Properties

Label 1008.3.d.a.449.1
Level $1008$
Weight $3$
Character 1008.449
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(449,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(2.57794i\) of defining polynomial
Character \(\chi\) \(=\) 1008.449
Dual form 1008.3.d.a.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.89753i q^{5} -2.64575 q^{7} +O(q^{10})\) \(q-8.89753i q^{5} -2.64575 q^{7} +17.7951i q^{11} +2.58301 q^{13} +25.8681i q^{17} -20.0000 q^{19} +17.7951i q^{23} -54.1660 q^{25} -11.9034i q^{29} +17.1660 q^{31} +23.5406i q^{35} +38.0000 q^{37} +15.7338i q^{41} +43.4980 q^{43} -16.9706i q^{47} +7.00000 q^{49} +85.5571i q^{53} +158.332 q^{55} +1.64899i q^{59} +100.332 q^{61} -22.9824i q^{65} -36.6640 q^{67} +17.7951i q^{71} +28.9150 q^{73} -47.0813i q^{77} -118.332 q^{79} -120.443i q^{83} +230.162 q^{85} +139.475i q^{89} -6.83399 q^{91} +177.951i q^{95} +44.4131 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{13} - 80 q^{19} - 132 q^{25} - 16 q^{31} + 152 q^{37} - 80 q^{43} + 28 q^{49} + 464 q^{55} + 232 q^{61} + 192 q^{67} - 96 q^{73} - 304 q^{79} + 328 q^{85} - 112 q^{91} - 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 8.89753i − 1.77951i −0.456443 0.889753i \(-0.650877\pi\)
0.456443 0.889753i \(-0.349123\pi\)
\(6\) 0 0
\(7\) −2.64575 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 17.7951i 1.61773i 0.587993 + 0.808866i \(0.299918\pi\)
−0.587993 + 0.808866i \(0.700082\pi\)
\(12\) 0 0
\(13\) 2.58301 0.198693 0.0993464 0.995053i \(-0.468325\pi\)
0.0993464 + 0.995053i \(0.468325\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25.8681i 1.52165i 0.648956 + 0.760826i \(0.275206\pi\)
−0.648956 + 0.760826i \(0.724794\pi\)
\(18\) 0 0
\(19\) −20.0000 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 17.7951i 0.773698i 0.922143 + 0.386849i \(0.126437\pi\)
−0.922143 + 0.386849i \(0.873563\pi\)
\(24\) 0 0
\(25\) −54.1660 −2.16664
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 11.9034i − 0.410463i −0.978713 0.205232i \(-0.934205\pi\)
0.978713 0.205232i \(-0.0657947\pi\)
\(30\) 0 0
\(31\) 17.1660 0.553742 0.276871 0.960907i \(-0.410702\pi\)
0.276871 + 0.960907i \(0.410702\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 23.5406i 0.672590i
\(36\) 0 0
\(37\) 38.0000 1.02703 0.513514 0.858082i \(-0.328344\pi\)
0.513514 + 0.858082i \(0.328344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 15.7338i 0.383752i 0.981419 + 0.191876i \(0.0614571\pi\)
−0.981419 + 0.191876i \(0.938543\pi\)
\(42\) 0 0
\(43\) 43.4980 1.01158 0.505791 0.862656i \(-0.331201\pi\)
0.505791 + 0.862656i \(0.331201\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 16.9706i − 0.361076i −0.983568 0.180538i \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 85.5571i 1.61429i 0.590356 + 0.807143i \(0.298987\pi\)
−0.590356 + 0.807143i \(0.701013\pi\)
\(54\) 0 0
\(55\) 158.332 2.87876
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.64899i 0.0279489i 0.999902 + 0.0139745i \(0.00444836\pi\)
−0.999902 + 0.0139745i \(0.995552\pi\)
\(60\) 0 0
\(61\) 100.332 1.64479 0.822394 0.568919i \(-0.192638\pi\)
0.822394 + 0.568919i \(0.192638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 22.9824i − 0.353575i
\(66\) 0 0
\(67\) −36.6640 −0.547225 −0.273612 0.961840i \(-0.588219\pi\)
−0.273612 + 0.961840i \(0.588219\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 17.7951i 0.250635i 0.992117 + 0.125317i \(0.0399949\pi\)
−0.992117 + 0.125317i \(0.960005\pi\)
\(72\) 0 0
\(73\) 28.9150 0.396096 0.198048 0.980192i \(-0.436540\pi\)
0.198048 + 0.980192i \(0.436540\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 47.0813i − 0.611445i
\(78\) 0 0
\(79\) −118.332 −1.49787 −0.748937 0.662641i \(-0.769435\pi\)
−0.748937 + 0.662641i \(0.769435\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 120.443i − 1.45112i −0.688159 0.725560i \(-0.741581\pi\)
0.688159 0.725560i \(-0.258419\pi\)
\(84\) 0 0
\(85\) 230.162 2.70779
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 139.475i 1.56713i 0.621309 + 0.783566i \(0.286601\pi\)
−0.621309 + 0.783566i \(0.713399\pi\)
\(90\) 0 0
\(91\) −6.83399 −0.0750988
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 177.951i 1.87316i
\(96\) 0 0
\(97\) 44.4131 0.457867 0.228933 0.973442i \(-0.426476\pi\)
0.228933 + 0.973442i \(0.426476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 31.8799i − 0.315642i −0.987468 0.157821i \(-0.949553\pi\)
0.987468 0.157821i \(-0.0504470\pi\)
\(102\) 0 0
\(103\) −4.50197 −0.0437084 −0.0218542 0.999761i \(-0.506957\pi\)
−0.0218542 + 0.999761i \(0.506957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 172.179i 1.60915i 0.593851 + 0.804575i \(0.297607\pi\)
−0.593851 + 0.804575i \(0.702393\pi\)
\(108\) 0 0
\(109\) −177.830 −1.63147 −0.815734 0.578427i \(-0.803667\pi\)
−0.815734 + 0.578427i \(0.803667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 31.3475i − 0.277411i −0.990334 0.138706i \(-0.955706\pi\)
0.990334 0.138706i \(-0.0442942\pi\)
\(114\) 0 0
\(115\) 158.332 1.37680
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 68.4405i − 0.575131i
\(120\) 0 0
\(121\) −195.664 −1.61706
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 259.505i 2.07604i
\(126\) 0 0
\(127\) −214.332 −1.68765 −0.843827 0.536616i \(-0.819703\pi\)
−0.843827 + 0.536616i \(0.819703\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 91.4488i 0.698082i 0.937107 + 0.349041i \(0.113493\pi\)
−0.937107 + 0.349041i \(0.886507\pi\)
\(132\) 0 0
\(133\) 52.9150 0.397857
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 106.891i 0.780223i 0.920768 + 0.390111i \(0.127563\pi\)
−0.920768 + 0.390111i \(0.872437\pi\)
\(138\) 0 0
\(139\) 121.328 0.872864 0.436432 0.899737i \(-0.356242\pi\)
0.436432 + 0.899737i \(0.356242\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 45.9647i 0.321432i
\(144\) 0 0
\(145\) −105.911 −0.730421
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.6749i 0.118623i 0.998240 + 0.0593117i \(0.0188906\pi\)
−0.998240 + 0.0593117i \(0.981109\pi\)
\(150\) 0 0
\(151\) 50.8340 0.336649 0.168324 0.985732i \(-0.446164\pi\)
0.168324 + 0.985732i \(0.446164\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 152.735i − 0.985388i
\(156\) 0 0
\(157\) 68.9961 0.439465 0.219733 0.975560i \(-0.429481\pi\)
0.219733 + 0.975560i \(0.429481\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 47.0813i − 0.292430i
\(162\) 0 0
\(163\) 166.996 1.02452 0.512258 0.858832i \(-0.328809\pi\)
0.512258 + 0.858832i \(0.328809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 120.443i 0.721215i 0.932718 + 0.360608i \(0.117431\pi\)
−0.932718 + 0.360608i \(0.882569\pi\)
\(168\) 0 0
\(169\) −162.328 −0.960521
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 91.8610i 0.530989i 0.964112 + 0.265494i \(0.0855351\pi\)
−0.964112 + 0.265494i \(0.914465\pi\)
\(174\) 0 0
\(175\) 143.310 0.818913
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 133.291i 0.744643i 0.928104 + 0.372321i \(0.121438\pi\)
−0.928104 + 0.372321i \(0.878562\pi\)
\(180\) 0 0
\(181\) 83.0850 0.459033 0.229517 0.973305i \(-0.426286\pi\)
0.229517 + 0.973305i \(0.426286\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 338.106i − 1.82760i
\(186\) 0 0
\(187\) −460.324 −2.46163
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 41.3616i 0.216553i 0.994121 + 0.108276i \(0.0345331\pi\)
−0.994121 + 0.108276i \(0.965467\pi\)
\(192\) 0 0
\(193\) 134.000 0.694301 0.347150 0.937810i \(-0.387149\pi\)
0.347150 + 0.937810i \(0.387149\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 68.8269i 0.349375i 0.984624 + 0.174688i \(0.0558916\pi\)
−0.984624 + 0.174688i \(0.944108\pi\)
\(198\) 0 0
\(199\) 278.494 1.39947 0.699734 0.714404i \(-0.253302\pi\)
0.699734 + 0.714404i \(0.253302\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 31.4935i 0.155140i
\(204\) 0 0
\(205\) 139.992 0.682888
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 355.901i − 1.70288i
\(210\) 0 0
\(211\) 211.498 1.00236 0.501180 0.865343i \(-0.332899\pi\)
0.501180 + 0.865343i \(0.332899\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 387.025i − 1.80012i
\(216\) 0 0
\(217\) −45.4170 −0.209295
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 66.8174i 0.302341i
\(222\) 0 0
\(223\) −222.494 −0.997731 −0.498866 0.866679i \(-0.666250\pi\)
−0.498866 + 0.866679i \(0.666250\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 101.823i 0.448561i 0.974525 + 0.224281i \(0.0720032\pi\)
−0.974525 + 0.224281i \(0.927997\pi\)
\(228\) 0 0
\(229\) −163.085 −0.712161 −0.356081 0.934455i \(-0.615887\pi\)
−0.356081 + 0.934455i \(0.615887\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 362.858i − 1.55733i −0.627441 0.778664i \(-0.715898\pi\)
0.627441 0.778664i \(-0.284102\pi\)
\(234\) 0 0
\(235\) −150.996 −0.642536
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 177.126i − 0.741113i −0.928810 0.370557i \(-0.879167\pi\)
0.928810 0.370557i \(-0.120833\pi\)
\(240\) 0 0
\(241\) 152.753 0.633830 0.316915 0.948454i \(-0.397353\pi\)
0.316915 + 0.948454i \(0.397353\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 62.2827i − 0.254215i
\(246\) 0 0
\(247\) −51.6601 −0.209150
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 356.382i 1.41985i 0.704278 + 0.709924i \(0.251271\pi\)
−0.704278 + 0.709924i \(0.748729\pi\)
\(252\) 0 0
\(253\) −316.664 −1.25164
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 59.5689i − 0.231786i −0.993262 0.115893i \(-0.963027\pi\)
0.993262 0.115893i \(-0.0369729\pi\)
\(258\) 0 0
\(259\) −100.539 −0.388180
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 7.42045i − 0.0282146i −0.999900 0.0141073i \(-0.995509\pi\)
0.999900 0.0141073i \(-0.00449065\pi\)
\(264\) 0 0
\(265\) 761.247 2.87263
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.207i 1.59928i 0.600477 + 0.799642i \(0.294977\pi\)
−0.600477 + 0.799642i \(0.705023\pi\)
\(270\) 0 0
\(271\) 41.1660 0.151904 0.0759520 0.997111i \(-0.475800\pi\)
0.0759520 + 0.997111i \(0.475800\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 963.887i − 3.50504i
\(276\) 0 0
\(277\) 32.0000 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 17.0907i − 0.0608211i −0.999537 0.0304106i \(-0.990319\pi\)
0.999537 0.0304106i \(-0.00968147\pi\)
\(282\) 0 0
\(283\) −439.660 −1.55357 −0.776785 0.629766i \(-0.783151\pi\)
−0.776785 + 0.629766i \(0.783151\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 41.6278i − 0.145045i
\(288\) 0 0
\(289\) −380.158 −1.31543
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 394.377i 1.34600i 0.739644 + 0.672998i \(0.234994\pi\)
−0.739644 + 0.672998i \(0.765006\pi\)
\(294\) 0 0
\(295\) 14.6719 0.0497353
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 45.9647i 0.153728i
\(300\) 0 0
\(301\) −115.085 −0.382342
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 892.707i − 2.92691i
\(306\) 0 0
\(307\) 23.3360 0.0760129 0.0380064 0.999277i \(-0.487899\pi\)
0.0380064 + 0.999277i \(0.487899\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 527.256i − 1.69536i −0.530511 0.847678i \(-0.678000\pi\)
0.530511 0.847678i \(-0.322000\pi\)
\(312\) 0 0
\(313\) −295.328 −0.943540 −0.471770 0.881722i \(-0.656385\pi\)
−0.471770 + 0.881722i \(0.656385\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 107.475i − 0.339037i −0.985527 0.169518i \(-0.945779\pi\)
0.985527 0.169518i \(-0.0542212\pi\)
\(318\) 0 0
\(319\) 211.822 0.664019
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 517.362i − 1.60174i
\(324\) 0 0
\(325\) −139.911 −0.430496
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 44.8999i 0.136474i
\(330\) 0 0
\(331\) 273.490 0.826254 0.413127 0.910673i \(-0.364437\pi\)
0.413127 + 0.910673i \(0.364437\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 326.219i 0.973789i
\(336\) 0 0
\(337\) −341.166 −1.01236 −0.506181 0.862427i \(-0.668943\pi\)
−0.506181 + 0.862427i \(0.668943\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 305.470i 0.895807i
\(342\) 0 0
\(343\) −18.5203 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 116.320i 0.335217i 0.985854 + 0.167609i \(0.0536045\pi\)
−0.985854 + 0.167609i \(0.946395\pi\)
\(348\) 0 0
\(349\) −158.324 −0.453651 −0.226825 0.973935i \(-0.572835\pi\)
−0.226825 + 0.973935i \(0.572835\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 230.339i 0.652519i 0.945280 + 0.326260i \(0.105788\pi\)
−0.945280 + 0.326260i \(0.894212\pi\)
\(354\) 0 0
\(355\) 158.332 0.446006
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 171.698i 0.478269i 0.970987 + 0.239134i \(0.0768636\pi\)
−0.970987 + 0.239134i \(0.923136\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 257.272i − 0.704856i
\(366\) 0 0
\(367\) −517.490 −1.41005 −0.705027 0.709180i \(-0.749065\pi\)
−0.705027 + 0.709180i \(0.749065\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 226.363i − 0.610143i
\(372\) 0 0
\(373\) −233.336 −0.625566 −0.312783 0.949825i \(-0.601261\pi\)
−0.312783 + 0.949825i \(0.601261\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 30.7466i − 0.0815560i
\(378\) 0 0
\(379\) −441.166 −1.16403 −0.582013 0.813179i \(-0.697735\pi\)
−0.582013 + 0.813179i \(0.697735\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 213.060i − 0.556292i −0.960539 0.278146i \(-0.910280\pi\)
0.960539 0.278146i \(-0.0897200\pi\)
\(384\) 0 0
\(385\) −418.907 −1.08807
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 565.096i 1.45269i 0.687331 + 0.726344i \(0.258782\pi\)
−0.687331 + 0.726344i \(0.741218\pi\)
\(390\) 0 0
\(391\) −460.324 −1.17730
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1052.86i 2.66547i
\(396\) 0 0
\(397\) 498.324 1.25522 0.627612 0.778526i \(-0.284032\pi\)
0.627612 + 0.778526i \(0.284032\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 193.392i 0.482275i 0.970491 + 0.241138i \(0.0775205\pi\)
−0.970491 + 0.241138i \(0.922480\pi\)
\(402\) 0 0
\(403\) 44.3399 0.110025
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 676.212i 1.66145i
\(408\) 0 0
\(409\) −454.243 −1.11062 −0.555309 0.831644i \(-0.687400\pi\)
−0.555309 + 0.831644i \(0.687400\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 4.36281i − 0.0105637i
\(414\) 0 0
\(415\) −1071.64 −2.58228
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 339.411i 0.810051i 0.914305 + 0.405025i \(0.132737\pi\)
−0.914305 + 0.405025i \(0.867263\pi\)
\(420\) 0 0
\(421\) 247.320 0.587459 0.293729 0.955889i \(-0.405104\pi\)
0.293729 + 0.955889i \(0.405104\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1401.17i − 3.29687i
\(426\) 0 0
\(427\) −265.454 −0.621671
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 456.419i − 1.05898i −0.848317 0.529489i \(-0.822384\pi\)
0.848317 0.529489i \(-0.177616\pi\)
\(432\) 0 0
\(433\) −637.984 −1.47340 −0.736702 0.676217i \(-0.763618\pi\)
−0.736702 + 0.676217i \(0.763618\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 355.901i − 0.814419i
\(438\) 0 0
\(439\) 784.146 1.78621 0.893105 0.449848i \(-0.148522\pi\)
0.893105 + 0.449848i \(0.148522\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 472.222i − 1.06596i −0.846127 0.532981i \(-0.821072\pi\)
0.846127 0.532981i \(-0.178928\pi\)
\(444\) 0 0
\(445\) 1240.98 2.78872
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 739.852i − 1.64778i −0.566752 0.823888i \(-0.691800\pi\)
0.566752 0.823888i \(-0.308200\pi\)
\(450\) 0 0
\(451\) −279.984 −0.620808
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 60.8056i 0.133639i
\(456\) 0 0
\(457\) −248.324 −0.543379 −0.271689 0.962385i \(-0.587582\pi\)
−0.271689 + 0.962385i \(0.587582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 355.970i 0.772168i 0.922464 + 0.386084i \(0.126173\pi\)
−0.922464 + 0.386084i \(0.873827\pi\)
\(462\) 0 0
\(463\) 6.33202 0.0136761 0.00683804 0.999977i \(-0.497823\pi\)
0.00683804 + 0.999977i \(0.497823\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 878.691i − 1.88156i −0.339011 0.940782i \(-0.610093\pi\)
0.339011 0.940782i \(-0.389907\pi\)
\(468\) 0 0
\(469\) 97.0039 0.206831
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 774.050i 1.63647i
\(474\) 0 0
\(475\) 1083.32 2.28067
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 224.396i − 0.468468i −0.972180 0.234234i \(-0.924742\pi\)
0.972180 0.234234i \(-0.0752581\pi\)
\(480\) 0 0
\(481\) 98.1542 0.204063
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 395.166i − 0.814776i
\(486\) 0 0
\(487\) 717.490 1.47329 0.736643 0.676282i \(-0.236410\pi\)
0.736643 + 0.676282i \(0.236410\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 274.002i − 0.558050i −0.960284 0.279025i \(-0.909989\pi\)
0.960284 0.279025i \(-0.0900112\pi\)
\(492\) 0 0
\(493\) 307.919 0.624582
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 47.0813i − 0.0947310i
\(498\) 0 0
\(499\) 728.810 1.46054 0.730271 0.683158i \(-0.239394\pi\)
0.730271 + 0.683158i \(0.239394\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 594.657i 1.18222i 0.806590 + 0.591111i \(0.201310\pi\)
−0.806590 + 0.591111i \(0.798690\pi\)
\(504\) 0 0
\(505\) −283.652 −0.561688
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 994.015i − 1.95288i −0.215795 0.976439i \(-0.569234\pi\)
0.215795 0.976439i \(-0.430766\pi\)
\(510\) 0 0
\(511\) −76.5020 −0.149710
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 40.0564i 0.0777794i
\(516\) 0 0
\(517\) 301.992 0.584124
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 40.8459i − 0.0783990i −0.999231 0.0391995i \(-0.987519\pi\)
0.999231 0.0391995i \(-0.0124808\pi\)
\(522\) 0 0
\(523\) 232.000 0.443595 0.221797 0.975093i \(-0.428808\pi\)
0.221797 + 0.975093i \(0.428808\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 444.052i 0.842603i
\(528\) 0 0
\(529\) 212.336 0.401391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 40.6405i 0.0762487i
\(534\) 0 0
\(535\) 1531.97 2.86349
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 124.565i 0.231105i
\(540\) 0 0
\(541\) 250.332 0.462721 0.231360 0.972868i \(-0.425682\pi\)
0.231360 + 0.972868i \(0.425682\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1582.25i 2.90321i
\(546\) 0 0
\(547\) −888.324 −1.62399 −0.811996 0.583662i \(-0.801619\pi\)
−0.811996 + 0.583662i \(0.801619\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 238.069i 0.432066i
\(552\) 0 0
\(553\) 313.077 0.566143
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 316.309i − 0.567879i −0.958842 0.283940i \(-0.908358\pi\)
0.958842 0.283940i \(-0.0916415\pi\)
\(558\) 0 0
\(559\) 112.356 0.200994
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 58.4690i 0.103853i 0.998651 + 0.0519263i \(0.0165361\pi\)
−0.998651 + 0.0519263i \(0.983464\pi\)
\(564\) 0 0
\(565\) −278.915 −0.493655
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 221.665i − 0.389570i −0.980846 0.194785i \(-0.937599\pi\)
0.980846 0.194785i \(-0.0624009\pi\)
\(570\) 0 0
\(571\) 487.644 0.854018 0.427009 0.904247i \(-0.359567\pi\)
0.427009 + 0.904247i \(0.359567\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 963.887i − 1.67633i
\(576\) 0 0
\(577\) −487.328 −0.844589 −0.422295 0.906459i \(-0.638775\pi\)
−0.422295 + 0.906459i \(0.638775\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 318.662i 0.548472i
\(582\) 0 0
\(583\) −1522.49 −2.61148
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 445.701i − 0.759286i −0.925133 0.379643i \(-0.876047\pi\)
0.925133 0.379643i \(-0.123953\pi\)
\(588\) 0 0
\(589\) −343.320 −0.582887
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 276.648i 0.466523i 0.972414 + 0.233261i \(0.0749397\pi\)
−0.972414 + 0.233261i \(0.925060\pi\)
\(594\) 0 0
\(595\) −608.952 −1.02345
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 82.3793i − 0.137528i −0.997633 0.0687641i \(-0.978094\pi\)
0.997633 0.0687641i \(-0.0219056\pi\)
\(600\) 0 0
\(601\) −418.000 −0.695507 −0.347754 0.937586i \(-0.613055\pi\)
−0.347754 + 0.937586i \(0.613055\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1740.93i 2.87756i
\(606\) 0 0
\(607\) −76.8419 −0.126593 −0.0632964 0.997995i \(-0.520161\pi\)
−0.0632964 + 0.997995i \(0.520161\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 43.8351i − 0.0717431i
\(612\) 0 0
\(613\) 59.3281 0.0967832 0.0483916 0.998828i \(-0.484590\pi\)
0.0483916 + 0.998828i \(0.484590\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 29.1143i − 0.0471869i −0.999722 0.0235935i \(-0.992489\pi\)
0.999722 0.0235935i \(-0.00751073\pi\)
\(618\) 0 0
\(619\) −455.644 −0.736098 −0.368049 0.929806i \(-0.619974\pi\)
−0.368049 + 0.929806i \(0.619974\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 369.015i − 0.592320i
\(624\) 0 0
\(625\) 954.806 1.52769
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 982.987i 1.56278i
\(630\) 0 0
\(631\) 45.0039 0.0713216 0.0356608 0.999364i \(-0.488646\pi\)
0.0356608 + 0.999364i \(0.488646\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1907.03i 3.00319i
\(636\) 0 0
\(637\) 18.0810 0.0283847
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 641.223i 1.00035i 0.865925 + 0.500174i \(0.166731\pi\)
−0.865925 + 0.500174i \(0.833269\pi\)
\(642\) 0 0
\(643\) 604.000 0.939347 0.469673 0.882840i \(-0.344372\pi\)
0.469673 + 0.882840i \(0.344372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 179.600i − 0.277588i −0.990321 0.138794i \(-0.955677\pi\)
0.990321 0.138794i \(-0.0443226\pi\)
\(648\) 0 0
\(649\) −29.3438 −0.0452139
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 392.092i − 0.600447i −0.953869 0.300224i \(-0.902939\pi\)
0.953869 0.300224i \(-0.0970613\pi\)
\(654\) 0 0
\(655\) 813.668 1.24224
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1266.54i 1.92191i 0.276701 + 0.960956i \(0.410759\pi\)
−0.276701 + 0.960956i \(0.589241\pi\)
\(660\) 0 0
\(661\) 917.644 1.38827 0.694133 0.719846i \(-0.255788\pi\)
0.694133 + 0.719846i \(0.255788\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 470.813i − 0.707989i
\(666\) 0 0
\(667\) 211.822 0.317574
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1785.41i 2.66083i
\(672\) 0 0
\(673\) −152.008 −0.225866 −0.112933 0.993603i \(-0.536025\pi\)
−0.112933 + 0.993603i \(0.536025\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 163.178i − 0.241031i −0.992711 0.120516i \(-0.961545\pi\)
0.992711 0.120516i \(-0.0384548\pi\)
\(678\) 0 0
\(679\) −117.506 −0.173057
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 324.914i − 0.475716i −0.971300 0.237858i \(-0.923555\pi\)
0.971300 0.237858i \(-0.0764453\pi\)
\(684\) 0 0
\(685\) 951.061 1.38841
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 220.995i 0.320747i
\(690\) 0 0
\(691\) −1218.98 −1.76408 −0.882041 0.471173i \(-0.843831\pi\)
−0.882041 + 0.471173i \(0.843831\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1079.52i − 1.55327i
\(696\) 0 0
\(697\) −407.004 −0.583937
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 427.202i 0.609417i 0.952446 + 0.304709i \(0.0985591\pi\)
−0.952446 + 0.304709i \(0.901441\pi\)
\(702\) 0 0
\(703\) −760.000 −1.08108
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 84.3463i 0.119302i
\(708\) 0 0
\(709\) 71.4980 0.100843 0.0504217 0.998728i \(-0.483943\pi\)
0.0504217 + 0.998728i \(0.483943\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 305.470i 0.428429i
\(714\) 0 0
\(715\) 408.972 0.571989
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 111.030i − 0.154422i −0.997015 0.0772112i \(-0.975398\pi\)
0.997015 0.0772112i \(-0.0246016\pi\)
\(720\) 0 0
\(721\) 11.9111 0.0165202
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 644.761i 0.889326i
\(726\) 0 0
\(727\) 1338.82 1.84157 0.920783 0.390076i \(-0.127551\pi\)
0.920783 + 0.390076i \(0.127551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1125.21i 1.53928i
\(732\) 0 0
\(733\) 49.0771 0.0669538 0.0334769 0.999439i \(-0.489342\pi\)
0.0334769 + 0.999439i \(0.489342\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 652.439i − 0.885263i
\(738\) 0 0
\(739\) −1430.32 −1.93548 −0.967738 0.251960i \(-0.918925\pi\)
−0.967738 + 0.251960i \(0.918925\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 875.736i 1.17865i 0.807896 + 0.589325i \(0.200606\pi\)
−0.807896 + 0.589325i \(0.799394\pi\)
\(744\) 0 0
\(745\) 157.263 0.211091
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 455.543i − 0.608202i
\(750\) 0 0
\(751\) −320.826 −0.427199 −0.213599 0.976921i \(-0.568519\pi\)
−0.213599 + 0.976921i \(0.568519\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 452.297i − 0.599069i
\(756\) 0 0
\(757\) 289.830 0.382867 0.191433 0.981506i \(-0.438686\pi\)
0.191433 + 0.981506i \(0.438686\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 704.657i 0.925962i 0.886368 + 0.462981i \(0.153220\pi\)
−0.886368 + 0.462981i \(0.846780\pi\)
\(762\) 0 0
\(763\) 470.494 0.616637
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.25934i 0.00555325i
\(768\) 0 0
\(769\) −117.320 −0.152562 −0.0762810 0.997086i \(-0.524305\pi\)
−0.0762810 + 0.997086i \(0.524305\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 658.005i − 0.851235i −0.904903 0.425618i \(-0.860057\pi\)
0.904903 0.425618i \(-0.139943\pi\)
\(774\) 0 0
\(775\) −929.814 −1.19976
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 314.676i − 0.403949i
\(780\) 0 0
\(781\) −316.664 −0.405460
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 613.894i − 0.782031i
\(786\) 0 0
\(787\) 1200.65 1.52560 0.762801 0.646634i \(-0.223824\pi\)
0.762801 + 0.646634i \(0.223824\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 82.9376i 0.104852i
\(792\) 0 0
\(793\) 259.158 0.326807
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 797.411i − 1.00052i −0.865876 0.500258i \(-0.833239\pi\)
0.865876 0.500258i \(-0.166761\pi\)
\(798\) 0 0
\(799\) 438.996 0.549432
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 514.545i 0.640778i
\(804\) 0 0
\(805\) −418.907 −0.520382
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 156.016i 0.192851i 0.995340 + 0.0964254i \(0.0307409\pi\)
−0.995340 + 0.0964254i \(0.969259\pi\)
\(810\) 0 0
\(811\) 598.316 0.737751 0.368876 0.929479i \(-0.379743\pi\)
0.368876 + 0.929479i \(0.379743\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1485.85i − 1.82313i
\(816\) 0 0
\(817\) −869.961 −1.06482
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 980.978i 1.19486i 0.801922 + 0.597429i \(0.203811\pi\)
−0.801922 + 0.597429i \(0.796189\pi\)
\(822\) 0 0
\(823\) 431.336 0.524102 0.262051 0.965054i \(-0.415601\pi\)
0.262051 + 0.965054i \(0.415601\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1219.41i − 1.47449i −0.675623 0.737247i \(-0.736125\pi\)
0.675623 0.737247i \(-0.263875\pi\)
\(828\) 0 0
\(829\) −770.081 −0.928928 −0.464464 0.885592i \(-0.653753\pi\)
−0.464464 + 0.885592i \(0.653753\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 181.077i 0.217379i
\(834\) 0 0
\(835\) 1071.64 1.28341
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1310.03i 1.56142i 0.624894 + 0.780710i \(0.285142\pi\)
−0.624894 + 0.780710i \(0.714858\pi\)
\(840\) 0 0
\(841\) 699.308 0.831520
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1444.32i 1.70925i
\(846\) 0 0
\(847\) 517.678 0.611191
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 676.212i 0.794609i
\(852\) 0 0
\(853\) 898.988 1.05391 0.526957 0.849892i \(-0.323333\pi\)
0.526957 + 0.849892i \(0.323333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 746.156i 0.870660i 0.900271 + 0.435330i \(0.143368\pi\)
−0.900271 + 0.435330i \(0.856632\pi\)
\(858\) 0 0
\(859\) 991.984 1.15481 0.577406 0.816457i \(-0.304065\pi\)
0.577406 + 0.816457i \(0.304065\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 209.418i − 0.242663i −0.992612 0.121332i \(-0.961284\pi\)
0.992612 0.121332i \(-0.0387164\pi\)
\(864\) 0 0
\(865\) 817.336 0.944897
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 2105.73i − 2.42316i
\(870\) 0 0
\(871\) −94.7034 −0.108730
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 686.587i − 0.784671i
\(876\) 0 0
\(877\) −865.304 −0.986664 −0.493332 0.869841i \(-0.664221\pi\)
−0.493332 + 0.869841i \(0.664221\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 995.046i 1.12945i 0.825279 + 0.564725i \(0.191018\pi\)
−0.825279 + 0.564725i \(0.808982\pi\)
\(882\) 0 0
\(883\) −101.474 −0.114920 −0.0574600 0.998348i \(-0.518300\pi\)
−0.0574600 + 0.998348i \(0.518300\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1074.09i − 1.21093i −0.795873 0.605464i \(-0.792988\pi\)
0.795873 0.605464i \(-0.207012\pi\)
\(888\) 0 0
\(889\) 567.069 0.637873
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 339.411i 0.380080i
\(894\) 0 0
\(895\) 1185.96 1.32510
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 204.334i − 0.227291i
\(900\) 0 0
\(901\) −2213.20 −2.45638
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 739.251i − 0.816852i
\(906\) 0 0
\(907\) 432.162 0.476474 0.238237 0.971207i \(-0.423430\pi\)
0.238237 + 0.971207i \(0.423430\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 104.984i 0.115241i 0.998339 + 0.0576205i \(0.0183513\pi\)
−0.998339 + 0.0576205i \(0.981649\pi\)
\(912\) 0 0
\(913\) 2143.29 2.34752
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 241.951i − 0.263850i
\(918\) 0 0
\(919\) 91.8379 0.0999325 0.0499662 0.998751i \(-0.484089\pi\)
0.0499662 + 0.998751i \(0.484089\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 45.9647i 0.0497993i
\(924\) 0 0
\(925\) −2058.31 −2.22520
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1309.00i − 1.40904i −0.709682 0.704522i \(-0.751162\pi\)
0.709682 0.704522i \(-0.248838\pi\)
\(930\) 0 0
\(931\) −140.000 −0.150376
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4095.75i 4.38048i
\(936\) 0 0
\(937\) 1262.00 1.34685 0.673426 0.739255i \(-0.264822\pi\)
0.673426 + 0.739255i \(0.264822\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1315.97i − 1.39849i −0.714884 0.699243i \(-0.753521\pi\)
0.714884 0.699243i \(-0.246479\pi\)
\(942\) 0 0
\(943\) −279.984 −0.296908
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 486.582i − 0.513814i −0.966436 0.256907i \(-0.917297\pi\)
0.966436 0.256907i \(-0.0827034\pi\)
\(948\) 0 0
\(949\) 74.6877 0.0787014
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 43.3711i 0.0455100i 0.999741 + 0.0227550i \(0.00724377\pi\)
−0.999741 + 0.0227550i \(0.992756\pi\)
\(954\) 0 0
\(955\) 368.016 0.385357
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 282.806i − 0.294896i
\(960\) 0 0
\(961\) −666.328 −0.693369
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1192.27i − 1.23551i
\(966\) 0 0
\(967\) −1648.99 −1.70526 −0.852631 0.522514i \(-0.824994\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 518.323i − 0.533803i −0.963724 0.266902i \(-0.914000\pi\)
0.963724 0.266902i \(-0.0859999\pi\)
\(972\) 0 0
\(973\) −321.004 −0.329912
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 109.948i 0.112536i 0.998416 + 0.0562682i \(0.0179202\pi\)
−0.998416 + 0.0562682i \(0.982080\pi\)
\(978\) 0 0
\(979\) −2481.96 −2.53520
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 589.710i 0.599909i 0.953954 + 0.299954i \(0.0969715\pi\)
−0.953954 + 0.299954i \(0.903029\pi\)
\(984\) 0 0
\(985\) 612.389 0.621715
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 774.050i 0.782659i
\(990\) 0 0
\(991\) −713.474 −0.719954 −0.359977 0.932961i \(-0.617215\pi\)
−0.359977 + 0.932961i \(0.617215\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 2477.91i − 2.49036i
\(996\) 0 0
\(997\) 1222.99 1.22667 0.613334 0.789824i \(-0.289828\pi\)
0.613334 + 0.789824i \(0.289828\pi\)
\(998\) 0 0
\(999\) 0 0
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 1008.3.d.a.449.1 4
3.2 odd 2 inner 1008.3.d.a.449.4 4
4.3 odd 2 126.3.b.a.71.3 yes 4
8.3 odd 2 4032.3.d.i.449.4 4
8.5 even 2 4032.3.d.j.449.4 4
12.11 even 2 126.3.b.a.71.2 4
20.3 even 4 3150.3.c.b.449.5 8
20.7 even 4 3150.3.c.b.449.3 8
20.19 odd 2 3150.3.e.e.701.1 4
24.5 odd 2 4032.3.d.j.449.1 4
24.11 even 2 4032.3.d.i.449.1 4
28.3 even 6 882.3.s.i.863.1 8
28.11 odd 6 882.3.s.e.863.2 8
28.19 even 6 882.3.s.i.557.4 8
28.23 odd 6 882.3.s.e.557.3 8
28.27 even 2 882.3.b.f.197.4 4
36.7 odd 6 1134.3.q.c.701.4 8
36.11 even 6 1134.3.q.c.701.1 8
36.23 even 6 1134.3.q.c.1079.4 8
36.31 odd 6 1134.3.q.c.1079.1 8
60.23 odd 4 3150.3.c.b.449.2 8
60.47 odd 4 3150.3.c.b.449.8 8
60.59 even 2 3150.3.e.e.701.3 4
84.11 even 6 882.3.s.e.863.3 8
84.23 even 6 882.3.s.e.557.2 8
84.47 odd 6 882.3.s.i.557.1 8
84.59 odd 6 882.3.s.i.863.4 8
84.83 odd 2 882.3.b.f.197.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.2 4 12.11 even 2
126.3.b.a.71.3 yes 4 4.3 odd 2
882.3.b.f.197.1 4 84.83 odd 2
882.3.b.f.197.4 4 28.27 even 2
882.3.s.e.557.2 8 84.23 even 6
882.3.s.e.557.3 8 28.23 odd 6
882.3.s.e.863.2 8 28.11 odd 6
882.3.s.e.863.3 8 84.11 even 6
882.3.s.i.557.1 8 84.47 odd 6
882.3.s.i.557.4 8 28.19 even 6
882.3.s.i.863.1 8 28.3 even 6
882.3.s.i.863.4 8 84.59 odd 6
1008.3.d.a.449.1 4 1.1 even 1 trivial
1008.3.d.a.449.4 4 3.2 odd 2 inner
1134.3.q.c.701.1 8 36.11 even 6
1134.3.q.c.701.4 8 36.7 odd 6
1134.3.q.c.1079.1 8 36.31 odd 6
1134.3.q.c.1079.4 8 36.23 even 6
3150.3.c.b.449.2 8 60.23 odd 4
3150.3.c.b.449.3 8 20.7 even 4
3150.3.c.b.449.5 8 20.3 even 4
3150.3.c.b.449.8 8 60.47 odd 4
3150.3.e.e.701.1 4 20.19 odd 2
3150.3.e.e.701.3 4 60.59 even 2
4032.3.d.i.449.1 4 24.11 even 2
4032.3.d.i.449.4 4 8.3 odd 2
4032.3.d.j.449.1 4 24.5 odd 2
4032.3.d.j.449.4 4 8.5 even 2