Properties

Label 2151.2.a.k.1.12
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(0.854685\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} +O(q^{10})\) \(q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} -2.41036 q^{10} -0.210277 q^{11} -4.81635 q^{13} +2.84700 q^{14} +0.150696 q^{16} +4.49202 q^{17} -4.63640 q^{19} +3.58025 q^{20} -0.179720 q^{22} +7.53504 q^{23} +2.95338 q^{25} -4.11646 q^{26} -4.22882 q^{28} +2.29153 q^{29} -7.14674 q^{31} +5.71760 q^{32} +3.83926 q^{34} -9.39416 q^{35} -1.94848 q^{37} -3.96266 q^{38} +7.88071 q^{40} +8.56569 q^{41} -1.06242 q^{43} +0.266949 q^{44} +6.44008 q^{46} +4.78606 q^{47} +4.09593 q^{49} +2.52421 q^{50} +6.11443 q^{52} +4.11306 q^{53} +0.593017 q^{55} -9.30831 q^{56} +1.95853 q^{58} +7.70829 q^{59} +8.53627 q^{61} -6.10820 q^{62} +4.58536 q^{64} +13.5830 q^{65} +16.0563 q^{67} -5.70268 q^{68} -8.02904 q^{70} -5.59508 q^{71} +11.2368 q^{73} -1.66533 q^{74} +5.88597 q^{76} -0.700443 q^{77} -2.49484 q^{79} -0.424988 q^{80} +7.32097 q^{82} +10.5444 q^{83} -12.6683 q^{85} -0.908034 q^{86} +0.587598 q^{88} +4.26861 q^{89} -16.0435 q^{91} -9.56584 q^{92} +4.09057 q^{94} +13.0755 q^{95} +1.01549 q^{97} +3.50072 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.854685 0.604353 0.302177 0.953252i \(-0.402287\pi\)
0.302177 + 0.953252i \(0.402287\pi\)
\(3\) 0 0
\(4\) −1.26951 −0.634757
\(5\) −2.82017 −1.26122 −0.630610 0.776100i \(-0.717195\pi\)
−0.630610 + 0.776100i \(0.717195\pi\)
\(6\) 0 0
\(7\) 3.33105 1.25902 0.629510 0.776992i \(-0.283256\pi\)
0.629510 + 0.776992i \(0.283256\pi\)
\(8\) −2.79440 −0.987971
\(9\) 0 0
\(10\) −2.41036 −0.762223
\(11\) −0.210277 −0.0634008 −0.0317004 0.999497i \(-0.510092\pi\)
−0.0317004 + 0.999497i \(0.510092\pi\)
\(12\) 0 0
\(13\) −4.81635 −1.33582 −0.667908 0.744244i \(-0.732810\pi\)
−0.667908 + 0.744244i \(0.732810\pi\)
\(14\) 2.84700 0.760893
\(15\) 0 0
\(16\) 0.150696 0.0376739
\(17\) 4.49202 1.08947 0.544737 0.838607i \(-0.316629\pi\)
0.544737 + 0.838607i \(0.316629\pi\)
\(18\) 0 0
\(19\) −4.63640 −1.06366 −0.531831 0.846850i \(-0.678496\pi\)
−0.531831 + 0.846850i \(0.678496\pi\)
\(20\) 3.58025 0.800569
\(21\) 0 0
\(22\) −0.179720 −0.0383165
\(23\) 7.53504 1.57116 0.785582 0.618758i \(-0.212364\pi\)
0.785582 + 0.618758i \(0.212364\pi\)
\(24\) 0 0
\(25\) 2.95338 0.590677
\(26\) −4.11646 −0.807305
\(27\) 0 0
\(28\) −4.22882 −0.799172
\(29\) 2.29153 0.425526 0.212763 0.977104i \(-0.431754\pi\)
0.212763 + 0.977104i \(0.431754\pi\)
\(30\) 0 0
\(31\) −7.14674 −1.28359 −0.641796 0.766876i \(-0.721810\pi\)
−0.641796 + 0.766876i \(0.721810\pi\)
\(32\) 5.71760 1.01074
\(33\) 0 0
\(34\) 3.83926 0.658427
\(35\) −9.39416 −1.58790
\(36\) 0 0
\(37\) −1.94848 −0.320327 −0.160164 0.987090i \(-0.551202\pi\)
−0.160164 + 0.987090i \(0.551202\pi\)
\(38\) −3.96266 −0.642828
\(39\) 0 0
\(40\) 7.88071 1.24605
\(41\) 8.56569 1.33774 0.668868 0.743381i \(-0.266779\pi\)
0.668868 + 0.743381i \(0.266779\pi\)
\(42\) 0 0
\(43\) −1.06242 −0.162018 −0.0810088 0.996713i \(-0.525814\pi\)
−0.0810088 + 0.996713i \(0.525814\pi\)
\(44\) 0.266949 0.0402441
\(45\) 0 0
\(46\) 6.44008 0.949538
\(47\) 4.78606 0.698119 0.349060 0.937101i \(-0.386501\pi\)
0.349060 + 0.937101i \(0.386501\pi\)
\(48\) 0 0
\(49\) 4.09593 0.585132
\(50\) 2.52421 0.356977
\(51\) 0 0
\(52\) 6.11443 0.847919
\(53\) 4.11306 0.564973 0.282486 0.959271i \(-0.408841\pi\)
0.282486 + 0.959271i \(0.408841\pi\)
\(54\) 0 0
\(55\) 0.593017 0.0799624
\(56\) −9.30831 −1.24388
\(57\) 0 0
\(58\) 1.95853 0.257168
\(59\) 7.70829 1.00353 0.501767 0.865003i \(-0.332684\pi\)
0.501767 + 0.865003i \(0.332684\pi\)
\(60\) 0 0
\(61\) 8.53627 1.09296 0.546479 0.837473i \(-0.315968\pi\)
0.546479 + 0.837473i \(0.315968\pi\)
\(62\) −6.10820 −0.775743
\(63\) 0 0
\(64\) 4.58536 0.573170
\(65\) 13.5830 1.68476
\(66\) 0 0
\(67\) 16.0563 1.96159 0.980793 0.195053i \(-0.0624879\pi\)
0.980793 + 0.195053i \(0.0624879\pi\)
\(68\) −5.70268 −0.691552
\(69\) 0 0
\(70\) −8.02904 −0.959654
\(71\) −5.59508 −0.664014 −0.332007 0.943277i \(-0.607726\pi\)
−0.332007 + 0.943277i \(0.607726\pi\)
\(72\) 0 0
\(73\) 11.2368 1.31517 0.657584 0.753381i \(-0.271578\pi\)
0.657584 + 0.753381i \(0.271578\pi\)
\(74\) −1.66533 −0.193591
\(75\) 0 0
\(76\) 5.88597 0.675168
\(77\) −0.700443 −0.0798229
\(78\) 0 0
\(79\) −2.49484 −0.280691 −0.140345 0.990103i \(-0.544821\pi\)
−0.140345 + 0.990103i \(0.544821\pi\)
\(80\) −0.424988 −0.0475151
\(81\) 0 0
\(82\) 7.32097 0.808465
\(83\) 10.5444 1.15740 0.578698 0.815542i \(-0.303561\pi\)
0.578698 + 0.815542i \(0.303561\pi\)
\(84\) 0 0
\(85\) −12.6683 −1.37407
\(86\) −0.908034 −0.0979158
\(87\) 0 0
\(88\) 0.587598 0.0626381
\(89\) 4.26861 0.452472 0.226236 0.974073i \(-0.427358\pi\)
0.226236 + 0.974073i \(0.427358\pi\)
\(90\) 0 0
\(91\) −16.0435 −1.68182
\(92\) −9.56584 −0.997308
\(93\) 0 0
\(94\) 4.09057 0.421911
\(95\) 13.0755 1.34151
\(96\) 0 0
\(97\) 1.01549 0.103107 0.0515536 0.998670i \(-0.483583\pi\)
0.0515536 + 0.998670i \(0.483583\pi\)
\(98\) 3.50072 0.353627
\(99\) 0 0
\(100\) −3.74936 −0.374936
\(101\) −15.6127 −1.55352 −0.776760 0.629797i \(-0.783138\pi\)
−0.776760 + 0.629797i \(0.783138\pi\)
\(102\) 0 0
\(103\) 13.4344 1.32373 0.661864 0.749624i \(-0.269766\pi\)
0.661864 + 0.749624i \(0.269766\pi\)
\(104\) 13.4588 1.31975
\(105\) 0 0
\(106\) 3.51537 0.341443
\(107\) −0.322242 −0.0311523 −0.0155762 0.999879i \(-0.504958\pi\)
−0.0155762 + 0.999879i \(0.504958\pi\)
\(108\) 0 0
\(109\) −10.4286 −0.998881 −0.499440 0.866348i \(-0.666461\pi\)
−0.499440 + 0.866348i \(0.666461\pi\)
\(110\) 0.506842 0.0483255
\(111\) 0 0
\(112\) 0.501975 0.0474322
\(113\) −3.87587 −0.364611 −0.182306 0.983242i \(-0.558356\pi\)
−0.182306 + 0.983242i \(0.558356\pi\)
\(114\) 0 0
\(115\) −21.2501 −1.98158
\(116\) −2.90913 −0.270106
\(117\) 0 0
\(118\) 6.58816 0.606489
\(119\) 14.9632 1.37167
\(120\) 0 0
\(121\) −10.9558 −0.995980
\(122\) 7.29582 0.660532
\(123\) 0 0
\(124\) 9.07288 0.814769
\(125\) 5.77181 0.516247
\(126\) 0 0
\(127\) 10.9530 0.971923 0.485962 0.873980i \(-0.338469\pi\)
0.485962 + 0.873980i \(0.338469\pi\)
\(128\) −7.51617 −0.664342
\(129\) 0 0
\(130\) 11.6091 1.01819
\(131\) 6.21012 0.542581 0.271290 0.962498i \(-0.412550\pi\)
0.271290 + 0.962498i \(0.412550\pi\)
\(132\) 0 0
\(133\) −15.4441 −1.33917
\(134\) 13.7230 1.18549
\(135\) 0 0
\(136\) −12.5525 −1.07637
\(137\) −5.68963 −0.486098 −0.243049 0.970014i \(-0.578148\pi\)
−0.243049 + 0.970014i \(0.578148\pi\)
\(138\) 0 0
\(139\) 21.7100 1.84142 0.920708 0.390253i \(-0.127612\pi\)
0.920708 + 0.390253i \(0.127612\pi\)
\(140\) 11.9260 1.00793
\(141\) 0 0
\(142\) −4.78203 −0.401299
\(143\) 1.01277 0.0846918
\(144\) 0 0
\(145\) −6.46251 −0.536683
\(146\) 9.60392 0.794826
\(147\) 0 0
\(148\) 2.47362 0.203330
\(149\) −1.75560 −0.143825 −0.0719123 0.997411i \(-0.522910\pi\)
−0.0719123 + 0.997411i \(0.522910\pi\)
\(150\) 0 0
\(151\) −3.96839 −0.322943 −0.161471 0.986877i \(-0.551624\pi\)
−0.161471 + 0.986877i \(0.551624\pi\)
\(152\) 12.9560 1.05087
\(153\) 0 0
\(154\) −0.598658 −0.0482412
\(155\) 20.1550 1.61889
\(156\) 0 0
\(157\) −0.806844 −0.0643932 −0.0321966 0.999482i \(-0.510250\pi\)
−0.0321966 + 0.999482i \(0.510250\pi\)
\(158\) −2.13230 −0.169636
\(159\) 0 0
\(160\) −16.1246 −1.27476
\(161\) 25.0996 1.97813
\(162\) 0 0
\(163\) −22.8673 −1.79111 −0.895554 0.444953i \(-0.853220\pi\)
−0.895554 + 0.444953i \(0.853220\pi\)
\(164\) −10.8743 −0.849138
\(165\) 0 0
\(166\) 9.01212 0.699476
\(167\) 11.2810 0.872951 0.436475 0.899716i \(-0.356227\pi\)
0.436475 + 0.899716i \(0.356227\pi\)
\(168\) 0 0
\(169\) 10.1973 0.784405
\(170\) −10.8274 −0.830422
\(171\) 0 0
\(172\) 1.34876 0.102842
\(173\) 12.4886 0.949491 0.474745 0.880123i \(-0.342540\pi\)
0.474745 + 0.880123i \(0.342540\pi\)
\(174\) 0 0
\(175\) 9.83789 0.743674
\(176\) −0.0316878 −0.00238856
\(177\) 0 0
\(178\) 3.64831 0.273453
\(179\) 2.96249 0.221427 0.110714 0.993852i \(-0.464686\pi\)
0.110714 + 0.993852i \(0.464686\pi\)
\(180\) 0 0
\(181\) −15.5039 −1.15239 −0.576197 0.817311i \(-0.695464\pi\)
−0.576197 + 0.817311i \(0.695464\pi\)
\(182\) −13.7122 −1.01641
\(183\) 0 0
\(184\) −21.0559 −1.55226
\(185\) 5.49504 0.404003
\(186\) 0 0
\(187\) −0.944566 −0.0690735
\(188\) −6.07598 −0.443136
\(189\) 0 0
\(190\) 11.1754 0.810748
\(191\) −23.3311 −1.68818 −0.844088 0.536205i \(-0.819858\pi\)
−0.844088 + 0.536205i \(0.819858\pi\)
\(192\) 0 0
\(193\) −6.69691 −0.482054 −0.241027 0.970518i \(-0.577484\pi\)
−0.241027 + 0.970518i \(0.577484\pi\)
\(194\) 0.867921 0.0623131
\(195\) 0 0
\(196\) −5.19984 −0.371417
\(197\) −3.29364 −0.234662 −0.117331 0.993093i \(-0.537434\pi\)
−0.117331 + 0.993093i \(0.537434\pi\)
\(198\) 0 0
\(199\) 21.4825 1.52285 0.761427 0.648251i \(-0.224499\pi\)
0.761427 + 0.648251i \(0.224499\pi\)
\(200\) −8.25295 −0.583572
\(201\) 0 0
\(202\) −13.3439 −0.938874
\(203\) 7.63321 0.535746
\(204\) 0 0
\(205\) −24.1568 −1.68718
\(206\) 11.4821 0.799999
\(207\) 0 0
\(208\) −0.725804 −0.0503254
\(209\) 0.974926 0.0674371
\(210\) 0 0
\(211\) 16.6132 1.14370 0.571850 0.820358i \(-0.306226\pi\)
0.571850 + 0.820358i \(0.306226\pi\)
\(212\) −5.22159 −0.358621
\(213\) 0 0
\(214\) −0.275415 −0.0188270
\(215\) 2.99621 0.204340
\(216\) 0 0
\(217\) −23.8062 −1.61607
\(218\) −8.91318 −0.603677
\(219\) 0 0
\(220\) −0.752844 −0.0507567
\(221\) −21.6351 −1.45534
\(222\) 0 0
\(223\) 7.93092 0.531094 0.265547 0.964098i \(-0.414448\pi\)
0.265547 + 0.964098i \(0.414448\pi\)
\(224\) 19.0457 1.27254
\(225\) 0 0
\(226\) −3.31265 −0.220354
\(227\) 28.7427 1.90772 0.953859 0.300254i \(-0.0970715\pi\)
0.953859 + 0.300254i \(0.0970715\pi\)
\(228\) 0 0
\(229\) 2.04148 0.134905 0.0674523 0.997723i \(-0.478513\pi\)
0.0674523 + 0.997723i \(0.478513\pi\)
\(230\) −18.1621 −1.19758
\(231\) 0 0
\(232\) −6.40346 −0.420408
\(233\) −5.14041 −0.336760 −0.168380 0.985722i \(-0.553854\pi\)
−0.168380 + 0.985722i \(0.553854\pi\)
\(234\) 0 0
\(235\) −13.4975 −0.880482
\(236\) −9.78578 −0.637000
\(237\) 0 0
\(238\) 12.7888 0.828973
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 20.7729 1.33810 0.669049 0.743219i \(-0.266702\pi\)
0.669049 + 0.743219i \(0.266702\pi\)
\(242\) −9.36374 −0.601924
\(243\) 0 0
\(244\) −10.8369 −0.693763
\(245\) −11.5512 −0.737981
\(246\) 0 0
\(247\) 22.3305 1.42086
\(248\) 19.9709 1.26815
\(249\) 0 0
\(250\) 4.93308 0.311995
\(251\) −20.7731 −1.31119 −0.655593 0.755114i \(-0.727581\pi\)
−0.655593 + 0.755114i \(0.727581\pi\)
\(252\) 0 0
\(253\) −1.58444 −0.0996131
\(254\) 9.36138 0.587385
\(255\) 0 0
\(256\) −15.5947 −0.974667
\(257\) −10.7431 −0.670134 −0.335067 0.942194i \(-0.608759\pi\)
−0.335067 + 0.942194i \(0.608759\pi\)
\(258\) 0 0
\(259\) −6.49048 −0.403299
\(260\) −17.2438 −1.06941
\(261\) 0 0
\(262\) 5.30769 0.327910
\(263\) −9.69482 −0.597808 −0.298904 0.954283i \(-0.596621\pi\)
−0.298904 + 0.954283i \(0.596621\pi\)
\(264\) 0 0
\(265\) −11.5996 −0.712555
\(266\) −13.1998 −0.809333
\(267\) 0 0
\(268\) −20.3837 −1.24513
\(269\) −18.8231 −1.14766 −0.573832 0.818973i \(-0.694544\pi\)
−0.573832 + 0.818973i \(0.694544\pi\)
\(270\) 0 0
\(271\) −16.2771 −0.988762 −0.494381 0.869245i \(-0.664605\pi\)
−0.494381 + 0.869245i \(0.664605\pi\)
\(272\) 0.676927 0.0410447
\(273\) 0 0
\(274\) −4.86284 −0.293775
\(275\) −0.621028 −0.0374494
\(276\) 0 0
\(277\) 31.7697 1.90886 0.954429 0.298438i \(-0.0964655\pi\)
0.954429 + 0.298438i \(0.0964655\pi\)
\(278\) 18.5552 1.11287
\(279\) 0 0
\(280\) 26.2511 1.56880
\(281\) −26.8446 −1.60142 −0.800709 0.599054i \(-0.795543\pi\)
−0.800709 + 0.599054i \(0.795543\pi\)
\(282\) 0 0
\(283\) −24.2791 −1.44324 −0.721620 0.692289i \(-0.756602\pi\)
−0.721620 + 0.692289i \(0.756602\pi\)
\(284\) 7.10303 0.421488
\(285\) 0 0
\(286\) 0.865596 0.0511838
\(287\) 28.5328 1.68424
\(288\) 0 0
\(289\) 3.17821 0.186954
\(290\) −5.52341 −0.324346
\(291\) 0 0
\(292\) −14.2653 −0.834813
\(293\) 6.67176 0.389768 0.194884 0.980826i \(-0.437567\pi\)
0.194884 + 0.980826i \(0.437567\pi\)
\(294\) 0 0
\(295\) −21.7387 −1.26568
\(296\) 5.44483 0.316474
\(297\) 0 0
\(298\) −1.50049 −0.0869208
\(299\) −36.2914 −2.09879
\(300\) 0 0
\(301\) −3.53898 −0.203983
\(302\) −3.39172 −0.195172
\(303\) 0 0
\(304\) −0.698685 −0.0400723
\(305\) −24.0738 −1.37846
\(306\) 0 0
\(307\) −6.54889 −0.373765 −0.186882 0.982382i \(-0.559838\pi\)
−0.186882 + 0.982382i \(0.559838\pi\)
\(308\) 0.889223 0.0506682
\(309\) 0 0
\(310\) 17.2262 0.978383
\(311\) 1.24119 0.0703815 0.0351907 0.999381i \(-0.488796\pi\)
0.0351907 + 0.999381i \(0.488796\pi\)
\(312\) 0 0
\(313\) −2.96978 −0.167862 −0.0839311 0.996472i \(-0.526748\pi\)
−0.0839311 + 0.996472i \(0.526748\pi\)
\(314\) −0.689597 −0.0389162
\(315\) 0 0
\(316\) 3.16723 0.178171
\(317\) 4.94899 0.277963 0.138982 0.990295i \(-0.455617\pi\)
0.138982 + 0.990295i \(0.455617\pi\)
\(318\) 0 0
\(319\) −0.481855 −0.0269787
\(320\) −12.9315 −0.722893
\(321\) 0 0
\(322\) 21.4523 1.19549
\(323\) −20.8268 −1.15883
\(324\) 0 0
\(325\) −14.2245 −0.789036
\(326\) −19.5444 −1.08246
\(327\) 0 0
\(328\) −23.9360 −1.32164
\(329\) 15.9426 0.878946
\(330\) 0 0
\(331\) 11.8841 0.653211 0.326605 0.945161i \(-0.394095\pi\)
0.326605 + 0.945161i \(0.394095\pi\)
\(332\) −13.3862 −0.734665
\(333\) 0 0
\(334\) 9.64170 0.527570
\(335\) −45.2815 −2.47399
\(336\) 0 0
\(337\) 23.8172 1.29741 0.648703 0.761042i \(-0.275312\pi\)
0.648703 + 0.761042i \(0.275312\pi\)
\(338\) 8.71545 0.474058
\(339\) 0 0
\(340\) 16.0826 0.872199
\(341\) 1.50279 0.0813807
\(342\) 0 0
\(343\) −9.67363 −0.522327
\(344\) 2.96883 0.160069
\(345\) 0 0
\(346\) 10.6738 0.573828
\(347\) −22.0441 −1.18339 −0.591696 0.806162i \(-0.701541\pi\)
−0.591696 + 0.806162i \(0.701541\pi\)
\(348\) 0 0
\(349\) −13.3807 −0.716254 −0.358127 0.933673i \(-0.616585\pi\)
−0.358127 + 0.933673i \(0.616585\pi\)
\(350\) 8.40829 0.449442
\(351\) 0 0
\(352\) −1.20228 −0.0640817
\(353\) 32.7460 1.74289 0.871447 0.490490i \(-0.163182\pi\)
0.871447 + 0.490490i \(0.163182\pi\)
\(354\) 0 0
\(355\) 15.7791 0.837468
\(356\) −5.41906 −0.287210
\(357\) 0 0
\(358\) 2.53200 0.133820
\(359\) 5.72619 0.302217 0.151108 0.988517i \(-0.451716\pi\)
0.151108 + 0.988517i \(0.451716\pi\)
\(360\) 0 0
\(361\) 2.49619 0.131378
\(362\) −13.2509 −0.696453
\(363\) 0 0
\(364\) 20.3675 1.06755
\(365\) −31.6898 −1.65872
\(366\) 0 0
\(367\) 5.40110 0.281935 0.140968 0.990014i \(-0.454979\pi\)
0.140968 + 0.990014i \(0.454979\pi\)
\(368\) 1.13550 0.0591919
\(369\) 0 0
\(370\) 4.69653 0.244161
\(371\) 13.7008 0.711312
\(372\) 0 0
\(373\) 13.1487 0.680813 0.340407 0.940278i \(-0.389435\pi\)
0.340407 + 0.940278i \(0.389435\pi\)
\(374\) −0.807306 −0.0417448
\(375\) 0 0
\(376\) −13.3742 −0.689721
\(377\) −11.0368 −0.568425
\(378\) 0 0
\(379\) −22.2643 −1.14364 −0.571821 0.820379i \(-0.693763\pi\)
−0.571821 + 0.820379i \(0.693763\pi\)
\(380\) −16.5995 −0.851535
\(381\) 0 0
\(382\) −19.9407 −1.02025
\(383\) 35.4884 1.81337 0.906686 0.421806i \(-0.138604\pi\)
0.906686 + 0.421806i \(0.138604\pi\)
\(384\) 0 0
\(385\) 1.97537 0.100674
\(386\) −5.72374 −0.291331
\(387\) 0 0
\(388\) −1.28918 −0.0654480
\(389\) 12.1136 0.614183 0.307091 0.951680i \(-0.400644\pi\)
0.307091 + 0.951680i \(0.400644\pi\)
\(390\) 0 0
\(391\) 33.8475 1.71174
\(392\) −11.4457 −0.578093
\(393\) 0 0
\(394\) −2.81502 −0.141819
\(395\) 7.03587 0.354013
\(396\) 0 0
\(397\) −13.8215 −0.693681 −0.346840 0.937924i \(-0.612745\pi\)
−0.346840 + 0.937924i \(0.612745\pi\)
\(398\) 18.3608 0.920341
\(399\) 0 0
\(400\) 0.445062 0.0222531
\(401\) 10.0053 0.499640 0.249820 0.968292i \(-0.419629\pi\)
0.249820 + 0.968292i \(0.419629\pi\)
\(402\) 0 0
\(403\) 34.4212 1.71464
\(404\) 19.8205 0.986107
\(405\) 0 0
\(406\) 6.52399 0.323780
\(407\) 0.409719 0.0203090
\(408\) 0 0
\(409\) −3.05615 −0.151117 −0.0755585 0.997141i \(-0.524074\pi\)
−0.0755585 + 0.997141i \(0.524074\pi\)
\(410\) −20.6464 −1.01965
\(411\) 0 0
\(412\) −17.0551 −0.840246
\(413\) 25.6767 1.26347
\(414\) 0 0
\(415\) −29.7370 −1.45973
\(416\) −27.5380 −1.35016
\(417\) 0 0
\(418\) 0.833255 0.0407558
\(419\) −12.7158 −0.621209 −0.310605 0.950539i \(-0.600531\pi\)
−0.310605 + 0.950539i \(0.600531\pi\)
\(420\) 0 0
\(421\) −38.6710 −1.88471 −0.942355 0.334616i \(-0.891393\pi\)
−0.942355 + 0.334616i \(0.891393\pi\)
\(422\) 14.1990 0.691199
\(423\) 0 0
\(424\) −11.4936 −0.558177
\(425\) 13.2667 0.643527
\(426\) 0 0
\(427\) 28.4348 1.37606
\(428\) 0.409091 0.0197742
\(429\) 0 0
\(430\) 2.56081 0.123493
\(431\) 15.2077 0.732531 0.366266 0.930510i \(-0.380636\pi\)
0.366266 + 0.930510i \(0.380636\pi\)
\(432\) 0 0
\(433\) −6.68681 −0.321348 −0.160674 0.987008i \(-0.551367\pi\)
−0.160674 + 0.987008i \(0.551367\pi\)
\(434\) −20.3468 −0.976676
\(435\) 0 0
\(436\) 13.2393 0.634047
\(437\) −34.9354 −1.67119
\(438\) 0 0
\(439\) 14.6690 0.700113 0.350057 0.936729i \(-0.386162\pi\)
0.350057 + 0.936729i \(0.386162\pi\)
\(440\) −1.65713 −0.0790005
\(441\) 0 0
\(442\) −18.4912 −0.879538
\(443\) −16.9264 −0.804198 −0.402099 0.915596i \(-0.631719\pi\)
−0.402099 + 0.915596i \(0.631719\pi\)
\(444\) 0 0
\(445\) −12.0382 −0.570667
\(446\) 6.77843 0.320968
\(447\) 0 0
\(448\) 15.2741 0.721632
\(449\) 28.4725 1.34370 0.671850 0.740687i \(-0.265500\pi\)
0.671850 + 0.740687i \(0.265500\pi\)
\(450\) 0 0
\(451\) −1.80117 −0.0848136
\(452\) 4.92047 0.231440
\(453\) 0 0
\(454\) 24.5659 1.15294
\(455\) 45.2456 2.12115
\(456\) 0 0
\(457\) −7.17324 −0.335550 −0.167775 0.985825i \(-0.553658\pi\)
−0.167775 + 0.985825i \(0.553658\pi\)
\(458\) 1.74482 0.0815300
\(459\) 0 0
\(460\) 26.9773 1.25782
\(461\) 17.1051 0.796666 0.398333 0.917241i \(-0.369589\pi\)
0.398333 + 0.917241i \(0.369589\pi\)
\(462\) 0 0
\(463\) 10.2474 0.476235 0.238117 0.971236i \(-0.423470\pi\)
0.238117 + 0.971236i \(0.423470\pi\)
\(464\) 0.345323 0.0160312
\(465\) 0 0
\(466\) −4.39343 −0.203522
\(467\) 31.6949 1.46666 0.733332 0.679871i \(-0.237964\pi\)
0.733332 + 0.679871i \(0.237964\pi\)
\(468\) 0 0
\(469\) 53.4843 2.46968
\(470\) −11.5361 −0.532122
\(471\) 0 0
\(472\) −21.5401 −0.991462
\(473\) 0.223402 0.0102720
\(474\) 0 0
\(475\) −13.6931 −0.628281
\(476\) −18.9959 −0.870677
\(477\) 0 0
\(478\) 0.854685 0.0390924
\(479\) 23.7240 1.08397 0.541987 0.840387i \(-0.317672\pi\)
0.541987 + 0.840387i \(0.317672\pi\)
\(480\) 0 0
\(481\) 9.38455 0.427899
\(482\) 17.7542 0.808683
\(483\) 0 0
\(484\) 13.9085 0.632206
\(485\) −2.86385 −0.130041
\(486\) 0 0
\(487\) 18.1842 0.824006 0.412003 0.911182i \(-0.364829\pi\)
0.412003 + 0.911182i \(0.364829\pi\)
\(488\) −23.8538 −1.07981
\(489\) 0 0
\(490\) −9.87265 −0.446001
\(491\) −27.2627 −1.23035 −0.615174 0.788392i \(-0.710914\pi\)
−0.615174 + 0.788392i \(0.710914\pi\)
\(492\) 0 0
\(493\) 10.2936 0.463600
\(494\) 19.0856 0.858700
\(495\) 0 0
\(496\) −1.07698 −0.0483579
\(497\) −18.6375 −0.836007
\(498\) 0 0
\(499\) −8.57191 −0.383732 −0.191866 0.981421i \(-0.561454\pi\)
−0.191866 + 0.981421i \(0.561454\pi\)
\(500\) −7.32740 −0.327691
\(501\) 0 0
\(502\) −17.7544 −0.792420
\(503\) −23.6499 −1.05450 −0.527249 0.849711i \(-0.676776\pi\)
−0.527249 + 0.849711i \(0.676776\pi\)
\(504\) 0 0
\(505\) 44.0305 1.95933
\(506\) −1.35420 −0.0602015
\(507\) 0 0
\(508\) −13.9050 −0.616935
\(509\) 25.3730 1.12464 0.562319 0.826920i \(-0.309909\pi\)
0.562319 + 0.826920i \(0.309909\pi\)
\(510\) 0 0
\(511\) 37.4304 1.65582
\(512\) 1.70383 0.0752992
\(513\) 0 0
\(514\) −9.18193 −0.404997
\(515\) −37.8873 −1.66951
\(516\) 0 0
\(517\) −1.00640 −0.0442613
\(518\) −5.54731 −0.243735
\(519\) 0 0
\(520\) −37.9563 −1.66449
\(521\) 15.8794 0.695688 0.347844 0.937553i \(-0.386914\pi\)
0.347844 + 0.937553i \(0.386914\pi\)
\(522\) 0 0
\(523\) 16.5175 0.722259 0.361129 0.932516i \(-0.382391\pi\)
0.361129 + 0.932516i \(0.382391\pi\)
\(524\) −7.88384 −0.344407
\(525\) 0 0
\(526\) −8.28601 −0.361287
\(527\) −32.1033 −1.39844
\(528\) 0 0
\(529\) 33.7768 1.46856
\(530\) −9.91396 −0.430635
\(531\) 0 0
\(532\) 19.6065 0.850050
\(533\) −41.2554 −1.78697
\(534\) 0 0
\(535\) 0.908779 0.0392899
\(536\) −44.8677 −1.93799
\(537\) 0 0
\(538\) −16.0878 −0.693594
\(539\) −0.861278 −0.0370979
\(540\) 0 0
\(541\) −43.3424 −1.86344 −0.931719 0.363181i \(-0.881691\pi\)
−0.931719 + 0.363181i \(0.881691\pi\)
\(542\) −13.9118 −0.597562
\(543\) 0 0
\(544\) 25.6836 1.10117
\(545\) 29.4105 1.25981
\(546\) 0 0
\(547\) 18.6320 0.796648 0.398324 0.917245i \(-0.369592\pi\)
0.398324 + 0.917245i \(0.369592\pi\)
\(548\) 7.22307 0.308554
\(549\) 0 0
\(550\) −0.530783 −0.0226327
\(551\) −10.6244 −0.452617
\(552\) 0 0
\(553\) −8.31043 −0.353396
\(554\) 27.1531 1.15362
\(555\) 0 0
\(556\) −27.5611 −1.16885
\(557\) −17.0235 −0.721309 −0.360654 0.932699i \(-0.617447\pi\)
−0.360654 + 0.932699i \(0.617447\pi\)
\(558\) 0 0
\(559\) 5.11699 0.216426
\(560\) −1.41566 −0.0598225
\(561\) 0 0
\(562\) −22.9437 −0.967822
\(563\) −19.5189 −0.822622 −0.411311 0.911495i \(-0.634929\pi\)
−0.411311 + 0.911495i \(0.634929\pi\)
\(564\) 0 0
\(565\) 10.9306 0.459855
\(566\) −20.7509 −0.872227
\(567\) 0 0
\(568\) 15.6349 0.656026
\(569\) 40.5460 1.69978 0.849888 0.526964i \(-0.176670\pi\)
0.849888 + 0.526964i \(0.176670\pi\)
\(570\) 0 0
\(571\) −7.45404 −0.311942 −0.155971 0.987762i \(-0.549851\pi\)
−0.155971 + 0.987762i \(0.549851\pi\)
\(572\) −1.28572 −0.0537588
\(573\) 0 0
\(574\) 24.3865 1.01787
\(575\) 22.2539 0.928050
\(576\) 0 0
\(577\) 3.39347 0.141272 0.0706360 0.997502i \(-0.477497\pi\)
0.0706360 + 0.997502i \(0.477497\pi\)
\(578\) 2.71637 0.112986
\(579\) 0 0
\(580\) 8.20425 0.340663
\(581\) 35.1239 1.45719
\(582\) 0 0
\(583\) −0.864882 −0.0358197
\(584\) −31.4002 −1.29935
\(585\) 0 0
\(586\) 5.70225 0.235558
\(587\) 8.97090 0.370269 0.185134 0.982713i \(-0.440728\pi\)
0.185134 + 0.982713i \(0.440728\pi\)
\(588\) 0 0
\(589\) 33.1351 1.36531
\(590\) −18.5797 −0.764916
\(591\) 0 0
\(592\) −0.293627 −0.0120680
\(593\) 29.9761 1.23097 0.615486 0.788148i \(-0.288960\pi\)
0.615486 + 0.788148i \(0.288960\pi\)
\(594\) 0 0
\(595\) −42.1987 −1.72998
\(596\) 2.22876 0.0912937
\(597\) 0 0
\(598\) −31.0177 −1.26841
\(599\) −33.9162 −1.38578 −0.692890 0.721044i \(-0.743663\pi\)
−0.692890 + 0.721044i \(0.743663\pi\)
\(600\) 0 0
\(601\) −28.1595 −1.14865 −0.574325 0.818628i \(-0.694735\pi\)
−0.574325 + 0.818628i \(0.694735\pi\)
\(602\) −3.02471 −0.123278
\(603\) 0 0
\(604\) 5.03793 0.204990
\(605\) 30.8972 1.25615
\(606\) 0 0
\(607\) −14.1651 −0.574942 −0.287471 0.957789i \(-0.592814\pi\)
−0.287471 + 0.957789i \(0.592814\pi\)
\(608\) −26.5091 −1.07509
\(609\) 0 0
\(610\) −20.5755 −0.833077
\(611\) −23.0514 −0.932559
\(612\) 0 0
\(613\) 21.2591 0.858648 0.429324 0.903151i \(-0.358752\pi\)
0.429324 + 0.903151i \(0.358752\pi\)
\(614\) −5.59724 −0.225886
\(615\) 0 0
\(616\) 1.95732 0.0788627
\(617\) 1.39712 0.0562458 0.0281229 0.999604i \(-0.491047\pi\)
0.0281229 + 0.999604i \(0.491047\pi\)
\(618\) 0 0
\(619\) −7.38678 −0.296900 −0.148450 0.988920i \(-0.547428\pi\)
−0.148450 + 0.988920i \(0.547428\pi\)
\(620\) −25.5871 −1.02760
\(621\) 0 0
\(622\) 1.06083 0.0425353
\(623\) 14.2190 0.569671
\(624\) 0 0
\(625\) −31.0444 −1.24178
\(626\) −2.53823 −0.101448
\(627\) 0 0
\(628\) 1.02430 0.0408740
\(629\) −8.75258 −0.348988
\(630\) 0 0
\(631\) 19.0951 0.760166 0.380083 0.924952i \(-0.375895\pi\)
0.380083 + 0.924952i \(0.375895\pi\)
\(632\) 6.97158 0.277314
\(633\) 0 0
\(634\) 4.22983 0.167988
\(635\) −30.8894 −1.22581
\(636\) 0 0
\(637\) −19.7274 −0.781629
\(638\) −0.411834 −0.0163047
\(639\) 0 0
\(640\) 21.1969 0.837882
\(641\) 41.5685 1.64186 0.820929 0.571030i \(-0.193456\pi\)
0.820929 + 0.571030i \(0.193456\pi\)
\(642\) 0 0
\(643\) 11.4830 0.452845 0.226423 0.974029i \(-0.427297\pi\)
0.226423 + 0.974029i \(0.427297\pi\)
\(644\) −31.8643 −1.25563
\(645\) 0 0
\(646\) −17.8003 −0.700344
\(647\) 39.2841 1.54442 0.772209 0.635369i \(-0.219152\pi\)
0.772209 + 0.635369i \(0.219152\pi\)
\(648\) 0 0
\(649\) −1.62087 −0.0636249
\(650\) −12.1575 −0.476856
\(651\) 0 0
\(652\) 29.0304 1.13692
\(653\) 42.2701 1.65416 0.827079 0.562086i \(-0.190001\pi\)
0.827079 + 0.562086i \(0.190001\pi\)
\(654\) 0 0
\(655\) −17.5136 −0.684314
\(656\) 1.29081 0.0503978
\(657\) 0 0
\(658\) 13.6259 0.531194
\(659\) −27.8321 −1.08419 −0.542093 0.840319i \(-0.682368\pi\)
−0.542093 + 0.840319i \(0.682368\pi\)
\(660\) 0 0
\(661\) −38.8101 −1.50954 −0.754769 0.655990i \(-0.772251\pi\)
−0.754769 + 0.655990i \(0.772251\pi\)
\(662\) 10.1572 0.394770
\(663\) 0 0
\(664\) −29.4652 −1.14347
\(665\) 43.5550 1.68899
\(666\) 0 0
\(667\) 17.2668 0.668572
\(668\) −14.3214 −0.554112
\(669\) 0 0
\(670\) −38.7014 −1.49516
\(671\) −1.79498 −0.0692944
\(672\) 0 0
\(673\) −25.1330 −0.968806 −0.484403 0.874845i \(-0.660963\pi\)
−0.484403 + 0.874845i \(0.660963\pi\)
\(674\) 20.3562 0.784091
\(675\) 0 0
\(676\) −12.9456 −0.497907
\(677\) −40.8531 −1.57011 −0.785055 0.619425i \(-0.787366\pi\)
−0.785055 + 0.619425i \(0.787366\pi\)
\(678\) 0 0
\(679\) 3.38264 0.129814
\(680\) 35.4003 1.35754
\(681\) 0 0
\(682\) 1.28441 0.0491827
\(683\) 4.28639 0.164014 0.0820070 0.996632i \(-0.473867\pi\)
0.0820070 + 0.996632i \(0.473867\pi\)
\(684\) 0 0
\(685\) 16.0458 0.613077
\(686\) −8.26790 −0.315670
\(687\) 0 0
\(688\) −0.160102 −0.00610383
\(689\) −19.8100 −0.754700
\(690\) 0 0
\(691\) 44.6869 1.69997 0.849985 0.526807i \(-0.176611\pi\)
0.849985 + 0.526807i \(0.176611\pi\)
\(692\) −15.8545 −0.602696
\(693\) 0 0
\(694\) −18.8408 −0.715186
\(695\) −61.2259 −2.32243
\(696\) 0 0
\(697\) 38.4772 1.45743
\(698\) −11.4363 −0.432870
\(699\) 0 0
\(700\) −12.4893 −0.472053
\(701\) 42.8739 1.61932 0.809662 0.586896i \(-0.199650\pi\)
0.809662 + 0.586896i \(0.199650\pi\)
\(702\) 0 0
\(703\) 9.03391 0.340720
\(704\) −0.964194 −0.0363394
\(705\) 0 0
\(706\) 27.9875 1.05332
\(707\) −52.0067 −1.95591
\(708\) 0 0
\(709\) 30.5886 1.14878 0.574389 0.818583i \(-0.305240\pi\)
0.574389 + 0.818583i \(0.305240\pi\)
\(710\) 13.4862 0.506126
\(711\) 0 0
\(712\) −11.9282 −0.447029
\(713\) −53.8509 −2.01673
\(714\) 0 0
\(715\) −2.85618 −0.106815
\(716\) −3.76093 −0.140553
\(717\) 0 0
\(718\) 4.89408 0.182646
\(719\) −7.37273 −0.274956 −0.137478 0.990505i \(-0.543900\pi\)
−0.137478 + 0.990505i \(0.543900\pi\)
\(720\) 0 0
\(721\) 44.7506 1.66660
\(722\) 2.13345 0.0793989
\(723\) 0 0
\(724\) 19.6824 0.731490
\(725\) 6.76777 0.251349
\(726\) 0 0
\(727\) 16.3311 0.605687 0.302843 0.953040i \(-0.402064\pi\)
0.302843 + 0.953040i \(0.402064\pi\)
\(728\) 44.8321 1.66159
\(729\) 0 0
\(730\) −27.0847 −1.00245
\(731\) −4.77241 −0.176514
\(732\) 0 0
\(733\) −24.3359 −0.898866 −0.449433 0.893314i \(-0.648374\pi\)
−0.449433 + 0.893314i \(0.648374\pi\)
\(734\) 4.61624 0.170389
\(735\) 0 0
\(736\) 43.0824 1.58804
\(737\) −3.37626 −0.124366
\(738\) 0 0
\(739\) −1.89263 −0.0696215 −0.0348107 0.999394i \(-0.511083\pi\)
−0.0348107 + 0.999394i \(0.511083\pi\)
\(740\) −6.97603 −0.256444
\(741\) 0 0
\(742\) 11.7099 0.429884
\(743\) 51.3232 1.88286 0.941432 0.337203i \(-0.109481\pi\)
0.941432 + 0.337203i \(0.109481\pi\)
\(744\) 0 0
\(745\) 4.95110 0.181394
\(746\) 11.2380 0.411452
\(747\) 0 0
\(748\) 1.19914 0.0438449
\(749\) −1.07341 −0.0392214
\(750\) 0 0
\(751\) −24.9517 −0.910499 −0.455249 0.890364i \(-0.650450\pi\)
−0.455249 + 0.890364i \(0.650450\pi\)
\(752\) 0.721239 0.0263009
\(753\) 0 0
\(754\) −9.43300 −0.343530
\(755\) 11.1915 0.407302
\(756\) 0 0
\(757\) −28.0172 −1.01830 −0.509151 0.860677i \(-0.670041\pi\)
−0.509151 + 0.860677i \(0.670041\pi\)
\(758\) −19.0290 −0.691163
\(759\) 0 0
\(760\) −36.5381 −1.32538
\(761\) −18.8832 −0.684514 −0.342257 0.939606i \(-0.611191\pi\)
−0.342257 + 0.939606i \(0.611191\pi\)
\(762\) 0 0
\(763\) −34.7383 −1.25761
\(764\) 29.6191 1.07158
\(765\) 0 0
\(766\) 30.3314 1.09592
\(767\) −37.1259 −1.34054
\(768\) 0 0
\(769\) 28.6249 1.03224 0.516121 0.856516i \(-0.327376\pi\)
0.516121 + 0.856516i \(0.327376\pi\)
\(770\) 1.68832 0.0608428
\(771\) 0 0
\(772\) 8.50182 0.305987
\(773\) −23.0387 −0.828643 −0.414322 0.910131i \(-0.635981\pi\)
−0.414322 + 0.910131i \(0.635981\pi\)
\(774\) 0 0
\(775\) −21.1071 −0.758188
\(776\) −2.83768 −0.101867
\(777\) 0 0
\(778\) 10.3533 0.371183
\(779\) −39.7140 −1.42290
\(780\) 0 0
\(781\) 1.17651 0.0420990
\(782\) 28.9289 1.03450
\(783\) 0 0
\(784\) 0.617238 0.0220442
\(785\) 2.27544 0.0812140
\(786\) 0 0
\(787\) 14.1619 0.504816 0.252408 0.967621i \(-0.418777\pi\)
0.252408 + 0.967621i \(0.418777\pi\)
\(788\) 4.18132 0.148953
\(789\) 0 0
\(790\) 6.01345 0.213949
\(791\) −12.9107 −0.459053
\(792\) 0 0
\(793\) −41.1137 −1.45999
\(794\) −11.8130 −0.419228
\(795\) 0 0
\(796\) −27.2723 −0.966642
\(797\) −20.7689 −0.735671 −0.367836 0.929891i \(-0.619901\pi\)
−0.367836 + 0.929891i \(0.619901\pi\)
\(798\) 0 0
\(799\) 21.4991 0.760583
\(800\) 16.8863 0.597020
\(801\) 0 0
\(802\) 8.55136 0.301959
\(803\) −2.36284 −0.0833828
\(804\) 0 0
\(805\) −70.7853 −2.49485
\(806\) 29.4193 1.03625
\(807\) 0 0
\(808\) 43.6281 1.53483
\(809\) −11.5690 −0.406746 −0.203373 0.979101i \(-0.565190\pi\)
−0.203373 + 0.979101i \(0.565190\pi\)
\(810\) 0 0
\(811\) −33.8449 −1.18846 −0.594228 0.804296i \(-0.702542\pi\)
−0.594228 + 0.804296i \(0.702542\pi\)
\(812\) −9.69047 −0.340069
\(813\) 0 0
\(814\) 0.350180 0.0122738
\(815\) 64.4899 2.25898
\(816\) 0 0
\(817\) 4.92580 0.172332
\(818\) −2.61205 −0.0913281
\(819\) 0 0
\(820\) 30.6673 1.07095
\(821\) 42.0603 1.46791 0.733957 0.679196i \(-0.237671\pi\)
0.733957 + 0.679196i \(0.237671\pi\)
\(822\) 0 0
\(823\) −0.495945 −0.0172876 −0.00864379 0.999963i \(-0.502751\pi\)
−0.00864379 + 0.999963i \(0.502751\pi\)
\(824\) −37.5410 −1.30780
\(825\) 0 0
\(826\) 21.9455 0.763582
\(827\) −35.9757 −1.25100 −0.625498 0.780226i \(-0.715104\pi\)
−0.625498 + 0.780226i \(0.715104\pi\)
\(828\) 0 0
\(829\) −55.6039 −1.93120 −0.965601 0.260028i \(-0.916268\pi\)
−0.965601 + 0.260028i \(0.916268\pi\)
\(830\) −25.4157 −0.882193
\(831\) 0 0
\(832\) −22.0847 −0.765649
\(833\) 18.3990 0.637486
\(834\) 0 0
\(835\) −31.8144 −1.10098
\(836\) −1.23768 −0.0428062
\(837\) 0 0
\(838\) −10.8680 −0.375430
\(839\) 0.160571 0.00554352 0.00277176 0.999996i \(-0.499118\pi\)
0.00277176 + 0.999996i \(0.499118\pi\)
\(840\) 0 0
\(841\) −23.7489 −0.818927
\(842\) −33.0515 −1.13903
\(843\) 0 0
\(844\) −21.0907 −0.725972
\(845\) −28.7581 −0.989308
\(846\) 0 0
\(847\) −36.4943 −1.25396
\(848\) 0.619821 0.0212847
\(849\) 0 0
\(850\) 11.3388 0.388918
\(851\) −14.6818 −0.503287
\(852\) 0 0
\(853\) 25.5042 0.873246 0.436623 0.899645i \(-0.356174\pi\)
0.436623 + 0.899645i \(0.356174\pi\)
\(854\) 24.3028 0.831624
\(855\) 0 0
\(856\) 0.900474 0.0307776
\(857\) 50.9080 1.73899 0.869493 0.493946i \(-0.164446\pi\)
0.869493 + 0.493946i \(0.164446\pi\)
\(858\) 0 0
\(859\) 18.4338 0.628955 0.314477 0.949265i \(-0.398171\pi\)
0.314477 + 0.949265i \(0.398171\pi\)
\(860\) −3.80373 −0.129706
\(861\) 0 0
\(862\) 12.9978 0.442708
\(863\) 31.1929 1.06182 0.530909 0.847429i \(-0.321851\pi\)
0.530909 + 0.847429i \(0.321851\pi\)
\(864\) 0 0
\(865\) −35.2200 −1.19752
\(866\) −5.71511 −0.194207
\(867\) 0 0
\(868\) 30.2223 1.02581
\(869\) 0.524606 0.0177960
\(870\) 0 0
\(871\) −77.3327 −2.62032
\(872\) 29.1418 0.986865
\(873\) 0 0
\(874\) −29.8588 −1.00999
\(875\) 19.2262 0.649965
\(876\) 0 0
\(877\) 18.1105 0.611548 0.305774 0.952104i \(-0.401085\pi\)
0.305774 + 0.952104i \(0.401085\pi\)
\(878\) 12.5374 0.423116
\(879\) 0 0
\(880\) 0.0893651 0.00301250
\(881\) 25.5190 0.859758 0.429879 0.902887i \(-0.358556\pi\)
0.429879 + 0.902887i \(0.358556\pi\)
\(882\) 0 0
\(883\) 27.9850 0.941770 0.470885 0.882195i \(-0.343935\pi\)
0.470885 + 0.882195i \(0.343935\pi\)
\(884\) 27.4661 0.923786
\(885\) 0 0
\(886\) −14.4667 −0.486020
\(887\) 36.2906 1.21852 0.609260 0.792971i \(-0.291467\pi\)
0.609260 + 0.792971i \(0.291467\pi\)
\(888\) 0 0
\(889\) 36.4851 1.22367
\(890\) −10.2889 −0.344884
\(891\) 0 0
\(892\) −10.0684 −0.337115
\(893\) −22.1901 −0.742563
\(894\) 0 0
\(895\) −8.35475 −0.279269
\(896\) −25.0368 −0.836420
\(897\) 0 0
\(898\) 24.3350 0.812069
\(899\) −16.3770 −0.546202
\(900\) 0 0
\(901\) 18.4760 0.615523
\(902\) −1.53943 −0.0512574
\(903\) 0 0
\(904\) 10.8307 0.360225
\(905\) 43.7236 1.45342
\(906\) 0 0
\(907\) 47.3395 1.57188 0.785941 0.618301i \(-0.212179\pi\)
0.785941 + 0.618301i \(0.212179\pi\)
\(908\) −36.4892 −1.21094
\(909\) 0 0
\(910\) 38.6707 1.28192
\(911\) −19.7190 −0.653319 −0.326660 0.945142i \(-0.605923\pi\)
−0.326660 + 0.945142i \(0.605923\pi\)
\(912\) 0 0
\(913\) −2.21724 −0.0733798
\(914\) −6.13086 −0.202791
\(915\) 0 0
\(916\) −2.59168 −0.0856317
\(917\) 20.6862 0.683120
\(918\) 0 0
\(919\) −11.8733 −0.391665 −0.195833 0.980637i \(-0.562741\pi\)
−0.195833 + 0.980637i \(0.562741\pi\)
\(920\) 59.3814 1.95775
\(921\) 0 0
\(922\) 14.6195 0.481468
\(923\) 26.9479 0.887000
\(924\) 0 0
\(925\) −5.75460 −0.189210
\(926\) 8.75825 0.287814
\(927\) 0 0
\(928\) 13.1021 0.430096
\(929\) 31.5247 1.03429 0.517146 0.855897i \(-0.326994\pi\)
0.517146 + 0.855897i \(0.326994\pi\)
\(930\) 0 0
\(931\) −18.9903 −0.622383
\(932\) 6.52583 0.213761
\(933\) 0 0
\(934\) 27.0891 0.886383
\(935\) 2.66384 0.0871170
\(936\) 0 0
\(937\) 7.82762 0.255717 0.127859 0.991792i \(-0.459190\pi\)
0.127859 + 0.991792i \(0.459190\pi\)
\(938\) 45.7122 1.49256
\(939\) 0 0
\(940\) 17.1353 0.558892
\(941\) 39.5843 1.29041 0.645206 0.764009i \(-0.276772\pi\)
0.645206 + 0.764009i \(0.276772\pi\)
\(942\) 0 0
\(943\) 64.5428 2.10180
\(944\) 1.16161 0.0378070
\(945\) 0 0
\(946\) 0.190938 0.00620794
\(947\) −20.0897 −0.652828 −0.326414 0.945227i \(-0.605840\pi\)
−0.326414 + 0.945227i \(0.605840\pi\)
\(948\) 0 0
\(949\) −54.1204 −1.75682
\(950\) −11.7033 −0.379704
\(951\) 0 0
\(952\) −41.8131 −1.35517
\(953\) 21.7639 0.705003 0.352501 0.935811i \(-0.385331\pi\)
0.352501 + 0.935811i \(0.385331\pi\)
\(954\) 0 0
\(955\) 65.7977 2.12916
\(956\) −1.26951 −0.0410590
\(957\) 0 0
\(958\) 20.2765 0.655104
\(959\) −18.9525 −0.612007
\(960\) 0 0
\(961\) 20.0758 0.647607
\(962\) 8.02083 0.258602
\(963\) 0 0
\(964\) −26.3714 −0.849367
\(965\) 18.8864 0.607976
\(966\) 0 0
\(967\) −6.19414 −0.199190 −0.0995950 0.995028i \(-0.531755\pi\)
−0.0995950 + 0.995028i \(0.531755\pi\)
\(968\) 30.6149 0.983999
\(969\) 0 0
\(970\) −2.44769 −0.0785906
\(971\) 39.2831 1.26065 0.630327 0.776330i \(-0.282921\pi\)
0.630327 + 0.776330i \(0.282921\pi\)
\(972\) 0 0
\(973\) 72.3171 2.31838
\(974\) 15.5418 0.497991
\(975\) 0 0
\(976\) 1.28638 0.0411760
\(977\) 23.6188 0.755633 0.377817 0.925880i \(-0.376675\pi\)
0.377817 + 0.925880i \(0.376675\pi\)
\(978\) 0 0
\(979\) −0.897589 −0.0286871
\(980\) 14.6644 0.468439
\(981\) 0 0
\(982\) −23.3010 −0.743564
\(983\) −50.3413 −1.60564 −0.802819 0.596223i \(-0.796668\pi\)
−0.802819 + 0.596223i \(0.796668\pi\)
\(984\) 0 0
\(985\) 9.28863 0.295960
\(986\) 8.79777 0.280178
\(987\) 0 0
\(988\) −28.3489 −0.901900
\(989\) −8.00537 −0.254556
\(990\) 0 0
\(991\) 17.5107 0.556247 0.278123 0.960545i \(-0.410288\pi\)
0.278123 + 0.960545i \(0.410288\pi\)
\(992\) −40.8622 −1.29738
\(993\) 0 0
\(994\) −15.9292 −0.505243
\(995\) −60.5844 −1.92065
\(996\) 0 0
\(997\) 19.0698 0.603945 0.301973 0.953317i \(-0.402355\pi\)
0.301973 + 0.953317i \(0.402355\pi\)
\(998\) −7.32628 −0.231909
\(999\) 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 2151.2.a.k.1.12 yes 20
3.2 odd 2 2151.2.a.j.1.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.9 20 3.2 odd 2
2151.2.a.k.1.12 yes 20 1.1 even 1 trivial