Properties

Label 1856.2.a.y.1.2
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,2,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1856.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.36333 q^{3} -4.14134 q^{5} -1.14134 q^{9} +O(q^{10})\) \(q+1.36333 q^{3} -4.14134 q^{5} -1.14134 q^{9} -5.64600 q^{11} +2.86799 q^{13} -5.64600 q^{15} +2.00000 q^{17} +4.28267 q^{19} +2.72666 q^{23} +12.1507 q^{25} -5.64600 q^{27} -1.00000 q^{29} +5.36333 q^{31} -7.69735 q^{33} +6.28267 q^{37} +3.91002 q^{39} +11.7360 q^{41} +2.91934 q^{43} +4.72666 q^{45} +4.19269 q^{47} -7.00000 q^{49} +2.72666 q^{51} -1.41468 q^{53} +23.3820 q^{55} +5.83869 q^{57} +1.27334 q^{59} -3.45331 q^{61} -11.8773 q^{65} +9.45331 q^{67} +3.71733 q^{69} -13.8387 q^{71} -7.73599 q^{73} +16.5653 q^{75} +14.9193 q^{79} -4.27334 q^{81} +9.27334 q^{83} -8.28267 q^{85} -1.36333 q^{87} -16.3013 q^{89} +7.31198 q^{93} -17.7360 q^{95} -10.2827 q^{97} +6.44398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 4 q^{5} + 5 q^{9} + 2 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + 7 q^{25} + 2 q^{27} - 3 q^{29} + 14 q^{31} - 2 q^{33} + 2 q^{37} + 18 q^{39} + 10 q^{41} - 6 q^{43} + 10 q^{45} + 2 q^{47} - 21 q^{49} + 4 q^{51} + 26 q^{55} - 12 q^{57} + 8 q^{59} - 2 q^{61} - 2 q^{65} + 20 q^{67} + 28 q^{69} - 12 q^{71} + 2 q^{73} + 16 q^{75} + 30 q^{79} - 17 q^{81} + 32 q^{83} - 8 q^{85} - 2 q^{87} + 10 q^{89} + 22 q^{93} - 28 q^{95} - 14 q^{97} + 32 q^{99}+O(q^{100}) \) 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
\(3\) 1.36333 0.787118 0.393559 0.919299i \(-0.371244\pi\)
0.393559 + 0.919299i \(0.371244\pi\)
\(4\) 0 0
\(5\) −4.14134 −1.85206 −0.926031 0.377448i \(-0.876802\pi\)
−0.926031 + 0.377448i \(0.876802\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −1.14134 −0.380445
\(10\) 0 0
\(11\) −5.64600 −1.70233 −0.851167 0.524896i \(-0.824104\pi\)
−0.851167 + 0.524896i \(0.824104\pi\)
\(12\) 0 0
\(13\) 2.86799 0.795438 0.397719 0.917507i \(-0.369802\pi\)
0.397719 + 0.917507i \(0.369802\pi\)
\(14\) 0 0
\(15\) −5.64600 −1.45779
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.28267 0.982512 0.491256 0.871015i \(-0.336538\pi\)
0.491256 + 0.871015i \(0.336538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.72666 0.568547 0.284274 0.958743i \(-0.408248\pi\)
0.284274 + 0.958743i \(0.408248\pi\)
\(24\) 0 0
\(25\) 12.1507 2.43013
\(26\) 0 0
\(27\) −5.64600 −1.08657
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 5.36333 0.963282 0.481641 0.876369i \(-0.340041\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(32\) 0 0
\(33\) −7.69735 −1.33994
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.28267 1.03286 0.516432 0.856328i \(-0.327260\pi\)
0.516432 + 0.856328i \(0.327260\pi\)
\(38\) 0 0
\(39\) 3.91002 0.626104
\(40\) 0 0
\(41\) 11.7360 1.83285 0.916426 0.400203i \(-0.131060\pi\)
0.916426 + 0.400203i \(0.131060\pi\)
\(42\) 0 0
\(43\) 2.91934 0.445196 0.222598 0.974910i \(-0.428546\pi\)
0.222598 + 0.974910i \(0.428546\pi\)
\(44\) 0 0
\(45\) 4.72666 0.704608
\(46\) 0 0
\(47\) 4.19269 0.611566 0.305783 0.952101i \(-0.401082\pi\)
0.305783 + 0.952101i \(0.401082\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 2.72666 0.381808
\(52\) 0 0
\(53\) −1.41468 −0.194321 −0.0971606 0.995269i \(-0.530976\pi\)
−0.0971606 + 0.995269i \(0.530976\pi\)
\(54\) 0 0
\(55\) 23.3820 3.15283
\(56\) 0 0
\(57\) 5.83869 0.773353
\(58\) 0 0
\(59\) 1.27334 0.165775 0.0828876 0.996559i \(-0.473586\pi\)
0.0828876 + 0.996559i \(0.473586\pi\)
\(60\) 0 0
\(61\) −3.45331 −0.442151 −0.221076 0.975257i \(-0.570957\pi\)
−0.221076 + 0.975257i \(0.570957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.8773 −1.47320
\(66\) 0 0
\(67\) 9.45331 1.15491 0.577453 0.816424i \(-0.304047\pi\)
0.577453 + 0.816424i \(0.304047\pi\)
\(68\) 0 0
\(69\) 3.71733 0.447514
\(70\) 0 0
\(71\) −13.8387 −1.64235 −0.821175 0.570676i \(-0.806681\pi\)
−0.821175 + 0.570676i \(0.806681\pi\)
\(72\) 0 0
\(73\) −7.73599 −0.905429 −0.452714 0.891656i \(-0.649544\pi\)
−0.452714 + 0.891656i \(0.649544\pi\)
\(74\) 0 0
\(75\) 16.5653 1.91280
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.9193 1.67856 0.839279 0.543701i \(-0.182978\pi\)
0.839279 + 0.543701i \(0.182978\pi\)
\(80\) 0 0
\(81\) −4.27334 −0.474816
\(82\) 0 0
\(83\) 9.27334 1.01788 0.508941 0.860801i \(-0.330037\pi\)
0.508941 + 0.860801i \(0.330037\pi\)
\(84\) 0 0
\(85\) −8.28267 −0.898382
\(86\) 0 0
\(87\) −1.36333 −0.146164
\(88\) 0 0
\(89\) −16.3013 −1.72794 −0.863969 0.503545i \(-0.832029\pi\)
−0.863969 + 0.503545i \(0.832029\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.31198 0.758217
\(94\) 0 0
\(95\) −17.7360 −1.81967
\(96\) 0 0
\(97\) −10.2827 −1.04405 −0.522024 0.852931i \(-0.674823\pi\)
−0.522024 + 0.852931i \(0.674823\pi\)
\(98\) 0 0
\(99\) 6.44398 0.647645
\(100\) 0 0
\(101\) 0.829359 0.0825243 0.0412622 0.999148i \(-0.486862\pi\)
0.0412622 + 0.999148i \(0.486862\pi\)
\(102\) 0 0
\(103\) 2.72666 0.268665 0.134333 0.990936i \(-0.457111\pi\)
0.134333 + 0.990936i \(0.457111\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0187 1.74193 0.870965 0.491346i \(-0.163495\pi\)
0.870965 + 0.491346i \(0.163495\pi\)
\(108\) 0 0
\(109\) 6.97070 0.667672 0.333836 0.942631i \(-0.391657\pi\)
0.333836 + 0.942631i \(0.391657\pi\)
\(110\) 0 0
\(111\) 8.56534 0.812987
\(112\) 0 0
\(113\) 10.5653 0.993904 0.496952 0.867778i \(-0.334452\pi\)
0.496952 + 0.867778i \(0.334452\pi\)
\(114\) 0 0
\(115\) −11.2920 −1.05298
\(116\) 0 0
\(117\) −3.27334 −0.302621
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 20.8773 1.89794
\(122\) 0 0
\(123\) 16.0000 1.44267
\(124\) 0 0
\(125\) −29.6133 −2.64869
\(126\) 0 0
\(127\) −5.73599 −0.508986 −0.254493 0.967075i \(-0.581909\pi\)
−0.254493 + 0.967075i \(0.581909\pi\)
\(128\) 0 0
\(129\) 3.98002 0.350422
\(130\) 0 0
\(131\) 3.71733 0.324784 0.162392 0.986726i \(-0.448079\pi\)
0.162392 + 0.986726i \(0.448079\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 23.3820 2.01240
\(136\) 0 0
\(137\) 6.28267 0.536765 0.268382 0.963312i \(-0.413511\pi\)
0.268382 + 0.963312i \(0.413511\pi\)
\(138\) 0 0
\(139\) −4.17997 −0.354540 −0.177270 0.984162i \(-0.556727\pi\)
−0.177270 + 0.984162i \(0.556727\pi\)
\(140\) 0 0
\(141\) 5.71601 0.481375
\(142\) 0 0
\(143\) −16.1927 −1.35410
\(144\) 0 0
\(145\) 4.14134 0.343919
\(146\) 0 0
\(147\) −9.54330 −0.787118
\(148\) 0 0
\(149\) −8.60398 −0.704865 −0.352433 0.935837i \(-0.614645\pi\)
−0.352433 + 0.935837i \(0.614645\pi\)
\(150\) 0 0
\(151\) −6.01866 −0.489791 −0.244896 0.969549i \(-0.578754\pi\)
−0.244896 + 0.969549i \(0.578754\pi\)
\(152\) 0 0
\(153\) −2.28267 −0.184543
\(154\) 0 0
\(155\) −22.2113 −1.78406
\(156\) 0 0
\(157\) −1.71733 −0.137058 −0.0685288 0.997649i \(-0.521831\pi\)
−0.0685288 + 0.997649i \(0.521831\pi\)
\(158\) 0 0
\(159\) −1.92867 −0.152954
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.1086 0.791770 0.395885 0.918300i \(-0.370438\pi\)
0.395885 + 0.918300i \(0.370438\pi\)
\(164\) 0 0
\(165\) 31.8773 2.48165
\(166\) 0 0
\(167\) −13.8387 −1.07087 −0.535435 0.844576i \(-0.679852\pi\)
−0.535435 + 0.844576i \(0.679852\pi\)
\(168\) 0 0
\(169\) −4.77462 −0.367278
\(170\) 0 0
\(171\) −4.88797 −0.373792
\(172\) 0 0
\(173\) 10.5653 0.803268 0.401634 0.915800i \(-0.368442\pi\)
0.401634 + 0.915800i \(0.368442\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.73599 0.130485
\(178\) 0 0
\(179\) −17.8387 −1.33333 −0.666663 0.745359i \(-0.732278\pi\)
−0.666663 + 0.745359i \(0.732278\pi\)
\(180\) 0 0
\(181\) −10.9707 −0.815445 −0.407723 0.913106i \(-0.633677\pi\)
−0.407723 + 0.913106i \(0.633677\pi\)
\(182\) 0 0
\(183\) −4.70800 −0.348025
\(184\) 0 0
\(185\) −26.0187 −1.91293
\(186\) 0 0
\(187\) −11.2920 −0.825753
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.3013 1.61367 0.806834 0.590778i \(-0.201179\pi\)
0.806834 + 0.590778i \(0.201179\pi\)
\(192\) 0 0
\(193\) −4.82936 −0.347625 −0.173812 0.984779i \(-0.555609\pi\)
−0.173812 + 0.984779i \(0.555609\pi\)
\(194\) 0 0
\(195\) −16.1927 −1.15958
\(196\) 0 0
\(197\) 18.5653 1.32273 0.661363 0.750066i \(-0.269978\pi\)
0.661363 + 0.750066i \(0.269978\pi\)
\(198\) 0 0
\(199\) 1.98134 0.140454 0.0702268 0.997531i \(-0.477628\pi\)
0.0702268 + 0.997531i \(0.477628\pi\)
\(200\) 0 0
\(201\) 12.8880 0.909047
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −48.6027 −3.39456
\(206\) 0 0
\(207\) −3.11203 −0.216301
\(208\) 0 0
\(209\) −24.1800 −1.67256
\(210\) 0 0
\(211\) 20.6553 1.42197 0.710986 0.703206i \(-0.248249\pi\)
0.710986 + 0.703206i \(0.248249\pi\)
\(212\) 0 0
\(213\) −18.8667 −1.29272
\(214\) 0 0
\(215\) −12.0900 −0.824530
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.5467 −0.712679
\(220\) 0 0
\(221\) 5.73599 0.385844
\(222\) 0 0
\(223\) 22.4040 1.50028 0.750142 0.661276i \(-0.229985\pi\)
0.750142 + 0.661276i \(0.229985\pi\)
\(224\) 0 0
\(225\) −13.8680 −0.924533
\(226\) 0 0
\(227\) 26.7640 1.77639 0.888194 0.459469i \(-0.151960\pi\)
0.888194 + 0.459469i \(0.151960\pi\)
\(228\) 0 0
\(229\) 0.829359 0.0548056 0.0274028 0.999624i \(-0.491276\pi\)
0.0274028 + 0.999624i \(0.491276\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.1507 1.51665 0.758325 0.651876i \(-0.226018\pi\)
0.758325 + 0.651876i \(0.226018\pi\)
\(234\) 0 0
\(235\) −17.3633 −1.13266
\(236\) 0 0
\(237\) 20.3400 1.32122
\(238\) 0 0
\(239\) −10.7267 −0.693850 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(240\) 0 0
\(241\) −0.603978 −0.0389056 −0.0194528 0.999811i \(-0.506192\pi\)
−0.0194528 + 0.999811i \(0.506192\pi\)
\(242\) 0 0
\(243\) 11.1120 0.712837
\(244\) 0 0
\(245\) 28.9894 1.85206
\(246\) 0 0
\(247\) 12.2827 0.781528
\(248\) 0 0
\(249\) 12.6426 0.801193
\(250\) 0 0
\(251\) 1.92867 0.121737 0.0608684 0.998146i \(-0.480613\pi\)
0.0608684 + 0.998146i \(0.480613\pi\)
\(252\) 0 0
\(253\) −15.3947 −0.967857
\(254\) 0 0
\(255\) −11.2920 −0.707133
\(256\) 0 0
\(257\) 21.5946 1.34704 0.673519 0.739170i \(-0.264782\pi\)
0.673519 + 0.739170i \(0.264782\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.14134 0.0706469
\(262\) 0 0
\(263\) −8.08998 −0.498850 −0.249425 0.968394i \(-0.580242\pi\)
−0.249425 + 0.968394i \(0.580242\pi\)
\(264\) 0 0
\(265\) 5.85866 0.359895
\(266\) 0 0
\(267\) −22.2241 −1.36009
\(268\) 0 0
\(269\) −11.6587 −0.710845 −0.355422 0.934706i \(-0.615663\pi\)
−0.355422 + 0.934706i \(0.615663\pi\)
\(270\) 0 0
\(271\) 2.07133 0.125824 0.0629121 0.998019i \(-0.479961\pi\)
0.0629121 + 0.998019i \(0.479961\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −68.6027 −4.13690
\(276\) 0 0
\(277\) 13.4720 0.809452 0.404726 0.914438i \(-0.367367\pi\)
0.404726 + 0.914438i \(0.367367\pi\)
\(278\) 0 0
\(279\) −6.12136 −0.366476
\(280\) 0 0
\(281\) −22.8680 −1.36419 −0.682095 0.731264i \(-0.738931\pi\)
−0.682095 + 0.731264i \(0.738931\pi\)
\(282\) 0 0
\(283\) 0.707999 0.0420862 0.0210431 0.999779i \(-0.493301\pi\)
0.0210431 + 0.999779i \(0.493301\pi\)
\(284\) 0 0
\(285\) −24.1800 −1.43230
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −14.0187 −0.821788
\(292\) 0 0
\(293\) 18.5653 1.08460 0.542300 0.840185i \(-0.317554\pi\)
0.542300 + 0.840185i \(0.317554\pi\)
\(294\) 0 0
\(295\) −5.27334 −0.307026
\(296\) 0 0
\(297\) 31.8773 1.84971
\(298\) 0 0
\(299\) 7.82003 0.452244
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.13069 0.0649564
\(304\) 0 0
\(305\) 14.3013 0.818892
\(306\) 0 0
\(307\) −8.55263 −0.488124 −0.244062 0.969760i \(-0.578480\pi\)
−0.244062 + 0.969760i \(0.578480\pi\)
\(308\) 0 0
\(309\) 3.71733 0.211471
\(310\) 0 0
\(311\) 13.1706 0.746839 0.373419 0.927663i \(-0.378185\pi\)
0.373419 + 0.927663i \(0.378185\pi\)
\(312\) 0 0
\(313\) 8.14134 0.460176 0.230088 0.973170i \(-0.426099\pi\)
0.230088 + 0.973170i \(0.426099\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.1120 −0.961107 −0.480554 0.876965i \(-0.659564\pi\)
−0.480554 + 0.876965i \(0.659564\pi\)
\(318\) 0 0
\(319\) 5.64600 0.316115
\(320\) 0 0
\(321\) 24.5653 1.37110
\(322\) 0 0
\(323\) 8.56534 0.476589
\(324\) 0 0
\(325\) 34.8480 1.93302
\(326\) 0 0
\(327\) 9.50335 0.525536
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.7487 0.975557 0.487778 0.872967i \(-0.337807\pi\)
0.487778 + 0.872967i \(0.337807\pi\)
\(332\) 0 0
\(333\) −7.17064 −0.392949
\(334\) 0 0
\(335\) −39.1493 −2.13896
\(336\) 0 0
\(337\) 12.5467 0.683462 0.341731 0.939798i \(-0.388987\pi\)
0.341731 + 0.939798i \(0.388987\pi\)
\(338\) 0 0
\(339\) 14.4040 0.782320
\(340\) 0 0
\(341\) −30.2814 −1.63983
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −15.3947 −0.828823
\(346\) 0 0
\(347\) 21.3107 1.14402 0.572008 0.820248i \(-0.306165\pi\)
0.572008 + 0.820248i \(0.306165\pi\)
\(348\) 0 0
\(349\) −20.3213 −1.08777 −0.543887 0.839158i \(-0.683048\pi\)
−0.543887 + 0.839158i \(0.683048\pi\)
\(350\) 0 0
\(351\) −16.1927 −0.864302
\(352\) 0 0
\(353\) −3.09337 −0.164644 −0.0823218 0.996606i \(-0.526234\pi\)
−0.0823218 + 0.996606i \(0.526234\pi\)
\(354\) 0 0
\(355\) 57.3107 3.04173
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.4847 0.817251 0.408625 0.912702i \(-0.366008\pi\)
0.408625 + 0.912702i \(0.366008\pi\)
\(360\) 0 0
\(361\) −0.658719 −0.0346694
\(362\) 0 0
\(363\) 28.4626 1.49390
\(364\) 0 0
\(365\) 32.0373 1.67691
\(366\) 0 0
\(367\) 32.8480 1.71465 0.857326 0.514773i \(-0.172124\pi\)
0.857326 + 0.514773i \(0.172124\pi\)
\(368\) 0 0
\(369\) −13.3947 −0.697300
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.706681 0.0365905 0.0182953 0.999833i \(-0.494176\pi\)
0.0182953 + 0.999833i \(0.494176\pi\)
\(374\) 0 0
\(375\) −40.3727 −2.08484
\(376\) 0 0
\(377\) −2.86799 −0.147709
\(378\) 0 0
\(379\) −15.3947 −0.790773 −0.395386 0.918515i \(-0.629389\pi\)
−0.395386 + 0.918515i \(0.629389\pi\)
\(380\) 0 0
\(381\) −7.82003 −0.400632
\(382\) 0 0
\(383\) −7.43466 −0.379893 −0.189947 0.981794i \(-0.560831\pi\)
−0.189947 + 0.981794i \(0.560831\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.33195 −0.169373
\(388\) 0 0
\(389\) 33.1893 1.68276 0.841382 0.540441i \(-0.181742\pi\)
0.841382 + 0.540441i \(0.181742\pi\)
\(390\) 0 0
\(391\) 5.45331 0.275786
\(392\) 0 0
\(393\) 5.06794 0.255644
\(394\) 0 0
\(395\) −61.7860 −3.10879
\(396\) 0 0
\(397\) −6.44267 −0.323348 −0.161674 0.986844i \(-0.551689\pi\)
−0.161674 + 0.986844i \(0.551689\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.32131 −0.215796 −0.107898 0.994162i \(-0.534412\pi\)
−0.107898 + 0.994162i \(0.534412\pi\)
\(402\) 0 0
\(403\) 15.3820 0.766231
\(404\) 0 0
\(405\) 17.6974 0.879388
\(406\) 0 0
\(407\) −35.4720 −1.75828
\(408\) 0 0
\(409\) −22.3599 −1.10563 −0.552814 0.833305i \(-0.686446\pi\)
−0.552814 + 0.833305i \(0.686446\pi\)
\(410\) 0 0
\(411\) 8.56534 0.422497
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −38.4040 −1.88518
\(416\) 0 0
\(417\) −5.69867 −0.279065
\(418\) 0 0
\(419\) 1.63328 0.0797911 0.0398955 0.999204i \(-0.487297\pi\)
0.0398955 + 0.999204i \(0.487297\pi\)
\(420\) 0 0
\(421\) −14.5653 −0.709871 −0.354936 0.934891i \(-0.615497\pi\)
−0.354936 + 0.934891i \(0.615497\pi\)
\(422\) 0 0
\(423\) −4.78527 −0.232668
\(424\) 0 0
\(425\) 24.3013 1.17879
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −22.0759 −1.06584
\(430\) 0 0
\(431\) 38.7640 1.86719 0.933597 0.358324i \(-0.116652\pi\)
0.933597 + 0.358324i \(0.116652\pi\)
\(432\) 0 0
\(433\) 3.73599 0.179540 0.0897700 0.995963i \(-0.471387\pi\)
0.0897700 + 0.995963i \(0.471387\pi\)
\(434\) 0 0
\(435\) 5.64600 0.270705
\(436\) 0 0
\(437\) 11.6774 0.558605
\(438\) 0 0
\(439\) 33.3107 1.58983 0.794915 0.606720i \(-0.207515\pi\)
0.794915 + 0.606720i \(0.207515\pi\)
\(440\) 0 0
\(441\) 7.98935 0.380445
\(442\) 0 0
\(443\) 12.8480 0.610428 0.305214 0.952284i \(-0.401272\pi\)
0.305214 + 0.952284i \(0.401272\pi\)
\(444\) 0 0
\(445\) 67.5093 3.20025
\(446\) 0 0
\(447\) −11.7300 −0.554812
\(448\) 0 0
\(449\) −0.341281 −0.0161061 −0.00805303 0.999968i \(-0.502563\pi\)
−0.00805303 + 0.999968i \(0.502563\pi\)
\(450\) 0 0
\(451\) −66.2614 −3.12013
\(452\) 0 0
\(453\) −8.20541 −0.385524
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.9066 −0.790859 −0.395429 0.918496i \(-0.629404\pi\)
−0.395429 + 0.918496i \(0.629404\pi\)
\(458\) 0 0
\(459\) −11.2920 −0.527065
\(460\) 0 0
\(461\) −26.0373 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(462\) 0 0
\(463\) −41.3107 −1.91987 −0.959935 0.280224i \(-0.909591\pi\)
−0.959935 + 0.280224i \(0.909591\pi\)
\(464\) 0 0
\(465\) −30.2814 −1.40426
\(466\) 0 0
\(467\) −33.1180 −1.53252 −0.766258 0.642532i \(-0.777884\pi\)
−0.766258 + 0.642532i \(0.777884\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.34128 −0.107881
\(472\) 0 0
\(473\) −16.4826 −0.757872
\(474\) 0 0
\(475\) 52.0373 2.38764
\(476\) 0 0
\(477\) 1.61462 0.0739286
\(478\) 0 0
\(479\) 24.6553 1.12653 0.563265 0.826276i \(-0.309545\pi\)
0.563265 + 0.826276i \(0.309545\pi\)
\(480\) 0 0
\(481\) 18.0187 0.821580
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 42.5840 1.93364
\(486\) 0 0
\(487\) −5.45331 −0.247113 −0.123557 0.992338i \(-0.539430\pi\)
−0.123557 + 0.992338i \(0.539430\pi\)
\(488\) 0 0
\(489\) 13.7814 0.623216
\(490\) 0 0
\(491\) −16.3727 −0.738888 −0.369444 0.929253i \(-0.620452\pi\)
−0.369444 + 0.929253i \(0.620452\pi\)
\(492\) 0 0
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −26.6867 −1.19948
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −25.6333 −1.14750 −0.573752 0.819029i \(-0.694513\pi\)
−0.573752 + 0.819029i \(0.694513\pi\)
\(500\) 0 0
\(501\) −18.8667 −0.842901
\(502\) 0 0
\(503\) −0.475360 −0.0211952 −0.0105976 0.999944i \(-0.503373\pi\)
−0.0105976 + 0.999944i \(0.503373\pi\)
\(504\) 0 0
\(505\) −3.43466 −0.152840
\(506\) 0 0
\(507\) −6.50937 −0.289091
\(508\) 0 0
\(509\) −31.9987 −1.41832 −0.709158 0.705049i \(-0.750925\pi\)
−0.709158 + 0.705049i \(0.750925\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −24.1800 −1.06757
\(514\) 0 0
\(515\) −11.2920 −0.497585
\(516\) 0 0
\(517\) −23.6719 −1.04109
\(518\) 0 0
\(519\) 14.4040 0.632267
\(520\) 0 0
\(521\) −26.7253 −1.17086 −0.585429 0.810724i \(-0.699074\pi\)
−0.585429 + 0.810724i \(0.699074\pi\)
\(522\) 0 0
\(523\) 9.45331 0.413365 0.206682 0.978408i \(-0.433733\pi\)
0.206682 + 0.978408i \(0.433733\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.7267 0.467260
\(528\) 0 0
\(529\) −15.5653 −0.676754
\(530\) 0 0
\(531\) −1.45331 −0.0630684
\(532\) 0 0
\(533\) 33.6587 1.45792
\(534\) 0 0
\(535\) −74.6213 −3.22616
\(536\) 0 0
\(537\) −24.3200 −1.04948
\(538\) 0 0
\(539\) 39.5220 1.70233
\(540\) 0 0
\(541\) 20.8667 0.897128 0.448564 0.893751i \(-0.351936\pi\)
0.448564 + 0.893751i \(0.351936\pi\)
\(542\) 0 0
\(543\) −14.9567 −0.641852
\(544\) 0 0
\(545\) −28.8680 −1.23657
\(546\) 0 0
\(547\) −12.5653 −0.537255 −0.268628 0.963244i \(-0.586570\pi\)
−0.268628 + 0.963244i \(0.586570\pi\)
\(548\) 0 0
\(549\) 3.94139 0.168214
\(550\) 0 0
\(551\) −4.28267 −0.182448
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −35.4720 −1.50570
\(556\) 0 0
\(557\) −28.3786 −1.20244 −0.601220 0.799084i \(-0.705318\pi\)
−0.601220 + 0.799084i \(0.705318\pi\)
\(558\) 0 0
\(559\) 8.37266 0.354126
\(560\) 0 0
\(561\) −15.3947 −0.649965
\(562\) 0 0
\(563\) 3.16470 0.133376 0.0666881 0.997774i \(-0.478757\pi\)
0.0666881 + 0.997774i \(0.478757\pi\)
\(564\) 0 0
\(565\) −43.7546 −1.84077
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.3599 0.434311 0.217156 0.976137i \(-0.430322\pi\)
0.217156 + 0.976137i \(0.430322\pi\)
\(570\) 0 0
\(571\) −18.5840 −0.777716 −0.388858 0.921298i \(-0.627130\pi\)
−0.388858 + 0.921298i \(0.627130\pi\)
\(572\) 0 0
\(573\) 30.4040 1.27015
\(574\) 0 0
\(575\) 33.1307 1.38165
\(576\) 0 0
\(577\) −28.0187 −1.16643 −0.583216 0.812317i \(-0.698206\pi\)
−0.583216 + 0.812317i \(0.698206\pi\)
\(578\) 0 0
\(579\) −6.58400 −0.273622
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7.98728 0.330799
\(584\) 0 0
\(585\) 13.5560 0.560472
\(586\) 0 0
\(587\) −45.8760 −1.89351 −0.946753 0.321962i \(-0.895658\pi\)
−0.946753 + 0.321962i \(0.895658\pi\)
\(588\) 0 0
\(589\) 22.9694 0.946437
\(590\) 0 0
\(591\) 25.3107 1.04114
\(592\) 0 0
\(593\) 42.8026 1.75769 0.878846 0.477105i \(-0.158314\pi\)
0.878846 + 0.477105i \(0.158314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.70122 0.110554
\(598\) 0 0
\(599\) 8.90069 0.363672 0.181836 0.983329i \(-0.441796\pi\)
0.181836 + 0.983329i \(0.441796\pi\)
\(600\) 0 0
\(601\) 13.7173 0.559541 0.279771 0.960067i \(-0.409742\pi\)
0.279771 + 0.960067i \(0.409742\pi\)
\(602\) 0 0
\(603\) −10.7894 −0.439379
\(604\) 0 0
\(605\) −86.4600 −3.51510
\(606\) 0 0
\(607\) −24.4754 −0.993424 −0.496712 0.867915i \(-0.665460\pi\)
−0.496712 + 0.867915i \(0.665460\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0246 0.486463
\(612\) 0 0
\(613\) 8.74663 0.353273 0.176637 0.984276i \(-0.443478\pi\)
0.176637 + 0.984276i \(0.443478\pi\)
\(614\) 0 0
\(615\) −66.2614 −2.67192
\(616\) 0 0
\(617\) 22.2827 0.897067 0.448533 0.893766i \(-0.351947\pi\)
0.448533 + 0.893766i \(0.351947\pi\)
\(618\) 0 0
\(619\) 14.9966 0.602765 0.301382 0.953503i \(-0.402552\pi\)
0.301382 + 0.953503i \(0.402552\pi\)
\(620\) 0 0
\(621\) −15.3947 −0.617768
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 61.8853 2.47541
\(626\) 0 0
\(627\) −32.9652 −1.31650
\(628\) 0 0
\(629\) 12.5653 0.501013
\(630\) 0 0
\(631\) 10.7267 0.427021 0.213511 0.976941i \(-0.431510\pi\)
0.213511 + 0.976941i \(0.431510\pi\)
\(632\) 0 0
\(633\) 28.1600 1.11926
\(634\) 0 0
\(635\) 23.7546 0.942674
\(636\) 0 0
\(637\) −20.0759 −0.795438
\(638\) 0 0
\(639\) 15.7946 0.624824
\(640\) 0 0
\(641\) −22.9253 −0.905494 −0.452747 0.891639i \(-0.649556\pi\)
−0.452747 + 0.891639i \(0.649556\pi\)
\(642\) 0 0
\(643\) 30.5467 1.20464 0.602322 0.798253i \(-0.294242\pi\)
0.602322 + 0.798253i \(0.294242\pi\)
\(644\) 0 0
\(645\) −16.4826 −0.649002
\(646\) 0 0
\(647\) −24.7453 −0.972839 −0.486419 0.873725i \(-0.661697\pi\)
−0.486419 + 0.873725i \(0.661697\pi\)
\(648\) 0 0
\(649\) −7.18930 −0.282205
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.4134 0.603171 0.301586 0.953439i \(-0.402484\pi\)
0.301586 + 0.953439i \(0.402484\pi\)
\(654\) 0 0
\(655\) −15.3947 −0.601521
\(656\) 0 0
\(657\) 8.82936 0.344466
\(658\) 0 0
\(659\) 18.1341 0.706403 0.353202 0.935547i \(-0.385093\pi\)
0.353202 + 0.935547i \(0.385093\pi\)
\(660\) 0 0
\(661\) −11.0934 −0.431482 −0.215741 0.976451i \(-0.569217\pi\)
−0.215741 + 0.976451i \(0.569217\pi\)
\(662\) 0 0
\(663\) 7.82003 0.303705
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.72666 −0.105577
\(668\) 0 0
\(669\) 30.5441 1.18090
\(670\) 0 0
\(671\) 19.4974 0.752689
\(672\) 0 0
\(673\) −35.7160 −1.37675 −0.688375 0.725355i \(-0.741676\pi\)
−0.688375 + 0.725355i \(0.741676\pi\)
\(674\) 0 0
\(675\) −68.6027 −2.64052
\(676\) 0 0
\(677\) −5.64006 −0.216765 −0.108383 0.994109i \(-0.534567\pi\)
−0.108383 + 0.994109i \(0.534567\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 36.4881 1.39823
\(682\) 0 0
\(683\) −26.0187 −0.995576 −0.497788 0.867299i \(-0.665854\pi\)
−0.497788 + 0.867299i \(0.665854\pi\)
\(684\) 0 0
\(685\) −26.0187 −0.994122
\(686\) 0 0
\(687\) 1.13069 0.0431385
\(688\) 0 0
\(689\) −4.05729 −0.154570
\(690\) 0 0
\(691\) 9.65872 0.367435 0.183717 0.982979i \(-0.441187\pi\)
0.183717 + 0.982979i \(0.441187\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.3107 0.656631
\(696\) 0 0
\(697\) 23.4720 0.889064
\(698\) 0 0
\(699\) 31.5620 1.19378
\(700\) 0 0
\(701\) 16.8867 0.637800 0.318900 0.947788i \(-0.396687\pi\)
0.318900 + 0.947788i \(0.396687\pi\)
\(702\) 0 0
\(703\) 26.9066 1.01480
\(704\) 0 0
\(705\) −23.6719 −0.891536
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36.2814 −1.36257 −0.681287 0.732016i \(-0.738580\pi\)
−0.681287 + 0.732016i \(0.738580\pi\)
\(710\) 0 0
\(711\) −17.0280 −0.638599
\(712\) 0 0
\(713\) 14.6240 0.547671
\(714\) 0 0
\(715\) 67.0594 2.50788
\(716\) 0 0
\(717\) −14.6240 −0.546142
\(718\) 0 0
\(719\) 47.3293 1.76509 0.882543 0.470232i \(-0.155830\pi\)
0.882543 + 0.470232i \(0.155830\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −0.823420 −0.0306233
\(724\) 0 0
\(725\) −12.1507 −0.451264
\(726\) 0 0
\(727\) 52.5254 1.94806 0.974029 0.226421i \(-0.0727026\pi\)
0.974029 + 0.226421i \(0.0727026\pi\)
\(728\) 0 0
\(729\) 27.9694 1.03590
\(730\) 0 0
\(731\) 5.83869 0.215952
\(732\) 0 0
\(733\) 29.4320 1.08710 0.543548 0.839378i \(-0.317081\pi\)
0.543548 + 0.839378i \(0.317081\pi\)
\(734\) 0 0
\(735\) 39.5220 1.45779
\(736\) 0 0
\(737\) −53.3734 −1.96603
\(738\) 0 0
\(739\) −4.08998 −0.150453 −0.0752263 0.997166i \(-0.523968\pi\)
−0.0752263 + 0.997166i \(0.523968\pi\)
\(740\) 0 0
\(741\) 16.7453 0.615154
\(742\) 0 0
\(743\) 18.8294 0.690782 0.345391 0.938459i \(-0.387746\pi\)
0.345391 + 0.938459i \(0.387746\pi\)
\(744\) 0 0
\(745\) 35.6320 1.30545
\(746\) 0 0
\(747\) −10.5840 −0.387248
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −22.6613 −0.826921 −0.413461 0.910522i \(-0.635680\pi\)
−0.413461 + 0.910522i \(0.635680\pi\)
\(752\) 0 0
\(753\) 2.62941 0.0958212
\(754\) 0 0
\(755\) 24.9253 0.907124
\(756\) 0 0
\(757\) −35.1680 −1.27820 −0.639101 0.769122i \(-0.720694\pi\)
−0.639101 + 0.769122i \(0.720694\pi\)
\(758\) 0 0
\(759\) −20.9880 −0.761817
\(760\) 0 0
\(761\) 13.4720 0.488359 0.244179 0.969730i \(-0.421481\pi\)
0.244179 + 0.969730i \(0.421481\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 9.45331 0.341785
\(766\) 0 0
\(767\) 3.65194 0.131864
\(768\) 0 0
\(769\) 44.0560 1.58870 0.794349 0.607461i \(-0.207812\pi\)
0.794349 + 0.607461i \(0.207812\pi\)
\(770\) 0 0
\(771\) 29.4406 1.06028
\(772\) 0 0
\(773\) −4.26401 −0.153366 −0.0766830 0.997056i \(-0.524433\pi\)
−0.0766830 + 0.997056i \(0.524433\pi\)
\(774\) 0 0
\(775\) 65.1680 2.34090
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 50.2614 1.80080
\(780\) 0 0
\(781\) 78.1332 2.79583
\(782\) 0 0
\(783\) 5.64600 0.201772
\(784\) 0 0
\(785\) 7.11203 0.253839
\(786\) 0 0
\(787\) 7.08660 0.252610 0.126305 0.991991i \(-0.459688\pi\)
0.126305 + 0.991991i \(0.459688\pi\)
\(788\) 0 0
\(789\) −11.0293 −0.392654
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.90408 −0.351704
\(794\) 0 0
\(795\) 7.98728 0.283280
\(796\) 0 0
\(797\) −19.4533 −0.689072 −0.344536 0.938773i \(-0.611964\pi\)
−0.344536 + 0.938773i \(0.611964\pi\)
\(798\) 0 0
\(799\) 8.38538 0.296653
\(800\) 0 0
\(801\) 18.6053 0.657386
\(802\) 0 0
\(803\) 43.6774 1.54134
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −15.8947 −0.559519
\(808\) 0 0
\(809\) −14.3200 −0.503464 −0.251732 0.967797i \(-0.581000\pi\)
−0.251732 + 0.967797i \(0.581000\pi\)
\(810\) 0 0
\(811\) 3.43466 0.120607 0.0603035 0.998180i \(-0.480793\pi\)
0.0603035 + 0.998180i \(0.480793\pi\)
\(812\) 0 0
\(813\) 2.82390 0.0990385
\(814\) 0 0
\(815\) −41.8633 −1.46641
\(816\) 0 0
\(817\) 12.5026 0.437410
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33.0921 1.15492 0.577460 0.816419i \(-0.304044\pi\)
0.577460 + 0.816419i \(0.304044\pi\)
\(822\) 0 0
\(823\) 9.20796 0.320969 0.160485 0.987038i \(-0.448694\pi\)
0.160485 + 0.987038i \(0.448694\pi\)
\(824\) 0 0
\(825\) −93.5279 −3.25623
\(826\) 0 0
\(827\) 20.8607 0.725399 0.362699 0.931906i \(-0.381855\pi\)
0.362699 + 0.931906i \(0.381855\pi\)
\(828\) 0 0
\(829\) 11.4160 0.396494 0.198247 0.980152i \(-0.436475\pi\)
0.198247 + 0.980152i \(0.436475\pi\)
\(830\) 0 0
\(831\) 18.3667 0.637134
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 57.3107 1.98332
\(836\) 0 0
\(837\) −30.2814 −1.04668
\(838\) 0 0
\(839\) −33.8260 −1.16780 −0.583901 0.811825i \(-0.698474\pi\)
−0.583901 + 0.811825i \(0.698474\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −31.1766 −1.07378
\(844\) 0 0
\(845\) 19.7733 0.680222
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.965235 0.0331268
\(850\) 0 0
\(851\) 17.1307 0.587232
\(852\) 0 0
\(853\) 17.4347 0.596951 0.298476 0.954417i \(-0.403522\pi\)
0.298476 + 0.954417i \(0.403522\pi\)
\(854\) 0 0
\(855\) 20.2427 0.692286
\(856\) 0 0
\(857\) −8.98935 −0.307070 −0.153535 0.988143i \(-0.549066\pi\)
−0.153535 + 0.988143i \(0.549066\pi\)
\(858\) 0 0
\(859\) −3.48469 −0.118896 −0.0594480 0.998231i \(-0.518934\pi\)
−0.0594480 + 0.998231i \(0.518934\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31.8947 −1.08571 −0.542853 0.839827i \(-0.682656\pi\)
−0.542853 + 0.839827i \(0.682656\pi\)
\(864\) 0 0
\(865\) −43.7546 −1.48770
\(866\) 0 0
\(867\) −17.7233 −0.601914
\(868\) 0 0
\(869\) −84.2346 −2.85746
\(870\) 0 0
\(871\) 27.1120 0.918656
\(872\) 0 0
\(873\) 11.7360 0.397203
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.5853 1.30293 0.651467 0.758677i \(-0.274154\pi\)
0.651467 + 0.758677i \(0.274154\pi\)
\(878\) 0 0
\(879\) 25.3107 0.853707
\(880\) 0 0
\(881\) −8.54669 −0.287945 −0.143973 0.989582i \(-0.545988\pi\)
−0.143973 + 0.989582i \(0.545988\pi\)
\(882\) 0 0
\(883\) −20.9253 −0.704192 −0.352096 0.935964i \(-0.614531\pi\)
−0.352096 + 0.935964i \(0.614531\pi\)
\(884\) 0 0
\(885\) −7.18930 −0.241666
\(886\) 0 0
\(887\) 38.2886 1.28561 0.642803 0.766032i \(-0.277771\pi\)
0.642803 + 0.766032i \(0.277771\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 24.1273 0.808295
\(892\) 0 0
\(893\) 17.9559 0.600871
\(894\) 0 0
\(895\) 73.8760 2.46940
\(896\) 0 0
\(897\) 10.6613 0.355969
\(898\) 0 0
\(899\) −5.36333 −0.178877
\(900\) 0 0
\(901\) −2.82936 −0.0942596
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 45.4333 1.51026
\(906\) 0 0
\(907\) −4.28267 −0.142204 −0.0711019 0.997469i \(-0.522652\pi\)
−0.0711019 + 0.997469i \(0.522652\pi\)
\(908\) 0 0
\(909\) −0.946578 −0.0313960
\(910\) 0 0
\(911\) −41.9660 −1.39040 −0.695198 0.718819i \(-0.744683\pi\)
−0.695198 + 0.718819i \(0.744683\pi\)
\(912\) 0 0
\(913\) −52.3573 −1.73277
\(914\) 0 0
\(915\) 19.4974 0.644564
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.06794 0.167176 0.0835879 0.996500i \(-0.473362\pi\)
0.0835879 + 0.996500i \(0.473362\pi\)
\(920\) 0 0
\(921\) −11.6600 −0.384211
\(922\) 0 0
\(923\) −39.6893 −1.30639
\(924\) 0 0
\(925\) 76.3386 2.51000
\(926\) 0 0
\(927\) −3.11203 −0.102213
\(928\) 0 0
\(929\) 27.6960 0.908677 0.454339 0.890829i \(-0.349876\pi\)
0.454339 + 0.890829i \(0.349876\pi\)
\(930\) 0 0
\(931\) −29.9787 −0.982512
\(932\) 0 0
\(933\) 17.9559 0.587850
\(934\) 0 0
\(935\) 46.7640 1.52935
\(936\) 0 0
\(937\) 14.6027 0.477048 0.238524 0.971137i \(-0.423336\pi\)
0.238524 + 0.971137i \(0.423336\pi\)
\(938\) 0 0
\(939\) 11.0993 0.362212
\(940\) 0 0
\(941\) −10.1600 −0.331206 −0.165603 0.986192i \(-0.552957\pi\)
−0.165603 + 0.986192i \(0.552957\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.2487 −0.593002 −0.296501 0.955033i \(-0.595820\pi\)
−0.296501 + 0.955033i \(0.595820\pi\)
\(948\) 0 0
\(949\) −22.1867 −0.720212
\(950\) 0 0
\(951\) −23.3293 −0.756505
\(952\) 0 0
\(953\) −32.5640 −1.05485 −0.527426 0.849601i \(-0.676843\pi\)
−0.527426 + 0.849601i \(0.676843\pi\)
\(954\) 0 0
\(955\) −92.3573 −2.98861
\(956\) 0 0
\(957\) 7.69735 0.248820
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.23471 −0.0720874
\(962\) 0 0
\(963\) −20.5653 −0.662709
\(964\) 0 0
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −15.1247 −0.486379 −0.243190 0.969979i \(-0.578194\pi\)
−0.243190 + 0.969979i \(0.578194\pi\)
\(968\) 0 0
\(969\) 11.6774 0.375131
\(970\) 0 0
\(971\) −8.81070 −0.282749 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 47.5093 1.52151
\(976\) 0 0
\(977\) −6.08273 −0.194604 −0.0973018 0.995255i \(-0.531021\pi\)
−0.0973018 + 0.995255i \(0.531021\pi\)
\(978\) 0 0
\(979\) 92.0373 2.94153
\(980\) 0 0
\(981\) −7.95591 −0.254013
\(982\) 0 0
\(983\) −0.335342 −0.0106958 −0.00534788 0.999986i \(-0.501702\pi\)
−0.00534788 + 0.999986i \(0.501702\pi\)
\(984\) 0 0
\(985\) −76.8853 −2.44977
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.96005 0.253115
\(990\) 0 0
\(991\) −22.1986 −0.705163 −0.352581 0.935781i \(-0.614696\pi\)
−0.352581 + 0.935781i \(0.614696\pi\)
\(992\) 0 0
\(993\) 24.1973 0.767878
\(994\) 0 0
\(995\) −8.20541 −0.260129
\(996\) 0 0
\(997\) −29.1893 −0.924434 −0.462217 0.886767i \(-0.652946\pi\)
−0.462217 + 0.886767i \(0.652946\pi\)
\(998\) 0 0
\(999\) −35.4720 −1.12228
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 1856.2.a.y.1.2 3
4.3 odd 2 1856.2.a.x.1.2 3
8.3 odd 2 232.2.a.d.1.2 3
8.5 even 2 464.2.a.j.1.2 3
24.5 odd 2 4176.2.a.bu.1.1 3
24.11 even 2 2088.2.a.s.1.1 3
40.19 odd 2 5800.2.a.p.1.2 3
232.115 odd 2 6728.2.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.2.a.d.1.2 3 8.3 odd 2
464.2.a.j.1.2 3 8.5 even 2
1856.2.a.x.1.2 3 4.3 odd 2
1856.2.a.y.1.2 3 1.1 even 1 trivial
2088.2.a.s.1.1 3 24.11 even 2
4176.2.a.bu.1.1 3 24.5 odd 2
5800.2.a.p.1.2 3 40.19 odd 2
6728.2.a.j.1.2 3 232.115 odd 2