Properties

Label 2960.2.a.o.1.1
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.00000 q^{5} -1.37228 q^{7} +1.00000 q^{9} +3.37228 q^{11} -4.74456 q^{13} -2.00000 q^{15} -5.37228 q^{17} +2.00000 q^{19} +2.74456 q^{21} -6.74456 q^{23} +1.00000 q^{25} +4.00000 q^{27} +8.11684 q^{29} +2.62772 q^{31} -6.74456 q^{33} -1.37228 q^{35} +1.00000 q^{37} +9.48913 q^{39} +5.37228 q^{41} -7.37228 q^{43} +1.00000 q^{45} +8.74456 q^{47} -5.11684 q^{49} +10.7446 q^{51} +1.37228 q^{53} +3.37228 q^{55} -4.00000 q^{57} -12.7446 q^{59} -5.37228 q^{61} -1.37228 q^{63} -4.74456 q^{65} -4.74456 q^{67} +13.4891 q^{69} +6.74456 q^{71} +8.74456 q^{73} -2.00000 q^{75} -4.62772 q^{77} +4.74456 q^{79} -11.0000 q^{81} +0.744563 q^{83} -5.37228 q^{85} -16.2337 q^{87} +10.0000 q^{89} +6.51087 q^{91} -5.25544 q^{93} +2.00000 q^{95} -0.116844 q^{97} +3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{5} + 3 q^{7} + 2 q^{9} + q^{11} + 2 q^{13} - 4 q^{15} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{25} + 8 q^{27} - q^{29} + 11 q^{31} - 2 q^{33} + 3 q^{35} + 2 q^{37} - 4 q^{39} + 5 q^{41} - 9 q^{43} + 2 q^{45} + 6 q^{47} + 7 q^{49} + 10 q^{51} - 3 q^{53} + q^{55} - 8 q^{57} - 14 q^{59} - 5 q^{61} + 3 q^{63} + 2 q^{65} + 2 q^{67} + 4 q^{69} + 2 q^{71} + 6 q^{73} - 4 q^{75} - 15 q^{77} - 2 q^{79} - 22 q^{81} - 10 q^{83} - 5 q^{85} + 2 q^{87} + 20 q^{89} + 36 q^{91} - 22 q^{93} + 4 q^{95} + 17 q^{97} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.37228 −0.518674 −0.259337 0.965787i \(-0.583504\pi\)
−0.259337 + 0.965787i \(0.583504\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.37228 1.01678 0.508391 0.861127i \(-0.330241\pi\)
0.508391 + 0.861127i \(0.330241\pi\)
\(12\) 0 0
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.74456 0.598913
\(22\) 0 0
\(23\) −6.74456 −1.40634 −0.703169 0.711022i \(-0.748232\pi\)
−0.703169 + 0.711022i \(0.748232\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 8.11684 1.50726 0.753630 0.657299i \(-0.228301\pi\)
0.753630 + 0.657299i \(0.228301\pi\)
\(30\) 0 0
\(31\) 2.62772 0.471952 0.235976 0.971759i \(-0.424171\pi\)
0.235976 + 0.971759i \(0.424171\pi\)
\(32\) 0 0
\(33\) −6.74456 −1.17408
\(34\) 0 0
\(35\) −1.37228 −0.231958
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 9.48913 1.51948
\(40\) 0 0
\(41\) 5.37228 0.839009 0.419505 0.907753i \(-0.362204\pi\)
0.419505 + 0.907753i \(0.362204\pi\)
\(42\) 0 0
\(43\) −7.37228 −1.12426 −0.562131 0.827048i \(-0.690018\pi\)
−0.562131 + 0.827048i \(0.690018\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 8.74456 1.27553 0.637763 0.770233i \(-0.279860\pi\)
0.637763 + 0.770233i \(0.279860\pi\)
\(48\) 0 0
\(49\) −5.11684 −0.730978
\(50\) 0 0
\(51\) 10.7446 1.50454
\(52\) 0 0
\(53\) 1.37228 0.188497 0.0942487 0.995549i \(-0.469955\pi\)
0.0942487 + 0.995549i \(0.469955\pi\)
\(54\) 0 0
\(55\) 3.37228 0.454718
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −12.7446 −1.65920 −0.829600 0.558358i \(-0.811432\pi\)
−0.829600 + 0.558358i \(0.811432\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 0 0
\(63\) −1.37228 −0.172891
\(64\) 0 0
\(65\) −4.74456 −0.588491
\(66\) 0 0
\(67\) −4.74456 −0.579641 −0.289820 0.957081i \(-0.593596\pi\)
−0.289820 + 0.957081i \(0.593596\pi\)
\(68\) 0 0
\(69\) 13.4891 1.62390
\(70\) 0 0
\(71\) 6.74456 0.800432 0.400216 0.916421i \(-0.368935\pi\)
0.400216 + 0.916421i \(0.368935\pi\)
\(72\) 0 0
\(73\) 8.74456 1.02347 0.511737 0.859142i \(-0.329002\pi\)
0.511737 + 0.859142i \(0.329002\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 0 0
\(77\) −4.62772 −0.527377
\(78\) 0 0
\(79\) 4.74456 0.533805 0.266903 0.963724i \(-0.414000\pi\)
0.266903 + 0.963724i \(0.414000\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 0.744563 0.0817264 0.0408632 0.999165i \(-0.486989\pi\)
0.0408632 + 0.999165i \(0.486989\pi\)
\(84\) 0 0
\(85\) −5.37228 −0.582706
\(86\) 0 0
\(87\) −16.2337 −1.74043
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 6.51087 0.682525
\(92\) 0 0
\(93\) −5.25544 −0.544963
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −0.116844 −0.0118637 −0.00593185 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) 0 0
\(99\) 3.37228 0.338927
\(100\) 0 0
\(101\) −11.4891 −1.14321 −0.571605 0.820529i \(-0.693679\pi\)
−0.571605 + 0.820529i \(0.693679\pi\)
\(102\) 0 0
\(103\) 9.48913 0.934991 0.467496 0.883995i \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) 0 0
\(105\) 2.74456 0.267842
\(106\) 0 0
\(107\) 3.48913 0.337306 0.168653 0.985675i \(-0.446058\pi\)
0.168653 + 0.985675i \(0.446058\pi\)
\(108\) 0 0
\(109\) 0.116844 0.0111916 0.00559581 0.999984i \(-0.498219\pi\)
0.00559581 + 0.999984i \(0.498219\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 17.3723 1.63425 0.817123 0.576463i \(-0.195567\pi\)
0.817123 + 0.576463i \(0.195567\pi\)
\(114\) 0 0
\(115\) −6.74456 −0.628934
\(116\) 0 0
\(117\) −4.74456 −0.438635
\(118\) 0 0
\(119\) 7.37228 0.675816
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 0 0
\(123\) −10.7446 −0.968805
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.7446 1.48584 0.742920 0.669380i \(-0.233440\pi\)
0.742920 + 0.669380i \(0.233440\pi\)
\(128\) 0 0
\(129\) 14.7446 1.29819
\(130\) 0 0
\(131\) 3.25544 0.284429 0.142214 0.989836i \(-0.454578\pi\)
0.142214 + 0.989836i \(0.454578\pi\)
\(132\) 0 0
\(133\) −2.74456 −0.237984
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 16.7446 1.43058 0.715292 0.698825i \(-0.246294\pi\)
0.715292 + 0.698825i \(0.246294\pi\)
\(138\) 0 0
\(139\) 1.88316 0.159727 0.0798636 0.996806i \(-0.474552\pi\)
0.0798636 + 0.996806i \(0.474552\pi\)
\(140\) 0 0
\(141\) −17.4891 −1.47285
\(142\) 0 0
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 8.11684 0.674067
\(146\) 0 0
\(147\) 10.2337 0.844060
\(148\) 0 0
\(149\) −11.4891 −0.941226 −0.470613 0.882340i \(-0.655967\pi\)
−0.470613 + 0.882340i \(0.655967\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) −5.37228 −0.434323
\(154\) 0 0
\(155\) 2.62772 0.211063
\(156\) 0 0
\(157\) 17.3723 1.38646 0.693229 0.720717i \(-0.256187\pi\)
0.693229 + 0.720717i \(0.256187\pi\)
\(158\) 0 0
\(159\) −2.74456 −0.217658
\(160\) 0 0
\(161\) 9.25544 0.729431
\(162\) 0 0
\(163\) −19.3723 −1.51735 −0.758677 0.651467i \(-0.774154\pi\)
−0.758677 + 0.651467i \(0.774154\pi\)
\(164\) 0 0
\(165\) −6.74456 −0.525063
\(166\) 0 0
\(167\) −1.48913 −0.115232 −0.0576160 0.998339i \(-0.518350\pi\)
−0.0576160 + 0.998339i \(0.518350\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 22.8614 1.73812 0.869060 0.494706i \(-0.164724\pi\)
0.869060 + 0.494706i \(0.164724\pi\)
\(174\) 0 0
\(175\) −1.37228 −0.103735
\(176\) 0 0
\(177\) 25.4891 1.91588
\(178\) 0 0
\(179\) 26.2337 1.96080 0.980399 0.197023i \(-0.0631272\pi\)
0.980399 + 0.197023i \(0.0631272\pi\)
\(180\) 0 0
\(181\) 7.48913 0.556662 0.278331 0.960485i \(-0.410219\pi\)
0.278331 + 0.960485i \(0.410219\pi\)
\(182\) 0 0
\(183\) 10.7446 0.794261
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −18.1168 −1.32483
\(188\) 0 0
\(189\) −5.48913 −0.399275
\(190\) 0 0
\(191\) −18.6277 −1.34785 −0.673927 0.738798i \(-0.735394\pi\)
−0.673927 + 0.738798i \(0.735394\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 9.48913 0.679530
\(196\) 0 0
\(197\) 27.4891 1.95852 0.979260 0.202610i \(-0.0649423\pi\)
0.979260 + 0.202610i \(0.0649423\pi\)
\(198\) 0 0
\(199\) 11.2554 0.797877 0.398938 0.916978i \(-0.369379\pi\)
0.398938 + 0.916978i \(0.369379\pi\)
\(200\) 0 0
\(201\) 9.48913 0.669311
\(202\) 0 0
\(203\) −11.1386 −0.781776
\(204\) 0 0
\(205\) 5.37228 0.375216
\(206\) 0 0
\(207\) −6.74456 −0.468780
\(208\) 0 0
\(209\) 6.74456 0.466531
\(210\) 0 0
\(211\) 14.1168 0.971844 0.485922 0.874002i \(-0.338484\pi\)
0.485922 + 0.874002i \(0.338484\pi\)
\(212\) 0 0
\(213\) −13.4891 −0.924260
\(214\) 0 0
\(215\) −7.37228 −0.502785
\(216\) 0 0
\(217\) −3.60597 −0.244789
\(218\) 0 0
\(219\) −17.4891 −1.18181
\(220\) 0 0
\(221\) 25.4891 1.71458
\(222\) 0 0
\(223\) 1.37228 0.0918948 0.0459474 0.998944i \(-0.485369\pi\)
0.0459474 + 0.998944i \(0.485369\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 11.3723 0.754805 0.377402 0.926049i \(-0.376817\pi\)
0.377402 + 0.926049i \(0.376817\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 9.25544 0.608963
\(232\) 0 0
\(233\) 6.23369 0.408382 0.204191 0.978931i \(-0.434544\pi\)
0.204191 + 0.978931i \(0.434544\pi\)
\(234\) 0 0
\(235\) 8.74456 0.570432
\(236\) 0 0
\(237\) −9.48913 −0.616385
\(238\) 0 0
\(239\) −17.6060 −1.13884 −0.569418 0.822048i \(-0.692831\pi\)
−0.569418 + 0.822048i \(0.692831\pi\)
\(240\) 0 0
\(241\) −10.2337 −0.659210 −0.329605 0.944119i \(-0.606916\pi\)
−0.329605 + 0.944119i \(0.606916\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −5.11684 −0.326903
\(246\) 0 0
\(247\) −9.48913 −0.603779
\(248\) 0 0
\(249\) −1.48913 −0.0943695
\(250\) 0 0
\(251\) −11.4891 −0.725187 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(252\) 0 0
\(253\) −22.7446 −1.42994
\(254\) 0 0
\(255\) 10.7446 0.672851
\(256\) 0 0
\(257\) −20.9783 −1.30859 −0.654294 0.756241i \(-0.727034\pi\)
−0.654294 + 0.756241i \(0.727034\pi\)
\(258\) 0 0
\(259\) −1.37228 −0.0852694
\(260\) 0 0
\(261\) 8.11684 0.502420
\(262\) 0 0
\(263\) −0.116844 −0.00720491 −0.00360245 0.999994i \(-0.501147\pi\)
−0.00360245 + 0.999994i \(0.501147\pi\)
\(264\) 0 0
\(265\) 1.37228 0.0842986
\(266\) 0 0
\(267\) −20.0000 −1.22398
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 6.74456 0.409703 0.204852 0.978793i \(-0.434329\pi\)
0.204852 + 0.978793i \(0.434329\pi\)
\(272\) 0 0
\(273\) −13.0217 −0.788112
\(274\) 0 0
\(275\) 3.37228 0.203356
\(276\) 0 0
\(277\) 0.744563 0.0447364 0.0223682 0.999750i \(-0.492879\pi\)
0.0223682 + 0.999750i \(0.492879\pi\)
\(278\) 0 0
\(279\) 2.62772 0.157317
\(280\) 0 0
\(281\) 24.7446 1.47614 0.738068 0.674726i \(-0.235738\pi\)
0.738068 + 0.674726i \(0.235738\pi\)
\(282\) 0 0
\(283\) −5.48913 −0.326295 −0.163147 0.986602i \(-0.552165\pi\)
−0.163147 + 0.986602i \(0.552165\pi\)
\(284\) 0 0
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) −7.37228 −0.435172
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) 0.233688 0.0136990
\(292\) 0 0
\(293\) −18.8614 −1.10190 −0.550948 0.834540i \(-0.685734\pi\)
−0.550948 + 0.834540i \(0.685734\pi\)
\(294\) 0 0
\(295\) −12.7446 −0.742017
\(296\) 0 0
\(297\) 13.4891 0.782718
\(298\) 0 0
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) 10.1168 0.583125
\(302\) 0 0
\(303\) 22.9783 1.32007
\(304\) 0 0
\(305\) −5.37228 −0.307616
\(306\) 0 0
\(307\) 23.4891 1.34060 0.670298 0.742092i \(-0.266166\pi\)
0.670298 + 0.742092i \(0.266166\pi\)
\(308\) 0 0
\(309\) −18.9783 −1.07963
\(310\) 0 0
\(311\) 1.37228 0.0778149 0.0389075 0.999243i \(-0.487612\pi\)
0.0389075 + 0.999243i \(0.487612\pi\)
\(312\) 0 0
\(313\) −3.48913 −0.197217 −0.0986085 0.995126i \(-0.531439\pi\)
−0.0986085 + 0.995126i \(0.531439\pi\)
\(314\) 0 0
\(315\) −1.37228 −0.0773193
\(316\) 0 0
\(317\) 25.3723 1.42505 0.712525 0.701647i \(-0.247552\pi\)
0.712525 + 0.701647i \(0.247552\pi\)
\(318\) 0 0
\(319\) 27.3723 1.53255
\(320\) 0 0
\(321\) −6.97825 −0.389488
\(322\) 0 0
\(323\) −10.7446 −0.597843
\(324\) 0 0
\(325\) −4.74456 −0.263181
\(326\) 0 0
\(327\) −0.233688 −0.0129230
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −23.4891 −1.29108 −0.645540 0.763727i \(-0.723367\pi\)
−0.645540 + 0.763727i \(0.723367\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) −4.74456 −0.259223
\(336\) 0 0
\(337\) −7.25544 −0.395229 −0.197614 0.980280i \(-0.563319\pi\)
−0.197614 + 0.980280i \(0.563319\pi\)
\(338\) 0 0
\(339\) −34.7446 −1.88707
\(340\) 0 0
\(341\) 8.86141 0.479872
\(342\) 0 0
\(343\) 16.6277 0.897812
\(344\) 0 0
\(345\) 13.4891 0.726230
\(346\) 0 0
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −18.9783 −1.01298
\(352\) 0 0
\(353\) −22.8614 −1.21679 −0.608395 0.793634i \(-0.708186\pi\)
−0.608395 + 0.793634i \(0.708186\pi\)
\(354\) 0 0
\(355\) 6.74456 0.357964
\(356\) 0 0
\(357\) −14.7446 −0.780365
\(358\) 0 0
\(359\) 30.9783 1.63497 0.817485 0.575950i \(-0.195368\pi\)
0.817485 + 0.575950i \(0.195368\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −0.744563 −0.0390794
\(364\) 0 0
\(365\) 8.74456 0.457711
\(366\) 0 0
\(367\) −3.88316 −0.202699 −0.101350 0.994851i \(-0.532316\pi\)
−0.101350 + 0.994851i \(0.532316\pi\)
\(368\) 0 0
\(369\) 5.37228 0.279670
\(370\) 0 0
\(371\) −1.88316 −0.0977686
\(372\) 0 0
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) −38.5109 −1.98341
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −33.4891 −1.71570
\(382\) 0 0
\(383\) 13.4891 0.689262 0.344631 0.938738i \(-0.388004\pi\)
0.344631 + 0.938738i \(0.388004\pi\)
\(384\) 0 0
\(385\) −4.62772 −0.235850
\(386\) 0 0
\(387\) −7.37228 −0.374754
\(388\) 0 0
\(389\) 14.8614 0.753503 0.376752 0.926314i \(-0.377041\pi\)
0.376752 + 0.926314i \(0.377041\pi\)
\(390\) 0 0
\(391\) 36.2337 1.83242
\(392\) 0 0
\(393\) −6.51087 −0.328430
\(394\) 0 0
\(395\) 4.74456 0.238725
\(396\) 0 0
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) 0 0
\(399\) 5.48913 0.274800
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −12.4674 −0.621044
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 3.37228 0.167158
\(408\) 0 0
\(409\) 6.23369 0.308236 0.154118 0.988052i \(-0.450746\pi\)
0.154118 + 0.988052i \(0.450746\pi\)
\(410\) 0 0
\(411\) −33.4891 −1.65190
\(412\) 0 0
\(413\) 17.4891 0.860584
\(414\) 0 0
\(415\) 0.744563 0.0365491
\(416\) 0 0
\(417\) −3.76631 −0.184437
\(418\) 0 0
\(419\) −13.4891 −0.658987 −0.329493 0.944158i \(-0.606878\pi\)
−0.329493 + 0.944158i \(0.606878\pi\)
\(420\) 0 0
\(421\) −7.48913 −0.364998 −0.182499 0.983206i \(-0.558419\pi\)
−0.182499 + 0.983206i \(0.558419\pi\)
\(422\) 0 0
\(423\) 8.74456 0.425175
\(424\) 0 0
\(425\) −5.37228 −0.260594
\(426\) 0 0
\(427\) 7.37228 0.356770
\(428\) 0 0
\(429\) 32.0000 1.54497
\(430\) 0 0
\(431\) 1.37228 0.0661005 0.0330502 0.999454i \(-0.489478\pi\)
0.0330502 + 0.999454i \(0.489478\pi\)
\(432\) 0 0
\(433\) 12.9783 0.623695 0.311847 0.950132i \(-0.399052\pi\)
0.311847 + 0.950132i \(0.399052\pi\)
\(434\) 0 0
\(435\) −16.2337 −0.778346
\(436\) 0 0
\(437\) −13.4891 −0.645272
\(438\) 0 0
\(439\) 13.3723 0.638224 0.319112 0.947717i \(-0.396615\pi\)
0.319112 + 0.947717i \(0.396615\pi\)
\(440\) 0 0
\(441\) −5.11684 −0.243659
\(442\) 0 0
\(443\) 16.9783 0.806661 0.403331 0.915054i \(-0.367852\pi\)
0.403331 + 0.915054i \(0.367852\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 22.9783 1.08683
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 18.1168 0.853089
\(452\) 0 0
\(453\) −40.0000 −1.87936
\(454\) 0 0
\(455\) 6.51087 0.305235
\(456\) 0 0
\(457\) 18.8614 0.882299 0.441150 0.897434i \(-0.354571\pi\)
0.441150 + 0.897434i \(0.354571\pi\)
\(458\) 0 0
\(459\) −21.4891 −1.00303
\(460\) 0 0
\(461\) −39.0951 −1.82084 −0.910420 0.413685i \(-0.864241\pi\)
−0.910420 + 0.413685i \(0.864241\pi\)
\(462\) 0 0
\(463\) 5.48913 0.255101 0.127551 0.991832i \(-0.459288\pi\)
0.127551 + 0.991832i \(0.459288\pi\)
\(464\) 0 0
\(465\) −5.25544 −0.243715
\(466\) 0 0
\(467\) −30.3505 −1.40446 −0.702228 0.711953i \(-0.747811\pi\)
−0.702228 + 0.711953i \(0.747811\pi\)
\(468\) 0 0
\(469\) 6.51087 0.300644
\(470\) 0 0
\(471\) −34.7446 −1.60094
\(472\) 0 0
\(473\) −24.8614 −1.14313
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 1.37228 0.0628324
\(478\) 0 0
\(479\) −3.25544 −0.148745 −0.0743724 0.997231i \(-0.523695\pi\)
−0.0743724 + 0.997231i \(0.523695\pi\)
\(480\) 0 0
\(481\) −4.74456 −0.216333
\(482\) 0 0
\(483\) −18.5109 −0.842274
\(484\) 0 0
\(485\) −0.116844 −0.00530561
\(486\) 0 0
\(487\) 37.7228 1.70938 0.854692 0.519136i \(-0.173746\pi\)
0.854692 + 0.519136i \(0.173746\pi\)
\(488\) 0 0
\(489\) 38.7446 1.75209
\(490\) 0 0
\(491\) −14.9783 −0.675959 −0.337979 0.941153i \(-0.609743\pi\)
−0.337979 + 0.941153i \(0.609743\pi\)
\(492\) 0 0
\(493\) −43.6060 −1.96391
\(494\) 0 0
\(495\) 3.37228 0.151573
\(496\) 0 0
\(497\) −9.25544 −0.415163
\(498\) 0 0
\(499\) 24.9783 1.11818 0.559090 0.829107i \(-0.311151\pi\)
0.559090 + 0.829107i \(0.311151\pi\)
\(500\) 0 0
\(501\) 2.97825 0.133058
\(502\) 0 0
\(503\) −29.4891 −1.31486 −0.657428 0.753518i \(-0.728355\pi\)
−0.657428 + 0.753518i \(0.728355\pi\)
\(504\) 0 0
\(505\) −11.4891 −0.511259
\(506\) 0 0
\(507\) −19.0217 −0.844786
\(508\) 0 0
\(509\) −11.2554 −0.498888 −0.249444 0.968389i \(-0.580248\pi\)
−0.249444 + 0.968389i \(0.580248\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 9.48913 0.418141
\(516\) 0 0
\(517\) 29.4891 1.29693
\(518\) 0 0
\(519\) −45.7228 −2.00701
\(520\) 0 0
\(521\) 27.0951 1.18706 0.593529 0.804813i \(-0.297734\pi\)
0.593529 + 0.804813i \(0.297734\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 2.74456 0.119783
\(526\) 0 0
\(527\) −14.1168 −0.614939
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) −12.7446 −0.553067
\(532\) 0 0
\(533\) −25.4891 −1.10406
\(534\) 0 0
\(535\) 3.48913 0.150848
\(536\) 0 0
\(537\) −52.4674 −2.26413
\(538\) 0 0
\(539\) −17.2554 −0.743244
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) −14.9783 −0.642778
\(544\) 0 0
\(545\) 0.116844 0.00500505
\(546\) 0 0
\(547\) −39.3723 −1.68344 −0.841719 0.539916i \(-0.818456\pi\)
−0.841719 + 0.539916i \(0.818456\pi\)
\(548\) 0 0
\(549\) −5.37228 −0.229283
\(550\) 0 0
\(551\) 16.2337 0.691578
\(552\) 0 0
\(553\) −6.51087 −0.276871
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 8.74456 0.370519 0.185260 0.982690i \(-0.440687\pi\)
0.185260 + 0.982690i \(0.440687\pi\)
\(558\) 0 0
\(559\) 34.9783 1.47942
\(560\) 0 0
\(561\) 36.2337 1.52979
\(562\) 0 0
\(563\) −33.0951 −1.39479 −0.697396 0.716686i \(-0.745658\pi\)
−0.697396 + 0.716686i \(0.745658\pi\)
\(564\) 0 0
\(565\) 17.3723 0.730857
\(566\) 0 0
\(567\) 15.0951 0.633934
\(568\) 0 0
\(569\) −32.9783 −1.38252 −0.691260 0.722606i \(-0.742944\pi\)
−0.691260 + 0.722606i \(0.742944\pi\)
\(570\) 0 0
\(571\) 27.6060 1.15527 0.577637 0.816294i \(-0.303975\pi\)
0.577637 + 0.816294i \(0.303975\pi\)
\(572\) 0 0
\(573\) 37.2554 1.55637
\(574\) 0 0
\(575\) −6.74456 −0.281268
\(576\) 0 0
\(577\) −7.48913 −0.311776 −0.155888 0.987775i \(-0.549824\pi\)
−0.155888 + 0.987775i \(0.549824\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) −1.02175 −0.0423893
\(582\) 0 0
\(583\) 4.62772 0.191661
\(584\) 0 0
\(585\) −4.74456 −0.196164
\(586\) 0 0
\(587\) −47.8397 −1.97455 −0.987277 0.159010i \(-0.949170\pi\)
−0.987277 + 0.159010i \(0.949170\pi\)
\(588\) 0 0
\(589\) 5.25544 0.216547
\(590\) 0 0
\(591\) −54.9783 −2.26150
\(592\) 0 0
\(593\) −10.2337 −0.420247 −0.210124 0.977675i \(-0.567387\pi\)
−0.210124 + 0.977675i \(0.567387\pi\)
\(594\) 0 0
\(595\) 7.37228 0.302234
\(596\) 0 0
\(597\) −22.5109 −0.921309
\(598\) 0 0
\(599\) −17.4891 −0.714586 −0.357293 0.933992i \(-0.616300\pi\)
−0.357293 + 0.933992i \(0.616300\pi\)
\(600\) 0 0
\(601\) −5.37228 −0.219140 −0.109570 0.993979i \(-0.534947\pi\)
−0.109570 + 0.993979i \(0.534947\pi\)
\(602\) 0 0
\(603\) −4.74456 −0.193214
\(604\) 0 0
\(605\) 0.372281 0.0151354
\(606\) 0 0
\(607\) −17.7228 −0.719347 −0.359673 0.933078i \(-0.617112\pi\)
−0.359673 + 0.933078i \(0.617112\pi\)
\(608\) 0 0
\(609\) 22.2772 0.902717
\(610\) 0 0
\(611\) −41.4891 −1.67847
\(612\) 0 0
\(613\) 43.0951 1.74059 0.870297 0.492527i \(-0.163927\pi\)
0.870297 + 0.492527i \(0.163927\pi\)
\(614\) 0 0
\(615\) −10.7446 −0.433263
\(616\) 0 0
\(617\) −31.7228 −1.27711 −0.638556 0.769575i \(-0.720468\pi\)
−0.638556 + 0.769575i \(0.720468\pi\)
\(618\) 0 0
\(619\) −30.1168 −1.21050 −0.605249 0.796036i \(-0.706926\pi\)
−0.605249 + 0.796036i \(0.706926\pi\)
\(620\) 0 0
\(621\) −26.9783 −1.08260
\(622\) 0 0
\(623\) −13.7228 −0.549793
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −13.4891 −0.538704
\(628\) 0 0
\(629\) −5.37228 −0.214207
\(630\) 0 0
\(631\) 19.0951 0.760164 0.380082 0.924953i \(-0.375896\pi\)
0.380082 + 0.924953i \(0.375896\pi\)
\(632\) 0 0
\(633\) −28.2337 −1.12219
\(634\) 0 0
\(635\) 16.7446 0.664488
\(636\) 0 0
\(637\) 24.2772 0.961897
\(638\) 0 0
\(639\) 6.74456 0.266811
\(640\) 0 0
\(641\) −49.6060 −1.95932 −0.979659 0.200670i \(-0.935688\pi\)
−0.979659 + 0.200670i \(0.935688\pi\)
\(642\) 0 0
\(643\) −3.37228 −0.132990 −0.0664949 0.997787i \(-0.521182\pi\)
−0.0664949 + 0.997787i \(0.521182\pi\)
\(644\) 0 0
\(645\) 14.7446 0.580567
\(646\) 0 0
\(647\) 13.4891 0.530312 0.265156 0.964205i \(-0.414577\pi\)
0.265156 + 0.964205i \(0.414577\pi\)
\(648\) 0 0
\(649\) −42.9783 −1.68704
\(650\) 0 0
\(651\) 7.21194 0.282658
\(652\) 0 0
\(653\) 0.510875 0.0199921 0.00999604 0.999950i \(-0.496818\pi\)
0.00999604 + 0.999950i \(0.496818\pi\)
\(654\) 0 0
\(655\) 3.25544 0.127200
\(656\) 0 0
\(657\) 8.74456 0.341158
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 4.35053 0.169216 0.0846080 0.996414i \(-0.473036\pi\)
0.0846080 + 0.996414i \(0.473036\pi\)
\(662\) 0 0
\(663\) −50.9783 −1.97983
\(664\) 0 0
\(665\) −2.74456 −0.106430
\(666\) 0 0
\(667\) −54.7446 −2.11972
\(668\) 0 0
\(669\) −2.74456 −0.106111
\(670\) 0 0
\(671\) −18.1168 −0.699393
\(672\) 0 0
\(673\) −8.51087 −0.328070 −0.164035 0.986455i \(-0.552451\pi\)
−0.164035 + 0.986455i \(0.552451\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 0.160343 0.00615339
\(680\) 0 0
\(681\) −22.7446 −0.871574
\(682\) 0 0
\(683\) −8.62772 −0.330130 −0.165065 0.986283i \(-0.552783\pi\)
−0.165065 + 0.986283i \(0.552783\pi\)
\(684\) 0 0
\(685\) 16.7446 0.639777
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) 0 0
\(689\) −6.51087 −0.248045
\(690\) 0 0
\(691\) −22.3505 −0.850254 −0.425127 0.905134i \(-0.639771\pi\)
−0.425127 + 0.905134i \(0.639771\pi\)
\(692\) 0 0
\(693\) −4.62772 −0.175792
\(694\) 0 0
\(695\) 1.88316 0.0714322
\(696\) 0 0
\(697\) −28.8614 −1.09320
\(698\) 0 0
\(699\) −12.4674 −0.471559
\(700\) 0 0
\(701\) −26.4674 −0.999659 −0.499829 0.866124i \(-0.666604\pi\)
−0.499829 + 0.866124i \(0.666604\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) −17.4891 −0.658679
\(706\) 0 0
\(707\) 15.7663 0.592953
\(708\) 0 0
\(709\) 40.1168 1.50662 0.753310 0.657666i \(-0.228456\pi\)
0.753310 + 0.657666i \(0.228456\pi\)
\(710\) 0 0
\(711\) 4.74456 0.177935
\(712\) 0 0
\(713\) −17.7228 −0.663725
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 35.2119 1.31501
\(718\) 0 0
\(719\) 34.7446 1.29575 0.647877 0.761745i \(-0.275657\pi\)
0.647877 + 0.761745i \(0.275657\pi\)
\(720\) 0 0
\(721\) −13.0217 −0.484955
\(722\) 0 0
\(723\) 20.4674 0.761190
\(724\) 0 0
\(725\) 8.11684 0.301452
\(726\) 0 0
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 39.6060 1.46488
\(732\) 0 0
\(733\) −29.3723 −1.08489 −0.542445 0.840091i \(-0.682501\pi\)
−0.542445 + 0.840091i \(0.682501\pi\)
\(734\) 0 0
\(735\) 10.2337 0.377475
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 36.8614 1.35597 0.677984 0.735076i \(-0.262854\pi\)
0.677984 + 0.735076i \(0.262854\pi\)
\(740\) 0 0
\(741\) 18.9783 0.697183
\(742\) 0 0
\(743\) 5.37228 0.197090 0.0985449 0.995133i \(-0.468581\pi\)
0.0985449 + 0.995133i \(0.468581\pi\)
\(744\) 0 0
\(745\) −11.4891 −0.420929
\(746\) 0 0
\(747\) 0.744563 0.0272421
\(748\) 0 0
\(749\) −4.78806 −0.174952
\(750\) 0 0
\(751\) −10.7446 −0.392075 −0.196037 0.980596i \(-0.562807\pi\)
−0.196037 + 0.980596i \(0.562807\pi\)
\(752\) 0 0
\(753\) 22.9783 0.837374
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 43.9565 1.59763 0.798813 0.601579i \(-0.205462\pi\)
0.798813 + 0.601579i \(0.205462\pi\)
\(758\) 0 0
\(759\) 45.4891 1.65115
\(760\) 0 0
\(761\) 5.37228 0.194745 0.0973725 0.995248i \(-0.468956\pi\)
0.0973725 + 0.995248i \(0.468956\pi\)
\(762\) 0 0
\(763\) −0.160343 −0.00580480
\(764\) 0 0
\(765\) −5.37228 −0.194235
\(766\) 0 0
\(767\) 60.4674 2.18335
\(768\) 0 0
\(769\) 46.2337 1.66723 0.833615 0.552346i \(-0.186267\pi\)
0.833615 + 0.552346i \(0.186267\pi\)
\(770\) 0 0
\(771\) 41.9565 1.51103
\(772\) 0 0
\(773\) −24.3505 −0.875828 −0.437914 0.899017i \(-0.644283\pi\)
−0.437914 + 0.899017i \(0.644283\pi\)
\(774\) 0 0
\(775\) 2.62772 0.0943904
\(776\) 0 0
\(777\) 2.74456 0.0984606
\(778\) 0 0
\(779\) 10.7446 0.384964
\(780\) 0 0
\(781\) 22.7446 0.813864
\(782\) 0 0
\(783\) 32.4674 1.16029
\(784\) 0 0
\(785\) 17.3723 0.620043
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 0 0
\(789\) 0.233688 0.00831951
\(790\) 0 0
\(791\) −23.8397 −0.847641
\(792\) 0 0
\(793\) 25.4891 0.905145
\(794\) 0 0
\(795\) −2.74456 −0.0973396
\(796\) 0 0
\(797\) −27.4891 −0.973715 −0.486857 0.873481i \(-0.661857\pi\)
−0.486857 + 0.873481i \(0.661857\pi\)
\(798\) 0 0
\(799\) −46.9783 −1.66197
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 29.4891 1.04065
\(804\) 0 0
\(805\) 9.25544 0.326211
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) −51.4891 −1.81026 −0.905131 0.425134i \(-0.860227\pi\)
−0.905131 + 0.425134i \(0.860227\pi\)
\(810\) 0 0
\(811\) −49.4891 −1.73780 −0.868899 0.494989i \(-0.835172\pi\)
−0.868899 + 0.494989i \(0.835172\pi\)
\(812\) 0 0
\(813\) −13.4891 −0.473084
\(814\) 0 0
\(815\) −19.3723 −0.678581
\(816\) 0 0
\(817\) −14.7446 −0.515847
\(818\) 0 0
\(819\) 6.51087 0.227508
\(820\) 0 0
\(821\) −32.7446 −1.14279 −0.571397 0.820674i \(-0.693598\pi\)
−0.571397 + 0.820674i \(0.693598\pi\)
\(822\) 0 0
\(823\) 39.7228 1.38465 0.692325 0.721586i \(-0.256586\pi\)
0.692325 + 0.721586i \(0.256586\pi\)
\(824\) 0 0
\(825\) −6.74456 −0.234816
\(826\) 0 0
\(827\) 54.3505 1.88995 0.944977 0.327138i \(-0.106084\pi\)
0.944977 + 0.327138i \(0.106084\pi\)
\(828\) 0 0
\(829\) −40.3505 −1.40143 −0.700716 0.713440i \(-0.747136\pi\)
−0.700716 + 0.713440i \(0.747136\pi\)
\(830\) 0 0
\(831\) −1.48913 −0.0516572
\(832\) 0 0
\(833\) 27.4891 0.952442
\(834\) 0 0
\(835\) −1.48913 −0.0515333
\(836\) 0 0
\(837\) 10.5109 0.363309
\(838\) 0 0
\(839\) 29.4891 1.01808 0.509039 0.860744i \(-0.330001\pi\)
0.509039 + 0.860744i \(0.330001\pi\)
\(840\) 0 0
\(841\) 36.8832 1.27183
\(842\) 0 0
\(843\) −49.4891 −1.70450
\(844\) 0 0
\(845\) 9.51087 0.327184
\(846\) 0 0
\(847\) −0.510875 −0.0175539
\(848\) 0 0
\(849\) 10.9783 0.376773
\(850\) 0 0
\(851\) −6.74456 −0.231201
\(852\) 0 0
\(853\) 35.4891 1.21512 0.607562 0.794272i \(-0.292148\pi\)
0.607562 + 0.794272i \(0.292148\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −29.1386 −0.995355 −0.497678 0.867362i \(-0.665814\pi\)
−0.497678 + 0.867362i \(0.665814\pi\)
\(858\) 0 0
\(859\) 19.7228 0.672934 0.336467 0.941695i \(-0.390768\pi\)
0.336467 + 0.941695i \(0.390768\pi\)
\(860\) 0 0
\(861\) 14.7446 0.502493
\(862\) 0 0
\(863\) −53.8397 −1.83272 −0.916362 0.400352i \(-0.868888\pi\)
−0.916362 + 0.400352i \(0.868888\pi\)
\(864\) 0 0
\(865\) 22.8614 0.777311
\(866\) 0 0
\(867\) −23.7228 −0.805669
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 22.5109 0.762752
\(872\) 0 0
\(873\) −0.116844 −0.00395457
\(874\) 0 0
\(875\) −1.37228 −0.0463916
\(876\) 0 0
\(877\) 49.3723 1.66718 0.833592 0.552381i \(-0.186281\pi\)
0.833592 + 0.552381i \(0.186281\pi\)
\(878\) 0 0
\(879\) 37.7228 1.27236
\(880\) 0 0
\(881\) 1.37228 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(882\) 0 0
\(883\) 11.3723 0.382708 0.191354 0.981521i \(-0.438712\pi\)
0.191354 + 0.981521i \(0.438712\pi\)
\(884\) 0 0
\(885\) 25.4891 0.856808
\(886\) 0 0
\(887\) 25.3723 0.851918 0.425959 0.904743i \(-0.359937\pi\)
0.425959 + 0.904743i \(0.359937\pi\)
\(888\) 0 0
\(889\) −22.9783 −0.770666
\(890\) 0 0
\(891\) −37.0951 −1.24273
\(892\) 0 0
\(893\) 17.4891 0.585251
\(894\) 0 0
\(895\) 26.2337 0.876895
\(896\) 0 0
\(897\) −64.0000 −2.13690
\(898\) 0 0
\(899\) 21.3288 0.711355
\(900\) 0 0
\(901\) −7.37228 −0.245606
\(902\) 0 0
\(903\) −20.2337 −0.673335
\(904\) 0 0
\(905\) 7.48913 0.248947
\(906\) 0 0
\(907\) −40.4674 −1.34370 −0.671849 0.740689i \(-0.734499\pi\)
−0.671849 + 0.740689i \(0.734499\pi\)
\(908\) 0 0
\(909\) −11.4891 −0.381070
\(910\) 0 0
\(911\) −17.7663 −0.588624 −0.294312 0.955709i \(-0.595091\pi\)
−0.294312 + 0.955709i \(0.595091\pi\)
\(912\) 0 0
\(913\) 2.51087 0.0830978
\(914\) 0 0
\(915\) 10.7446 0.355204
\(916\) 0 0
\(917\) −4.46738 −0.147526
\(918\) 0 0
\(919\) −7.72281 −0.254752 −0.127376 0.991854i \(-0.540656\pi\)
−0.127376 + 0.991854i \(0.540656\pi\)
\(920\) 0 0
\(921\) −46.9783 −1.54799
\(922\) 0 0
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 9.48913 0.311664
\(928\) 0 0
\(929\) 31.0951 1.02020 0.510098 0.860116i \(-0.329609\pi\)
0.510098 + 0.860116i \(0.329609\pi\)
\(930\) 0 0
\(931\) −10.2337 −0.335396
\(932\) 0 0
\(933\) −2.74456 −0.0898529
\(934\) 0 0
\(935\) −18.1168 −0.592484
\(936\) 0 0
\(937\) 36.9783 1.20803 0.604013 0.796974i \(-0.293567\pi\)
0.604013 + 0.796974i \(0.293567\pi\)
\(938\) 0 0
\(939\) 6.97825 0.227727
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −36.2337 −1.17993
\(944\) 0 0
\(945\) −5.48913 −0.178561
\(946\) 0 0
\(947\) −37.0951 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(948\) 0 0
\(949\) −41.4891 −1.34679
\(950\) 0 0
\(951\) −50.7446 −1.64551
\(952\) 0 0
\(953\) 46.2337 1.49766 0.748828 0.662764i \(-0.230617\pi\)
0.748828 + 0.662764i \(0.230617\pi\)
\(954\) 0 0
\(955\) −18.6277 −0.602779
\(956\) 0 0
\(957\) −54.7446 −1.76964
\(958\) 0 0
\(959\) −22.9783 −0.742006
\(960\) 0 0
\(961\) −24.0951 −0.777261
\(962\) 0 0
\(963\) 3.48913 0.112435
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) 0 0
\(969\) 21.4891 0.690330
\(970\) 0 0
\(971\) 19.6060 0.629185 0.314593 0.949227i \(-0.398132\pi\)
0.314593 + 0.949227i \(0.398132\pi\)
\(972\) 0 0
\(973\) −2.58422 −0.0828463
\(974\) 0 0
\(975\) 9.48913 0.303895
\(976\) 0 0
\(977\) −11.8832 −0.380176 −0.190088 0.981767i \(-0.560877\pi\)
−0.190088 + 0.981767i \(0.560877\pi\)
\(978\) 0 0
\(979\) 33.7228 1.07779
\(980\) 0 0
\(981\) 0.116844 0.00373054
\(982\) 0 0
\(983\) −30.8614 −0.984326 −0.492163 0.870503i \(-0.663794\pi\)
−0.492163 + 0.870503i \(0.663794\pi\)
\(984\) 0 0
\(985\) 27.4891 0.875876
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 49.7228 1.58109
\(990\) 0 0
\(991\) −25.6060 −0.813400 −0.406700 0.913562i \(-0.633321\pi\)
−0.406700 + 0.913562i \(0.633321\pi\)
\(992\) 0 0
\(993\) 46.9783 1.49081
\(994\) 0 0
\(995\) 11.2554 0.356821
\(996\) 0 0
\(997\) 26.2337 0.830829 0.415415 0.909632i \(-0.363637\pi\)
0.415415 + 0.909632i \(0.363637\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 2960.2.a.o.1.1 2
4.3 odd 2 370.2.a.f.1.2 2
12.11 even 2 3330.2.a.bb.1.2 2
20.3 even 4 1850.2.b.m.149.1 4
20.7 even 4 1850.2.b.m.149.4 4
20.19 odd 2 1850.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 4.3 odd 2
1850.2.a.q.1.1 2 20.19 odd 2
1850.2.b.m.149.1 4 20.3 even 4
1850.2.b.m.149.4 4 20.7 even 4
2960.2.a.o.1.1 2 1.1 even 1 trivial
3330.2.a.bb.1.2 2 12.11 even 2