Properties

Label 4026.2.a.w.1.4
Level $4026$
Weight $2$
Character 4026.1
Self dual yes
Analytic conductor $32.148$
Analytic rank $1$
Dimension $6$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.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: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.30998405.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 9x^{4} + 8x^{3} + 16x^{2} - 13x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.516938\) of defining polynomial
Character \(\chi\) \(=\) 4026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.407480 q^{5} +1.00000 q^{6} -3.39127 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.407480 q^{5} +1.00000 q^{6} -3.39127 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.407480 q^{10} +1.00000 q^{11} -1.00000 q^{12} -0.190339 q^{13} +3.39127 q^{14} -0.407480 q^{15} +1.00000 q^{16} +3.90821 q^{17} -1.00000 q^{18} -4.21451 q^{19} +0.407480 q^{20} +3.39127 q^{21} -1.00000 q^{22} +4.09855 q^{23} +1.00000 q^{24} -4.83396 q^{25} +0.190339 q^{26} -1.00000 q^{27} -3.39127 q^{28} +0.0390710 q^{29} +0.407480 q^{30} -2.91932 q^{31} -1.00000 q^{32} -1.00000 q^{33} -3.90821 q^{34} -1.38188 q^{35} +1.00000 q^{36} +6.51463 q^{37} +4.21451 q^{38} +0.190339 q^{39} -0.407480 q^{40} +8.27278 q^{41} -3.39127 q^{42} +2.67207 q^{43} +1.00000 q^{44} +0.407480 q^{45} -4.09855 q^{46} -4.59990 q^{47} -1.00000 q^{48} +4.50073 q^{49} +4.83396 q^{50} -3.90821 q^{51} -0.190339 q^{52} -2.73918 q^{53} +1.00000 q^{54} +0.407480 q^{55} +3.39127 q^{56} +4.21451 q^{57} -0.0390710 q^{58} -13.4343 q^{59} -0.407480 q^{60} +1.00000 q^{61} +2.91932 q^{62} -3.39127 q^{63} +1.00000 q^{64} -0.0775596 q^{65} +1.00000 q^{66} +8.45198 q^{67} +3.90821 q^{68} -4.09855 q^{69} +1.38188 q^{70} +1.56991 q^{71} -1.00000 q^{72} +11.6664 q^{73} -6.51463 q^{74} +4.83396 q^{75} -4.21451 q^{76} -3.39127 q^{77} -0.190339 q^{78} +9.38091 q^{79} +0.407480 q^{80} +1.00000 q^{81} -8.27278 q^{82} -11.5050 q^{83} +3.39127 q^{84} +1.59252 q^{85} -2.67207 q^{86} -0.0390710 q^{87} -1.00000 q^{88} -3.26264 q^{89} -0.407480 q^{90} +0.645493 q^{91} +4.09855 q^{92} +2.91932 q^{93} +4.59990 q^{94} -1.71733 q^{95} +1.00000 q^{96} -6.20116 q^{97} -4.50073 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - 6 q^{3} + 6 q^{4} - q^{5} + 6 q^{6} + 5 q^{7} - 6 q^{8} + 6 q^{9} + q^{10} + 6 q^{11} - 6 q^{12} + 2 q^{13} - 5 q^{14} + q^{15} + 6 q^{16} - 4 q^{17} - 6 q^{18} - 5 q^{19} - q^{20} - 5 q^{21} - 6 q^{22} - 6 q^{23} + 6 q^{24} - q^{25} - 2 q^{26} - 6 q^{27} + 5 q^{28} - 10 q^{29} - q^{30} - 15 q^{31} - 6 q^{32} - 6 q^{33} + 4 q^{34} - 21 q^{35} + 6 q^{36} - 3 q^{37} + 5 q^{38} - 2 q^{39} + q^{40} - 9 q^{41} + 5 q^{42} + 10 q^{43} + 6 q^{44} - q^{45} + 6 q^{46} - 14 q^{47} - 6 q^{48} + 3 q^{49} + q^{50} + 4 q^{51} + 2 q^{52} - 11 q^{53} + 6 q^{54} - q^{55} - 5 q^{56} + 5 q^{57} + 10 q^{58} - 20 q^{59} + q^{60} + 6 q^{61} + 15 q^{62} + 5 q^{63} + 6 q^{64} - 2 q^{65} + 6 q^{66} + 14 q^{67} - 4 q^{68} + 6 q^{69} + 21 q^{70} - 21 q^{71} - 6 q^{72} + 16 q^{73} + 3 q^{74} + q^{75} - 5 q^{76} + 5 q^{77} + 2 q^{78} - 6 q^{79} - q^{80} + 6 q^{81} + 9 q^{82} - 10 q^{83} - 5 q^{84} + 13 q^{85} - 10 q^{86} + 10 q^{87} - 6 q^{88} - 11 q^{89} + q^{90} - 21 q^{91} - 6 q^{92} + 15 q^{93} + 14 q^{94} + 9 q^{95} + 6 q^{96} + 2 q^{97} - 3 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0.407480 0.182231 0.0911154 0.995840i \(-0.470957\pi\)
0.0911154 + 0.995840i \(0.470957\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.39127 −1.28178 −0.640890 0.767632i \(-0.721435\pi\)
−0.640890 + 0.767632i \(0.721435\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.407480 −0.128857
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −0.190339 −0.0527907 −0.0263953 0.999652i \(-0.508403\pi\)
−0.0263953 + 0.999652i \(0.508403\pi\)
\(14\) 3.39127 0.906356
\(15\) −0.407480 −0.105211
\(16\) 1.00000 0.250000
\(17\) 3.90821 0.947880 0.473940 0.880557i \(-0.342831\pi\)
0.473940 + 0.880557i \(0.342831\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.21451 −0.966874 −0.483437 0.875379i \(-0.660612\pi\)
−0.483437 + 0.875379i \(0.660612\pi\)
\(20\) 0.407480 0.0911154
\(21\) 3.39127 0.740036
\(22\) −1.00000 −0.213201
\(23\) 4.09855 0.854607 0.427303 0.904108i \(-0.359464\pi\)
0.427303 + 0.904108i \(0.359464\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.83396 −0.966792
\(26\) 0.190339 0.0373286
\(27\) −1.00000 −0.192450
\(28\) −3.39127 −0.640890
\(29\) 0.0390710 0.00725529 0.00362765 0.999993i \(-0.498845\pi\)
0.00362765 + 0.999993i \(0.498845\pi\)
\(30\) 0.407480 0.0743954
\(31\) −2.91932 −0.524326 −0.262163 0.965024i \(-0.584436\pi\)
−0.262163 + 0.965024i \(0.584436\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −3.90821 −0.670253
\(35\) −1.38188 −0.233580
\(36\) 1.00000 0.166667
\(37\) 6.51463 1.07100 0.535500 0.844536i \(-0.320123\pi\)
0.535500 + 0.844536i \(0.320123\pi\)
\(38\) 4.21451 0.683683
\(39\) 0.190339 0.0304787
\(40\) −0.407480 −0.0644283
\(41\) 8.27278 1.29199 0.645996 0.763341i \(-0.276442\pi\)
0.645996 + 0.763341i \(0.276442\pi\)
\(42\) −3.39127 −0.523285
\(43\) 2.67207 0.407487 0.203743 0.979024i \(-0.434689\pi\)
0.203743 + 0.979024i \(0.434689\pi\)
\(44\) 1.00000 0.150756
\(45\) 0.407480 0.0607436
\(46\) −4.09855 −0.604298
\(47\) −4.59990 −0.670964 −0.335482 0.942047i \(-0.608899\pi\)
−0.335482 + 0.942047i \(0.608899\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.50073 0.642961
\(50\) 4.83396 0.683625
\(51\) −3.90821 −0.547259
\(52\) −0.190339 −0.0263953
\(53\) −2.73918 −0.376255 −0.188127 0.982145i \(-0.560242\pi\)
−0.188127 + 0.982145i \(0.560242\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.407480 0.0549447
\(56\) 3.39127 0.453178
\(57\) 4.21451 0.558225
\(58\) −0.0390710 −0.00513027
\(59\) −13.4343 −1.74900 −0.874501 0.485024i \(-0.838811\pi\)
−0.874501 + 0.485024i \(0.838811\pi\)
\(60\) −0.407480 −0.0526055
\(61\) 1.00000 0.128037
\(62\) 2.91932 0.370754
\(63\) −3.39127 −0.427260
\(64\) 1.00000 0.125000
\(65\) −0.0775596 −0.00962008
\(66\) 1.00000 0.123091
\(67\) 8.45198 1.03257 0.516287 0.856416i \(-0.327314\pi\)
0.516287 + 0.856416i \(0.327314\pi\)
\(68\) 3.90821 0.473940
\(69\) −4.09855 −0.493407
\(70\) 1.38188 0.165166
\(71\) 1.56991 0.186314 0.0931571 0.995651i \(-0.470304\pi\)
0.0931571 + 0.995651i \(0.470304\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.6664 1.36545 0.682724 0.730677i \(-0.260795\pi\)
0.682724 + 0.730677i \(0.260795\pi\)
\(74\) −6.51463 −0.757311
\(75\) 4.83396 0.558178
\(76\) −4.21451 −0.483437
\(77\) −3.39127 −0.386471
\(78\) −0.190339 −0.0215517
\(79\) 9.38091 1.05543 0.527717 0.849420i \(-0.323048\pi\)
0.527717 + 0.849420i \(0.323048\pi\)
\(80\) 0.407480 0.0455577
\(81\) 1.00000 0.111111
\(82\) −8.27278 −0.913576
\(83\) −11.5050 −1.26284 −0.631421 0.775441i \(-0.717528\pi\)
−0.631421 + 0.775441i \(0.717528\pi\)
\(84\) 3.39127 0.370018
\(85\) 1.59252 0.172733
\(86\) −2.67207 −0.288137
\(87\) −0.0390710 −0.00418885
\(88\) −1.00000 −0.106600
\(89\) −3.26264 −0.345839 −0.172920 0.984936i \(-0.555320\pi\)
−0.172920 + 0.984936i \(0.555320\pi\)
\(90\) −0.407480 −0.0429522
\(91\) 0.645493 0.0676660
\(92\) 4.09855 0.427303
\(93\) 2.91932 0.302720
\(94\) 4.59990 0.474443
\(95\) −1.71733 −0.176194
\(96\) 1.00000 0.102062
\(97\) −6.20116 −0.629633 −0.314816 0.949153i \(-0.601943\pi\)
−0.314816 + 0.949153i \(0.601943\pi\)
\(98\) −4.50073 −0.454642
\(99\) 1.00000 0.100504
\(100\) −4.83396 −0.483396
\(101\) −5.66027 −0.563218 −0.281609 0.959529i \(-0.590868\pi\)
−0.281609 + 0.959529i \(0.590868\pi\)
\(102\) 3.90821 0.386971
\(103\) −11.6542 −1.14833 −0.574163 0.818741i \(-0.694673\pi\)
−0.574163 + 0.818741i \(0.694673\pi\)
\(104\) 0.190339 0.0186643
\(105\) 1.38188 0.134857
\(106\) 2.73918 0.266052
\(107\) 15.0109 1.45116 0.725578 0.688140i \(-0.241572\pi\)
0.725578 + 0.688140i \(0.241572\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 9.88750 0.947050 0.473525 0.880780i \(-0.342981\pi\)
0.473525 + 0.880780i \(0.342981\pi\)
\(110\) −0.407480 −0.0388517
\(111\) −6.51463 −0.618342
\(112\) −3.39127 −0.320445
\(113\) −17.4789 −1.64428 −0.822139 0.569287i \(-0.807219\pi\)
−0.822139 + 0.569287i \(0.807219\pi\)
\(114\) −4.21451 −0.394725
\(115\) 1.67008 0.155736
\(116\) 0.0390710 0.00362765
\(117\) −0.190339 −0.0175969
\(118\) 13.4343 1.23673
\(119\) −13.2538 −1.21497
\(120\) 0.407480 0.0371977
\(121\) 1.00000 0.0909091
\(122\) −1.00000 −0.0905357
\(123\) −8.27278 −0.745931
\(124\) −2.91932 −0.262163
\(125\) −4.00715 −0.358410
\(126\) 3.39127 0.302119
\(127\) 13.0092 1.15438 0.577190 0.816610i \(-0.304149\pi\)
0.577190 + 0.816610i \(0.304149\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.67207 −0.235263
\(130\) 0.0775596 0.00680243
\(131\) −21.9677 −1.91933 −0.959665 0.281145i \(-0.909286\pi\)
−0.959665 + 0.281145i \(0.909286\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 14.2925 1.23932
\(134\) −8.45198 −0.730140
\(135\) −0.407480 −0.0350703
\(136\) −3.90821 −0.335126
\(137\) 9.51622 0.813025 0.406513 0.913645i \(-0.366745\pi\)
0.406513 + 0.913645i \(0.366745\pi\)
\(138\) 4.09855 0.348892
\(139\) −8.06367 −0.683952 −0.341976 0.939709i \(-0.611096\pi\)
−0.341976 + 0.939709i \(0.611096\pi\)
\(140\) −1.38188 −0.116790
\(141\) 4.59990 0.387381
\(142\) −1.56991 −0.131744
\(143\) −0.190339 −0.0159170
\(144\) 1.00000 0.0833333
\(145\) 0.0159206 0.00132214
\(146\) −11.6664 −0.965517
\(147\) −4.50073 −0.371214
\(148\) 6.51463 0.535500
\(149\) −1.61425 −0.132245 −0.0661223 0.997812i \(-0.521063\pi\)
−0.0661223 + 0.997812i \(0.521063\pi\)
\(150\) −4.83396 −0.394691
\(151\) −11.6620 −0.949040 −0.474520 0.880245i \(-0.657378\pi\)
−0.474520 + 0.880245i \(0.657378\pi\)
\(152\) 4.21451 0.341842
\(153\) 3.90821 0.315960
\(154\) 3.39127 0.273277
\(155\) −1.18957 −0.0955483
\(156\) 0.190339 0.0152394
\(157\) −15.5480 −1.24087 −0.620433 0.784259i \(-0.713043\pi\)
−0.620433 + 0.784259i \(0.713043\pi\)
\(158\) −9.38091 −0.746305
\(159\) 2.73918 0.217231
\(160\) −0.407480 −0.0322142
\(161\) −13.8993 −1.09542
\(162\) −1.00000 −0.0785674
\(163\) −4.35200 −0.340875 −0.170437 0.985369i \(-0.554518\pi\)
−0.170437 + 0.985369i \(0.554518\pi\)
\(164\) 8.27278 0.645996
\(165\) −0.407480 −0.0317223
\(166\) 11.5050 0.892963
\(167\) −11.0735 −0.856895 −0.428448 0.903567i \(-0.640939\pi\)
−0.428448 + 0.903567i \(0.640939\pi\)
\(168\) −3.39127 −0.261642
\(169\) −12.9638 −0.997213
\(170\) −1.59252 −0.122141
\(171\) −4.21451 −0.322291
\(172\) 2.67207 0.203743
\(173\) 2.93785 0.223361 0.111680 0.993744i \(-0.464377\pi\)
0.111680 + 0.993744i \(0.464377\pi\)
\(174\) 0.0390710 0.00296196
\(175\) 16.3933 1.23922
\(176\) 1.00000 0.0753778
\(177\) 13.4343 1.00979
\(178\) 3.26264 0.244545
\(179\) 0.675191 0.0504661 0.0252331 0.999682i \(-0.491967\pi\)
0.0252331 + 0.999682i \(0.491967\pi\)
\(180\) 0.407480 0.0303718
\(181\) −19.6035 −1.45711 −0.728557 0.684985i \(-0.759809\pi\)
−0.728557 + 0.684985i \(0.759809\pi\)
\(182\) −0.645493 −0.0478471
\(183\) −1.00000 −0.0739221
\(184\) −4.09855 −0.302149
\(185\) 2.65459 0.195169
\(186\) −2.91932 −0.214055
\(187\) 3.90821 0.285797
\(188\) −4.59990 −0.335482
\(189\) 3.39127 0.246679
\(190\) 1.71733 0.124588
\(191\) −1.78577 −0.129214 −0.0646069 0.997911i \(-0.520579\pi\)
−0.0646069 + 0.997911i \(0.520579\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.84397 0.636603 0.318301 0.947990i \(-0.396888\pi\)
0.318301 + 0.947990i \(0.396888\pi\)
\(194\) 6.20116 0.445218
\(195\) 0.0775596 0.00555416
\(196\) 4.50073 0.321481
\(197\) 15.8207 1.12718 0.563589 0.826055i \(-0.309420\pi\)
0.563589 + 0.826055i \(0.309420\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 2.49048 0.176545 0.0882727 0.996096i \(-0.471865\pi\)
0.0882727 + 0.996096i \(0.471865\pi\)
\(200\) 4.83396 0.341813
\(201\) −8.45198 −0.596156
\(202\) 5.66027 0.398255
\(203\) −0.132500 −0.00929969
\(204\) −3.90821 −0.273629
\(205\) 3.37100 0.235441
\(206\) 11.6542 0.811990
\(207\) 4.09855 0.284869
\(208\) −0.190339 −0.0131977
\(209\) −4.21451 −0.291524
\(210\) −1.38188 −0.0953586
\(211\) 6.86376 0.472520 0.236260 0.971690i \(-0.424078\pi\)
0.236260 + 0.971690i \(0.424078\pi\)
\(212\) −2.73918 −0.188127
\(213\) −1.56991 −0.107569
\(214\) −15.0109 −1.02612
\(215\) 1.08882 0.0742567
\(216\) 1.00000 0.0680414
\(217\) 9.90022 0.672071
\(218\) −9.88750 −0.669666
\(219\) −11.6664 −0.788341
\(220\) 0.407480 0.0274723
\(221\) −0.743887 −0.0500392
\(222\) 6.51463 0.437234
\(223\) 0.569295 0.0381228 0.0190614 0.999818i \(-0.493932\pi\)
0.0190614 + 0.999818i \(0.493932\pi\)
\(224\) 3.39127 0.226589
\(225\) −4.83396 −0.322264
\(226\) 17.4789 1.16268
\(227\) 10.8386 0.719381 0.359691 0.933072i \(-0.382882\pi\)
0.359691 + 0.933072i \(0.382882\pi\)
\(228\) 4.21451 0.279113
\(229\) −20.1015 −1.32834 −0.664171 0.747580i \(-0.731215\pi\)
−0.664171 + 0.747580i \(0.731215\pi\)
\(230\) −1.67008 −0.110122
\(231\) 3.39127 0.223129
\(232\) −0.0390710 −0.00256513
\(233\) −1.73528 −0.113682 −0.0568409 0.998383i \(-0.518103\pi\)
−0.0568409 + 0.998383i \(0.518103\pi\)
\(234\) 0.190339 0.0124429
\(235\) −1.87437 −0.122270
\(236\) −13.4343 −0.874501
\(237\) −9.38091 −0.609355
\(238\) 13.2538 0.859117
\(239\) −4.90179 −0.317070 −0.158535 0.987353i \(-0.550677\pi\)
−0.158535 + 0.987353i \(0.550677\pi\)
\(240\) −0.407480 −0.0263028
\(241\) 29.1570 1.87817 0.939084 0.343689i \(-0.111677\pi\)
0.939084 + 0.343689i \(0.111677\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 1.00000 0.0640184
\(245\) 1.83396 0.117167
\(246\) 8.27278 0.527453
\(247\) 0.802187 0.0510419
\(248\) 2.91932 0.185377
\(249\) 11.5050 0.729102
\(250\) 4.00715 0.253434
\(251\) 2.69434 0.170066 0.0850328 0.996378i \(-0.472901\pi\)
0.0850328 + 0.996378i \(0.472901\pi\)
\(252\) −3.39127 −0.213630
\(253\) 4.09855 0.257674
\(254\) −13.0092 −0.816269
\(255\) −1.59252 −0.0997274
\(256\) 1.00000 0.0625000
\(257\) −7.10517 −0.443209 −0.221604 0.975137i \(-0.571129\pi\)
−0.221604 + 0.975137i \(0.571129\pi\)
\(258\) 2.67207 0.166356
\(259\) −22.0929 −1.37279
\(260\) −0.0775596 −0.00481004
\(261\) 0.0390710 0.00241843
\(262\) 21.9677 1.35717
\(263\) −19.9870 −1.23245 −0.616227 0.787569i \(-0.711340\pi\)
−0.616227 + 0.787569i \(0.711340\pi\)
\(264\) 1.00000 0.0615457
\(265\) −1.11616 −0.0685652
\(266\) −14.2925 −0.876332
\(267\) 3.26264 0.199670
\(268\) 8.45198 0.516287
\(269\) 18.3489 1.11875 0.559377 0.828914i \(-0.311041\pi\)
0.559377 + 0.828914i \(0.311041\pi\)
\(270\) 0.407480 0.0247985
\(271\) 7.27120 0.441694 0.220847 0.975308i \(-0.429118\pi\)
0.220847 + 0.975308i \(0.429118\pi\)
\(272\) 3.90821 0.236970
\(273\) −0.645493 −0.0390670
\(274\) −9.51622 −0.574896
\(275\) −4.83396 −0.291499
\(276\) −4.09855 −0.246704
\(277\) −9.48724 −0.570033 −0.285017 0.958523i \(-0.591999\pi\)
−0.285017 + 0.958523i \(0.591999\pi\)
\(278\) 8.06367 0.483627
\(279\) −2.91932 −0.174775
\(280\) 1.38188 0.0825830
\(281\) −31.6176 −1.88615 −0.943073 0.332587i \(-0.892078\pi\)
−0.943073 + 0.332587i \(0.892078\pi\)
\(282\) −4.59990 −0.273920
\(283\) −20.9491 −1.24530 −0.622648 0.782502i \(-0.713943\pi\)
−0.622648 + 0.782502i \(0.713943\pi\)
\(284\) 1.56991 0.0931571
\(285\) 1.71733 0.101726
\(286\) 0.190339 0.0112550
\(287\) −28.0553 −1.65605
\(288\) −1.00000 −0.0589256
\(289\) −1.72589 −0.101523
\(290\) −0.0159206 −0.000934893 0
\(291\) 6.20116 0.363519
\(292\) 11.6664 0.682724
\(293\) −18.7636 −1.09618 −0.548091 0.836419i \(-0.684645\pi\)
−0.548091 + 0.836419i \(0.684645\pi\)
\(294\) 4.50073 0.262488
\(295\) −5.47423 −0.318722
\(296\) −6.51463 −0.378655
\(297\) −1.00000 −0.0580259
\(298\) 1.61425 0.0935111
\(299\) −0.780116 −0.0451153
\(300\) 4.83396 0.279089
\(301\) −9.06172 −0.522309
\(302\) 11.6620 0.671073
\(303\) 5.66027 0.325174
\(304\) −4.21451 −0.241719
\(305\) 0.407480 0.0233323
\(306\) −3.90821 −0.223418
\(307\) −12.8926 −0.735818 −0.367909 0.929862i \(-0.619926\pi\)
−0.367909 + 0.929862i \(0.619926\pi\)
\(308\) −3.39127 −0.193236
\(309\) 11.6542 0.662987
\(310\) 1.18957 0.0675629
\(311\) −26.0626 −1.47788 −0.738938 0.673773i \(-0.764673\pi\)
−0.738938 + 0.673773i \(0.764673\pi\)
\(312\) −0.190339 −0.0107758
\(313\) −9.41313 −0.532062 −0.266031 0.963964i \(-0.585712\pi\)
−0.266031 + 0.963964i \(0.585712\pi\)
\(314\) 15.5480 0.877425
\(315\) −1.38188 −0.0778600
\(316\) 9.38091 0.527717
\(317\) 8.50811 0.477863 0.238931 0.971036i \(-0.423203\pi\)
0.238931 + 0.971036i \(0.423203\pi\)
\(318\) −2.73918 −0.153605
\(319\) 0.0390710 0.00218755
\(320\) 0.407480 0.0227789
\(321\) −15.0109 −0.837826
\(322\) 13.8993 0.774578
\(323\) −16.4712 −0.916481
\(324\) 1.00000 0.0555556
\(325\) 0.920093 0.0510376
\(326\) 4.35200 0.241035
\(327\) −9.88750 −0.546780
\(328\) −8.27278 −0.456788
\(329\) 15.5995 0.860028
\(330\) 0.407480 0.0224311
\(331\) −34.5824 −1.90082 −0.950409 0.311003i \(-0.899335\pi\)
−0.950409 + 0.311003i \(0.899335\pi\)
\(332\) −11.5050 −0.631421
\(333\) 6.51463 0.357000
\(334\) 11.0735 0.605917
\(335\) 3.44402 0.188167
\(336\) 3.39127 0.185009
\(337\) 4.70730 0.256423 0.128212 0.991747i \(-0.459076\pi\)
0.128212 + 0.991747i \(0.459076\pi\)
\(338\) 12.9638 0.705136
\(339\) 17.4789 0.949324
\(340\) 1.59252 0.0863665
\(341\) −2.91932 −0.158090
\(342\) 4.21451 0.227894
\(343\) 8.47571 0.457645
\(344\) −2.67207 −0.144068
\(345\) −1.67008 −0.0899140
\(346\) −2.93785 −0.157940
\(347\) −22.0091 −1.18151 −0.590754 0.806851i \(-0.701170\pi\)
−0.590754 + 0.806851i \(0.701170\pi\)
\(348\) −0.0390710 −0.00209442
\(349\) 18.4244 0.986236 0.493118 0.869963i \(-0.335857\pi\)
0.493118 + 0.869963i \(0.335857\pi\)
\(350\) −16.3933 −0.876257
\(351\) 0.190339 0.0101596
\(352\) −1.00000 −0.0533002
\(353\) 12.4271 0.661429 0.330714 0.943731i \(-0.392710\pi\)
0.330714 + 0.943731i \(0.392710\pi\)
\(354\) −13.4343 −0.714027
\(355\) 0.639708 0.0339522
\(356\) −3.26264 −0.172920
\(357\) 13.2538 0.701466
\(358\) −0.675191 −0.0356849
\(359\) −19.6443 −1.03679 −0.518393 0.855143i \(-0.673469\pi\)
−0.518393 + 0.855143i \(0.673469\pi\)
\(360\) −0.407480 −0.0214761
\(361\) −1.23793 −0.0651544
\(362\) 19.6035 1.03034
\(363\) −1.00000 −0.0524864
\(364\) 0.645493 0.0338330
\(365\) 4.75382 0.248827
\(366\) 1.00000 0.0522708
\(367\) −1.14405 −0.0597188 −0.0298594 0.999554i \(-0.509506\pi\)
−0.0298594 + 0.999554i \(0.509506\pi\)
\(368\) 4.09855 0.213652
\(369\) 8.27278 0.430664
\(370\) −2.65459 −0.138005
\(371\) 9.28929 0.482276
\(372\) 2.91932 0.151360
\(373\) −17.9312 −0.928441 −0.464220 0.885720i \(-0.653665\pi\)
−0.464220 + 0.885720i \(0.653665\pi\)
\(374\) −3.90821 −0.202089
\(375\) 4.00715 0.206928
\(376\) 4.59990 0.237222
\(377\) −0.00743674 −0.000383012 0
\(378\) −3.39127 −0.174428
\(379\) −3.60125 −0.184984 −0.0924920 0.995713i \(-0.529483\pi\)
−0.0924920 + 0.995713i \(0.529483\pi\)
\(380\) −1.71733 −0.0880971
\(381\) −13.0092 −0.666481
\(382\) 1.78577 0.0913679
\(383\) −13.9835 −0.714524 −0.357262 0.934004i \(-0.616290\pi\)
−0.357262 + 0.934004i \(0.616290\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.38188 −0.0704270
\(386\) −8.84397 −0.450146
\(387\) 2.67207 0.135829
\(388\) −6.20116 −0.314816
\(389\) −8.68005 −0.440096 −0.220048 0.975489i \(-0.570621\pi\)
−0.220048 + 0.975489i \(0.570621\pi\)
\(390\) −0.0775596 −0.00392738
\(391\) 16.0180 0.810065
\(392\) −4.50073 −0.227321
\(393\) 21.9677 1.10813
\(394\) −15.8207 −0.797035
\(395\) 3.82254 0.192333
\(396\) 1.00000 0.0502519
\(397\) 4.05936 0.203733 0.101867 0.994798i \(-0.467519\pi\)
0.101867 + 0.994798i \(0.467519\pi\)
\(398\) −2.49048 −0.124836
\(399\) −14.2925 −0.715522
\(400\) −4.83396 −0.241698
\(401\) 2.25650 0.112684 0.0563422 0.998412i \(-0.482056\pi\)
0.0563422 + 0.998412i \(0.482056\pi\)
\(402\) 8.45198 0.421546
\(403\) 0.555662 0.0276795
\(404\) −5.66027 −0.281609
\(405\) 0.407480 0.0202479
\(406\) 0.132500 0.00657588
\(407\) 6.51463 0.322918
\(408\) 3.90821 0.193485
\(409\) −4.93027 −0.243786 −0.121893 0.992543i \(-0.538897\pi\)
−0.121893 + 0.992543i \(0.538897\pi\)
\(410\) −3.37100 −0.166482
\(411\) −9.51622 −0.469400
\(412\) −11.6542 −0.574163
\(413\) 45.5595 2.24184
\(414\) −4.09855 −0.201433
\(415\) −4.68807 −0.230129
\(416\) 0.190339 0.00933216
\(417\) 8.06367 0.394880
\(418\) 4.21451 0.206138
\(419\) −1.40235 −0.0685094 −0.0342547 0.999413i \(-0.510906\pi\)
−0.0342547 + 0.999413i \(0.510906\pi\)
\(420\) 1.38188 0.0674287
\(421\) −8.61008 −0.419629 −0.209815 0.977741i \(-0.567286\pi\)
−0.209815 + 0.977741i \(0.567286\pi\)
\(422\) −6.86376 −0.334122
\(423\) −4.59990 −0.223655
\(424\) 2.73918 0.133026
\(425\) −18.8921 −0.916403
\(426\) 1.56991 0.0760625
\(427\) −3.39127 −0.164115
\(428\) 15.0109 0.725578
\(429\) 0.190339 0.00918967
\(430\) −1.08882 −0.0525074
\(431\) −35.2866 −1.69969 −0.849847 0.527029i \(-0.823306\pi\)
−0.849847 + 0.527029i \(0.823306\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −0.934742 −0.0449208 −0.0224604 0.999748i \(-0.507150\pi\)
−0.0224604 + 0.999748i \(0.507150\pi\)
\(434\) −9.90022 −0.475226
\(435\) −0.0159206 −0.000763337 0
\(436\) 9.88750 0.473525
\(437\) −17.2734 −0.826297
\(438\) 11.6664 0.557441
\(439\) 0.742524 0.0354387 0.0177194 0.999843i \(-0.494359\pi\)
0.0177194 + 0.999843i \(0.494359\pi\)
\(440\) −0.407480 −0.0194259
\(441\) 4.50073 0.214320
\(442\) 0.743887 0.0353831
\(443\) −22.5191 −1.06991 −0.534957 0.844879i \(-0.679672\pi\)
−0.534957 + 0.844879i \(0.679672\pi\)
\(444\) −6.51463 −0.309171
\(445\) −1.32946 −0.0630225
\(446\) −0.569295 −0.0269569
\(447\) 1.61425 0.0763515
\(448\) −3.39127 −0.160223
\(449\) −30.8487 −1.45584 −0.727920 0.685663i \(-0.759513\pi\)
−0.727920 + 0.685663i \(0.759513\pi\)
\(450\) 4.83396 0.227875
\(451\) 8.27278 0.389550
\(452\) −17.4789 −0.822139
\(453\) 11.6620 0.547929
\(454\) −10.8386 −0.508679
\(455\) 0.263026 0.0123308
\(456\) −4.21451 −0.197362
\(457\) 11.1117 0.519785 0.259893 0.965638i \(-0.416313\pi\)
0.259893 + 0.965638i \(0.416313\pi\)
\(458\) 20.1015 0.939280
\(459\) −3.90821 −0.182420
\(460\) 1.67008 0.0778678
\(461\) −1.71664 −0.0799520 −0.0399760 0.999201i \(-0.512728\pi\)
−0.0399760 + 0.999201i \(0.512728\pi\)
\(462\) −3.39127 −0.157776
\(463\) 18.7085 0.869458 0.434729 0.900561i \(-0.356844\pi\)
0.434729 + 0.900561i \(0.356844\pi\)
\(464\) 0.0390710 0.00181382
\(465\) 1.18957 0.0551648
\(466\) 1.73528 0.0803852
\(467\) −15.4734 −0.716023 −0.358012 0.933717i \(-0.616545\pi\)
−0.358012 + 0.933717i \(0.616545\pi\)
\(468\) −0.190339 −0.00879844
\(469\) −28.6630 −1.32353
\(470\) 1.87437 0.0864581
\(471\) 15.5480 0.716415
\(472\) 13.4343 0.618366
\(473\) 2.67207 0.122862
\(474\) 9.38091 0.430879
\(475\) 20.3728 0.934766
\(476\) −13.2538 −0.607487
\(477\) −2.73918 −0.125418
\(478\) 4.90179 0.224202
\(479\) 6.51343 0.297606 0.148803 0.988867i \(-0.452458\pi\)
0.148803 + 0.988867i \(0.452458\pi\)
\(480\) 0.407480 0.0185989
\(481\) −1.23999 −0.0565387
\(482\) −29.1570 −1.32806
\(483\) 13.8993 0.632440
\(484\) 1.00000 0.0454545
\(485\) −2.52685 −0.114738
\(486\) 1.00000 0.0453609
\(487\) −16.7789 −0.760324 −0.380162 0.924920i \(-0.624132\pi\)
−0.380162 + 0.924920i \(0.624132\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 4.35200 0.196804
\(490\) −1.83396 −0.0828499
\(491\) −21.5232 −0.971329 −0.485665 0.874145i \(-0.661422\pi\)
−0.485665 + 0.874145i \(0.661422\pi\)
\(492\) −8.27278 −0.372966
\(493\) 0.152698 0.00687715
\(494\) −0.802187 −0.0360921
\(495\) 0.407480 0.0183149
\(496\) −2.91932 −0.131081
\(497\) −5.32400 −0.238814
\(498\) −11.5050 −0.515553
\(499\) 20.7732 0.929936 0.464968 0.885327i \(-0.346066\pi\)
0.464968 + 0.885327i \(0.346066\pi\)
\(500\) −4.00715 −0.179205
\(501\) 11.0735 0.494729
\(502\) −2.69434 −0.120254
\(503\) −9.49556 −0.423386 −0.211693 0.977336i \(-0.567898\pi\)
−0.211693 + 0.977336i \(0.567898\pi\)
\(504\) 3.39127 0.151059
\(505\) −2.30645 −0.102636
\(506\) −4.09855 −0.182203
\(507\) 12.9638 0.575741
\(508\) 13.0092 0.577190
\(509\) −20.7252 −0.918630 −0.459315 0.888273i \(-0.651905\pi\)
−0.459315 + 0.888273i \(0.651905\pi\)
\(510\) 1.59252 0.0705180
\(511\) −39.5639 −1.75020
\(512\) −1.00000 −0.0441942
\(513\) 4.21451 0.186075
\(514\) 7.10517 0.313396
\(515\) −4.74888 −0.209261
\(516\) −2.67207 −0.117631
\(517\) −4.59990 −0.202303
\(518\) 22.0929 0.970706
\(519\) −2.93785 −0.128957
\(520\) 0.0775596 0.00340121
\(521\) −3.13961 −0.137549 −0.0687745 0.997632i \(-0.521909\pi\)
−0.0687745 + 0.997632i \(0.521909\pi\)
\(522\) −0.0390710 −0.00171009
\(523\) 14.6829 0.642037 0.321019 0.947073i \(-0.395975\pi\)
0.321019 + 0.947073i \(0.395975\pi\)
\(524\) −21.9677 −0.959665
\(525\) −16.3933 −0.715461
\(526\) 19.9870 0.871476
\(527\) −11.4093 −0.496998
\(528\) −1.00000 −0.0435194
\(529\) −6.20189 −0.269647
\(530\) 1.11616 0.0484829
\(531\) −13.4343 −0.583001
\(532\) 14.2925 0.619660
\(533\) −1.57464 −0.0682051
\(534\) −3.26264 −0.141188
\(535\) 6.11664 0.264445
\(536\) −8.45198 −0.365070
\(537\) −0.675191 −0.0291366
\(538\) −18.3489 −0.791078
\(539\) 4.50073 0.193860
\(540\) −0.407480 −0.0175352
\(541\) −17.2033 −0.739629 −0.369815 0.929106i \(-0.620579\pi\)
−0.369815 + 0.929106i \(0.620579\pi\)
\(542\) −7.27120 −0.312325
\(543\) 19.6035 0.841266
\(544\) −3.90821 −0.167563
\(545\) 4.02896 0.172582
\(546\) 0.645493 0.0276245
\(547\) 14.2728 0.610259 0.305130 0.952311i \(-0.401300\pi\)
0.305130 + 0.952311i \(0.401300\pi\)
\(548\) 9.51622 0.406513
\(549\) 1.00000 0.0426790
\(550\) 4.83396 0.206121
\(551\) −0.164665 −0.00701496
\(552\) 4.09855 0.174446
\(553\) −31.8132 −1.35284
\(554\) 9.48724 0.403074
\(555\) −2.65459 −0.112681
\(556\) −8.06367 −0.341976
\(557\) −2.96521 −0.125640 −0.0628199 0.998025i \(-0.520009\pi\)
−0.0628199 + 0.998025i \(0.520009\pi\)
\(558\) 2.91932 0.123585
\(559\) −0.508600 −0.0215115
\(560\) −1.38188 −0.0583950
\(561\) −3.90821 −0.165005
\(562\) 31.6176 1.33371
\(563\) 16.8136 0.708609 0.354304 0.935130i \(-0.384718\pi\)
0.354304 + 0.935130i \(0.384718\pi\)
\(564\) 4.59990 0.193691
\(565\) −7.12232 −0.299638
\(566\) 20.9491 0.880558
\(567\) −3.39127 −0.142420
\(568\) −1.56991 −0.0658720
\(569\) −19.8262 −0.831157 −0.415579 0.909557i \(-0.636421\pi\)
−0.415579 + 0.909557i \(0.636421\pi\)
\(570\) −1.71733 −0.0719310
\(571\) −31.9143 −1.33557 −0.667786 0.744353i \(-0.732758\pi\)
−0.667786 + 0.744353i \(0.732758\pi\)
\(572\) −0.190339 −0.00795849
\(573\) 1.78577 0.0746016
\(574\) 28.0553 1.17100
\(575\) −19.8122 −0.826227
\(576\) 1.00000 0.0416667
\(577\) 5.45364 0.227038 0.113519 0.993536i \(-0.463788\pi\)
0.113519 + 0.993536i \(0.463788\pi\)
\(578\) 1.72589 0.0717875
\(579\) −8.84397 −0.367543
\(580\) 0.0159206 0.000661069 0
\(581\) 39.0167 1.61869
\(582\) −6.20116 −0.257047
\(583\) −2.73918 −0.113445
\(584\) −11.6664 −0.482758
\(585\) −0.0775596 −0.00320669
\(586\) 18.7636 0.775117
\(587\) 24.9575 1.03011 0.515054 0.857158i \(-0.327772\pi\)
0.515054 + 0.857158i \(0.327772\pi\)
\(588\) −4.50073 −0.185607
\(589\) 12.3035 0.506957
\(590\) 5.47423 0.225371
\(591\) −15.8207 −0.650776
\(592\) 6.51463 0.267750
\(593\) 4.75203 0.195143 0.0975713 0.995229i \(-0.468893\pi\)
0.0975713 + 0.995229i \(0.468893\pi\)
\(594\) 1.00000 0.0410305
\(595\) −5.40067 −0.221406
\(596\) −1.61425 −0.0661223
\(597\) −2.49048 −0.101929
\(598\) 0.780116 0.0319013
\(599\) −28.7186 −1.17341 −0.586704 0.809801i \(-0.699575\pi\)
−0.586704 + 0.809801i \(0.699575\pi\)
\(600\) −4.83396 −0.197346
\(601\) −33.1823 −1.35353 −0.676766 0.736198i \(-0.736619\pi\)
−0.676766 + 0.736198i \(0.736619\pi\)
\(602\) 9.06172 0.369328
\(603\) 8.45198 0.344191
\(604\) −11.6620 −0.474520
\(605\) 0.407480 0.0165664
\(606\) −5.66027 −0.229933
\(607\) 29.5058 1.19760 0.598802 0.800897i \(-0.295644\pi\)
0.598802 + 0.800897i \(0.295644\pi\)
\(608\) 4.21451 0.170921
\(609\) 0.132500 0.00536918
\(610\) −0.407480 −0.0164984
\(611\) 0.875542 0.0354206
\(612\) 3.90821 0.157980
\(613\) −0.815836 −0.0329513 −0.0164756 0.999864i \(-0.505245\pi\)
−0.0164756 + 0.999864i \(0.505245\pi\)
\(614\) 12.8926 0.520302
\(615\) −3.37100 −0.135932
\(616\) 3.39127 0.136638
\(617\) 34.3165 1.38153 0.690764 0.723080i \(-0.257274\pi\)
0.690764 + 0.723080i \(0.257274\pi\)
\(618\) −11.6542 −0.468803
\(619\) −1.13167 −0.0454855 −0.0227427 0.999741i \(-0.507240\pi\)
−0.0227427 + 0.999741i \(0.507240\pi\)
\(620\) −1.18957 −0.0477742
\(621\) −4.09855 −0.164469
\(622\) 26.0626 1.04502
\(623\) 11.0645 0.443290
\(624\) 0.190339 0.00761968
\(625\) 22.5370 0.901479
\(626\) 9.41313 0.376225
\(627\) 4.21451 0.168311
\(628\) −15.5480 −0.620433
\(629\) 25.4606 1.01518
\(630\) 1.38188 0.0550553
\(631\) 25.9961 1.03489 0.517445 0.855717i \(-0.326883\pi\)
0.517445 + 0.855717i \(0.326883\pi\)
\(632\) −9.38091 −0.373152
\(633\) −6.86376 −0.272810
\(634\) −8.50811 −0.337900
\(635\) 5.30099 0.210363
\(636\) 2.73918 0.108615
\(637\) −0.856666 −0.0339424
\(638\) −0.0390710 −0.00154683
\(639\) 1.56991 0.0621047
\(640\) −0.407480 −0.0161071
\(641\) −29.9358 −1.18239 −0.591196 0.806528i \(-0.701344\pi\)
−0.591196 + 0.806528i \(0.701344\pi\)
\(642\) 15.0109 0.592432
\(643\) 35.8982 1.41569 0.707844 0.706368i \(-0.249668\pi\)
0.707844 + 0.706368i \(0.249668\pi\)
\(644\) −13.8993 −0.547709
\(645\) −1.08882 −0.0428721
\(646\) 16.4712 0.648050
\(647\) 16.7452 0.658322 0.329161 0.944274i \(-0.393234\pi\)
0.329161 + 0.944274i \(0.393234\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −13.4343 −0.527344
\(650\) −0.920093 −0.0360890
\(651\) −9.90022 −0.388020
\(652\) −4.35200 −0.170437
\(653\) 22.5161 0.881122 0.440561 0.897723i \(-0.354780\pi\)
0.440561 + 0.897723i \(0.354780\pi\)
\(654\) 9.88750 0.386632
\(655\) −8.95143 −0.349761
\(656\) 8.27278 0.322998
\(657\) 11.6664 0.455149
\(658\) −15.5995 −0.608132
\(659\) −30.5059 −1.18834 −0.594170 0.804340i \(-0.702519\pi\)
−0.594170 + 0.804340i \(0.702519\pi\)
\(660\) −0.407480 −0.0158612
\(661\) −13.7892 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(662\) 34.5824 1.34408
\(663\) 0.743887 0.0288902
\(664\) 11.5050 0.446482
\(665\) 5.82393 0.225842
\(666\) −6.51463 −0.252437
\(667\) 0.160134 0.00620042
\(668\) −11.0735 −0.428448
\(669\) −0.569295 −0.0220102
\(670\) −3.44402 −0.133054
\(671\) 1.00000 0.0386046
\(672\) −3.39127 −0.130821
\(673\) 22.3861 0.862922 0.431461 0.902132i \(-0.357998\pi\)
0.431461 + 0.902132i \(0.357998\pi\)
\(674\) −4.70730 −0.181319
\(675\) 4.83396 0.186059
\(676\) −12.9638 −0.498607
\(677\) 45.6483 1.75441 0.877203 0.480120i \(-0.159407\pi\)
0.877203 + 0.480120i \(0.159407\pi\)
\(678\) −17.4789 −0.671274
\(679\) 21.0298 0.807051
\(680\) −1.59252 −0.0610703
\(681\) −10.8386 −0.415335
\(682\) 2.91932 0.111787
\(683\) 27.5327 1.05351 0.526754 0.850018i \(-0.323409\pi\)
0.526754 + 0.850018i \(0.323409\pi\)
\(684\) −4.21451 −0.161146
\(685\) 3.87767 0.148158
\(686\) −8.47571 −0.323604
\(687\) 20.1015 0.766919
\(688\) 2.67207 0.101872
\(689\) 0.521373 0.0198627
\(690\) 1.67008 0.0635788
\(691\) 18.4860 0.703239 0.351619 0.936143i \(-0.385631\pi\)
0.351619 + 0.936143i \(0.385631\pi\)
\(692\) 2.93785 0.111680
\(693\) −3.39127 −0.128824
\(694\) 22.0091 0.835453
\(695\) −3.28579 −0.124637
\(696\) 0.0390710 0.00148098
\(697\) 32.3318 1.22465
\(698\) −18.4244 −0.697374
\(699\) 1.73528 0.0656342
\(700\) 16.3933 0.619608
\(701\) 27.0939 1.02332 0.511661 0.859187i \(-0.329030\pi\)
0.511661 + 0.859187i \(0.329030\pi\)
\(702\) −0.190339 −0.00718390
\(703\) −27.4560 −1.03552
\(704\) 1.00000 0.0376889
\(705\) 1.87437 0.0705928
\(706\) −12.4271 −0.467701
\(707\) 19.1955 0.721922
\(708\) 13.4343 0.504893
\(709\) 44.8042 1.68266 0.841328 0.540525i \(-0.181774\pi\)
0.841328 + 0.540525i \(0.181774\pi\)
\(710\) −0.639708 −0.0240078
\(711\) 9.38091 0.351812
\(712\) 3.26264 0.122273
\(713\) −11.9650 −0.448092
\(714\) −13.2538 −0.496011
\(715\) −0.0775596 −0.00290056
\(716\) 0.675191 0.0252331
\(717\) 4.90179 0.183061
\(718\) 19.6443 0.733118
\(719\) −2.90008 −0.108155 −0.0540774 0.998537i \(-0.517222\pi\)
−0.0540774 + 0.998537i \(0.517222\pi\)
\(720\) 0.407480 0.0151859
\(721\) 39.5227 1.47190
\(722\) 1.23793 0.0460711
\(723\) −29.1570 −1.08436
\(724\) −19.6035 −0.728557
\(725\) −0.188867 −0.00701436
\(726\) 1.00000 0.0371135
\(727\) −17.2518 −0.639836 −0.319918 0.947445i \(-0.603655\pi\)
−0.319918 + 0.947445i \(0.603655\pi\)
\(728\) −0.645493 −0.0239236
\(729\) 1.00000 0.0370370
\(730\) −4.75382 −0.175947
\(731\) 10.4430 0.386249
\(732\) −1.00000 −0.0369611
\(733\) 16.7611 0.619086 0.309543 0.950885i \(-0.399824\pi\)
0.309543 + 0.950885i \(0.399824\pi\)
\(734\) 1.14405 0.0422276
\(735\) −1.83396 −0.0676466
\(736\) −4.09855 −0.151075
\(737\) 8.45198 0.311333
\(738\) −8.27278 −0.304525
\(739\) 36.8176 1.35436 0.677178 0.735819i \(-0.263203\pi\)
0.677178 + 0.735819i \(0.263203\pi\)
\(740\) 2.65459 0.0975845
\(741\) −0.802187 −0.0294691
\(742\) −9.28929 −0.341021
\(743\) 35.7438 1.31131 0.655657 0.755059i \(-0.272392\pi\)
0.655657 + 0.755059i \(0.272392\pi\)
\(744\) −2.91932 −0.107028
\(745\) −0.657776 −0.0240990
\(746\) 17.9312 0.656507
\(747\) −11.5050 −0.420947
\(748\) 3.90821 0.142898
\(749\) −50.9060 −1.86006
\(750\) −4.00715 −0.146320
\(751\) −0.0439324 −0.00160312 −0.000801558 1.00000i \(-0.500255\pi\)
−0.000801558 1.00000i \(0.500255\pi\)
\(752\) −4.59990 −0.167741
\(753\) −2.69434 −0.0981874
\(754\) 0.00743674 0.000270830 0
\(755\) −4.75204 −0.172944
\(756\) 3.39127 0.123339
\(757\) 23.2903 0.846501 0.423250 0.906013i \(-0.360889\pi\)
0.423250 + 0.906013i \(0.360889\pi\)
\(758\) 3.60125 0.130803
\(759\) −4.09855 −0.148768
\(760\) 1.71733 0.0622941
\(761\) −14.8136 −0.536994 −0.268497 0.963281i \(-0.586527\pi\)
−0.268497 + 0.963281i \(0.586527\pi\)
\(762\) 13.0092 0.471273
\(763\) −33.5312 −1.21391
\(764\) −1.78577 −0.0646069
\(765\) 1.59252 0.0575777
\(766\) 13.9835 0.505245
\(767\) 2.55708 0.0923310
\(768\) −1.00000 −0.0360844
\(769\) −25.8876 −0.933532 −0.466766 0.884381i \(-0.654581\pi\)
−0.466766 + 0.884381i \(0.654581\pi\)
\(770\) 1.38188 0.0497994
\(771\) 7.10517 0.255887
\(772\) 8.84397 0.318301
\(773\) −21.7017 −0.780556 −0.390278 0.920697i \(-0.627621\pi\)
−0.390278 + 0.920697i \(0.627621\pi\)
\(774\) −2.67207 −0.0960456
\(775\) 14.1119 0.506914
\(776\) 6.20116 0.222609
\(777\) 22.0929 0.792578
\(778\) 8.68005 0.311195
\(779\) −34.8657 −1.24919
\(780\) 0.0775596 0.00277708
\(781\) 1.56991 0.0561758
\(782\) −16.0180 −0.572802
\(783\) −0.0390710 −0.00139628
\(784\) 4.50073 0.160740
\(785\) −6.33551 −0.226124
\(786\) −21.9677 −0.783563
\(787\) −25.9523 −0.925100 −0.462550 0.886593i \(-0.653065\pi\)
−0.462550 + 0.886593i \(0.653065\pi\)
\(788\) 15.8207 0.563589
\(789\) 19.9870 0.711557
\(790\) −3.82254 −0.136000
\(791\) 59.2758 2.10760
\(792\) −1.00000 −0.0355335
\(793\) −0.190339 −0.00675915
\(794\) −4.05936 −0.144061
\(795\) 1.11616 0.0395861
\(796\) 2.49048 0.0882727
\(797\) 3.96050 0.140288 0.0701441 0.997537i \(-0.477654\pi\)
0.0701441 + 0.997537i \(0.477654\pi\)
\(798\) 14.2925 0.505950
\(799\) −17.9774 −0.635993
\(800\) 4.83396 0.170906
\(801\) −3.26264 −0.115280
\(802\) −2.25650 −0.0796799
\(803\) 11.6664 0.411698
\(804\) −8.45198 −0.298078
\(805\) −5.66369 −0.199619
\(806\) −0.555662 −0.0195724
\(807\) −18.3489 −0.645913
\(808\) 5.66027 0.199128
\(809\) 42.6688 1.50016 0.750078 0.661350i \(-0.230016\pi\)
0.750078 + 0.661350i \(0.230016\pi\)
\(810\) −0.407480 −0.0143174
\(811\) −5.56902 −0.195555 −0.0977774 0.995208i \(-0.531173\pi\)
−0.0977774 + 0.995208i \(0.531173\pi\)
\(812\) −0.132500 −0.00464985
\(813\) −7.27120 −0.255012
\(814\) −6.51463 −0.228338
\(815\) −1.77335 −0.0621179
\(816\) −3.90821 −0.136815
\(817\) −11.2615 −0.393989
\(818\) 4.93027 0.172383
\(819\) 0.645493 0.0225553
\(820\) 3.37100 0.117720
\(821\) −6.78716 −0.236873 −0.118437 0.992962i \(-0.537788\pi\)
−0.118437 + 0.992962i \(0.537788\pi\)
\(822\) 9.51622 0.331916
\(823\) −2.74207 −0.0955827 −0.0477913 0.998857i \(-0.515218\pi\)
−0.0477913 + 0.998857i \(0.515218\pi\)
\(824\) 11.6542 0.405995
\(825\) 4.83396 0.168297
\(826\) −45.5595 −1.58522
\(827\) −49.4575 −1.71980 −0.859902 0.510459i \(-0.829476\pi\)
−0.859902 + 0.510459i \(0.829476\pi\)
\(828\) 4.09855 0.142434
\(829\) 32.3937 1.12508 0.562540 0.826770i \(-0.309824\pi\)
0.562540 + 0.826770i \(0.309824\pi\)
\(830\) 4.68807 0.162725
\(831\) 9.48724 0.329109
\(832\) −0.190339 −0.00659883
\(833\) 17.5898 0.609451
\(834\) −8.06367 −0.279222
\(835\) −4.51225 −0.156153
\(836\) −4.21451 −0.145762
\(837\) 2.91932 0.100907
\(838\) 1.40235 0.0484435
\(839\) 53.0397 1.83113 0.915567 0.402166i \(-0.131742\pi\)
0.915567 + 0.402166i \(0.131742\pi\)
\(840\) −1.38188 −0.0476793
\(841\) −28.9985 −0.999947
\(842\) 8.61008 0.296723
\(843\) 31.6176 1.08897
\(844\) 6.86376 0.236260
\(845\) −5.28248 −0.181723
\(846\) 4.59990 0.158148
\(847\) −3.39127 −0.116526
\(848\) −2.73918 −0.0940637
\(849\) 20.9491 0.718973
\(850\) 18.8921 0.647995
\(851\) 26.7005 0.915283
\(852\) −1.56991 −0.0537843
\(853\) 33.5046 1.14718 0.573588 0.819144i \(-0.305551\pi\)
0.573588 + 0.819144i \(0.305551\pi\)
\(854\) 3.39127 0.116047
\(855\) −1.71733 −0.0587314
\(856\) −15.0109 −0.513061
\(857\) 16.7805 0.573210 0.286605 0.958049i \(-0.407473\pi\)
0.286605 + 0.958049i \(0.407473\pi\)
\(858\) −0.190339 −0.00649808
\(859\) −22.2763 −0.760057 −0.380029 0.924975i \(-0.624086\pi\)
−0.380029 + 0.924975i \(0.624086\pi\)
\(860\) 1.08882 0.0371283
\(861\) 28.0553 0.956120
\(862\) 35.2866 1.20187
\(863\) 36.0948 1.22868 0.614341 0.789041i \(-0.289422\pi\)
0.614341 + 0.789041i \(0.289422\pi\)
\(864\) 1.00000 0.0340207
\(865\) 1.19712 0.0407032
\(866\) 0.934742 0.0317638
\(867\) 1.72589 0.0586143
\(868\) 9.90022 0.336035
\(869\) 9.38091 0.318225
\(870\) 0.0159206 0.000539761 0
\(871\) −1.60874 −0.0545102
\(872\) −9.88750 −0.334833
\(873\) −6.20116 −0.209878
\(874\) 17.2734 0.584280
\(875\) 13.5893 0.459403
\(876\) −11.6664 −0.394171
\(877\) −0.184201 −0.00622003 −0.00311001 0.999995i \(-0.500990\pi\)
−0.00311001 + 0.999995i \(0.500990\pi\)
\(878\) −0.742524 −0.0250590
\(879\) 18.7636 0.632881
\(880\) 0.407480 0.0137362
\(881\) −37.7832 −1.27295 −0.636474 0.771298i \(-0.719608\pi\)
−0.636474 + 0.771298i \(0.719608\pi\)
\(882\) −4.50073 −0.151547
\(883\) −40.5121 −1.36334 −0.681671 0.731659i \(-0.738746\pi\)
−0.681671 + 0.731659i \(0.738746\pi\)
\(884\) −0.743887 −0.0250196
\(885\) 5.47423 0.184014
\(886\) 22.5191 0.756543
\(887\) −12.4836 −0.419158 −0.209579 0.977792i \(-0.567209\pi\)
−0.209579 + 0.977792i \(0.567209\pi\)
\(888\) 6.51463 0.218617
\(889\) −44.1177 −1.47966
\(890\) 1.32946 0.0445637
\(891\) 1.00000 0.0335013
\(892\) 0.569295 0.0190614
\(893\) 19.3863 0.648738
\(894\) −1.61425 −0.0539886
\(895\) 0.275127 0.00919648
\(896\) 3.39127 0.113294
\(897\) 0.780116 0.0260473
\(898\) 30.8487 1.02943
\(899\) −0.114061 −0.00380414
\(900\) −4.83396 −0.161132
\(901\) −10.7053 −0.356644
\(902\) −8.27278 −0.275453
\(903\) 9.06172 0.301555
\(904\) 17.4789 0.581340
\(905\) −7.98803 −0.265531
\(906\) −11.6620 −0.387444
\(907\) 31.8018 1.05596 0.527980 0.849257i \(-0.322949\pi\)
0.527980 + 0.849257i \(0.322949\pi\)
\(908\) 10.8386 0.359691
\(909\) −5.66027 −0.187739
\(910\) −0.263026 −0.00871922
\(911\) 43.8225 1.45190 0.725952 0.687745i \(-0.241399\pi\)
0.725952 + 0.687745i \(0.241399\pi\)
\(912\) 4.21451 0.139556
\(913\) −11.5050 −0.380761
\(914\) −11.1117 −0.367544
\(915\) −0.407480 −0.0134709
\(916\) −20.1015 −0.664171
\(917\) 74.4986 2.46016
\(918\) 3.90821 0.128990
\(919\) 38.8641 1.28201 0.641004 0.767537i \(-0.278518\pi\)
0.641004 + 0.767537i \(0.278518\pi\)
\(920\) −1.67008 −0.0550609
\(921\) 12.8926 0.424825
\(922\) 1.71664 0.0565346
\(923\) −0.298816 −0.00983565
\(924\) 3.39127 0.111565
\(925\) −31.4915 −1.03543
\(926\) −18.7085 −0.614800
\(927\) −11.6542 −0.382776
\(928\) −0.0390710 −0.00128257
\(929\) 47.7171 1.56555 0.782774 0.622306i \(-0.213804\pi\)
0.782774 + 0.622306i \(0.213804\pi\)
\(930\) −1.18957 −0.0390074
\(931\) −18.9684 −0.621663
\(932\) −1.73528 −0.0568409
\(933\) 26.0626 0.853253
\(934\) 15.4734 0.506305
\(935\) 1.59252 0.0520810
\(936\) 0.190339 0.00622144
\(937\) 37.2358 1.21644 0.608220 0.793769i \(-0.291884\pi\)
0.608220 + 0.793769i \(0.291884\pi\)
\(938\) 28.6630 0.935879
\(939\) 9.41313 0.307186
\(940\) −1.87437 −0.0611351
\(941\) −19.9915 −0.651704 −0.325852 0.945421i \(-0.605651\pi\)
−0.325852 + 0.945421i \(0.605651\pi\)
\(942\) −15.5480 −0.506582
\(943\) 33.9064 1.10414
\(944\) −13.4343 −0.437251
\(945\) 1.38188 0.0449525
\(946\) −2.67207 −0.0868765
\(947\) −10.9948 −0.357283 −0.178642 0.983914i \(-0.557170\pi\)
−0.178642 + 0.983914i \(0.557170\pi\)
\(948\) −9.38091 −0.304678
\(949\) −2.22057 −0.0720828
\(950\) −20.3728 −0.660979
\(951\) −8.50811 −0.275894
\(952\) 13.2538 0.429558
\(953\) 30.4978 0.987920 0.493960 0.869485i \(-0.335549\pi\)
0.493960 + 0.869485i \(0.335549\pi\)
\(954\) 2.73918 0.0886841
\(955\) −0.727666 −0.0235467
\(956\) −4.90179 −0.158535
\(957\) −0.0390710 −0.00126298
\(958\) −6.51343 −0.210439
\(959\) −32.2721 −1.04212
\(960\) −0.407480 −0.0131514
\(961\) −22.4776 −0.725083
\(962\) 1.23999 0.0399789
\(963\) 15.0109 0.483719
\(964\) 29.1570 0.939084
\(965\) 3.60375 0.116009
\(966\) −13.8993 −0.447203
\(967\) 16.1826 0.520396 0.260198 0.965555i \(-0.416212\pi\)
0.260198 + 0.965555i \(0.416212\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 16.4712 0.529131
\(970\) 2.52685 0.0811324
\(971\) −3.85576 −0.123737 −0.0618687 0.998084i \(-0.519706\pi\)
−0.0618687 + 0.998084i \(0.519706\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 27.3461 0.876676
\(974\) 16.7789 0.537630
\(975\) −0.920093 −0.0294666
\(976\) 1.00000 0.0320092
\(977\) 40.7503 1.30372 0.651860 0.758340i \(-0.273989\pi\)
0.651860 + 0.758340i \(0.273989\pi\)
\(978\) −4.35200 −0.139162
\(979\) −3.26264 −0.104274
\(980\) 1.83396 0.0585837
\(981\) 9.88750 0.315683
\(982\) 21.5232 0.686833
\(983\) 34.8851 1.11266 0.556331 0.830960i \(-0.312209\pi\)
0.556331 + 0.830960i \(0.312209\pi\)
\(984\) 8.27278 0.263727
\(985\) 6.44662 0.205407
\(986\) −0.152698 −0.00486288
\(987\) −15.5995 −0.496538
\(988\) 0.802187 0.0255210
\(989\) 10.9516 0.348241
\(990\) −0.407480 −0.0129506
\(991\) −12.8931 −0.409564 −0.204782 0.978808i \(-0.565649\pi\)
−0.204782 + 0.978808i \(0.565649\pi\)
\(992\) 2.91932 0.0926886
\(993\) 34.5824 1.09744
\(994\) 5.32400 0.168867
\(995\) 1.01482 0.0321720
\(996\) 11.5050 0.364551
\(997\) −58.4476 −1.85105 −0.925527 0.378682i \(-0.876377\pi\)
−0.925527 + 0.378682i \(0.876377\pi\)
\(998\) −20.7732 −0.657564
\(999\) −6.51463 −0.206114
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 4026.2.a.w.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.w.1.4 6 1.1 even 1 trivial