Properties

Label 531.8.a.b.1.16
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 1.16
Root \(19.7363\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.7363 q^{2} +261.522 q^{4} +90.5952 q^{5} -807.818 q^{7} +2635.23 q^{8} +O(q^{10})\) \(q+19.7363 q^{2} +261.522 q^{4} +90.5952 q^{5} -807.818 q^{7} +2635.23 q^{8} +1788.01 q^{10} -4971.97 q^{11} -2921.19 q^{13} -15943.4 q^{14} +18534.9 q^{16} +29331.0 q^{17} +8907.94 q^{19} +23692.6 q^{20} -98128.3 q^{22} +21320.0 q^{23} -69917.5 q^{25} -57653.6 q^{26} -211262. q^{28} -144361. q^{29} +124102. q^{31} +28500.9 q^{32} +578885. q^{34} -73184.5 q^{35} +249110. q^{37} +175810. q^{38} +238739. q^{40} -769960. q^{41} -214862. q^{43} -1.30028e6 q^{44} +420778. q^{46} -826532. q^{47} -170973. q^{49} -1.37991e6 q^{50} -763956. q^{52} -791662. q^{53} -450437. q^{55} -2.12879e6 q^{56} -2.84915e6 q^{58} -205379. q^{59} -2.23090e6 q^{61} +2.44931e6 q^{62} -1.80996e6 q^{64} -264646. q^{65} -60136.4 q^{67} +7.67069e6 q^{68} -1.44439e6 q^{70} -3.73452e6 q^{71} +476492. q^{73} +4.91652e6 q^{74} +2.32962e6 q^{76} +4.01645e6 q^{77} +1.82860e6 q^{79} +1.67917e6 q^{80} -1.51962e7 q^{82} -4.01281e6 q^{83} +2.65724e6 q^{85} -4.24058e6 q^{86} -1.31023e7 q^{88} -5.38781e6 q^{89} +2.35979e6 q^{91} +5.57565e6 q^{92} -1.63127e7 q^{94} +807017. q^{95} -1.52550e7 q^{97} -3.37437e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 19.7363 1.74446 0.872230 0.489096i \(-0.162673\pi\)
0.872230 + 0.489096i \(0.162673\pi\)
\(3\) 0 0
\(4\) 261.522 2.04314
\(5\) 90.5952 0.324123 0.162062 0.986781i \(-0.448186\pi\)
0.162062 + 0.986781i \(0.448186\pi\)
\(6\) 0 0
\(7\) −807.818 −0.890165 −0.445083 0.895490i \(-0.646826\pi\)
−0.445083 + 0.895490i \(0.646826\pi\)
\(8\) 2635.23 1.81971
\(9\) 0 0
\(10\) 1788.01 0.565420
\(11\) −4971.97 −1.12630 −0.563150 0.826355i \(-0.690411\pi\)
−0.563150 + 0.826355i \(0.690411\pi\)
\(12\) 0 0
\(13\) −2921.19 −0.368773 −0.184386 0.982854i \(-0.559030\pi\)
−0.184386 + 0.982854i \(0.559030\pi\)
\(14\) −15943.4 −1.55286
\(15\) 0 0
\(16\) 18534.9 1.13128
\(17\) 29331.0 1.44795 0.723977 0.689824i \(-0.242312\pi\)
0.723977 + 0.689824i \(0.242312\pi\)
\(18\) 0 0
\(19\) 8907.94 0.297948 0.148974 0.988841i \(-0.452403\pi\)
0.148974 + 0.988841i \(0.452403\pi\)
\(20\) 23692.6 0.662229
\(21\) 0 0
\(22\) −98128.3 −1.96479
\(23\) 21320.0 0.365376 0.182688 0.983171i \(-0.441520\pi\)
0.182688 + 0.983171i \(0.441520\pi\)
\(24\) 0 0
\(25\) −69917.5 −0.894944
\(26\) −57653.6 −0.643309
\(27\) 0 0
\(28\) −211262. −1.81873
\(29\) −144361. −1.09915 −0.549575 0.835444i \(-0.685210\pi\)
−0.549575 + 0.835444i \(0.685210\pi\)
\(30\) 0 0
\(31\) 124102. 0.748190 0.374095 0.927390i \(-0.377953\pi\)
0.374095 + 0.927390i \(0.377953\pi\)
\(32\) 28500.9 0.153757
\(33\) 0 0
\(34\) 578885. 2.52590
\(35\) −73184.5 −0.288523
\(36\) 0 0
\(37\) 249110. 0.808510 0.404255 0.914646i \(-0.367531\pi\)
0.404255 + 0.914646i \(0.367531\pi\)
\(38\) 175810. 0.519757
\(39\) 0 0
\(40\) 238739. 0.589812
\(41\) −769960. −1.74472 −0.872358 0.488867i \(-0.837410\pi\)
−0.872358 + 0.488867i \(0.837410\pi\)
\(42\) 0 0
\(43\) −214862. −0.412116 −0.206058 0.978540i \(-0.566064\pi\)
−0.206058 + 0.978540i \(0.566064\pi\)
\(44\) −1.30028e6 −2.30119
\(45\) 0 0
\(46\) 420778. 0.637383
\(47\) −826532. −1.16123 −0.580614 0.814179i \(-0.697187\pi\)
−0.580614 + 0.814179i \(0.697187\pi\)
\(48\) 0 0
\(49\) −170973. −0.207606
\(50\) −1.37991e6 −1.56119
\(51\) 0 0
\(52\) −763956. −0.753454
\(53\) −791662. −0.730423 −0.365211 0.930925i \(-0.619003\pi\)
−0.365211 + 0.930925i \(0.619003\pi\)
\(54\) 0 0
\(55\) −450437. −0.365060
\(56\) −2.12879e6 −1.61985
\(57\) 0 0
\(58\) −2.84915e6 −1.91742
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −2.23090e6 −1.25842 −0.629210 0.777236i \(-0.716621\pi\)
−0.629210 + 0.777236i \(0.716621\pi\)
\(62\) 2.44931e6 1.30519
\(63\) 0 0
\(64\) −1.80996e6 −0.863057
\(65\) −264646. −0.119528
\(66\) 0 0
\(67\) −60136.4 −0.0244273 −0.0122137 0.999925i \(-0.503888\pi\)
−0.0122137 + 0.999925i \(0.503888\pi\)
\(68\) 7.67069e6 2.95837
\(69\) 0 0
\(70\) −1.44439e6 −0.503317
\(71\) −3.73452e6 −1.23831 −0.619157 0.785268i \(-0.712525\pi\)
−0.619157 + 0.785268i \(0.712525\pi\)
\(72\) 0 0
\(73\) 476492. 0.143359 0.0716796 0.997428i \(-0.477164\pi\)
0.0716796 + 0.997428i \(0.477164\pi\)
\(74\) 4.91652e6 1.41041
\(75\) 0 0
\(76\) 2.32962e6 0.608748
\(77\) 4.01645e6 1.00259
\(78\) 0 0
\(79\) 1.82860e6 0.417277 0.208638 0.977993i \(-0.433097\pi\)
0.208638 + 0.977993i \(0.433097\pi\)
\(80\) 1.67917e6 0.366674
\(81\) 0 0
\(82\) −1.51962e7 −3.04359
\(83\) −4.01281e6 −0.770327 −0.385164 0.922848i \(-0.625855\pi\)
−0.385164 + 0.922848i \(0.625855\pi\)
\(84\) 0 0
\(85\) 2.65724e6 0.469316
\(86\) −4.24058e6 −0.718920
\(87\) 0 0
\(88\) −1.31023e7 −2.04955
\(89\) −5.38781e6 −0.810116 −0.405058 0.914291i \(-0.632749\pi\)
−0.405058 + 0.914291i \(0.632749\pi\)
\(90\) 0 0
\(91\) 2.35979e6 0.328269
\(92\) 5.57565e6 0.746514
\(93\) 0 0
\(94\) −1.63127e7 −2.02571
\(95\) 807017. 0.0965717
\(96\) 0 0
\(97\) −1.52550e7 −1.69711 −0.848556 0.529106i \(-0.822527\pi\)
−0.848556 + 0.529106i \(0.822527\pi\)
\(98\) −3.37437e6 −0.362160
\(99\) 0 0
\(100\) −1.82850e7 −1.82850
\(101\) 1.51903e7 1.46703 0.733517 0.679671i \(-0.237877\pi\)
0.733517 + 0.679671i \(0.237877\pi\)
\(102\) 0 0
\(103\) −7.35870e6 −0.663546 −0.331773 0.943359i \(-0.607647\pi\)
−0.331773 + 0.943359i \(0.607647\pi\)
\(104\) −7.69802e6 −0.671061
\(105\) 0 0
\(106\) −1.56245e7 −1.27419
\(107\) 3.56527e6 0.281352 0.140676 0.990056i \(-0.455072\pi\)
0.140676 + 0.990056i \(0.455072\pi\)
\(108\) 0 0
\(109\) 1.51208e7 1.11836 0.559179 0.829047i \(-0.311116\pi\)
0.559179 + 0.829047i \(0.311116\pi\)
\(110\) −8.88996e6 −0.636833
\(111\) 0 0
\(112\) −1.49728e7 −1.00703
\(113\) 2.38727e7 1.55642 0.778212 0.628002i \(-0.216127\pi\)
0.778212 + 0.628002i \(0.216127\pi\)
\(114\) 0 0
\(115\) 1.93149e6 0.118427
\(116\) −3.77536e7 −2.24572
\(117\) 0 0
\(118\) −4.05342e6 −0.227109
\(119\) −2.36941e7 −1.28892
\(120\) 0 0
\(121\) 5.23332e6 0.268552
\(122\) −4.40297e7 −2.19526
\(123\) 0 0
\(124\) 3.24553e7 1.52866
\(125\) −1.34119e7 −0.614195
\(126\) 0 0
\(127\) 6.26438e6 0.271372 0.135686 0.990752i \(-0.456676\pi\)
0.135686 + 0.990752i \(0.456676\pi\)
\(128\) −3.93701e7 −1.65933
\(129\) 0 0
\(130\) −5.22314e6 −0.208511
\(131\) 2.94087e7 1.14295 0.571474 0.820620i \(-0.306372\pi\)
0.571474 + 0.820620i \(0.306372\pi\)
\(132\) 0 0
\(133\) −7.19600e6 −0.265222
\(134\) −1.18687e6 −0.0426124
\(135\) 0 0
\(136\) 7.72938e7 2.63486
\(137\) 4.62523e6 0.153678 0.0768390 0.997044i \(-0.475517\pi\)
0.0768390 + 0.997044i \(0.475517\pi\)
\(138\) 0 0
\(139\) −1.29866e7 −0.410152 −0.205076 0.978746i \(-0.565744\pi\)
−0.205076 + 0.978746i \(0.565744\pi\)
\(140\) −1.91393e7 −0.589493
\(141\) 0 0
\(142\) −7.37056e7 −2.16019
\(143\) 1.45241e7 0.415349
\(144\) 0 0
\(145\) −1.30784e7 −0.356260
\(146\) 9.40419e6 0.250084
\(147\) 0 0
\(148\) 6.51478e7 1.65190
\(149\) 6.03191e7 1.49384 0.746918 0.664917i \(-0.231533\pi\)
0.746918 + 0.664917i \(0.231533\pi\)
\(150\) 0 0
\(151\) −9.08405e6 −0.214714 −0.107357 0.994221i \(-0.534239\pi\)
−0.107357 + 0.994221i \(0.534239\pi\)
\(152\) 2.34745e7 0.542180
\(153\) 0 0
\(154\) 7.92699e7 1.74898
\(155\) 1.12430e7 0.242506
\(156\) 0 0
\(157\) 5.23075e7 1.07874 0.539368 0.842070i \(-0.318663\pi\)
0.539368 + 0.842070i \(0.318663\pi\)
\(158\) 3.60898e7 0.727923
\(159\) 0 0
\(160\) 2.58205e6 0.0498361
\(161\) −1.72227e7 −0.325245
\(162\) 0 0
\(163\) 1.85380e7 0.335280 0.167640 0.985848i \(-0.446385\pi\)
0.167640 + 0.985848i \(0.446385\pi\)
\(164\) −2.01361e8 −3.56470
\(165\) 0 0
\(166\) −7.91980e7 −1.34380
\(167\) −1.04169e7 −0.173074 −0.0865369 0.996249i \(-0.527580\pi\)
−0.0865369 + 0.996249i \(0.527580\pi\)
\(168\) 0 0
\(169\) −5.42151e7 −0.864007
\(170\) 5.24442e7 0.818703
\(171\) 0 0
\(172\) −5.61911e7 −0.842011
\(173\) −7.37218e7 −1.08252 −0.541259 0.840856i \(-0.682052\pi\)
−0.541259 + 0.840856i \(0.682052\pi\)
\(174\) 0 0
\(175\) 5.64806e7 0.796648
\(176\) −9.21549e7 −1.27416
\(177\) 0 0
\(178\) −1.06335e8 −1.41321
\(179\) 9.01187e7 1.17444 0.587218 0.809429i \(-0.300223\pi\)
0.587218 + 0.809429i \(0.300223\pi\)
\(180\) 0 0
\(181\) 5.95539e7 0.746509 0.373255 0.927729i \(-0.378242\pi\)
0.373255 + 0.927729i \(0.378242\pi\)
\(182\) 4.65736e7 0.572651
\(183\) 0 0
\(184\) 5.61831e7 0.664880
\(185\) 2.25682e7 0.262057
\(186\) 0 0
\(187\) −1.45833e8 −1.63083
\(188\) −2.16156e8 −2.37255
\(189\) 0 0
\(190\) 1.59275e7 0.168465
\(191\) −8.79003e7 −0.912795 −0.456398 0.889776i \(-0.650861\pi\)
−0.456398 + 0.889776i \(0.650861\pi\)
\(192\) 0 0
\(193\) 9.19947e7 0.921112 0.460556 0.887631i \(-0.347650\pi\)
0.460556 + 0.887631i \(0.347650\pi\)
\(194\) −3.01077e8 −2.96054
\(195\) 0 0
\(196\) −4.47131e7 −0.424168
\(197\) 9.51132e7 0.886357 0.443179 0.896433i \(-0.353851\pi\)
0.443179 + 0.896433i \(0.353851\pi\)
\(198\) 0 0
\(199\) −9.84642e7 −0.885712 −0.442856 0.896593i \(-0.646035\pi\)
−0.442856 + 0.896593i \(0.646035\pi\)
\(200\) −1.84249e8 −1.62854
\(201\) 0 0
\(202\) 2.99800e8 2.55918
\(203\) 1.16618e8 0.978425
\(204\) 0 0
\(205\) −6.97547e7 −0.565503
\(206\) −1.45234e8 −1.15753
\(207\) 0 0
\(208\) −5.41440e7 −0.417185
\(209\) −4.42900e7 −0.335578
\(210\) 0 0
\(211\) −1.85405e8 −1.35873 −0.679364 0.733802i \(-0.737744\pi\)
−0.679364 + 0.733802i \(0.737744\pi\)
\(212\) −2.07037e8 −1.49236
\(213\) 0 0
\(214\) 7.03654e7 0.490807
\(215\) −1.94655e7 −0.133576
\(216\) 0 0
\(217\) −1.00252e8 −0.666013
\(218\) 2.98428e8 1.95093
\(219\) 0 0
\(220\) −1.17799e8 −0.745869
\(221\) −8.56814e7 −0.533966
\(222\) 0 0
\(223\) −5.33953e7 −0.322431 −0.161215 0.986919i \(-0.551541\pi\)
−0.161215 + 0.986919i \(0.551541\pi\)
\(224\) −2.30236e7 −0.136869
\(225\) 0 0
\(226\) 4.71160e8 2.71512
\(227\) 7.79437e7 0.442273 0.221137 0.975243i \(-0.429023\pi\)
0.221137 + 0.975243i \(0.429023\pi\)
\(228\) 0 0
\(229\) −2.50546e7 −0.137868 −0.0689340 0.997621i \(-0.521960\pi\)
−0.0689340 + 0.997621i \(0.521960\pi\)
\(230\) 3.81205e7 0.206591
\(231\) 0 0
\(232\) −3.80424e8 −2.00014
\(233\) −2.23037e8 −1.15513 −0.577567 0.816344i \(-0.695998\pi\)
−0.577567 + 0.816344i \(0.695998\pi\)
\(234\) 0 0
\(235\) −7.48799e7 −0.376381
\(236\) −5.37111e7 −0.265994
\(237\) 0 0
\(238\) −4.67634e8 −2.24847
\(239\) 1.46524e8 0.694249 0.347124 0.937819i \(-0.387158\pi\)
0.347124 + 0.937819i \(0.387158\pi\)
\(240\) 0 0
\(241\) −5.35245e7 −0.246316 −0.123158 0.992387i \(-0.539302\pi\)
−0.123158 + 0.992387i \(0.539302\pi\)
\(242\) 1.03286e8 0.468478
\(243\) 0 0
\(244\) −5.83429e8 −2.57113
\(245\) −1.54893e7 −0.0672900
\(246\) 0 0
\(247\) −2.60218e7 −0.109875
\(248\) 3.27037e8 1.36149
\(249\) 0 0
\(250\) −2.64702e8 −1.07144
\(251\) 3.32032e8 1.32532 0.662661 0.748919i \(-0.269427\pi\)
0.662661 + 0.748919i \(0.269427\pi\)
\(252\) 0 0
\(253\) −1.06002e8 −0.411523
\(254\) 1.23636e8 0.473398
\(255\) 0 0
\(256\) −5.45345e8 −2.03157
\(257\) −3.09916e8 −1.13888 −0.569441 0.822032i \(-0.692840\pi\)
−0.569441 + 0.822032i \(0.692840\pi\)
\(258\) 0 0
\(259\) −2.01236e8 −0.719708
\(260\) −6.92108e7 −0.244212
\(261\) 0 0
\(262\) 5.80419e8 1.99383
\(263\) 8.42828e7 0.285689 0.142845 0.989745i \(-0.454375\pi\)
0.142845 + 0.989745i \(0.454375\pi\)
\(264\) 0 0
\(265\) −7.17208e7 −0.236747
\(266\) −1.42022e8 −0.462670
\(267\) 0 0
\(268\) −1.57270e7 −0.0499084
\(269\) 1.91871e8 0.601001 0.300501 0.953782i \(-0.402846\pi\)
0.300501 + 0.953782i \(0.402846\pi\)
\(270\) 0 0
\(271\) −7.09645e6 −0.0216595 −0.0108298 0.999941i \(-0.503447\pi\)
−0.0108298 + 0.999941i \(0.503447\pi\)
\(272\) 5.43646e8 1.63804
\(273\) 0 0
\(274\) 9.12851e7 0.268085
\(275\) 3.47628e8 1.00798
\(276\) 0 0
\(277\) −2.59631e8 −0.733968 −0.366984 0.930227i \(-0.619610\pi\)
−0.366984 + 0.930227i \(0.619610\pi\)
\(278\) −2.56308e8 −0.715493
\(279\) 0 0
\(280\) −1.92858e8 −0.525030
\(281\) 4.92572e7 0.132433 0.0662167 0.997805i \(-0.478907\pi\)
0.0662167 + 0.997805i \(0.478907\pi\)
\(282\) 0 0
\(283\) −1.62830e8 −0.427053 −0.213526 0.976937i \(-0.568495\pi\)
−0.213526 + 0.976937i \(0.568495\pi\)
\(284\) −9.76658e8 −2.53005
\(285\) 0 0
\(286\) 2.86652e8 0.724559
\(287\) 6.21988e8 1.55309
\(288\) 0 0
\(289\) 4.49966e8 1.09657
\(290\) −2.58120e8 −0.621482
\(291\) 0 0
\(292\) 1.24613e8 0.292903
\(293\) 5.14011e8 1.19381 0.596905 0.802312i \(-0.296397\pi\)
0.596905 + 0.802312i \(0.296397\pi\)
\(294\) 0 0
\(295\) −1.86064e7 −0.0421973
\(296\) 6.56462e8 1.47126
\(297\) 0 0
\(298\) 1.19048e9 2.60594
\(299\) −6.22799e7 −0.134741
\(300\) 0 0
\(301\) 1.73569e8 0.366852
\(302\) −1.79286e8 −0.374560
\(303\) 0 0
\(304\) 1.65108e8 0.337062
\(305\) −2.02109e8 −0.407883
\(306\) 0 0
\(307\) 1.27895e8 0.252272 0.126136 0.992013i \(-0.459742\pi\)
0.126136 + 0.992013i \(0.459742\pi\)
\(308\) 1.05039e9 2.04844
\(309\) 0 0
\(310\) 2.21896e8 0.423042
\(311\) −6.02895e8 −1.13653 −0.568264 0.822846i \(-0.692385\pi\)
−0.568264 + 0.822846i \(0.692385\pi\)
\(312\) 0 0
\(313\) −8.09308e8 −1.49179 −0.745897 0.666061i \(-0.767979\pi\)
−0.745897 + 0.666061i \(0.767979\pi\)
\(314\) 1.03236e9 1.88181
\(315\) 0 0
\(316\) 4.78219e8 0.852555
\(317\) −9.39100e7 −0.165579 −0.0827893 0.996567i \(-0.526383\pi\)
−0.0827893 + 0.996567i \(0.526383\pi\)
\(318\) 0 0
\(319\) 7.17759e8 1.23797
\(320\) −1.63974e8 −0.279737
\(321\) 0 0
\(322\) −3.39912e8 −0.567376
\(323\) 2.61278e8 0.431415
\(324\) 0 0
\(325\) 2.04243e8 0.330031
\(326\) 3.65873e8 0.584882
\(327\) 0 0
\(328\) −2.02902e9 −3.17489
\(329\) 6.67688e8 1.03368
\(330\) 0 0
\(331\) 9.98609e8 1.51355 0.756776 0.653674i \(-0.226773\pi\)
0.756776 + 0.653674i \(0.226773\pi\)
\(332\) −1.04944e9 −1.57389
\(333\) 0 0
\(334\) −2.05591e8 −0.301920
\(335\) −5.44807e6 −0.00791746
\(336\) 0 0
\(337\) 1.07195e9 1.52570 0.762851 0.646575i \(-0.223799\pi\)
0.762851 + 0.646575i \(0.223799\pi\)
\(338\) −1.07001e9 −1.50722
\(339\) 0 0
\(340\) 6.94927e8 0.958878
\(341\) −6.17030e8 −0.842687
\(342\) 0 0
\(343\) 8.03388e8 1.07497
\(344\) −5.66210e8 −0.749934
\(345\) 0 0
\(346\) −1.45500e9 −1.88841
\(347\) 1.06764e9 1.37174 0.685869 0.727725i \(-0.259422\pi\)
0.685869 + 0.727725i \(0.259422\pi\)
\(348\) 0 0
\(349\) −1.09465e9 −1.37843 −0.689216 0.724556i \(-0.742045\pi\)
−0.689216 + 0.724556i \(0.742045\pi\)
\(350\) 1.11472e9 1.38972
\(351\) 0 0
\(352\) −1.41706e8 −0.173176
\(353\) 2.65164e8 0.320851 0.160426 0.987048i \(-0.448713\pi\)
0.160426 + 0.987048i \(0.448713\pi\)
\(354\) 0 0
\(355\) −3.38330e8 −0.401366
\(356\) −1.40903e9 −1.65518
\(357\) 0 0
\(358\) 1.77861e9 2.04876
\(359\) 7.74082e8 0.882992 0.441496 0.897263i \(-0.354448\pi\)
0.441496 + 0.897263i \(0.354448\pi\)
\(360\) 0 0
\(361\) −8.14520e8 −0.911227
\(362\) 1.17537e9 1.30226
\(363\) 0 0
\(364\) 6.17138e8 0.670699
\(365\) 4.31679e7 0.0464660
\(366\) 0 0
\(367\) 6.18045e8 0.652663 0.326331 0.945255i \(-0.394187\pi\)
0.326331 + 0.945255i \(0.394187\pi\)
\(368\) 3.95164e8 0.413342
\(369\) 0 0
\(370\) 4.45413e8 0.457148
\(371\) 6.39519e8 0.650197
\(372\) 0 0
\(373\) −2.27726e7 −0.0227212 −0.0113606 0.999935i \(-0.503616\pi\)
−0.0113606 + 0.999935i \(0.503616\pi\)
\(374\) −2.87820e9 −2.84492
\(375\) 0 0
\(376\) −2.17810e9 −2.11310
\(377\) 4.21707e8 0.405337
\(378\) 0 0
\(379\) −9.75855e8 −0.920763 −0.460382 0.887721i \(-0.652287\pi\)
−0.460382 + 0.887721i \(0.652287\pi\)
\(380\) 2.11053e8 0.197310
\(381\) 0 0
\(382\) −1.73483e9 −1.59233
\(383\) 8.32929e8 0.757552 0.378776 0.925488i \(-0.376345\pi\)
0.378776 + 0.925488i \(0.376345\pi\)
\(384\) 0 0
\(385\) 3.63871e8 0.324964
\(386\) 1.81564e9 1.60684
\(387\) 0 0
\(388\) −3.98951e9 −3.46744
\(389\) 1.77468e8 0.152861 0.0764304 0.997075i \(-0.475648\pi\)
0.0764304 + 0.997075i \(0.475648\pi\)
\(390\) 0 0
\(391\) 6.25336e8 0.529048
\(392\) −4.50552e8 −0.377784
\(393\) 0 0
\(394\) 1.87718e9 1.54621
\(395\) 1.65663e8 0.135249
\(396\) 0 0
\(397\) −1.50894e9 −1.21033 −0.605166 0.796099i \(-0.706893\pi\)
−0.605166 + 0.796099i \(0.706893\pi\)
\(398\) −1.94332e9 −1.54509
\(399\) 0 0
\(400\) −1.29591e9 −1.01243
\(401\) −1.77996e9 −1.37850 −0.689248 0.724525i \(-0.742059\pi\)
−0.689248 + 0.724525i \(0.742059\pi\)
\(402\) 0 0
\(403\) −3.62525e8 −0.275912
\(404\) 3.97258e9 2.99736
\(405\) 0 0
\(406\) 2.30160e9 1.70682
\(407\) −1.23857e9 −0.910625
\(408\) 0 0
\(409\) 1.01746e9 0.735335 0.367667 0.929957i \(-0.380157\pi\)
0.367667 + 0.929957i \(0.380157\pi\)
\(410\) −1.37670e9 −0.986497
\(411\) 0 0
\(412\) −1.92446e9 −1.35572
\(413\) 1.65909e8 0.115890
\(414\) 0 0
\(415\) −3.63541e8 −0.249681
\(416\) −8.32568e7 −0.0567013
\(417\) 0 0
\(418\) −8.74122e8 −0.585403
\(419\) 1.28968e9 0.856514 0.428257 0.903657i \(-0.359128\pi\)
0.428257 + 0.903657i \(0.359128\pi\)
\(420\) 0 0
\(421\) 5.35636e8 0.349851 0.174925 0.984582i \(-0.444032\pi\)
0.174925 + 0.984582i \(0.444032\pi\)
\(422\) −3.65921e9 −2.37025
\(423\) 0 0
\(424\) −2.08621e9 −1.32916
\(425\) −2.05075e9 −1.29584
\(426\) 0 0
\(427\) 1.80216e9 1.12020
\(428\) 9.32397e8 0.574841
\(429\) 0 0
\(430\) −3.84176e8 −0.233019
\(431\) 2.75089e9 1.65502 0.827509 0.561452i \(-0.189757\pi\)
0.827509 + 0.561452i \(0.189757\pi\)
\(432\) 0 0
\(433\) 8.00172e8 0.473670 0.236835 0.971550i \(-0.423890\pi\)
0.236835 + 0.971550i \(0.423890\pi\)
\(434\) −1.97860e9 −1.16183
\(435\) 0 0
\(436\) 3.95441e9 2.28496
\(437\) 1.89917e8 0.108863
\(438\) 0 0
\(439\) 7.72445e8 0.435754 0.217877 0.975976i \(-0.430087\pi\)
0.217877 + 0.975976i \(0.430087\pi\)
\(440\) −1.18700e9 −0.664305
\(441\) 0 0
\(442\) −1.69104e9 −0.931483
\(443\) −2.85561e9 −1.56058 −0.780289 0.625419i \(-0.784928\pi\)
−0.780289 + 0.625419i \(0.784928\pi\)
\(444\) 0 0
\(445\) −4.88110e8 −0.262577
\(446\) −1.05383e9 −0.562467
\(447\) 0 0
\(448\) 1.46212e9 0.768263
\(449\) −1.99990e9 −1.04267 −0.521334 0.853352i \(-0.674566\pi\)
−0.521334 + 0.853352i \(0.674566\pi\)
\(450\) 0 0
\(451\) 3.82822e9 1.96507
\(452\) 6.24324e9 3.17999
\(453\) 0 0
\(454\) 1.53832e9 0.771528
\(455\) 2.13786e8 0.106399
\(456\) 0 0
\(457\) −1.08512e9 −0.531829 −0.265915 0.963997i \(-0.585674\pi\)
−0.265915 + 0.963997i \(0.585674\pi\)
\(458\) −4.94485e8 −0.240505
\(459\) 0 0
\(460\) 5.05127e8 0.241962
\(461\) 3.43732e9 1.63405 0.817027 0.576600i \(-0.195621\pi\)
0.817027 + 0.576600i \(0.195621\pi\)
\(462\) 0 0
\(463\) −2.83083e9 −1.32550 −0.662750 0.748840i \(-0.730611\pi\)
−0.662750 + 0.748840i \(0.730611\pi\)
\(464\) −2.67572e9 −1.24345
\(465\) 0 0
\(466\) −4.40194e9 −2.01508
\(467\) 1.27289e9 0.578337 0.289169 0.957278i \(-0.406621\pi\)
0.289169 + 0.957278i \(0.406621\pi\)
\(468\) 0 0
\(469\) 4.85793e7 0.0217443
\(470\) −1.47785e9 −0.656581
\(471\) 0 0
\(472\) −5.41221e8 −0.236907
\(473\) 1.06829e9 0.464167
\(474\) 0 0
\(475\) −6.22821e8 −0.266646
\(476\) −6.19652e9 −2.63344
\(477\) 0 0
\(478\) 2.89184e9 1.21109
\(479\) 1.83875e9 0.764448 0.382224 0.924070i \(-0.375158\pi\)
0.382224 + 0.924070i \(0.375158\pi\)
\(480\) 0 0
\(481\) −7.27699e8 −0.298156
\(482\) −1.05638e9 −0.429689
\(483\) 0 0
\(484\) 1.36863e9 0.548689
\(485\) −1.38203e9 −0.550073
\(486\) 0 0
\(487\) −1.80837e9 −0.709473 −0.354736 0.934966i \(-0.615429\pi\)
−0.354736 + 0.934966i \(0.615429\pi\)
\(488\) −5.87893e9 −2.28996
\(489\) 0 0
\(490\) −3.05701e8 −0.117385
\(491\) −3.14661e9 −1.19966 −0.599829 0.800128i \(-0.704765\pi\)
−0.599829 + 0.800128i \(0.704765\pi\)
\(492\) 0 0
\(493\) −4.23425e9 −1.59152
\(494\) −5.13575e8 −0.191672
\(495\) 0 0
\(496\) 2.30021e9 0.846412
\(497\) 3.01681e9 1.10230
\(498\) 0 0
\(499\) 2.26779e9 0.817056 0.408528 0.912746i \(-0.366042\pi\)
0.408528 + 0.912746i \(0.366042\pi\)
\(500\) −3.50752e9 −1.25489
\(501\) 0 0
\(502\) 6.55308e9 2.31197
\(503\) −1.65622e9 −0.580270 −0.290135 0.956986i \(-0.593700\pi\)
−0.290135 + 0.956986i \(0.593700\pi\)
\(504\) 0 0
\(505\) 1.37616e9 0.475500
\(506\) −2.09210e9 −0.717885
\(507\) 0 0
\(508\) 1.63827e9 0.554451
\(509\) −5.27031e9 −1.77143 −0.885715 0.464229i \(-0.846332\pi\)
−0.885715 + 0.464229i \(0.846332\pi\)
\(510\) 0 0
\(511\) −3.84919e8 −0.127613
\(512\) −5.72373e9 −1.88466
\(513\) 0 0
\(514\) −6.11660e9 −1.98673
\(515\) −6.66663e8 −0.215071
\(516\) 0 0
\(517\) 4.10949e9 1.30789
\(518\) −3.97165e9 −1.25550
\(519\) 0 0
\(520\) −6.97403e8 −0.217507
\(521\) 1.06833e9 0.330957 0.165478 0.986213i \(-0.447083\pi\)
0.165478 + 0.986213i \(0.447083\pi\)
\(522\) 0 0
\(523\) −4.88919e9 −1.49445 −0.747225 0.664572i \(-0.768614\pi\)
−0.747225 + 0.664572i \(0.768614\pi\)
\(524\) 7.69102e9 2.33520
\(525\) 0 0
\(526\) 1.66343e9 0.498374
\(527\) 3.64002e9 1.08335
\(528\) 0 0
\(529\) −2.95028e9 −0.866500
\(530\) −1.41550e9 −0.412995
\(531\) 0 0
\(532\) −1.88191e9 −0.541887
\(533\) 2.24920e9 0.643404
\(534\) 0 0
\(535\) 3.22997e8 0.0911927
\(536\) −1.58473e8 −0.0444507
\(537\) 0 0
\(538\) 3.78682e9 1.04842
\(539\) 8.50070e8 0.233827
\(540\) 0 0
\(541\) 2.76671e9 0.751231 0.375615 0.926776i \(-0.377431\pi\)
0.375615 + 0.926776i \(0.377431\pi\)
\(542\) −1.40058e8 −0.0377842
\(543\) 0 0
\(544\) 8.35960e8 0.222633
\(545\) 1.36987e9 0.362486
\(546\) 0 0
\(547\) −5.33645e9 −1.39411 −0.697054 0.717018i \(-0.745506\pi\)
−0.697054 + 0.717018i \(0.745506\pi\)
\(548\) 1.20960e9 0.313986
\(549\) 0 0
\(550\) 6.86089e9 1.75837
\(551\) −1.28596e9 −0.327489
\(552\) 0 0
\(553\) −1.47718e9 −0.371445
\(554\) −5.12416e9 −1.28038
\(555\) 0 0
\(556\) −3.39629e9 −0.837998
\(557\) 4.61945e8 0.113265 0.0566326 0.998395i \(-0.481964\pi\)
0.0566326 + 0.998395i \(0.481964\pi\)
\(558\) 0 0
\(559\) 6.27653e8 0.151977
\(560\) −1.35647e9 −0.326400
\(561\) 0 0
\(562\) 9.72156e8 0.231025
\(563\) −5.71818e9 −1.35045 −0.675224 0.737612i \(-0.735953\pi\)
−0.675224 + 0.737612i \(0.735953\pi\)
\(564\) 0 0
\(565\) 2.16276e9 0.504473
\(566\) −3.21366e9 −0.744976
\(567\) 0 0
\(568\) −9.84131e9 −2.25338
\(569\) 5.60950e9 1.27653 0.638265 0.769816i \(-0.279652\pi\)
0.638265 + 0.769816i \(0.279652\pi\)
\(570\) 0 0
\(571\) 3.21411e9 0.722494 0.361247 0.932470i \(-0.382351\pi\)
0.361247 + 0.932470i \(0.382351\pi\)
\(572\) 3.79837e9 0.848615
\(573\) 0 0
\(574\) 1.22757e10 2.70930
\(575\) −1.49064e9 −0.326991
\(576\) 0 0
\(577\) 2.42295e9 0.525085 0.262543 0.964920i \(-0.415439\pi\)
0.262543 + 0.964920i \(0.415439\pi\)
\(578\) 8.88068e9 1.91293
\(579\) 0 0
\(580\) −3.42029e9 −0.727889
\(581\) 3.24162e9 0.685718
\(582\) 0 0
\(583\) 3.93612e9 0.822675
\(584\) 1.25567e9 0.260873
\(585\) 0 0
\(586\) 1.01447e10 2.08255
\(587\) 8.83936e9 1.80380 0.901898 0.431949i \(-0.142174\pi\)
0.901898 + 0.431949i \(0.142174\pi\)
\(588\) 0 0
\(589\) 1.10549e9 0.222921
\(590\) −3.67221e8 −0.0736114
\(591\) 0 0
\(592\) 4.61723e9 0.914651
\(593\) 2.57682e9 0.507450 0.253725 0.967276i \(-0.418344\pi\)
0.253725 + 0.967276i \(0.418344\pi\)
\(594\) 0 0
\(595\) −2.14657e9 −0.417769
\(596\) 1.57748e10 3.05211
\(597\) 0 0
\(598\) −1.22917e9 −0.235050
\(599\) 6.40768e9 1.21817 0.609084 0.793106i \(-0.291537\pi\)
0.609084 + 0.793106i \(0.291537\pi\)
\(600\) 0 0
\(601\) 2.70821e9 0.508887 0.254443 0.967088i \(-0.418108\pi\)
0.254443 + 0.967088i \(0.418108\pi\)
\(602\) 3.42562e9 0.639958
\(603\) 0 0
\(604\) −2.37568e9 −0.438691
\(605\) 4.74114e8 0.0870439
\(606\) 0 0
\(607\) −8.53113e9 −1.54827 −0.774134 0.633022i \(-0.781814\pi\)
−0.774134 + 0.633022i \(0.781814\pi\)
\(608\) 2.53885e8 0.0458114
\(609\) 0 0
\(610\) −3.98888e9 −0.711535
\(611\) 2.41446e9 0.428229
\(612\) 0 0
\(613\) −2.93110e9 −0.513947 −0.256973 0.966418i \(-0.582725\pi\)
−0.256973 + 0.966418i \(0.582725\pi\)
\(614\) 2.52418e9 0.440079
\(615\) 0 0
\(616\) 1.05843e10 1.82443
\(617\) −7.69933e9 −1.31964 −0.659819 0.751425i \(-0.729367\pi\)
−0.659819 + 0.751425i \(0.729367\pi\)
\(618\) 0 0
\(619\) 1.43713e9 0.243545 0.121772 0.992558i \(-0.461142\pi\)
0.121772 + 0.992558i \(0.461142\pi\)
\(620\) 2.94030e9 0.495473
\(621\) 0 0
\(622\) −1.18989e10 −1.98263
\(623\) 4.35237e9 0.721137
\(624\) 0 0
\(625\) 4.24725e9 0.695869
\(626\) −1.59728e10 −2.60237
\(627\) 0 0
\(628\) 1.36796e10 2.20401
\(629\) 7.30664e9 1.17069
\(630\) 0 0
\(631\) −7.31824e9 −1.15959 −0.579794 0.814763i \(-0.696867\pi\)
−0.579794 + 0.814763i \(0.696867\pi\)
\(632\) 4.81878e9 0.759325
\(633\) 0 0
\(634\) −1.85344e9 −0.288845
\(635\) 5.67523e8 0.0879580
\(636\) 0 0
\(637\) 4.99444e8 0.0765595
\(638\) 1.41659e10 2.15959
\(639\) 0 0
\(640\) −3.56674e9 −0.537826
\(641\) −1.21948e10 −1.82883 −0.914414 0.404779i \(-0.867348\pi\)
−0.914414 + 0.404779i \(0.867348\pi\)
\(642\) 0 0
\(643\) 1.07659e10 1.59702 0.798510 0.601982i \(-0.205622\pi\)
0.798510 + 0.601982i \(0.205622\pi\)
\(644\) −4.50411e9 −0.664521
\(645\) 0 0
\(646\) 5.15667e9 0.752585
\(647\) 1.11488e10 1.61831 0.809156 0.587594i \(-0.199925\pi\)
0.809156 + 0.587594i \(0.199925\pi\)
\(648\) 0 0
\(649\) 1.02114e9 0.146632
\(650\) 4.03100e9 0.575726
\(651\) 0 0
\(652\) 4.84810e9 0.685023
\(653\) −1.17825e10 −1.65593 −0.827965 0.560780i \(-0.810502\pi\)
−0.827965 + 0.560780i \(0.810502\pi\)
\(654\) 0 0
\(655\) 2.66429e9 0.370456
\(656\) −1.42711e10 −1.97376
\(657\) 0 0
\(658\) 1.31777e10 1.80322
\(659\) 1.18824e10 1.61736 0.808680 0.588249i \(-0.200182\pi\)
0.808680 + 0.588249i \(0.200182\pi\)
\(660\) 0 0
\(661\) −3.51347e9 −0.473185 −0.236592 0.971609i \(-0.576031\pi\)
−0.236592 + 0.971609i \(0.576031\pi\)
\(662\) 1.97089e10 2.64033
\(663\) 0 0
\(664\) −1.05747e10 −1.40178
\(665\) −6.51923e8 −0.0859648
\(666\) 0 0
\(667\) −3.07778e9 −0.401603
\(668\) −2.72425e9 −0.353614
\(669\) 0 0
\(670\) −1.07525e8 −0.0138117
\(671\) 1.10920e10 1.41736
\(672\) 0 0
\(673\) −7.05934e8 −0.0892712 −0.0446356 0.999003i \(-0.514213\pi\)
−0.0446356 + 0.999003i \(0.514213\pi\)
\(674\) 2.11563e10 2.66152
\(675\) 0 0
\(676\) −1.41784e10 −1.76529
\(677\) 1.39303e10 1.72544 0.862720 0.505681i \(-0.168759\pi\)
0.862720 + 0.505681i \(0.168759\pi\)
\(678\) 0 0
\(679\) 1.23232e10 1.51071
\(680\) 7.00245e9 0.854021
\(681\) 0 0
\(682\) −1.21779e10 −1.47003
\(683\) 2.00962e9 0.241347 0.120673 0.992692i \(-0.461495\pi\)
0.120673 + 0.992692i \(0.461495\pi\)
\(684\) 0 0
\(685\) 4.19024e8 0.0498106
\(686\) 1.58559e10 1.87524
\(687\) 0 0
\(688\) −3.98244e9 −0.466219
\(689\) 2.31260e9 0.269360
\(690\) 0 0
\(691\) −1.48060e9 −0.170712 −0.0853558 0.996351i \(-0.527203\pi\)
−0.0853558 + 0.996351i \(0.527203\pi\)
\(692\) −1.92799e10 −2.21173
\(693\) 0 0
\(694\) 2.10712e10 2.39294
\(695\) −1.17653e9 −0.132940
\(696\) 0 0
\(697\) −2.25837e10 −2.52627
\(698\) −2.16043e10 −2.40462
\(699\) 0 0
\(700\) 1.47709e10 1.62766
\(701\) 9.38403e9 1.02891 0.514454 0.857518i \(-0.327995\pi\)
0.514454 + 0.857518i \(0.327995\pi\)
\(702\) 0 0
\(703\) 2.21906e9 0.240894
\(704\) 8.99908e9 0.972061
\(705\) 0 0
\(706\) 5.23337e9 0.559712
\(707\) −1.22710e10 −1.30590
\(708\) 0 0
\(709\) 5.90748e9 0.622501 0.311251 0.950328i \(-0.399252\pi\)
0.311251 + 0.950328i \(0.399252\pi\)
\(710\) −6.67738e9 −0.700167
\(711\) 0 0
\(712\) −1.41981e10 −1.47418
\(713\) 2.64585e9 0.273371
\(714\) 0 0
\(715\) 1.31581e9 0.134624
\(716\) 2.35680e10 2.39954
\(717\) 0 0
\(718\) 1.52775e10 1.54034
\(719\) 1.54549e10 1.55065 0.775327 0.631560i \(-0.217585\pi\)
0.775327 + 0.631560i \(0.217585\pi\)
\(720\) 0 0
\(721\) 5.94450e9 0.590665
\(722\) −1.60756e10 −1.58960
\(723\) 0 0
\(724\) 1.55746e10 1.52522
\(725\) 1.00934e10 0.983678
\(726\) 0 0
\(727\) −1.56689e8 −0.0151240 −0.00756201 0.999971i \(-0.502407\pi\)
−0.00756201 + 0.999971i \(0.502407\pi\)
\(728\) 6.21860e9 0.597355
\(729\) 0 0
\(730\) 8.51975e8 0.0810581
\(731\) −6.30211e9 −0.596726
\(732\) 0 0
\(733\) 7.57137e9 0.710086 0.355043 0.934850i \(-0.384466\pi\)
0.355043 + 0.934850i \(0.384466\pi\)
\(734\) 1.21979e10 1.13854
\(735\) 0 0
\(736\) 6.07640e8 0.0561790
\(737\) 2.98996e8 0.0275125
\(738\) 0 0
\(739\) −7.72084e9 −0.703734 −0.351867 0.936050i \(-0.614453\pi\)
−0.351867 + 0.936050i \(0.614453\pi\)
\(740\) 5.90208e9 0.535419
\(741\) 0 0
\(742\) 1.26217e10 1.13424
\(743\) −5.80399e9 −0.519118 −0.259559 0.965727i \(-0.583577\pi\)
−0.259559 + 0.965727i \(0.583577\pi\)
\(744\) 0 0
\(745\) 5.46462e9 0.484187
\(746\) −4.49446e8 −0.0396362
\(747\) 0 0
\(748\) −3.81384e10 −3.33202
\(749\) −2.88009e9 −0.250450
\(750\) 0 0
\(751\) 1.00286e10 0.863975 0.431987 0.901880i \(-0.357813\pi\)
0.431987 + 0.901880i \(0.357813\pi\)
\(752\) −1.53197e10 −1.31367
\(753\) 0 0
\(754\) 8.32294e9 0.707093
\(755\) −8.22971e8 −0.0695938
\(756\) 0 0
\(757\) −1.32049e10 −1.10637 −0.553184 0.833059i \(-0.686587\pi\)
−0.553184 + 0.833059i \(0.686587\pi\)
\(758\) −1.92598e10 −1.60623
\(759\) 0 0
\(760\) 2.12667e9 0.175733
\(761\) −2.09392e10 −1.72232 −0.861161 0.508332i \(-0.830262\pi\)
−0.861161 + 0.508332i \(0.830262\pi\)
\(762\) 0 0
\(763\) −1.22148e10 −0.995523
\(764\) −2.29878e10 −1.86497
\(765\) 0 0
\(766\) 1.64390e10 1.32152
\(767\) 5.99952e8 0.0480101
\(768\) 0 0
\(769\) 1.74920e10 1.38706 0.693532 0.720426i \(-0.256054\pi\)
0.693532 + 0.720426i \(0.256054\pi\)
\(770\) 7.18147e9 0.566886
\(771\) 0 0
\(772\) 2.40586e10 1.88196
\(773\) −1.87442e10 −1.45962 −0.729808 0.683652i \(-0.760390\pi\)
−0.729808 + 0.683652i \(0.760390\pi\)
\(774\) 0 0
\(775\) −8.67689e9 −0.669589
\(776\) −4.02003e10 −3.08826
\(777\) 0 0
\(778\) 3.50256e9 0.266659
\(779\) −6.85876e9 −0.519834
\(780\) 0 0
\(781\) 1.85679e10 1.39471
\(782\) 1.23418e10 0.922902
\(783\) 0 0
\(784\) −3.16896e9 −0.234861
\(785\) 4.73881e9 0.349644
\(786\) 0 0
\(787\) −1.93622e10 −1.41594 −0.707968 0.706245i \(-0.750388\pi\)
−0.707968 + 0.706245i \(0.750388\pi\)
\(788\) 2.48742e10 1.81095
\(789\) 0 0
\(790\) 3.26957e9 0.235937
\(791\) −1.92848e10 −1.38547
\(792\) 0 0
\(793\) 6.51689e9 0.464071
\(794\) −2.97809e10 −2.11138
\(795\) 0 0
\(796\) −2.57505e10 −1.80963
\(797\) −2.19937e10 −1.53884 −0.769420 0.638744i \(-0.779454\pi\)
−0.769420 + 0.638744i \(0.779454\pi\)
\(798\) 0 0
\(799\) −2.42430e10 −1.68141
\(800\) −1.99271e9 −0.137604
\(801\) 0 0
\(802\) −3.51299e10 −2.40473
\(803\) −2.36910e9 −0.161465
\(804\) 0 0
\(805\) −1.56029e9 −0.105419
\(806\) −7.15491e9 −0.481318
\(807\) 0 0
\(808\) 4.00298e10 2.66958
\(809\) −9.05373e9 −0.601185 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(810\) 0 0
\(811\) −2.27877e9 −0.150012 −0.0750062 0.997183i \(-0.523898\pi\)
−0.0750062 + 0.997183i \(0.523898\pi\)
\(812\) 3.04980e10 1.99906
\(813\) 0 0
\(814\) −2.44448e10 −1.58855
\(815\) 1.67946e9 0.108672
\(816\) 0 0
\(817\) −1.91398e9 −0.122789
\(818\) 2.00809e10 1.28276
\(819\) 0 0
\(820\) −1.82424e10 −1.15540
\(821\) −1.34643e10 −0.849148 −0.424574 0.905393i \(-0.639576\pi\)
−0.424574 + 0.905393i \(0.639576\pi\)
\(822\) 0 0
\(823\) 1.64535e10 1.02887 0.514434 0.857530i \(-0.328002\pi\)
0.514434 + 0.857530i \(0.328002\pi\)
\(824\) −1.93919e10 −1.20746
\(825\) 0 0
\(826\) 3.27443e9 0.202165
\(827\) −1.52598e10 −0.938165 −0.469082 0.883154i \(-0.655415\pi\)
−0.469082 + 0.883154i \(0.655415\pi\)
\(828\) 0 0
\(829\) −1.06615e10 −0.649946 −0.324973 0.945723i \(-0.605355\pi\)
−0.324973 + 0.945723i \(0.605355\pi\)
\(830\) −7.17496e9 −0.435558
\(831\) 0 0
\(832\) 5.28725e9 0.318272
\(833\) −5.01479e9 −0.300604
\(834\) 0 0
\(835\) −9.43722e8 −0.0560972
\(836\) −1.15828e10 −0.685633
\(837\) 0 0
\(838\) 2.54536e10 1.49415
\(839\) 1.09099e10 0.637757 0.318878 0.947796i \(-0.396694\pi\)
0.318878 + 0.947796i \(0.396694\pi\)
\(840\) 0 0
\(841\) 3.59025e9 0.208132
\(842\) 1.05715e10 0.610300
\(843\) 0 0
\(844\) −4.84874e10 −2.77607
\(845\) −4.91163e9 −0.280045
\(846\) 0 0
\(847\) −4.22757e9 −0.239056
\(848\) −1.46734e10 −0.826312
\(849\) 0 0
\(850\) −4.04742e10 −2.26054
\(851\) 5.31103e9 0.295410
\(852\) 0 0
\(853\) −3.09781e10 −1.70897 −0.854483 0.519480i \(-0.826126\pi\)
−0.854483 + 0.519480i \(0.826126\pi\)
\(854\) 3.55680e10 1.95415
\(855\) 0 0
\(856\) 9.39531e9 0.511980
\(857\) −4.23395e9 −0.229781 −0.114890 0.993378i \(-0.536652\pi\)
−0.114890 + 0.993378i \(0.536652\pi\)
\(858\) 0 0
\(859\) 1.85088e10 0.996329 0.498165 0.867082i \(-0.334007\pi\)
0.498165 + 0.867082i \(0.334007\pi\)
\(860\) −5.09064e9 −0.272915
\(861\) 0 0
\(862\) 5.42924e10 2.88711
\(863\) 1.05929e10 0.561020 0.280510 0.959851i \(-0.409496\pi\)
0.280510 + 0.959851i \(0.409496\pi\)
\(864\) 0 0
\(865\) −6.67884e9 −0.350869
\(866\) 1.57924e10 0.826298
\(867\) 0 0
\(868\) −2.62180e10 −1.36076
\(869\) −9.09175e9 −0.469979
\(870\) 0 0
\(871\) 1.75670e8 0.00900812
\(872\) 3.98467e10 2.03509
\(873\) 0 0
\(874\) 3.74827e9 0.189907
\(875\) 1.08344e10 0.546735
\(876\) 0 0
\(877\) −9.90824e9 −0.496018 −0.248009 0.968758i \(-0.579776\pi\)
−0.248009 + 0.968758i \(0.579776\pi\)
\(878\) 1.52452e10 0.760156
\(879\) 0 0
\(880\) −8.34879e9 −0.412985
\(881\) −1.77972e9 −0.0876873 −0.0438437 0.999038i \(-0.513960\pi\)
−0.0438437 + 0.999038i \(0.513960\pi\)
\(882\) 0 0
\(883\) 3.07682e10 1.50397 0.751985 0.659180i \(-0.229096\pi\)
0.751985 + 0.659180i \(0.229096\pi\)
\(884\) −2.24076e10 −1.09097
\(885\) 0 0
\(886\) −5.63591e10 −2.72236
\(887\) 6.77910e9 0.326167 0.163083 0.986612i \(-0.447856\pi\)
0.163083 + 0.986612i \(0.447856\pi\)
\(888\) 0 0
\(889\) −5.06048e9 −0.241566
\(890\) −9.63348e9 −0.458056
\(891\) 0 0
\(892\) −1.39640e10 −0.658771
\(893\) −7.36270e9 −0.345985
\(894\) 0 0
\(895\) 8.16432e9 0.380662
\(896\) 3.18039e10 1.47707
\(897\) 0 0
\(898\) −3.94707e10 −1.81889
\(899\) −1.79155e10 −0.822374
\(900\) 0 0
\(901\) −2.32202e10 −1.05762
\(902\) 7.55549e10 3.42799
\(903\) 0 0
\(904\) 6.29101e10 2.83225
\(905\) 5.39530e9 0.241961
\(906\) 0 0
\(907\) 3.35163e10 1.49153 0.745763 0.666211i \(-0.232085\pi\)
0.745763 + 0.666211i \(0.232085\pi\)
\(908\) 2.03840e10 0.903626
\(909\) 0 0
\(910\) 4.21935e9 0.185610
\(911\) −2.19745e10 −0.962953 −0.481476 0.876459i \(-0.659899\pi\)
−0.481476 + 0.876459i \(0.659899\pi\)
\(912\) 0 0
\(913\) 1.99516e10 0.867619
\(914\) −2.14163e10 −0.927755
\(915\) 0 0
\(916\) −6.55233e9 −0.281683
\(917\) −2.37569e10 −1.01741
\(918\) 0 0
\(919\) 1.02526e10 0.435741 0.217871 0.975978i \(-0.430089\pi\)
0.217871 + 0.975978i \(0.430089\pi\)
\(920\) 5.08992e9 0.215503
\(921\) 0 0
\(922\) 6.78399e10 2.85054
\(923\) 1.09093e10 0.456656
\(924\) 0 0
\(925\) −1.74172e10 −0.723571
\(926\) −5.58701e10 −2.31228
\(927\) 0 0
\(928\) −4.11443e9 −0.169002
\(929\) 3.38955e10 1.38703 0.693517 0.720440i \(-0.256060\pi\)
0.693517 + 0.720440i \(0.256060\pi\)
\(930\) 0 0
\(931\) −1.52301e9 −0.0618557
\(932\) −5.83292e10 −2.36010
\(933\) 0 0
\(934\) 2.51221e10 1.00889
\(935\) −1.32117e10 −0.528591
\(936\) 0 0
\(937\) −5.25645e9 −0.208739 −0.104370 0.994539i \(-0.533283\pi\)
−0.104370 + 0.994539i \(0.533283\pi\)
\(938\) 9.58776e8 0.0379321
\(939\) 0 0
\(940\) −1.95827e10 −0.768999
\(941\) −2.78632e10 −1.09010 −0.545052 0.838402i \(-0.683490\pi\)
−0.545052 + 0.838402i \(0.683490\pi\)
\(942\) 0 0
\(943\) −1.64156e10 −0.637477
\(944\) −3.80668e9 −0.147280
\(945\) 0 0
\(946\) 2.10840e10 0.809720
\(947\) −2.23930e10 −0.856817 −0.428409 0.903585i \(-0.640926\pi\)
−0.428409 + 0.903585i \(0.640926\pi\)
\(948\) 0 0
\(949\) −1.39193e9 −0.0528670
\(950\) −1.22922e10 −0.465154
\(951\) 0 0
\(952\) −6.24393e10 −2.34546
\(953\) 4.95205e9 0.185336 0.0926680 0.995697i \(-0.470460\pi\)
0.0926680 + 0.995697i \(0.470460\pi\)
\(954\) 0 0
\(955\) −7.96334e9 −0.295858
\(956\) 3.83191e10 1.41845
\(957\) 0 0
\(958\) 3.62901e10 1.33355
\(959\) −3.73635e9 −0.136799
\(960\) 0 0
\(961\) −1.21114e10 −0.440211
\(962\) −1.43621e10 −0.520122
\(963\) 0 0
\(964\) −1.39978e10 −0.503258
\(965\) 8.33428e9 0.298554
\(966\) 0 0
\(967\) 2.63566e9 0.0937338 0.0468669 0.998901i \(-0.485076\pi\)
0.0468669 + 0.998901i \(0.485076\pi\)
\(968\) 1.37910e10 0.488688
\(969\) 0 0
\(970\) −2.72761e10 −0.959581
\(971\) −2.95178e10 −1.03471 −0.517353 0.855772i \(-0.673082\pi\)
−0.517353 + 0.855772i \(0.673082\pi\)
\(972\) 0 0
\(973\) 1.04908e10 0.365103
\(974\) −3.56905e10 −1.23765
\(975\) 0 0
\(976\) −4.13494e10 −1.42362
\(977\) −3.98464e9 −0.136697 −0.0683483 0.997662i \(-0.521773\pi\)
−0.0683483 + 0.997662i \(0.521773\pi\)
\(978\) 0 0
\(979\) 2.67880e10 0.912434
\(980\) −4.05079e9 −0.137483
\(981\) 0 0
\(982\) −6.21024e10 −2.09275
\(983\) 2.75939e10 0.926566 0.463283 0.886210i \(-0.346671\pi\)
0.463283 + 0.886210i \(0.346671\pi\)
\(984\) 0 0
\(985\) 8.61680e9 0.287289
\(986\) −8.35684e10 −2.77634
\(987\) 0 0
\(988\) −6.80528e9 −0.224490
\(989\) −4.58086e9 −0.150577
\(990\) 0 0
\(991\) 7.84374e8 0.0256015 0.0128008 0.999918i \(-0.495925\pi\)
0.0128008 + 0.999918i \(0.495925\pi\)
\(992\) 3.53702e9 0.115039
\(993\) 0 0
\(994\) 5.95408e10 1.92292
\(995\) −8.92038e9 −0.287080
\(996\) 0 0
\(997\) 8.21987e9 0.262683 0.131342 0.991337i \(-0.458072\pi\)
0.131342 + 0.991337i \(0.458072\pi\)
\(998\) 4.47579e10 1.42532
\(999\) 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.1.16 16
3.2 odd 2 177.8.a.a.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.1 16 3.2 odd 2
531.8.a.b.1.16 16 1.1 even 1 trivial