Properties

Label 4015.2.a.b.1.9
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $1$
Dimension $23$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 4015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.09501 q^{2} -2.94583 q^{3} -0.800943 q^{4} +1.00000 q^{5} +3.22573 q^{6} +2.47152 q^{7} +3.06707 q^{8} +5.67794 q^{9} +O(q^{10})\) \(q-1.09501 q^{2} -2.94583 q^{3} -0.800943 q^{4} +1.00000 q^{5} +3.22573 q^{6} +2.47152 q^{7} +3.06707 q^{8} +5.67794 q^{9} -1.09501 q^{10} +1.00000 q^{11} +2.35945 q^{12} +0.870712 q^{13} -2.70635 q^{14} -2.94583 q^{15} -1.75660 q^{16} +5.55777 q^{17} -6.21743 q^{18} +0.757543 q^{19} -0.800943 q^{20} -7.28069 q^{21} -1.09501 q^{22} -7.95898 q^{23} -9.03509 q^{24} +1.00000 q^{25} -0.953443 q^{26} -7.88878 q^{27} -1.97955 q^{28} -7.07445 q^{29} +3.22573 q^{30} +6.68841 q^{31} -4.21064 q^{32} -2.94583 q^{33} -6.08584 q^{34} +2.47152 q^{35} -4.54771 q^{36} -5.46020 q^{37} -0.829521 q^{38} -2.56497 q^{39} +3.06707 q^{40} -10.2168 q^{41} +7.97247 q^{42} -5.63905 q^{43} -0.800943 q^{44} +5.67794 q^{45} +8.71520 q^{46} +7.07320 q^{47} +5.17466 q^{48} -0.891581 q^{49} -1.09501 q^{50} -16.3723 q^{51} -0.697391 q^{52} -5.43075 q^{53} +8.63833 q^{54} +1.00000 q^{55} +7.58034 q^{56} -2.23160 q^{57} +7.74663 q^{58} +6.03850 q^{59} +2.35945 q^{60} +3.86324 q^{61} -7.32391 q^{62} +14.0332 q^{63} +8.12392 q^{64} +0.870712 q^{65} +3.22573 q^{66} +1.40262 q^{67} -4.45146 q^{68} +23.4458 q^{69} -2.70635 q^{70} -12.8543 q^{71} +17.4147 q^{72} -1.00000 q^{73} +5.97900 q^{74} -2.94583 q^{75} -0.606749 q^{76} +2.47152 q^{77} +2.80868 q^{78} -11.9602 q^{79} -1.75660 q^{80} +6.20521 q^{81} +11.1875 q^{82} +13.3277 q^{83} +5.83142 q^{84} +5.55777 q^{85} +6.17484 q^{86} +20.8402 q^{87} +3.06707 q^{88} -15.7768 q^{89} -6.21743 q^{90} +2.15198 q^{91} +6.37469 q^{92} -19.7030 q^{93} -7.74525 q^{94} +0.757543 q^{95} +12.4039 q^{96} +7.91533 q^{97} +0.976294 q^{98} +5.67794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 23 q - 5 q^{2} - 3 q^{3} + 15 q^{4} + 23 q^{5} - 15 q^{6} - 10 q^{7} - 12 q^{8} + 8 q^{9} - 5 q^{10} + 23 q^{11} + 4 q^{12} - 19 q^{13} - 5 q^{14} - 3 q^{15} - q^{16} - 26 q^{17} + 5 q^{18} - 34 q^{19} + 15 q^{20} - 26 q^{21} - 5 q^{22} - 4 q^{23} - 23 q^{24} + 23 q^{25} - 13 q^{26} - 3 q^{27} - 28 q^{28} - 36 q^{29} - 15 q^{30} - 24 q^{31} - 19 q^{32} - 3 q^{33} - 4 q^{34} - 10 q^{35} - 14 q^{36} + 4 q^{37} - 15 q^{38} - 34 q^{39} - 12 q^{40} - 74 q^{41} + 7 q^{42} - 15 q^{43} + 15 q^{44} + 8 q^{45} + 3 q^{46} + q^{47} + 29 q^{48} + q^{49} - 5 q^{50} - 47 q^{51} - 23 q^{52} - 6 q^{53} - 11 q^{54} + 23 q^{55} - 20 q^{56} - 19 q^{57} + 22 q^{58} - 17 q^{59} + 4 q^{60} - 59 q^{61} + 38 q^{62} - 21 q^{63} - 18 q^{64} - 19 q^{65} - 15 q^{66} + 12 q^{67} + 5 q^{68} - 8 q^{69} - 5 q^{70} - 34 q^{71} + 21 q^{72} - 23 q^{73} - 9 q^{74} - 3 q^{75} - 53 q^{76} - 10 q^{77} + 23 q^{78} - 62 q^{79} - q^{80} + 7 q^{81} + 24 q^{82} - 8 q^{83} + 46 q^{84} - 26 q^{85} - 11 q^{86} + q^{87} - 12 q^{88} - 77 q^{89} + 5 q^{90} - 6 q^{91} + 47 q^{92} - 32 q^{94} - 34 q^{95} - 53 q^{96} - 21 q^{97} - 3 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09501 −0.774292 −0.387146 0.922018i \(-0.626539\pi\)
−0.387146 + 0.922018i \(0.626539\pi\)
\(3\) −2.94583 −1.70078 −0.850389 0.526154i \(-0.823634\pi\)
−0.850389 + 0.526154i \(0.823634\pi\)
\(4\) −0.800943 −0.400472
\(5\) 1.00000 0.447214
\(6\) 3.22573 1.31690
\(7\) 2.47152 0.934147 0.467074 0.884218i \(-0.345308\pi\)
0.467074 + 0.884218i \(0.345308\pi\)
\(8\) 3.06707 1.08437
\(9\) 5.67794 1.89265
\(10\) −1.09501 −0.346274
\(11\) 1.00000 0.301511
\(12\) 2.35945 0.681113
\(13\) 0.870712 0.241492 0.120746 0.992683i \(-0.461471\pi\)
0.120746 + 0.992683i \(0.461471\pi\)
\(14\) −2.70635 −0.723303
\(15\) −2.94583 −0.760611
\(16\) −1.75660 −0.439151
\(17\) 5.55777 1.34796 0.673979 0.738750i \(-0.264584\pi\)
0.673979 + 0.738750i \(0.264584\pi\)
\(18\) −6.21743 −1.46546
\(19\) 0.757543 0.173792 0.0868962 0.996217i \(-0.472305\pi\)
0.0868962 + 0.996217i \(0.472305\pi\)
\(20\) −0.800943 −0.179096
\(21\) −7.28069 −1.58878
\(22\) −1.09501 −0.233458
\(23\) −7.95898 −1.65956 −0.829781 0.558089i \(-0.811535\pi\)
−0.829781 + 0.558089i \(0.811535\pi\)
\(24\) −9.03509 −1.84428
\(25\) 1.00000 0.200000
\(26\) −0.953443 −0.186986
\(27\) −7.88878 −1.51820
\(28\) −1.97955 −0.374099
\(29\) −7.07445 −1.31369 −0.656846 0.754024i \(-0.728110\pi\)
−0.656846 + 0.754024i \(0.728110\pi\)
\(30\) 3.22573 0.588935
\(31\) 6.68841 1.20127 0.600637 0.799522i \(-0.294914\pi\)
0.600637 + 0.799522i \(0.294914\pi\)
\(32\) −4.21064 −0.744343
\(33\) −2.94583 −0.512804
\(34\) −6.08584 −1.04371
\(35\) 2.47152 0.417763
\(36\) −4.54771 −0.757952
\(37\) −5.46020 −0.897652 −0.448826 0.893619i \(-0.648158\pi\)
−0.448826 + 0.893619i \(0.648158\pi\)
\(38\) −0.829521 −0.134566
\(39\) −2.56497 −0.410725
\(40\) 3.06707 0.484947
\(41\) −10.2168 −1.59559 −0.797796 0.602928i \(-0.794001\pi\)
−0.797796 + 0.602928i \(0.794001\pi\)
\(42\) 7.97247 1.23018
\(43\) −5.63905 −0.859946 −0.429973 0.902842i \(-0.641477\pi\)
−0.429973 + 0.902842i \(0.641477\pi\)
\(44\) −0.800943 −0.120747
\(45\) 5.67794 0.846418
\(46\) 8.71520 1.28499
\(47\) 7.07320 1.03173 0.515866 0.856669i \(-0.327470\pi\)
0.515866 + 0.856669i \(0.327470\pi\)
\(48\) 5.17466 0.746899
\(49\) −0.891581 −0.127369
\(50\) −1.09501 −0.154858
\(51\) −16.3723 −2.29258
\(52\) −0.697391 −0.0967107
\(53\) −5.43075 −0.745970 −0.372985 0.927837i \(-0.621666\pi\)
−0.372985 + 0.927837i \(0.621666\pi\)
\(54\) 8.63833 1.17553
\(55\) 1.00000 0.134840
\(56\) 7.58034 1.01297
\(57\) −2.23160 −0.295582
\(58\) 7.74663 1.01718
\(59\) 6.03850 0.786145 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(60\) 2.35945 0.304603
\(61\) 3.86324 0.494637 0.247319 0.968934i \(-0.420451\pi\)
0.247319 + 0.968934i \(0.420451\pi\)
\(62\) −7.32391 −0.930137
\(63\) 14.0332 1.76801
\(64\) 8.12392 1.01549
\(65\) 0.870712 0.107999
\(66\) 3.22573 0.397060
\(67\) 1.40262 0.171357 0.0856785 0.996323i \(-0.472694\pi\)
0.0856785 + 0.996323i \(0.472694\pi\)
\(68\) −4.45146 −0.539819
\(69\) 23.4458 2.82255
\(70\) −2.70635 −0.323471
\(71\) −12.8543 −1.52552 −0.762760 0.646682i \(-0.776156\pi\)
−0.762760 + 0.646682i \(0.776156\pi\)
\(72\) 17.4147 2.05234
\(73\) −1.00000 −0.117041
\(74\) 5.97900 0.695045
\(75\) −2.94583 −0.340156
\(76\) −0.606749 −0.0695989
\(77\) 2.47152 0.281656
\(78\) 2.80868 0.318021
\(79\) −11.9602 −1.34563 −0.672814 0.739812i \(-0.734914\pi\)
−0.672814 + 0.739812i \(0.734914\pi\)
\(80\) −1.75660 −0.196394
\(81\) 6.20521 0.689468
\(82\) 11.1875 1.23545
\(83\) 13.3277 1.46291 0.731455 0.681890i \(-0.238842\pi\)
0.731455 + 0.681890i \(0.238842\pi\)
\(84\) 5.83142 0.636260
\(85\) 5.55777 0.602825
\(86\) 6.17484 0.665850
\(87\) 20.8402 2.23430
\(88\) 3.06707 0.326951
\(89\) −15.7768 −1.67234 −0.836171 0.548469i \(-0.815211\pi\)
−0.836171 + 0.548469i \(0.815211\pi\)
\(90\) −6.21743 −0.655375
\(91\) 2.15198 0.225589
\(92\) 6.37469 0.664607
\(93\) −19.7030 −2.04310
\(94\) −7.74525 −0.798862
\(95\) 0.757543 0.0777223
\(96\) 12.4039 1.26596
\(97\) 7.91533 0.803680 0.401840 0.915710i \(-0.368371\pi\)
0.401840 + 0.915710i \(0.368371\pi\)
\(98\) 0.976294 0.0986206
\(99\) 5.67794 0.570655
\(100\) −0.800943 −0.0800943
\(101\) 2.19475 0.218385 0.109193 0.994021i \(-0.465173\pi\)
0.109193 + 0.994021i \(0.465173\pi\)
\(102\) 17.9279 1.77513
\(103\) −4.26799 −0.420537 −0.210269 0.977644i \(-0.567434\pi\)
−0.210269 + 0.977644i \(0.567434\pi\)
\(104\) 2.67054 0.261868
\(105\) −7.28069 −0.710523
\(106\) 5.94675 0.577599
\(107\) 15.1153 1.46125 0.730624 0.682780i \(-0.239229\pi\)
0.730624 + 0.682780i \(0.239229\pi\)
\(108\) 6.31846 0.607995
\(109\) −12.6267 −1.20942 −0.604711 0.796445i \(-0.706712\pi\)
−0.604711 + 0.796445i \(0.706712\pi\)
\(110\) −1.09501 −0.104406
\(111\) 16.0849 1.52671
\(112\) −4.34148 −0.410232
\(113\) −7.20649 −0.677929 −0.338965 0.940799i \(-0.610077\pi\)
−0.338965 + 0.940799i \(0.610077\pi\)
\(114\) 2.44363 0.228867
\(115\) −7.95898 −0.742179
\(116\) 5.66623 0.526097
\(117\) 4.94386 0.457060
\(118\) −6.61224 −0.608706
\(119\) 13.7362 1.25919
\(120\) −9.03509 −0.824787
\(121\) 1.00000 0.0909091
\(122\) −4.23030 −0.382994
\(123\) 30.0969 2.71375
\(124\) −5.35704 −0.481076
\(125\) 1.00000 0.0894427
\(126\) −15.3665 −1.36896
\(127\) −4.51562 −0.400696 −0.200348 0.979725i \(-0.564207\pi\)
−0.200348 + 0.979725i \(0.564207\pi\)
\(128\) −0.474530 −0.0419429
\(129\) 16.6117 1.46258
\(130\) −0.953443 −0.0836225
\(131\) 1.88324 0.164540 0.0822698 0.996610i \(-0.473783\pi\)
0.0822698 + 0.996610i \(0.473783\pi\)
\(132\) 2.35945 0.205363
\(133\) 1.87229 0.162348
\(134\) −1.53589 −0.132680
\(135\) −7.88878 −0.678958
\(136\) 17.0461 1.46169
\(137\) −20.4924 −1.75078 −0.875392 0.483414i \(-0.839397\pi\)
−0.875392 + 0.483414i \(0.839397\pi\)
\(138\) −25.6735 −2.18548
\(139\) 8.93621 0.757959 0.378980 0.925405i \(-0.376275\pi\)
0.378980 + 0.925405i \(0.376275\pi\)
\(140\) −1.97955 −0.167302
\(141\) −20.8365 −1.75475
\(142\) 14.0756 1.18120
\(143\) 0.870712 0.0728126
\(144\) −9.97390 −0.831158
\(145\) −7.07445 −0.587501
\(146\) 1.09501 0.0906240
\(147\) 2.62645 0.216626
\(148\) 4.37331 0.359484
\(149\) −1.90995 −0.156470 −0.0782348 0.996935i \(-0.524928\pi\)
−0.0782348 + 0.996935i \(0.524928\pi\)
\(150\) 3.22573 0.263380
\(151\) −18.6127 −1.51468 −0.757340 0.653021i \(-0.773501\pi\)
−0.757340 + 0.653021i \(0.773501\pi\)
\(152\) 2.32344 0.188456
\(153\) 31.5567 2.55121
\(154\) −2.70635 −0.218084
\(155\) 6.68841 0.537226
\(156\) 2.05440 0.164484
\(157\) −1.00577 −0.0802695 −0.0401347 0.999194i \(-0.512779\pi\)
−0.0401347 + 0.999194i \(0.512779\pi\)
\(158\) 13.0966 1.04191
\(159\) 15.9981 1.26873
\(160\) −4.21064 −0.332880
\(161\) −19.6708 −1.55028
\(162\) −6.79480 −0.533850
\(163\) 6.91444 0.541580 0.270790 0.962638i \(-0.412715\pi\)
0.270790 + 0.962638i \(0.412715\pi\)
\(164\) 8.18305 0.638989
\(165\) −2.94583 −0.229333
\(166\) −14.5941 −1.13272
\(167\) 12.2339 0.946685 0.473342 0.880879i \(-0.343047\pi\)
0.473342 + 0.880879i \(0.343047\pi\)
\(168\) −22.3304 −1.72283
\(169\) −12.2419 −0.941682
\(170\) −6.08584 −0.466763
\(171\) 4.30129 0.328928
\(172\) 4.51655 0.344384
\(173\) 3.10000 0.235689 0.117844 0.993032i \(-0.462402\pi\)
0.117844 + 0.993032i \(0.462402\pi\)
\(174\) −22.8203 −1.73000
\(175\) 2.47152 0.186829
\(176\) −1.75660 −0.132409
\(177\) −17.7884 −1.33706
\(178\) 17.2759 1.29488
\(179\) −5.92443 −0.442813 −0.221406 0.975182i \(-0.571065\pi\)
−0.221406 + 0.975182i \(0.571065\pi\)
\(180\) −4.54771 −0.338966
\(181\) −9.28992 −0.690514 −0.345257 0.938508i \(-0.612208\pi\)
−0.345257 + 0.938508i \(0.612208\pi\)
\(182\) −2.35645 −0.174672
\(183\) −11.3805 −0.841268
\(184\) −24.4108 −1.79959
\(185\) −5.46020 −0.401442
\(186\) 21.5750 1.58196
\(187\) 5.55777 0.406425
\(188\) −5.66523 −0.413179
\(189\) −19.4973 −1.41822
\(190\) −0.829521 −0.0601798
\(191\) 12.0638 0.872908 0.436454 0.899727i \(-0.356234\pi\)
0.436454 + 0.899727i \(0.356234\pi\)
\(192\) −23.9317 −1.72712
\(193\) 10.2839 0.740248 0.370124 0.928982i \(-0.379315\pi\)
0.370124 + 0.928982i \(0.379315\pi\)
\(194\) −8.66740 −0.622283
\(195\) −2.56497 −0.183682
\(196\) 0.714105 0.0510075
\(197\) 6.06425 0.432060 0.216030 0.976387i \(-0.430689\pi\)
0.216030 + 0.976387i \(0.430689\pi\)
\(198\) −6.21743 −0.441854
\(199\) −11.5157 −0.816327 −0.408164 0.912909i \(-0.633831\pi\)
−0.408164 + 0.912909i \(0.633831\pi\)
\(200\) 3.06707 0.216875
\(201\) −4.13188 −0.291440
\(202\) −2.40328 −0.169094
\(203\) −17.4847 −1.22718
\(204\) 13.1133 0.918113
\(205\) −10.2168 −0.713570
\(206\) 4.67351 0.325619
\(207\) −45.1906 −3.14097
\(208\) −1.52950 −0.106052
\(209\) 0.757543 0.0524004
\(210\) 7.97247 0.550152
\(211\) −7.92338 −0.545468 −0.272734 0.962089i \(-0.587928\pi\)
−0.272734 + 0.962089i \(0.587928\pi\)
\(212\) 4.34972 0.298740
\(213\) 37.8665 2.59457
\(214\) −16.5514 −1.13143
\(215\) −5.63905 −0.384580
\(216\) −24.1955 −1.64629
\(217\) 16.5306 1.12217
\(218\) 13.8265 0.936447
\(219\) 2.94583 0.199061
\(220\) −0.800943 −0.0539996
\(221\) 4.83922 0.325521
\(222\) −17.6132 −1.18212
\(223\) 9.00137 0.602776 0.301388 0.953502i \(-0.402550\pi\)
0.301388 + 0.953502i \(0.402550\pi\)
\(224\) −10.4067 −0.695326
\(225\) 5.67794 0.378530
\(226\) 7.89121 0.524915
\(227\) 1.98190 0.131544 0.0657718 0.997835i \(-0.479049\pi\)
0.0657718 + 0.997835i \(0.479049\pi\)
\(228\) 1.78738 0.118372
\(229\) 14.2877 0.944160 0.472080 0.881556i \(-0.343503\pi\)
0.472080 + 0.881556i \(0.343503\pi\)
\(230\) 8.71520 0.574663
\(231\) −7.28069 −0.479035
\(232\) −21.6979 −1.42453
\(233\) −12.9620 −0.849168 −0.424584 0.905388i \(-0.639580\pi\)
−0.424584 + 0.905388i \(0.639580\pi\)
\(234\) −5.41359 −0.353898
\(235\) 7.07320 0.461404
\(236\) −4.83649 −0.314829
\(237\) 35.2328 2.28861
\(238\) −15.0413 −0.974982
\(239\) 15.7979 1.02188 0.510942 0.859615i \(-0.329297\pi\)
0.510942 + 0.859615i \(0.329297\pi\)
\(240\) 5.17466 0.334023
\(241\) −17.6872 −1.13933 −0.569665 0.821877i \(-0.692927\pi\)
−0.569665 + 0.821877i \(0.692927\pi\)
\(242\) −1.09501 −0.0703902
\(243\) 5.38680 0.345564
\(244\) −3.09424 −0.198088
\(245\) −0.891581 −0.0569610
\(246\) −32.9566 −2.10123
\(247\) 0.659602 0.0419695
\(248\) 20.5139 1.30263
\(249\) −39.2613 −2.48808
\(250\) −1.09501 −0.0692548
\(251\) −12.5130 −0.789815 −0.394907 0.918721i \(-0.629223\pi\)
−0.394907 + 0.918721i \(0.629223\pi\)
\(252\) −11.2398 −0.708039
\(253\) −7.95898 −0.500377
\(254\) 4.94467 0.310256
\(255\) −16.3723 −1.02527
\(256\) −15.7282 −0.983014
\(257\) −3.00200 −0.187260 −0.0936299 0.995607i \(-0.529847\pi\)
−0.0936299 + 0.995607i \(0.529847\pi\)
\(258\) −18.1900 −1.13246
\(259\) −13.4950 −0.838539
\(260\) −0.697391 −0.0432504
\(261\) −40.1683 −2.48636
\(262\) −2.06218 −0.127402
\(263\) −30.8707 −1.90357 −0.951784 0.306767i \(-0.900753\pi\)
−0.951784 + 0.306767i \(0.900753\pi\)
\(264\) −9.03509 −0.556071
\(265\) −5.43075 −0.333608
\(266\) −2.05018 −0.125705
\(267\) 46.4760 2.84428
\(268\) −1.12342 −0.0686236
\(269\) −19.9040 −1.21357 −0.606786 0.794866i \(-0.707541\pi\)
−0.606786 + 0.794866i \(0.707541\pi\)
\(270\) 8.63833 0.525712
\(271\) 8.34396 0.506860 0.253430 0.967354i \(-0.418441\pi\)
0.253430 + 0.967354i \(0.418441\pi\)
\(272\) −9.76281 −0.591957
\(273\) −6.33939 −0.383677
\(274\) 22.4395 1.35562
\(275\) 1.00000 0.0603023
\(276\) −18.7788 −1.13035
\(277\) 14.3377 0.861471 0.430735 0.902478i \(-0.358254\pi\)
0.430735 + 0.902478i \(0.358254\pi\)
\(278\) −9.78528 −0.586882
\(279\) 37.9764 2.27359
\(280\) 7.58034 0.453012
\(281\) 6.41640 0.382770 0.191385 0.981515i \(-0.438702\pi\)
0.191385 + 0.981515i \(0.438702\pi\)
\(282\) 22.8162 1.35869
\(283\) 9.77983 0.581351 0.290675 0.956822i \(-0.406120\pi\)
0.290675 + 0.956822i \(0.406120\pi\)
\(284\) 10.2955 0.610927
\(285\) −2.23160 −0.132188
\(286\) −0.953443 −0.0563782
\(287\) −25.2510 −1.49052
\(288\) −23.9078 −1.40878
\(289\) 13.8889 0.816992
\(290\) 7.74663 0.454898
\(291\) −23.3172 −1.36688
\(292\) 0.800943 0.0468717
\(293\) 0.807120 0.0471524 0.0235762 0.999722i \(-0.492495\pi\)
0.0235762 + 0.999722i \(0.492495\pi\)
\(294\) −2.87600 −0.167732
\(295\) 6.03850 0.351575
\(296\) −16.7468 −0.973390
\(297\) −7.88878 −0.457753
\(298\) 2.09143 0.121153
\(299\) −6.92998 −0.400771
\(300\) 2.35945 0.136223
\(301\) −13.9370 −0.803317
\(302\) 20.3812 1.17280
\(303\) −6.46536 −0.371425
\(304\) −1.33070 −0.0763211
\(305\) 3.86324 0.221208
\(306\) −34.5551 −1.97538
\(307\) −19.8361 −1.13211 −0.566054 0.824368i \(-0.691531\pi\)
−0.566054 + 0.824368i \(0.691531\pi\)
\(308\) −1.97955 −0.112795
\(309\) 12.5728 0.715241
\(310\) −7.32391 −0.415970
\(311\) −21.9230 −1.24314 −0.621571 0.783358i \(-0.713505\pi\)
−0.621571 + 0.783358i \(0.713505\pi\)
\(312\) −7.86697 −0.445379
\(313\) −9.06793 −0.512550 −0.256275 0.966604i \(-0.582495\pi\)
−0.256275 + 0.966604i \(0.582495\pi\)
\(314\) 1.10134 0.0621520
\(315\) 14.0332 0.790679
\(316\) 9.57944 0.538885
\(317\) −8.58224 −0.482027 −0.241013 0.970522i \(-0.577480\pi\)
−0.241013 + 0.970522i \(0.577480\pi\)
\(318\) −17.5181 −0.982368
\(319\) −7.07445 −0.396093
\(320\) 8.12392 0.454141
\(321\) −44.5271 −2.48526
\(322\) 21.5398 1.20037
\(323\) 4.21026 0.234265
\(324\) −4.97002 −0.276112
\(325\) 0.870712 0.0482984
\(326\) −7.57141 −0.419342
\(327\) 37.1963 2.05696
\(328\) −31.3356 −1.73022
\(329\) 17.4816 0.963790
\(330\) 3.22573 0.177571
\(331\) −18.5141 −1.01762 −0.508812 0.860878i \(-0.669915\pi\)
−0.508812 + 0.860878i \(0.669915\pi\)
\(332\) −10.6748 −0.585854
\(333\) −31.0027 −1.69894
\(334\) −13.3963 −0.733011
\(335\) 1.40262 0.0766332
\(336\) 12.7893 0.697713
\(337\) 34.1933 1.86263 0.931315 0.364216i \(-0.118663\pi\)
0.931315 + 0.364216i \(0.118663\pi\)
\(338\) 13.4050 0.729137
\(339\) 21.2291 1.15301
\(340\) −4.45146 −0.241414
\(341\) 6.68841 0.362198
\(342\) −4.70997 −0.254686
\(343\) −19.5042 −1.05313
\(344\) −17.2954 −0.932504
\(345\) 23.4458 1.26228
\(346\) −3.39454 −0.182492
\(347\) −0.224774 −0.0120665 −0.00603325 0.999982i \(-0.501920\pi\)
−0.00603325 + 0.999982i \(0.501920\pi\)
\(348\) −16.6918 −0.894774
\(349\) −4.33673 −0.232140 −0.116070 0.993241i \(-0.537030\pi\)
−0.116070 + 0.993241i \(0.537030\pi\)
\(350\) −2.70635 −0.144661
\(351\) −6.86886 −0.366633
\(352\) −4.21064 −0.224428
\(353\) −26.8758 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(354\) 19.4786 1.03527
\(355\) −12.8543 −0.682233
\(356\) 12.6363 0.669725
\(357\) −40.4645 −2.14161
\(358\) 6.48734 0.342866
\(359\) −0.0594923 −0.00313988 −0.00156994 0.999999i \(-0.500500\pi\)
−0.00156994 + 0.999999i \(0.500500\pi\)
\(360\) 17.4147 0.917834
\(361\) −18.4261 −0.969796
\(362\) 10.1726 0.534660
\(363\) −2.94583 −0.154616
\(364\) −1.72362 −0.0903421
\(365\) −1.00000 −0.0523424
\(366\) 12.4618 0.651387
\(367\) 16.0978 0.840298 0.420149 0.907455i \(-0.361978\pi\)
0.420149 + 0.907455i \(0.361978\pi\)
\(368\) 13.9808 0.728798
\(369\) −58.0102 −3.01989
\(370\) 5.97900 0.310833
\(371\) −13.4222 −0.696846
\(372\) 15.7810 0.818204
\(373\) 27.9783 1.44866 0.724331 0.689452i \(-0.242149\pi\)
0.724331 + 0.689452i \(0.242149\pi\)
\(374\) −6.08584 −0.314692
\(375\) −2.94583 −0.152122
\(376\) 21.6940 1.11878
\(377\) −6.15981 −0.317246
\(378\) 21.3498 1.09812
\(379\) 32.4329 1.66597 0.832983 0.553298i \(-0.186631\pi\)
0.832983 + 0.553298i \(0.186631\pi\)
\(380\) −0.606749 −0.0311256
\(381\) 13.3023 0.681495
\(382\) −13.2101 −0.675886
\(383\) −27.2576 −1.39280 −0.696400 0.717654i \(-0.745216\pi\)
−0.696400 + 0.717654i \(0.745216\pi\)
\(384\) 1.39789 0.0713356
\(385\) 2.47152 0.125960
\(386\) −11.2610 −0.573168
\(387\) −32.0182 −1.62758
\(388\) −6.33973 −0.321851
\(389\) −37.2600 −1.88916 −0.944578 0.328288i \(-0.893528\pi\)
−0.944578 + 0.328288i \(0.893528\pi\)
\(390\) 2.80868 0.142223
\(391\) −44.2342 −2.23702
\(392\) −2.73454 −0.138115
\(393\) −5.54772 −0.279846
\(394\) −6.64045 −0.334541
\(395\) −11.9602 −0.601783
\(396\) −4.54771 −0.228531
\(397\) 24.5578 1.23252 0.616259 0.787543i \(-0.288647\pi\)
0.616259 + 0.787543i \(0.288647\pi\)
\(398\) 12.6099 0.632076
\(399\) −5.51544 −0.276118
\(400\) −1.75660 −0.0878302
\(401\) −13.0718 −0.652775 −0.326388 0.945236i \(-0.605831\pi\)
−0.326388 + 0.945236i \(0.605831\pi\)
\(402\) 4.52447 0.225660
\(403\) 5.82368 0.290098
\(404\) −1.75787 −0.0874572
\(405\) 6.20521 0.308340
\(406\) 19.1460 0.950198
\(407\) −5.46020 −0.270652
\(408\) −50.2150 −2.48601
\(409\) 13.2276 0.654063 0.327031 0.945014i \(-0.393952\pi\)
0.327031 + 0.945014i \(0.393952\pi\)
\(410\) 11.1875 0.552512
\(411\) 60.3672 2.97770
\(412\) 3.41842 0.168413
\(413\) 14.9243 0.734376
\(414\) 49.4844 2.43203
\(415\) 13.3277 0.654233
\(416\) −3.66626 −0.179753
\(417\) −26.3246 −1.28912
\(418\) −0.829521 −0.0405732
\(419\) 25.4228 1.24199 0.620993 0.783816i \(-0.286729\pi\)
0.620993 + 0.783816i \(0.286729\pi\)
\(420\) 5.83142 0.284544
\(421\) −14.2837 −0.696144 −0.348072 0.937468i \(-0.613163\pi\)
−0.348072 + 0.937468i \(0.613163\pi\)
\(422\) 8.67622 0.422352
\(423\) 40.1612 1.95270
\(424\) −16.6565 −0.808911
\(425\) 5.55777 0.269592
\(426\) −41.4644 −2.00896
\(427\) 9.54808 0.462064
\(428\) −12.1065 −0.585189
\(429\) −2.56497 −0.123838
\(430\) 6.17484 0.297777
\(431\) −4.69541 −0.226170 −0.113085 0.993585i \(-0.536073\pi\)
−0.113085 + 0.993585i \(0.536073\pi\)
\(432\) 13.8575 0.666717
\(433\) 26.9266 1.29401 0.647005 0.762486i \(-0.276021\pi\)
0.647005 + 0.762486i \(0.276021\pi\)
\(434\) −18.1012 −0.868885
\(435\) 20.8402 0.999209
\(436\) 10.1133 0.484339
\(437\) −6.02927 −0.288419
\(438\) −3.22573 −0.154131
\(439\) −24.6259 −1.17533 −0.587664 0.809105i \(-0.699952\pi\)
−0.587664 + 0.809105i \(0.699952\pi\)
\(440\) 3.06707 0.146217
\(441\) −5.06235 −0.241064
\(442\) −5.29902 −0.252049
\(443\) 26.0494 1.23765 0.618823 0.785530i \(-0.287610\pi\)
0.618823 + 0.785530i \(0.287610\pi\)
\(444\) −12.8831 −0.611403
\(445\) −15.7768 −0.747894
\(446\) −9.85663 −0.466725
\(447\) 5.62641 0.266120
\(448\) 20.0784 0.948617
\(449\) −12.9213 −0.609795 −0.304897 0.952385i \(-0.598622\pi\)
−0.304897 + 0.952385i \(0.598622\pi\)
\(450\) −6.21743 −0.293093
\(451\) −10.2168 −0.481089
\(452\) 5.77199 0.271491
\(453\) 54.8299 2.57613
\(454\) −2.17021 −0.101853
\(455\) 2.15198 0.100887
\(456\) −6.84448 −0.320522
\(457\) −37.2398 −1.74200 −0.871002 0.491279i \(-0.836529\pi\)
−0.871002 + 0.491279i \(0.836529\pi\)
\(458\) −15.6453 −0.731056
\(459\) −43.8441 −2.04647
\(460\) 6.37469 0.297221
\(461\) 23.9723 1.11650 0.558251 0.829672i \(-0.311473\pi\)
0.558251 + 0.829672i \(0.311473\pi\)
\(462\) 7.97247 0.370913
\(463\) 0.494605 0.0229863 0.0114931 0.999934i \(-0.496342\pi\)
0.0114931 + 0.999934i \(0.496342\pi\)
\(464\) 12.4270 0.576909
\(465\) −19.7030 −0.913703
\(466\) 14.1936 0.657504
\(467\) 10.7787 0.498777 0.249389 0.968404i \(-0.419770\pi\)
0.249389 + 0.968404i \(0.419770\pi\)
\(468\) −3.95975 −0.183039
\(469\) 3.46660 0.160073
\(470\) −7.74525 −0.357262
\(471\) 2.96284 0.136521
\(472\) 18.5205 0.852476
\(473\) −5.63905 −0.259284
\(474\) −38.5804 −1.77206
\(475\) 0.757543 0.0347585
\(476\) −11.0019 −0.504271
\(477\) −30.8355 −1.41186
\(478\) −17.2990 −0.791237
\(479\) 15.3091 0.699490 0.349745 0.936845i \(-0.386268\pi\)
0.349745 + 0.936845i \(0.386268\pi\)
\(480\) 12.4039 0.566156
\(481\) −4.75427 −0.216776
\(482\) 19.3677 0.882175
\(483\) 57.9469 2.63668
\(484\) −0.800943 −0.0364065
\(485\) 7.91533 0.359417
\(486\) −5.89863 −0.267567
\(487\) 35.3315 1.60102 0.800512 0.599317i \(-0.204561\pi\)
0.800512 + 0.599317i \(0.204561\pi\)
\(488\) 11.8488 0.536372
\(489\) −20.3688 −0.921109
\(490\) 0.976294 0.0441045
\(491\) −1.24511 −0.0561909 −0.0280955 0.999605i \(-0.508944\pi\)
−0.0280955 + 0.999605i \(0.508944\pi\)
\(492\) −24.1059 −1.08678
\(493\) −39.3182 −1.77080
\(494\) −0.722274 −0.0324967
\(495\) 5.67794 0.255205
\(496\) −11.7489 −0.527541
\(497\) −31.7696 −1.42506
\(498\) 42.9917 1.92650
\(499\) 37.9293 1.69795 0.848974 0.528434i \(-0.177221\pi\)
0.848974 + 0.528434i \(0.177221\pi\)
\(500\) −0.800943 −0.0358193
\(501\) −36.0389 −1.61010
\(502\) 13.7019 0.611547
\(503\) 27.9846 1.24777 0.623885 0.781516i \(-0.285553\pi\)
0.623885 + 0.781516i \(0.285553\pi\)
\(504\) 43.0407 1.91719
\(505\) 2.19475 0.0976649
\(506\) 8.71520 0.387438
\(507\) 36.0625 1.60159
\(508\) 3.61675 0.160467
\(509\) 0.0577389 0.00255923 0.00127962 0.999999i \(-0.499593\pi\)
0.00127962 + 0.999999i \(0.499593\pi\)
\(510\) 17.9279 0.793860
\(511\) −2.47152 −0.109334
\(512\) 18.1717 0.803083
\(513\) −5.97609 −0.263851
\(514\) 3.28724 0.144994
\(515\) −4.26799 −0.188070
\(516\) −13.3050 −0.585721
\(517\) 7.07320 0.311079
\(518\) 14.7772 0.649274
\(519\) −9.13209 −0.400854
\(520\) 2.67054 0.117111
\(521\) −14.8885 −0.652275 −0.326138 0.945322i \(-0.605747\pi\)
−0.326138 + 0.945322i \(0.605747\pi\)
\(522\) 43.9849 1.92517
\(523\) 18.5362 0.810533 0.405266 0.914199i \(-0.367179\pi\)
0.405266 + 0.914199i \(0.367179\pi\)
\(524\) −1.50837 −0.0658935
\(525\) −7.28069 −0.317756
\(526\) 33.8039 1.47392
\(527\) 37.1727 1.61927
\(528\) 5.17466 0.225198
\(529\) 40.3454 1.75415
\(530\) 5.94675 0.258310
\(531\) 34.2862 1.48790
\(532\) −1.49959 −0.0650156
\(533\) −8.89587 −0.385323
\(534\) −50.8918 −2.20231
\(535\) 15.1153 0.653490
\(536\) 4.30193 0.185815
\(537\) 17.4524 0.753127
\(538\) 21.7952 0.939659
\(539\) −0.891581 −0.0384031
\(540\) 6.31846 0.271903
\(541\) 2.55332 0.109776 0.0548878 0.998493i \(-0.482520\pi\)
0.0548878 + 0.998493i \(0.482520\pi\)
\(542\) −9.13676 −0.392457
\(543\) 27.3666 1.17441
\(544\) −23.4018 −1.00334
\(545\) −12.6267 −0.540870
\(546\) 6.94173 0.297078
\(547\) −38.4112 −1.64234 −0.821172 0.570680i \(-0.806680\pi\)
−0.821172 + 0.570680i \(0.806680\pi\)
\(548\) 16.4132 0.701139
\(549\) 21.9353 0.936174
\(550\) −1.09501 −0.0466916
\(551\) −5.35920 −0.228310
\(552\) 71.9101 3.06070
\(553\) −29.5599 −1.25701
\(554\) −15.7000 −0.667030
\(555\) 16.0849 0.682764
\(556\) −7.15739 −0.303541
\(557\) 6.71479 0.284515 0.142257 0.989830i \(-0.454564\pi\)
0.142257 + 0.989830i \(0.454564\pi\)
\(558\) −41.5847 −1.76042
\(559\) −4.90999 −0.207670
\(560\) −4.34148 −0.183461
\(561\) −16.3723 −0.691239
\(562\) −7.02605 −0.296376
\(563\) 7.27771 0.306719 0.153360 0.988170i \(-0.450991\pi\)
0.153360 + 0.988170i \(0.450991\pi\)
\(564\) 16.6888 0.702726
\(565\) −7.20649 −0.303179
\(566\) −10.7091 −0.450135
\(567\) 15.3363 0.644065
\(568\) −39.4249 −1.65423
\(569\) 30.3294 1.27147 0.635736 0.771906i \(-0.280697\pi\)
0.635736 + 0.771906i \(0.280697\pi\)
\(570\) 2.44363 0.102352
\(571\) −4.43487 −0.185593 −0.0927967 0.995685i \(-0.529581\pi\)
−0.0927967 + 0.995685i \(0.529581\pi\)
\(572\) −0.697391 −0.0291594
\(573\) −35.5380 −1.48462
\(574\) 27.6502 1.15410
\(575\) −7.95898 −0.331912
\(576\) 46.1272 1.92196
\(577\) −24.8176 −1.03317 −0.516586 0.856235i \(-0.672797\pi\)
−0.516586 + 0.856235i \(0.672797\pi\)
\(578\) −15.2085 −0.632590
\(579\) −30.2945 −1.25900
\(580\) 5.66623 0.235278
\(581\) 32.9398 1.36657
\(582\) 25.5327 1.05837
\(583\) −5.43075 −0.224919
\(584\) −3.06707 −0.126916
\(585\) 4.94386 0.204403
\(586\) −0.883808 −0.0365098
\(587\) −9.03596 −0.372954 −0.186477 0.982459i \(-0.559707\pi\)
−0.186477 + 0.982459i \(0.559707\pi\)
\(588\) −2.10364 −0.0867525
\(589\) 5.06676 0.208772
\(590\) −6.61224 −0.272222
\(591\) −17.8643 −0.734839
\(592\) 9.59141 0.394205
\(593\) −30.8139 −1.26538 −0.632688 0.774407i \(-0.718048\pi\)
−0.632688 + 0.774407i \(0.718048\pi\)
\(594\) 8.63833 0.354435
\(595\) 13.7362 0.563128
\(596\) 1.52977 0.0626616
\(597\) 33.9234 1.38839
\(598\) 7.58843 0.310314
\(599\) 34.8634 1.42448 0.712241 0.701935i \(-0.247680\pi\)
0.712241 + 0.701935i \(0.247680\pi\)
\(600\) −9.03509 −0.368856
\(601\) 26.0001 1.06057 0.530283 0.847821i \(-0.322086\pi\)
0.530283 + 0.847821i \(0.322086\pi\)
\(602\) 15.2612 0.622002
\(603\) 7.96399 0.324319
\(604\) 14.9077 0.606586
\(605\) 1.00000 0.0406558
\(606\) 7.07966 0.287592
\(607\) 10.3793 0.421284 0.210642 0.977563i \(-0.432445\pi\)
0.210642 + 0.977563i \(0.432445\pi\)
\(608\) −3.18974 −0.129361
\(609\) 51.5069 2.08717
\(610\) −4.23030 −0.171280
\(611\) 6.15872 0.249155
\(612\) −25.2751 −1.02169
\(613\) 22.2019 0.896726 0.448363 0.893851i \(-0.352007\pi\)
0.448363 + 0.893851i \(0.352007\pi\)
\(614\) 21.7209 0.876583
\(615\) 30.0969 1.21362
\(616\) 7.58034 0.305421
\(617\) −23.0135 −0.926490 −0.463245 0.886230i \(-0.653315\pi\)
−0.463245 + 0.886230i \(0.653315\pi\)
\(618\) −13.7674 −0.553805
\(619\) −17.4912 −0.703031 −0.351515 0.936182i \(-0.614334\pi\)
−0.351515 + 0.936182i \(0.614334\pi\)
\(620\) −5.35704 −0.215144
\(621\) 62.7866 2.51954
\(622\) 24.0060 0.962555
\(623\) −38.9928 −1.56221
\(624\) 4.50564 0.180370
\(625\) 1.00000 0.0400000
\(626\) 9.92951 0.396863
\(627\) −2.23160 −0.0891214
\(628\) 0.805567 0.0321456
\(629\) −30.3466 −1.21000
\(630\) −15.3665 −0.612217
\(631\) 1.27512 0.0507618 0.0253809 0.999678i \(-0.491920\pi\)
0.0253809 + 0.999678i \(0.491920\pi\)
\(632\) −36.6828 −1.45916
\(633\) 23.3410 0.927721
\(634\) 9.39768 0.373230
\(635\) −4.51562 −0.179197
\(636\) −12.8136 −0.508090
\(637\) −0.776310 −0.0307585
\(638\) 7.74663 0.306692
\(639\) −72.9857 −2.88727
\(640\) −0.474530 −0.0187574
\(641\) −25.5304 −1.00839 −0.504195 0.863590i \(-0.668211\pi\)
−0.504195 + 0.863590i \(0.668211\pi\)
\(642\) 48.7578 1.92432
\(643\) −32.1027 −1.26601 −0.633003 0.774149i \(-0.718178\pi\)
−0.633003 + 0.774149i \(0.718178\pi\)
\(644\) 15.7552 0.620841
\(645\) 16.6117 0.654085
\(646\) −4.61029 −0.181389
\(647\) −35.2256 −1.38486 −0.692431 0.721484i \(-0.743460\pi\)
−0.692431 + 0.721484i \(0.743460\pi\)
\(648\) 19.0318 0.747642
\(649\) 6.03850 0.237032
\(650\) −0.953443 −0.0373971
\(651\) −48.6963 −1.90856
\(652\) −5.53807 −0.216888
\(653\) −44.1898 −1.72928 −0.864640 0.502393i \(-0.832453\pi\)
−0.864640 + 0.502393i \(0.832453\pi\)
\(654\) −40.7305 −1.59269
\(655\) 1.88324 0.0735844
\(656\) 17.9468 0.700705
\(657\) −5.67794 −0.221518
\(658\) −19.1426 −0.746255
\(659\) −44.6080 −1.73768 −0.868841 0.495091i \(-0.835135\pi\)
−0.868841 + 0.495091i \(0.835135\pi\)
\(660\) 2.35945 0.0918413
\(661\) 25.8795 1.00659 0.503297 0.864113i \(-0.332120\pi\)
0.503297 + 0.864113i \(0.332120\pi\)
\(662\) 20.2732 0.787939
\(663\) −14.2556 −0.553640
\(664\) 40.8772 1.58634
\(665\) 1.87229 0.0726041
\(666\) 33.9484 1.31548
\(667\) 56.3054 2.18015
\(668\) −9.79863 −0.379120
\(669\) −26.5165 −1.02519
\(670\) −1.53589 −0.0593365
\(671\) 3.86324 0.149139
\(672\) 30.6564 1.18260
\(673\) 9.19811 0.354561 0.177281 0.984160i \(-0.443270\pi\)
0.177281 + 0.984160i \(0.443270\pi\)
\(674\) −37.4422 −1.44222
\(675\) −7.88878 −0.303639
\(676\) 9.80503 0.377117
\(677\) −45.8043 −1.76040 −0.880201 0.474602i \(-0.842592\pi\)
−0.880201 + 0.474602i \(0.842592\pi\)
\(678\) −23.2462 −0.892765
\(679\) 19.5629 0.750755
\(680\) 17.0461 0.653688
\(681\) −5.83836 −0.223726
\(682\) −7.32391 −0.280447
\(683\) −25.2811 −0.967356 −0.483678 0.875246i \(-0.660699\pi\)
−0.483678 + 0.875246i \(0.660699\pi\)
\(684\) −3.44509 −0.131726
\(685\) −20.4924 −0.782974
\(686\) 21.3574 0.815429
\(687\) −42.0893 −1.60581
\(688\) 9.90557 0.377646
\(689\) −4.72862 −0.180146
\(690\) −25.6735 −0.977375
\(691\) −33.3171 −1.26744 −0.633721 0.773562i \(-0.718473\pi\)
−0.633721 + 0.773562i \(0.718473\pi\)
\(692\) −2.48292 −0.0943866
\(693\) 14.0332 0.533076
\(694\) 0.246131 0.00934299
\(695\) 8.93621 0.338970
\(696\) 63.9183 2.42282
\(697\) −56.7825 −2.15079
\(698\) 4.74878 0.179744
\(699\) 38.1839 1.44425
\(700\) −1.97955 −0.0748199
\(701\) 31.3586 1.18440 0.592198 0.805792i \(-0.298260\pi\)
0.592198 + 0.805792i \(0.298260\pi\)
\(702\) 7.52150 0.283881
\(703\) −4.13634 −0.156005
\(704\) 8.12392 0.306182
\(705\) −20.8365 −0.784747
\(706\) 29.4294 1.10759
\(707\) 5.42436 0.204004
\(708\) 14.2475 0.535454
\(709\) 21.0375 0.790078 0.395039 0.918664i \(-0.370731\pi\)
0.395039 + 0.918664i \(0.370731\pi\)
\(710\) 14.0756 0.528248
\(711\) −67.9093 −2.54680
\(712\) −48.3887 −1.81344
\(713\) −53.2329 −1.99359
\(714\) 44.3092 1.65823
\(715\) 0.870712 0.0325628
\(716\) 4.74513 0.177334
\(717\) −46.5381 −1.73800
\(718\) 0.0651449 0.00243119
\(719\) 31.2347 1.16486 0.582428 0.812882i \(-0.302103\pi\)
0.582428 + 0.812882i \(0.302103\pi\)
\(720\) −9.97390 −0.371705
\(721\) −10.5484 −0.392844
\(722\) 20.1769 0.750906
\(723\) 52.1035 1.93775
\(724\) 7.44070 0.276531
\(725\) −7.07445 −0.262739
\(726\) 3.22573 0.119718
\(727\) 41.2639 1.53039 0.765196 0.643798i \(-0.222642\pi\)
0.765196 + 0.643798i \(0.222642\pi\)
\(728\) 6.60029 0.244623
\(729\) −34.4843 −1.27720
\(730\) 1.09501 0.0405283
\(731\) −31.3405 −1.15917
\(732\) 9.11511 0.336904
\(733\) 36.2052 1.33727 0.668636 0.743590i \(-0.266879\pi\)
0.668636 + 0.743590i \(0.266879\pi\)
\(734\) −17.6273 −0.650636
\(735\) 2.62645 0.0968781
\(736\) 33.5124 1.23528
\(737\) 1.40262 0.0516661
\(738\) 63.5221 2.33828
\(739\) −2.45772 −0.0904085 −0.0452043 0.998978i \(-0.514394\pi\)
−0.0452043 + 0.998978i \(0.514394\pi\)
\(740\) 4.37331 0.160766
\(741\) −1.94308 −0.0713808
\(742\) 14.6975 0.539563
\(743\) −49.8334 −1.82821 −0.914104 0.405479i \(-0.867105\pi\)
−0.914104 + 0.405479i \(0.867105\pi\)
\(744\) −60.4304 −2.21549
\(745\) −1.90995 −0.0699753
\(746\) −30.6367 −1.12169
\(747\) 75.6741 2.76877
\(748\) −4.45146 −0.162762
\(749\) 37.3577 1.36502
\(750\) 3.22573 0.117787
\(751\) −14.9540 −0.545679 −0.272840 0.962059i \(-0.587963\pi\)
−0.272840 + 0.962059i \(0.587963\pi\)
\(752\) −12.4248 −0.453086
\(753\) 36.8613 1.34330
\(754\) 6.74508 0.245641
\(755\) −18.6127 −0.677385
\(756\) 15.6162 0.567957
\(757\) 5.93691 0.215781 0.107890 0.994163i \(-0.465590\pi\)
0.107890 + 0.994163i \(0.465590\pi\)
\(758\) −35.5145 −1.28995
\(759\) 23.4458 0.851030
\(760\) 2.32344 0.0842801
\(761\) 18.3124 0.663824 0.331912 0.943310i \(-0.392306\pi\)
0.331912 + 0.943310i \(0.392306\pi\)
\(762\) −14.5662 −0.527677
\(763\) −31.2073 −1.12978
\(764\) −9.66244 −0.349575
\(765\) 31.5567 1.14094
\(766\) 29.8475 1.07843
\(767\) 5.25779 0.189848
\(768\) 46.3327 1.67189
\(769\) −6.43575 −0.232079 −0.116039 0.993245i \(-0.537020\pi\)
−0.116039 + 0.993245i \(0.537020\pi\)
\(770\) −2.70635 −0.0975302
\(771\) 8.84341 0.318488
\(772\) −8.23678 −0.296448
\(773\) 38.7052 1.39213 0.696065 0.717978i \(-0.254932\pi\)
0.696065 + 0.717978i \(0.254932\pi\)
\(774\) 35.0604 1.26022
\(775\) 6.68841 0.240255
\(776\) 24.2769 0.871490
\(777\) 39.7541 1.42617
\(778\) 40.8002 1.46276
\(779\) −7.73965 −0.277302
\(780\) 2.05440 0.0735593
\(781\) −12.8543 −0.459961
\(782\) 48.4371 1.73211
\(783\) 55.8088 1.99444
\(784\) 1.56615 0.0559341
\(785\) −1.00577 −0.0358976
\(786\) 6.07484 0.216682
\(787\) −9.41304 −0.335539 −0.167769 0.985826i \(-0.553656\pi\)
−0.167769 + 0.985826i \(0.553656\pi\)
\(788\) −4.85712 −0.173028
\(789\) 90.9400 3.23755
\(790\) 13.0966 0.465956
\(791\) −17.8110 −0.633286
\(792\) 17.4147 0.618803
\(793\) 3.36377 0.119451
\(794\) −26.8911 −0.954330
\(795\) 15.9981 0.567394
\(796\) 9.22343 0.326916
\(797\) 12.4775 0.441977 0.220988 0.975276i \(-0.429072\pi\)
0.220988 + 0.975276i \(0.429072\pi\)
\(798\) 6.03949 0.213796
\(799\) 39.3112 1.39073
\(800\) −4.21064 −0.148869
\(801\) −89.5800 −3.16515
\(802\) 14.3138 0.505439
\(803\) −1.00000 −0.0352892
\(804\) 3.30940 0.116714
\(805\) −19.6708 −0.693304
\(806\) −6.37702 −0.224621
\(807\) 58.6340 2.06402
\(808\) 6.73145 0.236812
\(809\) −13.4837 −0.474062 −0.237031 0.971502i \(-0.576174\pi\)
−0.237031 + 0.971502i \(0.576174\pi\)
\(810\) −6.79480 −0.238745
\(811\) 39.3284 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(812\) 14.0042 0.491452
\(813\) −24.5799 −0.862056
\(814\) 5.97900 0.209564
\(815\) 6.91444 0.242202
\(816\) 28.7596 1.00679
\(817\) −4.27182 −0.149452
\(818\) −14.4844 −0.506436
\(819\) 12.2188 0.426961
\(820\) 8.18305 0.285765
\(821\) 29.3360 1.02383 0.511917 0.859035i \(-0.328936\pi\)
0.511917 + 0.859035i \(0.328936\pi\)
\(822\) −66.1030 −2.30561
\(823\) −28.2826 −0.985868 −0.492934 0.870067i \(-0.664076\pi\)
−0.492934 + 0.870067i \(0.664076\pi\)
\(824\) −13.0902 −0.456020
\(825\) −2.94583 −0.102561
\(826\) −16.3423 −0.568621
\(827\) 8.63406 0.300236 0.150118 0.988668i \(-0.452035\pi\)
0.150118 + 0.988668i \(0.452035\pi\)
\(828\) 36.1951 1.25787
\(829\) 40.3031 1.39979 0.699893 0.714248i \(-0.253231\pi\)
0.699893 + 0.714248i \(0.253231\pi\)
\(830\) −14.5941 −0.506567
\(831\) −42.2366 −1.46517
\(832\) 7.07360 0.245233
\(833\) −4.95520 −0.171688
\(834\) 28.8258 0.998156
\(835\) 12.2339 0.423370
\(836\) −0.606749 −0.0209849
\(837\) −52.7634 −1.82377
\(838\) −27.8384 −0.961661
\(839\) −28.7110 −0.991215 −0.495607 0.868547i \(-0.665055\pi\)
−0.495607 + 0.868547i \(0.665055\pi\)
\(840\) −22.3304 −0.770473
\(841\) 21.0479 0.725788
\(842\) 15.6408 0.539019
\(843\) −18.9016 −0.651007
\(844\) 6.34618 0.218445
\(845\) −12.2419 −0.421133
\(846\) −43.9771 −1.51196
\(847\) 2.47152 0.0849225
\(848\) 9.53967 0.327594
\(849\) −28.8098 −0.988749
\(850\) −6.08584 −0.208743
\(851\) 43.4576 1.48971
\(852\) −30.3289 −1.03905
\(853\) −13.4424 −0.460258 −0.230129 0.973160i \(-0.573915\pi\)
−0.230129 + 0.973160i \(0.573915\pi\)
\(854\) −10.4553 −0.357773
\(855\) 4.30129 0.147101
\(856\) 46.3597 1.58454
\(857\) 2.90205 0.0991322 0.0495661 0.998771i \(-0.484216\pi\)
0.0495661 + 0.998771i \(0.484216\pi\)
\(858\) 2.80868 0.0958869
\(859\) −8.61521 −0.293947 −0.146974 0.989140i \(-0.546953\pi\)
−0.146974 + 0.989140i \(0.546953\pi\)
\(860\) 4.51655 0.154013
\(861\) 74.3852 2.53504
\(862\) 5.14154 0.175121
\(863\) 20.5742 0.700353 0.350177 0.936684i \(-0.386122\pi\)
0.350177 + 0.936684i \(0.386122\pi\)
\(864\) 33.2168 1.13006
\(865\) 3.10000 0.105403
\(866\) −29.4850 −1.00194
\(867\) −40.9143 −1.38952
\(868\) −13.2400 −0.449396
\(869\) −11.9602 −0.405722
\(870\) −22.8203 −0.773680
\(871\) 1.22128 0.0413814
\(872\) −38.7272 −1.31147
\(873\) 44.9428 1.52108
\(874\) 6.60214 0.223321
\(875\) 2.47152 0.0835527
\(876\) −2.35945 −0.0797183
\(877\) −21.8467 −0.737711 −0.368856 0.929487i \(-0.620250\pi\)
−0.368856 + 0.929487i \(0.620250\pi\)
\(878\) 26.9657 0.910048
\(879\) −2.37764 −0.0801958
\(880\) −1.75660 −0.0592151
\(881\) −10.7521 −0.362247 −0.181124 0.983460i \(-0.557973\pi\)
−0.181124 + 0.983460i \(0.557973\pi\)
\(882\) 5.54334 0.186654
\(883\) −21.1124 −0.710489 −0.355244 0.934773i \(-0.615602\pi\)
−0.355244 + 0.934773i \(0.615602\pi\)
\(884\) −3.87594 −0.130362
\(885\) −17.7884 −0.597951
\(886\) −28.5245 −0.958300
\(887\) 18.2536 0.612896 0.306448 0.951887i \(-0.400859\pi\)
0.306448 + 0.951887i \(0.400859\pi\)
\(888\) 49.3334 1.65552
\(889\) −11.1604 −0.374309
\(890\) 17.2759 0.579088
\(891\) 6.20521 0.207882
\(892\) −7.20958 −0.241395
\(893\) 5.35825 0.179307
\(894\) −6.16100 −0.206055
\(895\) −5.92443 −0.198032
\(896\) −1.17281 −0.0391808
\(897\) 20.4146 0.681623
\(898\) 14.1490 0.472159
\(899\) −47.3168 −1.57811
\(900\) −4.54771 −0.151590
\(901\) −30.1829 −1.00554
\(902\) 11.1875 0.372503
\(903\) 41.0562 1.36626
\(904\) −22.1028 −0.735129
\(905\) −9.28992 −0.308807
\(906\) −60.0396 −1.99468
\(907\) −18.5536 −0.616061 −0.308031 0.951376i \(-0.599670\pi\)
−0.308031 + 0.951376i \(0.599670\pi\)
\(908\) −1.58739 −0.0526794
\(909\) 12.4616 0.413327
\(910\) −2.35645 −0.0781157
\(911\) −44.2680 −1.46666 −0.733332 0.679871i \(-0.762036\pi\)
−0.733332 + 0.679871i \(0.762036\pi\)
\(912\) 3.92003 0.129805
\(913\) 13.3277 0.441084
\(914\) 40.7781 1.34882
\(915\) −11.3805 −0.376227
\(916\) −11.4437 −0.378109
\(917\) 4.65447 0.153704
\(918\) 48.0099 1.58456
\(919\) 9.35309 0.308530 0.154265 0.988030i \(-0.450699\pi\)
0.154265 + 0.988030i \(0.450699\pi\)
\(920\) −24.4108 −0.804799
\(921\) 58.4340 1.92547
\(922\) −26.2500 −0.864498
\(923\) −11.1924 −0.368401
\(924\) 5.83142 0.191840
\(925\) −5.46020 −0.179530
\(926\) −0.541600 −0.0177981
\(927\) −24.2334 −0.795929
\(928\) 29.7880 0.977838
\(929\) −13.5633 −0.444999 −0.222499 0.974933i \(-0.571422\pi\)
−0.222499 + 0.974933i \(0.571422\pi\)
\(930\) 21.5750 0.707473
\(931\) −0.675411 −0.0221357
\(932\) 10.3818 0.340068
\(933\) 64.5816 2.11431
\(934\) −11.8028 −0.386199
\(935\) 5.55777 0.181759
\(936\) 15.1632 0.495624
\(937\) −48.3686 −1.58013 −0.790067 0.613020i \(-0.789954\pi\)
−0.790067 + 0.613020i \(0.789954\pi\)
\(938\) −3.79598 −0.123943
\(939\) 26.7126 0.871734
\(940\) −5.66523 −0.184779
\(941\) 37.4129 1.21963 0.609813 0.792545i \(-0.291245\pi\)
0.609813 + 0.792545i \(0.291245\pi\)
\(942\) −3.24436 −0.105707
\(943\) 81.3151 2.64798
\(944\) −10.6072 −0.345236
\(945\) −19.4973 −0.634247
\(946\) 6.17484 0.200761
\(947\) −36.0948 −1.17292 −0.586461 0.809977i \(-0.699479\pi\)
−0.586461 + 0.809977i \(0.699479\pi\)
\(948\) −28.2194 −0.916525
\(949\) −0.870712 −0.0282645
\(950\) −0.829521 −0.0269132
\(951\) 25.2819 0.819821
\(952\) 42.1298 1.36544
\(953\) −32.6936 −1.05905 −0.529525 0.848294i \(-0.677630\pi\)
−0.529525 + 0.848294i \(0.677630\pi\)
\(954\) 33.7653 1.09319
\(955\) 12.0638 0.390376
\(956\) −12.6533 −0.409235
\(957\) 20.8402 0.673667
\(958\) −16.7637 −0.541610
\(959\) −50.6474 −1.63549
\(960\) −23.9317 −0.772393
\(961\) 13.7349 0.443060
\(962\) 5.20599 0.167848
\(963\) 85.8237 2.76563
\(964\) 14.1664 0.456269
\(965\) 10.2839 0.331049
\(966\) −63.4527 −2.04156
\(967\) 49.8269 1.60233 0.801163 0.598447i \(-0.204215\pi\)
0.801163 + 0.598447i \(0.204215\pi\)
\(968\) 3.06707 0.0985795
\(969\) −12.4027 −0.398433
\(970\) −8.66740 −0.278293
\(971\) −51.0389 −1.63792 −0.818958 0.573853i \(-0.805448\pi\)
−0.818958 + 0.573853i \(0.805448\pi\)
\(972\) −4.31452 −0.138388
\(973\) 22.0860 0.708046
\(974\) −38.6885 −1.23966
\(975\) −2.56497 −0.0821449
\(976\) −6.78618 −0.217220
\(977\) 3.92220 0.125482 0.0627412 0.998030i \(-0.480016\pi\)
0.0627412 + 0.998030i \(0.480016\pi\)
\(978\) 22.3041 0.713207
\(979\) −15.7768 −0.504230
\(980\) 0.714105 0.0228113
\(981\) −71.6939 −2.28901
\(982\) 1.36341 0.0435082
\(983\) −47.4469 −1.51332 −0.756660 0.653808i \(-0.773170\pi\)
−0.756660 + 0.653808i \(0.773170\pi\)
\(984\) 92.3094 2.94272
\(985\) 6.06425 0.193223
\(986\) 43.0540 1.37112
\(987\) −51.4978 −1.63919
\(988\) −0.528304 −0.0168076
\(989\) 44.8810 1.42713
\(990\) −6.21743 −0.197603
\(991\) −55.2816 −1.75608 −0.878039 0.478588i \(-0.841149\pi\)
−0.878039 + 0.478588i \(0.841149\pi\)
\(992\) −28.1625 −0.894160
\(993\) 54.5393 1.73075
\(994\) 34.7881 1.10341
\(995\) −11.5157 −0.365073
\(996\) 31.4461 0.996407
\(997\) −31.4861 −0.997176 −0.498588 0.866839i \(-0.666148\pi\)
−0.498588 + 0.866839i \(0.666148\pi\)
\(998\) −41.5331 −1.31471
\(999\) 43.0743 1.36281
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 4015.2.a.b.1.9 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.b.1.9 23 1.1 even 1 trivial