Properties

Label 648.4.i.t.433.2
Level $648$
Weight $4$
Character 648.433
Analytic conductor $38.233$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,4,Mod(217,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.217");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), 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: \(38.2332376837\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-67})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 16x^{2} - 17x + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.2
Root \(-3.29436 + 2.47935i\) of defining polynomial
Character \(\chi\) \(=\) 648.433
Dual form 648.4.i.t.217.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.08872 - 15.7421i) q^{5} +(0.411277 + 0.712352i) q^{7} +O(q^{10})\) \(q+(9.08872 - 15.7421i) q^{5} +(0.411277 + 0.712352i) q^{7} +(-32.7662 - 56.7527i) q^{11} +(-15.9113 + 27.5591i) q^{13} -124.420 q^{17} +27.8872 q^{19} +(-21.5887 + 37.3928i) q^{23} +(-102.710 - 177.899i) q^{25} +(19.9113 + 34.4873i) q^{29} +(-147.420 + 255.338i) q^{31} +14.9519 q^{35} -104.177 q^{37} +(153.823 - 266.428i) q^{41} +(180.541 + 312.706i) q^{43} +(198.823 + 344.371i) q^{47} +(171.162 - 296.461i) q^{49} -107.161 q^{53} -1191.21 q^{55} +(94.4677 - 163.623i) q^{59} +(49.5083 + 85.7509i) q^{61} +(289.226 + 500.955i) q^{65} +(-212.524 + 368.102i) q^{67} +445.532 q^{71} -499.806 q^{73} +(26.9519 - 46.6821i) q^{77} +(-285.138 - 493.873i) q^{79} +(654.371 + 1133.40i) q^{83} +(-1130.82 + 1958.63i) q^{85} -1323.00 q^{89} -26.1757 q^{91} +(253.459 - 439.004i) q^{95} +(-472.436 - 818.283i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 30 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 30 q^{7} - 46 q^{11} - 92 q^{13} - 44 q^{17} - 172 q^{19} - 58 q^{23} - 184 q^{25} + 108 q^{29} - 136 q^{31} - 564 q^{35} - 360 q^{37} + 672 q^{41} + 70 q^{43} + 852 q^{47} - 166 q^{49} - 1336 q^{53} - 2780 q^{55} + 548 q^{59} - 284 q^{61} - 34 q^{65} - 1162 q^{67} + 1612 q^{71} - 3020 q^{73} - 516 q^{77} + 22 q^{79} + 1540 q^{83} - 3304 q^{85} - 5292 q^{89} - 3564 q^{91} + 1666 q^{95} - 472 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 9.08872 15.7421i 0.812920 1.40802i −0.0978916 0.995197i \(-0.531210\pi\)
0.910812 0.412822i \(-0.135457\pi\)
\(6\) 0 0
\(7\) 0.411277 + 0.712352i 0.0222068 + 0.0384634i 0.876915 0.480645i \(-0.159597\pi\)
−0.854708 + 0.519108i \(0.826264\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −32.7662 56.7527i −0.898125 1.55560i −0.829889 0.557928i \(-0.811596\pi\)
−0.0682357 0.997669i \(-0.521737\pi\)
\(12\) 0 0
\(13\) −15.9113 + 27.5591i −0.339461 + 0.587964i −0.984331 0.176328i \(-0.943578\pi\)
0.644870 + 0.764292i \(0.276911\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −124.420 −1.77507 −0.887535 0.460741i \(-0.847584\pi\)
−0.887535 + 0.460741i \(0.847584\pi\)
\(18\) 0 0
\(19\) 27.8872 0.336725 0.168362 0.985725i \(-0.446152\pi\)
0.168362 + 0.985725i \(0.446152\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −21.5887 + 37.3928i −0.195720 + 0.338997i −0.947136 0.320831i \(-0.896038\pi\)
0.751416 + 0.659828i \(0.229371\pi\)
\(24\) 0 0
\(25\) −102.710 177.899i −0.821678 1.42319i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 19.9113 + 34.4873i 0.127498 + 0.220832i 0.922706 0.385503i \(-0.125972\pi\)
−0.795209 + 0.606336i \(0.792639\pi\)
\(30\) 0 0
\(31\) −147.420 + 255.338i −0.854108 + 1.47936i 0.0233627 + 0.999727i \(0.492563\pi\)
−0.877470 + 0.479631i \(0.840771\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 14.9519 0.0722096
\(36\) 0 0
\(37\) −104.177 −0.462883 −0.231441 0.972849i \(-0.574344\pi\)
−0.231441 + 0.972849i \(0.574344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 153.823 266.428i 0.585928 1.01486i −0.408831 0.912610i \(-0.634063\pi\)
0.994759 0.102247i \(-0.0326032\pi\)
\(42\) 0 0
\(43\) 180.541 + 312.706i 0.640283 + 1.10900i 0.985369 + 0.170432i \(0.0545164\pi\)
−0.345086 + 0.938571i \(0.612150\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 198.823 + 344.371i 0.617048 + 1.06876i 0.990022 + 0.140916i \(0.0450049\pi\)
−0.372974 + 0.927842i \(0.621662\pi\)
\(48\) 0 0
\(49\) 171.162 296.461i 0.499014 0.864317i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −107.161 −0.277730 −0.138865 0.990311i \(-0.544345\pi\)
−0.138865 + 0.990311i \(0.544345\pi\)
\(54\) 0 0
\(55\) −1191.21 −2.92041
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 94.4677 163.623i 0.208452 0.361049i −0.742775 0.669541i \(-0.766491\pi\)
0.951227 + 0.308492i \(0.0998243\pi\)
\(60\) 0 0
\(61\) 49.5083 + 85.7509i 0.103916 + 0.179988i 0.913295 0.407299i \(-0.133529\pi\)
−0.809379 + 0.587287i \(0.800196\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 289.226 + 500.955i 0.551910 + 0.955935i
\(66\) 0 0
\(67\) −212.524 + 368.102i −0.387522 + 0.671207i −0.992116 0.125327i \(-0.960002\pi\)
0.604594 + 0.796534i \(0.293335\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 445.532 0.744718 0.372359 0.928089i \(-0.378549\pi\)
0.372359 + 0.928089i \(0.378549\pi\)
\(72\) 0 0
\(73\) −499.806 −0.801341 −0.400670 0.916222i \(-0.631223\pi\)
−0.400670 + 0.916222i \(0.631223\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 26.9519 46.6821i 0.0398890 0.0690898i
\(78\) 0 0
\(79\) −285.138 493.873i −0.406082 0.703355i 0.588365 0.808596i \(-0.299772\pi\)
−0.994447 + 0.105241i \(0.966439\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 654.371 + 1133.40i 0.865381 + 1.49888i 0.866669 + 0.498884i \(0.166257\pi\)
−0.00128787 + 0.999999i \(0.500410\pi\)
\(84\) 0 0
\(85\) −1130.82 + 1958.63i −1.44299 + 2.49933i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1323.00 −1.57570 −0.787852 0.615864i \(-0.788807\pi\)
−0.787852 + 0.615864i \(0.788807\pi\)
\(90\) 0 0
\(91\) −26.1757 −0.0301534
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 253.459 439.004i 0.273730 0.474115i
\(96\) 0 0
\(97\) −472.436 818.283i −0.494522 0.856537i 0.505458 0.862851i \(-0.331323\pi\)
−0.999980 + 0.00631404i \(0.997990\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −699.050 1210.79i −0.688694 1.19285i −0.972261 0.233900i \(-0.924851\pi\)
0.283567 0.958952i \(-0.408482\pi\)
\(102\) 0 0
\(103\) 480.275 831.861i 0.459446 0.795784i −0.539486 0.841995i \(-0.681381\pi\)
0.998932 + 0.0462110i \(0.0147147\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1935.87 −1.74905 −0.874523 0.484983i \(-0.838826\pi\)
−0.874523 + 0.484983i \(0.838826\pi\)
\(108\) 0 0
\(109\) −1305.40 −1.14711 −0.573555 0.819167i \(-0.694436\pi\)
−0.573555 + 0.819167i \(0.694436\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 547.016 947.459i 0.455389 0.788756i −0.543322 0.839524i \(-0.682834\pi\)
0.998710 + 0.0507683i \(0.0161670\pi\)
\(114\) 0 0
\(115\) 392.428 + 679.705i 0.318209 + 0.551155i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −51.1709 88.6305i −0.0394187 0.0682752i
\(120\) 0 0
\(121\) −1481.74 + 2566.46i −1.11326 + 1.92822i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1461.82 −1.04600
\(126\) 0 0
\(127\) −1301.05 −0.909050 −0.454525 0.890734i \(-0.650191\pi\)
−0.454525 + 0.890734i \(0.650191\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 978.833 1695.39i 0.652832 1.13074i −0.329601 0.944120i \(-0.606914\pi\)
0.982433 0.186618i \(-0.0597526\pi\)
\(132\) 0 0
\(133\) 11.4694 + 19.8655i 0.00747760 + 0.0129516i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1276.98 2211.80i −0.796351 1.37932i −0.921978 0.387243i \(-0.873427\pi\)
0.125626 0.992078i \(-0.459906\pi\)
\(138\) 0 0
\(139\) −102.630 + 177.761i −0.0626258 + 0.108471i −0.895638 0.444783i \(-0.853281\pi\)
0.833013 + 0.553254i \(0.186614\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2085.41 1.21951
\(144\) 0 0
\(145\) 723.872 0.414582
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 82.1202 142.236i 0.0451513 0.0782044i −0.842567 0.538592i \(-0.818956\pi\)
0.887718 + 0.460388i \(0.152290\pi\)
\(150\) 0 0
\(151\) −1713.86 2968.49i −0.923654 1.59982i −0.793712 0.608294i \(-0.791854\pi\)
−0.129942 0.991522i \(-0.541479\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2679.71 + 4641.40i 1.38864 + 2.40520i
\(156\) 0 0
\(157\) 494.042 855.706i 0.251139 0.434986i −0.712700 0.701468i \(-0.752528\pi\)
0.963840 + 0.266482i \(0.0858614\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −35.5157 −0.0173853
\(162\) 0 0
\(163\) −2845.45 −1.36732 −0.683660 0.729801i \(-0.739613\pi\)
−0.683660 + 0.729801i \(0.739613\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1602.95 + 2776.38i −0.742752 + 1.28648i 0.208485 + 0.978026i \(0.433147\pi\)
−0.951238 + 0.308459i \(0.900187\pi\)
\(168\) 0 0
\(169\) 592.163 + 1025.66i 0.269532 + 0.466844i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 892.817 + 1546.40i 0.392368 + 0.679601i 0.992761 0.120104i \(-0.0383228\pi\)
−0.600394 + 0.799705i \(0.704989\pi\)
\(174\) 0 0
\(175\) 84.4843 146.331i 0.0364938 0.0632091i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1318.84 0.550696 0.275348 0.961345i \(-0.411207\pi\)
0.275348 + 0.961345i \(0.411207\pi\)
\(180\) 0 0
\(181\) 3462.97 1.42210 0.711051 0.703141i \(-0.248220\pi\)
0.711051 + 0.703141i \(0.248220\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −946.840 + 1639.97i −0.376287 + 0.651748i
\(186\) 0 0
\(187\) 4076.75 + 7061.14i 1.59423 + 2.76129i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 795.655 + 1378.12i 0.301422 + 0.522078i 0.976458 0.215706i \(-0.0692054\pi\)
−0.675036 + 0.737784i \(0.735872\pi\)
\(192\) 0 0
\(193\) 382.228 662.038i 0.142556 0.246915i −0.785902 0.618351i \(-0.787801\pi\)
0.928459 + 0.371436i \(0.121134\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1191.63 −0.430965 −0.215483 0.976508i \(-0.569132\pi\)
−0.215483 + 0.976508i \(0.569132\pi\)
\(198\) 0 0
\(199\) −2708.20 −0.964719 −0.482360 0.875973i \(-0.660220\pi\)
−0.482360 + 0.875973i \(0.660220\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −16.3781 + 28.3677i −0.00566264 + 0.00980798i
\(204\) 0 0
\(205\) −2796.10 4842.99i −0.952625 1.65000i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −913.758 1582.68i −0.302421 0.523808i
\(210\) 0 0
\(211\) 1649.48 2856.98i 0.538173 0.932144i −0.460829 0.887489i \(-0.652448\pi\)
0.999002 0.0446549i \(-0.0142188\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6563.54 2.08200
\(216\) 0 0
\(217\) −242.521 −0.0758682
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1979.67 3428.90i 0.602567 1.04368i
\(222\) 0 0
\(223\) 1260.83 + 2183.83i 0.378617 + 0.655784i 0.990861 0.134885i \(-0.0430664\pi\)
−0.612244 + 0.790669i \(0.709733\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1657.33 2870.59i −0.484586 0.839328i 0.515257 0.857036i \(-0.327696\pi\)
−0.999843 + 0.0177077i \(0.994363\pi\)
\(228\) 0 0
\(229\) 2064.83 3576.39i 0.595842 1.03203i −0.397585 0.917565i \(-0.630152\pi\)
0.993427 0.114464i \(-0.0365151\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −986.257 −0.277304 −0.138652 0.990341i \(-0.544277\pi\)
−0.138652 + 0.990341i \(0.544277\pi\)
\(234\) 0 0
\(235\) 7228.17 2.00644
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 179.499 310.902i 0.0485809 0.0841446i −0.840712 0.541482i \(-0.817863\pi\)
0.889293 + 0.457337i \(0.151197\pi\)
\(240\) 0 0
\(241\) −1021.02 1768.45i −0.272902 0.472680i 0.696702 0.717361i \(-0.254650\pi\)
−0.969604 + 0.244681i \(0.921317\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3111.28 5388.90i −0.811317 1.40524i
\(246\) 0 0
\(247\) −443.721 + 768.548i −0.114305 + 0.197982i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2609.69 0.656264 0.328132 0.944632i \(-0.393581\pi\)
0.328132 + 0.944632i \(0.393581\pi\)
\(252\) 0 0
\(253\) 2829.52 0.703124
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −439.419 + 761.096i −0.106654 + 0.184731i −0.914413 0.404783i \(-0.867347\pi\)
0.807759 + 0.589514i \(0.200681\pi\)
\(258\) 0 0
\(259\) −42.8457 74.2110i −0.0102792 0.0178040i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1608.67 2786.29i −0.377166 0.653271i 0.613483 0.789708i \(-0.289768\pi\)
−0.990649 + 0.136438i \(0.956435\pi\)
\(264\) 0 0
\(265\) −973.955 + 1686.94i −0.225772 + 0.391049i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7978.38 −1.80837 −0.904183 0.427146i \(-0.859519\pi\)
−0.904183 + 0.427146i \(0.859519\pi\)
\(270\) 0 0
\(271\) 4248.21 0.952251 0.476126 0.879377i \(-0.342041\pi\)
0.476126 + 0.879377i \(0.342041\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6730.81 + 11658.1i −1.47594 + 2.55640i
\(276\) 0 0
\(277\) −415.517 719.697i −0.0901301 0.156110i 0.817436 0.576020i \(-0.195395\pi\)
−0.907566 + 0.419910i \(0.862062\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1481.00 2565.17i −0.314410 0.544573i 0.664902 0.746930i \(-0.268473\pi\)
−0.979312 + 0.202357i \(0.935140\pi\)
\(282\) 0 0
\(283\) 2215.42 3837.22i 0.465346 0.806004i −0.533871 0.845566i \(-0.679263\pi\)
0.999217 + 0.0395625i \(0.0125964\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 253.054 0.0520465
\(288\) 0 0
\(289\) 10567.2 2.15087
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1711.43 + 2964.29i −0.341239 + 0.591043i −0.984663 0.174467i \(-0.944180\pi\)
0.643424 + 0.765510i \(0.277513\pi\)
\(294\) 0 0
\(295\) −1717.18 2974.24i −0.338909 0.587007i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −687.008 1189.93i −0.132879 0.230153i
\(300\) 0 0
\(301\) −148.504 + 257.217i −0.0284374 + 0.0492549i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1799.87 0.337902
\(306\) 0 0
\(307\) 139.734 0.0259774 0.0129887 0.999916i \(-0.495865\pi\)
0.0129887 + 0.999916i \(0.495865\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3799.94 6581.69i 0.692845 1.20004i −0.278057 0.960565i \(-0.589690\pi\)
0.970902 0.239478i \(-0.0769764\pi\)
\(312\) 0 0
\(313\) 1508.76 + 2613.25i 0.272460 + 0.471915i 0.969491 0.245126i \(-0.0788293\pi\)
−0.697031 + 0.717041i \(0.745496\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −960.603 1663.81i −0.170198 0.294792i 0.768291 0.640101i \(-0.221107\pi\)
−0.938489 + 0.345309i \(0.887774\pi\)
\(318\) 0 0
\(319\) 1304.83 2260.04i 0.229018 0.396670i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3469.72 −0.597710
\(324\) 0 0
\(325\) 6536.98 1.11571
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −163.542 + 283.263i −0.0274054 + 0.0474675i
\(330\) 0 0
\(331\) −456.206 790.172i −0.0757563 0.131214i 0.825659 0.564170i \(-0.190804\pi\)
−0.901415 + 0.432956i \(0.857470\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3863.14 + 6691.16i 0.630048 + 1.09128i
\(336\) 0 0
\(337\) 3007.31 5208.82i 0.486109 0.841965i −0.513764 0.857932i \(-0.671749\pi\)
0.999873 + 0.0159664i \(0.00508248\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 19321.5 3.06838
\(342\) 0 0
\(343\) 563.715 0.0887398
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1369.92 2372.77i 0.211934 0.367080i −0.740386 0.672182i \(-0.765357\pi\)
0.952320 + 0.305102i \(0.0986906\pi\)
\(348\) 0 0
\(349\) 3804.34 + 6589.32i 0.583501 + 1.01065i 0.995060 + 0.0992704i \(0.0316509\pi\)
−0.411560 + 0.911383i \(0.635016\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4362.56 7556.17i −0.657777 1.13930i −0.981190 0.193046i \(-0.938163\pi\)
0.323412 0.946258i \(-0.395170\pi\)
\(354\) 0 0
\(355\) 4049.32 7013.63i 0.605396 1.04858i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10470.0 1.53923 0.769614 0.638509i \(-0.220449\pi\)
0.769614 + 0.638509i \(0.220449\pi\)
\(360\) 0 0
\(361\) −6081.30 −0.886616
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4542.60 + 7868.01i −0.651426 + 1.12830i
\(366\) 0 0
\(367\) 3947.07 + 6836.53i 0.561405 + 0.972381i 0.997374 + 0.0724198i \(0.0230721\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −44.0727 76.3362i −0.00616750 0.0106824i
\(372\) 0 0
\(373\) 134.970 233.775i 0.0187359 0.0324515i −0.856505 0.516138i \(-0.827369\pi\)
0.875241 + 0.483686i \(0.160702\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1267.26 −0.173122
\(378\) 0 0
\(379\) −8580.72 −1.16296 −0.581480 0.813561i \(-0.697526\pi\)
−0.581480 + 0.813561i \(0.697526\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1707.46 2957.41i 0.227800 0.394561i −0.729356 0.684134i \(-0.760180\pi\)
0.957156 + 0.289574i \(0.0935135\pi\)
\(384\) 0 0
\(385\) −489.917 848.561i −0.0648532 0.112329i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1652.00 + 2861.35i 0.215321 + 0.372947i 0.953372 0.301798i \(-0.0975869\pi\)
−0.738051 + 0.674745i \(0.764254\pi\)
\(390\) 0 0
\(391\) 2686.06 4652.39i 0.347417 0.601743i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10366.1 −1.32045
\(396\) 0 0
\(397\) −12756.1 −1.61262 −0.806311 0.591492i \(-0.798539\pi\)
−0.806311 + 0.591492i \(0.798539\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2314.26 + 4008.42i −0.288201 + 0.499179i −0.973380 0.229195i \(-0.926391\pi\)
0.685179 + 0.728375i \(0.259724\pi\)
\(402\) 0 0
\(403\) −4691.27 8125.51i −0.579873 1.00437i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3413.50 + 5912.35i 0.415727 + 0.720060i
\(408\) 0 0
\(409\) 1482.78 2568.24i 0.179263 0.310492i −0.762365 0.647147i \(-0.775962\pi\)
0.941628 + 0.336654i \(0.109295\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 155.409 0.0185162
\(414\) 0 0
\(415\) 23789.6 2.81394
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −935.131 + 1619.69i −0.109031 + 0.188848i −0.915378 0.402595i \(-0.868108\pi\)
0.806347 + 0.591443i \(0.201442\pi\)
\(420\) 0 0
\(421\) 6325.31 + 10955.8i 0.732248 + 1.26829i 0.955920 + 0.293627i \(0.0948623\pi\)
−0.223672 + 0.974665i \(0.571804\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12779.1 + 22134.1i 1.45854 + 2.52626i
\(426\) 0 0
\(427\) −40.7232 + 70.5347i −0.00461530 + 0.00799394i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3266.47 −0.365059 −0.182530 0.983200i \(-0.558429\pi\)
−0.182530 + 0.983200i \(0.558429\pi\)
\(432\) 0 0
\(433\) 6788.95 0.753478 0.376739 0.926319i \(-0.377045\pi\)
0.376739 + 0.926319i \(0.377045\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −602.050 + 1042.78i −0.0659038 + 0.114149i
\(438\) 0 0
\(439\) −2459.27 4259.59i −0.267368 0.463096i 0.700813 0.713345i \(-0.252821\pi\)
−0.968181 + 0.250249i \(0.919487\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7016.74 12153.4i −0.752540 1.30344i −0.946588 0.322446i \(-0.895495\pi\)
0.194047 0.980992i \(-0.437838\pi\)
\(444\) 0 0
\(445\) −12024.4 + 20826.8i −1.28092 + 2.21862i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5374.04 0.564848 0.282424 0.959290i \(-0.408862\pi\)
0.282424 + 0.959290i \(0.408862\pi\)
\(450\) 0 0
\(451\) −20160.7 −2.10495
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −237.904 + 412.062i −0.0245123 + 0.0424566i
\(456\) 0 0
\(457\) −6084.62 10538.9i −0.622815 1.07875i −0.988959 0.148189i \(-0.952655\pi\)
0.366144 0.930558i \(-0.380678\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3093.04 5357.30i −0.312489 0.541246i 0.666412 0.745584i \(-0.267829\pi\)
−0.978900 + 0.204338i \(0.934496\pi\)
\(462\) 0 0
\(463\) −7598.89 + 13161.7i −0.762743 + 1.32111i 0.178688 + 0.983906i \(0.442815\pi\)
−0.941431 + 0.337204i \(0.890519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1981.82 −0.196377 −0.0981883 0.995168i \(-0.531305\pi\)
−0.0981883 + 0.995168i \(0.531305\pi\)
\(468\) 0 0
\(469\) −349.625 −0.0344225
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11831.3 20492.3i 1.15011 1.99205i
\(474\) 0 0
\(475\) −2864.29 4961.10i −0.276679 0.479223i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4889.76 8469.31i −0.466427 0.807876i 0.532837 0.846218i \(-0.321126\pi\)
−0.999265 + 0.0383417i \(0.987792\pi\)
\(480\) 0 0
\(481\) 1657.60 2871.04i 0.157131 0.272158i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17175.4 −1.60803
\(486\) 0 0
\(487\) 18668.9 1.73710 0.868549 0.495603i \(-0.165053\pi\)
0.868549 + 0.495603i \(0.165053\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8003.11 + 13861.8i −0.735591 + 1.27408i 0.218872 + 0.975753i \(0.429762\pi\)
−0.954464 + 0.298328i \(0.903571\pi\)
\(492\) 0 0
\(493\) −2477.35 4290.90i −0.226317 0.391993i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 183.237 + 317.376i 0.0165378 + 0.0286444i
\(498\) 0 0
\(499\) −4930.29 + 8539.50i −0.442304 + 0.766094i −0.997860 0.0653857i \(-0.979172\pi\)
0.555556 + 0.831479i \(0.312506\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10512.0 −0.931825 −0.465913 0.884831i \(-0.654274\pi\)
−0.465913 + 0.884831i \(0.654274\pi\)
\(504\) 0 0
\(505\) −25413.9 −2.23941
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8333.63 14434.3i 0.725700 1.25695i −0.232985 0.972480i \(-0.574849\pi\)
0.958685 0.284470i \(-0.0918175\pi\)
\(510\) 0 0
\(511\) −205.558 356.038i −0.0177952 0.0308223i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8730.18 15121.1i −0.746986 1.29382i
\(516\) 0 0
\(517\) 13029.3 22567.4i 1.10837 1.91976i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10923.9 −0.918586 −0.459293 0.888285i \(-0.651897\pi\)
−0.459293 + 0.888285i \(0.651897\pi\)
\(522\) 0 0
\(523\) −17488.0 −1.46213 −0.731067 0.682305i \(-0.760978\pi\)
−0.731067 + 0.682305i \(0.760978\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18341.9 31769.1i 1.51610 2.62596i
\(528\) 0 0
\(529\) 5151.35 + 8922.41i 0.423387 + 0.733328i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4895.03 + 8478.43i 0.397799 + 0.689009i
\(534\) 0 0
\(535\) −17594.6 + 30474.8i −1.42184 + 2.46269i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −22433.3 −1.79271
\(540\) 0 0
\(541\) −8724.14 −0.693309 −0.346655 0.937993i \(-0.612682\pi\)
−0.346655 + 0.937993i \(0.612682\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −11864.5 + 20549.9i −0.932510 + 1.61515i
\(546\) 0 0
\(547\) 479.917 + 831.241i 0.0375133 + 0.0649749i 0.884172 0.467161i \(-0.154723\pi\)
−0.846659 + 0.532136i \(0.821390\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 555.270 + 961.757i 0.0429316 + 0.0743597i
\(552\) 0 0
\(553\) 234.541 406.237i 0.0180356 0.0312386i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14212.0 1.08112 0.540558 0.841307i \(-0.318213\pi\)
0.540558 + 0.841307i \(0.318213\pi\)
\(558\) 0 0
\(559\) −11490.5 −0.869405
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3555.28 + 6157.93i −0.266141 + 0.460969i −0.967862 0.251483i \(-0.919082\pi\)
0.701721 + 0.712452i \(0.252415\pi\)
\(564\) 0 0
\(565\) −9943.35 17222.4i −0.740389 1.28239i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2429.59 + 4208.17i 0.179005 + 0.310045i 0.941540 0.336901i \(-0.109379\pi\)
−0.762535 + 0.646947i \(0.776046\pi\)
\(570\) 0 0
\(571\) −4290.14 + 7430.74i −0.314425 + 0.544601i −0.979315 0.202341i \(-0.935145\pi\)
0.664890 + 0.746941i \(0.268478\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8869.49 0.643276
\(576\) 0 0
\(577\) −1029.99 −0.0743136 −0.0371568 0.999309i \(-0.511830\pi\)
−0.0371568 + 0.999309i \(0.511830\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −538.255 + 932.286i −0.0384348 + 0.0665709i
\(582\) 0 0
\(583\) 3511.25 + 6081.66i 0.249436 + 0.432036i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7770.72 + 13459.3i 0.546391 + 0.946378i 0.998518 + 0.0544239i \(0.0173322\pi\)
−0.452126 + 0.891954i \(0.649334\pi\)
\(588\) 0 0
\(589\) −4111.12 + 7120.68i −0.287599 + 0.498136i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12643.5 0.875562 0.437781 0.899082i \(-0.355765\pi\)
0.437781 + 0.899082i \(0.355765\pi\)
\(594\) 0 0
\(595\) −1860.31 −0.128177
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8581.46 14863.5i 0.585357 1.01387i −0.409474 0.912322i \(-0.634288\pi\)
0.994831 0.101546i \(-0.0323789\pi\)
\(600\) 0 0
\(601\) 2005.09 + 3472.92i 0.136089 + 0.235713i 0.926013 0.377492i \(-0.123213\pi\)
−0.789924 + 0.613205i \(0.789880\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 26934.3 + 46651.6i 1.80998 + 3.13497i
\(606\) 0 0
\(607\) −8660.58 + 15000.6i −0.579114 + 1.00305i 0.416468 + 0.909151i \(0.363268\pi\)
−0.995581 + 0.0939037i \(0.970065\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12654.1 −0.837855
\(612\) 0 0
\(613\) 2474.79 0.163060 0.0815300 0.996671i \(-0.474019\pi\)
0.0815300 + 0.996671i \(0.474019\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5922.19 10257.5i 0.386416 0.669291i −0.605549 0.795808i \(-0.707046\pi\)
0.991964 + 0.126517i \(0.0403797\pi\)
\(618\) 0 0
\(619\) 3924.32 + 6797.12i 0.254817 + 0.441356i 0.964846 0.262817i \(-0.0846515\pi\)
−0.710029 + 0.704173i \(0.751318\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −544.119 942.442i −0.0349914 0.0606069i
\(624\) 0 0
\(625\) −447.377 + 774.880i −0.0286322 + 0.0495924i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12961.7 0.821649
\(630\) 0 0
\(631\) −11346.0 −0.715811 −0.357906 0.933758i \(-0.616509\pi\)
−0.357906 + 0.933758i \(0.616509\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11824.9 + 20481.3i −0.738985 + 1.27996i
\(636\) 0 0
\(637\) 5446.80 + 9434.14i 0.338791 + 0.586804i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8792.62 + 15229.3i 0.541791 + 0.938409i 0.998801 + 0.0489477i \(0.0155868\pi\)
−0.457011 + 0.889461i \(0.651080\pi\)
\(642\) 0 0
\(643\) 5322.44 9218.73i 0.326433 0.565399i −0.655368 0.755309i \(-0.727487\pi\)
0.981801 + 0.189911i \(0.0608199\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7872.89 −0.478385 −0.239193 0.970972i \(-0.576883\pi\)
−0.239193 + 0.970972i \(0.576883\pi\)
\(648\) 0 0
\(649\) −12381.4 −0.748862
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1360.54 + 2356.53i −0.0815347 + 0.141222i −0.903909 0.427724i \(-0.859315\pi\)
0.822375 + 0.568946i \(0.192649\pi\)
\(654\) 0 0
\(655\) −17792.7 30817.8i −1.06140 1.83840i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1000.38 1732.71i −0.0591338 0.102423i 0.834943 0.550336i \(-0.185501\pi\)
−0.894077 + 0.447914i \(0.852167\pi\)
\(660\) 0 0
\(661\) 13869.0 24021.8i 0.816100 1.41353i −0.0924356 0.995719i \(-0.529465\pi\)
0.908535 0.417808i \(-0.137201\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 416.968 0.0243148
\(666\) 0 0
\(667\) −1719.44 −0.0998153
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3244.39 5619.46i 0.186659 0.323304i
\(672\) 0 0
\(673\) 12346.5 + 21384.8i 0.707166 + 1.22485i 0.965904 + 0.258900i \(0.0833601\pi\)
−0.258738 + 0.965948i \(0.583307\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7382.36 12786.6i −0.419095 0.725893i 0.576754 0.816918i \(-0.304319\pi\)
−0.995849 + 0.0910246i \(0.970986\pi\)
\(678\) 0 0
\(679\) 388.604 673.082i 0.0219635 0.0380420i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −571.548 −0.0320200 −0.0160100 0.999872i \(-0.505096\pi\)
−0.0160100 + 0.999872i \(0.505096\pi\)
\(684\) 0 0
\(685\) −46424.6 −2.58948
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1705.07 2953.26i 0.0942784 0.163295i
\(690\) 0 0
\(691\) 1753.79 + 3037.66i 0.0965519 + 0.167233i 0.910255 0.414048i \(-0.135885\pi\)
−0.813703 + 0.581280i \(0.802552\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1865.56 + 3231.24i 0.101819 + 0.176357i
\(696\) 0 0
\(697\) −19138.5 + 33148.9i −1.04006 + 1.80144i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −21145.2 −1.13929 −0.569644 0.821891i \(-0.692919\pi\)
−0.569644 + 0.821891i \(0.692919\pi\)
\(702\) 0 0
\(703\) −2905.22 −0.155864
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 575.006 995.939i 0.0305874 0.0529790i
\(708\) 0 0
\(709\) 7907.61 + 13696.4i 0.418867 + 0.725499i 0.995826 0.0912745i \(-0.0290941\pi\)
−0.576959 + 0.816773i \(0.695761\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6365.20 11024.9i −0.334332 0.579080i
\(714\) 0 0
\(715\) 18953.7 32828.7i 0.991367 1.71710i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33309.8 1.72774 0.863871 0.503713i \(-0.168033\pi\)
0.863871 + 0.503713i \(0.168033\pi\)
\(720\) 0 0
\(721\) 790.104 0.0408114
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4090.17 7084.38i 0.209524 0.362906i
\(726\) 0 0
\(727\) −3518.65 6094.48i −0.179504 0.310910i 0.762207 0.647334i \(-0.224116\pi\)
−0.941711 + 0.336424i \(0.890783\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −22462.8 38906.7i −1.13655 1.96856i
\(732\) 0 0
\(733\) −6330.63 + 10965.0i −0.319000 + 0.552524i −0.980280 0.197615i \(-0.936680\pi\)
0.661280 + 0.750140i \(0.270014\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27854.4 1.39217
\(738\) 0 0
\(739\) 32709.4 1.62819 0.814096 0.580730i \(-0.197233\pi\)
0.814096 + 0.580730i \(0.197233\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14690.6 25444.8i 0.725364 1.25637i −0.233460 0.972366i \(-0.575005\pi\)
0.958824 0.284001i \(-0.0916619\pi\)
\(744\) 0 0
\(745\) −1492.74 2585.49i −0.0734089 0.127148i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −796.180 1379.02i −0.0388408 0.0672743i
\(750\) 0 0
\(751\) −1112.82 + 1927.46i −0.0540712 + 0.0936540i −0.891794 0.452442i \(-0.850553\pi\)
0.837723 + 0.546096i \(0.183886\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −62307.1 −3.00343
\(756\) 0 0
\(757\) 32628.1 1.56656 0.783281 0.621668i \(-0.213545\pi\)
0.783281 + 0.621668i \(0.213545\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9345.32 + 16186.6i −0.445161 + 0.771041i −0.998063 0.0622046i \(-0.980187\pi\)
0.552903 + 0.833246i \(0.313520\pi\)
\(762\) 0 0
\(763\) −536.882 929.908i −0.0254737 0.0441218i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3006.20 + 5206.89i 0.141522 + 0.245124i
\(768\) 0 0
\(769\) −3422.83 + 5928.52i −0.160508 + 0.278008i −0.935051 0.354513i \(-0.884647\pi\)
0.774543 + 0.632521i \(0.217980\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −161.873 −0.00753190 −0.00376595 0.999993i \(-0.501199\pi\)
−0.00376595 + 0.999993i \(0.501199\pi\)
\(774\) 0 0
\(775\) 60565.7 2.80721
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4289.69 7429.95i 0.197296 0.341727i
\(780\) 0 0
\(781\) −14598.4 25285.2i −0.668849 1.15848i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8980.43 15554.6i −0.408312 0.707218i
\(786\) 0 0
\(787\) −4979.56 + 8624.85i −0.225543 + 0.390651i −0.956482 0.291791i \(-0.905749\pi\)
0.730939 + 0.682442i \(0.239082\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 899.899 0.0404510
\(792\) 0 0
\(793\) −3150.96 −0.141102
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2328.18 4032.53i 0.103473 0.179221i −0.809640 0.586927i \(-0.800338\pi\)
0.913114 + 0.407705i \(0.133671\pi\)
\(798\) 0 0
\(799\) −24737.4 42846.5i −1.09530 1.89712i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16376.7 + 28365.3i 0.719704 + 1.24656i
\(804\) 0 0
\(805\) −322.793 + 559.093i −0.0141329 + 0.0244788i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 173.564 0.00754289 0.00377144 0.999993i \(-0.498800\pi\)
0.00377144 + 0.999993i \(0.498800\pi\)
\(810\) 0 0
\(811\) −22674.1 −0.981744 −0.490872 0.871232i \(-0.663322\pi\)
−0.490872 + 0.871232i \(0.663322\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −25861.5 + 44793.5i −1.11152 + 1.92521i
\(816\) 0 0
\(817\) 5034.78 + 8720.49i 0.215599 + 0.373429i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20416.1 35361.7i −0.867877 1.50321i −0.864162 0.503214i \(-0.832151\pi\)
−0.00371541 0.999993i \(-0.501183\pi\)
\(822\) 0 0
\(823\) 2440.51 4227.08i 0.103367 0.179036i −0.809703 0.586840i \(-0.800372\pi\)
0.913070 + 0.407804i \(0.133705\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22213.4 −0.934022 −0.467011 0.884251i \(-0.654669\pi\)
−0.467011 + 0.884251i \(0.654669\pi\)
\(828\) 0 0
\(829\) 35477.2 1.48634 0.743168 0.669104i \(-0.233322\pi\)
0.743168 + 0.669104i \(0.233322\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21295.9 + 36885.5i −0.885784 + 1.53422i
\(834\) 0 0
\(835\) 29137.5 + 50467.5i 1.20760 + 2.09162i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11752.6 20356.2i −0.483607 0.837631i 0.516216 0.856458i \(-0.327340\pi\)
−0.999823 + 0.0188271i \(0.994007\pi\)
\(840\) 0 0
\(841\) 11401.6 19748.1i 0.467489 0.809714i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 21528.0 0.876433
\(846\) 0 0
\(847\) −2437.63 −0.0988876
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2249.06 3895.48i 0.0905954 0.156916i
\(852\) 0 0
\(853\) 6217.04 + 10768.2i 0.249551 + 0.432236i 0.963401 0.268063i \(-0.0863835\pi\)
−0.713850 + 0.700299i \(0.753050\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −397.933 689.240i −0.0158613 0.0274726i 0.857986 0.513673i \(-0.171716\pi\)
−0.873847 + 0.486201i \(0.838382\pi\)
\(858\) 0 0
\(859\) 4291.98 7433.92i 0.170478 0.295276i −0.768109 0.640319i \(-0.778802\pi\)
0.938587 + 0.345043i \(0.112136\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37865.4 1.49357 0.746787 0.665063i \(-0.231595\pi\)
0.746787 + 0.665063i \(0.231595\pi\)
\(864\) 0 0
\(865\) 32458.3 1.27585
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18685.7 + 32364.6i −0.729425 + 1.26340i
\(870\) 0 0
\(871\) −6763.06 11714.0i −0.263097 0.455697i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −601.213 1041.33i −0.0232283 0.0402325i
\(876\) 0 0
\(877\) −2302.28 + 3987.66i −0.0886459 + 0.153539i −0.906939 0.421262i \(-0.861587\pi\)
0.818293 + 0.574801i \(0.194921\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −13021.9 −0.497979 −0.248990 0.968506i \(-0.580099\pi\)
−0.248990 + 0.968506i \(0.580099\pi\)
\(882\) 0 0
\(883\) −4301.10 −0.163923 −0.0819613 0.996636i \(-0.526118\pi\)
−0.0819613 + 0.996636i \(0.526118\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7597.89 13159.9i 0.287613 0.498160i −0.685627 0.727953i \(-0.740472\pi\)
0.973239 + 0.229794i \(0.0738051\pi\)
\(888\) 0 0
\(889\) −535.091 926.804i −0.0201871 0.0349651i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5544.61 + 9603.55i 0.207775 + 0.359877i
\(894\) 0 0
\(895\) 11986.6 20761.3i 0.447672 0.775391i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11741.2 −0.435587
\(900\) 0 0
\(901\) 13332.9 0.492990
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 31474.0 54514.5i 1.15606 2.00235i
\(906\) 0 0
\(907\) 2328.99 + 4033.94i 0.0852624 + 0.147679i 0.905503 0.424340i \(-0.139494\pi\)
−0.820241 + 0.572019i \(0.806160\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6446.40 11165.5i −0.234444 0.406069i 0.724667 0.689099i \(-0.241994\pi\)
−0.959111 + 0.283030i \(0.908660\pi\)
\(912\) 0 0
\(913\) 42882.5 74274.7i 1.55444 2.69237i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1610.28 0.0579894
\(918\) 0 0
\(919\) −36405.3 −1.30675 −0.653373 0.757036i \(-0.726647\pi\)
−0.653373 + 0.757036i \(0.726647\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7088.99 + 12278.5i −0.252803 + 0.437867i
\(924\) 0 0
\(925\) 10700.0 + 18533.0i 0.380341 + 0.658770i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2139.85 + 3706.33i 0.0755717 + 0.130894i 0.901335 0.433123i \(-0.142588\pi\)
−0.825763 + 0.564017i \(0.809255\pi\)
\(930\) 0 0
\(931\) 4773.23 8267.47i 0.168030 0.291037i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 148210. 5.18394
\(936\) 0 0
\(937\) 21006.6 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −6764.54 + 11716.5i −0.234344 + 0.405896i −0.959082 0.283129i \(-0.908628\pi\)
0.724738 + 0.689025i \(0.241961\pi\)
\(942\) 0 0
\(943\) 6641.67 + 11503.7i 0.229356 + 0.397256i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2076.57 3596.73i −0.0712561 0.123419i 0.828196 0.560439i \(-0.189367\pi\)
−0.899452 + 0.437019i \(0.856034\pi\)
\(948\) 0 0
\(949\) 7952.55 13774.2i 0.272024 0.471159i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18387.5 0.625005 0.312503 0.949917i \(-0.398833\pi\)
0.312503 + 0.949917i \(0.398833\pi\)
\(954\) 0 0
\(955\) 28926.0 0.980128
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1050.39 1819.32i 0.0353689 0.0612607i
\(960\) 0 0
\(961\) −28569.6 49483.9i −0.959000 1.66104i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6947.93 12034.2i −0.231774 0.401444i
\(966\) 0 0
\(967\) 17227.0 29838.1i 0.572889 0.992273i −0.423378 0.905953i \(-0.639156\pi\)
0.996267 0.0863204i \(-0.0275109\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2689.17 0.0888770 0.0444385 0.999012i \(-0.485850\pi\)
0.0444385 + 0.999012i \(0.485850\pi\)
\(972\) 0 0
\(973\) −168.838 −0.00556288
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 74.4796 129.002i 0.00243891 0.00422431i −0.864803 0.502111i \(-0.832557\pi\)
0.867242 + 0.497886i \(0.165890\pi\)
\(978\) 0 0
\(979\) 43349.6 + 75083.8i 1.41518 + 2.45116i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16110.9 + 27904.9i 0.522744 + 0.905419i 0.999650 + 0.0264649i \(0.00842501\pi\)
−0.476906 + 0.878955i \(0.658242\pi\)
\(984\) 0 0
\(985\) −10830.4 + 18758.8i −0.350340 + 0.606807i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15590.6 −0.501265
\(990\) 0 0
\(991\) 2979.37 0.0955023 0.0477512 0.998859i \(-0.484795\pi\)
0.0477512 + 0.998859i \(0.484795\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24614.1 + 42632.8i −0.784240 + 1.35834i
\(996\) 0 0
\(997\) 16046.9 + 27794.1i 0.509740 + 0.882896i 0.999936 + 0.0112838i \(0.00359182\pi\)
−0.490196 + 0.871612i \(0.663075\pi\)
\(998\) 0 0
\(999\) 0 0
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 648.4.i.t.433.2 4
3.2 odd 2 648.4.i.n.433.1 4
9.2 odd 6 648.4.i.n.217.1 4
9.4 even 3 648.4.a.c.1.1 2
9.5 odd 6 648.4.a.f.1.2 yes 2
9.7 even 3 inner 648.4.i.t.217.2 4
36.23 even 6 1296.4.a.q.1.2 2
36.31 odd 6 1296.4.a.m.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
648.4.a.c.1.1 2 9.4 even 3
648.4.a.f.1.2 yes 2 9.5 odd 6
648.4.i.n.217.1 4 9.2 odd 6
648.4.i.n.433.1 4 3.2 odd 2
648.4.i.t.217.2 4 9.7 even 3 inner
648.4.i.t.433.2 4 1.1 even 1 trivial
1296.4.a.m.1.1 2 36.31 odd 6
1296.4.a.q.1.2 2 36.23 even 6