Properties

Label 6030.2.a.bu.1.2
Level $6030$
Weight $2$
Character 6030.1
Self dual yes
Analytic conductor $48.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70292.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 10 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2010)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.59676\) of defining polynomial
Character \(\chi\) \(=\) 6030.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.45037 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.50940 q^{11} +4.95977 q^{13} -1.45037 q^{14} +1.00000 q^{16} +3.19351 q^{17} +3.19351 q^{19} +1.00000 q^{20} +3.50940 q^{22} -4.15328 q^{23} +1.00000 q^{25} +4.95977 q^{26} -1.45037 q^{28} +3.25254 q^{29} +6.95977 q^{31} +1.00000 q^{32} +3.19351 q^{34} -1.45037 q^{35} -4.70291 q^{37} +3.19351 q^{38} +1.00000 q^{40} -8.21231 q^{41} -7.01880 q^{43} +3.50940 q^{44} -4.15328 q^{46} +8.66700 q^{47} -4.89642 q^{49} +1.00000 q^{50} +4.95977 q^{52} -0.900743 q^{53} +3.50940 q^{55} -1.45037 q^{56} +3.25254 q^{58} -8.21231 q^{59} +0.549629 q^{61} +6.95977 q^{62} +1.00000 q^{64} +4.95977 q^{65} -1.00000 q^{67} +3.19351 q^{68} -1.45037 q^{70} +6.64388 q^{71} +7.40582 q^{73} -4.70291 q^{74} +3.19351 q^{76} -5.08993 q^{77} -12.3656 q^{79} +1.00000 q^{80} -8.21231 q^{82} +17.0899 q^{83} +3.19351 q^{85} -7.01880 q^{86} +3.50940 q^{88} -14.1087 q^{89} -7.19351 q^{91} -4.15328 q^{92} +8.66700 q^{94} +3.19351 q^{95} -3.50940 q^{97} -4.89642 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{4} + 4q^{5} + q^{7} + 4q^{8} + 4q^{10} + 3q^{11} + 2q^{13} + q^{14} + 4q^{16} + 2q^{17} + 2q^{19} + 4q^{20} + 3q^{22} + 12q^{23} + 4q^{25} + 2q^{26} + q^{28} - 2q^{29} + 10q^{31} + 4q^{32} + 2q^{34} + q^{35} + 3q^{37} + 2q^{38} + 4q^{40} - 6q^{43} + 3q^{44} + 12q^{46} + 14q^{47} + 13q^{49} + 4q^{50} + 2q^{52} + 10q^{53} + 3q^{55} + q^{56} - 2q^{58} + 9q^{61} + 10q^{62} + 4q^{64} + 2q^{65} - 4q^{67} + 2q^{68} + q^{70} + 9q^{71} - 14q^{73} + 3q^{74} + 2q^{76} + 23q^{77} + 12q^{79} + 4q^{80} + 25q^{83} + 2q^{85} - 6q^{86} + 3q^{88} + 9q^{89} - 18q^{91} + 12q^{92} + 14q^{94} + 2q^{95} - 3q^{97} + 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.45037 −0.548189 −0.274094 0.961703i \(-0.588378\pi\)
−0.274094 + 0.961703i \(0.588378\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.50940 1.05812 0.529062 0.848583i \(-0.322544\pi\)
0.529062 + 0.848583i \(0.322544\pi\)
\(12\) 0 0
\(13\) 4.95977 1.37559 0.687797 0.725903i \(-0.258578\pi\)
0.687797 + 0.725903i \(0.258578\pi\)
\(14\) −1.45037 −0.387628
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.19351 0.774540 0.387270 0.921966i \(-0.373418\pi\)
0.387270 + 0.921966i \(0.373418\pi\)
\(18\) 0 0
\(19\) 3.19351 0.732642 0.366321 0.930489i \(-0.380617\pi\)
0.366321 + 0.930489i \(0.380617\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 3.50940 0.748207
\(23\) −4.15328 −0.866019 −0.433010 0.901389i \(-0.642548\pi\)
−0.433010 + 0.901389i \(0.642548\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.95977 0.972691
\(27\) 0 0
\(28\) −1.45037 −0.274094
\(29\) 3.25254 0.603981 0.301991 0.953311i \(-0.402349\pi\)
0.301991 + 0.953311i \(0.402349\pi\)
\(30\) 0 0
\(31\) 6.95977 1.25001 0.625006 0.780620i \(-0.285097\pi\)
0.625006 + 0.780620i \(0.285097\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.19351 0.547683
\(35\) −1.45037 −0.245158
\(36\) 0 0
\(37\) −4.70291 −0.773154 −0.386577 0.922257i \(-0.626343\pi\)
−0.386577 + 0.922257i \(0.626343\pi\)
\(38\) 3.19351 0.518056
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −8.21231 −1.28255 −0.641274 0.767312i \(-0.721594\pi\)
−0.641274 + 0.767312i \(0.721594\pi\)
\(42\) 0 0
\(43\) −7.01880 −1.07036 −0.535178 0.844739i \(-0.679756\pi\)
−0.535178 + 0.844739i \(0.679756\pi\)
\(44\) 3.50940 0.529062
\(45\) 0 0
\(46\) −4.15328 −0.612368
\(47\) 8.66700 1.26421 0.632106 0.774882i \(-0.282191\pi\)
0.632106 + 0.774882i \(0.282191\pi\)
\(48\) 0 0
\(49\) −4.89642 −0.699489
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.95977 0.687797
\(53\) −0.900743 −0.123727 −0.0618633 0.998085i \(-0.519704\pi\)
−0.0618633 + 0.998085i \(0.519704\pi\)
\(54\) 0 0
\(55\) 3.50940 0.473207
\(56\) −1.45037 −0.193814
\(57\) 0 0
\(58\) 3.25254 0.427079
\(59\) −8.21231 −1.06915 −0.534576 0.845120i \(-0.679529\pi\)
−0.534576 + 0.845120i \(0.679529\pi\)
\(60\) 0 0
\(61\) 0.549629 0.0703727 0.0351864 0.999381i \(-0.488798\pi\)
0.0351864 + 0.999381i \(0.488798\pi\)
\(62\) 6.95977 0.883892
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.95977 0.615184
\(66\) 0 0
\(67\) −1.00000 −0.122169
\(68\) 3.19351 0.387270
\(69\) 0 0
\(70\) −1.45037 −0.173353
\(71\) 6.64388 0.788484 0.394242 0.919007i \(-0.371007\pi\)
0.394242 + 0.919007i \(0.371007\pi\)
\(72\) 0 0
\(73\) 7.40582 0.866786 0.433393 0.901205i \(-0.357316\pi\)
0.433393 + 0.901205i \(0.357316\pi\)
\(74\) −4.70291 −0.546702
\(75\) 0 0
\(76\) 3.19351 0.366321
\(77\) −5.08993 −0.580052
\(78\) 0 0
\(79\) −12.3656 −1.39124 −0.695619 0.718411i \(-0.744870\pi\)
−0.695619 + 0.718411i \(0.744870\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −8.21231 −0.906898
\(83\) 17.0899 1.87586 0.937932 0.346819i \(-0.112738\pi\)
0.937932 + 0.346819i \(0.112738\pi\)
\(84\) 0 0
\(85\) 3.19351 0.346385
\(86\) −7.01880 −0.756857
\(87\) 0 0
\(88\) 3.50940 0.374103
\(89\) −14.1087 −1.49552 −0.747761 0.663968i \(-0.768871\pi\)
−0.747761 + 0.663968i \(0.768871\pi\)
\(90\) 0 0
\(91\) −7.19351 −0.754085
\(92\) −4.15328 −0.433010
\(93\) 0 0
\(94\) 8.66700 0.893933
\(95\) 3.19351 0.327647
\(96\) 0 0
\(97\) −3.50940 −0.356326 −0.178163 0.984001i \(-0.557015\pi\)
−0.178163 + 0.984001i \(0.557015\pi\)
\(98\) −4.89642 −0.494613
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.68411 0.167575 0.0837877 0.996484i \(-0.473298\pi\)
0.0837877 + 0.996484i \(0.473298\pi\)
\(102\) 0 0
\(103\) −11.0188 −1.08571 −0.542857 0.839825i \(-0.682658\pi\)
−0.542857 + 0.839825i \(0.682658\pi\)
\(104\) 4.95977 0.486346
\(105\) 0 0
\(106\) −0.900743 −0.0874879
\(107\) 17.9195 1.73235 0.866174 0.499743i \(-0.166572\pi\)
0.866174 + 0.499743i \(0.166572\pi\)
\(108\) 0 0
\(109\) 5.83739 0.559121 0.279560 0.960128i \(-0.409811\pi\)
0.279560 + 0.960128i \(0.409811\pi\)
\(110\) 3.50940 0.334608
\(111\) 0 0
\(112\) −1.45037 −0.137047
\(113\) 8.99568 0.846242 0.423121 0.906073i \(-0.360934\pi\)
0.423121 + 0.906073i \(0.360934\pi\)
\(114\) 0 0
\(115\) −4.15328 −0.387296
\(116\) 3.25254 0.301991
\(117\) 0 0
\(118\) −8.21231 −0.756005
\(119\) −4.63178 −0.424594
\(120\) 0 0
\(121\) 1.31589 0.119626
\(122\) 0.549629 0.0497610
\(123\) 0 0
\(124\) 6.95977 0.625006
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 15.3254 1.35991 0.679953 0.733256i \(-0.262000\pi\)
0.679953 + 0.733256i \(0.262000\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.95977 0.435001
\(131\) 0.886946 0.0774928 0.0387464 0.999249i \(-0.487664\pi\)
0.0387464 + 0.999249i \(0.487664\pi\)
\(132\) 0 0
\(133\) −4.63178 −0.401626
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) 3.19351 0.273841
\(137\) 8.70291 0.743540 0.371770 0.928325i \(-0.378751\pi\)
0.371770 + 0.928325i \(0.378751\pi\)
\(138\) 0 0
\(139\) 18.6456 1.58150 0.790749 0.612141i \(-0.209692\pi\)
0.790749 + 0.612141i \(0.209692\pi\)
\(140\) −1.45037 −0.122579
\(141\) 0 0
\(142\) 6.64388 0.557542
\(143\) 17.4058 1.45555
\(144\) 0 0
\(145\) 3.25254 0.270109
\(146\) 7.40582 0.612910
\(147\) 0 0
\(148\) −4.70291 −0.386577
\(149\) 3.25254 0.266458 0.133229 0.991085i \(-0.457465\pi\)
0.133229 + 0.991085i \(0.457465\pi\)
\(150\) 0 0
\(151\) −14.2354 −1.15846 −0.579232 0.815163i \(-0.696647\pi\)
−0.579232 + 0.815163i \(0.696647\pi\)
\(152\) 3.19351 0.259028
\(153\) 0 0
\(154\) −5.08993 −0.410159
\(155\) 6.95977 0.559022
\(156\) 0 0
\(157\) −2.58554 −0.206348 −0.103174 0.994663i \(-0.532900\pi\)
−0.103174 + 0.994663i \(0.532900\pi\)
\(158\) −12.3656 −0.983754
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 6.02380 0.474742
\(162\) 0 0
\(163\) −14.7029 −1.15162 −0.575810 0.817583i \(-0.695313\pi\)
−0.575810 + 0.817583i \(0.695313\pi\)
\(164\) −8.21231 −0.641274
\(165\) 0 0
\(166\) 17.0899 1.32644
\(167\) 22.0728 1.70805 0.854023 0.520235i \(-0.174156\pi\)
0.854023 + 0.520235i \(0.174156\pi\)
\(168\) 0 0
\(169\) 11.5993 0.892256
\(170\) 3.19351 0.244931
\(171\) 0 0
\(172\) −7.01880 −0.535178
\(173\) 6.77058 0.514758 0.257379 0.966311i \(-0.417141\pi\)
0.257379 + 0.966311i \(0.417141\pi\)
\(174\) 0 0
\(175\) −1.45037 −0.109638
\(176\) 3.50940 0.264531
\(177\) 0 0
\(178\) −14.1087 −1.05749
\(179\) 13.6268 1.01851 0.509256 0.860615i \(-0.329920\pi\)
0.509256 + 0.860615i \(0.329920\pi\)
\(180\) 0 0
\(181\) −21.2073 −1.57633 −0.788163 0.615466i \(-0.788968\pi\)
−0.788163 + 0.615466i \(0.788968\pi\)
\(182\) −7.19351 −0.533219
\(183\) 0 0
\(184\) −4.15328 −0.306184
\(185\) −4.70291 −0.345765
\(186\) 0 0
\(187\) 11.2073 0.819560
\(188\) 8.66700 0.632106
\(189\) 0 0
\(190\) 3.19351 0.231682
\(191\) 4.98120 0.360427 0.180213 0.983628i \(-0.442321\pi\)
0.180213 + 0.983628i \(0.442321\pi\)
\(192\) 0 0
\(193\) 2.92455 0.210513 0.105257 0.994445i \(-0.466434\pi\)
0.105257 + 0.994445i \(0.466434\pi\)
\(194\) −3.50940 −0.251960
\(195\) 0 0
\(196\) −4.89642 −0.349744
\(197\) −7.91954 −0.564244 −0.282122 0.959379i \(-0.591038\pi\)
−0.282122 + 0.959379i \(0.591038\pi\)
\(198\) 0 0
\(199\) 9.08993 0.644368 0.322184 0.946677i \(-0.395583\pi\)
0.322184 + 0.946677i \(0.395583\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 1.68411 0.118494
\(203\) −4.71739 −0.331096
\(204\) 0 0
\(205\) −8.21231 −0.573573
\(206\) −11.0188 −0.767716
\(207\) 0 0
\(208\) 4.95977 0.343898
\(209\) 11.2073 0.775226
\(210\) 0 0
\(211\) −8.41083 −0.579025 −0.289513 0.957174i \(-0.593493\pi\)
−0.289513 + 0.957174i \(0.593493\pi\)
\(212\) −0.900743 −0.0618633
\(213\) 0 0
\(214\) 17.9195 1.22495
\(215\) −7.01880 −0.478678
\(216\) 0 0
\(217\) −10.0943 −0.685243
\(218\) 5.83739 0.395358
\(219\) 0 0
\(220\) 3.50940 0.236604
\(221\) 15.8391 1.06545
\(222\) 0 0
\(223\) −14.5513 −0.974429 −0.487214 0.873282i \(-0.661987\pi\)
−0.487214 + 0.873282i \(0.661987\pi\)
\(224\) −1.45037 −0.0969070
\(225\) 0 0
\(226\) 8.99568 0.598384
\(227\) 14.2690 0.947064 0.473532 0.880776i \(-0.342979\pi\)
0.473532 + 0.880776i \(0.342979\pi\)
\(228\) 0 0
\(229\) −7.56843 −0.500136 −0.250068 0.968228i \(-0.580453\pi\)
−0.250068 + 0.968228i \(0.580453\pi\)
\(230\) −4.15328 −0.273859
\(231\) 0 0
\(232\) 3.25254 0.213540
\(233\) −23.9102 −1.56641 −0.783205 0.621763i \(-0.786417\pi\)
−0.783205 + 0.621763i \(0.786417\pi\)
\(234\) 0 0
\(235\) 8.66700 0.565373
\(236\) −8.21231 −0.534576
\(237\) 0 0
\(238\) −4.63178 −0.300234
\(239\) 15.3254 0.991316 0.495658 0.868518i \(-0.334927\pi\)
0.495658 + 0.868518i \(0.334927\pi\)
\(240\) 0 0
\(241\) 17.9340 1.15523 0.577616 0.816309i \(-0.303983\pi\)
0.577616 + 0.816309i \(0.303983\pi\)
\(242\) 1.31589 0.0845885
\(243\) 0 0
\(244\) 0.549629 0.0351864
\(245\) −4.89642 −0.312821
\(246\) 0 0
\(247\) 15.8391 1.00782
\(248\) 6.95977 0.441946
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 14.7029 0.928040 0.464020 0.885825i \(-0.346407\pi\)
0.464020 + 0.885825i \(0.346407\pi\)
\(252\) 0 0
\(253\) −14.5755 −0.916356
\(254\) 15.3254 0.961599
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.0856 1.00339 0.501696 0.865044i \(-0.332709\pi\)
0.501696 + 0.865044i \(0.332709\pi\)
\(258\) 0 0
\(259\) 6.82097 0.423834
\(260\) 4.95977 0.307592
\(261\) 0 0
\(262\) 0.886946 0.0547957
\(263\) −8.27134 −0.510033 −0.255016 0.966937i \(-0.582081\pi\)
−0.255016 + 0.966937i \(0.582081\pi\)
\(264\) 0 0
\(265\) −0.900743 −0.0553322
\(266\) −4.63178 −0.283993
\(267\) 0 0
\(268\) −1.00000 −0.0610847
\(269\) −12.8655 −0.784424 −0.392212 0.919875i \(-0.628290\pi\)
−0.392212 + 0.919875i \(0.628290\pi\)
\(270\) 0 0
\(271\) −20.2475 −1.22995 −0.614975 0.788546i \(-0.710834\pi\)
−0.614975 + 0.788546i \(0.710834\pi\)
\(272\) 3.19351 0.193635
\(273\) 0 0
\(274\) 8.70291 0.525762
\(275\) 3.50940 0.211625
\(276\) 0 0
\(277\) −17.6413 −1.05996 −0.529980 0.848010i \(-0.677801\pi\)
−0.529980 + 0.848010i \(0.677801\pi\)
\(278\) 18.6456 1.11829
\(279\) 0 0
\(280\) −1.45037 −0.0866763
\(281\) 17.8391 1.06419 0.532095 0.846684i \(-0.321405\pi\)
0.532095 + 0.846684i \(0.321405\pi\)
\(282\) 0 0
\(283\) 20.1550 1.19809 0.599044 0.800716i \(-0.295547\pi\)
0.599044 + 0.800716i \(0.295547\pi\)
\(284\) 6.64388 0.394242
\(285\) 0 0
\(286\) 17.4058 1.02923
\(287\) 11.9109 0.703078
\(288\) 0 0
\(289\) −6.80149 −0.400087
\(290\) 3.25254 0.190996
\(291\) 0 0
\(292\) 7.40582 0.433393
\(293\) 10.6815 0.624019 0.312009 0.950079i \(-0.398998\pi\)
0.312009 + 0.950079i \(0.398998\pi\)
\(294\) 0 0
\(295\) −8.21231 −0.478139
\(296\) −4.70291 −0.273351
\(297\) 0 0
\(298\) 3.25254 0.188415
\(299\) −20.5993 −1.19129
\(300\) 0 0
\(301\) 10.1799 0.586758
\(302\) −14.2354 −0.819157
\(303\) 0 0
\(304\) 3.19351 0.183160
\(305\) 0.549629 0.0314716
\(306\) 0 0
\(307\) −6.75957 −0.385789 −0.192894 0.981220i \(-0.561787\pi\)
−0.192894 + 0.981220i \(0.561787\pi\)
\(308\) −5.08993 −0.290026
\(309\) 0 0
\(310\) 6.95977 0.395288
\(311\) 12.1181 0.687152 0.343576 0.939125i \(-0.388362\pi\)
0.343576 + 0.939125i \(0.388362\pi\)
\(312\) 0 0
\(313\) 13.9427 0.788086 0.394043 0.919092i \(-0.371076\pi\)
0.394043 + 0.919092i \(0.371076\pi\)
\(314\) −2.58554 −0.145910
\(315\) 0 0
\(316\) −12.3656 −0.695619
\(317\) −22.1584 −1.24454 −0.622271 0.782802i \(-0.713790\pi\)
−0.622271 + 0.782802i \(0.713790\pi\)
\(318\) 0 0
\(319\) 11.4145 0.639087
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 6.02380 0.335693
\(323\) 10.1985 0.567461
\(324\) 0 0
\(325\) 4.95977 0.275119
\(326\) −14.7029 −0.814319
\(327\) 0 0
\(328\) −8.21231 −0.453449
\(329\) −12.5704 −0.693027
\(330\) 0 0
\(331\) 16.0607 0.882777 0.441389 0.897316i \(-0.354486\pi\)
0.441389 + 0.897316i \(0.354486\pi\)
\(332\) 17.0899 0.937932
\(333\) 0 0
\(334\) 22.0728 1.20777
\(335\) −1.00000 −0.0546358
\(336\) 0 0
\(337\) −12.7636 −0.695279 −0.347640 0.937628i \(-0.613017\pi\)
−0.347640 + 0.937628i \(0.613017\pi\)
\(338\) 11.5993 0.630921
\(339\) 0 0
\(340\) 3.19351 0.173192
\(341\) 24.4246 1.32267
\(342\) 0 0
\(343\) 17.2542 0.931641
\(344\) −7.01880 −0.378428
\(345\) 0 0
\(346\) 6.77058 0.363989
\(347\) −35.2929 −1.89462 −0.947312 0.320313i \(-0.896212\pi\)
−0.947312 + 0.320313i \(0.896212\pi\)
\(348\) 0 0
\(349\) −8.03760 −0.430243 −0.215121 0.976587i \(-0.569015\pi\)
−0.215121 + 0.976587i \(0.569015\pi\)
\(350\) −1.45037 −0.0775256
\(351\) 0 0
\(352\) 3.50940 0.187052
\(353\) −24.0943 −1.28241 −0.641204 0.767371i \(-0.721565\pi\)
−0.641204 + 0.767371i \(0.721565\pi\)
\(354\) 0 0
\(355\) 6.64388 0.352621
\(356\) −14.1087 −0.747761
\(357\) 0 0
\(358\) 13.6268 0.720197
\(359\) −28.8700 −1.52370 −0.761850 0.647754i \(-0.775709\pi\)
−0.761850 + 0.647754i \(0.775709\pi\)
\(360\) 0 0
\(361\) −8.80149 −0.463236
\(362\) −21.2073 −1.11463
\(363\) 0 0
\(364\) −7.19351 −0.377042
\(365\) 7.40582 0.387638
\(366\) 0 0
\(367\) 25.4873 1.33043 0.665213 0.746654i \(-0.268341\pi\)
0.665213 + 0.746654i \(0.268341\pi\)
\(368\) −4.15328 −0.216505
\(369\) 0 0
\(370\) −4.70291 −0.244493
\(371\) 1.30641 0.0678255
\(372\) 0 0
\(373\) −10.7613 −0.557197 −0.278598 0.960408i \(-0.589870\pi\)
−0.278598 + 0.960408i \(0.589870\pi\)
\(374\) 11.2073 0.579516
\(375\) 0 0
\(376\) 8.66700 0.446967
\(377\) 16.1319 0.830833
\(378\) 0 0
\(379\) −24.4471 −1.25576 −0.627881 0.778310i \(-0.716077\pi\)
−0.627881 + 0.778310i \(0.716077\pi\)
\(380\) 3.19351 0.163824
\(381\) 0 0
\(382\) 4.98120 0.254860
\(383\) 4.50440 0.230164 0.115082 0.993356i \(-0.463287\pi\)
0.115082 + 0.993356i \(0.463287\pi\)
\(384\) 0 0
\(385\) −5.08993 −0.259407
\(386\) 2.92455 0.148855
\(387\) 0 0
\(388\) −3.50940 −0.178163
\(389\) −19.8125 −1.00453 −0.502267 0.864713i \(-0.667500\pi\)
−0.502267 + 0.864713i \(0.667500\pi\)
\(390\) 0 0
\(391\) −13.2636 −0.670767
\(392\) −4.89642 −0.247307
\(393\) 0 0
\(394\) −7.91954 −0.398981
\(395\) −12.3656 −0.622181
\(396\) 0 0
\(397\) −5.92023 −0.297128 −0.148564 0.988903i \(-0.547465\pi\)
−0.148564 + 0.988903i \(0.547465\pi\)
\(398\) 9.08993 0.455637
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −10.3784 −0.518272 −0.259136 0.965841i \(-0.583438\pi\)
−0.259136 + 0.965841i \(0.583438\pi\)
\(402\) 0 0
\(403\) 34.5189 1.71951
\(404\) 1.68411 0.0837877
\(405\) 0 0
\(406\) −4.71739 −0.234120
\(407\) −16.5044 −0.818093
\(408\) 0 0
\(409\) 0.0804569 0.00397834 0.00198917 0.999998i \(-0.499367\pi\)
0.00198917 + 0.999998i \(0.499367\pi\)
\(410\) −8.21231 −0.405577
\(411\) 0 0
\(412\) −11.0188 −0.542857
\(413\) 11.9109 0.586097
\(414\) 0 0
\(415\) 17.0899 0.838912
\(416\) 4.95977 0.243173
\(417\) 0 0
\(418\) 11.2073 0.548167
\(419\) −28.9145 −1.41257 −0.706284 0.707929i \(-0.749630\pi\)
−0.706284 + 0.707929i \(0.749630\pi\)
\(420\) 0 0
\(421\) 7.21731 0.351750 0.175875 0.984412i \(-0.443724\pi\)
0.175875 + 0.984412i \(0.443724\pi\)
\(422\) −8.41083 −0.409433
\(423\) 0 0
\(424\) −0.900743 −0.0437439
\(425\) 3.19351 0.154908
\(426\) 0 0
\(427\) −0.797166 −0.0385775
\(428\) 17.9195 0.866174
\(429\) 0 0
\(430\) −7.01880 −0.338477
\(431\) −14.8324 −0.714451 −0.357226 0.934018i \(-0.616277\pi\)
−0.357226 + 0.934018i \(0.616277\pi\)
\(432\) 0 0
\(433\) 1.83393 0.0881330 0.0440665 0.999029i \(-0.485969\pi\)
0.0440665 + 0.999029i \(0.485969\pi\)
\(434\) −10.0943 −0.484540
\(435\) 0 0
\(436\) 5.83739 0.279560
\(437\) −13.2636 −0.634482
\(438\) 0 0
\(439\) 6.07113 0.289759 0.144880 0.989449i \(-0.453720\pi\)
0.144880 + 0.989449i \(0.453720\pi\)
\(440\) 3.50940 0.167304
\(441\) 0 0
\(442\) 15.8391 0.753389
\(443\) 10.3870 0.493502 0.246751 0.969079i \(-0.420637\pi\)
0.246751 + 0.969079i \(0.420637\pi\)
\(444\) 0 0
\(445\) −14.1087 −0.668818
\(446\) −14.5513 −0.689025
\(447\) 0 0
\(448\) −1.45037 −0.0685236
\(449\) −18.3877 −0.867769 −0.433885 0.900968i \(-0.642857\pi\)
−0.433885 + 0.900968i \(0.642857\pi\)
\(450\) 0 0
\(451\) −28.8203 −1.35709
\(452\) 8.99568 0.423121
\(453\) 0 0
\(454\) 14.2690 0.669676
\(455\) −7.19351 −0.337237
\(456\) 0 0
\(457\) −9.31157 −0.435577 −0.217788 0.975996i \(-0.569884\pi\)
−0.217788 + 0.975996i \(0.569884\pi\)
\(458\) −7.56843 −0.353649
\(459\) 0 0
\(460\) −4.15328 −0.193648
\(461\) 24.2627 1.13003 0.565013 0.825082i \(-0.308871\pi\)
0.565013 + 0.825082i \(0.308871\pi\)
\(462\) 0 0
\(463\) 10.8193 0.502814 0.251407 0.967881i \(-0.419107\pi\)
0.251407 + 0.967881i \(0.419107\pi\)
\(464\) 3.25254 0.150995
\(465\) 0 0
\(466\) −23.9102 −1.10762
\(467\) 19.5232 0.903426 0.451713 0.892163i \(-0.350813\pi\)
0.451713 + 0.892163i \(0.350813\pi\)
\(468\) 0 0
\(469\) 1.45037 0.0669719
\(470\) 8.66700 0.399779
\(471\) 0 0
\(472\) −8.21231 −0.378002
\(473\) −24.6318 −1.13257
\(474\) 0 0
\(475\) 3.19351 0.146528
\(476\) −4.63178 −0.212297
\(477\) 0 0
\(478\) 15.3254 0.700966
\(479\) 23.8888 1.09151 0.545753 0.837946i \(-0.316244\pi\)
0.545753 + 0.837946i \(0.316244\pi\)
\(480\) 0 0
\(481\) −23.3254 −1.06355
\(482\) 17.9340 0.816872
\(483\) 0 0
\(484\) 1.31589 0.0598131
\(485\) −3.50940 −0.159354
\(486\) 0 0
\(487\) −20.4263 −0.925605 −0.462802 0.886462i \(-0.653156\pi\)
−0.462802 + 0.886462i \(0.653156\pi\)
\(488\) 0.549629 0.0248805
\(489\) 0 0
\(490\) −4.89642 −0.221198
\(491\) 8.04801 0.363202 0.181601 0.983372i \(-0.441872\pi\)
0.181601 + 0.983372i \(0.441872\pi\)
\(492\) 0 0
\(493\) 10.3870 0.467808
\(494\) 15.8391 0.712634
\(495\) 0 0
\(496\) 6.95977 0.312503
\(497\) −9.63610 −0.432238
\(498\) 0 0
\(499\) −35.5284 −1.59047 −0.795234 0.606303i \(-0.792652\pi\)
−0.795234 + 0.606303i \(0.792652\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 14.7029 0.656223
\(503\) −27.2456 −1.21482 −0.607410 0.794388i \(-0.707792\pi\)
−0.607410 + 0.794388i \(0.707792\pi\)
\(504\) 0 0
\(505\) 1.68411 0.0749420
\(506\) −14.5755 −0.647961
\(507\) 0 0
\(508\) 15.3254 0.679953
\(509\) −7.64820 −0.339001 −0.169500 0.985530i \(-0.554215\pi\)
−0.169500 + 0.985530i \(0.554215\pi\)
\(510\) 0 0
\(511\) −10.7412 −0.475162
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 16.0856 0.709506
\(515\) −11.0188 −0.485546
\(516\) 0 0
\(517\) 30.4160 1.33769
\(518\) 6.82097 0.299696
\(519\) 0 0
\(520\) 4.95977 0.217500
\(521\) −39.6557 −1.73735 −0.868675 0.495383i \(-0.835028\pi\)
−0.868675 + 0.495383i \(0.835028\pi\)
\(522\) 0 0
\(523\) −34.5882 −1.51244 −0.756219 0.654319i \(-0.772956\pi\)
−0.756219 + 0.654319i \(0.772956\pi\)
\(524\) 0.886946 0.0387464
\(525\) 0 0
\(526\) −8.27134 −0.360648
\(527\) 22.2261 0.968185
\(528\) 0 0
\(529\) −5.75024 −0.250011
\(530\) −0.900743 −0.0391258
\(531\) 0 0
\(532\) −4.63178 −0.200813
\(533\) −40.7312 −1.76426
\(534\) 0 0
\(535\) 17.9195 0.774729
\(536\) −1.00000 −0.0431934
\(537\) 0 0
\(538\) −12.8655 −0.554672
\(539\) −17.1835 −0.740146
\(540\) 0 0
\(541\) 35.0025 1.50488 0.752438 0.658663i \(-0.228878\pi\)
0.752438 + 0.658663i \(0.228878\pi\)
\(542\) −20.2475 −0.869706
\(543\) 0 0
\(544\) 3.19351 0.136921
\(545\) 5.83739 0.250046
\(546\) 0 0
\(547\) 31.3972 1.34245 0.671223 0.741255i \(-0.265769\pi\)
0.671223 + 0.741255i \(0.265769\pi\)
\(548\) 8.70291 0.371770
\(549\) 0 0
\(550\) 3.50940 0.149641
\(551\) 10.3870 0.442502
\(552\) 0 0
\(553\) 17.9347 0.762661
\(554\) −17.6413 −0.749505
\(555\) 0 0
\(556\) 18.6456 0.790749
\(557\) −36.6635 −1.55348 −0.776742 0.629819i \(-0.783129\pi\)
−0.776742 + 0.629819i \(0.783129\pi\)
\(558\) 0 0
\(559\) −34.8116 −1.47238
\(560\) −1.45037 −0.0612894
\(561\) 0 0
\(562\) 17.8391 0.752496
\(563\) 4.98120 0.209933 0.104966 0.994476i \(-0.466527\pi\)
0.104966 + 0.994476i \(0.466527\pi\)
\(564\) 0 0
\(565\) 8.99568 0.378451
\(566\) 20.1550 0.847177
\(567\) 0 0
\(568\) 6.64388 0.278771
\(569\) 13.8298 0.579774 0.289887 0.957061i \(-0.406382\pi\)
0.289887 + 0.957061i \(0.406382\pi\)
\(570\) 0 0
\(571\) −6.38702 −0.267289 −0.133644 0.991029i \(-0.542668\pi\)
−0.133644 + 0.991029i \(0.542668\pi\)
\(572\) 17.4058 0.727774
\(573\) 0 0
\(574\) 11.9109 0.497151
\(575\) −4.15328 −0.173204
\(576\) 0 0
\(577\) 26.3117 1.09537 0.547686 0.836684i \(-0.315509\pi\)
0.547686 + 0.836684i \(0.315509\pi\)
\(578\) −6.80149 −0.282905
\(579\) 0 0
\(580\) 3.25254 0.135054
\(581\) −24.7868 −1.02833
\(582\) 0 0
\(583\) −3.16107 −0.130918
\(584\) 7.40582 0.306455
\(585\) 0 0
\(586\) 10.6815 0.441248
\(587\) −15.7362 −0.649502 −0.324751 0.945800i \(-0.605281\pi\)
−0.324751 + 0.945800i \(0.605281\pi\)
\(588\) 0 0
\(589\) 22.2261 0.915811
\(590\) −8.21231 −0.338096
\(591\) 0 0
\(592\) −4.70291 −0.193288
\(593\) 33.1275 1.36038 0.680192 0.733034i \(-0.261896\pi\)
0.680192 + 0.733034i \(0.261896\pi\)
\(594\) 0 0
\(595\) −4.63178 −0.189884
\(596\) 3.25254 0.133229
\(597\) 0 0
\(598\) −20.5993 −0.842369
\(599\) −32.3528 −1.32190 −0.660950 0.750430i \(-0.729846\pi\)
−0.660950 + 0.750430i \(0.729846\pi\)
\(600\) 0 0
\(601\) 6.53684 0.266643 0.133322 0.991073i \(-0.457436\pi\)
0.133322 + 0.991073i \(0.457436\pi\)
\(602\) 10.1799 0.414900
\(603\) 0 0
\(604\) −14.2354 −0.579232
\(605\) 1.31589 0.0534985
\(606\) 0 0
\(607\) 38.7398 1.57240 0.786201 0.617971i \(-0.212045\pi\)
0.786201 + 0.617971i \(0.212045\pi\)
\(608\) 3.19351 0.129514
\(609\) 0 0
\(610\) 0.549629 0.0222538
\(611\) 42.9864 1.73904
\(612\) 0 0
\(613\) 17.6030 0.710977 0.355489 0.934681i \(-0.384314\pi\)
0.355489 + 0.934681i \(0.384314\pi\)
\(614\) −6.75957 −0.272794
\(615\) 0 0
\(616\) −5.08993 −0.205079
\(617\) −7.26533 −0.292491 −0.146246 0.989248i \(-0.546719\pi\)
−0.146246 + 0.989248i \(0.546719\pi\)
\(618\) 0 0
\(619\) −16.8065 −0.675510 −0.337755 0.941234i \(-0.609667\pi\)
−0.337755 + 0.941234i \(0.609667\pi\)
\(620\) 6.95977 0.279511
\(621\) 0 0
\(622\) 12.1181 0.485890
\(623\) 20.4629 0.819829
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 13.9427 0.557261
\(627\) 0 0
\(628\) −2.58554 −0.103174
\(629\) −15.0188 −0.598839
\(630\) 0 0
\(631\) −6.65321 −0.264860 −0.132430 0.991192i \(-0.542278\pi\)
−0.132430 + 0.991192i \(0.542278\pi\)
\(632\) −12.3656 −0.491877
\(633\) 0 0
\(634\) −22.1584 −0.880024
\(635\) 15.3254 0.608169
\(636\) 0 0
\(637\) −24.2851 −0.962212
\(638\) 11.4145 0.451903
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 33.4421 1.32088 0.660441 0.750878i \(-0.270369\pi\)
0.660441 + 0.750878i \(0.270369\pi\)
\(642\) 0 0
\(643\) −33.3029 −1.31334 −0.656670 0.754178i \(-0.728035\pi\)
−0.656670 + 0.754178i \(0.728035\pi\)
\(644\) 6.02380 0.237371
\(645\) 0 0
\(646\) 10.1985 0.401255
\(647\) 44.0645 1.73235 0.866177 0.499737i \(-0.166570\pi\)
0.866177 + 0.499737i \(0.166570\pi\)
\(648\) 0 0
\(649\) −28.8203 −1.13130
\(650\) 4.95977 0.194538
\(651\) 0 0
\(652\) −14.7029 −0.575810
\(653\) −20.8835 −0.817233 −0.408616 0.912706i \(-0.633989\pi\)
−0.408616 + 0.912706i \(0.633989\pi\)
\(654\) 0 0
\(655\) 0.886946 0.0346558
\(656\) −8.21231 −0.320637
\(657\) 0 0
\(658\) −12.5704 −0.490044
\(659\) −25.9676 −1.01155 −0.505776 0.862665i \(-0.668794\pi\)
−0.505776 + 0.862665i \(0.668794\pi\)
\(660\) 0 0
\(661\) −14.0352 −0.545907 −0.272954 0.962027i \(-0.588001\pi\)
−0.272954 + 0.962027i \(0.588001\pi\)
\(662\) 16.0607 0.624218
\(663\) 0 0
\(664\) 17.0899 0.663218
\(665\) −4.63178 −0.179613
\(666\) 0 0
\(667\) −13.5087 −0.523060
\(668\) 22.0728 0.854023
\(669\) 0 0
\(670\) −1.00000 −0.0386334
\(671\) 1.92887 0.0744631
\(672\) 0 0
\(673\) 19.8066 0.763490 0.381745 0.924268i \(-0.375323\pi\)
0.381745 + 0.924268i \(0.375323\pi\)
\(674\) −12.7636 −0.491637
\(675\) 0 0
\(676\) 11.5993 0.446128
\(677\) −21.8391 −0.839344 −0.419672 0.907676i \(-0.637855\pi\)
−0.419672 + 0.907676i \(0.637855\pi\)
\(678\) 0 0
\(679\) 5.08993 0.195334
\(680\) 3.19351 0.122466
\(681\) 0 0
\(682\) 24.4246 0.935267
\(683\) 13.5943 0.520173 0.260086 0.965585i \(-0.416249\pi\)
0.260086 + 0.965585i \(0.416249\pi\)
\(684\) 0 0
\(685\) 8.70291 0.332521
\(686\) 17.2542 0.658770
\(687\) 0 0
\(688\) −7.01880 −0.267589
\(689\) −4.46748 −0.170197
\(690\) 0 0
\(691\) −44.4060 −1.68928 −0.844641 0.535332i \(-0.820186\pi\)
−0.844641 + 0.535332i \(0.820186\pi\)
\(692\) 6.77058 0.257379
\(693\) 0 0
\(694\) −35.2929 −1.33970
\(695\) 18.6456 0.707267
\(696\) 0 0
\(697\) −26.2261 −0.993385
\(698\) −8.03760 −0.304228
\(699\) 0 0
\(700\) −1.45037 −0.0548189
\(701\) 2.38634 0.0901308 0.0450654 0.998984i \(-0.485650\pi\)
0.0450654 + 0.998984i \(0.485650\pi\)
\(702\) 0 0
\(703\) −15.0188 −0.566445
\(704\) 3.50940 0.132265
\(705\) 0 0
\(706\) −24.0943 −0.906799
\(707\) −2.44259 −0.0918629
\(708\) 0 0
\(709\) 12.1543 0.456464 0.228232 0.973607i \(-0.426706\pi\)
0.228232 + 0.973607i \(0.426706\pi\)
\(710\) 6.64388 0.249340
\(711\) 0 0
\(712\) −14.1087 −0.528747
\(713\) −28.9059 −1.08253
\(714\) 0 0
\(715\) 17.4058 0.650941
\(716\) 13.6268 0.509256
\(717\) 0 0
\(718\) −28.8700 −1.07742
\(719\) 1.62576 0.0606308 0.0303154 0.999540i \(-0.490349\pi\)
0.0303154 + 0.999540i \(0.490349\pi\)
\(720\) 0 0
\(721\) 15.9814 0.595177
\(722\) −8.80149 −0.327557
\(723\) 0 0
\(724\) −21.2073 −0.788163
\(725\) 3.25254 0.120796
\(726\) 0 0
\(727\) 9.18072 0.340494 0.170247 0.985401i \(-0.445543\pi\)
0.170247 + 0.985401i \(0.445543\pi\)
\(728\) −7.19351 −0.266609
\(729\) 0 0
\(730\) 7.40582 0.274102
\(731\) −22.4146 −0.829035
\(732\) 0 0
\(733\) −31.4220 −1.16060 −0.580299 0.814404i \(-0.697064\pi\)
−0.580299 + 0.814404i \(0.697064\pi\)
\(734\) 25.4873 0.940753
\(735\) 0 0
\(736\) −4.15328 −0.153092
\(737\) −3.50940 −0.129270
\(738\) 0 0
\(739\) −29.0326 −1.06798 −0.533991 0.845490i \(-0.679308\pi\)
−0.533991 + 0.845490i \(0.679308\pi\)
\(740\) −4.70291 −0.172882
\(741\) 0 0
\(742\) 1.30641 0.0479599
\(743\) 40.3452 1.48012 0.740060 0.672540i \(-0.234797\pi\)
0.740060 + 0.672540i \(0.234797\pi\)
\(744\) 0 0
\(745\) 3.25254 0.119164
\(746\) −10.7613 −0.393998
\(747\) 0 0
\(748\) 11.2073 0.409780
\(749\) −25.9900 −0.949654
\(750\) 0 0
\(751\) 10.0376 0.366277 0.183139 0.983087i \(-0.441374\pi\)
0.183139 + 0.983087i \(0.441374\pi\)
\(752\) 8.66700 0.316053
\(753\) 0 0
\(754\) 16.1319 0.587487
\(755\) −14.2354 −0.518080
\(756\) 0 0
\(757\) 7.71825 0.280524 0.140262 0.990114i \(-0.455205\pi\)
0.140262 + 0.990114i \(0.455205\pi\)
\(758\) −24.4471 −0.887957
\(759\) 0 0
\(760\) 3.19351 0.115841
\(761\) 37.4341 1.35699 0.678493 0.734607i \(-0.262634\pi\)
0.678493 + 0.734607i \(0.262634\pi\)
\(762\) 0 0
\(763\) −8.46639 −0.306504
\(764\) 4.98120 0.180213
\(765\) 0 0
\(766\) 4.50440 0.162750
\(767\) −40.7312 −1.47072
\(768\) 0 0
\(769\) 33.1645 1.19594 0.597970 0.801518i \(-0.295974\pi\)
0.597970 + 0.801518i \(0.295974\pi\)
\(770\) −5.08993 −0.183428
\(771\) 0 0
\(772\) 2.92455 0.105257
\(773\) 54.1967 1.94932 0.974660 0.223690i \(-0.0718104\pi\)
0.974660 + 0.223690i \(0.0718104\pi\)
\(774\) 0 0
\(775\) 6.95977 0.250002
\(776\) −3.50940 −0.125980
\(777\) 0 0
\(778\) −19.8125 −0.710313
\(779\) −26.2261 −0.939648
\(780\) 0 0
\(781\) 23.3160 0.834314
\(782\) −13.2636 −0.474304
\(783\) 0 0
\(784\) −4.89642 −0.174872
\(785\) −2.58554 −0.0922818
\(786\) 0 0
\(787\) −53.2021 −1.89645 −0.948224 0.317602i \(-0.897122\pi\)
−0.948224 + 0.317602i \(0.897122\pi\)
\(788\) −7.91954 −0.282122
\(789\) 0 0
\(790\) −12.3656 −0.439948
\(791\) −13.0471 −0.463901
\(792\) 0 0
\(793\) 2.72603 0.0968042
\(794\) −5.92023 −0.210101
\(795\) 0 0
\(796\) 9.08993 0.322184
\(797\) 17.8183 0.631158 0.315579 0.948899i \(-0.397801\pi\)
0.315579 + 0.948899i \(0.397801\pi\)
\(798\) 0 0
\(799\) 27.6782 0.979183
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −10.3784 −0.366473
\(803\) 25.9900 0.917167
\(804\) 0 0
\(805\) 6.02380 0.212311
\(806\) 34.5189 1.21588
\(807\) 0 0
\(808\) 1.68411 0.0592468
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 54.7819 1.92365 0.961826 0.273660i \(-0.0882344\pi\)
0.961826 + 0.273660i \(0.0882344\pi\)
\(812\) −4.71739 −0.165548
\(813\) 0 0
\(814\) −16.5044 −0.578479
\(815\) −14.7029 −0.515021
\(816\) 0 0
\(817\) −22.4146 −0.784188
\(818\) 0.0804569 0.00281311
\(819\) 0 0
\(820\) −8.21231 −0.286786
\(821\) 33.6672 1.17499 0.587496 0.809227i \(-0.300114\pi\)
0.587496 + 0.809227i \(0.300114\pi\)
\(822\) 0 0
\(823\) 39.4434 1.37491 0.687456 0.726226i \(-0.258728\pi\)
0.687456 + 0.726226i \(0.258728\pi\)
\(824\) −11.0188 −0.383858
\(825\) 0 0
\(826\) 11.9109 0.414433
\(827\) 40.2268 1.39882 0.699411 0.714719i \(-0.253446\pi\)
0.699411 + 0.714719i \(0.253446\pi\)
\(828\) 0 0
\(829\) −19.2878 −0.669892 −0.334946 0.942237i \(-0.608718\pi\)
−0.334946 + 0.942237i \(0.608718\pi\)
\(830\) 17.0899 0.593200
\(831\) 0 0
\(832\) 4.95977 0.171949
\(833\) −15.6368 −0.541782
\(834\) 0 0
\(835\) 22.0728 0.763862
\(836\) 11.2073 0.387613
\(837\) 0 0
\(838\) −28.9145 −0.998836
\(839\) −20.5079 −0.708010 −0.354005 0.935244i \(-0.615180\pi\)
−0.354005 + 0.935244i \(0.615180\pi\)
\(840\) 0 0
\(841\) −18.4210 −0.635206
\(842\) 7.21731 0.248725
\(843\) 0 0
\(844\) −8.41083 −0.289513
\(845\) 11.5993 0.399029
\(846\) 0 0
\(847\) −1.90853 −0.0655778
\(848\) −0.900743 −0.0309316
\(849\) 0 0
\(850\) 3.19351 0.109537
\(851\) 19.5325 0.669566
\(852\) 0 0
\(853\) −54.3342 −1.86037 −0.930183 0.367096i \(-0.880352\pi\)
−0.930183 + 0.367096i \(0.880352\pi\)
\(854\) −0.797166 −0.0272784
\(855\) 0 0
\(856\) 17.9195 0.612477
\(857\) 6.35526 0.217092 0.108546 0.994091i \(-0.465381\pi\)
0.108546 + 0.994091i \(0.465381\pi\)
\(858\) 0 0
\(859\) −20.7602 −0.708331 −0.354165 0.935183i \(-0.615235\pi\)
−0.354165 + 0.935183i \(0.615235\pi\)
\(860\) −7.01880 −0.239339
\(861\) 0 0
\(862\) −14.8324 −0.505193
\(863\) 41.0916 1.39878 0.699388 0.714743i \(-0.253456\pi\)
0.699388 + 0.714743i \(0.253456\pi\)
\(864\) 0 0
\(865\) 6.77058 0.230207
\(866\) 1.83393 0.0623195
\(867\) 0 0
\(868\) −10.0943 −0.342621
\(869\) −43.3958 −1.47210
\(870\) 0 0
\(871\) −4.95977 −0.168055
\(872\) 5.83739 0.197679
\(873\) 0 0
\(874\) −13.2636 −0.448646
\(875\) −1.45037 −0.0490315
\(876\) 0 0
\(877\) −36.5803 −1.23523 −0.617614 0.786481i \(-0.711901\pi\)
−0.617614 + 0.786481i \(0.711901\pi\)
\(878\) 6.07113 0.204891
\(879\) 0 0
\(880\) 3.50940 0.118302
\(881\) −18.8828 −0.636177 −0.318088 0.948061i \(-0.603041\pi\)
−0.318088 + 0.948061i \(0.603041\pi\)
\(882\) 0 0
\(883\) −24.1181 −0.811637 −0.405819 0.913954i \(-0.633014\pi\)
−0.405819 + 0.913954i \(0.633014\pi\)
\(884\) 15.8391 0.532726
\(885\) 0 0
\(886\) 10.3870 0.348959
\(887\) −40.1104 −1.34678 −0.673388 0.739289i \(-0.735162\pi\)
−0.673388 + 0.739289i \(0.735162\pi\)
\(888\) 0 0
\(889\) −22.2275 −0.745486
\(890\) −14.1087 −0.472926
\(891\) 0 0
\(892\) −14.5513 −0.487214
\(893\) 27.6782 0.926215
\(894\) 0 0
\(895\) 13.6268 0.455493
\(896\) −1.45037 −0.0484535
\(897\) 0 0
\(898\) −18.3877 −0.613606
\(899\) 22.6369 0.754984
\(900\) 0 0
\(901\) −2.87653 −0.0958312
\(902\) −28.8203 −0.959611
\(903\) 0 0
\(904\) 8.99568 0.299192
\(905\) −21.2073 −0.704955
\(906\) 0 0
\(907\) −13.5423 −0.449663 −0.224832 0.974398i \(-0.572183\pi\)
−0.224832 + 0.974398i \(0.572183\pi\)
\(908\) 14.2690 0.473532
\(909\) 0 0
\(910\) −7.19351 −0.238463
\(911\) 47.4931 1.57352 0.786759 0.617261i \(-0.211758\pi\)
0.786759 + 0.617261i \(0.211758\pi\)
\(912\) 0 0
\(913\) 59.9754 1.98490
\(914\) −9.31157 −0.307999
\(915\) 0 0
\(916\) −7.56843 −0.250068
\(917\) −1.28640 −0.0424807
\(918\) 0 0
\(919\) 4.48365 0.147902 0.0739510 0.997262i \(-0.476439\pi\)
0.0739510 + 0.997262i \(0.476439\pi\)
\(920\) −4.15328 −0.136930
\(921\) 0 0
\(922\) 24.2627 0.799049
\(923\) 32.9521 1.08463
\(924\) 0 0
\(925\) −4.70291 −0.154631
\(926\) 10.8193 0.355543
\(927\) 0 0
\(928\) 3.25254 0.106770
\(929\) −6.01516 −0.197351 −0.0986755 0.995120i \(-0.531461\pi\)
−0.0986755 + 0.995120i \(0.531461\pi\)
\(930\) 0 0
\(931\) −15.6368 −0.512475
\(932\) −23.9102 −0.783205
\(933\) 0 0
\(934\) 19.5232 0.638819
\(935\) 11.2073 0.366518
\(936\) 0 0
\(937\) 39.4017 1.28720 0.643598 0.765364i \(-0.277441\pi\)
0.643598 + 0.765364i \(0.277441\pi\)
\(938\) 1.45037 0.0473563
\(939\) 0 0
\(940\) 8.66700 0.282686
\(941\) 31.9900 1.04284 0.521422 0.853299i \(-0.325402\pi\)
0.521422 + 0.853299i \(0.325402\pi\)
\(942\) 0 0
\(943\) 34.1081 1.11071
\(944\) −8.21231 −0.267288
\(945\) 0 0
\(946\) −24.6318 −0.800848
\(947\) 17.7680 0.577381 0.288690 0.957422i \(-0.406780\pi\)
0.288690 + 0.957422i \(0.406780\pi\)
\(948\) 0 0
\(949\) 36.7312 1.19234
\(950\) 3.19351 0.103611
\(951\) 0 0
\(952\) −4.63178 −0.150117
\(953\) −41.0226 −1.32885 −0.664426 0.747354i \(-0.731324\pi\)
−0.664426 + 0.747354i \(0.731324\pi\)
\(954\) 0 0
\(955\) 4.98120 0.161188
\(956\) 15.3254 0.495658
\(957\) 0 0
\(958\) 23.8888 0.771812
\(959\) −12.6225 −0.407600
\(960\) 0 0
\(961\) 17.4384 0.562530
\(962\) −23.3254 −0.752040
\(963\) 0 0
\(964\) 17.9340 0.577616
\(965\) 2.92455 0.0941445
\(966\) 0 0
\(967\) 47.3081 1.52133 0.760663 0.649147i \(-0.224874\pi\)
0.760663 + 0.649147i \(0.224874\pi\)
\(968\) 1.31589 0.0422943
\(969\) 0 0
\(970\) −3.50940 −0.112680
\(971\) −0.494667 −0.0158746 −0.00793731 0.999968i \(-0.502527\pi\)
−0.00793731 + 0.999968i \(0.502527\pi\)
\(972\) 0 0
\(973\) −27.0430 −0.866959
\(974\) −20.4263 −0.654501
\(975\) 0 0
\(976\) 0.549629 0.0175932
\(977\) 29.2760 0.936622 0.468311 0.883564i \(-0.344863\pi\)
0.468311 + 0.883564i \(0.344863\pi\)
\(978\) 0 0
\(979\) −49.5132 −1.58245
\(980\) −4.89642 −0.156410
\(981\) 0 0
\(982\) 8.04801 0.256822
\(983\) −16.9626 −0.541021 −0.270511 0.962717i \(-0.587193\pi\)
−0.270511 + 0.962717i \(0.587193\pi\)
\(984\) 0 0
\(985\) −7.91954 −0.252338
\(986\) 10.3870 0.330790
\(987\) 0 0
\(988\) 15.8391 0.503908
\(989\) 29.1511 0.926950
\(990\) 0 0
\(991\) 37.6996 1.19757 0.598784 0.800911i \(-0.295651\pi\)
0.598784 + 0.800911i \(0.295651\pi\)
\(992\) 6.95977 0.220973
\(993\) 0 0
\(994\) −9.63610 −0.305638
\(995\) 9.08993 0.288170
\(996\) 0 0
\(997\) 27.4427 0.869120 0.434560 0.900643i \(-0.356904\pi\)
0.434560 + 0.900643i \(0.356904\pi\)
\(998\) −35.5284 −1.12463
\(999\) 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 6030.2.a.bu.1.2 4
3.2 odd 2 2010.2.a.r.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2010.2.a.r.1.2 4 3.2 odd 2
6030.2.a.bu.1.2 4 1.1 even 1 trivial