Properties

Label 201.5.b.a.133.13
Level $201$
Weight $5$
Character 201.133
Analytic conductor $20.777$
Analytic rank $0$
Dimension $46$
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] = [201,5,Mod(133,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.133");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 201.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: \(20.7773625799\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 133.13
Character \(\chi\) \(=\) 201.133
Dual form 201.5.b.a.133.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.05388i q^{2} -5.19615i q^{3} -0.433982 q^{4} +2.19375i q^{5} -21.0646 q^{6} +19.8533i q^{7} -63.1028i q^{8} -27.0000 q^{9} +O(q^{10})\) \(q-4.05388i q^{2} -5.19615i q^{3} -0.433982 q^{4} +2.19375i q^{5} -21.0646 q^{6} +19.8533i q^{7} -63.1028i q^{8} -27.0000 q^{9} +8.89321 q^{10} +61.3918i q^{11} +2.25504i q^{12} -236.159i q^{13} +80.4831 q^{14} +11.3991 q^{15} -262.755 q^{16} -483.140 q^{17} +109.455i q^{18} -407.088 q^{19} -0.952048i q^{20} +103.161 q^{21} +248.875 q^{22} -331.409 q^{23} -327.892 q^{24} +620.187 q^{25} -957.363 q^{26} +140.296i q^{27} -8.61599i q^{28} -174.542 q^{29} -46.2105i q^{30} -1557.14i q^{31} +55.5345i q^{32} +319.001 q^{33} +1958.59i q^{34} -43.5532 q^{35} +11.7175 q^{36} -1375.04 q^{37} +1650.29i q^{38} -1227.12 q^{39} +138.432 q^{40} +1731.82i q^{41} -418.202i q^{42} +806.745i q^{43} -26.6430i q^{44} -59.2312i q^{45} +1343.49i q^{46} +1725.60 q^{47} +1365.32i q^{48} +2006.85 q^{49} -2514.17i q^{50} +2510.47i q^{51} +102.489i q^{52} +491.806i q^{53} +568.744 q^{54} -134.678 q^{55} +1252.80 q^{56} +2115.29i q^{57} +707.572i q^{58} -5496.96 q^{59} -4.94699 q^{60} +3388.09i q^{61} -6312.45 q^{62} -536.040i q^{63} -3978.96 q^{64} +518.074 q^{65} -1293.19i q^{66} +(3236.63 - 3110.52i) q^{67} +209.674 q^{68} +1722.05i q^{69} +176.560i q^{70} +465.285 q^{71} +1703.78i q^{72} +8586.98 q^{73} +5574.26i q^{74} -3222.59i q^{75} +176.669 q^{76} -1218.83 q^{77} +4974.60i q^{78} -10742.8i q^{79} -576.419i q^{80} +729.000 q^{81} +7020.61 q^{82} -563.229 q^{83} -44.7700 q^{84} -1059.89i q^{85} +3270.45 q^{86} +906.945i q^{87} +3874.00 q^{88} -12456.6 q^{89} -240.117 q^{90} +4688.55 q^{91} +143.826 q^{92} -8091.11 q^{93} -6995.40i q^{94} -893.048i q^{95} +288.566 q^{96} +10955.7i q^{97} -8135.52i q^{98} -1657.58i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 396 q^{4} - 1242 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q - 396 q^{4} - 1242 q^{9} + 396 q^{10} + 792 q^{14} - 252 q^{15} + 3396 q^{16} + 462 q^{17} - 590 q^{19} - 936 q^{21} + 3184 q^{22} - 1446 q^{23} - 1404 q^{24} - 6278 q^{25} + 2700 q^{26} - 1014 q^{29} + 540 q^{33} + 9924 q^{35} + 10692 q^{36} - 386 q^{37} + 4968 q^{39} - 9988 q^{40} - 2754 q^{47} - 19062 q^{49} - 2320 q^{55} - 3396 q^{56} - 7098 q^{59} + 72 q^{60} - 21180 q^{62} - 75644 q^{64} + 18396 q^{65} + 8574 q^{67} + 9084 q^{68} - 23040 q^{71} - 22338 q^{73} + 28016 q^{76} + 45084 q^{77} + 33534 q^{81} + 17564 q^{82} + 35856 q^{83} + 40176 q^{84} + 31764 q^{86} - 19448 q^{88} - 14538 q^{89} - 10692 q^{90} + 13792 q^{91} - 67692 q^{92} + 22464 q^{93} + 22464 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.05388i 1.01347i −0.862102 0.506736i \(-0.830852\pi\)
0.862102 0.506736i \(-0.169148\pi\)
\(3\) 5.19615i 0.577350i
\(4\) −0.433982 −0.0271239
\(5\) 2.19375i 0.0877500i 0.999037 + 0.0438750i \(0.0139703\pi\)
−0.999037 + 0.0438750i \(0.986030\pi\)
\(6\) −21.0646 −0.585128
\(7\) 19.8533i 0.405170i 0.979265 + 0.202585i \(0.0649342\pi\)
−0.979265 + 0.202585i \(0.935066\pi\)
\(8\) 63.1028i 0.985982i
\(9\) −27.0000 −0.333333
\(10\) 8.89321 0.0889321
\(11\) 61.3918i 0.507371i 0.967287 + 0.253685i \(0.0816428\pi\)
−0.967287 + 0.253685i \(0.918357\pi\)
\(12\) 2.25504i 0.0156600i
\(13\) 236.159i 1.39739i −0.715418 0.698696i \(-0.753764\pi\)
0.715418 0.698696i \(-0.246236\pi\)
\(14\) 80.4831 0.410628
\(15\) 11.3991 0.0506625
\(16\) −262.755 −1.02639
\(17\) −483.140 −1.67176 −0.835882 0.548909i \(-0.815043\pi\)
−0.835882 + 0.548909i \(0.815043\pi\)
\(18\) 109.455i 0.337824i
\(19\) −407.088 −1.12767 −0.563833 0.825889i \(-0.690674\pi\)
−0.563833 + 0.825889i \(0.690674\pi\)
\(20\) 0.952048i 0.00238012i
\(21\) 103.161 0.233925
\(22\) 248.875 0.514206
\(23\) −331.409 −0.626482 −0.313241 0.949674i \(-0.601415\pi\)
−0.313241 + 0.949674i \(0.601415\pi\)
\(24\) −327.892 −0.569257
\(25\) 620.187 0.992300
\(26\) −957.363 −1.41622
\(27\) 140.296i 0.192450i
\(28\) 8.61599i 0.0109898i
\(29\) −174.542 −0.207541 −0.103770 0.994601i \(-0.533091\pi\)
−0.103770 + 0.994601i \(0.533091\pi\)
\(30\) 46.2105i 0.0513449i
\(31\) 1557.14i 1.62033i −0.586203 0.810164i \(-0.699378\pi\)
0.586203 0.810164i \(-0.300622\pi\)
\(32\) 55.5345i 0.0542329i
\(33\) 319.001 0.292931
\(34\) 1958.59i 1.69429i
\(35\) −43.5532 −0.0355536
\(36\) 11.7175 0.00904129
\(37\) −1375.04 −1.00441 −0.502207 0.864748i \(-0.667478\pi\)
−0.502207 + 0.864748i \(0.667478\pi\)
\(38\) 1650.29i 1.14286i
\(39\) −1227.12 −0.806785
\(40\) 138.432 0.0865199
\(41\) 1731.82i 1.03023i 0.857120 + 0.515117i \(0.172252\pi\)
−0.857120 + 0.515117i \(0.827748\pi\)
\(42\) 418.202i 0.237076i
\(43\) 806.745i 0.436314i 0.975914 + 0.218157i \(0.0700045\pi\)
−0.975914 + 0.218157i \(0.929995\pi\)
\(44\) 26.6430i 0.0137619i
\(45\) 59.2312i 0.0292500i
\(46\) 1343.49i 0.634922i
\(47\) 1725.60 0.781170 0.390585 0.920567i \(-0.372273\pi\)
0.390585 + 0.920567i \(0.372273\pi\)
\(48\) 1365.32i 0.592585i
\(49\) 2006.85 0.835837
\(50\) 2514.17i 1.00567i
\(51\) 2510.47i 0.965194i
\(52\) 102.489i 0.0379027i
\(53\) 491.806i 0.175082i 0.996161 + 0.0875412i \(0.0279009\pi\)
−0.996161 + 0.0875412i \(0.972099\pi\)
\(54\) 568.744 0.195043
\(55\) −134.678 −0.0445218
\(56\) 1252.80 0.399490
\(57\) 2115.29i 0.651059i
\(58\) 707.572i 0.210336i
\(59\) −5496.96 −1.57913 −0.789567 0.613665i \(-0.789695\pi\)
−0.789567 + 0.613665i \(0.789695\pi\)
\(60\) −4.94699 −0.00137416
\(61\) 3388.09i 0.910533i 0.890355 + 0.455267i \(0.150456\pi\)
−0.890355 + 0.455267i \(0.849544\pi\)
\(62\) −6312.45 −1.64216
\(63\) 536.040i 0.135057i
\(64\) −3978.96 −0.971425
\(65\) 518.074 0.122621
\(66\) 1293.19i 0.296877i
\(67\) 3236.63 3110.52i 0.721014 0.692920i
\(68\) 209.674 0.0453447
\(69\) 1722.05i 0.361700i
\(70\) 176.560i 0.0360326i
\(71\) 465.285 0.0923001 0.0461500 0.998935i \(-0.485305\pi\)
0.0461500 + 0.998935i \(0.485305\pi\)
\(72\) 1703.78i 0.328661i
\(73\) 8586.98 1.61137 0.805684 0.592345i \(-0.201798\pi\)
0.805684 + 0.592345i \(0.201798\pi\)
\(74\) 5574.26i 1.01794i
\(75\) 3222.59i 0.572905i
\(76\) 176.669 0.0305867
\(77\) −1218.83 −0.205571
\(78\) 4974.60i 0.817653i
\(79\) 10742.8i 1.72132i −0.509176 0.860662i \(-0.670050\pi\)
0.509176 0.860662i \(-0.329950\pi\)
\(80\) 576.419i 0.0900655i
\(81\) 729.000 0.111111
\(82\) 7020.61 1.04411
\(83\) −563.229 −0.0817577 −0.0408788 0.999164i \(-0.513016\pi\)
−0.0408788 + 0.999164i \(0.513016\pi\)
\(84\) −44.7700 −0.00634495
\(85\) 1059.89i 0.146697i
\(86\) 3270.45 0.442192
\(87\) 906.945i 0.119824i
\(88\) 3874.00 0.500258
\(89\) −12456.6 −1.57261 −0.786303 0.617841i \(-0.788007\pi\)
−0.786303 + 0.617841i \(0.788007\pi\)
\(90\) −240.117 −0.0296440
\(91\) 4688.55 0.566181
\(92\) 143.826 0.0169926
\(93\) −8091.11 −0.935497
\(94\) 6995.40i 0.791693i
\(95\) 893.048i 0.0989527i
\(96\) 288.566 0.0313114
\(97\) 10955.7i 1.16438i 0.813052 + 0.582191i \(0.197804\pi\)
−0.813052 + 0.582191i \(0.802196\pi\)
\(98\) 8135.52i 0.847097i
\(99\) 1657.58i 0.169124i
\(100\) −269.150 −0.0269150
\(101\) 15359.3i 1.50567i −0.658209 0.752835i \(-0.728686\pi\)
0.658209 0.752835i \(-0.271314\pi\)
\(102\) 10177.2 0.978196
\(103\) −4995.44 −0.470868 −0.235434 0.971890i \(-0.575651\pi\)
−0.235434 + 0.971890i \(0.575651\pi\)
\(104\) −14902.3 −1.37780
\(105\) 226.309i 0.0205269i
\(106\) 1993.73 0.177441
\(107\) 2433.49 0.212550 0.106275 0.994337i \(-0.466108\pi\)
0.106275 + 0.994337i \(0.466108\pi\)
\(108\) 60.8860i 0.00521999i
\(109\) 20839.0i 1.75397i −0.480514 0.876987i \(-0.659550\pi\)
0.480514 0.876987i \(-0.340450\pi\)
\(110\) 545.970i 0.0451215i
\(111\) 7144.93i 0.579898i
\(112\) 5216.57i 0.415862i
\(113\) 20065.6i 1.57143i −0.618589 0.785715i \(-0.712295\pi\)
0.618589 0.785715i \(-0.287705\pi\)
\(114\) 8575.14 0.659829
\(115\) 727.029i 0.0549738i
\(116\) 75.7479 0.00562931
\(117\) 6376.30i 0.465798i
\(118\) 22284.1i 1.60041i
\(119\) 9591.94i 0.677349i
\(120\) 719.313i 0.0499523i
\(121\) 10872.0 0.742575
\(122\) 13734.9 0.922799
\(123\) 8998.82 0.594806
\(124\) 675.769i 0.0439496i
\(125\) 2731.63i 0.174824i
\(126\) −2173.04 −0.136876
\(127\) −9265.62 −0.574470 −0.287235 0.957860i \(-0.592736\pi\)
−0.287235 + 0.957860i \(0.592736\pi\)
\(128\) 17018.8i 1.03874i
\(129\) 4191.97 0.251906
\(130\) 2100.21i 0.124273i
\(131\) 9285.00 0.541053 0.270526 0.962713i \(-0.412802\pi\)
0.270526 + 0.962713i \(0.412802\pi\)
\(132\) −138.441 −0.00794541
\(133\) 8082.04i 0.456897i
\(134\) −12609.7 13120.9i −0.702255 0.730727i
\(135\) −307.774 −0.0168875
\(136\) 30487.5i 1.64833i
\(137\) 34125.1i 1.81816i −0.416620 0.909081i \(-0.636785\pi\)
0.416620 0.909081i \(-0.363215\pi\)
\(138\) 6981.00 0.366572
\(139\) 6542.07i 0.338599i 0.985565 + 0.169300i \(0.0541505\pi\)
−0.985565 + 0.169300i \(0.945849\pi\)
\(140\) 18.9013 0.000964353
\(141\) 8966.50i 0.451009i
\(142\) 1886.21i 0.0935434i
\(143\) 14498.3 0.708996
\(144\) 7094.40 0.342129
\(145\) 382.900i 0.0182117i
\(146\) 34810.6i 1.63308i
\(147\) 10427.9i 0.482571i
\(148\) 596.744 0.0272436
\(149\) 5798.85 0.261198 0.130599 0.991435i \(-0.458310\pi\)
0.130599 + 0.991435i \(0.458310\pi\)
\(150\) −13064.0 −0.580622
\(151\) −11480.5 −0.503507 −0.251754 0.967791i \(-0.581007\pi\)
−0.251754 + 0.967791i \(0.581007\pi\)
\(152\) 25688.4i 1.11186i
\(153\) 13044.8 0.557255
\(154\) 4941.01i 0.208341i
\(155\) 3415.97 0.142184
\(156\) 532.548 0.0218831
\(157\) 32540.4 1.32015 0.660076 0.751199i \(-0.270524\pi\)
0.660076 + 0.751199i \(0.270524\pi\)
\(158\) −43550.0 −1.74451
\(159\) 2555.50 0.101084
\(160\) −121.829 −0.00475894
\(161\) 6579.57i 0.253832i
\(162\) 2955.28i 0.112608i
\(163\) 40283.9 1.51620 0.758099 0.652139i \(-0.226128\pi\)
0.758099 + 0.652139i \(0.226128\pi\)
\(164\) 751.580i 0.0279439i
\(165\) 699.809i 0.0257046i
\(166\) 2283.26i 0.0828591i
\(167\) −30195.4 −1.08270 −0.541349 0.840798i \(-0.682086\pi\)
−0.541349 + 0.840798i \(0.682086\pi\)
\(168\) 6509.75i 0.230646i
\(169\) −27210.2 −0.952706
\(170\) −4296.66 −0.148673
\(171\) 10991.4 0.375889
\(172\) 350.113i 0.0118345i
\(173\) 11899.2 0.397580 0.198790 0.980042i \(-0.436299\pi\)
0.198790 + 0.980042i \(0.436299\pi\)
\(174\) 3676.65 0.121438
\(175\) 12312.8i 0.402050i
\(176\) 16131.0i 0.520759i
\(177\) 28563.1i 0.911713i
\(178\) 50497.7i 1.59379i
\(179\) 47223.6i 1.47385i 0.675975 + 0.736924i \(0.263723\pi\)
−0.675975 + 0.736924i \(0.736277\pi\)
\(180\) 25.7053i 0.000793373i
\(181\) −39859.4 −1.21667 −0.608336 0.793680i \(-0.708163\pi\)
−0.608336 + 0.793680i \(0.708163\pi\)
\(182\) 19006.8i 0.573809i
\(183\) 17605.1 0.525697
\(184\) 20912.9i 0.617700i
\(185\) 3016.50i 0.0881372i
\(186\) 32800.4i 0.948099i
\(187\) 29660.9i 0.848204i
\(188\) −748.881 −0.0211884
\(189\) −2785.34 −0.0779750
\(190\) −3620.31 −0.100286
\(191\) 52143.3i 1.42933i 0.699469 + 0.714663i \(0.253420\pi\)
−0.699469 + 0.714663i \(0.746580\pi\)
\(192\) 20675.3i 0.560852i
\(193\) 31454.2 0.844431 0.422216 0.906495i \(-0.361252\pi\)
0.422216 + 0.906495i \(0.361252\pi\)
\(194\) 44413.0 1.18007
\(195\) 2691.99i 0.0707954i
\(196\) −870.935 −0.0226712
\(197\) 27651.1i 0.712492i −0.934392 0.356246i \(-0.884057\pi\)
0.934392 0.356246i \(-0.115943\pi\)
\(198\) −6719.64 −0.171402
\(199\) −53743.2 −1.35712 −0.678558 0.734547i \(-0.737395\pi\)
−0.678558 + 0.734547i \(0.737395\pi\)
\(200\) 39135.6i 0.978390i
\(201\) −16162.7 16818.0i −0.400058 0.416278i
\(202\) −62265.0 −1.52595
\(203\) 3465.23i 0.0840892i
\(204\) 1089.50i 0.0261798i
\(205\) −3799.19 −0.0904030
\(206\) 20250.9i 0.477211i
\(207\) 8948.05 0.208827
\(208\) 62052.1i 1.43427i
\(209\) 24991.9i 0.572145i
\(210\) 917.431 0.0208034
\(211\) 50734.7 1.13957 0.569784 0.821795i \(-0.307027\pi\)
0.569784 + 0.821795i \(0.307027\pi\)
\(212\) 213.435i 0.00474891i
\(213\) 2417.69i 0.0532895i
\(214\) 9865.08i 0.215413i
\(215\) −1769.80 −0.0382866
\(216\) 8853.08 0.189752
\(217\) 30914.3 0.656508
\(218\) −84478.8 −1.77760
\(219\) 44619.3i 0.930324i
\(220\) 58.4480 0.00120760
\(221\) 114098.i 2.33611i
\(222\) 28964.7 0.587710
\(223\) 73794.1 1.48393 0.741963 0.670441i \(-0.233895\pi\)
0.741963 + 0.670441i \(0.233895\pi\)
\(224\) −1102.54 −0.0219736
\(225\) −16745.1 −0.330767
\(226\) −81343.6 −1.59260
\(227\) −3603.56 −0.0699326 −0.0349663 0.999388i \(-0.511132\pi\)
−0.0349663 + 0.999388i \(0.511132\pi\)
\(228\) 917.998i 0.0176592i
\(229\) 13896.7i 0.264997i 0.991183 + 0.132498i \(0.0422999\pi\)
−0.991183 + 0.132498i \(0.957700\pi\)
\(230\) −2947.29 −0.0557144
\(231\) 6333.24i 0.118687i
\(232\) 11014.1i 0.204631i
\(233\) 68643.7i 1.26441i −0.774800 0.632206i \(-0.782150\pi\)
0.774800 0.632206i \(-0.217850\pi\)
\(234\) 25848.8 0.472072
\(235\) 3785.54i 0.0685476i
\(236\) 2385.58 0.0428322
\(237\) −55821.2 −0.993807
\(238\) −38884.6 −0.686473
\(239\) 10929.4i 0.191338i 0.995413 + 0.0956692i \(0.0304991\pi\)
−0.995413 + 0.0956692i \(0.969501\pi\)
\(240\) −2995.16 −0.0519994
\(241\) −86262.6 −1.48521 −0.742606 0.669728i \(-0.766411\pi\)
−0.742606 + 0.669728i \(0.766411\pi\)
\(242\) 44074.0i 0.752578i
\(243\) 3788.00i 0.0641500i
\(244\) 1470.37i 0.0246972i
\(245\) 4402.52i 0.0733447i
\(246\) 36480.2i 0.602819i
\(247\) 96137.5i 1.57579i
\(248\) −98259.7 −1.59761
\(249\) 2926.62i 0.0472028i
\(250\) 11073.7 0.177179
\(251\) 29959.7i 0.475544i −0.971321 0.237772i \(-0.923583\pi\)
0.971321 0.237772i \(-0.0764170\pi\)
\(252\) 232.632i 0.00366326i
\(253\) 20345.8i 0.317859i
\(254\) 37561.8i 0.582209i
\(255\) −5507.34 −0.0846957
\(256\) 5328.88 0.0813123
\(257\) 38523.2 0.583252 0.291626 0.956532i \(-0.405804\pi\)
0.291626 + 0.956532i \(0.405804\pi\)
\(258\) 16993.8i 0.255300i
\(259\) 27299.2i 0.406958i
\(260\) −224.835 −0.00332596
\(261\) 4712.62 0.0691802
\(262\) 37640.3i 0.548341i
\(263\) −53998.5 −0.780675 −0.390337 0.920672i \(-0.627642\pi\)
−0.390337 + 0.920672i \(0.627642\pi\)
\(264\) 20129.9i 0.288824i
\(265\) −1078.90 −0.0153635
\(266\) −32763.7 −0.463051
\(267\) 64726.5i 0.907944i
\(268\) −1404.64 + 1349.91i −0.0195567 + 0.0187947i
\(269\) 51631.8 0.713531 0.356766 0.934194i \(-0.383879\pi\)
0.356766 + 0.934194i \(0.383879\pi\)
\(270\) 1247.68i 0.0171150i
\(271\) 121695.i 1.65705i −0.559954 0.828524i \(-0.689181\pi\)
0.559954 0.828524i \(-0.310819\pi\)
\(272\) 126948. 1.71588
\(273\) 24362.4i 0.326885i
\(274\) −138339. −1.84265
\(275\) 38074.5i 0.503464i
\(276\) 747.340i 0.00981070i
\(277\) 30389.6 0.396065 0.198032 0.980196i \(-0.436545\pi\)
0.198032 + 0.980196i \(0.436545\pi\)
\(278\) 26520.8 0.343160
\(279\) 42042.7i 0.540110i
\(280\) 2748.33i 0.0350553i
\(281\) 64376.8i 0.815298i 0.913139 + 0.407649i \(0.133651\pi\)
−0.913139 + 0.407649i \(0.866349\pi\)
\(282\) −36349.2 −0.457084
\(283\) −43342.2 −0.541175 −0.270588 0.962695i \(-0.587218\pi\)
−0.270588 + 0.962695i \(0.587218\pi\)
\(284\) −201.925 −0.00250354
\(285\) −4640.41 −0.0571304
\(286\) 58774.3i 0.718547i
\(287\) −34382.5 −0.417420
\(288\) 1499.43i 0.0180776i
\(289\) 149903. 1.79480
\(290\) −1552.23 −0.0184570
\(291\) 56927.3 0.672256
\(292\) −3726.60 −0.0437066
\(293\) 77718.3 0.905291 0.452646 0.891691i \(-0.350480\pi\)
0.452646 + 0.891691i \(0.350480\pi\)
\(294\) −42273.4 −0.489072
\(295\) 12059.0i 0.138569i
\(296\) 86769.1i 0.990334i
\(297\) −8613.04 −0.0976435
\(298\) 23507.9i 0.264716i
\(299\) 78265.4i 0.875442i
\(300\) 1398.55i 0.0155394i
\(301\) −16016.6 −0.176781
\(302\) 46540.5i 0.510290i
\(303\) −79809.5 −0.869299
\(304\) 106964. 1.15742
\(305\) −7432.63 −0.0798992
\(306\) 52882.0i 0.564762i
\(307\) 98419.8 1.04425 0.522127 0.852868i \(-0.325139\pi\)
0.522127 + 0.852868i \(0.325139\pi\)
\(308\) 528.951 0.00557589
\(309\) 25957.1i 0.271856i
\(310\) 13847.9i 0.144099i
\(311\) 118257.i 1.22266i −0.791376 0.611329i \(-0.790635\pi\)
0.791376 0.611329i \(-0.209365\pi\)
\(312\) 77434.8i 0.795475i
\(313\) 94235.3i 0.961889i −0.876751 0.480945i \(-0.840294\pi\)
0.876751 0.480945i \(-0.159706\pi\)
\(314\) 131915.i 1.33794i
\(315\) 1175.94 0.0118512
\(316\) 4662.18i 0.0466890i
\(317\) 1669.41 0.0166128 0.00830642 0.999966i \(-0.497356\pi\)
0.00830642 + 0.999966i \(0.497356\pi\)
\(318\) 10359.7i 0.102446i
\(319\) 10715.4i 0.105300i
\(320\) 8728.83i 0.0852425i
\(321\) 12644.8i 0.122716i
\(322\) −26672.8 −0.257251
\(323\) 196680. 1.88519
\(324\) −316.373 −0.00301376
\(325\) 146463.i 1.38663i
\(326\) 163306.i 1.53662i
\(327\) −108282. −1.01266
\(328\) 109283. 1.01579
\(329\) 34259.0i 0.316507i
\(330\) 2836.95 0.0260509
\(331\) 26818.3i 0.244780i −0.992482 0.122390i \(-0.960944\pi\)
0.992482 0.122390i \(-0.0390559\pi\)
\(332\) 244.431 0.00221759
\(333\) 37126.1 0.334804
\(334\) 122408.i 1.09728i
\(335\) 6823.70 + 7100.36i 0.0608037 + 0.0632690i
\(336\) −27106.1 −0.240098
\(337\) 126882.i 1.11722i −0.829430 0.558611i \(-0.811335\pi\)
0.829430 0.558611i \(-0.188665\pi\)
\(338\) 110307.i 0.965540i
\(339\) −104264. −0.907265
\(340\) 459.972i 0.00397900i
\(341\) 95595.4 0.822107
\(342\) 44557.7i 0.380953i
\(343\) 87510.4i 0.743826i
\(344\) 50907.9 0.430198
\(345\) −3777.75 −0.0317391
\(346\) 48237.9i 0.402936i
\(347\) 215554.i 1.79018i 0.445886 + 0.895090i \(0.352889\pi\)
−0.445886 + 0.895090i \(0.647111\pi\)
\(348\) 393.598i 0.00325008i
\(349\) 71512.4 0.587125 0.293562 0.955940i \(-0.405159\pi\)
0.293562 + 0.955940i \(0.405159\pi\)
\(350\) 49914.6 0.407466
\(351\) 33132.2 0.268928
\(352\) −3409.37 −0.0275162
\(353\) 182287.i 1.46287i 0.681912 + 0.731434i \(0.261149\pi\)
−0.681912 + 0.731434i \(0.738851\pi\)
\(354\) 115791. 0.923995
\(355\) 1020.72i 0.00809933i
\(356\) 5405.95 0.0426552
\(357\) −49841.2 −0.391067
\(358\) 191439. 1.49370
\(359\) −55541.3 −0.430951 −0.215475 0.976509i \(-0.569130\pi\)
−0.215475 + 0.976509i \(0.569130\pi\)
\(360\) −3737.66 −0.0288400
\(361\) 35399.3 0.271632
\(362\) 161585.i 1.23306i
\(363\) 56492.8i 0.428726i
\(364\) −2034.75 −0.0153570
\(365\) 18837.7i 0.141398i
\(366\) 71368.9i 0.532778i
\(367\) 99043.5i 0.735350i 0.929954 + 0.367675i \(0.119846\pi\)
−0.929954 + 0.367675i \(0.880154\pi\)
\(368\) 87079.5 0.643014
\(369\) 46759.2i 0.343411i
\(370\) −12228.5 −0.0893246
\(371\) −9763.99 −0.0709381
\(372\) 3511.40 0.0253743
\(373\) 209435.i 1.50533i −0.658405 0.752664i \(-0.728769\pi\)
0.658405 0.752664i \(-0.271231\pi\)
\(374\) −120242. −0.859631
\(375\) 14194.0 0.100935
\(376\) 108891.i 0.770219i
\(377\) 41219.6i 0.290016i
\(378\) 11291.5i 0.0790254i
\(379\) 33726.9i 0.234800i 0.993085 + 0.117400i \(0.0374559\pi\)
−0.993085 + 0.117400i \(0.962544\pi\)
\(380\) 387.567i 0.00268398i
\(381\) 48145.6i 0.331670i
\(382\) 211383. 1.44858
\(383\) 29233.3i 0.199288i 0.995023 + 0.0996439i \(0.0317704\pi\)
−0.995023 + 0.0996439i \(0.968230\pi\)
\(384\) 88432.2 0.599719
\(385\) 2673.81i 0.0180389i
\(386\) 127512.i 0.855807i
\(387\) 21782.1i 0.145438i
\(388\) 4754.57i 0.0315826i
\(389\) 291228. 1.92457 0.962286 0.272040i \(-0.0876982\pi\)
0.962286 + 0.272040i \(0.0876982\pi\)
\(390\) −10913.0 −0.0717490
\(391\) 160117. 1.04733
\(392\) 126638.i 0.824121i
\(393\) 48246.3i 0.312377i
\(394\) −112094. −0.722090
\(395\) 23567.0 0.151046
\(396\) 719.360i 0.00458729i
\(397\) −266513. −1.69098 −0.845488 0.533995i \(-0.820690\pi\)
−0.845488 + 0.533995i \(0.820690\pi\)
\(398\) 217869.i 1.37540i
\(399\) −41995.5 −0.263789
\(400\) −162958. −1.01848
\(401\) 26541.3i 0.165057i 0.996589 + 0.0825283i \(0.0262995\pi\)
−0.996589 + 0.0825283i \(0.973701\pi\)
\(402\) −68178.4 + 65521.9i −0.421885 + 0.405447i
\(403\) −367732. −2.26424
\(404\) 6665.68i 0.0408396i
\(405\) 1599.24i 0.00975000i
\(406\) −14047.6 −0.0852220
\(407\) 84416.4i 0.509610i
\(408\) 158418. 0.951664
\(409\) 178774.i 1.06871i −0.845261 0.534354i \(-0.820555\pi\)
0.845261 0.534354i \(-0.179445\pi\)
\(410\) 15401.5i 0.0916208i
\(411\) −177319. −1.04972
\(412\) 2167.93 0.0127718
\(413\) 109133.i 0.639817i
\(414\) 36274.4i 0.211641i
\(415\) 1235.58i 0.00717423i
\(416\) 13115.0 0.0757847
\(417\) 33993.6 0.195490
\(418\) −101314. −0.579852
\(419\) −153113. −0.872137 −0.436069 0.899913i \(-0.643630\pi\)
−0.436069 + 0.899913i \(0.643630\pi\)
\(420\) 98.2141i 0.000556769i
\(421\) −235921. −1.33107 −0.665536 0.746366i \(-0.731797\pi\)
−0.665536 + 0.746366i \(0.731797\pi\)
\(422\) 205673.i 1.15492i
\(423\) −46591.3 −0.260390
\(424\) 31034.4 0.172628
\(425\) −299637. −1.65889
\(426\) −9801.03 −0.0540073
\(427\) −67264.9 −0.368921
\(428\) −1056.09 −0.00576519
\(429\) 75335.2i 0.409339i
\(430\) 7174.55i 0.0388023i
\(431\) 344575. 1.85494 0.927468 0.373903i \(-0.121981\pi\)
0.927468 + 0.373903i \(0.121981\pi\)
\(432\) 36863.6i 0.197528i
\(433\) 172115.i 0.918002i 0.888436 + 0.459001i \(0.151793\pi\)
−0.888436 + 0.459001i \(0.848207\pi\)
\(434\) 125323.i 0.665352i
\(435\) −1989.61 −0.0105145
\(436\) 9043.74i 0.0475746i
\(437\) 134913. 0.706463
\(438\) −180881. −0.942857
\(439\) −9717.32 −0.0504217 −0.0252108 0.999682i \(-0.508026\pi\)
−0.0252108 + 0.999682i \(0.508026\pi\)
\(440\) 8498.58i 0.0438976i
\(441\) −54184.8 −0.278612
\(442\) 462540. 2.36758
\(443\) 312138.i 1.59052i 0.606268 + 0.795261i \(0.292666\pi\)
−0.606268 + 0.795261i \(0.707334\pi\)
\(444\) 3100.77i 0.0157291i
\(445\) 27326.7i 0.137996i
\(446\) 299153.i 1.50392i
\(447\) 30131.7i 0.150803i
\(448\) 78995.5i 0.393592i
\(449\) −299938. −1.48778 −0.743889 0.668303i \(-0.767021\pi\)
−0.743889 + 0.668303i \(0.767021\pi\)
\(450\) 67882.6i 0.335222i
\(451\) −106320. −0.522710
\(452\) 8708.10i 0.0426233i
\(453\) 59654.3i 0.290700i
\(454\) 14608.4i 0.0708747i
\(455\) 10285.5i 0.0496824i
\(456\) 133481. 0.641932
\(457\) −81434.1 −0.389918 −0.194959 0.980811i \(-0.562457\pi\)
−0.194959 + 0.980811i \(0.562457\pi\)
\(458\) 56335.6 0.268566
\(459\) 67782.7i 0.321731i
\(460\) 315.517i 0.00149110i
\(461\) −350275. −1.64819 −0.824096 0.566450i \(-0.808316\pi\)
−0.824096 + 0.566450i \(0.808316\pi\)
\(462\) 25674.2 0.120286
\(463\) 9936.97i 0.0463545i 0.999731 + 0.0231772i \(0.00737821\pi\)
−0.999731 + 0.0231772i \(0.992622\pi\)
\(464\) 45861.7 0.213017
\(465\) 17749.9i 0.0820898i
\(466\) −278274. −1.28145
\(467\) 182819. 0.838278 0.419139 0.907922i \(-0.362332\pi\)
0.419139 + 0.907922i \(0.362332\pi\)
\(468\) 2767.20i 0.0126342i
\(469\) 61754.2 + 64257.9i 0.280750 + 0.292133i
\(470\) 15346.2 0.0694710
\(471\) 169085.i 0.762190i
\(472\) 346874.i 1.55700i
\(473\) −49527.6 −0.221373
\(474\) 226293.i 1.00720i
\(475\) −252471. −1.11898
\(476\) 4162.73i 0.0183723i
\(477\) 13278.8i 0.0583608i
\(478\) 44306.7 0.193916
\(479\) −199683. −0.870300 −0.435150 0.900358i \(-0.643305\pi\)
−0.435150 + 0.900358i \(0.643305\pi\)
\(480\) 633.041i 0.00274757i
\(481\) 324729.i 1.40356i
\(482\) 349699.i 1.50522i
\(483\) −34188.5 −0.146550
\(484\) −4718.27 −0.0201415
\(485\) −24034.0 −0.102175
\(486\) −15356.1 −0.0650142
\(487\) 120283.i 0.507160i −0.967314 0.253580i \(-0.918392\pi\)
0.967314 0.253580i \(-0.0816080\pi\)
\(488\) 213798. 0.897769
\(489\) 209321.i 0.875378i
\(490\) 17847.3 0.0743327
\(491\) −292583. −1.21363 −0.606814 0.794844i \(-0.707553\pi\)
−0.606814 + 0.794844i \(0.707553\pi\)
\(492\) −3905.33 −0.0161334
\(493\) 84328.0 0.346959
\(494\) 389731. 1.59702
\(495\) 3636.31 0.0148406
\(496\) 409146.i 1.66309i
\(497\) 9237.45i 0.0373972i
\(498\) 11864.2 0.0478387
\(499\) 237737.i 0.954762i −0.878696 0.477381i \(-0.841586\pi\)
0.878696 0.477381i \(-0.158414\pi\)
\(500\) 1185.48i 0.00474191i
\(501\) 156900.i 0.625096i
\(502\) −121453. −0.481950
\(503\) 101233.i 0.400115i −0.979784 0.200058i \(-0.935887\pi\)
0.979784 0.200058i \(-0.0641129\pi\)
\(504\) −33825.6 −0.133163
\(505\) 33694.6 0.132123
\(506\) −82479.6 −0.322141
\(507\) 141389.i 0.550045i
\(508\) 4021.11 0.0155819
\(509\) 132814. 0.512634 0.256317 0.966593i \(-0.417491\pi\)
0.256317 + 0.966593i \(0.417491\pi\)
\(510\) 22326.1i 0.0858367i
\(511\) 170480.i 0.652878i
\(512\) 250698.i 0.956336i
\(513\) 57112.8i 0.217020i
\(514\) 156169.i 0.591109i
\(515\) 10958.7i 0.0413187i
\(516\) −1819.24 −0.00683267
\(517\) 105938.i 0.396343i
\(518\) −110668. −0.412440
\(519\) 61829.9i 0.229543i
\(520\) 32692.0i 0.120902i
\(521\) 347656.i 1.28078i −0.768051 0.640389i \(-0.778773\pi\)
0.768051 0.640389i \(-0.221227\pi\)
\(522\) 19104.4i 0.0701121i
\(523\) −286879. −1.04881 −0.524404 0.851470i \(-0.675712\pi\)
−0.524404 + 0.851470i \(0.675712\pi\)
\(524\) −4029.53 −0.0146754
\(525\) 63979.1 0.232124
\(526\) 218904.i 0.791192i
\(527\) 752315.i 2.70881i
\(528\) −83819.3 −0.300660
\(529\) −170009. −0.607520
\(530\) 4373.74i 0.0155704i
\(531\) 148418. 0.526378
\(532\) 3507.46i 0.0123928i
\(533\) 408986. 1.43964
\(534\) 262394. 0.920176
\(535\) 5338.46i 0.0186513i
\(536\) −196283. 204241.i −0.683207 0.710907i
\(537\) 245381. 0.850927
\(538\) 209310.i 0.723144i
\(539\) 123204.i 0.424079i
\(540\) 133.569 0.000458054
\(541\) 17631.0i 0.0602397i 0.999546 + 0.0301198i \(0.00958889\pi\)
−0.999546 + 0.0301198i \(0.990411\pi\)
\(542\) −493338. −1.67937
\(543\) 207115.i 0.702446i
\(544\) 26830.9i 0.0906647i
\(545\) 45715.5 0.153911
\(546\) −98762.4 −0.331289
\(547\) 333837.i 1.11573i 0.829931 + 0.557866i \(0.188380\pi\)
−0.829931 + 0.557866i \(0.811620\pi\)
\(548\) 14809.7i 0.0493156i
\(549\) 91478.5i 0.303511i
\(550\) 154349. 0.510246
\(551\) 71053.7 0.234037
\(552\) 108666. 0.356629
\(553\) 213280. 0.697429
\(554\) 123196.i 0.401400i
\(555\) −15674.2 −0.0508861
\(556\) 2839.14i 0.00918412i
\(557\) −60637.0 −0.195446 −0.0977231 0.995214i \(-0.531156\pi\)
−0.0977231 + 0.995214i \(0.531156\pi\)
\(558\) 170436. 0.547385
\(559\) 190520. 0.609702
\(560\) 11443.8 0.0364918
\(561\) −154122. −0.489711
\(562\) 260976. 0.826281
\(563\) 57207.8i 0.180484i −0.995920 0.0902420i \(-0.971236\pi\)
0.995920 0.0902420i \(-0.0287640\pi\)
\(564\) 3891.30i 0.0122331i
\(565\) 44018.9 0.137893
\(566\) 175704.i 0.548465i
\(567\) 14473.1i 0.0450189i
\(568\) 29360.8i 0.0910062i
\(569\) 304897. 0.941734 0.470867 0.882204i \(-0.343941\pi\)
0.470867 + 0.882204i \(0.343941\pi\)
\(570\) 18811.7i 0.0579000i
\(571\) −354457. −1.08716 −0.543578 0.839359i \(-0.682931\pi\)
−0.543578 + 0.839359i \(0.682931\pi\)
\(572\) −6291.99 −0.0192307
\(573\) 270944. 0.825222
\(574\) 139382.i 0.423043i
\(575\) −205536. −0.621658
\(576\) 107432. 0.323808
\(577\) 462510.i 1.38922i 0.719389 + 0.694608i \(0.244422\pi\)
−0.719389 + 0.694608i \(0.755578\pi\)
\(578\) 607690.i 1.81897i
\(579\) 163441.i 0.487533i
\(580\) 166.172i 0.000493971i
\(581\) 11182.0i 0.0331258i
\(582\) 230777.i 0.681313i
\(583\) −30192.9 −0.0888317
\(584\) 541863.i 1.58878i
\(585\) −13988.0 −0.0408737
\(586\) 315061.i 0.917486i
\(587\) 211848.i 0.614820i 0.951577 + 0.307410i \(0.0994624\pi\)
−0.951577 + 0.307410i \(0.900538\pi\)
\(588\) 4525.51i 0.0130892i
\(589\) 633891.i 1.82719i
\(590\) −48885.6 −0.140436
\(591\) −143679. −0.411357
\(592\) 361300. 1.03092
\(593\) 182460.i 0.518871i −0.965760 0.259435i \(-0.916464\pi\)
0.965760 0.259435i \(-0.0835365\pi\)
\(594\) 34916.3i 0.0989589i
\(595\) 21042.3 0.0594373
\(596\) −2516.60 −0.00708470
\(597\) 279258.i 0.783531i
\(598\) 317279. 0.887235
\(599\) 127925.i 0.356534i −0.983982 0.178267i \(-0.942951\pi\)
0.983982 0.178267i \(-0.0570490\pi\)
\(600\) −203355. −0.564874
\(601\) 376156. 1.04140 0.520702 0.853738i \(-0.325670\pi\)
0.520702 + 0.853738i \(0.325670\pi\)
\(602\) 64929.3i 0.179163i
\(603\) −87389.1 + 83984.0i −0.240338 + 0.230973i
\(604\) 4982.32 0.0136571
\(605\) 23850.5i 0.0651609i
\(606\) 323539.i 0.881010i
\(607\) −552472. −1.49945 −0.749726 0.661748i \(-0.769815\pi\)
−0.749726 + 0.661748i \(0.769815\pi\)
\(608\) 22607.4i 0.0611567i
\(609\) −18005.9 −0.0485489
\(610\) 30131.0i 0.0809756i
\(611\) 407518.i 1.09160i
\(612\) −5661.20 −0.0151149
\(613\) 130656. 0.347703 0.173851 0.984772i \(-0.444379\pi\)
0.173851 + 0.984772i \(0.444379\pi\)
\(614\) 398983.i 1.05832i
\(615\) 19741.1i 0.0521942i
\(616\) 76911.8i 0.202690i
\(617\) 251274. 0.660051 0.330026 0.943972i \(-0.392943\pi\)
0.330026 + 0.943972i \(0.392943\pi\)
\(618\) 105227. 0.275518
\(619\) 101077. 0.263797 0.131899 0.991263i \(-0.457893\pi\)
0.131899 + 0.991263i \(0.457893\pi\)
\(620\) −1482.47 −0.00385658
\(621\) 46495.4i 0.120567i
\(622\) −479399. −1.23913
\(623\) 247305.i 0.637173i
\(624\) 322432. 0.828075
\(625\) 381625. 0.976959
\(626\) −382019. −0.974847
\(627\) −129862. −0.330328
\(628\) −14122.0 −0.0358076
\(629\) 664338. 1.67914
\(630\) 4767.11i 0.0120109i
\(631\) 89295.7i 0.224270i −0.993693 0.112135i \(-0.964231\pi\)
0.993693 0.112135i \(-0.0357690\pi\)
\(632\) −677900. −1.69720
\(633\) 263625.i 0.657930i
\(634\) 6767.58i 0.0168366i
\(635\) 20326.5i 0.0504097i
\(636\) −1109.04 −0.00274179
\(637\) 473935.i 1.16799i
\(638\) −43439.1 −0.106718
\(639\) −12562.7 −0.0307667
\(640\) −37334.9 −0.0911497
\(641\) 218971.i 0.532932i −0.963844 0.266466i \(-0.914144\pi\)
0.963844 0.266466i \(-0.0858559\pi\)
\(642\) −51260.4 −0.124369
\(643\) −286522. −0.693004 −0.346502 0.938049i \(-0.612631\pi\)
−0.346502 + 0.938049i \(0.612631\pi\)
\(644\) 2855.42i 0.00688490i
\(645\) 9196.13i 0.0221048i
\(646\) 797319.i 1.91059i
\(647\) 474638.i 1.13385i 0.823771 + 0.566923i \(0.191866\pi\)
−0.823771 + 0.566923i \(0.808134\pi\)
\(648\) 46002.0i 0.109554i
\(649\) 337469.i 0.801206i
\(650\) −593744. −1.40531
\(651\) 160636.i 0.379035i
\(652\) −17482.5 −0.0411252
\(653\) 392898.i 0.921412i 0.887553 + 0.460706i \(0.152404\pi\)
−0.887553 + 0.460706i \(0.847596\pi\)
\(654\) 438964.i 1.02630i
\(655\) 20369.0i 0.0474774i
\(656\) 455046.i 1.05742i
\(657\) −231849. −0.537123
\(658\) 138882. 0.320770
\(659\) −96696.5 −0.222659 −0.111329 0.993784i \(-0.535511\pi\)
−0.111329 + 0.993784i \(0.535511\pi\)
\(660\) 303.705i 0.000697210i
\(661\) 50931.1i 0.116568i 0.998300 + 0.0582841i \(0.0185629\pi\)
−0.998300 + 0.0582841i \(0.981437\pi\)
\(662\) −108718. −0.248077
\(663\) 592871. 1.34875
\(664\) 35541.3i 0.0806116i
\(665\) 17730.0 0.0400927
\(666\) 150505.i 0.339315i
\(667\) 57844.7 0.130021
\(668\) 13104.2 0.0293670
\(669\) 383445.i 0.856745i
\(670\) 28784.0 27662.5i 0.0641213 0.0616228i
\(671\) −208001. −0.461978
\(672\) 5728.99i 0.0126864i
\(673\) 596582.i 1.31716i −0.752509 0.658582i \(-0.771156\pi\)
0.752509 0.658582i \(-0.228844\pi\)
\(674\) −514364. −1.13227
\(675\) 87009.9i 0.190968i
\(676\) 11808.8 0.0258411
\(677\) 674717.i 1.47212i −0.676915 0.736062i \(-0.736683\pi\)
0.676915 0.736062i \(-0.263317\pi\)
\(678\) 422674.i 0.919487i
\(679\) −217507. −0.471773
\(680\) −66881.9 −0.144641
\(681\) 18724.6i 0.0403756i
\(682\) 387533.i 0.833182i
\(683\) 86363.0i 0.185134i 0.995706 + 0.0925671i \(0.0295073\pi\)
−0.995706 + 0.0925671i \(0.970493\pi\)
\(684\) −4770.06 −0.0101956
\(685\) 74861.8 0.159544
\(686\) 354757. 0.753846
\(687\) 72209.3 0.152996
\(688\) 211977.i 0.447828i
\(689\) 116145. 0.244659
\(690\) 15314.6i 0.0321667i
\(691\) −295298. −0.618450 −0.309225 0.950989i \(-0.600070\pi\)
−0.309225 + 0.950989i \(0.600070\pi\)
\(692\) −5164.03 −0.0107839
\(693\) 32908.5 0.0685238
\(694\) 873830. 1.81430
\(695\) −14351.7 −0.0297121
\(696\) 57230.8 0.118144
\(697\) 836713.i 1.72231i
\(698\) 289903.i 0.595034i
\(699\) −356683. −0.730009
\(700\) 5343.53i 0.0109052i
\(701\) 219670.i 0.447028i −0.974701 0.223514i \(-0.928247\pi\)
0.974701 0.223514i \(-0.0717529\pi\)
\(702\) 134314.i 0.272551i
\(703\) 559763. 1.13264
\(704\) 244275.i 0.492872i
\(705\) 19670.3 0.0395760
\(706\) 738969. 1.48257
\(707\) 304934. 0.610052
\(708\) 12395.9i 0.0247292i
\(709\) 315595. 0.627824 0.313912 0.949452i \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(710\) 4137.87 0.00820843
\(711\) 290055.i 0.573775i
\(712\) 786048.i 1.55056i
\(713\) 516049.i 1.01511i
\(714\) 202050.i 0.396336i
\(715\) 31805.5i 0.0622144i
\(716\) 20494.2i 0.0399765i
\(717\) 56791.0 0.110469
\(718\) 225158.i 0.436756i
\(719\) 249888. 0.483379 0.241689 0.970354i \(-0.422299\pi\)
0.241689 + 0.970354i \(0.422299\pi\)
\(720\) 15563.3i 0.0300218i
\(721\) 99176.1i 0.190782i
\(722\) 143505.i 0.275291i
\(723\) 448234.i 0.857488i
\(724\) 17298.3 0.0330009
\(725\) −108249. −0.205942
\(726\) −229015. −0.434501
\(727\) 377046.i 0.713387i 0.934221 + 0.356693i \(0.116096\pi\)
−0.934221 + 0.356693i \(0.883904\pi\)
\(728\) 295861.i 0.558245i
\(729\) −19683.0 −0.0370370
\(730\) 76365.8 0.143302
\(731\) 389771.i 0.729415i
\(732\) −7640.28 −0.0142589
\(733\) 64904.6i 0.120800i −0.998174 0.0604001i \(-0.980762\pi\)
0.998174 0.0604001i \(-0.0192377\pi\)
\(734\) 401511. 0.745256
\(735\) 22876.1 0.0423456
\(736\) 18404.7i 0.0339760i
\(737\) 190961. + 198703.i 0.351567 + 0.365821i
\(738\) −189557. −0.348037
\(739\) 309648.i 0.566996i 0.958973 + 0.283498i \(0.0914949\pi\)
−0.958973 + 0.283498i \(0.908505\pi\)
\(740\) 1309.11i 0.00239062i
\(741\) 499545. 0.909784
\(742\) 39582.1i 0.0718937i
\(743\) −917129. −1.66132 −0.830659 0.556782i \(-0.812036\pi\)
−0.830659 + 0.556782i \(0.812036\pi\)
\(744\) 510572.i 0.922383i
\(745\) 12721.2i 0.0229201i
\(746\) −849024. −1.52561
\(747\) 15207.2 0.0272526
\(748\) 12872.3i 0.0230066i
\(749\) 48312.8i 0.0861189i
\(750\) 57540.7i 0.102295i
\(751\) 531941. 0.943156 0.471578 0.881824i \(-0.343685\pi\)
0.471578 + 0.881824i \(0.343685\pi\)
\(752\) −453412. −0.801784
\(753\) −155675. −0.274555
\(754\) 167100. 0.293922
\(755\) 25185.3i 0.0441827i
\(756\) 1208.79 0.00211498
\(757\) 1.10930e6i 1.93578i 0.251380 + 0.967889i \(0.419116\pi\)
−0.251380 + 0.967889i \(0.580884\pi\)
\(758\) 136725. 0.237963
\(759\) −105720. −0.183516
\(760\) −56353.9 −0.0975656
\(761\) −770312. −1.33014 −0.665070 0.746781i \(-0.731598\pi\)
−0.665070 + 0.746781i \(0.731598\pi\)
\(762\) 195177. 0.336138
\(763\) 413723. 0.710657
\(764\) 22629.2i 0.0387689i
\(765\) 28617.0i 0.0488991i
\(766\) 118509. 0.201972
\(767\) 1.29816e6i 2.20667i
\(768\) 27689.7i 0.0469457i
\(769\) 336459.i 0.568956i 0.958683 + 0.284478i \(0.0918203\pi\)
−0.958683 + 0.284478i \(0.908180\pi\)
\(770\) −10839.3 −0.0182819
\(771\) 200172.i 0.336740i
\(772\) −13650.6 −0.0229043
\(773\) 575355. 0.962891 0.481445 0.876476i \(-0.340112\pi\)
0.481445 + 0.876476i \(0.340112\pi\)
\(774\) −88302.2 −0.147397
\(775\) 965716.i 1.60785i
\(776\) 691334. 1.14806
\(777\) −141851. −0.234957
\(778\) 1.18061e6i 1.95050i
\(779\) 705004.i 1.16176i
\(780\) 1168.28i 0.00192024i
\(781\) 28564.7i 0.0468303i
\(782\) 649096.i 1.06144i
\(783\) 24487.5i 0.0399412i
\(784\) −527309. −0.857894
\(785\) 71385.5i 0.115843i
\(786\) −195585. −0.316585
\(787\) 29692.7i 0.0479402i 0.999713 + 0.0239701i \(0.00763065\pi\)
−0.999713 + 0.0239701i \(0.992369\pi\)
\(788\) 12000.1i 0.0193255i
\(789\) 280584.i 0.450723i
\(790\) 95537.8i 0.153081i
\(791\) 398369. 0.636696
\(792\) −104598. −0.166753
\(793\) 800130. 1.27237
\(794\) 1.08041e6i 1.71375i
\(795\) 5606.13i 0.00887010i
\(796\) 23323.6 0.0368103
\(797\) 85596.4 0.134753 0.0673766 0.997728i \(-0.478537\pi\)
0.0673766 + 0.997728i \(0.478537\pi\)
\(798\) 170245.i 0.267343i
\(799\) −833708. −1.30593
\(800\) 34441.8i 0.0538153i
\(801\) 336329. 0.524202
\(802\) 107595. 0.167280
\(803\) 527171.i 0.817561i
\(804\) 7014.34 + 7298.73i 0.0108511 + 0.0112911i
\(805\) 14433.9 0.0222737
\(806\) 1.49074e6i 2.29474i
\(807\) 268287.i 0.411958i
\(808\) −969218. −1.48456
\(809\) 762900.i 1.16566i −0.812595 0.582828i \(-0.801946\pi\)
0.812595 0.582828i \(-0.198054\pi\)
\(810\) 6483.15 0.00988134
\(811\) 1.01687e6i 1.54606i 0.634371 + 0.773029i \(0.281259\pi\)
−0.634371 + 0.773029i \(0.718741\pi\)
\(812\) 1503.85i 0.00228083i
\(813\) −632347. −0.956697
\(814\) −342214. −0.516475
\(815\) 88372.7i 0.133046i
\(816\) 659639.i 0.990663i
\(817\) 328416.i 0.492017i
\(818\) −724731. −1.08310
\(819\) −126591. −0.188727
\(820\) 1648.78 0.00245208
\(821\) 278756. 0.413559 0.206779 0.978388i \(-0.433702\pi\)
0.206779 + 0.978388i \(0.433702\pi\)
\(822\) 718831.i 1.06386i
\(823\) 126757. 0.187142 0.0935711 0.995613i \(-0.470172\pi\)
0.0935711 + 0.995613i \(0.470172\pi\)
\(824\) 315227.i 0.464268i
\(825\) 197841. 0.290675
\(826\) −442413. −0.648436
\(827\) 717778. 1.04949 0.524746 0.851259i \(-0.324160\pi\)
0.524746 + 0.851259i \(0.324160\pi\)
\(828\) −3883.29 −0.00566421
\(829\) 564602. 0.821548 0.410774 0.911737i \(-0.365258\pi\)
0.410774 + 0.911737i \(0.365258\pi\)
\(830\) −5008.91 −0.00727088
\(831\) 157909.i 0.228668i
\(832\) 939668.i 1.35746i
\(833\) −969587. −1.39732
\(834\) 137806.i 0.198124i
\(835\) 66241.0i 0.0950067i
\(836\) 10846.0i 0.0155188i
\(837\) 218460. 0.311832
\(838\) 620704.i 0.883886i
\(839\) 700709. 0.995437 0.497719 0.867339i \(-0.334171\pi\)
0.497719 + 0.867339i \(0.334171\pi\)
\(840\) 14280.8 0.0202392
\(841\) −676816. −0.956927
\(842\) 956395.i 1.34900i
\(843\) 334511. 0.470713
\(844\) −22017.9 −0.0309095
\(845\) 59692.4i 0.0835999i
\(846\) 188876.i 0.263898i
\(847\) 215846.i 0.300869i
\(848\) 129225.i 0.179702i
\(849\) 225213.i 0.312448i
\(850\) 1.21470e6i 1.68124i
\(851\) 455702. 0.629247
\(852\) 1049.23i 0.00144542i
\(853\) 118032. 0.162219 0.0811093 0.996705i \(-0.474154\pi\)
0.0811093 + 0.996705i \(0.474154\pi\)
\(854\) 272684.i 0.373890i
\(855\) 24112.3i 0.0329842i
\(856\) 153560.i 0.209571i
\(857\) 1.06958e6i 1.45630i −0.685417 0.728151i \(-0.740380\pi\)
0.685417 0.728151i \(-0.259620\pi\)
\(858\) −305400. −0.414853
\(859\) −30306.7 −0.0410726 −0.0205363 0.999789i \(-0.506537\pi\)
−0.0205363 + 0.999789i \(0.506537\pi\)
\(860\) 768.060 0.00103848
\(861\) 178656.i 0.240997i
\(862\) 1.39687e6i 1.87992i
\(863\) 218235. 0.293024 0.146512 0.989209i \(-0.453195\pi\)
0.146512 + 0.989209i \(0.453195\pi\)
\(864\) −7791.28 −0.0104371
\(865\) 26103.8i 0.0348876i
\(866\) 697735. 0.930368
\(867\) 778920.i 1.03623i
\(868\) −13416.3 −0.0178071
\(869\) 659520. 0.873350
\(870\) 8065.65i 0.0106562i
\(871\) −734578. 764361.i −0.968282 1.00754i
\(872\) −1.31500e6 −1.72939
\(873\) 295803.i 0.388127i
\(874\) 546920.i 0.715980i
\(875\) −54231.9 −0.0708335
\(876\) 19364.0i 0.0252340i
\(877\) 40301.6 0.0523991 0.0261995 0.999657i \(-0.491659\pi\)
0.0261995 + 0.999657i \(0.491659\pi\)
\(878\) 39392.9i 0.0511009i
\(879\) 403836.i 0.522670i
\(880\) 35387.4 0.0456966
\(881\) −623690. −0.803557 −0.401779 0.915737i \(-0.631608\pi\)
−0.401779 + 0.915737i \(0.631608\pi\)
\(882\) 219659.i 0.282366i
\(883\) 435337.i 0.558346i 0.960241 + 0.279173i \(0.0900604\pi\)
−0.960241 + 0.279173i \(0.909940\pi\)
\(884\) 49516.5i 0.0633644i
\(885\) −62660.2 −0.0800028
\(886\) 1.26537e6 1.61195
\(887\) −1.34469e6 −1.70913 −0.854563 0.519348i \(-0.826175\pi\)
−0.854563 + 0.519348i \(0.826175\pi\)
\(888\) 450865. 0.571769
\(889\) 183953.i 0.232758i
\(890\) −110779. −0.139855
\(891\) 44754.7i 0.0563745i
\(892\) −32025.3 −0.0402498
\(893\) −702472. −0.880899
\(894\) −122151. −0.152834
\(895\) −103597. −0.129330
\(896\) −337879. −0.420868
\(897\) 406679. 0.505437
\(898\) 1.21591e6i 1.50782i
\(899\) 271785.i 0.336284i
\(900\) 7267.06 0.00897168
\(901\) 237611.i 0.292697i
\(902\) 431008.i 0.529752i
\(903\) 83224.5i 0.102065i
\(904\) −1.26620e6 −1.54940
\(905\) 87441.5i 0.106763i
\(906\) 241832. 0.294616
\(907\) −299833. −0.364472 −0.182236 0.983255i \(-0.558334\pi\)
−0.182236 + 0.983255i \(0.558334\pi\)
\(908\) 1563.88 0.00189684
\(909\) 414702.i 0.501890i
\(910\) 41696.2 0.0503517
\(911\) −406913. −0.490304 −0.245152 0.969485i \(-0.578838\pi\)
−0.245152 + 0.969485i \(0.578838\pi\)
\(912\) 555804.i 0.668239i
\(913\) 34577.7i 0.0414815i
\(914\) 330124.i 0.395171i
\(915\) 38621.1i 0.0461299i
\(916\) 6030.91i 0.00718774i
\(917\) 184338.i 0.219218i
\(918\) −274783. −0.326065
\(919\) 888551.i 1.05209i 0.850458 + 0.526043i \(0.176325\pi\)
−0.850458 + 0.526043i \(0.823675\pi\)
\(920\) −45877.6 −0.0542032
\(921\) 511404.i 0.602900i
\(922\) 1.41998e6i 1.67039i
\(923\) 109881.i 0.128979i
\(924\) 2748.51i 0.00321924i
\(925\) −852784. −0.996679
\(926\) 40283.3 0.0469789
\(927\) 134877. 0.156956
\(928\) 9693.08i 0.0112555i
\(929\) 287037.i 0.332588i −0.986076 0.166294i \(-0.946820\pi\)
0.986076 0.166294i \(-0.0531801\pi\)
\(930\) −71955.9 −0.0831957
\(931\) −816962. −0.942546
\(932\) 29790.1i 0.0342958i
\(933\) −614480. −0.705902
\(934\) 741128.i 0.849571i
\(935\) 65068.5 0.0744299
\(936\) 402363. 0.459268
\(937\) 1213.00i 0.00138160i 1.00000 0.000690800i \(0.000219888\pi\)
−1.00000 0.000690800i \(0.999780\pi\)
\(938\) 260494. 250344.i 0.296069 0.284533i
\(939\) −489661. −0.555347
\(940\) 1642.86i 0.00185928i
\(941\) 650986.i 0.735178i −0.929988 0.367589i \(-0.880183\pi\)
0.929988 0.367589i \(-0.119817\pi\)
\(942\) −685451. −0.772458
\(943\) 573942.i 0.645423i
\(944\) 1.44436e6 1.62080
\(945\) 6110.35i 0.00684230i
\(946\) 200779.i 0.224355i
\(947\) 324996. 0.362392 0.181196 0.983447i \(-0.442003\pi\)
0.181196 + 0.983447i \(0.442003\pi\)
\(948\) 24225.4 0.0269559
\(949\) 2.02790e6i 2.25171i
\(950\) 1.02349e6i 1.13406i
\(951\) 8674.49i 0.00959142i
\(952\) −605278. −0.667854
\(953\) 1.41035e6 1.55290 0.776448 0.630181i \(-0.217019\pi\)
0.776448 + 0.630181i \(0.217019\pi\)
\(954\) −53830.6 −0.0591470
\(955\) −114389. −0.125423
\(956\) 4743.18i 0.00518984i
\(957\) −55679.0 −0.0607950
\(958\) 809490.i 0.882024i
\(959\) 677496. 0.736664
\(960\) −45356.3 −0.0492148
\(961\) −1.50115e6 −1.62546
\(962\) 1.31641e6 1.42247
\(963\) −65704.1 −0.0708501
\(964\) 37436.4 0.0402847
\(965\) 69002.7i 0.0740988i
\(966\) 138596.i 0.148524i
\(967\) −197552. −0.211266 −0.105633 0.994405i \(-0.533687\pi\)
−0.105633 + 0.994405i \(0.533687\pi\)
\(968\) 686057.i 0.732166i
\(969\) 1.02198e6i 1.08842i
\(970\) 97431.1i 0.103551i
\(971\) −480283. −0.509400 −0.254700 0.967020i \(-0.581977\pi\)
−0.254700 + 0.967020i \(0.581977\pi\)
\(972\) 1643.92i 0.00174000i
\(973\) −129882. −0.137190
\(974\) −487611. −0.513992
\(975\) −761044. −0.800573
\(976\) 890240.i 0.934560i
\(977\) 1.54035e6 1.61373 0.806866 0.590735i \(-0.201162\pi\)
0.806866 + 0.590735i \(0.201162\pi\)
\(978\) −848564. −0.887170
\(979\) 764734.i 0.797894i
\(980\) 1910.61i 0.00198939i
\(981\) 562652.i 0.584658i
\(982\) 1.18610e6i 1.22998i
\(983\) 1.18578e6i 1.22715i 0.789636 + 0.613575i \(0.210269\pi\)
−0.789636 + 0.613575i \(0.789731\pi\)
\(984\) 567851.i 0.586468i
\(985\) 60659.6 0.0625211
\(986\) 341856.i 0.351633i
\(987\) 178015. 0.182735
\(988\) 41722.0i 0.0427416i
\(989\) 267363.i 0.273343i
\(990\) 14741.2i 0.0150405i
\(991\) 92429.5i 0.0941160i 0.998892 + 0.0470580i \(0.0149846\pi\)
−0.998892 + 0.0470580i \(0.985015\pi\)
\(992\) 86474.8 0.0878752
\(993\) −139352. −0.141324
\(994\) 37447.5 0.0379010
\(995\) 117899.i 0.119087i
\(996\) 1270.10i 0.00128032i
\(997\) −668935. −0.672967 −0.336484 0.941689i \(-0.609238\pi\)
−0.336484 + 0.941689i \(0.609238\pi\)
\(998\) −963758. −0.967624
\(999\) 192913.i 0.193299i
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 201.5.b.a.133.13 46
67.66 odd 2 inner 201.5.b.a.133.34 yes 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
201.5.b.a.133.13 46 1.1 even 1 trivial
201.5.b.a.133.34 yes 46 67.66 odd 2 inner