Properties

Label 2151.2.a.j.1.12
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
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: \(1\)
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 1.12
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 \(-0.502729\pi\)
−0.00857210 + 0.999963i \(0.502729\pi\)
\(3\) 0 0
\(4\) −1.99941 −0.999706
\(5\) 2.24441 1.00373 0.501866 0.864945i \(-0.332647\pi\)
0.501866 + 0.864945i \(0.332647\pi\)
\(6\) 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 \(-0.396206\pi\)
0.320331 + 0.947306i \(0.396206\pi\)
\(12\) 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 \(-0.312754\pi\)
0.554906 + 0.831913i \(0.312754\pi\)
\(18\) 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 \(-0.350949\pi\)
0.451333 + 0.892356i \(0.350949\pi\)
\(24\) 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 \(-0.327525\pi\)
0.515718 + 0.856758i \(0.327525\pi\)
\(30\) 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 \(-0.820446\pi\)
−0.845078 + 0.534643i \(0.820446\pi\)
\(42\) 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 \(-0.327857\pi\)
0.514826 + 0.857295i \(0.327857\pi\)
\(48\) 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 \(-0.770477\pi\)
−0.751101 + 0.660188i \(0.770477\pi\)
\(54\) 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 \(-0.880060\pi\)
−0.929846 + 0.367948i \(0.880060\pi\)
\(60\) 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 \(-0.614313\pi\)
−0.351456 + 0.936204i \(0.614313\pi\)
\(72\) 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 \(-0.112076\pi\)
0.938652 + 0.344867i \(0.112076\pi\)
\(84\) 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 \(-0.696265\pi\)
−0.578253 + 0.815858i \(0.696265\pi\)
\(90\) 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 \(-0.482214\pi\)
0.0558465 + 0.998439i \(0.482214\pi\)
\(102\) 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 \(-0.460341\pi\)
0.124271 + 0.992248i \(0.460341\pi\)
\(108\) 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 \(-0.866285\pi\)
−0.913057 + 0.407833i \(0.866285\pi\)
\(114\) 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 \(-0.486174\pi\)
0.0434212 + 0.999057i \(0.486174\pi\)
\(132\) 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 \(-0.355335\pi\)
0.438994 + 0.898490i \(0.355335\pi\)
\(138\) 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 \(-0.383120\pi\)
0.358995 + 0.933340i \(0.383120\pi\)
\(150\) 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 \(-0.189612\pi\)
0.827766 + 0.561074i \(0.189612\pi\)
\(168\) 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 \(-0.557628\pi\)
−0.180056 + 0.983656i \(0.557628\pi\)
\(174\) 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 \(-0.214918\pi\)
0.780591 + 0.625043i \(0.214918\pi\)
\(180\) 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 \(-0.675039\pi\)
−0.522603 + 0.852576i \(0.675039\pi\)
\(192\) 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 \(-0.323125\pi\)
0.527513 + 0.849547i \(0.323125\pi\)
\(198\) 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 \(-0.472745\pi\)
0.0855187 + 0.996337i \(0.472745\pi\)
\(228\) 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 \(-0.571486\pi\)
−0.222698 + 0.974888i \(0.571486\pi\)
\(234\) 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 \(-0.784359\pi\)
−0.779170 + 0.626813i \(0.784359\pi\)
\(252\) 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 \(-0.327121\pi\)
0.516805 + 0.856103i \(0.327121\pi\)
\(258\) 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 \(-0.404231\pi\)
0.296349 + 0.955080i \(0.404231\pi\)
\(264\) 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 \(-0.737581\pi\)
−0.678987 + 0.734151i \(0.737581\pi\)
\(270\) 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 \(-0.544072\pi\)
−0.138016 + 0.990430i \(0.544072\pi\)
\(282\) 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 \(-0.478008\pi\)
0.0690340 + 0.997614i \(0.478008\pi\)
\(294\) 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 \(-0.299473\pi\)
0.589125 + 0.808042i \(0.299473\pi\)
\(312\) 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 \(-0.486306\pi\)
0.0430069 + 0.999075i \(0.486306\pi\)
\(318\) 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 \(-0.0951637\pi\)
0.955642 + 0.294532i \(0.0951637\pi\)
\(348\) 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 \(-0.562768\pi\)
−0.195916 + 0.980621i \(0.562768\pi\)
\(354\) 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 \(-0.563093\pi\)
−0.196916 + 0.980420i \(0.563093\pi\)
\(360\) 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 \(-0.606580\pi\)
−0.328608 + 0.944466i \(0.606580\pi\)
\(384\) 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 \(-0.352162\pi\)
0.447928 + 0.894070i \(0.352162\pi\)
\(390\) 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 \(-0.671808\pi\)
−0.513921 + 0.857837i \(0.671808\pi\)
\(402\) 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 \(-0.538153\pi\)
−0.119576 + 0.992825i \(0.538153\pi\)
\(420\) 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 \(-0.246193\pi\)
0.715513 + 0.698600i \(0.246193\pi\)
\(432\) 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 \(-0.904528\pi\)
−0.955356 + 0.295456i \(0.904528\pi\)
\(444\) 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 \(-0.769179\pi\)
−0.748403 + 0.663244i \(0.769179\pi\)
\(450\) 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 \(-0.377564\pi\)
0.375230 + 0.926932i \(0.377564\pi\)
\(462\) 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 \(-0.481258\pi\)
0.0588453 + 0.998267i \(0.481258\pi\)
\(468\) 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 \(-0.429599\pi\)
0.219373 + 0.975641i \(0.429599\pi\)
\(480\) 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 \(-0.447569\pi\)
0.163975 + 0.986465i \(0.447569\pi\)
\(492\) 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 \(-0.223624\pi\)
0.763206 + 0.646156i \(0.223624\pi\)
\(504\) 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 \(-0.585899\pi\)
−0.266597 + 0.963808i \(0.585899\pi\)
\(510\) 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 \(-0.479446\pi\)
0.0645285 + 0.997916i \(0.479446\pi\)
\(522\) 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 \(-0.543598\pi\)
−0.136539 + 0.990635i \(0.543598\pi\)
\(558\) 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 \(-0.820323\pi\)
−0.844871 + 0.534970i \(0.820323\pi\)
\(564\) 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 \(-0.415404\pi\)
0.262648 + 0.964892i \(0.415404\pi\)
\(570\) 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 \(-0.400780\pi\)
0.306687 + 0.951811i \(0.400780\pi\)
\(588\) 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 \(-0.507056\pi\)
−0.0221638 + 0.999754i \(0.507056\pi\)
\(594\) 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 \(-0.532827\pi\)
−0.102948 + 0.994687i \(0.532827\pi\)
\(600\) 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 \(-0.263212\pi\)
0.677156 + 0.735840i \(0.263212\pi\)
\(618\) 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 \(-0.205086\pi\)
0.799522 + 0.600637i \(0.205086\pi\)
\(642\) 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 \(-0.560000\pi\)
−0.187380 + 0.982287i \(0.560000\pi\)
\(648\) 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 \(-0.291641\pi\)
0.608825 + 0.793305i \(0.291641\pi\)
\(654\) 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 \(-0.493710\pi\)
0.0197580 + 0.999805i \(0.493710\pi\)
\(660\) 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 \(-0.723274\pi\)
−0.645315 + 0.763916i \(0.723274\pi\)
\(678\) 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 \(-0.471296\pi\)
0.0900536 + 0.995937i \(0.471296\pi\)
\(684\) 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 \(-0.323695\pi\)
0.525991 + 0.850490i \(0.323695\pi\)
\(702\) 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 \(-0.322150\pi\)
0.530112 + 0.847928i \(0.322150\pi\)
\(720\) 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 \(-0.634087\pi\)
−0.408898 + 0.912580i \(0.634087\pi\)
\(744\) 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 \(-0.804780\pi\)
−0.817751 + 0.575572i \(0.804780\pi\)
\(762\) 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 \(-0.691451\pi\)
−0.565848 + 0.824510i \(0.691451\pi\)
\(774\) 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 \(-0.446675\pi\)
0.166743 + 0.986000i \(0.446675\pi\)
\(798\) 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 \(-0.594918\pi\)
−0.293794 + 0.955869i \(0.594918\pi\)
\(810\) 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 \(-0.277364\pi\)
0.643782 + 0.765209i \(0.277364\pi\)
\(822\) 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 \(-0.428028\pi\)
0.224185 + 0.974547i \(0.428028\pi\)
\(828\) 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 \(-0.344017\pi\)
0.470656 + 0.882317i \(0.344017\pi\)
\(840\) 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 \(-0.615961\pi\)
−0.356298 + 0.934372i \(0.615961\pi\)
\(858\) 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 \(-0.718865\pi\)
−0.634673 + 0.772781i \(0.718865\pi\)
\(864\) 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 \(-0.477895\pi\)
0.0693893 + 0.997590i \(0.477895\pi\)
\(882\) 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 \(-0.542746\pi\)
−0.133886 + 0.990997i \(0.542746\pi\)
\(888\) 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 \(-0.244777\pi\)
0.718613 + 0.695410i \(0.244777\pi\)
\(912\) 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 \(-0.380845\pi\)
0.365656 + 0.930750i \(0.380845\pi\)
\(930\) 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 \(-0.609325\pi\)
−0.336743 + 0.941597i \(0.609325\pi\)
\(942\) 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 \(-0.721067\pi\)
−0.640003 + 0.768373i \(0.721067\pi\)
\(948\) 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 \(-0.677884\pi\)
−0.530203 + 0.847871i \(0.677884\pi\)
\(954\) 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 \(-0.714114\pi\)
−0.623069 + 0.782167i \(0.714114\pi\)
\(972\) 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 \(-0.460399\pi\)
0.124091 + 0.992271i \(0.460399\pi\)
\(978\) 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 \(-0.504071\pi\)
−0.0127905 + 0.999918i \(0.504071\pi\)
\(984\) 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.j.1.12 20
3.2 odd 2 2151.2.a.k.1.9 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.12 20 1.1 even 1 trivial
2151.2.a.k.1.9 yes 20 3.2 odd 2