Properties

Label 531.8.a.b.1.13
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 1.13
Root \(15.0467\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+15.0467 q^{2} +98.4026 q^{4} -159.890 q^{5} +980.332 q^{7} -445.343 q^{8} +O(q^{10})\) \(q+15.0467 q^{2} +98.4026 q^{4} -159.890 q^{5} +980.332 q^{7} -445.343 q^{8} -2405.81 q^{10} +3390.91 q^{11} -7169.91 q^{13} +14750.7 q^{14} -19296.5 q^{16} +12242.5 q^{17} +19577.3 q^{19} -15733.6 q^{20} +51021.9 q^{22} -103217. q^{23} -52560.2 q^{25} -107883. q^{26} +96467.2 q^{28} +169076. q^{29} -37743.0 q^{31} -233344. q^{32} +184209. q^{34} -156745. q^{35} -29517.3 q^{37} +294573. q^{38} +71205.8 q^{40} -375081. q^{41} +719695. q^{43} +333674. q^{44} -1.55308e6 q^{46} +317727. q^{47} +137507. q^{49} -790857. q^{50} -705538. q^{52} -290607. q^{53} -542172. q^{55} -436584. q^{56} +2.54404e6 q^{58} -205379. q^{59} -2.85898e6 q^{61} -567906. q^{62} -1.04110e6 q^{64} +1.14640e6 q^{65} +1.89394e6 q^{67} +1.20469e6 q^{68} -2.35849e6 q^{70} -3.57204e6 q^{71} -2.76466e6 q^{73} -444137. q^{74} +1.92645e6 q^{76} +3.32421e6 q^{77} -4.56707e6 q^{79} +3.08531e6 q^{80} -5.64372e6 q^{82} +661014. q^{83} -1.95745e6 q^{85} +1.08290e7 q^{86} -1.51012e6 q^{88} +5.31858e6 q^{89} -7.02889e6 q^{91} -1.01568e7 q^{92} +4.78073e6 q^{94} -3.13021e6 q^{95} -1.84545e6 q^{97} +2.06903e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.0467 1.32995 0.664976 0.746865i \(-0.268442\pi\)
0.664976 + 0.746865i \(0.268442\pi\)
\(3\) 0 0
\(4\) 98.4026 0.768770
\(5\) −159.890 −0.572039 −0.286020 0.958224i \(-0.592332\pi\)
−0.286020 + 0.958224i \(0.592332\pi\)
\(6\) 0 0
\(7\) 980.332 1.08026 0.540132 0.841580i \(-0.318374\pi\)
0.540132 + 0.841580i \(0.318374\pi\)
\(8\) −445.343 −0.307524
\(9\) 0 0
\(10\) −2405.81 −0.760785
\(11\) 3390.91 0.768142 0.384071 0.923304i \(-0.374522\pi\)
0.384071 + 0.923304i \(0.374522\pi\)
\(12\) 0 0
\(13\) −7169.91 −0.905132 −0.452566 0.891731i \(-0.649491\pi\)
−0.452566 + 0.891731i \(0.649491\pi\)
\(14\) 14750.7 1.43670
\(15\) 0 0
\(16\) −19296.5 −1.17776
\(17\) 12242.5 0.604364 0.302182 0.953250i \(-0.402285\pi\)
0.302182 + 0.953250i \(0.402285\pi\)
\(18\) 0 0
\(19\) 19577.3 0.654808 0.327404 0.944884i \(-0.393826\pi\)
0.327404 + 0.944884i \(0.393826\pi\)
\(20\) −15733.6 −0.439767
\(21\) 0 0
\(22\) 51021.9 1.02159
\(23\) −103217. −1.76891 −0.884454 0.466628i \(-0.845469\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(24\) 0 0
\(25\) −52560.2 −0.672771
\(26\) −107883. −1.20378
\(27\) 0 0
\(28\) 96467.2 0.830475
\(29\) 169076. 1.28733 0.643665 0.765307i \(-0.277413\pi\)
0.643665 + 0.765307i \(0.277413\pi\)
\(30\) 0 0
\(31\) −37743.0 −0.227546 −0.113773 0.993507i \(-0.536294\pi\)
−0.113773 + 0.993507i \(0.536294\pi\)
\(32\) −233344. −1.25884
\(33\) 0 0
\(34\) 184209. 0.803774
\(35\) −156745. −0.617954
\(36\) 0 0
\(37\) −29517.3 −0.0958011 −0.0479006 0.998852i \(-0.515253\pi\)
−0.0479006 + 0.998852i \(0.515253\pi\)
\(38\) 294573. 0.870863
\(39\) 0 0
\(40\) 71205.8 0.175916
\(41\) −375081. −0.849926 −0.424963 0.905211i \(-0.639713\pi\)
−0.424963 + 0.905211i \(0.639713\pi\)
\(42\) 0 0
\(43\) 719695. 1.38041 0.690206 0.723613i \(-0.257520\pi\)
0.690206 + 0.723613i \(0.257520\pi\)
\(44\) 333674. 0.590525
\(45\) 0 0
\(46\) −1.55308e6 −2.35256
\(47\) 317727. 0.446387 0.223193 0.974774i \(-0.428352\pi\)
0.223193 + 0.974774i \(0.428352\pi\)
\(48\) 0 0
\(49\) 137507. 0.166970
\(50\) −790857. −0.894752
\(51\) 0 0
\(52\) −705538. −0.695839
\(53\) −290607. −0.268127 −0.134063 0.990973i \(-0.542803\pi\)
−0.134063 + 0.990973i \(0.542803\pi\)
\(54\) 0 0
\(55\) −542172. −0.439407
\(56\) −436584. −0.332208
\(57\) 0 0
\(58\) 2.54404e6 1.71209
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −2.85898e6 −1.61271 −0.806355 0.591432i \(-0.798563\pi\)
−0.806355 + 0.591432i \(0.798563\pi\)
\(62\) −567906. −0.302626
\(63\) 0 0
\(64\) −1.04110e6 −0.496436
\(65\) 1.14640e6 0.517771
\(66\) 0 0
\(67\) 1.89394e6 0.769316 0.384658 0.923059i \(-0.374319\pi\)
0.384658 + 0.923059i \(0.374319\pi\)
\(68\) 1.20469e6 0.464617
\(69\) 0 0
\(70\) −2.35849e6 −0.821848
\(71\) −3.57204e6 −1.18444 −0.592218 0.805778i \(-0.701748\pi\)
−0.592218 + 0.805778i \(0.701748\pi\)
\(72\) 0 0
\(73\) −2.76466e6 −0.831786 −0.415893 0.909413i \(-0.636531\pi\)
−0.415893 + 0.909413i \(0.636531\pi\)
\(74\) −444137. −0.127411
\(75\) 0 0
\(76\) 1.92645e6 0.503397
\(77\) 3.32421e6 0.829796
\(78\) 0 0
\(79\) −4.56707e6 −1.04218 −0.521091 0.853501i \(-0.674475\pi\)
−0.521091 + 0.853501i \(0.674475\pi\)
\(80\) 3.08531e6 0.673727
\(81\) 0 0
\(82\) −5.64372e6 −1.13036
\(83\) 661014. 0.126893 0.0634464 0.997985i \(-0.479791\pi\)
0.0634464 + 0.997985i \(0.479791\pi\)
\(84\) 0 0
\(85\) −1.95745e6 −0.345720
\(86\) 1.08290e7 1.83588
\(87\) 0 0
\(88\) −1.51012e6 −0.236222
\(89\) 5.31858e6 0.799707 0.399853 0.916579i \(-0.369061\pi\)
0.399853 + 0.916579i \(0.369061\pi\)
\(90\) 0 0
\(91\) −7.02889e6 −0.977782
\(92\) −1.01568e7 −1.35988
\(93\) 0 0
\(94\) 4.78073e6 0.593673
\(95\) −3.13021e6 −0.374576
\(96\) 0 0
\(97\) −1.84545e6 −0.205305 −0.102653 0.994717i \(-0.532733\pi\)
−0.102653 + 0.994717i \(0.532733\pi\)
\(98\) 2.06903e6 0.222062
\(99\) 0 0
\(100\) −5.17206e6 −0.517206
\(101\) −4.95032e6 −0.478088 −0.239044 0.971009i \(-0.576834\pi\)
−0.239044 + 0.971009i \(0.576834\pi\)
\(102\) 0 0
\(103\) −2.16194e7 −1.94946 −0.974728 0.223393i \(-0.928287\pi\)
−0.974728 + 0.223393i \(0.928287\pi\)
\(104\) 3.19307e6 0.278350
\(105\) 0 0
\(106\) −4.37267e6 −0.356595
\(107\) 1.82441e6 0.143973 0.0719863 0.997406i \(-0.477066\pi\)
0.0719863 + 0.997406i \(0.477066\pi\)
\(108\) 0 0
\(109\) 5.23038e6 0.386848 0.193424 0.981115i \(-0.438041\pi\)
0.193424 + 0.981115i \(0.438041\pi\)
\(110\) −8.15788e6 −0.584390
\(111\) 0 0
\(112\) −1.89169e7 −1.27229
\(113\) −2.44766e7 −1.59579 −0.797895 0.602796i \(-0.794053\pi\)
−0.797895 + 0.602796i \(0.794053\pi\)
\(114\) 0 0
\(115\) 1.65034e7 1.01188
\(116\) 1.66376e7 0.989661
\(117\) 0 0
\(118\) −3.09027e6 −0.173145
\(119\) 1.20017e7 0.652872
\(120\) 0 0
\(121\) −7.98892e6 −0.409958
\(122\) −4.30181e7 −2.14483
\(123\) 0 0
\(124\) −3.71400e6 −0.174931
\(125\) 2.08952e7 0.956891
\(126\) 0 0
\(127\) −2.70518e7 −1.17188 −0.585940 0.810355i \(-0.699275\pi\)
−0.585940 + 0.810355i \(0.699275\pi\)
\(128\) 1.42029e7 0.598606
\(129\) 0 0
\(130\) 1.72495e7 0.688611
\(131\) −1.34945e7 −0.524454 −0.262227 0.965006i \(-0.584457\pi\)
−0.262227 + 0.965006i \(0.584457\pi\)
\(132\) 0 0
\(133\) 1.91922e7 0.707366
\(134\) 2.84975e7 1.02315
\(135\) 0 0
\(136\) −5.45210e6 −0.185857
\(137\) −5.35174e7 −1.77817 −0.889084 0.457745i \(-0.848657\pi\)
−0.889084 + 0.457745i \(0.848657\pi\)
\(138\) 0 0
\(139\) 1.33665e7 0.422149 0.211075 0.977470i \(-0.432304\pi\)
0.211075 + 0.977470i \(0.432304\pi\)
\(140\) −1.54241e7 −0.475064
\(141\) 0 0
\(142\) −5.37473e7 −1.57524
\(143\) −2.43125e7 −0.695270
\(144\) 0 0
\(145\) −2.70336e7 −0.736404
\(146\) −4.15990e7 −1.10624
\(147\) 0 0
\(148\) −2.90458e6 −0.0736490
\(149\) 1.60776e7 0.398170 0.199085 0.979982i \(-0.436203\pi\)
0.199085 + 0.979982i \(0.436203\pi\)
\(150\) 0 0
\(151\) −5.18108e7 −1.22462 −0.612310 0.790618i \(-0.709760\pi\)
−0.612310 + 0.790618i \(0.709760\pi\)
\(152\) −8.71859e6 −0.201370
\(153\) 0 0
\(154\) 5.00184e7 1.10359
\(155\) 6.03472e6 0.130166
\(156\) 0 0
\(157\) −8.16318e7 −1.68349 −0.841746 0.539874i \(-0.818472\pi\)
−0.841746 + 0.539874i \(0.818472\pi\)
\(158\) −6.87193e7 −1.38605
\(159\) 0 0
\(160\) 3.73093e7 0.720107
\(161\) −1.01187e8 −1.91089
\(162\) 0 0
\(163\) 8.91769e6 0.161286 0.0806428 0.996743i \(-0.474303\pi\)
0.0806428 + 0.996743i \(0.474303\pi\)
\(164\) −3.69089e7 −0.653398
\(165\) 0 0
\(166\) 9.94606e6 0.168761
\(167\) 3.21164e7 0.533604 0.266802 0.963751i \(-0.414033\pi\)
0.266802 + 0.963751i \(0.414033\pi\)
\(168\) 0 0
\(169\) −1.13409e7 −0.180736
\(170\) −2.94531e7 −0.459790
\(171\) 0 0
\(172\) 7.08198e7 1.06122
\(173\) 5.06027e7 0.743040 0.371520 0.928425i \(-0.378837\pi\)
0.371520 + 0.928425i \(0.378837\pi\)
\(174\) 0 0
\(175\) −5.15265e7 −0.726770
\(176\) −6.54325e7 −0.904689
\(177\) 0 0
\(178\) 8.00270e7 1.06357
\(179\) 6.22045e7 0.810655 0.405327 0.914172i \(-0.367157\pi\)
0.405327 + 0.914172i \(0.367157\pi\)
\(180\) 0 0
\(181\) 6.29686e7 0.789313 0.394656 0.918829i \(-0.370864\pi\)
0.394656 + 0.918829i \(0.370864\pi\)
\(182\) −1.05761e8 −1.30040
\(183\) 0 0
\(184\) 4.59671e7 0.543982
\(185\) 4.71952e6 0.0548020
\(186\) 0 0
\(187\) 4.15131e7 0.464237
\(188\) 3.12651e7 0.343169
\(189\) 0 0
\(190\) −4.70992e7 −0.498168
\(191\) 7.18839e7 0.746474 0.373237 0.927736i \(-0.378248\pi\)
0.373237 + 0.927736i \(0.378248\pi\)
\(192\) 0 0
\(193\) −3.68734e7 −0.369201 −0.184600 0.982814i \(-0.559099\pi\)
−0.184600 + 0.982814i \(0.559099\pi\)
\(194\) −2.77678e7 −0.273046
\(195\) 0 0
\(196\) 1.35311e7 0.128362
\(197\) −1.75363e8 −1.63420 −0.817101 0.576495i \(-0.804420\pi\)
−0.817101 + 0.576495i \(0.804420\pi\)
\(198\) 0 0
\(199\) 8.16489e7 0.734454 0.367227 0.930131i \(-0.380307\pi\)
0.367227 + 0.930131i \(0.380307\pi\)
\(200\) 2.34073e7 0.206893
\(201\) 0 0
\(202\) −7.44858e7 −0.635834
\(203\) 1.65751e8 1.39066
\(204\) 0 0
\(205\) 5.99716e7 0.486191
\(206\) −3.25300e8 −2.59268
\(207\) 0 0
\(208\) 1.38354e8 1.06603
\(209\) 6.63846e7 0.502986
\(210\) 0 0
\(211\) 1.04928e7 0.0768955 0.0384478 0.999261i \(-0.487759\pi\)
0.0384478 + 0.999261i \(0.487759\pi\)
\(212\) −2.85965e7 −0.206128
\(213\) 0 0
\(214\) 2.74513e7 0.191476
\(215\) −1.15072e8 −0.789650
\(216\) 0 0
\(217\) −3.70006e7 −0.245810
\(218\) 7.86998e7 0.514489
\(219\) 0 0
\(220\) −5.33511e7 −0.337803
\(221\) −8.77775e7 −0.547029
\(222\) 0 0
\(223\) −2.52899e8 −1.52714 −0.763572 0.645723i \(-0.776556\pi\)
−0.763572 + 0.645723i \(0.776556\pi\)
\(224\) −2.28754e8 −1.35988
\(225\) 0 0
\(226\) −3.68291e8 −2.12232
\(227\) −457979. −0.00259869 −0.00129935 0.999999i \(-0.500414\pi\)
−0.00129935 + 0.999999i \(0.500414\pi\)
\(228\) 0 0
\(229\) −1.85147e8 −1.01881 −0.509405 0.860527i \(-0.670135\pi\)
−0.509405 + 0.860527i \(0.670135\pi\)
\(230\) 2.48321e8 1.34576
\(231\) 0 0
\(232\) −7.52970e7 −0.395886
\(233\) 3.11021e8 1.61081 0.805404 0.592726i \(-0.201948\pi\)
0.805404 + 0.592726i \(0.201948\pi\)
\(234\) 0 0
\(235\) −5.08013e7 −0.255351
\(236\) −2.02098e7 −0.100085
\(237\) 0 0
\(238\) 1.80586e8 0.868288
\(239\) 1.44383e8 0.684107 0.342053 0.939680i \(-0.388878\pi\)
0.342053 + 0.939680i \(0.388878\pi\)
\(240\) 0 0
\(241\) −9.66758e7 −0.444896 −0.222448 0.974945i \(-0.571405\pi\)
−0.222448 + 0.974945i \(0.571405\pi\)
\(242\) −1.20207e8 −0.545224
\(243\) 0 0
\(244\) −2.81331e8 −1.23980
\(245\) −2.19860e7 −0.0955136
\(246\) 0 0
\(247\) −1.40367e8 −0.592688
\(248\) 1.68086e7 0.0699761
\(249\) 0 0
\(250\) 3.14404e8 1.27262
\(251\) −1.46912e8 −0.586407 −0.293203 0.956050i \(-0.594721\pi\)
−0.293203 + 0.956050i \(0.594721\pi\)
\(252\) 0 0
\(253\) −3.50000e8 −1.35877
\(254\) −4.07040e8 −1.55854
\(255\) 0 0
\(256\) 3.46967e8 1.29255
\(257\) 4.93706e8 1.81427 0.907136 0.420837i \(-0.138264\pi\)
0.907136 + 0.420837i \(0.138264\pi\)
\(258\) 0 0
\(259\) −2.89367e7 −0.103490
\(260\) 1.12808e8 0.398047
\(261\) 0 0
\(262\) −2.03048e8 −0.697498
\(263\) 1.94462e8 0.659157 0.329578 0.944128i \(-0.393093\pi\)
0.329578 + 0.944128i \(0.393093\pi\)
\(264\) 0 0
\(265\) 4.64651e7 0.153379
\(266\) 2.88779e8 0.940762
\(267\) 0 0
\(268\) 1.86369e8 0.591427
\(269\) 5.97880e7 0.187275 0.0936377 0.995606i \(-0.470150\pi\)
0.0936377 + 0.995606i \(0.470150\pi\)
\(270\) 0 0
\(271\) −5.18384e8 −1.58219 −0.791096 0.611693i \(-0.790489\pi\)
−0.791096 + 0.611693i \(0.790489\pi\)
\(272\) −2.36237e8 −0.711797
\(273\) 0 0
\(274\) −8.05258e8 −2.36488
\(275\) −1.78227e8 −0.516783
\(276\) 0 0
\(277\) 2.35392e8 0.665444 0.332722 0.943025i \(-0.392033\pi\)
0.332722 + 0.943025i \(0.392033\pi\)
\(278\) 2.01122e8 0.561438
\(279\) 0 0
\(280\) 6.98053e7 0.190036
\(281\) 3.16518e8 0.850994 0.425497 0.904960i \(-0.360099\pi\)
0.425497 + 0.904960i \(0.360099\pi\)
\(282\) 0 0
\(283\) −2.45453e8 −0.643748 −0.321874 0.946782i \(-0.604313\pi\)
−0.321874 + 0.946782i \(0.604313\pi\)
\(284\) −3.51498e8 −0.910559
\(285\) 0 0
\(286\) −3.65822e8 −0.924675
\(287\) −3.67703e8 −0.918144
\(288\) 0 0
\(289\) −2.60460e8 −0.634745
\(290\) −4.06766e8 −0.979381
\(291\) 0 0
\(292\) −2.72050e8 −0.639452
\(293\) −1.97162e8 −0.457916 −0.228958 0.973436i \(-0.573532\pi\)
−0.228958 + 0.973436i \(0.573532\pi\)
\(294\) 0 0
\(295\) 3.28380e7 0.0744732
\(296\) 1.31453e7 0.0294612
\(297\) 0 0
\(298\) 2.41914e8 0.529546
\(299\) 7.40059e8 1.60110
\(300\) 0 0
\(301\) 7.05540e8 1.49121
\(302\) −7.79581e8 −1.62868
\(303\) 0 0
\(304\) −3.77772e8 −0.771209
\(305\) 4.57121e8 0.922534
\(306\) 0 0
\(307\) 1.34184e7 0.0264677 0.0132339 0.999912i \(-0.495787\pi\)
0.0132339 + 0.999912i \(0.495787\pi\)
\(308\) 3.27111e8 0.637922
\(309\) 0 0
\(310\) 9.08024e7 0.173114
\(311\) 7.42366e8 1.39945 0.699724 0.714413i \(-0.253306\pi\)
0.699724 + 0.714413i \(0.253306\pi\)
\(312\) 0 0
\(313\) 7.47946e8 1.37869 0.689343 0.724435i \(-0.257899\pi\)
0.689343 + 0.724435i \(0.257899\pi\)
\(314\) −1.22829e9 −2.23896
\(315\) 0 0
\(316\) −4.49412e8 −0.801198
\(317\) 7.65404e8 1.34953 0.674766 0.738031i \(-0.264244\pi\)
0.674766 + 0.738031i \(0.264244\pi\)
\(318\) 0 0
\(319\) 5.73323e8 0.988853
\(320\) 1.66462e8 0.283981
\(321\) 0 0
\(322\) −1.52253e9 −2.54139
\(323\) 2.39674e8 0.395742
\(324\) 0 0
\(325\) 3.76852e8 0.608947
\(326\) 1.34182e8 0.214502
\(327\) 0 0
\(328\) 1.67039e8 0.261373
\(329\) 3.11478e8 0.482216
\(330\) 0 0
\(331\) 3.13126e8 0.474592 0.237296 0.971437i \(-0.423739\pi\)
0.237296 + 0.971437i \(0.423739\pi\)
\(332\) 6.50455e7 0.0975515
\(333\) 0 0
\(334\) 4.83245e8 0.709667
\(335\) −3.02822e8 −0.440079
\(336\) 0 0
\(337\) 1.30356e9 1.85535 0.927675 0.373388i \(-0.121804\pi\)
0.927675 + 0.373388i \(0.121804\pi\)
\(338\) −1.70643e8 −0.240369
\(339\) 0 0
\(340\) −1.92618e8 −0.265779
\(341\) −1.27983e8 −0.174788
\(342\) 0 0
\(343\) −6.72543e8 −0.899892
\(344\) −3.20511e8 −0.424511
\(345\) 0 0
\(346\) 7.61402e8 0.988206
\(347\) 2.10675e8 0.270683 0.135341 0.990799i \(-0.456787\pi\)
0.135341 + 0.990799i \(0.456787\pi\)
\(348\) 0 0
\(349\) 6.27337e8 0.789972 0.394986 0.918687i \(-0.370749\pi\)
0.394986 + 0.918687i \(0.370749\pi\)
\(350\) −7.75302e8 −0.966569
\(351\) 0 0
\(352\) −7.91247e8 −0.966970
\(353\) 8.10514e8 0.980729 0.490364 0.871517i \(-0.336864\pi\)
0.490364 + 0.871517i \(0.336864\pi\)
\(354\) 0 0
\(355\) 5.71133e8 0.677544
\(356\) 5.23362e8 0.614791
\(357\) 0 0
\(358\) 9.35971e8 1.07813
\(359\) −4.90257e8 −0.559234 −0.279617 0.960112i \(-0.590207\pi\)
−0.279617 + 0.960112i \(0.590207\pi\)
\(360\) 0 0
\(361\) −5.10603e8 −0.571226
\(362\) 9.47468e8 1.04975
\(363\) 0 0
\(364\) −6.91661e8 −0.751689
\(365\) 4.42041e8 0.475815
\(366\) 0 0
\(367\) −6.65868e8 −0.703164 −0.351582 0.936157i \(-0.614356\pi\)
−0.351582 + 0.936157i \(0.614356\pi\)
\(368\) 1.99173e9 2.08335
\(369\) 0 0
\(370\) 7.10131e7 0.0728840
\(371\) −2.84891e8 −0.289648
\(372\) 0 0
\(373\) −1.50881e9 −1.50541 −0.752704 0.658360i \(-0.771251\pi\)
−0.752704 + 0.658360i \(0.771251\pi\)
\(374\) 6.24635e8 0.617412
\(375\) 0 0
\(376\) −1.41497e8 −0.137275
\(377\) −1.21226e9 −1.16520
\(378\) 0 0
\(379\) −1.17501e9 −1.10868 −0.554339 0.832291i \(-0.687029\pi\)
−0.554339 + 0.832291i \(0.687029\pi\)
\(380\) −3.08020e8 −0.287963
\(381\) 0 0
\(382\) 1.08161e9 0.992774
\(383\) −3.09605e8 −0.281587 −0.140794 0.990039i \(-0.544965\pi\)
−0.140794 + 0.990039i \(0.544965\pi\)
\(384\) 0 0
\(385\) −5.31508e8 −0.474676
\(386\) −5.54822e8 −0.491019
\(387\) 0 0
\(388\) −1.81597e8 −0.157833
\(389\) 1.81950e9 1.56721 0.783606 0.621258i \(-0.213378\pi\)
0.783606 + 0.621258i \(0.213378\pi\)
\(390\) 0 0
\(391\) −1.26364e9 −1.06906
\(392\) −6.12379e7 −0.0513474
\(393\) 0 0
\(394\) −2.63863e9 −2.17341
\(395\) 7.30229e8 0.596169
\(396\) 0 0
\(397\) 6.58203e8 0.527950 0.263975 0.964529i \(-0.414966\pi\)
0.263975 + 0.964529i \(0.414966\pi\)
\(398\) 1.22855e9 0.976788
\(399\) 0 0
\(400\) 1.01423e9 0.792364
\(401\) −7.74197e8 −0.599578 −0.299789 0.954005i \(-0.596916\pi\)
−0.299789 + 0.954005i \(0.596916\pi\)
\(402\) 0 0
\(403\) 2.70614e8 0.205960
\(404\) −4.87124e8 −0.367540
\(405\) 0 0
\(406\) 2.49400e9 1.84951
\(407\) −1.00090e8 −0.0735889
\(408\) 0 0
\(409\) 1.08898e9 0.787022 0.393511 0.919320i \(-0.371260\pi\)
0.393511 + 0.919320i \(0.371260\pi\)
\(410\) 9.02373e8 0.646610
\(411\) 0 0
\(412\) −2.12741e9 −1.49868
\(413\) −2.01340e8 −0.140638
\(414\) 0 0
\(415\) −1.05689e8 −0.0725877
\(416\) 1.67305e9 1.13942
\(417\) 0 0
\(418\) 9.98869e8 0.668946
\(419\) −1.86951e9 −1.24159 −0.620795 0.783973i \(-0.713190\pi\)
−0.620795 + 0.783973i \(0.713190\pi\)
\(420\) 0 0
\(421\) 2.61169e9 1.70583 0.852913 0.522053i \(-0.174834\pi\)
0.852913 + 0.522053i \(0.174834\pi\)
\(422\) 1.57881e8 0.102267
\(423\) 0 0
\(424\) 1.29420e8 0.0824555
\(425\) −6.43468e8 −0.406598
\(426\) 0 0
\(427\) −2.80275e9 −1.74215
\(428\) 1.79527e8 0.110682
\(429\) 0 0
\(430\) −1.73145e9 −1.05020
\(431\) −1.54835e9 −0.931534 −0.465767 0.884908i \(-0.654221\pi\)
−0.465767 + 0.884908i \(0.654221\pi\)
\(432\) 0 0
\(433\) −1.73247e9 −1.02555 −0.512776 0.858523i \(-0.671383\pi\)
−0.512776 + 0.858523i \(0.671383\pi\)
\(434\) −5.56736e8 −0.326916
\(435\) 0 0
\(436\) 5.14683e8 0.297397
\(437\) −2.02071e9 −1.15830
\(438\) 0 0
\(439\) −3.14201e9 −1.77248 −0.886239 0.463227i \(-0.846691\pi\)
−0.886239 + 0.463227i \(0.846691\pi\)
\(440\) 2.41452e8 0.135129
\(441\) 0 0
\(442\) −1.32076e9 −0.727522
\(443\) −1.87980e9 −1.02730 −0.513651 0.857999i \(-0.671707\pi\)
−0.513651 + 0.857999i \(0.671707\pi\)
\(444\) 0 0
\(445\) −8.50388e8 −0.457464
\(446\) −3.80529e9 −2.03103
\(447\) 0 0
\(448\) −1.02063e9 −0.536282
\(449\) −2.34987e9 −1.22513 −0.612564 0.790421i \(-0.709862\pi\)
−0.612564 + 0.790421i \(0.709862\pi\)
\(450\) 0 0
\(451\) −1.27186e9 −0.652864
\(452\) −2.40856e9 −1.22680
\(453\) 0 0
\(454\) −6.89106e6 −0.00345613
\(455\) 1.12385e9 0.559330
\(456\) 0 0
\(457\) −1.89675e8 −0.0929615 −0.0464808 0.998919i \(-0.514801\pi\)
−0.0464808 + 0.998919i \(0.514801\pi\)
\(458\) −2.78585e9 −1.35497
\(459\) 0 0
\(460\) 1.62398e9 0.777907
\(461\) 2.07603e9 0.986919 0.493459 0.869769i \(-0.335732\pi\)
0.493459 + 0.869769i \(0.335732\pi\)
\(462\) 0 0
\(463\) 3.00869e9 1.40878 0.704392 0.709811i \(-0.251220\pi\)
0.704392 + 0.709811i \(0.251220\pi\)
\(464\) −3.26258e9 −1.51617
\(465\) 0 0
\(466\) 4.67983e9 2.14230
\(467\) 1.33042e9 0.604479 0.302239 0.953232i \(-0.402266\pi\)
0.302239 + 0.953232i \(0.402266\pi\)
\(468\) 0 0
\(469\) 1.85669e9 0.831064
\(470\) −7.64391e8 −0.339604
\(471\) 0 0
\(472\) 9.14641e7 0.0400363
\(473\) 2.44042e9 1.06035
\(474\) 0 0
\(475\) −1.02898e9 −0.440536
\(476\) 1.18100e9 0.501909
\(477\) 0 0
\(478\) 2.17249e9 0.909829
\(479\) −2.48445e9 −1.03290 −0.516448 0.856318i \(-0.672746\pi\)
−0.516448 + 0.856318i \(0.672746\pi\)
\(480\) 0 0
\(481\) 2.11636e8 0.0867127
\(482\) −1.45465e9 −0.591689
\(483\) 0 0
\(484\) −7.86130e8 −0.315163
\(485\) 2.95068e8 0.117443
\(486\) 0 0
\(487\) −1.39919e9 −0.548940 −0.274470 0.961596i \(-0.588502\pi\)
−0.274470 + 0.961596i \(0.588502\pi\)
\(488\) 1.27323e9 0.495948
\(489\) 0 0
\(490\) −3.30816e8 −0.127028
\(491\) −1.98268e9 −0.755907 −0.377953 0.925825i \(-0.623372\pi\)
−0.377953 + 0.925825i \(0.623372\pi\)
\(492\) 0 0
\(493\) 2.06992e9 0.778016
\(494\) −2.11206e9 −0.788246
\(495\) 0 0
\(496\) 7.28305e8 0.267996
\(497\) −3.50178e9 −1.27950
\(498\) 0 0
\(499\) 3.50830e9 1.26400 0.631998 0.774970i \(-0.282235\pi\)
0.631998 + 0.774970i \(0.282235\pi\)
\(500\) 2.05615e9 0.735629
\(501\) 0 0
\(502\) −2.21054e9 −0.779893
\(503\) 3.10883e9 1.08920 0.544602 0.838694i \(-0.316681\pi\)
0.544602 + 0.838694i \(0.316681\pi\)
\(504\) 0 0
\(505\) 7.91505e8 0.273485
\(506\) −5.26634e9 −1.80710
\(507\) 0 0
\(508\) −2.66197e9 −0.900906
\(509\) −2.98895e9 −1.00463 −0.502315 0.864685i \(-0.667518\pi\)
−0.502315 + 0.864685i \(0.667518\pi\)
\(510\) 0 0
\(511\) −2.71028e9 −0.898549
\(512\) 3.40274e9 1.12043
\(513\) 0 0
\(514\) 7.42864e9 2.41289
\(515\) 3.45673e9 1.11517
\(516\) 0 0
\(517\) 1.07738e9 0.342888
\(518\) −4.35402e8 −0.137637
\(519\) 0 0
\(520\) −5.10540e8 −0.159227
\(521\) −2.53744e8 −0.0786075 −0.0393037 0.999227i \(-0.512514\pi\)
−0.0393037 + 0.999227i \(0.512514\pi\)
\(522\) 0 0
\(523\) 2.79711e9 0.854975 0.427487 0.904021i \(-0.359399\pi\)
0.427487 + 0.904021i \(0.359399\pi\)
\(524\) −1.32789e9 −0.403185
\(525\) 0 0
\(526\) 2.92600e9 0.876646
\(527\) −4.62067e8 −0.137521
\(528\) 0 0
\(529\) 7.24899e9 2.12903
\(530\) 6.99145e8 0.203987
\(531\) 0 0
\(532\) 1.88856e9 0.543802
\(533\) 2.68929e9 0.769295
\(534\) 0 0
\(535\) −2.91705e8 −0.0823580
\(536\) −8.43453e8 −0.236583
\(537\) 0 0
\(538\) 8.99610e8 0.249067
\(539\) 4.66274e8 0.128257
\(540\) 0 0
\(541\) −2.58195e9 −0.701064 −0.350532 0.936551i \(-0.613999\pi\)
−0.350532 + 0.936551i \(0.613999\pi\)
\(542\) −7.79996e9 −2.10424
\(543\) 0 0
\(544\) −2.85671e9 −0.760798
\(545\) −8.36284e8 −0.221292
\(546\) 0 0
\(547\) 1.44486e9 0.377459 0.188730 0.982029i \(-0.439563\pi\)
0.188730 + 0.982029i \(0.439563\pi\)
\(548\) −5.26625e9 −1.36700
\(549\) 0 0
\(550\) −2.68172e9 −0.687297
\(551\) 3.31005e9 0.842955
\(552\) 0 0
\(553\) −4.47725e9 −1.12583
\(554\) 3.54186e9 0.885008
\(555\) 0 0
\(556\) 1.31530e9 0.324536
\(557\) 4.77372e9 1.17048 0.585240 0.810860i \(-0.301000\pi\)
0.585240 + 0.810860i \(0.301000\pi\)
\(558\) 0 0
\(559\) −5.16015e9 −1.24946
\(560\) 3.02463e9 0.727803
\(561\) 0 0
\(562\) 4.76255e9 1.13178
\(563\) 3.69151e9 0.871816 0.435908 0.899991i \(-0.356427\pi\)
0.435908 + 0.899991i \(0.356427\pi\)
\(564\) 0 0
\(565\) 3.91356e9 0.912855
\(566\) −3.69326e9 −0.856154
\(567\) 0 0
\(568\) 1.59078e9 0.364243
\(569\) 2.15110e9 0.489516 0.244758 0.969584i \(-0.421291\pi\)
0.244758 + 0.969584i \(0.421291\pi\)
\(570\) 0 0
\(571\) 5.63827e8 0.126742 0.0633709 0.997990i \(-0.479815\pi\)
0.0633709 + 0.997990i \(0.479815\pi\)
\(572\) −2.39241e9 −0.534503
\(573\) 0 0
\(574\) −5.53271e9 −1.22109
\(575\) 5.42513e9 1.19007
\(576\) 0 0
\(577\) −3.12035e9 −0.676221 −0.338110 0.941106i \(-0.609788\pi\)
−0.338110 + 0.941106i \(0.609788\pi\)
\(578\) −3.91906e9 −0.844179
\(579\) 0 0
\(580\) −2.66018e9 −0.566125
\(581\) 6.48013e8 0.137078
\(582\) 0 0
\(583\) −9.85420e8 −0.205959
\(584\) 1.23122e9 0.255795
\(585\) 0 0
\(586\) −2.96663e9 −0.609006
\(587\) 4.80777e9 0.981094 0.490547 0.871415i \(-0.336797\pi\)
0.490547 + 0.871415i \(0.336797\pi\)
\(588\) 0 0
\(589\) −7.38903e8 −0.148999
\(590\) 4.94103e8 0.0990457
\(591\) 0 0
\(592\) 5.69579e8 0.112831
\(593\) 4.97973e8 0.0980651 0.0490326 0.998797i \(-0.484386\pi\)
0.0490326 + 0.998797i \(0.484386\pi\)
\(594\) 0 0
\(595\) −1.91895e9 −0.373469
\(596\) 1.58207e9 0.306101
\(597\) 0 0
\(598\) 1.11354e10 2.12938
\(599\) 1.25174e9 0.237968 0.118984 0.992896i \(-0.462036\pi\)
0.118984 + 0.992896i \(0.462036\pi\)
\(600\) 0 0
\(601\) −3.57133e9 −0.671073 −0.335536 0.942027i \(-0.608918\pi\)
−0.335536 + 0.942027i \(0.608918\pi\)
\(602\) 1.06160e10 1.98324
\(603\) 0 0
\(604\) −5.09832e9 −0.941451
\(605\) 1.27735e9 0.234512
\(606\) 0 0
\(607\) −2.73915e9 −0.497113 −0.248556 0.968617i \(-0.579956\pi\)
−0.248556 + 0.968617i \(0.579956\pi\)
\(608\) −4.56823e9 −0.824300
\(609\) 0 0
\(610\) 6.87816e9 1.22692
\(611\) −2.27807e9 −0.404039
\(612\) 0 0
\(613\) 2.74148e9 0.480700 0.240350 0.970686i \(-0.422738\pi\)
0.240350 + 0.970686i \(0.422738\pi\)
\(614\) 2.01903e8 0.0352008
\(615\) 0 0
\(616\) −1.48042e9 −0.255183
\(617\) 9.78150e9 1.67652 0.838258 0.545274i \(-0.183575\pi\)
0.838258 + 0.545274i \(0.183575\pi\)
\(618\) 0 0
\(619\) 8.34172e9 1.41364 0.706819 0.707394i \(-0.250130\pi\)
0.706819 + 0.707394i \(0.250130\pi\)
\(620\) 5.93832e8 0.100067
\(621\) 0 0
\(622\) 1.11701e10 1.86120
\(623\) 5.21397e9 0.863895
\(624\) 0 0
\(625\) 7.65329e8 0.125391
\(626\) 1.12541e10 1.83358
\(627\) 0 0
\(628\) −8.03278e9 −1.29422
\(629\) −3.61365e8 −0.0578987
\(630\) 0 0
\(631\) −4.64175e7 −0.00735494 −0.00367747 0.999993i \(-0.501171\pi\)
−0.00367747 + 0.999993i \(0.501171\pi\)
\(632\) 2.03391e9 0.320496
\(633\) 0 0
\(634\) 1.15168e10 1.79481
\(635\) 4.32531e9 0.670361
\(636\) 0 0
\(637\) −9.85914e8 −0.151130
\(638\) 8.62660e9 1.31513
\(639\) 0 0
\(640\) −2.27090e9 −0.342426
\(641\) 9.01725e9 1.35229 0.676147 0.736767i \(-0.263648\pi\)
0.676147 + 0.736767i \(0.263648\pi\)
\(642\) 0 0
\(643\) −6.73082e9 −0.998457 −0.499229 0.866470i \(-0.666383\pi\)
−0.499229 + 0.866470i \(0.666383\pi\)
\(644\) −9.95708e9 −1.46903
\(645\) 0 0
\(646\) 3.60630e9 0.526318
\(647\) −1.43491e9 −0.208286 −0.104143 0.994562i \(-0.533210\pi\)
−0.104143 + 0.994562i \(0.533210\pi\)
\(648\) 0 0
\(649\) −6.96421e8 −0.100004
\(650\) 5.67037e9 0.809869
\(651\) 0 0
\(652\) 8.77524e8 0.123992
\(653\) −3.13170e9 −0.440133 −0.220066 0.975485i \(-0.570627\pi\)
−0.220066 + 0.975485i \(0.570627\pi\)
\(654\) 0 0
\(655\) 2.15763e9 0.300008
\(656\) 7.23773e9 1.00101
\(657\) 0 0
\(658\) 4.68671e9 0.641323
\(659\) −8.36231e9 −1.13822 −0.569111 0.822260i \(-0.692713\pi\)
−0.569111 + 0.822260i \(0.692713\pi\)
\(660\) 0 0
\(661\) 1.18208e10 1.59200 0.795999 0.605298i \(-0.206946\pi\)
0.795999 + 0.605298i \(0.206946\pi\)
\(662\) 4.71150e9 0.631185
\(663\) 0 0
\(664\) −2.94378e8 −0.0390227
\(665\) −3.06864e9 −0.404641
\(666\) 0 0
\(667\) −1.74516e10 −2.27717
\(668\) 3.16034e9 0.410219
\(669\) 0 0
\(670\) −4.55647e9 −0.585284
\(671\) −9.69452e9 −1.23879
\(672\) 0 0
\(673\) 2.34562e9 0.296624 0.148312 0.988941i \(-0.452616\pi\)
0.148312 + 0.988941i \(0.452616\pi\)
\(674\) 1.96142e10 2.46753
\(675\) 0 0
\(676\) −1.11597e9 −0.138944
\(677\) 1.57414e10 1.94977 0.974884 0.222713i \(-0.0714912\pi\)
0.974884 + 0.222713i \(0.0714912\pi\)
\(678\) 0 0
\(679\) −1.80915e9 −0.221784
\(680\) 8.71736e8 0.106317
\(681\) 0 0
\(682\) −1.92572e9 −0.232459
\(683\) 6.44477e9 0.773989 0.386994 0.922082i \(-0.373513\pi\)
0.386994 + 0.922082i \(0.373513\pi\)
\(684\) 0 0
\(685\) 8.55688e9 1.01718
\(686\) −1.01195e10 −1.19681
\(687\) 0 0
\(688\) −1.38876e10 −1.62580
\(689\) 2.08362e9 0.242690
\(690\) 0 0
\(691\) 1.59645e10 1.84070 0.920349 0.391097i \(-0.127904\pi\)
0.920349 + 0.391097i \(0.127904\pi\)
\(692\) 4.97943e9 0.571227
\(693\) 0 0
\(694\) 3.16997e9 0.359995
\(695\) −2.13717e9 −0.241486
\(696\) 0 0
\(697\) −4.59192e9 −0.513664
\(698\) 9.43933e9 1.05062
\(699\) 0 0
\(700\) −5.07034e9 −0.558719
\(701\) −1.28238e10 −1.40606 −0.703028 0.711162i \(-0.748169\pi\)
−0.703028 + 0.711162i \(0.748169\pi\)
\(702\) 0 0
\(703\) −5.77868e8 −0.0627314
\(704\) −3.53028e9 −0.381333
\(705\) 0 0
\(706\) 1.21955e10 1.30432
\(707\) −4.85295e9 −0.516462
\(708\) 0 0
\(709\) 4.26207e8 0.0449117 0.0224558 0.999748i \(-0.492851\pi\)
0.0224558 + 0.999748i \(0.492851\pi\)
\(710\) 8.59365e9 0.901101
\(711\) 0 0
\(712\) −2.36859e9 −0.245929
\(713\) 3.89573e9 0.402509
\(714\) 0 0
\(715\) 3.88732e9 0.397722
\(716\) 6.12108e9 0.623207
\(717\) 0 0
\(718\) −7.37674e9 −0.743754
\(719\) −9.93256e9 −0.996576 −0.498288 0.867012i \(-0.666038\pi\)
−0.498288 + 0.867012i \(0.666038\pi\)
\(720\) 0 0
\(721\) −2.11942e10 −2.10593
\(722\) −7.68288e9 −0.759703
\(723\) 0 0
\(724\) 6.19627e9 0.606800
\(725\) −8.88670e9 −0.866079
\(726\) 0 0
\(727\) −7.03816e8 −0.0679342 −0.0339671 0.999423i \(-0.510814\pi\)
−0.0339671 + 0.999423i \(0.510814\pi\)
\(728\) 3.13027e9 0.300692
\(729\) 0 0
\(730\) 6.65125e9 0.632810
\(731\) 8.81085e9 0.834271
\(732\) 0 0
\(733\) 1.24906e10 1.17144 0.585718 0.810515i \(-0.300813\pi\)
0.585718 + 0.810515i \(0.300813\pi\)
\(734\) −1.00191e10 −0.935174
\(735\) 0 0
\(736\) 2.40851e10 2.22678
\(737\) 6.42218e9 0.590944
\(738\) 0 0
\(739\) −8.92221e9 −0.813237 −0.406618 0.913598i \(-0.633292\pi\)
−0.406618 + 0.913598i \(0.633292\pi\)
\(740\) 4.64413e8 0.0421302
\(741\) 0 0
\(742\) −4.28666e9 −0.385217
\(743\) −4.73069e9 −0.423120 −0.211560 0.977365i \(-0.567854\pi\)
−0.211560 + 0.977365i \(0.567854\pi\)
\(744\) 0 0
\(745\) −2.57064e9 −0.227769
\(746\) −2.27026e10 −2.00212
\(747\) 0 0
\(748\) 4.08500e9 0.356892
\(749\) 1.78853e9 0.155528
\(750\) 0 0
\(751\) 6.35674e9 0.547640 0.273820 0.961781i \(-0.411713\pi\)
0.273820 + 0.961781i \(0.411713\pi\)
\(752\) −6.13100e9 −0.525738
\(753\) 0 0
\(754\) −1.82405e10 −1.54967
\(755\) 8.28403e9 0.700531
\(756\) 0 0
\(757\) −3.90470e9 −0.327154 −0.163577 0.986531i \(-0.552303\pi\)
−0.163577 + 0.986531i \(0.552303\pi\)
\(758\) −1.76801e10 −1.47449
\(759\) 0 0
\(760\) 1.39401e9 0.115191
\(761\) −2.13395e10 −1.75525 −0.877623 0.479351i \(-0.840872\pi\)
−0.877623 + 0.479351i \(0.840872\pi\)
\(762\) 0 0
\(763\) 5.12750e9 0.417898
\(764\) 7.07356e9 0.573867
\(765\) 0 0
\(766\) −4.65853e9 −0.374497
\(767\) 1.47255e9 0.117838
\(768\) 0 0
\(769\) −1.05289e9 −0.0834910 −0.0417455 0.999128i \(-0.513292\pi\)
−0.0417455 + 0.999128i \(0.513292\pi\)
\(770\) −7.99743e9 −0.631296
\(771\) 0 0
\(772\) −3.62844e9 −0.283830
\(773\) −5.93967e9 −0.462524 −0.231262 0.972891i \(-0.574285\pi\)
−0.231262 + 0.972891i \(0.574285\pi\)
\(774\) 0 0
\(775\) 1.98378e9 0.153087
\(776\) 8.21856e8 0.0631364
\(777\) 0 0
\(778\) 2.73774e10 2.08432
\(779\) −7.34305e9 −0.556539
\(780\) 0 0
\(781\) −1.21124e10 −0.909815
\(782\) −1.90135e10 −1.42180
\(783\) 0 0
\(784\) −2.65340e9 −0.196651
\(785\) 1.30521e10 0.963023
\(786\) 0 0
\(787\) 1.31636e10 0.962640 0.481320 0.876545i \(-0.340158\pi\)
0.481320 + 0.876545i \(0.340158\pi\)
\(788\) −1.72562e10 −1.25633
\(789\) 0 0
\(790\) 1.09875e10 0.792875
\(791\) −2.39952e10 −1.72388
\(792\) 0 0
\(793\) 2.04986e10 1.45972
\(794\) 9.90377e9 0.702148
\(795\) 0 0
\(796\) 8.03446e9 0.564626
\(797\) 1.29607e10 0.906825 0.453412 0.891301i \(-0.350206\pi\)
0.453412 + 0.891301i \(0.350206\pi\)
\(798\) 0 0
\(799\) 3.88977e9 0.269780
\(800\) 1.22646e10 0.846912
\(801\) 0 0
\(802\) −1.16491e10 −0.797410
\(803\) −9.37471e9 −0.638930
\(804\) 0 0
\(805\) 1.61788e10 1.09310
\(806\) 4.07184e9 0.273916
\(807\) 0 0
\(808\) 2.20459e9 0.147024
\(809\) 1.62295e10 1.07767 0.538836 0.842411i \(-0.318864\pi\)
0.538836 + 0.842411i \(0.318864\pi\)
\(810\) 0 0
\(811\) 8.71515e9 0.573723 0.286861 0.957972i \(-0.407388\pi\)
0.286861 + 0.957972i \(0.407388\pi\)
\(812\) 1.63103e10 1.06910
\(813\) 0 0
\(814\) −1.50603e9 −0.0978696
\(815\) −1.42585e9 −0.0922618
\(816\) 0 0
\(817\) 1.40897e10 0.903906
\(818\) 1.63855e10 1.04670
\(819\) 0 0
\(820\) 5.90136e9 0.373769
\(821\) −2.43096e10 −1.53312 −0.766561 0.642171i \(-0.778034\pi\)
−0.766561 + 0.642171i \(0.778034\pi\)
\(822\) 0 0
\(823\) −1.50334e10 −0.940062 −0.470031 0.882650i \(-0.655757\pi\)
−0.470031 + 0.882650i \(0.655757\pi\)
\(824\) 9.62805e9 0.599506
\(825\) 0 0
\(826\) −3.02949e9 −0.187042
\(827\) 7.68810e9 0.472661 0.236331 0.971673i \(-0.424055\pi\)
0.236331 + 0.971673i \(0.424055\pi\)
\(828\) 0 0
\(829\) 4.74776e9 0.289433 0.144716 0.989473i \(-0.453773\pi\)
0.144716 + 0.989473i \(0.453773\pi\)
\(830\) −1.59028e9 −0.0965381
\(831\) 0 0
\(832\) 7.46461e9 0.449340
\(833\) 1.68343e9 0.100911
\(834\) 0 0
\(835\) −5.13509e9 −0.305243
\(836\) 6.53242e9 0.386680
\(837\) 0 0
\(838\) −2.81299e10 −1.65125
\(839\) 1.32986e10 0.777391 0.388696 0.921366i \(-0.372926\pi\)
0.388696 + 0.921366i \(0.372926\pi\)
\(840\) 0 0
\(841\) 1.13370e10 0.657221
\(842\) 3.92973e10 2.26867
\(843\) 0 0
\(844\) 1.03251e9 0.0591150
\(845\) 1.81329e9 0.103388
\(846\) 0 0
\(847\) −7.83179e9 −0.442863
\(848\) 5.60768e9 0.315790
\(849\) 0 0
\(850\) −9.68205e9 −0.540756
\(851\) 3.04670e9 0.169463
\(852\) 0 0
\(853\) 5.80154e9 0.320053 0.160027 0.987113i \(-0.448842\pi\)
0.160027 + 0.987113i \(0.448842\pi\)
\(854\) −4.21720e10 −2.31698
\(855\) 0 0
\(856\) −8.12489e8 −0.0442751
\(857\) 2.65557e10 1.44120 0.720601 0.693350i \(-0.243866\pi\)
0.720601 + 0.693350i \(0.243866\pi\)
\(858\) 0 0
\(859\) 4.83329e9 0.260176 0.130088 0.991502i \(-0.458474\pi\)
0.130088 + 0.991502i \(0.458474\pi\)
\(860\) −1.13234e10 −0.607060
\(861\) 0 0
\(862\) −2.32975e10 −1.23889
\(863\) −9.78741e9 −0.518358 −0.259179 0.965829i \(-0.583452\pi\)
−0.259179 + 0.965829i \(0.583452\pi\)
\(864\) 0 0
\(865\) −8.09085e9 −0.425048
\(866\) −2.60679e10 −1.36393
\(867\) 0 0
\(868\) −3.64096e9 −0.188972
\(869\) −1.54865e10 −0.800543
\(870\) 0 0
\(871\) −1.35794e10 −0.696333
\(872\) −2.32931e9 −0.118965
\(873\) 0 0
\(874\) −3.04050e10 −1.54048
\(875\) 2.04843e10 1.03369
\(876\) 0 0
\(877\) −1.33028e10 −0.665956 −0.332978 0.942935i \(-0.608053\pi\)
−0.332978 + 0.942935i \(0.608053\pi\)
\(878\) −4.72767e10 −2.35731
\(879\) 0 0
\(880\) 1.04620e10 0.517518
\(881\) −1.14650e10 −0.564885 −0.282443 0.959284i \(-0.591145\pi\)
−0.282443 + 0.959284i \(0.591145\pi\)
\(882\) 0 0
\(883\) −1.56517e10 −0.765067 −0.382533 0.923942i \(-0.624948\pi\)
−0.382533 + 0.923942i \(0.624948\pi\)
\(884\) −8.63753e9 −0.420540
\(885\) 0 0
\(886\) −2.82847e10 −1.36626
\(887\) −6.19771e9 −0.298194 −0.149097 0.988823i \(-0.547637\pi\)
−0.149097 + 0.988823i \(0.547637\pi\)
\(888\) 0 0
\(889\) −2.65197e10 −1.26594
\(890\) −1.27955e10 −0.608405
\(891\) 0 0
\(892\) −2.48859e10 −1.17402
\(893\) 6.22022e9 0.292298
\(894\) 0 0
\(895\) −9.94587e9 −0.463727
\(896\) 1.39235e10 0.646653
\(897\) 0 0
\(898\) −3.53577e10 −1.62936
\(899\) −6.38144e9 −0.292927
\(900\) 0 0
\(901\) −3.55775e9 −0.162046
\(902\) −1.91373e10 −0.868277
\(903\) 0 0
\(904\) 1.09005e10 0.490745
\(905\) −1.00680e10 −0.451518
\(906\) 0 0
\(907\) −2.84672e10 −1.26683 −0.633417 0.773811i \(-0.718348\pi\)
−0.633417 + 0.773811i \(0.718348\pi\)
\(908\) −4.50663e7 −0.00199780
\(909\) 0 0
\(910\) 1.69102e10 0.743881
\(911\) −1.41611e10 −0.620558 −0.310279 0.950646i \(-0.600422\pi\)
−0.310279 + 0.950646i \(0.600422\pi\)
\(912\) 0 0
\(913\) 2.24144e9 0.0974717
\(914\) −2.85398e9 −0.123634
\(915\) 0 0
\(916\) −1.82190e10 −0.783231
\(917\) −1.32291e10 −0.566549
\(918\) 0 0
\(919\) 1.43616e10 0.610379 0.305189 0.952292i \(-0.401280\pi\)
0.305189 + 0.952292i \(0.401280\pi\)
\(920\) −7.34968e9 −0.311179
\(921\) 0 0
\(922\) 3.12374e10 1.31255
\(923\) 2.56112e10 1.07207
\(924\) 0 0
\(925\) 1.55144e9 0.0644522
\(926\) 4.52709e10 1.87362
\(927\) 0 0
\(928\) −3.94529e10 −1.62055
\(929\) −2.83698e10 −1.16092 −0.580458 0.814290i \(-0.697127\pi\)
−0.580458 + 0.814290i \(0.697127\pi\)
\(930\) 0 0
\(931\) 2.69201e9 0.109334
\(932\) 3.06053e10 1.23834
\(933\) 0 0
\(934\) 2.00185e10 0.803928
\(935\) −6.63753e9 −0.265562
\(936\) 0 0
\(937\) −4.33315e10 −1.72074 −0.860370 0.509669i \(-0.829768\pi\)
−0.860370 + 0.509669i \(0.829768\pi\)
\(938\) 2.79370e10 1.10527
\(939\) 0 0
\(940\) −4.99898e9 −0.196306
\(941\) −3.82084e10 −1.49484 −0.747421 0.664351i \(-0.768708\pi\)
−0.747421 + 0.664351i \(0.768708\pi\)
\(942\) 0 0
\(943\) 3.87148e10 1.50344
\(944\) 3.96309e9 0.153332
\(945\) 0 0
\(946\) 3.67202e10 1.41022
\(947\) −1.03534e10 −0.396148 −0.198074 0.980187i \(-0.563469\pi\)
−0.198074 + 0.980187i \(0.563469\pi\)
\(948\) 0 0
\(949\) 1.98224e10 0.752877
\(950\) −1.54828e10 −0.585891
\(951\) 0 0
\(952\) −5.34487e9 −0.200774
\(953\) −2.92299e10 −1.09396 −0.546981 0.837145i \(-0.684223\pi\)
−0.546981 + 0.837145i \(0.684223\pi\)
\(954\) 0 0
\(955\) −1.14935e10 −0.427013
\(956\) 1.42077e10 0.525921
\(957\) 0 0
\(958\) −3.73828e10 −1.37370
\(959\) −5.24648e10 −1.92089
\(960\) 0 0
\(961\) −2.60881e10 −0.948223
\(962\) 3.18443e9 0.115324
\(963\) 0 0
\(964\) −9.51315e9 −0.342022
\(965\) 5.89568e9 0.211197
\(966\) 0 0
\(967\) 4.59588e9 0.163447 0.0817233 0.996655i \(-0.473958\pi\)
0.0817233 + 0.996655i \(0.473958\pi\)
\(968\) 3.55781e9 0.126072
\(969\) 0 0
\(970\) 4.43980e9 0.156193
\(971\) 5.52070e10 1.93520 0.967602 0.252481i \(-0.0812466\pi\)
0.967602 + 0.252481i \(0.0812466\pi\)
\(972\) 0 0
\(973\) 1.31036e10 0.456033
\(974\) −2.10531e10 −0.730063
\(975\) 0 0
\(976\) 5.51681e10 1.89939
\(977\) 4.44629e9 0.152534 0.0762671 0.997087i \(-0.475700\pi\)
0.0762671 + 0.997087i \(0.475700\pi\)
\(978\) 0 0
\(979\) 1.80348e10 0.614288
\(980\) −2.16348e9 −0.0734280
\(981\) 0 0
\(982\) −2.98328e10 −1.00532
\(983\) −3.90750e10 −1.31208 −0.656041 0.754725i \(-0.727770\pi\)
−0.656041 + 0.754725i \(0.727770\pi\)
\(984\) 0 0
\(985\) 2.80387e10 0.934828
\(986\) 3.11454e10 1.03472
\(987\) 0 0
\(988\) −1.38125e10 −0.455641
\(989\) −7.42850e10 −2.44182
\(990\) 0 0
\(991\) −1.52795e10 −0.498713 −0.249357 0.968412i \(-0.580219\pi\)
−0.249357 + 0.968412i \(0.580219\pi\)
\(992\) 8.80708e9 0.286445
\(993\) 0 0
\(994\) −5.26902e10 −1.70168
\(995\) −1.30548e10 −0.420137
\(996\) 0 0
\(997\) 1.22551e10 0.391637 0.195818 0.980640i \(-0.437264\pi\)
0.195818 + 0.980640i \(0.437264\pi\)
\(998\) 5.27883e10 1.68105
\(999\) 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.1.13 16
3.2 odd 2 177.8.a.a.1.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.4 16 3.2 odd 2
531.8.a.b.1.13 16 1.1 even 1 trivial