Properties

Label 138.3.b.a.91.5
Level $138$
Weight $3$
Character 138.91
Analytic conductor $3.760$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,3,Mod(91,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 138.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.76022764817\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1358954496.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.5
Root \(-1.21752i\) of defining polynomial
Character \(\chi\) \(=\) 138.91
Dual form 138.3.b.a.91.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -6.00464i q^{5} -2.44949 q^{6} -4.69017i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -6.00464i q^{5} -2.44949 q^{6} -4.69017i q^{7} +2.82843 q^{8} +3.00000 q^{9} -8.49185i q^{10} -6.16614i q^{11} -3.46410 q^{12} +8.71279 q^{13} -6.63290i q^{14} +10.4003i q^{15} +4.00000 q^{16} -14.0774i q^{17} +4.24264 q^{18} +14.6472i q^{19} -12.0093i q^{20} +8.12361i q^{21} -8.72025i q^{22} +(9.92304 + 20.7493i) q^{23} -4.89898 q^{24} -11.0557 q^{25} +12.3218 q^{26} -5.19615 q^{27} -9.38034i q^{28} +1.08244 q^{29} +14.7083i q^{30} -48.1396 q^{31} +5.65685 q^{32} +10.6801i q^{33} -19.9085i q^{34} -28.1628 q^{35} +6.00000 q^{36} +56.9392i q^{37} +20.7142i q^{38} -15.0910 q^{39} -16.9837i q^{40} +22.8431 q^{41} +11.4885i q^{42} +47.9731i q^{43} -12.3323i q^{44} -18.0139i q^{45} +(14.0333 + 29.3439i) q^{46} +7.39962 q^{47} -6.92820 q^{48} +27.0023 q^{49} -15.6352 q^{50} +24.3829i q^{51} +17.4256 q^{52} -86.2935i q^{53} -7.34847 q^{54} -37.0255 q^{55} -13.2658i q^{56} -25.3697i q^{57} +1.53081 q^{58} +87.6447 q^{59} +20.8007i q^{60} +0.0196058i q^{61} -68.0797 q^{62} -14.0705i q^{63} +8.00000 q^{64} -52.3172i q^{65} +15.1039i q^{66} +104.894i q^{67} -28.1549i q^{68} +(-17.1872 - 35.9388i) q^{69} -39.8282 q^{70} -7.78407 q^{71} +8.48528 q^{72} +19.7199 q^{73} +80.5242i q^{74} +19.1491 q^{75} +29.2943i q^{76} -28.9203 q^{77} -21.3419 q^{78} -46.2745i q^{79} -24.0186i q^{80} +9.00000 q^{81} +32.3050 q^{82} +16.9890i q^{83} +16.2472i q^{84} -84.5300 q^{85} +67.8442i q^{86} -1.87485 q^{87} -17.4405i q^{88} -88.0122i q^{89} -25.4755i q^{90} -40.8645i q^{91} +(19.8461 + 41.4986i) q^{92} +83.3803 q^{93} +10.4646 q^{94} +87.9510 q^{95} -9.79796 q^{96} -38.8992i q^{97} +38.1870 q^{98} -18.4984i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 24 q^{9} + 16 q^{13} + 32 q^{16} + 16 q^{23} + 72 q^{25} + 32 q^{26} - 144 q^{29} - 128 q^{31} - 112 q^{35} + 48 q^{36} + 48 q^{39} - 16 q^{41} - 80 q^{46} - 112 q^{47} + 40 q^{49} - 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} + 80 q^{59} - 96 q^{62} + 64 q^{64} - 72 q^{69} - 144 q^{70} + 32 q^{71} + 64 q^{73} + 48 q^{75} + 224 q^{77} - 144 q^{78} + 72 q^{81} + 48 q^{85} + 96 q^{87} + 32 q^{92} + 192 q^{93} - 16 q^{94} + 112 q^{95} + 224 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) −1.73205 −0.577350
\(4\) 2.00000 0.500000
\(5\) 6.00464i 1.20093i −0.799652 0.600464i \(-0.794982\pi\)
0.799652 0.600464i \(-0.205018\pi\)
\(6\) −2.44949 −0.408248
\(7\) 4.69017i 0.670024i −0.942214 0.335012i \(-0.891260\pi\)
0.942214 0.335012i \(-0.108740\pi\)
\(8\) 2.82843 0.353553
\(9\) 3.00000 0.333333
\(10\) 8.49185i 0.849185i
\(11\) 6.16614i 0.560559i −0.959918 0.280279i \(-0.909573\pi\)
0.959918 0.280279i \(-0.0904271\pi\)
\(12\) −3.46410 −0.288675
\(13\) 8.71279 0.670215 0.335107 0.942180i \(-0.391227\pi\)
0.335107 + 0.942180i \(0.391227\pi\)
\(14\) 6.63290i 0.473779i
\(15\) 10.4003i 0.693356i
\(16\) 4.00000 0.250000
\(17\) 14.0774i 0.828085i −0.910258 0.414043i \(-0.864116\pi\)
0.910258 0.414043i \(-0.135884\pi\)
\(18\) 4.24264 0.235702
\(19\) 14.6472i 0.770904i 0.922728 + 0.385452i \(0.125954\pi\)
−0.922728 + 0.385452i \(0.874046\pi\)
\(20\) 12.0093i 0.600464i
\(21\) 8.12361i 0.386839i
\(22\) 8.72025i 0.396375i
\(23\) 9.92304 + 20.7493i 0.431437 + 0.902143i
\(24\) −4.89898 −0.204124
\(25\) −11.0557 −0.442229
\(26\) 12.3218 0.473914
\(27\) −5.19615 −0.192450
\(28\) 9.38034i 0.335012i
\(29\) 1.08244 0.0373256 0.0186628 0.999826i \(-0.494059\pi\)
0.0186628 + 0.999826i \(0.494059\pi\)
\(30\) 14.7083i 0.490277i
\(31\) −48.1396 −1.55289 −0.776446 0.630184i \(-0.782979\pi\)
−0.776446 + 0.630184i \(0.782979\pi\)
\(32\) 5.65685 0.176777
\(33\) 10.6801i 0.323639i
\(34\) 19.9085i 0.585545i
\(35\) −28.1628 −0.804651
\(36\) 6.00000 0.166667
\(37\) 56.9392i 1.53890i 0.638708 + 0.769449i \(0.279469\pi\)
−0.638708 + 0.769449i \(0.720531\pi\)
\(38\) 20.7142i 0.545111i
\(39\) −15.0910 −0.386949
\(40\) 16.9837i 0.424592i
\(41\) 22.8431 0.557148 0.278574 0.960415i \(-0.410138\pi\)
0.278574 + 0.960415i \(0.410138\pi\)
\(42\) 11.4885i 0.273536i
\(43\) 47.9731i 1.11565i 0.829957 + 0.557827i \(0.188365\pi\)
−0.829957 + 0.557827i \(0.811635\pi\)
\(44\) 12.3323i 0.280279i
\(45\) 18.0139i 0.400309i
\(46\) 14.0333 + 29.3439i 0.305072 + 0.637912i
\(47\) 7.39962 0.157439 0.0787194 0.996897i \(-0.474917\pi\)
0.0787194 + 0.996897i \(0.474917\pi\)
\(48\) −6.92820 −0.144338
\(49\) 27.0023 0.551067
\(50\) −15.6352 −0.312703
\(51\) 24.3829i 0.478095i
\(52\) 17.4256 0.335107
\(53\) 86.2935i 1.62818i −0.580739 0.814090i \(-0.697236\pi\)
0.580739 0.814090i \(-0.302764\pi\)
\(54\) −7.34847 −0.136083
\(55\) −37.0255 −0.673191
\(56\) 13.2658i 0.236889i
\(57\) 25.3697i 0.445082i
\(58\) 1.53081 0.0263932
\(59\) 87.6447 1.48550 0.742751 0.669567i \(-0.233520\pi\)
0.742751 + 0.669567i \(0.233520\pi\)
\(60\) 20.8007i 0.346678i
\(61\) 0.0196058i 0.000321407i 1.00000 0.000160703i \(5.11535e-5\pi\)
−1.00000 0.000160703i \(0.999949\pi\)
\(62\) −68.0797 −1.09806
\(63\) 14.0705i 0.223341i
\(64\) 8.00000 0.125000
\(65\) 52.3172i 0.804880i
\(66\) 15.1039i 0.228847i
\(67\) 104.894i 1.56558i 0.622287 + 0.782789i \(0.286203\pi\)
−0.622287 + 0.782789i \(0.713797\pi\)
\(68\) 28.1549i 0.414043i
\(69\) −17.1872 35.9388i −0.249090 0.520853i
\(70\) −39.8282 −0.568974
\(71\) −7.78407 −0.109635 −0.0548174 0.998496i \(-0.517458\pi\)
−0.0548174 + 0.998496i \(0.517458\pi\)
\(72\) 8.48528 0.117851
\(73\) 19.7199 0.270135 0.135067 0.990836i \(-0.456875\pi\)
0.135067 + 0.990836i \(0.456875\pi\)
\(74\) 80.5242i 1.08817i
\(75\) 19.1491 0.255321
\(76\) 29.2943i 0.385452i
\(77\) −28.9203 −0.375588
\(78\) −21.3419 −0.273614
\(79\) 46.2745i 0.585753i −0.956150 0.292876i \(-0.905388\pi\)
0.956150 0.292876i \(-0.0946125\pi\)
\(80\) 24.0186i 0.300232i
\(81\) 9.00000 0.111111
\(82\) 32.3050 0.393963
\(83\) 16.9890i 0.204687i 0.994749 + 0.102344i \(0.0326341\pi\)
−0.994749 + 0.102344i \(0.967366\pi\)
\(84\) 16.2472i 0.193419i
\(85\) −84.5300 −0.994471
\(86\) 67.8442i 0.788886i
\(87\) −1.87485 −0.0215500
\(88\) 17.4405i 0.198187i
\(89\) 88.0122i 0.988901i −0.869206 0.494451i \(-0.835369\pi\)
0.869206 0.494451i \(-0.164631\pi\)
\(90\) 25.4755i 0.283062i
\(91\) 40.8645i 0.449060i
\(92\) 19.8461 + 41.4986i 0.215718 + 0.451072i
\(93\) 83.3803 0.896563
\(94\) 10.4646 0.111326
\(95\) 87.9510 0.925800
\(96\) −9.79796 −0.102062
\(97\) 38.8992i 0.401023i −0.979691 0.200511i \(-0.935740\pi\)
0.979691 0.200511i \(-0.0642604\pi\)
\(98\) 38.1870 0.389663
\(99\) 18.4984i 0.186853i
\(100\) −22.1115 −0.221115
\(101\) −5.54609 −0.0549118 −0.0274559 0.999623i \(-0.508741\pi\)
−0.0274559 + 0.999623i \(0.508741\pi\)
\(102\) 34.4826i 0.338064i
\(103\) 40.6192i 0.394361i −0.980367 0.197181i \(-0.936821\pi\)
0.980367 0.197181i \(-0.0631785\pi\)
\(104\) 24.6435 0.236957
\(105\) 48.7794 0.464566
\(106\) 122.037i 1.15130i
\(107\) 124.844i 1.16677i 0.812197 + 0.583383i \(0.198271\pi\)
−0.812197 + 0.583383i \(0.801729\pi\)
\(108\) −10.3923 −0.0962250
\(109\) 77.9912i 0.715515i −0.933814 0.357758i \(-0.883541\pi\)
0.933814 0.357758i \(-0.116459\pi\)
\(110\) −52.3620 −0.476018
\(111\) 98.6216i 0.888483i
\(112\) 18.7607i 0.167506i
\(113\) 68.3531i 0.604895i −0.953166 0.302448i \(-0.902196\pi\)
0.953166 0.302448i \(-0.0978037\pi\)
\(114\) 35.8781i 0.314720i
\(115\) 124.592 59.5843i 1.08341 0.518125i
\(116\) 2.16489 0.0186628
\(117\) 26.1384 0.223405
\(118\) 123.948 1.05041
\(119\) −66.0256 −0.554837
\(120\) 29.4166i 0.245138i
\(121\) 82.9787 0.685774
\(122\) 0.0277268i 0.000227269i
\(123\) −39.5653 −0.321669
\(124\) −96.2793 −0.776446
\(125\) 83.7304i 0.669843i
\(126\) 19.8987i 0.157926i
\(127\) −178.475 −1.40531 −0.702656 0.711530i \(-0.748003\pi\)
−0.702656 + 0.711530i \(0.748003\pi\)
\(128\) 11.3137 0.0883883
\(129\) 83.0919i 0.644123i
\(130\) 73.9877i 0.569136i
\(131\) −224.058 −1.71037 −0.855184 0.518325i \(-0.826556\pi\)
−0.855184 + 0.518325i \(0.826556\pi\)
\(132\) 21.3602i 0.161819i
\(133\) 68.6978 0.516525
\(134\) 148.342i 1.10703i
\(135\) 31.2010i 0.231119i
\(136\) 39.8170i 0.292772i
\(137\) 235.813i 1.72126i 0.509227 + 0.860632i \(0.329931\pi\)
−0.509227 + 0.860632i \(0.670069\pi\)
\(138\) −24.3064 50.8252i −0.176133 0.368298i
\(139\) 147.556 1.06156 0.530778 0.847511i \(-0.321900\pi\)
0.530778 + 0.847511i \(0.321900\pi\)
\(140\) −56.3256 −0.402326
\(141\) −12.8165 −0.0908973
\(142\) −11.0083 −0.0775235
\(143\) 53.7244i 0.375695i
\(144\) 12.0000 0.0833333
\(145\) 6.49969i 0.0448254i
\(146\) 27.8881 0.191014
\(147\) −46.7693 −0.318159
\(148\) 113.878i 0.769449i
\(149\) 238.929i 1.60355i 0.597625 + 0.801775i \(0.296111\pi\)
−0.597625 + 0.801775i \(0.703889\pi\)
\(150\) 27.0809 0.180539
\(151\) 196.451 1.30100 0.650500 0.759506i \(-0.274559\pi\)
0.650500 + 0.759506i \(0.274559\pi\)
\(152\) 41.4285i 0.272556i
\(153\) 42.2323i 0.276028i
\(154\) −40.8994 −0.265581
\(155\) 289.061i 1.86491i
\(156\) −30.1820 −0.193474
\(157\) 189.728i 1.20846i 0.796811 + 0.604229i \(0.206519\pi\)
−0.796811 + 0.604229i \(0.793481\pi\)
\(158\) 65.4420i 0.414190i
\(159\) 149.465i 0.940030i
\(160\) 33.9674i 0.212296i
\(161\) 97.3177 46.5408i 0.604458 0.289073i
\(162\) 12.7279 0.0785674
\(163\) −76.3170 −0.468202 −0.234101 0.972212i \(-0.575215\pi\)
−0.234101 + 0.972212i \(0.575215\pi\)
\(164\) 45.6861 0.278574
\(165\) 64.1300 0.388667
\(166\) 24.0261i 0.144736i
\(167\) −145.600 −0.871856 −0.435928 0.899982i \(-0.643580\pi\)
−0.435928 + 0.899982i \(0.643580\pi\)
\(168\) 22.9771i 0.136768i
\(169\) −93.0872 −0.550812
\(170\) −119.544 −0.703197
\(171\) 43.9415i 0.256968i
\(172\) 95.9462i 0.557827i
\(173\) −297.300 −1.71849 −0.859247 0.511560i \(-0.829068\pi\)
−0.859247 + 0.511560i \(0.829068\pi\)
\(174\) −2.65143 −0.0152381
\(175\) 51.8533i 0.296304i
\(176\) 24.6646i 0.140140i
\(177\) −151.805 −0.857656
\(178\) 124.468i 0.699259i
\(179\) −215.202 −1.20225 −0.601123 0.799157i \(-0.705280\pi\)
−0.601123 + 0.799157i \(0.705280\pi\)
\(180\) 36.0279i 0.200155i
\(181\) 81.8052i 0.451962i 0.974132 + 0.225981i \(0.0725588\pi\)
−0.974132 + 0.225981i \(0.927441\pi\)
\(182\) 57.7911i 0.317534i
\(183\) 0.0339583i 0.000185564i
\(184\) 28.0666 + 58.6879i 0.152536 + 0.318956i
\(185\) 341.900 1.84811
\(186\) 117.918 0.633965
\(187\) −86.8036 −0.464190
\(188\) 14.7992 0.0787194
\(189\) 24.3708i 0.128946i
\(190\) 124.382 0.654640
\(191\) 124.361i 0.651106i 0.945524 + 0.325553i \(0.105550\pi\)
−0.945524 + 0.325553i \(0.894450\pi\)
\(192\) −13.8564 −0.0721688
\(193\) −346.137 −1.79346 −0.896729 0.442581i \(-0.854063\pi\)
−0.896729 + 0.442581i \(0.854063\pi\)
\(194\) 55.0118i 0.283566i
\(195\) 90.6161i 0.464698i
\(196\) 54.0046 0.275534
\(197\) 213.943 1.08601 0.543003 0.839731i \(-0.317287\pi\)
0.543003 + 0.839731i \(0.317287\pi\)
\(198\) 26.1607i 0.132125i
\(199\) 374.684i 1.88284i −0.337243 0.941418i \(-0.609494\pi\)
0.337243 0.941418i \(-0.390506\pi\)
\(200\) −31.2703 −0.156352
\(201\) 181.681i 0.903887i
\(202\) −7.84336 −0.0388285
\(203\) 5.07685i 0.0250091i
\(204\) 48.7657i 0.239048i
\(205\) 137.164i 0.669095i
\(206\) 57.4442i 0.278855i
\(207\) 29.7691 + 62.2479i 0.143812 + 0.300714i
\(208\) 34.8512 0.167554
\(209\) 90.3166 0.432137
\(210\) 68.9845 0.328498
\(211\) −185.158 −0.877524 −0.438762 0.898603i \(-0.644583\pi\)
−0.438762 + 0.898603i \(0.644583\pi\)
\(212\) 172.587i 0.814090i
\(213\) 13.4824 0.0632977
\(214\) 176.556i 0.825028i
\(215\) 288.061 1.33982
\(216\) −14.6969 −0.0680414
\(217\) 225.783i 1.04048i
\(218\) 110.296i 0.505946i
\(219\) −34.1558 −0.155962
\(220\) −74.0510 −0.336595
\(221\) 122.654i 0.554995i
\(222\) 139.472i 0.628253i
\(223\) 290.940 1.30467 0.652333 0.757933i \(-0.273790\pi\)
0.652333 + 0.757933i \(0.273790\pi\)
\(224\) 26.5316i 0.118445i
\(225\) −33.1672 −0.147410
\(226\) 96.6659i 0.427725i
\(227\) 0.735161i 0.00323860i −0.999999 0.00161930i \(-0.999485\pi\)
0.999999 0.00161930i \(-0.000515439\pi\)
\(228\) 50.7393i 0.222541i
\(229\) 145.885i 0.637054i 0.947914 + 0.318527i \(0.103188\pi\)
−0.947914 + 0.318527i \(0.896812\pi\)
\(230\) 176.200 84.2650i 0.766086 0.366369i
\(231\) 50.0914 0.216846
\(232\) 3.06161 0.0131966
\(233\) 225.010 0.965708 0.482854 0.875701i \(-0.339600\pi\)
0.482854 + 0.875701i \(0.339600\pi\)
\(234\) 36.9653 0.157971
\(235\) 44.4321i 0.189073i
\(236\) 175.289 0.742751
\(237\) 80.1498i 0.338185i
\(238\) −93.3743 −0.392329
\(239\) −211.317 −0.884171 −0.442086 0.896973i \(-0.645761\pi\)
−0.442086 + 0.896973i \(0.645761\pi\)
\(240\) 41.6014i 0.173339i
\(241\) 158.875i 0.659231i −0.944115 0.329616i \(-0.893081\pi\)
0.944115 0.329616i \(-0.106919\pi\)
\(242\) 117.350 0.484915
\(243\) −15.5885 −0.0641500
\(244\) 0.0392116i 0.000160703i
\(245\) 162.139i 0.661792i
\(246\) −55.9538 −0.227455
\(247\) 127.618i 0.516671i
\(248\) −136.159 −0.549030
\(249\) 29.4259i 0.118176i
\(250\) 118.413i 0.473650i
\(251\) 322.852i 1.28626i −0.765756 0.643131i \(-0.777635\pi\)
0.765756 0.643131i \(-0.222365\pi\)
\(252\) 28.1410i 0.111671i
\(253\) 127.943 61.1869i 0.505704 0.241846i
\(254\) −252.401 −0.993706
\(255\) 146.410 0.574158
\(256\) 16.0000 0.0625000
\(257\) −349.762 −1.36094 −0.680471 0.732775i \(-0.738225\pi\)
−0.680471 + 0.732775i \(0.738225\pi\)
\(258\) 117.510i 0.455464i
\(259\) 267.055 1.03110
\(260\) 104.634i 0.402440i
\(261\) 3.24733 0.0124419
\(262\) −316.866 −1.20941
\(263\) 262.536i 0.998237i −0.866534 0.499118i \(-0.833657\pi\)
0.866534 0.499118i \(-0.166343\pi\)
\(264\) 30.2078i 0.114424i
\(265\) −518.162 −1.95533
\(266\) 97.1533 0.365238
\(267\) 152.442i 0.570942i
\(268\) 209.787i 0.782789i
\(269\) 76.8118 0.285546 0.142773 0.989755i \(-0.454398\pi\)
0.142773 + 0.989755i \(0.454398\pi\)
\(270\) 44.1249i 0.163426i
\(271\) 371.450 1.37067 0.685333 0.728230i \(-0.259657\pi\)
0.685333 + 0.728230i \(0.259657\pi\)
\(272\) 56.3098i 0.207021i
\(273\) 70.7794i 0.259265i
\(274\) 333.490i 1.21712i
\(275\) 68.1712i 0.247895i
\(276\) −34.3744 71.8777i −0.124545 0.260426i
\(277\) 259.023 0.935102 0.467551 0.883966i \(-0.345136\pi\)
0.467551 + 0.883966i \(0.345136\pi\)
\(278\) 208.676 0.750633
\(279\) −144.419 −0.517631
\(280\) −79.6564 −0.284487
\(281\) 250.908i 0.892912i −0.894806 0.446456i \(-0.852686\pi\)
0.894806 0.446456i \(-0.147314\pi\)
\(282\) −18.1253 −0.0642741
\(283\) 116.602i 0.412021i 0.978550 + 0.206011i \(0.0660481\pi\)
−0.978550 + 0.206011i \(0.933952\pi\)
\(284\) −15.5681 −0.0548174
\(285\) −152.336 −0.534511
\(286\) 75.9777i 0.265656i
\(287\) 107.138i 0.373303i
\(288\) 16.9706 0.0589256
\(289\) 90.8255 0.314275
\(290\) 9.19195i 0.0316964i
\(291\) 67.3754i 0.231531i
\(292\) 39.4397 0.135067
\(293\) 158.185i 0.539881i −0.962877 0.269940i \(-0.912996\pi\)
0.962877 0.269940i \(-0.0870040\pi\)
\(294\) −66.1418 −0.224972
\(295\) 526.275i 1.78398i
\(296\) 161.048i 0.544083i
\(297\) 32.0402i 0.107880i
\(298\) 337.897i 1.13388i
\(299\) 86.4574 + 180.784i 0.289155 + 0.604630i
\(300\) 38.2982 0.127661
\(301\) 225.002 0.747515
\(302\) 277.824 0.919946
\(303\) 9.60611 0.0317033
\(304\) 58.5887i 0.192726i
\(305\) 0.117726 0.000385987
\(306\) 59.7255i 0.195182i
\(307\) −76.3951 −0.248844 −0.124422 0.992229i \(-0.539708\pi\)
−0.124422 + 0.992229i \(0.539708\pi\)
\(308\) −57.8405 −0.187794
\(309\) 70.3545i 0.227685i
\(310\) 408.794i 1.31869i
\(311\) −41.9614 −0.134924 −0.0674620 0.997722i \(-0.521490\pi\)
−0.0674620 + 0.997722i \(0.521490\pi\)
\(312\) −42.6838 −0.136807
\(313\) 148.033i 0.472947i 0.971638 + 0.236474i \(0.0759917\pi\)
−0.971638 + 0.236474i \(0.924008\pi\)
\(314\) 268.316i 0.854509i
\(315\) −84.4884 −0.268217
\(316\) 92.5490i 0.292876i
\(317\) −569.463 −1.79641 −0.898206 0.439574i \(-0.855129\pi\)
−0.898206 + 0.439574i \(0.855129\pi\)
\(318\) 211.375i 0.664701i
\(319\) 6.67450i 0.0209232i
\(320\) 48.0371i 0.150116i
\(321\) 216.236i 0.673633i
\(322\) 137.628 65.8186i 0.427416 0.204406i
\(323\) 206.195 0.638374
\(324\) 18.0000 0.0555556
\(325\) −96.3263 −0.296389
\(326\) −107.929 −0.331069
\(327\) 135.085i 0.413103i
\(328\) 64.6099 0.196981
\(329\) 34.7055i 0.105488i
\(330\) 90.6936 0.274829
\(331\) −222.086 −0.670955 −0.335478 0.942048i \(-0.608898\pi\)
−0.335478 + 0.942048i \(0.608898\pi\)
\(332\) 33.9781i 0.102344i
\(333\) 170.818i 0.512966i
\(334\) −205.909 −0.616495
\(335\) 629.849 1.88015
\(336\) 32.4945i 0.0967097i
\(337\) 601.218i 1.78403i 0.452006 + 0.892015i \(0.350708\pi\)
−0.452006 + 0.892015i \(0.649292\pi\)
\(338\) −131.645 −0.389483
\(339\) 118.391i 0.349236i
\(340\) −169.060 −0.497235
\(341\) 296.836i 0.870487i
\(342\) 62.1427i 0.181704i
\(343\) 356.464i 1.03925i
\(344\) 135.688i 0.394443i
\(345\) −215.800 + 103.203i −0.625507 + 0.299139i
\(346\) −420.445 −1.21516
\(347\) −65.1900 −0.187867 −0.0939337 0.995578i \(-0.529944\pi\)
−0.0939337 + 0.995578i \(0.529944\pi\)
\(348\) −3.74970 −0.0107750
\(349\) −529.604 −1.51749 −0.758746 0.651387i \(-0.774187\pi\)
−0.758746 + 0.651387i \(0.774187\pi\)
\(350\) 73.3316i 0.209519i
\(351\) −45.2730 −0.128983
\(352\) 34.8810i 0.0990937i
\(353\) 403.821 1.14397 0.571984 0.820265i \(-0.306174\pi\)
0.571984 + 0.820265i \(0.306174\pi\)
\(354\) −214.685 −0.606454
\(355\) 46.7405i 0.131664i
\(356\) 176.024i 0.494451i
\(357\) 114.360 0.320335
\(358\) −304.341 −0.850116
\(359\) 244.100i 0.679945i −0.940435 0.339972i \(-0.889582\pi\)
0.940435 0.339972i \(-0.110418\pi\)
\(360\) 50.9511i 0.141531i
\(361\) 146.460 0.405707
\(362\) 115.690i 0.319586i
\(363\) −143.723 −0.395932
\(364\) 81.7290i 0.224530i
\(365\) 118.411i 0.324413i
\(366\) 0.0480242i 0.000131214i
\(367\) 653.421i 1.78044i −0.455532 0.890219i \(-0.650551\pi\)
0.455532 0.890219i \(-0.349449\pi\)
\(368\) 39.6922 + 82.9972i 0.107859 + 0.225536i
\(369\) 68.5292 0.185716
\(370\) 483.519 1.30681
\(371\) −404.731 −1.09092
\(372\) 166.761 0.448281
\(373\) 496.872i 1.33210i 0.745909 + 0.666048i \(0.232016\pi\)
−0.745909 + 0.666048i \(0.767984\pi\)
\(374\) −122.759 −0.328232
\(375\) 145.025i 0.386734i
\(376\) 20.9293 0.0556630
\(377\) 9.43111 0.0250162
\(378\) 34.4656i 0.0911788i
\(379\) 469.320i 1.23831i −0.785269 0.619155i \(-0.787475\pi\)
0.785269 0.619155i \(-0.212525\pi\)
\(380\) 175.902 0.462900
\(381\) 309.127 0.811357
\(382\) 175.873i 0.460401i
\(383\) 135.332i 0.353347i 0.984270 + 0.176673i \(0.0565336\pi\)
−0.984270 + 0.176673i \(0.943466\pi\)
\(384\) −19.5959 −0.0510310
\(385\) 173.656i 0.451054i
\(386\) −489.512 −1.26817
\(387\) 143.919i 0.371885i
\(388\) 77.7984i 0.200511i
\(389\) 455.240i 1.17028i 0.810931 + 0.585142i \(0.198961\pi\)
−0.810931 + 0.585142i \(0.801039\pi\)
\(390\) 128.150i 0.328591i
\(391\) 292.097 139.691i 0.747051 0.357266i
\(392\) 76.3740 0.194832
\(393\) 388.080 0.987481
\(394\) 302.561 0.767922
\(395\) −277.862 −0.703447
\(396\) 36.9969i 0.0934264i
\(397\) 368.101 0.927207 0.463603 0.886043i \(-0.346556\pi\)
0.463603 + 0.886043i \(0.346556\pi\)
\(398\) 529.884i 1.33137i
\(399\) −118.988 −0.298216
\(400\) −44.2229 −0.110557
\(401\) 442.864i 1.10440i 0.833712 + 0.552200i \(0.186212\pi\)
−0.833712 + 0.552200i \(0.813788\pi\)
\(402\) 256.936i 0.639144i
\(403\) −419.431 −1.04077
\(404\) −11.0922 −0.0274559
\(405\) 54.0418i 0.133436i
\(406\) 7.17975i 0.0176841i
\(407\) 351.096 0.862643
\(408\) 68.9651i 0.169032i
\(409\) 120.960 0.295747 0.147873 0.989006i \(-0.452757\pi\)
0.147873 + 0.989006i \(0.452757\pi\)
\(410\) 193.980i 0.473121i
\(411\) 408.441i 0.993773i
\(412\) 81.2384i 0.197181i
\(413\) 411.069i 0.995323i
\(414\) 42.0999 + 88.0318i 0.101691 + 0.212637i
\(415\) 102.013 0.245815
\(416\) 49.2870 0.118478
\(417\) −255.575 −0.612889
\(418\) 127.727 0.305567
\(419\) 546.060i 1.30325i 0.758543 + 0.651623i \(0.225911\pi\)
−0.758543 + 0.651623i \(0.774089\pi\)
\(420\) 97.5588 0.232283
\(421\) 794.995i 1.88835i −0.329447 0.944174i \(-0.606862\pi\)
0.329447 0.944174i \(-0.393138\pi\)
\(422\) −261.852 −0.620503
\(423\) 22.1989 0.0524796
\(424\) 244.075i 0.575648i
\(425\) 155.636i 0.366203i
\(426\) 19.0670 0.0447582
\(427\) 0.0919546 0.000215350
\(428\) 249.688i 0.583383i
\(429\) 93.0533i 0.216907i
\(430\) 407.380 0.947396
\(431\) 461.422i 1.07058i −0.844667 0.535292i \(-0.820201\pi\)
0.844667 0.535292i \(-0.179799\pi\)
\(432\) −20.7846 −0.0481125
\(433\) 583.616i 1.34784i 0.738803 + 0.673921i \(0.235391\pi\)
−0.738803 + 0.673921i \(0.764609\pi\)
\(434\) 319.306i 0.735727i
\(435\) 11.2578i 0.0258800i
\(436\) 155.982i 0.357758i
\(437\) −303.919 + 145.345i −0.695466 + 0.332596i
\(438\) −48.3036 −0.110282
\(439\) 517.764 1.17942 0.589708 0.807617i \(-0.299243\pi\)
0.589708 + 0.807617i \(0.299243\pi\)
\(440\) −104.724 −0.238009
\(441\) 81.0069 0.183689
\(442\) 173.459i 0.392441i
\(443\) −50.3232 −0.113596 −0.0567981 0.998386i \(-0.518089\pi\)
−0.0567981 + 0.998386i \(0.518089\pi\)
\(444\) 197.243i 0.444242i
\(445\) −528.482 −1.18760
\(446\) 411.452 0.922538
\(447\) 413.837i 0.925811i
\(448\) 37.5214i 0.0837531i
\(449\) −495.693 −1.10399 −0.551997 0.833846i \(-0.686134\pi\)
−0.551997 + 0.833846i \(0.686134\pi\)
\(450\) −46.9055 −0.104234
\(451\) 140.854i 0.312314i
\(452\) 136.706i 0.302448i
\(453\) −340.263 −0.751133
\(454\) 1.03968i 0.00229003i
\(455\) −245.377 −0.539289
\(456\) 71.7562i 0.157360i
\(457\) 184.249i 0.403170i 0.979471 + 0.201585i \(0.0646092\pi\)
−0.979471 + 0.201585i \(0.935391\pi\)
\(458\) 206.313i 0.450465i
\(459\) 73.1486i 0.159365i
\(460\) 249.184 119.169i 0.541705 0.259062i
\(461\) 160.194 0.347491 0.173746 0.984791i \(-0.444413\pi\)
0.173746 + 0.984791i \(0.444413\pi\)
\(462\) 70.8399 0.153333
\(463\) −458.195 −0.989621 −0.494811 0.869001i \(-0.664763\pi\)
−0.494811 + 0.869001i \(0.664763\pi\)
\(464\) 4.32977 0.00933141
\(465\) 500.669i 1.07671i
\(466\) 318.212 0.682859
\(467\) 804.257i 1.72218i 0.508455 + 0.861089i \(0.330217\pi\)
−0.508455 + 0.861089i \(0.669783\pi\)
\(468\) 52.2768 0.111702
\(469\) 491.969 1.04898
\(470\) 62.8364i 0.133695i
\(471\) 328.618i 0.697703i
\(472\) 247.897 0.525205
\(473\) 295.809 0.625389
\(474\) 113.349i 0.239133i
\(475\) 161.935i 0.340916i
\(476\) −132.051 −0.277419
\(477\) 258.881i 0.542726i
\(478\) −298.847 −0.625204
\(479\) 396.331i 0.827414i −0.910410 0.413707i \(-0.864234\pi\)
0.910410 0.413707i \(-0.135766\pi\)
\(480\) 58.8332i 0.122569i
\(481\) 496.100i 1.03139i
\(482\) 224.683i 0.466147i
\(483\) −168.559 + 80.6110i −0.348984 + 0.166896i
\(484\) 165.957 0.342887
\(485\) −233.576 −0.481600
\(486\) −22.0454 −0.0453609
\(487\) −636.665 −1.30732 −0.653660 0.756789i \(-0.726767\pi\)
−0.653660 + 0.756789i \(0.726767\pi\)
\(488\) 0.0554536i 0.000113634i
\(489\) 132.185 0.270317
\(490\) 229.299i 0.467958i
\(491\) 928.442 1.89092 0.945461 0.325736i \(-0.105612\pi\)
0.945461 + 0.325736i \(0.105612\pi\)
\(492\) −79.1307 −0.160835
\(493\) 15.2380i 0.0309088i
\(494\) 180.479i 0.365342i
\(495\) −111.076 −0.224397
\(496\) −192.559 −0.388223
\(497\) 36.5086i 0.0734580i
\(498\) 41.6144i 0.0835631i
\(499\) 457.999 0.917834 0.458917 0.888479i \(-0.348238\pi\)
0.458917 + 0.888479i \(0.348238\pi\)
\(500\) 167.461i 0.334921i
\(501\) 252.186 0.503366
\(502\) 456.581i 0.909524i
\(503\) 9.28932i 0.0184678i −0.999957 0.00923392i \(-0.997061\pi\)
0.999957 0.00923392i \(-0.00293929\pi\)
\(504\) 39.7974i 0.0789631i
\(505\) 33.3023i 0.0659451i
\(506\) 180.939 86.5314i 0.357587 0.171011i
\(507\) 161.232 0.318011
\(508\) −356.949 −0.702656
\(509\) −294.128 −0.577855 −0.288928 0.957351i \(-0.593299\pi\)
−0.288928 + 0.957351i \(0.593299\pi\)
\(510\) 207.055 0.405991
\(511\) 92.4895i 0.180997i
\(512\) 22.6274 0.0441942
\(513\) 76.1090i 0.148361i
\(514\) −494.638 −0.962331
\(515\) −243.904 −0.473600
\(516\) 166.184i 0.322061i
\(517\) 45.6271i 0.0882536i
\(518\) 377.672 0.729097
\(519\) 514.938 0.992174
\(520\) 147.975i 0.284568i
\(521\) 235.388i 0.451801i −0.974150 0.225901i \(-0.927468\pi\)
0.974150 0.225901i \(-0.0725325\pi\)
\(522\) 4.59242 0.00879774
\(523\) 512.874i 0.980639i 0.871543 + 0.490319i \(0.163120\pi\)
−0.871543 + 0.490319i \(0.836880\pi\)
\(524\) −448.116 −0.855184
\(525\) 89.8125i 0.171071i
\(526\) 371.282i 0.705860i
\(527\) 677.683i 1.28593i
\(528\) 42.7203i 0.0809097i
\(529\) −332.066 + 411.792i −0.627725 + 0.778435i
\(530\) −732.791 −1.38262
\(531\) 262.934 0.495168
\(532\) 137.396 0.258262
\(533\) 199.027 0.373409
\(534\) 215.585i 0.403717i
\(535\) 749.643 1.40120
\(536\) 296.684i 0.553515i
\(537\) 372.741 0.694117
\(538\) 108.628 0.201911
\(539\) 166.500i 0.308905i
\(540\) 62.4021i 0.115559i
\(541\) −221.929 −0.410220 −0.205110 0.978739i \(-0.565755\pi\)
−0.205110 + 0.978739i \(0.565755\pi\)
\(542\) 525.310 0.969207
\(543\) 141.691i 0.260941i
\(544\) 79.6341i 0.146386i
\(545\) −468.309 −0.859283
\(546\) 100.097i 0.183328i
\(547\) 378.530 0.692011 0.346006 0.938232i \(-0.387538\pi\)
0.346006 + 0.938232i \(0.387538\pi\)
\(548\) 471.627i 0.860632i
\(549\) 0.0588174i 0.000107136i
\(550\) 96.4086i 0.175288i
\(551\) 15.8547i 0.0287745i
\(552\) −48.6128 101.650i −0.0880666 0.184149i
\(553\) −217.035 −0.392469
\(554\) 366.314 0.661217
\(555\) −592.188 −1.06700
\(556\) 295.112 0.530778
\(557\) 256.556i 0.460603i 0.973119 + 0.230301i \(0.0739712\pi\)
−0.973119 + 0.230301i \(0.926029\pi\)
\(558\) −204.239 −0.366020
\(559\) 417.980i 0.747728i
\(560\) −112.651 −0.201163
\(561\) 150.348 0.268000
\(562\) 354.838i 0.631384i
\(563\) 298.314i 0.529864i −0.964267 0.264932i \(-0.914650\pi\)
0.964267 0.264932i \(-0.0853496\pi\)
\(564\) −25.6330 −0.0454486
\(565\) −410.436 −0.726436
\(566\) 164.900i 0.291343i
\(567\) 42.2115i 0.0744472i
\(568\) −22.0167 −0.0387617
\(569\) 361.429i 0.635200i −0.948225 0.317600i \(-0.897123\pi\)
0.948225 0.317600i \(-0.102877\pi\)
\(570\) −215.435 −0.377956
\(571\) 898.203i 1.57303i 0.617568 + 0.786517i \(0.288118\pi\)
−0.617568 + 0.786517i \(0.711882\pi\)
\(572\) 107.449i 0.187847i
\(573\) 215.400i 0.375916i
\(574\) 151.516i 0.263965i
\(575\) −109.706 229.399i −0.190794 0.398954i
\(576\) 24.0000 0.0416667
\(577\) −573.615 −0.994133 −0.497067 0.867712i \(-0.665590\pi\)
−0.497067 + 0.867712i \(0.665590\pi\)
\(578\) 128.447 0.222226
\(579\) 599.527 1.03545
\(580\) 12.9994i 0.0224127i
\(581\) 79.6814 0.137145
\(582\) 95.2832i 0.163717i
\(583\) −532.098 −0.912690
\(584\) 55.7762 0.0955071
\(585\) 156.952i 0.268293i
\(586\) 223.707i 0.381753i
\(587\) −223.626 −0.380965 −0.190482 0.981691i \(-0.561005\pi\)
−0.190482 + 0.981691i \(0.561005\pi\)
\(588\) −93.5387 −0.159079
\(589\) 705.110i 1.19713i
\(590\) 744.265i 1.26147i
\(591\) −370.560 −0.627006
\(592\) 227.757i 0.384725i
\(593\) −278.419 −0.469509 −0.234754 0.972055i \(-0.575429\pi\)
−0.234754 + 0.972055i \(0.575429\pi\)
\(594\) 45.3117i 0.0762824i
\(595\) 396.460i 0.666320i
\(596\) 477.858i 0.801775i
\(597\) 648.972i 1.08706i
\(598\) 122.269 + 255.668i 0.204464 + 0.427538i
\(599\) −292.918 −0.489012 −0.244506 0.969648i \(-0.578626\pi\)
−0.244506 + 0.969648i \(0.578626\pi\)
\(600\) 54.1618 0.0902696
\(601\) 39.6741 0.0660135 0.0330067 0.999455i \(-0.489492\pi\)
0.0330067 + 0.999455i \(0.489492\pi\)
\(602\) 318.201 0.528573
\(603\) 314.681i 0.521859i
\(604\) 392.902 0.650500
\(605\) 498.257i 0.823566i
\(606\) 13.5851 0.0224176
\(607\) 120.120 0.197892 0.0989458 0.995093i \(-0.468453\pi\)
0.0989458 + 0.995093i \(0.468453\pi\)
\(608\) 82.8569i 0.136278i
\(609\) 8.79336i 0.0144390i
\(610\) 0.166490 0.000272934
\(611\) 64.4714 0.105518
\(612\) 84.4647i 0.138014i
\(613\) 414.040i 0.675432i 0.941248 + 0.337716i \(0.109654\pi\)
−0.941248 + 0.337716i \(0.890346\pi\)
\(614\) −108.039 −0.175959
\(615\) 237.576i 0.386302i
\(616\) −81.7989 −0.132790
\(617\) 154.341i 0.250147i 0.992147 + 0.125073i \(0.0399167\pi\)
−0.992147 + 0.125073i \(0.960083\pi\)
\(618\) 99.4963i 0.160997i
\(619\) 879.766i 1.42127i −0.703561 0.710635i \(-0.748408\pi\)
0.703561 0.710635i \(-0.251592\pi\)
\(620\) 578.123i 0.932456i
\(621\) −51.5616 107.816i −0.0830300 0.173618i
\(622\) −59.3423 −0.0954057
\(623\) −412.792 −0.662588
\(624\) −60.3640 −0.0967372
\(625\) −779.164 −1.24666
\(626\) 209.350i 0.334424i
\(627\) −156.433 −0.249494
\(628\) 379.456i 0.604229i
\(629\) 801.559 1.27434
\(630\) −119.485 −0.189658
\(631\) 211.786i 0.335636i −0.985818 0.167818i \(-0.946328\pi\)
0.985818 0.167818i \(-0.0536720\pi\)
\(632\) 130.884i 0.207095i
\(633\) 320.702 0.506639
\(634\) −805.342 −1.27026
\(635\) 1071.68i 1.68768i
\(636\) 298.929i 0.470015i
\(637\) 235.265 0.369334
\(638\) 9.43917i 0.0147949i
\(639\) −23.3522 −0.0365449
\(640\) 67.9348i 0.106148i
\(641\) 106.224i 0.165715i −0.996561 0.0828576i \(-0.973595\pi\)
0.996561 0.0828576i \(-0.0264047\pi\)
\(642\) 305.804i 0.476330i
\(643\) 970.054i 1.50864i 0.656508 + 0.754319i \(0.272033\pi\)
−0.656508 + 0.754319i \(0.727967\pi\)
\(644\) 194.635 93.0815i 0.302229 0.144537i
\(645\) −498.937 −0.773546
\(646\) 291.603 0.451399
\(647\) −297.519 −0.459844 −0.229922 0.973209i \(-0.573847\pi\)
−0.229922 + 0.973209i \(0.573847\pi\)
\(648\) 25.4558 0.0392837
\(649\) 540.430i 0.832711i
\(650\) −136.226 −0.209578
\(651\) 391.068i 0.600719i
\(652\) −152.634 −0.234101
\(653\) 1191.05 1.82396 0.911981 0.410232i \(-0.134552\pi\)
0.911981 + 0.410232i \(0.134552\pi\)
\(654\) 191.039i 0.292108i
\(655\) 1345.39i 2.05403i
\(656\) 91.3722 0.139287
\(657\) 59.1596 0.0900450
\(658\) 49.0810i 0.0745911i
\(659\) 164.114i 0.249034i −0.992217 0.124517i \(-0.960262\pi\)
0.992217 0.124517i \(-0.0397382\pi\)
\(660\) 128.260 0.194333
\(661\) 221.664i 0.335346i −0.985843 0.167673i \(-0.946375\pi\)
0.985843 0.167673i \(-0.0536253\pi\)
\(662\) −314.077 −0.474437
\(663\) 212.443i 0.320427i
\(664\) 48.0522i 0.0723678i
\(665\) 412.505i 0.620309i
\(666\) 241.573i 0.362722i
\(667\) 10.7411 + 22.4599i 0.0161037 + 0.0336731i
\(668\) −291.200 −0.435928
\(669\) −503.924 −0.753249
\(670\) 890.741 1.32946
\(671\) 0.120892 0.000180167
\(672\) 45.9541i 0.0683841i
\(673\) 1194.29 1.77457 0.887286 0.461219i \(-0.152588\pi\)
0.887286 + 0.461219i \(0.152588\pi\)
\(674\) 850.251i 1.26150i
\(675\) 57.4472 0.0851070
\(676\) −186.174 −0.275406
\(677\) 1102.28i 1.62819i 0.580732 + 0.814095i \(0.302767\pi\)
−0.580732 + 0.814095i \(0.697233\pi\)
\(678\) 167.430i 0.246947i
\(679\) −182.444 −0.268695
\(680\) −239.087 −0.351599
\(681\) 1.27334i 0.00186980i
\(682\) 419.790i 0.615527i
\(683\) 305.671 0.447542 0.223771 0.974642i \(-0.428163\pi\)
0.223771 + 0.974642i \(0.428163\pi\)
\(684\) 87.8830i 0.128484i
\(685\) 1415.97 2.06712
\(686\) 504.116i 0.734863i
\(687\) 252.681i 0.367803i
\(688\) 191.892i 0.278913i
\(689\) 751.858i 1.09123i
\(690\) −305.187 + 145.951i −0.442300 + 0.211523i
\(691\) −471.859 −0.682863 −0.341432 0.939907i \(-0.610912\pi\)
−0.341432 + 0.939907i \(0.610912\pi\)
\(692\) −594.599 −0.859247
\(693\) −86.7608 −0.125196
\(694\) −92.1926 −0.132842
\(695\) 886.022i 1.27485i
\(696\) −5.30287 −0.00761907
\(697\) 321.572i 0.461366i
\(698\) −748.974 −1.07303
\(699\) −389.729 −0.557552
\(700\) 103.707i 0.148152i
\(701\) 77.3101i 0.110285i −0.998478 0.0551427i \(-0.982439\pi\)
0.998478 0.0551427i \(-0.0175614\pi\)
\(702\) −64.0257 −0.0912047
\(703\) −833.999 −1.18634
\(704\) 49.3292i 0.0700698i
\(705\) 76.9586i 0.109161i
\(706\) 571.088 0.808907
\(707\) 26.0121i 0.0367923i
\(708\) −303.610 −0.428828
\(709\) 474.562i 0.669341i −0.942335 0.334670i \(-0.891375\pi\)
0.942335 0.334670i \(-0.108625\pi\)
\(710\) 66.1011i 0.0931002i
\(711\) 138.823i 0.195251i
\(712\) 248.936i 0.349629i
\(713\) −477.692 998.864i −0.669974 1.40093i
\(714\) 161.729 0.226511
\(715\) −322.595 −0.451183
\(716\) −430.404 −0.601123
\(717\) 366.012 0.510477
\(718\) 345.210i 0.480794i
\(719\) −286.574 −0.398574 −0.199287 0.979941i \(-0.563863\pi\)
−0.199287 + 0.979941i \(0.563863\pi\)
\(720\) 72.0557i 0.100077i
\(721\) −190.511 −0.264232
\(722\) 207.126 0.286878
\(723\) 275.179i 0.380607i
\(724\) 163.610i 0.225981i
\(725\) −11.9672 −0.0165065
\(726\) −203.255 −0.279966
\(727\) 189.093i 0.260101i 0.991507 + 0.130050i \(0.0415139\pi\)
−0.991507 + 0.130050i \(0.958486\pi\)
\(728\) 115.582i 0.158767i
\(729\) 27.0000 0.0370370
\(730\) 167.458i 0.229394i
\(731\) 675.339 0.923856
\(732\) 0.0679165i 9.27821e-5i
\(733\) 777.152i 1.06023i −0.847924 0.530117i \(-0.822148\pi\)
0.847924 0.530117i \(-0.177852\pi\)
\(734\) 924.077i 1.25896i
\(735\) 280.833i 0.382086i
\(736\) 56.1332 + 117.376i 0.0762680 + 0.159478i
\(737\) 646.790 0.877598
\(738\) 96.9149 0.131321
\(739\) −343.771 −0.465184 −0.232592 0.972574i \(-0.574721\pi\)
−0.232592 + 0.972574i \(0.574721\pi\)
\(740\) 683.799 0.924053
\(741\) 221.041i 0.298300i
\(742\) −572.376 −0.771397
\(743\) 560.485i 0.754354i −0.926141 0.377177i \(-0.876895\pi\)
0.926141 0.377177i \(-0.123105\pi\)
\(744\) 235.835 0.316983
\(745\) 1434.68 1.92575
\(746\) 702.683i 0.941935i
\(747\) 50.9671i 0.0682290i
\(748\) −173.607 −0.232095
\(749\) 585.539 0.781762
\(750\) 205.097i 0.273462i
\(751\) 1220.79i 1.62555i −0.582576 0.812776i \(-0.697955\pi\)
0.582576 0.812776i \(-0.302045\pi\)
\(752\) 29.5985 0.0393597
\(753\) 559.195i 0.742623i
\(754\) 13.3376 0.0176891
\(755\) 1179.62i 1.56241i
\(756\) 48.7417i 0.0644731i
\(757\) 595.690i 0.786908i −0.919344 0.393454i \(-0.871280\pi\)
0.919344 0.393454i \(-0.128720\pi\)
\(758\) 663.718i 0.875618i
\(759\) −221.604 + 105.979i −0.291968 + 0.139630i
\(760\) 248.763 0.327320
\(761\) −387.684 −0.509440 −0.254720 0.967015i \(-0.581983\pi\)
−0.254720 + 0.967015i \(0.581983\pi\)
\(762\) 437.172 0.573716
\(763\) −365.792 −0.479413
\(764\) 248.722i 0.325553i
\(765\) −253.590 −0.331490
\(766\) 191.388i 0.249854i
\(767\) 763.630 0.995606
\(768\) −27.7128 −0.0360844
\(769\) 487.155i 0.633492i −0.948510 0.316746i \(-0.897410\pi\)
0.948510 0.316746i \(-0.102590\pi\)
\(770\) 245.587i 0.318944i
\(771\) 605.805 0.785740
\(772\) −692.274 −0.896729
\(773\) 1216.60i 1.57386i −0.617041 0.786931i \(-0.711669\pi\)
0.617041 0.786931i \(-0.288331\pi\)
\(774\) 203.533i 0.262962i
\(775\) 532.219 0.686734
\(776\) 110.024i 0.141783i
\(777\) −462.552 −0.595306
\(778\) 643.807i 0.827515i
\(779\) 334.586i 0.429507i
\(780\) 181.232i 0.232349i
\(781\) 47.9977i 0.0614567i
\(782\) 413.088 197.553i 0.528245 0.252625i
\(783\) −5.62454 −0.00718332
\(784\) 108.009 0.137767
\(785\) 1139.25 1.45127
\(786\) 548.828 0.698255
\(787\) 169.922i 0.215911i 0.994156 + 0.107955i \(0.0344303\pi\)
−0.994156 + 0.107955i \(0.965570\pi\)
\(788\) 427.886 0.543003
\(789\) 454.726i 0.576332i
\(790\) −392.956 −0.497412
\(791\) −320.588 −0.405294
\(792\) 52.3215i 0.0660625i
\(793\) 0.170821i 0.000215412i
\(794\) 520.574 0.655634
\(795\) 897.482 1.12891
\(796\) 749.369i 0.941418i
\(797\) 199.739i 0.250614i −0.992118 0.125307i \(-0.960008\pi\)
0.992118 0.125307i \(-0.0399916\pi\)
\(798\) −168.274 −0.210870
\(799\) 104.168i 0.130373i
\(800\) −62.5406 −0.0781758
\(801\) 264.037i 0.329634i
\(802\) 626.305i 0.780929i
\(803\) 121.595i 0.151426i
\(804\) 363.362i 0.451943i
\(805\) −279.461 584.358i −0.347156 0.725911i
\(806\) −593.165 −0.735936
\(807\) −133.042 −0.164860
\(808\) −15.6867 −0.0194143
\(809\) 1223.51 1.51238 0.756188 0.654354i \(-0.227059\pi\)
0.756188 + 0.654354i \(0.227059\pi\)
\(810\) 76.4266i 0.0943538i
\(811\) −825.239 −1.01756 −0.508779 0.860897i \(-0.669903\pi\)
−0.508779 + 0.860897i \(0.669903\pi\)
\(812\) 10.1537i 0.0125045i
\(813\) −643.371 −0.791354
\(814\) 496.524 0.609980
\(815\) 458.256i 0.562278i
\(816\) 97.5314i 0.119524i
\(817\) −702.671 −0.860062
\(818\) 171.064 0.209125
\(819\) 122.593i 0.149687i
\(820\) 274.329i 0.334547i
\(821\) −552.657 −0.673151 −0.336576 0.941656i \(-0.609269\pi\)
−0.336576 + 0.941656i \(0.609269\pi\)
\(822\) 577.622i 0.702703i
\(823\) 1203.74 1.46262 0.731310 0.682046i \(-0.238909\pi\)
0.731310 + 0.682046i \(0.238909\pi\)
\(824\) 114.888i 0.139428i
\(825\) 118.076i 0.143122i
\(826\) 581.339i 0.703800i
\(827\) 332.187i 0.401677i −0.979624 0.200839i \(-0.935633\pi\)
0.979624 0.200839i \(-0.0643667\pi\)
\(828\) 59.5383 + 124.496i 0.0719061 + 0.150357i
\(829\) 907.749 1.09499 0.547496 0.836808i \(-0.315581\pi\)
0.547496 + 0.836808i \(0.315581\pi\)
\(830\) 144.268 0.173817
\(831\) −448.642 −0.539882
\(832\) 69.7024 0.0837769
\(833\) 380.123i 0.456331i
\(834\) −361.437 −0.433378
\(835\) 874.275i 1.04704i
\(836\) 180.633 0.216068
\(837\) 250.141 0.298854
\(838\) 772.245i 0.921534i
\(839\) 35.1836i 0.0419352i −0.999780 0.0209676i \(-0.993325\pi\)
0.999780 0.0209676i \(-0.00667469\pi\)
\(840\) 137.969 0.164249
\(841\) −839.828 −0.998607
\(842\) 1124.29i 1.33526i
\(843\) 434.586i 0.515523i
\(844\) −370.315 −0.438762
\(845\) 558.955i 0.661486i
\(846\) 31.3939 0.0371087
\(847\) 389.184i 0.459485i
\(848\) 345.174i 0.407045i
\(849\) 201.961i 0.237881i
\(850\) 220.103i 0.258945i
\(851\) −1181.45 + 565.010i −1.38831 + 0.663937i
\(852\) 26.9648 0.0316488
\(853\) −1236.74 −1.44987 −0.724933 0.688819i \(-0.758129\pi\)
−0.724933 + 0.688819i \(0.758129\pi\)
\(854\) 0.130043 0.000152276
\(855\) 263.853 0.308600
\(856\) 353.112i 0.412514i
\(857\) −1604.20 −1.87188 −0.935938 0.352166i \(-0.885445\pi\)
−0.935938 + 0.352166i \(0.885445\pi\)
\(858\) 131.597i 0.153377i
\(859\) −465.805 −0.542265 −0.271132 0.962542i \(-0.587398\pi\)
−0.271132 + 0.962542i \(0.587398\pi\)
\(860\) 576.123 0.669910
\(861\) 185.568i 0.215526i
\(862\) 652.549i 0.757018i
\(863\) 703.235 0.814873 0.407437 0.913234i \(-0.366423\pi\)
0.407437 + 0.913234i \(0.366423\pi\)
\(864\) −29.3939 −0.0340207
\(865\) 1785.18i 2.06379i
\(866\) 825.358i 0.953069i
\(867\) −157.314 −0.181447
\(868\) 451.566i 0.520238i
\(869\) −285.335 −0.328349
\(870\) 15.9209i 0.0182999i
\(871\) 913.917i 1.04927i
\(872\) 220.592i 0.252973i
\(873\) 116.698i 0.133674i
\(874\) −429.806 + 205.548i −0.491769 + 0.235181i
\(875\) −392.710 −0.448811
\(876\) −68.3116 −0.0779812
\(877\) 1178.27 1.34352 0.671762 0.740767i \(-0.265538\pi\)
0.671762 + 0.740767i \(0.265538\pi\)
\(878\) 732.228 0.833973
\(879\) 273.984i 0.311700i
\(880\) −148.102 −0.168298
\(881\) 1220.58i 1.38545i 0.721203 + 0.692724i \(0.243590\pi\)
−0.721203 + 0.692724i \(0.756410\pi\)
\(882\) 114.561 0.129888
\(883\) 399.783 0.452756 0.226378 0.974040i \(-0.427312\pi\)
0.226378 + 0.974040i \(0.427312\pi\)
\(884\) 245.308i 0.277497i
\(885\) 911.535i 1.02998i
\(886\) −71.1677 −0.0803247
\(887\) −1467.55 −1.65451 −0.827256 0.561825i \(-0.810099\pi\)
−0.827256 + 0.561825i \(0.810099\pi\)
\(888\) 278.944i 0.314126i
\(889\) 837.077i 0.941593i
\(890\) −747.386 −0.839760
\(891\) 55.4953i 0.0622843i
\(892\) 581.881 0.652333
\(893\) 108.384i 0.121370i
\(894\) 585.254i 0.654647i
\(895\) 1292.21i 1.44381i
\(896\) 53.0632i 0.0592224i
\(897\) −149.749 313.128i −0.166944 0.349083i
\(898\) −701.016 −0.780641
\(899\) −52.1085 −0.0579627
\(900\) −66.3344 −0.0737048
\(901\) −1214.79 −1.34827
\(902\) 199.197i 0.220839i
\(903\) −389.715 −0.431578
\(904\) 193.332i 0.213863i
\(905\) 491.211 0.542775
\(906\) −481.205 −0.531131
\(907\) 0.658645i 0.000726179i 1.00000 0.000363090i \(0.000115575\pi\)
−1.00000 0.000363090i \(0.999884\pi\)
\(908\) 1.47032i 0.00161930i
\(909\) −16.6383 −0.0183039
\(910\) −347.015 −0.381335
\(911\) 912.515i 1.00166i 0.865545 + 0.500832i \(0.166972\pi\)
−0.865545 + 0.500832i \(0.833028\pi\)
\(912\) 101.479i 0.111270i
\(913\) 104.757 0.114739
\(914\) 260.567i 0.285084i
\(915\) −0.203907 −0.000222849
\(916\) 291.771i 0.318527i
\(917\) 1050.87i 1.14599i
\(918\) 103.448i 0.112688i
\(919\) 536.529i 0.583818i 0.956446 + 0.291909i \(0.0942904\pi\)
−0.956446 + 0.291909i \(0.905710\pi\)
\(920\) 352.400 168.530i 0.383043 0.183185i
\(921\) 132.320 0.143670
\(922\) 226.548 0.245714
\(923\) −67.8210 −0.0734789
\(924\) 100.183 0.108423
\(925\) 629.505i 0.680546i
\(926\) −647.985 −0.699768
\(927\) 121.858i 0.131454i
\(928\) 6.12323 0.00659830
\(929\) −1075.21 −1.15738 −0.578691 0.815547i \(-0.696436\pi\)
−0.578691 + 0.815547i \(0.696436\pi\)
\(930\) 708.053i 0.761347i
\(931\) 395.507i 0.424820i
\(932\) 450.020 0.482854
\(933\) 72.6792 0.0778984
\(934\) 1137.39i 1.21776i
\(935\) 521.224i 0.557459i
\(936\) 73.9305 0.0789856
\(937\) 429.773i 0.458669i −0.973348 0.229335i \(-0.926345\pi\)
0.973348 0.229335i \(-0.0736550\pi\)
\(938\) 695.750 0.741738
\(939\) 256.400i 0.273056i
\(940\) 88.8641i 0.0945363i
\(941\) 746.559i 0.793367i 0.917955 + 0.396684i \(0.129839\pi\)
−0.917955 + 0.396684i \(0.870161\pi\)
\(942\) 464.737i 0.493351i
\(943\) 226.673 + 473.977i 0.240374 + 0.502627i
\(944\) 350.579 0.371376
\(945\) 146.338 0.154855
\(946\) 418.337 0.442217
\(947\) 472.748 0.499206 0.249603 0.968348i \(-0.419700\pi\)
0.249603 + 0.968348i \(0.419700\pi\)
\(948\) 160.300i 0.169092i
\(949\) 171.815 0.181048
\(950\) 229.011i 0.241064i
\(951\) 986.338 1.03716
\(952\) −186.749 −0.196165
\(953\) 1583.70i 1.66181i −0.556417 0.830903i \(-0.687824\pi\)
0.556417 0.830903i \(-0.312176\pi\)
\(954\) 366.112i 0.383766i
\(955\) 746.744 0.781931
\(956\) −422.634 −0.442086
\(957\) 11.5606i 0.0120800i
\(958\) 560.497i 0.585070i
\(959\) 1106.00 1.15329
\(960\) 83.2028i 0.0866695i
\(961\) 1356.43 1.41147
\(962\) 701.591i 0.729305i
\(963\) 374.532i 0.388922i
\(964\) 317.750i 0.329616i
\(965\) 2078.43i 2.15381i
\(966\) −238.379 + 114.001i −0.246769 + 0.118014i
\(967\) 656.660 0.679069 0.339535 0.940594i \(-0.389730\pi\)
0.339535 + 0.940594i \(0.389730\pi\)
\(968\) 234.699 0.242458
\(969\) −357.140 −0.368565
\(970\) −330.326 −0.340542
\(971\) 963.625i 0.992405i 0.868207 + 0.496202i \(0.165273\pi\)
−0.868207 + 0.496202i \(0.834727\pi\)
\(972\) −31.1769 −0.0320750
\(973\) 692.064i 0.711268i
\(974\) −900.380 −0.924414
\(975\) 166.842 0.171120
\(976\) 0.0784233i 8.03517e-5i
\(977\) 313.286i 0.320661i 0.987063 + 0.160330i \(0.0512560\pi\)
−0.987063 + 0.160330i \(0.948744\pi\)
\(978\) 186.938 0.191143
\(979\) −542.696 −0.554337
\(980\) 324.278i 0.330896i
\(981\) 233.974i 0.238505i
\(982\) 1313.02 1.33708
\(983\) 671.993i 0.683614i −0.939770 0.341807i \(-0.888961\pi\)
0.939770 0.341807i \(-0.111039\pi\)
\(984\) −111.908 −0.113727
\(985\) 1284.65i 1.30422i
\(986\) 21.5498i 0.0218558i
\(987\) 60.1117i 0.0609034i
\(988\) 255.236i 0.258336i
\(989\) −995.408 + 476.039i −1.00648 + 0.481334i
\(990\) −157.086 −0.158673
\(991\) −1319.81 −1.33179 −0.665896 0.746044i \(-0.731951\pi\)
−0.665896 + 0.746044i \(0.731951\pi\)
\(992\) −272.319 −0.274515
\(993\) 384.665 0.387376
\(994\) 51.6310i 0.0519426i
\(995\) −2249.84 −2.26115
\(996\) 58.8517i 0.0590881i
\(997\) −828.619 −0.831112 −0.415556 0.909568i \(-0.636413\pi\)
−0.415556 + 0.909568i \(0.636413\pi\)
\(998\) 647.708 0.649006
\(999\) 295.865i 0.296161i
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 138.3.b.a.91.5 8
3.2 odd 2 414.3.b.c.91.4 8
4.3 odd 2 1104.3.c.c.1057.5 8
23.22 odd 2 inner 138.3.b.a.91.6 yes 8
69.68 even 2 414.3.b.c.91.1 8
92.91 even 2 1104.3.c.c.1057.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.3.b.a.91.5 8 1.1 even 1 trivial
138.3.b.a.91.6 yes 8 23.22 odd 2 inner
414.3.b.c.91.1 8 69.68 even 2
414.3.b.c.91.4 8 3.2 odd 2
1104.3.c.c.1057.5 8 4.3 odd 2
1104.3.c.c.1057.8 8 92.91 even 2