Properties

Label 4015.2.a.f.1.19
Level $4015$
Weight $2$
Character 4015.1
Self dual yes
Analytic conductor $32.060$
Analytic rank $1$
Dimension $31$
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] = [4015,2,Mod(1,4015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4015, 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("4015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4015 = 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4015.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: \(32.0599364115\)
Analytic rank: \(1\)
Dimension: \(31\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 4015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.416759 q^{2} +1.40268 q^{3} -1.82631 q^{4} -1.00000 q^{5} +0.584579 q^{6} -1.52897 q^{7} -1.59465 q^{8} -1.03249 q^{9} +O(q^{10})\) \(q+0.416759 q^{2} +1.40268 q^{3} -1.82631 q^{4} -1.00000 q^{5} +0.584579 q^{6} -1.52897 q^{7} -1.59465 q^{8} -1.03249 q^{9} -0.416759 q^{10} +1.00000 q^{11} -2.56173 q^{12} +2.05922 q^{13} -0.637213 q^{14} -1.40268 q^{15} +2.98804 q^{16} +6.91820 q^{17} -0.430297 q^{18} -1.02478 q^{19} +1.82631 q^{20} -2.14466 q^{21} +0.416759 q^{22} -0.555659 q^{23} -2.23678 q^{24} +1.00000 q^{25} +0.858198 q^{26} -5.65629 q^{27} +2.79238 q^{28} +8.69161 q^{29} -0.584579 q^{30} -9.18767 q^{31} +4.43459 q^{32} +1.40268 q^{33} +2.88322 q^{34} +1.52897 q^{35} +1.88564 q^{36} -10.0732 q^{37} -0.427087 q^{38} +2.88843 q^{39} +1.59465 q^{40} +3.73463 q^{41} -0.893806 q^{42} -7.10211 q^{43} -1.82631 q^{44} +1.03249 q^{45} -0.231576 q^{46} +12.0692 q^{47} +4.19127 q^{48} -4.66224 q^{49} +0.416759 q^{50} +9.70403 q^{51} -3.76078 q^{52} -9.47032 q^{53} -2.35731 q^{54} -1.00000 q^{55} +2.43817 q^{56} -1.43744 q^{57} +3.62230 q^{58} -10.1112 q^{59} +2.56173 q^{60} -3.95339 q^{61} -3.82904 q^{62} +1.57864 q^{63} -4.12793 q^{64} -2.05922 q^{65} +0.584579 q^{66} -1.23353 q^{67} -12.6348 q^{68} -0.779412 q^{69} +0.637213 q^{70} -3.25447 q^{71} +1.64645 q^{72} +1.00000 q^{73} -4.19811 q^{74} +1.40268 q^{75} +1.87157 q^{76} -1.52897 q^{77} +1.20378 q^{78} -8.38322 q^{79} -2.98804 q^{80} -4.83652 q^{81} +1.55644 q^{82} +15.2456 q^{83} +3.91682 q^{84} -6.91820 q^{85} -2.95986 q^{86} +12.1916 q^{87} -1.59465 q^{88} -6.12001 q^{89} +0.430297 q^{90} -3.14849 q^{91} +1.01481 q^{92} -12.8874 q^{93} +5.02996 q^{94} +1.02478 q^{95} +6.22031 q^{96} -12.1941 q^{97} -1.94303 q^{98} -1.03249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.416759 0.294693 0.147346 0.989085i \(-0.452927\pi\)
0.147346 + 0.989085i \(0.452927\pi\)
\(3\) 1.40268 0.809838 0.404919 0.914352i \(-0.367300\pi\)
0.404919 + 0.914352i \(0.367300\pi\)
\(4\) −1.82631 −0.913156
\(5\) −1.00000 −0.447214
\(6\) 0.584579 0.238654
\(7\) −1.52897 −0.577898 −0.288949 0.957345i \(-0.593306\pi\)
−0.288949 + 0.957345i \(0.593306\pi\)
\(8\) −1.59465 −0.563793
\(9\) −1.03249 −0.344162
\(10\) −0.416759 −0.131791
\(11\) 1.00000 0.301511
\(12\) −2.56173 −0.739509
\(13\) 2.05922 0.571125 0.285563 0.958360i \(-0.407820\pi\)
0.285563 + 0.958360i \(0.407820\pi\)
\(14\) −0.637213 −0.170302
\(15\) −1.40268 −0.362171
\(16\) 2.98804 0.747010
\(17\) 6.91820 1.67791 0.838955 0.544200i \(-0.183167\pi\)
0.838955 + 0.544200i \(0.183167\pi\)
\(18\) −0.430297 −0.101422
\(19\) −1.02478 −0.235101 −0.117551 0.993067i \(-0.537504\pi\)
−0.117551 + 0.993067i \(0.537504\pi\)
\(20\) 1.82631 0.408376
\(21\) −2.14466 −0.468004
\(22\) 0.416759 0.0888532
\(23\) −0.555659 −0.115863 −0.0579314 0.998321i \(-0.518450\pi\)
−0.0579314 + 0.998321i \(0.518450\pi\)
\(24\) −2.23678 −0.456581
\(25\) 1.00000 0.200000
\(26\) 0.858198 0.168307
\(27\) −5.65629 −1.08855
\(28\) 2.79238 0.527711
\(29\) 8.69161 1.61399 0.806996 0.590557i \(-0.201092\pi\)
0.806996 + 0.590557i \(0.201092\pi\)
\(30\) −0.584579 −0.106729
\(31\) −9.18767 −1.65015 −0.825077 0.565021i \(-0.808868\pi\)
−0.825077 + 0.565021i \(0.808868\pi\)
\(32\) 4.43459 0.783932
\(33\) 1.40268 0.244175
\(34\) 2.88322 0.494468
\(35\) 1.52897 0.258444
\(36\) 1.88564 0.314274
\(37\) −10.0732 −1.65603 −0.828015 0.560706i \(-0.810530\pi\)
−0.828015 + 0.560706i \(0.810530\pi\)
\(38\) −0.427087 −0.0692827
\(39\) 2.88843 0.462519
\(40\) 1.59465 0.252136
\(41\) 3.73463 0.583251 0.291625 0.956533i \(-0.405804\pi\)
0.291625 + 0.956533i \(0.405804\pi\)
\(42\) −0.893806 −0.137917
\(43\) −7.10211 −1.08306 −0.541531 0.840681i \(-0.682155\pi\)
−0.541531 + 0.840681i \(0.682155\pi\)
\(44\) −1.82631 −0.275327
\(45\) 1.03249 0.153914
\(46\) −0.231576 −0.0341440
\(47\) 12.0692 1.76048 0.880240 0.474529i \(-0.157381\pi\)
0.880240 + 0.474529i \(0.157381\pi\)
\(48\) 4.19127 0.604958
\(49\) −4.66224 −0.666034
\(50\) 0.416759 0.0589386
\(51\) 9.70403 1.35884
\(52\) −3.76078 −0.521527
\(53\) −9.47032 −1.30085 −0.650424 0.759571i \(-0.725409\pi\)
−0.650424 + 0.759571i \(0.725409\pi\)
\(54\) −2.35731 −0.320789
\(55\) −1.00000 −0.134840
\(56\) 2.43817 0.325815
\(57\) −1.43744 −0.190394
\(58\) 3.62230 0.475632
\(59\) −10.1112 −1.31637 −0.658183 0.752858i \(-0.728675\pi\)
−0.658183 + 0.752858i \(0.728675\pi\)
\(60\) 2.56173 0.330718
\(61\) −3.95339 −0.506180 −0.253090 0.967443i \(-0.581447\pi\)
−0.253090 + 0.967443i \(0.581447\pi\)
\(62\) −3.82904 −0.486288
\(63\) 1.57864 0.198890
\(64\) −4.12793 −0.515991
\(65\) −2.05922 −0.255415
\(66\) 0.584579 0.0719567
\(67\) −1.23353 −0.150700 −0.0753501 0.997157i \(-0.524007\pi\)
−0.0753501 + 0.997157i \(0.524007\pi\)
\(68\) −12.6348 −1.53219
\(69\) −0.779412 −0.0938302
\(70\) 0.637213 0.0761615
\(71\) −3.25447 −0.386235 −0.193117 0.981176i \(-0.561860\pi\)
−0.193117 + 0.981176i \(0.561860\pi\)
\(72\) 1.64645 0.194036
\(73\) 1.00000 0.117041
\(74\) −4.19811 −0.488020
\(75\) 1.40268 0.161968
\(76\) 1.87157 0.214684
\(77\) −1.52897 −0.174243
\(78\) 1.20378 0.136301
\(79\) −8.38322 −0.943186 −0.471593 0.881816i \(-0.656321\pi\)
−0.471593 + 0.881816i \(0.656321\pi\)
\(80\) −2.98804 −0.334073
\(81\) −4.83652 −0.537391
\(82\) 1.55644 0.171880
\(83\) 15.2456 1.67343 0.836714 0.547641i \(-0.184474\pi\)
0.836714 + 0.547641i \(0.184474\pi\)
\(84\) 3.91682 0.427360
\(85\) −6.91820 −0.750384
\(86\) −2.95986 −0.319170
\(87\) 12.1916 1.30707
\(88\) −1.59465 −0.169990
\(89\) −6.12001 −0.648720 −0.324360 0.945934i \(-0.605149\pi\)
−0.324360 + 0.945934i \(0.605149\pi\)
\(90\) 0.430297 0.0453573
\(91\) −3.14849 −0.330052
\(92\) 1.01481 0.105801
\(93\) −12.8874 −1.33636
\(94\) 5.02996 0.518801
\(95\) 1.02478 0.105141
\(96\) 6.22031 0.634858
\(97\) −12.1941 −1.23813 −0.619064 0.785341i \(-0.712488\pi\)
−0.619064 + 0.785341i \(0.712488\pi\)
\(98\) −1.94303 −0.196276
\(99\) −1.03249 −0.103769
\(100\) −1.82631 −0.182631
\(101\) 19.2090 1.91136 0.955682 0.294402i \(-0.0951206\pi\)
0.955682 + 0.294402i \(0.0951206\pi\)
\(102\) 4.04424 0.400439
\(103\) −1.01096 −0.0996125 −0.0498062 0.998759i \(-0.515860\pi\)
−0.0498062 + 0.998759i \(0.515860\pi\)
\(104\) −3.28373 −0.321997
\(105\) 2.14466 0.209298
\(106\) −3.94684 −0.383351
\(107\) 0.0132951 0.00128528 0.000642642 1.00000i \(-0.499795\pi\)
0.000642642 1.00000i \(0.499795\pi\)
\(108\) 10.3302 0.994020
\(109\) −17.2613 −1.65333 −0.826665 0.562694i \(-0.809765\pi\)
−0.826665 + 0.562694i \(0.809765\pi\)
\(110\) −0.416759 −0.0397364
\(111\) −14.1295 −1.34112
\(112\) −4.56863 −0.431695
\(113\) −11.6377 −1.09479 −0.547393 0.836876i \(-0.684380\pi\)
−0.547393 + 0.836876i \(0.684380\pi\)
\(114\) −0.599067 −0.0561078
\(115\) 0.555659 0.0518154
\(116\) −15.8736 −1.47383
\(117\) −2.12612 −0.196560
\(118\) −4.21393 −0.387924
\(119\) −10.5777 −0.969660
\(120\) 2.23678 0.204189
\(121\) 1.00000 0.0909091
\(122\) −1.64761 −0.149168
\(123\) 5.23849 0.472339
\(124\) 16.7795 1.50685
\(125\) −1.00000 −0.0894427
\(126\) 0.657913 0.0586116
\(127\) −15.5671 −1.38136 −0.690678 0.723163i \(-0.742688\pi\)
−0.690678 + 0.723163i \(0.742688\pi\)
\(128\) −10.5895 −0.935991
\(129\) −9.96199 −0.877104
\(130\) −0.858198 −0.0752690
\(131\) −2.08978 −0.182585 −0.0912927 0.995824i \(-0.529100\pi\)
−0.0912927 + 0.995824i \(0.529100\pi\)
\(132\) −2.56173 −0.222970
\(133\) 1.56687 0.135864
\(134\) −0.514086 −0.0444103
\(135\) 5.65629 0.486816
\(136\) −11.0321 −0.945995
\(137\) −18.4460 −1.57594 −0.787972 0.615711i \(-0.788869\pi\)
−0.787972 + 0.615711i \(0.788869\pi\)
\(138\) −0.324827 −0.0276511
\(139\) −16.7729 −1.42266 −0.711328 0.702860i \(-0.751906\pi\)
−0.711328 + 0.702860i \(0.751906\pi\)
\(140\) −2.79238 −0.235999
\(141\) 16.9293 1.42570
\(142\) −1.35633 −0.113821
\(143\) 2.05922 0.172201
\(144\) −3.08511 −0.257093
\(145\) −8.69161 −0.721799
\(146\) 0.416759 0.0344912
\(147\) −6.53964 −0.539380
\(148\) 18.3969 1.51221
\(149\) −15.0955 −1.23667 −0.618337 0.785913i \(-0.712193\pi\)
−0.618337 + 0.785913i \(0.712193\pi\)
\(150\) 0.584579 0.0477307
\(151\) 15.5459 1.26510 0.632552 0.774518i \(-0.282007\pi\)
0.632552 + 0.774518i \(0.282007\pi\)
\(152\) 1.63417 0.132549
\(153\) −7.14295 −0.577473
\(154\) −0.637213 −0.0513481
\(155\) 9.18767 0.737971
\(156\) −5.27518 −0.422352
\(157\) −13.4904 −1.07665 −0.538324 0.842738i \(-0.680942\pi\)
−0.538324 + 0.842738i \(0.680942\pi\)
\(158\) −3.49378 −0.277950
\(159\) −13.2838 −1.05348
\(160\) −4.43459 −0.350585
\(161\) 0.849587 0.0669569
\(162\) −2.01566 −0.158365
\(163\) 20.5026 1.60589 0.802944 0.596054i \(-0.203266\pi\)
0.802944 + 0.596054i \(0.203266\pi\)
\(164\) −6.82059 −0.532599
\(165\) −1.40268 −0.109199
\(166\) 6.35375 0.493147
\(167\) 12.7938 0.990011 0.495005 0.868890i \(-0.335166\pi\)
0.495005 + 0.868890i \(0.335166\pi\)
\(168\) 3.41998 0.263857
\(169\) −8.75961 −0.673816
\(170\) −2.88322 −0.221133
\(171\) 1.05807 0.0809129
\(172\) 12.9707 0.989004
\(173\) 17.6575 1.34247 0.671237 0.741242i \(-0.265763\pi\)
0.671237 + 0.741242i \(0.265763\pi\)
\(174\) 5.08094 0.385185
\(175\) −1.52897 −0.115580
\(176\) 2.98804 0.225232
\(177\) −14.1828 −1.06604
\(178\) −2.55057 −0.191173
\(179\) −12.9425 −0.967370 −0.483685 0.875242i \(-0.660702\pi\)
−0.483685 + 0.875242i \(0.660702\pi\)
\(180\) −1.88564 −0.140547
\(181\) 2.58085 0.191833 0.0959166 0.995389i \(-0.469422\pi\)
0.0959166 + 0.995389i \(0.469422\pi\)
\(182\) −1.31216 −0.0972639
\(183\) −5.54535 −0.409924
\(184\) 0.886080 0.0653227
\(185\) 10.0732 0.740599
\(186\) −5.37092 −0.393815
\(187\) 6.91820 0.505909
\(188\) −22.0422 −1.60759
\(189\) 8.64832 0.629073
\(190\) 0.427087 0.0309842
\(191\) 15.4394 1.11715 0.558577 0.829453i \(-0.311348\pi\)
0.558577 + 0.829453i \(0.311348\pi\)
\(192\) −5.79017 −0.417869
\(193\) −2.73093 −0.196577 −0.0982884 0.995158i \(-0.531337\pi\)
−0.0982884 + 0.995158i \(0.531337\pi\)
\(194\) −5.08201 −0.364867
\(195\) −2.88843 −0.206845
\(196\) 8.51471 0.608193
\(197\) −9.84862 −0.701685 −0.350842 0.936435i \(-0.614105\pi\)
−0.350842 + 0.936435i \(0.614105\pi\)
\(198\) −0.430297 −0.0305799
\(199\) 16.6457 1.17998 0.589991 0.807410i \(-0.299131\pi\)
0.589991 + 0.807410i \(0.299131\pi\)
\(200\) −1.59465 −0.112759
\(201\) −1.73026 −0.122043
\(202\) 8.00550 0.563265
\(203\) −13.2892 −0.932722
\(204\) −17.7226 −1.24083
\(205\) −3.73463 −0.260838
\(206\) −0.421325 −0.0293551
\(207\) 0.573710 0.0398756
\(208\) 6.15304 0.426636
\(209\) −1.02478 −0.0708857
\(210\) 0.893806 0.0616785
\(211\) 1.71690 0.118196 0.0590981 0.998252i \(-0.481178\pi\)
0.0590981 + 0.998252i \(0.481178\pi\)
\(212\) 17.2958 1.18788
\(213\) −4.56499 −0.312788
\(214\) 0.00554083 0.000378764 0
\(215\) 7.10211 0.484360
\(216\) 9.01980 0.613719
\(217\) 14.0477 0.953620
\(218\) −7.19378 −0.487225
\(219\) 1.40268 0.0947844
\(220\) 1.82631 0.123130
\(221\) 14.2461 0.958297
\(222\) −5.88860 −0.395217
\(223\) 6.28221 0.420688 0.210344 0.977627i \(-0.432542\pi\)
0.210344 + 0.977627i \(0.432542\pi\)
\(224\) −6.78037 −0.453032
\(225\) −1.03249 −0.0688324
\(226\) −4.85012 −0.322625
\(227\) 2.68694 0.178338 0.0891692 0.996016i \(-0.471579\pi\)
0.0891692 + 0.996016i \(0.471579\pi\)
\(228\) 2.62522 0.173859
\(229\) −20.8926 −1.38062 −0.690312 0.723512i \(-0.742527\pi\)
−0.690312 + 0.723512i \(0.742527\pi\)
\(230\) 0.231576 0.0152696
\(231\) −2.14466 −0.141108
\(232\) −13.8601 −0.909958
\(233\) −12.9315 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(234\) −0.886078 −0.0579247
\(235\) −12.0692 −0.787311
\(236\) 18.4662 1.20205
\(237\) −11.7590 −0.763828
\(238\) −4.40837 −0.285752
\(239\) 10.4220 0.674143 0.337071 0.941479i \(-0.390564\pi\)
0.337071 + 0.941479i \(0.390564\pi\)
\(240\) −4.19127 −0.270545
\(241\) −4.15242 −0.267481 −0.133740 0.991016i \(-0.542699\pi\)
−0.133740 + 0.991016i \(0.542699\pi\)
\(242\) 0.416759 0.0267903
\(243\) 10.1848 0.653354
\(244\) 7.22012 0.462221
\(245\) 4.66224 0.297860
\(246\) 2.18319 0.139195
\(247\) −2.11025 −0.134272
\(248\) 14.6511 0.930345
\(249\) 21.3848 1.35521
\(250\) −0.416759 −0.0263581
\(251\) −2.77030 −0.174860 −0.0874300 0.996171i \(-0.527865\pi\)
−0.0874300 + 0.996171i \(0.527865\pi\)
\(252\) −2.88310 −0.181618
\(253\) −0.555659 −0.0349340
\(254\) −6.48772 −0.407076
\(255\) −9.70403 −0.607690
\(256\) 3.84258 0.240161
\(257\) −9.79718 −0.611131 −0.305566 0.952171i \(-0.598846\pi\)
−0.305566 + 0.952171i \(0.598846\pi\)
\(258\) −4.15175 −0.258476
\(259\) 15.4017 0.957015
\(260\) 3.76078 0.233234
\(261\) −8.97396 −0.555474
\(262\) −0.870936 −0.0538066
\(263\) −5.96549 −0.367848 −0.183924 0.982940i \(-0.558880\pi\)
−0.183924 + 0.982940i \(0.558880\pi\)
\(264\) −2.23678 −0.137664
\(265\) 9.47032 0.581757
\(266\) 0.653005 0.0400383
\(267\) −8.58443 −0.525358
\(268\) 2.25282 0.137613
\(269\) −6.16244 −0.375731 −0.187865 0.982195i \(-0.560157\pi\)
−0.187865 + 0.982195i \(0.560157\pi\)
\(270\) 2.35731 0.143461
\(271\) 22.2281 1.35026 0.675130 0.737699i \(-0.264087\pi\)
0.675130 + 0.737699i \(0.264087\pi\)
\(272\) 20.6719 1.25342
\(273\) −4.41633 −0.267289
\(274\) −7.68751 −0.464420
\(275\) 1.00000 0.0603023
\(276\) 1.42345 0.0856816
\(277\) 21.8299 1.31163 0.655816 0.754921i \(-0.272325\pi\)
0.655816 + 0.754921i \(0.272325\pi\)
\(278\) −6.99023 −0.419246
\(279\) 9.48613 0.567920
\(280\) −2.43817 −0.145709
\(281\) −31.3683 −1.87128 −0.935639 0.352960i \(-0.885175\pi\)
−0.935639 + 0.352960i \(0.885175\pi\)
\(282\) 7.05543 0.420145
\(283\) −21.0095 −1.24888 −0.624442 0.781071i \(-0.714673\pi\)
−0.624442 + 0.781071i \(0.714673\pi\)
\(284\) 5.94368 0.352693
\(285\) 1.43744 0.0851468
\(286\) 0.858198 0.0507463
\(287\) −5.71014 −0.337059
\(288\) −4.57865 −0.269800
\(289\) 30.8615 1.81538
\(290\) −3.62230 −0.212709
\(291\) −17.1045 −1.00268
\(292\) −1.82631 −0.106877
\(293\) −0.276370 −0.0161457 −0.00807286 0.999967i \(-0.502570\pi\)
−0.00807286 + 0.999967i \(0.502570\pi\)
\(294\) −2.72545 −0.158951
\(295\) 10.1112 0.588697
\(296\) 16.0633 0.933658
\(297\) −5.65629 −0.328211
\(298\) −6.29119 −0.364439
\(299\) −1.14422 −0.0661722
\(300\) −2.56173 −0.147902
\(301\) 10.8589 0.625898
\(302\) 6.47887 0.372817
\(303\) 26.9440 1.54790
\(304\) −3.06209 −0.175623
\(305\) 3.95339 0.226370
\(306\) −2.97688 −0.170177
\(307\) −7.33762 −0.418780 −0.209390 0.977832i \(-0.567148\pi\)
−0.209390 + 0.977832i \(0.567148\pi\)
\(308\) 2.79238 0.159111
\(309\) −1.41805 −0.0806700
\(310\) 3.82904 0.217475
\(311\) −31.5817 −1.79083 −0.895417 0.445228i \(-0.853122\pi\)
−0.895417 + 0.445228i \(0.853122\pi\)
\(312\) −4.60603 −0.260765
\(313\) 21.3123 1.20464 0.602320 0.798255i \(-0.294243\pi\)
0.602320 + 0.798255i \(0.294243\pi\)
\(314\) −5.62222 −0.317280
\(315\) −1.57864 −0.0889465
\(316\) 15.3104 0.861276
\(317\) 29.1322 1.63623 0.818114 0.575056i \(-0.195020\pi\)
0.818114 + 0.575056i \(0.195020\pi\)
\(318\) −5.53615 −0.310452
\(319\) 8.69161 0.486637
\(320\) 4.12793 0.230758
\(321\) 0.0186487 0.00104087
\(322\) 0.354073 0.0197317
\(323\) −7.08965 −0.394479
\(324\) 8.83299 0.490722
\(325\) 2.05922 0.114225
\(326\) 8.54464 0.473244
\(327\) −24.2121 −1.33893
\(328\) −5.95542 −0.328833
\(329\) −18.4536 −1.01738
\(330\) −0.584579 −0.0321800
\(331\) 3.65757 0.201038 0.100519 0.994935i \(-0.467950\pi\)
0.100519 + 0.994935i \(0.467950\pi\)
\(332\) −27.8433 −1.52810
\(333\) 10.4005 0.569942
\(334\) 5.33191 0.291749
\(335\) 1.23353 0.0673952
\(336\) −6.40834 −0.349603
\(337\) −2.52764 −0.137689 −0.0688445 0.997627i \(-0.521931\pi\)
−0.0688445 + 0.997627i \(0.521931\pi\)
\(338\) −3.65064 −0.198569
\(339\) −16.3240 −0.886599
\(340\) 12.6348 0.685218
\(341\) −9.18767 −0.497540
\(342\) 0.440961 0.0238445
\(343\) 17.8313 0.962797
\(344\) 11.3254 0.610623
\(345\) 0.779412 0.0419621
\(346\) 7.35892 0.395618
\(347\) 28.6273 1.53679 0.768396 0.639974i \(-0.221055\pi\)
0.768396 + 0.639974i \(0.221055\pi\)
\(348\) −22.2656 −1.19356
\(349\) 5.58630 0.299028 0.149514 0.988760i \(-0.452229\pi\)
0.149514 + 0.988760i \(0.452229\pi\)
\(350\) −0.637213 −0.0340604
\(351\) −11.6476 −0.621701
\(352\) 4.43459 0.236364
\(353\) −31.2555 −1.66356 −0.831781 0.555103i \(-0.812679\pi\)
−0.831781 + 0.555103i \(0.812679\pi\)
\(354\) −5.91080 −0.314155
\(355\) 3.25447 0.172729
\(356\) 11.1771 0.592383
\(357\) −14.8372 −0.785268
\(358\) −5.39391 −0.285077
\(359\) −12.7143 −0.671036 −0.335518 0.942034i \(-0.608911\pi\)
−0.335518 + 0.942034i \(0.608911\pi\)
\(360\) −1.64645 −0.0867756
\(361\) −17.9498 −0.944727
\(362\) 1.07559 0.0565319
\(363\) 1.40268 0.0736217
\(364\) 5.75013 0.301389
\(365\) −1.00000 −0.0523424
\(366\) −2.31107 −0.120802
\(367\) 19.7409 1.03047 0.515234 0.857050i \(-0.327705\pi\)
0.515234 + 0.857050i \(0.327705\pi\)
\(368\) −1.66033 −0.0865508
\(369\) −3.85595 −0.200733
\(370\) 4.19811 0.218249
\(371\) 14.4799 0.751757
\(372\) 23.5364 1.22030
\(373\) −15.1098 −0.782357 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(374\) 2.88322 0.149088
\(375\) −1.40268 −0.0724341
\(376\) −19.2462 −0.992547
\(377\) 17.8979 0.921791
\(378\) 3.60426 0.185383
\(379\) 3.96090 0.203458 0.101729 0.994812i \(-0.467563\pi\)
0.101729 + 0.994812i \(0.467563\pi\)
\(380\) −1.87157 −0.0960097
\(381\) −21.8357 −1.11867
\(382\) 6.43449 0.329217
\(383\) 14.4766 0.739721 0.369861 0.929087i \(-0.379405\pi\)
0.369861 + 0.929087i \(0.379405\pi\)
\(384\) −14.8537 −0.758001
\(385\) 1.52897 0.0779237
\(386\) −1.13814 −0.0579297
\(387\) 7.33282 0.372748
\(388\) 22.2703 1.13060
\(389\) 0.420250 0.0213075 0.0106538 0.999943i \(-0.496609\pi\)
0.0106538 + 0.999943i \(0.496609\pi\)
\(390\) −1.20378 −0.0609557
\(391\) −3.84416 −0.194408
\(392\) 7.43464 0.375506
\(393\) −2.93130 −0.147865
\(394\) −4.10450 −0.206781
\(395\) 8.38322 0.421806
\(396\) 1.88564 0.0947571
\(397\) 4.84757 0.243293 0.121646 0.992574i \(-0.461183\pi\)
0.121646 + 0.992574i \(0.461183\pi\)
\(398\) 6.93724 0.347732
\(399\) 2.19781 0.110028
\(400\) 2.98804 0.149402
\(401\) 14.4688 0.722539 0.361269 0.932461i \(-0.382343\pi\)
0.361269 + 0.932461i \(0.382343\pi\)
\(402\) −0.721099 −0.0359651
\(403\) −18.9194 −0.942444
\(404\) −35.0816 −1.74537
\(405\) 4.83652 0.240328
\(406\) −5.53840 −0.274866
\(407\) −10.0732 −0.499312
\(408\) −15.4745 −0.766103
\(409\) −0.859207 −0.0424850 −0.0212425 0.999774i \(-0.506762\pi\)
−0.0212425 + 0.999774i \(0.506762\pi\)
\(410\) −1.55644 −0.0768670
\(411\) −25.8738 −1.27626
\(412\) 1.84632 0.0909617
\(413\) 15.4598 0.760725
\(414\) 0.239098 0.0117510
\(415\) −15.2456 −0.748379
\(416\) 9.13180 0.447723
\(417\) −23.5270 −1.15212
\(418\) −0.427087 −0.0208895
\(419\) −7.47353 −0.365106 −0.182553 0.983196i \(-0.558436\pi\)
−0.182553 + 0.983196i \(0.558436\pi\)
\(420\) −3.91682 −0.191121
\(421\) −20.9877 −1.02288 −0.511439 0.859320i \(-0.670887\pi\)
−0.511439 + 0.859320i \(0.670887\pi\)
\(422\) 0.715533 0.0348316
\(423\) −12.4613 −0.605890
\(424\) 15.1018 0.733410
\(425\) 6.91820 0.335582
\(426\) −1.90250 −0.0921763
\(427\) 6.04463 0.292520
\(428\) −0.0242810 −0.00117366
\(429\) 2.88843 0.139455
\(430\) 2.95986 0.142737
\(431\) 20.7998 1.00189 0.500947 0.865478i \(-0.332985\pi\)
0.500947 + 0.865478i \(0.332985\pi\)
\(432\) −16.9012 −0.813161
\(433\) −6.88854 −0.331042 −0.165521 0.986206i \(-0.552931\pi\)
−0.165521 + 0.986206i \(0.552931\pi\)
\(434\) 5.85450 0.281025
\(435\) −12.1916 −0.584540
\(436\) 31.5245 1.50975
\(437\) 0.569430 0.0272395
\(438\) 0.584579 0.0279323
\(439\) −26.9712 −1.28726 −0.643632 0.765335i \(-0.722573\pi\)
−0.643632 + 0.765335i \(0.722573\pi\)
\(440\) 1.59465 0.0760219
\(441\) 4.81370 0.229224
\(442\) 5.93719 0.282403
\(443\) 22.9865 1.09212 0.546062 0.837745i \(-0.316126\pi\)
0.546062 + 0.837745i \(0.316126\pi\)
\(444\) 25.8049 1.22465
\(445\) 6.12001 0.290116
\(446\) 2.61817 0.123974
\(447\) −21.1742 −1.00151
\(448\) 6.31149 0.298190
\(449\) −3.74464 −0.176721 −0.0883603 0.996089i \(-0.528163\pi\)
−0.0883603 + 0.996089i \(0.528163\pi\)
\(450\) −0.430297 −0.0202844
\(451\) 3.73463 0.175857
\(452\) 21.2541 0.999710
\(453\) 21.8059 1.02453
\(454\) 1.11980 0.0525550
\(455\) 3.14849 0.147604
\(456\) 2.29222 0.107343
\(457\) −4.95202 −0.231646 −0.115823 0.993270i \(-0.536951\pi\)
−0.115823 + 0.993270i \(0.536951\pi\)
\(458\) −8.70718 −0.406860
\(459\) −39.1314 −1.82650
\(460\) −1.01481 −0.0473156
\(461\) 33.6844 1.56884 0.784419 0.620232i \(-0.212961\pi\)
0.784419 + 0.620232i \(0.212961\pi\)
\(462\) −0.893806 −0.0415836
\(463\) −12.4478 −0.578498 −0.289249 0.957254i \(-0.593406\pi\)
−0.289249 + 0.957254i \(0.593406\pi\)
\(464\) 25.9709 1.20567
\(465\) 12.8874 0.597637
\(466\) −5.38929 −0.249654
\(467\) 33.2477 1.53852 0.769260 0.638935i \(-0.220625\pi\)
0.769260 + 0.638935i \(0.220625\pi\)
\(468\) 3.88295 0.179490
\(469\) 1.88604 0.0870893
\(470\) −5.02996 −0.232015
\(471\) −18.9227 −0.871910
\(472\) 16.1238 0.742158
\(473\) −7.10211 −0.326555
\(474\) −4.90066 −0.225095
\(475\) −1.02478 −0.0470203
\(476\) 19.3183 0.885451
\(477\) 9.77797 0.447702
\(478\) 4.34346 0.198665
\(479\) −23.1598 −1.05820 −0.529099 0.848560i \(-0.677470\pi\)
−0.529099 + 0.848560i \(0.677470\pi\)
\(480\) −6.22031 −0.283917
\(481\) −20.7430 −0.945800
\(482\) −1.73056 −0.0788247
\(483\) 1.19170 0.0542242
\(484\) −1.82631 −0.0830142
\(485\) 12.1941 0.553707
\(486\) 4.24460 0.192539
\(487\) −10.1003 −0.457690 −0.228845 0.973463i \(-0.573495\pi\)
−0.228845 + 0.973463i \(0.573495\pi\)
\(488\) 6.30427 0.285381
\(489\) 28.7586 1.30051
\(490\) 1.94303 0.0877771
\(491\) 27.5045 1.24126 0.620629 0.784104i \(-0.286877\pi\)
0.620629 + 0.784104i \(0.286877\pi\)
\(492\) −9.56712 −0.431319
\(493\) 60.1303 2.70813
\(494\) −0.879467 −0.0395691
\(495\) 1.03249 0.0464068
\(496\) −27.4531 −1.23268
\(497\) 4.97600 0.223204
\(498\) 8.91229 0.399369
\(499\) −8.62750 −0.386220 −0.193110 0.981177i \(-0.561857\pi\)
−0.193110 + 0.981177i \(0.561857\pi\)
\(500\) 1.82631 0.0816752
\(501\) 17.9456 0.801749
\(502\) −1.15455 −0.0515300
\(503\) −19.8170 −0.883597 −0.441799 0.897114i \(-0.645659\pi\)
−0.441799 + 0.897114i \(0.645659\pi\)
\(504\) −2.51738 −0.112133
\(505\) −19.2090 −0.854788
\(506\) −0.231576 −0.0102948
\(507\) −12.2869 −0.545682
\(508\) 28.4304 1.26139
\(509\) 42.0636 1.86444 0.932219 0.361895i \(-0.117870\pi\)
0.932219 + 0.361895i \(0.117870\pi\)
\(510\) −4.04424 −0.179082
\(511\) −1.52897 −0.0676378
\(512\) 22.7805 1.00676
\(513\) 5.79647 0.255920
\(514\) −4.08306 −0.180096
\(515\) 1.01096 0.0445480
\(516\) 18.1937 0.800933
\(517\) 12.0692 0.530805
\(518\) 6.41879 0.282026
\(519\) 24.7678 1.08719
\(520\) 3.28373 0.144001
\(521\) −20.8980 −0.915559 −0.457780 0.889066i \(-0.651355\pi\)
−0.457780 + 0.889066i \(0.651355\pi\)
\(522\) −3.73998 −0.163694
\(523\) −11.2086 −0.490118 −0.245059 0.969508i \(-0.578807\pi\)
−0.245059 + 0.969508i \(0.578807\pi\)
\(524\) 3.81660 0.166729
\(525\) −2.14466 −0.0936007
\(526\) −2.48617 −0.108402
\(527\) −63.5621 −2.76881
\(528\) 4.19127 0.182402
\(529\) −22.6912 −0.986576
\(530\) 3.94684 0.171440
\(531\) 10.4397 0.453043
\(532\) −2.86159 −0.124065
\(533\) 7.69042 0.333109
\(534\) −3.57763 −0.154819
\(535\) −0.0132951 −0.000574796 0
\(536\) 1.96705 0.0849638
\(537\) −18.1542 −0.783413
\(538\) −2.56825 −0.110725
\(539\) −4.66224 −0.200817
\(540\) −10.3302 −0.444539
\(541\) −25.7007 −1.10496 −0.552479 0.833527i \(-0.686318\pi\)
−0.552479 + 0.833527i \(0.686318\pi\)
\(542\) 9.26375 0.397912
\(543\) 3.62011 0.155354
\(544\) 30.6794 1.31537
\(545\) 17.2613 0.739392
\(546\) −1.84054 −0.0787680
\(547\) −19.8262 −0.847707 −0.423853 0.905731i \(-0.639323\pi\)
−0.423853 + 0.905731i \(0.639323\pi\)
\(548\) 33.6881 1.43908
\(549\) 4.08182 0.174208
\(550\) 0.416759 0.0177706
\(551\) −8.90701 −0.379451
\(552\) 1.24289 0.0529008
\(553\) 12.8177 0.545065
\(554\) 9.09780 0.386529
\(555\) 14.1295 0.599765
\(556\) 30.6325 1.29911
\(557\) 30.8244 1.30607 0.653036 0.757327i \(-0.273495\pi\)
0.653036 + 0.757327i \(0.273495\pi\)
\(558\) 3.95343 0.167362
\(559\) −14.6248 −0.618564
\(560\) 4.56863 0.193060
\(561\) 9.70403 0.409705
\(562\) −13.0730 −0.551452
\(563\) −17.0695 −0.719394 −0.359697 0.933069i \(-0.617120\pi\)
−0.359697 + 0.933069i \(0.617120\pi\)
\(564\) −30.9182 −1.30189
\(565\) 11.6377 0.489603
\(566\) −8.75588 −0.368037
\(567\) 7.39490 0.310557
\(568\) 5.18974 0.217757
\(569\) −24.1003 −1.01034 −0.505170 0.863020i \(-0.668570\pi\)
−0.505170 + 0.863020i \(0.668570\pi\)
\(570\) 0.599067 0.0250922
\(571\) 11.6950 0.489420 0.244710 0.969596i \(-0.421307\pi\)
0.244710 + 0.969596i \(0.421307\pi\)
\(572\) −3.76078 −0.157246
\(573\) 21.6565 0.904714
\(574\) −2.37975 −0.0993289
\(575\) −0.555659 −0.0231726
\(576\) 4.26203 0.177585
\(577\) 0.831618 0.0346207 0.0173104 0.999850i \(-0.494490\pi\)
0.0173104 + 0.999850i \(0.494490\pi\)
\(578\) 12.8618 0.534980
\(579\) −3.83062 −0.159195
\(580\) 15.8736 0.659115
\(581\) −23.3102 −0.967070
\(582\) −7.12844 −0.295483
\(583\) −9.47032 −0.392220
\(584\) −1.59465 −0.0659870
\(585\) 2.12612 0.0879041
\(586\) −0.115180 −0.00475803
\(587\) −36.3618 −1.50081 −0.750406 0.660978i \(-0.770142\pi\)
−0.750406 + 0.660978i \(0.770142\pi\)
\(588\) 11.9434 0.492538
\(589\) 9.41536 0.387953
\(590\) 4.21393 0.173485
\(591\) −13.8145 −0.568251
\(592\) −30.0992 −1.23707
\(593\) 37.0630 1.52199 0.760997 0.648756i \(-0.224710\pi\)
0.760997 + 0.648756i \(0.224710\pi\)
\(594\) −2.35731 −0.0967215
\(595\) 10.5777 0.433645
\(596\) 27.5691 1.12928
\(597\) 23.3486 0.955595
\(598\) −0.476865 −0.0195005
\(599\) 16.8635 0.689022 0.344511 0.938782i \(-0.388045\pi\)
0.344511 + 0.938782i \(0.388045\pi\)
\(600\) −2.23678 −0.0913163
\(601\) 3.44336 0.140457 0.0702287 0.997531i \(-0.477627\pi\)
0.0702287 + 0.997531i \(0.477627\pi\)
\(602\) 4.52555 0.184448
\(603\) 1.27361 0.0518653
\(604\) −28.3916 −1.15524
\(605\) −1.00000 −0.0406558
\(606\) 11.2292 0.456154
\(607\) 3.14805 0.127775 0.0638877 0.997957i \(-0.479650\pi\)
0.0638877 + 0.997957i \(0.479650\pi\)
\(608\) −4.54449 −0.184303
\(609\) −18.6406 −0.755354
\(610\) 1.64761 0.0667097
\(611\) 24.8532 1.00545
\(612\) 13.0452 0.527323
\(613\) −40.4271 −1.63283 −0.816417 0.577463i \(-0.804043\pi\)
−0.816417 + 0.577463i \(0.804043\pi\)
\(614\) −3.05801 −0.123411
\(615\) −5.23849 −0.211236
\(616\) 2.43817 0.0982369
\(617\) −46.9003 −1.88814 −0.944068 0.329750i \(-0.893036\pi\)
−0.944068 + 0.329750i \(0.893036\pi\)
\(618\) −0.590984 −0.0237729
\(619\) −26.1801 −1.05227 −0.526134 0.850401i \(-0.676359\pi\)
−0.526134 + 0.850401i \(0.676359\pi\)
\(620\) −16.7795 −0.673883
\(621\) 3.14297 0.126123
\(622\) −13.1619 −0.527746
\(623\) 9.35734 0.374894
\(624\) 8.63075 0.345507
\(625\) 1.00000 0.0400000
\(626\) 8.88206 0.354999
\(627\) −1.43744 −0.0574060
\(628\) 24.6376 0.983147
\(629\) −69.6887 −2.77867
\(630\) −0.657913 −0.0262119
\(631\) −19.7703 −0.787042 −0.393521 0.919316i \(-0.628743\pi\)
−0.393521 + 0.919316i \(0.628743\pi\)
\(632\) 13.3683 0.531762
\(633\) 2.40826 0.0957199
\(634\) 12.1411 0.482185
\(635\) 15.5671 0.617761
\(636\) 24.2604 0.961989
\(637\) −9.60059 −0.380389
\(638\) 3.62230 0.143408
\(639\) 3.36020 0.132927
\(640\) 10.5895 0.418588
\(641\) 7.79782 0.307995 0.153998 0.988071i \(-0.450785\pi\)
0.153998 + 0.988071i \(0.450785\pi\)
\(642\) 0.00777202 0.000306737 0
\(643\) −1.37087 −0.0540619 −0.0270309 0.999635i \(-0.508605\pi\)
−0.0270309 + 0.999635i \(0.508605\pi\)
\(644\) −1.55161 −0.0611421
\(645\) 9.96199 0.392253
\(646\) −2.95467 −0.116250
\(647\) 45.6510 1.79473 0.897363 0.441292i \(-0.145480\pi\)
0.897363 + 0.441292i \(0.145480\pi\)
\(648\) 7.71254 0.302977
\(649\) −10.1112 −0.396899
\(650\) 0.858198 0.0336613
\(651\) 19.7044 0.772278
\(652\) −37.4442 −1.46643
\(653\) −21.6563 −0.847476 −0.423738 0.905785i \(-0.639282\pi\)
−0.423738 + 0.905785i \(0.639282\pi\)
\(654\) −10.0906 −0.394573
\(655\) 2.08978 0.0816546
\(656\) 11.1592 0.435694
\(657\) −1.03249 −0.0402811
\(658\) −7.69068 −0.299814
\(659\) 22.5335 0.877780 0.438890 0.898541i \(-0.355372\pi\)
0.438890 + 0.898541i \(0.355372\pi\)
\(660\) 2.56173 0.0997153
\(661\) 11.5859 0.450639 0.225319 0.974285i \(-0.427657\pi\)
0.225319 + 0.974285i \(0.427657\pi\)
\(662\) 1.52432 0.0592445
\(663\) 19.9827 0.776066
\(664\) −24.3114 −0.943467
\(665\) −1.56687 −0.0607604
\(666\) 4.33449 0.167958
\(667\) −4.82957 −0.187002
\(668\) −23.3654 −0.904035
\(669\) 8.81194 0.340689
\(670\) 0.514086 0.0198609
\(671\) −3.95339 −0.152619
\(672\) −9.51069 −0.366883
\(673\) 26.4702 1.02035 0.510175 0.860070i \(-0.329580\pi\)
0.510175 + 0.860070i \(0.329580\pi\)
\(674\) −1.05341 −0.0405760
\(675\) −5.65629 −0.217711
\(676\) 15.9978 0.615299
\(677\) 0.369548 0.0142029 0.00710145 0.999975i \(-0.497740\pi\)
0.00710145 + 0.999975i \(0.497740\pi\)
\(678\) −6.80318 −0.261274
\(679\) 18.6445 0.715511
\(680\) 11.0321 0.423062
\(681\) 3.76892 0.144425
\(682\) −3.82904 −0.146621
\(683\) −38.6482 −1.47883 −0.739416 0.673249i \(-0.764898\pi\)
−0.739416 + 0.673249i \(0.764898\pi\)
\(684\) −1.93237 −0.0738861
\(685\) 18.4460 0.704784
\(686\) 7.43133 0.283729
\(687\) −29.3057 −1.11808
\(688\) −21.2214 −0.809058
\(689\) −19.5015 −0.742947
\(690\) 0.324827 0.0123659
\(691\) −3.42747 −0.130387 −0.0651936 0.997873i \(-0.520767\pi\)
−0.0651936 + 0.997873i \(0.520767\pi\)
\(692\) −32.2481 −1.22589
\(693\) 1.57864 0.0599677
\(694\) 11.9307 0.452882
\(695\) 16.7729 0.636231
\(696\) −19.4412 −0.736919
\(697\) 25.8369 0.978642
\(698\) 2.32814 0.0881213
\(699\) −18.1387 −0.686069
\(700\) 2.79238 0.105542
\(701\) −23.9261 −0.903677 −0.451838 0.892100i \(-0.649232\pi\)
−0.451838 + 0.892100i \(0.649232\pi\)
\(702\) −4.85422 −0.183211
\(703\) 10.3229 0.389335
\(704\) −4.12793 −0.155577
\(705\) −16.9293 −0.637594
\(706\) −13.0260 −0.490240
\(707\) −29.3700 −1.10457
\(708\) 25.9022 0.973464
\(709\) −32.3934 −1.21656 −0.608281 0.793722i \(-0.708140\pi\)
−0.608281 + 0.793722i \(0.708140\pi\)
\(710\) 1.35633 0.0509021
\(711\) 8.65556 0.324609
\(712\) 9.75927 0.365744
\(713\) 5.10521 0.191191
\(714\) −6.18353 −0.231413
\(715\) −2.05922 −0.0770105
\(716\) 23.6371 0.883360
\(717\) 14.6187 0.545947
\(718\) −5.29880 −0.197749
\(719\) 20.9020 0.779513 0.389757 0.920918i \(-0.372559\pi\)
0.389757 + 0.920918i \(0.372559\pi\)
\(720\) 3.08511 0.114975
\(721\) 1.54572 0.0575658
\(722\) −7.48074 −0.278404
\(723\) −5.82452 −0.216616
\(724\) −4.71344 −0.175174
\(725\) 8.69161 0.322798
\(726\) 0.584579 0.0216958
\(727\) −17.7799 −0.659420 −0.329710 0.944082i \(-0.606951\pi\)
−0.329710 + 0.944082i \(0.606951\pi\)
\(728\) 5.02074 0.186081
\(729\) 28.7956 1.06650
\(730\) −0.416759 −0.0154249
\(731\) −49.1338 −1.81728
\(732\) 10.1275 0.374324
\(733\) 18.1618 0.670822 0.335411 0.942072i \(-0.391125\pi\)
0.335411 + 0.942072i \(0.391125\pi\)
\(734\) 8.22720 0.303672
\(735\) 6.53964 0.241218
\(736\) −2.46412 −0.0908286
\(737\) −1.23353 −0.0454378
\(738\) −1.60700 −0.0591545
\(739\) 15.6914 0.577218 0.288609 0.957447i \(-0.406807\pi\)
0.288609 + 0.957447i \(0.406807\pi\)
\(740\) −18.3969 −0.676282
\(741\) −2.96001 −0.108739
\(742\) 6.03461 0.221537
\(743\) 2.72073 0.0998141 0.0499070 0.998754i \(-0.484107\pi\)
0.0499070 + 0.998754i \(0.484107\pi\)
\(744\) 20.5508 0.753429
\(745\) 15.0955 0.553057
\(746\) −6.29715 −0.230555
\(747\) −15.7409 −0.575930
\(748\) −12.6348 −0.461974
\(749\) −0.0203278 −0.000742762 0
\(750\) −0.584579 −0.0213458
\(751\) −18.6383 −0.680120 −0.340060 0.940404i \(-0.610447\pi\)
−0.340060 + 0.940404i \(0.610447\pi\)
\(752\) 36.0634 1.31510
\(753\) −3.88585 −0.141608
\(754\) 7.45912 0.271645
\(755\) −15.5459 −0.565772
\(756\) −15.7945 −0.574441
\(757\) −41.0573 −1.49225 −0.746126 0.665805i \(-0.768088\pi\)
−0.746126 + 0.665805i \(0.768088\pi\)
\(758\) 1.65074 0.0599575
\(759\) −0.779412 −0.0282909
\(760\) −1.63417 −0.0592775
\(761\) 27.6541 1.00246 0.501229 0.865314i \(-0.332881\pi\)
0.501229 + 0.865314i \(0.332881\pi\)
\(762\) −9.10020 −0.329665
\(763\) 26.3920 0.955456
\(764\) −28.1971 −1.02014
\(765\) 7.14295 0.258254
\(766\) 6.03326 0.217990
\(767\) −20.8212 −0.751810
\(768\) 5.38992 0.194492
\(769\) 39.9562 1.44086 0.720428 0.693530i \(-0.243945\pi\)
0.720428 + 0.693530i \(0.243945\pi\)
\(770\) 0.637213 0.0229636
\(771\) −13.7423 −0.494917
\(772\) 4.98753 0.179505
\(773\) 28.0949 1.01050 0.505251 0.862973i \(-0.331400\pi\)
0.505251 + 0.862973i \(0.331400\pi\)
\(774\) 3.05602 0.109846
\(775\) −9.18767 −0.330031
\(776\) 19.4454 0.698048
\(777\) 21.6037 0.775028
\(778\) 0.175143 0.00627918
\(779\) −3.82718 −0.137123
\(780\) 5.27518 0.188882
\(781\) −3.25447 −0.116454
\(782\) −1.60209 −0.0572905
\(783\) −49.1623 −1.75692
\(784\) −13.9310 −0.497535
\(785\) 13.4904 0.481491
\(786\) −1.22164 −0.0435746
\(787\) 44.8960 1.60037 0.800185 0.599753i \(-0.204735\pi\)
0.800185 + 0.599753i \(0.204735\pi\)
\(788\) 17.9866 0.640748
\(789\) −8.36768 −0.297897
\(790\) 3.49378 0.124303
\(791\) 17.7938 0.632674
\(792\) 1.64645 0.0585041
\(793\) −8.14091 −0.289092
\(794\) 2.02027 0.0716966
\(795\) 13.2838 0.471129
\(796\) −30.4003 −1.07751
\(797\) −48.8760 −1.73128 −0.865639 0.500669i \(-0.833087\pi\)
−0.865639 + 0.500669i \(0.833087\pi\)
\(798\) 0.915957 0.0324245
\(799\) 83.4975 2.95393
\(800\) 4.43459 0.156786
\(801\) 6.31883 0.223265
\(802\) 6.03001 0.212927
\(803\) 1.00000 0.0352892
\(804\) 3.15999 0.111444
\(805\) −0.849587 −0.0299440
\(806\) −7.88484 −0.277732
\(807\) −8.64394 −0.304281
\(808\) −30.6315 −1.07761
\(809\) 56.5694 1.98887 0.994437 0.105335i \(-0.0335914\pi\)
0.994437 + 0.105335i \(0.0335914\pi\)
\(810\) 2.01566 0.0708230
\(811\) 41.6209 1.46151 0.730753 0.682642i \(-0.239169\pi\)
0.730753 + 0.682642i \(0.239169\pi\)
\(812\) 24.2703 0.851721
\(813\) 31.1789 1.09349
\(814\) −4.19811 −0.147144
\(815\) −20.5026 −0.718175
\(816\) 28.9960 1.01506
\(817\) 7.27812 0.254629
\(818\) −0.358082 −0.0125200
\(819\) 3.25078 0.113591
\(820\) 6.82059 0.238185
\(821\) −40.4047 −1.41013 −0.705066 0.709141i \(-0.749083\pi\)
−0.705066 + 0.709141i \(0.749083\pi\)
\(822\) −10.7831 −0.376105
\(823\) −43.1125 −1.50281 −0.751403 0.659843i \(-0.770623\pi\)
−0.751403 + 0.659843i \(0.770623\pi\)
\(824\) 1.61212 0.0561608
\(825\) 1.40268 0.0488351
\(826\) 6.44298 0.224180
\(827\) −25.7392 −0.895041 −0.447521 0.894274i \(-0.647693\pi\)
−0.447521 + 0.894274i \(0.647693\pi\)
\(828\) −1.04777 −0.0364126
\(829\) 50.6863 1.76041 0.880204 0.474595i \(-0.157405\pi\)
0.880204 + 0.474595i \(0.157405\pi\)
\(830\) −6.35375 −0.220542
\(831\) 30.6204 1.06221
\(832\) −8.50032 −0.294696
\(833\) −32.2543 −1.11755
\(834\) −9.80507 −0.339522
\(835\) −12.7938 −0.442746
\(836\) 1.87157 0.0647297
\(837\) 51.9681 1.79628
\(838\) −3.11466 −0.107594
\(839\) −9.81138 −0.338726 −0.169363 0.985554i \(-0.554171\pi\)
−0.169363 + 0.985554i \(0.554171\pi\)
\(840\) −3.41998 −0.118001
\(841\) 46.5441 1.60497
\(842\) −8.74680 −0.301435
\(843\) −43.9997 −1.51543
\(844\) −3.13560 −0.107932
\(845\) 8.75961 0.301340
\(846\) −5.19336 −0.178552
\(847\) −1.52897 −0.0525361
\(848\) −28.2977 −0.971747
\(849\) −29.4696 −1.01139
\(850\) 2.88322 0.0988936
\(851\) 5.59728 0.191872
\(852\) 8.33709 0.285624
\(853\) −24.5647 −0.841078 −0.420539 0.907274i \(-0.638159\pi\)
−0.420539 + 0.907274i \(0.638159\pi\)
\(854\) 2.51915 0.0862035
\(855\) −1.05807 −0.0361854
\(856\) −0.0212010 −0.000724634 0
\(857\) −32.7436 −1.11850 −0.559250 0.828999i \(-0.688911\pi\)
−0.559250 + 0.828999i \(0.688911\pi\)
\(858\) 1.20378 0.0410963
\(859\) 19.4957 0.665184 0.332592 0.943071i \(-0.392077\pi\)
0.332592 + 0.943071i \(0.392077\pi\)
\(860\) −12.9707 −0.442296
\(861\) −8.00951 −0.272963
\(862\) 8.66851 0.295251
\(863\) −9.02807 −0.307319 −0.153660 0.988124i \(-0.549106\pi\)
−0.153660 + 0.988124i \(0.549106\pi\)
\(864\) −25.0833 −0.853352
\(865\) −17.6575 −0.600373
\(866\) −2.87086 −0.0975558
\(867\) 43.2889 1.47017
\(868\) −25.6555 −0.870804
\(869\) −8.38322 −0.284381
\(870\) −5.08094 −0.172260
\(871\) −2.54012 −0.0860687
\(872\) 27.5257 0.932137
\(873\) 12.5903 0.426116
\(874\) 0.237315 0.00802729
\(875\) 1.52897 0.0516887
\(876\) −2.56173 −0.0865530
\(877\) 29.9432 1.01111 0.505556 0.862794i \(-0.331288\pi\)
0.505556 + 0.862794i \(0.331288\pi\)
\(878\) −11.2405 −0.379347
\(879\) −0.387659 −0.0130754
\(880\) −2.98804 −0.100727
\(881\) −38.3300 −1.29137 −0.645686 0.763603i \(-0.723429\pi\)
−0.645686 + 0.763603i \(0.723429\pi\)
\(882\) 2.00615 0.0675506
\(883\) −0.271994 −0.00915331 −0.00457666 0.999990i \(-0.501457\pi\)
−0.00457666 + 0.999990i \(0.501457\pi\)
\(884\) −26.0178 −0.875075
\(885\) 14.1828 0.476749
\(886\) 9.57984 0.321841
\(887\) 22.6847 0.761679 0.380839 0.924641i \(-0.375635\pi\)
0.380839 + 0.924641i \(0.375635\pi\)
\(888\) 22.5316 0.756112
\(889\) 23.8017 0.798282
\(890\) 2.55057 0.0854952
\(891\) −4.83652 −0.162029
\(892\) −11.4733 −0.384154
\(893\) −12.3684 −0.413891
\(894\) −8.82453 −0.295136
\(895\) 12.9425 0.432621
\(896\) 16.1911 0.540907
\(897\) −1.60498 −0.0535888
\(898\) −1.56061 −0.0520783
\(899\) −79.8556 −2.66333
\(900\) 1.88564 0.0628547
\(901\) −65.5176 −2.18271
\(902\) 1.55644 0.0518237
\(903\) 15.2316 0.506876
\(904\) 18.5581 0.617233
\(905\) −2.58085 −0.0857904
\(906\) 9.08779 0.301922
\(907\) −17.7568 −0.589606 −0.294803 0.955558i \(-0.595254\pi\)
−0.294803 + 0.955558i \(0.595254\pi\)
\(908\) −4.90719 −0.162851
\(909\) −19.8330 −0.657818
\(910\) 1.31216 0.0434977
\(911\) −1.46318 −0.0484773 −0.0242386 0.999706i \(-0.507716\pi\)
−0.0242386 + 0.999706i \(0.507716\pi\)
\(912\) −4.29514 −0.142226
\(913\) 15.2456 0.504557
\(914\) −2.06380 −0.0682644
\(915\) 5.54535 0.183323
\(916\) 38.1565 1.26072
\(917\) 3.19522 0.105516
\(918\) −16.3083 −0.538255
\(919\) 32.9872 1.08815 0.544073 0.839038i \(-0.316881\pi\)
0.544073 + 0.839038i \(0.316881\pi\)
\(920\) −0.886080 −0.0292132
\(921\) −10.2923 −0.339144
\(922\) 14.0382 0.462325
\(923\) −6.70168 −0.220589
\(924\) 3.91682 0.128854
\(925\) −10.0732 −0.331206
\(926\) −5.18772 −0.170479
\(927\) 1.04380 0.0342828
\(928\) 38.5437 1.26526
\(929\) 14.7059 0.482486 0.241243 0.970465i \(-0.422445\pi\)
0.241243 + 0.970465i \(0.422445\pi\)
\(930\) 5.37092 0.176119
\(931\) 4.77778 0.156586
\(932\) 23.6169 0.773596
\(933\) −44.2991 −1.45029
\(934\) 13.8563 0.453391
\(935\) −6.91820 −0.226249
\(936\) 3.39041 0.110819
\(937\) −8.40476 −0.274572 −0.137286 0.990531i \(-0.543838\pi\)
−0.137286 + 0.990531i \(0.543838\pi\)
\(938\) 0.786024 0.0256646
\(939\) 29.8943 0.975563
\(940\) 22.0422 0.718938
\(941\) 4.27537 0.139373 0.0696865 0.997569i \(-0.477800\pi\)
0.0696865 + 0.997569i \(0.477800\pi\)
\(942\) −7.88618 −0.256946
\(943\) −2.07518 −0.0675771
\(944\) −30.2127 −0.983339
\(945\) −8.64832 −0.281330
\(946\) −2.95986 −0.0962335
\(947\) −0.339000 −0.0110160 −0.00550801 0.999985i \(-0.501753\pi\)
−0.00550801 + 0.999985i \(0.501753\pi\)
\(948\) 21.4756 0.697494
\(949\) 2.05922 0.0668452
\(950\) −0.427087 −0.0138565
\(951\) 40.8632 1.32508
\(952\) 16.8678 0.546688
\(953\) −12.6535 −0.409886 −0.204943 0.978774i \(-0.565701\pi\)
−0.204943 + 0.978774i \(0.565701\pi\)
\(954\) 4.07505 0.131935
\(955\) −15.4394 −0.499606
\(956\) −19.0338 −0.615598
\(957\) 12.1916 0.394097
\(958\) −9.65205 −0.311844
\(959\) 28.2034 0.910735
\(960\) 5.79017 0.186877
\(961\) 53.4132 1.72301
\(962\) −8.64483 −0.278721
\(963\) −0.0137270 −0.000442346 0
\(964\) 7.58362 0.244252
\(965\) 2.73093 0.0879118
\(966\) 0.496651 0.0159795
\(967\) −35.8135 −1.15169 −0.575843 0.817561i \(-0.695326\pi\)
−0.575843 + 0.817561i \(0.695326\pi\)
\(968\) −1.59465 −0.0512539
\(969\) −9.94452 −0.319464
\(970\) 5.08201 0.163174
\(971\) 35.3725 1.13516 0.567579 0.823319i \(-0.307880\pi\)
0.567579 + 0.823319i \(0.307880\pi\)
\(972\) −18.6006 −0.596615
\(973\) 25.6453 0.822149
\(974\) −4.20940 −0.134878
\(975\) 2.88843 0.0925038
\(976\) −11.8129 −0.378121
\(977\) −23.2763 −0.744676 −0.372338 0.928097i \(-0.621444\pi\)
−0.372338 + 0.928097i \(0.621444\pi\)
\(978\) 11.9854 0.383251
\(979\) −6.12001 −0.195596
\(980\) −8.51471 −0.271992
\(981\) 17.8220 0.569013
\(982\) 11.4627 0.365790
\(983\) 47.3065 1.50884 0.754422 0.656390i \(-0.227917\pi\)
0.754422 + 0.656390i \(0.227917\pi\)
\(984\) −8.35355 −0.266301
\(985\) 9.84862 0.313803
\(986\) 25.0598 0.798067
\(987\) −25.8844 −0.823911
\(988\) 3.85398 0.122612
\(989\) 3.94635 0.125487
\(990\) 0.430297 0.0136757
\(991\) 15.6916 0.498461 0.249231 0.968444i \(-0.419822\pi\)
0.249231 + 0.968444i \(0.419822\pi\)
\(992\) −40.7435 −1.29361
\(993\) 5.13040 0.162808
\(994\) 2.07379 0.0657767
\(995\) −16.6457 −0.527704
\(996\) −39.0553 −1.23751
\(997\) 26.7697 0.847804 0.423902 0.905708i \(-0.360660\pi\)
0.423902 + 0.905708i \(0.360660\pi\)
\(998\) −3.59558 −0.113816
\(999\) 56.9772 1.80268
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 4015.2.a.f.1.19 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4015.2.a.f.1.19 31 1.1 even 1 trivial