Properties

Label 531.8.a.b.1.15
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
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: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(19.0314\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} +O(q^{10})\) \(q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} -715.724 q^{10} -1618.00 q^{11} -7200.81 q^{13} +22048.6 q^{14} +8486.39 q^{16} -25827.4 q^{17} -45988.2 q^{19} -8807.49 q^{20} -30792.9 q^{22} +53080.0 q^{23} -76710.7 q^{25} -137042. q^{26} +271324. q^{28} -164417. q^{29} +69217.7 q^{31} -97185.7 q^{32} -491532. q^{34} -43569.7 q^{35} -382449. q^{37} -875221. q^{38} -76006.5 q^{40} +531606. q^{41} +562745. q^{43} -378929. q^{44} +1.01019e6 q^{46} -1.23094e6 q^{47} +518668. q^{49} -1.45991e6 q^{50} -1.68639e6 q^{52} -273502. q^{53} +60849.1 q^{55} +2.34146e6 q^{56} -3.12909e6 q^{58} -205379. q^{59} +2.63599e6 q^{61} +1.31731e6 q^{62} -2.93584e6 q^{64} +270804. q^{65} -4.66343e6 q^{67} -6.04865e6 q^{68} -829194. q^{70} +5.88720e6 q^{71} -3.00177e6 q^{73} -7.27855e6 q^{74} -1.07702e7 q^{76} -1.87452e6 q^{77} +5.62846e6 q^{79} -319152. q^{80} +1.01172e7 q^{82} +5.80414e6 q^{83} +971304. q^{85} +1.07098e7 q^{86} -3.27006e6 q^{88} +5.58369e6 q^{89} -8.34241e6 q^{91} +1.24311e7 q^{92} -2.34265e7 q^{94} +1.72950e6 q^{95} +3.69012e6 q^{97} +9.87100e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 19.0314 1.68216 0.841078 0.540914i \(-0.181921\pi\)
0.841078 + 0.540914i \(0.181921\pi\)
\(3\) 0 0
\(4\) 234.195 1.82965
\(5\) −37.6075 −0.134549 −0.0672743 0.997735i \(-0.521430\pi\)
−0.0672743 + 0.997735i \(0.521430\pi\)
\(6\) 0 0
\(7\) 1158.54 1.27664 0.638318 0.769773i \(-0.279630\pi\)
0.638318 + 0.769773i \(0.279630\pi\)
\(8\) 2021.05 1.39560
\(9\) 0 0
\(10\) −715.724 −0.226332
\(11\) −1618.00 −0.366526 −0.183263 0.983064i \(-0.558666\pi\)
−0.183263 + 0.983064i \(0.558666\pi\)
\(12\) 0 0
\(13\) −7200.81 −0.909033 −0.454516 0.890738i \(-0.650188\pi\)
−0.454516 + 0.890738i \(0.650188\pi\)
\(14\) 22048.6 2.14750
\(15\) 0 0
\(16\) 8486.39 0.517968
\(17\) −25827.4 −1.27500 −0.637499 0.770451i \(-0.720031\pi\)
−0.637499 + 0.770451i \(0.720031\pi\)
\(18\) 0 0
\(19\) −45988.2 −1.53819 −0.769093 0.639137i \(-0.779292\pi\)
−0.769093 + 0.639137i \(0.779292\pi\)
\(20\) −8807.49 −0.246177
\(21\) 0 0
\(22\) −30792.9 −0.616554
\(23\) 53080.0 0.909670 0.454835 0.890576i \(-0.349698\pi\)
0.454835 + 0.890576i \(0.349698\pi\)
\(24\) 0 0
\(25\) −76710.7 −0.981897
\(26\) −137042. −1.52913
\(27\) 0 0
\(28\) 271324. 2.33580
\(29\) −164417. −1.25185 −0.625926 0.779882i \(-0.715279\pi\)
−0.625926 + 0.779882i \(0.715279\pi\)
\(30\) 0 0
\(31\) 69217.7 0.417303 0.208651 0.977990i \(-0.433093\pi\)
0.208651 + 0.977990i \(0.433093\pi\)
\(32\) −97185.7 −0.524297
\(33\) 0 0
\(34\) −491532. −2.14475
\(35\) −43569.7 −0.171770
\(36\) 0 0
\(37\) −382449. −1.24127 −0.620637 0.784098i \(-0.713126\pi\)
−0.620637 + 0.784098i \(0.713126\pi\)
\(38\) −875221. −2.58747
\(39\) 0 0
\(40\) −76006.5 −0.187776
\(41\) 531606. 1.20461 0.602305 0.798266i \(-0.294249\pi\)
0.602305 + 0.798266i \(0.294249\pi\)
\(42\) 0 0
\(43\) 562745. 1.07937 0.539687 0.841866i \(-0.318543\pi\)
0.539687 + 0.841866i \(0.318543\pi\)
\(44\) −378929. −0.670615
\(45\) 0 0
\(46\) 1.01019e6 1.53021
\(47\) −1.23094e6 −1.72939 −0.864695 0.502298i \(-0.832488\pi\)
−0.864695 + 0.502298i \(0.832488\pi\)
\(48\) 0 0
\(49\) 518668. 0.629801
\(50\) −1.45991e6 −1.65170
\(51\) 0 0
\(52\) −1.68639e6 −1.66321
\(53\) −273502. −0.252345 −0.126173 0.992008i \(-0.540269\pi\)
−0.126173 + 0.992008i \(0.540269\pi\)
\(54\) 0 0
\(55\) 60849.1 0.0493156
\(56\) 2.34146e6 1.78167
\(57\) 0 0
\(58\) −3.12909e6 −2.10581
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 2.63599e6 1.48693 0.743464 0.668776i \(-0.233182\pi\)
0.743464 + 0.668776i \(0.233182\pi\)
\(62\) 1.31731e6 0.701968
\(63\) 0 0
\(64\) −2.93584e6 −1.39992
\(65\) 270804. 0.122309
\(66\) 0 0
\(67\) −4.66343e6 −1.89428 −0.947138 0.320825i \(-0.896040\pi\)
−0.947138 + 0.320825i \(0.896040\pi\)
\(68\) −6.04865e6 −2.33280
\(69\) 0 0
\(70\) −829194. −0.288944
\(71\) 5.88720e6 1.95211 0.976056 0.217521i \(-0.0697969\pi\)
0.976056 + 0.217521i \(0.0697969\pi\)
\(72\) 0 0
\(73\) −3.00177e6 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(74\) −7.27855e6 −2.08802
\(75\) 0 0
\(76\) −1.07702e7 −2.81434
\(77\) −1.87452e6 −0.467921
\(78\) 0 0
\(79\) 5.62846e6 1.28438 0.642192 0.766544i \(-0.278025\pi\)
0.642192 + 0.766544i \(0.278025\pi\)
\(80\) −319152. −0.0696919
\(81\) 0 0
\(82\) 1.01172e7 2.02634
\(83\) 5.80414e6 1.11420 0.557102 0.830444i \(-0.311913\pi\)
0.557102 + 0.830444i \(0.311913\pi\)
\(84\) 0 0
\(85\) 971304. 0.171549
\(86\) 1.07098e7 1.81568
\(87\) 0 0
\(88\) −3.27006e6 −0.511524
\(89\) 5.58369e6 0.839569 0.419784 0.907624i \(-0.362106\pi\)
0.419784 + 0.907624i \(0.362106\pi\)
\(90\) 0 0
\(91\) −8.34241e6 −1.16050
\(92\) 1.24311e7 1.66438
\(93\) 0 0
\(94\) −2.34265e7 −2.90910
\(95\) 1.72950e6 0.206961
\(96\) 0 0
\(97\) 3.69012e6 0.410525 0.205262 0.978707i \(-0.434195\pi\)
0.205262 + 0.978707i \(0.434195\pi\)
\(98\) 9.87100e6 1.05942
\(99\) 0 0
\(100\) −1.79653e7 −1.79653
\(101\) −1.00192e7 −0.967632 −0.483816 0.875170i \(-0.660750\pi\)
−0.483816 + 0.875170i \(0.660750\pi\)
\(102\) 0 0
\(103\) −8.73833e6 −0.787949 −0.393974 0.919121i \(-0.628900\pi\)
−0.393974 + 0.919121i \(0.628900\pi\)
\(104\) −1.45532e7 −1.26865
\(105\) 0 0
\(106\) −5.20513e6 −0.424484
\(107\) −3.46301e6 −0.273282 −0.136641 0.990621i \(-0.543631\pi\)
−0.136641 + 0.990621i \(0.543631\pi\)
\(108\) 0 0
\(109\) 108153. 0.00799916 0.00399958 0.999992i \(-0.498727\pi\)
0.00399958 + 0.999992i \(0.498727\pi\)
\(110\) 1.15804e6 0.0829566
\(111\) 0 0
\(112\) 9.83181e6 0.661257
\(113\) −1.27204e6 −0.0829328 −0.0414664 0.999140i \(-0.513203\pi\)
−0.0414664 + 0.999140i \(0.513203\pi\)
\(114\) 0 0
\(115\) −1.99621e6 −0.122395
\(116\) −3.85056e7 −2.29045
\(117\) 0 0
\(118\) −3.90866e6 −0.218998
\(119\) −2.99220e7 −1.62771
\(120\) 0 0
\(121\) −1.68692e7 −0.865658
\(122\) 5.01667e7 2.50124
\(123\) 0 0
\(124\) 1.62104e7 0.763518
\(125\) 5.82298e6 0.266662
\(126\) 0 0
\(127\) 1.46482e7 0.634560 0.317280 0.948332i \(-0.397230\pi\)
0.317280 + 0.948332i \(0.397230\pi\)
\(128\) −4.34335e7 −1.83058
\(129\) 0 0
\(130\) 5.15379e6 0.205743
\(131\) −3.44414e7 −1.33854 −0.669270 0.743019i \(-0.733393\pi\)
−0.669270 + 0.743019i \(0.733393\pi\)
\(132\) 0 0
\(133\) −5.32791e7 −1.96370
\(134\) −8.87517e7 −3.18647
\(135\) 0 0
\(136\) −5.21983e7 −1.77939
\(137\) 4.41452e7 1.46677 0.733385 0.679814i \(-0.237940\pi\)
0.733385 + 0.679814i \(0.237940\pi\)
\(138\) 0 0
\(139\) −1.22144e7 −0.385762 −0.192881 0.981222i \(-0.561783\pi\)
−0.192881 + 0.981222i \(0.561783\pi\)
\(140\) −1.02038e7 −0.314279
\(141\) 0 0
\(142\) 1.12042e8 3.28376
\(143\) 1.16509e7 0.333184
\(144\) 0 0
\(145\) 6.18330e6 0.168435
\(146\) −5.71279e7 −1.51920
\(147\) 0 0
\(148\) −8.95677e7 −2.27109
\(149\) −4.50817e7 −1.11647 −0.558237 0.829682i \(-0.688522\pi\)
−0.558237 + 0.829682i \(0.688522\pi\)
\(150\) 0 0
\(151\) −2.91236e7 −0.688376 −0.344188 0.938901i \(-0.611846\pi\)
−0.344188 + 0.938901i \(0.611846\pi\)
\(152\) −9.29443e7 −2.14669
\(153\) 0 0
\(154\) −3.56748e7 −0.787116
\(155\) −2.60310e6 −0.0561475
\(156\) 0 0
\(157\) 7.43102e7 1.53250 0.766249 0.642544i \(-0.222121\pi\)
0.766249 + 0.642544i \(0.222121\pi\)
\(158\) 1.07118e8 2.16053
\(159\) 0 0
\(160\) 3.65491e6 0.0705434
\(161\) 6.14952e7 1.16132
\(162\) 0 0
\(163\) −5.15149e7 −0.931700 −0.465850 0.884864i \(-0.654251\pi\)
−0.465850 + 0.884864i \(0.654251\pi\)
\(164\) 1.24500e8 2.20401
\(165\) 0 0
\(166\) 1.10461e8 1.87426
\(167\) −3.03652e7 −0.504508 −0.252254 0.967661i \(-0.581172\pi\)
−0.252254 + 0.967661i \(0.581172\pi\)
\(168\) 0 0
\(169\) −1.08969e7 −0.173660
\(170\) 1.84853e7 0.288573
\(171\) 0 0
\(172\) 1.31792e8 1.97488
\(173\) 9.23593e7 1.35619 0.678093 0.734976i \(-0.262807\pi\)
0.678093 + 0.734976i \(0.262807\pi\)
\(174\) 0 0
\(175\) −8.88723e7 −1.25353
\(176\) −1.37310e7 −0.189849
\(177\) 0 0
\(178\) 1.06266e8 1.41229
\(179\) −6.85422e7 −0.893249 −0.446624 0.894722i \(-0.647374\pi\)
−0.446624 + 0.894722i \(0.647374\pi\)
\(180\) 0 0
\(181\) −3.27553e7 −0.410588 −0.205294 0.978700i \(-0.565815\pi\)
−0.205294 + 0.978700i \(0.565815\pi\)
\(182\) −1.58768e8 −1.95215
\(183\) 0 0
\(184\) 1.07277e8 1.26954
\(185\) 1.43829e7 0.167012
\(186\) 0 0
\(187\) 4.17888e7 0.467320
\(188\) −2.88279e8 −3.16418
\(189\) 0 0
\(190\) 3.29149e7 0.348141
\(191\) −2.83438e7 −0.294335 −0.147167 0.989112i \(-0.547016\pi\)
−0.147167 + 0.989112i \(0.547016\pi\)
\(192\) 0 0
\(193\) −1.12860e8 −1.13003 −0.565013 0.825082i \(-0.691129\pi\)
−0.565013 + 0.825082i \(0.691129\pi\)
\(194\) 7.02282e7 0.690567
\(195\) 0 0
\(196\) 1.21470e8 1.15232
\(197\) 1.98897e7 0.185352 0.0926760 0.995696i \(-0.470458\pi\)
0.0926760 + 0.995696i \(0.470458\pi\)
\(198\) 0 0
\(199\) 1.48348e8 1.33443 0.667217 0.744863i \(-0.267485\pi\)
0.667217 + 0.744863i \(0.267485\pi\)
\(200\) −1.55036e8 −1.37034
\(201\) 0 0
\(202\) −1.90680e8 −1.62771
\(203\) −1.90483e8 −1.59816
\(204\) 0 0
\(205\) −1.99924e7 −0.162079
\(206\) −1.66303e8 −1.32545
\(207\) 0 0
\(208\) −6.11089e7 −0.470850
\(209\) 7.44091e7 0.563786
\(210\) 0 0
\(211\) −2.67425e8 −1.95981 −0.979905 0.199465i \(-0.936079\pi\)
−0.979905 + 0.199465i \(0.936079\pi\)
\(212\) −6.40528e7 −0.461703
\(213\) 0 0
\(214\) −6.59061e7 −0.459703
\(215\) −2.11634e7 −0.145228
\(216\) 0 0
\(217\) 8.01913e7 0.532744
\(218\) 2.05830e6 0.0134558
\(219\) 0 0
\(220\) 1.42506e7 0.0902303
\(221\) 1.85978e8 1.15901
\(222\) 0 0
\(223\) −2.62238e8 −1.58354 −0.791768 0.610822i \(-0.790839\pi\)
−0.791768 + 0.610822i \(0.790839\pi\)
\(224\) −1.12593e8 −0.669337
\(225\) 0 0
\(226\) −2.42087e7 −0.139506
\(227\) 3.00342e7 0.170422 0.0852109 0.996363i \(-0.472844\pi\)
0.0852109 + 0.996363i \(0.472844\pi\)
\(228\) 0 0
\(229\) −1.22290e8 −0.672925 −0.336463 0.941697i \(-0.609231\pi\)
−0.336463 + 0.941697i \(0.609231\pi\)
\(230\) −3.79907e7 −0.205887
\(231\) 0 0
\(232\) −3.32294e8 −1.74709
\(233\) 1.17471e8 0.608396 0.304198 0.952609i \(-0.401612\pi\)
0.304198 + 0.952609i \(0.401612\pi\)
\(234\) 0 0
\(235\) 4.62924e7 0.232687
\(236\) −4.80988e7 −0.238200
\(237\) 0 0
\(238\) −5.69459e8 −2.73806
\(239\) −1.02120e8 −0.483859 −0.241930 0.970294i \(-0.577780\pi\)
−0.241930 + 0.970294i \(0.577780\pi\)
\(240\) 0 0
\(241\) −1.01574e8 −0.467439 −0.233719 0.972304i \(-0.575090\pi\)
−0.233719 + 0.972304i \(0.575090\pi\)
\(242\) −3.21046e8 −1.45617
\(243\) 0 0
\(244\) 6.17337e8 2.72056
\(245\) −1.95058e7 −0.0847389
\(246\) 0 0
\(247\) 3.31152e8 1.39826
\(248\) 1.39892e8 0.582388
\(249\) 0 0
\(250\) 1.10820e8 0.448566
\(251\) 4.09066e8 1.63281 0.816405 0.577480i \(-0.195964\pi\)
0.816405 + 0.577480i \(0.195964\pi\)
\(252\) 0 0
\(253\) −8.58837e7 −0.333418
\(254\) 2.78777e8 1.06743
\(255\) 0 0
\(256\) −4.50813e8 −1.67941
\(257\) 5.16514e8 1.89809 0.949043 0.315147i \(-0.102054\pi\)
0.949043 + 0.315147i \(0.102054\pi\)
\(258\) 0 0
\(259\) −4.43082e8 −1.58465
\(260\) 6.34211e7 0.223783
\(261\) 0 0
\(262\) −6.55469e8 −2.25163
\(263\) 5.42714e8 1.83961 0.919805 0.392377i \(-0.128347\pi\)
0.919805 + 0.392377i \(0.128347\pi\)
\(264\) 0 0
\(265\) 1.02857e7 0.0339527
\(266\) −1.01398e9 −3.30326
\(267\) 0 0
\(268\) −1.09215e9 −3.46586
\(269\) −1.04494e8 −0.327309 −0.163655 0.986518i \(-0.552328\pi\)
−0.163655 + 0.986518i \(0.552328\pi\)
\(270\) 0 0
\(271\) 6.80419e7 0.207675 0.103837 0.994594i \(-0.466888\pi\)
0.103837 + 0.994594i \(0.466888\pi\)
\(272\) −2.19181e8 −0.660408
\(273\) 0 0
\(274\) 8.40147e8 2.46733
\(275\) 1.24118e8 0.359891
\(276\) 0 0
\(277\) −2.90966e8 −0.822553 −0.411276 0.911511i \(-0.634917\pi\)
−0.411276 + 0.911511i \(0.634917\pi\)
\(278\) −2.32457e8 −0.648913
\(279\) 0 0
\(280\) −8.80564e7 −0.239722
\(281\) 6.50736e7 0.174958 0.0874788 0.996166i \(-0.472119\pi\)
0.0874788 + 0.996166i \(0.472119\pi\)
\(282\) 0 0
\(283\) 5.18818e8 1.36070 0.680350 0.732887i \(-0.261828\pi\)
0.680350 + 0.732887i \(0.261828\pi\)
\(284\) 1.37875e9 3.57168
\(285\) 0 0
\(286\) 2.21734e8 0.560468
\(287\) 6.15886e8 1.53785
\(288\) 0 0
\(289\) 2.56716e8 0.625620
\(290\) 1.17677e8 0.283334
\(291\) 0 0
\(292\) −7.03000e8 −1.65240
\(293\) −2.50544e7 −0.0581900 −0.0290950 0.999577i \(-0.509263\pi\)
−0.0290950 + 0.999577i \(0.509263\pi\)
\(294\) 0 0
\(295\) 7.72379e6 0.0175167
\(296\) −7.72946e8 −1.73232
\(297\) 0 0
\(298\) −8.57969e8 −1.87808
\(299\) −3.82219e8 −0.826919
\(300\) 0 0
\(301\) 6.51961e8 1.37797
\(302\) −5.54264e8 −1.15796
\(303\) 0 0
\(304\) −3.90274e8 −0.796732
\(305\) −9.91331e7 −0.200064
\(306\) 0 0
\(307\) −3.47020e8 −0.684496 −0.342248 0.939610i \(-0.611188\pi\)
−0.342248 + 0.939610i \(0.611188\pi\)
\(308\) −4.39003e8 −0.856131
\(309\) 0 0
\(310\) −4.95408e7 −0.0944489
\(311\) 1.38483e8 0.261058 0.130529 0.991445i \(-0.458332\pi\)
0.130529 + 0.991445i \(0.458332\pi\)
\(312\) 0 0
\(313\) −3.52364e8 −0.649511 −0.324756 0.945798i \(-0.605282\pi\)
−0.324756 + 0.945798i \(0.605282\pi\)
\(314\) 1.41423e9 2.57790
\(315\) 0 0
\(316\) 1.31816e9 2.34997
\(317\) 31157.4 5.49355e−5 0 2.74678e−5 1.00000i \(-0.499991\pi\)
2.74678e−5 1.00000i \(0.499991\pi\)
\(318\) 0 0
\(319\) 2.66027e8 0.458837
\(320\) 1.10410e8 0.188357
\(321\) 0 0
\(322\) 1.17034e9 1.95352
\(323\) 1.18776e9 1.96118
\(324\) 0 0
\(325\) 5.52379e8 0.892576
\(326\) −9.80402e8 −1.56727
\(327\) 0 0
\(328\) 1.07440e9 1.68115
\(329\) −1.42609e9 −2.20780
\(330\) 0 0
\(331\) 1.16652e9 1.76805 0.884026 0.467438i \(-0.154823\pi\)
0.884026 + 0.467438i \(0.154823\pi\)
\(332\) 1.35930e9 2.03860
\(333\) 0 0
\(334\) −5.77893e8 −0.848662
\(335\) 1.75380e8 0.254872
\(336\) 0 0
\(337\) −4.15334e8 −0.591143 −0.295572 0.955321i \(-0.595510\pi\)
−0.295572 + 0.955321i \(0.595510\pi\)
\(338\) −2.07383e8 −0.292123
\(339\) 0 0
\(340\) 2.27475e8 0.313875
\(341\) −1.11994e8 −0.152952
\(342\) 0 0
\(343\) −3.53209e8 −0.472609
\(344\) 1.13733e9 1.50637
\(345\) 0 0
\(346\) 1.75773e9 2.28132
\(347\) −7.92787e8 −1.01860 −0.509300 0.860589i \(-0.670096\pi\)
−0.509300 + 0.860589i \(0.670096\pi\)
\(348\) 0 0
\(349\) 3.35615e7 0.0422622 0.0211311 0.999777i \(-0.493273\pi\)
0.0211311 + 0.999777i \(0.493273\pi\)
\(350\) −1.69137e9 −2.10863
\(351\) 0 0
\(352\) 1.57247e8 0.192169
\(353\) 8.45067e8 1.02254 0.511269 0.859421i \(-0.329175\pi\)
0.511269 + 0.859421i \(0.329175\pi\)
\(354\) 0 0
\(355\) −2.21403e8 −0.262654
\(356\) 1.30767e9 1.53612
\(357\) 0 0
\(358\) −1.30446e9 −1.50258
\(359\) 9.14770e8 1.04347 0.521736 0.853107i \(-0.325284\pi\)
0.521736 + 0.853107i \(0.325284\pi\)
\(360\) 0 0
\(361\) 1.22104e9 1.36602
\(362\) −6.23380e8 −0.690673
\(363\) 0 0
\(364\) −1.95375e9 −2.12332
\(365\) 1.12889e8 0.121514
\(366\) 0 0
\(367\) −1.55832e9 −1.64561 −0.822805 0.568324i \(-0.807592\pi\)
−0.822805 + 0.568324i \(0.807592\pi\)
\(368\) 4.50458e8 0.471180
\(369\) 0 0
\(370\) 2.73728e8 0.280940
\(371\) −3.16863e8 −0.322153
\(372\) 0 0
\(373\) 1.33970e9 1.33667 0.668337 0.743858i \(-0.267006\pi\)
0.668337 + 0.743858i \(0.267006\pi\)
\(374\) 7.95301e8 0.786106
\(375\) 0 0
\(376\) −2.48778e9 −2.41354
\(377\) 1.18393e9 1.13797
\(378\) 0 0
\(379\) −3.03138e8 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(380\) 4.05041e8 0.378666
\(381\) 0 0
\(382\) −5.39423e8 −0.495117
\(383\) 7.20550e8 0.655342 0.327671 0.944792i \(-0.393736\pi\)
0.327671 + 0.944792i \(0.393736\pi\)
\(384\) 0 0
\(385\) 7.04960e7 0.0629581
\(386\) −2.14788e9 −1.90088
\(387\) 0 0
\(388\) 8.64208e8 0.751117
\(389\) 5.84956e8 0.503848 0.251924 0.967747i \(-0.418937\pi\)
0.251924 + 0.967747i \(0.418937\pi\)
\(390\) 0 0
\(391\) −1.37092e9 −1.15983
\(392\) 1.04825e9 0.878951
\(393\) 0 0
\(394\) 3.78530e8 0.311791
\(395\) −2.11672e8 −0.172812
\(396\) 0 0
\(397\) 7.98494e8 0.640479 0.320240 0.947337i \(-0.396237\pi\)
0.320240 + 0.947337i \(0.396237\pi\)
\(398\) 2.82328e9 2.24473
\(399\) 0 0
\(400\) −6.50997e8 −0.508591
\(401\) 1.07381e9 0.831618 0.415809 0.909452i \(-0.363499\pi\)
0.415809 + 0.909452i \(0.363499\pi\)
\(402\) 0 0
\(403\) −4.98423e8 −0.379342
\(404\) −2.34646e9 −1.77043
\(405\) 0 0
\(406\) −3.62517e9 −2.68836
\(407\) 6.18804e8 0.454959
\(408\) 0 0
\(409\) −6.01255e8 −0.434537 −0.217269 0.976112i \(-0.569715\pi\)
−0.217269 + 0.976112i \(0.569715\pi\)
\(410\) −3.80483e8 −0.272642
\(411\) 0 0
\(412\) −2.04647e9 −1.44167
\(413\) −2.37939e8 −0.166204
\(414\) 0 0
\(415\) −2.18279e8 −0.149915
\(416\) 6.99815e8 0.476603
\(417\) 0 0
\(418\) 1.41611e9 0.948376
\(419\) −1.42631e9 −0.947251 −0.473626 0.880726i \(-0.657055\pi\)
−0.473626 + 0.880726i \(0.657055\pi\)
\(420\) 0 0
\(421\) −6.51184e8 −0.425320 −0.212660 0.977126i \(-0.568213\pi\)
−0.212660 + 0.977126i \(0.568213\pi\)
\(422\) −5.08948e9 −3.29671
\(423\) 0 0
\(424\) −5.52760e8 −0.352173
\(425\) 1.98124e9 1.25192
\(426\) 0 0
\(427\) 3.05390e9 1.89827
\(428\) −8.11021e8 −0.500010
\(429\) 0 0
\(430\) −4.02770e8 −0.244297
\(431\) −1.75470e9 −1.05568 −0.527841 0.849343i \(-0.676998\pi\)
−0.527841 + 0.849343i \(0.676998\pi\)
\(432\) 0 0
\(433\) 9.35278e8 0.553647 0.276824 0.960921i \(-0.410718\pi\)
0.276824 + 0.960921i \(0.410718\pi\)
\(434\) 1.52616e9 0.896158
\(435\) 0 0
\(436\) 2.53288e7 0.0146357
\(437\) −2.44106e9 −1.39924
\(438\) 0 0
\(439\) 8.68080e8 0.489704 0.244852 0.969560i \(-0.421261\pi\)
0.244852 + 0.969560i \(0.421261\pi\)
\(440\) 1.22979e8 0.0688249
\(441\) 0 0
\(442\) 3.53943e9 1.94964
\(443\) −8.22011e8 −0.449226 −0.224613 0.974448i \(-0.572112\pi\)
−0.224613 + 0.974448i \(0.572112\pi\)
\(444\) 0 0
\(445\) −2.09989e8 −0.112963
\(446\) −4.99076e9 −2.66376
\(447\) 0 0
\(448\) −3.40128e9 −1.78719
\(449\) −1.57362e9 −0.820422 −0.410211 0.911991i \(-0.634545\pi\)
−0.410211 + 0.911991i \(0.634545\pi\)
\(450\) 0 0
\(451\) −8.60140e8 −0.441521
\(452\) −2.97906e8 −0.151738
\(453\) 0 0
\(454\) 5.71593e8 0.286676
\(455\) 3.13737e8 0.156144
\(456\) 0 0
\(457\) 2.23657e9 1.09616 0.548081 0.836425i \(-0.315358\pi\)
0.548081 + 0.836425i \(0.315358\pi\)
\(458\) −2.32735e9 −1.13197
\(459\) 0 0
\(460\) −4.67502e8 −0.223940
\(461\) −3.55618e9 −1.69056 −0.845280 0.534324i \(-0.820566\pi\)
−0.845280 + 0.534324i \(0.820566\pi\)
\(462\) 0 0
\(463\) −3.48898e9 −1.63367 −0.816836 0.576870i \(-0.804274\pi\)
−0.816836 + 0.576870i \(0.804274\pi\)
\(464\) −1.39531e9 −0.648420
\(465\) 0 0
\(466\) 2.23565e9 1.02342
\(467\) 9.44654e8 0.429204 0.214602 0.976702i \(-0.431155\pi\)
0.214602 + 0.976702i \(0.431155\pi\)
\(468\) 0 0
\(469\) −5.40276e9 −2.41830
\(470\) 8.81010e8 0.391416
\(471\) 0 0
\(472\) −4.15080e8 −0.181692
\(473\) −9.10523e8 −0.395619
\(474\) 0 0
\(475\) 3.52779e9 1.51034
\(476\) −7.00760e9 −2.97814
\(477\) 0 0
\(478\) −1.94349e9 −0.813927
\(479\) −3.10871e8 −0.129243 −0.0646213 0.997910i \(-0.520584\pi\)
−0.0646213 + 0.997910i \(0.520584\pi\)
\(480\) 0 0
\(481\) 2.75394e9 1.12836
\(482\) −1.93311e9 −0.786305
\(483\) 0 0
\(484\) −3.95069e9 −1.58385
\(485\) −1.38776e8 −0.0552356
\(486\) 0 0
\(487\) −2.73852e9 −1.07440 −0.537198 0.843456i \(-0.680517\pi\)
−0.537198 + 0.843456i \(0.680517\pi\)
\(488\) 5.32746e9 2.07516
\(489\) 0 0
\(490\) −3.71224e8 −0.142544
\(491\) 3.51022e8 0.133829 0.0669143 0.997759i \(-0.478685\pi\)
0.0669143 + 0.997759i \(0.478685\pi\)
\(492\) 0 0
\(493\) 4.24646e9 1.59611
\(494\) 6.30230e9 2.35209
\(495\) 0 0
\(496\) 5.87408e8 0.216150
\(497\) 6.82055e9 2.49214
\(498\) 0 0
\(499\) −4.09829e9 −1.47656 −0.738280 0.674495i \(-0.764362\pi\)
−0.738280 + 0.674495i \(0.764362\pi\)
\(500\) 1.36371e9 0.487897
\(501\) 0 0
\(502\) 7.78511e9 2.74664
\(503\) 3.66315e9 1.28341 0.641707 0.766950i \(-0.278226\pi\)
0.641707 + 0.766950i \(0.278226\pi\)
\(504\) 0 0
\(505\) 3.76799e8 0.130194
\(506\) −1.63449e9 −0.560861
\(507\) 0 0
\(508\) 3.43055e9 1.16102
\(509\) 2.18735e9 0.735202 0.367601 0.929984i \(-0.380179\pi\)
0.367601 + 0.929984i \(0.380179\pi\)
\(510\) 0 0
\(511\) −3.47767e9 −1.15296
\(512\) −3.02013e9 −0.994446
\(513\) 0 0
\(514\) 9.82999e9 3.19288
\(515\) 3.28627e8 0.106017
\(516\) 0 0
\(517\) 1.99166e9 0.633867
\(518\) −8.43248e9 −2.66564
\(519\) 0 0
\(520\) 5.47308e8 0.170695
\(521\) −2.54763e9 −0.789231 −0.394615 0.918846i \(-0.629122\pi\)
−0.394615 + 0.918846i \(0.629122\pi\)
\(522\) 0 0
\(523\) 1.00934e9 0.308519 0.154259 0.988030i \(-0.450701\pi\)
0.154259 + 0.988030i \(0.450701\pi\)
\(524\) −8.06601e9 −2.44906
\(525\) 0 0
\(526\) 1.03286e10 3.09451
\(527\) −1.78771e9 −0.532060
\(528\) 0 0
\(529\) −5.87336e8 −0.172501
\(530\) 1.95752e8 0.0571138
\(531\) 0 0
\(532\) −1.24777e10 −3.59289
\(533\) −3.82799e9 −1.09503
\(534\) 0 0
\(535\) 1.30235e8 0.0367697
\(536\) −9.42500e9 −2.64365
\(537\) 0 0
\(538\) −1.98867e9 −0.550585
\(539\) −8.39207e8 −0.230839
\(540\) 0 0
\(541\) −9.25297e8 −0.251241 −0.125621 0.992078i \(-0.540092\pi\)
−0.125621 + 0.992078i \(0.540092\pi\)
\(542\) 1.29493e9 0.349341
\(543\) 0 0
\(544\) 2.51005e9 0.668477
\(545\) −4.06735e6 −0.00107628
\(546\) 0 0
\(547\) −5.27884e9 −1.37906 −0.689530 0.724257i \(-0.742183\pi\)
−0.689530 + 0.724257i \(0.742183\pi\)
\(548\) 1.03386e10 2.68367
\(549\) 0 0
\(550\) 2.36215e9 0.605393
\(551\) 7.56123e9 1.92558
\(552\) 0 0
\(553\) 6.52079e9 1.63969
\(554\) −5.53751e9 −1.38366
\(555\) 0 0
\(556\) −2.86055e9 −0.705810
\(557\) −3.38179e9 −0.829190 −0.414595 0.910006i \(-0.636077\pi\)
−0.414595 + 0.910006i \(0.636077\pi\)
\(558\) 0 0
\(559\) −4.05222e9 −0.981186
\(560\) −3.69750e8 −0.0889713
\(561\) 0 0
\(562\) 1.23844e9 0.294306
\(563\) 8.68855e7 0.0205196 0.0102598 0.999947i \(-0.496734\pi\)
0.0102598 + 0.999947i \(0.496734\pi\)
\(564\) 0 0
\(565\) 4.78383e7 0.0111585
\(566\) 9.87385e9 2.28891
\(567\) 0 0
\(568\) 1.18983e10 2.72437
\(569\) −3.35876e9 −0.764338 −0.382169 0.924092i \(-0.624823\pi\)
−0.382169 + 0.924092i \(0.624823\pi\)
\(570\) 0 0
\(571\) −4.94562e9 −1.11172 −0.555859 0.831277i \(-0.687610\pi\)
−0.555859 + 0.831277i \(0.687610\pi\)
\(572\) 2.72859e9 0.609611
\(573\) 0 0
\(574\) 1.17212e10 2.58690
\(575\) −4.07180e9 −0.893202
\(576\) 0 0
\(577\) −6.72074e9 −1.45647 −0.728235 0.685327i \(-0.759659\pi\)
−0.728235 + 0.685327i \(0.759659\pi\)
\(578\) 4.88567e9 1.05239
\(579\) 0 0
\(580\) 1.44810e9 0.308177
\(581\) 6.72432e9 1.42243
\(582\) 0 0
\(583\) 4.42527e8 0.0924911
\(584\) −6.06671e9 −1.26040
\(585\) 0 0
\(586\) −4.76822e8 −0.0978846
\(587\) 9.21166e9 1.87977 0.939885 0.341491i \(-0.110932\pi\)
0.939885 + 0.341491i \(0.110932\pi\)
\(588\) 0 0
\(589\) −3.18320e9 −0.641889
\(590\) 1.46995e8 0.0294659
\(591\) 0 0
\(592\) −3.24561e9 −0.642940
\(593\) 5.16913e9 1.01795 0.508974 0.860782i \(-0.330025\pi\)
0.508974 + 0.860782i \(0.330025\pi\)
\(594\) 0 0
\(595\) 1.12529e9 0.219006
\(596\) −1.05579e10 −2.04275
\(597\) 0 0
\(598\) −7.27417e9 −1.39101
\(599\) 2.29633e9 0.436557 0.218278 0.975887i \(-0.429956\pi\)
0.218278 + 0.975887i \(0.429956\pi\)
\(600\) 0 0
\(601\) 4.46476e9 0.838952 0.419476 0.907766i \(-0.362214\pi\)
0.419476 + 0.907766i \(0.362214\pi\)
\(602\) 1.24078e10 2.31796
\(603\) 0 0
\(604\) −6.82061e9 −1.25949
\(605\) 6.34410e8 0.116473
\(606\) 0 0
\(607\) −2.99837e9 −0.544158 −0.272079 0.962275i \(-0.587711\pi\)
−0.272079 + 0.962275i \(0.587711\pi\)
\(608\) 4.46939e9 0.806466
\(609\) 0 0
\(610\) −1.88664e9 −0.336539
\(611\) 8.86373e9 1.57207
\(612\) 0 0
\(613\) 8.71253e9 1.52768 0.763840 0.645405i \(-0.223312\pi\)
0.763840 + 0.645405i \(0.223312\pi\)
\(614\) −6.60429e9 −1.15143
\(615\) 0 0
\(616\) −3.78849e9 −0.653031
\(617\) −3.68960e9 −0.632385 −0.316192 0.948695i \(-0.602405\pi\)
−0.316192 + 0.948695i \(0.602405\pi\)
\(618\) 0 0
\(619\) 2.78943e9 0.472713 0.236357 0.971666i \(-0.424047\pi\)
0.236357 + 0.971666i \(0.424047\pi\)
\(620\) −6.09634e8 −0.102730
\(621\) 0 0
\(622\) 2.63554e9 0.439140
\(623\) 6.46892e9 1.07182
\(624\) 0 0
\(625\) 5.77403e9 0.946018
\(626\) −6.70599e9 −1.09258
\(627\) 0 0
\(628\) 1.74031e10 2.80393
\(629\) 9.87766e9 1.58262
\(630\) 0 0
\(631\) 1.40846e9 0.223173 0.111587 0.993755i \(-0.464407\pi\)
0.111587 + 0.993755i \(0.464407\pi\)
\(632\) 1.13754e10 1.79249
\(633\) 0 0
\(634\) 592969. 9.24101e−5 0
\(635\) −5.50884e8 −0.0853792
\(636\) 0 0
\(637\) −3.73483e9 −0.572510
\(638\) 5.06287e9 0.771835
\(639\) 0 0
\(640\) 1.63342e9 0.246303
\(641\) 8.08641e9 1.21270 0.606349 0.795199i \(-0.292633\pi\)
0.606349 + 0.795199i \(0.292633\pi\)
\(642\) 0 0
\(643\) −5.86883e9 −0.870590 −0.435295 0.900288i \(-0.643356\pi\)
−0.435295 + 0.900288i \(0.643356\pi\)
\(644\) 1.44019e10 2.12480
\(645\) 0 0
\(646\) 2.26047e10 3.29902
\(647\) −3.25260e9 −0.472135 −0.236067 0.971737i \(-0.575859\pi\)
−0.236067 + 0.971737i \(0.575859\pi\)
\(648\) 0 0
\(649\) 3.32304e8 0.0477177
\(650\) 1.05126e10 1.50145
\(651\) 0 0
\(652\) −1.20645e10 −1.70469
\(653\) 1.64182e9 0.230744 0.115372 0.993322i \(-0.463194\pi\)
0.115372 + 0.993322i \(0.463194\pi\)
\(654\) 0 0
\(655\) 1.29526e9 0.180099
\(656\) 4.51142e9 0.623950
\(657\) 0 0
\(658\) −2.71404e10 −3.71387
\(659\) −4.63845e9 −0.631355 −0.315678 0.948866i \(-0.602232\pi\)
−0.315678 + 0.948866i \(0.602232\pi\)
\(660\) 0 0
\(661\) 9.44802e9 1.27244 0.636218 0.771510i \(-0.280498\pi\)
0.636218 + 0.771510i \(0.280498\pi\)
\(662\) 2.22006e10 2.97414
\(663\) 0 0
\(664\) 1.17304e10 1.55498
\(665\) 2.00369e9 0.264214
\(666\) 0 0
\(667\) −8.72725e9 −1.13877
\(668\) −7.11138e9 −0.923073
\(669\) 0 0
\(670\) 3.33773e9 0.428735
\(671\) −4.26505e9 −0.544998
\(672\) 0 0
\(673\) 8.85543e9 1.11984 0.559921 0.828546i \(-0.310831\pi\)
0.559921 + 0.828546i \(0.310831\pi\)
\(674\) −7.90440e9 −0.994395
\(675\) 0 0
\(676\) −2.55200e9 −0.317737
\(677\) −1.90990e8 −0.0236565 −0.0118283 0.999930i \(-0.503765\pi\)
−0.0118283 + 0.999930i \(0.503765\pi\)
\(678\) 0 0
\(679\) 4.27515e9 0.524091
\(680\) 1.96305e9 0.239414
\(681\) 0 0
\(682\) −2.13141e9 −0.257290
\(683\) 3.13279e9 0.376235 0.188117 0.982147i \(-0.439761\pi\)
0.188117 + 0.982147i \(0.439761\pi\)
\(684\) 0 0
\(685\) −1.66019e9 −0.197352
\(686\) −6.72207e9 −0.795003
\(687\) 0 0
\(688\) 4.77567e9 0.559081
\(689\) 1.96944e9 0.229390
\(690\) 0 0
\(691\) 1.83456e9 0.211524 0.105762 0.994391i \(-0.466272\pi\)
0.105762 + 0.994391i \(0.466272\pi\)
\(692\) 2.16301e10 2.48135
\(693\) 0 0
\(694\) −1.50879e10 −1.71344
\(695\) 4.59352e8 0.0519038
\(696\) 0 0
\(697\) −1.37300e10 −1.53587
\(698\) 6.38723e8 0.0710917
\(699\) 0 0
\(700\) −2.08135e10 −2.29351
\(701\) −5.98660e9 −0.656398 −0.328199 0.944609i \(-0.606442\pi\)
−0.328199 + 0.944609i \(0.606442\pi\)
\(702\) 0 0
\(703\) 1.75881e10 1.90931
\(704\) 4.75020e9 0.513107
\(705\) 0 0
\(706\) 1.60828e10 1.72007
\(707\) −1.16077e10 −1.23531
\(708\) 0 0
\(709\) 6.41985e9 0.676493 0.338246 0.941058i \(-0.390166\pi\)
0.338246 + 0.941058i \(0.390166\pi\)
\(710\) −4.21361e9 −0.441825
\(711\) 0 0
\(712\) 1.12849e10 1.17170
\(713\) 3.67408e9 0.379608
\(714\) 0 0
\(715\) −4.38162e8 −0.0448295
\(716\) −1.60523e10 −1.63433
\(717\) 0 0
\(718\) 1.74094e10 1.75528
\(719\) 1.53454e10 1.53967 0.769836 0.638242i \(-0.220338\pi\)
0.769836 + 0.638242i \(0.220338\pi\)
\(720\) 0 0
\(721\) −1.01237e10 −1.00592
\(722\) 2.32382e10 2.29785
\(723\) 0 0
\(724\) −7.67113e9 −0.751232
\(725\) 1.26125e10 1.22919
\(726\) 0 0
\(727\) 5.06244e8 0.0488640 0.0244320 0.999701i \(-0.492222\pi\)
0.0244320 + 0.999701i \(0.492222\pi\)
\(728\) −1.68604e10 −1.61960
\(729\) 0 0
\(730\) 2.14844e9 0.204406
\(731\) −1.45342e10 −1.37620
\(732\) 0 0
\(733\) −1.29579e10 −1.21527 −0.607633 0.794218i \(-0.707881\pi\)
−0.607633 + 0.794218i \(0.707881\pi\)
\(734\) −2.96571e10 −2.76817
\(735\) 0 0
\(736\) −5.15862e9 −0.476937
\(737\) 7.54544e9 0.694302
\(738\) 0 0
\(739\) −3.46591e9 −0.315909 −0.157955 0.987446i \(-0.550490\pi\)
−0.157955 + 0.987446i \(0.550490\pi\)
\(740\) 3.36842e9 0.305573
\(741\) 0 0
\(742\) −6.03035e9 −0.541912
\(743\) −4.65023e9 −0.415924 −0.207962 0.978137i \(-0.566683\pi\)
−0.207962 + 0.978137i \(0.566683\pi\)
\(744\) 0 0
\(745\) 1.69541e9 0.150220
\(746\) 2.54963e10 2.24850
\(747\) 0 0
\(748\) 9.78674e9 0.855032
\(749\) −4.01204e9 −0.348882
\(750\) 0 0
\(751\) 2.20118e10 1.89634 0.948169 0.317766i \(-0.102933\pi\)
0.948169 + 0.317766i \(0.102933\pi\)
\(752\) −1.04462e10 −0.895769
\(753\) 0 0
\(754\) 2.25319e10 1.91425
\(755\) 1.09527e9 0.0926201
\(756\) 0 0
\(757\) 5.77225e9 0.483626 0.241813 0.970323i \(-0.422258\pi\)
0.241813 + 0.970323i \(0.422258\pi\)
\(758\) −5.76914e9 −0.481137
\(759\) 0 0
\(760\) 3.49540e9 0.288835
\(761\) −5.11102e9 −0.420398 −0.210199 0.977659i \(-0.567411\pi\)
−0.210199 + 0.977659i \(0.567411\pi\)
\(762\) 0 0
\(763\) 1.25299e8 0.0102120
\(764\) −6.63798e9 −0.538529
\(765\) 0 0
\(766\) 1.37131e10 1.10239
\(767\) 1.47889e9 0.118346
\(768\) 0 0
\(769\) 2.04941e9 0.162512 0.0812560 0.996693i \(-0.474107\pi\)
0.0812560 + 0.996693i \(0.474107\pi\)
\(770\) 1.34164e9 0.105905
\(771\) 0 0
\(772\) −2.64312e10 −2.06755
\(773\) −2.05915e10 −1.60346 −0.801732 0.597683i \(-0.796088\pi\)
−0.801732 + 0.597683i \(0.796088\pi\)
\(774\) 0 0
\(775\) −5.30974e9 −0.409748
\(776\) 7.45790e9 0.572929
\(777\) 0 0
\(778\) 1.11326e10 0.847552
\(779\) −2.44476e10 −1.85291
\(780\) 0 0
\(781\) −9.52551e9 −0.715500
\(782\) −2.60905e10 −1.95101
\(783\) 0 0
\(784\) 4.40162e9 0.326217
\(785\) −2.79462e9 −0.206196
\(786\) 0 0
\(787\) −1.82710e10 −1.33613 −0.668067 0.744101i \(-0.732878\pi\)
−0.668067 + 0.744101i \(0.732878\pi\)
\(788\) 4.65808e9 0.339129
\(789\) 0 0
\(790\) −4.02842e9 −0.290697
\(791\) −1.47371e9 −0.105875
\(792\) 0 0
\(793\) −1.89813e10 −1.35167
\(794\) 1.51965e10 1.07739
\(795\) 0 0
\(796\) 3.47425e10 2.44155
\(797\) −2.36801e10 −1.65684 −0.828418 0.560110i \(-0.810759\pi\)
−0.828418 + 0.560110i \(0.810759\pi\)
\(798\) 0 0
\(799\) 3.17919e10 2.20497
\(800\) 7.45518e9 0.514805
\(801\) 0 0
\(802\) 2.04362e10 1.39891
\(803\) 4.85687e9 0.331019
\(804\) 0 0
\(805\) −2.31268e9 −0.156254
\(806\) −9.48570e9 −0.638112
\(807\) 0 0
\(808\) −2.02493e10 −1.35043
\(809\) −1.22571e10 −0.813897 −0.406949 0.913451i \(-0.633407\pi\)
−0.406949 + 0.913451i \(0.633407\pi\)
\(810\) 0 0
\(811\) −7.84590e9 −0.516499 −0.258250 0.966078i \(-0.583146\pi\)
−0.258250 + 0.966078i \(0.583146\pi\)
\(812\) −4.46102e10 −2.92407
\(813\) 0 0
\(814\) 1.17767e10 0.765312
\(815\) 1.93735e9 0.125359
\(816\) 0 0
\(817\) −2.58796e10 −1.66028
\(818\) −1.14427e10 −0.730960
\(819\) 0 0
\(820\) −4.68212e9 −0.296547
\(821\) 8.10180e9 0.510952 0.255476 0.966815i \(-0.417768\pi\)
0.255476 + 0.966815i \(0.417768\pi\)
\(822\) 0 0
\(823\) 1.64265e9 0.102718 0.0513588 0.998680i \(-0.483645\pi\)
0.0513588 + 0.998680i \(0.483645\pi\)
\(824\) −1.76606e10 −1.09966
\(825\) 0 0
\(826\) −4.52833e9 −0.279581
\(827\) 2.72238e10 1.67371 0.836854 0.547427i \(-0.184393\pi\)
0.836854 + 0.547427i \(0.184393\pi\)
\(828\) 0 0
\(829\) 1.28481e10 0.783245 0.391623 0.920126i \(-0.371914\pi\)
0.391623 + 0.920126i \(0.371914\pi\)
\(830\) −4.15416e9 −0.252180
\(831\) 0 0
\(832\) 2.11404e10 1.27257
\(833\) −1.33959e10 −0.802995
\(834\) 0 0
\(835\) 1.14196e9 0.0678809
\(836\) 1.74263e10 1.03153
\(837\) 0 0
\(838\) −2.71447e10 −1.59342
\(839\) 2.14886e10 1.25615 0.628073 0.778154i \(-0.283844\pi\)
0.628073 + 0.778154i \(0.283844\pi\)
\(840\) 0 0
\(841\) 9.78301e9 0.567135
\(842\) −1.23930e10 −0.715455
\(843\) 0 0
\(844\) −6.26297e10 −3.58577
\(845\) 4.09805e8 0.0233657
\(846\) 0 0
\(847\) −1.95437e10 −1.10513
\(848\) −2.32105e9 −0.130707
\(849\) 0 0
\(850\) 3.77058e10 2.10592
\(851\) −2.03004e10 −1.12915
\(852\) 0 0
\(853\) 7.06458e9 0.389731 0.194865 0.980830i \(-0.437573\pi\)
0.194865 + 0.980830i \(0.437573\pi\)
\(854\) 5.81200e10 3.19318
\(855\) 0 0
\(856\) −6.99891e9 −0.381393
\(857\) 1.97164e10 1.07003 0.535015 0.844843i \(-0.320306\pi\)
0.535015 + 0.844843i \(0.320306\pi\)
\(858\) 0 0
\(859\) −1.48500e10 −0.799376 −0.399688 0.916651i \(-0.630882\pi\)
−0.399688 + 0.916651i \(0.630882\pi\)
\(860\) −4.95637e9 −0.265717
\(861\) 0 0
\(862\) −3.33945e10 −1.77582
\(863\) −1.20569e10 −0.638552 −0.319276 0.947662i \(-0.603440\pi\)
−0.319276 + 0.947662i \(0.603440\pi\)
\(864\) 0 0
\(865\) −3.47340e9 −0.182473
\(866\) 1.77997e10 0.931321
\(867\) 0 0
\(868\) 1.87804e10 0.974735
\(869\) −9.10687e9 −0.470760
\(870\) 0 0
\(871\) 3.35804e10 1.72196
\(872\) 2.18581e8 0.0111636
\(873\) 0 0
\(874\) −4.64568e10 −2.35374
\(875\) 6.74615e9 0.340430
\(876\) 0 0
\(877\) 3.18390e10 1.59390 0.796949 0.604046i \(-0.206446\pi\)
0.796949 + 0.604046i \(0.206446\pi\)
\(878\) 1.65208e10 0.823759
\(879\) 0 0
\(880\) 5.16389e8 0.0255439
\(881\) 2.68011e10 1.32050 0.660248 0.751048i \(-0.270451\pi\)
0.660248 + 0.751048i \(0.270451\pi\)
\(882\) 0 0
\(883\) −3.21062e10 −1.56937 −0.784686 0.619893i \(-0.787176\pi\)
−0.784686 + 0.619893i \(0.787176\pi\)
\(884\) 4.35552e10 2.12059
\(885\) 0 0
\(886\) −1.56440e10 −0.755667
\(887\) −1.26881e10 −0.610468 −0.305234 0.952277i \(-0.598735\pi\)
−0.305234 + 0.952277i \(0.598735\pi\)
\(888\) 0 0
\(889\) 1.69706e10 0.810103
\(890\) −3.99638e9 −0.190021
\(891\) 0 0
\(892\) −6.14148e10 −2.89732
\(893\) 5.66085e10 2.66012
\(894\) 0 0
\(895\) 2.57770e9 0.120185
\(896\) −5.03193e10 −2.33699
\(897\) 0 0
\(898\) −2.99482e10 −1.38008
\(899\) −1.13805e10 −0.522401
\(900\) 0 0
\(901\) 7.06385e9 0.321740
\(902\) −1.63697e10 −0.742708
\(903\) 0 0
\(904\) −2.57085e9 −0.115741
\(905\) 1.23184e9 0.0552441
\(906\) 0 0
\(907\) −1.35224e10 −0.601767 −0.300884 0.953661i \(-0.597282\pi\)
−0.300884 + 0.953661i \(0.597282\pi\)
\(908\) 7.03385e9 0.311812
\(909\) 0 0
\(910\) 5.97087e9 0.262659
\(911\) −2.02421e10 −0.887035 −0.443517 0.896266i \(-0.646270\pi\)
−0.443517 + 0.896266i \(0.646270\pi\)
\(912\) 0 0
\(913\) −9.39112e9 −0.408385
\(914\) 4.25650e10 1.84392
\(915\) 0 0
\(916\) −2.86397e10 −1.23122
\(917\) −3.99017e10 −1.70883
\(918\) 0 0
\(919\) 3.59827e10 1.52929 0.764644 0.644453i \(-0.222915\pi\)
0.764644 + 0.644453i \(0.222915\pi\)
\(920\) −4.03442e9 −0.170814
\(921\) 0 0
\(922\) −6.76791e10 −2.84378
\(923\) −4.23926e10 −1.77453
\(924\) 0 0
\(925\) 2.93379e10 1.21880
\(926\) −6.64002e10 −2.74809
\(927\) 0 0
\(928\) 1.59790e10 0.656342
\(929\) −3.99605e10 −1.63522 −0.817609 0.575774i \(-0.804701\pi\)
−0.817609 + 0.575774i \(0.804701\pi\)
\(930\) 0 0
\(931\) −2.38526e10 −0.968751
\(932\) 2.75112e10 1.11315
\(933\) 0 0
\(934\) 1.79781e10 0.721988
\(935\) −1.57157e9 −0.0628773
\(936\) 0 0
\(937\) 2.19504e10 0.871674 0.435837 0.900026i \(-0.356452\pi\)
0.435837 + 0.900026i \(0.356452\pi\)
\(938\) −1.02822e11 −4.06796
\(939\) 0 0
\(940\) 1.08415e10 0.425736
\(941\) 2.26436e10 0.885896 0.442948 0.896547i \(-0.353933\pi\)
0.442948 + 0.896547i \(0.353933\pi\)
\(942\) 0 0
\(943\) 2.82177e10 1.09580
\(944\) −1.74293e9 −0.0674337
\(945\) 0 0
\(946\) −1.73286e10 −0.665493
\(947\) −7.27658e9 −0.278421 −0.139211 0.990263i \(-0.544457\pi\)
−0.139211 + 0.990263i \(0.544457\pi\)
\(948\) 0 0
\(949\) 2.16152e10 0.820969
\(950\) 6.71388e10 2.54063
\(951\) 0 0
\(952\) −6.04738e10 −2.27163
\(953\) −4.70442e10 −1.76068 −0.880340 0.474343i \(-0.842686\pi\)
−0.880340 + 0.474343i \(0.842686\pi\)
\(954\) 0 0
\(955\) 1.06594e9 0.0396023
\(956\) −2.39161e10 −0.885293
\(957\) 0 0
\(958\) −5.91631e9 −0.217406
\(959\) 5.11439e10 1.87253
\(960\) 0 0
\(961\) −2.27215e10 −0.825859
\(962\) 5.24114e10 1.89807
\(963\) 0 0
\(964\) −2.37882e10 −0.855249
\(965\) 4.24437e9 0.152043
\(966\) 0 0
\(967\) −2.01441e8 −0.00716400 −0.00358200 0.999994i \(-0.501140\pi\)
−0.00358200 + 0.999994i \(0.501140\pi\)
\(968\) −3.40935e10 −1.20811
\(969\) 0 0
\(970\) −2.64111e9 −0.0929149
\(971\) −4.43321e10 −1.55400 −0.777001 0.629500i \(-0.783260\pi\)
−0.777001 + 0.629500i \(0.783260\pi\)
\(972\) 0 0
\(973\) −1.41508e10 −0.492478
\(974\) −5.21179e10 −1.80730
\(975\) 0 0
\(976\) 2.23701e10 0.770181
\(977\) −4.78091e10 −1.64013 −0.820067 0.572268i \(-0.806064\pi\)
−0.820067 + 0.572268i \(0.806064\pi\)
\(978\) 0 0
\(979\) −9.03443e9 −0.307724
\(980\) −4.56817e9 −0.155043
\(981\) 0 0
\(982\) 6.68045e9 0.225121
\(983\) 3.22996e10 1.08458 0.542288 0.840193i \(-0.317558\pi\)
0.542288 + 0.840193i \(0.317558\pi\)
\(984\) 0 0
\(985\) −7.48003e8 −0.0249389
\(986\) 8.08162e10 2.68491
\(987\) 0 0
\(988\) 7.75543e10 2.55833
\(989\) 2.98705e10 0.981873
\(990\) 0 0
\(991\) −5.62073e10 −1.83457 −0.917286 0.398230i \(-0.869625\pi\)
−0.917286 + 0.398230i \(0.869625\pi\)
\(992\) −6.72697e9 −0.218790
\(993\) 0 0
\(994\) 1.29805e11 4.19216
\(995\) −5.57901e9 −0.179546
\(996\) 0 0
\(997\) −6.21852e9 −0.198726 −0.0993629 0.995051i \(-0.531680\pi\)
−0.0993629 + 0.995051i \(0.531680\pi\)
\(998\) −7.79963e10 −2.48380
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.8.a.b.1.15 16
3.2 odd 2 177.8.a.a.1.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.2 16 3.2 odd 2
531.8.a.b.1.15 16 1.1 even 1 trivial