Properties

Label 91.8.a.c.1.7
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 1.7
Root \(10.4944\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.49443 q^{2} +45.4613 q^{3} -55.8447 q^{4} -130.405 q^{5} +386.168 q^{6} +343.000 q^{7} -1561.66 q^{8} -120.267 q^{9} +O(q^{10})\) \(q+8.49443 q^{2} +45.4613 q^{3} -55.8447 q^{4} -130.405 q^{5} +386.168 q^{6} +343.000 q^{7} -1561.66 q^{8} -120.267 q^{9} -1107.72 q^{10} +1042.49 q^{11} -2538.77 q^{12} -2197.00 q^{13} +2913.59 q^{14} -5928.40 q^{15} -6117.25 q^{16} -29763.8 q^{17} -1021.60 q^{18} +5796.19 q^{19} +7282.45 q^{20} +15593.2 q^{21} +8855.36 q^{22} -92127.5 q^{23} -70994.9 q^{24} -61119.4 q^{25} -18662.3 q^{26} -104891. q^{27} -19154.7 q^{28} -34842.8 q^{29} -50358.4 q^{30} +186908. q^{31} +147929. q^{32} +47393.0 q^{33} -252826. q^{34} -44729.1 q^{35} +6716.28 q^{36} +38460.8 q^{37} +49235.3 q^{38} -99878.5 q^{39} +203648. q^{40} +9960.74 q^{41} +132456. q^{42} -33656.3 q^{43} -58217.6 q^{44} +15683.5 q^{45} -782571. q^{46} +409530. q^{47} -278098. q^{48} +117649. q^{49} -519175. q^{50} -1.35310e6 q^{51} +122691. q^{52} -1.24539e6 q^{53} -890993. q^{54} -135946. q^{55} -535648. q^{56} +263502. q^{57} -295969. q^{58} -1.59698e6 q^{59} +331070. q^{60} +1.14496e6 q^{61} +1.58767e6 q^{62} -41251.6 q^{63} +2.03958e6 q^{64} +286501. q^{65} +402577. q^{66} -1.72469e6 q^{67} +1.66215e6 q^{68} -4.18824e6 q^{69} -379948. q^{70} -1.80050e6 q^{71} +187816. q^{72} +6.02038e6 q^{73} +326702. q^{74} -2.77857e6 q^{75} -323686. q^{76} +357574. q^{77} -848411. q^{78} +4.01302e6 q^{79} +797723. q^{80} -4.50548e6 q^{81} +84610.8 q^{82} +2.81688e6 q^{83} -870799. q^{84} +3.88136e6 q^{85} -285891. q^{86} -1.58400e6 q^{87} -1.62801e6 q^{88} -4.78900e6 q^{89} +133222. q^{90} -753571. q^{91} +5.14483e6 q^{92} +8.49707e6 q^{93} +3.47872e6 q^{94} -755854. q^{95} +6.72507e6 q^{96} -4.85052e6 q^{97} +999361. q^{98} -125377. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.49443 0.750808 0.375404 0.926861i \(-0.377504\pi\)
0.375404 + 0.926861i \(0.377504\pi\)
\(3\) 45.4613 0.972115 0.486058 0.873927i \(-0.338435\pi\)
0.486058 + 0.873927i \(0.338435\pi\)
\(4\) −55.8447 −0.436287
\(5\) −130.405 −0.466553 −0.233276 0.972410i \(-0.574945\pi\)
−0.233276 + 0.972410i \(0.574945\pi\)
\(6\) 386.168 0.729872
\(7\) 343.000 0.377964
\(8\) −1561.66 −1.07838
\(9\) −120.267 −0.0549919
\(10\) −1107.72 −0.350292
\(11\) 1042.49 0.236155 0.118078 0.993004i \(-0.462327\pi\)
0.118078 + 0.993004i \(0.462327\pi\)
\(12\) −2538.77 −0.424121
\(13\) −2197.00 −0.277350
\(14\) 2913.59 0.283779
\(15\) −5928.40 −0.453543
\(16\) −6117.25 −0.373367
\(17\) −29763.8 −1.46932 −0.734661 0.678435i \(-0.762659\pi\)
−0.734661 + 0.678435i \(0.762659\pi\)
\(18\) −1021.60 −0.0412884
\(19\) 5796.19 0.193867 0.0969337 0.995291i \(-0.469097\pi\)
0.0969337 + 0.995291i \(0.469097\pi\)
\(20\) 7282.45 0.203551
\(21\) 15593.2 0.367425
\(22\) 8855.36 0.177307
\(23\) −92127.5 −1.57885 −0.789427 0.613844i \(-0.789622\pi\)
−0.789427 + 0.613844i \(0.789622\pi\)
\(24\) −70994.9 −1.04831
\(25\) −61119.4 −0.782329
\(26\) −18662.3 −0.208237
\(27\) −104891. −1.02557
\(28\) −19154.7 −0.164901
\(29\) −34842.8 −0.265289 −0.132645 0.991164i \(-0.542347\pi\)
−0.132645 + 0.991164i \(0.542347\pi\)
\(30\) −50358.4 −0.340524
\(31\) 186908. 1.12684 0.563419 0.826172i \(-0.309486\pi\)
0.563419 + 0.826172i \(0.309486\pi\)
\(32\) 147929. 0.798049
\(33\) 47393.0 0.229570
\(34\) −252826. −1.10318
\(35\) −44729.1 −0.176340
\(36\) 6716.28 0.0239922
\(37\) 38460.8 0.124828 0.0624140 0.998050i \(-0.480120\pi\)
0.0624140 + 0.998050i \(0.480120\pi\)
\(38\) 49235.3 0.145557
\(39\) −99878.5 −0.269616
\(40\) 203648. 0.503119
\(41\) 9960.74 0.0225709 0.0112854 0.999936i \(-0.496408\pi\)
0.0112854 + 0.999936i \(0.496408\pi\)
\(42\) 132456. 0.275866
\(43\) −33656.3 −0.0645545 −0.0322773 0.999479i \(-0.510276\pi\)
−0.0322773 + 0.999479i \(0.510276\pi\)
\(44\) −58217.6 −0.103031
\(45\) 15683.5 0.0256566
\(46\) −782571. −1.18542
\(47\) 409530. 0.575364 0.287682 0.957726i \(-0.407115\pi\)
0.287682 + 0.957726i \(0.407115\pi\)
\(48\) −278098. −0.362956
\(49\) 117649. 0.142857
\(50\) −519175. −0.587379
\(51\) −1.35310e6 −1.42835
\(52\) 122691. 0.121004
\(53\) −1.24539e6 −1.14905 −0.574524 0.818488i \(-0.694813\pi\)
−0.574524 + 0.818488i \(0.694813\pi\)
\(54\) −890993. −0.770009
\(55\) −135946. −0.110179
\(56\) −535648. −0.407588
\(57\) 263502. 0.188461
\(58\) −295969. −0.199181
\(59\) −1.59698e6 −1.01232 −0.506161 0.862439i \(-0.668936\pi\)
−0.506161 + 0.862439i \(0.668936\pi\)
\(60\) 331070. 0.197875
\(61\) 1.14496e6 0.645856 0.322928 0.946424i \(-0.395333\pi\)
0.322928 + 0.946424i \(0.395333\pi\)
\(62\) 1.58767e6 0.846039
\(63\) −41251.6 −0.0207850
\(64\) 2.03958e6 0.972549
\(65\) 286501. 0.129398
\(66\) 402577. 0.172363
\(67\) −1.72469e6 −0.700566 −0.350283 0.936644i \(-0.613915\pi\)
−0.350283 + 0.936644i \(0.613915\pi\)
\(68\) 1.66215e6 0.641045
\(69\) −4.18824e6 −1.53483
\(70\) −379948. −0.132398
\(71\) −1.80050e6 −0.597019 −0.298509 0.954407i \(-0.596489\pi\)
−0.298509 + 0.954407i \(0.596489\pi\)
\(72\) 187816. 0.0593019
\(73\) 6.02038e6 1.81132 0.905658 0.424010i \(-0.139378\pi\)
0.905658 + 0.424010i \(0.139378\pi\)
\(74\) 326702. 0.0937219
\(75\) −2.77857e6 −0.760514
\(76\) −323686. −0.0845817
\(77\) 357574. 0.0892584
\(78\) −848411. −0.202430
\(79\) 4.01302e6 0.915748 0.457874 0.889017i \(-0.348611\pi\)
0.457874 + 0.889017i \(0.348611\pi\)
\(80\) 797723. 0.174195
\(81\) −4.50548e6 −0.941984
\(82\) 84610.8 0.0169464
\(83\) 2.81688e6 0.540749 0.270374 0.962755i \(-0.412853\pi\)
0.270374 + 0.962755i \(0.412853\pi\)
\(84\) −870799. −0.160303
\(85\) 3.88136e6 0.685516
\(86\) −285891. −0.0484681
\(87\) −1.58400e6 −0.257892
\(88\) −1.62801e6 −0.254664
\(89\) −4.78900e6 −0.720078 −0.360039 0.932937i \(-0.617237\pi\)
−0.360039 + 0.932937i \(0.617237\pi\)
\(90\) 133222. 0.0192632
\(91\) −753571. −0.104828
\(92\) 5.14483e6 0.688833
\(93\) 8.49707e6 1.09542
\(94\) 3.47872e6 0.431988
\(95\) −755854. −0.0904493
\(96\) 6.72507e6 0.775795
\(97\) −4.85052e6 −0.539620 −0.269810 0.962914i \(-0.586961\pi\)
−0.269810 + 0.962914i \(0.586961\pi\)
\(98\) 999361. 0.107258
\(99\) −125377. −0.0129866
\(100\) 3.41320e6 0.341320
\(101\) 1.01297e7 0.978298 0.489149 0.872200i \(-0.337307\pi\)
0.489149 + 0.872200i \(0.337307\pi\)
\(102\) −1.14938e7 −1.07242
\(103\) −7.80641e6 −0.703916 −0.351958 0.936016i \(-0.614484\pi\)
−0.351958 + 0.936016i \(0.614484\pi\)
\(104\) 3.43096e6 0.299088
\(105\) −2.03344e6 −0.171423
\(106\) −1.05788e7 −0.862715
\(107\) 1.89287e7 1.49375 0.746874 0.664966i \(-0.231554\pi\)
0.746874 + 0.664966i \(0.231554\pi\)
\(108\) 5.85763e6 0.447444
\(109\) 1.46063e7 1.08031 0.540154 0.841566i \(-0.318366\pi\)
0.540154 + 0.841566i \(0.318366\pi\)
\(110\) −1.15479e6 −0.0827233
\(111\) 1.74848e6 0.121347
\(112\) −2.09822e6 −0.141120
\(113\) 2.22062e6 0.144777 0.0723886 0.997377i \(-0.476938\pi\)
0.0723886 + 0.997377i \(0.476938\pi\)
\(114\) 2.23830e6 0.141498
\(115\) 1.20139e7 0.736619
\(116\) 1.94578e6 0.115742
\(117\) 264227. 0.0152520
\(118\) −1.35655e7 −0.760060
\(119\) −1.02090e7 −0.555351
\(120\) 9.25812e6 0.489090
\(121\) −1.84004e7 −0.944231
\(122\) 9.72577e6 0.484914
\(123\) 452829. 0.0219415
\(124\) −1.04378e7 −0.491624
\(125\) 1.81582e7 0.831550
\(126\) −350409. −0.0156055
\(127\) 1.81924e7 0.788092 0.394046 0.919091i \(-0.371075\pi\)
0.394046 + 0.919091i \(0.371075\pi\)
\(128\) −1.60986e6 −0.0678507
\(129\) −1.53006e6 −0.0627544
\(130\) 2.43366e6 0.0971534
\(131\) 1.42608e7 0.554236 0.277118 0.960836i \(-0.410621\pi\)
0.277118 + 0.960836i \(0.410621\pi\)
\(132\) −2.64665e6 −0.100158
\(133\) 1.98809e6 0.0732750
\(134\) −1.46503e7 −0.525991
\(135\) 1.36784e7 0.478484
\(136\) 4.64808e7 1.58448
\(137\) 3.67856e7 1.22224 0.611119 0.791539i \(-0.290720\pi\)
0.611119 + 0.791539i \(0.290720\pi\)
\(138\) −3.55767e7 −1.15236
\(139\) −4.81539e7 −1.52083 −0.760413 0.649440i \(-0.775003\pi\)
−0.760413 + 0.649440i \(0.775003\pi\)
\(140\) 2.49788e6 0.0769349
\(141\) 1.86178e7 0.559320
\(142\) −1.52942e7 −0.448247
\(143\) −2.29035e6 −0.0654977
\(144\) 735705. 0.0205322
\(145\) 4.54369e6 0.123771
\(146\) 5.11397e7 1.35995
\(147\) 5.34848e6 0.138874
\(148\) −2.14783e6 −0.0544608
\(149\) −4.45417e7 −1.10310 −0.551550 0.834142i \(-0.685964\pi\)
−0.551550 + 0.834142i \(0.685964\pi\)
\(150\) −2.36024e7 −0.571000
\(151\) −6.93213e7 −1.63850 −0.819252 0.573434i \(-0.805611\pi\)
−0.819252 + 0.573434i \(0.805611\pi\)
\(152\) −9.05164e6 −0.209062
\(153\) 3.57960e6 0.0808007
\(154\) 3.03739e6 0.0670159
\(155\) −2.43738e7 −0.525729
\(156\) 5.57769e6 0.117630
\(157\) −3.74846e6 −0.0773044 −0.0386522 0.999253i \(-0.512306\pi\)
−0.0386522 + 0.999253i \(0.512306\pi\)
\(158\) 3.40883e7 0.687551
\(159\) −5.66169e7 −1.11701
\(160\) −1.92908e7 −0.372332
\(161\) −3.15997e7 −0.596751
\(162\) −3.82715e7 −0.707250
\(163\) −1.74242e7 −0.315135 −0.157567 0.987508i \(-0.550365\pi\)
−0.157567 + 0.987508i \(0.550365\pi\)
\(164\) −556254. −0.00984737
\(165\) −6.18031e6 −0.107107
\(166\) 2.39278e7 0.405999
\(167\) −1.07364e8 −1.78382 −0.891909 0.452214i \(-0.850634\pi\)
−0.891909 + 0.452214i \(0.850634\pi\)
\(168\) −2.43513e7 −0.396222
\(169\) 4.82681e6 0.0769231
\(170\) 3.29699e7 0.514691
\(171\) −697091. −0.0106611
\(172\) 1.87952e6 0.0281643
\(173\) −891764. −0.0130945 −0.00654724 0.999979i \(-0.502084\pi\)
−0.00654724 + 0.999979i \(0.502084\pi\)
\(174\) −1.34552e7 −0.193627
\(175\) −2.09640e7 −0.295692
\(176\) −6.37718e6 −0.0881727
\(177\) −7.26011e7 −0.984094
\(178\) −4.06798e7 −0.540641
\(179\) −1.25293e8 −1.63283 −0.816414 0.577467i \(-0.804041\pi\)
−0.816414 + 0.577467i \(0.804041\pi\)
\(180\) −875840. −0.0111936
\(181\) −8.11620e6 −0.101737 −0.0508684 0.998705i \(-0.516199\pi\)
−0.0508684 + 0.998705i \(0.516199\pi\)
\(182\) −6.40115e6 −0.0787061
\(183\) 5.20514e7 0.627846
\(184\) 1.43871e8 1.70260
\(185\) −5.01549e6 −0.0582388
\(186\) 7.21778e7 0.822447
\(187\) −3.10285e7 −0.346988
\(188\) −2.28701e7 −0.251024
\(189\) −3.59778e7 −0.387630
\(190\) −6.42055e6 −0.0679101
\(191\) −2.83512e6 −0.0294411 −0.0147206 0.999892i \(-0.504686\pi\)
−0.0147206 + 0.999892i \(0.504686\pi\)
\(192\) 9.27222e7 0.945430
\(193\) −7.16084e7 −0.716990 −0.358495 0.933532i \(-0.616710\pi\)
−0.358495 + 0.933532i \(0.616710\pi\)
\(194\) −4.12024e7 −0.405151
\(195\) 1.30247e7 0.125790
\(196\) −6.57007e6 −0.0623267
\(197\) 1.95714e8 1.82386 0.911928 0.410350i \(-0.134593\pi\)
0.911928 + 0.410350i \(0.134593\pi\)
\(198\) −1.06501e6 −0.00975047
\(199\) −2.07334e8 −1.86502 −0.932512 0.361140i \(-0.882388\pi\)
−0.932512 + 0.361140i \(0.882388\pi\)
\(200\) 9.54475e7 0.843645
\(201\) −7.84067e7 −0.681031
\(202\) 8.60459e7 0.734515
\(203\) −1.19511e7 −0.100270
\(204\) 7.55635e7 0.623170
\(205\) −1.29893e6 −0.0105305
\(206\) −6.63110e7 −0.528506
\(207\) 1.10799e7 0.0868241
\(208\) 1.34396e7 0.103553
\(209\) 6.04247e6 0.0457828
\(210\) −1.72729e7 −0.128706
\(211\) 2.61035e8 1.91298 0.956488 0.291771i \(-0.0942446\pi\)
0.956488 + 0.291771i \(0.0942446\pi\)
\(212\) 6.95482e7 0.501315
\(213\) −8.18529e7 −0.580371
\(214\) 1.60788e8 1.12152
\(215\) 4.38896e6 0.0301181
\(216\) 1.63804e8 1.10595
\(217\) 6.41093e7 0.425904
\(218\) 1.24072e8 0.811105
\(219\) 2.73695e8 1.76081
\(220\) 7.59189e6 0.0480696
\(221\) 6.53910e7 0.407516
\(222\) 1.48523e7 0.0911085
\(223\) 5.01041e7 0.302556 0.151278 0.988491i \(-0.451661\pi\)
0.151278 + 0.988491i \(0.451661\pi\)
\(224\) 5.07398e7 0.301634
\(225\) 7.35066e6 0.0430217
\(226\) 1.88629e7 0.108700
\(227\) −2.39460e8 −1.35876 −0.679379 0.733788i \(-0.737751\pi\)
−0.679379 + 0.733788i \(0.737751\pi\)
\(228\) −1.47152e7 −0.0822232
\(229\) −1.85129e8 −1.01871 −0.509355 0.860557i \(-0.670116\pi\)
−0.509355 + 0.860557i \(0.670116\pi\)
\(230\) 1.02051e8 0.553059
\(231\) 1.62558e7 0.0867694
\(232\) 5.44124e7 0.286082
\(233\) −4.61056e7 −0.238786 −0.119393 0.992847i \(-0.538095\pi\)
−0.119393 + 0.992847i \(0.538095\pi\)
\(234\) 2.24446e6 0.0114513
\(235\) −5.34049e7 −0.268438
\(236\) 8.91831e7 0.441663
\(237\) 1.82437e8 0.890213
\(238\) −8.67194e7 −0.416962
\(239\) −6.41048e7 −0.303737 −0.151869 0.988401i \(-0.548529\pi\)
−0.151869 + 0.988401i \(0.548529\pi\)
\(240\) 3.62655e7 0.169338
\(241\) −1.61560e8 −0.743486 −0.371743 0.928336i \(-0.621240\pi\)
−0.371743 + 0.928336i \(0.621240\pi\)
\(242\) −1.56301e8 −0.708936
\(243\) 2.45724e7 0.109857
\(244\) −6.39399e7 −0.281778
\(245\) −1.53421e7 −0.0666504
\(246\) 3.84652e6 0.0164738
\(247\) −1.27342e7 −0.0537691
\(248\) −2.91885e8 −1.21515
\(249\) 1.28059e8 0.525670
\(250\) 1.54244e8 0.624335
\(251\) −1.95835e8 −0.781687 −0.390844 0.920457i \(-0.627817\pi\)
−0.390844 + 0.920457i \(0.627817\pi\)
\(252\) 2.30369e6 0.00906820
\(253\) −9.60421e7 −0.372855
\(254\) 1.54534e8 0.591706
\(255\) 1.76452e8 0.666400
\(256\) −2.74742e8 −1.02349
\(257\) −7.76172e7 −0.285228 −0.142614 0.989778i \(-0.545551\pi\)
−0.142614 + 0.989778i \(0.545551\pi\)
\(258\) −1.29970e7 −0.0471166
\(259\) 1.31920e7 0.0471806
\(260\) −1.59995e7 −0.0564548
\(261\) 4.19044e6 0.0145887
\(262\) 1.21138e8 0.416125
\(263\) 8.67584e7 0.294081 0.147040 0.989131i \(-0.453025\pi\)
0.147040 + 0.989131i \(0.453025\pi\)
\(264\) −7.40116e7 −0.247563
\(265\) 1.62405e8 0.536092
\(266\) 1.68877e7 0.0550155
\(267\) −2.17714e8 −0.699999
\(268\) 9.63148e7 0.305648
\(269\) 2.09851e8 0.657322 0.328661 0.944448i \(-0.393403\pi\)
0.328661 + 0.944448i \(0.393403\pi\)
\(270\) 1.16190e8 0.359250
\(271\) −5.33514e8 −1.62837 −0.814185 0.580605i \(-0.802816\pi\)
−0.814185 + 0.580605i \(0.802816\pi\)
\(272\) 1.82072e8 0.548596
\(273\) −3.42583e7 −0.101905
\(274\) 3.12473e8 0.917667
\(275\) −6.37164e7 −0.184751
\(276\) 2.33891e8 0.669625
\(277\) −4.09080e7 −0.115646 −0.0578228 0.998327i \(-0.518416\pi\)
−0.0578228 + 0.998327i \(0.518416\pi\)
\(278\) −4.09040e8 −1.14185
\(279\) −2.24789e7 −0.0619669
\(280\) 6.98514e7 0.190161
\(281\) 3.56604e8 0.958768 0.479384 0.877605i \(-0.340860\pi\)
0.479384 + 0.877605i \(0.340860\pi\)
\(282\) 1.58147e8 0.419942
\(283\) −3.28921e8 −0.862660 −0.431330 0.902194i \(-0.641955\pi\)
−0.431330 + 0.902194i \(0.641955\pi\)
\(284\) 1.00548e8 0.260471
\(285\) −3.43621e7 −0.0879272
\(286\) −1.94552e7 −0.0491762
\(287\) 3.41653e6 0.00853098
\(288\) −1.77910e7 −0.0438862
\(289\) 4.75543e8 1.15890
\(290\) 3.85960e7 0.0929286
\(291\) −2.20511e8 −0.524572
\(292\) −3.36206e8 −0.790253
\(293\) −3.19794e8 −0.742735 −0.371368 0.928486i \(-0.621111\pi\)
−0.371368 + 0.928486i \(0.621111\pi\)
\(294\) 4.54323e7 0.104267
\(295\) 2.08255e8 0.472301
\(296\) −6.00625e7 −0.134612
\(297\) −1.09348e8 −0.242195
\(298\) −3.78357e8 −0.828217
\(299\) 2.02404e8 0.437895
\(300\) 1.55168e8 0.331802
\(301\) −1.15441e7 −0.0243993
\(302\) −5.88845e8 −1.23020
\(303\) 4.60509e8 0.951019
\(304\) −3.54567e7 −0.0723837
\(305\) −1.49309e8 −0.301326
\(306\) 3.04067e7 0.0606658
\(307\) 6.60284e8 1.30241 0.651203 0.758903i \(-0.274265\pi\)
0.651203 + 0.758903i \(0.274265\pi\)
\(308\) −1.99686e7 −0.0389422
\(309\) −3.54890e8 −0.684288
\(310\) −2.07041e8 −0.394722
\(311\) −1.05295e9 −1.98493 −0.992467 0.122514i \(-0.960904\pi\)
−0.992467 + 0.122514i \(0.960904\pi\)
\(312\) 1.55976e8 0.290748
\(313\) 1.95767e8 0.360856 0.180428 0.983588i \(-0.442252\pi\)
0.180428 + 0.983588i \(0.442252\pi\)
\(314\) −3.18410e7 −0.0580408
\(315\) 5.37944e6 0.00969728
\(316\) −2.24106e8 −0.399529
\(317\) −1.66457e8 −0.293491 −0.146745 0.989174i \(-0.546880\pi\)
−0.146745 + 0.989174i \(0.546880\pi\)
\(318\) −4.80928e8 −0.838659
\(319\) −3.63233e7 −0.0626495
\(320\) −2.65973e8 −0.453745
\(321\) 8.60523e8 1.45210
\(322\) −2.68422e8 −0.448046
\(323\) −1.72516e8 −0.284853
\(324\) 2.51607e8 0.410975
\(325\) 1.34279e8 0.216979
\(326\) −1.48009e8 −0.236606
\(327\) 6.64023e8 1.05018
\(328\) −1.55552e7 −0.0243399
\(329\) 1.40469e8 0.217467
\(330\) −5.24982e7 −0.0804165
\(331\) 8.87071e8 1.34450 0.672249 0.740325i \(-0.265328\pi\)
0.672249 + 0.740325i \(0.265328\pi\)
\(332\) −1.57308e8 −0.235921
\(333\) −4.62557e6 −0.00686453
\(334\) −9.11995e8 −1.33931
\(335\) 2.24909e8 0.326851
\(336\) −9.53877e7 −0.137185
\(337\) 1.53476e8 0.218441 0.109221 0.994018i \(-0.465164\pi\)
0.109221 + 0.994018i \(0.465164\pi\)
\(338\) 4.10010e7 0.0577545
\(339\) 1.00952e8 0.140740
\(340\) −2.16753e8 −0.299081
\(341\) 1.94850e8 0.266109
\(342\) −5.92139e6 −0.00800446
\(343\) 4.03536e7 0.0539949
\(344\) 5.25595e7 0.0696140
\(345\) 5.46169e8 0.716078
\(346\) −7.57502e6 −0.00983145
\(347\) −1.15732e9 −1.48697 −0.743483 0.668754i \(-0.766828\pi\)
−0.743483 + 0.668754i \(0.766828\pi\)
\(348\) 8.84579e7 0.112515
\(349\) 5.73828e8 0.722592 0.361296 0.932451i \(-0.382334\pi\)
0.361296 + 0.932451i \(0.382334\pi\)
\(350\) −1.78077e8 −0.222008
\(351\) 2.30446e8 0.284443
\(352\) 1.54215e8 0.188464
\(353\) −1.05519e9 −1.27679 −0.638396 0.769708i \(-0.720402\pi\)
−0.638396 + 0.769708i \(0.720402\pi\)
\(354\) −6.16704e8 −0.738866
\(355\) 2.34794e8 0.278541
\(356\) 2.67440e8 0.314161
\(357\) −4.64113e8 −0.539865
\(358\) −1.06429e9 −1.22594
\(359\) 1.40542e9 1.60315 0.801577 0.597892i \(-0.203995\pi\)
0.801577 + 0.597892i \(0.203995\pi\)
\(360\) −2.44922e7 −0.0276675
\(361\) −8.60276e8 −0.962415
\(362\) −6.89425e7 −0.0763848
\(363\) −8.36506e8 −0.917901
\(364\) 4.20829e7 0.0457353
\(365\) −7.85090e8 −0.845074
\(366\) 4.42147e8 0.471392
\(367\) 2.16768e8 0.228909 0.114455 0.993428i \(-0.463488\pi\)
0.114455 + 0.993428i \(0.463488\pi\)
\(368\) 5.63567e8 0.589493
\(369\) −1.19795e6 −0.00124121
\(370\) −4.26038e7 −0.0437262
\(371\) −4.27167e8 −0.434300
\(372\) −4.74516e8 −0.477915
\(373\) −4.50926e8 −0.449909 −0.224954 0.974369i \(-0.572223\pi\)
−0.224954 + 0.974369i \(0.572223\pi\)
\(374\) −2.63569e8 −0.260522
\(375\) 8.25497e8 0.808363
\(376\) −6.39544e8 −0.620459
\(377\) 7.65496e7 0.0735780
\(378\) −3.05611e8 −0.291036
\(379\) 3.61271e8 0.340875 0.170438 0.985368i \(-0.445482\pi\)
0.170438 + 0.985368i \(0.445482\pi\)
\(380\) 4.22104e7 0.0394618
\(381\) 8.27051e8 0.766117
\(382\) −2.40827e7 −0.0221046
\(383\) −5.84010e8 −0.531159 −0.265580 0.964089i \(-0.585563\pi\)
−0.265580 + 0.964089i \(0.585563\pi\)
\(384\) −7.31866e7 −0.0659587
\(385\) −4.66296e7 −0.0416437
\(386\) −6.08272e8 −0.538322
\(387\) 4.04775e6 0.00354997
\(388\) 2.70876e8 0.235429
\(389\) −1.27226e9 −1.09585 −0.547926 0.836527i \(-0.684582\pi\)
−0.547926 + 0.836527i \(0.684582\pi\)
\(390\) 1.10637e8 0.0944443
\(391\) 2.74206e9 2.31984
\(392\) −1.83727e8 −0.154054
\(393\) 6.48316e8 0.538782
\(394\) 1.66248e9 1.36937
\(395\) −5.23319e8 −0.427245
\(396\) 7.00167e6 0.00566589
\(397\) 8.15517e8 0.654133 0.327067 0.945001i \(-0.393940\pi\)
0.327067 + 0.945001i \(0.393940\pi\)
\(398\) −1.76118e9 −1.40028
\(399\) 9.03813e7 0.0712317
\(400\) 3.73883e8 0.292096
\(401\) 6.46821e7 0.0500932 0.0250466 0.999686i \(-0.492027\pi\)
0.0250466 + 0.999686i \(0.492027\pi\)
\(402\) −6.66020e8 −0.511324
\(403\) −4.10636e8 −0.312528
\(404\) −5.65689e8 −0.426818
\(405\) 5.87539e8 0.439485
\(406\) −1.01517e8 −0.0752835
\(407\) 4.00950e7 0.0294788
\(408\) 2.11308e9 1.54030
\(409\) 8.76118e8 0.633186 0.316593 0.948562i \(-0.397461\pi\)
0.316593 + 0.948562i \(0.397461\pi\)
\(410\) −1.10337e7 −0.00790638
\(411\) 1.67232e9 1.18816
\(412\) 4.35946e8 0.307109
\(413\) −5.47766e8 −0.382622
\(414\) 9.41176e7 0.0651883
\(415\) −3.67337e8 −0.252288
\(416\) −3.25001e8 −0.221339
\(417\) −2.18914e9 −1.47842
\(418\) 5.13273e7 0.0343741
\(419\) 8.56812e8 0.569032 0.284516 0.958671i \(-0.408167\pi\)
0.284516 + 0.958671i \(0.408167\pi\)
\(420\) 1.13557e8 0.0747896
\(421\) 1.04415e9 0.681987 0.340994 0.940066i \(-0.389237\pi\)
0.340994 + 0.940066i \(0.389237\pi\)
\(422\) 2.21734e9 1.43628
\(423\) −4.92530e7 −0.0316403
\(424\) 1.94486e9 1.23911
\(425\) 1.81914e9 1.14949
\(426\) −6.95294e8 −0.435747
\(427\) 3.92721e8 0.244111
\(428\) −1.05707e9 −0.651702
\(429\) −1.04122e8 −0.0636713
\(430\) 3.72817e7 0.0226129
\(431\) −1.35707e9 −0.816456 −0.408228 0.912880i \(-0.633853\pi\)
−0.408228 + 0.912880i \(0.633853\pi\)
\(432\) 6.41647e8 0.382916
\(433\) 2.21225e9 1.30956 0.654781 0.755819i \(-0.272761\pi\)
0.654781 + 0.755819i \(0.272761\pi\)
\(434\) 5.44572e8 0.319773
\(435\) 2.06562e8 0.120320
\(436\) −8.15685e8 −0.471324
\(437\) −5.33988e8 −0.306088
\(438\) 2.32488e9 1.32203
\(439\) 4.10584e7 0.0231620 0.0115810 0.999933i \(-0.496314\pi\)
0.0115810 + 0.999933i \(0.496314\pi\)
\(440\) 2.12302e8 0.118814
\(441\) −1.41493e7 −0.00785598
\(442\) 5.55459e8 0.305967
\(443\) 1.19657e9 0.653920 0.326960 0.945038i \(-0.393976\pi\)
0.326960 + 0.945038i \(0.393976\pi\)
\(444\) −9.76432e7 −0.0529422
\(445\) 6.24511e8 0.335954
\(446\) 4.25606e8 0.227162
\(447\) −2.02493e9 −1.07234
\(448\) 6.99577e8 0.367589
\(449\) −8.48590e8 −0.442421 −0.221210 0.975226i \(-0.571001\pi\)
−0.221210 + 0.975226i \(0.571001\pi\)
\(450\) 6.24397e7 0.0323011
\(451\) 1.03840e7 0.00533023
\(452\) −1.24010e8 −0.0631644
\(453\) −3.15144e9 −1.59281
\(454\) −2.03407e9 −1.02017
\(455\) 9.82697e7 0.0489080
\(456\) −4.11500e8 −0.203232
\(457\) 1.07158e9 0.525192 0.262596 0.964906i \(-0.415421\pi\)
0.262596 + 0.964906i \(0.415421\pi\)
\(458\) −1.57257e9 −0.764856
\(459\) 3.12196e9 1.50690
\(460\) −6.70914e8 −0.321377
\(461\) −1.77089e8 −0.0841857 −0.0420929 0.999114i \(-0.513403\pi\)
−0.0420929 + 0.999114i \(0.513403\pi\)
\(462\) 1.38084e8 0.0651472
\(463\) −5.19209e7 −0.0243113 −0.0121557 0.999926i \(-0.503869\pi\)
−0.0121557 + 0.999926i \(0.503869\pi\)
\(464\) 2.13142e8 0.0990503
\(465\) −1.10806e9 −0.511069
\(466\) −3.91641e8 −0.179282
\(467\) −6.33313e8 −0.287746 −0.143873 0.989596i \(-0.545956\pi\)
−0.143873 + 0.989596i \(0.545956\pi\)
\(468\) −1.47557e7 −0.00665424
\(469\) −5.91569e8 −0.264789
\(470\) −4.53644e8 −0.201545
\(471\) −1.70410e8 −0.0751488
\(472\) 2.49394e9 1.09166
\(473\) −3.50864e7 −0.0152449
\(474\) 1.54970e9 0.668379
\(475\) −3.54260e8 −0.151668
\(476\) 5.70117e8 0.242292
\(477\) 1.49779e8 0.0631883
\(478\) −5.44533e8 −0.228048
\(479\) 2.66856e9 1.10944 0.554718 0.832038i \(-0.312826\pi\)
0.554718 + 0.832038i \(0.312826\pi\)
\(480\) −8.76985e8 −0.361949
\(481\) −8.44983e7 −0.0346211
\(482\) −1.37236e9 −0.558216
\(483\) −1.43657e9 −0.580111
\(484\) 1.02756e9 0.411955
\(485\) 6.32535e8 0.251761
\(486\) 2.08729e8 0.0824813
\(487\) −6.98548e7 −0.0274060 −0.0137030 0.999906i \(-0.504362\pi\)
−0.0137030 + 0.999906i \(0.504362\pi\)
\(488\) −1.78803e9 −0.696475
\(489\) −7.92128e8 −0.306347
\(490\) −1.30322e8 −0.0500417
\(491\) 2.78944e9 1.06348 0.531742 0.846906i \(-0.321538\pi\)
0.531742 + 0.846906i \(0.321538\pi\)
\(492\) −2.52881e7 −0.00957277
\(493\) 1.03705e9 0.389795
\(494\) −1.08170e8 −0.0403703
\(495\) 1.63499e7 0.00605894
\(496\) −1.14336e9 −0.420724
\(497\) −6.17570e8 −0.225652
\(498\) 1.08779e9 0.394678
\(499\) −3.61891e9 −1.30385 −0.651923 0.758286i \(-0.726037\pi\)
−0.651923 + 0.758286i \(0.726037\pi\)
\(500\) −1.01404e9 −0.362794
\(501\) −4.88091e9 −1.73408
\(502\) −1.66351e9 −0.586897
\(503\) −6.21040e8 −0.217586 −0.108793 0.994064i \(-0.534699\pi\)
−0.108793 + 0.994064i \(0.534699\pi\)
\(504\) 6.44209e7 0.0224140
\(505\) −1.32097e9 −0.456428
\(506\) −8.15823e8 −0.279943
\(507\) 2.19433e8 0.0747781
\(508\) −1.01595e9 −0.343834
\(509\) −6.97982e8 −0.234602 −0.117301 0.993096i \(-0.537424\pi\)
−0.117301 + 0.993096i \(0.537424\pi\)
\(510\) 1.49886e9 0.500339
\(511\) 2.06499e9 0.684613
\(512\) −2.12771e9 −0.700596
\(513\) −6.07970e8 −0.198825
\(514\) −6.59314e8 −0.214151
\(515\) 1.01800e9 0.328414
\(516\) 8.54457e7 0.0273789
\(517\) 4.26931e8 0.135875
\(518\) 1.12059e8 0.0354236
\(519\) −4.05408e7 −0.0127293
\(520\) −4.47415e8 −0.139540
\(521\) −3.31626e9 −1.02735 −0.513673 0.857986i \(-0.671715\pi\)
−0.513673 + 0.857986i \(0.671715\pi\)
\(522\) 3.55954e7 0.0109534
\(523\) 1.38363e9 0.422926 0.211463 0.977386i \(-0.432177\pi\)
0.211463 + 0.977386i \(0.432177\pi\)
\(524\) −7.96391e8 −0.241806
\(525\) −9.53050e8 −0.287447
\(526\) 7.36963e8 0.220798
\(527\) −5.56308e9 −1.65569
\(528\) −2.89915e8 −0.0857140
\(529\) 5.08266e9 1.49278
\(530\) 1.37954e9 0.402502
\(531\) 1.92065e8 0.0556695
\(532\) −1.11024e8 −0.0319689
\(533\) −2.18837e7 −0.00626003
\(534\) −1.84936e9 −0.525565
\(535\) −2.46840e9 −0.696912
\(536\) 2.69337e9 0.755474
\(537\) −5.69598e9 −1.58730
\(538\) 1.78257e9 0.493523
\(539\) 1.22648e8 0.0337365
\(540\) −7.63867e8 −0.208756
\(541\) 6.14028e9 1.66724 0.833620 0.552339i \(-0.186265\pi\)
0.833620 + 0.552339i \(0.186265\pi\)
\(542\) −4.53190e9 −1.22259
\(543\) −3.68973e8 −0.0988998
\(544\) −4.40294e9 −1.17259
\(545\) −1.90474e9 −0.504021
\(546\) −2.91005e8 −0.0765114
\(547\) 6.08930e9 1.59079 0.795393 0.606094i \(-0.207265\pi\)
0.795393 + 0.606094i \(0.207265\pi\)
\(548\) −2.05428e9 −0.533246
\(549\) −1.37701e8 −0.0355168
\(550\) −5.41235e8 −0.138713
\(551\) −2.01955e8 −0.0514309
\(552\) 6.54059e9 1.65512
\(553\) 1.37646e9 0.346120
\(554\) −3.47490e8 −0.0868278
\(555\) −2.28011e8 −0.0566149
\(556\) 2.68914e9 0.663516
\(557\) −4.90112e9 −1.20172 −0.600858 0.799356i \(-0.705174\pi\)
−0.600858 + 0.799356i \(0.705174\pi\)
\(558\) −1.90945e8 −0.0465253
\(559\) 7.39429e7 0.0179042
\(560\) 2.73619e8 0.0658397
\(561\) −1.41060e9 −0.337312
\(562\) 3.02914e9 0.719851
\(563\) 3.19645e9 0.754899 0.377450 0.926030i \(-0.376801\pi\)
0.377450 + 0.926030i \(0.376801\pi\)
\(564\) −1.03970e9 −0.244024
\(565\) −2.89581e8 −0.0675462
\(566\) −2.79400e9 −0.647692
\(567\) −1.54538e9 −0.356037
\(568\) 2.81175e9 0.643811
\(569\) −7.74869e9 −1.76334 −0.881668 0.471870i \(-0.843579\pi\)
−0.881668 + 0.471870i \(0.843579\pi\)
\(570\) −2.91887e8 −0.0660165
\(571\) 6.14695e9 1.38176 0.690881 0.722968i \(-0.257223\pi\)
0.690881 + 0.722968i \(0.257223\pi\)
\(572\) 1.27904e8 0.0285758
\(573\) −1.28888e8 −0.0286202
\(574\) 2.90215e7 0.00640513
\(575\) 5.63078e9 1.23518
\(576\) −2.45295e8 −0.0534823
\(577\) −3.50140e9 −0.758797 −0.379399 0.925233i \(-0.623869\pi\)
−0.379399 + 0.925233i \(0.623869\pi\)
\(578\) 4.03947e9 0.870115
\(579\) −3.25541e9 −0.696997
\(580\) −2.53741e8 −0.0539998
\(581\) 9.66191e8 0.204384
\(582\) −1.87312e9 −0.393853
\(583\) −1.29830e9 −0.271354
\(584\) −9.40176e9 −1.95328
\(585\) −3.44566e7 −0.00711586
\(586\) −2.71647e9 −0.557652
\(587\) 5.64465e9 1.15187 0.575935 0.817495i \(-0.304638\pi\)
0.575935 + 0.817495i \(0.304638\pi\)
\(588\) −2.98684e8 −0.0605887
\(589\) 1.08335e9 0.218457
\(590\) 1.76901e9 0.354608
\(591\) 8.89743e9 1.77300
\(592\) −2.35274e8 −0.0466067
\(593\) −8.67330e9 −1.70802 −0.854010 0.520256i \(-0.825836\pi\)
−0.854010 + 0.520256i \(0.825836\pi\)
\(594\) −9.28852e8 −0.181842
\(595\) 1.33131e9 0.259101
\(596\) 2.48742e9 0.481268
\(597\) −9.42567e9 −1.81302
\(598\) 1.71931e9 0.328776
\(599\) −4.08098e9 −0.775837 −0.387918 0.921694i \(-0.626806\pi\)
−0.387918 + 0.921694i \(0.626806\pi\)
\(600\) 4.33917e9 0.820120
\(601\) −3.62520e9 −0.681196 −0.340598 0.940209i \(-0.610629\pi\)
−0.340598 + 0.940209i \(0.610629\pi\)
\(602\) −9.80606e7 −0.0183192
\(603\) 2.07424e8 0.0385255
\(604\) 3.87123e9 0.714857
\(605\) 2.39951e9 0.440533
\(606\) 3.91176e9 0.714033
\(607\) 8.74425e9 1.58695 0.793473 0.608605i \(-0.208271\pi\)
0.793473 + 0.608605i \(0.208271\pi\)
\(608\) 8.57426e8 0.154716
\(609\) −5.43312e8 −0.0974739
\(610\) −1.26829e9 −0.226238
\(611\) −8.99737e8 −0.159577
\(612\) −1.99902e8 −0.0352523
\(613\) −7.68701e9 −1.34786 −0.673932 0.738794i \(-0.735396\pi\)
−0.673932 + 0.738794i \(0.735396\pi\)
\(614\) 5.60874e9 0.977858
\(615\) −5.90513e7 −0.0102369
\(616\) −5.58408e8 −0.0962541
\(617\) 6.45616e9 1.10656 0.553282 0.832994i \(-0.313375\pi\)
0.553282 + 0.832994i \(0.313375\pi\)
\(618\) −3.01458e9 −0.513769
\(619\) −3.22485e9 −0.546502 −0.273251 0.961943i \(-0.588099\pi\)
−0.273251 + 0.961943i \(0.588099\pi\)
\(620\) 1.36115e9 0.229368
\(621\) 9.66339e9 1.61923
\(622\) −8.94420e9 −1.49030
\(623\) −1.64263e9 −0.272164
\(624\) 6.10982e8 0.100666
\(625\) 2.40702e9 0.394367
\(626\) 1.66293e9 0.270934
\(627\) 2.74699e8 0.0445062
\(628\) 2.09332e8 0.0337269
\(629\) −1.14474e9 −0.183412
\(630\) 4.56953e7 0.00728080
\(631\) 3.24674e9 0.514452 0.257226 0.966351i \(-0.417192\pi\)
0.257226 + 0.966351i \(0.417192\pi\)
\(632\) −6.26695e9 −0.987521
\(633\) 1.18670e10 1.85963
\(634\) −1.41396e9 −0.220355
\(635\) −2.37239e9 −0.367686
\(636\) 3.16175e9 0.487336
\(637\) −2.58475e8 −0.0396214
\(638\) −3.08545e8 −0.0470378
\(639\) 2.16541e8 0.0328312
\(640\) 2.09935e8 0.0316559
\(641\) 6.88178e9 1.03204 0.516022 0.856576i \(-0.327412\pi\)
0.516022 + 0.856576i \(0.327412\pi\)
\(642\) 7.30965e9 1.09025
\(643\) 4.29735e9 0.637473 0.318737 0.947843i \(-0.396741\pi\)
0.318737 + 0.947843i \(0.396741\pi\)
\(644\) 1.76468e9 0.260354
\(645\) 1.99528e8 0.0292782
\(646\) −1.46543e9 −0.213870
\(647\) 1.11221e10 1.61444 0.807218 0.590254i \(-0.200972\pi\)
0.807218 + 0.590254i \(0.200972\pi\)
\(648\) 7.03601e9 1.01581
\(649\) −1.66484e9 −0.239065
\(650\) 1.14063e9 0.162910
\(651\) 2.91450e9 0.414028
\(652\) 9.73050e8 0.137489
\(653\) −1.26139e10 −1.77277 −0.886387 0.462945i \(-0.846793\pi\)
−0.886387 + 0.462945i \(0.846793\pi\)
\(654\) 5.64049e9 0.788488
\(655\) −1.85969e9 −0.258580
\(656\) −6.09323e7 −0.00842722
\(657\) −7.24054e8 −0.0996076
\(658\) 1.19320e9 0.163276
\(659\) −7.31188e9 −0.995245 −0.497623 0.867394i \(-0.665794\pi\)
−0.497623 + 0.867394i \(0.665794\pi\)
\(660\) 3.45137e8 0.0467292
\(661\) −8.30130e9 −1.11800 −0.558999 0.829168i \(-0.688815\pi\)
−0.558999 + 0.829168i \(0.688815\pi\)
\(662\) 7.53516e9 1.00946
\(663\) 2.97276e9 0.396153
\(664\) −4.39900e9 −0.583131
\(665\) −2.59258e8 −0.0341866
\(666\) −3.92916e7 −0.00515394
\(667\) 3.20998e9 0.418853
\(668\) 5.99570e9 0.778256
\(669\) 2.27780e9 0.294120
\(670\) 1.91047e9 0.245403
\(671\) 1.19361e9 0.152522
\(672\) 2.30670e9 0.293223
\(673\) 9.26631e9 1.17180 0.585901 0.810383i \(-0.300741\pi\)
0.585901 + 0.810383i \(0.300741\pi\)
\(674\) 1.30369e9 0.164008
\(675\) 6.41090e9 0.802336
\(676\) −2.69552e8 −0.0335605
\(677\) 1.22345e10 1.51539 0.757695 0.652609i \(-0.226326\pi\)
0.757695 + 0.652609i \(0.226326\pi\)
\(678\) 8.57534e8 0.105669
\(679\) −1.66373e9 −0.203957
\(680\) −6.06134e9 −0.739244
\(681\) −1.08862e10 −1.32087
\(682\) 1.65514e9 0.199797
\(683\) −6.37930e9 −0.766127 −0.383064 0.923722i \(-0.625131\pi\)
−0.383064 + 0.923722i \(0.625131\pi\)
\(684\) 3.89288e7 0.00465131
\(685\) −4.79704e9 −0.570238
\(686\) 3.42781e8 0.0405398
\(687\) −8.41622e9 −0.990303
\(688\) 2.05884e8 0.0241025
\(689\) 2.73611e9 0.318689
\(690\) 4.63940e9 0.537638
\(691\) −3.63823e9 −0.419486 −0.209743 0.977757i \(-0.567263\pi\)
−0.209743 + 0.977757i \(0.567263\pi\)
\(692\) 4.98003e7 0.00571295
\(693\) −4.30045e7 −0.00490848
\(694\) −9.83079e9 −1.11643
\(695\) 6.27953e9 0.709545
\(696\) 2.47366e9 0.278104
\(697\) −2.96469e8 −0.0331638
\(698\) 4.87434e9 0.542528
\(699\) −2.09602e9 −0.232127
\(700\) 1.17073e9 0.129007
\(701\) −1.08142e10 −1.18572 −0.592858 0.805307i \(-0.702001\pi\)
−0.592858 + 0.805307i \(0.702001\pi\)
\(702\) 1.95751e9 0.213562
\(703\) 2.22926e8 0.0242001
\(704\) 2.12625e9 0.229673
\(705\) −2.42786e9 −0.260952
\(706\) −8.96326e9 −0.958626
\(707\) 3.47448e9 0.369762
\(708\) 4.05438e9 0.429347
\(709\) −7.50891e9 −0.791252 −0.395626 0.918412i \(-0.629472\pi\)
−0.395626 + 0.918412i \(0.629472\pi\)
\(710\) 1.99444e9 0.209131
\(711\) −4.82634e8 −0.0503587
\(712\) 7.47877e9 0.776515
\(713\) −1.72193e10 −1.77911
\(714\) −3.94238e9 −0.405335
\(715\) 2.98674e8 0.0305581
\(716\) 6.99694e9 0.712381
\(717\) −2.91429e9 −0.295267
\(718\) 1.19382e10 1.20366
\(719\) 6.47250e9 0.649413 0.324707 0.945815i \(-0.394734\pi\)
0.324707 + 0.945815i \(0.394734\pi\)
\(720\) −9.59399e7 −0.00957933
\(721\) −2.67760e9 −0.266055
\(722\) −7.30755e9 −0.722590
\(723\) −7.34471e9 −0.722754
\(724\) 4.53247e8 0.0443864
\(725\) 2.12957e9 0.207543
\(726\) −7.10564e9 −0.689168
\(727\) −1.98504e10 −1.91602 −0.958009 0.286739i \(-0.907429\pi\)
−0.958009 + 0.286739i \(0.907429\pi\)
\(728\) 1.17682e9 0.113045
\(729\) 1.09706e10 1.04878
\(730\) −6.66889e9 −0.634489
\(731\) 1.00174e9 0.0948513
\(732\) −2.90679e9 −0.273921
\(733\) −1.49386e10 −1.40102 −0.700511 0.713642i \(-0.747045\pi\)
−0.700511 + 0.713642i \(0.747045\pi\)
\(734\) 1.84132e9 0.171867
\(735\) −6.97471e8 −0.0647918
\(736\) −1.36284e10 −1.26000
\(737\) −1.79797e9 −0.165443
\(738\) −1.01759e7 −0.000931914 0
\(739\) −1.03698e10 −0.945182 −0.472591 0.881282i \(-0.656681\pi\)
−0.472591 + 0.881282i \(0.656681\pi\)
\(740\) 2.80089e8 0.0254088
\(741\) −5.78915e8 −0.0522698
\(742\) −3.62854e9 −0.326076
\(743\) −6.40768e9 −0.573112 −0.286556 0.958063i \(-0.592511\pi\)
−0.286556 + 0.958063i \(0.592511\pi\)
\(744\) −1.32695e10 −1.18127
\(745\) 5.80848e9 0.514655
\(746\) −3.83036e9 −0.337795
\(747\) −3.38779e8 −0.0297368
\(748\) 1.73277e9 0.151386
\(749\) 6.49254e9 0.564584
\(750\) 7.01213e9 0.606925
\(751\) 3.61512e9 0.311446 0.155723 0.987801i \(-0.450229\pi\)
0.155723 + 0.987801i \(0.450229\pi\)
\(752\) −2.50520e9 −0.214822
\(753\) −8.90294e9 −0.759890
\(754\) 6.50245e8 0.0552430
\(755\) 9.03987e9 0.764448
\(756\) 2.00917e9 0.169118
\(757\) 4.97903e8 0.0417166 0.0208583 0.999782i \(-0.493360\pi\)
0.0208583 + 0.999782i \(0.493360\pi\)
\(758\) 3.06879e9 0.255932
\(759\) −4.36620e9 −0.362458
\(760\) 1.18038e9 0.0975384
\(761\) 2.27559e9 0.187175 0.0935874 0.995611i \(-0.470167\pi\)
0.0935874 + 0.995611i \(0.470167\pi\)
\(762\) 7.02532e9 0.575207
\(763\) 5.00997e9 0.408318
\(764\) 1.58326e8 0.0128448
\(765\) −4.66800e8 −0.0376978
\(766\) −4.96083e9 −0.398799
\(767\) 3.50858e9 0.280768
\(768\) −1.24901e10 −0.994952
\(769\) −1.17120e10 −0.928726 −0.464363 0.885645i \(-0.653717\pi\)
−0.464363 + 0.885645i \(0.653717\pi\)
\(770\) −3.96092e8 −0.0312665
\(771\) −3.52858e9 −0.277274
\(772\) 3.99895e9 0.312813
\(773\) 3.36168e9 0.261775 0.130887 0.991397i \(-0.458217\pi\)
0.130887 + 0.991397i \(0.458217\pi\)
\(774\) 3.43833e7 0.00266535
\(775\) −1.14237e10 −0.881557
\(776\) 7.57485e9 0.581913
\(777\) 5.99728e8 0.0458649
\(778\) −1.08071e10 −0.822775
\(779\) 5.77343e7 0.00437575
\(780\) −7.27361e8 −0.0548806
\(781\) −1.87700e9 −0.140989
\(782\) 2.32923e10 1.74176
\(783\) 3.65471e9 0.272074
\(784\) −7.19688e8 −0.0533382
\(785\) 4.88820e8 0.0360666
\(786\) 5.50707e9 0.404522
\(787\) 9.64652e9 0.705439 0.352719 0.935729i \(-0.385257\pi\)
0.352719 + 0.935729i \(0.385257\pi\)
\(788\) −1.09296e10 −0.795724
\(789\) 3.94415e9 0.285880
\(790\) −4.44529e9 −0.320779
\(791\) 7.61674e8 0.0547206
\(792\) 1.95796e8 0.0140045
\(793\) −2.51548e9 −0.179128
\(794\) 6.92735e9 0.491129
\(795\) 7.38315e9 0.521143
\(796\) 1.15785e10 0.813685
\(797\) −9.41347e9 −0.658636 −0.329318 0.944219i \(-0.606819\pi\)
−0.329318 + 0.944219i \(0.606819\pi\)
\(798\) 7.67737e8 0.0534814
\(799\) −1.21891e10 −0.845395
\(800\) −9.04136e9 −0.624336
\(801\) 5.75959e8 0.0395984
\(802\) 5.49437e8 0.0376104
\(803\) 6.27619e9 0.427752
\(804\) 4.37860e9 0.297125
\(805\) 4.12078e9 0.278416
\(806\) −3.48812e9 −0.234649
\(807\) 9.54011e9 0.638993
\(808\) −1.58191e10 −1.05497
\(809\) −1.90105e8 −0.0126233 −0.00631167 0.999980i \(-0.502009\pi\)
−0.00631167 + 0.999980i \(0.502009\pi\)
\(810\) 4.99081e9 0.329969
\(811\) 1.18714e10 0.781501 0.390750 0.920497i \(-0.372216\pi\)
0.390750 + 0.920497i \(0.372216\pi\)
\(812\) 6.67404e8 0.0437464
\(813\) −2.42543e10 −1.58296
\(814\) 3.40584e8 0.0221329
\(815\) 2.27221e9 0.147027
\(816\) 8.27725e9 0.533299
\(817\) −1.95078e8 −0.0125150
\(818\) 7.44212e9 0.475401
\(819\) 9.06299e7 0.00576471
\(820\) 7.25386e7 0.00459431
\(821\) 2.50076e10 1.57714 0.788571 0.614944i \(-0.210821\pi\)
0.788571 + 0.614944i \(0.210821\pi\)
\(822\) 1.42054e10 0.892078
\(823\) 2.48122e10 1.55155 0.775776 0.631009i \(-0.217359\pi\)
0.775776 + 0.631009i \(0.217359\pi\)
\(824\) 1.21909e10 0.759086
\(825\) −2.89663e9 −0.179599
\(826\) −4.65296e9 −0.287276
\(827\) 1.77670e10 1.09231 0.546155 0.837684i \(-0.316091\pi\)
0.546155 + 0.837684i \(0.316091\pi\)
\(828\) −6.18755e8 −0.0378802
\(829\) −9.96869e9 −0.607711 −0.303856 0.952718i \(-0.598274\pi\)
−0.303856 + 0.952718i \(0.598274\pi\)
\(830\) −3.12032e9 −0.189420
\(831\) −1.85973e9 −0.112421
\(832\) −4.48096e9 −0.269737
\(833\) −3.50168e9 −0.209903
\(834\) −1.85955e10 −1.11001
\(835\) 1.40008e10 0.832245
\(836\) −3.37440e8 −0.0199744
\(837\) −1.96050e10 −1.15565
\(838\) 7.27813e9 0.427234
\(839\) −2.61866e10 −1.53078 −0.765390 0.643567i \(-0.777454\pi\)
−0.765390 + 0.643567i \(0.777454\pi\)
\(840\) 3.17554e9 0.184859
\(841\) −1.60359e10 −0.929622
\(842\) 8.86947e9 0.512042
\(843\) 1.62117e10 0.932033
\(844\) −1.45774e10 −0.834606
\(845\) −6.29442e8 −0.0358887
\(846\) −4.18376e8 −0.0237558
\(847\) −6.31133e9 −0.356886
\(848\) 7.61834e9 0.429017
\(849\) −1.49532e10 −0.838605
\(850\) 1.54526e10 0.863048
\(851\) −3.54330e9 −0.197085
\(852\) 4.57105e9 0.253208
\(853\) 1.60269e9 0.0884151 0.0442076 0.999022i \(-0.485924\pi\)
0.0442076 + 0.999022i \(0.485924\pi\)
\(854\) 3.33594e9 0.183280
\(855\) 9.09044e7 0.00497398
\(856\) −2.95601e10 −1.61082
\(857\) 5.04497e9 0.273795 0.136898 0.990585i \(-0.456287\pi\)
0.136898 + 0.990585i \(0.456287\pi\)
\(858\) −8.84461e8 −0.0478050
\(859\) −6.78061e9 −0.365000 −0.182500 0.983206i \(-0.558419\pi\)
−0.182500 + 0.983206i \(0.558419\pi\)
\(860\) −2.45100e8 −0.0131401
\(861\) 1.55320e8 0.00829310
\(862\) −1.15276e10 −0.613002
\(863\) −7.09709e9 −0.375874 −0.187937 0.982181i \(-0.560180\pi\)
−0.187937 + 0.982181i \(0.560180\pi\)
\(864\) −1.55165e10 −0.818458
\(865\) 1.16291e8 0.00610927
\(866\) 1.87918e10 0.983230
\(867\) 2.16188e10 1.12659
\(868\) −3.58017e9 −0.185816
\(869\) 4.18353e9 0.216259
\(870\) 1.75463e9 0.0903373
\(871\) 3.78914e9 0.194302
\(872\) −2.28100e10 −1.16498
\(873\) 5.83359e8 0.0296747
\(874\) −4.53593e9 −0.229814
\(875\) 6.22827e9 0.314296
\(876\) −1.52844e10 −0.768217
\(877\) −6.12021e8 −0.0306385 −0.0153192 0.999883i \(-0.504876\pi\)
−0.0153192 + 0.999883i \(0.504876\pi\)
\(878\) 3.48768e8 0.0173903
\(879\) −1.45383e10 −0.722024
\(880\) 8.31619e8 0.0411372
\(881\) 2.62029e10 1.29102 0.645512 0.763750i \(-0.276644\pi\)
0.645512 + 0.763750i \(0.276644\pi\)
\(882\) −1.20190e8 −0.00589834
\(883\) 2.83780e10 1.38714 0.693569 0.720391i \(-0.256037\pi\)
0.693569 + 0.720391i \(0.256037\pi\)
\(884\) −3.65174e9 −0.177794
\(885\) 9.46757e9 0.459131
\(886\) 1.01642e10 0.490969
\(887\) −2.43273e10 −1.17047 −0.585235 0.810864i \(-0.698998\pi\)
−0.585235 + 0.810864i \(0.698998\pi\)
\(888\) −2.73052e9 −0.130858
\(889\) 6.23999e9 0.297871
\(890\) 5.30487e9 0.252237
\(891\) −4.69692e9 −0.222455
\(892\) −2.79805e9 −0.132001
\(893\) 2.37371e9 0.111544
\(894\) −1.72006e10 −0.805123
\(895\) 1.63389e10 0.761800
\(896\) −5.52183e8 −0.0256451
\(897\) 9.20156e9 0.425685
\(898\) −7.20828e9 −0.332173
\(899\) −6.51238e9 −0.298938
\(900\) −4.10495e8 −0.0187698
\(901\) 3.70674e10 1.68832
\(902\) 8.82060e7 0.00400198
\(903\) −5.24810e8 −0.0237189
\(904\) −3.46785e9 −0.156124
\(905\) 1.05840e9 0.0474655
\(906\) −2.67697e10 −1.19590
\(907\) 2.55913e10 1.13885 0.569425 0.822043i \(-0.307166\pi\)
0.569425 + 0.822043i \(0.307166\pi\)
\(908\) 1.33726e10 0.592808
\(909\) −1.21827e9 −0.0537984
\(910\) 8.34745e8 0.0367205
\(911\) 3.22614e9 0.141374 0.0706869 0.997499i \(-0.477481\pi\)
0.0706869 + 0.997499i \(0.477481\pi\)
\(912\) −1.61191e9 −0.0703653
\(913\) 2.93658e9 0.127701
\(914\) 9.10246e9 0.394319
\(915\) −6.78778e9 −0.292923
\(916\) 1.03385e10 0.444449
\(917\) 4.89146e9 0.209482
\(918\) 2.65193e10 1.13139
\(919\) −2.19319e10 −0.932122 −0.466061 0.884753i \(-0.654327\pi\)
−0.466061 + 0.884753i \(0.654327\pi\)
\(920\) −1.87616e10 −0.794352
\(921\) 3.00174e10 1.26609
\(922\) −1.50427e9 −0.0632073
\(923\) 3.95569e9 0.165583
\(924\) −9.07801e8 −0.0378563
\(925\) −2.35070e9 −0.0976566
\(926\) −4.41038e8 −0.0182531
\(927\) 9.38855e8 0.0387097
\(928\) −5.15427e9 −0.211714
\(929\) −7.69802e9 −0.315010 −0.157505 0.987518i \(-0.550345\pi\)
−0.157505 + 0.987518i \(0.550345\pi\)
\(930\) −9.41237e9 −0.383715
\(931\) 6.81915e8 0.0276953
\(932\) 2.57475e9 0.104179
\(933\) −4.78685e10 −1.92958
\(934\) −5.37963e9 −0.216042
\(935\) 4.04628e9 0.161888
\(936\) −4.12632e8 −0.0164474
\(937\) 1.96480e10 0.780241 0.390121 0.920764i \(-0.372433\pi\)
0.390121 + 0.920764i \(0.372433\pi\)
\(938\) −5.02504e9 −0.198806
\(939\) 8.89982e9 0.350794
\(940\) 2.98238e9 0.117116
\(941\) −8.52895e9 −0.333681 −0.166841 0.985984i \(-0.553357\pi\)
−0.166841 + 0.985984i \(0.553357\pi\)
\(942\) −1.44754e9 −0.0564224
\(943\) −9.17658e8 −0.0356361
\(944\) 9.76915e9 0.377968
\(945\) 4.69170e9 0.180850
\(946\) −2.98039e8 −0.0114460
\(947\) 7.92096e8 0.0303077 0.0151539 0.999885i \(-0.495176\pi\)
0.0151539 + 0.999885i \(0.495176\pi\)
\(948\) −1.01881e10 −0.388388
\(949\) −1.32268e10 −0.502368
\(950\) −3.00923e9 −0.113874
\(951\) −7.56735e9 −0.285307
\(952\) 1.59429e10 0.598877
\(953\) 1.74946e10 0.654755 0.327377 0.944894i \(-0.393835\pi\)
0.327377 + 0.944894i \(0.393835\pi\)
\(954\) 1.27229e9 0.0474423
\(955\) 3.69715e8 0.0137358
\(956\) 3.57991e9 0.132516
\(957\) −1.65130e9 −0.0609025
\(958\) 2.26679e10 0.832975
\(959\) 1.26175e10 0.461963
\(960\) −1.20915e10 −0.441093
\(961\) 7.42186e9 0.269762
\(962\) −7.17765e8 −0.0259938
\(963\) −2.27650e9 −0.0821440
\(964\) 9.02224e9 0.324373
\(965\) 9.33812e9 0.334514
\(966\) −1.22028e10 −0.435552
\(967\) −1.57615e9 −0.0560537 −0.0280269 0.999607i \(-0.508922\pi\)
−0.0280269 + 0.999607i \(0.508922\pi\)
\(968\) 2.87351e10 1.01824
\(969\) −7.84282e9 −0.276910
\(970\) 5.37302e9 0.189024
\(971\) 8.34171e9 0.292407 0.146204 0.989255i \(-0.453295\pi\)
0.146204 + 0.989255i \(0.453295\pi\)
\(972\) −1.37224e9 −0.0479290
\(973\) −1.65168e10 −0.574818
\(974\) −5.93377e8 −0.0205766
\(975\) 6.10452e9 0.210929
\(976\) −7.00400e9 −0.241141
\(977\) 4.51998e10 1.55062 0.775311 0.631580i \(-0.217593\pi\)
0.775311 + 0.631580i \(0.217593\pi\)
\(978\) −6.72867e9 −0.230008
\(979\) −4.99249e9 −0.170050
\(980\) 8.56773e8 0.0290787
\(981\) −1.75666e9 −0.0594082
\(982\) 2.36947e10 0.798473
\(983\) −4.03179e9 −0.135382 −0.0676909 0.997706i \(-0.521563\pi\)
−0.0676909 + 0.997706i \(0.521563\pi\)
\(984\) −7.07162e8 −0.0236612
\(985\) −2.55222e10 −0.850925
\(986\) 8.80916e9 0.292661
\(987\) 6.38589e9 0.211403
\(988\) 7.11139e8 0.0234588
\(989\) 3.10067e9 0.101922
\(990\) 1.38883e8 0.00454911
\(991\) −1.71004e10 −0.558146 −0.279073 0.960270i \(-0.590027\pi\)
−0.279073 + 0.960270i \(0.590027\pi\)
\(992\) 2.76491e10 0.899271
\(993\) 4.03274e10 1.30701
\(994\) −5.24590e9 −0.169421
\(995\) 2.70374e10 0.870132
\(996\) −7.15143e9 −0.229343
\(997\) 1.49544e10 0.477899 0.238949 0.971032i \(-0.423197\pi\)
0.238949 + 0.971032i \(0.423197\pi\)
\(998\) −3.07406e10 −0.978938
\(999\) −4.03421e9 −0.128020
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.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.c.1.7 10 1.1 even 1 trivial