Properties

Label 1024.2.b.g.513.5
Level $1024$
Weight $2$
Character 1024.513
Analytic conductor $8.177$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 513.5
Root \(-0.382683 - 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1024.513
Dual form 1024.2.b.g.513.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.08239i q^{3} -0.585786i q^{5} -3.69552 q^{7} +1.82843 q^{9} +O(q^{10})\) \(q+1.08239i q^{3} -0.585786i q^{5} -3.69552 q^{7} +1.82843 q^{9} -4.14386i q^{11} +3.41421i q^{13} +0.634051 q^{15} +2.82843 q^{17} -6.30864i q^{19} -4.00000i q^{21} +6.75699 q^{23} +4.65685 q^{25} +5.22625i q^{27} +7.41421i q^{29} +3.06147 q^{31} +4.48528 q^{33} +2.16478i q^{35} -9.07107i q^{37} -3.69552 q^{39} +4.00000 q^{41} +1.08239i q^{43} -1.07107i q^{45} +3.06147 q^{47} +6.65685 q^{49} +3.06147i q^{51} -4.58579i q^{53} -2.42742 q^{55} +6.82843 q^{57} -1.08239i q^{59} -1.07107i q^{61} -6.75699 q^{63} +2.00000 q^{65} +1.97908i q^{67} +7.31371i q^{69} +8.02509 q^{71} +6.48528 q^{73} +5.04054i q^{75} +15.3137i q^{77} -14.7821 q^{79} -0.171573 q^{81} +13.6997i q^{83} -1.65685i q^{85} -8.02509 q^{87} -4.82843 q^{89} -12.6173i q^{91} +3.31371i q^{93} -3.69552 q^{95} +5.17157 q^{97} -7.57675i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9} + O(q^{10}) \) \( 8 q - 8 q^{9} - 8 q^{25} - 32 q^{33} + 32 q^{41} + 8 q^{49} + 32 q^{57} + 16 q^{65} - 16 q^{73} - 24 q^{81} - 16 q^{89} + 64 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 1.08239i 0.624919i 0.949931 + 0.312460i \(0.101153\pi\)
−0.949931 + 0.312460i \(0.898847\pi\)
\(4\) 0 0
\(5\) − 0.585786i − 0.261972i −0.991384 0.130986i \(-0.958186\pi\)
0.991384 0.130986i \(-0.0418142\pi\)
\(6\) 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
\(11\) − 4.14386i − 1.24942i −0.780857 0.624710i \(-0.785217\pi\)
0.780857 0.624710i \(-0.214783\pi\)
\(12\) 0 0
\(13\) 3.41421i 0.946932i 0.880812 + 0.473466i \(0.156997\pi\)
−0.880812 + 0.473466i \(0.843003\pi\)
\(14\) 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
\(19\) − 6.30864i − 1.44730i −0.690166 0.723651i \(-0.742462\pi\)
0.690166 0.723651i \(-0.257538\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 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
\(27\) 5.22625i 1.00579i
\(28\) 0 0
\(29\) 7.41421i 1.37678i 0.725338 + 0.688392i \(0.241683\pi\)
−0.725338 + 0.688392i \(0.758317\pi\)
\(30\) 0 0
\(31\) 3.06147 0.549856 0.274928 0.961465i \(-0.411346\pi\)
0.274928 + 0.961465i \(0.411346\pi\)
\(32\) 0 0
\(33\) 4.48528 0.780787
\(34\) 0 0
\(35\) 2.16478i 0.365915i
\(36\) 0 0
\(37\) − 9.07107i − 1.49127i −0.666352 0.745637i \(-0.732145\pi\)
0.666352 0.745637i \(-0.267855\pi\)
\(38\) 0 0
\(39\) −3.69552 −0.591756
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 1.08239i 0.165063i 0.996588 + 0.0825316i \(0.0263005\pi\)
−0.996588 + 0.0825316i \(0.973699\pi\)
\(44\) 0 0
\(45\) − 1.07107i − 0.159665i
\(46\) 0 0
\(47\) 3.06147 0.446561 0.223280 0.974754i \(-0.428323\pi\)
0.223280 + 0.974754i \(0.428323\pi\)
\(48\) 0 0
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) 3.06147i 0.428691i
\(52\) 0 0
\(53\) − 4.58579i − 0.629906i −0.949107 0.314953i \(-0.898011\pi\)
0.949107 0.314953i \(-0.101989\pi\)
\(54\) 0 0
\(55\) −2.42742 −0.327313
\(56\) 0 0
\(57\) 6.82843 0.904447
\(58\) 0 0
\(59\) − 1.08239i − 0.140915i −0.997515 0.0704577i \(-0.977554\pi\)
0.997515 0.0704577i \(-0.0224460\pi\)
\(60\) 0 0
\(61\) − 1.07107i − 0.137136i −0.997646 0.0685681i \(-0.978157\pi\)
0.997646 0.0685681i \(-0.0218430\pi\)
\(62\) 0 0
\(63\) −6.75699 −0.851300
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 1.97908i 0.241783i 0.992666 + 0.120891i \(0.0385752\pi\)
−0.992666 + 0.120891i \(0.961425\pi\)
\(68\) 0 0
\(69\) 7.31371i 0.880467i
\(70\) 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 \(-0.376088\pi\)
0.379522 + 0.925183i \(0.376088\pi\)
\(74\) 0 0
\(75\) 5.04054i 0.582032i
\(76\) 0 0
\(77\) 15.3137i 1.74516i
\(78\) 0 0
\(79\) −14.7821 −1.66311 −0.831557 0.555440i \(-0.812550\pi\)
−0.831557 + 0.555440i \(0.812550\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) 13.6997i 1.50374i 0.659314 + 0.751868i \(0.270847\pi\)
−0.659314 + 0.751868i \(0.729153\pi\)
\(84\) 0 0
\(85\) − 1.65685i − 0.179711i
\(86\) 0 0
\(87\) −8.02509 −0.860380
\(88\) 0 0
\(89\) −4.82843 −0.511812 −0.255906 0.966702i \(-0.582374\pi\)
−0.255906 + 0.966702i \(0.582374\pi\)
\(90\) 0 0
\(91\) − 12.6173i − 1.32265i
\(92\) 0 0
\(93\) 3.31371i 0.343616i
\(94\) 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
\(99\) − 7.57675i − 0.761492i
\(100\) 0 0
\(101\) − 17.0711i − 1.69863i −0.527883 0.849317i \(-0.677014\pi\)
0.527883 0.849317i \(-0.322986\pi\)
\(102\) 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
\(107\) 1.97908i 0.191324i 0.995414 + 0.0956622i \(0.0304969\pi\)
−0.995414 + 0.0956622i \(0.969503\pi\)
\(108\) 0 0
\(109\) − 13.0711i − 1.25198i −0.779831 0.625991i \(-0.784695\pi\)
0.779831 0.625991i \(-0.215305\pi\)
\(110\) 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
\(115\) − 3.95815i − 0.369099i
\(116\) 0 0
\(117\) 6.24264i 0.577132i
\(118\) 0 0
\(119\) −10.4525 −0.958179
\(120\) 0 0
\(121\) −6.17157 −0.561052
\(122\) 0 0
\(123\) 4.32957i 0.390384i
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 11.7206 1.04004 0.520018 0.854155i \(-0.325925\pi\)
0.520018 + 0.854155i \(0.325925\pi\)
\(128\) 0 0
\(129\) −1.17157 −0.103151
\(130\) 0 0
\(131\) − 14.5964i − 1.27529i −0.770330 0.637645i \(-0.779909\pi\)
0.770330 0.637645i \(-0.220091\pi\)
\(132\) 0 0
\(133\) 23.3137i 2.02155i
\(134\) 0 0
\(135\) 3.06147 0.263489
\(136\) 0 0
\(137\) −12.9706 −1.10815 −0.554075 0.832467i \(-0.686928\pi\)
−0.554075 + 0.832467i \(0.686928\pi\)
\(138\) 0 0
\(139\) 20.1940i 1.71284i 0.516283 + 0.856418i \(0.327315\pi\)
−0.516283 + 0.856418i \(0.672685\pi\)
\(140\) 0 0
\(141\) 3.31371i 0.279065i
\(142\) 0 0
\(143\) 14.1480 1.18312
\(144\) 0 0
\(145\) 4.34315 0.360679
\(146\) 0 0
\(147\) 7.20533i 0.594285i
\(148\) 0 0
\(149\) − 1.75736i − 0.143968i −0.997406 0.0719842i \(-0.977067\pi\)
0.997406 0.0719842i \(-0.0229331\pi\)
\(150\) 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
\(155\) − 1.79337i − 0.144047i
\(156\) 0 0
\(157\) 2.24264i 0.178982i 0.995988 + 0.0894911i \(0.0285241\pi\)
−0.995988 + 0.0894911i \(0.971476\pi\)
\(158\) 0 0
\(159\) 4.96362 0.393641
\(160\) 0 0
\(161\) −24.9706 −1.96796
\(162\) 0 0
\(163\) 5.41196i 0.423898i 0.977281 + 0.211949i \(0.0679810\pi\)
−0.977281 + 0.211949i \(0.932019\pi\)
\(164\) 0 0
\(165\) − 2.62742i − 0.204544i
\(166\) 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
\(171\) − 11.5349i − 0.882096i
\(172\) 0 0
\(173\) − 19.2132i − 1.46075i −0.683045 0.730376i \(-0.739345\pi\)
0.683045 0.730376i \(-0.260655\pi\)
\(174\) 0 0
\(175\) −17.2095 −1.30092
\(176\) 0 0
\(177\) 1.17157 0.0880608
\(178\) 0 0
\(179\) − 11.5349i − 0.862159i −0.902314 0.431079i \(-0.858133\pi\)
0.902314 0.431079i \(-0.141867\pi\)
\(180\) 0 0
\(181\) 14.2426i 1.05865i 0.848420 + 0.529324i \(0.177554\pi\)
−0.848420 + 0.529324i \(0.822446\pi\)
\(182\) 0 0
\(183\) 1.15932 0.0856991
\(184\) 0 0
\(185\) −5.31371 −0.390672
\(186\) 0 0
\(187\) − 11.7206i − 0.857096i
\(188\) 0 0
\(189\) − 19.3137i − 1.40487i
\(190\) 0 0
\(191\) −11.7206 −0.848073 −0.424037 0.905645i \(-0.639387\pi\)
−0.424037 + 0.905645i \(0.639387\pi\)
\(192\) 0 0
\(193\) 0.485281 0.0349313 0.0174657 0.999847i \(-0.494440\pi\)
0.0174657 + 0.999847i \(0.494440\pi\)
\(194\) 0 0
\(195\) 2.16478i 0.155023i
\(196\) 0 0
\(197\) 10.7279i 0.764333i 0.924094 + 0.382166i \(0.124822\pi\)
−0.924094 + 0.382166i \(0.875178\pi\)
\(198\) 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
\(203\) − 27.3994i − 1.92306i
\(204\) 0 0
\(205\) − 2.34315i − 0.163652i
\(206\) 0 0
\(207\) 12.3547 0.858708
\(208\) 0 0
\(209\) −26.1421 −1.80829
\(210\) 0 0
\(211\) − 13.3283i − 0.917555i −0.888551 0.458778i \(-0.848287\pi\)
0.888551 0.458778i \(-0.151713\pi\)
\(212\) 0 0
\(213\) 8.68629i 0.595175i
\(214\) 0 0
\(215\) 0.634051 0.0432419
\(216\) 0 0
\(217\) −11.3137 −0.768025
\(218\) 0 0
\(219\) 7.01962i 0.474342i
\(220\) 0 0
\(221\) 9.65685i 0.649590i
\(222\) 0 0
\(223\) 3.06147 0.205011 0.102506 0.994732i \(-0.467314\pi\)
0.102506 + 0.994732i \(0.467314\pi\)
\(224\) 0 0
\(225\) 8.51472 0.567648
\(226\) 0 0
\(227\) 1.08239i 0.0718409i 0.999355 + 0.0359204i \(0.0114363\pi\)
−0.999355 + 0.0359204i \(0.988564\pi\)
\(228\) 0 0
\(229\) − 3.89949i − 0.257686i −0.991665 0.128843i \(-0.958874\pi\)
0.991665 0.128843i \(-0.0411263\pi\)
\(230\) 0 0
\(231\) −16.5754 −1.09058
\(232\) 0 0
\(233\) −20.8284 −1.36452 −0.682258 0.731112i \(-0.739002\pi\)
−0.682258 + 0.731112i \(0.739002\pi\)
\(234\) 0 0
\(235\) − 1.79337i − 0.116986i
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) −12.2459 −0.792119 −0.396060 0.918225i \(-0.629623\pi\)
−0.396060 + 0.918225i \(0.629623\pi\)
\(240\) 0 0
\(241\) −13.1716 −0.848456 −0.424228 0.905556i \(-0.639454\pi\)
−0.424228 + 0.905556i \(0.639454\pi\)
\(242\) 0 0
\(243\) 15.4930i 0.993879i
\(244\) 0 0
\(245\) − 3.89949i − 0.249130i
\(246\) 0 0
\(247\) 21.5391 1.37050
\(248\) 0 0
\(249\) −14.8284 −0.939713
\(250\) 0 0
\(251\) 18.0292i 1.13800i 0.822339 + 0.568998i \(0.192669\pi\)
−0.822339 + 0.568998i \(0.807331\pi\)
\(252\) 0 0
\(253\) − 28.0000i − 1.76034i
\(254\) 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
\(259\) 33.5223i 2.08297i
\(260\) 0 0
\(261\) 13.5563i 0.839117i
\(262\) 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
\(267\) − 5.22625i − 0.319841i
\(268\) 0 0
\(269\) 25.5563i 1.55820i 0.626901 + 0.779099i \(0.284323\pi\)
−0.626901 + 0.779099i \(0.715677\pi\)
\(270\) 0 0
\(271\) −5.59767 −0.340034 −0.170017 0.985441i \(-0.554382\pi\)
−0.170017 + 0.985441i \(0.554382\pi\)
\(272\) 0 0
\(273\) 13.6569 0.826550
\(274\) 0 0
\(275\) − 19.2974i − 1.16367i
\(276\) 0 0
\(277\) 3.41421i 0.205140i 0.994726 + 0.102570i \(0.0327066\pi\)
−0.994726 + 0.102570i \(0.967293\pi\)
\(278\) 0 0
\(279\) 5.59767 0.335124
\(280\) 0 0
\(281\) −17.7990 −1.06180 −0.530899 0.847435i \(-0.678146\pi\)
−0.530899 + 0.847435i \(0.678146\pi\)
\(282\) 0 0
\(283\) 14.5964i 0.867664i 0.900994 + 0.433832i \(0.142839\pi\)
−0.900994 + 0.433832i \(0.857161\pi\)
\(284\) 0 0
\(285\) − 4.00000i − 0.236940i
\(286\) 0 0
\(287\) −14.7821 −0.872558
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 5.59767i 0.328141i
\(292\) 0 0
\(293\) − 0.585786i − 0.0342220i −0.999854 0.0171110i \(-0.994553\pi\)
0.999854 0.0171110i \(-0.00544687\pi\)
\(294\) 0 0
\(295\) −0.634051 −0.0369159
\(296\) 0 0
\(297\) 21.6569 1.25666
\(298\) 0 0
\(299\) 23.0698i 1.33416i
\(300\) 0 0
\(301\) − 4.00000i − 0.230556i
\(302\) 0 0
\(303\) 18.4776 1.06151
\(304\) 0 0
\(305\) −0.627417 −0.0359258
\(306\) 0 0
\(307\) − 4.51528i − 0.257701i −0.991664 0.128850i \(-0.958871\pi\)
0.991664 0.128850i \(-0.0411287\pi\)
\(308\) 0 0
\(309\) 15.3137i 0.871166i
\(310\) 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 \(-0.154173\pi\)
0.884978 + 0.465633i \(0.154173\pi\)
\(314\) 0 0
\(315\) 3.95815i 0.223017i
\(316\) 0 0
\(317\) 15.4142i 0.865748i 0.901454 + 0.432874i \(0.142501\pi\)
−0.901454 + 0.432874i \(0.857499\pi\)
\(318\) 0 0
\(319\) 30.7235 1.72018
\(320\) 0 0
\(321\) −2.14214 −0.119562
\(322\) 0 0
\(323\) − 17.8435i − 0.992841i
\(324\) 0 0
\(325\) 15.8995i 0.881945i
\(326\) 0 0
\(327\) 14.1480 0.782387
\(328\) 0 0
\(329\) −11.3137 −0.623745
\(330\) 0 0
\(331\) − 8.47343i − 0.465742i −0.972508 0.232871i \(-0.925188\pi\)
0.972508 0.232871i \(-0.0748120\pi\)
\(332\) 0 0
\(333\) − 16.5858i − 0.908895i
\(334\) 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
\(339\) − 8.28772i − 0.450127i
\(340\) 0 0
\(341\) − 12.6863i − 0.687001i
\(342\) 0 0
\(343\) 1.26810 0.0684710
\(344\) 0 0
\(345\) 4.28427 0.230657
\(346\) 0 0
\(347\) 15.4930i 0.831710i 0.909431 + 0.415855i \(0.136518\pi\)
−0.909431 + 0.415855i \(0.863482\pi\)
\(348\) 0 0
\(349\) 2.72792i 0.146022i 0.997331 + 0.0730112i \(0.0232609\pi\)
−0.997331 + 0.0730112i \(0.976739\pi\)
\(350\) 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
\(355\) − 4.70099i − 0.249502i
\(356\) 0 0
\(357\) − 11.3137i − 0.598785i
\(358\) 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
\(363\) − 6.68006i − 0.350612i
\(364\) 0 0
\(365\) − 3.79899i − 0.198848i
\(366\) 0 0
\(367\) −9.18440 −0.479422 −0.239711 0.970844i \(-0.577053\pi\)
−0.239711 + 0.970844i \(0.577053\pi\)
\(368\) 0 0
\(369\) 7.31371 0.380736
\(370\) 0 0
\(371\) 16.9469i 0.879837i
\(372\) 0 0
\(373\) 6.72792i 0.348359i 0.984714 + 0.174179i \(0.0557272\pi\)
−0.984714 + 0.174179i \(0.944273\pi\)
\(374\) 0 0
\(375\) 6.12293 0.316187
\(376\) 0 0
\(377\) −25.3137 −1.30372
\(378\) 0 0
\(379\) − 26.3170i − 1.35181i −0.736988 0.675906i \(-0.763753\pi\)
0.736988 0.675906i \(-0.236247\pi\)
\(380\) 0 0
\(381\) 12.6863i 0.649938i
\(382\) 0 0
\(383\) −20.9050 −1.06820 −0.534098 0.845423i \(-0.679349\pi\)
−0.534098 + 0.845423i \(0.679349\pi\)
\(384\) 0 0
\(385\) 8.97056 0.457182
\(386\) 0 0
\(387\) 1.97908i 0.100602i
\(388\) 0 0
\(389\) − 20.3848i − 1.03355i −0.856121 0.516775i \(-0.827133\pi\)
0.856121 0.516775i \(-0.172867\pi\)
\(390\) 0 0
\(391\) 19.1116 0.966517
\(392\) 0 0
\(393\) 15.7990 0.796954
\(394\) 0 0
\(395\) 8.65914i 0.435688i
\(396\) 0 0
\(397\) − 5.07107i − 0.254510i −0.991870 0.127255i \(-0.959383\pi\)
0.991870 0.127255i \(-0.0406166\pi\)
\(398\) 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
\(403\) 10.4525i 0.520676i
\(404\) 0 0
\(405\) 0.100505i 0.00499414i
\(406\) 0 0
\(407\) −37.5892 −1.86323
\(408\) 0 0
\(409\) 31.3137 1.54836 0.774182 0.632964i \(-0.218162\pi\)
0.774182 + 0.632964i \(0.218162\pi\)
\(410\) 0 0
\(411\) − 14.0392i − 0.692504i
\(412\) 0 0
\(413\) 4.00000i 0.196827i
\(414\) 0 0
\(415\) 8.02509 0.393936
\(416\) 0 0
\(417\) −21.8579 −1.07038
\(418\) 0 0
\(419\) 12.4316i 0.607322i 0.952780 + 0.303661i \(0.0982091\pi\)
−0.952780 + 0.303661i \(0.901791\pi\)
\(420\) 0 0
\(421\) − 8.58579i − 0.418446i −0.977868 0.209223i \(-0.932907\pi\)
0.977868 0.209223i \(-0.0670934\pi\)
\(422\) 0 0
\(423\) 5.59767 0.272168
\(424\) 0 0
\(425\) 13.1716 0.638915
\(426\) 0 0
\(427\) 3.95815i 0.191548i
\(428\) 0 0
\(429\) 15.3137i 0.739353i
\(430\) 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
\(435\) 4.70099i 0.225395i
\(436\) 0 0
\(437\) − 42.6274i − 2.03915i
\(438\) 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
\(443\) − 1.97908i − 0.0940287i −0.998894 0.0470144i \(-0.985029\pi\)
0.998894 0.0470144i \(-0.0149707\pi\)
\(444\) 0 0
\(445\) 2.82843i 0.134080i
\(446\) 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
\(451\) − 16.5754i − 0.780507i
\(452\) 0 0
\(453\) 0.686292i 0.0322448i
\(454\) 0 0
\(455\) −7.39104 −0.346497
\(456\) 0 0
\(457\) 26.6274 1.24558 0.622789 0.782390i \(-0.285999\pi\)
0.622789 + 0.782390i \(0.285999\pi\)
\(458\) 0 0
\(459\) 14.7821i 0.689968i
\(460\) 0 0
\(461\) 38.2426i 1.78114i 0.454849 + 0.890569i \(0.349693\pi\)
−0.454849 + 0.890569i \(0.650307\pi\)
\(462\) 0 0
\(463\) 35.6871 1.65852 0.829260 0.558864i \(-0.188762\pi\)
0.829260 + 0.558864i \(0.188762\pi\)
\(464\) 0 0
\(465\) 1.94113 0.0900175
\(466\) 0 0
\(467\) 38.0376i 1.76017i 0.474817 + 0.880084i \(0.342514\pi\)
−0.474817 + 0.880084i \(0.657486\pi\)
\(468\) 0 0
\(469\) − 7.31371i − 0.337716i
\(470\) 0 0
\(471\) −2.42742 −0.111849
\(472\) 0 0
\(473\) 4.48528 0.206233
\(474\) 0 0
\(475\) − 29.3784i − 1.34798i
\(476\) 0 0
\(477\) − 8.38478i − 0.383913i
\(478\) 0 0
\(479\) 35.1618 1.60658 0.803292 0.595585i \(-0.203080\pi\)
0.803292 + 0.595585i \(0.203080\pi\)
\(480\) 0 0
\(481\) 30.9706 1.41214
\(482\) 0 0
\(483\) − 27.0279i − 1.22981i
\(484\) 0 0
\(485\) − 3.02944i − 0.137560i
\(486\) 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
\(491\) − 5.04054i − 0.227477i −0.993511 0.113738i \(-0.963717\pi\)
0.993511 0.113738i \(-0.0362825\pi\)
\(492\) 0 0
\(493\) 20.9706i 0.944467i
\(494\) 0 0
\(495\) −4.43835 −0.199489
\(496\) 0 0
\(497\) −29.6569 −1.33029
\(498\) 0 0
\(499\) − 17.6578i − 0.790473i −0.918579 0.395237i \(-0.870663\pi\)
0.918579 0.395237i \(-0.129337\pi\)
\(500\) 0 0
\(501\) − 4.00000i − 0.178707i
\(502\) 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
\(507\) 1.45381i 0.0645660i
\(508\) 0 0
\(509\) 26.2426i 1.16318i 0.813480 + 0.581592i \(0.197570\pi\)
−0.813480 + 0.581592i \(0.802430\pi\)
\(510\) 0 0
\(511\) −23.9665 −1.06021
\(512\) 0 0
\(513\) 32.9706 1.45569
\(514\) 0 0
\(515\) − 8.28772i − 0.365201i
\(516\) 0 0
\(517\) − 12.6863i − 0.557942i
\(518\) 0 0
\(519\) 20.7962 0.912853
\(520\) 0 0
\(521\) −6.34315 −0.277898 −0.138949 0.990300i \(-0.544372\pi\)
−0.138949 + 0.990300i \(0.544372\pi\)
\(522\) 0 0
\(523\) 14.2249i 0.622013i 0.950408 + 0.311007i \(0.100666\pi\)
−0.950408 + 0.311007i \(0.899334\pi\)
\(524\) 0 0
\(525\) − 18.6274i − 0.812967i
\(526\) 0 0
\(527\) 8.65914 0.377198
\(528\) 0 0
\(529\) 22.6569 0.985081
\(530\) 0 0
\(531\) − 1.97908i − 0.0858846i
\(532\) 0 0
\(533\) 13.6569i 0.591544i
\(534\) 0 0
\(535\) 1.15932 0.0501216
\(536\) 0 0
\(537\) 12.4853 0.538780
\(538\) 0 0
\(539\) − 27.5851i − 1.18817i
\(540\) 0 0
\(541\) 10.2426i 0.440366i 0.975459 + 0.220183i \(0.0706654\pi\)
−0.975459 + 0.220183i \(0.929335\pi\)
\(542\) 0 0
\(543\) −15.4161 −0.661569
\(544\) 0 0
\(545\) −7.65685 −0.327984
\(546\) 0 0
\(547\) − 18.9259i − 0.809214i −0.914491 0.404607i \(-0.867408\pi\)
0.914491 0.404607i \(-0.132592\pi\)
\(548\) 0 0
\(549\) − 1.95837i − 0.0835812i
\(550\) 0 0
\(551\) 46.7736 1.99262
\(552\) 0 0
\(553\) 54.6274 2.32299
\(554\) 0 0
\(555\) − 5.75152i − 0.244138i
\(556\) 0 0
\(557\) 11.4142i 0.483636i 0.970322 + 0.241818i \(0.0777437\pi\)
−0.970322 + 0.241818i \(0.922256\pi\)
\(558\) 0 0
\(559\) −3.69552 −0.156304
\(560\) 0 0
\(561\) 12.6863 0.535616
\(562\) 0 0
\(563\) − 14.9678i − 0.630817i −0.948956 0.315408i \(-0.897859\pi\)
0.948956 0.315408i \(-0.102141\pi\)
\(564\) 0 0
\(565\) 4.48528i 0.188697i
\(566\) 0 0
\(567\) 0.634051 0.0266276
\(568\) 0 0
\(569\) 25.6569 1.07559 0.537796 0.843075i \(-0.319257\pi\)
0.537796 + 0.843075i \(0.319257\pi\)
\(570\) 0 0
\(571\) 27.5851i 1.15440i 0.816603 + 0.577200i \(0.195855\pi\)
−0.816603 + 0.577200i \(0.804145\pi\)
\(572\) 0 0
\(573\) − 12.6863i − 0.529977i
\(574\) 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
\(579\) 0.525265i 0.0218293i
\(580\) 0 0
\(581\) − 50.6274i − 2.10038i
\(582\) 0 0
\(583\) −19.0029 −0.787018
\(584\) 0 0
\(585\) 3.65685 0.151192
\(586\) 0 0
\(587\) 18.5545i 0.765827i 0.923784 + 0.382913i \(0.125079\pi\)
−0.923784 + 0.382913i \(0.874921\pi\)
\(588\) 0 0
\(589\) − 19.3137i − 0.795807i
\(590\) 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
\(595\) 6.12293i 0.251016i
\(596\) 0 0
\(597\) − 13.3726i − 0.547303i
\(598\) 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 \(-0.639657\pi\)
−0.424805 + 0.905285i \(0.639657\pi\)
\(602\) 0 0
\(603\) 3.61859i 0.147361i
\(604\) 0 0
\(605\) 3.61522i 0.146980i
\(606\) 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
\(611\) 10.4525i 0.422863i
\(612\) 0 0
\(613\) 39.4142i 1.59193i 0.605346 + 0.795963i \(0.293035\pi\)
−0.605346 + 0.795963i \(0.706965\pi\)
\(614\) 0 0
\(615\) 2.53620 0.102270
\(616\) 0 0
\(617\) −38.4853 −1.54936 −0.774680 0.632354i \(-0.782089\pi\)
−0.774680 + 0.632354i \(0.782089\pi\)
\(618\) 0 0
\(619\) − 4.14386i − 0.166556i −0.996526 0.0832779i \(-0.973461\pi\)
0.996526 0.0832779i \(-0.0265389\pi\)
\(620\) 0 0
\(621\) 35.3137i 1.41709i
\(622\) 0 0
\(623\) 17.8435 0.714886
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) − 28.2960i − 1.13004i
\(628\) 0 0
\(629\) − 25.6569i − 1.02301i
\(630\) 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
\(635\) − 6.86577i − 0.272460i
\(636\) 0 0
\(637\) 22.7279i 0.900513i
\(638\) 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
\(643\) − 18.9259i − 0.746366i −0.927758 0.373183i \(-0.878266\pi\)
0.927758 0.373183i \(-0.121734\pi\)
\(644\) 0 0
\(645\) 0.686292i 0.0270227i
\(646\) 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
\(651\) − 12.2459i − 0.479953i
\(652\) 0 0
\(653\) − 6.44365i − 0.252160i −0.992020 0.126080i \(-0.959760\pi\)
0.992020 0.126080i \(-0.0402395\pi\)
\(654\) 0 0
\(655\) −8.55035 −0.334090
\(656\) 0 0
\(657\) 11.8579 0.462619
\(658\) 0 0
\(659\) 1.45381i 0.0566324i 0.999599 + 0.0283162i \(0.00901453\pi\)
−0.999599 + 0.0283162i \(0.990985\pi\)
\(660\) 0 0
\(661\) − 29.0711i − 1.13073i −0.824840 0.565367i \(-0.808735\pi\)
0.824840 0.565367i \(-0.191265\pi\)
\(662\) 0 0
\(663\) −10.4525 −0.405942
\(664\) 0 0
\(665\) 13.6569 0.529590
\(666\) 0 0
\(667\) 50.0977i 1.93979i
\(668\) 0 0
\(669\) 3.31371i 0.128115i
\(670\) 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
\(675\) 24.3379i 0.936766i
\(676\) 0 0
\(677\) 13.5563i 0.521013i 0.965472 + 0.260506i \(0.0838895\pi\)
−0.965472 + 0.260506i \(0.916110\pi\)
\(678\) 0 0
\(679\) −19.1116 −0.733437
\(680\) 0 0
\(681\) −1.17157 −0.0448948
\(682\) 0 0
\(683\) − 32.8113i − 1.25549i −0.778419 0.627745i \(-0.783978\pi\)
0.778419 0.627745i \(-0.216022\pi\)
\(684\) 0 0
\(685\) 7.59798i 0.290304i
\(686\) 0 0
\(687\) 4.22078 0.161033
\(688\) 0 0
\(689\) 15.6569 0.596479
\(690\) 0 0
\(691\) 40.7276i 1.54935i 0.632358 + 0.774676i \(0.282087\pi\)
−0.632358 + 0.774676i \(0.717913\pi\)
\(692\) 0 0
\(693\) 28.0000i 1.06363i
\(694\) 0 0
\(695\) 11.8294 0.448714
\(696\) 0 0
\(697\) 11.3137 0.428537
\(698\) 0 0
\(699\) − 22.5445i − 0.852712i
\(700\) 0 0
\(701\) 24.8701i 0.939329i 0.882845 + 0.469665i \(0.155625\pi\)
−0.882845 + 0.469665i \(0.844375\pi\)
\(702\) 0 0
\(703\) −57.2261 −2.15832
\(704\) 0 0
\(705\) 1.94113 0.0731070
\(706\) 0 0
\(707\) 63.0864i 2.37261i
\(708\) 0 0
\(709\) − 23.6985i − 0.890015i −0.895527 0.445008i \(-0.853201\pi\)
0.895527 0.445008i \(-0.146799\pi\)
\(710\) 0 0
\(711\) −27.0279 −1.01363
\(712\) 0 0
\(713\) 20.6863 0.774708
\(714\) 0 0
\(715\) − 8.28772i − 0.309943i
\(716\) 0 0
\(717\) − 13.2548i − 0.495011i
\(718\) 0 0
\(719\) −29.5641 −1.10256 −0.551278 0.834321i \(-0.685860\pi\)
−0.551278 + 0.834321i \(0.685860\pi\)
\(720\) 0 0
\(721\) −52.2843 −1.94717
\(722\) 0 0
\(723\) − 14.2568i − 0.530216i
\(724\) 0 0
\(725\) 34.5269i 1.28230i
\(726\) 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
\(731\) 3.06147i 0.113232i
\(732\) 0 0
\(733\) − 18.5269i − 0.684307i −0.939644 0.342154i \(-0.888844\pi\)
0.939644 0.342154i \(-0.111156\pi\)
\(734\) 0 0
\(735\) 4.22078 0.155686
\(736\) 0 0
\(737\) 8.20101 0.302088
\(738\) 0 0
\(739\) 49.7582i 1.83038i 0.403018 + 0.915192i \(0.367961\pi\)
−0.403018 + 0.915192i \(0.632039\pi\)
\(740\) 0 0
\(741\) 23.3137i 0.856450i
\(742\) 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
\(747\) 25.0489i 0.916490i
\(748\) 0 0
\(749\) − 7.31371i − 0.267237i
\(750\) 0 0
\(751\) 14.7821 0.539405 0.269703 0.962944i \(-0.413075\pi\)
0.269703 + 0.962944i \(0.413075\pi\)
\(752\) 0 0
\(753\) −19.5147 −0.711156
\(754\) 0 0
\(755\) − 0.371418i − 0.0135173i
\(756\) 0 0
\(757\) 18.9289i 0.687984i 0.938973 + 0.343992i \(0.111779\pi\)
−0.938973 + 0.343992i \(0.888221\pi\)
\(758\) 0 0
\(759\) 30.3070 1.10007
\(760\) 0 0
\(761\) −34.6274 −1.25524 −0.627621 0.778519i \(-0.715971\pi\)
−0.627621 + 0.778519i \(0.715971\pi\)
\(762\) 0 0
\(763\) 48.3044i 1.74874i
\(764\) 0 0
\(765\) − 3.02944i − 0.109530i
\(766\) 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
\(771\) − 16.2040i − 0.583574i
\(772\) 0 0
\(773\) − 39.6985i − 1.42786i −0.700219 0.713928i \(-0.746915\pi\)
0.700219 0.713928i \(-0.253085\pi\)
\(774\) 0 0
\(775\) 14.2568 0.512120
\(776\) 0 0
\(777\) −36.2843 −1.30169
\(778\) 0 0
\(779\) − 25.2346i − 0.904123i
\(780\) 0 0
\(781\) − 33.2548i − 1.18995i
\(782\) 0 0
\(783\) −38.7485 −1.38476
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) − 7.94816i − 0.283321i −0.989915 0.141661i \(-0.954756\pi\)
0.989915 0.141661i \(-0.0452442\pi\)
\(788\) 0 0
\(789\) − 13.9411i − 0.496317i
\(790\) 0 0
\(791\) 28.2960 1.00609
\(792\) 0 0
\(793\) 3.65685 0.129859
\(794\) 0 0
\(795\) − 2.90762i − 0.103123i
\(796\) 0 0
\(797\) − 39.6985i − 1.40619i −0.711095 0.703096i \(-0.751800\pi\)
0.711095 0.703096i \(-0.248200\pi\)
\(798\) 0 0
\(799\) 8.65914 0.306338
\(800\) 0 0
\(801\) −8.82843 −0.311937
\(802\) 0 0
\(803\) − 26.8741i − 0.948366i
\(804\) 0 0
\(805\) 14.6274i 0.515549i
\(806\) 0 0
\(807\) −27.6620 −0.973748
\(808\) 0 0
\(809\) 24.6863 0.867924 0.433962 0.900931i \(-0.357115\pi\)
0.433962 + 0.900931i \(0.357115\pi\)
\(810\) 0 0
\(811\) − 2.35049i − 0.0825370i −0.999148 0.0412685i \(-0.986860\pi\)
0.999148 0.0412685i \(-0.0131399\pi\)
\(812\) 0 0
\(813\) − 6.05887i − 0.212494i
\(814\) 0 0
\(815\) 3.17025 0.111049
\(816\) 0 0
\(817\) 6.82843 0.238896
\(818\) 0 0
\(819\) − 23.0698i − 0.806124i
\(820\) 0 0
\(821\) − 38.5269i − 1.34460i −0.740279 0.672299i \(-0.765307\pi\)
0.740279 0.672299i \(-0.234693\pi\)
\(822\) 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
\(827\) − 18.5545i − 0.645204i −0.946535 0.322602i \(-0.895443\pi\)
0.946535 0.322602i \(-0.104557\pi\)
\(828\) 0 0
\(829\) − 12.3848i − 0.430141i −0.976599 0.215071i \(-0.931002\pi\)
0.976599 0.215071i \(-0.0689981\pi\)
\(830\) 0 0
\(831\) −3.69552 −0.128196
\(832\) 0 0
\(833\) 18.8284 0.652366
\(834\) 0 0
\(835\) 2.16478i 0.0749155i
\(836\) 0 0
\(837\) 16.0000i 0.553041i
\(838\) 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
\(843\) − 19.2655i − 0.663539i
\(844\) 0 0
\(845\) − 0.786797i − 0.0270666i
\(846\) 0 0
\(847\) 22.8072 0.783663
\(848\) 0 0
\(849\) −15.7990 −0.542220
\(850\) 0 0
\(851\) − 61.2931i − 2.10110i
\(852\) 0 0
\(853\) 32.1005i 1.09910i 0.835461 + 0.549550i \(0.185201\pi\)
−0.835461 + 0.549550i \(0.814799\pi\)
\(854\) 0 0
\(855\) −6.75699 −0.231084
\(856\) 0 0
\(857\) −24.2843 −0.829535 −0.414767 0.909927i \(-0.636137\pi\)
−0.414767 + 0.909927i \(0.636137\pi\)
\(858\) 0 0
\(859\) 11.9063i 0.406238i 0.979154 + 0.203119i \(0.0651079\pi\)
−0.979154 + 0.203119i \(0.934892\pi\)
\(860\) 0 0
\(861\) − 16.0000i − 0.545279i
\(862\) 0 0
\(863\) 0.525265 0.0178802 0.00894011 0.999960i \(-0.497154\pi\)
0.00894011 + 0.999960i \(0.497154\pi\)
\(864\) 0 0
\(865\) −11.2548 −0.382676
\(866\) 0 0
\(867\) − 9.74153i − 0.330840i
\(868\) 0 0
\(869\) 61.2548i 2.07793i
\(870\) 0 0
\(871\) −6.75699 −0.228952
\(872\) 0 0
\(873\) 9.45584 0.320032
\(874\) 0 0
\(875\) 20.9050i 0.706718i
\(876\) 0 0
\(877\) 49.5563i 1.67340i 0.547662 + 0.836700i \(0.315518\pi\)
−0.547662 + 0.836700i \(0.684482\pi\)
\(878\) 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
\(883\) 29.3784i 0.988663i 0.869273 + 0.494332i \(0.164587\pi\)
−0.869273 + 0.494332i \(0.835413\pi\)
\(884\) 0 0
\(885\) − 0.686292i − 0.0230694i
\(886\) 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
\(891\) 0.710974i 0.0238185i
\(892\) 0 0
\(893\) − 19.3137i − 0.646309i
\(894\) 0 0
\(895\) −6.75699 −0.225861
\(896\) 0 0
\(897\) −24.9706 −0.833743
\(898\) 0 0
\(899\) 22.6984i 0.757033i
\(900\) 0 0
\(901\) − 12.9706i − 0.432112i
\(902\) 0 0
\(903\) 4.32957 0.144079
\(904\) 0 0
\(905\) 8.34315 0.277336
\(906\) 0 0
\(907\) − 45.0572i − 1.49610i −0.663643 0.748050i \(-0.730990\pi\)
0.663643 0.748050i \(-0.269010\pi\)
\(908\) 0 0
\(909\) − 31.2132i − 1.03528i
\(910\) 0 0
\(911\) 23.9665 0.794045 0.397022 0.917809i \(-0.370044\pi\)
0.397022 + 0.917809i \(0.370044\pi\)
\(912\) 0 0
\(913\) 56.7696 1.87880
\(914\) 0 0
\(915\) − 0.679111i − 0.0224507i
\(916\) 0 0
\(917\) 53.9411i 1.78129i
\(918\) 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
\(923\) 27.3994i 0.901861i
\(924\) 0 0
\(925\) − 42.2426i − 1.38893i
\(926\) 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
\(931\) − 41.9957i − 1.37635i
\(932\) 0 0
\(933\) 26.6274i 0.871742i
\(934\) 0 0
\(935\) −6.86577 −0.224535
\(936\) 0 0
\(937\) −43.4558 −1.41964 −0.709820 0.704383i \(-0.751224\pi\)
−0.709820 + 0.704383i \(0.751224\pi\)
\(938\) 0 0
\(939\) 33.8937i 1.10608i
\(940\) 0 0
\(941\) 2.92893i 0.0954805i 0.998860 + 0.0477402i \(0.0152020\pi\)
−0.998860 + 0.0477402i \(0.984798\pi\)
\(942\) 0 0
\(943\) 27.0279 0.880151
\(944\) 0 0
\(945\) −11.3137 −0.368035
\(946\) 0 0
\(947\) 21.4621i 0.697426i 0.937230 + 0.348713i \(0.113381\pi\)
−0.937230 + 0.348713i \(0.886619\pi\)
\(948\) 0 0
\(949\) 22.1421i 0.718764i
\(950\) 0 0
\(951\) −16.6842 −0.541023
\(952\) 0 0
\(953\) −11.0294 −0.357279 −0.178639 0.983915i \(-0.557170\pi\)
−0.178639 + 0.983915i \(0.557170\pi\)
\(954\) 0 0
\(955\) 6.86577i 0.222171i
\(956\) 0 0
\(957\) 33.2548i 1.07498i
\(958\) 0 0
\(959\) 47.9329 1.54784
\(960\) 0 0
\(961\) −21.6274 −0.697659
\(962\) 0 0
\(963\) 3.61859i 0.116608i
\(964\) 0 0
\(965\) − 0.284271i − 0.00915102i
\(966\) 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
\(971\) 57.6745i 1.85086i 0.378916 + 0.925431i \(0.376297\pi\)
−0.378916 + 0.925431i \(0.623703\pi\)
\(972\) 0 0
\(973\) − 74.6274i − 2.39245i
\(974\) 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
\(979\) 20.0083i 0.639469i
\(980\) 0 0
\(981\) − 23.8995i − 0.763052i
\(982\) 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
\(987\) − 12.2459i − 0.389790i
\(988\) 0 0
\(989\) 7.31371i 0.232562i
\(990\) 0 0
\(991\) −27.0279 −0.858571 −0.429285 0.903169i \(-0.641235\pi\)
−0.429285 + 0.903169i \(0.641235\pi\)
\(992\) 0 0
\(993\) 9.17157 0.291051
\(994\) 0 0
\(995\) 7.23719i 0.229434i
\(996\) 0 0
\(997\) − 28.3848i − 0.898955i −0.893292 0.449477i \(-0.851610\pi\)
0.893292 0.449477i \(-0.148390\pi\)
\(998\) 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.b.g.513.5 8
4.3 odd 2 inner 1024.2.b.g.513.3 8
8.3 odd 2 inner 1024.2.b.g.513.6 8
8.5 even 2 inner 1024.2.b.g.513.4 8
16.3 odd 4 1024.2.a.h.1.3 4
16.5 even 4 1024.2.a.i.1.3 4
16.11 odd 4 1024.2.a.i.1.2 4
16.13 even 4 1024.2.a.h.1.2 4
32.3 odd 8 512.2.e.i.385.3 yes 8
32.5 even 8 512.2.e.i.129.2 8
32.11 odd 8 512.2.e.j.129.2 yes 8
32.13 even 8 512.2.e.j.385.3 yes 8
32.19 odd 8 512.2.e.j.385.2 yes 8
32.21 even 8 512.2.e.j.129.3 yes 8
32.27 odd 8 512.2.e.i.129.3 yes 8
32.29 even 8 512.2.e.i.385.2 yes 8
48.5 odd 4 9216.2.a.w.1.4 4
48.11 even 4 9216.2.a.w.1.3 4
48.29 odd 4 9216.2.a.bp.1.2 4
48.35 even 4 9216.2.a.bp.1.1 4
96.5 odd 8 4608.2.k.bi.1153.1 8
96.11 even 8 4608.2.k.bd.1153.4 8
96.29 odd 8 4608.2.k.bi.3457.2 8
96.35 even 8 4608.2.k.bi.3457.1 8
96.53 odd 8 4608.2.k.bd.1153.3 8
96.59 even 8 4608.2.k.bi.1153.2 8
96.77 odd 8 4608.2.k.bd.3457.4 8
96.83 even 8 4608.2.k.bd.3457.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.e.i.129.2 8 32.5 even 8
512.2.e.i.129.3 yes 8 32.27 odd 8
512.2.e.i.385.2 yes 8 32.29 even 8
512.2.e.i.385.3 yes 8 32.3 odd 8
512.2.e.j.129.2 yes 8 32.11 odd 8
512.2.e.j.129.3 yes 8 32.21 even 8
512.2.e.j.385.2 yes 8 32.19 odd 8
512.2.e.j.385.3 yes 8 32.13 even 8
1024.2.a.h.1.2 4 16.13 even 4
1024.2.a.h.1.3 4 16.3 odd 4
1024.2.a.i.1.2 4 16.11 odd 4
1024.2.a.i.1.3 4 16.5 even 4
1024.2.b.g.513.3 8 4.3 odd 2 inner
1024.2.b.g.513.4 8 8.5 even 2 inner
1024.2.b.g.513.5 8 1.1 even 1 trivial
1024.2.b.g.513.6 8 8.3 odd 2 inner
4608.2.k.bd.1153.3 8 96.53 odd 8
4608.2.k.bd.1153.4 8 96.11 even 8
4608.2.k.bd.3457.3 8 96.83 even 8
4608.2.k.bd.3457.4 8 96.77 odd 8
4608.2.k.bi.1153.1 8 96.5 odd 8
4608.2.k.bi.1153.2 8 96.59 even 8
4608.2.k.bi.3457.1 8 96.35 even 8
4608.2.k.bi.3457.2 8 96.29 odd 8
9216.2.a.w.1.3 4 48.11 even 4
9216.2.a.w.1.4 4 48.5 odd 4
9216.2.a.bp.1.1 4 48.35 even 4
9216.2.a.bp.1.2 4 48.29 odd 4