Properties

Label 2151.2.a.k.1.16
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 about

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 1.16
Root \(1.66953\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} +O(q^{10})\) \(q+1.66953 q^{2} +0.787326 q^{4} -3.19534 q^{5} -2.48662 q^{7} -2.02459 q^{8} -5.33471 q^{10} +4.44538 q^{11} -0.0955662 q^{13} -4.15149 q^{14} -4.95477 q^{16} -0.658610 q^{17} +5.40875 q^{19} -2.51577 q^{20} +7.42169 q^{22} -0.635524 q^{23} +5.21017 q^{25} -0.159551 q^{26} -1.95778 q^{28} +9.38370 q^{29} +7.21262 q^{31} -4.22294 q^{32} -1.09957 q^{34} +7.94559 q^{35} -11.9620 q^{37} +9.03006 q^{38} +6.46926 q^{40} +6.90694 q^{41} +7.01304 q^{43} +3.49996 q^{44} -1.06103 q^{46} +7.53884 q^{47} -0.816716 q^{49} +8.69853 q^{50} -0.0752418 q^{52} +0.341267 q^{53} -14.2045 q^{55} +5.03440 q^{56} +15.6664 q^{58} +1.57718 q^{59} -1.38928 q^{61} +12.0417 q^{62} +2.85921 q^{64} +0.305366 q^{65} -0.142775 q^{67} -0.518541 q^{68} +13.2654 q^{70} -3.28506 q^{71} +0.402150 q^{73} -19.9709 q^{74} +4.25845 q^{76} -11.0540 q^{77} +6.98905 q^{79} +15.8322 q^{80} +11.5313 q^{82} +0.0483201 q^{83} +2.10448 q^{85} +11.7085 q^{86} -9.00009 q^{88} +5.95912 q^{89} +0.237637 q^{91} -0.500365 q^{92} +12.5863 q^{94} -17.2828 q^{95} -11.1375 q^{97} -1.36353 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.66953 1.18054 0.590268 0.807208i \(-0.299022\pi\)
0.590268 + 0.807208i \(0.299022\pi\)
\(3\) 0 0
\(4\) 0.787326 0.393663
\(5\) −3.19534 −1.42900 −0.714499 0.699637i \(-0.753345\pi\)
−0.714499 + 0.699637i \(0.753345\pi\)
\(6\) 0 0
\(7\) −2.48662 −0.939854 −0.469927 0.882705i \(-0.655720\pi\)
−0.469927 + 0.882705i \(0.655720\pi\)
\(8\) −2.02459 −0.715802
\(9\) 0 0
\(10\) −5.33471 −1.68698
\(11\) 4.44538 1.34033 0.670166 0.742211i \(-0.266223\pi\)
0.670166 + 0.742211i \(0.266223\pi\)
\(12\) 0 0
\(13\) −0.0955662 −0.0265053 −0.0132526 0.999912i \(-0.504219\pi\)
−0.0132526 + 0.999912i \(0.504219\pi\)
\(14\) −4.15149 −1.10953
\(15\) 0 0
\(16\) −4.95477 −1.23869
\(17\) −0.658610 −0.159736 −0.0798682 0.996805i \(-0.525450\pi\)
−0.0798682 + 0.996805i \(0.525450\pi\)
\(18\) 0 0
\(19\) 5.40875 1.24085 0.620426 0.784265i \(-0.286960\pi\)
0.620426 + 0.784265i \(0.286960\pi\)
\(20\) −2.51577 −0.562544
\(21\) 0 0
\(22\) 7.42169 1.58231
\(23\) −0.635524 −0.132516 −0.0662580 0.997803i \(-0.521106\pi\)
−0.0662580 + 0.997803i \(0.521106\pi\)
\(24\) 0 0
\(25\) 5.21017 1.04203
\(26\) −0.159551 −0.0312904
\(27\) 0 0
\(28\) −1.95778 −0.369986
\(29\) 9.38370 1.74251 0.871255 0.490831i \(-0.163307\pi\)
0.871255 + 0.490831i \(0.163307\pi\)
\(30\) 0 0
\(31\) 7.21262 1.29542 0.647712 0.761885i \(-0.275726\pi\)
0.647712 + 0.761885i \(0.275726\pi\)
\(32\) −4.22294 −0.746518
\(33\) 0 0
\(34\) −1.09957 −0.188574
\(35\) 7.94559 1.34305
\(36\) 0 0
\(37\) −11.9620 −1.96655 −0.983273 0.182140i \(-0.941698\pi\)
−0.983273 + 0.182140i \(0.941698\pi\)
\(38\) 9.03006 1.46487
\(39\) 0 0
\(40\) 6.46926 1.02288
\(41\) 6.90694 1.07868 0.539342 0.842087i \(-0.318673\pi\)
0.539342 + 0.842087i \(0.318673\pi\)
\(42\) 0 0
\(43\) 7.01304 1.06948 0.534739 0.845017i \(-0.320410\pi\)
0.534739 + 0.845017i \(0.320410\pi\)
\(44\) 3.49996 0.527639
\(45\) 0 0
\(46\) −1.06103 −0.156440
\(47\) 7.53884 1.09965 0.549827 0.835279i \(-0.314694\pi\)
0.549827 + 0.835279i \(0.314694\pi\)
\(48\) 0 0
\(49\) −0.816716 −0.116674
\(50\) 8.69853 1.23016
\(51\) 0 0
\(52\) −0.0752418 −0.0104342
\(53\) 0.341267 0.0468766 0.0234383 0.999725i \(-0.492539\pi\)
0.0234383 + 0.999725i \(0.492539\pi\)
\(54\) 0 0
\(55\) −14.2045 −1.91533
\(56\) 5.03440 0.672750
\(57\) 0 0
\(58\) 15.6664 2.05709
\(59\) 1.57718 0.205332 0.102666 0.994716i \(-0.467263\pi\)
0.102666 + 0.994716i \(0.467263\pi\)
\(60\) 0 0
\(61\) −1.38928 −0.177879 −0.0889396 0.996037i \(-0.528348\pi\)
−0.0889396 + 0.996037i \(0.528348\pi\)
\(62\) 12.0417 1.52929
\(63\) 0 0
\(64\) 2.85921 0.357402
\(65\) 0.305366 0.0378760
\(66\) 0 0
\(67\) −0.142775 −0.0174427 −0.00872136 0.999962i \(-0.502776\pi\)
−0.00872136 + 0.999962i \(0.502776\pi\)
\(68\) −0.518541 −0.0628823
\(69\) 0 0
\(70\) 13.2654 1.58552
\(71\) −3.28506 −0.389865 −0.194933 0.980817i \(-0.562449\pi\)
−0.194933 + 0.980817i \(0.562449\pi\)
\(72\) 0 0
\(73\) 0.402150 0.0470681 0.0235340 0.999723i \(-0.492508\pi\)
0.0235340 + 0.999723i \(0.492508\pi\)
\(74\) −19.9709 −2.32158
\(75\) 0 0
\(76\) 4.25845 0.488478
\(77\) −11.0540 −1.25972
\(78\) 0 0
\(79\) 6.98905 0.786330 0.393165 0.919468i \(-0.371380\pi\)
0.393165 + 0.919468i \(0.371380\pi\)
\(80\) 15.8322 1.77009
\(81\) 0 0
\(82\) 11.5313 1.27342
\(83\) 0.0483201 0.00530382 0.00265191 0.999996i \(-0.499156\pi\)
0.00265191 + 0.999996i \(0.499156\pi\)
\(84\) 0 0
\(85\) 2.10448 0.228263
\(86\) 11.7085 1.26256
\(87\) 0 0
\(88\) −9.00009 −0.959412
\(89\) 5.95912 0.631665 0.315833 0.948815i \(-0.397716\pi\)
0.315833 + 0.948815i \(0.397716\pi\)
\(90\) 0 0
\(91\) 0.237637 0.0249111
\(92\) −0.500365 −0.0521667
\(93\) 0 0
\(94\) 12.5863 1.29818
\(95\) −17.2828 −1.77317
\(96\) 0 0
\(97\) −11.1375 −1.13084 −0.565421 0.824802i \(-0.691286\pi\)
−0.565421 + 0.824802i \(0.691286\pi\)
\(98\) −1.36353 −0.137737
\(99\) 0 0
\(100\) 4.10211 0.410211
\(101\) −0.600219 −0.0597240 −0.0298620 0.999554i \(-0.509507\pi\)
−0.0298620 + 0.999554i \(0.509507\pi\)
\(102\) 0 0
\(103\) 0.225700 0.0222389 0.0111195 0.999938i \(-0.496460\pi\)
0.0111195 + 0.999938i \(0.496460\pi\)
\(104\) 0.193483 0.0189725
\(105\) 0 0
\(106\) 0.569755 0.0553395
\(107\) 8.77148 0.847971 0.423986 0.905669i \(-0.360631\pi\)
0.423986 + 0.905669i \(0.360631\pi\)
\(108\) 0 0
\(109\) 4.72372 0.452450 0.226225 0.974075i \(-0.427361\pi\)
0.226225 + 0.974075i \(0.427361\pi\)
\(110\) −23.7148 −2.26112
\(111\) 0 0
\(112\) 12.3206 1.16419
\(113\) −11.7376 −1.10418 −0.552089 0.833785i \(-0.686169\pi\)
−0.552089 + 0.833785i \(0.686169\pi\)
\(114\) 0 0
\(115\) 2.03071 0.189365
\(116\) 7.38803 0.685962
\(117\) 0 0
\(118\) 2.63315 0.242401
\(119\) 1.63771 0.150129
\(120\) 0 0
\(121\) 8.76140 0.796491
\(122\) −2.31944 −0.209993
\(123\) 0 0
\(124\) 5.67869 0.509961
\(125\) −0.671570 −0.0600671
\(126\) 0 0
\(127\) −0.436274 −0.0387130 −0.0193565 0.999813i \(-0.506162\pi\)
−0.0193565 + 0.999813i \(0.506162\pi\)
\(128\) 13.2194 1.16844
\(129\) 0 0
\(130\) 0.509817 0.0447139
\(131\) 11.9696 1.04579 0.522894 0.852398i \(-0.324853\pi\)
0.522894 + 0.852398i \(0.324853\pi\)
\(132\) 0 0
\(133\) −13.4495 −1.16622
\(134\) −0.238367 −0.0205917
\(135\) 0 0
\(136\) 1.33342 0.114340
\(137\) 18.5405 1.58402 0.792009 0.610509i \(-0.209035\pi\)
0.792009 + 0.610509i \(0.209035\pi\)
\(138\) 0 0
\(139\) 0.00950510 0.000806212 0 0.000403106 1.00000i \(-0.499872\pi\)
0.000403106 1.00000i \(0.499872\pi\)
\(140\) 6.25577 0.528709
\(141\) 0 0
\(142\) −5.48451 −0.460250
\(143\) −0.424828 −0.0355259
\(144\) 0 0
\(145\) −29.9841 −2.49004
\(146\) 0.671401 0.0555655
\(147\) 0 0
\(148\) −9.41802 −0.774156
\(149\) 18.0008 1.47468 0.737342 0.675519i \(-0.236081\pi\)
0.737342 + 0.675519i \(0.236081\pi\)
\(150\) 0 0
\(151\) −6.75284 −0.549538 −0.274769 0.961510i \(-0.588601\pi\)
−0.274769 + 0.961510i \(0.588601\pi\)
\(152\) −10.9505 −0.888204
\(153\) 0 0
\(154\) −18.4549 −1.48714
\(155\) −23.0467 −1.85116
\(156\) 0 0
\(157\) −3.73808 −0.298331 −0.149166 0.988812i \(-0.547659\pi\)
−0.149166 + 0.988812i \(0.547659\pi\)
\(158\) 11.6684 0.928290
\(159\) 0 0
\(160\) 13.4937 1.06677
\(161\) 1.58031 0.124546
\(162\) 0 0
\(163\) 14.3516 1.12410 0.562052 0.827102i \(-0.310012\pi\)
0.562052 + 0.827102i \(0.310012\pi\)
\(164\) 5.43802 0.424638
\(165\) 0 0
\(166\) 0.0806718 0.00626135
\(167\) 0.336460 0.0260360 0.0130180 0.999915i \(-0.495856\pi\)
0.0130180 + 0.999915i \(0.495856\pi\)
\(168\) 0 0
\(169\) −12.9909 −0.999297
\(170\) 3.51349 0.269472
\(171\) 0 0
\(172\) 5.52155 0.421014
\(173\) −11.3461 −0.862628 −0.431314 0.902202i \(-0.641950\pi\)
−0.431314 + 0.902202i \(0.641950\pi\)
\(174\) 0 0
\(175\) −12.9557 −0.979361
\(176\) −22.0258 −1.66026
\(177\) 0 0
\(178\) 9.94892 0.745703
\(179\) 4.38379 0.327660 0.163830 0.986489i \(-0.447615\pi\)
0.163830 + 0.986489i \(0.447615\pi\)
\(180\) 0 0
\(181\) −8.91623 −0.662738 −0.331369 0.943501i \(-0.607511\pi\)
−0.331369 + 0.943501i \(0.607511\pi\)
\(182\) 0.396742 0.0294084
\(183\) 0 0
\(184\) 1.28668 0.0948552
\(185\) 38.2227 2.81019
\(186\) 0 0
\(187\) −2.92777 −0.214100
\(188\) 5.93553 0.432893
\(189\) 0 0
\(190\) −28.8541 −2.09329
\(191\) 9.60112 0.694713 0.347356 0.937733i \(-0.387079\pi\)
0.347356 + 0.937733i \(0.387079\pi\)
\(192\) 0 0
\(193\) −20.1533 −1.45067 −0.725333 0.688398i \(-0.758314\pi\)
−0.725333 + 0.688398i \(0.758314\pi\)
\(194\) −18.5944 −1.33500
\(195\) 0 0
\(196\) −0.643022 −0.0459302
\(197\) −21.4790 −1.53032 −0.765158 0.643842i \(-0.777339\pi\)
−0.765158 + 0.643842i \(0.777339\pi\)
\(198\) 0 0
\(199\) 0.0533765 0.00378376 0.00189188 0.999998i \(-0.499398\pi\)
0.00189188 + 0.999998i \(0.499398\pi\)
\(200\) −10.5485 −0.745890
\(201\) 0 0
\(202\) −1.00208 −0.0705063
\(203\) −23.3337 −1.63771
\(204\) 0 0
\(205\) −22.0700 −1.54144
\(206\) 0.376813 0.0262538
\(207\) 0 0
\(208\) 0.473508 0.0328319
\(209\) 24.0439 1.66315
\(210\) 0 0
\(211\) −3.28901 −0.226425 −0.113212 0.993571i \(-0.536114\pi\)
−0.113212 + 0.993571i \(0.536114\pi\)
\(212\) 0.268688 0.0184536
\(213\) 0 0
\(214\) 14.6442 1.00106
\(215\) −22.4090 −1.52828
\(216\) 0 0
\(217\) −17.9351 −1.21751
\(218\) 7.88639 0.534134
\(219\) 0 0
\(220\) −11.1836 −0.753996
\(221\) 0.0629408 0.00423386
\(222\) 0 0
\(223\) −4.32252 −0.289458 −0.144729 0.989471i \(-0.546231\pi\)
−0.144729 + 0.989471i \(0.546231\pi\)
\(224\) 10.5009 0.701618
\(225\) 0 0
\(226\) −19.5962 −1.30352
\(227\) 22.3311 1.48217 0.741085 0.671411i \(-0.234312\pi\)
0.741085 + 0.671411i \(0.234312\pi\)
\(228\) 0 0
\(229\) 19.6537 1.29875 0.649376 0.760468i \(-0.275030\pi\)
0.649376 + 0.760468i \(0.275030\pi\)
\(230\) 3.39033 0.223552
\(231\) 0 0
\(232\) −18.9982 −1.24729
\(233\) 3.43729 0.225185 0.112592 0.993641i \(-0.464085\pi\)
0.112592 + 0.993641i \(0.464085\pi\)
\(234\) 0 0
\(235\) −24.0891 −1.57140
\(236\) 1.24176 0.0808315
\(237\) 0 0
\(238\) 2.73421 0.177232
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 8.22484 0.529808 0.264904 0.964275i \(-0.414660\pi\)
0.264904 + 0.964275i \(0.414660\pi\)
\(242\) 14.6274 0.940285
\(243\) 0 0
\(244\) −1.09382 −0.0700245
\(245\) 2.60968 0.166727
\(246\) 0 0
\(247\) −0.516893 −0.0328891
\(248\) −14.6026 −0.927268
\(249\) 0 0
\(250\) −1.12121 −0.0709113
\(251\) −15.3444 −0.968531 −0.484265 0.874921i \(-0.660913\pi\)
−0.484265 + 0.874921i \(0.660913\pi\)
\(252\) 0 0
\(253\) −2.82515 −0.177615
\(254\) −0.728371 −0.0457021
\(255\) 0 0
\(256\) 16.3518 1.02199
\(257\) 27.1321 1.69245 0.846225 0.532825i \(-0.178870\pi\)
0.846225 + 0.532825i \(0.178870\pi\)
\(258\) 0 0
\(259\) 29.7450 1.84827
\(260\) 0.240423 0.0149104
\(261\) 0 0
\(262\) 19.9836 1.23459
\(263\) −15.0146 −0.925840 −0.462920 0.886400i \(-0.653198\pi\)
−0.462920 + 0.886400i \(0.653198\pi\)
\(264\) 0 0
\(265\) −1.09046 −0.0669865
\(266\) −22.4543 −1.37676
\(267\) 0 0
\(268\) −0.112410 −0.00686655
\(269\) 24.3072 1.48204 0.741018 0.671485i \(-0.234343\pi\)
0.741018 + 0.671485i \(0.234343\pi\)
\(270\) 0 0
\(271\) −27.2074 −1.65273 −0.826366 0.563133i \(-0.809596\pi\)
−0.826366 + 0.563133i \(0.809596\pi\)
\(272\) 3.26326 0.197864
\(273\) 0 0
\(274\) 30.9538 1.86999
\(275\) 23.1612 1.39667
\(276\) 0 0
\(277\) −31.1450 −1.87132 −0.935660 0.352902i \(-0.885195\pi\)
−0.935660 + 0.352902i \(0.885195\pi\)
\(278\) 0.0158690 0.000951761 0
\(279\) 0 0
\(280\) −16.0866 −0.961358
\(281\) 21.5282 1.28426 0.642132 0.766594i \(-0.278050\pi\)
0.642132 + 0.766594i \(0.278050\pi\)
\(282\) 0 0
\(283\) 29.2986 1.74162 0.870809 0.491621i \(-0.163596\pi\)
0.870809 + 0.491621i \(0.163596\pi\)
\(284\) −2.58642 −0.153476
\(285\) 0 0
\(286\) −0.709263 −0.0419396
\(287\) −17.1750 −1.01381
\(288\) 0 0
\(289\) −16.5662 −0.974484
\(290\) −50.0593 −2.93958
\(291\) 0 0
\(292\) 0.316623 0.0185290
\(293\) 10.7054 0.625417 0.312709 0.949849i \(-0.398764\pi\)
0.312709 + 0.949849i \(0.398764\pi\)
\(294\) 0 0
\(295\) −5.03962 −0.293418
\(296\) 24.2182 1.40766
\(297\) 0 0
\(298\) 30.0529 1.74092
\(299\) 0.0607346 0.00351237
\(300\) 0 0
\(301\) −17.4388 −1.00515
\(302\) −11.2741 −0.648749
\(303\) 0 0
\(304\) −26.7991 −1.53703
\(305\) 4.43922 0.254189
\(306\) 0 0
\(307\) 21.3838 1.22044 0.610219 0.792233i \(-0.291081\pi\)
0.610219 + 0.792233i \(0.291081\pi\)
\(308\) −8.70308 −0.495904
\(309\) 0 0
\(310\) −38.4772 −2.18536
\(311\) 2.67787 0.151848 0.0759240 0.997114i \(-0.475809\pi\)
0.0759240 + 0.997114i \(0.475809\pi\)
\(312\) 0 0
\(313\) 0.666770 0.0376881 0.0188440 0.999822i \(-0.494001\pi\)
0.0188440 + 0.999822i \(0.494001\pi\)
\(314\) −6.24084 −0.352191
\(315\) 0 0
\(316\) 5.50266 0.309549
\(317\) −24.4769 −1.37476 −0.687379 0.726299i \(-0.741239\pi\)
−0.687379 + 0.726299i \(0.741239\pi\)
\(318\) 0 0
\(319\) 41.7141 2.33554
\(320\) −9.13615 −0.510726
\(321\) 0 0
\(322\) 2.63837 0.147031
\(323\) −3.56225 −0.198209
\(324\) 0 0
\(325\) −0.497916 −0.0276194
\(326\) 23.9604 1.32704
\(327\) 0 0
\(328\) −13.9838 −0.772124
\(329\) −18.7462 −1.03351
\(330\) 0 0
\(331\) −28.3306 −1.55719 −0.778595 0.627526i \(-0.784068\pi\)
−0.778595 + 0.627526i \(0.784068\pi\)
\(332\) 0.0380437 0.00208792
\(333\) 0 0
\(334\) 0.561729 0.0307364
\(335\) 0.456213 0.0249256
\(336\) 0 0
\(337\) 28.2985 1.54152 0.770758 0.637127i \(-0.219878\pi\)
0.770758 + 0.637127i \(0.219878\pi\)
\(338\) −21.6886 −1.17971
\(339\) 0 0
\(340\) 1.65691 0.0898587
\(341\) 32.0628 1.73630
\(342\) 0 0
\(343\) 19.4372 1.04951
\(344\) −14.1985 −0.765534
\(345\) 0 0
\(346\) −18.9426 −1.01836
\(347\) 36.1955 1.94307 0.971537 0.236887i \(-0.0761270\pi\)
0.971537 + 0.236887i \(0.0761270\pi\)
\(348\) 0 0
\(349\) 35.8094 1.91683 0.958416 0.285375i \(-0.0921180\pi\)
0.958416 + 0.285375i \(0.0921180\pi\)
\(350\) −21.6300 −1.15617
\(351\) 0 0
\(352\) −18.7726 −1.00058
\(353\) −27.6089 −1.46948 −0.734738 0.678351i \(-0.762695\pi\)
−0.734738 + 0.678351i \(0.762695\pi\)
\(354\) 0 0
\(355\) 10.4969 0.557117
\(356\) 4.69177 0.248663
\(357\) 0 0
\(358\) 7.31887 0.386814
\(359\) 16.6008 0.876154 0.438077 0.898937i \(-0.355660\pi\)
0.438077 + 0.898937i \(0.355660\pi\)
\(360\) 0 0
\(361\) 10.2545 0.539713
\(362\) −14.8859 −0.782386
\(363\) 0 0
\(364\) 0.187098 0.00980659
\(365\) −1.28500 −0.0672602
\(366\) 0 0
\(367\) 21.8482 1.14047 0.570234 0.821483i \(-0.306853\pi\)
0.570234 + 0.821483i \(0.306853\pi\)
\(368\) 3.14888 0.164147
\(369\) 0 0
\(370\) 63.8139 3.31753
\(371\) −0.848601 −0.0440572
\(372\) 0 0
\(373\) −26.2374 −1.35852 −0.679261 0.733897i \(-0.737699\pi\)
−0.679261 + 0.733897i \(0.737699\pi\)
\(374\) −4.88800 −0.252752
\(375\) 0 0
\(376\) −15.2631 −0.787134
\(377\) −0.896765 −0.0461857
\(378\) 0 0
\(379\) −23.8736 −1.22630 −0.613151 0.789966i \(-0.710098\pi\)
−0.613151 + 0.789966i \(0.710098\pi\)
\(380\) −13.6072 −0.698033
\(381\) 0 0
\(382\) 16.0294 0.820133
\(383\) 1.53922 0.0786507 0.0393254 0.999226i \(-0.487479\pi\)
0.0393254 + 0.999226i \(0.487479\pi\)
\(384\) 0 0
\(385\) 35.3212 1.80013
\(386\) −33.6465 −1.71256
\(387\) 0 0
\(388\) −8.76885 −0.445171
\(389\) 20.7468 1.05191 0.525953 0.850514i \(-0.323709\pi\)
0.525953 + 0.850514i \(0.323709\pi\)
\(390\) 0 0
\(391\) 0.418563 0.0211676
\(392\) 1.65352 0.0835153
\(393\) 0 0
\(394\) −35.8598 −1.80659
\(395\) −22.3324 −1.12366
\(396\) 0 0
\(397\) 0.280543 0.0140800 0.00704002 0.999975i \(-0.497759\pi\)
0.00704002 + 0.999975i \(0.497759\pi\)
\(398\) 0.0891135 0.00446686
\(399\) 0 0
\(400\) −25.8152 −1.29076
\(401\) 14.9965 0.748888 0.374444 0.927250i \(-0.377834\pi\)
0.374444 + 0.927250i \(0.377834\pi\)
\(402\) 0 0
\(403\) −0.689283 −0.0343356
\(404\) −0.472568 −0.0235112
\(405\) 0 0
\(406\) −38.9563 −1.93337
\(407\) −53.1757 −2.63582
\(408\) 0 0
\(409\) −2.64336 −0.130706 −0.0653528 0.997862i \(-0.520817\pi\)
−0.0653528 + 0.997862i \(0.520817\pi\)
\(410\) −36.8465 −1.81972
\(411\) 0 0
\(412\) 0.177700 0.00875464
\(413\) −3.92185 −0.192982
\(414\) 0 0
\(415\) −0.154399 −0.00757915
\(416\) 0.403571 0.0197867
\(417\) 0 0
\(418\) 40.1420 1.96341
\(419\) −12.1813 −0.595094 −0.297547 0.954707i \(-0.596169\pi\)
−0.297547 + 0.954707i \(0.596169\pi\)
\(420\) 0 0
\(421\) 6.22710 0.303490 0.151745 0.988420i \(-0.451511\pi\)
0.151745 + 0.988420i \(0.451511\pi\)
\(422\) −5.49110 −0.267302
\(423\) 0 0
\(424\) −0.690926 −0.0335544
\(425\) −3.43147 −0.166451
\(426\) 0 0
\(427\) 3.45461 0.167180
\(428\) 6.90602 0.333815
\(429\) 0 0
\(430\) −37.4125 −1.80419
\(431\) −22.4753 −1.08260 −0.541299 0.840830i \(-0.682067\pi\)
−0.541299 + 0.840830i \(0.682067\pi\)
\(432\) 0 0
\(433\) 11.0267 0.529908 0.264954 0.964261i \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(434\) −29.9431 −1.43731
\(435\) 0 0
\(436\) 3.71911 0.178113
\(437\) −3.43739 −0.164433
\(438\) 0 0
\(439\) 37.6999 1.79932 0.899659 0.436593i \(-0.143815\pi\)
0.899659 + 0.436593i \(0.143815\pi\)
\(440\) 28.7583 1.37100
\(441\) 0 0
\(442\) 0.105082 0.00499822
\(443\) 12.1550 0.577503 0.288751 0.957404i \(-0.406760\pi\)
0.288751 + 0.957404i \(0.406760\pi\)
\(444\) 0 0
\(445\) −19.0414 −0.902648
\(446\) −7.21658 −0.341715
\(447\) 0 0
\(448\) −7.10978 −0.335906
\(449\) −4.49986 −0.212362 −0.106181 0.994347i \(-0.533862\pi\)
−0.106181 + 0.994347i \(0.533862\pi\)
\(450\) 0 0
\(451\) 30.7040 1.44579
\(452\) −9.24130 −0.434674
\(453\) 0 0
\(454\) 37.2825 1.74975
\(455\) −0.759330 −0.0355979
\(456\) 0 0
\(457\) 3.25466 0.152246 0.0761232 0.997098i \(-0.475746\pi\)
0.0761232 + 0.997098i \(0.475746\pi\)
\(458\) 32.8124 1.53322
\(459\) 0 0
\(460\) 1.59883 0.0745460
\(461\) −38.0611 −1.77268 −0.886341 0.463033i \(-0.846761\pi\)
−0.886341 + 0.463033i \(0.846761\pi\)
\(462\) 0 0
\(463\) −9.16202 −0.425795 −0.212898 0.977075i \(-0.568290\pi\)
−0.212898 + 0.977075i \(0.568290\pi\)
\(464\) −46.4941 −2.15843
\(465\) 0 0
\(466\) 5.73866 0.265838
\(467\) 4.39258 0.203265 0.101632 0.994822i \(-0.467593\pi\)
0.101632 + 0.994822i \(0.467593\pi\)
\(468\) 0 0
\(469\) 0.355027 0.0163936
\(470\) −40.2175 −1.85509
\(471\) 0 0
\(472\) −3.19315 −0.146977
\(473\) 31.1756 1.43346
\(474\) 0 0
\(475\) 28.1805 1.29301
\(476\) 1.28941 0.0591002
\(477\) 0 0
\(478\) 1.66953 0.0763625
\(479\) 24.3906 1.11444 0.557219 0.830366i \(-0.311869\pi\)
0.557219 + 0.830366i \(0.311869\pi\)
\(480\) 0 0
\(481\) 1.14317 0.0521239
\(482\) 13.7316 0.625458
\(483\) 0 0
\(484\) 6.89808 0.313549
\(485\) 35.5881 1.61597
\(486\) 0 0
\(487\) −33.7875 −1.53106 −0.765530 0.643400i \(-0.777523\pi\)
−0.765530 + 0.643400i \(0.777523\pi\)
\(488\) 2.81273 0.127326
\(489\) 0 0
\(490\) 4.35694 0.196827
\(491\) 6.60713 0.298175 0.149088 0.988824i \(-0.452366\pi\)
0.149088 + 0.988824i \(0.452366\pi\)
\(492\) 0 0
\(493\) −6.18020 −0.278342
\(494\) −0.862968 −0.0388268
\(495\) 0 0
\(496\) −35.7369 −1.60463
\(497\) 8.16871 0.366417
\(498\) 0 0
\(499\) −7.45431 −0.333701 −0.166850 0.985982i \(-0.553360\pi\)
−0.166850 + 0.985982i \(0.553360\pi\)
\(500\) −0.528745 −0.0236462
\(501\) 0 0
\(502\) −25.6179 −1.14338
\(503\) −1.72248 −0.0768015 −0.0384008 0.999262i \(-0.512226\pi\)
−0.0384008 + 0.999262i \(0.512226\pi\)
\(504\) 0 0
\(505\) 1.91790 0.0853455
\(506\) −4.71666 −0.209681
\(507\) 0 0
\(508\) −0.343490 −0.0152399
\(509\) −6.17725 −0.273802 −0.136901 0.990585i \(-0.543714\pi\)
−0.136901 + 0.990585i \(0.543714\pi\)
\(510\) 0 0
\(511\) −0.999994 −0.0442371
\(512\) 0.860923 0.0380478
\(513\) 0 0
\(514\) 45.2977 1.99800
\(515\) −0.721189 −0.0317794
\(516\) 0 0
\(517\) 33.5130 1.47390
\(518\) 49.6602 2.18194
\(519\) 0 0
\(520\) −0.618242 −0.0271117
\(521\) 26.4121 1.15714 0.578568 0.815634i \(-0.303612\pi\)
0.578568 + 0.815634i \(0.303612\pi\)
\(522\) 0 0
\(523\) 30.6730 1.34124 0.670618 0.741803i \(-0.266029\pi\)
0.670618 + 0.741803i \(0.266029\pi\)
\(524\) 9.42397 0.411688
\(525\) 0 0
\(526\) −25.0673 −1.09299
\(527\) −4.75030 −0.206926
\(528\) 0 0
\(529\) −22.5961 −0.982440
\(530\) −1.82056 −0.0790800
\(531\) 0 0
\(532\) −10.5891 −0.459098
\(533\) −0.660070 −0.0285908
\(534\) 0 0
\(535\) −28.0278 −1.21175
\(536\) 0.289061 0.0124855
\(537\) 0 0
\(538\) 40.5816 1.74960
\(539\) −3.63061 −0.156382
\(540\) 0 0
\(541\) −8.59747 −0.369634 −0.184817 0.982773i \(-0.559169\pi\)
−0.184817 + 0.982773i \(0.559169\pi\)
\(542\) −45.4236 −1.95111
\(543\) 0 0
\(544\) 2.78127 0.119246
\(545\) −15.0939 −0.646551
\(546\) 0 0
\(547\) 20.3050 0.868178 0.434089 0.900870i \(-0.357070\pi\)
0.434089 + 0.900870i \(0.357070\pi\)
\(548\) 14.5974 0.623570
\(549\) 0 0
\(550\) 38.6683 1.64882
\(551\) 50.7541 2.16220
\(552\) 0 0
\(553\) −17.3791 −0.739035
\(554\) −51.9974 −2.20916
\(555\) 0 0
\(556\) 0.00748361 0.000317376 0
\(557\) −25.7890 −1.09272 −0.546358 0.837552i \(-0.683986\pi\)
−0.546358 + 0.837552i \(0.683986\pi\)
\(558\) 0 0
\(559\) −0.670209 −0.0283468
\(560\) −39.3686 −1.66363
\(561\) 0 0
\(562\) 35.9419 1.51612
\(563\) 10.6066 0.447016 0.223508 0.974702i \(-0.428249\pi\)
0.223508 + 0.974702i \(0.428249\pi\)
\(564\) 0 0
\(565\) 37.5055 1.57787
\(566\) 48.9148 2.05604
\(567\) 0 0
\(568\) 6.65092 0.279066
\(569\) −31.2790 −1.31128 −0.655641 0.755072i \(-0.727602\pi\)
−0.655641 + 0.755072i \(0.727602\pi\)
\(570\) 0 0
\(571\) −3.31115 −0.138567 −0.0692836 0.997597i \(-0.522071\pi\)
−0.0692836 + 0.997597i \(0.522071\pi\)
\(572\) −0.334478 −0.0139852
\(573\) 0 0
\(574\) −28.6741 −1.19683
\(575\) −3.31119 −0.138086
\(576\) 0 0
\(577\) 43.3143 1.80320 0.901599 0.432573i \(-0.142394\pi\)
0.901599 + 0.432573i \(0.142394\pi\)
\(578\) −27.6578 −1.15041
\(579\) 0 0
\(580\) −23.6073 −0.980238
\(581\) −0.120154 −0.00498482
\(582\) 0 0
\(583\) 1.51706 0.0628302
\(584\) −0.814190 −0.0336914
\(585\) 0 0
\(586\) 17.8730 0.738327
\(587\) −36.3210 −1.49913 −0.749565 0.661931i \(-0.769737\pi\)
−0.749565 + 0.661931i \(0.769737\pi\)
\(588\) 0 0
\(589\) 39.0112 1.60743
\(590\) −8.41380 −0.346391
\(591\) 0 0
\(592\) 59.2691 2.43594
\(593\) −25.2745 −1.03790 −0.518949 0.854805i \(-0.673677\pi\)
−0.518949 + 0.854805i \(0.673677\pi\)
\(594\) 0 0
\(595\) −5.23304 −0.214534
\(596\) 14.1725 0.580529
\(597\) 0 0
\(598\) 0.101398 0.00414648
\(599\) 28.3772 1.15946 0.579731 0.814808i \(-0.303158\pi\)
0.579731 + 0.814808i \(0.303158\pi\)
\(600\) 0 0
\(601\) 22.8258 0.931084 0.465542 0.885026i \(-0.345859\pi\)
0.465542 + 0.885026i \(0.345859\pi\)
\(602\) −29.1145 −1.18662
\(603\) 0 0
\(604\) −5.31669 −0.216333
\(605\) −27.9956 −1.13818
\(606\) 0 0
\(607\) 27.1407 1.10161 0.550803 0.834635i \(-0.314321\pi\)
0.550803 + 0.834635i \(0.314321\pi\)
\(608\) −22.8408 −0.926318
\(609\) 0 0
\(610\) 7.41140 0.300079
\(611\) −0.720458 −0.0291466
\(612\) 0 0
\(613\) 14.9443 0.603595 0.301797 0.953372i \(-0.402413\pi\)
0.301797 + 0.953372i \(0.402413\pi\)
\(614\) 35.7009 1.44077
\(615\) 0 0
\(616\) 22.3798 0.901708
\(617\) −6.00090 −0.241587 −0.120794 0.992678i \(-0.538544\pi\)
−0.120794 + 0.992678i \(0.538544\pi\)
\(618\) 0 0
\(619\) −28.7394 −1.15513 −0.577567 0.816343i \(-0.695998\pi\)
−0.577567 + 0.816343i \(0.695998\pi\)
\(620\) −18.1453 −0.728733
\(621\) 0 0
\(622\) 4.47078 0.179262
\(623\) −14.8181 −0.593673
\(624\) 0 0
\(625\) −23.9050 −0.956199
\(626\) 1.11319 0.0444921
\(627\) 0 0
\(628\) −2.94309 −0.117442
\(629\) 7.87831 0.314129
\(630\) 0 0
\(631\) −24.2406 −0.965003 −0.482501 0.875895i \(-0.660272\pi\)
−0.482501 + 0.875895i \(0.660272\pi\)
\(632\) −14.1500 −0.562856
\(633\) 0 0
\(634\) −40.8649 −1.62295
\(635\) 1.39404 0.0553208
\(636\) 0 0
\(637\) 0.0780505 0.00309247
\(638\) 69.6429 2.75719
\(639\) 0 0
\(640\) −42.2405 −1.66970
\(641\) −47.5182 −1.87686 −0.938428 0.345476i \(-0.887717\pi\)
−0.938428 + 0.345476i \(0.887717\pi\)
\(642\) 0 0
\(643\) −24.9865 −0.985370 −0.492685 0.870208i \(-0.663984\pi\)
−0.492685 + 0.870208i \(0.663984\pi\)
\(644\) 1.24422 0.0490291
\(645\) 0 0
\(646\) −5.94729 −0.233993
\(647\) −31.5472 −1.24025 −0.620125 0.784503i \(-0.712918\pi\)
−0.620125 + 0.784503i \(0.712918\pi\)
\(648\) 0 0
\(649\) 7.01117 0.275212
\(650\) −0.831286 −0.0326057
\(651\) 0 0
\(652\) 11.2994 0.442518
\(653\) 7.47349 0.292460 0.146230 0.989251i \(-0.453286\pi\)
0.146230 + 0.989251i \(0.453286\pi\)
\(654\) 0 0
\(655\) −38.2468 −1.49443
\(656\) −34.2223 −1.33616
\(657\) 0 0
\(658\) −31.2974 −1.22010
\(659\) −20.6225 −0.803340 −0.401670 0.915785i \(-0.631570\pi\)
−0.401670 + 0.915785i \(0.631570\pi\)
\(660\) 0 0
\(661\) 38.4744 1.49648 0.748240 0.663428i \(-0.230899\pi\)
0.748240 + 0.663428i \(0.230899\pi\)
\(662\) −47.2988 −1.83832
\(663\) 0 0
\(664\) −0.0978286 −0.00379649
\(665\) 42.9757 1.66653
\(666\) 0 0
\(667\) −5.96357 −0.230910
\(668\) 0.264903 0.0102494
\(669\) 0 0
\(670\) 0.761661 0.0294255
\(671\) −6.17588 −0.238417
\(672\) 0 0
\(673\) 23.5232 0.906754 0.453377 0.891319i \(-0.350219\pi\)
0.453377 + 0.891319i \(0.350219\pi\)
\(674\) 47.2451 1.81981
\(675\) 0 0
\(676\) −10.2281 −0.393387
\(677\) −8.45475 −0.324942 −0.162471 0.986713i \(-0.551946\pi\)
−0.162471 + 0.986713i \(0.551946\pi\)
\(678\) 0 0
\(679\) 27.6948 1.06283
\(680\) −4.26072 −0.163391
\(681\) 0 0
\(682\) 53.5298 2.04976
\(683\) −21.6625 −0.828894 −0.414447 0.910074i \(-0.636025\pi\)
−0.414447 + 0.910074i \(0.636025\pi\)
\(684\) 0 0
\(685\) −59.2430 −2.26356
\(686\) 32.4510 1.23898
\(687\) 0 0
\(688\) −34.7480 −1.32475
\(689\) −0.0326136 −0.00124248
\(690\) 0 0
\(691\) 2.61127 0.0993375 0.0496688 0.998766i \(-0.484183\pi\)
0.0496688 + 0.998766i \(0.484183\pi\)
\(692\) −8.93308 −0.339585
\(693\) 0 0
\(694\) 60.4294 2.29387
\(695\) −0.0303720 −0.00115207
\(696\) 0 0
\(697\) −4.54898 −0.172305
\(698\) 59.7848 2.26289
\(699\) 0 0
\(700\) −10.2004 −0.385538
\(701\) 34.6693 1.30944 0.654721 0.755870i \(-0.272786\pi\)
0.654721 + 0.755870i \(0.272786\pi\)
\(702\) 0 0
\(703\) −64.6996 −2.44019
\(704\) 12.7103 0.479037
\(705\) 0 0
\(706\) −46.0939 −1.73477
\(707\) 1.49252 0.0561319
\(708\) 0 0
\(709\) 27.0398 1.01550 0.507750 0.861504i \(-0.330477\pi\)
0.507750 + 0.861504i \(0.330477\pi\)
\(710\) 17.5249 0.657696
\(711\) 0 0
\(712\) −12.0648 −0.452147
\(713\) −4.58380 −0.171664
\(714\) 0 0
\(715\) 1.35747 0.0507664
\(716\) 3.45148 0.128988
\(717\) 0 0
\(718\) 27.7154 1.03433
\(719\) −30.6086 −1.14151 −0.570753 0.821122i \(-0.693349\pi\)
−0.570753 + 0.821122i \(0.693349\pi\)
\(720\) 0 0
\(721\) −0.561231 −0.0209013
\(722\) 17.1203 0.637150
\(723\) 0 0
\(724\) −7.01998 −0.260896
\(725\) 48.8907 1.81575
\(726\) 0 0
\(727\) −45.7934 −1.69838 −0.849192 0.528085i \(-0.822910\pi\)
−0.849192 + 0.528085i \(0.822910\pi\)
\(728\) −0.481118 −0.0178314
\(729\) 0 0
\(730\) −2.14535 −0.0794030
\(731\) −4.61885 −0.170834
\(732\) 0 0
\(733\) −23.1892 −0.856511 −0.428256 0.903658i \(-0.640872\pi\)
−0.428256 + 0.903658i \(0.640872\pi\)
\(734\) 36.4762 1.34636
\(735\) 0 0
\(736\) 2.68378 0.0989256
\(737\) −0.634688 −0.0233790
\(738\) 0 0
\(739\) 4.00843 0.147452 0.0737262 0.997279i \(-0.476511\pi\)
0.0737262 + 0.997279i \(0.476511\pi\)
\(740\) 30.0937 1.10627
\(741\) 0 0
\(742\) −1.41676 −0.0520110
\(743\) −35.7259 −1.31065 −0.655327 0.755345i \(-0.727469\pi\)
−0.655327 + 0.755345i \(0.727469\pi\)
\(744\) 0 0
\(745\) −57.5187 −2.10732
\(746\) −43.8041 −1.60378
\(747\) 0 0
\(748\) −2.30511 −0.0842832
\(749\) −21.8114 −0.796970
\(750\) 0 0
\(751\) −31.8260 −1.16135 −0.580674 0.814136i \(-0.697211\pi\)
−0.580674 + 0.814136i \(0.697211\pi\)
\(752\) −37.3532 −1.36213
\(753\) 0 0
\(754\) −1.49717 −0.0545239
\(755\) 21.5776 0.785288
\(756\) 0 0
\(757\) −36.5025 −1.32671 −0.663353 0.748306i \(-0.730867\pi\)
−0.663353 + 0.748306i \(0.730867\pi\)
\(758\) −39.8576 −1.44769
\(759\) 0 0
\(760\) 34.9906 1.26924
\(761\) 47.8246 1.73364 0.866819 0.498622i \(-0.166161\pi\)
0.866819 + 0.498622i \(0.166161\pi\)
\(762\) 0 0
\(763\) −11.7461 −0.425238
\(764\) 7.55922 0.273483
\(765\) 0 0
\(766\) 2.56978 0.0928499
\(767\) −0.150725 −0.00544237
\(768\) 0 0
\(769\) 23.5915 0.850732 0.425366 0.905021i \(-0.360145\pi\)
0.425366 + 0.905021i \(0.360145\pi\)
\(770\) 58.9697 2.12512
\(771\) 0 0
\(772\) −15.8672 −0.571074
\(773\) −22.1221 −0.795675 −0.397838 0.917456i \(-0.630239\pi\)
−0.397838 + 0.917456i \(0.630239\pi\)
\(774\) 0 0
\(775\) 37.5790 1.34988
\(776\) 22.5489 0.809459
\(777\) 0 0
\(778\) 34.6374 1.24181
\(779\) 37.3579 1.33849
\(780\) 0 0
\(781\) −14.6034 −0.522549
\(782\) 0.698802 0.0249891
\(783\) 0 0
\(784\) 4.04664 0.144523
\(785\) 11.9444 0.426315
\(786\) 0 0
\(787\) 40.1991 1.43294 0.716472 0.697615i \(-0.245756\pi\)
0.716472 + 0.697615i \(0.245756\pi\)
\(788\) −16.9110 −0.602429
\(789\) 0 0
\(790\) −37.2845 −1.32652
\(791\) 29.1869 1.03777
\(792\) 0 0
\(793\) 0.132768 0.00471474
\(794\) 0.468374 0.0166220
\(795\) 0 0
\(796\) 0.0420247 0.00148953
\(797\) 18.4680 0.654169 0.327084 0.944995i \(-0.393934\pi\)
0.327084 + 0.944995i \(0.393934\pi\)
\(798\) 0 0
\(799\) −4.96516 −0.175655
\(800\) −22.0023 −0.777897
\(801\) 0 0
\(802\) 25.0370 0.884089
\(803\) 1.78771 0.0630869
\(804\) 0 0
\(805\) −5.04962 −0.177976
\(806\) −1.15078 −0.0405344
\(807\) 0 0
\(808\) 1.21520 0.0427506
\(809\) 21.2475 0.747021 0.373510 0.927626i \(-0.378154\pi\)
0.373510 + 0.927626i \(0.378154\pi\)
\(810\) 0 0
\(811\) 50.7012 1.78036 0.890180 0.455609i \(-0.150579\pi\)
0.890180 + 0.455609i \(0.150579\pi\)
\(812\) −18.3712 −0.644704
\(813\) 0 0
\(814\) −88.7784 −3.11168
\(815\) −45.8582 −1.60634
\(816\) 0 0
\(817\) 37.9317 1.32706
\(818\) −4.41316 −0.154302
\(819\) 0 0
\(820\) −17.3763 −0.606807
\(821\) 27.4463 0.957883 0.478941 0.877847i \(-0.341021\pi\)
0.478941 + 0.877847i \(0.341021\pi\)
\(822\) 0 0
\(823\) 26.9936 0.940938 0.470469 0.882416i \(-0.344085\pi\)
0.470469 + 0.882416i \(0.344085\pi\)
\(824\) −0.456952 −0.0159187
\(825\) 0 0
\(826\) −6.54765 −0.227822
\(827\) −15.3186 −0.532682 −0.266341 0.963879i \(-0.585815\pi\)
−0.266341 + 0.963879i \(0.585815\pi\)
\(828\) 0 0
\(829\) −39.1591 −1.36005 −0.680026 0.733188i \(-0.738031\pi\)
−0.680026 + 0.733188i \(0.738031\pi\)
\(830\) −0.257774 −0.00894745
\(831\) 0 0
\(832\) −0.273244 −0.00947304
\(833\) 0.537897 0.0186370
\(834\) 0 0
\(835\) −1.07510 −0.0372054
\(836\) 18.9304 0.654722
\(837\) 0 0
\(838\) −20.3370 −0.702530
\(839\) 27.9558 0.965143 0.482571 0.875857i \(-0.339703\pi\)
0.482571 + 0.875857i \(0.339703\pi\)
\(840\) 0 0
\(841\) 59.0538 2.03634
\(842\) 10.3963 0.358281
\(843\) 0 0
\(844\) −2.58952 −0.0891351
\(845\) 41.5102 1.42799
\(846\) 0 0
\(847\) −21.7863 −0.748585
\(848\) −1.69090 −0.0580657
\(849\) 0 0
\(850\) −5.72894 −0.196501
\(851\) 7.60216 0.260599
\(852\) 0 0
\(853\) −27.0275 −0.925404 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(854\) 5.76758 0.197362
\(855\) 0 0
\(856\) −17.7587 −0.606980
\(857\) −44.6176 −1.52411 −0.762054 0.647514i \(-0.775809\pi\)
−0.762054 + 0.647514i \(0.775809\pi\)
\(858\) 0 0
\(859\) −26.4804 −0.903499 −0.451749 0.892145i \(-0.649200\pi\)
−0.451749 + 0.892145i \(0.649200\pi\)
\(860\) −17.6432 −0.601628
\(861\) 0 0
\(862\) −37.5232 −1.27804
\(863\) −13.0078 −0.442791 −0.221396 0.975184i \(-0.571061\pi\)
−0.221396 + 0.975184i \(0.571061\pi\)
\(864\) 0 0
\(865\) 36.2546 1.23269
\(866\) 18.4094 0.625575
\(867\) 0 0
\(868\) −14.1207 −0.479289
\(869\) 31.0690 1.05394
\(870\) 0 0
\(871\) 0.0136444 0.000462324 0
\(872\) −9.56362 −0.323865
\(873\) 0 0
\(874\) −5.73882 −0.194119
\(875\) 1.66994 0.0564543
\(876\) 0 0
\(877\) 42.0717 1.42066 0.710330 0.703869i \(-0.248546\pi\)
0.710330 + 0.703869i \(0.248546\pi\)
\(878\) 62.9411 2.12416
\(879\) 0 0
\(880\) 70.3799 2.37251
\(881\) −35.8562 −1.20802 −0.604012 0.796975i \(-0.706432\pi\)
−0.604012 + 0.796975i \(0.706432\pi\)
\(882\) 0 0
\(883\) −7.97760 −0.268468 −0.134234 0.990950i \(-0.542857\pi\)
−0.134234 + 0.990950i \(0.542857\pi\)
\(884\) 0.0495550 0.00166671
\(885\) 0 0
\(886\) 20.2932 0.681762
\(887\) −14.7497 −0.495246 −0.247623 0.968856i \(-0.579649\pi\)
−0.247623 + 0.968856i \(0.579649\pi\)
\(888\) 0 0
\(889\) 1.08485 0.0363846
\(890\) −31.7901 −1.06561
\(891\) 0 0
\(892\) −3.40324 −0.113949
\(893\) 40.7757 1.36451
\(894\) 0 0
\(895\) −14.0077 −0.468226
\(896\) −32.8717 −1.09817
\(897\) 0 0
\(898\) −7.51265 −0.250700
\(899\) 67.6811 2.25729
\(900\) 0 0
\(901\) −0.224762 −0.00748789
\(902\) 51.2612 1.70681
\(903\) 0 0
\(904\) 23.7638 0.790373
\(905\) 28.4903 0.947051
\(906\) 0 0
\(907\) 24.0644 0.799045 0.399522 0.916723i \(-0.369176\pi\)
0.399522 + 0.916723i \(0.369176\pi\)
\(908\) 17.5819 0.583476
\(909\) 0 0
\(910\) −1.26772 −0.0420246
\(911\) −12.7089 −0.421065 −0.210532 0.977587i \(-0.567520\pi\)
−0.210532 + 0.977587i \(0.567520\pi\)
\(912\) 0 0
\(913\) 0.214801 0.00710888
\(914\) 5.43375 0.179732
\(915\) 0 0
\(916\) 15.4739 0.511271
\(917\) −29.7638 −0.982888
\(918\) 0 0
\(919\) −14.4294 −0.475981 −0.237991 0.971267i \(-0.576489\pi\)
−0.237991 + 0.971267i \(0.576489\pi\)
\(920\) −4.11137 −0.135548
\(921\) 0 0
\(922\) −63.5441 −2.09271
\(923\) 0.313941 0.0103335
\(924\) 0 0
\(925\) −62.3242 −2.04921
\(926\) −15.2963 −0.502666
\(927\) 0 0
\(928\) −39.6268 −1.30081
\(929\) 43.1956 1.41720 0.708601 0.705609i \(-0.249326\pi\)
0.708601 + 0.705609i \(0.249326\pi\)
\(930\) 0 0
\(931\) −4.41741 −0.144775
\(932\) 2.70627 0.0886468
\(933\) 0 0
\(934\) 7.33354 0.239961
\(935\) 9.35521 0.305948
\(936\) 0 0
\(937\) 41.4858 1.35528 0.677642 0.735392i \(-0.263002\pi\)
0.677642 + 0.735392i \(0.263002\pi\)
\(938\) 0.592727 0.0193532
\(939\) 0 0
\(940\) −18.9660 −0.618603
\(941\) −40.5176 −1.32084 −0.660418 0.750898i \(-0.729621\pi\)
−0.660418 + 0.750898i \(0.729621\pi\)
\(942\) 0 0
\(943\) −4.38953 −0.142943
\(944\) −7.81457 −0.254343
\(945\) 0 0
\(946\) 52.0486 1.69224
\(947\) −40.6797 −1.32191 −0.660956 0.750424i \(-0.729849\pi\)
−0.660956 + 0.750424i \(0.729849\pi\)
\(948\) 0 0
\(949\) −0.0384319 −0.00124755
\(950\) 47.0482 1.52644
\(951\) 0 0
\(952\) −3.31570 −0.107463
\(953\) 28.0671 0.909183 0.454591 0.890700i \(-0.349785\pi\)
0.454591 + 0.890700i \(0.349785\pi\)
\(954\) 0 0
\(955\) −30.6788 −0.992743
\(956\) 0.787326 0.0254640
\(957\) 0 0
\(958\) 40.7209 1.31563
\(959\) −46.1031 −1.48875
\(960\) 0 0
\(961\) 21.0219 0.678125
\(962\) 1.90855 0.0615340
\(963\) 0 0
\(964\) 6.47563 0.208566
\(965\) 64.3966 2.07300
\(966\) 0 0
\(967\) 37.2824 1.19892 0.599461 0.800404i \(-0.295382\pi\)
0.599461 + 0.800404i \(0.295382\pi\)
\(968\) −17.7383 −0.570130
\(969\) 0 0
\(970\) 59.4153 1.90771
\(971\) −27.9856 −0.898100 −0.449050 0.893507i \(-0.648238\pi\)
−0.449050 + 0.893507i \(0.648238\pi\)
\(972\) 0 0
\(973\) −0.0236356 −0.000757722 0
\(974\) −56.4093 −1.80747
\(975\) 0 0
\(976\) 6.88357 0.220338
\(977\) 21.6395 0.692309 0.346154 0.938178i \(-0.387487\pi\)
0.346154 + 0.938178i \(0.387487\pi\)
\(978\) 0 0
\(979\) 26.4905 0.846641
\(980\) 2.05467 0.0656341
\(981\) 0 0
\(982\) 11.0308 0.352007
\(983\) 36.9976 1.18004 0.590021 0.807388i \(-0.299120\pi\)
0.590021 + 0.807388i \(0.299120\pi\)
\(984\) 0 0
\(985\) 68.6327 2.18682
\(986\) −10.3180 −0.328593
\(987\) 0 0
\(988\) −0.406964 −0.0129472
\(989\) −4.45695 −0.141723
\(990\) 0 0
\(991\) −32.9780 −1.04758 −0.523791 0.851847i \(-0.675483\pi\)
−0.523791 + 0.851847i \(0.675483\pi\)
\(992\) −30.4585 −0.967058
\(993\) 0 0
\(994\) 13.6379 0.432568
\(995\) −0.170556 −0.00540698
\(996\) 0 0
\(997\) −6.23842 −0.197573 −0.0987863 0.995109i \(-0.531496\pi\)
−0.0987863 + 0.995109i \(0.531496\pi\)
\(998\) −12.4452 −0.393945
\(999\) 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.1.16 yes 20
3.2 odd 2 2151.2.a.j.1.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.5 20 3.2 odd 2
2151.2.a.k.1.16 yes 20 1.1 even 1 trivial