Properties

Label 6018.2.a.s.1.1
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
Analytic rank $0$
Dimension $8$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 17x^{6} + 37x^{5} + 105x^{4} - 117x^{3} - 238x^{2} + 42x + 90 \) 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 1.1
Root \(-2.36472\) of defining polynomial
Character \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.85158 q^{5} +1.00000 q^{6} +4.10228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.85158 q^{5} +1.00000 q^{6} +4.10228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.85158 q^{10} -2.03357 q^{11} -1.00000 q^{12} +3.36769 q^{13} -4.10228 q^{14} +3.85158 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.19110 q^{19} -3.85158 q^{20} -4.10228 q^{21} +2.03357 q^{22} +0.906308 q^{23} +1.00000 q^{24} +9.83469 q^{25} -3.36769 q^{26} -1.00000 q^{27} +4.10228 q^{28} -2.02807 q^{29} -3.85158 q^{30} +6.66369 q^{31} -1.00000 q^{32} +2.03357 q^{33} +1.00000 q^{34} -15.8003 q^{35} +1.00000 q^{36} +0.914678 q^{37} +6.19110 q^{38} -3.36769 q^{39} +3.85158 q^{40} +4.76289 q^{41} +4.10228 q^{42} -2.58915 q^{43} -2.03357 q^{44} -3.85158 q^{45} -0.906308 q^{46} +7.74937 q^{47} -1.00000 q^{48} +9.82874 q^{49} -9.83469 q^{50} +1.00000 q^{51} +3.36769 q^{52} -8.72500 q^{53} +1.00000 q^{54} +7.83245 q^{55} -4.10228 q^{56} +6.19110 q^{57} +2.02807 q^{58} +1.00000 q^{59} +3.85158 q^{60} -10.5221 q^{61} -6.66369 q^{62} +4.10228 q^{63} +1.00000 q^{64} -12.9709 q^{65} -2.03357 q^{66} +5.05990 q^{67} -1.00000 q^{68} -0.906308 q^{69} +15.8003 q^{70} +7.71254 q^{71} -1.00000 q^{72} -2.79488 q^{73} -0.914678 q^{74} -9.83469 q^{75} -6.19110 q^{76} -8.34227 q^{77} +3.36769 q^{78} +0.640835 q^{79} -3.85158 q^{80} +1.00000 q^{81} -4.76289 q^{82} -6.57742 q^{83} -4.10228 q^{84} +3.85158 q^{85} +2.58915 q^{86} +2.02807 q^{87} +2.03357 q^{88} +12.8278 q^{89} +3.85158 q^{90} +13.8152 q^{91} +0.906308 q^{92} -6.66369 q^{93} -7.74937 q^{94} +23.8455 q^{95} +1.00000 q^{96} -5.13805 q^{97} -9.82874 q^{98} -2.03357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - q^{5} + 8 q^{6} + 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 8 q^{3} + 8 q^{4} - q^{5} + 8 q^{6} + 6 q^{7} - 8 q^{8} + 8 q^{9} + q^{10} - 8 q^{12} + 6 q^{13} - 6 q^{14} + q^{15} + 8 q^{16} - 8 q^{17} - 8 q^{18} - 7 q^{19} - q^{20} - 6 q^{21} - 5 q^{23} + 8 q^{24} + 9 q^{25} - 6 q^{26} - 8 q^{27} + 6 q^{28} - 15 q^{29} - q^{30} + 21 q^{31} - 8 q^{32} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} + 7 q^{38} - 6 q^{39} + q^{40} - q^{41} + 6 q^{42} + 14 q^{43} - q^{45} + 5 q^{46} - 8 q^{47} - 8 q^{48} + 2 q^{49} - 9 q^{50} + 8 q^{51} + 6 q^{52} + 8 q^{53} + 8 q^{54} + 24 q^{55} - 6 q^{56} + 7 q^{57} + 15 q^{58} + 8 q^{59} + q^{60} - 21 q^{62} + 6 q^{63} + 8 q^{64} + 6 q^{65} + 15 q^{67} - 8 q^{68} + 5 q^{69} + 2 q^{70} - 22 q^{71} - 8 q^{72} + 13 q^{73} - 7 q^{74} - 9 q^{75} - 7 q^{76} - 6 q^{77} + 6 q^{78} + 26 q^{79} - q^{80} + 8 q^{81} + q^{82} + 30 q^{83} - 6 q^{84} + q^{85} - 14 q^{86} + 15 q^{87} - 6 q^{89} + q^{90} + 3 q^{91} - 5 q^{92} - 21 q^{93} + 8 q^{94} + 37 q^{95} + 8 q^{96} + 23 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.85158 −1.72248 −0.861240 0.508198i \(-0.830312\pi\)
−0.861240 + 0.508198i \(0.830312\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.10228 1.55052 0.775259 0.631644i \(-0.217619\pi\)
0.775259 + 0.631644i \(0.217619\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.85158 1.21798
\(11\) −2.03357 −0.613144 −0.306572 0.951848i \(-0.599182\pi\)
−0.306572 + 0.951848i \(0.599182\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.36769 0.934029 0.467014 0.884250i \(-0.345330\pi\)
0.467014 + 0.884250i \(0.345330\pi\)
\(14\) −4.10228 −1.09638
\(15\) 3.85158 0.994474
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −6.19110 −1.42034 −0.710168 0.704033i \(-0.751381\pi\)
−0.710168 + 0.704033i \(0.751381\pi\)
\(20\) −3.85158 −0.861240
\(21\) −4.10228 −0.895192
\(22\) 2.03357 0.433558
\(23\) 0.906308 0.188978 0.0944891 0.995526i \(-0.469878\pi\)
0.0944891 + 0.995526i \(0.469878\pi\)
\(24\) 1.00000 0.204124
\(25\) 9.83469 1.96694
\(26\) −3.36769 −0.660458
\(27\) −1.00000 −0.192450
\(28\) 4.10228 0.775259
\(29\) −2.02807 −0.376603 −0.188301 0.982111i \(-0.560298\pi\)
−0.188301 + 0.982111i \(0.560298\pi\)
\(30\) −3.85158 −0.703200
\(31\) 6.66369 1.19683 0.598417 0.801185i \(-0.295797\pi\)
0.598417 + 0.801185i \(0.295797\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.03357 0.353999
\(34\) 1.00000 0.171499
\(35\) −15.8003 −2.67074
\(36\) 1.00000 0.166667
\(37\) 0.914678 0.150372 0.0751860 0.997170i \(-0.476045\pi\)
0.0751860 + 0.997170i \(0.476045\pi\)
\(38\) 6.19110 1.00433
\(39\) −3.36769 −0.539262
\(40\) 3.85158 0.608989
\(41\) 4.76289 0.743839 0.371919 0.928265i \(-0.378700\pi\)
0.371919 + 0.928265i \(0.378700\pi\)
\(42\) 4.10228 0.632996
\(43\) −2.58915 −0.394842 −0.197421 0.980319i \(-0.563257\pi\)
−0.197421 + 0.980319i \(0.563257\pi\)
\(44\) −2.03357 −0.306572
\(45\) −3.85158 −0.574160
\(46\) −0.906308 −0.133628
\(47\) 7.74937 1.13036 0.565181 0.824967i \(-0.308806\pi\)
0.565181 + 0.824967i \(0.308806\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.82874 1.40411
\(50\) −9.83469 −1.39084
\(51\) 1.00000 0.140028
\(52\) 3.36769 0.467014
\(53\) −8.72500 −1.19847 −0.599236 0.800573i \(-0.704529\pi\)
−0.599236 + 0.800573i \(0.704529\pi\)
\(54\) 1.00000 0.136083
\(55\) 7.83245 1.05613
\(56\) −4.10228 −0.548191
\(57\) 6.19110 0.820031
\(58\) 2.02807 0.266298
\(59\) 1.00000 0.130189
\(60\) 3.85158 0.497237
\(61\) −10.5221 −1.34721 −0.673606 0.739090i \(-0.735256\pi\)
−0.673606 + 0.739090i \(0.735256\pi\)
\(62\) −6.66369 −0.846289
\(63\) 4.10228 0.516839
\(64\) 1.00000 0.125000
\(65\) −12.9709 −1.60885
\(66\) −2.03357 −0.250315
\(67\) 5.05990 0.618165 0.309083 0.951035i \(-0.399978\pi\)
0.309083 + 0.951035i \(0.399978\pi\)
\(68\) −1.00000 −0.121268
\(69\) −0.906308 −0.109107
\(70\) 15.8003 1.88850
\(71\) 7.71254 0.915311 0.457655 0.889130i \(-0.348689\pi\)
0.457655 + 0.889130i \(0.348689\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.79488 −0.327116 −0.163558 0.986534i \(-0.552297\pi\)
−0.163558 + 0.986534i \(0.552297\pi\)
\(74\) −0.914678 −0.106329
\(75\) −9.83469 −1.13561
\(76\) −6.19110 −0.710168
\(77\) −8.34227 −0.950690
\(78\) 3.36769 0.381316
\(79\) 0.640835 0.0720996 0.0360498 0.999350i \(-0.488523\pi\)
0.0360498 + 0.999350i \(0.488523\pi\)
\(80\) −3.85158 −0.430620
\(81\) 1.00000 0.111111
\(82\) −4.76289 −0.525974
\(83\) −6.57742 −0.721966 −0.360983 0.932572i \(-0.617559\pi\)
−0.360983 + 0.932572i \(0.617559\pi\)
\(84\) −4.10228 −0.447596
\(85\) 3.85158 0.417763
\(86\) 2.58915 0.279195
\(87\) 2.02807 0.217432
\(88\) 2.03357 0.216779
\(89\) 12.8278 1.35975 0.679874 0.733329i \(-0.262034\pi\)
0.679874 + 0.733329i \(0.262034\pi\)
\(90\) 3.85158 0.405992
\(91\) 13.8152 1.44823
\(92\) 0.906308 0.0944891
\(93\) −6.66369 −0.690992
\(94\) −7.74937 −0.799286
\(95\) 23.8455 2.44650
\(96\) 1.00000 0.102062
\(97\) −5.13805 −0.521690 −0.260845 0.965381i \(-0.584001\pi\)
−0.260845 + 0.965381i \(0.584001\pi\)
\(98\) −9.82874 −0.992853
\(99\) −2.03357 −0.204381
\(100\) 9.83469 0.983469
\(101\) 6.29084 0.625962 0.312981 0.949759i \(-0.398672\pi\)
0.312981 + 0.949759i \(0.398672\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −8.32588 −0.820374 −0.410187 0.912001i \(-0.634537\pi\)
−0.410187 + 0.912001i \(0.634537\pi\)
\(104\) −3.36769 −0.330229
\(105\) 15.8003 1.54195
\(106\) 8.72500 0.847447
\(107\) −8.96644 −0.866819 −0.433409 0.901197i \(-0.642690\pi\)
−0.433409 + 0.901197i \(0.642690\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.10396 −0.201523 −0.100761 0.994911i \(-0.532128\pi\)
−0.100761 + 0.994911i \(0.532128\pi\)
\(110\) −7.83245 −0.746795
\(111\) −0.914678 −0.0868174
\(112\) 4.10228 0.387629
\(113\) 2.72031 0.255905 0.127953 0.991780i \(-0.459159\pi\)
0.127953 + 0.991780i \(0.459159\pi\)
\(114\) −6.19110 −0.579849
\(115\) −3.49072 −0.325511
\(116\) −2.02807 −0.188301
\(117\) 3.36769 0.311343
\(118\) −1.00000 −0.0920575
\(119\) −4.10228 −0.376056
\(120\) −3.85158 −0.351600
\(121\) −6.86460 −0.624055
\(122\) 10.5221 0.952623
\(123\) −4.76289 −0.429456
\(124\) 6.66369 0.598417
\(125\) −18.6212 −1.66553
\(126\) −4.10228 −0.365461
\(127\) −17.3191 −1.53682 −0.768410 0.639958i \(-0.778952\pi\)
−0.768410 + 0.639958i \(0.778952\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.58915 0.227962
\(130\) 12.9709 1.13763
\(131\) −13.5534 −1.18416 −0.592081 0.805878i \(-0.701694\pi\)
−0.592081 + 0.805878i \(0.701694\pi\)
\(132\) 2.03357 0.176999
\(133\) −25.3976 −2.20226
\(134\) −5.05990 −0.437109
\(135\) 3.85158 0.331491
\(136\) 1.00000 0.0857493
\(137\) −2.68219 −0.229155 −0.114578 0.993414i \(-0.536551\pi\)
−0.114578 + 0.993414i \(0.536551\pi\)
\(138\) 0.906308 0.0771501
\(139\) 14.7398 1.25022 0.625108 0.780538i \(-0.285055\pi\)
0.625108 + 0.780538i \(0.285055\pi\)
\(140\) −15.8003 −1.33537
\(141\) −7.74937 −0.652615
\(142\) −7.71254 −0.647222
\(143\) −6.84842 −0.572694
\(144\) 1.00000 0.0833333
\(145\) 7.81127 0.648691
\(146\) 2.79488 0.231306
\(147\) −9.82874 −0.810661
\(148\) 0.914678 0.0751860
\(149\) 3.89820 0.319353 0.159677 0.987169i \(-0.448955\pi\)
0.159677 + 0.987169i \(0.448955\pi\)
\(150\) 9.83469 0.802999
\(151\) −10.0378 −0.816869 −0.408434 0.912788i \(-0.633925\pi\)
−0.408434 + 0.912788i \(0.633925\pi\)
\(152\) 6.19110 0.502164
\(153\) −1.00000 −0.0808452
\(154\) 8.34227 0.672239
\(155\) −25.6657 −2.06152
\(156\) −3.36769 −0.269631
\(157\) 10.9642 0.875041 0.437521 0.899208i \(-0.355857\pi\)
0.437521 + 0.899208i \(0.355857\pi\)
\(158\) −0.640835 −0.0509821
\(159\) 8.72500 0.691938
\(160\) 3.85158 0.304494
\(161\) 3.71793 0.293014
\(162\) −1.00000 −0.0785674
\(163\) 24.7990 1.94241 0.971205 0.238245i \(-0.0765721\pi\)
0.971205 + 0.238245i \(0.0765721\pi\)
\(164\) 4.76289 0.371919
\(165\) −7.83245 −0.609756
\(166\) 6.57742 0.510507
\(167\) −9.47876 −0.733488 −0.366744 0.930322i \(-0.619528\pi\)
−0.366744 + 0.930322i \(0.619528\pi\)
\(168\) 4.10228 0.316498
\(169\) −1.65868 −0.127591
\(170\) −3.85158 −0.295403
\(171\) −6.19110 −0.473445
\(172\) −2.58915 −0.197421
\(173\) −4.53051 −0.344448 −0.172224 0.985058i \(-0.555095\pi\)
−0.172224 + 0.985058i \(0.555095\pi\)
\(174\) −2.02807 −0.153747
\(175\) 40.3447 3.04977
\(176\) −2.03357 −0.153286
\(177\) −1.00000 −0.0751646
\(178\) −12.8278 −0.961487
\(179\) 1.28226 0.0958406 0.0479203 0.998851i \(-0.484741\pi\)
0.0479203 + 0.998851i \(0.484741\pi\)
\(180\) −3.85158 −0.287080
\(181\) 2.18692 0.162553 0.0812763 0.996692i \(-0.474100\pi\)
0.0812763 + 0.996692i \(0.474100\pi\)
\(182\) −13.8152 −1.02405
\(183\) 10.5221 0.777814
\(184\) −0.906308 −0.0668139
\(185\) −3.52296 −0.259013
\(186\) 6.66369 0.488605
\(187\) 2.03357 0.148709
\(188\) 7.74937 0.565181
\(189\) −4.10228 −0.298397
\(190\) −23.8455 −1.72994
\(191\) −7.49899 −0.542608 −0.271304 0.962494i \(-0.587455\pi\)
−0.271304 + 0.962494i \(0.587455\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 4.84547 0.348784 0.174392 0.984676i \(-0.444204\pi\)
0.174392 + 0.984676i \(0.444204\pi\)
\(194\) 5.13805 0.368891
\(195\) 12.9709 0.928868
\(196\) 9.82874 0.702053
\(197\) 15.3061 1.09052 0.545258 0.838268i \(-0.316432\pi\)
0.545258 + 0.838268i \(0.316432\pi\)
\(198\) 2.03357 0.144519
\(199\) 6.62692 0.469770 0.234885 0.972023i \(-0.424529\pi\)
0.234885 + 0.972023i \(0.424529\pi\)
\(200\) −9.83469 −0.695418
\(201\) −5.05990 −0.356898
\(202\) −6.29084 −0.442622
\(203\) −8.31971 −0.583929
\(204\) 1.00000 0.0700140
\(205\) −18.3447 −1.28125
\(206\) 8.32588 0.580092
\(207\) 0.906308 0.0629928
\(208\) 3.36769 0.233507
\(209\) 12.5900 0.870869
\(210\) −15.8003 −1.09032
\(211\) 16.6354 1.14522 0.572612 0.819826i \(-0.305930\pi\)
0.572612 + 0.819826i \(0.305930\pi\)
\(212\) −8.72500 −0.599236
\(213\) −7.71254 −0.528455
\(214\) 8.96644 0.612933
\(215\) 9.97232 0.680107
\(216\) 1.00000 0.0680414
\(217\) 27.3363 1.85571
\(218\) 2.10396 0.142498
\(219\) 2.79488 0.188860
\(220\) 7.83245 0.528064
\(221\) −3.36769 −0.226535
\(222\) 0.914678 0.0613891
\(223\) −1.93856 −0.129815 −0.0649076 0.997891i \(-0.520675\pi\)
−0.0649076 + 0.997891i \(0.520675\pi\)
\(224\) −4.10228 −0.274095
\(225\) 9.83469 0.655646
\(226\) −2.72031 −0.180952
\(227\) −9.70323 −0.644026 −0.322013 0.946735i \(-0.604359\pi\)
−0.322013 + 0.946735i \(0.604359\pi\)
\(228\) 6.19110 0.410015
\(229\) −21.2594 −1.40486 −0.702431 0.711752i \(-0.747902\pi\)
−0.702431 + 0.711752i \(0.747902\pi\)
\(230\) 3.49072 0.230171
\(231\) 8.34227 0.548881
\(232\) 2.02807 0.133149
\(233\) 9.13446 0.598418 0.299209 0.954188i \(-0.403277\pi\)
0.299209 + 0.954188i \(0.403277\pi\)
\(234\) −3.36769 −0.220153
\(235\) −29.8473 −1.94703
\(236\) 1.00000 0.0650945
\(237\) −0.640835 −0.0416267
\(238\) 4.10228 0.265912
\(239\) −14.6133 −0.945253 −0.472626 0.881263i \(-0.656694\pi\)
−0.472626 + 0.881263i \(0.656694\pi\)
\(240\) 3.85158 0.248619
\(241\) −16.3402 −1.05257 −0.526283 0.850309i \(-0.676415\pi\)
−0.526283 + 0.850309i \(0.676415\pi\)
\(242\) 6.86460 0.441273
\(243\) −1.00000 −0.0641500
\(244\) −10.5221 −0.673606
\(245\) −37.8562 −2.41854
\(246\) 4.76289 0.303671
\(247\) −20.8497 −1.32663
\(248\) −6.66369 −0.423145
\(249\) 6.57742 0.416827
\(250\) 18.6212 1.17771
\(251\) −7.29327 −0.460347 −0.230174 0.973150i \(-0.573929\pi\)
−0.230174 + 0.973150i \(0.573929\pi\)
\(252\) 4.10228 0.258420
\(253\) −1.84304 −0.115871
\(254\) 17.3191 1.08670
\(255\) −3.85158 −0.241195
\(256\) 1.00000 0.0625000
\(257\) 12.3591 0.770937 0.385469 0.922721i \(-0.374040\pi\)
0.385469 + 0.922721i \(0.374040\pi\)
\(258\) −2.58915 −0.161193
\(259\) 3.75227 0.233155
\(260\) −12.9709 −0.804423
\(261\) −2.02807 −0.125534
\(262\) 13.5534 0.837329
\(263\) 20.2388 1.24798 0.623988 0.781434i \(-0.285512\pi\)
0.623988 + 0.781434i \(0.285512\pi\)
\(264\) −2.03357 −0.125157
\(265\) 33.6051 2.06434
\(266\) 25.3976 1.55723
\(267\) −12.8278 −0.785051
\(268\) 5.05990 0.309083
\(269\) 28.6320 1.74572 0.872862 0.487968i \(-0.162262\pi\)
0.872862 + 0.487968i \(0.162262\pi\)
\(270\) −3.85158 −0.234400
\(271\) 0.0160069 0.000972353 0 0.000486176 1.00000i \(-0.499845\pi\)
0.000486176 1.00000i \(0.499845\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −13.8152 −0.836135
\(274\) 2.68219 0.162037
\(275\) −19.9995 −1.20602
\(276\) −0.906308 −0.0545533
\(277\) 26.2831 1.57920 0.789598 0.613624i \(-0.210289\pi\)
0.789598 + 0.613624i \(0.210289\pi\)
\(278\) −14.7398 −0.884037
\(279\) 6.66369 0.398945
\(280\) 15.8003 0.944248
\(281\) 30.6341 1.82748 0.913740 0.406300i \(-0.133181\pi\)
0.913740 + 0.406300i \(0.133181\pi\)
\(282\) 7.74937 0.461468
\(283\) 0.234324 0.0139291 0.00696455 0.999976i \(-0.497783\pi\)
0.00696455 + 0.999976i \(0.497783\pi\)
\(284\) 7.71254 0.457655
\(285\) −23.8455 −1.41249
\(286\) 6.84842 0.404956
\(287\) 19.5387 1.15334
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −7.81127 −0.458694
\(291\) 5.13805 0.301198
\(292\) −2.79488 −0.163558
\(293\) −3.63443 −0.212326 −0.106163 0.994349i \(-0.533856\pi\)
−0.106163 + 0.994349i \(0.533856\pi\)
\(294\) 9.82874 0.573224
\(295\) −3.85158 −0.224248
\(296\) −0.914678 −0.0531646
\(297\) 2.03357 0.118000
\(298\) −3.89820 −0.225817
\(299\) 3.05216 0.176511
\(300\) −9.83469 −0.567806
\(301\) −10.6214 −0.612209
\(302\) 10.0378 0.577613
\(303\) −6.29084 −0.361399
\(304\) −6.19110 −0.355084
\(305\) 40.5266 2.32055
\(306\) 1.00000 0.0571662
\(307\) 15.7695 0.900011 0.450005 0.893026i \(-0.351422\pi\)
0.450005 + 0.893026i \(0.351422\pi\)
\(308\) −8.34227 −0.475345
\(309\) 8.32588 0.473643
\(310\) 25.6657 1.45772
\(311\) 31.5974 1.79173 0.895863 0.444331i \(-0.146559\pi\)
0.895863 + 0.444331i \(0.146559\pi\)
\(312\) 3.36769 0.190658
\(313\) 12.5346 0.708497 0.354249 0.935151i \(-0.384737\pi\)
0.354249 + 0.935151i \(0.384737\pi\)
\(314\) −10.9642 −0.618748
\(315\) −15.8003 −0.890245
\(316\) 0.640835 0.0360498
\(317\) 8.78766 0.493564 0.246782 0.969071i \(-0.420627\pi\)
0.246782 + 0.969071i \(0.420627\pi\)
\(318\) −8.72500 −0.489274
\(319\) 4.12421 0.230912
\(320\) −3.85158 −0.215310
\(321\) 8.96644 0.500458
\(322\) −3.71793 −0.207192
\(323\) 6.19110 0.344482
\(324\) 1.00000 0.0555556
\(325\) 33.1202 1.83718
\(326\) −24.7990 −1.37349
\(327\) 2.10396 0.116349
\(328\) −4.76289 −0.262987
\(329\) 31.7901 1.75265
\(330\) 7.83245 0.431162
\(331\) 28.1629 1.54797 0.773987 0.633202i \(-0.218260\pi\)
0.773987 + 0.633202i \(0.218260\pi\)
\(332\) −6.57742 −0.360983
\(333\) 0.914678 0.0501240
\(334\) 9.47876 0.518655
\(335\) −19.4886 −1.06478
\(336\) −4.10228 −0.223798
\(337\) −28.0093 −1.52576 −0.762882 0.646537i \(-0.776217\pi\)
−0.762882 + 0.646537i \(0.776217\pi\)
\(338\) 1.65868 0.0902201
\(339\) −2.72031 −0.147747
\(340\) 3.85158 0.208881
\(341\) −13.5511 −0.733831
\(342\) 6.19110 0.334776
\(343\) 11.6043 0.626573
\(344\) 2.58915 0.139598
\(345\) 3.49072 0.187934
\(346\) 4.53051 0.243562
\(347\) 29.5665 1.58721 0.793607 0.608431i \(-0.208201\pi\)
0.793607 + 0.608431i \(0.208201\pi\)
\(348\) 2.02807 0.108716
\(349\) 28.6964 1.53608 0.768042 0.640399i \(-0.221231\pi\)
0.768042 + 0.640399i \(0.221231\pi\)
\(350\) −40.3447 −2.15652
\(351\) −3.36769 −0.179754
\(352\) 2.03357 0.108390
\(353\) −5.29416 −0.281780 −0.140890 0.990025i \(-0.544996\pi\)
−0.140890 + 0.990025i \(0.544996\pi\)
\(354\) 1.00000 0.0531494
\(355\) −29.7055 −1.57660
\(356\) 12.8278 0.679874
\(357\) 4.10228 0.217116
\(358\) −1.28226 −0.0677696
\(359\) 1.73252 0.0914391 0.0457195 0.998954i \(-0.485442\pi\)
0.0457195 + 0.998954i \(0.485442\pi\)
\(360\) 3.85158 0.202996
\(361\) 19.3297 1.01735
\(362\) −2.18692 −0.114942
\(363\) 6.86460 0.360298
\(364\) 13.8152 0.724114
\(365\) 10.7647 0.563451
\(366\) −10.5221 −0.549997
\(367\) −28.1227 −1.46799 −0.733996 0.679153i \(-0.762347\pi\)
−0.733996 + 0.679153i \(0.762347\pi\)
\(368\) 0.906308 0.0472446
\(369\) 4.76289 0.247946
\(370\) 3.52296 0.183150
\(371\) −35.7924 −1.85825
\(372\) −6.66369 −0.345496
\(373\) −12.4392 −0.644079 −0.322040 0.946726i \(-0.604368\pi\)
−0.322040 + 0.946726i \(0.604368\pi\)
\(374\) −2.03357 −0.105153
\(375\) 18.6212 0.961595
\(376\) −7.74937 −0.399643
\(377\) −6.82990 −0.351758
\(378\) 4.10228 0.210999
\(379\) −6.21140 −0.319058 −0.159529 0.987193i \(-0.550998\pi\)
−0.159529 + 0.987193i \(0.550998\pi\)
\(380\) 23.8455 1.22325
\(381\) 17.3191 0.887283
\(382\) 7.49899 0.383682
\(383\) −5.50310 −0.281195 −0.140598 0.990067i \(-0.544902\pi\)
−0.140598 + 0.990067i \(0.544902\pi\)
\(384\) 1.00000 0.0510310
\(385\) 32.1310 1.63755
\(386\) −4.84547 −0.246628
\(387\) −2.58915 −0.131614
\(388\) −5.13805 −0.260845
\(389\) 24.9539 1.26522 0.632608 0.774473i \(-0.281985\pi\)
0.632608 + 0.774473i \(0.281985\pi\)
\(390\) −12.9709 −0.656809
\(391\) −0.906308 −0.0458340
\(392\) −9.82874 −0.496426
\(393\) 13.5534 0.683677
\(394\) −15.3061 −0.771111
\(395\) −2.46823 −0.124190
\(396\) −2.03357 −0.102191
\(397\) −12.0760 −0.606075 −0.303037 0.952979i \(-0.598001\pi\)
−0.303037 + 0.952979i \(0.598001\pi\)
\(398\) −6.62692 −0.332177
\(399\) 25.3976 1.27147
\(400\) 9.83469 0.491735
\(401\) 39.3071 1.96290 0.981452 0.191711i \(-0.0614035\pi\)
0.981452 + 0.191711i \(0.0614035\pi\)
\(402\) 5.05990 0.252365
\(403\) 22.4412 1.11788
\(404\) 6.29084 0.312981
\(405\) −3.85158 −0.191387
\(406\) 8.31971 0.412900
\(407\) −1.86006 −0.0921997
\(408\) −1.00000 −0.0495074
\(409\) 28.9306 1.43053 0.715263 0.698856i \(-0.246307\pi\)
0.715263 + 0.698856i \(0.246307\pi\)
\(410\) 18.3447 0.905979
\(411\) 2.68219 0.132303
\(412\) −8.32588 −0.410187
\(413\) 4.10228 0.201860
\(414\) −0.906308 −0.0445426
\(415\) 25.3335 1.24357
\(416\) −3.36769 −0.165114
\(417\) −14.7398 −0.721813
\(418\) −12.5900 −0.615798
\(419\) 16.0554 0.784360 0.392180 0.919889i \(-0.371721\pi\)
0.392180 + 0.919889i \(0.371721\pi\)
\(420\) 15.8003 0.770975
\(421\) 19.7480 0.962461 0.481231 0.876594i \(-0.340190\pi\)
0.481231 + 0.876594i \(0.340190\pi\)
\(422\) −16.6354 −0.809796
\(423\) 7.74937 0.376787
\(424\) 8.72500 0.423724
\(425\) −9.83469 −0.477053
\(426\) 7.71254 0.373674
\(427\) −43.1645 −2.08888
\(428\) −8.96644 −0.433409
\(429\) 6.84842 0.330645
\(430\) −9.97232 −0.480908
\(431\) 20.4445 0.984778 0.492389 0.870375i \(-0.336124\pi\)
0.492389 + 0.870375i \(0.336124\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.91123 0.476303 0.238152 0.971228i \(-0.423458\pi\)
0.238152 + 0.971228i \(0.423458\pi\)
\(434\) −27.3363 −1.31219
\(435\) −7.81127 −0.374522
\(436\) −2.10396 −0.100761
\(437\) −5.61104 −0.268412
\(438\) −2.79488 −0.133545
\(439\) −16.4570 −0.785452 −0.392726 0.919656i \(-0.628468\pi\)
−0.392726 + 0.919656i \(0.628468\pi\)
\(440\) −7.83245 −0.373398
\(441\) 9.82874 0.468035
\(442\) 3.36769 0.160185
\(443\) −22.6158 −1.07451 −0.537254 0.843420i \(-0.680538\pi\)
−0.537254 + 0.843420i \(0.680538\pi\)
\(444\) −0.914678 −0.0434087
\(445\) −49.4075 −2.34214
\(446\) 1.93856 0.0917933
\(447\) −3.89820 −0.184379
\(448\) 4.10228 0.193815
\(449\) −12.8141 −0.604734 −0.302367 0.953192i \(-0.597777\pi\)
−0.302367 + 0.953192i \(0.597777\pi\)
\(450\) −9.83469 −0.463612
\(451\) −9.68566 −0.456080
\(452\) 2.72031 0.127953
\(453\) 10.0378 0.471619
\(454\) 9.70323 0.455395
\(455\) −53.2104 −2.49454
\(456\) −6.19110 −0.289925
\(457\) 30.5625 1.42966 0.714828 0.699301i \(-0.246505\pi\)
0.714828 + 0.699301i \(0.246505\pi\)
\(458\) 21.2594 0.993387
\(459\) 1.00000 0.0466760
\(460\) −3.49072 −0.162756
\(461\) −8.23817 −0.383690 −0.191845 0.981425i \(-0.561447\pi\)
−0.191845 + 0.981425i \(0.561447\pi\)
\(462\) −8.34227 −0.388118
\(463\) −14.1573 −0.657946 −0.328973 0.944339i \(-0.606703\pi\)
−0.328973 + 0.944339i \(0.606703\pi\)
\(464\) −2.02807 −0.0941507
\(465\) 25.6657 1.19022
\(466\) −9.13446 −0.423146
\(467\) −9.80450 −0.453698 −0.226849 0.973930i \(-0.572842\pi\)
−0.226849 + 0.973930i \(0.572842\pi\)
\(468\) 3.36769 0.155671
\(469\) 20.7572 0.958477
\(470\) 29.8473 1.37675
\(471\) −10.9642 −0.505205
\(472\) −1.00000 −0.0460287
\(473\) 5.26521 0.242095
\(474\) 0.640835 0.0294345
\(475\) −60.8875 −2.79371
\(476\) −4.10228 −0.188028
\(477\) −8.72500 −0.399490
\(478\) 14.6133 0.668395
\(479\) 37.7151 1.72325 0.861624 0.507547i \(-0.169448\pi\)
0.861624 + 0.507547i \(0.169448\pi\)
\(480\) −3.85158 −0.175800
\(481\) 3.08035 0.140452
\(482\) 16.3402 0.744277
\(483\) −3.71793 −0.169172
\(484\) −6.86460 −0.312027
\(485\) 19.7896 0.898601
\(486\) 1.00000 0.0453609
\(487\) 25.9042 1.17383 0.586915 0.809648i \(-0.300342\pi\)
0.586915 + 0.809648i \(0.300342\pi\)
\(488\) 10.5221 0.476312
\(489\) −24.7990 −1.12145
\(490\) 37.8562 1.71017
\(491\) 9.43153 0.425639 0.212819 0.977092i \(-0.431735\pi\)
0.212819 + 0.977092i \(0.431735\pi\)
\(492\) −4.76289 −0.214728
\(493\) 2.02807 0.0913396
\(494\) 20.8497 0.938072
\(495\) 7.83245 0.352043
\(496\) 6.66369 0.299208
\(497\) 31.6391 1.41921
\(498\) −6.57742 −0.294741
\(499\) 15.0782 0.674994 0.337497 0.941327i \(-0.390420\pi\)
0.337497 + 0.941327i \(0.390420\pi\)
\(500\) −18.6212 −0.832766
\(501\) 9.47876 0.423480
\(502\) 7.29327 0.325515
\(503\) −6.89424 −0.307399 −0.153699 0.988118i \(-0.549119\pi\)
−0.153699 + 0.988118i \(0.549119\pi\)
\(504\) −4.10228 −0.182730
\(505\) −24.2297 −1.07821
\(506\) 1.84304 0.0819330
\(507\) 1.65868 0.0736644
\(508\) −17.3191 −0.768410
\(509\) −15.3575 −0.680707 −0.340354 0.940298i \(-0.610547\pi\)
−0.340354 + 0.940298i \(0.610547\pi\)
\(510\) 3.85158 0.170551
\(511\) −11.4654 −0.507199
\(512\) −1.00000 −0.0441942
\(513\) 6.19110 0.273344
\(514\) −12.3591 −0.545135
\(515\) 32.0678 1.41308
\(516\) 2.58915 0.113981
\(517\) −15.7589 −0.693074
\(518\) −3.75227 −0.164865
\(519\) 4.53051 0.198867
\(520\) 12.9709 0.568813
\(521\) −43.0445 −1.88581 −0.942907 0.333057i \(-0.891920\pi\)
−0.942907 + 0.333057i \(0.891920\pi\)
\(522\) 2.02807 0.0887661
\(523\) 37.4999 1.63976 0.819878 0.572538i \(-0.194041\pi\)
0.819878 + 0.572538i \(0.194041\pi\)
\(524\) −13.5534 −0.592081
\(525\) −40.3447 −1.76079
\(526\) −20.2388 −0.882452
\(527\) −6.66369 −0.290275
\(528\) 2.03357 0.0884997
\(529\) −22.1786 −0.964287
\(530\) −33.6051 −1.45971
\(531\) 1.00000 0.0433963
\(532\) −25.3976 −1.10113
\(533\) 16.0399 0.694767
\(534\) 12.8278 0.555115
\(535\) 34.5350 1.49308
\(536\) −5.05990 −0.218554
\(537\) −1.28226 −0.0553336
\(538\) −28.6320 −1.23441
\(539\) −19.9874 −0.860919
\(540\) 3.85158 0.165746
\(541\) −26.7279 −1.14912 −0.574561 0.818461i \(-0.694827\pi\)
−0.574561 + 0.818461i \(0.694827\pi\)
\(542\) −0.0160069 −0.000687557 0
\(543\) −2.18692 −0.0938498
\(544\) 1.00000 0.0428746
\(545\) 8.10357 0.347119
\(546\) 13.8152 0.591237
\(547\) 29.7929 1.27385 0.636926 0.770925i \(-0.280206\pi\)
0.636926 + 0.770925i \(0.280206\pi\)
\(548\) −2.68219 −0.114578
\(549\) −10.5221 −0.449071
\(550\) 19.9995 0.852782
\(551\) 12.5560 0.534902
\(552\) 0.906308 0.0385750
\(553\) 2.62889 0.111792
\(554\) −26.2831 −1.11666
\(555\) 3.52296 0.149541
\(556\) 14.7398 0.625108
\(557\) 19.5006 0.826265 0.413133 0.910671i \(-0.364435\pi\)
0.413133 + 0.910671i \(0.364435\pi\)
\(558\) −6.66369 −0.282096
\(559\) −8.71945 −0.368793
\(560\) −15.8003 −0.667684
\(561\) −2.03357 −0.0858573
\(562\) −30.6341 −1.29222
\(563\) 43.2795 1.82401 0.912006 0.410177i \(-0.134533\pi\)
0.912006 + 0.410177i \(0.134533\pi\)
\(564\) −7.74937 −0.326307
\(565\) −10.4775 −0.440792
\(566\) −0.234324 −0.00984937
\(567\) 4.10228 0.172280
\(568\) −7.71254 −0.323611
\(569\) −3.79414 −0.159059 −0.0795293 0.996833i \(-0.525342\pi\)
−0.0795293 + 0.996833i \(0.525342\pi\)
\(570\) 23.8455 0.998779
\(571\) −45.4622 −1.90254 −0.951268 0.308366i \(-0.900218\pi\)
−0.951268 + 0.308366i \(0.900218\pi\)
\(572\) −6.84842 −0.286347
\(573\) 7.49899 0.313275
\(574\) −19.5387 −0.815531
\(575\) 8.91326 0.371709
\(576\) 1.00000 0.0416667
\(577\) 16.7057 0.695469 0.347735 0.937593i \(-0.386951\pi\)
0.347735 + 0.937593i \(0.386951\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −4.84547 −0.201371
\(580\) 7.81127 0.324345
\(581\) −26.9825 −1.11942
\(582\) −5.13805 −0.212979
\(583\) 17.7429 0.734835
\(584\) 2.79488 0.115653
\(585\) −12.9709 −0.536282
\(586\) 3.63443 0.150137
\(587\) 1.65489 0.0683045 0.0341522 0.999417i \(-0.489127\pi\)
0.0341522 + 0.999417i \(0.489127\pi\)
\(588\) −9.82874 −0.405330
\(589\) −41.2555 −1.69991
\(590\) 3.85158 0.158567
\(591\) −15.3061 −0.629610
\(592\) 0.914678 0.0375930
\(593\) −15.5054 −0.636729 −0.318364 0.947968i \(-0.603134\pi\)
−0.318364 + 0.947968i \(0.603134\pi\)
\(594\) −2.03357 −0.0834383
\(595\) 15.8003 0.647749
\(596\) 3.89820 0.159677
\(597\) −6.62692 −0.271222
\(598\) −3.05216 −0.124812
\(599\) 13.5626 0.554152 0.277076 0.960848i \(-0.410635\pi\)
0.277076 + 0.960848i \(0.410635\pi\)
\(600\) 9.83469 0.401500
\(601\) 33.5262 1.36756 0.683782 0.729687i \(-0.260334\pi\)
0.683782 + 0.729687i \(0.260334\pi\)
\(602\) 10.6214 0.432897
\(603\) 5.05990 0.206055
\(604\) −10.0378 −0.408434
\(605\) 26.4396 1.07492
\(606\) 6.29084 0.255548
\(607\) −0.334109 −0.0135611 −0.00678053 0.999977i \(-0.502158\pi\)
−0.00678053 + 0.999977i \(0.502158\pi\)
\(608\) 6.19110 0.251082
\(609\) 8.31971 0.337132
\(610\) −40.5266 −1.64087
\(611\) 26.0975 1.05579
\(612\) −1.00000 −0.0404226
\(613\) −3.86596 −0.156145 −0.0780723 0.996948i \(-0.524876\pi\)
−0.0780723 + 0.996948i \(0.524876\pi\)
\(614\) −15.7695 −0.636404
\(615\) 18.3447 0.739729
\(616\) 8.34227 0.336120
\(617\) 8.72380 0.351207 0.175603 0.984461i \(-0.443812\pi\)
0.175603 + 0.984461i \(0.443812\pi\)
\(618\) −8.32588 −0.334916
\(619\) 26.7813 1.07643 0.538215 0.842807i \(-0.319099\pi\)
0.538215 + 0.842807i \(0.319099\pi\)
\(620\) −25.6657 −1.03076
\(621\) −0.906308 −0.0363689
\(622\) −31.5974 −1.26694
\(623\) 52.6234 2.10831
\(624\) −3.36769 −0.134815
\(625\) 22.5477 0.901908
\(626\) −12.5346 −0.500983
\(627\) −12.5900 −0.502797
\(628\) 10.9642 0.437521
\(629\) −0.914678 −0.0364706
\(630\) 15.8003 0.629499
\(631\) −32.8906 −1.30935 −0.654677 0.755909i \(-0.727195\pi\)
−0.654677 + 0.755909i \(0.727195\pi\)
\(632\) −0.640835 −0.0254911
\(633\) −16.6354 −0.661196
\(634\) −8.78766 −0.349003
\(635\) 66.7059 2.64714
\(636\) 8.72500 0.345969
\(637\) 33.1001 1.31147
\(638\) −4.12421 −0.163279
\(639\) 7.71254 0.305104
\(640\) 3.85158 0.152247
\(641\) 1.52225 0.0601253 0.0300626 0.999548i \(-0.490429\pi\)
0.0300626 + 0.999548i \(0.490429\pi\)
\(642\) −8.96644 −0.353877
\(643\) −0.936604 −0.0369360 −0.0184680 0.999829i \(-0.505879\pi\)
−0.0184680 + 0.999829i \(0.505879\pi\)
\(644\) 3.71793 0.146507
\(645\) −9.97232 −0.392660
\(646\) −6.19110 −0.243585
\(647\) −8.74507 −0.343804 −0.171902 0.985114i \(-0.554991\pi\)
−0.171902 + 0.985114i \(0.554991\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.03357 −0.0798245
\(650\) −33.1202 −1.29908
\(651\) −27.3363 −1.07140
\(652\) 24.7990 0.971205
\(653\) 46.5279 1.82078 0.910390 0.413752i \(-0.135782\pi\)
0.910390 + 0.413752i \(0.135782\pi\)
\(654\) −2.10396 −0.0822713
\(655\) 52.2019 2.03970
\(656\) 4.76289 0.185960
\(657\) −2.79488 −0.109039
\(658\) −31.7901 −1.23931
\(659\) 1.76409 0.0687192 0.0343596 0.999410i \(-0.489061\pi\)
0.0343596 + 0.999410i \(0.489061\pi\)
\(660\) −7.83245 −0.304878
\(661\) −16.6486 −0.647555 −0.323777 0.946133i \(-0.604953\pi\)
−0.323777 + 0.946133i \(0.604953\pi\)
\(662\) −28.1629 −1.09458
\(663\) 3.36769 0.130790
\(664\) 6.57742 0.255254
\(665\) 97.8211 3.79334
\(666\) −0.914678 −0.0354430
\(667\) −1.83805 −0.0711697
\(668\) −9.47876 −0.366744
\(669\) 1.93856 0.0749489
\(670\) 19.4886 0.752912
\(671\) 21.3973 0.826035
\(672\) 4.10228 0.158249
\(673\) −17.2731 −0.665827 −0.332914 0.942957i \(-0.608032\pi\)
−0.332914 + 0.942957i \(0.608032\pi\)
\(674\) 28.0093 1.07888
\(675\) −9.83469 −0.378537
\(676\) −1.65868 −0.0637953
\(677\) 27.4725 1.05586 0.527928 0.849289i \(-0.322969\pi\)
0.527928 + 0.849289i \(0.322969\pi\)
\(678\) 2.72031 0.104473
\(679\) −21.0778 −0.808890
\(680\) −3.85158 −0.147701
\(681\) 9.70323 0.371828
\(682\) 13.5511 0.518897
\(683\) −27.1264 −1.03796 −0.518981 0.854786i \(-0.673688\pi\)
−0.518981 + 0.854786i \(0.673688\pi\)
\(684\) −6.19110 −0.236723
\(685\) 10.3307 0.394715
\(686\) −11.6043 −0.443054
\(687\) 21.2594 0.811097
\(688\) −2.58915 −0.0987104
\(689\) −29.3831 −1.11941
\(690\) −3.49072 −0.132889
\(691\) −26.3483 −1.00234 −0.501168 0.865350i \(-0.667096\pi\)
−0.501168 + 0.865350i \(0.667096\pi\)
\(692\) −4.53051 −0.172224
\(693\) −8.34227 −0.316897
\(694\) −29.5665 −1.12233
\(695\) −56.7717 −2.15347
\(696\) −2.02807 −0.0768737
\(697\) −4.76289 −0.180407
\(698\) −28.6964 −1.08618
\(699\) −9.13446 −0.345497
\(700\) 40.3447 1.52489
\(701\) 17.7786 0.671488 0.335744 0.941953i \(-0.391012\pi\)
0.335744 + 0.941953i \(0.391012\pi\)
\(702\) 3.36769 0.127105
\(703\) −5.66286 −0.213579
\(704\) −2.03357 −0.0766430
\(705\) 29.8473 1.12412
\(706\) 5.29416 0.199248
\(707\) 25.8068 0.970565
\(708\) −1.00000 −0.0375823
\(709\) −13.8529 −0.520258 −0.260129 0.965574i \(-0.583765\pi\)
−0.260129 + 0.965574i \(0.583765\pi\)
\(710\) 29.7055 1.11483
\(711\) 0.640835 0.0240332
\(712\) −12.8278 −0.480744
\(713\) 6.03935 0.226176
\(714\) −4.10228 −0.153524
\(715\) 26.3773 0.986454
\(716\) 1.28226 0.0479203
\(717\) 14.6133 0.545742
\(718\) −1.73252 −0.0646572
\(719\) −8.30614 −0.309767 −0.154883 0.987933i \(-0.549500\pi\)
−0.154883 + 0.987933i \(0.549500\pi\)
\(720\) −3.85158 −0.143540
\(721\) −34.1551 −1.27200
\(722\) −19.3297 −0.719377
\(723\) 16.3402 0.607700
\(724\) 2.18692 0.0812763
\(725\) −19.9454 −0.740754
\(726\) −6.86460 −0.254769
\(727\) 7.62966 0.282968 0.141484 0.989941i \(-0.454813\pi\)
0.141484 + 0.989941i \(0.454813\pi\)
\(728\) −13.8152 −0.512026
\(729\) 1.00000 0.0370370
\(730\) −10.7647 −0.398420
\(731\) 2.58915 0.0957632
\(732\) 10.5221 0.388907
\(733\) −9.68544 −0.357740 −0.178870 0.983873i \(-0.557244\pi\)
−0.178870 + 0.983873i \(0.557244\pi\)
\(734\) 28.1227 1.03803
\(735\) 37.8562 1.39635
\(736\) −0.906308 −0.0334070
\(737\) −10.2897 −0.379024
\(738\) −4.76289 −0.175325
\(739\) 52.2739 1.92293 0.961463 0.274935i \(-0.0886563\pi\)
0.961463 + 0.274935i \(0.0886563\pi\)
\(740\) −3.52296 −0.129506
\(741\) 20.8497 0.765932
\(742\) 35.7924 1.31398
\(743\) −1.96054 −0.0719251 −0.0359626 0.999353i \(-0.511450\pi\)
−0.0359626 + 0.999353i \(0.511450\pi\)
\(744\) 6.66369 0.244303
\(745\) −15.0142 −0.550079
\(746\) 12.4392 0.455433
\(747\) −6.57742 −0.240655
\(748\) 2.03357 0.0743546
\(749\) −36.7829 −1.34402
\(750\) −18.6212 −0.679951
\(751\) 17.7692 0.648407 0.324203 0.945987i \(-0.394904\pi\)
0.324203 + 0.945987i \(0.394904\pi\)
\(752\) 7.74937 0.282590
\(753\) 7.29327 0.265782
\(754\) 6.82990 0.248730
\(755\) 38.6616 1.40704
\(756\) −4.10228 −0.149199
\(757\) 20.9295 0.760697 0.380348 0.924843i \(-0.375804\pi\)
0.380348 + 0.924843i \(0.375804\pi\)
\(758\) 6.21140 0.225608
\(759\) 1.84304 0.0668981
\(760\) −23.8455 −0.864968
\(761\) −53.7044 −1.94678 −0.973391 0.229151i \(-0.926405\pi\)
−0.973391 + 0.229151i \(0.926405\pi\)
\(762\) −17.3191 −0.627404
\(763\) −8.63104 −0.312465
\(764\) −7.49899 −0.271304
\(765\) 3.85158 0.139254
\(766\) 5.50310 0.198835
\(767\) 3.36769 0.121600
\(768\) −1.00000 −0.0360844
\(769\) −11.0674 −0.399100 −0.199550 0.979888i \(-0.563948\pi\)
−0.199550 + 0.979888i \(0.563948\pi\)
\(770\) −32.1310 −1.15792
\(771\) −12.3591 −0.445101
\(772\) 4.84547 0.174392
\(773\) −18.8995 −0.679767 −0.339883 0.940468i \(-0.610388\pi\)
−0.339883 + 0.940468i \(0.610388\pi\)
\(774\) 2.58915 0.0930651
\(775\) 65.5353 2.35410
\(776\) 5.13805 0.184445
\(777\) −3.75227 −0.134612
\(778\) −24.9539 −0.894642
\(779\) −29.4875 −1.05650
\(780\) 12.9709 0.464434
\(781\) −15.6840 −0.561217
\(782\) 0.906308 0.0324095
\(783\) 2.02807 0.0724772
\(784\) 9.82874 0.351026
\(785\) −42.2297 −1.50724
\(786\) −13.5534 −0.483432
\(787\) 36.5777 1.30385 0.651927 0.758282i \(-0.273961\pi\)
0.651927 + 0.758282i \(0.273961\pi\)
\(788\) 15.3061 0.545258
\(789\) −20.2388 −0.720519
\(790\) 2.46823 0.0878157
\(791\) 11.1595 0.396786
\(792\) 2.03357 0.0722597
\(793\) −35.4350 −1.25834
\(794\) 12.0760 0.428559
\(795\) −33.6051 −1.19185
\(796\) 6.62692 0.234885
\(797\) −38.4484 −1.36191 −0.680956 0.732324i \(-0.738435\pi\)
−0.680956 + 0.732324i \(0.738435\pi\)
\(798\) −25.3976 −0.899067
\(799\) −7.74937 −0.274153
\(800\) −9.83469 −0.347709
\(801\) 12.8278 0.453249
\(802\) −39.3071 −1.38798
\(803\) 5.68358 0.200569
\(804\) −5.05990 −0.178449
\(805\) −14.3199 −0.504711
\(806\) −22.4412 −0.790458
\(807\) −28.6320 −1.00789
\(808\) −6.29084 −0.221311
\(809\) 49.6867 1.74689 0.873446 0.486921i \(-0.161880\pi\)
0.873446 + 0.486921i \(0.161880\pi\)
\(810\) 3.85158 0.135331
\(811\) 23.5153 0.825733 0.412867 0.910792i \(-0.364528\pi\)
0.412867 + 0.910792i \(0.364528\pi\)
\(812\) −8.31971 −0.291965
\(813\) −0.0160069 −0.000561388 0
\(814\) 1.86006 0.0651950
\(815\) −95.5155 −3.34576
\(816\) 1.00000 0.0350070
\(817\) 16.0297 0.560807
\(818\) −28.9306 −1.01153
\(819\) 13.8152 0.482743
\(820\) −18.3447 −0.640624
\(821\) 25.6672 0.895791 0.447896 0.894086i \(-0.352174\pi\)
0.447896 + 0.894086i \(0.352174\pi\)
\(822\) −2.68219 −0.0935522
\(823\) −6.25093 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(824\) 8.32588 0.290046
\(825\) 19.9995 0.696293
\(826\) −4.10228 −0.142737
\(827\) 37.4425 1.30200 0.651002 0.759076i \(-0.274349\pi\)
0.651002 + 0.759076i \(0.274349\pi\)
\(828\) 0.906308 0.0314964
\(829\) −54.0389 −1.87685 −0.938424 0.345485i \(-0.887714\pi\)
−0.938424 + 0.345485i \(0.887714\pi\)
\(830\) −25.3335 −0.879338
\(831\) −26.2831 −0.911750
\(832\) 3.36769 0.116754
\(833\) −9.82874 −0.340546
\(834\) 14.7398 0.510399
\(835\) 36.5082 1.26342
\(836\) 12.5900 0.435435
\(837\) −6.66369 −0.230331
\(838\) −16.0554 −0.554626
\(839\) −11.7778 −0.406614 −0.203307 0.979115i \(-0.565169\pi\)
−0.203307 + 0.979115i \(0.565169\pi\)
\(840\) −15.8003 −0.545162
\(841\) −24.8869 −0.858170
\(842\) −19.7480 −0.680563
\(843\) −30.6341 −1.05510
\(844\) 16.6354 0.572612
\(845\) 6.38853 0.219772
\(846\) −7.74937 −0.266429
\(847\) −28.1606 −0.967608
\(848\) −8.72500 −0.299618
\(849\) −0.234324 −0.00804198
\(850\) 9.83469 0.337327
\(851\) 0.828980 0.0284171
\(852\) −7.71254 −0.264227
\(853\) 46.2275 1.58280 0.791399 0.611300i \(-0.209353\pi\)
0.791399 + 0.611300i \(0.209353\pi\)
\(854\) 43.1645 1.47706
\(855\) 23.8455 0.815500
\(856\) 8.96644 0.306467
\(857\) 42.8968 1.46533 0.732663 0.680591i \(-0.238277\pi\)
0.732663 + 0.680591i \(0.238277\pi\)
\(858\) −6.84842 −0.233801
\(859\) −6.90108 −0.235462 −0.117731 0.993046i \(-0.537562\pi\)
−0.117731 + 0.993046i \(0.537562\pi\)
\(860\) 9.97232 0.340053
\(861\) −19.5387 −0.665879
\(862\) −20.4445 −0.696343
\(863\) −32.3405 −1.10088 −0.550442 0.834874i \(-0.685541\pi\)
−0.550442 + 0.834874i \(0.685541\pi\)
\(864\) 1.00000 0.0340207
\(865\) 17.4496 0.593306
\(866\) −9.91123 −0.336797
\(867\) −1.00000 −0.0339618
\(868\) 27.3363 0.927856
\(869\) −1.30318 −0.0442074
\(870\) 7.81127 0.264827
\(871\) 17.0402 0.577384
\(872\) 2.10396 0.0712491
\(873\) −5.13805 −0.173897
\(874\) 5.61104 0.189796
\(875\) −76.3895 −2.58244
\(876\) 2.79488 0.0944302
\(877\) 47.7667 1.61297 0.806484 0.591256i \(-0.201368\pi\)
0.806484 + 0.591256i \(0.201368\pi\)
\(878\) 16.4570 0.555399
\(879\) 3.63443 0.122586
\(880\) 7.83245 0.264032
\(881\) −22.6017 −0.761470 −0.380735 0.924684i \(-0.624329\pi\)
−0.380735 + 0.924684i \(0.624329\pi\)
\(882\) −9.82874 −0.330951
\(883\) −12.5773 −0.423261 −0.211630 0.977350i \(-0.567877\pi\)
−0.211630 + 0.977350i \(0.567877\pi\)
\(884\) −3.36769 −0.113268
\(885\) 3.85158 0.129470
\(886\) 22.6158 0.759792
\(887\) 5.72503 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(888\) 0.914678 0.0306946
\(889\) −71.0478 −2.38287
\(890\) 49.4075 1.65614
\(891\) −2.03357 −0.0681271
\(892\) −1.93856 −0.0649076
\(893\) −47.9771 −1.60549
\(894\) 3.89820 0.130375
\(895\) −4.93873 −0.165084
\(896\) −4.10228 −0.137048
\(897\) −3.05216 −0.101909
\(898\) 12.8141 0.427612
\(899\) −13.5144 −0.450731
\(900\) 9.83469 0.327823
\(901\) 8.72500 0.290672
\(902\) 9.68566 0.322497
\(903\) 10.6214 0.353459
\(904\) −2.72031 −0.0904762
\(905\) −8.42311 −0.279994
\(906\) −10.0378 −0.333485
\(907\) 40.0786 1.33079 0.665394 0.746492i \(-0.268264\pi\)
0.665394 + 0.746492i \(0.268264\pi\)
\(908\) −9.70323 −0.322013
\(909\) 6.29084 0.208654
\(910\) 53.2104 1.76391
\(911\) −17.3584 −0.575109 −0.287555 0.957764i \(-0.592842\pi\)
−0.287555 + 0.957764i \(0.592842\pi\)
\(912\) 6.19110 0.205008
\(913\) 13.3756 0.442669
\(914\) −30.5625 −1.01092
\(915\) −40.5266 −1.33977
\(916\) −21.2594 −0.702431
\(917\) −55.5997 −1.83607
\(918\) −1.00000 −0.0330049
\(919\) 26.3032 0.867664 0.433832 0.900994i \(-0.357161\pi\)
0.433832 + 0.900994i \(0.357161\pi\)
\(920\) 3.49072 0.115086
\(921\) −15.7695 −0.519622
\(922\) 8.23817 0.271310
\(923\) 25.9734 0.854926
\(924\) 8.34227 0.274441
\(925\) 8.99557 0.295773
\(926\) 14.1573 0.465238
\(927\) −8.32588 −0.273458
\(928\) 2.02807 0.0665746
\(929\) −1.21778 −0.0399539 −0.0199770 0.999800i \(-0.506359\pi\)
−0.0199770 + 0.999800i \(0.506359\pi\)
\(930\) −25.6657 −0.841613
\(931\) −60.8507 −1.99430
\(932\) 9.13446 0.299209
\(933\) −31.5974 −1.03445
\(934\) 9.80450 0.320813
\(935\) −7.83245 −0.256149
\(936\) −3.36769 −0.110076
\(937\) 47.8832 1.56427 0.782137 0.623106i \(-0.214129\pi\)
0.782137 + 0.623106i \(0.214129\pi\)
\(938\) −20.7572 −0.677745
\(939\) −12.5346 −0.409051
\(940\) −29.8473 −0.973513
\(941\) −15.8978 −0.518255 −0.259127 0.965843i \(-0.583435\pi\)
−0.259127 + 0.965843i \(0.583435\pi\)
\(942\) 10.9642 0.357234
\(943\) 4.31665 0.140569
\(944\) 1.00000 0.0325472
\(945\) 15.8003 0.513983
\(946\) −5.26521 −0.171187
\(947\) 40.0018 1.29988 0.649942 0.759984i \(-0.274793\pi\)
0.649942 + 0.759984i \(0.274793\pi\)
\(948\) −0.640835 −0.0208134
\(949\) −9.41228 −0.305536
\(950\) 60.8875 1.97545
\(951\) −8.78766 −0.284959
\(952\) 4.10228 0.132956
\(953\) −30.1726 −0.977386 −0.488693 0.872456i \(-0.662526\pi\)
−0.488693 + 0.872456i \(0.662526\pi\)
\(954\) 8.72500 0.282482
\(955\) 28.8830 0.934631
\(956\) −14.6133 −0.472626
\(957\) −4.12421 −0.133317
\(958\) −37.7151 −1.21852
\(959\) −11.0031 −0.355309
\(960\) 3.85158 0.124309
\(961\) 13.4047 0.432411
\(962\) −3.08035 −0.0993144
\(963\) −8.96644 −0.288940
\(964\) −16.3402 −0.526283
\(965\) −18.6627 −0.600774
\(966\) 3.71793 0.119623
\(967\) 20.8432 0.670271 0.335135 0.942170i \(-0.391218\pi\)
0.335135 + 0.942170i \(0.391218\pi\)
\(968\) 6.86460 0.220637
\(969\) −6.19110 −0.198887
\(970\) −19.7896 −0.635407
\(971\) −20.0708 −0.644102 −0.322051 0.946722i \(-0.604372\pi\)
−0.322051 + 0.946722i \(0.604372\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 60.4670 1.93848
\(974\) −25.9042 −0.830024
\(975\) −33.1202 −1.06069
\(976\) −10.5221 −0.336803
\(977\) −38.2031 −1.22223 −0.611113 0.791543i \(-0.709278\pi\)
−0.611113 + 0.791543i \(0.709278\pi\)
\(978\) 24.7990 0.792986
\(979\) −26.0863 −0.833721
\(980\) −37.8562 −1.20927
\(981\) −2.10396 −0.0671743
\(982\) −9.43153 −0.300972
\(983\) 49.6362 1.58315 0.791574 0.611073i \(-0.209262\pi\)
0.791574 + 0.611073i \(0.209262\pi\)
\(984\) 4.76289 0.151835
\(985\) −58.9528 −1.87839
\(986\) −2.02807 −0.0645868
\(987\) −31.7901 −1.01189
\(988\) −20.8497 −0.663317
\(989\) −2.34657 −0.0746165
\(990\) −7.83245 −0.248932
\(991\) 60.5667 1.92397 0.961983 0.273109i \(-0.0880520\pi\)
0.961983 + 0.273109i \(0.0880520\pi\)
\(992\) −6.66369 −0.211572
\(993\) −28.1629 −0.893723
\(994\) −31.6391 −1.00353
\(995\) −25.5241 −0.809169
\(996\) 6.57742 0.208414
\(997\) −9.71941 −0.307817 −0.153908 0.988085i \(-0.549186\pi\)
−0.153908 + 0.988085i \(0.549186\pi\)
\(998\) −15.0782 −0.477293
\(999\) −0.914678 −0.0289391
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 6018.2.a.s.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.s.1.1 8 1.1 even 1 trivial