Properties

Label 2151.2.a.j.1.19
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
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: \(1\)
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 1.19
Root \(-2.12763\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.12763 q^{2} +2.52683 q^{4} -2.10198 q^{5} +0.789580 q^{7} +1.12089 q^{8} +O(q^{10})\) \(q+2.12763 q^{2} +2.52683 q^{4} -2.10198 q^{5} +0.789580 q^{7} +1.12089 q^{8} -4.47224 q^{10} -4.48898 q^{11} -4.31061 q^{13} +1.67994 q^{14} -2.66880 q^{16} +6.64316 q^{17} +3.39928 q^{19} -5.31133 q^{20} -9.55090 q^{22} -9.17525 q^{23} -0.581692 q^{25} -9.17139 q^{26} +1.99513 q^{28} -0.384188 q^{29} +2.78887 q^{31} -7.92002 q^{32} +14.1342 q^{34} -1.65968 q^{35} -0.131728 q^{37} +7.23242 q^{38} -2.35609 q^{40} -8.64441 q^{41} -3.94739 q^{43} -11.3429 q^{44} -19.5216 q^{46} -13.3918 q^{47} -6.37656 q^{49} -1.23763 q^{50} -10.8921 q^{52} -0.625549 q^{53} +9.43573 q^{55} +0.885034 q^{56} -0.817411 q^{58} +7.17980 q^{59} +6.78363 q^{61} +5.93370 q^{62} -11.5133 q^{64} +9.06079 q^{65} +0.301693 q^{67} +16.7861 q^{68} -3.53119 q^{70} -2.81573 q^{71} -2.28817 q^{73} -0.280269 q^{74} +8.58938 q^{76} -3.54441 q^{77} +14.2938 q^{79} +5.60977 q^{80} -18.3921 q^{82} +14.1111 q^{83} -13.9638 q^{85} -8.39860 q^{86} -5.03166 q^{88} -12.1171 q^{89} -3.40357 q^{91} -23.1843 q^{92} -28.4927 q^{94} -7.14521 q^{95} -10.1442 q^{97} -13.5670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12763 1.50446 0.752232 0.658898i \(-0.228977\pi\)
0.752232 + 0.658898i \(0.228977\pi\)
\(3\) 0 0
\(4\) 2.52683 1.26341
\(5\) −2.10198 −0.940033 −0.470016 0.882658i \(-0.655752\pi\)
−0.470016 + 0.882658i \(0.655752\pi\)
\(6\) 0 0
\(7\) 0.789580 0.298433 0.149217 0.988805i \(-0.452325\pi\)
0.149217 + 0.988805i \(0.452325\pi\)
\(8\) 1.12089 0.396295
\(9\) 0 0
\(10\) −4.47224 −1.41425
\(11\) −4.48898 −1.35348 −0.676739 0.736223i \(-0.736607\pi\)
−0.676739 + 0.736223i \(0.736607\pi\)
\(12\) 0 0
\(13\) −4.31061 −1.19555 −0.597773 0.801665i \(-0.703948\pi\)
−0.597773 + 0.801665i \(0.703948\pi\)
\(14\) 1.67994 0.448982
\(15\) 0 0
\(16\) −2.66880 −0.667201
\(17\) 6.64316 1.61120 0.805602 0.592458i \(-0.201842\pi\)
0.805602 + 0.592458i \(0.201842\pi\)
\(18\) 0 0
\(19\) 3.39928 0.779848 0.389924 0.920847i \(-0.372501\pi\)
0.389924 + 0.920847i \(0.372501\pi\)
\(20\) −5.31133 −1.18765
\(21\) 0 0
\(22\) −9.55090 −2.03626
\(23\) −9.17525 −1.91317 −0.956586 0.291451i \(-0.905862\pi\)
−0.956586 + 0.291451i \(0.905862\pi\)
\(24\) 0 0
\(25\) −0.581692 −0.116338
\(26\) −9.17139 −1.79866
\(27\) 0 0
\(28\) 1.99513 0.377045
\(29\) −0.384188 −0.0713418 −0.0356709 0.999364i \(-0.511357\pi\)
−0.0356709 + 0.999364i \(0.511357\pi\)
\(30\) 0 0
\(31\) 2.78887 0.500896 0.250448 0.968130i \(-0.419422\pi\)
0.250448 + 0.968130i \(0.419422\pi\)
\(32\) −7.92002 −1.40008
\(33\) 0 0
\(34\) 14.1342 2.42400
\(35\) −1.65968 −0.280537
\(36\) 0 0
\(37\) −0.131728 −0.0216560 −0.0108280 0.999941i \(-0.503447\pi\)
−0.0108280 + 0.999941i \(0.503447\pi\)
\(38\) 7.23242 1.17325
\(39\) 0 0
\(40\) −2.35609 −0.372530
\(41\) −8.64441 −1.35003 −0.675015 0.737804i \(-0.735863\pi\)
−0.675015 + 0.737804i \(0.735863\pi\)
\(42\) 0 0
\(43\) −3.94739 −0.601971 −0.300986 0.953629i \(-0.597316\pi\)
−0.300986 + 0.953629i \(0.597316\pi\)
\(44\) −11.3429 −1.71000
\(45\) 0 0
\(46\) −19.5216 −2.87830
\(47\) −13.3918 −1.95339 −0.976694 0.214637i \(-0.931143\pi\)
−0.976694 + 0.214637i \(0.931143\pi\)
\(48\) 0 0
\(49\) −6.37656 −0.910938
\(50\) −1.23763 −0.175027
\(51\) 0 0
\(52\) −10.8921 −1.51047
\(53\) −0.625549 −0.0859257 −0.0429629 0.999077i \(-0.513680\pi\)
−0.0429629 + 0.999077i \(0.513680\pi\)
\(54\) 0 0
\(55\) 9.43573 1.27231
\(56\) 0.885034 0.118268
\(57\) 0 0
\(58\) −0.817411 −0.107331
\(59\) 7.17980 0.934731 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(60\) 0 0
\(61\) 6.78363 0.868555 0.434278 0.900779i \(-0.357004\pi\)
0.434278 + 0.900779i \(0.357004\pi\)
\(62\) 5.93370 0.753580
\(63\) 0 0
\(64\) −11.5133 −1.43916
\(65\) 9.06079 1.12385
\(66\) 0 0
\(67\) 0.301693 0.0368576 0.0184288 0.999830i \(-0.494134\pi\)
0.0184288 + 0.999830i \(0.494134\pi\)
\(68\) 16.7861 2.03561
\(69\) 0 0
\(70\) −3.53119 −0.422058
\(71\) −2.81573 −0.334166 −0.167083 0.985943i \(-0.553435\pi\)
−0.167083 + 0.985943i \(0.553435\pi\)
\(72\) 0 0
\(73\) −2.28817 −0.267810 −0.133905 0.990994i \(-0.542752\pi\)
−0.133905 + 0.990994i \(0.542752\pi\)
\(74\) −0.280269 −0.0325807
\(75\) 0 0
\(76\) 8.58938 0.985270
\(77\) −3.54441 −0.403923
\(78\) 0 0
\(79\) 14.2938 1.60818 0.804090 0.594508i \(-0.202653\pi\)
0.804090 + 0.594508i \(0.202653\pi\)
\(80\) 5.60977 0.627191
\(81\) 0 0
\(82\) −18.3921 −2.03107
\(83\) 14.1111 1.54890 0.774450 0.632635i \(-0.218027\pi\)
0.774450 + 0.632635i \(0.218027\pi\)
\(84\) 0 0
\(85\) −13.9638 −1.51458
\(86\) −8.39860 −0.905644
\(87\) 0 0
\(88\) −5.03166 −0.536377
\(89\) −12.1171 −1.28441 −0.642207 0.766531i \(-0.721981\pi\)
−0.642207 + 0.766531i \(0.721981\pi\)
\(90\) 0 0
\(91\) −3.40357 −0.356791
\(92\) −23.1843 −2.41713
\(93\) 0 0
\(94\) −28.4927 −2.93880
\(95\) −7.14521 −0.733083
\(96\) 0 0
\(97\) −10.1442 −1.02999 −0.514993 0.857194i \(-0.672206\pi\)
−0.514993 + 0.857194i \(0.672206\pi\)
\(98\) −13.5670 −1.37047
\(99\) 0 0
\(100\) −1.46984 −0.146984
\(101\) −4.78785 −0.476409 −0.238205 0.971215i \(-0.576559\pi\)
−0.238205 + 0.971215i \(0.576559\pi\)
\(102\) 0 0
\(103\) 15.2517 1.50280 0.751399 0.659848i \(-0.229379\pi\)
0.751399 + 0.659848i \(0.229379\pi\)
\(104\) −4.83172 −0.473789
\(105\) 0 0
\(106\) −1.33094 −0.129272
\(107\) 13.0380 1.26043 0.630217 0.776419i \(-0.282966\pi\)
0.630217 + 0.776419i \(0.282966\pi\)
\(108\) 0 0
\(109\) 6.33358 0.606647 0.303324 0.952888i \(-0.401904\pi\)
0.303324 + 0.952888i \(0.401904\pi\)
\(110\) 20.0758 1.91415
\(111\) 0 0
\(112\) −2.10724 −0.199115
\(113\) 14.7090 1.38371 0.691854 0.722038i \(-0.256794\pi\)
0.691854 + 0.722038i \(0.256794\pi\)
\(114\) 0 0
\(115\) 19.2862 1.79844
\(116\) −0.970775 −0.0901342
\(117\) 0 0
\(118\) 15.2760 1.40627
\(119\) 5.24531 0.480837
\(120\) 0 0
\(121\) 9.15094 0.831903
\(122\) 14.4331 1.30671
\(123\) 0 0
\(124\) 7.04699 0.632838
\(125\) 11.7326 1.04939
\(126\) 0 0
\(127\) 12.3025 1.09167 0.545837 0.837892i \(-0.316212\pi\)
0.545837 + 0.837892i \(0.316212\pi\)
\(128\) −8.65603 −0.765093
\(129\) 0 0
\(130\) 19.2781 1.69080
\(131\) −18.2760 −1.59678 −0.798389 0.602142i \(-0.794314\pi\)
−0.798389 + 0.602142i \(0.794314\pi\)
\(132\) 0 0
\(133\) 2.68400 0.232733
\(134\) 0.641892 0.0554510
\(135\) 0 0
\(136\) 7.44626 0.638512
\(137\) −13.4590 −1.14988 −0.574938 0.818197i \(-0.694974\pi\)
−0.574938 + 0.818197i \(0.694974\pi\)
\(138\) 0 0
\(139\) −14.8779 −1.26192 −0.630962 0.775814i \(-0.717340\pi\)
−0.630962 + 0.775814i \(0.717340\pi\)
\(140\) −4.19372 −0.354434
\(141\) 0 0
\(142\) −5.99084 −0.502740
\(143\) 19.3502 1.61815
\(144\) 0 0
\(145\) 0.807554 0.0670637
\(146\) −4.86838 −0.402910
\(147\) 0 0
\(148\) −0.332854 −0.0273604
\(149\) 21.7502 1.78185 0.890923 0.454154i \(-0.150058\pi\)
0.890923 + 0.454154i \(0.150058\pi\)
\(150\) 0 0
\(151\) −9.40234 −0.765151 −0.382576 0.923924i \(-0.624963\pi\)
−0.382576 + 0.923924i \(0.624963\pi\)
\(152\) 3.81022 0.309050
\(153\) 0 0
\(154\) −7.54121 −0.607688
\(155\) −5.86214 −0.470859
\(156\) 0 0
\(157\) −3.15258 −0.251603 −0.125802 0.992055i \(-0.540150\pi\)
−0.125802 + 0.992055i \(0.540150\pi\)
\(158\) 30.4120 2.41945
\(159\) 0 0
\(160\) 16.6477 1.31612
\(161\) −7.24460 −0.570954
\(162\) 0 0
\(163\) 4.24712 0.332660 0.166330 0.986070i \(-0.446808\pi\)
0.166330 + 0.986070i \(0.446808\pi\)
\(164\) −21.8429 −1.70564
\(165\) 0 0
\(166\) 30.0234 2.33026
\(167\) −5.46371 −0.422794 −0.211397 0.977400i \(-0.567801\pi\)
−0.211397 + 0.977400i \(0.567801\pi\)
\(168\) 0 0
\(169\) 5.58132 0.429332
\(170\) −29.7098 −2.27864
\(171\) 0 0
\(172\) −9.97437 −0.760538
\(173\) −2.70195 −0.205426 −0.102713 0.994711i \(-0.532752\pi\)
−0.102713 + 0.994711i \(0.532752\pi\)
\(174\) 0 0
\(175\) −0.459293 −0.0347193
\(176\) 11.9802 0.903042
\(177\) 0 0
\(178\) −25.7808 −1.93235
\(179\) −21.4365 −1.60224 −0.801119 0.598505i \(-0.795762\pi\)
−0.801119 + 0.598505i \(0.795762\pi\)
\(180\) 0 0
\(181\) 25.4855 1.89432 0.947159 0.320763i \(-0.103939\pi\)
0.947159 + 0.320763i \(0.103939\pi\)
\(182\) −7.24155 −0.536779
\(183\) 0 0
\(184\) −10.2845 −0.758180
\(185\) 0.276890 0.0203573
\(186\) 0 0
\(187\) −29.8210 −2.18073
\(188\) −33.8386 −2.46793
\(189\) 0 0
\(190\) −15.2024 −1.10290
\(191\) −20.1181 −1.45570 −0.727849 0.685738i \(-0.759480\pi\)
−0.727849 + 0.685738i \(0.759480\pi\)
\(192\) 0 0
\(193\) −11.9500 −0.860180 −0.430090 0.902786i \(-0.641518\pi\)
−0.430090 + 0.902786i \(0.641518\pi\)
\(194\) −21.5831 −1.54958
\(195\) 0 0
\(196\) −16.1125 −1.15089
\(197\) 3.53147 0.251607 0.125803 0.992055i \(-0.459849\pi\)
0.125803 + 0.992055i \(0.459849\pi\)
\(198\) 0 0
\(199\) 1.98449 0.140677 0.0703383 0.997523i \(-0.477592\pi\)
0.0703383 + 0.997523i \(0.477592\pi\)
\(200\) −0.652014 −0.0461044
\(201\) 0 0
\(202\) −10.1868 −0.716741
\(203\) −0.303347 −0.0212908
\(204\) 0 0
\(205\) 18.1703 1.26907
\(206\) 32.4501 2.26091
\(207\) 0 0
\(208\) 11.5042 0.797670
\(209\) −15.2593 −1.05551
\(210\) 0 0
\(211\) −7.46169 −0.513684 −0.256842 0.966453i \(-0.582682\pi\)
−0.256842 + 0.966453i \(0.582682\pi\)
\(212\) −1.58065 −0.108560
\(213\) 0 0
\(214\) 27.7402 1.89628
\(215\) 8.29732 0.565873
\(216\) 0 0
\(217\) 2.20204 0.149484
\(218\) 13.4755 0.912679
\(219\) 0 0
\(220\) 23.8424 1.60746
\(221\) −28.6360 −1.92627
\(222\) 0 0
\(223\) 21.8418 1.46264 0.731318 0.682036i \(-0.238905\pi\)
0.731318 + 0.682036i \(0.238905\pi\)
\(224\) −6.25349 −0.417829
\(225\) 0 0
\(226\) 31.2954 2.08174
\(227\) −24.6350 −1.63508 −0.817542 0.575869i \(-0.804664\pi\)
−0.817542 + 0.575869i \(0.804664\pi\)
\(228\) 0 0
\(229\) −11.4551 −0.756975 −0.378488 0.925606i \(-0.623556\pi\)
−0.378488 + 0.925606i \(0.623556\pi\)
\(230\) 41.0339 2.70569
\(231\) 0 0
\(232\) −0.430633 −0.0282724
\(233\) −3.84650 −0.251992 −0.125996 0.992031i \(-0.540213\pi\)
−0.125996 + 0.992031i \(0.540213\pi\)
\(234\) 0 0
\(235\) 28.1492 1.83625
\(236\) 18.1421 1.18095
\(237\) 0 0
\(238\) 11.1601 0.723402
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −18.2874 −1.17799 −0.588997 0.808135i \(-0.700477\pi\)
−0.588997 + 0.808135i \(0.700477\pi\)
\(242\) 19.4698 1.25157
\(243\) 0 0
\(244\) 17.1411 1.09734
\(245\) 13.4034 0.856311
\(246\) 0 0
\(247\) −14.6529 −0.932345
\(248\) 3.12602 0.198503
\(249\) 0 0
\(250\) 24.9627 1.57878
\(251\) −4.28904 −0.270722 −0.135361 0.990796i \(-0.543219\pi\)
−0.135361 + 0.990796i \(0.543219\pi\)
\(252\) 0 0
\(253\) 41.1875 2.58944
\(254\) 26.1753 1.64238
\(255\) 0 0
\(256\) 4.60972 0.288108
\(257\) 18.0042 1.12307 0.561535 0.827453i \(-0.310211\pi\)
0.561535 + 0.827453i \(0.310211\pi\)
\(258\) 0 0
\(259\) −0.104010 −0.00646287
\(260\) 22.8950 1.41989
\(261\) 0 0
\(262\) −38.8845 −2.40229
\(263\) −7.38769 −0.455545 −0.227772 0.973714i \(-0.573144\pi\)
−0.227772 + 0.973714i \(0.573144\pi\)
\(264\) 0 0
\(265\) 1.31489 0.0807730
\(266\) 5.71058 0.350138
\(267\) 0 0
\(268\) 0.762325 0.0465664
\(269\) −15.8974 −0.969283 −0.484641 0.874713i \(-0.661050\pi\)
−0.484641 + 0.874713i \(0.661050\pi\)
\(270\) 0 0
\(271\) 19.7386 1.19903 0.599516 0.800363i \(-0.295360\pi\)
0.599516 + 0.800363i \(0.295360\pi\)
\(272\) −17.7293 −1.07500
\(273\) 0 0
\(274\) −28.6357 −1.72995
\(275\) 2.61121 0.157462
\(276\) 0 0
\(277\) 29.9558 1.79987 0.899935 0.436024i \(-0.143614\pi\)
0.899935 + 0.436024i \(0.143614\pi\)
\(278\) −31.6547 −1.89852
\(279\) 0 0
\(280\) −1.86032 −0.111175
\(281\) −4.27216 −0.254855 −0.127428 0.991848i \(-0.540672\pi\)
−0.127428 + 0.991848i \(0.540672\pi\)
\(282\) 0 0
\(283\) −27.6755 −1.64514 −0.822570 0.568664i \(-0.807460\pi\)
−0.822570 + 0.568664i \(0.807460\pi\)
\(284\) −7.11486 −0.422189
\(285\) 0 0
\(286\) 41.1702 2.43444
\(287\) −6.82546 −0.402894
\(288\) 0 0
\(289\) 27.1316 1.59598
\(290\) 1.71818 0.100895
\(291\) 0 0
\(292\) −5.78180 −0.338354
\(293\) −10.5839 −0.618320 −0.309160 0.951010i \(-0.600048\pi\)
−0.309160 + 0.951010i \(0.600048\pi\)
\(294\) 0 0
\(295\) −15.0918 −0.878677
\(296\) −0.147653 −0.00858216
\(297\) 0 0
\(298\) 46.2765 2.68072
\(299\) 39.5509 2.28729
\(300\) 0 0
\(301\) −3.11678 −0.179648
\(302\) −20.0047 −1.15114
\(303\) 0 0
\(304\) −9.07201 −0.520315
\(305\) −14.2590 −0.816470
\(306\) 0 0
\(307\) 25.6078 1.46152 0.730758 0.682637i \(-0.239167\pi\)
0.730758 + 0.682637i \(0.239167\pi\)
\(308\) −8.95611 −0.510322
\(309\) 0 0
\(310\) −12.4725 −0.708390
\(311\) 13.8656 0.786246 0.393123 0.919486i \(-0.371395\pi\)
0.393123 + 0.919486i \(0.371395\pi\)
\(312\) 0 0
\(313\) 4.30573 0.243374 0.121687 0.992569i \(-0.461170\pi\)
0.121687 + 0.992569i \(0.461170\pi\)
\(314\) −6.70753 −0.378528
\(315\) 0 0
\(316\) 36.1180 2.03179
\(317\) −2.12732 −0.119482 −0.0597410 0.998214i \(-0.519027\pi\)
−0.0597410 + 0.998214i \(0.519027\pi\)
\(318\) 0 0
\(319\) 1.72461 0.0965596
\(320\) 24.2007 1.35286
\(321\) 0 0
\(322\) −15.4139 −0.858980
\(323\) 22.5820 1.25649
\(324\) 0 0
\(325\) 2.50745 0.139088
\(326\) 9.03631 0.500475
\(327\) 0 0
\(328\) −9.68944 −0.535010
\(329\) −10.5739 −0.582956
\(330\) 0 0
\(331\) −13.3771 −0.735269 −0.367635 0.929970i \(-0.619832\pi\)
−0.367635 + 0.929970i \(0.619832\pi\)
\(332\) 35.6564 1.95690
\(333\) 0 0
\(334\) −11.6248 −0.636079
\(335\) −0.634151 −0.0346474
\(336\) 0 0
\(337\) 12.2307 0.666247 0.333123 0.942883i \(-0.391897\pi\)
0.333123 + 0.942883i \(0.391897\pi\)
\(338\) 11.8750 0.645915
\(339\) 0 0
\(340\) −35.2840 −1.91354
\(341\) −12.5192 −0.677952
\(342\) 0 0
\(343\) −10.5619 −0.570288
\(344\) −4.42460 −0.238558
\(345\) 0 0
\(346\) −5.74876 −0.309055
\(347\) 16.4824 0.884820 0.442410 0.896813i \(-0.354124\pi\)
0.442410 + 0.896813i \(0.354124\pi\)
\(348\) 0 0
\(349\) −7.08592 −0.379301 −0.189650 0.981852i \(-0.560735\pi\)
−0.189650 + 0.981852i \(0.560735\pi\)
\(350\) −0.977207 −0.0522339
\(351\) 0 0
\(352\) 35.5528 1.89497
\(353\) −33.6523 −1.79113 −0.895567 0.444927i \(-0.853230\pi\)
−0.895567 + 0.444927i \(0.853230\pi\)
\(354\) 0 0
\(355\) 5.91860 0.314127
\(356\) −30.6179 −1.62274
\(357\) 0 0
\(358\) −45.6090 −2.41051
\(359\) 11.2250 0.592435 0.296218 0.955120i \(-0.404275\pi\)
0.296218 + 0.955120i \(0.404275\pi\)
\(360\) 0 0
\(361\) −7.44491 −0.391837
\(362\) 54.2237 2.84994
\(363\) 0 0
\(364\) −8.60023 −0.450774
\(365\) 4.80968 0.251750
\(366\) 0 0
\(367\) 9.75843 0.509386 0.254693 0.967022i \(-0.418026\pi\)
0.254693 + 0.967022i \(0.418026\pi\)
\(368\) 24.4869 1.27647
\(369\) 0 0
\(370\) 0.589120 0.0306269
\(371\) −0.493921 −0.0256431
\(372\) 0 0
\(373\) 4.61889 0.239157 0.119579 0.992825i \(-0.461846\pi\)
0.119579 + 0.992825i \(0.461846\pi\)
\(374\) −63.4482 −3.28083
\(375\) 0 0
\(376\) −15.0107 −0.774118
\(377\) 1.65608 0.0852925
\(378\) 0 0
\(379\) 27.8720 1.43169 0.715845 0.698259i \(-0.246042\pi\)
0.715845 + 0.698259i \(0.246042\pi\)
\(380\) −18.0547 −0.926186
\(381\) 0 0
\(382\) −42.8040 −2.19005
\(383\) 22.8580 1.16799 0.583994 0.811758i \(-0.301489\pi\)
0.583994 + 0.811758i \(0.301489\pi\)
\(384\) 0 0
\(385\) 7.45027 0.379701
\(386\) −25.4252 −1.29411
\(387\) 0 0
\(388\) −25.6326 −1.30130
\(389\) 29.2780 1.48445 0.742227 0.670149i \(-0.233770\pi\)
0.742227 + 0.670149i \(0.233770\pi\)
\(390\) 0 0
\(391\) −60.9527 −3.08251
\(392\) −7.14743 −0.361000
\(393\) 0 0
\(394\) 7.51367 0.378533
\(395\) −30.0453 −1.51174
\(396\) 0 0
\(397\) −32.1482 −1.61347 −0.806735 0.590913i \(-0.798768\pi\)
−0.806735 + 0.590913i \(0.798768\pi\)
\(398\) 4.22226 0.211643
\(399\) 0 0
\(400\) 1.55242 0.0776212
\(401\) 8.54317 0.426626 0.213313 0.976984i \(-0.431575\pi\)
0.213313 + 0.976984i \(0.431575\pi\)
\(402\) 0 0
\(403\) −12.0217 −0.598845
\(404\) −12.0981 −0.601902
\(405\) 0 0
\(406\) −0.645411 −0.0320312
\(407\) 0.591325 0.0293109
\(408\) 0 0
\(409\) −4.37441 −0.216301 −0.108150 0.994135i \(-0.534493\pi\)
−0.108150 + 0.994135i \(0.534493\pi\)
\(410\) 38.6598 1.90927
\(411\) 0 0
\(412\) 38.5385 1.89865
\(413\) 5.66903 0.278955
\(414\) 0 0
\(415\) −29.6613 −1.45602
\(416\) 34.1401 1.67386
\(417\) 0 0
\(418\) −32.4662 −1.58797
\(419\) 6.49460 0.317282 0.158641 0.987336i \(-0.449289\pi\)
0.158641 + 0.987336i \(0.449289\pi\)
\(420\) 0 0
\(421\) 16.6912 0.813481 0.406741 0.913544i \(-0.366665\pi\)
0.406741 + 0.913544i \(0.366665\pi\)
\(422\) −15.8757 −0.772819
\(423\) 0 0
\(424\) −0.701172 −0.0340519
\(425\) −3.86428 −0.187445
\(426\) 0 0
\(427\) 5.35623 0.259206
\(428\) 32.9448 1.59245
\(429\) 0 0
\(430\) 17.6537 0.851335
\(431\) 13.7066 0.660226 0.330113 0.943941i \(-0.392913\pi\)
0.330113 + 0.943941i \(0.392913\pi\)
\(432\) 0 0
\(433\) −21.1110 −1.01453 −0.507264 0.861791i \(-0.669343\pi\)
−0.507264 + 0.861791i \(0.669343\pi\)
\(434\) 4.68513 0.224893
\(435\) 0 0
\(436\) 16.0039 0.766446
\(437\) −31.1892 −1.49198
\(438\) 0 0
\(439\) −36.5659 −1.74520 −0.872598 0.488438i \(-0.837567\pi\)
−0.872598 + 0.488438i \(0.837567\pi\)
\(440\) 10.5764 0.504212
\(441\) 0 0
\(442\) −60.9270 −2.89800
\(443\) −2.61320 −0.124157 −0.0620785 0.998071i \(-0.519773\pi\)
−0.0620785 + 0.998071i \(0.519773\pi\)
\(444\) 0 0
\(445\) 25.4699 1.20739
\(446\) 46.4714 2.20048
\(447\) 0 0
\(448\) −9.09067 −0.429494
\(449\) −21.8573 −1.03151 −0.515756 0.856736i \(-0.672489\pi\)
−0.515756 + 0.856736i \(0.672489\pi\)
\(450\) 0 0
\(451\) 38.8046 1.82724
\(452\) 37.1671 1.74819
\(453\) 0 0
\(454\) −52.4143 −2.45993
\(455\) 7.15423 0.335395
\(456\) 0 0
\(457\) −33.7368 −1.57814 −0.789070 0.614304i \(-0.789437\pi\)
−0.789070 + 0.614304i \(0.789437\pi\)
\(458\) −24.3723 −1.13884
\(459\) 0 0
\(460\) 48.7328 2.27218
\(461\) 12.2555 0.570795 0.285397 0.958409i \(-0.407874\pi\)
0.285397 + 0.958409i \(0.407874\pi\)
\(462\) 0 0
\(463\) −40.6275 −1.88812 −0.944061 0.329772i \(-0.893028\pi\)
−0.944061 + 0.329772i \(0.893028\pi\)
\(464\) 1.02532 0.0475994
\(465\) 0 0
\(466\) −8.18394 −0.379113
\(467\) −10.4900 −0.485419 −0.242710 0.970099i \(-0.578036\pi\)
−0.242710 + 0.970099i \(0.578036\pi\)
\(468\) 0 0
\(469\) 0.238211 0.0109995
\(470\) 59.8911 2.76257
\(471\) 0 0
\(472\) 8.04778 0.370429
\(473\) 17.7198 0.814755
\(474\) 0 0
\(475\) −1.97733 −0.0907263
\(476\) 13.2540 0.607495
\(477\) 0 0
\(478\) −2.12763 −0.0973157
\(479\) 27.6348 1.26267 0.631333 0.775512i \(-0.282508\pi\)
0.631333 + 0.775512i \(0.282508\pi\)
\(480\) 0 0
\(481\) 0.567828 0.0258907
\(482\) −38.9089 −1.77225
\(483\) 0 0
\(484\) 23.1228 1.05104
\(485\) 21.3229 0.968221
\(486\) 0 0
\(487\) −17.4271 −0.789696 −0.394848 0.918746i \(-0.629203\pi\)
−0.394848 + 0.918746i \(0.629203\pi\)
\(488\) 7.60372 0.344204
\(489\) 0 0
\(490\) 28.5175 1.28829
\(491\) −30.0841 −1.35768 −0.678838 0.734288i \(-0.737516\pi\)
−0.678838 + 0.734288i \(0.737516\pi\)
\(492\) 0 0
\(493\) −2.55222 −0.114946
\(494\) −31.1761 −1.40268
\(495\) 0 0
\(496\) −7.44295 −0.334198
\(497\) −2.22324 −0.0997262
\(498\) 0 0
\(499\) 11.5301 0.516157 0.258079 0.966124i \(-0.416911\pi\)
0.258079 + 0.966124i \(0.416911\pi\)
\(500\) 29.6462 1.32582
\(501\) 0 0
\(502\) −9.12552 −0.407292
\(503\) −7.70067 −0.343356 −0.171678 0.985153i \(-0.554919\pi\)
−0.171678 + 0.985153i \(0.554919\pi\)
\(504\) 0 0
\(505\) 10.0640 0.447840
\(506\) 87.6319 3.89571
\(507\) 0 0
\(508\) 31.0864 1.37923
\(509\) −33.1324 −1.46857 −0.734284 0.678842i \(-0.762482\pi\)
−0.734284 + 0.678842i \(0.762482\pi\)
\(510\) 0 0
\(511\) −1.80669 −0.0799234
\(512\) 27.1199 1.19854
\(513\) 0 0
\(514\) 38.3063 1.68962
\(515\) −32.0588 −1.41268
\(516\) 0 0
\(517\) 60.1153 2.64387
\(518\) −0.221295 −0.00972315
\(519\) 0 0
\(520\) 10.1562 0.445377
\(521\) −36.2158 −1.58664 −0.793321 0.608804i \(-0.791650\pi\)
−0.793321 + 0.608804i \(0.791650\pi\)
\(522\) 0 0
\(523\) −13.3239 −0.582613 −0.291306 0.956630i \(-0.594090\pi\)
−0.291306 + 0.956630i \(0.594090\pi\)
\(524\) −46.1801 −2.01739
\(525\) 0 0
\(526\) −15.7183 −0.685351
\(527\) 18.5269 0.807045
\(528\) 0 0
\(529\) 61.1852 2.66023
\(530\) 2.79760 0.121520
\(531\) 0 0
\(532\) 6.78201 0.294037
\(533\) 37.2626 1.61402
\(534\) 0 0
\(535\) −27.4056 −1.18485
\(536\) 0.338165 0.0146065
\(537\) 0 0
\(538\) −33.8239 −1.45825
\(539\) 28.6243 1.23293
\(540\) 0 0
\(541\) −15.2808 −0.656972 −0.328486 0.944509i \(-0.606538\pi\)
−0.328486 + 0.944509i \(0.606538\pi\)
\(542\) 41.9964 1.80390
\(543\) 0 0
\(544\) −52.6140 −2.25581
\(545\) −13.3130 −0.570268
\(546\) 0 0
\(547\) −24.9953 −1.06872 −0.534362 0.845256i \(-0.679448\pi\)
−0.534362 + 0.845256i \(0.679448\pi\)
\(548\) −34.0084 −1.45277
\(549\) 0 0
\(550\) 5.55569 0.236895
\(551\) −1.30596 −0.0556358
\(552\) 0 0
\(553\) 11.2861 0.479935
\(554\) 63.7350 2.70784
\(555\) 0 0
\(556\) −37.5938 −1.59433
\(557\) −12.6114 −0.534360 −0.267180 0.963647i \(-0.586092\pi\)
−0.267180 + 0.963647i \(0.586092\pi\)
\(558\) 0 0
\(559\) 17.0156 0.719685
\(560\) 4.42936 0.187175
\(561\) 0 0
\(562\) −9.08958 −0.383421
\(563\) 0.586176 0.0247044 0.0123522 0.999924i \(-0.496068\pi\)
0.0123522 + 0.999924i \(0.496068\pi\)
\(564\) 0 0
\(565\) −30.9180 −1.30073
\(566\) −58.8834 −2.47505
\(567\) 0 0
\(568\) −3.15613 −0.132428
\(569\) −41.0817 −1.72224 −0.861118 0.508406i \(-0.830235\pi\)
−0.861118 + 0.508406i \(0.830235\pi\)
\(570\) 0 0
\(571\) −7.37931 −0.308814 −0.154407 0.988007i \(-0.549347\pi\)
−0.154407 + 0.988007i \(0.549347\pi\)
\(572\) 48.8946 2.04439
\(573\) 0 0
\(574\) −14.5221 −0.606140
\(575\) 5.33717 0.222576
\(576\) 0 0
\(577\) −0.852487 −0.0354895 −0.0177447 0.999843i \(-0.505649\pi\)
−0.0177447 + 0.999843i \(0.505649\pi\)
\(578\) 57.7261 2.40109
\(579\) 0 0
\(580\) 2.04055 0.0847291
\(581\) 11.1419 0.462243
\(582\) 0 0
\(583\) 2.80807 0.116299
\(584\) −2.56479 −0.106132
\(585\) 0 0
\(586\) −22.5187 −0.930240
\(587\) −24.7875 −1.02309 −0.511544 0.859257i \(-0.670926\pi\)
−0.511544 + 0.859257i \(0.670926\pi\)
\(588\) 0 0
\(589\) 9.48015 0.390623
\(590\) −32.1098 −1.32194
\(591\) 0 0
\(592\) 0.351557 0.0144489
\(593\) −40.5797 −1.66641 −0.833204 0.552966i \(-0.813496\pi\)
−0.833204 + 0.552966i \(0.813496\pi\)
\(594\) 0 0
\(595\) −11.0255 −0.452002
\(596\) 54.9590 2.25121
\(597\) 0 0
\(598\) 84.1498 3.44114
\(599\) 21.9846 0.898267 0.449134 0.893465i \(-0.351733\pi\)
0.449134 + 0.893465i \(0.351733\pi\)
\(600\) 0 0
\(601\) 11.3425 0.462669 0.231335 0.972874i \(-0.425691\pi\)
0.231335 + 0.972874i \(0.425691\pi\)
\(602\) −6.63137 −0.270275
\(603\) 0 0
\(604\) −23.7581 −0.966702
\(605\) −19.2351 −0.782016
\(606\) 0 0
\(607\) −2.12766 −0.0863590 −0.0431795 0.999067i \(-0.513749\pi\)
−0.0431795 + 0.999067i \(0.513749\pi\)
\(608\) −26.9224 −1.09185
\(609\) 0 0
\(610\) −30.3380 −1.22835
\(611\) 57.7266 2.33537
\(612\) 0 0
\(613\) −45.6639 −1.84435 −0.922175 0.386773i \(-0.873590\pi\)
−0.922175 + 0.386773i \(0.873590\pi\)
\(614\) 54.4841 2.19880
\(615\) 0 0
\(616\) −3.97290 −0.160073
\(617\) −10.0287 −0.403742 −0.201871 0.979412i \(-0.564702\pi\)
−0.201871 + 0.979412i \(0.564702\pi\)
\(618\) 0 0
\(619\) −4.86175 −0.195410 −0.0977052 0.995215i \(-0.531150\pi\)
−0.0977052 + 0.995215i \(0.531150\pi\)
\(620\) −14.8126 −0.594889
\(621\) 0 0
\(622\) 29.5009 1.18288
\(623\) −9.56745 −0.383312
\(624\) 0 0
\(625\) −21.7532 −0.870127
\(626\) 9.16101 0.366148
\(627\) 0 0
\(628\) −7.96602 −0.317879
\(629\) −0.875092 −0.0348922
\(630\) 0 0
\(631\) −28.5095 −1.13495 −0.567473 0.823392i \(-0.692079\pi\)
−0.567473 + 0.823392i \(0.692079\pi\)
\(632\) 16.0218 0.637314
\(633\) 0 0
\(634\) −4.52615 −0.179756
\(635\) −25.8596 −1.02621
\(636\) 0 0
\(637\) 27.4868 1.08907
\(638\) 3.66934 0.145271
\(639\) 0 0
\(640\) 18.1948 0.719212
\(641\) −19.0522 −0.752517 −0.376258 0.926515i \(-0.622790\pi\)
−0.376258 + 0.926515i \(0.622790\pi\)
\(642\) 0 0
\(643\) 16.7496 0.660539 0.330269 0.943887i \(-0.392860\pi\)
0.330269 + 0.943887i \(0.392860\pi\)
\(644\) −18.3058 −0.721351
\(645\) 0 0
\(646\) 48.0461 1.89035
\(647\) −30.9651 −1.21736 −0.608682 0.793414i \(-0.708302\pi\)
−0.608682 + 0.793414i \(0.708302\pi\)
\(648\) 0 0
\(649\) −32.2300 −1.26514
\(650\) 5.33493 0.209253
\(651\) 0 0
\(652\) 10.7317 0.420287
\(653\) 36.1207 1.41351 0.706756 0.707458i \(-0.250158\pi\)
0.706756 + 0.707458i \(0.250158\pi\)
\(654\) 0 0
\(655\) 38.4156 1.50102
\(656\) 23.0702 0.900741
\(657\) 0 0
\(658\) −22.4973 −0.877037
\(659\) 23.2340 0.905069 0.452535 0.891747i \(-0.350520\pi\)
0.452535 + 0.891747i \(0.350520\pi\)
\(660\) 0 0
\(661\) 46.5878 1.81206 0.906028 0.423218i \(-0.139100\pi\)
0.906028 + 0.423218i \(0.139100\pi\)
\(662\) −28.4615 −1.10619
\(663\) 0 0
\(664\) 15.8171 0.613821
\(665\) −5.64171 −0.218776
\(666\) 0 0
\(667\) 3.52502 0.136489
\(668\) −13.8058 −0.534164
\(669\) 0 0
\(670\) −1.34924 −0.0521257
\(671\) −30.4516 −1.17557
\(672\) 0 0
\(673\) −5.17261 −0.199390 −0.0996948 0.995018i \(-0.531787\pi\)
−0.0996948 + 0.995018i \(0.531787\pi\)
\(674\) 26.0224 1.00234
\(675\) 0 0
\(676\) 14.1030 0.542424
\(677\) −14.8722 −0.571584 −0.285792 0.958292i \(-0.592257\pi\)
−0.285792 + 0.958292i \(0.592257\pi\)
\(678\) 0 0
\(679\) −8.00966 −0.307382
\(680\) −15.6519 −0.600222
\(681\) 0 0
\(682\) −26.6362 −1.01995
\(683\) 47.9804 1.83592 0.917960 0.396672i \(-0.129835\pi\)
0.917960 + 0.396672i \(0.129835\pi\)
\(684\) 0 0
\(685\) 28.2904 1.08092
\(686\) −22.4718 −0.857977
\(687\) 0 0
\(688\) 10.5348 0.401636
\(689\) 2.69649 0.102728
\(690\) 0 0
\(691\) −10.4497 −0.397524 −0.198762 0.980048i \(-0.563692\pi\)
−0.198762 + 0.980048i \(0.563692\pi\)
\(692\) −6.82736 −0.259537
\(693\) 0 0
\(694\) 35.0684 1.33118
\(695\) 31.2729 1.18625
\(696\) 0 0
\(697\) −57.4262 −2.17517
\(698\) −15.0762 −0.570644
\(699\) 0 0
\(700\) −1.16055 −0.0438648
\(701\) 22.4554 0.848129 0.424065 0.905632i \(-0.360603\pi\)
0.424065 + 0.905632i \(0.360603\pi\)
\(702\) 0 0
\(703\) −0.447781 −0.0168884
\(704\) 51.6829 1.94787
\(705\) 0 0
\(706\) −71.5999 −2.69470
\(707\) −3.78040 −0.142176
\(708\) 0 0
\(709\) −2.79774 −0.105071 −0.0525356 0.998619i \(-0.516730\pi\)
−0.0525356 + 0.998619i \(0.516730\pi\)
\(710\) 12.5926 0.472592
\(711\) 0 0
\(712\) −13.5820 −0.509007
\(713\) −25.5886 −0.958300
\(714\) 0 0
\(715\) −40.6737 −1.52111
\(716\) −54.1662 −2.02429
\(717\) 0 0
\(718\) 23.8828 0.891298
\(719\) 15.6650 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(720\) 0 0
\(721\) 12.0425 0.448485
\(722\) −15.8400 −0.589505
\(723\) 0 0
\(724\) 64.3973 2.39331
\(725\) 0.223479 0.00829980
\(726\) 0 0
\(727\) 15.9906 0.593058 0.296529 0.955024i \(-0.404171\pi\)
0.296529 + 0.955024i \(0.404171\pi\)
\(728\) −3.81503 −0.141395
\(729\) 0 0
\(730\) 10.2332 0.378749
\(731\) −26.2232 −0.969898
\(732\) 0 0
\(733\) −51.6646 −1.90828 −0.954138 0.299369i \(-0.903224\pi\)
−0.954138 + 0.299369i \(0.903224\pi\)
\(734\) 20.7624 0.766353
\(735\) 0 0
\(736\) 72.6682 2.67858
\(737\) −1.35429 −0.0498860
\(738\) 0 0
\(739\) −19.0256 −0.699866 −0.349933 0.936775i \(-0.613796\pi\)
−0.349933 + 0.936775i \(0.613796\pi\)
\(740\) 0.699652 0.0257197
\(741\) 0 0
\(742\) −1.05088 −0.0385791
\(743\) 12.4637 0.457247 0.228624 0.973515i \(-0.426578\pi\)
0.228624 + 0.973515i \(0.426578\pi\)
\(744\) 0 0
\(745\) −45.7184 −1.67499
\(746\) 9.82731 0.359803
\(747\) 0 0
\(748\) −75.3525 −2.75516
\(749\) 10.2946 0.376156
\(750\) 0 0
\(751\) 10.9426 0.399303 0.199651 0.979867i \(-0.436019\pi\)
0.199651 + 0.979867i \(0.436019\pi\)
\(752\) 35.7400 1.30330
\(753\) 0 0
\(754\) 3.52353 0.128320
\(755\) 19.7635 0.719267
\(756\) 0 0
\(757\) −4.14204 −0.150545 −0.0752725 0.997163i \(-0.523983\pi\)
−0.0752725 + 0.997163i \(0.523983\pi\)
\(758\) 59.3015 2.15393
\(759\) 0 0
\(760\) −8.00900 −0.290517
\(761\) −24.2607 −0.879449 −0.439725 0.898133i \(-0.644924\pi\)
−0.439725 + 0.898133i \(0.644924\pi\)
\(762\) 0 0
\(763\) 5.00087 0.181044
\(764\) −50.8350 −1.83915
\(765\) 0 0
\(766\) 48.6334 1.75720
\(767\) −30.9493 −1.11751
\(768\) 0 0
\(769\) −4.60195 −0.165951 −0.0829753 0.996552i \(-0.526442\pi\)
−0.0829753 + 0.996552i \(0.526442\pi\)
\(770\) 15.8514 0.571246
\(771\) 0 0
\(772\) −30.1956 −1.08676
\(773\) 24.5840 0.884227 0.442113 0.896959i \(-0.354229\pi\)
0.442113 + 0.896959i \(0.354229\pi\)
\(774\) 0 0
\(775\) −1.62227 −0.0582735
\(776\) −11.3705 −0.408179
\(777\) 0 0
\(778\) 62.2929 2.23331
\(779\) −29.3848 −1.05282
\(780\) 0 0
\(781\) 12.6397 0.452286
\(782\) −129.685 −4.63752
\(783\) 0 0
\(784\) 17.0178 0.607778
\(785\) 6.62665 0.236515
\(786\) 0 0
\(787\) 10.5232 0.375110 0.187555 0.982254i \(-0.439944\pi\)
0.187555 + 0.982254i \(0.439944\pi\)
\(788\) 8.92340 0.317883
\(789\) 0 0
\(790\) −63.9253 −2.27436
\(791\) 11.6140 0.412945
\(792\) 0 0
\(793\) −29.2416 −1.03840
\(794\) −68.3995 −2.42741
\(795\) 0 0
\(796\) 5.01445 0.177733
\(797\) 14.8664 0.526594 0.263297 0.964715i \(-0.415190\pi\)
0.263297 + 0.964715i \(0.415190\pi\)
\(798\) 0 0
\(799\) −88.9636 −3.14730
\(800\) 4.60702 0.162883
\(801\) 0 0
\(802\) 18.1767 0.641843
\(803\) 10.2715 0.362475
\(804\) 0 0
\(805\) 15.2280 0.536716
\(806\) −25.5778 −0.900940
\(807\) 0 0
\(808\) −5.36666 −0.188799
\(809\) 37.7784 1.32822 0.664108 0.747636i \(-0.268811\pi\)
0.664108 + 0.747636i \(0.268811\pi\)
\(810\) 0 0
\(811\) 17.9591 0.630629 0.315315 0.948987i \(-0.397890\pi\)
0.315315 + 0.948987i \(0.397890\pi\)
\(812\) −0.766505 −0.0268991
\(813\) 0 0
\(814\) 1.25812 0.0440972
\(815\) −8.92735 −0.312711
\(816\) 0 0
\(817\) −13.4183 −0.469446
\(818\) −9.30714 −0.325417
\(819\) 0 0
\(820\) 45.9133 1.60336
\(821\) −35.4571 −1.23746 −0.618730 0.785604i \(-0.712352\pi\)
−0.618730 + 0.785604i \(0.712352\pi\)
\(822\) 0 0
\(823\) −8.61910 −0.300443 −0.150221 0.988652i \(-0.547999\pi\)
−0.150221 + 0.988652i \(0.547999\pi\)
\(824\) 17.0955 0.595551
\(825\) 0 0
\(826\) 12.0616 0.419678
\(827\) 17.3968 0.604948 0.302474 0.953158i \(-0.402188\pi\)
0.302474 + 0.953158i \(0.402188\pi\)
\(828\) 0 0
\(829\) −0.772910 −0.0268443 −0.0134221 0.999910i \(-0.504273\pi\)
−0.0134221 + 0.999910i \(0.504273\pi\)
\(830\) −63.1084 −2.19052
\(831\) 0 0
\(832\) 49.6293 1.72059
\(833\) −42.3605 −1.46771
\(834\) 0 0
\(835\) 11.4846 0.397441
\(836\) −38.5576 −1.33354
\(837\) 0 0
\(838\) 13.8181 0.477339
\(839\) −26.3653 −0.910230 −0.455115 0.890433i \(-0.650402\pi\)
−0.455115 + 0.890433i \(0.650402\pi\)
\(840\) 0 0
\(841\) −28.8524 −0.994910
\(842\) 35.5129 1.22385
\(843\) 0 0
\(844\) −18.8544 −0.648995
\(845\) −11.7318 −0.403586
\(846\) 0 0
\(847\) 7.22540 0.248268
\(848\) 1.66947 0.0573297
\(849\) 0 0
\(850\) −8.22177 −0.282004
\(851\) 1.20864 0.0414316
\(852\) 0 0
\(853\) −0.382069 −0.0130818 −0.00654089 0.999979i \(-0.502082\pi\)
−0.00654089 + 0.999979i \(0.502082\pi\)
\(854\) 11.3961 0.389966
\(855\) 0 0
\(856\) 14.6142 0.499504
\(857\) 40.4862 1.38298 0.691492 0.722384i \(-0.256954\pi\)
0.691492 + 0.722384i \(0.256954\pi\)
\(858\) 0 0
\(859\) −4.24423 −0.144811 −0.0724056 0.997375i \(-0.523068\pi\)
−0.0724056 + 0.997375i \(0.523068\pi\)
\(860\) 20.9659 0.714931
\(861\) 0 0
\(862\) 29.1627 0.993286
\(863\) 1.91165 0.0650733 0.0325367 0.999471i \(-0.489641\pi\)
0.0325367 + 0.999471i \(0.489641\pi\)
\(864\) 0 0
\(865\) 5.67944 0.193107
\(866\) −44.9164 −1.52632
\(867\) 0 0
\(868\) 5.56417 0.188860
\(869\) −64.1646 −2.17664
\(870\) 0 0
\(871\) −1.30048 −0.0440650
\(872\) 7.09926 0.240411
\(873\) 0 0
\(874\) −66.3593 −2.24464
\(875\) 9.26382 0.313174
\(876\) 0 0
\(877\) 26.1285 0.882298 0.441149 0.897434i \(-0.354571\pi\)
0.441149 + 0.897434i \(0.354571\pi\)
\(878\) −77.7989 −2.62559
\(879\) 0 0
\(880\) −25.1821 −0.848889
\(881\) −24.0315 −0.809642 −0.404821 0.914396i \(-0.632666\pi\)
−0.404821 + 0.914396i \(0.632666\pi\)
\(882\) 0 0
\(883\) 21.4066 0.720389 0.360194 0.932877i \(-0.382710\pi\)
0.360194 + 0.932877i \(0.382710\pi\)
\(884\) −72.3583 −2.43367
\(885\) 0 0
\(886\) −5.55993 −0.186790
\(887\) 14.2082 0.477066 0.238533 0.971134i \(-0.423334\pi\)
0.238533 + 0.971134i \(0.423334\pi\)
\(888\) 0 0
\(889\) 9.71384 0.325792
\(890\) 54.1907 1.81648
\(891\) 0 0
\(892\) 55.1905 1.84791
\(893\) −45.5223 −1.52335
\(894\) 0 0
\(895\) 45.0590 1.50616
\(896\) −6.83464 −0.228329
\(897\) 0 0
\(898\) −46.5044 −1.55187
\(899\) −1.07145 −0.0357348
\(900\) 0 0
\(901\) −4.15562 −0.138444
\(902\) 82.5619 2.74901
\(903\) 0 0
\(904\) 16.4872 0.548356
\(905\) −53.5698 −1.78072
\(906\) 0 0
\(907\) 19.2537 0.639310 0.319655 0.947534i \(-0.396433\pi\)
0.319655 + 0.947534i \(0.396433\pi\)
\(908\) −62.2484 −2.06579
\(909\) 0 0
\(910\) 15.2216 0.504590
\(911\) 44.9443 1.48907 0.744536 0.667583i \(-0.232671\pi\)
0.744536 + 0.667583i \(0.232671\pi\)
\(912\) 0 0
\(913\) −63.3447 −2.09640
\(914\) −71.7795 −2.37425
\(915\) 0 0
\(916\) −28.9451 −0.956372
\(917\) −14.4303 −0.476532
\(918\) 0 0
\(919\) 43.4424 1.43303 0.716516 0.697571i \(-0.245736\pi\)
0.716516 + 0.697571i \(0.245736\pi\)
\(920\) 21.6177 0.712714
\(921\) 0 0
\(922\) 26.0752 0.858740
\(923\) 12.1375 0.399511
\(924\) 0 0
\(925\) 0.0766253 0.00251942
\(926\) −86.4405 −2.84061
\(927\) 0 0
\(928\) 3.04277 0.0998839
\(929\) 1.23801 0.0406177 0.0203089 0.999794i \(-0.493535\pi\)
0.0203089 + 0.999794i \(0.493535\pi\)
\(930\) 0 0
\(931\) −21.6757 −0.710393
\(932\) −9.71942 −0.318370
\(933\) 0 0
\(934\) −22.3189 −0.730296
\(935\) 62.6831 2.04996
\(936\) 0 0
\(937\) −42.3900 −1.38482 −0.692410 0.721504i \(-0.743451\pi\)
−0.692410 + 0.721504i \(0.743451\pi\)
\(938\) 0.506825 0.0165484
\(939\) 0 0
\(940\) 71.1280 2.31994
\(941\) 8.79386 0.286672 0.143336 0.989674i \(-0.454217\pi\)
0.143336 + 0.989674i \(0.454217\pi\)
\(942\) 0 0
\(943\) 79.3146 2.58284
\(944\) −19.1615 −0.623653
\(945\) 0 0
\(946\) 37.7011 1.22577
\(947\) 46.4504 1.50943 0.754717 0.656051i \(-0.227774\pi\)
0.754717 + 0.656051i \(0.227774\pi\)
\(948\) 0 0
\(949\) 9.86339 0.320179
\(950\) −4.20704 −0.136495
\(951\) 0 0
\(952\) 5.87942 0.190553
\(953\) 23.4320 0.759038 0.379519 0.925184i \(-0.376089\pi\)
0.379519 + 0.925184i \(0.376089\pi\)
\(954\) 0 0
\(955\) 42.2879 1.36840
\(956\) −2.52683 −0.0817234
\(957\) 0 0
\(958\) 58.7967 1.89963
\(959\) −10.6269 −0.343161
\(960\) 0 0
\(961\) −23.2222 −0.749103
\(962\) 1.20813 0.0389517
\(963\) 0 0
\(964\) −46.2091 −1.48829
\(965\) 25.1186 0.808597
\(966\) 0 0
\(967\) −30.8007 −0.990483 −0.495242 0.868755i \(-0.664921\pi\)
−0.495242 + 0.868755i \(0.664921\pi\)
\(968\) 10.2572 0.329679
\(969\) 0 0
\(970\) 45.3672 1.45665
\(971\) −43.0309 −1.38093 −0.690464 0.723367i \(-0.742594\pi\)
−0.690464 + 0.723367i \(0.742594\pi\)
\(972\) 0 0
\(973\) −11.7473 −0.376600
\(974\) −37.0784 −1.18807
\(975\) 0 0
\(976\) −18.1042 −0.579501
\(977\) 49.7231 1.59078 0.795392 0.606096i \(-0.207265\pi\)
0.795392 + 0.606096i \(0.207265\pi\)
\(978\) 0 0
\(979\) 54.3936 1.73843
\(980\) 33.8680 1.08187
\(981\) 0 0
\(982\) −64.0079 −2.04257
\(983\) −14.4551 −0.461046 −0.230523 0.973067i \(-0.574044\pi\)
−0.230523 + 0.973067i \(0.574044\pi\)
\(984\) 0 0
\(985\) −7.42306 −0.236518
\(986\) −5.43019 −0.172932
\(987\) 0 0
\(988\) −37.0254 −1.17794
\(989\) 36.2183 1.15167
\(990\) 0 0
\(991\) 13.1410 0.417438 0.208719 0.977976i \(-0.433071\pi\)
0.208719 + 0.977976i \(0.433071\pi\)
\(992\) −22.0879 −0.701292
\(993\) 0 0
\(994\) −4.73025 −0.150034
\(995\) −4.17135 −0.132241
\(996\) 0 0
\(997\) 9.68247 0.306647 0.153324 0.988176i \(-0.451002\pi\)
0.153324 + 0.988176i \(0.451002\pi\)
\(998\) 24.5318 0.776540
\(999\) 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.j.1.19 20
3.2 odd 2 2151.2.a.k.1.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.19 20 1.1 even 1 trivial
2151.2.a.k.1.2 yes 20 3.2 odd 2