Properties

Label 1078.2.a.r.1.2
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} -2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} -2.82843 q^{6} -1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} +2.82843 q^{12} -2.82843 q^{13} +1.00000 q^{16} +2.82843 q^{17} -5.00000 q^{18} +5.65685 q^{19} +1.00000 q^{22} +8.00000 q^{23} -2.82843 q^{24} -5.00000 q^{25} +2.82843 q^{26} +5.65685 q^{27} +2.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} -2.82843 q^{33} -2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} -5.65685 q^{38} -8.00000 q^{39} +2.82843 q^{41} -4.00000 q^{43} -1.00000 q^{44} -8.00000 q^{46} +2.82843 q^{47} +2.82843 q^{48} +5.00000 q^{50} +8.00000 q^{51} -2.82843 q^{52} +14.0000 q^{53} -5.65685 q^{54} +16.0000 q^{57} -2.00000 q^{58} -8.48528 q^{59} -8.48528 q^{61} -8.48528 q^{62} +1.00000 q^{64} +2.82843 q^{66} +4.00000 q^{67} +2.82843 q^{68} +22.6274 q^{69} -5.00000 q^{72} -14.1421 q^{73} -2.00000 q^{74} -14.1421 q^{75} +5.65685 q^{76} +8.00000 q^{78} -16.0000 q^{79} +1.00000 q^{81} -2.82843 q^{82} -16.9706 q^{83} +4.00000 q^{86} +5.65685 q^{87} +1.00000 q^{88} -11.3137 q^{89} +8.00000 q^{92} +24.0000 q^{93} -2.82843 q^{94} -2.82843 q^{96} -16.9706 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{18} + 2 q^{22} + 16 q^{23} - 10 q^{25} + 4 q^{29} - 2 q^{32} + 10 q^{36} + 4 q^{37} - 16 q^{39} - 8 q^{43} - 2 q^{44} - 16 q^{46} + 10 q^{50} + 16 q^{51} + 28 q^{53} + 32 q^{57} - 4 q^{58} + 2 q^{64} + 8 q^{67} - 10 q^{72} - 4 q^{74} + 16 q^{78} - 32 q^{79} + 2 q^{81} + 8 q^{86} + 2 q^{88} + 16 q^{92} + 48 q^{93} - 10 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.82843 1.63299 0.816497 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −2.82843 −1.15470
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.82843 0.816497
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) −5.00000 −1.17851
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −2.82843 −0.577350
\(25\) −5.00000 −1.00000
\(26\) 2.82843 0.554700
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.82843 −0.492366
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 5.00000 0.833333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −5.65685 −0.917663
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 2.82843 0.441726 0.220863 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 2.82843 0.408248
\(49\) 0 0
\(50\) 5.00000 0.707107
\(51\) 8.00000 1.12022
\(52\) −2.82843 −0.392232
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −5.65685 −0.769800
\(55\) 0 0
\(56\) 0 0
\(57\) 16.0000 2.11925
\(58\) −2.00000 −0.262613
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) −8.48528 −1.08643 −0.543214 0.839594i \(-0.682793\pi\)
−0.543214 + 0.839594i \(0.682793\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.82843 0.348155
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.82843 0.342997
\(69\) 22.6274 2.72402
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −5.00000 −0.589256
\(73\) −14.1421 −1.65521 −0.827606 0.561310i \(-0.810298\pi\)
−0.827606 + 0.561310i \(0.810298\pi\)
\(74\) −2.00000 −0.232495
\(75\) −14.1421 −1.63299
\(76\) 5.65685 0.648886
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.82843 −0.312348
\(83\) −16.9706 −1.86276 −0.931381 0.364047i \(-0.881395\pi\)
−0.931381 + 0.364047i \(0.881395\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 5.65685 0.606478
\(88\) 1.00000 0.106600
\(89\) −11.3137 −1.19925 −0.599625 0.800281i \(-0.704684\pi\)
−0.599625 + 0.800281i \(0.704684\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 24.0000 2.48868
\(94\) −2.82843 −0.291730
\(95\) 0 0
\(96\) −2.82843 −0.288675
\(97\) −16.9706 −1.72310 −0.861550 0.507673i \(-0.830506\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) −5.00000 −0.500000
\(101\) −2.82843 −0.281439 −0.140720 0.990050i \(-0.544942\pi\)
−0.140720 + 0.990050i \(0.544942\pi\)
\(102\) −8.00000 −0.792118
\(103\) 14.1421 1.39347 0.696733 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(104\) 2.82843 0.277350
\(105\) 0 0
\(106\) −14.0000 −1.35980
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 5.65685 0.544331
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 5.65685 0.536925
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −16.0000 −1.49854
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −14.1421 −1.30744
\(118\) 8.48528 0.781133
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.48528 0.768221
\(123\) 8.00000 0.721336
\(124\) 8.48528 0.762001
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) −2.82843 −0.246183
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −2.82843 −0.242536
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −22.6274 −1.92617
\(139\) −16.9706 −1.43942 −0.719712 0.694273i \(-0.755726\pi\)
−0.719712 + 0.694273i \(0.755726\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 2.82843 0.236525
\(144\) 5.00000 0.416667
\(145\) 0 0
\(146\) 14.1421 1.17041
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 14.1421 1.15470
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −5.65685 −0.458831
\(153\) 14.1421 1.14332
\(154\) 0 0
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) −11.3137 −0.902932 −0.451466 0.892288i \(-0.649099\pi\)
−0.451466 + 0.892288i \(0.649099\pi\)
\(158\) 16.0000 1.27289
\(159\) 39.5980 3.14032
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.82843 0.220863
\(165\) 0 0
\(166\) 16.9706 1.31717
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 28.2843 2.16295
\(172\) −4.00000 −0.304997
\(173\) 19.7990 1.50529 0.752645 0.658427i \(-0.228778\pi\)
0.752645 + 0.658427i \(0.228778\pi\)
\(174\) −5.65685 −0.428845
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −24.0000 −1.80395
\(178\) 11.3137 0.847998
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 5.65685 0.420471 0.210235 0.977651i \(-0.432577\pi\)
0.210235 + 0.977651i \(0.432577\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −24.0000 −1.75977
\(187\) −2.82843 −0.206835
\(188\) 2.82843 0.206284
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 2.82843 0.204124
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 16.9706 1.21842
\(195\) 0 0
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 5.00000 0.355335
\(199\) −19.7990 −1.40351 −0.701757 0.712417i \(-0.747601\pi\)
−0.701757 + 0.712417i \(0.747601\pi\)
\(200\) 5.00000 0.353553
\(201\) 11.3137 0.798007
\(202\) 2.82843 0.199007
\(203\) 0 0
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) −14.1421 −0.985329
\(207\) 40.0000 2.78019
\(208\) −2.82843 −0.196116
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −5.65685 −0.384900
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −40.0000 −2.70295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −5.65685 −0.379663
\(223\) 14.1421 0.947027 0.473514 0.880786i \(-0.342985\pi\)
0.473514 + 0.880786i \(0.342985\pi\)
\(224\) 0 0
\(225\) −25.0000 −1.66667
\(226\) −2.00000 −0.133038
\(227\) 11.3137 0.750917 0.375459 0.926839i \(-0.377485\pi\)
0.375459 + 0.926839i \(0.377485\pi\)
\(228\) 16.0000 1.05963
\(229\) −5.65685 −0.373815 −0.186908 0.982377i \(-0.559847\pi\)
−0.186908 + 0.982377i \(0.559847\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 14.1421 0.924500
\(235\) 0 0
\(236\) −8.48528 −0.552345
\(237\) −45.2548 −2.93962
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 14.1421 0.910975 0.455488 0.890242i \(-0.349465\pi\)
0.455488 + 0.890242i \(0.349465\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −14.1421 −0.907218
\(244\) −8.48528 −0.543214
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) −16.0000 −1.01806
\(248\) −8.48528 −0.538816
\(249\) −48.0000 −3.04188
\(250\) 0 0
\(251\) −14.1421 −0.892644 −0.446322 0.894873i \(-0.647266\pi\)
−0.446322 + 0.894873i \(0.647266\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.3137 −0.705730 −0.352865 0.935674i \(-0.614792\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 11.3137 0.704361
\(259\) 0 0
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) −11.3137 −0.698963
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 2.82843 0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) −32.0000 −1.95837
\(268\) 4.00000 0.244339
\(269\) −5.65685 −0.344904 −0.172452 0.985018i \(-0.555169\pi\)
−0.172452 + 0.985018i \(0.555169\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) 2.82843 0.171499
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 5.00000 0.301511
\(276\) 22.6274 1.36201
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 16.9706 1.01783
\(279\) 42.4264 2.54000
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −8.00000 −0.476393
\(283\) 11.3137 0.672530 0.336265 0.941767i \(-0.390836\pi\)
0.336265 + 0.941767i \(0.390836\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −48.0000 −2.81381
\(292\) −14.1421 −0.827606
\(293\) 14.1421 0.826192 0.413096 0.910687i \(-0.364447\pi\)
0.413096 + 0.910687i \(0.364447\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −5.65685 −0.328244
\(298\) −6.00000 −0.347571
\(299\) −22.6274 −1.30858
\(300\) −14.1421 −0.816497
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −8.00000 −0.459588
\(304\) 5.65685 0.324443
\(305\) 0 0
\(306\) −14.1421 −0.808452
\(307\) −22.6274 −1.29141 −0.645707 0.763585i \(-0.723437\pi\)
−0.645707 + 0.763585i \(0.723437\pi\)
\(308\) 0 0
\(309\) 40.0000 2.27552
\(310\) 0 0
\(311\) −31.1127 −1.76424 −0.882120 0.471025i \(-0.843884\pi\)
−0.882120 + 0.471025i \(0.843884\pi\)
\(312\) 8.00000 0.452911
\(313\) −11.3137 −0.639489 −0.319744 0.947504i \(-0.603597\pi\)
−0.319744 + 0.947504i \(0.603597\pi\)
\(314\) 11.3137 0.638470
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −39.5980 −2.22054
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 33.9411 1.89441
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 14.1421 0.784465
\(326\) 4.00000 0.221540
\(327\) 5.65685 0.312825
\(328\) −2.82843 −0.156174
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −16.9706 −0.931381
\(333\) 10.0000 0.547997
\(334\) 11.3137 0.619059
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 5.00000 0.271964
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) −8.48528 −0.459504
\(342\) −28.2843 −1.52944
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −19.7990 −1.06440
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 5.65685 0.303239
\(349\) 25.4558 1.36262 0.681310 0.731995i \(-0.261411\pi\)
0.681310 + 0.731995i \(0.261411\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 1.00000 0.0533002
\(353\) −11.3137 −0.602168 −0.301084 0.953598i \(-0.597348\pi\)
−0.301084 + 0.953598i \(0.597348\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) −11.3137 −0.599625
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) −5.65685 −0.297318
\(363\) 2.82843 0.148454
\(364\) 0 0
\(365\) 0 0
\(366\) 24.0000 1.25450
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 8.00000 0.417029
\(369\) 14.1421 0.736210
\(370\) 0 0
\(371\) 0 0
\(372\) 24.0000 1.24434
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) −2.82843 −0.145865
\(377\) −5.65685 −0.291343
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −2.82843 −0.144526 −0.0722629 0.997386i \(-0.523022\pi\)
−0.0722629 + 0.997386i \(0.523022\pi\)
\(384\) −2.82843 −0.144338
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −20.0000 −1.01666
\(388\) −16.9706 −0.861550
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 22.6274 1.14432
\(392\) 0 0
\(393\) 32.0000 1.61419
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) −22.6274 −1.13564 −0.567819 0.823154i \(-0.692213\pi\)
−0.567819 + 0.823154i \(0.692213\pi\)
\(398\) 19.7990 0.992434
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −11.3137 −0.564276
\(403\) −24.0000 −1.19553
\(404\) −2.82843 −0.140720
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) −8.00000 −0.396059
\(409\) 8.48528 0.419570 0.209785 0.977748i \(-0.432724\pi\)
0.209785 + 0.977748i \(0.432724\pi\)
\(410\) 0 0
\(411\) −5.65685 −0.279032
\(412\) 14.1421 0.696733
\(413\) 0 0
\(414\) −40.0000 −1.96589
\(415\) 0 0
\(416\) 2.82843 0.138675
\(417\) −48.0000 −2.35057
\(418\) 5.65685 0.276686
\(419\) 25.4558 1.24360 0.621800 0.783176i \(-0.286402\pi\)
0.621800 + 0.783176i \(0.286402\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −4.00000 −0.194717
\(423\) 14.1421 0.687614
\(424\) −14.0000 −0.679900
\(425\) −14.1421 −0.685994
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 5.65685 0.272166
\(433\) −16.9706 −0.815553 −0.407777 0.913082i \(-0.633696\pi\)
−0.407777 + 0.913082i \(0.633696\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 45.2548 2.16483
\(438\) 40.0000 1.91127
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 5.65685 0.268462
\(445\) 0 0
\(446\) −14.1421 −0.669650
\(447\) 16.9706 0.802680
\(448\) 0 0
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 25.0000 1.17851
\(451\) −2.82843 −0.133185
\(452\) 2.00000 0.0940721
\(453\) 45.2548 2.12626
\(454\) −11.3137 −0.530979
\(455\) 0 0
\(456\) −16.0000 −0.749269
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 5.65685 0.264327
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) −19.7990 −0.922131 −0.461065 0.887366i \(-0.652533\pi\)
−0.461065 + 0.887366i \(0.652533\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) −25.4558 −1.17796 −0.588978 0.808149i \(-0.700470\pi\)
−0.588978 + 0.808149i \(0.700470\pi\)
\(468\) −14.1421 −0.653720
\(469\) 0 0
\(470\) 0 0
\(471\) −32.0000 −1.47448
\(472\) 8.48528 0.390567
\(473\) 4.00000 0.183920
\(474\) 45.2548 2.07862
\(475\) −28.2843 −1.29777
\(476\) 0 0
\(477\) 70.0000 3.20508
\(478\) 16.0000 0.731823
\(479\) −16.9706 −0.775405 −0.387702 0.921785i \(-0.626731\pi\)
−0.387702 + 0.921785i \(0.626731\pi\)
\(480\) 0 0
\(481\) −5.65685 −0.257930
\(482\) −14.1421 −0.644157
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 14.1421 0.641500
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 8.48528 0.384111
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 8.00000 0.360668
\(493\) 5.65685 0.254772
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 48.0000 2.15093
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −32.0000 −1.42965
\(502\) 14.1421 0.631194
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) −14.1421 −0.628074
\(508\) 0 0
\(509\) −16.9706 −0.752207 −0.376103 0.926578i \(-0.622736\pi\)
−0.376103 + 0.926578i \(0.622736\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 32.0000 1.41283
\(514\) 11.3137 0.499026
\(515\) 0 0
\(516\) −11.3137 −0.498058
\(517\) −2.82843 −0.124394
\(518\) 0 0
\(519\) 56.0000 2.45813
\(520\) 0 0
\(521\) 39.5980 1.73482 0.867409 0.497595i \(-0.165783\pi\)
0.867409 + 0.497595i \(0.165783\pi\)
\(522\) −10.0000 −0.437688
\(523\) 33.9411 1.48414 0.742071 0.670321i \(-0.233844\pi\)
0.742071 + 0.670321i \(0.233844\pi\)
\(524\) 11.3137 0.494242
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 24.0000 1.04546
\(528\) −2.82843 −0.123091
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −42.4264 −1.84115
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 32.0000 1.38478
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 56.5685 2.44111
\(538\) 5.65685 0.243884
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 16.9706 0.728948
\(543\) 16.0000 0.686626
\(544\) −2.82843 −0.121268
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −42.4264 −1.81071
\(550\) −5.00000 −0.213201
\(551\) 11.3137 0.481980
\(552\) −22.6274 −0.963087
\(553\) 0 0
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −16.9706 −0.719712
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −42.4264 −1.79605
\(559\) 11.3137 0.478519
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 10.0000 0.421825
\(563\) 28.2843 1.19204 0.596020 0.802970i \(-0.296748\pi\)
0.596020 + 0.802970i \(0.296748\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −11.3137 −0.475551
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 2.82843 0.118262
\(573\) −22.6274 −0.945274
\(574\) 0 0
\(575\) −40.0000 −1.66812
\(576\) 5.00000 0.208333
\(577\) 39.5980 1.64849 0.824243 0.566237i \(-0.191601\pi\)
0.824243 + 0.566237i \(0.191601\pi\)
\(578\) 9.00000 0.374351
\(579\) −39.5980 −1.64564
\(580\) 0 0
\(581\) 0 0
\(582\) 48.0000 1.98966
\(583\) −14.0000 −0.579821
\(584\) 14.1421 0.585206
\(585\) 0 0
\(586\) −14.1421 −0.584206
\(587\) −14.1421 −0.583708 −0.291854 0.956463i \(-0.594272\pi\)
−0.291854 + 0.956463i \(0.594272\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) −28.2843 −1.16346
\(592\) 2.00000 0.0821995
\(593\) 42.4264 1.74224 0.871122 0.491067i \(-0.163393\pi\)
0.871122 + 0.491067i \(0.163393\pi\)
\(594\) 5.65685 0.232104
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −56.0000 −2.29193
\(598\) 22.6274 0.925304
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 14.1421 0.577350
\(601\) 25.4558 1.03837 0.519183 0.854663i \(-0.326236\pi\)
0.519183 + 0.854663i \(0.326236\pi\)
\(602\) 0 0
\(603\) 20.0000 0.814463
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) −33.9411 −1.37763 −0.688814 0.724938i \(-0.741868\pi\)
−0.688814 + 0.724938i \(0.741868\pi\)
\(608\) −5.65685 −0.229416
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 14.1421 0.571662
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 22.6274 0.913168
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) −40.0000 −1.60904
\(619\) −2.82843 −0.113684 −0.0568420 0.998383i \(-0.518103\pi\)
−0.0568420 + 0.998383i \(0.518103\pi\)
\(620\) 0 0
\(621\) 45.2548 1.81601
\(622\) 31.1127 1.24751
\(623\) 0 0
\(624\) −8.00000 −0.320256
\(625\) 25.0000 1.00000
\(626\) 11.3137 0.452187
\(627\) −16.0000 −0.638978
\(628\) −11.3137 −0.451466
\(629\) 5.65685 0.225554
\(630\) 0 0
\(631\) 48.0000 1.91085 0.955425 0.295234i \(-0.0953977\pi\)
0.955425 + 0.295234i \(0.0953977\pi\)
\(632\) 16.0000 0.636446
\(633\) 11.3137 0.449680
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 39.5980 1.57016
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) −33.9411 −1.33955
\(643\) −25.4558 −1.00388 −0.501940 0.864902i \(-0.667380\pi\)
−0.501940 + 0.864902i \(0.667380\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 8.48528 0.333076
\(650\) −14.1421 −0.554700
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −5.65685 −0.221201
\(655\) 0 0
\(656\) 2.82843 0.110432
\(657\) −70.7107 −2.75869
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 28.0000 1.08825
\(663\) −22.6274 −0.878776
\(664\) 16.9706 0.658586
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 16.0000 0.619522
\(668\) −11.3137 −0.437741
\(669\) 40.0000 1.54649
\(670\) 0 0
\(671\) 8.48528 0.327571
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) −28.2843 −1.08866
\(676\) −5.00000 −0.192308
\(677\) −2.82843 −0.108705 −0.0543526 0.998522i \(-0.517310\pi\)
−0.0543526 + 0.998522i \(0.517310\pi\)
\(678\) −5.65685 −0.217250
\(679\) 0 0
\(680\) 0 0
\(681\) 32.0000 1.22624
\(682\) 8.48528 0.324918
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 28.2843 1.08148
\(685\) 0 0
\(686\) 0 0
\(687\) −16.0000 −0.610438
\(688\) −4.00000 −0.152499
\(689\) −39.5980 −1.50856
\(690\) 0 0
\(691\) −8.48528 −0.322795 −0.161398 0.986889i \(-0.551600\pi\)
−0.161398 + 0.986889i \(0.551600\pi\)
\(692\) 19.7990 0.752645
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) −5.65685 −0.214423
\(697\) 8.00000 0.303022
\(698\) −25.4558 −0.963518
\(699\) 62.2254 2.35358
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 16.0000 0.603881
\(703\) 11.3137 0.426705
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 11.3137 0.425797
\(707\) 0 0
\(708\) −24.0000 −0.901975
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −80.0000 −3.00023
\(712\) 11.3137 0.423999
\(713\) 67.8823 2.54221
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) −45.2548 −1.69007
\(718\) −32.0000 −1.19423
\(719\) 19.7990 0.738378 0.369189 0.929354i \(-0.379636\pi\)
0.369189 + 0.929354i \(0.379636\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13.0000 −0.483810
\(723\) 40.0000 1.48762
\(724\) 5.65685 0.210235
\(725\) −10.0000 −0.371391
\(726\) −2.82843 −0.104973
\(727\) 19.7990 0.734304 0.367152 0.930161i \(-0.380333\pi\)
0.367152 + 0.930161i \(0.380333\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) −11.3137 −0.418453
\(732\) −24.0000 −0.887066
\(733\) 48.0833 1.77600 0.887998 0.459848i \(-0.152096\pi\)
0.887998 + 0.459848i \(0.152096\pi\)
\(734\) 8.48528 0.313197
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −4.00000 −0.147342
\(738\) −14.1421 −0.520579
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −45.2548 −1.66248
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −24.0000 −0.879883
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) −84.8528 −3.10460
\(748\) −2.82843 −0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 2.82843 0.103142
\(753\) −40.0000 −1.45768
\(754\) 5.65685 0.206010
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 12.0000 0.435860
\(759\) −22.6274 −0.821323
\(760\) 0 0
\(761\) 8.48528 0.307591 0.153796 0.988103i \(-0.450850\pi\)
0.153796 + 0.988103i \(0.450850\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 2.82843 0.102195
\(767\) 24.0000 0.866590
\(768\) 2.82843 0.102062
\(769\) −2.82843 −0.101996 −0.0509978 0.998699i \(-0.516240\pi\)
−0.0509978 + 0.998699i \(0.516240\pi\)
\(770\) 0 0
\(771\) −32.0000 −1.15245
\(772\) −14.0000 −0.503871
\(773\) 28.2843 1.01731 0.508657 0.860969i \(-0.330142\pi\)
0.508657 + 0.860969i \(0.330142\pi\)
\(774\) 20.0000 0.718885
\(775\) −42.4264 −1.52400
\(776\) 16.9706 0.609208
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) −22.6274 −0.809155
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) 0 0
\(786\) −32.0000 −1.14140
\(787\) −39.5980 −1.41152 −0.705758 0.708453i \(-0.749393\pi\)
−0.705758 + 0.708453i \(0.749393\pi\)
\(788\) −10.0000 −0.356235
\(789\) −45.2548 −1.61111
\(790\) 0 0
\(791\) 0 0
\(792\) 5.00000 0.177667
\(793\) 24.0000 0.852265
\(794\) 22.6274 0.803017
\(795\) 0 0
\(796\) −19.7990 −0.701757
\(797\) −11.3137 −0.400752 −0.200376 0.979719i \(-0.564216\pi\)
−0.200376 + 0.979719i \(0.564216\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 5.00000 0.176777
\(801\) −56.5685 −1.99875
\(802\) −30.0000 −1.05934
\(803\) 14.1421 0.499065
\(804\) 11.3137 0.399004
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) −16.0000 −0.563227
\(808\) 2.82843 0.0995037
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −22.6274 −0.794556 −0.397278 0.917698i \(-0.630045\pi\)
−0.397278 + 0.917698i \(0.630045\pi\)
\(812\) 0 0
\(813\) −48.0000 −1.68343
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) 8.00000 0.280056
\(817\) −22.6274 −0.791633
\(818\) −8.48528 −0.296681
\(819\) 0 0
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 5.65685 0.197305
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −14.1421 −0.492665
\(825\) 14.1421 0.492366
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 40.0000 1.39010
\(829\) 11.3137 0.392941 0.196471 0.980510i \(-0.437052\pi\)
0.196471 + 0.980510i \(0.437052\pi\)
\(830\) 0 0
\(831\) 62.2254 2.15858
\(832\) −2.82843 −0.0980581
\(833\) 0 0
\(834\) 48.0000 1.66210
\(835\) 0 0
\(836\) −5.65685 −0.195646
\(837\) 48.0000 1.65912
\(838\) −25.4558 −0.879358
\(839\) −2.82843 −0.0976481 −0.0488241 0.998807i \(-0.515547\pi\)
−0.0488241 + 0.998807i \(0.515547\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −14.0000 −0.482472
\(843\) −28.2843 −0.974162
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −14.1421 −0.486217
\(847\) 0 0
\(848\) 14.0000 0.480762
\(849\) 32.0000 1.09824
\(850\) 14.1421 0.485071
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 8.48528 0.290531 0.145265 0.989393i \(-0.453596\pi\)
0.145265 + 0.989393i \(0.453596\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.4264 1.44926 0.724629 0.689139i \(-0.242011\pi\)
0.724629 + 0.689139i \(0.242011\pi\)
\(858\) −8.00000 −0.273115
\(859\) −25.4558 −0.868542 −0.434271 0.900782i \(-0.642994\pi\)
−0.434271 + 0.900782i \(0.642994\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) −5.65685 −0.192450
\(865\) 0 0
\(866\) 16.9706 0.576683
\(867\) −25.4558 −0.864526
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −11.3137 −0.383350
\(872\) −2.00000 −0.0677285
\(873\) −84.8528 −2.87183
\(874\) −45.2548 −1.53077
\(875\) 0 0
\(876\) −40.0000 −1.35147
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) −16.9706 −0.572729
\(879\) 40.0000 1.34917
\(880\) 0 0
\(881\) −5.65685 −0.190584 −0.0952921 0.995449i \(-0.530379\pi\)
−0.0952921 + 0.995449i \(0.530379\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 28.2843 0.949693 0.474846 0.880069i \(-0.342504\pi\)
0.474846 + 0.880069i \(0.342504\pi\)
\(888\) −5.65685 −0.189832
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 14.1421 0.473514
\(893\) 16.0000 0.535420
\(894\) −16.9706 −0.567581
\(895\) 0 0
\(896\) 0 0
\(897\) −64.0000 −2.13690
\(898\) −10.0000 −0.333704
\(899\) 16.9706 0.566000
\(900\) −25.0000 −0.833333
\(901\) 39.5980 1.31920
\(902\) 2.82843 0.0941763
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) −45.2548 −1.50349
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 11.3137 0.375459
\(909\) −14.1421 −0.469065
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 16.0000 0.529813
\(913\) 16.9706 0.561644
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −5.65685 −0.186908
\(917\) 0 0
\(918\) −16.0000 −0.528079
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −64.0000 −2.10887
\(922\) 19.7990 0.652045
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 8.00000 0.262896
\(927\) 70.7107 2.32244
\(928\) −2.00000 −0.0656532
\(929\) −28.2843 −0.927977 −0.463988 0.885841i \(-0.653582\pi\)
−0.463988 + 0.885841i \(0.653582\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 22.0000 0.720634
\(933\) −88.0000 −2.88099
\(934\) 25.4558 0.832941
\(935\) 0 0
\(936\) 14.1421 0.462250
\(937\) 42.4264 1.38601 0.693005 0.720933i \(-0.256286\pi\)
0.693005 + 0.720933i \(0.256286\pi\)
\(938\) 0 0
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) 2.82843 0.0922041 0.0461020 0.998937i \(-0.485320\pi\)
0.0461020 + 0.998937i \(0.485320\pi\)
\(942\) 32.0000 1.04262
\(943\) 22.6274 0.736850
\(944\) −8.48528 −0.276172
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −45.2548 −1.46981
\(949\) 40.0000 1.29845
\(950\) 28.2843 0.917663
\(951\) −5.65685 −0.183436
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −70.0000 −2.26633
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −5.65685 −0.182860
\(958\) 16.9706 0.548294
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 5.65685 0.182384
\(963\) 60.0000 1.93347
\(964\) 14.1421 0.455488
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 45.2548 1.45379
\(970\) 0 0
\(971\) −36.7696 −1.17999 −0.589996 0.807406i \(-0.700871\pi\)
−0.589996 + 0.807406i \(0.700871\pi\)
\(972\) −14.1421 −0.453609
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 40.0000 1.28103
\(976\) −8.48528 −0.271607
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 11.3137 0.361773
\(979\) 11.3137 0.361588
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 28.0000 0.893516
\(983\) 2.82843 0.0902128 0.0451064 0.998982i \(-0.485637\pi\)
0.0451064 + 0.998982i \(0.485637\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) −5.65685 −0.180151
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −8.48528 −0.269408
\(993\) −79.1960 −2.51321
\(994\) 0 0
\(995\) 0 0
\(996\) −48.0000 −1.52094
\(997\) 42.4264 1.34366 0.671829 0.740706i \(-0.265509\pi\)
0.671829 + 0.740706i \(0.265509\pi\)
\(998\) 12.0000 0.379853
\(999\) 11.3137 0.357950
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 1078.2.a.r.1.2 yes 2
3.2 odd 2 9702.2.a.dn.1.1 2
4.3 odd 2 8624.2.a.by.1.1 2
7.2 even 3 1078.2.e.r.67.1 4
7.3 odd 6 1078.2.e.r.177.2 4
7.4 even 3 1078.2.e.r.177.1 4
7.5 odd 6 1078.2.e.r.67.2 4
7.6 odd 2 inner 1078.2.a.r.1.1 2
21.20 even 2 9702.2.a.dn.1.2 2
28.27 even 2 8624.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.r.1.1 2 7.6 odd 2 inner
1078.2.a.r.1.2 yes 2 1.1 even 1 trivial
1078.2.e.r.67.1 4 7.2 even 3
1078.2.e.r.67.2 4 7.5 odd 6
1078.2.e.r.177.1 4 7.4 even 3
1078.2.e.r.177.2 4 7.3 odd 6
8624.2.a.by.1.1 2 4.3 odd 2
8624.2.a.by.1.2 2 28.27 even 2
9702.2.a.dn.1.1 2 3.2 odd 2
9702.2.a.dn.1.2 2 21.20 even 2