Properties

Label 29.4.b.a.28.6
Level $29$
Weight $4$
Character 29.28
Analytic conductor $1.711$
Analytic rank $0$
Dimension $6$
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] = [29,4,Mod(28,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 29.b (of order \(2\), degree \(1\), 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: \(1.71105539017\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 38x^{4} + 301x^{2} + 560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 28.6
Root \(5.28644i\) of defining polynomial
Character \(\chi\) \(=\) 29.28
Dual form 29.4.b.a.28.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.28644i q^{2} +1.10381i q^{3} -19.9464 q^{4} +15.1112 q^{5} -5.83520 q^{6} -16.1112 q^{7} -63.1540i q^{8} +25.7816 q^{9} +O(q^{10})\) \(q+5.28644i q^{2} +1.10381i q^{3} -19.9464 q^{4} +15.1112 q^{5} -5.83520 q^{6} -16.1112 q^{7} -63.1540i q^{8} +25.7816 q^{9} +79.8845i q^{10} +43.9832i q^{11} -22.0170i q^{12} +28.0112 q^{13} -85.1709i q^{14} +16.6798i q^{15} +174.288 q^{16} -68.9056i q^{17} +136.293i q^{18} -96.2090i q^{19} -301.415 q^{20} -17.7837i q^{21} -232.515 q^{22} +29.2153 q^{23} +69.7097 q^{24} +103.349 q^{25} +148.079i q^{26} +58.2606i q^{27} +321.361 q^{28} +(-131.360 - 84.4602i) q^{29} -88.1769 q^{30} -204.543i q^{31} +416.132i q^{32} -48.5489 q^{33} +364.265 q^{34} -243.460 q^{35} -514.251 q^{36} +395.536i q^{37} +508.603 q^{38} +30.9189i q^{39} -954.333i q^{40} -255.250i q^{41} +94.0121 q^{42} +268.729i q^{43} -877.308i q^{44} +389.592 q^{45} +154.445i q^{46} +65.3058i q^{47} +192.380i q^{48} -83.4287 q^{49} +546.347i q^{50} +76.0584 q^{51} -558.723 q^{52} -124.644 q^{53} -307.991 q^{54} +664.640i q^{55} +1017.49i q^{56} +106.196 q^{57} +(446.494 - 694.427i) q^{58} -84.3946 q^{59} -332.703i q^{60} -110.352i q^{61} +1081.30 q^{62} -415.373 q^{63} -805.548 q^{64} +423.283 q^{65} -256.651i q^{66} -737.197 q^{67} +1374.42i q^{68} +32.2480i q^{69} -1287.04i q^{70} -20.2796 q^{71} -1628.21i q^{72} -139.307i q^{73} -2090.98 q^{74} +114.077i q^{75} +1919.02i q^{76} -708.624i q^{77} -163.451 q^{78} +77.1390i q^{79} +2633.71 q^{80} +631.795 q^{81} +1349.36 q^{82} -178.222 q^{83} +354.720i q^{84} -1041.25i q^{85} -1420.62 q^{86} +(93.2277 - 144.996i) q^{87} +2777.72 q^{88} -421.125i q^{89} +2059.55i q^{90} -451.294 q^{91} -582.740 q^{92} +225.776 q^{93} -345.235 q^{94} -1453.83i q^{95} -459.328 q^{96} +1171.11i q^{97} -441.040i q^{98} +1133.96i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{4} + 22 q^{5} - 12 q^{6} - 28 q^{7} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{4} + 22 q^{5} - 12 q^{6} - 28 q^{7} + 40 q^{9} + 30 q^{13} + 244 q^{16} - 548 q^{20} - 204 q^{22} - 76 q^{23} + 236 q^{24} + 24 q^{25} + 576 q^{28} - 54 q^{29} + 204 q^{30} + 166 q^{33} + 512 q^{34} - 796 q^{35} - 1184 q^{36} + 920 q^{38} - 192 q^{42} + 344 q^{45} - 1234 q^{49} + 1444 q^{51} - 1796 q^{52} + 990 q^{53} - 1204 q^{54} + 1028 q^{57} + 1600 q^{58} + 1260 q^{59} + 3076 q^{62} - 384 q^{63} - 3228 q^{64} - 1378 q^{65} - 664 q^{67} - 896 q^{71} - 3248 q^{74} - 3204 q^{78} + 6588 q^{80} - 1230 q^{81} + 4272 q^{82} - 2244 q^{83} - 780 q^{86} + 1772 q^{87} + 7932 q^{88} + 1348 q^{91} - 768 q^{92} - 3718 q^{93} - 3852 q^{94} + 1260 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.28644i 1.86904i 0.355914 + 0.934519i \(0.384170\pi\)
−0.355914 + 0.934519i \(0.615830\pi\)
\(3\) 1.10381i 0.212427i 0.994343 + 0.106214i \(0.0338728\pi\)
−0.994343 + 0.106214i \(0.966127\pi\)
\(4\) −19.9464 −2.49330
\(5\) 15.1112 1.35159 0.675794 0.737090i \(-0.263801\pi\)
0.675794 + 0.737090i \(0.263801\pi\)
\(6\) −5.83520 −0.397035
\(7\) −16.1112 −0.869924 −0.434962 0.900449i \(-0.643238\pi\)
−0.434962 + 0.900449i \(0.643238\pi\)
\(8\) 63.1540i 2.79104i
\(9\) 25.7816 0.954875
\(10\) 79.8845i 2.52617i
\(11\) 43.9832i 1.20559i 0.797898 + 0.602793i \(0.205946\pi\)
−0.797898 + 0.602793i \(0.794054\pi\)
\(12\) 22.0170i 0.529646i
\(13\) 28.0112 0.597608 0.298804 0.954314i \(-0.403412\pi\)
0.298804 + 0.954314i \(0.403412\pi\)
\(14\) 85.1709i 1.62592i
\(15\) 16.6798i 0.287114i
\(16\) 174.288 2.72325
\(17\) 68.9056i 0.983062i −0.870860 0.491531i \(-0.836437\pi\)
0.870860 0.491531i \(-0.163563\pi\)
\(18\) 136.293i 1.78470i
\(19\) 96.2090i 1.16168i −0.814019 0.580838i \(-0.802725\pi\)
0.814019 0.580838i \(-0.197275\pi\)
\(20\) −301.415 −3.36992
\(21\) 17.7837i 0.184796i
\(22\) −232.515 −2.25329
\(23\) 29.2153 0.264861 0.132431 0.991192i \(-0.457722\pi\)
0.132431 + 0.991192i \(0.457722\pi\)
\(24\) 69.7097 0.592893
\(25\) 103.349 0.826791
\(26\) 148.079i 1.11695i
\(27\) 58.2606i 0.415269i
\(28\) 321.361 2.16898
\(29\) −131.360 84.4602i −0.841136 0.540823i
\(30\) −88.1769 −0.536628
\(31\) 204.543i 1.18506i −0.805547 0.592532i \(-0.798128\pi\)
0.805547 0.592532i \(-0.201872\pi\)
\(32\) 416.132i 2.29882i
\(33\) −48.5489 −0.256100
\(34\) 364.265 1.83738
\(35\) −243.460 −1.17578
\(36\) −514.251 −2.38079
\(37\) 395.536i 1.75745i 0.477326 + 0.878726i \(0.341606\pi\)
−0.477326 + 0.878726i \(0.658394\pi\)
\(38\) 508.603 2.17122
\(39\) 30.9189i 0.126948i
\(40\) 954.333i 3.77233i
\(41\) 255.250i 0.972278i −0.873881 0.486139i \(-0.838405\pi\)
0.873881 0.486139i \(-0.161595\pi\)
\(42\) 94.0121 0.345390
\(43\) 268.729i 0.953042i 0.879163 + 0.476521i \(0.158102\pi\)
−0.879163 + 0.476521i \(0.841898\pi\)
\(44\) 877.308i 3.00589i
\(45\) 389.592 1.29060
\(46\) 154.445i 0.495036i
\(47\) 65.3058i 0.202677i 0.994852 + 0.101339i \(0.0323126\pi\)
−0.994852 + 0.101339i \(0.967687\pi\)
\(48\) 192.380i 0.578494i
\(49\) −83.4287 −0.243232
\(50\) 546.347i 1.54530i
\(51\) 76.0584 0.208829
\(52\) −558.723 −1.49002
\(53\) −124.644 −0.323040 −0.161520 0.986869i \(-0.551640\pi\)
−0.161520 + 0.986869i \(0.551640\pi\)
\(54\) −307.991 −0.776154
\(55\) 664.640i 1.62946i
\(56\) 1017.49i 2.42799i
\(57\) 106.196 0.246772
\(58\) 446.494 694.427i 1.01082 1.57212i
\(59\) −84.3946 −0.186224 −0.0931122 0.995656i \(-0.529682\pi\)
−0.0931122 + 0.995656i \(0.529682\pi\)
\(60\) 332.703i 0.715863i
\(61\) 110.352i 0.231626i −0.993271 0.115813i \(-0.963053\pi\)
0.993271 0.115813i \(-0.0369473\pi\)
\(62\) 1081.30 2.21493
\(63\) −415.373 −0.830668
\(64\) −805.548 −1.57334
\(65\) 423.283 0.807720
\(66\) 256.651i 0.478660i
\(67\) −737.197 −1.34422 −0.672112 0.740450i \(-0.734613\pi\)
−0.672112 + 0.740450i \(0.734613\pi\)
\(68\) 1374.42i 2.45107i
\(69\) 32.2480i 0.0562638i
\(70\) 1287.04i 2.19758i
\(71\) −20.2796 −0.0338978 −0.0169489 0.999856i \(-0.505395\pi\)
−0.0169489 + 0.999856i \(0.505395\pi\)
\(72\) 1628.21i 2.66509i
\(73\) 139.307i 0.223351i −0.993745 0.111676i \(-0.964378\pi\)
0.993745 0.111676i \(-0.0356218\pi\)
\(74\) −2090.98 −3.28474
\(75\) 114.077i 0.175633i
\(76\) 1919.02i 2.89641i
\(77\) 708.624i 1.04877i
\(78\) −163.451 −0.237271
\(79\) 77.1390i 0.109858i 0.998490 + 0.0549292i \(0.0174933\pi\)
−0.998490 + 0.0549292i \(0.982507\pi\)
\(80\) 2633.71 3.68072
\(81\) 631.795 0.866660
\(82\) 1349.36 1.81722
\(83\) −178.222 −0.235691 −0.117846 0.993032i \(-0.537599\pi\)
−0.117846 + 0.993032i \(0.537599\pi\)
\(84\) 354.720i 0.460752i
\(85\) 1041.25i 1.32870i
\(86\) −1420.62 −1.78127
\(87\) 93.2277 144.996i 0.114886 0.178680i
\(88\) 2777.72 3.36484
\(89\) 421.125i 0.501564i −0.968044 0.250782i \(-0.919312\pi\)
0.968044 0.250782i \(-0.0806877\pi\)
\(90\) 2059.55i 2.41218i
\(91\) −451.294 −0.519874
\(92\) −582.740 −0.660379
\(93\) 225.776 0.251740
\(94\) −345.235 −0.378812
\(95\) 1453.83i 1.57011i
\(96\) −459.328 −0.488333
\(97\) 1171.11i 1.22585i 0.790139 + 0.612927i \(0.210008\pi\)
−0.790139 + 0.612927i \(0.789992\pi\)
\(98\) 441.040i 0.454610i
\(99\) 1133.96i 1.15118i
\(100\) −2061.44 −2.06144
\(101\) 393.150i 0.387326i 0.981068 + 0.193663i \(0.0620368\pi\)
−0.981068 + 0.193663i \(0.937963\pi\)
\(102\) 402.078i 0.390310i
\(103\) −139.694 −0.133635 −0.0668176 0.997765i \(-0.521285\pi\)
−0.0668176 + 0.997765i \(0.521285\pi\)
\(104\) 1769.02i 1.66795i
\(105\) 268.733i 0.249768i
\(106\) 658.921i 0.603774i
\(107\) 990.473 0.894885 0.447442 0.894313i \(-0.352335\pi\)
0.447442 + 0.894313i \(0.352335\pi\)
\(108\) 1162.09i 1.03539i
\(109\) 1331.33 1.16990 0.584948 0.811071i \(-0.301115\pi\)
0.584948 + 0.811071i \(0.301115\pi\)
\(110\) −3513.58 −3.04551
\(111\) −436.595 −0.373331
\(112\) −2807.99 −2.36902
\(113\) 550.551i 0.458332i −0.973387 0.229166i \(-0.926400\pi\)
0.973387 0.229166i \(-0.0735998\pi\)
\(114\) 561.399i 0.461226i
\(115\) 441.479 0.357983
\(116\) 2620.16 + 1684.68i 2.09721 + 1.34844i
\(117\) 722.174 0.570641
\(118\) 446.147i 0.348060i
\(119\) 1110.15i 0.855190i
\(120\) 1053.40 0.801347
\(121\) −603.526 −0.453438
\(122\) 583.371 0.432917
\(123\) 281.747 0.206539
\(124\) 4079.90i 2.95472i
\(125\) −327.175 −0.234107
\(126\) 2195.84i 1.55255i
\(127\) 769.969i 0.537982i 0.963143 + 0.268991i \(0.0866902\pi\)
−0.963143 + 0.268991i \(0.913310\pi\)
\(128\) 929.427i 0.641801i
\(129\) −296.625 −0.202452
\(130\) 2237.66i 1.50966i
\(131\) 2325.57i 1.55104i −0.631326 0.775518i \(-0.717489\pi\)
0.631326 0.775518i \(-0.282511\pi\)
\(132\) 968.377 0.638534
\(133\) 1550.04i 1.01057i
\(134\) 3897.15i 2.51241i
\(135\) 880.389i 0.561273i
\(136\) −4351.66 −2.74376
\(137\) 1421.60i 0.886535i 0.896389 + 0.443267i \(0.146181\pi\)
−0.896389 + 0.443267i \(0.853819\pi\)
\(138\) −170.477 −0.105159
\(139\) 2368.97 1.44556 0.722782 0.691077i \(-0.242863\pi\)
0.722782 + 0.691077i \(0.242863\pi\)
\(140\) 4856.16 2.93157
\(141\) −72.0850 −0.0430542
\(142\) 107.207i 0.0633562i
\(143\) 1232.02i 0.720468i
\(144\) 4493.43 2.60036
\(145\) −1985.01 1276.30i −1.13687 0.730970i
\(146\) 736.437 0.417452
\(147\) 92.0890i 0.0516692i
\(148\) 7889.53i 4.38186i
\(149\) −675.196 −0.371236 −0.185618 0.982622i \(-0.559429\pi\)
−0.185618 + 0.982622i \(0.559429\pi\)
\(150\) −603.061 −0.328265
\(151\) −1634.01 −0.880621 −0.440310 0.897846i \(-0.645132\pi\)
−0.440310 + 0.897846i \(0.645132\pi\)
\(152\) −6075.98 −3.24228
\(153\) 1776.50i 0.938701i
\(154\) 3746.09 1.96019
\(155\) 3090.89i 1.60172i
\(156\) 616.721i 0.316521i
\(157\) 2359.77i 1.19956i 0.800166 + 0.599778i \(0.204745\pi\)
−0.800166 + 0.599778i \(0.795255\pi\)
\(158\) −407.791 −0.205330
\(159\) 137.582i 0.0686226i
\(160\) 6288.26i 3.10706i
\(161\) −470.694 −0.230409
\(162\) 3339.95i 1.61982i
\(163\) 223.466i 0.107382i 0.998558 + 0.0536909i \(0.0170986\pi\)
−0.998558 + 0.0536909i \(0.982901\pi\)
\(164\) 5091.33i 2.42418i
\(165\) −733.634 −0.346141
\(166\) 942.158i 0.440516i
\(167\) −3387.40 −1.56961 −0.784806 0.619742i \(-0.787237\pi\)
−0.784806 + 0.619742i \(0.787237\pi\)
\(168\) −1123.11 −0.515772
\(169\) −1412.37 −0.642865
\(170\) 5504.49 2.48338
\(171\) 2480.42i 1.10926i
\(172\) 5360.18i 2.37622i
\(173\) 1619.76 0.711839 0.355920 0.934517i \(-0.384168\pi\)
0.355920 + 0.934517i \(0.384168\pi\)
\(174\) 766.512 + 492.842i 0.333961 + 0.214726i
\(175\) −1665.08 −0.719245
\(176\) 7665.76i 3.28312i
\(177\) 93.1552i 0.0395592i
\(178\) 2226.25 0.937442
\(179\) 3919.41 1.63659 0.818296 0.574797i \(-0.194919\pi\)
0.818296 + 0.574797i \(0.194919\pi\)
\(180\) −7770.96 −3.21785
\(181\) −1658.73 −0.681172 −0.340586 0.940213i \(-0.610625\pi\)
−0.340586 + 0.940213i \(0.610625\pi\)
\(182\) 2385.74i 0.971663i
\(183\) 121.807 0.0492037
\(184\) 1845.06i 0.739238i
\(185\) 5977.03i 2.37535i
\(186\) 1193.55i 0.470512i
\(187\) 3030.69 1.18517
\(188\) 1302.62i 0.505336i
\(189\) 938.650i 0.361252i
\(190\) 7685.61 2.93459
\(191\) 1913.06i 0.724734i 0.932035 + 0.362367i \(0.118031\pi\)
−0.932035 + 0.362367i \(0.881969\pi\)
\(192\) 889.169i 0.334220i
\(193\) 3220.07i 1.20096i −0.799639 0.600480i \(-0.794976\pi\)
0.799639 0.600480i \(-0.205024\pi\)
\(194\) −6190.98 −2.29117
\(195\) 467.222i 0.171582i
\(196\) 1664.10 0.606452
\(197\) −3615.40 −1.30755 −0.653773 0.756690i \(-0.726815\pi\)
−0.653773 + 0.756690i \(0.726815\pi\)
\(198\) −5994.60 −2.15161
\(199\) −3888.43 −1.38514 −0.692571 0.721349i \(-0.743522\pi\)
−0.692571 + 0.721349i \(0.743522\pi\)
\(200\) 6526.89i 2.30761i
\(201\) 813.723i 0.285550i
\(202\) −2078.36 −0.723927
\(203\) 2116.37 + 1360.76i 0.731725 + 0.470475i
\(204\) −1517.09 −0.520675
\(205\) 3857.14i 1.31412i
\(206\) 738.482i 0.249769i
\(207\) 753.217 0.252909
\(208\) 4882.02 1.62744
\(209\) 4231.58 1.40050
\(210\) 1420.64 0.466825
\(211\) 2986.97i 0.974556i −0.873247 0.487278i \(-0.837990\pi\)
0.873247 0.487278i \(-0.162010\pi\)
\(212\) 2486.20 0.805437
\(213\) 22.3847i 0.00720081i
\(214\) 5236.07i 1.67257i
\(215\) 4060.83i 1.28812i
\(216\) 3679.39 1.15903
\(217\) 3295.44i 1.03092i
\(218\) 7038.01i 2.18658i
\(219\) 153.768 0.0474459
\(220\) 13257.2i 4.06273i
\(221\) 1930.13i 0.587486i
\(222\) 2308.03i 0.697770i
\(223\) 844.880 0.253710 0.126855 0.991921i \(-0.459512\pi\)
0.126855 + 0.991921i \(0.459512\pi\)
\(224\) 6704.39i 1.99980i
\(225\) 2664.50 0.789482
\(226\) 2910.45 0.856639
\(227\) 6128.60 1.79194 0.895969 0.444117i \(-0.146483\pi\)
0.895969 + 0.444117i \(0.146483\pi\)
\(228\) −2118.23 −0.615277
\(229\) 6219.40i 1.79471i 0.441306 + 0.897357i \(0.354515\pi\)
−0.441306 + 0.897357i \(0.645485\pi\)
\(230\) 2333.85i 0.669085i
\(231\) 782.183 0.222787
\(232\) −5334.00 + 8295.91i −1.50946 + 2.34764i
\(233\) −3816.41 −1.07305 −0.536526 0.843884i \(-0.680264\pi\)
−0.536526 + 0.843884i \(0.680264\pi\)
\(234\) 3817.73i 1.06655i
\(235\) 986.851i 0.273936i
\(236\) 1683.37 0.464314
\(237\) −85.1465 −0.0233370
\(238\) −5868.75 −1.59838
\(239\) 584.691 0.158245 0.0791224 0.996865i \(-0.474788\pi\)
0.0791224 + 0.996865i \(0.474788\pi\)
\(240\) 2907.10i 0.781885i
\(241\) −1296.46 −0.346523 −0.173262 0.984876i \(-0.555431\pi\)
−0.173262 + 0.984876i \(0.555431\pi\)
\(242\) 3190.50i 0.847492i
\(243\) 2270.42i 0.599371i
\(244\) 2201.13i 0.577513i
\(245\) −1260.71 −0.328750
\(246\) 1489.44i 0.386028i
\(247\) 2694.93i 0.694227i
\(248\) −12917.7 −3.30756
\(249\) 196.722i 0.0500673i
\(250\) 1729.59i 0.437555i
\(251\) 6848.45i 1.72219i −0.508441 0.861097i \(-0.669778\pi\)
0.508441 0.861097i \(-0.330222\pi\)
\(252\) 8285.21 2.07111
\(253\) 1284.98i 0.319313i
\(254\) −4070.39 −1.00551
\(255\) 1149.33 0.282251
\(256\) −1531.03 −0.373786
\(257\) 4908.41 1.19136 0.595678 0.803224i \(-0.296884\pi\)
0.595678 + 0.803224i \(0.296884\pi\)
\(258\) 1568.09i 0.378391i
\(259\) 6372.57i 1.52885i
\(260\) −8442.98 −2.01389
\(261\) −3386.67 2177.52i −0.803180 0.516418i
\(262\) 12294.0 2.89894
\(263\) 5483.04i 1.28555i −0.766057 0.642773i \(-0.777784\pi\)
0.766057 0.642773i \(-0.222216\pi\)
\(264\) 3066.06i 0.714784i
\(265\) −1883.52 −0.436617
\(266\) −8194.21 −1.88879
\(267\) 464.840 0.106546
\(268\) 14704.4 3.35156
\(269\) 3321.99i 0.752957i 0.926425 + 0.376479i \(0.122865\pi\)
−0.926425 + 0.376479i \(0.877135\pi\)
\(270\) −4654.12 −1.04904
\(271\) 3672.45i 0.823192i 0.911366 + 0.411596i \(0.135029\pi\)
−0.911366 + 0.411596i \(0.864971\pi\)
\(272\) 12009.4i 2.67713i
\(273\) 498.141i 0.110435i
\(274\) −7515.18 −1.65697
\(275\) 4545.62i 0.996768i
\(276\) 643.232i 0.140283i
\(277\) −3272.17 −0.709767 −0.354884 0.934910i \(-0.615480\pi\)
−0.354884 + 0.934910i \(0.615480\pi\)
\(278\) 12523.4i 2.70181i
\(279\) 5273.45i 1.13159i
\(280\) 15375.5i 3.28164i
\(281\) 374.989 0.0796083 0.0398042 0.999208i \(-0.487327\pi\)
0.0398042 + 0.999208i \(0.487327\pi\)
\(282\) 381.073i 0.0804700i
\(283\) 3491.52 0.733390 0.366695 0.930341i \(-0.380489\pi\)
0.366695 + 0.930341i \(0.380489\pi\)
\(284\) 404.504 0.0845173
\(285\) 1604.75 0.333534
\(286\) −6513.01 −1.34658
\(287\) 4112.39i 0.845808i
\(288\) 10728.5i 2.19509i
\(289\) 165.020 0.0335884
\(290\) 6747.06 10493.6i 1.36621 2.12485i
\(291\) −1292.67 −0.260405
\(292\) 2778.67i 0.556882i
\(293\) 4208.27i 0.839078i −0.907737 0.419539i \(-0.862192\pi\)
0.907737 0.419539i \(-0.137808\pi\)
\(294\) 486.823 0.0965717
\(295\) −1275.31 −0.251699
\(296\) 24979.7 4.90511
\(297\) −2562.49 −0.500643
\(298\) 3569.38i 0.693855i
\(299\) 818.355 0.158283
\(300\) 2275.43i 0.437906i
\(301\) 4329.55i 0.829074i
\(302\) 8638.08i 1.64591i
\(303\) −433.962 −0.0822787
\(304\) 16768.1i 3.16354i
\(305\) 1667.56i 0.313063i
\(306\) 9391.34 1.75447
\(307\) 4418.40i 0.821406i 0.911769 + 0.410703i \(0.134717\pi\)
−0.911769 + 0.410703i \(0.865283\pi\)
\(308\) 14134.5i 2.61490i
\(309\) 154.195i 0.0283878i
\(310\) 16339.8 2.99367
\(311\) 4608.82i 0.840329i 0.907448 + 0.420165i \(0.138028\pi\)
−0.907448 + 0.420165i \(0.861972\pi\)
\(312\) 1952.65 0.354318
\(313\) −4361.87 −0.787692 −0.393846 0.919176i \(-0.628856\pi\)
−0.393846 + 0.919176i \(0.628856\pi\)
\(314\) −12474.8 −2.24202
\(315\) −6276.79 −1.12272
\(316\) 1538.65i 0.273910i
\(317\) 1780.78i 0.315516i −0.987478 0.157758i \(-0.949573\pi\)
0.987478 0.157758i \(-0.0504265\pi\)
\(318\) 727.321 0.128258
\(319\) 3714.83 5777.64i 0.652009 1.01406i
\(320\) −12172.8 −2.12650
\(321\) 1093.29i 0.190098i
\(322\) 2488.29i 0.430643i
\(323\) −6629.34 −1.14200
\(324\) −12602.0 −2.16085
\(325\) 2894.93 0.494097
\(326\) −1181.34 −0.200701
\(327\) 1469.53i 0.248518i
\(328\) −16120.1 −2.71367
\(329\) 1052.16i 0.176314i
\(330\) 3878.31i 0.646951i
\(331\) 2653.52i 0.440636i 0.975428 + 0.220318i \(0.0707095\pi\)
−0.975428 + 0.220318i \(0.929290\pi\)
\(332\) 3554.89 0.587650
\(333\) 10197.6i 1.67815i
\(334\) 17907.3i 2.93366i
\(335\) −11140.0 −1.81684
\(336\) 3099.48i 0.503245i
\(337\) 7284.33i 1.17746i 0.808331 + 0.588728i \(0.200371\pi\)
−0.808331 + 0.588728i \(0.799629\pi\)
\(338\) 7466.42i 1.20154i
\(339\) 607.701 0.0973622
\(340\) 20769.2i 3.31284i
\(341\) 8996.46 1.42870
\(342\) 13112.6 2.07324
\(343\) 6870.29 1.08152
\(344\) 16971.3 2.65998
\(345\) 487.307i 0.0760455i
\(346\) 8562.77i 1.33045i
\(347\) −5506.92 −0.851951 −0.425976 0.904735i \(-0.640069\pi\)
−0.425976 + 0.904735i \(0.640069\pi\)
\(348\) −1859.56 + 2892.15i −0.286445 + 0.445504i
\(349\) 1176.85 0.180502 0.0902511 0.995919i \(-0.471233\pi\)
0.0902511 + 0.995919i \(0.471233\pi\)
\(350\) 8802.32i 1.34430i
\(351\) 1631.95i 0.248168i
\(352\) −18302.8 −2.77143
\(353\) 8886.53 1.33989 0.669947 0.742409i \(-0.266317\pi\)
0.669947 + 0.742409i \(0.266317\pi\)
\(354\) 492.459 0.0739376
\(355\) −306.449 −0.0458158
\(356\) 8399.94i 1.25055i
\(357\) −1225.39 −0.181666
\(358\) 20719.7i 3.05885i
\(359\) 10030.8i 1.47466i 0.675532 + 0.737331i \(0.263914\pi\)
−0.675532 + 0.737331i \(0.736086\pi\)
\(360\) 24604.3i 3.60211i
\(361\) −2397.17 −0.349493
\(362\) 8768.75i 1.27314i
\(363\) 666.175i 0.0963227i
\(364\) 9001.70 1.29620
\(365\) 2105.10i 0.301879i
\(366\) 643.928i 0.0919635i
\(367\) 3757.93i 0.534502i 0.963627 + 0.267251i \(0.0861153\pi\)
−0.963627 + 0.267251i \(0.913885\pi\)
\(368\) 5091.88 0.721284
\(369\) 6580.77i 0.928404i
\(370\) −31597.2 −4.43962
\(371\) 2008.16 0.281020
\(372\) −4503.41 −0.627664
\(373\) 10917.6 1.51553 0.757765 0.652528i \(-0.226292\pi\)
0.757765 + 0.652528i \(0.226292\pi\)
\(374\) 16021.6i 2.21512i
\(375\) 361.137i 0.0497308i
\(376\) 4124.32 0.565680
\(377\) −3679.55 2365.83i −0.502670 0.323200i
\(378\) 4962.11 0.675195
\(379\) 4846.58i 0.656866i −0.944527 0.328433i \(-0.893480\pi\)
0.944527 0.328433i \(-0.106520\pi\)
\(380\) 28998.8i 3.91475i
\(381\) −849.896 −0.114282
\(382\) −10113.3 −1.35456
\(383\) −4309.20 −0.574908 −0.287454 0.957794i \(-0.592809\pi\)
−0.287454 + 0.957794i \(0.592809\pi\)
\(384\) 1025.91 0.136336
\(385\) 10708.2i 1.41750i
\(386\) 17022.7 2.24464
\(387\) 6928.27i 0.910036i
\(388\) 23359.4i 3.05642i
\(389\) 393.410i 0.0512769i 0.999671 + 0.0256384i \(0.00816186\pi\)
−0.999671 + 0.0256384i \(0.991838\pi\)
\(390\) −2469.94 −0.320693
\(391\) 2013.10i 0.260375i
\(392\) 5268.85i 0.678870i
\(393\) 2566.97 0.329483
\(394\) 19112.6i 2.44385i
\(395\) 1165.66i 0.148483i
\(396\) 22618.4i 2.87025i
\(397\) −11339.5 −1.43353 −0.716766 0.697314i \(-0.754378\pi\)
−0.716766 + 0.697314i \(0.754378\pi\)
\(398\) 20555.9i 2.58888i
\(399\) −1710.95 −0.214673
\(400\) 18012.5 2.25156
\(401\) 3865.08 0.481329 0.240664 0.970608i \(-0.422635\pi\)
0.240664 + 0.970608i \(0.422635\pi\)
\(402\) 4301.69 0.533704
\(403\) 5729.49i 0.708204i
\(404\) 7841.94i 0.965721i
\(405\) 9547.19 1.17137
\(406\) −7193.56 + 11188.1i −0.879336 + 1.36762i
\(407\) −17397.0 −2.11876
\(408\) 4803.39i 0.582851i
\(409\) 13673.2i 1.65305i 0.562899 + 0.826526i \(0.309686\pi\)
−0.562899 + 0.826526i \(0.690314\pi\)
\(410\) 20390.5 2.45614
\(411\) −1569.17 −0.188324
\(412\) 2786.39 0.333193
\(413\) 1359.70 0.162001
\(414\) 3981.84i 0.472697i
\(415\) −2693.15 −0.318558
\(416\) 11656.3i 1.37380i
\(417\) 2614.88i 0.307077i
\(418\) 22370.0i 2.61759i
\(419\) 16494.7 1.92319 0.961597 0.274465i \(-0.0885007\pi\)
0.961597 + 0.274465i \(0.0885007\pi\)
\(420\) 5360.25i 0.622746i
\(421\) 14117.6i 1.63432i −0.576412 0.817159i \(-0.695548\pi\)
0.576412 0.817159i \(-0.304452\pi\)
\(422\) 15790.4 1.82148
\(423\) 1683.69i 0.193531i
\(424\) 7871.75i 0.901618i
\(425\) 7121.32i 0.812787i
\(426\) 118.335 0.0134586
\(427\) 1777.91i 0.201497i
\(428\) −19756.4 −2.23122
\(429\) −1359.91 −0.153047
\(430\) −21467.3 −2.40755
\(431\) −9592.11 −1.07201 −0.536004 0.844215i \(-0.680067\pi\)
−0.536004 + 0.844215i \(0.680067\pi\)
\(432\) 10154.1i 1.13088i
\(433\) 4582.98i 0.508646i 0.967119 + 0.254323i \(0.0818527\pi\)
−0.967119 + 0.254323i \(0.918147\pi\)
\(434\) −17421.1 −1.92682
\(435\) 1408.78 2191.07i 0.155278 0.241502i
\(436\) −26555.3 −2.91690
\(437\) 2810.77i 0.307683i
\(438\) 812.883i 0.0886782i
\(439\) −3068.67 −0.333621 −0.166810 0.985989i \(-0.553347\pi\)
−0.166810 + 0.985989i \(0.553347\pi\)
\(440\) 41974.7 4.54787
\(441\) −2150.93 −0.232256
\(442\) 10203.5 1.09803
\(443\) 1166.32i 0.125087i 0.998042 + 0.0625437i \(0.0199213\pi\)
−0.998042 + 0.0625437i \(0.980079\pi\)
\(444\) 8708.50 0.930827
\(445\) 6363.72i 0.677908i
\(446\) 4466.41i 0.474194i
\(447\) 745.285i 0.0788608i
\(448\) 12978.4 1.36868
\(449\) 17012.4i 1.78811i −0.447953 0.894057i \(-0.647847\pi\)
0.447953 0.894057i \(-0.352153\pi\)
\(450\) 14085.7i 1.47557i
\(451\) 11226.7 1.17217
\(452\) 10981.5i 1.14276i
\(453\) 1803.63i 0.187068i
\(454\) 32398.5i 3.34920i
\(455\) −6819.61 −0.702655
\(456\) 6706.70i 0.688750i
\(457\) 10704.5 1.09570 0.547851 0.836576i \(-0.315446\pi\)
0.547851 + 0.836576i \(0.315446\pi\)
\(458\) −32878.4 −3.35439
\(459\) 4014.48 0.408235
\(460\) −8805.92 −0.892561
\(461\) 4080.29i 0.412230i −0.978528 0.206115i \(-0.933918\pi\)
0.978528 0.206115i \(-0.0660822\pi\)
\(462\) 4134.96i 0.416398i
\(463\) 12486.5 1.25334 0.626671 0.779284i \(-0.284417\pi\)
0.626671 + 0.779284i \(0.284417\pi\)
\(464\) −22894.5 14720.4i −2.29063 1.47280i
\(465\) 3411.74 0.340249
\(466\) 20175.2i 2.00558i
\(467\) 11387.4i 1.12836i −0.825652 0.564180i \(-0.809192\pi\)
0.825652 0.564180i \(-0.190808\pi\)
\(468\) −14404.8 −1.42278
\(469\) 11877.1 1.16937
\(470\) −5216.92 −0.511997
\(471\) −2604.73 −0.254819
\(472\) 5329.85i 0.519759i
\(473\) −11819.6 −1.14897
\(474\) 450.122i 0.0436177i
\(475\) 9943.09i 0.960464i
\(476\) 22143.6i 2.13225i
\(477\) −3213.52 −0.308463
\(478\) 3090.93i 0.295766i
\(479\) 4606.66i 0.439423i 0.975565 + 0.219711i \(0.0705116\pi\)
−0.975565 + 0.219711i \(0.929488\pi\)
\(480\) −6941.01 −0.660026
\(481\) 11079.4i 1.05027i
\(482\) 6853.64i 0.647665i
\(483\) 519.555i 0.0489452i
\(484\) 12038.2 1.13056
\(485\) 17696.8i 1.65685i
\(486\) −12002.4 −1.12025
\(487\) 12954.0 1.20535 0.602673 0.797988i \(-0.294102\pi\)
0.602673 + 0.797988i \(0.294102\pi\)
\(488\) −6969.19 −0.646476
\(489\) −246.663 −0.0228109
\(490\) 6664.66i 0.614446i
\(491\) 15855.1i 1.45730i 0.684889 + 0.728648i \(0.259851\pi\)
−0.684889 + 0.728648i \(0.740149\pi\)
\(492\) −5619.84 −0.514963
\(493\) −5819.78 + 9051.44i −0.531663 + 0.826889i
\(494\) 14246.6 1.29754
\(495\) 17135.5i 1.55593i
\(496\) 35649.4i 3.22723i
\(497\) 326.728 0.0294885
\(498\) 1039.96 0.0935777
\(499\) −5174.21 −0.464188 −0.232094 0.972693i \(-0.574558\pi\)
−0.232094 + 0.972693i \(0.574558\pi\)
\(500\) 6525.96 0.583700
\(501\) 3739.03i 0.333429i
\(502\) 36203.9 3.21884
\(503\) 14427.3i 1.27889i −0.768835 0.639447i \(-0.779163\pi\)
0.768835 0.639447i \(-0.220837\pi\)
\(504\) 26232.5i 2.31843i
\(505\) 5940.98i 0.523505i
\(506\) −6792.98 −0.596808
\(507\) 1558.99i 0.136562i
\(508\) 15358.1i 1.34135i
\(509\) −7950.93 −0.692375 −0.346187 0.938165i \(-0.612524\pi\)
−0.346187 + 0.938165i \(0.612524\pi\)
\(510\) 6075.88i 0.527539i
\(511\) 2244.40i 0.194299i
\(512\) 15529.1i 1.34042i
\(513\) 5605.20 0.482408
\(514\) 25948.0i 2.22669i
\(515\) −2110.94 −0.180620
\(516\) 5916.60 0.504775
\(517\) −2872.36 −0.244345
\(518\) 33688.2 2.85748
\(519\) 1787.90i 0.151214i
\(520\) 26732.0i 2.25438i
\(521\) 1504.77 0.126536 0.0632680 0.997997i \(-0.479848\pi\)
0.0632680 + 0.997997i \(0.479848\pi\)
\(522\) 11511.3 17903.4i 0.965205 1.50117i
\(523\) 5742.09 0.480085 0.240042 0.970762i \(-0.422839\pi\)
0.240042 + 0.970762i \(0.422839\pi\)
\(524\) 46386.7i 3.86720i
\(525\) 1837.92i 0.152787i
\(526\) 28985.7 2.40273
\(527\) −14094.1 −1.16499
\(528\) −8461.51 −0.697424
\(529\) −11313.5 −0.929848
\(530\) 9957.10i 0.816055i
\(531\) −2175.83 −0.177821
\(532\) 30917.8i 2.51966i
\(533\) 7149.87i 0.581041i
\(534\) 2457.35i 0.199138i
\(535\) 14967.3 1.20952
\(536\) 46556.9i 3.75178i
\(537\) 4326.26i 0.347657i
\(538\) −17561.5 −1.40731
\(539\) 3669.46i 0.293237i
\(540\) 17560.6i 1.39942i
\(541\) 24646.5i 1.95866i 0.202270 + 0.979330i \(0.435168\pi\)
−0.202270 + 0.979330i \(0.564832\pi\)
\(542\) −19414.2 −1.53858
\(543\) 1830.91i 0.144700i
\(544\) 28673.8 2.25989
\(545\) 20118.1 1.58122
\(546\) 2633.39 0.206408
\(547\) −13689.9 −1.07009 −0.535045 0.844823i \(-0.679705\pi\)
−0.535045 + 0.844823i \(0.679705\pi\)
\(548\) 28355.8i 2.21040i
\(549\) 2845.06i 0.221173i
\(550\) −24030.1 −1.86300
\(551\) −8125.83 + 12638.0i −0.628262 + 0.977128i
\(552\) 2036.59 0.157034
\(553\) 1242.80i 0.0955685i
\(554\) 17298.1i 1.32658i
\(555\) −6597.48 −0.504590
\(556\) −47252.4 −3.60423
\(557\) 6626.46 0.504079 0.252040 0.967717i \(-0.418899\pi\)
0.252040 + 0.967717i \(0.418899\pi\)
\(558\) 27877.7 2.11498
\(559\) 7527.42i 0.569546i
\(560\) −42432.2 −3.20194
\(561\) 3345.29i 0.251762i
\(562\) 1982.35i 0.148791i
\(563\) 9393.71i 0.703193i −0.936152 0.351597i \(-0.885639\pi\)
0.936152 0.351597i \(-0.114361\pi\)
\(564\) 1437.84 0.107347
\(565\) 8319.49i 0.619476i
\(566\) 18457.7i 1.37073i
\(567\) −10179.0 −0.753928
\(568\) 1280.73i 0.0946099i
\(569\) 16343.1i 1.20411i −0.798454 0.602056i \(-0.794348\pi\)
0.798454 0.602056i \(-0.205652\pi\)
\(570\) 8483.41i 0.623388i
\(571\) 6948.21 0.509236 0.254618 0.967042i \(-0.418050\pi\)
0.254618 + 0.967042i \(0.418050\pi\)
\(572\) 24574.4i 1.79634i
\(573\) −2111.65 −0.153953
\(574\) −21739.9 −1.58085
\(575\) 3019.37 0.218985
\(576\) −20768.3 −1.50234
\(577\) 4998.50i 0.360642i −0.983608 0.180321i \(-0.942286\pi\)
0.983608 0.180321i \(-0.0577136\pi\)
\(578\) 872.367i 0.0627780i
\(579\) 3554.33 0.255117
\(580\) 39593.8 + 25457.5i 2.83456 + 1.82253i
\(581\) 2871.37 0.205034
\(582\) 6833.64i 0.486707i
\(583\) 5482.24i 0.389453i
\(584\) −8797.78 −0.623382
\(585\) 10912.9 0.771271
\(586\) 22246.8 1.56827
\(587\) 3270.44 0.229958 0.114979 0.993368i \(-0.463320\pi\)
0.114979 + 0.993368i \(0.463320\pi\)
\(588\) 1836.85i 0.128827i
\(589\) −19678.9 −1.37666
\(590\) 6741.82i 0.470434i
\(591\) 3990.70i 0.277759i
\(592\) 68937.3i 4.78599i
\(593\) −18552.1 −1.28473 −0.642363 0.766401i \(-0.722046\pi\)
−0.642363 + 0.766401i \(0.722046\pi\)
\(594\) 13546.5i 0.935720i
\(595\) 16775.8i 1.15586i
\(596\) 13467.7 0.925605
\(597\) 4292.07i 0.294242i
\(598\) 4326.18i 0.295837i
\(599\) 3093.91i 0.211041i −0.994417 0.105521i \(-0.966349\pi\)
0.994417 0.105521i \(-0.0336509\pi\)
\(600\) 7204.42 0.490199
\(601\) 18480.1i 1.25428i −0.778908 0.627138i \(-0.784226\pi\)
0.778908 0.627138i \(-0.215774\pi\)
\(602\) 22887.9 1.54957
\(603\) −19006.1 −1.28357
\(604\) 32592.6 2.19565
\(605\) −9120.01 −0.612861
\(606\) 2294.11i 0.153782i
\(607\) 25620.3i 1.71317i −0.516005 0.856586i \(-0.672581\pi\)
0.516005 0.856586i \(-0.327419\pi\)
\(608\) 40035.6 2.67049
\(609\) −1502.01 + 2336.06i −0.0999418 + 0.155438i
\(610\) 8815.44 0.585126
\(611\) 1829.29i 0.121122i
\(612\) 35434.8i 2.34047i
\(613\) −17878.4 −1.17798 −0.588990 0.808140i \(-0.700474\pi\)
−0.588990 + 0.808140i \(0.700474\pi\)
\(614\) −23357.6 −1.53524
\(615\) 4257.54 0.279155
\(616\) −44752.4 −2.92715
\(617\) 11940.2i 0.779086i 0.921008 + 0.389543i \(0.127367\pi\)
−0.921008 + 0.389543i \(0.872633\pi\)
\(618\) 815.140 0.0530578
\(619\) 1604.39i 0.104177i −0.998642 0.0520886i \(-0.983412\pi\)
0.998642 0.0520886i \(-0.0165878\pi\)
\(620\) 61652.2i 3.99357i
\(621\) 1702.10i 0.109989i
\(622\) −24364.3 −1.57061
\(623\) 6784.84i 0.436322i
\(624\) 5388.80i 0.345713i
\(625\) −17862.6 −1.14321
\(626\) 23058.8i 1.47223i
\(627\) 4670.85i 0.297505i
\(628\) 47069.0i 2.99086i
\(629\) 27254.6 1.72769
\(630\) 33181.9i 2.09841i
\(631\) 25956.1 1.63755 0.818775 0.574115i \(-0.194654\pi\)
0.818775 + 0.574115i \(0.194654\pi\)
\(632\) 4871.64 0.306619
\(633\) 3297.03 0.207022
\(634\) 9413.97 0.589711
\(635\) 11635.2i 0.727130i
\(636\) 2744.28i 0.171097i
\(637\) −2336.94 −0.145358
\(638\) 30543.1 + 19638.2i 1.89532 + 1.21863i
\(639\) −522.840 −0.0323681
\(640\) 14044.8i 0.867451i
\(641\) 9646.69i 0.594417i 0.954813 + 0.297209i \(0.0960557\pi\)
−0.954813 + 0.297209i \(0.903944\pi\)
\(642\) −5779.61 −0.355300
\(643\) 2551.86 0.156509 0.0782546 0.996933i \(-0.475065\pi\)
0.0782546 + 0.996933i \(0.475065\pi\)
\(644\) 9388.66 0.574480
\(645\) −4482.36 −0.273632
\(646\) 35045.6i 2.13444i
\(647\) 6430.59 0.390746 0.195373 0.980729i \(-0.437408\pi\)
0.195373 + 0.980729i \(0.437408\pi\)
\(648\) 39900.4i 2.41888i
\(649\) 3711.95i 0.224510i
\(650\) 15303.8i 0.923486i
\(651\) −3637.52 −0.218995
\(652\) 4457.35i 0.267735i
\(653\) 21430.7i 1.28430i 0.766580 + 0.642149i \(0.221957\pi\)
−0.766580 + 0.642149i \(0.778043\pi\)
\(654\) −7768.60 −0.464489
\(655\) 35142.1i 2.09636i
\(656\) 44487.1i 2.64776i
\(657\) 3591.56i 0.213272i
\(658\) 5562.16 0.329537
\(659\) 23348.6i 1.38017i −0.723729 0.690084i \(-0.757574\pi\)
0.723729 0.690084i \(-0.242426\pi\)
\(660\) 14633.4 0.863035
\(661\) 8442.89 0.496808 0.248404 0.968656i \(-0.420094\pi\)
0.248404 + 0.968656i \(0.420094\pi\)
\(662\) −14027.7 −0.823565
\(663\) 2130.49 0.124798
\(664\) 11255.4i 0.657823i
\(665\) 23423.1i 1.36588i
\(666\) −53908.7 −3.13652
\(667\) −3837.72 2467.53i −0.222784 0.143243i
\(668\) 67566.5 3.91351
\(669\) 932.584i 0.0538950i
\(670\) 58890.6i 3.39574i
\(671\) 4853.65 0.279245
\(672\) 7400.34 0.424813
\(673\) −17804.0 −1.01975 −0.509875 0.860248i \(-0.670308\pi\)
−0.509875 + 0.860248i \(0.670308\pi\)
\(674\) −38508.2 −2.20071
\(675\) 6021.17i 0.343341i
\(676\) 28171.8 1.60286
\(677\) 18135.2i 1.02953i 0.857330 + 0.514766i \(0.172121\pi\)
−0.857330 + 0.514766i \(0.827879\pi\)
\(678\) 3212.57i 0.181974i
\(679\) 18868.0i 1.06640i
\(680\) −65758.9 −3.70844
\(681\) 6764.79i 0.380657i
\(682\) 47559.2i 2.67029i
\(683\) 24553.7 1.37558 0.687789 0.725911i \(-0.258581\pi\)
0.687789 + 0.725911i \(0.258581\pi\)
\(684\) 49475.5i 2.76571i
\(685\) 21482.1i 1.19823i
\(686\) 36319.3i 2.02140i
\(687\) −6865.00 −0.381246
\(688\) 46836.3i 2.59537i
\(689\) −3491.42 −0.193052
\(690\) −2576.12 −0.142132
\(691\) 9633.82 0.530373 0.265187 0.964197i \(-0.414566\pi\)
0.265187 + 0.964197i \(0.414566\pi\)
\(692\) −32308.4 −1.77483
\(693\) 18269.5i 1.00144i
\(694\) 29112.0i 1.59233i
\(695\) 35798.0 1.95381
\(696\) −9157.07 5887.70i −0.498704 0.320650i
\(697\) −17588.2 −0.955810
\(698\) 6221.34i 0.337365i
\(699\) 4212.57i 0.227946i
\(700\) 33212.3 1.79330
\(701\) −3582.74 −0.193036 −0.0965180 0.995331i \(-0.530771\pi\)
−0.0965180 + 0.995331i \(0.530771\pi\)
\(702\) −8627.20 −0.463836
\(703\) 38054.1 2.04159
\(704\) 35430.6i 1.89679i
\(705\) −1089.29 −0.0581916
\(706\) 46978.1i 2.50431i
\(707\) 6334.13i 0.336944i
\(708\) 1858.11i 0.0986330i
\(709\) 16096.5 0.852631 0.426315 0.904575i \(-0.359811\pi\)
0.426315 + 0.904575i \(0.359811\pi\)
\(710\) 1620.02i 0.0856315i
\(711\) 1988.77i 0.104901i
\(712\) −26595.7 −1.39988
\(713\) 5975.78i 0.313878i
\(714\) 6477.96i 0.339540i
\(715\) 18617.4i 0.973776i
\(716\) −78178.1 −4.08052
\(717\) 645.385i 0.0336156i
\(718\) −53027.0 −2.75620
\(719\) 11121.1 0.576837 0.288419 0.957504i \(-0.406871\pi\)
0.288419 + 0.957504i \(0.406871\pi\)
\(720\) 67901.2 3.51462
\(721\) 2250.63 0.116252
\(722\) 12672.5i 0.653215i
\(723\) 1431.04i 0.0736111i
\(724\) 33085.7 1.69837
\(725\) −13575.9 8728.87i −0.695444 0.447148i
\(726\) 3521.69 0.180031
\(727\) 16605.7i 0.847139i 0.905864 + 0.423569i \(0.139223\pi\)
−0.905864 + 0.423569i \(0.860777\pi\)
\(728\) 28501.0i 1.45099i
\(729\) 14552.4 0.739337
\(730\) 11128.5 0.564223
\(731\) 18516.9 0.936900
\(732\) −2429.62 −0.122680
\(733\) 31570.9i 1.59086i 0.606047 + 0.795429i \(0.292754\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(734\) −19866.1 −0.999005
\(735\) 1391.58i 0.0698355i
\(736\) 12157.4i 0.608870i
\(737\) 32424.3i 1.62058i
\(738\) 34788.8 1.73522
\(739\) 18181.6i 0.905036i 0.891755 + 0.452518i \(0.149474\pi\)
−0.891755 + 0.452518i \(0.850526\pi\)
\(740\) 119220.i 5.92247i
\(741\) 2974.68 0.147473
\(742\) 10616.0i 0.525238i
\(743\) 5488.34i 0.270993i −0.990778 0.135496i \(-0.956737\pi\)
0.990778 0.135496i \(-0.0432629\pi\)
\(744\) 14258.6i 0.702616i
\(745\) −10203.0 −0.501759
\(746\) 57715.2i 2.83258i
\(747\) −4594.84 −0.225056
\(748\) −60451.4 −2.95498
\(749\) −15957.7 −0.778482
\(750\) 1909.13 0.0929487
\(751\) 3778.15i 0.183577i −0.995779 0.0917887i \(-0.970742\pi\)
0.995779 0.0917887i \(-0.0292584\pi\)
\(752\) 11382.0i 0.551942i
\(753\) 7559.36 0.365841
\(754\) 12506.8 19451.7i 0.604074 0.939509i
\(755\) −24691.9 −1.19024
\(756\) 18722.7i 0.900712i
\(757\) 34334.1i 1.64847i −0.566247 0.824236i \(-0.691605\pi\)
0.566247 0.824236i \(-0.308395\pi\)
\(758\) 25621.1 1.22771
\(759\) −1418.37 −0.0678309
\(760\) −91815.5 −4.38223
\(761\) −16919.9 −0.805974 −0.402987 0.915206i \(-0.632028\pi\)
−0.402987 + 0.915206i \(0.632028\pi\)
\(762\) 4492.92i 0.213598i
\(763\) −21449.4 −1.01772
\(764\) 38158.7i 1.80698i
\(765\) 26845.0i 1.26874i
\(766\) 22780.3i 1.07453i
\(767\) −2363.99 −0.111289
\(768\) 1689.96i 0.0794025i
\(769\) 15516.6i 0.727623i 0.931473 + 0.363811i \(0.118525\pi\)
−0.931473 + 0.363811i \(0.881475\pi\)
\(770\) 56608.0 2.64937
\(771\) 5417.93i 0.253077i
\(772\) 64228.8i 2.99436i
\(773\) 13036.6i 0.606588i 0.952897 + 0.303294i \(0.0980865\pi\)
−0.952897 + 0.303294i \(0.901914\pi\)
\(774\) −36625.9 −1.70089
\(775\) 21139.3i 0.979801i
\(776\) 73960.0 3.42141
\(777\) 7034.07 0.324770
\(778\) −2079.74 −0.0958384
\(779\) −24557.4 −1.12947
\(780\) 9319.41i 0.427806i
\(781\) 891.961i 0.0408667i
\(782\) 10642.1 0.486651
\(783\) 4920.71 7653.12i 0.224587 0.349298i
\(784\) −14540.6 −0.662383
\(785\) 35659.1i 1.62131i
\(786\) 13570.1i 0.615815i
\(787\) −14835.9 −0.671975 −0.335987 0.941867i \(-0.609070\pi\)
−0.335987 + 0.941867i \(0.609070\pi\)
\(788\) 72114.3 3.26011
\(789\) 6052.21 0.273085
\(790\) −6162.21 −0.277521
\(791\) 8870.05i 0.398714i
\(792\) 71614.0 3.21300
\(793\) 3091.10i 0.138421i
\(794\) 59945.4i 2.67932i
\(795\) 2079.04i 0.0927495i
\(796\) 77560.2 3.45358
\(797\) 38449.3i 1.70884i −0.519583 0.854420i \(-0.673913\pi\)
0.519583 0.854420i \(-0.326087\pi\)
\(798\) 9044.81i 0.401232i
\(799\) 4499.94 0.199245
\(800\) 43006.7i 1.90065i
\(801\) 10857.3i 0.478931i
\(802\) 20432.5i 0.899621i
\(803\) 6127.17 0.269269
\(804\) 16230.8i 0.711963i
\(805\) −7112.76 −0.311418
\(806\) 30288.6 1.32366
\(807\) −3666.83 −0.159949
\(808\) 24829.0 1.08104
\(809\) 2871.72i 0.124802i −0.998051 0.0624008i \(-0.980124\pi\)
0.998051 0.0624008i \(-0.0198757\pi\)
\(810\) 50470.6i 2.18933i
\(811\) −29135.9 −1.26153 −0.630764 0.775975i \(-0.717258\pi\)
−0.630764 + 0.775975i \(0.717258\pi\)
\(812\) −42214.0 27142.2i −1.82441 1.17304i
\(813\) −4053.67 −0.174869
\(814\) 91967.9i 3.96004i
\(815\) 3376.85i 0.145136i
\(816\) 13256.1 0.568695
\(817\) 25854.2 1.10713
\(818\) −72282.7 −3.08962
\(819\) −11635.1 −0.496414
\(820\) 76936.2i 3.27650i
\(821\) 46283.9 1.96750 0.983752 0.179534i \(-0.0574589\pi\)
0.983752 + 0.179534i \(0.0574589\pi\)
\(822\) 8295.30i 0.351985i
\(823\) 1912.88i 0.0810193i −0.999179 0.0405097i \(-0.987102\pi\)
0.999179 0.0405097i \(-0.0128982\pi\)
\(824\) 8822.21i 0.372981i
\(825\) −5017.48 −0.211741
\(826\) 7187.97i 0.302786i
\(827\) 15268.9i 0.642020i 0.947076 + 0.321010i \(0.104022\pi\)
−0.947076 + 0.321010i \(0.895978\pi\)
\(828\) −15024.0 −0.630579
\(829\) 18328.2i 0.767873i −0.923360 0.383936i \(-0.874568\pi\)
0.923360 0.383936i \(-0.125432\pi\)
\(830\) 14237.2i 0.595396i
\(831\) 3611.84i 0.150774i
\(832\) −22564.4 −0.940239
\(833\) 5748.70i 0.239113i
\(834\) −13823.4 −0.573939
\(835\) −51187.8 −2.12147
\(836\) −84404.9 −3.49187
\(837\) 11916.8 0.492120
\(838\) 87198.1i 3.59452i
\(839\) 19835.2i 0.816193i 0.912939 + 0.408096i \(0.133807\pi\)
−0.912939 + 0.408096i \(0.866193\pi\)
\(840\) −16971.5 −0.697111
\(841\) 10121.9 + 22189.4i 0.415021 + 0.909812i
\(842\) 74631.6 3.05460
\(843\) 413.914i 0.0169110i
\(844\) 59579.3i 2.42986i
\(845\) −21342.7 −0.868888
\(846\) −8900.72 −0.361718
\(847\) 9723.53 0.394456
\(848\) −21723.9 −0.879720
\(849\) 3853.96i 0.155792i
\(850\) 37646.4 1.51913
\(851\) 11555.7i 0.465481i
\(852\) 446.494i 0.0179538i
\(853\) 19348.3i 0.776639i −0.921525 0.388320i \(-0.873056\pi\)
0.921525 0.388320i \(-0.126944\pi\)
\(854\) −9398.81 −0.376605
\(855\) 37482.2i 1.49926i
\(856\) 62552.3i 2.49766i
\(857\) −5065.74 −0.201916 −0.100958 0.994891i \(-0.532191\pi\)
−0.100958 + 0.994891i \(0.532191\pi\)
\(858\) 7189.10i 0.286051i
\(859\) 15166.2i 0.602403i 0.953561 + 0.301201i \(0.0973876\pi\)
−0.953561 + 0.301201i \(0.902612\pi\)
\(860\) 80998.9i 3.21167i
\(861\) −4539.28 −0.179673
\(862\) 50708.1i 2.00362i
\(863\) 1090.76 0.0430241 0.0215120 0.999769i \(-0.493152\pi\)
0.0215120 + 0.999769i \(0.493152\pi\)
\(864\) −24244.1 −0.954630
\(865\) 24476.6 0.962114
\(866\) −24227.6 −0.950679
\(867\) 182.150i 0.00713510i
\(868\) 65732.1i 2.57038i
\(869\) −3392.83 −0.132444
\(870\) 11582.9 + 7447.45i 0.451377 + 0.290221i
\(871\) −20649.8 −0.803319
\(872\) 84079.0i 3.26522i
\(873\) 30193.0i 1.17054i
\(874\) 14859.0 0.575071
\(875\) 5271.18 0.203655
\(876\) −3067.12 −0.118297
\(877\) −29404.1 −1.13216 −0.566081 0.824350i \(-0.691541\pi\)
−0.566081 + 0.824350i \(0.691541\pi\)
\(878\) 16222.3i 0.623550i
\(879\) 4645.11 0.178243
\(880\) 115839.i 4.43742i
\(881\) 17945.9i 0.686279i −0.939284 0.343140i \(-0.888510\pi\)
0.939284 0.343140i \(-0.111490\pi\)
\(882\) 11370.7i 0.434096i
\(883\) −2077.31 −0.0791699 −0.0395849 0.999216i \(-0.512604\pi\)
−0.0395849 + 0.999216i \(0.512604\pi\)
\(884\) 38499.1i 1.46478i
\(885\) 1407.69i 0.0534677i
\(886\) −6165.70 −0.233793
\(887\) 19087.8i 0.722553i 0.932459 + 0.361277i \(0.117659\pi\)
−0.932459 + 0.361277i \(0.882341\pi\)
\(888\) 27572.7i 1.04198i
\(889\) 12405.1i 0.468004i
\(890\) 33641.4 1.26704
\(891\) 27788.4i 1.04483i
\(892\) −16852.3 −0.632576
\(893\) 6283.01 0.235446
\(894\) 3939.90 0.147394
\(895\) 59227.0 2.21200
\(896\) 14974.2i 0.558318i
\(897\) 903.305i 0.0336237i
\(898\) 89934.8 3.34205
\(899\) −17275.7 + 26868.8i −0.640910 + 0.996801i
\(900\) −53147.2 −1.96842
\(901\) 8588.65i 0.317569i
\(902\) 59349.4i 2.19082i
\(903\) 4778.99 0.176118
\(904\) −34769.5 −1.27922
\(905\) −25065.4 −0.920665
\(906\) 9534.76 0.349637
\(907\) 24351.0i 0.891468i −0.895166 0.445734i \(-0.852943\pi\)
0.895166 0.445734i \(-0.147057\pi\)
\(908\) −122244. −4.46784
\(909\) 10136.1i 0.369848i
\(910\) 36051.4i 1.31329i
\(911\) 5244.81i 0.190745i 0.995442 + 0.0953723i \(0.0304042\pi\)
−0.995442 + 0.0953723i \(0.969596\pi\)
\(912\) 18508.7 0.672023
\(913\) 7838.77i 0.284146i
\(914\) 56588.7i 2.04791i
\(915\) 1840.66 0.0665031
\(916\) 124055.i 4.47476i
\(917\) 37467.7i 1.34928i
\(918\) 21222.3i 0.763007i
\(919\) −44920.3 −1.61239 −0.806195 0.591651i \(-0.798477\pi\)
−0.806195 + 0.591651i \(0.798477\pi\)
\(920\) 27881.1i 0.999145i
\(921\) −4877.06 −0.174489
\(922\) 21570.2 0.770474
\(923\) −568.055 −0.0202576
\(924\) −15601.7 −0.555476
\(925\) 40878.2i 1.45305i
\(926\) 66009.1i 2.34254i
\(927\) −3601.53 −0.127605
\(928\) 35146.6 54663.1i 1.24326 1.93362i
\(929\) 23724.7 0.837870 0.418935 0.908016i \(-0.362404\pi\)
0.418935 + 0.908016i \(0.362404\pi\)
\(930\) 18036.0i 0.635938i
\(931\) 8026.59i 0.282557i
\(932\) 76123.7 2.67544
\(933\) −5087.24 −0.178509
\(934\) 60198.6 2.10895
\(935\) 45797.4 1.60186
\(936\) 45608.1i 1.59268i
\(937\) 11968.6 0.417288 0.208644 0.977992i \(-0.433095\pi\)
0.208644 + 0.977992i \(0.433095\pi\)
\(938\) 62787.8i 2.18560i
\(939\) 4814.66i 0.167327i
\(940\) 19684.1i 0.683006i
\(941\) 24728.4 0.856665 0.428333 0.903621i \(-0.359101\pi\)
0.428333 + 0.903621i \(0.359101\pi\)
\(942\) 13769.7i 0.476266i
\(943\) 7457.21i 0.257519i
\(944\) −14709.0 −0.507136
\(945\) 14184.1i 0.488265i
\(946\) 62483.5i 2.14748i
\(947\) 22561.4i 0.774177i 0.922043 + 0.387089i \(0.126519\pi\)
−0.922043 + 0.387089i \(0.873481\pi\)
\(948\) 1698.37 0.0581861
\(949\) 3902.15i 0.133477i
\(950\) 52563.5 1.79514
\(951\) 1965.63 0.0670242
\(952\) 70110.6 2.38687
\(953\) −2244.63 −0.0762968 −0.0381484 0.999272i \(-0.512146\pi\)
−0.0381484 + 0.999272i \(0.512146\pi\)
\(954\) 16988.1i 0.576529i
\(955\) 28908.7i 0.979542i
\(956\) −11662.5 −0.394552
\(957\) 6377.39 + 4100.46i 0.215415 + 0.138505i
\(958\) −24352.8 −0.821298
\(959\) 22903.7i 0.771218i
\(960\) 13436.4i 0.451728i
\(961\) −12046.8 −0.404377
\(962\) −58570.7 −1.96299
\(963\) 25536.0 0.854503
\(964\) 25859.7 0.863987
\(965\) 48659.1i 1.62320i
\(966\) 2746.59 0.0914805
\(967\) 12600.9i 0.419047i −0.977804 0.209524i \(-0.932809\pi\)
0.977804 0.209524i \(-0.0671913\pi\)
\(968\) 38115.0i 1.26556i
\(969\) 7317.50i 0.242592i
\(970\) −93553.3 −3.09672
\(971\) 51444.9i 1.70025i −0.526577 0.850127i \(-0.676525\pi\)
0.526577 0.850127i \(-0.323475\pi\)
\(972\) 45286.7i 1.49441i
\(973\) −38167.0 −1.25753
\(974\) 68480.7i 2.25284i
\(975\) 3195.43i 0.104960i
\(976\) 19233.1i 0.630775i
\(977\) 52349.5 1.71424 0.857118 0.515120i \(-0.172253\pi\)
0.857118 + 0.515120i \(0.172253\pi\)
\(978\) 1303.97i 0.0426344i
\(979\) 18522.5 0.604678
\(980\) 25146.6 0.819673
\(981\) 34323.9 1.11710
\(982\) −83817.1 −2.72374
\(983\) 43551.4i 1.41310i −0.707665 0.706548i \(-0.750251\pi\)
0.707665 0.706548i \(-0.249749\pi\)
\(984\) 17793.4i 0.576457i
\(985\) −54633.1 −1.76727
\(986\) −47849.9 30765.9i −1.54549 0.993698i
\(987\) 1161.38 0.0374539
\(988\) 53754.2i 1.73092i
\(989\) 7851.00i 0.252424i
\(990\) −90585.7 −2.90808
\(991\) −13980.1 −0.448126 −0.224063 0.974575i \(-0.571932\pi\)
−0.224063 + 0.974575i \(0.571932\pi\)
\(992\) 85116.8 2.72425
\(993\) −2928.97 −0.0936032
\(994\) 1727.23i 0.0551151i
\(995\) −58758.9 −1.87214
\(996\) 3923.90i 0.124833i
\(997\) 25485.2i 0.809553i 0.914416 + 0.404776i \(0.132651\pi\)
−0.914416 + 0.404776i \(0.867349\pi\)
\(998\) 27353.2i 0.867584i
\(999\) −23044.2 −0.729815
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 29.4.b.a.28.6 yes 6
3.2 odd 2 261.4.c.c.28.1 6
4.3 odd 2 464.4.e.a.289.3 6
29.12 odd 4 841.4.a.c.1.1 6
29.17 odd 4 841.4.a.c.1.6 6
29.28 even 2 inner 29.4.b.a.28.1 6
87.86 odd 2 261.4.c.c.28.6 6
116.115 odd 2 464.4.e.a.289.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.b.a.28.1 6 29.28 even 2 inner
29.4.b.a.28.6 yes 6 1.1 even 1 trivial
261.4.c.c.28.1 6 3.2 odd 2
261.4.c.c.28.6 6 87.86 odd 2
464.4.e.a.289.3 6 4.3 odd 2
464.4.e.a.289.4 6 116.115 odd 2
841.4.a.c.1.1 6 29.12 odd 4
841.4.a.c.1.6 6 29.17 odd 4