Properties

Label 2151.2.a.k.1.8
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.8
Root \(-0.510429\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} +O(q^{10})\) \(q-0.510429 q^{2} -1.73946 q^{4} +1.97336 q^{5} -1.30841 q^{7} +1.90873 q^{8} -1.00726 q^{10} -4.91929 q^{11} +1.24517 q^{13} +0.667852 q^{14} +2.50465 q^{16} -0.353562 q^{17} +3.33866 q^{19} -3.43259 q^{20} +2.51095 q^{22} -2.97413 q^{23} -1.10584 q^{25} -0.635571 q^{26} +2.27594 q^{28} +7.75908 q^{29} -3.52991 q^{31} -5.09591 q^{32} +0.180468 q^{34} -2.58198 q^{35} +7.67166 q^{37} -1.70415 q^{38} +3.76662 q^{40} +10.3589 q^{41} -11.1008 q^{43} +8.55693 q^{44} +1.51808 q^{46} +3.54382 q^{47} -5.28805 q^{49} +0.564451 q^{50} -2.16593 q^{52} +6.99423 q^{53} -9.70755 q^{55} -2.49741 q^{56} -3.96046 q^{58} -1.32123 q^{59} +13.0637 q^{61} +1.80177 q^{62} -2.40821 q^{64} +2.45717 q^{65} -12.2786 q^{67} +0.615007 q^{68} +1.31792 q^{70} +12.6828 q^{71} +11.4497 q^{73} -3.91584 q^{74} -5.80746 q^{76} +6.43647 q^{77} +4.16632 q^{79} +4.94259 q^{80} -5.28750 q^{82} +11.7987 q^{83} -0.697706 q^{85} +5.66617 q^{86} -9.38960 q^{88} -3.43216 q^{89} -1.62920 q^{91} +5.17339 q^{92} -1.80887 q^{94} +6.58838 q^{95} +0.321470 q^{97} +2.69918 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.510429 −0.360928 −0.180464 0.983582i \(-0.557760\pi\)
−0.180464 + 0.983582i \(0.557760\pi\)
\(3\) 0 0
\(4\) −1.73946 −0.869731
\(5\) 1.97336 0.882515 0.441257 0.897381i \(-0.354533\pi\)
0.441257 + 0.897381i \(0.354533\pi\)
\(6\) 0 0
\(7\) −1.30841 −0.494534 −0.247267 0.968947i \(-0.579532\pi\)
−0.247267 + 0.968947i \(0.579532\pi\)
\(8\) 1.90873 0.674838
\(9\) 0 0
\(10\) −1.00726 −0.318524
\(11\) −4.91929 −1.48322 −0.741611 0.670830i \(-0.765938\pi\)
−0.741611 + 0.670830i \(0.765938\pi\)
\(12\) 0 0
\(13\) 1.24517 0.345348 0.172674 0.984979i \(-0.444759\pi\)
0.172674 + 0.984979i \(0.444759\pi\)
\(14\) 0.667852 0.178491
\(15\) 0 0
\(16\) 2.50465 0.626163
\(17\) −0.353562 −0.0857513 −0.0428757 0.999080i \(-0.513652\pi\)
−0.0428757 + 0.999080i \(0.513652\pi\)
\(18\) 0 0
\(19\) 3.33866 0.765940 0.382970 0.923761i \(-0.374901\pi\)
0.382970 + 0.923761i \(0.374901\pi\)
\(20\) −3.43259 −0.767551
\(21\) 0 0
\(22\) 2.51095 0.535336
\(23\) −2.97413 −0.620149 −0.310075 0.950712i \(-0.600354\pi\)
−0.310075 + 0.950712i \(0.600354\pi\)
\(24\) 0 0
\(25\) −1.10584 −0.221167
\(26\) −0.635571 −0.124646
\(27\) 0 0
\(28\) 2.27594 0.430112
\(29\) 7.75908 1.44082 0.720412 0.693546i \(-0.243953\pi\)
0.720412 + 0.693546i \(0.243953\pi\)
\(30\) 0 0
\(31\) −3.52991 −0.633990 −0.316995 0.948427i \(-0.602674\pi\)
−0.316995 + 0.948427i \(0.602674\pi\)
\(32\) −5.09591 −0.900838
\(33\) 0 0
\(34\) 0.180468 0.0309500
\(35\) −2.58198 −0.436434
\(36\) 0 0
\(37\) 7.67166 1.26121 0.630607 0.776102i \(-0.282806\pi\)
0.630607 + 0.776102i \(0.282806\pi\)
\(38\) −1.70415 −0.276449
\(39\) 0 0
\(40\) 3.76662 0.595555
\(41\) 10.3589 1.61779 0.808897 0.587950i \(-0.200065\pi\)
0.808897 + 0.587950i \(0.200065\pi\)
\(42\) 0 0
\(43\) −11.1008 −1.69286 −0.846428 0.532503i \(-0.821252\pi\)
−0.846428 + 0.532503i \(0.821252\pi\)
\(44\) 8.55693 1.29001
\(45\) 0 0
\(46\) 1.51808 0.223829
\(47\) 3.54382 0.516919 0.258460 0.966022i \(-0.416785\pi\)
0.258460 + 0.966022i \(0.416785\pi\)
\(48\) 0 0
\(49\) −5.28805 −0.755436
\(50\) 0.564451 0.0798255
\(51\) 0 0
\(52\) −2.16593 −0.300360
\(53\) 6.99423 0.960731 0.480366 0.877068i \(-0.340504\pi\)
0.480366 + 0.877068i \(0.340504\pi\)
\(54\) 0 0
\(55\) −9.70755 −1.30897
\(56\) −2.49741 −0.333730
\(57\) 0 0
\(58\) −3.96046 −0.520034
\(59\) −1.32123 −0.172009 −0.0860047 0.996295i \(-0.527410\pi\)
−0.0860047 + 0.996295i \(0.527410\pi\)
\(60\) 0 0
\(61\) 13.0637 1.67263 0.836317 0.548246i \(-0.184704\pi\)
0.836317 + 0.548246i \(0.184704\pi\)
\(62\) 1.80177 0.228825
\(63\) 0 0
\(64\) −2.40821 −0.301026
\(65\) 2.45717 0.304775
\(66\) 0 0
\(67\) −12.2786 −1.50007 −0.750034 0.661400i \(-0.769963\pi\)
−0.750034 + 0.661400i \(0.769963\pi\)
\(68\) 0.615007 0.0745806
\(69\) 0 0
\(70\) 1.31792 0.157521
\(71\) 12.6828 1.50518 0.752588 0.658492i \(-0.228805\pi\)
0.752588 + 0.658492i \(0.228805\pi\)
\(72\) 0 0
\(73\) 11.4497 1.34008 0.670040 0.742325i \(-0.266277\pi\)
0.670040 + 0.742325i \(0.266277\pi\)
\(74\) −3.91584 −0.455207
\(75\) 0 0
\(76\) −5.80746 −0.666162
\(77\) 6.43647 0.733504
\(78\) 0 0
\(79\) 4.16632 0.468748 0.234374 0.972147i \(-0.424696\pi\)
0.234374 + 0.972147i \(0.424696\pi\)
\(80\) 4.94259 0.552598
\(81\) 0 0
\(82\) −5.28750 −0.583907
\(83\) 11.7987 1.29508 0.647541 0.762031i \(-0.275798\pi\)
0.647541 + 0.762031i \(0.275798\pi\)
\(84\) 0 0
\(85\) −0.697706 −0.0756768
\(86\) 5.66617 0.610999
\(87\) 0 0
\(88\) −9.38960 −1.00094
\(89\) −3.43216 −0.363809 −0.181904 0.983316i \(-0.558226\pi\)
−0.181904 + 0.983316i \(0.558226\pi\)
\(90\) 0 0
\(91\) −1.62920 −0.170786
\(92\) 5.17339 0.539363
\(93\) 0 0
\(94\) −1.80887 −0.186570
\(95\) 6.58838 0.675954
\(96\) 0 0
\(97\) 0.321470 0.0326403 0.0163202 0.999867i \(-0.494805\pi\)
0.0163202 + 0.999867i \(0.494805\pi\)
\(98\) 2.69918 0.272658
\(99\) 0 0
\(100\) 1.92356 0.192356
\(101\) −5.75176 −0.572321 −0.286161 0.958182i \(-0.592379\pi\)
−0.286161 + 0.958182i \(0.592379\pi\)
\(102\) 0 0
\(103\) −13.1895 −1.29960 −0.649799 0.760106i \(-0.725147\pi\)
−0.649799 + 0.760106i \(0.725147\pi\)
\(104\) 2.37669 0.233054
\(105\) 0 0
\(106\) −3.57006 −0.346755
\(107\) 4.87136 0.470932 0.235466 0.971883i \(-0.424338\pi\)
0.235466 + 0.971883i \(0.424338\pi\)
\(108\) 0 0
\(109\) −4.66428 −0.446757 −0.223379 0.974732i \(-0.571709\pi\)
−0.223379 + 0.974732i \(0.571709\pi\)
\(110\) 4.95502 0.472442
\(111\) 0 0
\(112\) −3.27712 −0.309659
\(113\) −2.06815 −0.194555 −0.0972775 0.995257i \(-0.531013\pi\)
−0.0972775 + 0.995257i \(0.531013\pi\)
\(114\) 0 0
\(115\) −5.86904 −0.547291
\(116\) −13.4966 −1.25313
\(117\) 0 0
\(118\) 0.674394 0.0620830
\(119\) 0.462605 0.0424069
\(120\) 0 0
\(121\) 13.1995 1.19995
\(122\) −6.66809 −0.603700
\(123\) 0 0
\(124\) 6.14014 0.551401
\(125\) −12.0490 −1.07770
\(126\) 0 0
\(127\) 19.8504 1.76144 0.880721 0.473635i \(-0.157058\pi\)
0.880721 + 0.473635i \(0.157058\pi\)
\(128\) 11.4210 1.00949
\(129\) 0 0
\(130\) −1.25421 −0.110002
\(131\) 16.1821 1.41384 0.706918 0.707295i \(-0.250085\pi\)
0.706918 + 0.707295i \(0.250085\pi\)
\(132\) 0 0
\(133\) −4.36834 −0.378783
\(134\) 6.26734 0.541416
\(135\) 0 0
\(136\) −0.674854 −0.0578682
\(137\) −5.15387 −0.440324 −0.220162 0.975463i \(-0.570659\pi\)
−0.220162 + 0.975463i \(0.570659\pi\)
\(138\) 0 0
\(139\) −9.66367 −0.819662 −0.409831 0.912162i \(-0.634412\pi\)
−0.409831 + 0.912162i \(0.634412\pi\)
\(140\) 4.49125 0.379580
\(141\) 0 0
\(142\) −6.47369 −0.543260
\(143\) −6.12535 −0.512228
\(144\) 0 0
\(145\) 15.3115 1.27155
\(146\) −5.84424 −0.483672
\(147\) 0 0
\(148\) −13.3446 −1.09692
\(149\) 18.4553 1.51192 0.755958 0.654620i \(-0.227171\pi\)
0.755958 + 0.654620i \(0.227171\pi\)
\(150\) 0 0
\(151\) 21.9774 1.78849 0.894246 0.447576i \(-0.147713\pi\)
0.894246 + 0.447576i \(0.147713\pi\)
\(152\) 6.37259 0.516885
\(153\) 0 0
\(154\) −3.28536 −0.264742
\(155\) −6.96579 −0.559506
\(156\) 0 0
\(157\) 17.6338 1.40733 0.703664 0.710532i \(-0.251546\pi\)
0.703664 + 0.710532i \(0.251546\pi\)
\(158\) −2.12661 −0.169184
\(159\) 0 0
\(160\) −10.0561 −0.795003
\(161\) 3.89139 0.306685
\(162\) 0 0
\(163\) 14.0892 1.10355 0.551775 0.833993i \(-0.313951\pi\)
0.551775 + 0.833993i \(0.313951\pi\)
\(164\) −18.0190 −1.40705
\(165\) 0 0
\(166\) −6.02242 −0.467431
\(167\) 15.5247 1.20133 0.600667 0.799499i \(-0.294902\pi\)
0.600667 + 0.799499i \(0.294902\pi\)
\(168\) 0 0
\(169\) −11.4496 −0.880735
\(170\) 0.356129 0.0273139
\(171\) 0 0
\(172\) 19.3094 1.47233
\(173\) 24.7262 1.87990 0.939951 0.341310i \(-0.110871\pi\)
0.939951 + 0.341310i \(0.110871\pi\)
\(174\) 0 0
\(175\) 1.44689 0.109375
\(176\) −12.3211 −0.928740
\(177\) 0 0
\(178\) 1.75188 0.131309
\(179\) 20.0246 1.49671 0.748354 0.663299i \(-0.230844\pi\)
0.748354 + 0.663299i \(0.230844\pi\)
\(180\) 0 0
\(181\) −9.47348 −0.704158 −0.352079 0.935970i \(-0.614525\pi\)
−0.352079 + 0.935970i \(0.614525\pi\)
\(182\) 0.831589 0.0616415
\(183\) 0 0
\(184\) −5.67681 −0.418500
\(185\) 15.1390 1.11304
\(186\) 0 0
\(187\) 1.73927 0.127188
\(188\) −6.16434 −0.449581
\(189\) 0 0
\(190\) −3.36290 −0.243970
\(191\) −11.6015 −0.839453 −0.419726 0.907651i \(-0.637874\pi\)
−0.419726 + 0.907651i \(0.637874\pi\)
\(192\) 0 0
\(193\) −19.7244 −1.41979 −0.709896 0.704307i \(-0.751258\pi\)
−0.709896 + 0.704307i \(0.751258\pi\)
\(194\) −0.164087 −0.0117808
\(195\) 0 0
\(196\) 9.19837 0.657026
\(197\) 3.37371 0.240367 0.120184 0.992752i \(-0.461652\pi\)
0.120184 + 0.992752i \(0.461652\pi\)
\(198\) 0 0
\(199\) 15.5040 1.09905 0.549524 0.835478i \(-0.314809\pi\)
0.549524 + 0.835478i \(0.314809\pi\)
\(200\) −2.11074 −0.149252
\(201\) 0 0
\(202\) 2.93586 0.206567
\(203\) −10.1521 −0.712537
\(204\) 0 0
\(205\) 20.4419 1.42773
\(206\) 6.73229 0.469061
\(207\) 0 0
\(208\) 3.11872 0.216244
\(209\) −16.4238 −1.13606
\(210\) 0 0
\(211\) 8.23128 0.566665 0.283332 0.959022i \(-0.408560\pi\)
0.283332 + 0.959022i \(0.408560\pi\)
\(212\) −12.1662 −0.835578
\(213\) 0 0
\(214\) −2.48648 −0.169973
\(215\) −21.9059 −1.49397
\(216\) 0 0
\(217\) 4.61858 0.313530
\(218\) 2.38078 0.161247
\(219\) 0 0
\(220\) 16.8859 1.13845
\(221\) −0.440244 −0.0296140
\(222\) 0 0
\(223\) 8.24532 0.552148 0.276074 0.961136i \(-0.410967\pi\)
0.276074 + 0.961136i \(0.410967\pi\)
\(224\) 6.66756 0.445495
\(225\) 0 0
\(226\) 1.05564 0.0702203
\(227\) −4.91650 −0.326319 −0.163160 0.986600i \(-0.552169\pi\)
−0.163160 + 0.986600i \(0.552169\pi\)
\(228\) 0 0
\(229\) 0.733030 0.0484400 0.0242200 0.999707i \(-0.492290\pi\)
0.0242200 + 0.999707i \(0.492290\pi\)
\(230\) 2.99573 0.197532
\(231\) 0 0
\(232\) 14.8100 0.972323
\(233\) 16.6162 1.08856 0.544281 0.838903i \(-0.316803\pi\)
0.544281 + 0.838903i \(0.316803\pi\)
\(234\) 0 0
\(235\) 6.99324 0.456189
\(236\) 2.29823 0.149602
\(237\) 0 0
\(238\) −0.236127 −0.0153058
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −9.85009 −0.634500 −0.317250 0.948342i \(-0.602759\pi\)
−0.317250 + 0.948342i \(0.602759\pi\)
\(242\) −6.73738 −0.433095
\(243\) 0 0
\(244\) −22.7238 −1.45474
\(245\) −10.4352 −0.666684
\(246\) 0 0
\(247\) 4.15719 0.264516
\(248\) −6.73764 −0.427841
\(249\) 0 0
\(250\) 6.15018 0.388971
\(251\) 13.2011 0.833249 0.416624 0.909079i \(-0.363213\pi\)
0.416624 + 0.909079i \(0.363213\pi\)
\(252\) 0 0
\(253\) 14.6306 0.919819
\(254\) −10.1322 −0.635753
\(255\) 0 0
\(256\) −1.01321 −0.0633257
\(257\) −17.8240 −1.11183 −0.555914 0.831240i \(-0.687632\pi\)
−0.555914 + 0.831240i \(0.687632\pi\)
\(258\) 0 0
\(259\) −10.0377 −0.623713
\(260\) −4.27416 −0.265072
\(261\) 0 0
\(262\) −8.25981 −0.510293
\(263\) −7.70672 −0.475217 −0.237608 0.971361i \(-0.576363\pi\)
−0.237608 + 0.971361i \(0.576363\pi\)
\(264\) 0 0
\(265\) 13.8022 0.847860
\(266\) 2.22973 0.136713
\(267\) 0 0
\(268\) 21.3581 1.30466
\(269\) 16.4022 1.00006 0.500029 0.866009i \(-0.333323\pi\)
0.500029 + 0.866009i \(0.333323\pi\)
\(270\) 0 0
\(271\) −29.1174 −1.76876 −0.884378 0.466772i \(-0.845417\pi\)
−0.884378 + 0.466772i \(0.845417\pi\)
\(272\) −0.885549 −0.0536943
\(273\) 0 0
\(274\) 2.63068 0.158925
\(275\) 5.43994 0.328041
\(276\) 0 0
\(277\) −14.5818 −0.876134 −0.438067 0.898942i \(-0.644337\pi\)
−0.438067 + 0.898942i \(0.644337\pi\)
\(278\) 4.93262 0.295839
\(279\) 0 0
\(280\) −4.92829 −0.294522
\(281\) −12.4685 −0.743807 −0.371903 0.928271i \(-0.621295\pi\)
−0.371903 + 0.928271i \(0.621295\pi\)
\(282\) 0 0
\(283\) −12.8582 −0.764343 −0.382172 0.924091i \(-0.624824\pi\)
−0.382172 + 0.924091i \(0.624824\pi\)
\(284\) −22.0613 −1.30910
\(285\) 0 0
\(286\) 3.12656 0.184877
\(287\) −13.5538 −0.800054
\(288\) 0 0
\(289\) −16.8750 −0.992647
\(290\) −7.81542 −0.458937
\(291\) 0 0
\(292\) −19.9162 −1.16551
\(293\) 16.1140 0.941389 0.470694 0.882296i \(-0.344003\pi\)
0.470694 + 0.882296i \(0.344003\pi\)
\(294\) 0 0
\(295\) −2.60726 −0.151801
\(296\) 14.6431 0.851115
\(297\) 0 0
\(298\) −9.42012 −0.545693
\(299\) −3.70330 −0.214167
\(300\) 0 0
\(301\) 14.5244 0.837175
\(302\) −11.2179 −0.645516
\(303\) 0 0
\(304\) 8.36217 0.479604
\(305\) 25.7794 1.47612
\(306\) 0 0
\(307\) −16.7468 −0.955791 −0.477896 0.878417i \(-0.658600\pi\)
−0.477896 + 0.878417i \(0.658600\pi\)
\(308\) −11.1960 −0.637951
\(309\) 0 0
\(310\) 3.55554 0.201941
\(311\) 13.2617 0.752004 0.376002 0.926619i \(-0.377299\pi\)
0.376002 + 0.926619i \(0.377299\pi\)
\(312\) 0 0
\(313\) −5.47215 −0.309304 −0.154652 0.987969i \(-0.549426\pi\)
−0.154652 + 0.987969i \(0.549426\pi\)
\(314\) −9.00079 −0.507944
\(315\) 0 0
\(316\) −7.24716 −0.407684
\(317\) −2.83295 −0.159114 −0.0795572 0.996830i \(-0.525351\pi\)
−0.0795572 + 0.996830i \(0.525351\pi\)
\(318\) 0 0
\(319\) −38.1692 −2.13706
\(320\) −4.75227 −0.265660
\(321\) 0 0
\(322\) −1.98628 −0.110691
\(323\) −1.18042 −0.0656804
\(324\) 0 0
\(325\) −1.37695 −0.0763797
\(326\) −7.19153 −0.398302
\(327\) 0 0
\(328\) 19.7724 1.09175
\(329\) −4.63678 −0.255634
\(330\) 0 0
\(331\) −18.6580 −1.02553 −0.512767 0.858528i \(-0.671380\pi\)
−0.512767 + 0.858528i \(0.671380\pi\)
\(332\) −20.5235 −1.12637
\(333\) 0 0
\(334\) −7.92424 −0.433595
\(335\) −24.2301 −1.32383
\(336\) 0 0
\(337\) 12.6388 0.688478 0.344239 0.938882i \(-0.388137\pi\)
0.344239 + 0.938882i \(0.388137\pi\)
\(338\) 5.84418 0.317882
\(339\) 0 0
\(340\) 1.21363 0.0658185
\(341\) 17.3647 0.940349
\(342\) 0 0
\(343\) 16.0779 0.868123
\(344\) −21.1884 −1.14240
\(345\) 0 0
\(346\) −12.6210 −0.678509
\(347\) −27.2616 −1.46348 −0.731739 0.681585i \(-0.761291\pi\)
−0.731739 + 0.681585i \(0.761291\pi\)
\(348\) 0 0
\(349\) −0.710696 −0.0380427 −0.0190213 0.999819i \(-0.506055\pi\)
−0.0190213 + 0.999819i \(0.506055\pi\)
\(350\) −0.738536 −0.0394764
\(351\) 0 0
\(352\) 25.0683 1.33614
\(353\) −10.8851 −0.579353 −0.289677 0.957125i \(-0.593548\pi\)
−0.289677 + 0.957125i \(0.593548\pi\)
\(354\) 0 0
\(355\) 25.0278 1.32834
\(356\) 5.97012 0.316416
\(357\) 0 0
\(358\) −10.2211 −0.540204
\(359\) −34.6231 −1.82734 −0.913669 0.406460i \(-0.866763\pi\)
−0.913669 + 0.406460i \(0.866763\pi\)
\(360\) 0 0
\(361\) −7.85338 −0.413336
\(362\) 4.83554 0.254150
\(363\) 0 0
\(364\) 2.83393 0.148538
\(365\) 22.5943 1.18264
\(366\) 0 0
\(367\) 21.6577 1.13052 0.565261 0.824912i \(-0.308775\pi\)
0.565261 + 0.824912i \(0.308775\pi\)
\(368\) −7.44917 −0.388315
\(369\) 0 0
\(370\) −7.72737 −0.401727
\(371\) −9.15135 −0.475114
\(372\) 0 0
\(373\) 27.2659 1.41177 0.705887 0.708324i \(-0.250548\pi\)
0.705887 + 0.708324i \(0.250548\pi\)
\(374\) −0.887776 −0.0459058
\(375\) 0 0
\(376\) 6.76419 0.348837
\(377\) 9.66137 0.497586
\(378\) 0 0
\(379\) 3.94609 0.202697 0.101348 0.994851i \(-0.467684\pi\)
0.101348 + 0.994851i \(0.467684\pi\)
\(380\) −11.4602 −0.587898
\(381\) 0 0
\(382\) 5.92172 0.302982
\(383\) 19.0059 0.971157 0.485578 0.874193i \(-0.338609\pi\)
0.485578 + 0.874193i \(0.338609\pi\)
\(384\) 0 0
\(385\) 12.7015 0.647328
\(386\) 10.0679 0.512442
\(387\) 0 0
\(388\) −0.559185 −0.0283883
\(389\) 19.7747 1.00262 0.501309 0.865268i \(-0.332852\pi\)
0.501309 + 0.865268i \(0.332852\pi\)
\(390\) 0 0
\(391\) 1.05154 0.0531786
\(392\) −10.0935 −0.509797
\(393\) 0 0
\(394\) −1.72204 −0.0867552
\(395\) 8.22166 0.413677
\(396\) 0 0
\(397\) −33.9206 −1.70243 −0.851213 0.524821i \(-0.824132\pi\)
−0.851213 + 0.524821i \(0.824132\pi\)
\(398\) −7.91368 −0.396677
\(399\) 0 0
\(400\) −2.76974 −0.138487
\(401\) 4.19358 0.209417 0.104709 0.994503i \(-0.466609\pi\)
0.104709 + 0.994503i \(0.466609\pi\)
\(402\) 0 0
\(403\) −4.39533 −0.218947
\(404\) 10.0050 0.497766
\(405\) 0 0
\(406\) 5.18192 0.257174
\(407\) −37.7392 −1.87066
\(408\) 0 0
\(409\) −16.3446 −0.808187 −0.404094 0.914718i \(-0.632413\pi\)
−0.404094 + 0.914718i \(0.632413\pi\)
\(410\) −10.4342 −0.515306
\(411\) 0 0
\(412\) 22.9426 1.13030
\(413\) 1.72871 0.0850645
\(414\) 0 0
\(415\) 23.2832 1.14293
\(416\) −6.34527 −0.311102
\(417\) 0 0
\(418\) 8.38320 0.410036
\(419\) 21.6479 1.05757 0.528786 0.848755i \(-0.322648\pi\)
0.528786 + 0.848755i \(0.322648\pi\)
\(420\) 0 0
\(421\) −0.857053 −0.0417702 −0.0208851 0.999782i \(-0.506648\pi\)
−0.0208851 + 0.999782i \(0.506648\pi\)
\(422\) −4.20148 −0.204525
\(423\) 0 0
\(424\) 13.3501 0.648338
\(425\) 0.390982 0.0189654
\(426\) 0 0
\(427\) −17.0927 −0.827174
\(428\) −8.47355 −0.409585
\(429\) 0 0
\(430\) 11.1814 0.539216
\(431\) −3.54552 −0.170782 −0.0853908 0.996348i \(-0.527214\pi\)
−0.0853908 + 0.996348i \(0.527214\pi\)
\(432\) 0 0
\(433\) −6.61178 −0.317742 −0.158871 0.987299i \(-0.550785\pi\)
−0.158871 + 0.987299i \(0.550785\pi\)
\(434\) −2.35746 −0.113162
\(435\) 0 0
\(436\) 8.11334 0.388559
\(437\) −9.92960 −0.474997
\(438\) 0 0
\(439\) −31.0403 −1.48147 −0.740736 0.671796i \(-0.765523\pi\)
−0.740736 + 0.671796i \(0.765523\pi\)
\(440\) −18.5291 −0.883340
\(441\) 0 0
\(442\) 0.224713 0.0106885
\(443\) 10.6671 0.506811 0.253406 0.967360i \(-0.418449\pi\)
0.253406 + 0.967360i \(0.418449\pi\)
\(444\) 0 0
\(445\) −6.77291 −0.321066
\(446\) −4.20865 −0.199285
\(447\) 0 0
\(448\) 3.15093 0.148868
\(449\) −8.35664 −0.394374 −0.197187 0.980366i \(-0.563181\pi\)
−0.197187 + 0.980366i \(0.563181\pi\)
\(450\) 0 0
\(451\) −50.9586 −2.39955
\(452\) 3.59746 0.169210
\(453\) 0 0
\(454\) 2.50952 0.117778
\(455\) −3.21500 −0.150721
\(456\) 0 0
\(457\) 1.46849 0.0686930 0.0343465 0.999410i \(-0.489065\pi\)
0.0343465 + 0.999410i \(0.489065\pi\)
\(458\) −0.374160 −0.0174833
\(459\) 0 0
\(460\) 10.2090 0.475996
\(461\) −42.2951 −1.96988 −0.984940 0.172896i \(-0.944687\pi\)
−0.984940 + 0.172896i \(0.944687\pi\)
\(462\) 0 0
\(463\) 26.1426 1.21495 0.607475 0.794339i \(-0.292182\pi\)
0.607475 + 0.794339i \(0.292182\pi\)
\(464\) 19.4338 0.902192
\(465\) 0 0
\(466\) −8.48138 −0.392892
\(467\) −19.1402 −0.885705 −0.442852 0.896595i \(-0.646033\pi\)
−0.442852 + 0.896595i \(0.646033\pi\)
\(468\) 0 0
\(469\) 16.0655 0.741834
\(470\) −3.56955 −0.164651
\(471\) 0 0
\(472\) −2.52187 −0.116078
\(473\) 54.6081 2.51088
\(474\) 0 0
\(475\) −3.69201 −0.169401
\(476\) −0.804684 −0.0368826
\(477\) 0 0
\(478\) −0.510429 −0.0233465
\(479\) 26.2033 1.19726 0.598629 0.801026i \(-0.295712\pi\)
0.598629 + 0.801026i \(0.295712\pi\)
\(480\) 0 0
\(481\) 9.55252 0.435557
\(482\) 5.02777 0.229009
\(483\) 0 0
\(484\) −22.9599 −1.04363
\(485\) 0.634377 0.0288056
\(486\) 0 0
\(487\) −40.7502 −1.84657 −0.923284 0.384119i \(-0.874505\pi\)
−0.923284 + 0.384119i \(0.874505\pi\)
\(488\) 24.9351 1.12876
\(489\) 0 0
\(490\) 5.32645 0.240625
\(491\) −6.89804 −0.311304 −0.155652 0.987812i \(-0.549748\pi\)
−0.155652 + 0.987812i \(0.549748\pi\)
\(492\) 0 0
\(493\) −2.74331 −0.123553
\(494\) −2.12195 −0.0954711
\(495\) 0 0
\(496\) −8.84120 −0.396982
\(497\) −16.5944 −0.744360
\(498\) 0 0
\(499\) 21.4957 0.962281 0.481140 0.876644i \(-0.340223\pi\)
0.481140 + 0.876644i \(0.340223\pi\)
\(500\) 20.9588 0.937308
\(501\) 0 0
\(502\) −6.73825 −0.300743
\(503\) −24.9171 −1.11100 −0.555499 0.831517i \(-0.687473\pi\)
−0.555499 + 0.831517i \(0.687473\pi\)
\(504\) 0 0
\(505\) −11.3503 −0.505082
\(506\) −7.46789 −0.331988
\(507\) 0 0
\(508\) −34.5291 −1.53198
\(509\) −36.1616 −1.60284 −0.801418 0.598105i \(-0.795921\pi\)
−0.801418 + 0.598105i \(0.795921\pi\)
\(510\) 0 0
\(511\) −14.9809 −0.662715
\(512\) −22.3249 −0.986630
\(513\) 0 0
\(514\) 9.09787 0.401290
\(515\) −26.0276 −1.14691
\(516\) 0 0
\(517\) −17.4331 −0.766706
\(518\) 5.12354 0.225115
\(519\) 0 0
\(520\) 4.69008 0.205673
\(521\) 27.6929 1.21325 0.606625 0.794988i \(-0.292523\pi\)
0.606625 + 0.794988i \(0.292523\pi\)
\(522\) 0 0
\(523\) 24.9136 1.08939 0.544697 0.838633i \(-0.316645\pi\)
0.544697 + 0.838633i \(0.316645\pi\)
\(524\) −28.1482 −1.22966
\(525\) 0 0
\(526\) 3.93373 0.171519
\(527\) 1.24804 0.0543655
\(528\) 0 0
\(529\) −14.1545 −0.615415
\(530\) −7.04502 −0.306016
\(531\) 0 0
\(532\) 7.59857 0.329440
\(533\) 12.8986 0.558702
\(534\) 0 0
\(535\) 9.61297 0.415605
\(536\) −23.4365 −1.01230
\(537\) 0 0
\(538\) −8.37214 −0.360949
\(539\) 26.0135 1.12048
\(540\) 0 0
\(541\) 38.8319 1.66951 0.834757 0.550619i \(-0.185608\pi\)
0.834757 + 0.550619i \(0.185608\pi\)
\(542\) 14.8624 0.638393
\(543\) 0 0
\(544\) 1.80172 0.0772480
\(545\) −9.20432 −0.394270
\(546\) 0 0
\(547\) 22.6886 0.970093 0.485046 0.874488i \(-0.338803\pi\)
0.485046 + 0.874488i \(0.338803\pi\)
\(548\) 8.96496 0.382964
\(549\) 0 0
\(550\) −2.77670 −0.118399
\(551\) 25.9049 1.10359
\(552\) 0 0
\(553\) −5.45127 −0.231812
\(554\) 7.44296 0.316221
\(555\) 0 0
\(556\) 16.8096 0.712885
\(557\) −42.1075 −1.78415 −0.892077 0.451883i \(-0.850752\pi\)
−0.892077 + 0.451883i \(0.850752\pi\)
\(558\) 0 0
\(559\) −13.8224 −0.584624
\(560\) −6.46696 −0.273279
\(561\) 0 0
\(562\) 6.36427 0.268461
\(563\) −39.8210 −1.67825 −0.839127 0.543936i \(-0.816933\pi\)
−0.839127 + 0.543936i \(0.816933\pi\)
\(564\) 0 0
\(565\) −4.08121 −0.171698
\(566\) 6.56322 0.275873
\(567\) 0 0
\(568\) 24.2081 1.01575
\(569\) 32.6387 1.36828 0.684142 0.729349i \(-0.260177\pi\)
0.684142 + 0.729349i \(0.260177\pi\)
\(570\) 0 0
\(571\) −9.33796 −0.390781 −0.195391 0.980725i \(-0.562597\pi\)
−0.195391 + 0.980725i \(0.562597\pi\)
\(572\) 10.6548 0.445501
\(573\) 0 0
\(574\) 6.91824 0.288762
\(575\) 3.28890 0.137157
\(576\) 0 0
\(577\) −21.3634 −0.889371 −0.444686 0.895687i \(-0.646685\pi\)
−0.444686 + 0.895687i \(0.646685\pi\)
\(578\) 8.61349 0.358274
\(579\) 0 0
\(580\) −26.6337 −1.10591
\(581\) −15.4376 −0.640462
\(582\) 0 0
\(583\) −34.4067 −1.42498
\(584\) 21.8543 0.904337
\(585\) 0 0
\(586\) −8.22504 −0.339773
\(587\) −34.4889 −1.42351 −0.711755 0.702427i \(-0.752100\pi\)
−0.711755 + 0.702427i \(0.752100\pi\)
\(588\) 0 0
\(589\) −11.7851 −0.485599
\(590\) 1.33082 0.0547891
\(591\) 0 0
\(592\) 19.2149 0.789726
\(593\) 38.2385 1.57027 0.785134 0.619326i \(-0.212594\pi\)
0.785134 + 0.619326i \(0.212594\pi\)
\(594\) 0 0
\(595\) 0.912888 0.0374247
\(596\) −32.1023 −1.31496
\(597\) 0 0
\(598\) 1.89027 0.0772989
\(599\) 1.50195 0.0613680 0.0306840 0.999529i \(-0.490231\pi\)
0.0306840 + 0.999529i \(0.490231\pi\)
\(600\) 0 0
\(601\) 1.49985 0.0611800 0.0305900 0.999532i \(-0.490261\pi\)
0.0305900 + 0.999532i \(0.490261\pi\)
\(602\) −7.41370 −0.302160
\(603\) 0 0
\(604\) −38.2288 −1.55551
\(605\) 26.0473 1.05897
\(606\) 0 0
\(607\) −12.2316 −0.496464 −0.248232 0.968701i \(-0.579849\pi\)
−0.248232 + 0.968701i \(0.579849\pi\)
\(608\) −17.0135 −0.689988
\(609\) 0 0
\(610\) −13.1586 −0.532774
\(611\) 4.41266 0.178517
\(612\) 0 0
\(613\) 23.9996 0.969335 0.484667 0.874698i \(-0.338941\pi\)
0.484667 + 0.874698i \(0.338941\pi\)
\(614\) 8.54806 0.344972
\(615\) 0 0
\(616\) 12.2855 0.494996
\(617\) −4.04588 −0.162881 −0.0814405 0.996678i \(-0.525952\pi\)
−0.0814405 + 0.996678i \(0.525952\pi\)
\(618\) 0 0
\(619\) 11.8427 0.476000 0.238000 0.971265i \(-0.423508\pi\)
0.238000 + 0.971265i \(0.423508\pi\)
\(620\) 12.1167 0.486620
\(621\) 0 0
\(622\) −6.76917 −0.271419
\(623\) 4.49069 0.179916
\(624\) 0 0
\(625\) −18.2479 −0.729918
\(626\) 2.79315 0.111637
\(627\) 0 0
\(628\) −30.6733 −1.22400
\(629\) −2.71241 −0.108151
\(630\) 0 0
\(631\) 41.6870 1.65953 0.829767 0.558110i \(-0.188473\pi\)
0.829767 + 0.558110i \(0.188473\pi\)
\(632\) 7.95238 0.316329
\(633\) 0 0
\(634\) 1.44602 0.0574288
\(635\) 39.1721 1.55450
\(636\) 0 0
\(637\) −6.58452 −0.260888
\(638\) 19.4827 0.771326
\(639\) 0 0
\(640\) 22.5378 0.890887
\(641\) −41.8667 −1.65363 −0.826817 0.562471i \(-0.809851\pi\)
−0.826817 + 0.562471i \(0.809851\pi\)
\(642\) 0 0
\(643\) −31.1939 −1.23017 −0.615085 0.788461i \(-0.710878\pi\)
−0.615085 + 0.788461i \(0.710878\pi\)
\(644\) −6.76893 −0.266733
\(645\) 0 0
\(646\) 0.602521 0.0237059
\(647\) 11.5784 0.455195 0.227597 0.973755i \(-0.426913\pi\)
0.227597 + 0.973755i \(0.426913\pi\)
\(648\) 0 0
\(649\) 6.49951 0.255128
\(650\) 0.702838 0.0275676
\(651\) 0 0
\(652\) −24.5076 −0.959792
\(653\) 3.04123 0.119012 0.0595062 0.998228i \(-0.481047\pi\)
0.0595062 + 0.998228i \(0.481047\pi\)
\(654\) 0 0
\(655\) 31.9332 1.24773
\(656\) 25.9455 1.01300
\(657\) 0 0
\(658\) 2.36675 0.0922655
\(659\) 15.2081 0.592423 0.296212 0.955122i \(-0.404277\pi\)
0.296212 + 0.955122i \(0.404277\pi\)
\(660\) 0 0
\(661\) 2.84857 0.110797 0.0553983 0.998464i \(-0.482357\pi\)
0.0553983 + 0.998464i \(0.482357\pi\)
\(662\) 9.52356 0.370144
\(663\) 0 0
\(664\) 22.5206 0.873970
\(665\) −8.62033 −0.334282
\(666\) 0 0
\(667\) −23.0765 −0.893526
\(668\) −27.0046 −1.04484
\(669\) 0 0
\(670\) 12.3677 0.477808
\(671\) −64.2641 −2.48089
\(672\) 0 0
\(673\) −34.2012 −1.31836 −0.659181 0.751985i \(-0.729097\pi\)
−0.659181 + 0.751985i \(0.729097\pi\)
\(674\) −6.45120 −0.248491
\(675\) 0 0
\(676\) 19.9161 0.766003
\(677\) −24.7943 −0.952922 −0.476461 0.879196i \(-0.658081\pi\)
−0.476461 + 0.879196i \(0.658081\pi\)
\(678\) 0 0
\(679\) −0.420616 −0.0161417
\(680\) −1.33173 −0.0510696
\(681\) 0 0
\(682\) −8.86342 −0.339398
\(683\) 2.51062 0.0960662 0.0480331 0.998846i \(-0.484705\pi\)
0.0480331 + 0.998846i \(0.484705\pi\)
\(684\) 0 0
\(685\) −10.1704 −0.388593
\(686\) −8.20661 −0.313330
\(687\) 0 0
\(688\) −27.8037 −1.06000
\(689\) 8.70900 0.331786
\(690\) 0 0
\(691\) 13.0442 0.496226 0.248113 0.968731i \(-0.420190\pi\)
0.248113 + 0.968731i \(0.420190\pi\)
\(692\) −43.0104 −1.63501
\(693\) 0 0
\(694\) 13.9151 0.528210
\(695\) −19.0699 −0.723364
\(696\) 0 0
\(697\) −3.66252 −0.138728
\(698\) 0.362760 0.0137307
\(699\) 0 0
\(700\) −2.51682 −0.0951267
\(701\) −8.84820 −0.334192 −0.167096 0.985941i \(-0.553439\pi\)
−0.167096 + 0.985941i \(0.553439\pi\)
\(702\) 0 0
\(703\) 25.6130 0.966014
\(704\) 11.8467 0.446489
\(705\) 0 0
\(706\) 5.55605 0.209105
\(707\) 7.52568 0.283032
\(708\) 0 0
\(709\) 29.5125 1.10837 0.554183 0.832395i \(-0.313031\pi\)
0.554183 + 0.832395i \(0.313031\pi\)
\(710\) −12.7749 −0.479435
\(711\) 0 0
\(712\) −6.55107 −0.245512
\(713\) 10.4984 0.393169
\(714\) 0 0
\(715\) −12.0875 −0.452049
\(716\) −34.8320 −1.30173
\(717\) 0 0
\(718\) 17.6726 0.659537
\(719\) −10.5343 −0.392861 −0.196431 0.980518i \(-0.562935\pi\)
−0.196431 + 0.980518i \(0.562935\pi\)
\(720\) 0 0
\(721\) 17.2573 0.642695
\(722\) 4.00859 0.149184
\(723\) 0 0
\(724\) 16.4788 0.612429
\(725\) −8.58028 −0.318664
\(726\) 0 0
\(727\) 13.6078 0.504684 0.252342 0.967638i \(-0.418799\pi\)
0.252342 + 0.967638i \(0.418799\pi\)
\(728\) −3.10970 −0.115253
\(729\) 0 0
\(730\) −11.5328 −0.426848
\(731\) 3.92482 0.145165
\(732\) 0 0
\(733\) 26.1557 0.966083 0.483041 0.875597i \(-0.339532\pi\)
0.483041 + 0.875597i \(0.339532\pi\)
\(734\) −11.0547 −0.408037
\(735\) 0 0
\(736\) 15.1559 0.558654
\(737\) 60.4019 2.22493
\(738\) 0 0
\(739\) 18.5120 0.680976 0.340488 0.940249i \(-0.389408\pi\)
0.340488 + 0.940249i \(0.389408\pi\)
\(740\) −26.3337 −0.968045
\(741\) 0 0
\(742\) 4.67111 0.171482
\(743\) 3.57989 0.131333 0.0656667 0.997842i \(-0.479083\pi\)
0.0656667 + 0.997842i \(0.479083\pi\)
\(744\) 0 0
\(745\) 36.4190 1.33429
\(746\) −13.9173 −0.509549
\(747\) 0 0
\(748\) −3.02540 −0.110620
\(749\) −6.37376 −0.232892
\(750\) 0 0
\(751\) 24.0523 0.877681 0.438841 0.898565i \(-0.355389\pi\)
0.438841 + 0.898565i \(0.355389\pi\)
\(752\) 8.87604 0.323676
\(753\) 0 0
\(754\) −4.93144 −0.179593
\(755\) 43.3693 1.57837
\(756\) 0 0
\(757\) 1.29824 0.0471855 0.0235927 0.999722i \(-0.492490\pi\)
0.0235927 + 0.999722i \(0.492490\pi\)
\(758\) −2.01420 −0.0731589
\(759\) 0 0
\(760\) 12.5754 0.456159
\(761\) −1.07425 −0.0389414 −0.0194707 0.999810i \(-0.506198\pi\)
−0.0194707 + 0.999810i \(0.506198\pi\)
\(762\) 0 0
\(763\) 6.10281 0.220937
\(764\) 20.1803 0.730098
\(765\) 0 0
\(766\) −9.70117 −0.350517
\(767\) −1.64515 −0.0594031
\(768\) 0 0
\(769\) 12.5505 0.452584 0.226292 0.974060i \(-0.427340\pi\)
0.226292 + 0.974060i \(0.427340\pi\)
\(770\) −6.48321 −0.233639
\(771\) 0 0
\(772\) 34.3098 1.23484
\(773\) −8.72300 −0.313744 −0.156872 0.987619i \(-0.550141\pi\)
−0.156872 + 0.987619i \(0.550141\pi\)
\(774\) 0 0
\(775\) 3.90350 0.140218
\(776\) 0.613599 0.0220269
\(777\) 0 0
\(778\) −10.0936 −0.361873
\(779\) 34.5849 1.23913
\(780\) 0 0
\(781\) −62.3906 −2.23251
\(782\) −0.536736 −0.0191936
\(783\) 0 0
\(784\) −13.2447 −0.473026
\(785\) 34.7979 1.24199
\(786\) 0 0
\(787\) −13.4078 −0.477937 −0.238969 0.971027i \(-0.576809\pi\)
−0.238969 + 0.971027i \(0.576809\pi\)
\(788\) −5.86845 −0.209055
\(789\) 0 0
\(790\) −4.19658 −0.149307
\(791\) 2.70599 0.0962140
\(792\) 0 0
\(793\) 16.2665 0.577641
\(794\) 17.3140 0.614453
\(795\) 0 0
\(796\) −26.9686 −0.955876
\(797\) 28.0679 0.994215 0.497107 0.867689i \(-0.334395\pi\)
0.497107 + 0.867689i \(0.334395\pi\)
\(798\) 0 0
\(799\) −1.25296 −0.0443265
\(800\) 5.63524 0.199236
\(801\) 0 0
\(802\) −2.14053 −0.0755846
\(803\) −56.3242 −1.98764
\(804\) 0 0
\(805\) 7.67913 0.270654
\(806\) 2.24351 0.0790241
\(807\) 0 0
\(808\) −10.9786 −0.386224
\(809\) 56.3769 1.98211 0.991053 0.133468i \(-0.0426112\pi\)
0.991053 + 0.133468i \(0.0426112\pi\)
\(810\) 0 0
\(811\) 39.6998 1.39405 0.697025 0.717047i \(-0.254507\pi\)
0.697025 + 0.717047i \(0.254507\pi\)
\(812\) 17.6592 0.619715
\(813\) 0 0
\(814\) 19.2632 0.675174
\(815\) 27.8031 0.973900
\(816\) 0 0
\(817\) −37.0617 −1.29663
\(818\) 8.34275 0.291697
\(819\) 0 0
\(820\) −35.5580 −1.24174
\(821\) 0.289440 0.0101015 0.00505077 0.999987i \(-0.498392\pi\)
0.00505077 + 0.999987i \(0.498392\pi\)
\(822\) 0 0
\(823\) 41.7663 1.45588 0.727942 0.685639i \(-0.240477\pi\)
0.727942 + 0.685639i \(0.240477\pi\)
\(824\) −25.1752 −0.877018
\(825\) 0 0
\(826\) −0.882386 −0.0307021
\(827\) 7.69068 0.267431 0.133716 0.991020i \(-0.457309\pi\)
0.133716 + 0.991020i \(0.457309\pi\)
\(828\) 0 0
\(829\) −5.46418 −0.189779 −0.0948894 0.995488i \(-0.530250\pi\)
−0.0948894 + 0.995488i \(0.530250\pi\)
\(830\) −11.8844 −0.412515
\(831\) 0 0
\(832\) −2.99863 −0.103959
\(833\) 1.86965 0.0647796
\(834\) 0 0
\(835\) 30.6358 1.06020
\(836\) 28.5686 0.988067
\(837\) 0 0
\(838\) −11.0497 −0.381707
\(839\) −45.3313 −1.56501 −0.782506 0.622643i \(-0.786059\pi\)
−0.782506 + 0.622643i \(0.786059\pi\)
\(840\) 0 0
\(841\) 31.2033 1.07598
\(842\) 0.437465 0.0150760
\(843\) 0 0
\(844\) −14.3180 −0.492846
\(845\) −22.5941 −0.777262
\(846\) 0 0
\(847\) −17.2703 −0.593416
\(848\) 17.5181 0.601575
\(849\) 0 0
\(850\) −0.199568 −0.00684514
\(851\) −22.8165 −0.782141
\(852\) 0 0
\(853\) −0.228143 −0.00781146 −0.00390573 0.999992i \(-0.501243\pi\)
−0.00390573 + 0.999992i \(0.501243\pi\)
\(854\) 8.72462 0.298550
\(855\) 0 0
\(856\) 9.29812 0.317803
\(857\) −44.4962 −1.51996 −0.759980 0.649946i \(-0.774791\pi\)
−0.759980 + 0.649946i \(0.774791\pi\)
\(858\) 0 0
\(859\) −32.6841 −1.11517 −0.557583 0.830121i \(-0.688271\pi\)
−0.557583 + 0.830121i \(0.688271\pi\)
\(860\) 38.1045 1.29935
\(861\) 0 0
\(862\) 1.80973 0.0616398
\(863\) −39.9025 −1.35830 −0.679148 0.734001i \(-0.737651\pi\)
−0.679148 + 0.734001i \(0.737651\pi\)
\(864\) 0 0
\(865\) 48.7939 1.65904
\(866\) 3.37484 0.114682
\(867\) 0 0
\(868\) −8.03385 −0.272687
\(869\) −20.4954 −0.695257
\(870\) 0 0
\(871\) −15.2889 −0.518045
\(872\) −8.90285 −0.301489
\(873\) 0 0
\(874\) 5.06835 0.171440
\(875\) 15.7651 0.532959
\(876\) 0 0
\(877\) −4.27574 −0.144382 −0.0721908 0.997391i \(-0.522999\pi\)
−0.0721908 + 0.997391i \(0.522999\pi\)
\(878\) 15.8439 0.534705
\(879\) 0 0
\(880\) −24.3141 −0.819627
\(881\) −38.9219 −1.31131 −0.655655 0.755060i \(-0.727608\pi\)
−0.655655 + 0.755060i \(0.727608\pi\)
\(882\) 0 0
\(883\) 8.70306 0.292881 0.146441 0.989219i \(-0.453218\pi\)
0.146441 + 0.989219i \(0.453218\pi\)
\(884\) 0.765788 0.0257562
\(885\) 0 0
\(886\) −5.44482 −0.182922
\(887\) −9.32121 −0.312976 −0.156488 0.987680i \(-0.550017\pi\)
−0.156488 + 0.987680i \(0.550017\pi\)
\(888\) 0 0
\(889\) −25.9726 −0.871093
\(890\) 3.45709 0.115882
\(891\) 0 0
\(892\) −14.3424 −0.480220
\(893\) 11.8316 0.395929
\(894\) 0 0
\(895\) 39.5158 1.32087
\(896\) −14.9434 −0.499225
\(897\) 0 0
\(898\) 4.26547 0.142341
\(899\) −27.3888 −0.913469
\(900\) 0 0
\(901\) −2.47289 −0.0823839
\(902\) 26.0108 0.866064
\(903\) 0 0
\(904\) −3.94753 −0.131293
\(905\) −18.6946 −0.621430
\(906\) 0 0
\(907\) 10.7219 0.356016 0.178008 0.984029i \(-0.443035\pi\)
0.178008 + 0.984029i \(0.443035\pi\)
\(908\) 8.55206 0.283810
\(909\) 0 0
\(910\) 1.64103 0.0543996
\(911\) 44.4272 1.47194 0.735970 0.677014i \(-0.236726\pi\)
0.735970 + 0.677014i \(0.236726\pi\)
\(912\) 0 0
\(913\) −58.0415 −1.92089
\(914\) −0.749560 −0.0247932
\(915\) 0 0
\(916\) −1.27508 −0.0421297
\(917\) −21.1729 −0.699190
\(918\) 0 0
\(919\) 11.3809 0.375421 0.187710 0.982224i \(-0.439893\pi\)
0.187710 + 0.982224i \(0.439893\pi\)
\(920\) −11.2024 −0.369333
\(921\) 0 0
\(922\) 21.5887 0.710984
\(923\) 15.7923 0.519809
\(924\) 0 0
\(925\) −8.48361 −0.278939
\(926\) −13.3440 −0.438509
\(927\) 0 0
\(928\) −39.5395 −1.29795
\(929\) −12.7353 −0.417831 −0.208916 0.977934i \(-0.566993\pi\)
−0.208916 + 0.977934i \(0.566993\pi\)
\(930\) 0 0
\(931\) −17.6550 −0.578619
\(932\) −28.9032 −0.946756
\(933\) 0 0
\(934\) 9.76974 0.319676
\(935\) 3.43222 0.112246
\(936\) 0 0
\(937\) −22.8143 −0.745312 −0.372656 0.927970i \(-0.621553\pi\)
−0.372656 + 0.927970i \(0.621553\pi\)
\(938\) −8.20028 −0.267749
\(939\) 0 0
\(940\) −12.1645 −0.396762
\(941\) 41.4690 1.35185 0.675925 0.736971i \(-0.263744\pi\)
0.675925 + 0.736971i \(0.263744\pi\)
\(942\) 0 0
\(943\) −30.8088 −1.00327
\(944\) −3.30922 −0.107706
\(945\) 0 0
\(946\) −27.8736 −0.906247
\(947\) 27.4984 0.893578 0.446789 0.894639i \(-0.352568\pi\)
0.446789 + 0.894639i \(0.352568\pi\)
\(948\) 0 0
\(949\) 14.2568 0.462794
\(950\) 1.88451 0.0611415
\(951\) 0 0
\(952\) 0.882988 0.0286178
\(953\) 22.5938 0.731887 0.365943 0.930637i \(-0.380746\pi\)
0.365943 + 0.930637i \(0.380746\pi\)
\(954\) 0 0
\(955\) −22.8939 −0.740829
\(956\) −1.73946 −0.0562582
\(957\) 0 0
\(958\) −13.3749 −0.432124
\(959\) 6.74339 0.217755
\(960\) 0 0
\(961\) −18.5397 −0.598056
\(962\) −4.87588 −0.157205
\(963\) 0 0
\(964\) 17.1339 0.551845
\(965\) −38.9233 −1.25299
\(966\) 0 0
\(967\) 1.80872 0.0581645 0.0290823 0.999577i \(-0.490742\pi\)
0.0290823 + 0.999577i \(0.490742\pi\)
\(968\) 25.1942 0.809772
\(969\) 0 0
\(970\) −0.323804 −0.0103967
\(971\) −16.7063 −0.536132 −0.268066 0.963401i \(-0.586384\pi\)
−0.268066 + 0.963401i \(0.586384\pi\)
\(972\) 0 0
\(973\) 12.6441 0.405351
\(974\) 20.8001 0.666477
\(975\) 0 0
\(976\) 32.7200 1.04734
\(977\) 40.8168 1.30585 0.652923 0.757424i \(-0.273543\pi\)
0.652923 + 0.757424i \(0.273543\pi\)
\(978\) 0 0
\(979\) 16.8838 0.539609
\(980\) 18.1517 0.579836
\(981\) 0 0
\(982\) 3.52096 0.112358
\(983\) 30.0618 0.958822 0.479411 0.877591i \(-0.340850\pi\)
0.479411 + 0.877591i \(0.340850\pi\)
\(984\) 0 0
\(985\) 6.65756 0.212128
\(986\) 1.40027 0.0445936
\(987\) 0 0
\(988\) −7.23128 −0.230058
\(989\) 33.0152 1.04982
\(990\) 0 0
\(991\) −1.64335 −0.0522028 −0.0261014 0.999659i \(-0.508309\pi\)
−0.0261014 + 0.999659i \(0.508309\pi\)
\(992\) 17.9881 0.571122
\(993\) 0 0
\(994\) 8.47026 0.268660
\(995\) 30.5950 0.969926
\(996\) 0 0
\(997\) −44.6421 −1.41383 −0.706915 0.707299i \(-0.749914\pi\)
−0.706915 + 0.707299i \(0.749914\pi\)
\(998\) −10.9720 −0.347314
\(999\) 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.1.8 yes 20
3.2 odd 2 2151.2.a.j.1.13 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.13 20 3.2 odd 2
2151.2.a.k.1.8 yes 20 1.1 even 1 trivial