Properties

Label 2151.2.a.k.1.9
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.9
Root \(0.0242456\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} +O(q^{10})\) \(q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} -0.0544170 q^{10} -2.12483 q^{11} -3.04809 q^{13} -0.102203 q^{14} +3.99647 q^{16} -4.57587 q^{17} -0.932781 q^{19} +4.48751 q^{20} -0.0515178 q^{22} -4.32903 q^{23} +0.0373862 q^{25} -0.0739027 q^{26} +8.42816 q^{28} -5.55445 q^{29} -1.27333 q^{31} +0.290833 q^{32} -0.110944 q^{34} +9.46091 q^{35} -8.37064 q^{37} -0.0226158 q^{38} +0.217636 q^{40} +10.8223 q^{41} -9.30747 q^{43} +4.24842 q^{44} -0.104960 q^{46} -7.05894 q^{47} +10.7689 q^{49} +0.000906450 q^{50} +6.09439 q^{52} +10.9362 q^{53} +4.76900 q^{55} +0.408751 q^{56} -0.134671 q^{58} +14.2846 q^{59} -4.93532 q^{61} -0.0308727 q^{62} -7.98590 q^{64} +6.84118 q^{65} +11.3100 q^{67} +9.14905 q^{68} +0.229385 q^{70} +5.92285 q^{71} -13.7138 q^{73} -0.202951 q^{74} +1.86501 q^{76} +8.95685 q^{77} +4.19498 q^{79} -8.96973 q^{80} +0.262392 q^{82} -17.1030 q^{83} +10.2701 q^{85} -0.225665 q^{86} +0.206041 q^{88} +10.9105 q^{89} +12.8487 q^{91} +8.65552 q^{92} -0.171148 q^{94} +2.09355 q^{95} -3.59611 q^{97} +0.261098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0242456 0.0171442 0.00857210 0.999963i \(-0.497271\pi\)
0.00857210 + 0.999963i \(0.497271\pi\)
\(3\) 0 0
\(4\) −1.99941 −0.999706
\(5\) −2.24441 −1.00373 −0.501866 0.864945i \(-0.667353\pi\)
−0.501866 + 0.864945i \(0.667353\pi\)
\(6\) 0 0
\(7\) −4.21532 −1.59324 −0.796620 0.604480i \(-0.793381\pi\)
−0.796620 + 0.604480i \(0.793381\pi\)
\(8\) −0.0969680 −0.0342834
\(9\) 0 0
\(10\) −0.0544170 −0.0172082
\(11\) −2.12483 −0.640661 −0.320331 0.947306i \(-0.603794\pi\)
−0.320331 + 0.947306i \(0.603794\pi\)
\(12\) 0 0
\(13\) −3.04809 −0.845389 −0.422694 0.906272i \(-0.638916\pi\)
−0.422694 + 0.906272i \(0.638916\pi\)
\(14\) −0.102203 −0.0273148
\(15\) 0 0
\(16\) 3.99647 0.999118
\(17\) −4.57587 −1.10981 −0.554906 0.831913i \(-0.687246\pi\)
−0.554906 + 0.831913i \(0.687246\pi\)
\(18\) 0 0
\(19\) −0.932781 −0.213995 −0.106997 0.994259i \(-0.534124\pi\)
−0.106997 + 0.994259i \(0.534124\pi\)
\(20\) 4.48751 1.00344
\(21\) 0 0
\(22\) −0.0515178 −0.0109836
\(23\) −4.32903 −0.902666 −0.451333 0.892356i \(-0.649051\pi\)
−0.451333 + 0.892356i \(0.649051\pi\)
\(24\) 0 0
\(25\) 0.0373862 0.00747724
\(26\) −0.0739027 −0.0144935
\(27\) 0 0
\(28\) 8.42816 1.59277
\(29\) −5.55445 −1.03144 −0.515718 0.856758i \(-0.672475\pi\)
−0.515718 + 0.856758i \(0.672475\pi\)
\(30\) 0 0
\(31\) −1.27333 −0.228697 −0.114349 0.993441i \(-0.536478\pi\)
−0.114349 + 0.993441i \(0.536478\pi\)
\(32\) 0.290833 0.0514125
\(33\) 0 0
\(34\) −0.110944 −0.0190268
\(35\) 9.46091 1.59919
\(36\) 0 0
\(37\) −8.37064 −1.37612 −0.688062 0.725652i \(-0.741538\pi\)
−0.688062 + 0.725652i \(0.741538\pi\)
\(38\) −0.0226158 −0.00366877
\(39\) 0 0
\(40\) 0.217636 0.0344113
\(41\) 10.8223 1.69016 0.845078 0.534643i \(-0.179554\pi\)
0.845078 + 0.534643i \(0.179554\pi\)
\(42\) 0 0
\(43\) −9.30747 −1.41938 −0.709688 0.704516i \(-0.751164\pi\)
−0.709688 + 0.704516i \(0.751164\pi\)
\(44\) 4.24842 0.640473
\(45\) 0 0
\(46\) −0.104960 −0.0154755
\(47\) −7.05894 −1.02965 −0.514826 0.857295i \(-0.672143\pi\)
−0.514826 + 0.857295i \(0.672143\pi\)
\(48\) 0 0
\(49\) 10.7689 1.53842
\(50\) 0.000906450 0 0.000128191 0
\(51\) 0 0
\(52\) 6.09439 0.845140
\(53\) 10.9362 1.50220 0.751101 0.660188i \(-0.229523\pi\)
0.751101 + 0.660188i \(0.229523\pi\)
\(54\) 0 0
\(55\) 4.76900 0.643052
\(56\) 0.408751 0.0546217
\(57\) 0 0
\(58\) −0.134671 −0.0176832
\(59\) 14.2846 1.85969 0.929846 0.367948i \(-0.119940\pi\)
0.929846 + 0.367948i \(0.119940\pi\)
\(60\) 0 0
\(61\) −4.93532 −0.631903 −0.315951 0.948775i \(-0.602324\pi\)
−0.315951 + 0.948775i \(0.602324\pi\)
\(62\) −0.0308727 −0.00392083
\(63\) 0 0
\(64\) −7.98590 −0.998237
\(65\) 6.84118 0.848544
\(66\) 0 0
\(67\) 11.3100 1.38174 0.690871 0.722978i \(-0.257227\pi\)
0.690871 + 0.722978i \(0.257227\pi\)
\(68\) 9.14905 1.10948
\(69\) 0 0
\(70\) 0.229385 0.0274168
\(71\) 5.92285 0.702913 0.351456 0.936204i \(-0.385687\pi\)
0.351456 + 0.936204i \(0.385687\pi\)
\(72\) 0 0
\(73\) −13.7138 −1.60508 −0.802540 0.596598i \(-0.796519\pi\)
−0.802540 + 0.596598i \(0.796519\pi\)
\(74\) −0.202951 −0.0235926
\(75\) 0 0
\(76\) 1.86501 0.213932
\(77\) 8.95685 1.02073
\(78\) 0 0
\(79\) 4.19498 0.471972 0.235986 0.971756i \(-0.424168\pi\)
0.235986 + 0.971756i \(0.424168\pi\)
\(80\) −8.96973 −1.00285
\(81\) 0 0
\(82\) 0.262392 0.0289764
\(83\) −17.1030 −1.87730 −0.938652 0.344867i \(-0.887924\pi\)
−0.938652 + 0.344867i \(0.887924\pi\)
\(84\) 0 0
\(85\) 10.2701 1.11395
\(86\) −0.225665 −0.0243341
\(87\) 0 0
\(88\) 0.206041 0.0219640
\(89\) 10.9105 1.15651 0.578253 0.815858i \(-0.303735\pi\)
0.578253 + 0.815858i \(0.303735\pi\)
\(90\) 0 0
\(91\) 12.8487 1.34691
\(92\) 8.65552 0.902400
\(93\) 0 0
\(94\) −0.171148 −0.0176526
\(95\) 2.09355 0.214793
\(96\) 0 0
\(97\) −3.59611 −0.365130 −0.182565 0.983194i \(-0.558440\pi\)
−0.182565 + 0.983194i \(0.558440\pi\)
\(98\) 0.261098 0.0263749
\(99\) 0 0
\(100\) −0.0747504 −0.00747504
\(101\) −1.12250 −0.111693 −0.0558465 0.998439i \(-0.517786\pi\)
−0.0558465 + 0.998439i \(0.517786\pi\)
\(102\) 0 0
\(103\) 10.3522 1.02003 0.510015 0.860165i \(-0.329640\pi\)
0.510015 + 0.860165i \(0.329640\pi\)
\(104\) 0.295567 0.0289828
\(105\) 0 0
\(106\) 0.265154 0.0257540
\(107\) −2.57093 −0.248541 −0.124271 0.992248i \(-0.539659\pi\)
−0.124271 + 0.992248i \(0.539659\pi\)
\(108\) 0 0
\(109\) −4.60875 −0.441438 −0.220719 0.975337i \(-0.570840\pi\)
−0.220719 + 0.975337i \(0.570840\pi\)
\(110\) 0.115627 0.0110246
\(111\) 0 0
\(112\) −16.8464 −1.59184
\(113\) 19.4118 1.82611 0.913057 0.407833i \(-0.133715\pi\)
0.913057 + 0.407833i \(0.133715\pi\)
\(114\) 0 0
\(115\) 9.71613 0.906034
\(116\) 11.1056 1.03113
\(117\) 0 0
\(118\) 0.346337 0.0318829
\(119\) 19.2887 1.76820
\(120\) 0 0
\(121\) −6.48509 −0.589553
\(122\) −0.119660 −0.0108335
\(123\) 0 0
\(124\) 2.54592 0.228630
\(125\) 11.1382 0.996227
\(126\) 0 0
\(127\) −19.0061 −1.68652 −0.843260 0.537506i \(-0.819367\pi\)
−0.843260 + 0.537506i \(0.819367\pi\)
\(128\) −0.775288 −0.0685264
\(129\) 0 0
\(130\) 0.165868 0.0145476
\(131\) −0.993956 −0.0868424 −0.0434212 0.999057i \(-0.513826\pi\)
−0.0434212 + 0.999057i \(0.513826\pi\)
\(132\) 0 0
\(133\) 3.93197 0.340945
\(134\) 0.274218 0.0236889
\(135\) 0 0
\(136\) 0.443713 0.0380481
\(137\) −10.2766 −0.877989 −0.438994 0.898490i \(-0.644665\pi\)
−0.438994 + 0.898490i \(0.644665\pi\)
\(138\) 0 0
\(139\) −9.12331 −0.773829 −0.386915 0.922116i \(-0.626459\pi\)
−0.386915 + 0.922116i \(0.626459\pi\)
\(140\) −18.9163 −1.59872
\(141\) 0 0
\(142\) 0.143603 0.0120509
\(143\) 6.47669 0.541608
\(144\) 0 0
\(145\) 12.4665 1.03529
\(146\) −0.332499 −0.0275178
\(147\) 0 0
\(148\) 16.7364 1.37572
\(149\) −8.76418 −0.717989 −0.358995 0.933340i \(-0.616880\pi\)
−0.358995 + 0.933340i \(0.616880\pi\)
\(150\) 0 0
\(151\) 2.25182 0.183251 0.0916253 0.995794i \(-0.470794\pi\)
0.0916253 + 0.995794i \(0.470794\pi\)
\(152\) 0.0904499 0.00733646
\(153\) 0 0
\(154\) 0.217164 0.0174996
\(155\) 2.85788 0.229551
\(156\) 0 0
\(157\) −9.16321 −0.731304 −0.365652 0.930752i \(-0.619154\pi\)
−0.365652 + 0.930752i \(0.619154\pi\)
\(158\) 0.101710 0.00809158
\(159\) 0 0
\(160\) −0.652749 −0.0516043
\(161\) 18.2483 1.43816
\(162\) 0 0
\(163\) 4.26808 0.334302 0.167151 0.985931i \(-0.446543\pi\)
0.167151 + 0.985931i \(0.446543\pi\)
\(164\) −21.6382 −1.68966
\(165\) 0 0
\(166\) −0.414673 −0.0321849
\(167\) −21.3942 −1.65553 −0.827766 0.561074i \(-0.810388\pi\)
−0.827766 + 0.561074i \(0.810388\pi\)
\(168\) 0 0
\(169\) −3.70913 −0.285318
\(170\) 0.249005 0.0190978
\(171\) 0 0
\(172\) 18.6095 1.41896
\(173\) 4.73652 0.360111 0.180056 0.983656i \(-0.442372\pi\)
0.180056 + 0.983656i \(0.442372\pi\)
\(174\) 0 0
\(175\) −0.157595 −0.0119130
\(176\) −8.49184 −0.640096
\(177\) 0 0
\(178\) 0.264530 0.0198274
\(179\) −20.8872 −1.56118 −0.780591 0.625043i \(-0.785082\pi\)
−0.780591 + 0.625043i \(0.785082\pi\)
\(180\) 0 0
\(181\) 2.27241 0.168907 0.0844535 0.996427i \(-0.473086\pi\)
0.0844535 + 0.996427i \(0.473086\pi\)
\(182\) 0.311524 0.0230917
\(183\) 0 0
\(184\) 0.419778 0.0309464
\(185\) 18.7872 1.38126
\(186\) 0 0
\(187\) 9.72295 0.711013
\(188\) 14.1137 1.02935
\(189\) 0 0
\(190\) 0.0507592 0.00368246
\(191\) 14.4450 1.04521 0.522603 0.852576i \(-0.324961\pi\)
0.522603 + 0.852576i \(0.324961\pi\)
\(192\) 0 0
\(193\) 13.2203 0.951618 0.475809 0.879549i \(-0.342155\pi\)
0.475809 + 0.879549i \(0.342155\pi\)
\(194\) −0.0871897 −0.00625986
\(195\) 0 0
\(196\) −21.5315 −1.53796
\(197\) −14.8080 −1.05503 −0.527513 0.849547i \(-0.676875\pi\)
−0.527513 + 0.849547i \(0.676875\pi\)
\(198\) 0 0
\(199\) −9.69564 −0.687306 −0.343653 0.939097i \(-0.611664\pi\)
−0.343653 + 0.939097i \(0.611664\pi\)
\(200\) −0.00362527 −0.000256345 0
\(201\) 0 0
\(202\) −0.0272156 −0.00191489
\(203\) 23.4138 1.64333
\(204\) 0 0
\(205\) −24.2896 −1.69646
\(206\) 0.250994 0.0174876
\(207\) 0 0
\(208\) −12.1816 −0.844644
\(209\) 1.98200 0.137098
\(210\) 0 0
\(211\) −3.95231 −0.272088 −0.136044 0.990703i \(-0.543439\pi\)
−0.136044 + 0.990703i \(0.543439\pi\)
\(212\) −21.8660 −1.50176
\(213\) 0 0
\(214\) −0.0623336 −0.00426104
\(215\) 20.8898 1.42467
\(216\) 0 0
\(217\) 5.36750 0.364370
\(218\) −0.111742 −0.00756810
\(219\) 0 0
\(220\) −9.53520 −0.642863
\(221\) 13.9477 0.938222
\(222\) 0 0
\(223\) −27.4364 −1.83728 −0.918639 0.395098i \(-0.870710\pi\)
−0.918639 + 0.395098i \(0.870710\pi\)
\(224\) −1.22595 −0.0819124
\(225\) 0 0
\(226\) 0.470651 0.0313073
\(227\) −2.57694 −0.171037 −0.0855187 0.996337i \(-0.527255\pi\)
−0.0855187 + 0.996337i \(0.527255\pi\)
\(228\) 0 0
\(229\) −14.8499 −0.981306 −0.490653 0.871355i \(-0.663242\pi\)
−0.490653 + 0.871355i \(0.663242\pi\)
\(230\) 0.235573 0.0155332
\(231\) 0 0
\(232\) 0.538604 0.0353611
\(233\) 6.79867 0.445396 0.222698 0.974888i \(-0.428514\pi\)
0.222698 + 0.974888i \(0.428514\pi\)
\(234\) 0 0
\(235\) 15.8432 1.03349
\(236\) −28.5607 −1.85915
\(237\) 0 0
\(238\) 0.467666 0.0303143
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −14.4562 −0.931208 −0.465604 0.884993i \(-0.654163\pi\)
−0.465604 + 0.884993i \(0.654163\pi\)
\(242\) −0.157235 −0.0101074
\(243\) 0 0
\(244\) 9.86774 0.631717
\(245\) −24.1699 −1.54416
\(246\) 0 0
\(247\) 2.84320 0.180909
\(248\) 0.123473 0.00784051
\(249\) 0 0
\(250\) 0.270051 0.0170795
\(251\) 24.6887 1.55834 0.779170 0.626813i \(-0.215641\pi\)
0.779170 + 0.626813i \(0.215641\pi\)
\(252\) 0 0
\(253\) 9.19847 0.578303
\(254\) −0.460814 −0.0289140
\(255\) 0 0
\(256\) 15.9530 0.997062
\(257\) −16.5700 −1.03361 −0.516805 0.856103i \(-0.672879\pi\)
−0.516805 + 0.856103i \(0.672879\pi\)
\(258\) 0 0
\(259\) 35.2849 2.19250
\(260\) −13.6783 −0.848294
\(261\) 0 0
\(262\) −0.0240990 −0.00148884
\(263\) −9.61196 −0.592699 −0.296349 0.955080i \(-0.595769\pi\)
−0.296349 + 0.955080i \(0.595769\pi\)
\(264\) 0 0
\(265\) −24.5453 −1.50781
\(266\) 0.0953328 0.00584523
\(267\) 0 0
\(268\) −22.6134 −1.38134
\(269\) 22.2724 1.35797 0.678987 0.734151i \(-0.262419\pi\)
0.678987 + 0.734151i \(0.262419\pi\)
\(270\) 0 0
\(271\) −8.41793 −0.511353 −0.255676 0.966762i \(-0.582298\pi\)
−0.255676 + 0.966762i \(0.582298\pi\)
\(272\) −18.2873 −1.10883
\(273\) 0 0
\(274\) −0.249162 −0.0150524
\(275\) −0.0794394 −0.00479038
\(276\) 0 0
\(277\) 12.8383 0.771382 0.385691 0.922628i \(-0.373963\pi\)
0.385691 + 0.922628i \(0.373963\pi\)
\(278\) −0.221200 −0.0132667
\(279\) 0 0
\(280\) −0.917406 −0.0548255
\(281\) 4.62713 0.276031 0.138016 0.990430i \(-0.455928\pi\)
0.138016 + 0.990430i \(0.455928\pi\)
\(282\) 0 0
\(283\) −31.0298 −1.84453 −0.922264 0.386560i \(-0.873663\pi\)
−0.922264 + 0.386560i \(0.873663\pi\)
\(284\) −11.8422 −0.702706
\(285\) 0 0
\(286\) 0.157031 0.00928543
\(287\) −45.6193 −2.69282
\(288\) 0 0
\(289\) 3.93857 0.231681
\(290\) 0.302257 0.0177491
\(291\) 0 0
\(292\) 27.4196 1.60461
\(293\) −2.36334 −0.138068 −0.0690340 0.997614i \(-0.521992\pi\)
−0.0690340 + 0.997614i \(0.521992\pi\)
\(294\) 0 0
\(295\) −32.0605 −1.86663
\(296\) 0.811684 0.0471782
\(297\) 0 0
\(298\) −0.212492 −0.0123094
\(299\) 13.1953 0.763104
\(300\) 0 0
\(301\) 39.2340 2.26141
\(302\) 0.0545967 0.00314169
\(303\) 0 0
\(304\) −3.72784 −0.213806
\(305\) 11.0769 0.634261
\(306\) 0 0
\(307\) 9.68688 0.552859 0.276430 0.961034i \(-0.410849\pi\)
0.276430 + 0.961034i \(0.410849\pi\)
\(308\) −17.9084 −1.02043
\(309\) 0 0
\(310\) 0.0692910 0.00393546
\(311\) −20.7787 −1.17825 −0.589125 0.808042i \(-0.700527\pi\)
−0.589125 + 0.808042i \(0.700527\pi\)
\(312\) 0 0
\(313\) 33.5440 1.89602 0.948009 0.318245i \(-0.103093\pi\)
0.948009 + 0.318245i \(0.103093\pi\)
\(314\) −0.222167 −0.0125376
\(315\) 0 0
\(316\) −8.38749 −0.471833
\(317\) −1.53143 −0.0860137 −0.0430069 0.999075i \(-0.513694\pi\)
−0.0430069 + 0.999075i \(0.513694\pi\)
\(318\) 0 0
\(319\) 11.8023 0.660801
\(320\) 17.9236 1.00196
\(321\) 0 0
\(322\) 0.442439 0.0246562
\(323\) 4.26828 0.237494
\(324\) 0 0
\(325\) −0.113957 −0.00632118
\(326\) 0.103482 0.00573134
\(327\) 0 0
\(328\) −1.04941 −0.0579442
\(329\) 29.7557 1.64048
\(330\) 0 0
\(331\) 17.6141 0.968160 0.484080 0.875024i \(-0.339154\pi\)
0.484080 + 0.875024i \(0.339154\pi\)
\(332\) 34.1960 1.87675
\(333\) 0 0
\(334\) −0.518714 −0.0283828
\(335\) −25.3844 −1.38690
\(336\) 0 0
\(337\) 1.96831 0.107221 0.0536103 0.998562i \(-0.482927\pi\)
0.0536103 + 0.998562i \(0.482927\pi\)
\(338\) −0.0899299 −0.00489154
\(339\) 0 0
\(340\) −20.5342 −1.11363
\(341\) 2.70562 0.146517
\(342\) 0 0
\(343\) −15.8872 −0.857827
\(344\) 0.902527 0.0486610
\(345\) 0 0
\(346\) 0.114840 0.00617382
\(347\) −35.6033 −1.91128 −0.955642 0.294532i \(-0.904836\pi\)
−0.955642 + 0.294532i \(0.904836\pi\)
\(348\) 0 0
\(349\) −21.4549 −1.14845 −0.574226 0.818697i \(-0.694697\pi\)
−0.574226 + 0.818697i \(0.694697\pi\)
\(350\) −0.00382097 −0.000204240 0
\(351\) 0 0
\(352\) −0.617971 −0.0329380
\(353\) 7.36184 0.391831 0.195916 0.980621i \(-0.437232\pi\)
0.195916 + 0.980621i \(0.437232\pi\)
\(354\) 0 0
\(355\) −13.2933 −0.705536
\(356\) −21.8145 −1.15617
\(357\) 0 0
\(358\) −0.506421 −0.0267652
\(359\) 7.46205 0.393832 0.196916 0.980420i \(-0.436907\pi\)
0.196916 + 0.980420i \(0.436907\pi\)
\(360\) 0 0
\(361\) −18.1299 −0.954206
\(362\) 0.0550959 0.00289577
\(363\) 0 0
\(364\) −25.6898 −1.34651
\(365\) 30.7794 1.61107
\(366\) 0 0
\(367\) 30.8625 1.61101 0.805505 0.592589i \(-0.201894\pi\)
0.805505 + 0.592589i \(0.201894\pi\)
\(368\) −17.3009 −0.901870
\(369\) 0 0
\(370\) 0.455505 0.0236806
\(371\) −46.0995 −2.39337
\(372\) 0 0
\(373\) 17.3634 0.899043 0.449521 0.893270i \(-0.351595\pi\)
0.449521 + 0.893270i \(0.351595\pi\)
\(374\) 0.235738 0.0121897
\(375\) 0 0
\(376\) 0.684491 0.0352999
\(377\) 16.9305 0.871965
\(378\) 0 0
\(379\) 14.2565 0.732306 0.366153 0.930555i \(-0.380675\pi\)
0.366153 + 0.930555i \(0.380675\pi\)
\(380\) −4.18586 −0.214730
\(381\) 0 0
\(382\) 0.350228 0.0179192
\(383\) 12.8620 0.657217 0.328608 0.944466i \(-0.393420\pi\)
0.328608 + 0.944466i \(0.393420\pi\)
\(384\) 0 0
\(385\) −20.1029 −1.02454
\(386\) 0.320534 0.0163147
\(387\) 0 0
\(388\) 7.19011 0.365022
\(389\) −17.6690 −0.895856 −0.447928 0.894070i \(-0.647838\pi\)
−0.447928 + 0.894070i \(0.647838\pi\)
\(390\) 0 0
\(391\) 19.8091 1.00179
\(392\) −1.04424 −0.0527421
\(393\) 0 0
\(394\) −0.359028 −0.0180876
\(395\) −9.41526 −0.473733
\(396\) 0 0
\(397\) 7.56134 0.379493 0.189747 0.981833i \(-0.439233\pi\)
0.189747 + 0.981833i \(0.439233\pi\)
\(398\) −0.235076 −0.0117833
\(399\) 0 0
\(400\) 0.149413 0.00747065
\(401\) 20.5825 1.02784 0.513921 0.857837i \(-0.328192\pi\)
0.513921 + 0.857837i \(0.328192\pi\)
\(402\) 0 0
\(403\) 3.88124 0.193338
\(404\) 2.24434 0.111660
\(405\) 0 0
\(406\) 0.567681 0.0281735
\(407\) 17.7862 0.881629
\(408\) 0 0
\(409\) 30.2928 1.49788 0.748940 0.662637i \(-0.230563\pi\)
0.748940 + 0.662637i \(0.230563\pi\)
\(410\) −0.588916 −0.0290845
\(411\) 0 0
\(412\) −20.6983 −1.01973
\(413\) −60.2140 −2.96294
\(414\) 0 0
\(415\) 38.3863 1.88431
\(416\) −0.886485 −0.0434635
\(417\) 0 0
\(418\) 0.0480548 0.00235044
\(419\) 4.89531 0.239152 0.119576 0.992825i \(-0.461847\pi\)
0.119576 + 0.992825i \(0.461847\pi\)
\(420\) 0 0
\(421\) −21.9505 −1.06980 −0.534900 0.844915i \(-0.679651\pi\)
−0.534900 + 0.844915i \(0.679651\pi\)
\(422\) −0.0958260 −0.00466474
\(423\) 0 0
\(424\) −1.06046 −0.0515005
\(425\) −0.171074 −0.00829832
\(426\) 0 0
\(427\) 20.8039 1.00677
\(428\) 5.14035 0.248468
\(429\) 0 0
\(430\) 0.506485 0.0244249
\(431\) −29.7089 −1.43103 −0.715513 0.698600i \(-0.753807\pi\)
−0.715513 + 0.698600i \(0.753807\pi\)
\(432\) 0 0
\(433\) −7.35974 −0.353686 −0.176843 0.984239i \(-0.556589\pi\)
−0.176843 + 0.984239i \(0.556589\pi\)
\(434\) 0.130138 0.00624683
\(435\) 0 0
\(436\) 9.21479 0.441308
\(437\) 4.03804 0.193166
\(438\) 0 0
\(439\) 19.8181 0.945865 0.472933 0.881099i \(-0.343195\pi\)
0.472933 + 0.881099i \(0.343195\pi\)
\(440\) −0.462440 −0.0220460
\(441\) 0 0
\(442\) 0.338169 0.0160851
\(443\) 40.2158 1.91071 0.955356 0.295456i \(-0.0954716\pi\)
0.955356 + 0.295456i \(0.0954716\pi\)
\(444\) 0 0
\(445\) −24.4876 −1.16082
\(446\) −0.665211 −0.0314987
\(447\) 0 0
\(448\) 33.6631 1.59043
\(449\) 31.7168 1.49681 0.748403 0.663244i \(-0.230821\pi\)
0.748403 + 0.663244i \(0.230821\pi\)
\(450\) 0 0
\(451\) −22.9955 −1.08282
\(452\) −38.8123 −1.82558
\(453\) 0 0
\(454\) −0.0624793 −0.00293230
\(455\) −28.8377 −1.35193
\(456\) 0 0
\(457\) 21.4539 1.00357 0.501786 0.864992i \(-0.332676\pi\)
0.501786 + 0.864992i \(0.332676\pi\)
\(458\) −0.360043 −0.0168237
\(459\) 0 0
\(460\) −19.4266 −0.905768
\(461\) −16.1130 −0.750459 −0.375230 0.926932i \(-0.622436\pi\)
−0.375230 + 0.926932i \(0.622436\pi\)
\(462\) 0 0
\(463\) −12.7421 −0.592178 −0.296089 0.955160i \(-0.595682\pi\)
−0.296089 + 0.955160i \(0.595682\pi\)
\(464\) −22.1982 −1.03053
\(465\) 0 0
\(466\) 0.164838 0.00763595
\(467\) −2.54331 −0.117691 −0.0588453 0.998267i \(-0.518742\pi\)
−0.0588453 + 0.998267i \(0.518742\pi\)
\(468\) 0 0
\(469\) −47.6755 −2.20145
\(470\) 0.384126 0.0177184
\(471\) 0 0
\(472\) −1.38515 −0.0637565
\(473\) 19.7768 0.909339
\(474\) 0 0
\(475\) −0.0348732 −0.00160009
\(476\) −38.5661 −1.76768
\(477\) 0 0
\(478\) 0.0242456 0.00110897
\(479\) −9.60241 −0.438745 −0.219373 0.975641i \(-0.570401\pi\)
−0.219373 + 0.975641i \(0.570401\pi\)
\(480\) 0 0
\(481\) 25.5145 1.16336
\(482\) −0.350500 −0.0159648
\(483\) 0 0
\(484\) 12.9664 0.589380
\(485\) 8.07115 0.366492
\(486\) 0 0
\(487\) −36.2997 −1.64490 −0.822448 0.568841i \(-0.807392\pi\)
−0.822448 + 0.568841i \(0.807392\pi\)
\(488\) 0.478568 0.0216638
\(489\) 0 0
\(490\) −0.586012 −0.0264733
\(491\) −7.26687 −0.327949 −0.163975 0.986465i \(-0.552431\pi\)
−0.163975 + 0.986465i \(0.552431\pi\)
\(492\) 0 0
\(493\) 25.4165 1.14470
\(494\) 0.0689351 0.00310154
\(495\) 0 0
\(496\) −5.08884 −0.228496
\(497\) −24.9667 −1.11991
\(498\) 0 0
\(499\) −26.2796 −1.17644 −0.588218 0.808703i \(-0.700170\pi\)
−0.588218 + 0.808703i \(0.700170\pi\)
\(500\) −22.2698 −0.995934
\(501\) 0 0
\(502\) 0.598592 0.0267165
\(503\) −34.2338 −1.52641 −0.763206 0.646156i \(-0.776376\pi\)
−0.763206 + 0.646156i \(0.776376\pi\)
\(504\) 0 0
\(505\) 2.51935 0.112110
\(506\) 0.223022 0.00991454
\(507\) 0 0
\(508\) 38.0011 1.68602
\(509\) 12.0294 0.533194 0.266597 0.963808i \(-0.414101\pi\)
0.266597 + 0.963808i \(0.414101\pi\)
\(510\) 0 0
\(511\) 57.8081 2.55728
\(512\) 1.93737 0.0856203
\(513\) 0 0
\(514\) −0.401750 −0.0177204
\(515\) −23.2346 −1.02384
\(516\) 0 0
\(517\) 14.9991 0.659658
\(518\) 0.855503 0.0375886
\(519\) 0 0
\(520\) −0.663375 −0.0290909
\(521\) −2.94578 −0.129057 −0.0645285 0.997916i \(-0.520554\pi\)
−0.0645285 + 0.997916i \(0.520554\pi\)
\(522\) 0 0
\(523\) −9.40392 −0.411205 −0.205602 0.978636i \(-0.565915\pi\)
−0.205602 + 0.978636i \(0.565915\pi\)
\(524\) 1.98733 0.0868168
\(525\) 0 0
\(526\) −0.233047 −0.0101613
\(527\) 5.82660 0.253811
\(528\) 0 0
\(529\) −4.25948 −0.185195
\(530\) −0.595115 −0.0258502
\(531\) 0 0
\(532\) −7.86163 −0.340845
\(533\) −32.9873 −1.42884
\(534\) 0 0
\(535\) 5.77023 0.249469
\(536\) −1.09671 −0.0473708
\(537\) 0 0
\(538\) 0.540007 0.0232814
\(539\) −22.8821 −0.985603
\(540\) 0 0
\(541\) 13.6171 0.585447 0.292723 0.956197i \(-0.405438\pi\)
0.292723 + 0.956197i \(0.405438\pi\)
\(542\) −0.204097 −0.00876674
\(543\) 0 0
\(544\) −1.33081 −0.0570581
\(545\) 10.3439 0.443085
\(546\) 0 0
\(547\) −20.3413 −0.869730 −0.434865 0.900496i \(-0.643204\pi\)
−0.434865 + 0.900496i \(0.643204\pi\)
\(548\) 20.5471 0.877731
\(549\) 0 0
\(550\) −0.00192605 −8.21272e−5 0
\(551\) 5.18109 0.220722
\(552\) 0 0
\(553\) −17.6832 −0.751965
\(554\) 0.311273 0.0132247
\(555\) 0 0
\(556\) 18.2413 0.773602
\(557\) 6.44485 0.273077 0.136539 0.990635i \(-0.456402\pi\)
0.136539 + 0.990635i \(0.456402\pi\)
\(558\) 0 0
\(559\) 28.3700 1.19993
\(560\) 37.8103 1.59778
\(561\) 0 0
\(562\) 0.112187 0.00473234
\(563\) 40.0936 1.68974 0.844871 0.534970i \(-0.179677\pi\)
0.844871 + 0.534970i \(0.179677\pi\)
\(564\) 0 0
\(565\) −43.5682 −1.83293
\(566\) −0.752334 −0.0316230
\(567\) 0 0
\(568\) −0.574327 −0.0240982
\(569\) −12.5303 −0.525296 −0.262648 0.964892i \(-0.584596\pi\)
−0.262648 + 0.964892i \(0.584596\pi\)
\(570\) 0 0
\(571\) −29.8993 −1.25125 −0.625623 0.780126i \(-0.715155\pi\)
−0.625623 + 0.780126i \(0.715155\pi\)
\(572\) −12.9496 −0.541449
\(573\) 0 0
\(574\) −1.10607 −0.0461663
\(575\) −0.161846 −0.00674945
\(576\) 0 0
\(577\) 20.1769 0.839974 0.419987 0.907530i \(-0.362035\pi\)
0.419987 + 0.907530i \(0.362035\pi\)
\(578\) 0.0954928 0.00397198
\(579\) 0 0
\(580\) −24.9256 −1.03498
\(581\) 72.0948 2.99100
\(582\) 0 0
\(583\) −23.2376 −0.962402
\(584\) 1.32980 0.0550276
\(585\) 0 0
\(586\) −0.0573006 −0.00236706
\(587\) −14.8609 −0.613373 −0.306687 0.951811i \(-0.599220\pi\)
−0.306687 + 0.951811i \(0.599220\pi\)
\(588\) 0 0
\(589\) 1.18774 0.0489400
\(590\) −0.777324 −0.0320019
\(591\) 0 0
\(592\) −33.4530 −1.37491
\(593\) 1.07945 0.0443276 0.0221638 0.999754i \(-0.492944\pi\)
0.0221638 + 0.999754i \(0.492944\pi\)
\(594\) 0 0
\(595\) −43.2919 −1.77479
\(596\) 17.5232 0.717778
\(597\) 0 0
\(598\) 0.319927 0.0130828
\(599\) 5.03917 0.205895 0.102948 0.994687i \(-0.467173\pi\)
0.102948 + 0.994687i \(0.467173\pi\)
\(600\) 0 0
\(601\) 16.2608 0.663291 0.331645 0.943404i \(-0.392396\pi\)
0.331645 + 0.943404i \(0.392396\pi\)
\(602\) 0.951250 0.0387700
\(603\) 0 0
\(604\) −4.50232 −0.183197
\(605\) 14.5552 0.591753
\(606\) 0 0
\(607\) 20.0565 0.814069 0.407034 0.913413i \(-0.366563\pi\)
0.407034 + 0.913413i \(0.366563\pi\)
\(608\) −0.271283 −0.0110020
\(609\) 0 0
\(610\) 0.268565 0.0108739
\(611\) 21.5163 0.870456
\(612\) 0 0
\(613\) 12.3418 0.498479 0.249240 0.968442i \(-0.419819\pi\)
0.249240 + 0.968442i \(0.419819\pi\)
\(614\) 0.234864 0.00947833
\(615\) 0 0
\(616\) −0.868527 −0.0349940
\(617\) −33.6404 −1.35431 −0.677156 0.735840i \(-0.736788\pi\)
−0.677156 + 0.735840i \(0.736788\pi\)
\(618\) 0 0
\(619\) 2.36911 0.0952227 0.0476113 0.998866i \(-0.484839\pi\)
0.0476113 + 0.998866i \(0.484839\pi\)
\(620\) −5.71409 −0.229483
\(621\) 0 0
\(622\) −0.503790 −0.0202002
\(623\) −45.9910 −1.84259
\(624\) 0 0
\(625\) −25.1855 −1.00742
\(626\) 0.813292 0.0325057
\(627\) 0 0
\(628\) 18.3210 0.731089
\(629\) 38.3029 1.52724
\(630\) 0 0
\(631\) −5.90443 −0.235052 −0.117526 0.993070i \(-0.537496\pi\)
−0.117526 + 0.993070i \(0.537496\pi\)
\(632\) −0.406779 −0.0161808
\(633\) 0 0
\(634\) −0.0371304 −0.00147464
\(635\) 42.6576 1.69281
\(636\) 0 0
\(637\) −32.8247 −1.30056
\(638\) 0.286153 0.0113289
\(639\) 0 0
\(640\) 1.74007 0.0687821
\(641\) −40.4846 −1.59904 −0.799522 0.600637i \(-0.794914\pi\)
−0.799522 + 0.600637i \(0.794914\pi\)
\(642\) 0 0
\(643\) −16.3242 −0.643762 −0.321881 0.946780i \(-0.604315\pi\)
−0.321881 + 0.946780i \(0.604315\pi\)
\(644\) −36.4858 −1.43774
\(645\) 0 0
\(646\) 0.103487 0.00407164
\(647\) 9.53248 0.374760 0.187380 0.982287i \(-0.440000\pi\)
0.187380 + 0.982287i \(0.440000\pi\)
\(648\) 0 0
\(649\) −30.3523 −1.19143
\(650\) −0.00276294 −0.000108372 0
\(651\) 0 0
\(652\) −8.53365 −0.334204
\(653\) −31.1156 −1.21765 −0.608825 0.793305i \(-0.708359\pi\)
−0.608825 + 0.793305i \(0.708359\pi\)
\(654\) 0 0
\(655\) 2.23085 0.0871664
\(656\) 43.2509 1.68867
\(657\) 0 0
\(658\) 0.721443 0.0281248
\(659\) −1.01441 −0.0395160 −0.0197580 0.999805i \(-0.506290\pi\)
−0.0197580 + 0.999805i \(0.506290\pi\)
\(660\) 0 0
\(661\) 8.64358 0.336196 0.168098 0.985770i \(-0.446237\pi\)
0.168098 + 0.985770i \(0.446237\pi\)
\(662\) 0.427065 0.0165983
\(663\) 0 0
\(664\) 1.65845 0.0643603
\(665\) −8.82496 −0.342217
\(666\) 0 0
\(667\) 24.0454 0.931042
\(668\) 42.7758 1.65505
\(669\) 0 0
\(670\) −0.615459 −0.0237773
\(671\) 10.4867 0.404835
\(672\) 0 0
\(673\) 2.37311 0.0914768 0.0457384 0.998953i \(-0.485436\pi\)
0.0457384 + 0.998953i \(0.485436\pi\)
\(674\) 0.0477227 0.00183821
\(675\) 0 0
\(676\) 7.41608 0.285234
\(677\) 33.5812 1.29063 0.645315 0.763916i \(-0.276726\pi\)
0.645315 + 0.763916i \(0.276726\pi\)
\(678\) 0 0
\(679\) 15.1588 0.581740
\(680\) −0.995874 −0.0381900
\(681\) 0 0
\(682\) 0.0655992 0.00251192
\(683\) −4.70697 −0.180107 −0.0900536 0.995937i \(-0.528704\pi\)
−0.0900536 + 0.995937i \(0.528704\pi\)
\(684\) 0 0
\(685\) 23.0649 0.881265
\(686\) −0.385193 −0.0147068
\(687\) 0 0
\(688\) −37.1971 −1.41812
\(689\) −33.3345 −1.26994
\(690\) 0 0
\(691\) 14.1647 0.538851 0.269426 0.963021i \(-0.413166\pi\)
0.269426 + 0.963021i \(0.413166\pi\)
\(692\) −9.47026 −0.360005
\(693\) 0 0
\(694\) −0.863221 −0.0327674
\(695\) 20.4765 0.776717
\(696\) 0 0
\(697\) −49.5213 −1.87575
\(698\) −0.520185 −0.0196893
\(699\) 0 0
\(700\) 0.315097 0.0119095
\(701\) −27.8527 −1.05198 −0.525991 0.850490i \(-0.676305\pi\)
−0.525991 + 0.850490i \(0.676305\pi\)
\(702\) 0 0
\(703\) 7.80798 0.294483
\(704\) 16.9687 0.639531
\(705\) 0 0
\(706\) 0.178492 0.00671764
\(707\) 4.73170 0.177954
\(708\) 0 0
\(709\) 2.50868 0.0942155 0.0471077 0.998890i \(-0.485000\pi\)
0.0471077 + 0.998890i \(0.485000\pi\)
\(710\) −0.322304 −0.0120959
\(711\) 0 0
\(712\) −1.05796 −0.0396489
\(713\) 5.51230 0.206437
\(714\) 0 0
\(715\) −14.5364 −0.543629
\(716\) 41.7621 1.56072
\(717\) 0 0
\(718\) 0.180922 0.00675193
\(719\) −28.4290 −1.06022 −0.530112 0.847928i \(-0.677850\pi\)
−0.530112 + 0.847928i \(0.677850\pi\)
\(720\) 0 0
\(721\) −43.6377 −1.62515
\(722\) −0.439570 −0.0163591
\(723\) 0 0
\(724\) −4.54348 −0.168857
\(725\) −0.207660 −0.00771230
\(726\) 0 0
\(727\) 44.1617 1.63787 0.818933 0.573888i \(-0.194566\pi\)
0.818933 + 0.573888i \(0.194566\pi\)
\(728\) −1.24591 −0.0461765
\(729\) 0 0
\(730\) 0.746265 0.0276205
\(731\) 42.5898 1.57524
\(732\) 0 0
\(733\) 48.2081 1.78061 0.890303 0.455368i \(-0.150492\pi\)
0.890303 + 0.455368i \(0.150492\pi\)
\(734\) 0.748279 0.0276195
\(735\) 0 0
\(736\) −1.25902 −0.0464083
\(737\) −24.0320 −0.885228
\(738\) 0 0
\(739\) 13.7530 0.505911 0.252956 0.967478i \(-0.418597\pi\)
0.252956 + 0.967478i \(0.418597\pi\)
\(740\) −37.5633 −1.38085
\(741\) 0 0
\(742\) −1.11771 −0.0410324
\(743\) 22.2915 0.817796 0.408898 0.912580i \(-0.365913\pi\)
0.408898 + 0.912580i \(0.365913\pi\)
\(744\) 0 0
\(745\) 19.6704 0.720669
\(746\) 0.420985 0.0154134
\(747\) 0 0
\(748\) −19.4402 −0.710804
\(749\) 10.8373 0.395986
\(750\) 0 0
\(751\) −0.504426 −0.0184068 −0.00920338 0.999958i \(-0.502930\pi\)
−0.00920338 + 0.999958i \(0.502930\pi\)
\(752\) −28.2109 −1.02874
\(753\) 0 0
\(754\) 0.410489 0.0149491
\(755\) −5.05401 −0.183934
\(756\) 0 0
\(757\) 27.5960 1.00299 0.501497 0.865159i \(-0.332783\pi\)
0.501497 + 0.865159i \(0.332783\pi\)
\(758\) 0.345656 0.0125548
\(759\) 0 0
\(760\) −0.203007 −0.00736384
\(761\) 45.1173 1.63550 0.817751 0.575572i \(-0.195220\pi\)
0.817751 + 0.575572i \(0.195220\pi\)
\(762\) 0 0
\(763\) 19.4273 0.703317
\(764\) −28.8816 −1.04490
\(765\) 0 0
\(766\) 0.311846 0.0112675
\(767\) −43.5407 −1.57216
\(768\) 0 0
\(769\) −35.6644 −1.28609 −0.643045 0.765829i \(-0.722329\pi\)
−0.643045 + 0.765829i \(0.722329\pi\)
\(770\) −0.487405 −0.0175649
\(771\) 0 0
\(772\) −26.4328 −0.951339
\(773\) 31.4644 1.13170 0.565848 0.824510i \(-0.308549\pi\)
0.565848 + 0.824510i \(0.308549\pi\)
\(774\) 0 0
\(775\) −0.0476051 −0.00171002
\(776\) 0.348708 0.0125179
\(777\) 0 0
\(778\) −0.428396 −0.0153587
\(779\) −10.0948 −0.361684
\(780\) 0 0
\(781\) −12.5851 −0.450329
\(782\) 0.480282 0.0171749
\(783\) 0 0
\(784\) 43.0377 1.53706
\(785\) 20.5660 0.734033
\(786\) 0 0
\(787\) −0.598640 −0.0213392 −0.0106696 0.999943i \(-0.503396\pi\)
−0.0106696 + 0.999943i \(0.503396\pi\)
\(788\) 29.6073 1.05472
\(789\) 0 0
\(790\) −0.228278 −0.00812178
\(791\) −81.8271 −2.90944
\(792\) 0 0
\(793\) 15.0433 0.534204
\(794\) 0.183329 0.00650610
\(795\) 0 0
\(796\) 19.3856 0.687104
\(797\) −9.41471 −0.333486 −0.166743 0.986000i \(-0.553325\pi\)
−0.166743 + 0.986000i \(0.553325\pi\)
\(798\) 0 0
\(799\) 32.3008 1.14272
\(800\) 0.0108731 0.000384423 0
\(801\) 0 0
\(802\) 0.499035 0.0176215
\(803\) 29.1395 1.02831
\(804\) 0 0
\(805\) −40.9566 −1.44353
\(806\) 0.0941028 0.00331463
\(807\) 0 0
\(808\) 0.108847 0.00382921
\(809\) 16.7127 0.587588 0.293794 0.955869i \(-0.405082\pi\)
0.293794 + 0.955869i \(0.405082\pi\)
\(810\) 0 0
\(811\) −33.3891 −1.17245 −0.586225 0.810148i \(-0.699387\pi\)
−0.586225 + 0.810148i \(0.699387\pi\)
\(812\) −46.8138 −1.64284
\(813\) 0 0
\(814\) 0.431237 0.0151148
\(815\) −9.57933 −0.335549
\(816\) 0 0
\(817\) 8.68184 0.303739
\(818\) 0.734465 0.0256800
\(819\) 0 0
\(820\) 48.5650 1.69596
\(821\) −36.8927 −1.28756 −0.643782 0.765209i \(-0.722636\pi\)
−0.643782 + 0.765209i \(0.722636\pi\)
\(822\) 0 0
\(823\) −17.1641 −0.598304 −0.299152 0.954205i \(-0.596704\pi\)
−0.299152 + 0.954205i \(0.596704\pi\)
\(824\) −1.00383 −0.0349701
\(825\) 0 0
\(826\) −1.45992 −0.0507972
\(827\) −12.8940 −0.448369 −0.224185 0.974547i \(-0.571972\pi\)
−0.224185 + 0.974547i \(0.571972\pi\)
\(828\) 0 0
\(829\) −8.56457 −0.297460 −0.148730 0.988878i \(-0.547518\pi\)
−0.148730 + 0.988878i \(0.547518\pi\)
\(830\) 0.930697 0.0323050
\(831\) 0 0
\(832\) 24.3418 0.843898
\(833\) −49.2771 −1.70735
\(834\) 0 0
\(835\) 48.0174 1.66171
\(836\) −3.96284 −0.137058
\(837\) 0 0
\(838\) 0.118690 0.00410007
\(839\) −27.2656 −0.941312 −0.470656 0.882317i \(-0.655983\pi\)
−0.470656 + 0.882317i \(0.655983\pi\)
\(840\) 0 0
\(841\) 1.85197 0.0638610
\(842\) −0.532202 −0.0183409
\(843\) 0 0
\(844\) 7.90230 0.272008
\(845\) 8.32481 0.286382
\(846\) 0 0
\(847\) 27.3367 0.939301
\(848\) 43.7062 1.50088
\(849\) 0 0
\(850\) −0.00414779 −0.000142268 0
\(851\) 36.2368 1.24218
\(852\) 0 0
\(853\) −17.4725 −0.598248 −0.299124 0.954214i \(-0.596694\pi\)
−0.299124 + 0.954214i \(0.596694\pi\)
\(854\) 0.504403 0.0172603
\(855\) 0 0
\(856\) 0.249298 0.00852083
\(857\) 20.8610 0.712597 0.356298 0.934372i \(-0.384039\pi\)
0.356298 + 0.934372i \(0.384039\pi\)
\(858\) 0 0
\(859\) −39.4728 −1.34679 −0.673397 0.739281i \(-0.735166\pi\)
−0.673397 + 0.739281i \(0.735166\pi\)
\(860\) −41.7673 −1.42425
\(861\) 0 0
\(862\) −0.720308 −0.0245338
\(863\) 37.2894 1.26935 0.634673 0.772781i \(-0.281135\pi\)
0.634673 + 0.772781i \(0.281135\pi\)
\(864\) 0 0
\(865\) −10.6307 −0.361455
\(866\) −0.178441 −0.00606367
\(867\) 0 0
\(868\) −10.7319 −0.364263
\(869\) −8.91362 −0.302374
\(870\) 0 0
\(871\) −34.4741 −1.16811
\(872\) 0.446901 0.0151340
\(873\) 0 0
\(874\) 0.0979046 0.00331167
\(875\) −46.9509 −1.58723
\(876\) 0 0
\(877\) −51.4108 −1.73602 −0.868010 0.496548i \(-0.834601\pi\)
−0.868010 + 0.496548i \(0.834601\pi\)
\(878\) 0.480501 0.0162161
\(879\) 0 0
\(880\) 19.0592 0.642485
\(881\) −4.11918 −0.138779 −0.0693893 0.997590i \(-0.522105\pi\)
−0.0693893 + 0.997590i \(0.522105\pi\)
\(882\) 0 0
\(883\) 55.1078 1.85453 0.927263 0.374410i \(-0.122155\pi\)
0.927263 + 0.374410i \(0.122155\pi\)
\(884\) −27.8871 −0.937946
\(885\) 0 0
\(886\) 0.975056 0.0327576
\(887\) 7.97495 0.267773 0.133886 0.990997i \(-0.457254\pi\)
0.133886 + 0.990997i \(0.457254\pi\)
\(888\) 0 0
\(889\) 80.1168 2.68703
\(890\) −0.593714 −0.0199014
\(891\) 0 0
\(892\) 54.8567 1.83674
\(893\) 6.58445 0.220340
\(894\) 0 0
\(895\) 46.8794 1.56701
\(896\) 3.26809 0.109179
\(897\) 0 0
\(898\) 0.768991 0.0256616
\(899\) 7.07267 0.235887
\(900\) 0 0
\(901\) −50.0426 −1.66716
\(902\) −0.557539 −0.0185640
\(903\) 0 0
\(904\) −1.88233 −0.0626053
\(905\) −5.10023 −0.169537
\(906\) 0 0
\(907\) −50.2402 −1.66820 −0.834099 0.551615i \(-0.814012\pi\)
−0.834099 + 0.551615i \(0.814012\pi\)
\(908\) 5.15236 0.170987
\(909\) 0 0
\(910\) −0.699187 −0.0231778
\(911\) −43.3795 −1.43723 −0.718613 0.695410i \(-0.755223\pi\)
−0.718613 + 0.695410i \(0.755223\pi\)
\(912\) 0 0
\(913\) 36.3411 1.20271
\(914\) 0.520163 0.0172054
\(915\) 0 0
\(916\) 29.6910 0.981018
\(917\) 4.18984 0.138361
\(918\) 0 0
\(919\) 9.92184 0.327291 0.163646 0.986519i \(-0.447675\pi\)
0.163646 + 0.986519i \(0.447675\pi\)
\(920\) −0.942154 −0.0310619
\(921\) 0 0
\(922\) −0.390670 −0.0128660
\(923\) −18.0534 −0.594235
\(924\) 0 0
\(925\) −0.312946 −0.0102896
\(926\) −0.308941 −0.0101524
\(927\) 0 0
\(928\) −1.61542 −0.0530287
\(929\) −22.2900 −0.731312 −0.365656 0.930750i \(-0.619155\pi\)
−0.365656 + 0.930750i \(0.619155\pi\)
\(930\) 0 0
\(931\) −10.0450 −0.329213
\(932\) −13.5933 −0.445265
\(933\) 0 0
\(934\) −0.0616641 −0.00201771
\(935\) −21.8223 −0.713666
\(936\) 0 0
\(937\) −46.5712 −1.52141 −0.760707 0.649096i \(-0.775147\pi\)
−0.760707 + 0.649096i \(0.775147\pi\)
\(938\) −1.15592 −0.0377421
\(939\) 0 0
\(940\) −31.6770 −1.03319
\(941\) 20.6597 0.673486 0.336743 0.941597i \(-0.390675\pi\)
0.336743 + 0.941597i \(0.390675\pi\)
\(942\) 0 0
\(943\) −46.8500 −1.52565
\(944\) 57.0879 1.85805
\(945\) 0 0
\(946\) 0.479500 0.0155899
\(947\) 39.3901 1.28001 0.640003 0.768373i \(-0.278933\pi\)
0.640003 + 0.768373i \(0.278933\pi\)
\(948\) 0 0
\(949\) 41.8010 1.35692
\(950\) −0.000845519 0 −2.74323e−5 0
\(951\) 0 0
\(952\) −1.87039 −0.0606197
\(953\) 32.7355 1.06041 0.530203 0.847871i \(-0.322116\pi\)
0.530203 + 0.847871i \(0.322116\pi\)
\(954\) 0 0
\(955\) −32.4206 −1.04911
\(956\) −1.99941 −0.0646656
\(957\) 0 0
\(958\) −0.232816 −0.00752194
\(959\) 43.3191 1.39885
\(960\) 0 0
\(961\) −29.3786 −0.947698
\(962\) 0.618613 0.0199449
\(963\) 0 0
\(964\) 28.9040 0.930934
\(965\) −29.6718 −0.955169
\(966\) 0 0
\(967\) −14.7746 −0.475118 −0.237559 0.971373i \(-0.576347\pi\)
−0.237559 + 0.971373i \(0.576347\pi\)
\(968\) 0.628846 0.0202119
\(969\) 0 0
\(970\) 0.195690 0.00628322
\(971\) 38.8307 1.24614 0.623069 0.782167i \(-0.285886\pi\)
0.623069 + 0.782167i \(0.285886\pi\)
\(972\) 0 0
\(973\) 38.4577 1.23290
\(974\) −0.880106 −0.0282004
\(975\) 0 0
\(976\) −19.7239 −0.631346
\(977\) −7.75740 −0.248181 −0.124091 0.992271i \(-0.539601\pi\)
−0.124091 + 0.992271i \(0.539601\pi\)
\(978\) 0 0
\(979\) −23.1829 −0.740928
\(980\) 48.3256 1.54370
\(981\) 0 0
\(982\) −0.176189 −0.00562242
\(983\) 0.802036 0.0255810 0.0127905 0.999918i \(-0.495929\pi\)
0.0127905 + 0.999918i \(0.495929\pi\)
\(984\) 0 0
\(985\) 33.2352 1.05896
\(986\) 0.616236 0.0196250
\(987\) 0 0
\(988\) −5.68474 −0.180856
\(989\) 40.2924 1.28122
\(990\) 0 0
\(991\) −13.3518 −0.424135 −0.212068 0.977255i \(-0.568020\pi\)
−0.212068 + 0.977255i \(0.568020\pi\)
\(992\) −0.370327 −0.0117579
\(993\) 0 0
\(994\) −0.605332 −0.0192000
\(995\) 21.7610 0.689870
\(996\) 0 0
\(997\) −4.98041 −0.157731 −0.0788655 0.996885i \(-0.525130\pi\)
−0.0788655 + 0.996885i \(0.525130\pi\)
\(998\) −0.637163 −0.0201690
\(999\) 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.1.9 yes 20
3.2 odd 2 2151.2.a.j.1.12 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.12 20 3.2 odd 2
2151.2.a.k.1.9 yes 20 1.1 even 1 trivial