Properties

Label 6013.2.a.f.1.13
Level $6013$
Weight $2$
Character 6013.1
Self dual yes
Analytic conductor $48.014$
Analytic rank $0$
Dimension $110$
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] = [6013,2,Mod(1,6013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6013, 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("6013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6013 = 7 \cdot 859 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6013.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: \(48.0140467354\)
Analytic rank: \(0\)
Dimension: \(110\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 6013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23147 q^{2} +2.27145 q^{3} +2.97947 q^{4} -3.38307 q^{5} -5.06867 q^{6} -1.00000 q^{7} -2.18565 q^{8} +2.15947 q^{9} +O(q^{10})\) \(q-2.23147 q^{2} +2.27145 q^{3} +2.97947 q^{4} -3.38307 q^{5} -5.06867 q^{6} -1.00000 q^{7} -2.18565 q^{8} +2.15947 q^{9} +7.54923 q^{10} -1.25111 q^{11} +6.76770 q^{12} -4.40536 q^{13} +2.23147 q^{14} -7.68446 q^{15} -1.08171 q^{16} -7.52986 q^{17} -4.81880 q^{18} -0.544700 q^{19} -10.0797 q^{20} -2.27145 q^{21} +2.79182 q^{22} +7.05915 q^{23} -4.96459 q^{24} +6.44516 q^{25} +9.83043 q^{26} -1.90922 q^{27} -2.97947 q^{28} +4.54232 q^{29} +17.1477 q^{30} -7.89947 q^{31} +6.78511 q^{32} -2.84183 q^{33} +16.8027 q^{34} +3.38307 q^{35} +6.43407 q^{36} -0.578275 q^{37} +1.21548 q^{38} -10.0065 q^{39} +7.39422 q^{40} -0.546589 q^{41} +5.06867 q^{42} -6.25284 q^{43} -3.72765 q^{44} -7.30564 q^{45} -15.7523 q^{46} +0.251029 q^{47} -2.45705 q^{48} +1.00000 q^{49} -14.3822 q^{50} -17.1037 q^{51} -13.1256 q^{52} -11.3467 q^{53} +4.26037 q^{54} +4.23260 q^{55} +2.18565 q^{56} -1.23726 q^{57} -10.1361 q^{58} -4.98797 q^{59} -22.8956 q^{60} -2.93734 q^{61} +17.6275 q^{62} -2.15947 q^{63} -12.9774 q^{64} +14.9036 q^{65} +6.34147 q^{66} +6.89699 q^{67} -22.4350 q^{68} +16.0345 q^{69} -7.54923 q^{70} +6.94602 q^{71} -4.71985 q^{72} -0.847622 q^{73} +1.29040 q^{74} +14.6398 q^{75} -1.62292 q^{76} +1.25111 q^{77} +22.3293 q^{78} -15.2146 q^{79} +3.65950 q^{80} -10.8151 q^{81} +1.21970 q^{82} +8.11205 q^{83} -6.76770 q^{84} +25.4740 q^{85} +13.9530 q^{86} +10.3176 q^{87} +2.73450 q^{88} +0.924246 q^{89} +16.3023 q^{90} +4.40536 q^{91} +21.0325 q^{92} -17.9432 q^{93} -0.560164 q^{94} +1.84276 q^{95} +15.4120 q^{96} -2.56900 q^{97} -2.23147 q^{98} -2.70174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 110 q + 16 q^{2} + 29 q^{3} + 118 q^{4} + 12 q^{6} - 110 q^{7} + 57 q^{8} + 127 q^{9} + 3 q^{10} + 52 q^{11} + 62 q^{12} - 9 q^{13} - 16 q^{14} + 39 q^{15} + 146 q^{16} + 11 q^{17} + 60 q^{18} + 14 q^{19} + 18 q^{20} - 29 q^{21} + 32 q^{22} + 73 q^{23} + 24 q^{24} + 132 q^{25} - 7 q^{26} + 116 q^{27} - 118 q^{28} + 35 q^{29} + 18 q^{30} + 36 q^{31} + 140 q^{32} + 42 q^{33} - 7 q^{34} + 180 q^{36} + 49 q^{37} + 45 q^{39} + 6 q^{40} - 14 q^{41} - 12 q^{42} + 58 q^{43} + 92 q^{44} + 17 q^{45} + 27 q^{46} + 87 q^{47} + 98 q^{48} + 110 q^{49} + 91 q^{50} + 42 q^{51} + 16 q^{52} + 95 q^{53} + 41 q^{54} + 8 q^{55} - 57 q^{56} + 61 q^{57} + 46 q^{58} + 114 q^{59} + 81 q^{60} - 47 q^{61} + 31 q^{62} - 127 q^{63} + 199 q^{64} + 62 q^{65} + 21 q^{66} + 95 q^{67} + 60 q^{68} - 39 q^{69} - 3 q^{70} + 131 q^{71} + 186 q^{72} + 31 q^{73} + 23 q^{74} + 121 q^{75} + 14 q^{76} - 52 q^{77} + 110 q^{78} + 9 q^{79} + 61 q^{80} + 194 q^{81} + 45 q^{82} + 73 q^{83} - 62 q^{84} + 59 q^{85} + 72 q^{86} + 64 q^{87} + 100 q^{88} - 17 q^{89} + 11 q^{90} + 9 q^{91} + 192 q^{92} + 85 q^{93} - 11 q^{94} + 108 q^{95} + 68 q^{96} + 32 q^{97} + 16 q^{98} + 160 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23147 −1.57789 −0.788944 0.614465i \(-0.789372\pi\)
−0.788944 + 0.614465i \(0.789372\pi\)
\(3\) 2.27145 1.31142 0.655710 0.755013i \(-0.272369\pi\)
0.655710 + 0.755013i \(0.272369\pi\)
\(4\) 2.97947 1.48973
\(5\) −3.38307 −1.51295 −0.756477 0.654020i \(-0.773081\pi\)
−0.756477 + 0.654020i \(0.773081\pi\)
\(6\) −5.06867 −2.06928
\(7\) −1.00000 −0.377964
\(8\) −2.18565 −0.772745
\(9\) 2.15947 0.719823
\(10\) 7.54923 2.38727
\(11\) −1.25111 −0.377224 −0.188612 0.982052i \(-0.560399\pi\)
−0.188612 + 0.982052i \(0.560399\pi\)
\(12\) 6.76770 1.95367
\(13\) −4.40536 −1.22183 −0.610913 0.791698i \(-0.709197\pi\)
−0.610913 + 0.791698i \(0.709197\pi\)
\(14\) 2.23147 0.596386
\(15\) −7.68446 −1.98412
\(16\) −1.08171 −0.270428
\(17\) −7.52986 −1.82626 −0.913130 0.407669i \(-0.866342\pi\)
−0.913130 + 0.407669i \(0.866342\pi\)
\(18\) −4.81880 −1.13580
\(19\) −0.544700 −0.124963 −0.0624814 0.998046i \(-0.519901\pi\)
−0.0624814 + 0.998046i \(0.519901\pi\)
\(20\) −10.0797 −2.25390
\(21\) −2.27145 −0.495670
\(22\) 2.79182 0.595218
\(23\) 7.05915 1.47193 0.735967 0.677017i \(-0.236728\pi\)
0.735967 + 0.677017i \(0.236728\pi\)
\(24\) −4.96459 −1.01339
\(25\) 6.44516 1.28903
\(26\) 9.83043 1.92791
\(27\) −1.90922 −0.367429
\(28\) −2.97947 −0.563066
\(29\) 4.54232 0.843487 0.421744 0.906715i \(-0.361418\pi\)
0.421744 + 0.906715i \(0.361418\pi\)
\(30\) 17.1477 3.13072
\(31\) −7.89947 −1.41879 −0.709394 0.704812i \(-0.751031\pi\)
−0.709394 + 0.704812i \(0.751031\pi\)
\(32\) 6.78511 1.19945
\(33\) −2.84183 −0.494700
\(34\) 16.8027 2.88163
\(35\) 3.38307 0.571843
\(36\) 6.43407 1.07234
\(37\) −0.578275 −0.0950678 −0.0475339 0.998870i \(-0.515136\pi\)
−0.0475339 + 0.998870i \(0.515136\pi\)
\(38\) 1.21548 0.197177
\(39\) −10.0065 −1.60233
\(40\) 7.39422 1.16913
\(41\) −0.546589 −0.0853629 −0.0426815 0.999089i \(-0.513590\pi\)
−0.0426815 + 0.999089i \(0.513590\pi\)
\(42\) 5.06867 0.782113
\(43\) −6.25284 −0.953549 −0.476774 0.879026i \(-0.658194\pi\)
−0.476774 + 0.879026i \(0.658194\pi\)
\(44\) −3.72765 −0.561964
\(45\) −7.30564 −1.08906
\(46\) −15.7523 −2.32255
\(47\) 0.251029 0.0366163 0.0183082 0.999832i \(-0.494172\pi\)
0.0183082 + 0.999832i \(0.494172\pi\)
\(48\) −2.45705 −0.354644
\(49\) 1.00000 0.142857
\(50\) −14.3822 −2.03395
\(51\) −17.1037 −2.39499
\(52\) −13.1256 −1.82020
\(53\) −11.3467 −1.55859 −0.779294 0.626659i \(-0.784422\pi\)
−0.779294 + 0.626659i \(0.784422\pi\)
\(54\) 4.26037 0.579763
\(55\) 4.23260 0.570723
\(56\) 2.18565 0.292070
\(57\) −1.23726 −0.163879
\(58\) −10.1361 −1.33093
\(59\) −4.98797 −0.649379 −0.324689 0.945821i \(-0.605260\pi\)
−0.324689 + 0.945821i \(0.605260\pi\)
\(60\) −22.8956 −2.95581
\(61\) −2.93734 −0.376088 −0.188044 0.982161i \(-0.560215\pi\)
−0.188044 + 0.982161i \(0.560215\pi\)
\(62\) 17.6275 2.23869
\(63\) −2.15947 −0.272068
\(64\) −12.9774 −1.62217
\(65\) 14.9036 1.84857
\(66\) 6.34147 0.780581
\(67\) 6.89699 0.842602 0.421301 0.906921i \(-0.361574\pi\)
0.421301 + 0.906921i \(0.361574\pi\)
\(68\) −22.4350 −2.72064
\(69\) 16.0345 1.93032
\(70\) −7.54923 −0.902305
\(71\) 6.94602 0.824341 0.412171 0.911107i \(-0.364771\pi\)
0.412171 + 0.911107i \(0.364771\pi\)
\(72\) −4.71985 −0.556240
\(73\) −0.847622 −0.0992066 −0.0496033 0.998769i \(-0.515796\pi\)
−0.0496033 + 0.998769i \(0.515796\pi\)
\(74\) 1.29040 0.150006
\(75\) 14.6398 1.69046
\(76\) −1.62292 −0.186161
\(77\) 1.25111 0.142577
\(78\) 22.3293 2.52829
\(79\) −15.2146 −1.71177 −0.855886 0.517165i \(-0.826987\pi\)
−0.855886 + 0.517165i \(0.826987\pi\)
\(80\) 3.65950 0.409145
\(81\) −10.8151 −1.20168
\(82\) 1.21970 0.134693
\(83\) 8.11205 0.890414 0.445207 0.895428i \(-0.353130\pi\)
0.445207 + 0.895428i \(0.353130\pi\)
\(84\) −6.76770 −0.738417
\(85\) 25.4740 2.76305
\(86\) 13.9530 1.50459
\(87\) 10.3176 1.10617
\(88\) 2.73450 0.291498
\(89\) 0.924246 0.0979699 0.0489849 0.998800i \(-0.484401\pi\)
0.0489849 + 0.998800i \(0.484401\pi\)
\(90\) 16.3023 1.71842
\(91\) 4.40536 0.461807
\(92\) 21.0325 2.19279
\(93\) −17.9432 −1.86063
\(94\) −0.560164 −0.0577765
\(95\) 1.84276 0.189063
\(96\) 15.4120 1.57298
\(97\) −2.56900 −0.260843 −0.130421 0.991459i \(-0.541633\pi\)
−0.130421 + 0.991459i \(0.541633\pi\)
\(98\) −2.23147 −0.225413
\(99\) −2.70174 −0.271535
\(100\) 19.2032 1.92032
\(101\) 1.13625 0.113061 0.0565307 0.998401i \(-0.481996\pi\)
0.0565307 + 0.998401i \(0.481996\pi\)
\(102\) 38.1664 3.77903
\(103\) 2.81818 0.277684 0.138842 0.990315i \(-0.455662\pi\)
0.138842 + 0.990315i \(0.455662\pi\)
\(104\) 9.62858 0.944160
\(105\) 7.68446 0.749927
\(106\) 25.3198 2.45928
\(107\) −5.60690 −0.542040 −0.271020 0.962574i \(-0.587361\pi\)
−0.271020 + 0.962574i \(0.587361\pi\)
\(108\) −5.68845 −0.547372
\(109\) 3.16393 0.303049 0.151525 0.988453i \(-0.451582\pi\)
0.151525 + 0.988453i \(0.451582\pi\)
\(110\) −9.44492 −0.900538
\(111\) −1.31352 −0.124674
\(112\) 1.08171 0.102212
\(113\) −2.00506 −0.188621 −0.0943103 0.995543i \(-0.530065\pi\)
−0.0943103 + 0.995543i \(0.530065\pi\)
\(114\) 2.76091 0.258583
\(115\) −23.8816 −2.22697
\(116\) 13.5337 1.25657
\(117\) −9.51324 −0.879499
\(118\) 11.1305 1.02465
\(119\) 7.52986 0.690261
\(120\) 16.7956 1.53322
\(121\) −9.43472 −0.857702
\(122\) 6.55460 0.593425
\(123\) −1.24155 −0.111947
\(124\) −23.5362 −2.11362
\(125\) −4.88909 −0.437293
\(126\) 4.81880 0.429293
\(127\) 5.38996 0.478281 0.239141 0.970985i \(-0.423134\pi\)
0.239141 + 0.970985i \(0.423134\pi\)
\(128\) 15.3884 1.36016
\(129\) −14.2030 −1.25050
\(130\) −33.2570 −2.91683
\(131\) 20.2237 1.76695 0.883475 0.468478i \(-0.155197\pi\)
0.883475 + 0.468478i \(0.155197\pi\)
\(132\) −8.46715 −0.736971
\(133\) 0.544700 0.0472315
\(134\) −15.3904 −1.32953
\(135\) 6.45902 0.555904
\(136\) 16.4577 1.41123
\(137\) 6.14206 0.524752 0.262376 0.964966i \(-0.415494\pi\)
0.262376 + 0.964966i \(0.415494\pi\)
\(138\) −35.7805 −3.04584
\(139\) 3.61692 0.306783 0.153392 0.988165i \(-0.450980\pi\)
0.153392 + 0.988165i \(0.450980\pi\)
\(140\) 10.0797 0.851894
\(141\) 0.570199 0.0480194
\(142\) −15.4999 −1.30072
\(143\) 5.51159 0.460902
\(144\) −2.33592 −0.194660
\(145\) −15.3670 −1.27616
\(146\) 1.89144 0.156537
\(147\) 2.27145 0.187346
\(148\) −1.72295 −0.141626
\(149\) −6.33298 −0.518818 −0.259409 0.965768i \(-0.583528\pi\)
−0.259409 + 0.965768i \(0.583528\pi\)
\(150\) −32.6684 −2.66736
\(151\) 8.40014 0.683593 0.341797 0.939774i \(-0.388965\pi\)
0.341797 + 0.939774i \(0.388965\pi\)
\(152\) 1.19053 0.0965644
\(153\) −16.2605 −1.31458
\(154\) −2.79182 −0.224971
\(155\) 26.7245 2.14656
\(156\) −29.8141 −2.38704
\(157\) −19.6202 −1.56587 −0.782933 0.622106i \(-0.786277\pi\)
−0.782933 + 0.622106i \(0.786277\pi\)
\(158\) 33.9509 2.70099
\(159\) −25.7734 −2.04396
\(160\) −22.9545 −1.81471
\(161\) −7.05915 −0.556339
\(162\) 24.1336 1.89611
\(163\) 20.9355 1.63979 0.819896 0.572512i \(-0.194031\pi\)
0.819896 + 0.572512i \(0.194031\pi\)
\(164\) −1.62855 −0.127168
\(165\) 9.61412 0.748458
\(166\) −18.1018 −1.40497
\(167\) 10.7398 0.831072 0.415536 0.909577i \(-0.363594\pi\)
0.415536 + 0.909577i \(0.363594\pi\)
\(168\) 4.96459 0.383027
\(169\) 6.40717 0.492859
\(170\) −56.8446 −4.35978
\(171\) −1.17626 −0.0899512
\(172\) −18.6301 −1.42053
\(173\) −22.4178 −1.70439 −0.852197 0.523221i \(-0.824730\pi\)
−0.852197 + 0.523221i \(0.824730\pi\)
\(174\) −23.0235 −1.74541
\(175\) −6.44516 −0.487209
\(176\) 1.35334 0.102012
\(177\) −11.3299 −0.851609
\(178\) −2.06243 −0.154586
\(179\) 11.4219 0.853709 0.426855 0.904320i \(-0.359622\pi\)
0.426855 + 0.904320i \(0.359622\pi\)
\(180\) −21.7669 −1.62241
\(181\) 8.95619 0.665709 0.332854 0.942978i \(-0.391988\pi\)
0.332854 + 0.942978i \(0.391988\pi\)
\(182\) −9.83043 −0.728680
\(183\) −6.67202 −0.493210
\(184\) −15.4288 −1.13743
\(185\) 1.95634 0.143833
\(186\) 40.0398 2.93586
\(187\) 9.42069 0.688909
\(188\) 0.747932 0.0545486
\(189\) 1.90922 0.138875
\(190\) −4.11207 −0.298321
\(191\) 0.506428 0.0366438 0.0183219 0.999832i \(-0.494168\pi\)
0.0183219 + 0.999832i \(0.494168\pi\)
\(192\) −29.4774 −2.12735
\(193\) 16.8603 1.21363 0.606814 0.794844i \(-0.292447\pi\)
0.606814 + 0.794844i \(0.292447\pi\)
\(194\) 5.73266 0.411581
\(195\) 33.8528 2.42425
\(196\) 2.97947 0.212819
\(197\) 23.7570 1.69262 0.846308 0.532694i \(-0.178820\pi\)
0.846308 + 0.532694i \(0.178820\pi\)
\(198\) 6.02885 0.428452
\(199\) −25.8468 −1.83223 −0.916116 0.400914i \(-0.868693\pi\)
−0.916116 + 0.400914i \(0.868693\pi\)
\(200\) −14.0869 −0.996094
\(201\) 15.6662 1.10501
\(202\) −2.53552 −0.178398
\(203\) −4.54232 −0.318808
\(204\) −50.9598 −3.56790
\(205\) 1.84915 0.129150
\(206\) −6.28869 −0.438154
\(207\) 15.2440 1.05953
\(208\) 4.76532 0.330416
\(209\) 0.681481 0.0471390
\(210\) −17.1477 −1.18330
\(211\) −22.9983 −1.58327 −0.791636 0.610993i \(-0.790770\pi\)
−0.791636 + 0.610993i \(0.790770\pi\)
\(212\) −33.8071 −2.32188
\(213\) 15.7775 1.08106
\(214\) 12.5116 0.855279
\(215\) 21.1538 1.44268
\(216\) 4.17289 0.283929
\(217\) 7.89947 0.536251
\(218\) −7.06021 −0.478178
\(219\) −1.92533 −0.130102
\(220\) 12.6109 0.850226
\(221\) 33.1717 2.23137
\(222\) 2.93108 0.196721
\(223\) 18.7442 1.25520 0.627602 0.778534i \(-0.284037\pi\)
0.627602 + 0.778534i \(0.284037\pi\)
\(224\) −6.78511 −0.453349
\(225\) 13.9181 0.927876
\(226\) 4.47424 0.297622
\(227\) 4.25365 0.282324 0.141162 0.989986i \(-0.454916\pi\)
0.141162 + 0.989986i \(0.454916\pi\)
\(228\) −3.68637 −0.244136
\(229\) −3.77045 −0.249159 −0.124579 0.992210i \(-0.539758\pi\)
−0.124579 + 0.992210i \(0.539758\pi\)
\(230\) 53.2911 3.51391
\(231\) 2.84183 0.186979
\(232\) −9.92793 −0.651801
\(233\) 4.52884 0.296694 0.148347 0.988935i \(-0.452605\pi\)
0.148347 + 0.988935i \(0.452605\pi\)
\(234\) 21.2285 1.38775
\(235\) −0.849248 −0.0553989
\(236\) −14.8615 −0.967402
\(237\) −34.5591 −2.24485
\(238\) −16.8027 −1.08916
\(239\) −3.40685 −0.220371 −0.110185 0.993911i \(-0.535144\pi\)
−0.110185 + 0.993911i \(0.535144\pi\)
\(240\) 8.31237 0.536561
\(241\) −0.898375 −0.0578694 −0.0289347 0.999581i \(-0.509211\pi\)
−0.0289347 + 0.999581i \(0.509211\pi\)
\(242\) 21.0533 1.35336
\(243\) −18.8383 −1.20848
\(244\) −8.75172 −0.560271
\(245\) −3.38307 −0.216136
\(246\) 2.77048 0.176639
\(247\) 2.39960 0.152683
\(248\) 17.2655 1.09636
\(249\) 18.4261 1.16771
\(250\) 10.9099 0.690000
\(251\) 21.7755 1.37446 0.687228 0.726442i \(-0.258827\pi\)
0.687228 + 0.726442i \(0.258827\pi\)
\(252\) −6.43407 −0.405308
\(253\) −8.83178 −0.555249
\(254\) −12.0275 −0.754675
\(255\) 57.8629 3.62352
\(256\) −8.38406 −0.524004
\(257\) 10.5212 0.656292 0.328146 0.944627i \(-0.393576\pi\)
0.328146 + 0.944627i \(0.393576\pi\)
\(258\) 31.6936 1.97316
\(259\) 0.578275 0.0359322
\(260\) 44.4049 2.75387
\(261\) 9.80900 0.607162
\(262\) −45.1286 −2.78805
\(263\) −11.4838 −0.708124 −0.354062 0.935222i \(-0.615200\pi\)
−0.354062 + 0.935222i \(0.615200\pi\)
\(264\) 6.21126 0.382277
\(265\) 38.3866 2.35807
\(266\) −1.21548 −0.0745261
\(267\) 2.09938 0.128480
\(268\) 20.5494 1.25525
\(269\) 5.98881 0.365144 0.182572 0.983192i \(-0.441558\pi\)
0.182572 + 0.983192i \(0.441558\pi\)
\(270\) −14.4131 −0.877155
\(271\) −7.82828 −0.475534 −0.237767 0.971322i \(-0.576415\pi\)
−0.237767 + 0.971322i \(0.576415\pi\)
\(272\) 8.14513 0.493871
\(273\) 10.0065 0.605623
\(274\) −13.7058 −0.828000
\(275\) −8.06362 −0.486254
\(276\) 47.7742 2.87567
\(277\) 31.3248 1.88212 0.941062 0.338233i \(-0.109829\pi\)
0.941062 + 0.338233i \(0.109829\pi\)
\(278\) −8.07106 −0.484070
\(279\) −17.0587 −1.02128
\(280\) −7.39422 −0.441889
\(281\) 29.9083 1.78418 0.892089 0.451861i \(-0.149240\pi\)
0.892089 + 0.451861i \(0.149240\pi\)
\(282\) −1.27238 −0.0757693
\(283\) −21.2362 −1.26236 −0.631181 0.775636i \(-0.717430\pi\)
−0.631181 + 0.775636i \(0.717430\pi\)
\(284\) 20.6954 1.22805
\(285\) 4.18573 0.247941
\(286\) −12.2990 −0.727253
\(287\) 0.546589 0.0322642
\(288\) 14.6522 0.863392
\(289\) 39.6988 2.33522
\(290\) 34.2910 2.01364
\(291\) −5.83535 −0.342074
\(292\) −2.52546 −0.147791
\(293\) −14.8950 −0.870173 −0.435086 0.900389i \(-0.643282\pi\)
−0.435086 + 0.900389i \(0.643282\pi\)
\(294\) −5.06867 −0.295611
\(295\) 16.8747 0.982481
\(296\) 1.26391 0.0734631
\(297\) 2.38865 0.138603
\(298\) 14.1319 0.818638
\(299\) −31.0981 −1.79845
\(300\) 43.6189 2.51834
\(301\) 6.25284 0.360408
\(302\) −18.7447 −1.07863
\(303\) 2.58094 0.148271
\(304\) 0.589208 0.0337934
\(305\) 9.93724 0.569004
\(306\) 36.2849 2.07427
\(307\) −1.43001 −0.0816149 −0.0408075 0.999167i \(-0.512993\pi\)
−0.0408075 + 0.999167i \(0.512993\pi\)
\(308\) 3.72765 0.212402
\(309\) 6.40135 0.364160
\(310\) −59.6349 −3.38704
\(311\) 19.0043 1.07763 0.538817 0.842423i \(-0.318871\pi\)
0.538817 + 0.842423i \(0.318871\pi\)
\(312\) 21.8708 1.23819
\(313\) 15.9971 0.904211 0.452106 0.891964i \(-0.350673\pi\)
0.452106 + 0.891964i \(0.350673\pi\)
\(314\) 43.7820 2.47076
\(315\) 7.30564 0.411626
\(316\) −45.3313 −2.55008
\(317\) 14.1550 0.795021 0.397511 0.917598i \(-0.369874\pi\)
0.397511 + 0.917598i \(0.369874\pi\)
\(318\) 57.5126 3.22515
\(319\) −5.68295 −0.318184
\(320\) 43.9033 2.45427
\(321\) −12.7358 −0.710842
\(322\) 15.7523 0.877841
\(323\) 4.10152 0.228215
\(324\) −32.2232 −1.79018
\(325\) −28.3932 −1.57497
\(326\) −46.7169 −2.58741
\(327\) 7.18669 0.397425
\(328\) 1.19465 0.0659638
\(329\) −0.251029 −0.0138397
\(330\) −21.4536 −1.18098
\(331\) −15.7787 −0.867278 −0.433639 0.901087i \(-0.642771\pi\)
−0.433639 + 0.901087i \(0.642771\pi\)
\(332\) 24.1696 1.32648
\(333\) −1.24877 −0.0684320
\(334\) −23.9656 −1.31134
\(335\) −23.3330 −1.27482
\(336\) 2.45705 0.134043
\(337\) −0.912483 −0.0497061 −0.0248530 0.999691i \(-0.507912\pi\)
−0.0248530 + 0.999691i \(0.507912\pi\)
\(338\) −14.2974 −0.777677
\(339\) −4.55440 −0.247361
\(340\) 75.8991 4.11621
\(341\) 9.88312 0.535201
\(342\) 2.62480 0.141933
\(343\) −1.00000 −0.0539949
\(344\) 13.6665 0.736850
\(345\) −54.2458 −2.92049
\(346\) 50.0247 2.68934
\(347\) −4.87937 −0.261938 −0.130969 0.991386i \(-0.541809\pi\)
−0.130969 + 0.991386i \(0.541809\pi\)
\(348\) 30.7411 1.64789
\(349\) −29.6797 −1.58872 −0.794358 0.607450i \(-0.792193\pi\)
−0.794358 + 0.607450i \(0.792193\pi\)
\(350\) 14.3822 0.768761
\(351\) 8.41079 0.448935
\(352\) −8.48893 −0.452462
\(353\) 5.94182 0.316251 0.158126 0.987419i \(-0.449455\pi\)
0.158126 + 0.987419i \(0.449455\pi\)
\(354\) 25.2824 1.34374
\(355\) −23.4989 −1.24719
\(356\) 2.75376 0.145949
\(357\) 17.1037 0.905223
\(358\) −25.4875 −1.34706
\(359\) −27.4000 −1.44612 −0.723059 0.690786i \(-0.757264\pi\)
−0.723059 + 0.690786i \(0.757264\pi\)
\(360\) 15.9676 0.841566
\(361\) −18.7033 −0.984384
\(362\) −19.9855 −1.05041
\(363\) −21.4305 −1.12481
\(364\) 13.1256 0.687969
\(365\) 2.86756 0.150095
\(366\) 14.8884 0.778230
\(367\) 0.373548 0.0194990 0.00974952 0.999952i \(-0.496897\pi\)
0.00974952 + 0.999952i \(0.496897\pi\)
\(368\) −7.63596 −0.398052
\(369\) −1.18034 −0.0614462
\(370\) −4.36553 −0.226953
\(371\) 11.3467 0.589091
\(372\) −53.4613 −2.77184
\(373\) −25.9007 −1.34109 −0.670545 0.741869i \(-0.733940\pi\)
−0.670545 + 0.741869i \(0.733940\pi\)
\(374\) −21.0220 −1.08702
\(375\) −11.1053 −0.573476
\(376\) −0.548662 −0.0282951
\(377\) −20.0105 −1.03059
\(378\) −4.26037 −0.219130
\(379\) −28.1494 −1.44594 −0.722970 0.690880i \(-0.757223\pi\)
−0.722970 + 0.690880i \(0.757223\pi\)
\(380\) 5.49044 0.281654
\(381\) 12.2430 0.627228
\(382\) −1.13008 −0.0578199
\(383\) 8.49187 0.433914 0.216957 0.976181i \(-0.430387\pi\)
0.216957 + 0.976181i \(0.430387\pi\)
\(384\) 34.9539 1.78374
\(385\) −4.23260 −0.215713
\(386\) −37.6232 −1.91497
\(387\) −13.5028 −0.686387
\(388\) −7.65426 −0.388586
\(389\) 6.17570 0.313120 0.156560 0.987668i \(-0.449959\pi\)
0.156560 + 0.987668i \(0.449959\pi\)
\(390\) −75.5416 −3.82520
\(391\) −53.1544 −2.68813
\(392\) −2.18565 −0.110392
\(393\) 45.9370 2.31721
\(394\) −53.0131 −2.67076
\(395\) 51.4719 2.58983
\(396\) −8.04974 −0.404515
\(397\) 29.2827 1.46966 0.734828 0.678253i \(-0.237263\pi\)
0.734828 + 0.678253i \(0.237263\pi\)
\(398\) 57.6764 2.89106
\(399\) 1.23726 0.0619404
\(400\) −6.97180 −0.348590
\(401\) −29.9593 −1.49610 −0.748048 0.663645i \(-0.769009\pi\)
−0.748048 + 0.663645i \(0.769009\pi\)
\(402\) −34.9586 −1.74358
\(403\) 34.8000 1.73351
\(404\) 3.38543 0.168431
\(405\) 36.5882 1.81808
\(406\) 10.1361 0.503044
\(407\) 0.723486 0.0358619
\(408\) 37.3827 1.85072
\(409\) −0.0225409 −0.00111458 −0.000557289 1.00000i \(-0.500177\pi\)
−0.000557289 1.00000i \(0.500177\pi\)
\(410\) −4.12633 −0.203785
\(411\) 13.9514 0.688170
\(412\) 8.39668 0.413675
\(413\) 4.98797 0.245442
\(414\) −34.0166 −1.67183
\(415\) −27.4436 −1.34716
\(416\) −29.8908 −1.46552
\(417\) 8.21565 0.402322
\(418\) −1.52071 −0.0743801
\(419\) 30.6438 1.49705 0.748523 0.663109i \(-0.230764\pi\)
0.748523 + 0.663109i \(0.230764\pi\)
\(420\) 22.8956 1.11719
\(421\) 20.7079 1.00924 0.504621 0.863341i \(-0.331632\pi\)
0.504621 + 0.863341i \(0.331632\pi\)
\(422\) 51.3202 2.49823
\(423\) 0.542089 0.0263573
\(424\) 24.7999 1.20439
\(425\) −48.5312 −2.35411
\(426\) −35.2071 −1.70579
\(427\) 2.93734 0.142148
\(428\) −16.7056 −0.807495
\(429\) 12.5193 0.604437
\(430\) −47.2041 −2.27638
\(431\) −2.86069 −0.137795 −0.0688973 0.997624i \(-0.521948\pi\)
−0.0688973 + 0.997624i \(0.521948\pi\)
\(432\) 2.06522 0.0993631
\(433\) −26.3432 −1.26597 −0.632987 0.774163i \(-0.718171\pi\)
−0.632987 + 0.774163i \(0.718171\pi\)
\(434\) −17.6275 −0.846145
\(435\) −34.9053 −1.67358
\(436\) 9.42682 0.451463
\(437\) −3.84512 −0.183937
\(438\) 4.29631 0.205286
\(439\) −11.9257 −0.569184 −0.284592 0.958649i \(-0.591858\pi\)
−0.284592 + 0.958649i \(0.591858\pi\)
\(440\) −9.25099 −0.441024
\(441\) 2.15947 0.102832
\(442\) −74.0218 −3.52086
\(443\) 21.0508 1.00015 0.500076 0.865982i \(-0.333305\pi\)
0.500076 + 0.865982i \(0.333305\pi\)
\(444\) −3.91359 −0.185731
\(445\) −3.12679 −0.148224
\(446\) −41.8271 −1.98057
\(447\) −14.3850 −0.680389
\(448\) 12.9774 0.613123
\(449\) 27.8890 1.31616 0.658081 0.752947i \(-0.271368\pi\)
0.658081 + 0.752947i \(0.271368\pi\)
\(450\) −31.0579 −1.46409
\(451\) 0.683844 0.0322010
\(452\) −5.97402 −0.280994
\(453\) 19.0805 0.896478
\(454\) −9.49189 −0.445477
\(455\) −14.9036 −0.698693
\(456\) 2.70422 0.126637
\(457\) 32.0010 1.49694 0.748471 0.663167i \(-0.230788\pi\)
0.748471 + 0.663167i \(0.230788\pi\)
\(458\) 8.41366 0.393145
\(459\) 14.3762 0.671021
\(460\) −71.1544 −3.31759
\(461\) 27.0463 1.25967 0.629837 0.776727i \(-0.283122\pi\)
0.629837 + 0.776727i \(0.283122\pi\)
\(462\) −6.34147 −0.295032
\(463\) 22.7241 1.05608 0.528038 0.849220i \(-0.322928\pi\)
0.528038 + 0.849220i \(0.322928\pi\)
\(464\) −4.91348 −0.228102
\(465\) 60.7032 2.81504
\(466\) −10.1060 −0.468151
\(467\) 14.5631 0.673899 0.336950 0.941523i \(-0.390605\pi\)
0.336950 + 0.941523i \(0.390605\pi\)
\(468\) −28.3444 −1.31022
\(469\) −6.89699 −0.318474
\(470\) 1.89507 0.0874132
\(471\) −44.5663 −2.05351
\(472\) 10.9020 0.501804
\(473\) 7.82300 0.359702
\(474\) 77.1176 3.54213
\(475\) −3.51068 −0.161081
\(476\) 22.4350 1.02831
\(477\) −24.5028 −1.12191
\(478\) 7.60229 0.347721
\(479\) −29.2601 −1.33693 −0.668465 0.743744i \(-0.733048\pi\)
−0.668465 + 0.743744i \(0.733048\pi\)
\(480\) −52.1400 −2.37985
\(481\) 2.54751 0.116156
\(482\) 2.00470 0.0913115
\(483\) −16.0345 −0.729594
\(484\) −28.1104 −1.27775
\(485\) 8.69112 0.394643
\(486\) 42.0371 1.90684
\(487\) −22.6707 −1.02731 −0.513654 0.857997i \(-0.671709\pi\)
−0.513654 + 0.857997i \(0.671709\pi\)
\(488\) 6.42001 0.290620
\(489\) 47.5538 2.15046
\(490\) 7.54923 0.341039
\(491\) 27.1569 1.22557 0.612787 0.790248i \(-0.290049\pi\)
0.612787 + 0.790248i \(0.290049\pi\)
\(492\) −3.69915 −0.166771
\(493\) −34.2030 −1.54043
\(494\) −5.35464 −0.240917
\(495\) 9.14017 0.410820
\(496\) 8.54495 0.383679
\(497\) −6.94602 −0.311572
\(498\) −41.1173 −1.84251
\(499\) 8.26118 0.369821 0.184911 0.982755i \(-0.440800\pi\)
0.184911 + 0.982755i \(0.440800\pi\)
\(500\) −14.5669 −0.651451
\(501\) 24.3949 1.08988
\(502\) −48.5914 −2.16874
\(503\) −32.7531 −1.46039 −0.730194 0.683240i \(-0.760570\pi\)
−0.730194 + 0.683240i \(0.760570\pi\)
\(504\) 4.71985 0.210239
\(505\) −3.84402 −0.171057
\(506\) 19.7079 0.876122
\(507\) 14.5535 0.646345
\(508\) 16.0592 0.712512
\(509\) −2.18381 −0.0967955 −0.0483978 0.998828i \(-0.515412\pi\)
−0.0483978 + 0.998828i \(0.515412\pi\)
\(510\) −129.120 −5.71751
\(511\) 0.847622 0.0374966
\(512\) −12.0680 −0.533336
\(513\) 1.03995 0.0459150
\(514\) −23.4777 −1.03556
\(515\) −9.53411 −0.420123
\(516\) −42.3173 −1.86292
\(517\) −0.314065 −0.0138126
\(518\) −1.29040 −0.0566971
\(519\) −50.9208 −2.23518
\(520\) −32.5742 −1.42847
\(521\) −20.9230 −0.916654 −0.458327 0.888784i \(-0.651551\pi\)
−0.458327 + 0.888784i \(0.651551\pi\)
\(522\) −21.8885 −0.958034
\(523\) 23.2523 1.01675 0.508377 0.861135i \(-0.330246\pi\)
0.508377 + 0.861135i \(0.330246\pi\)
\(524\) 60.2558 2.63229
\(525\) −14.6398 −0.638935
\(526\) 25.6259 1.11734
\(527\) 59.4819 2.59107
\(528\) 3.07404 0.133780
\(529\) 26.8316 1.16659
\(530\) −85.6587 −3.72078
\(531\) −10.7714 −0.467438
\(532\) 1.62292 0.0703624
\(533\) 2.40792 0.104299
\(534\) −4.68470 −0.202727
\(535\) 18.9685 0.820082
\(536\) −15.0744 −0.651116
\(537\) 25.9441 1.11957
\(538\) −13.3639 −0.576157
\(539\) −1.25111 −0.0538892
\(540\) 19.2444 0.828149
\(541\) −15.5941 −0.670441 −0.335221 0.942140i \(-0.608811\pi\)
−0.335221 + 0.942140i \(0.608811\pi\)
\(542\) 17.4686 0.750340
\(543\) 20.3435 0.873024
\(544\) −51.0910 −2.19051
\(545\) −10.7038 −0.458500
\(546\) −22.3293 −0.955606
\(547\) 16.9059 0.722842 0.361421 0.932403i \(-0.382292\pi\)
0.361421 + 0.932403i \(0.382292\pi\)
\(548\) 18.3001 0.781740
\(549\) −6.34310 −0.270717
\(550\) 17.9937 0.767255
\(551\) −2.47420 −0.105405
\(552\) −35.0458 −1.49165
\(553\) 15.2146 0.646989
\(554\) −69.9004 −2.96978
\(555\) 4.44373 0.188626
\(556\) 10.7765 0.457026
\(557\) −31.5914 −1.33857 −0.669286 0.743005i \(-0.733400\pi\)
−0.669286 + 0.743005i \(0.733400\pi\)
\(558\) 38.0660 1.61146
\(559\) 27.5460 1.16507
\(560\) −3.65950 −0.154642
\(561\) 21.3986 0.903450
\(562\) −66.7394 −2.81523
\(563\) −18.2500 −0.769147 −0.384573 0.923094i \(-0.625651\pi\)
−0.384573 + 0.923094i \(0.625651\pi\)
\(564\) 1.69889 0.0715361
\(565\) 6.78327 0.285374
\(566\) 47.3880 1.99187
\(567\) 10.8151 0.454191
\(568\) −15.1816 −0.637006
\(569\) −25.7110 −1.07786 −0.538930 0.842351i \(-0.681171\pi\)
−0.538930 + 0.842351i \(0.681171\pi\)
\(570\) −9.34034 −0.391224
\(571\) 11.3801 0.476244 0.238122 0.971235i \(-0.423468\pi\)
0.238122 + 0.971235i \(0.423468\pi\)
\(572\) 16.4216 0.686622
\(573\) 1.15032 0.0480555
\(574\) −1.21970 −0.0509092
\(575\) 45.4974 1.89737
\(576\) −28.0242 −1.16768
\(577\) 23.8859 0.994384 0.497192 0.867641i \(-0.334364\pi\)
0.497192 + 0.867641i \(0.334364\pi\)
\(578\) −88.5868 −3.68472
\(579\) 38.2972 1.59158
\(580\) −45.7854 −1.90114
\(581\) −8.11205 −0.336545
\(582\) 13.0214 0.539756
\(583\) 14.1960 0.587937
\(584\) 1.85261 0.0766614
\(585\) 32.1839 1.33064
\(586\) 33.2377 1.37304
\(587\) −12.7728 −0.527192 −0.263596 0.964633i \(-0.584909\pi\)
−0.263596 + 0.964633i \(0.584909\pi\)
\(588\) 6.76770 0.279095
\(589\) 4.30285 0.177296
\(590\) −37.6553 −1.55025
\(591\) 53.9628 2.21973
\(592\) 0.625526 0.0257090
\(593\) −39.5261 −1.62314 −0.811571 0.584254i \(-0.801387\pi\)
−0.811571 + 0.584254i \(0.801387\pi\)
\(594\) −5.33020 −0.218701
\(595\) −25.4740 −1.04433
\(596\) −18.8689 −0.772901
\(597\) −58.7096 −2.40283
\(598\) 69.3945 2.83775
\(599\) 11.7371 0.479564 0.239782 0.970827i \(-0.422924\pi\)
0.239782 + 0.970827i \(0.422924\pi\)
\(600\) −31.9976 −1.30630
\(601\) −23.6931 −0.966460 −0.483230 0.875493i \(-0.660537\pi\)
−0.483230 + 0.875493i \(0.660537\pi\)
\(602\) −13.9530 −0.568683
\(603\) 14.8939 0.606525
\(604\) 25.0279 1.01837
\(605\) 31.9183 1.29766
\(606\) −5.75929 −0.233955
\(607\) 42.0122 1.70522 0.852611 0.522546i \(-0.175018\pi\)
0.852611 + 0.522546i \(0.175018\pi\)
\(608\) −3.69585 −0.149887
\(609\) −10.3176 −0.418092
\(610\) −22.1747 −0.897826
\(611\) −1.10587 −0.0447388
\(612\) −48.4476 −1.95838
\(613\) 13.2379 0.534674 0.267337 0.963603i \(-0.413856\pi\)
0.267337 + 0.963603i \(0.413856\pi\)
\(614\) 3.19103 0.128779
\(615\) 4.20025 0.169370
\(616\) −2.73450 −0.110176
\(617\) 31.3521 1.26219 0.631095 0.775706i \(-0.282606\pi\)
0.631095 + 0.775706i \(0.282606\pi\)
\(618\) −14.2844 −0.574604
\(619\) 4.83758 0.194439 0.0972194 0.995263i \(-0.469005\pi\)
0.0972194 + 0.995263i \(0.469005\pi\)
\(620\) 79.6247 3.19780
\(621\) −13.4775 −0.540832
\(622\) −42.4076 −1.70039
\(623\) −0.924246 −0.0370291
\(624\) 10.8242 0.433314
\(625\) −15.6857 −0.627427
\(626\) −35.6972 −1.42675
\(627\) 1.54795 0.0618191
\(628\) −58.4578 −2.33272
\(629\) 4.35433 0.173618
\(630\) −16.3023 −0.649500
\(631\) 29.7327 1.18364 0.591820 0.806070i \(-0.298410\pi\)
0.591820 + 0.806070i \(0.298410\pi\)
\(632\) 33.2537 1.32276
\(633\) −52.2395 −2.07633
\(634\) −31.5864 −1.25446
\(635\) −18.2346 −0.723618
\(636\) −76.7910 −3.04496
\(637\) −4.40536 −0.174547
\(638\) 12.6813 0.502059
\(639\) 14.9997 0.593380
\(640\) −52.0601 −2.05785
\(641\) 6.19788 0.244802 0.122401 0.992481i \(-0.460941\pi\)
0.122401 + 0.992481i \(0.460941\pi\)
\(642\) 28.4195 1.12163
\(643\) 1.44309 0.0569098 0.0284549 0.999595i \(-0.490941\pi\)
0.0284549 + 0.999595i \(0.490941\pi\)
\(644\) −21.0325 −0.828797
\(645\) 48.0497 1.89195
\(646\) −9.15242 −0.360097
\(647\) −6.58237 −0.258780 −0.129390 0.991594i \(-0.541302\pi\)
−0.129390 + 0.991594i \(0.541302\pi\)
\(648\) 23.6381 0.928590
\(649\) 6.24051 0.244962
\(650\) 63.3587 2.48513
\(651\) 17.9432 0.703251
\(652\) 62.3765 2.44285
\(653\) −20.2903 −0.794019 −0.397010 0.917814i \(-0.629952\pi\)
−0.397010 + 0.917814i \(0.629952\pi\)
\(654\) −16.0369 −0.627092
\(655\) −68.4181 −2.67332
\(656\) 0.591252 0.0230845
\(657\) −1.83041 −0.0714112
\(658\) 0.560164 0.0218375
\(659\) 23.8434 0.928805 0.464403 0.885624i \(-0.346269\pi\)
0.464403 + 0.885624i \(0.346269\pi\)
\(660\) 28.6450 1.11500
\(661\) 44.4651 1.72949 0.864745 0.502211i \(-0.167480\pi\)
0.864745 + 0.502211i \(0.167480\pi\)
\(662\) 35.2098 1.36847
\(663\) 75.3478 2.92627
\(664\) −17.7301 −0.688063
\(665\) −1.84276 −0.0714592
\(666\) 2.78659 0.107978
\(667\) 32.0649 1.24156
\(668\) 31.9989 1.23808
\(669\) 42.5764 1.64610
\(670\) 52.0670 2.01152
\(671\) 3.67494 0.141870
\(672\) −15.4120 −0.594532
\(673\) 4.18712 0.161402 0.0807009 0.996738i \(-0.474284\pi\)
0.0807009 + 0.996738i \(0.474284\pi\)
\(674\) 2.03618 0.0784307
\(675\) −12.3052 −0.473628
\(676\) 19.0899 0.734228
\(677\) −3.59693 −0.138241 −0.0691206 0.997608i \(-0.522019\pi\)
−0.0691206 + 0.997608i \(0.522019\pi\)
\(678\) 10.1630 0.390308
\(679\) 2.56900 0.0985893
\(680\) −55.6774 −2.13513
\(681\) 9.66193 0.370246
\(682\) −22.0539 −0.844488
\(683\) 27.5424 1.05388 0.526939 0.849903i \(-0.323339\pi\)
0.526939 + 0.849903i \(0.323339\pi\)
\(684\) −3.50464 −0.134003
\(685\) −20.7790 −0.793926
\(686\) 2.23147 0.0851980
\(687\) −8.56439 −0.326752
\(688\) 6.76376 0.257866
\(689\) 49.9862 1.90432
\(690\) 121.048 4.60822
\(691\) −46.0906 −1.75337 −0.876685 0.481066i \(-0.840250\pi\)
−0.876685 + 0.481066i \(0.840250\pi\)
\(692\) −66.7931 −2.53909
\(693\) 2.70174 0.102631
\(694\) 10.8882 0.413310
\(695\) −12.2363 −0.464149
\(696\) −22.5508 −0.854785
\(697\) 4.11574 0.155895
\(698\) 66.2293 2.50682
\(699\) 10.2870 0.389091
\(700\) −19.2032 −0.725811
\(701\) −7.73041 −0.291974 −0.145987 0.989287i \(-0.546636\pi\)
−0.145987 + 0.989287i \(0.546636\pi\)
\(702\) −18.7684 −0.708369
\(703\) 0.314986 0.0118799
\(704\) 16.2361 0.611922
\(705\) −1.92902 −0.0726512
\(706\) −13.2590 −0.499009
\(707\) −1.13625 −0.0427332
\(708\) −33.7571 −1.26867
\(709\) 19.0508 0.715469 0.357735 0.933823i \(-0.383549\pi\)
0.357735 + 0.933823i \(0.383549\pi\)
\(710\) 52.4371 1.96793
\(711\) −32.8554 −1.23217
\(712\) −2.02008 −0.0757057
\(713\) −55.7636 −2.08836
\(714\) −38.1664 −1.42834
\(715\) −18.6461 −0.697325
\(716\) 34.0310 1.27180
\(717\) −7.73848 −0.288999
\(718\) 61.1424 2.28181
\(719\) −45.7794 −1.70728 −0.853642 0.520860i \(-0.825611\pi\)
−0.853642 + 0.520860i \(0.825611\pi\)
\(720\) 7.90259 0.294512
\(721\) −2.81818 −0.104955
\(722\) 41.7359 1.55325
\(723\) −2.04061 −0.0758911
\(724\) 26.6847 0.991728
\(725\) 29.2760 1.08728
\(726\) 47.8215 1.77482
\(727\) −29.3335 −1.08792 −0.543960 0.839111i \(-0.683076\pi\)
−0.543960 + 0.839111i \(0.683076\pi\)
\(728\) −9.62858 −0.356859
\(729\) −10.3448 −0.383141
\(730\) −6.39889 −0.236833
\(731\) 47.0830 1.74143
\(732\) −19.8791 −0.734751
\(733\) −21.3349 −0.788022 −0.394011 0.919106i \(-0.628913\pi\)
−0.394011 + 0.919106i \(0.628913\pi\)
\(734\) −0.833562 −0.0307673
\(735\) −7.68446 −0.283446
\(736\) 47.8971 1.76551
\(737\) −8.62891 −0.317850
\(738\) 2.63390 0.0969553
\(739\) −5.27844 −0.194171 −0.0970853 0.995276i \(-0.530952\pi\)
−0.0970853 + 0.995276i \(0.530952\pi\)
\(740\) 5.82886 0.214273
\(741\) 5.45056 0.200231
\(742\) −25.3198 −0.929520
\(743\) −24.6117 −0.902916 −0.451458 0.892292i \(-0.649096\pi\)
−0.451458 + 0.892292i \(0.649096\pi\)
\(744\) 39.2177 1.43779
\(745\) 21.4249 0.784949
\(746\) 57.7968 2.11609
\(747\) 17.5177 0.640941
\(748\) 28.0686 1.02629
\(749\) 5.60690 0.204872
\(750\) 24.7812 0.904881
\(751\) 43.6644 1.59334 0.796668 0.604418i \(-0.206594\pi\)
0.796668 + 0.604418i \(0.206594\pi\)
\(752\) −0.271541 −0.00990207
\(753\) 49.4619 1.80249
\(754\) 44.6529 1.62616
\(755\) −28.4182 −1.03425
\(756\) 5.68845 0.206887
\(757\) 30.6179 1.11283 0.556413 0.830906i \(-0.312177\pi\)
0.556413 + 0.830906i \(0.312177\pi\)
\(758\) 62.8147 2.28153
\(759\) −20.0609 −0.728165
\(760\) −4.02763 −0.146098
\(761\) 25.4933 0.924131 0.462065 0.886846i \(-0.347109\pi\)
0.462065 + 0.886846i \(0.347109\pi\)
\(762\) −27.3199 −0.989696
\(763\) −3.16393 −0.114542
\(764\) 1.50888 0.0545895
\(765\) 55.0104 1.98891
\(766\) −18.9494 −0.684669
\(767\) 21.9738 0.793428
\(768\) −19.0439 −0.687189
\(769\) −28.6493 −1.03312 −0.516560 0.856251i \(-0.672788\pi\)
−0.516560 + 0.856251i \(0.672788\pi\)
\(770\) 9.44492 0.340371
\(771\) 23.8983 0.860675
\(772\) 50.2346 1.80798
\(773\) −48.0951 −1.72986 −0.864929 0.501894i \(-0.832637\pi\)
−0.864929 + 0.501894i \(0.832637\pi\)
\(774\) 30.1312 1.08304
\(775\) −50.9134 −1.82886
\(776\) 5.61495 0.201565
\(777\) 1.31352 0.0471223
\(778\) −13.7809 −0.494069
\(779\) 0.297727 0.0106672
\(780\) 100.863 3.61149
\(781\) −8.69025 −0.310962
\(782\) 118.613 4.24158
\(783\) −8.67228 −0.309922
\(784\) −1.08171 −0.0386325
\(785\) 66.3766 2.36908
\(786\) −102.507 −3.65631
\(787\) −18.0545 −0.643575 −0.321787 0.946812i \(-0.604284\pi\)
−0.321787 + 0.946812i \(0.604284\pi\)
\(788\) 70.7832 2.52155
\(789\) −26.0849 −0.928648
\(790\) −114.858 −4.08647
\(791\) 2.00506 0.0712919
\(792\) 5.90506 0.209827
\(793\) 12.9400 0.459514
\(794\) −65.3435 −2.31896
\(795\) 87.1932 3.09242
\(796\) −77.0097 −2.72954
\(797\) 20.3503 0.720846 0.360423 0.932789i \(-0.382632\pi\)
0.360423 + 0.932789i \(0.382632\pi\)
\(798\) −2.76091 −0.0977350
\(799\) −1.89021 −0.0668709
\(800\) 43.7312 1.54613
\(801\) 1.99588 0.0705210
\(802\) 66.8533 2.36067
\(803\) 1.06047 0.0374231
\(804\) 46.6768 1.64616
\(805\) 23.8816 0.841716
\(806\) −77.6552 −2.73529
\(807\) 13.6033 0.478857
\(808\) −2.48345 −0.0873676
\(809\) 51.2660 1.80242 0.901209 0.433385i \(-0.142681\pi\)
0.901209 + 0.433385i \(0.142681\pi\)
\(810\) −81.6456 −2.86873
\(811\) −1.84372 −0.0647418 −0.0323709 0.999476i \(-0.510306\pi\)
−0.0323709 + 0.999476i \(0.510306\pi\)
\(812\) −13.5337 −0.474939
\(813\) −17.7815 −0.623625
\(814\) −1.61444 −0.0565861
\(815\) −70.8261 −2.48093
\(816\) 18.5012 0.647673
\(817\) 3.40592 0.119158
\(818\) 0.0502995 0.00175868
\(819\) 9.51324 0.332419
\(820\) 5.50948 0.192399
\(821\) 20.7700 0.724879 0.362439 0.932007i \(-0.381944\pi\)
0.362439 + 0.932007i \(0.381944\pi\)
\(822\) −31.1321 −1.08586
\(823\) −20.7678 −0.723921 −0.361961 0.932193i \(-0.617892\pi\)
−0.361961 + 0.932193i \(0.617892\pi\)
\(824\) −6.15957 −0.214579
\(825\) −18.3161 −0.637684
\(826\) −11.1305 −0.387281
\(827\) 29.0443 1.00997 0.504985 0.863128i \(-0.331498\pi\)
0.504985 + 0.863128i \(0.331498\pi\)
\(828\) 45.4191 1.57842
\(829\) 3.66633 0.127337 0.0636685 0.997971i \(-0.479720\pi\)
0.0636685 + 0.997971i \(0.479720\pi\)
\(830\) 61.2397 2.12566
\(831\) 71.1526 2.46826
\(832\) 57.1699 1.98201
\(833\) −7.52986 −0.260894
\(834\) −18.3330 −0.634819
\(835\) −36.3335 −1.25737
\(836\) 2.03045 0.0702246
\(837\) 15.0818 0.521304
\(838\) −68.3807 −2.36217
\(839\) −17.1701 −0.592777 −0.296389 0.955067i \(-0.595782\pi\)
−0.296389 + 0.955067i \(0.595782\pi\)
\(840\) −16.7956 −0.579502
\(841\) −8.36734 −0.288529
\(842\) −46.2091 −1.59247
\(843\) 67.9350 2.33981
\(844\) −68.5228 −2.35865
\(845\) −21.6759 −0.745673
\(846\) −1.20966 −0.0415889
\(847\) 9.43472 0.324181
\(848\) 12.2738 0.421485
\(849\) −48.2369 −1.65549
\(850\) 108.296 3.71452
\(851\) −4.08213 −0.139934
\(852\) 47.0086 1.61049
\(853\) 21.6515 0.741332 0.370666 0.928766i \(-0.379129\pi\)
0.370666 + 0.928766i \(0.379129\pi\)
\(854\) −6.55460 −0.224294
\(855\) 3.97938 0.136092
\(856\) 12.2547 0.418859
\(857\) 32.5424 1.11163 0.555814 0.831307i \(-0.312407\pi\)
0.555814 + 0.831307i \(0.312407\pi\)
\(858\) −27.9364 −0.953734
\(859\) 1.00000 0.0341196
\(860\) 63.0270 2.14920
\(861\) 1.24155 0.0423119
\(862\) 6.38355 0.217425
\(863\) 24.9076 0.847865 0.423933 0.905694i \(-0.360649\pi\)
0.423933 + 0.905694i \(0.360649\pi\)
\(864\) −12.9543 −0.440713
\(865\) 75.8410 2.57867
\(866\) 58.7841 1.99756
\(867\) 90.1737 3.06246
\(868\) 23.5362 0.798871
\(869\) 19.0351 0.645722
\(870\) 77.8902 2.64072
\(871\) −30.3837 −1.02951
\(872\) −6.91525 −0.234180
\(873\) −5.54769 −0.187761
\(874\) 8.58028 0.290232
\(875\) 4.88909 0.165281
\(876\) −5.73645 −0.193817
\(877\) 28.2725 0.954693 0.477347 0.878715i \(-0.341599\pi\)
0.477347 + 0.878715i \(0.341599\pi\)
\(878\) 26.6119 0.898108
\(879\) −33.8331 −1.14116
\(880\) −4.57845 −0.154339
\(881\) 9.56875 0.322379 0.161190 0.986923i \(-0.448467\pi\)
0.161190 + 0.986923i \(0.448467\pi\)
\(882\) −4.81880 −0.162257
\(883\) −15.1023 −0.508234 −0.254117 0.967174i \(-0.581785\pi\)
−0.254117 + 0.967174i \(0.581785\pi\)
\(884\) 98.8340 3.32415
\(885\) 38.3299 1.28845
\(886\) −46.9742 −1.57813
\(887\) −12.6415 −0.424461 −0.212231 0.977220i \(-0.568073\pi\)
−0.212231 + 0.977220i \(0.568073\pi\)
\(888\) 2.87090 0.0963411
\(889\) −5.38996 −0.180773
\(890\) 6.97734 0.233881
\(891\) 13.5309 0.453302
\(892\) 55.8477 1.86992
\(893\) −0.136736 −0.00457568
\(894\) 32.0998 1.07358
\(895\) −38.6409 −1.29162
\(896\) −15.3884 −0.514091
\(897\) −70.6376 −2.35852
\(898\) −62.2335 −2.07676
\(899\) −35.8819 −1.19673
\(900\) 41.4686 1.38229
\(901\) 85.4390 2.84639
\(902\) −1.52598 −0.0508096
\(903\) 14.2030 0.472646
\(904\) 4.38237 0.145756
\(905\) −30.2994 −1.00719
\(906\) −42.5775 −1.41454
\(907\) −16.7565 −0.556389 −0.278194 0.960525i \(-0.589736\pi\)
−0.278194 + 0.960525i \(0.589736\pi\)
\(908\) 12.6736 0.420588
\(909\) 2.45370 0.0813842
\(910\) 33.2570 1.10246
\(911\) −6.86506 −0.227450 −0.113725 0.993512i \(-0.536278\pi\)
−0.113725 + 0.993512i \(0.536278\pi\)
\(912\) 1.33836 0.0443174
\(913\) −10.1491 −0.335886
\(914\) −71.4093 −2.36201
\(915\) 22.5719 0.746204
\(916\) −11.2339 −0.371180
\(917\) −20.2237 −0.667844
\(918\) −32.0800 −1.05880
\(919\) −48.3632 −1.59535 −0.797677 0.603085i \(-0.793938\pi\)
−0.797677 + 0.603085i \(0.793938\pi\)
\(920\) 52.1969 1.72088
\(921\) −3.24819 −0.107031
\(922\) −60.3532 −1.98763
\(923\) −30.5997 −1.00720
\(924\) 8.46715 0.278549
\(925\) −3.72707 −0.122545
\(926\) −50.7081 −1.66637
\(927\) 6.08578 0.199883
\(928\) 30.8201 1.01172
\(929\) 0.608949 0.0199790 0.00998949 0.999950i \(-0.496820\pi\)
0.00998949 + 0.999950i \(0.496820\pi\)
\(930\) −135.458 −4.44183
\(931\) −0.544700 −0.0178518
\(932\) 13.4935 0.441996
\(933\) 43.1672 1.41323
\(934\) −32.4971 −1.06334
\(935\) −31.8709 −1.04229
\(936\) 20.7926 0.679628
\(937\) 50.2766 1.64246 0.821232 0.570595i \(-0.193287\pi\)
0.821232 + 0.570595i \(0.193287\pi\)
\(938\) 15.3904 0.502516
\(939\) 36.3366 1.18580
\(940\) −2.53031 −0.0825295
\(941\) 21.2326 0.692162 0.346081 0.938205i \(-0.387512\pi\)
0.346081 + 0.938205i \(0.387512\pi\)
\(942\) 99.4485 3.24021
\(943\) −3.85846 −0.125649
\(944\) 5.39555 0.175610
\(945\) −6.45902 −0.210112
\(946\) −17.4568 −0.567569
\(947\) −52.8484 −1.71734 −0.858672 0.512526i \(-0.828710\pi\)
−0.858672 + 0.512526i \(0.828710\pi\)
\(948\) −102.968 −3.34423
\(949\) 3.73407 0.121213
\(950\) 7.83399 0.254168
\(951\) 32.1522 1.04261
\(952\) −16.4577 −0.533396
\(953\) −15.8047 −0.511965 −0.255983 0.966681i \(-0.582399\pi\)
−0.255983 + 0.966681i \(0.582399\pi\)
\(954\) 54.6774 1.77025
\(955\) −1.71328 −0.0554404
\(956\) −10.1506 −0.328294
\(957\) −12.9085 −0.417273
\(958\) 65.2932 2.10953
\(959\) −6.14206 −0.198338
\(960\) 99.7241 3.21858
\(961\) 31.4017 1.01296
\(962\) −5.68469 −0.183282
\(963\) −12.1079 −0.390173
\(964\) −2.67668 −0.0862100
\(965\) −57.0395 −1.83617
\(966\) 35.7805 1.15122
\(967\) −37.3530 −1.20119 −0.600596 0.799552i \(-0.705070\pi\)
−0.600596 + 0.799552i \(0.705070\pi\)
\(968\) 20.6210 0.662785
\(969\) 9.31638 0.299285
\(970\) −19.3940 −0.622703
\(971\) −19.6029 −0.629087 −0.314543 0.949243i \(-0.601851\pi\)
−0.314543 + 0.949243i \(0.601851\pi\)
\(972\) −56.1280 −1.80031
\(973\) −3.61692 −0.115953
\(974\) 50.5891 1.62098
\(975\) −64.4937 −2.06545
\(976\) 3.17736 0.101705
\(977\) 5.52603 0.176793 0.0883967 0.996085i \(-0.471826\pi\)
0.0883967 + 0.996085i \(0.471826\pi\)
\(978\) −106.115 −3.39318
\(979\) −1.15633 −0.0369566
\(980\) −10.0797 −0.321986
\(981\) 6.83241 0.218142
\(982\) −60.5998 −1.93382
\(983\) 23.9019 0.762351 0.381175 0.924503i \(-0.375519\pi\)
0.381175 + 0.924503i \(0.375519\pi\)
\(984\) 2.71359 0.0865062
\(985\) −80.3716 −2.56085
\(986\) 76.3231 2.43062
\(987\) −0.570199 −0.0181496
\(988\) 7.14953 0.227457
\(989\) −44.1397 −1.40356
\(990\) −20.3960 −0.648228
\(991\) −9.78457 −0.310817 −0.155409 0.987850i \(-0.549669\pi\)
−0.155409 + 0.987850i \(0.549669\pi\)
\(992\) −53.5988 −1.70176
\(993\) −35.8406 −1.13737
\(994\) 15.4999 0.491625
\(995\) 87.4415 2.77208
\(996\) 54.9000 1.73957
\(997\) −34.6072 −1.09602 −0.548011 0.836471i \(-0.684615\pi\)
−0.548011 + 0.836471i \(0.684615\pi\)
\(998\) −18.4346 −0.583537
\(999\) 1.10405 0.0349307
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 6013.2.a.f.1.13 110
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6013.2.a.f.1.13 110 1.1 even 1 trivial