Properties

Label 1050.3.c.b.449.32
Level $1050$
Weight $3$
Character 1050.449
Analytic conductor $28.610$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(449,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.c (of order \(2\), degree \(1\), not 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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.32
Character \(\chi\) \(=\) 1050.449
Dual form 1050.3.c.b.449.31

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +(2.79495 + 1.09006i) q^{3} +2.00000 q^{4} +(3.95266 + 1.54158i) q^{6} -2.64575i q^{7} +2.82843 q^{8} +(6.62354 + 6.09333i) q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +(2.79495 + 1.09006i) q^{3} +2.00000 q^{4} +(3.95266 + 1.54158i) q^{6} -2.64575i q^{7} +2.82843 q^{8} +(6.62354 + 6.09333i) q^{9} +3.30912i q^{11} +(5.58991 + 2.18012i) q^{12} +13.2159i q^{13} -3.74166i q^{14} +4.00000 q^{16} +14.2264 q^{17} +(9.36710 + 8.61727i) q^{18} -7.45106 q^{19} +(2.88403 - 7.39475i) q^{21} +4.67981i q^{22} +7.07387 q^{23} +(7.90533 + 3.08315i) q^{24} +18.6902i q^{26} +(11.8704 + 24.2506i) q^{27} -5.29150i q^{28} +25.7409i q^{29} +41.5150 q^{31} +5.65685 q^{32} +(-3.60714 + 9.24885i) q^{33} +20.1192 q^{34} +(13.2471 + 12.1867i) q^{36} -27.7055i q^{37} -10.5374 q^{38} +(-14.4062 + 36.9380i) q^{39} +1.25580i q^{41} +(4.07863 - 10.4578i) q^{42} -2.12678i q^{43} +6.61825i q^{44} +10.0040 q^{46} -54.8353 q^{47} +(11.1798 + 4.36024i) q^{48} -7.00000 q^{49} +(39.7623 + 15.5077i) q^{51} +26.4319i q^{52} +62.3147 q^{53} +(16.7873 + 34.2956i) q^{54} -7.48331i q^{56} +(-20.8254 - 8.12210i) q^{57} +36.4032i q^{58} -21.7601i q^{59} +84.8047 q^{61} +58.7111 q^{62} +(16.1214 - 17.5242i) q^{63} +8.00000 q^{64} +(-5.10127 + 13.0799i) q^{66} +54.7450i q^{67} +28.4529 q^{68} +(19.7711 + 7.71093i) q^{69} -100.282i q^{71} +(18.7342 + 17.2345i) q^{72} -63.2034i q^{73} -39.1815i q^{74} -14.9021 q^{76} +8.75512 q^{77} +(-20.3734 + 52.2382i) q^{78} -116.942 q^{79} +(6.74263 + 80.7189i) q^{81} +1.77596i q^{82} -23.7433 q^{83} +(5.76805 - 14.7895i) q^{84} -3.00772i q^{86} +(-28.0592 + 71.9448i) q^{87} +9.35962i q^{88} -3.26308i q^{89} +34.9661 q^{91} +14.1477 q^{92} +(116.033 + 45.2538i) q^{93} -77.5489 q^{94} +(15.8107 + 6.16631i) q^{96} +55.5754i q^{97} -9.89949 q^{98} +(-20.1636 + 21.9181i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 64 q^{4} - 16 q^{6} - 16 q^{9} + 128 q^{16} + 48 q^{19} + 56 q^{21} - 32 q^{24} + 48 q^{31} + 256 q^{34} - 32 q^{36} + 192 q^{39} + 160 q^{46} - 224 q^{49} + 288 q^{51} - 80 q^{54} - 112 q^{61} + 256 q^{64} - 192 q^{66} + 344 q^{69} + 96 q^{76} - 256 q^{79} + 160 q^{81} + 112 q^{84} - 448 q^{91} + 416 q^{94} - 64 q^{96} - 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 2.79495 + 1.09006i 0.931652 + 0.363353i
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 3.95266 + 1.54158i 0.658777 + 0.256929i
\(7\) 2.64575i 0.377964i
\(8\) 2.82843 0.353553
\(9\) 6.62354 + 6.09333i 0.735949 + 0.677037i
\(10\) 0 0
\(11\) 3.30912i 0.300830i 0.988623 + 0.150415i \(0.0480609\pi\)
−0.988623 + 0.150415i \(0.951939\pi\)
\(12\) 5.58991 + 2.18012i 0.465826 + 0.181677i
\(13\) 13.2159i 1.01661i 0.861177 + 0.508306i \(0.169728\pi\)
−0.861177 + 0.508306i \(0.830272\pi\)
\(14\) 3.74166i 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 14.2264 0.836850 0.418425 0.908251i \(-0.362582\pi\)
0.418425 + 0.908251i \(0.362582\pi\)
\(18\) 9.36710 + 8.61727i 0.520395 + 0.478737i
\(19\) −7.45106 −0.392161 −0.196081 0.980588i \(-0.562821\pi\)
−0.196081 + 0.980588i \(0.562821\pi\)
\(20\) 0 0
\(21\) 2.88403 7.39475i 0.137335 0.352131i
\(22\) 4.67981i 0.212719i
\(23\) 7.07387 0.307559 0.153780 0.988105i \(-0.450855\pi\)
0.153780 + 0.988105i \(0.450855\pi\)
\(24\) 7.90533 + 3.08315i 0.329389 + 0.128465i
\(25\) 0 0
\(26\) 18.6902i 0.718853i
\(27\) 11.8704 + 24.2506i 0.439645 + 0.898172i
\(28\) 5.29150i 0.188982i
\(29\) 25.7409i 0.887619i 0.896121 + 0.443809i \(0.146373\pi\)
−0.896121 + 0.443809i \(0.853627\pi\)
\(30\) 0 0
\(31\) 41.5150 1.33919 0.669597 0.742724i \(-0.266467\pi\)
0.669597 + 0.742724i \(0.266467\pi\)
\(32\) 5.65685 0.176777
\(33\) −3.60714 + 9.24885i −0.109307 + 0.280268i
\(34\) 20.1192 0.591742
\(35\) 0 0
\(36\) 13.2471 + 12.1867i 0.367975 + 0.338518i
\(37\) 27.7055i 0.748797i −0.927268 0.374398i \(-0.877849\pi\)
0.927268 0.374398i \(-0.122151\pi\)
\(38\) −10.5374 −0.277300
\(39\) −14.4062 + 36.9380i −0.369389 + 0.947127i
\(40\) 0 0
\(41\) 1.25580i 0.0306292i 0.999883 + 0.0153146i \(0.00487497\pi\)
−0.999883 + 0.0153146i \(0.995125\pi\)
\(42\) 4.07863 10.4578i 0.0971102 0.248994i
\(43\) 2.12678i 0.0494599i −0.999694 0.0247300i \(-0.992127\pi\)
0.999694 0.0247300i \(-0.00787259\pi\)
\(44\) 6.61825i 0.150415i
\(45\) 0 0
\(46\) 10.0040 0.217477
\(47\) −54.8353 −1.16671 −0.583355 0.812218i \(-0.698260\pi\)
−0.583355 + 0.812218i \(0.698260\pi\)
\(48\) 11.1798 + 4.36024i 0.232913 + 0.0908383i
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) 39.7623 + 15.5077i 0.779653 + 0.304072i
\(52\) 26.4319i 0.508306i
\(53\) 62.3147 1.17575 0.587874 0.808952i \(-0.299965\pi\)
0.587874 + 0.808952i \(0.299965\pi\)
\(54\) 16.7873 + 34.2956i 0.310876 + 0.635103i
\(55\) 0 0
\(56\) 7.48331i 0.133631i
\(57\) −20.8254 8.12210i −0.365357 0.142493i
\(58\) 36.4032i 0.627641i
\(59\) 21.7601i 0.368815i −0.982850 0.184407i \(-0.940963\pi\)
0.982850 0.184407i \(-0.0590366\pi\)
\(60\) 0 0
\(61\) 84.8047 1.39024 0.695121 0.718893i \(-0.255351\pi\)
0.695121 + 0.718893i \(0.255351\pi\)
\(62\) 58.7111 0.946954
\(63\) 16.1214 17.5242i 0.255896 0.278163i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) −5.10127 + 13.0799i −0.0772919 + 0.198180i
\(67\) 54.7450i 0.817089i 0.912738 + 0.408545i \(0.133964\pi\)
−0.912738 + 0.408545i \(0.866036\pi\)
\(68\) 28.4529 0.418425
\(69\) 19.7711 + 7.71093i 0.286538 + 0.111753i
\(70\) 0 0
\(71\) 100.282i 1.41243i −0.707999 0.706213i \(-0.750402\pi\)
0.707999 0.706213i \(-0.249598\pi\)
\(72\) 18.7342 + 17.2345i 0.260197 + 0.239369i
\(73\) 63.2034i 0.865800i −0.901442 0.432900i \(-0.857490\pi\)
0.901442 0.432900i \(-0.142510\pi\)
\(74\) 39.1815i 0.529479i
\(75\) 0 0
\(76\) −14.9021 −0.196081
\(77\) 8.75512 0.113703
\(78\) −20.3734 + 52.2382i −0.261197 + 0.669720i
\(79\) −116.942 −1.48028 −0.740138 0.672455i \(-0.765240\pi\)
−0.740138 + 0.672455i \(0.765240\pi\)
\(80\) 0 0
\(81\) 6.74263 + 80.7189i 0.0832423 + 0.996529i
\(82\) 1.77596i 0.0216581i
\(83\) −23.7433 −0.286064 −0.143032 0.989718i \(-0.545685\pi\)
−0.143032 + 0.989718i \(0.545685\pi\)
\(84\) 5.76805 14.7895i 0.0686673 0.176066i
\(85\) 0 0
\(86\) 3.00772i 0.0349734i
\(87\) −28.0592 + 71.9448i −0.322519 + 0.826952i
\(88\) 9.35962i 0.106359i
\(89\) 3.26308i 0.0366638i −0.999832 0.0183319i \(-0.994164\pi\)
0.999832 0.0183319i \(-0.00583555\pi\)
\(90\) 0 0
\(91\) 34.9661 0.384243
\(92\) 14.1477 0.153780
\(93\) 116.033 + 45.2538i 1.24766 + 0.486600i
\(94\) −77.5489 −0.824988
\(95\) 0 0
\(96\) 15.8107 + 6.16631i 0.164694 + 0.0642323i
\(97\) 55.5754i 0.572942i 0.958089 + 0.286471i \(0.0924822\pi\)
−0.958089 + 0.286471i \(0.907518\pi\)
\(98\) −9.89949 −0.101015
\(99\) −20.1636 + 21.9181i −0.203673 + 0.221395i
\(100\) 0 0
\(101\) 124.810i 1.23574i −0.786280 0.617870i \(-0.787996\pi\)
0.786280 0.617870i \(-0.212004\pi\)
\(102\) 56.2324 + 21.9312i 0.551298 + 0.215011i
\(103\) 112.881i 1.09593i 0.836500 + 0.547967i \(0.184598\pi\)
−0.836500 + 0.547967i \(0.815402\pi\)
\(104\) 37.3803i 0.359426i
\(105\) 0 0
\(106\) 88.1263 0.831380
\(107\) −84.6754 −0.791359 −0.395680 0.918389i \(-0.629491\pi\)
−0.395680 + 0.918389i \(0.629491\pi\)
\(108\) 23.7408 + 48.5013i 0.219822 + 0.449086i
\(109\) −165.164 −1.51527 −0.757634 0.652680i \(-0.773645\pi\)
−0.757634 + 0.652680i \(0.773645\pi\)
\(110\) 0 0
\(111\) 30.2006 77.4356i 0.272078 0.697618i
\(112\) 10.5830i 0.0944911i
\(113\) −55.8626 −0.494359 −0.247179 0.968970i \(-0.579504\pi\)
−0.247179 + 0.968970i \(0.579504\pi\)
\(114\) −29.4515 11.4864i −0.258347 0.100758i
\(115\) 0 0
\(116\) 51.4819i 0.443809i
\(117\) −80.5291 + 87.5364i −0.688283 + 0.748174i
\(118\) 30.7734i 0.260791i
\(119\) 37.6396i 0.316300i
\(120\) 0 0
\(121\) 110.050 0.909502
\(122\) 119.932 0.983049
\(123\) −1.36889 + 3.50989i −0.0111292 + 0.0285357i
\(124\) 83.0301 0.669597
\(125\) 0 0
\(126\) 22.7992 24.7830i 0.180946 0.196691i
\(127\) 166.400i 1.31023i 0.755527 + 0.655117i \(0.227381\pi\)
−0.755527 + 0.655117i \(0.772619\pi\)
\(128\) 11.3137 0.0883883
\(129\) 2.31831 5.94424i 0.0179714 0.0460794i
\(130\) 0 0
\(131\) 229.177i 1.74944i −0.484626 0.874721i \(-0.661044\pi\)
0.484626 0.874721i \(-0.338956\pi\)
\(132\) −7.21428 + 18.4977i −0.0546537 + 0.140134i
\(133\) 19.7137i 0.148223i
\(134\) 77.4211i 0.577769i
\(135\) 0 0
\(136\) 40.2385 0.295871
\(137\) −123.559 −0.901889 −0.450945 0.892552i \(-0.648913\pi\)
−0.450945 + 0.892552i \(0.648913\pi\)
\(138\) 27.9606 + 10.9049i 0.202613 + 0.0790211i
\(139\) −169.795 −1.22155 −0.610774 0.791805i \(-0.709141\pi\)
−0.610774 + 0.791805i \(0.709141\pi\)
\(140\) 0 0
\(141\) −153.262 59.7738i −1.08697 0.423927i
\(142\) 141.821i 0.998736i
\(143\) −43.7332 −0.305827
\(144\) 26.4942 + 24.3733i 0.183987 + 0.169259i
\(145\) 0 0
\(146\) 89.3832i 0.612213i
\(147\) −19.5647 7.63041i −0.133093 0.0519076i
\(148\) 55.4110i 0.374398i
\(149\) 237.791i 1.59592i −0.602713 0.797958i \(-0.705914\pi\)
0.602713 0.797958i \(-0.294086\pi\)
\(150\) 0 0
\(151\) 117.067 0.775281 0.387640 0.921811i \(-0.373290\pi\)
0.387640 + 0.921811i \(0.373290\pi\)
\(152\) −21.0748 −0.138650
\(153\) 94.2295 + 86.6865i 0.615879 + 0.566578i
\(154\) 12.3816 0.0804001
\(155\) 0 0
\(156\) −28.8123 + 73.8759i −0.184694 + 0.473564i
\(157\) 31.6151i 0.201370i 0.994918 + 0.100685i \(0.0321035\pi\)
−0.994918 + 0.100685i \(0.967897\pi\)
\(158\) −165.381 −1.04671
\(159\) 174.167 + 67.9267i 1.09539 + 0.427212i
\(160\) 0 0
\(161\) 18.7157i 0.116247i
\(162\) 9.53552 + 114.154i 0.0588612 + 0.704653i
\(163\) 107.553i 0.659833i 0.944010 + 0.329917i \(0.107021\pi\)
−0.944010 + 0.329917i \(0.892979\pi\)
\(164\) 2.51159i 0.0153146i
\(165\) 0 0
\(166\) −33.5781 −0.202278
\(167\) 116.906 0.700036 0.350018 0.936743i \(-0.386175\pi\)
0.350018 + 0.936743i \(0.386175\pi\)
\(168\) 8.15726 20.9155i 0.0485551 0.124497i
\(169\) −5.66117 −0.0334980
\(170\) 0 0
\(171\) −49.3524 45.4018i −0.288611 0.265508i
\(172\) 4.25355i 0.0247300i
\(173\) −317.805 −1.83702 −0.918512 0.395392i \(-0.870609\pi\)
−0.918512 + 0.395392i \(0.870609\pi\)
\(174\) −39.6816 + 101.745i −0.228055 + 0.584743i
\(175\) 0 0
\(176\) 13.2365i 0.0752074i
\(177\) 23.7198 60.8184i 0.134010 0.343607i
\(178\) 4.61469i 0.0259252i
\(179\) 325.244i 1.81700i −0.417880 0.908502i \(-0.637227\pi\)
0.417880 0.908502i \(-0.362773\pi\)
\(180\) 0 0
\(181\) −34.5370 −0.190812 −0.0954060 0.995438i \(-0.530415\pi\)
−0.0954060 + 0.995438i \(0.530415\pi\)
\(182\) 49.4495 0.271701
\(183\) 237.025 + 92.4421i 1.29522 + 0.505148i
\(184\) 20.0079 0.108739
\(185\) 0 0
\(186\) 164.095 + 63.9986i 0.882231 + 0.344079i
\(187\) 47.0771i 0.251749i
\(188\) −109.671 −0.583355
\(189\) 64.1612 31.4062i 0.339477 0.166170i
\(190\) 0 0
\(191\) 80.5748i 0.421858i −0.977501 0.210929i \(-0.932351\pi\)
0.977501 0.210929i \(-0.0676489\pi\)
\(192\) 22.3596 + 8.72047i 0.116456 + 0.0454191i
\(193\) 102.042i 0.528717i −0.964424 0.264359i \(-0.914840\pi\)
0.964424 0.264359i \(-0.0851602\pi\)
\(194\) 78.5954i 0.405131i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) 50.5512 0.256605 0.128303 0.991735i \(-0.459047\pi\)
0.128303 + 0.991735i \(0.459047\pi\)
\(198\) −28.5156 + 30.9969i −0.144018 + 0.156550i
\(199\) 10.1649 0.0510800 0.0255400 0.999674i \(-0.491869\pi\)
0.0255400 + 0.999674i \(0.491869\pi\)
\(200\) 0 0
\(201\) −59.6752 + 153.010i −0.296892 + 0.761242i
\(202\) 176.508i 0.873801i
\(203\) 68.1041 0.335488
\(204\) 79.5246 + 31.0153i 0.389826 + 0.152036i
\(205\) 0 0
\(206\) 159.638i 0.774943i
\(207\) 46.8541 + 43.1034i 0.226348 + 0.208229i
\(208\) 52.8638i 0.254153i
\(209\) 24.6565i 0.117974i
\(210\) 0 0
\(211\) −305.386 −1.44733 −0.723665 0.690152i \(-0.757544\pi\)
−0.723665 + 0.690152i \(0.757544\pi\)
\(212\) 124.629 0.587874
\(213\) 109.314 280.284i 0.513209 1.31589i
\(214\) −119.749 −0.559575
\(215\) 0 0
\(216\) 33.5746 + 68.5912i 0.155438 + 0.317552i
\(217\) 109.838i 0.506168i
\(218\) −233.577 −1.07146
\(219\) 68.8955 176.651i 0.314591 0.806624i
\(220\) 0 0
\(221\) 188.016i 0.850751i
\(222\) 42.7101 109.510i 0.192388 0.493290i
\(223\) 103.856i 0.465724i 0.972510 + 0.232862i \(0.0748090\pi\)
−0.972510 + 0.232862i \(0.925191\pi\)
\(224\) 14.9666i 0.0668153i
\(225\) 0 0
\(226\) −79.0016 −0.349565
\(227\) 392.738 1.73012 0.865062 0.501664i \(-0.167279\pi\)
0.865062 + 0.501664i \(0.167279\pi\)
\(228\) −41.6508 16.2442i −0.182679 0.0712465i
\(229\) 427.149 1.86528 0.932641 0.360806i \(-0.117498\pi\)
0.932641 + 0.360806i \(0.117498\pi\)
\(230\) 0 0
\(231\) 24.4702 + 9.54360i 0.105931 + 0.0413143i
\(232\) 72.8064i 0.313821i
\(233\) −57.7328 −0.247780 −0.123890 0.992296i \(-0.539537\pi\)
−0.123890 + 0.992296i \(0.539537\pi\)
\(234\) −113.885 + 123.795i −0.486690 + 0.529039i
\(235\) 0 0
\(236\) 43.5201i 0.184407i
\(237\) −326.847 127.473i −1.37910 0.537863i
\(238\) 53.2305i 0.223658i
\(239\) 52.9710i 0.221636i 0.993841 + 0.110818i \(0.0353471\pi\)
−0.993841 + 0.110818i \(0.964653\pi\)
\(240\) 0 0
\(241\) −157.253 −0.652502 −0.326251 0.945283i \(-0.605785\pi\)
−0.326251 + 0.945283i \(0.605785\pi\)
\(242\) 155.634 0.643115
\(243\) −69.1430 + 232.955i −0.284539 + 0.958664i
\(244\) 169.609 0.695121
\(245\) 0 0
\(246\) −1.93591 + 4.96374i −0.00786953 + 0.0201778i
\(247\) 98.4728i 0.398675i
\(248\) 117.422 0.473477
\(249\) −66.3614 25.8816i −0.266512 0.103942i
\(250\) 0 0
\(251\) 150.522i 0.599688i −0.953988 0.299844i \(-0.903065\pi\)
0.953988 0.299844i \(-0.0969346\pi\)
\(252\) 32.2429 35.0485i 0.127948 0.139081i
\(253\) 23.4083i 0.0925230i
\(254\) 235.325i 0.926476i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −313.116 −1.21835 −0.609175 0.793036i \(-0.708499\pi\)
−0.609175 + 0.793036i \(0.708499\pi\)
\(258\) 3.27859 8.40643i 0.0127077 0.0325831i
\(259\) −73.3018 −0.283019
\(260\) 0 0
\(261\) −156.848 + 170.496i −0.600951 + 0.653242i
\(262\) 324.105i 1.23704i
\(263\) −413.301 −1.57149 −0.785744 0.618552i \(-0.787720\pi\)
−0.785744 + 0.618552i \(0.787720\pi\)
\(264\) −10.2025 + 26.1597i −0.0386460 + 0.0990898i
\(265\) 0 0
\(266\) 27.8793i 0.104809i
\(267\) 3.55695 9.12016i 0.0133219 0.0341579i
\(268\) 109.490i 0.408545i
\(269\) 77.5114i 0.288147i −0.989567 0.144073i \(-0.953980\pi\)
0.989567 0.144073i \(-0.0460201\pi\)
\(270\) 0 0
\(271\) 277.456 1.02382 0.511912 0.859038i \(-0.328938\pi\)
0.511912 + 0.859038i \(0.328938\pi\)
\(272\) 56.9058 0.209212
\(273\) 97.7287 + 38.1151i 0.357980 + 0.139616i
\(274\) −174.739 −0.637732
\(275\) 0 0
\(276\) 39.5423 + 15.4219i 0.143269 + 0.0558763i
\(277\) 64.2751i 0.232040i −0.993247 0.116020i \(-0.962986\pi\)
0.993247 0.116020i \(-0.0370137\pi\)
\(278\) −240.127 −0.863764
\(279\) 274.977 + 252.965i 0.985579 + 0.906684i
\(280\) 0 0
\(281\) 238.377i 0.848315i 0.905588 + 0.424158i \(0.139430\pi\)
−0.905588 + 0.424158i \(0.860570\pi\)
\(282\) −216.746 84.5329i −0.768601 0.299762i
\(283\) 295.213i 1.04315i −0.853204 0.521577i \(-0.825344\pi\)
0.853204 0.521577i \(-0.174656\pi\)
\(284\) 200.564i 0.706213i
\(285\) 0 0
\(286\) −61.8481 −0.216252
\(287\) 3.32252 0.0115767
\(288\) 37.4684 + 34.4691i 0.130099 + 0.119684i
\(289\) −86.6082 −0.299682
\(290\) 0 0
\(291\) −60.5804 + 155.331i −0.208180 + 0.533782i
\(292\) 126.407i 0.432900i
\(293\) 358.351 1.22304 0.611521 0.791228i \(-0.290558\pi\)
0.611521 + 0.791228i \(0.290558\pi\)
\(294\) −27.6686 10.7910i −0.0941110 0.0367042i
\(295\) 0 0
\(296\) 78.3629i 0.264740i
\(297\) −80.2484 + 39.2807i −0.270197 + 0.132258i
\(298\) 336.288i 1.12848i
\(299\) 93.4878i 0.312668i
\(300\) 0 0
\(301\) −5.62692 −0.0186941
\(302\) 165.558 0.548206
\(303\) 136.050 348.838i 0.449010 1.15128i
\(304\) −29.8042 −0.0980403
\(305\) 0 0
\(306\) 133.261 + 122.593i 0.435492 + 0.400631i
\(307\) 531.675i 1.73184i −0.500182 0.865920i \(-0.666733\pi\)
0.500182 0.865920i \(-0.333267\pi\)
\(308\) 17.5102 0.0568514
\(309\) −123.047 + 315.498i −0.398211 + 1.02103i
\(310\) 0 0
\(311\) 418.127i 1.34446i −0.740343 0.672229i \(-0.765337\pi\)
0.740343 0.672229i \(-0.234663\pi\)
\(312\) −40.7468 + 104.476i −0.130599 + 0.334860i
\(313\) 306.865i 0.980399i −0.871610 0.490199i \(-0.836924\pi\)
0.871610 0.490199i \(-0.163076\pi\)
\(314\) 44.7105i 0.142390i
\(315\) 0 0
\(316\) −233.884 −0.740138
\(317\) 83.8149 0.264400 0.132200 0.991223i \(-0.457796\pi\)
0.132200 + 0.991223i \(0.457796\pi\)
\(318\) 246.309 + 96.0628i 0.774556 + 0.302084i
\(319\) −85.1800 −0.267022
\(320\) 0 0
\(321\) −236.664 92.3012i −0.737271 0.287543i
\(322\) 26.4680i 0.0821987i
\(323\) −106.002 −0.328180
\(324\) 13.4853 + 161.438i 0.0416212 + 0.498265i
\(325\) 0 0
\(326\) 152.103i 0.466572i
\(327\) −461.626 180.039i −1.41170 0.550577i
\(328\) 3.55193i 0.0108290i
\(329\) 145.081i 0.440975i
\(330\) 0 0
\(331\) 145.013 0.438105 0.219052 0.975713i \(-0.429703\pi\)
0.219052 + 0.975713i \(0.429703\pi\)
\(332\) −47.4866 −0.143032
\(333\) 168.819 183.508i 0.506963 0.551076i
\(334\) 165.330 0.495000
\(335\) 0 0
\(336\) 11.5361 29.5790i 0.0343336 0.0880328i
\(337\) 610.894i 1.81274i 0.422483 + 0.906371i \(0.361159\pi\)
−0.422483 + 0.906371i \(0.638841\pi\)
\(338\) −8.00610 −0.0236867
\(339\) −156.133 60.8935i −0.460570 0.179627i
\(340\) 0 0
\(341\) 137.378i 0.402869i
\(342\) −69.7949 64.2078i −0.204079 0.187742i
\(343\) 18.5203i 0.0539949i
\(344\) 6.01543i 0.0174867i
\(345\) 0 0
\(346\) −449.445 −1.29897
\(347\) −70.0988 −0.202014 −0.101007 0.994886i \(-0.532206\pi\)
−0.101007 + 0.994886i \(0.532206\pi\)
\(348\) −56.1183 + 143.890i −0.161260 + 0.413476i
\(349\) 564.413 1.61723 0.808614 0.588340i \(-0.200218\pi\)
0.808614 + 0.588340i \(0.200218\pi\)
\(350\) 0 0
\(351\) −320.495 + 156.879i −0.913091 + 0.446948i
\(352\) 18.7192i 0.0531796i
\(353\) 1.65210 0.00468018 0.00234009 0.999997i \(-0.499255\pi\)
0.00234009 + 0.999997i \(0.499255\pi\)
\(354\) 33.5448 86.0102i 0.0947593 0.242967i
\(355\) 0 0
\(356\) 6.52616i 0.0183319i
\(357\) 41.0294 105.201i 0.114928 0.294681i
\(358\) 459.964i 1.28482i
\(359\) 396.146i 1.10347i −0.834020 0.551735i \(-0.813966\pi\)
0.834020 0.551735i \(-0.186034\pi\)
\(360\) 0 0
\(361\) −305.482 −0.846210
\(362\) −48.8427 −0.134925
\(363\) 307.584 + 119.961i 0.847339 + 0.330470i
\(364\) 69.9322 0.192121
\(365\) 0 0
\(366\) 335.204 + 130.733i 0.915859 + 0.357194i
\(367\) 363.378i 0.990131i 0.868856 + 0.495066i \(0.164856\pi\)
−0.868856 + 0.495066i \(0.835144\pi\)
\(368\) 28.2955 0.0768899
\(369\) −7.65198 + 8.31782i −0.0207371 + 0.0225415i
\(370\) 0 0
\(371\) 164.869i 0.444391i
\(372\) 232.065 + 90.5077i 0.623831 + 0.243300i
\(373\) 207.182i 0.555448i −0.960661 0.277724i \(-0.910420\pi\)
0.960661 0.277724i \(-0.0895800\pi\)
\(374\) 66.5771i 0.178014i
\(375\) 0 0
\(376\) −155.098 −0.412494
\(377\) −340.191 −0.902363
\(378\) 90.7376 44.4150i 0.240046 0.117500i
\(379\) −489.449 −1.29142 −0.645712 0.763581i \(-0.723439\pi\)
−0.645712 + 0.763581i \(0.723439\pi\)
\(380\) 0 0
\(381\) −181.386 + 465.080i −0.476078 + 1.22068i
\(382\) 113.950i 0.298299i
\(383\) −210.772 −0.550318 −0.275159 0.961399i \(-0.588731\pi\)
−0.275159 + 0.961399i \(0.588731\pi\)
\(384\) 31.6213 + 12.3326i 0.0823471 + 0.0321162i
\(385\) 0 0
\(386\) 144.310i 0.373859i
\(387\) 12.9592 14.0868i 0.0334862 0.0364000i
\(388\) 111.151i 0.286471i
\(389\) 328.979i 0.845704i −0.906199 0.422852i \(-0.861029\pi\)
0.906199 0.422852i \(-0.138971\pi\)
\(390\) 0 0
\(391\) 100.636 0.257381
\(392\) −19.7990 −0.0505076
\(393\) 249.816 640.539i 0.635665 1.62987i
\(394\) 71.4902 0.181447
\(395\) 0 0
\(396\) −40.3272 + 43.8363i −0.101836 + 0.110698i
\(397\) 32.3623i 0.0815172i 0.999169 + 0.0407586i \(0.0129775\pi\)
−0.999169 + 0.0407586i \(0.987023\pi\)
\(398\) 14.3754 0.0361190
\(399\) −21.4890 + 55.0988i −0.0538573 + 0.138092i
\(400\) 0 0
\(401\) 233.517i 0.582338i 0.956672 + 0.291169i \(0.0940441\pi\)
−0.956672 + 0.291169i \(0.905956\pi\)
\(402\) −84.3935 + 216.388i −0.209934 + 0.538280i
\(403\) 548.660i 1.36144i
\(404\) 249.620i 0.617870i
\(405\) 0 0
\(406\) 96.3138 0.237226
\(407\) 91.6809 0.225260
\(408\) 112.465 + 43.8623i 0.275649 + 0.107506i
\(409\) 30.2404 0.0739375 0.0369687 0.999316i \(-0.488230\pi\)
0.0369687 + 0.999316i \(0.488230\pi\)
\(410\) 0 0
\(411\) −345.341 134.686i −0.840246 0.327704i
\(412\) 225.763i 0.547967i
\(413\) −57.5717 −0.139399
\(414\) 66.2616 + 60.9574i 0.160052 + 0.147240i
\(415\) 0 0
\(416\) 74.7607i 0.179713i
\(417\) −474.570 185.087i −1.13806 0.443853i
\(418\) 34.8695i 0.0834200i
\(419\) 690.640i 1.64831i −0.566367 0.824153i \(-0.691652\pi\)
0.566367 0.824153i \(-0.308348\pi\)
\(420\) 0 0
\(421\) −637.548 −1.51437 −0.757183 0.653203i \(-0.773425\pi\)
−0.757183 + 0.653203i \(0.773425\pi\)
\(422\) −431.882 −1.02342
\(423\) −363.204 334.130i −0.858639 0.789905i
\(424\) 176.253 0.415690
\(425\) 0 0
\(426\) 154.593 396.382i 0.362894 0.930474i
\(427\) 224.372i 0.525462i
\(428\) −169.351 −0.395680
\(429\) −122.232 47.6718i −0.284924 0.111123i
\(430\) 0 0
\(431\) 474.481i 1.10088i 0.834874 + 0.550442i \(0.185541\pi\)
−0.834874 + 0.550442i \(0.814459\pi\)
\(432\) 47.4816 + 97.0025i 0.109911 + 0.224543i
\(433\) 66.7488i 0.154154i −0.997025 0.0770772i \(-0.975441\pi\)
0.997025 0.0770772i \(-0.0245588\pi\)
\(434\) 155.335i 0.357915i
\(435\) 0 0
\(436\) −330.328 −0.757634
\(437\) −52.7078 −0.120613
\(438\) 97.4329 249.822i 0.222450 0.570370i
\(439\) 361.090 0.822528 0.411264 0.911516i \(-0.365087\pi\)
0.411264 + 0.911516i \(0.365087\pi\)
\(440\) 0 0
\(441\) −46.3648 42.6533i −0.105136 0.0967195i
\(442\) 265.895i 0.601572i
\(443\) −415.163 −0.937163 −0.468581 0.883420i \(-0.655235\pi\)
−0.468581 + 0.883420i \(0.655235\pi\)
\(444\) 60.4012 154.871i 0.136039 0.348809i
\(445\) 0 0
\(446\) 146.875i 0.329316i
\(447\) 259.207 664.616i 0.579881 1.48684i
\(448\) 21.1660i 0.0472456i
\(449\) 538.703i 1.19978i 0.800081 + 0.599892i \(0.204790\pi\)
−0.800081 + 0.599892i \(0.795210\pi\)
\(450\) 0 0
\(451\) −4.15559 −0.00921416
\(452\) −111.725 −0.247179
\(453\) 327.198 + 127.610i 0.722292 + 0.281701i
\(454\) 555.416 1.22338
\(455\) 0 0
\(456\) −58.9031 22.9728i −0.129173 0.0503789i
\(457\) 666.562i 1.45856i −0.684216 0.729280i \(-0.739855\pi\)
0.684216 0.729280i \(-0.260145\pi\)
\(458\) 604.081 1.31895
\(459\) 168.874 + 345.000i 0.367917 + 0.751635i
\(460\) 0 0
\(461\) 246.725i 0.535196i −0.963531 0.267598i \(-0.913770\pi\)
0.963531 0.267598i \(-0.0862299\pi\)
\(462\) 34.6060 + 13.4967i 0.0749048 + 0.0292136i
\(463\) 278.772i 0.602099i −0.953609 0.301049i \(-0.902663\pi\)
0.953609 0.301049i \(-0.0973369\pi\)
\(464\) 102.964i 0.221905i
\(465\) 0 0
\(466\) −81.6465 −0.175207
\(467\) −263.485 −0.564208 −0.282104 0.959384i \(-0.591032\pi\)
−0.282104 + 0.959384i \(0.591032\pi\)
\(468\) −161.058 + 175.073i −0.344142 + 0.374087i
\(469\) 144.842 0.308831
\(470\) 0 0
\(471\) −34.4623 + 88.3628i −0.0731685 + 0.187607i
\(472\) 61.5468i 0.130396i
\(473\) 7.03777 0.0148790
\(474\) −462.231 180.275i −0.975172 0.380326i
\(475\) 0 0
\(476\) 75.2793i 0.158150i
\(477\) 412.744 + 379.704i 0.865291 + 0.796025i
\(478\) 74.9123i 0.156720i
\(479\) 163.187i 0.340683i 0.985385 + 0.170342i \(0.0544871\pi\)
−0.985385 + 0.170342i \(0.945513\pi\)
\(480\) 0 0
\(481\) 366.154 0.761235
\(482\) −222.389 −0.461388
\(483\) 20.4012 52.3095i 0.0422385 0.108301i
\(484\) 220.099 0.454751
\(485\) 0 0
\(486\) −97.7830 + 329.449i −0.201200 + 0.677878i
\(487\) 752.251i 1.54466i 0.635219 + 0.772332i \(0.280910\pi\)
−0.635219 + 0.772332i \(0.719090\pi\)
\(488\) 239.864 0.491524
\(489\) −117.239 + 300.605i −0.239752 + 0.614735i
\(490\) 0 0
\(491\) 67.2107i 0.136885i 0.997655 + 0.0684427i \(0.0218030\pi\)
−0.997655 + 0.0684427i \(0.978197\pi\)
\(492\) −2.73778 + 7.01978i −0.00556460 + 0.0142679i
\(493\) 366.202i 0.742804i
\(494\) 139.262i 0.281906i
\(495\) 0 0
\(496\) 166.060 0.334799
\(497\) −265.322 −0.533847
\(498\) −93.8492 36.6021i −0.188452 0.0734982i
\(499\) 247.715 0.496423 0.248212 0.968706i \(-0.420157\pi\)
0.248212 + 0.968706i \(0.420157\pi\)
\(500\) 0 0
\(501\) 326.747 + 127.435i 0.652190 + 0.254360i
\(502\) 212.870i 0.424043i
\(503\) −142.465 −0.283231 −0.141616 0.989922i \(-0.545230\pi\)
−0.141616 + 0.989922i \(0.545230\pi\)
\(504\) 45.5983 49.5661i 0.0904728 0.0983453i
\(505\) 0 0
\(506\) 33.1043i 0.0654236i
\(507\) −15.8227 6.17101i −0.0312085 0.0121716i
\(508\) 332.800i 0.655117i
\(509\) 541.364i 1.06358i 0.846875 + 0.531792i \(0.178481\pi\)
−0.846875 + 0.531792i \(0.821519\pi\)
\(510\) 0 0
\(511\) −167.221 −0.327242
\(512\) 22.6274 0.0441942
\(513\) −88.4471 180.693i −0.172412 0.352228i
\(514\) −442.813 −0.861504
\(515\) 0 0
\(516\) 4.63662 11.8885i 0.00898571 0.0230397i
\(517\) 181.457i 0.350981i
\(518\) −103.664 −0.200124
\(519\) −888.251 346.427i −1.71147 0.667489i
\(520\) 0 0
\(521\) 796.131i 1.52808i −0.645168 0.764041i \(-0.723213\pi\)
0.645168 0.764041i \(-0.276787\pi\)
\(522\) −221.817 + 241.118i −0.424936 + 0.461912i
\(523\) 382.162i 0.730711i 0.930868 + 0.365355i \(0.119053\pi\)
−0.930868 + 0.365355i \(0.880947\pi\)
\(524\) 458.354i 0.874721i
\(525\) 0 0
\(526\) −584.496 −1.11121
\(527\) 590.612 1.12071
\(528\) −14.4286 + 36.9954i −0.0273268 + 0.0700671i
\(529\) −478.960 −0.905407
\(530\) 0 0
\(531\) 132.591 144.129i 0.249701 0.271429i
\(532\) 39.4273i 0.0741115i
\(533\) −16.5965 −0.0311380
\(534\) 5.03029 12.8979i 0.00942001 0.0241533i
\(535\) 0 0
\(536\) 154.842i 0.288885i
\(537\) 354.535 909.042i 0.660214 1.69282i
\(538\) 109.618i 0.203750i
\(539\) 23.1639i 0.0429756i
\(540\) 0 0
\(541\) 781.905 1.44530 0.722648 0.691217i \(-0.242925\pi\)
0.722648 + 0.691217i \(0.242925\pi\)
\(542\) 392.382 0.723952
\(543\) −96.5293 37.6474i −0.177770 0.0693321i
\(544\) 80.4769 0.147936
\(545\) 0 0
\(546\) 138.209 + 53.9029i 0.253130 + 0.0987233i
\(547\) 836.992i 1.53015i 0.643942 + 0.765075i \(0.277298\pi\)
−0.643942 + 0.765075i \(0.722702\pi\)
\(548\) −247.118 −0.450945
\(549\) 561.708 + 516.743i 1.02315 + 0.941244i
\(550\) 0 0
\(551\) 191.797i 0.348090i
\(552\) 55.9212 + 21.8098i 0.101307 + 0.0395105i
\(553\) 309.399i 0.559492i
\(554\) 90.8988i 0.164077i
\(555\) 0 0
\(556\) −339.590 −0.610774
\(557\) −151.652 −0.272267 −0.136133 0.990691i \(-0.543468\pi\)
−0.136133 + 0.990691i \(0.543468\pi\)
\(558\) 388.876 + 357.746i 0.696910 + 0.641123i
\(559\) 28.1074 0.0502815
\(560\) 0 0
\(561\) −51.3168 + 131.578i −0.0914738 + 0.234542i
\(562\) 337.115i 0.599850i
\(563\) −830.912 −1.47586 −0.737932 0.674875i \(-0.764198\pi\)
−0.737932 + 0.674875i \(0.764198\pi\)
\(564\) −306.525 119.548i −0.543483 0.211964i
\(565\) 0 0
\(566\) 417.494i 0.737621i
\(567\) 213.562 17.8393i 0.376653 0.0314626i
\(568\) 283.641i 0.499368i
\(569\) 76.3013i 0.134097i 0.997750 + 0.0670486i \(0.0213583\pi\)
−0.997750 + 0.0670486i \(0.978642\pi\)
\(570\) 0 0
\(571\) −273.471 −0.478933 −0.239467 0.970905i \(-0.576973\pi\)
−0.239467 + 0.970905i \(0.576973\pi\)
\(572\) −87.4664 −0.152913
\(573\) 87.8313 225.203i 0.153283 0.393025i
\(574\) 4.69876 0.00818599
\(575\) 0 0
\(576\) 52.9883 + 48.7466i 0.0919936 + 0.0846296i
\(577\) 783.907i 1.35859i 0.733865 + 0.679295i \(0.237714\pi\)
−0.733865 + 0.679295i \(0.762286\pi\)
\(578\) −122.482 −0.211907
\(579\) 111.232 285.204i 0.192111 0.492580i
\(580\) 0 0
\(581\) 62.8189i 0.108122i
\(582\) −85.6736 + 219.671i −0.147206 + 0.377441i
\(583\) 206.207i 0.353700i
\(584\) 178.766i 0.306107i
\(585\) 0 0
\(586\) 506.786 0.864822
\(587\) 552.525 0.941269 0.470634 0.882328i \(-0.344025\pi\)
0.470634 + 0.882328i \(0.344025\pi\)
\(588\) −39.1294 15.2608i −0.0665465 0.0259538i
\(589\) −309.331 −0.525180
\(590\) 0 0
\(591\) 141.288 + 55.1038i 0.239067 + 0.0932382i
\(592\) 110.822i 0.187199i
\(593\) 236.428 0.398699 0.199349 0.979928i \(-0.436117\pi\)
0.199349 + 0.979928i \(0.436117\pi\)
\(594\) −113.488 + 55.5512i −0.191058 + 0.0935206i
\(595\) 0 0
\(596\) 475.583i 0.797958i
\(597\) 28.4105 + 11.0804i 0.0475888 + 0.0185601i
\(598\) 132.212i 0.221090i
\(599\) 1179.57i 1.96923i 0.174727 + 0.984617i \(0.444096\pi\)
−0.174727 + 0.984617i \(0.555904\pi\)
\(600\) 0 0
\(601\) 1090.04 1.81370 0.906851 0.421451i \(-0.138479\pi\)
0.906851 + 0.421451i \(0.138479\pi\)
\(602\) −7.95767 −0.0132187
\(603\) −333.579 + 362.606i −0.553199 + 0.601336i
\(604\) 234.135 0.387640
\(605\) 0 0
\(606\) 192.404 493.331i 0.317498 0.814078i
\(607\) 1077.37i 1.77491i −0.460892 0.887456i \(-0.652470\pi\)
0.460892 0.887456i \(-0.347530\pi\)
\(608\) −42.1496 −0.0693249
\(609\) 190.348 + 74.2375i 0.312558 + 0.121901i
\(610\) 0 0
\(611\) 724.701i 1.18609i
\(612\) 188.459 + 173.373i 0.307939 + 0.283289i
\(613\) 942.291i 1.53718i 0.639742 + 0.768590i \(0.279041\pi\)
−0.639742 + 0.768590i \(0.720959\pi\)
\(614\) 751.902i 1.22460i
\(615\) 0 0
\(616\) 24.7632 0.0402000
\(617\) 863.672 1.39979 0.699896 0.714245i \(-0.253230\pi\)
0.699896 + 0.714245i \(0.253230\pi\)
\(618\) −174.015 + 446.182i −0.281578 + 0.721977i
\(619\) −626.507 −1.01213 −0.506064 0.862496i \(-0.668900\pi\)
−0.506064 + 0.862496i \(0.668900\pi\)
\(620\) 0 0
\(621\) 83.9697 + 171.546i 0.135217 + 0.276241i
\(622\) 591.320i 0.950676i
\(623\) −8.63330 −0.0138576
\(624\) −57.6246 + 147.752i −0.0923472 + 0.236782i
\(625\) 0 0
\(626\) 433.972i 0.693247i
\(627\) 26.8770 68.9138i 0.0428661 0.109910i
\(628\) 63.2302i 0.100685i
\(629\) 394.151i 0.626630i
\(630\) 0 0
\(631\) 612.214 0.970228 0.485114 0.874451i \(-0.338778\pi\)
0.485114 + 0.874451i \(0.338778\pi\)
\(632\) −330.761 −0.523356
\(633\) −853.541 332.889i −1.34841 0.525891i
\(634\) 118.532 0.186959
\(635\) 0 0
\(636\) 348.333 + 135.853i 0.547694 + 0.213606i
\(637\) 92.5116i 0.145230i
\(638\) −120.463 −0.188813
\(639\) 611.053 664.224i 0.956264 1.03947i
\(640\) 0 0
\(641\) 804.983i 1.25582i 0.778284 + 0.627912i \(0.216090\pi\)
−0.778284 + 0.627912i \(0.783910\pi\)
\(642\) −334.693 130.534i −0.521329 0.203323i
\(643\) 71.5877i 0.111334i −0.998449 0.0556670i \(-0.982271\pi\)
0.998449 0.0556670i \(-0.0177285\pi\)
\(644\) 37.4314i 0.0581233i
\(645\) 0 0
\(646\) −149.910 −0.232058
\(647\) 579.969 0.896397 0.448199 0.893934i \(-0.352066\pi\)
0.448199 + 0.893934i \(0.352066\pi\)
\(648\) 19.0710 + 228.307i 0.0294306 + 0.352326i
\(649\) 72.0068 0.110950
\(650\) 0 0
\(651\) 119.730 306.994i 0.183918 0.471572i
\(652\) 215.106i 0.329917i
\(653\) 1063.75 1.62902 0.814511 0.580148i \(-0.197005\pi\)
0.814511 + 0.580148i \(0.197005\pi\)
\(654\) −652.838 254.613i −0.998223 0.389317i
\(655\) 0 0
\(656\) 5.02318i 0.00765729i
\(657\) 385.119 418.631i 0.586179 0.637185i
\(658\) 205.175i 0.311816i
\(659\) 227.741i 0.345586i −0.984958 0.172793i \(-0.944721\pi\)
0.984958 0.172793i \(-0.0552792\pi\)
\(660\) 0 0
\(661\) 87.7668 0.132779 0.0663894 0.997794i \(-0.478852\pi\)
0.0663894 + 0.997794i \(0.478852\pi\)
\(662\) 205.079 0.309787
\(663\) −204.948 + 525.496i −0.309123 + 0.792603i
\(664\) −67.1562 −0.101139
\(665\) 0 0
\(666\) 238.746 259.520i 0.358477 0.389670i
\(667\) 182.088i 0.272996i
\(668\) 233.812 0.350018
\(669\) −113.210 + 290.274i −0.169222 + 0.433892i
\(670\) 0 0
\(671\) 280.629i 0.418226i
\(672\) 16.3145 41.8311i 0.0242775 0.0622486i
\(673\) 286.806i 0.426160i 0.977035 + 0.213080i \(0.0683495\pi\)
−0.977035 + 0.213080i \(0.931650\pi\)
\(674\) 863.935i 1.28180i
\(675\) 0 0
\(676\) −11.3223 −0.0167490
\(677\) 739.076 1.09169 0.545847 0.837885i \(-0.316208\pi\)
0.545847 + 0.837885i \(0.316208\pi\)
\(678\) −220.806 86.1164i −0.325672 0.127015i
\(679\) 147.039 0.216552
\(680\) 0 0
\(681\) 1097.69 + 428.108i 1.61187 + 0.628646i
\(682\) 194.282i 0.284872i
\(683\) −384.541 −0.563017 −0.281509 0.959559i \(-0.590835\pi\)
−0.281509 + 0.959559i \(0.590835\pi\)
\(684\) −98.7048 90.8036i −0.144305 0.132754i
\(685\) 0 0
\(686\) 26.1916i 0.0381802i
\(687\) 1193.86 + 465.618i 1.73779 + 0.677756i
\(688\) 8.50711i 0.0123650i
\(689\) 823.547i 1.19528i
\(690\) 0 0
\(691\) 1158.11 1.67599 0.837994 0.545680i \(-0.183729\pi\)
0.837994 + 0.545680i \(0.183729\pi\)
\(692\) −635.611 −0.918512
\(693\) 57.9899 + 53.3479i 0.0836795 + 0.0769810i
\(694\) −99.1347 −0.142845
\(695\) 0 0
\(696\) −79.3633 + 203.491i −0.114028 + 0.292372i
\(697\) 17.8655i 0.0256320i
\(698\) 798.200 1.14355
\(699\) −161.360 62.9321i −0.230845 0.0900317i
\(700\) 0 0
\(701\) 1033.07i 1.47371i 0.676049 + 0.736856i \(0.263691\pi\)
−0.676049 + 0.736856i \(0.736309\pi\)
\(702\) −453.248 + 221.860i −0.645653 + 0.316040i
\(703\) 206.435i 0.293649i
\(704\) 26.4730i 0.0376037i
\(705\) 0 0
\(706\) 2.33643 0.00330939
\(707\) −330.216 −0.467066
\(708\) 47.4395 121.637i 0.0670050 0.171803i
\(709\) 978.655 1.38033 0.690166 0.723651i \(-0.257538\pi\)
0.690166 + 0.723651i \(0.257538\pi\)
\(710\) 0 0
\(711\) −774.569 712.565i −1.08941 1.00220i
\(712\) 9.22938i 0.0129626i
\(713\) 293.672 0.411882
\(714\) 58.0244 148.777i 0.0812666 0.208371i
\(715\) 0 0
\(716\) 650.488i 0.908502i
\(717\) −57.7415 + 148.052i −0.0805321 + 0.206488i
\(718\) 560.235i 0.780271i
\(719\) 1081.91i 1.50474i −0.658739 0.752372i \(-0.728910\pi\)
0.658739 0.752372i \(-0.271090\pi\)
\(720\) 0 0
\(721\) 298.656 0.414224
\(722\) −432.016 −0.598361
\(723\) −439.515 171.415i −0.607904 0.237088i
\(724\) −69.0740 −0.0954060
\(725\) 0 0
\(726\) 434.989 + 169.650i 0.599159 + 0.233678i
\(727\) 416.273i 0.572589i −0.958142 0.286295i \(-0.907576\pi\)
0.958142 0.286295i \(-0.0924237\pi\)
\(728\) 98.8991 0.135850
\(729\) −447.187 + 575.730i −0.613425 + 0.789753i
\(730\) 0 0
\(731\) 30.2565i 0.0413905i
\(732\) 474.051 + 184.884i 0.647610 + 0.252574i
\(733\) 338.733i 0.462118i −0.972940 0.231059i \(-0.925781\pi\)
0.972940 0.231059i \(-0.0742191\pi\)
\(734\) 513.894i 0.700128i
\(735\) 0 0
\(736\) 40.0158 0.0543693
\(737\) −181.158 −0.245805
\(738\) −10.8215 + 11.7632i −0.0146633 + 0.0159393i
\(739\) −324.854 −0.439586 −0.219793 0.975547i \(-0.570538\pi\)
−0.219793 + 0.975547i \(0.570538\pi\)
\(740\) 0 0
\(741\) 107.341 275.227i 0.144860 0.371426i
\(742\) 233.160i 0.314232i
\(743\) −857.635 −1.15429 −0.577143 0.816643i \(-0.695833\pi\)
−0.577143 + 0.816643i \(0.695833\pi\)
\(744\) 328.190 + 127.997i 0.441115 + 0.172039i
\(745\) 0 0
\(746\) 293.000i 0.392761i
\(747\) −157.265 144.676i −0.210528 0.193676i
\(748\) 94.1542i 0.125875i
\(749\) 224.030i 0.299106i
\(750\) 0 0
\(751\) −356.621 −0.474861 −0.237431 0.971404i \(-0.576305\pi\)
−0.237431 + 0.971404i \(0.576305\pi\)
\(752\) −219.341 −0.291677
\(753\) 164.077 420.701i 0.217898 0.558700i
\(754\) −481.103 −0.638067
\(755\) 0 0
\(756\) 128.322 62.8123i 0.169739 0.0830851i
\(757\) 1356.47i 1.79191i 0.444147 + 0.895954i \(0.353507\pi\)
−0.444147 + 0.895954i \(0.646493\pi\)
\(758\) −692.186 −0.913174
\(759\) −25.5164 + 65.4252i −0.0336185 + 0.0861992i
\(760\) 0 0
\(761\) 106.889i 0.140459i 0.997531 + 0.0702294i \(0.0223731\pi\)
−0.997531 + 0.0702294i \(0.977627\pi\)
\(762\) −256.518 + 657.722i −0.336638 + 0.863153i
\(763\) 436.983i 0.572717i
\(764\) 161.150i 0.210929i
\(765\) 0 0
\(766\) −298.076 −0.389134
\(767\) 287.580 0.374941
\(768\) 44.7193 + 17.4409i 0.0582282 + 0.0227096i
\(769\) −500.283 −0.650563 −0.325282 0.945617i \(-0.605459\pi\)
−0.325282 + 0.945617i \(0.605459\pi\)
\(770\) 0 0
\(771\) −875.145 341.315i −1.13508 0.442691i
\(772\) 204.085i 0.264359i
\(773\) 1146.06 1.48262 0.741309 0.671164i \(-0.234206\pi\)
0.741309 + 0.671164i \(0.234206\pi\)
\(774\) 18.3270 19.9217i 0.0236783 0.0257387i
\(775\) 0 0
\(776\) 157.191i 0.202566i
\(777\) −204.875 79.9033i −0.263675 0.102836i
\(778\) 465.246i 0.598003i
\(779\) 9.35701i 0.0120116i
\(780\) 0 0
\(781\) 331.846 0.424899
\(782\) 142.321 0.181996
\(783\) −624.234 + 305.556i −0.797234 + 0.390237i
\(784\) −28.0000 −0.0357143
\(785\) 0 0
\(786\) 353.294 905.859i 0.449483 1.15249i
\(787\) 488.016i 0.620097i 0.950721 + 0.310048i \(0.100345\pi\)
−0.950721 + 0.310048i \(0.899655\pi\)
\(788\) 101.102 0.128303
\(789\) −1155.16 450.523i −1.46408 0.571005i
\(790\) 0 0
\(791\) 147.798i 0.186850i
\(792\) −57.0313 + 61.9938i −0.0720092 + 0.0782750i
\(793\) 1120.77i 1.41333i
\(794\) 45.7672i 0.0576413i
\(795\) 0 0
\(796\) 20.3299 0.0255400
\(797\) 197.490 0.247792 0.123896 0.992295i \(-0.460461\pi\)
0.123896 + 0.992295i \(0.460461\pi\)
\(798\) −30.3901 + 77.9214i −0.0380828 + 0.0976459i
\(799\) −780.112 −0.976361
\(800\) 0 0
\(801\) 19.8830 21.6131i 0.0248228 0.0269827i
\(802\) 330.243i 0.411775i
\(803\) 209.148 0.260458
\(804\) −119.350 + 306.019i −0.148446 + 0.380621i
\(805\) 0 0
\(806\) 775.923i 0.962684i
\(807\) 84.4920 216.641i 0.104699 0.268452i
\(808\) 353.015i 0.436900i
\(809\) 982.627i 1.21462i 0.794465 + 0.607310i \(0.207751\pi\)
−0.794465 + 0.607310i \(0.792249\pi\)
\(810\) 0 0
\(811\) 344.348 0.424597 0.212298 0.977205i \(-0.431905\pi\)
0.212298 + 0.977205i \(0.431905\pi\)
\(812\) 136.208 0.167744
\(813\) 775.477 + 302.444i 0.953846 + 0.372009i
\(814\) 129.656 0.159283
\(815\) 0 0
\(816\) 159.049 + 62.0307i 0.194913 + 0.0760180i
\(817\) 15.8467i 0.0193963i
\(818\) 42.7664 0.0522817
\(819\) 231.599 + 213.060i 0.282783 + 0.260147i
\(820\) 0 0
\(821\) 616.490i 0.750901i −0.926842 0.375451i \(-0.877488\pi\)
0.926842 0.375451i \(-0.122512\pi\)
\(822\) −488.386 190.475i −0.594144 0.231722i
\(823\) 638.840i 0.776234i 0.921610 + 0.388117i \(0.126874\pi\)
−0.921610 + 0.388117i \(0.873126\pi\)
\(824\) 319.276i 0.387471i
\(825\) 0 0
\(826\) −81.4187 −0.0985699
\(827\) −1066.42 −1.28951 −0.644753 0.764391i \(-0.723040\pi\)
−0.644753 + 0.764391i \(0.723040\pi\)
\(828\) 93.7081 + 86.2068i 0.113174 + 0.104115i
\(829\) 625.285 0.754265 0.377132 0.926159i \(-0.376910\pi\)
0.377132 + 0.926159i \(0.376910\pi\)
\(830\) 0 0
\(831\) 70.0637 179.646i 0.0843125 0.216181i
\(832\) 105.728i 0.127076i
\(833\) −99.5851 −0.119550
\(834\) −671.143 261.752i −0.804727 0.313851i
\(835\) 0 0
\(836\) 49.3130i 0.0589868i
\(837\) 492.801 + 1006.77i 0.588770 + 1.20283i
\(838\) 976.713i 1.16553i
\(839\) 387.376i 0.461712i −0.972988 0.230856i \(-0.925847\pi\)
0.972988 0.230856i \(-0.0741526\pi\)
\(840\) 0 0
\(841\) 178.404 0.212133
\(842\) −901.629 −1.07082
\(843\) −259.845 + 666.252i −0.308238 + 0.790334i
\(844\) −610.773 −0.723665
\(845\) 0 0
\(846\) −513.648 472.531i −0.607149 0.558547i
\(847\) 291.164i 0.343759i
\(848\) 249.259 0.293937
\(849\) 321.799 825.106i 0.379033 0.971856i
\(850\) 0 0
\(851\) 195.985i 0.230299i
\(852\) 218.627 560.569i 0.256605 0.657944i
\(853\) 830.771i 0.973940i 0.873419 + 0.486970i \(0.161898\pi\)
−0.873419 + 0.486970i \(0.838102\pi\)
\(854\) 317.310i 0.371558i
\(855\) 0 0
\(856\) −239.498 −0.279788
\(857\) 292.550 0.341365 0.170683 0.985326i \(-0.445403\pi\)
0.170683 + 0.985326i \(0.445403\pi\)
\(858\) −172.863 67.4181i −0.201472 0.0785758i
\(859\) 672.801 0.783237 0.391619 0.920128i \(-0.371915\pi\)
0.391619 + 0.920128i \(0.371915\pi\)
\(860\) 0 0
\(861\) 9.28630 + 3.62175i 0.0107855 + 0.00420644i
\(862\) 671.017i 0.778442i
\(863\) 699.281 0.810291 0.405145 0.914252i \(-0.367221\pi\)
0.405145 + 0.914252i \(0.367221\pi\)
\(864\) 67.1492 + 137.182i 0.0777190 + 0.158776i
\(865\) 0 0
\(866\) 94.3971i 0.109004i
\(867\) −242.066 94.4080i −0.279199 0.108890i
\(868\) 219.677i 0.253084i
\(869\) 386.975i 0.445311i
\(870\) 0 0
\(871\) −723.506 −0.830662
\(872\) −467.155 −0.535728
\(873\) −338.639 + 368.106i −0.387903 + 0.421656i
\(874\) −74.5401 −0.0852862
\(875\) 0 0
\(876\) 137.791 353.301i 0.157296 0.403312i
\(877\) 358.626i 0.408924i −0.978874 0.204462i \(-0.934456\pi\)
0.978874 0.204462i \(-0.0655445\pi\)
\(878\) 510.658 0.581615
\(879\) 1001.58 + 390.624i 1.13945 + 0.444396i
\(880\) 0 0
\(881\) 1550.04i 1.75941i 0.475522 + 0.879704i \(0.342259\pi\)
−0.475522 + 0.879704i \(0.657741\pi\)
\(882\) −65.5697 60.3209i −0.0743421 0.0683910i
\(883\) 1595.94i 1.80741i −0.428154 0.903706i \(-0.640836\pi\)
0.428154 0.903706i \(-0.359164\pi\)
\(884\) 376.032i 0.425375i
\(885\) 0 0
\(886\) −587.129 −0.662674
\(887\) 134.533 0.151671 0.0758357 0.997120i \(-0.475838\pi\)
0.0758357 + 0.997120i \(0.475838\pi\)
\(888\) 85.4202 219.021i 0.0961939 0.246645i
\(889\) 440.253 0.495222
\(890\) 0 0
\(891\) −267.109 + 22.3122i −0.299785 + 0.0250418i
\(892\) 207.713i 0.232862i
\(893\) 408.581 0.457538
\(894\) 366.574 939.909i 0.410038 1.05135i
\(895\) 0 0
\(896\) 29.9333i 0.0334077i
\(897\) −101.907 + 261.294i −0.113609 + 0.291298i
\(898\) 761.841i 0.848375i
\(899\) 1068.64i 1.18869i
\(900\) 0 0
\(901\) 886.517 0.983925
\(902\) −5.87688 −0.00651539
\(903\) −15.7270 6.13368i −0.0174164 0.00679256i
\(904\) −158.003 −0.174782
\(905\) 0 0
\(906\) 462.728 + 180.468i 0.510737 + 0.199192i
\(907\) 879.668i 0.969865i 0.874551 + 0.484933i \(0.161156\pi\)
−0.874551 + 0.484933i \(0.838844\pi\)
\(908\) 785.477 0.865062
\(909\) 760.507 826.683i 0.836642 0.909442i
\(910\) 0 0
\(911\) 1522.40i 1.67113i 0.549391 + 0.835565i \(0.314860\pi\)
−0.549391 + 0.835565i \(0.685140\pi\)
\(912\) −83.3015 32.4884i −0.0913394 0.0356232i
\(913\) 78.5695i 0.0860564i
\(914\) 942.661i 1.03136i
\(915\) 0 0
\(916\) 854.299 0.932641
\(917\) −606.345 −0.661227
\(918\) 238.824 + 487.904i 0.260156 + 0.531486i
\(919\) −1178.73 −1.28262 −0.641309 0.767282i \(-0.721608\pi\)
−0.641309 + 0.767282i \(0.721608\pi\)
\(920\) 0 0
\(921\) 579.557 1486.01i 0.629270 1.61347i
\(922\) 348.922i 0.378441i
\(923\) 1325.32 1.43589
\(924\) 48.9403 + 19.0872i 0.0529657 + 0.0206571i
\(925\) 0 0
\(926\) 394.243i 0.425748i
\(927\) −687.823 + 747.674i −0.741988 + 0.806552i
\(928\) 145.613i 0.156910i
\(929\) 534.346i 0.575184i 0.957753 + 0.287592i \(0.0928547\pi\)
−0.957753 + 0.287592i \(0.907145\pi\)
\(930\) 0 0
\(931\) 52.1574 0.0560230
\(932\) −115.466 −0.123890
\(933\) 455.783 1168.65i 0.488513 1.25257i
\(934\) −372.624 −0.398955
\(935\) 0 0
\(936\) −227.771 + 247.590i −0.243345 + 0.264519i
\(937\) 1383.17i 1.47616i 0.674711 + 0.738082i \(0.264268\pi\)
−0.674711 + 0.738082i \(0.735732\pi\)
\(938\) 204.837 0.218376
\(939\) 334.501 857.673i 0.356231 0.913390i
\(940\) 0 0
\(941\) 562.997i 0.598296i 0.954207 + 0.299148i \(0.0967024\pi\)
−0.954207 + 0.299148i \(0.903298\pi\)
\(942\) −48.7371 + 124.964i −0.0517379 + 0.132658i
\(943\) 8.88333i 0.00942029i
\(944\) 87.0403i 0.0922037i
\(945\) 0 0
\(946\) 9.95291 0.0105210
\(947\) 722.442 0.762874 0.381437 0.924395i \(-0.375429\pi\)
0.381437 + 0.924395i \(0.375429\pi\)
\(948\) −653.694 254.947i −0.689551 0.268931i
\(949\) 835.293 0.880182
\(950\) 0 0
\(951\) 234.259 + 91.3632i 0.246329 + 0.0960707i
\(952\) 106.461i 0.111829i
\(953\) 1567.96 1.64529 0.822644 0.568557i \(-0.192498\pi\)
0.822644 + 0.568557i \(0.192498\pi\)
\(954\) 583.708 + 536.983i 0.611853 + 0.562875i
\(955\) 0 0
\(956\) 105.942i 0.110818i
\(957\) −238.074 92.8512i −0.248771 0.0970232i
\(958\) 230.782i 0.240899i
\(959\) 326.906i 0.340882i
\(960\) 0 0
\(961\) 762.498 0.793443
\(962\) 517.820 0.538274
\(963\) −560.851 515.955i −0.582400 0.535779i
\(964\) −314.506 −0.326251
\(965\) 0 0
\(966\) 28.8517 73.9768i 0.0298672 0.0765806i
\(967\) 1182.12i 1.22246i −0.791453 0.611230i \(-0.790675\pi\)
0.791453 0.611230i \(-0.209325\pi\)
\(968\) 311.268 0.321557
\(969\) −296.271 115.549i −0.305749 0.119245i
\(970\) 0 0
\(971\) 1672.57i 1.72252i −0.508165 0.861260i \(-0.669676\pi\)
0.508165 0.861260i \(-0.330324\pi\)
\(972\) −138.286 + 465.911i −0.142270 + 0.479332i
\(973\) 449.236i 0.461702i
\(974\) 1063.84i 1.09224i
\(975\) 0 0
\(976\) 339.219 0.347560
\(977\) −383.549 −0.392578 −0.196289 0.980546i \(-0.562889\pi\)
−0.196289 + 0.980546i \(0.562889\pi\)
\(978\) −165.801 + 425.120i −0.169531 + 0.434683i
\(979\) 10.7979 0.0110296
\(980\) 0 0
\(981\) −1093.97 1006.40i −1.11516 1.02589i
\(982\) 95.0503i 0.0967925i
\(983\) −68.4583 −0.0696422 −0.0348211 0.999394i \(-0.511086\pi\)
−0.0348211 + 0.999394i \(0.511086\pi\)
\(984\) −3.87181 + 9.92748i −0.00393477 + 0.0100889i
\(985\) 0 0
\(986\) 517.888i 0.525242i
\(987\) −158.147 + 405.494i −0.160229 + 0.410835i
\(988\) 196.946i 0.199338i
\(989\) 15.0445i 0.0152119i
\(990\) 0 0
\(991\) −700.214 −0.706573 −0.353287 0.935515i \(-0.614936\pi\)
−0.353287 + 0.935515i \(0.614936\pi\)
\(992\) 234.845 0.236738
\(993\) 405.304 + 158.072i 0.408161 + 0.159187i
\(994\) −375.222 −0.377487
\(995\) 0 0
\(996\) −132.723 51.7632i −0.133256 0.0519711i
\(997\) 1678.07i 1.68312i 0.540166 + 0.841558i \(0.318361\pi\)
−0.540166 + 0.841558i \(0.681639\pi\)
\(998\) 350.322 0.351024
\(999\) 671.875 328.875i 0.672548 0.329205i
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 1050.3.c.b.449.32 32
3.2 odd 2 inner 1050.3.c.b.449.2 32
5.2 odd 4 1050.3.e.c.701.13 yes 16
5.3 odd 4 1050.3.e.b.701.4 16
5.4 even 2 inner 1050.3.c.b.449.1 32
15.2 even 4 1050.3.e.c.701.5 yes 16
15.8 even 4 1050.3.e.b.701.12 yes 16
15.14 odd 2 inner 1050.3.c.b.449.31 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.3.c.b.449.1 32 5.4 even 2 inner
1050.3.c.b.449.2 32 3.2 odd 2 inner
1050.3.c.b.449.31 32 15.14 odd 2 inner
1050.3.c.b.449.32 32 1.1 even 1 trivial
1050.3.e.b.701.4 16 5.3 odd 4
1050.3.e.b.701.12 yes 16 15.8 even 4
1050.3.e.c.701.5 yes 16 15.2 even 4
1050.3.e.c.701.13 yes 16 5.2 odd 4