Properties

Label 531.8.a.d.1.14
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 1.14
Root \(-15.8998\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.8998 q^{2} +192.403 q^{4} -255.355 q^{5} -1072.82 q^{7} +1152.80 q^{8} +O(q^{10})\) \(q+17.8998 q^{2} +192.403 q^{4} -255.355 q^{5} -1072.82 q^{7} +1152.80 q^{8} -4570.81 q^{10} -4742.70 q^{11} +5721.05 q^{13} -19203.3 q^{14} -3992.71 q^{16} +12850.6 q^{17} -52123.1 q^{19} -49131.1 q^{20} -84893.4 q^{22} +9824.17 q^{23} -12918.7 q^{25} +102406. q^{26} -206414. q^{28} +196436. q^{29} +272544. q^{31} -219027. q^{32} +230023. q^{34} +273951. q^{35} +309341. q^{37} -932994. q^{38} -294373. q^{40} +421229. q^{41} -406292. q^{43} -912509. q^{44} +175851. q^{46} +261121. q^{47} +327408. q^{49} -231242. q^{50} +1.10075e6 q^{52} -738907. q^{53} +1.21107e6 q^{55} -1.23675e6 q^{56} +3.51616e6 q^{58} +205379. q^{59} +233563. q^{61} +4.87848e6 q^{62} -3.40947e6 q^{64} -1.46090e6 q^{65} +3.54145e6 q^{67} +2.47250e6 q^{68} +4.90367e6 q^{70} -1.03036e6 q^{71} -5.88757e6 q^{73} +5.53715e6 q^{74} -1.00286e7 q^{76} +5.08808e6 q^{77} +4.34224e6 q^{79} +1.01956e6 q^{80} +7.53991e6 q^{82} +4.12014e6 q^{83} -3.28147e6 q^{85} -7.27255e6 q^{86} -5.46737e6 q^{88} +7.32690e6 q^{89} -6.13768e6 q^{91} +1.89020e6 q^{92} +4.67402e6 q^{94} +1.33099e7 q^{95} +1.46546e7 q^{97} +5.86054e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.8998 1.58213 0.791067 0.611730i \(-0.209526\pi\)
0.791067 + 0.611730i \(0.209526\pi\)
\(3\) 0 0
\(4\) 192.403 1.50315
\(5\) −255.355 −0.913587 −0.456793 0.889573i \(-0.651002\pi\)
−0.456793 + 0.889573i \(0.651002\pi\)
\(6\) 0 0
\(7\) −1072.82 −1.18218 −0.591092 0.806604i \(-0.701303\pi\)
−0.591092 + 0.806604i \(0.701303\pi\)
\(8\) 1152.80 0.796046
\(9\) 0 0
\(10\) −4570.81 −1.44542
\(11\) −4742.70 −1.07436 −0.537182 0.843467i \(-0.680511\pi\)
−0.537182 + 0.843467i \(0.680511\pi\)
\(12\) 0 0
\(13\) 5721.05 0.722227 0.361114 0.932522i \(-0.382397\pi\)
0.361114 + 0.932522i \(0.382397\pi\)
\(14\) −19203.3 −1.87037
\(15\) 0 0
\(16\) −3992.71 −0.243695
\(17\) 12850.6 0.634385 0.317192 0.948361i \(-0.397260\pi\)
0.317192 + 0.948361i \(0.397260\pi\)
\(18\) 0 0
\(19\) −52123.1 −1.74338 −0.871692 0.490055i \(-0.836977\pi\)
−0.871692 + 0.490055i \(0.836977\pi\)
\(20\) −49131.1 −1.37326
\(21\) 0 0
\(22\) −84893.4 −1.69979
\(23\) 9824.17 0.168364 0.0841818 0.996450i \(-0.473172\pi\)
0.0841818 + 0.996450i \(0.473172\pi\)
\(24\) 0 0
\(25\) −12918.7 −0.165359
\(26\) 102406. 1.14266
\(27\) 0 0
\(28\) −206414. −1.77700
\(29\) 196436. 1.49564 0.747821 0.663901i \(-0.231100\pi\)
0.747821 + 0.663901i \(0.231100\pi\)
\(30\) 0 0
\(31\) 272544. 1.64312 0.821562 0.570119i \(-0.193103\pi\)
0.821562 + 0.570119i \(0.193103\pi\)
\(32\) −219027. −1.18161
\(33\) 0 0
\(34\) 230023. 1.00368
\(35\) 273951. 1.08003
\(36\) 0 0
\(37\) 309341. 1.00400 0.501998 0.864869i \(-0.332599\pi\)
0.501998 + 0.864869i \(0.332599\pi\)
\(38\) −932994. −2.75827
\(39\) 0 0
\(40\) −294373. −0.727258
\(41\) 421229. 0.954497 0.477249 0.878768i \(-0.341634\pi\)
0.477249 + 0.878768i \(0.341634\pi\)
\(42\) 0 0
\(43\) −406292. −0.779289 −0.389645 0.920965i \(-0.627402\pi\)
−0.389645 + 0.920965i \(0.627402\pi\)
\(44\) −912509. −1.61493
\(45\) 0 0
\(46\) 175851. 0.266374
\(47\) 261121. 0.366859 0.183430 0.983033i \(-0.441280\pi\)
0.183430 + 0.983033i \(0.441280\pi\)
\(48\) 0 0
\(49\) 327408. 0.397560
\(50\) −231242. −0.261620
\(51\) 0 0
\(52\) 1.10075e6 1.08561
\(53\) −738907. −0.681748 −0.340874 0.940109i \(-0.610723\pi\)
−0.340874 + 0.940109i \(0.610723\pi\)
\(54\) 0 0
\(55\) 1.21107e6 0.981524
\(56\) −1.23675e6 −0.941074
\(57\) 0 0
\(58\) 3.51616e6 2.36630
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) 233563. 0.131750 0.0658748 0.997828i \(-0.479016\pi\)
0.0658748 + 0.997828i \(0.479016\pi\)
\(62\) 4.87848e6 2.59964
\(63\) 0 0
\(64\) −3.40947e6 −1.62576
\(65\) −1.46090e6 −0.659817
\(66\) 0 0
\(67\) 3.54145e6 1.43853 0.719265 0.694736i \(-0.244479\pi\)
0.719265 + 0.694736i \(0.244479\pi\)
\(68\) 2.47250e6 0.953574
\(69\) 0 0
\(70\) 4.90367e6 1.70875
\(71\) −1.03036e6 −0.341652 −0.170826 0.985301i \(-0.554644\pi\)
−0.170826 + 0.985301i \(0.554644\pi\)
\(72\) 0 0
\(73\) −5.88757e6 −1.77136 −0.885679 0.464298i \(-0.846307\pi\)
−0.885679 + 0.464298i \(0.846307\pi\)
\(74\) 5.53715e6 1.58845
\(75\) 0 0
\(76\) −1.00286e7 −2.62056
\(77\) 5.08808e6 1.27010
\(78\) 0 0
\(79\) 4.34224e6 0.990875 0.495437 0.868644i \(-0.335008\pi\)
0.495437 + 0.868644i \(0.335008\pi\)
\(80\) 1.01956e6 0.222637
\(81\) 0 0
\(82\) 7.53991e6 1.51014
\(83\) 4.12014e6 0.790931 0.395465 0.918481i \(-0.370583\pi\)
0.395465 + 0.918481i \(0.370583\pi\)
\(84\) 0 0
\(85\) −3.28147e6 −0.579566
\(86\) −7.27255e6 −1.23294
\(87\) 0 0
\(88\) −5.46737e6 −0.855243
\(89\) 7.32690e6 1.10168 0.550840 0.834611i \(-0.314308\pi\)
0.550840 + 0.834611i \(0.314308\pi\)
\(90\) 0 0
\(91\) −6.13768e6 −0.853806
\(92\) 1.89020e6 0.253075
\(93\) 0 0
\(94\) 4.67402e6 0.580421
\(95\) 1.33099e7 1.59273
\(96\) 0 0
\(97\) 1.46546e7 1.63032 0.815158 0.579239i \(-0.196650\pi\)
0.815158 + 0.579239i \(0.196650\pi\)
\(98\) 5.86054e6 0.628994
\(99\) 0 0
\(100\) −2.48559e6 −0.248559
\(101\) −3.05329e6 −0.294878 −0.147439 0.989071i \(-0.547103\pi\)
−0.147439 + 0.989071i \(0.547103\pi\)
\(102\) 0 0
\(103\) −1.49285e7 −1.34613 −0.673063 0.739585i \(-0.735022\pi\)
−0.673063 + 0.739585i \(0.735022\pi\)
\(104\) 6.59521e6 0.574926
\(105\) 0 0
\(106\) −1.32263e7 −1.07862
\(107\) 2.22015e7 1.75202 0.876011 0.482291i \(-0.160195\pi\)
0.876011 + 0.482291i \(0.160195\pi\)
\(108\) 0 0
\(109\) −2.47303e6 −0.182910 −0.0914549 0.995809i \(-0.529152\pi\)
−0.0914549 + 0.995809i \(0.529152\pi\)
\(110\) 2.16780e7 1.55290
\(111\) 0 0
\(112\) 4.28347e6 0.288093
\(113\) 1.16610e7 0.760260 0.380130 0.924933i \(-0.375879\pi\)
0.380130 + 0.924933i \(0.375879\pi\)
\(114\) 0 0
\(115\) −2.50865e6 −0.153815
\(116\) 3.77948e7 2.24817
\(117\) 0 0
\(118\) 3.67624e6 0.205976
\(119\) −1.37864e7 −0.749960
\(120\) 0 0
\(121\) 3.00601e6 0.154256
\(122\) 4.18073e6 0.208446
\(123\) 0 0
\(124\) 5.24382e7 2.46986
\(125\) 2.32485e7 1.06466
\(126\) 0 0
\(127\) −3.47578e7 −1.50570 −0.752852 0.658190i \(-0.771322\pi\)
−0.752852 + 0.658190i \(0.771322\pi\)
\(128\) −3.29934e7 −1.39057
\(129\) 0 0
\(130\) −2.61498e7 −1.04392
\(131\) 1.06271e7 0.413016 0.206508 0.978445i \(-0.433790\pi\)
0.206508 + 0.978445i \(0.433790\pi\)
\(132\) 0 0
\(133\) 5.59189e7 2.06100
\(134\) 6.33912e7 2.27595
\(135\) 0 0
\(136\) 1.48142e7 0.505000
\(137\) 4.64453e7 1.54319 0.771596 0.636112i \(-0.219459\pi\)
0.771596 + 0.636112i \(0.219459\pi\)
\(138\) 0 0
\(139\) −1.75461e7 −0.554151 −0.277076 0.960848i \(-0.589365\pi\)
−0.277076 + 0.960848i \(0.589365\pi\)
\(140\) 5.27090e7 1.62344
\(141\) 0 0
\(142\) −1.84432e7 −0.540539
\(143\) −2.71332e7 −0.775934
\(144\) 0 0
\(145\) −5.01609e7 −1.36640
\(146\) −1.05386e8 −2.80253
\(147\) 0 0
\(148\) 5.95181e7 1.50915
\(149\) −6.90672e7 −1.71049 −0.855244 0.518225i \(-0.826593\pi\)
−0.855244 + 0.518225i \(0.826593\pi\)
\(150\) 0 0
\(151\) 9.93956e6 0.234935 0.117468 0.993077i \(-0.462522\pi\)
0.117468 + 0.993077i \(0.462522\pi\)
\(152\) −6.00875e7 −1.38781
\(153\) 0 0
\(154\) 9.10756e7 2.00946
\(155\) −6.95955e7 −1.50114
\(156\) 0 0
\(157\) −1.73655e7 −0.358128 −0.179064 0.983837i \(-0.557307\pi\)
−0.179064 + 0.983837i \(0.557307\pi\)
\(158\) 7.77252e7 1.56770
\(159\) 0 0
\(160\) 5.59297e7 1.07950
\(161\) −1.05396e7 −0.199037
\(162\) 0 0
\(163\) −8.66389e7 −1.56695 −0.783477 0.621421i \(-0.786556\pi\)
−0.783477 + 0.621421i \(0.786556\pi\)
\(164\) 8.10456e7 1.43475
\(165\) 0 0
\(166\) 7.37497e7 1.25136
\(167\) 2.60451e7 0.432732 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(168\) 0 0
\(169\) −3.00181e7 −0.478388
\(170\) −5.87377e7 −0.916950
\(171\) 0 0
\(172\) −7.81717e7 −1.17139
\(173\) 2.86328e7 0.420438 0.210219 0.977654i \(-0.432582\pi\)
0.210219 + 0.977654i \(0.432582\pi\)
\(174\) 0 0
\(175\) 1.38595e7 0.195485
\(176\) 1.89362e7 0.261817
\(177\) 0 0
\(178\) 1.31150e8 1.74300
\(179\) −9.04329e6 −0.117853 −0.0589266 0.998262i \(-0.518768\pi\)
−0.0589266 + 0.998262i \(0.518768\pi\)
\(180\) 0 0
\(181\) 9.04918e7 1.13432 0.567158 0.823609i \(-0.308043\pi\)
0.567158 + 0.823609i \(0.308043\pi\)
\(182\) −1.09863e8 −1.35084
\(183\) 0 0
\(184\) 1.13253e7 0.134025
\(185\) −7.89919e7 −0.917237
\(186\) 0 0
\(187\) −6.09466e7 −0.681559
\(188\) 5.02405e7 0.551444
\(189\) 0 0
\(190\) 2.38245e8 2.51992
\(191\) 7.52927e7 0.781873 0.390936 0.920418i \(-0.372151\pi\)
0.390936 + 0.920418i \(0.372151\pi\)
\(192\) 0 0
\(193\) 3.03390e7 0.303774 0.151887 0.988398i \(-0.451465\pi\)
0.151887 + 0.988398i \(0.451465\pi\)
\(194\) 2.62314e8 2.57938
\(195\) 0 0
\(196\) 6.29942e7 0.597592
\(197\) −5.75348e7 −0.536165 −0.268083 0.963396i \(-0.586390\pi\)
−0.268083 + 0.963396i \(0.586390\pi\)
\(198\) 0 0
\(199\) 3.30510e6 0.0297303 0.0148651 0.999890i \(-0.495268\pi\)
0.0148651 + 0.999890i \(0.495268\pi\)
\(200\) −1.48926e7 −0.131633
\(201\) 0 0
\(202\) −5.46532e7 −0.466537
\(203\) −2.10741e8 −1.76812
\(204\) 0 0
\(205\) −1.07563e8 −0.872016
\(206\) −2.67217e8 −2.12975
\(207\) 0 0
\(208\) −2.28425e7 −0.176003
\(209\) 2.47204e8 1.87303
\(210\) 0 0
\(211\) 7.47470e7 0.547779 0.273890 0.961761i \(-0.411690\pi\)
0.273890 + 0.961761i \(0.411690\pi\)
\(212\) −1.42168e8 −1.02477
\(213\) 0 0
\(214\) 3.97403e8 2.77193
\(215\) 1.03749e8 0.711948
\(216\) 0 0
\(217\) −2.92391e8 −1.94248
\(218\) −4.42668e7 −0.289388
\(219\) 0 0
\(220\) 2.33014e8 1.47538
\(221\) 7.35190e7 0.458170
\(222\) 0 0
\(223\) −2.95302e8 −1.78320 −0.891599 0.452826i \(-0.850416\pi\)
−0.891599 + 0.452826i \(0.850416\pi\)
\(224\) 2.34977e8 1.39688
\(225\) 0 0
\(226\) 2.08730e8 1.20283
\(227\) 4.08714e6 0.0231915 0.0115957 0.999933i \(-0.496309\pi\)
0.0115957 + 0.999933i \(0.496309\pi\)
\(228\) 0 0
\(229\) 2.00230e8 1.10181 0.550903 0.834570i \(-0.314284\pi\)
0.550903 + 0.834570i \(0.314284\pi\)
\(230\) −4.49044e7 −0.243356
\(231\) 0 0
\(232\) 2.26451e8 1.19060
\(233\) 9.85532e7 0.510417 0.255208 0.966886i \(-0.417856\pi\)
0.255208 + 0.966886i \(0.417856\pi\)
\(234\) 0 0
\(235\) −6.66787e7 −0.335158
\(236\) 3.95155e7 0.195693
\(237\) 0 0
\(238\) −2.46775e8 −1.18654
\(239\) 3.06770e8 1.45352 0.726758 0.686894i \(-0.241026\pi\)
0.726758 + 0.686894i \(0.241026\pi\)
\(240\) 0 0
\(241\) 3.30373e8 1.52035 0.760176 0.649717i \(-0.225112\pi\)
0.760176 + 0.649717i \(0.225112\pi\)
\(242\) 5.38071e7 0.244054
\(243\) 0 0
\(244\) 4.49382e7 0.198039
\(245\) −8.36054e7 −0.363206
\(246\) 0 0
\(247\) −2.98199e8 −1.25912
\(248\) 3.14188e8 1.30800
\(249\) 0 0
\(250\) 4.16143e8 1.68443
\(251\) 4.29400e8 1.71397 0.856987 0.515339i \(-0.172334\pi\)
0.856987 + 0.515339i \(0.172334\pi\)
\(252\) 0 0
\(253\) −4.65931e7 −0.180884
\(254\) −6.22158e8 −2.38222
\(255\) 0 0
\(256\) −1.54163e8 −0.574302
\(257\) 3.01963e8 1.10966 0.554828 0.831965i \(-0.312784\pi\)
0.554828 + 0.831965i \(0.312784\pi\)
\(258\) 0 0
\(259\) −3.31869e8 −1.18691
\(260\) −2.81081e8 −0.991803
\(261\) 0 0
\(262\) 1.90224e8 0.653447
\(263\) 7.22951e7 0.245055 0.122528 0.992465i \(-0.460900\pi\)
0.122528 + 0.992465i \(0.460900\pi\)
\(264\) 0 0
\(265\) 1.88684e8 0.622836
\(266\) 1.00094e9 3.26078
\(267\) 0 0
\(268\) 6.81384e8 2.16232
\(269\) −1.46096e8 −0.457622 −0.228811 0.973471i \(-0.573484\pi\)
−0.228811 + 0.973471i \(0.573484\pi\)
\(270\) 0 0
\(271\) −2.87297e8 −0.876876 −0.438438 0.898761i \(-0.644468\pi\)
−0.438438 + 0.898761i \(0.644468\pi\)
\(272\) −5.13087e7 −0.154597
\(273\) 0 0
\(274\) 8.31362e8 2.44154
\(275\) 6.12693e7 0.177656
\(276\) 0 0
\(277\) 3.14837e8 0.890033 0.445016 0.895522i \(-0.353198\pi\)
0.445016 + 0.895522i \(0.353198\pi\)
\(278\) −3.14071e8 −0.876741
\(279\) 0 0
\(280\) 3.15811e8 0.859753
\(281\) −7.28235e8 −1.95794 −0.978971 0.204002i \(-0.934605\pi\)
−0.978971 + 0.204002i \(0.934605\pi\)
\(282\) 0 0
\(283\) −2.37367e8 −0.622541 −0.311271 0.950321i \(-0.600755\pi\)
−0.311271 + 0.950321i \(0.600755\pi\)
\(284\) −1.98244e8 −0.513553
\(285\) 0 0
\(286\) −4.85679e8 −1.22763
\(287\) −4.51904e8 −1.12839
\(288\) 0 0
\(289\) −2.45200e8 −0.597556
\(290\) −8.97870e8 −2.16182
\(291\) 0 0
\(292\) −1.13279e9 −2.66261
\(293\) 1.12840e7 0.0262076 0.0131038 0.999914i \(-0.495829\pi\)
0.0131038 + 0.999914i \(0.495829\pi\)
\(294\) 0 0
\(295\) −5.24446e7 −0.118939
\(296\) 3.56608e8 0.799227
\(297\) 0 0
\(298\) −1.23629e9 −2.70622
\(299\) 5.62045e7 0.121597
\(300\) 0 0
\(301\) 4.35880e8 0.921264
\(302\) 1.77916e8 0.371699
\(303\) 0 0
\(304\) 2.08112e8 0.424855
\(305\) −5.96416e7 −0.120365
\(306\) 0 0
\(307\) 3.01581e8 0.594867 0.297433 0.954743i \(-0.403869\pi\)
0.297433 + 0.954743i \(0.403869\pi\)
\(308\) 9.78961e8 1.90914
\(309\) 0 0
\(310\) −1.24575e9 −2.37500
\(311\) −7.94458e8 −1.49765 −0.748824 0.662769i \(-0.769381\pi\)
−0.748824 + 0.662769i \(0.769381\pi\)
\(312\) 0 0
\(313\) 9.42174e8 1.73670 0.868352 0.495948i \(-0.165179\pi\)
0.868352 + 0.495948i \(0.165179\pi\)
\(314\) −3.10839e8 −0.566607
\(315\) 0 0
\(316\) 8.35459e8 1.48943
\(317\) 9.46323e7 0.166852 0.0834261 0.996514i \(-0.473414\pi\)
0.0834261 + 0.996514i \(0.473414\pi\)
\(318\) 0 0
\(319\) −9.31635e8 −1.60686
\(320\) 8.70626e8 1.48527
\(321\) 0 0
\(322\) −1.88657e8 −0.314903
\(323\) −6.69814e8 −1.10598
\(324\) 0 0
\(325\) −7.39083e7 −0.119427
\(326\) −1.55082e9 −2.47913
\(327\) 0 0
\(328\) 4.85592e8 0.759824
\(329\) −2.80137e8 −0.433696
\(330\) 0 0
\(331\) 2.56756e8 0.389156 0.194578 0.980887i \(-0.437666\pi\)
0.194578 + 0.980887i \(0.437666\pi\)
\(332\) 7.92726e8 1.18889
\(333\) 0 0
\(334\) 4.66202e8 0.684639
\(335\) −9.04327e8 −1.31422
\(336\) 0 0
\(337\) 9.60763e7 0.136745 0.0683725 0.997660i \(-0.478219\pi\)
0.0683725 + 0.997660i \(0.478219\pi\)
\(338\) −5.37318e8 −0.756873
\(339\) 0 0
\(340\) −6.31365e8 −0.871172
\(341\) −1.29259e9 −1.76531
\(342\) 0 0
\(343\) 5.32265e8 0.712195
\(344\) −4.68373e8 −0.620350
\(345\) 0 0
\(346\) 5.12521e8 0.665189
\(347\) −4.17126e8 −0.535937 −0.267969 0.963428i \(-0.586352\pi\)
−0.267969 + 0.963428i \(0.586352\pi\)
\(348\) 0 0
\(349\) −5.98395e8 −0.753527 −0.376764 0.926309i \(-0.622963\pi\)
−0.376764 + 0.926309i \(0.622963\pi\)
\(350\) 2.48081e8 0.309283
\(351\) 0 0
\(352\) 1.03878e9 1.26947
\(353\) 1.21639e9 1.47184 0.735921 0.677067i \(-0.236749\pi\)
0.735921 + 0.677067i \(0.236749\pi\)
\(354\) 0 0
\(355\) 2.63107e8 0.312129
\(356\) 1.40972e9 1.65599
\(357\) 0 0
\(358\) −1.61873e8 −0.186459
\(359\) 1.46927e8 0.167599 0.0837993 0.996483i \(-0.473295\pi\)
0.0837993 + 0.996483i \(0.473295\pi\)
\(360\) 0 0
\(361\) 1.82295e9 2.03939
\(362\) 1.61979e9 1.79464
\(363\) 0 0
\(364\) −1.18091e9 −1.28340
\(365\) 1.50342e9 1.61829
\(366\) 0 0
\(367\) −1.54258e9 −1.62899 −0.814494 0.580172i \(-0.802985\pi\)
−0.814494 + 0.580172i \(0.802985\pi\)
\(368\) −3.92250e7 −0.0410295
\(369\) 0 0
\(370\) −1.41394e9 −1.45119
\(371\) 7.92717e8 0.805952
\(372\) 0 0
\(373\) 3.08393e8 0.307697 0.153849 0.988094i \(-0.450833\pi\)
0.153849 + 0.988094i \(0.450833\pi\)
\(374\) −1.09093e9 −1.07832
\(375\) 0 0
\(376\) 3.01020e8 0.292037
\(377\) 1.12382e9 1.08019
\(378\) 0 0
\(379\) −8.49811e8 −0.801835 −0.400918 0.916114i \(-0.631309\pi\)
−0.400918 + 0.916114i \(0.631309\pi\)
\(380\) 2.56087e9 2.39411
\(381\) 0 0
\(382\) 1.34772e9 1.23703
\(383\) −8.83511e8 −0.803556 −0.401778 0.915737i \(-0.631608\pi\)
−0.401778 + 0.915737i \(0.631608\pi\)
\(384\) 0 0
\(385\) −1.29927e9 −1.16034
\(386\) 5.43062e8 0.480612
\(387\) 0 0
\(388\) 2.81958e9 2.45061
\(389\) 1.07810e9 0.928616 0.464308 0.885674i \(-0.346303\pi\)
0.464308 + 0.885674i \(0.346303\pi\)
\(390\) 0 0
\(391\) 1.26247e8 0.106807
\(392\) 3.77435e8 0.316476
\(393\) 0 0
\(394\) −1.02986e9 −0.848285
\(395\) −1.10881e9 −0.905250
\(396\) 0 0
\(397\) −2.25182e9 −1.80620 −0.903101 0.429427i \(-0.858715\pi\)
−0.903101 + 0.429427i \(0.858715\pi\)
\(398\) 5.91607e7 0.0470373
\(399\) 0 0
\(400\) 5.15804e7 0.0402972
\(401\) −2.61153e8 −0.202250 −0.101125 0.994874i \(-0.532244\pi\)
−0.101125 + 0.994874i \(0.532244\pi\)
\(402\) 0 0
\(403\) 1.55924e9 1.18671
\(404\) −5.87461e8 −0.443246
\(405\) 0 0
\(406\) −3.77222e9 −2.79741
\(407\) −1.46711e9 −1.07866
\(408\) 0 0
\(409\) 3.01019e8 0.217552 0.108776 0.994066i \(-0.465307\pi\)
0.108776 + 0.994066i \(0.465307\pi\)
\(410\) −1.92536e9 −1.37965
\(411\) 0 0
\(412\) −2.87228e9 −2.02343
\(413\) −2.20335e8 −0.153907
\(414\) 0 0
\(415\) −1.05210e9 −0.722584
\(416\) −1.25306e9 −0.853387
\(417\) 0 0
\(418\) 4.42491e9 2.96338
\(419\) 2.40890e9 1.59982 0.799908 0.600123i \(-0.204882\pi\)
0.799908 + 0.600123i \(0.204882\pi\)
\(420\) 0 0
\(421\) 1.27032e9 0.829711 0.414855 0.909887i \(-0.363832\pi\)
0.414855 + 0.909887i \(0.363832\pi\)
\(422\) 1.33796e9 0.866660
\(423\) 0 0
\(424\) −8.51810e8 −0.542703
\(425\) −1.66013e8 −0.104901
\(426\) 0 0
\(427\) −2.50572e8 −0.155752
\(428\) 4.27164e9 2.63355
\(429\) 0 0
\(430\) 1.85708e9 1.12640
\(431\) −2.24959e9 −1.35342 −0.676710 0.736250i \(-0.736595\pi\)
−0.676710 + 0.736250i \(0.736595\pi\)
\(432\) 0 0
\(433\) 1.18429e9 0.701053 0.350527 0.936553i \(-0.386003\pi\)
0.350527 + 0.936553i \(0.386003\pi\)
\(434\) −5.23375e9 −3.07326
\(435\) 0 0
\(436\) −4.75819e8 −0.274940
\(437\) −5.12066e8 −0.293522
\(438\) 0 0
\(439\) 1.33626e9 0.753818 0.376909 0.926250i \(-0.376987\pi\)
0.376909 + 0.926250i \(0.376987\pi\)
\(440\) 1.39612e9 0.781339
\(441\) 0 0
\(442\) 1.31598e9 0.724886
\(443\) −1.47906e8 −0.0808299 −0.0404149 0.999183i \(-0.512868\pi\)
−0.0404149 + 0.999183i \(0.512868\pi\)
\(444\) 0 0
\(445\) −1.87096e9 −1.00648
\(446\) −5.28585e9 −2.82126
\(447\) 0 0
\(448\) 3.65776e9 1.92195
\(449\) −2.18342e9 −1.13835 −0.569174 0.822217i \(-0.692737\pi\)
−0.569174 + 0.822217i \(0.692737\pi\)
\(450\) 0 0
\(451\) −1.99776e9 −1.02548
\(452\) 2.24362e9 1.14278
\(453\) 0 0
\(454\) 7.31589e7 0.0366920
\(455\) 1.56729e9 0.780026
\(456\) 0 0
\(457\) −9.95575e8 −0.487941 −0.243971 0.969783i \(-0.578450\pi\)
−0.243971 + 0.969783i \(0.578450\pi\)
\(458\) 3.58408e9 1.74320
\(459\) 0 0
\(460\) −4.82672e8 −0.231206
\(461\) 3.16168e9 1.50302 0.751509 0.659722i \(-0.229326\pi\)
0.751509 + 0.659722i \(0.229326\pi\)
\(462\) 0 0
\(463\) 3.01390e9 1.41122 0.705612 0.708599i \(-0.250672\pi\)
0.705612 + 0.708599i \(0.250672\pi\)
\(464\) −7.84310e8 −0.364481
\(465\) 0 0
\(466\) 1.76408e9 0.807547
\(467\) −8.11607e8 −0.368754 −0.184377 0.982856i \(-0.559027\pi\)
−0.184377 + 0.982856i \(0.559027\pi\)
\(468\) 0 0
\(469\) −3.79935e9 −1.70061
\(470\) −1.19354e9 −0.530265
\(471\) 0 0
\(472\) 2.36761e8 0.103636
\(473\) 1.92692e9 0.837240
\(474\) 0 0
\(475\) 6.73361e8 0.288284
\(476\) −2.65255e9 −1.12730
\(477\) 0 0
\(478\) 5.49112e9 2.29966
\(479\) 2.44233e9 1.01538 0.507692 0.861538i \(-0.330499\pi\)
0.507692 + 0.861538i \(0.330499\pi\)
\(480\) 0 0
\(481\) 1.76976e9 0.725113
\(482\) 5.91360e9 2.40540
\(483\) 0 0
\(484\) 5.78366e8 0.231870
\(485\) −3.74212e9 −1.48944
\(486\) 0 0
\(487\) −1.11140e8 −0.0436032 −0.0218016 0.999762i \(-0.506940\pi\)
−0.0218016 + 0.999762i \(0.506940\pi\)
\(488\) 2.69251e8 0.104879
\(489\) 0 0
\(490\) −1.49652e9 −0.574640
\(491\) 1.10060e9 0.419607 0.209804 0.977744i \(-0.432718\pi\)
0.209804 + 0.977744i \(0.432718\pi\)
\(492\) 0 0
\(493\) 2.52432e9 0.948812
\(494\) −5.33770e9 −1.99209
\(495\) 0 0
\(496\) −1.08819e9 −0.400422
\(497\) 1.10539e9 0.403896
\(498\) 0 0
\(499\) 1.19831e9 0.431735 0.215867 0.976423i \(-0.430742\pi\)
0.215867 + 0.976423i \(0.430742\pi\)
\(500\) 4.47307e9 1.60034
\(501\) 0 0
\(502\) 7.68617e9 2.71173
\(503\) 4.77486e8 0.167291 0.0836456 0.996496i \(-0.473344\pi\)
0.0836456 + 0.996496i \(0.473344\pi\)
\(504\) 0 0
\(505\) 7.79673e8 0.269397
\(506\) −8.34006e8 −0.286182
\(507\) 0 0
\(508\) −6.68750e9 −2.26329
\(509\) 1.61594e9 0.543140 0.271570 0.962419i \(-0.412457\pi\)
0.271570 + 0.962419i \(0.412457\pi\)
\(510\) 0 0
\(511\) 6.31633e9 2.09407
\(512\) 1.46367e9 0.481945
\(513\) 0 0
\(514\) 5.40509e9 1.75562
\(515\) 3.81207e9 1.22980
\(516\) 0 0
\(517\) −1.23842e9 −0.394140
\(518\) −5.94038e9 −1.87785
\(519\) 0 0
\(520\) −1.68412e9 −0.525245
\(521\) −2.28015e9 −0.706368 −0.353184 0.935554i \(-0.614901\pi\)
−0.353184 + 0.935554i \(0.614901\pi\)
\(522\) 0 0
\(523\) −4.17653e8 −0.127661 −0.0638307 0.997961i \(-0.520332\pi\)
−0.0638307 + 0.997961i \(0.520332\pi\)
\(524\) 2.04469e9 0.620824
\(525\) 0 0
\(526\) 1.29407e9 0.387710
\(527\) 3.50235e9 1.04237
\(528\) 0 0
\(529\) −3.30831e9 −0.971654
\(530\) 3.37740e9 0.985410
\(531\) 0 0
\(532\) 1.07590e10 3.09799
\(533\) 2.40987e9 0.689364
\(534\) 0 0
\(535\) −5.66928e9 −1.60062
\(536\) 4.08257e9 1.14514
\(537\) 0 0
\(538\) −2.61510e9 −0.724019
\(539\) −1.55280e9 −0.427124
\(540\) 0 0
\(541\) 7.14439e9 1.93988 0.969940 0.243346i \(-0.0782450\pi\)
0.969940 + 0.243346i \(0.0782450\pi\)
\(542\) −5.14255e9 −1.38734
\(543\) 0 0
\(544\) −2.81463e9 −0.749592
\(545\) 6.31502e8 0.167104
\(546\) 0 0
\(547\) −1.90559e9 −0.497822 −0.248911 0.968526i \(-0.580073\pi\)
−0.248911 + 0.968526i \(0.580073\pi\)
\(548\) 8.93622e9 2.31965
\(549\) 0 0
\(550\) 1.09671e9 0.281075
\(551\) −1.02388e10 −2.60748
\(552\) 0 0
\(553\) −4.65846e9 −1.17140
\(554\) 5.63551e9 1.40815
\(555\) 0 0
\(556\) −3.37592e9 −0.832971
\(557\) 8.13950e9 1.99574 0.997872 0.0652056i \(-0.0207703\pi\)
0.997872 + 0.0652056i \(0.0207703\pi\)
\(558\) 0 0
\(559\) −2.32442e9 −0.562824
\(560\) −1.09381e9 −0.263198
\(561\) 0 0
\(562\) −1.30353e10 −3.09772
\(563\) −3.38360e9 −0.799098 −0.399549 0.916712i \(-0.630833\pi\)
−0.399549 + 0.916712i \(0.630833\pi\)
\(564\) 0 0
\(565\) −2.97771e9 −0.694564
\(566\) −4.24883e9 −0.984944
\(567\) 0 0
\(568\) −1.18779e9 −0.271971
\(569\) −1.41168e9 −0.321250 −0.160625 0.987016i \(-0.551351\pi\)
−0.160625 + 0.987016i \(0.551351\pi\)
\(570\) 0 0
\(571\) 3.06389e9 0.688726 0.344363 0.938837i \(-0.388095\pi\)
0.344363 + 0.938837i \(0.388095\pi\)
\(572\) −5.22051e9 −1.16634
\(573\) 0 0
\(574\) −8.08900e9 −1.78527
\(575\) −1.26915e8 −0.0278404
\(576\) 0 0
\(577\) 5.11879e9 1.10931 0.554653 0.832082i \(-0.312851\pi\)
0.554653 + 0.832082i \(0.312851\pi\)
\(578\) −4.38904e9 −0.945414
\(579\) 0 0
\(580\) −9.65110e9 −2.05390
\(581\) −4.42018e9 −0.935026
\(582\) 0 0
\(583\) 3.50441e9 0.732445
\(584\) −6.78718e9 −1.41008
\(585\) 0 0
\(586\) 2.01982e8 0.0414639
\(587\) −5.18384e9 −1.05784 −0.528918 0.848673i \(-0.677402\pi\)
−0.528918 + 0.848673i \(0.677402\pi\)
\(588\) 0 0
\(589\) −1.42058e10 −2.86459
\(590\) −9.38748e8 −0.188177
\(591\) 0 0
\(592\) −1.23511e9 −0.244669
\(593\) −5.88946e9 −1.15980 −0.579902 0.814686i \(-0.696909\pi\)
−0.579902 + 0.814686i \(0.696909\pi\)
\(594\) 0 0
\(595\) 3.52044e9 0.685153
\(596\) −1.32887e10 −2.57112
\(597\) 0 0
\(598\) 1.00605e9 0.192382
\(599\) −1.29167e9 −0.245561 −0.122780 0.992434i \(-0.539181\pi\)
−0.122780 + 0.992434i \(0.539181\pi\)
\(600\) 0 0
\(601\) −4.79647e9 −0.901282 −0.450641 0.892705i \(-0.648805\pi\)
−0.450641 + 0.892705i \(0.648805\pi\)
\(602\) 7.80216e9 1.45756
\(603\) 0 0
\(604\) 1.91240e9 0.353142
\(605\) −7.67602e8 −0.140926
\(606\) 0 0
\(607\) −3.30008e9 −0.598914 −0.299457 0.954110i \(-0.596805\pi\)
−0.299457 + 0.954110i \(0.596805\pi\)
\(608\) 1.14164e10 2.05999
\(609\) 0 0
\(610\) −1.06757e9 −0.190433
\(611\) 1.49389e9 0.264956
\(612\) 0 0
\(613\) 2.21730e9 0.388788 0.194394 0.980924i \(-0.437726\pi\)
0.194394 + 0.980924i \(0.437726\pi\)
\(614\) 5.39824e9 0.941159
\(615\) 0 0
\(616\) 5.86553e9 1.01105
\(617\) 3.08716e9 0.529128 0.264564 0.964368i \(-0.414772\pi\)
0.264564 + 0.964368i \(0.414772\pi\)
\(618\) 0 0
\(619\) 4.78027e9 0.810094 0.405047 0.914296i \(-0.367255\pi\)
0.405047 + 0.914296i \(0.367255\pi\)
\(620\) −1.33904e10 −2.25643
\(621\) 0 0
\(622\) −1.42206e10 −2.36948
\(623\) −7.86047e9 −1.30239
\(624\) 0 0
\(625\) −4.92735e9 −0.807298
\(626\) 1.68647e10 2.74770
\(627\) 0 0
\(628\) −3.34117e9 −0.538319
\(629\) 3.97522e9 0.636919
\(630\) 0 0
\(631\) −4.87262e9 −0.772075 −0.386037 0.922483i \(-0.626156\pi\)
−0.386037 + 0.922483i \(0.626156\pi\)
\(632\) 5.00573e9 0.788782
\(633\) 0 0
\(634\) 1.69390e9 0.263982
\(635\) 8.87559e9 1.37559
\(636\) 0 0
\(637\) 1.87312e9 0.287129
\(638\) −1.66761e10 −2.54227
\(639\) 0 0
\(640\) 8.42504e9 1.27040
\(641\) −2.85862e9 −0.428700 −0.214350 0.976757i \(-0.568763\pi\)
−0.214350 + 0.976757i \(0.568763\pi\)
\(642\) 0 0
\(643\) −7.22075e9 −1.07113 −0.535567 0.844493i \(-0.679902\pi\)
−0.535567 + 0.844493i \(0.679902\pi\)
\(644\) −2.02785e9 −0.299182
\(645\) 0 0
\(646\) −1.19895e10 −1.74980
\(647\) 4.74606e8 0.0688919 0.0344460 0.999407i \(-0.489033\pi\)
0.0344460 + 0.999407i \(0.489033\pi\)
\(648\) 0 0
\(649\) −9.74051e8 −0.139870
\(650\) −1.32294e9 −0.188949
\(651\) 0 0
\(652\) −1.66696e10 −2.35536
\(653\) 2.58487e9 0.363281 0.181640 0.983365i \(-0.441859\pi\)
0.181640 + 0.983365i \(0.441859\pi\)
\(654\) 0 0
\(655\) −2.71370e9 −0.377326
\(656\) −1.68184e9 −0.232607
\(657\) 0 0
\(658\) −5.01440e9 −0.686164
\(659\) −2.68455e9 −0.365404 −0.182702 0.983168i \(-0.558484\pi\)
−0.182702 + 0.983168i \(0.558484\pi\)
\(660\) 0 0
\(661\) −9.19497e9 −1.23835 −0.619177 0.785251i \(-0.712534\pi\)
−0.619177 + 0.785251i \(0.712534\pi\)
\(662\) 4.59589e9 0.615696
\(663\) 0 0
\(664\) 4.74969e9 0.629618
\(665\) −1.42792e10 −1.88290
\(666\) 0 0
\(667\) 1.92982e9 0.251812
\(668\) 5.01115e9 0.650459
\(669\) 0 0
\(670\) −1.61873e10 −2.07928
\(671\) −1.10772e9 −0.141547
\(672\) 0 0
\(673\) 1.23253e10 1.55863 0.779317 0.626630i \(-0.215566\pi\)
0.779317 + 0.626630i \(0.215566\pi\)
\(674\) 1.71975e9 0.216349
\(675\) 0 0
\(676\) −5.77557e9 −0.719087
\(677\) −1.51222e10 −1.87307 −0.936537 0.350569i \(-0.885988\pi\)
−0.936537 + 0.350569i \(0.885988\pi\)
\(678\) 0 0
\(679\) −1.57218e10 −1.92733
\(680\) −3.78288e9 −0.461361
\(681\) 0 0
\(682\) −2.31371e10 −2.79296
\(683\) 2.68739e9 0.322745 0.161372 0.986894i \(-0.448408\pi\)
0.161372 + 0.986894i \(0.448408\pi\)
\(684\) 0 0
\(685\) −1.18601e10 −1.40984
\(686\) 9.52744e9 1.12679
\(687\) 0 0
\(688\) 1.62220e9 0.189909
\(689\) −4.22732e9 −0.492377
\(690\) 0 0
\(691\) −9.07091e9 −1.04587 −0.522935 0.852373i \(-0.675163\pi\)
−0.522935 + 0.852373i \(0.675163\pi\)
\(692\) 5.50903e9 0.631980
\(693\) 0 0
\(694\) −7.46647e9 −0.847924
\(695\) 4.48048e9 0.506265
\(696\) 0 0
\(697\) 5.41305e9 0.605518
\(698\) −1.07111e10 −1.19218
\(699\) 0 0
\(700\) 2.66660e9 0.293842
\(701\) −2.72996e9 −0.299326 −0.149663 0.988737i \(-0.547819\pi\)
−0.149663 + 0.988737i \(0.547819\pi\)
\(702\) 0 0
\(703\) −1.61238e10 −1.75035
\(704\) 1.61701e10 1.74666
\(705\) 0 0
\(706\) 2.17731e10 2.32865
\(707\) 3.27564e9 0.348601
\(708\) 0 0
\(709\) −9.03726e8 −0.0952302 −0.0476151 0.998866i \(-0.515162\pi\)
−0.0476151 + 0.998866i \(0.515162\pi\)
\(710\) 4.70957e9 0.493829
\(711\) 0 0
\(712\) 8.44644e9 0.876988
\(713\) 2.67752e9 0.276642
\(714\) 0 0
\(715\) 6.92861e9 0.708883
\(716\) −1.73996e9 −0.177151
\(717\) 0 0
\(718\) 2.62996e9 0.265164
\(719\) 6.51161e9 0.653337 0.326668 0.945139i \(-0.394074\pi\)
0.326668 + 0.945139i \(0.394074\pi\)
\(720\) 0 0
\(721\) 1.60156e10 1.59137
\(722\) 3.26304e10 3.22658
\(723\) 0 0
\(724\) 1.74109e10 1.70505
\(725\) −2.53769e9 −0.247318
\(726\) 0 0
\(727\) −6.14813e9 −0.593435 −0.296717 0.954965i \(-0.595892\pi\)
−0.296717 + 0.954965i \(0.595892\pi\)
\(728\) −7.07550e9 −0.679669
\(729\) 0 0
\(730\) 2.69110e10 2.56035
\(731\) −5.22110e9 −0.494369
\(732\) 0 0
\(733\) 2.71313e9 0.254452 0.127226 0.991874i \(-0.459393\pi\)
0.127226 + 0.991874i \(0.459393\pi\)
\(734\) −2.76120e10 −2.57728
\(735\) 0 0
\(736\) −2.15176e9 −0.198939
\(737\) −1.67960e10 −1.54550
\(738\) 0 0
\(739\) 1.10553e10 1.00766 0.503831 0.863802i \(-0.331924\pi\)
0.503831 + 0.863802i \(0.331924\pi\)
\(740\) −1.51983e10 −1.37874
\(741\) 0 0
\(742\) 1.41895e10 1.27512
\(743\) −1.67697e10 −1.49991 −0.749954 0.661490i \(-0.769925\pi\)
−0.749954 + 0.661490i \(0.769925\pi\)
\(744\) 0 0
\(745\) 1.76367e10 1.56268
\(746\) 5.52017e9 0.486818
\(747\) 0 0
\(748\) −1.17263e10 −1.02448
\(749\) −2.38183e10 −2.07121
\(750\) 0 0
\(751\) 7.82494e9 0.674126 0.337063 0.941482i \(-0.390566\pi\)
0.337063 + 0.941482i \(0.390566\pi\)
\(752\) −1.04258e9 −0.0894020
\(753\) 0 0
\(754\) 2.01161e10 1.70901
\(755\) −2.53812e9 −0.214634
\(756\) 0 0
\(757\) 1.40665e10 1.17856 0.589279 0.807930i \(-0.299412\pi\)
0.589279 + 0.807930i \(0.299412\pi\)
\(758\) −1.52115e10 −1.26861
\(759\) 0 0
\(760\) 1.53437e10 1.26789
\(761\) 4.85071e9 0.398987 0.199493 0.979899i \(-0.436070\pi\)
0.199493 + 0.979899i \(0.436070\pi\)
\(762\) 0 0
\(763\) 2.65313e9 0.216233
\(764\) 1.44865e10 1.17527
\(765\) 0 0
\(766\) −1.58147e10 −1.27133
\(767\) 1.17498e9 0.0940260
\(768\) 0 0
\(769\) 9.60096e9 0.761329 0.380665 0.924713i \(-0.375695\pi\)
0.380665 + 0.924713i \(0.375695\pi\)
\(770\) −2.32566e10 −1.83582
\(771\) 0 0
\(772\) 5.83731e9 0.456617
\(773\) 2.04008e10 1.58862 0.794309 0.607514i \(-0.207833\pi\)
0.794309 + 0.607514i \(0.207833\pi\)
\(774\) 0 0
\(775\) −3.52090e9 −0.271705
\(776\) 1.68938e10 1.29781
\(777\) 0 0
\(778\) 1.92978e10 1.46919
\(779\) −2.19558e10 −1.66405
\(780\) 0 0
\(781\) 4.88668e9 0.367058
\(782\) 2.25979e9 0.168983
\(783\) 0 0
\(784\) −1.30724e9 −0.0968836
\(785\) 4.43437e9 0.327181
\(786\) 0 0
\(787\) 1.60432e10 1.17322 0.586608 0.809871i \(-0.300463\pi\)
0.586608 + 0.809871i \(0.300463\pi\)
\(788\) −1.10699e10 −0.805935
\(789\) 0 0
\(790\) −1.98475e10 −1.43223
\(791\) −1.25102e10 −0.898768
\(792\) 0 0
\(793\) 1.33623e9 0.0951532
\(794\) −4.03071e10 −2.85765
\(795\) 0 0
\(796\) 6.35911e8 0.0446890
\(797\) 1.85146e10 1.29542 0.647711 0.761886i \(-0.275727\pi\)
0.647711 + 0.761886i \(0.275727\pi\)
\(798\) 0 0
\(799\) 3.35557e9 0.232730
\(800\) 2.82953e9 0.195389
\(801\) 0 0
\(802\) −4.67458e9 −0.319987
\(803\) 2.79230e10 1.90308
\(804\) 0 0
\(805\) 2.69134e9 0.181838
\(806\) 2.79100e10 1.87753
\(807\) 0 0
\(808\) −3.51983e9 −0.234737
\(809\) −1.01950e9 −0.0676968 −0.0338484 0.999427i \(-0.510776\pi\)
−0.0338484 + 0.999427i \(0.510776\pi\)
\(810\) 0 0
\(811\) 2.10382e10 1.38496 0.692478 0.721439i \(-0.256519\pi\)
0.692478 + 0.721439i \(0.256519\pi\)
\(812\) −4.05472e10 −2.65775
\(813\) 0 0
\(814\) −2.62610e10 −1.70658
\(815\) 2.21237e10 1.43155
\(816\) 0 0
\(817\) 2.11772e10 1.35860
\(818\) 5.38818e9 0.344196
\(819\) 0 0
\(820\) −2.06954e10 −1.31077
\(821\) −4.64433e9 −0.292902 −0.146451 0.989218i \(-0.546785\pi\)
−0.146451 + 0.989218i \(0.546785\pi\)
\(822\) 0 0
\(823\) −5.01145e9 −0.313375 −0.156687 0.987648i \(-0.550081\pi\)
−0.156687 + 0.987648i \(0.550081\pi\)
\(824\) −1.72095e10 −1.07158
\(825\) 0 0
\(826\) −3.94396e9 −0.243502
\(827\) 2.72122e10 1.67300 0.836498 0.547970i \(-0.184599\pi\)
0.836498 + 0.547970i \(0.184599\pi\)
\(828\) 0 0
\(829\) 3.17275e10 1.93417 0.967086 0.254450i \(-0.0818943\pi\)
0.967086 + 0.254450i \(0.0818943\pi\)
\(830\) −1.88324e10 −1.14322
\(831\) 0 0
\(832\) −1.95057e10 −1.17417
\(833\) 4.20739e9 0.252206
\(834\) 0 0
\(835\) −6.65076e9 −0.395338
\(836\) 4.75628e10 2.81544
\(837\) 0 0
\(838\) 4.31189e10 2.53112
\(839\) 3.12839e9 0.182875 0.0914374 0.995811i \(-0.470854\pi\)
0.0914374 + 0.995811i \(0.470854\pi\)
\(840\) 0 0
\(841\) 2.13371e10 1.23694
\(842\) 2.27385e10 1.31271
\(843\) 0 0
\(844\) 1.43815e10 0.823393
\(845\) 7.66529e9 0.437049
\(846\) 0 0
\(847\) −3.22492e9 −0.182359
\(848\) 2.95024e9 0.166139
\(849\) 0 0
\(850\) −2.97160e9 −0.165968
\(851\) 3.03902e9 0.169036
\(852\) 0 0
\(853\) −3.10920e10 −1.71525 −0.857623 0.514278i \(-0.828060\pi\)
−0.857623 + 0.514278i \(0.828060\pi\)
\(854\) −4.48519e9 −0.246421
\(855\) 0 0
\(856\) 2.55939e10 1.39469
\(857\) 6.63199e9 0.359924 0.179962 0.983674i \(-0.442403\pi\)
0.179962 + 0.983674i \(0.442403\pi\)
\(858\) 0 0
\(859\) 2.04251e9 0.109948 0.0549741 0.998488i \(-0.482492\pi\)
0.0549741 + 0.998488i \(0.482492\pi\)
\(860\) 1.99616e10 1.07016
\(861\) 0 0
\(862\) −4.02672e10 −2.14129
\(863\) −6.09687e8 −0.0322901 −0.0161450 0.999870i \(-0.505139\pi\)
−0.0161450 + 0.999870i \(0.505139\pi\)
\(864\) 0 0
\(865\) −7.31153e9 −0.384107
\(866\) 2.11986e10 1.10916
\(867\) 0 0
\(868\) −5.62569e10 −2.91983
\(869\) −2.05939e10 −1.06456
\(870\) 0 0
\(871\) 2.02608e10 1.03895
\(872\) −2.85091e9 −0.145605
\(873\) 0 0
\(874\) −9.16589e9 −0.464392
\(875\) −2.49415e10 −1.25862
\(876\) 0 0
\(877\) −1.17372e10 −0.587580 −0.293790 0.955870i \(-0.594917\pi\)
−0.293790 + 0.955870i \(0.594917\pi\)
\(878\) 2.39189e10 1.19264
\(879\) 0 0
\(880\) −4.83546e9 −0.239193
\(881\) −3.53359e10 −1.74101 −0.870504 0.492162i \(-0.836207\pi\)
−0.870504 + 0.492162i \(0.836207\pi\)
\(882\) 0 0
\(883\) 2.20915e10 1.07985 0.539924 0.841714i \(-0.318453\pi\)
0.539924 + 0.841714i \(0.318453\pi\)
\(884\) 1.41453e10 0.688697
\(885\) 0 0
\(886\) −2.64748e9 −0.127884
\(887\) 2.64826e10 1.27417 0.637085 0.770793i \(-0.280140\pi\)
0.637085 + 0.770793i \(0.280140\pi\)
\(888\) 0 0
\(889\) 3.72890e10 1.78002
\(890\) −3.34899e10 −1.59239
\(891\) 0 0
\(892\) −5.68170e10 −2.68041
\(893\) −1.36105e10 −0.639577
\(894\) 0 0
\(895\) 2.30925e9 0.107669
\(896\) 3.53961e10 1.64391
\(897\) 0 0
\(898\) −3.90828e10 −1.80102
\(899\) 5.35373e10 2.45752
\(900\) 0 0
\(901\) −9.49541e9 −0.432491
\(902\) −3.57595e10 −1.62244
\(903\) 0 0
\(904\) 1.34428e10 0.605202
\(905\) −2.31076e10 −1.03630
\(906\) 0 0
\(907\) −1.70454e10 −0.758547 −0.379274 0.925285i \(-0.623826\pi\)
−0.379274 + 0.925285i \(0.623826\pi\)
\(908\) 7.86377e8 0.0348602
\(909\) 0 0
\(910\) 2.80542e10 1.23411
\(911\) 3.21064e10 1.40695 0.703473 0.710722i \(-0.251632\pi\)
0.703473 + 0.710722i \(0.251632\pi\)
\(912\) 0 0
\(913\) −1.95406e10 −0.849747
\(914\) −1.78206e10 −0.771988
\(915\) 0 0
\(916\) 3.85248e10 1.65618
\(917\) −1.14011e10 −0.488261
\(918\) 0 0
\(919\) 9.83116e8 0.0417831 0.0208915 0.999782i \(-0.493350\pi\)
0.0208915 + 0.999782i \(0.493350\pi\)
\(920\) −2.89197e9 −0.122444
\(921\) 0 0
\(922\) 5.65934e10 2.37798
\(923\) −5.89473e9 −0.246750
\(924\) 0 0
\(925\) −3.99628e9 −0.166020
\(926\) 5.39483e10 2.23274
\(927\) 0 0
\(928\) −4.30247e10 −1.76726
\(929\) −1.58139e10 −0.647119 −0.323559 0.946208i \(-0.604880\pi\)
−0.323559 + 0.946208i \(0.604880\pi\)
\(930\) 0 0
\(931\) −1.70655e10 −0.693100
\(932\) 1.89619e10 0.767231
\(933\) 0 0
\(934\) −1.45276e10 −0.583418
\(935\) 1.55630e10 0.622664
\(936\) 0 0
\(937\) 1.93290e10 0.767576 0.383788 0.923421i \(-0.374619\pi\)
0.383788 + 0.923421i \(0.374619\pi\)
\(938\) −6.80076e10 −2.69059
\(939\) 0 0
\(940\) −1.28292e10 −0.503792
\(941\) 1.84380e9 0.0721356 0.0360678 0.999349i \(-0.488517\pi\)
0.0360678 + 0.999349i \(0.488517\pi\)
\(942\) 0 0
\(943\) 4.13822e9 0.160703
\(944\) −8.20018e8 −0.0317264
\(945\) 0 0
\(946\) 3.44915e10 1.32463
\(947\) 3.45923e10 1.32359 0.661797 0.749683i \(-0.269794\pi\)
0.661797 + 0.749683i \(0.269794\pi\)
\(948\) 0 0
\(949\) −3.36831e10 −1.27932
\(950\) 1.20530e10 0.456104
\(951\) 0 0
\(952\) −1.58930e10 −0.597003
\(953\) 2.99406e10 1.12056 0.560281 0.828303i \(-0.310693\pi\)
0.560281 + 0.828303i \(0.310693\pi\)
\(954\) 0 0
\(955\) −1.92264e10 −0.714309
\(956\) 5.90234e10 2.18485
\(957\) 0 0
\(958\) 4.37173e10 1.60647
\(959\) −4.98277e10 −1.82434
\(960\) 0 0
\(961\) 4.67675e10 1.69986
\(962\) 3.16783e10 1.14723
\(963\) 0 0
\(964\) 6.35646e10 2.28531
\(965\) −7.74723e9 −0.277524
\(966\) 0 0
\(967\) 3.35194e9 0.119207 0.0596037 0.998222i \(-0.481016\pi\)
0.0596037 + 0.998222i \(0.481016\pi\)
\(968\) 3.46533e9 0.122795
\(969\) 0 0
\(970\) −6.69832e10 −2.35649
\(971\) −3.76669e10 −1.32036 −0.660180 0.751107i \(-0.729520\pi\)
−0.660180 + 0.751107i \(0.729520\pi\)
\(972\) 0 0
\(973\) 1.88239e10 0.655109
\(974\) −1.98938e9 −0.0689861
\(975\) 0 0
\(976\) −9.32548e8 −0.0321068
\(977\) −2.34371e10 −0.804030 −0.402015 0.915633i \(-0.631690\pi\)
−0.402015 + 0.915633i \(0.631690\pi\)
\(978\) 0 0
\(979\) −3.47493e10 −1.18360
\(980\) −1.60859e10 −0.545952
\(981\) 0 0
\(982\) 1.97005e10 0.663875
\(983\) −3.82382e10 −1.28399 −0.641993 0.766710i \(-0.721892\pi\)
−0.641993 + 0.766710i \(0.721892\pi\)
\(984\) 0 0
\(985\) 1.46918e10 0.489833
\(986\) 4.51848e10 1.50115
\(987\) 0 0
\(988\) −5.73743e10 −1.89264
\(989\) −3.99148e9 −0.131204
\(990\) 0 0
\(991\) 5.46278e10 1.78302 0.891510 0.453001i \(-0.149647\pi\)
0.891510 + 0.453001i \(0.149647\pi\)
\(992\) −5.96944e10 −1.94152
\(993\) 0 0
\(994\) 1.97863e10 0.639017
\(995\) −8.43975e8 −0.0271612
\(996\) 0 0
\(997\) −2.14509e10 −0.685507 −0.342753 0.939425i \(-0.611360\pi\)
−0.342753 + 0.939425i \(0.611360\pi\)
\(998\) 2.14495e10 0.683062
\(999\) 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.1.14 17
3.2 odd 2 177.8.a.b.1.4 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.4 17 3.2 odd 2
531.8.a.d.1.14 17 1.1 even 1 trivial