Properties

Label 121.4.c.a.3.1
Level $121$
Weight $4$
Character 121.3
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 3.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 121.3
Dual form 121.4.c.a.81.1

$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(3\) 2.47214 + 7.60845i 0.475763 + 1.46425i 0.844926 + 0.534883i \(0.179644\pi\)
−0.369163 + 0.929364i \(0.620356\pi\)
\(4\) −2.47214 + 7.60845i −0.309017 + 0.951057i
\(5\) −14.5623 10.5801i −1.30249 0.946316i −0.302516 0.953144i \(-0.597826\pi\)
−0.999977 + 0.00682845i \(0.997826\pi\)
\(6\) 0 0
\(7\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(8\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(14\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(18\) 0 0
\(19\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(20\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(30\) 0 0
\(31\) −275.066 + 199.847i −1.59365 + 1.15786i −0.695175 + 0.718840i \(0.744673\pi\)
−0.898479 + 0.439017i \(0.855327\pi\)
\(32\) 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 −0.0456733 + 0.998956i \(0.514543\pi\)
−0.550221 + 0.835019i \(0.685457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 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 0.932309 0.361663i \(-0.117791\pi\)
−0.966834 + 0.255405i \(0.917791\pi\)
\(48\) 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 0.601913 0.798562i \(-0.294405\pi\)
0.945479 0.325684i \(-0.105595\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −222.492 + 684.761i −0.490950 + 1.51099i 0.332226 + 0.943200i \(0.392200\pi\)
−0.823175 + 0.567787i \(0.807800\pi\)
\(60\) 931.988 + 677.129i 2.00532 + 1.45695i
\(61\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(62\) 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 0.0912779 0.995825i \(-0.470905\pi\)
−0.918880 + 0.394537i \(0.870905\pi\)
\(72\) 0 0
\(73\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(74\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(80\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 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 −0.573296 0.819348i \(-0.694335\pi\)
0.602088 + 0.798429i \(0.294335\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1592.00 −1.59200
\(101\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(102\) 0 0
\(103\) 362.168 1114.64i 0.346461 1.06630i −0.614336 0.789044i \(-0.710576\pi\)
0.960797 0.277252i \(-0.0894238\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(108\) 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 0.706175 + 0.708037i \(0.250419\pi\)
−0.155134 + 0.987893i \(0.549581\pi\)
\(114\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(128\) 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 0.240419 0.970669i \(-0.422715\pi\)
−0.848868 + 0.528605i \(0.822715\pi\)
\(138\) 0 0
\(139\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(140\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(150\) 0 0
\(151\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(152\) 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 −0.999496 0.0317528i \(-0.989891\pi\)
0.789945 0.613177i \(-0.210109\pi\)
\(158\) 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 −0.463281 + 0.886212i \(0.653328\pi\)
−0.985999 + 0.166751i \(0.946672\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(168\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(174\) 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 −0.992775 0.119990i \(-0.961714\pi\)
0.732643 0.680613i \(-0.238286\pi\)
\(180\) −1646.44 + 5067.23i −0.681770 + 2.09827i
\(181\) 1658.48 + 1204.96i 0.681073 + 0.494828i 0.873713 0.486441i \(-0.161705\pi\)
−0.192641 + 0.981269i \(0.561705\pi\)
\(182\) 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 0.163343 0.986569i \(-0.447772\pi\)
0.447743 0.894162i \(-0.352228\pi\)
\(192\) 3313.73 + 2407.57i 1.24556 + 0.904955i
\(193\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(194\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(212\) 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 0.977254 0.212070i \(-0.0680206\pi\)
−0.915267 + 0.402847i \(0.868021\pi\)
\(224\) 0 0
\(225\) −5956.79 4327.86i −1.76498 1.28233i
\(226\) 0 0
\(227\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(228\) 0 0
\(229\) −2677.85 + 1945.57i −0.772738 + 0.561427i −0.902791 0.430080i \(-0.858485\pi\)
0.130053 + 0.991507i \(0.458485\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(234\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(240\) −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 −0.651747 0.758436i \(-0.725964\pi\)
0.519915 + 0.854218i \(0.325964\pi\)
\(252\) 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 −0.0965586 + 0.995327i \(0.530784\pi\)
−0.506921 + 0.861992i \(0.669216\pi\)
\(258\) 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 0.301828 0.953362i \(-0.402403\pi\)
−0.813431 + 0.581661i \(0.802403\pi\)
\(270\) 0 0
\(271\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 6912.00 1.50744
\(277\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(278\) 0 0
\(279\) 3887.43 11964.3i 0.834174 2.56732i
\(280\) 0 0
\(281\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(282\) 0 0
\(283\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(284\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(294\) 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 0.746958 + 0.664871i \(0.231514\pi\)
−0.213500 + 0.976943i \(0.568486\pi\)
\(312\) 0 0
\(313\) 8884.62 + 6455.06i 1.60444 + 1.16569i 0.878260 + 0.478183i \(0.158705\pi\)
0.726177 + 0.687508i \(0.241295\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4849.25 + 3523.18i −0.859183 + 0.624233i −0.927662 0.373420i \(-0.878185\pi\)
0.0684798 + 0.997653i \(0.478185\pi\)
\(318\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(338\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(348\) 0 0
\(349\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(350\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(360\) 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 0.456398 + 0.889776i \(0.349139\pi\)
−0.892231 + 0.451579i \(0.850861\pi\)
\(368\) 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 −0.994229 0.107276i \(-0.965787\pi\)
−0.409259 0.912418i \(-0.634213\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11737.2 + 8527.59i −1.56591 + 1.13770i −0.634967 + 0.772540i \(0.718986\pi\)
−0.930944 + 0.365161i \(0.881014\pi\)
\(384\) 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 −0.0862809 0.996271i \(-0.527498\pi\)
0.655396 0.755285i \(-0.272502\pi\)
\(390\) 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 0.942477 0.334270i \(-0.108490\pi\)
−0.0266682 + 0.999644i \(0.508490\pi\)
\(402\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(410\) 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 −0.911314 0.411712i \(-0.864931\pi\)
0.495270 0.868739i \(-0.335069\pi\)
\(422\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(432\) 1582.17 + 4869.41i 0.176208 + 0.542314i
\(433\) −4104.36 + 12631.9i −0.455527 + 1.40197i 0.414989 + 0.909827i \(0.363786\pi\)
−0.870516 + 0.492141i \(0.836214\pi\)
\(434\) 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 −0.312736 0.949840i \(-0.601245\pi\)
−0.305293 0.952258i \(-0.598755\pi\)
\(444\) 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 −0.837654 0.546201i \(-0.816073\pi\)
0.260619 + 0.965442i \(0.416073\pi\)
\(450\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(458\) 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 0.407684 0.913123i \(-0.366337\pi\)
−0.742451 + 0.669901i \(0.766337\pi\)
\(468\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(480\) 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 0.839483 + 0.543386i \(0.182858\pi\)
−0.359761 + 0.933044i \(0.617142\pi\)
\(488\) 0 0
\(489\) −24128.1 17530.1i −2.23131 1.62114i
\(490\) 0 0
\(491\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(492\) 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 −0.877566 0.479455i \(-0.840834\pi\)
0.991783 + 0.127933i \(0.0408343\pi\)
\(500\) 8620.89 + 6263.44i 0.771075 + 0.560219i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(504\) 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 −0.509728 0.860336i \(-0.670254\pi\)
−0.0933141 0.995637i \(-0.529746\pi\)
\(510\) 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 −0.249569 0.968357i \(-0.580289\pi\)
0.771091 0.636725i \(-0.219711\pi\)
\(522\) 0 0
\(523\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(524\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(542\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(548\) 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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(564\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(570\) 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 0.999997 + 0.00246060i \(0.000783233\pi\)
0.311356 + 0.950293i \(0.399217\pi\)
\(578\) 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 0.0976537 + 0.995220i \(0.468866\pi\)
−0.663979 + 0.747751i \(0.731134\pi\)
\(588\) −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 −0.0342312 0.999414i \(-0.510898\pi\)
−0.961077 + 0.276280i \(0.910898\pi\)
\(600\) 0 0
\(601\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(602\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(614\) 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 0.967949 0.251147i \(-0.0808078\pi\)
−0.930708 + 0.365764i \(0.880808\pi\)
\(620\) −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 0.742821 + 0.669490i \(0.233487\pi\)
−0.994471 + 0.105010i \(0.966513\pi\)
\(632\) 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 0.897799 0.440405i \(-0.145165\pi\)
−0.985198 + 0.171418i \(0.945165\pi\)
\(642\) 0 0
\(643\) 8226.08 + 5976.60i 0.504518 + 0.366554i 0.810740 0.585406i \(-0.199065\pi\)
−0.306222 + 0.951960i \(0.599065\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −26474.3 + 19234.7i −1.60867 + 1.16877i −0.741199 + 0.671285i \(0.765742\pi\)
−0.867473 + 0.497483i \(0.834258\pi\)
\(648\) 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 0.119055 0.992888i \(-0.462014\pi\)
0.487287 0.873242i \(-0.337986\pi\)
\(654\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(674\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(678\) 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 0.351807 0.936073i \(-0.385567\pi\)
0.998972 0.0453259i \(-0.0144326\pi\)
\(692\) 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 \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(702\) 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 0.988473 + 0.151394i \(0.0483763\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(710\) 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 −0.588311 0.808635i \(-0.700207\pi\)
0.951257 0.308399i \(-0.0997931\pi\)
\(720\) −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 \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(734\) 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 \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(740\) 19312.3 + 59437.2i 0.959372 + 2.95264i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(744\) 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 0.552872 + 0.833267i \(0.313532\pi\)
0.0424993 + 0.999096i \(0.486468\pi\)
\(752\) −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 −0.996157 0.0875898i \(-0.972084\pi\)
−0.224526 + 0.974468i \(0.572084\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(762\) 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 0.860817 + 0.508914i \(0.169953\pi\)
−0.397284 + 0.917696i \(0.630047\pi\)
\(774\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
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