Properties

Label 588.4.k.e.509.1
Level $588$
Weight $4$
Character 588.509
Analytic conductor $34.693$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,4,Mod(509,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.509");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.k (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: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 509.1
Character \(\chi\) \(=\) 588.509
Dual form 588.4.k.e.521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-5.19498 + 0.110261i) q^{3} +(4.87287 - 8.44005i) q^{5} +(26.9757 - 1.14560i) q^{9} +O(q^{10})\) \(q+(-5.19498 + 0.110261i) q^{3} +(4.87287 - 8.44005i) q^{5} +(26.9757 - 1.14560i) q^{9} +(-42.9817 + 24.8155i) q^{11} -2.47499i q^{13} +(-24.3838 + 44.3832i) q^{15} +(-24.7602 - 42.8858i) q^{17} +(96.0631 + 55.4620i) q^{19} +(-149.767 - 86.4683i) q^{23} +(15.0103 + 25.9987i) q^{25} +(-140.012 + 8.92575i) q^{27} +134.538i q^{29} +(2.18514 - 1.26159i) q^{31} +(220.553 - 133.655i) q^{33} +(58.5515 - 101.414i) q^{37} +(0.272894 + 12.8575i) q^{39} +160.696 q^{41} -442.678 q^{43} +(121.780 - 233.259i) q^{45} +(155.814 - 269.879i) q^{47} +(133.357 + 220.061i) q^{51} +(248.907 - 143.706i) q^{53} +483.690i q^{55} +(-505.161 - 277.532i) q^{57} +(276.360 + 478.670i) q^{59} +(504.937 + 291.525i) q^{61} +(-20.8890 - 12.0603i) q^{65} +(450.723 + 780.675i) q^{67} +(787.573 + 432.688i) q^{69} +984.717i q^{71} +(178.030 - 102.786i) q^{73} +(-80.8451 - 133.408i) q^{75} +(-321.643 + 557.103i) q^{79} +(726.375 - 61.8069i) q^{81} +351.902 q^{83} -482.612 q^{85} +(-14.8343 - 698.924i) q^{87} +(-544.376 + 942.887i) q^{89} +(-11.2126 + 6.79487i) q^{93} +(936.205 - 540.518i) q^{95} +1365.09i q^{97} +(-1131.03 + 718.654i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 64 q^{9} - 192 q^{15} - 456 q^{25} + 432 q^{37} - 688 q^{39} + 1248 q^{43} + 1536 q^{51} - 2720 q^{57} + 528 q^{67} - 3744 q^{79} - 3408 q^{81} + 13824 q^{85} + 5088 q^{93} - 15472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −5.19498 + 0.110261i −0.999775 + 0.0212197i
\(4\) 0 0
\(5\) 4.87287 8.44005i 0.435842 0.754901i −0.561522 0.827462i \(-0.689784\pi\)
0.997364 + 0.0725609i \(0.0231172\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 26.9757 1.14560i 0.999099 0.0424298i
\(10\) 0 0
\(11\) −42.9817 + 24.8155i −1.17813 + 0.680195i −0.955582 0.294726i \(-0.904772\pi\)
−0.222551 + 0.974921i \(0.571438\pi\)
\(12\) 0 0
\(13\) 2.47499i 0.0528029i −0.999651 0.0264015i \(-0.991595\pi\)
0.999651 0.0264015i \(-0.00840482\pi\)
\(14\) 0 0
\(15\) −24.3838 + 44.3832i −0.419726 + 0.763980i
\(16\) 0 0
\(17\) −24.7602 42.8858i −0.353248 0.611844i 0.633568 0.773687i \(-0.281590\pi\)
−0.986817 + 0.161843i \(0.948256\pi\)
\(18\) 0 0
\(19\) 96.0631 + 55.4620i 1.15991 + 0.669677i 0.951284 0.308317i \(-0.0997658\pi\)
0.208631 + 0.977994i \(0.433099\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −149.767 86.4683i −1.35777 0.783908i −0.368446 0.929649i \(-0.620110\pi\)
−0.989323 + 0.145741i \(0.953443\pi\)
\(24\) 0 0
\(25\) 15.0103 + 25.9987i 0.120083 + 0.207989i
\(26\) 0 0
\(27\) −140.012 + 8.92575i −0.997974 + 0.0636208i
\(28\) 0 0
\(29\) 134.538i 0.861488i 0.902474 + 0.430744i \(0.141749\pi\)
−0.902474 + 0.430744i \(0.858251\pi\)
\(30\) 0 0
\(31\) 2.18514 1.26159i 0.0126601 0.00730929i −0.493657 0.869657i \(-0.664340\pi\)
0.506317 + 0.862348i \(0.331007\pi\)
\(32\) 0 0
\(33\) 220.553 133.655i 1.16343 0.705042i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 58.5515 101.414i 0.260157 0.450605i −0.706127 0.708086i \(-0.749559\pi\)
0.966283 + 0.257481i \(0.0828924\pi\)
\(38\) 0 0
\(39\) 0.272894 + 12.8575i 0.00112046 + 0.0527910i
\(40\) 0 0
\(41\) 160.696 0.612111 0.306055 0.952014i \(-0.400991\pi\)
0.306055 + 0.952014i \(0.400991\pi\)
\(42\) 0 0
\(43\) −442.678 −1.56995 −0.784973 0.619530i \(-0.787323\pi\)
−0.784973 + 0.619530i \(0.787323\pi\)
\(44\) 0 0
\(45\) 121.780 233.259i 0.403420 0.772714i
\(46\) 0 0
\(47\) 155.814 269.879i 0.483572 0.837571i −0.516250 0.856438i \(-0.672673\pi\)
0.999822 + 0.0188668i \(0.00600584\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 133.357 + 220.061i 0.366152 + 0.604210i
\(52\) 0 0
\(53\) 248.907 143.706i 0.645093 0.372445i −0.141480 0.989941i \(-0.545186\pi\)
0.786574 + 0.617496i \(0.211853\pi\)
\(54\) 0 0
\(55\) 483.690i 1.18583i
\(56\) 0 0
\(57\) −505.161 277.532i −1.17386 0.644913i
\(58\) 0 0
\(59\) 276.360 + 478.670i 0.609814 + 1.05623i 0.991271 + 0.131842i \(0.0420892\pi\)
−0.381457 + 0.924387i \(0.624577\pi\)
\(60\) 0 0
\(61\) 504.937 + 291.525i 1.05984 + 0.611901i 0.925389 0.379018i \(-0.123738\pi\)
0.134455 + 0.990920i \(0.457072\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −20.8890 12.0603i −0.0398610 0.0230138i
\(66\) 0 0
\(67\) 450.723 + 780.675i 0.821859 + 1.42350i 0.904296 + 0.426906i \(0.140396\pi\)
−0.0824369 + 0.996596i \(0.526270\pi\)
\(68\) 0 0
\(69\) 787.573 + 432.688i 1.37410 + 0.754920i
\(70\) 0 0
\(71\) 984.717i 1.64598i 0.568058 + 0.822988i \(0.307695\pi\)
−0.568058 + 0.822988i \(0.692305\pi\)
\(72\) 0 0
\(73\) 178.030 102.786i 0.285436 0.164797i −0.350446 0.936583i \(-0.613970\pi\)
0.635882 + 0.771786i \(0.280637\pi\)
\(74\) 0 0
\(75\) −80.8451 133.408i −0.124469 0.205394i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −321.643 + 557.103i −0.458072 + 0.793405i −0.998859 0.0477551i \(-0.984793\pi\)
0.540787 + 0.841160i \(0.318127\pi\)
\(80\) 0 0
\(81\) 726.375 61.8069i 0.996399 0.0847832i
\(82\) 0 0
\(83\) 351.902 0.465376 0.232688 0.972551i \(-0.425248\pi\)
0.232688 + 0.972551i \(0.425248\pi\)
\(84\) 0 0
\(85\) −482.612 −0.615842
\(86\) 0 0
\(87\) −14.8343 698.924i −0.0182805 0.861294i
\(88\) 0 0
\(89\) −544.376 + 942.887i −0.648357 + 1.12299i 0.335159 + 0.942162i \(0.391210\pi\)
−0.983515 + 0.180825i \(0.942123\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −11.2126 + 6.79487i −0.0125021 + 0.00757629i
\(94\) 0 0
\(95\) 936.205 540.518i 1.01108 0.583747i
\(96\) 0 0
\(97\) 1365.09i 1.42891i 0.699683 + 0.714454i \(0.253325\pi\)
−0.699683 + 0.714454i \(0.746675\pi\)
\(98\) 0 0
\(99\) −1131.03 + 718.654i −1.14821 + 0.729571i
\(100\) 0 0
\(101\) −651.926 1129.17i −0.642268 1.11244i −0.984925 0.172980i \(-0.944660\pi\)
0.342658 0.939460i \(-0.388673\pi\)
\(102\) 0 0
\(103\) 1094.23 + 631.752i 1.04677 + 0.604354i 0.921744 0.387800i \(-0.126765\pi\)
0.125027 + 0.992153i \(0.460098\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1267.25 + 731.647i 1.14495 + 0.661037i 0.947652 0.319306i \(-0.103450\pi\)
0.197299 + 0.980343i \(0.436783\pi\)
\(108\) 0 0
\(109\) 1119.89 + 1939.71i 0.984093 + 1.70450i 0.645900 + 0.763422i \(0.276482\pi\)
0.338193 + 0.941077i \(0.390184\pi\)
\(110\) 0 0
\(111\) −292.992 + 533.300i −0.250536 + 0.456024i
\(112\) 0 0
\(113\) 281.355i 0.234227i −0.993119 0.117114i \(-0.962636\pi\)
0.993119 0.117114i \(-0.0373642\pi\)
\(114\) 0 0
\(115\) −1459.59 + 842.697i −1.18355 + 0.683321i
\(116\) 0 0
\(117\) −2.83536 66.7645i −0.00224042 0.0527554i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 566.115 980.541i 0.425331 0.736695i
\(122\) 0 0
\(123\) −834.814 + 17.7185i −0.611973 + 0.0129888i
\(124\) 0 0
\(125\) 1510.79 1.08103
\(126\) 0 0
\(127\) −576.095 −0.402521 −0.201261 0.979538i \(-0.564504\pi\)
−0.201261 + 0.979538i \(0.564504\pi\)
\(128\) 0 0
\(129\) 2299.70 48.8099i 1.56959 0.0333137i
\(130\) 0 0
\(131\) 1142.71 1979.23i 0.762130 1.32005i −0.179620 0.983736i \(-0.557487\pi\)
0.941750 0.336313i \(-0.109180\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −606.925 + 1225.20i −0.386932 + 0.781101i
\(136\) 0 0
\(137\) −892.802 + 515.459i −0.556768 + 0.321450i −0.751847 0.659337i \(-0.770837\pi\)
0.195079 + 0.980787i \(0.437504\pi\)
\(138\) 0 0
\(139\) 234.497i 0.143092i 0.997437 + 0.0715460i \(0.0227933\pi\)
−0.997437 + 0.0715460i \(0.977207\pi\)
\(140\) 0 0
\(141\) −779.697 + 1419.19i −0.465690 + 0.847644i
\(142\) 0 0
\(143\) 61.4180 + 106.379i 0.0359163 + 0.0622088i
\(144\) 0 0
\(145\) 1135.51 + 655.587i 0.650338 + 0.375473i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2015.55 1163.68i −1.10819 0.639814i −0.169830 0.985473i \(-0.554322\pi\)
−0.938360 + 0.345660i \(0.887655\pi\)
\(150\) 0 0
\(151\) −527.571 913.779i −0.284325 0.492466i 0.688120 0.725597i \(-0.258436\pi\)
−0.972445 + 0.233131i \(0.925103\pi\)
\(152\) 0 0
\(153\) −717.052 1128.51i −0.378891 0.596305i
\(154\) 0 0
\(155\) 24.5902i 0.0127428i
\(156\) 0 0
\(157\) −38.3884 + 22.1635i −0.0195142 + 0.0112665i −0.509725 0.860337i \(-0.670253\pi\)
0.490211 + 0.871604i \(0.336920\pi\)
\(158\) 0 0
\(159\) −1277.22 + 773.996i −0.637045 + 0.386050i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 126.057 218.337i 0.0605738 0.104917i −0.834148 0.551540i \(-0.814040\pi\)
0.894722 + 0.446623i \(0.147374\pi\)
\(164\) 0 0
\(165\) −53.3320 2512.76i −0.0251630 1.18556i
\(166\) 0 0
\(167\) −1657.35 −0.767962 −0.383981 0.923341i \(-0.625447\pi\)
−0.383981 + 0.923341i \(0.625447\pi\)
\(168\) 0 0
\(169\) 2190.87 0.997212
\(170\) 0 0
\(171\) 2654.90 + 1386.08i 1.18728 + 0.619859i
\(172\) 0 0
\(173\) 1663.55 2881.36i 0.731084 1.26627i −0.225336 0.974281i \(-0.572348\pi\)
0.956420 0.291994i \(-0.0943186\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1488.46 2456.21i −0.632090 1.04305i
\(178\) 0 0
\(179\) −372.577 + 215.108i −0.155574 + 0.0898206i −0.575766 0.817615i \(-0.695296\pi\)
0.420192 + 0.907435i \(0.361963\pi\)
\(180\) 0 0
\(181\) 2230.49i 0.915971i −0.888960 0.457986i \(-0.848571\pi\)
0.888960 0.457986i \(-0.151429\pi\)
\(182\) 0 0
\(183\) −2655.28 1458.79i −1.07259 0.589274i
\(184\) 0 0
\(185\) −570.627 988.355i −0.226775 0.392785i
\(186\) 0 0
\(187\) 2128.47 + 1228.87i 0.832346 + 0.480555i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1003.30 + 579.256i 0.380086 + 0.219442i 0.677855 0.735195i \(-0.262910\pi\)
−0.297770 + 0.954638i \(0.596243\pi\)
\(192\) 0 0
\(193\) 1127.84 + 1953.48i 0.420641 + 0.728572i 0.996002 0.0893276i \(-0.0284718\pi\)
−0.575361 + 0.817899i \(0.695138\pi\)
\(194\) 0 0
\(195\) 109.848 + 60.3497i 0.0403404 + 0.0221627i
\(196\) 0 0
\(197\) 3280.44i 1.18641i −0.805053 0.593203i \(-0.797863\pi\)
0.805053 0.593203i \(-0.202137\pi\)
\(198\) 0 0
\(199\) −3818.10 + 2204.38i −1.36009 + 0.785248i −0.989635 0.143603i \(-0.954131\pi\)
−0.370454 + 0.928851i \(0.620798\pi\)
\(200\) 0 0
\(201\) −2427.58 4005.90i −0.851880 1.40574i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 783.051 1356.28i 0.266784 0.462083i
\(206\) 0 0
\(207\) −4139.14 2160.97i −1.38981 0.725592i
\(208\) 0 0
\(209\) −5505.27 −1.82204
\(210\) 0 0
\(211\) 5620.46 1.83378 0.916892 0.399135i \(-0.130689\pi\)
0.916892 + 0.399135i \(0.130689\pi\)
\(212\) 0 0
\(213\) −108.576 5115.59i −0.0349271 1.64561i
\(214\) 0 0
\(215\) −2157.11 + 3736.22i −0.684249 + 1.18515i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −913.530 + 553.600i −0.281875 + 0.170816i
\(220\) 0 0
\(221\) −106.142 + 61.2811i −0.0323072 + 0.0186525i
\(222\) 0 0
\(223\) 1134.63i 0.340718i −0.985382 0.170359i \(-0.945507\pi\)
0.985382 0.170359i \(-0.0544928\pi\)
\(224\) 0 0
\(225\) 434.699 + 684.136i 0.128800 + 0.202707i
\(226\) 0 0
\(227\) 1061.60 + 1838.75i 0.310400 + 0.537629i 0.978449 0.206488i \(-0.0662035\pi\)
−0.668049 + 0.744118i \(0.732870\pi\)
\(228\) 0 0
\(229\) −2299.77 1327.77i −0.663638 0.383152i 0.130023 0.991511i \(-0.458495\pi\)
−0.793662 + 0.608359i \(0.791828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2734.14 + 1578.56i 0.768754 + 0.443840i 0.832430 0.554130i \(-0.186949\pi\)
−0.0636761 + 0.997971i \(0.520282\pi\)
\(234\) 0 0
\(235\) −1518.53 2630.16i −0.421522 0.730098i
\(236\) 0 0
\(237\) 1609.51 2929.60i 0.441133 0.802946i
\(238\) 0 0
\(239\) 2040.14i 0.552157i 0.961135 + 0.276079i \(0.0890351\pi\)
−0.961135 + 0.276079i \(0.910965\pi\)
\(240\) 0 0
\(241\) −3958.64 + 2285.52i −1.05809 + 0.610886i −0.924902 0.380205i \(-0.875853\pi\)
−0.133184 + 0.991091i \(0.542520\pi\)
\(242\) 0 0
\(243\) −3766.69 + 401.177i −0.994376 + 0.105907i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 137.268 237.755i 0.0353609 0.0612469i
\(248\) 0 0
\(249\) −1828.12 + 38.8009i −0.465272 + 0.00987513i
\(250\) 0 0
\(251\) 5848.44 1.47072 0.735359 0.677678i \(-0.237014\pi\)
0.735359 + 0.677678i \(0.237014\pi\)
\(252\) 0 0
\(253\) 8583.01 2.13284
\(254\) 0 0
\(255\) 2507.16 53.2131i 0.615704 0.0130680i
\(256\) 0 0
\(257\) −3884.44 + 6728.05i −0.942820 + 1.63301i −0.182762 + 0.983157i \(0.558504\pi\)
−0.760058 + 0.649855i \(0.774830\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 154.128 + 3629.26i 0.0365527 + 0.860712i
\(262\) 0 0
\(263\) −4077.64 + 2354.23i −0.956038 + 0.551969i −0.894951 0.446164i \(-0.852790\pi\)
−0.0610865 + 0.998132i \(0.519457\pi\)
\(264\) 0 0
\(265\) 2801.05i 0.649309i
\(266\) 0 0
\(267\) 2724.06 4958.30i 0.624381 1.13649i
\(268\) 0 0
\(269\) 2054.29 + 3558.14i 0.465622 + 0.806481i 0.999229 0.0392514i \(-0.0124973\pi\)
−0.533607 + 0.845732i \(0.679164\pi\)
\(270\) 0 0
\(271\) −700.809 404.612i −0.157089 0.0906953i 0.419395 0.907804i \(-0.362242\pi\)
−0.576484 + 0.817109i \(0.695576\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1290.34 744.978i −0.282947 0.163359i
\(276\) 0 0
\(277\) 3727.43 + 6456.10i 0.808518 + 1.40039i 0.913890 + 0.405962i \(0.133063\pi\)
−0.105372 + 0.994433i \(0.533603\pi\)
\(278\) 0 0
\(279\) 57.5003 36.5355i 0.0123385 0.00783988i
\(280\) 0 0
\(281\) 1289.89i 0.273837i −0.990582 0.136919i \(-0.956280\pi\)
0.990582 0.136919i \(-0.0437198\pi\)
\(282\) 0 0
\(283\) −7466.08 + 4310.54i −1.56824 + 0.905424i −0.571867 + 0.820346i \(0.693781\pi\)
−0.996374 + 0.0850780i \(0.972886\pi\)
\(284\) 0 0
\(285\) −4803.97 + 2911.21i −0.998466 + 0.605071i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1230.37 2131.06i 0.250431 0.433760i
\(290\) 0 0
\(291\) −150.516 7091.62i −0.0303209 1.42859i
\(292\) 0 0
\(293\) −6187.72 −1.23376 −0.616878 0.787059i \(-0.711603\pi\)
−0.616878 + 0.787059i \(0.711603\pi\)
\(294\) 0 0
\(295\) 5386.66 1.06313
\(296\) 0 0
\(297\) 5796.45 3858.10i 1.13247 0.753771i
\(298\) 0 0
\(299\) −214.008 + 370.673i −0.0413926 + 0.0716941i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3511.25 + 5794.13i 0.665729 + 1.09856i
\(304\) 0 0
\(305\) 4920.98 2841.13i 0.923850 0.533385i
\(306\) 0 0
\(307\) 3785.51i 0.703747i 0.936048 + 0.351873i \(0.114455\pi\)
−0.936048 + 0.351873i \(0.885545\pi\)
\(308\) 0 0
\(309\) −5754.15 3161.29i −1.05936 0.582005i
\(310\) 0 0
\(311\) −2998.76 5194.00i −0.546765 0.947025i −0.998494 0.0548698i \(-0.982526\pi\)
0.451728 0.892156i \(-0.350808\pi\)
\(312\) 0 0
\(313\) −4960.97 2864.22i −0.895881 0.517237i −0.0200193 0.999800i \(-0.506373\pi\)
−0.875861 + 0.482563i \(0.839706\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5818.71 3359.43i −1.03095 0.595219i −0.113693 0.993516i \(-0.536268\pi\)
−0.917257 + 0.398297i \(0.869602\pi\)
\(318\) 0 0
\(319\) −3338.63 5782.68i −0.585980 1.01495i
\(320\) 0 0
\(321\) −6664.01 3661.17i −1.15872 0.636593i
\(322\) 0 0
\(323\) 5493.00i 0.946249i
\(324\) 0 0
\(325\) 64.3464 37.1504i 0.0109825 0.00634072i
\(326\) 0 0
\(327\) −6031.69 9953.27i −1.02004 1.68323i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −845.184 + 1463.90i −0.140349 + 0.243092i −0.927628 0.373505i \(-0.878156\pi\)
0.787279 + 0.616597i \(0.211489\pi\)
\(332\) 0 0
\(333\) 1463.29 2802.79i 0.240803 0.461237i
\(334\) 0 0
\(335\) 8785.25 1.43280
\(336\) 0 0
\(337\) 4257.51 0.688194 0.344097 0.938934i \(-0.388185\pi\)
0.344097 + 0.938934i \(0.388185\pi\)
\(338\) 0 0
\(339\) 31.0224 + 1461.64i 0.00497023 + 0.234174i
\(340\) 0 0
\(341\) −62.6139 + 108.450i −0.00994349 + 0.0172226i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 7489.65 4538.73i 1.16878 0.708281i
\(346\) 0 0
\(347\) −1092.59 + 630.806i −0.169029 + 0.0975891i −0.582128 0.813097i \(-0.697780\pi\)
0.413099 + 0.910686i \(0.364446\pi\)
\(348\) 0 0
\(349\) 10981.9i 1.68438i −0.539179 0.842191i \(-0.681265\pi\)
0.539179 0.842191i \(-0.318735\pi\)
\(350\) 0 0
\(351\) 22.0911 + 346.528i 0.00335936 + 0.0526960i
\(352\) 0 0
\(353\) 3677.09 + 6368.90i 0.554424 + 0.960290i 0.997948 + 0.0640280i \(0.0203947\pi\)
−0.443524 + 0.896262i \(0.646272\pi\)
\(354\) 0 0
\(355\) 8311.06 + 4798.39i 1.24255 + 0.717387i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −432.312 249.595i −0.0635559 0.0366940i 0.467885 0.883789i \(-0.345016\pi\)
−0.531441 + 0.847095i \(0.678349\pi\)
\(360\) 0 0
\(361\) 2722.58 + 4715.64i 0.396935 + 0.687511i
\(362\) 0 0
\(363\) −2832.84 + 5156.31i −0.409603 + 0.745554i
\(364\) 0 0
\(365\) 2003.44i 0.287302i
\(366\) 0 0
\(367\) −282.908 + 163.337i −0.0402390 + 0.0232320i −0.519985 0.854176i \(-0.674062\pi\)
0.479746 + 0.877408i \(0.340729\pi\)
\(368\) 0 0
\(369\) 4334.89 184.094i 0.611559 0.0259717i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 137.986 239.000i 0.0191546 0.0331767i −0.856289 0.516497i \(-0.827236\pi\)
0.875444 + 0.483320i \(0.160569\pi\)
\(374\) 0 0
\(375\) −7848.53 + 166.581i −1.08079 + 0.0229392i
\(376\) 0 0
\(377\) 332.981 0.0454891
\(378\) 0 0
\(379\) 508.854 0.0689659 0.0344829 0.999405i \(-0.489022\pi\)
0.0344829 + 0.999405i \(0.489022\pi\)
\(380\) 0 0
\(381\) 2992.80 63.5206i 0.402430 0.00854137i
\(382\) 0 0
\(383\) 2212.79 3832.66i 0.295217 0.511331i −0.679818 0.733381i \(-0.737941\pi\)
0.975035 + 0.222049i \(0.0712746\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11941.5 + 507.133i −1.56853 + 0.0666125i
\(388\) 0 0
\(389\) −1558.98 + 900.077i −0.203196 + 0.117315i −0.598146 0.801388i \(-0.704096\pi\)
0.394949 + 0.918703i \(0.370762\pi\)
\(390\) 0 0
\(391\) 8563.87i 1.10766i
\(392\) 0 0
\(393\) −5718.13 + 10408.1i −0.733948 + 1.33592i
\(394\) 0 0
\(395\) 3134.65 + 5429.38i 0.399295 + 0.691599i
\(396\) 0 0
\(397\) 5601.29 + 3233.91i 0.708113 + 0.408829i 0.810362 0.585930i \(-0.199271\pi\)
−0.102249 + 0.994759i \(0.532604\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2974.16 1717.13i −0.370381 0.213839i 0.303244 0.952913i \(-0.401930\pi\)
−0.673625 + 0.739074i \(0.735264\pi\)
\(402\) 0 0
\(403\) −3.12242 5.40819i −0.000385952 0.000668489i
\(404\) 0 0
\(405\) 3017.88 6431.82i 0.370270 0.789135i
\(406\) 0 0
\(407\) 5811.93i 0.707829i
\(408\) 0 0
\(409\) −817.228 + 471.827i −0.0988003 + 0.0570424i −0.548586 0.836094i \(-0.684834\pi\)
0.449786 + 0.893136i \(0.351500\pi\)
\(410\) 0 0
\(411\) 4581.26 2776.24i 0.549822 0.333192i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1714.77 2970.07i 0.202831 0.351313i
\(416\) 0 0
\(417\) −25.8558 1218.21i −0.00303637 0.143060i
\(418\) 0 0
\(419\) −10148.4 −1.18325 −0.591626 0.806213i \(-0.701514\pi\)
−0.591626 + 0.806213i \(0.701514\pi\)
\(420\) 0 0
\(421\) −6775.40 −0.784354 −0.392177 0.919890i \(-0.628278\pi\)
−0.392177 + 0.919890i \(0.628278\pi\)
\(422\) 0 0
\(423\) 3894.03 7458.66i 0.447598 0.857335i
\(424\) 0 0
\(425\) 743.317 1287.46i 0.0848381 0.146944i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −330.795 545.865i −0.0372283 0.0614327i
\(430\) 0 0
\(431\) −14439.3 + 8336.51i −1.61372 + 0.931684i −0.625227 + 0.780443i \(0.714993\pi\)
−0.988497 + 0.151241i \(0.951673\pi\)
\(432\) 0 0
\(433\) 2848.46i 0.316140i 0.987428 + 0.158070i \(0.0505271\pi\)
−0.987428 + 0.158070i \(0.949473\pi\)
\(434\) 0 0
\(435\) −5971.24 3280.56i −0.658159 0.361588i
\(436\) 0 0
\(437\) −9591.41 16612.8i −1.04993 1.81853i
\(438\) 0 0
\(439\) 8042.46 + 4643.32i 0.874364 + 0.504814i 0.868796 0.495170i \(-0.164894\pi\)
0.00556785 + 0.999984i \(0.498228\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4281.76 + 2472.08i 0.459216 + 0.265128i 0.711714 0.702469i \(-0.247919\pi\)
−0.252499 + 0.967597i \(0.581252\pi\)
\(444\) 0 0
\(445\) 5305.34 + 9189.12i 0.565163 + 0.978890i
\(446\) 0 0
\(447\) 10599.1 + 5823.05i 1.12152 + 0.616154i
\(448\) 0 0
\(449\) 15964.7i 1.67799i 0.544136 + 0.838997i \(0.316858\pi\)
−0.544136 + 0.838997i \(0.683142\pi\)
\(450\) 0 0
\(451\) −6906.99 + 3987.75i −0.721147 + 0.416355i
\(452\) 0 0
\(453\) 2841.47 + 4688.90i 0.294711 + 0.486321i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4133.41 7159.28i 0.423092 0.732816i −0.573148 0.819452i \(-0.694278\pi\)
0.996240 + 0.0866354i \(0.0276115\pi\)
\(458\) 0 0
\(459\) 3849.50 + 5783.53i 0.391459 + 0.588130i
\(460\) 0 0
\(461\) −3468.42 −0.350413 −0.175207 0.984532i \(-0.556059\pi\)
−0.175207 + 0.984532i \(0.556059\pi\)
\(462\) 0 0
\(463\) −9918.45 −0.995572 −0.497786 0.867300i \(-0.665853\pi\)
−0.497786 + 0.867300i \(0.665853\pi\)
\(464\) 0 0
\(465\) 2.71133 + 127.746i 0.000270398 + 0.0127399i
\(466\) 0 0
\(467\) 8430.55 14602.1i 0.835373 1.44691i −0.0583529 0.998296i \(-0.518585\pi\)
0.893726 0.448613i \(-0.148082\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 196.983 119.372i 0.0192707 0.0116781i
\(472\) 0 0
\(473\) 19027.0 10985.3i 1.84960 1.06787i
\(474\) 0 0
\(475\) 3330.02i 0.321667i
\(476\) 0 0
\(477\) 6549.80 4161.72i 0.628710 0.399481i
\(478\) 0 0
\(479\) −1578.16 2733.46i −0.150539 0.260741i 0.780887 0.624672i \(-0.214768\pi\)
−0.931426 + 0.363932i \(0.881434\pi\)
\(480\) 0 0
\(481\) −250.999 144.914i −0.0237932 0.0137370i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11521.4 + 6651.91i 1.07868 + 0.622778i
\(486\) 0 0
\(487\) −1214.17 2103.01i −0.112976 0.195681i 0.803993 0.594639i \(-0.202705\pi\)
−0.916969 + 0.398958i \(0.869372\pi\)
\(488\) 0 0
\(489\) −630.789 + 1148.15i −0.0583339 + 0.106179i
\(490\) 0 0
\(491\) 1304.86i 0.119933i −0.998200 0.0599667i \(-0.980901\pi\)
0.998200 0.0599667i \(-0.0190995\pi\)
\(492\) 0 0
\(493\) 5769.79 3331.19i 0.527096 0.304319i
\(494\) 0 0
\(495\) 554.117 + 13047.9i 0.0503146 + 1.18476i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1272.72 + 2204.42i −0.114178 + 0.197762i −0.917451 0.397849i \(-0.869757\pi\)
0.803273 + 0.595611i \(0.203090\pi\)
\(500\) 0 0
\(501\) 8609.91 182.741i 0.767789 0.0162959i
\(502\) 0 0
\(503\) −787.994 −0.0698507 −0.0349253 0.999390i \(-0.511119\pi\)
−0.0349253 + 0.999390i \(0.511119\pi\)
\(504\) 0 0
\(505\) −12707.0 −1.11971
\(506\) 0 0
\(507\) −11381.6 + 241.567i −0.996987 + 0.0211605i
\(508\) 0 0
\(509\) 1538.89 2665.44i 0.134008 0.232109i −0.791210 0.611544i \(-0.790549\pi\)
0.925218 + 0.379436i \(0.123882\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −13945.0 6907.91i −1.20017 0.594526i
\(514\) 0 0
\(515\) 10664.0 6156.89i 0.912454 0.526806i
\(516\) 0 0
\(517\) 15466.4i 1.31569i
\(518\) 0 0
\(519\) −8324.43 + 15152.0i −0.704050 + 1.28150i
\(520\) 0 0
\(521\) −663.114 1148.55i −0.0557611 0.0965811i 0.836797 0.547513i \(-0.184425\pi\)
−0.892559 + 0.450931i \(0.851092\pi\)
\(522\) 0 0
\(523\) 5799.32 + 3348.24i 0.484869 + 0.279940i 0.722444 0.691430i \(-0.243019\pi\)
−0.237574 + 0.971369i \(0.576352\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −108.209 62.4743i −0.00894429 0.00516399i
\(528\) 0 0
\(529\) 8870.03 + 15363.3i 0.729023 + 1.26271i
\(530\) 0 0
\(531\) 8003.37 + 12595.8i 0.654080 + 1.02940i
\(532\) 0 0
\(533\) 397.721i 0.0323212i
\(534\) 0 0
\(535\) 12350.3 7130.44i 0.998036 0.576216i
\(536\) 0 0
\(537\) 1911.81 1158.56i 0.153633 0.0931016i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9558.74 + 16556.2i −0.759634 + 1.31573i 0.183403 + 0.983038i \(0.441289\pi\)
−0.943037 + 0.332687i \(0.892045\pi\)
\(542\) 0 0
\(543\) 245.935 + 11587.3i 0.0194366 + 0.915765i
\(544\) 0 0
\(545\) 21828.3 1.71564
\(546\) 0 0
\(547\) −12400.5 −0.969297 −0.484649 0.874709i \(-0.661053\pi\)
−0.484649 + 0.874709i \(0.661053\pi\)
\(548\) 0 0
\(549\) 13955.0 + 7285.64i 1.08485 + 0.566381i
\(550\) 0 0
\(551\) −7461.77 + 12924.2i −0.576918 + 0.999252i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3073.37 + 5071.57i 0.235058 + 0.387885i
\(556\) 0 0
\(557\) −6062.22 + 3500.02i −0.461157 + 0.266249i −0.712531 0.701641i \(-0.752451\pi\)
0.251374 + 0.967890i \(0.419118\pi\)
\(558\) 0 0
\(559\) 1095.62i 0.0828978i
\(560\) 0 0
\(561\) −11192.8 6149.27i −0.842356 0.462785i
\(562\) 0 0
\(563\) 10239.9 + 17736.1i 0.766538 + 1.32768i 0.939430 + 0.342742i \(0.111356\pi\)
−0.172891 + 0.984941i \(0.555311\pi\)
\(564\) 0 0
\(565\) −2374.65 1371.01i −0.176818 0.102086i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8603.00 + 4966.94i 0.633843 + 0.365949i 0.782239 0.622979i \(-0.214078\pi\)
−0.148396 + 0.988928i \(0.547411\pi\)
\(570\) 0 0
\(571\) −2929.14 5073.42i −0.214677 0.371832i 0.738496 0.674258i \(-0.235537\pi\)
−0.953173 + 0.302427i \(0.902203\pi\)
\(572\) 0 0
\(573\) −5276.00 2898.60i −0.384656 0.211328i
\(574\) 0 0
\(575\) 5191.68i 0.376535i
\(576\) 0 0
\(577\) 15287.5 8826.26i 1.10300 0.636815i 0.165989 0.986128i \(-0.446918\pi\)
0.937006 + 0.349313i \(0.113585\pi\)
\(578\) 0 0
\(579\) −6074.50 10023.9i −0.436006 0.719482i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −7132.28 + 12353.5i −0.506670 + 0.877579i
\(584\) 0 0
\(585\) −577.312 301.404i −0.0408016 0.0213017i
\(586\) 0 0
\(587\) 25424.1 1.78768 0.893838 0.448390i \(-0.148002\pi\)
0.893838 + 0.448390i \(0.148002\pi\)
\(588\) 0 0
\(589\) 279.881 0.0195795
\(590\) 0 0
\(591\) 361.704 + 17041.8i 0.0251751 + 1.18614i
\(592\) 0 0
\(593\) 3832.38 6637.89i 0.265392 0.459672i −0.702275 0.711906i \(-0.747832\pi\)
0.967666 + 0.252235i \(0.0811654\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19591.9 11872.7i 1.34312 0.813932i
\(598\) 0 0
\(599\) 23961.5 13834.2i 1.63446 0.943655i 0.651764 0.758422i \(-0.274029\pi\)
0.982695 0.185233i \(-0.0593041\pi\)
\(600\) 0 0
\(601\) 11624.0i 0.788937i 0.918909 + 0.394469i \(0.129071\pi\)
−0.918909 + 0.394469i \(0.870929\pi\)
\(602\) 0 0
\(603\) 13052.9 + 20542.9i 0.881518 + 1.38735i
\(604\) 0 0
\(605\) −5517.21 9556.09i −0.370754 0.642166i
\(606\) 0 0
\(607\) −2675.97 1544.97i −0.178936 0.103309i 0.407857 0.913046i \(-0.366276\pi\)
−0.586793 + 0.809737i \(0.699610\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −667.946 385.639i −0.0442262 0.0255340i
\(612\) 0 0
\(613\) 1623.78 + 2812.47i 0.106989 + 0.185310i 0.914549 0.404475i \(-0.132546\pi\)
−0.807560 + 0.589785i \(0.799213\pi\)
\(614\) 0 0
\(615\) −3918.39 + 7132.21i −0.256918 + 0.467640i
\(616\) 0 0
\(617\) 10686.9i 0.697307i −0.937252 0.348654i \(-0.886639\pi\)
0.937252 0.348654i \(-0.113361\pi\)
\(618\) 0 0
\(619\) −4946.00 + 2855.58i −0.321158 + 0.185420i −0.651908 0.758298i \(-0.726031\pi\)
0.330751 + 0.943718i \(0.392698\pi\)
\(620\) 0 0
\(621\) 21741.0 + 10769.8i 1.40489 + 0.695938i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5485.59 9501.31i 0.351077 0.608084i
\(626\) 0 0
\(627\) 28599.8 607.014i 1.82163 0.0386632i
\(628\) 0 0
\(629\) −5798.97 −0.367600
\(630\) 0 0
\(631\) −13708.4 −0.864854 −0.432427 0.901669i \(-0.642343\pi\)
−0.432427 + 0.901669i \(0.642343\pi\)
\(632\) 0 0
\(633\) −29198.2 + 619.716i −1.83337 + 0.0389123i
\(634\) 0 0
\(635\) −2807.23 + 4862.27i −0.175436 + 0.303864i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1128.10 + 26563.4i 0.0698385 + 1.64449i
\(640\) 0 0
\(641\) −14083.3 + 8131.00i −0.867796 + 0.501022i −0.866615 0.498977i \(-0.833709\pi\)
−0.00118054 + 0.999999i \(0.500376\pi\)
\(642\) 0 0
\(643\) 13805.8i 0.846729i −0.905959 0.423365i \(-0.860849\pi\)
0.905959 0.423365i \(-0.139151\pi\)
\(644\) 0 0
\(645\) 10794.2 19647.4i 0.658947 1.19941i
\(646\) 0 0
\(647\) −9180.08 15900.4i −0.557815 0.966163i −0.997679 0.0680988i \(-0.978307\pi\)
0.439864 0.898064i \(-0.355027\pi\)
\(648\) 0 0
\(649\) −23756.8 13716.0i −1.43688 0.829585i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −16302.6 9412.30i −0.976983 0.564061i −0.0756247 0.997136i \(-0.524095\pi\)
−0.901358 + 0.433075i \(0.857428\pi\)
\(654\) 0 0
\(655\) −11136.6 19289.1i −0.664338 1.15067i
\(656\) 0 0
\(657\) 4684.73 2976.67i 0.278187 0.176759i
\(658\) 0 0
\(659\) 3472.55i 0.205268i −0.994719 0.102634i \(-0.967273\pi\)
0.994719 0.102634i \(-0.0327270\pi\)
\(660\) 0 0
\(661\) 11328.2 6540.36i 0.666592 0.384857i −0.128192 0.991749i \(-0.540917\pi\)
0.794784 + 0.606892i \(0.207584\pi\)
\(662\) 0 0
\(663\) 544.649 330.057i 0.0319041 0.0193339i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 11633.3 20149.5i 0.675327 1.16970i
\(668\) 0 0
\(669\) 125.105 + 5894.36i 0.00722993 + 0.340641i
\(670\) 0 0
\(671\) −28937.3 −1.66485
\(672\) 0 0
\(673\) −13701.5 −0.784775 −0.392388 0.919800i \(-0.628351\pi\)
−0.392388 + 0.919800i \(0.628351\pi\)
\(674\) 0 0
\(675\) −2333.68 3506.15i −0.133072 0.199928i
\(676\) 0 0
\(677\) 1975.54 3421.74i 0.112151 0.194251i −0.804486 0.593971i \(-0.797559\pi\)
0.916637 + 0.399720i \(0.130893\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −5717.74 9435.20i −0.321739 0.530922i
\(682\) 0 0
\(683\) −22216.3 + 12826.6i −1.24463 + 0.718589i −0.970034 0.242970i \(-0.921878\pi\)
−0.274598 + 0.961559i \(0.588545\pi\)
\(684\) 0 0
\(685\) 10047.1i 0.560406i
\(686\) 0 0
\(687\) 12093.7 + 6644.19i 0.671619 + 0.368983i
\(688\) 0 0
\(689\) −355.671 616.041i −0.0196662 0.0340628i
\(690\) 0 0
\(691\) 7142.53 + 4123.74i 0.393219 + 0.227025i 0.683554 0.729900i \(-0.260433\pi\)
−0.290335 + 0.956925i \(0.593767\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1979.17 + 1142.67i 0.108020 + 0.0623655i
\(696\) 0 0
\(697\) −3978.86 6891.59i −0.216227 0.374516i
\(698\) 0 0
\(699\) −14377.9 7899.11i −0.777999 0.427428i
\(700\) 0 0
\(701\) 25364.7i 1.36663i 0.730122 + 0.683317i \(0.239463\pi\)
−0.730122 + 0.683317i \(0.760537\pi\)
\(702\) 0 0
\(703\) 11249.3 6494.77i 0.603519 0.348442i
\(704\) 0 0
\(705\) 8178.72 + 13496.2i 0.436920 + 0.720989i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14695.0 + 25452.5i −0.778397 + 1.34822i 0.154469 + 0.987998i \(0.450633\pi\)
−0.932865 + 0.360225i \(0.882700\pi\)
\(710\) 0 0
\(711\) −8038.33 + 15396.7i −0.423996 + 0.812126i
\(712\) 0 0
\(713\) −436.350 −0.0229193
\(714\) 0 0
\(715\) 1197.13 0.0626154
\(716\) 0 0
\(717\) −224.947 10598.5i −0.0117166 0.552033i
\(718\) 0 0
\(719\) −1802.02 + 3121.19i −0.0934687 + 0.161893i −0.908968 0.416865i \(-0.863129\pi\)
0.815500 + 0.578757i \(0.196462\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20313.1 12309.7i 1.04488 0.633201i
\(724\) 0 0
\(725\) −3497.82 + 2019.47i −0.179180 + 0.103450i
\(726\) 0 0
\(727\) 35634.9i 1.81792i −0.416887 0.908958i \(-0.636879\pi\)
0.416887 0.908958i \(-0.363121\pi\)
\(728\) 0 0
\(729\) 19523.7 2499.42i 0.991905 0.126984i
\(730\) 0 0
\(731\) 10960.8 + 18984.6i 0.554581 + 0.960562i
\(732\) 0 0
\(733\) −10831.6 6253.60i −0.545802 0.315119i 0.201625 0.979463i \(-0.435378\pi\)
−0.747427 + 0.664344i \(0.768711\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −38745.6 22369.8i −1.93652 1.11805i
\(738\) 0 0
\(739\) −4972.17 8612.05i −0.247502 0.428687i 0.715330 0.698787i \(-0.246276\pi\)
−0.962832 + 0.270100i \(0.912943\pi\)
\(740\) 0 0
\(741\) −686.889 + 1250.27i −0.0340533 + 0.0619835i
\(742\) 0 0
\(743\) 12175.9i 0.601198i 0.953751 + 0.300599i \(0.0971866\pi\)
−0.953751 + 0.300599i \(0.902813\pi\)
\(744\) 0 0
\(745\) −19643.0 + 11340.9i −0.965992 + 0.557716i
\(746\) 0 0
\(747\) 9492.79 403.140i 0.464957 0.0197458i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7950.95 13771.4i 0.386331 0.669144i −0.605622 0.795752i \(-0.707076\pi\)
0.991953 + 0.126608i \(0.0404091\pi\)
\(752\) 0 0
\(753\) −30382.5 + 644.853i −1.47039 + 0.0312081i
\(754\) 0 0
\(755\) −10283.1 −0.495684
\(756\) 0 0
\(757\) 39209.6 1.88256 0.941279 0.337630i \(-0.109625\pi\)
0.941279 + 0.337630i \(0.109625\pi\)
\(758\) 0 0
\(759\) −44588.6 + 946.368i −2.13236 + 0.0452582i
\(760\) 0 0
\(761\) −5888.86 + 10199.8i −0.280514 + 0.485864i −0.971511 0.236993i \(-0.923838\pi\)
0.690998 + 0.722857i \(0.257171\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −13018.8 + 552.882i −0.615288 + 0.0261301i
\(766\) 0 0
\(767\) 1184.70 683.988i 0.0557720 0.0322000i
\(768\) 0 0
\(769\) 5932.67i 0.278202i −0.990278 0.139101i \(-0.955579\pi\)
0.990278 0.139101i \(-0.0444213\pi\)
\(770\) 0 0
\(771\) 19437.8 35380.4i 0.907956 1.65265i
\(772\) 0 0
\(773\) 12154.4 + 21052.1i 0.565543 + 0.979550i 0.996999 + 0.0774157i \(0.0246668\pi\)
−0.431456 + 0.902134i \(0.642000\pi\)
\(774\) 0 0
\(775\) 65.5993 + 37.8738i 0.00304051 + 0.00175544i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 15437.0 + 8912.54i 0.709996 + 0.409916i
\(780\) 0 0
\(781\) −24436.2 42324.8i −1.11959 1.93918i
\(782\) 0 0
\(783\) −1200.86 18837.0i −0.0548085 0.859742i
\(784\) 0 0
\(785\) 432.000i 0.0196417i
\(786\) 0 0
\(787\) 32642.3 18846.0i 1.47849 0.853606i 0.478785 0.877932i \(-0.341077\pi\)
0.999704 + 0.0243259i \(0.00774392\pi\)
\(788\) 0 0
\(789\) 20923.7 12679.8i 0.944110 0.572131i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 721.521 1249.71i 0.0323102 0.0559629i
\(794\) 0 0
\(795\) 308.845 + 14551.4i 0.0137781 + 0.649163i
\(796\) 0 0
\(797\) 5364.15 0.238404 0.119202 0.992870i \(-0.461966\pi\)
0.119202 + 0.992870i \(0.461966\pi\)
\(798\) 0 0
\(799\) −15432.0 −0.683284
\(800\) 0 0
\(801\) −13604.7 + 26058.7i −0.600125 + 1.14948i
\(802\) 0 0
\(803\) −5101.35 + 8835.80i −0.224188 + 0.388305i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11064.3 18257.9i −0.482630 0.796419i
\(808\) 0 0
\(809\) −10538.2 + 6084.21i −0.457975 + 0.264412i −0.711193 0.702997i \(-0.751845\pi\)
0.253217 + 0.967409i \(0.418511\pi\)
\(810\) 0 0
\(811\) 24850.5i 1.07598i −0.842952 0.537989i \(-0.819184\pi\)
0.842952 0.537989i \(-0.180816\pi\)
\(812\) 0 0
\(813\) 3685.30 + 2024.68i 0.158978 + 0.0873415i
\(814\) 0 0
\(815\) −1228.52 2127.85i −0.0528013 0.0914545i
\(816\) 0 0
\(817\) −42525.0 24551.8i −1.82100 1.05136i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33792.8 + 19510.3i 1.43651 + 0.829372i 0.997606 0.0691595i \(-0.0220317\pi\)
0.438909 + 0.898532i \(0.355365\pi\)
\(822\) 0 0
\(823\) −7694.67 13327.6i −0.325905 0.564483i 0.655790 0.754943i \(-0.272335\pi\)
−0.981695 + 0.190460i \(0.939002\pi\)
\(824\) 0 0
\(825\) 6785.43 + 3727.87i 0.286350 + 0.157319i
\(826\) 0 0
\(827\) 21256.9i 0.893805i −0.894583 0.446902i \(-0.852527\pi\)
0.894583 0.446902i \(-0.147473\pi\)
\(828\) 0 0
\(829\) 25539.6 14745.3i 1.07000 0.617763i 0.141817 0.989893i \(-0.454706\pi\)
0.928181 + 0.372130i \(0.121372\pi\)
\(830\) 0 0
\(831\) −20075.8 33128.3i −0.838052 1.38292i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −8076.05 + 13988.1i −0.334711 + 0.579736i
\(836\) 0 0
\(837\) −294.685 + 196.142i −0.0121694 + 0.00809993i
\(838\) 0 0
\(839\) −39336.9 −1.61867 −0.809333 0.587350i \(-0.800171\pi\)
−0.809333 + 0.587350i \(0.800171\pi\)
\(840\) 0 0
\(841\) 6288.44 0.257839
\(842\) 0 0
\(843\) 142.224 + 6700.94i 0.00581073 + 0.273775i
\(844\) 0 0
\(845\) 10675.8 18491.1i 0.434627 0.752796i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 38310.9 23216.4i 1.54868 0.938498i
\(850\) 0 0
\(851\) −17538.2 + 10125.7i −0.706465 + 0.407878i
\(852\) 0 0
\(853\) 12458.2i 0.500072i −0.968237 0.250036i \(-0.919558\pi\)
0.968237 0.250036i \(-0.0804424\pi\)
\(854\) 0 0
\(855\) 24635.6 15653.4i 0.985401 0.626122i
\(856\) 0 0
\(857\) 1310.45 + 2269.77i 0.0522337 + 0.0904714i 0.890960 0.454082i \(-0.150033\pi\)
−0.838726 + 0.544553i \(0.816699\pi\)
\(858\) 0 0
\(859\) −17583.3 10151.7i −0.698410 0.403227i 0.108345 0.994113i \(-0.465445\pi\)
−0.806755 + 0.590886i \(0.798778\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15233.4 8795.00i −0.600870 0.346912i 0.168514 0.985699i \(-0.446103\pi\)
−0.769384 + 0.638787i \(0.779437\pi\)
\(864\) 0 0
\(865\) −16212.5 28080.9i −0.637275 1.10379i
\(866\) 0 0
\(867\) −6156.77 + 11206.5i −0.241171 + 0.438976i
\(868\) 0 0
\(869\) 31926.9i 1.24631i
\(870\) 0 0
\(871\) 1932.16 1115.53i 0.0751651 0.0433966i
\(872\) 0 0
\(873\) 1563.85 + 36824.3i 0.0606282 + 1.42762i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9083.05 + 15732.3i −0.349729 + 0.605749i −0.986201 0.165551i \(-0.947060\pi\)
0.636472 + 0.771300i \(0.280393\pi\)
\(878\) 0 0
\(879\) 32145.1 682.262i 1.23348 0.0261799i
\(880\) 0 0
\(881\) 7990.91 0.305585 0.152793 0.988258i \(-0.451173\pi\)
0.152793 + 0.988258i \(0.451173\pi\)
\(882\) 0 0
\(883\) 2454.35 0.0935397 0.0467699 0.998906i \(-0.485107\pi\)
0.0467699 + 0.998906i \(0.485107\pi\)
\(884\) 0 0
\(885\) −27983.6 + 593.937i −1.06289 + 0.0225593i
\(886\) 0 0
\(887\) −11813.7 + 20461.9i −0.447197 + 0.774568i −0.998202 0.0599335i \(-0.980911\pi\)
0.551005 + 0.834502i \(0.314244\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −29687.0 + 20681.9i −1.11622 + 0.777632i
\(892\) 0 0
\(893\) 29936.0 17283.6i 1.12180 0.647674i
\(894\) 0 0
\(895\) 4192.76i 0.156591i
\(896\) 0 0
\(897\) 1070.90 1949.23i 0.0398620 0.0725563i
\(898\) 0 0
\(899\) 169.732 + 293.985i 0.00629687 + 0.0109065i
\(900\) 0 0
\(901\) −12325.9 7116.38i −0.455756 0.263131i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18825.4 10868.9i −0.691468 0.399219i
\(906\) 0 0
\(907\) 4609.53 + 7983.94i 0.168751 + 0.292285i 0.937981 0.346687i \(-0.112693\pi\)
−0.769230 + 0.638972i \(0.779360\pi\)
\(908\) 0 0
\(909\) −18879.7 29713.2i −0.688890 1.08419i
\(910\) 0 0
\(911\) 19404.0i 0.705688i −0.935682 0.352844i \(-0.885215\pi\)
0.935682 0.352844i \(-0.114785\pi\)
\(912\) 0 0
\(913\) −15125.3 + 8732.61i −0.548275 + 0.316547i
\(914\) 0 0
\(915\) −25251.1 + 15302.2i −0.912324 + 0.552869i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 13656.8 23654.2i 0.490201 0.849053i −0.509735 0.860331i \(-0.670257\pi\)
0.999936 + 0.0112781i \(0.00359002\pi\)
\(920\) 0 0
\(921\) −417.392 19665.6i −0.0149333 0.703588i
\(922\) 0 0
\(923\) 2437.16 0.0869124
\(924\) 0 0
\(925\) 3515.51 0.124961
\(926\) 0 0
\(927\) 30241.3 + 15788.4i 1.07147 + 0.559395i
\(928\) 0 0
\(929\) −8806.15 + 15252.7i −0.311002 + 0.538671i −0.978579 0.205869i \(-0.933998\pi\)
0.667578 + 0.744540i \(0.267331\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 16151.2 + 26652.1i 0.566738 + 0.935210i
\(934\) 0 0
\(935\) 20743.5 11976.2i 0.725544 0.418893i
\(936\) 0 0
\(937\) 18758.1i 0.654004i 0.945024 + 0.327002i \(0.106038\pi\)
−0.945024 + 0.327002i \(0.893962\pi\)
\(938\) 0 0
\(939\) 26088.0 + 14332.6i 0.906655 + 0.498110i
\(940\) 0 0
\(941\) 2123.19 + 3677.48i 0.0735538 + 0.127399i 0.900456 0.434946i \(-0.143233\pi\)
−0.826903 + 0.562345i \(0.809899\pi\)
\(942\) 0 0
\(943\) −24067.1 13895.1i −0.831104 0.479838i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −27111.0 15652.6i −0.930296 0.537106i −0.0433905 0.999058i \(-0.513816\pi\)
−0.886905 + 0.461952i \(0.847149\pi\)
\(948\) 0 0
\(949\) −254.393 440.622i −0.00870175 0.0150719i
\(950\) 0 0
\(951\) 30598.5 + 16810.6i 1.04335 + 0.573209i
\(952\) 0 0
\(953\) 33871.8i 1.15133i −0.817687 0.575663i \(-0.804744\pi\)
0.817687 0.575663i \(-0.195256\pi\)
\(954\) 0 0
\(955\) 9777.91 5645.28i 0.331315 0.191285i
\(956\) 0 0
\(957\) 17981.7 + 29672.8i 0.607385 + 1.00228i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14892.3 + 25794.2i −0.499893 + 0.865840i
\(962\) 0 0
\(963\) 35023.1 + 18284.9i 1.17197 + 0.611862i
\(964\) 0 0
\(965\) 21983.3 0.733333
\(966\) 0 0
\(967\) 34214.9 1.13782 0.568912 0.822398i \(-0.307364\pi\)
0.568912 + 0.822398i \(0.307364\pi\)
\(968\) 0 0
\(969\) 605.661 + 28536.0i 0.0200791 + 0.946036i
\(970\) 0 0
\(971\) −6266.70 + 10854.2i −0.207114 + 0.358732i −0.950804 0.309792i \(-0.899740\pi\)
0.743690 + 0.668525i \(0.233074\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −330.182 + 200.091i −0.0108454 + 0.00657234i
\(976\) 0 0
\(977\) −33448.2 + 19311.4i −1.09530 + 0.632369i −0.934981 0.354697i \(-0.884584\pi\)
−0.160314 + 0.987066i \(0.551251\pi\)
\(978\) 0 0
\(979\) 54035.8i 1.76404i
\(980\) 0 0
\(981\) 32432.0 + 51042.0i 1.05553 + 1.66121i
\(982\) 0 0
\(983\) −17134.5 29677.9i −0.555958 0.962947i −0.997828 0.0658685i \(-0.979018\pi\)
0.441870 0.897079i \(-0.354315\pi\)
\(984\) 0 0
\(985\) −27687.1 15985.2i −0.895619 0.517086i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 66298.7 + 38277.6i 2.13162 + 1.23069i
\(990\) 0 0
\(991\) −7547.62 13072.9i −0.241935 0.419044i 0.719330 0.694668i \(-0.244449\pi\)
−0.961266 + 0.275624i \(0.911116\pi\)
\(992\) 0 0
\(993\) 4229.31 7698.13i 0.135159 0.246015i
\(994\) 0 0
\(995\) 42966.6i 1.36898i
\(996\) 0 0
\(997\) 1996.82 1152.87i 0.0634303 0.0366215i −0.467949 0.883755i \(-0.655007\pi\)
0.531380 + 0.847134i \(0.321674\pi\)
\(998\) 0 0
\(999\) −7292.70 + 14721.8i −0.230962 + 0.466243i
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 588.4.k.e.509.1 48
3.2 odd 2 inner 588.4.k.e.509.9 48
7.2 even 3 588.4.f.d.293.17 yes 24
7.3 odd 6 inner 588.4.k.e.521.9 48
7.4 even 3 inner 588.4.k.e.521.16 48
7.5 odd 6 588.4.f.d.293.8 yes 24
7.6 odd 2 inner 588.4.k.e.509.24 48
21.2 odd 6 588.4.f.d.293.7 24
21.5 even 6 588.4.f.d.293.18 yes 24
21.11 odd 6 inner 588.4.k.e.521.24 48
21.17 even 6 inner 588.4.k.e.521.1 48
21.20 even 2 inner 588.4.k.e.509.16 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.f.d.293.7 24 21.2 odd 6
588.4.f.d.293.8 yes 24 7.5 odd 6
588.4.f.d.293.17 yes 24 7.2 even 3
588.4.f.d.293.18 yes 24 21.5 even 6
588.4.k.e.509.1 48 1.1 even 1 trivial
588.4.k.e.509.9 48 3.2 odd 2 inner
588.4.k.e.509.16 48 21.20 even 2 inner
588.4.k.e.509.24 48 7.6 odd 2 inner
588.4.k.e.521.1 48 21.17 even 6 inner
588.4.k.e.521.9 48 7.3 odd 6 inner
588.4.k.e.521.16 48 7.4 even 3 inner
588.4.k.e.521.24 48 21.11 odd 6 inner