Properties

Label 667.2.a.d.1.7
Level $667$
Weight $2$
Character 667.1
Self dual yes
Analytic conductor $5.326$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [667,2,Mod(1,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.a (trivial)

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: yes
Analytic conductor: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + 1860 x^{8} - 5877 x^{7} - 2496 x^{6} + 6612 x^{5} + 1842 x^{4} - 3011 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.319955\) of defining polynomial
Character \(\chi\) \(=\) 667.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.319955 q^{2} +2.34279 q^{3} -1.89763 q^{4} +1.26502 q^{5} -0.749588 q^{6} +1.17463 q^{7} +1.24707 q^{8} +2.48866 q^{9} +O(q^{10})\) \(q-0.319955 q^{2} +2.34279 q^{3} -1.89763 q^{4} +1.26502 q^{5} -0.749588 q^{6} +1.17463 q^{7} +1.24707 q^{8} +2.48866 q^{9} -0.404751 q^{10} +1.82172 q^{11} -4.44574 q^{12} -2.29601 q^{13} -0.375829 q^{14} +2.96368 q^{15} +3.39625 q^{16} +1.44346 q^{17} -0.796260 q^{18} +6.82924 q^{19} -2.40055 q^{20} +2.75191 q^{21} -0.582869 q^{22} +1.00000 q^{23} +2.92162 q^{24} -3.39971 q^{25} +0.734622 q^{26} -1.19797 q^{27} -2.22901 q^{28} -1.00000 q^{29} -0.948247 q^{30} -0.0171270 q^{31} -3.58078 q^{32} +4.26790 q^{33} -0.461842 q^{34} +1.48593 q^{35} -4.72255 q^{36} +11.3129 q^{37} -2.18505 q^{38} -5.37908 q^{39} +1.57757 q^{40} -1.72398 q^{41} -0.880488 q^{42} +6.23644 q^{43} -3.45695 q^{44} +3.14821 q^{45} -0.319955 q^{46} +8.65063 q^{47} +7.95670 q^{48} -5.62025 q^{49} +1.08776 q^{50} +3.38172 q^{51} +4.35698 q^{52} -10.1287 q^{53} +0.383296 q^{54} +2.30452 q^{55} +1.46484 q^{56} +15.9995 q^{57} +0.319955 q^{58} -9.22383 q^{59} -5.62397 q^{60} -6.92452 q^{61} +0.00547988 q^{62} +2.92325 q^{63} -5.64681 q^{64} -2.90451 q^{65} -1.36554 q^{66} +1.95263 q^{67} -2.73915 q^{68} +2.34279 q^{69} -0.475433 q^{70} +14.6951 q^{71} +3.10352 q^{72} -8.10458 q^{73} -3.61961 q^{74} -7.96481 q^{75} -12.9594 q^{76} +2.13984 q^{77} +1.72106 q^{78} -11.4108 q^{79} +4.29634 q^{80} -10.2726 q^{81} +0.551596 q^{82} -5.50317 q^{83} -5.22210 q^{84} +1.82601 q^{85} -1.99538 q^{86} -2.34279 q^{87} +2.27181 q^{88} +10.2159 q^{89} -1.00729 q^{90} -2.69697 q^{91} -1.89763 q^{92} -0.0401250 q^{93} -2.76782 q^{94} +8.63915 q^{95} -8.38902 q^{96} +5.50080 q^{97} +1.79823 q^{98} +4.53364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} + 5 q^{3} + 21 q^{4} + 16 q^{5} + 6 q^{6} + q^{7} + 9 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} + 5 q^{3} + 21 q^{4} + 16 q^{5} + 6 q^{6} + q^{7} + 9 q^{8} + 23 q^{9} - 14 q^{10} + 4 q^{11} + 3 q^{12} + 15 q^{13} + 8 q^{14} + 8 q^{15} + 23 q^{16} + 20 q^{17} + 2 q^{18} - 4 q^{19} + 25 q^{20} + 5 q^{21} + 13 q^{22} + 16 q^{23} - 24 q^{24} + 30 q^{25} + 25 q^{26} + 8 q^{27} - 13 q^{28} - 16 q^{29} - 45 q^{30} - 19 q^{32} + q^{33} - 23 q^{34} + 5 q^{35} + 37 q^{36} + 5 q^{37} + 38 q^{38} - 10 q^{39} - 20 q^{40} + 7 q^{41} + 14 q^{42} - 17 q^{43} - 21 q^{44} + 48 q^{45} + 3 q^{46} + 29 q^{47} + 35 q^{48} + 31 q^{49} - 44 q^{50} - 14 q^{51} + 20 q^{52} + 63 q^{53} - 13 q^{54} + q^{55} - 19 q^{56} - 22 q^{57} - 3 q^{58} + 11 q^{59} - 3 q^{60} + 33 q^{62} - 33 q^{63} + 29 q^{64} + 53 q^{65} - 43 q^{66} - 13 q^{67} + 63 q^{68} + 5 q^{69} - 46 q^{70} - 23 q^{71} + 46 q^{72} - 38 q^{73} - 47 q^{74} + 37 q^{75} - 56 q^{76} + 97 q^{77} - 24 q^{78} - 27 q^{79} + 8 q^{80} + 40 q^{81} + 9 q^{82} + 36 q^{83} + 22 q^{84} + 6 q^{85} - 11 q^{86} - 5 q^{87} - 24 q^{88} - 16 q^{89} - 95 q^{90} - 47 q^{91} + 21 q^{92} + 62 q^{93} + 37 q^{94} - 12 q^{95} - 74 q^{96} - 30 q^{97} - 27 q^{98} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.319955 −0.226243 −0.113121 0.993581i \(-0.536085\pi\)
−0.113121 + 0.993581i \(0.536085\pi\)
\(3\) 2.34279 1.35261 0.676305 0.736622i \(-0.263580\pi\)
0.676305 + 0.736622i \(0.263580\pi\)
\(4\) −1.89763 −0.948814
\(5\) 1.26502 0.565736 0.282868 0.959159i \(-0.408714\pi\)
0.282868 + 0.959159i \(0.408714\pi\)
\(6\) −0.749588 −0.306018
\(7\) 1.17463 0.443968 0.221984 0.975050i \(-0.428747\pi\)
0.221984 + 0.975050i \(0.428747\pi\)
\(8\) 1.24707 0.440905
\(9\) 2.48866 0.829553
\(10\) −0.404751 −0.127994
\(11\) 1.82172 0.549269 0.274635 0.961549i \(-0.411443\pi\)
0.274635 + 0.961549i \(0.411443\pi\)
\(12\) −4.44574 −1.28338
\(13\) −2.29601 −0.636800 −0.318400 0.947956i \(-0.603145\pi\)
−0.318400 + 0.947956i \(0.603145\pi\)
\(14\) −0.375829 −0.100445
\(15\) 2.96368 0.765220
\(16\) 3.39625 0.849063
\(17\) 1.44346 0.350090 0.175045 0.984560i \(-0.443993\pi\)
0.175045 + 0.984560i \(0.443993\pi\)
\(18\) −0.796260 −0.187680
\(19\) 6.82924 1.56673 0.783367 0.621559i \(-0.213500\pi\)
0.783367 + 0.621559i \(0.213500\pi\)
\(20\) −2.40055 −0.536778
\(21\) 2.75191 0.600515
\(22\) −0.582869 −0.124268
\(23\) 1.00000 0.208514
\(24\) 2.92162 0.596372
\(25\) −3.39971 −0.679943
\(26\) 0.734622 0.144071
\(27\) −1.19797 −0.230549
\(28\) −2.22901 −0.421243
\(29\) −1.00000 −0.185695
\(30\) −0.948247 −0.173125
\(31\) −0.0171270 −0.00307610 −0.00153805 0.999999i \(-0.500490\pi\)
−0.00153805 + 0.999999i \(0.500490\pi\)
\(32\) −3.58078 −0.632999
\(33\) 4.26790 0.742947
\(34\) −0.461842 −0.0792053
\(35\) 1.48593 0.251169
\(36\) −4.72255 −0.787092
\(37\) 11.3129 1.85982 0.929911 0.367785i \(-0.119884\pi\)
0.929911 + 0.367785i \(0.119884\pi\)
\(38\) −2.18505 −0.354462
\(39\) −5.37908 −0.861341
\(40\) 1.57757 0.249436
\(41\) −1.72398 −0.269240 −0.134620 0.990897i \(-0.542981\pi\)
−0.134620 + 0.990897i \(0.542981\pi\)
\(42\) −0.880488 −0.135862
\(43\) 6.23644 0.951049 0.475524 0.879703i \(-0.342258\pi\)
0.475524 + 0.879703i \(0.342258\pi\)
\(44\) −3.45695 −0.521154
\(45\) 3.14821 0.469308
\(46\) −0.319955 −0.0471749
\(47\) 8.65063 1.26182 0.630912 0.775854i \(-0.282681\pi\)
0.630912 + 0.775854i \(0.282681\pi\)
\(48\) 7.95670 1.14845
\(49\) −5.62025 −0.802892
\(50\) 1.08776 0.153832
\(51\) 3.38172 0.473535
\(52\) 4.35698 0.604205
\(53\) −10.1287 −1.39129 −0.695643 0.718388i \(-0.744880\pi\)
−0.695643 + 0.718388i \(0.744880\pi\)
\(54\) 0.383296 0.0521599
\(55\) 2.30452 0.310741
\(56\) 1.46484 0.195748
\(57\) 15.9995 2.11918
\(58\) 0.319955 0.0420122
\(59\) −9.22383 −1.20084 −0.600420 0.799684i \(-0.705000\pi\)
−0.600420 + 0.799684i \(0.705000\pi\)
\(60\) −5.62397 −0.726052
\(61\) −6.92452 −0.886594 −0.443297 0.896375i \(-0.646191\pi\)
−0.443297 + 0.896375i \(0.646191\pi\)
\(62\) 0.00547988 0.000695946 0
\(63\) 2.92325 0.368295
\(64\) −5.64681 −0.705851
\(65\) −2.90451 −0.360261
\(66\) −1.36554 −0.168086
\(67\) 1.95263 0.238551 0.119276 0.992861i \(-0.461943\pi\)
0.119276 + 0.992861i \(0.461943\pi\)
\(68\) −2.73915 −0.332171
\(69\) 2.34279 0.282039
\(70\) −0.475433 −0.0568251
\(71\) 14.6951 1.74399 0.871997 0.489512i \(-0.162825\pi\)
0.871997 + 0.489512i \(0.162825\pi\)
\(72\) 3.10352 0.365754
\(73\) −8.10458 −0.948569 −0.474285 0.880372i \(-0.657293\pi\)
−0.474285 + 0.880372i \(0.657293\pi\)
\(74\) −3.61961 −0.420771
\(75\) −7.96481 −0.919697
\(76\) −12.9594 −1.48654
\(77\) 2.13984 0.243858
\(78\) 1.72106 0.194872
\(79\) −11.4108 −1.28382 −0.641909 0.766780i \(-0.721858\pi\)
−0.641909 + 0.766780i \(0.721858\pi\)
\(80\) 4.29634 0.480345
\(81\) −10.2726 −1.14139
\(82\) 0.551596 0.0609136
\(83\) −5.50317 −0.604052 −0.302026 0.953300i \(-0.597663\pi\)
−0.302026 + 0.953300i \(0.597663\pi\)
\(84\) −5.22210 −0.569778
\(85\) 1.82601 0.198059
\(86\) −1.99538 −0.215168
\(87\) −2.34279 −0.251173
\(88\) 2.27181 0.242175
\(89\) 10.2159 1.08288 0.541440 0.840739i \(-0.317879\pi\)
0.541440 + 0.840739i \(0.317879\pi\)
\(90\) −1.00729 −0.106177
\(91\) −2.69697 −0.282719
\(92\) −1.89763 −0.197841
\(93\) −0.0401250 −0.00416077
\(94\) −2.76782 −0.285478
\(95\) 8.63915 0.886358
\(96\) −8.38902 −0.856201
\(97\) 5.50080 0.558522 0.279261 0.960215i \(-0.409911\pi\)
0.279261 + 0.960215i \(0.409911\pi\)
\(98\) 1.79823 0.181648
\(99\) 4.53364 0.455648
\(100\) 6.45139 0.645139
\(101\) 11.6974 1.16393 0.581967 0.813212i \(-0.302283\pi\)
0.581967 + 0.813212i \(0.302283\pi\)
\(102\) −1.08200 −0.107134
\(103\) −1.00443 −0.0989694 −0.0494847 0.998775i \(-0.515758\pi\)
−0.0494847 + 0.998775i \(0.515758\pi\)
\(104\) −2.86328 −0.280768
\(105\) 3.48123 0.339733
\(106\) 3.24074 0.314768
\(107\) −18.3779 −1.77666 −0.888331 0.459203i \(-0.848135\pi\)
−0.888331 + 0.459203i \(0.848135\pi\)
\(108\) 2.27329 0.218748
\(109\) 4.25095 0.407167 0.203584 0.979058i \(-0.434741\pi\)
0.203584 + 0.979058i \(0.434741\pi\)
\(110\) −0.737343 −0.0703029
\(111\) 26.5036 2.51561
\(112\) 3.98934 0.376957
\(113\) −14.6737 −1.38038 −0.690191 0.723627i \(-0.742474\pi\)
−0.690191 + 0.723627i \(0.742474\pi\)
\(114\) −5.11911 −0.479449
\(115\) 1.26502 0.117964
\(116\) 1.89763 0.176190
\(117\) −5.71399 −0.528259
\(118\) 2.95122 0.271681
\(119\) 1.69553 0.155429
\(120\) 3.69591 0.337389
\(121\) −7.68134 −0.698304
\(122\) 2.21554 0.200585
\(123\) −4.03891 −0.364177
\(124\) 0.0325007 0.00291865
\(125\) −10.6258 −0.950404
\(126\) −0.935310 −0.0833240
\(127\) 7.03113 0.623912 0.311956 0.950097i \(-0.399016\pi\)
0.311956 + 0.950097i \(0.399016\pi\)
\(128\) 8.96830 0.792693
\(129\) 14.6107 1.28640
\(130\) 0.929315 0.0815063
\(131\) 6.42121 0.561023 0.280512 0.959851i \(-0.409496\pi\)
0.280512 + 0.959851i \(0.409496\pi\)
\(132\) −8.09889 −0.704918
\(133\) 8.02182 0.695580
\(134\) −0.624753 −0.0539704
\(135\) −1.51546 −0.130430
\(136\) 1.80009 0.154356
\(137\) −9.61591 −0.821542 −0.410771 0.911738i \(-0.634740\pi\)
−0.410771 + 0.911738i \(0.634740\pi\)
\(138\) −0.749588 −0.0638092
\(139\) −13.1215 −1.11295 −0.556476 0.830864i \(-0.687847\pi\)
−0.556476 + 0.830864i \(0.687847\pi\)
\(140\) −2.81975 −0.238312
\(141\) 20.2666 1.70676
\(142\) −4.70179 −0.394566
\(143\) −4.18269 −0.349774
\(144\) 8.45211 0.704342
\(145\) −1.26502 −0.105055
\(146\) 2.59310 0.214607
\(147\) −13.1670 −1.08600
\(148\) −21.4676 −1.76463
\(149\) 8.06760 0.660923 0.330462 0.943819i \(-0.392796\pi\)
0.330462 + 0.943819i \(0.392796\pi\)
\(150\) 2.54838 0.208075
\(151\) −19.8483 −1.61523 −0.807617 0.589708i \(-0.799243\pi\)
−0.807617 + 0.589708i \(0.799243\pi\)
\(152\) 8.51652 0.690781
\(153\) 3.59228 0.290418
\(154\) −0.684655 −0.0551711
\(155\) −0.0216661 −0.00174026
\(156\) 10.2075 0.817253
\(157\) −24.9057 −1.98769 −0.993844 0.110786i \(-0.964663\pi\)
−0.993844 + 0.110786i \(0.964663\pi\)
\(158\) 3.65096 0.290455
\(159\) −23.7294 −1.88187
\(160\) −4.52978 −0.358110
\(161\) 1.17463 0.0925737
\(162\) 3.28676 0.258232
\(163\) −13.6889 −1.07220 −0.536099 0.844155i \(-0.680103\pi\)
−0.536099 + 0.844155i \(0.680103\pi\)
\(164\) 3.27147 0.255459
\(165\) 5.39900 0.420312
\(166\) 1.76077 0.136662
\(167\) 7.67936 0.594246 0.297123 0.954839i \(-0.403973\pi\)
0.297123 + 0.954839i \(0.403973\pi\)
\(168\) 3.43181 0.264770
\(169\) −7.72832 −0.594486
\(170\) −0.584242 −0.0448093
\(171\) 16.9956 1.29969
\(172\) −11.8345 −0.902369
\(173\) −12.1140 −0.921012 −0.460506 0.887657i \(-0.652332\pi\)
−0.460506 + 0.887657i \(0.652332\pi\)
\(174\) 0.749588 0.0568261
\(175\) −3.99340 −0.301873
\(176\) 6.18702 0.466364
\(177\) −21.6095 −1.62427
\(178\) −3.26862 −0.244994
\(179\) −2.89201 −0.216159 −0.108079 0.994142i \(-0.534470\pi\)
−0.108079 + 0.994142i \(0.534470\pi\)
\(180\) −5.97414 −0.445286
\(181\) 6.58438 0.489413 0.244707 0.969597i \(-0.421308\pi\)
0.244707 + 0.969597i \(0.421308\pi\)
\(182\) 0.862909 0.0639630
\(183\) −16.2227 −1.19922
\(184\) 1.24707 0.0919350
\(185\) 14.3110 1.05217
\(186\) 0.0128382 0.000941343 0
\(187\) 2.62958 0.192294
\(188\) −16.4157 −1.19724
\(189\) −1.40716 −0.102356
\(190\) −2.76414 −0.200532
\(191\) 3.98219 0.288141 0.144070 0.989567i \(-0.453981\pi\)
0.144070 + 0.989567i \(0.453981\pi\)
\(192\) −13.2293 −0.954741
\(193\) 18.0090 1.29632 0.648160 0.761504i \(-0.275539\pi\)
0.648160 + 0.761504i \(0.275539\pi\)
\(194\) −1.76001 −0.126361
\(195\) −6.80466 −0.487292
\(196\) 10.6651 0.761796
\(197\) −0.368271 −0.0262382 −0.0131191 0.999914i \(-0.504176\pi\)
−0.0131191 + 0.999914i \(0.504176\pi\)
\(198\) −1.45056 −0.103087
\(199\) 13.2645 0.940296 0.470148 0.882588i \(-0.344201\pi\)
0.470148 + 0.882588i \(0.344201\pi\)
\(200\) −4.23967 −0.299790
\(201\) 4.57459 0.322667
\(202\) −3.74264 −0.263332
\(203\) −1.17463 −0.0824428
\(204\) −6.41725 −0.449297
\(205\) −2.18087 −0.152319
\(206\) 0.321373 0.0223911
\(207\) 2.48866 0.172974
\(208\) −7.79784 −0.540683
\(209\) 12.4410 0.860559
\(210\) −1.11384 −0.0768621
\(211\) −1.81668 −0.125066 −0.0625328 0.998043i \(-0.519918\pi\)
−0.0625328 + 0.998043i \(0.519918\pi\)
\(212\) 19.2205 1.32007
\(213\) 34.4276 2.35894
\(214\) 5.88012 0.401957
\(215\) 7.88925 0.538043
\(216\) −1.49394 −0.101650
\(217\) −0.0201179 −0.00136569
\(218\) −1.36011 −0.0921186
\(219\) −18.9873 −1.28304
\(220\) −4.37312 −0.294836
\(221\) −3.31420 −0.222937
\(222\) −8.47998 −0.569139
\(223\) −26.9316 −1.80348 −0.901738 0.432282i \(-0.857708\pi\)
−0.901738 + 0.432282i \(0.857708\pi\)
\(224\) −4.20609 −0.281031
\(225\) −8.46073 −0.564048
\(226\) 4.69492 0.312301
\(227\) 27.2783 1.81053 0.905263 0.424851i \(-0.139674\pi\)
0.905263 + 0.424851i \(0.139674\pi\)
\(228\) −30.3610 −2.01071
\(229\) −9.32732 −0.616367 −0.308183 0.951327i \(-0.599721\pi\)
−0.308183 + 0.951327i \(0.599721\pi\)
\(230\) −0.404751 −0.0266885
\(231\) 5.01320 0.329845
\(232\) −1.24707 −0.0818740
\(233\) 5.70467 0.373725 0.186863 0.982386i \(-0.440168\pi\)
0.186863 + 0.982386i \(0.440168\pi\)
\(234\) 1.82822 0.119515
\(235\) 10.9433 0.713859
\(236\) 17.5034 1.13938
\(237\) −26.7332 −1.73651
\(238\) −0.542494 −0.0351646
\(239\) −5.65400 −0.365727 −0.182863 0.983138i \(-0.558537\pi\)
−0.182863 + 0.983138i \(0.558537\pi\)
\(240\) 10.0654 0.649720
\(241\) −7.16163 −0.461321 −0.230661 0.973034i \(-0.574089\pi\)
−0.230661 + 0.973034i \(0.574089\pi\)
\(242\) 2.45769 0.157986
\(243\) −20.4725 −1.31331
\(244\) 13.1402 0.841213
\(245\) −7.10975 −0.454225
\(246\) 1.29227 0.0823923
\(247\) −15.6800 −0.997696
\(248\) −0.0213586 −0.00135627
\(249\) −12.8928 −0.817046
\(250\) 3.39980 0.215022
\(251\) 2.75657 0.173993 0.0869966 0.996209i \(-0.472273\pi\)
0.0869966 + 0.996209i \(0.472273\pi\)
\(252\) −5.54724 −0.349444
\(253\) 1.82172 0.114531
\(254\) −2.24965 −0.141155
\(255\) 4.27796 0.267896
\(256\) 8.42417 0.526510
\(257\) 5.17910 0.323064 0.161532 0.986868i \(-0.448357\pi\)
0.161532 + 0.986868i \(0.448357\pi\)
\(258\) −4.67476 −0.291038
\(259\) 13.2884 0.825701
\(260\) 5.51169 0.341820
\(261\) −2.48866 −0.154044
\(262\) −2.05450 −0.126927
\(263\) 15.2747 0.941878 0.470939 0.882166i \(-0.343915\pi\)
0.470939 + 0.882166i \(0.343915\pi\)
\(264\) 5.32236 0.327569
\(265\) −12.8131 −0.787100
\(266\) −2.56663 −0.157370
\(267\) 23.9336 1.46471
\(268\) −3.70536 −0.226341
\(269\) 9.13142 0.556753 0.278376 0.960472i \(-0.410204\pi\)
0.278376 + 0.960472i \(0.410204\pi\)
\(270\) 0.484878 0.0295087
\(271\) 15.3194 0.930589 0.465295 0.885156i \(-0.345948\pi\)
0.465295 + 0.885156i \(0.345948\pi\)
\(272\) 4.90235 0.297249
\(273\) −6.31842 −0.382408
\(274\) 3.07666 0.185868
\(275\) −6.19332 −0.373472
\(276\) −4.44574 −0.267602
\(277\) −8.47793 −0.509390 −0.254695 0.967021i \(-0.581975\pi\)
−0.254695 + 0.967021i \(0.581975\pi\)
\(278\) 4.19829 0.251797
\(279\) −0.0426233 −0.00255179
\(280\) 1.85306 0.110742
\(281\) −16.6184 −0.991369 −0.495685 0.868503i \(-0.665083\pi\)
−0.495685 + 0.868503i \(0.665083\pi\)
\(282\) −6.48441 −0.386141
\(283\) −18.9824 −1.12838 −0.564192 0.825643i \(-0.690812\pi\)
−0.564192 + 0.825643i \(0.690812\pi\)
\(284\) −27.8859 −1.65473
\(285\) 20.2397 1.19890
\(286\) 1.33828 0.0791339
\(287\) −2.02503 −0.119534
\(288\) −8.91135 −0.525106
\(289\) −14.9164 −0.877437
\(290\) 0.404751 0.0237678
\(291\) 12.8872 0.755462
\(292\) 15.3795 0.900016
\(293\) 3.35531 0.196020 0.0980098 0.995185i \(-0.468752\pi\)
0.0980098 + 0.995185i \(0.468752\pi\)
\(294\) 4.21287 0.245699
\(295\) −11.6684 −0.679359
\(296\) 14.1079 0.820004
\(297\) −2.18236 −0.126633
\(298\) −2.58127 −0.149529
\(299\) −2.29601 −0.132782
\(300\) 15.1143 0.872622
\(301\) 7.32551 0.422235
\(302\) 6.35058 0.365435
\(303\) 27.4045 1.57435
\(304\) 23.1938 1.33026
\(305\) −8.75968 −0.501578
\(306\) −1.14937 −0.0657050
\(307\) 12.5939 0.718770 0.359385 0.933189i \(-0.382986\pi\)
0.359385 + 0.933189i \(0.382986\pi\)
\(308\) −4.06063 −0.231376
\(309\) −2.35317 −0.133867
\(310\) 0.00693219 0.000393722 0
\(311\) 14.8454 0.841806 0.420903 0.907106i \(-0.361713\pi\)
0.420903 + 0.907106i \(0.361713\pi\)
\(312\) −6.70807 −0.379770
\(313\) 1.46525 0.0828209 0.0414105 0.999142i \(-0.486815\pi\)
0.0414105 + 0.999142i \(0.486815\pi\)
\(314\) 7.96870 0.449700
\(315\) 3.69798 0.208358
\(316\) 21.6535 1.21811
\(317\) 31.5934 1.77446 0.887230 0.461327i \(-0.152627\pi\)
0.887230 + 0.461327i \(0.152627\pi\)
\(318\) 7.59236 0.425758
\(319\) −1.82172 −0.101997
\(320\) −7.14335 −0.399326
\(321\) −43.0556 −2.40313
\(322\) −0.375829 −0.0209441
\(323\) 9.85772 0.548498
\(324\) 19.4935 1.08297
\(325\) 7.80579 0.432987
\(326\) 4.37984 0.242577
\(327\) 9.95908 0.550738
\(328\) −2.14992 −0.118709
\(329\) 10.1613 0.560210
\(330\) −1.72744 −0.0950924
\(331\) 14.8236 0.814781 0.407390 0.913254i \(-0.366439\pi\)
0.407390 + 0.913254i \(0.366439\pi\)
\(332\) 10.4430 0.573133
\(333\) 28.1538 1.54282
\(334\) −2.45705 −0.134444
\(335\) 2.47012 0.134957
\(336\) 9.34617 0.509875
\(337\) 6.06225 0.330232 0.165116 0.986274i \(-0.447200\pi\)
0.165116 + 0.986274i \(0.447200\pi\)
\(338\) 2.47272 0.134498
\(339\) −34.3773 −1.86712
\(340\) −3.46509 −0.187921
\(341\) −0.0312006 −0.00168961
\(342\) −5.43785 −0.294045
\(343\) −14.8241 −0.800427
\(344\) 7.77727 0.419322
\(345\) 2.96368 0.159559
\(346\) 3.87595 0.208372
\(347\) 16.8175 0.902811 0.451406 0.892319i \(-0.350923\pi\)
0.451406 + 0.892319i \(0.350923\pi\)
\(348\) 4.44574 0.238317
\(349\) −6.17253 −0.330408 −0.165204 0.986259i \(-0.552828\pi\)
−0.165204 + 0.986259i \(0.552828\pi\)
\(350\) 1.27771 0.0682965
\(351\) 2.75055 0.146813
\(352\) −6.52318 −0.347687
\(353\) 24.5929 1.30895 0.654474 0.756085i \(-0.272890\pi\)
0.654474 + 0.756085i \(0.272890\pi\)
\(354\) 6.91407 0.367479
\(355\) 18.5897 0.986640
\(356\) −19.3859 −1.02745
\(357\) 3.97227 0.210235
\(358\) 0.925313 0.0489043
\(359\) 8.25123 0.435483 0.217742 0.976006i \(-0.430131\pi\)
0.217742 + 0.976006i \(0.430131\pi\)
\(360\) 3.92603 0.206920
\(361\) 27.6385 1.45466
\(362\) −2.10671 −0.110726
\(363\) −17.9958 −0.944532
\(364\) 5.11784 0.268248
\(365\) −10.2525 −0.536640
\(366\) 5.19053 0.271314
\(367\) −3.55943 −0.185801 −0.0929004 0.995675i \(-0.529614\pi\)
−0.0929004 + 0.995675i \(0.529614\pi\)
\(368\) 3.39625 0.177042
\(369\) −4.29039 −0.223349
\(370\) −4.57889 −0.238045
\(371\) −11.8975 −0.617686
\(372\) 0.0761423 0.00394780
\(373\) 25.2920 1.30957 0.654785 0.755815i \(-0.272759\pi\)
0.654785 + 0.755815i \(0.272759\pi\)
\(374\) −0.841347 −0.0435050
\(375\) −24.8941 −1.28553
\(376\) 10.7879 0.556344
\(377\) 2.29601 0.118251
\(378\) 0.450230 0.0231573
\(379\) 2.33559 0.119971 0.0599855 0.998199i \(-0.480895\pi\)
0.0599855 + 0.998199i \(0.480895\pi\)
\(380\) −16.3939 −0.840989
\(381\) 16.4725 0.843909
\(382\) −1.27412 −0.0651898
\(383\) 13.9569 0.713165 0.356582 0.934264i \(-0.383942\pi\)
0.356582 + 0.934264i \(0.383942\pi\)
\(384\) 21.0108 1.07220
\(385\) 2.70695 0.137959
\(386\) −5.76209 −0.293283
\(387\) 15.5204 0.788945
\(388\) −10.4385 −0.529934
\(389\) 7.42722 0.376575 0.188287 0.982114i \(-0.439706\pi\)
0.188287 + 0.982114i \(0.439706\pi\)
\(390\) 2.17719 0.110246
\(391\) 1.44346 0.0729988
\(392\) −7.00883 −0.353999
\(393\) 15.0435 0.758846
\(394\) 0.117830 0.00593620
\(395\) −14.4350 −0.726303
\(396\) −8.60316 −0.432325
\(397\) 15.4152 0.773667 0.386834 0.922150i \(-0.373569\pi\)
0.386834 + 0.922150i \(0.373569\pi\)
\(398\) −4.24405 −0.212735
\(399\) 18.7934 0.940848
\(400\) −11.5463 −0.577314
\(401\) 33.0274 1.64931 0.824655 0.565636i \(-0.191369\pi\)
0.824655 + 0.565636i \(0.191369\pi\)
\(402\) −1.46366 −0.0730009
\(403\) 0.0393239 0.00195886
\(404\) −22.1973 −1.10436
\(405\) −12.9950 −0.645728
\(406\) 0.375829 0.0186521
\(407\) 20.6088 1.02154
\(408\) 4.21723 0.208784
\(409\) −4.65642 −0.230245 −0.115123 0.993351i \(-0.536726\pi\)
−0.115123 + 0.993351i \(0.536726\pi\)
\(410\) 0.697782 0.0344610
\(411\) −22.5280 −1.11123
\(412\) 1.90604 0.0939036
\(413\) −10.8346 −0.533135
\(414\) −0.796260 −0.0391340
\(415\) −6.96165 −0.341734
\(416\) 8.22153 0.403094
\(417\) −30.7409 −1.50539
\(418\) −3.98055 −0.194695
\(419\) −18.0795 −0.883242 −0.441621 0.897202i \(-0.645596\pi\)
−0.441621 + 0.897202i \(0.645596\pi\)
\(420\) −6.60608 −0.322344
\(421\) 8.63962 0.421069 0.210535 0.977586i \(-0.432480\pi\)
0.210535 + 0.977586i \(0.432480\pi\)
\(422\) 0.581257 0.0282952
\(423\) 21.5285 1.04675
\(424\) −12.6312 −0.613424
\(425\) −4.90735 −0.238041
\(426\) −11.0153 −0.533693
\(427\) −8.13374 −0.393619
\(428\) 34.8745 1.68572
\(429\) −9.79917 −0.473108
\(430\) −2.52421 −0.121728
\(431\) 22.3508 1.07660 0.538299 0.842754i \(-0.319067\pi\)
0.538299 + 0.842754i \(0.319067\pi\)
\(432\) −4.06859 −0.195750
\(433\) 13.1909 0.633914 0.316957 0.948440i \(-0.397339\pi\)
0.316957 + 0.948440i \(0.397339\pi\)
\(434\) 0.00643683 0.000308978 0
\(435\) −2.96368 −0.142098
\(436\) −8.06672 −0.386326
\(437\) 6.82924 0.326687
\(438\) 6.07509 0.290279
\(439\) −30.5025 −1.45580 −0.727902 0.685682i \(-0.759504\pi\)
−0.727902 + 0.685682i \(0.759504\pi\)
\(440\) 2.87389 0.137007
\(441\) −13.9869 −0.666042
\(442\) 1.06040 0.0504379
\(443\) 18.1792 0.863719 0.431860 0.901941i \(-0.357858\pi\)
0.431860 + 0.901941i \(0.357858\pi\)
\(444\) −50.2940 −2.38685
\(445\) 12.9233 0.612624
\(446\) 8.61693 0.408023
\(447\) 18.9007 0.893971
\(448\) −6.63291 −0.313375
\(449\) 15.4772 0.730415 0.365207 0.930926i \(-0.380998\pi\)
0.365207 + 0.930926i \(0.380998\pi\)
\(450\) 2.70706 0.127612
\(451\) −3.14060 −0.147885
\(452\) 27.8452 1.30973
\(453\) −46.5004 −2.18478
\(454\) −8.72785 −0.409618
\(455\) −3.41173 −0.159944
\(456\) 19.9524 0.934357
\(457\) −13.8500 −0.647876 −0.323938 0.946078i \(-0.605007\pi\)
−0.323938 + 0.946078i \(0.605007\pi\)
\(458\) 2.98433 0.139448
\(459\) −1.72921 −0.0807128
\(460\) −2.40055 −0.111926
\(461\) 6.58340 0.306619 0.153310 0.988178i \(-0.451007\pi\)
0.153310 + 0.988178i \(0.451007\pi\)
\(462\) −1.60400 −0.0746249
\(463\) 40.5550 1.88475 0.942374 0.334561i \(-0.108588\pi\)
0.942374 + 0.334561i \(0.108588\pi\)
\(464\) −3.39625 −0.157667
\(465\) −0.0507591 −0.00235390
\(466\) −1.82524 −0.0845526
\(467\) 14.8827 0.688688 0.344344 0.938844i \(-0.388101\pi\)
0.344344 + 0.938844i \(0.388101\pi\)
\(468\) 10.8430 0.501220
\(469\) 2.29361 0.105909
\(470\) −3.50135 −0.161505
\(471\) −58.3487 −2.68857
\(472\) −11.5027 −0.529457
\(473\) 11.3611 0.522382
\(474\) 8.55342 0.392872
\(475\) −23.2175 −1.06529
\(476\) −3.21748 −0.147473
\(477\) −25.2069 −1.15414
\(478\) 1.80903 0.0827430
\(479\) −28.0463 −1.28147 −0.640733 0.767764i \(-0.721369\pi\)
−0.640733 + 0.767764i \(0.721369\pi\)
\(480\) −10.6123 −0.484384
\(481\) −25.9745 −1.18433
\(482\) 2.29140 0.104371
\(483\) 2.75191 0.125216
\(484\) 14.5763 0.662560
\(485\) 6.95865 0.315976
\(486\) 6.55030 0.297127
\(487\) −18.2380 −0.826440 −0.413220 0.910631i \(-0.635596\pi\)
−0.413220 + 0.910631i \(0.635596\pi\)
\(488\) −8.63534 −0.390903
\(489\) −32.0702 −1.45027
\(490\) 2.27480 0.102765
\(491\) −5.07847 −0.229188 −0.114594 0.993412i \(-0.536557\pi\)
−0.114594 + 0.993412i \(0.536557\pi\)
\(492\) 7.66436 0.345536
\(493\) −1.44346 −0.0650101
\(494\) 5.01691 0.225721
\(495\) 5.73516 0.257776
\(496\) −0.0581677 −0.00261181
\(497\) 17.2613 0.774277
\(498\) 4.12511 0.184851
\(499\) −14.0668 −0.629717 −0.314859 0.949139i \(-0.601957\pi\)
−0.314859 + 0.949139i \(0.601957\pi\)
\(500\) 20.1639 0.901757
\(501\) 17.9911 0.803783
\(502\) −0.881980 −0.0393647
\(503\) −23.3184 −1.03972 −0.519859 0.854252i \(-0.674016\pi\)
−0.519859 + 0.854252i \(0.674016\pi\)
\(504\) 3.64549 0.162383
\(505\) 14.7975 0.658479
\(506\) −0.582869 −0.0259117
\(507\) −18.1058 −0.804108
\(508\) −13.3425 −0.591977
\(509\) 2.42877 0.107654 0.0538268 0.998550i \(-0.482858\pi\)
0.0538268 + 0.998550i \(0.482858\pi\)
\(510\) −1.36876 −0.0606095
\(511\) −9.51987 −0.421134
\(512\) −20.6319 −0.911812
\(513\) −8.18119 −0.361208
\(514\) −1.65708 −0.0730908
\(515\) −1.27063 −0.0559906
\(516\) −27.7256 −1.22055
\(517\) 15.7590 0.693081
\(518\) −4.25170 −0.186809
\(519\) −28.3806 −1.24577
\(520\) −3.62212 −0.158841
\(521\) 17.0976 0.749058 0.374529 0.927215i \(-0.377805\pi\)
0.374529 + 0.927215i \(0.377805\pi\)
\(522\) 0.796260 0.0348513
\(523\) −33.4942 −1.46460 −0.732298 0.680984i \(-0.761552\pi\)
−0.732298 + 0.680984i \(0.761552\pi\)
\(524\) −12.1851 −0.532307
\(525\) −9.35570 −0.408316
\(526\) −4.88722 −0.213093
\(527\) −0.0247222 −0.00107691
\(528\) 14.4949 0.630808
\(529\) 1.00000 0.0434783
\(530\) 4.09961 0.178076
\(531\) −22.9550 −0.996161
\(532\) −15.2224 −0.659976
\(533\) 3.95828 0.171452
\(534\) −7.65769 −0.331381
\(535\) −23.2485 −1.00512
\(536\) 2.43506 0.105178
\(537\) −6.77536 −0.292378
\(538\) −2.92165 −0.125961
\(539\) −10.2385 −0.441004
\(540\) 2.87577 0.123753
\(541\) 30.4452 1.30894 0.654471 0.756087i \(-0.272891\pi\)
0.654471 + 0.756087i \(0.272891\pi\)
\(542\) −4.90154 −0.210539
\(543\) 15.4258 0.661985
\(544\) −5.16871 −0.221607
\(545\) 5.37755 0.230349
\(546\) 2.02161 0.0865170
\(547\) −37.8141 −1.61681 −0.808406 0.588625i \(-0.799669\pi\)
−0.808406 + 0.588625i \(0.799669\pi\)
\(548\) 18.2474 0.779491
\(549\) −17.2328 −0.735476
\(550\) 1.98159 0.0844952
\(551\) −6.82924 −0.290935
\(552\) 2.92162 0.124352
\(553\) −13.4035 −0.569975
\(554\) 2.71256 0.115246
\(555\) 33.5277 1.42317
\(556\) 24.8997 1.05598
\(557\) −17.3216 −0.733939 −0.366970 0.930233i \(-0.619605\pi\)
−0.366970 + 0.930233i \(0.619605\pi\)
\(558\) 0.0136376 0.000577324 0
\(559\) −14.3190 −0.605628
\(560\) 5.04661 0.213258
\(561\) 6.16054 0.260098
\(562\) 5.31714 0.224290
\(563\) −45.6216 −1.92272 −0.961362 0.275289i \(-0.911227\pi\)
−0.961362 + 0.275289i \(0.911227\pi\)
\(564\) −38.4585 −1.61939
\(565\) −18.5625 −0.780932
\(566\) 6.07351 0.255289
\(567\) −12.0664 −0.506743
\(568\) 18.3258 0.768935
\(569\) 1.42662 0.0598069 0.0299035 0.999553i \(-0.490480\pi\)
0.0299035 + 0.999553i \(0.490480\pi\)
\(570\) −6.47580 −0.271242
\(571\) −25.9511 −1.08602 −0.543009 0.839727i \(-0.682715\pi\)
−0.543009 + 0.839727i \(0.682715\pi\)
\(572\) 7.93720 0.331871
\(573\) 9.32942 0.389742
\(574\) 0.647921 0.0270437
\(575\) −3.39971 −0.141778
\(576\) −14.0530 −0.585541
\(577\) 31.8837 1.32734 0.663669 0.748027i \(-0.268998\pi\)
0.663669 + 0.748027i \(0.268998\pi\)
\(578\) 4.77259 0.198514
\(579\) 42.1914 1.75341
\(580\) 2.40055 0.0996772
\(581\) −6.46419 −0.268180
\(582\) −4.12334 −0.170918
\(583\) −18.4517 −0.764190
\(584\) −10.1070 −0.418229
\(585\) −7.22834 −0.298855
\(586\) −1.07355 −0.0443480
\(587\) −23.2980 −0.961611 −0.480806 0.876827i \(-0.659656\pi\)
−0.480806 + 0.876827i \(0.659656\pi\)
\(588\) 24.9862 1.03041
\(589\) −0.116965 −0.00481944
\(590\) 3.73336 0.153700
\(591\) −0.862781 −0.0354901
\(592\) 38.4213 1.57911
\(593\) −9.93025 −0.407787 −0.203893 0.978993i \(-0.565360\pi\)
−0.203893 + 0.978993i \(0.565360\pi\)
\(594\) 0.698257 0.0286498
\(595\) 2.14488 0.0879317
\(596\) −15.3093 −0.627093
\(597\) 31.0759 1.27185
\(598\) 0.734622 0.0300409
\(599\) 30.8661 1.26116 0.630578 0.776126i \(-0.282818\pi\)
0.630578 + 0.776126i \(0.282818\pi\)
\(600\) −9.93266 −0.405499
\(601\) 12.4372 0.507322 0.253661 0.967293i \(-0.418365\pi\)
0.253661 + 0.967293i \(0.418365\pi\)
\(602\) −2.34384 −0.0955276
\(603\) 4.85942 0.197891
\(604\) 37.6647 1.53256
\(605\) −9.71708 −0.395055
\(606\) −8.76822 −0.356185
\(607\) −26.6894 −1.08329 −0.541645 0.840607i \(-0.682198\pi\)
−0.541645 + 0.840607i \(0.682198\pi\)
\(608\) −24.4540 −0.991742
\(609\) −2.75191 −0.111513
\(610\) 2.80271 0.113478
\(611\) −19.8620 −0.803529
\(612\) −6.81681 −0.275553
\(613\) 3.68416 0.148802 0.0744010 0.997228i \(-0.476296\pi\)
0.0744010 + 0.997228i \(0.476296\pi\)
\(614\) −4.02948 −0.162617
\(615\) −5.10932 −0.206028
\(616\) 2.66853 0.107518
\(617\) 17.9696 0.723428 0.361714 0.932289i \(-0.382192\pi\)
0.361714 + 0.932289i \(0.382192\pi\)
\(618\) 0.752909 0.0302864
\(619\) 23.1024 0.928564 0.464282 0.885687i \(-0.346312\pi\)
0.464282 + 0.885687i \(0.346312\pi\)
\(620\) 0.0411142 0.00165119
\(621\) −1.19797 −0.0480727
\(622\) −4.74987 −0.190452
\(623\) 11.9999 0.480764
\(624\) −18.2687 −0.731333
\(625\) 3.55662 0.142265
\(626\) −0.468815 −0.0187376
\(627\) 29.1465 1.16400
\(628\) 47.2617 1.88595
\(629\) 16.3296 0.651105
\(630\) −1.18319 −0.0471394
\(631\) 42.7618 1.70232 0.851160 0.524906i \(-0.175900\pi\)
0.851160 + 0.524906i \(0.175900\pi\)
\(632\) −14.2301 −0.566042
\(633\) −4.25610 −0.169165
\(634\) −10.1085 −0.401459
\(635\) 8.89455 0.352969
\(636\) 45.0296 1.78554
\(637\) 12.9042 0.511282
\(638\) 0.582869 0.0230760
\(639\) 36.5712 1.44673
\(640\) 11.3451 0.448455
\(641\) −37.1432 −1.46707 −0.733535 0.679652i \(-0.762131\pi\)
−0.733535 + 0.679652i \(0.762131\pi\)
\(642\) 13.7759 0.543691
\(643\) 7.50458 0.295952 0.147976 0.988991i \(-0.452724\pi\)
0.147976 + 0.988991i \(0.452724\pi\)
\(644\) −2.22901 −0.0878353
\(645\) 18.4829 0.727762
\(646\) −3.15403 −0.124094
\(647\) 40.6186 1.59688 0.798440 0.602074i \(-0.205659\pi\)
0.798440 + 0.602074i \(0.205659\pi\)
\(648\) −12.8106 −0.503247
\(649\) −16.8032 −0.659585
\(650\) −2.49751 −0.0979602
\(651\) −0.0471320 −0.00184725
\(652\) 25.9765 1.01732
\(653\) −16.0997 −0.630031 −0.315016 0.949087i \(-0.602010\pi\)
−0.315016 + 0.949087i \(0.602010\pi\)
\(654\) −3.18646 −0.124600
\(655\) 8.12298 0.317391
\(656\) −5.85506 −0.228602
\(657\) −20.1695 −0.786888
\(658\) −3.25116 −0.126743
\(659\) −28.4479 −1.10817 −0.554087 0.832459i \(-0.686933\pi\)
−0.554087 + 0.832459i \(0.686933\pi\)
\(660\) −10.2453 −0.398798
\(661\) 1.18518 0.0460982 0.0230491 0.999734i \(-0.492663\pi\)
0.0230491 + 0.999734i \(0.492663\pi\)
\(662\) −4.74290 −0.184338
\(663\) −7.76447 −0.301547
\(664\) −6.86283 −0.266329
\(665\) 10.1478 0.393515
\(666\) −9.00797 −0.349052
\(667\) −1.00000 −0.0387202
\(668\) −14.5726 −0.563829
\(669\) −63.0952 −2.43940
\(670\) −0.790328 −0.0305330
\(671\) −12.6145 −0.486978
\(672\) −9.85399 −0.380126
\(673\) 29.9312 1.15376 0.576882 0.816828i \(-0.304269\pi\)
0.576882 + 0.816828i \(0.304269\pi\)
\(674\) −1.93965 −0.0747126
\(675\) 4.07274 0.156760
\(676\) 14.6655 0.564057
\(677\) −10.1849 −0.391438 −0.195719 0.980660i \(-0.562704\pi\)
−0.195719 + 0.980660i \(0.562704\pi\)
\(678\) 10.9992 0.422422
\(679\) 6.46140 0.247966
\(680\) 2.27716 0.0873250
\(681\) 63.9074 2.44894
\(682\) 0.00998281 0.000382262 0
\(683\) −35.9185 −1.37438 −0.687192 0.726476i \(-0.741157\pi\)
−0.687192 + 0.726476i \(0.741157\pi\)
\(684\) −32.2514 −1.23316
\(685\) −12.1644 −0.464776
\(686\) 4.74305 0.181091
\(687\) −21.8519 −0.833704
\(688\) 21.1805 0.807500
\(689\) 23.2557 0.885970
\(690\) −0.948247 −0.0360991
\(691\) −7.29905 −0.277669 −0.138835 0.990316i \(-0.544336\pi\)
−0.138835 + 0.990316i \(0.544336\pi\)
\(692\) 22.9879 0.873869
\(693\) 5.32534 0.202293
\(694\) −5.38085 −0.204254
\(695\) −16.5990 −0.629636
\(696\) −2.92162 −0.110744
\(697\) −2.48849 −0.0942583
\(698\) 1.97493 0.0747523
\(699\) 13.3648 0.505504
\(700\) 7.57800 0.286421
\(701\) 4.13360 0.156124 0.0780620 0.996949i \(-0.475127\pi\)
0.0780620 + 0.996949i \(0.475127\pi\)
\(702\) −0.880052 −0.0332154
\(703\) 77.2582 2.91385
\(704\) −10.2869 −0.387702
\(705\) 25.6377 0.965573
\(706\) −7.86863 −0.296140
\(707\) 13.7401 0.516750
\(708\) 41.0068 1.54113
\(709\) 0.363807 0.0136631 0.00683153 0.999977i \(-0.497825\pi\)
0.00683153 + 0.999977i \(0.497825\pi\)
\(710\) −5.94788 −0.223220
\(711\) −28.3977 −1.06500
\(712\) 12.7399 0.477447
\(713\) −0.0171270 −0.000641412 0
\(714\) −1.27095 −0.0475640
\(715\) −5.29121 −0.197880
\(716\) 5.48795 0.205094
\(717\) −13.2461 −0.494686
\(718\) −2.64003 −0.0985249
\(719\) −21.2848 −0.793789 −0.396894 0.917864i \(-0.629912\pi\)
−0.396894 + 0.917864i \(0.629912\pi\)
\(720\) 10.6921 0.398472
\(721\) −1.17983 −0.0439393
\(722\) −8.84308 −0.329105
\(723\) −16.7782 −0.623988
\(724\) −12.4947 −0.464362
\(725\) 3.39971 0.126262
\(726\) 5.75784 0.213693
\(727\) −16.7296 −0.620466 −0.310233 0.950661i \(-0.600407\pi\)
−0.310233 + 0.950661i \(0.600407\pi\)
\(728\) −3.36330 −0.124652
\(729\) −17.1451 −0.635005
\(730\) 3.28034 0.121411
\(731\) 9.00205 0.332953
\(732\) 30.7846 1.13783
\(733\) −4.74919 −0.175415 −0.0877077 0.996146i \(-0.527954\pi\)
−0.0877077 + 0.996146i \(0.527954\pi\)
\(734\) 1.13886 0.0420361
\(735\) −16.6566 −0.614389
\(736\) −3.58078 −0.131989
\(737\) 3.55714 0.131029
\(738\) 1.37273 0.0505310
\(739\) −13.9169 −0.511943 −0.255972 0.966684i \(-0.582395\pi\)
−0.255972 + 0.966684i \(0.582395\pi\)
\(740\) −27.1570 −0.998312
\(741\) −36.7350 −1.34949
\(742\) 3.80666 0.139747
\(743\) 0.115038 0.00422033 0.00211016 0.999998i \(-0.499328\pi\)
0.00211016 + 0.999998i \(0.499328\pi\)
\(744\) −0.0500386 −0.00183450
\(745\) 10.2057 0.373908
\(746\) −8.09231 −0.296281
\(747\) −13.6955 −0.501093
\(748\) −4.98996 −0.182451
\(749\) −21.5873 −0.788781
\(750\) 7.96500 0.290841
\(751\) −26.3015 −0.959754 −0.479877 0.877336i \(-0.659319\pi\)
−0.479877 + 0.877336i \(0.659319\pi\)
\(752\) 29.3797 1.07137
\(753\) 6.45806 0.235345
\(754\) −0.734622 −0.0267534
\(755\) −25.1086 −0.913795
\(756\) 2.67028 0.0971170
\(757\) −19.5102 −0.709109 −0.354554 0.935035i \(-0.615367\pi\)
−0.354554 + 0.935035i \(0.615367\pi\)
\(758\) −0.747284 −0.0271426
\(759\) 4.26790 0.154915
\(760\) 10.7736 0.390800
\(761\) −25.8077 −0.935529 −0.467765 0.883853i \(-0.654941\pi\)
−0.467765 + 0.883853i \(0.654941\pi\)
\(762\) −5.27045 −0.190928
\(763\) 4.99329 0.180769
\(764\) −7.55671 −0.273392
\(765\) 4.54432 0.164300
\(766\) −4.46559 −0.161348
\(767\) 21.1781 0.764695
\(768\) 19.7360 0.712163
\(769\) 6.83296 0.246403 0.123201 0.992382i \(-0.460684\pi\)
0.123201 + 0.992382i \(0.460684\pi\)
\(770\) −0.866105 −0.0312123
\(771\) 12.1335 0.436979
\(772\) −34.1745 −1.22997
\(773\) 33.1309 1.19164 0.595819 0.803119i \(-0.296828\pi\)
0.595819 + 0.803119i \(0.296828\pi\)
\(774\) −4.96583 −0.178493
\(775\) 0.0582270 0.00209157
\(776\) 6.85987 0.246255
\(777\) 31.1319 1.11685
\(778\) −2.37638 −0.0851973
\(779\) −11.7734 −0.421828
\(780\) 12.9127 0.462349
\(781\) 26.7704 0.957922
\(782\) −0.461842 −0.0165155
\(783\) 1.19797 0.0428118
\(784\) −19.0878 −0.681706
\(785\) −31.5063 −1.12451
\(786\) −4.81326 −0.171683
\(787\) −36.8898 −1.31498 −0.657490 0.753464i \(-0.728382\pi\)
−0.657490 + 0.753464i \(0.728382\pi\)
\(788\) 0.698841 0.0248952
\(789\) 35.7854 1.27399
\(790\) 4.61855 0.164321
\(791\) −17.2361 −0.612846
\(792\) 5.65375 0.200897
\(793\) 15.8988 0.564583
\(794\) −4.93218 −0.175037
\(795\) −30.0183 −1.06464
\(796\) −25.1711 −0.892166
\(797\) −9.88897 −0.350285 −0.175143 0.984543i \(-0.556039\pi\)
−0.175143 + 0.984543i \(0.556039\pi\)
\(798\) −6.01306 −0.212860
\(799\) 12.4868 0.441752
\(800\) 12.1736 0.430403
\(801\) 25.4238 0.898306
\(802\) −10.5673 −0.373144
\(803\) −14.7643 −0.521020
\(804\) −8.68087 −0.306151
\(805\) 1.48593 0.0523723
\(806\) −0.0125819 −0.000443178 0
\(807\) 21.3930 0.753069
\(808\) 14.5874 0.513184
\(809\) −16.8507 −0.592438 −0.296219 0.955120i \(-0.595726\pi\)
−0.296219 + 0.955120i \(0.595726\pi\)
\(810\) 4.15783 0.146091
\(811\) −33.4678 −1.17521 −0.587607 0.809146i \(-0.699930\pi\)
−0.587607 + 0.809146i \(0.699930\pi\)
\(812\) 2.22901 0.0782229
\(813\) 35.8902 1.25872
\(814\) −6.59391 −0.231116
\(815\) −17.3168 −0.606581
\(816\) 11.4852 0.402061
\(817\) 42.5902 1.49004
\(818\) 1.48985 0.0520913
\(819\) −6.71182 −0.234530
\(820\) 4.13849 0.144522
\(821\) 53.1757 1.85584 0.927922 0.372774i \(-0.121593\pi\)
0.927922 + 0.372774i \(0.121593\pi\)
\(822\) 7.20797 0.251407
\(823\) 5.95055 0.207423 0.103712 0.994607i \(-0.466928\pi\)
0.103712 + 0.994607i \(0.466928\pi\)
\(824\) −1.25259 −0.0436361
\(825\) −14.5096 −0.505161
\(826\) 3.46658 0.120618
\(827\) −20.3489 −0.707602 −0.353801 0.935321i \(-0.615111\pi\)
−0.353801 + 0.935321i \(0.615111\pi\)
\(828\) −4.72255 −0.164120
\(829\) −38.1095 −1.32360 −0.661799 0.749682i \(-0.730207\pi\)
−0.661799 + 0.749682i \(0.730207\pi\)
\(830\) 2.22742 0.0773148
\(831\) −19.8620 −0.689005
\(832\) 12.9652 0.449486
\(833\) −8.11259 −0.281085
\(834\) 9.83572 0.340583
\(835\) 9.71457 0.336187
\(836\) −23.6083 −0.816510
\(837\) 0.0205176 0.000709191 0
\(838\) 5.78464 0.199827
\(839\) −14.9607 −0.516502 −0.258251 0.966078i \(-0.583146\pi\)
−0.258251 + 0.966078i \(0.583146\pi\)
\(840\) 4.34133 0.149790
\(841\) 1.00000 0.0344828
\(842\) −2.76429 −0.0952638
\(843\) −38.9333 −1.34094
\(844\) 3.44739 0.118664
\(845\) −9.77651 −0.336322
\(846\) −6.88815 −0.236819
\(847\) −9.02272 −0.310024
\(848\) −34.3996 −1.18129
\(849\) −44.4717 −1.52626
\(850\) 1.57013 0.0538551
\(851\) 11.3129 0.387800
\(852\) −65.3308 −2.23820
\(853\) −29.0903 −0.996034 −0.498017 0.867167i \(-0.665938\pi\)
−0.498017 + 0.867167i \(0.665938\pi\)
\(854\) 2.60243 0.0890535
\(855\) 21.4999 0.735281
\(856\) −22.9185 −0.783339
\(857\) 33.7905 1.15426 0.577130 0.816652i \(-0.304172\pi\)
0.577130 + 0.816652i \(0.304172\pi\)
\(858\) 3.13530 0.107037
\(859\) 50.1146 1.70989 0.854944 0.518721i \(-0.173592\pi\)
0.854944 + 0.518721i \(0.173592\pi\)
\(860\) −14.9709 −0.510502
\(861\) −4.74423 −0.161683
\(862\) −7.15125 −0.243572
\(863\) −13.8990 −0.473128 −0.236564 0.971616i \(-0.576021\pi\)
−0.236564 + 0.971616i \(0.576021\pi\)
\(864\) 4.28965 0.145937
\(865\) −15.3245 −0.521049
\(866\) −4.22050 −0.143418
\(867\) −34.9460 −1.18683
\(868\) 0.0381763 0.00129579
\(869\) −20.7873 −0.705162
\(870\) 0.948247 0.0321486
\(871\) −4.48326 −0.151909
\(872\) 5.30122 0.179522
\(873\) 13.6896 0.463323
\(874\) −2.18505 −0.0739105
\(875\) −12.4814 −0.421949
\(876\) 36.0309 1.21737
\(877\) 57.5237 1.94244 0.971218 0.238190i \(-0.0765543\pi\)
0.971218 + 0.238190i \(0.0765543\pi\)
\(878\) 9.75943 0.329365
\(879\) 7.86079 0.265138
\(880\) 7.82672 0.263839
\(881\) −58.4376 −1.96881 −0.984406 0.175912i \(-0.943712\pi\)
−0.984406 + 0.175912i \(0.943712\pi\)
\(882\) 4.47518 0.150687
\(883\) −1.51349 −0.0509330 −0.0254665 0.999676i \(-0.508107\pi\)
−0.0254665 + 0.999676i \(0.508107\pi\)
\(884\) 6.28912 0.211526
\(885\) −27.3365 −0.918907
\(886\) −5.81653 −0.195410
\(887\) −29.0687 −0.976032 −0.488016 0.872835i \(-0.662279\pi\)
−0.488016 + 0.872835i \(0.662279\pi\)
\(888\) 33.0518 1.10915
\(889\) 8.25897 0.276997
\(890\) −4.13489 −0.138602
\(891\) −18.7137 −0.626933
\(892\) 51.1063 1.71116
\(893\) 59.0772 1.97694
\(894\) −6.04737 −0.202254
\(895\) −3.65846 −0.122289
\(896\) 10.5344 0.351930
\(897\) −5.37908 −0.179602
\(898\) −4.95202 −0.165251
\(899\) 0.0171270 0.000571218 0
\(900\) 16.0553 0.535177
\(901\) −14.6204 −0.487075
\(902\) 1.00485 0.0334579
\(903\) 17.1621 0.571120
\(904\) −18.2990 −0.608617
\(905\) 8.32941 0.276879
\(906\) 14.8781 0.494290
\(907\) 12.3587 0.410362 0.205181 0.978724i \(-0.434222\pi\)
0.205181 + 0.978724i \(0.434222\pi\)
\(908\) −51.7641 −1.71785
\(909\) 29.1108 0.965545
\(910\) 1.09160 0.0361862
\(911\) −38.6517 −1.28059 −0.640294 0.768130i \(-0.721188\pi\)
−0.640294 + 0.768130i \(0.721188\pi\)
\(912\) 54.3382 1.79932
\(913\) −10.0252 −0.331787
\(914\) 4.43138 0.146577
\(915\) −20.5221 −0.678439
\(916\) 17.6998 0.584818
\(917\) 7.54254 0.249077
\(918\) 0.553271 0.0182607
\(919\) 33.6053 1.10854 0.554269 0.832338i \(-0.312998\pi\)
0.554269 + 0.832338i \(0.312998\pi\)
\(920\) 1.57757 0.0520110
\(921\) 29.5048 0.972216
\(922\) −2.10639 −0.0693704
\(923\) −33.7403 −1.11057
\(924\) −9.51320 −0.312961
\(925\) −38.4605 −1.26457
\(926\) −12.9758 −0.426410
\(927\) −2.49968 −0.0821004
\(928\) 3.58078 0.117545
\(929\) 26.8238 0.880061 0.440031 0.897983i \(-0.354968\pi\)
0.440031 + 0.897983i \(0.354968\pi\)
\(930\) 0.0162406 0.000532552 0
\(931\) −38.3820 −1.25792
\(932\) −10.8253 −0.354596
\(933\) 34.7797 1.13864
\(934\) −4.76179 −0.155811
\(935\) 3.32648 0.108787
\(936\) −7.12574 −0.232912
\(937\) 2.02573 0.0661778 0.0330889 0.999452i \(-0.489466\pi\)
0.0330889 + 0.999452i \(0.489466\pi\)
\(938\) −0.733853 −0.0239612
\(939\) 3.43278 0.112024
\(940\) −20.7662 −0.677320
\(941\) −45.9973 −1.49947 −0.749734 0.661739i \(-0.769819\pi\)
−0.749734 + 0.661739i \(0.769819\pi\)
\(942\) 18.6690 0.608268
\(943\) −1.72398 −0.0561404
\(944\) −31.3265 −1.01959
\(945\) −1.78010 −0.0579066
\(946\) −3.63503 −0.118185
\(947\) 28.2766 0.918866 0.459433 0.888212i \(-0.348053\pi\)
0.459433 + 0.888212i \(0.348053\pi\)
\(948\) 50.7296 1.64762
\(949\) 18.6082 0.604049
\(950\) 7.42855 0.241014
\(951\) 74.0166 2.40015
\(952\) 2.11444 0.0685293
\(953\) 22.0521 0.714338 0.357169 0.934040i \(-0.383742\pi\)
0.357169 + 0.934040i \(0.383742\pi\)
\(954\) 8.06508 0.261117
\(955\) 5.03756 0.163012
\(956\) 10.7292 0.347007
\(957\) −4.26790 −0.137962
\(958\) 8.97355 0.289922
\(959\) −11.2951 −0.364739
\(960\) −16.7354 −0.540132
\(961\) −30.9997 −0.999991
\(962\) 8.31067 0.267947
\(963\) −45.7364 −1.47384
\(964\) 13.5901 0.437708
\(965\) 22.7819 0.733375
\(966\) −0.880488 −0.0283292
\(967\) −50.2308 −1.61531 −0.807656 0.589654i \(-0.799264\pi\)
−0.807656 + 0.589654i \(0.799264\pi\)
\(968\) −9.57915 −0.307885
\(969\) 23.0946 0.741904
\(970\) −2.22646 −0.0714872
\(971\) 18.4011 0.590520 0.295260 0.955417i \(-0.404594\pi\)
0.295260 + 0.955417i \(0.404594\pi\)
\(972\) 38.8493 1.24609
\(973\) −15.4129 −0.494115
\(974\) 5.83533 0.186976
\(975\) 18.2873 0.585663
\(976\) −23.5174 −0.752774
\(977\) 20.0251 0.640660 0.320330 0.947306i \(-0.396206\pi\)
0.320330 + 0.947306i \(0.396206\pi\)
\(978\) 10.2610 0.328112
\(979\) 18.6104 0.594793
\(980\) 13.4917 0.430975
\(981\) 10.5792 0.337767
\(982\) 1.62489 0.0518522
\(983\) 58.8340 1.87651 0.938256 0.345942i \(-0.112441\pi\)
0.938256 + 0.345942i \(0.112441\pi\)
\(984\) −5.03680 −0.160567
\(985\) −0.465872 −0.0148439
\(986\) 0.461842 0.0147081
\(987\) 23.8057 0.757745
\(988\) 29.7549 0.946628
\(989\) 6.23644 0.198307
\(990\) −1.83500 −0.0583200
\(991\) −1.41402 −0.0449178 −0.0224589 0.999748i \(-0.507149\pi\)
−0.0224589 + 0.999748i \(0.507149\pi\)
\(992\) 0.0613282 0.00194717
\(993\) 34.7286 1.10208
\(994\) −5.52286 −0.175175
\(995\) 16.7799 0.531959
\(996\) 24.4657 0.775225
\(997\) 51.9389 1.64492 0.822460 0.568823i \(-0.192601\pi\)
0.822460 + 0.568823i \(0.192601\pi\)
\(998\) 4.50075 0.142469
\(999\) −13.5524 −0.428779
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 667.2.a.d.1.7 16
3.2 odd 2 6003.2.a.q.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
667.2.a.d.1.7 16 1.1 even 1 trivial
6003.2.a.q.1.10 16 3.2 odd 2