Properties

Label 121.4.c.a.27.1
Level $121$
Weight $4$
Character 121.27
Analytic conductor $7.139$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,4,Mod(3,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label 27.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 121.27
Dual form 121.4.c.a.9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.47214 - 4.70228i) q^{3} +(6.47214 - 4.70228i) q^{4} +(5.56231 - 17.1190i) q^{5} +(11.4336 + 35.1891i) q^{9} +O(q^{10})\) \(q+(-6.47214 - 4.70228i) q^{3} +(6.47214 - 4.70228i) q^{4} +(5.56231 - 17.1190i) q^{5} +(11.4336 + 35.1891i) q^{9} -64.0000 q^{12} +(-116.498 + 84.6411i) q^{15} +(19.7771 - 60.8676i) q^{16} +(-44.4984 - 136.952i) q^{20} -108.000 q^{23} +(-160.994 - 116.969i) q^{25} +(24.7214 - 76.0845i) q^{27} +(105.066 + 323.359i) q^{31} +(239.469 + 173.984i) q^{36} +(351.113 - 255.099i) q^{37} +666.000 q^{45} +(29.1246 + 21.1603i) q^{47} +(-414.217 + 300.946i) q^{48} +(-105.993 + 326.212i) q^{49} +(-228.055 - 701.880i) q^{53} +(582.492 - 423.205i) q^{59} +(-355.988 + 1095.62i) q^{60} +(-158.217 - 486.941i) q^{64} -416.000 q^{67} +(698.991 + 507.846i) q^{69} +(189.118 - 582.047i) q^{71} +(491.955 + 1514.08i) q^{75} +(-931.988 - 677.129i) q^{80} +(290.437 - 211.015i) q^{81} +1674.00 q^{89} +(-698.991 + 507.846i) q^{92} +(840.526 - 2586.87i) q^{93} +(-10.5066 - 32.3359i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9} - 256 q^{12} - 144 q^{15} - 64 q^{16} + 144 q^{20} - 432 q^{23} - 199 q^{25} - 80 q^{27} - 340 q^{31} + 296 q^{36} + 434 q^{37} + 2664 q^{45} + 36 q^{47} - 512 q^{48} + 343 q^{49} + 738 q^{53} + 720 q^{59} + 1152 q^{60} + 512 q^{64} - 1664 q^{67} + 864 q^{69} - 612 q^{71} - 1592 q^{75} - 1152 q^{80} + 359 q^{81} + 6696 q^{89} - 864 q^{92} - 2720 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\)

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 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(3\) −6.47214 4.70228i −1.24556 0.904955i −0.247607 0.968860i \(-0.579644\pi\)
−0.997956 + 0.0639059i \(0.979644\pi\)
\(4\) 6.47214 4.70228i 0.809017 0.587785i
\(5\) 5.56231 17.1190i 0.497508 1.53117i −0.315504 0.948924i \(-0.602174\pi\)
0.813012 0.582247i \(-0.197826\pi\)
\(6\) 0 0
\(7\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(8\) 0 0
\(9\) 11.4336 + 35.1891i 0.423468 + 1.30330i
\(10\) 0 0
\(11\) 0 0
\(12\) −64.0000 −1.53960
\(13\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(14\) 0 0
\(15\) −116.498 + 84.6411i −2.00532 + 1.45695i
\(16\) 19.7771 60.8676i 0.309017 0.951057i
\(17\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(18\) 0 0
\(19\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(20\) −44.4984 136.952i −0.497508 1.53117i
\(21\) 0 0
\(22\) 0 0
\(23\) −108.000 −0.979111 −0.489556 0.871972i \(-0.662841\pi\)
−0.489556 + 0.871972i \(0.662841\pi\)
\(24\) 0 0
\(25\) −160.994 116.969i −1.28796 0.935754i
\(26\) 0 0
\(27\) 24.7214 76.0845i 0.176208 0.542314i
\(28\) 0 0
\(29\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(30\) 0 0
\(31\) 105.066 + 323.359i 0.608722 + 1.87345i 0.468837 + 0.883285i \(0.344673\pi\)
0.139885 + 0.990168i \(0.455327\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 239.469 + 173.984i 1.10865 + 0.805483i
\(37\) 351.113 255.099i 1.56007 1.13346i 0.624122 0.781327i \(-0.285457\pi\)
0.935950 0.352132i \(-0.114543\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 666.000 2.20625
\(46\) 0 0
\(47\) 29.1246 + 21.1603i 0.0903885 + 0.0656711i 0.632062 0.774918i \(-0.282209\pi\)
−0.541673 + 0.840589i \(0.682209\pi\)
\(48\) −414.217 + 300.946i −1.24556 + 0.904955i
\(49\) −105.993 + 326.212i −0.309017 + 0.951057i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −228.055 701.880i −0.591051 1.81907i −0.573476 0.819222i \(-0.694405\pi\)
−0.0175748 0.999846i \(-0.505595\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 582.492 423.205i 1.28532 0.933841i 0.285623 0.958342i \(-0.407800\pi\)
0.999700 + 0.0245007i \(0.00779958\pi\)
\(60\) −355.988 + 1095.62i −0.765963 + 2.35739i
\(61\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −158.217 486.941i −0.309017 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) −416.000 −0.758545 −0.379272 0.925285i \(-0.623826\pi\)
−0.379272 + 0.925285i \(0.623826\pi\)
\(68\) 0 0
\(69\) 698.991 + 507.846i 1.21954 + 0.886051i
\(70\) 0 0
\(71\) 189.118 582.047i 0.316116 0.972905i −0.659177 0.751988i \(-0.729095\pi\)
0.975293 0.220917i \(-0.0709049\pi\)
\(72\) 0 0
\(73\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(74\) 0 0
\(75\) 491.955 + 1514.08i 0.757414 + 2.33108i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(80\) −931.988 677.129i −1.30249 0.946316i
\(81\) 290.437 211.015i 0.398405 0.289458i
\(82\) 0 0
\(83\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1674.00 1.99375 0.996874 0.0790026i \(-0.0251735\pi\)
0.996874 + 0.0790026i \(0.0251735\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −698.991 + 507.846i −0.792118 + 0.575507i
\(93\) 840.526 2586.87i 0.937188 2.88437i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.5066 32.3359i −0.0109977 0.0338476i 0.945407 0.325892i \(-0.105665\pi\)
−0.956405 + 0.292044i \(0.905665\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1592.00 −1.59200
\(101\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(102\) 0 0
\(103\) −948.168 + 688.884i −0.907046 + 0.659008i −0.940266 0.340440i \(-0.889424\pi\)
0.0332199 + 0.999448i \(0.489424\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(108\) −197.771 608.676i −0.176208 0.542314i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −3472.00 −2.96890
\(112\) 0 0
\(113\) −1732.91 1259.04i −1.44264 1.04814i −0.987481 0.157735i \(-0.949581\pi\)
−0.455163 0.890408i \(-0.650419\pi\)
\(114\) 0 0
\(115\) −600.729 + 1848.85i −0.487115 + 1.49919i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 2200.53 + 1598.78i 1.59365 + 1.15786i
\(125\) −1077.61 + 782.930i −0.771075 + 0.560219i
\(126\) 0 0
\(127\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1164.98 846.411i −0.742710 0.539611i
\(136\) 0 0
\(137\) 372.674 1146.97i 0.232407 0.715275i −0.765048 0.643973i \(-0.777285\pi\)
0.997455 0.0713013i \(-0.0227152\pi\)
\(138\) 0 0
\(139\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(140\) 0 0
\(141\) −88.9969 273.904i −0.0531553 0.163595i
\(142\) 0 0
\(143\) 0 0
\(144\) 2368.00 1.37037
\(145\) 0 0
\(146\) 0 0
\(147\) 2219.94 1612.88i 1.24556 0.904955i
\(148\) 1072.91 3302.07i 0.595895 1.83398i
\(149\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(150\) 0 0
\(151\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6120.00 3.17142
\(156\) 0 0
\(157\) 1079.23 + 784.106i 0.548610 + 0.398589i 0.827273 0.561800i \(-0.189891\pi\)
−0.278663 + 0.960389i \(0.589891\pi\)
\(158\) 0 0
\(159\) −1824.44 + 5615.04i −0.909982 + 2.80064i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1152.02 + 3545.54i 0.553576 + 1.70373i 0.699676 + 0.714460i \(0.253328\pi\)
−0.146100 + 0.989270i \(0.546672\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(168\) 0 0
\(169\) 1777.41 1291.36i 0.809017 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5760.00 −2.44603
\(178\) 0 0
\(179\) 1630.98 + 1184.98i 0.681034 + 0.494800i 0.873700 0.486464i \(-0.161714\pi\)
−0.192667 + 0.981264i \(0.561714\pi\)
\(180\) 4310.44 3131.72i 1.78490 1.29680i
\(181\) −633.485 + 1949.67i −0.260147 + 0.800649i 0.732625 + 0.680632i \(0.238295\pi\)
−0.992772 + 0.120017i \(0.961705\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2414.04 7429.65i −0.959372 2.95264i
\(186\) 0 0
\(187\) 0 0
\(188\) 288.000 0.111726
\(189\) 0 0
\(190\) 0 0
\(191\) −4223.07 + 3068.24i −1.59985 + 1.16236i −0.712038 + 0.702141i \(0.752228\pi\)
−0.887807 + 0.460215i \(0.847772\pi\)
\(192\) −1265.73 + 3895.53i −0.475763 + 1.46425i
\(193\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 847.943 + 2609.70i 0.309017 + 0.951057i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 3940.00 1.40351 0.701757 0.712417i \(-0.252399\pi\)
0.701757 + 0.712417i \(0.252399\pi\)
\(200\) 0 0
\(201\) 2692.41 + 1956.15i 0.944815 + 0.686448i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1234.83 3800.42i −0.414622 1.27608i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(212\) −4776.44 3470.28i −1.54739 1.12425i
\(213\) −3960.95 + 2877.80i −1.27418 + 0.925744i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −540.423 392.641i −0.162284 0.117907i 0.503679 0.863891i \(-0.331979\pi\)
−0.665964 + 0.745984i \(0.731979\pi\)
\(224\) 0 0
\(225\) 2275.29 7002.63i 0.674161 2.07485i
\(226\) 0 0
\(227\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(228\) 0 0
\(229\) 1022.85 + 3148.00i 0.295160 + 0.908408i 0.983168 + 0.182705i \(0.0584854\pi\)
−0.688008 + 0.725703i \(0.741515\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(234\) 0 0
\(235\) 524.243 380.885i 0.145523 0.105728i
\(236\) 1779.94 5478.09i 0.490950 1.51099i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(240\) 2847.90 + 8764.94i 0.765963 + 2.35739i
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −5032.00 −1.32841
\(244\) 0 0
\(245\) 4994.87 + 3628.99i 1.30249 + 0.946316i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 200.243 + 616.285i 0.0503555 + 0.154978i 0.973072 0.230500i \(-0.0740363\pi\)
−0.922717 + 0.385479i \(0.874036\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −3313.73 2407.57i −0.809017 0.587785i
\(257\) 6509.35 4729.32i 1.57993 1.14789i 0.663156 0.748481i \(-0.269216\pi\)
0.916774 0.399406i \(-0.130784\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −13284.0 −3.07936
\(266\) 0 0
\(267\) −10834.4 7871.62i −2.48334 1.80425i
\(268\) −2692.41 + 1956.15i −0.613675 + 0.445861i
\(269\) 862.157 2653.45i 0.195415 0.601426i −0.804556 0.593876i \(-0.797597\pi\)
0.999972 0.00754961i \(-0.00240314\pi\)
\(270\) 0 0
\(271\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 6912.00 1.50744
\(277\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(278\) 0 0
\(279\) −10177.4 + 7394.34i −2.18390 + 1.58669i
\(280\) 0 0
\(281\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(282\) 0 0
\(283\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(284\) −1512.95 4656.37i −0.316116 0.972905i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3974.70 + 2887.79i 0.809017 + 0.587785i
\(290\) 0 0
\(291\) −84.0526 + 258.687i −0.0169321 + 0.0521118i
\(292\) 0 0
\(293\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(294\) 0 0
\(295\) −4004.86 12325.7i −0.790413 2.43264i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 10303.6 + 7486.03i 1.98294 + 1.44069i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 9376.00 1.72616
\(310\) 0 0
\(311\) −7659.77 5565.15i −1.39661 1.01470i −0.995103 0.0988411i \(-0.968486\pi\)
−0.401507 0.915856i \(-0.631514\pi\)
\(312\) 0 0
\(313\) −3393.62 + 10444.5i −0.612840 + 1.88613i −0.183382 + 0.983042i \(0.558705\pi\)
−0.429458 + 0.903087i \(0.641295\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1852.25 + 5700.63i 0.328179 + 1.01003i 0.969985 + 0.243163i \(0.0781851\pi\)
−0.641807 + 0.766866i \(0.721815\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −9216.00 −1.60997
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 887.497 2731.43i 0.152177 0.468353i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8120.00 1.34839 0.674193 0.738555i \(-0.264492\pi\)
0.674193 + 0.738555i \(0.264492\pi\)
\(332\) 0 0
\(333\) 12991.2 + 9438.66i 2.13788 + 1.55326i
\(334\) 0 0
\(335\) −2313.92 + 7121.51i −0.377382 + 1.16146i
\(336\) 0 0
\(337\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(338\) 0 0
\(339\) 5295.32 + 16297.3i 0.848384 + 2.61106i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 12581.8 9141.24i 1.96343 1.42651i
\(346\) 0 0
\(347\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(348\) 0 0
\(349\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8802.00 1.32715 0.663574 0.748111i \(-0.269039\pi\)
0.663574 + 0.748111i \(0.269039\pi\)
\(354\) 0 0
\(355\) −8912.13 6475.04i −1.33241 0.968055i
\(356\) 10834.4 7871.62i 1.61298 1.17190i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(360\) 0 0
\(361\) −2119.55 6523.30i −0.309017 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8022.21 5828.48i 1.14102 0.829003i 0.153763 0.988108i \(-0.450861\pi\)
0.987262 + 0.159105i \(0.0508608\pi\)
\(368\) −2135.93 + 6573.70i −0.302562 + 0.931190i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −6724.21 20695.0i −0.937188 2.88437i
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 10656.0 1.46740
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3955.42 12173.5i 0.536085 1.64990i −0.205209 0.978718i \(-0.565787\pi\)
0.741293 0.671181i \(-0.234213\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4483.22 + 13797.9i 0.598125 + 1.84084i 0.538513 + 0.842617i \(0.318986\pi\)
0.0596113 + 0.998222i \(0.481014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −220.053 159.878i −0.0287925 0.0209190i
\(389\) −11431.4 + 8305.41i −1.48996 + 1.08252i −0.515791 + 0.856715i \(0.672502\pi\)
−0.974172 + 0.225807i \(0.927498\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2374.00 −0.300120 −0.150060 0.988677i \(-0.547947\pi\)
−0.150060 + 0.988677i \(0.547947\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −10303.6 + 7486.03i −1.28796 + 0.935754i
\(401\) −2808.96 + 8645.10i −0.349808 + 1.07660i 0.609151 + 0.793054i \(0.291510\pi\)
−0.958959 + 0.283544i \(0.908490\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1996.87 6145.73i −0.245000 0.754034i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(410\) 0 0
\(411\) −7805.40 + 5670.95i −0.936768 + 0.680602i
\(412\) −2897.34 + 8917.11i −0.346461 + 1.06630i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16344.0 −1.90562 −0.952812 0.303560i \(-0.901825\pi\)
−0.952812 + 0.303560i \(0.901825\pi\)
\(420\) 0 0
\(421\) 9408.87 + 6835.94i 1.08922 + 0.791362i 0.979267 0.202575i \(-0.0649310\pi\)
0.109950 + 0.993937i \(0.464931\pi\)
\(422\) 0 0
\(423\) −411.611 + 1266.81i −0.0473125 + 0.145613i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(432\) −4142.17 3009.46i −0.461320 0.335168i
\(433\) 10745.4 7806.96i 1.19258 0.866464i 0.199050 0.979989i \(-0.436214\pi\)
0.993535 + 0.113526i \(0.0362145\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −12691.0 −1.37037
\(442\) 0 0
\(443\) 15086.5 + 10961.0i 1.61802 + 1.17556i 0.811311 + 0.584615i \(0.198755\pi\)
0.806711 + 0.590946i \(0.201245\pi\)
\(444\) −22471.3 + 16326.3i −2.40189 + 1.74507i
\(445\) 9311.30 28657.2i 0.991906 3.05277i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2096.99 + 6453.87i 0.220408 + 0.678345i 0.998725 + 0.0504744i \(0.0160733\pi\)
−0.778318 + 0.627871i \(0.783927\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −17136.0 −1.78321
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 4805.83 + 14790.8i 0.487115 + 1.49919i
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 13268.0 1.33178 0.665892 0.746048i \(-0.268051\pi\)
0.665892 + 0.746048i \(0.268051\pi\)
\(464\) 0 0
\(465\) −39609.5 28778.0i −3.95020 2.86999i
\(466\) 0 0
\(467\) 1290.45 3971.61i 0.127870 0.393542i −0.866543 0.499102i \(-0.833663\pi\)
0.994413 + 0.105559i \(0.0336633\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3297.83 10149.7i −0.322624 0.992935i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 22091.0 16050.1i 2.12050 1.54063i
\(478\) 0 0
\(479\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −612.000 −0.0572979
\(486\) 0 0
\(487\) −13497.6 9806.61i −1.25593 0.912484i −0.257377 0.966311i \(-0.582858\pi\)
−0.998550 + 0.0538268i \(0.982858\pi\)
\(488\) 0 0
\(489\) 9216.12 28364.3i 0.852285 2.62306i
\(490\) 0 0
\(491\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 21760.0 1.96986
\(497\) 0 0
\(498\) 0 0
\(499\) −3333.15 + 2421.68i −0.299023 + 0.217253i −0.727172 0.686455i \(-0.759166\pi\)
0.428149 + 0.903708i \(0.359166\pi\)
\(500\) −3292.89 + 10134.5i −0.294525 + 0.906454i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −17576.0 −1.53960
\(508\) 0 0
\(509\) 18130.1 + 13172.3i 1.57878 + 1.14705i 0.918071 + 0.396416i \(0.129746\pi\)
0.660713 + 0.750638i \(0.270254\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6519.02 + 20063.5i 0.557791 + 1.71670i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16237.0 + 11796.9i −1.36536 + 0.991995i −0.367281 + 0.930110i \(0.619711\pi\)
−0.998083 + 0.0618848i \(0.980289\pi\)
\(522\) 0 0
\(523\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −503.000 −0.0413413
\(530\) 0 0
\(531\) 21552.2 + 15658.6i 1.76137 + 1.27971i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4983.83 15338.6i −0.400499 1.23261i
\(538\) 0 0
\(539\) 0 0
\(540\) −11520.0 −0.918040
\(541\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(542\) 0 0
\(543\) 13267.9 9639.68i 1.04858 0.761838i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(548\) −2981.40 9175.79i −0.232407 0.715275i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −19312.3 + 59437.2i −1.47705 + 4.54589i
\(556\) 0 0
\(557\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(564\) −1863.98 1354.26i −0.139162 0.101107i
\(565\) −31192.5 + 22662.6i −2.32261 + 1.68748i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 41760.0 3.04459
\(574\) 0 0
\(575\) 17387.4 + 12632.7i 1.26105 + 0.916207i
\(576\) 15326.0 11135.0i 1.10865 0.805483i
\(577\) −6942.38 + 21366.4i −0.500892 + 1.54159i 0.306676 + 0.951814i \(0.400783\pi\)
−0.807568 + 0.589774i \(0.799217\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21086.2 15320.0i 1.48266 1.07722i 0.505972 0.862550i \(-0.331134\pi\)
0.976688 0.214666i \(-0.0688662\pi\)
\(588\) 6783.54 20877.6i 0.475763 1.46425i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8583.26 26416.5i −0.595895 1.83398i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −25500.2 18527.0i −1.74816 1.27012i
\(598\) 0 0
\(599\) 5573.43 17153.3i 0.380174 1.17006i −0.559747 0.828663i \(-0.689102\pi\)
0.939921 0.341392i \(-0.110898\pi\)
\(600\) 0 0
\(601\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(602\) 0 0
\(603\) −4756.39 14638.7i −0.321219 0.988611i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3654.00 0.238419 0.119209 0.992869i \(-0.461964\pi\)
0.119209 + 0.992869i \(0.461964\pi\)
\(618\) 0 0
\(619\) −1501.54 1090.93i −0.0974989 0.0708371i 0.537968 0.842965i \(-0.319192\pi\)
−0.635467 + 0.772128i \(0.719192\pi\)
\(620\) 39609.5 28778.0i 2.56573 1.86411i
\(621\) −2669.91 + 8217.13i −0.172528 + 0.530986i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −277.806 855.000i −0.0177796 0.0547200i
\(626\) 0 0
\(627\) 0 0
\(628\) 10672.0 0.678120
\(629\) 0 0
\(630\) 0 0
\(631\) 10442.8 7587.13i 0.658829 0.478667i −0.207438 0.978248i \(-0.566513\pi\)
0.866267 + 0.499581i \(0.166513\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 14595.5 + 44920.3i 0.909982 + 2.80064i
\(637\) 0 0
\(638\) 0 0
\(639\) 22644.0 1.40185
\(640\) 0 0
\(641\) 3713.39 + 2697.93i 0.228814 + 0.166243i 0.696285 0.717765i \(-0.254835\pi\)
−0.467471 + 0.884008i \(0.654835\pi\)
\(642\) 0 0
\(643\) −3142.08 + 9670.34i −0.192709 + 0.593097i 0.807287 + 0.590159i \(0.200935\pi\)
−0.999996 + 0.00293742i \(0.999065\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10112.3 + 31122.4i 0.614458 + 1.89111i 0.409387 + 0.912361i \(0.365742\pi\)
0.205071 + 0.978747i \(0.434258\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 24128.1 + 17530.1i 1.44928 + 1.05296i
\(653\) −26488.8 + 19245.3i −1.58742 + 1.15333i −0.679922 + 0.733284i \(0.737986\pi\)
−0.907502 + 0.420047i \(0.862014\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −23582.0 −1.38765 −0.693823 0.720146i \(-0.744075\pi\)
−0.693823 + 0.720146i \(0.744075\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1651.39 + 5082.45i 0.0954354 + 0.293720i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(674\) 0 0
\(675\) −12879.6 + 9357.54i −0.734421 + 0.533588i
\(676\) 5431.28 16715.8i 0.309017 0.951057i
\(677\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −35352.0 −1.98054 −0.990268 0.139170i \(-0.955556\pi\)
−0.990268 + 0.139170i \(0.955556\pi\)
\(684\) 0 0
\(685\) −17562.1 12759.6i −0.979584 0.711709i
\(686\) 0 0
\(687\) 8182.77 25184.0i 0.454428 1.39859i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −9371.87 28843.6i −0.515952 1.58794i −0.781544 0.623851i \(-0.785567\pi\)
0.265592 0.964086i \(-0.414433\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −5184.00 −0.276937
\(706\) 0 0
\(707\) 0 0
\(708\) −37279.5 + 27085.1i −1.97888 + 1.43774i
\(709\) −10368.8 + 31911.8i −0.549234 + 1.69037i 0.161471 + 0.986877i \(0.448376\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −11347.1 34922.8i −0.596006 1.83432i
\(714\) 0 0
\(715\) 0 0
\(716\) 16128.0 0.841804
\(717\) 0 0
\(718\) 0 0
\(719\) −18319.4 + 13309.8i −0.950205 + 0.690365i −0.950855 0.309635i \(-0.899793\pi\)
0.000650107 1.00000i \(0.499793\pi\)
\(720\) 13171.5 40537.8i 0.681770 2.09827i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 5067.88 + 15597.3i 0.260147 + 0.800649i
\(725\) 0 0
\(726\) 0 0
\(727\) −33284.0 −1.69799 −0.848993 0.528405i \(-0.822790\pi\)
−0.848993 + 0.528405i \(0.822790\pi\)
\(728\) 0 0
\(729\) 24726.0 + 17964.5i 1.25621 + 0.912690i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(734\) 0 0
\(735\) −15263.0 46974.6i −0.765963 2.35739i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(740\) −50560.3 36734.2i −2.51167 1.82483i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −32079.1 23306.9i −1.55870 1.13246i −0.937064 0.349157i \(-0.886468\pi\)
−0.621637 0.783306i \(-0.713532\pi\)
\(752\) 1863.98 1354.26i 0.0903885 0.0656711i
\(753\) 1601.94 4930.28i 0.0775273 0.238605i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9711.17 + 29887.9i 0.466259 + 1.43500i 0.857392 + 0.514665i \(0.172084\pi\)
−0.391132 + 0.920335i \(0.627916\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12904.5 + 39716.1i −0.611087 + 1.88073i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 10125.9 + 31164.2i 0.475763 + 1.46425i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −64368.0 −3.00669
\(772\) 0 0
\(773\) −26081.1 18949.0i −1.21355 0.881693i −0.217999 0.975949i \(-0.569953\pi\)
−0.995548 + 0.0942558i \(0.969953\pi\)
\(774\) 0 0
\(775\) 20908.1 64348.5i 0.969085 2.98254i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 17759.5 + 12903.1i 0.809017 + 0.587785i
\(785\) 19426.1 14113.9i 0.883246 0.641716i
\(786\) 0 0
\(787\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0