Properties

Label 4002.2.a.ba.1.4
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
Dimension $4$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.18398\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59819 q^{10} +0.260186 q^{11} -1.00000 q^{12} +5.91704 q^{13} -1.23023 q^{14} -3.59819 q^{15} +1.00000 q^{16} -4.05866 q^{17} -1.00000 q^{18} +6.94699 q^{19} +3.59819 q^{20} -1.23023 q^{21} -0.260186 q^{22} +1.00000 q^{23} +1.00000 q^{24} +7.94699 q^{25} -5.91704 q^{26} -1.00000 q^{27} +1.23023 q^{28} -1.00000 q^{29} +3.59819 q^{30} +1.73981 q^{31} -1.00000 q^{32} -0.260186 q^{33} +4.05866 q^{34} +4.42662 q^{35} +1.00000 q^{36} +1.95089 q^{37} -6.94699 q^{38} -5.91704 q^{39} -3.59819 q^{40} +1.49042 q^{41} +1.23023 q^{42} -2.42662 q^{43} +0.260186 q^{44} +3.59819 q^{45} -1.00000 q^{46} +2.76977 q^{47} -1.00000 q^{48} -5.48652 q^{49} -7.94699 q^{50} +4.05866 q^{51} +5.91704 q^{52} -2.88833 q^{53} +1.00000 q^{54} +0.936200 q^{55} -1.23023 q^{56} -6.94699 q^{57} +1.00000 q^{58} +0.426620 q^{59} -3.59819 q^{60} -5.37361 q^{61} -1.73981 q^{62} +1.23023 q^{63} +1.00000 q^{64} +21.2907 q^{65} +0.260186 q^{66} +15.5643 q^{67} -4.05866 q^{68} -1.00000 q^{69} -4.42662 q^{70} +12.9362 q^{71} -1.00000 q^{72} +1.07906 q^{73} -1.95089 q^{74} -7.94699 q^{75} +6.94699 q^{76} +0.320090 q^{77} +5.91704 q^{78} +5.02871 q^{79} +3.59819 q^{80} +1.00000 q^{81} -1.49042 q^{82} +14.5452 q^{83} -1.23023 q^{84} -14.6038 q^{85} +2.42662 q^{86} +1.00000 q^{87} -0.260186 q^{88} -10.4279 q^{89} -3.59819 q^{90} +7.27935 q^{91} +1.00000 q^{92} -1.73981 q^{93} -2.76977 q^{94} +24.9966 q^{95} +1.00000 q^{96} -11.3488 q^{97} +5.48652 q^{98} +0.260186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.59819 1.60916 0.804580 0.593844i \(-0.202390\pi\)
0.804580 + 0.593844i \(0.202390\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.23023 0.464985 0.232492 0.972598i \(-0.425312\pi\)
0.232492 + 0.972598i \(0.425312\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.59819 −1.13785
\(11\) 0.260186 0.0784491 0.0392245 0.999230i \(-0.487511\pi\)
0.0392245 + 0.999230i \(0.487511\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.91704 1.64109 0.820546 0.571581i \(-0.193670\pi\)
0.820546 + 0.571581i \(0.193670\pi\)
\(14\) −1.23023 −0.328794
\(15\) −3.59819 −0.929049
\(16\) 1.00000 0.250000
\(17\) −4.05866 −0.984370 −0.492185 0.870491i \(-0.663802\pi\)
−0.492185 + 0.870491i \(0.663802\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.94699 1.59375 0.796875 0.604145i \(-0.206485\pi\)
0.796875 + 0.604145i \(0.206485\pi\)
\(20\) 3.59819 0.804580
\(21\) −1.23023 −0.268459
\(22\) −0.260186 −0.0554719
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 7.94699 1.58940
\(26\) −5.91704 −1.16043
\(27\) −1.00000 −0.192450
\(28\) 1.23023 0.232492
\(29\) −1.00000 −0.185695
\(30\) 3.59819 0.656937
\(31\) 1.73981 0.312480 0.156240 0.987719i \(-0.450063\pi\)
0.156240 + 0.987719i \(0.450063\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.260186 −0.0452926
\(34\) 4.05866 0.696055
\(35\) 4.42662 0.748235
\(36\) 1.00000 0.166667
\(37\) 1.95089 0.320724 0.160362 0.987058i \(-0.448734\pi\)
0.160362 + 0.987058i \(0.448734\pi\)
\(38\) −6.94699 −1.12695
\(39\) −5.91704 −0.947485
\(40\) −3.59819 −0.568924
\(41\) 1.49042 0.232765 0.116382 0.993204i \(-0.462870\pi\)
0.116382 + 0.993204i \(0.462870\pi\)
\(42\) 1.23023 0.189829
\(43\) −2.42662 −0.370056 −0.185028 0.982733i \(-0.559238\pi\)
−0.185028 + 0.982733i \(0.559238\pi\)
\(44\) 0.260186 0.0392245
\(45\) 3.59819 0.536387
\(46\) −1.00000 −0.147442
\(47\) 2.76977 0.404012 0.202006 0.979384i \(-0.435254\pi\)
0.202006 + 0.979384i \(0.435254\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.48652 −0.783789
\(50\) −7.94699 −1.12387
\(51\) 4.05866 0.568326
\(52\) 5.91704 0.820546
\(53\) −2.88833 −0.396743 −0.198371 0.980127i \(-0.563565\pi\)
−0.198371 + 0.980127i \(0.563565\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.936200 0.126237
\(56\) −1.23023 −0.164397
\(57\) −6.94699 −0.920152
\(58\) 1.00000 0.131306
\(59\) 0.426620 0.0555412 0.0277706 0.999614i \(-0.491159\pi\)
0.0277706 + 0.999614i \(0.491159\pi\)
\(60\) −3.59819 −0.464525
\(61\) −5.37361 −0.688021 −0.344010 0.938966i \(-0.611786\pi\)
−0.344010 + 0.938966i \(0.611786\pi\)
\(62\) −1.73981 −0.220957
\(63\) 1.23023 0.154995
\(64\) 1.00000 0.125000
\(65\) 21.2907 2.64078
\(66\) 0.260186 0.0320267
\(67\) 15.5643 1.90149 0.950744 0.309978i \(-0.100322\pi\)
0.950744 + 0.309978i \(0.100322\pi\)
\(68\) −4.05866 −0.492185
\(69\) −1.00000 −0.120386
\(70\) −4.42662 −0.529082
\(71\) 12.9362 1.53524 0.767622 0.640903i \(-0.221440\pi\)
0.767622 + 0.640903i \(0.221440\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.07906 0.126295 0.0631474 0.998004i \(-0.479886\pi\)
0.0631474 + 0.998004i \(0.479886\pi\)
\(74\) −1.95089 −0.226786
\(75\) −7.94699 −0.917640
\(76\) 6.94699 0.796875
\(77\) 0.320090 0.0364776
\(78\) 5.91704 0.669973
\(79\) 5.02871 0.565774 0.282887 0.959153i \(-0.408708\pi\)
0.282887 + 0.959153i \(0.408708\pi\)
\(80\) 3.59819 0.402290
\(81\) 1.00000 0.111111
\(82\) −1.49042 −0.164589
\(83\) 14.5452 1.59654 0.798271 0.602299i \(-0.205748\pi\)
0.798271 + 0.602299i \(0.205748\pi\)
\(84\) −1.23023 −0.134230
\(85\) −14.6038 −1.58401
\(86\) 2.42662 0.261669
\(87\) 1.00000 0.107211
\(88\) −0.260186 −0.0277359
\(89\) −10.4279 −1.10535 −0.552676 0.833396i \(-0.686393\pi\)
−0.552676 + 0.833396i \(0.686393\pi\)
\(90\) −3.59819 −0.379283
\(91\) 7.27935 0.763083
\(92\) 1.00000 0.104257
\(93\) −1.73981 −0.180410
\(94\) −2.76977 −0.285680
\(95\) 24.9966 2.56460
\(96\) 1.00000 0.102062
\(97\) −11.3488 −1.15230 −0.576148 0.817345i \(-0.695445\pi\)
−0.576148 + 0.817345i \(0.695445\pi\)
\(98\) 5.48652 0.554223
\(99\) 0.260186 0.0261497
\(100\) 7.94699 0.794699
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −4.05866 −0.401867
\(103\) −8.54394 −0.841860 −0.420930 0.907093i \(-0.638296\pi\)
−0.420930 + 0.907093i \(0.638296\pi\)
\(104\) −5.91704 −0.580214
\(105\) −4.42662 −0.431994
\(106\) 2.88833 0.280539
\(107\) 1.16078 0.112217 0.0561084 0.998425i \(-0.482131\pi\)
0.0561084 + 0.998425i \(0.482131\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −8.54519 −0.818480 −0.409240 0.912427i \(-0.634206\pi\)
−0.409240 + 0.912427i \(0.634206\pi\)
\(110\) −0.936200 −0.0892631
\(111\) −1.95089 −0.185170
\(112\) 1.23023 0.116246
\(113\) −10.9718 −1.03214 −0.516070 0.856546i \(-0.672606\pi\)
−0.516070 + 0.856546i \(0.672606\pi\)
\(114\) 6.94699 0.650645
\(115\) 3.59819 0.335533
\(116\) −1.00000 −0.0928477
\(117\) 5.91704 0.547031
\(118\) −0.426620 −0.0392736
\(119\) −4.99310 −0.457717
\(120\) 3.59819 0.328469
\(121\) −10.9323 −0.993846
\(122\) 5.37361 0.486504
\(123\) −1.49042 −0.134387
\(124\) 1.73981 0.156240
\(125\) 10.6038 0.948437
\(126\) −1.23023 −0.109598
\(127\) 21.3736 1.89660 0.948301 0.317373i \(-0.102801\pi\)
0.948301 + 0.317373i \(0.102801\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.42662 0.213652
\(130\) −21.2907 −1.86731
\(131\) −20.1542 −1.76088 −0.880439 0.474159i \(-0.842752\pi\)
−0.880439 + 0.474159i \(0.842752\pi\)
\(132\) −0.260186 −0.0226463
\(133\) 8.54643 0.741069
\(134\) −15.5643 −1.34455
\(135\) −3.59819 −0.309683
\(136\) 4.05866 0.348027
\(137\) −16.9380 −1.44711 −0.723554 0.690268i \(-0.757493\pi\)
−0.723554 + 0.690268i \(0.757493\pi\)
\(138\) 1.00000 0.0851257
\(139\) −9.53953 −0.809133 −0.404566 0.914509i \(-0.632577\pi\)
−0.404566 + 0.914509i \(0.632577\pi\)
\(140\) 4.42662 0.374118
\(141\) −2.76977 −0.233256
\(142\) −12.9362 −1.08558
\(143\) 1.53953 0.128742
\(144\) 1.00000 0.0833333
\(145\) −3.59819 −0.298814
\(146\) −1.07906 −0.0893039
\(147\) 5.48652 0.452521
\(148\) 1.95089 0.160362
\(149\) 8.75626 0.717341 0.358670 0.933464i \(-0.383230\pi\)
0.358670 + 0.933464i \(0.383230\pi\)
\(150\) 7.94699 0.648869
\(151\) −13.5999 −1.10675 −0.553374 0.832933i \(-0.686660\pi\)
−0.553374 + 0.832933i \(0.686660\pi\)
\(152\) −6.94699 −0.563475
\(153\) −4.05866 −0.328123
\(154\) −0.320090 −0.0257936
\(155\) 6.26019 0.502830
\(156\) −5.91704 −0.473742
\(157\) 7.49042 0.597801 0.298900 0.954284i \(-0.403380\pi\)
0.298900 + 0.954284i \(0.403380\pi\)
\(158\) −5.02871 −0.400063
\(159\) 2.88833 0.229060
\(160\) −3.59819 −0.284462
\(161\) 1.23023 0.0969560
\(162\) −1.00000 −0.0785674
\(163\) 9.66765 0.757229 0.378614 0.925555i \(-0.376401\pi\)
0.378614 + 0.925555i \(0.376401\pi\)
\(164\) 1.49042 0.116382
\(165\) −0.936200 −0.0728831
\(166\) −14.5452 −1.12893
\(167\) −14.8149 −1.14641 −0.573207 0.819411i \(-0.694301\pi\)
−0.573207 + 0.819411i \(0.694301\pi\)
\(168\) 1.23023 0.0949146
\(169\) 22.0114 1.69318
\(170\) 14.6038 1.12006
\(171\) 6.94699 0.531250
\(172\) −2.42662 −0.185028
\(173\) 14.3437 1.09053 0.545264 0.838264i \(-0.316429\pi\)
0.545264 + 0.838264i \(0.316429\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 9.77666 0.739046
\(176\) 0.260186 0.0196123
\(177\) −0.426620 −0.0320667
\(178\) 10.4279 0.781601
\(179\) −16.2371 −1.21362 −0.606810 0.794847i \(-0.707551\pi\)
−0.606810 + 0.794847i \(0.707551\pi\)
\(180\) 3.59819 0.268193
\(181\) 6.80537 0.505839 0.252920 0.967487i \(-0.418609\pi\)
0.252920 + 0.967487i \(0.418609\pi\)
\(182\) −7.27935 −0.539581
\(183\) 5.37361 0.397229
\(184\) −1.00000 −0.0737210
\(185\) 7.01967 0.516097
\(186\) 1.73981 0.127569
\(187\) −1.05601 −0.0772229
\(188\) 2.76977 0.202006
\(189\) −1.23023 −0.0894864
\(190\) −24.9966 −1.81345
\(191\) −10.0739 −0.728924 −0.364462 0.931218i \(-0.618747\pi\)
−0.364462 + 0.931218i \(0.618747\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −8.10953 −0.583737 −0.291868 0.956458i \(-0.594277\pi\)
−0.291868 + 0.956458i \(0.594277\pi\)
\(194\) 11.3488 0.814796
\(195\) −21.2907 −1.52466
\(196\) −5.48652 −0.391895
\(197\) 11.0300 0.785852 0.392926 0.919570i \(-0.371463\pi\)
0.392926 + 0.919570i \(0.371463\pi\)
\(198\) −0.260186 −0.0184906
\(199\) −7.93230 −0.562306 −0.281153 0.959663i \(-0.590717\pi\)
−0.281153 + 0.959663i \(0.590717\pi\)
\(200\) −7.94699 −0.561937
\(201\) −15.5643 −1.09782
\(202\) 10.0000 0.703598
\(203\) −1.23023 −0.0863455
\(204\) 4.05866 0.284163
\(205\) 5.36282 0.374556
\(206\) 8.54394 0.595285
\(207\) 1.00000 0.0695048
\(208\) 5.91704 0.410273
\(209\) 1.80751 0.125028
\(210\) 4.42662 0.305466
\(211\) 5.76587 0.396939 0.198469 0.980107i \(-0.436403\pi\)
0.198469 + 0.980107i \(0.436403\pi\)
\(212\) −2.88833 −0.198371
\(213\) −12.9362 −0.886374
\(214\) −1.16078 −0.0793493
\(215\) −8.73145 −0.595480
\(216\) 1.00000 0.0680414
\(217\) 2.14038 0.145298
\(218\) 8.54519 0.578753
\(219\) −1.07906 −0.0729163
\(220\) 0.936200 0.0631186
\(221\) −24.0153 −1.61544
\(222\) 1.95089 0.130935
\(223\) −13.6569 −0.914531 −0.457265 0.889330i \(-0.651171\pi\)
−0.457265 + 0.889330i \(0.651171\pi\)
\(224\) −1.23023 −0.0821985
\(225\) 7.94699 0.529799
\(226\) 10.9718 0.729834
\(227\) 4.14162 0.274889 0.137445 0.990509i \(-0.456111\pi\)
0.137445 + 0.990509i \(0.456111\pi\)
\(228\) −6.94699 −0.460076
\(229\) 20.9623 1.38522 0.692612 0.721310i \(-0.256460\pi\)
0.692612 + 0.721310i \(0.256460\pi\)
\(230\) −3.59819 −0.237258
\(231\) −0.320090 −0.0210604
\(232\) 1.00000 0.0656532
\(233\) 11.9401 0.782222 0.391111 0.920344i \(-0.372091\pi\)
0.391111 + 0.920344i \(0.372091\pi\)
\(234\) −5.91704 −0.386809
\(235\) 9.96615 0.650120
\(236\) 0.426620 0.0277706
\(237\) −5.02871 −0.326650
\(238\) 4.99310 0.323655
\(239\) 7.88268 0.509888 0.254944 0.966956i \(-0.417943\pi\)
0.254944 + 0.966956i \(0.417943\pi\)
\(240\) −3.59819 −0.232262
\(241\) −2.22634 −0.143411 −0.0717055 0.997426i \(-0.522844\pi\)
−0.0717055 + 0.997426i \(0.522844\pi\)
\(242\) 10.9323 0.702755
\(243\) −1.00000 −0.0641500
\(244\) −5.37361 −0.344010
\(245\) −19.7416 −1.26124
\(246\) 1.49042 0.0950257
\(247\) 41.1056 2.61549
\(248\) −1.73981 −0.110478
\(249\) −14.5452 −0.921764
\(250\) −10.6038 −0.670646
\(251\) 24.1095 1.52178 0.760890 0.648881i \(-0.224763\pi\)
0.760890 + 0.648881i \(0.224763\pi\)
\(252\) 1.23023 0.0774975
\(253\) 0.260186 0.0163578
\(254\) −21.3736 −1.34110
\(255\) 14.6038 0.914528
\(256\) 1.00000 0.0625000
\(257\) −4.23860 −0.264397 −0.132198 0.991223i \(-0.542204\pi\)
−0.132198 + 0.991223i \(0.542204\pi\)
\(258\) −2.42662 −0.151075
\(259\) 2.40005 0.149132
\(260\) 21.2907 1.32039
\(261\) −1.00000 −0.0618984
\(262\) 20.1542 1.24513
\(263\) −23.5649 −1.45307 −0.726536 0.687129i \(-0.758871\pi\)
−0.726536 + 0.687129i \(0.758871\pi\)
\(264\) 0.260186 0.0160133
\(265\) −10.3928 −0.638423
\(266\) −8.54643 −0.524015
\(267\) 10.4279 0.638175
\(268\) 15.5643 0.950744
\(269\) 10.4374 0.636380 0.318190 0.948027i \(-0.396925\pi\)
0.318190 + 0.948027i \(0.396925\pi\)
\(270\) 3.59819 0.218979
\(271\) 25.4718 1.54730 0.773652 0.633611i \(-0.218428\pi\)
0.773652 + 0.633611i \(0.218428\pi\)
\(272\) −4.05866 −0.246092
\(273\) −7.27935 −0.440566
\(274\) 16.9380 1.02326
\(275\) 2.06770 0.124687
\(276\) −1.00000 −0.0601929
\(277\) 1.93863 0.116481 0.0582404 0.998303i \(-0.481451\pi\)
0.0582404 + 0.998303i \(0.481451\pi\)
\(278\) 9.53953 0.572143
\(279\) 1.73981 0.104160
\(280\) −4.42662 −0.264541
\(281\) −21.8341 −1.30251 −0.651256 0.758858i \(-0.725758\pi\)
−0.651256 + 0.758858i \(0.725758\pi\)
\(282\) 2.76977 0.164937
\(283\) 17.4240 1.03575 0.517874 0.855457i \(-0.326724\pi\)
0.517874 + 0.855457i \(0.326724\pi\)
\(284\) 12.9362 0.767622
\(285\) −24.9966 −1.48067
\(286\) −1.53953 −0.0910344
\(287\) 1.83357 0.108232
\(288\) −1.00000 −0.0589256
\(289\) −0.527268 −0.0310158
\(290\) 3.59819 0.211293
\(291\) 11.3488 0.665278
\(292\) 1.07906 0.0631474
\(293\) −21.9436 −1.28196 −0.640980 0.767558i \(-0.721472\pi\)
−0.640980 + 0.767558i \(0.721472\pi\)
\(294\) −5.48652 −0.319981
\(295\) 1.53506 0.0893747
\(296\) −1.95089 −0.113393
\(297\) −0.260186 −0.0150975
\(298\) −8.75626 −0.507237
\(299\) 5.91704 0.342191
\(300\) −7.94699 −0.458820
\(301\) −2.98531 −0.172070
\(302\) 13.5999 0.782589
\(303\) 10.0000 0.574485
\(304\) 6.94699 0.398437
\(305\) −19.3353 −1.10714
\(306\) 4.05866 0.232018
\(307\) 14.3928 0.821439 0.410719 0.911762i \(-0.365278\pi\)
0.410719 + 0.911762i \(0.365278\pi\)
\(308\) 0.320090 0.0182388
\(309\) 8.54394 0.486048
\(310\) −6.26019 −0.355555
\(311\) 2.31709 0.131390 0.0656951 0.997840i \(-0.479074\pi\)
0.0656951 + 0.997840i \(0.479074\pi\)
\(312\) 5.91704 0.334986
\(313\) 0.452161 0.0255576 0.0127788 0.999918i \(-0.495932\pi\)
0.0127788 + 0.999918i \(0.495932\pi\)
\(314\) −7.49042 −0.422709
\(315\) 4.42662 0.249412
\(316\) 5.02871 0.282887
\(317\) 20.7625 1.16614 0.583069 0.812423i \(-0.301852\pi\)
0.583069 + 0.812423i \(0.301852\pi\)
\(318\) −2.88833 −0.161970
\(319\) −0.260186 −0.0145676
\(320\) 3.59819 0.201145
\(321\) −1.16078 −0.0647884
\(322\) −1.23023 −0.0685583
\(323\) −28.1955 −1.56884
\(324\) 1.00000 0.0555556
\(325\) 47.0227 2.60835
\(326\) −9.66765 −0.535441
\(327\) 8.54519 0.472550
\(328\) −1.49042 −0.0822947
\(329\) 3.40746 0.187859
\(330\) 0.936200 0.0515361
\(331\) 29.3142 1.61126 0.805628 0.592422i \(-0.201828\pi\)
0.805628 + 0.592422i \(0.201828\pi\)
\(332\) 14.5452 0.798271
\(333\) 1.95089 0.106908
\(334\) 14.8149 0.810636
\(335\) 56.0035 3.05980
\(336\) −1.23023 −0.0671148
\(337\) 21.6739 1.18065 0.590326 0.807165i \(-0.298999\pi\)
0.590326 + 0.807165i \(0.298999\pi\)
\(338\) −22.0114 −1.19726
\(339\) 10.9718 0.595907
\(340\) −14.6038 −0.792005
\(341\) 0.452675 0.0245137
\(342\) −6.94699 −0.375650
\(343\) −15.3613 −0.829435
\(344\) 2.42662 0.130835
\(345\) −3.59819 −0.193720
\(346\) −14.3437 −0.771120
\(347\) −9.91653 −0.532347 −0.266173 0.963925i \(-0.585759\pi\)
−0.266173 + 0.963925i \(0.585759\pi\)
\(348\) 1.00000 0.0536056
\(349\) −11.6107 −0.621509 −0.310754 0.950490i \(-0.600582\pi\)
−0.310754 + 0.950490i \(0.600582\pi\)
\(350\) −9.77666 −0.522585
\(351\) −5.91704 −0.315828
\(352\) −0.260186 −0.0138680
\(353\) 3.08686 0.164297 0.0821484 0.996620i \(-0.473822\pi\)
0.0821484 + 0.996620i \(0.473822\pi\)
\(354\) 0.426620 0.0226746
\(355\) 46.5469 2.47046
\(356\) −10.4279 −0.552676
\(357\) 4.99310 0.264263
\(358\) 16.2371 0.858159
\(359\) −18.1451 −0.957664 −0.478832 0.877907i \(-0.658940\pi\)
−0.478832 + 0.877907i \(0.658940\pi\)
\(360\) −3.59819 −0.189641
\(361\) 29.2607 1.54004
\(362\) −6.80537 −0.357682
\(363\) 10.9323 0.573797
\(364\) 7.27935 0.381541
\(365\) 3.88268 0.203229
\(366\) −5.37361 −0.280883
\(367\) −26.8359 −1.40082 −0.700411 0.713740i \(-0.747000\pi\)
−0.700411 + 0.713740i \(0.747000\pi\)
\(368\) 1.00000 0.0521286
\(369\) 1.49042 0.0775882
\(370\) −7.01967 −0.364935
\(371\) −3.55332 −0.184479
\(372\) −1.73981 −0.0902051
\(373\) −17.9404 −0.928918 −0.464459 0.885595i \(-0.653751\pi\)
−0.464459 + 0.885595i \(0.653751\pi\)
\(374\) 1.05601 0.0546048
\(375\) −10.6038 −0.547580
\(376\) −2.76977 −0.142840
\(377\) −5.91704 −0.304743
\(378\) 1.23023 0.0632764
\(379\) 0.913144 0.0469050 0.0234525 0.999725i \(-0.492534\pi\)
0.0234525 + 0.999725i \(0.492534\pi\)
\(380\) 24.9966 1.28230
\(381\) −21.3736 −1.09500
\(382\) 10.0739 0.515427
\(383\) −27.2829 −1.39409 −0.697044 0.717028i \(-0.745502\pi\)
−0.697044 + 0.717028i \(0.745502\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.15174 0.0586984
\(386\) 8.10953 0.412764
\(387\) −2.42662 −0.123352
\(388\) −11.3488 −0.576148
\(389\) 24.9054 1.26275 0.631376 0.775477i \(-0.282491\pi\)
0.631376 + 0.775477i \(0.282491\pi\)
\(390\) 21.2907 1.07809
\(391\) −4.05866 −0.205255
\(392\) 5.48652 0.277111
\(393\) 20.1542 1.01664
\(394\) −11.0300 −0.555681
\(395\) 18.0943 0.910421
\(396\) 0.260186 0.0130748
\(397\) 36.5966 1.83673 0.918365 0.395735i \(-0.129510\pi\)
0.918365 + 0.395735i \(0.129510\pi\)
\(398\) 7.93230 0.397610
\(399\) −8.54643 −0.427857
\(400\) 7.94699 0.397350
\(401\) −6.97056 −0.348093 −0.174047 0.984737i \(-0.555684\pi\)
−0.174047 + 0.984737i \(0.555684\pi\)
\(402\) 15.5643 0.776279
\(403\) 10.2945 0.512808
\(404\) −10.0000 −0.497519
\(405\) 3.59819 0.178796
\(406\) 1.23023 0.0610555
\(407\) 0.507594 0.0251605
\(408\) −4.05866 −0.200934
\(409\) −22.9578 −1.13519 −0.567595 0.823308i \(-0.692126\pi\)
−0.567595 + 0.823308i \(0.692126\pi\)
\(410\) −5.36282 −0.264851
\(411\) 16.9380 0.835488
\(412\) −8.54394 −0.420930
\(413\) 0.524843 0.0258258
\(414\) −1.00000 −0.0491473
\(415\) 52.3364 2.56909
\(416\) −5.91704 −0.290107
\(417\) 9.53953 0.467153
\(418\) −1.80751 −0.0884082
\(419\) 18.2512 0.891627 0.445814 0.895126i \(-0.352914\pi\)
0.445814 + 0.895126i \(0.352914\pi\)
\(420\) −4.42662 −0.215997
\(421\) −25.9897 −1.26666 −0.633331 0.773881i \(-0.718313\pi\)
−0.633331 + 0.773881i \(0.718313\pi\)
\(422\) −5.76587 −0.280678
\(423\) 2.76977 0.134671
\(424\) 2.88833 0.140270
\(425\) −32.2542 −1.56456
\(426\) 12.9362 0.626761
\(427\) −6.61080 −0.319919
\(428\) 1.16078 0.0561084
\(429\) −1.53953 −0.0743293
\(430\) 8.73145 0.421068
\(431\) 4.08296 0.196669 0.0983346 0.995153i \(-0.468648\pi\)
0.0983346 + 0.995153i \(0.468648\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 8.66646 0.416484 0.208242 0.978077i \(-0.433226\pi\)
0.208242 + 0.978077i \(0.433226\pi\)
\(434\) −2.14038 −0.102741
\(435\) 3.59819 0.172520
\(436\) −8.54519 −0.409240
\(437\) 6.94699 0.332320
\(438\) 1.07906 0.0515596
\(439\) 17.8336 0.851150 0.425575 0.904923i \(-0.360072\pi\)
0.425575 + 0.904923i \(0.360072\pi\)
\(440\) −0.936200 −0.0446316
\(441\) −5.48652 −0.261263
\(442\) 24.0153 1.14229
\(443\) 26.3467 1.25177 0.625884 0.779916i \(-0.284738\pi\)
0.625884 + 0.779916i \(0.284738\pi\)
\(444\) −1.95089 −0.0925851
\(445\) −37.5215 −1.77869
\(446\) 13.6569 0.646671
\(447\) −8.75626 −0.414157
\(448\) 1.23023 0.0581231
\(449\) 9.77718 0.461413 0.230707 0.973023i \(-0.425896\pi\)
0.230707 + 0.973023i \(0.425896\pi\)
\(450\) −7.94699 −0.374625
\(451\) 0.387787 0.0182602
\(452\) −10.9718 −0.516070
\(453\) 13.5999 0.638981
\(454\) −4.14162 −0.194376
\(455\) 26.1925 1.22792
\(456\) 6.94699 0.325323
\(457\) 35.8371 1.67639 0.838194 0.545372i \(-0.183612\pi\)
0.838194 + 0.545372i \(0.183612\pi\)
\(458\) −20.9623 −0.979502
\(459\) 4.05866 0.189442
\(460\) 3.59819 0.167767
\(461\) −12.3775 −0.576478 −0.288239 0.957558i \(-0.593070\pi\)
−0.288239 + 0.957558i \(0.593070\pi\)
\(462\) 0.320090 0.0148919
\(463\) −7.31619 −0.340012 −0.170006 0.985443i \(-0.554379\pi\)
−0.170006 + 0.985443i \(0.554379\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −6.26019 −0.290309
\(466\) −11.9401 −0.553114
\(467\) 8.42713 0.389961 0.194981 0.980807i \(-0.437536\pi\)
0.194981 + 0.980807i \(0.437536\pi\)
\(468\) 5.91704 0.273515
\(469\) 19.1478 0.884163
\(470\) −9.96615 −0.459704
\(471\) −7.49042 −0.345140
\(472\) −0.426620 −0.0196368
\(473\) −0.631373 −0.0290306
\(474\) 5.02871 0.230976
\(475\) 55.2077 2.53310
\(476\) −4.99310 −0.228859
\(477\) −2.88833 −0.132248
\(478\) −7.88268 −0.360545
\(479\) 2.15241 0.0983463 0.0491731 0.998790i \(-0.484341\pi\)
0.0491731 + 0.998790i \(0.484341\pi\)
\(480\) 3.59819 0.164234
\(481\) 11.5435 0.526338
\(482\) 2.22634 0.101407
\(483\) −1.23023 −0.0559776
\(484\) −10.9323 −0.496923
\(485\) −40.8352 −1.85423
\(486\) 1.00000 0.0453609
\(487\) −33.4837 −1.51729 −0.758645 0.651505i \(-0.774138\pi\)
−0.758645 + 0.651505i \(0.774138\pi\)
\(488\) 5.37361 0.243252
\(489\) −9.66765 −0.437186
\(490\) 19.7416 0.891833
\(491\) −0.652958 −0.0294676 −0.0147338 0.999891i \(-0.504690\pi\)
−0.0147338 + 0.999891i \(0.504690\pi\)
\(492\) −1.49042 −0.0671933
\(493\) 4.05866 0.182793
\(494\) −41.1056 −1.84943
\(495\) 0.936200 0.0420790
\(496\) 1.73981 0.0781200
\(497\) 15.9146 0.713865
\(498\) 14.5452 0.651785
\(499\) 2.14286 0.0959277 0.0479639 0.998849i \(-0.484727\pi\)
0.0479639 + 0.998849i \(0.484727\pi\)
\(500\) 10.6038 0.474218
\(501\) 14.8149 0.661882
\(502\) −24.1095 −1.07606
\(503\) 0.272736 0.0121607 0.00608034 0.999982i \(-0.498065\pi\)
0.00608034 + 0.999982i \(0.498065\pi\)
\(504\) −1.23023 −0.0547990
\(505\) −35.9819 −1.60117
\(506\) −0.260186 −0.0115667
\(507\) −22.0114 −0.977559
\(508\) 21.3736 0.948301
\(509\) 10.0413 0.445074 0.222537 0.974924i \(-0.428566\pi\)
0.222537 + 0.974924i \(0.428566\pi\)
\(510\) −14.6038 −0.646669
\(511\) 1.32750 0.0587252
\(512\) −1.00000 −0.0441942
\(513\) −6.94699 −0.306717
\(514\) 4.23860 0.186957
\(515\) −30.7428 −1.35469
\(516\) 2.42662 0.106826
\(517\) 0.720655 0.0316943
\(518\) −2.40005 −0.105452
\(519\) −14.3437 −0.629617
\(520\) −21.2907 −0.933657
\(521\) 27.9308 1.22367 0.611836 0.790985i \(-0.290431\pi\)
0.611836 + 0.790985i \(0.290431\pi\)
\(522\) 1.00000 0.0437688
\(523\) 37.3385 1.63270 0.816350 0.577558i \(-0.195994\pi\)
0.816350 + 0.577558i \(0.195994\pi\)
\(524\) −20.1542 −0.880439
\(525\) −9.77666 −0.426688
\(526\) 23.5649 1.02748
\(527\) −7.06132 −0.307596
\(528\) −0.260186 −0.0113231
\(529\) 1.00000 0.0434783
\(530\) 10.3928 0.451433
\(531\) 0.426620 0.0185137
\(532\) 8.54643 0.370535
\(533\) 8.81888 0.381988
\(534\) −10.4279 −0.451258
\(535\) 4.17671 0.180575
\(536\) −15.5643 −0.672277
\(537\) 16.2371 0.700684
\(538\) −10.4374 −0.449989
\(539\) −1.42752 −0.0614875
\(540\) −3.59819 −0.154842
\(541\) 34.0757 1.46503 0.732514 0.680752i \(-0.238347\pi\)
0.732514 + 0.680752i \(0.238347\pi\)
\(542\) −25.4718 −1.09411
\(543\) −6.80537 −0.292046
\(544\) 4.05866 0.174014
\(545\) −30.7472 −1.31707
\(546\) 7.27935 0.311527
\(547\) −22.2945 −0.953246 −0.476623 0.879108i \(-0.658139\pi\)
−0.476623 + 0.879108i \(0.658139\pi\)
\(548\) −16.9380 −0.723554
\(549\) −5.37361 −0.229340
\(550\) −2.06770 −0.0881669
\(551\) −6.94699 −0.295952
\(552\) 1.00000 0.0425628
\(553\) 6.18649 0.263076
\(554\) −1.93863 −0.0823643
\(555\) −7.01967 −0.297969
\(556\) −9.53953 −0.404566
\(557\) 30.6071 1.29686 0.648432 0.761273i \(-0.275425\pi\)
0.648432 + 0.761273i \(0.275425\pi\)
\(558\) −1.73981 −0.0736522
\(559\) −14.3584 −0.607296
\(560\) 4.42662 0.187059
\(561\) 1.05601 0.0445847
\(562\) 21.8341 0.921015
\(563\) 21.0982 0.889182 0.444591 0.895734i \(-0.353349\pi\)
0.444591 + 0.895734i \(0.353349\pi\)
\(564\) −2.76977 −0.116628
\(565\) −39.4787 −1.66088
\(566\) −17.4240 −0.732384
\(567\) 1.23023 0.0516650
\(568\) −12.9362 −0.542791
\(569\) 6.02034 0.252386 0.126193 0.992006i \(-0.459724\pi\)
0.126193 + 0.992006i \(0.459724\pi\)
\(570\) 24.9966 1.04699
\(571\) −12.7071 −0.531778 −0.265889 0.964004i \(-0.585665\pi\)
−0.265889 + 0.964004i \(0.585665\pi\)
\(572\) 1.53953 0.0643710
\(573\) 10.0739 0.420844
\(574\) −1.83357 −0.0765316
\(575\) 7.94699 0.331412
\(576\) 1.00000 0.0416667
\(577\) 4.66075 0.194030 0.0970148 0.995283i \(-0.469071\pi\)
0.0970148 + 0.995283i \(0.469071\pi\)
\(578\) 0.527268 0.0219315
\(579\) 8.10953 0.337021
\(580\) −3.59819 −0.149407
\(581\) 17.8940 0.742368
\(582\) −11.3488 −0.470423
\(583\) −0.751504 −0.0311241
\(584\) −1.07906 −0.0446520
\(585\) 21.2907 0.880260
\(586\) 21.9436 0.906482
\(587\) 10.8871 0.449358 0.224679 0.974433i \(-0.427867\pi\)
0.224679 + 0.974433i \(0.427867\pi\)
\(588\) 5.48652 0.226260
\(589\) 12.0865 0.498014
\(590\) −1.53506 −0.0631975
\(591\) −11.0300 −0.453712
\(592\) 1.95089 0.0801810
\(593\) 16.2945 0.669137 0.334568 0.942371i \(-0.391409\pi\)
0.334568 + 0.942371i \(0.391409\pi\)
\(594\) 0.260186 0.0106756
\(595\) −17.9662 −0.736540
\(596\) 8.75626 0.358670
\(597\) 7.93230 0.324648
\(598\) −5.91704 −0.241966
\(599\) −17.4693 −0.713778 −0.356889 0.934147i \(-0.616163\pi\)
−0.356889 + 0.934147i \(0.616163\pi\)
\(600\) 7.94699 0.324435
\(601\) 2.30234 0.0939145 0.0469572 0.998897i \(-0.485048\pi\)
0.0469572 + 0.998897i \(0.485048\pi\)
\(602\) 2.98531 0.121672
\(603\) 15.5643 0.633829
\(604\) −13.5999 −0.553374
\(605\) −39.3365 −1.59926
\(606\) −10.0000 −0.406222
\(607\) 26.0241 1.05629 0.528143 0.849156i \(-0.322889\pi\)
0.528143 + 0.849156i \(0.322889\pi\)
\(608\) −6.94699 −0.281738
\(609\) 1.23023 0.0498516
\(610\) 19.3353 0.782863
\(611\) 16.3888 0.663021
\(612\) −4.05866 −0.164062
\(613\) 14.2697 0.576349 0.288175 0.957578i \(-0.406952\pi\)
0.288175 + 0.957578i \(0.406952\pi\)
\(614\) −14.3928 −0.580845
\(615\) −5.36282 −0.216250
\(616\) −0.320090 −0.0128968
\(617\) −43.8250 −1.76433 −0.882165 0.470941i \(-0.843915\pi\)
−0.882165 + 0.470941i \(0.843915\pi\)
\(618\) −8.54394 −0.343688
\(619\) 6.40752 0.257540 0.128770 0.991674i \(-0.458897\pi\)
0.128770 + 0.991674i \(0.458897\pi\)
\(620\) 6.26019 0.251415
\(621\) −1.00000 −0.0401286
\(622\) −2.31709 −0.0929069
\(623\) −12.8287 −0.513972
\(624\) −5.91704 −0.236871
\(625\) −1.58028 −0.0632110
\(626\) −0.452161 −0.0180720
\(627\) −1.80751 −0.0721850
\(628\) 7.49042 0.298900
\(629\) −7.91800 −0.315711
\(630\) −4.42662 −0.176361
\(631\) −21.9176 −0.872524 −0.436262 0.899820i \(-0.643698\pi\)
−0.436262 + 0.899820i \(0.643698\pi\)
\(632\) −5.02871 −0.200031
\(633\) −5.76587 −0.229173
\(634\) −20.7625 −0.824584
\(635\) 76.9064 3.05194
\(636\) 2.88833 0.114530
\(637\) −32.4640 −1.28627
\(638\) 0.260186 0.0103009
\(639\) 12.9362 0.511748
\(640\) −3.59819 −0.142231
\(641\) −16.9605 −0.669899 −0.334950 0.942236i \(-0.608719\pi\)
−0.334950 + 0.942236i \(0.608719\pi\)
\(642\) 1.16078 0.0458123
\(643\) 30.3474 1.19678 0.598392 0.801203i \(-0.295807\pi\)
0.598392 + 0.801203i \(0.295807\pi\)
\(644\) 1.23023 0.0484780
\(645\) 8.73145 0.343800
\(646\) 28.1955 1.10934
\(647\) 25.7398 1.01194 0.505968 0.862552i \(-0.331135\pi\)
0.505968 + 0.862552i \(0.331135\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0.111001 0.00435715
\(650\) −47.0227 −1.84438
\(651\) −2.14038 −0.0838881
\(652\) 9.66765 0.378614
\(653\) −32.7663 −1.28225 −0.641123 0.767438i \(-0.721531\pi\)
−0.641123 + 0.767438i \(0.721531\pi\)
\(654\) −8.54519 −0.334143
\(655\) −72.5186 −2.83354
\(656\) 1.49042 0.0581911
\(657\) 1.07906 0.0420983
\(658\) −3.40746 −0.132837
\(659\) −31.4041 −1.22333 −0.611666 0.791116i \(-0.709500\pi\)
−0.611666 + 0.791116i \(0.709500\pi\)
\(660\) −0.936200 −0.0364415
\(661\) 14.3218 0.557055 0.278528 0.960428i \(-0.410154\pi\)
0.278528 + 0.960428i \(0.410154\pi\)
\(662\) −29.3142 −1.13933
\(663\) 24.0153 0.932676
\(664\) −14.5452 −0.564463
\(665\) 30.7517 1.19250
\(666\) −1.95089 −0.0755954
\(667\) −1.00000 −0.0387202
\(668\) −14.8149 −0.573207
\(669\) 13.6569 0.528004
\(670\) −56.0035 −2.16360
\(671\) −1.39814 −0.0539746
\(672\) 1.23023 0.0474573
\(673\) 0.669058 0.0257903 0.0128952 0.999917i \(-0.495895\pi\)
0.0128952 + 0.999917i \(0.495895\pi\)
\(674\) −21.6739 −0.834846
\(675\) −7.94699 −0.305880
\(676\) 22.0114 0.846591
\(677\) −25.5892 −0.983471 −0.491736 0.870745i \(-0.663637\pi\)
−0.491736 + 0.870745i \(0.663637\pi\)
\(678\) −10.9718 −0.421370
\(679\) −13.9617 −0.535800
\(680\) 14.6038 0.560032
\(681\) −4.14162 −0.158707
\(682\) −0.452675 −0.0173338
\(683\) −19.3182 −0.739190 −0.369595 0.929193i \(-0.620503\pi\)
−0.369595 + 0.929193i \(0.620503\pi\)
\(684\) 6.94699 0.265625
\(685\) −60.9460 −2.32863
\(686\) 15.3613 0.586499
\(687\) −20.9623 −0.799760
\(688\) −2.42662 −0.0925140
\(689\) −17.0904 −0.651091
\(690\) 3.59819 0.136981
\(691\) −39.2116 −1.49168 −0.745841 0.666124i \(-0.767952\pi\)
−0.745841 + 0.666124i \(0.767952\pi\)
\(692\) 14.3437 0.545264
\(693\) 0.320090 0.0121592
\(694\) 9.91653 0.376426
\(695\) −34.3251 −1.30202
\(696\) −1.00000 −0.0379049
\(697\) −6.04911 −0.229126
\(698\) 11.6107 0.439473
\(699\) −11.9401 −0.451616
\(700\) 9.77666 0.369523
\(701\) −25.5634 −0.965516 −0.482758 0.875754i \(-0.660365\pi\)
−0.482758 + 0.875754i \(0.660365\pi\)
\(702\) 5.91704 0.223324
\(703\) 13.5528 0.511154
\(704\) 0.260186 0.00980613
\(705\) −9.96615 −0.375347
\(706\) −3.08686 −0.116175
\(707\) −12.3023 −0.462677
\(708\) −0.426620 −0.0160334
\(709\) −37.0999 −1.39332 −0.696658 0.717403i \(-0.745331\pi\)
−0.696658 + 0.717403i \(0.745331\pi\)
\(710\) −46.5469 −1.74688
\(711\) 5.02871 0.188591
\(712\) 10.4279 0.390801
\(713\) 1.73981 0.0651565
\(714\) −4.99310 −0.186862
\(715\) 5.53953 0.207167
\(716\) −16.2371 −0.606810
\(717\) −7.88268 −0.294384
\(718\) 18.1451 0.677170
\(719\) 17.2907 0.644833 0.322416 0.946598i \(-0.395505\pi\)
0.322416 + 0.946598i \(0.395505\pi\)
\(720\) 3.59819 0.134097
\(721\) −10.5111 −0.391452
\(722\) −29.2607 −1.08897
\(723\) 2.22634 0.0827984
\(724\) 6.80537 0.252920
\(725\) −7.94699 −0.295144
\(726\) −10.9323 −0.405736
\(727\) −19.3410 −0.717318 −0.358659 0.933469i \(-0.616766\pi\)
−0.358659 + 0.933469i \(0.616766\pi\)
\(728\) −7.27935 −0.269790
\(729\) 1.00000 0.0370370
\(730\) −3.88268 −0.143704
\(731\) 9.84883 0.364272
\(732\) 5.37361 0.198614
\(733\) 1.32150 0.0488108 0.0244054 0.999702i \(-0.492231\pi\)
0.0244054 + 0.999702i \(0.492231\pi\)
\(734\) 26.8359 0.990531
\(735\) 19.7416 0.728179
\(736\) −1.00000 −0.0368605
\(737\) 4.04963 0.149170
\(738\) −1.49042 −0.0548631
\(739\) −40.9741 −1.50726 −0.753628 0.657301i \(-0.771698\pi\)
−0.753628 + 0.657301i \(0.771698\pi\)
\(740\) 7.01967 0.258048
\(741\) −41.1056 −1.51005
\(742\) 3.55332 0.130447
\(743\) −33.7691 −1.23887 −0.619434 0.785049i \(-0.712638\pi\)
−0.619434 + 0.785049i \(0.712638\pi\)
\(744\) 1.73981 0.0637847
\(745\) 31.5067 1.15432
\(746\) 17.9404 0.656844
\(747\) 14.5452 0.532180
\(748\) −1.05601 −0.0386114
\(749\) 1.42803 0.0521791
\(750\) 10.6038 0.387198
\(751\) 0.658994 0.0240470 0.0120235 0.999928i \(-0.496173\pi\)
0.0120235 + 0.999928i \(0.496173\pi\)
\(752\) 2.76977 0.101003
\(753\) −24.1095 −0.878600
\(754\) 5.91704 0.215486
\(755\) −48.9352 −1.78094
\(756\) −1.23023 −0.0447432
\(757\) 28.5833 1.03888 0.519438 0.854508i \(-0.326141\pi\)
0.519438 + 0.854508i \(0.326141\pi\)
\(758\) −0.913144 −0.0331669
\(759\) −0.260186 −0.00944416
\(760\) −24.9966 −0.906723
\(761\) −24.1159 −0.874199 −0.437099 0.899413i \(-0.643994\pi\)
−0.437099 + 0.899413i \(0.643994\pi\)
\(762\) 21.3736 0.774284
\(763\) −10.5126 −0.380581
\(764\) −10.0739 −0.364462
\(765\) −14.6038 −0.528003
\(766\) 27.2829 0.985770
\(767\) 2.52433 0.0911482
\(768\) −1.00000 −0.0360844
\(769\) −23.1676 −0.835446 −0.417723 0.908575i \(-0.637172\pi\)
−0.417723 + 0.908575i \(0.637172\pi\)
\(770\) −1.15174 −0.0415060
\(771\) 4.23860 0.152649
\(772\) −8.10953 −0.291868
\(773\) −10.0241 −0.360541 −0.180271 0.983617i \(-0.557697\pi\)
−0.180271 + 0.983617i \(0.557697\pi\)
\(774\) 2.42662 0.0872231
\(775\) 13.8263 0.496655
\(776\) 11.3488 0.407398
\(777\) −2.40005 −0.0861013
\(778\) −24.9054 −0.892900
\(779\) 10.3539 0.370968
\(780\) −21.2907 −0.762328
\(781\) 3.36582 0.120438
\(782\) 4.05866 0.145137
\(783\) 1.00000 0.0357371
\(784\) −5.48652 −0.195947
\(785\) 26.9520 0.961957
\(786\) −20.1542 −0.718875
\(787\) 2.86279 0.102047 0.0510237 0.998697i \(-0.483752\pi\)
0.0510237 + 0.998697i \(0.483752\pi\)
\(788\) 11.0300 0.392926
\(789\) 23.5649 0.838931
\(790\) −18.0943 −0.643765
\(791\) −13.4979 −0.479930
\(792\) −0.260186 −0.00924531
\(793\) −31.7959 −1.12910
\(794\) −36.5966 −1.29876
\(795\) 10.3928 0.368594
\(796\) −7.93230 −0.281153
\(797\) −53.3899 −1.89117 −0.945584 0.325379i \(-0.894508\pi\)
−0.945584 + 0.325379i \(0.894508\pi\)
\(798\) 8.54643 0.302540
\(799\) −11.2415 −0.397697
\(800\) −7.94699 −0.280969
\(801\) −10.4279 −0.368450
\(802\) 6.97056 0.246139
\(803\) 0.280757 0.00990771
\(804\) −15.5643 −0.548912
\(805\) 4.42662 0.156018
\(806\) −10.2945 −0.362610
\(807\) −10.4374 −0.367414
\(808\) 10.0000 0.351799
\(809\) −8.89404 −0.312698 −0.156349 0.987702i \(-0.549972\pi\)
−0.156349 + 0.987702i \(0.549972\pi\)
\(810\) −3.59819 −0.126428
\(811\) −36.1424 −1.26913 −0.634566 0.772869i \(-0.718821\pi\)
−0.634566 + 0.772869i \(0.718821\pi\)
\(812\) −1.23023 −0.0431728
\(813\) −25.4718 −0.893336
\(814\) −0.507594 −0.0177912
\(815\) 34.7861 1.21850
\(816\) 4.05866 0.142082
\(817\) −16.8577 −0.589777
\(818\) 22.9578 0.802700
\(819\) 7.27935 0.254361
\(820\) 5.36282 0.187278
\(821\) −17.6491 −0.615956 −0.307978 0.951393i \(-0.599652\pi\)
−0.307978 + 0.951393i \(0.599652\pi\)
\(822\) −16.9380 −0.590779
\(823\) −11.0177 −0.384052 −0.192026 0.981390i \(-0.561506\pi\)
−0.192026 + 0.981390i \(0.561506\pi\)
\(824\) 8.54394 0.297642
\(825\) −2.06770 −0.0719880
\(826\) −0.524843 −0.0182616
\(827\) 21.4807 0.746956 0.373478 0.927639i \(-0.378165\pi\)
0.373478 + 0.927639i \(0.378165\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.3284 −1.33120 −0.665600 0.746309i \(-0.731824\pi\)
−0.665600 + 0.746309i \(0.731824\pi\)
\(830\) −52.3364 −1.81662
\(831\) −1.93863 −0.0672502
\(832\) 5.91704 0.205136
\(833\) 22.2679 0.771538
\(834\) −9.53953 −0.330327
\(835\) −53.3069 −1.84476
\(836\) 1.80751 0.0625141
\(837\) −1.73981 −0.0601368
\(838\) −18.2512 −0.630476
\(839\) −32.5560 −1.12396 −0.561979 0.827152i \(-0.689960\pi\)
−0.561979 + 0.827152i \(0.689960\pi\)
\(840\) 4.42662 0.152733
\(841\) 1.00000 0.0344828
\(842\) 25.9897 0.895665
\(843\) 21.8341 0.752006
\(844\) 5.76587 0.198469
\(845\) 79.2011 2.72460
\(846\) −2.76977 −0.0952265
\(847\) −13.4493 −0.462123
\(848\) −2.88833 −0.0991857
\(849\) −17.4240 −0.597989
\(850\) 32.2542 1.10631
\(851\) 1.95089 0.0668756
\(852\) −12.9362 −0.443187
\(853\) 12.3858 0.424082 0.212041 0.977261i \(-0.431989\pi\)
0.212041 + 0.977261i \(0.431989\pi\)
\(854\) 6.61080 0.226217
\(855\) 24.9966 0.854866
\(856\) −1.16078 −0.0396747
\(857\) 5.65833 0.193285 0.0966424 0.995319i \(-0.469190\pi\)
0.0966424 + 0.995319i \(0.469190\pi\)
\(858\) 1.53953 0.0525587
\(859\) −48.8675 −1.66734 −0.833670 0.552264i \(-0.813764\pi\)
−0.833670 + 0.552264i \(0.813764\pi\)
\(860\) −8.73145 −0.297740
\(861\) −1.83357 −0.0624878
\(862\) −4.08296 −0.139066
\(863\) 37.6019 1.27998 0.639992 0.768381i \(-0.278937\pi\)
0.639992 + 0.768381i \(0.278937\pi\)
\(864\) 1.00000 0.0340207
\(865\) 51.6113 1.75484
\(866\) −8.66646 −0.294498
\(867\) 0.527268 0.0179070
\(868\) 2.14038 0.0726492
\(869\) 1.30840 0.0443844
\(870\) −3.59819 −0.121990
\(871\) 92.0949 3.12052
\(872\) 8.54519 0.289376
\(873\) −11.3488 −0.384099
\(874\) −6.94699 −0.234986
\(875\) 13.0452 0.441009
\(876\) −1.07906 −0.0364582
\(877\) 20.8111 0.702740 0.351370 0.936237i \(-0.385716\pi\)
0.351370 + 0.936237i \(0.385716\pi\)
\(878\) −17.8336 −0.601854
\(879\) 21.9436 0.740140
\(880\) 0.936200 0.0315593
\(881\) 39.5870 1.33372 0.666859 0.745184i \(-0.267638\pi\)
0.666859 + 0.745184i \(0.267638\pi\)
\(882\) 5.48652 0.184741
\(883\) 15.8518 0.533456 0.266728 0.963772i \(-0.414057\pi\)
0.266728 + 0.963772i \(0.414057\pi\)
\(884\) −24.0153 −0.807721
\(885\) −1.53506 −0.0516005
\(886\) −26.3467 −0.885133
\(887\) 16.0471 0.538810 0.269405 0.963027i \(-0.413173\pi\)
0.269405 + 0.963027i \(0.413173\pi\)
\(888\) 1.95089 0.0654675
\(889\) 26.2945 0.881891
\(890\) 37.5215 1.25772
\(891\) 0.260186 0.00871656
\(892\) −13.6569 −0.457265
\(893\) 19.2415 0.643894
\(894\) 8.75626 0.292853
\(895\) −58.4243 −1.95291
\(896\) −1.23023 −0.0410992
\(897\) −5.91704 −0.197564
\(898\) −9.77718 −0.326269
\(899\) −1.73981 −0.0580260
\(900\) 7.94699 0.264900
\(901\) 11.7228 0.390542
\(902\) −0.387787 −0.0129119
\(903\) 2.98531 0.0993449
\(904\) 10.9718 0.364917
\(905\) 24.4870 0.813977
\(906\) −13.5999 −0.451828
\(907\) −50.3850 −1.67301 −0.836503 0.547963i \(-0.815404\pi\)
−0.836503 + 0.547963i \(0.815404\pi\)
\(908\) 4.14162 0.137445
\(909\) −10.0000 −0.331679
\(910\) −26.1925 −0.868273
\(911\) 27.7391 0.919037 0.459518 0.888168i \(-0.348022\pi\)
0.459518 + 0.888168i \(0.348022\pi\)
\(912\) −6.94699 −0.230038
\(913\) 3.78445 0.125247
\(914\) −35.8371 −1.18539
\(915\) 19.3353 0.639205
\(916\) 20.9623 0.692612
\(917\) −24.7944 −0.818782
\(918\) −4.05866 −0.133956
\(919\) 48.3840 1.59604 0.798021 0.602630i \(-0.205880\pi\)
0.798021 + 0.602630i \(0.205880\pi\)
\(920\) −3.59819 −0.118629
\(921\) −14.3928 −0.474258
\(922\) 12.3775 0.407632
\(923\) 76.5440 2.51948
\(924\) −0.320090 −0.0105302
\(925\) 15.5037 0.509758
\(926\) 7.31619 0.240425
\(927\) −8.54394 −0.280620
\(928\) 1.00000 0.0328266
\(929\) −26.9976 −0.885762 −0.442881 0.896580i \(-0.646044\pi\)
−0.442881 + 0.896580i \(0.646044\pi\)
\(930\) 6.26019 0.205280
\(931\) −38.1148 −1.24916
\(932\) 11.9401 0.391111
\(933\) −2.31709 −0.0758581
\(934\) −8.42713 −0.275744
\(935\) −3.79972 −0.124264
\(936\) −5.91704 −0.193405
\(937\) 40.6191 1.32697 0.663484 0.748190i \(-0.269077\pi\)
0.663484 + 0.748190i \(0.269077\pi\)
\(938\) −19.1478 −0.625197
\(939\) −0.452161 −0.0147557
\(940\) 9.96615 0.325060
\(941\) 17.9404 0.584840 0.292420 0.956290i \(-0.405540\pi\)
0.292420 + 0.956290i \(0.405540\pi\)
\(942\) 7.49042 0.244051
\(943\) 1.49042 0.0485348
\(944\) 0.426620 0.0138853
\(945\) −4.42662 −0.143998
\(946\) 0.631373 0.0205277
\(947\) 17.6721 0.574267 0.287133 0.957891i \(-0.407298\pi\)
0.287133 + 0.957891i \(0.407298\pi\)
\(948\) −5.02871 −0.163325
\(949\) 6.38486 0.207261
\(950\) −55.2077 −1.79117
\(951\) −20.7625 −0.673270
\(952\) 4.99310 0.161827
\(953\) 13.3623 0.432848 0.216424 0.976300i \(-0.430561\pi\)
0.216424 + 0.976300i \(0.430561\pi\)
\(954\) 2.88833 0.0935132
\(955\) −36.2479 −1.17296
\(956\) 7.88268 0.254944
\(957\) 0.260186 0.00841062
\(958\) −2.15241 −0.0695413
\(959\) −20.8377 −0.672883
\(960\) −3.59819 −0.116131
\(961\) −27.9730 −0.902356
\(962\) −11.5435 −0.372177
\(963\) 1.16078 0.0374056
\(964\) −2.22634 −0.0717055
\(965\) −29.1797 −0.939326
\(966\) 1.23023 0.0395821
\(967\) −15.7511 −0.506522 −0.253261 0.967398i \(-0.581503\pi\)
−0.253261 + 0.967398i \(0.581503\pi\)
\(968\) 10.9323 0.351378
\(969\) 28.1955 0.905770
\(970\) 40.8352 1.31114
\(971\) −21.5165 −0.690497 −0.345248 0.938511i \(-0.612205\pi\)
−0.345248 + 0.938511i \(0.612205\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −11.7359 −0.376234
\(974\) 33.4837 1.07289
\(975\) −47.0227 −1.50593
\(976\) −5.37361 −0.172005
\(977\) −5.18112 −0.165759 −0.0828794 0.996560i \(-0.526412\pi\)
−0.0828794 + 0.996560i \(0.526412\pi\)
\(978\) 9.66765 0.309137
\(979\) −2.71318 −0.0867138
\(980\) −19.7416 −0.630621
\(981\) −8.54519 −0.272827
\(982\) 0.652958 0.0208367
\(983\) 4.54270 0.144890 0.0724448 0.997372i \(-0.476920\pi\)
0.0724448 + 0.997372i \(0.476920\pi\)
\(984\) 1.49042 0.0475129
\(985\) 39.6879 1.26456
\(986\) −4.05866 −0.129254
\(987\) −3.40746 −0.108461
\(988\) 41.1056 1.30774
\(989\) −2.42662 −0.0771620
\(990\) −0.936200 −0.0297544
\(991\) −23.7556 −0.754621 −0.377311 0.926087i \(-0.623151\pi\)
−0.377311 + 0.926087i \(0.623151\pi\)
\(992\) −1.73981 −0.0552391
\(993\) −29.3142 −0.930259
\(994\) −15.9146 −0.504779
\(995\) −28.5420 −0.904841
\(996\) −14.5452 −0.460882
\(997\) 25.9278 0.821143 0.410571 0.911828i \(-0.365329\pi\)
0.410571 + 0.911828i \(0.365329\pi\)
\(998\) −2.14286 −0.0678312
\(999\) −1.95089 −0.0617234
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 4002.2.a.ba.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.ba.1.4 4 1.1 even 1 trivial