Properties

Label 2151.2.a.k.1.20
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 1.20
Root \(2.67404\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +O(q^{10})\) \(q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +1.38826 q^{10} +5.14954 q^{11} -2.29568 q^{13} +2.73005 q^{14} +12.2266 q^{16} -1.92182 q^{17} +3.15077 q^{19} +2.67395 q^{20} +13.7701 q^{22} -6.68812 q^{23} -4.73047 q^{25} -6.13875 q^{26} +5.25837 q^{28} -7.68981 q^{29} -1.20467 q^{31} +15.8454 q^{32} -5.13902 q^{34} +0.530037 q^{35} -0.791859 q^{37} +8.42528 q^{38} +4.37372 q^{40} +2.49631 q^{41} +1.15197 q^{43} +26.5227 q^{44} -17.8843 q^{46} +8.92862 q^{47} -5.95767 q^{49} -12.6495 q^{50} -11.8239 q^{52} -7.62842 q^{53} +2.67345 q^{55} +8.60100 q^{56} -20.5629 q^{58} +9.24454 q^{59} +2.23081 q^{61} -3.22134 q^{62} +17.9179 q^{64} -1.19183 q^{65} +14.8406 q^{67} -9.89831 q^{68} +1.41734 q^{70} -4.46542 q^{71} -2.29621 q^{73} -2.11746 q^{74} +16.2280 q^{76} +5.25739 q^{77} -9.20506 q^{79} +6.34761 q^{80} +6.67524 q^{82} +13.9566 q^{83} -0.997737 q^{85} +3.08042 q^{86} +43.3826 q^{88} +7.81208 q^{89} -2.34376 q^{91} -34.4471 q^{92} +23.8755 q^{94} +1.63576 q^{95} -4.25925 q^{97} -15.9311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67404 1.89083 0.945416 0.325865i \(-0.105656\pi\)
0.945416 + 0.325865i \(0.105656\pi\)
\(3\) 0 0
\(4\) 5.15050 2.57525
\(5\) 0.519163 0.232177 0.116088 0.993239i \(-0.462964\pi\)
0.116088 + 0.993239i \(0.462964\pi\)
\(6\) 0 0
\(7\) 1.02094 0.385881 0.192940 0.981211i \(-0.438198\pi\)
0.192940 + 0.981211i \(0.438198\pi\)
\(8\) 8.42455 2.97853
\(9\) 0 0
\(10\) 1.38826 0.439008
\(11\) 5.14954 1.55265 0.776323 0.630336i \(-0.217083\pi\)
0.776323 + 0.630336i \(0.217083\pi\)
\(12\) 0 0
\(13\) −2.29568 −0.636708 −0.318354 0.947972i \(-0.603130\pi\)
−0.318354 + 0.947972i \(0.603130\pi\)
\(14\) 2.73005 0.729635
\(15\) 0 0
\(16\) 12.2266 3.05665
\(17\) −1.92182 −0.466109 −0.233054 0.972464i \(-0.574872\pi\)
−0.233054 + 0.972464i \(0.574872\pi\)
\(18\) 0 0
\(19\) 3.15077 0.722835 0.361418 0.932404i \(-0.382293\pi\)
0.361418 + 0.932404i \(0.382293\pi\)
\(20\) 2.67395 0.597913
\(21\) 0 0
\(22\) 13.7701 2.93579
\(23\) −6.68812 −1.39457 −0.697284 0.716795i \(-0.745609\pi\)
−0.697284 + 0.716795i \(0.745609\pi\)
\(24\) 0 0
\(25\) −4.73047 −0.946094
\(26\) −6.13875 −1.20391
\(27\) 0 0
\(28\) 5.25837 0.993738
\(29\) −7.68981 −1.42796 −0.713981 0.700165i \(-0.753110\pi\)
−0.713981 + 0.700165i \(0.753110\pi\)
\(30\) 0 0
\(31\) −1.20467 −0.216365 −0.108182 0.994131i \(-0.534503\pi\)
−0.108182 + 0.994131i \(0.534503\pi\)
\(32\) 15.8454 2.80109
\(33\) 0 0
\(34\) −5.13902 −0.881334
\(35\) 0.530037 0.0895926
\(36\) 0 0
\(37\) −0.791859 −0.130181 −0.0650904 0.997879i \(-0.520734\pi\)
−0.0650904 + 0.997879i \(0.520734\pi\)
\(38\) 8.42528 1.36676
\(39\) 0 0
\(40\) 4.37372 0.691546
\(41\) 2.49631 0.389858 0.194929 0.980817i \(-0.437552\pi\)
0.194929 + 0.980817i \(0.437552\pi\)
\(42\) 0 0
\(43\) 1.15197 0.175674 0.0878370 0.996135i \(-0.472005\pi\)
0.0878370 + 0.996135i \(0.472005\pi\)
\(44\) 26.5227 3.99845
\(45\) 0 0
\(46\) −17.8843 −2.63690
\(47\) 8.92862 1.30237 0.651187 0.758918i \(-0.274272\pi\)
0.651187 + 0.758918i \(0.274272\pi\)
\(48\) 0 0
\(49\) −5.95767 −0.851096
\(50\) −12.6495 −1.78891
\(51\) 0 0
\(52\) −11.8239 −1.63968
\(53\) −7.62842 −1.04784 −0.523922 0.851766i \(-0.675532\pi\)
−0.523922 + 0.851766i \(0.675532\pi\)
\(54\) 0 0
\(55\) 2.67345 0.360488
\(56\) 8.60100 1.14936
\(57\) 0 0
\(58\) −20.5629 −2.70004
\(59\) 9.24454 1.20354 0.601768 0.798671i \(-0.294463\pi\)
0.601768 + 0.798671i \(0.294463\pi\)
\(60\) 0 0
\(61\) 2.23081 0.285626 0.142813 0.989750i \(-0.454385\pi\)
0.142813 + 0.989750i \(0.454385\pi\)
\(62\) −3.22134 −0.409110
\(63\) 0 0
\(64\) 17.9179 2.23974
\(65\) −1.19183 −0.147829
\(66\) 0 0
\(67\) 14.8406 1.81307 0.906535 0.422130i \(-0.138717\pi\)
0.906535 + 0.422130i \(0.138717\pi\)
\(68\) −9.89831 −1.20035
\(69\) 0 0
\(70\) 1.41734 0.169405
\(71\) −4.46542 −0.529948 −0.264974 0.964256i \(-0.585363\pi\)
−0.264974 + 0.964256i \(0.585363\pi\)
\(72\) 0 0
\(73\) −2.29621 −0.268751 −0.134376 0.990930i \(-0.542903\pi\)
−0.134376 + 0.990930i \(0.542903\pi\)
\(74\) −2.11746 −0.246150
\(75\) 0 0
\(76\) 16.2280 1.86148
\(77\) 5.25739 0.599136
\(78\) 0 0
\(79\) −9.20506 −1.03565 −0.517825 0.855487i \(-0.673258\pi\)
−0.517825 + 0.855487i \(0.673258\pi\)
\(80\) 6.34761 0.709684
\(81\) 0 0
\(82\) 6.67524 0.737157
\(83\) 13.9566 1.53193 0.765967 0.642880i \(-0.222261\pi\)
0.765967 + 0.642880i \(0.222261\pi\)
\(84\) 0 0
\(85\) −0.997737 −0.108220
\(86\) 3.08042 0.332170
\(87\) 0 0
\(88\) 43.3826 4.62460
\(89\) 7.81208 0.828079 0.414039 0.910259i \(-0.364118\pi\)
0.414039 + 0.910259i \(0.364118\pi\)
\(90\) 0 0
\(91\) −2.34376 −0.245693
\(92\) −34.4471 −3.59136
\(93\) 0 0
\(94\) 23.8755 2.46257
\(95\) 1.63576 0.167826
\(96\) 0 0
\(97\) −4.25925 −0.432461 −0.216231 0.976342i \(-0.569376\pi\)
−0.216231 + 0.976342i \(0.569376\pi\)
\(98\) −15.9311 −1.60928
\(99\) 0 0
\(100\) −24.3643 −2.43643
\(101\) 16.5592 1.64770 0.823851 0.566807i \(-0.191821\pi\)
0.823851 + 0.566807i \(0.191821\pi\)
\(102\) 0 0
\(103\) −13.0011 −1.28103 −0.640517 0.767944i \(-0.721280\pi\)
−0.640517 + 0.767944i \(0.721280\pi\)
\(104\) −19.3401 −1.89645
\(105\) 0 0
\(106\) −20.3987 −1.98130
\(107\) −5.26519 −0.509005 −0.254503 0.967072i \(-0.581912\pi\)
−0.254503 + 0.967072i \(0.581912\pi\)
\(108\) 0 0
\(109\) −10.9722 −1.05095 −0.525475 0.850809i \(-0.676112\pi\)
−0.525475 + 0.850809i \(0.676112\pi\)
\(110\) 7.14892 0.681623
\(111\) 0 0
\(112\) 12.4827 1.17950
\(113\) −8.33557 −0.784144 −0.392072 0.919934i \(-0.628242\pi\)
−0.392072 + 0.919934i \(0.628242\pi\)
\(114\) 0 0
\(115\) −3.47223 −0.323787
\(116\) −39.6064 −3.67736
\(117\) 0 0
\(118\) 24.7203 2.27569
\(119\) −1.96207 −0.179862
\(120\) 0 0
\(121\) 15.5178 1.41071
\(122\) 5.96527 0.540070
\(123\) 0 0
\(124\) −6.20464 −0.557193
\(125\) −5.05170 −0.451838
\(126\) 0 0
\(127\) 10.2324 0.907979 0.453990 0.891007i \(-0.350000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(128\) 16.2225 1.43388
\(129\) 0 0
\(130\) −3.18702 −0.279520
\(131\) −11.8436 −1.03478 −0.517389 0.855750i \(-0.673096\pi\)
−0.517389 + 0.855750i \(0.673096\pi\)
\(132\) 0 0
\(133\) 3.21676 0.278928
\(134\) 39.6844 3.42821
\(135\) 0 0
\(136\) −16.1904 −1.38832
\(137\) −18.1801 −1.55323 −0.776614 0.629977i \(-0.783064\pi\)
−0.776614 + 0.629977i \(0.783064\pi\)
\(138\) 0 0
\(139\) 3.89676 0.330519 0.165259 0.986250i \(-0.447154\pi\)
0.165259 + 0.986250i \(0.447154\pi\)
\(140\) 2.72995 0.230723
\(141\) 0 0
\(142\) −11.9407 −1.00204
\(143\) −11.8217 −0.988582
\(144\) 0 0
\(145\) −3.99227 −0.331540
\(146\) −6.14016 −0.508163
\(147\) 0 0
\(148\) −4.07846 −0.335248
\(149\) 2.58671 0.211911 0.105956 0.994371i \(-0.466210\pi\)
0.105956 + 0.994371i \(0.466210\pi\)
\(150\) 0 0
\(151\) −17.0193 −1.38501 −0.692506 0.721412i \(-0.743494\pi\)
−0.692506 + 0.721412i \(0.743494\pi\)
\(152\) 26.5438 2.15299
\(153\) 0 0
\(154\) 14.0585 1.13286
\(155\) −0.625420 −0.0502350
\(156\) 0 0
\(157\) −9.59695 −0.765920 −0.382960 0.923765i \(-0.625095\pi\)
−0.382960 + 0.923765i \(0.625095\pi\)
\(158\) −24.6147 −1.95824
\(159\) 0 0
\(160\) 8.22633 0.650349
\(161\) −6.82819 −0.538137
\(162\) 0 0
\(163\) −1.07845 −0.0844708 −0.0422354 0.999108i \(-0.513448\pi\)
−0.0422354 + 0.999108i \(0.513448\pi\)
\(164\) 12.8572 1.00398
\(165\) 0 0
\(166\) 37.3205 2.89663
\(167\) 11.9566 0.925230 0.462615 0.886559i \(-0.346911\pi\)
0.462615 + 0.886559i \(0.346911\pi\)
\(168\) 0 0
\(169\) −7.72984 −0.594603
\(170\) −2.66799 −0.204625
\(171\) 0 0
\(172\) 5.93322 0.452404
\(173\) 16.3039 1.23956 0.619780 0.784776i \(-0.287222\pi\)
0.619780 + 0.784776i \(0.287222\pi\)
\(174\) 0 0
\(175\) −4.82954 −0.365079
\(176\) 62.9615 4.74590
\(177\) 0 0
\(178\) 20.8898 1.56576
\(179\) −0.105075 −0.00785365 −0.00392682 0.999992i \(-0.501250\pi\)
−0.00392682 + 0.999992i \(0.501250\pi\)
\(180\) 0 0
\(181\) −18.0099 −1.33866 −0.669332 0.742963i \(-0.733420\pi\)
−0.669332 + 0.742963i \(0.733420\pi\)
\(182\) −6.26732 −0.464565
\(183\) 0 0
\(184\) −56.3444 −4.15376
\(185\) −0.411104 −0.0302250
\(186\) 0 0
\(187\) −9.89647 −0.723702
\(188\) 45.9868 3.35393
\(189\) 0 0
\(190\) 4.37410 0.317330
\(191\) −1.54242 −0.111606 −0.0558028 0.998442i \(-0.517772\pi\)
−0.0558028 + 0.998442i \(0.517772\pi\)
\(192\) 0 0
\(193\) 9.49555 0.683505 0.341752 0.939790i \(-0.388980\pi\)
0.341752 + 0.939790i \(0.388980\pi\)
\(194\) −11.3894 −0.817712
\(195\) 0 0
\(196\) −30.6850 −2.19178
\(197\) 4.42454 0.315235 0.157618 0.987500i \(-0.449619\pi\)
0.157618 + 0.987500i \(0.449619\pi\)
\(198\) 0 0
\(199\) −14.1893 −1.00585 −0.502927 0.864329i \(-0.667744\pi\)
−0.502927 + 0.864329i \(0.667744\pi\)
\(200\) −39.8521 −2.81797
\(201\) 0 0
\(202\) 44.2800 3.11553
\(203\) −7.85087 −0.551023
\(204\) 0 0
\(205\) 1.29599 0.0905161
\(206\) −34.7654 −2.42222
\(207\) 0 0
\(208\) −28.0684 −1.94620
\(209\) 16.2250 1.12231
\(210\) 0 0
\(211\) 0.0479471 0.00330081 0.00165041 0.999999i \(-0.499475\pi\)
0.00165041 + 0.999999i \(0.499475\pi\)
\(212\) −39.2901 −2.69846
\(213\) 0 0
\(214\) −14.0793 −0.962444
\(215\) 0.598061 0.0407874
\(216\) 0 0
\(217\) −1.22990 −0.0834910
\(218\) −29.3402 −1.98717
\(219\) 0 0
\(220\) 13.7696 0.928347
\(221\) 4.41188 0.296775
\(222\) 0 0
\(223\) −22.1434 −1.48283 −0.741415 0.671047i \(-0.765845\pi\)
−0.741415 + 0.671047i \(0.765845\pi\)
\(224\) 16.1772 1.08089
\(225\) 0 0
\(226\) −22.2896 −1.48269
\(227\) 25.9911 1.72509 0.862546 0.505978i \(-0.168868\pi\)
0.862546 + 0.505978i \(0.168868\pi\)
\(228\) 0 0
\(229\) −16.4825 −1.08919 −0.544596 0.838699i \(-0.683317\pi\)
−0.544596 + 0.838699i \(0.683317\pi\)
\(230\) −9.28487 −0.612226
\(231\) 0 0
\(232\) −64.7833 −4.25323
\(233\) 8.28328 0.542656 0.271328 0.962487i \(-0.412537\pi\)
0.271328 + 0.962487i \(0.412537\pi\)
\(234\) 0 0
\(235\) 4.63541 0.302381
\(236\) 47.6140 3.09940
\(237\) 0 0
\(238\) −5.24665 −0.340090
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 18.7517 1.20790 0.603951 0.797021i \(-0.293592\pi\)
0.603951 + 0.797021i \(0.293592\pi\)
\(242\) 41.4952 2.66741
\(243\) 0 0
\(244\) 11.4898 0.735557
\(245\) −3.09301 −0.197605
\(246\) 0 0
\(247\) −7.23316 −0.460235
\(248\) −10.1488 −0.644450
\(249\) 0 0
\(250\) −13.5085 −0.854350
\(251\) 8.36672 0.528103 0.264051 0.964509i \(-0.414941\pi\)
0.264051 + 0.964509i \(0.414941\pi\)
\(252\) 0 0
\(253\) −34.4407 −2.16527
\(254\) 27.3619 1.71684
\(255\) 0 0
\(256\) 7.54385 0.471491
\(257\) −8.28841 −0.517017 −0.258509 0.966009i \(-0.583231\pi\)
−0.258509 + 0.966009i \(0.583231\pi\)
\(258\) 0 0
\(259\) −0.808443 −0.0502342
\(260\) −6.13854 −0.380696
\(261\) 0 0
\(262\) −31.6702 −1.95659
\(263\) −12.7232 −0.784543 −0.392272 0.919849i \(-0.628311\pi\)
−0.392272 + 0.919849i \(0.628311\pi\)
\(264\) 0 0
\(265\) −3.96040 −0.243285
\(266\) 8.60174 0.527406
\(267\) 0 0
\(268\) 76.4366 4.66911
\(269\) 7.72368 0.470921 0.235461 0.971884i \(-0.424340\pi\)
0.235461 + 0.971884i \(0.424340\pi\)
\(270\) 0 0
\(271\) 23.2037 1.40953 0.704764 0.709442i \(-0.251053\pi\)
0.704764 + 0.709442i \(0.251053\pi\)
\(272\) −23.4973 −1.42473
\(273\) 0 0
\(274\) −48.6142 −2.93689
\(275\) −24.3597 −1.46895
\(276\) 0 0
\(277\) −5.69405 −0.342123 −0.171061 0.985260i \(-0.554720\pi\)
−0.171061 + 0.985260i \(0.554720\pi\)
\(278\) 10.4201 0.624956
\(279\) 0 0
\(280\) 4.46532 0.266854
\(281\) 26.2890 1.56827 0.784134 0.620592i \(-0.213108\pi\)
0.784134 + 0.620592i \(0.213108\pi\)
\(282\) 0 0
\(283\) 21.4629 1.27584 0.637918 0.770104i \(-0.279796\pi\)
0.637918 + 0.770104i \(0.279796\pi\)
\(284\) −22.9991 −1.36475
\(285\) 0 0
\(286\) −31.6118 −1.86924
\(287\) 2.54859 0.150439
\(288\) 0 0
\(289\) −13.3066 −0.782742
\(290\) −10.6755 −0.626887
\(291\) 0 0
\(292\) −11.8266 −0.692101
\(293\) −15.8224 −0.924356 −0.462178 0.886787i \(-0.652932\pi\)
−0.462178 + 0.886787i \(0.652932\pi\)
\(294\) 0 0
\(295\) 4.79943 0.279433
\(296\) −6.67106 −0.387747
\(297\) 0 0
\(298\) 6.91696 0.400689
\(299\) 15.3538 0.887933
\(300\) 0 0
\(301\) 1.17610 0.0677891
\(302\) −45.5104 −2.61883
\(303\) 0 0
\(304\) 38.5232 2.20946
\(305\) 1.15815 0.0663157
\(306\) 0 0
\(307\) −11.3738 −0.649137 −0.324569 0.945862i \(-0.605219\pi\)
−0.324569 + 0.945862i \(0.605219\pi\)
\(308\) 27.0782 1.54292
\(309\) 0 0
\(310\) −1.67240 −0.0949859
\(311\) 19.7338 1.11900 0.559501 0.828830i \(-0.310993\pi\)
0.559501 + 0.828830i \(0.310993\pi\)
\(312\) 0 0
\(313\) 7.24433 0.409474 0.204737 0.978817i \(-0.434366\pi\)
0.204737 + 0.978817i \(0.434366\pi\)
\(314\) −25.6626 −1.44823
\(315\) 0 0
\(316\) −47.4106 −2.66705
\(317\) 11.0779 0.622199 0.311099 0.950377i \(-0.399303\pi\)
0.311099 + 0.950377i \(0.399303\pi\)
\(318\) 0 0
\(319\) −39.5990 −2.21712
\(320\) 9.30233 0.520016
\(321\) 0 0
\(322\) −18.2589 −1.01753
\(323\) −6.05519 −0.336920
\(324\) 0 0
\(325\) 10.8597 0.602386
\(326\) −2.88382 −0.159720
\(327\) 0 0
\(328\) 21.0303 1.16120
\(329\) 9.11562 0.502560
\(330\) 0 0
\(331\) 11.7234 0.644374 0.322187 0.946676i \(-0.395582\pi\)
0.322187 + 0.946676i \(0.395582\pi\)
\(332\) 71.8833 3.94511
\(333\) 0 0
\(334\) 31.9724 1.74945
\(335\) 7.70471 0.420953
\(336\) 0 0
\(337\) −28.6376 −1.55999 −0.779995 0.625786i \(-0.784778\pi\)
−0.779995 + 0.625786i \(0.784778\pi\)
\(338\) −20.6699 −1.12429
\(339\) 0 0
\(340\) −5.13884 −0.278693
\(341\) −6.20349 −0.335938
\(342\) 0 0
\(343\) −13.2291 −0.714302
\(344\) 9.70484 0.523250
\(345\) 0 0
\(346\) 43.5972 2.34380
\(347\) −1.96621 −0.105552 −0.0527760 0.998606i \(-0.516807\pi\)
−0.0527760 + 0.998606i \(0.516807\pi\)
\(348\) 0 0
\(349\) −1.25673 −0.0672713 −0.0336357 0.999434i \(-0.510709\pi\)
−0.0336357 + 0.999434i \(0.510709\pi\)
\(350\) −12.9144 −0.690304
\(351\) 0 0
\(352\) 81.5963 4.34910
\(353\) 21.0991 1.12299 0.561495 0.827480i \(-0.310226\pi\)
0.561495 + 0.827480i \(0.310226\pi\)
\(354\) 0 0
\(355\) −2.31828 −0.123042
\(356\) 40.2361 2.13251
\(357\) 0 0
\(358\) −0.280974 −0.0148499
\(359\) 17.5335 0.925382 0.462691 0.886520i \(-0.346884\pi\)
0.462691 + 0.886520i \(0.346884\pi\)
\(360\) 0 0
\(361\) −9.07267 −0.477509
\(362\) −48.1592 −2.53119
\(363\) 0 0
\(364\) −12.0715 −0.632721
\(365\) −1.19211 −0.0623978
\(366\) 0 0
\(367\) 28.4166 1.48334 0.741668 0.670767i \(-0.234035\pi\)
0.741668 + 0.670767i \(0.234035\pi\)
\(368\) −81.7730 −4.26271
\(369\) 0 0
\(370\) −1.09931 −0.0571503
\(371\) −7.78819 −0.404343
\(372\) 0 0
\(373\) 22.4497 1.16240 0.581201 0.813760i \(-0.302583\pi\)
0.581201 + 0.813760i \(0.302583\pi\)
\(374\) −26.4636 −1.36840
\(375\) 0 0
\(376\) 75.2197 3.87916
\(377\) 17.6534 0.909195
\(378\) 0 0
\(379\) 16.8689 0.866499 0.433250 0.901274i \(-0.357367\pi\)
0.433250 + 0.901274i \(0.357367\pi\)
\(380\) 8.42499 0.432193
\(381\) 0 0
\(382\) −4.12449 −0.211027
\(383\) −15.0924 −0.771184 −0.385592 0.922669i \(-0.626003\pi\)
−0.385592 + 0.922669i \(0.626003\pi\)
\(384\) 0 0
\(385\) 2.72945 0.139105
\(386\) 25.3915 1.29239
\(387\) 0 0
\(388\) −21.9373 −1.11370
\(389\) 4.28146 0.217079 0.108539 0.994092i \(-0.465383\pi\)
0.108539 + 0.994092i \(0.465383\pi\)
\(390\) 0 0
\(391\) 12.8533 0.650021
\(392\) −50.1907 −2.53502
\(393\) 0 0
\(394\) 11.8314 0.596057
\(395\) −4.77893 −0.240454
\(396\) 0 0
\(397\) −26.2285 −1.31637 −0.658186 0.752855i \(-0.728676\pi\)
−0.658186 + 0.752855i \(0.728676\pi\)
\(398\) −37.9429 −1.90190
\(399\) 0 0
\(400\) −57.8376 −2.89188
\(401\) −1.34368 −0.0671001 −0.0335500 0.999437i \(-0.510681\pi\)
−0.0335500 + 0.999437i \(0.510681\pi\)
\(402\) 0 0
\(403\) 2.76554 0.137761
\(404\) 85.2881 4.24324
\(405\) 0 0
\(406\) −20.9935 −1.04189
\(407\) −4.07771 −0.202124
\(408\) 0 0
\(409\) 14.8377 0.733677 0.366838 0.930285i \(-0.380440\pi\)
0.366838 + 0.930285i \(0.380440\pi\)
\(410\) 3.46554 0.171151
\(411\) 0 0
\(412\) −66.9620 −3.29898
\(413\) 9.43816 0.464421
\(414\) 0 0
\(415\) 7.24575 0.355680
\(416\) −36.3759 −1.78348
\(417\) 0 0
\(418\) 43.3863 2.12209
\(419\) 33.9802 1.66004 0.830021 0.557731i \(-0.188328\pi\)
0.830021 + 0.557731i \(0.188328\pi\)
\(420\) 0 0
\(421\) 40.1479 1.95669 0.978344 0.206985i \(-0.0663652\pi\)
0.978344 + 0.206985i \(0.0663652\pi\)
\(422\) 0.128213 0.00624129
\(423\) 0 0
\(424\) −64.2660 −3.12103
\(425\) 9.09109 0.440983
\(426\) 0 0
\(427\) 2.27753 0.110217
\(428\) −27.1183 −1.31081
\(429\) 0 0
\(430\) 1.59924 0.0771222
\(431\) 25.2309 1.21533 0.607665 0.794194i \(-0.292106\pi\)
0.607665 + 0.794194i \(0.292106\pi\)
\(432\) 0 0
\(433\) −31.2782 −1.50313 −0.751567 0.659657i \(-0.770702\pi\)
−0.751567 + 0.659657i \(0.770702\pi\)
\(434\) −3.28880 −0.157868
\(435\) 0 0
\(436\) −56.5124 −2.70645
\(437\) −21.0727 −1.00804
\(438\) 0 0
\(439\) −4.15467 −0.198291 −0.0991457 0.995073i \(-0.531611\pi\)
−0.0991457 + 0.995073i \(0.531611\pi\)
\(440\) 22.5227 1.07373
\(441\) 0 0
\(442\) 11.7976 0.561152
\(443\) −20.0339 −0.951840 −0.475920 0.879489i \(-0.657885\pi\)
−0.475920 + 0.879489i \(0.657885\pi\)
\(444\) 0 0
\(445\) 4.05575 0.192261
\(446\) −59.2123 −2.80378
\(447\) 0 0
\(448\) 18.2932 0.864272
\(449\) 15.7631 0.743906 0.371953 0.928252i \(-0.378688\pi\)
0.371953 + 0.928252i \(0.378688\pi\)
\(450\) 0 0
\(451\) 12.8549 0.605312
\(452\) −42.9323 −2.01937
\(453\) 0 0
\(454\) 69.5014 3.26186
\(455\) −1.21680 −0.0570443
\(456\) 0 0
\(457\) 14.3920 0.673227 0.336614 0.941643i \(-0.390718\pi\)
0.336614 + 0.941643i \(0.390718\pi\)
\(458\) −44.0748 −2.05948
\(459\) 0 0
\(460\) −17.8837 −0.833831
\(461\) 9.25187 0.430903 0.215451 0.976515i \(-0.430878\pi\)
0.215451 + 0.976515i \(0.430878\pi\)
\(462\) 0 0
\(463\) 9.95525 0.462660 0.231330 0.972875i \(-0.425692\pi\)
0.231330 + 0.972875i \(0.425692\pi\)
\(464\) −94.0204 −4.36479
\(465\) 0 0
\(466\) 22.1498 1.02607
\(467\) −18.9730 −0.877964 −0.438982 0.898496i \(-0.644661\pi\)
−0.438982 + 0.898496i \(0.644661\pi\)
\(468\) 0 0
\(469\) 15.1514 0.699629
\(470\) 12.3953 0.571752
\(471\) 0 0
\(472\) 77.8811 3.58477
\(473\) 5.93212 0.272759
\(474\) 0 0
\(475\) −14.9046 −0.683870
\(476\) −10.1056 −0.463190
\(477\) 0 0
\(478\) 2.67404 0.122308
\(479\) −33.1598 −1.51511 −0.757555 0.652771i \(-0.773606\pi\)
−0.757555 + 0.652771i \(0.773606\pi\)
\(480\) 0 0
\(481\) 1.81786 0.0828871
\(482\) 50.1428 2.28394
\(483\) 0 0
\(484\) 79.9242 3.63292
\(485\) −2.21125 −0.100408
\(486\) 0 0
\(487\) −37.0030 −1.67677 −0.838384 0.545080i \(-0.816499\pi\)
−0.838384 + 0.545080i \(0.816499\pi\)
\(488\) 18.7936 0.850745
\(489\) 0 0
\(490\) −8.27083 −0.373638
\(491\) −15.5770 −0.702980 −0.351490 0.936192i \(-0.614325\pi\)
−0.351490 + 0.936192i \(0.614325\pi\)
\(492\) 0 0
\(493\) 14.7784 0.665586
\(494\) −19.3418 −0.870227
\(495\) 0 0
\(496\) −14.7290 −0.661353
\(497\) −4.55895 −0.204497
\(498\) 0 0
\(499\) −3.63382 −0.162672 −0.0813360 0.996687i \(-0.525919\pi\)
−0.0813360 + 0.996687i \(0.525919\pi\)
\(500\) −26.0188 −1.16360
\(501\) 0 0
\(502\) 22.3730 0.998554
\(503\) −42.5761 −1.89838 −0.949188 0.314710i \(-0.898093\pi\)
−0.949188 + 0.314710i \(0.898093\pi\)
\(504\) 0 0
\(505\) 8.59693 0.382558
\(506\) −92.0959 −4.09416
\(507\) 0 0
\(508\) 52.7020 2.33827
\(509\) 35.1080 1.55614 0.778068 0.628180i \(-0.216200\pi\)
0.778068 + 0.628180i \(0.216200\pi\)
\(510\) 0 0
\(511\) −2.34430 −0.103706
\(512\) −12.2725 −0.542371
\(513\) 0 0
\(514\) −22.1636 −0.977593
\(515\) −6.74968 −0.297426
\(516\) 0 0
\(517\) 45.9783 2.02212
\(518\) −2.16181 −0.0949845
\(519\) 0 0
\(520\) −10.0407 −0.440313
\(521\) 36.0412 1.57899 0.789497 0.613754i \(-0.210341\pi\)
0.789497 + 0.613754i \(0.210341\pi\)
\(522\) 0 0
\(523\) 0.568975 0.0248795 0.0124398 0.999923i \(-0.496040\pi\)
0.0124398 + 0.999923i \(0.496040\pi\)
\(524\) −61.0003 −2.66481
\(525\) 0 0
\(526\) −34.0222 −1.48344
\(527\) 2.31515 0.100850
\(528\) 0 0
\(529\) 21.7309 0.944822
\(530\) −10.5903 −0.460012
\(531\) 0 0
\(532\) 16.5679 0.718309
\(533\) −5.73074 −0.248226
\(534\) 0 0
\(535\) −2.73349 −0.118179
\(536\) 125.026 5.40029
\(537\) 0 0
\(538\) 20.6534 0.890433
\(539\) −30.6793 −1.32145
\(540\) 0 0
\(541\) 12.1080 0.520565 0.260283 0.965533i \(-0.416184\pi\)
0.260283 + 0.965533i \(0.416184\pi\)
\(542\) 62.0478 2.66518
\(543\) 0 0
\(544\) −30.4519 −1.30561
\(545\) −5.69638 −0.244006
\(546\) 0 0
\(547\) 40.4919 1.73131 0.865654 0.500642i \(-0.166903\pi\)
0.865654 + 0.500642i \(0.166903\pi\)
\(548\) −93.6363 −3.99995
\(549\) 0 0
\(550\) −65.1390 −2.77753
\(551\) −24.2288 −1.03218
\(552\) 0 0
\(553\) −9.39785 −0.399637
\(554\) −15.2261 −0.646896
\(555\) 0 0
\(556\) 20.0702 0.851168
\(557\) −26.3904 −1.11820 −0.559098 0.829101i \(-0.688853\pi\)
−0.559098 + 0.829101i \(0.688853\pi\)
\(558\) 0 0
\(559\) −2.64456 −0.111853
\(560\) 6.48055 0.273853
\(561\) 0 0
\(562\) 70.2977 2.96533
\(563\) −30.2542 −1.27506 −0.637532 0.770424i \(-0.720045\pi\)
−0.637532 + 0.770424i \(0.720045\pi\)
\(564\) 0 0
\(565\) −4.32752 −0.182060
\(566\) 57.3927 2.41239
\(567\) 0 0
\(568\) −37.6192 −1.57847
\(569\) 4.56920 0.191551 0.0957754 0.995403i \(-0.469467\pi\)
0.0957754 + 0.995403i \(0.469467\pi\)
\(570\) 0 0
\(571\) −10.4991 −0.439375 −0.219687 0.975570i \(-0.570504\pi\)
−0.219687 + 0.975570i \(0.570504\pi\)
\(572\) −60.8877 −2.54584
\(573\) 0 0
\(574\) 6.81505 0.284455
\(575\) 31.6379 1.31939
\(576\) 0 0
\(577\) −3.62882 −0.151070 −0.0755349 0.997143i \(-0.524066\pi\)
−0.0755349 + 0.997143i \(0.524066\pi\)
\(578\) −35.5825 −1.48003
\(579\) 0 0
\(580\) −20.5622 −0.853798
\(581\) 14.2489 0.591143
\(582\) 0 0
\(583\) −39.2829 −1.62693
\(584\) −19.3446 −0.800483
\(585\) 0 0
\(586\) −42.3098 −1.74780
\(587\) 22.5623 0.931244 0.465622 0.884984i \(-0.345831\pi\)
0.465622 + 0.884984i \(0.345831\pi\)
\(588\) 0 0
\(589\) −3.79563 −0.156396
\(590\) 12.8339 0.528362
\(591\) 0 0
\(592\) −9.68175 −0.397917
\(593\) −26.6646 −1.09498 −0.547492 0.836811i \(-0.684417\pi\)
−0.547492 + 0.836811i \(0.684417\pi\)
\(594\) 0 0
\(595\) −1.01863 −0.0417599
\(596\) 13.3228 0.545724
\(597\) 0 0
\(598\) 41.0567 1.67893
\(599\) 29.4480 1.20321 0.601605 0.798793i \(-0.294528\pi\)
0.601605 + 0.798793i \(0.294528\pi\)
\(600\) 0 0
\(601\) −33.9235 −1.38377 −0.691884 0.722009i \(-0.743219\pi\)
−0.691884 + 0.722009i \(0.743219\pi\)
\(602\) 3.14493 0.128178
\(603\) 0 0
\(604\) −87.6579 −3.56675
\(605\) 8.05626 0.327534
\(606\) 0 0
\(607\) 0.484162 0.0196515 0.00982577 0.999952i \(-0.496872\pi\)
0.00982577 + 0.999952i \(0.496872\pi\)
\(608\) 49.9250 2.02473
\(609\) 0 0
\(610\) 3.09695 0.125392
\(611\) −20.4973 −0.829231
\(612\) 0 0
\(613\) −20.8655 −0.842750 −0.421375 0.906886i \(-0.638452\pi\)
−0.421375 + 0.906886i \(0.638452\pi\)
\(614\) −30.4140 −1.22741
\(615\) 0 0
\(616\) 44.2912 1.78454
\(617\) −13.7272 −0.552637 −0.276319 0.961066i \(-0.589115\pi\)
−0.276319 + 0.961066i \(0.589115\pi\)
\(618\) 0 0
\(619\) −45.3078 −1.82107 −0.910536 0.413429i \(-0.864331\pi\)
−0.910536 + 0.413429i \(0.864331\pi\)
\(620\) −3.22122 −0.129367
\(621\) 0 0
\(622\) 52.7690 2.11585
\(623\) 7.97569 0.319539
\(624\) 0 0
\(625\) 21.0297 0.841187
\(626\) 19.3716 0.774247
\(627\) 0 0
\(628\) −49.4290 −1.97243
\(629\) 1.52181 0.0606784
\(630\) 0 0
\(631\) 27.0302 1.07605 0.538027 0.842927i \(-0.319170\pi\)
0.538027 + 0.842927i \(0.319170\pi\)
\(632\) −77.5485 −3.08471
\(633\) 0 0
\(634\) 29.6228 1.17647
\(635\) 5.31229 0.210812
\(636\) 0 0
\(637\) 13.6769 0.541900
\(638\) −105.889 −4.19220
\(639\) 0 0
\(640\) 8.42214 0.332914
\(641\) 44.9994 1.77737 0.888684 0.458520i \(-0.151620\pi\)
0.888684 + 0.458520i \(0.151620\pi\)
\(642\) 0 0
\(643\) −43.2140 −1.70419 −0.852097 0.523384i \(-0.824669\pi\)
−0.852097 + 0.523384i \(0.824669\pi\)
\(644\) −35.1686 −1.38584
\(645\) 0 0
\(646\) −16.1918 −0.637059
\(647\) 17.6690 0.694642 0.347321 0.937746i \(-0.387091\pi\)
0.347321 + 0.937746i \(0.387091\pi\)
\(648\) 0 0
\(649\) 47.6051 1.86867
\(650\) 29.0392 1.13901
\(651\) 0 0
\(652\) −5.55456 −0.217533
\(653\) 38.1713 1.49376 0.746879 0.664960i \(-0.231551\pi\)
0.746879 + 0.664960i \(0.231551\pi\)
\(654\) 0 0
\(655\) −6.14876 −0.240252
\(656\) 30.5214 1.19166
\(657\) 0 0
\(658\) 24.3755 0.950258
\(659\) 13.7251 0.534655 0.267328 0.963606i \(-0.413859\pi\)
0.267328 + 0.963606i \(0.413859\pi\)
\(660\) 0 0
\(661\) 31.8679 1.23952 0.619759 0.784793i \(-0.287231\pi\)
0.619759 + 0.784793i \(0.287231\pi\)
\(662\) 31.3488 1.21840
\(663\) 0 0
\(664\) 117.578 4.56291
\(665\) 1.67002 0.0647607
\(666\) 0 0
\(667\) 51.4304 1.99139
\(668\) 61.5824 2.38270
\(669\) 0 0
\(670\) 20.6027 0.795952
\(671\) 11.4876 0.443475
\(672\) 0 0
\(673\) −33.3721 −1.28640 −0.643200 0.765698i \(-0.722394\pi\)
−0.643200 + 0.765698i \(0.722394\pi\)
\(674\) −76.5781 −2.94968
\(675\) 0 0
\(676\) −39.8125 −1.53125
\(677\) −36.3936 −1.39872 −0.699359 0.714770i \(-0.746531\pi\)
−0.699359 + 0.714770i \(0.746531\pi\)
\(678\) 0 0
\(679\) −4.34846 −0.166878
\(680\) −8.40549 −0.322336
\(681\) 0 0
\(682\) −16.5884 −0.635203
\(683\) 22.4689 0.859748 0.429874 0.902889i \(-0.358558\pi\)
0.429874 + 0.902889i \(0.358558\pi\)
\(684\) 0 0
\(685\) −9.43842 −0.360624
\(686\) −35.3750 −1.35063
\(687\) 0 0
\(688\) 14.0847 0.536974
\(689\) 17.5124 0.667171
\(690\) 0 0
\(691\) −19.7002 −0.749430 −0.374715 0.927140i \(-0.622260\pi\)
−0.374715 + 0.927140i \(0.622260\pi\)
\(692\) 83.9730 3.19217
\(693\) 0 0
\(694\) −5.25774 −0.199581
\(695\) 2.02305 0.0767388
\(696\) 0 0
\(697\) −4.79745 −0.181716
\(698\) −3.36055 −0.127199
\(699\) 0 0
\(700\) −24.8745 −0.940169
\(701\) −26.1013 −0.985833 −0.492917 0.870077i \(-0.664069\pi\)
−0.492917 + 0.870077i \(0.664069\pi\)
\(702\) 0 0
\(703\) −2.49496 −0.0940992
\(704\) 92.2690 3.47752
\(705\) 0 0
\(706\) 56.4197 2.12339
\(707\) 16.9060 0.635816
\(708\) 0 0
\(709\) 38.7834 1.45654 0.728270 0.685291i \(-0.240325\pi\)
0.728270 + 0.685291i \(0.240325\pi\)
\(710\) −6.19919 −0.232651
\(711\) 0 0
\(712\) 65.8133 2.46646
\(713\) 8.05697 0.301736
\(714\) 0 0
\(715\) −6.13740 −0.229526
\(716\) −0.541186 −0.0202251
\(717\) 0 0
\(718\) 46.8852 1.74974
\(719\) −28.2950 −1.05522 −0.527612 0.849485i \(-0.676913\pi\)
−0.527612 + 0.849485i \(0.676913\pi\)
\(720\) 0 0
\(721\) −13.2734 −0.494326
\(722\) −24.2607 −0.902890
\(723\) 0 0
\(724\) −92.7598 −3.44739
\(725\) 36.3764 1.35099
\(726\) 0 0
\(727\) −5.95426 −0.220831 −0.110416 0.993886i \(-0.535218\pi\)
−0.110416 + 0.993886i \(0.535218\pi\)
\(728\) −19.7452 −0.731805
\(729\) 0 0
\(730\) −3.18775 −0.117984
\(731\) −2.21388 −0.0818832
\(732\) 0 0
\(733\) 38.6216 1.42652 0.713260 0.700900i \(-0.247218\pi\)
0.713260 + 0.700900i \(0.247218\pi\)
\(734\) 75.9873 2.80474
\(735\) 0 0
\(736\) −105.976 −3.90631
\(737\) 76.4224 2.81506
\(738\) 0 0
\(739\) 52.1755 1.91931 0.959654 0.281183i \(-0.0907266\pi\)
0.959654 + 0.281183i \(0.0907266\pi\)
\(740\) −2.11739 −0.0778368
\(741\) 0 0
\(742\) −20.8259 −0.764544
\(743\) 10.6094 0.389223 0.194611 0.980880i \(-0.437655\pi\)
0.194611 + 0.980880i \(0.437655\pi\)
\(744\) 0 0
\(745\) 1.34292 0.0492009
\(746\) 60.0315 2.19791
\(747\) 0 0
\(748\) −50.9717 −1.86371
\(749\) −5.37546 −0.196415
\(750\) 0 0
\(751\) 1.96839 0.0718277 0.0359139 0.999355i \(-0.488566\pi\)
0.0359139 + 0.999355i \(0.488566\pi\)
\(752\) 109.167 3.98090
\(753\) 0 0
\(754\) 47.2059 1.71914
\(755\) −8.83581 −0.321568
\(756\) 0 0
\(757\) 42.9685 1.56172 0.780859 0.624707i \(-0.214782\pi\)
0.780859 + 0.624707i \(0.214782\pi\)
\(758\) 45.1082 1.63840
\(759\) 0 0
\(760\) 13.7806 0.499874
\(761\) 4.94581 0.179285 0.0896427 0.995974i \(-0.471427\pi\)
0.0896427 + 0.995974i \(0.471427\pi\)
\(762\) 0 0
\(763\) −11.2020 −0.405541
\(764\) −7.94423 −0.287412
\(765\) 0 0
\(766\) −40.3576 −1.45818
\(767\) −21.2225 −0.766301
\(768\) 0 0
\(769\) −20.4312 −0.736769 −0.368385 0.929674i \(-0.620089\pi\)
−0.368385 + 0.929674i \(0.620089\pi\)
\(770\) 7.29865 0.263025
\(771\) 0 0
\(772\) 48.9068 1.76019
\(773\) −5.93494 −0.213465 −0.106733 0.994288i \(-0.534039\pi\)
−0.106733 + 0.994288i \(0.534039\pi\)
\(774\) 0 0
\(775\) 5.69865 0.204702
\(776\) −35.8823 −1.28810
\(777\) 0 0
\(778\) 11.4488 0.410460
\(779\) 7.86529 0.281803
\(780\) 0 0
\(781\) −22.9949 −0.822821
\(782\) 34.3703 1.22908
\(783\) 0 0
\(784\) −72.8422 −2.60151
\(785\) −4.98238 −0.177829
\(786\) 0 0
\(787\) 26.9085 0.959184 0.479592 0.877492i \(-0.340785\pi\)
0.479592 + 0.877492i \(0.340785\pi\)
\(788\) 22.7886 0.811809
\(789\) 0 0
\(790\) −12.7790 −0.454658
\(791\) −8.51015 −0.302586
\(792\) 0 0
\(793\) −5.12123 −0.181860
\(794\) −70.1362 −2.48904
\(795\) 0 0
\(796\) −73.0821 −2.59033
\(797\) −0.527733 −0.0186933 −0.00934663 0.999956i \(-0.502975\pi\)
−0.00934663 + 0.999956i \(0.502975\pi\)
\(798\) 0 0
\(799\) −17.1592 −0.607048
\(800\) −74.9560 −2.65009
\(801\) 0 0
\(802\) −3.59305 −0.126875
\(803\) −11.8244 −0.417275
\(804\) 0 0
\(805\) −3.54495 −0.124943
\(806\) 7.39517 0.260484
\(807\) 0 0
\(808\) 139.504 4.90773
\(809\) 29.8806 1.05055 0.525273 0.850934i \(-0.323963\pi\)
0.525273 + 0.850934i \(0.323963\pi\)
\(810\) 0 0
\(811\) −0.241148 −0.00846783 −0.00423392 0.999991i \(-0.501348\pi\)
−0.00423392 + 0.999991i \(0.501348\pi\)
\(812\) −40.4359 −1.41902
\(813\) 0 0
\(814\) −10.9040 −0.382184
\(815\) −0.559893 −0.0196122
\(816\) 0 0
\(817\) 3.62959 0.126983
\(818\) 39.6766 1.38726
\(819\) 0 0
\(820\) 6.67501 0.233101
\(821\) −12.4693 −0.435183 −0.217592 0.976040i \(-0.569820\pi\)
−0.217592 + 0.976040i \(0.569820\pi\)
\(822\) 0 0
\(823\) −17.5737 −0.612582 −0.306291 0.951938i \(-0.599088\pi\)
−0.306291 + 0.951938i \(0.599088\pi\)
\(824\) −109.528 −3.81560
\(825\) 0 0
\(826\) 25.2380 0.878143
\(827\) 27.0166 0.939458 0.469729 0.882811i \(-0.344352\pi\)
0.469729 + 0.882811i \(0.344352\pi\)
\(828\) 0 0
\(829\) −34.3187 −1.19194 −0.595969 0.803008i \(-0.703232\pi\)
−0.595969 + 0.803008i \(0.703232\pi\)
\(830\) 19.3754 0.672531
\(831\) 0 0
\(832\) −41.1339 −1.42606
\(833\) 11.4496 0.396704
\(834\) 0 0
\(835\) 6.20743 0.214817
\(836\) 83.5668 2.89022
\(837\) 0 0
\(838\) 90.8645 3.13886
\(839\) −57.5893 −1.98820 −0.994101 0.108454i \(-0.965410\pi\)
−0.994101 + 0.108454i \(0.965410\pi\)
\(840\) 0 0
\(841\) 30.1332 1.03908
\(842\) 107.357 3.69977
\(843\) 0 0
\(844\) 0.246951 0.00850042
\(845\) −4.01305 −0.138053
\(846\) 0 0
\(847\) 15.8428 0.544364
\(848\) −93.2697 −3.20290
\(849\) 0 0
\(850\) 24.3100 0.833825
\(851\) 5.29604 0.181546
\(852\) 0 0
\(853\) 4.45708 0.152607 0.0763037 0.997085i \(-0.475688\pi\)
0.0763037 + 0.997085i \(0.475688\pi\)
\(854\) 6.09021 0.208403
\(855\) 0 0
\(856\) −44.3569 −1.51609
\(857\) 14.3007 0.488503 0.244252 0.969712i \(-0.421458\pi\)
0.244252 + 0.969712i \(0.421458\pi\)
\(858\) 0 0
\(859\) 56.6289 1.93215 0.966077 0.258256i \(-0.0831477\pi\)
0.966077 + 0.258256i \(0.0831477\pi\)
\(860\) 3.08031 0.105038
\(861\) 0 0
\(862\) 67.4685 2.29799
\(863\) 21.5070 0.732106 0.366053 0.930594i \(-0.380709\pi\)
0.366053 + 0.930594i \(0.380709\pi\)
\(864\) 0 0
\(865\) 8.46437 0.287797
\(866\) −83.6392 −2.84217
\(867\) 0 0
\(868\) −6.33459 −0.215010
\(869\) −47.4018 −1.60800
\(870\) 0 0
\(871\) −34.0694 −1.15440
\(872\) −92.4362 −3.13028
\(873\) 0 0
\(874\) −56.3492 −1.90604
\(875\) −5.15751 −0.174356
\(876\) 0 0
\(877\) −9.70284 −0.327642 −0.163821 0.986490i \(-0.552382\pi\)
−0.163821 + 0.986490i \(0.552382\pi\)
\(878\) −11.1098 −0.374936
\(879\) 0 0
\(880\) 32.6873 1.10189
\(881\) 10.7139 0.360960 0.180480 0.983579i \(-0.442235\pi\)
0.180480 + 0.983579i \(0.442235\pi\)
\(882\) 0 0
\(883\) 16.9553 0.570592 0.285296 0.958440i \(-0.407908\pi\)
0.285296 + 0.958440i \(0.407908\pi\)
\(884\) 22.7234 0.764270
\(885\) 0 0
\(886\) −53.5715 −1.79977
\(887\) 13.5417 0.454687 0.227343 0.973815i \(-0.426996\pi\)
0.227343 + 0.973815i \(0.426996\pi\)
\(888\) 0 0
\(889\) 10.4467 0.350371
\(890\) 10.8452 0.363533
\(891\) 0 0
\(892\) −114.049 −3.81865
\(893\) 28.1320 0.941401
\(894\) 0 0
\(895\) −0.0545509 −0.00182344
\(896\) 16.5623 0.553307
\(897\) 0 0
\(898\) 42.1511 1.40660
\(899\) 9.26368 0.308961
\(900\) 0 0
\(901\) 14.6604 0.488409
\(902\) 34.3744 1.14454
\(903\) 0 0
\(904\) −70.2234 −2.33560
\(905\) −9.35007 −0.310807
\(906\) 0 0
\(907\) 6.47529 0.215008 0.107504 0.994205i \(-0.465714\pi\)
0.107504 + 0.994205i \(0.465714\pi\)
\(908\) 133.867 4.44254
\(909\) 0 0
\(910\) −3.25376 −0.107861
\(911\) 32.6698 1.08240 0.541199 0.840895i \(-0.317971\pi\)
0.541199 + 0.840895i \(0.317971\pi\)
\(912\) 0 0
\(913\) 71.8700 2.37855
\(914\) 38.4847 1.27296
\(915\) 0 0
\(916\) −84.8928 −2.80494
\(917\) −12.0916 −0.399301
\(918\) 0 0
\(919\) −1.45226 −0.0479055 −0.0239527 0.999713i \(-0.507625\pi\)
−0.0239527 + 0.999713i \(0.507625\pi\)
\(920\) −29.2520 −0.964408
\(921\) 0 0
\(922\) 24.7399 0.814765
\(923\) 10.2512 0.337422
\(924\) 0 0
\(925\) 3.74586 0.123163
\(926\) 26.6207 0.874812
\(927\) 0 0
\(928\) −121.848 −3.99985
\(929\) −37.4415 −1.22842 −0.614208 0.789144i \(-0.710524\pi\)
−0.614208 + 0.789144i \(0.710524\pi\)
\(930\) 0 0
\(931\) −18.7712 −0.615202
\(932\) 42.6630 1.39747
\(933\) 0 0
\(934\) −50.7345 −1.66008
\(935\) −5.13789 −0.168027
\(936\) 0 0
\(937\) −15.6947 −0.512724 −0.256362 0.966581i \(-0.582524\pi\)
−0.256362 + 0.966581i \(0.582524\pi\)
\(938\) 40.5156 1.32288
\(939\) 0 0
\(940\) 23.8747 0.778706
\(941\) 25.7719 0.840140 0.420070 0.907492i \(-0.362006\pi\)
0.420070 + 0.907492i \(0.362006\pi\)
\(942\) 0 0
\(943\) −16.6956 −0.543684
\(944\) 113.029 3.67879
\(945\) 0 0
\(946\) 15.8627 0.515742
\(947\) −53.4780 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(948\) 0 0
\(949\) 5.27137 0.171116
\(950\) −39.8555 −1.29308
\(951\) 0 0
\(952\) −16.5295 −0.535725
\(953\) −24.6066 −0.797085 −0.398542 0.917150i \(-0.630484\pi\)
−0.398542 + 0.917150i \(0.630484\pi\)
\(954\) 0 0
\(955\) −0.800768 −0.0259122
\(956\) 5.15050 0.166579
\(957\) 0 0
\(958\) −88.6707 −2.86482
\(959\) −18.5608 −0.599360
\(960\) 0 0
\(961\) −29.5488 −0.953186
\(962\) 4.86102 0.156726
\(963\) 0 0
\(964\) 96.5805 3.11065
\(965\) 4.92974 0.158694
\(966\) 0 0
\(967\) −22.5510 −0.725192 −0.362596 0.931946i \(-0.618110\pi\)
−0.362596 + 0.931946i \(0.618110\pi\)
\(968\) 130.730 4.20183
\(969\) 0 0
\(970\) −5.91297 −0.189854
\(971\) 2.61563 0.0839396 0.0419698 0.999119i \(-0.486637\pi\)
0.0419698 + 0.999119i \(0.486637\pi\)
\(972\) 0 0
\(973\) 3.97837 0.127541
\(974\) −98.9477 −3.17049
\(975\) 0 0
\(976\) 27.2752 0.873059
\(977\) 44.8697 1.43551 0.717755 0.696296i \(-0.245170\pi\)
0.717755 + 0.696296i \(0.245170\pi\)
\(978\) 0 0
\(979\) 40.2286 1.28571
\(980\) −15.9305 −0.508882
\(981\) 0 0
\(982\) −41.6536 −1.32922
\(983\) 27.0373 0.862355 0.431177 0.902267i \(-0.358098\pi\)
0.431177 + 0.902267i \(0.358098\pi\)
\(984\) 0 0
\(985\) 2.29706 0.0731903
\(986\) 39.5181 1.25851
\(987\) 0 0
\(988\) −37.2544 −1.18522
\(989\) −7.70452 −0.244989
\(990\) 0 0
\(991\) −29.3172 −0.931293 −0.465646 0.884971i \(-0.654178\pi\)
−0.465646 + 0.884971i \(0.654178\pi\)
\(992\) −19.0884 −0.606058
\(993\) 0 0
\(994\) −12.1908 −0.386669
\(995\) −7.36658 −0.233536
\(996\) 0 0
\(997\) 57.2679 1.81369 0.906847 0.421460i \(-0.138482\pi\)
0.906847 + 0.421460i \(0.138482\pi\)
\(998\) −9.71698 −0.307585
\(999\) 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.1.20 yes 20
3.2 odd 2 2151.2.a.j.1.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.1 20 3.2 odd 2
2151.2.a.k.1.20 yes 20 1.1 even 1 trivial