Properties

Label 121.4.c.a.81.1
Level $121$
Weight $4$
Character 121.81
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 81.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 121.81
Dual form 121.4.c.a.3.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.47214 - 7.60845i) q^{3} +(-2.47214 - 7.60845i) q^{4} +(-14.5623 + 10.5801i) q^{5} +(-29.9336 - 21.7481i) q^{9} +O(q^{10})\) \(q+(2.47214 - 7.60845i) q^{3} +(-2.47214 - 7.60845i) q^{4} +(-14.5623 + 10.5801i) q^{5} +(-29.9336 - 21.7481i) q^{9} -64.0000 q^{12} +(44.4984 + 136.952i) q^{15} +(-51.7771 + 37.6183i) q^{16} +(116.498 + 84.6411i) q^{20} -108.000 q^{23} +(61.4944 - 189.260i) q^{25} +(-64.7214 + 47.0228i) q^{27} +(-275.066 - 199.847i) q^{31} +(-91.4690 + 281.513i) q^{36} +(-134.113 - 412.759i) q^{37} +666.000 q^{45} +(-11.1246 + 34.2380i) q^{47} +(158.217 + 486.941i) q^{48} +(277.493 - 201.610i) q^{49} +(597.055 + 433.786i) q^{53} +(-222.492 - 684.761i) q^{59} +(931.988 - 677.129i) q^{60} +(414.217 + 300.946i) q^{64} -416.000 q^{67} +(-266.991 + 821.713i) q^{69} +(-495.118 + 359.725i) q^{71} +(-1287.96 - 935.754i) q^{75} +(355.988 - 1095.62i) q^{80} +(-110.937 - 341.429i) q^{81} +1674.00 q^{89} +(266.991 + 821.713i) q^{92} +(-2200.53 + 1598.78i) q^{93} +(27.5066 + 19.9847i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} + 8 q^{4} - 18 q^{5} - 37 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(3\) 2.47214 7.60845i 0.475763 1.46425i −0.369163 0.929364i \(-0.620356\pi\)
0.844926 0.534883i \(-0.179644\pi\)
\(4\) −2.47214 7.60845i −0.309017 0.951057i
\(5\) −14.5623 + 10.5801i −1.30249 + 0.946316i −0.999977 0.00682845i \(-0.997826\pi\)
−0.302516 + 0.953144i \(0.597826\pi\)
\(6\) 0 0
\(7\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(8\) 0 0
\(9\) −29.9336 21.7481i −1.10865 0.805483i
\(10\) 0 0
\(11\) 0 0
\(12\) −64.0000 −1.53960
\(13\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) 44.4984 + 136.952i 0.765963 + 2.35739i
\(16\) −51.7771 + 37.6183i −0.809017 + 0.587785i
\(17\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(20\) 116.498 + 84.6411i 1.30249 + 0.946316i
\(21\) 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
\(25\) 61.4944 189.260i 0.491955 1.51408i
\(26\) 0 0
\(27\) −64.7214 + 47.0228i −0.461320 + 0.335168i
\(28\) 0 0
\(29\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(30\) 0 0
\(31\) −275.066 199.847i −1.59365 1.15786i −0.898479 0.439017i \(-0.855327\pi\)
−0.695175 0.718840i \(-0.744673\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −91.4690 + 281.513i −0.423468 + 1.30330i
\(37\) −134.113 412.759i −0.595895 1.83398i −0.550221 0.835019i \(-0.685457\pi\)
−0.0456733 0.998956i \(-0.514543\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(42\) 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
\(47\) −11.1246 + 34.2380i −0.0345253 + 0.106258i −0.966834 0.255405i \(-0.917791\pi\)
0.932309 + 0.361663i \(0.117791\pi\)
\(48\) 158.217 + 486.941i 0.475763 + 1.46425i
\(49\) 277.493 201.610i 0.809017 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 597.055 + 433.786i 1.54739 + 1.12425i 0.945479 + 0.325684i \(0.105595\pi\)
0.601913 + 0.798562i \(0.294405\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −222.492 684.761i −0.490950 1.51099i −0.823175 0.567787i \(-0.807800\pi\)
0.332226 0.943200i \(-0.392200\pi\)
\(60\) 931.988 677.129i 2.00532 1.45695i
\(61\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 414.217 + 300.946i 0.809017 + 0.587785i
\(65\) 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
\(69\) −266.991 + 821.713i −0.465825 + 1.43366i
\(70\) 0 0
\(71\) −495.118 + 359.725i −0.827602 + 0.601288i −0.918880 0.394537i \(-0.870905\pi\)
0.0912779 + 0.995825i \(0.470905\pi\)
\(72\) 0 0
\(73\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) 0 0
\(75\) −1287.96 935.754i −1.98294 1.44069i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(80\) 355.988 1095.62i 0.497508 1.53117i
\(81\) −110.937 341.429i −0.152177 0.468353i
\(82\) 0 0
\(83\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(84\) 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
\(92\) 266.991 + 821.713i 0.302562 + 0.931190i
\(93\) −2200.53 + 1598.78i −2.45359 + 1.78264i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 27.5066 + 19.9847i 0.0287925 + 0.0209190i 0.602088 0.798429i \(-0.294335\pi\)
−0.573296 + 0.819348i \(0.694335\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1592.00 −1.59200
\(101\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(102\) 0 0
\(103\) 362.168 + 1114.64i 0.346461 + 1.06630i 0.960797 + 0.277252i \(0.0894238\pi\)
−0.614336 + 0.789044i \(0.710576\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(108\) 517.771 + 376.183i 0.461320 + 0.335168i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −3472.00 −2.96890
\(112\) 0 0
\(113\) 661.914 2037.16i 0.551041 1.69593i −0.155134 0.987893i \(-0.549581\pi\)
0.706175 0.708037i \(-0.250419\pi\)
\(114\) 0 0
\(115\) 1572.73 1142.65i 1.27528 0.926549i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −840.526 + 2586.87i −0.608722 + 1.87345i
\(125\) 411.611 + 1266.81i 0.294525 + 0.906454i
\(126\) 0 0
\(127\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(128\) 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
\(135\) 444.984 1369.52i 0.283690 0.873108i
\(136\) 0 0
\(137\) −975.674 + 708.869i −0.608449 + 0.442064i −0.848868 0.528605i \(-0.822715\pi\)
0.240419 + 0.970669i \(0.422715\pi\)
\(138\) 0 0
\(139\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(140\) 0 0
\(141\) 232.997 + 169.282i 0.139162 + 0.101107i
\(142\) 0 0
\(143\) 0 0
\(144\) 2368.00 1.37037
\(145\) 0 0
\(146\) 0 0
\(147\) −847.943 2609.70i −0.475763 1.46425i
\(148\) −2808.91 + 2040.79i −1.56007 + 1.13346i
\(149\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6120.00 3.17142
\(156\) 0 0
\(157\) −412.229 + 1268.71i −0.209551 + 0.644930i 0.789945 + 0.613177i \(0.210109\pi\)
−0.999496 + 0.0317528i \(0.989891\pi\)
\(158\) 0 0
\(159\) 4776.44 3470.28i 2.38237 1.73089i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3016.02 2191.26i −1.44928 1.05296i −0.985999 0.166751i \(-0.946672\pi\)
−0.463281 0.886212i \(-0.653328\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(168\) 0 0
\(169\) −678.910 2089.47i −0.309017 0.951057i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5760.00 −2.44603
\(178\) 0 0
\(179\) −622.978 + 1917.33i −0.260132 + 0.800603i 0.732643 + 0.680613i \(0.238286\pi\)
−0.992775 + 0.119990i \(0.961714\pi\)
\(180\) −1646.44 5067.23i −0.681770 2.09827i
\(181\) 1658.48 1204.96i 0.681073 0.494828i −0.192641 0.981269i \(-0.561705\pi\)
0.873713 + 0.486441i \(0.161705\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6320.04 + 4591.78i 2.51167 + 1.82483i
\(186\) 0 0
\(187\) 0 0
\(188\) 288.000 0.111726
\(189\) 0 0
\(190\) 0 0
\(191\) 1613.07 + 4964.52i 0.611087 + 1.88073i 0.447743 + 0.894162i \(0.352228\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(192\) 3313.73 2407.57i 1.24556 0.904955i
\(193\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2219.94 1612.88i −0.809017 0.587785i
\(197\) 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
\(201\) −1028.41 + 3165.12i −0.360887 + 1.11070i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3232.83 + 2348.79i 1.08549 + 0.788658i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(212\) 1824.44 5615.04i 0.591051 1.81907i
\(213\) 1512.95 + 4656.37i 0.486692 + 1.49788i
\(214\) 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
\(223\) 206.423 635.306i 0.0619871 0.190777i −0.915267 0.402847i \(-0.868021\pi\)
0.977254 + 0.212070i \(0.0680206\pi\)
\(224\) 0 0
\(225\) −5956.79 + 4327.86i −1.76498 + 1.28233i
\(226\) 0 0
\(227\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(228\) 0 0
\(229\) −2677.85 1945.57i −0.772738 0.561427i 0.130053 0.991507i \(-0.458485\pi\)
−0.902791 + 0.430080i \(0.858485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(234\) 0 0
\(235\) −200.243 616.285i −0.0555847 0.171072i
\(236\) −4659.94 + 3385.64i −1.28532 + 0.933841i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(240\) −7455.90 5417.03i −2.00532 1.45695i
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −5032.00 −1.32841
\(244\) 0 0
\(245\) −1907.87 + 5871.82i −0.497508 + 1.53117i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −524.243 380.885i −0.131832 0.0957818i 0.519915 0.854218i \(-0.325964\pi\)
−0.651747 + 0.758436i \(0.725964\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1265.73 3895.53i 0.309017 0.951057i
\(257\) −2486.35 7652.20i −0.603480 1.85732i −0.506921 0.861992i \(-0.669216\pi\)
−0.0965586 0.995327i \(-0.530784\pi\)
\(258\) 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
\(267\) 4138.36 12736.5i 0.948551 2.91934i
\(268\) 1028.41 + 3165.12i 0.234403 + 0.721419i
\(269\) −2257.16 + 1639.92i −0.511603 + 0.371702i −0.813431 0.581661i \(-0.802403\pi\)
0.301828 + 0.953362i \(0.402403\pi\)
\(270\) 0 0
\(271\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 6912.00 1.50744
\(277\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(278\) 0 0
\(279\) 3887.43 + 11964.3i 0.834174 + 2.56732i
\(280\) 0 0
\(281\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(284\) 3960.95 + 2877.80i 0.827602 + 0.601288i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1518.20 + 4672.54i −0.309017 + 0.951057i
\(290\) 0 0
\(291\) 220.053 159.878i 0.0443289 0.0322068i
\(292\) 0 0
\(293\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(294\) 0 0
\(295\) 10484.9 + 7617.70i 2.06933 + 1.50346i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −3935.64 + 12112.7i −0.757414 + 2.33108i
\(301\) 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
\(311\) 2925.77 9004.60i 0.533458 1.64181i −0.213500 0.976943i \(-0.568486\pi\)
0.746958 0.664871i \(-0.231514\pi\)
\(312\) 0 0
\(313\) 8884.62 6455.06i 1.60444 1.16569i 0.726177 0.687508i \(-0.241295\pi\)
0.878260 0.478183i \(-0.158705\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4849.25 3523.18i −0.859183 0.624233i 0.0684798 0.997653i \(-0.478185\pi\)
−0.927662 + 0.373420i \(0.878185\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −9216.00 −1.60997
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2323.50 + 1688.12i −0.398405 + 0.289458i
\(325\) 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
\(333\) −4962.19 + 15272.1i −0.816596 + 2.51323i
\(334\) 0 0
\(335\) 6057.92 4401.34i 0.987998 0.717823i
\(336\) 0 0
\(337\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(338\) 0 0
\(339\) −13863.3 10072.3i −2.22110 1.61372i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4805.83 14790.8i −0.749963 2.30815i
\(346\) 0 0
\(347\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(350\) 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
\(355\) 3404.13 10476.8i 0.508937 1.56635i
\(356\) −4138.36 12736.5i −0.616102 1.89617i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(360\) 0 0
\(361\) 5549.05 + 4031.62i 0.809017 + 0.587785i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3064.21 9430.68i −0.435833 1.34136i −0.892231 0.451579i \(-0.850861\pi\)
0.456398 0.889776i \(-0.349139\pi\)
\(368\) 5591.93 4062.77i 0.792118 0.575507i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 17604.2 + 12790.2i 2.45359 + 1.78264i
\(373\) 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
\(379\) −10355.4 + 7523.65i −1.40349 + 1.01969i −0.409259 + 0.912418i \(0.634213\pi\)
−0.994229 + 0.107276i \(0.965787\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11737.2 8527.59i −1.56591 1.13770i −0.930944 0.365161i \(-0.881014\pi\)
−0.634967 0.772540i \(-0.718986\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 84.0526 258.687i 0.0109977 0.0338476i
\(389\) 4366.41 + 13438.4i 0.569115 + 1.75156i 0.655396 + 0.755285i \(0.272502\pi\)
−0.0862809 + 0.996271i \(0.527498\pi\)
\(390\) 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
\(400\) 3935.64 + 12112.7i 0.491955 + 1.51408i
\(401\) 7353.96 5342.97i 0.915809 0.665374i −0.0266682 0.999644i \(-0.508490\pi\)
0.942477 + 0.334270i \(0.108490\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 5227.87 + 3798.27i 0.641419 + 0.466018i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(410\) 0 0
\(411\) 2981.40 + 9175.79i 0.357814 + 1.10124i
\(412\) 7585.34 5511.07i 0.907046 0.659008i
\(413\) 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
\(421\) −3593.87 + 11060.8i −0.416044 + 1.28045i 0.495270 + 0.868739i \(0.335069\pi\)
−0.911314 + 0.411712i \(0.864931\pi\)
\(422\) 0 0
\(423\) 1077.61 782.930i 0.123866 0.0899938i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(432\) 1582.17 4869.41i 0.176208 0.542314i
\(433\) −4104.36 12631.9i −0.455527 1.40197i −0.870516 0.492141i \(-0.836214\pi\)
0.414989 0.909827i \(-0.363786\pi\)
\(434\) 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
\(443\) −5762.55 + 17735.3i −0.618029 + 1.90210i −0.305293 + 0.952258i \(0.598755\pi\)
−0.312736 + 0.949840i \(0.601245\pi\)
\(444\) 8583.26 + 26416.5i 0.917440 + 2.82359i
\(445\) −24377.3 + 17711.1i −2.59684 + 1.88672i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5489.99 3988.71i −0.577035 0.419240i 0.260619 0.965442i \(-0.416073\pi\)
−0.837654 + 0.546201i \(0.816073\pi\)
\(450\) 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 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −12581.8 9141.24i −1.27528 0.926549i
\(461\) 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
\(465\) 15129.5 46563.7i 1.50884 4.64374i
\(466\) 0 0
\(467\) −3378.45 + 2454.59i −0.334767 + 0.243223i −0.742451 0.669901i \(-0.766337\pi\)
0.407684 + 0.913123i \(0.366337\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 8633.83 + 6272.84i 0.844641 + 0.613668i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8438.02 25969.5i −0.809959 2.49280i
\(478\) 0 0
\(479\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −612.000 −0.0572979
\(486\) 0 0
\(487\) 5155.64 15867.4i 0.479721 1.47643i −0.359761 0.933044i \(-0.617142\pi\)
0.839483 0.543386i \(-0.182858\pi\)
\(488\) 0 0
\(489\) −24128.1 + 17530.1i −2.23131 + 1.62114i
\(490\) 0 0
\(491\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 21760.0 1.96986
\(497\) 0 0
\(498\) 0 0
\(499\) 1273.15 + 3918.35i 0.114216 + 0.351522i 0.991783 0.127933i \(-0.0408343\pi\)
−0.877566 + 0.479455i \(0.840834\pi\)
\(500\) 8620.89 6263.44i 0.771075 0.560219i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −17576.0 −1.53960
\(508\) 0 0
\(509\) −6925.07 + 21313.2i −0.603042 + 1.85597i −0.0933141 + 0.995637i \(0.529746\pi\)
−0.509728 + 0.860336i \(0.670254\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17067.0 12399.9i −1.46032 1.06098i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6201.97 + 19087.7i 0.521523 + 1.60508i 0.771091 + 0.636725i \(0.219711\pi\)
−0.249569 + 0.968357i \(0.580289\pi\)
\(522\) 0 0
\(523\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −503.000 −0.0413413
\(530\) 0 0
\(531\) −8232.21 + 25336.1i −0.672783 + 2.07061i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13047.8 + 9479.80i 1.04852 + 0.761794i
\(538\) 0 0
\(539\) 0 0
\(540\) −11520.0 −0.918040
\(541\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(542\) 0 0
\(543\) −5067.88 15597.3i −0.400522 1.23268i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(548\) 7805.40 + 5670.95i 0.608449 + 0.442064i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 50560.3 36734.2i 3.86697 2.80952i
\(556\) 0 0
\(557\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(564\) 711.975 2191.23i 0.0531553 0.163595i
\(565\) 11914.5 + 36668.9i 0.887159 + 2.73040i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(570\) 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
\(575\) −6641.39 + 20440.1i −0.481679 + 1.48245i
\(576\) −5854.02 18016.8i −0.423468 1.30330i
\(577\) 18175.4 13205.2i 1.31135 0.952754i 0.311356 0.950293i \(-0.399217\pi\)
0.999997 0.00246060i \(-0.000783233\pi\)
\(578\) 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
\(587\) −8054.22 24788.3i −0.566326 1.74297i −0.663979 0.747751i \(-0.731134\pi\)
0.0976537 0.995220i \(-0.468866\pi\)
\(588\) −17759.5 + 12903.1i −1.24556 + 0.904955i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 22471.3 + 16326.3i 1.56007 + 1.13346i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 9740.22 29977.3i 0.667740 2.05509i
\(598\) 0 0
\(599\) −14591.4 + 10601.3i −0.995308 + 0.723134i −0.961077 0.276280i \(-0.910898\pi\)
−0.0342312 + 0.999414i \(0.510898\pi\)
\(600\) 0 0
\(601\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(602\) 0 0
\(603\) 12452.4 + 9047.19i 0.840963 + 0.610995i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(614\) 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
\(619\) 573.536 1765.16i 0.0372413 0.114617i −0.930708 0.365764i \(-0.880808\pi\)
0.967949 + 0.251147i \(0.0808078\pi\)
\(620\) −15129.5 46563.7i −0.980023 3.01620i
\(621\) 6989.91 5078.46i 0.451683 0.328167i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 727.306 + 528.419i 0.0465476 + 0.0338188i
\(626\) 0 0
\(627\) 0 0
\(628\) 10672.0 0.678120
\(629\) 0 0
\(630\) 0 0
\(631\) −3988.79 12276.2i −0.251650 0.774500i −0.994471 0.105010i \(-0.966513\pi\)
0.742821 0.669490i \(-0.233487\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −38211.5 27762.3i −2.38237 1.73089i
\(637\) 0 0
\(638\) 0 0
\(639\) 22644.0 1.40185
\(640\) 0 0
\(641\) −1418.39 + 4365.35i −0.0873993 + 0.268987i −0.985198 0.171418i \(-0.945165\pi\)
0.897799 + 0.440405i \(0.145165\pi\)
\(642\) 0 0
\(643\) 8226.08 5976.60i 0.504518 0.366554i −0.306222 0.951960i \(-0.599065\pi\)
0.810740 + 0.585406i \(0.199065\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26474.3 19234.7i −1.60867 1.16877i −0.867473 0.497483i \(-0.834258\pi\)
−0.741199 0.671285i \(-0.765742\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −9216.12 + 28364.3i −0.553576 + 1.70373i
\(653\) 10117.8 + 31139.5i 0.606342 + 1.86613i 0.487287 + 0.873242i \(0.337986\pi\)
0.119055 + 0.992888i \(0.462014\pi\)
\(654\) 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
\(669\) −4323.39 3141.12i −0.249853 0.181529i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(674\) 0 0
\(675\) 4919.55 + 15140.8i 0.280524 + 0.863364i
\(676\) −14219.3 + 10330.9i −0.809017 + 0.587785i
\(677\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(678\) 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
\(685\) 6708.14 20645.5i 0.374168 1.15157i
\(686\) 0 0
\(687\) −21422.8 + 15564.6i −1.18971 + 0.864374i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 24535.9 + 17826.4i 1.35078 + 0.981398i 0.998972 + 0.0453259i \(0.0144326\pi\)
0.351807 + 0.936073i \(0.385567\pi\)
\(692\) 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 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −5184.00 −0.276937
\(706\) 0 0
\(707\) 0 0
\(708\) 14239.5 + 43824.7i 0.755866 + 2.32632i
\(709\) 27145.8 19722.5i 1.43791 1.04470i 0.449439 0.893311i \(-0.351624\pi\)
0.988473 0.151394i \(-0.0483763\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29707.1 + 21583.5i 1.56036 + 1.13367i
\(714\) 0 0
\(715\) 0 0
\(716\) 16128.0 0.841804
\(717\) 0 0
\(718\) 0 0
\(719\) 6997.38 + 21535.7i 0.362946 + 1.11703i 0.951257 + 0.308399i \(0.0997931\pi\)
−0.588311 + 0.808635i \(0.700207\pi\)
\(720\) −34483.5 + 25053.8i −1.78490 + 1.29680i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −13267.9 9639.68i −0.681073 0.494828i
\(725\) 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
\(729\) −9444.49 + 29067.1i −0.479830 + 1.47676i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(734\) 0 0
\(735\) 39959.0 + 29031.9i 2.00532 + 1.45695i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(740\) 19312.3 59437.2i 0.959372 2.95264i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12253.1 37711.3i 0.595371 1.83236i 0.0424993 0.999096i \(-0.486468\pi\)
0.552872 0.833267i \(-0.313532\pi\)
\(752\) −711.975 2191.23i −0.0345253 0.106258i
\(753\) −4193.94 + 3047.08i −0.202969 + 0.147466i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −25424.2 18471.7i −1.22068 0.886878i −0.224526 0.974468i \(-0.572084\pi\)
−0.996157 + 0.0875898i \(0.972084\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 33784.5 24545.9i 1.59985 1.16236i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −26509.9 19260.5i −1.24556 0.904955i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −64368.0 −3.00669
\(772\) 0 0
\(773\) 9962.09 30660.2i 0.463534 1.42661i −0.397284 0.917696i \(-0.630047\pi\)
0.860817 0.508914i \(-0.169953\pi\)
\(774\) 0 0
\(775\) −54738.1 + 39769.6i −2.53710 + 1.84331i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −6783.54 + 20877.6i −0.309017 + 0.951057i
\(785\) −7420.12 22836.8i −0.337370 1.03832i
\(786\) 0 0
\(787\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
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