Properties

Label 4410.2.a.bx.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -3.41421 q^{11} -1.17157 q^{13} +1.00000 q^{16} -3.41421 q^{17} -4.82843 q^{19} +1.00000 q^{20} -3.41421 q^{22} +3.65685 q^{23} +1.00000 q^{25} -1.17157 q^{26} -5.41421 q^{29} -7.41421 q^{31} +1.00000 q^{32} -3.41421 q^{34} -3.41421 q^{37} -4.82843 q^{38} +1.00000 q^{40} +3.65685 q^{41} -7.07107 q^{43} -3.41421 q^{44} +3.65685 q^{46} -2.58579 q^{47} +1.00000 q^{50} -1.17157 q^{52} -6.48528 q^{53} -3.41421 q^{55} -5.41421 q^{58} +0.828427 q^{59} +1.65685 q^{61} -7.41421 q^{62} +1.00000 q^{64} -1.17157 q^{65} +5.89949 q^{67} -3.41421 q^{68} +10.4853 q^{71} -12.8284 q^{73} -3.41421 q^{74} -4.82843 q^{76} +13.6569 q^{79} +1.00000 q^{80} +3.65685 q^{82} -0.485281 q^{83} -3.41421 q^{85} -7.07107 q^{86} -3.41421 q^{88} -14.4853 q^{89} +3.65685 q^{92} -2.58579 q^{94} -4.82843 q^{95} -14.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} - 4 q^{11} - 8 q^{13} + 2 q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} - 8 q^{26} - 8 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} - 4 q^{37} - 4 q^{38} + 2 q^{40} - 4 q^{41} - 4 q^{44} - 4 q^{46} - 8 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} - 4 q^{55} - 8 q^{58} - 4 q^{59} - 8 q^{61} - 12 q^{62} + 2 q^{64} - 8 q^{65} - 8 q^{67} - 4 q^{68} + 4 q^{71} - 20 q^{73} - 4 q^{74} - 4 q^{76} + 16 q^{79} + 2 q^{80} - 4 q^{82} + 16 q^{83} - 4 q^{85} - 4 q^{88} - 12 q^{89} - 4 q^{92} - 8 q^{94} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.41421 −1.02942 −0.514712 0.857363i \(-0.672101\pi\)
−0.514712 + 0.857363i \(0.672101\pi\)
\(12\) 0 0
\(13\) −1.17157 −0.324936 −0.162468 0.986714i \(-0.551945\pi\)
−0.162468 + 0.986714i \(0.551945\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.41421 −0.828068 −0.414034 0.910261i \(-0.635881\pi\)
−0.414034 + 0.910261i \(0.635881\pi\)
\(18\) 0 0
\(19\) −4.82843 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.41421 −0.727913
\(23\) 3.65685 0.762507 0.381253 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.17157 −0.229764
\(27\) 0 0
\(28\) 0 0
\(29\) −5.41421 −1.00539 −0.502697 0.864463i \(-0.667659\pi\)
−0.502697 + 0.864463i \(0.667659\pi\)
\(30\) 0 0
\(31\) −7.41421 −1.33163 −0.665816 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.41421 −0.585533
\(35\) 0 0
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) −4.82843 −0.783274
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.65685 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) −7.07107 −1.07833 −0.539164 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(44\) −3.41421 −0.514712
\(45\) 0 0
\(46\) 3.65685 0.539174
\(47\) −2.58579 −0.377176 −0.188588 0.982056i \(-0.560391\pi\)
−0.188588 + 0.982056i \(0.560391\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −1.17157 −0.162468
\(53\) −6.48528 −0.890822 −0.445411 0.895326i \(-0.646942\pi\)
−0.445411 + 0.895326i \(0.646942\pi\)
\(54\) 0 0
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) 0 0
\(58\) −5.41421 −0.710921
\(59\) 0.828427 0.107852 0.0539260 0.998545i \(-0.482826\pi\)
0.0539260 + 0.998545i \(0.482826\pi\)
\(60\) 0 0
\(61\) 1.65685 0.212138 0.106069 0.994359i \(-0.466173\pi\)
0.106069 + 0.994359i \(0.466173\pi\)
\(62\) −7.41421 −0.941606
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.17157 −0.145316
\(66\) 0 0
\(67\) 5.89949 0.720738 0.360369 0.932810i \(-0.382651\pi\)
0.360369 + 0.932810i \(0.382651\pi\)
\(68\) −3.41421 −0.414034
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) −12.8284 −1.50145 −0.750727 0.660613i \(-0.770297\pi\)
−0.750727 + 0.660613i \(0.770297\pi\)
\(74\) −3.41421 −0.396894
\(75\) 0 0
\(76\) −4.82843 −0.553859
\(77\) 0 0
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 3.65685 0.403832
\(83\) −0.485281 −0.0532666 −0.0266333 0.999645i \(-0.508479\pi\)
−0.0266333 + 0.999645i \(0.508479\pi\)
\(84\) 0 0
\(85\) −3.41421 −0.370323
\(86\) −7.07107 −0.762493
\(87\) 0 0
\(88\) −3.41421 −0.363956
\(89\) −14.4853 −1.53544 −0.767718 0.640787i \(-0.778608\pi\)
−0.767718 + 0.640787i \(0.778608\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.65685 0.381253
\(93\) 0 0
\(94\) −2.58579 −0.266704
\(95\) −4.82843 −0.495386
\(96\) 0 0
\(97\) −14.4853 −1.47076 −0.735379 0.677656i \(-0.762996\pi\)
−0.735379 + 0.677656i \(0.762996\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −9.65685 −0.960893 −0.480446 0.877024i \(-0.659525\pi\)
−0.480446 + 0.877024i \(0.659525\pi\)
\(102\) 0 0
\(103\) −5.65685 −0.557386 −0.278693 0.960380i \(-0.589901\pi\)
−0.278693 + 0.960380i \(0.589901\pi\)
\(104\) −1.17157 −0.114882
\(105\) 0 0
\(106\) −6.48528 −0.629906
\(107\) 10.8284 1.04682 0.523412 0.852080i \(-0.324659\pi\)
0.523412 + 0.852080i \(0.324659\pi\)
\(108\) 0 0
\(109\) 14.4853 1.38744 0.693719 0.720246i \(-0.255971\pi\)
0.693719 + 0.720246i \(0.255971\pi\)
\(110\) −3.41421 −0.325532
\(111\) 0 0
\(112\) 0 0
\(113\) 4.34315 0.408569 0.204284 0.978912i \(-0.434513\pi\)
0.204284 + 0.978912i \(0.434513\pi\)
\(114\) 0 0
\(115\) 3.65685 0.341003
\(116\) −5.41421 −0.502697
\(117\) 0 0
\(118\) 0.828427 0.0762629
\(119\) 0 0
\(120\) 0 0
\(121\) 0.656854 0.0597140
\(122\) 1.65685 0.150005
\(123\) 0 0
\(124\) −7.41421 −0.665816
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −14.1421 −1.25491 −0.627456 0.778652i \(-0.715904\pi\)
−0.627456 + 0.778652i \(0.715904\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −1.17157 −0.102754
\(131\) 6.48528 0.566622 0.283311 0.959028i \(-0.408567\pi\)
0.283311 + 0.959028i \(0.408567\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 5.89949 0.509639
\(135\) 0 0
\(136\) −3.41421 −0.292766
\(137\) −4.48528 −0.383203 −0.191602 0.981473i \(-0.561368\pi\)
−0.191602 + 0.981473i \(0.561368\pi\)
\(138\) 0 0
\(139\) 1.31371 0.111427 0.0557137 0.998447i \(-0.482257\pi\)
0.0557137 + 0.998447i \(0.482257\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.4853 0.879905
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −5.41421 −0.449626
\(146\) −12.8284 −1.06169
\(147\) 0 0
\(148\) −3.41421 −0.280647
\(149\) 12.7279 1.04271 0.521356 0.853339i \(-0.325426\pi\)
0.521356 + 0.853339i \(0.325426\pi\)
\(150\) 0 0
\(151\) 3.17157 0.258099 0.129049 0.991638i \(-0.458807\pi\)
0.129049 + 0.991638i \(0.458807\pi\)
\(152\) −4.82843 −0.391637
\(153\) 0 0
\(154\) 0 0
\(155\) −7.41421 −0.595524
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 13.6569 1.08648
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 7.07107 0.553849 0.276924 0.960892i \(-0.410685\pi\)
0.276924 + 0.960892i \(0.410685\pi\)
\(164\) 3.65685 0.285552
\(165\) 0 0
\(166\) −0.485281 −0.0376651
\(167\) 6.58579 0.509623 0.254812 0.966991i \(-0.417987\pi\)
0.254812 + 0.966991i \(0.417987\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) −3.41421 −0.261858
\(171\) 0 0
\(172\) −7.07107 −0.539164
\(173\) 20.1421 1.53138 0.765689 0.643211i \(-0.222398\pi\)
0.765689 + 0.643211i \(0.222398\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.41421 −0.257356
\(177\) 0 0
\(178\) −14.4853 −1.08572
\(179\) −10.7279 −0.801843 −0.400921 0.916113i \(-0.631310\pi\)
−0.400921 + 0.916113i \(0.631310\pi\)
\(180\) 0 0
\(181\) −3.51472 −0.261247 −0.130623 0.991432i \(-0.541698\pi\)
−0.130623 + 0.991432i \(0.541698\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.65685 0.269587
\(185\) −3.41421 −0.251018
\(186\) 0 0
\(187\) 11.6569 0.852434
\(188\) −2.58579 −0.188588
\(189\) 0 0
\(190\) −4.82843 −0.350291
\(191\) 9.31371 0.673916 0.336958 0.941520i \(-0.390602\pi\)
0.336958 + 0.941520i \(0.390602\pi\)
\(192\) 0 0
\(193\) 20.1421 1.44986 0.724931 0.688821i \(-0.241871\pi\)
0.724931 + 0.688821i \(0.241871\pi\)
\(194\) −14.4853 −1.03998
\(195\) 0 0
\(196\) 0 0
\(197\) 0.343146 0.0244481 0.0122241 0.999925i \(-0.496109\pi\)
0.0122241 + 0.999925i \(0.496109\pi\)
\(198\) 0 0
\(199\) −18.2426 −1.29319 −0.646593 0.762835i \(-0.723807\pi\)
−0.646593 + 0.762835i \(0.723807\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −9.65685 −0.679454
\(203\) 0 0
\(204\) 0 0
\(205\) 3.65685 0.255406
\(206\) −5.65685 −0.394132
\(207\) 0 0
\(208\) −1.17157 −0.0812340
\(209\) 16.4853 1.14031
\(210\) 0 0
\(211\) −23.7990 −1.63839 −0.819195 0.573515i \(-0.805579\pi\)
−0.819195 + 0.573515i \(0.805579\pi\)
\(212\) −6.48528 −0.445411
\(213\) 0 0
\(214\) 10.8284 0.740216
\(215\) −7.07107 −0.482243
\(216\) 0 0
\(217\) 0 0
\(218\) 14.4853 0.981067
\(219\) 0 0
\(220\) −3.41421 −0.230186
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 10.8284 0.725125 0.362563 0.931959i \(-0.381902\pi\)
0.362563 + 0.931959i \(0.381902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.34315 0.288902
\(227\) 24.9706 1.65735 0.828677 0.559727i \(-0.189094\pi\)
0.828677 + 0.559727i \(0.189094\pi\)
\(228\) 0 0
\(229\) 5.17157 0.341747 0.170874 0.985293i \(-0.445341\pi\)
0.170874 + 0.985293i \(0.445341\pi\)
\(230\) 3.65685 0.241126
\(231\) 0 0
\(232\) −5.41421 −0.355461
\(233\) 23.6569 1.54981 0.774906 0.632076i \(-0.217797\pi\)
0.774906 + 0.632076i \(0.217797\pi\)
\(234\) 0 0
\(235\) −2.58579 −0.168678
\(236\) 0.828427 0.0539260
\(237\) 0 0
\(238\) 0 0
\(239\) −19.6569 −1.27150 −0.635748 0.771897i \(-0.719308\pi\)
−0.635748 + 0.771897i \(0.719308\pi\)
\(240\) 0 0
\(241\) −19.0711 −1.22848 −0.614238 0.789121i \(-0.710536\pi\)
−0.614238 + 0.789121i \(0.710536\pi\)
\(242\) 0.656854 0.0422242
\(243\) 0 0
\(244\) 1.65685 0.106069
\(245\) 0 0
\(246\) 0 0
\(247\) 5.65685 0.359937
\(248\) −7.41421 −0.470803
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 14.3431 0.905331 0.452666 0.891680i \(-0.350473\pi\)
0.452666 + 0.891680i \(0.350473\pi\)
\(252\) 0 0
\(253\) −12.4853 −0.784943
\(254\) −14.1421 −0.887357
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.89949 −0.243244 −0.121622 0.992577i \(-0.538810\pi\)
−0.121622 + 0.992577i \(0.538810\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.17157 −0.0726579
\(261\) 0 0
\(262\) 6.48528 0.400662
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) 0 0
\(265\) −6.48528 −0.398388
\(266\) 0 0
\(267\) 0 0
\(268\) 5.89949 0.360369
\(269\) 14.9706 0.912771 0.456386 0.889782i \(-0.349144\pi\)
0.456386 + 0.889782i \(0.349144\pi\)
\(270\) 0 0
\(271\) −1.75736 −0.106752 −0.0533760 0.998574i \(-0.516998\pi\)
−0.0533760 + 0.998574i \(0.516998\pi\)
\(272\) −3.41421 −0.207017
\(273\) 0 0
\(274\) −4.48528 −0.270966
\(275\) −3.41421 −0.205885
\(276\) 0 0
\(277\) −31.4142 −1.88750 −0.943749 0.330664i \(-0.892727\pi\)
−0.943749 + 0.330664i \(0.892727\pi\)
\(278\) 1.31371 0.0787910
\(279\) 0 0
\(280\) 0 0
\(281\) −21.4558 −1.27995 −0.639974 0.768396i \(-0.721055\pi\)
−0.639974 + 0.768396i \(0.721055\pi\)
\(282\) 0 0
\(283\) 14.9706 0.889908 0.444954 0.895554i \(-0.353220\pi\)
0.444954 + 0.895554i \(0.353220\pi\)
\(284\) 10.4853 0.622187
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −5.34315 −0.314303
\(290\) −5.41421 −0.317934
\(291\) 0 0
\(292\) −12.8284 −0.750727
\(293\) −25.3137 −1.47884 −0.739421 0.673243i \(-0.764901\pi\)
−0.739421 + 0.673243i \(0.764901\pi\)
\(294\) 0 0
\(295\) 0.828427 0.0482329
\(296\) −3.41421 −0.198447
\(297\) 0 0
\(298\) 12.7279 0.737309
\(299\) −4.28427 −0.247766
\(300\) 0 0
\(301\) 0 0
\(302\) 3.17157 0.182504
\(303\) 0 0
\(304\) −4.82843 −0.276929
\(305\) 1.65685 0.0948712
\(306\) 0 0
\(307\) −26.6274 −1.51971 −0.759853 0.650094i \(-0.774729\pi\)
−0.759853 + 0.650094i \(0.774729\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −7.41421 −0.421099
\(311\) −21.4558 −1.21665 −0.608325 0.793688i \(-0.708158\pi\)
−0.608325 + 0.793688i \(0.708158\pi\)
\(312\) 0 0
\(313\) −7.65685 −0.432791 −0.216395 0.976306i \(-0.569430\pi\)
−0.216395 + 0.976306i \(0.569430\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 13.6569 0.768258
\(317\) 14.9706 0.840831 0.420415 0.907332i \(-0.361884\pi\)
0.420415 + 0.907332i \(0.361884\pi\)
\(318\) 0 0
\(319\) 18.4853 1.03498
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) 16.4853 0.917266
\(324\) 0 0
\(325\) −1.17157 −0.0649872
\(326\) 7.07107 0.391630
\(327\) 0 0
\(328\) 3.65685 0.201916
\(329\) 0 0
\(330\) 0 0
\(331\) 15.7990 0.868391 0.434196 0.900819i \(-0.357033\pi\)
0.434196 + 0.900819i \(0.357033\pi\)
\(332\) −0.485281 −0.0266333
\(333\) 0 0
\(334\) 6.58579 0.360358
\(335\) 5.89949 0.322324
\(336\) 0 0
\(337\) −26.4853 −1.44275 −0.721373 0.692547i \(-0.756488\pi\)
−0.721373 + 0.692547i \(0.756488\pi\)
\(338\) −11.6274 −0.632448
\(339\) 0 0
\(340\) −3.41421 −0.185162
\(341\) 25.3137 1.37081
\(342\) 0 0
\(343\) 0 0
\(344\) −7.07107 −0.381246
\(345\) 0 0
\(346\) 20.1421 1.08285
\(347\) 17.6569 0.947870 0.473935 0.880560i \(-0.342833\pi\)
0.473935 + 0.880560i \(0.342833\pi\)
\(348\) 0 0
\(349\) 20.4853 1.09655 0.548276 0.836297i \(-0.315284\pi\)
0.548276 + 0.836297i \(0.315284\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.41421 −0.181978
\(353\) −11.8995 −0.633346 −0.316673 0.948535i \(-0.602566\pi\)
−0.316673 + 0.948535i \(0.602566\pi\)
\(354\) 0 0
\(355\) 10.4853 0.556501
\(356\) −14.4853 −0.767718
\(357\) 0 0
\(358\) −10.7279 −0.566988
\(359\) −32.8284 −1.73262 −0.866309 0.499508i \(-0.833514\pi\)
−0.866309 + 0.499508i \(0.833514\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) −3.51472 −0.184730
\(363\) 0 0
\(364\) 0 0
\(365\) −12.8284 −0.671471
\(366\) 0 0
\(367\) 0.485281 0.0253315 0.0126657 0.999920i \(-0.495968\pi\)
0.0126657 + 0.999920i \(0.495968\pi\)
\(368\) 3.65685 0.190627
\(369\) 0 0
\(370\) −3.41421 −0.177497
\(371\) 0 0
\(372\) 0 0
\(373\) 21.7574 1.12655 0.563277 0.826268i \(-0.309541\pi\)
0.563277 + 0.826268i \(0.309541\pi\)
\(374\) 11.6569 0.602762
\(375\) 0 0
\(376\) −2.58579 −0.133352
\(377\) 6.34315 0.326689
\(378\) 0 0
\(379\) 4.48528 0.230393 0.115197 0.993343i \(-0.463250\pi\)
0.115197 + 0.993343i \(0.463250\pi\)
\(380\) −4.82843 −0.247693
\(381\) 0 0
\(382\) 9.31371 0.476531
\(383\) 29.6985 1.51752 0.758761 0.651369i \(-0.225805\pi\)
0.758761 + 0.651369i \(0.225805\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.1421 1.02521
\(387\) 0 0
\(388\) −14.4853 −0.735379
\(389\) 16.0416 0.813343 0.406671 0.913574i \(-0.366689\pi\)
0.406671 + 0.913574i \(0.366689\pi\)
\(390\) 0 0
\(391\) −12.4853 −0.631408
\(392\) 0 0
\(393\) 0 0
\(394\) 0.343146 0.0172874
\(395\) 13.6569 0.687151
\(396\) 0 0
\(397\) −8.34315 −0.418730 −0.209365 0.977838i \(-0.567140\pi\)
−0.209365 + 0.977838i \(0.567140\pi\)
\(398\) −18.2426 −0.914421
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 4.97056 0.248218 0.124109 0.992269i \(-0.460393\pi\)
0.124109 + 0.992269i \(0.460393\pi\)
\(402\) 0 0
\(403\) 8.68629 0.432695
\(404\) −9.65685 −0.480446
\(405\) 0 0
\(406\) 0 0
\(407\) 11.6569 0.577809
\(408\) 0 0
\(409\) −26.8701 −1.32864 −0.664319 0.747449i \(-0.731279\pi\)
−0.664319 + 0.747449i \(0.731279\pi\)
\(410\) 3.65685 0.180599
\(411\) 0 0
\(412\) −5.65685 −0.278693
\(413\) 0 0
\(414\) 0 0
\(415\) −0.485281 −0.0238215
\(416\) −1.17157 −0.0574411
\(417\) 0 0
\(418\) 16.4853 0.806321
\(419\) 40.2843 1.96802 0.984008 0.178126i \(-0.0570034\pi\)
0.984008 + 0.178126i \(0.0570034\pi\)
\(420\) 0 0
\(421\) 8.82843 0.430271 0.215136 0.976584i \(-0.430981\pi\)
0.215136 + 0.976584i \(0.430981\pi\)
\(422\) −23.7990 −1.15852
\(423\) 0 0
\(424\) −6.48528 −0.314953
\(425\) −3.41421 −0.165614
\(426\) 0 0
\(427\) 0 0
\(428\) 10.8284 0.523412
\(429\) 0 0
\(430\) −7.07107 −0.340997
\(431\) −16.1421 −0.777539 −0.388770 0.921335i \(-0.627100\pi\)
−0.388770 + 0.921335i \(0.627100\pi\)
\(432\) 0 0
\(433\) −32.8284 −1.57763 −0.788817 0.614628i \(-0.789306\pi\)
−0.788817 + 0.614628i \(0.789306\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.4853 0.693719
\(437\) −17.6569 −0.844642
\(438\) 0 0
\(439\) −14.2426 −0.679764 −0.339882 0.940468i \(-0.610387\pi\)
−0.339882 + 0.940468i \(0.610387\pi\)
\(440\) −3.41421 −0.162766
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) −0.970563 −0.0461128 −0.0230564 0.999734i \(-0.507340\pi\)
−0.0230564 + 0.999734i \(0.507340\pi\)
\(444\) 0 0
\(445\) −14.4853 −0.686668
\(446\) 10.8284 0.512741
\(447\) 0 0
\(448\) 0 0
\(449\) −38.8284 −1.83243 −0.916213 0.400691i \(-0.868770\pi\)
−0.916213 + 0.400691i \(0.868770\pi\)
\(450\) 0 0
\(451\) −12.4853 −0.587909
\(452\) 4.34315 0.204284
\(453\) 0 0
\(454\) 24.9706 1.17193
\(455\) 0 0
\(456\) 0 0
\(457\) 0.142136 0.00664882 0.00332441 0.999994i \(-0.498942\pi\)
0.00332441 + 0.999994i \(0.498942\pi\)
\(458\) 5.17157 0.241652
\(459\) 0 0
\(460\) 3.65685 0.170502
\(461\) 2.68629 0.125113 0.0625565 0.998041i \(-0.480075\pi\)
0.0625565 + 0.998041i \(0.480075\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) −5.41421 −0.251349
\(465\) 0 0
\(466\) 23.6569 1.09588
\(467\) 38.8284 1.79677 0.898383 0.439214i \(-0.144743\pi\)
0.898383 + 0.439214i \(0.144743\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2.58579 −0.119273
\(471\) 0 0
\(472\) 0.828427 0.0381314
\(473\) 24.1421 1.11006
\(474\) 0 0
\(475\) −4.82843 −0.221543
\(476\) 0 0
\(477\) 0 0
\(478\) −19.6569 −0.899084
\(479\) 30.8284 1.40859 0.704293 0.709909i \(-0.251264\pi\)
0.704293 + 0.709909i \(0.251264\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) −19.0711 −0.868663
\(483\) 0 0
\(484\) 0.656854 0.0298570
\(485\) −14.4853 −0.657743
\(486\) 0 0
\(487\) −9.17157 −0.415604 −0.207802 0.978171i \(-0.566631\pi\)
−0.207802 + 0.978171i \(0.566631\pi\)
\(488\) 1.65685 0.0750023
\(489\) 0 0
\(490\) 0 0
\(491\) 18.9289 0.854251 0.427125 0.904192i \(-0.359526\pi\)
0.427125 + 0.904192i \(0.359526\pi\)
\(492\) 0 0
\(493\) 18.4853 0.832535
\(494\) 5.65685 0.254514
\(495\) 0 0
\(496\) −7.41421 −0.332908
\(497\) 0 0
\(498\) 0 0
\(499\) −31.1127 −1.39280 −0.696398 0.717656i \(-0.745215\pi\)
−0.696398 + 0.717656i \(0.745215\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 14.3431 0.640166
\(503\) 25.4142 1.13316 0.566582 0.824005i \(-0.308265\pi\)
0.566582 + 0.824005i \(0.308265\pi\)
\(504\) 0 0
\(505\) −9.65685 −0.429724
\(506\) −12.4853 −0.555038
\(507\) 0 0
\(508\) −14.1421 −0.627456
\(509\) −35.5980 −1.57785 −0.788926 0.614488i \(-0.789363\pi\)
−0.788926 + 0.614488i \(0.789363\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −3.89949 −0.171999
\(515\) −5.65685 −0.249271
\(516\) 0 0
\(517\) 8.82843 0.388274
\(518\) 0 0
\(519\) 0 0
\(520\) −1.17157 −0.0513769
\(521\) 20.8284 0.912510 0.456255 0.889849i \(-0.349190\pi\)
0.456255 + 0.889849i \(0.349190\pi\)
\(522\) 0 0
\(523\) −24.6274 −1.07688 −0.538441 0.842663i \(-0.680986\pi\)
−0.538441 + 0.842663i \(0.680986\pi\)
\(524\) 6.48528 0.283311
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) 25.3137 1.10268
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) −6.48528 −0.281703
\(531\) 0 0
\(532\) 0 0
\(533\) −4.28427 −0.185572
\(534\) 0 0
\(535\) 10.8284 0.468154
\(536\) 5.89949 0.254819
\(537\) 0 0
\(538\) 14.9706 0.645427
\(539\) 0 0
\(540\) 0 0
\(541\) −43.9411 −1.88918 −0.944588 0.328258i \(-0.893539\pi\)
−0.944588 + 0.328258i \(0.893539\pi\)
\(542\) −1.75736 −0.0754850
\(543\) 0 0
\(544\) −3.41421 −0.146383
\(545\) 14.4853 0.620481
\(546\) 0 0
\(547\) −7.27208 −0.310932 −0.155466 0.987841i \(-0.549688\pi\)
−0.155466 + 0.987841i \(0.549688\pi\)
\(548\) −4.48528 −0.191602
\(549\) 0 0
\(550\) −3.41421 −0.145583
\(551\) 26.1421 1.11369
\(552\) 0 0
\(553\) 0 0
\(554\) −31.4142 −1.33466
\(555\) 0 0
\(556\) 1.31371 0.0557137
\(557\) −6.68629 −0.283307 −0.141654 0.989916i \(-0.545242\pi\)
−0.141654 + 0.989916i \(0.545242\pi\)
\(558\) 0 0
\(559\) 8.28427 0.350387
\(560\) 0 0
\(561\) 0 0
\(562\) −21.4558 −0.905060
\(563\) −15.7990 −0.665848 −0.332924 0.942954i \(-0.608035\pi\)
−0.332924 + 0.942954i \(0.608035\pi\)
\(564\) 0 0
\(565\) 4.34315 0.182718
\(566\) 14.9706 0.629260
\(567\) 0 0
\(568\) 10.4853 0.439953
\(569\) −0.686292 −0.0287708 −0.0143854 0.999897i \(-0.504579\pi\)
−0.0143854 + 0.999897i \(0.504579\pi\)
\(570\) 0 0
\(571\) −4.68629 −0.196115 −0.0980576 0.995181i \(-0.531263\pi\)
−0.0980576 + 0.995181i \(0.531263\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 3.65685 0.152501
\(576\) 0 0
\(577\) 36.6274 1.52482 0.762410 0.647095i \(-0.224016\pi\)
0.762410 + 0.647095i \(0.224016\pi\)
\(578\) −5.34315 −0.222246
\(579\) 0 0
\(580\) −5.41421 −0.224813
\(581\) 0 0
\(582\) 0 0
\(583\) 22.1421 0.917034
\(584\) −12.8284 −0.530844
\(585\) 0 0
\(586\) −25.3137 −1.04570
\(587\) −10.1421 −0.418611 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(588\) 0 0
\(589\) 35.7990 1.47507
\(590\) 0.828427 0.0341058
\(591\) 0 0
\(592\) −3.41421 −0.140323
\(593\) 39.6985 1.63022 0.815111 0.579305i \(-0.196676\pi\)
0.815111 + 0.579305i \(0.196676\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.7279 0.521356
\(597\) 0 0
\(598\) −4.28427 −0.175197
\(599\) −32.6274 −1.33312 −0.666560 0.745451i \(-0.732234\pi\)
−0.666560 + 0.745451i \(0.732234\pi\)
\(600\) 0 0
\(601\) 46.3848 1.89207 0.946037 0.324058i \(-0.105047\pi\)
0.946037 + 0.324058i \(0.105047\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.17157 0.129049
\(605\) 0.656854 0.0267049
\(606\) 0 0
\(607\) 31.1127 1.26283 0.631413 0.775447i \(-0.282475\pi\)
0.631413 + 0.775447i \(0.282475\pi\)
\(608\) −4.82843 −0.195819
\(609\) 0 0
\(610\) 1.65685 0.0670841
\(611\) 3.02944 0.122558
\(612\) 0 0
\(613\) 19.4142 0.784133 0.392066 0.919937i \(-0.371760\pi\)
0.392066 + 0.919937i \(0.371760\pi\)
\(614\) −26.6274 −1.07460
\(615\) 0 0
\(616\) 0 0
\(617\) −39.1127 −1.57462 −0.787309 0.616559i \(-0.788526\pi\)
−0.787309 + 0.616559i \(0.788526\pi\)
\(618\) 0 0
\(619\) −0.828427 −0.0332973 −0.0166486 0.999861i \(-0.505300\pi\)
−0.0166486 + 0.999861i \(0.505300\pi\)
\(620\) −7.41421 −0.297762
\(621\) 0 0
\(622\) −21.4558 −0.860301
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −7.65685 −0.306029
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 11.6569 0.464789
\(630\) 0 0
\(631\) 15.4558 0.615287 0.307644 0.951502i \(-0.400460\pi\)
0.307644 + 0.951502i \(0.400460\pi\)
\(632\) 13.6569 0.543240
\(633\) 0 0
\(634\) 14.9706 0.594557
\(635\) −14.1421 −0.561214
\(636\) 0 0
\(637\) 0 0
\(638\) 18.4853 0.731839
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 18.6274 0.735739 0.367869 0.929877i \(-0.380087\pi\)
0.367869 + 0.929877i \(0.380087\pi\)
\(642\) 0 0
\(643\) −6.97056 −0.274892 −0.137446 0.990509i \(-0.543889\pi\)
−0.137446 + 0.990509i \(0.543889\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.4853 0.648605
\(647\) 13.6152 0.535270 0.267635 0.963520i \(-0.413758\pi\)
0.267635 + 0.963520i \(0.413758\pi\)
\(648\) 0 0
\(649\) −2.82843 −0.111025
\(650\) −1.17157 −0.0459529
\(651\) 0 0
\(652\) 7.07107 0.276924
\(653\) 45.1127 1.76540 0.882698 0.469940i \(-0.155725\pi\)
0.882698 + 0.469940i \(0.155725\pi\)
\(654\) 0 0
\(655\) 6.48528 0.253401
\(656\) 3.65685 0.142776
\(657\) 0 0
\(658\) 0 0
\(659\) −28.5858 −1.11354 −0.556772 0.830665i \(-0.687960\pi\)
−0.556772 + 0.830665i \(0.687960\pi\)
\(660\) 0 0
\(661\) 22.3431 0.869048 0.434524 0.900660i \(-0.356917\pi\)
0.434524 + 0.900660i \(0.356917\pi\)
\(662\) 15.7990 0.614045
\(663\) 0 0
\(664\) −0.485281 −0.0188326
\(665\) 0 0
\(666\) 0 0
\(667\) −19.7990 −0.766620
\(668\) 6.58579 0.254812
\(669\) 0 0
\(670\) 5.89949 0.227917
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) 10.9706 0.422884 0.211442 0.977391i \(-0.432184\pi\)
0.211442 + 0.977391i \(0.432184\pi\)
\(674\) −26.4853 −1.02017
\(675\) 0 0
\(676\) −11.6274 −0.447208
\(677\) 15.4558 0.594016 0.297008 0.954875i \(-0.404011\pi\)
0.297008 + 0.954875i \(0.404011\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −3.41421 −0.130929
\(681\) 0 0
\(682\) 25.3137 0.969312
\(683\) 26.3431 1.00799 0.503996 0.863706i \(-0.331863\pi\)
0.503996 + 0.863706i \(0.331863\pi\)
\(684\) 0 0
\(685\) −4.48528 −0.171374
\(686\) 0 0
\(687\) 0 0
\(688\) −7.07107 −0.269582
\(689\) 7.59798 0.289460
\(690\) 0 0
\(691\) 1.02944 0.0391616 0.0195808 0.999808i \(-0.493767\pi\)
0.0195808 + 0.999808i \(0.493767\pi\)
\(692\) 20.1421 0.765689
\(693\) 0 0
\(694\) 17.6569 0.670245
\(695\) 1.31371 0.0498318
\(696\) 0 0
\(697\) −12.4853 −0.472914
\(698\) 20.4853 0.775379
\(699\) 0 0
\(700\) 0 0
\(701\) 7.75736 0.292991 0.146496 0.989211i \(-0.453201\pi\)
0.146496 + 0.989211i \(0.453201\pi\)
\(702\) 0 0
\(703\) 16.4853 0.621754
\(704\) −3.41421 −0.128678
\(705\) 0 0
\(706\) −11.8995 −0.447843
\(707\) 0 0
\(708\) 0 0
\(709\) 21.1127 0.792904 0.396452 0.918055i \(-0.370241\pi\)
0.396452 + 0.918055i \(0.370241\pi\)
\(710\) 10.4853 0.393506
\(711\) 0 0
\(712\) −14.4853 −0.542859
\(713\) −27.1127 −1.01538
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −10.7279 −0.400921
\(717\) 0 0
\(718\) −32.8284 −1.22515
\(719\) −33.4558 −1.24769 −0.623846 0.781547i \(-0.714431\pi\)
−0.623846 + 0.781547i \(0.714431\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 4.31371 0.160540
\(723\) 0 0
\(724\) −3.51472 −0.130623
\(725\) −5.41421 −0.201079
\(726\) 0 0
\(727\) −13.4558 −0.499050 −0.249525 0.968368i \(-0.580274\pi\)
−0.249525 + 0.968368i \(0.580274\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −12.8284 −0.474801
\(731\) 24.1421 0.892929
\(732\) 0 0
\(733\) 6.97056 0.257464 0.128732 0.991679i \(-0.458909\pi\)
0.128732 + 0.991679i \(0.458909\pi\)
\(734\) 0.485281 0.0179121
\(735\) 0 0
\(736\) 3.65685 0.134793
\(737\) −20.1421 −0.741945
\(738\) 0 0
\(739\) −2.82843 −0.104045 −0.0520227 0.998646i \(-0.516567\pi\)
−0.0520227 + 0.998646i \(0.516567\pi\)
\(740\) −3.41421 −0.125509
\(741\) 0 0
\(742\) 0 0
\(743\) 45.6569 1.67499 0.837494 0.546447i \(-0.184020\pi\)
0.837494 + 0.546447i \(0.184020\pi\)
\(744\) 0 0
\(745\) 12.7279 0.466315
\(746\) 21.7574 0.796594
\(747\) 0 0
\(748\) 11.6569 0.426217
\(749\) 0 0
\(750\) 0 0
\(751\) −41.7990 −1.52527 −0.762633 0.646831i \(-0.776094\pi\)
−0.762633 + 0.646831i \(0.776094\pi\)
\(752\) −2.58579 −0.0942939
\(753\) 0 0
\(754\) 6.34315 0.231004
\(755\) 3.17157 0.115425
\(756\) 0 0
\(757\) 21.3553 0.776173 0.388087 0.921623i \(-0.373136\pi\)
0.388087 + 0.921623i \(0.373136\pi\)
\(758\) 4.48528 0.162913
\(759\) 0 0
\(760\) −4.82843 −0.175145
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 9.31371 0.336958
\(765\) 0 0
\(766\) 29.6985 1.07305
\(767\) −0.970563 −0.0350450
\(768\) 0 0
\(769\) 12.9289 0.466229 0.233115 0.972449i \(-0.425108\pi\)
0.233115 + 0.972449i \(0.425108\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.1421 0.724931
\(773\) −18.9706 −0.682324 −0.341162 0.940005i \(-0.610820\pi\)
−0.341162 + 0.940005i \(0.610820\pi\)
\(774\) 0 0
\(775\) −7.41421 −0.266326
\(776\) −14.4853 −0.519991
\(777\) 0 0
\(778\) 16.0416 0.575120
\(779\) −17.6569 −0.632622
\(780\) 0 0
\(781\) −35.7990 −1.28099
\(782\) −12.4853 −0.446473
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) 3.02944 0.107988 0.0539939 0.998541i \(-0.482805\pi\)
0.0539939 + 0.998541i \(0.482805\pi\)
\(788\) 0.343146 0.0122241
\(789\) 0 0
\(790\) 13.6569 0.485889
\(791\) 0 0
\(792\) 0 0
\(793\) −1.94113 −0.0689314
\(794\) −8.34315 −0.296087
\(795\) 0 0
\(796\) −18.2426 −0.646593
\(797\) −7.85786 −0.278340 −0.139170 0.990269i \(-0.544443\pi\)
−0.139170 + 0.990269i \(0.544443\pi\)
\(798\) 0 0
\(799\) 8.82843 0.312327
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 4.97056 0.175517
\(803\) 43.7990 1.54563
\(804\) 0 0
\(805\) 0 0
\(806\) 8.68629 0.305962
\(807\) 0 0
\(808\) −9.65685 −0.339727
\(809\) −16.4853 −0.579592 −0.289796 0.957088i \(-0.593587\pi\)
−0.289796 + 0.957088i \(0.593587\pi\)
\(810\) 0 0
\(811\) 24.6274 0.864786 0.432393 0.901685i \(-0.357669\pi\)
0.432393 + 0.901685i \(0.357669\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 11.6569 0.408573
\(815\) 7.07107 0.247689
\(816\) 0 0
\(817\) 34.1421 1.19448
\(818\) −26.8701 −0.939490
\(819\) 0 0
\(820\) 3.65685 0.127703
\(821\) 28.9289 1.00963 0.504813 0.863229i \(-0.331561\pi\)
0.504813 + 0.863229i \(0.331561\pi\)
\(822\) 0 0
\(823\) 10.3431 0.360539 0.180270 0.983617i \(-0.442303\pi\)
0.180270 + 0.983617i \(0.442303\pi\)
\(824\) −5.65685 −0.197066
\(825\) 0 0
\(826\) 0 0
\(827\) 23.7990 0.827572 0.413786 0.910374i \(-0.364206\pi\)
0.413786 + 0.910374i \(0.364206\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) −0.485281 −0.0168444
\(831\) 0 0
\(832\) −1.17157 −0.0406170
\(833\) 0 0
\(834\) 0 0
\(835\) 6.58579 0.227911
\(836\) 16.4853 0.570155
\(837\) 0 0
\(838\) 40.2843 1.39160
\(839\) 24.2843 0.838386 0.419193 0.907897i \(-0.362313\pi\)
0.419193 + 0.907897i \(0.362313\pi\)
\(840\) 0 0
\(841\) 0.313708 0.0108175
\(842\) 8.82843 0.304248
\(843\) 0 0
\(844\) −23.7990 −0.819195
\(845\) −11.6274 −0.399995
\(846\) 0 0
\(847\) 0 0
\(848\) −6.48528 −0.222705
\(849\) 0 0
\(850\) −3.41421 −0.117107
\(851\) −12.4853 −0.427990
\(852\) 0 0
\(853\) 51.6569 1.76870 0.884349 0.466827i \(-0.154603\pi\)
0.884349 + 0.466827i \(0.154603\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10.8284 0.370108
\(857\) −51.8995 −1.77285 −0.886426 0.462869i \(-0.846820\pi\)
−0.886426 + 0.462869i \(0.846820\pi\)
\(858\) 0 0
\(859\) −30.9706 −1.05670 −0.528351 0.849026i \(-0.677189\pi\)
−0.528351 + 0.849026i \(0.677189\pi\)
\(860\) −7.07107 −0.241121
\(861\) 0 0
\(862\) −16.1421 −0.549803
\(863\) 28.9706 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(864\) 0 0
\(865\) 20.1421 0.684853
\(866\) −32.8284 −1.11556
\(867\) 0 0
\(868\) 0 0
\(869\) −46.6274 −1.58173
\(870\) 0 0
\(871\) −6.91169 −0.234194
\(872\) 14.4853 0.490534
\(873\) 0 0
\(874\) −17.6569 −0.597252
\(875\) 0 0
\(876\) 0 0
\(877\) −41.5563 −1.40326 −0.701629 0.712542i \(-0.747544\pi\)
−0.701629 + 0.712542i \(0.747544\pi\)
\(878\) −14.2426 −0.480666
\(879\) 0 0
\(880\) −3.41421 −0.115093
\(881\) −52.1421 −1.75671 −0.878357 0.478006i \(-0.841360\pi\)
−0.878357 + 0.478006i \(0.841360\pi\)
\(882\) 0 0
\(883\) 20.5269 0.690786 0.345393 0.938458i \(-0.387746\pi\)
0.345393 + 0.938458i \(0.387746\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −0.970563 −0.0326067
\(887\) −45.8995 −1.54115 −0.770577 0.637347i \(-0.780032\pi\)
−0.770577 + 0.637347i \(0.780032\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −14.4853 −0.485548
\(891\) 0 0
\(892\) 10.8284 0.362563
\(893\) 12.4853 0.417804
\(894\) 0 0
\(895\) −10.7279 −0.358595
\(896\) 0 0
\(897\) 0 0
\(898\) −38.8284 −1.29572
\(899\) 40.1421 1.33882
\(900\) 0 0
\(901\) 22.1421 0.737661
\(902\) −12.4853 −0.415714
\(903\) 0 0
\(904\) 4.34315 0.144451
\(905\) −3.51472 −0.116833
\(906\) 0 0
\(907\) −56.3259 −1.87027 −0.935135 0.354290i \(-0.884722\pi\)
−0.935135 + 0.354290i \(0.884722\pi\)
\(908\) 24.9706 0.828677
\(909\) 0 0
\(910\) 0 0
\(911\) 25.5980 0.848099 0.424049 0.905639i \(-0.360608\pi\)
0.424049 + 0.905639i \(0.360608\pi\)
\(912\) 0 0
\(913\) 1.65685 0.0548339
\(914\) 0.142136 0.00470143
\(915\) 0 0
\(916\) 5.17157 0.170874
\(917\) 0 0
\(918\) 0 0
\(919\) −19.4558 −0.641789 −0.320895 0.947115i \(-0.603984\pi\)
−0.320895 + 0.947115i \(0.603984\pi\)
\(920\) 3.65685 0.120563
\(921\) 0 0
\(922\) 2.68629 0.0884683
\(923\) −12.2843 −0.404342
\(924\) 0 0
\(925\) −3.41421 −0.112259
\(926\) 36.0000 1.18303
\(927\) 0 0
\(928\) −5.41421 −0.177730
\(929\) −52.9117 −1.73598 −0.867988 0.496585i \(-0.834587\pi\)
−0.867988 + 0.496585i \(0.834587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 23.6569 0.774906
\(933\) 0 0
\(934\) 38.8284 1.27050
\(935\) 11.6569 0.381220
\(936\) 0 0
\(937\) 18.9706 0.619741 0.309871 0.950779i \(-0.399714\pi\)
0.309871 + 0.950779i \(0.399714\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.58579 −0.0843391
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) 13.3726 0.435471
\(944\) 0.828427 0.0269630
\(945\) 0 0
\(946\) 24.1421 0.784929
\(947\) −38.6274 −1.25522 −0.627611 0.778527i \(-0.715967\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(948\) 0 0
\(949\) 15.0294 0.487876
\(950\) −4.82843 −0.156655
\(951\) 0 0
\(952\) 0 0
\(953\) 45.1716 1.46325 0.731625 0.681707i \(-0.238762\pi\)
0.731625 + 0.681707i \(0.238762\pi\)
\(954\) 0 0
\(955\) 9.31371 0.301385
\(956\) −19.6569 −0.635748
\(957\) 0 0
\(958\) 30.8284 0.996021
\(959\) 0 0
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 4.00000 0.128965
\(963\) 0 0
\(964\) −19.0711 −0.614238
\(965\) 20.1421 0.648398
\(966\) 0 0
\(967\) 40.9706 1.31752 0.658762 0.752351i \(-0.271080\pi\)
0.658762 + 0.752351i \(0.271080\pi\)
\(968\) 0.656854 0.0211121
\(969\) 0 0
\(970\) −14.4853 −0.465094
\(971\) −9.65685 −0.309903 −0.154952 0.987922i \(-0.549522\pi\)
−0.154952 + 0.987922i \(0.549522\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −9.17157 −0.293876
\(975\) 0 0
\(976\) 1.65685 0.0530346
\(977\) −29.3137 −0.937829 −0.468914 0.883244i \(-0.655355\pi\)
−0.468914 + 0.883244i \(0.655355\pi\)
\(978\) 0 0
\(979\) 49.4558 1.58062
\(980\) 0 0
\(981\) 0 0
\(982\) 18.9289 0.604046
\(983\) −14.3848 −0.458803 −0.229402 0.973332i \(-0.573677\pi\)
−0.229402 + 0.973332i \(0.573677\pi\)
\(984\) 0 0
\(985\) 0.343146 0.0109335
\(986\) 18.4853 0.588691
\(987\) 0 0
\(988\) 5.65685 0.179969
\(989\) −25.8579 −0.822232
\(990\) 0 0
\(991\) −12.1421 −0.385708 −0.192854 0.981227i \(-0.561774\pi\)
−0.192854 + 0.981227i \(0.561774\pi\)
\(992\) −7.41421 −0.235402
\(993\) 0 0
\(994\) 0 0
\(995\) −18.2426 −0.578331
\(996\) 0 0
\(997\) 8.34315 0.264230 0.132115 0.991234i \(-0.457823\pi\)
0.132115 + 0.991234i \(0.457823\pi\)
\(998\) −31.1127 −0.984855
\(999\) 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 4410.2.a.bx.1.1 yes 2
3.2 odd 2 4410.2.a.bp.1.2 2
7.6 odd 2 4410.2.a.bu.1.1 yes 2
21.20 even 2 4410.2.a.bs.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4410.2.a.bp.1.2 2 3.2 odd 2
4410.2.a.bs.1.2 yes 2 21.20 even 2
4410.2.a.bu.1.1 yes 2 7.6 odd 2
4410.2.a.bx.1.1 yes 2 1.1 even 1 trivial