Properties

Label 4400.2.b.p.4049.1
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4400,2,Mod(4049,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4400.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-3.37228i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.p.4049.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.37228i q^{3} -3.37228i q^{7} -8.37228 q^{9} +1.00000 q^{11} -2.00000i q^{13} +1.37228i q^{17} +0.627719 q^{19} -11.3723 q^{21} +2.74456i q^{23} +18.1168i q^{27} -1.37228 q^{29} -3.37228 q^{31} -3.37228i q^{33} +9.37228i q^{37} -6.74456 q^{39} -11.4891 q^{41} -4.00000i q^{43} -2.74456i q^{47} -4.37228 q^{49} +4.62772 q^{51} +4.11684i q^{53} -2.11684i q^{57} -2.74456 q^{59} -5.37228 q^{61} +28.2337i q^{63} -8.00000i q^{67} +9.25544 q^{69} -10.1168 q^{71} +15.4891i q^{73} -3.37228i q^{77} -1.25544 q^{79} +35.9783 q^{81} -2.74456i q^{83} +4.62772i q^{87} +1.37228 q^{89} -6.74456 q^{91} +11.3723i q^{93} -12.7446i q^{97} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{9} + 4 q^{11} + 14 q^{19} - 34 q^{21} + 6 q^{29} - 2 q^{31} - 4 q^{39} - 6 q^{49} + 30 q^{51} + 12 q^{59} - 10 q^{61} + 60 q^{69} - 6 q^{71} - 28 q^{79} + 52 q^{81} - 6 q^{89} - 4 q^{91} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 3.37228i − 1.94699i −0.228714 0.973494i \(-0.573452\pi\)
0.228714 0.973494i \(-0.426548\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.37228i − 1.27460i −0.770615 0.637301i \(-0.780051\pi\)
0.770615 0.637301i \(-0.219949\pi\)
\(8\) 0 0
\(9\) −8.37228 −2.79076
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.37228i 0.332827i 0.986056 + 0.166414i \(0.0532187\pi\)
−0.986056 + 0.166414i \(0.946781\pi\)
\(18\) 0 0
\(19\) 0.627719 0.144009 0.0720043 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 0 0
\(21\) −11.3723 −2.48164
\(22\) 0 0
\(23\) 2.74456i 0.572281i 0.958188 + 0.286140i \(0.0923724\pi\)
−0.958188 + 0.286140i \(0.907628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 18.1168i 3.48659i
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) 0 0
\(33\) − 3.37228i − 0.587039i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.37228i 1.54079i 0.637565 + 0.770397i \(0.279942\pi\)
−0.637565 + 0.770397i \(0.720058\pi\)
\(38\) 0 0
\(39\) −6.74456 −1.07999
\(40\) 0 0
\(41\) −11.4891 −1.79430 −0.897150 0.441726i \(-0.854366\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.74456i − 0.400336i −0.979762 0.200168i \(-0.935851\pi\)
0.979762 0.200168i \(-0.0641487\pi\)
\(48\) 0 0
\(49\) −4.37228 −0.624612
\(50\) 0 0
\(51\) 4.62772 0.648010
\(52\) 0 0
\(53\) 4.11684i 0.565492i 0.959195 + 0.282746i \(0.0912454\pi\)
−0.959195 + 0.282746i \(0.908755\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.11684i − 0.280383i
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 0 0
\(63\) 28.2337i 3.55711i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 9.25544 1.11422
\(70\) 0 0
\(71\) −10.1168 −1.20065 −0.600324 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(72\) 0 0
\(73\) 15.4891i 1.81286i 0.422351 + 0.906432i \(0.361205\pi\)
−0.422351 + 0.906432i \(0.638795\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.37228i − 0.384307i
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) − 2.74456i − 0.301255i −0.988591 0.150627i \(-0.951871\pi\)
0.988591 0.150627i \(-0.0481294\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.62772i 0.496144i
\(88\) 0 0
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) −6.74456 −0.707022
\(92\) 0 0
\(93\) 11.3723i 1.17925i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.7446i − 1.29401i −0.762484 0.647007i \(-0.776020\pi\)
0.762484 0.647007i \(-0.223980\pi\)
\(98\) 0 0
\(99\) −8.37228 −0.841446
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 9.48913i − 0.934991i −0.883995 0.467496i \(-0.845156\pi\)
0.883995 0.467496i \(-0.154844\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 15.4891 1.48359 0.741795 0.670627i \(-0.233975\pi\)
0.741795 + 0.670627i \(0.233975\pi\)
\(110\) 0 0
\(111\) 31.6060 2.99991
\(112\) 0 0
\(113\) − 3.25544i − 0.306246i −0.988207 0.153123i \(-0.951067\pi\)
0.988207 0.153123i \(-0.0489330\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.7446i 1.54804i
\(118\) 0 0
\(119\) 4.62772 0.424222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 38.7446i 3.49348i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) −13.4891 −1.18765
\(130\) 0 0
\(131\) −22.1168 −1.93236 −0.966179 0.257873i \(-0.916978\pi\)
−0.966179 + 0.257873i \(0.916978\pi\)
\(132\) 0 0
\(133\) − 2.11684i − 0.183554i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.74456i 0.747098i 0.927610 + 0.373549i \(0.121859\pi\)
−0.927610 + 0.373549i \(0.878141\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −9.25544 −0.779448
\(142\) 0 0
\(143\) − 2.00000i − 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 14.7446i 1.21611i
\(148\) 0 0
\(149\) −21.6060 −1.77003 −0.885015 0.465563i \(-0.845852\pi\)
−0.885015 + 0.465563i \(0.845852\pi\)
\(150\) 0 0
\(151\) 12.2337 0.995563 0.497782 0.867302i \(-0.334148\pi\)
0.497782 + 0.867302i \(0.334148\pi\)
\(152\) 0 0
\(153\) − 11.4891i − 0.928841i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.37228i 0.747989i 0.927431 + 0.373995i \(0.122012\pi\)
−0.927431 + 0.373995i \(0.877988\pi\)
\(158\) 0 0
\(159\) 13.8832 1.10101
\(160\) 0 0
\(161\) 9.25544 0.729431
\(162\) 0 0
\(163\) − 5.88316i − 0.460804i −0.973095 0.230402i \(-0.925996\pi\)
0.973095 0.230402i \(-0.0740042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.62772i 0.358104i 0.983840 + 0.179052i \(0.0573030\pi\)
−0.983840 + 0.179052i \(0.942697\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −5.25544 −0.401893
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.25544i 0.695681i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 18.1168i 1.33924i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.37228i 0.100351i
\(188\) 0 0
\(189\) 61.0951 4.44401
\(190\) 0 0
\(191\) 5.48913 0.397179 0.198590 0.980083i \(-0.436364\pi\)
0.198590 + 0.980083i \(0.436364\pi\)
\(192\) 0 0
\(193\) − 14.8614i − 1.06975i −0.844932 0.534874i \(-0.820359\pi\)
0.844932 0.534874i \(-0.179641\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 20.7446i − 1.47799i −0.673712 0.738994i \(-0.735301\pi\)
0.673712 0.738994i \(-0.264699\pi\)
\(198\) 0 0
\(199\) 18.1168 1.28427 0.642135 0.766592i \(-0.278049\pi\)
0.642135 + 0.766592i \(0.278049\pi\)
\(200\) 0 0
\(201\) −26.9783 −1.90290
\(202\) 0 0
\(203\) 4.62772i 0.324802i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 22.9783i − 1.59710i
\(208\) 0 0
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) −6.11684 −0.421101 −0.210550 0.977583i \(-0.567526\pi\)
−0.210550 + 0.977583i \(0.567526\pi\)
\(212\) 0 0
\(213\) 34.1168i 2.33765i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 11.3723i 0.772001i
\(218\) 0 0
\(219\) 52.2337 3.52963
\(220\) 0 0
\(221\) 2.74456 0.184619
\(222\) 0 0
\(223\) − 18.7446i − 1.25523i −0.778524 0.627614i \(-0.784031\pi\)
0.778524 0.627614i \(-0.215969\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.74456i 0.182163i 0.995843 + 0.0910815i \(0.0290324\pi\)
−0.995843 + 0.0910815i \(0.970968\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −11.3723 −0.748241
\(232\) 0 0
\(233\) − 1.37228i − 0.0899011i −0.998989 0.0449506i \(-0.985687\pi\)
0.998989 0.0449506i \(-0.0143130\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.23369i 0.275008i
\(238\) 0 0
\(239\) −14.7446 −0.953746 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) − 66.9783i − 4.29666i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.25544i − 0.0798816i
\(248\) 0 0
\(249\) −9.25544 −0.586540
\(250\) 0 0
\(251\) −2.74456 −0.173235 −0.0866176 0.996242i \(-0.527606\pi\)
−0.0866176 + 0.996242i \(0.527606\pi\)
\(252\) 0 0
\(253\) 2.74456i 0.172549i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 31.6060 1.96390
\(260\) 0 0
\(261\) 11.4891 0.711159
\(262\) 0 0
\(263\) − 24.8614i − 1.53302i −0.642232 0.766510i \(-0.721992\pi\)
0.642232 0.766510i \(-0.278008\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.62772i − 0.283212i
\(268\) 0 0
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 22.7446i 1.37656i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.7446i − 0.765747i −0.923801 0.382873i \(-0.874935\pi\)
0.923801 0.382873i \(-0.125065\pi\)
\(278\) 0 0
\(279\) 28.2337 1.69031
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 5.25544i 0.312403i 0.987725 + 0.156202i \(0.0499250\pi\)
−0.987725 + 0.156202i \(0.950075\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 38.7446i 2.28702i
\(288\) 0 0
\(289\) 15.1168 0.889226
\(290\) 0 0
\(291\) −42.9783 −2.51943
\(292\) 0 0
\(293\) − 23.4891i − 1.37225i −0.727484 0.686125i \(-0.759310\pi\)
0.727484 0.686125i \(-0.240690\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.1168i 1.05125i
\(298\) 0 0
\(299\) 5.48913 0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 0 0
\(303\) − 20.2337i − 1.16240i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5.25544i − 0.299944i −0.988690 0.149972i \(-0.952082\pi\)
0.988690 0.149972i \(-0.0479183\pi\)
\(308\) 0 0
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) −19.3723 −1.09850 −0.549251 0.835658i \(-0.685087\pi\)
−0.549251 + 0.835658i \(0.685087\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 24.3505i − 1.36766i −0.729640 0.683831i \(-0.760313\pi\)
0.729640 0.683831i \(-0.239687\pi\)
\(318\) 0 0
\(319\) −1.37228 −0.0768330
\(320\) 0 0
\(321\) 40.4674 2.25867
\(322\) 0 0
\(323\) 0.861407i 0.0479299i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 52.2337i − 2.88853i
\(328\) 0 0
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) −30.9783 −1.70272 −0.851359 0.524583i \(-0.824221\pi\)
−0.851359 + 0.524583i \(0.824221\pi\)
\(332\) 0 0
\(333\) − 78.4674i − 4.29999i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.1168i 1.31373i 0.754009 + 0.656864i \(0.228117\pi\)
−0.754009 + 0.656864i \(0.771883\pi\)
\(338\) 0 0
\(339\) −10.9783 −0.596257
\(340\) 0 0
\(341\) −3.37228 −0.182619
\(342\) 0 0
\(343\) − 8.86141i − 0.478471i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.2337i 1.73040i 0.501431 + 0.865198i \(0.332807\pi\)
−0.501431 + 0.865198i \(0.667193\pi\)
\(348\) 0 0
\(349\) −19.4891 −1.04323 −0.521614 0.853181i \(-0.674670\pi\)
−0.521614 + 0.853181i \(0.674670\pi\)
\(350\) 0 0
\(351\) 36.2337 1.93401
\(352\) 0 0
\(353\) 0.510875i 0.0271911i 0.999908 + 0.0135956i \(0.00432773\pi\)
−0.999908 + 0.0135956i \(0.995672\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 15.6060i − 0.825955i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) 0 0
\(363\) − 3.37228i − 0.176999i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) 0 0
\(369\) 96.1902 5.00746
\(370\) 0 0
\(371\) 13.8832 0.720778
\(372\) 0 0
\(373\) − 31.4891i − 1.63045i −0.579148 0.815223i \(-0.696615\pi\)
0.579148 0.815223i \(-0.303385\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.74456i 0.141352i
\(378\) 0 0
\(379\) −0.233688 −0.0120037 −0.00600187 0.999982i \(-0.501910\pi\)
−0.00600187 + 0.999982i \(0.501910\pi\)
\(380\) 0 0
\(381\) −26.9783 −1.38214
\(382\) 0 0
\(383\) 32.2337i 1.64706i 0.567269 + 0.823532i \(0.308000\pi\)
−0.567269 + 0.823532i \(0.692000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 33.4891i 1.70235i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −3.76631 −0.190471
\(392\) 0 0
\(393\) 74.5842i 3.76228i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 24.9783i 1.25362i 0.779171 + 0.626811i \(0.215640\pi\)
−0.779171 + 0.626811i \(0.784360\pi\)
\(398\) 0 0
\(399\) −7.13859 −0.357377
\(400\) 0 0
\(401\) 13.3723 0.667780 0.333890 0.942612i \(-0.391639\pi\)
0.333890 + 0.942612i \(0.391639\pi\)
\(402\) 0 0
\(403\) 6.74456i 0.335971i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.37228i 0.464567i
\(408\) 0 0
\(409\) 1.76631 0.0873385 0.0436693 0.999046i \(-0.486095\pi\)
0.0436693 + 0.999046i \(0.486095\pi\)
\(410\) 0 0
\(411\) 29.4891 1.45459
\(412\) 0 0
\(413\) 9.25544i 0.455430i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.4891i 0.660565i
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 10.2337 0.498759 0.249380 0.968406i \(-0.419773\pi\)
0.249380 + 0.968406i \(0.419773\pi\)
\(422\) 0 0
\(423\) 22.9783i 1.11724i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.1168i 0.876736i
\(428\) 0 0
\(429\) −6.74456 −0.325631
\(430\) 0 0
\(431\) 34.9783 1.68484 0.842422 0.538819i \(-0.181129\pi\)
0.842422 + 0.538819i \(0.181129\pi\)
\(432\) 0 0
\(433\) − 27.7228i − 1.33227i −0.745830 0.666137i \(-0.767947\pi\)
0.745830 0.666137i \(-0.232053\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.72281i 0.0824133i
\(438\) 0 0
\(439\) 18.9783 0.905782 0.452891 0.891566i \(-0.350393\pi\)
0.452891 + 0.891566i \(0.350393\pi\)
\(440\) 0 0
\(441\) 36.6060 1.74314
\(442\) 0 0
\(443\) 29.4891i 1.40107i 0.713618 + 0.700535i \(0.247055\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 72.8614i 3.44623i
\(448\) 0 0
\(449\) −28.9783 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(450\) 0 0
\(451\) −11.4891 −0.541002
\(452\) 0 0
\(453\) − 41.2554i − 1.93835i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16.3505i − 0.764846i −0.923987 0.382423i \(-0.875090\pi\)
0.923987 0.382423i \(-0.124910\pi\)
\(458\) 0 0
\(459\) −24.8614 −1.16043
\(460\) 0 0
\(461\) 16.1168 0.750636 0.375318 0.926896i \(-0.377533\pi\)
0.375318 + 0.926896i \(0.377533\pi\)
\(462\) 0 0
\(463\) − 0.233688i − 0.0108604i −0.999985 0.00543020i \(-0.998272\pi\)
0.999985 0.00543020i \(-0.00172849\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.3723i 0.896442i 0.893923 + 0.448221i \(0.147942\pi\)
−0.893923 + 0.448221i \(0.852058\pi\)
\(468\) 0 0
\(469\) −26.9783 −1.24574
\(470\) 0 0
\(471\) 31.6060 1.45633
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 34.4674i − 1.57815i
\(478\) 0 0
\(479\) 5.48913 0.250805 0.125402 0.992106i \(-0.459978\pi\)
0.125402 + 0.992106i \(0.459978\pi\)
\(480\) 0 0
\(481\) 18.7446 0.854678
\(482\) 0 0
\(483\) − 31.2119i − 1.42019i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 0 0
\(489\) −19.8397 −0.897180
\(490\) 0 0
\(491\) 7.37228 0.332706 0.166353 0.986066i \(-0.446801\pi\)
0.166353 + 0.986066i \(0.446801\pi\)
\(492\) 0 0
\(493\) − 1.88316i − 0.0848131i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.1168i 1.53035i
\(498\) 0 0
\(499\) −33.4891 −1.49918 −0.749590 0.661903i \(-0.769749\pi\)
−0.749590 + 0.661903i \(0.769749\pi\)
\(500\) 0 0
\(501\) 15.6060 0.697223
\(502\) 0 0
\(503\) − 34.9783i − 1.55960i −0.626027 0.779802i \(-0.715320\pi\)
0.626027 0.779802i \(-0.284680\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 30.3505i − 1.34791i
\(508\) 0 0
\(509\) −9.76631 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(510\) 0 0
\(511\) 52.2337 2.31068
\(512\) 0 0
\(513\) 11.3723i 0.502098i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.74456i − 0.120706i
\(518\) 0 0
\(519\) 20.2337 0.888160
\(520\) 0 0
\(521\) 12.5109 0.548111 0.274056 0.961714i \(-0.411635\pi\)
0.274056 + 0.961714i \(0.411635\pi\)
\(522\) 0 0
\(523\) 30.9783i 1.35458i 0.735714 + 0.677292i \(0.236847\pi\)
−0.735714 + 0.677292i \(0.763153\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.62772i − 0.201587i
\(528\) 0 0
\(529\) 15.4674 0.672495
\(530\) 0 0
\(531\) 22.9783 0.997171
\(532\) 0 0
\(533\) 22.9783i 0.995299i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 40.4674i 1.74630i
\(538\) 0 0
\(539\) −4.37228 −0.188327
\(540\) 0 0
\(541\) −20.1168 −0.864891 −0.432445 0.901660i \(-0.642349\pi\)
−0.432445 + 0.901660i \(0.642349\pi\)
\(542\) 0 0
\(543\) 33.7228i 1.44718i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20.0000i − 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 44.9783 1.91962
\(550\) 0 0
\(551\) −0.861407 −0.0366972
\(552\) 0 0
\(553\) 4.23369i 0.180035i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.97825i 0.210935i 0.994423 + 0.105468i \(0.0336339\pi\)
−0.994423 + 0.105468i \(0.966366\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 4.62772 0.195382
\(562\) 0 0
\(563\) − 8.23369i − 0.347009i −0.984833 0.173504i \(-0.944491\pi\)
0.984833 0.173504i \(-0.0555090\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 121.329i − 5.09533i
\(568\) 0 0
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) −15.3723 −0.643310 −0.321655 0.946857i \(-0.604239\pi\)
−0.321655 + 0.946857i \(0.604239\pi\)
\(572\) 0 0
\(573\) − 18.5109i − 0.773303i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 36.9783i 1.53942i 0.638391 + 0.769712i \(0.279600\pi\)
−0.638391 + 0.769712i \(0.720400\pi\)
\(578\) 0 0
\(579\) −50.1168 −2.08278
\(580\) 0 0
\(581\) −9.25544 −0.383980
\(582\) 0 0
\(583\) 4.11684i 0.170502i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 24.8614i − 1.02614i −0.858347 0.513070i \(-0.828508\pi\)
0.858347 0.513070i \(-0.171492\pi\)
\(588\) 0 0
\(589\) −2.11684 −0.0872230
\(590\) 0 0
\(591\) −69.9565 −2.87763
\(592\) 0 0
\(593\) 12.5109i 0.513760i 0.966443 + 0.256880i \(0.0826945\pi\)
−0.966443 + 0.256880i \(0.917305\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 61.0951i − 2.50046i
\(598\) 0 0
\(599\) −39.6060 −1.61826 −0.809128 0.587632i \(-0.800060\pi\)
−0.809128 + 0.587632i \(0.800060\pi\)
\(600\) 0 0
\(601\) −16.5109 −0.673493 −0.336746 0.941595i \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) 0 0
\(603\) 66.9783i 2.72757i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.88316i 0.238790i 0.992847 + 0.119395i \(0.0380955\pi\)
−0.992847 + 0.119395i \(0.961905\pi\)
\(608\) 0 0
\(609\) 15.6060 0.632386
\(610\) 0 0
\(611\) −5.48913 −0.222066
\(612\) 0 0
\(613\) − 20.5109i − 0.828426i −0.910180 0.414213i \(-0.864057\pi\)
0.910180 0.414213i \(-0.135943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.23369i − 0.0899249i −0.998989 0.0449624i \(-0.985683\pi\)
0.998989 0.0449624i \(-0.0143168\pi\)
\(618\) 0 0
\(619\) −44.4674 −1.78729 −0.893647 0.448770i \(-0.851862\pi\)
−0.893647 + 0.448770i \(0.851862\pi\)
\(620\) 0 0
\(621\) −49.7228 −1.99531
\(622\) 0 0
\(623\) − 4.62772i − 0.185406i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 2.11684i − 0.0845386i
\(628\) 0 0
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) −42.1168 −1.67665 −0.838323 0.545175i \(-0.816463\pi\)
−0.838323 + 0.545175i \(0.816463\pi\)
\(632\) 0 0
\(633\) 20.6277i 0.819878i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.74456i 0.346472i
\(638\) 0 0
\(639\) 84.7011 3.35072
\(640\) 0 0
\(641\) −27.0951 −1.07019 −0.535096 0.844791i \(-0.679725\pi\)
−0.535096 + 0.844791i \(0.679725\pi\)
\(642\) 0 0
\(643\) − 5.88316i − 0.232009i −0.993249 0.116005i \(-0.962991\pi\)
0.993249 0.116005i \(-0.0370087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 37.7228i 1.48304i 0.670933 + 0.741518i \(0.265894\pi\)
−0.670933 + 0.741518i \(0.734106\pi\)
\(648\) 0 0
\(649\) −2.74456 −0.107734
\(650\) 0 0
\(651\) 38.3505 1.50308
\(652\) 0 0
\(653\) − 10.6277i − 0.415895i −0.978140 0.207947i \(-0.933322\pi\)
0.978140 0.207947i \(-0.0666783\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 129.679i − 5.05927i
\(658\) 0 0
\(659\) 12.8614 0.501009 0.250505 0.968115i \(-0.419403\pi\)
0.250505 + 0.968115i \(0.419403\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) 0 0
\(663\) − 9.25544i − 0.359451i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.76631i − 0.145832i
\(668\) 0 0
\(669\) −63.2119 −2.44391
\(670\) 0 0
\(671\) −5.37228 −0.207395
\(672\) 0 0
\(673\) − 14.8614i − 0.572865i −0.958100 0.286433i \(-0.907531\pi\)
0.958100 0.286433i \(-0.0924694\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.25544i 0.125117i 0.998041 + 0.0625583i \(0.0199259\pi\)
−0.998041 + 0.0625583i \(0.980074\pi\)
\(678\) 0 0
\(679\) −42.9783 −1.64935
\(680\) 0 0
\(681\) 9.25544 0.354669
\(682\) 0 0
\(683\) − 28.6277i − 1.09541i −0.836672 0.547705i \(-0.815502\pi\)
0.836672 0.547705i \(-0.184498\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 33.7228i − 1.28661i
\(688\) 0 0
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) −40.2337 −1.53056 −0.765281 0.643697i \(-0.777400\pi\)
−0.765281 + 0.643697i \(0.777400\pi\)
\(692\) 0 0
\(693\) 28.2337i 1.07251i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 15.7663i − 0.597192i
\(698\) 0 0
\(699\) −4.62772 −0.175036
\(700\) 0 0
\(701\) −37.3723 −1.41153 −0.705766 0.708445i \(-0.749397\pi\)
−0.705766 + 0.708445i \(0.749397\pi\)
\(702\) 0 0
\(703\) 5.88316i 0.221887i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 20.2337i − 0.760966i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 10.5109 0.394189
\(712\) 0 0
\(713\) − 9.25544i − 0.346619i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 49.7228i 1.85693i
\(718\) 0 0
\(719\) 13.8832 0.517754 0.258877 0.965910i \(-0.416648\pi\)
0.258877 + 0.965910i \(0.416648\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 74.1902i 2.75916i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 24.2337i 0.898778i 0.893336 + 0.449389i \(0.148358\pi\)
−0.893336 + 0.449389i \(0.851642\pi\)
\(728\) 0 0
\(729\) −117.935 −4.36795
\(730\) 0 0
\(731\) 5.48913 0.203023
\(732\) 0 0
\(733\) − 46.2337i − 1.70768i −0.520535 0.853840i \(-0.674268\pi\)
0.520535 0.853840i \(-0.325732\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.00000i − 0.294684i
\(738\) 0 0
\(739\) −20.4674 −0.752905 −0.376452 0.926436i \(-0.622856\pi\)
−0.376452 + 0.926436i \(0.622856\pi\)
\(740\) 0 0
\(741\) −4.23369 −0.155528
\(742\) 0 0
\(743\) − 4.62772i − 0.169775i −0.996391 0.0848873i \(-0.972947\pi\)
0.996391 0.0848873i \(-0.0270530\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.9783i 0.840730i
\(748\) 0 0
\(749\) 40.4674 1.47865
\(750\) 0 0
\(751\) −8.86141 −0.323357 −0.161679 0.986843i \(-0.551691\pi\)
−0.161679 + 0.986843i \(0.551691\pi\)
\(752\) 0 0
\(753\) 9.25544i 0.337287i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 20.9783i − 0.762467i −0.924479 0.381234i \(-0.875499\pi\)
0.924479 0.381234i \(-0.124501\pi\)
\(758\) 0 0
\(759\) 9.25544 0.335951
\(760\) 0 0
\(761\) 4.97825 0.180461 0.0902307 0.995921i \(-0.471240\pi\)
0.0902307 + 0.995921i \(0.471240\pi\)
\(762\) 0 0
\(763\) − 52.2337i − 1.89099i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.48913i 0.198201i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 60.7011 2.18610
\(772\) 0 0
\(773\) 33.6060i 1.20872i 0.796710 + 0.604361i \(0.206572\pi\)
−0.796710 + 0.604361i \(0.793428\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 106.584i − 3.82369i
\(778\) 0 0
\(779\) −7.21194 −0.258395
\(780\) 0 0
\(781\) −10.1168 −0.362009
\(782\) 0 0
\(783\) − 24.8614i − 0.888474i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 44.4674i 1.58509i 0.609813 + 0.792545i \(0.291245\pi\)
−0.609813 + 0.792545i \(0.708755\pi\)
\(788\) 0 0
\(789\) −83.8397 −2.98477
\(790\) 0 0
\(791\) −10.9783 −0.390342
\(792\) 0 0
\(793\) 10.7446i 0.381551i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.4891i 0.406966i 0.979079 + 0.203483i \(0.0652261\pi\)
−0.979079 + 0.203483i \(0.934774\pi\)
\(798\) 0 0
\(799\) 3.76631 0.133243
\(800\) 0 0
\(801\) −11.4891 −0.405948
\(802\) 0 0
\(803\) 15.4891i 0.546599i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 29.4891i − 1.03807i
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −44.8614 −1.57530 −0.787649 0.616125i \(-0.788702\pi\)
−0.787649 + 0.616125i \(0.788702\pi\)
\(812\) 0 0
\(813\) − 53.9565i − 1.89234i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 2.51087i − 0.0878444i
\(818\) 0 0
\(819\) 56.4674 1.97313
\(820\) 0 0
\(821\) 11.4891 0.400973 0.200487 0.979696i \(-0.435748\pi\)
0.200487 + 0.979696i \(0.435748\pi\)
\(822\) 0 0
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 46.9783i − 1.63359i −0.576925 0.816797i \(-0.695748\pi\)
0.576925 0.816797i \(-0.304252\pi\)
\(828\) 0 0
\(829\) 24.7446 0.859414 0.429707 0.902968i \(-0.358617\pi\)
0.429707 + 0.902968i \(0.358617\pi\)
\(830\) 0 0
\(831\) −42.9783 −1.49090
\(832\) 0 0
\(833\) − 6.00000i − 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 61.0951i − 2.11176i
\(838\) 0 0
\(839\) −10.9783 −0.379011 −0.189506 0.981880i \(-0.560689\pi\)
−0.189506 + 0.981880i \(0.560689\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) − 60.7011i − 2.09066i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 3.37228i − 0.115873i
\(848\) 0 0
\(849\) 17.7228 0.608245
\(850\) 0 0
\(851\) −25.7228 −0.881767
\(852\) 0 0
\(853\) 38.4674i 1.31710i 0.752538 + 0.658549i \(0.228829\pi\)
−0.752538 + 0.658549i \(0.771171\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 36.3505i 1.24171i 0.783925 + 0.620855i \(0.213215\pi\)
−0.783925 + 0.620855i \(0.786785\pi\)
\(858\) 0 0
\(859\) −42.7446 −1.45843 −0.729213 0.684287i \(-0.760114\pi\)
−0.729213 + 0.684287i \(0.760114\pi\)
\(860\) 0 0
\(861\) 130.658 4.45280
\(862\) 0 0
\(863\) 21.2554i 0.723544i 0.932267 + 0.361772i \(0.117828\pi\)
−0.932267 + 0.361772i \(0.882172\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 50.9783i − 1.73131i
\(868\) 0 0
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 106.701i 3.61128i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 36.9783i 1.24867i 0.781158 + 0.624333i \(0.214629\pi\)
−0.781158 + 0.624333i \(0.785371\pi\)
\(878\) 0 0
\(879\) −79.2119 −2.67175
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 3.37228i 0.113486i 0.998389 + 0.0567432i \(0.0180716\pi\)
−0.998389 + 0.0567432i \(0.981928\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.9783i 0.368614i 0.982869 + 0.184307i \(0.0590040\pi\)
−0.982869 + 0.184307i \(0.940996\pi\)
\(888\) 0 0
\(889\) −26.9783 −0.904821
\(890\) 0 0
\(891\) 35.9783 1.20532
\(892\) 0 0
\(893\) − 1.72281i − 0.0576517i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 18.5109i − 0.618060i
\(898\) 0 0
\(899\) 4.62772 0.154343
\(900\) 0 0
\(901\) −5.64947 −0.188211
\(902\) 0 0
\(903\) 45.4891i 1.51378i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.394031i 0.0130836i 0.999979 + 0.00654179i \(0.00208233\pi\)
−0.999979 + 0.00654179i \(0.997918\pi\)
\(908\) 0 0
\(909\) −50.2337 −1.66615
\(910\) 0 0
\(911\) 8.39403 0.278107 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(912\) 0 0
\(913\) − 2.74456i − 0.0908318i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 74.5842i 2.46299i
\(918\) 0 0
\(919\) 18.9783 0.626035 0.313017 0.949747i \(-0.398660\pi\)
0.313017 + 0.949747i \(0.398660\pi\)
\(920\) 0 0
\(921\) −17.7228 −0.583987
\(922\) 0 0
\(923\) 20.2337i 0.666000i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 79.4456i 2.60934i
\(928\) 0 0
\(929\) −24.3505 −0.798915 −0.399458 0.916752i \(-0.630801\pi\)
−0.399458 + 0.916752i \(0.630801\pi\)
\(930\) 0 0
\(931\) −2.74456 −0.0899494
\(932\) 0 0
\(933\) 65.3288i 2.13877i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 28.5109i − 0.931410i −0.884940 0.465705i \(-0.845801\pi\)
0.884940 0.465705i \(-0.154199\pi\)
\(938\) 0 0
\(939\) 74.1902 2.42111
\(940\) 0 0
\(941\) −15.0951 −0.492086 −0.246043 0.969259i \(-0.579130\pi\)
−0.246043 + 0.969259i \(0.579130\pi\)
\(942\) 0 0
\(943\) − 31.5326i − 1.02684i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 48.8614i − 1.58778i −0.608060 0.793891i \(-0.708052\pi\)
0.608060 0.793891i \(-0.291948\pi\)
\(948\) 0 0
\(949\) 30.9783 1.00560
\(950\) 0 0
\(951\) −82.1168 −2.66282
\(952\) 0 0
\(953\) − 40.1168i − 1.29951i −0.760143 0.649756i \(-0.774871\pi\)
0.760143 0.649756i \(-0.225129\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.62772i 0.149593i
\(958\) 0 0
\(959\) 29.4891 0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 0 0
\(963\) − 100.467i − 3.23752i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 47.6060i − 1.53090i −0.643493 0.765452i \(-0.722515\pi\)
0.643493 0.765452i \(-0.277485\pi\)
\(968\) 0 0
\(969\) 2.90491 0.0933190
\(970\) 0 0
\(971\) 1.02175 0.0327895 0.0163947 0.999866i \(-0.494781\pi\)
0.0163947 + 0.999866i \(0.494781\pi\)
\(972\) 0 0
\(973\) 13.4891i 0.432442i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.2337i 0.455376i 0.973734 + 0.227688i \(0.0731166\pi\)
−0.973734 + 0.227688i \(0.926883\pi\)
\(978\) 0 0
\(979\) 1.37228 0.0438583
\(980\) 0 0
\(981\) −129.679 −4.14034
\(982\) 0 0
\(983\) − 13.7228i − 0.437690i −0.975760 0.218845i \(-0.929771\pi\)
0.975760 0.218845i \(-0.0702289\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 31.2119i 0.993487i
\(988\) 0 0
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 104.467i 3.31517i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.2337i 0.704148i 0.935972 + 0.352074i \(0.114523\pi\)
−0.935972 + 0.352074i \(0.885477\pi\)
\(998\) 0 0
\(999\) −169.796 −5.37211
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 4400.2.b.p.4049.1 4
4.3 odd 2 550.2.b.f.199.2 4
5.2 odd 4 4400.2.a.bl.1.1 2
5.3 odd 4 880.2.a.n.1.2 2
5.4 even 2 inner 4400.2.b.p.4049.4 4
12.11 even 2 4950.2.c.bc.199.4 4
15.8 even 4 7920.2.a.bq.1.1 2
20.3 even 4 110.2.a.d.1.1 2
20.7 even 4 550.2.a.n.1.2 2
20.19 odd 2 550.2.b.f.199.3 4
40.3 even 4 3520.2.a.bq.1.2 2
40.13 odd 4 3520.2.a.bj.1.1 2
55.43 even 4 9680.2.a.bt.1.2 2
60.23 odd 4 990.2.a.m.1.2 2
60.47 odd 4 4950.2.a.bw.1.1 2
60.59 even 2 4950.2.c.bc.199.1 4
140.83 odd 4 5390.2.a.bp.1.2 2
220.43 odd 4 1210.2.a.r.1.1 2
220.87 odd 4 6050.2.a.cb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 20.3 even 4
550.2.a.n.1.2 2 20.7 even 4
550.2.b.f.199.2 4 4.3 odd 2
550.2.b.f.199.3 4 20.19 odd 2
880.2.a.n.1.2 2 5.3 odd 4
990.2.a.m.1.2 2 60.23 odd 4
1210.2.a.r.1.1 2 220.43 odd 4
3520.2.a.bj.1.1 2 40.13 odd 4
3520.2.a.bq.1.2 2 40.3 even 4
4400.2.a.bl.1.1 2 5.2 odd 4
4400.2.b.p.4049.1 4 1.1 even 1 trivial
4400.2.b.p.4049.4 4 5.4 even 2 inner
4950.2.a.bw.1.1 2 60.47 odd 4
4950.2.c.bc.199.1 4 60.59 even 2
4950.2.c.bc.199.4 4 12.11 even 2
5390.2.a.bp.1.2 2 140.83 odd 4
6050.2.a.cb.1.2 2 220.87 odd 4
7920.2.a.bq.1.1 2 15.8 even 4
9680.2.a.bt.1.2 2 55.43 even 4