Properties

Label 201.4.j.a.5.1
Level $201$
Weight $4$
Character 201.5
Analytic conductor $11.859$
Analytic rank $0$
Dimension $20$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(5,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.j (of order \(22\), degree \(10\), 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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label 5.1
Root \(0.928368 - 0.371662i\) of defining polynomial
Character \(\chi\) \(=\) 201.5
Dual form 201.4.j.a.161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.72659 + 2.15856i) q^{3} +(1.13852 + 7.91857i) q^{4} +(18.3144 + 15.8695i) q^{7} +(17.6812 - 20.4052i) q^{9} +O(q^{10})\) \(q+(-4.72659 + 2.15856i) q^{3} +(1.13852 + 7.91857i) q^{4} +(18.3144 + 15.8695i) q^{7} +(17.6812 - 20.4052i) q^{9} +(-22.4740 - 34.9703i) q^{12} +(50.6773 + 78.8554i) q^{13} +(-61.4076 + 18.0309i) q^{16} +(-77.6235 - 89.5823i) q^{19} +(-120.820 - 35.4758i) q^{21} +(-105.157 + 67.5801i) q^{25} +(-39.5260 + 134.613i) q^{27} +(-104.812 + 163.091i) q^{28} +(-33.9504 + 52.8278i) q^{31} +(181.711 + 116.778i) q^{36} -328.515 q^{37} +(-409.744 - 263.327i) q^{39} +(459.997 + 66.1376i) q^{43} +(251.327 - 217.776i) q^{48} +(34.7613 + 241.770i) q^{49} +(-566.725 + 491.070i) q^{52} +(560.263 + 255.863i) q^{57} +(62.8229 - 213.955i) q^{61} +(647.641 - 93.1167i) q^{63} +(-212.692 - 465.732i) q^{64} +(547.134 + 37.5088i) q^{67} +(-994.413 - 291.986i) q^{73} +(351.157 - 546.410i) q^{75} +(620.988 - 716.658i) q^{76} +(237.905 + 370.187i) q^{79} +(-103.748 - 721.580i) q^{81} +(143.363 - 997.109i) q^{84} +(-323.272 + 2248.41i) q^{91} +(46.4374 - 322.979i) q^{93} -639.951i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{22}\right)\)

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 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(3\) −4.72659 + 2.15856i −0.909632 + 0.415415i
\(4\) 1.13852 + 7.91857i 0.142315 + 0.989821i
\(5\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(6\) 0 0
\(7\) 18.3144 + 15.8695i 0.988882 + 0.856871i 0.989704 0.143128i \(-0.0457161\pi\)
−0.000821854 1.00000i \(0.500262\pi\)
\(8\) 0 0
\(9\) 17.6812 20.4052i 0.654861 0.755750i
\(10\) 0 0
\(11\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(12\) −22.4740 34.9703i −0.540641 0.841254i
\(13\) 50.6773 + 78.8554i 1.08118 + 1.68235i 0.552109 + 0.833772i \(0.313823\pi\)
0.529071 + 0.848578i \(0.322541\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −61.4076 + 18.0309i −0.959493 + 0.281733i
\(17\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(18\) 0 0
\(19\) −77.6235 89.5823i −0.937266 1.08166i −0.996515 0.0834192i \(-0.973416\pi\)
0.0592487 0.998243i \(-0.481129\pi\)
\(20\) 0 0
\(21\) −120.820 35.4758i −1.25548 0.368641i
\(22\) 0 0
\(23\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(24\) 0 0
\(25\) −105.157 + 67.5801i −0.841254 + 0.540641i
\(26\) 0 0
\(27\) −39.5260 + 134.613i −0.281733 + 0.959493i
\(28\) −104.812 + 163.091i −0.707417 + 1.10076i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −33.9504 + 52.8278i −0.196699 + 0.306069i −0.925566 0.378585i \(-0.876411\pi\)
0.728868 + 0.684655i \(0.240047\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 181.711 + 116.778i 0.841254 + 0.540641i
\(37\) −328.515 −1.45966 −0.729832 0.683627i \(-0.760402\pi\)
−0.729832 + 0.683627i \(0.760402\pi\)
\(38\) 0 0
\(39\) −409.744 263.327i −1.68235 1.08118i
\(40\) 0 0
\(41\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(42\) 0 0
\(43\) 459.997 + 66.1376i 1.63137 + 0.234555i 0.896304 0.443441i \(-0.146242\pi\)
0.735065 + 0.677996i \(0.237152\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(48\) 251.327 217.776i 0.755750 0.654861i
\(49\) 34.7613 + 241.770i 0.101345 + 0.704868i
\(50\) 0 0
\(51\) 0 0
\(52\) −566.725 + 491.070i −1.51136 + 1.30960i
\(53\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 560.263 + 255.863i 1.30191 + 0.594560i
\(58\) 0 0
\(59\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(60\) 0 0
\(61\) 62.8229 213.955i 0.131863 0.449084i −0.866918 0.498451i \(-0.833902\pi\)
0.998781 + 0.0493670i \(0.0157204\pi\)
\(62\) 0 0
\(63\) 647.641 93.1167i 1.29516 0.186216i
\(64\) −212.692 465.732i −0.415415 0.909632i
\(65\) 0 0
\(66\) 0 0
\(67\) 547.134 + 37.5088i 0.997658 + 0.0683944i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(72\) 0 0
\(73\) −994.413 291.986i −1.59435 0.468142i −0.640380 0.768058i \(-0.721223\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) 351.157 546.410i 0.540641 0.841254i
\(76\) 620.988 716.658i 0.937266 1.08166i
\(77\) 0 0
\(78\) 0 0
\(79\) 237.905 + 370.187i 0.338815 + 0.527207i 0.968295 0.249810i \(-0.0803682\pi\)
−0.629480 + 0.777017i \(0.716732\pi\)
\(80\) 0 0
\(81\) −103.748 721.580i −0.142315 0.989821i
\(82\) 0 0
\(83\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(84\) 143.363 997.109i 0.186216 1.29516i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(90\) 0 0
\(91\) −323.272 + 2248.41i −0.372397 + 2.59008i
\(92\) 0 0
\(93\) 46.4374 322.979i 0.0517778 0.360122i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 639.951i 0.669868i −0.942242 0.334934i \(-0.891286\pi\)
0.942242 0.334934i \(-0.108714\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −654.861 755.750i −0.654861 0.755750i
\(101\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(102\) 0 0
\(103\) 1656.76 + 1064.73i 1.58490 + 1.01856i 0.973914 + 0.226917i \(0.0728647\pi\)
0.610990 + 0.791638i \(0.290772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(108\) −1110.94 159.730i −0.989821 0.142315i
\(109\) 1230.49 + 1914.68i 1.08128 + 1.68250i 0.542482 + 0.840068i \(0.317485\pi\)
0.538797 + 0.842435i \(0.318879\pi\)
\(110\) 0 0
\(111\) 1552.75 709.119i 1.32776 0.606366i
\(112\) −1410.78 644.282i −1.19023 0.543562i
\(113\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2505.10 + 360.179i 1.97946 + 0.284603i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1119.71 + 719.593i −0.841254 + 0.540641i
\(122\) 0 0
\(123\) 0 0
\(124\) −456.974 208.693i −0.330947 0.151139i
\(125\) 0 0
\(126\) 0 0
\(127\) 1213.80 1400.80i 0.848091 0.978749i −0.151862 0.988402i \(-0.548527\pi\)
0.999953 + 0.00965233i \(0.00307248\pi\)
\(128\) 0 0
\(129\) −2316.98 + 680.326i −1.58138 + 0.464336i
\(130\) 0 0
\(131\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(132\) 0 0
\(133\) 2872.49i 1.87275i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(138\) 0 0
\(139\) 897.332 + 3056.03i 0.547559 + 1.86482i 0.500148 + 0.865940i \(0.333279\pi\)
0.0474117 + 0.998875i \(0.484903\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −717.837 + 1571.84i −0.415415 + 0.909632i
\(145\) 0 0
\(146\) 0 0
\(147\) −686.177 1067.71i −0.384999 0.599071i
\(148\) −374.021 2601.37i −0.207732 1.44481i
\(149\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(150\) 0 0
\(151\) 369.943 2573.01i 0.199374 1.38668i −0.606732 0.794907i \(-0.707520\pi\)
0.806106 0.591771i \(-0.201571\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1618.67 3544.39i 0.830752 1.81909i
\(157\) 1301.84 + 2850.63i 0.661770 + 1.44908i 0.880863 + 0.473371i \(0.156963\pi\)
−0.219093 + 0.975704i \(0.570310\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1892.32 −0.909311 −0.454656 0.890667i \(-0.650238\pi\)
−0.454656 + 0.890667i \(0.650238\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(168\) 0 0
\(169\) −2737.32 + 5993.89i −1.24593 + 2.72821i
\(170\) 0 0
\(171\) −3200.43 −1.43124
\(172\) 3717.82i 1.64814i
\(173\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(174\) 0 0
\(175\) −2998.34 431.096i −1.29516 0.186216i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(180\) 0 0
\(181\) 1892.65 + 4144.33i 0.777236 + 1.70191i 0.710031 + 0.704171i \(0.248681\pi\)
0.0672055 + 0.997739i \(0.478592\pi\)
\(182\) 0 0
\(183\) 164.897 + 1146.88i 0.0666095 + 0.463279i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2860.13 + 1838.10i −1.10076 + 0.707417i
\(190\) 0 0
\(191\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(192\) 2010.62 + 1742.21i 0.755750 + 0.654861i
\(193\) −2696.48 1732.92i −1.00568 0.646314i −0.0694118 0.997588i \(-0.522112\pi\)
−0.936273 + 0.351274i \(0.885749\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1874.90 + 550.519i −0.683271 + 0.200626i
\(197\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(198\) 0 0
\(199\) 1801.54 2079.09i 0.641749 0.740618i −0.337934 0.941170i \(-0.609728\pi\)
0.979683 + 0.200552i \(0.0642736\pi\)
\(200\) 0 0
\(201\) −2667.04 + 1003.73i −0.935914 + 0.352228i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −4533.80 3928.56i −1.51136 1.30960i
\(209\) 0 0
\(210\) 0 0
\(211\) 2531.99 5544.29i 0.826111 1.80893i 0.316241 0.948679i \(-0.397579\pi\)
0.509870 0.860252i \(-0.329694\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1460.13 + 428.732i −0.456774 + 0.134121i
\(218\) 0 0
\(219\) 5330.45 766.403i 1.64474 0.236478i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1647.37 3607.23i 0.494690 1.08322i −0.483469 0.875362i \(-0.660623\pi\)
0.978159 0.207858i \(-0.0666493\pi\)
\(224\) 0 0
\(225\) −480.313 + 3340.65i −0.142315 + 0.989821i
\(226\) 0 0
\(227\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(228\) −1388.20 + 4727.79i −0.403228 + 1.37327i
\(229\) −1105.18 + 1719.69i −0.318919 + 0.496247i −0.963289 0.268466i \(-0.913483\pi\)
0.644370 + 0.764714i \(0.277120\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1923.55 1236.19i −0.527207 0.338815i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 6602.26 + 1938.60i 1.76468 + 0.518158i 0.993028 0.117881i \(-0.0376101\pi\)
0.771657 + 0.636039i \(0.219428\pi\)
\(242\) 0 0
\(243\) 2047.94 + 3186.66i 0.540641 + 0.841254i
\(244\) 1765.74 + 253.876i 0.463279 + 0.0666095i
\(245\) 0 0
\(246\) 0 0
\(247\) 3130.30 10660.8i 0.806381 2.74628i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(252\) 1474.70 + 5022.38i 0.368641 + 1.25548i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 3445.77 2214.46i 0.841254 0.540641i
\(257\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(258\) 0 0
\(259\) −6016.54 5213.36i −1.44344 1.25074i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 325.907 + 4375.23i 0.0742833 + 0.997237i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 4734.67 2162.25i 1.06130 0.484677i 0.193243 0.981151i \(-0.438099\pi\)
0.868052 + 0.496474i \(0.165372\pi\)
\(272\) 0 0
\(273\) −3325.35 11325.1i −0.737213 2.51072i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5751.10 6637.13i 1.24747 1.43966i 0.393517 0.919317i \(-0.371258\pi\)
0.853956 0.520344i \(-0.174196\pi\)
\(278\) 0 0
\(279\) 477.679 + 1626.83i 0.102501 + 0.349088i
\(280\) 0 0
\(281\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(282\) 0 0
\(283\) 4939.42 + 5700.39i 1.03752 + 1.19736i 0.979996 + 0.199016i \(0.0637747\pi\)
0.0575227 + 0.998344i \(0.481680\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4713.99 1384.15i −0.959493 0.281733i
\(290\) 0 0
\(291\) 1381.37 + 3024.78i 0.278273 + 0.609333i
\(292\) 1179.95 8206.76i 0.236478 1.64474i
\(293\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 4726.59 + 2158.56i 0.909632 + 0.415415i
\(301\) 7374.98 + 8511.18i 1.41225 + 1.62982i
\(302\) 0 0
\(303\) 0 0
\(304\) 6381.92 + 4101.41i 1.20404 + 0.773789i
\(305\) 0 0
\(306\) 0 0
\(307\) 50.1460 + 32.2269i 0.00932242 + 0.00599115i 0.545294 0.838245i \(-0.316418\pi\)
−0.535971 + 0.844236i \(0.680054\pi\)
\(308\) 0 0
\(309\) −10129.1 1456.34i −1.86480 0.268118i
\(310\) 0 0
\(311\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(312\) 0 0
\(313\) 9704.29 + 4431.80i 1.75246 + 0.800320i 0.987770 + 0.155917i \(0.0498332\pi\)
0.764686 + 0.644403i \(0.222894\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2660.50 + 2305.33i −0.473622 + 0.410396i
\(317\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5595.76 1643.06i 0.959493 0.281733i
\(325\) −10658.1 4867.39i −1.81909 0.830752i
\(326\) 0 0
\(327\) −9948.96 6393.81i −1.68250 1.08128i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5828.69 838.039i 0.967897 0.139163i 0.359803 0.933028i \(-0.382844\pi\)
0.608093 + 0.793866i \(0.291935\pi\)
\(332\) 0 0
\(333\) −5808.55 + 6703.43i −0.955876 + 1.10314i
\(334\) 0 0
\(335\) 0 0
\(336\) 8058.90 1.30848
\(337\) −9229.39 7997.32i −1.49186 1.29270i −0.851150 0.524923i \(-0.824094\pi\)
−0.640711 0.767782i \(-0.721360\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1293.70 2013.03i 0.203654 0.316891i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(348\) 0 0
\(349\) 540.296 + 3757.84i 0.0828692 + 0.576368i 0.988375 + 0.152035i \(0.0485825\pi\)
−0.905506 + 0.424334i \(0.860508\pi\)
\(350\) 0 0
\(351\) −12618.0 + 3704.99i −1.91881 + 0.563412i
\(352\) 0 0
\(353\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(360\) 0 0
\(361\) −1023.44 + 7118.20i −0.149212 + 1.03779i
\(362\) 0 0
\(363\) 3739.11 5818.18i 0.540641 0.841254i
\(364\) −18172.2 −2.61671
\(365\) 0 0
\(366\) 0 0
\(367\) 12785.6 + 5839.01i 1.81854 + 0.830500i 0.920972 + 0.389629i \(0.127397\pi\)
0.897570 + 0.440871i \(0.145330\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2610.40 0.363825
\(373\) 14192.6i 1.97015i −0.172116 0.985077i \(-0.555060\pi\)
0.172116 0.985077i \(-0.444940\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −13051.1 + 5960.26i −1.76885 + 0.807805i −0.787301 + 0.616569i \(0.788522\pi\)
−0.981544 + 0.191236i \(0.938750\pi\)
\(380\) 0 0
\(381\) −2713.43 + 9241.08i −0.364864 + 1.24261i
\(382\) 0 0
\(383\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9482.87 8216.95i 1.24558 1.07931i
\(388\) 5067.50 728.596i 0.663049 0.0953321i
\(389\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14819.9 4351.53i 1.87353 0.550118i 0.875808 0.482660i \(-0.160329\pi\)
0.997722 0.0674577i \(-0.0214888\pi\)
\(398\) 0 0
\(399\) 6200.43 + 13577.1i 0.777970 + 1.70352i
\(400\) 5238.89 6046.00i 0.654861 0.755750i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −5886.27 −0.727583
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −12155.2 10532.5i −1.46952 1.27335i −0.888018 0.459808i \(-0.847918\pi\)
−0.581506 0.813542i \(-0.697536\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6544.92 + 14331.4i −0.782633 + 1.71373i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −10837.9 12507.7i −1.27275 1.46883i
\(418\) 0 0
\(419\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(420\) 0 0
\(421\) 8668.73 + 10004.2i 1.00353 + 1.15814i 0.987396 + 0.158271i \(0.0505919\pi\)
0.0161389 + 0.999870i \(0.494863\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4545.92 2921.48i 0.515205 0.331102i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 8978.95i 1.00000i
\(433\) −7353.78 + 11442.7i −0.816167 + 1.26998i 0.143727 + 0.989617i \(0.454091\pi\)
−0.959894 + 0.280363i \(0.909545\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13760.6 + 11923.6i −1.51150 + 1.30972i
\(437\) 0 0
\(438\) 0 0
\(439\) −8522.65 −0.926569 −0.463285 0.886210i \(-0.653329\pi\)
−0.463285 + 0.886210i \(0.653329\pi\)
\(440\) 0 0
\(441\) 5547.99 + 3565.48i 0.599071 + 0.384999i
\(442\) 0 0
\(443\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(444\) 7383.05 + 11488.3i 0.789154 + 1.22795i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 3495.59 11904.9i 0.368641 1.25548i
\(449\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3805.43 + 12960.1i 0.394690 + 1.34419i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16437.1 + 10563.5i −1.68248 + 1.08126i −0.838121 + 0.545484i \(0.816346\pi\)
−0.844358 + 0.535780i \(0.820018\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(462\) 0 0
\(463\) −4654.45 + 15851.6i −0.467193 + 1.59111i 0.302795 + 0.953056i \(0.402080\pi\)
−0.769988 + 0.638058i \(0.779738\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(468\) 20246.9i 1.99981i
\(469\) 9425.17 + 9369.69i 0.927961 + 0.922499i
\(470\) 0 0
\(471\) −12306.5 10663.6i −1.20394 1.04322i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 14216.6 + 4174.37i 1.37327 + 0.403228i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(480\) 0 0
\(481\) −16648.2 25905.2i −1.57816 2.45566i
\(482\) 0 0
\(483\) 0 0
\(484\) −6972.96 8047.22i −0.654861 0.755750i
\(485\) 0 0
\(486\) 0 0
\(487\) −19769.6 + 2842.44i −1.83952 + 0.264483i −0.972351 0.233526i \(-0.924974\pi\)
−0.867172 + 0.498010i \(0.834064\pi\)
\(488\) 0 0
\(489\) 8944.20 4084.68i 0.827138 0.377741i
\(490\) 0 0
\(491\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1132.28 3856.18i 0.102501 0.349088i
\(497\) 0 0
\(498\) 0 0
\(499\) 5586.40i 0.501165i 0.968095 + 0.250583i \(0.0806222\pi\)
−0.968095 + 0.250583i \(0.919378\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 34239.3i 2.99925i
\(508\) 12474.3 + 8016.74i 1.08948 + 0.700168i
\(509\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(510\) 0 0
\(511\) −13578.4 21128.3i −1.17548 1.82909i
\(512\) 0 0
\(513\) 15127.1 6908.31i 1.30191 0.594560i
\(514\) 0 0
\(515\) 0 0
\(516\) −8025.13 17572.6i −0.684664 1.49920i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.755750 0.654861i \(-0.227273\pi\)
−0.755750 + 0.654861i \(0.772727\pi\)
\(522\) 0 0
\(523\) 6508.81 4182.96i 0.544189 0.349729i −0.239487 0.970900i \(-0.576979\pi\)
0.783675 + 0.621171i \(0.213343\pi\)
\(524\) 0 0
\(525\) 15102.5 4434.48i 1.25548 0.368641i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7967.69 + 9195.21i −0.654861 + 0.755750i
\(530\) 0 0
\(531\) 0 0
\(532\) 22746.0 3270.38i 1.85369 0.266521i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6761.61 + 23027.9i 0.537346 + 1.83003i 0.557414 + 0.830234i \(0.311793\pi\)
−0.0200682 + 0.999799i \(0.506388\pi\)
\(542\) 0 0
\(543\) −17891.6 15503.1i −1.41400 1.22524i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1481.11 + 5044.21i 0.115773 + 0.394287i 0.996908 0.0785783i \(-0.0250381\pi\)
−0.881135 + 0.472865i \(0.843220\pi\)
\(548\) 0 0
\(549\) −3255.02 5064.91i −0.253043 0.393743i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1517.60 + 10555.2i −0.116700 + 0.811667i
\(554\) 0 0
\(555\) 0 0
\(556\) −23177.8 + 10584.9i −1.76791 + 0.807377i
\(557\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(558\) 0 0
\(559\) 18096.1 + 39624.9i 1.36920 + 2.99813i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.281733 0.959493i \(-0.409091\pi\)
−0.281733 + 0.959493i \(0.590909\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9551.03 14861.7i 0.707417 1.10076i
\(568\) 0 0
\(569\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(570\) 0 0
\(571\) 9383.28 20546.5i 0.687703 1.50586i −0.166568 0.986030i \(-0.553268\pi\)
0.854270 0.519829i \(-0.174004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −13264.0 3894.67i −0.959493 0.281733i
\(577\) 25191.6 + 3622.01i 1.81758 + 0.261328i 0.965183 0.261575i \(-0.0842417\pi\)
0.852393 + 0.522902i \(0.175151\pi\)
\(578\) 0 0
\(579\) 16485.8 + 2370.30i 1.18329 + 0.170132i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(588\) 7673.53 6649.15i 0.538182 0.466337i
\(589\) 7367.78 1059.33i 0.515423 0.0741066i
\(590\) 0 0
\(591\) 0 0
\(592\) 20173.3 5923.42i 1.40054 0.411235i
\(593\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4027.31 + 13715.7i −0.276092 + 0.940282i
\(598\) 0 0
\(599\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(600\) 0 0
\(601\) 17836.3 20584.1i 1.21058 1.39708i 0.316843 0.948478i \(-0.397377\pi\)
0.893732 0.448601i \(-0.148077\pi\)
\(602\) 0 0
\(603\) 10439.4 10501.2i 0.705016 0.709191i
\(604\) 20795.7 1.40094
\(605\) 0 0
\(606\) 0 0
\(607\) −1646.03 11448.4i −0.110066 0.765529i −0.967852 0.251521i \(-0.919069\pi\)
0.857785 0.514008i \(-0.171840\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 9201.89 20149.3i 0.606299 1.32761i −0.318778 0.947829i \(-0.603273\pi\)
0.925077 0.379779i \(-0.124000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(618\) 0 0
\(619\) −28712.0 + 8430.59i −1.86435 + 0.547422i −0.865424 + 0.501040i \(0.832951\pi\)
−0.998924 + 0.0463820i \(0.985231\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 29909.4 + 8782.20i 1.91881 + 0.563412i
\(625\) 6490.86 14213.0i 0.415415 0.909632i
\(626\) 0 0
\(627\) 0 0
\(628\) −21090.7 + 13554.2i −1.34015 + 0.861259i
\(629\) 0 0
\(630\) 0 0
\(631\) 6176.61 9610.99i 0.389678 0.606351i −0.589885 0.807487i \(-0.700827\pi\)
0.979563 + 0.201136i \(0.0644634\pi\)
\(632\) 0 0
\(633\) 31671.0i 1.98864i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −17303.2 + 14993.3i −1.07626 + 0.932587i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 26539.3 + 7792.65i 1.62770 + 0.477935i 0.963072 0.269242i \(-0.0867732\pi\)
0.664625 + 0.747177i \(0.268591\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5975.98 5178.22i 0.359781 0.311752i
\(652\) −2154.44 14984.4i −0.129408 0.900056i
\(653\) 0 0 −0.989821 0.142315i \(-0.954545\pi\)
0.989821 + 0.142315i \(0.0454545\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −23540.5 + 15128.6i −1.39787 + 0.898358i
\(658\) 0 0
\(659\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(660\) 0 0
\(661\) −18239.0 15804.2i −1.07325 0.929972i −0.0755037 0.997146i \(-0.524056\pi\)
−0.997741 + 0.0671734i \(0.978602\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20605.8i 1.19083i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −15931.8 + 7275.80i −0.912518 + 0.416733i −0.815638 0.578563i \(-0.803614\pi\)
−0.0968804 + 0.995296i \(0.530886\pi\)
\(674\) 0 0
\(675\) −4940.75 16826.6i −0.281733 0.959493i
\(676\) −50579.5 14851.5i −2.87776 0.844986i
\(677\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(678\) 0 0
\(679\) 10155.7 11720.3i 0.573990 0.662420i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.540641 0.841254i \(-0.681818\pi\)
0.540641 + 0.841254i \(0.318182\pi\)
\(684\) −3643.75 25342.8i −0.203687 1.41668i
\(685\) 0 0
\(686\) 0 0
\(687\) 1511.67 10513.9i 0.0839501 0.583886i
\(688\) −29439.8 + 4232.80i −1.63137 + 0.234555i
\(689\) 0 0
\(690\) 0 0
\(691\) 3927.57 + 1153.24i 0.216226 + 0.0634896i 0.388051 0.921638i \(-0.373148\pi\)
−0.171826 + 0.985127i \(0.554967\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 24233.4i 1.30848i
\(701\) 0 0 0.540641 0.841254i \(-0.318182\pi\)
−0.540641 + 0.841254i \(0.681818\pi\)
\(702\) 0 0
\(703\) 25500.5 + 29429.1i 1.36809 + 1.57886i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −20982.1 13484.4i −1.11142 0.714267i −0.149819 0.988713i \(-0.547869\pi\)
−0.961602 + 0.274446i \(0.911505\pi\)
\(710\) 0 0
\(711\) 11760.2 + 1690.86i 0.620313 + 0.0891876i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(720\) 0 0
\(721\) 13445.7 + 45791.8i 0.694512 + 2.36529i
\(722\) 0 0
\(723\) −35390.7 + 5088.42i −1.82046 + 0.261743i
\(724\) −30662.4 + 19705.5i −1.57397 + 1.01153i
\(725\) 0 0
\(726\) 0 0
\(727\) 14585.5 + 6660.99i 0.744081 + 0.339811i 0.751121 0.660165i \(-0.229514\pi\)
−0.00703957 + 0.999975i \(0.502241\pi\)
\(728\) 0 0
\(729\) −16558.4 10641.4i −0.841254 0.540641i
\(730\) 0 0
\(731\) 0 0
\(732\) −8893.95 + 2611.50i −0.449084 + 0.131863i
\(733\) −12547.6 + 1804.08i −0.632274 + 0.0909073i −0.450998 0.892525i \(-0.648932\pi\)
−0.181276 + 0.983432i \(0.558023\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 26170.4 + 22676.8i 1.30270 + 1.12879i 0.983461 + 0.181118i \(0.0579715\pi\)
0.319235 + 0.947675i \(0.396574\pi\)
\(740\) 0 0
\(741\) 8216.38 + 57146.2i 0.407336 + 2.83309i
\(742\) 0 0
\(743\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −5240.06 36445.4i −0.254611 1.77086i −0.569757 0.821813i \(-0.692963\pi\)
0.315147 0.949043i \(-0.397946\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −17811.4 20555.5i −0.856871 0.988882i
\(757\) 14056.2 6419.25i 0.674876 0.308206i −0.0483365 0.998831i \(-0.515392\pi\)
0.723212 + 0.690626i \(0.242665\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(762\) 0 0
\(763\) −7849.33 + 54593.3i −0.372431 + 2.59031i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −11506.7 + 17904.8i −0.540641 + 0.841254i
\(769\) −34711.4 15852.2i −1.62773 0.743360i −0.628330 0.777947i \(-0.716261\pi\)
−0.999402 + 0.0345865i \(0.988989\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10652.3 23325.3i 0.496612 1.08743i
\(773\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(774\) 0 0
\(775\) 7849.57i 0.363825i
\(776\) 0 0
\(777\) 39691.1 + 11654.3i 1.83257 + 0.538092i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −6493.93 14219.7i −0.295824 0.647764i
\(785\) 0 0
\(786\) 0 0
\(787\) 16449.2 + 2365.04i 0.745047 + 0.107122i 0.504375 0.863485i \(-0.331723\pi\)
0.240672 + 0.970606i \(0.422632\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20055.2 5888.74i 0.898085 0.263701i
\(794\) 0 0
\(795\) 0 0
\(796\) 18514.5 + 11898.6i 0.824410 + 0.529816i
\(797\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −10984.6 19976.4i −0.481838 0.876260i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.281733 0.959493i \(-0.590909\pi\)
0.281733 + 0.959493i \(0.409091\pi\)
\(810\) 0 0
\(811\) −723.621 627.022i −0.0313314 0.0271488i 0.639054 0.769162i \(-0.279326\pi\)
−0.670386 + 0.742013i \(0.733871\pi\)
\(812\) 0 0
\(813\) −17711.5 + 20440.2i −0.764046 + 0.881756i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −29781.8 46341.4i −1.27532 1.98443i
\(818\) 0 0
\(819\) 40163.4 + 46351.1i 1.71358 + 1.97758i
\(820\) 0 0
\(821\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(822\) 0 0
\(823\) −17333.3 20003.7i −0.734145 0.847249i 0.258786 0.965935i \(-0.416677\pi\)
−0.992932 + 0.118686i \(0.962132\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(828\) 0 0
\(829\) −35080.7 + 22545.0i −1.46972 + 0.944534i −0.471696 + 0.881761i \(0.656358\pi\)
−0.998028 + 0.0627735i \(0.980005\pi\)
\(830\) 0 0
\(831\) −12856.5 + 43785.1i −0.536685 + 1.82778i
\(832\) 25946.8 40373.9i 1.08118 1.68235i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5769.39 6658.24i −0.238255 0.274961i
\(838\) 0 0
\(839\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(840\) 0 0
\(841\) 24389.0 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 46785.5 + 13737.5i 1.90809 + 0.560265i
\(845\) 0 0
\(846\) 0 0
\(847\) −31926.3 4590.31i −1.29516 0.186216i
\(848\) 0 0
\(849\) −35651.2 16281.4i −1.44116 0.658157i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 6928.38 + 48188.0i 0.278105 + 1.93426i 0.349777 + 0.936833i \(0.386257\pi\)
−0.0716721 + 0.997428i \(0.522834\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.989821 0.142315i \(-0.0454545\pi\)
−0.989821 + 0.142315i \(0.954545\pi\)
\(858\) 0 0
\(859\) 4216.68 2709.90i 0.167487 0.107637i −0.454212 0.890894i \(-0.650079\pi\)
0.621699 + 0.783256i \(0.286443\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 25268.9 3633.11i 0.989821 0.142315i
\(868\) −5057.33 11074.0i −0.197762 0.433037i
\(869\) 0 0
\(870\) 0 0
\(871\) 24769.5 + 45045.3i 0.963585 + 1.75236i
\(872\) 0 0
\(873\) −13058.3 11315.1i −0.506252 0.438670i
\(874\) 0 0
\(875\) 0 0
\(876\) 12137.6 + 41337.0i 0.468142 + 1.59435i
\(877\) −21699.3 6371.49i −0.835500 0.245325i −0.164122 0.986440i \(-0.552479\pi\)
−0.671378 + 0.741115i \(0.734297\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(882\) 0 0
\(883\) −21002.2 32680.1i −0.800432 1.24550i −0.965806 0.259266i \(-0.916519\pi\)
0.165374 0.986231i \(-0.447117\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(888\) 0 0
\(889\) 44460.0 6392.39i 1.67732 0.241163i
\(890\) 0 0
\(891\) 0 0
\(892\) 30439.7 + 8937.89i 1.14260 + 0.335496i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −27000.0 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −53230.3 24309.5i −1.96168 0.895868i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −43590.4 28013.8i −1.59581 1.02556i −0.969216 0.246212i \(-0.920814\pi\)
−0.626589 0.779350i \(-0.715550\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(912\) −39017.8 5609.91i −1.41668 0.203687i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −14875.8 6793.55i −0.536583 0.245049i
\(917\) 0 0
\(918\) 0 0
\(919\) 39767.6 34458.8i 1.42743 1.23688i 0.498369 0.866965i \(-0.333933\pi\)
0.929066 0.369914i \(-0.120613\pi\)
\(920\) 0 0
\(921\) −306.583 44.0800i −0.0109688 0.00157707i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 34545.6 22201.1i 1.22795 0.789154i
\(926\) 0 0
\(927\) 51019.6 14980.7i 1.80766 0.530778i
\(928\) 0 0
\(929\) 0 0 −0.755750 0.654861i \(-0.772727\pi\)
0.755750 + 0.654861i \(0.227273\pi\)
\(930\) 0 0
\(931\) 18960.0 21881.0i 0.667443 0.770270i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 57003.7i 1.98744i −0.111899 0.993720i \(-0.535693\pi\)
0.111899 0.993720i \(-0.464307\pi\)
\(938\) 0 0
\(939\) −55434.5 −1.92656
\(940\) 0 0
\(941\) 0 0 0.909632 0.415415i \(-0.136364\pi\)
−0.909632 + 0.415415i \(0.863636\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(948\) 7598.87 16639.2i 0.260337 0.570059i
\(949\) −27369.5 93211.8i −0.936196 3.18839i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 10737.5 + 23511.8i 0.360427 + 0.789225i
\(962\) 0 0
\(963\) 0 0
\(964\) −7834.13 + 54487.6i −0.261743 + 1.82046i
\(965\) 0 0
\(966\) 0 0
\(967\) 51152.1 1.70108 0.850538 0.525913i \(-0.176276\pi\)
0.850538 + 0.525913i \(0.176276\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(972\) −22902.2 + 19844.9i −0.755750 + 0.654861i
\(973\) −32063.6 + 70209.5i −1.05644 + 2.31327i
\(974\) 0 0
\(975\) 60883.0 1.99981
\(976\) 14271.2i 0.468043i
\(977\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 60826.0 + 8745.46i 1.97964 + 0.284629i
\(982\) 0 0
\(983\) 0 0 −0.909632 0.415415i \(-0.863636\pi\)
0.909632 + 0.415415i \(0.136364\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 87982.3 + 12649.9i 2.83309 + 0.407336i
\(989\) 0 0
\(990\) 0 0
\(991\) −53141.1 + 7640.53i −1.70341 + 0.244914i −0.924217 0.381867i \(-0.875281\pi\)
−0.779196 + 0.626781i \(0.784372\pi\)
\(992\) 0 0
\(993\) −25740.9 + 16542.6i −0.822620 + 0.528665i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 32697.3 + 21013.3i 1.03865 + 0.667499i 0.944652 0.328073i \(-0.106399\pi\)
0.0939968 + 0.995572i \(0.470036\pi\)
\(998\) 0 0
\(999\) 12984.9 44222.4i 0.411235 1.40054i
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 201.4.j.a.5.1 20
3.2 odd 2 CM 201.4.j.a.5.1 20
67.27 odd 22 inner 201.4.j.a.161.1 yes 20
201.161 even 22 inner 201.4.j.a.161.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.4.j.a.5.1 20 1.1 even 1 trivial
201.4.j.a.5.1 20 3.2 odd 2 CM
201.4.j.a.161.1 yes 20 67.27 odd 22 inner
201.4.j.a.161.1 yes 20 201.161 even 22 inner