 2.79694e9 0.374697
\(663\) 0 0
\(664\) 1.32182e9 0.175220
\(665\) −3.27768e8 −0.0432207
\(666\) 0 0
\(667\) 4.58030e9 0.597660
\(668\) −5.91800e9 −0.768170
\(669\) 0 0
\(670\) −5.15404e9 −0.662043
\(671\) −1.34299e9 −0.171610
\(672\) 0 0
\(673\) −5.59964e9 −0.708121 −0.354060 0.935223i \(-0.615199\pi\)
−0.354060 + 0.935223i \(0.615199\pi\)
\(674\) 9.48014e9 1.19263
\(675\) 0 0
\(676\) 7.31331e9 0.910543
\(677\) −2.66208e9 −0.329731 −0.164866 0.986316i \(-0.552719\pi\)
−0.164866 + 0.986316i \(0.552719\pi\)
\(678\) 0 0
\(679\) 9.52749e8 0.116798
\(680\) 1.03438e8 0.0126153
\(681\) 0 0
\(682\) 1.41017e9 0.170226
\(683\) 3.08492e9 0.370485 0.185243 0.982693i \(-0.440693\pi\)
0.185243 + 0.982693i \(0.440693\pi\)
\(684\) 0 0
\(685\) −5.27065e9 −0.626538
\(686\) 2.67422e9 0.316274
\(687\) 0 0
\(688\) −1.12407e10 −1.31593
\(689\) 1.68777e10 1.96583
\(690\) 0 0
\(691\) 1.50276e9 0.173267 0.0866336 0.996240i \(-0.472389\pi\)
0.0866336 + 0.996240i \(0.472389\pi\)
\(692\) 1.19188e10 1.36730
\(693\) 0 0
\(694\) −3.07152e9 −0.348815
\(695\) −1.04793e8 −0.0118409
\(696\) 0 0
\(697\) 8.70033e8 0.0973243
\(698\) 1.40022e10 1.55849
\(699\) 0 0
\(700\) −6.90354e8 −0.0760726
\(701\) 8.60128e9 0.943083 0.471541 0.881844i \(-0.343698\pi\)
0.471541 + 0.881844i \(0.343698\pi\)
\(702\) 0 0
\(703\) 2.87747e9 0.312368
\(704\) −8.78171e8 −0.0948582
\(705\) 0 0
\(706\) −1.33251e10 −1.42512
\(707\) −1.84160e8 −0.0195987
\(708\) 0 0
\(709\) 1.22113e10 1.28677 0.643384 0.765544i \(-0.277530\pi\)
0.643384 + 0.765544i \(0.277530\pi\)
\(710\) 1.16186e8 0.0121828
\(711\) 0 0
\(712\) 1.16608e9 0.121073
\(713\) 4.67485e9 0.483008
\(714\) 0 0
\(715\) −8.89059e8 −0.0909619
\(716\) 8.75188e9 0.891057
\(717\) 0 0
\(718\) −7.38882e9 −0.744972
\(719\) −3.71959e9 −0.373202 −0.186601 0.982436i \(-0.559747\pi\)
−0.186601 + 0.982436i \(0.559747\pi\)
\(720\) 0 0
\(721\) 1.77748e9 0.176616
\(722\) 6.39927e9 0.632777
\(723\) 0 0
\(724\) 1.63214e10 1.59835
\(725\) 9.29737e9 0.906102
\(726\) 0 0
\(727\) 1.15890e10 1.11861 0.559303 0.828963i \(-0.311069\pi\)
0.559303 + 0.828963i \(0.311069\pi\)
\(728\) −2.76036e8 −0.0265159
\(729\) 0 0
\(730\) −2.47829e9 −0.235788
\(731\) 1.95274e9 0.184898
\(732\) 0 0
\(733\) 1.55996e10 1.46302 0.731510 0.681830i \(-0.238816\pi\)
0.731510 + 0.681830i \(0.238816\pi\)
\(734\) −1.11450e10 −1.04027
\(735\) 0 0
\(736\) −7.16958e9 −0.662859
\(737\) 1.29941e9 0.119566
\(738\) 0 0
\(739\) −1.06655e10 −0.972137 −0.486069 0.873921i \(-0.661569\pi\)
−0.486069 + 0.873921i \(0.661569\pi\)
\(740\) −2.10221e9 −0.190706
\(741\) 0 0
\(742\) 2.44381e9 0.219611
\(743\) 7.49599e9 0.670453 0.335226 0.942138i \(-0.391187\pi\)
0.335226 + 0.942138i \(0.391187\pi\)
\(744\) 0 0
\(745\) 1.01595e10 0.900173
\(746\) 9.09743e9 0.802292
\(747\) 0 0
\(748\) 1.97071e8 0.0172174
\(749\) −8.10077e8 −0.0704433
\(750\) 0 0
\(751\) 1.56605e10 1.34916 0.674582 0.738200i \(-0.264324\pi\)
0.674582 + 0.738200i \(0.264324\pi\)
\(752\) −2.51986e10 −2.16080
\(753\) 0 0
\(754\) −2.80932e10 −2.38672
\(755\) −4.96394e9 −0.419771
\(756\) 0 0
\(757\) 2.02856e10 1.69962 0.849810 0.527090i \(-0.176717\pi\)
0.849810 + 0.527090i \(0.176717\pi\)
\(758\) −2.70857e9 −0.225890
\(759\) 0 0
\(760\) −7.22997e8 −0.0597432
\(761\) 1.72538e10 1.41919 0.709594 0.704611i \(-0.248879\pi\)
0.709594 + 0.704611i \(0.248879\pi\)
\(762\) 0 0
\(763\) −2.43884e9 −0.198769
\(764\) −2.07084e10 −1.68004
\(765\) 0 0
\(766\) −9.70235e9 −0.779967
\(767\) −2.31855e9 −0.185538
\(768\) 0 0
\(769\) 2.62229e9 0.207940 0.103970 0.994580i \(-0.466845\pi\)
0.103970 + 0.994580i \(0.466845\pi\)
\(770\) −1.28731e8 −0.0101617
\(771\) 0 0
\(772\) 1.02573e10 0.802366
\(773\) 1.36251e10 1.06099 0.530497 0.847687i \(-0.322005\pi\)
0.530497 + 0.847687i \(0.322005\pi\)
\(774\) 0 0
\(775\) 9.48929e9 0.732281
\(776\) 2.10159e9 0.161448
\(777\) 0 0
\(778\) −3.18280e9 −0.242315
\(779\) −6.08125e9 −0.460905
\(780\) 0 0
\(781\) −2.92921e7 −0.00220025
\(782\) 1.39307e9 0.104172
\(783\) 0 0
\(784\) 1.46851e10 1.08836
\(785\) −5.45067e9 −0.402166
\(786\) 0 0
\(787\) −1.27830e9 −0.0934803 −0.0467402 0.998907i \(-0.514883\pi\)
−0.0467402 + 0.998907i \(0.514883\pi\)
\(788\) −1.04820e10 −0.763140
\(789\) 0 0
\(790\) −7.34287e9 −0.529872
\(791\) −2.12387e9 −0.152585
\(792\) 0 0
\(793\) 2.73071e10 1.94455
\(794\) 3.05403e10 2.16522
\(795\) 0 0
\(796\) 8.98409e9 0.631362
\(797\) 1.05141e10 0.735642 0.367821 0.929897i \(-0.380104\pi\)
0.367821 + 0.929897i \(0.380104\pi\)
\(798\) 0 0
\(799\) 4.37752e9 0.303609
\(800\) −1.45532e10 −1.00495
\(801\) 0 0
\(802\) −3.24754e10 −2.22303
\(803\) 6.24812e8 0.0425838
\(804\) 0 0
\(805\) −4.26758e8 −0.0288334
\(806\) −2.86731e10 −1.92886
\(807\) 0 0
\(808\) −4.06222e8 −0.0270909
\(809\) 6.76915e7 0.00449484 0.00224742 0.999997i \(-0.499285\pi\)
0.00224742 + 0.999997i \(0.499285\pi\)
\(810\) 0 0
\(811\) −1.33239e10 −0.877121 −0.438560 0.898702i \(-0.644511\pi\)
−0.438560 + 0.898702i \(0.644511\pi\)
\(812\) −1.90766e9 −0.125041
\(813\) 0 0
\(814\) 1.13013e9 0.0734416
\(815\) −6.66343e9 −0.431168
\(816\) 0 0
\(817\) −1.36490e10 −0.875637
\(818\) −5.69488e9 −0.363788
\(819\) 0 0
\(820\) 4.44281e9 0.281391
\(821\) −1.93379e10 −1.21957 −0.609787 0.792565i \(-0.708745\pi\)
−0.609787 + 0.792565i \(0.708745\pi\)
\(822\) 0 0
\(823\) −1.01992e10 −0.637774 −0.318887 0.947793i \(-0.603309\pi\)
−0.318887 + 0.947793i \(0.603309\pi\)
\(824\) 3.92079e9 0.244134
\(825\) 0 0
\(826\) −3.35715e8 −0.0207272
\(827\) 2.20862e9 0.135785 0.0678925 0.997693i \(-0.478372\pi\)
0.0678925 + 0.997693i \(0.478372\pi\)
\(828\) 0 0
\(829\) 5.49783e9 0.335159 0.167579 0.985859i \(-0.446405\pi\)
0.167579 + 0.985859i \(0.446405\pi\)
\(830\) −1.25341e10 −0.760887
\(831\) 0 0
\(832\) 1.78559e10 1.07486
\(833\) −2.55111e9 −0.152923
\(834\) 0 0
\(835\) −7.42593e9 −0.441416
\(836\) −1.37746e9 −0.0815376
\(837\) 0 0
\(838\) 1.97647e10 1.16021
\(839\) −4.86845e9 −0.284593 −0.142297 0.989824i \(-0.545449\pi\)
−0.142297 + 0.989824i \(0.545449\pi\)
\(840\) 0 0
\(841\) 8.44158e9 0.489371
\(842\) 1.08413e10 0.625876
\(843\) 0 0
\(844\) 1.43235e10 0.820070
\(845\) 9.17677e9 0.523229
\(846\) 0 0
\(847\) −2.01926e9 −0.114183
\(848\) 2.70228e10 1.52176
\(849\) 0 0
\(850\) 2.82774e9 0.157933
\(851\) 3.74649e9 0.208387
\(852\) 0 0
\(853\) 2.22806e10 1.22915 0.614577 0.788857i \(-0.289327\pi\)
0.614577 + 0.788857i \(0.289327\pi\)
\(854\) 3.95394e9 0.217234
\(855\) 0 0
\(856\) −1.78688e9 −0.0973727
\(857\) −3.09258e10 −1.67837 −0.839186 0.543844i \(-0.816968\pi\)
−0.839186 + 0.543844i \(0.816968\pi\)
\(858\) 0 0
\(859\) 6.06715e9 0.326594 0.163297 0.986577i \(-0.447787\pi\)
0.163297 + 0.986577i \(0.447787\pi\)
\(860\) 9.97163e9 0.534591
\(861\) 0 0
\(862\) 4.09674e9 0.217853
\(863\) 5.47212e8 0.0289813 0.0144906 0.999895i \(-0.495387\pi\)
0.0144906 + 0.999895i \(0.495387\pi\)
\(864\) 0 0
\(865\) 1.49558e10 0.785694
\(866\) −3.32052e10 −1.73738
\(867\) 0 0
\(868\) −1.94703e9 −0.101054
\(869\) 1.85124e9 0.0956961
\(870\) 0 0
\(871\) −2.64210e10 −1.35483
\(872\) −5.37964e9 −0.274755
\(873\) 0 0
\(874\) −9.73714e9 −0.493334
\(875\) −2.03299e9 −0.102591
\(876\) 0 0
\(877\) −1.05471e10 −0.528001 −0.264001 0.964523i \(-0.585042\pi\)
−0.264001 + 0.964523i \(0.585042\pi\)
\(878\) −4.35310e10 −2.17054
\(879\) 0 0
\(880\) −1.42347e9 −0.0704138
\(881\) 2.80065e10 1.37989 0.689943 0.723864i \(-0.257636\pi\)
0.689943 + 0.723864i \(0.257636\pi\)
\(882\) 0 0
\(883\) −1.61306e10 −0.788477 −0.394238 0.919008i \(-0.628992\pi\)
−0.394238 + 0.919008i \(0.628992\pi\)
\(884\) −4.00706e9 −0.195094
\(885\) 0 0
\(886\) 3.89940e10 1.88356
\(887\) −8.47554e9 −0.407788 −0.203894 0.978993i \(-0.565360\pi\)
−0.203894 + 0.978993i \(0.565360\pi\)
\(888\) 0 0
\(889\) 1.46407e8 0.00698886
\(890\) −1.10573e10 −0.525757
\(891\) 0 0
\(892\) 1.22505e10 0.577933
\(893\) −3.05974e10 −1.43782
\(894\) 0 0
\(895\) 1.09819e10 0.512031
\(896\) −7.95756e8 −0.0369574
\(897\) 0 0
\(898\) 4.55357e10 2.09838
\(899\) 2.62218e10 1.20366
\(900\) 0 0
\(901\) −4.69442e9 −0.213818
\(902\) −2.38842e9 −0.108365
\(903\) 0 0
\(904\) −4.68487e9 −0.210915
\(905\) 2.04802e10 0.918466
\(906\) 0 0
\(907\) 1.59071e10 0.707892 0.353946 0.935266i \(-0.384840\pi\)
0.353946 + 0.935266i \(0.384840\pi\)
\(908\) −2.06406e10 −0.914999
\(909\) 0 0
\(910\) 2.61751e9 0.115144
\(911\) 7.44930e9 0.326438 0.163219 0.986590i \(-0.447812\pi\)
0.163219 + 0.986590i \(0.447812\pi\)
\(912\) 0 0
\(913\) 3.16003e9 0.137418
\(914\) 5.12923e10 2.22198
\(915\) 0 0
\(916\) 1.82268e10 0.783567
\(917\) 4.38337e8 0.0187722
\(918\) 0 0
\(919\) −2.63715e10 −1.12080 −0.560402 0.828221i \(-0.689353\pi\)
−0.560402 + 0.828221i \(0.689353\pi\)
\(920\) −9.41349e8 −0.0398560
\(921\) 0 0
\(922\) 9.75990e8 0.0410098
\(923\) 5.95599e8 0.0249315
\(924\) 0 0
\(925\) 7.60485e9 0.315933
\(926\) 3.22057e10 1.33289
\(927\) 0 0
\(928\) −4.02150e10 −1.65185
\(929\) −1.48686e9 −0.0608437 −0.0304219 0.999537i \(-0.509685\pi\)
−0.0304219 + 0.999537i \(0.509685\pi\)
\(930\) 0 0
\(931\) 1.78314e10 0.724207
\(932\) 1.53968e10 0.622983
\(933\) 0 0
\(934\) −2.90022e10 −1.16471
\(935\) 2.47285e8 0.00989368
\(936\) 0 0
\(937\) 1.58188e10 0.628182 0.314091 0.949393i \(-0.398300\pi\)
0.314091 + 0.949393i \(0.398300\pi\)
\(938\) −3.82563e9 −0.151354
\(939\) 0 0
\(940\) 2.23537e10 0.877814
\(941\) 7.02941e9 0.275014 0.137507 0.990501i \(-0.456091\pi\)
0.137507 + 0.990501i \(0.456091\pi\)
\(942\) 0 0
\(943\) −7.91785e9 −0.307480
\(944\) −3.71221e9 −0.143625
\(945\) 0 0
\(946\) −5.36066e9 −0.205873
\(947\) −4.03890e10 −1.54539 −0.772695 0.634778i \(-0.781092\pi\)
−0.772695 + 0.634778i \(0.781092\pi\)
\(948\) 0 0
\(949\) −1.27044e10 −0.482526
\(950\) −1.97650e10 −0.747936
\(951\) 0 0
\(952\) 7.67775e7 0.00288406
\(953\) −2.02572e10 −0.758148 −0.379074 0.925366i \(-0.623757\pi\)
−0.379074 + 0.925366i \(0.623757\pi\)
\(954\) 0 0
\(955\) −2.59850e10 −0.965406
\(956\) 1.50123e10 0.555705
\(957\) 0 0
\(958\) 1.14837e10 0.421989
\(959\) −3.91219e9 −0.143237
\(960\) 0 0
\(961\) −7.49560e8 −0.0272442
\(962\) −2.29790e10 −0.832182
\(963\) 0 0
\(964\) −2.06766e10 −0.743378
\(965\) 1.28709e10 0.461066
\(966\) 0 0
\(967\) 1.14238e10 0.406273 0.203137 0.979150i \(-0.434886\pi\)
0.203137 + 0.979150i \(0.434886\pi\)
\(968\) −4.45411e9 −0.157833
\(969\) 0 0
\(970\) −1.99283e10 −0.701081
\(971\) −2.91162e10 −1.02063 −0.510314 0.859988i \(-0.670471\pi\)
−0.510314 + 0.859988i \(0.670471\pi\)
\(972\) 0 0
\(973\) −7.77832e7 −0.00270701
\(974\) 4.91992e10 1.70609
\(975\) 0 0
\(976\) 4.37212e10 1.50528
\(977\) −4.46068e10 −1.53028 −0.765139 0.643865i \(-0.777330\pi\)
−0.765139 + 0.643865i \(0.777330\pi\)
\(978\) 0 0
\(979\) 2.78771e9 0.0949529
\(980\) −1.30272e10 −0.442141
\(981\) 0 0
\(982\) −2.76378e10 −0.931351
\(983\) 6.33957e9 0.212874 0.106437 0.994319i \(-0.466056\pi\)
0.106437 + 0.994319i \(0.466056\pi\)
\(984\) 0 0
\(985\) −1.31529e10 −0.438526
\(986\) 7.81391e9 0.259597
\(987\) 0 0
\(988\) 2.80081e10 0.923919
\(989\) −1.77712e10 −0.584156
\(990\) 0 0
\(991\) −2.38700e10 −0.779103 −0.389551 0.921005i \(-0.627370\pi\)
−0.389551 + 0.921005i \(0.627370\pi\)
\(992\) −4.10451e10 −1.33497
\(993\) 0 0
\(994\) 8.62397e7 0.00278519
\(995\) 1.12733e10 0.362802
\(996\) 0 0
\(997\) −1.70894e9 −0.0546127 −0.0273063 0.999627i \(-0.508693\pi\)
−0.0273063 + 0.999627i \(0.508693\pi\)
\(998\) 2.01966e10 0.643163
\(999\) 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.1.3 17
3.2 odd 2 177.8.a.b.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.15 17 3.2 odd 2
531.8.a.d.1.3 17 1.1 even 1 trivial