Properties

Label 3150.2.a.bt.1.1
Level 3150
Weight 2
Character 3150.1
Self dual yes
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\)
Character \(\chi\) = 3150.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} -4.89898 q^{11} -0.449490 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.44949 q^{19} -4.89898 q^{22} +6.89898 q^{23} -0.449490 q^{26} -1.00000 q^{28} +2.89898 q^{29} -0.898979 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} +6.44949 q^{38} +10.8990 q^{41} +8.89898 q^{43} -4.89898 q^{44} +6.89898 q^{46} +0.898979 q^{47} +1.00000 q^{49} -0.449490 q^{52} +1.10102 q^{53} -1.00000 q^{56} +2.89898 q^{58} +6.44949 q^{59} +8.44949 q^{61} -0.898979 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} +10.8990 q^{71} -6.89898 q^{73} +2.00000 q^{74} +6.44949 q^{76} +4.89898 q^{77} -2.89898 q^{79} +10.8990 q^{82} -2.44949 q^{83} +8.89898 q^{86} -4.89898 q^{88} +10.0000 q^{89} +0.449490 q^{91} +6.89898 q^{92} +0.898979 q^{94} -3.79796 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{7} + 2q^{8} + 4q^{13} - 2q^{14} + 2q^{16} - 4q^{17} + 8q^{19} + 4q^{23} + 4q^{26} - 2q^{28} - 4q^{29} + 8q^{31} + 2q^{32} - 4q^{34} + 4q^{37} + 8q^{38} + 12q^{41} + 8q^{43} + 4q^{46} - 8q^{47} + 2q^{49} + 4q^{52} + 12q^{53} - 2q^{56} - 4q^{58} + 8q^{59} + 12q^{61} + 8q^{62} + 2q^{64} - 16q^{67} - 4q^{68} + 12q^{71} - 4q^{73} + 4q^{74} + 8q^{76} + 4q^{79} + 12q^{82} + 8q^{86} + 20q^{89} - 4q^{91} + 4q^{92} - 8q^{94} + 12q^{97} + 2q^{98} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) −0.449490 −0.124666 −0.0623330 0.998055i \(-0.519854\pi\)
−0.0623330 + 0.998055i \(0.519854\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.44949 1.47961 0.739807 0.672819i \(-0.234917\pi\)
0.739807 + 0.672819i \(0.234917\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.89898 −1.04447
\(23\) 6.89898 1.43854 0.719268 0.694732i \(-0.244477\pi\)
0.719268 + 0.694732i \(0.244477\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.449490 −0.0881522
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 2.89898 0.538327 0.269163 0.963095i \(-0.413253\pi\)
0.269163 + 0.963095i \(0.413253\pi\)
\(30\) 0 0
\(31\) −0.898979 −0.161461 −0.0807307 0.996736i \(-0.525725\pi\)
−0.0807307 + 0.996736i \(0.525725\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.44949 1.04625
\(39\) 0 0
\(40\) 0 0
\(41\) 10.8990 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(42\) 0 0
\(43\) 8.89898 1.35708 0.678541 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) −4.89898 −0.738549
\(45\) 0 0
\(46\) 6.89898 1.01720
\(47\) 0.898979 0.131130 0.0655648 0.997848i \(-0.479115\pi\)
0.0655648 + 0.997848i \(0.479115\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −0.449490 −0.0623330
\(53\) 1.10102 0.151237 0.0756184 0.997137i \(-0.475907\pi\)
0.0756184 + 0.997137i \(0.475907\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.89898 0.380655
\(59\) 6.44949 0.839652 0.419826 0.907605i \(-0.362091\pi\)
0.419826 + 0.907605i \(0.362091\pi\)
\(60\) 0 0
\(61\) 8.44949 1.08185 0.540923 0.841072i \(-0.318075\pi\)
0.540923 + 0.841072i \(0.318075\pi\)
\(62\) −0.898979 −0.114171
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 10.8990 1.29347 0.646735 0.762714i \(-0.276134\pi\)
0.646735 + 0.762714i \(0.276134\pi\)
\(72\) 0 0
\(73\) −6.89898 −0.807464 −0.403732 0.914877i \(-0.632287\pi\)
−0.403732 + 0.914877i \(0.632287\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 6.44949 0.739807
\(77\) 4.89898 0.558291
\(78\) 0 0
\(79\) −2.89898 −0.326161 −0.163080 0.986613i \(-0.552143\pi\)
−0.163080 + 0.986613i \(0.552143\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.8990 1.20359
\(83\) −2.44949 −0.268866 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.89898 0.959602
\(87\) 0 0
\(88\) −4.89898 −0.522233
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0.449490 0.0471193
\(92\) 6.89898 0.719268
\(93\) 0 0
\(94\) 0.898979 0.0927227
\(95\) 0 0
\(96\) 0 0
\(97\) −3.79796 −0.385624 −0.192812 0.981236i \(-0.561761\pi\)
−0.192812 + 0.981236i \(0.561761\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) −8.44949 −0.840756 −0.420378 0.907349i \(-0.638102\pi\)
−0.420378 + 0.907349i \(0.638102\pi\)
\(102\) 0 0
\(103\) 3.10102 0.305553 0.152776 0.988261i \(-0.451179\pi\)
0.152776 + 0.988261i \(0.451179\pi\)
\(104\) −0.449490 −0.0440761
\(105\) 0 0
\(106\) 1.10102 0.106941
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −2.89898 −0.277672 −0.138836 0.990315i \(-0.544336\pi\)
−0.138836 + 0.990315i \(0.544336\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −0.202041 −0.0190064 −0.00950321 0.999955i \(-0.503025\pi\)
−0.00950321 + 0.999955i \(0.503025\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.89898 0.269163
\(117\) 0 0
\(118\) 6.44949 0.593724
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 8.44949 0.764981
\(123\) 0 0
\(124\) −0.898979 −0.0807307
\(125\) 0 0
\(126\) 0 0
\(127\) −5.10102 −0.452642 −0.226321 0.974053i \(-0.572670\pi\)
−0.226321 + 0.974053i \(0.572670\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 1.55051 0.135469 0.0677344 0.997703i \(-0.478423\pi\)
0.0677344 + 0.997703i \(0.478423\pi\)
\(132\) 0 0
\(133\) −6.44949 −0.559242
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −17.7980 −1.52058 −0.760291 0.649582i \(-0.774944\pi\)
−0.760291 + 0.649582i \(0.774944\pi\)
\(138\) 0 0
\(139\) 6.44949 0.547039 0.273519 0.961867i \(-0.411812\pi\)
0.273519 + 0.961867i \(0.411812\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.8990 0.914622
\(143\) 2.20204 0.184144
\(144\) 0 0
\(145\) 0 0
\(146\) −6.89898 −0.570964
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) −15.7980 −1.29422 −0.647110 0.762397i \(-0.724022\pi\)
−0.647110 + 0.762397i \(0.724022\pi\)
\(150\) 0 0
\(151\) −19.5959 −1.59469 −0.797347 0.603522i \(-0.793764\pi\)
−0.797347 + 0.603522i \(0.793764\pi\)
\(152\) 6.44949 0.523123
\(153\) 0 0
\(154\) 4.89898 0.394771
\(155\) 0 0
\(156\) 0 0
\(157\) 8.44949 0.674343 0.337171 0.941443i \(-0.390530\pi\)
0.337171 + 0.941443i \(0.390530\pi\)
\(158\) −2.89898 −0.230630
\(159\) 0 0
\(160\) 0 0
\(161\) −6.89898 −0.543716
\(162\) 0 0
\(163\) −16.8990 −1.32363 −0.661815 0.749667i \(-0.730214\pi\)
−0.661815 + 0.749667i \(0.730214\pi\)
\(164\) 10.8990 0.851067
\(165\) 0 0
\(166\) −2.44949 −0.190117
\(167\) −4.89898 −0.379094 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(168\) 0 0
\(169\) −12.7980 −0.984458
\(170\) 0 0
\(171\) 0 0
\(172\) 8.89898 0.678541
\(173\) −18.2474 −1.38733 −0.693664 0.720299i \(-0.744005\pi\)
−0.693664 + 0.720299i \(0.744005\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.89898 −0.369274
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 5.79796 0.433360 0.216680 0.976243i \(-0.430477\pi\)
0.216680 + 0.976243i \(0.430477\pi\)
\(180\) 0 0
\(181\) 14.2474 1.05900 0.529502 0.848309i \(-0.322379\pi\)
0.529502 + 0.848309i \(0.322379\pi\)
\(182\) 0.449490 0.0333184
\(183\) 0 0
\(184\) 6.89898 0.508600
\(185\) 0 0
\(186\) 0 0
\(187\) 9.79796 0.716498
\(188\) 0.898979 0.0655648
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6969 1.20815 0.604074 0.796928i \(-0.293543\pi\)
0.604074 + 0.796928i \(0.293543\pi\)
\(192\) 0 0
\(193\) 17.5959 1.26658 0.633291 0.773914i \(-0.281704\pi\)
0.633291 + 0.773914i \(0.281704\pi\)
\(194\) −3.79796 −0.272678
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.10102 −0.648421 −0.324210 0.945985i \(-0.605099\pi\)
−0.324210 + 0.945985i \(0.605099\pi\)
\(198\) 0 0
\(199\) 7.10102 0.503378 0.251689 0.967808i \(-0.419014\pi\)
0.251689 + 0.967808i \(0.419014\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.44949 −0.594504
\(203\) −2.89898 −0.203468
\(204\) 0 0
\(205\) 0 0
\(206\) 3.10102 0.216058
\(207\) 0 0
\(208\) −0.449490 −0.0311665
\(209\) −31.5959 −2.18554
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 1.10102 0.0756184
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) 0.898979 0.0610267
\(218\) −2.89898 −0.196344
\(219\) 0 0
\(220\) 0 0
\(221\) 0.898979 0.0604719
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −0.202041 −0.0134396
\(227\) 7.34847 0.487735 0.243868 0.969809i \(-0.421584\pi\)
0.243868 + 0.969809i \(0.421584\pi\)
\(228\) 0 0
\(229\) 15.1464 1.00090 0.500452 0.865764i \(-0.333167\pi\)
0.500452 + 0.865764i \(0.333167\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.89898 0.190327
\(233\) −10.2020 −0.668358 −0.334179 0.942510i \(-0.608459\pi\)
−0.334179 + 0.942510i \(0.608459\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.44949 0.419826
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 25.7980 1.66873 0.834366 0.551211i \(-0.185834\pi\)
0.834366 + 0.551211i \(0.185834\pi\)
\(240\) 0 0
\(241\) 20.6969 1.33321 0.666604 0.745412i \(-0.267747\pi\)
0.666604 + 0.745412i \(0.267747\pi\)
\(242\) 13.0000 0.835672
\(243\) 0 0
\(244\) 8.44949 0.540923
\(245\) 0 0
\(246\) 0 0
\(247\) −2.89898 −0.184458
\(248\) −0.898979 −0.0570853
\(249\) 0 0
\(250\) 0 0
\(251\) 1.55051 0.0978673 0.0489337 0.998802i \(-0.484418\pi\)
0.0489337 + 0.998802i \(0.484418\pi\)
\(252\) 0 0
\(253\) −33.7980 −2.12486
\(254\) −5.10102 −0.320066
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.6969 −1.29104 −0.645520 0.763744i \(-0.723359\pi\)
−0.645520 + 0.763744i \(0.723359\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) 1.55051 0.0957908
\(263\) 9.79796 0.604168 0.302084 0.953281i \(-0.402318\pi\)
0.302084 + 0.953281i \(0.402318\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.44949 −0.395444
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 15.1464 0.923494 0.461747 0.887012i \(-0.347223\pi\)
0.461747 + 0.887012i \(0.347223\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −17.7980 −1.07521
\(275\) 0 0
\(276\) 0 0
\(277\) −5.10102 −0.306491 −0.153245 0.988188i \(-0.548972\pi\)
−0.153245 + 0.988188i \(0.548972\pi\)
\(278\) 6.44949 0.386815
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 28.2474 1.67914 0.839568 0.543254i \(-0.182808\pi\)
0.839568 + 0.543254i \(0.182808\pi\)
\(284\) 10.8990 0.646735
\(285\) 0 0
\(286\) 2.20204 0.130209
\(287\) −10.8990 −0.643346
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −6.89898 −0.403732
\(293\) 6.24745 0.364980 0.182490 0.983208i \(-0.441584\pi\)
0.182490 + 0.983208i \(0.441584\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) −15.7980 −0.915151
\(299\) −3.10102 −0.179337
\(300\) 0 0
\(301\) −8.89898 −0.512929
\(302\) −19.5959 −1.12762
\(303\) 0 0
\(304\) 6.44949 0.369904
\(305\) 0 0
\(306\) 0 0
\(307\) 4.24745 0.242415 0.121207 0.992627i \(-0.461323\pi\)
0.121207 + 0.992627i \(0.461323\pi\)
\(308\) 4.89898 0.279145
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 17.5959 0.994580 0.497290 0.867584i \(-0.334328\pi\)
0.497290 + 0.867584i \(0.334328\pi\)
\(314\) 8.44949 0.476832
\(315\) 0 0
\(316\) −2.89898 −0.163080
\(317\) −26.4949 −1.48810 −0.744051 0.668123i \(-0.767098\pi\)
−0.744051 + 0.668123i \(0.767098\pi\)
\(318\) 0 0
\(319\) −14.2020 −0.795162
\(320\) 0 0
\(321\) 0 0
\(322\) −6.89898 −0.384465
\(323\) −12.8990 −0.717718
\(324\) 0 0
\(325\) 0 0
\(326\) −16.8990 −0.935948
\(327\) 0 0
\(328\) 10.8990 0.601795
\(329\) −0.898979 −0.0495623
\(330\) 0 0
\(331\) 10.6969 0.587957 0.293978 0.955812i \(-0.405021\pi\)
0.293978 + 0.955812i \(0.405021\pi\)
\(332\) −2.44949 −0.134433
\(333\) 0 0
\(334\) −4.89898 −0.268060
\(335\) 0 0
\(336\) 0 0
\(337\) −29.5959 −1.61219 −0.806096 0.591785i \(-0.798424\pi\)
−0.806096 + 0.591785i \(0.798424\pi\)
\(338\) −12.7980 −0.696117
\(339\) 0 0
\(340\) 0 0
\(341\) 4.40408 0.238494
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.89898 0.479801
\(345\) 0 0
\(346\) −18.2474 −0.980989
\(347\) −19.1010 −1.02540 −0.512698 0.858569i \(-0.671354\pi\)
−0.512698 + 0.858569i \(0.671354\pi\)
\(348\) 0 0
\(349\) −3.55051 −0.190054 −0.0950272 0.995475i \(-0.530294\pi\)
−0.0950272 + 0.995475i \(0.530294\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.89898 −0.261116
\(353\) −13.1010 −0.697297 −0.348648 0.937254i \(-0.613359\pi\)
−0.348648 + 0.937254i \(0.613359\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 5.79796 0.306432
\(359\) −11.5959 −0.612009 −0.306005 0.952030i \(-0.598992\pi\)
−0.306005 + 0.952030i \(0.598992\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 14.2474 0.748829
\(363\) 0 0
\(364\) 0.449490 0.0235597
\(365\) 0 0
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 6.89898 0.359634
\(369\) 0 0
\(370\) 0 0
\(371\) −1.10102 −0.0571621
\(372\) 0 0
\(373\) 24.6969 1.27876 0.639380 0.768891i \(-0.279191\pi\)
0.639380 + 0.768891i \(0.279191\pi\)
\(374\) 9.79796 0.506640
\(375\) 0 0
\(376\) 0.898979 0.0463613
\(377\) −1.30306 −0.0671111
\(378\) 0 0
\(379\) 1.30306 0.0669338 0.0334669 0.999440i \(-0.489345\pi\)
0.0334669 + 0.999440i \(0.489345\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.6969 0.854290
\(383\) 16.8990 0.863498 0.431749 0.901994i \(-0.357897\pi\)
0.431749 + 0.901994i \(0.357897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.5959 0.895609
\(387\) 0 0
\(388\) −3.79796 −0.192812
\(389\) −22.8990 −1.16102 −0.580512 0.814252i \(-0.697148\pi\)
−0.580512 + 0.814252i \(0.697148\pi\)
\(390\) 0 0
\(391\) −13.7980 −0.697793
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −9.10102 −0.458503
\(395\) 0 0
\(396\) 0 0
\(397\) −17.3485 −0.870695 −0.435347 0.900263i \(-0.643374\pi\)
−0.435347 + 0.900263i \(0.643374\pi\)
\(398\) 7.10102 0.355942
\(399\) 0 0
\(400\) 0 0
\(401\) −29.3939 −1.46786 −0.733930 0.679225i \(-0.762316\pi\)
−0.733930 + 0.679225i \(0.762316\pi\)
\(402\) 0 0
\(403\) 0.404082 0.0201288
\(404\) −8.44949 −0.420378
\(405\) 0 0
\(406\) −2.89898 −0.143874
\(407\) −9.79796 −0.485667
\(408\) 0 0
\(409\) −14.4949 −0.716727 −0.358363 0.933582i \(-0.616665\pi\)
−0.358363 + 0.933582i \(0.616665\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.10102 0.152776
\(413\) −6.44949 −0.317359
\(414\) 0 0
\(415\) 0 0
\(416\) −0.449490 −0.0220380
\(417\) 0 0
\(418\) −31.5959 −1.54541
\(419\) 6.44949 0.315078 0.157539 0.987513i \(-0.449644\pi\)
0.157539 + 0.987513i \(0.449644\pi\)
\(420\) 0 0
\(421\) −23.7980 −1.15984 −0.579921 0.814673i \(-0.696917\pi\)
−0.579921 + 0.814673i \(0.696917\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 1.10102 0.0534703
\(425\) 0 0
\(426\) 0 0
\(427\) −8.44949 −0.408899
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −17.7980 −0.857298 −0.428649 0.903471i \(-0.641010\pi\)
−0.428649 + 0.903471i \(0.641010\pi\)
\(432\) 0 0
\(433\) −19.7980 −0.951429 −0.475715 0.879600i \(-0.657810\pi\)
−0.475715 + 0.879600i \(0.657810\pi\)
\(434\) 0.898979 0.0431524
\(435\) 0 0
\(436\) −2.89898 −0.138836
\(437\) 44.4949 2.12848
\(438\) 0 0
\(439\) 37.3939 1.78471 0.892356 0.451332i \(-0.149051\pi\)
0.892356 + 0.451332i \(0.149051\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.898979 0.0427601
\(443\) 9.79796 0.465515 0.232758 0.972535i \(-0.425225\pi\)
0.232758 + 0.972535i \(0.425225\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) −53.3939 −2.51422
\(452\) −0.202041 −0.00950321
\(453\) 0 0
\(454\) 7.34847 0.344881
\(455\) 0 0
\(456\) 0 0
\(457\) −9.59592 −0.448878 −0.224439 0.974488i \(-0.572055\pi\)
−0.224439 + 0.974488i \(0.572055\pi\)
\(458\) 15.1464 0.707746
\(459\) 0 0
\(460\) 0 0
\(461\) −2.65153 −0.123494 −0.0617470 0.998092i \(-0.519667\pi\)
−0.0617470 + 0.998092i \(0.519667\pi\)
\(462\) 0 0
\(463\) −35.5959 −1.65428 −0.827141 0.561994i \(-0.810034\pi\)
−0.827141 + 0.561994i \(0.810034\pi\)
\(464\) 2.89898 0.134582
\(465\) 0 0
\(466\) −10.2020 −0.472600
\(467\) −5.55051 −0.256847 −0.128423 0.991719i \(-0.540992\pi\)
−0.128423 + 0.991719i \(0.540992\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 6.44949 0.296862
\(473\) −43.5959 −2.00454
\(474\) 0 0
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 25.7980 1.17997
\(479\) −38.6969 −1.76811 −0.884054 0.467385i \(-0.845196\pi\)
−0.884054 + 0.467385i \(0.845196\pi\)
\(480\) 0 0
\(481\) −0.898979 −0.0409899
\(482\) 20.6969 0.942720
\(483\) 0 0
\(484\) 13.0000 0.590909
\(485\) 0 0
\(486\) 0 0
\(487\) −36.6969 −1.66290 −0.831449 0.555602i \(-0.812488\pi\)
−0.831449 + 0.555602i \(0.812488\pi\)
\(488\) 8.44949 0.382490
\(489\) 0 0
\(490\) 0 0
\(491\) 19.5959 0.884351 0.442176 0.896928i \(-0.354207\pi\)
0.442176 + 0.896928i \(0.354207\pi\)
\(492\) 0 0
\(493\) −5.79796 −0.261127
\(494\) −2.89898 −0.130431
\(495\) 0 0
\(496\) −0.898979 −0.0403654
\(497\) −10.8990 −0.488886
\(498\) 0 0
\(499\) 25.7980 1.15488 0.577438 0.816435i \(-0.304053\pi\)
0.577438 + 0.816435i \(0.304053\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.55051 0.0692027
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −33.7980 −1.50250
\(507\) 0 0
\(508\) −5.10102 −0.226321
\(509\) 36.4495 1.61560 0.807798 0.589460i \(-0.200659\pi\)
0.807798 + 0.589460i \(0.200659\pi\)
\(510\) 0 0
\(511\) 6.89898 0.305193
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −20.6969 −0.912903
\(515\) 0 0
\(516\) 0 0
\(517\) −4.40408 −0.193691
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 0 0
\(521\) −3.30306 −0.144710 −0.0723549 0.997379i \(-0.523051\pi\)
−0.0723549 + 0.997379i \(0.523051\pi\)
\(522\) 0 0
\(523\) 1.14643 0.0501298 0.0250649 0.999686i \(-0.492021\pi\)
0.0250649 + 0.999686i \(0.492021\pi\)
\(524\) 1.55051 0.0677344
\(525\) 0 0
\(526\) 9.79796 0.427211
\(527\) 1.79796 0.0783203
\(528\) 0 0
\(529\) 24.5959 1.06939
\(530\) 0 0
\(531\) 0 0
\(532\) −6.44949 −0.279621
\(533\) −4.89898 −0.212198
\(534\) 0 0
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) 15.1464 0.653009
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) −29.5959 −1.27243 −0.636214 0.771513i \(-0.719500\pi\)
−0.636214 + 0.771513i \(0.719500\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 10.6969 0.457368 0.228684 0.973501i \(-0.426558\pi\)
0.228684 + 0.973501i \(0.426558\pi\)
\(548\) −17.7980 −0.760291
\(549\) 0 0
\(550\) 0 0
\(551\) 18.6969 0.796516
\(552\) 0 0
\(553\) 2.89898 0.123277
\(554\) −5.10102 −0.216722
\(555\) 0 0
\(556\) 6.44949 0.273519
\(557\) 16.6969 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −14.0454 −0.591943 −0.295972 0.955197i \(-0.595643\pi\)
−0.295972 + 0.955197i \(0.595643\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.2474 1.18733
\(567\) 0 0
\(568\) 10.8990 0.457311
\(569\) −14.2020 −0.595381 −0.297690 0.954663i \(-0.596216\pi\)
−0.297690 + 0.954663i \(0.596216\pi\)
\(570\) 0 0
\(571\) −20.8990 −0.874595 −0.437298 0.899317i \(-0.644064\pi\)
−0.437298 + 0.899317i \(0.644064\pi\)
\(572\) 2.20204 0.0920720
\(573\) 0 0
\(574\) −10.8990 −0.454915
\(575\) 0 0
\(576\) 0 0
\(577\) 46.4949 1.93561 0.967804 0.251705i \(-0.0809913\pi\)
0.967804 + 0.251705i \(0.0809913\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 2.44949 0.101622
\(582\) 0 0
\(583\) −5.39388 −0.223392
\(584\) −6.89898 −0.285482
\(585\) 0 0
\(586\) 6.24745 0.258080
\(587\) 33.1464 1.36810 0.684050 0.729435i \(-0.260217\pi\)
0.684050 + 0.729435i \(0.260217\pi\)
\(588\) 0 0
\(589\) −5.79796 −0.238901
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 1.10102 0.0452135 0.0226067 0.999744i \(-0.492803\pi\)
0.0226067 + 0.999744i \(0.492803\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.7980 −0.647110
\(597\) 0 0
\(598\) −3.10102 −0.126810
\(599\) 22.8990 0.935627 0.467813 0.883827i \(-0.345042\pi\)
0.467813 + 0.883827i \(0.345042\pi\)
\(600\) 0 0
\(601\) 19.3939 0.791093 0.395546 0.918446i \(-0.370555\pi\)
0.395546 + 0.918446i \(0.370555\pi\)
\(602\) −8.89898 −0.362695
\(603\) 0 0
\(604\) −19.5959 −0.797347
\(605\) 0 0
\(606\) 0 0
\(607\) −25.3939 −1.03071 −0.515353 0.856978i \(-0.672339\pi\)
−0.515353 + 0.856978i \(0.672339\pi\)
\(608\) 6.44949 0.261561
\(609\) 0 0
\(610\) 0 0
\(611\) −0.404082 −0.0163474
\(612\) 0 0
\(613\) −8.20204 −0.331277 −0.165639 0.986187i \(-0.552969\pi\)
−0.165639 + 0.986187i \(0.552969\pi\)
\(614\) 4.24745 0.171413
\(615\) 0 0
\(616\) 4.89898 0.197386
\(617\) 9.59592 0.386317 0.193159 0.981168i \(-0.438127\pi\)
0.193159 + 0.981168i \(0.438127\pi\)
\(618\) 0 0
\(619\) −46.4495 −1.86696 −0.933481 0.358626i \(-0.883245\pi\)
−0.933481 + 0.358626i \(0.883245\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) −10.0000 −0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) 17.5959 0.703274
\(627\) 0 0
\(628\) 8.44949 0.337171
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 6.49490 0.258558 0.129279 0.991608i \(-0.458734\pi\)
0.129279 + 0.991608i \(0.458734\pi\)
\(632\) −2.89898 −0.115315
\(633\) 0 0
\(634\) −26.4949 −1.05225
\(635\) 0 0
\(636\) 0 0
\(637\) −0.449490 −0.0178094
\(638\) −14.2020 −0.562264
\(639\) 0 0
\(640\) 0 0
\(641\) −6.20204 −0.244966 −0.122483 0.992471i \(-0.539086\pi\)
−0.122483 + 0.992471i \(0.539086\pi\)
\(642\) 0 0
\(643\) −9.14643 −0.360700 −0.180350 0.983603i \(-0.557723\pi\)
−0.180350 + 0.983603i \(0.557723\pi\)
\(644\) −6.89898 −0.271858
\(645\) 0 0
\(646\) −12.8990 −0.507504
\(647\) −22.2929 −0.876423 −0.438211 0.898872i \(-0.644388\pi\)
−0.438211 + 0.898872i \(0.644388\pi\)
\(648\) 0 0
\(649\) −31.5959 −1.24025
\(650\) 0 0
\(651\) 0 0
\(652\) −16.8990 −0.661815
\(653\) 39.7980 1.55741 0.778707 0.627387i \(-0.215876\pi\)
0.778707 + 0.627387i \(0.215876\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 10.8990 0.425534
\(657\) 0 0
\(658\) −0.898979 −0.0350459
\(659\) −7.10102 −0.276616 −0.138308 0.990389i \(-0.544166\pi\)
−0.138308 + 0.990389i \(0.544166\pi\)
\(660\) 0 0
\(661\) 12.9444 0.503478 0.251739 0.967795i \(-0.418997\pi\)
0.251739 + 0.967795i \(0.418997\pi\)
\(662\) 10.6969 0.415748
\(663\) 0 0
\(664\) −2.44949 −0.0950586
\(665\) 0 0
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) −4.89898 −0.189547
\(669\) 0 0
\(670\) 0 0
\(671\) −41.3939 −1.59799
\(672\) 0 0
\(673\) 1.79796 0.0693062 0.0346531 0.999399i \(-0.488967\pi\)
0.0346531 + 0.999399i \(0.488967\pi\)
\(674\) −29.5959 −1.13999
\(675\) 0 0
\(676\) −12.7980 −0.492229
\(677\) 31.5505 1.21258 0.606292 0.795242i \(-0.292656\pi\)
0.606292 + 0.795242i \(0.292656\pi\)
\(678\) 0 0
\(679\) 3.79796 0.145752
\(680\) 0 0
\(681\) 0 0
\(682\) 4.40408 0.168641
\(683\) 35.5959 1.36204 0.681020 0.732265i \(-0.261537\pi\)
0.681020 + 0.732265i \(0.261537\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 8.89898 0.339270
\(689\) −0.494897 −0.0188541
\(690\) 0 0
\(691\) −13.1464 −0.500114 −0.250057 0.968231i \(-0.580449\pi\)
−0.250057 + 0.968231i \(0.580449\pi\)
\(692\) −18.2474 −0.693664
\(693\) 0 0
\(694\) −19.1010 −0.725065
\(695\) 0 0
\(696\) 0 0
\(697\) −21.7980 −0.825657
\(698\) −3.55051 −0.134389
\(699\) 0 0
\(700\) 0 0
\(701\) −40.6969 −1.53710 −0.768551 0.639788i \(-0.779022\pi\)
−0.768551 + 0.639788i \(0.779022\pi\)
\(702\) 0 0
\(703\) 12.8990 0.486494
\(704\) −4.89898 −0.184637
\(705\) 0 0
\(706\) −13.1010 −0.493063
\(707\) 8.44949 0.317776
\(708\) 0 0
\(709\) 40.2929 1.51323 0.756615 0.653861i \(-0.226852\pi\)
0.756615 + 0.653861i \(0.226852\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −6.20204 −0.232268
\(714\) 0 0
\(715\) 0 0
\(716\) 5.79796 0.216680
\(717\) 0 0
\(718\) −11.5959 −0.432756
\(719\) −44.4949 −1.65938 −0.829690 0.558225i \(-0.811483\pi\)
−0.829690 + 0.558225i \(0.811483\pi\)
\(720\) 0 0
\(721\) −3.10102 −0.115488
\(722\) 22.5959 0.840933
\(723\) 0 0
\(724\) 14.2474 0.529502
\(725\) 0 0
\(726\) 0 0
\(727\) −6.69694 −0.248376 −0.124188 0.992259i \(-0.539633\pi\)
−0.124188 + 0.992259i \(0.539633\pi\)
\(728\) 0.449490 0.0166592
\(729\) 0 0
\(730\) 0 0
\(731\) −17.7980 −0.658281
\(732\) 0 0
\(733\) −43.6413 −1.61193 −0.805965 0.591964i \(-0.798353\pi\)
−0.805965 + 0.591964i \(0.798353\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 6.89898 0.254300
\(737\) 39.1918 1.44365
\(738\) 0 0
\(739\) −44.4949 −1.63677 −0.818386 0.574669i \(-0.805131\pi\)
−0.818386 + 0.574669i \(0.805131\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.10102 −0.0404197
\(743\) 15.3031 0.561415 0.280707 0.959793i \(-0.409431\pi\)
0.280707 + 0.959793i \(0.409431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 24.6969 0.904219
\(747\) 0 0
\(748\) 9.79796 0.358249
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −22.2020 −0.810164 −0.405082 0.914280i \(-0.632757\pi\)
−0.405082 + 0.914280i \(0.632757\pi\)
\(752\) 0.898979 0.0327824
\(753\) 0 0
\(754\) −1.30306 −0.0474547
\(755\) 0 0
\(756\) 0 0
\(757\) −32.2020 −1.17040 −0.585202 0.810888i \(-0.698985\pi\)
−0.585202 + 0.810888i \(0.698985\pi\)
\(758\) 1.30306 0.0473293
\(759\) 0 0
\(760\) 0 0
\(761\) 30.8990 1.12009 0.560044 0.828463i \(-0.310784\pi\)
0.560044 + 0.828463i \(0.310784\pi\)
\(762\) 0 0
\(763\) 2.89898 0.104950
\(764\) 16.6969 0.604074
\(765\) 0 0
\(766\) 16.8990 0.610585
\(767\) −2.89898 −0.104676
\(768\) 0 0
\(769\) −11.3031 −0.407599 −0.203799 0.979013i \(-0.565329\pi\)
−0.203799 + 0.979013i \(0.565329\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17.5959 0.633291
\(773\) 13.3485 0.480111 0.240056 0.970759i \(-0.422834\pi\)
0.240056 + 0.970759i \(0.422834\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.79796 −0.136339
\(777\) 0 0
\(778\) −22.8990 −0.820968
\(779\) 70.2929 2.51850
\(780\) 0 0
\(781\) −53.3939 −1.91058
\(782\) −13.7980 −0.493414
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 45.5505 1.62370 0.811850 0.583866i \(-0.198461\pi\)
0.811850 + 0.583866i \(0.198461\pi\)
\(788\) −9.10102 −0.324210
\(789\) 0 0
\(790\) 0 0
\(791\) 0.202041 0.00718375
\(792\) 0 0
\(793\) −3.79796 −0.134869
\(794\) −17.3485 −0.615674
\(795\) 0 0
\(796\) 7.10102 0.251689
\(797\) −52.9444 −1.87539 −0.937693 0.347464i \(-0.887043\pi\)
−0.937693 + 0.347464i \(0.887043\pi\)
\(798\) 0 0
\(799\) −1.79796 −0.0636072
\(800\) 0 0
\(801\) 0 0
\(802\) −29.3939 −1.03793
\(803\) 33.7980 1.19270
\(804\) 0 0
\(805\) 0 0
\(806\) 0.404082 0.0142332
\(807\) 0 0
\(808\) −8.44949 −0.297252
\(809\) −8.40408 −0.295472 −0.147736 0.989027i \(-0.547199\pi\)
−0.147736 + 0.989027i \(0.547199\pi\)
\(810\) 0 0
\(811\) −38.9444 −1.36752 −0.683761 0.729706i \(-0.739657\pi\)
−0.683761 + 0.729706i \(0.739657\pi\)
\(812\) −2.89898 −0.101734
\(813\) 0 0
\(814\) −9.79796 −0.343418
\(815\) 0 0
\(816\) 0 0
\(817\) 57.3939 2.00796
\(818\) −14.4949 −0.506802
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7980 −0.970155 −0.485078 0.874471i \(-0.661209\pi\)
−0.485078 + 0.874471i \(0.661209\pi\)
\(822\) 0 0
\(823\) 39.1918 1.36614 0.683071 0.730352i \(-0.260644\pi\)
0.683071 + 0.730352i \(0.260644\pi\)
\(824\) 3.10102 0.108029
\(825\) 0 0
\(826\) −6.44949 −0.224406
\(827\) −23.5959 −0.820510 −0.410255 0.911971i \(-0.634560\pi\)
−0.410255 + 0.911971i \(0.634560\pi\)
\(828\) 0 0
\(829\) 39.6413 1.37680 0.688400 0.725331i \(-0.258313\pi\)
0.688400 + 0.725331i \(0.258313\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.449490 −0.0155833
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) −31.5959 −1.09277
\(837\) 0 0
\(838\) 6.44949 0.222794
\(839\) −27.1010 −0.935631 −0.467816 0.883826i \(-0.654959\pi\)
−0.467816 + 0.883826i \(0.654959\pi\)
\(840\) 0 0
\(841\) −20.5959 −0.710204
\(842\) −23.7980 −0.820132
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) −13.0000 −0.446685
\(848\) 1.10102 0.0378092
\(849\) 0 0
\(850\) 0 0
\(851\) 13.7980 0.472988
\(852\) 0 0
\(853\) 29.8434 1.02182 0.510909 0.859635i \(-0.329309\pi\)
0.510909 + 0.859635i \(0.329309\pi\)
\(854\) −8.44949 −0.289136
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −25.1918 −0.860537 −0.430268 0.902701i \(-0.641581\pi\)
−0.430268 + 0.902701i \(0.641581\pi\)
\(858\) 0 0
\(859\) −29.6413 −1.01135 −0.505674 0.862724i \(-0.668756\pi\)
−0.505674 + 0.862724i \(0.668756\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −17.7980 −0.606201
\(863\) −13.3939 −0.455933 −0.227966 0.973669i \(-0.573208\pi\)
−0.227966 + 0.973669i \(0.573208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −19.7980 −0.672762
\(867\) 0 0
\(868\) 0.898979 0.0305134
\(869\) 14.2020 0.481771
\(870\) 0 0
\(871\) 3.59592 0.121843
\(872\) −2.89898 −0.0981718
\(873\) 0 0
\(874\) 44.4949 1.50506
\(875\) 0 0
\(876\) 0 0
\(877\) 19.3939 0.654885 0.327442 0.944871i \(-0.393813\pi\)
0.327442 + 0.944871i \(0.393813\pi\)
\(878\) 37.3939 1.26198
\(879\) 0 0
\(880\) 0 0
\(881\) −27.7980 −0.936537 −0.468269 0.883586i \(-0.655122\pi\)
−0.468269 + 0.883586i \(0.655122\pi\)
\(882\) 0 0
\(883\) 41.7980 1.40661 0.703307 0.710887i \(-0.251706\pi\)
0.703307 + 0.710887i \(0.251706\pi\)
\(884\) 0.898979 0.0302360
\(885\) 0 0
\(886\) 9.79796 0.329169
\(887\) 26.6969 0.896395 0.448198 0.893934i \(-0.352066\pi\)
0.448198 + 0.893934i \(0.352066\pi\)
\(888\) 0 0
\(889\) 5.10102 0.171083
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 5.79796 0.194021
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −2.60612 −0.0869191
\(900\) 0 0
\(901\) −2.20204 −0.0733606
\(902\) −53.3939 −1.77782
\(903\) 0 0
\(904\) −0.202041 −0.00671978
\(905\) 0 0
\(906\) 0 0
\(907\) −22.2020 −0.737207 −0.368603 0.929587i \(-0.620164\pi\)
−0.368603 + 0.929587i \(0.620164\pi\)
\(908\) 7.34847 0.243868
\(909\) 0 0
\(910\) 0 0
\(911\) −3.59592 −0.119138 −0.0595690 0.998224i \(-0.518973\pi\)
−0.0595690 + 0.998224i \(0.518973\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −9.59592 −0.317405
\(915\) 0 0
\(916\) 15.1464 0.500452
\(917\) −1.55051 −0.0512024
\(918\) 0 0
\(919\) −17.1010 −0.564111 −0.282055 0.959398i \(-0.591016\pi\)
−0.282055 + 0.959398i \(0.591016\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.65153 −0.0873235
\(923\) −4.89898 −0.161252
\(924\) 0 0
\(925\) 0 0
\(926\) −35.5959 −1.16975
\(927\) 0 0
\(928\) 2.89898 0.0951637
\(929\) −40.2929 −1.32197 −0.660983 0.750401i \(-0.729860\pi\)
−0.660983 + 0.750401i \(0.729860\pi\)
\(930\) 0 0
\(931\) 6.44949 0.211373
\(932\) −10.2020 −0.334179
\(933\) 0 0
\(934\) −5.55051 −0.181618
\(935\) 0 0
\(936\) 0 0
\(937\) −50.8990 −1.66280 −0.831399 0.555677i \(-0.812459\pi\)
−0.831399 + 0.555677i \(0.812459\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) 24.4495 0.797031 0.398515 0.917162i \(-0.369526\pi\)
0.398515 + 0.917162i \(0.369526\pi\)
\(942\) 0 0
\(943\) 75.1918 2.44858
\(944\) 6.44949 0.209913
\(945\) 0 0
\(946\) −43.5959 −1.41743
\(947\) 44.0908 1.43276 0.716379 0.697711i \(-0.245798\pi\)
0.716379 + 0.697711i \(0.245798\pi\)
\(948\) 0 0
\(949\) 3.10102 0.100663
\(950\) 0 0
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) −21.7980 −0.706105 −0.353053 0.935603i \(-0.614856\pi\)
−0.353053 + 0.935603i \(0.614856\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 25.7980 0.834366
\(957\) 0 0
\(958\) −38.6969 −1.25024
\(959\) 17.7980 0.574726
\(960\) 0 0
\(961\) −30.1918 −0.973930
\(962\) −0.898979 −0.0289843
\(963\) 0 0
\(964\) 20.6969 0.666604
\(965\) 0 0
\(966\) 0 0
\(967\) 32.2929 1.03847 0.519234 0.854632i \(-0.326217\pi\)
0.519234 + 0.854632i \(0.326217\pi\)
\(968\) 13.0000 0.417836
\(969\) 0 0
\(970\) 0 0
\(971\) 14.4495 0.463706 0.231853 0.972751i \(-0.425521\pi\)
0.231853 + 0.972751i \(0.425521\pi\)
\(972\) 0 0
\(973\) −6.44949 −0.206761
\(974\) −36.6969 −1.17585
\(975\) 0 0
\(976\) 8.44949 0.270462
\(977\) −29.3939 −0.940393 −0.470197 0.882562i \(-0.655817\pi\)
−0.470197 + 0.882562i \(0.655817\pi\)
\(978\) 0 0
\(979\) −48.9898 −1.56572
\(980\) 0 0
\(981\) 0 0
\(982\) 19.5959 0.625331
\(983\) 42.6969 1.36182 0.680910 0.732367i \(-0.261584\pi\)
0.680910 + 0.732367i \(0.261584\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.79796 −0.184645
\(987\) 0 0
\(988\) −2.89898 −0.0922288
\(989\) 61.3939 1.95221
\(990\) 0 0
\(991\) 60.6969 1.92810 0.964051 0.265718i \(-0.0856089\pi\)
0.964051 + 0.265718i \(0.0856089\pi\)
\(992\) −0.898979 −0.0285426
\(993\) 0 0
\(994\) −10.8990 −0.345695
\(995\) 0 0
\(996\) 0 0
\(997\) 42.6515 1.35079 0.675394 0.737457i \(-0.263974\pi\)
0.675394 + 0.737457i \(0.263974\pi\)
\(998\) 25.7980 0.816620
\(999\) 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 3150.2.a.bt.1.1 2
3.2 odd 2 350.2.a.g.1.2 2
5.2 odd 4 630.2.g.g.379.4 4
5.3 odd 4 630.2.g.g.379.2 4
5.4 even 2 3150.2.a.bs.1.1 2
12.11 even 2 2800.2.a.bm.1.1 2
15.2 even 4 70.2.c.a.29.1 4
15.8 even 4 70.2.c.a.29.4 yes 4
15.14 odd 2 350.2.a.h.1.1 2
20.3 even 4 5040.2.t.t.1009.4 4
20.7 even 4 5040.2.t.t.1009.3 4
21.20 even 2 2450.2.a.bl.1.1 2
60.23 odd 4 560.2.g.e.449.1 4
60.47 odd 4 560.2.g.e.449.3 4
60.59 even 2 2800.2.a.bl.1.2 2
105.2 even 12 490.2.i.c.459.2 8
105.17 odd 12 490.2.i.f.79.4 8
105.23 even 12 490.2.i.c.459.3 8
105.32 even 12 490.2.i.c.79.3 8
105.38 odd 12 490.2.i.f.79.1 8
105.47 odd 12 490.2.i.f.459.1 8
105.53 even 12 490.2.i.c.79.2 8
105.62 odd 4 490.2.c.e.99.2 4
105.68 odd 12 490.2.i.f.459.4 8
105.83 odd 4 490.2.c.e.99.3 4
105.104 even 2 2450.2.a.bq.1.2 2
120.53 even 4 2240.2.g.j.449.2 4
120.77 even 4 2240.2.g.j.449.4 4
120.83 odd 4 2240.2.g.i.449.4 4
120.107 odd 4 2240.2.g.i.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.c.a.29.1 4 15.2 even 4
70.2.c.a.29.4 yes 4 15.8 even 4
350.2.a.g.1.2 2 3.2 odd 2
350.2.a.h.1.1 2 15.14 odd 2
490.2.c.e.99.2 4 105.62 odd 4
490.2.c.e.99.3 4 105.83 odd 4
490.2.i.c.79.2 8 105.53 even 12
490.2.i.c.79.3 8 105.32 even 12
490.2.i.c.459.2 8 105.2 even 12
490.2.i.c.459.3 8 105.23 even 12
490.2.i.f.79.1 8 105.38 odd 12
490.2.i.f.79.4 8 105.17 odd 12
490.2.i.f.459.1 8 105.47 odd 12
490.2.i.f.459.4 8 105.68 odd 12
560.2.g.e.449.1 4 60.23 odd 4
560.2.g.e.449.3 4 60.47 odd 4
630.2.g.g.379.2 4 5.3 odd 4
630.2.g.g.379.4 4 5.2 odd 4
2240.2.g.i.449.2 4 120.107 odd 4
2240.2.g.i.449.4 4 120.83 odd 4
2240.2.g.j.449.2 4 120.53 even 4
2240.2.g.j.449.4 4 120.77 even 4
2450.2.a.bl.1.1 2 21.20 even 2
2450.2.a.bq.1.2 2 105.104 even 2
2800.2.a.bl.1.2 2 60.59 even 2
2800.2.a.bm.1.1 2 12.11 even 2
3150.2.a.bs.1.1 2 5.4 even 2
3150.2.a.bt.1.1 2 1.1 even 1 trivial
5040.2.t.t.1009.3 4 20.7 even 4
5040.2.t.t.1009.4 4 20.3 even 4