Properties

Label 2151.2.a.k.1.13
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 1.13
Root \(0.946263\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.946263 q^{2} -1.10459 q^{4} +4.23503 q^{5} +2.19590 q^{7} -2.93776 q^{8} +O(q^{10})\) \(q+0.946263 q^{2} -1.10459 q^{4} +4.23503 q^{5} +2.19590 q^{7} -2.93776 q^{8} +4.00746 q^{10} -2.38110 q^{11} -5.96438 q^{13} +2.07790 q^{14} -0.570718 q^{16} +4.21712 q^{17} +5.07539 q^{19} -4.67796 q^{20} -2.25315 q^{22} +0.949546 q^{23} +12.9355 q^{25} -5.64388 q^{26} -2.42556 q^{28} +0.816897 q^{29} +9.04308 q^{31} +5.33546 q^{32} +3.99051 q^{34} +9.29972 q^{35} +9.83456 q^{37} +4.80266 q^{38} -12.4415 q^{40} +4.51363 q^{41} -4.33172 q^{43} +2.63013 q^{44} +0.898520 q^{46} -6.16222 q^{47} -2.17802 q^{49} +12.2404 q^{50} +6.58817 q^{52} -8.87566 q^{53} -10.0840 q^{55} -6.45102 q^{56} +0.773000 q^{58} +11.5005 q^{59} -2.93819 q^{61} +8.55714 q^{62} +6.19019 q^{64} -25.2594 q^{65} +9.58957 q^{67} -4.65817 q^{68} +8.79998 q^{70} -3.41462 q^{71} -7.39326 q^{73} +9.30609 q^{74} -5.60621 q^{76} -5.22867 q^{77} -8.43221 q^{79} -2.41701 q^{80} +4.27108 q^{82} -1.97759 q^{83} +17.8596 q^{85} -4.09895 q^{86} +6.99510 q^{88} +15.1171 q^{89} -13.0972 q^{91} -1.04885 q^{92} -5.83108 q^{94} +21.4945 q^{95} +3.33650 q^{97} -2.06098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.946263 0.669109 0.334555 0.942376i \(-0.391414\pi\)
0.334555 + 0.942376i \(0.391414\pi\)
\(3\) 0 0
\(4\) −1.10459 −0.552293
\(5\) 4.23503 1.89396 0.946982 0.321286i \(-0.104115\pi\)
0.946982 + 0.321286i \(0.104115\pi\)
\(6\) 0 0
\(7\) 2.19590 0.829973 0.414986 0.909828i \(-0.363786\pi\)
0.414986 + 0.909828i \(0.363786\pi\)
\(8\) −2.93776 −1.03865
\(9\) 0 0
\(10\) 4.00746 1.26727
\(11\) −2.38110 −0.717930 −0.358965 0.933351i \(-0.616870\pi\)
−0.358965 + 0.933351i \(0.616870\pi\)
\(12\) 0 0
\(13\) −5.96438 −1.65422 −0.827111 0.562038i \(-0.810017\pi\)
−0.827111 + 0.562038i \(0.810017\pi\)
\(14\) 2.07790 0.555342
\(15\) 0 0
\(16\) −0.570718 −0.142680
\(17\) 4.21712 1.02280 0.511401 0.859342i \(-0.329127\pi\)
0.511401 + 0.859342i \(0.329127\pi\)
\(18\) 0 0
\(19\) 5.07539 1.16438 0.582188 0.813054i \(-0.302197\pi\)
0.582188 + 0.813054i \(0.302197\pi\)
\(20\) −4.67796 −1.04602
\(21\) 0 0
\(22\) −2.25315 −0.480373
\(23\) 0.949546 0.197994 0.0989970 0.995088i \(-0.468437\pi\)
0.0989970 + 0.995088i \(0.468437\pi\)
\(24\) 0 0
\(25\) 12.9355 2.58710
\(26\) −5.64388 −1.10686
\(27\) 0 0
\(28\) −2.42556 −0.458388
\(29\) 0.816897 0.151694 0.0758470 0.997119i \(-0.475834\pi\)
0.0758470 + 0.997119i \(0.475834\pi\)
\(30\) 0 0
\(31\) 9.04308 1.62419 0.812093 0.583528i \(-0.198328\pi\)
0.812093 + 0.583528i \(0.198328\pi\)
\(32\) 5.33546 0.943185
\(33\) 0 0
\(34\) 3.99051 0.684366
\(35\) 9.29972 1.57194
\(36\) 0 0
\(37\) 9.83456 1.61679 0.808396 0.588639i \(-0.200336\pi\)
0.808396 + 0.588639i \(0.200336\pi\)
\(38\) 4.80266 0.779094
\(39\) 0 0
\(40\) −12.4415 −1.96717
\(41\) 4.51363 0.704911 0.352455 0.935829i \(-0.385347\pi\)
0.352455 + 0.935829i \(0.385347\pi\)
\(42\) 0 0
\(43\) −4.33172 −0.660581 −0.330290 0.943879i \(-0.607147\pi\)
−0.330290 + 0.943879i \(0.607147\pi\)
\(44\) 2.63013 0.396507
\(45\) 0 0
\(46\) 0.898520 0.132480
\(47\) −6.16222 −0.898852 −0.449426 0.893317i \(-0.648372\pi\)
−0.449426 + 0.893317i \(0.648372\pi\)
\(48\) 0 0
\(49\) −2.17802 −0.311145
\(50\) 12.2404 1.73105
\(51\) 0 0
\(52\) 6.58817 0.913615
\(53\) −8.87566 −1.21917 −0.609583 0.792722i \(-0.708663\pi\)
−0.609583 + 0.792722i \(0.708663\pi\)
\(54\) 0 0
\(55\) −10.0840 −1.35973
\(56\) −6.45102 −0.862054
\(57\) 0 0
\(58\) 0.773000 0.101500
\(59\) 11.5005 1.49723 0.748616 0.663004i \(-0.230719\pi\)
0.748616 + 0.663004i \(0.230719\pi\)
\(60\) 0 0
\(61\) −2.93819 −0.376197 −0.188098 0.982150i \(-0.560232\pi\)
−0.188098 + 0.982150i \(0.560232\pi\)
\(62\) 8.55714 1.08676
\(63\) 0 0
\(64\) 6.19019 0.773773
\(65\) −25.2594 −3.13304
\(66\) 0 0
\(67\) 9.58957 1.17155 0.585776 0.810473i \(-0.300790\pi\)
0.585776 + 0.810473i \(0.300790\pi\)
\(68\) −4.65817 −0.564886
\(69\) 0 0
\(70\) 8.79998 1.05180
\(71\) −3.41462 −0.405241 −0.202620 0.979257i \(-0.564946\pi\)
−0.202620 + 0.979257i \(0.564946\pi\)
\(72\) 0 0
\(73\) −7.39326 −0.865315 −0.432658 0.901558i \(-0.642424\pi\)
−0.432658 + 0.901558i \(0.642424\pi\)
\(74\) 9.30609 1.08181
\(75\) 0 0
\(76\) −5.60621 −0.643076
\(77\) −5.22867 −0.595862
\(78\) 0 0
\(79\) −8.43221 −0.948697 −0.474349 0.880337i \(-0.657316\pi\)
−0.474349 + 0.880337i \(0.657316\pi\)
\(80\) −2.41701 −0.270230
\(81\) 0 0
\(82\) 4.27108 0.471662
\(83\) −1.97759 −0.217069 −0.108535 0.994093i \(-0.534616\pi\)
−0.108535 + 0.994093i \(0.534616\pi\)
\(84\) 0 0
\(85\) 17.8596 1.93715
\(86\) −4.09895 −0.442001
\(87\) 0 0
\(88\) 6.99510 0.745680
\(89\) 15.1171 1.60241 0.801206 0.598388i \(-0.204192\pi\)
0.801206 + 0.598388i \(0.204192\pi\)
\(90\) 0 0
\(91\) −13.0972 −1.37296
\(92\) −1.04885 −0.109351
\(93\) 0 0
\(94\) −5.83108 −0.601430
\(95\) 21.4945 2.20529
\(96\) 0 0
\(97\) 3.33650 0.338770 0.169385 0.985550i \(-0.445822\pi\)
0.169385 + 0.985550i \(0.445822\pi\)
\(98\) −2.06098 −0.208190
\(99\) 0 0
\(100\) −14.2884 −1.42884
\(101\) 13.4637 1.33968 0.669842 0.742503i \(-0.266362\pi\)
0.669842 + 0.742503i \(0.266362\pi\)
\(102\) 0 0
\(103\) −19.6498 −1.93615 −0.968075 0.250659i \(-0.919353\pi\)
−0.968075 + 0.250659i \(0.919353\pi\)
\(104\) 17.5219 1.71816
\(105\) 0 0
\(106\) −8.39871 −0.815755
\(107\) −6.99976 −0.676693 −0.338346 0.941022i \(-0.609868\pi\)
−0.338346 + 0.941022i \(0.609868\pi\)
\(108\) 0 0
\(109\) 6.41477 0.614423 0.307212 0.951641i \(-0.400604\pi\)
0.307212 + 0.951641i \(0.400604\pi\)
\(110\) −9.54217 −0.909810
\(111\) 0 0
\(112\) −1.25324 −0.118420
\(113\) −1.55854 −0.146615 −0.0733077 0.997309i \(-0.523356\pi\)
−0.0733077 + 0.997309i \(0.523356\pi\)
\(114\) 0 0
\(115\) 4.02136 0.374994
\(116\) −0.902333 −0.0837795
\(117\) 0 0
\(118\) 10.8825 1.00181
\(119\) 9.26038 0.848898
\(120\) 0 0
\(121\) −5.33035 −0.484577
\(122\) −2.78030 −0.251717
\(123\) 0 0
\(124\) −9.98886 −0.897026
\(125\) 33.6071 3.00591
\(126\) 0 0
\(127\) −11.3630 −1.00830 −0.504150 0.863616i \(-0.668194\pi\)
−0.504150 + 0.863616i \(0.668194\pi\)
\(128\) −4.81338 −0.425446
\(129\) 0 0
\(130\) −23.9020 −2.09634
\(131\) 16.0798 1.40490 0.702451 0.711732i \(-0.252089\pi\)
0.702451 + 0.711732i \(0.252089\pi\)
\(132\) 0 0
\(133\) 11.1451 0.966400
\(134\) 9.07426 0.783896
\(135\) 0 0
\(136\) −12.3889 −1.06234
\(137\) 15.6125 1.33386 0.666932 0.745119i \(-0.267607\pi\)
0.666932 + 0.745119i \(0.267607\pi\)
\(138\) 0 0
\(139\) −15.7408 −1.33512 −0.667558 0.744557i \(-0.732661\pi\)
−0.667558 + 0.744557i \(0.732661\pi\)
\(140\) −10.2723 −0.868171
\(141\) 0 0
\(142\) −3.23113 −0.271150
\(143\) 14.2018 1.18762
\(144\) 0 0
\(145\) 3.45959 0.287303
\(146\) −6.99597 −0.578990
\(147\) 0 0
\(148\) −10.8631 −0.892943
\(149\) −22.7671 −1.86515 −0.932576 0.360973i \(-0.882445\pi\)
−0.932576 + 0.360973i \(0.882445\pi\)
\(150\) 0 0
\(151\) −11.1715 −0.909123 −0.454561 0.890715i \(-0.650204\pi\)
−0.454561 + 0.890715i \(0.650204\pi\)
\(152\) −14.9103 −1.20938
\(153\) 0 0
\(154\) −4.94770 −0.398697
\(155\) 38.2978 3.07615
\(156\) 0 0
\(157\) −0.305290 −0.0243648 −0.0121824 0.999926i \(-0.503878\pi\)
−0.0121824 + 0.999926i \(0.503878\pi\)
\(158\) −7.97909 −0.634782
\(159\) 0 0
\(160\) 22.5959 1.78636
\(161\) 2.08511 0.164330
\(162\) 0 0
\(163\) 6.47753 0.507360 0.253680 0.967288i \(-0.418359\pi\)
0.253680 + 0.967288i \(0.418359\pi\)
\(164\) −4.98569 −0.389317
\(165\) 0 0
\(166\) −1.87132 −0.145243
\(167\) 0.771956 0.0597358 0.0298679 0.999554i \(-0.490491\pi\)
0.0298679 + 0.999554i \(0.490491\pi\)
\(168\) 0 0
\(169\) 22.5739 1.73645
\(170\) 16.8999 1.29617
\(171\) 0 0
\(172\) 4.78475 0.364834
\(173\) −11.2432 −0.854803 −0.427401 0.904062i \(-0.640571\pi\)
−0.427401 + 0.904062i \(0.640571\pi\)
\(174\) 0 0
\(175\) 28.4051 2.14722
\(176\) 1.35894 0.102434
\(177\) 0 0
\(178\) 14.3048 1.07219
\(179\) −5.71268 −0.426986 −0.213493 0.976945i \(-0.568484\pi\)
−0.213493 + 0.976945i \(0.568484\pi\)
\(180\) 0 0
\(181\) −7.76105 −0.576875 −0.288437 0.957499i \(-0.593136\pi\)
−0.288437 + 0.957499i \(0.593136\pi\)
\(182\) −12.3934 −0.918660
\(183\) 0 0
\(184\) −2.78953 −0.205647
\(185\) 41.6497 3.06215
\(186\) 0 0
\(187\) −10.0414 −0.734300
\(188\) 6.80670 0.496430
\(189\) 0 0
\(190\) 20.3394 1.47558
\(191\) 19.9452 1.44318 0.721592 0.692319i \(-0.243411\pi\)
0.721592 + 0.692319i \(0.243411\pi\)
\(192\) 0 0
\(193\) −8.59376 −0.618592 −0.309296 0.950966i \(-0.600093\pi\)
−0.309296 + 0.950966i \(0.600093\pi\)
\(194\) 3.15721 0.226674
\(195\) 0 0
\(196\) 2.40581 0.171843
\(197\) −7.93505 −0.565349 −0.282674 0.959216i \(-0.591222\pi\)
−0.282674 + 0.959216i \(0.591222\pi\)
\(198\) 0 0
\(199\) −18.9031 −1.34001 −0.670004 0.742357i \(-0.733708\pi\)
−0.670004 + 0.742357i \(0.733708\pi\)
\(200\) −38.0014 −2.68710
\(201\) 0 0
\(202\) 12.7402 0.896395
\(203\) 1.79383 0.125902
\(204\) 0 0
\(205\) 19.1154 1.33508
\(206\) −18.5939 −1.29550
\(207\) 0 0
\(208\) 3.40398 0.236024
\(209\) −12.0850 −0.835939
\(210\) 0 0
\(211\) 7.18073 0.494342 0.247171 0.968972i \(-0.420499\pi\)
0.247171 + 0.968972i \(0.420499\pi\)
\(212\) 9.80393 0.673337
\(213\) 0 0
\(214\) −6.62362 −0.452781
\(215\) −18.3450 −1.25112
\(216\) 0 0
\(217\) 19.8577 1.34803
\(218\) 6.07006 0.411116
\(219\) 0 0
\(220\) 11.1387 0.750971
\(221\) −25.1525 −1.69194
\(222\) 0 0
\(223\) −16.8849 −1.13070 −0.565348 0.824852i \(-0.691258\pi\)
−0.565348 + 0.824852i \(0.691258\pi\)
\(224\) 11.7161 0.782818
\(225\) 0 0
\(226\) −1.47479 −0.0981018
\(227\) −18.1109 −1.20207 −0.601033 0.799224i \(-0.705244\pi\)
−0.601033 + 0.799224i \(0.705244\pi\)
\(228\) 0 0
\(229\) 28.4187 1.87796 0.938980 0.343973i \(-0.111773\pi\)
0.938980 + 0.343973i \(0.111773\pi\)
\(230\) 3.80526 0.250912
\(231\) 0 0
\(232\) −2.39984 −0.157558
\(233\) 17.1517 1.12364 0.561821 0.827259i \(-0.310101\pi\)
0.561821 + 0.827259i \(0.310101\pi\)
\(234\) 0 0
\(235\) −26.0972 −1.70239
\(236\) −12.7032 −0.826910
\(237\) 0 0
\(238\) 8.76276 0.568005
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 2.97567 0.191680 0.0958399 0.995397i \(-0.469446\pi\)
0.0958399 + 0.995397i \(0.469446\pi\)
\(242\) −5.04391 −0.324235
\(243\) 0 0
\(244\) 3.24548 0.207771
\(245\) −9.22397 −0.589298
\(246\) 0 0
\(247\) −30.2716 −1.92614
\(248\) −26.5664 −1.68697
\(249\) 0 0
\(250\) 31.8012 2.01128
\(251\) −15.0788 −0.951767 −0.475883 0.879508i \(-0.657872\pi\)
−0.475883 + 0.879508i \(0.657872\pi\)
\(252\) 0 0
\(253\) −2.26097 −0.142146
\(254\) −10.7524 −0.674663
\(255\) 0 0
\(256\) −16.9351 −1.05844
\(257\) 2.15365 0.134341 0.0671704 0.997742i \(-0.478603\pi\)
0.0671704 + 0.997742i \(0.478603\pi\)
\(258\) 0 0
\(259\) 21.5957 1.34189
\(260\) 27.9011 1.73036
\(261\) 0 0
\(262\) 15.2158 0.940033
\(263\) −5.70756 −0.351943 −0.175972 0.984395i \(-0.556307\pi\)
−0.175972 + 0.984395i \(0.556307\pi\)
\(264\) 0 0
\(265\) −37.5887 −2.30906
\(266\) 10.5462 0.646627
\(267\) 0 0
\(268\) −10.5925 −0.647040
\(269\) −5.16679 −0.315025 −0.157513 0.987517i \(-0.550347\pi\)
−0.157513 + 0.987517i \(0.550347\pi\)
\(270\) 0 0
\(271\) 3.13749 0.190589 0.0952943 0.995449i \(-0.469621\pi\)
0.0952943 + 0.995449i \(0.469621\pi\)
\(272\) −2.40679 −0.145933
\(273\) 0 0
\(274\) 14.7735 0.892500
\(275\) −30.8008 −1.85736
\(276\) 0 0
\(277\) −18.9634 −1.13940 −0.569701 0.821852i \(-0.692941\pi\)
−0.569701 + 0.821852i \(0.692941\pi\)
\(278\) −14.8949 −0.893339
\(279\) 0 0
\(280\) −27.3203 −1.63270
\(281\) 16.1906 0.965853 0.482927 0.875661i \(-0.339574\pi\)
0.482927 + 0.875661i \(0.339574\pi\)
\(282\) 0 0
\(283\) −5.53384 −0.328953 −0.164476 0.986381i \(-0.552593\pi\)
−0.164476 + 0.986381i \(0.552593\pi\)
\(284\) 3.77174 0.223812
\(285\) 0 0
\(286\) 13.4387 0.794644
\(287\) 9.91149 0.585057
\(288\) 0 0
\(289\) 0.784104 0.0461238
\(290\) 3.27368 0.192237
\(291\) 0 0
\(292\) 8.16649 0.477907
\(293\) −30.0543 −1.75579 −0.877896 0.478851i \(-0.841053\pi\)
−0.877896 + 0.478851i \(0.841053\pi\)
\(294\) 0 0
\(295\) 48.7048 2.83570
\(296\) −28.8915 −1.67929
\(297\) 0 0
\(298\) −21.5437 −1.24799
\(299\) −5.66346 −0.327526
\(300\) 0 0
\(301\) −9.51203 −0.548264
\(302\) −10.5712 −0.608302
\(303\) 0 0
\(304\) −2.89662 −0.166133
\(305\) −12.4433 −0.712504
\(306\) 0 0
\(307\) 12.9061 0.736591 0.368296 0.929709i \(-0.379941\pi\)
0.368296 + 0.929709i \(0.379941\pi\)
\(308\) 5.77551 0.329090
\(309\) 0 0
\(310\) 36.2398 2.05828
\(311\) 19.6943 1.11676 0.558381 0.829585i \(-0.311423\pi\)
0.558381 + 0.829585i \(0.311423\pi\)
\(312\) 0 0
\(313\) −14.4786 −0.818380 −0.409190 0.912449i \(-0.634189\pi\)
−0.409190 + 0.912449i \(0.634189\pi\)
\(314\) −0.288885 −0.0163027
\(315\) 0 0
\(316\) 9.31410 0.523959
\(317\) 0.816418 0.0458546 0.0229273 0.999737i \(-0.492701\pi\)
0.0229273 + 0.999737i \(0.492701\pi\)
\(318\) 0 0
\(319\) −1.94512 −0.108906
\(320\) 26.2157 1.46550
\(321\) 0 0
\(322\) 1.97306 0.109954
\(323\) 21.4036 1.19093
\(324\) 0 0
\(325\) −77.1523 −4.27964
\(326\) 6.12945 0.339479
\(327\) 0 0
\(328\) −13.2599 −0.732158
\(329\) −13.5316 −0.746023
\(330\) 0 0
\(331\) −10.6673 −0.586326 −0.293163 0.956063i \(-0.594708\pi\)
−0.293163 + 0.956063i \(0.594708\pi\)
\(332\) 2.18442 0.119886
\(333\) 0 0
\(334\) 0.730474 0.0399698
\(335\) 40.6121 2.21888
\(336\) 0 0
\(337\) 27.6962 1.50871 0.754355 0.656466i \(-0.227950\pi\)
0.754355 + 0.656466i \(0.227950\pi\)
\(338\) 21.3608 1.16188
\(339\) 0 0
\(340\) −19.7275 −1.06987
\(341\) −21.5325 −1.16605
\(342\) 0 0
\(343\) −20.1540 −1.08821
\(344\) 12.7255 0.686114
\(345\) 0 0
\(346\) −10.6390 −0.571956
\(347\) 18.4956 0.992898 0.496449 0.868066i \(-0.334637\pi\)
0.496449 + 0.868066i \(0.334637\pi\)
\(348\) 0 0
\(349\) −24.7763 −1.32625 −0.663123 0.748511i \(-0.730769\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(350\) 26.8787 1.43673
\(351\) 0 0
\(352\) −12.7043 −0.677140
\(353\) −0.00689704 −0.000367092 0 −0.000183546 1.00000i \(-0.500058\pi\)
−0.000183546 1.00000i \(0.500058\pi\)
\(354\) 0 0
\(355\) −14.4610 −0.767512
\(356\) −16.6982 −0.885001
\(357\) 0 0
\(358\) −5.40570 −0.285700
\(359\) −14.7259 −0.777202 −0.388601 0.921406i \(-0.627041\pi\)
−0.388601 + 0.921406i \(0.627041\pi\)
\(360\) 0 0
\(361\) 6.75963 0.355770
\(362\) −7.34400 −0.385992
\(363\) 0 0
\(364\) 14.4670 0.758276
\(365\) −31.3107 −1.63888
\(366\) 0 0
\(367\) −10.2522 −0.535158 −0.267579 0.963536i \(-0.586224\pi\)
−0.267579 + 0.963536i \(0.586224\pi\)
\(368\) −0.541923 −0.0282497
\(369\) 0 0
\(370\) 39.4116 2.04891
\(371\) −19.4901 −1.01187
\(372\) 0 0
\(373\) 22.7071 1.17573 0.587866 0.808959i \(-0.299968\pi\)
0.587866 + 0.808959i \(0.299968\pi\)
\(374\) −9.50181 −0.491327
\(375\) 0 0
\(376\) 18.1031 0.933596
\(377\) −4.87229 −0.250936
\(378\) 0 0
\(379\) −22.6893 −1.16547 −0.582736 0.812661i \(-0.698018\pi\)
−0.582736 + 0.812661i \(0.698018\pi\)
\(380\) −23.7425 −1.21796
\(381\) 0 0
\(382\) 18.8734 0.965647
\(383\) −28.2456 −1.44328 −0.721640 0.692268i \(-0.756611\pi\)
−0.721640 + 0.692268i \(0.756611\pi\)
\(384\) 0 0
\(385\) −22.1436 −1.12854
\(386\) −8.13196 −0.413906
\(387\) 0 0
\(388\) −3.68545 −0.187100
\(389\) 30.2010 1.53125 0.765627 0.643285i \(-0.222429\pi\)
0.765627 + 0.643285i \(0.222429\pi\)
\(390\) 0 0
\(391\) 4.00435 0.202509
\(392\) 6.39848 0.323172
\(393\) 0 0
\(394\) −7.50864 −0.378280
\(395\) −35.7107 −1.79680
\(396\) 0 0
\(397\) −11.0056 −0.552354 −0.276177 0.961107i \(-0.589068\pi\)
−0.276177 + 0.961107i \(0.589068\pi\)
\(398\) −17.8873 −0.896612
\(399\) 0 0
\(400\) −7.38253 −0.369127
\(401\) 20.8723 1.04232 0.521158 0.853460i \(-0.325500\pi\)
0.521158 + 0.853460i \(0.325500\pi\)
\(402\) 0 0
\(403\) −53.9364 −2.68676
\(404\) −14.8718 −0.739898
\(405\) 0 0
\(406\) 1.69743 0.0842421
\(407\) −23.4171 −1.16074
\(408\) 0 0
\(409\) 28.2317 1.39597 0.697984 0.716113i \(-0.254081\pi\)
0.697984 + 0.716113i \(0.254081\pi\)
\(410\) 18.0882 0.893311
\(411\) 0 0
\(412\) 21.7049 1.06932
\(413\) 25.2539 1.24266
\(414\) 0 0
\(415\) −8.37517 −0.411121
\(416\) −31.8227 −1.56024
\(417\) 0 0
\(418\) −11.4356 −0.559335
\(419\) −17.3156 −0.845922 −0.422961 0.906148i \(-0.639009\pi\)
−0.422961 + 0.906148i \(0.639009\pi\)
\(420\) 0 0
\(421\) −14.0043 −0.682530 −0.341265 0.939967i \(-0.610855\pi\)
−0.341265 + 0.939967i \(0.610855\pi\)
\(422\) 6.79486 0.330769
\(423\) 0 0
\(424\) 26.0745 1.26629
\(425\) 54.5506 2.64609
\(426\) 0 0
\(427\) −6.45198 −0.312233
\(428\) 7.73184 0.373733
\(429\) 0 0
\(430\) −17.3592 −0.837133
\(431\) −24.8909 −1.19895 −0.599476 0.800393i \(-0.704624\pi\)
−0.599476 + 0.800393i \(0.704624\pi\)
\(432\) 0 0
\(433\) 18.6159 0.894624 0.447312 0.894378i \(-0.352381\pi\)
0.447312 + 0.894378i \(0.352381\pi\)
\(434\) 18.7906 0.901979
\(435\) 0 0
\(436\) −7.08566 −0.339342
\(437\) 4.81932 0.230539
\(438\) 0 0
\(439\) −32.6083 −1.55631 −0.778155 0.628072i \(-0.783844\pi\)
−0.778155 + 0.628072i \(0.783844\pi\)
\(440\) 29.6245 1.41229
\(441\) 0 0
\(442\) −23.8009 −1.13209
\(443\) 18.3996 0.874191 0.437096 0.899415i \(-0.356007\pi\)
0.437096 + 0.899415i \(0.356007\pi\)
\(444\) 0 0
\(445\) 64.0216 3.03491
\(446\) −15.9776 −0.756559
\(447\) 0 0
\(448\) 13.5930 0.642211
\(449\) 20.7087 0.977304 0.488652 0.872479i \(-0.337489\pi\)
0.488652 + 0.872479i \(0.337489\pi\)
\(450\) 0 0
\(451\) −10.7474 −0.506076
\(452\) 1.72155 0.0809747
\(453\) 0 0
\(454\) −17.1377 −0.804313
\(455\) −55.4671 −2.60034
\(456\) 0 0
\(457\) −12.5865 −0.588770 −0.294385 0.955687i \(-0.595115\pi\)
−0.294385 + 0.955687i \(0.595115\pi\)
\(458\) 26.8916 1.25656
\(459\) 0 0
\(460\) −4.44193 −0.207106
\(461\) −32.5030 −1.51382 −0.756909 0.653521i \(-0.773291\pi\)
−0.756909 + 0.653521i \(0.773291\pi\)
\(462\) 0 0
\(463\) −2.40154 −0.111609 −0.0558045 0.998442i \(-0.517772\pi\)
−0.0558045 + 0.998442i \(0.517772\pi\)
\(464\) −0.466218 −0.0216436
\(465\) 0 0
\(466\) 16.2300 0.751840
\(467\) −26.0409 −1.20503 −0.602514 0.798109i \(-0.705834\pi\)
−0.602514 + 0.798109i \(0.705834\pi\)
\(468\) 0 0
\(469\) 21.0578 0.972357
\(470\) −24.6948 −1.13909
\(471\) 0 0
\(472\) −33.7855 −1.55510
\(473\) 10.3143 0.474250
\(474\) 0 0
\(475\) 65.6528 3.01236
\(476\) −10.2289 −0.468840
\(477\) 0 0
\(478\) 0.946263 0.0432811
\(479\) 18.4214 0.841695 0.420847 0.907131i \(-0.361733\pi\)
0.420847 + 0.907131i \(0.361733\pi\)
\(480\) 0 0
\(481\) −58.6571 −2.67453
\(482\) 2.81577 0.128255
\(483\) 0 0
\(484\) 5.88783 0.267629
\(485\) 14.1302 0.641619
\(486\) 0 0
\(487\) 24.5135 1.11081 0.555406 0.831579i \(-0.312563\pi\)
0.555406 + 0.831579i \(0.312563\pi\)
\(488\) 8.63169 0.390738
\(489\) 0 0
\(490\) −8.72830 −0.394305
\(491\) 0.364931 0.0164691 0.00823454 0.999966i \(-0.497379\pi\)
0.00823454 + 0.999966i \(0.497379\pi\)
\(492\) 0 0
\(493\) 3.44495 0.155153
\(494\) −28.6449 −1.28880
\(495\) 0 0
\(496\) −5.16105 −0.231738
\(497\) −7.49817 −0.336339
\(498\) 0 0
\(499\) 2.24220 0.100375 0.0501874 0.998740i \(-0.484018\pi\)
0.0501874 + 0.998740i \(0.484018\pi\)
\(500\) −37.1220 −1.66014
\(501\) 0 0
\(502\) −14.2685 −0.636836
\(503\) −17.8943 −0.797868 −0.398934 0.916980i \(-0.630620\pi\)
−0.398934 + 0.916980i \(0.630620\pi\)
\(504\) 0 0
\(505\) 57.0191 2.53732
\(506\) −2.13947 −0.0951110
\(507\) 0 0
\(508\) 12.5514 0.556877
\(509\) −6.29346 −0.278953 −0.139476 0.990225i \(-0.544542\pi\)
−0.139476 + 0.990225i \(0.544542\pi\)
\(510\) 0 0
\(511\) −16.2349 −0.718188
\(512\) −6.39831 −0.282768
\(513\) 0 0
\(514\) 2.03792 0.0898887
\(515\) −83.2175 −3.66700
\(516\) 0 0
\(517\) 14.6729 0.645313
\(518\) 20.4352 0.897873
\(519\) 0 0
\(520\) 74.2058 3.25414
\(521\) 39.0853 1.71236 0.856178 0.516681i \(-0.172833\pi\)
0.856178 + 0.516681i \(0.172833\pi\)
\(522\) 0 0
\(523\) 15.0192 0.656745 0.328372 0.944548i \(-0.393500\pi\)
0.328372 + 0.944548i \(0.393500\pi\)
\(524\) −17.7616 −0.775917
\(525\) 0 0
\(526\) −5.40085 −0.235488
\(527\) 38.1358 1.66122
\(528\) 0 0
\(529\) −22.0984 −0.960798
\(530\) −35.5688 −1.54501
\(531\) 0 0
\(532\) −12.3107 −0.533736
\(533\) −26.9210 −1.16608
\(534\) 0 0
\(535\) −29.6442 −1.28163
\(536\) −28.1718 −1.21684
\(537\) 0 0
\(538\) −4.88915 −0.210786
\(539\) 5.18608 0.223380
\(540\) 0 0
\(541\) 3.27190 0.140670 0.0703349 0.997523i \(-0.477593\pi\)
0.0703349 + 0.997523i \(0.477593\pi\)
\(542\) 2.96889 0.127525
\(543\) 0 0
\(544\) 22.5003 0.964692
\(545\) 27.1668 1.16370
\(546\) 0 0
\(547\) −34.6602 −1.48196 −0.740981 0.671526i \(-0.765639\pi\)
−0.740981 + 0.671526i \(0.765639\pi\)
\(548\) −17.2453 −0.736684
\(549\) 0 0
\(550\) −29.1456 −1.24277
\(551\) 4.14608 0.176629
\(552\) 0 0
\(553\) −18.5163 −0.787393
\(554\) −17.9444 −0.762385
\(555\) 0 0
\(556\) 17.3871 0.737376
\(557\) 19.1211 0.810186 0.405093 0.914276i \(-0.367239\pi\)
0.405093 + 0.914276i \(0.367239\pi\)
\(558\) 0 0
\(559\) 25.8360 1.09275
\(560\) −5.30752 −0.224284
\(561\) 0 0
\(562\) 15.3206 0.646261
\(563\) 37.7459 1.59080 0.795399 0.606086i \(-0.207261\pi\)
0.795399 + 0.606086i \(0.207261\pi\)
\(564\) 0 0
\(565\) −6.60049 −0.277685
\(566\) −5.23647 −0.220105
\(567\) 0 0
\(568\) 10.0313 0.420905
\(569\) −3.37306 −0.141406 −0.0707031 0.997497i \(-0.522524\pi\)
−0.0707031 + 0.997497i \(0.522524\pi\)
\(570\) 0 0
\(571\) −15.5252 −0.649709 −0.324855 0.945764i \(-0.605315\pi\)
−0.324855 + 0.945764i \(0.605315\pi\)
\(572\) −15.6871 −0.655911
\(573\) 0 0
\(574\) 9.37888 0.391467
\(575\) 12.2829 0.512230
\(576\) 0 0
\(577\) −30.3253 −1.26246 −0.631229 0.775596i \(-0.717449\pi\)
−0.631229 + 0.775596i \(0.717449\pi\)
\(578\) 0.741969 0.0308618
\(579\) 0 0
\(580\) −3.82141 −0.158675
\(581\) −4.34260 −0.180161
\(582\) 0 0
\(583\) 21.1339 0.875275
\(584\) 21.7196 0.898763
\(585\) 0 0
\(586\) −28.4393 −1.17482
\(587\) −32.7853 −1.35319 −0.676597 0.736354i \(-0.736546\pi\)
−0.676597 + 0.736354i \(0.736546\pi\)
\(588\) 0 0
\(589\) 45.8972 1.89116
\(590\) 46.0876 1.89739
\(591\) 0 0
\(592\) −5.61277 −0.230683
\(593\) 35.8084 1.47047 0.735237 0.677810i \(-0.237071\pi\)
0.735237 + 0.677810i \(0.237071\pi\)
\(594\) 0 0
\(595\) 39.2180 1.60778
\(596\) 25.1482 1.03011
\(597\) 0 0
\(598\) −5.35912 −0.219151
\(599\) −0.743135 −0.0303637 −0.0151818 0.999885i \(-0.504833\pi\)
−0.0151818 + 0.999885i \(0.504833\pi\)
\(600\) 0 0
\(601\) −2.26360 −0.0923341 −0.0461671 0.998934i \(-0.514701\pi\)
−0.0461671 + 0.998934i \(0.514701\pi\)
\(602\) −9.00088 −0.366848
\(603\) 0 0
\(604\) 12.3399 0.502102
\(605\) −22.5742 −0.917772
\(606\) 0 0
\(607\) 34.9036 1.41670 0.708348 0.705864i \(-0.249441\pi\)
0.708348 + 0.705864i \(0.249441\pi\)
\(608\) 27.0796 1.09822
\(609\) 0 0
\(610\) −11.7747 −0.476743
\(611\) 36.7539 1.48690
\(612\) 0 0
\(613\) 14.0779 0.568603 0.284301 0.958735i \(-0.408238\pi\)
0.284301 + 0.958735i \(0.408238\pi\)
\(614\) 12.2126 0.492860
\(615\) 0 0
\(616\) 15.3605 0.618894
\(617\) −8.38378 −0.337518 −0.168759 0.985657i \(-0.553976\pi\)
−0.168759 + 0.985657i \(0.553976\pi\)
\(618\) 0 0
\(619\) −43.7765 −1.75952 −0.879762 0.475414i \(-0.842298\pi\)
−0.879762 + 0.475414i \(0.842298\pi\)
\(620\) −42.3032 −1.69894
\(621\) 0 0
\(622\) 18.6360 0.747235
\(623\) 33.1957 1.32996
\(624\) 0 0
\(625\) 77.6498 3.10599
\(626\) −13.7006 −0.547585
\(627\) 0 0
\(628\) 0.337219 0.0134565
\(629\) 41.4735 1.65366
\(630\) 0 0
\(631\) 43.4013 1.72778 0.863890 0.503681i \(-0.168021\pi\)
0.863890 + 0.503681i \(0.168021\pi\)
\(632\) 24.7718 0.985368
\(633\) 0 0
\(634\) 0.772546 0.0306817
\(635\) −48.1225 −1.90969
\(636\) 0 0
\(637\) 12.9905 0.514703
\(638\) −1.84059 −0.0728698
\(639\) 0 0
\(640\) −20.3848 −0.805780
\(641\) 25.0623 0.989903 0.494951 0.868921i \(-0.335186\pi\)
0.494951 + 0.868921i \(0.335186\pi\)
\(642\) 0 0
\(643\) −10.5472 −0.415941 −0.207970 0.978135i \(-0.566686\pi\)
−0.207970 + 0.978135i \(0.566686\pi\)
\(644\) −2.30318 −0.0907581
\(645\) 0 0
\(646\) 20.2534 0.796859
\(647\) 9.65606 0.379619 0.189809 0.981821i \(-0.439213\pi\)
0.189809 + 0.981821i \(0.439213\pi\)
\(648\) 0 0
\(649\) −27.3838 −1.07491
\(650\) −73.0064 −2.86355
\(651\) 0 0
\(652\) −7.15499 −0.280211
\(653\) 24.7538 0.968692 0.484346 0.874876i \(-0.339058\pi\)
0.484346 + 0.874876i \(0.339058\pi\)
\(654\) 0 0
\(655\) 68.0986 2.66083
\(656\) −2.57601 −0.100576
\(657\) 0 0
\(658\) −12.8045 −0.499171
\(659\) 14.0114 0.545807 0.272904 0.962041i \(-0.412016\pi\)
0.272904 + 0.962041i \(0.412016\pi\)
\(660\) 0 0
\(661\) 31.6444 1.23082 0.615411 0.788206i \(-0.288990\pi\)
0.615411 + 0.788206i \(0.288990\pi\)
\(662\) −10.0940 −0.392316
\(663\) 0 0
\(664\) 5.80969 0.225460
\(665\) 47.1997 1.83033
\(666\) 0 0
\(667\) 0.775681 0.0300345
\(668\) −0.852692 −0.0329916
\(669\) 0 0
\(670\) 38.4298 1.48467
\(671\) 6.99614 0.270083
\(672\) 0 0
\(673\) −16.6489 −0.641769 −0.320885 0.947118i \(-0.603980\pi\)
−0.320885 + 0.947118i \(0.603980\pi\)
\(674\) 26.2079 1.00949
\(675\) 0 0
\(676\) −24.9348 −0.959030
\(677\) −30.6653 −1.17856 −0.589281 0.807928i \(-0.700589\pi\)
−0.589281 + 0.807928i \(0.700589\pi\)
\(678\) 0 0
\(679\) 7.32662 0.281170
\(680\) −52.4673 −2.01203
\(681\) 0 0
\(682\) −20.3754 −0.780215
\(683\) 16.6909 0.638660 0.319330 0.947644i \(-0.396542\pi\)
0.319330 + 0.947644i \(0.396542\pi\)
\(684\) 0 0
\(685\) 66.1194 2.52629
\(686\) −19.0710 −0.728135
\(687\) 0 0
\(688\) 2.47219 0.0942514
\(689\) 52.9378 2.01677
\(690\) 0 0
\(691\) −17.8963 −0.680807 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(692\) 12.4191 0.472102
\(693\) 0 0
\(694\) 17.5017 0.664357
\(695\) −66.6628 −2.52866
\(696\) 0 0
\(697\) 19.0345 0.720984
\(698\) −23.4449 −0.887403
\(699\) 0 0
\(700\) −31.3759 −1.18590
\(701\) 20.7781 0.784780 0.392390 0.919799i \(-0.371648\pi\)
0.392390 + 0.919799i \(0.371648\pi\)
\(702\) 0 0
\(703\) 49.9143 1.88255
\(704\) −14.7395 −0.555515
\(705\) 0 0
\(706\) −0.00652641 −0.000245625 0
\(707\) 29.5649 1.11190
\(708\) 0 0
\(709\) −28.7680 −1.08041 −0.540203 0.841535i \(-0.681652\pi\)
−0.540203 + 0.841535i \(0.681652\pi\)
\(710\) −13.6839 −0.513549
\(711\) 0 0
\(712\) −44.4104 −1.66435
\(713\) 8.58682 0.321579
\(714\) 0 0
\(715\) 60.1451 2.24930
\(716\) 6.31014 0.235821
\(717\) 0 0
\(718\) −13.9345 −0.520033
\(719\) −11.7648 −0.438754 −0.219377 0.975640i \(-0.570402\pi\)
−0.219377 + 0.975640i \(0.570402\pi\)
\(720\) 0 0
\(721\) −43.1490 −1.60695
\(722\) 6.39639 0.238049
\(723\) 0 0
\(724\) 8.57275 0.318604
\(725\) 10.5670 0.392448
\(726\) 0 0
\(727\) 5.53649 0.205337 0.102668 0.994716i \(-0.467262\pi\)
0.102668 + 0.994716i \(0.467262\pi\)
\(728\) 38.4764 1.42603
\(729\) 0 0
\(730\) −29.6282 −1.09659
\(731\) −18.2674 −0.675643
\(732\) 0 0
\(733\) −6.44675 −0.238116 −0.119058 0.992887i \(-0.537987\pi\)
−0.119058 + 0.992887i \(0.537987\pi\)
\(734\) −9.70124 −0.358079
\(735\) 0 0
\(736\) 5.06626 0.186745
\(737\) −22.8337 −0.841092
\(738\) 0 0
\(739\) −25.2860 −0.930161 −0.465081 0.885268i \(-0.653975\pi\)
−0.465081 + 0.885268i \(0.653975\pi\)
\(740\) −46.0057 −1.69120
\(741\) 0 0
\(742\) −18.4427 −0.677054
\(743\) −13.3483 −0.489700 −0.244850 0.969561i \(-0.578739\pi\)
−0.244850 + 0.969561i \(0.578739\pi\)
\(744\) 0 0
\(745\) −96.4194 −3.53253
\(746\) 21.4869 0.786692
\(747\) 0 0
\(748\) 11.0916 0.405549
\(749\) −15.3708 −0.561637
\(750\) 0 0
\(751\) 29.7601 1.08596 0.542981 0.839745i \(-0.317296\pi\)
0.542981 + 0.839745i \(0.317296\pi\)
\(752\) 3.51689 0.128248
\(753\) 0 0
\(754\) −4.61047 −0.167903
\(755\) −47.3116 −1.72185
\(756\) 0 0
\(757\) 9.87390 0.358873 0.179436 0.983770i \(-0.442573\pi\)
0.179436 + 0.983770i \(0.442573\pi\)
\(758\) −21.4701 −0.779828
\(759\) 0 0
\(760\) −63.1455 −2.29053
\(761\) −45.0481 −1.63299 −0.816495 0.577352i \(-0.804086\pi\)
−0.816495 + 0.577352i \(0.804086\pi\)
\(762\) 0 0
\(763\) 14.0862 0.509955
\(764\) −22.0312 −0.797060
\(765\) 0 0
\(766\) −26.7277 −0.965712
\(767\) −68.5931 −2.47675
\(768\) 0 0
\(769\) 35.2285 1.27037 0.635186 0.772359i \(-0.280923\pi\)
0.635186 + 0.772359i \(0.280923\pi\)
\(770\) −20.9537 −0.755117
\(771\) 0 0
\(772\) 9.49254 0.341644
\(773\) 47.4525 1.70675 0.853374 0.521300i \(-0.174553\pi\)
0.853374 + 0.521300i \(0.174553\pi\)
\(774\) 0 0
\(775\) 116.977 4.20193
\(776\) −9.80182 −0.351865
\(777\) 0 0
\(778\) 28.5781 1.02458
\(779\) 22.9085 0.820781
\(780\) 0 0
\(781\) 8.13056 0.290934
\(782\) 3.78917 0.135500
\(783\) 0 0
\(784\) 1.24303 0.0443941
\(785\) −1.29291 −0.0461461
\(786\) 0 0
\(787\) −5.48849 −0.195643 −0.0978217 0.995204i \(-0.531188\pi\)
−0.0978217 + 0.995204i \(0.531188\pi\)
\(788\) 8.76494 0.312238
\(789\) 0 0
\(790\) −33.7917 −1.20225
\(791\) −3.42241 −0.121687
\(792\) 0 0
\(793\) 17.5245 0.622313
\(794\) −10.4142 −0.369585
\(795\) 0 0
\(796\) 20.8801 0.740077
\(797\) −25.3956 −0.899558 −0.449779 0.893140i \(-0.648497\pi\)
−0.449779 + 0.893140i \(0.648497\pi\)
\(798\) 0 0
\(799\) −25.9868 −0.919348
\(800\) 69.0169 2.44012
\(801\) 0 0
\(802\) 19.7507 0.697423
\(803\) 17.6041 0.621235
\(804\) 0 0
\(805\) 8.83051 0.311234
\(806\) −51.0381 −1.79774
\(807\) 0 0
\(808\) −39.5530 −1.39147
\(809\) −44.6599 −1.57016 −0.785079 0.619396i \(-0.787378\pi\)
−0.785079 + 0.619396i \(0.787378\pi\)
\(810\) 0 0
\(811\) −29.4164 −1.03295 −0.516476 0.856302i \(-0.672756\pi\)
−0.516476 + 0.856302i \(0.672756\pi\)
\(812\) −1.98144 −0.0695347
\(813\) 0 0
\(814\) −22.1587 −0.776664
\(815\) 27.4326 0.960921
\(816\) 0 0
\(817\) −21.9852 −0.769164
\(818\) 26.7146 0.934055
\(819\) 0 0
\(820\) −21.1146 −0.737353
\(821\) −30.4469 −1.06260 −0.531302 0.847182i \(-0.678297\pi\)
−0.531302 + 0.847182i \(0.678297\pi\)
\(822\) 0 0
\(823\) −15.6616 −0.545928 −0.272964 0.962024i \(-0.588004\pi\)
−0.272964 + 0.962024i \(0.588004\pi\)
\(824\) 57.7263 2.01099
\(825\) 0 0
\(826\) 23.8968 0.831476
\(827\) −14.6578 −0.509701 −0.254850 0.966980i \(-0.582026\pi\)
−0.254850 + 0.966980i \(0.582026\pi\)
\(828\) 0 0
\(829\) −40.4386 −1.40449 −0.702246 0.711934i \(-0.747819\pi\)
−0.702246 + 0.711934i \(0.747819\pi\)
\(830\) −7.92512 −0.275085
\(831\) 0 0
\(832\) −36.9207 −1.27999
\(833\) −9.18496 −0.318240
\(834\) 0 0
\(835\) 3.26926 0.113137
\(836\) 13.3490 0.461683
\(837\) 0 0
\(838\) −16.3851 −0.566014
\(839\) −2.18300 −0.0753655 −0.0376828 0.999290i \(-0.511998\pi\)
−0.0376828 + 0.999290i \(0.511998\pi\)
\(840\) 0 0
\(841\) −28.3327 −0.976989
\(842\) −13.2518 −0.456687
\(843\) 0 0
\(844\) −7.93173 −0.273021
\(845\) 95.6011 3.28878
\(846\) 0 0
\(847\) −11.7049 −0.402186
\(848\) 5.06550 0.173950
\(849\) 0 0
\(850\) 51.6192 1.77052
\(851\) 9.33837 0.320115
\(852\) 0 0
\(853\) 19.6098 0.671428 0.335714 0.941964i \(-0.391023\pi\)
0.335714 + 0.941964i \(0.391023\pi\)
\(854\) −6.10527 −0.208918
\(855\) 0 0
\(856\) 20.5636 0.702849
\(857\) 21.1183 0.721387 0.360693 0.932684i \(-0.382540\pi\)
0.360693 + 0.932684i \(0.382540\pi\)
\(858\) 0 0
\(859\) −21.6952 −0.740231 −0.370116 0.928986i \(-0.620682\pi\)
−0.370116 + 0.928986i \(0.620682\pi\)
\(860\) 20.2636 0.690983
\(861\) 0 0
\(862\) −23.5533 −0.802230
\(863\) −17.3555 −0.590788 −0.295394 0.955375i \(-0.595451\pi\)
−0.295394 + 0.955375i \(0.595451\pi\)
\(864\) 0 0
\(865\) −47.6152 −1.61897
\(866\) 17.6156 0.598601
\(867\) 0 0
\(868\) −21.9346 −0.744508
\(869\) 20.0780 0.681098
\(870\) 0 0
\(871\) −57.1959 −1.93801
\(872\) −18.8450 −0.638173
\(873\) 0 0
\(874\) 4.56034 0.154256
\(875\) 73.7980 2.49483
\(876\) 0 0
\(877\) −28.1907 −0.951932 −0.475966 0.879464i \(-0.657902\pi\)
−0.475966 + 0.879464i \(0.657902\pi\)
\(878\) −30.8561 −1.04134
\(879\) 0 0
\(880\) 5.75515 0.194006
\(881\) −24.1851 −0.814818 −0.407409 0.913246i \(-0.633568\pi\)
−0.407409 + 0.913246i \(0.633568\pi\)
\(882\) 0 0
\(883\) −53.3377 −1.79496 −0.897479 0.441058i \(-0.854603\pi\)
−0.897479 + 0.441058i \(0.854603\pi\)
\(884\) 27.7831 0.934448
\(885\) 0 0
\(886\) 17.4109 0.584929
\(887\) −17.7250 −0.595147 −0.297574 0.954699i \(-0.596177\pi\)
−0.297574 + 0.954699i \(0.596177\pi\)
\(888\) 0 0
\(889\) −24.9520 −0.836862
\(890\) 60.5812 2.03069
\(891\) 0 0
\(892\) 18.6508 0.624476
\(893\) −31.2757 −1.04660
\(894\) 0 0
\(895\) −24.1934 −0.808695
\(896\) −10.5697 −0.353109
\(897\) 0 0
\(898\) 19.5959 0.653923
\(899\) 7.38727 0.246379
\(900\) 0 0
\(901\) −37.4297 −1.24696
\(902\) −10.1699 −0.338620
\(903\) 0 0
\(904\) 4.57862 0.152283
\(905\) −32.8683 −1.09258
\(906\) 0 0
\(907\) −35.4045 −1.17559 −0.587793 0.809011i \(-0.700003\pi\)
−0.587793 + 0.809011i \(0.700003\pi\)
\(908\) 20.0051 0.663892
\(909\) 0 0
\(910\) −52.4865 −1.73991
\(911\) 15.1901 0.503270 0.251635 0.967822i \(-0.419032\pi\)
0.251635 + 0.967822i \(0.419032\pi\)
\(912\) 0 0
\(913\) 4.70885 0.155840
\(914\) −11.9101 −0.393952
\(915\) 0 0
\(916\) −31.3909 −1.03718
\(917\) 35.3097 1.16603
\(918\) 0 0
\(919\) −15.7199 −0.518550 −0.259275 0.965804i \(-0.583484\pi\)
−0.259275 + 0.965804i \(0.583484\pi\)
\(920\) −11.8138 −0.389488
\(921\) 0 0
\(922\) −30.7564 −1.01291
\(923\) 20.3661 0.670358
\(924\) 0 0
\(925\) 127.215 4.18280
\(926\) −2.27249 −0.0746786
\(927\) 0 0
\(928\) 4.35852 0.143076
\(929\) −22.1517 −0.726773 −0.363386 0.931638i \(-0.618380\pi\)
−0.363386 + 0.931638i \(0.618380\pi\)
\(930\) 0 0
\(931\) −11.0543 −0.362290
\(932\) −18.9455 −0.620580
\(933\) 0 0
\(934\) −24.6415 −0.806295
\(935\) −42.5257 −1.39074
\(936\) 0 0
\(937\) 17.2669 0.564086 0.282043 0.959402i \(-0.408988\pi\)
0.282043 + 0.959402i \(0.408988\pi\)
\(938\) 19.9262 0.650613
\(939\) 0 0
\(940\) 28.8266 0.940220
\(941\) 45.5870 1.48609 0.743047 0.669239i \(-0.233380\pi\)
0.743047 + 0.669239i \(0.233380\pi\)
\(942\) 0 0
\(943\) 4.28590 0.139568
\(944\) −6.56352 −0.213624
\(945\) 0 0
\(946\) 9.76001 0.317325
\(947\) 36.0414 1.17119 0.585595 0.810604i \(-0.300861\pi\)
0.585595 + 0.810604i \(0.300861\pi\)
\(948\) 0 0
\(949\) 44.0962 1.43142
\(950\) 62.1248 2.01560
\(951\) 0 0
\(952\) −27.2047 −0.881711
\(953\) 23.8618 0.772960 0.386480 0.922298i \(-0.373691\pi\)
0.386480 + 0.922298i \(0.373691\pi\)
\(954\) 0 0
\(955\) 84.4685 2.73334
\(956\) −1.10459 −0.0357249
\(957\) 0 0
\(958\) 17.4315 0.563186
\(959\) 34.2835 1.10707
\(960\) 0 0
\(961\) 50.7774 1.63798
\(962\) −55.5051 −1.78956
\(963\) 0 0
\(964\) −3.28688 −0.105863
\(965\) −36.3949 −1.17159
\(966\) 0 0
\(967\) 41.1614 1.32366 0.661830 0.749654i \(-0.269780\pi\)
0.661830 + 0.749654i \(0.269780\pi\)
\(968\) 15.6593 0.503308
\(969\) 0 0
\(970\) 13.3709 0.429313
\(971\) 37.7773 1.21233 0.606165 0.795339i \(-0.292707\pi\)
0.606165 + 0.795339i \(0.292707\pi\)
\(972\) 0 0
\(973\) −34.5652 −1.10811
\(974\) 23.1962 0.743255
\(975\) 0 0
\(976\) 1.67688 0.0536756
\(977\) −2.87111 −0.0918549 −0.0459274 0.998945i \(-0.514624\pi\)
−0.0459274 + 0.998945i \(0.514624\pi\)
\(978\) 0 0
\(979\) −35.9954 −1.15042
\(980\) 10.1887 0.325465
\(981\) 0 0
\(982\) 0.345320 0.0110196
\(983\) 8.67353 0.276643 0.138321 0.990387i \(-0.455829\pi\)
0.138321 + 0.990387i \(0.455829\pi\)
\(984\) 0 0
\(985\) −33.6052 −1.07075
\(986\) 3.25983 0.103814
\(987\) 0 0
\(988\) 33.4376 1.06379
\(989\) −4.11316 −0.130791
\(990\) 0 0
\(991\) −37.7021 −1.19765 −0.598823 0.800881i \(-0.704365\pi\)
−0.598823 + 0.800881i \(0.704365\pi\)
\(992\) 48.2490 1.53191
\(993\) 0 0
\(994\) −7.09524 −0.225047
\(995\) −80.0554 −2.53793
\(996\) 0 0
\(997\) −31.1876 −0.987720 −0.493860 0.869541i \(-0.664415\pi\)
−0.493860 + 0.869541i \(0.664415\pi\)
\(998\) 2.12172 0.0671618
\(999\) 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.1.13 yes 20
3.2 odd 2 2151.2.a.j.1.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.8 20 3.2 odd 2
2151.2.a.k.1.13 yes 20 1.1 even 1 trivial