Properties

Label 2052.3.bl.a.145.3
Level $2052$
Weight $3$
Character 2052.145
Analytic conductor $55.913$
Analytic rank $0$
Dimension $80$
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] = [2052,3,Mod(145,2052)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2052, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2052.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2052.bl (of order \(6\), 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: \(55.9129502467\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.3
Character \(\chi\) \(=\) 2052.145
Dual form 2052.3.bl.a.1585.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.35221 q^{5} +(0.352343 + 0.610277i) q^{7} +O(q^{10})\) \(q-8.35221 q^{5} +(0.352343 + 0.610277i) q^{7} +(10.3266 + 17.8862i) q^{11} +(-7.44006 + 4.29552i) q^{13} +(-2.72428 - 4.71858i) q^{17} +(0.849123 - 18.9810i) q^{19} +(11.8696 + 20.5587i) q^{23} +44.7594 q^{25} +2.37991i q^{29} +(2.32115 + 1.34012i) q^{31} +(-2.94285 - 5.09716i) q^{35} +19.8089i q^{37} +75.4429i q^{41} +(-5.44837 + 9.43685i) q^{43} -30.8259 q^{47} +(24.2517 - 42.0052i) q^{49} +(76.8522 + 44.3706i) q^{53} +(-86.2498 - 149.389i) q^{55} -91.5455i q^{59} -10.0853 q^{61} +(62.1410 - 35.8771i) q^{65} +(13.9419 - 8.04937i) q^{67} +(-67.0947 + 38.7371i) q^{71} +(-39.6851 - 68.7366i) q^{73} +(-7.27700 + 12.6041i) q^{77} +(-78.3707 - 45.2473i) q^{79} +(31.5302 + 54.6119i) q^{83} +(22.7537 + 39.4106i) q^{85} +(65.0621 + 37.5636i) q^{89} +(-5.24291 - 3.02700i) q^{91} +(-7.09205 + 158.533i) q^{95} +(-65.4754 - 37.8022i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1027\) \(1217\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −8.35221 −1.67044 −0.835221 0.549914i \(-0.814660\pi\)
−0.835221 + 0.549914i \(0.814660\pi\)
\(6\) 0 0
\(7\) 0.352343 + 0.610277i 0.0503348 + 0.0871824i 0.890095 0.455775i \(-0.150638\pi\)
−0.839760 + 0.542957i \(0.817305\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.3266 + 17.8862i 0.938780 + 1.62601i 0.767751 + 0.640749i \(0.221376\pi\)
0.171029 + 0.985266i \(0.445291\pi\)
\(12\) 0 0
\(13\) −7.44006 + 4.29552i −0.572312 + 0.330425i −0.758072 0.652170i \(-0.773859\pi\)
0.185760 + 0.982595i \(0.440525\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.72428 4.71858i −0.160251 0.277564i 0.774707 0.632320i \(-0.217897\pi\)
−0.934959 + 0.354756i \(0.884564\pi\)
\(18\) 0 0
\(19\) 0.849123 18.9810i 0.0446907 0.999001i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 11.8696 + 20.5587i 0.516069 + 0.893857i 0.999826 + 0.0186548i \(0.00593836\pi\)
−0.483757 + 0.875202i \(0.660728\pi\)
\(24\) 0 0
\(25\) 44.7594 1.79038
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.37991i 0.0820658i 0.999158 + 0.0410329i \(0.0130648\pi\)
−0.999158 + 0.0410329i \(0.986935\pi\)
\(30\) 0 0
\(31\) 2.32115 + 1.34012i 0.0748757 + 0.0432295i 0.536970 0.843601i \(-0.319569\pi\)
−0.462095 + 0.886831i \(0.652902\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.94285 5.09716i −0.0840813 0.145633i
\(36\) 0 0
\(37\) 19.8089i 0.535376i 0.963506 + 0.267688i \(0.0862596\pi\)
−0.963506 + 0.267688i \(0.913740\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 75.4429i 1.84007i 0.391834 + 0.920036i \(0.371841\pi\)
−0.391834 + 0.920036i \(0.628159\pi\)
\(42\) 0 0
\(43\) −5.44837 + 9.43685i −0.126706 + 0.219462i −0.922399 0.386239i \(-0.873774\pi\)
0.795692 + 0.605701i \(0.207107\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −30.8259 −0.655871 −0.327936 0.944700i \(-0.606353\pi\)
−0.327936 + 0.944700i \(0.606353\pi\)
\(48\) 0 0
\(49\) 24.2517 42.0052i 0.494933 0.857249i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 76.8522 + 44.3706i 1.45004 + 0.837182i 0.998483 0.0550603i \(-0.0175351\pi\)
0.451558 + 0.892242i \(0.350868\pi\)
\(54\) 0 0
\(55\) −86.2498 149.389i −1.56818 2.71616i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 91.5455i 1.55162i −0.630967 0.775809i \(-0.717342\pi\)
0.630967 0.775809i \(-0.282658\pi\)
\(60\) 0 0
\(61\) −10.0853 −0.165332 −0.0826662 0.996577i \(-0.526344\pi\)
−0.0826662 + 0.996577i \(0.526344\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 62.1410 35.8771i 0.956015 0.551955i
\(66\) 0 0
\(67\) 13.9419 8.04937i 0.208088 0.120140i −0.392334 0.919823i \(-0.628332\pi\)
0.600423 + 0.799683i \(0.294999\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −67.0947 + 38.7371i −0.944995 + 0.545593i −0.891523 0.452976i \(-0.850362\pi\)
−0.0534727 + 0.998569i \(0.517029\pi\)
\(72\) 0 0
\(73\) −39.6851 68.7366i −0.543631 0.941597i −0.998692 0.0511363i \(-0.983716\pi\)
0.455061 0.890460i \(-0.349618\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.27700 + 12.6041i −0.0945065 + 0.163690i
\(78\) 0 0
\(79\) −78.3707 45.2473i −0.992034 0.572751i −0.0861523 0.996282i \(-0.527457\pi\)
−0.905882 + 0.423531i \(0.860791\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 31.5302 + 54.6119i 0.379882 + 0.657975i 0.991045 0.133530i \(-0.0426313\pi\)
−0.611163 + 0.791505i \(0.709298\pi\)
\(84\) 0 0
\(85\) 22.7537 + 39.4106i 0.267691 + 0.463654i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 65.0621 + 37.5636i 0.731035 + 0.422063i 0.818801 0.574078i \(-0.194639\pi\)
−0.0877659 + 0.996141i \(0.527973\pi\)
\(90\) 0 0
\(91\) −5.24291 3.02700i −0.0576144 0.0332637i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.09205 + 158.533i −0.0746532 + 1.66877i
\(96\) 0 0
\(97\) −65.4754 37.8022i −0.675004 0.389714i 0.122966 0.992411i \(-0.460759\pi\)
−0.797970 + 0.602697i \(0.794093\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −84.9133 −0.840726 −0.420363 0.907356i \(-0.638097\pi\)
−0.420363 + 0.907356i \(0.638097\pi\)
\(102\) 0 0
\(103\) 24.5788 + 14.1906i 0.238629 + 0.137772i 0.614546 0.788881i \(-0.289339\pi\)
−0.375918 + 0.926653i \(0.622672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 148.821i 1.39085i −0.718600 0.695423i \(-0.755217\pi\)
0.718600 0.695423i \(-0.244783\pi\)
\(108\) 0 0
\(109\) −49.6931 + 28.6903i −0.455900 + 0.263214i −0.710319 0.703880i \(-0.751449\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −59.1471 34.1486i −0.523426 0.302200i 0.214909 0.976634i \(-0.431054\pi\)
−0.738335 + 0.674434i \(0.764388\pi\)
\(114\) 0 0
\(115\) −99.1372 171.711i −0.862063 1.49314i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.91976 3.32512i 0.0161324 0.0279422i
\(120\) 0 0
\(121\) −152.776 + 264.617i −1.26262 + 2.18691i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −165.035 −1.32028
\(126\) 0 0
\(127\) −45.0028 25.9824i −0.354353 0.204586i 0.312248 0.950001i \(-0.398918\pi\)
−0.666601 + 0.745415i \(0.732251\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −90.7869 −0.693030 −0.346515 0.938044i \(-0.612635\pi\)
−0.346515 + 0.938044i \(0.612635\pi\)
\(132\) 0 0
\(133\) 11.8829 6.16964i 0.0893448 0.0463882i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −232.188 −1.69481 −0.847403 0.530951i \(-0.821835\pi\)
−0.847403 + 0.530951i \(0.821835\pi\)
\(138\) 0 0
\(139\) 13.1914 + 22.8482i 0.0949022 + 0.164375i 0.909568 0.415556i \(-0.136413\pi\)
−0.814666 + 0.579931i \(0.803080\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −153.661 88.7161i −1.07455 0.620392i
\(144\) 0 0
\(145\) 19.8775i 0.137086i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −100.867 −0.676959 −0.338479 0.940974i \(-0.609913\pi\)
−0.338479 + 0.940974i \(0.609913\pi\)
\(150\) 0 0
\(151\) −201.351 + 116.250i −1.33345 + 0.769868i −0.985827 0.167765i \(-0.946345\pi\)
−0.347624 + 0.937634i \(0.613012\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −19.3867 11.1929i −0.125076 0.0722124i
\(156\) 0 0
\(157\) 197.395 1.25729 0.628647 0.777691i \(-0.283609\pi\)
0.628647 + 0.777691i \(0.283609\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.36433 + 14.4875i −0.0519524 + 0.0899842i
\(162\) 0 0
\(163\) −238.025 −1.46027 −0.730137 0.683301i \(-0.760544\pi\)
−0.730137 + 0.683301i \(0.760544\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −217.700 + 125.689i −1.30360 + 0.752631i −0.981019 0.193912i \(-0.937882\pi\)
−0.322576 + 0.946543i \(0.604549\pi\)
\(168\) 0 0
\(169\) −47.5970 + 82.4404i −0.281639 + 0.487813i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −238.033 137.428i −1.37591 0.794384i −0.384249 0.923229i \(-0.625540\pi\)
−0.991665 + 0.128845i \(0.958873\pi\)
\(174\) 0 0
\(175\) 15.7707 + 27.3156i 0.0901182 + 0.156089i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 106.008i 0.592221i −0.955154 0.296111i \(-0.904310\pi\)
0.955154 0.296111i \(-0.0956897\pi\)
\(180\) 0 0
\(181\) 155.686 + 89.8854i 0.860144 + 0.496604i 0.864061 0.503388i \(-0.167913\pi\)
−0.00391638 + 0.999992i \(0.501247\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 165.448i 0.894314i
\(186\) 0 0
\(187\) 56.2649 97.4536i 0.300882 0.521142i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 60.2220 + 104.308i 0.315298 + 0.546113i 0.979501 0.201440i \(-0.0645621\pi\)
−0.664203 + 0.747553i \(0.731229\pi\)
\(192\) 0 0
\(193\) 117.258i 0.607555i 0.952743 + 0.303778i \(0.0982480\pi\)
−0.952743 + 0.303778i \(0.901752\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −216.141 −1.09716 −0.548581 0.836098i \(-0.684832\pi\)
−0.548581 + 0.836098i \(0.684832\pi\)
\(198\) 0 0
\(199\) 129.858 224.920i 0.652551 1.13025i −0.329950 0.943998i \(-0.607032\pi\)
0.982502 0.186254i \(-0.0596347\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.45240 + 0.838545i −0.00715469 + 0.00413076i
\(204\) 0 0
\(205\) 630.115i 3.07373i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 348.266 180.821i 1.66634 0.865174i
\(210\) 0 0
\(211\) 41.9123i 0.198636i 0.995056 + 0.0993182i \(0.0316662\pi\)
−0.995056 + 0.0993182i \(0.968334\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 45.5059 78.8185i 0.211655 0.366598i
\(216\) 0 0
\(217\) 1.88872i 0.00870379i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 40.5375 + 23.4044i 0.183428 + 0.105902i
\(222\) 0 0
\(223\) −91.4323 52.7885i −0.410010 0.236720i 0.280784 0.959771i \(-0.409406\pi\)
−0.690794 + 0.723051i \(0.742739\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 384.737 222.128i 1.69488 0.978537i 0.744403 0.667730i \(-0.232734\pi\)
0.950473 0.310807i \(-0.100599\pi\)
\(228\) 0 0
\(229\) 118.302 204.906i 0.516604 0.894784i −0.483210 0.875504i \(-0.660529\pi\)
0.999814 0.0192797i \(-0.00613729\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 85.0481 + 147.308i 0.365013 + 0.632222i 0.988778 0.149390i \(-0.0477310\pi\)
−0.623765 + 0.781612i \(0.714398\pi\)
\(234\) 0 0
\(235\) 257.465 1.09559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −221.528 + 383.698i −0.926896 + 1.60543i −0.138413 + 0.990375i \(0.544200\pi\)
−0.788483 + 0.615057i \(0.789133\pi\)
\(240\) 0 0
\(241\) 153.683i 0.637690i −0.947807 0.318845i \(-0.896705\pi\)
0.947807 0.318845i \(-0.103295\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −202.555 + 350.836i −0.826757 + 1.43198i
\(246\) 0 0
\(247\) 75.2158 + 144.867i 0.304518 + 0.586507i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 84.3664 146.127i 0.336121 0.582179i −0.647578 0.761999i \(-0.724218\pi\)
0.983700 + 0.179820i \(0.0575515\pi\)
\(252\) 0 0
\(253\) −245.144 + 424.602i −0.968950 + 1.67827i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −279.805 + 161.545i −1.08873 + 0.628581i −0.933239 0.359256i \(-0.883030\pi\)
−0.155495 + 0.987837i \(0.549697\pi\)
\(258\) 0 0
\(259\) −12.0889 + 6.97953i −0.0466753 + 0.0269480i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 107.182 185.645i 0.407537 0.705875i −0.587076 0.809532i \(-0.699721\pi\)
0.994613 + 0.103657i \(0.0330544\pi\)
\(264\) 0 0
\(265\) −641.885 370.593i −2.42221 1.39846i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 339.682 196.116i 1.26276 0.729054i 0.289152 0.957283i \(-0.406627\pi\)
0.973608 + 0.228229i \(0.0732935\pi\)
\(270\) 0 0
\(271\) −89.8653 155.651i −0.331606 0.574359i 0.651221 0.758888i \(-0.274257\pi\)
−0.982827 + 0.184529i \(0.940924\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 462.212 + 800.574i 1.68077 + 2.91118i
\(276\) 0 0
\(277\) 112.854 + 195.468i 0.407414 + 0.705662i 0.994599 0.103791i \(-0.0330973\pi\)
−0.587185 + 0.809453i \(0.699764\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 138.809i 0.493982i −0.969018 0.246991i \(-0.920558\pi\)
0.969018 0.246991i \(-0.0794418\pi\)
\(282\) 0 0
\(283\) 45.9017 0.162197 0.0810984 0.996706i \(-0.474157\pi\)
0.0810984 + 0.996706i \(0.474157\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −46.0411 + 26.5818i −0.160422 + 0.0926196i
\(288\) 0 0
\(289\) 129.657 224.572i 0.448639 0.777065i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −357.501 206.404i −1.22014 0.704449i −0.255193 0.966890i \(-0.582139\pi\)
−0.964948 + 0.262441i \(0.915472\pi\)
\(294\) 0 0
\(295\) 764.607i 2.59189i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −176.621 101.972i −0.590705 0.341044i
\(300\) 0 0
\(301\) −7.67878 −0.0255109
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 84.2343 0.276178
\(306\) 0 0
\(307\) 48.2502 27.8573i 0.157167 0.0907403i −0.419354 0.907823i \(-0.637743\pi\)
0.576521 + 0.817083i \(0.304410\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −143.848 + 249.152i −0.462534 + 0.801133i −0.999086 0.0427343i \(-0.986393\pi\)
0.536552 + 0.843867i \(0.319726\pi\)
\(312\) 0 0
\(313\) 507.327 1.62085 0.810426 0.585841i \(-0.199236\pi\)
0.810426 + 0.585841i \(0.199236\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 50.1009i 0.158047i 0.996873 + 0.0790235i \(0.0251802\pi\)
−0.996873 + 0.0790235i \(0.974820\pi\)
\(318\) 0 0
\(319\) −42.5674 + 24.5763i −0.133440 + 0.0770417i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −91.8767 + 47.7029i −0.284448 + 0.147687i
\(324\) 0 0
\(325\) −333.013 + 192.265i −1.02465 + 0.591585i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.8613 18.8124i −0.0330131 0.0571804i
\(330\) 0 0
\(331\) 269.005 155.310i 0.812705 0.469215i −0.0351896 0.999381i \(-0.511204\pi\)
0.847894 + 0.530165i \(0.177870\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −116.446 + 67.2301i −0.347600 + 0.200687i
\(336\) 0 0
\(337\) 31.2347i 0.0926845i 0.998926 + 0.0463423i \(0.0147565\pi\)
−0.998926 + 0.0463423i \(0.985244\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 55.3552i 0.162332i
\(342\) 0 0
\(343\) 68.7094 0.200319
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −369.111 −1.06372 −0.531861 0.846832i \(-0.678507\pi\)
−0.531861 + 0.846832i \(0.678507\pi\)
\(348\) 0 0
\(349\) 182.826 + 316.663i 0.523856 + 0.907345i 0.999614 + 0.0277692i \(0.00884035\pi\)
−0.475758 + 0.879576i \(0.657826\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −289.028 500.611i −0.818776 1.41816i −0.906584 0.422025i \(-0.861319\pi\)
0.0878078 0.996137i \(-0.472014\pi\)
\(354\) 0 0
\(355\) 560.389 323.541i 1.57856 0.911382i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −311.123 538.882i −0.866639 1.50106i −0.865410 0.501064i \(-0.832942\pi\)
−0.00122868 0.999999i \(-0.500391\pi\)
\(360\) 0 0
\(361\) −359.558 32.2344i −0.996005 0.0892921i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 331.458 + 574.102i 0.908104 + 1.57288i
\(366\) 0 0
\(367\) 242.198 0.659940 0.329970 0.943991i \(-0.392961\pi\)
0.329970 + 0.943991i \(0.392961\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 62.5348i 0.168557i
\(372\) 0 0
\(373\) −290.835 167.914i −0.779718 0.450170i 0.0566124 0.998396i \(-0.481970\pi\)
−0.836330 + 0.548226i \(0.815303\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.2229 17.7067i −0.0271166 0.0469673i
\(378\) 0 0
\(379\) 83.3649i 0.219960i 0.993934 + 0.109980i \(0.0350787\pi\)
−0.993934 + 0.109980i \(0.964921\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.6495i 0.0852467i 0.999091 + 0.0426234i \(0.0135715\pi\)
−0.999091 + 0.0426234i \(0.986428\pi\)
\(384\) 0 0
\(385\) 60.7791 105.272i 0.157868 0.273435i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 253.262 0.651058 0.325529 0.945532i \(-0.394458\pi\)
0.325529 + 0.945532i \(0.394458\pi\)
\(390\) 0 0
\(391\) 64.6720 112.015i 0.165402 0.286484i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 654.568 + 377.915i 1.65714 + 0.956747i
\(396\) 0 0
\(397\) 131.950 + 228.543i 0.332367 + 0.575676i 0.982975 0.183737i \(-0.0588195\pi\)
−0.650609 + 0.759413i \(0.725486\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 744.298i 1.85610i −0.372450 0.928052i \(-0.621482\pi\)
0.372450 0.928052i \(-0.378518\pi\)
\(402\) 0 0
\(403\) −23.0260 −0.0571364
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −354.305 + 204.558i −0.870529 + 0.502600i
\(408\) 0 0
\(409\) −40.6985 + 23.4973i −0.0995074 + 0.0574506i −0.548928 0.835870i \(-0.684964\pi\)
0.449420 + 0.893320i \(0.351631\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 55.8681 32.2555i 0.135274 0.0781004i
\(414\) 0 0
\(415\) −263.347 456.130i −0.634571 1.09911i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 35.3570 61.2401i 0.0843842 0.146158i −0.820744 0.571295i \(-0.806441\pi\)
0.905129 + 0.425138i \(0.139774\pi\)
\(420\) 0 0
\(421\) 110.483 + 63.7872i 0.262429 + 0.151513i 0.625442 0.780271i \(-0.284919\pi\)
−0.363013 + 0.931784i \(0.618252\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −121.937 211.201i −0.286911 0.496944i
\(426\) 0 0
\(427\) −3.55348 6.15481i −0.00832197 0.0144141i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 541.149 + 312.432i 1.25557 + 0.724901i 0.972209 0.234113i \(-0.0752187\pi\)
0.283357 + 0.959015i \(0.408552\pi\)
\(432\) 0 0
\(433\) 78.5687 + 45.3617i 0.181452 + 0.104761i 0.587975 0.808879i \(-0.299925\pi\)
−0.406523 + 0.913641i \(0.633259\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 400.304 207.840i 0.916027 0.475606i
\(438\) 0 0
\(439\) −130.860 75.5521i −0.298087 0.172101i 0.343496 0.939154i \(-0.388389\pi\)
−0.641583 + 0.767054i \(0.721722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −871.228 −1.96666 −0.983328 0.181842i \(-0.941794\pi\)
−0.983328 + 0.181842i \(0.941794\pi\)
\(444\) 0 0
\(445\) −543.412 313.739i −1.22115 0.705032i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 174.763i 0.389228i −0.980880 0.194614i \(-0.937655\pi\)
0.980880 0.194614i \(-0.0623454\pi\)
\(450\) 0 0
\(451\) −1349.38 + 779.067i −2.99198 + 1.72742i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 43.7899 + 25.2821i 0.0962416 + 0.0555651i
\(456\) 0 0
\(457\) −370.554 641.818i −0.810840 1.40442i −0.912277 0.409574i \(-0.865677\pi\)
0.101437 0.994842i \(-0.467656\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −282.972 + 490.122i −0.613822 + 1.06317i 0.376768 + 0.926308i \(0.377035\pi\)
−0.990590 + 0.136863i \(0.956298\pi\)
\(462\) 0 0
\(463\) 178.485 309.146i 0.385497 0.667701i −0.606341 0.795205i \(-0.707363\pi\)
0.991838 + 0.127504i \(0.0406965\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 427.161 0.914691 0.457345 0.889289i \(-0.348800\pi\)
0.457345 + 0.889289i \(0.348800\pi\)
\(468\) 0 0
\(469\) 9.82469 + 5.67229i 0.0209482 + 0.0120944i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −225.052 −0.475797
\(474\) 0 0
\(475\) 38.0062 849.579i 0.0800132 1.78859i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 329.200 0.687266 0.343633 0.939104i \(-0.388342\pi\)
0.343633 + 0.939104i \(0.388342\pi\)
\(480\) 0 0
\(481\) −85.0895 147.379i −0.176901 0.306402i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 546.864 + 315.732i 1.12756 + 0.650994i
\(486\) 0 0
\(487\) 788.052i 1.61818i 0.587687 + 0.809088i \(0.300039\pi\)
−0.587687 + 0.809088i \(0.699961\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 781.176 1.59099 0.795495 0.605960i \(-0.207211\pi\)
0.795495 + 0.605960i \(0.207211\pi\)
\(492\) 0 0
\(493\) 11.2298 6.48352i 0.0227785 0.0131512i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −47.2807 27.2975i −0.0951323 0.0549246i
\(498\) 0 0
\(499\) 233.501 0.467938 0.233969 0.972244i \(-0.424829\pi\)
0.233969 + 0.972244i \(0.424829\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −402.390 + 696.960i −0.799980 + 1.38561i 0.119649 + 0.992816i \(0.461823\pi\)
−0.919628 + 0.392789i \(0.871510\pi\)
\(504\) 0 0
\(505\) 709.214 1.40438
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 98.8113 57.0487i 0.194128 0.112080i −0.399785 0.916609i \(-0.630915\pi\)
0.593914 + 0.804529i \(0.297582\pi\)
\(510\) 0 0
\(511\) 27.9655 48.4377i 0.0547271 0.0947901i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −205.287 118.523i −0.398616 0.230141i
\(516\) 0 0
\(517\) −318.327 551.358i −0.615719 1.06646i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 330.381i 0.634130i −0.948404 0.317065i \(-0.897303\pi\)
0.948404 0.317065i \(-0.102697\pi\)
\(522\) 0 0
\(523\) 223.871 + 129.252i 0.428052 + 0.247136i 0.698517 0.715594i \(-0.253844\pi\)
−0.270464 + 0.962730i \(0.587177\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.6034i 0.0277104i
\(528\) 0 0
\(529\) −17.2737 + 29.9189i −0.0326535 + 0.0565575i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −324.067 561.300i −0.608005 1.05310i
\(534\) 0 0
\(535\) 1242.98i 2.32333i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1001.75 1.85853
\(540\) 0 0
\(541\) 307.180 532.051i 0.567800 0.983458i −0.428983 0.903312i \(-0.641128\pi\)
0.996783 0.0801456i \(-0.0255385\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 415.047 239.628i 0.761555 0.439684i
\(546\) 0 0
\(547\) 321.852i 0.588394i 0.955745 + 0.294197i \(0.0950522\pi\)
−0.955745 + 0.294197i \(0.904948\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 45.1731 + 2.02083i 0.0819838 + 0.00366758i
\(552\) 0 0
\(553\) 63.7704i 0.115317i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 257.207 445.496i 0.461772 0.799813i −0.537277 0.843406i \(-0.680547\pi\)
0.999049 + 0.0435927i \(0.0138804\pi\)
\(558\) 0 0
\(559\) 93.6143i 0.167467i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −378.199 218.353i −0.671756 0.387839i 0.124986 0.992159i \(-0.460112\pi\)
−0.796742 + 0.604320i \(0.793445\pi\)
\(564\) 0 0
\(565\) 494.009 + 285.216i 0.874353 + 0.504808i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −734.219 + 423.901i −1.29037 + 0.744994i −0.978719 0.205204i \(-0.934214\pi\)
−0.311648 + 0.950198i \(0.600881\pi\)
\(570\) 0 0
\(571\) −393.820 + 682.116i −0.689702 + 1.19460i 0.282232 + 0.959346i \(0.408925\pi\)
−0.971934 + 0.235253i \(0.924408\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 531.275 + 920.196i 0.923957 + 1.60034i
\(576\) 0 0
\(577\) 291.421 0.505062 0.252531 0.967589i \(-0.418737\pi\)
0.252531 + 0.967589i \(0.418737\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.2189 + 38.4843i −0.0382425 + 0.0662380i
\(582\) 0 0
\(583\) 1832.79i 3.14372i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 449.526 778.603i 0.765803 1.32641i −0.174018 0.984743i \(-0.555675\pi\)
0.939821 0.341667i \(-0.110992\pi\)
\(588\) 0 0
\(589\) 27.4077 42.9198i 0.0465326 0.0728690i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −272.320 + 471.672i −0.459224 + 0.795400i −0.998920 0.0464601i \(-0.985206\pi\)
0.539696 + 0.841860i \(0.318539\pi\)
\(594\) 0 0
\(595\) −16.0342 + 27.7721i −0.0269483 + 0.0466758i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −372.742 + 215.203i −0.622274 + 0.359270i −0.777754 0.628569i \(-0.783641\pi\)
0.155480 + 0.987839i \(0.450308\pi\)
\(600\) 0 0
\(601\) −640.942 + 370.048i −1.06646 + 0.615721i −0.927212 0.374537i \(-0.877802\pi\)
−0.139247 + 0.990258i \(0.544468\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1276.02 2210.13i 2.10913 3.65311i
\(606\) 0 0
\(607\) −452.418 261.204i −0.745334 0.430319i 0.0786715 0.996901i \(-0.474932\pi\)
−0.824006 + 0.566582i \(0.808266\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 229.347 132.413i 0.375363 0.216716i
\(612\) 0 0
\(613\) −129.627 224.520i −0.211463 0.366264i 0.740710 0.671825i \(-0.234489\pi\)
−0.952172 + 0.305561i \(0.901156\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25.6423 + 44.4138i 0.0415596 + 0.0719834i 0.886057 0.463576i \(-0.153434\pi\)
−0.844497 + 0.535560i \(0.820101\pi\)
\(618\) 0 0
\(619\) −338.509 586.315i −0.546865 0.947197i −0.998487 0.0549882i \(-0.982488\pi\)
0.451622 0.892209i \(-0.350845\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 52.9412i 0.0849778i
\(624\) 0 0
\(625\) 259.420 0.415072
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 93.4699 53.9649i 0.148601 0.0857947i
\(630\) 0 0
\(631\) −174.679 + 302.553i −0.276829 + 0.479481i −0.970595 0.240719i \(-0.922617\pi\)
0.693766 + 0.720200i \(0.255950\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 375.873 + 217.010i 0.591926 + 0.341748i
\(636\) 0 0
\(637\) 416.695i 0.654152i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 289.836 + 167.337i 0.452162 + 0.261056i 0.708743 0.705467i \(-0.249263\pi\)
−0.256581 + 0.966523i \(0.582596\pi\)
\(642\) 0 0
\(643\) −59.1079 −0.0919252 −0.0459626 0.998943i \(-0.514636\pi\)
−0.0459626 + 0.998943i \(0.514636\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −157.684 −0.243716 −0.121858 0.992548i \(-0.538885\pi\)
−0.121858 + 0.992548i \(0.538885\pi\)
\(648\) 0 0
\(649\) 1637.40 945.352i 2.52295 1.45663i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −199.103 + 344.856i −0.304905 + 0.528111i −0.977240 0.212136i \(-0.931958\pi\)
0.672335 + 0.740247i \(0.265291\pi\)
\(654\) 0 0
\(655\) 758.271 1.15767
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1062.19i 1.61182i 0.592041 + 0.805908i \(0.298322\pi\)
−0.592041 + 0.805908i \(0.701678\pi\)
\(660\) 0 0
\(661\) −352.363 + 203.437i −0.533076 + 0.307772i −0.742268 0.670103i \(-0.766250\pi\)
0.209192 + 0.977875i \(0.432917\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −99.2481 + 51.5301i −0.149245 + 0.0774889i
\(666\) 0 0
\(667\) −48.9278 + 28.2485i −0.0733551 + 0.0423516i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −104.146 180.387i −0.155211 0.268833i
\(672\) 0 0
\(673\) 376.962 217.639i 0.560122 0.323387i −0.193072 0.981185i \(-0.561845\pi\)
0.753195 + 0.657798i \(0.228512\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −396.408 + 228.866i −0.585536 + 0.338059i −0.763330 0.646008i \(-0.776437\pi\)
0.177794 + 0.984068i \(0.443104\pi\)
\(678\) 0 0
\(679\) 53.2775i 0.0784646i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 332.751i 0.487190i −0.969877 0.243595i \(-0.921673\pi\)
0.969877 0.243595i \(-0.0783267\pi\)
\(684\) 0 0
\(685\) 1939.29 2.83107
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −762.380 −1.10650
\(690\) 0 0
\(691\) 42.6760 + 73.9170i 0.0617598 + 0.106971i 0.895252 0.445560i \(-0.146995\pi\)
−0.833492 + 0.552531i \(0.813662\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −110.177 190.833i −0.158529 0.274580i
\(696\) 0 0
\(697\) 355.984 205.527i 0.510737 0.294874i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 264.113 + 457.458i 0.376766 + 0.652579i 0.990590 0.136865i \(-0.0437026\pi\)
−0.613823 + 0.789443i \(0.710369\pi\)
\(702\) 0 0
\(703\) 375.993 + 16.8202i 0.534841 + 0.0239263i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.9187 51.8206i −0.0423178 0.0732965i
\(708\) 0 0
\(709\) −1202.26 −1.69571 −0.847854 0.530229i \(-0.822106\pi\)
−0.847854 + 0.530229i \(0.822106\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 63.6264i 0.0892376i
\(714\) 0 0
\(715\) 1283.41 + 740.975i 1.79497 + 1.03633i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −346.447 600.064i −0.481846 0.834581i 0.517937 0.855419i \(-0.326700\pi\)
−0.999783 + 0.0208376i \(0.993367\pi\)
\(720\) 0 0
\(721\) 19.9998i 0.0277390i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 106.523i 0.146929i
\(726\) 0 0
\(727\) −654.043 + 1132.84i −0.899646 + 1.55823i −0.0716995 + 0.997426i \(0.522842\pi\)
−0.827947 + 0.560807i \(0.810491\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 59.3714 0.0812194
\(732\) 0 0
\(733\) −216.509 + 375.004i −0.295374 + 0.511602i −0.975072 0.221890i \(-0.928777\pi\)
0.679698 + 0.733492i \(0.262111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 287.945 + 166.245i 0.390698 + 0.225570i
\(738\) 0 0
\(739\) 617.926 + 1070.28i 0.836165 + 1.44828i 0.893078 + 0.449901i \(0.148541\pi\)
−0.0569131 + 0.998379i \(0.518126\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 947.212i 1.27485i 0.770513 + 0.637424i \(0.220000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(744\) 0 0
\(745\) 842.461 1.13082
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 90.8217 52.4359i 0.121257 0.0700079i
\(750\) 0 0
\(751\) 1219.16 703.884i 1.62339 0.937263i 0.637382 0.770548i \(-0.280017\pi\)
0.986005 0.166715i \(-0.0533160\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1681.73 970.946i 2.22745 1.28602i
\(756\) 0 0
\(757\) −325.685 564.102i −0.430231 0.745182i 0.566662 0.823950i \(-0.308234\pi\)
−0.996893 + 0.0787687i \(0.974901\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −261.899 + 453.622i −0.344151 + 0.596087i −0.985199 0.171414i \(-0.945167\pi\)
0.641048 + 0.767501i \(0.278500\pi\)
\(762\) 0 0
\(763\) −35.0181 20.2177i −0.0458952 0.0264976i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 393.236 + 681.104i 0.512693 + 0.888010i
\(768\) 0 0
\(769\) 92.2444 + 159.772i 0.119954 + 0.207766i 0.919749 0.392507i \(-0.128392\pi\)
−0.799795 + 0.600273i \(0.795059\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −152.283 87.9206i −0.197002 0.113739i 0.398254 0.917275i \(-0.369616\pi\)
−0.595257 + 0.803536i \(0.702950\pi\)
\(774\) 0 0
\(775\) 103.893 + 59.9828i 0.134056 + 0.0773971i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1431.98 + 64.0603i 1.83823 + 0.0822341i
\(780\) 0 0
\(781\) −1385.72 800.044i −1.77429 1.02438i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1648.68 −2.10024
\(786\) 0 0
\(787\) −80.0272 46.2037i −0.101686 0.0587087i 0.448294 0.893886i \(-0.352032\pi\)
−0.549981 + 0.835177i \(0.685365\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 48.1281i 0.0608447i
\(792\) 0 0
\(793\) 75.0350 43.3215i 0.0946217 0.0546299i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 769.440 + 444.236i 0.965420 + 0.557385i 0.897837 0.440328i \(-0.145138\pi\)
0.0675830 + 0.997714i \(0.478471\pi\)
\(798\) 0 0
\(799\) 83.9783 + 145.455i 0.105104 + 0.182046i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 819.622 1419.63i 1.02070 1.76790i
\(804\) 0 0
\(805\) 69.8607 121.002i 0.0867834 0.150313i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 939.940 1.16185 0.580927 0.813956i \(-0.302690\pi\)
0.580927 + 0.813956i \(0.302690\pi\)
\(810\) 0 0
\(811\) 361.756 + 208.860i 0.446062 + 0.257534i 0.706166 0.708047i \(-0.250423\pi\)
−0.260104 + 0.965581i \(0.583757\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1988.03 2.43930
\(816\) 0 0
\(817\) 174.495 + 111.429i 0.213580 + 0.136387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 560.135 0.682260 0.341130 0.940016i \(-0.389190\pi\)
0.341130 + 0.940016i \(0.389190\pi\)
\(822\) 0 0
\(823\) 623.961 + 1080.73i 0.758154 + 1.31316i 0.943791 + 0.330542i \(0.107232\pi\)
−0.185637 + 0.982618i \(0.559435\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −496.644 286.737i −0.600537 0.346720i 0.168716 0.985665i \(-0.446038\pi\)
−0.769253 + 0.638945i \(0.779371\pi\)
\(828\) 0 0
\(829\) 921.847i 1.11200i −0.831183 0.555999i \(-0.812336\pi\)
0.831183 0.555999i \(-0.187664\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −264.273 −0.317255
\(834\) 0 0
\(835\) 1818.28 1049.78i 2.17758 1.25723i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 431.746 + 249.269i 0.514596 + 0.297102i 0.734721 0.678369i \(-0.237313\pi\)
−0.220125 + 0.975472i \(0.570646\pi\)
\(840\) 0 0
\(841\) 835.336 0.993265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 397.540 688.560i 0.470462 0.814864i
\(846\) 0 0
\(847\) −215.319 −0.254214
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −407.245 + 235.123i −0.478549 + 0.276291i
\(852\) 0 0
\(853\) −681.852 + 1181.00i −0.799358 + 1.38453i 0.120677 + 0.992692i \(0.461493\pi\)
−0.920035 + 0.391837i \(0.871840\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1159.34 669.347i −1.35279 0.781036i −0.364153 0.931339i \(-0.618642\pi\)
−0.988640 + 0.150303i \(0.951975\pi\)
\(858\) 0 0
\(859\) −316.285 547.822i −0.368202 0.637744i 0.621083 0.783745i \(-0.286693\pi\)
−0.989284 + 0.146001i \(0.953360\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 731.113i 0.847176i −0.905855 0.423588i \(-0.860770\pi\)
0.905855 0.423588i \(-0.139230\pi\)
\(864\) 0 0
\(865\) 1988.10 + 1147.83i 2.29838 + 1.32697i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1869.00i 2.15075i
\(870\) 0 0
\(871\) −69.1525 + 119.776i −0.0793944 + 0.137515i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −58.1489 100.717i −0.0664559 0.115105i
\(876\) 0 0
\(877\) 1157.72i 1.32009i −0.751224 0.660047i \(-0.770536\pi\)
0.751224 0.660047i \(-0.229464\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0365 −0.0204728 −0.0102364 0.999948i \(-0.503258\pi\)
−0.0102364 + 0.999948i \(0.503258\pi\)
\(882\) 0 0
\(883\) 494.516 856.527i 0.560041 0.970019i −0.437451 0.899242i \(-0.644119\pi\)
0.997492 0.0707772i \(-0.0225479\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1048.59 + 605.403i −1.18217 + 0.682528i −0.956516 0.291678i \(-0.905786\pi\)
−0.225657 + 0.974207i \(0.572453\pi\)
\(888\) 0 0
\(889\) 36.6189i 0.0411911i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −26.1750 + 585.108i −0.0293113 + 0.655216i
\(894\) 0 0
\(895\) 885.398i 0.989271i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.18935 + 5.52412i −0.00354767 + 0.00614474i
\(900\) 0 0
\(901\) 483.511i 0.536638i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1300.32 750.742i −1.43682 0.829549i
\(906\) 0 0
\(907\) 279.508 + 161.374i 0.308168 + 0.177921i 0.646106 0.763247i \(-0.276396\pi\)
−0.337938 + 0.941168i \(0.609730\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −567.038 + 327.380i −0.622435 + 0.359363i −0.777816 0.628492i \(-0.783673\pi\)
0.155382 + 0.987855i \(0.450339\pi\)
\(912\) 0 0
\(913\) −651.198 + 1127.91i −0.713251 + 1.23539i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −31.9882 55.4051i −0.0348835 0.0604200i
\(918\) 0 0
\(919\) −1299.09 −1.41359 −0.706796 0.707417i \(-0.749860\pi\)
−0.706796 + 0.707417i \(0.749860\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 332.792 576.413i 0.360555 0.624500i
\(924\) 0 0
\(925\) 886.635i 0.958524i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 260.771 451.669i 0.280701 0.486188i −0.690857 0.722992i \(-0.742767\pi\)
0.971558 + 0.236804i \(0.0760998\pi\)
\(930\) 0 0
\(931\) −776.709 495.990i −0.834273 0.532749i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −469.936 + 813.953i −0.502606 + 0.870538i
\(936\) 0 0
\(937\) −625.318 + 1083.08i −0.667362 + 1.15590i 0.311277 + 0.950319i \(0.399243\pi\)
−0.978639 + 0.205586i \(0.934090\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −165.450 + 95.5225i −0.175823 + 0.101512i −0.585329 0.810796i \(-0.699035\pi\)
0.409505 + 0.912308i \(0.365701\pi\)
\(942\) 0 0
\(943\) −1551.01 + 895.476i −1.64476 + 0.949603i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −292.243 + 506.180i −0.308599 + 0.534509i −0.978056 0.208342i \(-0.933193\pi\)
0.669457 + 0.742851i \(0.266527\pi\)
\(948\) 0 0
\(949\) 590.519 + 340.936i 0.622254 + 0.359258i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −221.454 + 127.856i −0.232375 + 0.134162i −0.611667 0.791115i \(-0.709501\pi\)
0.379292 + 0.925277i \(0.376168\pi\)
\(954\) 0 0
\(955\) −502.987 871.198i −0.526688 0.912250i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −81.8100 141.699i −0.0853076 0.147757i
\(960\) 0 0
\(961\) −476.908 826.029i −0.496262 0.859552i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 979.365i 1.01489i
\(966\) 0 0
\(967\) −775.857 −0.802334 −0.401167 0.916005i \(-0.631395\pi\)
−0.401167 + 0.916005i \(0.631395\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −618.028 + 356.819i −0.636486 + 0.367476i −0.783260 0.621695i \(-0.786445\pi\)
0.146773 + 0.989170i \(0.453111\pi\)
\(972\) 0 0
\(973\) −9.29581 + 16.1008i −0.00955376 + 0.0165476i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 749.812 + 432.904i 0.767463 + 0.443095i 0.831969 0.554822i \(-0.187214\pi\)
−0.0645057 + 0.997917i \(0.520547\pi\)
\(978\) 0 0
\(979\) 1551.61i 1.58490i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1367.55 789.558i −1.39120 0.803212i −0.397756 0.917491i \(-0.630211\pi\)
−0.993449 + 0.114279i \(0.963544\pi\)
\(984\) 0 0
\(985\) 1805.25 1.83274
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −258.679 −0.261556
\(990\) 0 0
\(991\) 208.038 120.111i 0.209927 0.121202i −0.391350 0.920242i \(-0.627992\pi\)
0.601278 + 0.799040i \(0.294659\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1084.60 + 1878.58i −1.09005 + 1.88802i
\(996\) 0 0
\(997\) −454.623 −0.455991 −0.227995 0.973662i \(-0.573217\pi\)
−0.227995 + 0.973662i \(0.573217\pi\)
\(998\) 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 2052.3.bl.a.145.3 80
3.2 odd 2 684.3.bl.a.373.11 yes 80
9.2 odd 6 684.3.s.a.601.24 yes 80
9.7 even 3 2052.3.s.a.829.38 80
19.8 odd 6 2052.3.s.a.901.38 80
57.8 even 6 684.3.s.a.445.24 80
171.65 even 6 684.3.bl.a.673.11 yes 80
171.160 odd 6 inner 2052.3.bl.a.1585.3 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.s.a.445.24 80 57.8 even 6
684.3.s.a.601.24 yes 80 9.2 odd 6
684.3.bl.a.373.11 yes 80 3.2 odd 2
684.3.bl.a.673.11 yes 80 171.65 even 6
2052.3.s.a.829.38 80 9.7 even 3
2052.3.s.a.901.38 80 19.8 odd 6
2052.3.bl.a.145.3 80 1.1 even 1 trivial
2052.3.bl.a.1585.3 80 171.160 odd 6 inner