Properties

Label 1100.4.b.c.749.2
Level $1100$
Weight $4$
Character 1100.749
Analytic conductor $64.902$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,4,Mod(749,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.749");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1100.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: \(64.9021010063\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.4.b.c.749.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000i q^{3} -26.0000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+5.00000i q^{3} -26.0000i q^{7} +2.00000 q^{9} -11.0000 q^{11} -52.0000i q^{13} +46.0000i q^{17} +96.0000 q^{19} +130.000 q^{21} -27.0000i q^{23} +145.000i q^{27} -16.0000 q^{29} -293.000 q^{31} -55.0000i q^{33} -29.0000i q^{37} +260.000 q^{39} -472.000 q^{41} +110.000i q^{43} -224.000i q^{47} -333.000 q^{49} -230.000 q^{51} -754.000i q^{53} +480.000i q^{57} -825.000 q^{59} -548.000 q^{61} -52.0000i q^{63} -123.000i q^{67} +135.000 q^{69} +1001.00 q^{71} +1020.00i q^{73} +286.000i q^{77} -526.000 q^{79} -671.000 q^{81} +158.000i q^{83} -80.0000i q^{87} +1217.00 q^{89} -1352.00 q^{91} -1465.00i q^{93} -263.000i q^{97} -22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 22 q^{11} + 192 q^{19} + 260 q^{21} - 32 q^{29} - 586 q^{31} + 520 q^{39} - 944 q^{41} - 666 q^{49} - 460 q^{51} - 1650 q^{59} - 1096 q^{61} + 270 q^{69} + 2002 q^{71} - 1052 q^{79} - 1342 q^{81} + 2434 q^{89} - 2704 q^{91} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 5.00000i 0.962250i 0.876652 + 0.481125i \(0.159772\pi\)
−0.876652 + 0.481125i \(0.840228\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 26.0000i − 1.40387i −0.712242 0.701934i \(-0.752320\pi\)
0.712242 0.701934i \(-0.247680\pi\)
\(8\) 0 0
\(9\) 2.00000 0.0740741
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) − 52.0000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46.0000i 0.656273i 0.944630 + 0.328136i \(0.106421\pi\)
−0.944630 + 0.328136i \(0.893579\pi\)
\(18\) 0 0
\(19\) 96.0000 1.15915 0.579577 0.814918i \(-0.303218\pi\)
0.579577 + 0.814918i \(0.303218\pi\)
\(20\) 0 0
\(21\) 130.000 1.35087
\(22\) 0 0
\(23\) − 27.0000i − 0.244778i −0.992482 0.122389i \(-0.960944\pi\)
0.992482 0.122389i \(-0.0390555\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 145.000i 1.03353i
\(28\) 0 0
\(29\) −16.0000 −0.102453 −0.0512263 0.998687i \(-0.516313\pi\)
−0.0512263 + 0.998687i \(0.516313\pi\)
\(30\) 0 0
\(31\) −293.000 −1.69756 −0.848780 0.528746i \(-0.822662\pi\)
−0.848780 + 0.528746i \(0.822662\pi\)
\(32\) 0 0
\(33\) − 55.0000i − 0.290129i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 29.0000i − 0.128853i −0.997922 0.0644266i \(-0.979478\pi\)
0.997922 0.0644266i \(-0.0205218\pi\)
\(38\) 0 0
\(39\) 260.000 1.06752
\(40\) 0 0
\(41\) −472.000 −1.79790 −0.898951 0.438048i \(-0.855670\pi\)
−0.898951 + 0.438048i \(0.855670\pi\)
\(42\) 0 0
\(43\) 110.000i 0.390113i 0.980792 + 0.195056i \(0.0624890\pi\)
−0.980792 + 0.195056i \(0.937511\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 224.000i − 0.695186i −0.937645 0.347593i \(-0.886999\pi\)
0.937645 0.347593i \(-0.113001\pi\)
\(48\) 0 0
\(49\) −333.000 −0.970845
\(50\) 0 0
\(51\) −230.000 −0.631499
\(52\) 0 0
\(53\) − 754.000i − 1.95415i −0.212899 0.977074i \(-0.568291\pi\)
0.212899 0.977074i \(-0.431709\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 480.000i 1.11540i
\(58\) 0 0
\(59\) −825.000 −1.82044 −0.910219 0.414127i \(-0.864087\pi\)
−0.910219 + 0.414127i \(0.864087\pi\)
\(60\) 0 0
\(61\) −548.000 −1.15023 −0.575116 0.818072i \(-0.695043\pi\)
−0.575116 + 0.818072i \(0.695043\pi\)
\(62\) 0 0
\(63\) − 52.0000i − 0.103990i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 123.000i − 0.224281i −0.993692 0.112141i \(-0.964229\pi\)
0.993692 0.112141i \(-0.0357707\pi\)
\(68\) 0 0
\(69\) 135.000 0.235538
\(70\) 0 0
\(71\) 1001.00 1.67319 0.836597 0.547818i \(-0.184541\pi\)
0.836597 + 0.547818i \(0.184541\pi\)
\(72\) 0 0
\(73\) 1020.00i 1.63537i 0.575666 + 0.817685i \(0.304743\pi\)
−0.575666 + 0.817685i \(0.695257\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 286.000i 0.423282i
\(78\) 0 0
\(79\) −526.000 −0.749109 −0.374555 0.927205i \(-0.622204\pi\)
−0.374555 + 0.927205i \(0.622204\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 0 0
\(83\) 158.000i 0.208949i 0.994528 + 0.104474i \(0.0333160\pi\)
−0.994528 + 0.104474i \(0.966684\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 80.0000i − 0.0985851i
\(88\) 0 0
\(89\) 1217.00 1.44946 0.724729 0.689034i \(-0.241965\pi\)
0.724729 + 0.689034i \(0.241965\pi\)
\(90\) 0 0
\(91\) −1352.00 −1.55745
\(92\) 0 0
\(93\) − 1465.00i − 1.63348i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 263.000i − 0.275295i −0.990481 0.137647i \(-0.956046\pi\)
0.990481 0.137647i \(-0.0439541\pi\)
\(98\) 0 0
\(99\) −22.0000 −0.0223342
\(100\) 0 0
\(101\) −814.000 −0.801941 −0.400970 0.916091i \(-0.631327\pi\)
−0.400970 + 0.916091i \(0.631327\pi\)
\(102\) 0 0
\(103\) − 376.000i − 0.359693i −0.983695 0.179847i \(-0.942440\pi\)
0.983695 0.179847i \(-0.0575601\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1494.00i − 1.34982i −0.737901 0.674909i \(-0.764183\pi\)
0.737901 0.674909i \(-0.235817\pi\)
\(108\) 0 0
\(109\) −842.000 −0.739899 −0.369949 0.929052i \(-0.620625\pi\)
−0.369949 + 0.929052i \(0.620625\pi\)
\(110\) 0 0
\(111\) 145.000 0.123989
\(112\) 0 0
\(113\) − 1281.00i − 1.06643i −0.845980 0.533214i \(-0.820984\pi\)
0.845980 0.533214i \(-0.179016\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 104.000i − 0.0821778i
\(118\) 0 0
\(119\) 1196.00 0.921321
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) − 2360.00i − 1.73003i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1736.00i 1.21295i 0.795101 + 0.606477i \(0.207418\pi\)
−0.795101 + 0.606477i \(0.792582\pi\)
\(128\) 0 0
\(129\) −550.000 −0.375386
\(130\) 0 0
\(131\) 6.00000 0.00400170 0.00200085 0.999998i \(-0.499363\pi\)
0.00200085 + 0.999998i \(0.499363\pi\)
\(132\) 0 0
\(133\) − 2496.00i − 1.62730i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1439.00i − 0.897387i −0.893686 0.448694i \(-0.851889\pi\)
0.893686 0.448694i \(-0.148111\pi\)
\(138\) 0 0
\(139\) 318.000 0.194046 0.0970231 0.995282i \(-0.469068\pi\)
0.0970231 + 0.995282i \(0.469068\pi\)
\(140\) 0 0
\(141\) 1120.00 0.668943
\(142\) 0 0
\(143\) 572.000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 1665.00i − 0.934196i
\(148\) 0 0
\(149\) 922.000 0.506934 0.253467 0.967344i \(-0.418429\pi\)
0.253467 + 0.967344i \(0.418429\pi\)
\(150\) 0 0
\(151\) −1030.00 −0.555101 −0.277550 0.960711i \(-0.589523\pi\)
−0.277550 + 0.960711i \(0.589523\pi\)
\(152\) 0 0
\(153\) 92.0000i 0.0486128i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1017.00i 0.516977i 0.966014 + 0.258489i \(0.0832245\pi\)
−0.966014 + 0.258489i \(0.916776\pi\)
\(158\) 0 0
\(159\) 3770.00 1.88038
\(160\) 0 0
\(161\) −702.000 −0.343636
\(162\) 0 0
\(163\) 2444.00i 1.17441i 0.809438 + 0.587205i \(0.199772\pi\)
−0.809438 + 0.587205i \(0.800228\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1452.00i − 0.672809i −0.941718 0.336405i \(-0.890789\pi\)
0.941718 0.336405i \(-0.109211\pi\)
\(168\) 0 0
\(169\) −507.000 −0.230769
\(170\) 0 0
\(171\) 192.000 0.0858632
\(172\) 0 0
\(173\) − 1914.00i − 0.841149i −0.907258 0.420574i \(-0.861829\pi\)
0.907258 0.420574i \(-0.138171\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4125.00i − 1.75172i
\(178\) 0 0
\(179\) −1293.00 −0.539907 −0.269954 0.962873i \(-0.587008\pi\)
−0.269954 + 0.962873i \(0.587008\pi\)
\(180\) 0 0
\(181\) 455.000 0.186850 0.0934251 0.995626i \(-0.470218\pi\)
0.0934251 + 0.995626i \(0.470218\pi\)
\(182\) 0 0
\(183\) − 2740.00i − 1.10681i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 506.000i − 0.197874i
\(188\) 0 0
\(189\) 3770.00 1.45094
\(190\) 0 0
\(191\) −1115.00 −0.422401 −0.211200 0.977443i \(-0.567737\pi\)
−0.211200 + 0.977443i \(0.567737\pi\)
\(192\) 0 0
\(193\) − 5012.00i − 1.86928i −0.355591 0.934642i \(-0.615720\pi\)
0.355591 0.934642i \(-0.384280\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 4146.00i − 1.49944i −0.661753 0.749721i \(-0.730187\pi\)
0.661753 0.749721i \(-0.269813\pi\)
\(198\) 0 0
\(199\) 1240.00 0.441715 0.220857 0.975306i \(-0.429114\pi\)
0.220857 + 0.975306i \(0.429114\pi\)
\(200\) 0 0
\(201\) 615.000 0.215815
\(202\) 0 0
\(203\) 416.000i 0.143830i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 54.0000i − 0.0181317i
\(208\) 0 0
\(209\) −1056.00 −0.349498
\(210\) 0 0
\(211\) −2820.00 −0.920080 −0.460040 0.887898i \(-0.652165\pi\)
−0.460040 + 0.887898i \(0.652165\pi\)
\(212\) 0 0
\(213\) 5005.00i 1.61003i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7618.00i 2.38315i
\(218\) 0 0
\(219\) −5100.00 −1.57363
\(220\) 0 0
\(221\) 2392.00 0.728069
\(222\) 0 0
\(223\) − 3695.00i − 1.10958i −0.831992 0.554788i \(-0.812799\pi\)
0.831992 0.554788i \(-0.187201\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 486.000i − 0.142101i −0.997473 0.0710506i \(-0.977365\pi\)
0.997473 0.0710506i \(-0.0226352\pi\)
\(228\) 0 0
\(229\) −4231.00 −1.22093 −0.610464 0.792044i \(-0.709017\pi\)
−0.610464 + 0.792044i \(0.709017\pi\)
\(230\) 0 0
\(231\) −1430.00 −0.407303
\(232\) 0 0
\(233\) 3336.00i 0.937977i 0.883204 + 0.468988i \(0.155381\pi\)
−0.883204 + 0.468988i \(0.844619\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 2630.00i − 0.720831i
\(238\) 0 0
\(239\) −3610.00 −0.977036 −0.488518 0.872554i \(-0.662462\pi\)
−0.488518 + 0.872554i \(0.662462\pi\)
\(240\) 0 0
\(241\) −6408.00 −1.71276 −0.856381 0.516345i \(-0.827292\pi\)
−0.856381 + 0.516345i \(0.827292\pi\)
\(242\) 0 0
\(243\) 560.000i 0.147835i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4992.00i − 1.28596i
\(248\) 0 0
\(249\) −790.000 −0.201061
\(250\) 0 0
\(251\) −1787.00 −0.449380 −0.224690 0.974430i \(-0.572137\pi\)
−0.224690 + 0.974430i \(0.572137\pi\)
\(252\) 0 0
\(253\) 297.000i 0.0738033i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 354.000i − 0.0859218i −0.999077 0.0429609i \(-0.986321\pi\)
0.999077 0.0429609i \(-0.0136791\pi\)
\(258\) 0 0
\(259\) −754.000 −0.180893
\(260\) 0 0
\(261\) −32.0000 −0.00758908
\(262\) 0 0
\(263\) 2026.00i 0.475013i 0.971386 + 0.237507i \(0.0763302\pi\)
−0.971386 + 0.237507i \(0.923670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6085.00i 1.39474i
\(268\) 0 0
\(269\) 8750.00 1.98326 0.991630 0.129112i \(-0.0412128\pi\)
0.991630 + 0.129112i \(0.0412128\pi\)
\(270\) 0 0
\(271\) −5036.00 −1.12884 −0.564419 0.825488i \(-0.690900\pi\)
−0.564419 + 0.825488i \(0.690900\pi\)
\(272\) 0 0
\(273\) − 6760.00i − 1.49866i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4306.00i − 0.934016i −0.884253 0.467008i \(-0.845332\pi\)
0.884253 0.467008i \(-0.154668\pi\)
\(278\) 0 0
\(279\) −586.000 −0.125745
\(280\) 0 0
\(281\) −1202.00 −0.255179 −0.127590 0.991827i \(-0.540724\pi\)
−0.127590 + 0.991827i \(0.540724\pi\)
\(282\) 0 0
\(283\) 6620.00i 1.39052i 0.718757 + 0.695262i \(0.244712\pi\)
−0.718757 + 0.695262i \(0.755288\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12272.0i 2.52402i
\(288\) 0 0
\(289\) 2797.00 0.569306
\(290\) 0 0
\(291\) 1315.00 0.264903
\(292\) 0 0
\(293\) − 6968.00i − 1.38933i −0.719331 0.694667i \(-0.755552\pi\)
0.719331 0.694667i \(-0.244448\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1595.00i − 0.311620i
\(298\) 0 0
\(299\) −1404.00 −0.271557
\(300\) 0 0
\(301\) 2860.00 0.547667
\(302\) 0 0
\(303\) − 4070.00i − 0.771668i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7640.00i 1.42032i 0.704041 + 0.710159i \(0.251377\pi\)
−0.704041 + 0.710159i \(0.748623\pi\)
\(308\) 0 0
\(309\) 1880.00 0.346115
\(310\) 0 0
\(311\) −652.000 −0.118880 −0.0594398 0.998232i \(-0.518931\pi\)
−0.0594398 + 0.998232i \(0.518931\pi\)
\(312\) 0 0
\(313\) − 8055.00i − 1.45462i −0.686310 0.727309i \(-0.740771\pi\)
0.686310 0.727309i \(-0.259229\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 5675.00i − 1.00549i −0.864435 0.502744i \(-0.832324\pi\)
0.864435 0.502744i \(-0.167676\pi\)
\(318\) 0 0
\(319\) 176.000 0.0308906
\(320\) 0 0
\(321\) 7470.00 1.29886
\(322\) 0 0
\(323\) 4416.00i 0.760721i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 4210.00i − 0.711968i
\(328\) 0 0
\(329\) −5824.00 −0.975950
\(330\) 0 0
\(331\) −2245.00 −0.372799 −0.186399 0.982474i \(-0.559682\pi\)
−0.186399 + 0.982474i \(0.559682\pi\)
\(332\) 0 0
\(333\) − 58.0000i − 0.00954469i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 694.000i − 0.112180i −0.998426 0.0560899i \(-0.982137\pi\)
0.998426 0.0560899i \(-0.0178633\pi\)
\(338\) 0 0
\(339\) 6405.00 1.02617
\(340\) 0 0
\(341\) 3223.00 0.511834
\(342\) 0 0
\(343\) − 260.000i − 0.0409291i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4092.00i 0.633055i 0.948583 + 0.316527i \(0.102517\pi\)
−0.948583 + 0.316527i \(0.897483\pi\)
\(348\) 0 0
\(349\) −334.000 −0.0512281 −0.0256141 0.999672i \(-0.508154\pi\)
−0.0256141 + 0.999672i \(0.508154\pi\)
\(350\) 0 0
\(351\) 7540.00 1.14660
\(352\) 0 0
\(353\) − 891.000i − 0.134343i −0.997741 0.0671716i \(-0.978603\pi\)
0.997741 0.0671716i \(-0.0213975\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5980.00i 0.886541i
\(358\) 0 0
\(359\) −2476.00 −0.364006 −0.182003 0.983298i \(-0.558258\pi\)
−0.182003 + 0.983298i \(0.558258\pi\)
\(360\) 0 0
\(361\) 2357.00 0.343636
\(362\) 0 0
\(363\) 605.000i 0.0874773i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1379.00i 0.196140i 0.995180 + 0.0980698i \(0.0312668\pi\)
−0.995180 + 0.0980698i \(0.968733\pi\)
\(368\) 0 0
\(369\) −944.000 −0.133178
\(370\) 0 0
\(371\) −19604.0 −2.74337
\(372\) 0 0
\(373\) 6266.00i 0.869816i 0.900475 + 0.434908i \(0.143219\pi\)
−0.900475 + 0.434908i \(0.856781\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 832.000i 0.113661i
\(378\) 0 0
\(379\) −151.000 −0.0204653 −0.0102327 0.999948i \(-0.503257\pi\)
−0.0102327 + 0.999948i \(0.503257\pi\)
\(380\) 0 0
\(381\) −8680.00 −1.16717
\(382\) 0 0
\(383\) 1989.00i 0.265361i 0.991159 + 0.132680i \(0.0423584\pi\)
−0.991159 + 0.132680i \(0.957642\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 220.000i 0.0288972i
\(388\) 0 0
\(389\) −6817.00 −0.888523 −0.444262 0.895897i \(-0.646534\pi\)
−0.444262 + 0.895897i \(0.646534\pi\)
\(390\) 0 0
\(391\) 1242.00 0.160641
\(392\) 0 0
\(393\) 30.0000i 0.00385064i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 4170.00i − 0.527170i −0.964636 0.263585i \(-0.915095\pi\)
0.964636 0.263585i \(-0.0849049\pi\)
\(398\) 0 0
\(399\) 12480.0 1.56587
\(400\) 0 0
\(401\) 10914.0 1.35915 0.679575 0.733606i \(-0.262164\pi\)
0.679575 + 0.733606i \(0.262164\pi\)
\(402\) 0 0
\(403\) 15236.0i 1.88327i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 319.000i 0.0388507i
\(408\) 0 0
\(409\) 1102.00 0.133228 0.0666142 0.997779i \(-0.478780\pi\)
0.0666142 + 0.997779i \(0.478780\pi\)
\(410\) 0 0
\(411\) 7195.00 0.863511
\(412\) 0 0
\(413\) 21450.0i 2.55565i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1590.00i 0.186721i
\(418\) 0 0
\(419\) −11028.0 −1.28581 −0.642903 0.765947i \(-0.722270\pi\)
−0.642903 + 0.765947i \(0.722270\pi\)
\(420\) 0 0
\(421\) 2622.00 0.303536 0.151768 0.988416i \(-0.451503\pi\)
0.151768 + 0.988416i \(0.451503\pi\)
\(422\) 0 0
\(423\) − 448.000i − 0.0514953i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 14248.0i 1.61478i
\(428\) 0 0
\(429\) −2860.00 −0.321870
\(430\) 0 0
\(431\) 16598.0 1.85498 0.927491 0.373845i \(-0.121961\pi\)
0.927491 + 0.373845i \(0.121961\pi\)
\(432\) 0 0
\(433\) 5763.00i 0.639612i 0.947483 + 0.319806i \(0.103618\pi\)
−0.947483 + 0.319806i \(0.896382\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2592.00i − 0.283735i
\(438\) 0 0
\(439\) 3128.00 0.340071 0.170036 0.985438i \(-0.445612\pi\)
0.170036 + 0.985438i \(0.445612\pi\)
\(440\) 0 0
\(441\) −666.000 −0.0719145
\(442\) 0 0
\(443\) − 6369.00i − 0.683071i −0.939869 0.341535i \(-0.889053\pi\)
0.939869 0.341535i \(-0.110947\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4610.00i 0.487798i
\(448\) 0 0
\(449\) −8691.00 −0.913483 −0.456741 0.889600i \(-0.650983\pi\)
−0.456741 + 0.889600i \(0.650983\pi\)
\(450\) 0 0
\(451\) 5192.00 0.542088
\(452\) 0 0
\(453\) − 5150.00i − 0.534146i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2260.00i 0.231331i 0.993288 + 0.115666i \(0.0369001\pi\)
−0.993288 + 0.115666i \(0.963100\pi\)
\(458\) 0 0
\(459\) −6670.00 −0.678277
\(460\) 0 0
\(461\) −12756.0 −1.28873 −0.644367 0.764717i \(-0.722879\pi\)
−0.644367 + 0.764717i \(0.722879\pi\)
\(462\) 0 0
\(463\) 6887.00i 0.691287i 0.938366 + 0.345644i \(0.112339\pi\)
−0.938366 + 0.345644i \(0.887661\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1535.00i − 0.152101i −0.997104 0.0760507i \(-0.975769\pi\)
0.997104 0.0760507i \(-0.0242311\pi\)
\(468\) 0 0
\(469\) −3198.00 −0.314861
\(470\) 0 0
\(471\) −5085.00 −0.497462
\(472\) 0 0
\(473\) − 1210.00i − 0.117623i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1508.00i − 0.144752i
\(478\) 0 0
\(479\) 17564.0 1.67541 0.837703 0.546126i \(-0.183898\pi\)
0.837703 + 0.546126i \(0.183898\pi\)
\(480\) 0 0
\(481\) −1508.00 −0.142950
\(482\) 0 0
\(483\) − 3510.00i − 0.330664i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 7541.00i − 0.701674i −0.936437 0.350837i \(-0.885897\pi\)
0.936437 0.350837i \(-0.114103\pi\)
\(488\) 0 0
\(489\) −12220.0 −1.13008
\(490\) 0 0
\(491\) 12552.0 1.15369 0.576847 0.816852i \(-0.304283\pi\)
0.576847 + 0.816852i \(0.304283\pi\)
\(492\) 0 0
\(493\) − 736.000i − 0.0672369i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 26026.0i − 2.34894i
\(498\) 0 0
\(499\) 8396.00 0.753220 0.376610 0.926372i \(-0.377090\pi\)
0.376610 + 0.926372i \(0.377090\pi\)
\(500\) 0 0
\(501\) 7260.00 0.647411
\(502\) 0 0
\(503\) 12194.0i 1.08092i 0.841369 + 0.540461i \(0.181750\pi\)
−0.841369 + 0.540461i \(0.818250\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2535.00i − 0.222058i
\(508\) 0 0
\(509\) −18295.0 −1.59315 −0.796573 0.604542i \(-0.793356\pi\)
−0.796573 + 0.604542i \(0.793356\pi\)
\(510\) 0 0
\(511\) 26520.0 2.29584
\(512\) 0 0
\(513\) 13920.0i 1.19802i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2464.00i 0.209607i
\(518\) 0 0
\(519\) 9570.00 0.809396
\(520\) 0 0
\(521\) 7101.00 0.597122 0.298561 0.954391i \(-0.403493\pi\)
0.298561 + 0.954391i \(0.403493\pi\)
\(522\) 0 0
\(523\) 4912.00i 0.410682i 0.978690 + 0.205341i \(0.0658304\pi\)
−0.978690 + 0.205341i \(0.934170\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 13478.0i − 1.11406i
\(528\) 0 0
\(529\) 11438.0 0.940084
\(530\) 0 0
\(531\) −1650.00 −0.134847
\(532\) 0 0
\(533\) 24544.0i 1.99459i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6465.00i − 0.519526i
\(538\) 0 0
\(539\) 3663.00 0.292721
\(540\) 0 0
\(541\) 11496.0 0.913589 0.456794 0.889572i \(-0.348997\pi\)
0.456794 + 0.889572i \(0.348997\pi\)
\(542\) 0 0
\(543\) 2275.00i 0.179797i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 19048.0i 1.48891i 0.667673 + 0.744455i \(0.267291\pi\)
−0.667673 + 0.744455i \(0.732709\pi\)
\(548\) 0 0
\(549\) −1096.00 −0.0852024
\(550\) 0 0
\(551\) −1536.00 −0.118758
\(552\) 0 0
\(553\) 13676.0i 1.05165i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10718.0i 0.815325i 0.913133 + 0.407663i \(0.133656\pi\)
−0.913133 + 0.407663i \(0.866344\pi\)
\(558\) 0 0
\(559\) 5720.00 0.432791
\(560\) 0 0
\(561\) 2530.00 0.190404
\(562\) 0 0
\(563\) − 6660.00i − 0.498553i −0.968432 0.249277i \(-0.919807\pi\)
0.968432 0.249277i \(-0.0801929\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 17446.0i 1.29217i
\(568\) 0 0
\(569\) 10960.0 0.807499 0.403750 0.914870i \(-0.367707\pi\)
0.403750 + 0.914870i \(0.367707\pi\)
\(570\) 0 0
\(571\) 18596.0 1.36290 0.681452 0.731863i \(-0.261349\pi\)
0.681452 + 0.731863i \(0.261349\pi\)
\(572\) 0 0
\(573\) − 5575.00i − 0.406455i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 21119.0i − 1.52374i −0.647733 0.761868i \(-0.724283\pi\)
0.647733 0.761868i \(-0.275717\pi\)
\(578\) 0 0
\(579\) 25060.0 1.79872
\(580\) 0 0
\(581\) 4108.00 0.293337
\(582\) 0 0
\(583\) 8294.00i 0.589198i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 2836.00i − 0.199411i −0.995017 0.0997055i \(-0.968210\pi\)
0.995017 0.0997055i \(-0.0317901\pi\)
\(588\) 0 0
\(589\) −28128.0 −1.96773
\(590\) 0 0
\(591\) 20730.0 1.44284
\(592\) 0 0
\(593\) − 18044.0i − 1.24954i −0.780808 0.624771i \(-0.785192\pi\)
0.780808 0.624771i \(-0.214808\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6200.00i 0.425040i
\(598\) 0 0
\(599\) 9264.00 0.631914 0.315957 0.948773i \(-0.397674\pi\)
0.315957 + 0.948773i \(0.397674\pi\)
\(600\) 0 0
\(601\) −19326.0 −1.31169 −0.655844 0.754897i \(-0.727687\pi\)
−0.655844 + 0.754897i \(0.727687\pi\)
\(602\) 0 0
\(603\) − 246.000i − 0.0166134i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10082.0i 0.674161i 0.941476 + 0.337081i \(0.109439\pi\)
−0.941476 + 0.337081i \(0.890561\pi\)
\(608\) 0 0
\(609\) −2080.00 −0.138400
\(610\) 0 0
\(611\) −11648.0 −0.771240
\(612\) 0 0
\(613\) − 13088.0i − 0.862348i −0.902269 0.431174i \(-0.858100\pi\)
0.902269 0.431174i \(-0.141900\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23426.0i 1.52852i 0.644910 + 0.764259i \(0.276895\pi\)
−0.644910 + 0.764259i \(0.723105\pi\)
\(618\) 0 0
\(619\) −23587.0 −1.53157 −0.765785 0.643097i \(-0.777649\pi\)
−0.765785 + 0.643097i \(0.777649\pi\)
\(620\) 0 0
\(621\) 3915.00 0.252985
\(622\) 0 0
\(623\) − 31642.0i − 2.03485i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 5280.00i − 0.336304i
\(628\) 0 0
\(629\) 1334.00 0.0845629
\(630\) 0 0
\(631\) 19683.0 1.24179 0.620894 0.783895i \(-0.286770\pi\)
0.620894 + 0.783895i \(0.286770\pi\)
\(632\) 0 0
\(633\) − 14100.0i − 0.885347i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17316.0i 1.07706i
\(638\) 0 0
\(639\) 2002.00 0.123940
\(640\) 0 0
\(641\) 375.000 0.0231070 0.0115535 0.999933i \(-0.496322\pi\)
0.0115535 + 0.999933i \(0.496322\pi\)
\(642\) 0 0
\(643\) 21055.0i 1.29133i 0.763619 + 0.645667i \(0.223421\pi\)
−0.763619 + 0.645667i \(0.776579\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 6427.00i − 0.390528i −0.980751 0.195264i \(-0.937444\pi\)
0.980751 0.195264i \(-0.0625563\pi\)
\(648\) 0 0
\(649\) 9075.00 0.548883
\(650\) 0 0
\(651\) −38090.0 −2.29319
\(652\) 0 0
\(653\) 7617.00i 0.456472i 0.973606 + 0.228236i \(0.0732958\pi\)
−0.973606 + 0.228236i \(0.926704\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2040.00i 0.121138i
\(658\) 0 0
\(659\) 17630.0 1.04214 0.521068 0.853515i \(-0.325534\pi\)
0.521068 + 0.853515i \(0.325534\pi\)
\(660\) 0 0
\(661\) 4605.00 0.270974 0.135487 0.990779i \(-0.456740\pi\)
0.135487 + 0.990779i \(0.456740\pi\)
\(662\) 0 0
\(663\) 11960.0i 0.700585i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 432.000i 0.0250781i
\(668\) 0 0
\(669\) 18475.0 1.06769
\(670\) 0 0
\(671\) 6028.00 0.346808
\(672\) 0 0
\(673\) 2818.00i 0.161406i 0.996738 + 0.0807028i \(0.0257165\pi\)
−0.996738 + 0.0807028i \(0.974284\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8438.00i 0.479023i 0.970894 + 0.239512i \(0.0769873\pi\)
−0.970894 + 0.239512i \(0.923013\pi\)
\(678\) 0 0
\(679\) −6838.00 −0.386478
\(680\) 0 0
\(681\) 2430.00 0.136737
\(682\) 0 0
\(683\) 17344.0i 0.971669i 0.874051 + 0.485834i \(0.161484\pi\)
−0.874051 + 0.485834i \(0.838516\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 21155.0i − 1.17484i
\(688\) 0 0
\(689\) −39208.0 −2.16793
\(690\) 0 0
\(691\) −3947.00 −0.217295 −0.108648 0.994080i \(-0.534652\pi\)
−0.108648 + 0.994080i \(0.534652\pi\)
\(692\) 0 0
\(693\) 572.000i 0.0313542i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 21712.0i − 1.17991i
\(698\) 0 0
\(699\) −16680.0 −0.902569
\(700\) 0 0
\(701\) −7998.00 −0.430928 −0.215464 0.976512i \(-0.569126\pi\)
−0.215464 + 0.976512i \(0.569126\pi\)
\(702\) 0 0
\(703\) − 2784.00i − 0.149361i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 21164.0i 1.12582i
\(708\) 0 0
\(709\) 881.000 0.0466666 0.0233333 0.999728i \(-0.492572\pi\)
0.0233333 + 0.999728i \(0.492572\pi\)
\(710\) 0 0
\(711\) −1052.00 −0.0554896
\(712\) 0 0
\(713\) 7911.00i 0.415525i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 18050.0i − 0.940153i
\(718\) 0 0
\(719\) 26093.0 1.35341 0.676707 0.736252i \(-0.263406\pi\)
0.676707 + 0.736252i \(0.263406\pi\)
\(720\) 0 0
\(721\) −9776.00 −0.504962
\(722\) 0 0
\(723\) − 32040.0i − 1.64811i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7481.00i − 0.381644i −0.981625 0.190822i \(-0.938885\pi\)
0.981625 0.190822i \(-0.0611153\pi\)
\(728\) 0 0
\(729\) −20917.0 −1.06269
\(730\) 0 0
\(731\) −5060.00 −0.256020
\(732\) 0 0
\(733\) − 17788.0i − 0.896337i −0.893949 0.448168i \(-0.852077\pi\)
0.893949 0.448168i \(-0.147923\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1353.00i 0.0676233i
\(738\) 0 0
\(739\) 32182.0 1.60194 0.800970 0.598704i \(-0.204317\pi\)
0.800970 + 0.598704i \(0.204317\pi\)
\(740\) 0 0
\(741\) 24960.0 1.23742
\(742\) 0 0
\(743\) 32044.0i 1.58221i 0.611682 + 0.791104i \(0.290493\pi\)
−0.611682 + 0.791104i \(0.709507\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 316.000i 0.0154777i
\(748\) 0 0
\(749\) −38844.0 −1.89497
\(750\) 0 0
\(751\) −33779.0 −1.64130 −0.820648 0.571434i \(-0.806387\pi\)
−0.820648 + 0.571434i \(0.806387\pi\)
\(752\) 0 0
\(753\) − 8935.00i − 0.432416i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 15630.0i − 0.750439i −0.926936 0.375219i \(-0.877567\pi\)
0.926936 0.375219i \(-0.122433\pi\)
\(758\) 0 0
\(759\) −1485.00 −0.0710172
\(760\) 0 0
\(761\) 1948.00 0.0927923 0.0463962 0.998923i \(-0.485226\pi\)
0.0463962 + 0.998923i \(0.485226\pi\)
\(762\) 0 0
\(763\) 21892.0i 1.03872i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 42900.0i 2.01959i
\(768\) 0 0
\(769\) 17420.0 0.816881 0.408440 0.912785i \(-0.366073\pi\)
0.408440 + 0.912785i \(0.366073\pi\)
\(770\) 0 0
\(771\) 1770.00 0.0826783
\(772\) 0 0
\(773\) − 11122.0i − 0.517504i −0.965944 0.258752i \(-0.916689\pi\)
0.965944 0.258752i \(-0.0833112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3770.00i − 0.174064i
\(778\) 0 0
\(779\) −45312.0 −2.08404
\(780\) 0 0
\(781\) −11011.0 −0.504487
\(782\) 0 0
\(783\) − 2320.00i − 0.105888i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29648.0i 1.34287i 0.741064 + 0.671434i \(0.234321\pi\)
−0.741064 + 0.671434i \(0.765679\pi\)
\(788\) 0 0
\(789\) −10130.0 −0.457082
\(790\) 0 0
\(791\) −33306.0 −1.49712
\(792\) 0 0
\(793\) 28496.0i 1.27607i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 23219.0i − 1.03194i −0.856606 0.515972i \(-0.827431\pi\)
0.856606 0.515972i \(-0.172569\pi\)
\(798\) 0 0
\(799\) 10304.0 0.456232
\(800\) 0 0
\(801\) 2434.00 0.107367
\(802\) 0 0
\(803\) − 11220.0i − 0.493082i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 43750.0i 1.90839i
\(808\) 0 0
\(809\) −9232.00 −0.401211 −0.200606 0.979672i \(-0.564291\pi\)
−0.200606 + 0.979672i \(0.564291\pi\)
\(810\) 0 0
\(811\) −32286.0 −1.39792 −0.698961 0.715160i \(-0.746354\pi\)
−0.698961 + 0.715160i \(0.746354\pi\)
\(812\) 0 0
\(813\) − 25180.0i − 1.08623i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 10560.0i 0.452200i
\(818\) 0 0
\(819\) −2704.00 −0.115367
\(820\) 0 0
\(821\) −31706.0 −1.34780 −0.673902 0.738821i \(-0.735383\pi\)
−0.673902 + 0.738821i \(0.735383\pi\)
\(822\) 0 0
\(823\) − 41139.0i − 1.74242i −0.490906 0.871212i \(-0.663334\pi\)
0.490906 0.871212i \(-0.336666\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 15252.0i − 0.641311i −0.947196 0.320655i \(-0.896097\pi\)
0.947196 0.320655i \(-0.103903\pi\)
\(828\) 0 0
\(829\) −369.000 −0.0154595 −0.00772973 0.999970i \(-0.502460\pi\)
−0.00772973 + 0.999970i \(0.502460\pi\)
\(830\) 0 0
\(831\) 21530.0 0.898757
\(832\) 0 0
\(833\) − 15318.0i − 0.637140i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 42485.0i − 1.75448i
\(838\) 0 0
\(839\) 10257.0 0.422063 0.211032 0.977479i \(-0.432318\pi\)
0.211032 + 0.977479i \(0.432318\pi\)
\(840\) 0 0
\(841\) −24133.0 −0.989503
\(842\) 0 0
\(843\) − 6010.00i − 0.245546i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 3146.00i − 0.127624i
\(848\) 0 0
\(849\) −33100.0 −1.33803
\(850\) 0 0
\(851\) −783.000 −0.0315404
\(852\) 0 0
\(853\) 34386.0i 1.38025i 0.723690 + 0.690126i \(0.242445\pi\)
−0.723690 + 0.690126i \(0.757555\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 23464.0i − 0.935257i −0.883925 0.467628i \(-0.845109\pi\)
0.883925 0.467628i \(-0.154891\pi\)
\(858\) 0 0
\(859\) 22475.0 0.892709 0.446355 0.894856i \(-0.352722\pi\)
0.446355 + 0.894856i \(0.352722\pi\)
\(860\) 0 0
\(861\) −61360.0 −2.42874
\(862\) 0 0
\(863\) − 2880.00i − 0.113599i −0.998386 0.0567997i \(-0.981910\pi\)
0.998386 0.0567997i \(-0.0180897\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13985.0i 0.547815i
\(868\) 0 0
\(869\) 5786.00 0.225865
\(870\) 0 0
\(871\) −6396.00 −0.248818
\(872\) 0 0
\(873\) − 526.000i − 0.0203922i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 11084.0i − 0.426773i −0.976968 0.213387i \(-0.931551\pi\)
0.976968 0.213387i \(-0.0684494\pi\)
\(878\) 0 0
\(879\) 34840.0 1.33689
\(880\) 0 0
\(881\) 41797.0 1.59838 0.799192 0.601076i \(-0.205261\pi\)
0.799192 + 0.601076i \(0.205261\pi\)
\(882\) 0 0
\(883\) − 23780.0i − 0.906298i −0.891435 0.453149i \(-0.850301\pi\)
0.891435 0.453149i \(-0.149699\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 14334.0i − 0.542603i −0.962494 0.271301i \(-0.912546\pi\)
0.962494 0.271301i \(-0.0874540\pi\)
\(888\) 0 0
\(889\) 45136.0 1.70283
\(890\) 0 0
\(891\) 7381.00 0.277523
\(892\) 0 0
\(893\) − 21504.0i − 0.805827i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7020.00i − 0.261305i
\(898\) 0 0
\(899\) 4688.00 0.173919
\(900\) 0 0
\(901\) 34684.0 1.28245
\(902\) 0 0
\(903\) 14300.0i 0.526992i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 25716.0i 0.941440i 0.882283 + 0.470720i \(0.156006\pi\)
−0.882283 + 0.470720i \(0.843994\pi\)
\(908\) 0 0
\(909\) −1628.00 −0.0594030
\(910\) 0 0
\(911\) −28300.0 −1.02922 −0.514611 0.857424i \(-0.672064\pi\)
−0.514611 + 0.857424i \(0.672064\pi\)
\(912\) 0 0
\(913\) − 1738.00i − 0.0630004i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 156.000i − 0.00561786i
\(918\) 0 0
\(919\) −21410.0 −0.768499 −0.384250 0.923229i \(-0.625540\pi\)
−0.384250 + 0.923229i \(0.625540\pi\)
\(920\) 0 0
\(921\) −38200.0 −1.36670
\(922\) 0 0
\(923\) − 52052.0i − 1.85624i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 752.000i − 0.0266439i
\(928\) 0 0
\(929\) −19938.0 −0.704138 −0.352069 0.935974i \(-0.614522\pi\)
−0.352069 + 0.935974i \(0.614522\pi\)
\(930\) 0 0
\(931\) −31968.0 −1.12536
\(932\) 0 0
\(933\) − 3260.00i − 0.114392i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26728.0i 0.931874i 0.884818 + 0.465937i \(0.154283\pi\)
−0.884818 + 0.465937i \(0.845717\pi\)
\(938\) 0 0
\(939\) 40275.0 1.39971
\(940\) 0 0
\(941\) 8634.00 0.299108 0.149554 0.988754i \(-0.452216\pi\)
0.149554 + 0.988754i \(0.452216\pi\)
\(942\) 0 0
\(943\) 12744.0i 0.440087i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24841.0i 0.852401i 0.904629 + 0.426201i \(0.140148\pi\)
−0.904629 + 0.426201i \(0.859852\pi\)
\(948\) 0 0
\(949\) 53040.0 1.81428
\(950\) 0 0
\(951\) 28375.0 0.967531
\(952\) 0 0
\(953\) − 51234.0i − 1.74148i −0.491742 0.870741i \(-0.663640\pi\)
0.491742 0.870741i \(-0.336360\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 880.000i 0.0297245i
\(958\) 0 0
\(959\) −37414.0 −1.25981
\(960\) 0 0
\(961\) 56058.0 1.88171
\(962\) 0 0
\(963\) − 2988.00i − 0.0999865i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 26752.0i − 0.889645i −0.895619 0.444822i \(-0.853267\pi\)
0.895619 0.444822i \(-0.146733\pi\)
\(968\) 0 0
\(969\) −22080.0 −0.732004
\(970\) 0 0
\(971\) 21155.0 0.699172 0.349586 0.936904i \(-0.386322\pi\)
0.349586 + 0.936904i \(0.386322\pi\)
\(972\) 0 0
\(973\) − 8268.00i − 0.272415i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 56323.0i − 1.84435i −0.386770 0.922176i \(-0.626409\pi\)
0.386770 0.922176i \(-0.373591\pi\)
\(978\) 0 0
\(979\) −13387.0 −0.437028
\(980\) 0 0
\(981\) −1684.00 −0.0548073
\(982\) 0 0
\(983\) − 24683.0i − 0.800880i −0.916323 0.400440i \(-0.868857\pi\)
0.916323 0.400440i \(-0.131143\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 29120.0i − 0.939108i
\(988\) 0 0
\(989\) 2970.00 0.0954909
\(990\) 0 0
\(991\) −26816.0 −0.859574 −0.429787 0.902930i \(-0.641411\pi\)
−0.429787 + 0.902930i \(0.641411\pi\)
\(992\) 0 0
\(993\) − 11225.0i − 0.358726i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 18614.0i 0.591285i 0.955299 + 0.295643i \(0.0955337\pi\)
−0.955299 + 0.295643i \(0.904466\pi\)
\(998\) 0 0
\(999\) 4205.00 0.133173
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 1100.4.b.c.749.2 2
5.2 odd 4 1100.4.a.d.1.1 1
5.3 odd 4 44.4.a.a.1.1 1
5.4 even 2 inner 1100.4.b.c.749.1 2
15.8 even 4 396.4.a.e.1.1 1
20.3 even 4 176.4.a.e.1.1 1
35.13 even 4 2156.4.a.b.1.1 1
40.3 even 4 704.4.a.c.1.1 1
40.13 odd 4 704.4.a.j.1.1 1
55.43 even 4 484.4.a.a.1.1 1
60.23 odd 4 1584.4.a.p.1.1 1
220.43 odd 4 1936.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.a.1.1 1 5.3 odd 4
176.4.a.e.1.1 1 20.3 even 4
396.4.a.e.1.1 1 15.8 even 4
484.4.a.a.1.1 1 55.43 even 4
704.4.a.c.1.1 1 40.3 even 4
704.4.a.j.1.1 1 40.13 odd 4
1100.4.a.d.1.1 1 5.2 odd 4
1100.4.b.c.749.1 2 5.4 even 2 inner
1100.4.b.c.749.2 2 1.1 even 1 trivial
1584.4.a.p.1.1 1 60.23 odd 4
1936.4.a.m.1.1 1 220.43 odd 4
2156.4.a.b.1.1 1 35.13 even 4