Properties

Label 4031.2.a.b.1.3
Level $4031$
Weight $2$
Character 4031.1
Self dual yes
Analytic conductor $32.188$
Analytic rank $1$
Dimension $59$
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] = [4031,2,Mod(1,4031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, 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("4031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4031 = 29 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4031.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: \(32.1876970548\)
Analytic rank: \(1\)
Dimension: \(59\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 4031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.49699 q^{2} -1.64487 q^{3} +4.23496 q^{4} +3.35969 q^{5} +4.10722 q^{6} +2.65574 q^{7} -5.58068 q^{8} -0.294408 q^{9} +O(q^{10})\) \(q-2.49699 q^{2} -1.64487 q^{3} +4.23496 q^{4} +3.35969 q^{5} +4.10722 q^{6} +2.65574 q^{7} -5.58068 q^{8} -0.294408 q^{9} -8.38911 q^{10} -1.44302 q^{11} -6.96596 q^{12} -1.58926 q^{13} -6.63137 q^{14} -5.52624 q^{15} +5.46498 q^{16} -2.75463 q^{17} +0.735133 q^{18} -1.18144 q^{19} +14.2281 q^{20} -4.36835 q^{21} +3.60320 q^{22} +1.11583 q^{23} +9.17948 q^{24} +6.28750 q^{25} +3.96838 q^{26} +5.41887 q^{27} +11.2470 q^{28} +1.00000 q^{29} +13.7990 q^{30} -3.87621 q^{31} -2.48464 q^{32} +2.37357 q^{33} +6.87829 q^{34} +8.92247 q^{35} -1.24681 q^{36} -10.3550 q^{37} +2.95005 q^{38} +2.61413 q^{39} -18.7493 q^{40} +7.07077 q^{41} +10.9077 q^{42} +11.3569 q^{43} -6.11111 q^{44} -0.989118 q^{45} -2.78621 q^{46} -12.8863 q^{47} -8.98917 q^{48} +0.0529764 q^{49} -15.6998 q^{50} +4.53101 q^{51} -6.73047 q^{52} +10.5507 q^{53} -13.5309 q^{54} -4.84808 q^{55} -14.8209 q^{56} +1.94332 q^{57} -2.49699 q^{58} -1.08776 q^{59} -23.4034 q^{60} -6.20476 q^{61} +9.67885 q^{62} -0.781872 q^{63} -4.72583 q^{64} -5.33943 q^{65} -5.92678 q^{66} +0.654700 q^{67} -11.6658 q^{68} -1.83539 q^{69} -22.2793 q^{70} +1.51803 q^{71} +1.64299 q^{72} -15.9517 q^{73} +25.8562 q^{74} -10.3421 q^{75} -5.00337 q^{76} -3.83228 q^{77} -6.52746 q^{78} -4.67693 q^{79} +18.3606 q^{80} -8.03010 q^{81} -17.6556 q^{82} -8.32610 q^{83} -18.4998 q^{84} -9.25470 q^{85} -28.3581 q^{86} -1.64487 q^{87} +8.05300 q^{88} +6.66528 q^{89} +2.46982 q^{90} -4.22068 q^{91} +4.72548 q^{92} +6.37585 q^{93} +32.1769 q^{94} -3.96928 q^{95} +4.08691 q^{96} +10.6505 q^{97} -0.132282 q^{98} +0.424835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.49699 −1.76564 −0.882819 0.469712i \(-0.844358\pi\)
−0.882819 + 0.469712i \(0.844358\pi\)
\(3\) −1.64487 −0.949665 −0.474833 0.880076i \(-0.657491\pi\)
−0.474833 + 0.880076i \(0.657491\pi\)
\(4\) 4.23496 2.11748
\(5\) 3.35969 1.50250 0.751249 0.660019i \(-0.229452\pi\)
0.751249 + 0.660019i \(0.229452\pi\)
\(6\) 4.10722 1.67677
\(7\) 2.65574 1.00378 0.501888 0.864932i \(-0.332639\pi\)
0.501888 + 0.864932i \(0.332639\pi\)
\(8\) −5.58068 −1.97307
\(9\) −0.294408 −0.0981359
\(10\) −8.38911 −2.65287
\(11\) −1.44302 −0.435085 −0.217543 0.976051i \(-0.569804\pi\)
−0.217543 + 0.976051i \(0.569804\pi\)
\(12\) −6.96596 −2.01090
\(13\) −1.58926 −0.440783 −0.220391 0.975412i \(-0.570733\pi\)
−0.220391 + 0.975412i \(0.570733\pi\)
\(14\) −6.63137 −1.77231
\(15\) −5.52624 −1.42687
\(16\) 5.46498 1.36624
\(17\) −2.75463 −0.668096 −0.334048 0.942556i \(-0.608415\pi\)
−0.334048 + 0.942556i \(0.608415\pi\)
\(18\) 0.735133 0.173273
\(19\) −1.18144 −0.271042 −0.135521 0.990774i \(-0.543271\pi\)
−0.135521 + 0.990774i \(0.543271\pi\)
\(20\) 14.2281 3.18151
\(21\) −4.36835 −0.953252
\(22\) 3.60320 0.768204
\(23\) 1.11583 0.232666 0.116333 0.993210i \(-0.462886\pi\)
0.116333 + 0.993210i \(0.462886\pi\)
\(24\) 9.17948 1.87375
\(25\) 6.28750 1.25750
\(26\) 3.96838 0.778263
\(27\) 5.41887 1.04286
\(28\) 11.2470 2.12548
\(29\) 1.00000 0.185695
\(30\) 13.7990 2.51934
\(31\) −3.87621 −0.696187 −0.348094 0.937460i \(-0.613171\pi\)
−0.348094 + 0.937460i \(0.613171\pi\)
\(32\) −2.48464 −0.439227
\(33\) 2.37357 0.413186
\(34\) 6.87829 1.17962
\(35\) 8.92247 1.50817
\(36\) −1.24681 −0.207801
\(37\) −10.3550 −1.70234 −0.851172 0.524887i \(-0.824108\pi\)
−0.851172 + 0.524887i \(0.824108\pi\)
\(38\) 2.95005 0.478562
\(39\) 2.61413 0.418596
\(40\) −18.7493 −2.96453
\(41\) 7.07077 1.10427 0.552134 0.833755i \(-0.313814\pi\)
0.552134 + 0.833755i \(0.313814\pi\)
\(42\) 10.9077 1.68310
\(43\) 11.3569 1.73191 0.865955 0.500122i \(-0.166711\pi\)
0.865955 + 0.500122i \(0.166711\pi\)
\(44\) −6.11111 −0.921285
\(45\) −0.989118 −0.147449
\(46\) −2.78621 −0.410804
\(47\) −12.8863 −1.87965 −0.939827 0.341649i \(-0.889014\pi\)
−0.939827 + 0.341649i \(0.889014\pi\)
\(48\) −8.98917 −1.29747
\(49\) 0.0529764 0.00756806
\(50\) −15.6998 −2.22029
\(51\) 4.53101 0.634468
\(52\) −6.73047 −0.933349
\(53\) 10.5507 1.44925 0.724623 0.689145i \(-0.242014\pi\)
0.724623 + 0.689145i \(0.242014\pi\)
\(54\) −13.5309 −1.84132
\(55\) −4.84808 −0.653715
\(56\) −14.8209 −1.98052
\(57\) 1.94332 0.257399
\(58\) −2.49699 −0.327871
\(59\) −1.08776 −0.141614 −0.0708069 0.997490i \(-0.522557\pi\)
−0.0708069 + 0.997490i \(0.522557\pi\)
\(60\) −23.4034 −3.02137
\(61\) −6.20476 −0.794438 −0.397219 0.917724i \(-0.630025\pi\)
−0.397219 + 0.917724i \(0.630025\pi\)
\(62\) 9.67885 1.22922
\(63\) −0.781872 −0.0985066
\(64\) −4.72583 −0.590729
\(65\) −5.33943 −0.662275
\(66\) −5.92678 −0.729536
\(67\) 0.654700 0.0799844 0.0399922 0.999200i \(-0.487267\pi\)
0.0399922 + 0.999200i \(0.487267\pi\)
\(68\) −11.6658 −1.41468
\(69\) −1.83539 −0.220955
\(70\) −22.2793 −2.66289
\(71\) 1.51803 0.180157 0.0900784 0.995935i \(-0.471288\pi\)
0.0900784 + 0.995935i \(0.471288\pi\)
\(72\) 1.64299 0.193629
\(73\) −15.9517 −1.86700 −0.933501 0.358575i \(-0.883263\pi\)
−0.933501 + 0.358575i \(0.883263\pi\)
\(74\) 25.8562 3.00573
\(75\) −10.3421 −1.19420
\(76\) −5.00337 −0.573926
\(77\) −3.83228 −0.436729
\(78\) −6.52746 −0.739089
\(79\) −4.67693 −0.526195 −0.263098 0.964769i \(-0.584744\pi\)
−0.263098 + 0.964769i \(0.584744\pi\)
\(80\) 18.3606 2.05278
\(81\) −8.03010 −0.892233
\(82\) −17.6556 −1.94974
\(83\) −8.32610 −0.913908 −0.456954 0.889490i \(-0.651060\pi\)
−0.456954 + 0.889490i \(0.651060\pi\)
\(84\) −18.4998 −2.01849
\(85\) −9.25470 −1.00381
\(86\) −28.3581 −3.05793
\(87\) −1.64487 −0.176348
\(88\) 8.05300 0.858453
\(89\) 6.66528 0.706518 0.353259 0.935526i \(-0.385073\pi\)
0.353259 + 0.935526i \(0.385073\pi\)
\(90\) 2.46982 0.260342
\(91\) −4.22068 −0.442447
\(92\) 4.72548 0.492665
\(93\) 6.37585 0.661145
\(94\) 32.1769 3.31879
\(95\) −3.96928 −0.407240
\(96\) 4.08691 0.417118
\(97\) 10.6505 1.08139 0.540696 0.841218i \(-0.318161\pi\)
0.540696 + 0.841218i \(0.318161\pi\)
\(98\) −0.132282 −0.0133625
\(99\) 0.424835 0.0426975
\(100\) 26.6273 2.66273
\(101\) −8.52962 −0.848729 −0.424365 0.905491i \(-0.639503\pi\)
−0.424365 + 0.905491i \(0.639503\pi\)
\(102\) −11.3139 −1.12024
\(103\) 5.68028 0.559694 0.279847 0.960045i \(-0.409716\pi\)
0.279847 + 0.960045i \(0.409716\pi\)
\(104\) 8.86917 0.869694
\(105\) −14.6763 −1.43226
\(106\) −26.3449 −2.55885
\(107\) −17.1341 −1.65642 −0.828208 0.560421i \(-0.810639\pi\)
−0.828208 + 0.560421i \(0.810639\pi\)
\(108\) 22.9487 2.20824
\(109\) −6.77580 −0.649004 −0.324502 0.945885i \(-0.605197\pi\)
−0.324502 + 0.945885i \(0.605197\pi\)
\(110\) 12.1056 1.15422
\(111\) 17.0325 1.61666
\(112\) 14.5136 1.37140
\(113\) 8.89649 0.836911 0.418456 0.908237i \(-0.362572\pi\)
0.418456 + 0.908237i \(0.362572\pi\)
\(114\) −4.85245 −0.454474
\(115\) 3.74883 0.349580
\(116\) 4.23496 0.393206
\(117\) 0.467892 0.0432566
\(118\) 2.71612 0.250039
\(119\) −7.31560 −0.670620
\(120\) 30.8402 2.81531
\(121\) −8.91771 −0.810701
\(122\) 15.4932 1.40269
\(123\) −11.6305 −1.04868
\(124\) −16.4156 −1.47416
\(125\) 4.32561 0.386894
\(126\) 1.95233 0.173927
\(127\) −12.2028 −1.08283 −0.541413 0.840757i \(-0.682111\pi\)
−0.541413 + 0.840757i \(0.682111\pi\)
\(128\) 16.7696 1.48224
\(129\) −18.6806 −1.64473
\(130\) 13.3325 1.16934
\(131\) 1.65229 0.144361 0.0721804 0.997392i \(-0.477004\pi\)
0.0721804 + 0.997392i \(0.477004\pi\)
\(132\) 10.0520 0.874912
\(133\) −3.13761 −0.272065
\(134\) −1.63478 −0.141224
\(135\) 18.2057 1.56690
\(136\) 15.3727 1.31820
\(137\) 10.0485 0.858503 0.429251 0.903185i \(-0.358777\pi\)
0.429251 + 0.903185i \(0.358777\pi\)
\(138\) 4.58294 0.390126
\(139\) 1.00000 0.0848189
\(140\) 37.7863 3.19353
\(141\) 21.1962 1.78504
\(142\) −3.79050 −0.318092
\(143\) 2.29333 0.191778
\(144\) −1.60893 −0.134078
\(145\) 3.35969 0.279007
\(146\) 39.8312 3.29645
\(147\) −0.0871393 −0.00718712
\(148\) −43.8528 −3.60468
\(149\) 6.18242 0.506483 0.253242 0.967403i \(-0.418503\pi\)
0.253242 + 0.967403i \(0.418503\pi\)
\(150\) 25.8242 2.10853
\(151\) −4.03288 −0.328191 −0.164096 0.986444i \(-0.552471\pi\)
−0.164096 + 0.986444i \(0.552471\pi\)
\(152\) 6.59326 0.534784
\(153\) 0.810985 0.0655642
\(154\) 9.56916 0.771105
\(155\) −13.0228 −1.04602
\(156\) 11.0707 0.886369
\(157\) 9.95214 0.794268 0.397134 0.917761i \(-0.370005\pi\)
0.397134 + 0.917761i \(0.370005\pi\)
\(158\) 11.6782 0.929071
\(159\) −17.3545 −1.37630
\(160\) −8.34762 −0.659937
\(161\) 2.96335 0.233545
\(162\) 20.0511 1.57536
\(163\) −14.2973 −1.11985 −0.559924 0.828544i \(-0.689170\pi\)
−0.559924 + 0.828544i \(0.689170\pi\)
\(164\) 29.9444 2.33827
\(165\) 7.97445 0.620810
\(166\) 20.7902 1.61363
\(167\) 16.9733 1.31343 0.656716 0.754138i \(-0.271945\pi\)
0.656716 + 0.754138i \(0.271945\pi\)
\(168\) 24.3784 1.88083
\(169\) −10.4742 −0.805711
\(170\) 23.1089 1.77237
\(171\) 0.347826 0.0265989
\(172\) 48.0960 3.66729
\(173\) −7.31591 −0.556218 −0.278109 0.960549i \(-0.589708\pi\)
−0.278109 + 0.960549i \(0.589708\pi\)
\(174\) 4.10722 0.311368
\(175\) 16.6980 1.26225
\(176\) −7.88605 −0.594433
\(177\) 1.78922 0.134486
\(178\) −16.6431 −1.24746
\(179\) 20.9837 1.56840 0.784199 0.620509i \(-0.213074\pi\)
0.784199 + 0.620509i \(0.213074\pi\)
\(180\) −4.18888 −0.312220
\(181\) −11.4341 −0.849890 −0.424945 0.905219i \(-0.639707\pi\)
−0.424945 + 0.905219i \(0.639707\pi\)
\(182\) 10.5390 0.781202
\(183\) 10.2060 0.754450
\(184\) −6.22706 −0.459065
\(185\) −34.7894 −2.55777
\(186\) −15.9204 −1.16734
\(187\) 3.97498 0.290679
\(188\) −54.5728 −3.98013
\(189\) 14.3911 1.04680
\(190\) 9.91126 0.719038
\(191\) −27.3743 −1.98073 −0.990366 0.138472i \(-0.955781\pi\)
−0.990366 + 0.138472i \(0.955781\pi\)
\(192\) 7.77337 0.560995
\(193\) 20.9509 1.50808 0.754038 0.656830i \(-0.228103\pi\)
0.754038 + 0.656830i \(0.228103\pi\)
\(194\) −26.5941 −1.90935
\(195\) 8.78266 0.628939
\(196\) 0.224353 0.0160252
\(197\) −12.9317 −0.921342 −0.460671 0.887571i \(-0.652391\pi\)
−0.460671 + 0.887571i \(0.652391\pi\)
\(198\) −1.06081 −0.0753884
\(199\) 7.24092 0.513295 0.256647 0.966505i \(-0.417382\pi\)
0.256647 + 0.966505i \(0.417382\pi\)
\(200\) −35.0885 −2.48113
\(201\) −1.07690 −0.0759584
\(202\) 21.2984 1.49855
\(203\) 2.65574 0.186397
\(204\) 19.1886 1.34347
\(205\) 23.7556 1.65916
\(206\) −14.1836 −0.988218
\(207\) −0.328508 −0.0228329
\(208\) −8.68529 −0.602217
\(209\) 1.70484 0.117926
\(210\) 36.6466 2.52885
\(211\) −20.9654 −1.44331 −0.721657 0.692251i \(-0.756619\pi\)
−0.721657 + 0.692251i \(0.756619\pi\)
\(212\) 44.6817 3.06875
\(213\) −2.49696 −0.171089
\(214\) 42.7837 2.92463
\(215\) 38.1556 2.60219
\(216\) −30.2410 −2.05764
\(217\) −10.2942 −0.698817
\(218\) 16.9191 1.14591
\(219\) 26.2384 1.77303
\(220\) −20.5314 −1.38423
\(221\) 4.37784 0.294485
\(222\) −42.5301 −2.85443
\(223\) 2.39949 0.160682 0.0803408 0.996767i \(-0.474399\pi\)
0.0803408 + 0.996767i \(0.474399\pi\)
\(224\) −6.59857 −0.440886
\(225\) −1.85109 −0.123406
\(226\) −22.2145 −1.47768
\(227\) 18.2496 1.21127 0.605633 0.795744i \(-0.292920\pi\)
0.605633 + 0.795744i \(0.292920\pi\)
\(228\) 8.22988 0.545037
\(229\) −25.3716 −1.67661 −0.838303 0.545205i \(-0.816452\pi\)
−0.838303 + 0.545205i \(0.816452\pi\)
\(230\) −9.36078 −0.617232
\(231\) 6.30359 0.414746
\(232\) −5.58068 −0.366389
\(233\) −18.4927 −1.21150 −0.605749 0.795655i \(-0.707127\pi\)
−0.605749 + 0.795655i \(0.707127\pi\)
\(234\) −1.16832 −0.0763755
\(235\) −43.2938 −2.82418
\(236\) −4.60661 −0.299865
\(237\) 7.69293 0.499710
\(238\) 18.2670 1.18407
\(239\) 15.4825 1.00148 0.500741 0.865597i \(-0.333061\pi\)
0.500741 + 0.865597i \(0.333061\pi\)
\(240\) −30.2008 −1.94945
\(241\) −21.3856 −1.37757 −0.688785 0.724965i \(-0.741856\pi\)
−0.688785 + 0.724965i \(0.741856\pi\)
\(242\) 22.2674 1.43140
\(243\) −3.04814 −0.195538
\(244\) −26.2769 −1.68221
\(245\) 0.177984 0.0113710
\(246\) 29.0412 1.85160
\(247\) 1.87763 0.119470
\(248\) 21.6319 1.37362
\(249\) 13.6953 0.867907
\(250\) −10.8010 −0.683115
\(251\) −9.77403 −0.616931 −0.308466 0.951235i \(-0.599816\pi\)
−0.308466 + 0.951235i \(0.599816\pi\)
\(252\) −3.31120 −0.208586
\(253\) −1.61015 −0.101229
\(254\) 30.4704 1.91188
\(255\) 15.2228 0.953287
\(256\) −32.4220 −2.02637
\(257\) −10.6832 −0.666402 −0.333201 0.942856i \(-0.608129\pi\)
−0.333201 + 0.942856i \(0.608129\pi\)
\(258\) 46.6453 2.90401
\(259\) −27.5001 −1.70877
\(260\) −22.6123 −1.40235
\(261\) −0.294408 −0.0182234
\(262\) −4.12574 −0.254889
\(263\) 25.2549 1.55728 0.778642 0.627469i \(-0.215909\pi\)
0.778642 + 0.627469i \(0.215909\pi\)
\(264\) −13.2461 −0.815243
\(265\) 35.4470 2.17749
\(266\) 7.83459 0.480369
\(267\) −10.9635 −0.670956
\(268\) 2.77263 0.169365
\(269\) −24.5680 −1.49794 −0.748969 0.662605i \(-0.769451\pi\)
−0.748969 + 0.662605i \(0.769451\pi\)
\(270\) −45.4595 −2.76658
\(271\) 7.38985 0.448902 0.224451 0.974485i \(-0.427941\pi\)
0.224451 + 0.974485i \(0.427941\pi\)
\(272\) −15.0540 −0.912783
\(273\) 6.94246 0.420177
\(274\) −25.0911 −1.51581
\(275\) −9.07296 −0.547120
\(276\) −7.77279 −0.467867
\(277\) −2.42883 −0.145934 −0.0729671 0.997334i \(-0.523247\pi\)
−0.0729671 + 0.997334i \(0.523247\pi\)
\(278\) −2.49699 −0.149760
\(279\) 1.14119 0.0683210
\(280\) −49.7934 −2.97573
\(281\) −18.8033 −1.12171 −0.560856 0.827913i \(-0.689528\pi\)
−0.560856 + 0.827913i \(0.689528\pi\)
\(282\) −52.9267 −3.15174
\(283\) 24.9836 1.48512 0.742559 0.669780i \(-0.233612\pi\)
0.742559 + 0.669780i \(0.233612\pi\)
\(284\) 6.42879 0.381479
\(285\) 6.52895 0.386741
\(286\) −5.72643 −0.338611
\(287\) 18.7781 1.10844
\(288\) 0.731498 0.0431039
\(289\) −9.41201 −0.553647
\(290\) −8.38911 −0.492625
\(291\) −17.5186 −1.02696
\(292\) −67.5547 −3.95334
\(293\) −6.76619 −0.395285 −0.197643 0.980274i \(-0.563329\pi\)
−0.197643 + 0.980274i \(0.563329\pi\)
\(294\) 0.217586 0.0126899
\(295\) −3.65452 −0.212775
\(296\) 57.7877 3.35884
\(297\) −7.81951 −0.453734
\(298\) −15.4374 −0.894267
\(299\) −1.77334 −0.102555
\(300\) −43.7985 −2.52871
\(301\) 30.1610 1.73845
\(302\) 10.0701 0.579467
\(303\) 14.0301 0.806009
\(304\) −6.45656 −0.370309
\(305\) −20.8461 −1.19364
\(306\) −2.02502 −0.115763
\(307\) −5.40587 −0.308529 −0.154265 0.988030i \(-0.549301\pi\)
−0.154265 + 0.988030i \(0.549301\pi\)
\(308\) −16.2296 −0.924765
\(309\) −9.34331 −0.531522
\(310\) 32.5179 1.84689
\(311\) 16.1425 0.915359 0.457680 0.889117i \(-0.348681\pi\)
0.457680 + 0.889117i \(0.348681\pi\)
\(312\) −14.5886 −0.825918
\(313\) −6.76520 −0.382392 −0.191196 0.981552i \(-0.561237\pi\)
−0.191196 + 0.981552i \(0.561237\pi\)
\(314\) −24.8504 −1.40239
\(315\) −2.62684 −0.148006
\(316\) −19.8066 −1.11421
\(317\) 22.2330 1.24873 0.624365 0.781133i \(-0.285358\pi\)
0.624365 + 0.781133i \(0.285358\pi\)
\(318\) 43.3340 2.43005
\(319\) −1.44302 −0.0807933
\(320\) −15.8773 −0.887569
\(321\) 28.1833 1.57304
\(322\) −7.39945 −0.412355
\(323\) 3.25444 0.181082
\(324\) −34.0072 −1.88929
\(325\) −9.99250 −0.554284
\(326\) 35.7001 1.97725
\(327\) 11.1453 0.616337
\(328\) −39.4597 −2.17880
\(329\) −34.2226 −1.88675
\(330\) −19.9121 −1.09613
\(331\) 12.6863 0.697299 0.348650 0.937253i \(-0.386640\pi\)
0.348650 + 0.937253i \(0.386640\pi\)
\(332\) −35.2607 −1.93518
\(333\) 3.04858 0.167061
\(334\) −42.3821 −2.31905
\(335\) 2.19959 0.120176
\(336\) −23.8729 −1.30238
\(337\) 22.3956 1.21997 0.609984 0.792414i \(-0.291176\pi\)
0.609984 + 0.792414i \(0.291176\pi\)
\(338\) 26.1541 1.42259
\(339\) −14.6336 −0.794786
\(340\) −39.1933 −2.12556
\(341\) 5.59342 0.302901
\(342\) −0.868519 −0.0469641
\(343\) −18.4495 −0.996180
\(344\) −63.3792 −3.41718
\(345\) −6.16633 −0.331984
\(346\) 18.2678 0.982080
\(347\) −14.0204 −0.752654 −0.376327 0.926487i \(-0.622813\pi\)
−0.376327 + 0.926487i \(0.622813\pi\)
\(348\) −6.96596 −0.373414
\(349\) −0.480731 −0.0257329 −0.0128665 0.999917i \(-0.504096\pi\)
−0.0128665 + 0.999917i \(0.504096\pi\)
\(350\) −41.6947 −2.22868
\(351\) −8.61201 −0.459675
\(352\) 3.58537 0.191101
\(353\) 23.1003 1.22951 0.614753 0.788720i \(-0.289256\pi\)
0.614753 + 0.788720i \(0.289256\pi\)
\(354\) −4.46766 −0.237453
\(355\) 5.10010 0.270685
\(356\) 28.2272 1.49604
\(357\) 12.0332 0.636864
\(358\) −52.3962 −2.76922
\(359\) 8.93834 0.471748 0.235874 0.971784i \(-0.424205\pi\)
0.235874 + 0.971784i \(0.424205\pi\)
\(360\) 5.51995 0.290927
\(361\) −17.6042 −0.926536
\(362\) 28.5508 1.50060
\(363\) 14.6685 0.769894
\(364\) −17.8744 −0.936874
\(365\) −53.5926 −2.80517
\(366\) −25.4843 −1.33209
\(367\) −0.224077 −0.0116967 −0.00584836 0.999983i \(-0.501862\pi\)
−0.00584836 + 0.999983i \(0.501862\pi\)
\(368\) 6.09796 0.317878
\(369\) −2.08169 −0.108368
\(370\) 86.8688 4.51610
\(371\) 28.0199 1.45472
\(372\) 27.0015 1.39996
\(373\) −3.45908 −0.179105 −0.0895523 0.995982i \(-0.528544\pi\)
−0.0895523 + 0.995982i \(0.528544\pi\)
\(374\) −9.92548 −0.513234
\(375\) −7.11505 −0.367420
\(376\) 71.9141 3.70869
\(377\) −1.58926 −0.0818513
\(378\) −35.9345 −1.84827
\(379\) −14.0565 −0.722035 −0.361018 0.932559i \(-0.617571\pi\)
−0.361018 + 0.932559i \(0.617571\pi\)
\(380\) −16.8098 −0.862322
\(381\) 20.0721 1.02832
\(382\) 68.3533 3.49726
\(383\) 15.2901 0.781285 0.390643 0.920542i \(-0.372253\pi\)
0.390643 + 0.920542i \(0.372253\pi\)
\(384\) −27.5838 −1.40763
\(385\) −12.8753 −0.656184
\(386\) −52.3141 −2.66272
\(387\) −3.34356 −0.169963
\(388\) 45.1043 2.28983
\(389\) −26.2883 −1.33287 −0.666436 0.745563i \(-0.732181\pi\)
−0.666436 + 0.745563i \(0.732181\pi\)
\(390\) −21.9302 −1.11048
\(391\) −3.07369 −0.155443
\(392\) −0.295644 −0.0149323
\(393\) −2.71779 −0.137094
\(394\) 32.2902 1.62676
\(395\) −15.7130 −0.790608
\(396\) 1.79916 0.0904112
\(397\) 5.62692 0.282407 0.141203 0.989981i \(-0.454903\pi\)
0.141203 + 0.989981i \(0.454903\pi\)
\(398\) −18.0805 −0.906293
\(399\) 5.16096 0.258371
\(400\) 34.3611 1.71805
\(401\) −27.8936 −1.39294 −0.696470 0.717586i \(-0.745247\pi\)
−0.696470 + 0.717586i \(0.745247\pi\)
\(402\) 2.68900 0.134115
\(403\) 6.16031 0.306867
\(404\) −36.1226 −1.79717
\(405\) −26.9786 −1.34058
\(406\) −6.63137 −0.329109
\(407\) 14.9424 0.740665
\(408\) −25.2861 −1.25185
\(409\) 4.43537 0.219315 0.109657 0.993969i \(-0.465025\pi\)
0.109657 + 0.993969i \(0.465025\pi\)
\(410\) −59.3174 −2.92948
\(411\) −16.5285 −0.815290
\(412\) 24.0558 1.18514
\(413\) −2.88880 −0.142149
\(414\) 0.820281 0.0403146
\(415\) −27.9731 −1.37315
\(416\) 3.94875 0.193603
\(417\) −1.64487 −0.0805496
\(418\) −4.25697 −0.208215
\(419\) −12.8489 −0.627712 −0.313856 0.949471i \(-0.601621\pi\)
−0.313856 + 0.949471i \(0.601621\pi\)
\(420\) −62.1535 −3.03278
\(421\) −6.89588 −0.336085 −0.168042 0.985780i \(-0.553745\pi\)
−0.168042 + 0.985780i \(0.553745\pi\)
\(422\) 52.3503 2.54837
\(423\) 3.79382 0.184462
\(424\) −58.8799 −2.85946
\(425\) −17.3198 −0.840131
\(426\) 6.23488 0.302081
\(427\) −16.4783 −0.797439
\(428\) −72.5622 −3.50743
\(429\) −3.77223 −0.182125
\(430\) −95.2742 −4.59453
\(431\) −21.3752 −1.02961 −0.514803 0.857308i \(-0.672135\pi\)
−0.514803 + 0.857308i \(0.672135\pi\)
\(432\) 29.6140 1.42480
\(433\) −31.3707 −1.50758 −0.753791 0.657115i \(-0.771777\pi\)
−0.753791 + 0.657115i \(0.771777\pi\)
\(434\) 25.7045 1.23386
\(435\) −5.52624 −0.264963
\(436\) −28.6953 −1.37425
\(437\) −1.31829 −0.0630621
\(438\) −65.5170 −3.13053
\(439\) 18.8567 0.899982 0.449991 0.893033i \(-0.351427\pi\)
0.449991 + 0.893033i \(0.351427\pi\)
\(440\) 27.0556 1.28982
\(441\) −0.0155967 −0.000742699 0
\(442\) −10.9314 −0.519954
\(443\) 10.9523 0.520362 0.260181 0.965560i \(-0.416218\pi\)
0.260181 + 0.965560i \(0.416218\pi\)
\(444\) 72.1322 3.42324
\(445\) 22.3933 1.06154
\(446\) −5.99150 −0.283706
\(447\) −10.1693 −0.480990
\(448\) −12.5506 −0.592960
\(449\) −20.3161 −0.958778 −0.479389 0.877603i \(-0.659142\pi\)
−0.479389 + 0.877603i \(0.659142\pi\)
\(450\) 4.62215 0.217890
\(451\) −10.2032 −0.480451
\(452\) 37.6763 1.77214
\(453\) 6.63356 0.311672
\(454\) −45.5690 −2.13866
\(455\) −14.1802 −0.664776
\(456\) −10.8450 −0.507866
\(457\) −26.6303 −1.24571 −0.622857 0.782336i \(-0.714028\pi\)
−0.622857 + 0.782336i \(0.714028\pi\)
\(458\) 63.3527 2.96028
\(459\) −14.9270 −0.696732
\(460\) 15.8761 0.740229
\(461\) 22.3445 1.04069 0.520343 0.853958i \(-0.325804\pi\)
0.520343 + 0.853958i \(0.325804\pi\)
\(462\) −15.7400 −0.732292
\(463\) −6.37362 −0.296207 −0.148104 0.988972i \(-0.547317\pi\)
−0.148104 + 0.988972i \(0.547317\pi\)
\(464\) 5.46498 0.253705
\(465\) 21.4209 0.993369
\(466\) 46.1762 2.13907
\(467\) 22.7290 1.05177 0.525885 0.850556i \(-0.323734\pi\)
0.525885 + 0.850556i \(0.323734\pi\)
\(468\) 1.98150 0.0915950
\(469\) 1.73872 0.0802865
\(470\) 108.104 4.98648
\(471\) −16.3700 −0.754288
\(472\) 6.07042 0.279414
\(473\) −16.3882 −0.753529
\(474\) −19.2092 −0.882307
\(475\) −7.42833 −0.340835
\(476\) −30.9813 −1.42002
\(477\) −3.10620 −0.142223
\(478\) −38.6597 −1.76825
\(479\) −3.31497 −0.151465 −0.0757323 0.997128i \(-0.524129\pi\)
−0.0757323 + 0.997128i \(0.524129\pi\)
\(480\) 13.7307 0.626719
\(481\) 16.4568 0.750364
\(482\) 53.3998 2.43229
\(483\) −4.87432 −0.221789
\(484\) −37.7661 −1.71664
\(485\) 35.7822 1.62479
\(486\) 7.61118 0.345250
\(487\) 32.1780 1.45812 0.729062 0.684448i \(-0.239957\pi\)
0.729062 + 0.684448i \(0.239957\pi\)
\(488\) 34.6268 1.56748
\(489\) 23.5171 1.06348
\(490\) −0.444425 −0.0200771
\(491\) 1.71274 0.0772949 0.0386474 0.999253i \(-0.487695\pi\)
0.0386474 + 0.999253i \(0.487695\pi\)
\(492\) −49.2546 −2.22057
\(493\) −2.75463 −0.124062
\(494\) −4.68841 −0.210942
\(495\) 1.42731 0.0641529
\(496\) −21.1834 −0.951162
\(497\) 4.03150 0.180837
\(498\) −34.1971 −1.53241
\(499\) 1.56043 0.0698546 0.0349273 0.999390i \(-0.488880\pi\)
0.0349273 + 0.999390i \(0.488880\pi\)
\(500\) 18.3188 0.819240
\(501\) −27.9188 −1.24732
\(502\) 24.4057 1.08928
\(503\) −4.72499 −0.210677 −0.105338 0.994436i \(-0.533593\pi\)
−0.105338 + 0.994436i \(0.533593\pi\)
\(504\) 4.36337 0.194360
\(505\) −28.6569 −1.27521
\(506\) 4.02054 0.178735
\(507\) 17.2287 0.765156
\(508\) −51.6785 −2.29286
\(509\) −5.24997 −0.232701 −0.116350 0.993208i \(-0.537120\pi\)
−0.116350 + 0.993208i \(0.537120\pi\)
\(510\) −38.0111 −1.68316
\(511\) −42.3636 −1.87405
\(512\) 47.4181 2.09560
\(513\) −6.40209 −0.282659
\(514\) 26.6759 1.17662
\(515\) 19.0840 0.840940
\(516\) −79.1116 −3.48269
\(517\) 18.5951 0.817810
\(518\) 68.6675 3.01708
\(519\) 12.0337 0.528221
\(520\) 29.7976 1.30671
\(521\) 9.72129 0.425898 0.212949 0.977063i \(-0.431693\pi\)
0.212949 + 0.977063i \(0.431693\pi\)
\(522\) 0.735133 0.0321759
\(523\) 12.5361 0.548167 0.274083 0.961706i \(-0.411626\pi\)
0.274083 + 0.961706i \(0.411626\pi\)
\(524\) 6.99737 0.305681
\(525\) −27.4660 −1.19871
\(526\) −63.0612 −2.74960
\(527\) 10.6775 0.465120
\(528\) 12.9715 0.564512
\(529\) −21.7549 −0.945867
\(530\) −88.5108 −3.84466
\(531\) 0.320244 0.0138974
\(532\) −13.2877 −0.576093
\(533\) −11.2373 −0.486742
\(534\) 27.3758 1.18467
\(535\) −57.5652 −2.48876
\(536\) −3.65367 −0.157815
\(537\) −34.5155 −1.48945
\(538\) 61.3461 2.64482
\(539\) −0.0764458 −0.00329275
\(540\) 77.1005 3.31788
\(541\) 12.9027 0.554730 0.277365 0.960765i \(-0.410539\pi\)
0.277365 + 0.960765i \(0.410539\pi\)
\(542\) −18.4524 −0.792598
\(543\) 18.8076 0.807111
\(544\) 6.84427 0.293446
\(545\) −22.7646 −0.975127
\(546\) −17.3353 −0.741881
\(547\) −18.9601 −0.810676 −0.405338 0.914167i \(-0.632846\pi\)
−0.405338 + 0.914167i \(0.632846\pi\)
\(548\) 42.5551 1.81786
\(549\) 1.82673 0.0779629
\(550\) 22.6551 0.966017
\(551\) −1.18144 −0.0503312
\(552\) 10.2427 0.435958
\(553\) −12.4207 −0.528183
\(554\) 6.06476 0.257667
\(555\) 57.2240 2.42902
\(556\) 4.23496 0.179602
\(557\) −7.97679 −0.337987 −0.168994 0.985617i \(-0.554052\pi\)
−0.168994 + 0.985617i \(0.554052\pi\)
\(558\) −2.84953 −0.120630
\(559\) −18.0491 −0.763396
\(560\) 48.7611 2.06053
\(561\) −6.53831 −0.276048
\(562\) 46.9517 1.98054
\(563\) 24.3492 1.02620 0.513098 0.858330i \(-0.328498\pi\)
0.513098 + 0.858330i \(0.328498\pi\)
\(564\) 89.7651 3.77979
\(565\) 29.8894 1.25746
\(566\) −62.3837 −2.62218
\(567\) −21.3259 −0.895603
\(568\) −8.47163 −0.355462
\(569\) 9.25964 0.388184 0.194092 0.980983i \(-0.437824\pi\)
0.194092 + 0.980983i \(0.437824\pi\)
\(570\) −16.3027 −0.682846
\(571\) 22.7673 0.952781 0.476391 0.879234i \(-0.341945\pi\)
0.476391 + 0.879234i \(0.341945\pi\)
\(572\) 9.71217 0.406086
\(573\) 45.0271 1.88103
\(574\) −46.8888 −1.95710
\(575\) 7.01576 0.292577
\(576\) 1.39132 0.0579717
\(577\) −18.1896 −0.757243 −0.378622 0.925552i \(-0.623602\pi\)
−0.378622 + 0.925552i \(0.623602\pi\)
\(578\) 23.5017 0.977541
\(579\) −34.4614 −1.43217
\(580\) 14.2281 0.590792
\(581\) −22.1120 −0.917360
\(582\) 43.7438 1.81324
\(583\) −15.2248 −0.630546
\(584\) 89.0212 3.68372
\(585\) 1.57197 0.0649929
\(586\) 16.8951 0.697931
\(587\) 30.8811 1.27460 0.637300 0.770616i \(-0.280051\pi\)
0.637300 + 0.770616i \(0.280051\pi\)
\(588\) −0.369031 −0.0152186
\(589\) 4.57952 0.188696
\(590\) 9.12531 0.375683
\(591\) 21.2709 0.874967
\(592\) −56.5896 −2.32582
\(593\) −16.5289 −0.678762 −0.339381 0.940649i \(-0.610218\pi\)
−0.339381 + 0.940649i \(0.610218\pi\)
\(594\) 19.5252 0.801130
\(595\) −24.5781 −1.00760
\(596\) 26.1823 1.07247
\(597\) −11.9104 −0.487458
\(598\) 4.42802 0.181075
\(599\) 2.42945 0.0992647 0.0496324 0.998768i \(-0.484195\pi\)
0.0496324 + 0.998768i \(0.484195\pi\)
\(600\) 57.7160 2.35625
\(601\) −11.4692 −0.467836 −0.233918 0.972256i \(-0.575155\pi\)
−0.233918 + 0.972256i \(0.575155\pi\)
\(602\) −75.3117 −3.06948
\(603\) −0.192749 −0.00784934
\(604\) −17.0791 −0.694938
\(605\) −29.9607 −1.21808
\(606\) −35.0330 −1.42312
\(607\) 39.2906 1.59476 0.797378 0.603480i \(-0.206220\pi\)
0.797378 + 0.603480i \(0.206220\pi\)
\(608\) 2.93546 0.119049
\(609\) −4.36835 −0.177014
\(610\) 52.0524 2.10754
\(611\) 20.4797 0.828519
\(612\) 3.43449 0.138831
\(613\) −39.0765 −1.57828 −0.789142 0.614211i \(-0.789474\pi\)
−0.789142 + 0.614211i \(0.789474\pi\)
\(614\) 13.4984 0.544751
\(615\) −39.0748 −1.57565
\(616\) 21.3867 0.861695
\(617\) 27.6791 1.11432 0.557158 0.830406i \(-0.311892\pi\)
0.557158 + 0.830406i \(0.311892\pi\)
\(618\) 23.3302 0.938477
\(619\) −39.4805 −1.58686 −0.793428 0.608664i \(-0.791706\pi\)
−0.793428 + 0.608664i \(0.791706\pi\)
\(620\) −55.1512 −2.21493
\(621\) 6.04651 0.242638
\(622\) −40.3078 −1.61619
\(623\) 17.7013 0.709187
\(624\) 14.2862 0.571904
\(625\) −16.9048 −0.676193
\(626\) 16.8926 0.675165
\(627\) −2.80424 −0.111991
\(628\) 42.1469 1.68185
\(629\) 28.5241 1.13733
\(630\) 6.55921 0.261325
\(631\) 33.0853 1.31710 0.658552 0.752535i \(-0.271169\pi\)
0.658552 + 0.752535i \(0.271169\pi\)
\(632\) 26.1004 1.03822
\(633\) 34.4852 1.37067
\(634\) −55.5156 −2.20481
\(635\) −40.9977 −1.62694
\(636\) −73.4955 −2.91429
\(637\) −0.0841935 −0.00333587
\(638\) 3.60320 0.142652
\(639\) −0.446919 −0.0176799
\(640\) 56.3407 2.22706
\(641\) 8.24486 0.325652 0.162826 0.986655i \(-0.447939\pi\)
0.162826 + 0.986655i \(0.447939\pi\)
\(642\) −70.3735 −2.77742
\(643\) −30.5515 −1.20483 −0.602416 0.798182i \(-0.705795\pi\)
−0.602416 + 0.798182i \(0.705795\pi\)
\(644\) 12.5497 0.494526
\(645\) −62.7610 −2.47121
\(646\) −8.12631 −0.319725
\(647\) 35.3839 1.39108 0.695541 0.718486i \(-0.255165\pi\)
0.695541 + 0.718486i \(0.255165\pi\)
\(648\) 44.8134 1.76044
\(649\) 1.56965 0.0616141
\(650\) 24.9512 0.978666
\(651\) 16.9326 0.663642
\(652\) −60.5484 −2.37126
\(653\) −23.8978 −0.935192 −0.467596 0.883942i \(-0.654880\pi\)
−0.467596 + 0.883942i \(0.654880\pi\)
\(654\) −27.8297 −1.08823
\(655\) 5.55116 0.216902
\(656\) 38.6416 1.50870
\(657\) 4.69630 0.183220
\(658\) 85.4536 3.33133
\(659\) 22.7142 0.884819 0.442409 0.896813i \(-0.354124\pi\)
0.442409 + 0.896813i \(0.354124\pi\)
\(660\) 33.7715 1.31455
\(661\) 17.2869 0.672382 0.336191 0.941794i \(-0.390861\pi\)
0.336191 + 0.941794i \(0.390861\pi\)
\(662\) −31.6775 −1.23118
\(663\) −7.20097 −0.279662
\(664\) 46.4653 1.80320
\(665\) −10.5414 −0.408778
\(666\) −7.61227 −0.294970
\(667\) 1.11583 0.0432049
\(668\) 71.8812 2.78117
\(669\) −3.94684 −0.152594
\(670\) −5.49235 −0.212188
\(671\) 8.95356 0.345649
\(672\) 10.8538 0.418694
\(673\) −46.8739 −1.80686 −0.903428 0.428741i \(-0.858957\pi\)
−0.903428 + 0.428741i \(0.858957\pi\)
\(674\) −55.9217 −2.15402
\(675\) 34.0711 1.31140
\(676\) −44.3580 −1.70608
\(677\) −34.2492 −1.31631 −0.658153 0.752885i \(-0.728662\pi\)
−0.658153 + 0.752885i \(0.728662\pi\)
\(678\) 36.5399 1.40330
\(679\) 28.2849 1.08548
\(680\) 51.6475 1.98059
\(681\) −30.0181 −1.15030
\(682\) −13.9667 −0.534814
\(683\) 22.4911 0.860596 0.430298 0.902687i \(-0.358408\pi\)
0.430298 + 0.902687i \(0.358408\pi\)
\(684\) 1.47303 0.0563227
\(685\) 33.7599 1.28990
\(686\) 46.0683 1.75889
\(687\) 41.7330 1.59221
\(688\) 62.0652 2.36621
\(689\) −16.7678 −0.638803
\(690\) 15.3973 0.586164
\(691\) 19.8073 0.753505 0.376753 0.926314i \(-0.377041\pi\)
0.376753 + 0.926314i \(0.377041\pi\)
\(692\) −30.9826 −1.17778
\(693\) 1.12825 0.0428588
\(694\) 35.0088 1.32892
\(695\) 3.35969 0.127440
\(696\) 9.17948 0.347947
\(697\) −19.4774 −0.737757
\(698\) 1.20038 0.0454350
\(699\) 30.4181 1.15052
\(700\) 70.7154 2.67279
\(701\) −15.2918 −0.577564 −0.288782 0.957395i \(-0.593250\pi\)
−0.288782 + 0.957395i \(0.593250\pi\)
\(702\) 21.5041 0.811620
\(703\) 12.2338 0.461406
\(704\) 6.81945 0.257018
\(705\) 71.2127 2.68202
\(706\) −57.6813 −2.17086
\(707\) −22.6525 −0.851935
\(708\) 7.57727 0.284771
\(709\) 10.4939 0.394107 0.197054 0.980393i \(-0.436863\pi\)
0.197054 + 0.980393i \(0.436863\pi\)
\(710\) −12.7349 −0.477933
\(711\) 1.37692 0.0516387
\(712\) −37.1968 −1.39401
\(713\) −4.32517 −0.161979
\(714\) −30.0468 −1.12447
\(715\) 7.70488 0.288146
\(716\) 88.8653 3.32105
\(717\) −25.4667 −0.951072
\(718\) −22.3190 −0.832936
\(719\) −35.0305 −1.30642 −0.653208 0.757179i \(-0.726577\pi\)
−0.653208 + 0.757179i \(0.726577\pi\)
\(720\) −5.40551 −0.201451
\(721\) 15.0854 0.561808
\(722\) 43.9575 1.63593
\(723\) 35.1766 1.30823
\(724\) −48.4230 −1.79963
\(725\) 6.28750 0.233512
\(726\) −36.6270 −1.35936
\(727\) −16.6871 −0.618890 −0.309445 0.950917i \(-0.600143\pi\)
−0.309445 + 0.950917i \(0.600143\pi\)
\(728\) 23.5542 0.872979
\(729\) 29.1041 1.07793
\(730\) 133.820 4.95291
\(731\) −31.2841 −1.15708
\(732\) 43.2221 1.59753
\(733\) 26.9810 0.996566 0.498283 0.867014i \(-0.333964\pi\)
0.498283 + 0.867014i \(0.333964\pi\)
\(734\) 0.559518 0.0206522
\(735\) −0.292761 −0.0107986
\(736\) −2.77243 −0.102193
\(737\) −0.944742 −0.0348000
\(738\) 5.19796 0.191339
\(739\) −13.6493 −0.502099 −0.251049 0.967974i \(-0.580776\pi\)
−0.251049 + 0.967974i \(0.580776\pi\)
\(740\) −147.332 −5.41603
\(741\) −3.08845 −0.113457
\(742\) −69.9654 −2.56851
\(743\) −29.8524 −1.09518 −0.547589 0.836747i \(-0.684454\pi\)
−0.547589 + 0.836747i \(0.684454\pi\)
\(744\) −35.5816 −1.30448
\(745\) 20.7710 0.760990
\(746\) 8.63730 0.316234
\(747\) 2.45127 0.0896872
\(748\) 16.8339 0.615507
\(749\) −45.5038 −1.66267
\(750\) 17.7662 0.648731
\(751\) 19.3906 0.707574 0.353787 0.935326i \(-0.384894\pi\)
0.353787 + 0.935326i \(0.384894\pi\)
\(752\) −70.4232 −2.56807
\(753\) 16.0770 0.585878
\(754\) 3.96838 0.144520
\(755\) −13.5492 −0.493106
\(756\) 60.9459 2.21658
\(757\) −46.5885 −1.69329 −0.846644 0.532159i \(-0.821381\pi\)
−0.846644 + 0.532159i \(0.821381\pi\)
\(758\) 35.0990 1.27485
\(759\) 2.64849 0.0961341
\(760\) 22.1513 0.803512
\(761\) −12.3401 −0.447330 −0.223665 0.974666i \(-0.571802\pi\)
−0.223665 + 0.974666i \(0.571802\pi\)
\(762\) −50.1197 −1.81565
\(763\) −17.9948 −0.651455
\(764\) −115.929 −4.19416
\(765\) 2.72466 0.0985101
\(766\) −38.1791 −1.37947
\(767\) 1.72873 0.0624209
\(768\) 53.3299 1.92438
\(769\) −46.5874 −1.67999 −0.839993 0.542598i \(-0.817441\pi\)
−0.839993 + 0.542598i \(0.817441\pi\)
\(770\) 32.1494 1.15858
\(771\) 17.5725 0.632859
\(772\) 88.7261 3.19332
\(773\) −19.2581 −0.692665 −0.346333 0.938112i \(-0.612573\pi\)
−0.346333 + 0.938112i \(0.612573\pi\)
\(774\) 8.34883 0.300093
\(775\) −24.3717 −0.875456
\(776\) −59.4368 −2.13366
\(777\) 45.2341 1.62276
\(778\) 65.6417 2.35337
\(779\) −8.35371 −0.299303
\(780\) 37.1942 1.33177
\(781\) −2.19054 −0.0783836
\(782\) 7.67497 0.274456
\(783\) 5.41887 0.193655
\(784\) 0.289515 0.0103398
\(785\) 33.4361 1.19339
\(786\) 6.78630 0.242059
\(787\) −34.5240 −1.23065 −0.615324 0.788274i \(-0.710975\pi\)
−0.615324 + 0.788274i \(0.710975\pi\)
\(788\) −54.7651 −1.95092
\(789\) −41.5410 −1.47890
\(790\) 39.2353 1.39593
\(791\) 23.6268 0.840072
\(792\) −2.37087 −0.0842451
\(793\) 9.86100 0.350174
\(794\) −14.0504 −0.498629
\(795\) −58.3056 −2.06789
\(796\) 30.6650 1.08689
\(797\) 20.9223 0.741107 0.370554 0.928811i \(-0.379168\pi\)
0.370554 + 0.928811i \(0.379168\pi\)
\(798\) −12.8869 −0.456190
\(799\) 35.4969 1.25579
\(800\) −15.6222 −0.552328
\(801\) −1.96231 −0.0693348
\(802\) 69.6500 2.45943
\(803\) 23.0185 0.812305
\(804\) −4.56061 −0.160840
\(805\) 9.95592 0.350900
\(806\) −15.3822 −0.541817
\(807\) 40.4112 1.42254
\(808\) 47.6011 1.67460
\(809\) 33.3318 1.17188 0.585942 0.810353i \(-0.300724\pi\)
0.585942 + 0.810353i \(0.300724\pi\)
\(810\) 67.3654 2.36698
\(811\) −14.7125 −0.516627 −0.258314 0.966061i \(-0.583167\pi\)
−0.258314 + 0.966061i \(0.583167\pi\)
\(812\) 11.2470 0.394691
\(813\) −12.1553 −0.426306
\(814\) −37.3109 −1.30775
\(815\) −48.0343 −1.68257
\(816\) 24.7618 0.866838
\(817\) −13.4175 −0.469420
\(818\) −11.0751 −0.387231
\(819\) 1.24260 0.0434200
\(820\) 100.604 3.51324
\(821\) −7.56726 −0.264099 −0.132050 0.991243i \(-0.542156\pi\)
−0.132050 + 0.991243i \(0.542156\pi\)
\(822\) 41.2715 1.43951
\(823\) −32.0914 −1.11864 −0.559318 0.828953i \(-0.688937\pi\)
−0.559318 + 0.828953i \(0.688937\pi\)
\(824\) −31.6998 −1.10432
\(825\) 14.9238 0.519581
\(826\) 7.21332 0.250983
\(827\) −31.6708 −1.10130 −0.550650 0.834736i \(-0.685620\pi\)
−0.550650 + 0.834736i \(0.685620\pi\)
\(828\) −1.39122 −0.0483482
\(829\) 36.7716 1.27713 0.638566 0.769567i \(-0.279528\pi\)
0.638566 + 0.769567i \(0.279528\pi\)
\(830\) 69.8486 2.42448
\(831\) 3.99511 0.138589
\(832\) 7.51059 0.260383
\(833\) −0.145931 −0.00505619
\(834\) 4.10722 0.142221
\(835\) 57.0249 1.97343
\(836\) 7.21994 0.249707
\(837\) −21.0046 −0.726027
\(838\) 32.0837 1.10831
\(839\) −32.8907 −1.13551 −0.567757 0.823196i \(-0.692189\pi\)
−0.567757 + 0.823196i \(0.692189\pi\)
\(840\) 81.9037 2.82594
\(841\) 1.00000 0.0344828
\(842\) 17.2189 0.593404
\(843\) 30.9290 1.06525
\(844\) −88.7875 −3.05619
\(845\) −35.1902 −1.21058
\(846\) −9.47312 −0.325693
\(847\) −23.6831 −0.813763
\(848\) 57.6592 1.98003
\(849\) −41.0947 −1.41037
\(850\) 43.2473 1.48337
\(851\) −11.5543 −0.396077
\(852\) −10.5745 −0.362277
\(853\) 19.8295 0.678950 0.339475 0.940615i \(-0.389751\pi\)
0.339475 + 0.940615i \(0.389751\pi\)
\(854\) 41.1461 1.40799
\(855\) 1.16859 0.0399648
\(856\) 95.6199 3.26822
\(857\) 35.2039 1.20254 0.601271 0.799045i \(-0.294661\pi\)
0.601271 + 0.799045i \(0.294661\pi\)
\(858\) 9.41922 0.321567
\(859\) −45.2210 −1.54292 −0.771460 0.636278i \(-0.780473\pi\)
−0.771460 + 0.636278i \(0.780473\pi\)
\(860\) 161.588 5.51009
\(861\) −30.8876 −1.05265
\(862\) 53.3737 1.81791
\(863\) 19.7302 0.671625 0.335812 0.941929i \(-0.390989\pi\)
0.335812 + 0.941929i \(0.390989\pi\)
\(864\) −13.4639 −0.458053
\(865\) −24.5792 −0.835717
\(866\) 78.3324 2.66184
\(867\) 15.4815 0.525780
\(868\) −43.5956 −1.47973
\(869\) 6.74888 0.228940
\(870\) 13.7990 0.467829
\(871\) −1.04049 −0.0352557
\(872\) 37.8136 1.28053
\(873\) −3.13558 −0.106123
\(874\) 3.29175 0.111345
\(875\) 11.4877 0.388355
\(876\) 111.119 3.75435
\(877\) 11.1899 0.377856 0.188928 0.981991i \(-0.439499\pi\)
0.188928 + 0.981991i \(0.439499\pi\)
\(878\) −47.0850 −1.58904
\(879\) 11.1295 0.375388
\(880\) −26.4947 −0.893135
\(881\) 10.5144 0.354238 0.177119 0.984189i \(-0.443322\pi\)
0.177119 + 0.984189i \(0.443322\pi\)
\(882\) 0.0389447 0.00131134
\(883\) −42.7272 −1.43789 −0.718943 0.695069i \(-0.755374\pi\)
−0.718943 + 0.695069i \(0.755374\pi\)
\(884\) 18.5400 0.623567
\(885\) 6.01121 0.202065
\(886\) −27.3479 −0.918771
\(887\) 21.7025 0.728700 0.364350 0.931262i \(-0.381291\pi\)
0.364350 + 0.931262i \(0.381291\pi\)
\(888\) −95.0531 −3.18977
\(889\) −32.4076 −1.08692
\(890\) −55.9157 −1.87430
\(891\) 11.5876 0.388198
\(892\) 10.1617 0.340240
\(893\) 15.2244 0.509465
\(894\) 25.3926 0.849254
\(895\) 70.4988 2.35652
\(896\) 44.5359 1.48784
\(897\) 2.91691 0.0973929
\(898\) 50.7292 1.69286
\(899\) −3.87621 −0.129279
\(900\) −7.83929 −0.261310
\(901\) −29.0632 −0.968236
\(902\) 25.4773 0.848303
\(903\) −49.6109 −1.65095
\(904\) −49.6485 −1.65128
\(905\) −38.4150 −1.27696
\(906\) −16.5639 −0.550300
\(907\) 49.2469 1.63522 0.817609 0.575774i \(-0.195299\pi\)
0.817609 + 0.575774i \(0.195299\pi\)
\(908\) 77.2862 2.56483
\(909\) 2.51119 0.0832908
\(910\) 35.4077 1.17375
\(911\) 2.33396 0.0773274 0.0386637 0.999252i \(-0.487690\pi\)
0.0386637 + 0.999252i \(0.487690\pi\)
\(912\) 10.6202 0.351670
\(913\) 12.0147 0.397628
\(914\) 66.4956 2.19948
\(915\) 34.2890 1.13356
\(916\) −107.448 −3.55018
\(917\) 4.38805 0.144906
\(918\) 37.2725 1.23018
\(919\) −42.1536 −1.39052 −0.695259 0.718759i \(-0.744710\pi\)
−0.695259 + 0.718759i \(0.744710\pi\)
\(920\) −20.9210 −0.689745
\(921\) 8.89194 0.293000
\(922\) −55.7939 −1.83747
\(923\) −2.41255 −0.0794100
\(924\) 26.6955 0.878217
\(925\) −65.1068 −2.14070
\(926\) 15.9149 0.522995
\(927\) −1.67232 −0.0549261
\(928\) −2.48464 −0.0815623
\(929\) 10.1722 0.333739 0.166870 0.985979i \(-0.446634\pi\)
0.166870 + 0.985979i \(0.446634\pi\)
\(930\) −53.4877 −1.75393
\(931\) −0.0625887 −0.00205126
\(932\) −78.3160 −2.56533
\(933\) −26.5523 −0.869285
\(934\) −56.7540 −1.85705
\(935\) 13.3547 0.436745
\(936\) −2.61115 −0.0853482
\(937\) −18.9928 −0.620468 −0.310234 0.950660i \(-0.600407\pi\)
−0.310234 + 0.950660i \(0.600407\pi\)
\(938\) −4.34156 −0.141757
\(939\) 11.1279 0.363144
\(940\) −183.348 −5.98014
\(941\) −59.9551 −1.95448 −0.977241 0.212132i \(-0.931959\pi\)
−0.977241 + 0.212132i \(0.931959\pi\)
\(942\) 40.8757 1.33180
\(943\) 7.88974 0.256925
\(944\) −5.94457 −0.193479
\(945\) 48.3497 1.57282
\(946\) 40.9211 1.33046
\(947\) 42.8168 1.39136 0.695679 0.718353i \(-0.255104\pi\)
0.695679 + 0.718353i \(0.255104\pi\)
\(948\) 32.5793 1.05813
\(949\) 25.3514 0.822942
\(950\) 18.5485 0.601792
\(951\) −36.5704 −1.18588
\(952\) 40.8260 1.32318
\(953\) −28.0572 −0.908861 −0.454430 0.890782i \(-0.650157\pi\)
−0.454430 + 0.890782i \(0.650157\pi\)
\(954\) 7.75615 0.251115
\(955\) −91.9690 −2.97605
\(956\) 65.5679 2.12062
\(957\) 2.37357 0.0767266
\(958\) 8.27744 0.267432
\(959\) 26.6863 0.861745
\(960\) 26.1161 0.842893
\(961\) −15.9750 −0.515323
\(962\) −41.0924 −1.32487
\(963\) 5.04441 0.162554
\(964\) −90.5674 −2.91698
\(965\) 70.3884 2.26588
\(966\) 12.1711 0.391599
\(967\) −46.6731 −1.50091 −0.750453 0.660924i \(-0.770165\pi\)
−0.750453 + 0.660924i \(0.770165\pi\)
\(968\) 49.7669 1.59957
\(969\) −5.35313 −0.171967
\(970\) −89.3479 −2.86879
\(971\) 5.43855 0.174531 0.0872657 0.996185i \(-0.472187\pi\)
0.0872657 + 0.996185i \(0.472187\pi\)
\(972\) −12.9088 −0.414049
\(973\) 2.65574 0.0851392
\(974\) −80.3481 −2.57452
\(975\) 16.4363 0.526384
\(976\) −33.9089 −1.08540
\(977\) 50.3371 1.61043 0.805213 0.592986i \(-0.202051\pi\)
0.805213 + 0.592986i \(0.202051\pi\)
\(978\) −58.7220 −1.87772
\(979\) −9.61810 −0.307396
\(980\) 0.753757 0.0240779
\(981\) 1.99485 0.0636906
\(982\) −4.27669 −0.136475
\(983\) −48.8725 −1.55879 −0.779395 0.626533i \(-0.784473\pi\)
−0.779395 + 0.626533i \(0.784473\pi\)
\(984\) 64.9060 2.06913
\(985\) −43.4463 −1.38431
\(986\) 6.87829 0.219049
\(987\) 56.2917 1.79178
\(988\) 7.95167 0.252976
\(989\) 12.6723 0.402956
\(990\) −3.56399 −0.113271
\(991\) −23.1052 −0.733961 −0.366980 0.930229i \(-0.619608\pi\)
−0.366980 + 0.930229i \(0.619608\pi\)
\(992\) 9.63098 0.305784
\(993\) −20.8672 −0.662201
\(994\) −10.0666 −0.319293
\(995\) 24.3272 0.771225
\(996\) 57.9992 1.83778
\(997\) 25.9052 0.820425 0.410213 0.911990i \(-0.365455\pi\)
0.410213 + 0.911990i \(0.365455\pi\)
\(998\) −3.89639 −0.123338
\(999\) −56.1121 −1.77531
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 4031.2.a.b.1.3 59
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4031.2.a.b.1.3 59 1.1 even 1 trivial