Properties

Label 8006.2.a.a.1.14
Level $8006$
Weight $2$
Character 8006.1
Self dual yes
Analytic conductor $63.928$
Analytic rank $1$
Dimension $69$
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] = [8006,2,Mod(1,8006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8006 = 2 \cdot 4003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8006.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: \(63.9282318582\)
Analytic rank: \(1\)
Dimension: \(69\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 8006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.16991 q^{3} +1.00000 q^{4} -2.35323 q^{5} -2.16991 q^{6} -2.98482 q^{7} +1.00000 q^{8} +1.70849 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.16991 q^{3} +1.00000 q^{4} -2.35323 q^{5} -2.16991 q^{6} -2.98482 q^{7} +1.00000 q^{8} +1.70849 q^{9} -2.35323 q^{10} +3.26147 q^{11} -2.16991 q^{12} -0.832404 q^{13} -2.98482 q^{14} +5.10629 q^{15} +1.00000 q^{16} -0.928806 q^{17} +1.70849 q^{18} +1.16473 q^{19} -2.35323 q^{20} +6.47678 q^{21} +3.26147 q^{22} -3.48217 q^{23} -2.16991 q^{24} +0.537703 q^{25} -0.832404 q^{26} +2.80246 q^{27} -2.98482 q^{28} -3.05692 q^{29} +5.10629 q^{30} +5.74043 q^{31} +1.00000 q^{32} -7.07708 q^{33} -0.928806 q^{34} +7.02397 q^{35} +1.70849 q^{36} -0.186459 q^{37} +1.16473 q^{38} +1.80624 q^{39} -2.35323 q^{40} -2.25855 q^{41} +6.47678 q^{42} +3.59422 q^{43} +3.26147 q^{44} -4.02047 q^{45} -3.48217 q^{46} +9.11758 q^{47} -2.16991 q^{48} +1.90915 q^{49} +0.537703 q^{50} +2.01542 q^{51} -0.832404 q^{52} +6.06820 q^{53} +2.80246 q^{54} -7.67500 q^{55} -2.98482 q^{56} -2.52734 q^{57} -3.05692 q^{58} -13.0104 q^{59} +5.10629 q^{60} +6.73229 q^{61} +5.74043 q^{62} -5.09953 q^{63} +1.00000 q^{64} +1.95884 q^{65} -7.07708 q^{66} +2.28839 q^{67} -0.928806 q^{68} +7.55597 q^{69} +7.02397 q^{70} +6.03999 q^{71} +1.70849 q^{72} +3.41169 q^{73} -0.186459 q^{74} -1.16676 q^{75} +1.16473 q^{76} -9.73490 q^{77} +1.80624 q^{78} -2.81530 q^{79} -2.35323 q^{80} -11.2065 q^{81} -2.25855 q^{82} +10.5789 q^{83} +6.47678 q^{84} +2.18570 q^{85} +3.59422 q^{86} +6.63323 q^{87} +3.26147 q^{88} +17.8579 q^{89} -4.02047 q^{90} +2.48457 q^{91} -3.48217 q^{92} -12.4562 q^{93} +9.11758 q^{94} -2.74087 q^{95} -2.16991 q^{96} -5.59501 q^{97} +1.90915 q^{98} +5.57219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.16991 −1.25280 −0.626398 0.779504i \(-0.715471\pi\)
−0.626398 + 0.779504i \(0.715471\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.35323 −1.05240 −0.526199 0.850362i \(-0.676383\pi\)
−0.526199 + 0.850362i \(0.676383\pi\)
\(6\) −2.16991 −0.885860
\(7\) −2.98482 −1.12816 −0.564078 0.825722i \(-0.690768\pi\)
−0.564078 + 0.825722i \(0.690768\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.70849 0.569496
\(10\) −2.35323 −0.744157
\(11\) 3.26147 0.983371 0.491685 0.870773i \(-0.336381\pi\)
0.491685 + 0.870773i \(0.336381\pi\)
\(12\) −2.16991 −0.626398
\(13\) −0.832404 −0.230867 −0.115434 0.993315i \(-0.536826\pi\)
−0.115434 + 0.993315i \(0.536826\pi\)
\(14\) −2.98482 −0.797727
\(15\) 5.10629 1.31844
\(16\) 1.00000 0.250000
\(17\) −0.928806 −0.225268 −0.112634 0.993637i \(-0.535929\pi\)
−0.112634 + 0.993637i \(0.535929\pi\)
\(18\) 1.70849 0.402695
\(19\) 1.16473 0.267206 0.133603 0.991035i \(-0.457345\pi\)
0.133603 + 0.991035i \(0.457345\pi\)
\(20\) −2.35323 −0.526199
\(21\) 6.47678 1.41335
\(22\) 3.26147 0.695348
\(23\) −3.48217 −0.726082 −0.363041 0.931773i \(-0.618261\pi\)
−0.363041 + 0.931773i \(0.618261\pi\)
\(24\) −2.16991 −0.442930
\(25\) 0.537703 0.107541
\(26\) −0.832404 −0.163248
\(27\) 2.80246 0.539333
\(28\) −2.98482 −0.564078
\(29\) −3.05692 −0.567656 −0.283828 0.958875i \(-0.591604\pi\)
−0.283828 + 0.958875i \(0.591604\pi\)
\(30\) 5.10629 0.932277
\(31\) 5.74043 1.03101 0.515505 0.856886i \(-0.327604\pi\)
0.515505 + 0.856886i \(0.327604\pi\)
\(32\) 1.00000 0.176777
\(33\) −7.07708 −1.23196
\(34\) −0.928806 −0.159289
\(35\) 7.02397 1.18727
\(36\) 1.70849 0.284748
\(37\) −0.186459 −0.0306537 −0.0153269 0.999883i \(-0.504879\pi\)
−0.0153269 + 0.999883i \(0.504879\pi\)
\(38\) 1.16473 0.188943
\(39\) 1.80624 0.289229
\(40\) −2.35323 −0.372079
\(41\) −2.25855 −0.352726 −0.176363 0.984325i \(-0.556433\pi\)
−0.176363 + 0.984325i \(0.556433\pi\)
\(42\) 6.47678 0.999388
\(43\) 3.59422 0.548114 0.274057 0.961714i \(-0.411634\pi\)
0.274057 + 0.961714i \(0.411634\pi\)
\(44\) 3.26147 0.491685
\(45\) −4.02047 −0.599337
\(46\) −3.48217 −0.513417
\(47\) 9.11758 1.32994 0.664968 0.746872i \(-0.268445\pi\)
0.664968 + 0.746872i \(0.268445\pi\)
\(48\) −2.16991 −0.313199
\(49\) 1.90915 0.272735
\(50\) 0.537703 0.0760426
\(51\) 2.01542 0.282215
\(52\) −0.832404 −0.115434
\(53\) 6.06820 0.833531 0.416765 0.909014i \(-0.363164\pi\)
0.416765 + 0.909014i \(0.363164\pi\)
\(54\) 2.80246 0.381366
\(55\) −7.67500 −1.03490
\(56\) −2.98482 −0.398863
\(57\) −2.52734 −0.334755
\(58\) −3.05692 −0.401393
\(59\) −13.0104 −1.69381 −0.846907 0.531741i \(-0.821538\pi\)
−0.846907 + 0.531741i \(0.821538\pi\)
\(60\) 5.10629 0.659219
\(61\) 6.73229 0.861981 0.430991 0.902356i \(-0.358164\pi\)
0.430991 + 0.902356i \(0.358164\pi\)
\(62\) 5.74043 0.729035
\(63\) −5.09953 −0.642481
\(64\) 1.00000 0.125000
\(65\) 1.95884 0.242964
\(66\) −7.07708 −0.871129
\(67\) 2.28839 0.279571 0.139786 0.990182i \(-0.455359\pi\)
0.139786 + 0.990182i \(0.455359\pi\)
\(68\) −0.928806 −0.112634
\(69\) 7.55597 0.909632
\(70\) 7.02397 0.839525
\(71\) 6.03999 0.716815 0.358408 0.933565i \(-0.383320\pi\)
0.358408 + 0.933565i \(0.383320\pi\)
\(72\) 1.70849 0.201347
\(73\) 3.41169 0.399308 0.199654 0.979866i \(-0.436018\pi\)
0.199654 + 0.979866i \(0.436018\pi\)
\(74\) −0.186459 −0.0216755
\(75\) −1.16676 −0.134726
\(76\) 1.16473 0.133603
\(77\) −9.73490 −1.10940
\(78\) 1.80624 0.204516
\(79\) −2.81530 −0.316746 −0.158373 0.987379i \(-0.550625\pi\)
−0.158373 + 0.987379i \(0.550625\pi\)
\(80\) −2.35323 −0.263099
\(81\) −11.2065 −1.24517
\(82\) −2.25855 −0.249415
\(83\) 10.5789 1.16118 0.580591 0.814195i \(-0.302822\pi\)
0.580591 + 0.814195i \(0.302822\pi\)
\(84\) 6.47678 0.706674
\(85\) 2.18570 0.237072
\(86\) 3.59422 0.387575
\(87\) 6.63323 0.711157
\(88\) 3.26147 0.347674
\(89\) 17.8579 1.89294 0.946469 0.322793i \(-0.104622\pi\)
0.946469 + 0.322793i \(0.104622\pi\)
\(90\) −4.02047 −0.423795
\(91\) 2.48457 0.260454
\(92\) −3.48217 −0.363041
\(93\) −12.4562 −1.29165
\(94\) 9.11758 0.940407
\(95\) −2.74087 −0.281207
\(96\) −2.16991 −0.221465
\(97\) −5.59501 −0.568087 −0.284044 0.958811i \(-0.591676\pi\)
−0.284044 + 0.958811i \(0.591676\pi\)
\(98\) 1.90915 0.192853
\(99\) 5.57219 0.560026
\(100\) 0.537703 0.0537703
\(101\) −6.64581 −0.661283 −0.330641 0.943756i \(-0.607265\pi\)
−0.330641 + 0.943756i \(0.607265\pi\)
\(102\) 2.01542 0.199556
\(103\) 9.91051 0.976511 0.488256 0.872701i \(-0.337633\pi\)
0.488256 + 0.872701i \(0.337633\pi\)
\(104\) −0.832404 −0.0816239
\(105\) −15.2414 −1.48740
\(106\) 6.06820 0.589395
\(107\) 4.50760 0.435766 0.217883 0.975975i \(-0.430085\pi\)
0.217883 + 0.975975i \(0.430085\pi\)
\(108\) 2.80246 0.269666
\(109\) −2.92956 −0.280601 −0.140301 0.990109i \(-0.544807\pi\)
−0.140301 + 0.990109i \(0.544807\pi\)
\(110\) −7.67500 −0.731783
\(111\) 0.404599 0.0384029
\(112\) −2.98482 −0.282039
\(113\) 1.44266 0.135714 0.0678570 0.997695i \(-0.478384\pi\)
0.0678570 + 0.997695i \(0.478384\pi\)
\(114\) −2.52734 −0.236707
\(115\) 8.19434 0.764127
\(116\) −3.05692 −0.283828
\(117\) −1.42215 −0.131478
\(118\) −13.0104 −1.19771
\(119\) 2.77232 0.254138
\(120\) 5.10629 0.466139
\(121\) −0.362804 −0.0329822
\(122\) 6.73229 0.609513
\(123\) 4.90084 0.441894
\(124\) 5.74043 0.515505
\(125\) 10.5008 0.939222
\(126\) −5.09953 −0.454302
\(127\) −18.3308 −1.62660 −0.813299 0.581847i \(-0.802330\pi\)
−0.813299 + 0.581847i \(0.802330\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.79912 −0.686674
\(130\) 1.95884 0.171802
\(131\) −1.69597 −0.148178 −0.0740889 0.997252i \(-0.523605\pi\)
−0.0740889 + 0.997252i \(0.523605\pi\)
\(132\) −7.07708 −0.615981
\(133\) −3.47649 −0.301450
\(134\) 2.28839 0.197687
\(135\) −6.59483 −0.567593
\(136\) −0.928806 −0.0796444
\(137\) −16.6391 −1.42158 −0.710789 0.703406i \(-0.751662\pi\)
−0.710789 + 0.703406i \(0.751662\pi\)
\(138\) 7.55597 0.643207
\(139\) −22.2620 −1.88824 −0.944118 0.329608i \(-0.893083\pi\)
−0.944118 + 0.329608i \(0.893083\pi\)
\(140\) 7.02397 0.593634
\(141\) −19.7843 −1.66614
\(142\) 6.03999 0.506865
\(143\) −2.71486 −0.227028
\(144\) 1.70849 0.142374
\(145\) 7.19364 0.597400
\(146\) 3.41169 0.282354
\(147\) −4.14267 −0.341682
\(148\) −0.186459 −0.0153269
\(149\) −2.19544 −0.179858 −0.0899288 0.995948i \(-0.528664\pi\)
−0.0899288 + 0.995948i \(0.528664\pi\)
\(150\) −1.16676 −0.0952658
\(151\) 21.6601 1.76267 0.881336 0.472490i \(-0.156645\pi\)
0.881336 + 0.472490i \(0.156645\pi\)
\(152\) 1.16473 0.0944717
\(153\) −1.58685 −0.128290
\(154\) −9.73490 −0.784461
\(155\) −13.5086 −1.08503
\(156\) 1.80624 0.144615
\(157\) 8.58779 0.685380 0.342690 0.939448i \(-0.388662\pi\)
0.342690 + 0.939448i \(0.388662\pi\)
\(158\) −2.81530 −0.223973
\(159\) −13.1674 −1.04424
\(160\) −2.35323 −0.186039
\(161\) 10.3936 0.819133
\(162\) −11.2065 −0.880468
\(163\) −12.2050 −0.955973 −0.477987 0.878367i \(-0.658633\pi\)
−0.477987 + 0.878367i \(0.658633\pi\)
\(164\) −2.25855 −0.176363
\(165\) 16.6540 1.29651
\(166\) 10.5789 0.821080
\(167\) 10.2158 0.790525 0.395263 0.918568i \(-0.370654\pi\)
0.395263 + 0.918568i \(0.370654\pi\)
\(168\) 6.47678 0.499694
\(169\) −12.3071 −0.946700
\(170\) 2.18570 0.167635
\(171\) 1.98992 0.152173
\(172\) 3.59422 0.274057
\(173\) −5.95829 −0.453000 −0.226500 0.974011i \(-0.572728\pi\)
−0.226500 + 0.974011i \(0.572728\pi\)
\(174\) 6.63323 0.502864
\(175\) −1.60495 −0.121322
\(176\) 3.26147 0.245843
\(177\) 28.2314 2.12200
\(178\) 17.8579 1.33851
\(179\) −25.5355 −1.90861 −0.954306 0.298830i \(-0.903404\pi\)
−0.954306 + 0.298830i \(0.903404\pi\)
\(180\) −4.02047 −0.299668
\(181\) −0.390180 −0.0290019 −0.0145009 0.999895i \(-0.504616\pi\)
−0.0145009 + 0.999895i \(0.504616\pi\)
\(182\) 2.48457 0.184169
\(183\) −14.6084 −1.07989
\(184\) −3.48217 −0.256709
\(185\) 0.438782 0.0322599
\(186\) −12.4562 −0.913331
\(187\) −3.02927 −0.221522
\(188\) 9.11758 0.664968
\(189\) −8.36483 −0.608452
\(190\) −2.74087 −0.198844
\(191\) −19.5953 −1.41786 −0.708932 0.705276i \(-0.750823\pi\)
−0.708932 + 0.705276i \(0.750823\pi\)
\(192\) −2.16991 −0.156599
\(193\) 27.3634 1.96966 0.984831 0.173518i \(-0.0555134\pi\)
0.984831 + 0.173518i \(0.0555134\pi\)
\(194\) −5.59501 −0.401698
\(195\) −4.25050 −0.304384
\(196\) 1.90915 0.136368
\(197\) −4.92877 −0.351160 −0.175580 0.984465i \(-0.556180\pi\)
−0.175580 + 0.984465i \(0.556180\pi\)
\(198\) 5.57219 0.395998
\(199\) −4.46565 −0.316561 −0.158281 0.987394i \(-0.550595\pi\)
−0.158281 + 0.987394i \(0.550595\pi\)
\(200\) 0.537703 0.0380213
\(201\) −4.96559 −0.350246
\(202\) −6.64581 −0.467598
\(203\) 9.12436 0.640404
\(204\) 2.01542 0.141108
\(205\) 5.31489 0.371208
\(206\) 9.91051 0.690498
\(207\) −5.94924 −0.413501
\(208\) −0.832404 −0.0577168
\(209\) 3.79872 0.262763
\(210\) −15.2414 −1.05175
\(211\) −26.5620 −1.82860 −0.914300 0.405037i \(-0.867259\pi\)
−0.914300 + 0.405037i \(0.867259\pi\)
\(212\) 6.06820 0.416765
\(213\) −13.1062 −0.898023
\(214\) 4.50760 0.308133
\(215\) −8.45804 −0.576833
\(216\) 2.80246 0.190683
\(217\) −17.1341 −1.16314
\(218\) −2.92956 −0.198415
\(219\) −7.40305 −0.500251
\(220\) −7.67500 −0.517448
\(221\) 0.773141 0.0520071
\(222\) 0.404599 0.0271549
\(223\) 4.85876 0.325367 0.162683 0.986678i \(-0.447985\pi\)
0.162683 + 0.986678i \(0.447985\pi\)
\(224\) −2.98482 −0.199432
\(225\) 0.918659 0.0612439
\(226\) 1.44266 0.0959642
\(227\) 11.9983 0.796356 0.398178 0.917308i \(-0.369643\pi\)
0.398178 + 0.917308i \(0.369643\pi\)
\(228\) −2.52734 −0.167377
\(229\) −28.6718 −1.89469 −0.947344 0.320217i \(-0.896244\pi\)
−0.947344 + 0.320217i \(0.896244\pi\)
\(230\) 8.19434 0.540319
\(231\) 21.1238 1.38985
\(232\) −3.05692 −0.200697
\(233\) 25.9732 1.70156 0.850780 0.525521i \(-0.176130\pi\)
0.850780 + 0.525521i \(0.176130\pi\)
\(234\) −1.42215 −0.0929690
\(235\) −21.4558 −1.39962
\(236\) −13.0104 −0.846907
\(237\) 6.10894 0.396818
\(238\) 2.77232 0.179703
\(239\) −7.37644 −0.477142 −0.238571 0.971125i \(-0.576679\pi\)
−0.238571 + 0.971125i \(0.576679\pi\)
\(240\) 5.10629 0.329610
\(241\) 3.64527 0.234813 0.117406 0.993084i \(-0.462542\pi\)
0.117406 + 0.993084i \(0.462542\pi\)
\(242\) −0.362804 −0.0233219
\(243\) 15.9097 1.02061
\(244\) 6.73229 0.430991
\(245\) −4.49267 −0.287026
\(246\) 4.90084 0.312466
\(247\) −0.969522 −0.0616892
\(248\) 5.74043 0.364517
\(249\) −22.9551 −1.45472
\(250\) 10.5008 0.664130
\(251\) −11.4270 −0.721266 −0.360633 0.932708i \(-0.617439\pi\)
−0.360633 + 0.932708i \(0.617439\pi\)
\(252\) −5.09953 −0.321240
\(253\) −11.3570 −0.714007
\(254\) −18.3308 −1.15018
\(255\) −4.74275 −0.297003
\(256\) 1.00000 0.0625000
\(257\) −1.00591 −0.0627468 −0.0313734 0.999508i \(-0.509988\pi\)
−0.0313734 + 0.999508i \(0.509988\pi\)
\(258\) −7.79912 −0.485552
\(259\) 0.556548 0.0345822
\(260\) 1.95884 0.121482
\(261\) −5.22272 −0.323278
\(262\) −1.69597 −0.104777
\(263\) −26.2139 −1.61642 −0.808209 0.588896i \(-0.799563\pi\)
−0.808209 + 0.588896i \(0.799563\pi\)
\(264\) −7.07708 −0.435564
\(265\) −14.2799 −0.877206
\(266\) −3.47649 −0.213158
\(267\) −38.7501 −2.37147
\(268\) 2.28839 0.139786
\(269\) 12.8463 0.783251 0.391626 0.920125i \(-0.371913\pi\)
0.391626 + 0.920125i \(0.371913\pi\)
\(270\) −6.59483 −0.401349
\(271\) −11.0513 −0.671318 −0.335659 0.941984i \(-0.608959\pi\)
−0.335659 + 0.941984i \(0.608959\pi\)
\(272\) −0.928806 −0.0563171
\(273\) −5.39129 −0.326296
\(274\) −16.6391 −1.00521
\(275\) 1.75370 0.105752
\(276\) 7.55597 0.454816
\(277\) −6.45607 −0.387907 −0.193954 0.981011i \(-0.562131\pi\)
−0.193954 + 0.981011i \(0.562131\pi\)
\(278\) −22.2620 −1.33518
\(279\) 9.80745 0.587157
\(280\) 7.02397 0.419763
\(281\) −12.1267 −0.723417 −0.361708 0.932291i \(-0.617806\pi\)
−0.361708 + 0.932291i \(0.617806\pi\)
\(282\) −19.7843 −1.17814
\(283\) 25.0839 1.49108 0.745541 0.666459i \(-0.232191\pi\)
0.745541 + 0.666459i \(0.232191\pi\)
\(284\) 6.03999 0.358408
\(285\) 5.94743 0.352295
\(286\) −2.71486 −0.160533
\(287\) 6.74136 0.397930
\(288\) 1.70849 0.100674
\(289\) −16.1373 −0.949254
\(290\) 7.19364 0.422425
\(291\) 12.1406 0.711697
\(292\) 3.41169 0.199654
\(293\) 1.76172 0.102921 0.0514604 0.998675i \(-0.483612\pi\)
0.0514604 + 0.998675i \(0.483612\pi\)
\(294\) −4.14267 −0.241605
\(295\) 30.6166 1.78257
\(296\) −0.186459 −0.0108377
\(297\) 9.14013 0.530364
\(298\) −2.19544 −0.127178
\(299\) 2.89857 0.167628
\(300\) −1.16676 −0.0673631
\(301\) −10.7281 −0.618358
\(302\) 21.6601 1.24640
\(303\) 14.4208 0.828452
\(304\) 1.16473 0.0668016
\(305\) −15.8426 −0.907147
\(306\) −1.58685 −0.0907144
\(307\) 17.7922 1.01546 0.507729 0.861517i \(-0.330485\pi\)
0.507729 + 0.861517i \(0.330485\pi\)
\(308\) −9.73490 −0.554698
\(309\) −21.5049 −1.22337
\(310\) −13.5086 −0.767234
\(311\) −11.1608 −0.632868 −0.316434 0.948614i \(-0.602486\pi\)
−0.316434 + 0.948614i \(0.602486\pi\)
\(312\) 1.80624 0.102258
\(313\) −22.1853 −1.25399 −0.626995 0.779023i \(-0.715715\pi\)
−0.626995 + 0.779023i \(0.715715\pi\)
\(314\) 8.58779 0.484637
\(315\) 12.0004 0.676145
\(316\) −2.81530 −0.158373
\(317\) −8.19846 −0.460472 −0.230236 0.973135i \(-0.573950\pi\)
−0.230236 + 0.973135i \(0.573950\pi\)
\(318\) −13.1674 −0.738392
\(319\) −9.97006 −0.558216
\(320\) −2.35323 −0.131550
\(321\) −9.78106 −0.545925
\(322\) 10.3936 0.579215
\(323\) −1.08180 −0.0601931
\(324\) −11.2065 −0.622585
\(325\) −0.447586 −0.0248276
\(326\) −12.2050 −0.675975
\(327\) 6.35687 0.351536
\(328\) −2.25855 −0.124707
\(329\) −27.2143 −1.50038
\(330\) 16.6540 0.916774
\(331\) −32.7930 −1.80247 −0.901234 0.433333i \(-0.857338\pi\)
−0.901234 + 0.433333i \(0.857338\pi\)
\(332\) 10.5789 0.580591
\(333\) −0.318564 −0.0174572
\(334\) 10.2158 0.558986
\(335\) −5.38512 −0.294220
\(336\) 6.47678 0.353337
\(337\) 17.8495 0.972325 0.486163 0.873868i \(-0.338396\pi\)
0.486163 + 0.873868i \(0.338396\pi\)
\(338\) −12.3071 −0.669418
\(339\) −3.13043 −0.170022
\(340\) 2.18570 0.118536
\(341\) 18.7222 1.01387
\(342\) 1.98992 0.107603
\(343\) 15.1953 0.820468
\(344\) 3.59422 0.193787
\(345\) −17.7810 −0.957294
\(346\) −5.95829 −0.320320
\(347\) −23.4732 −1.26011 −0.630053 0.776552i \(-0.716967\pi\)
−0.630053 + 0.776552i \(0.716967\pi\)
\(348\) 6.63323 0.355578
\(349\) −5.79132 −0.310002 −0.155001 0.987914i \(-0.549538\pi\)
−0.155001 + 0.987914i \(0.549538\pi\)
\(350\) −1.60495 −0.0857879
\(351\) −2.33277 −0.124514
\(352\) 3.26147 0.173837
\(353\) −3.57572 −0.190316 −0.0951581 0.995462i \(-0.530336\pi\)
−0.0951581 + 0.995462i \(0.530336\pi\)
\(354\) 28.2314 1.50048
\(355\) −14.2135 −0.754375
\(356\) 17.8579 0.946469
\(357\) −6.01567 −0.318383
\(358\) −25.5355 −1.34959
\(359\) −10.1988 −0.538271 −0.269136 0.963102i \(-0.586738\pi\)
−0.269136 + 0.963102i \(0.586738\pi\)
\(360\) −4.02047 −0.211897
\(361\) −17.6434 −0.928601
\(362\) −0.390180 −0.0205074
\(363\) 0.787251 0.0413200
\(364\) 2.48457 0.130227
\(365\) −8.02850 −0.420231
\(366\) −14.6084 −0.763595
\(367\) −13.0245 −0.679876 −0.339938 0.940448i \(-0.610406\pi\)
−0.339938 + 0.940448i \(0.610406\pi\)
\(368\) −3.48217 −0.181520
\(369\) −3.85871 −0.200876
\(370\) 0.438782 0.0228112
\(371\) −18.1125 −0.940353
\(372\) −12.4562 −0.645823
\(373\) −10.3512 −0.535965 −0.267983 0.963424i \(-0.586357\pi\)
−0.267983 + 0.963424i \(0.586357\pi\)
\(374\) −3.02927 −0.156640
\(375\) −22.7858 −1.17665
\(376\) 9.11758 0.470204
\(377\) 2.54459 0.131053
\(378\) −8.36483 −0.430240
\(379\) 13.3344 0.684943 0.342472 0.939528i \(-0.388736\pi\)
0.342472 + 0.939528i \(0.388736\pi\)
\(380\) −2.74087 −0.140604
\(381\) 39.7761 2.03779
\(382\) −19.5953 −1.00258
\(383\) −15.1494 −0.774098 −0.387049 0.922059i \(-0.626506\pi\)
−0.387049 + 0.922059i \(0.626506\pi\)
\(384\) −2.16991 −0.110733
\(385\) 22.9085 1.16752
\(386\) 27.3634 1.39276
\(387\) 6.14069 0.312149
\(388\) −5.59501 −0.284044
\(389\) −12.4338 −0.630420 −0.315210 0.949022i \(-0.602075\pi\)
−0.315210 + 0.949022i \(0.602075\pi\)
\(390\) −4.25050 −0.215232
\(391\) 3.23425 0.163563
\(392\) 1.90915 0.0964266
\(393\) 3.68010 0.185636
\(394\) −4.92877 −0.248308
\(395\) 6.62506 0.333343
\(396\) 5.57219 0.280013
\(397\) −22.5317 −1.13083 −0.565417 0.824805i \(-0.691285\pi\)
−0.565417 + 0.824805i \(0.691285\pi\)
\(398\) −4.46565 −0.223843
\(399\) 7.54366 0.377656
\(400\) 0.537703 0.0268851
\(401\) 33.7936 1.68757 0.843786 0.536679i \(-0.180322\pi\)
0.843786 + 0.536679i \(0.180322\pi\)
\(402\) −4.96559 −0.247661
\(403\) −4.77835 −0.238027
\(404\) −6.64581 −0.330641
\(405\) 26.3716 1.31041
\(406\) 9.12436 0.452834
\(407\) −0.608132 −0.0301440
\(408\) 2.01542 0.0997782
\(409\) −30.0250 −1.48464 −0.742319 0.670046i \(-0.766274\pi\)
−0.742319 + 0.670046i \(0.766274\pi\)
\(410\) 5.31489 0.262484
\(411\) 36.1054 1.78095
\(412\) 9.91051 0.488256
\(413\) 38.8338 1.91089
\(414\) −5.94924 −0.292389
\(415\) −24.8945 −1.22202
\(416\) −0.832404 −0.0408119
\(417\) 48.3064 2.36557
\(418\) 3.79872 0.185801
\(419\) 22.2807 1.08849 0.544243 0.838928i \(-0.316817\pi\)
0.544243 + 0.838928i \(0.316817\pi\)
\(420\) −15.2414 −0.743702
\(421\) 27.1606 1.32373 0.661864 0.749624i \(-0.269765\pi\)
0.661864 + 0.749624i \(0.269765\pi\)
\(422\) −26.5620 −1.29302
\(423\) 15.5773 0.757394
\(424\) 6.06820 0.294698
\(425\) −0.499421 −0.0242255
\(426\) −13.1062 −0.634998
\(427\) −20.0947 −0.972449
\(428\) 4.50760 0.217883
\(429\) 5.89099 0.284420
\(430\) −8.45804 −0.407883
\(431\) 8.37480 0.403400 0.201700 0.979447i \(-0.435353\pi\)
0.201700 + 0.979447i \(0.435353\pi\)
\(432\) 2.80246 0.134833
\(433\) 33.4393 1.60699 0.803496 0.595310i \(-0.202971\pi\)
0.803496 + 0.595310i \(0.202971\pi\)
\(434\) −17.1341 −0.822465
\(435\) −15.6095 −0.748420
\(436\) −2.92956 −0.140301
\(437\) −4.05577 −0.194014
\(438\) −7.40305 −0.353731
\(439\) 13.8872 0.662799 0.331400 0.943490i \(-0.392479\pi\)
0.331400 + 0.943490i \(0.392479\pi\)
\(440\) −7.67500 −0.365891
\(441\) 3.26176 0.155322
\(442\) 0.773141 0.0367746
\(443\) −13.5757 −0.645002 −0.322501 0.946569i \(-0.604524\pi\)
−0.322501 + 0.946569i \(0.604524\pi\)
\(444\) 0.404599 0.0192014
\(445\) −42.0239 −1.99212
\(446\) 4.85876 0.230069
\(447\) 4.76390 0.225325
\(448\) −2.98482 −0.141019
\(449\) −25.2855 −1.19329 −0.596647 0.802504i \(-0.703501\pi\)
−0.596647 + 0.802504i \(0.703501\pi\)
\(450\) 0.918659 0.0433060
\(451\) −7.36619 −0.346860
\(452\) 1.44266 0.0678570
\(453\) −47.0003 −2.20827
\(454\) 11.9983 0.563109
\(455\) −5.84678 −0.274101
\(456\) −2.52734 −0.118354
\(457\) 28.0984 1.31439 0.657193 0.753722i \(-0.271744\pi\)
0.657193 + 0.753722i \(0.271744\pi\)
\(458\) −28.6718 −1.33975
\(459\) −2.60294 −0.121495
\(460\) 8.19434 0.382063
\(461\) 7.28760 0.339418 0.169709 0.985494i \(-0.445717\pi\)
0.169709 + 0.985494i \(0.445717\pi\)
\(462\) 21.1238 0.982769
\(463\) −24.3859 −1.13331 −0.566654 0.823956i \(-0.691762\pi\)
−0.566654 + 0.823956i \(0.691762\pi\)
\(464\) −3.05692 −0.141914
\(465\) 29.3123 1.35932
\(466\) 25.9732 1.20319
\(467\) −29.9631 −1.38653 −0.693263 0.720684i \(-0.743828\pi\)
−0.693263 + 0.720684i \(0.743828\pi\)
\(468\) −1.42215 −0.0657390
\(469\) −6.83043 −0.315400
\(470\) −21.4558 −0.989682
\(471\) −18.6347 −0.858642
\(472\) −13.0104 −0.598854
\(473\) 11.7225 0.538999
\(474\) 6.10894 0.280593
\(475\) 0.626276 0.0287355
\(476\) 2.77232 0.127069
\(477\) 10.3674 0.474693
\(478\) −7.37644 −0.337391
\(479\) 14.9194 0.681686 0.340843 0.940120i \(-0.389288\pi\)
0.340843 + 0.940120i \(0.389288\pi\)
\(480\) 5.10629 0.233069
\(481\) 0.155209 0.00707694
\(482\) 3.64527 0.166038
\(483\) −22.5532 −1.02621
\(484\) −0.362804 −0.0164911
\(485\) 13.1664 0.597854
\(486\) 15.9097 0.721681
\(487\) 22.0249 0.998045 0.499023 0.866589i \(-0.333692\pi\)
0.499023 + 0.866589i \(0.333692\pi\)
\(488\) 6.73229 0.304756
\(489\) 26.4838 1.19764
\(490\) −4.49267 −0.202958
\(491\) −18.9968 −0.857313 −0.428656 0.903468i \(-0.641013\pi\)
−0.428656 + 0.903468i \(0.641013\pi\)
\(492\) 4.90084 0.220947
\(493\) 2.83928 0.127875
\(494\) −0.969522 −0.0436208
\(495\) −13.1127 −0.589370
\(496\) 5.74043 0.257753
\(497\) −18.0283 −0.808679
\(498\) −22.9551 −1.02864
\(499\) −14.1049 −0.631422 −0.315711 0.948855i \(-0.602243\pi\)
−0.315711 + 0.948855i \(0.602243\pi\)
\(500\) 10.5008 0.469611
\(501\) −22.1674 −0.990367
\(502\) −11.4270 −0.510012
\(503\) −24.6559 −1.09935 −0.549676 0.835378i \(-0.685249\pi\)
−0.549676 + 0.835378i \(0.685249\pi\)
\(504\) −5.09953 −0.227151
\(505\) 15.6391 0.695933
\(506\) −11.3570 −0.504879
\(507\) 26.7053 1.18602
\(508\) −18.3308 −0.813299
\(509\) 12.2854 0.544540 0.272270 0.962221i \(-0.412226\pi\)
0.272270 + 0.962221i \(0.412226\pi\)
\(510\) −4.74275 −0.210013
\(511\) −10.1833 −0.450482
\(512\) 1.00000 0.0441942
\(513\) 3.26409 0.144113
\(514\) −1.00591 −0.0443687
\(515\) −23.3217 −1.02768
\(516\) −7.79912 −0.343337
\(517\) 29.7367 1.30782
\(518\) 0.556548 0.0244533
\(519\) 12.9289 0.567517
\(520\) 1.95884 0.0859008
\(521\) 19.7011 0.863123 0.431561 0.902084i \(-0.357963\pi\)
0.431561 + 0.902084i \(0.357963\pi\)
\(522\) −5.22272 −0.228592
\(523\) 20.6282 0.902007 0.451004 0.892522i \(-0.351066\pi\)
0.451004 + 0.892522i \(0.351066\pi\)
\(524\) −1.69597 −0.0740889
\(525\) 3.48258 0.151992
\(526\) −26.2139 −1.14298
\(527\) −5.33174 −0.232254
\(528\) −7.07708 −0.307991
\(529\) −10.8745 −0.472805
\(530\) −14.2799 −0.620278
\(531\) −22.2282 −0.964621
\(532\) −3.47649 −0.150725
\(533\) 1.88002 0.0814329
\(534\) −38.7501 −1.67688
\(535\) −10.6074 −0.458599
\(536\) 2.28839 0.0988434
\(537\) 55.4096 2.39110
\(538\) 12.8463 0.553842
\(539\) 6.22663 0.268200
\(540\) −6.59483 −0.283796
\(541\) 21.1184 0.907949 0.453975 0.891015i \(-0.350006\pi\)
0.453975 + 0.891015i \(0.350006\pi\)
\(542\) −11.0513 −0.474694
\(543\) 0.846654 0.0363334
\(544\) −0.928806 −0.0398222
\(545\) 6.89394 0.295304
\(546\) −5.39129 −0.230726
\(547\) 3.00351 0.128421 0.0642104 0.997936i \(-0.479547\pi\)
0.0642104 + 0.997936i \(0.479547\pi\)
\(548\) −16.6391 −0.710789
\(549\) 11.5020 0.490895
\(550\) 1.75370 0.0747781
\(551\) −3.56047 −0.151681
\(552\) 7.55597 0.321603
\(553\) 8.40317 0.357339
\(554\) −6.45607 −0.274292
\(555\) −0.952116 −0.0404151
\(556\) −22.2620 −0.944118
\(557\) 15.8627 0.672125 0.336063 0.941840i \(-0.390905\pi\)
0.336063 + 0.941840i \(0.390905\pi\)
\(558\) 9.80745 0.415183
\(559\) −2.99184 −0.126541
\(560\) 7.02397 0.296817
\(561\) 6.57323 0.277522
\(562\) −12.1267 −0.511533
\(563\) −39.4995 −1.66470 −0.832352 0.554247i \(-0.813006\pi\)
−0.832352 + 0.554247i \(0.813006\pi\)
\(564\) −19.7843 −0.833069
\(565\) −3.39491 −0.142825
\(566\) 25.0839 1.05435
\(567\) 33.4495 1.40475
\(568\) 6.03999 0.253432
\(569\) 45.4813 1.90668 0.953338 0.301906i \(-0.0976229\pi\)
0.953338 + 0.301906i \(0.0976229\pi\)
\(570\) 5.94743 0.249110
\(571\) −0.956957 −0.0400474 −0.0200237 0.999800i \(-0.506374\pi\)
−0.0200237 + 0.999800i \(0.506374\pi\)
\(572\) −2.71486 −0.113514
\(573\) 42.5199 1.77629
\(574\) 6.74136 0.281379
\(575\) −1.87237 −0.0780832
\(576\) 1.70849 0.0711870
\(577\) −38.7347 −1.61255 −0.806273 0.591544i \(-0.798519\pi\)
−0.806273 + 0.591544i \(0.798519\pi\)
\(578\) −16.1373 −0.671224
\(579\) −59.3760 −2.46758
\(580\) 7.19364 0.298700
\(581\) −31.5760 −1.30999
\(582\) 12.1406 0.503246
\(583\) 19.7912 0.819670
\(584\) 3.41169 0.141177
\(585\) 3.34666 0.138367
\(586\) 1.76172 0.0727760
\(587\) 43.3425 1.78893 0.894467 0.447133i \(-0.147555\pi\)
0.894467 + 0.447133i \(0.147555\pi\)
\(588\) −4.14267 −0.170841
\(589\) 6.68602 0.275493
\(590\) 30.6166 1.26046
\(591\) 10.6950 0.439932
\(592\) −0.186459 −0.00766343
\(593\) −7.45527 −0.306151 −0.153076 0.988214i \(-0.548918\pi\)
−0.153076 + 0.988214i \(0.548918\pi\)
\(594\) 9.14013 0.375024
\(595\) −6.52391 −0.267454
\(596\) −2.19544 −0.0899288
\(597\) 9.69003 0.396587
\(598\) 2.89857 0.118531
\(599\) −44.8807 −1.83378 −0.916888 0.399144i \(-0.869307\pi\)
−0.916888 + 0.399144i \(0.869307\pi\)
\(600\) −1.16676 −0.0476329
\(601\) 12.3336 0.503097 0.251549 0.967845i \(-0.419060\pi\)
0.251549 + 0.967845i \(0.419060\pi\)
\(602\) −10.7281 −0.437245
\(603\) 3.90969 0.159215
\(604\) 21.6601 0.881336
\(605\) 0.853763 0.0347104
\(606\) 14.4208 0.585804
\(607\) −30.5734 −1.24094 −0.620468 0.784232i \(-0.713057\pi\)
−0.620468 + 0.784232i \(0.713057\pi\)
\(608\) 1.16473 0.0472358
\(609\) −19.7990 −0.802296
\(610\) −15.8426 −0.641450
\(611\) −7.58951 −0.307039
\(612\) −1.58685 −0.0641448
\(613\) 28.6355 1.15658 0.578289 0.815832i \(-0.303721\pi\)
0.578289 + 0.815832i \(0.303721\pi\)
\(614\) 17.7922 0.718037
\(615\) −11.5328 −0.465048
\(616\) −9.73490 −0.392230
\(617\) 12.6450 0.509069 0.254534 0.967064i \(-0.418078\pi\)
0.254534 + 0.967064i \(0.418078\pi\)
\(618\) −21.5049 −0.865053
\(619\) 6.60021 0.265285 0.132642 0.991164i \(-0.457654\pi\)
0.132642 + 0.991164i \(0.457654\pi\)
\(620\) −13.5086 −0.542517
\(621\) −9.75862 −0.391600
\(622\) −11.1608 −0.447505
\(623\) −53.3028 −2.13553
\(624\) 1.80624 0.0723074
\(625\) −27.3994 −1.09598
\(626\) −22.1853 −0.886705
\(627\) −8.24286 −0.329188
\(628\) 8.58779 0.342690
\(629\) 0.173185 0.00690532
\(630\) 12.0004 0.478107
\(631\) −25.2557 −1.00541 −0.502707 0.864457i \(-0.667662\pi\)
−0.502707 + 0.864457i \(0.667662\pi\)
\(632\) −2.81530 −0.111987
\(633\) 57.6369 2.29086
\(634\) −8.19846 −0.325603
\(635\) 43.1367 1.71183
\(636\) −13.1674 −0.522122
\(637\) −1.58918 −0.0629657
\(638\) −9.97006 −0.394718
\(639\) 10.3193 0.408224
\(640\) −2.35323 −0.0930197
\(641\) −38.2841 −1.51213 −0.756066 0.654496i \(-0.772881\pi\)
−0.756066 + 0.654496i \(0.772881\pi\)
\(642\) −9.78106 −0.386028
\(643\) 17.6205 0.694884 0.347442 0.937701i \(-0.387050\pi\)
0.347442 + 0.937701i \(0.387050\pi\)
\(644\) 10.3936 0.409567
\(645\) 18.3531 0.722654
\(646\) −1.08180 −0.0425630
\(647\) −47.2984 −1.85949 −0.929745 0.368203i \(-0.879973\pi\)
−0.929745 + 0.368203i \(0.879973\pi\)
\(648\) −11.2065 −0.440234
\(649\) −42.4331 −1.66565
\(650\) −0.447586 −0.0175557
\(651\) 37.1795 1.45718
\(652\) −12.2050 −0.477987
\(653\) 17.6773 0.691767 0.345883 0.938278i \(-0.387579\pi\)
0.345883 + 0.938278i \(0.387579\pi\)
\(654\) 6.35687 0.248573
\(655\) 3.99102 0.155942
\(656\) −2.25855 −0.0881815
\(657\) 5.82884 0.227405
\(658\) −27.2143 −1.06093
\(659\) 41.7114 1.62485 0.812424 0.583068i \(-0.198148\pi\)
0.812424 + 0.583068i \(0.198148\pi\)
\(660\) 16.6540 0.648257
\(661\) 47.3867 1.84313 0.921565 0.388224i \(-0.126911\pi\)
0.921565 + 0.388224i \(0.126911\pi\)
\(662\) −32.7930 −1.27454
\(663\) −1.67764 −0.0651543
\(664\) 10.5789 0.410540
\(665\) 8.18100 0.317246
\(666\) −0.318564 −0.0123441
\(667\) 10.6447 0.412165
\(668\) 10.2158 0.395263
\(669\) −10.5430 −0.407618
\(670\) −5.38512 −0.208045
\(671\) 21.9572 0.847647
\(672\) 6.47678 0.249847
\(673\) −12.6966 −0.489417 −0.244708 0.969597i \(-0.578692\pi\)
−0.244708 + 0.969597i \(0.578692\pi\)
\(674\) 17.8495 0.687538
\(675\) 1.50689 0.0580001
\(676\) −12.3071 −0.473350
\(677\) −24.3140 −0.934463 −0.467232 0.884135i \(-0.654749\pi\)
−0.467232 + 0.884135i \(0.654749\pi\)
\(678\) −3.13043 −0.120224
\(679\) 16.7001 0.640891
\(680\) 2.18570 0.0838176
\(681\) −26.0352 −0.997672
\(682\) 18.7222 0.716911
\(683\) 5.69273 0.217826 0.108913 0.994051i \(-0.465263\pi\)
0.108913 + 0.994051i \(0.465263\pi\)
\(684\) 1.98992 0.0760865
\(685\) 39.1558 1.49606
\(686\) 15.1953 0.580158
\(687\) 62.2152 2.37366
\(688\) 3.59422 0.137028
\(689\) −5.05119 −0.192435
\(690\) −17.7810 −0.676909
\(691\) 26.6737 1.01472 0.507358 0.861735i \(-0.330622\pi\)
0.507358 + 0.861735i \(0.330622\pi\)
\(692\) −5.95829 −0.226500
\(693\) −16.6320 −0.631797
\(694\) −23.4732 −0.891029
\(695\) 52.3876 1.98717
\(696\) 6.63323 0.251432
\(697\) 2.09775 0.0794580
\(698\) −5.79132 −0.219205
\(699\) −56.3594 −2.13171
\(700\) −1.60495 −0.0606612
\(701\) 38.7652 1.46414 0.732072 0.681228i \(-0.238554\pi\)
0.732072 + 0.681228i \(0.238554\pi\)
\(702\) −2.33277 −0.0880449
\(703\) −0.217174 −0.00819087
\(704\) 3.26147 0.122921
\(705\) 46.5570 1.75344
\(706\) −3.57572 −0.134574
\(707\) 19.8365 0.746030
\(708\) 28.2314 1.06100
\(709\) −24.4641 −0.918769 −0.459385 0.888238i \(-0.651930\pi\)
−0.459385 + 0.888238i \(0.651930\pi\)
\(710\) −14.2135 −0.533423
\(711\) −4.80991 −0.180386
\(712\) 17.8579 0.669255
\(713\) −19.9891 −0.748598
\(714\) −6.01567 −0.225131
\(715\) 6.38870 0.238924
\(716\) −25.5355 −0.954306
\(717\) 16.0062 0.597762
\(718\) −10.1988 −0.380615
\(719\) 20.9300 0.780557 0.390278 0.920697i \(-0.372379\pi\)
0.390278 + 0.920697i \(0.372379\pi\)
\(720\) −4.02047 −0.149834
\(721\) −29.5811 −1.10166
\(722\) −17.6434 −0.656620
\(723\) −7.90990 −0.294172
\(724\) −0.390180 −0.0145009
\(725\) −1.64371 −0.0610460
\(726\) 0.787251 0.0292176
\(727\) 41.2124 1.52848 0.764241 0.644930i \(-0.223114\pi\)
0.764241 + 0.644930i \(0.223114\pi\)
\(728\) 2.48457 0.0920845
\(729\) −0.903045 −0.0334461
\(730\) −8.02850 −0.297148
\(731\) −3.33833 −0.123473
\(732\) −14.6084 −0.539943
\(733\) −14.4327 −0.533083 −0.266542 0.963823i \(-0.585881\pi\)
−0.266542 + 0.963823i \(0.585881\pi\)
\(734\) −13.0245 −0.480745
\(735\) 9.74867 0.359585
\(736\) −3.48217 −0.128354
\(737\) 7.46352 0.274922
\(738\) −3.85871 −0.142041
\(739\) −10.8270 −0.398276 −0.199138 0.979971i \(-0.563814\pi\)
−0.199138 + 0.979971i \(0.563814\pi\)
\(740\) 0.438782 0.0161300
\(741\) 2.10377 0.0772839
\(742\) −18.1125 −0.664930
\(743\) −18.9790 −0.696272 −0.348136 0.937444i \(-0.613185\pi\)
−0.348136 + 0.937444i \(0.613185\pi\)
\(744\) −12.4562 −0.456666
\(745\) 5.16638 0.189282
\(746\) −10.3512 −0.378985
\(747\) 18.0739 0.661289
\(748\) −3.02927 −0.110761
\(749\) −13.4544 −0.491612
\(750\) −22.7858 −0.832019
\(751\) −33.1438 −1.20943 −0.604717 0.796441i \(-0.706714\pi\)
−0.604717 + 0.796441i \(0.706714\pi\)
\(752\) 9.11758 0.332484
\(753\) 24.7955 0.903599
\(754\) 2.54459 0.0926686
\(755\) −50.9712 −1.85503
\(756\) −8.36483 −0.304226
\(757\) 34.4302 1.25139 0.625693 0.780070i \(-0.284816\pi\)
0.625693 + 0.780070i \(0.284816\pi\)
\(758\) 13.3344 0.484328
\(759\) 24.6436 0.894505
\(760\) −2.74087 −0.0994218
\(761\) 13.3847 0.485194 0.242597 0.970127i \(-0.422001\pi\)
0.242597 + 0.970127i \(0.422001\pi\)
\(762\) 39.7761 1.44094
\(763\) 8.74421 0.316562
\(764\) −19.5953 −0.708932
\(765\) 3.73424 0.135012
\(766\) −15.1494 −0.547370
\(767\) 10.8299 0.391046
\(768\) −2.16991 −0.0782997
\(769\) −46.3553 −1.67162 −0.835808 0.549022i \(-0.815000\pi\)
−0.835808 + 0.549022i \(0.815000\pi\)
\(770\) 22.9085 0.825565
\(771\) 2.18273 0.0786089
\(772\) 27.3634 0.984831
\(773\) 15.7375 0.566037 0.283019 0.959114i \(-0.408664\pi\)
0.283019 + 0.959114i \(0.408664\pi\)
\(774\) 6.14069 0.220722
\(775\) 3.08664 0.110875
\(776\) −5.59501 −0.200849
\(777\) −1.20766 −0.0433244
\(778\) −12.4338 −0.445775
\(779\) −2.63059 −0.0942506
\(780\) −4.25050 −0.152192
\(781\) 19.6993 0.704895
\(782\) 3.23425 0.115657
\(783\) −8.56689 −0.306156
\(784\) 1.90915 0.0681839
\(785\) −20.2091 −0.721293
\(786\) 3.68010 0.131265
\(787\) −1.78885 −0.0637657 −0.0318828 0.999492i \(-0.510150\pi\)
−0.0318828 + 0.999492i \(0.510150\pi\)
\(788\) −4.92877 −0.175580
\(789\) 56.8817 2.02504
\(790\) 6.62506 0.235709
\(791\) −4.30608 −0.153106
\(792\) 5.57219 0.197999
\(793\) −5.60398 −0.199003
\(794\) −22.5317 −0.799620
\(795\) 30.9860 1.09896
\(796\) −4.46565 −0.158281
\(797\) 1.11191 0.0393857 0.0196929 0.999806i \(-0.493731\pi\)
0.0196929 + 0.999806i \(0.493731\pi\)
\(798\) 7.54366 0.267043
\(799\) −8.46846 −0.299593
\(800\) 0.537703 0.0190107
\(801\) 30.5101 1.07802
\(802\) 33.7936 1.19329
\(803\) 11.1271 0.392668
\(804\) −4.96559 −0.175123
\(805\) −24.4586 −0.862054
\(806\) −4.77835 −0.168310
\(807\) −27.8752 −0.981254
\(808\) −6.64581 −0.233799
\(809\) −55.6931 −1.95806 −0.979032 0.203706i \(-0.934701\pi\)
−0.979032 + 0.203706i \(0.934701\pi\)
\(810\) 26.3716 0.926603
\(811\) 9.26811 0.325447 0.162724 0.986672i \(-0.447972\pi\)
0.162724 + 0.986672i \(0.447972\pi\)
\(812\) 9.12436 0.320202
\(813\) 23.9803 0.841024
\(814\) −0.608132 −0.0213150
\(815\) 28.7213 1.00606
\(816\) 2.01542 0.0705538
\(817\) 4.18628 0.146459
\(818\) −30.0250 −1.04980
\(819\) 4.24487 0.148328
\(820\) 5.31489 0.185604
\(821\) 40.8524 1.42576 0.712879 0.701287i \(-0.247391\pi\)
0.712879 + 0.701287i \(0.247391\pi\)
\(822\) 36.1054 1.25932
\(823\) 17.2494 0.601276 0.300638 0.953738i \(-0.402800\pi\)
0.300638 + 0.953738i \(0.402800\pi\)
\(824\) 9.91051 0.345249
\(825\) −3.80537 −0.132486
\(826\) 38.8338 1.35120
\(827\) 22.7270 0.790296 0.395148 0.918618i \(-0.370693\pi\)
0.395148 + 0.918618i \(0.370693\pi\)
\(828\) −5.94924 −0.206750
\(829\) 51.0696 1.77372 0.886860 0.462039i \(-0.152882\pi\)
0.886860 + 0.462039i \(0.152882\pi\)
\(830\) −24.8945 −0.864102
\(831\) 14.0091 0.485969
\(832\) −0.832404 −0.0288584
\(833\) −1.77323 −0.0614387
\(834\) 48.3064 1.67271
\(835\) −24.0402 −0.831947
\(836\) 3.79872 0.131381
\(837\) 16.0873 0.556058
\(838\) 22.2807 0.769676
\(839\) −27.2821 −0.941882 −0.470941 0.882165i \(-0.656086\pi\)
−0.470941 + 0.882165i \(0.656086\pi\)
\(840\) −15.2414 −0.525877
\(841\) −19.6552 −0.677767
\(842\) 27.1606 0.936018
\(843\) 26.3137 0.906293
\(844\) −26.5620 −0.914300
\(845\) 28.9615 0.996305
\(846\) 15.5773 0.535558
\(847\) 1.08291 0.0372091
\(848\) 6.06820 0.208383
\(849\) −54.4297 −1.86802
\(850\) −0.499421 −0.0171300
\(851\) 0.649282 0.0222571
\(852\) −13.1062 −0.449011
\(853\) 39.8085 1.36302 0.681508 0.731811i \(-0.261325\pi\)
0.681508 + 0.731811i \(0.261325\pi\)
\(854\) −20.0947 −0.687625
\(855\) −4.68275 −0.160147
\(856\) 4.50760 0.154066
\(857\) −5.20800 −0.177902 −0.0889510 0.996036i \(-0.528351\pi\)
−0.0889510 + 0.996036i \(0.528351\pi\)
\(858\) 5.89099 0.201115
\(859\) 24.7829 0.845583 0.422791 0.906227i \(-0.361050\pi\)
0.422791 + 0.906227i \(0.361050\pi\)
\(860\) −8.45804 −0.288417
\(861\) −14.6281 −0.498525
\(862\) 8.37480 0.285247
\(863\) −4.66071 −0.158652 −0.0793261 0.996849i \(-0.525277\pi\)
−0.0793261 + 0.996849i \(0.525277\pi\)
\(864\) 2.80246 0.0953415
\(865\) 14.0212 0.476737
\(866\) 33.4393 1.13631
\(867\) 35.0165 1.18922
\(868\) −17.1341 −0.581570
\(869\) −9.18203 −0.311479
\(870\) −15.6095 −0.529213
\(871\) −1.90486 −0.0645439
\(872\) −2.92956 −0.0992075
\(873\) −9.55902 −0.323524
\(874\) −4.05577 −0.137188
\(875\) −31.3431 −1.05959
\(876\) −7.40305 −0.250126
\(877\) −23.7311 −0.801342 −0.400671 0.916222i \(-0.631223\pi\)
−0.400671 + 0.916222i \(0.631223\pi\)
\(878\) 13.8872 0.468670
\(879\) −3.82277 −0.128939
\(880\) −7.67500 −0.258724
\(881\) −37.2553 −1.25516 −0.627582 0.778551i \(-0.715955\pi\)
−0.627582 + 0.778551i \(0.715955\pi\)
\(882\) 3.26176 0.109829
\(883\) −40.1527 −1.35125 −0.675623 0.737248i \(-0.736125\pi\)
−0.675623 + 0.737248i \(0.736125\pi\)
\(884\) 0.773141 0.0260035
\(885\) −66.4351 −2.23319
\(886\) −13.5757 −0.456085
\(887\) −41.1548 −1.38184 −0.690921 0.722930i \(-0.742795\pi\)
−0.690921 + 0.722930i \(0.742795\pi\)
\(888\) 0.404599 0.0135775
\(889\) 54.7142 1.83505
\(890\) −42.0239 −1.40864
\(891\) −36.5498 −1.22446
\(892\) 4.85876 0.162683
\(893\) 10.6195 0.355367
\(894\) 4.76390 0.159329
\(895\) 60.0910 2.00862
\(896\) −2.98482 −0.0997158
\(897\) −6.28962 −0.210004
\(898\) −25.2855 −0.843787
\(899\) −17.5480 −0.585259
\(900\) 0.918659 0.0306220
\(901\) −5.63617 −0.187768
\(902\) −7.36619 −0.245267
\(903\) 23.2790 0.774676
\(904\) 1.44266 0.0479821
\(905\) 0.918184 0.0305215
\(906\) −47.0003 −1.56148
\(907\) 12.9280 0.429267 0.214634 0.976695i \(-0.431144\pi\)
0.214634 + 0.976695i \(0.431144\pi\)
\(908\) 11.9983 0.398178
\(909\) −11.3543 −0.376598
\(910\) −5.84678 −0.193819
\(911\) −17.8206 −0.590421 −0.295211 0.955432i \(-0.595390\pi\)
−0.295211 + 0.955432i \(0.595390\pi\)
\(912\) −2.52734 −0.0836887
\(913\) 34.5027 1.14187
\(914\) 28.0984 0.929412
\(915\) 34.3770 1.13647
\(916\) −28.6718 −0.947344
\(917\) 5.06217 0.167168
\(918\) −2.60294 −0.0859097
\(919\) −31.9855 −1.05510 −0.527551 0.849523i \(-0.676890\pi\)
−0.527551 + 0.849523i \(0.676890\pi\)
\(920\) 8.19434 0.270160
\(921\) −38.6075 −1.27216
\(922\) 7.28760 0.240004
\(923\) −5.02771 −0.165489
\(924\) 21.1238 0.694923
\(925\) −0.100260 −0.00329652
\(926\) −24.3859 −0.801370
\(927\) 16.9320 0.556120
\(928\) −3.05692 −0.100348
\(929\) 0.249465 0.00818468 0.00409234 0.999992i \(-0.498697\pi\)
0.00409234 + 0.999992i \(0.498697\pi\)
\(930\) 29.3123 0.961188
\(931\) 2.22363 0.0728766
\(932\) 25.9732 0.850780
\(933\) 24.2178 0.792854
\(934\) −29.9631 −0.980422
\(935\) 7.12858 0.233130
\(936\) −1.42215 −0.0464845
\(937\) −36.1938 −1.18240 −0.591201 0.806525i \(-0.701346\pi\)
−0.591201 + 0.806525i \(0.701346\pi\)
\(938\) −6.83043 −0.223022
\(939\) 48.1401 1.57099
\(940\) −21.4558 −0.699811
\(941\) −37.3473 −1.21749 −0.608743 0.793367i \(-0.708326\pi\)
−0.608743 + 0.793367i \(0.708326\pi\)
\(942\) −18.6347 −0.607151
\(943\) 7.86464 0.256108
\(944\) −13.0104 −0.423453
\(945\) 19.6844 0.640333
\(946\) 11.7225 0.381130
\(947\) −17.7454 −0.576649 −0.288324 0.957533i \(-0.593098\pi\)
−0.288324 + 0.957533i \(0.593098\pi\)
\(948\) 6.10894 0.198409
\(949\) −2.83990 −0.0921872
\(950\) 0.626276 0.0203191
\(951\) 17.7899 0.576877
\(952\) 2.77232 0.0898513
\(953\) 51.6298 1.67245 0.836226 0.548385i \(-0.184757\pi\)
0.836226 + 0.548385i \(0.184757\pi\)
\(954\) 10.3674 0.335659
\(955\) 46.1123 1.49216
\(956\) −7.37644 −0.238571
\(957\) 21.6341 0.699331
\(958\) 14.9194 0.482025
\(959\) 49.6648 1.60376
\(960\) 5.10629 0.164805
\(961\) 1.95248 0.0629834
\(962\) 0.155209 0.00500415
\(963\) 7.70118 0.248167
\(964\) 3.64527 0.117406
\(965\) −64.3925 −2.07287
\(966\) −22.5532 −0.725637
\(967\) −27.0761 −0.870708 −0.435354 0.900259i \(-0.643377\pi\)
−0.435354 + 0.900259i \(0.643377\pi\)
\(968\) −0.362804 −0.0116610
\(969\) 2.34741 0.0754097
\(970\) 13.1664 0.422746
\(971\) −32.7197 −1.05003 −0.525013 0.851094i \(-0.675940\pi\)
−0.525013 + 0.851094i \(0.675940\pi\)
\(972\) 15.9097 0.510305
\(973\) 66.4480 2.13022
\(974\) 22.0249 0.705724
\(975\) 0.971218 0.0311039
\(976\) 6.73229 0.215495
\(977\) −53.1152 −1.69931 −0.849653 0.527342i \(-0.823189\pi\)
−0.849653 + 0.527342i \(0.823189\pi\)
\(978\) 26.4838 0.846858
\(979\) 58.2432 1.86146
\(980\) −4.49267 −0.143513
\(981\) −5.00512 −0.159801
\(982\) −18.9968 −0.606212
\(983\) 16.0977 0.513438 0.256719 0.966486i \(-0.417359\pi\)
0.256719 + 0.966486i \(0.417359\pi\)
\(984\) 4.90084 0.156233
\(985\) 11.5985 0.369560
\(986\) 2.83928 0.0904212
\(987\) 59.0526 1.87966
\(988\) −0.969522 −0.0308446
\(989\) −12.5157 −0.397975
\(990\) −13.1127 −0.416748
\(991\) 35.2799 1.12070 0.560352 0.828255i \(-0.310666\pi\)
0.560352 + 0.828255i \(0.310666\pi\)
\(992\) 5.74043 0.182259
\(993\) 71.1578 2.25812
\(994\) −18.0283 −0.571823
\(995\) 10.5087 0.333148
\(996\) −22.9551 −0.727362
\(997\) 36.7985 1.16542 0.582711 0.812680i \(-0.301992\pi\)
0.582711 + 0.812680i \(0.301992\pi\)
\(998\) −14.1049 −0.446483
\(999\) −0.522544 −0.0165326
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 8006.2.a.a.1.14 69
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8006.2.a.a.1.14 69 1.1 even 1 trivial