Properties

Label 5520.2.a.bj.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.87298\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +2.87298 q^{11} +4.87298 q^{13} -1.00000 q^{15} -3.87298 q^{17} -4.87298 q^{19} +3.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.87298 q^{29} -3.00000 q^{31} -2.87298 q^{33} -3.00000 q^{35} +1.00000 q^{37} -4.87298 q^{39} +1.87298 q^{41} +11.7460 q^{43} +1.00000 q^{45} -0.872983 q^{47} +2.00000 q^{49} +3.87298 q^{51} +3.87298 q^{53} +2.87298 q^{55} +4.87298 q^{57} +1.87298 q^{59} +1.12702 q^{61} -3.00000 q^{63} +4.87298 q^{65} +4.74597 q^{67} -1.00000 q^{69} -9.61895 q^{71} +4.87298 q^{73} -1.00000 q^{75} -8.61895 q^{77} -4.00000 q^{79} +1.00000 q^{81} +7.87298 q^{83} -3.87298 q^{85} -1.87298 q^{87} -13.7460 q^{89} -14.6190 q^{91} +3.00000 q^{93} -4.87298 q^{95} +8.00000 q^{97} +2.87298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{19} + 6 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 6 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} - 2 q^{39} - 4 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{47} + 4 q^{49} - 2 q^{55} + 2 q^{57} - 4 q^{59} + 10 q^{61} - 6 q^{63} + 2 q^{65} - 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 6 q^{77} - 8 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{87} - 12 q^{89} - 6 q^{91} + 6 q^{93} - 2 q^{95} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.87298 0.866237 0.433119 0.901337i \(-0.357413\pi\)
0.433119 + 0.901337i \(0.357413\pi\)
\(12\) 0 0
\(13\) 4.87298 1.35152 0.675761 0.737121i \(-0.263815\pi\)
0.675761 + 0.737121i \(0.263815\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −3.87298 −0.939336 −0.469668 0.882843i \(-0.655626\pi\)
−0.469668 + 0.882843i \(0.655626\pi\)
\(18\) 0 0
\(19\) −4.87298 −1.11794 −0.558970 0.829188i \(-0.688803\pi\)
−0.558970 + 0.829188i \(0.688803\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.87298 0.347804 0.173902 0.984763i \(-0.444362\pi\)
0.173902 + 0.984763i \(0.444362\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −2.87298 −0.500122
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) −4.87298 −0.780302
\(40\) 0 0
\(41\) 1.87298 0.292511 0.146255 0.989247i \(-0.453278\pi\)
0.146255 + 0.989247i \(0.453278\pi\)
\(42\) 0 0
\(43\) 11.7460 1.79124 0.895622 0.444817i \(-0.146731\pi\)
0.895622 + 0.444817i \(0.146731\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −0.872983 −0.127338 −0.0636689 0.997971i \(-0.520280\pi\)
−0.0636689 + 0.997971i \(0.520280\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 3.87298 0.542326
\(52\) 0 0
\(53\) 3.87298 0.531995 0.265998 0.963974i \(-0.414299\pi\)
0.265998 + 0.963974i \(0.414299\pi\)
\(54\) 0 0
\(55\) 2.87298 0.387393
\(56\) 0 0
\(57\) 4.87298 0.645442
\(58\) 0 0
\(59\) 1.87298 0.243842 0.121921 0.992540i \(-0.461095\pi\)
0.121921 + 0.992540i \(0.461095\pi\)
\(60\) 0 0
\(61\) 1.12702 0.144300 0.0721498 0.997394i \(-0.477014\pi\)
0.0721498 + 0.997394i \(0.477014\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 4.87298 0.604419
\(66\) 0 0
\(67\) 4.74597 0.579812 0.289906 0.957055i \(-0.406376\pi\)
0.289906 + 0.957055i \(0.406376\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −9.61895 −1.14156 −0.570780 0.821103i \(-0.693359\pi\)
−0.570780 + 0.821103i \(0.693359\pi\)
\(72\) 0 0
\(73\) 4.87298 0.570340 0.285170 0.958477i \(-0.407950\pi\)
0.285170 + 0.958477i \(0.407950\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −8.61895 −0.982221
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.87298 0.864172 0.432086 0.901832i \(-0.357778\pi\)
0.432086 + 0.901832i \(0.357778\pi\)
\(84\) 0 0
\(85\) −3.87298 −0.420084
\(86\) 0 0
\(87\) −1.87298 −0.200805
\(88\) 0 0
\(89\) −13.7460 −1.45707 −0.728535 0.685009i \(-0.759798\pi\)
−0.728535 + 0.685009i \(0.759798\pi\)
\(90\) 0 0
\(91\) −14.6190 −1.53248
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −4.87298 −0.499958
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 2.87298 0.288746
\(100\) 0 0
\(101\) 3.87298 0.385376 0.192688 0.981260i \(-0.438279\pi\)
0.192688 + 0.981260i \(0.438279\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) −17.6190 −1.70329 −0.851644 0.524121i \(-0.824394\pi\)
−0.851644 + 0.524121i \(0.824394\pi\)
\(108\) 0 0
\(109\) −6.61895 −0.633980 −0.316990 0.948429i \(-0.602672\pi\)
−0.316990 + 0.948429i \(0.602672\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) 3.87298 0.364340 0.182170 0.983267i \(-0.441688\pi\)
0.182170 + 0.983267i \(0.441688\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 4.87298 0.450507
\(118\) 0 0
\(119\) 11.6190 1.06511
\(120\) 0 0
\(121\) −2.74597 −0.249633
\(122\) 0 0
\(123\) −1.87298 −0.168881
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.6190 −1.47469 −0.737347 0.675515i \(-0.763922\pi\)
−0.737347 + 0.675515i \(0.763922\pi\)
\(128\) 0 0
\(129\) −11.7460 −1.03417
\(130\) 0 0
\(131\) 19.4919 1.70302 0.851509 0.524340i \(-0.175688\pi\)
0.851509 + 0.524340i \(0.175688\pi\)
\(132\) 0 0
\(133\) 14.6190 1.26762
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) −10.4919 −0.889914 −0.444957 0.895552i \(-0.646781\pi\)
−0.444957 + 0.895552i \(0.646781\pi\)
\(140\) 0 0
\(141\) 0.872983 0.0735185
\(142\) 0 0
\(143\) 14.0000 1.17074
\(144\) 0 0
\(145\) 1.87298 0.155543
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −3.12702 −0.256175 −0.128088 0.991763i \(-0.540884\pi\)
−0.128088 + 0.991763i \(0.540884\pi\)
\(150\) 0 0
\(151\) 11.7460 0.955873 0.477937 0.878394i \(-0.341385\pi\)
0.477937 + 0.878394i \(0.341385\pi\)
\(152\) 0 0
\(153\) −3.87298 −0.313112
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) −3.87298 −0.307148
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) −2.87298 −0.223661
\(166\) 0 0
\(167\) 16.8730 1.30567 0.652835 0.757500i \(-0.273579\pi\)
0.652835 + 0.757500i \(0.273579\pi\)
\(168\) 0 0
\(169\) 10.7460 0.826613
\(170\) 0 0
\(171\) −4.87298 −0.372646
\(172\) 0 0
\(173\) −7.49193 −0.569601 −0.284801 0.958587i \(-0.591927\pi\)
−0.284801 + 0.958587i \(0.591927\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) −1.87298 −0.140782
\(178\) 0 0
\(179\) 0.254033 0.0189873 0.00949367 0.999955i \(-0.496978\pi\)
0.00949367 + 0.999955i \(0.496978\pi\)
\(180\) 0 0
\(181\) −6.25403 −0.464859 −0.232429 0.972613i \(-0.574667\pi\)
−0.232429 + 0.972613i \(0.574667\pi\)
\(182\) 0 0
\(183\) −1.12702 −0.0833115
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −11.1270 −0.813688
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 11.1270 0.805123 0.402561 0.915393i \(-0.368120\pi\)
0.402561 + 0.915393i \(0.368120\pi\)
\(192\) 0 0
\(193\) −5.74597 −0.413604 −0.206802 0.978383i \(-0.566306\pi\)
−0.206802 + 0.978383i \(0.566306\pi\)
\(194\) 0 0
\(195\) −4.87298 −0.348962
\(196\) 0 0
\(197\) 19.7460 1.40684 0.703421 0.710774i \(-0.251655\pi\)
0.703421 + 0.710774i \(0.251655\pi\)
\(198\) 0 0
\(199\) 27.2379 1.93084 0.965422 0.260693i \(-0.0839510\pi\)
0.965422 + 0.260693i \(0.0839510\pi\)
\(200\) 0 0
\(201\) −4.74597 −0.334755
\(202\) 0 0
\(203\) −5.61895 −0.394373
\(204\) 0 0
\(205\) 1.87298 0.130815
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −14.0000 −0.968400
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 9.61895 0.659080
\(214\) 0 0
\(215\) 11.7460 0.801068
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) −4.87298 −0.329286
\(220\) 0 0
\(221\) −18.8730 −1.26953
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 23.4919 1.55921 0.779607 0.626269i \(-0.215419\pi\)
0.779607 + 0.626269i \(0.215419\pi\)
\(228\) 0 0
\(229\) 9.74597 0.644032 0.322016 0.946734i \(-0.395640\pi\)
0.322016 + 0.946734i \(0.395640\pi\)
\(230\) 0 0
\(231\) 8.61895 0.567085
\(232\) 0 0
\(233\) −15.4919 −1.01491 −0.507455 0.861678i \(-0.669414\pi\)
−0.507455 + 0.861678i \(0.669414\pi\)
\(234\) 0 0
\(235\) −0.872983 −0.0569472
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 10.1270 0.655062 0.327531 0.944840i \(-0.393783\pi\)
0.327531 + 0.944840i \(0.393783\pi\)
\(240\) 0 0
\(241\) −12.8730 −0.829222 −0.414611 0.909999i \(-0.636082\pi\)
−0.414611 + 0.909999i \(0.636082\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −23.7460 −1.51092
\(248\) 0 0
\(249\) −7.87298 −0.498930
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 2.87298 0.180623
\(254\) 0 0
\(255\) 3.87298 0.242536
\(256\) 0 0
\(257\) 6.61895 0.412879 0.206439 0.978459i \(-0.433812\pi\)
0.206439 + 0.978459i \(0.433812\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 1.87298 0.115935
\(262\) 0 0
\(263\) 14.1270 0.871109 0.435555 0.900162i \(-0.356552\pi\)
0.435555 + 0.900162i \(0.356552\pi\)
\(264\) 0 0
\(265\) 3.87298 0.237915
\(266\) 0 0
\(267\) 13.7460 0.841239
\(268\) 0 0
\(269\) 15.3649 0.936816 0.468408 0.883512i \(-0.344828\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(270\) 0 0
\(271\) 22.7460 1.38172 0.690860 0.722989i \(-0.257232\pi\)
0.690860 + 0.722989i \(0.257232\pi\)
\(272\) 0 0
\(273\) 14.6190 0.884779
\(274\) 0 0
\(275\) 2.87298 0.173247
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 22.3649 1.33418 0.667090 0.744978i \(-0.267540\pi\)
0.667090 + 0.744978i \(0.267540\pi\)
\(282\) 0 0
\(283\) 24.2379 1.44079 0.720397 0.693562i \(-0.243960\pi\)
0.720397 + 0.693562i \(0.243960\pi\)
\(284\) 0 0
\(285\) 4.87298 0.288651
\(286\) 0 0
\(287\) −5.61895 −0.331676
\(288\) 0 0
\(289\) −2.00000 −0.117647
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 9.61895 0.561945 0.280973 0.959716i \(-0.409343\pi\)
0.280973 + 0.959716i \(0.409343\pi\)
\(294\) 0 0
\(295\) 1.87298 0.109049
\(296\) 0 0
\(297\) −2.87298 −0.166707
\(298\) 0 0
\(299\) 4.87298 0.281812
\(300\) 0 0
\(301\) −35.2379 −2.03108
\(302\) 0 0
\(303\) −3.87298 −0.222497
\(304\) 0 0
\(305\) 1.12702 0.0645328
\(306\) 0 0
\(307\) −16.8730 −0.962992 −0.481496 0.876448i \(-0.659906\pi\)
−0.481496 + 0.876448i \(0.659906\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 7.00000 0.395663 0.197832 0.980236i \(-0.436610\pi\)
0.197832 + 0.980236i \(0.436610\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −16.3649 −0.919145 −0.459573 0.888140i \(-0.651997\pi\)
−0.459573 + 0.888140i \(0.651997\pi\)
\(318\) 0 0
\(319\) 5.38105 0.301281
\(320\) 0 0
\(321\) 17.6190 0.983394
\(322\) 0 0
\(323\) 18.8730 1.05012
\(324\) 0 0
\(325\) 4.87298 0.270304
\(326\) 0 0
\(327\) 6.61895 0.366029
\(328\) 0 0
\(329\) 2.61895 0.144387
\(330\) 0 0
\(331\) −24.2379 −1.33224 −0.666118 0.745847i \(-0.732045\pi\)
−0.666118 + 0.745847i \(0.732045\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) 4.74597 0.259300
\(336\) 0 0
\(337\) 29.4919 1.60653 0.803264 0.595623i \(-0.203095\pi\)
0.803264 + 0.595623i \(0.203095\pi\)
\(338\) 0 0
\(339\) −3.87298 −0.210352
\(340\) 0 0
\(341\) −8.61895 −0.466742
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 10.2540 0.550465 0.275233 0.961378i \(-0.411245\pi\)
0.275233 + 0.961378i \(0.411245\pi\)
\(348\) 0 0
\(349\) 0.745967 0.0399307 0.0199653 0.999801i \(-0.493644\pi\)
0.0199653 + 0.999801i \(0.493644\pi\)
\(350\) 0 0
\(351\) −4.87298 −0.260101
\(352\) 0 0
\(353\) −8.87298 −0.472261 −0.236131 0.971721i \(-0.575879\pi\)
−0.236131 + 0.971721i \(0.575879\pi\)
\(354\) 0 0
\(355\) −9.61895 −0.510521
\(356\) 0 0
\(357\) −11.6190 −0.614940
\(358\) 0 0
\(359\) 28.8730 1.52386 0.761929 0.647661i \(-0.224253\pi\)
0.761929 + 0.647661i \(0.224253\pi\)
\(360\) 0 0
\(361\) 4.74597 0.249788
\(362\) 0 0
\(363\) 2.74597 0.144126
\(364\) 0 0
\(365\) 4.87298 0.255064
\(366\) 0 0
\(367\) 35.9839 1.87834 0.939171 0.343449i \(-0.111595\pi\)
0.939171 + 0.343449i \(0.111595\pi\)
\(368\) 0 0
\(369\) 1.87298 0.0975036
\(370\) 0 0
\(371\) −11.6190 −0.603226
\(372\) 0 0
\(373\) −13.7460 −0.711739 −0.355870 0.934536i \(-0.615815\pi\)
−0.355870 + 0.934536i \(0.615815\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 9.12702 0.470065
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 16.6190 0.851415
\(382\) 0 0
\(383\) 2.38105 0.121666 0.0608330 0.998148i \(-0.480624\pi\)
0.0608330 + 0.998148i \(0.480624\pi\)
\(384\) 0 0
\(385\) −8.61895 −0.439262
\(386\) 0 0
\(387\) 11.7460 0.597081
\(388\) 0 0
\(389\) −11.7460 −0.595544 −0.297772 0.954637i \(-0.596244\pi\)
−0.297772 + 0.954637i \(0.596244\pi\)
\(390\) 0 0
\(391\) −3.87298 −0.195865
\(392\) 0 0
\(393\) −19.4919 −0.983238
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −11.4919 −0.576764 −0.288382 0.957516i \(-0.593117\pi\)
−0.288382 + 0.957516i \(0.593117\pi\)
\(398\) 0 0
\(399\) −14.6190 −0.731863
\(400\) 0 0
\(401\) −19.2379 −0.960695 −0.480347 0.877078i \(-0.659489\pi\)
−0.480347 + 0.877078i \(0.659489\pi\)
\(402\) 0 0
\(403\) −14.6190 −0.728222
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 2.87298 0.142408
\(408\) 0 0
\(409\) −24.2379 −1.19849 −0.599244 0.800567i \(-0.704532\pi\)
−0.599244 + 0.800567i \(0.704532\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) −5.61895 −0.276490
\(414\) 0 0
\(415\) 7.87298 0.386470
\(416\) 0 0
\(417\) 10.4919 0.513792
\(418\) 0 0
\(419\) 20.6190 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(420\) 0 0
\(421\) −31.1270 −1.51704 −0.758519 0.651651i \(-0.774077\pi\)
−0.758519 + 0.651651i \(0.774077\pi\)
\(422\) 0 0
\(423\) −0.872983 −0.0424459
\(424\) 0 0
\(425\) −3.87298 −0.187867
\(426\) 0 0
\(427\) −3.38105 −0.163620
\(428\) 0 0
\(429\) −14.0000 −0.675926
\(430\) 0 0
\(431\) 9.74597 0.469447 0.234723 0.972062i \(-0.424582\pi\)
0.234723 + 0.972062i \(0.424582\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 0 0
\(435\) −1.87298 −0.0898027
\(436\) 0 0
\(437\) −4.87298 −0.233106
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 1.38105 0.0656157 0.0328078 0.999462i \(-0.489555\pi\)
0.0328078 + 0.999462i \(0.489555\pi\)
\(444\) 0 0
\(445\) −13.7460 −0.651621
\(446\) 0 0
\(447\) 3.12702 0.147903
\(448\) 0 0
\(449\) −14.1270 −0.666695 −0.333348 0.942804i \(-0.608178\pi\)
−0.333348 + 0.942804i \(0.608178\pi\)
\(450\) 0 0
\(451\) 5.38105 0.253384
\(452\) 0 0
\(453\) −11.7460 −0.551874
\(454\) 0 0
\(455\) −14.6190 −0.685347
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 3.87298 0.180775
\(460\) 0 0
\(461\) 16.2540 0.757026 0.378513 0.925596i \(-0.376436\pi\)
0.378513 + 0.925596i \(0.376436\pi\)
\(462\) 0 0
\(463\) −27.1270 −1.26070 −0.630350 0.776311i \(-0.717088\pi\)
−0.630350 + 0.776311i \(0.717088\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) −17.6190 −0.815308 −0.407654 0.913137i \(-0.633653\pi\)
−0.407654 + 0.913137i \(0.633653\pi\)
\(468\) 0 0
\(469\) −14.2379 −0.657445
\(470\) 0 0
\(471\) −17.0000 −0.783319
\(472\) 0 0
\(473\) 33.7460 1.55164
\(474\) 0 0
\(475\) −4.87298 −0.223588
\(476\) 0 0
\(477\) 3.87298 0.177332
\(478\) 0 0
\(479\) 27.1270 1.23947 0.619733 0.784813i \(-0.287241\pi\)
0.619733 + 0.784813i \(0.287241\pi\)
\(480\) 0 0
\(481\) 4.87298 0.222189
\(482\) 0 0
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −28.1109 −1.27383 −0.636913 0.770936i \(-0.719789\pi\)
−0.636913 + 0.770936i \(0.719789\pi\)
\(488\) 0 0
\(489\) −22.0000 −0.994874
\(490\) 0 0
\(491\) −35.8730 −1.61893 −0.809463 0.587172i \(-0.800241\pi\)
−0.809463 + 0.587172i \(0.800241\pi\)
\(492\) 0 0
\(493\) −7.25403 −0.326705
\(494\) 0 0
\(495\) 2.87298 0.129131
\(496\) 0 0
\(497\) 28.8569 1.29441
\(498\) 0 0
\(499\) −18.7460 −0.839185 −0.419592 0.907713i \(-0.637827\pi\)
−0.419592 + 0.907713i \(0.637827\pi\)
\(500\) 0 0
\(501\) −16.8730 −0.753829
\(502\) 0 0
\(503\) −17.1109 −0.762937 −0.381468 0.924382i \(-0.624581\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(504\) 0 0
\(505\) 3.87298 0.172345
\(506\) 0 0
\(507\) −10.7460 −0.477245
\(508\) 0 0
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −14.6190 −0.646704
\(512\) 0 0
\(513\) 4.87298 0.215147
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) −2.50807 −0.110305
\(518\) 0 0
\(519\) 7.49193 0.328859
\(520\) 0 0
\(521\) 33.1270 1.45132 0.725660 0.688053i \(-0.241534\pi\)
0.725660 + 0.688053i \(0.241534\pi\)
\(522\) 0 0
\(523\) −42.9839 −1.87955 −0.939777 0.341789i \(-0.888967\pi\)
−0.939777 + 0.341789i \(0.888967\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 11.6190 0.506129
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.87298 0.0812806
\(532\) 0 0
\(533\) 9.12702 0.395335
\(534\) 0 0
\(535\) −17.6190 −0.761734
\(536\) 0 0
\(537\) −0.254033 −0.0109623
\(538\) 0 0
\(539\) 5.74597 0.247496
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) 6.25403 0.268386
\(544\) 0 0
\(545\) −6.61895 −0.283525
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 0 0
\(549\) 1.12702 0.0480999
\(550\) 0 0
\(551\) −9.12702 −0.388824
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) −1.00000 −0.0424476
\(556\) 0 0
\(557\) 35.8730 1.51999 0.759994 0.649931i \(-0.225202\pi\)
0.759994 + 0.649931i \(0.225202\pi\)
\(558\) 0 0
\(559\) 57.2379 2.42091
\(560\) 0 0
\(561\) 11.1270 0.469783
\(562\) 0 0
\(563\) 21.1109 0.889718 0.444859 0.895601i \(-0.353254\pi\)
0.444859 + 0.895601i \(0.353254\pi\)
\(564\) 0 0
\(565\) 3.87298 0.162938
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 19.7460 0.827794 0.413897 0.910324i \(-0.364167\pi\)
0.413897 + 0.910324i \(0.364167\pi\)
\(570\) 0 0
\(571\) 2.36492 0.0989687 0.0494843 0.998775i \(-0.484242\pi\)
0.0494843 + 0.998775i \(0.484242\pi\)
\(572\) 0 0
\(573\) −11.1270 −0.464838
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 0 0
\(579\) 5.74597 0.238794
\(580\) 0 0
\(581\) −23.6190 −0.979879
\(582\) 0 0
\(583\) 11.1270 0.460834
\(584\) 0 0
\(585\) 4.87298 0.201473
\(586\) 0 0
\(587\) 17.4919 0.721969 0.360985 0.932572i \(-0.382441\pi\)
0.360985 + 0.932572i \(0.382441\pi\)
\(588\) 0 0
\(589\) 14.6190 0.602363
\(590\) 0 0
\(591\) −19.7460 −0.812241
\(592\) 0 0
\(593\) 21.3810 0.878014 0.439007 0.898484i \(-0.355330\pi\)
0.439007 + 0.898484i \(0.355330\pi\)
\(594\) 0 0
\(595\) 11.6190 0.476331
\(596\) 0 0
\(597\) −27.2379 −1.11477
\(598\) 0 0
\(599\) −6.25403 −0.255533 −0.127766 0.991804i \(-0.540781\pi\)
−0.127766 + 0.991804i \(0.540781\pi\)
\(600\) 0 0
\(601\) 2.74597 0.112010 0.0560052 0.998430i \(-0.482164\pi\)
0.0560052 + 0.998430i \(0.482164\pi\)
\(602\) 0 0
\(603\) 4.74597 0.193271
\(604\) 0 0
\(605\) −2.74597 −0.111639
\(606\) 0 0
\(607\) −38.8730 −1.57781 −0.788903 0.614518i \(-0.789351\pi\)
−0.788903 + 0.614518i \(0.789351\pi\)
\(608\) 0 0
\(609\) 5.61895 0.227691
\(610\) 0 0
\(611\) −4.25403 −0.172100
\(612\) 0 0
\(613\) −12.9839 −0.524413 −0.262207 0.965012i \(-0.584450\pi\)
−0.262207 + 0.965012i \(0.584450\pi\)
\(614\) 0 0
\(615\) −1.87298 −0.0755260
\(616\) 0 0
\(617\) 5.61895 0.226210 0.113105 0.993583i \(-0.463920\pi\)
0.113105 + 0.993583i \(0.463920\pi\)
\(618\) 0 0
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 41.2379 1.65216
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 14.0000 0.559106
\(628\) 0 0
\(629\) −3.87298 −0.154426
\(630\) 0 0
\(631\) 6.36492 0.253383 0.126692 0.991942i \(-0.459564\pi\)
0.126692 + 0.991942i \(0.459564\pi\)
\(632\) 0 0
\(633\) 1.00000 0.0397464
\(634\) 0 0
\(635\) −16.6190 −0.659503
\(636\) 0 0
\(637\) 9.74597 0.386149
\(638\) 0 0
\(639\) −9.61895 −0.380520
\(640\) 0 0
\(641\) 44.1109 1.74228 0.871138 0.491039i \(-0.163383\pi\)
0.871138 + 0.491039i \(0.163383\pi\)
\(642\) 0 0
\(643\) −8.49193 −0.334889 −0.167445 0.985881i \(-0.553552\pi\)
−0.167445 + 0.985881i \(0.553552\pi\)
\(644\) 0 0
\(645\) −11.7460 −0.462497
\(646\) 0 0
\(647\) 41.8569 1.64556 0.822781 0.568358i \(-0.192421\pi\)
0.822781 + 0.568358i \(0.192421\pi\)
\(648\) 0 0
\(649\) 5.38105 0.211225
\(650\) 0 0
\(651\) −9.00000 −0.352738
\(652\) 0 0
\(653\) 44.1109 1.72619 0.863096 0.505040i \(-0.168522\pi\)
0.863096 + 0.505040i \(0.168522\pi\)
\(654\) 0 0
\(655\) 19.4919 0.761613
\(656\) 0 0
\(657\) 4.87298 0.190113
\(658\) 0 0
\(659\) −21.1270 −0.822992 −0.411496 0.911412i \(-0.634994\pi\)
−0.411496 + 0.911412i \(0.634994\pi\)
\(660\) 0 0
\(661\) 3.23790 0.125940 0.0629699 0.998015i \(-0.479943\pi\)
0.0629699 + 0.998015i \(0.479943\pi\)
\(662\) 0 0
\(663\) 18.8730 0.732966
\(664\) 0 0
\(665\) 14.6190 0.566899
\(666\) 0 0
\(667\) 1.87298 0.0725222
\(668\) 0 0
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) 3.23790 0.124998
\(672\) 0 0
\(673\) 26.8730 1.03588 0.517939 0.855418i \(-0.326700\pi\)
0.517939 + 0.855418i \(0.326700\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 47.1109 1.81062 0.905309 0.424753i \(-0.139639\pi\)
0.905309 + 0.424753i \(0.139639\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) −23.4919 −0.900213
\(682\) 0 0
\(683\) −24.8730 −0.951738 −0.475869 0.879516i \(-0.657866\pi\)
−0.475869 + 0.879516i \(0.657866\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 0 0
\(687\) −9.74597 −0.371832
\(688\) 0 0
\(689\) 18.8730 0.719003
\(690\) 0 0
\(691\) 30.7298 1.16902 0.584509 0.811387i \(-0.301287\pi\)
0.584509 + 0.811387i \(0.301287\pi\)
\(692\) 0 0
\(693\) −8.61895 −0.327407
\(694\) 0 0
\(695\) −10.4919 −0.397982
\(696\) 0 0
\(697\) −7.25403 −0.274766
\(698\) 0 0
\(699\) 15.4919 0.585959
\(700\) 0 0
\(701\) −39.8569 −1.50537 −0.752686 0.658379i \(-0.771242\pi\)
−0.752686 + 0.658379i \(0.771242\pi\)
\(702\) 0 0
\(703\) −4.87298 −0.183788
\(704\) 0 0
\(705\) 0.872983 0.0328785
\(706\) 0 0
\(707\) −11.6190 −0.436976
\(708\) 0 0
\(709\) 16.6190 0.624138 0.312069 0.950059i \(-0.398978\pi\)
0.312069 + 0.950059i \(0.398978\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −3.00000 −0.112351
\(714\) 0 0
\(715\) 14.0000 0.523570
\(716\) 0 0
\(717\) −10.1270 −0.378200
\(718\) 0 0
\(719\) 0.635083 0.0236846 0.0118423 0.999930i \(-0.496230\pi\)
0.0118423 + 0.999930i \(0.496230\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) 12.8730 0.478751
\(724\) 0 0
\(725\) 1.87298 0.0695609
\(726\) 0 0
\(727\) −26.7460 −0.991953 −0.495976 0.868336i \(-0.665190\pi\)
−0.495976 + 0.868336i \(0.665190\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −45.4919 −1.68258
\(732\) 0 0
\(733\) 7.25403 0.267934 0.133967 0.990986i \(-0.457228\pi\)
0.133967 + 0.990986i \(0.457228\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) 13.6351 0.502255
\(738\) 0 0
\(739\) −28.7460 −1.05744 −0.528719 0.848797i \(-0.677327\pi\)
−0.528719 + 0.848797i \(0.677327\pi\)
\(740\) 0 0
\(741\) 23.7460 0.872330
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −3.12702 −0.114565
\(746\) 0 0
\(747\) 7.87298 0.288057
\(748\) 0 0
\(749\) 52.8569 1.93135
\(750\) 0 0
\(751\) −39.8569 −1.45440 −0.727199 0.686427i \(-0.759178\pi\)
−0.727199 + 0.686427i \(0.759178\pi\)
\(752\) 0 0
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 11.7460 0.427479
\(756\) 0 0
\(757\) 10.7460 0.390569 0.195284 0.980747i \(-0.437437\pi\)
0.195284 + 0.980747i \(0.437437\pi\)
\(758\) 0 0
\(759\) −2.87298 −0.104283
\(760\) 0 0
\(761\) 54.8569 1.98856 0.994280 0.106808i \(-0.0340631\pi\)
0.994280 + 0.106808i \(0.0340631\pi\)
\(762\) 0 0
\(763\) 19.8569 0.718866
\(764\) 0 0
\(765\) −3.87298 −0.140028
\(766\) 0 0
\(767\) 9.12702 0.329557
\(768\) 0 0
\(769\) 21.1270 0.761860 0.380930 0.924604i \(-0.375604\pi\)
0.380930 + 0.924604i \(0.375604\pi\)
\(770\) 0 0
\(771\) −6.61895 −0.238376
\(772\) 0 0
\(773\) 19.7460 0.710213 0.355107 0.934826i \(-0.384445\pi\)
0.355107 + 0.934826i \(0.384445\pi\)
\(774\) 0 0
\(775\) −3.00000 −0.107763
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) −9.12702 −0.327009
\(780\) 0 0
\(781\) −27.6351 −0.988861
\(782\) 0 0
\(783\) −1.87298 −0.0669350
\(784\) 0 0
\(785\) 17.0000 0.606756
\(786\) 0 0
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 0 0
\(789\) −14.1270 −0.502935
\(790\) 0 0
\(791\) −11.6190 −0.413122
\(792\) 0 0
\(793\) 5.49193 0.195024
\(794\) 0 0
\(795\) −3.87298 −0.137361
\(796\) 0 0
\(797\) 4.12702 0.146186 0.0730932 0.997325i \(-0.476713\pi\)
0.0730932 + 0.997325i \(0.476713\pi\)
\(798\) 0 0
\(799\) 3.38105 0.119613
\(800\) 0 0
\(801\) −13.7460 −0.485690
\(802\) 0 0
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) −15.3649 −0.540871
\(808\) 0 0
\(809\) −44.8569 −1.57708 −0.788541 0.614982i \(-0.789163\pi\)
−0.788541 + 0.614982i \(0.789163\pi\)
\(810\) 0 0
\(811\) 2.49193 0.0875036 0.0437518 0.999042i \(-0.486069\pi\)
0.0437518 + 0.999042i \(0.486069\pi\)
\(812\) 0 0
\(813\) −22.7460 −0.797736
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) −57.2379 −2.00250
\(818\) 0 0
\(819\) −14.6190 −0.510827
\(820\) 0 0
\(821\) 27.4919 0.959475 0.479738 0.877412i \(-0.340732\pi\)
0.479738 + 0.877412i \(0.340732\pi\)
\(822\) 0 0
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 0 0
\(825\) −2.87298 −0.100024
\(826\) 0 0
\(827\) −39.1109 −1.36002 −0.680009 0.733203i \(-0.738024\pi\)
−0.680009 + 0.733203i \(0.738024\pi\)
\(828\) 0 0
\(829\) 34.4919 1.19795 0.598977 0.800766i \(-0.295574\pi\)
0.598977 + 0.800766i \(0.295574\pi\)
\(830\) 0 0
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) −7.74597 −0.268382
\(834\) 0 0
\(835\) 16.8730 0.583914
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −25.4919 −0.879032
\(842\) 0 0
\(843\) −22.3649 −0.770289
\(844\) 0 0
\(845\) 10.7460 0.369672
\(846\) 0 0
\(847\) 8.23790 0.283058
\(848\) 0 0
\(849\) −24.2379 −0.831843
\(850\) 0 0
\(851\) 1.00000 0.0342796
\(852\) 0 0
\(853\) −28.2540 −0.967400 −0.483700 0.875234i \(-0.660707\pi\)
−0.483700 + 0.875234i \(0.660707\pi\)
\(854\) 0 0
\(855\) −4.87298 −0.166653
\(856\) 0 0
\(857\) −45.4919 −1.55397 −0.776987 0.629516i \(-0.783253\pi\)
−0.776987 + 0.629516i \(0.783253\pi\)
\(858\) 0 0
\(859\) −29.0000 −0.989467 −0.494734 0.869045i \(-0.664734\pi\)
−0.494734 + 0.869045i \(0.664734\pi\)
\(860\) 0 0
\(861\) 5.61895 0.191493
\(862\) 0 0
\(863\) −23.7460 −0.808322 −0.404161 0.914688i \(-0.632436\pi\)
−0.404161 + 0.914688i \(0.632436\pi\)
\(864\) 0 0
\(865\) −7.49193 −0.254733
\(866\) 0 0
\(867\) 2.00000 0.0679236
\(868\) 0 0
\(869\) −11.4919 −0.389837
\(870\) 0 0
\(871\) 23.1270 0.783629
\(872\) 0 0
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −22.7298 −0.767532 −0.383766 0.923430i \(-0.625373\pi\)
−0.383766 + 0.923430i \(0.625373\pi\)
\(878\) 0 0
\(879\) −9.61895 −0.324439
\(880\) 0 0
\(881\) −50.1109 −1.68828 −0.844139 0.536124i \(-0.819888\pi\)
−0.844139 + 0.536124i \(0.819888\pi\)
\(882\) 0 0
\(883\) 45.3488 1.52611 0.763054 0.646335i \(-0.223699\pi\)
0.763054 + 0.646335i \(0.223699\pi\)
\(884\) 0 0
\(885\) −1.87298 −0.0629596
\(886\) 0 0
\(887\) −4.25403 −0.142836 −0.0714182 0.997446i \(-0.522752\pi\)
−0.0714182 + 0.997446i \(0.522752\pi\)
\(888\) 0 0
\(889\) 49.8569 1.67215
\(890\) 0 0
\(891\) 2.87298 0.0962486
\(892\) 0 0
\(893\) 4.25403 0.142356
\(894\) 0 0
\(895\) 0.254033 0.00849140
\(896\) 0 0
\(897\) −4.87298 −0.162704
\(898\) 0 0
\(899\) −5.61895 −0.187402
\(900\) 0 0
\(901\) −15.0000 −0.499722
\(902\) 0 0
\(903\) 35.2379 1.17264
\(904\) 0 0
\(905\) −6.25403 −0.207891
\(906\) 0 0
\(907\) −15.2540 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(908\) 0 0
\(909\) 3.87298 0.128459
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 22.6190 0.748578
\(914\) 0 0
\(915\) −1.12702 −0.0372580
\(916\) 0 0
\(917\) −58.4758 −1.93104
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 16.8730 0.555984
\(922\) 0 0
\(923\) −46.8730 −1.54284
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) −7.36492 −0.241635 −0.120818 0.992675i \(-0.538552\pi\)
−0.120818 + 0.992675i \(0.538552\pi\)
\(930\) 0 0
\(931\) −9.74597 −0.319411
\(932\) 0 0
\(933\) −4.00000 −0.130954
\(934\) 0 0
\(935\) −11.1270 −0.363892
\(936\) 0 0
\(937\) −38.4758 −1.25695 −0.628475 0.777830i \(-0.716320\pi\)
−0.628475 + 0.777830i \(0.716320\pi\)
\(938\) 0 0
\(939\) −7.00000 −0.228436
\(940\) 0 0
\(941\) 38.1109 1.24238 0.621190 0.783660i \(-0.286650\pi\)
0.621190 + 0.783660i \(0.286650\pi\)
\(942\) 0 0
\(943\) 1.87298 0.0609927
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) −21.2379 −0.690139 −0.345070 0.938577i \(-0.612145\pi\)
−0.345070 + 0.938577i \(0.612145\pi\)
\(948\) 0 0
\(949\) 23.7460 0.770827
\(950\) 0 0
\(951\) 16.3649 0.530669
\(952\) 0 0
\(953\) −52.9839 −1.71632 −0.858158 0.513386i \(-0.828391\pi\)
−0.858158 + 0.513386i \(0.828391\pi\)
\(954\) 0 0
\(955\) 11.1270 0.360062
\(956\) 0 0
\(957\) −5.38105 −0.173945
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −17.6190 −0.567763
\(964\) 0 0
\(965\) −5.74597 −0.184969
\(966\) 0 0
\(967\) −44.6190 −1.43485 −0.717424 0.696636i \(-0.754679\pi\)
−0.717424 + 0.696636i \(0.754679\pi\)
\(968\) 0 0
\(969\) −18.8730 −0.606288
\(970\) 0 0
\(971\) −38.2540 −1.22763 −0.613815 0.789450i \(-0.710366\pi\)
−0.613815 + 0.789450i \(0.710366\pi\)
\(972\) 0 0
\(973\) 31.4758 1.00907
\(974\) 0 0
\(975\) −4.87298 −0.156060
\(976\) 0 0
\(977\) −44.1270 −1.41175 −0.705874 0.708337i \(-0.749446\pi\)
−0.705874 + 0.708337i \(0.749446\pi\)
\(978\) 0 0
\(979\) −39.4919 −1.26217
\(980\) 0 0
\(981\) −6.61895 −0.211327
\(982\) 0 0
\(983\) 31.3649 1.00039 0.500193 0.865914i \(-0.333262\pi\)
0.500193 + 0.865914i \(0.333262\pi\)
\(984\) 0 0
\(985\) 19.7460 0.629159
\(986\) 0 0
\(987\) −2.61895 −0.0833621
\(988\) 0 0
\(989\) 11.7460 0.373500
\(990\) 0 0
\(991\) 42.4919 1.34980 0.674900 0.737909i \(-0.264187\pi\)
0.674900 + 0.737909i \(0.264187\pi\)
\(992\) 0 0
\(993\) 24.2379 0.769167
\(994\) 0 0
\(995\) 27.2379 0.863499
\(996\) 0 0
\(997\) −26.2540 −0.831474 −0.415737 0.909485i \(-0.636476\pi\)
−0.415737 + 0.909485i \(0.636476\pi\)
\(998\) 0 0
\(999\) −1.00000 −0.0316386
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 5520.2.a.bj.1.2 2
4.3 odd 2 1380.2.a.i.1.1 2
12.11 even 2 4140.2.a.p.1.2 2
20.3 even 4 6900.2.f.o.6349.3 4
20.7 even 4 6900.2.f.o.6349.1 4
20.19 odd 2 6900.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.i.1.1 2 4.3 odd 2
4140.2.a.p.1.2 2 12.11 even 2
5520.2.a.bj.1.2 2 1.1 even 1 trivial
6900.2.a.j.1.1 2 20.19 odd 2
6900.2.f.o.6349.1 4 20.7 even 4
6900.2.f.o.6349.3 4 20.3 even 4