Properties

Label 7728.2.a.bh.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -4.37228 q^{13} -4.37228 q^{15} +6.74456 q^{17} -4.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +14.1168 q^{25} +1.00000 q^{27} +3.62772 q^{29} +4.74456 q^{31} -4.00000 q^{33} -4.37228 q^{35} +8.37228 q^{37} -4.37228 q^{39} +9.11684 q^{41} +11.1168 q^{43} -4.37228 q^{45} -10.3723 q^{47} +1.00000 q^{49} +6.74456 q^{51} -6.74456 q^{53} +17.4891 q^{55} -4.00000 q^{57} -8.74456 q^{59} -2.00000 q^{61} +1.00000 q^{63} +19.1168 q^{65} +4.00000 q^{67} +1.00000 q^{69} -3.25544 q^{71} -1.25544 q^{73} +14.1168 q^{75} -4.00000 q^{77} -9.48913 q^{79} +1.00000 q^{81} -4.00000 q^{83} -29.4891 q^{85} +3.62772 q^{87} -15.4891 q^{89} -4.37228 q^{91} +4.74456 q^{93} +17.4891 q^{95} -3.62772 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} - 3 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} + 2 q^{23} + 11 q^{25} + 2 q^{27} + 13 q^{29} - 2 q^{31} - 8 q^{33} - 3 q^{35} + 11 q^{37} - 3 q^{39} + q^{41} + 5 q^{43} - 3 q^{45} - 15 q^{47} + 2 q^{49} + 2 q^{51} - 2 q^{53} + 12 q^{55} - 8 q^{57} - 6 q^{59} - 4 q^{61} + 2 q^{63} + 21 q^{65} + 8 q^{67} + 2 q^{69} - 18 q^{71} - 14 q^{73} + 11 q^{75} - 8 q^{77} + 4 q^{79} + 2 q^{81} - 8 q^{83} - 36 q^{85} + 13 q^{87} - 8 q^{89} - 3 q^{91} - 2 q^{93} + 12 q^{95} - 13 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −4.37228 −1.95534 −0.977672 0.210138i \(-0.932609\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.37228 −1.21265 −0.606326 0.795216i \(-0.707357\pi\)
−0.606326 + 0.795216i \(0.707357\pi\)
\(14\) 0 0
\(15\) −4.37228 −1.12892
\(16\) 0 0
\(17\) 6.74456 1.63580 0.817898 0.575363i \(-0.195139\pi\)
0.817898 + 0.575363i \(0.195139\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 14.1168 2.82337
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.62772 0.673650 0.336825 0.941567i \(-0.390647\pi\)
0.336825 + 0.941567i \(0.390647\pi\)
\(30\) 0 0
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −4.37228 −0.739050
\(36\) 0 0
\(37\) 8.37228 1.37639 0.688197 0.725524i \(-0.258402\pi\)
0.688197 + 0.725524i \(0.258402\pi\)
\(38\) 0 0
\(39\) −4.37228 −0.700125
\(40\) 0 0
\(41\) 9.11684 1.42381 0.711906 0.702275i \(-0.247832\pi\)
0.711906 + 0.702275i \(0.247832\pi\)
\(42\) 0 0
\(43\) 11.1168 1.69530 0.847651 0.530554i \(-0.178016\pi\)
0.847651 + 0.530554i \(0.178016\pi\)
\(44\) 0 0
\(45\) −4.37228 −0.651781
\(46\) 0 0
\(47\) −10.3723 −1.51295 −0.756476 0.654021i \(-0.773081\pi\)
−0.756476 + 0.654021i \(0.773081\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.74456 0.944428
\(52\) 0 0
\(53\) −6.74456 −0.926437 −0.463218 0.886244i \(-0.653305\pi\)
−0.463218 + 0.886244i \(0.653305\pi\)
\(54\) 0 0
\(55\) 17.4891 2.35823
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −8.74456 −1.13845 −0.569223 0.822183i \(-0.692756\pi\)
−0.569223 + 0.822183i \(0.692756\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 19.1168 2.37115
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −3.25544 −0.386349 −0.193175 0.981164i \(-0.561878\pi\)
−0.193175 + 0.981164i \(0.561878\pi\)
\(72\) 0 0
\(73\) −1.25544 −0.146938 −0.0734689 0.997298i \(-0.523407\pi\)
−0.0734689 + 0.997298i \(0.523407\pi\)
\(74\) 0 0
\(75\) 14.1168 1.63007
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −9.48913 −1.06761 −0.533805 0.845608i \(-0.679238\pi\)
−0.533805 + 0.845608i \(0.679238\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −29.4891 −3.19854
\(86\) 0 0
\(87\) 3.62772 0.388932
\(88\) 0 0
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) 0 0
\(91\) −4.37228 −0.458340
\(92\) 0 0
\(93\) 4.74456 0.491988
\(94\) 0 0
\(95\) 17.4891 1.79435
\(96\) 0 0
\(97\) −3.62772 −0.368339 −0.184170 0.982894i \(-0.558960\pi\)
−0.184170 + 0.982894i \(0.558960\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 7.48913 0.745196 0.372598 0.927993i \(-0.378467\pi\)
0.372598 + 0.927993i \(0.378467\pi\)
\(102\) 0 0
\(103\) −7.11684 −0.701243 −0.350622 0.936517i \(-0.614030\pi\)
−0.350622 + 0.936517i \(0.614030\pi\)
\(104\) 0 0
\(105\) −4.37228 −0.426691
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −9.11684 −0.873235 −0.436618 0.899647i \(-0.643824\pi\)
−0.436618 + 0.899647i \(0.643824\pi\)
\(110\) 0 0
\(111\) 8.37228 0.794662
\(112\) 0 0
\(113\) 1.11684 0.105064 0.0525319 0.998619i \(-0.483271\pi\)
0.0525319 + 0.998619i \(0.483271\pi\)
\(114\) 0 0
\(115\) −4.37228 −0.407717
\(116\) 0 0
\(117\) −4.37228 −0.404218
\(118\) 0 0
\(119\) 6.74456 0.618273
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 9.11684 0.822038
\(124\) 0 0
\(125\) −39.8614 −3.56531
\(126\) 0 0
\(127\) 5.62772 0.499379 0.249690 0.968326i \(-0.419671\pi\)
0.249690 + 0.968326i \(0.419671\pi\)
\(128\) 0 0
\(129\) 11.1168 0.978784
\(130\) 0 0
\(131\) −7.25544 −0.633911 −0.316955 0.948440i \(-0.602661\pi\)
−0.316955 + 0.948440i \(0.602661\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) −4.37228 −0.376306
\(136\) 0 0
\(137\) 9.11684 0.778905 0.389452 0.921047i \(-0.372664\pi\)
0.389452 + 0.921047i \(0.372664\pi\)
\(138\) 0 0
\(139\) −14.3723 −1.21904 −0.609520 0.792770i \(-0.708638\pi\)
−0.609520 + 0.792770i \(0.708638\pi\)
\(140\) 0 0
\(141\) −10.3723 −0.873504
\(142\) 0 0
\(143\) 17.4891 1.46251
\(144\) 0 0
\(145\) −15.8614 −1.31722
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −16.2337 −1.32992 −0.664958 0.746881i \(-0.731550\pi\)
−0.664958 + 0.746881i \(0.731550\pi\)
\(150\) 0 0
\(151\) −2.37228 −0.193054 −0.0965268 0.995330i \(-0.530773\pi\)
−0.0965268 + 0.995330i \(0.530773\pi\)
\(152\) 0 0
\(153\) 6.74456 0.545266
\(154\) 0 0
\(155\) −20.7446 −1.66624
\(156\) 0 0
\(157\) −22.7446 −1.81521 −0.907607 0.419821i \(-0.862093\pi\)
−0.907607 + 0.419821i \(0.862093\pi\)
\(158\) 0 0
\(159\) −6.74456 −0.534879
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 17.4891 1.36153
\(166\) 0 0
\(167\) −17.4891 −1.35335 −0.676675 0.736282i \(-0.736580\pi\)
−0.676675 + 0.736282i \(0.736580\pi\)
\(168\) 0 0
\(169\) 6.11684 0.470526
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 15.4891 1.17762 0.588808 0.808273i \(-0.299597\pi\)
0.588808 + 0.808273i \(0.299597\pi\)
\(174\) 0 0
\(175\) 14.1168 1.06713
\(176\) 0 0
\(177\) −8.74456 −0.657282
\(178\) 0 0
\(179\) −7.86141 −0.587589 −0.293795 0.955869i \(-0.594918\pi\)
−0.293795 + 0.955869i \(0.594918\pi\)
\(180\) 0 0
\(181\) 24.9783 1.85662 0.928309 0.371809i \(-0.121262\pi\)
0.928309 + 0.371809i \(0.121262\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −36.6060 −2.69132
\(186\) 0 0
\(187\) −26.9783 −1.97285
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −25.4891 −1.84433 −0.922164 0.386799i \(-0.873581\pi\)
−0.922164 + 0.386799i \(0.873581\pi\)
\(192\) 0 0
\(193\) 17.1168 1.23210 0.616049 0.787708i \(-0.288732\pi\)
0.616049 + 0.787708i \(0.288732\pi\)
\(194\) 0 0
\(195\) 19.1168 1.36899
\(196\) 0 0
\(197\) −15.6277 −1.11343 −0.556714 0.830704i \(-0.687938\pi\)
−0.556714 + 0.830704i \(0.687938\pi\)
\(198\) 0 0
\(199\) −11.8614 −0.840833 −0.420416 0.907331i \(-0.638116\pi\)
−0.420416 + 0.907331i \(0.638116\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 3.62772 0.254616
\(204\) 0 0
\(205\) −39.8614 −2.78404
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −3.25544 −0.223059
\(214\) 0 0
\(215\) −48.6060 −3.31490
\(216\) 0 0
\(217\) 4.74456 0.322082
\(218\) 0 0
\(219\) −1.25544 −0.0848346
\(220\) 0 0
\(221\) −29.4891 −1.98365
\(222\) 0 0
\(223\) 20.7446 1.38916 0.694579 0.719416i \(-0.255591\pi\)
0.694579 + 0.719416i \(0.255591\pi\)
\(224\) 0 0
\(225\) 14.1168 0.941123
\(226\) 0 0
\(227\) 11.1168 0.737851 0.368925 0.929459i \(-0.379726\pi\)
0.368925 + 0.929459i \(0.379726\pi\)
\(228\) 0 0
\(229\) 10.7446 0.710021 0.355010 0.934862i \(-0.384477\pi\)
0.355010 + 0.934862i \(0.384477\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 0.510875 0.0334685 0.0167343 0.999860i \(-0.494673\pi\)
0.0167343 + 0.999860i \(0.494673\pi\)
\(234\) 0 0
\(235\) 45.3505 2.95834
\(236\) 0 0
\(237\) −9.48913 −0.616385
\(238\) 0 0
\(239\) 9.48913 0.613800 0.306900 0.951742i \(-0.400708\pi\)
0.306900 + 0.951742i \(0.400708\pi\)
\(240\) 0 0
\(241\) −8.37228 −0.539306 −0.269653 0.962958i \(-0.586909\pi\)
−0.269653 + 0.962958i \(0.586909\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.37228 −0.279335
\(246\) 0 0
\(247\) 17.4891 1.11281
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 19.1168 1.20664 0.603322 0.797498i \(-0.293843\pi\)
0.603322 + 0.797498i \(0.293843\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −29.4891 −1.84668
\(256\) 0 0
\(257\) −16.9783 −1.05907 −0.529537 0.848287i \(-0.677634\pi\)
−0.529537 + 0.848287i \(0.677634\pi\)
\(258\) 0 0
\(259\) 8.37228 0.520228
\(260\) 0 0
\(261\) 3.62772 0.224550
\(262\) 0 0
\(263\) 7.11684 0.438843 0.219422 0.975630i \(-0.429583\pi\)
0.219422 + 0.975630i \(0.429583\pi\)
\(264\) 0 0
\(265\) 29.4891 1.81150
\(266\) 0 0
\(267\) −15.4891 −0.947919
\(268\) 0 0
\(269\) 24.9783 1.52295 0.761475 0.648194i \(-0.224475\pi\)
0.761475 + 0.648194i \(0.224475\pi\)
\(270\) 0 0
\(271\) −18.9783 −1.15285 −0.576423 0.817151i \(-0.695552\pi\)
−0.576423 + 0.817151i \(0.695552\pi\)
\(272\) 0 0
\(273\) −4.37228 −0.264623
\(274\) 0 0
\(275\) −56.4674 −3.40511
\(276\) 0 0
\(277\) 17.2554 1.03678 0.518389 0.855145i \(-0.326532\pi\)
0.518389 + 0.855145i \(0.326532\pi\)
\(278\) 0 0
\(279\) 4.74456 0.284050
\(280\) 0 0
\(281\) −11.6277 −0.693652 −0.346826 0.937930i \(-0.612740\pi\)
−0.346826 + 0.937930i \(0.612740\pi\)
\(282\) 0 0
\(283\) 24.7446 1.47091 0.735456 0.677573i \(-0.236968\pi\)
0.735456 + 0.677573i \(0.236968\pi\)
\(284\) 0 0
\(285\) 17.4891 1.03597
\(286\) 0 0
\(287\) 9.11684 0.538150
\(288\) 0 0
\(289\) 28.4891 1.67583
\(290\) 0 0
\(291\) −3.62772 −0.212661
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 38.2337 2.22605
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −4.37228 −0.252856
\(300\) 0 0
\(301\) 11.1168 0.640764
\(302\) 0 0
\(303\) 7.48913 0.430239
\(304\) 0 0
\(305\) 8.74456 0.500712
\(306\) 0 0
\(307\) −1.62772 −0.0928988 −0.0464494 0.998921i \(-0.514791\pi\)
−0.0464494 + 0.998921i \(0.514791\pi\)
\(308\) 0 0
\(309\) −7.11684 −0.404863
\(310\) 0 0
\(311\) 17.4891 0.991717 0.495859 0.868403i \(-0.334853\pi\)
0.495859 + 0.868403i \(0.334853\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −4.37228 −0.246350
\(316\) 0 0
\(317\) −1.11684 −0.0627282 −0.0313641 0.999508i \(-0.509985\pi\)
−0.0313641 + 0.999508i \(0.509985\pi\)
\(318\) 0 0
\(319\) −14.5109 −0.812453
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) −26.9783 −1.50111
\(324\) 0 0
\(325\) −61.7228 −3.42377
\(326\) 0 0
\(327\) −9.11684 −0.504163
\(328\) 0 0
\(329\) −10.3723 −0.571842
\(330\) 0 0
\(331\) 13.4891 0.741429 0.370715 0.928747i \(-0.379113\pi\)
0.370715 + 0.928747i \(0.379113\pi\)
\(332\) 0 0
\(333\) 8.37228 0.458798
\(334\) 0 0
\(335\) −17.4891 −0.955533
\(336\) 0 0
\(337\) −12.2337 −0.666411 −0.333206 0.942854i \(-0.608130\pi\)
−0.333206 + 0.942854i \(0.608130\pi\)
\(338\) 0 0
\(339\) 1.11684 0.0606586
\(340\) 0 0
\(341\) −18.9783 −1.02773
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.37228 −0.235396
\(346\) 0 0
\(347\) −22.3723 −1.20101 −0.600503 0.799622i \(-0.705033\pi\)
−0.600503 + 0.799622i \(0.705033\pi\)
\(348\) 0 0
\(349\) 12.5109 0.669692 0.334846 0.942273i \(-0.391316\pi\)
0.334846 + 0.942273i \(0.391316\pi\)
\(350\) 0 0
\(351\) −4.37228 −0.233375
\(352\) 0 0
\(353\) 18.8832 1.00505 0.502524 0.864563i \(-0.332405\pi\)
0.502524 + 0.864563i \(0.332405\pi\)
\(354\) 0 0
\(355\) 14.2337 0.755446
\(356\) 0 0
\(357\) 6.74456 0.356960
\(358\) 0 0
\(359\) −13.6277 −0.719243 −0.359622 0.933098i \(-0.617094\pi\)
−0.359622 + 0.933098i \(0.617094\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 5.48913 0.287314
\(366\) 0 0
\(367\) 5.62772 0.293765 0.146882 0.989154i \(-0.453076\pi\)
0.146882 + 0.989154i \(0.453076\pi\)
\(368\) 0 0
\(369\) 9.11684 0.474604
\(370\) 0 0
\(371\) −6.74456 −0.350160
\(372\) 0 0
\(373\) −8.51087 −0.440676 −0.220338 0.975424i \(-0.570716\pi\)
−0.220338 + 0.975424i \(0.570716\pi\)
\(374\) 0 0
\(375\) −39.8614 −2.05843
\(376\) 0 0
\(377\) −15.8614 −0.816904
\(378\) 0 0
\(379\) −9.62772 −0.494543 −0.247271 0.968946i \(-0.579534\pi\)
−0.247271 + 0.968946i \(0.579534\pi\)
\(380\) 0 0
\(381\) 5.62772 0.288317
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 17.4891 0.891328
\(386\) 0 0
\(387\) 11.1168 0.565101
\(388\) 0 0
\(389\) −11.4891 −0.582522 −0.291261 0.956644i \(-0.594075\pi\)
−0.291261 + 0.956644i \(0.594075\pi\)
\(390\) 0 0
\(391\) 6.74456 0.341087
\(392\) 0 0
\(393\) −7.25544 −0.365988
\(394\) 0 0
\(395\) 41.4891 2.08754
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 20.9783 1.04760 0.523802 0.851840i \(-0.324513\pi\)
0.523802 + 0.851840i \(0.324513\pi\)
\(402\) 0 0
\(403\) −20.7446 −1.03336
\(404\) 0 0
\(405\) −4.37228 −0.217260
\(406\) 0 0
\(407\) −33.4891 −1.65999
\(408\) 0 0
\(409\) −10.7446 −0.531284 −0.265642 0.964072i \(-0.585584\pi\)
−0.265642 + 0.964072i \(0.585584\pi\)
\(410\) 0 0
\(411\) 9.11684 0.449701
\(412\) 0 0
\(413\) −8.74456 −0.430292
\(414\) 0 0
\(415\) 17.4891 0.858507
\(416\) 0 0
\(417\) −14.3723 −0.703814
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 1.86141 0.0907194 0.0453597 0.998971i \(-0.485557\pi\)
0.0453597 + 0.998971i \(0.485557\pi\)
\(422\) 0 0
\(423\) −10.3723 −0.504318
\(424\) 0 0
\(425\) 95.2119 4.61846
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) 17.4891 0.844383
\(430\) 0 0
\(431\) −19.8614 −0.956690 −0.478345 0.878172i \(-0.658763\pi\)
−0.478345 + 0.878172i \(0.658763\pi\)
\(432\) 0 0
\(433\) −17.8614 −0.858364 −0.429182 0.903218i \(-0.641198\pi\)
−0.429182 + 0.903218i \(0.641198\pi\)
\(434\) 0 0
\(435\) −15.8614 −0.760496
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −19.2554 −0.919012 −0.459506 0.888175i \(-0.651974\pi\)
−0.459506 + 0.888175i \(0.651974\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 14.3723 0.682848 0.341424 0.939909i \(-0.389091\pi\)
0.341424 + 0.939909i \(0.389091\pi\)
\(444\) 0 0
\(445\) 67.7228 3.21037
\(446\) 0 0
\(447\) −16.2337 −0.767827
\(448\) 0 0
\(449\) −16.9783 −0.801253 −0.400627 0.916241i \(-0.631208\pi\)
−0.400627 + 0.916241i \(0.631208\pi\)
\(450\) 0 0
\(451\) −36.4674 −1.71718
\(452\) 0 0
\(453\) −2.37228 −0.111459
\(454\) 0 0
\(455\) 19.1168 0.896211
\(456\) 0 0
\(457\) 24.2337 1.13360 0.566802 0.823854i \(-0.308180\pi\)
0.566802 + 0.823854i \(0.308180\pi\)
\(458\) 0 0
\(459\) 6.74456 0.314809
\(460\) 0 0
\(461\) −35.4891 −1.65289 −0.826447 0.563015i \(-0.809641\pi\)
−0.826447 + 0.563015i \(0.809641\pi\)
\(462\) 0 0
\(463\) −15.1168 −0.702539 −0.351270 0.936274i \(-0.614250\pi\)
−0.351270 + 0.936274i \(0.614250\pi\)
\(464\) 0 0
\(465\) −20.7446 −0.962006
\(466\) 0 0
\(467\) −0.138593 −0.00641334 −0.00320667 0.999995i \(-0.501021\pi\)
−0.00320667 + 0.999995i \(0.501021\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −22.7446 −1.04801
\(472\) 0 0
\(473\) −44.4674 −2.04461
\(474\) 0 0
\(475\) −56.4674 −2.59090
\(476\) 0 0
\(477\) −6.74456 −0.308812
\(478\) 0 0
\(479\) −4.74456 −0.216785 −0.108392 0.994108i \(-0.534570\pi\)
−0.108392 + 0.994108i \(0.534570\pi\)
\(480\) 0 0
\(481\) −36.6060 −1.66909
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 15.8614 0.720229
\(486\) 0 0
\(487\) −23.1168 −1.04752 −0.523762 0.851865i \(-0.675472\pi\)
−0.523762 + 0.851865i \(0.675472\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −14.9783 −0.675959 −0.337979 0.941153i \(-0.609743\pi\)
−0.337979 + 0.941153i \(0.609743\pi\)
\(492\) 0 0
\(493\) 24.4674 1.10196
\(494\) 0 0
\(495\) 17.4891 0.786078
\(496\) 0 0
\(497\) −3.25544 −0.146026
\(498\) 0 0
\(499\) 35.7228 1.59917 0.799586 0.600551i \(-0.205052\pi\)
0.799586 + 0.600551i \(0.205052\pi\)
\(500\) 0 0
\(501\) −17.4891 −0.781356
\(502\) 0 0
\(503\) 12.7446 0.568252 0.284126 0.958787i \(-0.408297\pi\)
0.284126 + 0.958787i \(0.408297\pi\)
\(504\) 0 0
\(505\) −32.7446 −1.45711
\(506\) 0 0
\(507\) 6.11684 0.271659
\(508\) 0 0
\(509\) −8.23369 −0.364952 −0.182476 0.983210i \(-0.558411\pi\)
−0.182476 + 0.983210i \(0.558411\pi\)
\(510\) 0 0
\(511\) −1.25544 −0.0555373
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 31.1168 1.37117
\(516\) 0 0
\(517\) 41.4891 1.82469
\(518\) 0 0
\(519\) 15.4891 0.679897
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 5.76631 0.252143 0.126072 0.992021i \(-0.459763\pi\)
0.126072 + 0.992021i \(0.459763\pi\)
\(524\) 0 0
\(525\) 14.1168 0.616110
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.74456 −0.379482
\(532\) 0 0
\(533\) −39.8614 −1.72659
\(534\) 0 0
\(535\) 17.4891 0.756121
\(536\) 0 0
\(537\) −7.86141 −0.339245
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −16.2337 −0.697941 −0.348970 0.937134i \(-0.613469\pi\)
−0.348970 + 0.937134i \(0.613469\pi\)
\(542\) 0 0
\(543\) 24.9783 1.07192
\(544\) 0 0
\(545\) 39.8614 1.70748
\(546\) 0 0
\(547\) 32.7446 1.40006 0.700028 0.714115i \(-0.253171\pi\)
0.700028 + 0.714115i \(0.253171\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −14.5109 −0.618184
\(552\) 0 0
\(553\) −9.48913 −0.403519
\(554\) 0 0
\(555\) −36.6060 −1.55384
\(556\) 0 0
\(557\) 20.2337 0.857329 0.428664 0.903464i \(-0.358984\pi\)
0.428664 + 0.903464i \(0.358984\pi\)
\(558\) 0 0
\(559\) −48.6060 −2.05581
\(560\) 0 0
\(561\) −26.9783 −1.13902
\(562\) 0 0
\(563\) −30.3723 −1.28004 −0.640020 0.768359i \(-0.721074\pi\)
−0.640020 + 0.768359i \(0.721074\pi\)
\(564\) 0 0
\(565\) −4.88316 −0.205436
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −37.1168 −1.55602 −0.778010 0.628252i \(-0.783770\pi\)
−0.778010 + 0.628252i \(0.783770\pi\)
\(570\) 0 0
\(571\) −22.9783 −0.961610 −0.480805 0.876828i \(-0.659655\pi\)
−0.480805 + 0.876828i \(0.659655\pi\)
\(572\) 0 0
\(573\) −25.4891 −1.06482
\(574\) 0 0
\(575\) 14.1168 0.588713
\(576\) 0 0
\(577\) −26.4674 −1.10185 −0.550926 0.834554i \(-0.685725\pi\)
−0.550926 + 0.834554i \(0.685725\pi\)
\(578\) 0 0
\(579\) 17.1168 0.711352
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 26.9783 1.11732
\(584\) 0 0
\(585\) 19.1168 0.790384
\(586\) 0 0
\(587\) −29.4891 −1.21715 −0.608573 0.793498i \(-0.708258\pi\)
−0.608573 + 0.793498i \(0.708258\pi\)
\(588\) 0 0
\(589\) −18.9783 −0.781985
\(590\) 0 0
\(591\) −15.6277 −0.642838
\(592\) 0 0
\(593\) −8.37228 −0.343808 −0.171904 0.985114i \(-0.554992\pi\)
−0.171904 + 0.985114i \(0.554992\pi\)
\(594\) 0 0
\(595\) −29.4891 −1.20894
\(596\) 0 0
\(597\) −11.8614 −0.485455
\(598\) 0 0
\(599\) −20.4674 −0.836274 −0.418137 0.908384i \(-0.637317\pi\)
−0.418137 + 0.908384i \(0.637317\pi\)
\(600\) 0 0
\(601\) −4.23369 −0.172696 −0.0863479 0.996265i \(-0.527520\pi\)
−0.0863479 + 0.996265i \(0.527520\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −21.8614 −0.888793
\(606\) 0 0
\(607\) −11.2554 −0.456844 −0.228422 0.973562i \(-0.573357\pi\)
−0.228422 + 0.973562i \(0.573357\pi\)
\(608\) 0 0
\(609\) 3.62772 0.147003
\(610\) 0 0
\(611\) 45.3505 1.83469
\(612\) 0 0
\(613\) 45.1168 1.82225 0.911126 0.412128i \(-0.135214\pi\)
0.911126 + 0.412128i \(0.135214\pi\)
\(614\) 0 0
\(615\) −39.8614 −1.60737
\(616\) 0 0
\(617\) −12.5109 −0.503669 −0.251834 0.967770i \(-0.581034\pi\)
−0.251834 + 0.967770i \(0.581034\pi\)
\(618\) 0 0
\(619\) −23.2554 −0.934715 −0.467357 0.884068i \(-0.654794\pi\)
−0.467357 + 0.884068i \(0.654794\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −15.4891 −0.620559
\(624\) 0 0
\(625\) 103.701 4.14804
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) 56.4674 2.25150
\(630\) 0 0
\(631\) −10.9783 −0.437037 −0.218519 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) −24.6060 −0.976458
\(636\) 0 0
\(637\) −4.37228 −0.173236
\(638\) 0 0
\(639\) −3.25544 −0.128783
\(640\) 0 0
\(641\) 31.3505 1.23827 0.619136 0.785284i \(-0.287483\pi\)
0.619136 + 0.785284i \(0.287483\pi\)
\(642\) 0 0
\(643\) 42.2337 1.66553 0.832767 0.553624i \(-0.186755\pi\)
0.832767 + 0.553624i \(0.186755\pi\)
\(644\) 0 0
\(645\) −48.6060 −1.91386
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 34.9783 1.37302
\(650\) 0 0
\(651\) 4.74456 0.185954
\(652\) 0 0
\(653\) 32.0951 1.25598 0.627989 0.778222i \(-0.283878\pi\)
0.627989 + 0.778222i \(0.283878\pi\)
\(654\) 0 0
\(655\) 31.7228 1.23951
\(656\) 0 0
\(657\) −1.25544 −0.0489793
\(658\) 0 0
\(659\) 22.9783 0.895106 0.447553 0.894258i \(-0.352296\pi\)
0.447553 + 0.894258i \(0.352296\pi\)
\(660\) 0 0
\(661\) −19.4891 −0.758039 −0.379020 0.925389i \(-0.623739\pi\)
−0.379020 + 0.925389i \(0.623739\pi\)
\(662\) 0 0
\(663\) −29.4891 −1.14526
\(664\) 0 0
\(665\) 17.4891 0.678199
\(666\) 0 0
\(667\) 3.62772 0.140466
\(668\) 0 0
\(669\) 20.7446 0.802031
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 26.6060 1.02558 0.512792 0.858513i \(-0.328611\pi\)
0.512792 + 0.858513i \(0.328611\pi\)
\(674\) 0 0
\(675\) 14.1168 0.543358
\(676\) 0 0
\(677\) −12.9783 −0.498795 −0.249397 0.968401i \(-0.580233\pi\)
−0.249397 + 0.968401i \(0.580233\pi\)
\(678\) 0 0
\(679\) −3.62772 −0.139219
\(680\) 0 0
\(681\) 11.1168 0.425998
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −39.8614 −1.52303
\(686\) 0 0
\(687\) 10.7446 0.409931
\(688\) 0 0
\(689\) 29.4891 1.12345
\(690\) 0 0
\(691\) 42.8397 1.62970 0.814849 0.579674i \(-0.196820\pi\)
0.814849 + 0.579674i \(0.196820\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) 62.8397 2.38364
\(696\) 0 0
\(697\) 61.4891 2.32907
\(698\) 0 0
\(699\) 0.510875 0.0193231
\(700\) 0 0
\(701\) −40.2337 −1.51961 −0.759803 0.650154i \(-0.774704\pi\)
−0.759803 + 0.650154i \(0.774704\pi\)
\(702\) 0 0
\(703\) −33.4891 −1.26307
\(704\) 0 0
\(705\) 45.3505 1.70800
\(706\) 0 0
\(707\) 7.48913 0.281658
\(708\) 0 0
\(709\) −27.4891 −1.03238 −0.516188 0.856475i \(-0.672649\pi\)
−0.516188 + 0.856475i \(0.672649\pi\)
\(710\) 0 0
\(711\) −9.48913 −0.355870
\(712\) 0 0
\(713\) 4.74456 0.177685
\(714\) 0 0
\(715\) −76.4674 −2.85972
\(716\) 0 0
\(717\) 9.48913 0.354378
\(718\) 0 0
\(719\) 5.62772 0.209878 0.104939 0.994479i \(-0.466535\pi\)
0.104939 + 0.994479i \(0.466535\pi\)
\(720\) 0 0
\(721\) −7.11684 −0.265045
\(722\) 0 0
\(723\) −8.37228 −0.311368
\(724\) 0 0
\(725\) 51.2119 1.90196
\(726\) 0 0
\(727\) −33.4891 −1.24204 −0.621021 0.783794i \(-0.713282\pi\)
−0.621021 + 0.783794i \(0.713282\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 74.9783 2.77317
\(732\) 0 0
\(733\) −16.2337 −0.599605 −0.299802 0.954001i \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(734\) 0 0
\(735\) −4.37228 −0.161274
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 16.7446 0.615959 0.307979 0.951393i \(-0.400347\pi\)
0.307979 + 0.951393i \(0.400347\pi\)
\(740\) 0 0
\(741\) 17.4891 0.642479
\(742\) 0 0
\(743\) 45.9565 1.68598 0.842990 0.537929i \(-0.180793\pi\)
0.842990 + 0.537929i \(0.180793\pi\)
\(744\) 0 0
\(745\) 70.9783 2.60044
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 9.48913 0.346263 0.173132 0.984899i \(-0.444611\pi\)
0.173132 + 0.984899i \(0.444611\pi\)
\(752\) 0 0
\(753\) 19.1168 0.696657
\(754\) 0 0
\(755\) 10.3723 0.377486
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −40.9783 −1.48546 −0.742730 0.669591i \(-0.766469\pi\)
−0.742730 + 0.669591i \(0.766469\pi\)
\(762\) 0 0
\(763\) −9.11684 −0.330052
\(764\) 0 0
\(765\) −29.4891 −1.06618
\(766\) 0 0
\(767\) 38.2337 1.38054
\(768\) 0 0
\(769\) −45.1168 −1.62696 −0.813478 0.581596i \(-0.802428\pi\)
−0.813478 + 0.581596i \(0.802428\pi\)
\(770\) 0 0
\(771\) −16.9783 −0.611457
\(772\) 0 0
\(773\) −1.39403 −0.0501398 −0.0250699 0.999686i \(-0.507981\pi\)
−0.0250699 + 0.999686i \(0.507981\pi\)
\(774\) 0 0
\(775\) 66.9783 2.40593
\(776\) 0 0
\(777\) 8.37228 0.300354
\(778\) 0 0
\(779\) −36.4674 −1.30658
\(780\) 0 0
\(781\) 13.0217 0.465955
\(782\) 0 0
\(783\) 3.62772 0.129644
\(784\) 0 0
\(785\) 99.4456 3.54937
\(786\) 0 0
\(787\) −29.4891 −1.05117 −0.525587 0.850740i \(-0.676154\pi\)
−0.525587 + 0.850740i \(0.676154\pi\)
\(788\) 0 0
\(789\) 7.11684 0.253366
\(790\) 0 0
\(791\) 1.11684 0.0397104
\(792\) 0 0
\(793\) 8.74456 0.310529
\(794\) 0 0
\(795\) 29.4891 1.04587
\(796\) 0 0
\(797\) −21.8614 −0.774371 −0.387185 0.922002i \(-0.626553\pi\)
−0.387185 + 0.922002i \(0.626553\pi\)
\(798\) 0 0
\(799\) −69.9565 −2.47488
\(800\) 0 0
\(801\) −15.4891 −0.547281
\(802\) 0 0
\(803\) 5.02175 0.177214
\(804\) 0 0
\(805\) −4.37228 −0.154103
\(806\) 0 0
\(807\) 24.9783 0.879276
\(808\) 0 0
\(809\) −1.25544 −0.0441388 −0.0220694 0.999756i \(-0.507025\pi\)
−0.0220694 + 0.999756i \(0.507025\pi\)
\(810\) 0 0
\(811\) 20.6060 0.723573 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(812\) 0 0
\(813\) −18.9783 −0.665596
\(814\) 0 0
\(815\) 17.4891 0.612617
\(816\) 0 0
\(817\) −44.4674 −1.55572
\(818\) 0 0
\(819\) −4.37228 −0.152780
\(820\) 0 0
\(821\) −0.510875 −0.0178297 −0.00891483 0.999960i \(-0.502838\pi\)
−0.00891483 + 0.999960i \(0.502838\pi\)
\(822\) 0 0
\(823\) −5.35053 −0.186508 −0.0932539 0.995642i \(-0.529727\pi\)
−0.0932539 + 0.995642i \(0.529727\pi\)
\(824\) 0 0
\(825\) −56.4674 −1.96594
\(826\) 0 0
\(827\) −18.2337 −0.634047 −0.317024 0.948418i \(-0.602683\pi\)
−0.317024 + 0.948418i \(0.602683\pi\)
\(828\) 0 0
\(829\) 47.4891 1.64937 0.824683 0.565596i \(-0.191354\pi\)
0.824683 + 0.565596i \(0.191354\pi\)
\(830\) 0 0
\(831\) 17.2554 0.598584
\(832\) 0 0
\(833\) 6.74456 0.233685
\(834\) 0 0
\(835\) 76.4674 2.64626
\(836\) 0 0
\(837\) 4.74456 0.163996
\(838\) 0 0
\(839\) −44.7446 −1.54475 −0.772377 0.635164i \(-0.780932\pi\)
−0.772377 + 0.635164i \(0.780932\pi\)
\(840\) 0 0
\(841\) −15.8397 −0.546195
\(842\) 0 0
\(843\) −11.6277 −0.400480
\(844\) 0 0
\(845\) −26.7446 −0.920041
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) 24.7446 0.849231
\(850\) 0 0
\(851\) 8.37228 0.286998
\(852\) 0 0
\(853\) −45.5842 −1.56077 −0.780387 0.625297i \(-0.784978\pi\)
−0.780387 + 0.625297i \(0.784978\pi\)
\(854\) 0 0
\(855\) 17.4891 0.598115
\(856\) 0 0
\(857\) −6.88316 −0.235124 −0.117562 0.993066i \(-0.537508\pi\)
−0.117562 + 0.993066i \(0.537508\pi\)
\(858\) 0 0
\(859\) −4.88316 −0.166611 −0.0833056 0.996524i \(-0.526548\pi\)
−0.0833056 + 0.996524i \(0.526548\pi\)
\(860\) 0 0
\(861\) 9.11684 0.310701
\(862\) 0 0
\(863\) 9.48913 0.323014 0.161507 0.986872i \(-0.448365\pi\)
0.161507 + 0.986872i \(0.448365\pi\)
\(864\) 0 0
\(865\) −67.7228 −2.30264
\(866\) 0 0
\(867\) 28.4891 0.967541
\(868\) 0 0
\(869\) 37.9565 1.28759
\(870\) 0 0
\(871\) −17.4891 −0.592596
\(872\) 0 0
\(873\) −3.62772 −0.122780
\(874\) 0 0
\(875\) −39.8614 −1.34756
\(876\) 0 0
\(877\) 12.2337 0.413102 0.206551 0.978436i \(-0.433776\pi\)
0.206551 + 0.978436i \(0.433776\pi\)
\(878\) 0 0
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) 4.97825 0.167722 0.0838608 0.996477i \(-0.473275\pi\)
0.0838608 + 0.996477i \(0.473275\pi\)
\(882\) 0 0
\(883\) 32.7446 1.10194 0.550971 0.834524i \(-0.314257\pi\)
0.550971 + 0.834524i \(0.314257\pi\)
\(884\) 0 0
\(885\) 38.2337 1.28521
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 5.62772 0.188748
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 41.4891 1.38838
\(894\) 0 0
\(895\) 34.3723 1.14894
\(896\) 0 0
\(897\) −4.37228 −0.145986
\(898\) 0 0
\(899\) 17.2119 0.574050
\(900\) 0 0
\(901\) −45.4891 −1.51546
\(902\) 0 0
\(903\) 11.1168 0.369945
\(904\) 0 0
\(905\) −109.212 −3.63033
\(906\) 0 0
\(907\) −22.0951 −0.733656 −0.366828 0.930289i \(-0.619556\pi\)
−0.366828 + 0.930289i \(0.619556\pi\)
\(908\) 0 0
\(909\) 7.48913 0.248399
\(910\) 0 0
\(911\) 48.3288 1.60120 0.800602 0.599196i \(-0.204513\pi\)
0.800602 + 0.599196i \(0.204513\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) 8.74456 0.289086
\(916\) 0 0
\(917\) −7.25544 −0.239596
\(918\) 0 0
\(919\) −45.9565 −1.51597 −0.757983 0.652275i \(-0.773815\pi\)
−0.757983 + 0.652275i \(0.773815\pi\)
\(920\) 0 0
\(921\) −1.62772 −0.0536352
\(922\) 0 0
\(923\) 14.2337 0.468508
\(924\) 0 0
\(925\) 118.190 3.88607
\(926\) 0 0
\(927\) −7.11684 −0.233748
\(928\) 0 0
\(929\) 33.1168 1.08653 0.543264 0.839562i \(-0.317188\pi\)
0.543264 + 0.839562i \(0.317188\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) 17.4891 0.572568
\(934\) 0 0
\(935\) 117.957 3.85759
\(936\) 0 0
\(937\) −27.6277 −0.902558 −0.451279 0.892383i \(-0.649032\pi\)
−0.451279 + 0.892383i \(0.649032\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −60.3723 −1.96808 −0.984040 0.177947i \(-0.943054\pi\)
−0.984040 + 0.177947i \(0.943054\pi\)
\(942\) 0 0
\(943\) 9.11684 0.296885
\(944\) 0 0
\(945\) −4.37228 −0.142230
\(946\) 0 0
\(947\) 57.3505 1.86364 0.931821 0.362918i \(-0.118220\pi\)
0.931821 + 0.362918i \(0.118220\pi\)
\(948\) 0 0
\(949\) 5.48913 0.178185
\(950\) 0 0
\(951\) −1.11684 −0.0362161
\(952\) 0 0
\(953\) 0.510875 0.0165489 0.00827443 0.999966i \(-0.497366\pi\)
0.00827443 + 0.999966i \(0.497366\pi\)
\(954\) 0 0
\(955\) 111.446 3.60630
\(956\) 0 0
\(957\) −14.5109 −0.469070
\(958\) 0 0
\(959\) 9.11684 0.294398
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −74.8397 −2.40917
\(966\) 0 0
\(967\) 30.5109 0.981164 0.490582 0.871395i \(-0.336784\pi\)
0.490582 + 0.871395i \(0.336784\pi\)
\(968\) 0 0
\(969\) −26.9783 −0.866666
\(970\) 0 0
\(971\) −40.4674 −1.29866 −0.649330 0.760507i \(-0.724951\pi\)
−0.649330 + 0.760507i \(0.724951\pi\)
\(972\) 0 0
\(973\) −14.3723 −0.460754
\(974\) 0 0
\(975\) −61.7228 −1.97671
\(976\) 0 0
\(977\) −43.3505 −1.38691 −0.693453 0.720502i \(-0.743912\pi\)
−0.693453 + 0.720502i \(0.743912\pi\)
\(978\) 0 0
\(979\) 61.9565 1.98014
\(980\) 0 0
\(981\) −9.11684 −0.291078
\(982\) 0 0
\(983\) 38.2337 1.21947 0.609733 0.792607i \(-0.291277\pi\)
0.609733 + 0.792607i \(0.291277\pi\)
\(984\) 0 0
\(985\) 68.3288 2.17714
\(986\) 0 0
\(987\) −10.3723 −0.330153
\(988\) 0 0
\(989\) 11.1168 0.353495
\(990\) 0 0
\(991\) −9.48913 −0.301432 −0.150716 0.988577i \(-0.548158\pi\)
−0.150716 + 0.988577i \(0.548158\pi\)
\(992\) 0 0
\(993\) 13.4891 0.428064
\(994\) 0 0
\(995\) 51.8614 1.64412
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 0 0
\(999\) 8.37228 0.264887
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 7728.2.a.bh.1.1 2
4.3 odd 2 966.2.a.o.1.1 2
12.11 even 2 2898.2.a.x.1.2 2
28.27 even 2 6762.2.a.cd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.o.1.1 2 4.3 odd 2
2898.2.a.x.1.2 2 12.11 even 2
6762.2.a.cd.1.2 2 28.27 even 2
7728.2.a.bh.1.1 2 1.1 even 1 trivial