Properties

Label 531.8.a.b.1.4
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 1.4
Root \(-14.0604\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.0604 q^{2} +69.6949 q^{4} -153.219 q^{5} -215.221 q^{7} +819.793 q^{8} +O(q^{10})\) \(q-14.0604 q^{2} +69.6949 q^{4} -153.219 q^{5} -215.221 q^{7} +819.793 q^{8} +2154.32 q^{10} -7869.22 q^{11} -14239.1 q^{13} +3026.10 q^{14} -20447.6 q^{16} +14439.7 q^{17} +52274.2 q^{19} -10678.6 q^{20} +110644. q^{22} +20917.2 q^{23} -54648.9 q^{25} +200207. q^{26} -14999.8 q^{28} +136874. q^{29} +9325.59 q^{31} +182567. q^{32} -203028. q^{34} +32976.0 q^{35} -525467. q^{37} -734996. q^{38} -125608. q^{40} -197767. q^{41} +410661. q^{43} -548444. q^{44} -294105. q^{46} +946517. q^{47} -777223. q^{49} +768386. q^{50} -992389. q^{52} +1.55829e6 q^{53} +1.20571e6 q^{55} -176437. q^{56} -1.92450e6 q^{58} -205379. q^{59} -659307. q^{61} -131121. q^{62} +50317.0 q^{64} +2.18169e6 q^{65} -2.30296e6 q^{67} +1.00637e6 q^{68} -463656. q^{70} +1.68822e6 q^{71} +2.53442e6 q^{73} +7.38828e6 q^{74} +3.64324e6 q^{76} +1.69362e6 q^{77} -1.26627e6 q^{79} +3.13296e6 q^{80} +2.78068e6 q^{82} -2.22258e6 q^{83} -2.21243e6 q^{85} -5.77406e6 q^{86} -6.45113e6 q^{88} +1.04798e7 q^{89} +3.06455e6 q^{91} +1.45782e6 q^{92} -1.33084e7 q^{94} -8.00940e6 q^{95} +5.41316e6 q^{97} +1.09281e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.0604 −1.24278 −0.621388 0.783503i \(-0.713431\pi\)
−0.621388 + 0.783503i \(0.713431\pi\)
\(3\) 0 0
\(4\) 69.6949 0.544491
\(5\) −153.219 −0.548173 −0.274087 0.961705i \(-0.588375\pi\)
−0.274087 + 0.961705i \(0.588375\pi\)
\(6\) 0 0
\(7\) −215.221 −0.237160 −0.118580 0.992944i \(-0.537834\pi\)
−0.118580 + 0.992944i \(0.537834\pi\)
\(8\) 819.793 0.566095
\(9\) 0 0
\(10\) 2154.32 0.681256
\(11\) −7869.22 −1.78261 −0.891307 0.453401i \(-0.850211\pi\)
−0.891307 + 0.453401i \(0.850211\pi\)
\(12\) 0 0
\(13\) −14239.1 −1.79754 −0.898772 0.438416i \(-0.855540\pi\)
−0.898772 + 0.438416i \(0.855540\pi\)
\(14\) 3026.10 0.294737
\(15\) 0 0
\(16\) −20447.6 −1.24802
\(17\) 14439.7 0.712830 0.356415 0.934328i \(-0.383999\pi\)
0.356415 + 0.934328i \(0.383999\pi\)
\(18\) 0 0
\(19\) 52274.2 1.74844 0.874218 0.485533i \(-0.161375\pi\)
0.874218 + 0.485533i \(0.161375\pi\)
\(20\) −10678.6 −0.298475
\(21\) 0 0
\(22\) 110644. 2.21539
\(23\) 20917.2 0.358473 0.179237 0.983806i \(-0.442637\pi\)
0.179237 + 0.983806i \(0.442637\pi\)
\(24\) 0 0
\(25\) −54648.9 −0.699506
\(26\) 200207. 2.23394
\(27\) 0 0
\(28\) −14999.8 −0.129132
\(29\) 136874. 1.04214 0.521070 0.853514i \(-0.325533\pi\)
0.521070 + 0.853514i \(0.325533\pi\)
\(30\) 0 0
\(31\) 9325.59 0.0562225 0.0281113 0.999605i \(-0.491051\pi\)
0.0281113 + 0.999605i \(0.491051\pi\)
\(32\) 182567. 0.984914
\(33\) 0 0
\(34\) −203028. −0.885888
\(35\) 32976.0 0.130005
\(36\) 0 0
\(37\) −525467. −1.70545 −0.852726 0.522358i \(-0.825052\pi\)
−0.852726 + 0.522358i \(0.825052\pi\)
\(38\) −734996. −2.17291
\(39\) 0 0
\(40\) −125608. −0.310318
\(41\) −197767. −0.448136 −0.224068 0.974574i \(-0.571934\pi\)
−0.224068 + 0.974574i \(0.571934\pi\)
\(42\) 0 0
\(43\) 410661. 0.787669 0.393835 0.919181i \(-0.371148\pi\)
0.393835 + 0.919181i \(0.371148\pi\)
\(44\) −548444. −0.970617
\(45\) 0 0
\(46\) −294105. −0.445502
\(47\) 946517. 1.32980 0.664899 0.746933i \(-0.268475\pi\)
0.664899 + 0.746933i \(0.268475\pi\)
\(48\) 0 0
\(49\) −777223. −0.943755
\(50\) 768386. 0.869329
\(51\) 0 0
\(52\) −992389. −0.978747
\(53\) 1.55829e6 1.43775 0.718873 0.695141i \(-0.244658\pi\)
0.718873 + 0.695141i \(0.244658\pi\)
\(54\) 0 0
\(55\) 1.20571e6 0.977181
\(56\) −176437. −0.134255
\(57\) 0 0
\(58\) −1.92450e6 −1.29515
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −659307. −0.371906 −0.185953 0.982559i \(-0.559537\pi\)
−0.185953 + 0.982559i \(0.559537\pi\)
\(62\) −131121. −0.0698720
\(63\) 0 0
\(64\) 50317.0 0.0239930
\(65\) 2.18169e6 0.985365
\(66\) 0 0
\(67\) −2.30296e6 −0.935457 −0.467729 0.883872i \(-0.654928\pi\)
−0.467729 + 0.883872i \(0.654928\pi\)
\(68\) 1.00637e6 0.388130
\(69\) 0 0
\(70\) −463656. −0.161567
\(71\) 1.68822e6 0.559788 0.279894 0.960031i \(-0.409701\pi\)
0.279894 + 0.960031i \(0.409701\pi\)
\(72\) 0 0
\(73\) 2.53442e6 0.762514 0.381257 0.924469i \(-0.375491\pi\)
0.381257 + 0.924469i \(0.375491\pi\)
\(74\) 7.38828e6 2.11949
\(75\) 0 0
\(76\) 3.64324e6 0.952008
\(77\) 1.69362e6 0.422765
\(78\) 0 0
\(79\) −1.26627e6 −0.288957 −0.144478 0.989508i \(-0.546150\pi\)
−0.144478 + 0.989508i \(0.546150\pi\)
\(80\) 3.13296e6 0.684131
\(81\) 0 0
\(82\) 2.78068e6 0.556932
\(83\) −2.22258e6 −0.426662 −0.213331 0.976980i \(-0.568431\pi\)
−0.213331 + 0.976980i \(0.568431\pi\)
\(84\) 0 0
\(85\) −2.21243e6 −0.390754
\(86\) −5.77406e6 −0.978896
\(87\) 0 0
\(88\) −6.45113e6 −1.00913
\(89\) 1.04798e7 1.57575 0.787876 0.615834i \(-0.211181\pi\)
0.787876 + 0.615834i \(0.211181\pi\)
\(90\) 0 0
\(91\) 3.06455e6 0.426306
\(92\) 1.45782e6 0.195186
\(93\) 0 0
\(94\) −1.33084e7 −1.65264
\(95\) −8.00940e6 −0.958446
\(96\) 0 0
\(97\) 5.41316e6 0.602212 0.301106 0.953591i \(-0.402644\pi\)
0.301106 + 0.953591i \(0.402644\pi\)
\(98\) 1.09281e7 1.17288
\(99\) 0 0
\(100\) −3.80875e6 −0.380875
\(101\) 8.00726e6 0.773320 0.386660 0.922222i \(-0.373629\pi\)
0.386660 + 0.922222i \(0.373629\pi\)
\(102\) 0 0
\(103\) 2.41569e6 0.217826 0.108913 0.994051i \(-0.465263\pi\)
0.108913 + 0.994051i \(0.465263\pi\)
\(104\) −1.16731e7 −1.01758
\(105\) 0 0
\(106\) −2.19102e7 −1.78680
\(107\) −1.50841e7 −1.19035 −0.595175 0.803596i \(-0.702917\pi\)
−0.595175 + 0.803596i \(0.702917\pi\)
\(108\) 0 0
\(109\) 1.30931e7 0.968386 0.484193 0.874961i \(-0.339113\pi\)
0.484193 + 0.874961i \(0.339113\pi\)
\(110\) −1.69528e7 −1.21442
\(111\) 0 0
\(112\) 4.40075e6 0.295981
\(113\) 8.64628e6 0.563709 0.281854 0.959457i \(-0.409050\pi\)
0.281854 + 0.959457i \(0.409050\pi\)
\(114\) 0 0
\(115\) −3.20492e6 −0.196505
\(116\) 9.53938e6 0.567437
\(117\) 0 0
\(118\) 2.88771e6 0.161796
\(119\) −3.10772e6 −0.169055
\(120\) 0 0
\(121\) 4.24374e7 2.17771
\(122\) 9.27012e6 0.462196
\(123\) 0 0
\(124\) 649946. 0.0306127
\(125\) 2.03435e7 0.931624
\(126\) 0 0
\(127\) 2.77631e7 1.20269 0.601347 0.798988i \(-0.294631\pi\)
0.601347 + 0.798988i \(0.294631\pi\)
\(128\) −2.40761e7 −1.01473
\(129\) 0 0
\(130\) −3.06755e7 −1.22459
\(131\) −2.08878e7 −0.811788 −0.405894 0.913920i \(-0.633040\pi\)
−0.405894 + 0.913920i \(0.633040\pi\)
\(132\) 0 0
\(133\) −1.12505e7 −0.414660
\(134\) 3.23805e7 1.16256
\(135\) 0 0
\(136\) 1.18375e7 0.403530
\(137\) 1.47003e7 0.488432 0.244216 0.969721i \(-0.421469\pi\)
0.244216 + 0.969721i \(0.421469\pi\)
\(138\) 0 0
\(139\) −3.38613e7 −1.06943 −0.534715 0.845033i \(-0.679581\pi\)
−0.534715 + 0.845033i \(0.679581\pi\)
\(140\) 2.29826e6 0.0707865
\(141\) 0 0
\(142\) −2.37370e7 −0.695691
\(143\) 1.12050e8 3.20433
\(144\) 0 0
\(145\) −2.09716e7 −0.571274
\(146\) −3.56349e7 −0.947634
\(147\) 0 0
\(148\) −3.66224e7 −0.928604
\(149\) 3.52382e7 0.872694 0.436347 0.899778i \(-0.356272\pi\)
0.436347 + 0.899778i \(0.356272\pi\)
\(150\) 0 0
\(151\) 4.31481e7 1.01986 0.509932 0.860215i \(-0.329671\pi\)
0.509932 + 0.860215i \(0.329671\pi\)
\(152\) 4.28541e7 0.989781
\(153\) 0 0
\(154\) −2.38130e7 −0.525402
\(155\) −1.42886e6 −0.0308197
\(156\) 0 0
\(157\) 7.00495e6 0.144463 0.0722314 0.997388i \(-0.476988\pi\)
0.0722314 + 0.997388i \(0.476988\pi\)
\(158\) 1.78043e7 0.359108
\(159\) 0 0
\(160\) −2.79728e7 −0.539904
\(161\) −4.50183e6 −0.0850156
\(162\) 0 0
\(163\) 5.15447e7 0.932240 0.466120 0.884722i \(-0.345652\pi\)
0.466120 + 0.884722i \(0.345652\pi\)
\(164\) −1.37833e7 −0.244006
\(165\) 0 0
\(166\) 3.12503e7 0.530245
\(167\) 8.79382e6 0.146107 0.0730533 0.997328i \(-0.476726\pi\)
0.0730533 + 0.997328i \(0.476726\pi\)
\(168\) 0 0
\(169\) 1.40002e8 2.23116
\(170\) 3.11077e7 0.485620
\(171\) 0 0
\(172\) 2.86210e7 0.428879
\(173\) −5.52334e7 −0.811037 −0.405518 0.914087i \(-0.632909\pi\)
−0.405518 + 0.914087i \(0.632909\pi\)
\(174\) 0 0
\(175\) 1.17616e7 0.165895
\(176\) 1.60906e8 2.22474
\(177\) 0 0
\(178\) −1.47350e8 −1.95831
\(179\) −1.14162e8 −1.48776 −0.743882 0.668311i \(-0.767018\pi\)
−0.743882 + 0.668311i \(0.767018\pi\)
\(180\) 0 0
\(181\) 7.45147e6 0.0934044 0.0467022 0.998909i \(-0.485129\pi\)
0.0467022 + 0.998909i \(0.485129\pi\)
\(182\) −4.30888e7 −0.529803
\(183\) 0 0
\(184\) 1.71478e7 0.202930
\(185\) 8.05116e7 0.934883
\(186\) 0 0
\(187\) −1.13629e8 −1.27070
\(188\) 6.59674e7 0.724063
\(189\) 0 0
\(190\) 1.12615e8 1.19113
\(191\) 2.65020e7 0.275208 0.137604 0.990487i \(-0.456060\pi\)
0.137604 + 0.990487i \(0.456060\pi\)
\(192\) 0 0
\(193\) −7.95433e7 −0.796440 −0.398220 0.917290i \(-0.630372\pi\)
−0.398220 + 0.917290i \(0.630372\pi\)
\(194\) −7.61112e7 −0.748415
\(195\) 0 0
\(196\) −5.41685e7 −0.513866
\(197\) −6.69673e7 −0.624067 −0.312033 0.950071i \(-0.601010\pi\)
−0.312033 + 0.950071i \(0.601010\pi\)
\(198\) 0 0
\(199\) −5.47720e7 −0.492689 −0.246345 0.969182i \(-0.579229\pi\)
−0.246345 + 0.969182i \(0.579229\pi\)
\(200\) −4.48008e7 −0.395987
\(201\) 0 0
\(202\) −1.12585e8 −0.961063
\(203\) −2.94581e7 −0.247154
\(204\) 0 0
\(205\) 3.03016e7 0.245656
\(206\) −3.39655e7 −0.270709
\(207\) 0 0
\(208\) 2.91154e8 2.24337
\(209\) −4.11357e8 −3.11679
\(210\) 0 0
\(211\) −6.45829e7 −0.473292 −0.236646 0.971596i \(-0.576048\pi\)
−0.236646 + 0.971596i \(0.576048\pi\)
\(212\) 1.08605e8 0.782840
\(213\) 0 0
\(214\) 2.12088e8 1.47934
\(215\) −6.29211e7 −0.431779
\(216\) 0 0
\(217\) −2.00706e6 −0.0133337
\(218\) −1.84094e8 −1.20349
\(219\) 0 0
\(220\) 8.40321e7 0.532066
\(221\) −2.05607e8 −1.28134
\(222\) 0 0
\(223\) −2.02022e8 −1.21992 −0.609961 0.792431i \(-0.708815\pi\)
−0.609961 + 0.792431i \(0.708815\pi\)
\(224\) −3.92924e7 −0.233583
\(225\) 0 0
\(226\) −1.21570e8 −0.700564
\(227\) 2.76130e8 1.56683 0.783416 0.621497i \(-0.213475\pi\)
0.783416 + 0.621497i \(0.213475\pi\)
\(228\) 0 0
\(229\) −1.40164e8 −0.771280 −0.385640 0.922649i \(-0.626019\pi\)
−0.385640 + 0.922649i \(0.626019\pi\)
\(230\) 4.50624e7 0.244212
\(231\) 0 0
\(232\) 1.12208e8 0.589951
\(233\) −8.67302e7 −0.449185 −0.224592 0.974453i \(-0.572105\pi\)
−0.224592 + 0.974453i \(0.572105\pi\)
\(234\) 0 0
\(235\) −1.45024e8 −0.728960
\(236\) −1.43139e7 −0.0708867
\(237\) 0 0
\(238\) 4.36958e7 0.210097
\(239\) −2.22648e8 −1.05494 −0.527469 0.849574i \(-0.676859\pi\)
−0.527469 + 0.849574i \(0.676859\pi\)
\(240\) 0 0
\(241\) 4.27798e8 1.96870 0.984350 0.176226i \(-0.0563888\pi\)
0.984350 + 0.176226i \(0.0563888\pi\)
\(242\) −5.96687e8 −2.70641
\(243\) 0 0
\(244\) −4.59503e7 −0.202500
\(245\) 1.19085e8 0.517341
\(246\) 0 0
\(247\) −7.44335e8 −3.14289
\(248\) 7.64505e6 0.0318273
\(249\) 0 0
\(250\) −2.86038e8 −1.15780
\(251\) −4.79689e7 −0.191471 −0.0957353 0.995407i \(-0.530520\pi\)
−0.0957353 + 0.995407i \(0.530520\pi\)
\(252\) 0 0
\(253\) −1.64602e8 −0.639019
\(254\) −3.90360e8 −1.49468
\(255\) 0 0
\(256\) 3.32079e8 1.23709
\(257\) −1.81282e8 −0.666175 −0.333087 0.942896i \(-0.608090\pi\)
−0.333087 + 0.942896i \(0.608090\pi\)
\(258\) 0 0
\(259\) 1.13092e8 0.404466
\(260\) 1.52053e8 0.536523
\(261\) 0 0
\(262\) 2.93691e8 1.00887
\(263\) 2.00422e8 0.679359 0.339680 0.940541i \(-0.389681\pi\)
0.339680 + 0.940541i \(0.389681\pi\)
\(264\) 0 0
\(265\) −2.38759e8 −0.788134
\(266\) 1.58187e8 0.515329
\(267\) 0 0
\(268\) −1.60504e8 −0.509348
\(269\) −2.62947e8 −0.823637 −0.411819 0.911266i \(-0.635106\pi\)
−0.411819 + 0.911266i \(0.635106\pi\)
\(270\) 0 0
\(271\) 3.69328e8 1.12725 0.563625 0.826031i \(-0.309406\pi\)
0.563625 + 0.826031i \(0.309406\pi\)
\(272\) −2.95256e8 −0.889627
\(273\) 0 0
\(274\) −2.06692e8 −0.607011
\(275\) 4.30044e8 1.24695
\(276\) 0 0
\(277\) −2.94313e8 −0.832014 −0.416007 0.909361i \(-0.636571\pi\)
−0.416007 + 0.909361i \(0.636571\pi\)
\(278\) 4.76104e8 1.32906
\(279\) 0 0
\(280\) 2.70335e7 0.0735951
\(281\) −4.33874e8 −1.16652 −0.583258 0.812287i \(-0.698223\pi\)
−0.583258 + 0.812287i \(0.698223\pi\)
\(282\) 0 0
\(283\) −8.12492e7 −0.213092 −0.106546 0.994308i \(-0.533979\pi\)
−0.106546 + 0.994308i \(0.533979\pi\)
\(284\) 1.17660e8 0.304800
\(285\) 0 0
\(286\) −1.57547e9 −3.98226
\(287\) 4.25636e7 0.106280
\(288\) 0 0
\(289\) −2.01835e8 −0.491873
\(290\) 2.94869e8 0.709965
\(291\) 0 0
\(292\) 1.76636e8 0.415182
\(293\) 6.42476e8 1.49218 0.746089 0.665847i \(-0.231929\pi\)
0.746089 + 0.665847i \(0.231929\pi\)
\(294\) 0 0
\(295\) 3.14680e7 0.0713661
\(296\) −4.30775e8 −0.965448
\(297\) 0 0
\(298\) −4.95463e8 −1.08456
\(299\) −2.97842e8 −0.644372
\(300\) 0 0
\(301\) −8.83829e7 −0.186804
\(302\) −6.06679e8 −1.26746
\(303\) 0 0
\(304\) −1.06888e9 −2.18208
\(305\) 1.01018e8 0.203869
\(306\) 0 0
\(307\) −6.93624e8 −1.36817 −0.684084 0.729403i \(-0.739798\pi\)
−0.684084 + 0.729403i \(0.739798\pi\)
\(308\) 1.18037e8 0.230192
\(309\) 0 0
\(310\) 2.00903e7 0.0383019
\(311\) 1.86724e8 0.351997 0.175999 0.984390i \(-0.443685\pi\)
0.175999 + 0.984390i \(0.443685\pi\)
\(312\) 0 0
\(313\) 4.50446e8 0.830304 0.415152 0.909752i \(-0.363728\pi\)
0.415152 + 0.909752i \(0.363728\pi\)
\(314\) −9.84924e7 −0.179535
\(315\) 0 0
\(316\) −8.82527e7 −0.157334
\(317\) 3.97194e8 0.700318 0.350159 0.936690i \(-0.386128\pi\)
0.350159 + 0.936690i \(0.386128\pi\)
\(318\) 0 0
\(319\) −1.07709e9 −1.85773
\(320\) −7.70953e6 −0.0131523
\(321\) 0 0
\(322\) 6.32976e7 0.105655
\(323\) 7.54822e8 1.24634
\(324\) 0 0
\(325\) 7.78149e8 1.25739
\(326\) −7.24740e8 −1.15856
\(327\) 0 0
\(328\) −1.62128e8 −0.253687
\(329\) −2.03710e8 −0.315375
\(330\) 0 0
\(331\) −7.65695e8 −1.16053 −0.580267 0.814426i \(-0.697052\pi\)
−0.580267 + 0.814426i \(0.697052\pi\)
\(332\) −1.54902e8 −0.232314
\(333\) 0 0
\(334\) −1.23645e8 −0.181578
\(335\) 3.52857e8 0.512793
\(336\) 0 0
\(337\) 1.34277e8 0.191116 0.0955581 0.995424i \(-0.469536\pi\)
0.0955581 + 0.995424i \(0.469536\pi\)
\(338\) −1.96849e9 −2.77284
\(339\) 0 0
\(340\) −1.54195e8 −0.212762
\(341\) −7.33851e7 −0.100223
\(342\) 0 0
\(343\) 3.44519e8 0.460981
\(344\) 3.36657e8 0.445896
\(345\) 0 0
\(346\) 7.76604e8 1.00794
\(347\) −1.31108e9 −1.68451 −0.842257 0.539076i \(-0.818773\pi\)
−0.842257 + 0.539076i \(0.818773\pi\)
\(348\) 0 0
\(349\) −1.04850e9 −1.32032 −0.660160 0.751125i \(-0.729511\pi\)
−0.660160 + 0.751125i \(0.729511\pi\)
\(350\) −1.65373e8 −0.206170
\(351\) 0 0
\(352\) −1.43666e9 −1.75572
\(353\) 1.06498e9 1.28863 0.644317 0.764758i \(-0.277142\pi\)
0.644317 + 0.764758i \(0.277142\pi\)
\(354\) 0 0
\(355\) −2.58667e8 −0.306861
\(356\) 7.30388e8 0.857983
\(357\) 0 0
\(358\) 1.60516e9 1.84896
\(359\) 8.86207e8 1.01089 0.505446 0.862858i \(-0.331328\pi\)
0.505446 + 0.862858i \(0.331328\pi\)
\(360\) 0 0
\(361\) 1.83872e9 2.05703
\(362\) −1.04771e8 −0.116081
\(363\) 0 0
\(364\) 2.13583e8 0.232120
\(365\) −3.88321e8 −0.417990
\(366\) 0 0
\(367\) −9.92035e8 −1.04760 −0.523801 0.851841i \(-0.675486\pi\)
−0.523801 + 0.851841i \(0.675486\pi\)
\(368\) −4.27707e8 −0.447382
\(369\) 0 0
\(370\) −1.13203e9 −1.16185
\(371\) −3.35377e8 −0.340976
\(372\) 0 0
\(373\) 9.84405e8 0.982184 0.491092 0.871108i \(-0.336598\pi\)
0.491092 + 0.871108i \(0.336598\pi\)
\(374\) 1.59767e9 1.57920
\(375\) 0 0
\(376\) 7.75948e8 0.752792
\(377\) −1.94895e9 −1.87329
\(378\) 0 0
\(379\) −1.03216e9 −0.973888 −0.486944 0.873433i \(-0.661888\pi\)
−0.486944 + 0.873433i \(0.661888\pi\)
\(380\) −5.58214e8 −0.521865
\(381\) 0 0
\(382\) −3.72629e8 −0.342022
\(383\) −1.11996e9 −1.01861 −0.509303 0.860587i \(-0.670097\pi\)
−0.509303 + 0.860587i \(0.670097\pi\)
\(384\) 0 0
\(385\) −2.59495e8 −0.231748
\(386\) 1.11841e9 0.989796
\(387\) 0 0
\(388\) 3.77269e8 0.327899
\(389\) 2.19361e9 1.88945 0.944725 0.327865i \(-0.106329\pi\)
0.944725 + 0.327865i \(0.106329\pi\)
\(390\) 0 0
\(391\) 3.02038e8 0.255531
\(392\) −6.37162e8 −0.534255
\(393\) 0 0
\(394\) 9.41587e8 0.775575
\(395\) 1.94017e8 0.158398
\(396\) 0 0
\(397\) −1.87970e9 −1.50772 −0.753862 0.657033i \(-0.771811\pi\)
−0.753862 + 0.657033i \(0.771811\pi\)
\(398\) 7.70117e8 0.612302
\(399\) 0 0
\(400\) 1.11744e9 0.872998
\(401\) −1.82808e9 −1.41576 −0.707881 0.706332i \(-0.750349\pi\)
−0.707881 + 0.706332i \(0.750349\pi\)
\(402\) 0 0
\(403\) −1.32788e8 −0.101062
\(404\) 5.58065e8 0.421066
\(405\) 0 0
\(406\) 4.14192e8 0.307157
\(407\) 4.13502e9 3.04016
\(408\) 0 0
\(409\) −2.05110e9 −1.48237 −0.741183 0.671303i \(-0.765735\pi\)
−0.741183 + 0.671303i \(0.765735\pi\)
\(410\) −4.26053e8 −0.305295
\(411\) 0 0
\(412\) 1.68361e8 0.118604
\(413\) 4.42019e7 0.0308756
\(414\) 0 0
\(415\) 3.40541e8 0.233884
\(416\) −2.59959e9 −1.77043
\(417\) 0 0
\(418\) 5.78385e9 3.87347
\(419\) 1.07209e8 0.0712005 0.0356002 0.999366i \(-0.488666\pi\)
0.0356002 + 0.999366i \(0.488666\pi\)
\(420\) 0 0
\(421\) −2.81280e9 −1.83718 −0.918589 0.395213i \(-0.870671\pi\)
−0.918589 + 0.395213i \(0.870671\pi\)
\(422\) 9.08061e8 0.588195
\(423\) 0 0
\(424\) 1.27747e9 0.813901
\(425\) −7.89113e8 −0.498629
\(426\) 0 0
\(427\) 1.41897e8 0.0882013
\(428\) −1.05128e9 −0.648135
\(429\) 0 0
\(430\) 8.84696e8 0.536604
\(431\) −1.62988e9 −0.980583 −0.490292 0.871558i \(-0.663110\pi\)
−0.490292 + 0.871558i \(0.663110\pi\)
\(432\) 0 0
\(433\) −8.30649e8 −0.491711 −0.245856 0.969306i \(-0.579069\pi\)
−0.245856 + 0.969306i \(0.579069\pi\)
\(434\) 2.82201e7 0.0165709
\(435\) 0 0
\(436\) 9.12520e8 0.527278
\(437\) 1.09343e9 0.626768
\(438\) 0 0
\(439\) 2.22387e9 1.25454 0.627270 0.778802i \(-0.284172\pi\)
0.627270 + 0.778802i \(0.284172\pi\)
\(440\) 9.88436e8 0.553177
\(441\) 0 0
\(442\) 2.89092e9 1.59242
\(443\) −1.30013e9 −0.710516 −0.355258 0.934768i \(-0.615607\pi\)
−0.355258 + 0.934768i \(0.615607\pi\)
\(444\) 0 0
\(445\) −1.60570e9 −0.863785
\(446\) 2.84052e9 1.51609
\(447\) 0 0
\(448\) −1.08293e7 −0.00569019
\(449\) −1.41883e9 −0.739720 −0.369860 0.929088i \(-0.620594\pi\)
−0.369860 + 0.929088i \(0.620594\pi\)
\(450\) 0 0
\(451\) 1.55627e9 0.798852
\(452\) 6.02602e8 0.306935
\(453\) 0 0
\(454\) −3.88249e9 −1.94722
\(455\) −4.69547e8 −0.233689
\(456\) 0 0
\(457\) −2.30832e9 −1.13133 −0.565664 0.824636i \(-0.691380\pi\)
−0.565664 + 0.824636i \(0.691380\pi\)
\(458\) 1.97076e9 0.958528
\(459\) 0 0
\(460\) −2.23366e8 −0.106995
\(461\) 2.65874e8 0.126393 0.0631965 0.998001i \(-0.479871\pi\)
0.0631965 + 0.998001i \(0.479871\pi\)
\(462\) 0 0
\(463\) −1.76799e9 −0.827840 −0.413920 0.910313i \(-0.635841\pi\)
−0.413920 + 0.910313i \(0.635841\pi\)
\(464\) −2.79873e9 −1.30061
\(465\) 0 0
\(466\) 1.21946e9 0.558236
\(467\) −3.74007e9 −1.69930 −0.849652 0.527344i \(-0.823188\pi\)
−0.849652 + 0.527344i \(0.823188\pi\)
\(468\) 0 0
\(469\) 4.95645e8 0.221853
\(470\) 2.03910e9 0.905933
\(471\) 0 0
\(472\) −1.68368e8 −0.0736993
\(473\) −3.23158e9 −1.40411
\(474\) 0 0
\(475\) −2.85673e9 −1.22304
\(476\) −2.16592e8 −0.0920490
\(477\) 0 0
\(478\) 3.13052e9 1.31105
\(479\) 4.49040e8 0.186686 0.0933428 0.995634i \(-0.470245\pi\)
0.0933428 + 0.995634i \(0.470245\pi\)
\(480\) 0 0
\(481\) 7.48216e9 3.06563
\(482\) −6.01502e9 −2.44665
\(483\) 0 0
\(484\) 2.95767e9 1.18574
\(485\) −8.29399e8 −0.330117
\(486\) 0 0
\(487\) −3.09706e9 −1.21506 −0.607531 0.794296i \(-0.707840\pi\)
−0.607531 + 0.794296i \(0.707840\pi\)
\(488\) −5.40495e8 −0.210534
\(489\) 0 0
\(490\) −1.67439e9 −0.642939
\(491\) 2.15480e9 0.821527 0.410763 0.911742i \(-0.365262\pi\)
0.410763 + 0.911742i \(0.365262\pi\)
\(492\) 0 0
\(493\) 1.97641e9 0.742869
\(494\) 1.04657e10 3.90591
\(495\) 0 0
\(496\) −1.90686e8 −0.0701669
\(497\) −3.63340e8 −0.132759
\(498\) 0 0
\(499\) 5.11219e9 1.84185 0.920926 0.389738i \(-0.127434\pi\)
0.920926 + 0.389738i \(0.127434\pi\)
\(500\) 1.41784e9 0.507261
\(501\) 0 0
\(502\) 6.74462e8 0.237955
\(503\) 3.11720e9 1.09214 0.546069 0.837740i \(-0.316124\pi\)
0.546069 + 0.837740i \(0.316124\pi\)
\(504\) 0 0
\(505\) −1.22687e9 −0.423913
\(506\) 2.31437e9 0.794158
\(507\) 0 0
\(508\) 1.93495e9 0.654856
\(509\) 3.33182e9 1.11988 0.559938 0.828535i \(-0.310825\pi\)
0.559938 + 0.828535i \(0.310825\pi\)
\(510\) 0 0
\(511\) −5.45460e8 −0.180838
\(512\) −1.58742e9 −0.522695
\(513\) 0 0
\(514\) 2.54889e9 0.827905
\(515\) −3.70129e8 −0.119406
\(516\) 0 0
\(517\) −7.44834e9 −2.37052
\(518\) −1.59011e9 −0.502660
\(519\) 0 0
\(520\) 1.78854e9 0.557811
\(521\) 1.20322e8 0.0372745 0.0186373 0.999826i \(-0.494067\pi\)
0.0186373 + 0.999826i \(0.494067\pi\)
\(522\) 0 0
\(523\) 1.56158e9 0.477318 0.238659 0.971103i \(-0.423292\pi\)
0.238659 + 0.971103i \(0.423292\pi\)
\(524\) −1.45577e9 −0.442012
\(525\) 0 0
\(526\) −2.81801e9 −0.844291
\(527\) 1.34658e8 0.0400771
\(528\) 0 0
\(529\) −2.96729e9 −0.871497
\(530\) 3.35705e9 0.979473
\(531\) 0 0
\(532\) −7.84103e8 −0.225779
\(533\) 2.81601e9 0.805543
\(534\) 0 0
\(535\) 2.31116e9 0.652518
\(536\) −1.88795e9 −0.529558
\(537\) 0 0
\(538\) 3.69715e9 1.02360
\(539\) 6.11614e9 1.68235
\(540\) 0 0
\(541\) 2.94601e9 0.799914 0.399957 0.916534i \(-0.369025\pi\)
0.399957 + 0.916534i \(0.369025\pi\)
\(542\) −5.19290e9 −1.40092
\(543\) 0 0
\(544\) 2.63621e9 0.702077
\(545\) −2.00611e9 −0.530843
\(546\) 0 0
\(547\) 5.07000e9 1.32450 0.662251 0.749282i \(-0.269601\pi\)
0.662251 + 0.749282i \(0.269601\pi\)
\(548\) 1.02453e9 0.265947
\(549\) 0 0
\(550\) −6.04659e9 −1.54968
\(551\) 7.15495e9 1.82212
\(552\) 0 0
\(553\) 2.72529e8 0.0685290
\(554\) 4.13816e9 1.03401
\(555\) 0 0
\(556\) −2.35996e9 −0.582295
\(557\) −7.03628e9 −1.72524 −0.862621 0.505851i \(-0.831179\pi\)
−0.862621 + 0.505851i \(0.831179\pi\)
\(558\) 0 0
\(559\) −5.84742e9 −1.41587
\(560\) −6.74279e8 −0.162249
\(561\) 0 0
\(562\) 6.10044e9 1.44972
\(563\) 4.94078e9 1.16685 0.583426 0.812166i \(-0.301712\pi\)
0.583426 + 0.812166i \(0.301712\pi\)
\(564\) 0 0
\(565\) −1.32478e9 −0.309010
\(566\) 1.14240e9 0.264825
\(567\) 0 0
\(568\) 1.38399e9 0.316893
\(569\) −2.66776e9 −0.607090 −0.303545 0.952817i \(-0.598170\pi\)
−0.303545 + 0.952817i \(0.598170\pi\)
\(570\) 0 0
\(571\) 6.77696e9 1.52338 0.761690 0.647941i \(-0.224370\pi\)
0.761690 + 0.647941i \(0.224370\pi\)
\(572\) 7.80933e9 1.74473
\(573\) 0 0
\(574\) −5.98461e8 −0.132082
\(575\) −1.14310e9 −0.250754
\(576\) 0 0
\(577\) −5.53458e9 −1.19941 −0.599707 0.800219i \(-0.704716\pi\)
−0.599707 + 0.800219i \(0.704716\pi\)
\(578\) 2.83787e9 0.611288
\(579\) 0 0
\(580\) −1.46162e9 −0.311053
\(581\) 4.78346e8 0.101187
\(582\) 0 0
\(583\) −1.22625e10 −2.56294
\(584\) 2.07770e9 0.431655
\(585\) 0 0
\(586\) −9.03347e9 −1.85444
\(587\) −5.22919e9 −1.06709 −0.533545 0.845771i \(-0.679141\pi\)
−0.533545 + 0.845771i \(0.679141\pi\)
\(588\) 0 0
\(589\) 4.87488e8 0.0983015
\(590\) −4.42452e8 −0.0886920
\(591\) 0 0
\(592\) 1.07445e10 2.12844
\(593\) −8.33930e8 −0.164224 −0.0821122 0.996623i \(-0.526167\pi\)
−0.0821122 + 0.996623i \(0.526167\pi\)
\(594\) 0 0
\(595\) 4.76162e8 0.0926714
\(596\) 2.45592e9 0.475174
\(597\) 0 0
\(598\) 4.18777e9 0.800809
\(599\) −3.80759e9 −0.723863 −0.361931 0.932205i \(-0.617883\pi\)
−0.361931 + 0.932205i \(0.617883\pi\)
\(600\) 0 0
\(601\) 8.81584e9 1.65654 0.828272 0.560326i \(-0.189324\pi\)
0.828272 + 0.560326i \(0.189324\pi\)
\(602\) 1.24270e9 0.232155
\(603\) 0 0
\(604\) 3.00720e9 0.555307
\(605\) −6.50222e9 −1.19376
\(606\) 0 0
\(607\) 3.12869e9 0.567809 0.283904 0.958853i \(-0.408370\pi\)
0.283904 + 0.958853i \(0.408370\pi\)
\(608\) 9.54357e9 1.72206
\(609\) 0 0
\(610\) −1.42036e9 −0.253363
\(611\) −1.34775e10 −2.39037
\(612\) 0 0
\(613\) 6.07990e9 1.06607 0.533034 0.846094i \(-0.321052\pi\)
0.533034 + 0.846094i \(0.321052\pi\)
\(614\) 9.75263e9 1.70033
\(615\) 0 0
\(616\) 1.38842e9 0.239325
\(617\) 1.45081e9 0.248664 0.124332 0.992241i \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(618\) 0 0
\(619\) −3.93245e9 −0.666416 −0.333208 0.942853i \(-0.608131\pi\)
−0.333208 + 0.942853i \(0.608131\pi\)
\(620\) −9.95841e7 −0.0167810
\(621\) 0 0
\(622\) −2.62542e9 −0.437454
\(623\) −2.25547e9 −0.373706
\(624\) 0 0
\(625\) 1.15244e9 0.188815
\(626\) −6.33345e9 −1.03188
\(627\) 0 0
\(628\) 4.88209e8 0.0786588
\(629\) −7.58757e9 −1.21570
\(630\) 0 0
\(631\) 9.82173e9 1.55627 0.778135 0.628097i \(-0.216166\pi\)
0.778135 + 0.628097i \(0.216166\pi\)
\(632\) −1.03808e9 −0.163577
\(633\) 0 0
\(634\) −5.58470e9 −0.870338
\(635\) −4.25383e9 −0.659284
\(636\) 0 0
\(637\) 1.10669e10 1.69644
\(638\) 1.51443e10 2.30875
\(639\) 0 0
\(640\) 3.68892e9 0.556249
\(641\) 7.96630e9 1.19469 0.597343 0.801986i \(-0.296223\pi\)
0.597343 + 0.801986i \(0.296223\pi\)
\(642\) 0 0
\(643\) 7.54836e9 1.11973 0.559866 0.828583i \(-0.310852\pi\)
0.559866 + 0.828583i \(0.310852\pi\)
\(644\) −3.13755e8 −0.0462903
\(645\) 0 0
\(646\) −1.06131e10 −1.54892
\(647\) −1.12209e10 −1.62878 −0.814388 0.580321i \(-0.802927\pi\)
−0.814388 + 0.580321i \(0.802927\pi\)
\(648\) 0 0
\(649\) 1.61617e9 0.232076
\(650\) −1.09411e10 −1.56266
\(651\) 0 0
\(652\) 3.59240e9 0.507596
\(653\) 6.06010e9 0.851694 0.425847 0.904795i \(-0.359976\pi\)
0.425847 + 0.904795i \(0.359976\pi\)
\(654\) 0 0
\(655\) 3.20041e9 0.445001
\(656\) 4.04385e9 0.559282
\(657\) 0 0
\(658\) 2.86425e9 0.391941
\(659\) 1.39437e10 1.89792 0.948960 0.315398i \(-0.102138\pi\)
0.948960 + 0.315398i \(0.102138\pi\)
\(660\) 0 0
\(661\) −8.46218e9 −1.13966 −0.569832 0.821761i \(-0.692992\pi\)
−0.569832 + 0.821761i \(0.692992\pi\)
\(662\) 1.07660e10 1.44228
\(663\) 0 0
\(664\) −1.82205e9 −0.241531
\(665\) 1.72379e9 0.227305
\(666\) 0 0
\(667\) 2.86302e9 0.373580
\(668\) 6.12884e8 0.0795538
\(669\) 0 0
\(670\) −4.96131e9 −0.637286
\(671\) 5.18823e9 0.662965
\(672\) 0 0
\(673\) 2.59986e9 0.328773 0.164387 0.986396i \(-0.447436\pi\)
0.164387 + 0.986396i \(0.447436\pi\)
\(674\) −1.88799e9 −0.237514
\(675\) 0 0
\(676\) 9.75744e9 1.21485
\(677\) −2.24286e8 −0.0277807 −0.0138903 0.999904i \(-0.504422\pi\)
−0.0138903 + 0.999904i \(0.504422\pi\)
\(678\) 0 0
\(679\) −1.16503e9 −0.142821
\(680\) −1.81374e9 −0.221204
\(681\) 0 0
\(682\) 1.03182e9 0.124555
\(683\) 1.29921e9 0.156029 0.0780146 0.996952i \(-0.475142\pi\)
0.0780146 + 0.996952i \(0.475142\pi\)
\(684\) 0 0
\(685\) −2.25236e9 −0.267745
\(686\) −4.84407e9 −0.572896
\(687\) 0 0
\(688\) −8.39702e9 −0.983027
\(689\) −2.21886e10 −2.58441
\(690\) 0 0
\(691\) −1.09890e10 −1.26703 −0.633514 0.773732i \(-0.718388\pi\)
−0.633514 + 0.773732i \(0.718388\pi\)
\(692\) −3.84949e9 −0.441602
\(693\) 0 0
\(694\) 1.84342e10 2.09347
\(695\) 5.18820e9 0.586232
\(696\) 0 0
\(697\) −2.85568e9 −0.319445
\(698\) 1.47423e10 1.64086
\(699\) 0 0
\(700\) 8.19724e8 0.0903284
\(701\) −1.12676e10 −1.23543 −0.617715 0.786402i \(-0.711941\pi\)
−0.617715 + 0.786402i \(0.711941\pi\)
\(702\) 0 0
\(703\) −2.74684e10 −2.98188
\(704\) −3.95956e8 −0.0427703
\(705\) 0 0
\(706\) −1.49740e10 −1.60148
\(707\) −1.72333e9 −0.183401
\(708\) 0 0
\(709\) −8.10979e9 −0.854571 −0.427285 0.904117i \(-0.640530\pi\)
−0.427285 + 0.904117i \(0.640530\pi\)
\(710\) 3.63696e9 0.381359
\(711\) 0 0
\(712\) 8.59127e9 0.892026
\(713\) 1.95065e8 0.0201543
\(714\) 0 0
\(715\) −1.71682e10 −1.75653
\(716\) −7.95647e9 −0.810075
\(717\) 0 0
\(718\) −1.24604e10 −1.25631
\(719\) −1.37914e10 −1.38375 −0.691873 0.722019i \(-0.743214\pi\)
−0.691873 + 0.722019i \(0.743214\pi\)
\(720\) 0 0
\(721\) −5.19907e8 −0.0516597
\(722\) −2.58532e10 −2.55643
\(723\) 0 0
\(724\) 5.19330e8 0.0508579
\(725\) −7.47999e9 −0.728984
\(726\) 0 0
\(727\) −8.25616e9 −0.796907 −0.398454 0.917189i \(-0.630453\pi\)
−0.398454 + 0.917189i \(0.630453\pi\)
\(728\) 2.51230e9 0.241330
\(729\) 0 0
\(730\) 5.45994e9 0.519467
\(731\) 5.92981e9 0.561474
\(732\) 0 0
\(733\) 1.14387e10 1.07279 0.536395 0.843967i \(-0.319786\pi\)
0.536395 + 0.843967i \(0.319786\pi\)
\(734\) 1.39484e10 1.30193
\(735\) 0 0
\(736\) 3.81881e9 0.353065
\(737\) 1.81225e10 1.66756
\(738\) 0 0
\(739\) 1.48024e10 1.34920 0.674598 0.738185i \(-0.264317\pi\)
0.674598 + 0.738185i \(0.264317\pi\)
\(740\) 5.61125e9 0.509036
\(741\) 0 0
\(742\) 4.71553e9 0.423757
\(743\) −1.21140e10 −1.08349 −0.541746 0.840542i \(-0.682236\pi\)
−0.541746 + 0.840542i \(0.682236\pi\)
\(744\) 0 0
\(745\) −5.39917e9 −0.478387
\(746\) −1.38411e10 −1.22063
\(747\) 0 0
\(748\) −7.91935e9 −0.691885
\(749\) 3.24641e9 0.282304
\(750\) 0 0
\(751\) −3.71223e9 −0.319812 −0.159906 0.987132i \(-0.551119\pi\)
−0.159906 + 0.987132i \(0.551119\pi\)
\(752\) −1.93540e10 −1.65962
\(753\) 0 0
\(754\) 2.74030e10 2.32808
\(755\) −6.61111e9 −0.559062
\(756\) 0 0
\(757\) 2.68089e9 0.224617 0.112309 0.993673i \(-0.464175\pi\)
0.112309 + 0.993673i \(0.464175\pi\)
\(758\) 1.45126e10 1.21032
\(759\) 0 0
\(760\) −6.56606e9 −0.542572
\(761\) 8.48234e9 0.697701 0.348850 0.937178i \(-0.386572\pi\)
0.348850 + 0.937178i \(0.386572\pi\)
\(762\) 0 0
\(763\) −2.81791e9 −0.229663
\(764\) 1.84705e9 0.149849
\(765\) 0 0
\(766\) 1.57471e10 1.26590
\(767\) 2.92440e9 0.234020
\(768\) 0 0
\(769\) 4.34767e9 0.344758 0.172379 0.985031i \(-0.444855\pi\)
0.172379 + 0.985031i \(0.444855\pi\)
\(770\) 3.64861e9 0.288011
\(771\) 0 0
\(772\) −5.54376e9 −0.433655
\(773\) 1.06536e10 0.829601 0.414800 0.909912i \(-0.363851\pi\)
0.414800 + 0.909912i \(0.363851\pi\)
\(774\) 0 0
\(775\) −5.09633e8 −0.0393280
\(776\) 4.43767e9 0.340910
\(777\) 0 0
\(778\) −3.08430e10 −2.34816
\(779\) −1.03381e10 −0.783537
\(780\) 0 0
\(781\) −1.32849e10 −0.997886
\(782\) −4.24678e9 −0.317567
\(783\) 0 0
\(784\) 1.58923e10 1.17783
\(785\) −1.07329e9 −0.0791907
\(786\) 0 0
\(787\) 2.49085e9 0.182153 0.0910763 0.995844i \(-0.470969\pi\)
0.0910763 + 0.995844i \(0.470969\pi\)
\(788\) −4.66728e9 −0.339799
\(789\) 0 0
\(790\) −2.72796e9 −0.196853
\(791\) −1.86086e9 −0.133689
\(792\) 0 0
\(793\) 9.38791e9 0.668517
\(794\) 2.64293e10 1.87376
\(795\) 0 0
\(796\) −3.81733e9 −0.268265
\(797\) −6.71594e9 −0.469897 −0.234949 0.972008i \(-0.575492\pi\)
−0.234949 + 0.972008i \(0.575492\pi\)
\(798\) 0 0
\(799\) 1.36674e10 0.947920
\(800\) −9.97712e9 −0.688954
\(801\) 0 0
\(802\) 2.57036e10 1.75947
\(803\) −1.99439e10 −1.35927
\(804\) 0 0
\(805\) 6.89766e8 0.0466033
\(806\) 1.86705e9 0.125598
\(807\) 0 0
\(808\) 6.56430e9 0.437773
\(809\) 5.26318e9 0.349485 0.174743 0.984614i \(-0.444091\pi\)
0.174743 + 0.984614i \(0.444091\pi\)
\(810\) 0 0
\(811\) 1.05418e10 0.693971 0.346985 0.937870i \(-0.387205\pi\)
0.346985 + 0.937870i \(0.387205\pi\)
\(812\) −2.05308e9 −0.134573
\(813\) 0 0
\(814\) −5.81400e10 −3.77824
\(815\) −7.89763e9 −0.511029
\(816\) 0 0
\(817\) 2.14670e10 1.37719
\(818\) 2.88393e10 1.84225
\(819\) 0 0
\(820\) 2.11187e9 0.133757
\(821\) −2.59089e10 −1.63399 −0.816994 0.576646i \(-0.804361\pi\)
−0.816994 + 0.576646i \(0.804361\pi\)
\(822\) 0 0
\(823\) −8.31491e9 −0.519946 −0.259973 0.965616i \(-0.583714\pi\)
−0.259973 + 0.965616i \(0.583714\pi\)
\(824\) 1.98036e9 0.123310
\(825\) 0 0
\(826\) −6.21497e8 −0.0383715
\(827\) 6.36168e9 0.391113 0.195557 0.980692i \(-0.437349\pi\)
0.195557 + 0.980692i \(0.437349\pi\)
\(828\) 0 0
\(829\) −2.31191e10 −1.40938 −0.704692 0.709513i \(-0.748915\pi\)
−0.704692 + 0.709513i \(0.748915\pi\)
\(830\) −4.78815e9 −0.290666
\(831\) 0 0
\(832\) −7.16467e8 −0.0431285
\(833\) −1.12228e10 −0.672737
\(834\) 0 0
\(835\) −1.34738e9 −0.0800918
\(836\) −2.86695e10 −1.69706
\(837\) 0 0
\(838\) −1.50740e9 −0.0884862
\(839\) −1.96736e10 −1.15005 −0.575026 0.818135i \(-0.695008\pi\)
−0.575026 + 0.818135i \(0.695008\pi\)
\(840\) 0 0
\(841\) 1.48448e9 0.0860574
\(842\) 3.95491e10 2.28320
\(843\) 0 0
\(844\) −4.50110e9 −0.257703
\(845\) −2.14510e10 −1.22306
\(846\) 0 0
\(847\) −9.13343e9 −0.516466
\(848\) −3.18632e10 −1.79434
\(849\) 0 0
\(850\) 1.10952e10 0.619684
\(851\) −1.09913e10 −0.611359
\(852\) 0 0
\(853\) −2.05583e10 −1.13413 −0.567067 0.823671i \(-0.691922\pi\)
−0.567067 + 0.823671i \(0.691922\pi\)
\(854\) −1.99513e9 −0.109614
\(855\) 0 0
\(856\) −1.23658e10 −0.673852
\(857\) 1.22990e10 0.667476 0.333738 0.942666i \(-0.391690\pi\)
0.333738 + 0.942666i \(0.391690\pi\)
\(858\) 0 0
\(859\) 6.11395e9 0.329114 0.164557 0.986368i \(-0.447381\pi\)
0.164557 + 0.986368i \(0.447381\pi\)
\(860\) −4.38528e9 −0.235100
\(861\) 0 0
\(862\) 2.29167e10 1.21864
\(863\) 5.85317e9 0.309994 0.154997 0.987915i \(-0.450463\pi\)
0.154997 + 0.987915i \(0.450463\pi\)
\(864\) 0 0
\(865\) 8.46281e9 0.444589
\(866\) 1.16793e10 0.611087
\(867\) 0 0
\(868\) −1.39882e8 −0.00726011
\(869\) 9.96458e9 0.515098
\(870\) 0 0
\(871\) 3.27919e10 1.68153
\(872\) 1.07336e10 0.548199
\(873\) 0 0
\(874\) −1.53741e10 −0.778932
\(875\) −4.37835e9 −0.220944
\(876\) 0 0
\(877\) −5.35250e9 −0.267953 −0.133976 0.990985i \(-0.542775\pi\)
−0.133976 + 0.990985i \(0.542775\pi\)
\(878\) −3.12686e10 −1.55911
\(879\) 0 0
\(880\) −2.46539e10 −1.21954
\(881\) 9.28115e9 0.457284 0.228642 0.973511i \(-0.426571\pi\)
0.228642 + 0.973511i \(0.426571\pi\)
\(882\) 0 0
\(883\) −1.19702e10 −0.585114 −0.292557 0.956248i \(-0.594506\pi\)
−0.292557 + 0.956248i \(0.594506\pi\)
\(884\) −1.43298e10 −0.697680
\(885\) 0 0
\(886\) 1.82804e10 0.883012
\(887\) −2.48052e10 −1.19347 −0.596734 0.802439i \(-0.703535\pi\)
−0.596734 + 0.802439i \(0.703535\pi\)
\(888\) 0 0
\(889\) −5.97521e9 −0.285231
\(890\) 2.25769e10 1.07349
\(891\) 0 0
\(892\) −1.40799e10 −0.664237
\(893\) 4.94784e10 2.32507
\(894\) 0 0
\(895\) 1.74917e10 0.815552
\(896\) 5.18169e9 0.240654
\(897\) 0 0
\(898\) 1.99493e10 0.919306
\(899\) 1.27643e9 0.0585918
\(900\) 0 0
\(901\) 2.25012e10 1.02487
\(902\) −2.18818e10 −0.992794
\(903\) 0 0
\(904\) 7.08816e9 0.319113
\(905\) −1.14171e9 −0.0512018
\(906\) 0 0
\(907\) 9.90452e9 0.440766 0.220383 0.975413i \(-0.429269\pi\)
0.220383 + 0.975413i \(0.429269\pi\)
\(908\) 1.92448e10 0.853127
\(909\) 0 0
\(910\) 6.60202e9 0.290424
\(911\) 8.15910e8 0.0357543 0.0178771 0.999840i \(-0.494309\pi\)
0.0178771 + 0.999840i \(0.494309\pi\)
\(912\) 0 0
\(913\) 1.74899e10 0.760573
\(914\) 3.24559e10 1.40599
\(915\) 0 0
\(916\) −9.76871e9 −0.419955
\(917\) 4.49549e9 0.192524
\(918\) 0 0
\(919\) −3.65375e9 −0.155287 −0.0776434 0.996981i \(-0.524740\pi\)
−0.0776434 + 0.996981i \(0.524740\pi\)
\(920\) −2.62737e9 −0.111241
\(921\) 0 0
\(922\) −3.73830e9 −0.157078
\(923\) −2.40386e10 −1.00624
\(924\) 0 0
\(925\) 2.87162e10 1.19297
\(926\) 2.48587e10 1.02882
\(927\) 0 0
\(928\) 2.49886e10 1.02642
\(929\) −3.46180e10 −1.41660 −0.708299 0.705913i \(-0.750537\pi\)
−0.708299 + 0.705913i \(0.750537\pi\)
\(930\) 0 0
\(931\) −4.06287e10 −1.65010
\(932\) −6.04465e9 −0.244577
\(933\) 0 0
\(934\) 5.25869e10 2.11185
\(935\) 1.74101e10 0.696564
\(936\) 0 0
\(937\) −1.64978e10 −0.655143 −0.327572 0.944826i \(-0.606230\pi\)
−0.327572 + 0.944826i \(0.606230\pi\)
\(938\) −6.96897e9 −0.275714
\(939\) 0 0
\(940\) −1.01075e10 −0.396912
\(941\) 2.07806e10 0.813009 0.406504 0.913649i \(-0.366748\pi\)
0.406504 + 0.913649i \(0.366748\pi\)
\(942\) 0 0
\(943\) −4.13673e9 −0.160645
\(944\) 4.19950e9 0.162478
\(945\) 0 0
\(946\) 4.54373e10 1.74499
\(947\) 3.39662e10 1.29964 0.649818 0.760090i \(-0.274845\pi\)
0.649818 + 0.760090i \(0.274845\pi\)
\(948\) 0 0
\(949\) −3.60877e10 −1.37065
\(950\) 4.01668e10 1.51997
\(951\) 0 0
\(952\) −2.54769e9 −0.0957012
\(953\) −1.25066e9 −0.0468074 −0.0234037 0.999726i \(-0.507450\pi\)
−0.0234037 + 0.999726i \(0.507450\pi\)
\(954\) 0 0
\(955\) −4.06061e9 −0.150862
\(956\) −1.55174e10 −0.574404
\(957\) 0 0
\(958\) −6.31369e9 −0.232008
\(959\) −3.16381e9 −0.115837
\(960\) 0 0
\(961\) −2.74256e10 −0.996839
\(962\) −1.05202e11 −3.80989
\(963\) 0 0
\(964\) 2.98154e10 1.07194
\(965\) 1.21876e10 0.436587
\(966\) 0 0
\(967\) 1.99659e10 0.710063 0.355032 0.934854i \(-0.384470\pi\)
0.355032 + 0.934854i \(0.384470\pi\)
\(968\) 3.47899e10 1.23279
\(969\) 0 0
\(970\) 1.16617e10 0.410261
\(971\) 5.02413e10 1.76114 0.880569 0.473919i \(-0.157161\pi\)
0.880569 + 0.473919i \(0.157161\pi\)
\(972\) 0 0
\(973\) 7.28767e9 0.253626
\(974\) 4.35459e10 1.51005
\(975\) 0 0
\(976\) 1.34812e10 0.464146
\(977\) 5.76202e10 1.97671 0.988357 0.152155i \(-0.0486213\pi\)
0.988357 + 0.152155i \(0.0486213\pi\)
\(978\) 0 0
\(979\) −8.24678e10 −2.80896
\(980\) 8.29964e9 0.281688
\(981\) 0 0
\(982\) −3.02974e10 −1.02097
\(983\) −1.60520e10 −0.539003 −0.269502 0.963000i \(-0.586859\pi\)
−0.269502 + 0.963000i \(0.586859\pi\)
\(984\) 0 0
\(985\) 1.02607e10 0.342096
\(986\) −2.77891e10 −0.923220
\(987\) 0 0
\(988\) −5.18764e10 −1.71128
\(989\) 8.58989e9 0.282358
\(990\) 0 0
\(991\) −8.99460e9 −0.293578 −0.146789 0.989168i \(-0.546894\pi\)
−0.146789 + 0.989168i \(0.546894\pi\)
\(992\) 1.70255e9 0.0553744
\(993\) 0 0
\(994\) 5.10870e9 0.164990
\(995\) 8.39212e9 0.270079
\(996\) 0 0
\(997\) 1.62638e9 0.0519744 0.0259872 0.999662i \(-0.491727\pi\)
0.0259872 + 0.999662i \(0.491727\pi\)
\(998\) −7.18794e10 −2.28901
\(999\) 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.1.4 16
3.2 odd 2 177.8.a.a.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.13 16 3.2 odd 2
531.8.a.b.1.4 16 1.1 even 1 trivial