Properties

Label 91.8.a.d.1.4
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(8.95987\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.95987 q^{2} -21.0756 q^{3} -47.7208 q^{4} -55.0053 q^{5} +188.835 q^{6} -343.000 q^{7} +1574.43 q^{8} -1742.82 q^{9} +O(q^{10})\) \(q-8.95987 q^{2} -21.0756 q^{3} -47.7208 q^{4} -55.0053 q^{5} +188.835 q^{6} -343.000 q^{7} +1574.43 q^{8} -1742.82 q^{9} +492.840 q^{10} -6313.17 q^{11} +1005.75 q^{12} -2197.00 q^{13} +3073.23 q^{14} +1159.27 q^{15} -7998.46 q^{16} -32863.9 q^{17} +15615.4 q^{18} -39983.1 q^{19} +2624.89 q^{20} +7228.94 q^{21} +56565.1 q^{22} +90391.6 q^{23} -33182.2 q^{24} -75099.4 q^{25} +19684.8 q^{26} +82823.4 q^{27} +16368.2 q^{28} -221283. q^{29} -10386.9 q^{30} +71297.6 q^{31} -129862. q^{32} +133054. q^{33} +294456. q^{34} +18866.8 q^{35} +83168.6 q^{36} +39592.1 q^{37} +358243. q^{38} +46303.2 q^{39} -86602.2 q^{40} +379303. q^{41} -64770.4 q^{42} +302309. q^{43} +301269. q^{44} +95864.1 q^{45} -809897. q^{46} +489575. q^{47} +168573. q^{48} +117649. q^{49} +672881. q^{50} +692628. q^{51} +104843. q^{52} +1.15378e6 q^{53} -742087. q^{54} +347258. q^{55} -540031. q^{56} +842669. q^{57} +1.98267e6 q^{58} -2.19644e6 q^{59} -55321.3 q^{60} +2.03500e6 q^{61} -638817. q^{62} +597786. q^{63} +2.18735e6 q^{64} +120847. q^{65} -1.19215e6 q^{66} +2.15818e6 q^{67} +1.56829e6 q^{68} -1.90506e6 q^{69} -169044. q^{70} -4.49029e6 q^{71} -2.74395e6 q^{72} -789659. q^{73} -354740. q^{74} +1.58277e6 q^{75} +1.90802e6 q^{76} +2.16542e6 q^{77} -414870. q^{78} -6.59926e6 q^{79} +439958. q^{80} +2.06598e6 q^{81} -3.39850e6 q^{82} +659464. q^{83} -344971. q^{84} +1.80769e6 q^{85} -2.70865e6 q^{86} +4.66368e6 q^{87} -9.93967e6 q^{88} -7.89272e6 q^{89} -858930. q^{90} +753571. q^{91} -4.31356e6 q^{92} -1.50264e6 q^{93} -4.38653e6 q^{94} +2.19928e6 q^{95} +2.73694e6 q^{96} +3.32359e6 q^{97} -1.05412e6 q^{98} +1.10027e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + 2548 q^{10} + 451 q^{11} - 16241 q^{12} - 21970 q^{13} + 1029 q^{14} + 27184 q^{15} + 11897 q^{16} - 8654 q^{17} + 159348 q^{18} + 10130 q^{19} - 82012 q^{20} + 34643 q^{21} - 57863 q^{22} - 52155 q^{23} - 49227 q^{24} + 47190 q^{25} + 6591 q^{26} - 155171 q^{27} - 123823 q^{28} + 520154 q^{29} + 1070236 q^{30} + 692605 q^{31} + 149835 q^{32} + 436053 q^{33} + 1059060 q^{34} - 77518 q^{35} + 2843742 q^{36} - 20511 q^{37} + 1905286 q^{38} + 221897 q^{39} + 636320 q^{40} + 355049 q^{41} - 379015 q^{42} + 1256772 q^{43} - 687913 q^{44} + 1259926 q^{45} + 4043075 q^{46} + 1260721 q^{47} + 1128551 q^{48} + 1176490 q^{49} + 609035 q^{50} + 1411976 q^{51} - 793117 q^{52} + 928854 q^{53} + 6642607 q^{54} + 3423196 q^{55} - 99813 q^{56} + 3014966 q^{57} + 1612588 q^{58} + 3144446 q^{59} + 7738848 q^{60} + 6322923 q^{61} + 6545331 q^{62} - 4200721 q^{63} - 6629943 q^{64} - 496522 q^{65} - 14343317 q^{66} + 3944507 q^{67} - 1787356 q^{68} - 148281 q^{69} - 873964 q^{70} + 6032248 q^{71} + 9760866 q^{72} + 1248533 q^{73} - 8263279 q^{74} + 1573413 q^{75} + 1788254 q^{76} - 154693 q^{77} - 2427685 q^{78} - 14947605 q^{79} - 9147616 q^{80} + 25716334 q^{81} - 6987095 q^{82} - 14177784 q^{83} + 5570663 q^{84} - 11788444 q^{85} + 8748840 q^{86} - 29484448 q^{87} - 15390723 q^{88} + 6734836 q^{89} + 5994972 q^{90} + 7535710 q^{91} - 24493215 q^{92} + 17307847 q^{93} - 22760149 q^{94} - 9329708 q^{95} - 36488483 q^{96} - 12365397 q^{97} - 352947 q^{98} - 43198042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.95987 −0.791948 −0.395974 0.918262i \(-0.629593\pi\)
−0.395974 + 0.918262i \(0.629593\pi\)
\(3\) −21.0756 −0.450668 −0.225334 0.974282i \(-0.572347\pi\)
−0.225334 + 0.974282i \(0.572347\pi\)
\(4\) −47.7208 −0.372819
\(5\) −55.0053 −0.196793 −0.0983964 0.995147i \(-0.531371\pi\)
−0.0983964 + 0.995147i \(0.531371\pi\)
\(6\) 188.835 0.356905
\(7\) −343.000 −0.377964
\(8\) 1574.43 1.08720
\(9\) −1742.82 −0.796899
\(10\) 492.840 0.155850
\(11\) −6313.17 −1.43012 −0.715061 0.699062i \(-0.753601\pi\)
−0.715061 + 0.699062i \(0.753601\pi\)
\(12\) 1005.75 0.168017
\(13\) −2197.00 −0.277350
\(14\) 3073.23 0.299328
\(15\) 1159.27 0.0886882
\(16\) −7998.46 −0.488188
\(17\) −32863.9 −1.62236 −0.811181 0.584795i \(-0.801175\pi\)
−0.811181 + 0.584795i \(0.801175\pi\)
\(18\) 15615.4 0.631102
\(19\) −39983.1 −1.33733 −0.668665 0.743564i \(-0.733134\pi\)
−0.668665 + 0.743564i \(0.733134\pi\)
\(20\) 2624.89 0.0733680
\(21\) 7228.94 0.170336
\(22\) 56565.1 1.13258
\(23\) 90391.6 1.54910 0.774552 0.632510i \(-0.217975\pi\)
0.774552 + 0.632510i \(0.217975\pi\)
\(24\) −33182.2 −0.489966
\(25\) −75099.4 −0.961273
\(26\) 19684.8 0.219647
\(27\) 82823.4 0.809804
\(28\) 16368.2 0.140912
\(29\) −221283. −1.68483 −0.842414 0.538831i \(-0.818866\pi\)
−0.842414 + 0.538831i \(0.818866\pi\)
\(30\) −10386.9 −0.0702364
\(31\) 71297.6 0.429842 0.214921 0.976631i \(-0.431051\pi\)
0.214921 + 0.976631i \(0.431051\pi\)
\(32\) −129862. −0.700582
\(33\) 133054. 0.644510
\(34\) 294456. 1.28483
\(35\) 18866.8 0.0743807
\(36\) 83168.6 0.297099
\(37\) 39592.1 0.128500 0.0642500 0.997934i \(-0.479535\pi\)
0.0642500 + 0.997934i \(0.479535\pi\)
\(38\) 358243. 1.05910
\(39\) 46303.2 0.124993
\(40\) −86602.2 −0.213953
\(41\) 379303. 0.859493 0.429747 0.902949i \(-0.358603\pi\)
0.429747 + 0.902949i \(0.358603\pi\)
\(42\) −64770.4 −0.134898
\(43\) 302309. 0.579844 0.289922 0.957050i \(-0.406371\pi\)
0.289922 + 0.957050i \(0.406371\pi\)
\(44\) 301269. 0.533176
\(45\) 95864.1 0.156824
\(46\) −809897. −1.22681
\(47\) 489575. 0.687823 0.343911 0.939002i \(-0.388248\pi\)
0.343911 + 0.939002i \(0.388248\pi\)
\(48\) 168573. 0.220010
\(49\) 117649. 0.142857
\(50\) 672881. 0.761278
\(51\) 692628. 0.731146
\(52\) 104843. 0.103401
\(53\) 1.15378e6 1.06453 0.532267 0.846577i \(-0.321340\pi\)
0.532267 + 0.846577i \(0.321340\pi\)
\(54\) −742087. −0.641323
\(55\) 347258. 0.281438
\(56\) −540031. −0.410923
\(57\) 842669. 0.602692
\(58\) 1.98267e6 1.33430
\(59\) −2.19644e6 −1.39232 −0.696158 0.717889i \(-0.745109\pi\)
−0.696158 + 0.717889i \(0.745109\pi\)
\(60\) −55321.3 −0.0330646
\(61\) 2.03500e6 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(62\) −638817. −0.340413
\(63\) 597786. 0.301199
\(64\) 2.18735e6 1.04301
\(65\) 120847. 0.0545805
\(66\) −1.19215e6 −0.510418
\(67\) 2.15818e6 0.876650 0.438325 0.898817i \(-0.355572\pi\)
0.438325 + 0.898817i \(0.355572\pi\)
\(68\) 1.56829e6 0.604847
\(69\) −1.90506e6 −0.698131
\(70\) −169044. −0.0589056
\(71\) −4.49029e6 −1.48892 −0.744458 0.667670i \(-0.767292\pi\)
−0.744458 + 0.667670i \(0.767292\pi\)
\(72\) −2.74395e6 −0.866389
\(73\) −789659. −0.237580 −0.118790 0.992919i \(-0.537902\pi\)
−0.118790 + 0.992919i \(0.537902\pi\)
\(74\) −354740. −0.101765
\(75\) 1.58277e6 0.433214
\(76\) 1.90802e6 0.498582
\(77\) 2.16542e6 0.540535
\(78\) −414870. −0.0989877
\(79\) −6.59926e6 −1.50592 −0.752958 0.658069i \(-0.771373\pi\)
−0.752958 + 0.658069i \(0.771373\pi\)
\(80\) 439958. 0.0960718
\(81\) 2.06598e6 0.431946
\(82\) −3.39850e6 −0.680674
\(83\) 659464. 0.126595 0.0632977 0.997995i \(-0.479838\pi\)
0.0632977 + 0.997995i \(0.479838\pi\)
\(84\) −344971. −0.0635046
\(85\) 1.80769e6 0.319269
\(86\) −2.70865e6 −0.459206
\(87\) 4.66368e6 0.759297
\(88\) −9.93967e6 −1.55483
\(89\) −7.89272e6 −1.18676 −0.593378 0.804924i \(-0.702206\pi\)
−0.593378 + 0.804924i \(0.702206\pi\)
\(90\) −858930. −0.124196
\(91\) 753571. 0.104828
\(92\) −4.31356e6 −0.577535
\(93\) −1.50264e6 −0.193716
\(94\) −4.38653e6 −0.544720
\(95\) 2.19928e6 0.263177
\(96\) 2.73694e6 0.315730
\(97\) 3.32359e6 0.369749 0.184874 0.982762i \(-0.440812\pi\)
0.184874 + 0.982762i \(0.440812\pi\)
\(98\) −1.05412e6 −0.113135
\(99\) 1.10027e7 1.13966
\(100\) 3.58380e6 0.358380
\(101\) 5.68139e6 0.548693 0.274347 0.961631i \(-0.411538\pi\)
0.274347 + 0.961631i \(0.411538\pi\)
\(102\) −6.20585e6 −0.579030
\(103\) −9.71526e6 −0.876041 −0.438020 0.898965i \(-0.644320\pi\)
−0.438020 + 0.898965i \(0.644320\pi\)
\(104\) −3.45903e6 −0.301535
\(105\) −397630. −0.0335210
\(106\) −1.03378e7 −0.843055
\(107\) 2.32695e6 0.183630 0.0918150 0.995776i \(-0.470733\pi\)
0.0918150 + 0.995776i \(0.470733\pi\)
\(108\) −3.95240e6 −0.301910
\(109\) −1.71300e7 −1.26696 −0.633482 0.773757i \(-0.718375\pi\)
−0.633482 + 0.773757i \(0.718375\pi\)
\(110\) −3.11138e6 −0.222884
\(111\) −834430. −0.0579108
\(112\) 2.74347e6 0.184518
\(113\) −1.35164e7 −0.881222 −0.440611 0.897698i \(-0.645238\pi\)
−0.440611 + 0.897698i \(0.645238\pi\)
\(114\) −7.55020e6 −0.477300
\(115\) −4.97202e6 −0.304853
\(116\) 1.05598e7 0.628135
\(117\) 3.82897e6 0.221020
\(118\) 1.96798e7 1.10264
\(119\) 1.12723e7 0.613195
\(120\) 1.82520e6 0.0964218
\(121\) 2.03689e7 1.04525
\(122\) −1.82334e7 −0.909091
\(123\) −7.99405e6 −0.387346
\(124\) −3.40238e6 −0.160253
\(125\) 8.42815e6 0.385964
\(126\) −5.35609e6 −0.238534
\(127\) −3.70618e6 −0.160551 −0.0802755 0.996773i \(-0.525580\pi\)
−0.0802755 + 0.996773i \(0.525580\pi\)
\(128\) −2.97600e6 −0.125429
\(129\) −6.37135e6 −0.261317
\(130\) −1.08277e6 −0.0432249
\(131\) 4.35220e7 1.69145 0.845726 0.533617i \(-0.179168\pi\)
0.845726 + 0.533617i \(0.179168\pi\)
\(132\) −6.34945e6 −0.240285
\(133\) 1.37142e7 0.505463
\(134\) −1.93370e7 −0.694261
\(135\) −4.55572e6 −0.159364
\(136\) −5.17421e7 −1.76383
\(137\) 5.59159e7 1.85786 0.928930 0.370255i \(-0.120730\pi\)
0.928930 + 0.370255i \(0.120730\pi\)
\(138\) 1.70691e7 0.552884
\(139\) 1.61731e7 0.510788 0.255394 0.966837i \(-0.417795\pi\)
0.255394 + 0.966837i \(0.417795\pi\)
\(140\) −900339. −0.0277305
\(141\) −1.03181e7 −0.309980
\(142\) 4.02324e7 1.17914
\(143\) 1.38700e7 0.396644
\(144\) 1.39399e7 0.389036
\(145\) 1.21717e7 0.331562
\(146\) 7.07524e6 0.188151
\(147\) −2.47953e6 −0.0643811
\(148\) −1.88937e6 −0.0479072
\(149\) 5.07725e7 1.25741 0.628705 0.777644i \(-0.283585\pi\)
0.628705 + 0.777644i \(0.283585\pi\)
\(150\) −1.41814e7 −0.343083
\(151\) −7.64411e7 −1.80679 −0.903395 0.428809i \(-0.858933\pi\)
−0.903395 + 0.428809i \(0.858933\pi\)
\(152\) −6.29508e7 −1.45395
\(153\) 5.72758e7 1.29286
\(154\) −1.94018e7 −0.428076
\(155\) −3.92175e6 −0.0845899
\(156\) −2.20962e6 −0.0465996
\(157\) −6.98540e7 −1.44060 −0.720298 0.693664i \(-0.755995\pi\)
−0.720298 + 0.693664i \(0.755995\pi\)
\(158\) 5.91285e7 1.19261
\(159\) −2.43168e7 −0.479751
\(160\) 7.14312e6 0.137869
\(161\) −3.10043e7 −0.585507
\(162\) −1.85109e7 −0.342079
\(163\) 1.50400e7 0.272014 0.136007 0.990708i \(-0.456573\pi\)
0.136007 + 0.990708i \(0.456573\pi\)
\(164\) −1.81006e7 −0.320435
\(165\) −7.31868e6 −0.126835
\(166\) −5.90871e6 −0.100257
\(167\) 8.73370e7 1.45108 0.725538 0.688182i \(-0.241591\pi\)
0.725538 + 0.688182i \(0.241591\pi\)
\(168\) 1.13815e7 0.185190
\(169\) 4.82681e6 0.0769231
\(170\) −1.61966e7 −0.252845
\(171\) 6.96832e7 1.06572
\(172\) −1.44264e7 −0.216177
\(173\) 3.18347e7 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(174\) −4.17860e7 −0.601324
\(175\) 2.57591e7 0.363327
\(176\) 5.04957e7 0.698168
\(177\) 4.62914e7 0.627472
\(178\) 7.07177e7 0.939849
\(179\) −1.09583e7 −0.142809 −0.0714045 0.997447i \(-0.522748\pi\)
−0.0714045 + 0.997447i \(0.522748\pi\)
\(180\) −4.57471e6 −0.0584669
\(181\) −1.01099e8 −1.26727 −0.633637 0.773631i \(-0.718439\pi\)
−0.633637 + 0.773631i \(0.718439\pi\)
\(182\) −6.75190e6 −0.0830187
\(183\) −4.28890e7 −0.517329
\(184\) 1.42316e8 1.68419
\(185\) −2.17778e6 −0.0252879
\(186\) 1.34635e7 0.153413
\(187\) 2.07475e8 2.32018
\(188\) −2.33629e7 −0.256433
\(189\) −2.84084e7 −0.306077
\(190\) −1.97053e7 −0.208422
\(191\) 5.05592e6 0.0525029 0.0262514 0.999655i \(-0.491643\pi\)
0.0262514 + 0.999655i \(0.491643\pi\)
\(192\) −4.60999e7 −0.470052
\(193\) 7.76809e7 0.777792 0.388896 0.921282i \(-0.372856\pi\)
0.388896 + 0.921282i \(0.372856\pi\)
\(194\) −2.97790e7 −0.292822
\(195\) −2.54692e6 −0.0245977
\(196\) −5.61430e6 −0.0532598
\(197\) −1.26008e8 −1.17427 −0.587134 0.809490i \(-0.699744\pi\)
−0.587134 + 0.809490i \(0.699744\pi\)
\(198\) −9.85827e7 −0.902553
\(199\) −1.55002e8 −1.39428 −0.697141 0.716934i \(-0.745545\pi\)
−0.697141 + 0.716934i \(0.745545\pi\)
\(200\) −1.18239e8 −1.04510
\(201\) −4.54851e7 −0.395078
\(202\) −5.09045e7 −0.434536
\(203\) 7.59001e7 0.636805
\(204\) −3.30528e7 −0.272585
\(205\) −2.08637e7 −0.169142
\(206\) 8.70475e7 0.693778
\(207\) −1.57536e8 −1.23448
\(208\) 1.75726e7 0.135399
\(209\) 2.52420e8 1.91255
\(210\) 3.56271e6 0.0265469
\(211\) 1.07390e8 0.787003 0.393502 0.919324i \(-0.371264\pi\)
0.393502 + 0.919324i \(0.371264\pi\)
\(212\) −5.50595e7 −0.396878
\(213\) 9.46357e7 0.671006
\(214\) −2.08491e7 −0.145425
\(215\) −1.66286e7 −0.114109
\(216\) 1.30400e8 0.880420
\(217\) −2.44551e7 −0.162465
\(218\) 1.53482e8 1.00337
\(219\) 1.66426e7 0.107070
\(220\) −1.65714e7 −0.104925
\(221\) 7.22020e7 0.449962
\(222\) 7.47638e6 0.0458623
\(223\) −5.43261e7 −0.328051 −0.164026 0.986456i \(-0.552448\pi\)
−0.164026 + 0.986456i \(0.552448\pi\)
\(224\) 4.45428e7 0.264795
\(225\) 1.30885e8 0.766037
\(226\) 1.21105e8 0.697882
\(227\) −2.42395e8 −1.37541 −0.687707 0.725989i \(-0.741383\pi\)
−0.687707 + 0.725989i \(0.741383\pi\)
\(228\) −4.02128e7 −0.224695
\(229\) −1.02369e8 −0.563304 −0.281652 0.959517i \(-0.590882\pi\)
−0.281652 + 0.959517i \(0.590882\pi\)
\(230\) 4.45486e7 0.241427
\(231\) −4.56375e7 −0.243602
\(232\) −3.48396e8 −1.83175
\(233\) 2.28645e8 1.18418 0.592088 0.805873i \(-0.298304\pi\)
0.592088 + 0.805873i \(0.298304\pi\)
\(234\) −3.43071e7 −0.175036
\(235\) −2.69292e7 −0.135359
\(236\) 1.04816e8 0.519081
\(237\) 1.39084e8 0.678667
\(238\) −1.00998e8 −0.485619
\(239\) −4.10470e7 −0.194486 −0.0972431 0.995261i \(-0.531002\pi\)
−0.0972431 + 0.995261i \(0.531002\pi\)
\(240\) −9.27239e6 −0.0432965
\(241\) −1.33632e8 −0.614967 −0.307483 0.951553i \(-0.599487\pi\)
−0.307483 + 0.951553i \(0.599487\pi\)
\(242\) −1.82503e8 −0.827782
\(243\) −2.24677e8 −1.00447
\(244\) −9.71120e7 −0.427965
\(245\) −6.47131e6 −0.0281133
\(246\) 7.16256e7 0.306758
\(247\) 8.78429e7 0.370909
\(248\) 1.12253e8 0.467325
\(249\) −1.38986e7 −0.0570524
\(250\) −7.55151e7 −0.305664
\(251\) −3.88436e8 −1.55046 −0.775232 0.631676i \(-0.782367\pi\)
−0.775232 + 0.631676i \(0.782367\pi\)
\(252\) −2.85268e7 −0.112293
\(253\) −5.70658e8 −2.21541
\(254\) 3.32068e7 0.127148
\(255\) −3.80982e7 −0.143884
\(256\) −2.53317e8 −0.943678
\(257\) −3.88086e8 −1.42614 −0.713070 0.701093i \(-0.752696\pi\)
−0.713070 + 0.701093i \(0.752696\pi\)
\(258\) 5.70865e7 0.206949
\(259\) −1.35801e7 −0.0485684
\(260\) −5.76689e6 −0.0203486
\(261\) 3.85656e8 1.34264
\(262\) −3.89952e8 −1.33954
\(263\) −9.75048e7 −0.330507 −0.165254 0.986251i \(-0.552844\pi\)
−0.165254 + 0.986251i \(0.552844\pi\)
\(264\) 2.09485e8 0.700711
\(265\) −6.34643e7 −0.209493
\(266\) −1.22877e8 −0.400301
\(267\) 1.66344e8 0.534833
\(268\) −1.02990e8 −0.326832
\(269\) 4.03040e8 1.26245 0.631227 0.775598i \(-0.282552\pi\)
0.631227 + 0.775598i \(0.282552\pi\)
\(270\) 4.08187e7 0.126208
\(271\) 5.09554e7 0.155524 0.0777621 0.996972i \(-0.475223\pi\)
0.0777621 + 0.996972i \(0.475223\pi\)
\(272\) 2.62861e8 0.792017
\(273\) −1.58820e7 −0.0472428
\(274\) −5.00999e8 −1.47133
\(275\) 4.74115e8 1.37474
\(276\) 9.09110e7 0.260276
\(277\) −8.41430e7 −0.237869 −0.118935 0.992902i \(-0.537948\pi\)
−0.118935 + 0.992902i \(0.537948\pi\)
\(278\) −1.44909e8 −0.404518
\(279\) −1.24259e8 −0.342541
\(280\) 2.97046e7 0.0808667
\(281\) 6.00251e8 1.61384 0.806921 0.590659i \(-0.201132\pi\)
0.806921 + 0.590659i \(0.201132\pi\)
\(282\) 9.24488e7 0.245488
\(283\) −5.69977e8 −1.49488 −0.747438 0.664332i \(-0.768716\pi\)
−0.747438 + 0.664332i \(0.768716\pi\)
\(284\) 2.14280e8 0.555095
\(285\) −4.63512e7 −0.118605
\(286\) −1.24274e8 −0.314122
\(287\) −1.30101e8 −0.324858
\(288\) 2.26327e8 0.558293
\(289\) 6.69698e8 1.63206
\(290\) −1.09057e8 −0.262580
\(291\) −7.00469e7 −0.166634
\(292\) 3.76832e7 0.0885742
\(293\) 3.51911e8 0.817327 0.408663 0.912685i \(-0.365995\pi\)
0.408663 + 0.912685i \(0.365995\pi\)
\(294\) 2.22162e7 0.0509865
\(295\) 1.20816e8 0.273998
\(296\) 6.23352e7 0.139705
\(297\) −5.22878e8 −1.15812
\(298\) −4.54915e8 −0.995803
\(299\) −1.98590e8 −0.429644
\(300\) −7.55310e7 −0.161510
\(301\) −1.03692e8 −0.219160
\(302\) 6.84902e8 1.43088
\(303\) −1.19739e8 −0.247278
\(304\) 3.19803e8 0.652868
\(305\) −1.11936e8 −0.225902
\(306\) −5.13183e8 −1.02388
\(307\) 6.45364e8 1.27298 0.636488 0.771286i \(-0.280386\pi\)
0.636488 + 0.771286i \(0.280386\pi\)
\(308\) −1.03335e8 −0.201522
\(309\) 2.04755e8 0.394803
\(310\) 3.51383e7 0.0669908
\(311\) 3.79660e8 0.715703 0.357852 0.933778i \(-0.383509\pi\)
0.357852 + 0.933778i \(0.383509\pi\)
\(312\) 7.29013e7 0.135892
\(313\) −6.75218e8 −1.24463 −0.622313 0.782769i \(-0.713807\pi\)
−0.622313 + 0.782769i \(0.713807\pi\)
\(314\) 6.25882e8 1.14088
\(315\) −3.28814e7 −0.0592739
\(316\) 3.14922e8 0.561433
\(317\) 7.42853e8 1.30977 0.654886 0.755728i \(-0.272717\pi\)
0.654886 + 0.755728i \(0.272717\pi\)
\(318\) 2.17875e8 0.379938
\(319\) 1.39700e9 2.40951
\(320\) −1.20316e8 −0.205257
\(321\) −4.90419e7 −0.0827561
\(322\) 2.77795e8 0.463691
\(323\) 1.31400e9 2.16964
\(324\) −9.85904e7 −0.161038
\(325\) 1.64993e8 0.266609
\(326\) −1.34756e8 −0.215421
\(327\) 3.61025e8 0.570980
\(328\) 5.97188e8 0.934442
\(329\) −1.67924e8 −0.259973
\(330\) 6.55744e7 0.100447
\(331\) −4.52244e8 −0.685449 −0.342724 0.939436i \(-0.611350\pi\)
−0.342724 + 0.939436i \(0.611350\pi\)
\(332\) −3.14701e7 −0.0471971
\(333\) −6.90019e7 −0.102401
\(334\) −7.82528e8 −1.14918
\(335\) −1.18711e8 −0.172518
\(336\) −5.78205e7 −0.0831561
\(337\) −6.95318e7 −0.0989643 −0.0494822 0.998775i \(-0.515757\pi\)
−0.0494822 + 0.998775i \(0.515757\pi\)
\(338\) −4.32476e7 −0.0609191
\(339\) 2.84866e8 0.397138
\(340\) −8.62643e7 −0.119030
\(341\) −4.50114e8 −0.614727
\(342\) −6.24352e8 −0.843992
\(343\) −4.03536e7 −0.0539949
\(344\) 4.75966e8 0.630407
\(345\) 1.04788e8 0.137387
\(346\) −2.85234e8 −0.370199
\(347\) −1.02087e9 −1.31165 −0.655827 0.754911i \(-0.727680\pi\)
−0.655827 + 0.754911i \(0.727680\pi\)
\(348\) −2.22555e8 −0.283080
\(349\) 5.69930e8 0.717682 0.358841 0.933399i \(-0.383172\pi\)
0.358841 + 0.933399i \(0.383172\pi\)
\(350\) −2.30798e8 −0.287736
\(351\) −1.81963e8 −0.224599
\(352\) 8.19844e8 1.00192
\(353\) −5.92530e7 −0.0716966 −0.0358483 0.999357i \(-0.511413\pi\)
−0.0358483 + 0.999357i \(0.511413\pi\)
\(354\) −4.14765e8 −0.496925
\(355\) 2.46989e8 0.293008
\(356\) 3.76647e8 0.442445
\(357\) −2.37571e8 −0.276347
\(358\) 9.81845e7 0.113097
\(359\) −2.50331e8 −0.285551 −0.142775 0.989755i \(-0.545603\pi\)
−0.142775 + 0.989755i \(0.545603\pi\)
\(360\) 1.50932e8 0.170499
\(361\) 7.04776e8 0.788453
\(362\) 9.05831e8 1.00361
\(363\) −4.29288e8 −0.471060
\(364\) −3.59610e7 −0.0390820
\(365\) 4.34354e7 0.0467540
\(366\) 3.84280e8 0.409698
\(367\) −1.29540e8 −0.136796 −0.0683981 0.997658i \(-0.521789\pi\)
−0.0683981 + 0.997658i \(0.521789\pi\)
\(368\) −7.22994e8 −0.756254
\(369\) −6.61055e8 −0.684929
\(370\) 1.95126e7 0.0200267
\(371\) −3.95748e8 −0.402356
\(372\) 7.17073e7 0.0722210
\(373\) −9.94161e8 −0.991918 −0.495959 0.868346i \(-0.665183\pi\)
−0.495959 + 0.868346i \(0.665183\pi\)
\(374\) −1.85895e9 −1.83746
\(375\) −1.77629e8 −0.173942
\(376\) 7.70804e8 0.747802
\(377\) 4.86159e8 0.467287
\(378\) 2.54536e8 0.242397
\(379\) 8.11711e8 0.765886 0.382943 0.923772i \(-0.374911\pi\)
0.382943 + 0.923772i \(0.374911\pi\)
\(380\) −1.04951e8 −0.0981173
\(381\) 7.81100e7 0.0723551
\(382\) −4.53003e7 −0.0415795
\(383\) −1.05588e9 −0.960331 −0.480165 0.877178i \(-0.659423\pi\)
−0.480165 + 0.877178i \(0.659423\pi\)
\(384\) 6.27211e7 0.0565268
\(385\) −1.19109e8 −0.106373
\(386\) −6.96010e8 −0.615971
\(387\) −5.26869e8 −0.462077
\(388\) −1.58605e8 −0.137849
\(389\) 2.90409e8 0.250142 0.125071 0.992148i \(-0.460084\pi\)
0.125071 + 0.992148i \(0.460084\pi\)
\(390\) 2.28201e7 0.0194801
\(391\) −2.97062e9 −2.51321
\(392\) 1.85231e8 0.155314
\(393\) −9.17255e8 −0.762283
\(394\) 1.12902e9 0.929958
\(395\) 3.62994e8 0.296353
\(396\) −5.25058e8 −0.424887
\(397\) 5.09106e8 0.408358 0.204179 0.978934i \(-0.434548\pi\)
0.204179 + 0.978934i \(0.434548\pi\)
\(398\) 1.38879e9 1.10420
\(399\) −2.89036e8 −0.227796
\(400\) 6.00680e8 0.469281
\(401\) 1.36139e9 1.05433 0.527166 0.849762i \(-0.323254\pi\)
0.527166 + 0.849762i \(0.323254\pi\)
\(402\) 4.07540e8 0.312881
\(403\) −1.56641e8 −0.119217
\(404\) −2.71120e8 −0.204563
\(405\) −1.13640e8 −0.0850039
\(406\) −6.80055e8 −0.504316
\(407\) −2.49952e8 −0.183771
\(408\) 1.09050e9 0.794903
\(409\) −2.07113e9 −1.49684 −0.748419 0.663226i \(-0.769187\pi\)
−0.748419 + 0.663226i \(0.769187\pi\)
\(410\) 1.86936e8 0.133952
\(411\) −1.17846e9 −0.837278
\(412\) 4.63620e8 0.326604
\(413\) 7.53380e8 0.526246
\(414\) 1.41150e9 0.977643
\(415\) −3.62740e7 −0.0249131
\(416\) 2.85308e8 0.194306
\(417\) −3.40858e8 −0.230196
\(418\) −2.26165e9 −1.51464
\(419\) −2.56819e9 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(420\) 1.89752e7 0.0124972
\(421\) −5.09167e8 −0.332562 −0.166281 0.986078i \(-0.553176\pi\)
−0.166281 + 0.986078i \(0.553176\pi\)
\(422\) −9.62203e8 −0.623266
\(423\) −8.53240e8 −0.548125
\(424\) 1.81656e9 1.15736
\(425\) 2.46806e9 1.55953
\(426\) −8.47923e8 −0.531402
\(427\) −6.98006e8 −0.433872
\(428\) −1.11044e8 −0.0684607
\(429\) −2.92320e8 −0.178755
\(430\) 1.48990e8 0.0903685
\(431\) 2.94447e9 1.77148 0.885742 0.464178i \(-0.153650\pi\)
0.885742 + 0.464178i \(0.153650\pi\)
\(432\) −6.62460e8 −0.395336
\(433\) 8.41535e8 0.498155 0.249078 0.968484i \(-0.419873\pi\)
0.249078 + 0.968484i \(0.419873\pi\)
\(434\) 2.19114e8 0.128664
\(435\) −2.56527e8 −0.149424
\(436\) 8.17457e8 0.472348
\(437\) −3.61414e9 −2.07167
\(438\) −1.49115e8 −0.0847935
\(439\) −3.11478e9 −1.75712 −0.878560 0.477632i \(-0.841495\pi\)
−0.878560 + 0.477632i \(0.841495\pi\)
\(440\) 5.46734e8 0.305979
\(441\) −2.05041e8 −0.113843
\(442\) −6.46920e8 −0.356347
\(443\) 1.41246e9 0.771906 0.385953 0.922518i \(-0.373873\pi\)
0.385953 + 0.922518i \(0.373873\pi\)
\(444\) 3.98196e7 0.0215902
\(445\) 4.34141e8 0.233545
\(446\) 4.86755e8 0.259799
\(447\) −1.07006e9 −0.566674
\(448\) −7.50262e8 −0.394221
\(449\) 9.76669e8 0.509196 0.254598 0.967047i \(-0.418057\pi\)
0.254598 + 0.967047i \(0.418057\pi\)
\(450\) −1.17271e9 −0.606661
\(451\) −2.39460e9 −1.22918
\(452\) 6.45012e8 0.328536
\(453\) 1.61105e9 0.814262
\(454\) 2.17183e9 1.08926
\(455\) −4.14504e7 −0.0206295
\(456\) 1.32673e9 0.655247
\(457\) −1.13924e8 −0.0558352 −0.0279176 0.999610i \(-0.508888\pi\)
−0.0279176 + 0.999610i \(0.508888\pi\)
\(458\) 9.17209e8 0.446107
\(459\) −2.72190e9 −1.31380
\(460\) 2.37269e8 0.113655
\(461\) 3.73061e9 1.77348 0.886741 0.462266i \(-0.152963\pi\)
0.886741 + 0.462266i \(0.152963\pi\)
\(462\) 4.08906e8 0.192920
\(463\) 2.18894e9 1.02495 0.512473 0.858703i \(-0.328730\pi\)
0.512473 + 0.858703i \(0.328730\pi\)
\(464\) 1.76993e9 0.822512
\(465\) 8.26533e7 0.0381219
\(466\) −2.04863e9 −0.937805
\(467\) −2.01447e9 −0.915273 −0.457637 0.889139i \(-0.651304\pi\)
−0.457637 + 0.889139i \(0.651304\pi\)
\(468\) −1.82721e8 −0.0824004
\(469\) −7.40256e8 −0.331343
\(470\) 2.41282e8 0.107197
\(471\) 1.47222e9 0.649230
\(472\) −3.45816e9 −1.51373
\(473\) −1.90853e9 −0.829248
\(474\) −1.24617e9 −0.537469
\(475\) 3.00271e9 1.28554
\(476\) −5.37924e8 −0.228611
\(477\) −2.01084e9 −0.848325
\(478\) 3.67776e8 0.154023
\(479\) 1.94687e9 0.809398 0.404699 0.914450i \(-0.367376\pi\)
0.404699 + 0.914450i \(0.367376\pi\)
\(480\) −1.50546e8 −0.0621333
\(481\) −8.69839e7 −0.0356395
\(482\) 1.19733e9 0.487022
\(483\) 6.53436e8 0.263869
\(484\) −9.72022e8 −0.389688
\(485\) −1.82815e8 −0.0727639
\(486\) 2.01307e9 0.795486
\(487\) −1.02425e9 −0.401843 −0.200922 0.979607i \(-0.564394\pi\)
−0.200922 + 0.979607i \(0.564394\pi\)
\(488\) 3.20398e9 1.24802
\(489\) −3.16978e8 −0.122588
\(490\) 5.79821e7 0.0222642
\(491\) −2.79896e9 −1.06711 −0.533557 0.845764i \(-0.679145\pi\)
−0.533557 + 0.845764i \(0.679145\pi\)
\(492\) 3.81482e8 0.144410
\(493\) 7.27223e9 2.73340
\(494\) −7.87060e8 −0.293740
\(495\) −6.05206e8 −0.224277
\(496\) −5.70272e8 −0.209844
\(497\) 1.54017e9 0.562757
\(498\) 1.24530e8 0.0451826
\(499\) 1.60143e9 0.576972 0.288486 0.957484i \(-0.406848\pi\)
0.288486 + 0.957484i \(0.406848\pi\)
\(500\) −4.02198e8 −0.143895
\(501\) −1.84068e9 −0.653953
\(502\) 3.48034e9 1.22789
\(503\) −2.26818e9 −0.794675 −0.397337 0.917673i \(-0.630066\pi\)
−0.397337 + 0.917673i \(0.630066\pi\)
\(504\) 9.41176e8 0.327464
\(505\) −3.12506e8 −0.107979
\(506\) 5.11302e9 1.75449
\(507\) −1.01728e8 −0.0346667
\(508\) 1.76862e8 0.0598564
\(509\) −5.34607e9 −1.79689 −0.898447 0.439083i \(-0.855303\pi\)
−0.898447 + 0.439083i \(0.855303\pi\)
\(510\) 3.41355e8 0.113949
\(511\) 2.70853e8 0.0897968
\(512\) 2.65061e9 0.872773
\(513\) −3.31154e9 −1.08298
\(514\) 3.47720e9 1.12943
\(515\) 5.34391e8 0.172399
\(516\) 3.04046e8 0.0974239
\(517\) −3.09077e9 −0.983670
\(518\) 1.21676e8 0.0384636
\(519\) −6.70936e8 −0.210667
\(520\) 1.90265e8 0.0593400
\(521\) 1.81151e9 0.561190 0.280595 0.959826i \(-0.409468\pi\)
0.280595 + 0.959826i \(0.409468\pi\)
\(522\) −3.45543e9 −1.06330
\(523\) −1.49653e9 −0.457435 −0.228717 0.973493i \(-0.573453\pi\)
−0.228717 + 0.973493i \(0.573453\pi\)
\(524\) −2.07691e9 −0.630605
\(525\) −5.42890e8 −0.163740
\(526\) 8.73630e8 0.261745
\(527\) −2.34312e9 −0.697360
\(528\) −1.06423e9 −0.314642
\(529\) 4.76582e9 1.39973
\(530\) 5.68631e8 0.165907
\(531\) 3.82800e9 1.10953
\(532\) −6.54453e8 −0.188446
\(533\) −8.33328e8 −0.238381
\(534\) −1.49042e9 −0.423560
\(535\) −1.27994e8 −0.0361371
\(536\) 3.39792e9 0.953095
\(537\) 2.30952e8 0.0643594
\(538\) −3.61119e9 −0.999798
\(539\) −7.42738e8 −0.204303
\(540\) 2.17403e8 0.0594137
\(541\) 2.25044e9 0.611052 0.305526 0.952184i \(-0.401168\pi\)
0.305526 + 0.952184i \(0.401168\pi\)
\(542\) −4.56554e8 −0.123167
\(543\) 2.13072e9 0.571119
\(544\) 4.26779e9 1.13660
\(545\) 9.42239e8 0.249329
\(546\) 1.42301e8 0.0374138
\(547\) 4.26901e9 1.11525 0.557624 0.830094i \(-0.311713\pi\)
0.557624 + 0.830094i \(0.311713\pi\)
\(548\) −2.66835e9 −0.692645
\(549\) −3.54664e9 −0.914774
\(550\) −4.24801e9 −1.08872
\(551\) 8.84759e9 2.25317
\(552\) −2.99940e9 −0.759009
\(553\) 2.26355e9 0.569182
\(554\) 7.53910e8 0.188380
\(555\) 4.58980e7 0.0113964
\(556\) −7.71793e8 −0.190432
\(557\) 2.19978e9 0.539369 0.269685 0.962949i \(-0.413081\pi\)
0.269685 + 0.962949i \(0.413081\pi\)
\(558\) 1.11334e9 0.271274
\(559\) −6.64172e8 −0.160820
\(560\) −1.50905e8 −0.0363117
\(561\) −4.37268e9 −1.04563
\(562\) −5.37817e9 −1.27808
\(563\) 4.69544e9 1.10891 0.554456 0.832213i \(-0.312927\pi\)
0.554456 + 0.832213i \(0.312927\pi\)
\(564\) 4.92388e8 0.115566
\(565\) 7.43471e8 0.173418
\(566\) 5.10692e9 1.18386
\(567\) −7.08633e8 −0.163260
\(568\) −7.06966e9 −1.61875
\(569\) 6.31379e9 1.43680 0.718402 0.695629i \(-0.244874\pi\)
0.718402 + 0.695629i \(0.244874\pi\)
\(570\) 4.15301e8 0.0939293
\(571\) 6.12408e9 1.37662 0.688311 0.725416i \(-0.258352\pi\)
0.688311 + 0.725416i \(0.258352\pi\)
\(572\) −6.61889e8 −0.147876
\(573\) −1.06557e8 −0.0236613
\(574\) 1.16569e9 0.257271
\(575\) −6.78836e9 −1.48911
\(576\) −3.81216e9 −0.831175
\(577\) −3.41672e9 −0.740446 −0.370223 0.928943i \(-0.620719\pi\)
−0.370223 + 0.928943i \(0.620719\pi\)
\(578\) −6.00040e9 −1.29251
\(579\) −1.63717e9 −0.350526
\(580\) −5.80845e8 −0.123612
\(581\) −2.26196e8 −0.0478486
\(582\) 6.27611e8 0.131965
\(583\) −7.28404e9 −1.52241
\(584\) −1.24327e9 −0.258297
\(585\) −2.10614e8 −0.0434951
\(586\) −3.15307e9 −0.647280
\(587\) 4.56212e9 0.930964 0.465482 0.885057i \(-0.345881\pi\)
0.465482 + 0.885057i \(0.345881\pi\)
\(588\) 1.18325e8 0.0240025
\(589\) −2.85070e9 −0.574841
\(590\) −1.08249e9 −0.216992
\(591\) 2.65570e9 0.529204
\(592\) −3.16676e8 −0.0627321
\(593\) 7.49087e9 1.47517 0.737583 0.675256i \(-0.235967\pi\)
0.737583 + 0.675256i \(0.235967\pi\)
\(594\) 4.68492e9 0.917169
\(595\) −6.20037e8 −0.120672
\(596\) −2.42291e9 −0.468786
\(597\) 3.26676e9 0.628358
\(598\) 1.77934e9 0.340256
\(599\) 1.23635e9 0.235043 0.117521 0.993070i \(-0.462505\pi\)
0.117521 + 0.993070i \(0.462505\pi\)
\(600\) 2.49197e9 0.470991
\(601\) −5.00324e9 −0.940135 −0.470068 0.882630i \(-0.655770\pi\)
−0.470068 + 0.882630i \(0.655770\pi\)
\(602\) 9.29066e8 0.173564
\(603\) −3.76132e9 −0.698601
\(604\) 3.64783e9 0.673605
\(605\) −1.12040e9 −0.205697
\(606\) 1.07284e9 0.195832
\(607\) −2.75374e9 −0.499761 −0.249881 0.968277i \(-0.580391\pi\)
−0.249881 + 0.968277i \(0.580391\pi\)
\(608\) 5.19230e9 0.936909
\(609\) −1.59964e9 −0.286987
\(610\) 1.00293e9 0.178903
\(611\) −1.07560e9 −0.190768
\(612\) −2.73325e9 −0.482002
\(613\) 5.52898e9 0.969468 0.484734 0.874662i \(-0.338916\pi\)
0.484734 + 0.874662i \(0.338916\pi\)
\(614\) −5.78238e9 −1.00813
\(615\) 4.39715e8 0.0762269
\(616\) 3.40931e9 0.587670
\(617\) −7.43691e8 −0.127466 −0.0637330 0.997967i \(-0.520301\pi\)
−0.0637330 + 0.997967i \(0.520301\pi\)
\(618\) −1.83458e9 −0.312664
\(619\) −2.46617e9 −0.417932 −0.208966 0.977923i \(-0.567010\pi\)
−0.208966 + 0.977923i \(0.567010\pi\)
\(620\) 1.87149e8 0.0315367
\(621\) 7.48654e9 1.25447
\(622\) −3.40170e9 −0.566800
\(623\) 2.70720e9 0.448552
\(624\) −3.70354e8 −0.0610199
\(625\) 5.40355e9 0.885318
\(626\) 6.04986e9 0.985678
\(627\) −5.31991e9 −0.861922
\(628\) 3.33349e9 0.537081
\(629\) −1.30115e9 −0.208473
\(630\) 2.94613e8 0.0469418
\(631\) 3.59113e9 0.569020 0.284510 0.958673i \(-0.408169\pi\)
0.284510 + 0.958673i \(0.408169\pi\)
\(632\) −1.03901e10 −1.63723
\(633\) −2.26332e9 −0.354677
\(634\) −6.65586e9 −1.03727
\(635\) 2.03859e8 0.0315953
\(636\) 1.16041e9 0.178860
\(637\) −2.58475e8 −0.0396214
\(638\) −1.25169e10 −1.90820
\(639\) 7.82575e9 1.18651
\(640\) 1.63696e8 0.0246835
\(641\) −1.01250e10 −1.51841 −0.759207 0.650849i \(-0.774413\pi\)
−0.759207 + 0.650849i \(0.774413\pi\)
\(642\) 4.39409e8 0.0655385
\(643\) −3.41022e9 −0.505876 −0.252938 0.967483i \(-0.581397\pi\)
−0.252938 + 0.967483i \(0.581397\pi\)
\(644\) 1.47955e9 0.218288
\(645\) 3.50458e8 0.0514253
\(646\) −1.17733e10 −1.71824
\(647\) 7.62526e9 1.10685 0.553426 0.832898i \(-0.313320\pi\)
0.553426 + 0.832898i \(0.313320\pi\)
\(648\) 3.25276e9 0.469612
\(649\) 1.38665e10 1.99118
\(650\) −1.47832e9 −0.211140
\(651\) 5.15407e8 0.0732178
\(652\) −7.17721e8 −0.101412
\(653\) −8.05772e8 −0.113244 −0.0566221 0.998396i \(-0.518033\pi\)
−0.0566221 + 0.998396i \(0.518033\pi\)
\(654\) −3.23474e9 −0.452186
\(655\) −2.39394e9 −0.332866
\(656\) −3.03384e9 −0.419594
\(657\) 1.37623e9 0.189327
\(658\) 1.50458e9 0.205885
\(659\) −1.78678e8 −0.0243205 −0.0121602 0.999926i \(-0.503871\pi\)
−0.0121602 + 0.999926i \(0.503871\pi\)
\(660\) 3.49253e8 0.0472864
\(661\) 7.73824e9 1.04217 0.521083 0.853506i \(-0.325528\pi\)
0.521083 + 0.853506i \(0.325528\pi\)
\(662\) 4.05205e9 0.542839
\(663\) −1.52170e9 −0.202784
\(664\) 1.03828e9 0.137635
\(665\) −7.54353e8 −0.0994716
\(666\) 6.18247e8 0.0810966
\(667\) −2.00021e10 −2.60997
\(668\) −4.16779e9 −0.540989
\(669\) 1.14496e9 0.147842
\(670\) 1.06364e9 0.136626
\(671\) −1.28473e10 −1.64166
\(672\) −9.38769e8 −0.119335
\(673\) 2.14358e9 0.271073 0.135537 0.990772i \(-0.456724\pi\)
0.135537 + 0.990772i \(0.456724\pi\)
\(674\) 6.22995e8 0.0783746
\(675\) −6.21999e9 −0.778443
\(676\) −2.30339e8 −0.0286784
\(677\) −5.36251e8 −0.0664214 −0.0332107 0.999448i \(-0.510573\pi\)
−0.0332107 + 0.999448i \(0.510573\pi\)
\(678\) −2.55236e9 −0.314513
\(679\) −1.13999e9 −0.139752
\(680\) 2.84609e9 0.347110
\(681\) 5.10863e9 0.619854
\(682\) 4.03296e9 0.486832
\(683\) 1.21466e10 1.45875 0.729377 0.684112i \(-0.239810\pi\)
0.729377 + 0.684112i \(0.239810\pi\)
\(684\) −3.32534e9 −0.397319
\(685\) −3.07567e9 −0.365614
\(686\) 3.61563e8 0.0427612
\(687\) 2.15748e9 0.253863
\(688\) −2.41801e9 −0.283073
\(689\) −2.53487e9 −0.295248
\(690\) −9.38890e8 −0.108804
\(691\) −1.22397e10 −1.41123 −0.705615 0.708595i \(-0.749329\pi\)
−0.705615 + 0.708595i \(0.749329\pi\)
\(692\) −1.51918e9 −0.174276
\(693\) −3.77393e9 −0.430752
\(694\) 9.14690e9 1.03876
\(695\) −8.89605e8 −0.100520
\(696\) 7.34267e9 0.825508
\(697\) −1.24654e10 −1.39441
\(698\) −5.10649e9 −0.568367
\(699\) −4.81884e9 −0.533670
\(700\) −1.22924e9 −0.135455
\(701\) 1.38399e10 1.51746 0.758732 0.651403i \(-0.225819\pi\)
0.758732 + 0.651403i \(0.225819\pi\)
\(702\) 1.63036e9 0.177871
\(703\) −1.58302e9 −0.171847
\(704\) −1.38091e10 −1.49163
\(705\) 5.67550e8 0.0610017
\(706\) 5.30899e8 0.0567800
\(707\) −1.94872e9 −0.207387
\(708\) −2.20906e9 −0.233933
\(709\) 9.16180e9 0.965426 0.482713 0.875779i \(-0.339651\pi\)
0.482713 + 0.875779i \(0.339651\pi\)
\(710\) −2.21299e9 −0.232047
\(711\) 1.15013e10 1.20006
\(712\) −1.24266e10 −1.29024
\(713\) 6.44471e9 0.665871
\(714\) 2.12861e9 0.218853
\(715\) −7.62925e8 −0.0780568
\(716\) 5.22937e8 0.0532419
\(717\) 8.65092e8 0.0876486
\(718\) 2.24293e9 0.226141
\(719\) 1.05119e10 1.05470 0.527352 0.849647i \(-0.323185\pi\)
0.527352 + 0.849647i \(0.323185\pi\)
\(720\) −7.66766e8 −0.0765595
\(721\) 3.33234e9 0.331112
\(722\) −6.31470e9 −0.624414
\(723\) 2.81639e9 0.277146
\(724\) 4.82451e9 0.472463
\(725\) 1.66182e10 1.61958
\(726\) 3.84637e9 0.373055
\(727\) 4.67722e9 0.451458 0.225729 0.974190i \(-0.427524\pi\)
0.225729 + 0.974190i \(0.427524\pi\)
\(728\) 1.18645e9 0.113970
\(729\) 2.16898e8 0.0207352
\(730\) −3.89176e8 −0.0370267
\(731\) −9.93505e9 −0.940717
\(732\) 2.04670e9 0.192870
\(733\) 4.12754e9 0.387103 0.193552 0.981090i \(-0.437999\pi\)
0.193552 + 0.981090i \(0.437999\pi\)
\(734\) 1.16066e9 0.108335
\(735\) 1.36387e8 0.0126697
\(736\) −1.17385e10 −1.08527
\(737\) −1.36250e10 −1.25372
\(738\) 5.92297e9 0.542428
\(739\) 1.29359e9 0.117907 0.0589535 0.998261i \(-0.481224\pi\)
0.0589535 + 0.998261i \(0.481224\pi\)
\(740\) 1.03925e8 0.00942779
\(741\) −1.85134e9 −0.167157
\(742\) 3.54585e9 0.318645
\(743\) −9.59715e9 −0.858383 −0.429192 0.903213i \(-0.641202\pi\)
−0.429192 + 0.903213i \(0.641202\pi\)
\(744\) −2.36581e9 −0.210608
\(745\) −2.79276e9 −0.247449
\(746\) 8.90755e9 0.785547
\(747\) −1.14933e9 −0.100884
\(748\) −9.90089e9 −0.865005
\(749\) −7.98143e8 −0.0694056
\(750\) 1.59153e9 0.137753
\(751\) −1.95130e10 −1.68106 −0.840532 0.541762i \(-0.817757\pi\)
−0.840532 + 0.541762i \(0.817757\pi\)
\(752\) −3.91585e9 −0.335787
\(753\) 8.18655e9 0.698744
\(754\) −4.35592e9 −0.370067
\(755\) 4.20466e9 0.355563
\(756\) 1.35567e9 0.114111
\(757\) 1.68500e10 1.41177 0.705886 0.708325i \(-0.250549\pi\)
0.705886 + 0.708325i \(0.250549\pi\)
\(758\) −7.27282e9 −0.606542
\(759\) 1.20270e10 0.998413
\(760\) 3.46262e9 0.286126
\(761\) 1.27730e10 1.05062 0.525310 0.850911i \(-0.323949\pi\)
0.525310 + 0.850911i \(0.323949\pi\)
\(762\) −6.99855e8 −0.0573015
\(763\) 5.87559e9 0.478867
\(764\) −2.41272e8 −0.0195741
\(765\) −3.15047e9 −0.254425
\(766\) 9.46059e9 0.760532
\(767\) 4.82558e9 0.386159
\(768\) 5.33881e9 0.425285
\(769\) 1.00489e9 0.0796851 0.0398426 0.999206i \(-0.487314\pi\)
0.0398426 + 0.999206i \(0.487314\pi\)
\(770\) 1.06720e9 0.0842422
\(771\) 8.17917e9 0.642715
\(772\) −3.70699e9 −0.289976
\(773\) −1.07528e10 −0.837323 −0.418661 0.908142i \(-0.637501\pi\)
−0.418661 + 0.908142i \(0.637501\pi\)
\(774\) 4.72068e9 0.365941
\(775\) −5.35441e9 −0.413196
\(776\) 5.23278e9 0.401991
\(777\) 2.86209e8 0.0218882
\(778\) −2.60203e9 −0.198100
\(779\) −1.51657e10 −1.14943
\(780\) 1.21541e8 0.00917047
\(781\) 2.83479e10 2.12933
\(782\) 2.66164e10 1.99033
\(783\) −1.83274e10 −1.36438
\(784\) −9.41011e8 −0.0697411
\(785\) 3.84234e9 0.283499
\(786\) 8.21848e9 0.603688
\(787\) 2.22554e10 1.62751 0.813755 0.581208i \(-0.197420\pi\)
0.813755 + 0.581208i \(0.197420\pi\)
\(788\) 6.01321e9 0.437789
\(789\) 2.05498e9 0.148949
\(790\) −3.25238e9 −0.234696
\(791\) 4.63611e9 0.333071
\(792\) 1.73230e10 1.23904
\(793\) −4.47090e9 −0.318375
\(794\) −4.56152e9 −0.323398
\(795\) 1.33755e9 0.0944115
\(796\) 7.39680e9 0.519814
\(797\) −2.35507e10 −1.64778 −0.823892 0.566746i \(-0.808202\pi\)
−0.823892 + 0.566746i \(0.808202\pi\)
\(798\) 2.58972e9 0.180403
\(799\) −1.60893e10 −1.11590
\(800\) 9.75260e9 0.673450
\(801\) 1.37556e10 0.945725
\(802\) −1.21979e10 −0.834976
\(803\) 4.98525e9 0.339768
\(804\) 2.17058e9 0.147292
\(805\) 1.70540e9 0.115223
\(806\) 1.40348e9 0.0944135
\(807\) −8.49434e9 −0.568947
\(808\) 8.94498e9 0.596540
\(809\) 1.02800e9 0.0682611 0.0341306 0.999417i \(-0.489134\pi\)
0.0341306 + 0.999417i \(0.489134\pi\)
\(810\) 1.01820e9 0.0673186
\(811\) −1.89814e9 −0.124956 −0.0624778 0.998046i \(-0.519900\pi\)
−0.0624778 + 0.998046i \(0.519900\pi\)
\(812\) −3.62201e9 −0.237413
\(813\) −1.07392e9 −0.0700897
\(814\) 2.23953e9 0.145537
\(815\) −8.27280e8 −0.0535304
\(816\) −5.53996e9 −0.356937
\(817\) −1.20872e10 −0.775443
\(818\) 1.85570e10 1.18542
\(819\) −1.31334e9 −0.0835377
\(820\) 9.95630e8 0.0630593
\(821\) −2.48565e9 −0.156762 −0.0783808 0.996923i \(-0.524975\pi\)
−0.0783808 + 0.996923i \(0.524975\pi\)
\(822\) 1.05589e10 0.663080
\(823\) 5.91392e9 0.369808 0.184904 0.982757i \(-0.440803\pi\)
0.184904 + 0.982757i \(0.440803\pi\)
\(824\) −1.52960e10 −0.952432
\(825\) −9.99228e9 −0.619549
\(826\) −6.75018e9 −0.416759
\(827\) −8.94815e9 −0.550128 −0.275064 0.961426i \(-0.588699\pi\)
−0.275064 + 0.961426i \(0.588699\pi\)
\(828\) 7.51775e9 0.460237
\(829\) 2.96249e10 1.80599 0.902996 0.429648i \(-0.141362\pi\)
0.902996 + 0.429648i \(0.141362\pi\)
\(830\) 3.25010e8 0.0197298
\(831\) 1.77337e9 0.107200
\(832\) −4.80562e9 −0.289279
\(833\) −3.86641e9 −0.231766
\(834\) 3.05404e9 0.182303
\(835\) −4.80399e9 −0.285562
\(836\) −1.20457e10 −0.713033
\(837\) 5.90511e9 0.348088
\(838\) 2.30107e10 1.35075
\(839\) −3.03954e10 −1.77681 −0.888404 0.459063i \(-0.848185\pi\)
−0.888404 + 0.459063i \(0.848185\pi\)
\(840\) −6.26043e8 −0.0364440
\(841\) 3.17164e10 1.83864
\(842\) 4.56207e9 0.263372
\(843\) −1.26507e10 −0.727306
\(844\) −5.12475e9 −0.293410
\(845\) −2.65500e8 −0.0151379
\(846\) 7.64491e9 0.434086
\(847\) −6.98654e9 −0.395067
\(848\) −9.22851e9 −0.519692
\(849\) 1.20126e10 0.673692
\(850\) −2.21135e10 −1.23507
\(851\) 3.57880e9 0.199060
\(852\) −4.51609e9 −0.250164
\(853\) 3.02337e10 1.66790 0.833950 0.551840i \(-0.186074\pi\)
0.833950 + 0.551840i \(0.186074\pi\)
\(854\) 6.25404e9 0.343604
\(855\) −3.83294e9 −0.209725
\(856\) 3.66363e9 0.199643
\(857\) 6.83987e9 0.371206 0.185603 0.982625i \(-0.440576\pi\)
0.185603 + 0.982625i \(0.440576\pi\)
\(858\) 2.61915e9 0.141564
\(859\) 2.91105e9 0.156702 0.0783509 0.996926i \(-0.475035\pi\)
0.0783509 + 0.996926i \(0.475035\pi\)
\(860\) 7.93529e8 0.0425420
\(861\) 2.74196e9 0.146403
\(862\) −2.63821e10 −1.40292
\(863\) −2.34751e10 −1.24328 −0.621641 0.783303i \(-0.713534\pi\)
−0.621641 + 0.783303i \(0.713534\pi\)
\(864\) −1.07557e10 −0.567334
\(865\) −1.75108e9 −0.0919916
\(866\) −7.54004e9 −0.394513
\(867\) −1.41143e10 −0.735517
\(868\) 1.16702e9 0.0605700
\(869\) 4.16623e10 2.15364
\(870\) 2.29845e9 0.118336
\(871\) −4.74153e9 −0.243139
\(872\) −2.69700e10 −1.37744
\(873\) −5.79242e9 −0.294652
\(874\) 3.23822e10 1.64065
\(875\) −2.89086e9 −0.145881
\(876\) −7.94197e8 −0.0399175
\(877\) −2.17287e10 −1.08777 −0.543883 0.839161i \(-0.683046\pi\)
−0.543883 + 0.839161i \(0.683046\pi\)
\(878\) 2.79080e10 1.39155
\(879\) −7.41674e9 −0.368343
\(880\) −2.77753e9 −0.137394
\(881\) −2.61248e10 −1.28717 −0.643586 0.765374i \(-0.722554\pi\)
−0.643586 + 0.765374i \(0.722554\pi\)
\(882\) 1.83714e9 0.0901574
\(883\) −4.27316e9 −0.208875 −0.104437 0.994531i \(-0.533304\pi\)
−0.104437 + 0.994531i \(0.533304\pi\)
\(884\) −3.44554e9 −0.167754
\(885\) −2.54627e9 −0.123482
\(886\) −1.26555e10 −0.611309
\(887\) −2.94661e10 −1.41772 −0.708859 0.705350i \(-0.750790\pi\)
−0.708859 + 0.705350i \(0.750790\pi\)
\(888\) −1.31376e9 −0.0629606
\(889\) 1.27122e9 0.0606826
\(890\) −3.88985e9 −0.184956
\(891\) −1.30429e10 −0.617736
\(892\) 2.59249e9 0.122304
\(893\) −1.95747e10 −0.919847
\(894\) 9.58763e9 0.448776
\(895\) 6.02762e8 0.0281038
\(896\) 1.02077e9 0.0474077
\(897\) 4.18542e9 0.193627
\(898\) −8.75082e9 −0.403257
\(899\) −1.57770e10 −0.724210
\(900\) −6.24592e9 −0.285593
\(901\) −3.79179e10 −1.72706
\(902\) 2.14553e10 0.973447
\(903\) 2.18537e9 0.0987685
\(904\) −2.12806e10 −0.958065
\(905\) 5.56096e9 0.249390
\(906\) −1.44348e10 −0.644853
\(907\) −5.35206e9 −0.238175 −0.119087 0.992884i \(-0.537997\pi\)
−0.119087 + 0.992884i \(0.537997\pi\)
\(908\) 1.15673e10 0.512780
\(909\) −9.90162e9 −0.437253
\(910\) 3.71390e8 0.0163375
\(911\) −9.37687e9 −0.410907 −0.205454 0.978667i \(-0.565867\pi\)
−0.205454 + 0.978667i \(0.565867\pi\)
\(912\) −6.74006e9 −0.294227
\(913\) −4.16331e9 −0.181047
\(914\) 1.02074e9 0.0442186
\(915\) 2.35912e9 0.101807
\(916\) 4.88511e9 0.210010
\(917\) −1.49281e10 −0.639309
\(918\) 2.43879e10 1.04046
\(919\) −8.37753e9 −0.356051 −0.178025 0.984026i \(-0.556971\pi\)
−0.178025 + 0.984026i \(0.556971\pi\)
\(920\) −7.82811e9 −0.331436
\(921\) −1.36015e10 −0.573690
\(922\) −3.34258e10 −1.40451
\(923\) 9.86516e9 0.412951
\(924\) 2.17786e9 0.0908193
\(925\) −2.97335e9 −0.123523
\(926\) −1.96126e10 −0.811704
\(927\) 1.69319e10 0.698116
\(928\) 2.87364e10 1.18036
\(929\) −7.25975e9 −0.297075 −0.148538 0.988907i \(-0.547457\pi\)
−0.148538 + 0.988907i \(0.547457\pi\)
\(930\) −7.40563e8 −0.0301906
\(931\) −4.70397e9 −0.191047
\(932\) −1.09111e10 −0.441483
\(933\) −8.00157e9 −0.322544
\(934\) 1.80493e10 0.724849
\(935\) −1.14122e10 −0.456594
\(936\) 6.02846e9 0.240293
\(937\) −7.83263e9 −0.311042 −0.155521 0.987833i \(-0.549706\pi\)
−0.155521 + 0.987833i \(0.549706\pi\)
\(938\) 6.63260e9 0.262406
\(939\) 1.42306e10 0.560912
\(940\) 1.28508e9 0.0504642
\(941\) 5.59550e9 0.218915 0.109457 0.993991i \(-0.465089\pi\)
0.109457 + 0.993991i \(0.465089\pi\)
\(942\) −1.31909e10 −0.514157
\(943\) 3.42858e10 1.33145
\(944\) 1.75682e10 0.679711
\(945\) 1.56261e9 0.0602338
\(946\) 1.71001e10 0.656721
\(947\) 2.79447e10 1.06924 0.534620 0.845093i \(-0.320455\pi\)
0.534620 + 0.845093i \(0.320455\pi\)
\(948\) −6.63718e9 −0.253020
\(949\) 1.73488e9 0.0658928
\(950\) −2.69039e10 −1.01808
\(951\) −1.56561e10 −0.590271
\(952\) 1.77475e10 0.666667
\(953\) −1.05088e10 −0.393304 −0.196652 0.980473i \(-0.563007\pi\)
−0.196652 + 0.980473i \(0.563007\pi\)
\(954\) 1.80168e10 0.671829
\(955\) −2.78102e8 −0.0103322
\(956\) 1.95880e9 0.0725081
\(957\) −2.94426e10 −1.08589
\(958\) −1.74437e10 −0.641001
\(959\) −1.91791e10 −0.702205
\(960\) 2.53574e9 0.0925028
\(961\) −2.24293e10 −0.815236
\(962\) 7.79364e8 0.0282246
\(963\) −4.05545e9 −0.146334
\(964\) 6.37704e9 0.229271
\(965\) −4.27286e9 −0.153064
\(966\) −5.85470e9 −0.208970
\(967\) −1.23602e10 −0.439575 −0.219788 0.975548i \(-0.570536\pi\)
−0.219788 + 0.975548i \(0.570536\pi\)
\(968\) 3.20696e10 1.13639
\(969\) −2.76934e10 −0.977784
\(970\) 1.63800e9 0.0576252
\(971\) 1.64550e10 0.576806 0.288403 0.957509i \(-0.406876\pi\)
0.288403 + 0.957509i \(0.406876\pi\)
\(972\) 1.07218e10 0.374485
\(973\) −5.54737e9 −0.193060
\(974\) 9.17718e9 0.318239
\(975\) −3.47734e9 −0.120152
\(976\) −1.62769e10 −0.560399
\(977\) 1.77865e10 0.610182 0.305091 0.952323i \(-0.401313\pi\)
0.305091 + 0.952323i \(0.401313\pi\)
\(978\) 2.84008e9 0.0970833
\(979\) 4.98281e10 1.69721
\(980\) 3.08816e8 0.0104811
\(981\) 2.98544e10 1.00964
\(982\) 2.50783e10 0.845099
\(983\) 8.74911e9 0.293783 0.146892 0.989153i \(-0.453073\pi\)
0.146892 + 0.989153i \(0.453073\pi\)
\(984\) −1.25861e10 −0.421123
\(985\) 6.93111e9 0.231087
\(986\) −6.51582e10 −2.16471
\(987\) 3.53911e9 0.117161
\(988\) −4.19193e9 −0.138282
\(989\) 2.73262e10 0.898239
\(990\) 5.42257e9 0.177616
\(991\) −3.28407e10 −1.07190 −0.535951 0.844249i \(-0.680047\pi\)
−0.535951 + 0.844249i \(0.680047\pi\)
\(992\) −9.25889e9 −0.301140
\(993\) 9.53133e9 0.308909
\(994\) −1.37997e10 −0.445674
\(995\) 8.52590e9 0.274385
\(996\) 6.63254e8 0.0212702
\(997\) 2.19468e10 0.701356 0.350678 0.936496i \(-0.385951\pi\)
0.350678 + 0.936496i \(0.385951\pi\)
\(998\) −1.43486e10 −0.456932
\(999\) 3.27916e9 0.104060
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 91.8.a.d.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.d.1.4 10 1.1 even 1 trivial