Properties

Label 1078.2.a.u.1.1
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: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} -4.24264 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} -4.24264 q^{5} -1.41421 q^{6} +1.00000 q^{8} -1.00000 q^{9} -4.24264 q^{10} -1.00000 q^{11} -1.41421 q^{12} +6.00000 q^{15} +1.00000 q^{16} +5.65685 q^{17} -1.00000 q^{18} -4.24264 q^{20} -1.00000 q^{22} +6.00000 q^{23} -1.41421 q^{24} +13.0000 q^{25} +5.65685 q^{27} +2.00000 q^{29} +6.00000 q^{30} +1.41421 q^{31} +1.00000 q^{32} +1.41421 q^{33} +5.65685 q^{34} -1.00000 q^{36} -10.0000 q^{37} -4.24264 q^{40} +11.3137 q^{41} -8.00000 q^{43} -1.00000 q^{44} +4.24264 q^{45} +6.00000 q^{46} -4.24264 q^{47} -1.41421 q^{48} +13.0000 q^{50} -8.00000 q^{51} +8.00000 q^{53} +5.65685 q^{54} +4.24264 q^{55} +2.00000 q^{58} -1.41421 q^{59} +6.00000 q^{60} +2.82843 q^{61} +1.41421 q^{62} +1.00000 q^{64} +1.41421 q^{66} +2.00000 q^{67} +5.65685 q^{68} -8.48528 q^{69} -2.00000 q^{71} -1.00000 q^{72} -8.48528 q^{73} -10.0000 q^{74} -18.3848 q^{75} +16.0000 q^{79} -4.24264 q^{80} -5.00000 q^{81} +11.3137 q^{82} +16.9706 q^{83} -24.0000 q^{85} -8.00000 q^{86} -2.82843 q^{87} -1.00000 q^{88} +7.07107 q^{89} +4.24264 q^{90} +6.00000 q^{92} -2.00000 q^{93} -4.24264 q^{94} -1.41421 q^{96} +9.89949 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 2 q^{11} + 12 q^{15} + 2 q^{16} - 2 q^{18} - 2 q^{22} + 12 q^{23} + 26 q^{25} + 4 q^{29} + 12 q^{30} + 2 q^{32} - 2 q^{36} - 20 q^{37} - 16 q^{43} - 2 q^{44} + 12 q^{46} + 26 q^{50} - 16 q^{51} + 16 q^{53} + 4 q^{58} + 12 q^{60} + 2 q^{64} + 4 q^{67} - 4 q^{71} - 2 q^{72} - 20 q^{74} + 32 q^{79} - 10 q^{81} - 48 q^{85} - 16 q^{86} - 2 q^{88} + 12 q^{92} - 4 q^{93} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.24264 −1.89737 −0.948683 0.316228i \(-0.897584\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −1.00000 −0.333333
\(10\) −4.24264 −1.34164
\(11\) −1.00000 −0.301511
\(12\) −1.41421 −0.408248
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 1.00000 0.250000
\(17\) 5.65685 1.37199 0.685994 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.24264 −0.948683
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.41421 −0.288675
\(25\) 13.0000 2.60000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 6.00000 1.09545
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.41421 0.246183
\(34\) 5.65685 0.970143
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −4.24264 −0.670820
\(41\) 11.3137 1.76690 0.883452 0.468521i \(-0.155213\pi\)
0.883452 + 0.468521i \(0.155213\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 4.24264 0.632456
\(46\) 6.00000 0.884652
\(47\) −4.24264 −0.618853 −0.309426 0.950923i \(-0.600137\pi\)
−0.309426 + 0.950923i \(0.600137\pi\)
\(48\) −1.41421 −0.204124
\(49\) 0 0
\(50\) 13.0000 1.83848
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 5.65685 0.769800
\(55\) 4.24264 0.572078
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −1.41421 −0.184115 −0.0920575 0.995754i \(-0.529344\pi\)
−0.0920575 + 0.995754i \(0.529344\pi\)
\(60\) 6.00000 0.774597
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.41421 0.174078
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 5.65685 0.685994
\(69\) −8.48528 −1.02151
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) −10.0000 −1.16248
\(75\) −18.3848 −2.12289
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −4.24264 −0.474342
\(81\) −5.00000 −0.555556
\(82\) 11.3137 1.24939
\(83\) 16.9706 1.86276 0.931381 0.364047i \(-0.118605\pi\)
0.931381 + 0.364047i \(0.118605\pi\)
\(84\) 0 0
\(85\) −24.0000 −2.60317
\(86\) −8.00000 −0.862662
\(87\) −2.82843 −0.303239
\(88\) −1.00000 −0.106600
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 4.24264 0.447214
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −2.00000 −0.207390
\(94\) −4.24264 −0.437595
\(95\) 0 0
\(96\) −1.41421 −0.144338
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 13.0000 1.30000
\(101\) −5.65685 −0.562878 −0.281439 0.959579i \(-0.590812\pi\)
−0.281439 + 0.959579i \(0.590812\pi\)
\(102\) −8.00000 −0.792118
\(103\) −18.3848 −1.81151 −0.905753 0.423806i \(-0.860694\pi\)
−0.905753 + 0.423806i \(0.860694\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 5.65685 0.544331
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.24264 0.404520
\(111\) 14.1421 1.34231
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −25.4558 −2.37377
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −1.41421 −0.130189
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) 1.00000 0.0909091
\(122\) 2.82843 0.256074
\(123\) −16.0000 −1.44267
\(124\) 1.41421 0.127000
\(125\) −33.9411 −3.03579
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.3137 0.996116
\(130\) 0 0
\(131\) 19.7990 1.72985 0.864923 0.501905i \(-0.167367\pi\)
0.864923 + 0.501905i \(0.167367\pi\)
\(132\) 1.41421 0.123091
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) −24.0000 −2.06559
\(136\) 5.65685 0.485071
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −8.48528 −0.722315
\(139\) −11.3137 −0.959616 −0.479808 0.877373i \(-0.659294\pi\)
−0.479808 + 0.877373i \(0.659294\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) −8.48528 −0.704664
\(146\) −8.48528 −0.702247
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −18.3848 −1.50111
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) −5.65685 −0.457330
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −4.24264 −0.338600 −0.169300 0.985565i \(-0.554151\pi\)
−0.169300 + 0.985565i \(0.554151\pi\)
\(158\) 16.0000 1.27289
\(159\) −11.3137 −0.897235
\(160\) −4.24264 −0.335410
\(161\) 0 0
\(162\) −5.00000 −0.392837
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 11.3137 0.883452
\(165\) −6.00000 −0.467099
\(166\) 16.9706 1.31717
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −24.0000 −1.84072
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 11.3137 0.860165 0.430083 0.902790i \(-0.358484\pi\)
0.430083 + 0.902790i \(0.358484\pi\)
\(174\) −2.82843 −0.214423
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 2.00000 0.150329
\(178\) 7.07107 0.529999
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 4.24264 0.316228
\(181\) 7.07107 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 6.00000 0.442326
\(185\) 42.4264 3.11925
\(186\) −2.00000 −0.146647
\(187\) −5.65685 −0.413670
\(188\) −4.24264 −0.309426
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.41421 −0.102062
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 9.89949 0.710742
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 1.00000 0.0710669
\(199\) 1.41421 0.100251 0.0501255 0.998743i \(-0.484038\pi\)
0.0501255 + 0.998743i \(0.484038\pi\)
\(200\) 13.0000 0.919239
\(201\) −2.82843 −0.199502
\(202\) −5.65685 −0.398015
\(203\) 0 0
\(204\) −8.00000 −0.560112
\(205\) −48.0000 −3.35247
\(206\) −18.3848 −1.28093
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 8.00000 0.549442
\(213\) 2.82843 0.193801
\(214\) 16.0000 1.09374
\(215\) 33.9411 2.31477
\(216\) 5.65685 0.384900
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 12.0000 0.810885
\(220\) 4.24264 0.286039
\(221\) 0 0
\(222\) 14.1421 0.949158
\(223\) 21.2132 1.42054 0.710271 0.703929i \(-0.248573\pi\)
0.710271 + 0.703929i \(0.248573\pi\)
\(224\) 0 0
\(225\) −13.0000 −0.866667
\(226\) −2.00000 −0.133038
\(227\) 14.1421 0.938647 0.469323 0.883026i \(-0.344498\pi\)
0.469323 + 0.883026i \(0.344498\pi\)
\(228\) 0 0
\(229\) −9.89949 −0.654177 −0.327089 0.944994i \(-0.606068\pi\)
−0.327089 + 0.944994i \(0.606068\pi\)
\(230\) −25.4558 −1.67851
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) −1.41421 −0.0920575
\(237\) −22.6274 −1.46981
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 6.00000 0.387298
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 1.00000 0.0642824
\(243\) −9.89949 −0.635053
\(244\) 2.82843 0.181071
\(245\) 0 0
\(246\) −16.0000 −1.02012
\(247\) 0 0
\(248\) 1.41421 0.0898027
\(249\) −24.0000 −1.52094
\(250\) −33.9411 −2.14663
\(251\) −18.3848 −1.16044 −0.580218 0.814461i \(-0.697033\pi\)
−0.580218 + 0.814461i \(0.697033\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 16.0000 1.00393
\(255\) 33.9411 2.12548
\(256\) 1.00000 0.0625000
\(257\) −1.41421 −0.0882162 −0.0441081 0.999027i \(-0.514045\pi\)
−0.0441081 + 0.999027i \(0.514045\pi\)
\(258\) 11.3137 0.704361
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 19.7990 1.22319
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 1.41421 0.0870388
\(265\) −33.9411 −2.08499
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 2.00000 0.122169
\(269\) −18.3848 −1.12094 −0.560470 0.828175i \(-0.689379\pi\)
−0.560470 + 0.828175i \(0.689379\pi\)
\(270\) −24.0000 −1.46059
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) 5.65685 0.342997
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −13.0000 −0.783929
\(276\) −8.48528 −0.510754
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −11.3137 −0.678551
\(279\) −1.41421 −0.0846668
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 6.00000 0.357295
\(283\) 19.7990 1.17693 0.588464 0.808523i \(-0.299733\pi\)
0.588464 + 0.808523i \(0.299733\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 15.0000 0.882353
\(290\) −8.48528 −0.498273
\(291\) −14.0000 −0.820695
\(292\) −8.48528 −0.496564
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) −10.0000 −0.581238
\(297\) −5.65685 −0.328244
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) −18.3848 −1.06145
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) −5.65685 −0.323381
\(307\) 25.4558 1.45284 0.726421 0.687250i \(-0.241182\pi\)
0.726421 + 0.687250i \(0.241182\pi\)
\(308\) 0 0
\(309\) 26.0000 1.47909
\(310\) −6.00000 −0.340777
\(311\) −18.3848 −1.04251 −0.521253 0.853402i \(-0.674535\pi\)
−0.521253 + 0.853402i \(0.674535\pi\)
\(312\) 0 0
\(313\) 12.7279 0.719425 0.359712 0.933063i \(-0.382875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(314\) −4.24264 −0.239426
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −11.3137 −0.634441
\(319\) −2.00000 −0.111979
\(320\) −4.24264 −0.237171
\(321\) −22.6274 −1.26294
\(322\) 0 0
\(323\) 0 0
\(324\) −5.00000 −0.277778
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 2.82843 0.156412
\(328\) 11.3137 0.624695
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 16.9706 0.931381
\(333\) 10.0000 0.547997
\(334\) −5.65685 −0.309529
\(335\) −8.48528 −0.463600
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −13.0000 −0.707107
\(339\) 2.82843 0.153619
\(340\) −24.0000 −1.30158
\(341\) −1.41421 −0.0765840
\(342\) 0 0
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 36.0000 1.93817
\(346\) 11.3137 0.608229
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −2.82843 −0.151620
\(349\) −14.1421 −0.757011 −0.378506 0.925599i \(-0.623562\pi\)
−0.378506 + 0.925599i \(0.623562\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −1.41421 −0.0752710 −0.0376355 0.999292i \(-0.511983\pi\)
−0.0376355 + 0.999292i \(0.511983\pi\)
\(354\) 2.00000 0.106299
\(355\) 8.48528 0.450352
\(356\) 7.07107 0.374766
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 4.24264 0.223607
\(361\) −19.0000 −1.00000
\(362\) 7.07107 0.371647
\(363\) −1.41421 −0.0742270
\(364\) 0 0
\(365\) 36.0000 1.88433
\(366\) −4.00000 −0.209083
\(367\) 21.2132 1.10732 0.553660 0.832743i \(-0.313231\pi\)
0.553660 + 0.832743i \(0.313231\pi\)
\(368\) 6.00000 0.312772
\(369\) −11.3137 −0.588968
\(370\) 42.4264 2.20564
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −5.65685 −0.292509
\(375\) 48.0000 2.47871
\(376\) −4.24264 −0.218797
\(377\) 0 0
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) −22.6274 −1.15924
\(382\) 16.0000 0.818631
\(383\) −15.5563 −0.794892 −0.397446 0.917625i \(-0.630103\pi\)
−0.397446 + 0.917625i \(0.630103\pi\)
\(384\) −1.41421 −0.0721688
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) 8.00000 0.406663
\(388\) 9.89949 0.502571
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 33.9411 1.71648
\(392\) 0 0
\(393\) −28.0000 −1.41241
\(394\) 22.0000 1.10834
\(395\) −67.8823 −3.41553
\(396\) 1.00000 0.0502519
\(397\) −12.7279 −0.638796 −0.319398 0.947621i \(-0.603481\pi\)
−0.319398 + 0.947621i \(0.603481\pi\)
\(398\) 1.41421 0.0708881
\(399\) 0 0
\(400\) 13.0000 0.650000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −2.82843 −0.141069
\(403\) 0 0
\(404\) −5.65685 −0.281439
\(405\) 21.2132 1.05409
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) −8.00000 −0.396059
\(409\) 2.82843 0.139857 0.0699284 0.997552i \(-0.477723\pi\)
0.0699284 + 0.997552i \(0.477723\pi\)
\(410\) −48.0000 −2.37055
\(411\) 25.4558 1.25564
\(412\) −18.3848 −0.905753
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) −72.0000 −3.53434
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 8.00000 0.389434
\(423\) 4.24264 0.206284
\(424\) 8.00000 0.388514
\(425\) 73.5391 3.56717
\(426\) 2.82843 0.137038
\(427\) 0 0
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 33.9411 1.63679
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 5.65685 0.272166
\(433\) 12.7279 0.611665 0.305832 0.952085i \(-0.401065\pi\)
0.305832 + 0.952085i \(0.401065\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 12.0000 0.573382
\(439\) 25.4558 1.21494 0.607471 0.794342i \(-0.292184\pi\)
0.607471 + 0.794342i \(0.292184\pi\)
\(440\) 4.24264 0.202260
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 14.1421 0.671156
\(445\) −30.0000 −1.42214
\(446\) 21.2132 1.00447
\(447\) 14.1421 0.668900
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) −13.0000 −0.612826
\(451\) −11.3137 −0.532742
\(452\) −2.00000 −0.0940721
\(453\) −5.65685 −0.265782
\(454\) 14.1421 0.663723
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −9.89949 −0.462573
\(459\) 32.0000 1.49363
\(460\) −25.4558 −1.18688
\(461\) −2.82843 −0.131733 −0.0658665 0.997828i \(-0.520981\pi\)
−0.0658665 + 0.997828i \(0.520981\pi\)
\(462\) 0 0
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 2.00000 0.0928477
\(465\) 8.48528 0.393496
\(466\) 14.0000 0.648537
\(467\) 4.24264 0.196326 0.0981630 0.995170i \(-0.468703\pi\)
0.0981630 + 0.995170i \(0.468703\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) 6.00000 0.276465
\(472\) −1.41421 −0.0650945
\(473\) 8.00000 0.367840
\(474\) −22.6274 −1.03931
\(475\) 0 0
\(476\) 0 0
\(477\) −8.00000 −0.366295
\(478\) −12.0000 −0.548867
\(479\) −2.82843 −0.129234 −0.0646171 0.997910i \(-0.520583\pi\)
−0.0646171 + 0.997910i \(0.520583\pi\)
\(480\) 6.00000 0.273861
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −42.0000 −1.90712
\(486\) −9.89949 −0.449050
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 2.82843 0.128037
\(489\) 14.1421 0.639529
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −16.0000 −0.721336
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) −4.24264 −0.190693
\(496\) 1.41421 0.0635001
\(497\) 0 0
\(498\) −24.0000 −1.07547
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) −33.9411 −1.51789
\(501\) 8.00000 0.357414
\(502\) −18.3848 −0.820553
\(503\) −31.1127 −1.38725 −0.693623 0.720338i \(-0.743987\pi\)
−0.693623 + 0.720338i \(0.743987\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) −6.00000 −0.266733
\(507\) 18.3848 0.816497
\(508\) 16.0000 0.709885
\(509\) −12.7279 −0.564155 −0.282078 0.959392i \(-0.591024\pi\)
−0.282078 + 0.959392i \(0.591024\pi\)
\(510\) 33.9411 1.50294
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −1.41421 −0.0623783
\(515\) 78.0000 3.43709
\(516\) 11.3137 0.498058
\(517\) 4.24264 0.186591
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) −1.41421 −0.0619578 −0.0309789 0.999520i \(-0.509862\pi\)
−0.0309789 + 0.999520i \(0.509862\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −8.48528 −0.371035 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(524\) 19.7990 0.864923
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 8.00000 0.348485
\(528\) 1.41421 0.0615457
\(529\) 13.0000 0.565217
\(530\) −33.9411 −1.47431
\(531\) 1.41421 0.0613716
\(532\) 0 0
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) −67.8823 −2.93481
\(536\) 2.00000 0.0863868
\(537\) −16.9706 −0.732334
\(538\) −18.3848 −0.792624
\(539\) 0 0
\(540\) −24.0000 −1.03280
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 8.48528 0.364474
\(543\) −10.0000 −0.429141
\(544\) 5.65685 0.242536
\(545\) 8.48528 0.363470
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −18.0000 −0.768922
\(549\) −2.82843 −0.120714
\(550\) −13.0000 −0.554322
\(551\) 0 0
\(552\) −8.48528 −0.361158
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −60.0000 −2.54686
\(556\) −11.3137 −0.479808
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −1.41421 −0.0598684
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −14.0000 −0.590554
\(563\) −22.6274 −0.953632 −0.476816 0.879003i \(-0.658209\pi\)
−0.476816 + 0.879003i \(0.658209\pi\)
\(564\) 6.00000 0.252646
\(565\) 8.48528 0.356978
\(566\) 19.7990 0.832214
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) −22.6274 −0.945274
\(574\) 0 0
\(575\) 78.0000 3.25282
\(576\) −1.00000 −0.0416667
\(577\) 7.07107 0.294372 0.147186 0.989109i \(-0.452978\pi\)
0.147186 + 0.989109i \(0.452978\pi\)
\(578\) 15.0000 0.623918
\(579\) 8.48528 0.352636
\(580\) −8.48528 −0.352332
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) −8.00000 −0.331326
\(584\) −8.48528 −0.351123
\(585\) 0 0
\(586\) 8.48528 0.350524
\(587\) 26.8701 1.10905 0.554523 0.832168i \(-0.312901\pi\)
0.554523 + 0.832168i \(0.312901\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 6.00000 0.247016
\(591\) −31.1127 −1.27981
\(592\) −10.0000 −0.410997
\(593\) −42.4264 −1.74224 −0.871122 0.491067i \(-0.836607\pi\)
−0.871122 + 0.491067i \(0.836607\pi\)
\(594\) −5.65685 −0.232104
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −2.00000 −0.0818546
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) −18.3848 −0.750555
\(601\) −5.65685 −0.230748 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 4.00000 0.162758
\(605\) −4.24264 −0.172488
\(606\) 8.00000 0.324978
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −12.0000 −0.485866
\(611\) 0 0
\(612\) −5.65685 −0.228665
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 25.4558 1.02731
\(615\) 67.8823 2.73728
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 26.0000 1.04587
\(619\) −9.89949 −0.397894 −0.198947 0.980010i \(-0.563752\pi\)
−0.198947 + 0.980010i \(0.563752\pi\)
\(620\) −6.00000 −0.240966
\(621\) 33.9411 1.36201
\(622\) −18.3848 −0.737162
\(623\) 0 0
\(624\) 0 0
\(625\) 79.0000 3.16000
\(626\) 12.7279 0.508710
\(627\) 0 0
\(628\) −4.24264 −0.169300
\(629\) −56.5685 −2.25554
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 16.0000 0.636446
\(633\) −11.3137 −0.449680
\(634\) 30.0000 1.19145
\(635\) −67.8823 −2.69382
\(636\) −11.3137 −0.448618
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) 2.00000 0.0791188
\(640\) −4.24264 −0.167705
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) −22.6274 −0.893033
\(643\) 38.1838 1.50582 0.752910 0.658123i \(-0.228649\pi\)
0.752910 + 0.658123i \(0.228649\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) 15.5563 0.611583 0.305792 0.952098i \(-0.401079\pi\)
0.305792 + 0.952098i \(0.401079\pi\)
\(648\) −5.00000 −0.196419
\(649\) 1.41421 0.0555127
\(650\) 0 0
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 2.82843 0.110600
\(655\) −84.0000 −3.28215
\(656\) 11.3137 0.441726
\(657\) 8.48528 0.331042
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −6.00000 −0.233550
\(661\) −12.7279 −0.495059 −0.247529 0.968880i \(-0.579619\pi\)
−0.247529 + 0.968880i \(0.579619\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 16.9706 0.658586
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 12.0000 0.464642
\(668\) −5.65685 −0.218870
\(669\) −30.0000 −1.15987
\(670\) −8.48528 −0.327815
\(671\) −2.82843 −0.109190
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 2.00000 0.0770371
\(675\) 73.5391 2.83052
\(676\) −13.0000 −0.500000
\(677\) 39.5980 1.52187 0.760937 0.648826i \(-0.224740\pi\)
0.760937 + 0.648826i \(0.224740\pi\)
\(678\) 2.82843 0.108625
\(679\) 0 0
\(680\) −24.0000 −0.920358
\(681\) −20.0000 −0.766402
\(682\) −1.41421 −0.0541530
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 76.3675 2.91785
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −8.00000 −0.304997
\(689\) 0 0
\(690\) 36.0000 1.37050
\(691\) −15.5563 −0.591791 −0.295896 0.955220i \(-0.595618\pi\)
−0.295896 + 0.955220i \(0.595618\pi\)
\(692\) 11.3137 0.430083
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 48.0000 1.82074
\(696\) −2.82843 −0.107211
\(697\) 64.0000 2.42417
\(698\) −14.1421 −0.535288
\(699\) −19.7990 −0.748867
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) −25.4558 −0.958723
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) 2.00000 0.0751646
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 8.48528 0.318447
\(711\) −16.0000 −0.600047
\(712\) 7.07107 0.264999
\(713\) 8.48528 0.317776
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 16.9706 0.633777
\(718\) −12.0000 −0.447836
\(719\) −18.3848 −0.685636 −0.342818 0.939402i \(-0.611381\pi\)
−0.342818 + 0.939402i \(0.611381\pi\)
\(720\) 4.24264 0.158114
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) 7.07107 0.262794
\(725\) 26.0000 0.965616
\(726\) −1.41421 −0.0524864
\(727\) −12.7279 −0.472052 −0.236026 0.971747i \(-0.575845\pi\)
−0.236026 + 0.971747i \(0.575845\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 36.0000 1.33242
\(731\) −45.2548 −1.67381
\(732\) −4.00000 −0.147844
\(733\) −14.1421 −0.522352 −0.261176 0.965291i \(-0.584110\pi\)
−0.261176 + 0.965291i \(0.584110\pi\)
\(734\) 21.2132 0.782994
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −2.00000 −0.0736709
\(738\) −11.3137 −0.416463
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 42.4264 1.55963
\(741\) 0 0
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 42.4264 1.55438
\(746\) −10.0000 −0.366126
\(747\) −16.9706 −0.620920
\(748\) −5.65685 −0.206835
\(749\) 0 0
\(750\) 48.0000 1.75271
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) −4.24264 −0.154713
\(753\) 26.0000 0.947493
\(754\) 0 0
\(755\) −16.9706 −0.617622
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) −6.00000 −0.217930
\(759\) 8.48528 0.307996
\(760\) 0 0
\(761\) 42.4264 1.53796 0.768978 0.639275i \(-0.220766\pi\)
0.768978 + 0.639275i \(0.220766\pi\)
\(762\) −22.6274 −0.819705
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) 24.0000 0.867722
\(766\) −15.5563 −0.562074
\(767\) 0 0
\(768\) −1.41421 −0.0510310
\(769\) −19.7990 −0.713970 −0.356985 0.934110i \(-0.616195\pi\)
−0.356985 + 0.934110i \(0.616195\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) −6.00000 −0.215945
\(773\) −1.41421 −0.0508657 −0.0254329 0.999677i \(-0.508096\pi\)
−0.0254329 + 0.999677i \(0.508096\pi\)
\(774\) 8.00000 0.287554
\(775\) 18.3848 0.660401
\(776\) 9.89949 0.355371
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 0 0
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 33.9411 1.21373
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) −28.0000 −0.998727
\(787\) 5.65685 0.201645 0.100823 0.994904i \(-0.467853\pi\)
0.100823 + 0.994904i \(0.467853\pi\)
\(788\) 22.0000 0.783718
\(789\) 11.3137 0.402779
\(790\) −67.8823 −2.41514
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −12.7279 −0.451697
\(795\) 48.0000 1.70238
\(796\) 1.41421 0.0501255
\(797\) −7.07107 −0.250470 −0.125235 0.992127i \(-0.539968\pi\)
−0.125235 + 0.992127i \(0.539968\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 13.0000 0.459619
\(801\) −7.07107 −0.249844
\(802\) 12.0000 0.423735
\(803\) 8.48528 0.299439
\(804\) −2.82843 −0.0997509
\(805\) 0 0
\(806\) 0 0
\(807\) 26.0000 0.915243
\(808\) −5.65685 −0.199007
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 21.2132 0.745356
\(811\) −36.7696 −1.29115 −0.645577 0.763695i \(-0.723383\pi\)
−0.645577 + 0.763695i \(0.723383\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 10.0000 0.350500
\(815\) 42.4264 1.48613
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 2.82843 0.0988936
\(819\) 0 0
\(820\) −48.0000 −1.67623
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) 25.4558 0.887875
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −18.3848 −0.640464
\(825\) 18.3848 0.640076
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) −6.00000 −0.208514
\(829\) 4.24264 0.147353 0.0736765 0.997282i \(-0.476527\pi\)
0.0736765 + 0.997282i \(0.476527\pi\)
\(830\) −72.0000 −2.49916
\(831\) 2.82843 0.0981170
\(832\) 0 0
\(833\) 0 0
\(834\) 16.0000 0.554035
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −24.0416 −0.830504
\(839\) −1.41421 −0.0488241 −0.0244120 0.999702i \(-0.507771\pi\)
−0.0244120 + 0.999702i \(0.507771\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −20.0000 −0.689246
\(843\) 19.7990 0.681913
\(844\) 8.00000 0.275371
\(845\) 55.1543 1.89737
\(846\) 4.24264 0.145865
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) −28.0000 −0.960958
\(850\) 73.5391 2.52237
\(851\) −60.0000 −2.05677
\(852\) 2.82843 0.0969003
\(853\) −28.2843 −0.968435 −0.484218 0.874948i \(-0.660896\pi\)
−0.484218 + 0.874948i \(0.660896\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 25.4558 0.869555 0.434778 0.900538i \(-0.356827\pi\)
0.434778 + 0.900538i \(0.356827\pi\)
\(858\) 0 0
\(859\) −12.7279 −0.434271 −0.217136 0.976141i \(-0.569671\pi\)
−0.217136 + 0.976141i \(0.569671\pi\)
\(860\) 33.9411 1.15738
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 5.65685 0.192450
\(865\) −48.0000 −1.63205
\(866\) 12.7279 0.432512
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 12.0000 0.406838
\(871\) 0 0
\(872\) −2.00000 −0.0677285
\(873\) −9.89949 −0.335047
\(874\) 0 0
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 25.4558 0.859093
\(879\) −12.0000 −0.404750
\(880\) 4.24264 0.143019
\(881\) 29.6985 1.00057 0.500284 0.865862i \(-0.333229\pi\)
0.500284 + 0.865862i \(0.333229\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) −8.48528 −0.285230
\(886\) 36.0000 1.20944
\(887\) −36.7696 −1.23460 −0.617300 0.786728i \(-0.711774\pi\)
−0.617300 + 0.786728i \(0.711774\pi\)
\(888\) 14.1421 0.474579
\(889\) 0 0
\(890\) −30.0000 −1.00560
\(891\) 5.00000 0.167506
\(892\) 21.2132 0.710271
\(893\) 0 0
\(894\) 14.1421 0.472984
\(895\) −50.9117 −1.70179
\(896\) 0 0
\(897\) 0 0
\(898\) −20.0000 −0.667409
\(899\) 2.82843 0.0943333
\(900\) −13.0000 −0.433333
\(901\) 45.2548 1.50766
\(902\) −11.3137 −0.376705
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −30.0000 −0.997234
\(906\) −5.65685 −0.187936
\(907\) −14.0000 −0.464862 −0.232431 0.972613i \(-0.574668\pi\)
−0.232431 + 0.972613i \(0.574668\pi\)
\(908\) 14.1421 0.469323
\(909\) 5.65685 0.187626
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) −16.9706 −0.561644
\(914\) 14.0000 0.463079
\(915\) 16.9706 0.561029
\(916\) −9.89949 −0.327089
\(917\) 0 0
\(918\) 32.0000 1.05616
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −25.4558 −0.839254
\(921\) −36.0000 −1.18624
\(922\) −2.82843 −0.0931493
\(923\) 0 0
\(924\) 0 0
\(925\) −130.000 −4.27437
\(926\) 26.0000 0.854413
\(927\) 18.3848 0.603835
\(928\) 2.00000 0.0656532
\(929\) 32.5269 1.06717 0.533587 0.845745i \(-0.320844\pi\)
0.533587 + 0.845745i \(0.320844\pi\)
\(930\) 8.48528 0.278243
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 26.0000 0.851202
\(934\) 4.24264 0.138823
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −25.4558 −0.831606 −0.415803 0.909455i \(-0.636499\pi\)
−0.415803 + 0.909455i \(0.636499\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 18.0000 0.587095
\(941\) −31.1127 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(942\) 6.00000 0.195491
\(943\) 67.8823 2.21055
\(944\) −1.41421 −0.0460287
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −22.6274 −0.734904
\(949\) 0 0
\(950\) 0 0
\(951\) −42.4264 −1.37577
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) −8.00000 −0.259010
\(955\) −67.8823 −2.19662
\(956\) −12.0000 −0.388108
\(957\) 2.82843 0.0914301
\(958\) −2.82843 −0.0913823
\(959\) 0 0
\(960\) 6.00000 0.193649
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) −16.0000 −0.515593
\(964\) 0 0
\(965\) 25.4558 0.819453
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −42.0000 −1.34854
\(971\) −52.3259 −1.67922 −0.839609 0.543191i \(-0.817216\pi\)
−0.839609 + 0.543191i \(0.817216\pi\)
\(972\) −9.89949 −0.317526
\(973\) 0 0
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 2.82843 0.0905357
\(977\) 52.0000 1.66363 0.831814 0.555055i \(-0.187303\pi\)
0.831814 + 0.555055i \(0.187303\pi\)
\(978\) 14.1421 0.452216
\(979\) −7.07107 −0.225992
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) −1.41421 −0.0451064 −0.0225532 0.999746i \(-0.507180\pi\)
−0.0225532 + 0.999746i \(0.507180\pi\)
\(984\) −16.0000 −0.510061
\(985\) −93.3381 −2.97400
\(986\) 11.3137 0.360302
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) −4.24264 −0.134840
\(991\) −46.0000 −1.46124 −0.730619 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) 1.41421 0.0449013
\(993\) −28.2843 −0.897574
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) −24.0000 −0.760469
\(997\) −16.9706 −0.537463 −0.268732 0.963215i \(-0.586604\pi\)
−0.268732 + 0.963215i \(0.586604\pi\)
\(998\) 6.00000 0.189927
\(999\) −56.5685 −1.78975
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.u.1.1 2
3.2 odd 2 9702.2.a.cp.1.2 2
4.3 odd 2 8624.2.a.bs.1.2 2
7.2 even 3 1078.2.e.p.67.2 4
7.3 odd 6 1078.2.e.p.177.1 4
7.4 even 3 1078.2.e.p.177.2 4
7.5 odd 6 1078.2.e.p.67.1 4
7.6 odd 2 inner 1078.2.a.u.1.2 yes 2
21.20 even 2 9702.2.a.cp.1.1 2
28.27 even 2 8624.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.u.1.1 2 1.1 even 1 trivial
1078.2.a.u.1.2 yes 2 7.6 odd 2 inner
1078.2.e.p.67.1 4 7.5 odd 6
1078.2.e.p.67.2 4 7.2 even 3
1078.2.e.p.177.1 4 7.3 odd 6
1078.2.e.p.177.2 4 7.4 even 3
8624.2.a.bs.1.1 2 28.27 even 2
8624.2.a.bs.1.2 2 4.3 odd 2
9702.2.a.cp.1.1 2 21.20 even 2
9702.2.a.cp.1.2 2 3.2 odd 2