Properties

Label 177.8.a.a.1.2
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(19.0314\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} -715.724 q^{10} +1618.00 q^{11} -6323.27 q^{12} -7200.81 q^{13} -22048.6 q^{14} -1015.40 q^{15} +8486.39 q^{16} +25827.4 q^{17} -13873.9 q^{18} -45988.2 q^{19} +8807.49 q^{20} -31280.5 q^{21} -30792.9 q^{22} -53080.0 q^{23} +54568.2 q^{24} -76710.7 q^{25} +137042. q^{26} -19683.0 q^{27} +271324. q^{28} +164417. q^{29} +19324.6 q^{30} +69217.7 q^{31} +97185.7 q^{32} -43686.1 q^{33} -491532. q^{34} +43569.7 q^{35} +170728. q^{36} -382449. q^{37} +875221. q^{38} +194422. q^{39} -76006.5 q^{40} -531606. q^{41} +595313. q^{42} +562745. q^{43} +378929. q^{44} +27415.9 q^{45} +1.01019e6 q^{46} +1.23094e6 q^{47} -229133. q^{48} +518668. q^{49} +1.45991e6 q^{50} -697340. q^{51} -1.68639e6 q^{52} +273502. q^{53} +374596. q^{54} +60849.1 q^{55} -2.34146e6 q^{56} +1.24168e6 q^{57} -3.12909e6 q^{58} +205379. q^{59} -237802. q^{60} +2.63599e6 q^{61} -1.31731e6 q^{62} +844575. q^{63} -2.93584e6 q^{64} -270804. q^{65} +831409. q^{66} -4.66343e6 q^{67} +6.04865e6 q^{68} +1.43316e6 q^{69} -829194. q^{70} -5.88720e6 q^{71} -1.47334e6 q^{72} -3.00177e6 q^{73} +7.27855e6 q^{74} +2.07119e6 q^{75} -1.07702e7 q^{76} +1.87452e6 q^{77} -3.70012e6 q^{78} +5.62846e6 q^{79} +319152. q^{80} +531441. q^{81} +1.01172e7 q^{82} -5.80414e6 q^{83} -7.32575e6 q^{84} +971304. q^{85} -1.07098e7 q^{86} -4.43925e6 q^{87} -3.27006e6 q^{88} -5.58369e6 q^{89} -521763. q^{90} -8.34241e6 q^{91} -1.24311e7 q^{92} -1.86888e6 q^{93} -2.34265e7 q^{94} -1.72950e6 q^{95} -2.62401e6 q^{96} +3.69012e6 q^{97} -9.87100e6 q^{98} +1.17952e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.0314 −1.68216 −0.841078 0.540914i \(-0.818079\pi\)
−0.841078 + 0.540914i \(0.818079\pi\)
\(3\) −27.0000 −0.577350
\(4\) 234.195 1.82965
\(5\) 37.6075 0.134549 0.0672743 0.997735i \(-0.478570\pi\)
0.0672743 + 0.997735i \(0.478570\pi\)
\(6\) 513.848 0.971193
\(7\) 1158.54 1.27664 0.638318 0.769773i \(-0.279630\pi\)
0.638318 + 0.769773i \(0.279630\pi\)
\(8\) −2021.05 −1.39560
\(9\) 729.000 0.333333
\(10\) −715.724 −0.226332
\(11\) 1618.00 0.366526 0.183263 0.983064i \(-0.441334\pi\)
0.183263 + 0.983064i \(0.441334\pi\)
\(12\) −6323.27 −1.05635
\(13\) −7200.81 −0.909033 −0.454516 0.890738i \(-0.650188\pi\)
−0.454516 + 0.890738i \(0.650188\pi\)
\(14\) −22048.6 −2.14750
\(15\) −1015.40 −0.0776817
\(16\) 8486.39 0.517968
\(17\) 25827.4 1.27500 0.637499 0.770451i \(-0.279969\pi\)
0.637499 + 0.770451i \(0.279969\pi\)
\(18\) −13873.9 −0.560719
\(19\) −45988.2 −1.53819 −0.769093 0.639137i \(-0.779292\pi\)
−0.769093 + 0.639137i \(0.779292\pi\)
\(20\) 8807.49 0.246177
\(21\) −31280.5 −0.737067
\(22\) −30792.9 −0.616554
\(23\) −53080.0 −0.909670 −0.454835 0.890576i \(-0.650302\pi\)
−0.454835 + 0.890576i \(0.650302\pi\)
\(24\) 54568.2 0.805750
\(25\) −76710.7 −0.981897
\(26\) 137042. 1.52913
\(27\) −19683.0 −0.192450
\(28\) 271324. 2.33580
\(29\) 164417. 1.25185 0.625926 0.779882i \(-0.284721\pi\)
0.625926 + 0.779882i \(0.284721\pi\)
\(30\) 19324.6 0.130673
\(31\) 69217.7 0.417303 0.208651 0.977990i \(-0.433093\pi\)
0.208651 + 0.977990i \(0.433093\pi\)
\(32\) 97185.7 0.524297
\(33\) −43686.1 −0.211614
\(34\) −491532. −2.14475
\(35\) 43569.7 0.171770
\(36\) 170728. 0.609883
\(37\) −382449. −1.24127 −0.620637 0.784098i \(-0.713126\pi\)
−0.620637 + 0.784098i \(0.713126\pi\)
\(38\) 875221. 2.58747
\(39\) 194422. 0.524830
\(40\) −76006.5 −0.187776
\(41\) −531606. −1.20461 −0.602305 0.798266i \(-0.705751\pi\)
−0.602305 + 0.798266i \(0.705751\pi\)
\(42\) 595313. 1.23986
\(43\) 562745. 1.07937 0.539687 0.841866i \(-0.318543\pi\)
0.539687 + 0.841866i \(0.318543\pi\)
\(44\) 378929. 0.670615
\(45\) 27415.9 0.0448496
\(46\) 1.01019e6 1.53021
\(47\) 1.23094e6 1.72939 0.864695 0.502298i \(-0.167512\pi\)
0.864695 + 0.502298i \(0.167512\pi\)
\(48\) −229133. −0.299049
\(49\) 518668. 0.629801
\(50\) 1.45991e6 1.65170
\(51\) −697340. −0.736120
\(52\) −1.68639e6 −1.66321
\(53\) 273502. 0.252345 0.126173 0.992008i \(-0.459731\pi\)
0.126173 + 0.992008i \(0.459731\pi\)
\(54\) 374596. 0.323731
\(55\) 60849.1 0.0493156
\(56\) −2.34146e6 −1.78167
\(57\) 1.24168e6 0.888072
\(58\) −3.12909e6 −2.10581
\(59\) 205379. 0.130189
\(60\) −237802. −0.142130
\(61\) 2.63599e6 1.48693 0.743464 0.668776i \(-0.233182\pi\)
0.743464 + 0.668776i \(0.233182\pi\)
\(62\) −1.31731e6 −0.701968
\(63\) 844575. 0.425546
\(64\) −2.93584e6 −1.39992
\(65\) −270804. −0.122309
\(66\) 831409. 0.355968
\(67\) −4.66343e6 −1.89428 −0.947138 0.320825i \(-0.896040\pi\)
−0.947138 + 0.320825i \(0.896040\pi\)
\(68\) 6.04865e6 2.33280
\(69\) 1.43316e6 0.525198
\(70\) −829194. −0.288944
\(71\) −5.88720e6 −1.95211 −0.976056 0.217521i \(-0.930203\pi\)
−0.976056 + 0.217521i \(0.930203\pi\)
\(72\) −1.47334e6 −0.465200
\(73\) −3.00177e6 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(74\) 7.27855e6 2.08802
\(75\) 2.07119e6 0.566898
\(76\) −1.07702e7 −2.81434
\(77\) 1.87452e6 0.467921
\(78\) −3.70012e6 −0.882846
\(79\) 5.62846e6 1.28438 0.642192 0.766544i \(-0.278025\pi\)
0.642192 + 0.766544i \(0.278025\pi\)
\(80\) 319152. 0.0696919
\(81\) 531441. 0.111111
\(82\) 1.01172e7 2.02634
\(83\) −5.80414e6 −1.11420 −0.557102 0.830444i \(-0.688087\pi\)
−0.557102 + 0.830444i \(0.688087\pi\)
\(84\) −7.32575e6 −1.34857
\(85\) 971304. 0.171549
\(86\) −1.07098e7 −1.81568
\(87\) −4.43925e6 −0.722757
\(88\) −3.27006e6 −0.511524
\(89\) −5.58369e6 −0.839569 −0.419784 0.907624i \(-0.637894\pi\)
−0.419784 + 0.907624i \(0.637894\pi\)
\(90\) −521763. −0.0754440
\(91\) −8.34241e6 −1.16050
\(92\) −1.24311e7 −1.66438
\(93\) −1.86888e6 −0.240930
\(94\) −2.34265e7 −2.90910
\(95\) −1.72950e6 −0.206961
\(96\) −2.62401e6 −0.302703
\(97\) 3.69012e6 0.410525 0.205262 0.978707i \(-0.434195\pi\)
0.205262 + 0.978707i \(0.434195\pi\)
\(98\) −9.87100e6 −1.05942
\(99\) 1.17952e6 0.122175
\(100\) −1.79653e7 −1.79653
\(101\) 1.00192e7 0.967632 0.483816 0.875170i \(-0.339250\pi\)
0.483816 + 0.875170i \(0.339250\pi\)
\(102\) 1.32714e7 1.23827
\(103\) −8.73833e6 −0.787949 −0.393974 0.919121i \(-0.628900\pi\)
−0.393974 + 0.919121i \(0.628900\pi\)
\(104\) 1.45532e7 1.26865
\(105\) −1.17638e6 −0.0991713
\(106\) −5.20513e6 −0.424484
\(107\) 3.46301e6 0.273282 0.136641 0.990621i \(-0.456369\pi\)
0.136641 + 0.990621i \(0.456369\pi\)
\(108\) −4.60966e6 −0.352116
\(109\) 108153. 0.00799916 0.00399958 0.999992i \(-0.498727\pi\)
0.00399958 + 0.999992i \(0.498727\pi\)
\(110\) −1.15804e6 −0.0829566
\(111\) 1.03261e7 0.716649
\(112\) 9.83181e6 0.661257
\(113\) 1.27204e6 0.0829328 0.0414664 0.999140i \(-0.486797\pi\)
0.0414664 + 0.999140i \(0.486797\pi\)
\(114\) −2.36310e7 −1.49388
\(115\) −1.99621e6 −0.122395
\(116\) 3.85056e7 2.29045
\(117\) −5.24939e6 −0.303011
\(118\) −3.90866e6 −0.218998
\(119\) 2.99220e7 1.62771
\(120\) 2.05217e6 0.108413
\(121\) −1.68692e7 −0.865658
\(122\) −5.01667e7 −2.50124
\(123\) 1.43534e7 0.695482
\(124\) 1.62104e7 0.763518
\(125\) −5.82298e6 −0.266662
\(126\) −1.60735e7 −0.715834
\(127\) 1.46482e7 0.634560 0.317280 0.948332i \(-0.397230\pi\)
0.317280 + 0.948332i \(0.397230\pi\)
\(128\) 4.34335e7 1.83058
\(129\) −1.51941e7 −0.623177
\(130\) 5.15379e6 0.205743
\(131\) 3.44414e7 1.33854 0.669270 0.743019i \(-0.266607\pi\)
0.669270 + 0.743019i \(0.266607\pi\)
\(132\) −1.02311e7 −0.387180
\(133\) −5.32791e7 −1.96370
\(134\) 8.87517e7 3.18647
\(135\) −740228. −0.0258939
\(136\) −5.21983e7 −1.77939
\(137\) −4.41452e7 −1.46677 −0.733385 0.679814i \(-0.762060\pi\)
−0.733385 + 0.679814i \(0.762060\pi\)
\(138\) −2.72751e7 −0.883465
\(139\) −1.22144e7 −0.385762 −0.192881 0.981222i \(-0.561783\pi\)
−0.192881 + 0.981222i \(0.561783\pi\)
\(140\) 1.02038e7 0.314279
\(141\) −3.32353e7 −0.998463
\(142\) 1.12042e8 3.28376
\(143\) −1.16509e7 −0.333184
\(144\) 6.18658e6 0.172656
\(145\) 6.18330e6 0.168435
\(146\) 5.71279e7 1.51920
\(147\) −1.40040e7 −0.363616
\(148\) −8.95677e7 −2.27109
\(149\) 4.50817e7 1.11647 0.558237 0.829682i \(-0.311478\pi\)
0.558237 + 0.829682i \(0.311478\pi\)
\(150\) −3.94177e7 −0.953612
\(151\) −2.91236e7 −0.688376 −0.344188 0.938901i \(-0.611846\pi\)
−0.344188 + 0.938901i \(0.611846\pi\)
\(152\) 9.29443e7 2.14669
\(153\) 1.88282e7 0.424999
\(154\) −3.56748e7 −0.787116
\(155\) 2.60310e6 0.0561475
\(156\) 4.55326e7 0.960255
\(157\) 7.43102e7 1.53250 0.766249 0.642544i \(-0.222121\pi\)
0.766249 + 0.642544i \(0.222121\pi\)
\(158\) −1.07118e8 −2.16053
\(159\) −7.38455e6 −0.145692
\(160\) 3.65491e6 0.0705434
\(161\) −6.14952e7 −1.16132
\(162\) −1.01141e7 −0.186906
\(163\) −5.15149e7 −0.931700 −0.465850 0.884864i \(-0.654251\pi\)
−0.465850 + 0.884864i \(0.654251\pi\)
\(164\) −1.24500e8 −2.20401
\(165\) −1.64292e6 −0.0284724
\(166\) 1.10461e8 1.87426
\(167\) 3.03652e7 0.504508 0.252254 0.967661i \(-0.418828\pi\)
0.252254 + 0.967661i \(0.418828\pi\)
\(168\) 6.32194e7 1.02865
\(169\) −1.08969e7 −0.173660
\(170\) −1.84853e7 −0.288573
\(171\) −3.35254e7 −0.512729
\(172\) 1.31792e8 1.97488
\(173\) −9.23593e7 −1.35619 −0.678093 0.734976i \(-0.737193\pi\)
−0.678093 + 0.734976i \(0.737193\pi\)
\(174\) 8.44853e7 1.21579
\(175\) −8.88723e7 −1.25353
\(176\) 1.37310e7 0.189849
\(177\) −5.54523e6 −0.0751646
\(178\) 1.06266e8 1.41229
\(179\) 6.85422e7 0.893249 0.446624 0.894722i \(-0.352626\pi\)
0.446624 + 0.894722i \(0.352626\pi\)
\(180\) 6.42066e6 0.0820590
\(181\) −3.27553e7 −0.410588 −0.205294 0.978700i \(-0.565815\pi\)
−0.205294 + 0.978700i \(0.565815\pi\)
\(182\) 1.58768e8 1.95215
\(183\) −7.11718e7 −0.858478
\(184\) 1.07277e8 1.26954
\(185\) −1.43829e7 −0.167012
\(186\) 3.55674e7 0.405282
\(187\) 4.17888e7 0.467320
\(188\) 2.88279e8 3.16418
\(189\) −2.28035e7 −0.245689
\(190\) 3.29149e7 0.348141
\(191\) 2.83438e7 0.294335 0.147167 0.989112i \(-0.452984\pi\)
0.147167 + 0.989112i \(0.452984\pi\)
\(192\) 7.92677e7 0.808243
\(193\) −1.12860e8 −1.13003 −0.565013 0.825082i \(-0.691129\pi\)
−0.565013 + 0.825082i \(0.691129\pi\)
\(194\) −7.02282e7 −0.690567
\(195\) 7.31172e6 0.0706152
\(196\) 1.21470e8 1.15232
\(197\) −1.98897e7 −0.185352 −0.0926760 0.995696i \(-0.529542\pi\)
−0.0926760 + 0.995696i \(0.529542\pi\)
\(198\) −2.24480e7 −0.205518
\(199\) 1.48348e8 1.33443 0.667217 0.744863i \(-0.267485\pi\)
0.667217 + 0.744863i \(0.267485\pi\)
\(200\) 1.55036e8 1.37034
\(201\) 1.25913e8 1.09366
\(202\) −1.90680e8 −1.62771
\(203\) 1.90483e8 1.59816
\(204\) −1.63314e8 −1.34684
\(205\) −1.99924e7 −0.162079
\(206\) 1.66303e8 1.32545
\(207\) −3.86953e7 −0.303223
\(208\) −6.11089e7 −0.470850
\(209\) −7.44091e7 −0.563786
\(210\) 2.23882e7 0.166822
\(211\) −2.67425e8 −1.95981 −0.979905 0.199465i \(-0.936079\pi\)
−0.979905 + 0.199465i \(0.936079\pi\)
\(212\) 6.40528e7 0.461703
\(213\) 1.58954e8 1.12705
\(214\) −6.59061e7 −0.459703
\(215\) 2.11634e7 0.145228
\(216\) 3.97802e7 0.268583
\(217\) 8.01913e7 0.532744
\(218\) −2.05830e6 −0.0134558
\(219\) 8.10478e7 0.521419
\(220\) 1.42506e7 0.0902303
\(221\) −1.85978e8 −1.15901
\(222\) −1.96521e8 −1.20552
\(223\) −2.62238e8 −1.58354 −0.791768 0.610822i \(-0.790839\pi\)
−0.791768 + 0.610822i \(0.790839\pi\)
\(224\) 1.12593e8 0.669337
\(225\) −5.59221e7 −0.327299
\(226\) −2.42087e7 −0.139506
\(227\) −3.00342e7 −0.170422 −0.0852109 0.996363i \(-0.527156\pi\)
−0.0852109 + 0.996363i \(0.527156\pi\)
\(228\) 2.90796e8 1.62486
\(229\) −1.22290e8 −0.672925 −0.336463 0.941697i \(-0.609231\pi\)
−0.336463 + 0.941697i \(0.609231\pi\)
\(230\) 3.79907e7 0.205887
\(231\) −5.06120e7 −0.270154
\(232\) −3.32294e8 −1.74709
\(233\) −1.17471e8 −0.608396 −0.304198 0.952609i \(-0.598388\pi\)
−0.304198 + 0.952609i \(0.598388\pi\)
\(234\) 9.99033e7 0.509712
\(235\) 4.62924e7 0.232687
\(236\) 4.80988e7 0.238200
\(237\) −1.51968e8 −0.741539
\(238\) −5.69459e8 −2.73806
\(239\) 1.02120e8 0.483859 0.241930 0.970294i \(-0.422220\pi\)
0.241930 + 0.970294i \(0.422220\pi\)
\(240\) −8.61710e6 −0.0402367
\(241\) −1.01574e8 −0.467439 −0.233719 0.972304i \(-0.575090\pi\)
−0.233719 + 0.972304i \(0.575090\pi\)
\(242\) 3.21046e8 1.45617
\(243\) −1.43489e7 −0.0641500
\(244\) 6.17337e8 2.72056
\(245\) 1.95058e7 0.0847389
\(246\) −2.73165e8 −1.16991
\(247\) 3.31152e8 1.39826
\(248\) −1.39892e8 −0.582388
\(249\) 1.56712e8 0.643286
\(250\) 1.10820e8 0.448566
\(251\) −4.09066e8 −1.63281 −0.816405 0.577480i \(-0.804036\pi\)
−0.816405 + 0.577480i \(0.804036\pi\)
\(252\) 1.97795e8 0.778599
\(253\) −8.58837e7 −0.333418
\(254\) −2.78777e8 −1.06743
\(255\) −2.62252e7 −0.0990440
\(256\) −4.50813e8 −1.67941
\(257\) −5.16514e8 −1.89809 −0.949043 0.315147i \(-0.897946\pi\)
−0.949043 + 0.315147i \(0.897946\pi\)
\(258\) 2.89166e8 1.04828
\(259\) −4.43082e8 −1.58465
\(260\) −6.34211e7 −0.223783
\(261\) 1.19860e8 0.417284
\(262\) −6.55469e8 −2.25163
\(263\) −5.42714e8 −1.83961 −0.919805 0.392377i \(-0.871653\pi\)
−0.919805 + 0.392377i \(0.871653\pi\)
\(264\) 8.82916e7 0.295329
\(265\) 1.02857e7 0.0339527
\(266\) 1.01398e9 3.30326
\(267\) 1.50760e8 0.484725
\(268\) −1.09215e9 −3.46586
\(269\) 1.04494e8 0.327309 0.163655 0.986518i \(-0.447672\pi\)
0.163655 + 0.986518i \(0.447672\pi\)
\(270\) 1.40876e7 0.0435576
\(271\) 6.80419e7 0.207675 0.103837 0.994594i \(-0.466888\pi\)
0.103837 + 0.994594i \(0.466888\pi\)
\(272\) 2.19181e8 0.660408
\(273\) 2.25245e8 0.670017
\(274\) 8.40147e8 2.46733
\(275\) −1.24118e8 −0.359891
\(276\) 3.35639e8 0.960928
\(277\) −2.90966e8 −0.822553 −0.411276 0.911511i \(-0.634917\pi\)
−0.411276 + 0.911511i \(0.634917\pi\)
\(278\) 2.32457e8 0.648913
\(279\) 5.04597e7 0.139101
\(280\) −8.80564e7 −0.239722
\(281\) −6.50736e7 −0.174958 −0.0874788 0.996166i \(-0.527881\pi\)
−0.0874788 + 0.996166i \(0.527881\pi\)
\(282\) 6.32514e8 1.67957
\(283\) 5.18818e8 1.36070 0.680350 0.732887i \(-0.261828\pi\)
0.680350 + 0.732887i \(0.261828\pi\)
\(284\) −1.37875e9 −3.57168
\(285\) 4.66965e7 0.119489
\(286\) 2.21734e8 0.560468
\(287\) −6.15886e8 −1.53785
\(288\) 7.08483e7 0.174766
\(289\) 2.56716e8 0.625620
\(290\) −1.17677e8 −0.283334
\(291\) −9.96332e7 −0.237017
\(292\) −7.03000e8 −1.65240
\(293\) 2.50544e7 0.0581900 0.0290950 0.999577i \(-0.490737\pi\)
0.0290950 + 0.999577i \(0.490737\pi\)
\(294\) 2.66517e8 0.611659
\(295\) 7.72379e6 0.0175167
\(296\) 7.72946e8 1.73232
\(297\) −3.18472e7 −0.0705380
\(298\) −8.57969e8 −1.87808
\(299\) 3.82219e8 0.826919
\(300\) 4.85062e8 1.03723
\(301\) 6.51961e8 1.37797
\(302\) 5.54264e8 1.15796
\(303\) −2.70520e8 −0.558662
\(304\) −3.90274e8 −0.796732
\(305\) 9.91331e7 0.200064
\(306\) −3.58327e8 −0.714915
\(307\) −3.47020e8 −0.684496 −0.342248 0.939610i \(-0.611188\pi\)
−0.342248 + 0.939610i \(0.611188\pi\)
\(308\) 4.39003e8 0.856131
\(309\) 2.35935e8 0.454922
\(310\) −4.95408e7 −0.0944489
\(311\) −1.38483e8 −0.261058 −0.130529 0.991445i \(-0.541668\pi\)
−0.130529 + 0.991445i \(0.541668\pi\)
\(312\) −3.92935e8 −0.732453
\(313\) −3.52364e8 −0.649511 −0.324756 0.945798i \(-0.605282\pi\)
−0.324756 + 0.945798i \(0.605282\pi\)
\(314\) −1.41423e9 −2.57790
\(315\) 3.17623e7 0.0572566
\(316\) 1.31816e9 2.34997
\(317\) −31157.4 −5.49355e−5 0 −2.74678e−5 1.00000i \(-0.500009\pi\)
−2.74678e−5 1.00000i \(0.500009\pi\)
\(318\) 1.40539e8 0.245076
\(319\) 2.66027e8 0.458837
\(320\) −1.10410e8 −0.188357
\(321\) −9.35014e7 −0.157779
\(322\) 1.17034e9 1.95352
\(323\) −1.18776e9 −1.96118
\(324\) 1.24461e8 0.203294
\(325\) 5.52379e8 0.892576
\(326\) 9.80402e8 1.56727
\(327\) −2.92012e6 −0.00461832
\(328\) 1.07440e9 1.68115
\(329\) 1.42609e9 2.20780
\(330\) 3.12672e7 0.0478950
\(331\) 1.16652e9 1.76805 0.884026 0.467438i \(-0.154823\pi\)
0.884026 + 0.467438i \(0.154823\pi\)
\(332\) −1.35930e9 −2.03860
\(333\) −2.78805e8 −0.413758
\(334\) −5.77893e8 −0.848662
\(335\) −1.75380e8 −0.254872
\(336\) −2.65459e8 −0.381777
\(337\) −4.15334e8 −0.591143 −0.295572 0.955321i \(-0.595510\pi\)
−0.295572 + 0.955321i \(0.595510\pi\)
\(338\) 2.07383e8 0.292123
\(339\) −3.43451e7 −0.0478813
\(340\) 2.27475e8 0.313875
\(341\) 1.11994e8 0.152952
\(342\) 6.38036e8 0.862490
\(343\) −3.53209e8 −0.472609
\(344\) −1.13733e9 −1.50637
\(345\) 5.38976e7 0.0706647
\(346\) 1.75773e9 2.28132
\(347\) 7.92787e8 1.01860 0.509300 0.860589i \(-0.329904\pi\)
0.509300 + 0.860589i \(0.329904\pi\)
\(348\) −1.03965e9 −1.32239
\(349\) 3.35615e7 0.0422622 0.0211311 0.999777i \(-0.493273\pi\)
0.0211311 + 0.999777i \(0.493273\pi\)
\(350\) 1.69137e9 2.10863
\(351\) 1.41733e8 0.174943
\(352\) 1.57247e8 0.192169
\(353\) −8.45067e8 −1.02254 −0.511269 0.859421i \(-0.670825\pi\)
−0.511269 + 0.859421i \(0.670825\pi\)
\(354\) 1.05534e8 0.126439
\(355\) −2.21403e8 −0.262654
\(356\) −1.30767e9 −1.53612
\(357\) −8.07895e8 −0.939758
\(358\) −1.30446e9 −1.50258
\(359\) −9.14770e8 −1.04347 −0.521736 0.853107i \(-0.674716\pi\)
−0.521736 + 0.853107i \(0.674716\pi\)
\(360\) −5.54087e7 −0.0625921
\(361\) 1.22104e9 1.36602
\(362\) 6.23380e8 0.690673
\(363\) 4.55469e8 0.499788
\(364\) −1.95375e9 −2.12332
\(365\) −1.12889e8 −0.121514
\(366\) 1.35450e9 1.44409
\(367\) −1.55832e9 −1.64561 −0.822805 0.568324i \(-0.807592\pi\)
−0.822805 + 0.568324i \(0.807592\pi\)
\(368\) −4.50458e8 −0.471180
\(369\) −3.87541e8 −0.401537
\(370\) 2.73728e8 0.280940
\(371\) 3.16863e8 0.322153
\(372\) −4.37682e8 −0.440817
\(373\) 1.33970e9 1.33667 0.668337 0.743858i \(-0.267006\pi\)
0.668337 + 0.743858i \(0.267006\pi\)
\(374\) −7.95301e8 −0.786106
\(375\) 1.57221e8 0.153957
\(376\) −2.48778e9 −2.41354
\(377\) −1.18393e9 −1.13797
\(378\) 4.33983e8 0.413287
\(379\) −3.03138e8 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(380\) −4.05041e8 −0.378666
\(381\) −3.95503e8 −0.366363
\(382\) −5.39423e8 −0.495117
\(383\) −7.20550e8 −0.655342 −0.327671 0.944792i \(-0.606264\pi\)
−0.327671 + 0.944792i \(0.606264\pi\)
\(384\) −1.17270e9 −1.05689
\(385\) 7.04960e7 0.0629581
\(386\) 2.14788e9 1.90088
\(387\) 4.10241e8 0.359791
\(388\) 8.64208e8 0.751117
\(389\) −5.84956e8 −0.503848 −0.251924 0.967747i \(-0.581063\pi\)
−0.251924 + 0.967747i \(0.581063\pi\)
\(390\) −1.39152e8 −0.118786
\(391\) −1.37092e9 −1.15983
\(392\) −1.04825e9 −0.878951
\(393\) −9.29918e8 −0.772807
\(394\) 3.78530e8 0.311791
\(395\) 2.11672e8 0.172812
\(396\) 2.76239e8 0.223538
\(397\) 7.98494e8 0.640479 0.320240 0.947337i \(-0.396237\pi\)
0.320240 + 0.947337i \(0.396237\pi\)
\(398\) −2.82328e9 −2.24473
\(399\) 1.43854e9 1.13375
\(400\) −6.50997e8 −0.508591
\(401\) −1.07381e9 −0.831618 −0.415809 0.909452i \(-0.636501\pi\)
−0.415809 + 0.909452i \(0.636501\pi\)
\(402\) −2.39630e9 −1.83971
\(403\) −4.98423e8 −0.379342
\(404\) 2.34646e9 1.77043
\(405\) 1.99862e7 0.0149499
\(406\) −3.62517e9 −2.68836
\(407\) −6.18804e8 −0.454959
\(408\) 1.40936e9 1.02733
\(409\) −6.01255e8 −0.434537 −0.217269 0.976112i \(-0.569715\pi\)
−0.217269 + 0.976112i \(0.569715\pi\)
\(410\) 3.80483e8 0.272642
\(411\) 1.19192e9 0.846840
\(412\) −2.04647e9 −1.44167
\(413\) 2.37939e8 0.166204
\(414\) 7.36427e8 0.510069
\(415\) −2.18279e8 −0.149915
\(416\) −6.99815e8 −0.476603
\(417\) 3.29788e8 0.222720
\(418\) 1.41611e9 0.948376
\(419\) 1.42631e9 0.947251 0.473626 0.880726i \(-0.342945\pi\)
0.473626 + 0.880726i \(0.342945\pi\)
\(420\) −2.75503e8 −0.181449
\(421\) −6.51184e8 −0.425320 −0.212660 0.977126i \(-0.568213\pi\)
−0.212660 + 0.977126i \(0.568213\pi\)
\(422\) 5.08948e9 3.29671
\(423\) 8.97352e8 0.576463
\(424\) −5.52760e8 −0.352173
\(425\) −1.98124e9 −1.25192
\(426\) −3.02513e9 −1.89588
\(427\) 3.05390e9 1.89827
\(428\) 8.11021e8 0.500010
\(429\) 3.14575e8 0.192364
\(430\) −4.02770e8 −0.244297
\(431\) 1.75470e9 1.05568 0.527841 0.849343i \(-0.323002\pi\)
0.527841 + 0.849343i \(0.323002\pi\)
\(432\) −1.67038e8 −0.0996830
\(433\) 9.35278e8 0.553647 0.276824 0.960921i \(-0.410718\pi\)
0.276824 + 0.960921i \(0.410718\pi\)
\(434\) −1.52616e9 −0.896158
\(435\) −1.66949e8 −0.0972461
\(436\) 2.53288e7 0.0146357
\(437\) 2.44106e9 1.39924
\(438\) −1.54245e9 −0.877108
\(439\) 8.68080e8 0.489704 0.244852 0.969560i \(-0.421261\pi\)
0.244852 + 0.969560i \(0.421261\pi\)
\(440\) −1.22979e8 −0.0688249
\(441\) 3.78109e8 0.209934
\(442\) 3.53943e9 1.94964
\(443\) 8.22011e8 0.449226 0.224613 0.974448i \(-0.427888\pi\)
0.224613 + 0.974448i \(0.427888\pi\)
\(444\) 2.41833e9 1.31122
\(445\) −2.09989e8 −0.112963
\(446\) 4.99076e9 2.66376
\(447\) −1.21721e9 −0.644596
\(448\) −3.40128e9 −1.78719
\(449\) 1.57362e9 0.820422 0.410211 0.911991i \(-0.365455\pi\)
0.410211 + 0.911991i \(0.365455\pi\)
\(450\) 1.06428e9 0.550568
\(451\) −8.60140e8 −0.441521
\(452\) 2.97906e8 0.151738
\(453\) 7.86338e8 0.397434
\(454\) 5.71593e8 0.286676
\(455\) −3.13737e8 −0.156144
\(456\) −2.50950e9 −1.23939
\(457\) 2.23657e9 1.09616 0.548081 0.836425i \(-0.315358\pi\)
0.548081 + 0.836425i \(0.315358\pi\)
\(458\) 2.32735e9 1.13197
\(459\) −5.08361e8 −0.245373
\(460\) −4.67502e8 −0.223940
\(461\) 3.55618e9 1.69056 0.845280 0.534324i \(-0.179434\pi\)
0.845280 + 0.534324i \(0.179434\pi\)
\(462\) 9.63219e8 0.454442
\(463\) −3.48898e9 −1.63367 −0.816836 0.576870i \(-0.804274\pi\)
−0.816836 + 0.576870i \(0.804274\pi\)
\(464\) 1.39531e9 0.648420
\(465\) −7.02838e7 −0.0324168
\(466\) 2.23565e9 1.02342
\(467\) −9.44654e8 −0.429204 −0.214602 0.976702i \(-0.568845\pi\)
−0.214602 + 0.976702i \(0.568845\pi\)
\(468\) −1.22938e9 −0.554404
\(469\) −5.40276e9 −2.41830
\(470\) −8.81010e8 −0.391416
\(471\) −2.00638e9 −0.884788
\(472\) −4.15080e8 −0.181692
\(473\) 9.10523e8 0.395619
\(474\) 2.89217e9 1.24738
\(475\) 3.52779e9 1.51034
\(476\) 7.00760e9 2.97814
\(477\) 1.99383e8 0.0841150
\(478\) −1.94349e9 −0.813927
\(479\) 3.10871e8 0.129243 0.0646213 0.997910i \(-0.479416\pi\)
0.0646213 + 0.997910i \(0.479416\pi\)
\(480\) −9.86825e7 −0.0407283
\(481\) 2.75394e9 1.12836
\(482\) 1.93311e9 0.786305
\(483\) 1.66037e9 0.670487
\(484\) −3.95069e9 −1.58385
\(485\) 1.38776e8 0.0552356
\(486\) 2.73080e8 0.107910
\(487\) −2.73852e9 −1.07440 −0.537198 0.843456i \(-0.680517\pi\)
−0.537198 + 0.843456i \(0.680517\pi\)
\(488\) −5.32746e9 −2.07516
\(489\) 1.39090e9 0.537917
\(490\) −3.71224e8 −0.142544
\(491\) −3.51022e8 −0.133829 −0.0669143 0.997759i \(-0.521315\pi\)
−0.0669143 + 0.997759i \(0.521315\pi\)
\(492\) 3.36149e9 1.27249
\(493\) 4.24646e9 1.59611
\(494\) −6.30230e9 −2.35209
\(495\) 4.43590e7 0.0164385
\(496\) 5.87408e8 0.216150
\(497\) −6.82055e9 −2.49214
\(498\) −2.98245e9 −1.08211
\(499\) −4.09829e9 −1.47656 −0.738280 0.674495i \(-0.764362\pi\)
−0.738280 + 0.674495i \(0.764362\pi\)
\(500\) −1.36371e9 −0.487897
\(501\) −8.19860e8 −0.291278
\(502\) 7.78511e9 2.74664
\(503\) −3.66315e9 −1.28341 −0.641707 0.766950i \(-0.721774\pi\)
−0.641707 + 0.766950i \(0.721774\pi\)
\(504\) −1.70692e9 −0.593892
\(505\) 3.76799e8 0.130194
\(506\) 1.63449e9 0.560861
\(507\) 2.94216e8 0.100263
\(508\) 3.43055e9 1.16102
\(509\) −2.18735e9 −0.735202 −0.367601 0.929984i \(-0.619821\pi\)
−0.367601 + 0.929984i \(0.619821\pi\)
\(510\) 4.99103e8 0.166608
\(511\) −3.47767e9 −1.15296
\(512\) 3.02013e9 0.994446
\(513\) 9.05186e8 0.296024
\(514\) 9.82999e9 3.19288
\(515\) −3.28627e8 −0.106017
\(516\) −3.55839e9 −1.14019
\(517\) 1.99166e9 0.633867
\(518\) 8.43248e9 2.66564
\(519\) 2.49370e9 0.782994
\(520\) 5.47308e8 0.170695
\(521\) 2.54763e9 0.789231 0.394615 0.918846i \(-0.370878\pi\)
0.394615 + 0.918846i \(0.370878\pi\)
\(522\) −2.28110e9 −0.701937
\(523\) 1.00934e9 0.308519 0.154259 0.988030i \(-0.450701\pi\)
0.154259 + 0.988030i \(0.450701\pi\)
\(524\) 8.06601e9 2.44906
\(525\) 2.39955e9 0.723723
\(526\) 1.03286e10 3.09451
\(527\) 1.78771e9 0.532060
\(528\) −3.70737e8 −0.109609
\(529\) −5.87336e8 −0.172501
\(530\) −1.95752e8 −0.0571138
\(531\) 1.49721e8 0.0433963
\(532\) −1.24777e10 −3.59289
\(533\) 3.82799e9 1.09503
\(534\) −2.86917e9 −0.815384
\(535\) 1.30235e8 0.0367697
\(536\) 9.42500e9 2.64365
\(537\) −1.85064e9 −0.515717
\(538\) −1.98867e9 −0.550585
\(539\) 8.39207e8 0.230839
\(540\) −1.73358e8 −0.0473768
\(541\) −9.25297e8 −0.251241 −0.125621 0.992078i \(-0.540092\pi\)
−0.125621 + 0.992078i \(0.540092\pi\)
\(542\) −1.29493e9 −0.349341
\(543\) 8.84393e8 0.237053
\(544\) 2.51005e9 0.668477
\(545\) 4.06735e6 0.00107628
\(546\) −4.28674e9 −1.12707
\(547\) −5.27884e9 −1.37906 −0.689530 0.724257i \(-0.742183\pi\)
−0.689530 + 0.724257i \(0.742183\pi\)
\(548\) −1.03386e10 −2.68367
\(549\) 1.92164e9 0.495642
\(550\) 2.36215e9 0.605393
\(551\) −7.56123e9 −1.92558
\(552\) −2.89648e9 −0.732967
\(553\) 6.52079e9 1.63969
\(554\) 5.53751e9 1.38366
\(555\) 3.88339e8 0.0964242
\(556\) −2.86055e9 −0.705810
\(557\) 3.38179e9 0.829190 0.414595 0.910006i \(-0.363923\pi\)
0.414595 + 0.910006i \(0.363923\pi\)
\(558\) −9.60320e8 −0.233989
\(559\) −4.05222e9 −0.981186
\(560\) 3.69750e8 0.0889713
\(561\) −1.12830e9 −0.269807
\(562\) 1.23844e9 0.294306
\(563\) −8.68855e7 −0.0205196 −0.0102598 0.999947i \(-0.503266\pi\)
−0.0102598 + 0.999947i \(0.503266\pi\)
\(564\) −7.78354e9 −1.82684
\(565\) 4.78383e7 0.0111585
\(566\) −9.87385e9 −2.28891
\(567\) 6.15695e8 0.141849
\(568\) 1.18983e10 2.72437
\(569\) 3.35876e9 0.764338 0.382169 0.924092i \(-0.375177\pi\)
0.382169 + 0.924092i \(0.375177\pi\)
\(570\) −8.88702e8 −0.200999
\(571\) −4.94562e9 −1.11172 −0.555859 0.831277i \(-0.687610\pi\)
−0.555859 + 0.831277i \(0.687610\pi\)
\(572\) −2.72859e9 −0.609611
\(573\) −7.65283e8 −0.169934
\(574\) 1.17212e10 2.58690
\(575\) 4.07180e9 0.893202
\(576\) −2.14023e9 −0.466639
\(577\) −6.72074e9 −1.45647 −0.728235 0.685327i \(-0.759659\pi\)
−0.728235 + 0.685327i \(0.759659\pi\)
\(578\) −4.88567e9 −1.05239
\(579\) 3.04721e9 0.652421
\(580\) 1.44810e9 0.308177
\(581\) −6.72432e9 −1.42243
\(582\) 1.89616e9 0.398699
\(583\) 4.42527e8 0.0924911
\(584\) 6.06671e9 1.26040
\(585\) −1.97416e8 −0.0407697
\(586\) −4.76822e8 −0.0978846
\(587\) −9.21166e9 −1.87977 −0.939885 0.341491i \(-0.889068\pi\)
−0.939885 + 0.341491i \(0.889068\pi\)
\(588\) −3.27968e9 −0.665290
\(589\) −3.18320e9 −0.641889
\(590\) −1.46995e8 −0.0294659
\(591\) 5.37023e8 0.107013
\(592\) −3.24561e9 −0.642940
\(593\) −5.16913e9 −1.01795 −0.508974 0.860782i \(-0.669975\pi\)
−0.508974 + 0.860782i \(0.669975\pi\)
\(594\) 6.06097e8 0.118656
\(595\) 1.12529e9 0.219006
\(596\) 1.05579e10 2.04275
\(597\) −4.00541e9 −0.770436
\(598\) −7.27417e9 −1.39101
\(599\) −2.29633e9 −0.436557 −0.218278 0.975887i \(-0.570044\pi\)
−0.218278 + 0.975887i \(0.570044\pi\)
\(600\) −4.18597e9 −0.791163
\(601\) 4.46476e9 0.838952 0.419476 0.907766i \(-0.362214\pi\)
0.419476 + 0.907766i \(0.362214\pi\)
\(602\) −1.24078e10 −2.31796
\(603\) −3.39964e9 −0.631426
\(604\) −6.82061e9 −1.25949
\(605\) −6.34410e8 −0.116473
\(606\) 5.14837e9 0.939757
\(607\) −2.99837e9 −0.544158 −0.272079 0.962275i \(-0.587711\pi\)
−0.272079 + 0.962275i \(0.587711\pi\)
\(608\) −4.46939e9 −0.806466
\(609\) −5.14305e9 −0.922699
\(610\) −1.88664e9 −0.336539
\(611\) −8.86373e9 −1.57207
\(612\) 4.40947e9 0.777600
\(613\) 8.71253e9 1.52768 0.763840 0.645405i \(-0.223312\pi\)
0.763840 + 0.645405i \(0.223312\pi\)
\(614\) 6.60429e9 1.15143
\(615\) 5.39794e8 0.0935761
\(616\) −3.78849e9 −0.653031
\(617\) 3.68960e9 0.632385 0.316192 0.948695i \(-0.397595\pi\)
0.316192 + 0.948695i \(0.397595\pi\)
\(618\) −4.49018e9 −0.765251
\(619\) 2.78943e9 0.472713 0.236357 0.971666i \(-0.424047\pi\)
0.236357 + 0.971666i \(0.424047\pi\)
\(620\) 6.09634e8 0.102730
\(621\) 1.04477e9 0.175066
\(622\) 2.63554e9 0.439140
\(623\) −6.46892e9 −1.07182
\(624\) 1.64994e9 0.271845
\(625\) 5.77403e9 0.946018
\(626\) 6.70599e9 1.09258
\(627\) 2.00905e9 0.325502
\(628\) 1.74031e10 2.80393
\(629\) −9.87766e9 −1.58262
\(630\) −6.04483e8 −0.0963145
\(631\) 1.40846e9 0.223173 0.111587 0.993755i \(-0.464407\pi\)
0.111587 + 0.993755i \(0.464407\pi\)
\(632\) −1.13754e10 −1.79249
\(633\) 7.22048e9 1.13150
\(634\) 592969. 9.24101e−5 0
\(635\) 5.50884e8 0.0853792
\(636\) −1.72943e9 −0.266564
\(637\) −3.73483e9 −0.572510
\(638\) −5.06287e9 −0.771835
\(639\) −4.29177e9 −0.650704
\(640\) 1.63342e9 0.246303
\(641\) −8.08641e9 −1.21270 −0.606349 0.795199i \(-0.707367\pi\)
−0.606349 + 0.795199i \(0.707367\pi\)
\(642\) 1.77946e9 0.265410
\(643\) −5.86883e9 −0.870590 −0.435295 0.900288i \(-0.643356\pi\)
−0.435295 + 0.900288i \(0.643356\pi\)
\(644\) −1.44019e10 −2.12480
\(645\) −5.71412e8 −0.0838476
\(646\) 2.26047e10 3.29902
\(647\) 3.25260e9 0.472135 0.236067 0.971737i \(-0.424141\pi\)
0.236067 + 0.971737i \(0.424141\pi\)
\(648\) −1.07407e9 −0.155067
\(649\) 3.32304e8 0.0477177
\(650\) −1.05126e10 −1.50145
\(651\) −2.16517e9 −0.307580
\(652\) −1.20645e10 −1.70469
\(653\) −1.64182e9 −0.230744 −0.115372 0.993322i \(-0.536806\pi\)
−0.115372 + 0.993322i \(0.536806\pi\)
\(654\) 5.55741e7 0.00776874
\(655\) 1.29526e9 0.180099
\(656\) −4.51142e9 −0.623950
\(657\) −2.18829e9 −0.301041
\(658\) −2.71404e10 −3.71387
\(659\) 4.63845e9 0.631355 0.315678 0.948866i \(-0.397768\pi\)
0.315678 + 0.948866i \(0.397768\pi\)
\(660\) −3.84765e8 −0.0520945
\(661\) 9.44802e9 1.27244 0.636218 0.771510i \(-0.280498\pi\)
0.636218 + 0.771510i \(0.280498\pi\)
\(662\) −2.22006e10 −2.97414
\(663\) 5.02141e9 0.669157
\(664\) 1.17304e10 1.55498
\(665\) −2.00369e9 −0.264214
\(666\) 5.30606e9 0.696005
\(667\) −8.72725e9 −1.13877
\(668\) 7.11138e9 0.923073
\(669\) 7.08042e9 0.914255
\(670\) 3.33773e9 0.428735
\(671\) 4.26505e9 0.544998
\(672\) −3.04002e9 −0.386442
\(673\) 8.85543e9 1.11984 0.559921 0.828546i \(-0.310831\pi\)
0.559921 + 0.828546i \(0.310831\pi\)
\(674\) 7.90440e9 0.994395
\(675\) 1.50990e9 0.188966
\(676\) −2.55200e9 −0.317737
\(677\) 1.90990e8 0.0236565 0.0118283 0.999930i \(-0.496235\pi\)
0.0118283 + 0.999930i \(0.496235\pi\)
\(678\) 6.53636e8 0.0805438
\(679\) 4.27515e9 0.524091
\(680\) −1.96305e9 −0.239414
\(681\) 8.10922e8 0.0983930
\(682\) −2.13141e9 −0.257290
\(683\) −3.13279e9 −0.376235 −0.188117 0.982147i \(-0.560239\pi\)
−0.188117 + 0.982147i \(0.560239\pi\)
\(684\) −7.85149e9 −0.938114
\(685\) −1.66019e9 −0.197352
\(686\) 6.72207e9 0.795003
\(687\) 3.30183e9 0.388513
\(688\) 4.77567e9 0.559081
\(689\) −1.96944e9 −0.229390
\(690\) −1.02575e9 −0.118869
\(691\) 1.83456e9 0.211524 0.105762 0.994391i \(-0.466272\pi\)
0.105762 + 0.994391i \(0.466272\pi\)
\(692\) −2.16301e10 −2.48135
\(693\) 1.36652e9 0.155974
\(694\) −1.50879e10 −1.71344
\(695\) −4.59352e8 −0.0519038
\(696\) 8.97193e9 1.00868
\(697\) −1.37300e10 −1.53587
\(698\) −6.38723e8 −0.0710917
\(699\) 3.17173e9 0.351258
\(700\) −2.08135e10 −2.29351
\(701\) 5.98660e9 0.656398 0.328199 0.944609i \(-0.393558\pi\)
0.328199 + 0.944609i \(0.393558\pi\)
\(702\) −2.69739e9 −0.294282
\(703\) 1.75881e10 1.90931
\(704\) −4.75020e9 −0.513107
\(705\) −1.24989e9 −0.134342
\(706\) 1.60828e10 1.72007
\(707\) 1.16077e10 1.23531
\(708\) −1.29867e9 −0.137525
\(709\) 6.41985e9 0.676493 0.338246 0.941058i \(-0.390166\pi\)
0.338246 + 0.941058i \(0.390166\pi\)
\(710\) 4.21361e9 0.441825
\(711\) 4.10315e9 0.428128
\(712\) 1.12849e10 1.17170
\(713\) −3.67408e9 −0.379608
\(714\) 1.53754e10 1.58082
\(715\) −4.38162e8 −0.0448295
\(716\) 1.60523e10 1.63433
\(717\) −2.75725e9 −0.279356
\(718\) 1.74094e10 1.75528
\(719\) −1.53454e10 −1.53967 −0.769836 0.638242i \(-0.779662\pi\)
−0.769836 + 0.638242i \(0.779662\pi\)
\(720\) 2.32662e8 0.0232306
\(721\) −1.01237e10 −1.00592
\(722\) −2.32382e10 −2.29785
\(723\) 2.74251e9 0.269876
\(724\) −7.67113e9 −0.751232
\(725\) −1.26125e10 −1.22919
\(726\) −8.66823e9 −0.840722
\(727\) 5.06244e8 0.0488640 0.0244320 0.999701i \(-0.492222\pi\)
0.0244320 + 0.999701i \(0.492222\pi\)
\(728\) 1.68604e10 1.61960
\(729\) 3.87420e8 0.0370370
\(730\) 2.14844e9 0.204406
\(731\) 1.45342e10 1.37620
\(732\) −1.66681e10 −1.57071
\(733\) −1.29579e10 −1.21527 −0.607633 0.794218i \(-0.707881\pi\)
−0.607633 + 0.794218i \(0.707881\pi\)
\(734\) 2.96571e10 2.76817
\(735\) −5.26657e8 −0.0489240
\(736\) −5.15862e9 −0.476937
\(737\) −7.54544e9 −0.694302
\(738\) 7.37545e9 0.675447
\(739\) −3.46591e9 −0.315909 −0.157955 0.987446i \(-0.550490\pi\)
−0.157955 + 0.987446i \(0.550490\pi\)
\(740\) −3.36842e9 −0.305573
\(741\) −8.94111e9 −0.807287
\(742\) −6.03035e9 −0.541912
\(743\) 4.65023e9 0.415924 0.207962 0.978137i \(-0.433317\pi\)
0.207962 + 0.978137i \(0.433317\pi\)
\(744\) 3.77709e9 0.336242
\(745\) 1.69541e9 0.150220
\(746\) −2.54963e10 −2.24850
\(747\) −4.23122e9 −0.371401
\(748\) 9.78674e9 0.855032
\(749\) 4.01204e9 0.348882
\(750\) −2.99213e9 −0.258980
\(751\) 2.20118e10 1.89634 0.948169 0.317766i \(-0.102933\pi\)
0.948169 + 0.317766i \(0.102933\pi\)
\(752\) 1.04462e10 0.895769
\(753\) 1.10448e10 0.942703
\(754\) 2.25319e10 1.91425
\(755\) −1.09527e9 −0.0926201
\(756\) −5.34047e9 −0.449525
\(757\) 5.77225e9 0.483626 0.241813 0.970323i \(-0.422258\pi\)
0.241813 + 0.970323i \(0.422258\pi\)
\(758\) 5.76914e9 0.481137
\(759\) 2.31886e9 0.192499
\(760\) 3.49540e9 0.288835
\(761\) 5.11102e9 0.420398 0.210199 0.977659i \(-0.432589\pi\)
0.210199 + 0.977659i \(0.432589\pi\)
\(762\) 7.52698e9 0.616280
\(763\) 1.25299e8 0.0102120
\(764\) 6.63798e9 0.538529
\(765\) 7.08081e8 0.0571831
\(766\) 1.37131e10 1.10239
\(767\) −1.47889e9 −0.118346
\(768\) 1.21720e10 0.969607
\(769\) 2.04941e9 0.162512 0.0812560 0.996693i \(-0.474107\pi\)
0.0812560 + 0.996693i \(0.474107\pi\)
\(770\) −1.34164e9 −0.105905
\(771\) 1.39459e10 1.09586
\(772\) −2.64312e10 −2.06755
\(773\) 2.05915e10 1.60346 0.801732 0.597683i \(-0.203912\pi\)
0.801732 + 0.597683i \(0.203912\pi\)
\(774\) −7.80747e9 −0.605225
\(775\) −5.30974e9 −0.409748
\(776\) −7.45790e9 −0.572929
\(777\) 1.19632e10 0.914901
\(778\) 1.11326e10 0.847552
\(779\) 2.44476e10 1.85291
\(780\) 1.71237e9 0.129201
\(781\) −9.52551e9 −0.715500
\(782\) 2.60905e10 1.95101
\(783\) −3.23622e9 −0.240919
\(784\) 4.40162e9 0.326217
\(785\) 2.79462e9 0.206196
\(786\) 1.76977e10 1.29998
\(787\) −1.82710e10 −1.33613 −0.668067 0.744101i \(-0.732878\pi\)
−0.668067 + 0.744101i \(0.732878\pi\)
\(788\) −4.65808e9 −0.339129
\(789\) 1.46533e10 1.06210
\(790\) −4.02842e9 −0.290697
\(791\) 1.47371e9 0.105875
\(792\) −2.38387e9 −0.170508
\(793\) −1.89813e10 −1.35167
\(794\) −1.51965e10 −1.07739
\(795\) −2.77715e8 −0.0196026
\(796\) 3.47425e10 2.44155
\(797\) 2.36801e10 1.65684 0.828418 0.560110i \(-0.189241\pi\)
0.828418 + 0.560110i \(0.189241\pi\)
\(798\) −2.73774e10 −1.90714
\(799\) 3.17919e10 2.20497
\(800\) −7.45518e9 −0.514805
\(801\) −4.07051e9 −0.279856
\(802\) 2.04362e10 1.39891
\(803\) −4.85687e9 −0.331019
\(804\) 2.94881e10 2.00102
\(805\) −2.31268e9 −0.156254
\(806\) 9.48570e9 0.638112
\(807\) −2.82134e9 −0.188972
\(808\) −2.02493e10 −1.35043
\(809\) 1.22571e10 0.813897 0.406949 0.913451i \(-0.366593\pi\)
0.406949 + 0.913451i \(0.366593\pi\)
\(810\) −3.80365e8 −0.0251480
\(811\) −7.84590e9 −0.516499 −0.258250 0.966078i \(-0.583146\pi\)
−0.258250 + 0.966078i \(0.583146\pi\)
\(812\) 4.46102e10 2.92407
\(813\) −1.83713e9 −0.119901
\(814\) 1.17767e10 0.765312
\(815\) −1.93735e9 −0.125359
\(816\) −5.91790e9 −0.381287
\(817\) −2.58796e10 −1.66028
\(818\) 1.14427e10 0.730960
\(819\) −6.08162e9 −0.386835
\(820\) −4.68212e9 −0.296547
\(821\) −8.10180e9 −0.510952 −0.255476 0.966815i \(-0.582232\pi\)
−0.255476 + 0.966815i \(0.582232\pi\)
\(822\) −2.26840e10 −1.42452
\(823\) 1.64265e9 0.102718 0.0513588 0.998680i \(-0.483645\pi\)
0.0513588 + 0.998680i \(0.483645\pi\)
\(824\) 1.76606e10 1.09966
\(825\) 3.35119e9 0.207783
\(826\) −4.52833e9 −0.279581
\(827\) −2.72238e10 −1.67371 −0.836854 0.547427i \(-0.815607\pi\)
−0.836854 + 0.547427i \(0.815607\pi\)
\(828\) −9.06226e9 −0.554792
\(829\) 1.28481e10 0.783245 0.391623 0.920126i \(-0.371914\pi\)
0.391623 + 0.920126i \(0.371914\pi\)
\(830\) 4.15416e9 0.252180
\(831\) 7.85609e9 0.474901
\(832\) 2.11404e10 1.27257
\(833\) 1.33959e10 0.802995
\(834\) −6.27634e9 −0.374650
\(835\) 1.14196e9 0.0678809
\(836\) −1.74263e10 −1.03153
\(837\) −1.36241e9 −0.0803099
\(838\) −2.71447e10 −1.59342
\(839\) −2.14886e10 −1.25615 −0.628073 0.778154i \(-0.716156\pi\)
−0.628073 + 0.778154i \(0.716156\pi\)
\(840\) 2.37752e9 0.138404
\(841\) 9.78301e9 0.567135
\(842\) 1.23930e10 0.715455
\(843\) 1.75699e9 0.101012
\(844\) −6.26297e10 −3.58577
\(845\) −4.09805e8 −0.0233657
\(846\) −1.70779e10 −0.969701
\(847\) −1.95437e10 −1.10513
\(848\) 2.32105e9 0.130707
\(849\) −1.40081e10 −0.785601
\(850\) 3.77058e10 2.10592
\(851\) 2.03004e10 1.12915
\(852\) 3.72264e10 2.06211
\(853\) 7.06458e9 0.389731 0.194865 0.980830i \(-0.437573\pi\)
0.194865 + 0.980830i \(0.437573\pi\)
\(854\) −5.81200e10 −3.19318
\(855\) −1.26081e9 −0.0689870
\(856\) −6.99891e9 −0.381393
\(857\) −1.97164e10 −1.07003 −0.535015 0.844843i \(-0.679694\pi\)
−0.535015 + 0.844843i \(0.679694\pi\)
\(858\) −5.98681e9 −0.323586
\(859\) −1.48500e10 −0.799376 −0.399688 0.916651i \(-0.630882\pi\)
−0.399688 + 0.916651i \(0.630882\pi\)
\(860\) 4.95637e9 0.265717
\(861\) 1.66289e10 0.887877
\(862\) −3.33945e10 −1.77582
\(863\) 1.20569e10 0.638552 0.319276 0.947662i \(-0.396560\pi\)
0.319276 + 0.947662i \(0.396560\pi\)
\(864\) −1.91291e9 −0.100901
\(865\) −3.47340e9 −0.182473
\(866\) −1.77997e10 −0.931321
\(867\) −6.93133e9 −0.361202
\(868\) 1.87804e10 0.974735
\(869\) 9.10687e9 0.470760
\(870\) 3.17728e9 0.163583
\(871\) 3.35804e10 1.72196
\(872\) −2.18581e8 −0.0111636
\(873\) 2.69010e9 0.136842
\(874\) −4.64568e10 −2.35374
\(875\) −6.74615e9 −0.340430
\(876\) 1.89810e10 0.954014
\(877\) 3.18390e10 1.59390 0.796949 0.604046i \(-0.206446\pi\)
0.796949 + 0.604046i \(0.206446\pi\)
\(878\) −1.65208e10 −0.823759
\(879\) −6.76470e8 −0.0335960
\(880\) 5.16389e8 0.0255439
\(881\) −2.68011e10 −1.32050 −0.660248 0.751048i \(-0.729549\pi\)
−0.660248 + 0.751048i \(0.729549\pi\)
\(882\) −7.19596e9 −0.353141
\(883\) −3.21062e10 −1.56937 −0.784686 0.619893i \(-0.787176\pi\)
−0.784686 + 0.619893i \(0.787176\pi\)
\(884\) −4.35552e10 −2.12059
\(885\) −2.08542e8 −0.0101133
\(886\) −1.56440e10 −0.755667
\(887\) 1.26881e10 0.610468 0.305234 0.952277i \(-0.401265\pi\)
0.305234 + 0.952277i \(0.401265\pi\)
\(888\) −2.08696e10 −1.00016
\(889\) 1.69706e10 0.810103
\(890\) 3.99638e9 0.190021
\(891\) 8.59874e8 0.0407251
\(892\) −6.14148e10 −2.89732
\(893\) −5.66085e10 −2.66012
\(894\) 2.31652e10 1.08431
\(895\) 2.57770e9 0.120185
\(896\) 5.03193e10 2.33699
\(897\) −1.03199e10 −0.477422
\(898\) −2.99482e10 −1.38008
\(899\) 1.13805e10 0.522401
\(900\) −1.30967e10 −0.598842
\(901\) 7.06385e9 0.321740
\(902\) 1.63697e10 0.742708
\(903\) −1.76030e10 −0.795570
\(904\) −2.57085e9 −0.115741
\(905\) −1.23184e9 −0.0552441
\(906\) −1.49651e10 −0.668546
\(907\) −1.35224e10 −0.601767 −0.300884 0.953661i \(-0.597282\pi\)
−0.300884 + 0.953661i \(0.597282\pi\)
\(908\) −7.03385e9 −0.311812
\(909\) 7.30403e9 0.322544
\(910\) 5.97087e9 0.262659
\(911\) 2.02421e10 0.887035 0.443517 0.896266i \(-0.353730\pi\)
0.443517 + 0.896266i \(0.353730\pi\)
\(912\) 1.05374e10 0.459993
\(913\) −9.39112e9 −0.408385
\(914\) −4.25650e10 −1.84392
\(915\) −2.67659e9 −0.115507
\(916\) −2.86397e10 −1.23122
\(917\) 3.99017e10 1.70883
\(918\) 9.67483e9 0.412756
\(919\) 3.59827e10 1.52929 0.764644 0.644453i \(-0.222915\pi\)
0.764644 + 0.644453i \(0.222915\pi\)
\(920\) 4.03442e9 0.170814
\(921\) 9.36955e9 0.395194
\(922\) −6.76791e10 −2.84378
\(923\) 4.23926e10 1.77453
\(924\) −1.18531e10 −0.494288
\(925\) 2.93379e10 1.21880
\(926\) 6.64002e10 2.74809
\(927\) −6.37024e9 −0.262650
\(928\) 1.59790e10 0.656342
\(929\) 3.99605e10 1.63522 0.817609 0.575774i \(-0.195299\pi\)
0.817609 + 0.575774i \(0.195299\pi\)
\(930\) 1.33760e9 0.0545301
\(931\) −2.38526e10 −0.968751
\(932\) −2.75112e10 −1.11315
\(933\) 3.73905e9 0.150722
\(934\) 1.79781e10 0.721988
\(935\) 1.57157e9 0.0628773
\(936\) 1.06093e10 0.422882
\(937\) 2.19504e10 0.871674 0.435837 0.900026i \(-0.356452\pi\)
0.435837 + 0.900026i \(0.356452\pi\)
\(938\) 1.02822e11 4.06796
\(939\) 9.51384e9 0.374996
\(940\) 1.08415e10 0.425736
\(941\) −2.26436e10 −0.885896 −0.442948 0.896547i \(-0.646067\pi\)
−0.442948 + 0.896547i \(0.646067\pi\)
\(942\) 3.81842e10 1.48835
\(943\) 2.82177e10 1.09580
\(944\) 1.74293e9 0.0674337
\(945\) −8.57583e8 −0.0330571
\(946\) −1.73286e10 −0.665493
\(947\) 7.27658e9 0.278421 0.139211 0.990263i \(-0.455543\pi\)
0.139211 + 0.990263i \(0.455543\pi\)
\(948\) −3.55903e10 −1.35676
\(949\) 2.16152e10 0.820969
\(950\) −6.71388e10 −2.54063
\(951\) 841249. 3.17170e−5 0
\(952\) −6.04738e10 −2.27163
\(953\) 4.70442e10 1.76068 0.880340 0.474343i \(-0.157314\pi\)
0.880340 + 0.474343i \(0.157314\pi\)
\(954\) −3.79454e9 −0.141495
\(955\) 1.06594e9 0.0396023
\(956\) 2.39161e10 0.885293
\(957\) −7.18273e9 −0.264910
\(958\) −5.91631e9 −0.217406
\(959\) −5.11439e10 −1.87253
\(960\) 2.98106e9 0.108748
\(961\) −2.27215e10 −0.825859
\(962\) −5.24114e10 −1.89807
\(963\) 2.52454e9 0.0910940
\(964\) −2.37882e10 −0.855249
\(965\) −4.24437e9 −0.152043
\(966\) −3.15992e10 −1.12786
\(967\) −2.01441e8 −0.00716400 −0.00358200 0.999994i \(-0.501140\pi\)
−0.00358200 + 0.999994i \(0.501140\pi\)
\(968\) 3.40935e10 1.20811
\(969\) 3.20694e10 1.13229
\(970\) −2.64111e9 −0.0929149
\(971\) 4.43321e10 1.55400 0.777001 0.629500i \(-0.216740\pi\)
0.777001 + 0.629500i \(0.216740\pi\)
\(972\) −3.36044e9 −0.117372
\(973\) −1.41508e10 −0.492478
\(974\) 5.21179e10 1.80730
\(975\) −1.49142e10 −0.515329
\(976\) 2.23701e10 0.770181
\(977\) 4.78091e10 1.64013 0.820067 0.572268i \(-0.193936\pi\)
0.820067 + 0.572268i \(0.193936\pi\)
\(978\) −2.64709e10 −0.904861
\(979\) −9.03443e9 −0.307724
\(980\) 4.56817e9 0.155043
\(981\) 7.88433e7 0.00266639
\(982\) 6.68045e9 0.225121
\(983\) −3.22996e10 −1.08458 −0.542288 0.840193i \(-0.682442\pi\)
−0.542288 + 0.840193i \(0.682442\pi\)
\(984\) −2.90088e10 −0.970615
\(985\) −7.48003e8 −0.0249389
\(986\) −8.08162e10 −2.68491
\(987\) −3.85043e10 −1.27467
\(988\) 7.75543e10 2.55833
\(989\) −2.98705e10 −0.981873
\(990\) −8.44215e8 −0.0276522
\(991\) −5.62073e10 −1.83457 −0.917286 0.398230i \(-0.869625\pi\)
−0.917286 + 0.398230i \(0.869625\pi\)
\(992\) 6.72697e9 0.218790
\(993\) −3.14961e10 −1.02079
\(994\) 1.29805e11 4.19216
\(995\) 5.57901e9 0.179546
\(996\) 3.67011e10 1.17699
\(997\) −6.21852e9 −0.198726 −0.0993629 0.995051i \(-0.531680\pi\)
−0.0993629 + 0.995051i \(0.531680\pi\)
\(998\) 7.79963e10 2.48380
\(999\) 7.52774e9 0.238883
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 177.8.a.a.1.2 16
3.2 odd 2 531.8.a.b.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.2 16 1.1 even 1 trivial
531.8.a.b.1.15 16 3.2 odd 2