Properties

Label 2151.2.a.j.1.8
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
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: \(1\)
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 1.8
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 \(-0.608586\pi\)
−0.334555 + 0.942376i \(0.608586\pi\)
\(3\) 0 0
\(4\) −1.10459 −0.552293
\(5\) −4.23503 −1.89396 −0.946982 0.321286i \(-0.895885\pi\)
−0.946982 + 0.321286i \(0.895885\pi\)
\(6\) 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 \(-0.383130\pi\)
0.358965 + 0.933351i \(0.383130\pi\)
\(12\) 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 \(-0.670873\pi\)
−0.511401 + 0.859342i \(0.670873\pi\)
\(18\) 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 \(-0.531563\pi\)
−0.0989970 + 0.995088i \(0.531563\pi\)
\(24\) 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 \(-0.524166\pi\)
−0.0758470 + 0.997119i \(0.524166\pi\)
\(30\) 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 \(-0.614653\pi\)
−0.352455 + 0.935829i \(0.614653\pi\)
\(42\) 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 \(-0.351628\pi\)
0.449426 + 0.893317i \(0.351628\pi\)
\(48\) 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 \(-0.291337\pi\)
0.609583 + 0.792722i \(0.291337\pi\)
\(54\) 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 \(-0.769281\pi\)
−0.748616 + 0.663004i \(0.769281\pi\)
\(60\) 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 \(-0.435054\pi\)
0.202620 + 0.979257i \(0.435054\pi\)
\(72\) 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 \(-0.465384\pi\)
0.108535 + 0.994093i \(0.465384\pi\)
\(84\) 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 \(-0.795808\pi\)
−0.801206 + 0.598388i \(0.795808\pi\)
\(90\) 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 \(-0.733638\pi\)
−0.669842 + 0.742503i \(0.733638\pi\)
\(102\) 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 \(-0.390132\pi\)
0.338346 + 0.941022i \(0.390132\pi\)
\(108\) 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 \(-0.476644\pi\)
0.0733077 + 0.997309i \(0.476644\pi\)
\(114\) 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 \(-0.747911\pi\)
−0.702451 + 0.711732i \(0.747911\pi\)
\(132\) 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 \(-0.732393\pi\)
−0.666932 + 0.745119i \(0.732393\pi\)
\(138\) 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 \(-0.117555\pi\)
0.932576 + 0.360973i \(0.117555\pi\)
\(150\) 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 \(-0.509509\pi\)
−0.0298679 + 0.999554i \(0.509509\pi\)
\(168\) 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 \(-0.359429\pi\)
0.427401 + 0.904062i \(0.359429\pi\)
\(174\) 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 \(-0.431516\pi\)
0.213493 + 0.976945i \(0.431516\pi\)
\(180\) 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 \(-0.756589\pi\)
−0.721592 + 0.692319i \(0.756589\pi\)
\(192\) 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 \(-0.408778\pi\)
0.282674 + 0.959216i \(0.408778\pi\)
\(198\) 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 \(-0.294756\pi\)
0.601033 + 0.799224i \(0.294756\pi\)
\(228\) 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 \(-0.689899\pi\)
−0.561821 + 0.827259i \(0.689899\pi\)
\(234\) 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 \(-0.342128\pi\)
0.475883 + 0.879508i \(0.342128\pi\)
\(252\) 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 \(-0.521397\pi\)
−0.0671704 + 0.997742i \(0.521397\pi\)
\(258\) 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 \(-0.443693\pi\)
0.175972 + 0.984395i \(0.443693\pi\)
\(264\) 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 \(-0.449653\pi\)
0.157513 + 0.987517i \(0.449653\pi\)
\(270\) 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 \(-0.660426\pi\)
−0.482927 + 0.875661i \(0.660426\pi\)
\(282\) 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 \(-0.158947\pi\)
0.877896 + 0.478851i \(0.158947\pi\)
\(294\) 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 \(-0.688577\pi\)
−0.558381 + 0.829585i \(0.688577\pi\)
\(312\) 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 \(-0.507299\pi\)
−0.0229273 + 0.999737i \(0.507299\pi\)
\(318\) 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 \(-0.665363\pi\)
−0.496449 + 0.868066i \(0.665363\pi\)
\(348\) 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 \(-0.499942\pi\)
0.000183546 1.00000i \(0.499942\pi\)
\(354\) 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 \(-0.372959\pi\)
0.388601 + 0.921406i \(0.372959\pi\)
\(360\) 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 \(-0.243389\pi\)
0.721640 + 0.692268i \(0.243389\pi\)
\(384\) 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 \(-0.777571\pi\)
−0.765627 + 0.643285i \(0.777571\pi\)
\(390\) 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 \(-0.674500\pi\)
−0.521158 + 0.853460i \(0.674500\pi\)
\(402\) 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 \(-0.360991\pi\)
0.422961 + 0.906148i \(0.360991\pi\)
\(420\) 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 \(-0.295376\pi\)
0.599476 + 0.800393i \(0.295376\pi\)
\(432\) 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 \(-0.643993\pi\)
−0.437096 + 0.899415i \(0.643993\pi\)
\(444\) 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 \(-0.662511\pi\)
−0.488652 + 0.872479i \(0.662511\pi\)
\(450\) 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 \(-0.226709\pi\)
0.756909 + 0.653521i \(0.226709\pi\)
\(462\) 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 \(-0.294166\pi\)
0.602514 + 0.798109i \(0.294166\pi\)
\(468\) 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 \(-0.638267\pi\)
−0.420847 + 0.907131i \(0.638267\pi\)
\(480\) 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 \(-0.502621\pi\)
−0.00823454 + 0.999966i \(0.502621\pi\)
\(492\) 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 \(-0.369380\pi\)
0.398934 + 0.916980i \(0.369380\pi\)
\(504\) 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 \(-0.455458\pi\)
0.139476 + 0.990225i \(0.455458\pi\)
\(510\) 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 \(-0.827167\pi\)
−0.856178 + 0.516681i \(0.827167\pi\)
\(522\) 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 \(-0.632761\pi\)
−0.405093 + 0.914276i \(0.632761\pi\)
\(558\) 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 \(-0.792739\pi\)
−0.795399 + 0.606086i \(0.792739\pi\)
\(564\) 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 \(-0.477476\pi\)
0.0707031 + 0.997497i \(0.477476\pi\)
\(570\) 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 \(-0.263454\pi\)
0.676597 + 0.736354i \(0.263454\pi\)
\(588\) 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 \(-0.762929\pi\)
−0.735237 + 0.677810i \(0.762929\pi\)
\(594\) 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 \(-0.495167\pi\)
0.0151818 + 0.999885i \(0.495167\pi\)
\(600\) 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 \(-0.446024\pi\)
0.168759 + 0.985657i \(0.446024\pi\)
\(618\) 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 \(-0.664814\pi\)
−0.494951 + 0.868921i \(0.664814\pi\)
\(642\) 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 \(-0.560787\pi\)
−0.189809 + 0.981821i \(0.560787\pi\)
\(648\) 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 \(-0.660942\pi\)
−0.484346 + 0.874876i \(0.660942\pi\)
\(654\) 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 \(-0.587984\pi\)
−0.272904 + 0.962041i \(0.587984\pi\)
\(660\) 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 \(-0.299411\pi\)
0.589281 + 0.807928i \(0.299411\pi\)
\(678\) 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 \(-0.603458\pi\)
−0.319330 + 0.947644i \(0.603458\pi\)
\(684\) 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 \(-0.628352\pi\)
−0.392390 + 0.919799i \(0.628352\pi\)
\(702\) 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 \(-0.429598\pi\)
0.219377 + 0.975640i \(0.429598\pi\)
\(720\) 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 \(-0.421261\pi\)
0.244850 + 0.969561i \(0.421261\pi\)
\(744\) 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 \(-0.195914\pi\)
0.816495 + 0.577352i \(0.195914\pi\)
\(762\) 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 \(-0.825447\pi\)
−0.853374 + 0.521300i \(0.825447\pi\)
\(774\) 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 \(-0.351503\pi\)
0.449779 + 0.893140i \(0.351503\pi\)
\(798\) 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 \(-0.212622\pi\)
0.785079 + 0.619396i \(0.212622\pi\)
\(810\) 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 \(-0.321703\pi\)
0.531302 + 0.847182i \(0.321703\pi\)
\(822\) 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 \(-0.417974\pi\)
0.254850 + 0.966980i \(0.417974\pi\)
\(828\) 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 \(-0.488002\pi\)
0.0376828 + 0.999290i \(0.488002\pi\)
\(840\) 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 \(-0.617460\pi\)
−0.360693 + 0.932684i \(0.617460\pi\)
\(858\) 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 \(-0.404549\pi\)
0.295394 + 0.955375i \(0.404549\pi\)
\(864\) 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 \(-0.366432\pi\)
0.407409 + 0.913246i \(0.366432\pi\)
\(882\) 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 \(-0.403823\pi\)
0.297574 + 0.954699i \(0.403823\pi\)
\(888\) 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 \(-0.580968\pi\)
−0.251635 + 0.967822i \(0.580968\pi\)
\(912\) 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 \(-0.381620\pi\)
0.363386 + 0.931638i \(0.381620\pi\)
\(930\) 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 \(-0.766620\pi\)
−0.743047 + 0.669239i \(0.766620\pi\)
\(942\) 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 \(-0.699139\pi\)
−0.585595 + 0.810604i \(0.699139\pi\)
\(948\) 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 \(-0.626309\pi\)
−0.386480 + 0.922298i \(0.626309\pi\)
\(954\) 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 \(-0.707293\pi\)
−0.606165 + 0.795339i \(0.707293\pi\)
\(972\) 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 \(-0.485376\pi\)
0.0459274 + 0.998945i \(0.485376\pi\)
\(978\) 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 \(-0.544171\pi\)
−0.138321 + 0.990387i \(0.544171\pi\)
\(984\) 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.j.1.8 20
3.2 odd 2 2151.2.a.k.1.13 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.8 20 1.1 even 1 trivial
2151.2.a.k.1.13 yes 20 3.2 odd 2