Properties

Label 1024.2.a.i.1.2
Level $1024$
Weight $2$
Character 1024.1
Self dual yes
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.765367\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.08239 q^{3} +0.585786 q^{5} -3.69552 q^{7} -1.82843 q^{9} +O(q^{10})\) \(q-1.08239 q^{3} +0.585786 q^{5} -3.69552 q^{7} -1.82843 q^{9} -4.14386 q^{11} +3.41421 q^{13} -0.634051 q^{15} +2.82843 q^{17} +6.30864 q^{19} +4.00000 q^{21} +6.75699 q^{23} -4.65685 q^{25} +5.22625 q^{27} +7.41421 q^{29} -3.06147 q^{31} +4.48528 q^{33} -2.16478 q^{35} +9.07107 q^{37} -3.69552 q^{39} -4.00000 q^{41} +1.08239 q^{43} -1.07107 q^{45} -3.06147 q^{47} +6.65685 q^{49} -3.06147 q^{51} +4.58579 q^{53} -2.42742 q^{55} -6.82843 q^{57} -1.08239 q^{59} -1.07107 q^{61} +6.75699 q^{63} +2.00000 q^{65} -1.97908 q^{67} -7.31371 q^{69} +8.02509 q^{71} -6.48528 q^{73} +5.04054 q^{75} +15.3137 q^{77} +14.7821 q^{79} -0.171573 q^{81} -13.6997 q^{83} +1.65685 q^{85} -8.02509 q^{87} +4.82843 q^{89} -12.6173 q^{91} +3.31371 q^{93} +3.69552 q^{95} +5.17157 q^{97} +7.57675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 4 q^{9} + O(q^{10}) \) \( 4 q + 8 q^{5} + 4 q^{9} + 8 q^{13} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 16 q^{33} + 8 q^{37} - 16 q^{41} + 24 q^{45} + 4 q^{49} + 24 q^{53} - 16 q^{57} + 24 q^{61} + 8 q^{65} + 16 q^{69} + 8 q^{73} + 16 q^{77} - 12 q^{81} - 16 q^{85} + 8 q^{89} - 32 q^{93} + 32 q^{97} + O(q^{100}) \)

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
\(3\) −1.08239 −0.624919 −0.312460 0.949931i \(-0.601153\pi\)
−0.312460 + 0.949931i \(0.601153\pi\)
\(4\) 0 0
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 0 0
\(7\) −3.69552 −1.39677 −0.698387 0.715720i \(-0.746099\pi\)
−0.698387 + 0.715720i \(0.746099\pi\)
\(8\) 0 0
\(9\) −1.82843 −0.609476
\(10\) 0 0
\(11\) −4.14386 −1.24942 −0.624710 0.780857i \(-0.714783\pi\)
−0.624710 + 0.780857i \(0.714783\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) −0.634051 −0.163711
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 6.30864 1.44730 0.723651 0.690166i \(-0.242462\pi\)
0.723651 + 0.690166i \(0.242462\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 6.75699 1.40893 0.704464 0.709739i \(-0.251187\pi\)
0.704464 + 0.709739i \(0.251187\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 5.22625 1.00579
\(28\) 0 0
\(29\) 7.41421 1.37678 0.688392 0.725338i \(-0.258317\pi\)
0.688392 + 0.725338i \(0.258317\pi\)
\(30\) 0 0
\(31\) −3.06147 −0.549856 −0.274928 0.961465i \(-0.588654\pi\)
−0.274928 + 0.961465i \(0.588654\pi\)
\(32\) 0 0
\(33\) 4.48528 0.780787
\(34\) 0 0
\(35\) −2.16478 −0.365915
\(36\) 0 0
\(37\) 9.07107 1.49127 0.745637 0.666352i \(-0.232145\pi\)
0.745637 + 0.666352i \(0.232145\pi\)
\(38\) 0 0
\(39\) −3.69552 −0.591756
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 1.08239 0.165063 0.0825316 0.996588i \(-0.473699\pi\)
0.0825316 + 0.996588i \(0.473699\pi\)
\(44\) 0 0
\(45\) −1.07107 −0.159665
\(46\) 0 0
\(47\) −3.06147 −0.446561 −0.223280 0.974754i \(-0.571677\pi\)
−0.223280 + 0.974754i \(0.571677\pi\)
\(48\) 0 0
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) −3.06147 −0.428691
\(52\) 0 0
\(53\) 4.58579 0.629906 0.314953 0.949107i \(-0.398011\pi\)
0.314953 + 0.949107i \(0.398011\pi\)
\(54\) 0 0
\(55\) −2.42742 −0.327313
\(56\) 0 0
\(57\) −6.82843 −0.904447
\(58\) 0 0
\(59\) −1.08239 −0.140915 −0.0704577 0.997515i \(-0.522446\pi\)
−0.0704577 + 0.997515i \(0.522446\pi\)
\(60\) 0 0
\(61\) −1.07107 −0.137136 −0.0685681 0.997646i \(-0.521843\pi\)
−0.0685681 + 0.997646i \(0.521843\pi\)
\(62\) 0 0
\(63\) 6.75699 0.851300
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −1.97908 −0.241783 −0.120891 0.992666i \(-0.538575\pi\)
−0.120891 + 0.992666i \(0.538575\pi\)
\(68\) 0 0
\(69\) −7.31371 −0.880467
\(70\) 0 0
\(71\) 8.02509 0.952403 0.476201 0.879336i \(-0.342013\pi\)
0.476201 + 0.879336i \(0.342013\pi\)
\(72\) 0 0
\(73\) −6.48528 −0.759045 −0.379522 0.925183i \(-0.623912\pi\)
−0.379522 + 0.925183i \(0.623912\pi\)
\(74\) 0 0
\(75\) 5.04054 0.582032
\(76\) 0 0
\(77\) 15.3137 1.74516
\(78\) 0 0
\(79\) 14.7821 1.66311 0.831557 0.555440i \(-0.187450\pi\)
0.831557 + 0.555440i \(0.187450\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) −13.6997 −1.50374 −0.751868 0.659314i \(-0.770847\pi\)
−0.751868 + 0.659314i \(0.770847\pi\)
\(84\) 0 0
\(85\) 1.65685 0.179711
\(86\) 0 0
\(87\) −8.02509 −0.860380
\(88\) 0 0
\(89\) 4.82843 0.511812 0.255906 0.966702i \(-0.417626\pi\)
0.255906 + 0.966702i \(0.417626\pi\)
\(90\) 0 0
\(91\) −12.6173 −1.32265
\(92\) 0 0
\(93\) 3.31371 0.343616
\(94\) 0 0
\(95\) 3.69552 0.379152
\(96\) 0 0
\(97\) 5.17157 0.525094 0.262547 0.964919i \(-0.415438\pi\)
0.262547 + 0.964919i \(0.415438\pi\)
\(98\) 0 0
\(99\) 7.57675 0.761492
\(100\) 0 0
\(101\) 17.0711 1.69863 0.849317 0.527883i \(-0.177014\pi\)
0.849317 + 0.527883i \(0.177014\pi\)
\(102\) 0 0
\(103\) 14.1480 1.39405 0.697023 0.717049i \(-0.254508\pi\)
0.697023 + 0.717049i \(0.254508\pi\)
\(104\) 0 0
\(105\) 2.34315 0.228668
\(106\) 0 0
\(107\) 1.97908 0.191324 0.0956622 0.995414i \(-0.469503\pi\)
0.0956622 + 0.995414i \(0.469503\pi\)
\(108\) 0 0
\(109\) −13.0711 −1.25198 −0.625991 0.779831i \(-0.715305\pi\)
−0.625991 + 0.779831i \(0.715305\pi\)
\(110\) 0 0
\(111\) −9.81845 −0.931926
\(112\) 0 0
\(113\) −7.65685 −0.720296 −0.360148 0.932895i \(-0.617274\pi\)
−0.360148 + 0.932895i \(0.617274\pi\)
\(114\) 0 0
\(115\) 3.95815 0.369099
\(116\) 0 0
\(117\) −6.24264 −0.577132
\(118\) 0 0
\(119\) −10.4525 −0.958179
\(120\) 0 0
\(121\) 6.17157 0.561052
\(122\) 0 0
\(123\) 4.32957 0.390384
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −11.7206 −1.04004 −0.520018 0.854155i \(-0.674075\pi\)
−0.520018 + 0.854155i \(0.674075\pi\)
\(128\) 0 0
\(129\) −1.17157 −0.103151
\(130\) 0 0
\(131\) 14.5964 1.27529 0.637645 0.770330i \(-0.279909\pi\)
0.637645 + 0.770330i \(0.279909\pi\)
\(132\) 0 0
\(133\) −23.3137 −2.02155
\(134\) 0 0
\(135\) 3.06147 0.263489
\(136\) 0 0
\(137\) 12.9706 1.10815 0.554075 0.832467i \(-0.313072\pi\)
0.554075 + 0.832467i \(0.313072\pi\)
\(138\) 0 0
\(139\) 20.1940 1.71284 0.856418 0.516283i \(-0.172685\pi\)
0.856418 + 0.516283i \(0.172685\pi\)
\(140\) 0 0
\(141\) 3.31371 0.279065
\(142\) 0 0
\(143\) −14.1480 −1.18312
\(144\) 0 0
\(145\) 4.34315 0.360679
\(146\) 0 0
\(147\) −7.20533 −0.594285
\(148\) 0 0
\(149\) 1.75736 0.143968 0.0719842 0.997406i \(-0.477067\pi\)
0.0719842 + 0.997406i \(0.477067\pi\)
\(150\) 0 0
\(151\) 0.634051 0.0515983 0.0257992 0.999667i \(-0.491787\pi\)
0.0257992 + 0.999667i \(0.491787\pi\)
\(152\) 0 0
\(153\) −5.17157 −0.418097
\(154\) 0 0
\(155\) −1.79337 −0.144047
\(156\) 0 0
\(157\) 2.24264 0.178982 0.0894911 0.995988i \(-0.471476\pi\)
0.0894911 + 0.995988i \(0.471476\pi\)
\(158\) 0 0
\(159\) −4.96362 −0.393641
\(160\) 0 0
\(161\) −24.9706 −1.96796
\(162\) 0 0
\(163\) −5.41196 −0.423898 −0.211949 0.977281i \(-0.567981\pi\)
−0.211949 + 0.977281i \(0.567981\pi\)
\(164\) 0 0
\(165\) 2.62742 0.204544
\(166\) 0 0
\(167\) −3.69552 −0.285968 −0.142984 0.989725i \(-0.545670\pi\)
−0.142984 + 0.989725i \(0.545670\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) −11.5349 −0.882096
\(172\) 0 0
\(173\) −19.2132 −1.46075 −0.730376 0.683045i \(-0.760655\pi\)
−0.730376 + 0.683045i \(0.760655\pi\)
\(174\) 0 0
\(175\) 17.2095 1.30092
\(176\) 0 0
\(177\) 1.17157 0.0880608
\(178\) 0 0
\(179\) 11.5349 0.862159 0.431079 0.902314i \(-0.358133\pi\)
0.431079 + 0.902314i \(0.358133\pi\)
\(180\) 0 0
\(181\) −14.2426 −1.05865 −0.529324 0.848420i \(-0.677554\pi\)
−0.529324 + 0.848420i \(0.677554\pi\)
\(182\) 0 0
\(183\) 1.15932 0.0856991
\(184\) 0 0
\(185\) 5.31371 0.390672
\(186\) 0 0
\(187\) −11.7206 −0.857096
\(188\) 0 0
\(189\) −19.3137 −1.40487
\(190\) 0 0
\(191\) 11.7206 0.848073 0.424037 0.905645i \(-0.360613\pi\)
0.424037 + 0.905645i \(0.360613\pi\)
\(192\) 0 0
\(193\) 0.485281 0.0349313 0.0174657 0.999847i \(-0.494440\pi\)
0.0174657 + 0.999847i \(0.494440\pi\)
\(194\) 0 0
\(195\) −2.16478 −0.155023
\(196\) 0 0
\(197\) −10.7279 −0.764333 −0.382166 0.924094i \(-0.624822\pi\)
−0.382166 + 0.924094i \(0.624822\pi\)
\(198\) 0 0
\(199\) −12.3547 −0.875798 −0.437899 0.899024i \(-0.644277\pi\)
−0.437899 + 0.899024i \(0.644277\pi\)
\(200\) 0 0
\(201\) 2.14214 0.151095
\(202\) 0 0
\(203\) −27.3994 −1.92306
\(204\) 0 0
\(205\) −2.34315 −0.163652
\(206\) 0 0
\(207\) −12.3547 −0.858708
\(208\) 0 0
\(209\) −26.1421 −1.80829
\(210\) 0 0
\(211\) 13.3283 0.917555 0.458778 0.888551i \(-0.348287\pi\)
0.458778 + 0.888551i \(0.348287\pi\)
\(212\) 0 0
\(213\) −8.68629 −0.595175
\(214\) 0 0
\(215\) 0.634051 0.0432419
\(216\) 0 0
\(217\) 11.3137 0.768025
\(218\) 0 0
\(219\) 7.01962 0.474342
\(220\) 0 0
\(221\) 9.65685 0.649590
\(222\) 0 0
\(223\) −3.06147 −0.205011 −0.102506 0.994732i \(-0.532686\pi\)
−0.102506 + 0.994732i \(0.532686\pi\)
\(224\) 0 0
\(225\) 8.51472 0.567648
\(226\) 0 0
\(227\) −1.08239 −0.0718409 −0.0359204 0.999355i \(-0.511436\pi\)
−0.0359204 + 0.999355i \(0.511436\pi\)
\(228\) 0 0
\(229\) 3.89949 0.257686 0.128843 0.991665i \(-0.458874\pi\)
0.128843 + 0.991665i \(0.458874\pi\)
\(230\) 0 0
\(231\) −16.5754 −1.09058
\(232\) 0 0
\(233\) 20.8284 1.36452 0.682258 0.731112i \(-0.260998\pi\)
0.682258 + 0.731112i \(0.260998\pi\)
\(234\) 0 0
\(235\) −1.79337 −0.116986
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 12.2459 0.792119 0.396060 0.918225i \(-0.370377\pi\)
0.396060 + 0.918225i \(0.370377\pi\)
\(240\) 0 0
\(241\) −13.1716 −0.848456 −0.424228 0.905556i \(-0.639454\pi\)
−0.424228 + 0.905556i \(0.639454\pi\)
\(242\) 0 0
\(243\) −15.4930 −0.993879
\(244\) 0 0
\(245\) 3.89949 0.249130
\(246\) 0 0
\(247\) 21.5391 1.37050
\(248\) 0 0
\(249\) 14.8284 0.939713
\(250\) 0 0
\(251\) 18.0292 1.13800 0.568998 0.822339i \(-0.307331\pi\)
0.568998 + 0.822339i \(0.307331\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) 0 0
\(255\) −1.79337 −0.112305
\(256\) 0 0
\(257\) −14.9706 −0.933838 −0.466919 0.884300i \(-0.654636\pi\)
−0.466919 + 0.884300i \(0.654636\pi\)
\(258\) 0 0
\(259\) −33.5223 −2.08297
\(260\) 0 0
\(261\) −13.5563 −0.839117
\(262\) 0 0
\(263\) −12.8799 −0.794210 −0.397105 0.917773i \(-0.629985\pi\)
−0.397105 + 0.917773i \(0.629985\pi\)
\(264\) 0 0
\(265\) 2.68629 0.165018
\(266\) 0 0
\(267\) −5.22625 −0.319841
\(268\) 0 0
\(269\) 25.5563 1.55820 0.779099 0.626901i \(-0.215677\pi\)
0.779099 + 0.626901i \(0.215677\pi\)
\(270\) 0 0
\(271\) 5.59767 0.340034 0.170017 0.985441i \(-0.445618\pi\)
0.170017 + 0.985441i \(0.445618\pi\)
\(272\) 0 0
\(273\) 13.6569 0.826550
\(274\) 0 0
\(275\) 19.2974 1.16367
\(276\) 0 0
\(277\) −3.41421 −0.205140 −0.102570 0.994726i \(-0.532707\pi\)
−0.102570 + 0.994726i \(0.532707\pi\)
\(278\) 0 0
\(279\) 5.59767 0.335124
\(280\) 0 0
\(281\) 17.7990 1.06180 0.530899 0.847435i \(-0.321854\pi\)
0.530899 + 0.847435i \(0.321854\pi\)
\(282\) 0 0
\(283\) 14.5964 0.867664 0.433832 0.900994i \(-0.357161\pi\)
0.433832 + 0.900994i \(0.357161\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 14.7821 0.872558
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −5.59767 −0.328141
\(292\) 0 0
\(293\) 0.585786 0.0342220 0.0171110 0.999854i \(-0.494553\pi\)
0.0171110 + 0.999854i \(0.494553\pi\)
\(294\) 0 0
\(295\) −0.634051 −0.0369159
\(296\) 0 0
\(297\) −21.6569 −1.25666
\(298\) 0 0
\(299\) 23.0698 1.33416
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −18.4776 −1.06151
\(304\) 0 0
\(305\) −0.627417 −0.0359258
\(306\) 0 0
\(307\) 4.51528 0.257701 0.128850 0.991664i \(-0.458871\pi\)
0.128850 + 0.991664i \(0.458871\pi\)
\(308\) 0 0
\(309\) −15.3137 −0.871166
\(310\) 0 0
\(311\) 24.6005 1.39497 0.697484 0.716600i \(-0.254303\pi\)
0.697484 + 0.716600i \(0.254303\pi\)
\(312\) 0 0
\(313\) −31.3137 −1.76996 −0.884978 0.465633i \(-0.845827\pi\)
−0.884978 + 0.465633i \(0.845827\pi\)
\(314\) 0 0
\(315\) 3.95815 0.223017
\(316\) 0 0
\(317\) 15.4142 0.865748 0.432874 0.901454i \(-0.357499\pi\)
0.432874 + 0.901454i \(0.357499\pi\)
\(318\) 0 0
\(319\) −30.7235 −1.72018
\(320\) 0 0
\(321\) −2.14214 −0.119562
\(322\) 0 0
\(323\) 17.8435 0.992841
\(324\) 0 0
\(325\) −15.8995 −0.881945
\(326\) 0 0
\(327\) 14.1480 0.782387
\(328\) 0 0
\(329\) 11.3137 0.623745
\(330\) 0 0
\(331\) −8.47343 −0.465742 −0.232871 0.972508i \(-0.574812\pi\)
−0.232871 + 0.972508i \(0.574812\pi\)
\(332\) 0 0
\(333\) −16.5858 −0.908895
\(334\) 0 0
\(335\) −1.15932 −0.0633402
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 8.28772 0.450127
\(340\) 0 0
\(341\) 12.6863 0.687001
\(342\) 0 0
\(343\) 1.26810 0.0684710
\(344\) 0 0
\(345\) −4.28427 −0.230657
\(346\) 0 0
\(347\) 15.4930 0.831710 0.415855 0.909431i \(-0.363482\pi\)
0.415855 + 0.909431i \(0.363482\pi\)
\(348\) 0 0
\(349\) 2.72792 0.146022 0.0730112 0.997331i \(-0.476739\pi\)
0.0730112 + 0.997331i \(0.476739\pi\)
\(350\) 0 0
\(351\) 17.8435 0.952418
\(352\) 0 0
\(353\) 22.9706 1.22260 0.611300 0.791399i \(-0.290647\pi\)
0.611300 + 0.791399i \(0.290647\pi\)
\(354\) 0 0
\(355\) 4.70099 0.249502
\(356\) 0 0
\(357\) 11.3137 0.598785
\(358\) 0 0
\(359\) −0.634051 −0.0334639 −0.0167320 0.999860i \(-0.505326\pi\)
−0.0167320 + 0.999860i \(0.505326\pi\)
\(360\) 0 0
\(361\) 20.7990 1.09468
\(362\) 0 0
\(363\) −6.68006 −0.350612
\(364\) 0 0
\(365\) −3.79899 −0.198848
\(366\) 0 0
\(367\) 9.18440 0.479422 0.239711 0.970844i \(-0.422947\pi\)
0.239711 + 0.970844i \(0.422947\pi\)
\(368\) 0 0
\(369\) 7.31371 0.380736
\(370\) 0 0
\(371\) −16.9469 −0.879837
\(372\) 0 0
\(373\) −6.72792 −0.348359 −0.174179 0.984714i \(-0.555727\pi\)
−0.174179 + 0.984714i \(0.555727\pi\)
\(374\) 0 0
\(375\) 6.12293 0.316187
\(376\) 0 0
\(377\) 25.3137 1.30372
\(378\) 0 0
\(379\) −26.3170 −1.35181 −0.675906 0.736988i \(-0.736247\pi\)
−0.675906 + 0.736988i \(0.736247\pi\)
\(380\) 0 0
\(381\) 12.6863 0.649938
\(382\) 0 0
\(383\) 20.9050 1.06820 0.534098 0.845423i \(-0.320651\pi\)
0.534098 + 0.845423i \(0.320651\pi\)
\(384\) 0 0
\(385\) 8.97056 0.457182
\(386\) 0 0
\(387\) −1.97908 −0.100602
\(388\) 0 0
\(389\) 20.3848 1.03355 0.516775 0.856121i \(-0.327133\pi\)
0.516775 + 0.856121i \(0.327133\pi\)
\(390\) 0 0
\(391\) 19.1116 0.966517
\(392\) 0 0
\(393\) −15.7990 −0.796954
\(394\) 0 0
\(395\) 8.65914 0.435688
\(396\) 0 0
\(397\) −5.07107 −0.254510 −0.127255 0.991870i \(-0.540617\pi\)
−0.127255 + 0.991870i \(0.540617\pi\)
\(398\) 0 0
\(399\) 25.2346 1.26331
\(400\) 0 0
\(401\) 7.51472 0.375267 0.187634 0.982239i \(-0.439918\pi\)
0.187634 + 0.982239i \(0.439918\pi\)
\(402\) 0 0
\(403\) −10.4525 −0.520676
\(404\) 0 0
\(405\) −0.100505 −0.00499414
\(406\) 0 0
\(407\) −37.5892 −1.86323
\(408\) 0 0
\(409\) −31.3137 −1.54836 −0.774182 0.632964i \(-0.781838\pi\)
−0.774182 + 0.632964i \(0.781838\pi\)
\(410\) 0 0
\(411\) −14.0392 −0.692504
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −8.02509 −0.393936
\(416\) 0 0
\(417\) −21.8579 −1.07038
\(418\) 0 0
\(419\) −12.4316 −0.607322 −0.303661 0.952780i \(-0.598209\pi\)
−0.303661 + 0.952780i \(0.598209\pi\)
\(420\) 0 0
\(421\) 8.58579 0.418446 0.209223 0.977868i \(-0.432907\pi\)
0.209223 + 0.977868i \(0.432907\pi\)
\(422\) 0 0
\(423\) 5.59767 0.272168
\(424\) 0 0
\(425\) −13.1716 −0.638915
\(426\) 0 0
\(427\) 3.95815 0.191548
\(428\) 0 0
\(429\) 15.3137 0.739353
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 25.4558 1.22333 0.611665 0.791117i \(-0.290500\pi\)
0.611665 + 0.791117i \(0.290500\pi\)
\(434\) 0 0
\(435\) −4.70099 −0.225395
\(436\) 0 0
\(437\) 42.6274 2.03915
\(438\) 0 0
\(439\) −38.1145 −1.81911 −0.909553 0.415588i \(-0.863576\pi\)
−0.909553 + 0.415588i \(0.863576\pi\)
\(440\) 0 0
\(441\) −12.1716 −0.579599
\(442\) 0 0
\(443\) −1.97908 −0.0940287 −0.0470144 0.998894i \(-0.514971\pi\)
−0.0470144 + 0.998894i \(0.514971\pi\)
\(444\) 0 0
\(445\) 2.82843 0.134080
\(446\) 0 0
\(447\) −1.90215 −0.0899687
\(448\) 0 0
\(449\) −6.14214 −0.289865 −0.144933 0.989442i \(-0.546297\pi\)
−0.144933 + 0.989442i \(0.546297\pi\)
\(450\) 0 0
\(451\) 16.5754 0.780507
\(452\) 0 0
\(453\) −0.686292 −0.0322448
\(454\) 0 0
\(455\) −7.39104 −0.346497
\(456\) 0 0
\(457\) −26.6274 −1.24558 −0.622789 0.782390i \(-0.714001\pi\)
−0.622789 + 0.782390i \(0.714001\pi\)
\(458\) 0 0
\(459\) 14.7821 0.689968
\(460\) 0 0
\(461\) 38.2426 1.78114 0.890569 0.454849i \(-0.150307\pi\)
0.890569 + 0.454849i \(0.150307\pi\)
\(462\) 0 0
\(463\) −35.6871 −1.65852 −0.829260 0.558864i \(-0.811238\pi\)
−0.829260 + 0.558864i \(0.811238\pi\)
\(464\) 0 0
\(465\) 1.94113 0.0900175
\(466\) 0 0
\(467\) −38.0376 −1.76017 −0.880084 0.474817i \(-0.842514\pi\)
−0.880084 + 0.474817i \(0.842514\pi\)
\(468\) 0 0
\(469\) 7.31371 0.337716
\(470\) 0 0
\(471\) −2.42742 −0.111849
\(472\) 0 0
\(473\) −4.48528 −0.206233
\(474\) 0 0
\(475\) −29.3784 −1.34798
\(476\) 0 0
\(477\) −8.38478 −0.383913
\(478\) 0 0
\(479\) −35.1618 −1.60658 −0.803292 0.595585i \(-0.796920\pi\)
−0.803292 + 0.595585i \(0.796920\pi\)
\(480\) 0 0
\(481\) 30.9706 1.41214
\(482\) 0 0
\(483\) 27.0279 1.22981
\(484\) 0 0
\(485\) 3.02944 0.137560
\(486\) 0 0
\(487\) −6.23172 −0.282386 −0.141193 0.989982i \(-0.545094\pi\)
−0.141193 + 0.989982i \(0.545094\pi\)
\(488\) 0 0
\(489\) 5.85786 0.264902
\(490\) 0 0
\(491\) −5.04054 −0.227477 −0.113738 0.993511i \(-0.536283\pi\)
−0.113738 + 0.993511i \(0.536283\pi\)
\(492\) 0 0
\(493\) 20.9706 0.944467
\(494\) 0 0
\(495\) 4.43835 0.199489
\(496\) 0 0
\(497\) −29.6569 −1.33029
\(498\) 0 0
\(499\) 17.6578 0.790473 0.395237 0.918579i \(-0.370663\pi\)
0.395237 + 0.918579i \(0.370663\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) 0 0
\(503\) −22.8072 −1.01692 −0.508460 0.861085i \(-0.669785\pi\)
−0.508460 + 0.861085i \(0.669785\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 1.45381 0.0645660
\(508\) 0 0
\(509\) 26.2426 1.16318 0.581592 0.813480i \(-0.302430\pi\)
0.581592 + 0.813480i \(0.302430\pi\)
\(510\) 0 0
\(511\) 23.9665 1.06021
\(512\) 0 0
\(513\) 32.9706 1.45569
\(514\) 0 0
\(515\) 8.28772 0.365201
\(516\) 0 0
\(517\) 12.6863 0.557942
\(518\) 0 0
\(519\) 20.7962 0.912853
\(520\) 0 0
\(521\) 6.34315 0.277898 0.138949 0.990300i \(-0.455628\pi\)
0.138949 + 0.990300i \(0.455628\pi\)
\(522\) 0 0
\(523\) 14.2249 0.622013 0.311007 0.950408i \(-0.399334\pi\)
0.311007 + 0.950408i \(0.399334\pi\)
\(524\) 0 0
\(525\) −18.6274 −0.812967
\(526\) 0 0
\(527\) −8.65914 −0.377198
\(528\) 0 0
\(529\) 22.6569 0.985081
\(530\) 0 0
\(531\) 1.97908 0.0858846
\(532\) 0 0
\(533\) −13.6569 −0.591544
\(534\) 0 0
\(535\) 1.15932 0.0501216
\(536\) 0 0
\(537\) −12.4853 −0.538780
\(538\) 0 0
\(539\) −27.5851 −1.18817
\(540\) 0 0
\(541\) 10.2426 0.440366 0.220183 0.975459i \(-0.429335\pi\)
0.220183 + 0.975459i \(0.429335\pi\)
\(542\) 0 0
\(543\) 15.4161 0.661569
\(544\) 0 0
\(545\) −7.65685 −0.327984
\(546\) 0 0
\(547\) 18.9259 0.809214 0.404607 0.914491i \(-0.367408\pi\)
0.404607 + 0.914491i \(0.367408\pi\)
\(548\) 0 0
\(549\) 1.95837 0.0835812
\(550\) 0 0
\(551\) 46.7736 1.99262
\(552\) 0 0
\(553\) −54.6274 −2.32299
\(554\) 0 0
\(555\) −5.75152 −0.244138
\(556\) 0 0
\(557\) 11.4142 0.483636 0.241818 0.970322i \(-0.422256\pi\)
0.241818 + 0.970322i \(0.422256\pi\)
\(558\) 0 0
\(559\) 3.69552 0.156304
\(560\) 0 0
\(561\) 12.6863 0.535616
\(562\) 0 0
\(563\) 14.9678 0.630817 0.315408 0.948956i \(-0.397859\pi\)
0.315408 + 0.948956i \(0.397859\pi\)
\(564\) 0 0
\(565\) −4.48528 −0.188697
\(566\) 0 0
\(567\) 0.634051 0.0266276
\(568\) 0 0
\(569\) −25.6569 −1.07559 −0.537796 0.843075i \(-0.680743\pi\)
−0.537796 + 0.843075i \(0.680743\pi\)
\(570\) 0 0
\(571\) 27.5851 1.15440 0.577200 0.816603i \(-0.304145\pi\)
0.577200 + 0.816603i \(0.304145\pi\)
\(572\) 0 0
\(573\) −12.6863 −0.529977
\(574\) 0 0
\(575\) −31.4663 −1.31224
\(576\) 0 0
\(577\) 17.3137 0.720779 0.360390 0.932802i \(-0.382644\pi\)
0.360390 + 0.932802i \(0.382644\pi\)
\(578\) 0 0
\(579\) −0.525265 −0.0218293
\(580\) 0 0
\(581\) 50.6274 2.10038
\(582\) 0 0
\(583\) −19.0029 −0.787018
\(584\) 0 0
\(585\) −3.65685 −0.151192
\(586\) 0 0
\(587\) 18.5545 0.765827 0.382913 0.923784i \(-0.374921\pi\)
0.382913 + 0.923784i \(0.374921\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 0 0
\(591\) 11.6118 0.477646
\(592\) 0 0
\(593\) 9.31371 0.382468 0.191234 0.981544i \(-0.438751\pi\)
0.191234 + 0.981544i \(0.438751\pi\)
\(594\) 0 0
\(595\) −6.12293 −0.251016
\(596\) 0 0
\(597\) 13.3726 0.547303
\(598\) 0 0
\(599\) 45.5055 1.85931 0.929653 0.368437i \(-0.120107\pi\)
0.929653 + 0.368437i \(0.120107\pi\)
\(600\) 0 0
\(601\) 20.8284 0.849609 0.424805 0.905285i \(-0.360343\pi\)
0.424805 + 0.905285i \(0.360343\pi\)
\(602\) 0 0
\(603\) 3.61859 0.147361
\(604\) 0 0
\(605\) 3.61522 0.146980
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 29.6569 1.20176
\(610\) 0 0
\(611\) −10.4525 −0.422863
\(612\) 0 0
\(613\) −39.4142 −1.59193 −0.795963 0.605346i \(-0.793035\pi\)
−0.795963 + 0.605346i \(0.793035\pi\)
\(614\) 0 0
\(615\) 2.53620 0.102270
\(616\) 0 0
\(617\) 38.4853 1.54936 0.774680 0.632354i \(-0.217911\pi\)
0.774680 + 0.632354i \(0.217911\pi\)
\(618\) 0 0
\(619\) −4.14386 −0.166556 −0.0832779 0.996526i \(-0.526539\pi\)
−0.0832779 + 0.996526i \(0.526539\pi\)
\(620\) 0 0
\(621\) 35.3137 1.41709
\(622\) 0 0
\(623\) −17.8435 −0.714886
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 28.2960 1.13004
\(628\) 0 0
\(629\) 25.6569 1.02301
\(630\) 0 0
\(631\) −14.1480 −0.563224 −0.281612 0.959528i \(-0.590869\pi\)
−0.281612 + 0.959528i \(0.590869\pi\)
\(632\) 0 0
\(633\) −14.4264 −0.573398
\(634\) 0 0
\(635\) −6.86577 −0.272460
\(636\) 0 0
\(637\) 22.7279 0.900513
\(638\) 0 0
\(639\) −14.6733 −0.580466
\(640\) 0 0
\(641\) −38.1421 −1.50652 −0.753262 0.657721i \(-0.771521\pi\)
−0.753262 + 0.657721i \(0.771521\pi\)
\(642\) 0 0
\(643\) 18.9259 0.746366 0.373183 0.927758i \(-0.378266\pi\)
0.373183 + 0.927758i \(0.378266\pi\)
\(644\) 0 0
\(645\) −0.686292 −0.0270227
\(646\) 0 0
\(647\) −21.5391 −0.846788 −0.423394 0.905946i \(-0.639161\pi\)
−0.423394 + 0.905946i \(0.639161\pi\)
\(648\) 0 0
\(649\) 4.48528 0.176063
\(650\) 0 0
\(651\) −12.2459 −0.479953
\(652\) 0 0
\(653\) −6.44365 −0.252160 −0.126080 0.992020i \(-0.540240\pi\)
−0.126080 + 0.992020i \(0.540240\pi\)
\(654\) 0 0
\(655\) 8.55035 0.334090
\(656\) 0 0
\(657\) 11.8579 0.462619
\(658\) 0 0
\(659\) −1.45381 −0.0566324 −0.0283162 0.999599i \(-0.509015\pi\)
−0.0283162 + 0.999599i \(0.509015\pi\)
\(660\) 0 0
\(661\) 29.0711 1.13073 0.565367 0.824840i \(-0.308735\pi\)
0.565367 + 0.824840i \(0.308735\pi\)
\(662\) 0 0
\(663\) −10.4525 −0.405942
\(664\) 0 0
\(665\) −13.6569 −0.529590
\(666\) 0 0
\(667\) 50.0977 1.93979
\(668\) 0 0
\(669\) 3.31371 0.128115
\(670\) 0 0
\(671\) 4.43835 0.171341
\(672\) 0 0
\(673\) −26.8284 −1.03416 −0.517080 0.855937i \(-0.672981\pi\)
−0.517080 + 0.855937i \(0.672981\pi\)
\(674\) 0 0
\(675\) −24.3379 −0.936766
\(676\) 0 0
\(677\) −13.5563 −0.521013 −0.260506 0.965472i \(-0.583890\pi\)
−0.260506 + 0.965472i \(0.583890\pi\)
\(678\) 0 0
\(679\) −19.1116 −0.733437
\(680\) 0 0
\(681\) 1.17157 0.0448948
\(682\) 0 0
\(683\) −32.8113 −1.25549 −0.627745 0.778419i \(-0.716022\pi\)
−0.627745 + 0.778419i \(0.716022\pi\)
\(684\) 0 0
\(685\) 7.59798 0.290304
\(686\) 0 0
\(687\) −4.22078 −0.161033
\(688\) 0 0
\(689\) 15.6569 0.596479
\(690\) 0 0
\(691\) −40.7276 −1.54935 −0.774676 0.632358i \(-0.782087\pi\)
−0.774676 + 0.632358i \(0.782087\pi\)
\(692\) 0 0
\(693\) −28.0000 −1.06363
\(694\) 0 0
\(695\) 11.8294 0.448714
\(696\) 0 0
\(697\) −11.3137 −0.428537
\(698\) 0 0
\(699\) −22.5445 −0.852712
\(700\) 0 0
\(701\) 24.8701 0.939329 0.469665 0.882845i \(-0.344375\pi\)
0.469665 + 0.882845i \(0.344375\pi\)
\(702\) 0 0
\(703\) 57.2261 2.15832
\(704\) 0 0
\(705\) 1.94113 0.0731070
\(706\) 0 0
\(707\) −63.0864 −2.37261
\(708\) 0 0
\(709\) 23.6985 0.890015 0.445008 0.895527i \(-0.353201\pi\)
0.445008 + 0.895527i \(0.353201\pi\)
\(710\) 0 0
\(711\) −27.0279 −1.01363
\(712\) 0 0
\(713\) −20.6863 −0.774708
\(714\) 0 0
\(715\) −8.28772 −0.309943
\(716\) 0 0
\(717\) −13.2548 −0.495011
\(718\) 0 0
\(719\) 29.5641 1.10256 0.551278 0.834321i \(-0.314140\pi\)
0.551278 + 0.834321i \(0.314140\pi\)
\(720\) 0 0
\(721\) −52.2843 −1.94717
\(722\) 0 0
\(723\) 14.2568 0.530216
\(724\) 0 0
\(725\) −34.5269 −1.28230
\(726\) 0 0
\(727\) 22.0643 0.818320 0.409160 0.912463i \(-0.365822\pi\)
0.409160 + 0.912463i \(0.365822\pi\)
\(728\) 0 0
\(729\) 17.2843 0.640158
\(730\) 0 0
\(731\) 3.06147 0.113232
\(732\) 0 0
\(733\) −18.5269 −0.684307 −0.342154 0.939644i \(-0.611156\pi\)
−0.342154 + 0.939644i \(0.611156\pi\)
\(734\) 0 0
\(735\) −4.22078 −0.155686
\(736\) 0 0
\(737\) 8.20101 0.302088
\(738\) 0 0
\(739\) −49.7582 −1.83038 −0.915192 0.403018i \(-0.867961\pi\)
−0.915192 + 0.403018i \(0.867961\pi\)
\(740\) 0 0
\(741\) −23.3137 −0.856450
\(742\) 0 0
\(743\) −27.1367 −0.995550 −0.497775 0.867306i \(-0.665849\pi\)
−0.497775 + 0.867306i \(0.665849\pi\)
\(744\) 0 0
\(745\) 1.02944 0.0377157
\(746\) 0 0
\(747\) 25.0489 0.916490
\(748\) 0 0
\(749\) −7.31371 −0.267237
\(750\) 0 0
\(751\) −14.7821 −0.539405 −0.269703 0.962944i \(-0.586925\pi\)
−0.269703 + 0.962944i \(0.586925\pi\)
\(752\) 0 0
\(753\) −19.5147 −0.711156
\(754\) 0 0
\(755\) 0.371418 0.0135173
\(756\) 0 0
\(757\) −18.9289 −0.687984 −0.343992 0.938973i \(-0.611779\pi\)
−0.343992 + 0.938973i \(0.611779\pi\)
\(758\) 0 0
\(759\) 30.3070 1.10007
\(760\) 0 0
\(761\) 34.6274 1.25524 0.627621 0.778519i \(-0.284029\pi\)
0.627621 + 0.778519i \(0.284029\pi\)
\(762\) 0 0
\(763\) 48.3044 1.74874
\(764\) 0 0
\(765\) −3.02944 −0.109530
\(766\) 0 0
\(767\) −3.69552 −0.133437
\(768\) 0 0
\(769\) 5.17157 0.186492 0.0932458 0.995643i \(-0.470276\pi\)
0.0932458 + 0.995643i \(0.470276\pi\)
\(770\) 0 0
\(771\) 16.2040 0.583574
\(772\) 0 0
\(773\) 39.6985 1.42786 0.713928 0.700219i \(-0.246915\pi\)
0.713928 + 0.700219i \(0.246915\pi\)
\(774\) 0 0
\(775\) 14.2568 0.512120
\(776\) 0 0
\(777\) 36.2843 1.30169
\(778\) 0 0
\(779\) −25.2346 −0.904123
\(780\) 0 0
\(781\) −33.2548 −1.18995
\(782\) 0 0
\(783\) 38.7485 1.38476
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) 7.94816 0.283321 0.141661 0.989915i \(-0.454756\pi\)
0.141661 + 0.989915i \(0.454756\pi\)
\(788\) 0 0
\(789\) 13.9411 0.496317
\(790\) 0 0
\(791\) 28.2960 1.00609
\(792\) 0 0
\(793\) −3.65685 −0.129859
\(794\) 0 0
\(795\) −2.90762 −0.103123
\(796\) 0 0
\(797\) −39.6985 −1.40619 −0.703096 0.711095i \(-0.748200\pi\)
−0.703096 + 0.711095i \(0.748200\pi\)
\(798\) 0 0
\(799\) −8.65914 −0.306338
\(800\) 0 0
\(801\) −8.82843 −0.311937
\(802\) 0 0
\(803\) 26.8741 0.948366
\(804\) 0 0
\(805\) −14.6274 −0.515549
\(806\) 0 0
\(807\) −27.6620 −0.973748
\(808\) 0 0
\(809\) −24.6863 −0.867924 −0.433962 0.900931i \(-0.642885\pi\)
−0.433962 + 0.900931i \(0.642885\pi\)
\(810\) 0 0
\(811\) −2.35049 −0.0825370 −0.0412685 0.999148i \(-0.513140\pi\)
−0.0412685 + 0.999148i \(0.513140\pi\)
\(812\) 0 0
\(813\) −6.05887 −0.212494
\(814\) 0 0
\(815\) −3.17025 −0.111049
\(816\) 0 0
\(817\) 6.82843 0.238896
\(818\) 0 0
\(819\) 23.0698 0.806124
\(820\) 0 0
\(821\) 38.5269 1.34460 0.672299 0.740279i \(-0.265307\pi\)
0.672299 + 0.740279i \(0.265307\pi\)
\(822\) 0 0
\(823\) 22.0643 0.769114 0.384557 0.923101i \(-0.374354\pi\)
0.384557 + 0.923101i \(0.374354\pi\)
\(824\) 0 0
\(825\) −20.8873 −0.727203
\(826\) 0 0
\(827\) −18.5545 −0.645204 −0.322602 0.946535i \(-0.604557\pi\)
−0.322602 + 0.946535i \(0.604557\pi\)
\(828\) 0 0
\(829\) −12.3848 −0.430141 −0.215071 0.976599i \(-0.568998\pi\)
−0.215071 + 0.976599i \(0.568998\pi\)
\(830\) 0 0
\(831\) 3.69552 0.128196
\(832\) 0 0
\(833\) 18.8284 0.652366
\(834\) 0 0
\(835\) −2.16478 −0.0749155
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) 29.4554 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(840\) 0 0
\(841\) 25.9706 0.895537
\(842\) 0 0
\(843\) −19.2655 −0.663539
\(844\) 0 0
\(845\) −0.786797 −0.0270666
\(846\) 0 0
\(847\) −22.8072 −0.783663
\(848\) 0 0
\(849\) −15.7990 −0.542220
\(850\) 0 0
\(851\) 61.2931 2.10110
\(852\) 0 0
\(853\) −32.1005 −1.09910 −0.549550 0.835461i \(-0.685201\pi\)
−0.549550 + 0.835461i \(0.685201\pi\)
\(854\) 0 0
\(855\) −6.75699 −0.231084
\(856\) 0 0
\(857\) 24.2843 0.829535 0.414767 0.909927i \(-0.363863\pi\)
0.414767 + 0.909927i \(0.363863\pi\)
\(858\) 0 0
\(859\) 11.9063 0.406238 0.203119 0.979154i \(-0.434892\pi\)
0.203119 + 0.979154i \(0.434892\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) −0.525265 −0.0178802 −0.00894011 0.999960i \(-0.502846\pi\)
−0.00894011 + 0.999960i \(0.502846\pi\)
\(864\) 0 0
\(865\) −11.2548 −0.382676
\(866\) 0 0
\(867\) 9.74153 0.330840
\(868\) 0 0
\(869\) −61.2548 −2.07793
\(870\) 0 0
\(871\) −6.75699 −0.228952
\(872\) 0 0
\(873\) −9.45584 −0.320032
\(874\) 0 0
\(875\) 20.9050 0.706718
\(876\) 0 0
\(877\) 49.5563 1.67340 0.836700 0.547662i \(-0.184482\pi\)
0.836700 + 0.547662i \(0.184482\pi\)
\(878\) 0 0
\(879\) −0.634051 −0.0213860
\(880\) 0 0
\(881\) −39.9411 −1.34565 −0.672825 0.739801i \(-0.734919\pi\)
−0.672825 + 0.739801i \(0.734919\pi\)
\(882\) 0 0
\(883\) −29.3784 −0.988663 −0.494332 0.869273i \(-0.664587\pi\)
−0.494332 + 0.869273i \(0.664587\pi\)
\(884\) 0 0
\(885\) 0.686292 0.0230694
\(886\) 0 0
\(887\) −17.2095 −0.577838 −0.288919 0.957354i \(-0.593296\pi\)
−0.288919 + 0.957354i \(0.593296\pi\)
\(888\) 0 0
\(889\) 43.3137 1.45270
\(890\) 0 0
\(891\) 0.710974 0.0238185
\(892\) 0 0
\(893\) −19.3137 −0.646309
\(894\) 0 0
\(895\) 6.75699 0.225861
\(896\) 0 0
\(897\) −24.9706 −0.833743
\(898\) 0 0
\(899\) −22.6984 −0.757033
\(900\) 0 0
\(901\) 12.9706 0.432112
\(902\) 0 0
\(903\) 4.32957 0.144079
\(904\) 0 0
\(905\) −8.34315 −0.277336
\(906\) 0 0
\(907\) −45.0572 −1.49610 −0.748050 0.663643i \(-0.769010\pi\)
−0.748050 + 0.663643i \(0.769010\pi\)
\(908\) 0 0
\(909\) −31.2132 −1.03528
\(910\) 0 0
\(911\) −23.9665 −0.794045 −0.397022 0.917809i \(-0.629956\pi\)
−0.397022 + 0.917809i \(0.629956\pi\)
\(912\) 0 0
\(913\) 56.7696 1.87880
\(914\) 0 0
\(915\) 0.679111 0.0224507
\(916\) 0 0
\(917\) −53.9411 −1.78129
\(918\) 0 0
\(919\) 19.0029 0.626846 0.313423 0.949614i \(-0.398524\pi\)
0.313423 + 0.949614i \(0.398524\pi\)
\(920\) 0 0
\(921\) −4.88730 −0.161042
\(922\) 0 0
\(923\) 27.3994 0.901861
\(924\) 0 0
\(925\) −42.2426 −1.38893
\(926\) 0 0
\(927\) −25.8686 −0.849637
\(928\) 0 0
\(929\) −42.8284 −1.40516 −0.702578 0.711607i \(-0.747968\pi\)
−0.702578 + 0.711607i \(0.747968\pi\)
\(930\) 0 0
\(931\) 41.9957 1.37635
\(932\) 0 0
\(933\) −26.6274 −0.871742
\(934\) 0 0
\(935\) −6.86577 −0.224535
\(936\) 0 0
\(937\) 43.4558 1.41964 0.709820 0.704383i \(-0.248776\pi\)
0.709820 + 0.704383i \(0.248776\pi\)
\(938\) 0 0
\(939\) 33.8937 1.10608
\(940\) 0 0
\(941\) 2.92893 0.0954805 0.0477402 0.998860i \(-0.484798\pi\)
0.0477402 + 0.998860i \(0.484798\pi\)
\(942\) 0 0
\(943\) −27.0279 −0.880151
\(944\) 0 0
\(945\) −11.3137 −0.368035
\(946\) 0 0
\(947\) −21.4621 −0.697426 −0.348713 0.937230i \(-0.613381\pi\)
−0.348713 + 0.937230i \(0.613381\pi\)
\(948\) 0 0
\(949\) −22.1421 −0.718764
\(950\) 0 0
\(951\) −16.6842 −0.541023
\(952\) 0 0
\(953\) 11.0294 0.357279 0.178639 0.983915i \(-0.442830\pi\)
0.178639 + 0.983915i \(0.442830\pi\)
\(954\) 0 0
\(955\) 6.86577 0.222171
\(956\) 0 0
\(957\) 33.2548 1.07498
\(958\) 0 0
\(959\) −47.9329 −1.54784
\(960\) 0 0
\(961\) −21.6274 −0.697659
\(962\) 0 0
\(963\) −3.61859 −0.116608
\(964\) 0 0
\(965\) 0.284271 0.00915102
\(966\) 0 0
\(967\) 11.0866 0.356520 0.178260 0.983983i \(-0.442953\pi\)
0.178260 + 0.983983i \(0.442953\pi\)
\(968\) 0 0
\(969\) −19.3137 −0.620446
\(970\) 0 0
\(971\) 57.6745 1.85086 0.925431 0.378916i \(-0.123703\pi\)
0.925431 + 0.378916i \(0.123703\pi\)
\(972\) 0 0
\(973\) −74.6274 −2.39245
\(974\) 0 0
\(975\) 17.2095 0.551145
\(976\) 0 0
\(977\) −47.1127 −1.50727 −0.753634 0.657294i \(-0.771701\pi\)
−0.753634 + 0.657294i \(0.771701\pi\)
\(978\) 0 0
\(979\) −20.0083 −0.639469
\(980\) 0 0
\(981\) 23.8995 0.763052
\(982\) 0 0
\(983\) −46.7736 −1.49185 −0.745924 0.666031i \(-0.767992\pi\)
−0.745924 + 0.666031i \(0.767992\pi\)
\(984\) 0 0
\(985\) −6.28427 −0.200234
\(986\) 0 0
\(987\) −12.2459 −0.389790
\(988\) 0 0
\(989\) 7.31371 0.232562
\(990\) 0 0
\(991\) 27.0279 0.858571 0.429285 0.903169i \(-0.358765\pi\)
0.429285 + 0.903169i \(0.358765\pi\)
\(992\) 0 0
\(993\) 9.17157 0.291051
\(994\) 0 0
\(995\) −7.23719 −0.229434
\(996\) 0 0
\(997\) 28.3848 0.898955 0.449477 0.893292i \(-0.351610\pi\)
0.449477 + 0.893292i \(0.351610\pi\)
\(998\) 0 0
\(999\) 47.4077 1.49991
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 1024.2.a.i.1.2 4
3.2 odd 2 9216.2.a.w.1.3 4
4.3 odd 2 inner 1024.2.a.i.1.3 4
8.3 odd 2 1024.2.a.h.1.2 4
8.5 even 2 1024.2.a.h.1.3 4
12.11 even 2 9216.2.a.w.1.4 4
16.3 odd 4 1024.2.b.g.513.5 8
16.5 even 4 1024.2.b.g.513.6 8
16.11 odd 4 1024.2.b.g.513.4 8
16.13 even 4 1024.2.b.g.513.3 8
24.5 odd 2 9216.2.a.bp.1.1 4
24.11 even 2 9216.2.a.bp.1.2 4
32.3 odd 8 512.2.e.j.129.3 yes 8
32.5 even 8 512.2.e.i.385.3 yes 8
32.11 odd 8 512.2.e.j.385.3 yes 8
32.13 even 8 512.2.e.i.129.3 yes 8
32.19 odd 8 512.2.e.i.129.2 8
32.21 even 8 512.2.e.j.385.2 yes 8
32.27 odd 8 512.2.e.i.385.2 yes 8
32.29 even 8 512.2.e.j.129.2 yes 8
96.5 odd 8 4608.2.k.bi.3457.1 8
96.11 even 8 4608.2.k.bd.3457.4 8
96.29 odd 8 4608.2.k.bd.1153.4 8
96.35 even 8 4608.2.k.bd.1153.3 8
96.53 odd 8 4608.2.k.bd.3457.3 8
96.59 even 8 4608.2.k.bi.3457.2 8
96.77 odd 8 4608.2.k.bi.1153.2 8
96.83 even 8 4608.2.k.bi.1153.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.i.129.2 8 32.19 odd 8
512.2.e.i.129.3 yes 8 32.13 even 8
512.2.e.i.385.2 yes 8 32.27 odd 8
512.2.e.i.385.3 yes 8 32.5 even 8
512.2.e.j.129.2 yes 8 32.29 even 8
512.2.e.j.129.3 yes 8 32.3 odd 8
512.2.e.j.385.2 yes 8 32.21 even 8
512.2.e.j.385.3 yes 8 32.11 odd 8
1024.2.a.h.1.2 4 8.3 odd 2
1024.2.a.h.1.3 4 8.5 even 2
1024.2.a.i.1.2 4 1.1 even 1 trivial
1024.2.a.i.1.3 4 4.3 odd 2 inner
1024.2.b.g.513.3 8 16.13 even 4
1024.2.b.g.513.4 8 16.11 odd 4
1024.2.b.g.513.5 8 16.3 odd 4
1024.2.b.g.513.6 8 16.5 even 4
4608.2.k.bd.1153.3 8 96.35 even 8
4608.2.k.bd.1153.4 8 96.29 odd 8
4608.2.k.bd.3457.3 8 96.53 odd 8
4608.2.k.bd.3457.4 8 96.11 even 8
4608.2.k.bi.1153.1 8 96.83 even 8
4608.2.k.bi.1153.2 8 96.77 odd 8
4608.2.k.bi.3457.1 8 96.5 odd 8
4608.2.k.bi.3457.2 8 96.59 even 8
9216.2.a.w.1.3 4 3.2 odd 2
9216.2.a.w.1.4 4 12.11 even 2
9216.2.a.bp.1.1 4 24.5 odd 2
9216.2.a.bp.1.2 4 24.11 even 2