Properties

Label 91.8.a.c.1.5
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.80018\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.80018 q^{2} -76.5407 q^{3} -113.559 q^{4} -339.464 q^{5} +290.868 q^{6} +343.000 q^{7} +917.967 q^{8} +3671.47 q^{9} +O(q^{10})\) \(q-3.80018 q^{2} -76.5407 q^{3} -113.559 q^{4} -339.464 q^{5} +290.868 q^{6} +343.000 q^{7} +917.967 q^{8} +3671.47 q^{9} +1290.03 q^{10} +3399.92 q^{11} +8691.85 q^{12} -2197.00 q^{13} -1303.46 q^{14} +25982.8 q^{15} +11047.1 q^{16} +8096.39 q^{17} -13952.3 q^{18} -49643.5 q^{19} +38549.1 q^{20} -26253.4 q^{21} -12920.3 q^{22} +20612.4 q^{23} -70261.8 q^{24} +37111.1 q^{25} +8349.00 q^{26} -113623. q^{27} -38950.6 q^{28} +138539. q^{29} -98739.5 q^{30} +92549.8 q^{31} -159481. q^{32} -260232. q^{33} -30767.8 q^{34} -116436. q^{35} -416927. q^{36} +40464.8 q^{37} +188654. q^{38} +168160. q^{39} -311617. q^{40} +52424.4 q^{41} +99767.9 q^{42} +696019. q^{43} -386090. q^{44} -1.24633e6 q^{45} -78330.9 q^{46} +77394.2 q^{47} -845550. q^{48} +117649. q^{49} -141029. q^{50} -619703. q^{51} +249488. q^{52} -682352. q^{53} +431786. q^{54} -1.15415e6 q^{55} +314863. q^{56} +3.79975e6 q^{57} -526472. q^{58} -1.73469e6 q^{59} -2.95057e6 q^{60} +2.70508e6 q^{61} -351706. q^{62} +1.25932e6 q^{63} -807969. q^{64} +745803. q^{65} +988929. q^{66} -4.71791e6 q^{67} -919415. q^{68} -1.57769e6 q^{69} +442479. q^{70} +2.79941e6 q^{71} +3.37029e6 q^{72} -3.16613e6 q^{73} -153774. q^{74} -2.84051e6 q^{75} +5.63745e6 q^{76} +1.16617e6 q^{77} -639038. q^{78} +2.13983e6 q^{79} -3.75008e6 q^{80} +667235. q^{81} -199222. q^{82} +5.08233e6 q^{83} +2.98131e6 q^{84} -2.74844e6 q^{85} -2.64500e6 q^{86} -1.06038e7 q^{87} +3.12101e6 q^{88} -1.26416e7 q^{89} +4.73630e6 q^{90} -753571. q^{91} -2.34072e6 q^{92} -7.08382e6 q^{93} -294112. q^{94} +1.68522e7 q^{95} +1.22067e7 q^{96} +1.03648e7 q^{97} -447088. q^{98} +1.24827e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.80018 −0.335892 −0.167946 0.985796i \(-0.553713\pi\)
−0.167946 + 0.985796i \(0.553713\pi\)
\(3\) −76.5407 −1.63670 −0.818348 0.574724i \(-0.805110\pi\)
−0.818348 + 0.574724i \(0.805110\pi\)
\(4\) −113.559 −0.887177
\(5\) −339.464 −1.21450 −0.607252 0.794509i \(-0.707728\pi\)
−0.607252 + 0.794509i \(0.707728\pi\)
\(6\) 290.868 0.549753
\(7\) 343.000 0.377964
\(8\) 917.967 0.633887
\(9\) 3671.47 1.67877
\(10\) 1290.03 0.407942
\(11\) 3399.92 0.770184 0.385092 0.922878i \(-0.374170\pi\)
0.385092 + 0.922878i \(0.374170\pi\)
\(12\) 8691.85 1.45204
\(13\) −2197.00 −0.277350
\(14\) −1303.46 −0.126955
\(15\) 25982.8 1.98777
\(16\) 11047.1 0.674259
\(17\) 8096.39 0.399687 0.199844 0.979828i \(-0.435957\pi\)
0.199844 + 0.979828i \(0.435957\pi\)
\(18\) −13952.3 −0.563886
\(19\) −49643.5 −1.66045 −0.830223 0.557431i \(-0.811787\pi\)
−0.830223 + 0.557431i \(0.811787\pi\)
\(20\) 38549.1 1.07748
\(21\) −26253.4 −0.618613
\(22\) −12920.3 −0.258698
\(23\) 20612.4 0.353249 0.176625 0.984278i \(-0.443482\pi\)
0.176625 + 0.984278i \(0.443482\pi\)
\(24\) −70261.8 −1.03748
\(25\) 37111.1 0.475022
\(26\) 8349.00 0.0931596
\(27\) −113623. −1.11094
\(28\) −38950.6 −0.335321
\(29\) 138539. 1.05482 0.527409 0.849611i \(-0.323163\pi\)
0.527409 + 0.849611i \(0.323163\pi\)
\(30\) −98739.5 −0.667677
\(31\) 92549.8 0.557968 0.278984 0.960296i \(-0.410002\pi\)
0.278984 + 0.960296i \(0.410002\pi\)
\(32\) −159481. −0.860365
\(33\) −260232. −1.26056
\(34\) −30767.8 −0.134252
\(35\) −116436. −0.459040
\(36\) −416927. −1.48937
\(37\) 40464.8 0.131332 0.0656662 0.997842i \(-0.479083\pi\)
0.0656662 + 0.997842i \(0.479083\pi\)
\(38\) 188654. 0.557730
\(39\) 168160. 0.453938
\(40\) −311617. −0.769859
\(41\) 52424.4 0.118793 0.0593963 0.998234i \(-0.481082\pi\)
0.0593963 + 0.998234i \(0.481082\pi\)
\(42\) 99767.9 0.207787
\(43\) 696019. 1.33500 0.667501 0.744609i \(-0.267364\pi\)
0.667501 + 0.744609i \(0.267364\pi\)
\(44\) −386090. −0.683289
\(45\) −1.24633e6 −2.03888
\(46\) −78330.9 −0.118654
\(47\) 77394.2 0.108734 0.0543670 0.998521i \(-0.482686\pi\)
0.0543670 + 0.998521i \(0.482686\pi\)
\(48\) −845550. −1.10356
\(49\) 117649. 0.142857
\(50\) −141029. −0.159556
\(51\) −619703. −0.654166
\(52\) 249488. 0.246059
\(53\) −682352. −0.629568 −0.314784 0.949163i \(-0.601932\pi\)
−0.314784 + 0.949163i \(0.601932\pi\)
\(54\) 431786. 0.373156
\(55\) −1.15415e6 −0.935392
\(56\) 314863. 0.239587
\(57\) 3.79975e6 2.71765
\(58\) −526472. −0.354305
\(59\) −1.73469e6 −1.09961 −0.549807 0.835292i \(-0.685299\pi\)
−0.549807 + 0.835292i \(0.685299\pi\)
\(60\) −2.95057e6 −1.76351
\(61\) 2.70508e6 1.52590 0.762950 0.646458i \(-0.223750\pi\)
0.762950 + 0.646458i \(0.223750\pi\)
\(62\) −351706. −0.187417
\(63\) 1.25932e6 0.634516
\(64\) −807969. −0.385270
\(65\) 745803. 0.336843
\(66\) 988929. 0.423410
\(67\) −4.71791e6 −1.91641 −0.958203 0.286090i \(-0.907644\pi\)
−0.958203 + 0.286090i \(0.907644\pi\)
\(68\) −919415. −0.354593
\(69\) −1.57769e6 −0.578162
\(70\) 442479. 0.154188
\(71\) 2.79941e6 0.928245 0.464122 0.885771i \(-0.346370\pi\)
0.464122 + 0.885771i \(0.346370\pi\)
\(72\) 3.37029e6 1.06415
\(73\) −3.16613e6 −0.952573 −0.476287 0.879290i \(-0.658018\pi\)
−0.476287 + 0.879290i \(0.658018\pi\)
\(74\) −153774. −0.0441134
\(75\) −2.84051e6 −0.777466
\(76\) 5.63745e6 1.47311
\(77\) 1.16617e6 0.291102
\(78\) −639038. −0.152474
\(79\) 2.13983e6 0.488298 0.244149 0.969738i \(-0.421491\pi\)
0.244149 + 0.969738i \(0.421491\pi\)
\(80\) −3.75008e6 −0.818891
\(81\) 667235. 0.139502
\(82\) −199222. −0.0399015
\(83\) 5.08233e6 0.975640 0.487820 0.872944i \(-0.337792\pi\)
0.487820 + 0.872944i \(0.337792\pi\)
\(84\) 2.98131e6 0.548819
\(85\) −2.74844e6 −0.485422
\(86\) −2.64500e6 −0.448416
\(87\) −1.06038e7 −1.72642
\(88\) 3.12101e6 0.488210
\(89\) −1.26416e7 −1.90081 −0.950403 0.311020i \(-0.899329\pi\)
−0.950403 + 0.311020i \(0.899329\pi\)
\(90\) 4.73630e6 0.684842
\(91\) −753571. −0.104828
\(92\) −2.34072e6 −0.313395
\(93\) −7.08382e6 −0.913224
\(94\) −294112. −0.0365229
\(95\) 1.68522e7 2.01662
\(96\) 1.22067e7 1.40816
\(97\) 1.03648e7 1.15308 0.576541 0.817069i \(-0.304402\pi\)
0.576541 + 0.817069i \(0.304402\pi\)
\(98\) −447088. −0.0479845
\(99\) 1.24827e7 1.29296
\(100\) −4.21428e6 −0.421428
\(101\) 4.83651e6 0.467097 0.233549 0.972345i \(-0.424966\pi\)
0.233549 + 0.972345i \(0.424966\pi\)
\(102\) 2.35499e6 0.219729
\(103\) 1.25305e7 1.12990 0.564948 0.825126i \(-0.308896\pi\)
0.564948 + 0.825126i \(0.308896\pi\)
\(104\) −2.01677e6 −0.175809
\(105\) 8.91211e6 0.751308
\(106\) 2.59306e6 0.211467
\(107\) −2.03751e6 −0.160789 −0.0803946 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(108\) 1.29028e7 0.985602
\(109\) 1.59876e7 1.18247 0.591235 0.806499i \(-0.298641\pi\)
0.591235 + 0.806499i \(0.298641\pi\)
\(110\) 4.38599e6 0.314190
\(111\) −3.09720e6 −0.214951
\(112\) 3.78914e6 0.254846
\(113\) 3.34020e6 0.217770 0.108885 0.994054i \(-0.465272\pi\)
0.108885 + 0.994054i \(0.465272\pi\)
\(114\) −1.44397e7 −0.912835
\(115\) −6.99718e6 −0.429023
\(116\) −1.57322e7 −0.935810
\(117\) −8.06623e6 −0.465607
\(118\) 6.59214e6 0.369351
\(119\) 2.77706e6 0.151068
\(120\) 2.38514e7 1.26002
\(121\) −7.92771e6 −0.406817
\(122\) −1.02798e7 −0.512537
\(123\) −4.01259e6 −0.194427
\(124\) −1.05098e7 −0.495016
\(125\) 1.39228e7 0.637588
\(126\) −4.78563e6 −0.213129
\(127\) −4.66494e6 −0.202085 −0.101042 0.994882i \(-0.532218\pi\)
−0.101042 + 0.994882i \(0.532218\pi\)
\(128\) 2.34839e7 0.989774
\(129\) −5.32738e7 −2.18499
\(130\) −2.83419e6 −0.113143
\(131\) −5.09818e7 −1.98137 −0.990684 0.136177i \(-0.956518\pi\)
−0.990684 + 0.136177i \(0.956518\pi\)
\(132\) 2.95516e7 1.11834
\(133\) −1.70277e7 −0.627590
\(134\) 1.79289e7 0.643705
\(135\) 3.85708e7 1.34924
\(136\) 7.43222e6 0.253357
\(137\) 5.34985e6 0.177754 0.0888771 0.996043i \(-0.471672\pi\)
0.0888771 + 0.996043i \(0.471672\pi\)
\(138\) 5.99550e6 0.194200
\(139\) −5.72070e7 −1.80675 −0.903374 0.428854i \(-0.858917\pi\)
−0.903374 + 0.428854i \(0.858917\pi\)
\(140\) 1.32223e7 0.407249
\(141\) −5.92380e6 −0.177965
\(142\) −1.06383e7 −0.311790
\(143\) −7.46962e6 −0.213611
\(144\) 4.05590e7 1.13193
\(145\) −4.70289e7 −1.28108
\(146\) 1.20319e7 0.319962
\(147\) −9.00493e6 −0.233814
\(148\) −4.59513e6 −0.116515
\(149\) −5.34532e7 −1.32380 −0.661899 0.749593i \(-0.730249\pi\)
−0.661899 + 0.749593i \(0.730249\pi\)
\(150\) 1.07944e7 0.261144
\(151\) −53852.3 −0.00127287 −0.000636436 1.00000i \(-0.500203\pi\)
−0.000636436 1.00000i \(0.500203\pi\)
\(152\) −4.55711e7 −1.05254
\(153\) 2.97257e7 0.670984
\(154\) −4.43167e6 −0.0977788
\(155\) −3.14173e7 −0.677655
\(156\) −1.90960e7 −0.402723
\(157\) −7.46784e7 −1.54009 −0.770045 0.637989i \(-0.779766\pi\)
−0.770045 + 0.637989i \(0.779766\pi\)
\(158\) −8.13175e6 −0.164015
\(159\) 5.22277e7 1.03041
\(160\) 5.41380e7 1.04492
\(161\) 7.07006e6 0.133516
\(162\) −2.53562e6 −0.0468577
\(163\) 1.75999e7 0.318312 0.159156 0.987253i \(-0.449123\pi\)
0.159156 + 0.987253i \(0.449123\pi\)
\(164\) −5.95324e6 −0.105390
\(165\) 8.83395e7 1.53095
\(166\) −1.93138e7 −0.327709
\(167\) 6.95017e7 1.15475 0.577375 0.816479i \(-0.304077\pi\)
0.577375 + 0.816479i \(0.304077\pi\)
\(168\) −2.40998e7 −0.392131
\(169\) 4.82681e6 0.0769231
\(170\) 1.04446e7 0.163049
\(171\) −1.82265e8 −2.78751
\(172\) −7.90390e7 −1.18438
\(173\) −5.15949e6 −0.0757610 −0.0378805 0.999282i \(-0.512061\pi\)
−0.0378805 + 0.999282i \(0.512061\pi\)
\(174\) 4.02965e7 0.579889
\(175\) 1.27291e7 0.179541
\(176\) 3.75591e7 0.519303
\(177\) 1.32774e8 1.79973
\(178\) 4.80405e7 0.638465
\(179\) −8.80205e7 −1.14709 −0.573546 0.819173i \(-0.694433\pi\)
−0.573546 + 0.819173i \(0.694433\pi\)
\(180\) 1.41532e8 1.80884
\(181\) −3.23048e7 −0.404941 −0.202470 0.979288i \(-0.564897\pi\)
−0.202470 + 0.979288i \(0.564897\pi\)
\(182\) 2.86371e6 0.0352110
\(183\) −2.07049e8 −2.49743
\(184\) 1.89215e7 0.223920
\(185\) −1.37364e7 −0.159504
\(186\) 2.69198e7 0.306744
\(187\) 2.75271e7 0.307833
\(188\) −8.78877e6 −0.0964664
\(189\) −3.89725e7 −0.419897
\(190\) −6.40414e7 −0.677366
\(191\) −2.06430e7 −0.214366 −0.107183 0.994239i \(-0.534183\pi\)
−0.107183 + 0.994239i \(0.534183\pi\)
\(192\) 6.18425e7 0.630569
\(193\) −3.31896e7 −0.332316 −0.166158 0.986099i \(-0.553136\pi\)
−0.166158 + 0.986099i \(0.553136\pi\)
\(194\) −3.93881e7 −0.387311
\(195\) −5.70843e7 −0.551309
\(196\) −1.33601e7 −0.126740
\(197\) −1.95827e8 −1.82490 −0.912452 0.409185i \(-0.865813\pi\)
−0.912452 + 0.409185i \(0.865813\pi\)
\(198\) −4.74366e7 −0.434295
\(199\) 2.03382e8 1.82947 0.914737 0.404049i \(-0.132397\pi\)
0.914737 + 0.404049i \(0.132397\pi\)
\(200\) 3.40667e7 0.301110
\(201\) 3.61112e8 3.13657
\(202\) −1.83796e7 −0.156894
\(203\) 4.75187e7 0.398684
\(204\) 7.03727e7 0.580361
\(205\) −1.77962e7 −0.144274
\(206\) −4.76183e7 −0.379523
\(207\) 7.56779e7 0.593025
\(208\) −2.42704e7 −0.187006
\(209\) −1.68784e8 −1.27885
\(210\) −3.38676e7 −0.252358
\(211\) 6.29200e7 0.461105 0.230553 0.973060i \(-0.425947\pi\)
0.230553 + 0.973060i \(0.425947\pi\)
\(212\) 7.74869e7 0.558538
\(213\) −2.14269e8 −1.51925
\(214\) 7.74292e6 0.0540078
\(215\) −2.36274e8 −1.62137
\(216\) −1.04302e8 −0.704212
\(217\) 3.17446e7 0.210892
\(218\) −6.07557e7 −0.397182
\(219\) 2.42338e8 1.55907
\(220\) 1.31064e8 0.829858
\(221\) −1.77878e7 −0.110853
\(222\) 1.17699e7 0.0722003
\(223\) 1.12770e8 0.680970 0.340485 0.940250i \(-0.389409\pi\)
0.340485 + 0.940250i \(0.389409\pi\)
\(224\) −5.47018e7 −0.325188
\(225\) 1.36252e8 0.797453
\(226\) −1.26934e7 −0.0731471
\(227\) 1.06587e7 0.0604804 0.0302402 0.999543i \(-0.490373\pi\)
0.0302402 + 0.999543i \(0.490373\pi\)
\(228\) −4.31494e8 −2.41103
\(229\) −2.35163e8 −1.29403 −0.647016 0.762477i \(-0.723983\pi\)
−0.647016 + 0.762477i \(0.723983\pi\)
\(230\) 2.65906e7 0.144105
\(231\) −8.92596e7 −0.476445
\(232\) 1.27174e8 0.668636
\(233\) 3.48398e8 1.80439 0.902193 0.431332i \(-0.141956\pi\)
0.902193 + 0.431332i \(0.141956\pi\)
\(234\) 3.06531e7 0.156394
\(235\) −2.62726e7 −0.132058
\(236\) 1.96989e8 0.975551
\(237\) −1.63784e8 −0.799194
\(238\) −1.05533e7 −0.0507424
\(239\) 9.32918e7 0.442029 0.221015 0.975270i \(-0.429063\pi\)
0.221015 + 0.975270i \(0.429063\pi\)
\(240\) 2.87034e8 1.34028
\(241\) 2.36583e8 1.08874 0.544369 0.838846i \(-0.316769\pi\)
0.544369 + 0.838846i \(0.316769\pi\)
\(242\) 3.01268e7 0.136647
\(243\) 1.97422e8 0.882619
\(244\) −3.07185e8 −1.35374
\(245\) −3.99376e7 −0.173501
\(246\) 1.52486e7 0.0653066
\(247\) 1.09067e8 0.460525
\(248\) 8.49576e7 0.353689
\(249\) −3.89005e8 −1.59682
\(250\) −5.29090e7 −0.214161
\(251\) 1.65859e8 0.662034 0.331017 0.943625i \(-0.392608\pi\)
0.331017 + 0.943625i \(0.392608\pi\)
\(252\) −1.43006e8 −0.562928
\(253\) 7.00806e7 0.272067
\(254\) 1.77276e7 0.0678786
\(255\) 2.10367e8 0.794488
\(256\) 1.41767e7 0.0528125
\(257\) −3.95785e8 −1.45443 −0.727217 0.686408i \(-0.759187\pi\)
−0.727217 + 0.686408i \(0.759187\pi\)
\(258\) 2.02450e8 0.733920
\(259\) 1.38794e7 0.0496390
\(260\) −8.46924e7 −0.298839
\(261\) 5.08641e8 1.77080
\(262\) 1.93740e8 0.665526
\(263\) −5.23975e7 −0.177609 −0.0888046 0.996049i \(-0.528305\pi\)
−0.0888046 + 0.996049i \(0.528305\pi\)
\(264\) −2.38884e8 −0.799050
\(265\) 2.31634e8 0.764614
\(266\) 6.47085e7 0.210802
\(267\) 9.67599e8 3.11104
\(268\) 5.35759e8 1.70019
\(269\) −2.80171e8 −0.877586 −0.438793 0.898588i \(-0.644594\pi\)
−0.438793 + 0.898588i \(0.644594\pi\)
\(270\) −1.46576e8 −0.453200
\(271\) 3.35906e8 1.02524 0.512620 0.858616i \(-0.328675\pi\)
0.512620 + 0.858616i \(0.328675\pi\)
\(272\) 8.94414e7 0.269493
\(273\) 5.76788e7 0.171572
\(274\) −2.03304e7 −0.0597062
\(275\) 1.26175e8 0.365854
\(276\) 1.79160e8 0.512932
\(277\) 5.40118e8 1.52689 0.763447 0.645870i \(-0.223505\pi\)
0.763447 + 0.645870i \(0.223505\pi\)
\(278\) 2.17397e8 0.606872
\(279\) 3.39794e8 0.936701
\(280\) −1.06885e8 −0.290979
\(281\) −4.45375e8 −1.19744 −0.598719 0.800959i \(-0.704324\pi\)
−0.598719 + 0.800959i \(0.704324\pi\)
\(282\) 2.25115e7 0.0597768
\(283\) 3.95539e7 0.103738 0.0518689 0.998654i \(-0.483482\pi\)
0.0518689 + 0.998654i \(0.483482\pi\)
\(284\) −3.17897e8 −0.823517
\(285\) −1.28988e9 −3.30059
\(286\) 2.83859e7 0.0717500
\(287\) 1.79816e7 0.0448994
\(288\) −5.85529e8 −1.44436
\(289\) −3.44787e8 −0.840250
\(290\) 1.78718e8 0.430305
\(291\) −7.93329e8 −1.88724
\(292\) 3.59541e8 0.845101
\(293\) −6.83430e8 −1.58729 −0.793647 0.608379i \(-0.791820\pi\)
−0.793647 + 0.608379i \(0.791820\pi\)
\(294\) 3.42204e7 0.0785361
\(295\) 5.88866e8 1.33549
\(296\) 3.71454e7 0.0832499
\(297\) −3.86308e8 −0.855630
\(298\) 2.03132e8 0.444653
\(299\) −4.52855e7 −0.0979738
\(300\) 3.22564e8 0.689750
\(301\) 2.38735e8 0.504583
\(302\) 204648. 0.000427547 0
\(303\) −3.70190e8 −0.764496
\(304\) −5.48415e8 −1.11957
\(305\) −9.18279e8 −1.85321
\(306\) −1.12963e8 −0.225378
\(307\) 4.62744e8 0.912760 0.456380 0.889785i \(-0.349146\pi\)
0.456380 + 0.889785i \(0.349146\pi\)
\(308\) −1.32429e8 −0.258259
\(309\) −9.59094e8 −1.84930
\(310\) 1.19392e8 0.227619
\(311\) −2.11511e8 −0.398723 −0.199361 0.979926i \(-0.563887\pi\)
−0.199361 + 0.979926i \(0.563887\pi\)
\(312\) 1.54365e8 0.287745
\(313\) −1.07114e9 −1.97442 −0.987211 0.159419i \(-0.949038\pi\)
−0.987211 + 0.159419i \(0.949038\pi\)
\(314\) 2.83792e8 0.517304
\(315\) −4.27493e8 −0.770623
\(316\) −2.42996e8 −0.433206
\(317\) −9.20889e8 −1.62368 −0.811839 0.583881i \(-0.801533\pi\)
−0.811839 + 0.583881i \(0.801533\pi\)
\(318\) −1.98475e8 −0.346107
\(319\) 4.71020e8 0.812404
\(320\) 2.74277e8 0.467912
\(321\) 1.55953e8 0.263163
\(322\) −2.68675e7 −0.0448468
\(323\) −4.01933e8 −0.663659
\(324\) −7.57703e7 −0.123763
\(325\) −8.15331e7 −0.131747
\(326\) −6.68828e7 −0.106919
\(327\) −1.22370e9 −1.93534
\(328\) 4.81238e7 0.0753011
\(329\) 2.65462e7 0.0410976
\(330\) −3.35706e8 −0.514234
\(331\) 1.06044e9 1.60727 0.803635 0.595123i \(-0.202897\pi\)
0.803635 + 0.595123i \(0.202897\pi\)
\(332\) −5.77142e8 −0.865565
\(333\) 1.48566e8 0.220477
\(334\) −2.64119e8 −0.387871
\(335\) 1.60156e9 2.32748
\(336\) −2.90023e8 −0.417105
\(337\) 3.67084e8 0.522469 0.261234 0.965275i \(-0.415870\pi\)
0.261234 + 0.965275i \(0.415870\pi\)
\(338\) −1.83428e7 −0.0258378
\(339\) −2.55661e8 −0.356423
\(340\) 3.12109e8 0.430655
\(341\) 3.14662e8 0.429738
\(342\) 6.92640e8 0.936302
\(343\) 4.03536e7 0.0539949
\(344\) 6.38923e8 0.846240
\(345\) 5.35569e8 0.702180
\(346\) 1.96070e7 0.0254475
\(347\) 4.72177e8 0.606669 0.303334 0.952884i \(-0.401900\pi\)
0.303334 + 0.952884i \(0.401900\pi\)
\(348\) 1.20416e9 1.53164
\(349\) −1.30323e9 −1.64108 −0.820542 0.571587i \(-0.806328\pi\)
−0.820542 + 0.571587i \(0.806328\pi\)
\(350\) −4.83729e7 −0.0603065
\(351\) 2.49629e8 0.308120
\(352\) −5.42221e8 −0.662639
\(353\) −3.12446e8 −0.378063 −0.189031 0.981971i \(-0.560535\pi\)
−0.189031 + 0.981971i \(0.560535\pi\)
\(354\) −5.04567e8 −0.604515
\(355\) −9.50301e8 −1.12736
\(356\) 1.43557e9 1.68635
\(357\) −2.12558e8 −0.247252
\(358\) 3.34494e8 0.385299
\(359\) −2.07105e8 −0.236244 −0.118122 0.992999i \(-0.537687\pi\)
−0.118122 + 0.992999i \(0.537687\pi\)
\(360\) −1.14409e9 −1.29242
\(361\) 1.57061e9 1.75708
\(362\) 1.22764e8 0.136016
\(363\) 6.06793e8 0.665836
\(364\) 8.55745e7 0.0930014
\(365\) 1.07479e9 1.15690
\(366\) 7.86823e8 0.838867
\(367\) −8.40606e8 −0.887690 −0.443845 0.896104i \(-0.646386\pi\)
−0.443845 + 0.896104i \(0.646386\pi\)
\(368\) 2.27707e8 0.238182
\(369\) 1.92475e8 0.199426
\(370\) 5.22007e7 0.0535760
\(371\) −2.34047e8 −0.237954
\(372\) 8.04429e8 0.810191
\(373\) 1.82593e9 1.82181 0.910903 0.412621i \(-0.135387\pi\)
0.910903 + 0.412621i \(0.135387\pi\)
\(374\) −1.04608e8 −0.103398
\(375\) −1.06566e9 −1.04354
\(376\) 7.10453e7 0.0689251
\(377\) −3.04369e8 −0.292554
\(378\) 1.48103e8 0.141040
\(379\) −1.61714e8 −0.152584 −0.0762922 0.997086i \(-0.524308\pi\)
−0.0762922 + 0.997086i \(0.524308\pi\)
\(380\) −1.91371e9 −1.78910
\(381\) 3.57058e8 0.330751
\(382\) 7.84472e7 0.0720038
\(383\) −2.65588e8 −0.241553 −0.120777 0.992680i \(-0.538539\pi\)
−0.120777 + 0.992680i \(0.538539\pi\)
\(384\) −1.79748e9 −1.61996
\(385\) −3.95874e8 −0.353545
\(386\) 1.26126e8 0.111622
\(387\) 2.55542e9 2.24116
\(388\) −1.17701e9 −1.02299
\(389\) −1.41972e8 −0.122287 −0.0611434 0.998129i \(-0.519475\pi\)
−0.0611434 + 0.998129i \(0.519475\pi\)
\(390\) 2.16931e8 0.185180
\(391\) 1.66886e8 0.141189
\(392\) 1.07998e8 0.0905553
\(393\) 3.90218e9 3.24290
\(394\) 7.44177e8 0.612970
\(395\) −7.26396e8 −0.593040
\(396\) −1.41752e9 −1.14709
\(397\) −7.22873e8 −0.579823 −0.289912 0.957053i \(-0.593626\pi\)
−0.289912 + 0.957053i \(0.593626\pi\)
\(398\) −7.72888e8 −0.614506
\(399\) 1.30331e9 1.02717
\(400\) 4.09968e8 0.320288
\(401\) −8.71880e8 −0.675229 −0.337615 0.941284i \(-0.609620\pi\)
−0.337615 + 0.941284i \(0.609620\pi\)
\(402\) −1.37229e9 −1.05355
\(403\) −2.03332e8 −0.154753
\(404\) −5.49228e8 −0.414398
\(405\) −2.26503e8 −0.169426
\(406\) −1.80580e8 −0.133915
\(407\) 1.37577e8 0.101150
\(408\) −5.68867e8 −0.414668
\(409\) −1.03396e9 −0.747263 −0.373632 0.927577i \(-0.621888\pi\)
−0.373632 + 0.927577i \(0.621888\pi\)
\(410\) 6.76288e7 0.0484605
\(411\) −4.09481e8 −0.290929
\(412\) −1.42295e9 −1.00242
\(413\) −5.94999e8 −0.415615
\(414\) −2.87590e8 −0.199192
\(415\) −1.72527e9 −1.18492
\(416\) 3.50379e8 0.238622
\(417\) 4.37866e9 2.95709
\(418\) 6.41410e8 0.429555
\(419\) 9.75703e8 0.647990 0.323995 0.946059i \(-0.394974\pi\)
0.323995 + 0.946059i \(0.394974\pi\)
\(420\) −1.01205e9 −0.666543
\(421\) −1.34931e9 −0.881299 −0.440649 0.897679i \(-0.645252\pi\)
−0.440649 + 0.897679i \(0.645252\pi\)
\(422\) −2.39107e8 −0.154881
\(423\) 2.84151e8 0.182540
\(424\) −6.26376e8 −0.399075
\(425\) 3.00466e8 0.189860
\(426\) 8.14261e8 0.510305
\(427\) 9.27843e8 0.576736
\(428\) 2.31377e8 0.142649
\(429\) 5.71730e8 0.349615
\(430\) 8.97883e8 0.544603
\(431\) 1.24186e8 0.0747143 0.0373572 0.999302i \(-0.488106\pi\)
0.0373572 + 0.999302i \(0.488106\pi\)
\(432\) −1.25520e9 −0.749063
\(433\) 1.58640e9 0.939083 0.469542 0.882910i \(-0.344419\pi\)
0.469542 + 0.882910i \(0.344419\pi\)
\(434\) −1.20635e8 −0.0708369
\(435\) 3.59962e9 2.09674
\(436\) −1.81553e9 −1.04906
\(437\) −1.02327e9 −0.586552
\(438\) −9.20927e8 −0.523680
\(439\) −5.98873e8 −0.337838 −0.168919 0.985630i \(-0.554028\pi\)
−0.168919 + 0.985630i \(0.554028\pi\)
\(440\) −1.05947e9 −0.592933
\(441\) 4.31945e8 0.239825
\(442\) 6.75968e7 0.0372347
\(443\) 3.01012e9 1.64502 0.822508 0.568754i \(-0.192574\pi\)
0.822508 + 0.568754i \(0.192574\pi\)
\(444\) 3.51714e8 0.190700
\(445\) 4.29138e9 2.30854
\(446\) −4.28548e8 −0.228732
\(447\) 4.09134e9 2.16665
\(448\) −2.77133e8 −0.145618
\(449\) −3.22007e9 −1.67881 −0.839407 0.543503i \(-0.817098\pi\)
−0.839407 + 0.543503i \(0.817098\pi\)
\(450\) −5.17784e8 −0.267858
\(451\) 1.78239e8 0.0914922
\(452\) −3.79308e8 −0.193200
\(453\) 4.12189e6 0.00208330
\(454\) −4.05051e7 −0.0203149
\(455\) 2.55811e8 0.127315
\(456\) 3.48804e9 1.72268
\(457\) 2.83525e9 1.38958 0.694792 0.719211i \(-0.255496\pi\)
0.694792 + 0.719211i \(0.255496\pi\)
\(458\) 8.93662e8 0.434654
\(459\) −9.19933e8 −0.444030
\(460\) 7.94590e8 0.380619
\(461\) −3.20816e9 −1.52512 −0.762558 0.646920i \(-0.776057\pi\)
−0.762558 + 0.646920i \(0.776057\pi\)
\(462\) 3.39203e8 0.160034
\(463\) −8.94081e8 −0.418642 −0.209321 0.977847i \(-0.567125\pi\)
−0.209321 + 0.977847i \(0.567125\pi\)
\(464\) 1.53044e9 0.711221
\(465\) 2.40470e9 1.10911
\(466\) −1.32397e9 −0.606079
\(467\) −7.58084e8 −0.344436 −0.172218 0.985059i \(-0.555093\pi\)
−0.172218 + 0.985059i \(0.555093\pi\)
\(468\) 9.15990e8 0.413076
\(469\) −1.61824e9 −0.724333
\(470\) 9.98405e7 0.0443572
\(471\) 5.71594e9 2.52066
\(472\) −1.59239e9 −0.697031
\(473\) 2.36641e9 1.02820
\(474\) 6.22409e8 0.268443
\(475\) −1.84232e9 −0.788748
\(476\) −3.15359e8 −0.134024
\(477\) −2.50524e9 −1.05690
\(478\) −3.54526e8 −0.148474
\(479\) −4.43815e9 −1.84513 −0.922567 0.385838i \(-0.873912\pi\)
−0.922567 + 0.385838i \(0.873912\pi\)
\(480\) −4.14376e9 −1.71021
\(481\) −8.89012e7 −0.0364250
\(482\) −8.99057e8 −0.365698
\(483\) −5.41147e8 −0.218525
\(484\) 9.00260e8 0.360919
\(485\) −3.51848e9 −1.40042
\(486\) −7.50239e8 −0.296465
\(487\) 3.04352e9 1.19405 0.597027 0.802221i \(-0.296348\pi\)
0.597027 + 0.802221i \(0.296348\pi\)
\(488\) 2.48318e9 0.967248
\(489\) −1.34711e9 −0.520980
\(490\) 1.51770e8 0.0582775
\(491\) −5.11802e9 −1.95127 −0.975634 0.219402i \(-0.929589\pi\)
−0.975634 + 0.219402i \(0.929589\pi\)
\(492\) 4.55665e8 0.172491
\(493\) 1.12166e9 0.421598
\(494\) −4.14474e8 −0.154687
\(495\) −4.23744e9 −1.57031
\(496\) 1.02240e9 0.376215
\(497\) 9.60198e8 0.350844
\(498\) 1.47829e9 0.536360
\(499\) −1.85528e8 −0.0668434 −0.0334217 0.999441i \(-0.510640\pi\)
−0.0334217 + 0.999441i \(0.510640\pi\)
\(500\) −1.58105e9 −0.565654
\(501\) −5.31971e9 −1.88997
\(502\) −6.30294e8 −0.222372
\(503\) 4.86650e8 0.170502 0.0852509 0.996360i \(-0.472831\pi\)
0.0852509 + 0.996360i \(0.472831\pi\)
\(504\) 1.15601e9 0.402212
\(505\) −1.64182e9 −0.567292
\(506\) −2.66319e8 −0.0913851
\(507\) −3.69447e8 −0.125900
\(508\) 5.29744e8 0.179285
\(509\) 4.07911e9 1.37105 0.685525 0.728049i \(-0.259573\pi\)
0.685525 + 0.728049i \(0.259573\pi\)
\(510\) −7.99434e8 −0.266862
\(511\) −1.08598e9 −0.360039
\(512\) −3.05982e9 −1.00751
\(513\) 5.64062e9 1.84466
\(514\) 1.50406e9 0.488532
\(515\) −4.25367e9 −1.37227
\(516\) 6.04970e9 1.93847
\(517\) 2.63134e8 0.0837452
\(518\) −5.27444e7 −0.0166733
\(519\) 3.94911e8 0.123998
\(520\) 6.84623e8 0.213520
\(521\) −1.01981e9 −0.315927 −0.157963 0.987445i \(-0.550493\pi\)
−0.157963 + 0.987445i \(0.550493\pi\)
\(522\) −1.93293e9 −0.594797
\(523\) −4.78076e9 −1.46131 −0.730653 0.682749i \(-0.760784\pi\)
−0.730653 + 0.682749i \(0.760784\pi\)
\(524\) 5.78942e9 1.75782
\(525\) −9.74294e8 −0.293855
\(526\) 1.99120e8 0.0596575
\(527\) 7.49319e8 0.223013
\(528\) −2.87480e9 −0.849941
\(529\) −2.97995e9 −0.875215
\(530\) −8.80252e8 −0.256827
\(531\) −6.36887e9 −1.84600
\(532\) 1.93364e9 0.556783
\(533\) −1.15176e8 −0.0329472
\(534\) −3.67705e9 −1.04497
\(535\) 6.91663e8 0.195279
\(536\) −4.33088e9 −1.21478
\(537\) 6.73715e9 1.87744
\(538\) 1.06470e9 0.294774
\(539\) 3.99997e8 0.110026
\(540\) −4.38005e9 −1.19702
\(541\) −5.96266e8 −0.161901 −0.0809505 0.996718i \(-0.525796\pi\)
−0.0809505 + 0.996718i \(0.525796\pi\)
\(542\) −1.27650e9 −0.344370
\(543\) 2.47263e9 0.662765
\(544\) −1.29122e9 −0.343877
\(545\) −5.42722e9 −1.43612
\(546\) −2.19190e8 −0.0576297
\(547\) −3.57892e9 −0.934967 −0.467484 0.884002i \(-0.654839\pi\)
−0.467484 + 0.884002i \(0.654839\pi\)
\(548\) −6.07522e8 −0.157699
\(549\) 9.93164e9 2.56164
\(550\) −4.79487e8 −0.122887
\(551\) −6.87754e9 −1.75147
\(552\) −1.44827e9 −0.366489
\(553\) 7.33962e8 0.184559
\(554\) −2.05255e9 −0.512871
\(555\) 1.05139e9 0.261059
\(556\) 6.49635e9 1.60290
\(557\) −1.71021e7 −0.00419331 −0.00209665 0.999998i \(-0.500667\pi\)
−0.00209665 + 0.999998i \(0.500667\pi\)
\(558\) −1.29128e9 −0.314630
\(559\) −1.52915e9 −0.370263
\(560\) −1.28628e9 −0.309512
\(561\) −2.10694e9 −0.503828
\(562\) 1.69250e9 0.402210
\(563\) 5.27373e9 1.24548 0.622742 0.782427i \(-0.286019\pi\)
0.622742 + 0.782427i \(0.286019\pi\)
\(564\) 6.72699e8 0.157886
\(565\) −1.13388e9 −0.264482
\(566\) −1.50312e8 −0.0348447
\(567\) 2.28862e8 0.0527269
\(568\) 2.56977e9 0.588402
\(569\) −6.49997e9 −1.47917 −0.739585 0.673063i \(-0.764978\pi\)
−0.739585 + 0.673063i \(0.764978\pi\)
\(570\) 4.90178e9 1.10864
\(571\) −5.10445e8 −0.114742 −0.0573710 0.998353i \(-0.518272\pi\)
−0.0573710 + 0.998353i \(0.518272\pi\)
\(572\) 8.48240e8 0.189510
\(573\) 1.58003e9 0.350852
\(574\) −6.83332e7 −0.0150813
\(575\) 7.64949e8 0.167801
\(576\) −2.96644e9 −0.646780
\(577\) 6.35150e9 1.37645 0.688226 0.725496i \(-0.258390\pi\)
0.688226 + 0.725496i \(0.258390\pi\)
\(578\) 1.31025e9 0.282233
\(579\) 2.54035e9 0.543900
\(580\) 5.34054e9 1.13655
\(581\) 1.74324e9 0.368757
\(582\) 3.01479e9 0.633909
\(583\) −2.31994e9 −0.484883
\(584\) −2.90640e9 −0.603824
\(585\) 2.73820e9 0.565482
\(586\) 2.59716e9 0.533159
\(587\) 1.23731e7 0.00252491 0.00126245 0.999999i \(-0.499598\pi\)
0.00126245 + 0.999999i \(0.499598\pi\)
\(588\) 1.02259e9 0.207434
\(589\) −4.59450e9 −0.926476
\(590\) −2.23780e9 −0.448579
\(591\) 1.49887e10 2.98681
\(592\) 4.47017e8 0.0885520
\(593\) −1.09632e7 −0.00215898 −0.00107949 0.999999i \(-0.500344\pi\)
−0.00107949 + 0.999999i \(0.500344\pi\)
\(594\) 1.46804e9 0.287399
\(595\) −9.42714e8 −0.183472
\(596\) 6.07007e9 1.17444
\(597\) −1.55670e10 −2.99429
\(598\) 1.72093e8 0.0329086
\(599\) −4.01930e9 −0.764111 −0.382055 0.924139i \(-0.624784\pi\)
−0.382055 + 0.924139i \(0.624784\pi\)
\(600\) −2.60749e9 −0.492826
\(601\) 1.98736e9 0.373436 0.186718 0.982414i \(-0.440215\pi\)
0.186718 + 0.982414i \(0.440215\pi\)
\(602\) −9.07235e8 −0.169485
\(603\) −1.73217e10 −3.21721
\(604\) 6.11539e6 0.00112926
\(605\) 2.69118e9 0.494081
\(606\) 1.40679e9 0.256788
\(607\) 7.82981e9 1.42099 0.710495 0.703703i \(-0.248471\pi\)
0.710495 + 0.703703i \(0.248471\pi\)
\(608\) 7.91718e9 1.42859
\(609\) −3.63712e9 −0.652524
\(610\) 3.48963e9 0.622479
\(611\) −1.70035e8 −0.0301574
\(612\) −3.37561e9 −0.595281
\(613\) −8.53356e9 −1.49630 −0.748150 0.663530i \(-0.769058\pi\)
−0.748150 + 0.663530i \(0.769058\pi\)
\(614\) −1.75851e9 −0.306589
\(615\) 1.36213e9 0.236133
\(616\) 1.07051e9 0.184526
\(617\) 7.29223e9 1.24986 0.624932 0.780680i \(-0.285127\pi\)
0.624932 + 0.780680i \(0.285127\pi\)
\(618\) 3.64473e9 0.621164
\(619\) 2.06975e9 0.350752 0.175376 0.984501i \(-0.443886\pi\)
0.175376 + 0.984501i \(0.443886\pi\)
\(620\) 3.56771e9 0.601200
\(621\) −2.34204e9 −0.392440
\(622\) 8.03779e8 0.133928
\(623\) −4.33608e9 −0.718437
\(624\) 1.85767e9 0.306072
\(625\) −7.62559e9 −1.24938
\(626\) 4.07052e9 0.663192
\(627\) 1.29188e10 2.09309
\(628\) 8.48038e9 1.36633
\(629\) 3.27619e8 0.0524919
\(630\) 1.62455e9 0.258846
\(631\) −4.58145e9 −0.725940 −0.362970 0.931801i \(-0.618237\pi\)
−0.362970 + 0.931801i \(0.618237\pi\)
\(632\) 1.96429e9 0.309526
\(633\) −4.81594e9 −0.754689
\(634\) 3.49955e9 0.545380
\(635\) 1.58358e9 0.245433
\(636\) −5.93090e9 −0.914157
\(637\) −2.58475e8 −0.0396214
\(638\) −1.78996e9 −0.272880
\(639\) 1.02780e10 1.55831
\(640\) −7.97196e9 −1.20209
\(641\) −6.97916e9 −1.04665 −0.523323 0.852134i \(-0.675308\pi\)
−0.523323 + 0.852134i \(0.675308\pi\)
\(642\) −5.92648e8 −0.0883943
\(643\) 1.79438e9 0.266180 0.133090 0.991104i \(-0.457510\pi\)
0.133090 + 0.991104i \(0.457510\pi\)
\(644\) −8.02866e8 −0.118452
\(645\) 1.80846e10 2.65368
\(646\) 1.52742e9 0.222918
\(647\) −8.25623e9 −1.19844 −0.599221 0.800584i \(-0.704523\pi\)
−0.599221 + 0.800584i \(0.704523\pi\)
\(648\) 6.12500e8 0.0884288
\(649\) −5.89781e9 −0.846904
\(650\) 3.09840e8 0.0442529
\(651\) −2.42975e9 −0.345166
\(652\) −1.99862e9 −0.282399
\(653\) −3.61339e9 −0.507830 −0.253915 0.967227i \(-0.581718\pi\)
−0.253915 + 0.967227i \(0.581718\pi\)
\(654\) 4.65028e9 0.650066
\(655\) 1.73065e10 2.40638
\(656\) 5.79135e8 0.0800970
\(657\) −1.16244e10 −1.59915
\(658\) −1.00880e8 −0.0138044
\(659\) −7.46935e9 −1.01668 −0.508339 0.861157i \(-0.669740\pi\)
−0.508339 + 0.861157i \(0.669740\pi\)
\(660\) −1.00317e10 −1.35822
\(661\) 9.47073e8 0.127549 0.0637747 0.997964i \(-0.479686\pi\)
0.0637747 + 0.997964i \(0.479686\pi\)
\(662\) −4.02987e9 −0.539869
\(663\) 1.36149e9 0.181433
\(664\) 4.66541e9 0.618445
\(665\) 5.78031e9 0.762211
\(666\) −5.64576e8 −0.0740564
\(667\) 2.85561e9 0.372614
\(668\) −7.89252e9 −1.02447
\(669\) −8.63152e9 −1.11454
\(670\) −6.08622e9 −0.781783
\(671\) 9.19706e9 1.17522
\(672\) 4.18692e9 0.532233
\(673\) 3.21500e9 0.406563 0.203282 0.979120i \(-0.434839\pi\)
0.203282 + 0.979120i \(0.434839\pi\)
\(674\) −1.39498e9 −0.175493
\(675\) −4.21666e9 −0.527722
\(676\) −5.48126e8 −0.0682444
\(677\) 1.46998e10 1.82075 0.910375 0.413785i \(-0.135793\pi\)
0.910375 + 0.413785i \(0.135793\pi\)
\(678\) 9.71558e8 0.119719
\(679\) 3.55513e9 0.435824
\(680\) −2.52297e9 −0.307703
\(681\) −8.15826e8 −0.0989881
\(682\) −1.19577e9 −0.144345
\(683\) 5.76686e9 0.692575 0.346288 0.938128i \(-0.387442\pi\)
0.346288 + 0.938128i \(0.387442\pi\)
\(684\) 2.06977e10 2.47301
\(685\) −1.81608e9 −0.215883
\(686\) −1.53351e8 −0.0181365
\(687\) 1.79995e10 2.11793
\(688\) 7.68897e9 0.900137
\(689\) 1.49913e9 0.174611
\(690\) −2.03526e9 −0.235857
\(691\) −1.39416e10 −1.60746 −0.803728 0.594997i \(-0.797153\pi\)
−0.803728 + 0.594997i \(0.797153\pi\)
\(692\) 5.85905e8 0.0672134
\(693\) 4.28157e9 0.488694
\(694\) −1.79436e9 −0.203775
\(695\) 1.94197e10 2.19430
\(696\) −9.73397e9 −1.09435
\(697\) 4.24448e8 0.0474799
\(698\) 4.95249e9 0.551226
\(699\) −2.66666e10 −2.95323
\(700\) −1.44550e9 −0.159285
\(701\) −3.04175e9 −0.333511 −0.166756 0.985998i \(-0.553329\pi\)
−0.166756 + 0.985998i \(0.553329\pi\)
\(702\) −9.48635e8 −0.103495
\(703\) −2.00882e9 −0.218070
\(704\) −2.74703e9 −0.296728
\(705\) 2.01092e9 0.216139
\(706\) 1.18735e9 0.126988
\(707\) 1.65892e9 0.176546
\(708\) −1.50777e10 −1.59668
\(709\) −1.01521e10 −1.06978 −0.534892 0.844920i \(-0.679648\pi\)
−0.534892 + 0.844920i \(0.679648\pi\)
\(710\) 3.61132e9 0.378670
\(711\) 7.85633e9 0.819740
\(712\) −1.16046e10 −1.20490
\(713\) 1.90767e9 0.197102
\(714\) 8.07760e8 0.0830498
\(715\) 2.53567e9 0.259431
\(716\) 9.99549e9 1.01767
\(717\) −7.14062e9 −0.723467
\(718\) 7.87038e8 0.0793524
\(719\) 8.36403e9 0.839198 0.419599 0.907709i \(-0.362171\pi\)
0.419599 + 0.907709i \(0.362171\pi\)
\(720\) −1.37683e10 −1.37473
\(721\) 4.29797e9 0.427061
\(722\) −5.96859e9 −0.590190
\(723\) −1.81082e10 −1.78193
\(724\) 3.66848e9 0.359254
\(725\) 5.14132e9 0.501062
\(726\) −2.30592e9 −0.223649
\(727\) −4.99911e8 −0.0482528 −0.0241264 0.999709i \(-0.507680\pi\)
−0.0241264 + 0.999709i \(0.507680\pi\)
\(728\) −6.91753e8 −0.0664494
\(729\) −1.65701e10 −1.58408
\(730\) −4.08439e9 −0.388595
\(731\) 5.63525e9 0.533583
\(732\) 2.35122e10 2.21566
\(733\) −1.16182e8 −0.0108962 −0.00544811 0.999985i \(-0.501734\pi\)
−0.00544811 + 0.999985i \(0.501734\pi\)
\(734\) 3.19446e9 0.298168
\(735\) 3.05685e9 0.283968
\(736\) −3.28728e9 −0.303924
\(737\) −1.60405e10 −1.47598
\(738\) −7.31439e8 −0.0669855
\(739\) 6.85800e9 0.625090 0.312545 0.949903i \(-0.398819\pi\)
0.312545 + 0.949903i \(0.398819\pi\)
\(740\) 1.55988e9 0.141508
\(741\) −8.34805e9 −0.753739
\(742\) 8.89420e8 0.0799269
\(743\) −1.01255e10 −0.905643 −0.452822 0.891601i \(-0.649583\pi\)
−0.452822 + 0.891601i \(0.649583\pi\)
\(744\) −6.50271e9 −0.578881
\(745\) 1.81455e10 1.60776
\(746\) −6.93885e9 −0.611930
\(747\) 1.86596e10 1.63788
\(748\) −3.12594e9 −0.273102
\(749\) −6.98867e8 −0.0607726
\(750\) 4.04969e9 0.350516
\(751\) −1.28118e10 −1.10375 −0.551873 0.833928i \(-0.686087\pi\)
−0.551873 + 0.833928i \(0.686087\pi\)
\(752\) 8.54978e8 0.0733150
\(753\) −1.26949e10 −1.08355
\(754\) 1.15666e9 0.0982665
\(755\) 1.82809e7 0.00154591
\(756\) 4.42567e9 0.372523
\(757\) 3.94851e9 0.330824 0.165412 0.986225i \(-0.447105\pi\)
0.165412 + 0.986225i \(0.447105\pi\)
\(758\) 6.14542e8 0.0512518
\(759\) −5.36401e9 −0.445291
\(760\) 1.54698e10 1.27831
\(761\) 8.86888e9 0.729495 0.364748 0.931106i \(-0.381155\pi\)
0.364748 + 0.931106i \(0.381155\pi\)
\(762\) −1.35688e9 −0.111097
\(763\) 5.48374e9 0.446932
\(764\) 2.34419e9 0.190181
\(765\) −1.00908e10 −0.814913
\(766\) 1.00928e9 0.0811358
\(767\) 3.81111e9 0.304978
\(768\) −1.08510e9 −0.0864380
\(769\) −2.95572e9 −0.234380 −0.117190 0.993110i \(-0.537389\pi\)
−0.117190 + 0.993110i \(0.537389\pi\)
\(770\) 1.50439e9 0.118753
\(771\) 3.02937e10 2.38046
\(772\) 3.76896e9 0.294823
\(773\) −2.21264e10 −1.72299 −0.861493 0.507769i \(-0.830470\pi\)
−0.861493 + 0.507769i \(0.830470\pi\)
\(774\) −9.71105e9 −0.752788
\(775\) 3.43462e9 0.265047
\(776\) 9.51454e9 0.730923
\(777\) −1.06234e9 −0.0812438
\(778\) 5.39520e8 0.0410751
\(779\) −2.60253e9 −0.197249
\(780\) 6.48241e9 0.489109
\(781\) 9.51778e9 0.714919
\(782\) −6.34198e8 −0.0474243
\(783\) −1.57411e10 −1.17184
\(784\) 1.29968e9 0.0963227
\(785\) 2.53507e10 1.87045
\(786\) −1.48290e10 −1.08926
\(787\) −1.42586e10 −1.04272 −0.521359 0.853338i \(-0.674575\pi\)
−0.521359 + 0.853338i \(0.674575\pi\)
\(788\) 2.22378e10 1.61901
\(789\) 4.01054e9 0.290692
\(790\) 2.76044e9 0.199197
\(791\) 1.14569e9 0.0823092
\(792\) 1.14587e10 0.819592
\(793\) −5.94307e9 −0.423208
\(794\) 2.74705e9 0.194758
\(795\) −1.77294e10 −1.25144
\(796\) −2.30958e10 −1.62307
\(797\) 1.04574e10 0.731678 0.365839 0.930678i \(-0.380782\pi\)
0.365839 + 0.930678i \(0.380782\pi\)
\(798\) −4.95283e9 −0.345019
\(799\) 6.26614e8 0.0434596
\(800\) −5.91850e9 −0.408692
\(801\) −4.64134e10 −3.19102
\(802\) 3.31330e9 0.226804
\(803\) −1.07646e10 −0.733656
\(804\) −4.10073e10 −2.78269
\(805\) −2.40003e9 −0.162156
\(806\) 7.72698e8 0.0519801
\(807\) 2.14445e10 1.43634
\(808\) 4.43976e9 0.296087
\(809\) 1.34180e10 0.890977 0.445489 0.895288i \(-0.353030\pi\)
0.445489 + 0.895288i \(0.353030\pi\)
\(810\) 8.60751e8 0.0569089
\(811\) −2.09601e9 −0.137981 −0.0689905 0.997617i \(-0.521978\pi\)
−0.0689905 + 0.997617i \(0.521978\pi\)
\(812\) −5.39616e9 −0.353703
\(813\) −2.57105e10 −1.67801
\(814\) −5.22818e8 −0.0339755
\(815\) −5.97454e9 −0.386592
\(816\) −6.84590e9 −0.441078
\(817\) −3.45528e10 −2.21670
\(818\) 3.92925e9 0.251000
\(819\) −2.76672e9 −0.175983
\(820\) 2.02091e9 0.127997
\(821\) −2.80182e10 −1.76701 −0.883505 0.468421i \(-0.844823\pi\)
−0.883505 + 0.468421i \(0.844823\pi\)
\(822\) 1.55610e9 0.0977208
\(823\) −1.74157e10 −1.08903 −0.544516 0.838750i \(-0.683287\pi\)
−0.544516 + 0.838750i \(0.683287\pi\)
\(824\) 1.15026e10 0.716227
\(825\) −9.65750e9 −0.598792
\(826\) 2.26110e9 0.139602
\(827\) 1.17896e9 0.0724818 0.0362409 0.999343i \(-0.488462\pi\)
0.0362409 + 0.999343i \(0.488462\pi\)
\(828\) −8.59388e9 −0.526118
\(829\) 2.62554e10 1.60058 0.800292 0.599610i \(-0.204678\pi\)
0.800292 + 0.599610i \(0.204678\pi\)
\(830\) 6.55634e9 0.398005
\(831\) −4.13410e10 −2.49906
\(832\) 1.77511e9 0.106855
\(833\) 9.52533e8 0.0570982
\(834\) −1.66397e10 −0.993264
\(835\) −2.35934e10 −1.40245
\(836\) 1.91669e10 1.13456
\(837\) −1.05157e10 −0.619870
\(838\) −3.70785e9 −0.217655
\(839\) 1.55887e10 0.911264 0.455632 0.890168i \(-0.349413\pi\)
0.455632 + 0.890168i \(0.349413\pi\)
\(840\) 8.18102e9 0.476245
\(841\) 1.94306e9 0.112642
\(842\) 5.12761e9 0.296021
\(843\) 3.40893e10 1.95984
\(844\) −7.14511e9 −0.409082
\(845\) −1.63853e9 −0.0934234
\(846\) −1.07982e9 −0.0613136
\(847\) −2.71921e9 −0.153762
\(848\) −7.53798e9 −0.424492
\(849\) −3.02749e9 −0.169787
\(850\) −1.14183e9 −0.0637725
\(851\) 8.34078e8 0.0463931
\(852\) 2.43321e10 1.34785
\(853\) −2.08797e9 −0.115187 −0.0575933 0.998340i \(-0.518343\pi\)
−0.0575933 + 0.998340i \(0.518343\pi\)
\(854\) −3.52597e9 −0.193721
\(855\) 6.18724e10 3.38544
\(856\) −1.87037e9 −0.101922
\(857\) 1.54418e10 0.838039 0.419020 0.907977i \(-0.362374\pi\)
0.419020 + 0.907977i \(0.362374\pi\)
\(858\) −2.17268e9 −0.117433
\(859\) −1.06055e9 −0.0570896 −0.0285448 0.999593i \(-0.509087\pi\)
−0.0285448 + 0.999593i \(0.509087\pi\)
\(860\) 2.68309e10 1.43844
\(861\) −1.37632e9 −0.0734866
\(862\) −4.71931e8 −0.0250959
\(863\) 9.61556e9 0.509257 0.254628 0.967039i \(-0.418047\pi\)
0.254628 + 0.967039i \(0.418047\pi\)
\(864\) 1.81206e10 0.955816
\(865\) 1.75146e9 0.0920121
\(866\) −6.02860e9 −0.315430
\(867\) 2.63902e10 1.37523
\(868\) −3.60487e9 −0.187099
\(869\) 7.27525e9 0.376079
\(870\) −1.36792e10 −0.704278
\(871\) 1.03652e10 0.531515
\(872\) 1.46761e10 0.749553
\(873\) 3.80541e10 1.93576
\(874\) 3.88862e9 0.197018
\(875\) 4.77551e9 0.240986
\(876\) −2.75195e10 −1.38317
\(877\) −1.91278e9 −0.0957563 −0.0478781 0.998853i \(-0.515246\pi\)
−0.0478781 + 0.998853i \(0.515246\pi\)
\(878\) 2.27582e9 0.113477
\(879\) 5.23102e10 2.59792
\(880\) −1.27500e10 −0.630696
\(881\) −1.14488e10 −0.564086 −0.282043 0.959402i \(-0.591012\pi\)
−0.282043 + 0.959402i \(0.591012\pi\)
\(882\) −1.64147e9 −0.0805551
\(883\) −9.61596e8 −0.0470035 −0.0235017 0.999724i \(-0.507482\pi\)
−0.0235017 + 0.999724i \(0.507482\pi\)
\(884\) 2.01996e9 0.0983465
\(885\) −4.50722e10 −2.18578
\(886\) −1.14390e10 −0.552547
\(887\) 2.68603e10 1.29234 0.646171 0.763192i \(-0.276369\pi\)
0.646171 + 0.763192i \(0.276369\pi\)
\(888\) −2.84313e9 −0.136255
\(889\) −1.60008e9 −0.0763808
\(890\) −1.63080e10 −0.775419
\(891\) 2.26855e9 0.107442
\(892\) −1.28060e10 −0.604141
\(893\) −3.84212e9 −0.180547
\(894\) −1.55479e10 −0.727762
\(895\) 2.98798e10 1.39315
\(896\) 8.05499e9 0.374099
\(897\) 3.46618e9 0.160353
\(898\) 1.22368e10 0.563900
\(899\) 1.28217e10 0.588555
\(900\) −1.54726e10 −0.707482
\(901\) −5.52459e9 −0.251630
\(902\) −6.77339e8 −0.0307315
\(903\) −1.82729e10 −0.825849
\(904\) 3.06619e9 0.138041
\(905\) 1.09663e10 0.491803
\(906\) −1.56639e7 −0.000699764 0
\(907\) 3.05483e10 1.35945 0.679723 0.733469i \(-0.262100\pi\)
0.679723 + 0.733469i \(0.262100\pi\)
\(908\) −1.21039e9 −0.0536568
\(909\) 1.77571e10 0.784150
\(910\) −9.72127e8 −0.0427640
\(911\) −2.29762e10 −1.00685 −0.503423 0.864040i \(-0.667926\pi\)
−0.503423 + 0.864040i \(0.667926\pi\)
\(912\) 4.19760e10 1.83240
\(913\) 1.72795e10 0.751422
\(914\) −1.07745e10 −0.466750
\(915\) 7.02857e10 3.03314
\(916\) 2.67048e10 1.14803
\(917\) −1.74867e10 −0.748887
\(918\) 3.49591e9 0.149146
\(919\) −8.52757e9 −0.362427 −0.181214 0.983444i \(-0.558003\pi\)
−0.181214 + 0.983444i \(0.558003\pi\)
\(920\) −6.42318e9 −0.271952
\(921\) −3.54188e10 −1.49391
\(922\) 1.21916e10 0.512274
\(923\) −6.15031e9 −0.257449
\(924\) 1.01362e10 0.422691
\(925\) 1.50169e9 0.0623857
\(926\) 3.39767e9 0.140619
\(927\) 4.60055e10 1.89684
\(928\) −2.20942e10 −0.907529
\(929\) −8.26112e8 −0.0338052 −0.0169026 0.999857i \(-0.505381\pi\)
−0.0169026 + 0.999857i \(0.505381\pi\)
\(930\) −9.13832e9 −0.372543
\(931\) −5.84051e9 −0.237207
\(932\) −3.95636e10 −1.60081
\(933\) 1.61892e10 0.652588
\(934\) 2.88086e9 0.115693
\(935\) −9.34447e9 −0.373864
\(936\) −7.40453e9 −0.295143
\(937\) −4.34340e10 −1.72481 −0.862404 0.506220i \(-0.831042\pi\)
−0.862404 + 0.506220i \(0.831042\pi\)
\(938\) 6.14961e9 0.243298
\(939\) 8.19856e10 3.23153
\(940\) 2.98348e9 0.117159
\(941\) −8.99511e9 −0.351919 −0.175960 0.984397i \(-0.556303\pi\)
−0.175960 + 0.984397i \(0.556303\pi\)
\(942\) −2.17216e10 −0.846669
\(943\) 1.08059e9 0.0419634
\(944\) −1.91632e10 −0.741424
\(945\) 1.32298e10 0.509967
\(946\) −8.99279e9 −0.345363
\(947\) −1.79765e10 −0.687829 −0.343914 0.939001i \(-0.611753\pi\)
−0.343914 + 0.939001i \(0.611753\pi\)
\(948\) 1.85991e10 0.709027
\(949\) 6.95598e9 0.264196
\(950\) 7.00117e9 0.264934
\(951\) 7.04855e10 2.65747
\(952\) 2.54925e9 0.0957598
\(953\) −2.08985e10 −0.782150 −0.391075 0.920359i \(-0.627897\pi\)
−0.391075 + 0.920359i \(0.627897\pi\)
\(954\) 9.52035e9 0.355004
\(955\) 7.00757e9 0.260349
\(956\) −1.05941e10 −0.392158
\(957\) −3.60522e10 −1.32966
\(958\) 1.68658e10 0.619765
\(959\) 1.83500e9 0.0671848
\(960\) −2.09933e10 −0.765829
\(961\) −1.89472e10 −0.688672
\(962\) 3.37841e8 0.0122349
\(963\) −7.48067e9 −0.269928
\(964\) −2.68660e10 −0.965902
\(965\) 1.12667e10 0.403599
\(966\) 2.05646e9 0.0734006
\(967\) −2.58736e10 −0.920161 −0.460080 0.887877i \(-0.652179\pi\)
−0.460080 + 0.887877i \(0.652179\pi\)
\(968\) −7.27738e9 −0.257876
\(969\) 3.07643e10 1.08621
\(970\) 1.33709e10 0.470390
\(971\) −1.57586e10 −0.552395 −0.276197 0.961101i \(-0.589074\pi\)
−0.276197 + 0.961101i \(0.589074\pi\)
\(972\) −2.24190e10 −0.783039
\(973\) −1.96220e10 −0.682886
\(974\) −1.15659e10 −0.401073
\(975\) 6.24059e9 0.215630
\(976\) 2.98832e10 1.02885
\(977\) 8.83147e9 0.302972 0.151486 0.988459i \(-0.451594\pi\)
0.151486 + 0.988459i \(0.451594\pi\)
\(978\) 5.11926e9 0.174993
\(979\) −4.29805e10 −1.46397
\(980\) 4.53526e9 0.153926
\(981\) 5.86980e10 1.98510
\(982\) 1.94494e10 0.655415
\(983\) −7.60217e9 −0.255270 −0.127635 0.991821i \(-0.540739\pi\)
−0.127635 + 0.991821i \(0.540739\pi\)
\(984\) −3.68343e9 −0.123245
\(985\) 6.64761e10 2.21635
\(986\) −4.26252e9 −0.141611
\(987\) −2.03186e9 −0.0672643
\(988\) −1.23855e10 −0.408567
\(989\) 1.43466e10 0.471588
\(990\) 1.61030e10 0.527454
\(991\) −5.10651e10 −1.66673 −0.833367 0.552720i \(-0.813590\pi\)
−0.833367 + 0.552720i \(0.813590\pi\)
\(992\) −1.47599e10 −0.480056
\(993\) −8.11669e10 −2.63061
\(994\) −3.64893e9 −0.117845
\(995\) −6.90409e10 −2.22191
\(996\) 4.41749e10 1.41667
\(997\) −1.53172e10 −0.489492 −0.244746 0.969587i \(-0.578705\pi\)
−0.244746 + 0.969587i \(0.578705\pi\)
\(998\) 7.05041e8 0.0224521
\(999\) −4.59772e9 −0.145903
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.c.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.c.1.5 10 1.1 even 1 trivial