Properties

Label 4416.2.a.bs.1.2
Level $4416$
Weight $2$
Character 4416.1
Self dual yes
Analytic conductor $35.262$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4416.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2619375326\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 4416.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.07838 q^{5} -4.34017 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.07838 q^{5} -4.34017 q^{7} +1.00000 q^{9} -3.41855 q^{11} -2.00000 q^{13} -1.07838 q^{15} -2.34017 q^{17} +0.921622 q^{19} -4.34017 q^{21} +1.00000 q^{23} -3.83710 q^{25} +1.00000 q^{27} -6.68035 q^{29} +8.68035 q^{31} -3.41855 q^{33} +4.68035 q^{35} -7.26180 q^{37} -2.00000 q^{39} +8.83710 q^{41} +7.75872 q^{43} -1.07838 q^{45} +12.6803 q^{47} +11.8371 q^{49} -2.34017 q^{51} +11.9155 q^{53} +3.68649 q^{55} +0.921622 q^{57} -1.84324 q^{59} -4.73820 q^{61} -4.34017 q^{63} +2.15676 q^{65} -7.75872 q^{67} +1.00000 q^{69} -2.52359 q^{71} +2.00000 q^{73} -3.83710 q^{75} +14.8371 q^{77} +12.3402 q^{79} +1.00000 q^{81} -0.894960 q^{83} +2.52359 q^{85} -6.68035 q^{87} -8.49693 q^{89} +8.68035 q^{91} +8.68035 q^{93} -0.993857 q^{95} +8.15676 q^{97} -3.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 2q^{7} + 3q^{9} + 4q^{11} - 6q^{13} + 4q^{17} + 6q^{19} - 2q^{21} + 3q^{23} + 17q^{25} + 3q^{27} + 2q^{29} + 4q^{31} + 4q^{33} - 8q^{35} - 14q^{37} - 6q^{39} - 2q^{41} - 2q^{43} + 16q^{47} + 7q^{49} + 4q^{51} + 4q^{53} + 24q^{55} + 6q^{57} - 12q^{59} - 22q^{61} - 2q^{63} + 2q^{67} + 3q^{69} + 8q^{71} + 6q^{73} + 17q^{75} + 16q^{77} + 26q^{79} + 3q^{81} - 4q^{83} - 8q^{85} + 2q^{87} - 8q^{89} + 4q^{91} + 4q^{93} + 32q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.07838 −0.482265 −0.241133 0.970492i \(-0.577519\pi\)
−0.241133 + 0.970492i \(0.577519\pi\)
\(6\) 0 0
\(7\) −4.34017 −1.64043 −0.820216 0.572055i \(-0.806147\pi\)
−0.820216 + 0.572055i \(0.806147\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.41855 −1.03073 −0.515366 0.856970i \(-0.672344\pi\)
−0.515366 + 0.856970i \(0.672344\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.07838 −0.278436
\(16\) 0 0
\(17\) −2.34017 −0.567575 −0.283788 0.958887i \(-0.591591\pi\)
−0.283788 + 0.958887i \(0.591591\pi\)
\(18\) 0 0
\(19\) 0.921622 0.211435 0.105717 0.994396i \(-0.466286\pi\)
0.105717 + 0.994396i \(0.466286\pi\)
\(20\) 0 0
\(21\) −4.34017 −0.947103
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.83710 −0.767420
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) 8.68035 1.55904 0.779518 0.626380i \(-0.215464\pi\)
0.779518 + 0.626380i \(0.215464\pi\)
\(32\) 0 0
\(33\) −3.41855 −0.595093
\(34\) 0 0
\(35\) 4.68035 0.791123
\(36\) 0 0
\(37\) −7.26180 −1.19383 −0.596916 0.802304i \(-0.703607\pi\)
−0.596916 + 0.802304i \(0.703607\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 8.83710 1.38012 0.690062 0.723751i \(-0.257583\pi\)
0.690062 + 0.723751i \(0.257583\pi\)
\(42\) 0 0
\(43\) 7.75872 1.18319 0.591597 0.806234i \(-0.298498\pi\)
0.591597 + 0.806234i \(0.298498\pi\)
\(44\) 0 0
\(45\) −1.07838 −0.160755
\(46\) 0 0
\(47\) 12.6803 1.84962 0.924809 0.380431i \(-0.124224\pi\)
0.924809 + 0.380431i \(0.124224\pi\)
\(48\) 0 0
\(49\) 11.8371 1.69101
\(50\) 0 0
\(51\) −2.34017 −0.327690
\(52\) 0 0
\(53\) 11.9155 1.63672 0.818358 0.574708i \(-0.194884\pi\)
0.818358 + 0.574708i \(0.194884\pi\)
\(54\) 0 0
\(55\) 3.68649 0.497086
\(56\) 0 0
\(57\) 0.921622 0.122072
\(58\) 0 0
\(59\) −1.84324 −0.239970 −0.119985 0.992776i \(-0.538285\pi\)
−0.119985 + 0.992776i \(0.538285\pi\)
\(60\) 0 0
\(61\) −4.73820 −0.606665 −0.303332 0.952885i \(-0.598099\pi\)
−0.303332 + 0.952885i \(0.598099\pi\)
\(62\) 0 0
\(63\) −4.34017 −0.546810
\(64\) 0 0
\(65\) 2.15676 0.267513
\(66\) 0 0
\(67\) −7.75872 −0.947879 −0.473939 0.880557i \(-0.657168\pi\)
−0.473939 + 0.880557i \(0.657168\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.52359 −0.299495 −0.149748 0.988724i \(-0.547846\pi\)
−0.149748 + 0.988724i \(0.547846\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −3.83710 −0.443070
\(76\) 0 0
\(77\) 14.8371 1.69084
\(78\) 0 0
\(79\) 12.3402 1.38838 0.694189 0.719793i \(-0.255763\pi\)
0.694189 + 0.719793i \(0.255763\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.894960 −0.0982347 −0.0491173 0.998793i \(-0.515641\pi\)
−0.0491173 + 0.998793i \(0.515641\pi\)
\(84\) 0 0
\(85\) 2.52359 0.273722
\(86\) 0 0
\(87\) −6.68035 −0.716208
\(88\) 0 0
\(89\) −8.49693 −0.900673 −0.450336 0.892859i \(-0.648696\pi\)
−0.450336 + 0.892859i \(0.648696\pi\)
\(90\) 0 0
\(91\) 8.68035 0.909948
\(92\) 0 0
\(93\) 8.68035 0.900110
\(94\) 0 0
\(95\) −0.993857 −0.101968
\(96\) 0 0
\(97\) 8.15676 0.828193 0.414097 0.910233i \(-0.364098\pi\)
0.414097 + 0.910233i \(0.364098\pi\)
\(98\) 0 0
\(99\) −3.41855 −0.343577
\(100\) 0 0
\(101\) 6.31351 0.628218 0.314109 0.949387i \(-0.398294\pi\)
0.314109 + 0.949387i \(0.398294\pi\)
\(102\) 0 0
\(103\) 1.81658 0.178993 0.0894966 0.995987i \(-0.471474\pi\)
0.0894966 + 0.995987i \(0.471474\pi\)
\(104\) 0 0
\(105\) 4.68035 0.456755
\(106\) 0 0
\(107\) −5.94214 −0.574448 −0.287224 0.957863i \(-0.592732\pi\)
−0.287224 + 0.957863i \(0.592732\pi\)
\(108\) 0 0
\(109\) −14.7792 −1.41559 −0.707797 0.706416i \(-0.750311\pi\)
−0.707797 + 0.706416i \(0.750311\pi\)
\(110\) 0 0
\(111\) −7.26180 −0.689259
\(112\) 0 0
\(113\) −4.18342 −0.393543 −0.196771 0.980449i \(-0.563046\pi\)
−0.196771 + 0.980449i \(0.563046\pi\)
\(114\) 0 0
\(115\) −1.07838 −0.100559
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 10.1568 0.931068
\(120\) 0 0
\(121\) 0.686489 0.0624081
\(122\) 0 0
\(123\) 8.83710 0.796815
\(124\) 0 0
\(125\) 9.52973 0.852365
\(126\) 0 0
\(127\) 20.9939 1.86290 0.931452 0.363865i \(-0.118543\pi\)
0.931452 + 0.363865i \(0.118543\pi\)
\(128\) 0 0
\(129\) 7.75872 0.683118
\(130\) 0 0
\(131\) 10.8371 0.946842 0.473421 0.880836i \(-0.343019\pi\)
0.473421 + 0.880836i \(0.343019\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) −1.07838 −0.0928120
\(136\) 0 0
\(137\) −1.65983 −0.141809 −0.0709043 0.997483i \(-0.522588\pi\)
−0.0709043 + 0.997483i \(0.522588\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 12.6803 1.06788
\(142\) 0 0
\(143\) 6.83710 0.571747
\(144\) 0 0
\(145\) 7.20394 0.598254
\(146\) 0 0
\(147\) 11.8371 0.976308
\(148\) 0 0
\(149\) 3.23513 0.265032 0.132516 0.991181i \(-0.457694\pi\)
0.132516 + 0.991181i \(0.457694\pi\)
\(150\) 0 0
\(151\) 17.3607 1.41279 0.706397 0.707816i \(-0.250320\pi\)
0.706397 + 0.707816i \(0.250320\pi\)
\(152\) 0 0
\(153\) −2.34017 −0.189192
\(154\) 0 0
\(155\) −9.36069 −0.751869
\(156\) 0 0
\(157\) 5.78539 0.461724 0.230862 0.972986i \(-0.425845\pi\)
0.230862 + 0.972986i \(0.425845\pi\)
\(158\) 0 0
\(159\) 11.9155 0.944959
\(160\) 0 0
\(161\) −4.34017 −0.342054
\(162\) 0 0
\(163\) 2.15676 0.168930 0.0844651 0.996426i \(-0.473082\pi\)
0.0844651 + 0.996426i \(0.473082\pi\)
\(164\) 0 0
\(165\) 3.68649 0.286993
\(166\) 0 0
\(167\) −5.47641 −0.423777 −0.211889 0.977294i \(-0.567961\pi\)
−0.211889 + 0.977294i \(0.567961\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0.921622 0.0704782
\(172\) 0 0
\(173\) 9.31965 0.708560 0.354280 0.935139i \(-0.384726\pi\)
0.354280 + 0.935139i \(0.384726\pi\)
\(174\) 0 0
\(175\) 16.6537 1.25890
\(176\) 0 0
\(177\) −1.84324 −0.138547
\(178\) 0 0
\(179\) −4.36683 −0.326393 −0.163196 0.986594i \(-0.552180\pi\)
−0.163196 + 0.986594i \(0.552180\pi\)
\(180\) 0 0
\(181\) 3.94214 0.293017 0.146509 0.989209i \(-0.453196\pi\)
0.146509 + 0.989209i \(0.453196\pi\)
\(182\) 0 0
\(183\) −4.73820 −0.350258
\(184\) 0 0
\(185\) 7.83096 0.575744
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) −4.34017 −0.315701
\(190\) 0 0
\(191\) −15.5174 −1.12280 −0.561402 0.827544i \(-0.689738\pi\)
−0.561402 + 0.827544i \(0.689738\pi\)
\(192\) 0 0
\(193\) −16.8371 −1.21196 −0.605981 0.795479i \(-0.707219\pi\)
−0.605981 + 0.795479i \(0.707219\pi\)
\(194\) 0 0
\(195\) 2.15676 0.154448
\(196\) 0 0
\(197\) −9.20394 −0.655753 −0.327877 0.944721i \(-0.606333\pi\)
−0.327877 + 0.944721i \(0.606333\pi\)
\(198\) 0 0
\(199\) −26.6947 −1.89234 −0.946169 0.323672i \(-0.895083\pi\)
−0.946169 + 0.323672i \(0.895083\pi\)
\(200\) 0 0
\(201\) −7.75872 −0.547258
\(202\) 0 0
\(203\) 28.9939 2.03497
\(204\) 0 0
\(205\) −9.52973 −0.665585
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.15061 −0.217932
\(210\) 0 0
\(211\) 19.5174 1.34364 0.671818 0.740716i \(-0.265514\pi\)
0.671818 + 0.740716i \(0.265514\pi\)
\(212\) 0 0
\(213\) −2.52359 −0.172914
\(214\) 0 0
\(215\) −8.36683 −0.570613
\(216\) 0 0
\(217\) −37.6742 −2.55749
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 4.68035 0.314834
\(222\) 0 0
\(223\) −29.6742 −1.98713 −0.993566 0.113256i \(-0.963872\pi\)
−0.993566 + 0.113256i \(0.963872\pi\)
\(224\) 0 0
\(225\) −3.83710 −0.255807
\(226\) 0 0
\(227\) 13.2618 0.880216 0.440108 0.897945i \(-0.354940\pi\)
0.440108 + 0.897945i \(0.354940\pi\)
\(228\) 0 0
\(229\) 11.9421 0.789159 0.394579 0.918862i \(-0.370890\pi\)
0.394579 + 0.918862i \(0.370890\pi\)
\(230\) 0 0
\(231\) 14.8371 0.976210
\(232\) 0 0
\(233\) 5.68649 0.372534 0.186267 0.982499i \(-0.440361\pi\)
0.186267 + 0.982499i \(0.440361\pi\)
\(234\) 0 0
\(235\) −13.6742 −0.892007
\(236\) 0 0
\(237\) 12.3402 0.801580
\(238\) 0 0
\(239\) 21.6742 1.40199 0.700994 0.713167i \(-0.252740\pi\)
0.700994 + 0.713167i \(0.252740\pi\)
\(240\) 0 0
\(241\) 16.1568 1.04075 0.520374 0.853938i \(-0.325793\pi\)
0.520374 + 0.853938i \(0.325793\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −12.7649 −0.815517
\(246\) 0 0
\(247\) −1.84324 −0.117283
\(248\) 0 0
\(249\) −0.894960 −0.0567158
\(250\) 0 0
\(251\) 28.0989 1.77359 0.886793 0.462166i \(-0.152928\pi\)
0.886793 + 0.462166i \(0.152928\pi\)
\(252\) 0 0
\(253\) −3.41855 −0.214922
\(254\) 0 0
\(255\) 2.52359 0.158033
\(256\) 0 0
\(257\) −0.523590 −0.0326607 −0.0163303 0.999867i \(-0.505198\pi\)
−0.0163303 + 0.999867i \(0.505198\pi\)
\(258\) 0 0
\(259\) 31.5174 1.95840
\(260\) 0 0
\(261\) −6.68035 −0.413503
\(262\) 0 0
\(263\) −4.99386 −0.307934 −0.153967 0.988076i \(-0.549205\pi\)
−0.153967 + 0.988076i \(0.549205\pi\)
\(264\) 0 0
\(265\) −12.8494 −0.789332
\(266\) 0 0
\(267\) −8.49693 −0.520004
\(268\) 0 0
\(269\) 3.84324 0.234327 0.117163 0.993113i \(-0.462620\pi\)
0.117163 + 0.993113i \(0.462620\pi\)
\(270\) 0 0
\(271\) 24.6803 1.49922 0.749612 0.661877i \(-0.230240\pi\)
0.749612 + 0.661877i \(0.230240\pi\)
\(272\) 0 0
\(273\) 8.68035 0.525358
\(274\) 0 0
\(275\) 13.1173 0.791005
\(276\) 0 0
\(277\) −25.8843 −1.55524 −0.777618 0.628737i \(-0.783572\pi\)
−0.777618 + 0.628737i \(0.783572\pi\)
\(278\) 0 0
\(279\) 8.68035 0.519679
\(280\) 0 0
\(281\) 29.1773 1.74057 0.870285 0.492548i \(-0.163934\pi\)
0.870285 + 0.492548i \(0.163934\pi\)
\(282\) 0 0
\(283\) −2.28231 −0.135669 −0.0678347 0.997697i \(-0.521609\pi\)
−0.0678347 + 0.997697i \(0.521609\pi\)
\(284\) 0 0
\(285\) −0.993857 −0.0588710
\(286\) 0 0
\(287\) −38.3545 −2.26400
\(288\) 0 0
\(289\) −11.5236 −0.677858
\(290\) 0 0
\(291\) 8.15676 0.478157
\(292\) 0 0
\(293\) 7.60197 0.444112 0.222056 0.975034i \(-0.428723\pi\)
0.222056 + 0.975034i \(0.428723\pi\)
\(294\) 0 0
\(295\) 1.98771 0.115729
\(296\) 0 0
\(297\) −3.41855 −0.198364
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −33.6742 −1.94095
\(302\) 0 0
\(303\) 6.31351 0.362702
\(304\) 0 0
\(305\) 5.10957 0.292573
\(306\) 0 0
\(307\) −10.8904 −0.621549 −0.310775 0.950484i \(-0.600588\pi\)
−0.310775 + 0.950484i \(0.600588\pi\)
\(308\) 0 0
\(309\) 1.81658 0.103342
\(310\) 0 0
\(311\) 11.5174 0.653095 0.326547 0.945181i \(-0.394115\pi\)
0.326547 + 0.945181i \(0.394115\pi\)
\(312\) 0 0
\(313\) −4.21008 −0.237968 −0.118984 0.992896i \(-0.537964\pi\)
−0.118984 + 0.992896i \(0.537964\pi\)
\(314\) 0 0
\(315\) 4.68035 0.263708
\(316\) 0 0
\(317\) −13.3197 −0.748106 −0.374053 0.927407i \(-0.622032\pi\)
−0.374053 + 0.927407i \(0.622032\pi\)
\(318\) 0 0
\(319\) 22.8371 1.27863
\(320\) 0 0
\(321\) −5.94214 −0.331658
\(322\) 0 0
\(323\) −2.15676 −0.120005
\(324\) 0 0
\(325\) 7.67420 0.425688
\(326\) 0 0
\(327\) −14.7792 −0.817294
\(328\) 0 0
\(329\) −55.0349 −3.03417
\(330\) 0 0
\(331\) 2.15676 0.118546 0.0592730 0.998242i \(-0.481122\pi\)
0.0592730 + 0.998242i \(0.481122\pi\)
\(332\) 0 0
\(333\) −7.26180 −0.397944
\(334\) 0 0
\(335\) 8.36683 0.457129
\(336\) 0 0
\(337\) 24.3545 1.32668 0.663338 0.748320i \(-0.269139\pi\)
0.663338 + 0.748320i \(0.269139\pi\)
\(338\) 0 0
\(339\) −4.18342 −0.227212
\(340\) 0 0
\(341\) −29.6742 −1.60695
\(342\) 0 0
\(343\) −20.9939 −1.13356
\(344\) 0 0
\(345\) −1.07838 −0.0580579
\(346\) 0 0
\(347\) 26.0410 1.39796 0.698978 0.715143i \(-0.253638\pi\)
0.698978 + 0.715143i \(0.253638\pi\)
\(348\) 0 0
\(349\) 0.523590 0.0280272 0.0140136 0.999902i \(-0.495539\pi\)
0.0140136 + 0.999902i \(0.495539\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 28.0410 1.49247 0.746237 0.665680i \(-0.231859\pi\)
0.746237 + 0.665680i \(0.231859\pi\)
\(354\) 0 0
\(355\) 2.72138 0.144436
\(356\) 0 0
\(357\) 10.1568 0.537553
\(358\) 0 0
\(359\) 9.84324 0.519507 0.259753 0.965675i \(-0.416359\pi\)
0.259753 + 0.965675i \(0.416359\pi\)
\(360\) 0 0
\(361\) −18.1506 −0.955295
\(362\) 0 0
\(363\) 0.686489 0.0360313
\(364\) 0 0
\(365\) −2.15676 −0.112890
\(366\) 0 0
\(367\) −17.3340 −0.904829 −0.452414 0.891808i \(-0.649437\pi\)
−0.452414 + 0.891808i \(0.649437\pi\)
\(368\) 0 0
\(369\) 8.83710 0.460041
\(370\) 0 0
\(371\) −51.7152 −2.68492
\(372\) 0 0
\(373\) −23.2618 −1.20445 −0.602225 0.798326i \(-0.705719\pi\)
−0.602225 + 0.798326i \(0.705719\pi\)
\(374\) 0 0
\(375\) 9.52973 0.492113
\(376\) 0 0
\(377\) 13.3607 0.688111
\(378\) 0 0
\(379\) −25.7998 −1.32524 −0.662622 0.748954i \(-0.730557\pi\)
−0.662622 + 0.748954i \(0.730557\pi\)
\(380\) 0 0
\(381\) 20.9939 1.07555
\(382\) 0 0
\(383\) 6.35455 0.324702 0.162351 0.986733i \(-0.448092\pi\)
0.162351 + 0.986733i \(0.448092\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 7.75872 0.394398
\(388\) 0 0
\(389\) −3.54864 −0.179923 −0.0899617 0.995945i \(-0.528674\pi\)
−0.0899617 + 0.995945i \(0.528674\pi\)
\(390\) 0 0
\(391\) −2.34017 −0.118348
\(392\) 0 0
\(393\) 10.8371 0.546659
\(394\) 0 0
\(395\) −13.3074 −0.669566
\(396\) 0 0
\(397\) 10.6270 0.533355 0.266677 0.963786i \(-0.414074\pi\)
0.266677 + 0.963786i \(0.414074\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 36.0677 1.80113 0.900567 0.434716i \(-0.143151\pi\)
0.900567 + 0.434716i \(0.143151\pi\)
\(402\) 0 0
\(403\) −17.3607 −0.864798
\(404\) 0 0
\(405\) −1.07838 −0.0535850
\(406\) 0 0
\(407\) 24.8248 1.23052
\(408\) 0 0
\(409\) −29.1506 −1.44141 −0.720703 0.693244i \(-0.756181\pi\)
−0.720703 + 0.693244i \(0.756181\pi\)
\(410\) 0 0
\(411\) −1.65983 −0.0818732
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 0.965105 0.0473752
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 12.7792 0.624307 0.312153 0.950032i \(-0.398950\pi\)
0.312153 + 0.950032i \(0.398950\pi\)
\(420\) 0 0
\(421\) −28.2557 −1.37710 −0.688548 0.725191i \(-0.741752\pi\)
−0.688548 + 0.725191i \(0.741752\pi\)
\(422\) 0 0
\(423\) 12.6803 0.616540
\(424\) 0 0
\(425\) 8.97948 0.435569
\(426\) 0 0
\(427\) 20.5646 0.995192
\(428\) 0 0
\(429\) 6.83710 0.330098
\(430\) 0 0
\(431\) −37.8720 −1.82423 −0.912115 0.409935i \(-0.865552\pi\)
−0.912115 + 0.409935i \(0.865552\pi\)
\(432\) 0 0
\(433\) −12.8371 −0.616912 −0.308456 0.951239i \(-0.599812\pi\)
−0.308456 + 0.951239i \(0.599812\pi\)
\(434\) 0 0
\(435\) 7.20394 0.345402
\(436\) 0 0
\(437\) 0.921622 0.0440872
\(438\) 0 0
\(439\) 25.3607 1.21040 0.605200 0.796074i \(-0.293093\pi\)
0.605200 + 0.796074i \(0.293093\pi\)
\(440\) 0 0
\(441\) 11.8371 0.563671
\(442\) 0 0
\(443\) 37.9877 1.80485 0.902425 0.430846i \(-0.141785\pi\)
0.902425 + 0.430846i \(0.141785\pi\)
\(444\) 0 0
\(445\) 9.16290 0.434363
\(446\) 0 0
\(447\) 3.23513 0.153017
\(448\) 0 0
\(449\) −38.1978 −1.80267 −0.901333 0.433128i \(-0.857410\pi\)
−0.901333 + 0.433128i \(0.857410\pi\)
\(450\) 0 0
\(451\) −30.2101 −1.42254
\(452\) 0 0
\(453\) 17.3607 0.815676
\(454\) 0 0
\(455\) −9.36069 −0.438836
\(456\) 0 0
\(457\) 11.8432 0.554004 0.277002 0.960869i \(-0.410659\pi\)
0.277002 + 0.960869i \(0.410659\pi\)
\(458\) 0 0
\(459\) −2.34017 −0.109230
\(460\) 0 0
\(461\) 39.1917 1.82534 0.912669 0.408700i \(-0.134018\pi\)
0.912669 + 0.408700i \(0.134018\pi\)
\(462\) 0 0
\(463\) −25.3607 −1.17861 −0.589306 0.807910i \(-0.700599\pi\)
−0.589306 + 0.807910i \(0.700599\pi\)
\(464\) 0 0
\(465\) −9.36069 −0.434092
\(466\) 0 0
\(467\) −13.2618 −0.613683 −0.306841 0.951761i \(-0.599272\pi\)
−0.306841 + 0.951761i \(0.599272\pi\)
\(468\) 0 0
\(469\) 33.6742 1.55493
\(470\) 0 0
\(471\) 5.78539 0.266577
\(472\) 0 0
\(473\) −26.5236 −1.21956
\(474\) 0 0
\(475\) −3.53636 −0.162259
\(476\) 0 0
\(477\) 11.9155 0.545572
\(478\) 0 0
\(479\) 11.6865 0.533969 0.266985 0.963701i \(-0.413973\pi\)
0.266985 + 0.963701i \(0.413973\pi\)
\(480\) 0 0
\(481\) 14.5236 0.662219
\(482\) 0 0
\(483\) −4.34017 −0.197485
\(484\) 0 0
\(485\) −8.79606 −0.399409
\(486\) 0 0
\(487\) 5.04718 0.228710 0.114355 0.993440i \(-0.463520\pi\)
0.114355 + 0.993440i \(0.463520\pi\)
\(488\) 0 0
\(489\) 2.15676 0.0975319
\(490\) 0 0
\(491\) 0.680346 0.0307036 0.0153518 0.999882i \(-0.495113\pi\)
0.0153518 + 0.999882i \(0.495113\pi\)
\(492\) 0 0
\(493\) 15.6332 0.704082
\(494\) 0 0
\(495\) 3.68649 0.165695
\(496\) 0 0
\(497\) 10.9528 0.491301
\(498\) 0 0
\(499\) −25.1917 −1.12773 −0.563867 0.825866i \(-0.690687\pi\)
−0.563867 + 0.825866i \(0.690687\pi\)
\(500\) 0 0
\(501\) −5.47641 −0.244668
\(502\) 0 0
\(503\) 19.8843 0.886596 0.443298 0.896374i \(-0.353808\pi\)
0.443298 + 0.896374i \(0.353808\pi\)
\(504\) 0 0
\(505\) −6.80835 −0.302968
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −33.4017 −1.48051 −0.740253 0.672329i \(-0.765294\pi\)
−0.740253 + 0.672329i \(0.765294\pi\)
\(510\) 0 0
\(511\) −8.68035 −0.383996
\(512\) 0 0
\(513\) 0.921622 0.0406906
\(514\) 0 0
\(515\) −1.95896 −0.0863222
\(516\) 0 0
\(517\) −43.3484 −1.90646
\(518\) 0 0
\(519\) 9.31965 0.409087
\(520\) 0 0
\(521\) 27.3874 1.19986 0.599931 0.800052i \(-0.295195\pi\)
0.599931 + 0.800052i \(0.295195\pi\)
\(522\) 0 0
\(523\) −28.0722 −1.22751 −0.613757 0.789495i \(-0.710342\pi\)
−0.613757 + 0.789495i \(0.710342\pi\)
\(524\) 0 0
\(525\) 16.6537 0.726826
\(526\) 0 0
\(527\) −20.3135 −0.884870
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.84324 −0.0799900
\(532\) 0 0
\(533\) −17.6742 −0.765555
\(534\) 0 0
\(535\) 6.40787 0.277036
\(536\) 0 0
\(537\) −4.36683 −0.188443
\(538\) 0 0
\(539\) −40.4657 −1.74298
\(540\) 0 0
\(541\) 41.8843 1.80075 0.900373 0.435119i \(-0.143294\pi\)
0.900373 + 0.435119i \(0.143294\pi\)
\(542\) 0 0
\(543\) 3.94214 0.169173
\(544\) 0 0
\(545\) 15.9376 0.682692
\(546\) 0 0
\(547\) −31.2039 −1.33418 −0.667092 0.744975i \(-0.732461\pi\)
−0.667092 + 0.744975i \(0.732461\pi\)
\(548\) 0 0
\(549\) −4.73820 −0.202222
\(550\) 0 0
\(551\) −6.15676 −0.262287
\(552\) 0 0
\(553\) −53.5585 −2.27754
\(554\) 0 0
\(555\) 7.83096 0.332406
\(556\) 0 0
\(557\) −6.60811 −0.279995 −0.139997 0.990152i \(-0.544709\pi\)
−0.139997 + 0.990152i \(0.544709\pi\)
\(558\) 0 0
\(559\) −15.5174 −0.656318
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 29.2618 1.23324 0.616619 0.787262i \(-0.288502\pi\)
0.616619 + 0.787262i \(0.288502\pi\)
\(564\) 0 0
\(565\) 4.51130 0.189792
\(566\) 0 0
\(567\) −4.34017 −0.182270
\(568\) 0 0
\(569\) 11.3340 0.475147 0.237574 0.971370i \(-0.423648\pi\)
0.237574 + 0.971370i \(0.423648\pi\)
\(570\) 0 0
\(571\) 34.9627 1.46314 0.731571 0.681765i \(-0.238788\pi\)
0.731571 + 0.681765i \(0.238788\pi\)
\(572\) 0 0
\(573\) −15.5174 −0.648251
\(574\) 0 0
\(575\) −3.83710 −0.160018
\(576\) 0 0
\(577\) 1.15061 0.0479006 0.0239503 0.999713i \(-0.492376\pi\)
0.0239503 + 0.999713i \(0.492376\pi\)
\(578\) 0 0
\(579\) −16.8371 −0.699726
\(580\) 0 0
\(581\) 3.88428 0.161147
\(582\) 0 0
\(583\) −40.7337 −1.68702
\(584\) 0 0
\(585\) 2.15676 0.0891709
\(586\) 0 0
\(587\) 16.5113 0.681494 0.340747 0.940155i \(-0.389320\pi\)
0.340747 + 0.940155i \(0.389320\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −9.20394 −0.378599
\(592\) 0 0
\(593\) 11.1629 0.458405 0.229203 0.973379i \(-0.426388\pi\)
0.229203 + 0.973379i \(0.426388\pi\)
\(594\) 0 0
\(595\) −10.9528 −0.449022
\(596\) 0 0
\(597\) −26.6947 −1.09254
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 41.1506 1.67857 0.839284 0.543693i \(-0.182974\pi\)
0.839284 + 0.543693i \(0.182974\pi\)
\(602\) 0 0
\(603\) −7.75872 −0.315960
\(604\) 0 0
\(605\) −0.740294 −0.0300973
\(606\) 0 0
\(607\) 24.0000 0.974130 0.487065 0.873366i \(-0.338067\pi\)
0.487065 + 0.873366i \(0.338067\pi\)
\(608\) 0 0
\(609\) 28.9939 1.17489
\(610\) 0 0
\(611\) −25.3607 −1.02598
\(612\) 0 0
\(613\) −4.73820 −0.191374 −0.0956871 0.995411i \(-0.530505\pi\)
−0.0956871 + 0.995411i \(0.530505\pi\)
\(614\) 0 0
\(615\) −9.52973 −0.384276
\(616\) 0 0
\(617\) −34.4846 −1.38830 −0.694150 0.719831i \(-0.744219\pi\)
−0.694150 + 0.719831i \(0.744219\pi\)
\(618\) 0 0
\(619\) 41.0661 1.65059 0.825293 0.564705i \(-0.191010\pi\)
0.825293 + 0.564705i \(0.191010\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 36.8781 1.47749
\(624\) 0 0
\(625\) 8.90885 0.356354
\(626\) 0 0
\(627\) −3.15061 −0.125823
\(628\) 0 0
\(629\) 16.9939 0.677589
\(630\) 0 0
\(631\) 22.8638 0.910192 0.455096 0.890442i \(-0.349605\pi\)
0.455096 + 0.890442i \(0.349605\pi\)
\(632\) 0 0
\(633\) 19.5174 0.775749
\(634\) 0 0
\(635\) −22.6393 −0.898414
\(636\) 0 0
\(637\) −23.6742 −0.938006
\(638\) 0 0
\(639\) −2.52359 −0.0998317
\(640\) 0 0
\(641\) 34.7070 1.37084 0.685422 0.728146i \(-0.259618\pi\)
0.685422 + 0.728146i \(0.259618\pi\)
\(642\) 0 0
\(643\) −2.96266 −0.116836 −0.0584180 0.998292i \(-0.518606\pi\)
−0.0584180 + 0.998292i \(0.518606\pi\)
\(644\) 0 0
\(645\) −8.36683 −0.329444
\(646\) 0 0
\(647\) 14.4703 0.568885 0.284442 0.958693i \(-0.408192\pi\)
0.284442 + 0.958693i \(0.408192\pi\)
\(648\) 0 0
\(649\) 6.30122 0.247345
\(650\) 0 0
\(651\) −37.6742 −1.47657
\(652\) 0 0
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) −11.6865 −0.456629
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 35.1338 1.36862 0.684309 0.729192i \(-0.260104\pi\)
0.684309 + 0.729192i \(0.260104\pi\)
\(660\) 0 0
\(661\) 42.2967 1.64515 0.822575 0.568656i \(-0.192537\pi\)
0.822575 + 0.568656i \(0.192537\pi\)
\(662\) 0 0
\(663\) 4.68035 0.181770
\(664\) 0 0
\(665\) 4.31351 0.167271
\(666\) 0 0
\(667\) −6.68035 −0.258664
\(668\) 0 0
\(669\) −29.6742 −1.14727
\(670\) 0 0
\(671\) 16.1978 0.625309
\(672\) 0 0
\(673\) 9.15061 0.352730 0.176365 0.984325i \(-0.443566\pi\)
0.176365 + 0.984325i \(0.443566\pi\)
\(674\) 0 0
\(675\) −3.83710 −0.147690
\(676\) 0 0
\(677\) 43.0037 1.65277 0.826383 0.563108i \(-0.190395\pi\)
0.826383 + 0.563108i \(0.190395\pi\)
\(678\) 0 0
\(679\) −35.4017 −1.35859
\(680\) 0 0
\(681\) 13.2618 0.508193
\(682\) 0 0
\(683\) −44.1978 −1.69118 −0.845591 0.533832i \(-0.820752\pi\)
−0.845591 + 0.533832i \(0.820752\pi\)
\(684\) 0 0
\(685\) 1.78992 0.0683893
\(686\) 0 0
\(687\) 11.9421 0.455621
\(688\) 0 0
\(689\) −23.8310 −0.907887
\(690\) 0 0
\(691\) −22.4703 −0.854809 −0.427405 0.904060i \(-0.640572\pi\)
−0.427405 + 0.904060i \(0.640572\pi\)
\(692\) 0 0
\(693\) 14.8371 0.563615
\(694\) 0 0
\(695\) 4.31351 0.163621
\(696\) 0 0
\(697\) −20.6803 −0.783324
\(698\) 0 0
\(699\) 5.68649 0.215083
\(700\) 0 0
\(701\) 24.4801 0.924601 0.462300 0.886723i \(-0.347024\pi\)
0.462300 + 0.886723i \(0.347024\pi\)
\(702\) 0 0
\(703\) −6.69263 −0.252417
\(704\) 0 0
\(705\) −13.6742 −0.515000
\(706\) 0 0
\(707\) −27.4017 −1.03055
\(708\) 0 0
\(709\) −32.5692 −1.22316 −0.611580 0.791182i \(-0.709466\pi\)
−0.611580 + 0.791182i \(0.709466\pi\)
\(710\) 0 0
\(711\) 12.3402 0.462793
\(712\) 0 0
\(713\) 8.68035 0.325082
\(714\) 0 0
\(715\) −7.37298 −0.275734
\(716\) 0 0
\(717\) 21.6742 0.809438
\(718\) 0 0
\(719\) −7.20394 −0.268661 −0.134331 0.990937i \(-0.542888\pi\)
−0.134331 + 0.990937i \(0.542888\pi\)
\(720\) 0 0
\(721\) −7.88428 −0.293626
\(722\) 0 0
\(723\) 16.1568 0.600876
\(724\) 0 0
\(725\) 25.6332 0.951992
\(726\) 0 0
\(727\) −16.6537 −0.617651 −0.308825 0.951119i \(-0.599936\pi\)
−0.308825 + 0.951119i \(0.599936\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.1568 −0.671552
\(732\) 0 0
\(733\) −46.2434 −1.70804 −0.854019 0.520242i \(-0.825842\pi\)
−0.854019 + 0.520242i \(0.825842\pi\)
\(734\) 0 0
\(735\) −12.7649 −0.470839
\(736\) 0 0
\(737\) 26.5236 0.977009
\(738\) 0 0
\(739\) 12.7337 0.468416 0.234208 0.972187i \(-0.424750\pi\)
0.234208 + 0.972187i \(0.424750\pi\)
\(740\) 0 0
\(741\) −1.84324 −0.0677133
\(742\) 0 0
\(743\) −20.3668 −0.747187 −0.373593 0.927593i \(-0.621874\pi\)
−0.373593 + 0.927593i \(0.621874\pi\)
\(744\) 0 0
\(745\) −3.48870 −0.127816
\(746\) 0 0
\(747\) −0.894960 −0.0327449
\(748\) 0 0
\(749\) 25.7899 0.942343
\(750\) 0 0
\(751\) 3.85762 0.140767 0.0703833 0.997520i \(-0.477578\pi\)
0.0703833 + 0.997520i \(0.477578\pi\)
\(752\) 0 0
\(753\) 28.0989 1.02398
\(754\) 0 0
\(755\) −18.7214 −0.681341
\(756\) 0 0
\(757\) 1.61634 0.0587470 0.0293735 0.999569i \(-0.490649\pi\)
0.0293735 + 0.999569i \(0.490649\pi\)
\(758\) 0 0
\(759\) −3.41855 −0.124086
\(760\) 0 0
\(761\) 20.0410 0.726487 0.363244 0.931694i \(-0.381669\pi\)
0.363244 + 0.931694i \(0.381669\pi\)
\(762\) 0 0
\(763\) 64.1445 2.32219
\(764\) 0 0
\(765\) 2.52359 0.0912406
\(766\) 0 0
\(767\) 3.68649 0.133111
\(768\) 0 0
\(769\) −23.1629 −0.835275 −0.417638 0.908614i \(-0.637142\pi\)
−0.417638 + 0.908614i \(0.637142\pi\)
\(770\) 0 0
\(771\) −0.523590 −0.0188566
\(772\) 0 0
\(773\) −41.2762 −1.48460 −0.742300 0.670067i \(-0.766265\pi\)
−0.742300 + 0.670067i \(0.766265\pi\)
\(774\) 0 0
\(775\) −33.3074 −1.19644
\(776\) 0 0
\(777\) 31.5174 1.13068
\(778\) 0 0
\(779\) 8.14447 0.291806
\(780\) 0 0
\(781\) 8.62702 0.308699
\(782\) 0 0
\(783\) −6.68035 −0.238736
\(784\) 0 0
\(785\) −6.23883 −0.222673
\(786\) 0 0
\(787\) 13.4329 0.478832 0.239416 0.970917i \(-0.423044\pi\)
0.239416 + 0.970917i \(0.423044\pi\)
\(788\) 0 0
\(789\) −4.99386 −0.177786
\(790\) 0 0
\(791\) 18.1568 0.645580
\(792\) 0 0
\(793\) 9.47641 0.336517
\(794\) 0 0
\(795\) −12.8494 −0.455721
\(796\) 0 0
\(797\) 5.07838 0.179885 0.0899427 0.995947i \(-0.471332\pi\)
0.0899427 + 0.995947i \(0.471332\pi\)
\(798\) 0 0
\(799\) −29.6742 −1.04980
\(800\) 0 0
\(801\) −8.49693 −0.300224
\(802\) 0 0
\(803\) −6.83710 −0.241276
\(804\) 0 0
\(805\) 4.68035 0.164961
\(806\) 0 0
\(807\) 3.84324 0.135289
\(808\) 0 0
\(809\) 8.35455 0.293730 0.146865 0.989157i \(-0.453082\pi\)
0.146865 + 0.989157i \(0.453082\pi\)
\(810\) 0 0
\(811\) 44.8781 1.57588 0.787942 0.615749i \(-0.211146\pi\)
0.787942 + 0.615749i \(0.211146\pi\)
\(812\) 0 0
\(813\) 24.6803 0.865578
\(814\) 0 0
\(815\) −2.32580 −0.0814691
\(816\) 0 0
\(817\) 7.15061 0.250168
\(818\) 0 0
\(819\) 8.68035 0.303316
\(820\) 0 0
\(821\) −49.3484 −1.72227 −0.861136 0.508375i \(-0.830246\pi\)
−0.861136 + 0.508375i \(0.830246\pi\)
\(822\) 0 0
\(823\) −10.0410 −0.350009 −0.175004 0.984568i \(-0.555994\pi\)
−0.175004 + 0.984568i \(0.555994\pi\)
\(824\) 0 0
\(825\) 13.1173 0.456687
\(826\) 0 0
\(827\) 54.0866 1.88078 0.940388 0.340104i \(-0.110462\pi\)
0.940388 + 0.340104i \(0.110462\pi\)
\(828\) 0 0
\(829\) 20.8371 0.723702 0.361851 0.932236i \(-0.382145\pi\)
0.361851 + 0.932236i \(0.382145\pi\)
\(830\) 0 0
\(831\) −25.8843 −0.897916
\(832\) 0 0
\(833\) −27.7009 −0.959778
\(834\) 0 0
\(835\) 5.90564 0.204373
\(836\) 0 0
\(837\) 8.68035 0.300037
\(838\) 0 0
\(839\) −8.68035 −0.299679 −0.149839 0.988710i \(-0.547876\pi\)
−0.149839 + 0.988710i \(0.547876\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 0 0
\(843\) 29.1773 1.00492
\(844\) 0 0
\(845\) 9.70540 0.333876
\(846\) 0 0
\(847\) −2.97948 −0.102376
\(848\) 0 0
\(849\) −2.28231 −0.0783288
\(850\) 0 0
\(851\) −7.26180 −0.248931
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) −0.993857 −0.0339892
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −38.3545 −1.30712
\(862\) 0 0
\(863\) −29.8432 −1.01588 −0.507938 0.861394i \(-0.669592\pi\)
−0.507938 + 0.861394i \(0.669592\pi\)
\(864\) 0 0
\(865\) −10.0501 −0.341714
\(866\) 0 0
\(867\) −11.5236 −0.391362
\(868\) 0 0
\(869\) −42.1855 −1.43105
\(870\) 0 0
\(871\) 15.5174 0.525789
\(872\) 0 0
\(873\) 8.15676 0.276064
\(874\) 0 0
\(875\) −41.3607 −1.39825
\(876\) 0 0
\(877\) −11.4764 −0.387531 −0.193765 0.981048i \(-0.562070\pi\)
−0.193765 + 0.981048i \(0.562070\pi\)
\(878\) 0 0
\(879\) 7.60197 0.256408
\(880\) 0 0
\(881\) −26.8227 −0.903681 −0.451840 0.892099i \(-0.649232\pi\)
−0.451840 + 0.892099i \(0.649232\pi\)
\(882\) 0 0
\(883\) −2.15676 −0.0725806 −0.0362903 0.999341i \(-0.511554\pi\)
−0.0362903 + 0.999341i \(0.511554\pi\)
\(884\) 0 0
\(885\) 1.98771 0.0668163
\(886\) 0 0
\(887\) −29.4140 −0.987626 −0.493813 0.869568i \(-0.664397\pi\)
−0.493813 + 0.869568i \(0.664397\pi\)
\(888\) 0 0
\(889\) −91.1170 −3.05597
\(890\) 0 0
\(891\) −3.41855 −0.114526
\(892\) 0 0
\(893\) 11.6865 0.391073
\(894\) 0 0
\(895\) 4.70910 0.157408
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) −57.9877 −1.93400
\(900\) 0 0
\(901\) −27.8843 −0.928960
\(902\) 0 0
\(903\) −33.6742 −1.12061
\(904\) 0 0
\(905\) −4.25112 −0.141312
\(906\) 0 0
\(907\) −44.2700 −1.46996 −0.734981 0.678088i \(-0.762809\pi\)
−0.734981 + 0.678088i \(0.762809\pi\)
\(908\) 0 0
\(909\) 6.31351 0.209406
\(910\) 0 0
\(911\) −20.5113 −0.679570 −0.339785 0.940503i \(-0.610354\pi\)
−0.339785 + 0.940503i \(0.610354\pi\)
\(912\) 0 0
\(913\) 3.05947 0.101254
\(914\) 0 0
\(915\) 5.10957 0.168917
\(916\) 0 0
\(917\) −47.0349 −1.55323
\(918\) 0 0
\(919\) 2.01438 0.0664481 0.0332241 0.999448i \(-0.489423\pi\)
0.0332241 + 0.999448i \(0.489423\pi\)
\(920\) 0 0
\(921\) −10.8904 −0.358852
\(922\) 0 0
\(923\) 5.04718 0.166130
\(924\) 0 0
\(925\) 27.8642 0.916171
\(926\) 0 0
\(927\) 1.81658 0.0596644
\(928\) 0 0
\(929\) −4.21008 −0.138128 −0.0690641 0.997612i \(-0.522001\pi\)
−0.0690641 + 0.997612i \(0.522001\pi\)
\(930\) 0 0
\(931\) 10.9093 0.357539
\(932\) 0 0
\(933\) 11.5174 0.377064
\(934\) 0 0
\(935\) −8.62702 −0.282134
\(936\) 0 0
\(937\) −20.6393 −0.674257 −0.337128 0.941459i \(-0.609456\pi\)
−0.337128 + 0.941459i \(0.609456\pi\)
\(938\) 0 0
\(939\) −4.21008 −0.137391
\(940\) 0 0
\(941\) 52.5958 1.71457 0.857287 0.514838i \(-0.172148\pi\)
0.857287 + 0.514838i \(0.172148\pi\)
\(942\) 0 0
\(943\) 8.83710 0.287776
\(944\) 0 0
\(945\) 4.68035 0.152252
\(946\) 0 0
\(947\) 44.0288 1.43074 0.715371 0.698745i \(-0.246258\pi\)
0.715371 + 0.698745i \(0.246258\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −13.3197 −0.431919
\(952\) 0 0
\(953\) 6.39350 0.207106 0.103553 0.994624i \(-0.466979\pi\)
0.103553 + 0.994624i \(0.466979\pi\)
\(954\) 0 0
\(955\) 16.7337 0.541489
\(956\) 0 0
\(957\) 22.8371 0.738219
\(958\) 0 0
\(959\) 7.20394 0.232627
\(960\) 0 0
\(961\) 44.3484 1.43059
\(962\) 0 0
\(963\) −5.94214 −0.191483
\(964\) 0 0
\(965\) 18.1568 0.584487
\(966\) 0 0
\(967\) 39.0882 1.25699 0.628496 0.777813i \(-0.283671\pi\)
0.628496 + 0.777813i \(0.283671\pi\)
\(968\) 0 0
\(969\) −2.15676 −0.0692850
\(970\) 0 0
\(971\) −54.1933 −1.73914 −0.869572 0.493806i \(-0.835605\pi\)
−0.869572 + 0.493806i \(0.835605\pi\)
\(972\) 0 0
\(973\) 17.3607 0.556558
\(974\) 0 0
\(975\) 7.67420 0.245771
\(976\) 0 0
\(977\) 47.7009 1.52609 0.763043 0.646348i \(-0.223704\pi\)
0.763043 + 0.646348i \(0.223704\pi\)
\(978\) 0 0
\(979\) 29.0472 0.928352
\(980\) 0 0
\(981\) −14.7792 −0.471865
\(982\) 0 0
\(983\) −49.5585 −1.58067 −0.790335 0.612675i \(-0.790094\pi\)
−0.790335 + 0.612675i \(0.790094\pi\)
\(984\) 0 0
\(985\) 9.92532 0.316247
\(986\) 0 0
\(987\) −55.0349 −1.75178
\(988\) 0 0
\(989\) 7.75872 0.246713
\(990\) 0 0
\(991\) 20.2602 0.643586 0.321793 0.946810i \(-0.395714\pi\)
0.321793 + 0.946810i \(0.395714\pi\)
\(992\) 0 0
\(993\) 2.15676 0.0684426
\(994\) 0 0
\(995\) 28.7870 0.912609
\(996\) 0 0
\(997\) −1.57077 −0.0497468 −0.0248734 0.999691i \(-0.507918\pi\)
−0.0248734 + 0.999691i \(0.507918\pi\)
\(998\) 0 0
\(999\) −7.26180 −0.229753
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 4416.2.a.bs.1.2 3
4.3 odd 2 4416.2.a.bp.1.2 3
8.3 odd 2 552.2.a.g.1.2 3
8.5 even 2 1104.2.a.o.1.2 3
24.5 odd 2 3312.2.a.bf.1.2 3
24.11 even 2 1656.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.a.g.1.2 3 8.3 odd 2
1104.2.a.o.1.2 3 8.5 even 2
1656.2.a.n.1.2 3 24.11 even 2
3312.2.a.bf.1.2 3 24.5 odd 2
4416.2.a.bp.1.2 3 4.3 odd 2
4416.2.a.bs.1.2 3 1.1 even 1 trivial