Properties

Label 8001.2.a.z.1.26
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.26
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.66842 q^{2} +0.783626 q^{4} +3.76494 q^{5} -1.00000 q^{7} -2.02942 q^{8} +O(q^{10})\) \(q+1.66842 q^{2} +0.783626 q^{4} +3.76494 q^{5} -1.00000 q^{7} -2.02942 q^{8} +6.28150 q^{10} -3.16270 q^{11} -3.02168 q^{13} -1.66842 q^{14} -4.95318 q^{16} +1.05689 q^{17} +3.14601 q^{19} +2.95030 q^{20} -5.27672 q^{22} -7.12860 q^{23} +9.17476 q^{25} -5.04143 q^{26} -0.783626 q^{28} +2.67282 q^{29} -8.88393 q^{31} -4.20514 q^{32} +1.76334 q^{34} -3.76494 q^{35} +0.943469 q^{37} +5.24887 q^{38} -7.64065 q^{40} +9.90868 q^{41} -10.8524 q^{43} -2.47838 q^{44} -11.8935 q^{46} +4.25536 q^{47} +1.00000 q^{49} +15.3074 q^{50} -2.36787 q^{52} -5.14637 q^{53} -11.9074 q^{55} +2.02942 q^{56} +4.45938 q^{58} -4.15250 q^{59} +1.23055 q^{61} -14.8221 q^{62} +2.89042 q^{64} -11.3764 q^{65} -10.7930 q^{67} +0.828207 q^{68} -6.28150 q^{70} -15.1564 q^{71} +7.29623 q^{73} +1.57410 q^{74} +2.46530 q^{76} +3.16270 q^{77} +5.09038 q^{79} -18.6484 q^{80} +16.5318 q^{82} -8.93203 q^{83} +3.97913 q^{85} -18.1064 q^{86} +6.41846 q^{88} +5.97445 q^{89} +3.02168 q^{91} -5.58616 q^{92} +7.09974 q^{94} +11.8445 q^{95} -9.75834 q^{97} +1.66842 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.66842 1.17975 0.589876 0.807494i \(-0.299177\pi\)
0.589876 + 0.807494i \(0.299177\pi\)
\(3\) 0 0
\(4\) 0.783626 0.391813
\(5\) 3.76494 1.68373 0.841866 0.539687i \(-0.181457\pi\)
0.841866 + 0.539687i \(0.181457\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.02942 −0.717509
\(9\) 0 0
\(10\) 6.28150 1.98638
\(11\) −3.16270 −0.953591 −0.476795 0.879014i \(-0.658202\pi\)
−0.476795 + 0.879014i \(0.658202\pi\)
\(12\) 0 0
\(13\) −3.02168 −0.838063 −0.419031 0.907972i \(-0.637630\pi\)
−0.419031 + 0.907972i \(0.637630\pi\)
\(14\) −1.66842 −0.445904
\(15\) 0 0
\(16\) −4.95318 −1.23830
\(17\) 1.05689 0.256333 0.128167 0.991753i \(-0.459091\pi\)
0.128167 + 0.991753i \(0.459091\pi\)
\(18\) 0 0
\(19\) 3.14601 0.721744 0.360872 0.932615i \(-0.382479\pi\)
0.360872 + 0.932615i \(0.382479\pi\)
\(20\) 2.95030 0.659708
\(21\) 0 0
\(22\) −5.27672 −1.12500
\(23\) −7.12860 −1.48642 −0.743208 0.669060i \(-0.766697\pi\)
−0.743208 + 0.669060i \(0.766697\pi\)
\(24\) 0 0
\(25\) 9.17476 1.83495
\(26\) −5.04143 −0.988705
\(27\) 0 0
\(28\) −0.783626 −0.148091
\(29\) 2.67282 0.496329 0.248165 0.968718i \(-0.420173\pi\)
0.248165 + 0.968718i \(0.420173\pi\)
\(30\) 0 0
\(31\) −8.88393 −1.59560 −0.797801 0.602921i \(-0.794003\pi\)
−0.797801 + 0.602921i \(0.794003\pi\)
\(32\) −4.20514 −0.743372
\(33\) 0 0
\(34\) 1.76334 0.302410
\(35\) −3.76494 −0.636391
\(36\) 0 0
\(37\) 0.943469 0.155105 0.0775527 0.996988i \(-0.475289\pi\)
0.0775527 + 0.996988i \(0.475289\pi\)
\(38\) 5.24887 0.851479
\(39\) 0 0
\(40\) −7.64065 −1.20809
\(41\) 9.90868 1.54748 0.773738 0.633506i \(-0.218385\pi\)
0.773738 + 0.633506i \(0.218385\pi\)
\(42\) 0 0
\(43\) −10.8524 −1.65498 −0.827489 0.561482i \(-0.810232\pi\)
−0.827489 + 0.561482i \(0.810232\pi\)
\(44\) −2.47838 −0.373629
\(45\) 0 0
\(46\) −11.8935 −1.75360
\(47\) 4.25536 0.620709 0.310354 0.950621i \(-0.399552\pi\)
0.310354 + 0.950621i \(0.399552\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 15.3074 2.16479
\(51\) 0 0
\(52\) −2.36787 −0.328364
\(53\) −5.14637 −0.706908 −0.353454 0.935452i \(-0.614993\pi\)
−0.353454 + 0.935452i \(0.614993\pi\)
\(54\) 0 0
\(55\) −11.9074 −1.60559
\(56\) 2.02942 0.271193
\(57\) 0 0
\(58\) 4.45938 0.585545
\(59\) −4.15250 −0.540609 −0.270305 0.962775i \(-0.587124\pi\)
−0.270305 + 0.962775i \(0.587124\pi\)
\(60\) 0 0
\(61\) 1.23055 0.157556 0.0787778 0.996892i \(-0.474898\pi\)
0.0787778 + 0.996892i \(0.474898\pi\)
\(62\) −14.8221 −1.88241
\(63\) 0 0
\(64\) 2.89042 0.361302
\(65\) −11.3764 −1.41107
\(66\) 0 0
\(67\) −10.7930 −1.31857 −0.659285 0.751893i \(-0.729141\pi\)
−0.659285 + 0.751893i \(0.729141\pi\)
\(68\) 0.828207 0.100435
\(69\) 0 0
\(70\) −6.28150 −0.750783
\(71\) −15.1564 −1.79874 −0.899369 0.437191i \(-0.855973\pi\)
−0.899369 + 0.437191i \(0.855973\pi\)
\(72\) 0 0
\(73\) 7.29623 0.853959 0.426980 0.904261i \(-0.359578\pi\)
0.426980 + 0.904261i \(0.359578\pi\)
\(74\) 1.57410 0.182986
\(75\) 0 0
\(76\) 2.46530 0.282789
\(77\) 3.16270 0.360423
\(78\) 0 0
\(79\) 5.09038 0.572713 0.286357 0.958123i \(-0.407556\pi\)
0.286357 + 0.958123i \(0.407556\pi\)
\(80\) −18.6484 −2.08496
\(81\) 0 0
\(82\) 16.5318 1.82564
\(83\) −8.93203 −0.980417 −0.490209 0.871605i \(-0.663079\pi\)
−0.490209 + 0.871605i \(0.663079\pi\)
\(84\) 0 0
\(85\) 3.97913 0.431597
\(86\) −18.1064 −1.95246
\(87\) 0 0
\(88\) 6.41846 0.684210
\(89\) 5.97445 0.633291 0.316645 0.948544i \(-0.397443\pi\)
0.316645 + 0.948544i \(0.397443\pi\)
\(90\) 0 0
\(91\) 3.02168 0.316758
\(92\) −5.58616 −0.582397
\(93\) 0 0
\(94\) 7.09974 0.732282
\(95\) 11.8445 1.21522
\(96\) 0 0
\(97\) −9.75834 −0.990809 −0.495405 0.868662i \(-0.664980\pi\)
−0.495405 + 0.868662i \(0.664980\pi\)
\(98\) 1.66842 0.168536
\(99\) 0 0
\(100\) 7.18958 0.718958
\(101\) −11.6815 −1.16235 −0.581174 0.813779i \(-0.697407\pi\)
−0.581174 + 0.813779i \(0.697407\pi\)
\(102\) 0 0
\(103\) 13.9511 1.37465 0.687323 0.726352i \(-0.258786\pi\)
0.687323 + 0.726352i \(0.258786\pi\)
\(104\) 6.13226 0.601318
\(105\) 0 0
\(106\) −8.58631 −0.833976
\(107\) −16.0629 −1.55286 −0.776432 0.630202i \(-0.782972\pi\)
−0.776432 + 0.630202i \(0.782972\pi\)
\(108\) 0 0
\(109\) −1.53849 −0.147360 −0.0736802 0.997282i \(-0.523474\pi\)
−0.0736802 + 0.997282i \(0.523474\pi\)
\(110\) −19.8665 −1.89420
\(111\) 0 0
\(112\) 4.95318 0.468032
\(113\) −15.3117 −1.44040 −0.720202 0.693765i \(-0.755951\pi\)
−0.720202 + 0.693765i \(0.755951\pi\)
\(114\) 0 0
\(115\) −26.8387 −2.50273
\(116\) 2.09449 0.194468
\(117\) 0 0
\(118\) −6.92811 −0.637784
\(119\) −1.05689 −0.0968849
\(120\) 0 0
\(121\) −0.997312 −0.0906647
\(122\) 2.05307 0.185876
\(123\) 0 0
\(124\) −6.96168 −0.625178
\(125\) 15.7177 1.40583
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 13.2327 1.16962
\(129\) 0 0
\(130\) −18.9807 −1.66471
\(131\) −3.78335 −0.330553 −0.165277 0.986247i \(-0.552852\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(132\) 0 0
\(133\) −3.14601 −0.272794
\(134\) −18.0072 −1.55559
\(135\) 0 0
\(136\) −2.14488 −0.183922
\(137\) −21.8842 −1.86969 −0.934847 0.355050i \(-0.884464\pi\)
−0.934847 + 0.355050i \(0.884464\pi\)
\(138\) 0 0
\(139\) 9.85829 0.836169 0.418085 0.908408i \(-0.362702\pi\)
0.418085 + 0.908408i \(0.362702\pi\)
\(140\) −2.95030 −0.249346
\(141\) 0 0
\(142\) −25.2873 −2.12206
\(143\) 9.55667 0.799169
\(144\) 0 0
\(145\) 10.0630 0.835685
\(146\) 12.1732 1.00746
\(147\) 0 0
\(148\) 0.739327 0.0607723
\(149\) 1.98780 0.162847 0.0814233 0.996680i \(-0.474053\pi\)
0.0814233 + 0.996680i \(0.474053\pi\)
\(150\) 0 0
\(151\) −8.05456 −0.655471 −0.327735 0.944770i \(-0.606285\pi\)
−0.327735 + 0.944770i \(0.606285\pi\)
\(152\) −6.38459 −0.517858
\(153\) 0 0
\(154\) 5.27672 0.425210
\(155\) −33.4475 −2.68656
\(156\) 0 0
\(157\) −9.72055 −0.775784 −0.387892 0.921705i \(-0.626797\pi\)
−0.387892 + 0.921705i \(0.626797\pi\)
\(158\) 8.49290 0.675659
\(159\) 0 0
\(160\) −15.8321 −1.25164
\(161\) 7.12860 0.561812
\(162\) 0 0
\(163\) 4.77862 0.374290 0.187145 0.982332i \(-0.440077\pi\)
0.187145 + 0.982332i \(0.440077\pi\)
\(164\) 7.76470 0.606321
\(165\) 0 0
\(166\) −14.9024 −1.15665
\(167\) 20.9276 1.61943 0.809713 0.586827i \(-0.199623\pi\)
0.809713 + 0.586827i \(0.199623\pi\)
\(168\) 0 0
\(169\) −3.86947 −0.297651
\(170\) 6.63885 0.509177
\(171\) 0 0
\(172\) −8.50424 −0.648442
\(173\) 13.7591 1.04609 0.523044 0.852306i \(-0.324796\pi\)
0.523044 + 0.852306i \(0.324796\pi\)
\(174\) 0 0
\(175\) −9.17476 −0.693547
\(176\) 15.6654 1.18083
\(177\) 0 0
\(178\) 9.96790 0.747126
\(179\) −9.24850 −0.691265 −0.345633 0.938370i \(-0.612336\pi\)
−0.345633 + 0.938370i \(0.612336\pi\)
\(180\) 0 0
\(181\) 14.1657 1.05293 0.526465 0.850197i \(-0.323517\pi\)
0.526465 + 0.850197i \(0.323517\pi\)
\(182\) 5.04143 0.373696
\(183\) 0 0
\(184\) 14.4669 1.06652
\(185\) 3.55210 0.261156
\(186\) 0 0
\(187\) −3.34263 −0.244437
\(188\) 3.33461 0.243202
\(189\) 0 0
\(190\) 19.7617 1.43366
\(191\) −3.34589 −0.242100 −0.121050 0.992646i \(-0.538626\pi\)
−0.121050 + 0.992646i \(0.538626\pi\)
\(192\) 0 0
\(193\) 6.91295 0.497605 0.248802 0.968554i \(-0.419963\pi\)
0.248802 + 0.968554i \(0.419963\pi\)
\(194\) −16.2810 −1.16891
\(195\) 0 0
\(196\) 0.783626 0.0559733
\(197\) 17.8080 1.26877 0.634385 0.773017i \(-0.281253\pi\)
0.634385 + 0.773017i \(0.281253\pi\)
\(198\) 0 0
\(199\) −16.6199 −1.17815 −0.589077 0.808077i \(-0.700509\pi\)
−0.589077 + 0.808077i \(0.700509\pi\)
\(200\) −18.6195 −1.31659
\(201\) 0 0
\(202\) −19.4896 −1.37128
\(203\) −2.67282 −0.187595
\(204\) 0 0
\(205\) 37.3056 2.60553
\(206\) 23.2764 1.62174
\(207\) 0 0
\(208\) 14.9669 1.03777
\(209\) −9.94990 −0.688249
\(210\) 0 0
\(211\) −14.6764 −1.01037 −0.505184 0.863012i \(-0.668575\pi\)
−0.505184 + 0.863012i \(0.668575\pi\)
\(212\) −4.03283 −0.276976
\(213\) 0 0
\(214\) −26.7997 −1.83199
\(215\) −40.8587 −2.78654
\(216\) 0 0
\(217\) 8.88393 0.603081
\(218\) −2.56685 −0.173849
\(219\) 0 0
\(220\) −9.33094 −0.629092
\(221\) −3.19358 −0.214823
\(222\) 0 0
\(223\) 17.2818 1.15728 0.578639 0.815584i \(-0.303584\pi\)
0.578639 + 0.815584i \(0.303584\pi\)
\(224\) 4.20514 0.280968
\(225\) 0 0
\(226\) −25.5464 −1.69932
\(227\) 7.13775 0.473749 0.236875 0.971540i \(-0.423877\pi\)
0.236875 + 0.971540i \(0.423877\pi\)
\(228\) 0 0
\(229\) 8.65153 0.571709 0.285855 0.958273i \(-0.407723\pi\)
0.285855 + 0.958273i \(0.407723\pi\)
\(230\) −44.7783 −2.95259
\(231\) 0 0
\(232\) −5.42427 −0.356121
\(233\) 2.76388 0.181068 0.0905340 0.995893i \(-0.471143\pi\)
0.0905340 + 0.995893i \(0.471143\pi\)
\(234\) 0 0
\(235\) 16.0212 1.04511
\(236\) −3.25401 −0.211818
\(237\) 0 0
\(238\) −1.76334 −0.114300
\(239\) −23.8172 −1.54061 −0.770304 0.637677i \(-0.779896\pi\)
−0.770304 + 0.637677i \(0.779896\pi\)
\(240\) 0 0
\(241\) −18.6840 −1.20354 −0.601770 0.798670i \(-0.705537\pi\)
−0.601770 + 0.798670i \(0.705537\pi\)
\(242\) −1.66394 −0.106962
\(243\) 0 0
\(244\) 0.964290 0.0617324
\(245\) 3.76494 0.240533
\(246\) 0 0
\(247\) −9.50623 −0.604867
\(248\) 18.0293 1.14486
\(249\) 0 0
\(250\) 26.2237 1.65854
\(251\) −19.4770 −1.22938 −0.614690 0.788769i \(-0.710719\pi\)
−0.614690 + 0.788769i \(0.710719\pi\)
\(252\) 0 0
\(253\) 22.5456 1.41743
\(254\) −1.66842 −0.104686
\(255\) 0 0
\(256\) 16.2969 1.01856
\(257\) 9.85146 0.614517 0.307259 0.951626i \(-0.400588\pi\)
0.307259 + 0.951626i \(0.400588\pi\)
\(258\) 0 0
\(259\) −0.943469 −0.0586243
\(260\) −8.91487 −0.552877
\(261\) 0 0
\(262\) −6.31222 −0.389971
\(263\) 13.8741 0.855514 0.427757 0.903894i \(-0.359304\pi\)
0.427757 + 0.903894i \(0.359304\pi\)
\(264\) 0 0
\(265\) −19.3758 −1.19024
\(266\) −5.24887 −0.321829
\(267\) 0 0
\(268\) −8.45765 −0.516633
\(269\) 26.2278 1.59914 0.799568 0.600575i \(-0.205062\pi\)
0.799568 + 0.600575i \(0.205062\pi\)
\(270\) 0 0
\(271\) 19.6970 1.19651 0.598255 0.801306i \(-0.295861\pi\)
0.598255 + 0.801306i \(0.295861\pi\)
\(272\) −5.23497 −0.317417
\(273\) 0 0
\(274\) −36.5121 −2.20577
\(275\) −29.0170 −1.74979
\(276\) 0 0
\(277\) −1.14232 −0.0686351 −0.0343176 0.999411i \(-0.510926\pi\)
−0.0343176 + 0.999411i \(0.510926\pi\)
\(278\) 16.4478 0.986472
\(279\) 0 0
\(280\) 7.64065 0.456616
\(281\) −16.7929 −1.00178 −0.500891 0.865511i \(-0.666994\pi\)
−0.500891 + 0.865511i \(0.666994\pi\)
\(282\) 0 0
\(283\) 13.1223 0.780042 0.390021 0.920806i \(-0.372468\pi\)
0.390021 + 0.920806i \(0.372468\pi\)
\(284\) −11.8770 −0.704769
\(285\) 0 0
\(286\) 15.9445 0.942820
\(287\) −9.90868 −0.584891
\(288\) 0 0
\(289\) −15.8830 −0.934293
\(290\) 16.7893 0.985901
\(291\) 0 0
\(292\) 5.71752 0.334593
\(293\) 8.84515 0.516739 0.258370 0.966046i \(-0.416815\pi\)
0.258370 + 0.966046i \(0.416815\pi\)
\(294\) 0 0
\(295\) −15.6339 −0.910241
\(296\) −1.91470 −0.111290
\(297\) 0 0
\(298\) 3.31648 0.192118
\(299\) 21.5403 1.24571
\(300\) 0 0
\(301\) 10.8524 0.625523
\(302\) −13.4384 −0.773292
\(303\) 0 0
\(304\) −15.5828 −0.893733
\(305\) 4.63294 0.265281
\(306\) 0 0
\(307\) 8.38864 0.478765 0.239382 0.970925i \(-0.423055\pi\)
0.239382 + 0.970925i \(0.423055\pi\)
\(308\) 2.47838 0.141219
\(309\) 0 0
\(310\) −55.8044 −3.16948
\(311\) 3.66921 0.208062 0.104031 0.994574i \(-0.466826\pi\)
0.104031 + 0.994574i \(0.466826\pi\)
\(312\) 0 0
\(313\) 27.7568 1.56891 0.784453 0.620188i \(-0.212944\pi\)
0.784453 + 0.620188i \(0.212944\pi\)
\(314\) −16.2180 −0.915232
\(315\) 0 0
\(316\) 3.98896 0.224397
\(317\) 8.76792 0.492455 0.246228 0.969212i \(-0.420809\pi\)
0.246228 + 0.969212i \(0.420809\pi\)
\(318\) 0 0
\(319\) −8.45332 −0.473295
\(320\) 10.8822 0.608335
\(321\) 0 0
\(322\) 11.8935 0.662799
\(323\) 3.32499 0.185007
\(324\) 0 0
\(325\) −27.7232 −1.53780
\(326\) 7.97274 0.441569
\(327\) 0 0
\(328\) −20.1089 −1.11033
\(329\) −4.25536 −0.234606
\(330\) 0 0
\(331\) 26.3095 1.44610 0.723051 0.690795i \(-0.242739\pi\)
0.723051 + 0.690795i \(0.242739\pi\)
\(332\) −6.99937 −0.384140
\(333\) 0 0
\(334\) 34.9160 1.91052
\(335\) −40.6348 −2.22012
\(336\) 0 0
\(337\) 0.194439 0.0105917 0.00529587 0.999986i \(-0.498314\pi\)
0.00529587 + 0.999986i \(0.498314\pi\)
\(338\) −6.45589 −0.351154
\(339\) 0 0
\(340\) 3.11815 0.169105
\(341\) 28.0972 1.52155
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 22.0241 1.18746
\(345\) 0 0
\(346\) 22.9560 1.23412
\(347\) 20.8704 1.12038 0.560191 0.828363i \(-0.310728\pi\)
0.560191 + 0.828363i \(0.310728\pi\)
\(348\) 0 0
\(349\) −6.06183 −0.324483 −0.162241 0.986751i \(-0.551872\pi\)
−0.162241 + 0.986751i \(0.551872\pi\)
\(350\) −15.3074 −0.818212
\(351\) 0 0
\(352\) 13.2996 0.708872
\(353\) 18.0447 0.960424 0.480212 0.877152i \(-0.340560\pi\)
0.480212 + 0.877152i \(0.340560\pi\)
\(354\) 0 0
\(355\) −57.0630 −3.02859
\(356\) 4.68174 0.248132
\(357\) 0 0
\(358\) −15.4304 −0.815521
\(359\) 34.6535 1.82894 0.914472 0.404650i \(-0.132606\pi\)
0.914472 + 0.404650i \(0.132606\pi\)
\(360\) 0 0
\(361\) −9.10261 −0.479085
\(362\) 23.6344 1.24220
\(363\) 0 0
\(364\) 2.36787 0.124110
\(365\) 27.4699 1.43784
\(366\) 0 0
\(367\) −11.3162 −0.590699 −0.295349 0.955389i \(-0.595436\pi\)
−0.295349 + 0.955389i \(0.595436\pi\)
\(368\) 35.3093 1.84062
\(369\) 0 0
\(370\) 5.92640 0.308099
\(371\) 5.14637 0.267186
\(372\) 0 0
\(373\) −3.90440 −0.202162 −0.101081 0.994878i \(-0.532230\pi\)
−0.101081 + 0.994878i \(0.532230\pi\)
\(374\) −5.57691 −0.288375
\(375\) 0 0
\(376\) −8.63593 −0.445364
\(377\) −8.07639 −0.415955
\(378\) 0 0
\(379\) 23.6160 1.21307 0.606537 0.795055i \(-0.292558\pi\)
0.606537 + 0.795055i \(0.292558\pi\)
\(380\) 9.28169 0.476141
\(381\) 0 0
\(382\) −5.58235 −0.285618
\(383\) −25.6133 −1.30878 −0.654388 0.756159i \(-0.727074\pi\)
−0.654388 + 0.756159i \(0.727074\pi\)
\(384\) 0 0
\(385\) 11.9074 0.606856
\(386\) 11.5337 0.587050
\(387\) 0 0
\(388\) −7.64689 −0.388212
\(389\) 15.4262 0.782140 0.391070 0.920361i \(-0.372105\pi\)
0.391070 + 0.920361i \(0.372105\pi\)
\(390\) 0 0
\(391\) −7.53415 −0.381018
\(392\) −2.02942 −0.102501
\(393\) 0 0
\(394\) 29.7113 1.49683
\(395\) 19.1650 0.964295
\(396\) 0 0
\(397\) −5.68868 −0.285507 −0.142753 0.989758i \(-0.545596\pi\)
−0.142753 + 0.989758i \(0.545596\pi\)
\(398\) −27.7290 −1.38993
\(399\) 0 0
\(400\) −45.4443 −2.27221
\(401\) 22.9927 1.14820 0.574101 0.818785i \(-0.305352\pi\)
0.574101 + 0.818785i \(0.305352\pi\)
\(402\) 0 0
\(403\) 26.8444 1.33721
\(404\) −9.15390 −0.455423
\(405\) 0 0
\(406\) −4.45938 −0.221315
\(407\) −2.98391 −0.147907
\(408\) 0 0
\(409\) −18.0347 −0.891761 −0.445880 0.895093i \(-0.647109\pi\)
−0.445880 + 0.895093i \(0.647109\pi\)
\(410\) 62.2414 3.07388
\(411\) 0 0
\(412\) 10.9325 0.538604
\(413\) 4.15250 0.204331
\(414\) 0 0
\(415\) −33.6285 −1.65076
\(416\) 12.7066 0.622992
\(417\) 0 0
\(418\) −16.6006 −0.811962
\(419\) 13.0874 0.639360 0.319680 0.947525i \(-0.396425\pi\)
0.319680 + 0.947525i \(0.396425\pi\)
\(420\) 0 0
\(421\) −40.1201 −1.95533 −0.977667 0.210160i \(-0.932602\pi\)
−0.977667 + 0.210160i \(0.932602\pi\)
\(422\) −24.4865 −1.19198
\(423\) 0 0
\(424\) 10.4442 0.507213
\(425\) 9.69671 0.470360
\(426\) 0 0
\(427\) −1.23055 −0.0595504
\(428\) −12.5873 −0.608432
\(429\) 0 0
\(430\) −68.1695 −3.28742
\(431\) 16.9188 0.814951 0.407476 0.913216i \(-0.366409\pi\)
0.407476 + 0.913216i \(0.366409\pi\)
\(432\) 0 0
\(433\) −8.66554 −0.416439 −0.208220 0.978082i \(-0.566767\pi\)
−0.208220 + 0.978082i \(0.566767\pi\)
\(434\) 14.8221 0.711485
\(435\) 0 0
\(436\) −1.20560 −0.0577378
\(437\) −22.4267 −1.07281
\(438\) 0 0
\(439\) −25.4905 −1.21660 −0.608298 0.793709i \(-0.708147\pi\)
−0.608298 + 0.793709i \(0.708147\pi\)
\(440\) 24.1651 1.15203
\(441\) 0 0
\(442\) −5.32823 −0.253438
\(443\) −34.9265 −1.65941 −0.829703 0.558205i \(-0.811490\pi\)
−0.829703 + 0.558205i \(0.811490\pi\)
\(444\) 0 0
\(445\) 22.4934 1.06629
\(446\) 28.8334 1.36530
\(447\) 0 0
\(448\) −2.89042 −0.136559
\(449\) −29.0615 −1.37150 −0.685749 0.727838i \(-0.740525\pi\)
−0.685749 + 0.727838i \(0.740525\pi\)
\(450\) 0 0
\(451\) −31.3382 −1.47566
\(452\) −11.9987 −0.564369
\(453\) 0 0
\(454\) 11.9088 0.558906
\(455\) 11.3764 0.533335
\(456\) 0 0
\(457\) 16.6346 0.778133 0.389066 0.921210i \(-0.372798\pi\)
0.389066 + 0.921210i \(0.372798\pi\)
\(458\) 14.4344 0.674475
\(459\) 0 0
\(460\) −21.0315 −0.980601
\(461\) 1.69635 0.0790067 0.0395034 0.999219i \(-0.487422\pi\)
0.0395034 + 0.999219i \(0.487422\pi\)
\(462\) 0 0
\(463\) −33.3579 −1.55027 −0.775136 0.631795i \(-0.782318\pi\)
−0.775136 + 0.631795i \(0.782318\pi\)
\(464\) −13.2389 −0.614602
\(465\) 0 0
\(466\) 4.61132 0.213615
\(467\) −22.2821 −1.03109 −0.515546 0.856862i \(-0.672411\pi\)
−0.515546 + 0.856862i \(0.672411\pi\)
\(468\) 0 0
\(469\) 10.7930 0.498373
\(470\) 26.7301 1.23297
\(471\) 0 0
\(472\) 8.42717 0.387892
\(473\) 34.3230 1.57817
\(474\) 0 0
\(475\) 28.8639 1.32437
\(476\) −0.828207 −0.0379608
\(477\) 0 0
\(478\) −39.7371 −1.81753
\(479\) 10.4324 0.476667 0.238333 0.971183i \(-0.423399\pi\)
0.238333 + 0.971183i \(0.423399\pi\)
\(480\) 0 0
\(481\) −2.85086 −0.129988
\(482\) −31.1727 −1.41988
\(483\) 0 0
\(484\) −0.781520 −0.0355236
\(485\) −36.7395 −1.66826
\(486\) 0 0
\(487\) 4.27431 0.193687 0.0968437 0.995300i \(-0.469125\pi\)
0.0968437 + 0.995300i \(0.469125\pi\)
\(488\) −2.49730 −0.113048
\(489\) 0 0
\(490\) 6.28150 0.283769
\(491\) −12.2377 −0.552281 −0.276141 0.961117i \(-0.589056\pi\)
−0.276141 + 0.961117i \(0.589056\pi\)
\(492\) 0 0
\(493\) 2.82487 0.127226
\(494\) −15.8604 −0.713593
\(495\) 0 0
\(496\) 44.0037 1.97583
\(497\) 15.1564 0.679859
\(498\) 0 0
\(499\) 18.0798 0.809361 0.404681 0.914458i \(-0.367383\pi\)
0.404681 + 0.914458i \(0.367383\pi\)
\(500\) 12.3168 0.550824
\(501\) 0 0
\(502\) −32.4959 −1.45036
\(503\) −15.4918 −0.690747 −0.345373 0.938465i \(-0.612248\pi\)
−0.345373 + 0.938465i \(0.612248\pi\)
\(504\) 0 0
\(505\) −43.9800 −1.95708
\(506\) 37.6156 1.67222
\(507\) 0 0
\(508\) −0.783626 −0.0347678
\(509\) −28.3379 −1.25606 −0.628029 0.778190i \(-0.716138\pi\)
−0.628029 + 0.778190i \(0.716138\pi\)
\(510\) 0 0
\(511\) −7.29623 −0.322766
\(512\) 0.724649 0.0320253
\(513\) 0 0
\(514\) 16.4364 0.724978
\(515\) 52.5251 2.31453
\(516\) 0 0
\(517\) −13.4584 −0.591902
\(518\) −1.57410 −0.0691621
\(519\) 0 0
\(520\) 23.0876 1.01246
\(521\) 24.5238 1.07441 0.537205 0.843452i \(-0.319480\pi\)
0.537205 + 0.843452i \(0.319480\pi\)
\(522\) 0 0
\(523\) 36.5848 1.59974 0.799870 0.600173i \(-0.204902\pi\)
0.799870 + 0.600173i \(0.204902\pi\)
\(524\) −2.96474 −0.129515
\(525\) 0 0
\(526\) 23.1478 1.00929
\(527\) −9.38934 −0.409006
\(528\) 0 0
\(529\) 27.8169 1.20943
\(530\) −32.3269 −1.40419
\(531\) 0 0
\(532\) −2.46530 −0.106884
\(533\) −29.9408 −1.29688
\(534\) 0 0
\(535\) −60.4760 −2.61460
\(536\) 21.9035 0.946087
\(537\) 0 0
\(538\) 43.7590 1.88658
\(539\) −3.16270 −0.136227
\(540\) 0 0
\(541\) −26.7095 −1.14833 −0.574166 0.818739i \(-0.694674\pi\)
−0.574166 + 0.818739i \(0.694674\pi\)
\(542\) 32.8629 1.41158
\(543\) 0 0
\(544\) −4.44438 −0.190551
\(545\) −5.79231 −0.248115
\(546\) 0 0
\(547\) 44.3462 1.89611 0.948054 0.318109i \(-0.103048\pi\)
0.948054 + 0.318109i \(0.103048\pi\)
\(548\) −17.1490 −0.732571
\(549\) 0 0
\(550\) −48.4126 −2.06432
\(551\) 8.40871 0.358223
\(552\) 0 0
\(553\) −5.09038 −0.216465
\(554\) −1.90586 −0.0809724
\(555\) 0 0
\(556\) 7.72521 0.327622
\(557\) 25.7466 1.09092 0.545460 0.838137i \(-0.316355\pi\)
0.545460 + 0.838137i \(0.316355\pi\)
\(558\) 0 0
\(559\) 32.7925 1.38698
\(560\) 18.6484 0.788040
\(561\) 0 0
\(562\) −28.0176 −1.18185
\(563\) 28.0882 1.18378 0.591888 0.806021i \(-0.298383\pi\)
0.591888 + 0.806021i \(0.298383\pi\)
\(564\) 0 0
\(565\) −57.6476 −2.42525
\(566\) 21.8936 0.920256
\(567\) 0 0
\(568\) 30.7588 1.29061
\(569\) 19.8532 0.832289 0.416144 0.909299i \(-0.363381\pi\)
0.416144 + 0.909299i \(0.363381\pi\)
\(570\) 0 0
\(571\) −20.5202 −0.858745 −0.429372 0.903128i \(-0.641265\pi\)
−0.429372 + 0.903128i \(0.641265\pi\)
\(572\) 7.48886 0.313125
\(573\) 0 0
\(574\) −16.5318 −0.690026
\(575\) −65.4032 −2.72750
\(576\) 0 0
\(577\) −15.0348 −0.625908 −0.312954 0.949768i \(-0.601319\pi\)
−0.312954 + 0.949768i \(0.601319\pi\)
\(578\) −26.4995 −1.10223
\(579\) 0 0
\(580\) 7.88562 0.327432
\(581\) 8.93203 0.370563
\(582\) 0 0
\(583\) 16.2764 0.674101
\(584\) −14.8071 −0.612724
\(585\) 0 0
\(586\) 14.7574 0.609624
\(587\) 39.4649 1.62889 0.814444 0.580242i \(-0.197042\pi\)
0.814444 + 0.580242i \(0.197042\pi\)
\(588\) 0 0
\(589\) −27.9489 −1.15162
\(590\) −26.0839 −1.07386
\(591\) 0 0
\(592\) −4.67317 −0.192066
\(593\) 7.71037 0.316627 0.158314 0.987389i \(-0.449394\pi\)
0.158314 + 0.987389i \(0.449394\pi\)
\(594\) 0 0
\(595\) −3.97913 −0.163128
\(596\) 1.55769 0.0638054
\(597\) 0 0
\(598\) 35.9383 1.46963
\(599\) −45.7827 −1.87063 −0.935315 0.353817i \(-0.884884\pi\)
−0.935315 + 0.353817i \(0.884884\pi\)
\(600\) 0 0
\(601\) −46.7608 −1.90741 −0.953706 0.300739i \(-0.902767\pi\)
−0.953706 + 0.300739i \(0.902767\pi\)
\(602\) 18.1064 0.737962
\(603\) 0 0
\(604\) −6.31176 −0.256822
\(605\) −3.75482 −0.152655
\(606\) 0 0
\(607\) 43.1360 1.75084 0.875419 0.483366i \(-0.160586\pi\)
0.875419 + 0.483366i \(0.160586\pi\)
\(608\) −13.2294 −0.536524
\(609\) 0 0
\(610\) 7.72969 0.312966
\(611\) −12.8583 −0.520193
\(612\) 0 0
\(613\) 19.6862 0.795120 0.397560 0.917576i \(-0.369857\pi\)
0.397560 + 0.917576i \(0.369857\pi\)
\(614\) 13.9958 0.564824
\(615\) 0 0
\(616\) −6.41846 −0.258607
\(617\) −5.42454 −0.218384 −0.109192 0.994021i \(-0.534826\pi\)
−0.109192 + 0.994021i \(0.534826\pi\)
\(618\) 0 0
\(619\) −23.7411 −0.954235 −0.477118 0.878839i \(-0.658318\pi\)
−0.477118 + 0.878839i \(0.658318\pi\)
\(620\) −26.2103 −1.05263
\(621\) 0 0
\(622\) 6.12178 0.245461
\(623\) −5.97445 −0.239361
\(624\) 0 0
\(625\) 13.3024 0.532096
\(626\) 46.3100 1.85092
\(627\) 0 0
\(628\) −7.61728 −0.303962
\(629\) 0.997143 0.0397587
\(630\) 0 0
\(631\) 29.7531 1.18445 0.592226 0.805772i \(-0.298249\pi\)
0.592226 + 0.805772i \(0.298249\pi\)
\(632\) −10.3305 −0.410927
\(633\) 0 0
\(634\) 14.6286 0.580975
\(635\) −3.76494 −0.149407
\(636\) 0 0
\(637\) −3.02168 −0.119723
\(638\) −14.1037 −0.558370
\(639\) 0 0
\(640\) 49.8204 1.96932
\(641\) −6.06350 −0.239494 −0.119747 0.992804i \(-0.538208\pi\)
−0.119747 + 0.992804i \(0.538208\pi\)
\(642\) 0 0
\(643\) 15.0189 0.592287 0.296144 0.955143i \(-0.404299\pi\)
0.296144 + 0.955143i \(0.404299\pi\)
\(644\) 5.58616 0.220125
\(645\) 0 0
\(646\) 5.54748 0.218263
\(647\) 8.33725 0.327771 0.163886 0.986479i \(-0.447597\pi\)
0.163886 + 0.986479i \(0.447597\pi\)
\(648\) 0 0
\(649\) 13.1331 0.515520
\(650\) −46.2539 −1.81423
\(651\) 0 0
\(652\) 3.74465 0.146652
\(653\) −2.97846 −0.116556 −0.0582781 0.998300i \(-0.518561\pi\)
−0.0582781 + 0.998300i \(0.518561\pi\)
\(654\) 0 0
\(655\) −14.2441 −0.556563
\(656\) −49.0795 −1.91623
\(657\) 0 0
\(658\) −7.09974 −0.276776
\(659\) 0.632856 0.0246526 0.0123263 0.999924i \(-0.496076\pi\)
0.0123263 + 0.999924i \(0.496076\pi\)
\(660\) 0 0
\(661\) −32.4745 −1.26311 −0.631557 0.775330i \(-0.717584\pi\)
−0.631557 + 0.775330i \(0.717584\pi\)
\(662\) 43.8953 1.70604
\(663\) 0 0
\(664\) 18.1269 0.703459
\(665\) −11.8445 −0.459311
\(666\) 0 0
\(667\) −19.0534 −0.737752
\(668\) 16.3994 0.634512
\(669\) 0 0
\(670\) −67.7960 −2.61919
\(671\) −3.89186 −0.150244
\(672\) 0 0
\(673\) −9.45807 −0.364582 −0.182291 0.983245i \(-0.558351\pi\)
−0.182291 + 0.983245i \(0.558351\pi\)
\(674\) 0.324405 0.0124956
\(675\) 0 0
\(676\) −3.03221 −0.116624
\(677\) −10.2613 −0.394372 −0.197186 0.980366i \(-0.563180\pi\)
−0.197186 + 0.980366i \(0.563180\pi\)
\(678\) 0 0
\(679\) 9.75834 0.374491
\(680\) −8.07533 −0.309675
\(681\) 0 0
\(682\) 46.8780 1.79505
\(683\) 1.93744 0.0741339 0.0370670 0.999313i \(-0.488199\pi\)
0.0370670 + 0.999313i \(0.488199\pi\)
\(684\) 0 0
\(685\) −82.3927 −3.14806
\(686\) −1.66842 −0.0637006
\(687\) 0 0
\(688\) 53.7540 2.04935
\(689\) 15.5507 0.592433
\(690\) 0 0
\(691\) 27.6381 1.05140 0.525702 0.850669i \(-0.323803\pi\)
0.525702 + 0.850669i \(0.323803\pi\)
\(692\) 10.7820 0.409871
\(693\) 0 0
\(694\) 34.8206 1.32177
\(695\) 37.1158 1.40788
\(696\) 0 0
\(697\) 10.4724 0.396670
\(698\) −10.1137 −0.382809
\(699\) 0 0
\(700\) −7.18958 −0.271741
\(701\) −18.7994 −0.710043 −0.355022 0.934858i \(-0.615526\pi\)
−0.355022 + 0.934858i \(0.615526\pi\)
\(702\) 0 0
\(703\) 2.96816 0.111946
\(704\) −9.14153 −0.344534
\(705\) 0 0
\(706\) 30.1062 1.13306
\(707\) 11.6815 0.439327
\(708\) 0 0
\(709\) −36.4454 −1.36874 −0.684369 0.729136i \(-0.739922\pi\)
−0.684369 + 0.729136i \(0.739922\pi\)
\(710\) −95.2051 −3.57298
\(711\) 0 0
\(712\) −12.1247 −0.454392
\(713\) 63.3300 2.37173
\(714\) 0 0
\(715\) 35.9803 1.34559
\(716\) −7.24737 −0.270847
\(717\) 0 0
\(718\) 57.8167 2.15770
\(719\) −25.7559 −0.960533 −0.480267 0.877123i \(-0.659460\pi\)
−0.480267 + 0.877123i \(0.659460\pi\)
\(720\) 0 0
\(721\) −13.9511 −0.519567
\(722\) −15.1870 −0.565201
\(723\) 0 0
\(724\) 11.1006 0.412552
\(725\) 24.5224 0.910740
\(726\) 0 0
\(727\) 0.347207 0.0128772 0.00643859 0.999979i \(-0.497951\pi\)
0.00643859 + 0.999979i \(0.497951\pi\)
\(728\) −6.13226 −0.227277
\(729\) 0 0
\(730\) 45.8313 1.69629
\(731\) −11.4698 −0.424226
\(732\) 0 0
\(733\) −35.7358 −1.31993 −0.659966 0.751295i \(-0.729429\pi\)
−0.659966 + 0.751295i \(0.729429\pi\)
\(734\) −18.8801 −0.696878
\(735\) 0 0
\(736\) 29.9768 1.10496
\(737\) 34.1349 1.25738
\(738\) 0 0
\(739\) −45.6054 −1.67762 −0.838811 0.544422i \(-0.816749\pi\)
−0.838811 + 0.544422i \(0.816749\pi\)
\(740\) 2.78352 0.102324
\(741\) 0 0
\(742\) 8.58631 0.315213
\(743\) 32.1925 1.18103 0.590513 0.807028i \(-0.298925\pi\)
0.590513 + 0.807028i \(0.298925\pi\)
\(744\) 0 0
\(745\) 7.48393 0.274190
\(746\) −6.51418 −0.238501
\(747\) 0 0
\(748\) −2.61937 −0.0957737
\(749\) 16.0629 0.586927
\(750\) 0 0
\(751\) −38.9426 −1.42104 −0.710518 0.703679i \(-0.751540\pi\)
−0.710518 + 0.703679i \(0.751540\pi\)
\(752\) −21.0776 −0.768621
\(753\) 0 0
\(754\) −13.4748 −0.490723
\(755\) −30.3249 −1.10364
\(756\) 0 0
\(757\) 17.7030 0.643427 0.321714 0.946837i \(-0.395741\pi\)
0.321714 + 0.946837i \(0.395741\pi\)
\(758\) 39.4015 1.43113
\(759\) 0 0
\(760\) −24.0376 −0.871934
\(761\) −13.0843 −0.474305 −0.237153 0.971472i \(-0.576214\pi\)
−0.237153 + 0.971472i \(0.576214\pi\)
\(762\) 0 0
\(763\) 1.53849 0.0556970
\(764\) −2.62193 −0.0948580
\(765\) 0 0
\(766\) −42.7337 −1.54403
\(767\) 12.5475 0.453064
\(768\) 0 0
\(769\) 19.4773 0.702371 0.351185 0.936306i \(-0.385779\pi\)
0.351185 + 0.936306i \(0.385779\pi\)
\(770\) 19.8665 0.715939
\(771\) 0 0
\(772\) 5.41717 0.194968
\(773\) 13.8332 0.497546 0.248773 0.968562i \(-0.419973\pi\)
0.248773 + 0.968562i \(0.419973\pi\)
\(774\) 0 0
\(775\) −81.5079 −2.92785
\(776\) 19.8038 0.710915
\(777\) 0 0
\(778\) 25.7374 0.922731
\(779\) 31.1728 1.11688
\(780\) 0 0
\(781\) 47.9353 1.71526
\(782\) −12.5701 −0.449507
\(783\) 0 0
\(784\) −4.95318 −0.176899
\(785\) −36.5973 −1.30621
\(786\) 0 0
\(787\) −7.07202 −0.252090 −0.126045 0.992025i \(-0.540228\pi\)
−0.126045 + 0.992025i \(0.540228\pi\)
\(788\) 13.9548 0.497121
\(789\) 0 0
\(790\) 31.9752 1.13763
\(791\) 15.3117 0.544421
\(792\) 0 0
\(793\) −3.71832 −0.132041
\(794\) −9.49112 −0.336827
\(795\) 0 0
\(796\) −13.0238 −0.461617
\(797\) −14.5337 −0.514809 −0.257405 0.966304i \(-0.582867\pi\)
−0.257405 + 0.966304i \(0.582867\pi\)
\(798\) 0 0
\(799\) 4.49745 0.159108
\(800\) −38.5812 −1.36405
\(801\) 0 0
\(802\) 38.3615 1.35459
\(803\) −23.0758 −0.814328
\(804\) 0 0
\(805\) 26.8387 0.945941
\(806\) 44.7877 1.57758
\(807\) 0 0
\(808\) 23.7066 0.833996
\(809\) 31.3659 1.10277 0.551383 0.834252i \(-0.314100\pi\)
0.551383 + 0.834252i \(0.314100\pi\)
\(810\) 0 0
\(811\) −18.3232 −0.643413 −0.321707 0.946839i \(-0.604257\pi\)
−0.321707 + 0.946839i \(0.604257\pi\)
\(812\) −2.09449 −0.0735021
\(813\) 0 0
\(814\) −4.97842 −0.174494
\(815\) 17.9912 0.630204
\(816\) 0 0
\(817\) −34.1418 −1.19447
\(818\) −30.0895 −1.05206
\(819\) 0 0
\(820\) 29.2336 1.02088
\(821\) 9.46589 0.330362 0.165181 0.986263i \(-0.447179\pi\)
0.165181 + 0.986263i \(0.447179\pi\)
\(822\) 0 0
\(823\) 11.5049 0.401036 0.200518 0.979690i \(-0.435738\pi\)
0.200518 + 0.979690i \(0.435738\pi\)
\(824\) −28.3127 −0.986321
\(825\) 0 0
\(826\) 6.92811 0.241060
\(827\) 15.4743 0.538094 0.269047 0.963127i \(-0.413291\pi\)
0.269047 + 0.963127i \(0.413291\pi\)
\(828\) 0 0
\(829\) 14.4745 0.502720 0.251360 0.967894i \(-0.419122\pi\)
0.251360 + 0.967894i \(0.419122\pi\)
\(830\) −56.1065 −1.94749
\(831\) 0 0
\(832\) −8.73390 −0.302794
\(833\) 1.05689 0.0366191
\(834\) 0 0
\(835\) 78.7911 2.72668
\(836\) −7.79700 −0.269665
\(837\) 0 0
\(838\) 21.8353 0.754286
\(839\) 24.9719 0.862127 0.431063 0.902322i \(-0.358139\pi\)
0.431063 + 0.902322i \(0.358139\pi\)
\(840\) 0 0
\(841\) −21.8561 −0.753657
\(842\) −66.9372 −2.30681
\(843\) 0 0
\(844\) −11.5008 −0.395875
\(845\) −14.5683 −0.501165
\(846\) 0 0
\(847\) 0.997312 0.0342680
\(848\) 25.4909 0.875361
\(849\) 0 0
\(850\) 16.1782 0.554907
\(851\) −6.72561 −0.230551
\(852\) 0 0
\(853\) 0.627948 0.0215005 0.0107503 0.999942i \(-0.496578\pi\)
0.0107503 + 0.999942i \(0.496578\pi\)
\(854\) −2.05307 −0.0702547
\(855\) 0 0
\(856\) 32.5985 1.11419
\(857\) −37.4706 −1.27997 −0.639986 0.768387i \(-0.721060\pi\)
−0.639986 + 0.768387i \(0.721060\pi\)
\(858\) 0 0
\(859\) 0.464206 0.0158385 0.00791926 0.999969i \(-0.497479\pi\)
0.00791926 + 0.999969i \(0.497479\pi\)
\(860\) −32.0179 −1.09180
\(861\) 0 0
\(862\) 28.2277 0.961440
\(863\) −13.3762 −0.455333 −0.227666 0.973739i \(-0.573110\pi\)
−0.227666 + 0.973739i \(0.573110\pi\)
\(864\) 0 0
\(865\) 51.8023 1.76133
\(866\) −14.4578 −0.491295
\(867\) 0 0
\(868\) 6.96168 0.236295
\(869\) −16.0994 −0.546134
\(870\) 0 0
\(871\) 32.6129 1.10504
\(872\) 3.12224 0.105733
\(873\) 0 0
\(874\) −37.4171 −1.26565
\(875\) −15.7177 −0.531356
\(876\) 0 0
\(877\) −4.07242 −0.137516 −0.0687580 0.997633i \(-0.521904\pi\)
−0.0687580 + 0.997633i \(0.521904\pi\)
\(878\) −42.5289 −1.43528
\(879\) 0 0
\(880\) 58.9794 1.98820
\(881\) 4.54811 0.153230 0.0766149 0.997061i \(-0.475589\pi\)
0.0766149 + 0.997061i \(0.475589\pi\)
\(882\) 0 0
\(883\) −22.3985 −0.753771 −0.376885 0.926260i \(-0.623005\pi\)
−0.376885 + 0.926260i \(0.623005\pi\)
\(884\) −2.50257 −0.0841707
\(885\) 0 0
\(886\) −58.2720 −1.95769
\(887\) −4.37232 −0.146808 −0.0734040 0.997302i \(-0.523386\pi\)
−0.0734040 + 0.997302i \(0.523386\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 37.5285 1.25796
\(891\) 0 0
\(892\) 13.5425 0.453436
\(893\) 13.3874 0.447993
\(894\) 0 0
\(895\) −34.8200 −1.16391
\(896\) −13.2327 −0.442074
\(897\) 0 0
\(898\) −48.4868 −1.61803
\(899\) −23.7451 −0.791944
\(900\) 0 0
\(901\) −5.43915 −0.181204
\(902\) −52.2853 −1.74091
\(903\) 0 0
\(904\) 31.0739 1.03350
\(905\) 53.3331 1.77285
\(906\) 0 0
\(907\) 22.6987 0.753697 0.376849 0.926275i \(-0.377008\pi\)
0.376849 + 0.926275i \(0.377008\pi\)
\(908\) 5.59333 0.185621
\(909\) 0 0
\(910\) 18.9807 0.629203
\(911\) 26.5960 0.881165 0.440582 0.897712i \(-0.354772\pi\)
0.440582 + 0.897712i \(0.354772\pi\)
\(912\) 0 0
\(913\) 28.2493 0.934917
\(914\) 27.7535 0.918003
\(915\) 0 0
\(916\) 6.77957 0.224003
\(917\) 3.78335 0.124937
\(918\) 0 0
\(919\) −12.3633 −0.407828 −0.203914 0.978989i \(-0.565366\pi\)
−0.203914 + 0.978989i \(0.565366\pi\)
\(920\) 54.4671 1.79573
\(921\) 0 0
\(922\) 2.83022 0.0932083
\(923\) 45.7979 1.50745
\(924\) 0 0
\(925\) 8.65610 0.284611
\(926\) −55.6549 −1.82893
\(927\) 0 0
\(928\) −11.2396 −0.368957
\(929\) 40.0029 1.31245 0.656227 0.754563i \(-0.272151\pi\)
0.656227 + 0.754563i \(0.272151\pi\)
\(930\) 0 0
\(931\) 3.14601 0.103106
\(932\) 2.16585 0.0709448
\(933\) 0 0
\(934\) −37.1759 −1.21643
\(935\) −12.5848 −0.411567
\(936\) 0 0
\(937\) −51.5727 −1.68481 −0.842403 0.538848i \(-0.818860\pi\)
−0.842403 + 0.538848i \(0.818860\pi\)
\(938\) 18.0072 0.587956
\(939\) 0 0
\(940\) 12.5546 0.409487
\(941\) 43.1695 1.40729 0.703644 0.710553i \(-0.251555\pi\)
0.703644 + 0.710553i \(0.251555\pi\)
\(942\) 0 0
\(943\) −70.6350 −2.30019
\(944\) 20.5681 0.669434
\(945\) 0 0
\(946\) 57.2652 1.86185
\(947\) −32.9398 −1.07040 −0.535200 0.844726i \(-0.679764\pi\)
−0.535200 + 0.844726i \(0.679764\pi\)
\(948\) 0 0
\(949\) −22.0469 −0.715671
\(950\) 48.1571 1.56242
\(951\) 0 0
\(952\) 2.14488 0.0695158
\(953\) 23.7506 0.769357 0.384679 0.923051i \(-0.374312\pi\)
0.384679 + 0.923051i \(0.374312\pi\)
\(954\) 0 0
\(955\) −12.5971 −0.407632
\(956\) −18.6638 −0.603631
\(957\) 0 0
\(958\) 17.4056 0.562348
\(959\) 21.8842 0.706678
\(960\) 0 0
\(961\) 47.9242 1.54594
\(962\) −4.75643 −0.153353
\(963\) 0 0
\(964\) −14.6412 −0.471563
\(965\) 26.0268 0.837833
\(966\) 0 0
\(967\) 1.31829 0.0423934 0.0211967 0.999775i \(-0.493252\pi\)
0.0211967 + 0.999775i \(0.493252\pi\)
\(968\) 2.02397 0.0650528
\(969\) 0 0
\(970\) −61.2970 −1.96813
\(971\) 46.5792 1.49480 0.747398 0.664376i \(-0.231303\pi\)
0.747398 + 0.664376i \(0.231303\pi\)
\(972\) 0 0
\(973\) −9.85829 −0.316042
\(974\) 7.13134 0.228503
\(975\) 0 0
\(976\) −6.09513 −0.195100
\(977\) −46.4171 −1.48501 −0.742507 0.669839i \(-0.766363\pi\)
−0.742507 + 0.669839i \(0.766363\pi\)
\(978\) 0 0
\(979\) −18.8954 −0.603900
\(980\) 2.95030 0.0942440
\(981\) 0 0
\(982\) −20.4177 −0.651554
\(983\) 56.9936 1.81781 0.908907 0.416998i \(-0.136918\pi\)
0.908907 + 0.416998i \(0.136918\pi\)
\(984\) 0 0
\(985\) 67.0462 2.13627
\(986\) 4.71307 0.150095
\(987\) 0 0
\(988\) −7.44933 −0.236995
\(989\) 77.3626 2.45999
\(990\) 0 0
\(991\) 6.27979 0.199484 0.0997421 0.995013i \(-0.468198\pi\)
0.0997421 + 0.995013i \(0.468198\pi\)
\(992\) 37.3582 1.18612
\(993\) 0 0
\(994\) 25.2873 0.802064
\(995\) −62.5730 −1.98370
\(996\) 0 0
\(997\) 16.8353 0.533179 0.266589 0.963810i \(-0.414103\pi\)
0.266589 + 0.963810i \(0.414103\pi\)
\(998\) 30.1646 0.954845
\(999\) 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 8001.2.a.z.1.26 yes 32
3.2 odd 2 inner 8001.2.a.z.1.7 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.7 32 3.2 odd 2 inner
8001.2.a.z.1.26 yes 32 1.1 even 1 trivial